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+{"_id": "0", "title": "Closed Form for Triangular Numbers", "text": "The closed-form expression for the $n$th triangular number is: :$\\displaystyle T_n = \\sum_{i \\mathop = 1}^n i = \\frac {n \\paren {n + 1} } 2$"}
+{"_id": "1", "title": "Sum of Sequence of Squares", "text": ":$\\displaystyle \\forall n \\in \\N: \\sum_{i \\mathop = 1}^n i^2 = \\frac {n \\paren {n + 1} \\paren {2 n + 1} } 6$"}
+{"_id": "2", "title": "Union is Associative", "text": "Set union is associative: :$A \\cup \\paren {B \\cup C} = \\paren {A \\cup B} \\cup C$"}
+{"_id": "3", "title": "Pythagoras's Theorem", "text": "Let $\\triangle ABC$ be a right triangle with $c$ as the hypotenuse. Then: :$a^2 + b^2 = c^2$"}
+{"_id": "4", "title": "Euclid's Theorem", "text": "For any finite set of prime numbers, there exists a prime number not in that set. {{:Euclid:Proposition/IX/20}}"}
+{"_id": "5", "title": "Limsup Squeeze Theorem", "text": "Let $\\left \\langle {x_n} \\right \\rangle$ and $\\left \\langle {y_n} \\right \\rangle$ be sequences in $\\R$. Let: : $(1): \\quad \\forall n \\ge n_0: \\left|{x_n}\\right| \\le y_n$ : $(2): \\quad \\displaystyle \\limsup_{n \\mathop \\to \\infty} \\left({y_n}\\right) = 0$, where $\\limsup$ denotes the limit superior. Then: : $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = 0$"}
+{"_id": "6", "title": "Square Root of Prime is Irrational", "text": "The square root of any prime number is irrational."}
+{"_id": "7", "title": "Derivative of Exponential Function", "text": "Let $\\exp$ be the exponential function. Then: :$\\map {\\dfrac \\d {\\d x} } {\\exp x} = \\exp x$"}
+{"_id": "8", "title": "Derivative of Sine Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\sin x} = \\cos x$"}
+{"_id": "9", "title": "Cauchy-Bunyakovsky-Schwarz Inequality", "text": "=== Semi-Inner Product Spaces === {{:Cauchy-Bunyakovsky-Schwarz Inequality/Inner Product Spaces}}"}
+{"_id": "10", "title": "0.999...=1", "text": ":$0.999 \\ldots = 1$"}
+{"_id": "12", "title": "Sum of Sequence of Cubes", "text": ":$\\displaystyle \\sum_{i \\mathop = 1}^n i^3 = \\paren {\\sum_{i \\mathop = 1}^n i}^2 = \\frac {n^2 \\paren {n + 1}^2} 4$"}
+{"_id": "13", "title": "Law of Cosines", "text": "Let $\\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$. Then: :$c^2 = a^2 + b^2 - 2 a b \\cos C$"}
+{"_id": "14", "title": "Euler's Formula", "text": "Let $z \\in \\C$ be a complex number. Then: :$e^{i z} = \\cos z + i \\sin z$"}
+{"_id": "16", "title": "Lagrange's Theorem (Group Theory)", "text": "Let $G$ be a finite group. Let $H$ be a subgroup of $G$. Then: : $\\order H$ divides $\\order G$ where $\\order G$ and $\\order H$ are the order of $G$ and $H$ respectively. In fact: :$\\index G H = \\dfrac {\\order G} {\\order H}$ where $\\index G H$ is the index of $H$ in $G$. When $G$ is an infinite group, we can still interpret this theorem sensibly: :A subgroup of finite index in an infinite group is itself an infinite group. :A finite subgroup of an infinite group has infinite index."}
+{"_id": "17", "title": "ProofWiki:Sandbox", "text": "$\\forall n \\in \\Z_{>0}: \\displaystyle \\sum_{k \\mathop = 0}^{n} \\binom {2n} {2k} E_{2n - 2k } = 0$ where $E_k$ denotes the $k$th Euler number."}
+{"_id": "18", "title": "Carathéodory's Theorem (Analysis)", "text": "Let $I \\subseteq \\R$. Let $c \\in I$ be an interior point of $I$. {{Disambiguate|Definition:Interior Point}} {{explain|In this case, there appears not to be a definition for \"interior point\" which appropriately captures the gist of this. It is clearly a point inside a real interval, but the concept has not yet been defined.}} Let $f : I \\to \\R$ be a real function. Then $f$ is differentiable at $c$ {{iff}}: :There exists a real function $\\varphi : I \\to \\R$ that is continuous at $c$ and satisfies: :$(1): \\quad \\forall x \\in I: f \\left({x}\\right) - f \\left({c}\\right) = \\varphi \\left({x}\\right) \\left({x - c}\\right)$ :$(2): \\quad \\varphi \\left({c}\\right) = f' \\left({c}\\right)$"}
+{"_id": "19", "title": "One-Step Subgroup Test", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H$ be a subset of $G$. Then $\\struct {H, \\circ}$ is a subgroup of $\\struct {G, \\circ}$ {{iff}}: : $(1): \\quad H \\ne \\O$, that is, $H$ is non-empty : $(2): \\quad \\forall a, b \\in H: a \\circ b^{-1} \\in H$."}
+{"_id": "20", "title": "Two-Step Subgroup Test", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H$ be a subset of $G$. Then $\\struct {H, \\circ}$ is a subgroup of $\\struct {G, \\circ}$ {{iff}}: :$(1): \\quad H \\ne \\O$, that is, $H$ is non-empty :$(2): \\quad a, b \\in H \\implies a \\circ b \\in H$ :$(3): \\quad a \\in H \\implies a^{-1} \\in H$. That is, $\\struct {H, \\circ}$ is a subgroup of $\\struct {G, \\circ}$ {{iff}} $\\struct {H, \\circ}$ is a $H$ be a nonempty subset of $G$ which is: :closed under its operation and: :closed under inversion."}
+{"_id": "21", "title": "Fundamental Theorem of Arithmetic", "text": "For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquely up to the order in which they appear."}
+{"_id": "22", "title": "Euclid's Lemma", "text": "Let $a, b, c \\in \\Z$. Let $a \\divides b c$, where $\\divides$ denotes divisibility. Let $a \\perp b$, where $\\perp$ denotes relative primeness. Then $a \\divides c$."}
+{"_id": "23", "title": "Fermat's Little Theorem", "text": "Let $p$ be a prime number. Let $n \\in \\Z_{>0}$ be a positive integer such that $p$ is not a divisor of $n$. Then: :$n^{p - 1} \\equiv 1 \\pmod p$"}
+{"_id": "24", "title": "De Morgan's Laws (Set Theory)", "text": "{{:De Morgan's Laws (Set Theory)/Set Difference}}"}
+{"_id": "25", "title": "De Moivre's Formula", "text": "Let $z \\in \\C$ be a complex number expressed in complex form: :$z = r \\paren {\\cos x + i \\sin x}$ Then: :$\\forall \\omega \\in \\C: \\paren {r \\paren {\\cos x + i \\sin x} }^\\omega = r^\\omega \\paren {\\map \\cos {\\omega x} + i \\, \\map \\sin {\\omega x} }$"}
+{"_id": "26", "title": "First Isomorphism Theorem", "text": "=== Groups === {{:First Isomorphism Theorem/Groups}} === Rings === {{:First Isomorphism Theorem/Rings}}"}
+{"_id": "27", "title": "Power Rule for Derivatives", "text": "Let $n \\in \\R$. Let $f: \\R \\to \\R$ be the real function defined as $\\map f x = x^n$. Then: :$\\map {f'} x = n x^{n - 1}$ everywhere that $\\map f x = x^n$ is defined. When $x = 0$ and $n = 0$, $\\map {f'} x$ is undefined."}
+{"_id": "28", "title": "Basel Problem", "text": ":$\\displaystyle \\map \\zeta 2 = \\sum_{n \\mathop = 1}^\\infty {\\frac 1 {n^2} } = \\frac {\\pi^2} 6$ where $\\zeta$ denotes the Riemann zeta function."}
+{"_id": "30", "title": "1+2+...+n+(n-1)+...+1 = n^2", "text": ":$\\forall n \\in \\N: 1 + 2 + \\cdots + n + \\paren {n - 1} + \\cdots + 1 = n^2$"}
+{"_id": "31", "title": "Necessary Conditions for Existence of Skolem Sequence", "text": "A Skolem sequence of order $n$ can only exist if $n \\equiv 0, 1 \\pmod 4$."}
+{"_id": "32", "title": "Divisibility by 9", "text": "A number expressed in decimal notation is divisible by $9$ {{iff}} the sum of its digits is divisible by $9$. That is: :$N = \\sqbrk {a_0 a_1 a_2 \\ldots a_n}_{10} = a_0 + a_1 10 + a_2 10^2 + \\cdots + a_n 10^n$ is divisible by $9$  {{iff}}: :$a_0 + a_1 + \\ldots + a_n$ is divisible by $9$."}
+{"_id": "33", "title": "Existence of Rational Powers of Irrational Numbers", "text": "There exist irrational numbers $a$ and $b$ such that $a^b$ is rational."}
+{"_id": "34", "title": "Intersection is Associative", "text": "Set intersection is associative: :$A \\cap \\paren {B \\cap C} = \\paren {A \\cap B} \\cap C$"}
+{"_id": "35", "title": "Center of Symmetric Group is Trivial", "text": "Let $n \\in \\N$ be a natural number. Let $S_n$ denote the symmetric group of order $n$. Let $n \\ge 3$. Then the center $\\map Z {S_n}$ of $S_n$ is trivial."}
+{"_id": "36", "title": "Square Root of 2 is Irrational", "text": ":$\\sqrt 2$ is irrational."}
+{"_id": "38", "title": "Empty Set is Unique", "text": "The empty set is unique."}
+{"_id": "39", "title": "Principle of Non-Contradiction", "text": "The '''Principle of Non-Contradiction''' is a valid deduction sequent in propositional logic."}
+{"_id": "40", "title": "Rule of Substitution", "text": "Let $S$ be a sequent that has been proved. Then a proof can be found for any substitution instance of $S$."}
+{"_id": "41", "title": "Rule of Sequent Introduction", "text": "Let the statements $P_1, P_2, \\ldots, P_n$ be conclusions in a proof, on various assumptions. Let $P_1, P_2, \\ldots, P_n \\vdash Q$ be a substitution instance of a sequent for which we already have a proof. {{explain|Question the use of \"substitution instance\": can we not \"just\" allow for $P_1, \\ldots, P_n$ to be \"just\" statements?}} Then we may introduce, at any stage of a proof (citing '''SI'''), either: :The conclusion $Q$ of the sequent already proved or: :A substitution instance of such a conclusion, together with a reference to the sequent that is being cited. This conclusion depend upon the pool of assumptions upon which $P_1, P_2, \\ldots, P_n \\vdash Q$ rests."}
+{"_id": "42", "title": "Law of Identity", "text": "Every proposition entails itself:"}
+{"_id": "43", "title": "Rule of Idempotence", "text": "The '''rule of idempotence''' is two-fold:"}
+{"_id": "46", "title": "Hypothetical Syllogism", "text": "The '''(rule of the) hypothetical syllogism''' is a valid deduction sequent in propositional logic: :If we can conclude that $p$ implies $q$, and if we can also conclude that $q$ implies $r$, then we may infer that $p$ implies $r$."}
+{"_id": "49", "title": "Extended Rule of Implication", "text": "Any sequent can be expressed as a theorem. That is: :$P_1, P_2, P_3, \\ldots, P_n \\vdash Q$ means the same thing as: :$\\vdash P_1 \\implies \\paren {P_2 \\implies \\paren {P_3 \\implies \\paren {\\ldots \\implies \\paren {P_n \\implies Q} \\ldots} } }$ The latter expression is known as the corresponding conditional of the former. Thus every sequent containing the symbol $\\vdash$ can, if so desired, be expressed in the form of a theorem which has $\\implies$."}
+{"_id": "51", "title": "Equivalences are Interderivable", "text": "If two propositional formulas are interderivable, they are equivalent: :$\\paren {p \\dashv \\vdash q} \\dashv \\vdash \\paren {p \\iff q}$"}
+{"_id": "52", "title": "Union is Commutative", "text": "Set union is commutative: :$S \\cup T = T \\cup S$"}
+{"_id": "53", "title": "Intersection is Commutative", "text": "Set intersection is commutative: :$S \\cap T = T \\cap S$"}
+{"_id": "54", "title": "Russell's Paradox", "text": "The comprehension principle leads to a contradiction."}
+{"_id": "57", "title": "De Morgan's Laws (Logic)", "text": "{{:De Morgan's Laws (Logic)/Disjunction of Negations}}"}
+{"_id": "62", "title": "Solution to Quadratic Equation", "text": "The quadratic equation of the form $a x^2 + b x + c = 0$ has solutions: :$x = \\dfrac {-b \\pm \\sqrt {b^2 - 4 a c} } {2 a}$"}
+{"_id": "63", "title": "Normal Subgroup Test", "text": "Let $G$ be a group and $H \\le G$. Then $H$ is a normal subgroup of $G$ {{iff}}: :$\\forall x \\in G: x H x^{-1} \\subseteq H$."}
+{"_id": "64", "title": "Equality of Ordered Pairs", "text": "Two ordered pairs are equal {{iff}} corresponding coordinates are equal: :$\\tuple {a, b} = \\tuple {c, d} \\iff a = c \\land b = d$"}
+{"_id": "65", "title": "Identity is Unique", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure that has an identity element $e \\in S$. Then $e$ is unique."}
+{"_id": "66", "title": "Cancellation Laws", "text": "Let $G$ be a group. Let $a, b, c \\in G$. Then the following hold: ;Right cancellation law :$b a = c a \\implies b = c$ ;Left cancellation law :$a b = a c \\implies b = c$"}
+{"_id": "67", "title": "Center of Group is Normal Subgroup", "text": "The center $\\map Z G$ of any group $G$ is a normal subgroup of $G$ which is abelian."}
+{"_id": "68", "title": "Composition of Relations is Associative", "text": "The composition of relations is an associative binary operation: :$\\paren {\\RR_3 \\circ \\RR_2} \\circ \\RR_1 = \\RR_3 \\circ \\paren {\\RR_2 \\circ \\RR_1}$"}
+{"_id": "69", "title": "Intersection of Subsemigroups", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroup. Let $\\left({T_1, \\circ}\\right)$ and $\\left({T_2, \\circ}\\right)$ be subsemigroups of $\\left({S, \\circ}\\right)$. Then the intersection of $\\left({T_1, \\circ}\\right)$ and $\\left({T_2, \\circ}\\right)$ is itself a subsemigroup of that $\\left({S, \\circ}\\right)$. If $\\left({T, \\circ}\\right)$ is that intersection of $\\left({T_1, \\circ}\\right)$ and $\\left({T_2, \\circ}\\right)$, it follows that $\\left({T, \\circ}\\right)$ is also a subsemigroup of both $\\left({T_1, \\circ}\\right)$ and $\\left({T_2, \\circ}\\right)$."}
+{"_id": "70", "title": "Cancellable Elements of Semigroup form Subsemigroup", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Let $C$ be the set of cancellable elements of $\\struct {S, \\circ}$. Then $\\struct {C, \\circ}$ is a subsemigroup of $\\struct {S, \\circ}$."}
+{"_id": "71", "title": "Subgroup of Cyclic Group is Cyclic", "text": "Let $G$ be a cyclic group. Let $H$ be a subgroup of $G$. Then $H$ is cyclic."}
+{"_id": "72", "title": "Empty Set is Subset of All Sets", "text": "The empty set $\\O$ is a subset of every set (including itself). That is: :$\\forall S: \\O \\subseteq S$"}
+{"_id": "73", "title": "Relation Reflexivity", "text": "Every relation has exactly one of these properties: it is either: :reflexive, :antireflexive or :non-reflexive."}
+{"_id": "74", "title": "Union is Idempotent", "text": "Set union is idempotent: :$S \\cup S = S$"}
+{"_id": "75", "title": "Intersection is Idempotent", "text": "Set intersection is idempotent: :$S \\cap S = S$"}
+{"_id": "76", "title": "Set is Subset of Union", "text": "The union of two sets is a superset of each: :$S \\subseteq S \\cup T$ :$T \\subseteq S \\cup T$"}
+{"_id": "77", "title": "Set is Subset of Itself", "text": "Every set is a subset of itself: :$\\forall S: S \\subseteq S$ Thus, by definition, the relation '''is a subset of''' is reflexive."}
+{"_id": "78", "title": "Subset of Set with Propositional Function", "text": "Let $S$ be a set. Let $P: S \\to \\set {\\text{true}, \\text{false} }$ be a propositional function on $S$. Then: :$\\set {x \\in S: \\map P x} \\subseteq S$"}
+{"_id": "80", "title": "Universal Generalisation", "text": "Let $\\mathbf a$ be any arbitrarily selected object in the universe of discourse. Then: :$\\map P {\\mathbf a} \\vdash \\forall x: \\map P x$ In natural language: :''Suppose $P$ is true of any arbitrarily selected $\\mathbf a$'' in the universe of discourse.'' :''Then $P$ is true of everything in the universe of discourse.''"}
+{"_id": "82", "title": "Existential Instantiation", "text": ":$\\exists x: P \\left({x}\\right), P \\left({\\mathbf a}\\right) \\implies y \\vdash y$ Suppose we have the following: : From our universe of discourse, ''any'' arbitrarily selected object $\\mathbf a$ which has the property $P$ implies a conclusion $y$ : $\\mathbf a$ is not free in $y$ : It is known that there ''does'' actually exists an object that has $P$. Then we may infer $y$. This is called the '''Rule of Existential Instantiation''' and often appears in a proof with its abbreviation '''EI'''. When using this rule of existential instantiation: :$\\exists x: P \\left({x}\\right), P \\left({\\mathbf a}\\right) \\implies y \\vdash y$ the instance of $P \\left({\\mathbf a}\\right)$ is referred to as the '''typical disjunct'''."}
+{"_id": "83", "title": "Singleton of Element is Subset", "text": "Let $S$ be a set. Let $\\set x$ be the singleton of $x$. Then: :$x \\in S \\iff \\set x \\subseteq S$"}
+{"_id": "84", "title": "Subset Relation is Transitive", "text": "The relation \"is a subset of\" is transitive: :$\\paren {R \\subseteq S} \\land \\paren {S \\subseteq T} \\implies R \\subseteq T$"}
+{"_id": "85", "title": "Set Inequality", "text": ":$S \\ne T \\iff \\left({S \\nsubseteq T}\\right) \\lor \\left({T \\nsubseteq S}\\right)$"}
+{"_id": "86", "title": "Set Equals Itself", "text": "All sets are equal to themselves: :$\\forall S: S = S$"}
+{"_id": "87", "title": "Union is Smallest Superset", "text": "Let $S_1$ and $S_2$ be sets. Then $S_1 \\cup S_2$ is the smallest set containing both $S_1$ and $S_2$. That is: :$\\paren {S_1 \\subseteq T} \\land \\paren {S_2 \\subseteq T} \\iff \\paren {S_1 \\cup S_2} \\subseteq T$"}
+{"_id": "88", "title": "Union with Empty Set", "text": "The union of any set with the empty set is the set itself: :$S \\cup \\O = S$"}
+{"_id": "89", "title": "Intersection is Subset", "text": "The intersection of two sets is a subset of each: :$S \\cap T \\subseteq S$ :$S \\cap T \\subseteq T$"}
+{"_id": "90", "title": "Intersection with Empty Set", "text": "The intersection of any set with the empty set is itself the empty set: :$S \\cap \\O = \\O$"}
+{"_id": "91", "title": "Intersection is Largest Subset", "text": "Let $T_1$ and $T_2$ be sets. Then $T_1 \\cap T_2$ is the largest set contained in both $T_1$ and $T_2$. That is: :$S \\subseteq T_1 \\land S \\subseteq T_2 \\iff S \\subseteq T_1 \\cap T_2$"}
+{"_id": "92", "title": "Intersection is Subset of Union", "text": "The intersection of two sets is a subset of their union: :$S \\cap T \\subseteq S \\cup T$"}
+{"_id": "93", "title": "Absorption Laws (Set Theory)/Union with Intersection", "text": ":$S \\cup \\paren {S \\cap T} = S$"}
+{"_id": "94", "title": "Absorption Laws (Set Theory)/Intersection with Union", "text": ":$S \\cap \\paren {S \\cup T} = S$"}
+{"_id": "95", "title": "Union Distributes over Intersection", "text": "Set union is distributive over set intersection: :$R \\cup \\paren {S \\cap T} = \\paren {R \\cup S} \\cap \\paren {R \\cup T}$"}
+{"_id": "96", "title": "Intersection Distributes over Union", "text": "Set intersection is distributive over set union: :$R \\cap \\paren {S \\cup T} = \\paren {R \\cap S} \\cup \\paren {R \\cap T}$"}
+{"_id": "97", "title": "Set Difference is Subset", "text": ":$S \\setminus T \\subseteq S$"}
+{"_id": "98", "title": "Set Difference with Empty Set is Self", "text": "The set difference between a set and the empty set is the set itself: :$S \\setminus \\O = S$"}
+{"_id": "99", "title": "Set Difference with Self is Empty Set", "text": "The set difference of a set with itself is the empty set: :$S \\setminus S = \\O$"}
+{"_id": "100", "title": "Set Difference Equals First Set iff Empty Intersection", "text": ":$S \\setminus T = S \\iff S \\cap T = \\O$"}
+{"_id": "101", "title": "Equal Set Differences iff Equal Intersections", "text": ":$R \\setminus S = R \\setminus T \\iff R \\cap S = R \\cap T$"}
+{"_id": "102", "title": "Euler's Number is Irrational", "text": "Euler's number $e$ is irrational."}
+{"_id": "104", "title": "Results Concerning Set Difference with Union", "text": "Let: * $S \\setminus T$ denote set difference * $S \\cup T$ denote set union * $S \\cap T$ denote set intersection."}
+{"_id": "105", "title": "Set Difference with Intersection", "text": "Let $S$ and $T$ be sets."}
+{"_id": "106", "title": "Results Concerning Set Difference with Intersection", "text": "Let: :$S \\setminus T$ denote set difference :$S \\cap T$ denote set intersection."}
+{"_id": "107", "title": "Pascal's Rule", "text": "For positive integers $n, k$ with $1 \\le k \\le n$: :$\\dbinom n {k - 1} + \\dbinom n k = \\dbinom {n + 1} k$ This is also valid for the real number definition: :$\\forall r \\in \\R, k \\in \\Z: \\dbinom r {k - 1} + \\dbinom r k = \\dbinom {r + 1} k$"}
+{"_id": "108", "title": "Set Difference Union Intersection", "text": ":$S = \\paren {S \\setminus T} \\cup \\paren {S \\cap T}$"}
+{"_id": "109", "title": "Absorption Laws (Logic)", "text": "For any two propositions $p$ and $q$, we have:"}
+{"_id": "110", "title": "Set Difference with Set Difference", "text": ":$S \\setminus \\paren {S \\setminus T} = S \\cap T = T \\setminus \\paren {T \\setminus S}$"}
+{"_id": "112", "title": "Dirichlet's Test for Uniform Convergence", "text": "Let $D$ be a set. Let $\\struct {V, \\norm {\\,\\cdot\\,} }$ be a normed vector space. Let $a_i, b_i$ be mappings from $D \\to M$. Let the following conditions be satisfied: :$(1): \\quad$ The sequence of partial sums of $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\map {a_n} x$ be bounded on $D$ :$(2): \\quad \\sequence {\\map {b_n} x}$ be monotonic for each $x \\in D$ :$(3): \\quad \\map {b_n} x \\to 0$ converge uniformly on $D$. Then: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\map {a_n} x \\, \\map {b_n} x$ converges uniformly on $D$."}
+{"_id": "114", "title": "Relative Complement of Empty Set", "text": "The relative complement of the empty set is the set itself: :$\\relcomp S \\O = S$"}
+{"_id": "115", "title": "Relative Complement with Self is Empty Set", "text": "The relative complement of a set in itself is the empty set: :$\\relcomp S S = \\O$"}
+{"_id": "116", "title": "Relative Complement of Relative Complement", "text": ":$\\relcomp S {\\relcomp S T} = T$"}
+{"_id": "117", "title": "Intersection with Relative Complement is Empty", "text": "The intersection of a set and its relative complement is the empty set: :$T \\cap \\relcomp S T = \\O$"}
+{"_id": "118", "title": "Union with Relative Complement", "text": "The union of a set $T$ and its relative complement in $S$ is the set $S$: :$\\relcomp S T \\cup T = S$"}
+{"_id": "119", "title": "Set Difference as Intersection with Relative Complement", "text": "Let $A, B \\subseteq S$. Then the set difference between $A$ and $B$ can be expressed as the intersection with the relative complement with respect to $S$: :$A \\setminus B = A \\cap \\relcomp S B$"}
+{"_id": "120", "title": "Equivalence of Axiom Schemata for Groups", "text": "In the definition of a group, the axioms for the existence of an identity element and for closure under taking inverses can be replaced by the following two axioms: : Given a group $G$, there exists at least one element $e \\in G$ such that $e$ is a '''left identity'''; : For any element $g$ in a group $G$, there exists at least one '''left inverse''' of $g$. Alternatively, we can also replace the aforementioned axioms with the following two: : Given a group $G$, there exists at least one element $e \\in G$ such that $e$ is a '''right identity'''; : For any element $g$ in a group $G$, there exists at least one '''right inverse''' of $g$. Thus we can formulate the group axioms as either of the following:"}
+{"_id": "121", "title": "Group has Latin Square Property", "text": "Let $\\struct {G, \\circ}$ be a group. Then $G$ satisfies the Latin square property. That is, for all $a, b \\in G$, there exists a unique $g \\in G$ such that $a \\circ g = b$. Similarly, there exists a unique $h \\in G$ such that $h \\circ a = b$."}
+{"_id": "122", "title": "Symmetric Difference is Commutative", "text": "Symmetric difference is commutative: :$S * T = T * S$"}
+{"_id": "124", "title": "Symmetric Difference of Equal Sets", "text": "The symmetric difference of two equal sets is the empty set: :$S = T \\iff S * T = \\O$"}
+{"_id": "125", "title": "Union is Empty iff Sets are Empty", "text": ":$S \\cup T = \\O \\iff S = \\O \\land T = \\O$"}
+{"_id": "126", "title": "Symmetric Difference with Empty Set", "text": ":$S * \\O = S$ where $*$ denotes the symmetric difference."}
+{"_id": "127", "title": "Intersection Distributes over Symmetric Difference", "text": "Intersection is distributive over symmetric difference: :$\\paren {R * S} \\cap T = \\paren {R \\cap T} * \\paren {S \\cap T}$ :$T \\cap \\paren {R * S} = \\paren {T \\cap R} * \\paren {T \\cap S}$"}
+{"_id": "128", "title": "Symmetric Difference of Unions", "text": "Let $R$, $S$ and $T$ be sets. Then: :$\\paren {R \\cup T} * \\paren {S \\cup T} = \\paren {R * S} \\setminus T$ where: :$*$ denotes the symmetric difference :$\\setminus$ denotes set difference :$\\cup$ denotes set union"}
+{"_id": "129", "title": "Intersection with Universe", "text": "The intersection of a set with the universe is the set itself: :$\\mathbb U \\cap S = S$"}
+{"_id": "130", "title": "Union with Universe", "text": "The union of a set with the universe is the universe: :$\\mathbb U \\cup S = \\mathbb U$"}
+{"_id": "131", "title": "Complement of Empty Set is Universe", "text": "The complement of the empty set is the universe: :$\\map \\complement \\O = \\mathbb U$"}
+{"_id": "132", "title": "Complement of Universe is Empty Set", "text": "The complement of the universe is the empty set: :$\\map \\complement {\\mathbb U} = \\O$"}
+{"_id": "133", "title": "Complement of Complement", "text": "The complement of the complement of a set is the set itself: :$\\map \\complement {\\map \\complement S} = S$"}
+{"_id": "134", "title": "Intersection with Complement", "text": "The intersection of a set and its complement is the empty set: :$S \\cap \\map \\complement S = \\O$"}
+{"_id": "135", "title": "Union with Complement", "text": "The union of a set and its complement is the universe: :$S \\cup \\map \\complement S = \\mathbb U$"}
+{"_id": "137", "title": "Set Difference as Intersection with Complement", "text": "Set difference can be expressed as the intersection with the set complement: :$A \\setminus B = A \\cap \\map \\complement B$"}
+{"_id": "138", "title": "Set Difference of Complements", "text": ":$\\complement \\left({S}\\right) \\setminus \\complement \\left({T}\\right) = T \\setminus S$"}
+{"_id": "139", "title": "Symmetric Difference of Complements", "text": "The symmetric difference of two sets equals the symmetric difference of their complements: :$\\map \\complement S * \\map \\complement T = S * T$"}
+{"_id": "141", "title": "Symmetric Difference with Self is Empty Set", "text": "The symmetric difference of a set with itself is the empty set: :$S * S = \\O$"}
+{"_id": "142", "title": "Symmetric Difference with Complement", "text": "The symmetric difference of a set with its complement is the universe: :$S * \\relcomp {} S = \\mathbb U$"}
+{"_id": "143", "title": "Symmetric Difference is Associative", "text": "Symmetric difference is associative: :$R * \\paren {S * T} = \\paren {R * S} * T$"}
+{"_id": "144", "title": "Cartesian Product is Empty iff Factor is Empty", "text": ":$S \\times T = \\O \\iff S = \\O \\lor T = \\O$ Thus: :$S \\times \\O = \\O = \\O \\times T$"}
+{"_id": "146", "title": "Cartesian Product of Subsets", "text": "Let $A, B, S, T$ be sets such that $A \\subseteq B$ and $S \\subseteq T$. Then: :$A \\times S \\subseteq B \\times T$ In addition, if $A, S \\ne \\O$, then: :$A \\times S \\subseteq B \\times T \\iff A \\subseteq B \\land S \\subseteq T$"}
+{"_id": "147", "title": "Cartesian Product of Intersections", "text": ":$\\paren {S_1 \\cap S_2} \\times \\paren {T_1 \\cap T_2} = \\paren {S_1 \\times T_1} \\cap \\paren {S_2 \\times T_2}$ where $S_1, S_2, T_1, T_2$ are sets."}
+{"_id": "148", "title": "Cartesian Product Distributes over Union", "text": "Cartesian product is distributive over union: :$A \\times \\paren {B \\cup C} = \\paren {A \\times B} \\cup \\paren {A \\times C}$ :$\\paren {B \\cup C} \\times A = \\paren {B \\times A} \\cup \\paren {C \\times A}$"}
+{"_id": "149", "title": "Cartesian Product of Unions", "text": ":$\\paren {S_1 \\cup S_2} \\times \\paren {T_1 \\cup T_2} = \\paren {S_1 \\times T_1} \\cup \\paren {S_2 \\times T_2} \\cup \\paren {S_1 \\times T_2} \\cup \\paren {S_2 \\times T_1}$"}
+{"_id": "150", "title": "Cartesian Product Distributes over Set Difference", "text": "Cartesian product is distributive over set difference: :$(1): \\quad S \\times \\paren {T_1 \\setminus T_2} = \\paren {S \\times T_1} \\setminus \\paren {S \\times T_2}$ :$(2): \\quad \\paren {T_1 \\setminus T_2} \\times S = \\paren {T_1 \\times S} \\setminus \\paren {T_2 \\times S}$"}
+{"_id": "151", "title": "Set Difference of Cartesian Products", "text": ":$\\paren {S_1 \\times S_2} \\setminus \\paren {T_1 \\times T_2} = \\paren {S_1 \\times \\paren {S_2 \\setminus T_2} } \\cup \\paren {\\paren {S_1 \\setminus T_1} \\times S_2}$"}
+{"_id": "152", "title": "Inverse of Inverse Relation", "text": "The inverse of an inverse relation is the relation itself: :$\\paren {\\RR^{-1} }^{-1} = \\RR$"}
+{"_id": "153", "title": "Diagonal Relation is Equivalence", "text": "The diagonal relation $\\Delta_S$ on a set $S$ is always an equivalence in $S$."}
+{"_id": "154", "title": "Trivial Relation is Equivalence", "text": "The trivial relation on $S$: :$\\RR = S \\times S$ is always an equivalence in $S$."}
+{"_id": "155", "title": "Equality of Relations", "text": "Let $\\RR_1$ and $\\RR_2$ be relations on $S_1 \\times T_1$ and $S_2 \\times T_2$ respectively. Then $\\RR_1$ and $\\RR_2$ are '''equal''' {{iff}}: :$S_1 = S_2$ :$T_1 = T_2$ :$\\tuple {s, t} \\in \\RR_1 \\iff \\tuple {s, t} \\in \\RR_2$ It is worth labouring the point that for two relations to be '''equal''', not only must their domains be equal, but so must their codomains."}
+{"_id": "156", "title": "Image of Singleton under Relation", "text": "Let $\\RR \\subseteq S \\times T$ be a relation. Then the image of an element of $S$ is equal to the image of a singleton containing that element, the singleton being a subset of $S$: :$\\forall s \\in S: \\map \\RR s = \\RR \\sqbrk {\\set s}$"}
+{"_id": "157", "title": "Image of Subset under Relation is Subset of Image", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation from $S$ to $T$. Let $A, B \\subseteq S$ such that $A \\subseteq B$. Then the image of $A$ is a subset of the image of $B$: :$A \\subseteq B \\implies \\RR \\sqbrk A \\subseteq \\RR \\sqbrk B$ In the notation of direct image mappings, this can be written: :$A \\subseteq B \\implies \\map {\\RR^\\to} A \\subseteq \\map {\\RR^\\to} B$"}
+{"_id": "158", "title": "Image of Element is Subset", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation. Let $A \\subseteq S$. Then: :$s \\in A \\implies \\map \\RR s \\subseteq \\RR \\sqbrk A$"}
+{"_id": "159", "title": "Image is Subset of Codomain", "text": "Let $\\RR = S \\times T$ be a relation. For all subsets $A$ of the domain of $\\RR$, the image of $A$ is a subset of the codomain of $\\RR$: :$\\forall A \\subseteq \\Dom \\RR: \\RR \\sqbrk A \\subseteq T$ In the notation of direct image mappings, this can be written as: :$\\forall A \\in \\powerset S: \\map {\\RR^\\to} A \\in \\powerset T$"}
+{"_id": "160", "title": "Image of Empty Set is Empty Set", "text": "Let $\\RR \\subseteq S \\times T$ be a relation. The image of the empty set is the empty set: :$\\RR \\sqbrk \\O = \\O$"}
+{"_id": "161", "title": "Domain of Composite Relation", "text": "Let $\\RR_2 \\circ \\RR_1$ be a composite relation. Then the domain of $\\RR_2 \\circ \\RR_1$ is the domain of $\\RR_1$: :$\\Dom {\\RR_2 \\circ \\RR_1} = \\Dom {\\RR_1}$"}
+{"_id": "162", "title": "Codomain of Composite Relation", "text": "Let $\\RR_2 \\circ \\RR_1$ be a composite relation. Then the codomain of $\\RR_2 \\circ \\RR_1$ is the codomain of $\\RR_2$: :$\\Cdm {\\RR_2 \\circ \\RR_1} = \\Cdm {\\RR_2}$"}
+{"_id": "163", "title": "Preimage of Mapping equals Domain", "text": "The preimage of a mapping is the same set as its domain: :$\\Preimg f = \\Dom f$"}
+{"_id": "164", "title": "Preimage of Relation is Subset of Domain", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation. Then the preimage of $\\mathcal R$ is a subset of its domain: :$\\Preimg {\\mathcal R} \\subseteq S$"}
+{"_id": "165", "title": "Preimage of Image under Left-Total Relation is Superset", "text": "Let $\\RR \\subseteq S \\times T$ be a left-total relation. Then: :$A \\subseteq S \\implies A \\subseteq \\paren {\\RR^{-1} \\circ \\RR} \\sqbrk A$ where: :$\\RR \\sqbrk A$ denotes the image of $A$ under $\\RR$ :$\\RR^{-1} \\sqbrk A$ denotes the preimage of $A$ under $\\RR$ :$\\RR^{-1} \\circ \\RR$ denotes composition of $\\RR^{-1}$ and $\\RR$. This can be expressed in the language and notation of direct image mappings and inverse image mappings as: :$\\forall A \\in \\powerset S: A \\subseteq \\map {\\paren {\\RR^\\gets \\circ \\RR^\\to} } A$"}
+{"_id": "166", "title": "Inverse of Composite Relation", "text": "Let $\\RR_2 \\circ \\RR_1 \\subseteq S_1 \\times S_3$ be the composite of the two relations $\\RR_1 \\subseteq S_1 \\times S_2$ and $\\RR_2 \\subseteq S_2 \\times S_3$. Then: :$\\paren {\\RR_2 \\circ \\RR_1}^{-1} = \\RR_1^{-1} \\circ \\RR_2^{-1}$"}
+{"_id": "167", "title": "Image of Union under Relation", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation. Let $S_1$ and $S_2$ be subsets of $S$. Then: :$\\RR \\sqbrk {S_1 \\cup S_2} = \\RR \\sqbrk {S_1} \\cup \\RR \\sqbrk {S_2}$ That is, the image of the union of subsets of $S$ is equal to the union of their images."}
+{"_id": "168", "title": "Image of Intersection under Relation", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation. Let $S_1$ and $S_2$ be subsets of $S$. Then: :$\\RR \\sqbrk {S_1 \\cap S_2} \\subseteq \\RR \\sqbrk {S_1} \\cap \\RR \\sqbrk {S_2}$ That is, the image of the intersection of subsets of $S$ is a subset of the intersection of their images."}
+{"_id": "169", "title": "Image of Set Difference under Relation", "text": "Let $\\RR \\subseteq S \\times T$ be a relation. Let $A$ and $B$ be subsets of $S$. Then: :$\\RR \\sqbrk A \\setminus \\RR \\sqbrk B \\subseteq \\RR \\sqbrk {A \\setminus B}$ where: :$\\setminus$ denotes set difference :$\\RR \\sqbrk A$ denotes image of $A$ under $\\RR$."}
+{"_id": "170", "title": "Preimage of Union under Relation", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation. Let $T_1$ and $T_2$ be subsets of $T$. Then: :$\\RR^{-1} \\sqbrk {T_1 \\cup T_2} = \\RR^{-1} \\sqbrk {T_1} \\cup \\RR^{-1} \\sqbrk {T_2}$"}
+{"_id": "171", "title": "Preimage of Intersection under Relation", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R \\subseteq S \\times T$ be a relation. Let $C$ and $D$ be subsets of $T$. Then: :$\\mathcal R^{-1} \\left[{C \\cap D}\\right] \\subseteq \\mathcal R^{-1} \\left[{C}\\right] \\cap \\mathcal R^{-1} \\left[{D}\\right]$"}
+{"_id": "173", "title": "Restriction is Subset of Relation", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation. Let $X \\subseteq S$. Then the restriction of $\\mathcal R$ to $X$ is a subset of $\\mathcal R$."}
+{"_id": "174", "title": "Null Relation is Antireflexive, Symmetric and Transitive", "text": "Let $S$ be a set which is non-empty. Let $\\RR \\subseteq S \\times S$ be the null relation. Then $\\RR$ is antireflexive, symmetric and transitive. If $S = \\O$ then Relation on Empty Set is Equivalence applies."}
+{"_id": "175", "title": "Symmetry of Relations is Symmetric", "text": "Let $\\mathcal R$ be a relation on $S$ which is symmetric. Then: :$\\left({x, y}\\right) \\in \\mathcal R \\iff \\left({y, x}\\right) \\in \\mathcal R$."}
+{"_id": "176", "title": "Relation is Symmetric and Antisymmetric iff Coreflexive", "text": "Let $S$ be a set. Let $\\RR \\subseteq S \\times S$ be a relation in $S$. Then: :$\\RR$ is both symmetric and antisymmetric {{iff}}: :$\\RR$ is coreflexive."}
+{"_id": "177", "title": "Relation both Symmetric and Asymmetric is Null", "text": "Let $\\RR$ be a relation in $S$ which is both symmetric and asymmetric. Then: :$\\RR = \\O$"}
+{"_id": "178", "title": "Equivalent Characterizations of Abelian Group", "text": "Let $G$ be a group. {{TFAE}} {{begin-eqn}} {{eqn | n = 1       | o =       | c = $G$ is abelian }} {{eqn | n = 2       | o =       | c = $\\forall a, b \\in G: \\paren {a b}^{-1} = a^{-1} b^{-1}$ }} {{eqn | n = 3       | o =       | c = Cross cancellation property: $\\forall a, b, c \\in G: a b = c a \\implies b = c$ }} {{eqn | n = 4       | o =       | c = Middle cancellation property: $\\forall a, b, c, d, x \\in G: a x b = c x d \\implies a b = c d$  }} {{end-eqn}}"}
+{"_id": "180", "title": "Asymmetric Relation is Antisymmetric", "text": "Let $\\RR$ be an asymmetric relation. Then $\\RR$ is also antisymmetric."}
+{"_id": "181", "title": "Asymmetric Relation is Antireflexive", "text": "Let $S$ be a set. Let $\\RR \\subseteq S \\times S$ be a relation on $S$. Let $\\RR$ be asymmetric. Then $\\RR$ is also antireflexive."}
+{"_id": "182", "title": "Antireflexive and Transitive Relation is Asymmetric", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation which is not null. Let $\\mathcal R$ be antireflexive and transitive. Then $\\mathcal R$ is also asymmetric."}
+{"_id": "183", "title": "Antitransitive Relation is Antireflexive", "text": "Let $S$ be a set. Let $\\mathcal R \\subseteq S \\times S$ be a relation on $S$. Let $\\mathcal R$ be antitransitive. Then $\\mathcal R$ is also antireflexive."}
+{"_id": "184", "title": "Symmetric Transitive and Serial Relation is Reflexive", "text": "Let $\\RR$ be a relation which is: :symmetric :transitive :serial. Then $\\RR$ is reflexive. Thus such a relation is an equivalence."}
+{"_id": "185", "title": "Inverse Relation Properties", "text": "Let $\\mathcal R$ be a relation on a set $S$. If $\\mathcal R$ has any of the properties: * Reflexive * Antireflexive * Non-reflexive * Symmetric * Asymmetric * Antisymmetric * Non-symmetric * Transitive * Antitransitive * Non-transitive ... then its inverse $\\mathcal R^{-1}$ has the same properties."}
+{"_id": "186", "title": "Properties of Restriction of Relation", "text": "Let $S$ be a set. Let $\\mathcal R \\subseteq S \\times S$ be a relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\mathcal R \\restriction_T \\ \\subseteq T \\times T$ be the restriction of $\\mathcal R$ to $T$. If $\\mathcal R$ on $S$ has any of the properties: * Reflexive * Antireflexive * Symmetric * Antisymmetric * Asymmetric * Transitive * Antitransitive * Connected ... then $\\mathcal R \\restriction_T$ on $T$ has the same properties."}
+{"_id": "188", "title": "Relation is Antireflexive iff Disjoint from Diagonal Relation", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation on a set $S$. Then: : $\\mathcal R$ is antireflexive iff : $\\Delta_S \\cap \\mathcal R = \\varnothing$ where $\\Delta_S$ is the diagonal relation."}
+{"_id": "189", "title": "Inverse Relation Equal iff Subset", "text": "If a relation $\\RR$ is a subset or superset of its inverse, then it equals its inverse. That is, the following are equivalent: {{begin-axiom}} {{axiom | n = 1         | m = \\RR \\subseteq \\RR^{-1} }} {{axiom | n = 2         | m = \\RR^{-1} \\subseteq \\RR }} {{axiom | n = 3         | m = \\RR = \\RR^{-1} }} {{end-axiom}}"}
+{"_id": "191", "title": "Relation is Symmetric iff Inverse is Symmetric", "text": "A relation $\\mathcal R$ is symmetric {{iff}} its inverse $\\mathcal R^{-1}$ is also symmetric."}
+{"_id": "194", "title": "Inverse of Many-to-One Relation is One-to-Many", "text": "The inverse of a many-to-one relation is a one-to-many relation, and vice versa."}
+{"_id": "195", "title": "Many-to-One Relation Composite with Inverse is Transitive", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation which is many-to-one. Then the composites (both ways) of $\\mathcal R$ and its inverse $\\mathcal R^{-1}$, that is, both $\\mathcal R^{-1} \\circ \\mathcal R$ and $\\mathcal R \\circ \\mathcal R^{-1}$, are transitive."}
+{"_id": "196", "title": "One-to-Many Relation Composite with Inverse is Coreflexive", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation which is one-to-many. Then the composite of $\\mathcal R$ with its inverse is a coreflexive relation: :$\\mathcal R^{-1} \\circ \\mathcal R \\subseteq \\Delta_X$ {{refactor|This is a separate result.}} That is, by Relation is Symmetric and Antisymmetric iff Coreflexive, $\\mathcal R^{-1} \\circ \\mathcal R$ is both symmetric and antisymmetric."}
+{"_id": "197", "title": "Element in its own Equivalence Class", "text": "Let $\\mathcal R$ be an equivalence relation on a set $S$. Then every element of $S$ is in its own $\\mathcal R$-class: :$\\forall x \\in S: x \\in \\eqclass x {\\mathcal R}$"}
+{"_id": "198", "title": "Equivalence Class of Element is Subset", "text": "Let $\\mathcal R$ be an equivalence relation on a set $S$. The $\\mathcal R$-class of every element of $S$ is a subset of the set the element is in: :$\\forall x \\in S: \\eqclass x {\\mathcal R} \\subseteq S$"}
+{"_id": "199", "title": "Equivalence Class is not Empty", "text": "Let $\\mathcal R$ be an equivalence relation on a set $S$. Then no $\\mathcal R$-class is empty."}
+{"_id": "200", "title": "Handshake Lemma", "text": "Let $G$ be a $\\tuple {p, q}$-undirected graph, which may be a multigraph or a loop-graph, or both. Let $V = \\set {v_1, v_2, \\ldots, v_p}$ be the vertex set of $G$. Then: :$\\displaystyle \\sum_{i \\mathop = 1}^p \\map {\\deg_G} {v_i} = 2 q$ where $\\map {\\deg_G} {v_i}$ is the degree of vertex $v_i$. That is, the sum of all the degrees of all the vertices of an graph is equal to twice its size. This result is known as the '''Handshake Lemma''' or '''Handshaking Lemma'''."}
+{"_id": "201", "title": "Equivalence Class holds Equivalent Elements", "text": "Let $\\RR$ be an equivalence relation on a set $S$. Then: :$\\tuple {x, y} \\in \\RR \\iff \\eqclass x \\RR = \\eqclass y \\RR$"}
+{"_id": "202", "title": "Equivalence Classes are Disjoint", "text": "Let $\\RR$ be an equivalence relation on a set $S$. Then all $\\RR$-classes are pairwise disjoint: :$\\tuple {x, y} \\notin \\RR \\iff \\eqclass x \\RR \\cap \\eqclass y \\RR = \\O$"}
+{"_id": "203", "title": "Fundamental Theorem on Equivalence Relations", "text": "Let $\\RR \\subseteq S \\times S$ be an equivalence on a set $S$. Then the quotient $S / \\RR$ of $S$ by $\\RR$ forms a partition of $S$."}
+{"_id": "204", "title": "Equivalence Class is Unique", "text": "Let $\\mathcal R$ be an equivalence relation on $S$. For each $x \\in S$, the one and only one $\\mathcal R$-class to which $x$ belongs is $\\eqclass x {\\mathcal R}$."}
+{"_id": "205", "title": "Equivalence Class Equivalent Statements", "text": "Let $\\mathcal R$ be an equivalence on $S$. {{refactor|Use TFAE template}} Then $\\forall x, y \\in S$, the following statements are all equivalent: :$(1): \\quad x$ and $y$ are in the same $\\mathcal R$-class :$(2): \\quad \\eqclass x {\\mathcal R} = \\eqclass y {\\mathcal R}$ :$(3): \\quad x \\mathrel {\\mathcal R} y$ :$(4): \\quad x \\in \\eqclass y {\\mathcal R}$ :$(5): \\quad y \\in \\eqclass x {\\mathcal R}$ :$(6): \\quad \\eqclass x {\\mathcal R} \\cap \\eqclass y {\\mathcal R} \\ne \\O$"}
+{"_id": "206", "title": "Relation Induced by Partition is Equivalence", "text": "Let $\\mathbb S$ be a partition of a set $S$. Let $\\RR$ be the relation induced by $\\mathbb S$. Then: :$(1): \\quad \\RR$ is unique :$(2): \\quad \\RR$ is an equivalence relation on $S$. Hence $\\mathbb S$ is the quotient set of $S$ by $\\RR$, that is: :$\\mathbb S = S / \\RR$"}
+{"_id": "207", "title": "Relation Partitions Set iff Equivalence", "text": "Let $\\RR$ be a relation on a set $S$. Then $S$ can be partitioned into subsets by $\\RR$ {{iff}} $\\RR$ is an equivalence relation on $S$. The partition of $S$ defined by $\\RR$ is the quotient set $S / \\RR$."}
+{"_id": "208", "title": "Intersection of Equivalences", "text": "The intersection of two equivalence relations is itself an equivalence relation."}
+{"_id": "210", "title": "Equivalence of Definitions of Equivalence Relation", "text": "Let $\\RR$ be a relation on a set $S$. {{TFAE|def = Equivalence Relation}}"}
+{"_id": "211", "title": "Equivalence iff Diagonal and Inverse Composite", "text": "Let $\\RR$ be a relation on $S$. Then $\\RR$ is an equivalence relation on $S$ {{iff}}: :$\\Delta_S \\subseteq \\RR$ and: :$\\RR = \\RR \\circ \\RR^{-1}$"}
+{"_id": "212", "title": "Equality of Elements in Range of Mapping", "text": "Let $f: S \\to T$ be a mapping. Then: :$\\exists y \\in \\Rng f: \\tuple {x_1, y} \\in f \\land \\tuple {x_2, y} \\in f \\iff \\map f {x_1} = \\map f {x_2}$"}
+{"_id": "213", "title": "Inverse of Mapping is One-to-Many Relation", "text": "Let $f$ be a mapping. Then its inverse $f^{-1}$ is a one-to-many relation."}
+{"_id": "214", "title": "Equality of Mappings", "text": "Two mappings $f_1: S_1 \\to T_1, f_2: S_2 \\to T_2$ are equal {{iff}}: :$(1): \\quad S_1 = S_2$ :$(2): \\quad T_1 = T_2$ :$(3): \\quad \\forall x \\in S_1: \\map {f_1} x = \\map {f_2} x$"}
+{"_id": "215", "title": "Mapping is Constant iff Image is Singleton", "text": "A mapping is a constant mapping {{iff}} its image is a singleton."}
+{"_id": "216", "title": "Diagonal Relation is Right Identity", "text": "Let $\\RR \\subseteq S \\times T$ be a relation on $S \\times T$. Then: :$\\RR \\circ \\Delta_S = \\RR$ where $\\Delta_S$ is the diagonal relation on $S$, and $\\circ$ signifies composition of relations."}
+{"_id": "217", "title": "Identity Mapping is Left Identity", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Then: :$I_T \\circ f = f$ where $I_T$ is the identity mapping on $T$, and $\\circ$ signifies composition of mappings."}
+{"_id": "218", "title": "Image of Element under Inverse Mapping", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping such that its inverse $f^{-1}: T \\to S$ is also a mapping. Then: :$\\forall x \\in S, y \\in T: \\map f x = y \\iff \\map {f^{-1} } y = x$"}
+{"_id": "219", "title": "Injection iff Left Cancellable", "text": "A mapping $f$ is an injection {{iff}} $f$ is left cancellable."}
+{"_id": "221", "title": "Identity Mapping is Injection", "text": "On any set $S$, the identity mapping $I_S: S \\to S$ is an injection."}
+{"_id": "222", "title": "Composite of Injections is Injection", "text": "A composite of injections is an injection. That is: :If $f$ and $g$ are injections, then so is $f \\circ g$."}
+{"_id": "223", "title": "Injection if Composite is Injection", "text": "Let $f$ and $g$ be mappings such that their composite $g \\circ f$ is an injection. Then $f$ is an injection."}
+{"_id": "224", "title": "Injection iff Left Inverse", "text": "A mapping $f: S \\to T, S \\ne \\O$ is an injection {{iff}}: :$\\exists g: T \\to S: g \\circ f = I_S$ where $g$ is a mapping. That is, {{iff}} $f$ has a left inverse."}
+{"_id": "225", "title": "Inclusion Mapping is Injection", "text": "Let $S, T$ be sets such that $S$ is a subset of $T$. Then the inclusion mapping $i_S: S \\to T$ defined as: :$\\forall x \\in S: \\map {i_S} x = x$ is an injection. For this reason the inclusion mapping can be known as the '''canonical injection of $S$ to $T$'''."}
+{"_id": "226", "title": "Surjection iff Right Cancellable", "text": "Let $f$ be a mapping. Then $f$ is a surjection {{iff}} $f$ is right cancellable."}
+{"_id": "227", "title": "Identity Mapping is Surjection", "text": "On any set $S$, the identity mapping $I_S: S \\to S$ is a surjection."}
+{"_id": "228", "title": "Restriction of Mapping to Image is Surjection", "text": "Let $f: S \\to T$ be a mapping. Let $g: S \\to \\Img f$ be the restriction of $f$ to $S \\times \\Img f$. Then $g$ is a surjective restriction of $f$."}
+{"_id": "229", "title": "Composite of Surjections is Surjection", "text": "A composite of surjections is a surjection. That is: :If $g$ and $f$ are surjections, then so is $g \\circ f$."}
+{"_id": "230", "title": "Surjection if Composite is Surjection", "text": "Let $f: S_1 \\to S_2$ and $g: S_2 \\to S_3$ be mappings such that $g \\circ f$ is a surjection. Then $g$ is a surjection."}
+{"_id": "231", "title": "Surjection iff Right Inverse", "text": "A mapping $f: S \\to T, S \\ne \\O$ is a surjection {{iff}}: :$\\exists g: T \\to S: f \\circ g = I_T$ where: :$g$ is a mapping :$I_T$ is the identity mapping on $T$. That is, {{iff}} $f$ has a right inverse."}
+{"_id": "232", "title": "Identity Mapping is Bijection", "text": "The identity mapping $I_S: S \\to S$ on the set $S$ is a bijection."}
+{"_id": "233", "title": "Identity Mapping is Permutation", "text": "The identity mapping $I_S: S \\to S$ on the set $S$ is a permutation."}
+{"_id": "234", "title": "Bijection iff Left and Right Inverse", "text": "Let $f: S \\to T$ be a mapping. $f$ is a bijection {{iff}}: :$(1): \\quad \\exists g_1: T \\to S: g_1 \\circ f = I_S$ :$(2): \\quad \\exists g_2: T \\to S: f \\circ g_2 = I_T$ where both $g_1$ and $g_2$ are mappings. It also follows that it is necessarily the case that $g_1 = g_2$ for such to be possible."}
+{"_id": "235", "title": "Inverse of Bijection is Bijection", "text": "Let $f: S \\to T$ be a bijection in the sense that: :$(1): \\quad f$ is an injection :$(2): \\quad f$ is a surjection. Then the inverse $f^{-1}$ of $f$ is itself a bijection by the same definition."}
+{"_id": "236", "title": "Inverse Element of Bijection", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a bijection. Then: :$\\map {f^{-1} } y = x \\iff \\map f x = y$ where $f^{-1}$ is the inverse mapping of $f$."}
+{"_id": "237", "title": "Composite of Bijection with Inverse is Identity Mapping", "text": "Let $f: S \\to T$ be a bijection. Then: : $f^{-1} \\circ f = I_S$ : $f \\circ f^{-1} = I_T$ where $I_S$ and $I_T$ are the identity mappings on $S$ and $T$ respectively."}
+{"_id": "238", "title": "Inverse of Inverse of Bijection", "text": "Let $f: S \\to T$ be a bijection. Then: :$\\paren {f^{-1} }^{-1} = f$ where $f^{-1}$ is the inverse of $f$."}
+{"_id": "239", "title": "Composite of Bijections is Bijection", "text": "Let $f$ and $g$ be mappings such that $\\Dom f = \\Cdm g$. Then: :If $f$ and $g$ are both bijections, then so is $f \\circ g$ where $f \\circ g$ is the composite mapping of $f$ with $g$."}
+{"_id": "240", "title": "Inverse of Composite Bijection", "text": "Let $f$ and $g$ be bijections such that $\\Dom g = \\Cdm f$. Then: :$\\paren {g \\circ f}^{-1} = f^{-1} \\circ g^{-1}$ and $f^{-1} \\circ g^{-1}$ is itself a bijection."}
+{"_id": "241", "title": "Composite of Permutations is Permutation", "text": "Let $f, g$ are permutations of a set $S$. Then their composite $g \\circ f$ is also a permutation of $S$."}
+{"_id": "242", "title": "Inverse of Permutation is Permutation", "text": "If $f$ is a permutation of $S$, then so is its inverse $f^{-1}$."}
+{"_id": "243", "title": "Left Inverse Mapping is Surjection", "text": "Let $f: S \\to T$ be an injection. Let $g: T \\to S$ be a left inverse of $f$. Then $g$ is a surjection."}
+{"_id": "244", "title": "Right Inverse Mapping is Injection", "text": "Let $f: S \\to T$ be a mapping. Let $g: T \\to S$ be a right inverse of $f$. Then $g$ is an injection."}
+{"_id": "245", "title": "Set Equivalence is Equivalence Relation", "text": "Set equivalence is an equivalence relation."}
+{"_id": "246", "title": "Set Equivalence Less One Element", "text": "Let $S$ and $T$ be sets which are equivalent: :$S \\sim T$ Let $a \\in S$ and $b \\in T$. Then: : $S \\setminus \\left\\{{a}\\right\\} \\sim T \\setminus \\left\\{{b}\\right\\}$ where $\\setminus$ denotes set difference."}
+{"_id": "247", "title": "Image of Union under Mapping", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let $A$ and $B$ be subsets of $S$. Then: :$f \\sqbrk {A \\cup B} = f \\sqbrk A \\cup f \\sqbrk B$ This can be expressed in the language and notation of direct image mappings as: :$\\forall A, B \\in \\powerset S: \\map {f^\\to} {A \\cup B} = \\map {f^\\to} A \\cup \\map {f^\\to} B$"}
+{"_id": "248", "title": "Image of Intersection under Mapping", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let $S_1$ and $S_2$ be subsets of $S$. Then: :$f \\sqbrk {S_1 \\cap S_2} \\subseteq f \\sqbrk {S_1} \\cap f \\sqbrk {S_2}$ This can be expressed in the language and notation of direct image mappings as: :$\\forall S_1, S_2 \\in \\powerset S: \\map {f^\\to} {S_1 \\cap S_2} \\subseteq \\map {f^\\to} {S_1} \\cap \\map {f^\\to} {S_2}$"}
+{"_id": "249", "title": "Image of Intersection under Injection", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Then: :$\\forall A, B \\subseteq S: f \\sqbrk {A \\cap B} = f \\sqbrk A \\cap f \\sqbrk B$ {{iff}} $f$ is an injection."}
+{"_id": "250", "title": "Image of Intersection under One-to-Many Relation", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation. Then: :$\\forall S_1, S_2 \\subseteq S: \\RR \\sqbrk {S_1 \\cap S_2} = \\RR \\sqbrk {S_1} \\cap \\RR \\sqbrk {S_2}$  {{iff}} $\\RR$ is one-to-many."}
+{"_id": "251", "title": "One-to-Many Image of Set Difference", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation. Let $A$ and $B$ be subsets of $S$. Then: :$(1): \\quad \\mathcal R \\left[{A}\\right] \\setminus \\mathcal R \\left[{B}\\right] = \\mathcal R \\left[{A \\setminus B}\\right]$ {{iff}} $\\mathcal R$ is one-to-many."}
+{"_id": "252", "title": "Preimage of Union under Mapping", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let $T_1$ and $T_2$ be subsets of $T$. Then: :$f^{-1} \\sqbrk {T_1 \\cup T_2} = f^{-1} \\sqbrk {T_1} \\cup f^{-1} \\sqbrk {T_2}$ This can be expressed in the language and notation of inverse image mappings as: :$\\forall T_1, T_2 \\in \\powerset T: \\map {f^\\gets} {T_1 \\cup T_2} = \\map {f^\\gets} {T_1} \\cup \\map {f^\\gets} {T_2}$"}
+{"_id": "253", "title": "Preimage of Intersection under Mapping", "text": "Let $f: S \\to T$ be a mapping. Let $T_1$ and $T_2$ be subsets of $T$. Then: :$f^{-1} \\sqbrk {T_1 \\cap T_2} = f^{-1} \\sqbrk {T_1} \\cap f^{-1} \\sqbrk {T_2}$ This can be expressed in the language and notation of inverse image mappings as: :$\\forall T_1, T_2 \\in \\powerset T: \\map {f^\\gets} {T_1 \\cap T_2} = \\map {f^\\gets} {T_1} \\cap \\map {f^\\gets} {T_2}$"}
+{"_id": "254", "title": "Preimage of Set Difference under Mapping", "text": "Let $f: S \\to T$ be a mapping. Let $T_1$ and $T_2$ be subsets of $T$. Then: :$f^{-1} \\sqbrk {T_1 \\setminus T_2} = f^{-1} \\sqbrk {T_1} \\setminus f^{-1} \\sqbrk {T_2}$ where: :$\\setminus$ denotes set difference :$f^{-1} \\sqbrk {T_1}$ denotes preimage."}
+{"_id": "255", "title": "Projection is Surjection", "text": "Let $S$ and $T$ be non-empty sets. Let $S \\times T$ be the Cartesian product of $S$ and $T$. Let $\\pr_1: S \\times T \\to T$ and $\\pr_2: S \\times T \\to T$ be the first projection and second projection respectively on $S \\times T$. Then $\\pr_1$ and $\\pr_2$ are both surjections."}
+{"_id": "256", "title": "Sum Rule for Derivatives", "text": "Let $\\map f x, \\map j x, \\map k x$ be real functions defined on the open interval $I$. Let $\\xi \\in I$ be a point in $I$ at which both $j$ and $k$ are differentiable. Let $\\map f x = \\map j x + \\map k x$. Then $f$ is differentiable at $\\xi$ and: :$\\map {f'} \\xi = \\map {j'} \\xi + \\map {k'} \\xi$ It follows from the definition of derivative that if $j$ and $k$ are both differentiable on the interval $I$, then: :$\\forall x \\in I: \\map {f'} x = \\map {j'} x + \\map {k'} x$"}
+{"_id": "257", "title": "Quotient Mapping is Surjection", "text": "Let $\\RR$ be an equivalence relation on $S$. Then the quotient mapping $q_\\RR: S \\to S / \\RR$ is a surjection."}
+{"_id": "258", "title": "Trivial Quotient is a Bijection", "text": "Let $\\Delta_S$ be the diagonal relation on a set $S$. Let $q_{\\Delta_S}: S \\to S / \\Delta_S$ be the trivial quotient of $S$. Then $q_{\\Delta_S}: S \\to S / \\Delta_S$ is a bijection."}
+{"_id": "260", "title": "Renaming Mapping is Well-Defined", "text": "Let $f: S \\to T$ be a mapping. Let $r: S / \\RR_f \\to \\Img f$ be the renaming mapping,  defined as: :$r: S / \\RR_f \\to \\Img f: \\map r {\\eqclass x {\\RR_f} } = \\map f x$ where: :$\\RR_f$ is the equivalence induced by the mapping $f$ :$S / \\RR_f$ is the quotient set of $S$ determined by $\\RR_f$ :$\\eqclass x {\\RR_f}$ is the equivalence class of $x$ under $\\RR_f$. The renaming mapping is always well-defined."}
+{"_id": "261", "title": "Powers of Group Elements", "text": "Let $\\left({G, \\circ}\\right)$ be a group whose identity is $e$. Let $a \\in G$. Then the following results hold:"}
+{"_id": "262", "title": "Renaming Mapping is Bijection", "text": "Let $f: S \\to T$ be a mapping. Let $r: S / \\RR_f \\to \\Img f$ be the renaming mapping, defined as: :$r: S / \\RR_f \\to \\Img f: \\map r {\\eqclass x {\\RR_f} } = \\map f x$ where: :$\\RR_f$ is the equivalence induced by the mapping $f$ :$S / \\RR_f$ is the quotient set of $S$ determined by $\\RR_f$ :$\\eqclass x {\\RR_f}$ is the equivalence class of $x$ under $\\RR_f$. The renaming mapping is a bijection."}
+{"_id": "263", "title": "Composite of Mapping with Inverse", "text": "Let $f: S \\to T$ be a mapping. Then: :$\\forall x \\in S: \\map {f^{-1} \\circ f} x = \\eqclass x {\\RR_f}$ where: :$\\RR_f$ is the equivalence induced by $f$ :$\\eqclass x {\\RR_f}$ is the $\\RR_f$-equivalence class of $x$."}
+{"_id": "264", "title": "Factoring Mapping into Surjection and Inclusion", "text": "Every mapping $f:S \\to T$ can be uniquely '''factored into''' a surjection $g$ followed by the inclusion mapping $i_T$. That is, $f = i_T \\circ g$ where: :$g: S \\to \\Img f: \\map g x = \\map f x$ :$i_T: \\Img f \\to T: \\map {i_T} x = x$ This can be illustrated using a commutative diagram as follows: ::$\\begin{xy}\\xymatrix@L+2mu@+1em {  S \\ar@{-->}[r]^*{g}      \\ar[rd]_*{f = i_T \\circ g} &  \\Img f \\ar@{-->}[d]^*{i_T} \\\\ &  T }\\end{xy}$"}
+{"_id": "265", "title": "Quotient Theorem for Surjections", "text": "Let $f: S \\to T$ be a surjection. Then there is one and only one bijection $r: S / \\RR_f \\to T$ such that: :$r \\circ q_{\\RR_f} = f$ where: :$\\RR_f$ is the equivalence induced by $f$ :$r: S / \\RR_f \\to T$ is the renaming mapping :$q_{\\RR_f}: S \\to S / \\RR_f$ is the quotient mapping induced by $\\RR_f$. This can be illustrated using a commutative diagram as follows: ::$\\begin {xy} \\xymatrix@L + 2mu@ + 1em {  S \\ar@{-->}[rr]^*{f = r \\circ q_{\\RR_f} }    \\ar[dd]_*{q_{\\RR_f} } && T \\\\ \\\\ S / \\RR_f \\ar[urur]_*{r}  } \\end {xy}$"}
+{"_id": "266", "title": "Quotient Theorem for Sets", "text": "A mapping $f: S \\to T$ can be uniquely '''factored into''' a surjection, followed by a bijection, followed by an injection. Thus: :$f = i \\circ r \\circ q_{\\RR_f}$ where: {{begin-eqn}} {{eqn | ll= q_{\\RR_f}:       | lo= S \\to S / \\RR_f:       | l = \\map {q_{\\RR_f} } s       | r = \\eqclass s {\\RR_f}       | c = Quotient Mapping }} {{eqn | ll= r:       | lo= S / \\RR_f \\to \\Img f:       | l = \\map r {\\eqclass s {\\RR_f} }       | r = \\map f s       | c = Renaming Mapping }} {{eqn | ll= i:       | lo= \\Img f \\to T:       | l = \\map i t       | r = t       | c = Inclusion Mapping }} {{end-eqn}} where: :$\\RR_f$ is the equivalence induced by $f$ :$S / \\RR_f$ is the quotient set of $S$ induced by $\\RR_f$ This can be illustrated using a commutative diagram as follows: ::$\\begin {xy} \\xymatrix@L + 2mu@ + 1em {  S \\ar@{-->}[rrr]^*{f = i_T \\circ r \\circ q_{\\RR_f} }    \\ar[d]_*{q_{\\RR_f} } & & & T \\\\ S / \\RR_f \\ar[rrr]_*{r} & & & \\Img f \\ar[u]_*{i_T} } \\end {xy}$"}
+{"_id": "267", "title": "Empty Set is Element of Power Set", "text": "The empty set is an element of all power sets: :$\\forall S: \\O \\in \\powerset S$"}
+{"_id": "268", "title": "Direct Image Mapping of Relation is Mapping", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R \\subseteq S \\times T$ be a relation on $S \\times T$. Let $\\mathcal R^\\to: \\powerset S \\to \\powerset T$ be the direct image mapping of $\\mathcal R$: :$\\forall X \\in \\powerset S: \\map {\\mathcal R^\\to} X = \\set {t \\in T: \\exists s \\in X: \\tuple {s, t} \\in \\mathcal R}$ Then $\\mathcal R^\\to$ is indeed a mapping."}
+{"_id": "269", "title": "Inverse Image Mapping of Relation is Mapping", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R \\subseteq S \\times T$ be a relation on $S \\times T$. Let $\\mathcal R^\\gets$ be the inverse image mapping of $\\mathcal R$: :$\\mathcal R^\\gets: \\powerset T \\to \\powerset S: \\map {\\mathcal R^\\gets} Y = \\mathcal R^{-1} \\sqbrk Y$ Then $\\mathcal R^\\gets$ is indeed a mapping."}
+{"_id": "270", "title": "Direct Image Mapping of Injection is Injection", "text": "Let $f: S \\to T$ be an injection. Then the direct image mapping of $f$: :$f^\\to: \\powerset S \\to \\powerset T$ is an injection."}
+{"_id": "271", "title": "Direct Image Mapping of Surjection is Surjection", "text": "Let $f: S \\to T$ be a surjection. Then the direct image mapping of $f$: :$f^\\to: \\powerset S \\to \\powerset T$ is a surjection."}
+{"_id": "272", "title": "Cantor's Theorem", "text": "There is no surjection from a set $S$ to its power set for any set $S$. That is, $S$ is strictly smaller than its power set."}
+{"_id": "273", "title": "Surjection Induced by Powerset is Induced by Surjection", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation. Let $\\mathcal R^\\to: \\powerset S \\to \\powerset T$ be the direct image mapping $\\mathcal R$. Let $\\mathcal R^\\to$ be a surjection. Let $X = \\Preimg {\\mathcal R}$, that is, the preimage of $\\mathcal R$. Then $\\mathcal R {\\restriction_X} \\subseteq X \\times T$, that is, the restriction of $\\mathcal R$ to $X$, is a surjection."}
+{"_id": "274", "title": "Mapping is Injection if its Direct Image Mapping is Injection", "text": "Let $f: S \\to T$ be a mapping. Let $f^\\to: \\powerset S \\to \\powerset T$ be the direct image mapping of $f$. Let $f^\\to$ be an injection. Then $f: S \\to T$ is also an injection."}
+{"_id": "275", "title": "Direct Image Mapping is Bijection iff Mapping is Bijection", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation. Let $\\mathcal R^\\to: \\powerset S \\to \\powerset T$ be the direct image mapping of $\\mathcal R$. Then $\\mathcal R \\subseteq S \\times T$ is a bijection {{iff}} $\\mathcal R^\\to: \\powerset S \\to \\powerset T$ is a bijection."}
+{"_id": "276", "title": "No Bijection from Set to its Power Set", "text": "Let $S$ be a set. Let $\\powerset S$ denote the power set of $S$. There is no bijection $f: S \\to \\powerset S$."}
+{"_id": "278", "title": "Union of Power Sets", "text": "The union of the power sets of two sets $S$ and $T$ is a subset of the power set of their union: :$\\powerset S \\cup \\powerset T \\subseteq \\powerset {S \\cup T}$"}
+{"_id": "279", "title": "Intersection of Power Sets", "text": "The intersection of the power sets of two sets $S$ and $T$ is equal to the power set of their intersection: :$\\powerset S \\cap \\powerset T = \\powerset {S \\cap T}$"}
+{"_id": "280", "title": "Cantor-Bernstein-Schröder Theorem", "text": "If a subset of one set is equivalent to the other, and a subset of the other is equivalent to the first, then the two sets are themselves equivalent: :$\\forall S, T: T \\sim S_1 \\subseteq S \\land S \\sim T_1 \\subseteq T \\implies S \\sim T$"}
+{"_id": "281", "title": "Trivial Ordering is Universally Compatible", "text": "Let $S$ be a set. Let $\\RR$ be the trivial ordering on $S$. Then $\\RR$ is universally compatible."}
+{"_id": "282", "title": "Identity Mapping is Order Isomorphism", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. The identity mapping $I_S$ is an order isomorphism from $\\left({S, \\preceq}\\right)$ to itself."}
+{"_id": "283", "title": "Inverse of Order Isomorphism is Order Isomorphism", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be ordered sets. Let $\\phi$ be a bijection from $\\left({S, \\preceq_1}\\right)$ to $\\left({T, \\preceq_2}\\right)$. Then: : $\\phi: \\left({S, \\preceq_1}\\right) \\to \\left({T, \\preceq_2}\\right)$ is an order isomorphism {{iff}}: : $\\phi^{-1}: \\left({T, \\preceq_2}\\right) \\to \\left({S, \\preceq_1}\\right)$ is also an order isomorphism."}
+{"_id": "284", "title": "Composite of Order Isomorphisms is Order Isomorphism", "text": "Let $\\left({S_1, \\preceq_1}\\right)$, $\\left({S_2, \\preceq_2}\\right)$ and $\\left({S_3, \\preceq_3}\\right)$ be ordered sets. Let: : $\\phi: \\left({S_1, \\preceq_1}\\right) \\to \\left({S_2, \\preceq_2}\\right)$ and: : $\\psi: \\left({S_2, \\preceq_2}\\right) \\to \\left({S_3, \\preceq_3}\\right)$ be order isomorphisms. Then $\\psi \\circ \\phi: \\left({S_1, \\preceq_1}\\right) \\to \\left({S_3, \\preceq_3}\\right)$ is also an order isomorphism."}
+{"_id": "285", "title": "Order Isomorphism is Equivalence Relation", "text": "Order isomorphism between ordered sets is an equivalence relation. So any given family of ordered sets can be partitioned into disjoint classes of isomorphic sets."}
+{"_id": "286", "title": "Dual Ordering is Ordering", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $\\succeq$ denote the dual ordering of $\\preceq$. Then $\\succeq$ is an ordering on $S$."}
+{"_id": "288", "title": "Power Set is Complete Lattice", "text": "Let $S$ be a set. Let $\\left({\\mathcal P \\left({S}\\right), \\subseteq}\\right)$ be the relational structure defined on $\\mathcal P \\left({S}\\right)$ by the relation $\\subseteq$. Then $\\left({\\mathcal P \\left({S}\\right), \\subseteq}\\right)$ is a complete lattice."}
+{"_id": "289", "title": "Totally Ordered Set is Lattice", "text": "Every totally ordered set is a lattice."}
+{"_id": "290", "title": "Trichotomy Law (Ordering)", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then $\\preceq$ is a total ordering {{iff}}: :$\\forall a, b \\in S: \\paren {a \\prec b} \\lor \\paren {a = b} \\lor \\paren {a \\succ b}$ That is, every element either strictly precedes, is the same as, or strictly succeeds, every other element. In other words, {{iff}} $\\prec$ is a trichotomy."}
+{"_id": "291", "title": "Complement of Reflexive Relation", "text": "Let $\\RR \\subseteq S \\times S$ be a relation. Then $\\RR$ is reflexive {{iff}} its complement $\\relcomp {S \\times S} \\RR \\subseteq S \\times S$ is antireflexive. Likewise, $\\RR$ is antireflexive {{iff}} its complement $\\relcomp {S \\times S} \\RR \\subseteq S \\times S$ is reflexive."}
+{"_id": "293", "title": "Strictly Increasing Mapping is Increasing", "text": "A mapping that is strictly increasing is an increasing mapping."}
+{"_id": "294", "title": "Strictly Decreasing Mapping is Decreasing", "text": "A mapping that is strictly decreasing is a decreasing mapping."}
+{"_id": "297", "title": "Strictly Monotone Mapping with Totally Ordered Domain is Injective", "text": "Let $\\struct {S, \\preceq_1}$ be a totally ordered set. Let $\\struct {T, \\preceq_2}$ be an ordered set. Let $\\phi: \\struct {S, \\preceq_1} \\to \\struct {T, \\preceq_2}$ be a strictly monotone mapping. Then $\\phi$ is injective."}
+{"_id": "298", "title": "Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing", "text": "Let $\\struct {S, \\preceq_1}$ be a totally ordered set. Let $\\struct {T, \\preceq_2}$ be an ordered set. Let $\\phi: S \\to T$ be a mapping. Then $\\phi$ is an order embedding {{iff}} $\\phi$ is strictly increasing. That is: :$\\forall x, y \\in S: x \\preceq_1 y \\iff \\map \\phi x \\preceq_2 \\map \\phi y$ {{iff}}: :$\\forall x, y \\in S: x \\prec_1 y \\implies \\map \\phi x \\prec_2 \\map \\phi y$"}
+{"_id": "299", "title": "Finite Totally Ordered Set is Well-Ordered", "text": "Every finite totally ordered set is well-ordered."}
+{"_id": "300", "title": "Subset of Well-Ordered Set is Well-Ordered", "text": "Every non-empty subset of a well-ordered set is itself well-ordered."}
+{"_id": "301", "title": "Element Commutes with Product of Commuting Elements", "text": "Let $(S, \\circ)$ be a semigroup. Let $x, y, z \\in S$. If $x$ commutes with both $y$ and $z$, then $x$ commutes with $y \\circ z$."}
+{"_id": "302", "title": "Associative Idempotent Anticommutative", "text": "Let $\\circ$ be a binary operation on a set $S$. Let $\\circ$ be associative. Then $\\circ$ is anticommutative {{iff}}: : $(1): \\quad \\circ$ is idempotent and: : $(2): \\quad \\forall a, b \\in S: a \\circ b \\circ a = a$."}
+{"_id": "303", "title": "Associative and Anticommutative", "text": "Let $\\circ$ be a binary operation on a set $S$. Let $\\circ$ be both associative and anticommutative. Then: :$\\forall x, y, z \\in S: x \\circ y \\circ z = x \\circ z$"}
+{"_id": "304", "title": "Constant Operation is Commutative", "text": "Let $S$ be a set. Let $x \\sqbrk c y = c$ be a constant operation on $S$. Then $\\sqbrk c$ is a commutative operation: :$\\forall x, y \\in S: x \\sqbrk c y = y \\sqbrk c x$"}
+{"_id": "305", "title": "Constant Operation is Associative", "text": "Let $S$ be a set. Let $x \\left[{c}\\right] y = c$ be a constant operation on $S$. Then $\\left[{c}\\right]$ is an associative operation: :$\\forall x, y, z \\in S: \\left({x \\left[{c}\\right] y}\\right) \\left[{c}\\right] z = x \\left[{c}\\right] \\left({y \\left[{c}\\right] z}\\right)$"}
+{"_id": "306", "title": "Left Operation is Idempotent", "text": "The left operation is idempotent: :$\\forall x: x \\leftarrow x = x$"}
+{"_id": "310", "title": "Left Operation is Associative", "text": "The left operation is associative: :$\\forall x, y, z: \\paren {x \\leftarrow y} \\leftarrow z = x \\leftarrow \\paren {y \\leftarrow z}$"}
+{"_id": "311", "title": "Right Operation is Associative", "text": "The right operation is associative: :$\\forall x, y, z: \\paren {x \\rightarrow y} \\rightarrow z = x \\rightarrow \\paren {y \\rightarrow z}$"}
+{"_id": "312", "title": "Max and Min are Commutative", "text": "The Max and Min operations are commutative: : $\\max \\left({x, y}\\right) = \\max \\left({y, x}\\right)$ : $\\min \\left({x, y}\\right) = \\min \\left({y, x}\\right)$"}
+{"_id": "313", "title": "Max and Min are Associative", "text": "The Max and Min operations are associative: : $\\max \\left({\\max \\left({x, y}\\right), z}\\right) = \\max \\left({x, \\max \\left({y, z}\\right)}\\right)$ : $\\min \\left({\\min \\left({x, y}\\right), z}\\right) = \\min \\left({x, \\min \\left({y, z}\\right)}\\right)$ Thus we are justified in writing $\\max \\left({x, y, z}\\right)$ and $\\min \\left({x, y, z}\\right)$."}
+{"_id": "314", "title": "Max and Min are Idempotent", "text": "The Max and Min operations are idempotent: : $\\max \\left({x, x}\\right) = x$ : $\\min \\left({x, x}\\right) = x$"}
+{"_id": "315", "title": "Max and Min Operations are Distributive over Each Other", "text": "The Max and Min operations are distributive over each other: :$\\max \\set {x, \\min \\set {y, z} } = \\min \\set {\\max \\set {x, y}, \\max \\set {x, z} }$ :$\\max \\set {\\min \\set {x, y}, z} = \\min \\set {\\max \\set {x, z}, \\max \\set {y, z} }$ :$\\min \\set {x, \\max \\set {y, z} } = \\max \\set {\\min \\set {x, y}, \\min \\set {x, z} }$ :$\\min \\set {\\max \\set {x, y}, z} = \\max \\set {\\min \\set {x, z}, \\min \\set {y, z} }$"}
+{"_id": "316", "title": "Equality of Algebraic Structures", "text": "Two algebraic structures $\\struct {S, \\circ}$ and $\\struct {T, *}$ are equal {{iff}}: :$S = T$ :$\\forall a, b \\in S: a \\circ b = a * b$"}
+{"_id": "317", "title": "Restriction of Associative Operation is Associative", "text": "Let $\\struct {S, \\circ}$ be an semigroup. Let $T \\subseteq S$. Let $T$ be closed under $\\circ$. Then $\\struct {T, \\circ {\\restriction_T} }$ is also a semigroup, where $\\circ {\\restriction_T}$ is the restriction of $\\circ$ to $T$."}
+{"_id": "318", "title": "Restriction of Commutative Operation is Commutative", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $T \\subseteq S$. Let the operation $\\circ$ be commutative on $\\struct {S, \\circ}$. Then the restriction $\\circ {\\restriction_T}$ of $\\circ$ to $T$ is also commutative."}
+{"_id": "319", "title": "Idempotent Magma Element forms Singleton Submagma", "text": "Let $\\struct {S, \\circ}$ be a magma. Let $x \\in S$ be an idempotent element of $\\struct {S, \\circ}$. Then  $\\struct {\\set x, \\circ}$ is a submagma of $\\struct {S, \\circ}$."}
+{"_id": "320", "title": "Magma Subset Product with Self", "text": "Let $\\left({S, \\circ}\\right)$ be a magma. Let $T \\subseteq S$. Then $\\left({T, \\circ}\\right)$ is a magma {{iff}} $T \\circ T \\subseteq T$, where $T \\circ T$ is the subset product of $T$ with itself."}
+{"_id": "321", "title": "Subset Product within Semigroup is Associative", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Then the operation $\\circ_\\PP$ induced on the power set of $S$ is also associative."}
+{"_id": "322", "title": "Subset Product within Commutative Structure is Commutative", "text": "Let $\\struct {S, \\circ}$ be a magma. If $\\circ$ is commutative, then the operation $\\circ_\\mathcal P$ induced on the power set of $S$ is also commutative."}
+{"_id": "323", "title": "Subset of Subset Product", "text": "Let $\\struct {S, \\circ}$ be a magma. Let $\\powerset S$ be the power set of $S$. Let $X, Y, Z \\in \\powerset S$. Then: :$X \\subseteq Y \\implies \\paren {X \\circ Z} \\subseteq \\paren {Y \\circ Z}$ :$X \\subseteq Y \\implies \\paren {Z \\circ X} \\subseteq \\paren {Z \\circ Y}$ where $X \\circ Z$ etc. denotes subset product."}
+{"_id": "324", "title": "Cancellable Element is Cancellable in Subset", "text": "Let $\\left ({S, \\circ}\\right)$ be an algebraic structure. Let $\\left ({T, \\circ}\\right) \\subseteq \\left ({S, \\circ}\\right)$. Let $x \\in T$ be cancellable in $S$. Then $x$ is also cancellable in $T$."}
+{"_id": "325", "title": "Subsemigroup Closure Test", "text": "To show that an algebraic structure $\\left({T, \\circ}\\right)$ is a subsemigroup of a semigroup $\\struct {S, \\circ}$, we need to show only that: : $(1): \\quad T \\subseteq S$ : $(2): \\quad \\circ$ is a closed operation in $T$."}
+{"_id": "326", "title": "Cancellable iff Regular Representations Injective", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Then $a \\in S$ is cancellable {{iff}}: ::the left regular representation $\\map {\\lambda_a} x$ is injective and ::the right regular representation $\\map {\\rho_a} x$ is injective."}
+{"_id": "328", "title": "More than one Left Identity then no Right Identity", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. If $\\struct {S, \\circ}$ has more than one left identity, then it has no right identity."}
+{"_id": "329", "title": "Element under Left Operation is Right Identity", "text": "Let $\\struct {S, \\leftarrow}$ be an algebraic structure in which the operation $\\leftarrow$ is the left operation. Then no matter what $S$ is, $\\struct {S, \\leftarrow}$ is a semigroup all of whose elements are right identities. Thus it can be seen that any right identity in a semigroup is not necessarily unique."}
+{"_id": "330", "title": "Element under Right Operation is Left Identity", "text": "Let $\\struct {S, \\rightarrow}$ be an algebraic structure in which the operation $\\rightarrow$ is the right operation. Then $\\struct {S, \\rightarrow}$ is a semigroup all of whose elements are left identities. Thus it can be seen that any left identity in a semigroup is not necessarily unique."}
+{"_id": "331", "title": "Left and Right Identity are the Same", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $e_L \\in S$ be a left identity, and $e_R \\in S$ be a right identity. Then: :$e_L = e_R$ that is, both the left identity and right identity are the same, and are therefore an identity $e$. Furthermore, $e$ is the only left identity and right identity for $\\circ$."}
+{"_id": "332", "title": "Identity Property in Semigroup", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Let $s \\in S$ be such that: :$\\forall a \\in S: \\exists x, y \\in S: s \\circ x = a = y \\circ s$ Then $\\struct {S, \\circ}$ has an identity."}
+{"_id": "333", "title": "Identity of Monoid is Cancellable", "text": "The identity of a monoid is cancellable."}
+{"_id": "334", "title": "Identity is only Idempotent Cancellable Element", "text": "Let $e_S$ is the identity of an algebraic structure $\\left({S, \\circ}\\right)$. Then $e_S$ is the only cancellable element of $\\left({S, \\circ}\\right)$ that is idempotent."}
+{"_id": "335", "title": "Set of all Self-Maps is Monoid", "text": "Let $S$ be a set. Let $S^S$ be the set of all mappings from $S$ to itself. Let the operation $\\circ$ represent composition of mappings. Then the algebraic structure $\\struct {S^S, \\circ}$ is a monoid whose identity element is the identity mapping on $S$."}
+{"_id": "336", "title": "Test for Submonoid", "text": "To show that $\\left({T, \\circ}\\right)$ is a submonoid of a monoid $\\left({S, \\circ}\\right)$, we need to show that: :$(1): \\quad T \\subseteq S$ :$(2): \\quad \\left({T, \\circ}\\right)$ is a magma (i.e. that it is closed) :$(3): \\quad \\left({T, \\circ}\\right)$ has an identity."}
+{"_id": "337", "title": "Identity of Cancellable Monoid is Identity of Submonoid", "text": "Let $\\struct {S, \\circ}$ be a monoid, all of whose elements are cancellable. Let $\\struct {T, \\circ}$ be a submonoid of $\\struct {S, \\circ}$. Then the identity of $\\struct {T, \\circ}$ is the same element as the identity of $\\struct {S, \\circ}$."}
+{"_id": "338", "title": "Cancellable Elements of Monoid form Submonoid", "text": "The cancellable elements of a monoid $\\struct {S, \\circ}$ form a submonoid of $\\struct {S, \\circ}$."}
+{"_id": "339", "title": "Product of Semigroup Element with Left Inverse is Idempotent", "text": "Let $\\struct {S, \\circ}$ be a semigroup with a left identity $e_L$. Let $x \\in S$ such that $\\exists x_L: x_L \\circ x = e_L$, that is $x$ has a left inverse with respect to the left identity. Then: :$\\paren {x \\circ x_L} \\circ \\paren {x \\circ x_L} = x \\circ x_L$ That is, $x \\circ x_L$ is idempotent."}
+{"_id": "340", "title": "Product of Semigroup Element with Right Inverse is Idempotent", "text": "Let $\\struct {S, \\circ}$ be a semigroup with a right identity $e_R$. Let $x \\in S$ such that $\\exists x_R: x \\circ x_R = e_R$, i.e. $x$ has a right inverse with respect to the right identity. Then: :$\\paren {x_R \\circ x} \\circ \\paren {x_R \\circ x} = x_R \\circ x$ That is, $x_R \\circ x$ is idempotent."}
+{"_id": "341", "title": "Left Inverse for All is Right Inverse", "text": "Let $\\struct {S, \\circ}$ be a semigroup with a left identity $e_L$ such that: :$\\forall x \\in S: \\exists x_L: x_L \\circ x = e_L$ That is, every element of $S$ has a left inverse with respect to the left identity. Then $x \\circ x_L = e_L$, that is, $x_L$ is also a right inverse with respect to the left identity."}
+{"_id": "342", "title": "Left Inverse and Right Inverse is Inverse", "text": "Let $\\struct {S, \\circ}$ be a monoid with identity element $e_S$. Let $x \\in S$ such that $x$ has both a left inverse and a right inverse. That is: :$\\exists x_L \\in S: x_L \\circ x = e_S$ :$\\exists x_R \\in S: x \\circ x_R = e_S$ Then $x_L = x_R$, that is, $x$ has an inverse. Furthermore, that element is the ''only'' inverse (both right and left) for $x$"}
+{"_id": "343", "title": "Left and Right Inverses of Product", "text": "Let $\\struct {S, \\circ}$ be a monoid whose identity is $e_S$. Let $x, y \\in S$. Let: :$(1): \\quad x \\circ y$ have a left inverse for $\\circ$ :$(2): \\quad y \\circ x$ have a right inverse for $\\circ$. Then both $x$ and $y$ are invertible for $\\circ$."}
+{"_id": "345", "title": "Equivalence of Definitions of Self-Inverse", "text": "Let $\\left({S, \\circ}\\right)$ be a monoid whose identity is $e_S$. Let $x \\in S$. {{TFAE|def = Self-Inverse Element|context = Abstract Algebra}}"}
+{"_id": "346", "title": "Inverse of Identity Element is Itself", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure with an identity element $e$. Let the inverse of $e$ be $e^{-1}$. Then: : $e^{-1} = e$ That is, $e$ is self-inverse."}
+{"_id": "347", "title": "Invertible Element of Associative Structure is Cancellable", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure where $\\circ$ is associative. Let $\\left({S, \\circ}\\right)$ have an identity element $e_S$. An element of $\\left({S, \\circ}\\right)$ which is invertible is also cancellable."}
+{"_id": "348", "title": "Regular Representation of Invertible Element is Permutation", "text": "Let $\\struct {S, \\circ}$ be a monoid. Let $a \\in S$ be invertible. Then the left regular representation $\\lambda_a$ and the right regular representation $\\rho_a$ are permutations of $S$."}
+{"_id": "349", "title": "Group is Inverse Semigroup with Identity", "text": "A group is an inverse semigroup with an identity."}
+{"_id": "350", "title": "Invertible Elements of Monoid form Subgroup of Cancellable Elements", "text": "Let $\\struct {S, \\circ}$ be an monoid whose identity is $e_S$. Let $C$ be the set of all cancellable elements of $S$. Let $T$ be the set of all invertible elements of $S$. Then $\\struct {T, \\circ}$ is a subgroup of $\\struct {C, \\circ}$."}
+{"_id": "351", "title": "Structure Induced by Associative Operation is Associative", "text": "Let $\\struct {T, \\circ}$ be an algebraic structure, and let $S$ be a set. Let $\\struct {T^S, \\oplus}$ be the structure on $T^S$ induced by $\\circ$. Let $\\circ$ be associative. Then the pointwise operation $\\oplus$ induced on $T^S$ by $\\circ$ is also associative."}
+{"_id": "352", "title": "Structure Induced by Commutative Operation is Commutative", "text": "Let $\\struct {T, \\circ}$ be an algebraic structure, and let $S$ be a set. Let $\\struct {T^S, \\oplus}$ be the structure on $T^S$ induced by $\\circ$. Let $\\circ$ be a commutative operation. Then the pointwise operation $\\oplus$ induced on $T^S$ by $\\circ$ is also commutative."}
+{"_id": "353", "title": "Induced Structure Identity", "text": "Let $\\struct {T, \\circ}$ be an algebraic structure, and let $S$ be a set. Let $\\struct {T^S, \\oplus}$ be the structure on $T^S$ induced by $\\circ$. Let $e$ be an identity for $\\circ$. Then the constant mapping $f_e: S \\to T$ defined as: :$\\forall x \\in S: \\map {f_e} x = e$ is the identity for the pointwise operation $\\oplus$ induced on $T^S$ by $\\circ$."}
+{"_id": "354", "title": "Structure Induced by Abelian Group Operation is Abelian Group", "text": "Let $\\struct {G, \\circ}$ be an abelian group whose identity is $e$. Let $S$ be a set. Let $\\struct {G^S, \\oplus}$ be the structure on $G^S$ induced by $\\circ$. Then $\\struct {G^S, \\oplus}$ is an abelian group."}
+{"_id": "355", "title": "Power Set with Union is Commutative Monoid", "text": "Let $S$ be a set and let $\\powerset S$ be its power set. Then $\\struct {\\powerset S, \\cup}$ is a commutative monoid whose identity is $\\O$. The only invertible element of this structure is $\\O$. Thus (except in the degenerate case $S = \\O$) $\\struct {\\powerset S, \\cup}$ cannot be a group."}
+{"_id": "356", "title": "Power Set with Intersection is Commutative Monoid", "text": "Let $S$ be a set and let $\\powerset S$ be its power set. Then $\\struct {\\powerset S, \\cap}$ is a commutative monoid whose identity is $S$. The only invertible element of this structure is $S$. Thus (except in the degenerate case $S = \\O$) $\\struct {\\powerset S, \\cap}$ cannot be a group."}
+{"_id": "357", "title": "Diagonal Relation is Universally Compatible", "text": "The diagonal relation $\\Delta_S$ on a set $S$ is universally compatible with every operation on $S$."}
+{"_id": "358", "title": "Trivial Relation is Universally Congruent", "text": "The trivial relation $\\mathcal R = S \\times S$ on a set $S$ is universally congruent with every closed operation on $S$."}
+{"_id": "360", "title": "Quotient Structure is Well-Defined", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\RR$ be a congruence relation on $\\struct {S, \\circ}$. Let $S / \\RR$ be the quotient set of $S$ by $\\RR$. Let $\\circ_\\RR$ be the operation induced on $S / \\RR$ by $\\circ$. Then $\\circ_\\RR$ is a well-defined operation in the quotient structure $\\struct {S / \\RR, \\circ_\\RR}$."}
+{"_id": "362", "title": "External Direct Product Closure", "text": "Let $\\struct {S, \\circ_1}$ and $\\struct {T, \\circ_2}$ be algebraic structures. Let $\\struct {S \\times T, \\circ}$ be the external direct product of $\\struct {S, \\circ_1}$ and $\\struct {T, \\circ_2}$. Let $\\struct {S, \\circ_1}$ and $\\struct {T, \\circ_2}$ be closed. Then $\\struct {S \\times T, \\circ}$ is also closed."}
+{"_id": "363", "title": "External Direct Product Associativity", "text": "Let $\\struct {S \\times T, \\circ}$ be the external direct product of the two algebraic structures $\\struct {S, \\circ_1}$ and $\\struct {T, \\circ_2}$. Let $\\circ_1$ and $\\circ_2$ be associative. Then $\\circ$ is also associative."}
+{"_id": "364", "title": "External Direct Product of Semigroups", "text": "The external direct product of two semigroups is itself a semigroup."}
+{"_id": "365", "title": "External Direct Product Commutativity", "text": "Let $\\left({S \\times T, \\circ}\\right)$ be the external direct product of the two algebraic structures $\\left({S, \\circ_1}\\right)$ and $\\left({T, \\circ_2}\\right)$. Let $\\circ_1$ and $\\circ_2$ be commutative operations. Then $\\circ$ is also a commutative operation."}
+{"_id": "366", "title": "External Direct Product Identity", "text": "Let $\\struct {S \\times T, \\circ}$ be the external direct product of the two monoids $\\struct {S, \\circ_1}$ and $\\struct {T, \\circ_2}$. Let: : $e_S$ be the identity for $\\struct {S, \\circ_1}$ and: : $e_T$ be the identity for $\\struct {T, \\circ_2}$. Then $\\tuple {e_S, e_T}$ is the identity for $\\struct {S \\times T, \\circ}$."}
+{"_id": "367", "title": "External Direct Product Inverses", "text": "Let $\\struct {S \\times T, \\circ}$ be the external direct product of the two monoids $\\struct {S, \\circ_1}$ and $\\struct {T, \\circ_2}$. Let: :$s^{-1}$ be an inverse of $s \\in \\struct {S, \\circ_1}$ and: :$t^{-1}$ be an inverse of $t \\in \\struct {T, \\circ_2}$. Then $\\tuple {s^{-1}, t^{-1} }$ is an inverse of $\\tuple {s, t} \\in \\struct {S \\times T, \\circ}$."}
+{"_id": "368", "title": "Morphism Property Preserves Closure", "text": "Let $\\phi: \\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right) \\to \\left({T, *_1, *_2, \\ldots, *_n}\\right)$ be a mapping from one algebraic structure $\\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right)$ to another $\\left({T, *_1, *_2, \\ldots, *_n}\\right)$. Let $\\circ_k$ have the morphism property under $\\phi$ for some operation $\\circ_k$ in $\\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right)$. Then the following properties hold: : If $S' \\subseteq S$ is closed under $\\circ_k$, then $\\phi \\left[{S'}\\right]$ is closed under $*_k$ : If $T' \\subseteq T$ is closed under $*_k$, then $\\phi^{-1} \\left[{T'}\\right]$ is closed under $\\circ_k$ where $\\phi \\left[{S'}\\right]$ denotes the image of $S'$."}
+{"_id": "370", "title": "Morphism Property Preserves Cancellability", "text": "Let: : $\\phi: \\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right) \\to \\left({T, *_1, *_2, \\ldots, *_n}\\right)$ be a mapping from one algebraic structure: : $\\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right)$ to another: : $\\left({T, *_1, *_2, \\ldots, *_n}\\right)$ Let $\\circ_k$ have the morphism property under $\\phi$ for some operation $\\circ_k$ in $\\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right)$. Then if an element $a \\in S$ is either left cancellable or right cancellable under $\\circ_k$, then $\\phi \\left({a}\\right)$ is correspondingly left cancellable or right cancellable under $*_k$. Thus, the morphism property is seen to preserve cancellability."}
+{"_id": "371", "title": "Quotient Mapping on Structure is Canonical Epimorphism", "text": "Let $\\mathcal R$ be a congruence relation on an algebraic structure $\\struct {S, \\circ}$. Then the quotient mapping from $\\struct {S, \\circ}$ to the quotient structure $\\struct {S / \\mathcal R, \\circ_\\mathcal R}$ is an epimorphism: :$q_\\mathcal R: \\struct {S, \\circ} \\to \\struct {S / \\mathcal R, \\circ_\\mathcal R}: \\forall x, y \\in S: \\map {q_\\mathcal R} {x \\circ y} = \\map {q_\\mathcal R} x \\circ_\\mathcal R \\map {q_\\mathcal R} y$ This is sometimes called the '''canonical epimorphism''' from $\\struct {S, \\circ}$ to $\\struct {S / \\mathcal R, \\circ_\\mathcal R}$."}
+{"_id": "372", "title": "Restriction of Homomorphism to Image is Epimorphism", "text": "Let $S$ and $T$ be algebraic structures. Let $\\phi: S \\to T$ be a homomorphism. Then a surjective restriction of $\\phi$ can be produced by limiting the codomain of $\\phi$ to its image $\\operatorname{Im} \\left({\\phi}\\right)$."}
+{"_id": "373", "title": "Epimorphism Preserves Associativity", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be algebraic structures. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be an epimorphism. Let $\\circ$ be an associative operation. Then $*$ is also an associative operation."}
+{"_id": "374", "title": "Epimorphism Preserves Semigroups", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be algebraic structures. Let $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ be an epimorphism. Let $\\left({S, \\circ}\\right)$ be a semigroup. Then $\\left({T, *}\\right)$ is also a semigroup."}
+{"_id": "375", "title": "Homomorphism Preserves Subsemigroups", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be semigroups. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be a homomorphism. Let $S'$ be a subsemigroup of $S$. Then $\\phi \\paren {S'}$ is a subsemigroup of $T$."}
+{"_id": "376", "title": "Epimorphism Preserves Commutativity", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be algebraic structures. Let $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ be an epimorphism. Let $\\circ$ be a commutative operation. Then $*$ is also a commutative operation."}
+{"_id": "377", "title": "Epimorphism Preserves Identity", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be algebraic structures. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be an epimorphism. Let $\\struct {S, \\circ}$ have an identity element $e_S$. Then $\\struct {T, *}$ has the identity element $\\map \\phi {e_S}$."}
+{"_id": "378", "title": "Epimorphism Preserves Inverses", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be algebraic structures. Let $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ be an epimorphism. Let $\\left({S, \\circ}\\right)$ have an identity $e_S$. Let $x^{-1}$ be an inverse element of $x$ for $\\circ$. Then $\\phi \\left({x^{-1}}\\right)$ is an inverse element of $\\phi \\left({x}\\right)$ for $*$. That is: : $\\phi \\left({x^{-1}}\\right) = \\left({\\phi \\left({x}\\right)}\\right)^{-1}$"}
+{"_id": "379", "title": "Homomorphism with Cancellable Codomain Preserves Identity", "text": "Let $\\struct{S, \\circ}$ and $\\struct{T, *}$ be algebraic structures. Let $\\phi: \\struct{S, \\circ} \\to \\struct{T, *}$ be a homomorphism. Let $\\struct{S, \\circ}$ have an identity $e_S$. Let $\\struct{T, *}$ have an identity $e_T$. Let every element of $\\struct{T, *}$ be cancellable. Then $\\map \\phi {e_S}$ is the identity $e_T$."}
+{"_id": "380", "title": "Homomorphism with Identity Preserves Inverses", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be algebraic structures. Let $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ be a homomorphism. Let $\\left({S, \\circ}\\right)$ have an identity $e_S$. Let $\\left({T, *}\\right)$ also have an identity $e_T = \\phi \\left({e_S}\\right)$. If $x^{-1}$ is an inverse of $x$ for $\\circ$, then $\\phi \\left({x^{-1}}\\right)$ is an inverse of $\\phi \\left({x}\\right)$ for $*$. That is: : $\\phi \\left({x^{-1}}\\right) = \\left({\\phi \\left({x}\\right)}\\right)^{-1}$"}
+{"_id": "381", "title": "Homomorphism to Group Preserves Identity", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\struct {T, *}$ be a group. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be a homomorphism. Let $\\struct {S, \\circ}$ have an identity $e_S$. Then: :$\\map \\phi {e_S} = e_T$"}
+{"_id": "382", "title": "Homomorphism of External Direct Products", "text": "Let: : $\\left({S_1 \\times S_2, \\circ}\\right)$ be the external direct product of two algebraic structures $\\left({S_1, \\circ_1}\\right)$ and $\\left({S_2, \\circ_2}\\right)$ : $\\left({T_1 \\times T_2, *}\\right)$ be the external direct product of two algebraic structures $\\left({T_1, *_1}\\right)$ and $\\left({T_2, *_2}\\right)$ : $\\phi_1$ be a homomorphism from $\\left({S_1, \\circ_1}\\right)$ onto $\\left({T_1, *_1}\\right)$ : $\\phi_2$ be a homomorphism from $\\left({S_2, \\circ_2}\\right)$ onto $\\left({T_2, *_2}\\right)$. Then the mapping $\\phi_1 \\times \\phi_2: \\left({S_1 \\times S_2, \\circ}\\right) \\to \\left({T_1 \\times T_2, *}\\right)$ defined as: :$\\left({\\phi_1 \\times \\phi_2}\\right) \\left({\\left({x, y}\\right)}\\right) = \\left({\\phi_1 \\left({x}\\right), \\phi_2 \\left({y}\\right)}\\right)$ is a homomorphism from $\\left({S_1 \\times S_2, \\circ}\\right)$ to $\\left({T_1 \\times T_2, *}\\right)$."}
+{"_id": "383", "title": "Monomorphism Image is Isomorphic to Domain", "text": "The image of a monomorphism is isomorphic to its domain. That is, if $\\phi: S_1 \\to S_2$ is a monomorphism, then: :$\\phi: S_1 \\to \\Img \\phi$ is an isomorphism."}
+{"_id": "384", "title": "Inverse of Algebraic Structure Isomorphism is Isomorphism", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be algebraic structures. Let $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ be a mapping. Then $\\phi$ is an isomorphism iff $\\phi^{-1}: \\left({T, *}\\right) \\to \\left({S, \\circ}\\right)$ is also an isomorphism."}
+{"_id": "385", "title": "Isomorphism is Equivalence Relation", "text": "Isomorphism is an equivalence on a set of magmas. This result applies to all magmas: rings, groups, R-algebraic structures etc."}
+{"_id": "386", "title": "Projection is Epimorphism", "text": "Let $\\struct {S, \\circ}$ be the external direct product of the algebraic structures $\\struct {S_1, \\circ_1}$ and $\\struct {S_2, \\circ_2}$. Then: :$\\pr_1$ is an epimorphism from $\\struct {S, \\circ}$ to $\\struct {S_1, \\circ_1}$ :$\\pr_2$ is an epimorphism from $\\struct {S, \\circ}$ to $\\struct {S_2, \\circ_2}$ where $\\pr_1$ and $\\pr_2$ are the first and second projection respectively of $\\struct {S, \\circ}$."}
+{"_id": "387", "title": "Quotient Theorem for Epimorphisms", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be algebraic structures. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be an epimorphism. Let $\\mathcal R_\\phi$ be the equivalence induced by $\\phi$. Let $S / \\mathcal R_\\phi$ be the quotient of $S$ by $\\mathcal R_\\phi$. Let $q_{\\mathcal R_\\phi}: S \\to S / \\mathcal R_\\phi$ be the quotient mapping induced by $\\mathcal R_\\phi$. Let $\\struct {S / \\mathcal R_\\phi}, {\\circ_{\\mathcal R_\\phi} }$ be the quotient structure defined by $\\mathcal R_\\phi$. Then: :The induced equivalence $\\mathcal R_\\phi$ is a congruence relation for $\\circ$ :There is one and only one isomorphism $\\psi: \\struct {S / \\mathcal R_\\phi}, {\\circ_{\\mathcal R_\\phi} } \\to \\struct {T, *}$ which satisfies $\\psi \\bullet q_{\\mathcal R_\\phi} = \\phi$. where, in order not to cause notational confusion, $\\bullet$ is used as the symbol to denote composition of mappings."}
+{"_id": "388", "title": "Homomorphism on Induced Structure", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\struct {T, \\oplus}$ be a commutative semigroup. Let $T^S$ be the set of all mappings from $S$ to $T$. Let $f$ and $g$ be homomorphisms from $S$ into $T$. Let $f \\oplus' g$ be the pointwise operation on $T^S$ induced by $\\oplus$. Then $f \\oplus' g$ is a homomorphism from $\\struct {S, \\circ}$ into $\\struct {T, \\oplus}$."}
+{"_id": "389", "title": "Inverse Mapping in Induced Structure", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure. Let $\\left({T, \\oplus}\\right)$ be an abelian group. Let $f$ be a homomorphism from $S$ into $T$. Let $f^*$ be the induced structure inverse of $f$. Then $f^*$ is a homomorphism from $\\left({S, \\circ}\\right)$ into $\\left({T, \\oplus}\\right)$."}
+{"_id": "390", "title": "Set of Homomorphisms is Subgroup of All Mappings", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\struct {T, \\oplus}$ be an abelian group. Let $\\struct {T^S, \\oplus}$ be the algebraic structure on $T^S$ induced by $\\oplus$. Then the set of all homomorphisms from $\\struct {S, \\circ}$ into $\\struct {T, \\oplus}$ is a subgroup of $\\struct {T^S, \\oplus}$."}
+{"_id": "391", "title": "Transplanting Theorem", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure. Let $f: S \\to T$ be a bijection. Then there exists one and only one operation $\\oplus$ such that $f: \\left({S, \\circ}\\right) \\to \\left({T, \\oplus}\\right)$ is an isomorphism. The operation $\\oplus$ is defined by: :$\\forall x, y \\in T: x \\oplus y = f \\left({f^{-1} \\left({x}\\right) \\circ f^{-1} \\left({y}\\right)}\\right)$ The operation $\\oplus$ is called the '''transplant of $\\circ$ under $f$'''."}
+{"_id": "392", "title": "Exists Bijection to a Disjoint Set", "text": "Let $S$ and $T$ be sets. Then there exists a bijection from $T$ onto a set $T'$ disjoint from $S$."}
+{"_id": "393", "title": "Embedding Theorem", "text": "Let: :$(1): \\quad \\left({T_2, \\oplus_2}\\right)$ be a submagma of $\\left({S_2, *_2}\\right)$ :$(2): \\quad f: \\left({T_1, \\oplus_1}\\right) \\to \\left({T_2, \\oplus_2}\\right)$ be an isomorphism then there exists: :$(1): \\quad$ a magma $\\left({S_1, *_1}\\right)$ which algebraically contains $\\left({T_1, \\oplus_1}\\right)$ :$(2): \\quad g: \\left({S_1, *_1}\\right) \\to \\left({S_2, *_2}\\right)$ where $g$ is an isomorphism which extends $f$. {{Expand|Add the corollary that extends this theorem to structures with two operations.}}"}
+{"_id": "396", "title": "Subset Product defining Inverse Completion of Commutative Semigroup is Commutative Semigroup", "text": "Let $\\left({S, \\circ}\\right)$ be a commutative semigroup. Let $\\left ({C, \\circ}\\right) \\subseteq \\left({S, \\circ}\\right)$ be the subsemigroup of cancellable elements of $\\left({S, \\circ}\\right)$. Let $\\left({T, \\circ'}\\right)$ be an inverse completion of $\\left({S, \\circ}\\right)$. Then: :$S \\circ' C^{-1}$ is a commutative semigroup where $S \\circ' C^{-1}$ is the subset product of $S$ with $C^{-1}$ under $\\circ'$ in $T$."}
+{"_id": "397", "title": "Inverse Completion of Commutative Semigroup is Inverse Completion of Itself", "text": "Let $\\left({S, \\circ}\\right)$ be a commutative semigroup. Let $\\left ({C, \\circ}\\right) \\subseteq \\left({S, \\circ}\\right)$ be the subsemigroup of cancellable elements of $\\left({S, \\circ}\\right)$. Let $\\left({T, \\circ'}\\right)$ be an inverse completion of $\\left({S, \\circ}\\right)$. Then $\\left({T, \\circ'}\\right)$ is its own inverse completion."}
+{"_id": "398", "title": "Identity of Inverse Completion of Commutative Monoid", "text": "Let $\\left({S, \\circ}\\right)$ be a commutative monoid whose identity is $e$. Let $\\left ({C, \\circ}\\right) \\subseteq \\left({S, \\circ}\\right)$ be the subsemigroup of cancellable elements of $\\left({S, \\circ}\\right)$. Let $\\left({T, \\circ'}\\right)$ be an inverse completion of $\\left({S, \\circ}\\right)$. Then $e \\in T$ is the identity for $\\circ'$."}
+{"_id": "399", "title": "Inverse Completion of Commutative Semigroup is Abelian Group", "text": "Let $\\struct {S, \\circ}$ be a commutative semigroup Let all the elements of $\\struct {S, \\circ}$ be cancellable. Then an inverse completion of $\\struct {S, \\circ}$ is an abelian group."}
+{"_id": "400", "title": "Taylor's Theorem", "text": "Every infinitely differentiable function can be approximated by a series of polynomials."}
+{"_id": "401", "title": "Inverse Completion Theorem", "text": "Every commutative semigroup containing cancellable elements admits an inverse completion."}
+{"_id": "402", "title": "Extension Theorem for Homomorphisms", "text": "Let $\\struct {S, \\circ}$ be a commutative semigroup with cancellable elements Let $\\struct {C, \\circ} \\subseteq \\struct {S, \\circ}$ be the subsemigroup of all cancellable elements of $S$ Let $\\struct {S', \\circ'}$ be an inverse completion of $\\struct {S, \\circ}$ Let $\\phi$ be a (semigroup) homomorphism from $\\struct {S, \\circ}$ into a semigroup $\\struct {T, *}$ such that $\\map \\phi y$ is invertible for all $y \\in C$. Then: :$(1): \\quad$ There is one and only one homomorphism $\\psi$ from $\\struct {S', \\circ'}$ into $\\struct {T, *}$ extending $\\phi$ :$(2): \\quad \\forall x \\in S, y \\in C: \\map \\psi {x \\circ' y^{-1} } = \\map \\phi * \\paren {\\map \\phi y}^{-1}$ :$(3): \\quad$ If $\\phi$ is a monomorphism, then so is $\\psi$."}
+{"_id": "403", "title": "Extension Theorem for Isomorphisms", "text": "Let the following conditions be fulfilled: : Let $\\left({S, \\circ}\\right)$ be a commutative semigroup with cancellable elements : Let $\\phi$ be an isomorphism from $\\left({S, \\circ}\\right)$ into a semigroup $\\left({T, *}\\right)$ : Let $\\left({S', \\circ'}\\right)$ be an inverse completion of $\\left({S, \\circ}\\right)$ : Let $\\left({T', \\circ'}\\right)$ be an inverse completion of $\\left({T, \\circ}\\right)$. Then there is a unique isomorphism $\\phi': S' \\to T'$ extending $\\phi$."}
+{"_id": "404", "title": "Inverse Completion is Unique", "text": "An inverse completion of a commutative semigroup is unique up to isomorphism."}
+{"_id": "405", "title": "Extension Theorem for Distributive Operations", "text": "Let $\\struct {R, *}$ be a commutative semigroup, all of whose elements are cancellable. Let $\\struct {T, *}$ be an inverse completion of $\\struct {R, *}$. Let $\\circ$ be an operation on $R$ which distributes over $*$. Then: :$(1): \\quad$ There is a unique operation $\\circ'$ on $T$ which distributes over $*$ in $T$ and induces on $R$ the operation $\\circ$ :$(2): \\quad$ If $\\circ$ is associative, then so is $\\circ'$ :$(3): \\quad$ If $\\circ$ is commutative, then so is $\\circ'$ :$(4): \\quad$ If $e$ is an identity for $\\circ$, then $e$ is also an identity for $\\circ'$ :$(5): \\quad$ Every element cancellable for $\\circ$ is also cancellable for $\\circ'$."}
+{"_id": "406", "title": "Monoid is not Empty", "text": "A monoid cannot be empty."}
+{"_id": "407", "title": "Group is not Empty", "text": "A group cannot be empty."}
+{"_id": "408", "title": "Identity is only Idempotent Element in Group", "text": "Every group has exactly one idempotent element: the identity."}
+{"_id": "409", "title": "Group Product Identity therefore Inverses", "text": "Let $g$ and $h$ be elements of a group $G$ whose identity element is $e$. Then if either: :$g h = e$ or: :$h g = e$ it follows that: :$g = h^{-1}$ and: :$h = g^{-1}$"}
+{"_id": "410", "title": "Self-Inverse Elements Commute iff Product is Self-Inverse", "text": "Let $\\struct {G, \\circ}$ be a group. Let $x, y \\in \\struct {G, \\circ}$, such that $x$ and $y$ are self-inverse. Then $x$ and $y$ commute {{iff}} $x \\circ y$ is also self-inverse."}
+{"_id": "411", "title": "Power Set of Group under Induced Operation is Semigroup", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\struct {\\powerset G, \\circ_\\mathcal P}$ be the algebraic structure consisting of the power set of $G$ and the operation induced on $\\powerset G$ by $\\circ$. Then $\\struct {\\powerset G, \\circ_\\mathcal P}$ is a semigroup."}
+{"_id": "412", "title": "Inverse of Product of Subsets of Group", "text": "Let $\\struct {G, \\circ}$ be a group. Let $X, Y \\subseteq G$. Then: :$\\paren {X \\circ Y}^{-1} = Y^{-1} \\circ X^{-1}$ where $X^{-1}$ is the inverse of $X$."}
+{"_id": "413", "title": "Regular Representations in Group are Permutations", "text": "Let $\\struct {G, \\circ}$ be a group. Let $a \\in G$ be any element of $G$. Then the left regular representation $\\lambda_a$ and the right regular representation $\\rho_a$ are permutations of $G$."}
+{"_id": "414", "title": "Set Equivalence of Regular Representations", "text": "If $S$ is a finite subset of a group $G$, then: :$\\card {a \\circ S} = \\card S = \\left|{S \\circ a}\\right|$ That is, $a \\circ S$, $S$ and $S \\circ a$ are equivalent: $a \\circ S \\sim S \\sim S \\circ a$."}
+{"_id": "415", "title": "Composition of Regular Representations", "text": "Let $\\left({S, *}\\right)$ be a semigroup. Let $\\lambda_x, \\rho_x$ be the left and right regular representations of $\\left({S, *}\\right)$ with respect to $x$. Let $\\lambda_x \\circ \\lambda_y$, $\\rho_x \\circ \\rho_y$ etc. be defined as the composition of the mappings $\\lambda_x$ and $\\lambda_y$ etc. Then $\\forall x, y \\in S$, the following results hold:"}
+{"_id": "416", "title": "All Elements Self-Inverse then Abelian", "text": "Let $\\struct {G, \\circ}$ be a group. Suppose that every element of $G$ is self-inverse. Then $G$ is abelian."}
+{"_id": "417", "title": "Commutation Property in Group", "text": "Let $\\struct {G, \\circ}$ be a group. Then $x$ and $y$ commute {{iff}} $x \\circ y \\circ x^{-1} = y$."}
+{"_id": "418", "title": "Identity Mapping is Automorphism", "text": "The identity mapping $I_S: \\left({S, \\circ}\\right) \\to \\left({S, \\circ}\\right)$ on the algebraic structure $\\left({S, \\circ}\\right)$ is an automorphism. Its image is $S$."}
+{"_id": "419", "title": "Group Homomorphism of Product with Inverse", "text": "Let $\\phi: \\struct {G, \\circ} \\to \\struct {H, *}$ be a group homomorphism. Then: :$(1): \\quad \\forall x, y \\in G: \\map \\phi {x \\circ y^{-1} } = \\map \\phi x * \\paren {\\map \\phi y}^{-1}$ :$(2): \\quad \\forall x, y \\in G: \\map \\phi {y^{-1} \\circ x} = \\paren {\\map \\phi y}^{-1} * \\map \\phi x$"}
+{"_id": "420", "title": "Mapping to Square is Endomorphism iff Abelian", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\phi: G \\to G$ be defined as: :$\\forall g \\in G: \\map \\phi g = g \\circ g$ Then $\\struct {G, \\circ}$ is abelian {{iff}} $\\phi$ is a (group) endomorphism."}
+{"_id": "421", "title": "Induced Group Product is Homomorphism iff Commutative", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H_1, H_2$ be subgroups of $G$. Let $\\phi: H_1 \\times H_2 \\to G$ be defined such that: :$\\forall \\tuple {h_1, h_2} \\in H_1 \\times H_2: \\map \\phi {h_1, h_2} = h_1 \\circ h_2$ Then $\\phi$ is a homomorphism {{iff}} every element of $H_1$ commutes with every element of $H_2$."}
+{"_id": "422", "title": "Isomorphism of Abelian Groups", "text": "Let $\\phi: \\struct {G, \\circ} \\to \\struct {H, *}$ be a group isomorphism. Then $\\struct {G, \\circ}$ is abelian {{iff}} $\\struct {H, *}$ is abelian."}
+{"_id": "424", "title": "Group Example: x inv c y", "text": "Let $\\struct {G, \\circ}$ be a group. Let $c \\in G$. We define a new operation $*$ on $G$ as: :$\\forall x, y \\in G: x * y = x \\circ c^{-1} \\circ y$ Then $\\struct {G, *}$ is a group."}
+{"_id": "425", "title": "Symmetric Difference on Power Set forms Abelian Group", "text": "Let $S$ be a set such that $\\O \\subset S$ (that is, $S$ is non-empty). Let $A * B$ be defined as the symmetric difference between $A$ and $B$. Let $\\powerset S$ denote the power set of $S$. Then the algebraic structure $\\struct {\\powerset S, *}$ is an abelian group."}
+{"_id": "426", "title": "Group is Subgroup of Itself", "text": "Let $\\struct {G, \\circ}$ be a group. Then: :$\\struct {G, \\circ} \\le \\struct {G, \\circ}$ That is, a group is always a subgroup of itself."}
+{"_id": "427", "title": "Identity of Subgroup", "text": "Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of group $G$. Then the identity of $H$ is also $e$."}
+{"_id": "428", "title": "Inverses in Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then for each $h \\in H$, the inverse of $h$ in $H$ is the same as the inverse of $h$ in $G$."}
+{"_id": "429", "title": "Subgroup of Abelian Group is Abelian", "text": "A subgroup of an abelian group is itself abelian."}
+{"_id": "430", "title": "Intersection of Subgroups is Subgroup", "text": "The intersection of two subgroups of a group is itself a subgroup of that group: :$\\forall H_1, H_2 \\le \\struct {G, \\circ}: H_1 \\cap H_2 \\le G$ It also follows that $H_1 \\cap H_2 \\le H_1$ and $H_1 \\cap H_2 \\le H_2$."}
+{"_id": "431", "title": "Union of Subgroups", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H, K \\le G$ be subgroups of $G$. Let neither $H \\subseteq K$ nor $K \\subseteq H$. Then $H \\cup K$ is ''not'' a subgroup of $G$."}
+{"_id": "432", "title": "Elements of Group with Equal Images under Homomorphisms form Subgroup", "text": "Let $\\struct {G, \\circ}$ and $\\struct {H, *}$ be groups. Let $f: G \\to H$ and $g: G \\to H$ be group homomorphisms. Then the set: :$S = \\set {x \\in G: \\map f x = \\map g x}$ is a subgroup of $G$."}
+{"_id": "433", "title": "Product of Subgroup with Itself", "text": "Let $\\struct {G, \\circ}$ be a group. Then: : $\\forall H \\le \\struct {G, \\circ}: H \\circ H = H$"}
+{"_id": "434", "title": "Inverse of Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H$ be a subgroup of $G$. Then: :$H^{-1} = H$ where $H^{-1}$ is the inverse of $H$."}
+{"_id": "435", "title": "Subset Product of Subgroups", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H, K$ be subgroups of $G$. Then $H \\circ K$ is a subgroup of $G$ {{iff}} $H$ and $K$ are permutable. That is: $H \\circ K$ is a subgroup of $G$ {{iff}}: :$H \\circ K = K \\circ H$ where $H \\circ K$ denotes subset product."}
+{"_id": "436", "title": "Group Homomorphism Preserves Subgroups", "text": "Let $\\struct {G_1, \\circ}$ and $\\struct {G_2, *}$ be groups. Let $\\phi: \\struct {G_1, \\circ} \\to \\struct {G_2, *}$ be a group homomorphism.  Then: :$H \\le G_1 \\implies \\phi \\sqbrk H \\le G_2$ where: :$\\phi \\sqbrk H$ denotes the image of $H$ under $\\phi$ :$\\le$ denotes subgroup. That is, group homomorphism preserves subgroups."}
+{"_id": "437", "title": "Image of Group Homomorphism is Subgroup", "text": "Let $\\phi: G_1 \\to G_2$ be a group homomorphism. Then: :$\\Img \\phi \\le G_2$ where $\\le$ denotes the relation of being a subgroup."}
+{"_id": "438", "title": "Conjugacy is Equivalence Relation", "text": "Conjugacy of group elements is an equivalence relation."}
+{"_id": "439", "title": "Kernel of Group Homomorphism is Subgroup", "text": "The kernel of a group homomorphism is a subgroup of its domain: :$\\map \\ker \\phi \\le \\Dom \\phi$"}
+{"_id": "441", "title": "Commutative Semigroup is Entropic Structure", "text": "A commutative semigroup is an entropic structure."}
+{"_id": "442", "title": "Abelian Group Induces Entropic Structure", "text": "Let $\\struct {G, \\circ}$ be an abelian group. Let the operation $*$ be defined on $G$ such that: :$\\forall x, y \\in G: x * y = x \\circ y^{-1}$ Then $\\struct {G, *}$ is an entropic structure."}
+{"_id": "443", "title": "Cancellable Semiring with Unity is Additive Semiring", "text": "Let $\\struct {S, *, \\circ}$ be a cancellable semiring with unity $1_S$. Then the distributand $*$ is commutative. That is to say, $\\struct {S, *, \\circ}$ is also an additive semiring."}
+{"_id": "444", "title": "Ring is not Empty", "text": "A ring cannot be empty."}
+{"_id": "445", "title": "Ring Product with Zero", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Then: :$\\forall x \\in R: 0_R \\circ x = 0_R = x \\circ 0_R$ That is, the zero is a zero element for the ring product, thereby justifying its name."}
+{"_id": "446", "title": "Product with Ring Negative", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Then: :$\\forall x, y \\in \\struct  {R, +, \\circ}: \\paren {-x} \\circ y = -\\paren {x \\circ y} = x \\circ \\paren {-y}$ where $\\paren {-x}$ denotes the negative of $x$."}
+{"_id": "447", "title": "Unity of Ring is Unique", "text": "A ring can have no more than one unity."}
+{"_id": "449", "title": "Null Ring iff Zero and Unity Coincide", "text": "The null ring is the only ring in which the unity and zero coincide."}
+{"_id": "450", "title": "Unity is Unit", "text": "The unity in a ring is a unit."}
+{"_id": "451", "title": "Unity and Negative form Subgroup of Units", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity. Then: :$\\struct {\\set {1_R, -1_R}, \\circ} \\le U_R$ That is, the set consisting of the unity and its negative forms a subgroup of the group of units."}
+{"_id": "452", "title": "Negative of Product Inverse", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring with unity. Let $z \\in U_R$, where $U_R$ is the set of units. Then: :$\\left({- z}\\right)^{-1} = - \\left({z^{-1}}\\right)$. where $z^{-1}$ is the ring product inverse of $z$."}
+{"_id": "453", "title": "Product of Negative with Product Inverse", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring with unity. Let $z \\in U_R$, where $U_R$ is the set of units. Then: : $(1): \\quad \\forall x \\in R: -\\left({x \\circ z^{-1}}\\right) = \\left({- x}\\right) \\circ z^{-1} = x \\circ \\left({\\left({- z}\\right)^{-1}}\\right)$ : $(2): \\quad \\forall x \\in R: -\\left({z^{-1} \\circ x}\\right) = z^{-1} \\circ \\left({- x}\\right) = \\left({\\left({- z}\\right)^{-1}}\\right) \\circ x$"}
+{"_id": "454", "title": "Negative of Division Product", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $\\struct {U_R, \\circ}$ be the group of units of $\\struct {R, +, \\circ}$. Then: :$\\displaystyle \\forall x \\in R: -\\frac x z = \\frac {-x} z = \\frac x {-z}$ where $\\dfrac x z$ is defined as $x \\circ \\paren {z^{-1} }$, that is the division product of $x$ by $z$."}
+{"_id": "455", "title": "Addition of Division Products", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $\\struct {U_R, \\circ}$ be the group of units of $\\struct {R, +, \\circ}$. Let $a, c \\in R, b, d \\in U_R$. Then: :$\\dfrac a b + \\dfrac c d = \\dfrac {a \\circ d + b \\circ c} {b \\circ d}$ where $\\dfrac x z$ is defined as $x \\circ \\paren {z^{-1} }$, that is, $x$ divided by $z$. The operation $+$ is well-defined. That is: :$\\dfrac a b = \\dfrac {a'} {b'}, \\dfrac c d = \\dfrac {c'} {d'} \\implies \\dfrac a b + \\dfrac c d = \\dfrac {a'} {b'} + \\dfrac {c'} {d'}$ {{questionable|This is an existing operation and thus already well-defined. See talk.}}"}
+{"_id": "456", "title": "Equality of Division Products", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $\\struct {U_R, \\circ}$ be the group of units of $\\struct {R, +, \\circ}$. Let $a, b \\in R, c, d \\in U_R$. Then: :$\\dfrac a c = \\dfrac b d \\iff a \\circ d = b \\circ c$ where $\\dfrac x z$ is defined as $x \\circ \\paren {z^{-1} }$, that is, $x$ divided by $z$."}
+{"_id": "457", "title": "Product of Division Products", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $\\struct {U_R, \\circ}$ be the group of units of $\\struct {R, +, \\circ}$. Let $a, b \\in R, c, d \\in U_R$. Then: :$\\dfrac a c \\circ \\dfrac b d = \\dfrac {a \\circ b} {c \\circ d}$ where $\\dfrac x z$ is defined as $x \\circ \\paren {z^{-1} }$, that is, $x$ divided by $z$."}
+{"_id": "458", "title": "Inverse of Division Product", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $\\struct {U_R, \\circ}$ be the group of units of $\\struct {R, +, \\circ}$. Let $a, b \\in U_R$. Then: :$\\paren {\\dfrac a b}^{-1} = \\dfrac {1_R} {\\paren {a / b}} = \\dfrac b a$ where $\\dfrac x z$ is defined as $x \\circ \\paren {z^{-1} }$, that is, $x$ divided by $z$."}
+{"_id": "459", "title": "Zero Product with Proper Zero Divisor is with Zero Divisor", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $x \\in R$ be a proper zero divisor of $R$. Then: :$\\paren {x \\divides 0_R} \\land \\paren {x \\circ y = 0_R} \\land \\paren {y \\ne 0_R} \\implies y \\divides 0_R$ That is, if $x$ is a proper zero divisor, then whatever non-zero element you form the product with it by to get zero must itself be a zero divisor."}
+{"_id": "460", "title": "Unit Not Zero Divisor", "text": "A unit of a ring is not a zero divisor."}
+{"_id": "461", "title": "Zero Divisor Product is Zero Divisor", "text": "The ring product of a zero divisor with any ring element is a zero divisor."}
+{"_id": "462", "title": "Product is Zero Divisor means Zero Divisor", "text": "If the ring product of two elements of a ring is a zero divisor, then one of the two elements must be a zero divisor."}
+{"_id": "463", "title": "Ring Element is Zero Divisor iff not Cancellable", "text": "Let $\\struct {R, +, \\circ}$ be a ring which is not null. Let $z \\in R^*$. Then $z$ is a zero divisor {{iff}} $z$ is not cancellable for $\\circ$."}
+{"_id": "464", "title": "Ring Less Zero is Semigroup for Product iff No Proper Zero Divisors", "text": "Let $\\struct {R, +, \\circ}$ be a non-null ring. Then $R$ has no zero divisors {{iff}} $\\struct {R^*, \\circ}$ is a semigroup."}
+{"_id": "465", "title": "Idempotent Elements of Ring with No Proper Zero Divisors", "text": "Let $\\left({R, +, \\circ}\\right)$ be a non-null ring with no (proper) zero divisors. Let $x \\in R$. Then: :$x \\circ x = x \\iff x \\in \\left\\{{0_R, 1_R}\\right\\}$ That is, the only elements of $R$ that are idempotent are zero and unity."}
+{"_id": "466", "title": "Non-Zero Elements of Division Ring form Group", "text": "Let $\\struct {R, +, \\circ}$ be a division ring. Then $\\struct {R^*, \\circ}$ is a group."}
+{"_id": "467", "title": "Null Ring and Ring Itself Subrings", "text": "In any ring $R$, the null ring and $R$ itself are subrings of $R$."}
+{"_id": "468", "title": "Subring Test", "text": "Let $S$ be a subset of a ring $\\struct {R, +, \\circ}$. Then $\\struct {S, +, \\circ}$ is a subring of $\\struct {R, +, \\circ}$ {{iff}} these all hold: :$(1): \\quad S \\ne \\O$ :$(2): \\quad \\forall x, y \\in S: x + \\paren {-y} \\in S$ :$(3): \\quad \\forall x, y \\in S: x \\circ y \\in S$"}
+{"_id": "469", "title": "Subdomain Test", "text": "Let $S$ be a subset of an integral domain $\\struct {R, +, \\circ}$. Then $\\struct {S, +\\restriction_S, \\circ \\restriction_S}$ is a subdomain of $\\struct {R, +, \\circ}$ {{iff}} these conditions hold: :$(1): \\quad$ $\\struct {S, + \\restriction_S, \\circ \\restriction_S}$ is a subring of $\\struct {R, +, \\circ}$ :$(2): \\quad$ The unity of $R$ is also in $S$, that is $1_R = 1_S$."}
+{"_id": "470", "title": "Centralizer of Ring Subset is Subring", "text": "Let $S$ be a subset of a ring $\\struct {R, +, \\circ}$ Then $\\map {C_R} S$, the centralizer of $S$ in $R$, is a subring of $R$."}
+{"_id": "471", "title": "Center of Ring is Commutative Subring", "text": "The center $\\map Z R$ of a ring $R$ is a commutative subring of $R$."}
+{"_id": "472", "title": "Ideal is Subring", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring, and let $J$ be an ideal of $R$. Then $J$ is a subring of $R$."}
+{"_id": "473", "title": "Ring is Ideal of Itself", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Then $R$ is an ideal of $R$."}
+{"_id": "474", "title": "Ideal of Unit is Whole Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity. Let $J$ be an ideal of $R$. If $J$ contains a unit of $R$, then $J = R$."}
+{"_id": "475", "title": "Test for Ideal", "text": "Let $J$ be a subset of a ring $\\struct {R, +, \\circ}$. Then $J$ is an ideal of $\\struct {R, +, \\circ}$ {{iff}} these all hold: : $(1): \\quad J \\ne \\O$ : $(2): \\quad \\forall x, y \\in J: x + \\paren {-y} \\in J$ : $(3): \\quad \\forall j \\in J, r \\in R: r \\circ j \\in J, j \\circ r \\in J$"}
+{"_id": "476", "title": "Epimorphism Preserves Rings", "text": "Let $\\struct {R_1, +_1, \\circ_1}$ be a ring, and $\\struct {R_2, +_2, \\circ_2}$ be a closed algebraic structure. Let $\\phi: R_1 \\to R_2$ be an epimorphism. Then $\\struct {R_2, +_2, \\circ_2}$ is a ring."}
+{"_id": "477", "title": "Ring Homomorphism of Addition is Group Homomorphism", "text": "Let $\\phi: \\left({R_1, +_1, \\circ_1}\\right) \\to \\left({R_2, +_2, \\circ_2}\\right)$ be a ring homomorphism. Then $\\phi: \\left({R_1, +_1}\\right) \\to \\left({R_2, +_2}\\right)$ is a group homomorphism."}
+{"_id": "478", "title": "Element of Integral Domain Divides Zero", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose zero is $0_D$. Then every element of $D$ is a divisor of zero: :$\\forall x \\in D: x \\divides 0_D$"}
+{"_id": "479", "title": "Unity Divides All Elements", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose unity is $1_D$. Then unity is a divisor of every element of $D$: :$\\forall x \\in D: 1_D \\divides x$ Also: :$\\forall x \\in D: -1_D \\divides x$"}
+{"_id": "480", "title": "Element of Integral Domain is Divisor of Itself", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose unity is $1_D$. Then every element of $D$ is a divisor of itself: :$\\forall x \\in D: x \\divides x$"}
+{"_id": "481", "title": "Unit of Integral Domain divides all Elements", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose unity is $1_D$. Let $\\struct {U_D, \\circ}$ be the group of units of $\\struct {D, +, \\circ}$. Then: :$\\forall x \\in D: \\forall u \\in U_D: u \\divides x$ That is, every unit of $D$ is a divisor of every element of $D$."}
+{"_id": "484", "title": "Divisor Relation in Integral Domain is Transitive", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain. Let $x, y, z \\in D$. Then: :$x \\divides y \\land y \\divides z \\implies x \\divides z$"}
+{"_id": "485", "title": "Integers form Unique Factorization Domain", "text": "The integers $\\struct {\\Z, +, \\times}$ form a unique factorization domain."}
+{"_id": "486", "title": "Trivial Ordering Compatibility in Boolean Ring", "text": "Let $\\struct {S, +, \\circ}$ be a Boolean ring. Then the trivial ordering is the only ordering on $S$ compatible with both its operations."}
+{"_id": "488", "title": "Field is Integral Domain", "text": "Every field is an integral domain."}
+{"_id": "489", "title": "Finite Integral Domain is Galois Field", "text": "A finite integral domain is a Galois field."}
+{"_id": "491", "title": "Center of Division Ring is Subfield", "text": "Let $\\struct {K, +, \\circ}$ be an division ring. Then $\\map Z K$, the center of $K$, is a subfield of $K$."}
+{"_id": "492", "title": "Ideals of Field", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Then $\\struct {R, +, \\circ}$ is a field {{iff}} the only ideals of $\\struct {R, +, \\circ}$ are $\\struct {R, +, \\circ}$ and $\\set {0_R}$."}
+{"_id": "493", "title": "Epimorphism from Division Ring to Ring", "text": "Let $\\left({K, +, \\circ}\\right)$ be a division ring whose zero is $0_K$. Let $\\left({R, +, \\circ}\\right)$ be a ring whose zero is $0_R$. Let $\\phi: K \\to R$ be a ring epimorphism. Then one of the following applies: :$(1): \\quad R$ is a null ring :$(2): \\quad R$ is a division ring and $\\phi$ is a ring isomorphism."}
+{"_id": "494", "title": "Congruence Class Modulo Subgroup is Coset", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$."}
+{"_id": "495", "title": "Cosets are Equivalent", "text": "All left cosets of a group $G$ with respect to a subgroup $H$ are equivalent. That is, any two left cosets are in one-to-one correspondence. The same applies to right cosets. As a special case of this: :$\\forall x \\in G: \\order {x H} = \\order H = \\order {H x}$ where $H$ is a subgroup of $G$."}
+{"_id": "496", "title": "Left and Right Coset Spaces are Equivalent", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H$ be a subgroup of $G$. Let: :$x H$ denote the left coset of $H$ by $x$ :$H y$ denote the right coset of $H$ by $y$. Then: :$\\order {\\set {x H: x \\in G} } = \\order {\\set {H y: y \\in G} }$"}
+{"_id": "497", "title": "Cosets in Abelian Group", "text": "Let $G$ be an abelian group. Then every right coset modulo $H$ is a left coset modulo $H$. That is: :$\\forall x \\in G: x H = H x$ In an abelian group, therefore, we can talk about '''congruence modulo $H$''' and not worry about whether it is left or right."}
+{"_id": "498", "title": "Conjugate of Set by Identity", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $S \\subseteq G$. Then the conjugate of $S$ by $e$ is $S$: :$S^e = S$"}
+{"_id": "499", "title": "Conjugate of Set by Group Product", "text": "Let $\\struct {G, \\circ}$ be a group. Let $S \\subseteq G$. Let $S^a$ denote the $G$-conjugate of $S$ by $a$ as: :$S^a := \\set {y \\in G: \\exists x \\in S: y = a \\circ x \\circ a^{-1} } = a \\circ S \\circ a^{-1}$ Then: :$\\paren {S^a}^b = S^{b \\circ a}$"}
+{"_id": "500", "title": "Conjugate of Subgroup is Subgroup", "text": "Let $G$ be a group. Let $H \\le G$ be a subgroup of $G$. Then the conjugate of $H$ by $a$ is a subgroup of $G$: :$\\forall H \\le G, a \\in G: H^a \\le G$"}
+{"_id": "501", "title": "Inner Automorphisms form Normal Subgroup of Automorphism Group", "text": "Let $G$ be a group. Then the set $\\Inn G$ of all inner automorphisms of $G$ is a normal subgroup of the automorphism group $\\Aut G$ of $G$: :$\\Inn G \\lhd \\Aut G$"}
+{"_id": "503", "title": "Subgroup of Abelian Group is Normal", "text": "Every subgroup of an abelian group is normal."}
+{"_id": "505", "title": "Intersection with Normal Subgroup is Normal", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$, and let $N$ be a normal subgroup of $G$. Then $H \\cap N$ is a normal subgroup of $H$."}
+{"_id": "506", "title": "Kernel is Normal Subgroup of Domain", "text": "Let $\\phi$ be a group homomorphism. Then the kernel of $\\phi$ is a normal subgroup of the domain of $\\phi$: :$\\map \\ker \\phi \\lhd \\Dom \\phi$"}
+{"_id": "507", "title": "Coset Product is Well-Defined", "text": "Let $\\struct {G, \\circ}$ be a group. Let $N$ be a normal subgroup of $G$. Let $a, b \\in G$. Then the coset product: :$\\paren {a \\circ N} \\circ \\paren {b \\circ N} = \\paren {a \\circ b} \\circ N$ is well-defined."}
+{"_id": "508", "title": "Quotient Group of Abelian Group is Abelian", "text": "Let $G$ be an abelian group. Let $N \\le G$. Then the quotient group $G / N$ is abelian."}
+{"_id": "509", "title": "Quotient Theorem for Group Epimorphisms", "text": "Let $\\struct {G, \\oplus}$ and $\\struct {H, \\odot}$ be groups. Let $\\phi: \\struct {G, \\oplus} \\to \\struct {H, \\odot}$ be a group epimorphism. Let $e_G$ and $e_H$ be the identities of $G$ and $H$ respectively. Let $K = \\map \\ker \\phi$ be the kernel of $\\phi$. There is one and only one group isomorphism $\\psi: G / K \\to H$ satisfying: :$\\psi \\circ q_K = \\phi$ where $q_K$ is the quotient epimorphism from $G$ to $G / K$."}
+{"_id": "510", "title": "Congruence Relation induces Normal Subgroup", "text": "Let $\\left({G, \\circ}\\right)$ be a group whose identity is $e$. Let $\\mathcal R$ be a congruence relation for $\\circ$. Let $H = \\left[\\!\\left[{e}\\right]\\!\\right]_\\mathcal R$, where $\\left[\\!\\left[{e}\\right]\\!\\right]_\\mathcal R$ is the equivalence class of $e$ under $\\mathcal R$. Then: : $(1): \\quad \\left({H, \\circ \\restriction_H}\\right)$ is a normal subgroup of $G$ : $(2): \\quad \\mathcal R$ is the equivalence relation $\\mathcal R_H$ defined by $H$ : $(3): \\quad \\left({G / \\mathcal R, \\circ_\\mathcal R}\\right)$ is the subgroup $\\left({G / H, \\circ_H}\\right)$ of the semigroup $\\left({\\mathcal P \\left({G}\\right), \\circ_\\mathcal P}\\right)$."}
+{"_id": "511", "title": "Preimage of Normal Subgroup of Quotient Group under Quotient Epimorphism is Normal", "text": "Let $G$ be a group. Let $H \\lhd G$ where $\\lhd$ denotes that $H$ is a normal subgroup of $G$. Let $K \\lhd G/H$ and $L = q_H^{-1} \\left[{K}\\right]$, where: :$q_H: G \\to G/H$ is the quotient epimorphism from $G$ to the quotient group $G/H$ :$q_H^{-1} \\left[{K}\\right]$ is the preimage of $K$ under $q_H$. Then: :$L \\lhd G$"}
+{"_id": "512", "title": "Trivial Quotient Group is Quotient Group", "text": "Let $G$ be a group. Then the trivial quotient group: :$G / \\set {e_G} \\cong G$ where: :$\\cong$ denotes group isomorphism :$e_G$ denotes the identity element of $G$ is a quotient group."}
+{"_id": "513", "title": "Correspondence Theorem (Group Theory)", "text": "Let $G$ be a group. Let $N \\lhd G$ be a normal subgroup of $G$. Then every subgroup of the quotient group $G / N$ is of the form $H / N = \\set {h N: h \\in H}$, where $N \\le H \\le G$. Conversely, if $N \\le H \\le G$ then $H / N \\le G / N$. The correspondence between subgroups of $G / N$ and subgroups of $G$ containing $N$ is a bijection. This bijection maps normal subgroups of $G / N$ onto normal subgroups of $G$ which contain $N$."}
+{"_id": "514", "title": "Centralizer of Group Element is Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group and let $a \\in G$. Then $\\map {C_G} a$, the centralizer of $a$ in $G$, is a subgroup of $G$."}
+{"_id": "515", "title": "Centralizer in Subgroup is Intersection", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then: :$\\forall x \\in G: \\map {C_H} x = \\map {C_G} x \\cap H$ That is, the centralizer of an element in a subgroup is the intersection of that subgroup with the centralizer of the element in the group."}
+{"_id": "516", "title": "Kernel of Inner Automorphism Group is Center", "text": "Let the mapping $\\kappa: G \\to \\Inn G$ from a group $G$ to its inner automorphism group $\\Inn G$ be defined as: :$\\map \\kappa a = \\kappa_a$ where $\\kappa_a$ is the inner automorphism of $G$ given by $a$. Then $\\kappa$ is a group epimorphism, and its kernel is the center of $G$: :$\\map \\ker \\kappa = \\map Z G$"}
+{"_id": "517", "title": "Group equals Center iff Abelian", "text": "Let $G$ be a group. Then $G$ is abelian {{iff}} $\\map Z G = G$, that is, {{iff}} $G$ equals its center."}
+{"_id": "518", "title": "Normalizer is Subgroup", "text": "Let $G$ be a group. The normalizer of a subset $S \\subseteq G$ is a subgroup of $G$. :$S \\subseteq G \\implies \\map {N_G} S \\le G$"}
+{"_id": "519", "title": "Subgroup is Normal Subgroup of Normalizer", "text": "Let $G$ be a group. A subgroup $H \\le G$ is a normal subgroup of its normalizer: :$H \\le G \\implies H \\lhd \\map {N_G} H$"}
+{"_id": "520", "title": "Normalizer of Subgroup is Largest Subgroup containing that Subgroup as Normal Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $\\map {N_G} H$, the normalizer of $H$ in $G$, is the largest subgroup of $G$ containing $H$ as a normal subgroup."}
+{"_id": "521", "title": "Normal Subgroup iff Normalizer is Group", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is normal in $G$ {{iff}} the normalizer of $H$ is equal to $G$: :$H \\lhd G \\iff \\map {N_G} H = G$"}
+{"_id": "522", "title": "Normalizer of Conjugate is Conjugate of Normalizer", "text": "The normalizer of a conjugate is the conjugate of the normalizer: :$S \\subseteq G \\implies \\map {N_G} {S^a} = \\paren {\\map {N_G} S}^a$"}
+{"_id": "525", "title": "Quotient Ring is Ring/Quotient Ring Product is Well-Defined", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let $\\struct {R / J, +, \\circ}$ be the quotient ring of $R$ by $J$. Then $\\circ$ is well-defined on $R / J$, that is: :$x_1 + J = x_2 + J, y_1 + J = y_2 + J \\implies x_1 \\circ y_1 + J = x_2 \\circ y_2 + J$"}
+{"_id": "526", "title": "Quotient Ring of Commutative Ring is Commutative", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let $\\struct {R / J, +, \\circ}$ be the quotient ring defined by $J$. If $\\struct {R, +, \\circ}$ is a commutative ring, then so is $\\struct {R / J, +, \\circ}$."}
+{"_id": "527", "title": "Quotient Ring of Ring with Unity is Ring with Unity", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let $\\struct {R / J, +, \\circ}$ be the quotient ring defined by $J$. Then $\\struct {R / J, +, \\circ}$ is a ring with unity, and its unity is $1_R + J$."}
+{"_id": "528", "title": "Ring Epimorphism with Trivial Kernel is Isomorphism", "text": "Let $\\phi: \\left({R_1, +_1, \\circ_1}\\right) \\to \\left({R_2, +_2, \\circ_2}\\right)$ be a ring epimorphism. Then $\\phi$ is a ring isomorphism iff $\\ker \\left({\\phi}\\right) = \\left\\{ {0_{R_1} }\\right\\}$."}
+{"_id": "529", "title": "Quotient Epimorphism is Epimorphism/Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let $\\struct {R / J, +, \\circ}$ be the quotient ring defined by $J$. Let $\\phi: R \\to R / J$ be the quotient (ring) epimorphism from $R$ to $R / J$: :$x \\in R: \\map \\phi x = x + J$ Then $\\phi$ is a ring epimorphism whose kernel is $J$."}
+{"_id": "530", "title": "Ring Homomorphism Preserves Subrings", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring homomorphism. If $S$ is a subring of $R_1$, then $\\phi \\sqbrk S$ is a subring of $R_2$."}
+{"_id": "531", "title": "Kernel of Ring Homomorphism is Subring", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring homomorphism. Then the kernel of $\\phi$ is a subring of $R_1$."}
+{"_id": "532", "title": "Kernel of Ring Homomorphism is Ideal", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring homomorphism. Then the kernel of $\\phi$ is an ideal of $R_1$."}
+{"_id": "533", "title": "Ideals of Division Ring", "text": "Let $\\struct {R, +, \\circ}$ be a division ring whose zero is $0_R$. The only ideals of $\\struct {R, +, \\circ}$ are $\\set {0_R}$ and $R$ itself. That is, $\\struct {R, +, \\circ}$ has no non-null proper ideals."}
+{"_id": "534", "title": "Quotient Ring of Kernel of Ring Epimorphism", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring epimorphism. Let $K = \\map \\ker \\phi$. Then there is a unique ring isomorphism $g: R_1 / K \\to R_2$ such that: :$g \\circ q_K = \\phi$ $\\phi$ is an isomorphism {{iff}} $K = \\set {0_{R_1} }$."}
+{"_id": "535", "title": "Ring Epimorphism Preserves Ideals", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring epimorphism. Let $J$ be an ideal of $R_1$. Then $\\phi \\sqbrk J$ is an ideal of $R_2$."}
+{"_id": "536", "title": "Preimage of Image of Subring under Ring Homomorphism", "text": "Let $\\phi: \\left({R_1, +_1, \\circ_1}\\right) \\to \\left({R_2, +_2, \\circ_2}\\right)$ be a ring homomorphism. Let $K = \\ker \\left({\\phi}\\right)$, where $\\ker \\left({\\phi}\\right)$ is the kernel of $\\phi$.  Let $J$ be a subring of $R_1$. Then: :$\\phi^{-1} \\left[{\\phi \\left[{J}\\right]}\\right] = J + K$"}
+{"_id": "537", "title": "Preimage of Subring under Ring Homomorphism is Subring", "text": "Let $\\phi: \\left({R_1, +_1, \\circ_1}\\right) \\to \\left({R_2, +_2, \\circ_2}\\right)$ be a ring homomorphism. Let $S_2$ be a subring of $R_2$. Then $S_1 = \\phi^{-1} \\left[{S_2}\\right]$ is a subring of $R_1$ such that $\\ker \\left({\\phi}\\right) \\subseteq S_1$."}
+{"_id": "538", "title": "Preimage of Ideal under Ring Homomorphism is Ideal", "text": "Let $\\phi: \\left({R_1, +_1, \\circ_1}\\right) \\to \\left({R_2, +_2, \\circ_2}\\right)$ be a ring homomorphism. Let $S_2$ be an ideal of $R_2$. Then $S_1 = \\phi^{-1} \\left[{S_2}\\right]$ is an ideal of $R_1$ such that $\\ker \\left({\\phi}\\right) \\subseteq S_1$."}
+{"_id": "539", "title": "Image of Preimage of Subring under Ring Epimorphism", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring epimorphism. Let $S_2$ be a subring of $R_2$. Then: :$\\phi \\sqbrk {\\phi^{-1} \\sqbrk {S_2} } = S_2$"}
+{"_id": "541", "title": "Properties of Ordered Ring", "text": "Let $\\struct {R, +, \\circ, \\le}$ be an ordered ring whose zero is $0_R$ and whose unity is $1_R$. Let $U_R$ be the group of units of $R$. Let $x, y, z \\in \\struct {R, +, \\circ, \\le}$. Then the following properties hold: : $(1): \\quad x < y \\iff x + z < y + z$. Hence $x \\le y \\iff x + z \\le y + z$ (because $\\struct {R, +, \\le}$ is an ordered group). : $(2): \\quad x < y \\iff 0 < y + \\paren {-x}$. Hence $x \\le y \\iff 0 \\le y + \\paren {-x}$ : $(3): \\quad 0 < x \\iff \\paren {-x} < 0$. Hence $0 \\le x \\iff \\paren {-x} \\le 0$ : $(4): \\quad x < 0 \\iff 0 < \\paren {-x}$. Hence $x \\le 0 \\iff 0 \\le \\paren {-x}$ : $(5): \\quad \\forall n \\in \\Z_{>0}: x > 0 \\implies n \\cdot x > 0$ : $(6): \\quad x \\le y, 0 \\le z: x \\circ z \\le y \\circ z, z \\circ x \\le z \\circ y$ : $(7): \\quad x \\le y, z \\le 0: y \\circ z \\le x \\circ z, z \\circ y \\le z \\circ x$"}
+{"_id": "542", "title": "Positive Elements of Ordered Ring", "text": "Let $\\struct {R, +, \\circ, \\le}$ be an ordered ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $P$ be the set of positive elements of $R$ , that is, $P = R_{\\ge 0}$. Then: : $(1): \\quad P + P \\subseteq P$ : $(2): \\quad P \\cap \\paren {-P} = \\set {0_R}$ : $(3): \\quad P \\circ P \\subseteq P$ If $\\le$ is a total ordering, that is, if $\\struct {R, +, \\circ, \\le}$ is a totally ordered ring, then: : $(4): \\quad P \\cup \\paren {-P} = R$ The converse is also true: Let $\\struct {R, +, \\circ}$ be a ring. Let $P \\subseteq R$ such that $(1)$, $(2)$ and $(3)$ are satisfied. Then there is one and only one ordering $\\le$ compatible with the ring structure of $R$ such that $P = R_{\\ge 0}$. Also, if $(4)$ is also satisfied, then $\\le$ is a total ordering."}
+{"_id": "543", "title": "Symmetric Difference with Intersection forms Ring", "text": "Let $S$ be a set. Then $\\struct {\\powerset S, *, \\cap}$ is a commutative ring with unity, in which the unity is $S$. This ring is not an integral domain."}
+{"_id": "544", "title": "Subfield Test", "text": "Let $\\struct {F, +, \\circ}$ be a field whose zero is $0_F$. Let $K$ be a subset of $F$. Then $\\struct {K, +, \\circ}$ is a subfield of $\\struct {F, +, \\circ}$ {{iff}} these all hold: :$(1): \\quad K^* \\ne \\O$ :$(2): \\quad \\forall x, y \\in K: x + \\paren {-y} \\in K$ :$(3): \\quad \\forall x, y \\in K: x \\circ y \\in K$ :$(4): \\quad x \\in K^* \\implies x^{-1} \\in K^*$ where $K^*$ denotes $K \\setminus \\set {0_F}$."}
+{"_id": "545", "title": "Field of Quotients of Subdomain", "text": "Let $\\struct {F, +, \\circ}$ be a field whose unity is $1_F$. Let $\\struct {D, +, \\circ}$ be a subdomain of $\\struct {F, +, \\circ}$ whose unity is $1_D$. Let: :$K = \\set {\\dfrac x y: x \\in D, y \\in D^*}$ where $\\dfrac x y$ is the division product of $x$ by $y$. Then $\\struct {K, +, \\circ}$ is a field of quotients of $\\struct {D, +, \\circ}$."}
+{"_id": "546", "title": "Existence of Field of Quotients", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain. Then there exists a field of quotients of $\\struct {D, +, \\circ}$."}
+{"_id": "547", "title": "Quotient Theorem for Monomorphisms", "text": "Let $K, L$ be fields of quotients of integral domains $\\struct {R, +_R, \\circ_R}, \\struct {S, +_S, \\circ_S}$ respectively. Let $\\phi: R \\to S$ be a monomorphism. Then there is one and only one monomorphism $\\psi: K \\to L$ extending $\\phi$, and: :$\\forall x \\in R, y \\in R^*: \\map \\psi {\\dfrac x y} = \\dfrac {\\map \\phi x} {\\map \\phi y}$ Also, if $\\phi$ is a ring isomorphism, then so is $\\psi$."}
+{"_id": "548", "title": "Field of Quotients is Unique", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain. Let $K, L$ be field of quotients of $\\struct {D, +, \\circ}$. Then there is one and only one (field) isomorphism $\\phi: K \\to L$ satisfying: :$\\forall x \\in D: \\map \\phi x = x$"}
+{"_id": "549", "title": "Divided by Positive Element of Field of Quotients", "text": "Let $\\struct {K, +, \\circ}$ be the field of quotients of a totally ordered integral domain $\\struct {D, +, \\circ, \\le}$. Then: :$\\forall z \\in K: \\exists x, y \\in D: z = \\dfrac x y, y \\in D_{>0}$"}
+{"_id": "550", "title": "Total Ordering on Field of Quotients is Unique", "text": "Let $\\struct {K, +, \\circ}$ be a field of quotients of an ordered integral domain $\\struct {D, +, \\circ, \\le}$. Then there is one and only one total ordering $\\le'$ on $K$ which is compatible with its ring structure and induces on $D$ its given total ordering $\\le$. That ordering is the one defined by: :$P = \\set {\\dfrac x y \\in K: x \\in D_+, y \\in D_+^*}$"}
+{"_id": "551", "title": "Order Embedding between Quotient Fields is Unique", "text": "Let $\\struct {R_1, +_1, \\circ_1, \\le_1}$ and $\\struct {S, +_2, \\circ_2, \\le_2}$ be totally ordered integral domains. Let $K, L$ be totally ordered fields of quotients of $\\struct {R_1, +_1, \\circ_1, \\le_1}$ and $\\struct {S, +_2, \\circ_2, \\le_2}$ respectively. Let $\\phi: R \\to S$ be a order embedding. Then there is one and only one order embedding $\\psi: K \\to L$ extending $\\phi$. Also: :$\\forall x \\in R, y \\in R_{\\ne 0}: \\map \\psi {\\dfrac x y} = \\dfrac {\\map \\phi x} {\\map \\phi y}$ If $\\phi: R \\to S$ is an order isomorphism, then so is $\\psi$."}
+{"_id": "552", "title": "Strict Ordering Preserved under Product with Cancellable Element", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be an ordered semigroup. Let $x, y, z \\in S$ be such that: :$(1): \\quad z$ is cancellable for $\\circ$ :$(2): \\quad x \\prec y$ Then: : $x \\circ z \\prec y \\circ z$ : $z \\circ x \\prec z \\circ y$"}
+{"_id": "553", "title": "Ordering of Inverses in Ordered Monoid", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be an ordered monoid whose identity is $e$. Let $x, y \\in S$ be invertible. Then: :$x \\prec y \\iff y^{-1} \\prec x^{-1}$"}
+{"_id": "555", "title": "Monomorphism from Total Ordering", "text": "Let the following conditions hold: : $(1): \\quad$ Let $\\left({S, \\circ, \\preceq}\\right)$ and $\\left({T, *, \\preccurlyeq}\\right)$ be ordered semigroups. : $(2): \\quad$ Let $\\phi: S \\to T$ be a mapping. : $(3): \\quad$ Let $\\preceq$ be a total ordering on $S$. Then $\\phi \\left({S, \\circ, \\preceq}\\right) \\to \\left({T, *, \\preccurlyeq}\\right)$ is a (structure) monomorphism iff: : $(1): \\quad \\phi$ is strictly increasing from $\\left({S, \\preceq}\\right)$ into $\\left({T, \\preccurlyeq}\\right)$; : $(2): \\quad \\phi$ is a homomorphism from $\\left({S, \\circ}\\right)$ into $\\left({T, *}\\right)$."}
+{"_id": "556", "title": "Extension Theorem for Total Orderings", "text": "Let the following conditions be fulfilled: :$(1):\\quad$ Let $\\struct {S, \\circ, \\preceq}$ be a totally ordered commutative semigroup :$(2):\\quad$ Let all the elements of $\\struct {S, \\circ, \\preceq}$ be cancellable :$(3):\\quad$ Let $\\struct {T, \\circ}$ be an inverse completion of $\\struct {S, \\circ}$. Then: :$(1):\\quad$ The relation $\\preceq'$ on $T$ satisfying $\\forall x_1, x_2, y_1, y_2 \\in S: x_1 \\circ \\paren {y_1}^{-1} \\preceq' x_2 \\circ \\paren {y_2}^{-1} \\iff x_1 \\circ y_2 \\preceq x_2 \\circ y_1$ is a well-defined relation :$(2):\\quad$ $\\preceq'$ is the only total ordering on $T$ compatible with $\\circ$ :$(3):\\quad$ $\\preceq'$ is the only total ordering on $T$ that induces the given ordering $\\preceq$ on $S$."}
+{"_id": "557", "title": "Cancellability in Naturally Ordered Semigroup", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a naturally ordered semigroup. Then:"}
+{"_id": "558", "title": "Strict Lower Closure of Sum with One", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a naturally ordered semigroup. Then: :$\\forall n \\in \\left({S, \\circ, \\preceq}\\right): \\left({n \\circ 1}\\right)^\\prec = n^\\prec \\cup \\left\\{{n}\\right\\}$ where $n^\\prec$ is defined as the strict lower closure of $n$, that is, the set of elements strictly preceding $n$."}
+{"_id": "559", "title": "Closed Interval of Naturally Ordered Semigroup with Successor equals Union with Successor", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a naturally ordered semigroup. Then: :$\\forall m, n \\in \\left({S, \\circ, \\preceq}\\right): m \\preceq n \\implies \\left[{m \\,.\\,.\\, n \\circ 1}\\right] = \\left[{m \\,.\\,.\\, n}\\right] \\cup \\left\\{{n \\circ 1}\\right\\}$ where $\\left[{m \\,.\\,.\\, n}\\right]$ is the closed interval between $m$ and $n$."}
+{"_id": "561", "title": "Homomorphism of Powers", "text": "Let $\\struct {T_1, \\odot}$ and $\\struct {T_2, \\oplus}$ be semigroups. Let $\\phi: \\struct {T_1, \\odot} \\to \\struct {T_2, \\oplus}$ be a (semigroup) homomorphism."}
+{"_id": "562", "title": "Naturally Ordered Semigroup is Unique", "text": "Let $\\struct {S, \\circ, \\preceq}$ and $\\struct {S', \\circ', \\preceq'}$ be naturally ordered semigroups. Let: :$0'$ be the smallest element of $S'$ :$1'$ be the smallest element of $S' \\setminus \\set {0'} = S'^*$. Then the mapping $g: S \\to S'$ defined as: :$\\forall a \\in S: \\map g a = \\circ'^a 1'$ is an isomorphism from $\\struct {S, \\circ, \\preceq}$ to $\\struct {S', \\circ', \\preceq'}$. This isomorphism is unique. Thus, up to isomorphism, there is only one naturally ordered semigroup."}
+{"_id": "563", "title": "Consecutive Subsets of N", "text": "Let $\\N_k$ denote the initial segment of the natural numbers determined by $k$: :$\\N_k = \\left\\{{0, 1, 2, 3, \\ldots, k - 1}\\right\\}$ Then: :$\\N_k = \\N_{k + 1} \\setminus \\left\\{{k}\\right\\}$ In particular: :$\\N_{k - 1} = \\N_k \\setminus \\left\\{{k - 1}\\right\\}$"}
+{"_id": "564", "title": "Well-Ordering Principle", "text": "Every non-empty subset of $\\N$ has a smallest (or '''first''') element. This is called the '''well-ordering principle'''. The '''well-ordering principle''' also holds for $\\N_{\\ne 0}$."}
+{"_id": "565", "title": "Equality of Natural Numbers", "text": "Let $m, n \\in \\N$. Then: : $\\N_m \\sim \\N_n \\iff m = n$ where $\\sim$ denotes set equivalence and $\\N_n$ denotes the set of all natural numbers less than $n$."}
+{"_id": "567", "title": "Cardinality Less One", "text": "Let $S$ be a finite set. Let: :$\\left\\lvert{S}\\right\\rvert = n + 1$ where $\\left\\lvert{S}\\right\\rvert$ is the cardinality of $S$. Let $a \\in S$. Then: :$\\left\\lvert{S \\setminus \\left\\{ {a}\\right\\} }\\right\\rvert = n$ where $\\setminus$ denotes set difference."}
+{"_id": "568", "title": "Cardinality of Empty Set", "text": ":$\\card S = 0 \\iff S = \\O$ That is, the empty set is finite, and has a cardinality of zero."}
+{"_id": "569", "title": "Cardinality of Subset of Finite Set", "text": "Let $A$ and $B$ be finite sets such that $A \\subseteq B$. Let :$\\card B = n$ where $\\card {\\, \\cdot \\,}$ denotes cardinality. Then $\\card A \\le n$."}
+{"_id": "570", "title": "Cardinality of Surjection", "text": "Let $S$ be a set. Let: :$\\card S = n$ where $\\card S$ denotes the cardinality of $S$. Let $f: S \\to T$ be a surjection. Then $\\card T \\le n$. The equality: :$\\card T = n$ occurs {{iff}} $f$ is a bijection."}
+{"_id": "571", "title": "Equivalence of Mappings between Sets of Same Cardinality", "text": "Let $S$ and $T$ be finite sets such that $\\card S = \\card T$. Let $f: S \\to T$ be a mapping. Then the following statements are equivalent: : $(1): \\quad f$ is bijective : $(2): \\quad f$ is injective : $(3): \\quad f$ is surjective."}
+{"_id": "572", "title": "Natural Numbers are Infinite", "text": "The set $\\N$ of natural numbers is infinite."}
+{"_id": "573", "title": "Finite Non-Empty Subset of Totally Ordered Set has Smallest and Greatest Elements", "text": "Let $\\left({S, \\preceq}\\right)$ be a totally ordered set. Then every finite $T$ such that $\\varnothing \\subset T \\subseteq S$ has both a smallest and a greatest element."}
+{"_id": "574", "title": "Unique Isomorphism between Finite Totally Ordered Sets", "text": "Let $S$ and $T$ be finite sets such that: :$\\card S = \\card T$ Let $\\struct {S, \\preceq}$ and $\\struct {T, \\preccurlyeq}$ be totally ordered sets. Then there is exactly one order isomorphism from $\\struct {S, \\preceq}$ to $\\struct {T, \\preccurlyeq}$."}
+{"_id": "575", "title": "Isomorphism to Closed Interval", "text": "Let $m, n \\in \\N$ such that $m < n$. Then: : $\\left|{\\left[{m + 1 \\,.\\,.\\, n}\\right]}\\right| = n - m$ Let $h: \\N_{n - m} \\to \\left[{m + 1 \\,.\\,.\\, n}\\right]$ be the mapping defined as: :$\\forall x \\in \\N_{n - m}: h \\left({x}\\right) = x + m + 1$ Let the orderings on $\\left[{m + 1 \\,.\\,.\\, n}\\right]$ and $\\N_{n - m}$ be those induced by the ordering of $\\N$. Then $h$ a unique order isomorphism."}
+{"_id": "576", "title": "Regular Representation wrt Cancellable Element on Finite Semigroup is Bijection", "text": "Let $\\left({S, \\circ}\\right)$ be a finite semigroup. Let $a \\in S$ be cancellable. Then: :the left regular representation $\\lambda_a$ and: :the right regular representation $\\rho_a$ of $\\left({S, \\circ}\\right)$ with respect to $a$ are both bijections."}
+{"_id": "577", "title": "Power Set of Natural Numbers is not Countable", "text": "The power set $\\powerset \\N$ of the natural numbers $\\N$ is not countable."}
+{"_id": "578", "title": "Subset of Countably Infinite Set is Countable", "text": "Every subset of a countably infinite set is countable."}
+{"_id": "580", "title": "Infinite Set has Countably Infinite Subset", "text": "Every infinite set has a countably infinite subset."}
+{"_id": "581", "title": "No Bijection between Finite Set and Proper Subset", "text": "A finite set can not be in one-to-one correspondence with one of its proper subsets. That is, a finite set is not Dedekind-infinite."}
+{"_id": "582", "title": "Infinite Set is Equivalent to Proper Subset", "text": "A set is infinite {{iff}} it is equivalent to one of its proper subsets."}
+{"_id": "583", "title": "Cartesian Product of Countable Sets is Countable", "text": "The cartesian product of two countable sets is countable."}
+{"_id": "584", "title": "Composition of Sequence with Mapping", "text": "Let $\\left \\langle {a_j} \\right \\rangle_{j \\in B}$ be a sequence. Let $\\sigma: A \\to B$ be a mapping, where $A \\subseteq \\N$. Then $\\left \\langle {a_j} \\right \\rangle \\circ \\sigma$ is a sequence whose value at each $k \\in A$ is $a_{\\sigma \\left({k}\\right)}$. Thus $\\left \\langle {a_j} \\right \\rangle \\circ \\sigma$ is denoted $\\left \\langle {a_{\\sigma \\left({k}\\right)}} \\right \\rangle_{k \\in A}$."}
+{"_id": "585", "title": "General Operation from Binary Operation", "text": "Let $\\left({S, \\oplus}\\right)$ be a magma. Then there a unique sequence $\\left \\langle {\\oplus_k} \\right \\rangle_{k \\mathop \\ge 1}$ such that: :$(1): \\quad \\forall n \\in \\N_{>0}: \\oplus_n$ is an $n$-ary operation on $S$ such that: :$(2): \\quad \\forall \\left({a_1, \\ldots, a_k}\\right) \\in S^k: \\oplus_k \\left({a_1, \\ldots, a_k}\\right) = \\begin{cases} a: & k = 1 \\\\ \\oplus_n \\left({a_1, \\ldots, a_n}\\right) \\oplus a_{n+1}: & k = n + 1 \\end{cases}$ In particular, $\\oplus_2$ is the same as the given binary operation $\\oplus$. The $n$th term $\\oplus_n$ of the sequence $\\left \\langle {\\oplus} \\right \\rangle$ is called the '''$n$-ary operation defined by $\\oplus$'''."}
+{"_id": "586", "title": "Strictly Increasing Sequence induces Partition", "text": "Let $\\left \\langle {r_k} \\right \\rangle_{0 \\mathop \\le k \\mathop \\le n}$ be a strictly increasing finite sequence of natural numbers. Let: :$\\forall k \\in \\left[{1 \\,.\\,.\\, n}\\right]: A_k := \\left[{r_{k-1} + 1 \\,.\\,.\\, r_k}\\right]$ Then: :$\\left\\{{A_k: k \\in \\left[{1 \\,.\\,.\\, n}\\right]}\\right\\}$ is a partition of $\\left[{r_0 + 1 \\,.\\,.\\, r_n}\\right]$."}
+{"_id": "587", "title": "Fundamental Principle of Counting", "text": "Let $A$ be a finite set. Let $\\sequence {B_n}$ be a sequence of distinct finite subsets of $A$ which form a partition of $A$. Let $p_k = \\size {B_k}$ for each $k \\in \\closedint 1 n$. Then: :$\\displaystyle \\size A = \\sum_{k \\mathop = 1}^n p_k$ That is, the sum of the numbers of elements in the subsets of a partition of a set is equal to the total number of elements in the set."}
+{"_id": "588", "title": "Odd Number Theorem", "text": ":$\\displaystyle \\sum_{j \\mathop = 1}^n \\paren {2 j - 1} = n^2$ That is, the sum of the first $n$ odd numbers is the $n$th square number."}
+{"_id": "589", "title": "General Associativity Theorem", "text": "If an operation is associative on $3$ entities, then it is associative on any number of them."}
+{"_id": "590", "title": "General Commutativity Theorem", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Let $\\family {a_k}_{1 \\mathop \\le k \\mathop \\le n}$ be a sequence of elements of $S$. Suppose that: :$\\forall i, j \\in \\closedint 1 n: a_i \\circ a_j = a_j \\circ a_i$ Then for every permutation $\\sigma: \\N_n \\to \\N_n$: :$a_{\\map \\sigma 1} \\circ \\cdots \\circ a_{\\map \\sigma n} = a_1 \\circ \\cdots \\circ a_n$ where $\\N_n$ is used here to denote the initial segment of $\\N_{>0}$: :$\\N_n = \\set {1, 2, \\ldots, n}$"}
+{"_id": "591", "title": "General Distributivity Theorem", "text": ":$\\displaystyle \\paren {\\sum_{i \\mathop = 1}^m a_i} * \\paren {\\sum_{j \\mathop = 1}^n b_j} = \\sum_{ {1 \\mathop \\le i \\mathop \\le m} \\atop {1 \\mathop \\le j \\mathop \\le n} } \\paren {a_i * b_j}$"}
+{"_id": "592", "title": "Associativity on Indexing Set", "text": "Let $\\left({S, \\circ}\\right)$ be a commutative semigroup. Let $\\left \\langle {x_\\alpha} \\right \\rangle_{\\alpha \\mathop \\in A}$ be a family of terms of $S$ indexed by a finite non-empty set $A$. Let $\\left \\langle {B_k} \\right \\rangle_{1 \\mathop \\le k \\mathop \\le n}$ be a family of distinct subsets of $A$ forming a partition of $A$. Then: : $\\displaystyle \\prod_{k \\mathop = 1}^n \\left({\\prod_{a \\mathop \\in B_k} x_\\alpha}\\right) = \\prod_{\\alpha \\mathop \\in A} x_\\alpha$"}
+{"_id": "593", "title": "External Direct Product of Groups is Group", "text": "Let $\\struct {G_1, \\circ_1}$ and $\\struct {G_2, \\circ_2}$ be groups whose identity elements are $e_1$ and $e_2$ respectively. Let $\\struct {G_1 \\times G_2, \\circ}$ be the external direct product of $G_1$ and $G_2$. Then $\\struct {G_1 \\times G_2, \\circ}$ is a group whose identity element is $\\tuple {e_1, e_2}$."}
+{"_id": "594", "title": "External Direct Product of Projection with Canonical Injection", "text": "Let $\\struct {S_1, \\circ_1}$ and $\\struct {S_2, \\circ_2}$ be algebraic structures with identity elements $e_1$ and $e_2$ respectively. Let $\\struct {S_1 \\times S_2, \\circ}$ be the external direct product of $\\struct {S_1, \\circ_1}$ and $\\struct {S_2, \\circ_2}$ Let: :$\\pr_1: \\struct {S_1 \\times S_2, \\circ} \\to \\struct {S_1, \\circ_1}$ be the first projection from $\\struct {S_1 \\times S_2, \\circ}$ to $\\struct {S_1, \\circ_1}$ :$\\pr_2: \\struct {S_1 \\times S_2, \\circ} \\to \\struct {S_2, \\circ_2}$ be the second projection from $\\struct {S_1 \\times S_2, \\circ}$ to $\\struct {S_2, \\circ_2}$. Let: :$\\inj_1: \\struct {S_1, \\circ_1} \\to \\struct {S_1 \\times S_2, \\circ}$ be the canonical injection from $\\struct {S_1, \\circ_1}$ to $\\struct {S_1 \\times S_2, \\circ}$ :$\\inj_2: \\struct {S_2, \\circ_2} \\to \\struct {S_1 \\times S_2, \\circ}$ be the canonical injection from $\\struct {S_2, \\circ_2}$ to $\\struct {S_1 \\times S_2, \\circ}$. Then: :$(1): \\quad \\pr_1 \\circ \\inj_1 = I_{S_1}$ :$(2): \\quad \\pr_2 \\circ \\inj_2 = I_{S_2}$ where $I_{S_1}$ and $I_{S_2}$ are the identity mappings on $S_1$ and $S_2$ respectively."}
+{"_id": "595", "title": "Inverse Completion of Natural Numbers", "text": "There exists an inverse completion of the natural numbers under addition."}
+{"_id": "596", "title": "L'Hôpital's Rule", "text": "Let $f$ and $g$ be real functions which are continuous on the closed interval $\\closedint a b$ and differentiable on the open interval $\\openint a b$. Let: :$\\forall x \\in \\openint a b: \\map {g'} x \\ne 0$ where $g'$ denotes the derivative of $g$ {{WRT|Differentiation}} $x$. Let: :$\\map f a = \\map g a = 0$ Then: :$\\displaystyle \\lim_{x \\mathop \\to a^+} \\frac {\\map f x} {\\map g x} = \\lim_{x \\mathop \\to a^+} \\frac {\\map {f'} x} {\\map {g'} x}$ provided that the second limit exists."}
+{"_id": "597", "title": "Natural Numbers under Addition form Commutative Monoid", "text": "The algebraic structure $\\struct {\\N, +}$ consisting of the set of natural numbers $\\N$ under addition $+$ is a commutative monoid whose identity is zero."}
+{"_id": "598", "title": "Natural Numbers form Commutative Semiring", "text": "The semiring of natural numbers $\\struct {\\N, +, \\times}$ forms a commutative semiring."}
+{"_id": "600", "title": "Integer Multiplication is Closed", "text": "The set of integers is closed under multiplication: :$\\forall a, b \\in \\Z: a \\times b \\in \\Z$"}
+{"_id": "601", "title": "Integer Multiplication is Commutative", "text": "The operation of multiplication on the set of integers $\\Z$ is commutative: :$\\forall x, y \\in \\Z: x \\times y = y \\times x$"}
+{"_id": "602", "title": "Integer Multiplication is Associative", "text": "The operation of multiplication on the set of integers $\\Z$ is associative: :$\\forall x, y, z \\in \\Z: x \\times \\paren {y \\times z} = \\paren {x \\times y} \\times z$"}
+{"_id": "603", "title": "Integer Multiplication Distributes over Addition", "text": "The operation of multiplication on the set of integers $\\Z$ is distributive over addition: :$\\forall x, y, z \\in \\Z: x \\times \\paren {y + z} = \\paren {x \\times y} + \\paren {x \\times z}$ :$\\forall x, y, z \\in \\Z: \\paren {y + z} \\times x = \\paren {y \\times x} + \\paren {z \\times x}$"}
+{"_id": "604", "title": "Construction of Inverse Completion", "text": "This page consists of a series of linked theorems, each of which builds towards one result. To access the proofs for the individual theorems, click on the links which form the titles of each major section."}
+{"_id": "605", "title": "Integer Multiplication Identity is One", "text": "The identity of integer multiplication is $1$: :$\\exists 1 \\in \\Z: \\forall a \\in \\Z: a \\times 1 = a = 1 \\times a$"}
+{"_id": "606", "title": "Integer Multiplication has Zero", "text": "The set of integers under multiplication $\\struct {\\Z, \\times}$ has a zero element, which is $0$."}
+{"_id": "607", "title": "Ring of Integers has no Zero Divisors", "text": "The integers have no zero divisors: :$\\forall x, y, \\in \\Z: x \\times y = 0 \\implies x = 0 \\lor y = 0$"}
+{"_id": "608", "title": "Integers form Integral Domain", "text": "The integers $\\Z$ form an integral domain under addition and multiplication."}
+{"_id": "610", "title": "Natural Numbers are Non-Negative Integers", "text": "Let $m \\in \\Z$. Then: :$(1): \\quad m \\in \\N \\iff 0 \\le m$ :$(2): \\quad m \\in \\N_{> 0} \\iff 0 < m$ :$(3): \\quad m \\notin \\N \\iff -m \\in \\N_{> 0}$ That is, the natural numbers are precisely those integers which are greater than or equal to zero."}
+{"_id": "611", "title": "Subtraction on Integers is Extension of Natural Numbers", "text": "Integer subtraction is an extension of the definition of subtraction on the natural numbers."}
+{"_id": "613", "title": "Multiplicative Ordering on Integers", "text": "Let $x, y, z \\in \\Z$ such that $z > 0$. Then: :$x < y \\iff z x < z y$ :$x \\le y \\iff z x \\le z y$"}
+{"_id": "614", "title": "Invertible Integers under Multiplication", "text": "The only invertible elements of $\\Z$ for multiplication (that is, units of $\\Z$) are $1$ and $-1$."}
+{"_id": "615", "title": "Index Laws for Monoids", "text": "These results are an extension of the results in Index Laws for Semigroup in which the domain of the indices is extended to include all integers. Let $\\struct {S, \\circ}$ be a monoid whose identity is $e$. Let $a \\in S$ be invertible for $\\circ$. Let $n \\in \\N$. Let $a^n$ be the $n$th power of $a$: :$a^n = \\begin{cases} e : & n = 0 \\\\ a^{n - 1} \\circ a : & n > 0 \\\\ \\paren {a^{-n}}^{-1} : & n < 0 \\end{cases}$ Then we have the following results:"}
+{"_id": "617", "title": "Totally Ordered Abelian Group Isomorphism", "text": "Let $\\left({\\Z', +', \\le'}\\right)$ be a totally ordered abelian group. Let $0'$ be the identity of $\\left({\\Z', +', \\le'}\\right)$. Let $\\N' = \\left\\{{x \\in \\Z': x \\ge' 0'}\\right\\}$. Let $\\Z'$ contain at least two elements. Let $\\N'$ be well-ordered for the ordering induced on $\\N'$ by $\\le'$. Then the mapping $g: \\Z \\to \\Z'$ defined by: :$\\forall n \\in \\Z: g \\left({n}\\right) = \\left({+'}\\right)^n 1'$ is an isomorphism from $\\left({\\Z, +, \\le}\\right)$ onto $\\left({\\Z', +', \\le'}\\right)$, where $1'$ is the smallest element of $\\N' \\setminus \\left\\{{0'}\\right\\}$."}
+{"_id": "618", "title": "Integers under Addition form Totally Ordered Group", "text": "Then the ordered structure $\\struct {\\Z, +, \\le}$ is a totally ordered group."}
+{"_id": "619", "title": "Integers form Totally Ordered Ring", "text": "The structure $\\struct {\\Z, +, \\times, \\le}$ is a totally ordered ring."}
+{"_id": "620", "title": "Congruences on Rational Numbers", "text": "There are only two congruence relations on the field of rational numbers $\\left({\\Q, +, \\times}\\right)$: :$(1): \\quad$ The diagonal relation $\\Delta_\\Q$ :$(2): \\quad$ The trivial relation $\\Q \\times \\Q$."}
+{"_id": "621", "title": "Number of Elements in Partition", "text": "Let $S$ be a set. Let there be a partition on $S$ of $n$ subsets, each of which has $m$ elements. Then: :$\\card S = n m$"}
+{"_id": "622", "title": "Cardinality of Complement", "text": "Let $T \\subseteq S$ such that $\\card S = n, \\card T = m$. Then: :$\\card {\\relcomp S T} = \\card {S \\setminus T} = n - m$ where: :$\\relcomp S T$ denotes the complement of $T$ relative to $S$ :$S \\setminus T$ denotes the difference between $S$ and $T$."}
+{"_id": "623", "title": "Cardinality of Cartesian Product", "text": "Let $S \\times T$ be the cartesian product of two finite sets $S$ and $T$. Then: :$\\card {S \\times T} = \\card S \\times \\card T$ where $\\card S$ denotes cardinality."}
+{"_id": "624", "title": "Cardinality of Set of All Mappings", "text": "Let $S$ and $T$ be sets. The cardinality of the set of all mappings from $S$ to $T$ (that is, the total number of mappings from $S$ to $T$) is: :$\\card {T^S} = \\card T^{\\card S}$"}
+{"_id": "626", "title": "Cardinality of Power Set of Finite Set", "text": "Let $S$ be a set such that: :$\\card S = n$ where $\\card S$ denotes the cardinality of $S$, Then: :$\\card {\\powerset S} = 2^n$ where $\\powerset S$ denotes the power set of $S$."}
+{"_id": "627", "title": "Cardinality of Set of Injections", "text": "Let $S$ and $T$ be finite sets. The number of injections from $S$ to $T$, where $\\card S = m, \\card T = n$ is often denoted ${}^m P_n$, and is: :${}^m P_n = \\begin{cases} \\dfrac {n!} {\\paren {n - m}!} & : m \\le n \\\\ 0 & : m > n \\end{cases}$"}
+{"_id": "628", "title": "Cardinality of Set of Bijections", "text": "Let $S$ and $T$ be sets such that $\\size S = \\size T = n$. Then there are $n!$ bijections from $S$ to $T$."}
+{"_id": "629", "title": "Cardinality of Set of Subsets", "text": "Let $S$ be a set such that $\\card S = n$. Let $m \\le n$. Then the number of subsets $T$ of $S$ such that $\\card T = m$ is: : ${}^m C_n = \\dfrac {n!} {m! \\paren {n - m}!}$"}
+{"_id": "630", "title": "Set of Integers Bounded Below by Integer has Smallest Element", "text": "Let $\\Z$ be the set of integers. Let $\\le$ be the ordering on the integers. Let $\\O \\subset S \\subseteq \\Z$ such that $S$ is bounded below in $\\struct {\\Z, \\le}$. Then $S$ has a smallest element."}
+{"_id": "631", "title": "Set of Integers Bounded Above by Integer has Greatest Element", "text": "Let $\\Z$ be the set of integers. Let $\\le$ be the ordering on the integers. Let $\\O \\subset S \\subseteq \\Z$ such that $S$ is bounded above in $\\struct {\\Z, \\le}$. Then $S$ has a greatest element."}
+{"_id": "632", "title": "Principle of Least Counterexample", "text": "Suppose $P \\paren n$ is a condition on $n \\in \\set {x \\in \\Z: x \\ge m \\in \\Z}$. Suppose next that: $\\neg \\paren {\\forall n \\ge m: P \\paren n}$. (That is, not all $n \\ge m$ satisfy $P \\paren n$.) Then there is a '''least counterexample''', that is a smallest integral value of $n$ for which $\\neg P \\paren n$."}
+{"_id": "633", "title": "Absolute Value is Bounded Below by Zero", "text": "Let $x \\in \\R$ be a real number. Then the absolute value $\\size x$ of $x$ is bounded below by $0$."}
+{"_id": "634", "title": "Division Theorem", "text": "For every pair of integers $a, b$ where $b \\ne 0$, there exist unique integers $q, r$ such that $a = q b + r$ and $0 \\le r < \\size b$: :$\\forall a, b \\in \\Z, b \\ne 0: \\exists! q, r \\in \\Z: a = q b + r, 0 \\le r < \\size b$"}
+{"_id": "635", "title": "Odd Integer 2n + 1", "text": "Let $m$ be an odd integer. Then there exists exactly one integer $n$ such that $2 n + 1 = m$."}
+{"_id": "636", "title": "Integer Divisor Results", "text": "Let $m, n \\in \\Z$ be integers. Let $m \\divides n$ denote that $m$ is a divisor of $n$. The following results all hold:"}
+{"_id": "637", "title": "Zero Divides Zero", "text": "Let $n \\in \\Z$ be an integer. Then: :$0 \\divides n \\implies n = 0$ That is, zero is the only integer divisible by zero."}
+{"_id": "638", "title": "Absolute Value of Integer is not less than Divisors", "text": "A (non-zero) integer is greater than or equal to its divisors in magnitude: :$\\forall c \\in \\Z_{\\ne 0}: a \\divides c \\implies a \\le \\size a \\le \\size c$"}
+{"_id": "639", "title": "Divisor Relation on Positive Integers is Partial Ordering", "text": "The divisor relation is a partial ordering of $\\Z_{>0}$."}
+{"_id": "640", "title": "Common Divisor in Integral Domain Divides Linear Combination", "text": "Let $\\struct {D, +, \\times}$ be an integral domain. Let $c$ be a common divisor of two elements $a$ and $b$ of $D$. That is: :$a, b, c \\in D: c \\divides a \\land c \\divides b$ Then: :$\\forall p, q \\in D: c \\divides \\paren {p \\times a + q \\times b}$"}
+{"_id": "642", "title": "Greatest Common Divisor is at least 1", "text": "Let $a, b \\in \\Z$ be integers. The greatest common divisor of $a$ and $b$ is at least $1$: :$\\forall a, b \\in \\Z_{\\ne 0}: \\gcd \\set {a, b} \\ge 1$"}
+{"_id": "643", "title": "GCD of Integer and Divisor", "text": "Let $a, b \\in \\Z_{>0}$, i.e. integers such that $a, b > 0$. Then: : $a \\divides b \\implies \\gcd \\set {a, b} = a$"}
+{"_id": "644", "title": "GCD for Negative Integers", "text": ":$\\gcd \\set {a, b} = \\gcd \\set {\\size a, b} = \\gcd \\set {a, \\size b} = \\gcd \\set {\\size a, \\size b}$ Alternatively, this can be put: :$\\gcd \\set {a, b} = \\gcd \\set {-a, b} = \\gcd \\set {a, -b} = \\gcd \\set {-a, -b}$   which follows directly from the above."}
+{"_id": "645", "title": "GCD with Zero", "text": "Let $a \\in \\Z$ be an integer such that $a \\ne 0$. Then: :$\\gcd \\left\\{{a, 0}\\right\\} = \\left\\lvert{a}\\right\\rvert$ where $\\gcd$ denotes greatest common divisor (GCD)."}
+{"_id": "646", "title": "Set of Integer Combinations equals Set of Multiples of GCD", "text": "The set of all integer combinations of $a$ and $b$ is precisely the set of all integer multiples of the GCD of $a$ and $b$: :$\\gcd \\set {a, b} \\divides c \\iff \\exists x, y \\in \\Z: c = x a + y b$"}
+{"_id": "647", "title": "GCD with Remainder", "text": "Let $a, b \\in \\Z$. Let $q, r \\in \\Z$ such that $a = q b + r$. Then: :$\\gcd \\set {a, b} = \\gcd \\set {b, r}$ where $\\gcd \\set {a, b}$ is the greatest common divisor of $a$ and $b$."}
+{"_id": "648", "title": "Integer Combination of Coprime Integers", "text": "Two integers are coprime {{iff}} there exists an integer combination of them equal to $1$: :$\\forall a, b \\in \\Z: a \\perp b \\iff \\exists m, n \\in \\Z: m a + n b = 1$"}
+{"_id": "650", "title": "Integers Divided by GCD are Coprime", "text": "Any pair of integers, not both zero, can be reduced to a pair of coprime ones by dividing them by their GCD: :$\\gcd \\set {a, b} = d \\iff \\dfrac a d, \\dfrac b d \\in \\Z \\land \\gcd \\set {\\dfrac a d, \\dfrac b d} = 1$ That is: :$\\dfrac a {\\gcd \\set {a, b} } \\perp \\dfrac b {\\gcd \\set {a, b} }$"}
+{"_id": "651", "title": "Product of Coprime Factors", "text": "Let $a, b, c \\in \\Z$ such that $a$ and $b$ are coprime. Let both $a$ and $b$ be divisors of $c$. Then $a b$ is also a divisor of $c$. That is: :$a \\perp b \\land a \\divides c \\land b \\divides c \\implies a b \\divides c$"}
+{"_id": "652", "title": "Existence of Lowest Common Multiple", "text": "Let $a, b \\in \\Z: a b \\ne 0$. The lowest common multiple of $a$ and $b$, denoted $\\lcm \\set {a, b}$, always exists."}
+{"_id": "653", "title": "Product of GCD and LCM", "text": ":$\\lcm \\set {a, b} \\times \\gcd \\set {a, b} = \\size {a b}$ where: :$\\lcm \\set {a, b}$ denotes the lowest common multiple of $a$ and $b$ :$\\gcd \\set {a, b}$ denotes the greatest common divisor of $a$ and $b$."}
+{"_id": "654", "title": "Congruent to Zero if Modulo is Divisor", "text": "Let $a, z \\in \\R$. Then $a$ is congruent to $0$ modulo $z$ {{iff}} $a$ is an integer multiple of $z$. :$\\exists k \\in \\Z: k z = a \\iff a \\equiv 0 \\pmod z$ If $z \\in \\Z$, then further: :$z \\divides a \\iff a \\equiv 0 \\pmod z$"}
+{"_id": "655", "title": "Integer is Congruent Modulo Divisor to Remainder", "text": "Let $a \\in \\Z$. Let $a$ have a remainder $r$ on division by $m$. Then: : $a \\equiv r \\pmod m$ where the notation denotes that $a$ and $r$ are congruent modulo $m$."}
+{"_id": "656", "title": "Integer is Congruent to Integer less than Modulus", "text": "Let $m \\in \\Z$. Then each integer is congruent (modulo $m$) to precisely one of the integers $0, 1, \\ldots, m - 1$."}
+{"_id": "657", "title": "Modulo Addition is Well-Defined", "text": "Let $m \\in \\Z$ be an integer. Let $\\Z_m$ be the set of integers modulo $m$. The modulo addition operation on $\\Z_m$, defined by the rule: :$\\eqclass a m +_m \\eqclass b m = \\eqclass {a + b} m$ is a well-defined operation. That is: :If $a \\equiv b \\pmod m$ and $x \\equiv y \\pmod m$, then $a + x \\equiv b + y \\pmod m$."}
+{"_id": "658", "title": "Modulo Multiplication is Well-Defined", "text": "The multiplication modulo $m$ operation on $\\Z_m$, the set of integers modulo $m$, defined by the rule: :$\\eqclass x m \\times_m \\eqclass y m = \\eqclass {x y} m$ is a well-defined operation. That is: :If $a \\equiv b \\pmod m$ and $x \\equiv y \\pmod m$, then $a x \\equiv b y \\pmod m$."}
+{"_id": "659", "title": "Congruence of Product", "text": "Let $a, b, z \\in \\R$. Let $a$ be congruent to $b$ modulo $z$, that is: :$a \\equiv b \\pmod z$ Then: :$\\forall m \\in \\Z: m a \\equiv m b \\pmod z$"}
+{"_id": "660", "title": "Congruence of Powers", "text": "Let $a, b \\in \\R$ and $m \\in \\Z$. Let $a$ be congruent to $b$ modulo $m$, that is: :$a \\equiv b \\pmod m$ Then: :$\\forall n \\in \\Z_{\\ge 0}: a^n \\equiv b^n \\pmod m$"}
+{"_id": "662", "title": "Modulo Addition is Associative", "text": "Addition modulo $m$ is associative: :$\\forall \\eqclass x m, \\eqclass y m, \\eqclass z m \\in \\Z_m: \\paren {\\eqclass x m +_m \\eqclass y m} +_m \\eqclass z m = \\eqclass x m +_m \\paren {\\eqclass y m +_m \\eqclass z m}$ where $\\Z_m$ is the set of integers modulo $m$. That is: :$\\forall x, y, z \\in \\Z: \\paren {x + y} + z \\equiv x + \\paren {y + z} \\pmod m$"}
+{"_id": "663", "title": "Modulo Addition is Commutative", "text": "Modulo addition is commutative: :$\\forall x, y, z \\in \\Z: x + y \\pmod m = y + x \\pmod m$"}
+{"_id": "664", "title": "Modulo Addition has Identity", "text": "Let $m \\in \\Z$ be an integer. Then addition modulo $m$ has an identity: :$\\forall \\eqclass x m \\in \\Z_m: \\eqclass x m +_m \\eqclass 0 m = \\eqclass x m = \\eqclass 0 m +_m \\eqclass x m$ That is: :$\\forall a \\in \\Z: a + 0 \\equiv a \\equiv 0 + a \\pmod m$"}
+{"_id": "665", "title": "Modulo Addition has Inverses", "text": "Let $m \\in \\Z$ be an integer. Then addition modulo $m$ has inverses: For each element $\\eqclass x m \\in \\Z_m$, there exists the element $\\eqclass {-x} m \\in \\Z_m$ with the property: :$\\eqclass x m +_m \\eqclass {-x} m = \\eqclass 0 m = \\eqclass {-x} m +_m \\eqclass x m$ where $\\Z_m$ is the set of integers modulo $m$. That is: :$\\forall a \\in \\Z: a + \\paren {-a} \\equiv 0 \\equiv \\paren {-a} + a \\pmod m$"}
+{"_id": "666", "title": "Modulo Multiplication is Closed", "text": "Multiplication modulo $m$ is closed on the set of integers modulo $m$: :$\\forall \\eqclass x m, \\eqclass y m \\in \\Z_m: \\eqclass x m \\times_m \\eqclass y m \\in \\Z_m$."}
+{"_id": "667", "title": "Modulo Multiplication is Associative", "text": "Multiplication modulo $m$ is associative: :$\\forall \\eqclass x m, \\eqclass y m, \\eqclass z m \\in \\Z_m: \\paren {\\eqclass x m \\times_m \\eqclass y m} \\times_m \\eqclass z m = \\eqclass x m \\times_m \\paren {\\eqclass y m \\times_m \\eqclass z m}$ That is: :$\\forall x, y, z \\in \\Z_m: \\paren {x \\cdot_m y} \\cdot_m z = x \\cdot_m \\paren {y \\cdot_m z}$"}
+{"_id": "668", "title": "Modulo Multiplication is Commutative", "text": "Multiplication modulo $m$ is commutative: :$\\forall \\eqclass x m, \\eqclass y m \\in \\Z_m: \\eqclass x m \\times_m \\eqclass y m = \\eqclass y m \\times_m \\eqclass x m$"}
+{"_id": "669", "title": "Modulo Multiplication has Identity", "text": "Multiplication modulo $m$ has an identity: :$\\forall \\eqclass x m \\in \\Z_m: \\eqclass x m \\times_m \\eqclass 1 m = \\eqclass x m = \\eqclass 1 m \\times_m \\eqclass x m$"}
+{"_id": "670", "title": "Modulo Multiplication Distributes over Modulo Addition", "text": "Multiplication modulo $m$ is distributive over addition modulo $m$: :$\\forall \\eqclass x m, \\eqclass y m, \\eqclass z m \\in \\Z_m$: :: $\\eqclass x m \\times_m \\paren {\\eqclass y m +_m \\eqclass z m} = \\paren {\\eqclass x m \\times_m \\eqclass y m} +_m \\paren {\\eqclass x m \\times_m \\eqclass z m}$ :: $\\paren {\\eqclass x m +_m \\eqclass y m} \\times_m \\eqclass z m = \\paren {\\eqclass x m \\times_m \\eqclass z m} +_m \\paren {\\eqclass y m \\times_m \\eqclass z m}$ where $\\Z_m$ is the set of integers modulo $m$. That is, $\\forall x, y, z, m \\in \\Z$: : $x \\paren {y + z} \\equiv x y + x z \\pmod m$ : $\\paren {x + y} z \\equiv x z + y z \\pmod m$"}
+{"_id": "671", "title": "Intersection of Congruence Classes", "text": "Let $\\mathcal R_m$ denote congruence modulo $m$ on the set of integers $\\Z$. Then: :$\\mathcal R_m \\cap \\mathcal R_n = \\mathcal R_{\\lcm \\set {m, n} }$ where $\\lcm \\set {m, n}$ is the lowest common multiple of $m$ and $n$. In the language of modulo arithmetic, this is equivalent to: :$a \\equiv b \\pmod m, a \\equiv b \\pmod n \\implies a \\equiv b \\pmod {\\lcm \\set {m, n} }$"}
+{"_id": "672", "title": "Mappings Between Residue Classes", "text": "Let $\\eqclass a m$ be the residue class of $a$ (modulo $m$). Let $\\phi: \\Z_m \\to \\Z_n$ be a mapping given by: :$\\map \\phi {\\eqclass x m} = \\eqclass x n$ Then $\\phi$ is well defined {{iff}} $m$ is a divisor of $n$."}
+{"_id": "674", "title": "Prime Number has 4 Integral Divisors", "text": "Let $p$ be an integer. Then $p$ is a prime number {{iff}} $p$ has exactly four integral divisors: $1, -1, p, -p$."}
+{"_id": "675", "title": "Prime not Divisor implies Coprime", "text": "Let $p, a \\in \\Z$. If $p$ is a prime number then: : $p \\nmid a \\implies p \\perp a$ where: : $p \\nmid a$ denotes that $p$ does not divide $a$ : $p \\perp a$ denotes that $p$ and $a$ are coprime. It follows directly that if $p$ and $q$ are primes, then: : $p \\divides q \\implies p = q$ : $p \\ne q \\implies p \\perp q$."}
+{"_id": "676", "title": "Composite Number has Two Divisors Less Than It", "text": "Let $n \\in \\Z_{> 1}$ such that $n \\notin \\mathbb P$. Then: :$\\exists a, b \\in \\Z: 1 < a < n, 1 < b < n: n = a b$ That is, a non-prime number greater than $1$ can be expressed as the product of two positive integers strictly greater than $1$ and less than $n$. Note that these two numbers are not necessarily distinct."}
+{"_id": "678", "title": "Exponents of Primes in Prime Decomposition are Less iff Divisor", "text": "Let $a, b \\in \\Z_{>0}$. Then $a \\divides b$ {{iff}}: :$(1): \\quad$ every prime $p_i$ in the prime decomposition of $a$ appears in the prime decomposition of $b$ and: :$(2): \\quad$ the exponent of each $p_i$ in $a$ is less than or equal to its exponent in $b$."}
+{"_id": "679", "title": "Not Coprime means Common Prime Factor", "text": "Let $a, b \\in \\Z$. If $d \\divides a$ and $d \\divides b$ such that $d > 1$, then $a$ and $b$ have a common divisor which is prime."}
+{"_id": "680", "title": "Set of Divisors of Integer", "text": "Let $n \\in \\Z_{>1}$. Let $n$ be expressed in its prime decomposition: :$n = p_1^{k_1} p_2^{k_2} \\dotsm p_r^{k_r}$ where $p_1 < p_2 < \\dotsb < p_r$ are distinct primes and $k_1, k_2, \\ldots, k_r$ are positive integers. The set of divisors of $n$ is: :$\\set {p_1^{h_1} p_2^{h_2} \\dotsm p_r^{h_r}: 0 \\le h_i \\le k_i, i = 1, 2, \\ldots, r}$"}
+{"_id": "681", "title": "Sum Less Minimum is Maximum", "text": "For all numbers $a, b$ where $a, b$ in $\\N, \\Z, \\Q$ or $\\R$: :$a + b - \\min \\left({a, b}\\right) = \\max \\left({a, b}\\right)$"}
+{"_id": "682", "title": "Sum Less Maximum is Minimum", "text": "For all numbers $a, b$ where $a, b$ in $\\N, \\Z, \\Q$ or $\\R$: :$a + b - \\max \\left({a, b}\\right) = \\min \\left({a, b}\\right)$."}
+{"_id": "683", "title": "GCD and LCM from Prime Decomposition", "text": "Let $m, n \\in \\Z$. Let: :$m = p_1^{k_1} p_2^{k_2} \\dotsm p_r^{k_r}$ :$n = p_1^{l_1} p_2^{l_2} \\dotsm p_r^{l_r}$ :$p_i \\divides m \\lor p_i \\divides n, 1 \\le i \\le r$. That is, the primes given in these prime decompositions may be divisors of ''either'' of the numbers $m$ or $n$. Note that if one of the primes $p_i$ does not appear in the prime decompositions of either one of $m$ or $n$, then its corresponding index $k_i$ or $l_i$ will be zero. Then the following results apply: :$\\gcd \\set {m, n} = p_1^{\\min \\set {k_1, l_1} } p_2^{\\min \\set {k_2, l_2} } \\ldots p_r^{\\min \\set {k_r, l_r} }$ :$\\lcm \\set {m, n} = p_1^{\\max \\set {k_1, l_1} } p_2^{\\max \\set {k_2, l_2} } \\ldots p_r^{\\max \\set {k_r, l_r} }$"}
+{"_id": "684", "title": "GCD and LCM Distribute Over Each Other", "text": "Let $a, b, c \\in \\Z$. Then: : $\\lcm \\set {a, \\gcd \\set {b, c} } = \\gcd \\set {\\lcm \\set {a, b}, \\lcm \\set {a, c} }$ : $\\gcd \\set {a, \\lcm \\set {b, c} } = \\lcm \\set {\\gcd \\set {a, b}, \\gcd \\set {a, c} }$ That is, greatest common divisor and lowest common multiple are distributive over each other."}
+{"_id": "685", "title": "Tau Function from Prime Decomposition", "text": "Let $n$ be an integer such that $n \\ge 2$. Let the prime decomposition of $n$ be: :$n = p_1^{k_1} p_2^{k_2} \\cdots p_r^{k_r}$ Let $\\map \\tau n$ be the tau function of $n$. Then: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$"}
+{"_id": "686", "title": "N less than M to the N", "text": ":$\\forall m, n \\in \\Z_{>0}: m > 1 \\implies n < m^n$"}
+{"_id": "687", "title": "Basis Representation Theorem", "text": "Let $b \\in \\Z: b > 1$. For every $n \\in \\Z_{> 0}$, there exists one and only one sequence $\\sequence {r_j}_{0 \\mathop \\le j \\mathop \\le t}$ such that: : $(1): \\quad \\displaystyle n = \\sum_{k \\mathop = 0}^t r_k b^k$ : $(2): \\quad \\displaystyle \\forall k \\in \\closedint 0 t: r_k \\in \\N_b$ : $(3): \\quad r_t \\ne 0$ This unique sequence is called the '''representation of $n$ to the base $b$''', or, informally, we can say '''$n$ is (written) in base $b$'''."}
+{"_id": "688", "title": "Sum of Geometric Sequence", "text": "Let $x$ be an element of one of the standard number fields: $\\Q, \\R, \\C$ such that $x \\ne 1$. Let $n \\in \\N_{>0}$. Then: :$\\displaystyle \\sum_{j \\mathop = 0}^{n - 1} x^j = \\frac {x^n - 1} {x - 1}$"}
+{"_id": "689", "title": "Congruence of Sum of Digits to Base Less 1", "text": "Let $x \\in \\Z$, and $b \\in \\N, b > 1$. Let $x$ be written in base $b$: :$x = \\sqbrk {r_m r_{m - 1} \\ldots r_2 r_1 r_0}_b$ Then: :$\\displaystyle \\map {s_b} x = \\sum_{j \\mathop = 0}^m r_j \\equiv x \\pmod {b - 1}$ where $\\map {s_b} x$ is the digit sum of $x$ in base $b$ notation. That is, the digit sum of any integer $x$ in base $b$ notation is congruent to $x$ modulo $b - 1$."}
+{"_id": "690", "title": "Euler Phi Function of Integer", "text": "Let $n \\in \\Z_{>0}$, that is, a (strictly) positive integer. Let $\\phi: \\Z_{>0} \\to \\Z_{>0}$ be the Euler $\\phi$-function. Then for any $n \\in \\Z_{>0}$, we have: :$\\map \\phi n = n \\paren {1 - \\dfrac 1 {p_1} } \\paren {1 - \\dfrac 1 {p_2} } \\cdots \\paren {1 - \\dfrac 1 {p_r} }$ where $p_1, p_2, \\ldots, p_r$ are the distinct primes dividing $n$. Or, more compactly: :$\\displaystyle \\map \\phi n = n \\prod_{p \\mathop \\divides n} \\paren {1 - \\frac 1 p}$ where $p \\divides n$ denotes the primes which divide $n$."}
+{"_id": "691", "title": "Euler Phi Function is Multiplicative", "text": "The Euler $\\phi$ function is a multiplicative function: :$m \\perp n \\implies \\map \\phi {m n} = \\map \\phi m \\, \\map \\phi n$ where $m, n \\in \\Z_{>0}$."}
+{"_id": "692", "title": "Euler Phi Function of Prime Power", "text": "Let $p^n$ be a prime power for some prime number $p > 1$. Then: :$\\map \\phi {p^n} = p^n \\paren {1 - \\dfrac 1 p} = \\paren {p - 1} p^{n - 1}$ where $\\phi: \\Z_{>0} \\to \\Z_{>0}$ is the Euler $\\phi$ function."}
+{"_id": "694", "title": "Möbius Function is Multiplicative", "text": "The Möbius function $\\mu$  is a multiplicative function: :$m \\perp n \\implies \\map \\mu {m n} = \\map \\mu m \\map \\mu n$ where $m, n \\in \\Z_{>0}$."}
+{"_id": "695", "title": "Sum of Möbius Function over Divisors", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive integer. Then: :$\\displaystyle \\sum_{d \\mathop \\divides n} \\map \\mu d \\frac n d = \\map \\phi n$ where: :$\\displaystyle \\sum_{d \\mathop \\divides n}$ denotes the sum over all of the divisors of $n$ :$\\map \\phi n$ is the Euler $\\phi$ function, the number of integers less than $n$ that are prime to $n$ :$\\map \\mu d$ is the Möbius function. Equivalently, this says that: :$\\phi = \\mu * I_{\\Z_{>0} }$ where: :$*$ denotes Dirichlet convolution :$I_{\\Z_{>0} }$ denotes the identity mapping on $\\Z_{>0}$, that is: ::$\\forall n \\in \\Z_{>0}: I_{\\Z_{>0} }: n \\mapsto n$ {{wtd|Add a link to a page proving this equivalence.}}"}
+{"_id": "696", "title": "Binomial Coefficient of Prime", "text": "Let $p$ be a prime number. Then: :$\\forall k \\in \\Z: 0 < k < p: \\dbinom p k \\equiv 0 \\pmod p$ where $\\dbinom p k$ is defined as a binomial coefficient."}
+{"_id": "697", "title": "Prime Power of Sum Modulo Prime", "text": "Let $p$ be a prime number. Then: :$\\forall n \\in \\N_{> 0}: \\paren {a + b}^{p^n} \\equiv a^{p^n} + b^{p^n} \\pmod p$"}
+{"_id": "698", "title": "Binomial Coefficient involving Power of Prime", "text": ": $\\dbinom {p^n k} {p^n} \\equiv k \\pmod p$ where $\\dbinom {p^n k} {p^n}$ is a binomial coefficient."}
+{"_id": "699", "title": "Cassini's Identity", "text": ":$F_{n + 1} F_{n - 1} - F_n^2 = \\paren {-1}^n$"}
+{"_id": "700", "title": "Rational Numbers are Countably Infinite", "text": "The set $\\Q$ of rational numbers is countably infinite."}
+{"_id": "701", "title": "Real Numbers form Ordered Field", "text": "The set of real numbers $\\R$ forms an ordered field under addition and multiplication: $\\struct {\\R, +, \\times, \\le}$."}
+{"_id": "702", "title": "Rational Numbers form Subfield of Real Numbers", "text": "The (ordered) field $\\struct {\\Q, +, \\times, \\le}$ of rational numbers forms a subfield of the field of real numbers $\\struct {\\R, +, \\times, \\le}$. That is, the field of real numbers $\\struct {\\R, +, \\times, \\le}$ is an extension of the rational numbers $\\struct {\\Q, +, \\times, \\le}$."}
+{"_id": "703", "title": "Definition:Rational Number/Canonical Form", "text": "Let $r \\in \\Q$ be a rational number. The '''canonical form of $r$''' is the expression $\\dfrac p q$, where: :$r = \\dfrac p q: p \\in \\Z, q \\in \\Z_{>0}, p \\perp q$ where $p \\perp q$ denotes that $p$ and $q$ have no common divisor except $1$."}
+{"_id": "704", "title": "Ordering Properties of Real Numbers", "text": "=== Trichotomy Law === {{:Trichotomy Law for Real Numbers}} === Ordering is Transitive === {{:Real Number Ordering is Transitive}} === Ordering is Compatible with Addition === {{:Real Number Ordering is Compatible with Addition}} === Ordering is Compatible with Multiplication === {{:Real Number Ordering is Compatible with Multiplication}}"}
+{"_id": "705", "title": "Order is Preserved on Positive Reals by Squaring", "text": ":$x < y \\iff x^2 < y^2$"}
+{"_id": "706", "title": "Real Plus Epsilon", "text": "Let $a, b \\in \\R$, such that: :$\\forall \\epsilon \\in \\R_{>0}: a < b + \\epsilon$ where $\\R_{>0}$ is the set of strictly positive real numbers. That is: :$\\epsilon > 0$ Then: :$a \\le b$"}
+{"_id": "707", "title": "Difference of Two Squares", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring whose zero is $0_R$. Let $x, y \\in R$. Then: : $x \\circ x + \\paren {- \\paren {y \\circ y} } = \\paren {x + y} \\circ \\paren {x + \\paren {-y} }$ When $R$ is one of the standard sets of numbers, that is $\\Z, \\Q, \\R$, and so on, then this translates into: :$x^2 - y^2 = \\paren {x + y} \\paren {x - y}$"}
+{"_id": "709", "title": "Mediant is Between", "text": "Let $a, b, c, d$ be ''any'' real numbers such that $b > 0, d > 0$. Let $r = \\dfrac a b < \\dfrac c d = s$. Then: :$r < \\dfrac {a + c} {b + d} < s$"}
+{"_id": "710", "title": "Real Numbers Between Epsilons", "text": "Let $a, b \\in \\R$ such that $\\forall \\epsilon \\in \\R_{>0}: a - \\epsilon < b < a + \\epsilon$. Then $a = b$."}
+{"_id": "711", "title": "Exists Integer Below Any Real Number", "text": "Let $x$ be a real number. Then there exists an integer less than $x$: :$\\forall x \\in \\R: \\exists n \\in \\Z: n < x$"}
+{"_id": "712", "title": "Archimedean Principle", "text": "Let $x$ be a real number. Then there exists a natural number greater than $x$. :$\\forall x \\in \\R: \\exists n \\in \\N: n > x$ That is, the set of natural numbers is unbounded above."}
+{"_id": "713", "title": "Real Number is between Floor Functions", "text": ":$\\forall x \\in \\R: \\floor x \\le x < \\floor {x + 1}$"}
+{"_id": "714", "title": "Real Number is between Ceiling Functions", "text": ":$\\forall x \\in \\R: \\left \\lceil {x - 1} \\right \\rceil \\le x < \\left \\lceil {x} \\right \\rceil$"}
+{"_id": "715", "title": "Real Number minus Floor", "text": ":$x - \\floor x \\in \\hointr 0 1$"}
+{"_id": "716", "title": "Ceiling minus Real Number", "text": ":$\\forall x \\in \\R: \\left \\lceil {x} \\right \\rceil - x \\in \\left[{0 \\,.\\,.\\, 1}\\right)$"}
+{"_id": "717", "title": "Real Number is Floor plus Difference", "text": ":There exists an integer $n \\in \\Z$ such that for some $t \\in \\hointr 0 1$: ::$x = n + t$ {{iff}}: :$n = \\floor x$"}
+{"_id": "718", "title": "Floor plus One", "text": "Let $x \\in \\R$. Then: :$\\left \\lfloor {x + 1} \\right \\rfloor = \\left \\lfloor {x} \\right \\rfloor + 1$ where $\\left \\lfloor {x} \\right \\rfloor$ is the floor function of $x$."}
+{"_id": "719", "title": "Real Number is Integer iff equals Floor", "text": ":$x = \\floor x \\iff x \\in \\Z$"}
+{"_id": "720", "title": "Sum of Floor and Floor of Negative", "text": "Let $x \\in \\R$. Then: :$\\floor x + \\floor {-x} = \\begin{cases} 0 & : x \\in \\Z \\\\ -1 & : x \\notin \\Z \\end{cases}$ where $\\floor x$ denotes the floor of $x$."}
+{"_id": "722", "title": "Ceiling defines Equivalence Relation", "text": "Let $\\mathcal R$ be the relation defined on $\\R$ such that: :$\\forall x, y, \\in \\R: \\left({x, y}\\right) \\in \\mathcal R \\iff \\left \\lceil {x}\\right \\rceil = \\left \\lceil {y}\\right \\rceil$ where $\\left \\lceil {x}\\right \\rceil$ is the ceiling of $x$. Then $\\mathcal R$ is an equivalence, and $\\forall n \\in \\Z$, the $\\mathcal R$-class of $n$ is the half-open interval $\\left({n - 1 \\,.\\,.\\, n}\\right]$."}
+{"_id": "723", "title": "Real Number is Ceiling minus Difference", "text": "Let $n$ be a integer. {{TFAE}} :$(1): \\quad$ There exists $t \\in \\hointr 0 1$ such that $x = n - t$ :$(2): \\quad n = \\ceiling x$"}
+{"_id": "724", "title": "Cauchy's Inequality", "text": ":$\\displaystyle \\sum {r_i^2} \\sum {s_i^2} \\ge \\left({\\sum {r_i s_i}}\\right)^2$"}
+{"_id": "725", "title": "Cancellable Finite Semigroup is Group", "text": "Let $\\struct {S, \\circ}$ be a non-empty finite semigroup in which all elements are cancellable. Then $\\struct {S, \\circ}$ is a group."}
+{"_id": "726", "title": "Finite Semigroup Equal Elements for Different Powers", "text": "Let $\\left({S, \\circ}\\right)$ be a finite semigroup. Then: : $\\forall x \\in S: \\exists m, n \\in \\N: m \\ne n: x^m = x^n$"}
+{"_id": "727", "title": "Element has Idempotent Power in Finite Semigroup", "text": "Let $\\struct {S, \\circ}$ be a finite semigroup. For every element in $\\struct {S, \\circ}$, there is a power of that element which is idempotent. That is: :$\\forall x \\in S: \\exists i \\in \\N: x^i = x^i \\circ x^i$"}
+{"_id": "728", "title": "Powers of Ring Elements", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Let $n \\cdot x$ be an integral multiple of $x$: :$n \\cdot x = \\begin {cases} 0_R & : n = 0 \\\\ x & : n = 1 \\\\ \\paren {n - 1} \\cdot x + x & : n > 1 \\end {cases}$ that is $n \\cdot x = x + x + \\cdots \\paren n \\cdots x$. For $n < 0$ we use: :$-n \\cdot x = n \\cdot \\paren {-x}$ Then: :$\\forall n \\in \\Z: \\forall x \\in R: \\paren {n \\cdot x} \\circ x = n \\cdot \\paren {x \\circ x} = x \\circ \\paren {n \\cdot x}$"}
+{"_id": "729", "title": "Power of Conjugate equals Conjugate of Power", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $x, y \\in G$ such that $\\exists a \\in G: x \\circ a = a \\circ y$. That is, let $x$ and $y$ be conjugate. Then: : $\\forall n \\in \\Z: y^n = \\paren {a^{-1} \\circ x \\circ a}^n = a^{-1} \\circ x^n \\circ a$ It follows directly that: : $\\exists b \\in G: \\forall n \\in \\Z: y^n = b \\circ x^n \\circ b^{-1}$ In particular: : $y^{-1} = \\paren {a^{-1} \\circ x \\circ a}^{-1} = a^{-1} \\circ x^{-1} \\circ a$"}
+{"_id": "730", "title": "Product of Conjugates equals Conjugate of Products", "text": "Let $\\struct {G, \\circ}$ be a group. Then: :$\\forall a, x, y \\in G: \\paren {a \\circ x \\circ a^{-1} } \\circ \\paren {a \\circ y \\circ a^{-1} } = a \\circ \\paren {x \\circ y} \\circ a^{-1}$ That is, the product of conjugates is equal to the conjugate of the product."}
+{"_id": "731", "title": "Power of Product with Inverse", "text": "Let $G$ be a group whose identity is $e$. Let $a, b \\in G: a b = b a^{-1}$. Then: : $\\forall n \\in \\Z: a^n b = b a^{-n}$"}
+{"_id": "732", "title": "Powers of Elements in Group Direct Product", "text": "Let $\\left({G, \\circ_1}\\right)$ and $\\left({H, \\circ_2}\\right)$ be group whose identities are $e_G$ and $e_H$. Let $\\left({G \\times H, \\circ}\\right)$ be the group direct product (either external or internal) of $G$ and $H$. Then: : $\\forall n \\in \\Z: \\forall g \\in G, h \\in H: \\left({g, h}\\right)^n = \\left({g^n, h^n}\\right)$"}
+{"_id": "733", "title": "Powers of Commutative Elements in Semigroups", "text": "Let $\\left ({S, \\circ}\\right)$ be a semigroup. Let $a, b \\in S$ both be cancellable elements of $S$. Then the following results hold:"}
+{"_id": "734", "title": "Powers of Commutative Elements in Monoids", "text": "These results are an extension of the results in Powers of Commutative Elements in Semigroups in which the domain of the indices is extended to include all integers. Let $\\left ({S, \\circ}\\right)$ be a monoid whose identity is $e_S$. Let $a, b \\in S$ be invertible elements for $\\circ$ that also commute. Then the following results hold."}
+{"_id": "735", "title": "Powers of Commutative Elements in Groups", "text": "Let $\\struct {G, \\circ}$ be a group. Let $a, b \\in G$ such that $a$ and $b$ commute. Then the following results hold:"}
+{"_id": "736", "title": "General Morphism Property for Semigroups", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be semigroups. Let $\\phi: S \\to T$ be a homomorphism. Then: :$\\forall s_k \\in S: \\map \\phi {s_1 \\circ s_2 \\circ \\cdots \\circ s_n} = \\map \\phi {s_1} * \\map \\phi {s_2} * \\cdots * \\map \\phi {s_n}$ Hence it follows that: :$\\forall n \\in \\N_{>0}: \\forall s \\in S: \\map \\phi {s^n} = \\paren {\\map \\phi s}^n$"}
+{"_id": "737", "title": "Homomorphism of Power of Group Element", "text": "Let $\\struct {G, \\circ}$ and $\\struct {H, \\ast}$ be groups. Let $\\phi: S \\to T$ be a group homomorphism. Then: : $\\forall n \\in \\Z: \\forall g \\in G: \\map \\phi {g^n} = \\paren {\\map \\phi g}^n$"}
+{"_id": "739", "title": "Finite Subgroup Test", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H$ be a non-empty finite subset of $G$. Then: :$H$ is a subgroup of $G$ {{iff}}: :$\\forall a, b \\in H: a \\circ b \\in H$ That is, a non-empty finite subset of $G$ is a subgroup {{iff}} it is closed."}
+{"_id": "740", "title": "Powers of Element form Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group. Then: :$\\forall a \\in G: H = \\set {a^n: n \\in \\Z} \\le G$ That is, the subset of $G$ comprising all elements possible as powers of $a \\in G$ is a subgroup of $G$."}
+{"_id": "742", "title": "Homomorphism of Generated Group", "text": "Let $\\left({G, \\circ}\\right)$ and $\\left({H, \\circ}\\right)$ be groups. Let $\\phi: G \\to H$ and $\\psi: G \\to H$ be homomorphisms.  Let $\\left \\langle {S} \\right \\rangle = G$ be the group generated by $S$. Let $\\forall x \\in S: \\phi \\left({x}\\right) = \\psi \\left({x}\\right)$ Then $\\phi = \\psi$."}
+{"_id": "743", "title": "Set of Words Generates Group", "text": "Let $S \\subseteq G$ where $G$ is a group. Let $\\hat S$ be defined as $S \\cup S^{-1}$, where $S^{-1}$ is the set of all the inverses of all the elements of $S$. Then $\\gen S = \\map W {\\hat S}$, where $\\map W {\\hat S}$ is the set of words of $\\hat S$."}
+{"_id": "744", "title": "Subset Product is Subset of Generator", "text": "Let $\\struct {G, \\circ}$ be a group. Let $X, Y \\subseteq \\struct {G, \\circ}$. Then $X \\circ Y \\subseteq \\gen {X, Y}$ where: :$X \\circ Y$ is the Subset Product of $X$ and $Y$ in $G$. :$\\gen {X, Y}$ is the subgroup of $G$ generated by $X$ and $Y$."}
+{"_id": "745", "title": "Order of Subset Product with Singleton", "text": "Let $\\struct {G, \\circ}$ be a group. Let $X, Y \\subseteq \\struct {G, \\circ}$ such that $X$ is a singleton: :$X = \\set x$ Then: :$\\order {X \\circ Y} = \\order Y = \\order {Y \\circ X}$ where $\\order S$ is defined as the order of $S$."}
+{"_id": "746", "title": "Product of Subset with Intersection", "text": "Let $\\struct {G, \\circ}$ be an algebraic structure. Let $X, Y, Z \\subseteq G$. Then: :$X \\circ \\paren {Y \\cap Z} \\subseteq \\paren {X \\circ Y} \\cap \\paren {X \\circ Z}$ :$\\paren {Y \\cap Z} \\circ X \\subseteq \\paren {Y \\circ X} \\cap \\paren {Z \\circ X}$ where $X \\circ Y$ denotes the subset product of $X$ and $Y$."}
+{"_id": "747", "title": "Order of Subgroup Product", "text": "Let $G$ be a group. Let $H$ and $K$ be subgroups of $G$. Then: :$\\order {H K} = \\dfrac {\\order H \\order K} {\\order {H \\cap K} }$ where: :$H K$ denotes subset product :$\\order H$ denotes the order of $H$."}
+{"_id": "748", "title": "Index of Intersection of Subgroups", "text": "Let $G$ be a group. Let $H, K$ be subgroups of finite index of $G$. Then: :$\\index G {H \\cap K} \\le \\index G H \\index G K$ where $\\index G H$ denotes the index of $H$ in $G$. Note that here the symbol $\\le$ is being used with its meaning '''less than or equal to'''. Equality holds {{iff}} $H K = \\set {h k: h \\in H, k \\in K} = G$."}
+{"_id": "749", "title": "Intersection of Subgroups of Prime Order", "text": "Let $G$ be a group whose identity is $e$. Let $H$ and $K$ be subsets of $G$ such that: :$\\order H = \\order K = p$ :$H \\ne K$ :$p$ is prime. Then: : $H \\cap K = \\set e$ That is, the intersection of two unequal subgroups of a group, both of whose order is the same prime, consists solely of the identity."}
+{"_id": "750", "title": "Tower Law for Subgroups", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H$ be a subgroup of $G$ with finite index. Let $K$ be a subgroup of $H$. Then: :$\\index G K = \\index G H \\index H K$ where $\\index G H$ denotes the index of $H$ in $G$."}
+{"_id": "751", "title": "Morphism from Integers to Group", "text": "Let $G$ be a group whose identity is $e$. Let $g \\in G$. Let $\\phi: \\Z \\to G$ be the mapping defined as: :$\\forall n \\in \\Z: \\map \\phi n = g^n$. Then: : If $g$ has infinite order, then $\\phi$ is a group isomorphism from $\\struct {\\Z, +}$ to $\\gen g$. : If $g$ has finite order such that $\\order g = m$, then $\\phi$ is a group epimorphism from $\\struct {\\Z, +}$ to $\\gen g$ whose kernel is the principal ideal $\\paren m$. :Thus $\\gen g$ is isomorphic to $\\struct {\\Z, +}$, and $m$ is the smallest (strictly) positive integer such that $g^m = e$."}
+{"_id": "752", "title": "Identity is Only Group Element of Order 1", "text": "In every group, the identity, and only the identity, has order $1$."}
+{"_id": "753", "title": "Group Element is Self-Inverse iff Order 2", "text": "Let $\\struct {S, \\circ}$ be a group whose identity is $e$. An element $x \\in \\struct {S, \\circ}$ is self-inverse {{iff}}: :$\\order x = 2$"}
+{"_id": "754", "title": "Powers of Infinite Order Element", "text": "Let $G$ be a group whose identity is $e$. Let $a \\in G$ have infinite order in $G$. Then: :$\\forall m, n \\in \\Z: m \\ne n \\implies a^m \\ne a^n$"}
+{"_id": "755", "title": "Element of Finite Group is of Finite Order", "text": "In any finite group, each element has finite order."}
+{"_id": "756", "title": "Inverse Element is Power of Order Less 1", "text": "Let $G$ be a group whose identity is $e$. Let $g \\in G$ be of finite order. Then: : $\\order g = n \\implies g^{n - 1} = g^{-1}$"}
+{"_id": "757", "title": "Equal Powers of Finite Order Element", "text": ":$g^r = g^s \\iff k \\divides \\paren {r - s}$"}
+{"_id": "758", "title": "Order of Element Divides Order of Finite Group", "text": "In a finite group, the order of a group element divides the order of its group: :$\\forall x \\in G: \\order x \\divides \\order G$"}
+{"_id": "760", "title": "Element to Power of Group Order is Identity", "text": "Let $G$ be a group whose identity is $e$ and whose order is $n$. Then: :$\\forall g \\in G: g^n = e$"}
+{"_id": "761", "title": "Boolean Group is Abelian", "text": "Let $G$ be a Boolean group. Then $G$ is abelian."}
+{"_id": "762", "title": "Order of Group Element equals Order of Inverse", "text": "Let $G$ be a group whose identity is $e$. Then: : $\\forall x \\in G: \\order x = \\order {x^{-1} }$ where $\\order x$ denotes the order of $x$."}
+{"_id": "764", "title": "Order of Conjugate Element equals Order of Element", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Then :$\\forall a, x \\in \\struct {G, \\circ}: \\order {x \\circ a \\circ x^{-1} } = \\order a$ where $\\order a$ denotes the order of $a$ in $G$."}
+{"_id": "765", "title": "Order of Homomorphic Image of Group Element", "text": "Let $G$ and $H$ be groups whose identities are $e_G$ and $e_H$ respectively. Let $\\phi: G \\to H$ be a homomorphism. Let $g \\in G$ be of finite order. Then: :$\\forall g \\in G: \\order {\\map \\phi g} \\divides \\order g$ where $\\divides$ denotes divisibility."}
+{"_id": "769", "title": "Non-Trivial Group has Non-Trivial Cyclic Subgroup", "text": "Let $G$ be a group whose identity element is $e$. Let $g \\in G$. If $g$ has infinite order, then $\\gen g$ is an infinite cyclic group. If $\\order g = n$, then $\\gen g$ is a cyclic group with $n$ elements. Thus, every group which is non-trivial has at least one cyclic subgroup which is also non-trivial. In the case that $G$ is itself cyclic, that cyclic subgroup may of course be itself."}
+{"_id": "770", "title": "Epimorphism from Integers to Cyclic Group", "text": "Let $\\gen a = \\struct {G, \\circ}$ be a cyclic group. Let $f: \\Z \\to G$ be a mapping defined as: $\\forall n \\in \\Z: \\map f n = a^n$. Then $f$ is a (group) epimorphism from $\\struct {\\Z, +}$ onto $\\gen a$."}
+{"_id": "771", "title": "Cyclic Group is Abelian", "text": "Let $G$ be a cyclic group. Then $G$ is abelian."}
+{"_id": "772", "title": "Cyclic Groups of Same Order are Isomorphic", "text": "Two cyclic groups of the same order are isomorphic to each other."}
+{"_id": "773", "title": "Order of Subgroup of Cyclic Group", "text": "Let $C_n = \\gen g$ be the cyclic group of order $n$ which is generated by $g$ whose identity is $e$. Let $a \\in C_n: a = g^i$. Let $H = \\gen a$. Then: :$\\order H = \\dfrac n {\\gcd \\set {n, i} }$ where: :$\\order H$ denotes the order of $H$ :$\\gcd \\set {n, i}$ denotes the greatest common divisor of $n$ and $i$."}
+{"_id": "774", "title": "Number of Powers of Cyclic Group Element", "text": "Let $G$ be a cyclic group of order $n$, generated by $g$. Let $d \\divides n$. Then the element $g^{n/d}$ has $d$ distinct powers."}
+{"_id": "775", "title": "Subgroup of Finite Cyclic Group is Determined by Order", "text": "Let $G = \\gen g$ be a cyclic group whose order is $n$ and whose identity is $e$. Let $d \\divides n$, where $\\divides$ denotes divisibility. Then there exists exactly one subgroup $G_d = \\gen {g^{n / d} }$ of $G$ with $d$ elements."}
+{"_id": "777", "title": "Prime Group is Cyclic", "text": "Let $p$ be a prime number. Let $G$ be a group whose order is $p$. Then $G$ is cyclic."}
+{"_id": "778", "title": "Group of Order less than 6 is Abelian", "text": "All groups with less than $6$ elements are abelian."}
+{"_id": "779", "title": "Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order", "text": "Let $C_n$ be the cyclic group of order $n$. Let $C_n = \\gen a$, that is, that $C_n$ is generated by $a$. Then: :$C_n = \\gen {a^k} \\iff k \\perp n$ That is, $C_n$ is also generated by $a^k$ {{iff}} $k$ is coprime to $n$."}
+{"_id": "780", "title": "Order of Conjugate of Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$ such that $H$ is of finite order. Then $\\order {H^a} = \\order H$."}
+{"_id": "781", "title": "Subgroup of Index 2 is Normal", "text": "A subgroup of index $2$ is always normal."}
+{"_id": "782", "title": "Intersection of Normal Subgroups is Normal", "text": "Let $G$ be a group. Let $I$ be an indexing set. Let $\\family {N_i}_{i \\mathop \\in I}$ be a non-empty indexed family of normal subgroups of $G$. Then $\\displaystyle \\bigcap_{i \\mathop \\in I} N_i$ is a normal subgroup of $G$."}
+{"_id": "783", "title": "Union of Conjugacy Classes is Normal", "text": "Let $G$ be a group. Let $H \\le G$. Then $H$ is normal in $G$ {{iff}} $H$ is a union of conjugacy classes of $G$."}
+{"_id": "784", "title": "Unique Subgroup of a Given Order is Normal", "text": "Let a group $G$ have only one subgroup of a given order. Then that subgroup is normal."}
+{"_id": "786", "title": "Smallest Normal Subgroup containing Set", "text": "Let $S \\subseteq G$ where $G$ is a group. Then there exists a unique smallest normal subgroup of $G$ which contains $S$."}
+{"_id": "787", "title": "Conjugate of Set with Inverse Closed for Inverses", "text": "Let $G$ be a group. Let $S \\subseteq G$. Let $\\hat S = S \\cup S^{-1}$, where $S^{-1}$ is the set of all the inverses of all the elements of $S$. Let $\\tilde S = \\set {a s a^{-1}: s \\in \\hat S, a \\in G}$. That is, $\\tilde S$ is the set containing all the conjugates of the elements of $S$ and all their inverses. Then: : $\\forall x \\in \\tilde S: x^{-1} \\in \\tilde S$"}
+{"_id": "788", "title": "Conjugate of Set with Inverse is Closed", "text": "Let $G$ be a group. Let $S \\subseteq G$. Let $\\hat S = S \\cup S$. Let $\\tilde S = \\left\\{{a s a^{-1}: s \\in \\hat S, a \\in G}\\right\\}$. Let $W \\left({\\tilde S}\\right)$ be the set of words of $\\tilde S$. Then $\\forall w \\in W \\left({\\tilde S}\\right): \\forall a \\in G: a w a^{-1} \\in W \\left({\\tilde S}\\right)$."}
+{"_id": "789", "title": "Generator of Normal Subgroup", "text": "Let $G$ be a group. Let $S \\subseteq G$. Let $\\hat S = S \\cup S^{-1}$, where $S^{-1}$ is the set of all the inverses of all the elements of $S$. Let $\\tilde S = \\left\\{{a s a^{-1}: s \\in \\hat S, a \\in G}\\right\\}$. Let $W \\left({\\tilde S}\\right)$ be the set of words of $\\tilde S$. Let $N$ be the smallest normal subgroup of $G$ that contains $S$. Then $N = \\left \\langle {S} \\right \\rangle = W \\left({\\tilde S}\\right)$."}
+{"_id": "790", "title": "Subset Product with Normal Subgroup as Generator", "text": "Let $G$ be a group whose identity is $e$. Let: :$H$ be a subgroup of $G$ :$N$ be a normal subgroup of $G$. Then: :$N \\lhd \\gen {N, H} = N H = H N \\le G$ where: :$\\le$ denotes subgroup :$\\lhd$ denotes normal subgroup :$\\gen {N, H}$ denotes a subgroup generator :$N H$ denotes subset product."}
+{"_id": "791", "title": "Subset Product of Normal Subgroups is Normal", "text": "Let $\\struct {G, \\circ}$ be a group. Let $N$ and $N'$ be normal subgroups of $G$. Then $N N'$ is also a normal subgroup of $G$."}
+{"_id": "792", "title": "Prime Group is Simple", "text": "Groups of prime order are simple."}
+{"_id": "793", "title": "Prime Group has no Proper Subgroups", "text": "A nontrivial group $G$ has no proper subgroups except the trivial group {{iff}} $G$ is finite and its order is prime."}
+{"_id": "794", "title": "Quotient Group of Cyclic Group", "text": "Let $G$ be a cyclic group which is generated by $g$. Let $H$ be a subgroup of $G$. Then $g H$ generates $G / H$."}
+{"_id": "795", "title": "Order of Element in Quotient Group", "text": "Let $G$ be a group, and let $H$ be a normal subgroup of $G$. Let $G / H$ be the quotient group of $G$ by $H$. The order of $a H \\in G / H$ divides the order of $a \\in G$."}
+{"_id": "799", "title": "Quotient of Group by Center Cyclic implies Abelian", "text": "Let $G$ be a group. Let $\\map Z G$ be the center of $G$. Let $G / \\map Z G$ be the quotient group of $G$ by $\\map Z G$. Let $G / \\map Z G$ be cyclic. Then $G$ is abelian, so $G = \\map Z G$. That is, the group $G / \\map Z G$ cannot be a cyclic group which is non-trivial."}
+{"_id": "800", "title": "Centralizer is Normal Subgroup of Normalizer", "text": "Let $G$ be a group. Let $H \\le G$ be a subgroup of $G$.  Let $\\map {C_G} H$ be the centralizer of $H$ in $G$. Let $\\map {N_G} H$ be the normalizer of $H$ in $G$. Let $\\Aut H$ be the automorphism group of $H$. Then: :$(1): \\quad \\map {C_G} H \\lhd \\map {N_G} H$ :$(2): \\quad \\map {N_G} H / \\map {C_G} H \\cong K$ where: :$\\map {N_G} H / \\map {C_G} H$ is the quotient group of $\\map {N_G} H$ by $\\map {C_G} H$ :$K$ is a subgroup of $\\Aut H$."}
+{"_id": "801", "title": "Number of Distinct Conjugate Subsets is Index of Normalizer", "text": "Let $G$ be a group. Let $S$ be a subset of $G$. Let $\\map {N_G} S$ be the normalizer of $S$ in $G$. Let $\\index G {\\map {N_G} S}$ be the index of $\\map {N_G} S$ in $G$. The number of distinct subsets of $G$ which are conjugates of $S \\subseteq G$ is $\\index G {\\map {N_G} S}$."}
+{"_id": "803", "title": "Element of Group Not Conjugate to Proper Subgroup", "text": "Let $G$ be a finite group. Let $H$ be a proper subgroup of $G$. Then there is at least one element of $G$ not contained in $H$ or in any of its conjugates."}
+{"_id": "804", "title": "Second Isomorphism Theorem", "text": "=== Groups === {{:Second Isomorphism Theorem/Groups}} === Rings === {{:Second Isomorphism Theorem/Rings}} This result is also referred to by some sources as the '''first isomorphism theorem'''."}
+{"_id": "805", "title": "Third Isomorphism Theorem", "text": "=== Groups === {{:Third Isomorphism Theorem/Groups}} === Rings === {{:Third Isomorphism Theorem/Rings}}"}
+{"_id": "807", "title": "Inverse of Inner Automorphism", "text": "Let $G$ be a group. Let $x \\in G$. Let $\\kappa_x$ be the inner automorphism of $G$ given by $x$. Then: : $\\paren {\\kappa_x}^{-1} = \\kappa_{x^{-1} }$"}
+{"_id": "809", "title": "Conjugates of Elements in Centralizer", "text": "Let $G$ be a group. Let $\\map {C_G} a$ be the centralizer of $a$ in $G$. Then $\\forall g, h \\in G: g a g^{-1} = h a h^{-1}$ {{iff}} $g$ and $h$ belong to the same left coset of $\\map {C_G} a$."}
+{"_id": "810", "title": "Number of Conjugates is Number of Cosets of Centralizer", "text": "Let $G$ be a group. Let $\\map {C_G} a$ be the centralizer of $a$ in $G$. Then the number of different conjugates of $a$ in $G$ equals the number of different (left) cosets of $\\map {C_G} a$: :$\\card {\\conjclass a} = \\index G {\\map {C_G} a}$ where: :$\\conjclass a$ is the conjugacy class of $a$ in $G$ :$\\index G {\\map {C_G} a}$ is the index of $\\map {C_G} a$ in $G$. Consequently: :$\\card {\\conjclass a} \\divides \\order G$"}
+{"_id": "811", "title": "Size of Conjugacy Class is Index of Normalizer", "text": "Let $G$ be a group. Let $x \\in G$. Let $\\conjclass x$ be the conjugacy class of $x$ in $G$. Let $\\map {N_G} x$ be the normalizer of $x$ in $G$. Let $\\index G {\\map {N_G} x}$ be the index of $\\map {N_G} x$ in $G$. The number of elements in $\\conjclass x$ is $\\index G {\\map {N_G} x}$."}
+{"_id": "812", "title": "Conjugacy Class of Element of Center is Singleton", "text": "Let $G$ be a group. Let $\\map Z G$ denote the center of $G$. The elements of $\\map Z G$ form singleton conjugacy classes, and the elements of $G \\setminus \\map Z G$ belong to multi-element conjugacy classes."}
+{"_id": "813", "title": "Conjugacy Class Equation", "text": "Let $G$ be a group. Let $\\order G$ denote the order of $G$. Let $\\map Z G$ denote the center of $G$. Let $x \\in G$. Let $\\map {N_G} x$ denote the normalizer of $x$ in $G$. Let $\\index G {\\map {N_G} x}$ denote the index of $\\map {N_G} x$ in $G$. Let $m$ be the number of non-singleton conjugacy classes of $G$. Let $x_j: j \\in \\set {1, 2, \\ldots, m}$ be arbitrary elements of those conjugacy classes. Then: :$\\displaystyle \\order G = \\order {\\map Z G} + \\sum_{j \\mathop = 1}^m \\index G {\\map {N_G} {x_j} }$"}
+{"_id": "814", "title": "Group of Order Prime Squared is Abelian", "text": "A group whose order is the square of a prime is abelian."}
+{"_id": "815", "title": "Center of Group of Prime Power Order is Non-Trivial", "text": "Let $G$ be a group whose order is the power of a prime. Then the center of $G$ is non-trivial: :$\\forall G: \\order G = p^r: p \\in \\mathbb P, r \\in \\N_{>0}: \\map Z G \\ne \\set e$"}
+{"_id": "816", "title": "Center of Group of Order Prime Cubed", "text": "Let $G$ be a group of order $p^3$, where $p$ is a prime. Let $\\map Z G$ be the center of $G$. Then $\\order {\\map Z G} \\ne p^2$."}
+{"_id": "817", "title": "Prime Power Group has Non-Trivial Proper Normal Subgroup", "text": "Let $G$ be a group, whose identity is $e$, such that $\\order G = p^n: n > 1, p \\in \\mathbb P$. Then $G$ has a proper normal subgroup which is non-trivial."}
+{"_id": "818", "title": "Composition Series of Group of Prime Power Order", "text": "Let $G$ be a group whose identity is $e$, and whose order is a prime power: :$\\order G = p^n, p \\in \\mathbb P, n \\ge 1$ Then $G$ has a composition series: :$\\set e = G_0 \\subset G_1 \\subset \\ldots \\subset G_n = G$ such that $\\order {G_k} = p^k$, $G_k \\lhd G_{k + 1}$ and $G_{k + 1} / G_k$ is cyclic and of order $p$."}
+{"_id": "819", "title": "Sum Rule for Counting", "text": "Let there be: : $r_1$ different objects in the set $S_1$ : $r_2$ different objects in the set $S_2$ : $\\ldots$ : $r_m$ different objects in the set $S_m$. Let $\\displaystyle \\bigcap_{i \\mathop = 1}^m S_i = \\varnothing$. Then the number of ways to select an object from one of the $m$ sets is $\\displaystyle \\sum_{i \\mathop = 1}^m r_i$."}
+{"_id": "820", "title": "Product Rule for Counting", "text": "Let it be possible to choose an element $\\alpha$ from a given set $S$ in $m$ different ways. Let it be possible to choose an element $\\beta$ from a given set $T$ in $n$ different ways. Then the ordered pair $\\tuple {\\alpha, \\beta}$ can be chosen from the cartesian product $S \\times T$ in $m n$ different ways."}
+{"_id": "822", "title": "External Direct Product of Abelian Groups is Abelian Group", "text": "Let $\\struct {G, \\circ_1}$ and $\\struct {H, \\circ_2}$ be groups. Then the group direct product $\\struct {G \\times H, \\circ}$ is abelian {{iff}} both $\\struct {G, \\circ_1}$ and $\\struct {H, \\circ_2}$ are abelian."}
+{"_id": "824", "title": "Group Direct Product of Cyclic Groups", "text": "Let $G$ and $H$ both be finite cyclic groups with orders $n = \\order G$ and $m = \\order H$ respectively. Then their group direct product $G \\times H$ is cyclic {{iff}} $g$ and $h$ are coprime, that is, $g \\perp h$."}
+{"_id": "825", "title": "Group Direct Product of Infinite Cyclic Groups", "text": "The group direct product of two infinite cyclic groups is not cyclic."}
+{"_id": "826", "title": "Order of Group Element in Group Direct Product", "text": "Let $G$ and $H$ be finite groups. Let $g \\in G: \\order g = m, h \\in H: \\order h = n$. Then $\\order {\\tuple {g, h} }$ in $G \\times H$ is $\\lcm \\set {m, n}$."}
+{"_id": "827", "title": "Subgroup Product is Internal Group Direct Product iff Surjective", "text": "Let $G$ be a group. Let $\\sequence {H_n}$ be a sequence of subgroups of $G$. Let $\\displaystyle \\phi: \\prod_{k \\mathop = 1}^n H_k \\to G$ be a mapping defined by: :$\\displaystyle \\map \\phi {h_1, h_2, \\ldots, h_n} = \\prod_{k \\mathop = 1}^n h_k$ Then $\\phi$ is surjective {{iff}}: : $\\displaystyle G = \\prod_{k \\mathop = 1}^n H_k$ That is, {{iff}} $G$ is the internal group direct product of $H_1, H_2, \\ldots, H_n$."}
+{"_id": "828", "title": "Internal Group Direct Product is Injective", "text": "Let $G$ be a group whose identity is $e$. Let $H_1, H_2$ be subgroups of $G$. Let $\\phi: H_1 \\times H_2 \\to G$ be a mapping defined by: :$\\map \\phi {h_1, h_2} = h_1 h_2$ Then $\\phi$ is injective {{iff}}: :$H_1 \\cap H_2 = \\set e$"}
+{"_id": "829", "title": "Internal Group Direct Product Isomorphism", "text": "Let $G$ be a group. Let $H_1, H_2$ be subgroups of $G$. Let $\\phi: H_1 \\times H_2 \\to G$ be the mapping defined by $\\map \\phi {h_1, h_2} := h_1 h_2$. If $\\phi$ is a (group) isomorphism, then both $H_1$ and $H_2$ are normal subgroups of $G$."}
+{"_id": "830", "title": "Internal Group Direct Product of Normal Subgroups", "text": "Let $G$ be a group whose identity is $e$. Let $H_1, H_2$ be subgroups of $G$. Let $\\phi: H_1 \\times H_2 \\to G$ be a mapping defined by $\\map \\phi {h_1, h_2} = h_1 h_2$. Let $H_1$ and $H_2$ be normal subgroups of $G$, and let $H_1 \\cap H_2 = \\set e$. Then $\\phi$ is a (group) homomorphism."}
+{"_id": "831", "title": "Internal Direct Product Theorem", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $H_1, H_2 \\le G$. Then $G$ is the internal group direct product of $H_1$ and $H_2$ {{iff}}: :$(1): \\quad G = H_1 \\circ H_2$ :$(2): \\quad H_1 \\cap H_2 = \\set e$ :$(3): \\quad H_1, H_2 \\lhd G$ where $H_1 \\lhd G$ denotes that $H_1$ is a normal subgroup of $G$."}
+{"_id": "832", "title": "Inclusion Mapping is Surjection iff Identity", "text": "Let $T$ be a set. Let $S\\subseteq T$ be a subset. Let $i_S: S \\to T$ be the inclusion mapping. Then: :$i_S: S \\to T$ is surjective {{iff}} $i_S: S \\to T = I_S: S \\to S$ where $I_S: S \\to S$ denotes the identity mapping on $S$. Alternatively, this theorem can be worded as: :$i_S: S \\to S = I_S: S \\to S$ It follows directly that from Surjection by Restriction of Codomain, the surjective restriction of $i_S: S \\to T$ to $i_S: S \\to \\Img {i_S}$ is itself the identity mapping."}
+{"_id": "834", "title": "Internal Direct Product Generated by Subgroups", "text": "Let $G$ be a group whose identity is $e$. Let $\\sequence {H_n}$ be a sequence of subgroups of $G$. Then: :the subgroup generated by $\\displaystyle \\bigcup_{k \\mathop = 1}^n H_k$ is the internal group direct product of $\\sequence {H_n}$ {{iff}}: :$\\sequence {H_n}$ is an independent sequence of subgroups such that every element of $H_i$ commutes with every element of $H_j$ whenever $1 \\le i < j \\le n$."}
+{"_id": "835", "title": "Internal Group Direct Product Commutativity", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $H_1, H_2 \\le G$. Let $\\struct {G, \\circ}$ be the internal group direct product of $H_1$ and $H_2$. Then: :$\\forall h_1 \\in H_1, h_2 \\in H_2: h_1 \\circ h_2 = h_2 \\circ h_1$"}
+{"_id": "836", "title": "Internal and External Group Direct Products are Isomorphic", "text": "Let $G$ be a group whose identity is $e$. Then $G$ is the (external) group direct product of $G_1, G_2, \\ldots, G_n$ {{iff}} $G$ is the internal group direct product of $N_1, N_2, \\ldots, N_n$ such that: :$\\forall i \\in \\N_n: N_i \\cong G_i$ where: :$\\cong$ denotes (group) isomorphism :$\\N_n$ denotes $\\set {1, 2, \\ldots, n}$"}
+{"_id": "838", "title": "Pullback of Quotient Group Isomorphism is Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group whose identity element is $e_G$. Let $\\struct {H, *}$ be a group whose identity element is $e_H$. Let $N \\lhd G, K \\lhd H$ be normal subgroups of $G$ and $H$ respectively. Let: :$G / N \\cong H / K$ where: :$G / N$ denotes the quotient of $G$ by $N$ :$\\cong$ denotes group isomorphism. Let $\\theta: G / N \\to H / K$ be such a group isomorphism. Let $G \\times^\\theta H$ be the pullback of $G$ and $H$ via $\\theta$. Then $G \\times^\\theta H$ is a subgroup of $G \\times H$."}
+{"_id": "839", "title": "Group/Examples/x+y over 1+xy", "text": "Let $G = \\set {x \\in \\R: -1 < x < 1}$ be the set of all real numbers whose absolute value is less than $1$. Let $\\circ: G \\times G \\to \\R$ be the binary operation defined as: :$\\forall x, y \\in G: x \\circ y = \\dfrac {x + y} {1 + x y}$ The algebraic structure $\\struct {G, \\circ}$ is a group."}
+{"_id": "840", "title": "Group/Examples/inv x = 1 - x", "text": "Let $S = \\set {x \\in \\R: 0 < x < 1}$. Then an operation $\\circ$ can be found such that $\\struct {S, \\circ}$ is a group such that the inverse of $x \\in S$ is $1 - x$."}
+{"_id": "841", "title": "Group/Examples/Self-Inverse and Cancellable Elements", "text": "Let $S$ be a set with an operation which assigns to each $\\tuple {a, b} \\in S \\times S$ an element $a \\ast b \\in S$ such that: : $(1): \\quad \\exists e \\in S: a \\ast b = e \\iff a = b$ : $(2): \\quad \\forall a, b, c \\in S: \\paren {a \\ast c} \\ast \\paren {b \\ast c} = a \\ast b$ Then $\\struct {S, \\circ}$ is a group, where $\\circ$ is defined as $a \\circ b = a \\ast \\paren {e \\ast b}$."}
+{"_id": "842", "title": "Complex Numbers under Addition form Abelian Group", "text": "Let $\\C$ be the set of complex numbers. The structure $\\struct {\\C, +}$ is an infinite abelian group."}
+{"_id": "843", "title": "Non-Zero Complex Numbers under Multiplication form Abelian Group", "text": "Let $\\C_{\\ne 0}$ be the set of complex numbers without zero, that is: :$\\C_{\\ne 0} = \\C \\setminus \\set 0$ The structure $\\struct {\\C_{\\ne 0}, \\times}$ is an infinite abelian group."}
+{"_id": "844", "title": "Real Numbers under Addition form Abelian Group", "text": "Let $\\R$ be the set of real numbers. The structure $\\struct {\\R, +}$ is an infinite abelian group."}
+{"_id": "845", "title": "Non-Zero Real Numbers under Multiplication form Abelian Group", "text": "Let $\\R_{\\ne 0}$ be the set of real numbers without zero: :$\\R_{\\ne 0} = \\R \\setminus \\set 0$ The structure $\\struct {\\R_{\\ne 0}, \\times}$ is an uncountable abelian group."}
+{"_id": "846", "title": "Rational Numbers under Addition form Abelian Group", "text": "Let $\\Q$ be the set of rational numbers. The structure $\\struct {\\Q, +}$ is a countably infinite abelian group."}
+{"_id": "847", "title": "Non-Zero Rational Numbers under Multiplication form Abelian Group", "text": "Let $\\Q_{\\ne 0}$ be the set of non-zero rational numbers: :$\\Q_{\\ne 0} = \\Q \\setminus \\set 0$ The structure $\\struct {\\Q_{\\ne 0}, \\times}$ is a countably infinite abelian group."}
+{"_id": "848", "title": "Integers under Multiplication form Countably Infinite Commutative Monoid", "text": "The set of integers under multiplication $\\struct {\\Z, \\times}$ is a countably infinite commutative monoid."}
+{"_id": "849", "title": "Additive Group of Rationals is Subgroup of Reals", "text": "Let $\\struct {\\Q, +}$ be the additive group of rational numbers. Let $\\struct {\\R, +}$ be the additive group of real numbers. Then $\\struct {\\Q, +}$ is a normal subgroup of $\\struct {\\R, +}$."}
+{"_id": "850", "title": "Additive Group of Integers is Subgroup of Rationals", "text": "Let $\\struct {\\Z, +}$ be the additive group of integers. Let $\\struct {\\Q, +}$ be the additive group of rational numbers. Then $\\struct {\\Z, +}$ is a subgroup of $\\struct {\\Q, +}$."}
+{"_id": "852", "title": "Multiplicative Group of Reals is Subgroup of Complex", "text": "Let $\\struct {\\R_{\\ne 0}, \\times}$ be the multiplicative group of real numbers. Let $\\struct {\\C_{\\ne 0}, \\times}$ be the multiplicative group of complex numbers. Then $\\struct {\\R_{\\ne 0}, \\times}$ is a normal subgroup of $\\struct {\\C_{\\ne 0}, \\times}$."}
+{"_id": "853", "title": "Multiplicative Group of Rationals is Subgroup of Reals", "text": "Let $\\struct {\\Q_{\\ne 0}, \\times}$ be the multiplicative group of rational numbers. Let $\\struct {\\R_{\\ne 0}, \\times}$ be the multiplicative group of real numbers. Then $\\struct {\\Q_{\\ne 0}, \\times}$ is a normal subgroup of $\\left({\\R_{\\ne 0}, \\times}\\right)$."}
+{"_id": "854", "title": "Circle Group is Infinite Abelian Group", "text": "The circle group $\\struct {K, \\times}$ is an uncountably infinite abelian group under the operation of complex multiplication."}
+{"_id": "855", "title": "Homomorphism from Reals to Circle Group", "text": "Let $\\struct {\\R, +}$ be the additive group of real numbers. Let $\\struct {K, \\times}$ be the circle group. Let $\\phi: \\struct {\\R, +} \\to \\struct {K, \\times}$ be the mapping defined as: :$\\forall x \\in \\R: \\map \\phi x = e^{i x}$ Then $\\phi$ is a (group) homomorphism."}
+{"_id": "856", "title": "Integers under Addition form Infinite Cyclic Group", "text": "The additive group of integers $\\struct {\\Z, +}$ is an infinite cyclic group which is generated by the element $1 \\in \\Z$."}
+{"_id": "858", "title": "Inverse of Generator of Cyclic Group is Generator", "text": "Let $\\gen g = G$ be a cyclic group. Then: :$G = \\gen {g^{-1} }$ where $g^{-1}$ denotes the inverse of $g$. Thus, in general, a generator of a cyclic group is not unique."}
+{"_id": "859", "title": "Generators of Infinite Cyclic Group", "text": "Let $\\gen g = G$ be an infinite cyclic group. Then the only other generator of $G$ is $g^{-1}$. Thus an infinite cyclic group has exactly $2$ generators."}
+{"_id": "860", "title": "Subgroup of Integers is Ideal", "text": "Let $\\struct {\\Z, +}$ be the additive group of integers. Every subgroup of $\\struct {\\Z, +}$ is an ideal of the ring $\\struct {\\Z, +, \\times}$."}
+{"_id": "861", "title": "Additive Group of Integers is Subgroup of Reals", "text": "Let $\\struct {\\Z, +}$ be the additive group of integers. Let $\\struct {\\R, +}$ be the additive group of real numbers. Then $\\struct {\\Z, +}$ is a subgroup of $\\struct {\\R, +}$."}
+{"_id": "862", "title": "Quotient Group of Reals by Integers is Circle Group", "text": "Let $\\struct {\\Z, +}$ be the additive group of integers. Let $\\struct {\\R, +}$ be the additive group of real numbers. Let $K$ be the circle group. Then the quotient group of $\\struct {\\R, +}$ by $\\struct {\\Z, +}$ is isomorphic to $K$."}
+{"_id": "863", "title": "Integers Modulo m under Addition form Cyclic Group", "text": "Let $\\Z_m$ be the set of integers modulo $m$. Let $+_m$ be the operation of addition modulo $m$. Let $\\struct {\\Z_m, +_m}$ denote the additive group of integers modulo $m$. Then $\\struct {\\Z_m, +_m}$ is a cyclic group of order $m$, generated by the element $\\eqclass 1 m \\in \\Z_m$."}
+{"_id": "864", "title": "Integers Modulo m under Multiplication form Commutative Monoid", "text": "The structure: :$\\struct {\\Z_m, \\times}$ (where $\\Z_m$ is the set of integers modulo $m$) is a commutative monoid."}
+{"_id": "865", "title": "Multiplicative Inverse in Ring of Integers Modulo m", "text": "Let $\\struct {\\Z_m, +_m, \\times_m}$ be the ring of integers modulo $m$. Then $\\eqclass k m \\in \\Z_m$ has an inverse in $\\struct {\\Z_m, \\times_m}$ {{iff}} $k \\perp m$."}
+{"_id": "866", "title": "Reduced Residue System is Subset of Set of All Residue Classes", "text": "Let $\\Z_m$ be the set of set of residue classes modulo $m$. Let $\\Z'_m$ be the reduced residue system modulo $m$. Then: :$\\forall m \\in \\Z_{> 1}: \\O \\subset \\Z'_m \\subset \\Z_m$"}
+{"_id": "867", "title": "Reduced Residue System under Multiplication forms Abelian Group", "text": "Let $\\Z_m$ be the set of set of residue classes modulo $m$. Let $\\struct {\\Z'_m, \\times}$ denote the multiplicative group of reduced residues modulo $m$. Then $\\struct {\\Z'_m, \\times}$ is an abelian group, precisely equal to the group of units of $\\Z_m$."}
+{"_id": "868", "title": "Ring of Integers Modulo Prime is Field", "text": "Let $m \\in \\Z: m \\ge 2$. Let $\\struct {\\Z_m, +, \\times}$ be the ring of integers modulo $m$. Then: :$m$ is prime {{iff}}: :$\\struct {\\Z_m, +, \\times}$ is a field."}
+{"_id": "869", "title": "Subgroups of Additive Group of Integers", "text": "Let $\\struct {\\Z, +}$ be the additive group of integers. Let $n \\Z$ be the additive group of integer multiples of $n$. Every non-trivial subgroup of $\\struct {\\Z, +}$ has the form $n \\Z$."}
+{"_id": "870", "title": "Integer Multiples under Addition form Infinite Cyclic Group", "text": "Let $n \\Z$ be the set of integer multiples of $n$. Then $\\struct {n \\Z, +}$ is a countably infinite cyclic group. It is generated by $n$ and $-n$: :$n \\Z = \\gen n$ :$n \\Z = \\gen {-n}$ Hence $\\struct {n \\Z, +}$ can be justifiably referred to as the additive group of integer multiples."}
+{"_id": "871", "title": "Quotient Group of Integers by Multiples", "text": "Let $\\struct {\\Z, +}$ be the additive group of integers. Let $\\struct {m \\Z, +}$ be the additive group of integer multiples of $m$. Let $\\struct {\\Z_m, +_m}$ be the additive group of integers modulo $m$. Then the quotient group of $\\struct {\\Z, +}$ by $\\struct {m \\Z, +}$ is $\\struct {\\Z_m, +_m}$. Thus: :$\\index \\Z {m \\Z} = m$"}
+{"_id": "872", "title": "Euler's Theorem", "text": "Let $a, m \\in \\Z$ be coprime integers: $a \\perp m$. Let $\\map \\phi m$ be the Euler $\\phi$ function of $m$. Then: :$a^{\\map \\phi m} \\equiv 1 \\pmod m$"}
+{"_id": "873", "title": "Symmetry Group is Group", "text": "Let $P$ be a geometric figure. Let $S_P$ be the set of all symmetries of $P$. Let $\\circ$ denote composition of mappings. The symmetry group $\\struct {S_P, \\circ}$ is indeed a group."}
+{"_id": "875", "title": "Area of Square", "text": "A square has an area of $L^2$ where $L$ is the length of a side of the square. Thus we have that the area is a function of the length of the side: :$\\forall L \\in \\R_{\\ge 0}: \\map \\Area L = L^2$ where it is noted that the domain of $L$ is the set of non-negative real numbers."}
+{"_id": "876", "title": "Order of Symmetric Group", "text": "Let $S$ be a finite set of cardinality $n$. Let $\\struct {\\map \\Gamma S, \\circ}$ be the symmetric group on $S$. Then $\\struct {\\map \\Gamma S, \\circ}$ has $n!$ elements (see factorial)."}
+{"_id": "877", "title": "Powers of Permutation Element", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\pi \\in S_n$, and let $i \\in \\N^*_n$. Let $k \\in \\Z: k > 0$ be the smallest such that: : $\\pi^k \\left({i}\\right) \\in \\left\\{{i, \\pi \\left({i}\\right), \\pi^2 \\left({i}\\right), \\ldots, \\pi^{k-1} \\left({i}\\right)}\\right\\}$ Then $\\pi^k \\left({i}\\right) = i$."}
+{"_id": "879", "title": "Equality of Cycles", "text": "Let $S_n$ denote the symmetric group on $n$ letters, realised as the permutations of $\\left\\{{1, \\ldots, n}\\right\\}$. Let $\\rho = \\begin{bmatrix} a_0 & \\cdots & a_{k-1} \\end{bmatrix}$, $\\sigma = \\begin{bmatrix} b_0 & \\cdots & b_{k-1} \\end{bmatrix} \\in S_n$ be $k$-cycles of $S_n$. For any $d \\in \\Z$, by Integer is Congruent to Integer less than Modulus we can associate to $d$ a unique integer $\\tilde d \\in \\left\\{{0, \\ldots, k-1}\\right\\}$ such that $d \\equiv \\tilde d \\pmod k$. Define $a_d$ and $b_d$ for any $d \\in \\Z$ by $a_d = a_{\\tilde d}$ and $b_d = b_{\\tilde d}$ Choose $i, j \\in \\left\\{{1, \\ldots, k}\\right\\}$ such that: :$\\displaystyle a_i = \\min\\left\\{ {a_0, \\ldots, a_{k-1} }\\right\\}$ :$\\displaystyle b_j = \\min\\left\\{ {b_0, \\ldots, b_{k-1} }\\right\\}$ Then $\\rho = \\sigma$ {{iff}} for all $d \\in \\Z$, $a_{i + d} = b_{j + d}$. That is, $\\rho = \\sigma$ {{iff}} they are identical when written with the lowest element first."}
+{"_id": "880", "title": "Identity Permutation is Disjoint from All", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $e \\in S_n$ be the identity permutation on $S_n$. Then $e$ is disjoint from every permutation $\\pi$ on $S_n$ (including itself)."}
+{"_id": "881", "title": "Disjoint Permutations Commute", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\rho, \\sigma \\in S_n$ such that $\\rho$ and $\\sigma$ are disjoint. Then $\\rho \\sigma = \\sigma \\rho$."}
+{"_id": "882", "title": "Permutation Induces Equivalence Relation", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\pi \\in S_n$. Let $\\mathcal R_\\pi$ be the relation defined by: :$i \\mathrel {\\mathcal R_\\pi} j \\iff \\exists k \\in \\Z: \\map {\\pi^k} i = j$ Then $\\mathcal R_\\pi$ is an equivalence relation."}
+{"_id": "883", "title": "Existence and Uniqueness of Cycle Decomposition", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Every element of $S_n$ may be uniquely expressed as a cycle decomposition, up to the order of factors."}
+{"_id": "885", "title": "Order of Product of Disjoint Permutations", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\pi$ be a product of disjoint permutations of orders $k_1, k_2, \\ldots, k_r$. Then: :$\\order \\pi = \\lcm \\set {k_1, k_2, \\ldots, k_r}$ where: :$\\order \\pi$ denotes the order of $\\pi$ in $S_n$ :$\\lcm$ denotes lowest common multiple."}
+{"_id": "886", "title": "Group Action defines Permutation Representation", "text": "Let $\\map \\Gamma X$ be the set of permutations on a set $X$. Let $G$ be a group. Let $\\phi: G \\times X \\to X$ be a group action. For $g \\in G$, let $\\phi_g: X \\to X$ be the mapping defined as: :$\\map {\\phi_g} x = \\map \\phi {g, x}$ Let $\\tilde \\phi: G \\to \\map \\Gamma X$ be the mapping associated to $\\phi$, defined by: :$\\map {\\tilde \\phi} g := \\phi_g$ Then $\\tilde \\phi$ is a group homomorphism."}
+{"_id": "887", "title": "Group Action determines Bijection", "text": "Let $*$ be a group action of $G$ on $X$. Then each $g \\in G$ determines a bijection $\\phi_g: X \\to X$ given by: :$\\map {\\phi_g} x = g * x$ Its inverse is: :$\\phi_{g^{-1} }: X \\to X$. These bijection are sometimes called '''transformations''' of $X$."}
+{"_id": "888", "title": "Group Action Induces Equivalence Relation", "text": "Let $G$ be a group whose identity is $e$. Let $X$ be a set. Let $*: G \\times S \\to S$ be a group action. Let $\\mathcal R_G$ be the relation induced by $G$, that is: :$x \\mathrel {\\mathcal R_G} y \\iff y \\in \\Orb x$ where: :$\\Orb x$ denotes the orbit of $x \\in X$. Then: :$\\mathcal R_G$ is an equivalence relation. :The equivalence class of an element is its orbit."}
+{"_id": "889", "title": "Partition Equation", "text": "Let group $G$ act on a finite set $X$. Let the distinct orbits of $X$ under the action of $G$ be: :$\\Orb {x_1}, \\Orb {x_2}, \\ldots, \\Orb {x_s}$ Then: :$\\card X = \\card {\\Orb {x_1} } + \\card {\\Orb {x_2} } + \\cdots + \\card {\\Orb {x_s} }$"}
+{"_id": "890", "title": "Definition:Stabilizer", "text": "Let $G$ be a group. Let $X$ be a set. Let $*: G \\times X \\to X$ be a group action. For each $x \\in X$, the '''stabilizer of $x$ by $G$''' is defined as: :$\\Stab x := \\set {g \\in G: g * x = x}$ where $*$ denotes the group action."}
+{"_id": "891", "title": "Stabilizer is Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group which acts on a set $X$. Let $\\Stab x$ be the stabilizer of $x$ by $G$. Then for each $x \\in X$, $\\Stab x$ is a subgroup of $G$."}
+{"_id": "892", "title": "Orbit-Stabilizer Theorem", "text": "Let $G$ be a group which acts on a finite set $X$. Let $x \\in X$. Let $\\Orb x$ denote  the orbit of $x$. Let $\\Stab x$ denote the stabilizer of $x$ by $G$. Let $\\index G {\\Stab x}$ denote the index of $\\Stab x$ in $G$. Then: :$\\order {\\Orb x} = \\index G {\\Stab x} = \\dfrac {\\order G} {\\order {\\Stab x} }$"}
+{"_id": "894", "title": "Action of Group on Coset Space is Group Action", "text": "Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$. Let $*: G \\times G / H \\to G / H$ be the action on the (left) coset space: :$\\forall g \\in G, \\forall g' H \\in G / H: g * \\paren {g' H} := \\paren {g g'} H$ Then $G$ is a group action."}
+{"_id": "895", "title": "Group Action on Sets with k Elements", "text": "Let $\\struct {G, \\circ}$ be a finite group whose identity is $e$. Let $\\Bbb S = \\set {S \\subseteq G: \\card S = k}$, that is, the set of all of subsets of $G$ which have exactly $k$ elements. Let $G$ act on $\\Bbb S$ by the rule: :$\\forall S \\in \\Bbb S: g * S = g S = \\set {x \\in G: x = g s: s \\in S}$ This is a group action, and: : $\\forall S \\in \\Bbb S: \\order {\\Stab S} \\divides \\card S$ where $\\Stab S$ denotes the stabilizer of $S$ by $G$."}
+{"_id": "898", "title": "Quotient of Transformation Group acts Effectively", "text": "Let $G$ be a transformation group (which may or may not be effective) acting on $X$. Then the quotient group $G / G_0$, where $G_0$ is the kernel, ''does'' act effectively on $X$."}
+{"_id": "899", "title": "Condition for Group to Act Effectively on Left Coset Space", "text": "Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$. Then $G$ acts effectively on the left coset space $G / H$ {{iff}}: :$\\ds \\bigcap_{a \\mathop \\in G} H^a = \\set e$ where $H^a$ denotes the conjugate of $H$ by $a$."}
+{"_id": "900", "title": "Set of Permutations is Largest Effective Transformation Group", "text": "The set of permutations of a set $X$ forms the largest effective transformation group of $X$."}
+{"_id": "901", "title": "Conjugacy Action is Group Action", "text": "Let $\\left({G, \\circ}\\right)$ be a group whose identity is $e$."}
+{"_id": "902", "title": "Conjugacy Action on Identity", "text": "Let $G$ be a group whose identity is $e$. For the conjugacy action: :$\\order {\\Orb e} = 1$ and thus: :$\\Stab e = G$"}
+{"_id": "903", "title": "Cauchy's Lemma (Group Theory)", "text": "Let $\\struct {G, \\circ}$ be a group of finite order whose identity is $e$. Let $p$ be a prime number which divides the order of $G$. Then $\\struct {G, \\circ}$ has an element of order $p$."}
+{"_id": "904", "title": "Transposition is Self-Inverse", "text": "All transpositions are self-inverse."}
+{"_id": "905", "title": "Conjugates of Transpositions", "text": "Let $k_1, k_2, k_3 \\in \\left\\{{1, 2, \\ldots, n}\\right\\}$. Then: : $(1): \\quad \\begin{bmatrix} k_1 & k_2 \\end{bmatrix} = \\begin{bmatrix} k_3 & k_2 \\end{bmatrix} \\begin{bmatrix} k_1 & k_3 \\end{bmatrix} \\begin{bmatrix} k_3 & k_2 \\end{bmatrix}$ : $(2): \\quad \\begin{bmatrix} k_1 & k_2 \\end{bmatrix} = \\begin{bmatrix} k_3 & k_1 \\end{bmatrix} \\begin{bmatrix} k_3 & k_2 \\end{bmatrix} \\begin{bmatrix} k_3 & k_1 \\end{bmatrix}$"}
+{"_id": "906", "title": "K-Cycle can be Factored into Transpositions", "text": "Every $k$-cycle can be factorised into the product of $k - 1$ transpositions. This factorisation is not unique."}
+{"_id": "907", "title": "Sign of Permutation is Plus or Minus Unity", "text": "Let $n \\in \\N$ be a natural number. Let $\\N_n$ denote the set of natural numbers $\\set {1, 2, \\ldots, n}$. Let $S_n$ denote the symmetric group on $n$ letters. Let $\\sequence {x_k}_{k \\mathop \\in \\N_n}$ be a finite sequence in $\\R$. Let $\\pi \\in S_n$. Let $\\map {\\Delta_n} {x_1, x_2, \\ldots, x_n}$ be the product of differences of $\\tuple {x_1, x_2, \\ldots, x_n}$. Let $\\map \\sgn \\pi$ be the sign of $\\pi$. Let $\\pi \\cdot \\map {\\Delta_n} {x_1, x_2, \\ldots, x_n}$ be defined as: :$\\pi \\cdot \\map {\\Delta_n} {x_1, x_2, \\ldots, x_n} := \\map {\\Delta_n} {x_{\\map \\pi 1}, x_{\\map \\pi 2}, \\ldots, x_{\\map \\pi n} }$ Then either: :$\\pi \\cdot \\Delta_n = \\Delta_n$ or: :$\\pi \\cdot \\Delta_n = -\\Delta_n$ That is: :$\\map \\sgn \\pi = \\begin{cases} 1 & :\\pi \\cdot \\Delta_n = \\Delta_n \\\\ -1 & : \\pi \\cdot \\Delta_n = -\\Delta_n \\end{cases}$ Thus: :$\\pi \\cdot \\Delta_n = \\map \\sgn \\pi \\Delta_n$"}
+{"_id": "908", "title": "Parity Function is Homomorphism", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\pi \\in S_n$. Let $\\map \\sgn \\pi$ be the sign of $\\pi$. Let the parity function of $\\pi$ be defined as: :Parity of $\\pi = \\begin{cases} \\mathrm {Even} & : \\map \\sgn \\pi = 1 \\\\ \\mathrm {Odd} & : \\map \\sgn \\pi = -1 \\end{cases}$ The mapping $\\sgn: S_n \\to C_2$, where $C_2$ is the cyclic group of order 2, is a homomorphism."}
+{"_id": "909", "title": "Permutation on Polynomial is Group Action", "text": "Let $n \\in \\Z: n > 0$. Let $F_n$ be the set of all polynomials in $n$ variables $x_1, x_2, \\ldots, x_n$: :$F = \\set {\\map f {x_1, x_2, \\ldots, x_n}: f \\text{ is a polynomial in $n$ variables} }$ Let $S_n$ denote the symmetric group on $n$ letters. Let $*: S_n \\times F \\to F$ be the mapping defined as: :$\\forall \\pi \\in S_n, f \\in F: \\pi * \\map f {x_1, x_2, \\ldots, x_n} = \\map f {x_{\\map \\pi 1}, x_{\\map \\pi 2}, \\ldots, x_{\\map \\pi n} }$ Then $*$ is a group action."}
+{"_id": "910", "title": "Parity of Inverse of Permutation", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Then: :$\\forall \\pi \\in S_n: \\map \\sgn \\pi = \\map \\sgn {\\pi^{-1} }$"}
+{"_id": "911", "title": "Parity of Conjugate of Permutation", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Then: :$\\forall \\pi, \\rho \\in S_n: \\map \\sgn {\\pi \\rho \\pi^{-1} } = \\map \\sgn \\rho$ where $\\map \\sgn \\pi$ is the sign of $\\pi$."}
+{"_id": "912", "title": "Parity of K-Cycle", "text": "Let $\\pi$ be a $k$-cycle. Then: :$\\map \\sgn \\pi = \\begin{cases} 1 & : k \\ \\text {odd} \\\\ -1 & : k \\ \\text {even} \\end{cases}$ Thus: :$\\map \\sgn \\pi = \\paren {-1}^{k - 1}$ or equivalently: :$\\map \\sgn \\pi = \\paren {-1}^{k + 1}$"}
+{"_id": "913", "title": "Alternating Group is Normal Subgroup of Symmetric Group", "text": "Let $n \\ge 2$ be a natural number. Let $S_n$ denote the symmetric group on $n$ letters. Let $A_n$ be the alternating group on $n$ letters. Then $A_n$ is a normal subgroup of $S_n$ whose index is $2$."}
+{"_id": "914", "title": "Group of Permutations either All or Half Even", "text": "Let $G$ be a group of permutations. Then either ''exactly half'' of the permutations in $G$ are even, or they are ''all'' even."}
+{"_id": "915", "title": "Cycle Decomposition of Conjugate", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\pi, \\rho \\in S_n$. The cycle decomposition of the permutation $\\pi \\rho \\pi^{-1}$ can be obtained from that of $\\rho$ by replacing each $i$ in the cycle decomposition of $\\rho$ with $\\map \\pi i$."}
+{"_id": "916", "title": "Conjugate Permutations have Same Cycle Type", "text": "Let $n \\ge 1$ be a natural number. Let $G$ be a subgroup of the symmetric group on $n$ letters $S_n$. Let $\\sigma, \\rho \\in G$. Then $\\sigma$ and $\\rho$ are conjugate {{iff}} they have the same cycle type."}
+{"_id": "917", "title": "Transpositions of Adjacent Elements generate Symmetric Group", "text": "Let $n \\in \\Z: n > 1$. Let $S_n$ denote the symmetric group on $n$ letters. Then the transpositions $a_k = \\begin{pmatrix} k & k + 1 \\end{pmatrix}$ for $1 \\le k < n$ are a set of generators for $S_n$. They satisfy the relations: {{begin-eqn}} {{eqn | l = a_k^2       | r = e       | c = (for $1 \\le k < n$) }} {{eqn | l = \\paren {a_k a_{k + 1} }^3       | r = e       | c = (for $1 \\le k < n - 1$) }} {{eqn | l = \\paren {a_i a_j}^2       | r = e       | c = (for $1 \\le i, j < n, \\size {i - j} > 1$) }} {{end-eqn}}"}
+{"_id": "919", "title": "Permutation of Cosets", "text": "Let $G$ be a group and let $H \\le G$. Let $\\mathbb S$ be the set of all distinct left cosets of $H$ in $G$. Then: :$(1): \\quad$ For any $g \\in G$, the mapping $\\theta_g: \\mathbb S \\to \\mathbb S$ defined by $\\map {\\theta_g} {x H} = g x H$ is a permutation of $\\mathbb S$. :$(2): \\quad$ The mapping $\\theta$ defined by $\\map \\theta g = \\theta_g$ is a homomorphism from $G$ into the symmetric group on $\\mathbb S$. :$(3): \\quad$ The kernel of $\\theta$ is the subgroup $\\displaystyle \\bigcap_{x \\mathop \\in G} x H x^{-1}$."}
+{"_id": "921", "title": "Basic Results about Modules", "text": "Let $\\struct {G, +_G}$ be an abelian group whose identity is $e$. Let $\\struct {R, +_R, \\times_R}$ be a ring whose zero is $0_R$. Let $\\struct {G, +_G, \\circ}_R$ be an $R$-module. Let $x \\in G, \\lambda \\in R, n \\in \\Z$. Let $\\sequence {x_m}$ be a sequence of elements of $G$. Let $\\sequence {\\lambda_m}$ be a sequence of elements of $R$ that is, scalars. Then:"}
+{"_id": "922", "title": "Basic Results about Unitary Modules", "text": "Let $\\struct {G, +_G}$ be an abelian group whose identity is $e$. Let $\\struct {R, +_R, \\times_R}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\\struct {G, +_G, \\circ}_R$ be an unitary $R$-module. Let $x \\in G, n \\in \\Z$. Then:"}
+{"_id": "923", "title": "Epimorphism preserves Modules", "text": "Let $\\left({G, +_G, \\circ}\\right)_R$ be an $R$-module. Let $\\left({H, +_H, \\circ}\\right)_R$ be an $R$-algebraic structure. Let $\\phi: G \\to H$ be an epimorphism. Then $H$ is an $R$-module."}
+{"_id": "924", "title": "Condition for Linear Transformation", "text": "Let $G$ be a unitary $R$-module, and let $H$ be an $R$-module. Let $\\phi: G \\to H$ be a mapping. Then $\\phi$ is a linear transformation {{iff}}: :$\\forall x, y \\in G: \\forall \\lambda, \\mu \\in R: \\map \\phi {\\lambda x + \\mu y} = \\lambda \\map \\phi x + \\mu \\map \\phi y$"}
+{"_id": "925", "title": "Module of All Mappings is Module", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {G, +_G, \\circ}_R$ be an $R$-module. Let $S$ be a set. Let $\\struct {G^S, +_G', \\circ}_R$ be the module of all mappings from $S$ to $G$. Then $\\struct {G^S, +_G', \\circ}_R$ is an $R$-module."}
+{"_id": "926", "title": "Kuratowski's Theorem", "text": "The following conditions on a graph $\\Gamma$ are equivalent: :$(1): \\quad \\Gamma$ is planar :$(2): \\quad \\Gamma$ contains no subdivision of either the complete graph $K_5$ or the complete bipartite graph $K_{3, 3}$."}
+{"_id": "927", "title": "Fundamental Theorem of Finite Abelian Groups", "text": "Every finite abelian group is an internal group direct product of cyclic groups whose orders are prime powers. The number of terms in the product and the orders of the cyclic groups are uniquely determined by the group."}
+{"_id": "928", "title": "Finite Direct Product of Modules is Module", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {G_1, +_1, \\circ_1}_R, \\struct {G_2, +_2, \\circ_2}_R, \\ldots, \\struct {G_n, +_n, \\circ_n}_R$ be $R$-modules. Let: :$\\ds G = \\prod_{k \\mathop = 1}^n G_k$ be their direct product. Then $G$ is a module."}
+{"_id": "929", "title": "Subring Module", "text": "Let $\\struct {R, +, \\times}$ be a ring. Let $\\struct {S, +_S, \\times_S}$ be a subring of $R$. Let $\\struct {G, +_G, \\circ}_R$ be an $R$-module. Let $\\circ_S$ be the restriction of $\\circ$ to $S \\times G$. Then $\\struct {G, +_G, \\circ_S}_S$ is an $S$-module. The module $\\struct {G, +_G, \\circ_S}_S$ is called the '''$S$-module obtained from $\\struct {G, +_G, \\circ}_R$ by restricting scalar multiplication'''. {{refactor|Extract the below into its own page}} If $\\struct {G, +_G, \\circ}_R$ is a unitary $R$-module and $1_R \\in S$, then $\\struct{G, +_G, \\circ_S}_S$ is also unitary."}
+{"_id": "931", "title": "Projection on Cartesian Product of Modules", "text": "Let $G$ be the cartesian product of a sequence $\\sequence {G_n}$ of $R$-modules. Then for each $j \\in \\closedint 1 n$, the projection $\\pr_j$ on the $j$th co-ordinate is an epimorphism from $G$ onto $G_j$."}
+{"_id": "933", "title": "Submodule Test", "text": "Let $\\left({G, +, \\circ}\\right)_R$ be a unitary $R$-module. Let $H$ be a non-empty subset of $G$. Then $\\left({H, +, \\circ}\\right)_R$ is a submodule of $G$ iff: :$\\forall x, y \\in H: \\forall \\lambda \\in R: x + y \\in H, \\lambda \\circ x \\in H$"}
+{"_id": "934", "title": "Module is Submodule of Itself", "text": "Let $\\left({G, +_G, \\circ}\\right)_R$ be an $R$-module. Then $\\left({G, +_G, \\circ}\\right)_R$ is a submodule of itself."}
+{"_id": "935", "title": "Null Module Submodule of All", "text": "Let $\\left({G, +_G, \\circ}\\right)_R$ be an $R$-module. Then the null module: :$\\left({\\left\\{{e_G}\\right\\}, +_G, \\circ}\\right)_R$ is a submodule of $\\left({G, +_G, \\circ}\\right)_R$."}
+{"_id": "937", "title": "First Sylow Theorem", "text": "Let $p$ be a prime number. Let $G$ be a group such that: :$\\order G = k p^n$ where: :$\\order G$ denotes the order of $G$ :$p$ is not a divisor of $k$. Then $G$ has at least one Sylow $p$-subgroup."}
+{"_id": "938", "title": "H-Cobordism Theorem", "text": "Let $X^n, Y^n$ be two simply connected manifolds. Let $n \\in \\N: n \\ge 5$ and $\\exists W$ such that $W$ is an h-cobordism between $X$ and $Y$. Then $\\exists \\psi: W \\to X \\times \\closedint 0 1$ such that $\\psi$ is a diffeomorphism. In particular, $X$ and $Y$ are diffeomorphic."}
+{"_id": "939", "title": "Group has Subgroups of All Prime Power Factors", "text": "Let $p$ be a prime. Let $G$ be a finite group of order $n$. If $p^k \\divides n$ then $G$ has at least one subgroup of order $p^k$."}
+{"_id": "940", "title": "Normalizer of Sylow p-Subgroup", "text": "Let $P$ be a Sylow $p$-subgroup of a finite group $G$. Let $\\map {N_G} P$ be the normalizer of $P$. Then any $p$-subgroup of $\\map {N_G} P$ is contained in $P$. In particular, $P$ is the unique Sylow $p$-subgroup of $\\map {N_G} P$."}
+{"_id": "941", "title": "Second Sylow Theorem", "text": "Let $P$ be a Sylow $p$-subgroup of the finite group $G$. Let $Q$ be any $p$-subgroup of $G$. Then $Q$ is a subset of a conjugate of $P$."}
+{"_id": "942", "title": "Third Sylow Theorem", "text": "All the Sylow $p$-subgroups of a finite group are conjugate."}
+{"_id": "943", "title": "Fourth Sylow Theorem", "text": "The number of Sylow $p$-subgroups of a finite group is congruent to $1 \\pmod p$."}
+{"_id": "944", "title": "Sylow p-Subgroup is Unique iff Normal", "text": "A group $G$ has exactly one Sylow $p$-subgroup $P$ {{iff}} $P$ is normal."}
+{"_id": "945", "title": "Intersection of Normal Subgroup with Sylow P-Subgroup", "text": "Let $P$ be a Sylow $p$-subgroup of a finite group $G$. Let $N$ be a normal subgroup of $G$. Then $P \\cap N$ is a Sylow $p$-subgroup of $N$."}
+{"_id": "946", "title": "Quotient of Sylow P-Subgroup", "text": "Let $P$ be a Sylow $p$-subgroup of a finite group $G$. Let $N$ be a normal subgroup of $G$. Then $P N / N$ is a Sylow $p$-subgroup of $G / N$."}
+{"_id": "947", "title": "Fifth Sylow Theorem", "text": "The number of Sylow $p$-subgroups of a finite group is a divisor of their common index."}
+{"_id": "948", "title": "Finite Submodule of Function Space", "text": "Let $\\left({G, +}\\right)$ be a group whose identity is $e$. Let $R$ be a ring. Let $\\left({G, +, \\circ}\\right)_R$ be an $R$-module. Let $S$ be a set. Let $G^S$ the set of all mappings $f: S \\to G$. Let $G^{\\left({S}\\right)}$ be the set of all mappings $f: S \\to G$ such that $f \\left({x}\\right) = e$ for all but finitely many elements $x$ of $S$. Then: : $\\left({G^{\\left({S}\\right)}, +', \\circ}\\right)_R$ is a submodule of $\\left({G^S, +, \\circ}\\right)_R$ where $+'$ is the operation induced on $G^{\\left({S}\\right)}$ by $+$."}
+{"_id": "949", "title": "Polynomial Functions form Submodule of All Functions", "text": "Let $K$ be a commutative ring with unity. Let $K^K$ be the $K$-module mappings $f: K \\to K$. Let $P \\left({K}\\right) \\subseteq K^K$ be the set of all polynomial functions on $K$. Then $P \\left({K}\\right)$ is a $K$-submodule of $K^K$."}
+{"_id": "950", "title": "Intersection and Sum of Submodules", "text": "Let $\\left({G, +, \\circ}\\right)_R$ be an $R$-module. Let $H$ and $K$ be submodules of $G$. Then $H + K$ and $H \\cap K$ are also submodules of $G$. The intersection of any set of submodules of $G$ is a submodule. Thus if $S \\subseteq G$, the intersection of all submodules of $G$ containing $S$ is the smallest submodule of $G$ containing $S$."}
+{"_id": "951", "title": "Linear Transformation of Submodule", "text": "Let $G$ and $H$ be $R$-modules. Let $\\phi: G \\to H$ be a linear transformation. Then: :$(1): \\quad$ If $M$ is a submodule of $G$, $\\phi \\sqbrk M$ is a submodule of $H$ :$(2): \\quad$ If $N$ is a submodule of $H$, $\\phi^{-1} \\sqbrk N$ is a submodule of $G$ :$(3): \\quad$ The codomain of $\\phi$ is a submodule of $H$ :$(4): \\quad$ The kernel of $\\phi$ is a submodule of $G$."}
+{"_id": "953", "title": "Poincaré Conjecture", "text": "Let $\\Sigma^m$ be a smooth $m$-manifold. Let $\\Sigma^m$ satisfy: :$H_0 \\struct {\\Sigma; \\Z} = 0$ and: :$H_m \\struct {\\Sigma; \\Z} = \\Z$ {{explain|Definition of the notation $H_0 \\struct {\\Sigma; \\Z}$, nature of $H_0$ and $H_m$}} Then $\\Sigma^m$ is homeomorphic to the $m$-sphere $\\Bbb S^m$."}
+{"_id": "954", "title": "Linear Transformation of Generated Module", "text": "Let $G$ and $H$ be $R$-modules. Let $\\phi$ and $\\psi$ be linear transformations $G$ into $H$.  Let $S$ be a generator for $G$. Suppose that $\\forall x \\in S: \\map \\phi x = \\map \\psi x$. Then $\\phi = \\psi$."}
+{"_id": "955", "title": "Generated Submodule is Linear Combinations", "text": "Let $G$ be a unitary $R$-module. Let $S \\subseteq G$. Then the submodule $H$ generated by $S$ is the set of all linear combinations of $S$."}
+{"_id": "956", "title": "Empty Set is Linearly Independent", "text": "The empty set is a linearly independent set."}
+{"_id": "957", "title": "Subset of Module Containing Identity is Linearly Dependent", "text": "Let $G$ be a group whose identity is $e$. Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Let $\\struct {G, +_G, \\circ}_R$ be an $R$-module. Let $H \\subseteq G$ such that $e \\in H$. Then $H$ is a linearly dependent set."}
+{"_id": "958", "title": "Subset of Linearly Independent Set is Linearly Independent", "text": "A subset of a linearly independent set is also linearly independent."}
+{"_id": "959", "title": "Superset of Linearly Dependent Set", "text": "Any set containing a linearly dependent set is also linearly dependent."}
+{"_id": "960", "title": "Number of Ordered Bases from a Basis", "text": "Each basis of $n$ elements determines $n!$ ordered bases."}
+{"_id": "961", "title": "Classification of Compact Two-Manifolds", "text": "Any smooth, compact, path-connected manifold of dimension $2$ is diffeomorphic to the sphere $\\mathbb S^2$, a connected sum of tori $\\mathbb T^2$, or a connected sum of projective spaces $\\mathbb{RP}^2$. Any such $2$-manifold with boundary is diffeomorphic to the sphere $\\mathbb S^2$, a connected sum of tori $\\mathbb T^2$, or a connected sum of projective spaces $\\mathbb{RP}^2$, with a number of open disks removed. The Euler characteristic, orientability, and number of boundary curves suffice to describe a surface."}
+{"_id": "962", "title": "Standard Ordered Basis is Basis", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n$ be a positive integer. For each $j \\in \\closedint 1 n$, let $e_j$ be the ordered $n$-tuple of elements of $R$ whose $j$th entry is $1_R$ and all of whose other entries is $0_R$. Then $\\sequence {e_n}$ is an ordered basis of the $R$-module $R^n$. This ordered basis is called the '''standard ordered basis of $R^n$'''. The corresponding set $\\set {e_1, e_2, \\ldots, e_n}$ is called the '''standard basis of $R^n$'''."}
+{"_id": "964", "title": "Unique Representation by Ordered Basis", "text": "Let $G$ be a unitary $R$-module. Then $\\sequence {a_n}$ is an ordered basis of $G$ {{Iff}}: :For every $x \\in G$ there exists one and only one sequence $\\sequence {\\lambda_n}$ of scalars such that $\\displaystyle x = \\sum_{k \\mathop = 1}^n \\lambda_k a_k$."}
+{"_id": "965", "title": "Isomorphism from R^n via n-Term Sequence", "text": "Let $G$ be a unitary $R$-module. Let $\\sequence {a_k}_{1 \\mathop \\le k \\mathop \\le n}$ be an ordered basis of $G$. Let $R^n$ be the $R$-module $R^n$. Let $\\psi: R^n \\to G$ be defined as: :$\\displaystyle \\map \\psi {\\sequence {\\lambda_k}_{1 \\mathop \\le k \\mathop \\le n} } = \\sum_{k \\mathop = 1}^n \\lambda_k a_k$ Then $\\psi$ is an isomorphism."}
+{"_id": "966", "title": "Unitary R-Modules with n-Element Bases Isomorphic", "text": "Any two unitary $R$-modules having bases of $n$ elements are isomorphic."}
+{"_id": "967", "title": "R-Module R^n is n-Dimensional", "text": "The $R$-module $R^n$ is $n$-dimensional."}
+{"_id": "968", "title": "Unique Linear Transformation Between Modules", "text": "Let $G$ and $H$ be unitary $R$-modules. Let $\\left \\langle {a_n} \\right \\rangle$ be an ordered basis of $G$. Let $\\left \\langle {b_n} \\right \\rangle$ be a sequence of elements of $H$. Then there is a unique linear transformation $\\phi: G \\to H$ satisfying $\\forall k \\in \\left[{1 \\,.\\,.\\, n}\\right]: \\phi \\left({a_k}\\right) = b_k$"}
+{"_id": "969", "title": "Linear Transformation from Ordered Basis less Kernel", "text": "Let $G$ and $H$ be unitary $R$-modules. Let $\\phi: G \\to H$ be a non-zero linear transformation. Let $G$ be $n$-dimensional. Let $\\left \\langle {a_n} \\right \\rangle$ be any ordered basis of $G$ such that $\\left\\{{a_k: r + 1 \\le k \\le n}\\right\\}$ is the basis of the kernel of $\\phi$. Then $\\left \\langle {\\phi \\left({a_r}\\right)} \\right \\rangle$ is an ordered basis of the image of $\\phi$."}
+{"_id": "970", "title": "Addition of Linear Transformations", "text": "Let $\\left({G, +_G, \\circ}\\right)_R$ and $\\left({H, +_H, \\circ}\\right)_R$ be $R$-modules. Let $\\phi: G \\to H$ and $\\psi: G \\to H$ be linear transformations. Let $\\phi +_H \\psi$ be the operation on $H^G$ induced by $+_H$ as defined in Induced Structure. Then $\\phi +_H \\psi: G \\to H$ is a linear transformation."}
+{"_id": "971", "title": "Negative Linear Transformation", "text": "Let $\\left({G, +_G, \\circ}\\right)_R$ and $\\left({H, +_H, \\circ}\\right)_R$ be $R$-modules. Let $\\phi: G \\to H$ be a linear transformation. Let $- \\phi$ be the negative of $\\phi$ as defined in Induced Structure Inverse. Then $- \\phi: G \\to H$ is also a linear transformation."}
+{"_id": "973", "title": "Morse-Sard Theorem", "text": "Let $f: X \\to Y$ be any smooth map of manifolds. Then almost every point in $Y$ is a regular value of $f$."}
+{"_id": "974", "title": "Linear Transformation from Center of Scalar Ring", "text": "Let $\\struct {G, +_G, \\circ}_R$ and $\\struct {H, +_H, \\circ}_R$ be $R$-modules. Let $\\phi: G \\to H$ be a linear transformation. Let $\\map Z R$ be the center of the scalar ring $R$. Let $\\lambda \\in \\map Z R$. Then $\\lambda \\circ \\phi$ is a linear transformation."}
+{"_id": "976", "title": "Product of Linear Transformations", "text": "Let $R$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\\struct {G, +_G, \\circ}_R$ be a unitary $R$-module such that $\\map \\dim G = n$. Let $\\struct {H, +_H, \\circ}_R$ be a unitary $R$-module such that $\\map \\dim H = m$. Let $\\map {\\LL_R} {G, H}$ be the set of all linear transformations from $G$ to $H$. Then: :$\\map \\dim {\\map {\\LL_R} {G, H} } = n m$ Let $\\sequence {a_n}$ be an ordered basis for $G$. Let $\\sequence {b_m}$ be an ordered basis for $H$. Let $\\phi_{i j}: G \\to H$ be the unique linear transformation defined for each $i \\in \\closedint 1 n, j \\in \\closedint 1 m$ which satisfies: :$\\forall k \\in \\closedint 1 n: \\map {\\phi_{i j} } {a_k} = \\delta_{i k} b_j$ where $\\delta$ is the Kronecker delta. Then: :$\\set {\\phi_{i j}: i \\in \\closedint 1 n, j \\in \\closedint 1 m}$ is a basis for $\\map \\dim {\\map {\\LL_R} {G, H} }$."}
+{"_id": "978", "title": "Dimension of Algebraic Dual", "text": "Let $G$ be an $n$-dimensional $R$-module. Let $G^*$ be the algebraic dual of $G$. Let $G^{**}$ be the algebraic dual of $G^*$. Then $G^*$ and $G^{**}$ are also $n$-dimensional."}
+{"_id": "979", "title": "Annihilator is Submodule of Algebraic Dual", "text": "Let $R$ be a commutative ring. Let $G$ be a module over $R$. Let $M$ be a submodule of $G$. Let $G^*$ be the algebraic dual of $G$. Then the annihilator $M^\\circ$ of $M$ is a submodule of $G^*$. Similarly, let $N$ be a submodule of $G^*$. Let $G^{**}$ be the algebraic dual of $G^*$. Then the annihilator $N^\\circ$ of $N$ is a submodule of $G^{**}$."}
+{"_id": "981", "title": "Properties of Evaluation Linear Transformation", "text": "Let $R$ be a commutative ring. Let $G$ be an $R$-module. Let $G^*$ be the algebraic dual of $G$. Let $\\left \\langle {x, t'} \\right \\rangle$ be the evaluation linear transformation from $G$ to $G^{**}$. Then the mapping $\\phi: G \\times G^* \\to R$ defined as $\\forall \\left({x, t'}\\right) \\in G \\times G^*: \\phi \\left({x, t'}\\right) = \\left \\langle {x, t'} \\right \\rangle$ satisfies the following properties: : $(1): \\quad \\forall x, y \\in G: \\forall t' \\in G^*: \\left \\langle {x + y, t'} \\right \\rangle = \\left \\langle {x, t'} \\right \\rangle + \\left \\langle {y, t'} \\right \\rangle$ : $(2): \\quad \\forall x \\in G: \\forall s', t' \\in G^*: \\left \\langle {x, s' + t'} \\right \\rangle = \\left \\langle {x, s'} \\right \\rangle + \\left \\langle {x, t'} \\right \\rangle$ : $(3): \\quad \\forall x \\in G: \\forall s', t' \\in G^*: \\forall \\lambda \\in R: \\left \\langle {\\lambda x, t'} \\right \\rangle = \\lambda \\left \\langle {x, t'} \\right \\rangle = \\left \\langle {x, \\lambda t'} \\right \\rangle$"}
+{"_id": "982", "title": "Zero Vector Space Product iff Factor is Zero", "text": "Let $F$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $\\struct {\\mathbf V, +, \\circ}_F$ be a vector space over $F$, as defined by the vector space axioms. Let $\\mathbf v \\in \\mathbf V, \\lambda \\in F$. Then: :$\\lambda \\circ \\mathbf v = \\bszero \\iff \\paren {\\lambda = 0_F \\lor x = \\bszero}$"}
+{"_id": "983", "title": "Homomorphic Image of Vector Space", "text": "Let $\\struct {K, +_K, \\times_K}$ be a division ring. Let $\\struct {V, +_V, \\circ_V}_K$ be a $K$-vector space. Let $\\struct {W, +_W, \\circ_W}_K$ be a $K$-algebraic structure. Let $\\phi: V \\to W$ be a homomorphism, i.e. a linear transformation. Then the homomorphic image of $\\phi$ is a $K$-vector space."}
+{"_id": "984", "title": "Direct Product of Vector Spaces is Vector Space", "text": "Let $K$ be a field. Let $V_1, V_2, \\ldots, V_n$ be $K$-vector spaces. Let $\\struct {V, + , \\circ}_K$ be their direct product. Then $\\struct {V, + , \\circ}_K$ is a $K$-vector space."}
+{"_id": "985", "title": "Definition:Vector Space on Cartesian Product", "text": "Let $\\struct {K, +, \\circ}$ be a division ring. Let $n \\in \\N_{>0}$. Let $+: K^n \\times K^n \\to K^n$ be defined as: :$\\tuple {\\alpha_1, \\ldots, \\alpha_n} + \\tuple {\\beta_1, \\ldots, \\beta_n} = \\tuple {\\alpha_1 +_R \\beta_1, \\ldots, \\alpha_n +_R \\beta_n}$ Let $\\times: K \\times K^n \\to K^n$ be defined as: :$\\lambda \\times \\tuple {\\alpha_1, \\ldots, \\alpha_n} = \\tuple {\\lambda \\times_R \\alpha_1, \\ldots, \\lambda \\times_R \\alpha_n}$ Then $\\struct {K^n, +, \\times}_K$ is '''the $K$-vector space $K^n$'''."}
+{"_id": "986", "title": "Definition:Vector Space over Division Subring", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose unity is $1_R$. Let $S$ be a division subring of $R$, such that $1_R \\in S$. Then $\\struct {R, +, \\circ_S}_S$, where $\\circ_S$ is the restriction of $\\circ$ to $S \\times R$, is the '''vector space on $R$ over the division subring $S$'''."}
+{"_id": "988", "title": "Division Ring is Vector Space over Prime Subfield", "text": "Let $\\struct {K, +, \\times}$ be a division ring. Let $\\struct {S, +, \\times}$ be the prime subfield of $K$ Then $\\struct {K, +, \\times_S}_S$ is an $S$-vector space, where $\\times_S$ is the restriction of $\\times$ to $S \\times K$."}
+{"_id": "989", "title": "Vector Space over Subring", "text": "Let $K$ be a division subring of the division ring $\\struct {L, +_L, \\times_L}$. Let $\\struct {G, +_G, \\circ}_L$ be a $L$-vector space. Then $\\struct {G, +_G, \\circ_K}_K$ is a $K$-vector space, where $\\circ_K$ is the restriction of $\\circ$ to $K \\times G$. The $K$-vector space $\\struct {G, +_G, \\circ_K}_K$ is called the '''$K$-vector space obtained from $\\struct {L, +_L, \\times_L}$ by restricting scalar multiplication'''."}
+{"_id": "990", "title": "Subspaces of Dimension 2 Real Vector Space", "text": "Take the $\\R$-vector space $\\left({\\R^2, +, \\times}\\right)_\\R$. Let $S$ be a subspace of $\\left({\\R^2, +, \\times}\\right)_\\R$. Then $S$ is one of: : $(1): \\quad \\left({\\R^2, +, \\times}\\right)_\\R$ : $(2): \\quad \\left\\{{0}\\right\\}$ : $(3): \\quad$ A line through the origin."}
+{"_id": "991", "title": "Subspace of Real Continuous Functions", "text": "Let $\\mathbb J = \\set {x \\in \\R: a \\le x \\le b}$ be a closed interval of the real number line $\\R$. Let $\\map \\CC {\\mathbb J}$ be the set of all continuous real functions on $\\mathbb J$. Then $\\struct {\\map \\CC {\\mathbb J}, +, \\times}_\\R$ is a subspace of the $\\R$-vector space $\\struct {\\R^{\\mathbb J}, +, \\times}_\\R$."}
+{"_id": "992", "title": "Singleton is Linearly Independent", "text": "Let $K$ be a division ring. Let $\\struct {G, +_G}$ be a group whose identity is $e$. Let $\\struct {G, +_G, \\circ}_K$ be a $K$-vector space whose zero is $0_K$. Let $x \\in G: x \\ne e$. Then $\\set x$ is a linearly independent subset of $G$."}
+{"_id": "993", "title": "Linearly Dependent Sequence of Vector Space", "text": "Let $\\struct {G, +}$ be a group whose identity is $\\mathbf 0$. Let $\\struct {G, +, \\circ}_K$ be a $K$-vector space. Let $\\sequence {a_k}_{1 \\mathop \\le k \\mathop \\le n}$ be a sequence of distinct non-zero vectors of $G$. Then $\\sequence {a_k}_{1 \\mathop \\le k \\mathop \\le n}$ is linearly dependent {{iff}}: :$\\exists p \\in \\closedint 2 n: a_p$ is a linear combination of $\\sequence {a_k}_{1 \\mathop \\le k \\mathop \\le p - 1}$"}
+{"_id": "994", "title": "Vector Space has Basis Between Linearly Independent Set and Finite Spanning Set", "text": "Let $K$ be a division ring. Let $G$ be a finitely generated $K$-vector space. Let $H$ be a linearly independent subset of $G$. Let $F$ be a finite generator for $G$ such that $H \\subseteq F$. Then there is a basis $B$ for $G$ such that $H \\subseteq B \\subseteq F$."}
+{"_id": "997", "title": "Bases of Finitely Generated Vector Space have Equal Cardinality", "text": "Let $K$ be a division ring. Let $G$ be a finitely generated $K$-vector space. Then any two bases of $G$ are finite and equivalent. {{improve|I still think there's a simpler way of saying \"they have the same number of elements\" than bringing in all that top-heavy technical set theoretic language of \"set equivalence\" and \"cardinality\" and so on. Since we are talking about a finite set, the complexities which arise with regard to transfinites do not arise, so there is no direct need to go into such depth.}}"}
+{"_id": "998", "title": "Linearly Independent Subset also Independent in Generated Subspace", "text": "Let $G$ be a finitely generated $K$-vector space. Let $S$ be a linearly independent subset of $G$. Let $M$ be the subspace of $G$ generated by $S$. If $M \\ne G$, then $\\forall b \\in G: b \\notin M$, the set $S \\cup \\set b$ is linearly independent."}
+{"_id": "999", "title": "Sufficient Conditions for Basis of Finite Dimensional Vector Space", "text": "Let $K$ be a division ring. Let $n \\ge 0$ be a natural number. Let $E$ be an $n$-dimensional vector space over $K$. Let $B \\subseteq E$ be a subset such that $\\card B = n$. {{TFAE}} : $(1): \\quad$ $B$ is a basis of $E$. : $(2): \\quad$ $B$ is linearly independent. : $(3): \\quad$ $B$ is a generator for $E$."}
+{"_id": "1000", "title": "Dimension of Proper Subspace is Less Than its Superspace", "text": "Let $G$ be a vector space whose dimension is $n$. Let $H$ be a subspace of $G$. Then $H$ is finite dimensional, and $\\map \\dim H \\le \\map \\dim G$. If $H$ is a proper subspace of $G$, then $\\map \\dim H < \\map \\dim G$."}
+{"_id": "1001", "title": "Results concerning Generators and Bases of Vector Spaces", "text": "Let $E$ be a vector space of $n$ dimensions. Let $G$ be a generator for $E$. Then $G$ has the following properties:"}
+{"_id": "1002", "title": "Grassmann's Identity", "text": "Let $K$ be a division ring. Let $\\struct {G, +_G, \\circ}_K$ be a $K$-vector space. Let $M$ and $N$ be finite-dimensional subspaces of $G$. Then the sum $M + N$ and intersection $M \\cap N$ are finite-dimensional, and: :$\\map \\dim {M + N} + \\map \\dim {M \\cap N} = \\map \\dim M + \\map \\dim N$"}
+{"_id": "1003", "title": "Rank Plus Nullity Theorem", "text": "Let $G$ be an $n$-dimensional vector space. Let $H$ be a vector space. Let $\\phi: G \\to H$ be a linear transformation. Let $\\map \\rho \\phi$ and $\\map \\nu \\phi$ be the rank and nullity respectively of $\\phi$. Then the image of $\\phi$ is finite-dimensional, and: :$\\map \\rho \\phi + \\map \\nu \\phi = n$ By definition of rank and nullity, it can be seen that this is equivalent to the alternative way of stating this result: :$\\map \\dim {\\Img \\phi} + \\map \\dim {\\map \\ker \\phi} = \\map \\dim G$ {{wtd|and the theorem is applicable to matrices}}"}
+{"_id": "1004", "title": "Linear Transformation of Vector Space Monomorphism", "text": "Let $G$ and $H$ be a $K$-vector space. Let $\\phi: G \\to H$ be a linear transformation. Then $\\phi$ is a monomorphism {{iff}} for every linearly independent sequence $\\sequence {a_n}$ of vectors of $G$, $\\sequence {\\map \\phi {a_n} }$ is a linearly independent sequence of vectors of $H$."}
+{"_id": "1005", "title": "Linear Transformation of Vector Space Equivalent Statements", "text": "Let $G$ and $H$ be $n$-dimensional vector spaces. Let $\\phi: G \\to H$ be a linear transformation. Then these statements are equivalent: : $(1): \\quad \\phi$ is an isomorphism. : $(2): \\quad \\phi$ is a monomorphism. : $(3): \\quad \\phi$ is an epimorphism. : $(4): \\quad \\phi \\left({B}\\right)$ is a basis of $H$ for every basis $B$ of $G$. : $(5): \\quad \\phi \\left({B}\\right)$ is a basis of $H$ for some basis $B$ of $G$."}
+{"_id": "1006", "title": "Results Concerning Annihilator of Vector Subspace", "text": "Let $G$ be an $n$-dimensional vector space over a field. Let $J: G \\to G^{**}$ be the evaluation isomorphism. Let $G^*$ be the algebraic dual of $G$. Let $G^{**}$ be the algebraic dual of $G^*$. Let $M$ be an $m$-dimensional subspace of $G$. Let $N$ be a $p$-dimensional subspace of $G^*$. Let $M^\\circ$ be the annihilator of $M$. Then: :$(1): \\quad M^\\circ$ is an $\\paren {n - m}$-dimensional subspace of $G^*$, and $M^{\\circ \\circ} = \\map J M$ :$(2): \\map {\\quad J^{-1} } {N^\\circ}$ is an $\\paren {n - p}$-dimensional subspace of $G$ :$(3): \\quad$ The mapping $M \\to M^\\circ$ is a bijection from the set of all $m$-dimensional subspaces of $G$ onto the set of all $\\paren {n - m}$-dimensional subspaces of $G^*$ :$(4): \\quad$ Its inverse is the bijection $N \\to \\map {J^{-1} } {N^\\circ}$."}
+{"_id": "1007", "title": "Vector Subspace Dimension One Less", "text": "Let $K$ be a field. Let $M$ be a subspace of the $n$-dimensional vector space $K^n$. The following statements are equivalent: :$(1): \\quad \\map \\dim M = n - 1$ :$(2): \\quad M$ is the kernel of a nonzero linear form :$(3): \\quad$ There exists a sequence $\\sequence {\\alpha_n} $ of scalars, not all of which are zero, such that: :::$M = \\set {\\tuple {\\lambda_1, \\ldots, \\lambda_n} \\in K^n: \\alpha_1 \\lambda_1 + \\cdots + \\alpha_n \\lambda_n = 0}$ Also, suppose the above hold. Let $\\sequence {\\beta_n}$ be a sequence of scalars such that: :$M = \\set {\\tuple {\\lambda_1, \\ldots, \\lambda_n} \\in K^n: \\beta_1 \\lambda_1 + \\cdots + \\beta_n \\lambda_n = 0}$ Then there is a non-zero scalar $\\gamma$ such that: :$\\forall k \\in \\closedint 1 n: \\beta_k = \\gamma \\alpha_k$"}
+{"_id": "1008", "title": "Rank and Nullity of Transpose", "text": "Let $G$ and $H$ be $n$-dimensional vector spaces over a field. Let $\\mathcal L \\left({G, H}\\right)$ be the set of all linear transformations from $G$ to $H$. Let $u \\in \\mathcal L \\left({G, H}\\right)$. Let $u^t$ be the transpose of $u$. Then: : $(1): \\quad$ $u$ and $u^t$ have the same rank and nullity : $(2): \\quad$ $\\ker \\left({u^t}\\right)$ is the annihilator of the image of $u$ : $(3): \\quad$ The image of $u^t$ is the annihilator of $\\ker \\left({u}\\right)$."}
+{"_id": "1009", "title": "Linear Operator on the Plane", "text": "Let $\\phi$ be a linear operator on the real vector space of two dimensions $\\R^2$. Then $\\phi$ is completely determined by an ordered tuple of $4$ real numbers."}
+{"_id": "1010", "title": "Similarity Mapping of Plane is Linear Operator", "text": "Let $G$ be a vector space over a field $K$. Let $\\beta \\in K$. Then the mapping: :$s_\\beta: G \\to G$ defined by $\\map {s_\\beta} {\\mathbf x} = \\beta \\mathbf x$ is a linear operator on $G$. If $\\beta \\ne 0$ then $s_\\beta$ is an automorphism of $G$, and $\\paren {s_\\beta}^{-1} = s_{\\beta^{-1} }$ The linear operators $s_\\beta$, where $\\beta \\ne 0$, are called '''similarities of $G$'''."}
+{"_id": "1011", "title": "Cantor-Dedekind Hypothesis", "text": "The points on an infinite straight line are in one-to-one correspondence with the set $\\R$ of real numbers. Hence the set of all points on an infinite straight line and $\\R$ are equinumerous."}
+{"_id": "1013", "title": "Full Angle measures 2 Pi Radians", "text": "One full angle is equal to $2 \\pi$ radians. :$2 \\pi \\approx 6 \\cdotp 28318 \\, 53071 \\, 79586 \\, 4769 \\ldots$ {{OEIS|A019692}}"}
+{"_id": "1014", "title": "Rotation of Plane about Origin is Linear Operator", "text": "Let $r_\\alpha$ be the plane rotation of the plane about the origin through an angle of $\\alpha$. That is, let $r_\\alpha: \\R^2 \\to \\R^2$ be the mapping defined as: :$\\forall x \\in \\R^2: \\map {r_\\alpha} x = \\text { the point into which a rotation of } \\alpha \\text{ carries } x$ Then $r_\\alpha$ is a linear operator determined by the ordered sequence: : $\\tuple {\\cos \\alpha  -\\sin \\alpha, \\sin \\alpha + \\cos \\alpha}$"}
+{"_id": "1015", "title": "Stretching and Contraction Mappings of Plane are Linear Operators", "text": "Let $s_\\beta: \\R^2 \\to \\R^2$ be a similarity of $\\R^2$. Then $s_{-1}$ is the same as the rotation $r_{\\pi}$ of the plane about the origin one half turn. If $\\beta \\ge 1$, then $s_\\beta$ is called a '''stretching''', and if $0 < \\beta \\le 1$, $s_\\beta$ is called a '''contraction'''. If $\\beta < 0$, then $s_\\beta$ is a stretching or contraction followed by a rotation one half turn. It is also the same as a rotation one half turn followed by a stretching or contraction."}
+{"_id": "1016", "title": "Reflection of Plane in Line through Origin is Linear Operator", "text": "Let $M$ be a straight line in the plane passing through the origin. Then the '''reflection''' $s_M$ of $\\R^2$ in $M$ is the rotation of the plane in space through one half turn about $M$ as an axis. :$s_M \\circ s_M = I_{\\R^2}$ and hence: :$s_M = s_M^{-1}$ If $M$ is the $x$-axis then $\\map {s_M} {\\lambda_1, \\lambda_2} = \\tuple {\\lambda_1, -\\lambda_2}$. If $M$ is the $y$-axis then $\\map {s_M} {\\lambda_1, \\lambda_2} = \\tuple {-\\lambda_1, \\lambda_2}$. In general, $s_M$ is a linear operator for every straight line $M$ through the origin."}
+{"_id": "1017", "title": "Projection of Straight Line on Another in Plane", "text": "Let $M$ and $N$ be distinct straight lines through the plane through the origin. Let $\\operatorname{pr}_{M, N}$ be the projection on $M$ along $N$. $M$ and $N$ are respectively the codomain and kernel of $\\operatorname{pr}_{M, N}$. {{explain|As the kernel is a concept defined in relation to a homomorphism, it needs to be clarified what homomorphism is being considered.}} :$\\operatorname{pr}_{M, N} \\left({x}\\right) = x \\iff x \\in M$ If $M$ is the $x$-axis and $N$ is the $y$-axis, then $\\operatorname{pr}_{M, N} \\left({\\lambda_1, \\lambda_2}\\right) = \\left({\\lambda_1, 0}\\right)$. If $M$ is the $y$-axis and $N$ is the $x$-axis, then $\\operatorname{pr}_{M, N} \\left({\\lambda_1, \\lambda_2}\\right) = \\left({0, \\lambda_2}\\right)$. Any such projection is a linear operator."}
+{"_id": "1019", "title": "Condition for Straight Lines in Plane to be Parallel", "text": "Let $L: \\alpha_1 x + \\alpha_2 y = \\beta$ be a straight line in $\\R^2$. Then the straight line $L'$ is parallel to $L$ iff there is a $\\beta' \\in \\R^2$ such that: :$L' = \\set {\\tuple {x, y} \\in \\R^2: \\alpha_1 x + \\alpha_2 y = \\beta'}$"}
+{"_id": "1020", "title": "Equation of Plane", "text": "A plane $P$ is the set of all $\\tuple {x, y, z} \\in \\R^3$, where: :$\\alpha_1 x + \\alpha_2 y + \\alpha_3 z = \\gamma$ where $\\alpha_1, \\alpha_2, \\alpha_3, \\gamma \\in \\R$ are given, and not all of $\\alpha_1, \\alpha_2, \\alpha_3$ are zero."}
+{"_id": "1024", "title": "Planes are Subspaces of Space", "text": "The two-dimensional subspaces of $\\R^3$ are precisely the homogeneous planes of solid analytic geometry."}
+{"_id": "1025", "title": "Matrix Space Semigroup under Hadamard Product", "text": "Let $\\map {\\MM_S} {m, n}$ be the matrix space over a semigroup $\\struct {S, \\cdot}$. Then the algebraic structure $\\struct {\\map {\\MM_S} {m, n}, \\circ}$, where $\\circ$ is the Hadamard product, is also a semigroup. If $\\struct {S, \\cdot}$ is a commutative semigroup then so is $\\struct {\\map {\\MM_S} {m, n}, \\circ}$. If $\\struct {S, \\cdot}$ is a monoid then so is $\\struct {\\map {\\MM_S} {m, n}, \\circ}$."}
+{"_id": "1026", "title": "Hadamard Product over Group forms Group", "text": "Let $\\struct {G, \\cdot}$ be a group whose identity is $e$. Let $\\map {\\MM_G} {m, n}$ be a $m \\times n$ matrix space over $\\struct {G, \\cdot}$. Then $\\struct {\\map {\\MM_G} {m, n}, \\circ}$, where $\\circ$ is Hadamard product, is also a group."}
+{"_id": "1027", "title": "Matrix Multiplication is Associative", "text": "Let $R$ be a ring. Matrix multiplication (conventional) is associative."}
+{"_id": "1028", "title": "Linear Transformations Isomorphic to Matrix Space", "text": "Let $R$ be a ring with unity. Let $F$, $G$ and $H$ be free $R$-modules of finite dimension $p,n,m>0$ respectively. Let $\\left \\langle {a_p} \\right \\rangle$, $\\left \\langle {b_n} \\right \\rangle$ and $\\left \\langle {c_m} \\right \\rangle$ be ordered bases  Let $\\operatorname{Hom} \\left({G, H}\\right)$ be the set of all linear transformations from $G$ to $H$. Let $\\mathcal M_R \\left({m, n}\\right)$ be the $m \\times n$ matrix space over $R$. Let $\\left[{u; \\left \\langle {c_m} \\right \\rangle, \\left \\langle {b_n} \\right \\rangle}\\right]$ be the matrix of $u$ relative to $\\left \\langle {b_n} \\right \\rangle$ and $\\left \\langle {c_m} \\right \\rangle$. Let $M: \\operatorname{Hom} \\left({G, H}\\right) \\to \\mathcal M_R \\left({m, n}\\right)$ be defined as: :$\\forall u \\in \\operatorname{Hom} \\left({G, H}\\right): M \\left({u}\\right) = \\left[{u; \\left \\langle {c_m} \\right \\rangle, \\left \\langle {b_n} \\right \\rangle}\\right]$ Then $M$ is an isomorphism of modules, and: :$\\forall u \\in \\operatorname{Hom} \\left({F, G}\\right), v \\in \\operatorname{Hom} \\left({G, H}\\right): \\left[{v \\circ u; \\left \\langle {c_m} \\right \\rangle, \\left \\langle {a_p} \\right \\rangle}\\right] = \\left[{v; \\left \\langle {c_m} \\right \\rangle, \\left \\langle {b_n} \\right \\rangle}\\right] \\left[{u; \\left \\langle {b_n} \\right \\rangle, \\left \\langle {a_p} \\right \\rangle}\\right]$"}
+{"_id": "1029", "title": "Matrix Multiplication Distributes over Matrix Addition", "text": "Matrix multiplication (conventional) is distributive over matrix entrywise addition."}
+{"_id": "1030", "title": "Unit Matrix is Unity of Ring of Square Matrices", "text": "Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\struct {\\map {\\MM_R} n, +, \\times}$ denote the ring of square matrices of order $n$ over $R$. The unit matrix over $R$: :$\\mathbf I_n = \\begin {pmatrix} 1_R & 0_R & 0_R & \\cdots & 0_R \\\\ 0_R & 1_R & 0_R & \\cdots & 0_R \\\\ 0_R & 0_R & 1_R & \\cdots & 0_R \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0_R & 0_R & 0_R & \\cdots & 1_R \\end {pmatrix}$ is the identity element of $\\struct {\\map {\\MM_R} n, +, \\times}$."}
+{"_id": "1031", "title": "Matrix Multiplication is Closed", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\map {\\MM_R} n$ be a $n \\times n$ matrix space over $R$. Then matrix multiplication (conventional) over $\\map {\\MM_R} n$ is closed."}
+{"_id": "1032", "title": "Square Matrices over Real Numbers under Multiplication form Monoid", "text": "Let $\\map {\\mathcal M_\\R} n$ be a $n \\times n$ matrix space over the set of real numbers $\\R$. Then the set of all $n \\times n$ real matrices $\\map {\\mathcal M_\\R} n$ under matrix multiplication (conventional) forms a monoid."}
+{"_id": "1033", "title": "Ring of Square Matrices over Commutative Ring with Unity", "text": "Let $R$ be a commutative ring with unity. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\struct {\\map {\\MM_R} n, +, \\times}$ denote the ring of square matrices of order $n$ over $R$. Then $\\struct {\\map {\\MM_R} n, +, \\times}$ is a ring with unity. However, for $n \\ge 2$, $\\struct {\\map {\\MM_R} n, +, \\times}$ is not a commutative ring."}
+{"_id": "1034", "title": "Invertible Matrix corresponds to Automorphism", "text": "Let $R$ be a ring with unity. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $G$ be an $n$-dimensional $R$-module. Let $\\map {\\mathcal M_R} n$ be the $n \\times n$ matrix space over $R$. Let $\\map {\\mathcal L_R} G$ be the set of all linear operators on $G$. Then the invertible elements of the ring of square matrices $\\struct {\\map {\\mathcal M_R} n, +, \\times}$ correspond directly to automorphisms of $\\map {\\mathcal L_R} G$."}
+{"_id": "1035", "title": "Change of Basis is Invertible", "text": "Let $R$ be a ring with unity. Let $M$ be a free $R$-module of finite dimension $n>0$. Let $\\mathcal A$ and $\\mathcal B$ be ordered bases of $M$. Let $\\mathbf P$ be the change of basis matrix from $\\mathcal A$ to $\\mathcal B$. Then $\\mathbf P$ is invertible, and its inverse $\\mathbf P^{-1}$ is the change of basis matrix from $\\mathcal B$ to $\\mathcal A$."}
+{"_id": "1037", "title": "Matrix Corresponding to Change of Basis under Linear Transformation", "text": "Let $R$ be a ring with unity. Let $G$ and $H$ be free $R$-modules of finite dimensions $n,m>0$ respectively. Let $\\left \\langle {a_n} \\right \\rangle$ and $\\left \\langle {{a_n}'} \\right \\rangle$ be ordered bases of $G$. Let $\\left \\langle {b_m} \\right \\rangle$ and $\\left \\langle {{b_m}'} \\right \\rangle$ be ordered bases of $H$. Let $u: G \\to H$ be a linear transformation, and let $\\left[{u; \\left \\langle {b_m} \\right \\rangle, \\left \\langle {a_n} \\right \\rangle}\\right]$ be the matrix of $u$ relative to $\\left \\langle {a_n} \\right \\rangle$ and $\\left \\langle {b_m} \\right \\rangle$. Let: : $\\mathbf A = \\left[{u; \\left \\langle {b_m} \\right \\rangle, \\left \\langle {a_n} \\right \\rangle}\\right]$ : $\\mathbf B = \\left[{u; \\left \\langle {{b_m}'} \\right \\rangle, \\left \\langle {{a_n}'} \\right \\rangle}\\right]$ Then: :$\\mathbf B = \\mathbf Q^{-1} \\mathbf A \\mathbf P$ where: : $\\mathbf P$ is the matrix corresponding to the change of basis from $\\left \\langle {a_n} \\right \\rangle$ to $\\left \\langle {{a_n}'} \\right \\rangle$ : $\\mathbf Q$ is the matrix corresponding to the change of basis from $\\left \\langle {b_m} \\right \\rangle$ to $\\left \\langle {{b_m}'} \\right \\rangle$."}
+{"_id": "1038", "title": "Matrix Equivalence is Equivalence Relation", "text": "Matrix equivalence is an equivalence relation."}
+{"_id": "1041", "title": "Equivalent Matrices have Equal Rank", "text": "Let $\\mathbf A$ and $\\mathbf B$ be $m \\times n$ matrices over a field $K$. Let $\\map \\phi {\\mathbf A}$ denote the rank of $\\mathbf A$. Let $\\mathbf A \\equiv \\mathbf B$ denote that $\\mathbf A$ and $\\mathbf B$ are matrix equivalent. Then: :$\\mathbf A \\equiv \\mathbf B$ {{iff}}: :$\\map \\phi {\\mathbf A} = \\map \\phi {\\mathbf B}$"}
+{"_id": "1043", "title": "Transpose of Matrix Product", "text": "Let $\\mathbf A$ and $\\mathbf B$ be matrices over a commutative ring such that $\\mathbf A \\mathbf B$ is defined. Then $\\mathbf B^\\intercal \\mathbf A^\\intercal$ is defined, and: :$\\paren {\\mathbf A \\mathbf B}^\\intercal = \\mathbf B^\\intercal \\mathbf A^\\intercal$ where $\\mathbf X^\\intercal$ is the transpose of $\\mathbf X$."}
+{"_id": "1045", "title": "General Linear Group is Group", "text": "Let $K$ be a field. Let $\\GL {n, K}$ be the general linear group of order $n$ over $K$. Then $\\GL {n, K}$ is a group."}
+{"_id": "1046", "title": "Transpose of Row Matrix is Column Matrix", "text": "Let $\\mathbf x = \\sqbrk x_{1 n} = \\begin {bmatrix} x_1 & x_2 & \\cdots & x_n \\end {bmatrix}$ be a row matrix. Then $\\mathbf x^\\intercal$, the transpose of $\\mathbf x$, is a column matrix: :$\\begin {bmatrix} x_1 & x_2 & \\cdots & x_n \\end{bmatrix}^\\intercal = \\begin {bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end {bmatrix}$"}
+{"_id": "1047", "title": "Transpose of Transpose of Matrix", "text": "Let $\\mathbf A$ be a matrix. Let $\\mathbf A^\\intercal$ be the transpose of $\\mathbf A$. Then: :$\\paren {\\mathbf A^\\intercal}^\\intercal = \\mathbf A$"}
+{"_id": "1048", "title": "Solution to Simultaneous Linear Equations", "text": "Let $\\displaystyle \\forall i \\in \\closedint 1 m: \\sum _{j \\mathop = 1}^n {\\alpha_{i j} x_j} = \\beta_i$ be a system of simultaneous linear equations. where all of $\\alpha_1, \\ldots, a_n, x_1, \\ldots x_n, \\beta_i, \\ldots, \\beta_m$ are elements of a field $K$. Then $x = \\tuple {x_1, x_2, \\ldots, x_n}$ is a solution of this system {{iff}}: :$\\sqbrk \\alpha_{m n} \\sqbrk x_{n 1} = \\sqbrk \\beta_{m 1}$ where $\\sqbrk a_{m n}$ is an $m \\times n$ matrix."}
+{"_id": "1049", "title": "Infinite Cyclic Group is Isomorphic to Integers", "text": "Let $G$ be an infinite cyclic group. Then $G$ is isomorphic to the additive group of integers: $G \\cong \\struct {\\Z, +}$."}
+{"_id": "1050", "title": "Subgroup of Infinite Cyclic Group is Infinite Cyclic Group", "text": "Let $G = \\gen a$ be an infinite cyclic group generated by $a$, whose identity is $e$. Let $g \\in G, g \\ne e: \\exists k \\in \\Z, k \\ne 0: g = a^k$. Let $H = \\gen g$. Then $H \\le G$ and $H \\cong G$. Thus, all non-trivial subgroups of an infinite cyclic group are themselves infinite cyclic groups. A subgroup of $G = \\gen a$ is denoted as follows: :$n G := \\gen {a^n}$ This notation is usually used in the context of $\\struct {\\Z, +}$, where $n \\Z$ is (informally) understood as '''the set of integer multiples of $n$'''."}
+{"_id": "1051", "title": "Quotient Group of Infinite Cyclic Group by Subgroup", "text": "Let $C_n$ be the cyclic group of order $n$. Then: :$C_n \\cong \\dfrac {\\struct {\\Z, +} } {\\struct {n \\Z, +} } = \\dfrac \\Z {n \\Z}$ where: :$\\Z$ is the additive group of integers :$n \\Z$ is the additive group of integer multiples :$\\Z / n \\Z$ is the quotient group of $\\Z$ by $n \\Z$. Thus, every cyclic group is isomorphic to one of: :$\\Z, \\dfrac \\Z \\Z, \\dfrac \\Z {2 \\Z}, \\dfrac \\Z {3 \\Z}, \\dfrac \\Z {4 \\Z}, \\ldots$"}
+{"_id": "1052", "title": "Ring Operations on Coset Space of Ideal", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\powerset R$ be the power set of $R$. Let $J$ be an ideal of $R$. Let $X$ and $Y$ be cosets of $J$. Let $X +_\\mathcal P Y$ be the sum of $X$ and $Y$, where $+_\\mathcal P$ is the operation induced on $\\powerset R$ by $+$. Similarly, let $X \\circ_\\mathcal P Y$ be the product of $X$ and $Y$, where $\\circ_\\mathcal P$ is the operation induced on $\\powerset R$ by $\\circ$. Then: * The sum $X +_\\mathcal P Y$ in $\\powerset R$ is also their sum in the quotient ring $R / J$. * The product $X \\circ_\\mathcal P Y$ in $\\powerset R$ may be a proper subset of their product in $R / J$."}
+{"_id": "1053", "title": "Property of Being an Ideal is not Transitive", "text": "Let $J_1$ be an ideal of a ring $R$. Let $J_2$ be an ideal of $J_1$. Then $J_2$ need not necessarily be an ideal of $R$."}
+{"_id": "1054", "title": "Ideals Containing Ideal Form Lattice", "text": "Let $J$ be an ideal of a ring $R$. Let $\\mathbb L_J$ be the set of all ideal of $R$ which contain $J$. Then the ordered set $\\struct {\\mathbb L_J, \\subseteq}$ is a lattice."}
+{"_id": "1055", "title": "Ideals Containing Ideal Isomorphic to Quotient Ring", "text": "Let $J$ be an ideal of a ring $R$. Let $\\mathbb L_J$ be the set of all ideals of $R$ which contain $J$. Let the ordered set $\\left({\\mathbb L \\left({R / J}\\right), \\subseteq}\\right)$ be the set of all ideals of $R / J$. Let the mapping $\\Phi_J: \\left({\\mathbb L_J, \\subseteq}\\right) \\to \\left({\\mathbb L \\left({R / J}\\right), \\subseteq}\\right)$ be defined as: :$\\forall a \\in \\mathbb L_J: \\Phi_J \\left({a}\\right) = q_J \\left({a}\\right)$ where $q_J: a \\to a / J$ is the quotient epimorphism from $a$ to $a / J$ from the definition of quotient ring. Then $\\Phi_J$ is an isomorphism."}
+{"_id": "1056", "title": "Ring of Integers is Principal Ideal Domain", "text": "The integers $\\Z$ form a principal ideal domain."}
+{"_id": "1057", "title": "Principal Ideals of Integers", "text": "Let $J$ be a non-zero ideal of $\\Z$. Then $J = \\ideal b$ where $b$ is the smallest strictly positive integer belonging to $J$."}
+{"_id": "1058", "title": "Natural Numbers Set Equivalent to Ideals of Integers", "text": "Let the mapping $\\psi: \\N \\to$ the set of all ideals of $\\Z$ be defined as: :$\\forall b \\in \\N: \\psi \\left({b}\\right) = \\left({b}\\right)$ where $\\left({b}\\right)$ is the principal ideal of $\\Z$ generated by $b$. Then $\\psi$ is a bijection."}
+{"_id": "1059", "title": "Canonical Epimorphism from Integers by Principal Ideal", "text": "Let $m$ be a strictly positive integer. Let $\\left({m}\\right)$ be the principal ideal of $\\Z$ generated by $m$. The restriction to $\\N_m$ of the canonical epimorphism $q_m$ from the ring $\\left({\\Z, +, \\times}\\right)$ onto $\\left({\\Z, +, \\times}\\right) / \\left({m}\\right)$ is an isomorphism from the ring $\\left({\\N_m, +_m, \\times_m}\\right)$ of integers modulo $m$ onto the quotient ring $\\left({\\Z, +, \\times}\\right) / \\left({m}\\right)$. In particular, $\\left({\\Z, +, \\times}\\right) / \\left({m}\\right)$ has $m$ elements."}
+{"_id": "1060", "title": "Integer Divisor is Equivalent to Subset of Ideal", "text": "Let $\\Z$ be the set of all integers. Let $\\Z_{>0}$ be the set of strictly positive integers. Let $m \\in \\Z_{>0}$ and let $n \\in \\Z$. Let $\\ideal m$ be the principal ideal of $\\Z$ generated by $m$. Then: :$m \\divides n \\iff \\ideal n \\subseteq \\ideal m$"}
+{"_id": "1061", "title": "Principal Ideals in Integral Domain", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain. Let $U_D$ be the group of units of $D$. Let $\\ideal x$ be the principal ideal of $D$ generated by $x$. Let $x, y \\in \\struct {D, +, \\circ}$. Then:"}
+{"_id": "1062", "title": "Principal Ideal Domain is Unique Factorization Domain", "text": "Every principal ideal domain is a unique factorization domain."}
+{"_id": "1063", "title": "Maximal Ideal iff Quotient Ring is Field", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. The following are equivalent: :$(1): \\quad$ $J$ is a maximal ideal. :$(2): \\quad$ The quotient ring $R / J$ is a field."}
+{"_id": "1064", "title": "Prime Number iff Generates Principal Maximal Ideal", "text": "Let $\\Z_{>0}$ be the set of strictly positive integers. Let $p \\in \\Z_{>0}$. Let $\\ideal p$ be the principal ideal of $\\Z$ generated by $p$.  Then $p$ is prime {{iff}} $\\ideal p$ is a maximal ideal of $\\Z$."}
+{"_id": "1065", "title": "Integral Domain of Prime Order is Field", "text": "Let $\\left({\\Z_p, +_p, \\times_p}\\right)$ be the ring of integers modulo $p$. The following statements are equivalent: : $(1): \\quad p$ is a prime. : $(2): \\quad \\left({\\Z_p, +_p, \\times_p}\\right)$ is an integral domain. : $(3): \\quad \\left({\\Z_p, +_p, \\times_p}\\right)$ is a field."}
+{"_id": "1066", "title": "Quotient Ring of Integers and Zero", "text": "Let $\\struct {\\Z, +, \\times}$ be the integral domain of integers. Let $\\ideal 0$ be the principal ideal of $\\struct {\\Z, +, \\times}$ generated by $0$. The quotient ring $\\struct {\\Z / \\ideal 0, +, \\times}$ is isomorphic to $\\struct {\\Z, +, \\times}$."}
+{"_id": "1067", "title": "Quotient Ring of Integers and Principal Ideal from Unity", "text": "Let $\\left({\\Z, +, \\times}\\right)$ be the integral domain of integers. Let $\\left({1}\\right)$ be the principal ideal of $\\left({\\Z, +, \\times}\\right)$ generated by $1$. The quotient ring $\\left({\\Z, +, \\times}\\right) / \\left({1}\\right)$ is isomorphic to the null ring."}
+{"_id": "1068", "title": "Principal Ideal of Principal Ideal Domain is of Irreducible Element iff Maximal", "text": "Let $\\struct {D, +, \\circ}$ be a principal ideal domain. Let $\\ideal p$ be the principal ideal of $D$ generated by $p$. Then $p$ is irreducible {{iff}} $\\ideal p$ is a maximal ideal of $D$."}
+{"_id": "1069", "title": "Subring Generated by Unity of Ring with Unity", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let the mapping $g: \\Z \\to R$ be defined as $\\forall n \\in \\Z: g \\left({n}\\right) = n 1_R$, where $n 1_R$ the $n$th power of $1_R$. Let $\\left({x}\\right)$ be the principal ideal of $\\left({R, +, \\circ}\\right)$ generated by $x$. Then $g$ is an epimorphism from $\\Z$ onto the subring $S$ of $R$ generated by $1_R$. If $R$ has no proper zero divisors, then $g$ is the only nonzero homomorphism from $\\Z$ into $R$. The kernel of $g$ is either: : $(1): \\quad \\left({0_R}\\right)$, in which case $g$ is an isomorphism from $\\Z$ onto $S$ or: : $(2): \\quad \\left({p}\\right)$ for some prime $p$, in which case $S$ is isomorphic to the field $\\Z_p$."}
+{"_id": "1070", "title": "Null Ring iff Characteristic is One", "text": "The only ring whose characteristic is $1$ is the null ring."}
+{"_id": "1071", "title": "Characteristic of Finite Ring with No Zero Divisors", "text": "Let $\\struct {R, +, \\circ}$ be a finite ring with unity with no proper zero divisors whose zero is $0_R$ and whose unity is $1_R$. Let $n \\ne 0$ be the characteristic of $R$. Then: :$(1): \\quad n$ must be a prime number :$(2): \\quad n$ is the order of all non-zero elements in $\\struct {R, +}$. It follows that $\\struct {R, +} \\cong C_n$, where $C_n$ is the cyclic group of order $n$."}
+{"_id": "1072", "title": "Integral Domain with Characteristic Zero", "text": "In an integral domain with characteristic zero, every non-zero element has infinite order under ring addition."}
+{"_id": "1073", "title": "Characteristic of Field is Zero or Prime", "text": "Let $F$ be a field. Then the characteristic of $F$ is either zero or a prime number."}
+{"_id": "1074", "title": "Field of Characteristic Zero has Unique Prime Subfield", "text": "Let $F$ be a field, whose zero is $0_F$ and whose unity is $1_F$, with characteristic zero. Then there exists a unique $P \\subseteq F$ such that: :$(1): \\quad P$ is a subfield of $F$ :$(2): \\quad P$ is isomorphic to the field of rational numbers $\\struct {\\Q, +, \\times}$. That is, $P \\cong \\Q$ is a unique minimal subfield of $F$, and all other subfields of $F$ contain $P$. This field $P$ is called the prime subfield of $F$."}
+{"_id": "1075", "title": "Field of Prime Characteristic has Unique Prime Subfield", "text": "Let $F$ be a field whose characteristic is $p$. Then there exists a unique $P \\subseteq F$ such that: :$(1): \\quad P$ is a subfield of $F$ :$(2): \\quad P \\cong \\Z_p$. That is, $P \\cong \\Z_p$ is a unique minimal subfield of $F$, and all other subfields of $F$ contain $P$. This field $P$ is called the prime subfield of $F$."}
+{"_id": "1076", "title": "Intersection of All Division Subrings is Prime Subfield", "text": "Let $\\struct {K, +, \\circ}$ be a division ring. Let $P$ be the intersection of the set of all division subrings of $K$. Then $P$ is the prime subfield of $K$."}
+{"_id": "1077", "title": "Characteristic of Ordered Integral Domain is Zero", "text": "Let $\\left({D, +, \\circ}\\right)$ be an ordered integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $\\operatorname{Char} \\left({D}\\right)$ be the characteristic of $D$. Then $\\operatorname{Char} \\left({D}\\right) = 0$. Let $g: \\Z \\to D$ be the mapping defined as: :$\\forall n \\in \\Z: g \\left({n}\\right) = n \\cdot 1_D$ where $n \\cdot 1_D$ is defined as the $n$th power of $1_D$. Then $g$ is the only monomorphism from the ordered ring $\\Z$ onto the ordered ring $D$."}
+{"_id": "1078", "title": "Monomorphism from Rational Numbers to Totally Ordered Field", "text": "Let $\\struct {F, +, \\circ, \\le}$ be a totally ordered field. There is one and only one (ring) monomorphism from the totally ordered field $\\Q$ onto $F$. Its image is the prime subfield of $F$."}
+{"_id": "1079", "title": "Set of Polynomials over Integral Domain is Subring", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring. Let $\\struct {D, +, \\circ}$ be an integral subdomain of $R$. Then $\\forall x \\in R$, the set $D \\sqbrk x$ of polynomials in $x$ over $D$ is a subring of $R$."}
+{"_id": "1081", "title": "Polynomials Closed under Ring Product", "text": "Let $\\left({R, +, \\circ}\\right)$ be a commutative ring. Let $\\displaystyle f = \\sum_{k \\in Z} a_k \\mathbf X^k$, $\\displaystyle g =  \\sum_{k \\in Z} b_k \\mathbf X^k$ be polynomials in the indeterminates $\\left\\{{X_j: j \\in J}\\right\\}$ over $R$, where $Z$ is the set of all multiindices indexed by $\\left\\{{X_j: j \\in J}\\right\\}$. Define the product :$\\displaystyle f \\otimes g = \\sum_{k \\in Z} c_k \\mathbf X^k$ where :$\\displaystyle c_k = \\sum_{\\substack{p + q = k \\\\ p, q \\in Z}} a_p b_q$ Then $f \\otimes g$ is a polynomial."}
+{"_id": "1083", "title": "Ring of Polynomial Forms is Integral Domain", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $\\struct {D, +, \\circ}$ be an integral subdomain of $R$. Let $X \\in R$ be transcendental over $D$. Let $D \\sqbrk X$ be the ring of polynomials in $X$ over $D$. Then $D \\sqbrk X$ is an integral domain."}
+{"_id": "1085", "title": "Injection is Bijection iff Inverse is Injection", "text": "Let $\\phi: S \\to T$ be an injection. Then $\\phi$ is a bijection {{iff}} its inverse $\\phi^{-1}$ is also an injection."}
+{"_id": "1086", "title": "Epimorphism from Polynomial Forms to Polynomial Functions", "text": "Let $D$ be an integral domain. Let $D \\sqbrk X$ be the ring of polynomial forms in $X$ over $D$. Let $\\map P D$ be the ring of polynomial functions over $D$. The mapping $\\kappa: D \\sqbrk X \\to \\map P D$ given by: :$\\displaystyle \\map \\kappa {\\sum_{k \\mathop = 0}^n {a_k \\circ X^k} } = f$ where $\\displaystyle f = \\sum_{k \\mathop = 0}^n {a_k \\circ x^k}, x \\in D$ is a ring epimorphism."}
+{"_id": "1087", "title": "Division Theorem for Polynomial Forms over Field", "text": "Let $\\struct {F, +, \\circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $X$ be transcendental over $F$. Let $F \\sqbrk X$ be the ring of polynomials in $X$ over $F$. Let $d$ be an element of $F \\sqbrk X$ of degree $n \\ge 1$. Then $\\forall f \\in F \\sqbrk X: \\exists q, r \\in F \\sqbrk X: f = q \\circ d + r$ such that either: :$(1): \\quad r = 0_F$ or: :$(2): \\quad r \\ne 0_F$ and $r$ has degree that is less than $n$."}
+{"_id": "1088", "title": "Polynomial Forms over Field form Principal Ideal Domain", "text": "Let $\\struct {F, +, \\circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $X$ be transcendental over $F$. Let $F \\sqbrk X$ be the ring of polynomials in $X$ over $F$. Then $F \\sqbrk X$ is a principal ideal domain."}
+{"_id": "1089", "title": "Equal Consecutive Prime Number Gaps are Multiples of Six", "text": "If you list the gaps between consecutive primes greater than $5$: :$2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, \\ldots$ you will notice that consecutive gaps that are equal are of the form $6 x$. This is ''always'' the case. {{OEIS|A001223}}"}
+{"_id": "1090", "title": "Standard Discrete Metric is Metric", "text": "The standard discrete metric is a metric."}
+{"_id": "1091", "title": "Derivative of Constant", "text": "Let $\\map {f_c} x$ be the constant function on $\\R$, where $c \\in \\R$. Then: :$\\map {f_c'} x = 0$"}
+{"_id": "1092", "title": "Derivative of Identity Function", "text": "Let $X$ be either set of either the real numbers $\\R$ or the complex numbers $\\C$. Let $I_X: X \\to X$ be the identity function. Then: :$\\map {I_X'} x = 1$ where $\\map {I_X'} x$ denotes the derivative of $I_X$ {{WRT|Differentiation}} $x$. This can be presented for each of $\\R$ and $\\C$:"}
+{"_id": "1093", "title": "Product Rule for Derivatives", "text": "Let $\\map f x, \\map j x, \\map k x$ be real functions defined on the open interval $I$. Let $\\xi \\in I$ be a point in $I$ at which both $j$ and $k$ are differentiable. Let $\\map f x = \\map j x \\map k x$. Then: :$\\map {f'} \\xi = \\map j \\xi \\map {k'} \\xi + \\map {j'} \\xi \\map k \\xi$ It follows from the definition of derivative that if $j$ and $k$ are both differentiable on the interval $I$, then: :$\\forall x \\in I: \\map {f'} x = \\map j x \\map {k'} x + \\map {j'} x \\map k x$ Using Leibniz's notation for derivatives, this can be written as: :$\\map {\\dfrac \\d {\\d x} } {y \\, z} = y \\dfrac {\\d z} {\\d x} + \\dfrac {\\d y} {\\d x} z$ where $y$ and $z$ represent functions of $x$."}
+{"_id": "1094", "title": "Derivative of Composite Function", "text": "Let $f, g, h$ be continuous real functions such that: :$\\forall x \\in \\R: \\map h x = \\map {f \\circ g} x = \\map f {\\map g x}$ Then: :$\\map {h'} x = \\map {f'} {\\map g x} \\map {g'} x$ where $h'$ denotes the derivative of $h$. Using the $D_x$ notation: :$\\map {D_x} {\\map f {\\map g x} } = \\map {D_{\\map g x} } {\\map f {\\map g x} } \\, \\map {D_x} {\\map g x}$ This is often informally referred to as the '''chain rule (for differentiation)'''."}
+{"_id": "1095", "title": "Derivative of Inverse Function", "text": "Let $I = \\closedint a b$ and $J = \\closedint c d$ be closed real intervals. Let $I^o = \\openint a b$ and $J^o = \\openint c d$ be the corresponding open real intervals. Let $f: I \\to J$ be a real function which is continuous on $I$ and differentiable on $I^o$ such that $J = f \\sqbrk I$. Let either: :$\\forall x \\in I^o: D \\map f x > 0$ or: :$\\forall x \\in I^o: D \\map f x < 0$ Then: :$f^{-1}: J \\to I$ exists and is continuous on $J$ :$f^{-1}$ is differentiable on $J^o$ :$\\forall y \\in J^o: D \\map {f^{-1} } y = \\dfrac 1 {D \\map f x}$"}
+{"_id": "1096", "title": "Upper Sum Never Smaller than Lower Sum", "text": "Let $\\closedint a b$ be a closed interval of the set $\\R$ of real numbers. Let $P = \\set {x_0, x_1, x_2, \\ldots, x_{n - 1}, x_n}$ be a finite subdivision of $\\closedint a b$. Let $f: \\R \\to \\R$ be a real function. Let $f$ be bounded on $\\closedint a b$. Let $\\map L P$ be the lower sum of $\\map f x$ on $\\closedint a b$ belonging to the subdivision $P$. Let $\\map U P$ be the upper sum of $\\map f x$ on $\\closedint a b$ belonging to the subdivision $P$. Then $\\map L P \\le \\map U P$."}
+{"_id": "1097", "title": "Dedekind's Theorem", "text": "Let $\\tuple {L, R}$ be a Dedekind cut of the set of real numbers $\\R$. Then there exists a unique real number which is a producer of $\\tuple {L, R}$."}
+{"_id": "1098", "title": "Complex Addition is Closed", "text": "The set of complex numbers $\\C$ is closed under addition: :$\\forall z, w \\in \\C: z + w \\in \\C$"}
+{"_id": "1099", "title": "Complex Addition is Associative", "text": "The operation of addition on the set of complex numbers $\\C$ is associative: :$\\forall z_1, z_2, z_3 \\in \\C: z_1 + \\paren {z_2 + z_3} = \\paren {z_1 + z_2} + z_3$"}
+{"_id": "1100", "title": "Complex Addition is Commutative", "text": "The operation of addition on the set of complex numbers is commutative: :$\\forall z, w \\in \\C: z + w = w + z$"}
+{"_id": "1101", "title": "Integers are Countably Infinite", "text": "The set $\\Z$ of integers is countably infinite."}
+{"_id": "1102", "title": "Permutation of Determinant Indices", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$ over a field. Let $\\lambda: \\N_{> 0} \\to \\N_{> 0}$ be any fixed permutation on $\\N_{> 0}$. Let $\\map \\det {\\mathbf A}$ be the determinant of $\\mathbf A$. Let $\\struct {S_n, \\circ}$ be the symmetric group of $n$ letters.  Then: :$\\displaystyle \\map \\det {\\mathbf A} = \\sum_{\\mu \\mathop \\in S_n} \\paren {\\map \\sgn \\mu \\map \\sgn \\lambda \\prod_{k \\mathop = 1}^n a_{\\map \\lambda k, \\map \\mu k} }$ :$\\displaystyle \\map \\det {\\mathbf A} = \\sum_{\\mu \\mathop \\in S_n} \\paren {\\map \\sgn \\mu \\map \\sgn \\lambda \\prod_{k \\mathop = 1}^n a_{\\map \\mu k, \\map \\lambda k} }$ where: :the summation $\\displaystyle \\sum_{\\mu \\mathop \\in S_n}$ goes over all the $n!$ permutations of $\\set {1, 2, \\ldots, n}$ :$\\map \\sgn \\mu$ is the sign of the permutation $\\mu$."}
+{"_id": "1103", "title": "Determinant of Transpose", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $\\map \\det {\\mathbf A}$ be the determinant of $\\mathbf A$. Let $\\mathbf A^\\intercal$ be the transpose of $\\mathbf A$. Then: :$\\map \\det {\\mathbf A} = \\map \\det {\\mathbf A^\\intercal}$"}
+{"_id": "1104", "title": "Determinant with Rows Transposed", "text": "If two rows of a matrix with determinant $D$ are transposed, its determinant becomes $-D$."}
+{"_id": "1105", "title": "Square Matrix with Duplicate Rows has Zero Determinant", "text": "If two rows of a square matrix over a commutative ring $\\struct {R, +, \\circ}$ are the same, then its determinant is zero."}
+{"_id": "1106", "title": "Determinant with Row Multiplied by Constant", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $\\map \\det {\\mathbf A}$ be the determinant of $\\mathbf A$. Let $\\mathbf B$ be the matrix resulting from one row of $\\mathbf A$ having been multiplied by a constant $c$. Then: :$\\map \\det {\\mathbf B} = c \\map \\det {\\mathbf A}$ That is, multiplying one row of a square matrix by a constant multiplies its determinant by that constant."}
+{"_id": "1107", "title": "Determinant as Sum of Determinants", "text": "Let $\\begin{vmatrix}   a_{11} & \\cdots & a_{1s} & \\cdots & a_{1n} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{r1} & \\cdots & a_{rs} & \\cdots & a_{rn} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{n1} & \\cdots & a_{ns} & \\cdots & a_{nn} \\end{vmatrix}$ be a determinant. Then $\\begin{vmatrix}   a_{11} & \\cdots & a_{1s} & \\cdots & a_{1n} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{r1} + a'_{r1} & \\cdots & a_{rs} + a'_{rs} & \\cdots & a_{rn} + a'_{rn} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{n1} & \\cdots & a_{ns} & \\cdots & a_{nn} \\end{vmatrix} = \\begin{vmatrix}   a_{11} & \\cdots & a_{1s} & \\cdots & a_{1n} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{r1} & \\cdots & a_{rs} & \\cdots & a_{rn} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{n1} & \\cdots & a_{ns} & \\cdots & a_{nn} \\end{vmatrix} + \\begin{vmatrix}   a_{11} & \\cdots & a_{1s} & \\cdots & a_{1n} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a'_{r1} & \\cdots & a'_{rs} & \\cdots & a'_{rn} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{n1} & \\cdots & a_{ns} & \\cdots & a_{nn} \\end{vmatrix}$. Similarly: Then $\\begin{vmatrix}   a_{11} & \\cdots & a_{1s} + a'_{1s} & \\cdots & a_{1n} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{r1} & \\cdots & a_{rs} + a'_{rs} & \\cdots & a_{rn} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{n1} & \\cdots & a_{ns} + a'_{ns} & \\cdots & a_{nn} \\end{vmatrix} = \\begin{vmatrix}   a_{11} & \\cdots & a_{1s} & \\cdots & a_{1n} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{r1} & \\cdots & a_{rs} & \\cdots & a_{rn} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{n1} & \\cdots & a_{ns} & \\cdots & a_{nn} \\end{vmatrix} + \\begin{vmatrix}   a_{11} & \\cdots & a'_{1s} & \\cdots & a_{1n} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{r1} & \\cdots & a'_{rs} & \\cdots & a_{rn} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{n1} & \\cdots & a'_{ns} & \\cdots & a_{nn} \\end{vmatrix}$."}
+{"_id": "1108", "title": "Multiple of Row Added to Row of Determinant", "text": "Let $\\mathbf A = \\begin {bmatrix} a_{1 1} & a_{1 2} & \\cdots & a_{1 n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ a_{r 1} & a_{r 2} & \\cdots & a_{r n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ a_{s 1} & a_{s 2} & \\cdots & a_{s n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ a_{n 1} & a_{n 2} & \\cdots & a_{n n} \\\\ \\end {bmatrix}$ be a square matrix of order $n$. Let $\\map \\det {\\mathbf A}$ denote the determinant of $\\mathbf A$. Let $\\mathbf B = \\begin{bmatrix} a_{1 1}             & a_{1 2}             & \\cdots &             a_{1 n} \\\\  \\vdots             & \\vdots              & \\ddots &              \\vdots \\\\ a_{r 1} + k a_{s 1} & a_{r 2} + k a_{s 2} & \\cdots & a_{r n} + k a_{s n} \\\\  \\vdots             & \\vdots              & \\ddots &              \\vdots \\\\ a_{s 1}             & a_{s 2}             & \\cdots &             a_{s n} \\\\  \\vdots             & \\vdots              & \\ddots &              \\vdots \\\\ a_{n 1}             & a_{n 2}             & \\cdots &             a_{n n} \\\\ \\end{bmatrix}$. Then $\\map \\det {\\mathbf B} = \\map \\det {\\mathbf A}$. That is, the value of a determinant remains unchanged if a constant multiple of any row is added to any other row."}
+{"_id": "1109", "title": "Determinant of Matrix Product", "text": "Let $\\mathbf A = \\sqbrk a_n$ and $\\mathbf B = \\sqbrk b_n$ be a square matrices of order $n$. Let $\\map \\det {\\mathbf A}$ be the determinant of $\\mathbf A$. Let $\\mathbf A \\mathbf B$ be the (conventional) matrix product of $\\mathbf A$ and $\\mathbf B$. Then: :$\\map \\det {\\mathbf A \\mathbf B} = \\map \\det {\\mathbf A} \\map \\det {\\mathbf B}$ That is, the determinant of the product is equal to the product of the determinants."}
+{"_id": "1110", "title": "Expansion Theorem for Determinants", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $D = \\map \\det {\\mathbf A}$ be the determinant of $\\mathbf A$: :$\\displaystyle \\map \\det {\\mathbf A} := \\sum_{\\lambda} \\paren {\\map \\sgn \\lambda \\prod_{k \\mathop = 1}^n a_{k \\map \\lambda k} } = \\sum_\\lambda \\map \\sgn \\lambda a_{1 \\map \\lambda 1} a_{2 \\map \\lambda 2} \\cdots a_{n \\map \\lambda n}$ where: :the summation $\\displaystyle \\sum_\\lambda$ goes over all the $n!$ permutations of $\\set {1, 2, \\ldots, n}$ :$\\map \\sgn \\lambda$ is the sign of the permutation $\\lambda$. Let $a_{p q}$ be an element of $\\mathbf A$. Let $A_{p q}$ be the cofactor of $a_{p q}$ in $D$. Then: :$(1): \\quad \\displaystyle \\forall r \\in \\closedint 1 n: D = \\sum_{k \\mathop = 1}^n a_{r k} A_{r k}$ :$(2): \\quad \\displaystyle \\forall r \\in \\closedint 1 n: D = \\sum_{k \\mathop = 1}^n a_{k r} A_{k r}$ Thus the value of a determinant can be found either by: :multiplying all the elements in a row by their cofactors and adding up the products or: :multiplying all the elements in a column by their cofactors and adding up the products. The identity: :$\\displaystyle D = \\sum_{k \\mathop = 1}^n a_{r k} A_{r k}$ is known as the '''expansion of $D$ in terms of row $r$''', while: :$\\displaystyle D = \\sum_{k \\mathop = 1}^n a_{k r} A_{k r}$ is known as the '''expansion of $D$ in terms of column $r$'''."}
+{"_id": "1111", "title": "Determinant with Unit Element in Otherwise Zero Row", "text": "Let $D$ be the determinant: :$D = \\begin {vmatrix}       1 &       0 & \\cdots &       0 \\\\ b_{2 1} & b_{2 2} & \\cdots & b_{2 n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ b_{n 1} & b_{n 2} & \\cdots & b_{n n} \\end {vmatrix}$ Then: :$D = \\begin {vmatrix} b_{2 2} & \\cdots & b_{2 n} \\\\  \\vdots & \\ddots &  \\vdots \\\\ b_{n 2} & \\cdots & b_{n n} \\end {vmatrix}$"}
+{"_id": "1112", "title": "Vandermonde Determinant", "text": "The '''Vandermonde determinant of order $n$''' is the determinant defined as follows: :$V_n = \\begin {vmatrix} 1 & x_1 & x_1^2 & \\cdots & x_1^{n - 2} & x_1^{n - 1} \\\\ 1 & x_2 & x_2^2 & \\cdots & x_2^{n - 2} & x_2^{n - 1} \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 1 & x_n & x_n^2 & \\cdots & x_n^{n - 2} & x_n^{n - 1} \\end {vmatrix}$ Its value is given by: :$\\displaystyle V_n = \\prod_{1 \\mathop \\le i \\mathop < j \\mathop \\le n} \\paren {x_j - x_i}$"}
+{"_id": "1114", "title": "Equality of Polynomials", "text": "$f$ and $g$ are equal as polynomials {{iff}} $f$ and $g$ are equal as functions. Thus we can say $f = g$ without ambiguity as to what it means. {{explain|In the exposition, the term was \"equal as forms\", but it has now morphed into \"equal as polynomials\". Needs to be resolved.}}"}
+{"_id": "1115", "title": "Value of Adjugate of Determinant", "text": "Let $D$ be the determinant of order $n$. Let $D^*$ be the adjugate of $D$. Then $D^* = D^{n-1}$."}
+{"_id": "1116", "title": "Determinant of Diagonal Matrix", "text": "Let $\\mathbf A = \\begin{bmatrix} a_{11} & 0 & \\cdots & 0 \\\\ 0 & a_{22} & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & a_{nn} \\\\ \\end{bmatrix}$ be a diagonal matrix. Then the determinant of $\\mathbf A$ is the product of the elements of $\\mathbf A$. That is: :$\\ds \\map \\det {\\mathbf A} = \\prod_{i \\mathop = 1}^n a_{ii}$"}
+{"_id": "1119", "title": "Matrix is Row Equivalent to Reduced Echelon Matrix", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be a matrix of order $m \\times n$ over a field $F$. Then $A$ is row equivalent to a reduced echelon matrix of order $m \\times n$."}
+{"_id": "1120", "title": "Square Matrix is Row Equivalent to Triangular Matrix", "text": "Let $\\mathbf A = \\left[{a}\\right]_n$ be a square matrix of order $n$ over a commutative ring $R$. Then $\\mathbf A$ can be converted to an upper or lower triangular matrix by elementary row operations."}
+{"_id": "1121", "title": "Product of Triangular Matrices", "text": "Let $\\mathbf A = \\sqbrk a_n, \\mathbf B = \\sqbrk b_n$ be upper triangular matrices of order $n$. Let $\\mathbf C = \\mathbf A \\mathbf B$. Then :$(1): \\quad$ the diagonal elements of $\\mathbf C$ are given by: ::::$\\forall j \\in \\closedint 1 n: c_{j j} = a_{j j} b_{j j}$ :::That is, the diagonal elements of $\\mathbf C$ are those of the factor matrices multiplied together. :$(2): \\quad$ The matrix $\\mathbf C$ is itself upper triangular. The same applies if both $\\mathbf A$ and $\\mathbf B$ are lower triangular matrices."}
+{"_id": "1122", "title": "Effect of Elementary Row Operations on Determinant", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $\\map \\det {\\mathbf A}$ denote the determinant of $\\mathbf A$. Take the elementary row operations: {{begin-axiom}} {{axiom | n = \\text {ERO} 1         | t = For some $\\lambda$, multiply row $i$ by $\\lambda$         | m = r_i \\to \\lambda r_i }} {{axiom | n = \\text {ERO} 2         | t = For some $\\lambda$, add $\\lambda$ times row $j$ to row $i$         | m = r_i \\to r_i + \\lambda r_j }} {{axiom | n = \\text {ERO} 3         | t = Exchange rows $i$ and $j$         | m = r_i \\leftrightarrow r_j }} {{end-axiom}} Applying $\\text {ERO} 1$ has the effect of multiplying $\\map \\det {\\mathbf A}$ by $\\lambda$. Applying $\\text {ERO} 2$ has no effect on $\\map \\det {\\mathbf A}$. Applying $\\text {ERO} 3$ has the effect of multiplying $\\map \\det {\\mathbf A}$ by $-1$."}
+{"_id": "1123", "title": "Modulus in Terms of Conjugate", "text": "Let $z = a + i b$ be a complex number. Let $\\cmod z$ be the modulus of $z$. Let $\\overline z$ be the conjugate of $z$. Then: :$\\cmod z^2 = z \\overline z$"}
+{"_id": "1124", "title": "Existence and Uniqueness of Positive Root of Positive Real Number", "text": "Let $x \\in \\R$ be a real number such that $x \\ge 0$. Let $n \\in \\Z$ be an integer such that $n \\ne 0$. Then there always exists a unique $y \\in \\R: \\paren {y \\ge 0} \\land \\paren {y^n = x}$. Hence the justification for the terminology '''the positive $n$th root of $x$''' and the notation $x^{1/n}$."}
+{"_id": "1125", "title": "Triangle Inequality", "text": "=== Geometry === {{:Triangle Inequality/Geometry}} === Real Numbers === {{:Triangle Inequality/Real Numbers}} === Complex Numbers === {{:Triangle Inequality/Complex Numbers}}"}
+{"_id": "1126", "title": "Even Power is Non-Negative", "text": "Let $x \\in \\R$ be a real number. Let $n \\in \\Z$ be an even integer. Then $x^n \\ge 0$. That is, all even powers are positive."}
+{"_id": "1127", "title": "Sign of Odd Power", "text": "Let $x \\in \\R$ be a real number. Let $n \\in \\Z$ be an odd integer. Then: :$x^n = 0 \\iff x = 0$ :$x^n > 0 \\iff x > 0$ :$x^n < 0 \\iff x < 0$ That is, the sign of an odd power matches the number it is a power of."}
+{"_id": "1128", "title": "Product of Absolute Values on Ordered Integral Domain", "text": "Let $\\struct {D, +, \\times, \\le}$ be an ordered integral domain whose zero is denoted by $0_D$. For all $a \\in D$, let $\\size a$ denote the absolute value of $a$. Then: :$\\size a \\times \\size b = \\size {a \\times b}$"}
+{"_id": "1129", "title": "Negative of Absolute Value", "text": "Let $x \\in \\R$ be a real number. Let $\\size x$ denote the absolute value of $x$. Then: :$-\\size x \\le x \\le \\size x$"}
+{"_id": "1130", "title": "Order of Squares in Ordered Ring", "text": "Let $\\struct {R, +, \\circ, \\le}$ be an ordered ring whose zero is $0_R$ and whose unity is $1_R$. Let $x, y \\in \\struct {R, +, \\circ, \\le}$ such that $0_R \\le x, y$. Then: :$x \\le y \\implies x \\circ x \\le y \\circ y$ When $R$ is one of the standard sets of numbers, that is $\\Z, \\Q, \\R$, then this translates into: :If $x, y$ are positive then $x \\le y \\implies x^2 \\le y^2$."}
+{"_id": "1131", "title": "Continuum Property", "text": "Let $S \\subset \\R$ be a non-empty subset of the set of real numbers such that $S$ is bounded above. Then $S$ admits a supremum in $\\R$. This is known as the '''least upper bound property''' of the real numbers. Similarly, let $S \\subset \\R$ be a non-empty subset of the set of real numbers such that $S$ is bounded below. Then $S$ admits an infimum in $\\R$. This is sometimes called the '''greatest lower bound property''' of the real numbers. The two properties taken together are called the '''continuum property of $\\R$'''. This can also be stated as: :The set $\\R$ of real numbers is Dedekind complete."}
+{"_id": "1132", "title": "Complex Multiplication is Closed", "text": "The set of complex numbers $\\C$ is closed under multiplication: :$\\forall z, w \\in \\C: z \\times w \\in \\C$"}
+{"_id": "1133", "title": "Complex Multiplication is Associative", "text": "The operation of multiplication on the set of complex numbers $\\C$ is associative: :$\\forall z_1, z_2, z_3 \\in \\C: z_1 \\paren {z_2 z_3} = \\paren {z_1 z_2} z_3$"}
+{"_id": "1134", "title": "Arithmetic Mean is Never Less than Harmonic Mean", "text": "Let $x_1, x_2, \\ldots, x_n \\in \\R_{> 0}$ be strictly positive real numbers. Let $A_n $ be the arithmetic mean of $x_1, x_2, \\ldots, x_n$. Let $H_n$ be the harmonic mean of $x_1, x_2, \\ldots, x_n$. Then $A_n \\ge H_n$."}
+{"_id": "1135", "title": "Multiple of Supremum", "text": "Let $S \\subseteq \\R: S \\ne \\varnothing$ be a non-empty subset of the set of real numbers $\\R$. Let $S$ be bounded above. Let $z \\in \\R: z > 0$ be a positive real number. Then: :$\\displaystyle \\map {\\sup_{x \\mathop \\in S} } {z x} = z \\map {\\sup_{x \\mathop \\in S} } x$"}
+{"_id": "1136", "title": "Cauchy's Mean Theorem", "text": "Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers which are all positive. Let $A_n$ be the arithmetic mean of $x_1, x_2, \\ldots, x_n$. Let $G_n$ be the geometric mean of $x_1, x_2, \\ldots, x_n$. Then: :$A_n \\ge G_n$ with equality holding {{iff}}: :$\\forall i, j \\in \\set {1, 2, \\ldots, n}: x_i = x_j$ That is, {{iff}} all terms are equal."}
+{"_id": "1137", "title": "Distance on Real Numbers is Metric", "text": "Let $x, y \\in \\R$ be real numbers. Let $\\map d {x, y}$ be the distance between $x$ and $y$: :$\\map d {x, y} = \\size {x - y}$ Then $\\map d {x, y}$ is a metric on $\\R$. Thus it follows that $\\tuple {\\R, d}$ is a metric space."}
+{"_id": "1138", "title": "Combination Theorem for Sequences/Real", "text": "Let $\\sequence {x_n}$ and $\\sequence {y_n}$ be sequences in $\\R$. Let $\\sequence {x_n}$ and $\\sequence {y_n}$ be convergent to the following limits: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ :$\\displaystyle \\lim_{n \\mathop \\to \\infty} y_n = m$ Let $\\lambda, \\mu \\in \\R$. Then the following results hold: === Sum Rule === {{:Combination Theorem for Sequences/Real/Sum Rule}} === Difference Rule === {{:Combination Theorem for Sequences/Real/Difference Rule}} === Multiple Rule === {{:Combination Theorem for Sequences/Real/Multiple Rule}} === Combined Sum Rule === {{:Combination Theorem for Sequences/Real/Combined Sum Rule}} === Product Rule === {{:Combination Theorem for Sequences/Real/Product Rule}} === Quotient Rule === {{:Combination Theorem for Sequences/Real/Quotient Rule}}"}
+{"_id": "1139", "title": "Convergent Sequence in Metric Space is Bounded", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $\\sequence {x_n}$ be a sequence in $M$ which is convergent, and so $x_n \\to l$ as $n \\to \\infty$. Then $\\sequence {x_n}$ is bounded."}
+{"_id": "1140", "title": "Convergent Sequence Minus Limit", "text": "Let $X$ be  one of the standard number fields $\\Q, \\R, \\C$. Let $\\left \\langle {x_n} \\right \\rangle$ be a sequence in $X$ which converges to $l$. That is: : $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ Then: : $\\displaystyle \\lim_{n \\mathop \\to \\infty} \\left|{x_n - l}\\right| = 0$"}
+{"_id": "1141", "title": "Monotone Convergence Theorem (Real Analysis)", "text": "Every bounded monotone sequence is convergent."}
+{"_id": "1142", "title": "One Plus Reciprocal to the Nth", "text": "Let $\\sequence {x_n}$ be the sequence in $\\R$ defined as $x_n = \\paren {1 + \\dfrac 1 n}^n$. Then $\\sequence {x_n}$ converges to a limit as $n$ increases without bound."}
+{"_id": "1143", "title": "Between two Real Numbers exists Rational Number", "text": "Let $a, b \\in \\R$ be real numbers such that $a < b$. Then: : $\\exists r \\in \\Q: a < r < b$"}
+{"_id": "1144", "title": "Power over Factorial", "text": "Let $x \\in \\R: x > 0$ be a positive real number. Let $\\sequence {x_n}$ be the sequence in $\\R$ defined as $x_n = \\dfrac {x^n} {n!}$. Then $\\sequence {x_n}$ converges to zero."}
+{"_id": "1145", "title": "Sequence of Powers of Reciprocals is Null Sequence", "text": "Let $r \\in \\Q_{>0}$ be a strictly positive rational number. Let $\\sequence {x_n}$ be the sequence in $\\R$ defined as: : $x_n = \\dfrac 1 {n^r}$ Then $\\sequence {x_n}$ is a null sequence."}
+{"_id": "1146", "title": "Euler's Number: Limit of Sequence implies Limit of Series", "text": "Let Euler's number $e$ be defined as: :$\\displaystyle e := \\lim_{n \\to \\infty} \\left({1 + \\frac 1 n}\\right)^n$ Then: :$\\displaystyle e = \\sum_{k \\mathop \\ge 0} \\frac 1 {k!}$ That is: :$\\displaystyle e = \\frac 1 {0!} + \\frac 1 {1!} + \\frac 1 {2!} + \\frac 1 {3!} + \\frac 1 {4!} \\cdots$"}
+{"_id": "1147", "title": "Degree of Field Extensions is Multiplicative", "text": "Let $E, K$ and $F$ be fields. Let $E / K$ and $K / F$ be finite field extensions. Then $E / F$ is a finite field extension, and: :$\\index E F = \\index E K \\index K F$ where $\\index E F$ denotes the degree of $E / F$"}
+{"_id": "1148", "title": "Convergent Sequence in Metric Space has Unique Limit", "text": "Let $\\left({X, d}\\right)$ be a metric space. Let $\\left \\langle {x_n} \\right \\rangle$ be a sequence in $\\left({X, d}\\right)$. Then $\\left \\langle {x_n} \\right \\rangle$ can have at most one limit."}
+{"_id": "1149", "title": "Lower and Upper Bounds for Sequences", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let $x_n \\to l$ as $n \\to \\infty$. Then: :$(1): \\quad \\forall n \\in \\N: x_n \\ge a \\implies l \\ge a$ :$(2): \\quad \\forall n \\in \\N: x_n \\le b \\implies l \\le b$"}
+{"_id": "1151", "title": "Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism", "text": "Let $\\struct {R, +_R, \\circ}$ and $\\struct {S, +_S, *}$ be rings whose zeros are $0_R$ and $0_S$ respectively. Let $\\phi: R \\to S$ be a ring homomorphism. If $R$ is a division ring, then either: : $(1): \\quad \\phi$ is a monomorphism (that is, $\\phi$ is injective) : $(2): \\quad \\phi$ is the zero homomorphism (that is, $\\forall a \\in R: \\map \\phi a = 0_S$)."}
+{"_id": "1152", "title": "Kernel of Ring Epimorphism is Ideal", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring epimorphism. Then: :The kernel of $\\phi$ is an ideal of $R_1$. :There is a unique ring isomorphism $g: R_1 / K \\to R_2$ such that: ::$g \\circ q_K = \\phi$ :$\\phi$ is a ring isomorphism {{iff}} $K = \\set {0_{R_1} }$."}
+{"_id": "1153", "title": "Unbounded Monotone Sequence Diverges to Infinity", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let $\\sequence {x_n}$ be monotone, that is either increasing or decreasing."}
+{"_id": "1154", "title": "Reciprocal of Null Sequence", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let $\\forall n \\in \\N: x_n > 0$. Then: :$x_n \\to 0$ as $n \\to \\infty$ {{iff}} $\\size {\\dfrac 1 {x_n} } \\to \\infty$ as $n \\to \\infty$"}
+{"_id": "1155", "title": "Modulus of Limit", "text": "Let $X$ be one of the standard number fields $\\Q, \\R, \\C$. Let $\\sequence {x_n}$ be a sequence in $X$. Let $\\sequence {x_n}$ be convergent to the limit $l$. That is, let $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$. Then :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\cmod {x_n} = \\cmod l$ where $\\cmod {x_n}$ is the modulus of $x_n$."}
+{"_id": "1157", "title": "Strictly Increasing Sequence of Natural Numbers", "text": "Let $\\N_{>0}$ be the set of natural numbers without zero: : $\\N_{>0} = \\left\\{{1, 2, 3, \\ldots}\\right\\}$ Let $\\left \\langle {n_r} \\right \\rangle$ be strictly increasing sequence in $\\N_{>0}$. Then: : $\\forall r \\in \\N_{>0}: n_r \\ge r$"}
+{"_id": "1158", "title": "Limit of Subsequence equals Limit of Sequence", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\sequence {x_n}$ be a sequence in $T$. Let $l \\in S$ such that: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ Let $\\sequence {x_{n_r} }$ be a subsequence of $\\sequence {x_n}$. Then: :$\\displaystyle \\lim_{r \\mathop \\to \\infty} x_{n_r} = l$ That is, the limit of a convergent sequence in a topological space equals the limit of any subsequence of it."}
+{"_id": "1159", "title": "Root of Number Greater than One", "text": "Let $x \\in \\R$ be a real number. Let $n \\in \\N^*$ be a natural number such that $n > 0$. Then $x \\ge 1 \\implies x^{1/n} \\ge 1$ where $x^{1/n}$ is the $n$th root of $x$."}
+{"_id": "1160", "title": "Limit of Root of Positive Real Number", "text": "Let $x \\in \\R: x > 0$ be a real number. Let $\\sequence {x_n}$ be the sequence in $\\R$ defined as: :$x_n = x^{1 / n}$ Then $x_n \\to 1$ as $n \\to \\infty$."}
+{"_id": "1161", "title": "Hero's Method", "text": "Let $a \\in \\R$ be a real number such that $a > 0$. Let $x_1 \\in \\R$ be a real number such that $x_1 > 0$. Let $\\sequence {x_n}$ be the sequence in $\\R$ defined recursively by: :$\\forall n \\in \\N_{>0}: x_{n + 1} = \\dfrac {x_n + \\dfrac a {x_n} } 2$ Then $x_n \\to \\sqrt a$ as $n \\to \\infty$."}
+{"_id": "1162", "title": "Limit of Integer to Reciprocal Power", "text": "Let $\\sequence {x_n}$ be the real sequence defined as $x_n = n^{1/n}$, using exponentiation. Then $\\sequence {x_n}$ converges with a limit of $1$."}
+{"_id": "1163", "title": "Difference Between Adjacent Square Roots Converges", "text": "Let $\\sequence {x_n}$ be the sequence in $\\R$ defined as $x_n = \\sqrt {n + 1} - \\sqrt n$. Then $\\sequence {x_n}$ converges to a zero limit."}
+{"_id": "1164", "title": "Peak Point Lemma", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$ which is infinite. Then $\\sequence {x_n}$ has an infinite subsequence which is monotone."}
+{"_id": "1165", "title": "Bolzano-Weierstrass Theorem", "text": "Every bounded sequence of real numbers has a convergent subsequence."}
+{"_id": "1166", "title": "Existence of Maximum and Minimum of Bounded Sequence", "text": "Let $\\sequence {x_n}$ be a bounded sequence in $\\R$ (which may or may not be convergent). Let $L$ be the set of all real numbers which are the limit of some subsequence of $\\sequence {x_n}$. Then $L$ has both a maximum and a minimum."}
+{"_id": "1170", "title": "Interval Defined by Absolute Value", "text": "Let $\\xi, \\delta \\in \\R$ be real numbers. Let $\\delta > 0$. Then:"}
+{"_id": "1171", "title": "Difference of Two Powers", "text": "Let $\\mathbb F$ denote one of the standard number systems, that is $\\Z$, $\\Q$, $\\R$ and $\\C$. Let $n \\in \\N$ such that $n \\ge 2$. Then for all $a, b \\in \\mathbb F$: {{begin-eqn}} {{eqn | l = a^n - b^n       | r = \\paren {a - b} \\sum_{j \\mathop = 0}^{n - 1} a^{n - j - 1} b^j       | c =  }} {{eqn | r = \\paren {a - b} \\paren {a^{n - 1} + a^{n - 2} b + a^{n - 3} b^2 + \\dotsb + a b^{n - 2} + b^{n - 1} }       | c =  }} {{end-eqn}}"}
+{"_id": "1172", "title": "Polynomial Factor Theorem", "text": "Let $P \\paren x$ be a polynomial in $x$ over a field $K$ of degree $n$. Then: :$\\xi \\in K: P \\paren \\xi = 0 \\iff \\map P x = \\paren {x - \\xi} \\map Q x$ where $Q$ is a polynomial of degree $n - 1$. Hence, if $\\xi_1, \\xi_2, \\ldots, \\xi_n \\in K$ such that all are different, and $\\map P {\\xi_1} = \\map P {\\xi_2} = \\dotsb = \\map P {\\xi_n} = 0$, then: :$\\displaystyle \\map P x = k \\prod_{j \\mathop = 1}^n \\paren {x - \\xi_j}$ where $k \\in K$."}
+{"_id": "1173", "title": "Telescoping Series/Example 1", "text": "Let $\\left \\langle {b_n} \\right \\rangle$ be a sequence in $\\R$. Let $\\left \\langle {a_n} \\right \\rangle$ be a sequence whose terms are defined as: :$a_k = b_k - b_{k + 1}$ Then: :$\\displaystyle \\sum_{k \\mathop = 1}^n a_k = b_1 - b_{n + 1}$"}
+{"_id": "1174", "title": "Terms in Convergent Series Converge to Zero", "text": "Let $\\sequence {a_n}$ be a sequence in any of the standard number fields $\\Q$, $\\R$, or $\\C$. Suppose that the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ converges in any of the standard number fields $\\Q$, $\\R$, or $\\C$. Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} a_n = 0$ {{expand|Expand (on a different page) to Banach spaces}}"}
+{"_id": "1175", "title": "Linear Combination of Convergent Series", "text": "Let $\\sequence {a_n}_{n \\mathop \\ge 1}$ and $\\sequence {b_n}_{n \\mathop \\ge 1}$ be sequences of real numbers. Let the two series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ and $\\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n$ converge to $\\alpha$ and $\\beta$ respectively. Let $\\lambda, \\mu \\in \\R$ be real numbers. Then the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\paren {\\lambda a_n + \\mu b_n}$ converges to $\\lambda \\alpha + \\mu \\beta$."}
+{"_id": "1176", "title": "Abel's Theorem", "text": "Let $\\displaystyle \\sum_{k \\mathop = 0}^\\infty a_k$ be a convergent series in $\\R$. Then: :$\\displaystyle \\lim_{x \\mathop \\to 1^-} \\paren {\\sum_{k \\mathop = 0}^\\infty a_k x^k} = \\sum_{k \\mathop = 0}^\\infty a_k$ where $\\displaystyle \\lim_{x \\mathop \\to 1^-}$ denotes the limit from the left."}
+{"_id": "1177", "title": "Alternating Series Test", "text": "Let $\\sequence {a_n}_{N \\mathop \\ge 0}$ be a decreasing sequence of positive terms in $\\R$ which converges with a limit of zero. That is, let $\\forall n \\in \\N: a_n \\ge 0, a_{n + 1} \\le a_n, a_n \\to 0$ as $n \\to \\infty$ Then the series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\paren {-1}^{n - 1} a_n = a_1 - a_2 + a_3 - a_4 + \\dotsb$  converges."}
+{"_id": "1178", "title": "Euler's Identity", "text": ":$e^{i \\pi} + 1 = 0$"}
+{"_id": "1179", "title": "Intermediate Value Theorem", "text": "Let $f: S \\to \\R$ be a real function on some subset $S$ of $\\R$. Let $I \\subseteq S$ be a real interval. Let $f: I \\to \\R$ be continuous on $I$. Let $a, b \\in I$. Let $k \\in \\R$ lie between $\\map f a$ and $\\map f b$. That is, either: :$\\map f a < k < \\map f b$ or: :$\\map f b < k < \\map f a$ Then $\\exists c \\in \\openint a b$ such that $\\map f c = k$."}
+{"_id": "1180", "title": "Tail of Convergent Series tends to Zero", "text": "Let $\\sequence {a_n}_{n \\mathop \\ge 1}$ be a sequence of real numbers. Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be a convergent series. Let $N \\in \\N_{\\ge 1}$ be a natural number. Let $\\displaystyle \\sum_{n \\mathop = N}^\\infty a_n$ be the tail of the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$. Then: :$\\displaystyle \\sum_{n \\mathop = N}^\\infty a_n$ is convergent :$\\displaystyle \\sum_{n \\mathop = N}^\\infty a_n \\to 0$ as $N \\to \\infty$. That is, the tail of a convergent series tends to zero."}
+{"_id": "1181", "title": "Comparison Test", "text": "Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n$ be a convergent series of positive real numbers. Let $\\sequence {a_n}$ be a sequence $\\R$ or sequence in $\\C$. Let $\\forall n \\in \\N_{>0}: \\cmod {a_n} \\le b_n$. Then the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ converges absolutely."}
+{"_id": "1182", "title": "Ratio Test", "text": "Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be a series of real numbers in $\\R$, or a series of complex numbers in $\\C$. Let the sequence $\\sequence {a_n}$ satisfy: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\size {\\frac {a_{n + 1} } {a_n} } = l$ :If $l > 1 $, then $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ diverges. :If $l < 1 $, then $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ converges absolutely."}
+{"_id": "1183", "title": "Limit of Subsequence of Bounded Sequence", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let $\\sequence {x_n}$ be bounded. Let $b \\in \\R$ be a real number. Suppose that $\\forall N: \\exists n > N: x_n \\ge b$. Then $\\sequence {x_n}$ has a subsequence which converges to a limit $l \\ge b$."}
+{"_id": "1184", "title": "Terms of Bounded Sequence Within Bounds", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let $\\sequence {x_n}$ be bounded. Let the limit superior of $\\sequence {x_n}$ be $\\overline l$. Let the limit inferior of $\\sequence {x_n}$ be $\\underline l$. Then: :$\\forall \\epsilon > 0: \\exists N: \\forall n > N: x_n < \\overline l + \\epsilon$ :$\\forall \\epsilon > 0: \\exists N: \\forall n > N: x_n > \\underline l - \\epsilon$"}
+{"_id": "1185", "title": "Convergence of Limsup and Liminf", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let the limit superior of $\\sequence {x_n}$ be $\\overline l$. Let the limit inferior of $\\sequence {x_n}$ be $\\underline l$. Then $\\left \\langle {x_n} \\right \\rangle$ converges to a limit $l$ {{iff}} $\\overline l = \\underline l = l$. Hence a bounded real sequence converges {{iff}} all its convergent subsequences have the same limit."}
+{"_id": "1186", "title": "Limsup and Liminf are Limits of Bounds", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let $\\sequence {x_n}$ be bounded. Let $\\displaystyle \\overline l = \\limsup_{n \\mathop \\to \\infty} x_n$ be the limit superior and $\\displaystyle \\liminf_{n \\mathop \\to \\infty} x_n$ the limit inferior of $\\sequence {x_n}$. Then: :$\\displaystyle \\overline l = \\limsup_{n \\mathop \\to \\infty} x_n = \\map {\\lim_{n \\mathop \\to \\infty} } {\\sup_{k \\mathop \\ge n} x_k}$ :$\\displaystyle \\underline l = \\liminf_{n \\mathop \\to \\infty} x_n = \\map {\\lim_{n \\mathop \\to \\infty} } {\\inf_{k \\mathop \\ge n} x_k}$"}
+{"_id": "1187", "title": "Nth Root Test", "text": "Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be a series of real numbers $\\R$ or complex numbers $\\C$. Let the sequence $\\sequence {a_n}$ be such that the limit superior $\\displaystyle \\limsup_{n \\mathop \\to \\infty} \\size {a_n}^{1/n} = l$. Then: :If $l > 1$, the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ diverges. :If $l < 1$, the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ converges absolutely."}
+{"_id": "1188", "title": "Series of Power over Factorial Converges", "text": "The series $\\displaystyle \\sum_{n \\mathop = 0}^\\infty \\frac {x^n} {n!}$ converges for all real values of $x$."}
+{"_id": "1189", "title": "Area of Triangle in Terms of Side and Altitude", "text": "The area of a triangle $\\triangle ABC$ is given by: :$\\dfrac {c \\cdot h_c} 2 = \\dfrac {b \\cdot h_b} 2 = \\dfrac {a \\cdot h_a} 2$ where: :$a, b, c$ are the sides :$h_a, h_b, h_c$ are the altitudes from $A$, $B$ and $C$ respectively."}
+{"_id": "1190", "title": "Absolutely Convergent Series is Convergent", "text": "Let $V$ be a Banach space with norm $\\norm {\\, \\cdot \\,}$. Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be an absolutely convergent series in $V$. Then $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ is convergent."}
+{"_id": "1191", "title": "Limit Comparison Test", "text": "Let $\\left \\langle {a_n} \\right \\rangle$ and $\\left \\langle {b_n} \\right \\rangle$ be sequences in $\\R$. Let $\\displaystyle \\frac {a_n}{b_n} \\to l$ as $n \\to \\infty$ where $l \\in \\R_{>0}$. Then the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ and $\\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n$ are either both convergent or both divergent."}
+{"_id": "1192", "title": "Area of Parallelogram", "text": "The area of a parallelogram equals the product of one of its bases and the associated altitude."}
+{"_id": "1193", "title": "Area of Triangle in Terms of Inradius and Exradii", "text": "The area of a $\\triangle ABC$ is given by the formula: :$(ABC) = \\rho_a \\left({s - a}\\right) = \\rho_b \\left({s - b}\\right) = \\rho_c \\left({s - c}\\right) = \\rho s = \\sqrt {\\rho_a \\rho_b \\rho_c \\rho}$ where: :$s$ is the semiperimeter :$I$ is the incenter :$\\rho$ is the inradius :$I_a, I_b, I_c$ are the excenters :$\\rho_a, \\rho_b, \\rho_c$ are the exradii from $I_a, I_b, I_c$, respectively."}
+{"_id": "1194", "title": "Area of Triangle in Terms of Circumradius", "text": "Let $\\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$. Then the area $\\AA$ of $\\triangle ABC$ is given by: :$\\AA = \\dfrac {a b c} {4 R}$ where $R$ is the circumradius of $\\triangle ABC$."}
+{"_id": "1195", "title": "Stewart's Theorem", "text": "Let $\\triangle ABC$ be a triangle with sides $a, b, c$. Let $CP$ be a cevian from $C$ to $P$. :400px Then: :$a^2 \\cdot AP + b^2 \\cdot PB = c \\paren {CP^2 + AP \\cdot PB}$"}
+{"_id": "1196", "title": "Length of Median of Triangle", "text": "Let $\\triangle ABC$ be a triangle. Let $CD$ be the median of $\\triangle ABC$ which bisects $AB$. :400px The length $m_c$ of $CD$ is given by: :${m_c}^2 = \\dfrac {a^2 + b^2} 2 - \\dfrac {c^2} 4$"}
+{"_id": "1198", "title": "Length of Angle Bisector", "text": "Let $\\triangle ABC$ be a triangle. Let $AD$ be the angle bisector of $\\angle BAC$ in $\\triangle ABC$. :300px Let $d$ be the length of $AD$. Then $d$ is given by: :$d^2 = \\dfrac {b c} {\\paren {b + c}^2} \\paren {\\paren {b + c}^2 - a^2}$ where $a$, $b$, and $c$ are the sides opposite $A$, $B$ and $C$ respectively."}
+{"_id": "1199", "title": "Supremum of Subset", "text": "Let $\\left({U, \\preceq}\\right)$ be an ordered set. Let $S \\subseteq U$. Let $T \\subseteq S$. Let $S$ admit a supremum (in $U$). If $T$ also admits a supremum (in $U$), then $\\sup \\left({T}\\right) \\preceq\\sup \\left({S}\\right)$."}
+{"_id": "1200", "title": "Area of Triangle", "text": "This page gathers a variety of formulas for the area of a triangle."}
+{"_id": "1201", "title": "Intersecting Chord Theorem", "text": "Let $AC$ and $BD$ both be chords of the same circle. Let $AC$ and $BD$ intersect at $E$. Then $AE \\cdot EC = DE \\cdot EB$. {{EuclidSaid}} :''If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.'' {{EuclidPropRef|III|35}}"}
+{"_id": "1202", "title": "Supremum Plus Constant", "text": "Let $S$ be a subset of the set of real numbers $\\R$. Let $S$ be bounded above. Let $\\xi \\in \\R$. Then: :$\\displaystyle \\map {\\sup_{x \\mathop \\in S} } {x + \\xi} = \\xi + \\map {\\sup_{x \\mathop \\in S} } x$ where $\\sup$ denotes supremum."}
+{"_id": "1203", "title": "Suprema and Infima of Combined Bounded Functions", "text": "Let $f$ and $g$ be real functions. Let $c$ be a constant."}
+{"_id": "1204", "title": "Multiple of Infimum", "text": "Let $T \\subseteq \\R: T \\ne \\varnothing$ be a non-empty subset of the set of real numbers $\\R$. Let $T$ be bounded below. Let $z \\in \\R: z > 0$ be a (strictly) positive real number. Then: :$\\displaystyle \\map {\\inf_{x \\mathop \\in T} } {z x} = z \\, \\map {\\inf_{x \\mathop \\in T} } x$ where $\\inf$ denotes infimum."}
+{"_id": "1205", "title": "Law of Sines", "text": "For any triangle $\\triangle ABC$: :$\\dfrac a {\\sin A} = \\dfrac b {\\sin B} = \\dfrac c {\\sin C} = 2 R$ where: :$a$, $b$, and $c$ are the sides opposite $A$, $B$ and $C$ respectively :$R$ is the circumradius of $\\triangle ABC$."}
+{"_id": "1207", "title": "Negative of Infimum is Supremum of Negatives", "text": "Let $T$ be a non-empty subset of the real numbers $\\R$. Let $T$ be bounded below. Then: : $(1): \\quad \\set {x \\in \\R: -x \\in T}$ is bounded above : $(2): \\quad \\displaystyle -\\inf_{x \\mathop \\in T} x = \\map {\\sup_{x \\mathop \\in T} } {-x}$ where $\\sup$ and $\\inf$ denote the supremum and infimum respectively."}
+{"_id": "1208", "title": "Infimum Plus Constant", "text": "Let $T$ be a subset of the set of real numbers. Let $T$ be bounded below. Let $\\xi \\in \\R$. Then: :$\\displaystyle \\map {\\inf_{x \\mathop \\in T} } {x + \\xi} = \\xi + \\map {\\inf_{x \\mathop \\in T} } x$ where $\\inf$ denotes infimum."}
+{"_id": "1209", "title": "Infimum of Subset", "text": "Let $\\struct {U, \\preceq}$ be an ordered set. Let $S \\subseteq U$. Let $T \\subseteq S$. Let $\\struct {S, \\preceq}$ admit an infimum in $U$. If $T$ also admits an infimum in $U$, then $\\map \\inf S \\preceq \\map \\inf T$."}
+{"_id": "1210", "title": "Construction of Equilateral Triangle", "text": "On a given straight line segment, it is possible to construct an equilateral triangle. {{:Euclid:Proposition/I/1}}"}
+{"_id": "1211", "title": "Construction of Equal Straight Line", "text": "At a given point, it is possible to construct a straight line segment of length equal to that of any given straight line segment. The given point will be an endpoint of the constructed straight line segment. {{:Euclid:Proposition/I/2}}"}
+{"_id": "1212", "title": "Construction of Equal Straight Lines from Unequal", "text": "Given two unequal straight line segments, it is possible to cut off from the greater a straight line segment equal to the lesser. {{:Euclid:Proposition/I/3}}"}
+{"_id": "1213", "title": "Triangle Side-Angle-Side Equality", "text": "If $2$ triangles have: : $2$ sides equal to $2$ sides respectively : the angles contained by the equal straight lines equal they will also have: : their third sides equal : the remaining two angles equal to their respective remaining angles, namely, those which the equal sides subtend."}
+{"_id": "1214", "title": "Isosceles Triangle has Two Equal Angles", "text": "In isosceles triangles, the angles at the base are equal to each other. Also, if the equal straight lines are extended, the angles under the base will also be equal to each other. {{:Euclid:Proposition/I/5}}"}
+{"_id": "1215", "title": "Triangle with Two Equal Angles is Isosceles", "text": "If a triangle has two angles equal to each other, the sides which subtend the equal angles will also be equal to one another. Hence, by definition, such a triangle will be isosceles. {{:Euclid:Proposition/I/6}}"}
+{"_id": "1216", "title": "Two Lines Meet at Unique Point", "text": "Let two straight line segments be constructed on a straight line segment from its endpoints so that they meet at a point. Then there cannot be two other straight line segments equal to the former two respectively, constructed on the same straight line segment and on the same side of it, meeting at a different point. {{:Euclid:Proposition/I/7}}"}
+{"_id": "1217", "title": "Triangle Side-Side-Side Equality", "text": "Let two triangles have all $3$ sides equal. Then they also have all $3$ angles equal. Thus two triangles whose sides are all equal are themselves congruent."}
+{"_id": "1218", "title": "Bisection of Angle", "text": "It is possible to bisect any given rectilineal angle. {{:Euclid:Proposition/I/9}}"}
+{"_id": "1219", "title": "Bisection of Straight Line", "text": "It is possible to bisect a straight line segment. {{:Euclid:Proposition/I/10}}"}
+{"_id": "1225", "title": "External Angle of Triangle Greater than Internal Opposite", "text": "The external angle of a triangle is greater than either of the opposite internal angles. {{:Euclid:Proposition/I/16}}"}
+{"_id": "1229", "title": "Limit iff Limits from Left and Right", "text": "Let $f$ be a real function defined on an open interval $\\openint a b$ except possibly at a point $c \\in \\openint a b$. Then: :$\\map f x \\to l$ as $x \\to c$ {{iff}}: :$\\map f x \\to l$ as $x \\to c^-$ and :$\\map f x \\to l$ as $x \\to c^+$"}
+{"_id": "1230", "title": "Limit of Function by Convergent Sequences", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $S \\subseteq A_1$ be an open set of $M_1$. Let $f$ be a mapping defined on $S$, except possibly at the point $c \\in S$. Then $\\displaystyle \\lim_{x \\mathop \\to c} f \\left({x}\\right) = l$ iff: : for each sequence $\\left \\langle {x_n} \\right \\rangle$ of points of $S$ such that $\\forall n \\in \\N_{>0}: x_n \\ne c$ and $\\displaystyle \\lim_{n \\to \\infty} x_n = c$ it is true that: : $\\displaystyle \\lim_{n \\to \\infty} f \\left({x_n}\\right) = l$"}
+{"_id": "1231", "title": "Combination Theorem for Limits of Functions", "text": "Let $X$ be one of the standard number fields $\\Q, \\R, \\C$. Let $f$ and $g$  be functions defined on an open subset $S \\subseteq X$, except possibly at the point $c \\in S$. Let $f$ and $g$ tend to the following limits: :$\\displaystyle \\lim_{x \\to c} \\ f \\left({x}\\right) = l$ :$\\displaystyle \\lim_{x \\to c} \\ g \\left({x}\\right) = m$ Let $\\lambda, \\mu \\in X$ be arbitrary numbers in $X$. Then the following results hold:"}
+{"_id": "1232", "title": "Real Polynomial Function is Continuous", "text": "A (real) polynomial function is continuous at every point. Thus a (real) polynomial function is continuous on every interval of $\\R$."}
+{"_id": "1233", "title": "Linear Function is Continuous", "text": "Let $\\alpha, \\beta \\in \\R$ be real numbers. Let $f : \\R \\to \\R$ be the real function with: :$\\map f x = \\alpha x + \\beta$ for all $x \\in \\R$. Then $f$ is continuous at every real number $c \\in \\R$."}
+{"_id": "1234", "title": "Real Rational Function is Continuous", "text": "A real rational function is continuous at every point at which it is defined. Thus a real rational function is continuous on every interval of $\\R$ not containing a root of the denominator of the function."}
+{"_id": "1235", "title": "Limit of Composite Function", "text": "Let $f$ and $g$ be real functions. Let: :$\\displaystyle \\lim_{y \\mathop \\to \\eta} \\map f y = l$ :$\\displaystyle \\lim_{x \\mathop \\to \\xi} \\map g x = \\eta$ Then, if either: :'''Hypothesis 1:''' $f$ is continuous at $\\eta$ (that is $l = \\map f \\eta$) or: :'''Hypothesis 2:''' for some open interval $I$ containing $\\xi$, it is true that $\\map g x \\ne \\eta$ for any $x \\in I$ except possibly $x = \\xi$ then: : $\\displaystyle \\lim_{x \\mathop \\to \\xi} \\map f {\\map g x} = l$"}
+{"_id": "1237", "title": "Limit of Image of Sequence", "text": "Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \\to A_2$ be a mapping which is continuous at $a \\in A_1$. Let $\\sequence {x_n}$ be a sequence of points in $A_1$ such that: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = a$ where $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n$ is the limit of $x_n$. Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map f {x_n} = \\map f a$ That is: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map f {x_n} = \\map f {\\lim_{n \\mathop \\to \\infty} x_n}$ That is, for a continuous mapping, the limit and function symbols commute."}
+{"_id": "1238", "title": "Image of Interval by Continuous Function is Interval", "text": "Let $I$ be a real interval. Let $f: I \\to \\R$ be a continuous real function. Then the image of $f$ is a real interval."}
+{"_id": "1239", "title": "Interval Divided into Subsets", "text": "Let $\\mathbb I$ be a real interval. Let $S$ and $T$ be non-empty subsets of $\\mathbb I$ such that $\\mathbb I \\subseteq S \\cup T$. Then one of $S$ or $T$ contains an element at zero distance from the other."}
+{"_id": "1240", "title": "Distance from Subset of Real Numbers", "text": "Let $S$ be a subset of the set of real numbers $\\R$. Let $x \\in \\R$ be a real number. Let $\\map d {x, S}$ be the distance between $x$ and $S$. Then:"}
+{"_id": "1241", "title": "Limit of Sequence to Zero Distance Point", "text": "Let $S$ be a non-empty subset of $\\R$. Let the distance $\\map d {\\xi, S} = 0$ for some $\\xi \\in \\R$. Then there exists a sequence $\\sequence {x_n}$ in $S$ such that $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = \\xi$."}
+{"_id": "1242", "title": "Image of Closed Real Interval is Bounded", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$. Then $f$ is bounded on $\\closedint a b$."}
+{"_id": "1243", "title": "Convergent Subsequence in Closed Interval", "text": "Let $\\closedint a b$ be a closed real interval. Then every sequence of points of $\\closedint a b$ contains a subsequence which converges to a point in $\\closedint a b$."}
+{"_id": "1244", "title": "Max and Min of Function on Closed Real Interval", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$. Then $f$ reaches a maximum and a minimum on $\\closedint a b$."}
+{"_id": "1245", "title": "Continuous Image of Closed Interval is Closed Interval", "text": "Let $f$ be a real function which is continuous on the closed interval $\\left[{a \\,.\\,.\\, b}\\right]$. Then the image of $\\left[{a \\,.\\,.\\, b}\\right]$ under $f$ is also a closed interval."}
+{"_id": "1247", "title": "Retraction Theorem", "text": "Let $M$ be a compact manifold with boundary $\\partial M$. Then there is no smooth mapping $f: M \\to \\partial M$ such that $\\partial f: \\partial M \\to \\partial M$ is the identity."}
+{"_id": "1248", "title": "Classification of Compact One-Manifolds", "text": "Every compact one-dimensional manifold is diffeomorphic to either a circle or a closed interval."}
+{"_id": "1249", "title": "Differentiable Function is Continuous", "text": "Let $f$ be a real function defined on an interval $I$. Let $x_0 \\in I$ such that $f$ is differentiable at $x_0$. Then $f$ is continuous at $x_0$."}
+{"_id": "1250", "title": "Quotient Rule for Derivatives", "text": "Let $\\map j x, \\map k x$ be real functions defined on the open interval $I$. Let $\\xi \\in I$ be a point in $I$ at which both $j$ and $k$ are differentiable. Define the real function $f$ on $I$ by: :$\\displaystyle \\map f x = \\begin{cases} \\dfrac {\\map j x} {\\map k x} & : \\map k x \\ne 0 \\\\ 0 & : \\text{otherwise} \\end{cases}$ Then, if $\\map k \\xi \\ne 0$, $f$ is differentiable at $\\xi$, and furthermore: :$\\map {f'} \\xi = \\dfrac {\\map {j'} \\xi \\map k \\xi - \\map j \\xi \\map {k'} \\xi} {\\paren {\\map k \\xi}^2}$ It follows from the definition of derivative that if $j$ and $k$ are both differentiable on the interval $I$, then: :$\\displaystyle \\forall x \\in I: \\map k x \\ne 0 \\implies \\map {f'} x = \\frac {\\map {j'} x \\map k x - \\map j x \\map {k'} x} {\\paren {\\map k x}^2}$"}
+{"_id": "1252", "title": "Preimage Theorem", "text": "Let $y$ be a regular value of a smooth submersion $f:X \\to Y$. Then the preimage $f^{-1}(y)$ is a smooth submanifold of $X$, with $\\dim f^{-1}(y) = \\dim X - \\dim Y$."}
+{"_id": "1255", "title": "Backwards Induction", "text": "Let $P$ be a propositional function on the natural numbers $\\N$. Suppose that: :$(1): \\quad \\forall n \\in \\N: \\map P {2^n}$ holds. :$(2): \\quad \\map P n \\implies \\map P {n - 1}$. Then $\\map P n$ holds for all $\\forall n \\in \\N$. The proof technique based on this result is called '''backwards induction'''."}
+{"_id": "1256", "title": "Inequalities Concerning Roots", "text": "Let $\\closedint X Y$ be a closed real interval such that $0 < X \\le Y$. Let $x, y \\in \\closedint X Y$. Then: :$\\forall n \\in \\N_{> 0}: X Y^{1/n} \\size {x - y} \\le n X Y \\size {x^{1/n} - y^{1/n} } \\le Y X^{1/n} \\size {x - y}$"}
+{"_id": "1257", "title": "Continuity of Root Function", "text": "Let $n \\in \\N_{>0}$ be a non-zero natural number. Let $f: \\hointr 0 \\infty \\to \\R$ be the real function defined by $\\map f x = x^{1/n}$. Then $f$ is continuous at each $\\xi > 0$ and continuous on the right at $\\xi = 0$."}
+{"_id": "1259", "title": "P-adic Norm is Norm", "text": "The $p$-adic norm forms a norm on the rational numbers $\\Q$."}
+{"_id": "1260", "title": "Fundamental Theorem of Algebra", "text": "Every non-constant polynomial with coefficients in $\\C$ has a root in $\\C$."}
+{"_id": "1261", "title": "Extendability Theorem for Intersection Numbers", "text": "Let $X = \\partial W$ be a smooth manifold which is the boundary of a smooth compact manifold $W$. Let $Y$ be a smooth manifold, $Z$ be a closed smooth submanifold of $Y$, and $f: X \\to Y$ a smooth map. Let there exist a smooth map $g: W \\to Y$ such that $g \\restriction_X = f$. Then: :$I \\left({f, Z}\\right) = 0$ where $I \\left({f, Z}\\right)$ is the intersection number. {{explain|what $I \\left({f, Z}\\right)$ is the intersection number of: presumably the words will go something like \"... the intersection number of $f$ with respect to $Z$\", or something.}}"}
+{"_id": "1264", "title": "Homotopy Group is Group", "text": "The set of all homotopy classes of continuous mappings: :$c: \\closedint 0 1^n \\to X$ satisfying: :$\\map c {\\partial \\closedint 0 1^n} = x_0$ in a space $X$ at a base point $x_0$, under the operation of concatenation on class members, forms a group. {{finish|This operation needs to be shown well-defined; probably a nasty proof}} This group is called the $n$th homotopy group."}
+{"_id": "1265", "title": "Derivative at Maximum or Minimum", "text": "Let $f$ be a real function which is differentiable on the open interval $\\openint a b$. Let $f$ have a local minimum or local maximum at $\\xi \\in \\openint a b$. Then: :$\\map {f'} \\xi = 0$"}
+{"_id": "1266", "title": "Behaviour of Function Near Limit", "text": "Let $f$ be a real function. Let $f \\left({x}\\right) \\to l$ as $x \\to \\xi$. Then: * If $l > 0$, then $\\exists h > 0: \\forall x: \\xi - h < x < \\xi + h, x \\ne \\xi: f \\left({x}\\right) > 0$ * If $l < 0$, then $\\exists h > 0: \\forall x: \\xi - h < x < \\xi + h, x \\ne \\xi: f \\left({x}\\right) < 0$"}
+{"_id": "1267", "title": "Fundamental Group is Independent of Base Point for Path-Connected Space", "text": "Let $X$ be a path-connected space. For $x \\in X$ let $\\pi_1(X,x)$ denote the fundamental group. For $x, y \\in X$, there is an isomorphism: :$\\phi: \\pi_1 \\left({X, x}\\right) \\to \\pi_1 \\left({X, y}\\right)$"}
+{"_id": "1268", "title": "List of Fundamental Groups for 2-Manifolds", "text": "For the following two-manifolds, the fundamental group for any point in $X$, written $\\pi_1 \\left({X}\\right)$ is isomorphic to the listed group: :$\\pi_1 \\left({\\Bbb S^1 \\times \\left[{0\\,.\\,.\\,1}\\right]}\\right) = \\Z$ :$\\pi_1 \\left({\\Bbb S^2}\\right) = \\left\\{{e}\\right\\}$, the trivial group. :$\\pi_1 \\left({\\Bbb T^2}\\right) = \\pi_1 \\left({\\Bbb S^1 \\times \\Bbb S^1}\\right) = \\Z \\times \\Z$ :$\\pi_1 \\left({\\Bbb {RP}^2}\\right) = \\Z_2$"}
+{"_id": "1269", "title": "Rolle's Theorem", "text": "Let $f$ be a real function which is: :continuous on the closed interval $\\closedint a b$ and: :differentiable on the open interval $\\openint a b$. Let $\\map f a = \\map f b$. Then: :$\\exists \\xi \\in \\openint a b: \\map {f'} \\xi = 0$"}
+{"_id": "1270", "title": "Mean Value Theorem", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$ and differentiable on the open interval $\\openint a b$. Then: :$\\exists \\xi \\in \\openint a b: \\map {f'} \\xi = \\dfrac {\\map f b - \\map f a} {b - a}$"}
+{"_id": "1271", "title": "Zero Derivative implies Constant Function", "text": "Let $f$ be a real function which is continuous on the closed interval $\\left[{a \\,.\\,.\\, b}\\right]$ and differentiable on the open interval $\\left({a \\,.\\,.\\, b}\\right)$. Suppose that: :$\\forall x \\in \\left({a \\,.\\,.\\, b}\\right): f' \\left({x}\\right) = 0$ Then $f$ is constant on $\\left[{a \\,.\\,.\\, b}\\right]$."}
+{"_id": "1272", "title": "Ostrowski's Theorem", "text": "Every non-trivial norm on the rational numbers $\\Q$ is equivalent to either: :the $p$-adic norm $\\norm {\\, \\cdot \\,}_p$ for some prime $p$ or: :the absolute value, $\\size {\\, \\cdot \\,}$."}
+{"_id": "1273", "title": "Cauchy Mean Value Theorem", "text": "Let $f$ and $g$ be a real functions which are continuous on the closed interval $\\left[{a \\,.\\,.\\, b}\\right]$ and differentiable on the open interval $\\left({a \\,.\\,.\\, b}\\right)$. Suppose that: :$\\forall x \\in \\left({a \\,.\\,.\\, b}\\right): g' \\left({x}\\right) \\ne 0$ Then: :$\\exists \\xi \\in \\left({a \\,.\\,.\\, b}\\right): \\dfrac {f' \\left({\\xi}\\right)} {g' \\left({\\xi}\\right)} = \\dfrac {f \\left({b}\\right) - f \\left({a}\\right)} {g \\left({b}\\right) - g \\left({a}\\right)}$"}
+{"_id": "1274", "title": "Homology Group is Group", "text": "The $p^{th}$ singular homology group of a space $X$ is a group."}
+{"_id": "1275", "title": "Limit of Absolute Value", "text": "Let $x, \\xi \\in \\R$ be real numbers. Then: :$\\left\\vert{x - \\xi}\\right\\vert \\to 0$ as $x \\to \\xi$ where $\\left\\vert{x - \\xi}\\right\\vert$ denotes the Absolute Value."}
+{"_id": "1276", "title": "Limit of Function in Interval", "text": "Let $f$ be a real function which is defined on the open interval $\\openint a b$. Let $\\xi \\in \\openint a b$ Suppose that, $\\forall x \\in \\openint a b$, either: :$\\xi \\le \\map f x \\le x$ or: :$x \\le \\map f x \\le \\xi$ Then $\\map f x \\to \\xi$ as $x \\to \\xi$."}
+{"_id": "1278", "title": "Derivative of Monotone Function", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$ and differentiable on the open interval $\\openint a b$."}
+{"_id": "1279", "title": "Strictly Monotone Real Function is Bijective", "text": "Let $f$ be a real function which is defined on $I \\subseteq \\R$. Let $f$ be strictly monotone on $I$. Let the image of $f$ be $J$.  Then $f: I \\to J$ is a bijection."}
+{"_id": "1280", "title": "Inverse of Strictly Monotone Function", "text": "Let $f$ be a real function which is defined on $I \\subseteq \\R$. Let $f$ be strictly monotone on $I$. Let the image of $f$ be $J$.  Then $f$ always has an inverse function $f^{-1}$ and: : if $f$ is strictly increasing then so is $f^{-1}$ : if $f$ is strictly decreasing then so is $f^{-1}$."}
+{"_id": "1281", "title": "Rokhlin's Theorem (Intersection Forms)", "text": "Let $M$ be a smooth 4-manifold. Then: :$\\map {\\omega_2} {\\map T M} = 0 \\implies \\operatorname {sign} Q_M = 0 \\pmod {16}$ where: :$Q_M$ is the intersection form :$\\map T M$ is the tangent bundle :$\\omega_2$ is the second Stiefel-Whitney class."}
+{"_id": "1283", "title": "Convex Real Function is Continuous", "text": "Let $f$ be a real function which is convex on the open interval $\\left({a \\,.\\,.\\, b}\\right)$. Then $f$ is continuous on $\\left({a \\,.\\,.\\, b}\\right)$."}
+{"_id": "1284", "title": "Convex Real Function is Left-Hand and Right-Hand Differentiable", "text": "Let $f$ be a real function which is either convex on the open interval $\\left({a \\,.\\,.\\, b}\\right)$. Then the left-hand derivative $f'_- \\left({x}\\right)$ and right-hand derivative $f'_+ \\left({x}\\right)$ both exist for all $x \\in \\left({a \\,.\\,.\\, b}\\right)$."}
+{"_id": "1285", "title": "Real Function is Convex iff Derivative is Increasing", "text": "Let $f$ be a real function which is differentiable on the open interval $\\openint a b$. Then: :$f$ is convex on $\\openint a b$ {{iff}}: :its derivative $f'$ is increasing on $\\openint a b$. Thus the intuitive result that a convex function \"gets steeper\"."}
+{"_id": "1286", "title": "Inverse of Strictly Increasing Convex Real Function is Concave", "text": "Let $f$ be a real function which is convex on the open interval $I$. Let $J = f \\left[{I}\\right]$. If $f$ be strictly increasing on $I$, then $f^{-1}$ is concave on $J$."}
+{"_id": "1287", "title": "Upper and Lower Bounds of Integral", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$. Let $\\displaystyle \\int_a^b \\map f x \\rd x$ be the definite integral of $\\map f x$ over $\\closedint a b$. Then: :$\\displaystyle m \\paren {b - a} \\le \\int_a^b \\map f x \\rd x \\le M \\paren {b - a}$ where: :$M$ is the maximum of $f$ :$m$ is the minimum of $f$ on $\\closedint a b$."}
+{"_id": "1288", "title": "Wilson's Theorem", "text": "A (strictly) positive integer $p$ is a prime {{iff}}: :$\\paren {p - 1}! \\equiv -1 \\pmod p$"}
+{"_id": "1289", "title": "Integral of Constant", "text": "Let $c$ be a constant."}
+{"_id": "1290", "title": "Sum of Integrals on Adjacent Intervals for Continuous Functions", "text": "Let $f$ be a real function which is continuous on any closed interval $I$. Let $a, b, c \\in I$. Then: :$\\ds \\int_a^c \\map f t \\rd t + \\int_c^b \\map f t \\rd t = \\int_a^b \\map f t \\rd t$"}
+{"_id": "1291", "title": "Primitives which Differ by Constant", "text": "Let $F$ be a primitive for a real function $f$ on the closed interval $\\closedint a b$. Let $G$ be a real function defined on $\\closedint a b$. Then $G$ is a primitive for $f$ on $\\closedint a b$ {{iff}}: :$\\exists c \\in \\R: \\forall x \\in \\closedint a b: \\map G x = \\map F x + c$ That is, {{iff}} $F$ and $G$ differ by a constant on the whole interval."}
+{"_id": "1293", "title": "Definite Integral of Function plus Constant", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$. Let $c$ be a constant. Then: :$\\ds \\int_a^b \\paren {\\map f t + c} \\rd t = \\int_a^b \\map f t \\rd t + c \\paren {b - a}$"}
+{"_id": "1294", "title": "Continuous Real Function is Darboux Integrable", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$. Then $f$ is Darboux integrable on $\\closedint a b$."}
+{"_id": "1295", "title": "Definite Integral on Zero Interval", "text": "Let $f$ be a real function which is defined on the closed interval $\\Bbb I := \\closedint a b$, where $a < b$. Then: :$\\displaystyle \\forall c \\in \\Bbb I: \\int_c^c \\map f t \\rd t = 0$"}
+{"_id": "1296", "title": "Linear Combination of Derivatives", "text": "Let $f \\left({x}\\right), g \\left({x}\\right)$ be real functions defined on the open interval $I$. Let $\\xi \\in I$ be a point in $I$ at which both $f$ and $g$ are differentiable. Then: :$D \\left({\\lambda f + \\mu g}\\right) = \\lambda D f + \\mu D g$ at the point $\\xi$. It follows from the definition of derivative that if $f$ and $g$ are both differentiable on the interval $I$, then: :$\\forall x \\in I: D \\left({\\lambda f \\left({x}\\right) + \\mu g \\left({x}\\right)}\\right) = \\lambda D f \\left({x}\\right) + \\mu D g \\left({x}\\right)$"}
+{"_id": "1297", "title": "Linear Combination of Integrals", "text": "Let $f$ and $g$ be real functions which are integrable on the closed interval $\\closedint a b$. Let $\\lambda$ and $\\mu$ be real numbers. Then the following results hold:"}
+{"_id": "1298", "title": "Integration by Parts", "text": "Let $f$ and $g$ be real functions which are continuous on the closed interval $\\closedint a b$. Let $f$ and $g$ have primitives $F$ and $G$ respectively on $\\closedint a b$. Then:"}
+{"_id": "1299", "title": "Integration by Substitution", "text": "Let $\\phi$ be a real function which has a derivative on the closed interval $\\closedint a b$. Let $I$ be an open interval which contains the image of $\\closedint a b$ under $\\phi$. Let $f$ be a real function which is continuous on $I$."}
+{"_id": "1300", "title": "Relative Sizes of Definite Integrals", "text": "Let $f$ and $g$ be real functions which are continuous on the closed interval $\\closedint a b$, where $a < b$. If: :$\\forall t \\in \\closedint a b: \\map f t \\le \\map g t$ then: :$\\displaystyle \\int_a^b \\map f t \\rd t \\le \\int_a^b \\map g t \\rd t$ Similarly, if: :$\\forall t \\in \\closedint a b: \\map f t < \\map g t$ then: :$\\displaystyle \\int_a^b \\map f t \\rd t < \\int_a^b \\map g t \\rd t$"}
+{"_id": "1301", "title": "Absolute Value of Definite Integral", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$. Then: :$\\displaystyle \\size {\\int_a^b \\map f t \\rd t} \\le \\int_a^b \\size {\\map f t} \\rd t$"}
+{"_id": "1302", "title": "Integral Test", "text": "Let $f$ be a real function which is continuous, positive and decreasing on the interval $\\hointr 1 {+\\infty}$. Let the sequence $\\sequence {\\Delta_n}$ be defined as: :$\\displaystyle \\Delta_n = \\sum_{k \\mathop = 1}^n \\map f k - \\int_1^n \\map f x \\rd x$ Then $\\sequence {\\Delta_n} $ is decreasing and bounded below by zero. Hence it converges."}
+{"_id": "1304", "title": "Properties of Natural Logarithm", "text": "Let $x \\in \\R$ be a real number such that $x > 0$. Let $\\ln x$ be the natural logarithm of $x$. Then:"}
+{"_id": "1306", "title": "Harmonic Series is Divergent", "text": "The harmonic series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\frac 1 n$  diverges."}
+{"_id": "1307", "title": "Sum of Reciprocals of Primes is Divergent", "text": "Let $n \\in \\N: n \\ge 1$. There exists a (strictly) positive real number $C \\in \\R_{>0}$ such that: :$(1): \\quad \\displaystyle \\sum_{\\substack {p \\mathop \\in \\Bbb P \\\\ p \\mathop \\le n} } \\frac 1 p > \\map \\ln {\\ln n} - C$ where $\\Bbb P$ is the set of all prime numbers. :$(2): \\quad \\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {\\map \\ln {\\ln n} - C} = +\\infty$"}
+{"_id": "1308", "title": "Measure of Interval is Length", "text": "Let $I$ be a real interval whose endpoints are $a$ and $b$. Then $I$ is Lebesgue measurable, and the value of the measure is the length of the interval $b - a$."}
+{"_id": "1309", "title": "Measurable Sets form Algebra of Sets", "text": "Let $\\mu^*$ be an outer measure on a set $X$. Then the set of $\\mu^*$-measurable sets is an algebra of sets."}
+{"_id": "1310", "title": "Classification of Compact Three-Manifolds Supporting Zero-Curvature Geometry", "text": "Every closed, orientable, path connected $3$-dimensional Riemannian manifold which supports a geometry of zero curvature is homeomorphic to one of the following: * Torus $\\mathbb T^3$ * Half-Twist Cube * Quarter-Twist Cube * Hantschze-Wendt Manifold * $\\frac 1 6$-Twist Hexagonal Prism * $\\frac 1 3$-Twist Hexagonal Prism The $3$-torus is described on the torus page. The other manifolds can be described using quotient spaces on familiar prisms, with the equivalence relations described below. <gallery perRow=\"5\"> File:Halftwistcube.JPG|The Half-Twist Cube File:Hantschzewendt.JPG|The Hantschze-Wendt Manifold File:Quartertwistcube.JPG|The Quarter-Twist Cube File:Sixthtwisthexagon.JPG|The $\\frac 1 6$-Twist Hexagonal Prism File:Thirdtwisthexagon.JPG|The $\\frac 1 3$-Twist Hexagonal Prism </gallery>"}
+{"_id": "1311", "title": "Sum of Logarithms", "text": "{{:Sum of Logarithms/General Logarithm}}"}
+{"_id": "1312", "title": "Heine-Borel Theorem/Real Line", "text": "Let $\\R$ be the real number line considered as a Euclidean space. Let $C \\subseteq \\R$. Then $C$ is closed and bounded in $\\R$ {{iff}} $C$ is compact."}
+{"_id": "1314", "title": "Measurable Image", "text": "Let $\\mathfrak M$ be the set of measurable sets of $\\R$. For any extended real-valued function $f: \\R \\to \\R \\cup \\left\\{{-\\infty \\,.\\,.\\, +\\infty}\\right\\}$ whose domain is measurable, the following statements are equivalent: : $(1): \\quad \\forall \\alpha \\in \\R: \\left\\{{x: f \\left({x}\\right) > \\alpha}\\right\\} \\in \\mathfrak M$ : $(2): \\quad \\forall \\alpha \\in \\R: \\left\\{{x: f \\left({x}\\right) \\ge \\alpha}\\right\\} \\in \\mathfrak M$ : $(3): \\quad \\forall \\alpha \\in \\R: \\left\\{{x: f \\left({x}\\right) < \\alpha}\\right\\} \\in \\mathfrak M$ : $(4): \\quad \\forall \\alpha \\in \\R: \\left\\{{x: f \\left({x}\\right) \\le \\alpha}\\right\\} \\in \\mathfrak M$ These statements imply: : $(5): \\quad \\forall \\alpha \\in \\R \\cup \\left\\{{-\\infty \\,.\\,.\\, +\\infty}\\right\\}: \\left\\{{x:f \\left({x}\\right) = \\alpha}\\right\\} \\in \\mathfrak M$ {{refactor}} {{proofread}}"}
+{"_id": "1315", "title": "Lebesgue Integral is Extension of Darboux Integral", "text": "Let $f: \\closedint a b \\to \\R$ be a Darboux integrable function. Then it is also Lebesgue integrable, and furthermore: :$\\displaystyle R \\int_a^b \\map f x \\rd x = \\int_{\\closedint a b} f \\rd \\lambda$ where $\\displaystyle R \\int_a^b$ is the Darboux integral and $\\displaystyle \\int_{\\closedint a b}$ is the Lebesgue integral."}
+{"_id": "1316", "title": "Properties of Exponential Function", "text": "Let $x \\in \\R$ be a real number. Let $\\exp x$ be the exponential of $x$. Then:"}
+{"_id": "1317", "title": "Euclidean Metric on Real Vector Space is Metric", "text": "The Euclidean metric on a real vector space $\\R^n$ is a metric."}
+{"_id": "1318", "title": "Metric Induces Topology", "text": "Let $M = \\struct {A, d}$ be a metric space. Then the topology $\\tau$ induced by the metric $d$ is a topology on $M$."}
+{"_id": "1319", "title": "Properties of Algebras of Sets", "text": "Let $X$ be a set. Let $\\mathfrak A$ be an algebra of sets on $X$. Then the following hold: :$(1): \\quad$ The intersection of two sets in $\\mathfrak A$ is in $\\mathfrak A$. :$(2): \\quad$ The difference of two sets in $\\mathfrak A$ is in $\\mathfrak A$. :$(3): \\quad$ $X \\in \\mathfrak A$. :$(4): \\quad$ The empty set $\\O$ is in $\\mathfrak A$."}
+{"_id": "1320", "title": "Countable Sets Have Measure Zero", "text": "Let $S$ be a countable set. {{explain|Is it assumed that $S \\subseteq \\R$?}} Then the measure of $S$ is $\\map m S = 0$."}
+{"_id": "1321", "title": "Mean Value of Convex Real Function", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$ and differentiable on the open interval $\\openint a b$. Let $f$ be convex on $\\openint a b$. Then: :$\\forall \\xi \\in \\openint a b: \\map f x - \\map f \\xi \\ge \\map {f'} \\xi \\paren {x - \\xi}$"}
+{"_id": "1322", "title": "Upper Bound of Natural Logarithm", "text": "Let $\\ln y$ be the natural logarithm of $y$ where $y \\in \\R_{>0}$. Then: :$\\ln y \\le y - 1$"}
+{"_id": "1323", "title": "Exponential of Sum/Real Numbers", "text": "Let $x, y \\in \\R$ be real numbers. Let $\\exp x$ be the exponential of $x$. Then: :$\\map \\exp {x + y} = \\paren {\\exp x} \\paren {\\exp y}$"}
+{"_id": "1324", "title": "Exponential of Product", "text": "Let $x, y \\in \\R$ be real numbers. Let $\\exp x$ be the exponential of $x$. Then: :$\\map \\exp {x y} = \\paren {\\exp y}^x$"}
+{"_id": "1325", "title": "Exponent Combination Laws", "text": "Let $a, b \\in \\R_{>0}$ be strictly positive real numbers. Let $x, y \\in \\R$ be real numbers. Let $a^x$ be defined as $a$ to the power of $x$. Then:"}
+{"_id": "1326", "title": "Derivative of Exponential at Zero", "text": "Let $\\exp x$ be the exponential of $x$ for real $x$. Then: : $\\displaystyle \\lim_{x \\mathop \\to 0} \\frac {\\exp x - 1} x = 1$"}
+{"_id": "1327", "title": "Classification of Groups of Order up to 15", "text": "Up to isomorphism, every group of order $\\order G \\le 15$ is one of the below: {| class=\"sortable wikitable\" |- bgcolor=\"#ececec\" ! Order !! Abelian !! Non-Abelian |- |1 || $\\Z_1$ ||  |- |2 || $\\Z_2$ || |- |3 || $\\Z_3$ || |- |4 || $\\Z_4, \\Z_2 \\oplus \\Z_2$ ||  |- |5 || $\\Z_5$ || |- |6 || $\\Z_6$ || $D_3 = S_3$ |-  |7 || $\\Z_7$ || |- |8 || $\\Z_8, \\Z_4 \\oplus \\Z_2, \\Z_2 \\oplus \\Z_2 \\oplus \\Z_2$ || $D_4, \\Dic 2$ |- |9 || $\\Z_9, \\Z_3 \\oplus \\Z_3$ ||  |- |10 || $\\Z_{10}$ || $D_5$ |- |11 || $\\Z_{11}$ || |- |12 || $\\Z_{12}, \\Z_6 \\oplus \\Z_2$ || $D_6, A_4, \\Dic 3$ |- |13 || $\\Z_{13}$ || |- |14 || $\\Z_{14}$ || $D_7$ |- |15 || $\\Z_{15}$ ||  |} where: : $D_n$ is the dihedral group of order $2 n$ : $S_n$ is the $n$th symmetric group : $A_n$ is the alternating group on $n$ points : $\\Dic 2$ is the dicyclic group of order $4 n$."}
+{"_id": "1329", "title": "Cyclic Groups of Order p q", "text": "Let $p, q$ be primes such that $p < q$ and $p$ does not divide $q - 1$. Let $G$ be a group of order $p q$. Then $G$ is cyclic."}
+{"_id": "1330", "title": "Derivative of Logarithm at One", "text": "Let $\\ln x$ be the natural logarithm of $x$ for real $x$ where $x > 0$. Then: :$\\displaystyle \\lim_{x \\mathop \\to 0} \\frac {\\map \\ln {1 + x} } x = 1$"}
+{"_id": "1331", "title": "Dicyclic Group is Group", "text": "The dicyclic group $Q_n$ is a non-abelian group on two generators."}
+{"_id": "1332", "title": "Existence of Interval of Convergence of Power Series", "text": "Let $\\xi \\in \\R$ be a real number. Let $\\displaystyle \\map S x = \\sum_{n \\mathop = 0}^\\infty a_n \\paren {x - \\xi}^n$ be a power series about $\\xi$. Then the interval of convergence of $\\map S x$ is a real interval whose midpoint is $\\xi$."}
+{"_id": "1334", "title": "Power Series is Differentiable on Interval of Convergence", "text": "Let $\\xi \\in \\R$ be a real number. Let $\\displaystyle \\map f x = \\sum_{n \\mathop = 0}^\\infty a_n \\paren {x - \\xi}^n$ be a power series about $\\xi$. Let $\\map f x$ have an interval of convergence $I$. Then $\\map f x$ is continuous on $I$, and differentiable on $I$ except possibly at its endpoints. Also: :$\\displaystyle \\map {D_x} {\\map f x} = \\sum_{n \\mathop = 1}^\\infty n a_n \\paren {x - \\xi}^{n - 1}$"}
+{"_id": "1336", "title": "Power Series Expansion for Exponential Function", "text": "Let $\\exp x$ be the exponential function. Then: {{begin-eqn}} {{eqn | ll= \\forall x \\in \\R:       | l = \\exp x       | r = \\sum_{n \\mathop = 0}^\\infty \\frac {x^n} {n!}       | c =  }} {{eqn | r = 1 + x + \\frac {x^2} {2!} + \\frac {x^3} {3!} + \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "1337", "title": "Properties of Real Sine Function", "text": "Let $x \\in \\R$ be a real number. Let $\\sin x$ be the sine of $x$. Then:"}
+{"_id": "1338", "title": "Properties of Real Cosine Function", "text": "Let $x \\in \\R$ be a real number. Let $\\cos x$ be the cosine of $x$. Then:"}
+{"_id": "1340", "title": "Derivative of Cosine Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\cos x} = -\\sin x$"}
+{"_id": "1341", "title": "Limit of Sine of X over X", "text": ":$\\displaystyle \\lim_{x \\mathop \\to 0} \\frac {\\sin x} x = 1$"}
+{"_id": "1342", "title": "Sum of Squares of Sine and Cosine", "text": ":$\\cos^2 x + \\sin^2 x = 1$"}
+{"_id": "1343", "title": "Cosine of Sum", "text": ":$\\map \\cos {a + b} = \\cos a \\cos b - \\sin a \\sin b$"}
+{"_id": "1344", "title": "Equivalence of Definitions of Sine and Cosine", "text": "The definitions for sine and cosine are equivalent. That is: : $\\displaystyle \\sin x = \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {x^{2 n + 1} } {\\paren {2 n + 1}!} \\iff \\sin x = \\frac {\\text{Opposite}} {\\text{Hypotenuse}}$ : $\\displaystyle \\cos x = \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {x^{2 n} } {\\paren {2 n}!} \\iff \\cos x = \\frac {\\text{Adjacent}} {\\text{Hypotenuse}}$"}
+{"_id": "1345", "title": "Real Cosine Function is Bounded", "text": ": $\\size {\\cos x} \\le 1$"}
+{"_id": "1346", "title": "Boundedness of Sine X over X", "text": "Let $x \\in \\R$. Then: :$\\size {\\dfrac {\\sin x} x} \\le 1$"}
+{"_id": "1347", "title": "Sine and Cosine are Periodic on Reals", "text": "The sine and cosine functions are periodic on the set of real numbers $\\R$: :$(1): \\quad \\map \\cos {x + 2 \\pi} = \\cos x$ :$(2): \\quad \\map \\sin {x + 2 \\pi} = \\sin x$ :800px"}
+{"_id": "1348", "title": "Differentiable Bounded Convex Real Function is Constant", "text": "Let $f$ be a real function which is: :$(1): \\quad$ Differentiable on $\\R$ :$(2): \\quad$ Bounded on $\\R$ :$(3): \\quad$ Convex on $\\R$. Then $f$ is constant."}
+{"_id": "1349", "title": "Banach-Tarski Paradox", "text": "The unit ball $\\mathbb D^3 \\subset \\R^3$ is equidecomposable to the union of two unit balls."}
+{"_id": "1350", "title": "Equidecomposability is Equivalence Relation", "text": "The property of being equidecomposable is an equivalence relation on the power set $\\mathcal P \\left({\\R^n}\\right)$."}
+{"_id": "1351", "title": "Equidecomposability Unaffected by Union", "text": "{{explain|I guess the sets should be disjoint, in the sense that $S_i \\cap S_j$ is empty for $i \\neq j$. Similar for $T$.}} Let $\\left\\{{S_1, \\ldots, S_m}\\right\\}, \\left\\{{T_1, \\ldots, T_m }\\right\\}$ be sets of sets in $\\R^n$ such that: : for each $k \\in \\left\\{{1, \\dots, m}\\right\\}, S_k$ and $T_k$ are equidecomposable.  Then the set $\\displaystyle S = \\bigcup_{i \\mathop = 1}^m S_i$ is equidecomposable with $\\displaystyle T = \\bigcup_{i \\mathop = 1}^m T_i$."}
+{"_id": "1352", "title": "Subsets of Equidecomposable Subsets are Equidecomposable", "text": "Let $A, B \\subseteq \\R^n$ be equidecomposable. Let $S \\subseteq A$. Then there exists $T \\subseteq B$ such that $S$ and $T$ are equidecomposable."}
+{"_id": "1353", "title": "Shape of Cosine Function", "text": "The cosine function is: :$(1): \\quad$ strictly decreasing on the interval $\\closedint 0 \\pi$ :$(2): \\quad$ strictly increasing on the interval $\\closedint \\pi {2 \\pi}$ :$(3): \\quad$ concave on the interval $\\closedint {-\\dfrac \\pi 2} {\\dfrac \\pi 2}$ :$(4): \\quad$ convex on the interval $\\closedint {\\dfrac \\pi 2} {\\dfrac {3 \\pi} 2}$"}
+{"_id": "1354", "title": "Tangent Function is Periodic on Reals", "text": "The tangent function is periodic on the set of real numbers $\\R$ with period $\\pi$. This can be written: :$\\tan x = \\tan \\left({x \\bmod \\pi}\\right)$ where $x \\bmod \\pi$ denotes the modulo operation."}
+{"_id": "1355", "title": "Derivative of Tangent Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\tan x} = \\sec^2 x = \\dfrac 1 {\\cos^2 x}$ when $\\cos x \\ne 0$."}
+{"_id": "1356", "title": "Stirling's Formula", "text": "The factorial function can be approximated by the formula: :$n! \\sim \\sqrt {2 \\pi n} \\paren {\\dfrac n e}^n$ where $\\sim$ denotes asymptotically equal."}
+{"_id": "1357", "title": "Derivative of Cotangent Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\cot x} = -\\csc^2 x = \\dfrac {-1} {\\sin^2 x}$ where $\\sin x \\ne 0$."}
+{"_id": "1358", "title": "Shape of Sine Function", "text": "The sine function is: :$(1): \\quad$ strictly increasing on the interval $\\closedint {-\\dfrac \\pi 2} {\\dfrac \\pi 2}$ :$(2): \\quad$ strictly decreasing on the interval $\\closedint {\\dfrac \\pi 2} {\\dfrac {3 \\pi} 2}$ :$(3): \\quad$ concave on the interval $\\closedint 0 \\pi$ :$(4): \\quad$ convex on the interval $\\closedint \\pi {2 \\pi}$"}
+{"_id": "1359", "title": "Derivative of Secant Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\sec x} = \\sec x \\tan x$ where $\\cos x \\ne 0$."}
+{"_id": "1360", "title": "Derivative of Cosecant Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\csc x} = -\\csc x \\cot x$ where $\\sin x \\ne 0$."}
+{"_id": "1361", "title": "Shape of Tangent Function", "text": "The nature of the tangent function on the set of real numbers $\\R$ is as follows: : $\\tan x$ is continuous and strictly increasing on the interval $\\left({-\\dfrac \\pi 2 \\,.\\,.\\, \\dfrac \\pi 2}\\right)$ : $\\tan x \\to + \\infty$ as $x \\to \\dfrac \\pi 2 ^-$ : $\\tan x \\to - \\infty$ as $x \\to -\\dfrac \\pi 2 ^+$ : $\\tan x$ is not defined on $\\forall n \\in \\Z: x = \\left({n + \\dfrac 1 2}\\right) \\pi$, at which points it is discontinuous : $\\forall n \\in \\Z: \\tan \\left({n \\pi}\\right) = 0$."}
+{"_id": "1362", "title": "Derivative of Arcsine Function", "text": ":$\\dfrac {\\map \\d {\\arcsin x} } {\\d x} = \\dfrac 1 {\\sqrt {1 - x^2} }$"}
+{"_id": "1363", "title": "Derivative of Arccosine Function", "text": ":$\\map {D_x} {\\arccos x} = \\dfrac {-1} {\\sqrt {1 - x^2}}$"}
+{"_id": "1364", "title": "Sum of Arcsine and Arccosine", "text": "Let $x \\in \\R$ be a real number such that $-1 \\le x \\le 1$. Then: : $\\arcsin x + \\arccos x = \\dfrac \\pi 2$ where $\\arcsin$ and $\\arccos$ denote arcsine and arccosine respectively."}
+{"_id": "1365", "title": "Derivative of Arctangent Function", "text": ":$\\dfrac {\\map \\d {\\arctan x} } {\\d x} = \\dfrac 1 {1 + x^2}$"}
+{"_id": "1367", "title": "Complex Numbers form Field", "text": "Consider the algebraic structure $\\struct {\\C, +, \\times}$, where: :$\\C$ is the set of all complex numbers :$+$ is the operation of complex addition :$\\times$ is the operation of complex multiplication Then $\\struct {\\C, +, \\times}$ forms a field."}
+{"_id": "1368", "title": "Complex Multiplication is Commutative", "text": "The operation of multiplication on the set of complex numbers $\\C$ is commutative: :$\\forall z_1, z_2 \\in \\C: z_1 z_2 = z_2 z_1$"}
+{"_id": "1369", "title": "Complex Multiplication Distributes over Addition", "text": "The operation of multiplication on the set of complex numbers $\\C$ is distributive over the operation of addition. :$\\forall z_1, z_2, z_3 \\in \\C:$ ::$z_1 \\paren {z_2 + z_3} = z_1 z_2 + z_1 z_3$ ::$\\paren {z_2 + z_3} z_1 = z_2 z_1 + z_3 z_1$"}
+{"_id": "1370", "title": "Complex Modulus of Product of Complex Numbers", "text": "Let $z_1, z_2 \\in \\C$ be complex numbers. Let $\\cmod z$ be the modulus of $z$. Then: :$\\cmod {z_1 z_2} = \\cmod {z_1} \\cdot \\cmod {z_2}$"}
+{"_id": "1371", "title": "Sum of Complex Conjugates", "text": "Let $z_1, z_2 \\in \\C$ be complex numbers. Let $\\overline z$ denote the complex conjugate of the complex number $z$. Then: :$\\overline {z_1 + z_2} = \\overline {z_1} + \\overline {z_2}$"}
+{"_id": "1372", "title": "Product of Complex Conjugates", "text": "Let $z_1, z_2 \\in \\C$ be complex numbers. Let $\\overline z$ denote the complex conjugate of the complex number $z$. Then: :$\\overline {z_1 z_2} = \\overline {z_1} \\cdot \\overline {z_2}$"}
+{"_id": "1373", "title": "Sum of Complex Number with Conjugate", "text": "Let $z \\in \\C$ be a complex number. Let $\\overline z$ be the complex conjugate of $z$. Let $\\map \\Re z$ be the real part of $z$. Then: :$z + \\overline z = 2 \\, \\map \\Re z$"}
+{"_id": "1374", "title": "Difference of Complex Number with Conjugate", "text": "Let $z \\in \\C$ be a complex number. Let $\\overline z$ be the complex conjugate of $z$. Let $\\map \\Im z$ be the imaginary part of $z$. Then :$z - \\overline z = 2 i \\, \\map \\Im z$"}
+{"_id": "1375", "title": "Complex Number equals Conjugate iff Wholly Real", "text": "Let $z \\in \\C$ be a complex number. Let $\\overline z$ be the complex conjugate of $z$. Then $z = \\overline z$ {{iff}} $z$ is wholly real."}
+{"_id": "1376", "title": "Complex Numbers cannot be Ordered Compatibly with Ring Structure", "text": "Let $\\struct {\\C, +, \\times}$ be the field of complex numbers. There exists no total ordering on $\\struct {\\C, +, \\times}$ which is compatible with the structure of $\\struct {\\C, +, \\times}$."}
+{"_id": "1378", "title": "Complex Plane is Metric Space", "text": "Let $\\C$ be the set of all complex numbers. Let $d: \\C \\times \\C \\to \\R$ be the function defined as: :$\\map d {z_1, z_2} = \\size {z_1 - z_2}$ where $\\size z$ is the modulus of $z$. Then $d$ is a metric on $\\C$ and so $\\struct {\\C, d}$ is a metric space."}
+{"_id": "1379", "title": "Real Number Line is Metric Space", "text": "Let $\\R$ be the real number line. Let $d: \\R \\times \\R \\to \\R$ be defined as: :$\\map d {x_1, x_2} = \\size {x_1 - x_2}$ where $\\size x$ is the absolute value of $x$. Then $d$ is a metric on $\\R$ and so $\\struct {\\R, d}$ is a metric space."}
+{"_id": "1380", "title": "P-Product Metric on Real Vector Space is Metric", "text": "Let $\\R^n$ be an $n$-dimensional real vector space. Let $p \\in \\R_{\\ge 1}$. Let $d_p: \\R^n \\times \\R^n \\to \\R$ be the $p$-product metric on $\\R^n$: : $\\displaystyle d_p \\left({x, y}\\right) :=  \\left({\\sum_{i \\mathop = 1}^n \\left \\vert {x_i - y_i} \\right \\vert^p}\\right)^{\\frac 1 p}$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({y_1, y_2, \\ldots, y_n}\\right) \\in \\R^n$. Then $d_p$ is a metric."}
+{"_id": "1381", "title": "Taxicab Metric is Metric", "text": "The taxicab metric is a metric."}
+{"_id": "1382", "title": "Zero and One are the only Consecutive Perfect Squares", "text": "If $n$ is a perfect square other than $0$, then $n+1$ is not a perfect square."}
+{"_id": "1383", "title": "ProofWiki:Jokes", "text": "If I cannot open these cans of food, I will die."}
+{"_id": "1385", "title": "P-Product Metric is Metric", "text": "Let $M_{1'} = \\left({A_{1'}, d_{1'}}\\right), M_{2'} = \\left({A_{2'}, d_{2'}}\\right), \\ldots, M_{n'} = \\left({A_{n'}, d_{n'}}\\right)$ be metric spaces. Let $\\displaystyle \\mathcal A = \\prod_{i \\mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \\ldots, A_{n'}$. Let $p \\in \\R_{\\ge 1}$. Let $d_p: \\mathcal A \\times \\mathcal A \\to \\R$ be the $p$-product metric on $\\mathcal A$: : $\\displaystyle d_p \\left({x, y}\\right) := \\left({\\sum_{i \\mathop = 1}^n \\left({d_{i'} \\left({x_i, y_i}\\right)}\\right)^p}\\right)^{\\frac 1 p}$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({y_1, y_2, \\ldots, y_n}\\right) \\in \\mathcal A$. Then $d_p$ is a metric."}
+{"_id": "1386", "title": "Euler Triangle Formula", "text": "Let $d$ be the distance between the incenter and the circumcenter of a triangle. Then: :$d^2 = R \\left({R - 2 \\rho}\\right)$ where: :$R$ is the circumradius :$\\rho$ is the inradius."}
+{"_id": "1387", "title": "Open Ball of Point Inside Open Ball", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $B_\\epsilon \\left({x}\\right)$ be an open $\\epsilon$-ball in $M = \\left({A, d}\\right)$. Let $y \\in B_\\epsilon \\left({x}\\right)$. Then: : $\\exists \\delta \\in \\R: B_\\delta \\left({y}\\right) \\subseteq B_\\epsilon \\left({x}\\right)$ That is, for every point in an open $\\epsilon$-ball in a metric space, there exists an open $\\delta$-ball of that point entirely contained within that open $\\epsilon$-ball."}
+{"_id": "1388", "title": "Open Sets in Metric Space", "text": "Let $M = \\struct {A, d}$ be a metric space. Then $\\O$ and $A$ are both open in $M$."}
+{"_id": "1389", "title": "Equivalence of Definitions of Continuity on Metric Spaces", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f: A_1 \\to A_2$ be a mapping from $A_1$ to $A_2$. {{TFAE|def = Continuous on Metric Space|view = continuity|context = Metric Space|contextview = metric spaces}}"}
+{"_id": "1390", "title": "Finite Intersection of Open Sets of Metric Space is Open", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $U_1, U_2, \\ldots, U_n$ be open in $M$. Then $\\ds \\bigcap_{i \\mathop = 1}^n U_i$ is open in $M$. That is, a finite intersection of open subsets is open."}
+{"_id": "1391", "title": "Union of Open Sets of Metric Space is Open", "text": "Let $M = \\struct {A, d}$ be a metric space. The union of a set of open sets of $M$ is open in $M$."}
+{"_id": "1392", "title": "Lipschitz Equivalent Metric Spaces are Homeomorphic", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $M_1$ and $M_2$ be Lipschitz equivalent. Then $M_1$ and $M_2$ are homeomorphic."}
+{"_id": "1393", "title": "P-Product Metrics on Real Vector Space are Topologically Equivalent", "text": "For $n \\in \\N$, let $\\R^n$ be an Euclidean space. Let $p \\in \\R_{\\ge 1}$. Let $d_p$ be the $p$-product metric on $\\R^n$. Let $d_\\infty$ be the Chebyshev distance on $\\R^n$. Then $d_p$ and $d_\\infty$ are topologically equivalent."}
+{"_id": "1394", "title": "Derivative of Function of Constant Multiple", "text": "Let $f$ be a real function which is differentiable on $\\R$. Let $c \\in \\R$ be a constant. Then: :$\\map {D_x} {\\map f {c x} } = c \\map {D_{c x} } {\\map f {c x} }$"}
+{"_id": "1395", "title": "Cotangent Function is Periodic on Reals", "text": "The cotangent function is periodic on the set of real numbers $\\R$ with period $\\pi$."}
+{"_id": "1396", "title": "Shape of Cotangent Function", "text": "The nature of the cotangent function on the set of real numbers $\\R$ is as follows: : $\\cot x$ is continuous and strictly decreasing on the interval $\\left({0 \\,.\\,.\\, \\pi}\\right)$ : $\\cot x \\to + \\infty$ as $x \\to 0^+$ : $\\cot x \\to - \\infty$ as $x \\to \\pi^-$ : $\\cot x$ is not defined on $\\forall n \\in \\Z: x = n \\pi$, at which points it is discontinuous : $\\forall n \\in \\Z: \\cot \\left({n + \\dfrac 1 2}\\right) \\pi = 0$"}
+{"_id": "1398", "title": "Change of Base of Logarithm", "text": ":$\\log_b x = \\dfrac {\\log_a x} {\\log_a b}$"}
+{"_id": "1399", "title": "Derivative of Function to Power of Function", "text": "Let $\\map u x, \\map v x$ be real functions which are differentiable on $\\R$. Then: :$\\map {\\dfrac \\d {\\d x} } {u^v} = v u^{v - 1} \\map {\\dfrac \\d {\\d x} } u + u^v \\paren {\\ln u} \\map {\\dfrac \\d {\\d x} } v$"}
+{"_id": "1400", "title": "Liouville's Theorem (Complex Analysis)", "text": "Let $f: \\C \\to \\C$ be a bounded entire function. Then $f$ is constant."}
+{"_id": "1401", "title": "Riemann Removable Singularities Theorem", "text": "Let $U \\subset \\C$ be a domain, let $z_0 \\in U$, and let $f: U \\setminus \\left\\{ {z_0}\\right\\} \\to \\C$ be holomorphic. Then the following are equivalent: :$(1): \\quad f$ extends to a holomorphic function $f: U \\to \\C$. :$(2): \\quad f$ extends to a continuous function $f: U \\to \\C$. :$(3): \\quad f$ is bounded in a neighborhood of $z_0$. :$(4): \\quad f \\left({z}\\right) = o \\left({\\dfrac 1 {\\left|{z - z_0}\\right|} }\\right)$ as $z \\to z_0$."}
+{"_id": "1402", "title": "Gaussian Integral", "text": ":$\\displaystyle \\int_{-\\infty}^\\infty e^{-x^2} \\rd x = \\sqrt \\pi$"}
+{"_id": "1403", "title": "Indiscrete Topology is not Metrizable", "text": "Let $S$ be a set with more than one element. The indiscrete topology on $S$ is not metrizable."}
+{"_id": "1404", "title": "Composite of Continuous Mappings is Continuous", "text": "Let $T_1, T_2, T_3$ be topological spaces. Let $f: T_1 \\to T_2$ and $g: T_2 \\to T_3$ be continuous mappings. Then the composite mapping $g \\circ f: T_1 \\to T_3$ is continuous."}
+{"_id": "1405", "title": "Continuous Mapping is Continuous on Induced Topological Spaces", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $\\tau_{d_1}$ and $\\tau_{d_2}$ be the topologies induced by the metrics $d_1$ and $d_2$. Let $T_1 = \\left({A_1, \\tau_{d_1}}\\right)$ and $T_2 = \\left({A_2, \\tau_{d_2}}\\right)$ be the resulting topological spaces. Let $f: A_1 \\to A_2$ be a mapping. Then $f$ is $\\left({d_1, d_2}\\right)$-continuous {{iff}} $f$ is $\\left({\\tau_{d_1}, \\tau_{d_2}}\\right)$-continuous."}
+{"_id": "1407", "title": "Synthetic Basis and Analytic Basis are Compatible", "text": "Let $\\struct {S, \\tau}$ be a topological space. Then $\\BB$ is an analytic basis for $\\tau$ {{iff}} $\\tau$ is the topology on $S$ generated by the synthetic basis $\\BB$."}
+{"_id": "1408", "title": "Picard's Existence Theorem", "text": "Let $\\map f {x, y} : \\R^2 \\to \\R$ be continuous in a region $D \\subseteq \\R^2$. Let $\\exists M \\in \\R: \\forall x, y \\in D: \\size {\\map f {x, y} } < M$. Let $\\map f {x, y}$ satisfy in $D$ the Lipschitz condition in $y$: :$\\size{\\map f {x, y_1} - \\map f {x, y_2} } \\le A \\size {y_1 - y_2}$ where $A$ is independent of $x, y_1, y_2$. Let the rectangle $R$ be defined as $\\set {\\tuple {x, y} \\in \\R^2: \\size {x - a} \\le h, \\size {y - b} \\le k}$ such that $M h \\le k$. Let $R \\subseteq D$. Then $\\forall x \\in \\R: \\size {x - a} \\le h$, the first order ordinary differential equation: :$y' = \\map f {x, y}$ has one and only one solution $y = \\map y x$ for which $b = \\map y a$."}
+{"_id": "1409", "title": "Continuity Test using Basis", "text": "Let $\\left({X_1, \\tau_1}\\right)$ and $\\left({X_2, \\tau_2}\\right)$ be topological spaces. Let $f: X_1 \\to X_2$ be a mapping. Let $\\mathcal B$ be an analytic basis for $\\tau_2$. Suppose that: :$\\forall B \\in \\mathcal B: f^{-1} \\left({B}\\right) \\in \\tau_1$ where $f^{-1} \\left({B}\\right)$ denotes the preimage of $B$ under $f$. Then $f$ is continuous."}
+{"_id": "1410", "title": "Sub-Basis for Real Number Line", "text": "Let the real number line $\\R$ be considered as a topology under the usual (Euclidean) metric. Then: :$\\mathcal B := \\left\\{{\\left({-\\infty \\,.\\,.\\, a}\\right), \\left({b \\,.\\,.\\, \\infty}\\right): a, b \\in \\R}\\right\\}$ is a sub-basis for $\\R$."}
+{"_id": "1411", "title": "Synthetic Basis formed from Synthetic Sub-Basis", "text": "Let $X$ be a set. Let $\\SS$ be a synthetic sub-basis on $X$. Define: :$\\displaystyle \\BB = \\set {\\bigcap \\FF: \\FF \\subseteq \\SS, \\text{$\\FF$ is finite} }$ Then $\\BB$ is a synthetic basis on $X$."}
+{"_id": "1412", "title": "Epimorphism Preserves Distributivity", "text": "Let $\\struct {R_1, +_1, \\circ_1}$ and $\\struct {R_2, +_2, \\circ_2}$ be algebraic structures. Let $\\phi: R_1 \\to R_2$ be an epimorphism. :If $\\circ_1$ is left distributive over $+_1$, then $\\circ_2$ is left distributive over $+_2$. :If $\\circ_1$ is right distributive over $+_1$, then $\\circ_2$ is right distributive over $+_2$. Consequently, if $\\circ_1$ is distributive over $+_1$, then $\\circ_2$ is distributive over $+_2$. That is, epimorphism preserves distributivity."}
+{"_id": "1413", "title": "Epimorphism Preserves Groups", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be algebraic structures. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be an epimorphism. Let $\\struct {S, \\circ}$ be a group. Then $\\struct {T, *}$ is also a group."}
+{"_id": "1415", "title": "Continuity of Composite with Inclusion", "text": "Let $T = \\left({A, \\tau}\\right)$ and $T' = \\left({A', \\tau'}\\right)$ be topological spaces. Let $H \\subseteq A$. Let $T_H = \\left({H, \\tau_H}\\right)$ be a topological subspace of $T$. Let $i: H \\to A$ be the inclusion mapping. Let $f: A \\to A'$ and $g: A' \\to H$ be mappings. Then the following apply:"}
+{"_id": "1416", "title": "Totally Bounded Metric Space is Bounded", "text": "Let $M = \\struct {A, d}$ be a totally bounded metric space. Then $M$ is bounded."}
+{"_id": "1417", "title": "Convergent Subsequence of Cauchy Sequence", "text": "Let $\\left({A, d}\\right)$ be a metric space. Let $\\left\\langle{x_n}\\right\\rangle_{n \\in \\N}$ be a Cauchy sequence in $A$.  Let $x \\in A$. Then $\\left\\langle{x_n}\\right\\rangle$ converges to $x$ {{iff}} it has a subsequence that converges to $x$."}
+{"_id": "1418", "title": "Existence and Uniqueness of Sigma-Algebra Generated by Collection of Subsets", "text": "Let $X$ be a set. Let $\\mathcal G \\subseteq \\powerset X$ be a collection of subsets of $X$. Then $\\map \\sigma {\\mathcal G}$, the $\\sigma$-algebra generated by $\\mathcal G$, exists and is unique."}
+{"_id": "1420", "title": "Projection from Product Topology is Continuous", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ and $T_2 = \\struct {S_2, \\tau_2}$ be topological spaces. Let $T = \\struct {T_1 \\times T_2, \\tau}$ be the product space of $T_1$ and $T_2$, where $\\tau$ is the Tychonoff topology on $S$. Let $\\pr_1: T \\to T_1$ and $\\pr_2: T \\to T_2$ be the first and second projections from $T$ onto its factors. Then both $\\pr_1$ and $\\pr_2$ are are continuous."}
+{"_id": "1421", "title": "Continuous Mapping to Topological Product", "text": "Let $T = T_1 \\times T_2$ be a product space of two topological spaces $T_1$ and $T_2$. Let $\\pr_1: T \\to T_1$ and $\\pr_2: T \\to T_2$ be the first and second projections from $T$ onto its factors. Let $T'$ be a topological space. Let $f: T' \\to T$ be a mapping. Then $f$ is continuous {{iff}} $\\pr_1 \\circ f$ and $\\pr_2 \\circ f$ are continuous."}
+{"_id": "1422", "title": "Open and Closed Sets in Topological Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then $S$ and $\\O$ are both both open and closed in $T$."}
+{"_id": "1423", "title": "Absolutely Continuous Real Function is Continuous", "text": "Let $I \\subseteq \\R$ be a real interval. Let $f : I \\to \\R$ be an absolutely continuous function. Then $f$ is continuous."}
+{"_id": "1425", "title": "Closed Set in Topological Subspace", "text": "Let $T$ be a topological space. Let $T' \\subseteq T$ be a subspace of $T$. Then $V \\subseteq T'$ is closed in $T'$ {{iff}} $V = T' \\cap W$ for some $W$ closed in $T$."}
+{"_id": "1426", "title": "Continuity Defined from Closed Sets", "text": "Let $T_1$ and $T_2$ be topological spaces. Let $f: T_1 \\to T_2$ be a mapping. Then $f$ is continuous {{iff}} for all $V$ closed in $T_2$, $f^{-1} \\sqbrk V$ is closed in $T_1$."}
+{"_id": "1427", "title": "Real Numbers are Uncountable", "text": "The set of real numbers $\\R$ is uncountably infinite."}
+{"_id": "1428", "title": "Condition for Point being in Closure", "text": "Let $T$ be a topological space. Let $H \\subseteq T$. Let $x \\in T$. Then $x \\in H^-$ {{iff}} every open set of $T$ which contains $x$ contains a point in $H$."}
+{"_id": "1430", "title": "Topological Closure of Subset is Subset of Topological Closure", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq K$ and $K \\subseteq S$. Then: :$\\map \\cl H \\subseteq \\map \\cl K$ where $\\map \\cl H$ denotes the closure of $H$."}
+{"_id": "1431", "title": "Closure of Topological Closure equals Closure", "text": "Let $T$ be a topological space. Let $H \\subseteq T$. Then: : $\\left({H^-}\\right)^- = H^-$ where $H^-$ denotes the closure of $H$."}
+{"_id": "1432", "title": "Topological Closure is Closed", "text": "Let $T$ be a topological space. Let $H \\subseteq T$. Then the closure $\\map \\cl H$ of $H$ is closed in $T$."}
+{"_id": "1433", "title": "Equivalence of Definitions of Closure of Topological Subspace", "text": "{{TFAE|def = Closure (Topology)|view = Closure|context = Topology (Mathematical Branch)|contextview = Topology}} Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$."}
+{"_id": "1434", "title": "Closure of Intersection is Subset of Intersection of Closures", "text": "Let $T$ be a topological space. Let $I$ be an indexing set. Let $\\forall i \\in I: H_i \\subseteq T$. Then: : $\\displaystyle \\left({\\bigcap_I H_i}\\right)^- \\subseteq \\bigcap_I H_i^-$ where $H_i^-$ denotes the closure of $H_i$."}
+{"_id": "1435", "title": "Closure of Finite Union equals Union of Closures", "text": "Let $T$ be a topological space. Let $n \\in \\N$. Let: :$\\forall i \\in \\set {1, 2, \\ldots, n}: H_i \\subseteq T$ Then: :$\\displaystyle \\bigcup_{i \\mathop = 1}^n \\map \\cl {H_i} = \\map \\cl {\\bigcup_{i \\mathop = 1}^n H_i}$"}
+{"_id": "1438", "title": "Closure of Open Ball in Metric Space", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $\\map {B_\\epsilon} x$ be an open $\\epsilon$-ball in $M$. Let $y \\in \\map \\cl {\\map {B_\\epsilon} x}$, where $\\cl$ denotes the closure of $\\map {B_\\epsilon} x$. Then $\\map d {x, y} \\le \\epsilon$."}
+{"_id": "1439", "title": "Nowhere Dense iff Complement of Closure is Everywhere Dense", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$. Then $H$ is nowhere dense in $T$ {{iff}} $S \\setminus H^-$ is everywhere dense in $T$, where $H^-$ denotes the closure of $H$."}
+{"_id": "1442", "title": "Totally Bounded Metric Space is Separable", "text": "A totally bounded metric space is separable."}
+{"_id": "1443", "title": "Convergent Sequence in Hausdorff Space has Unique Limit", "text": "Let $T = \\struct {S, \\tau}$ be a Hausdorff space. Let $\\sequence {x_n}$ be a convergent sequence in $T$. Then $\\sequence {x_n}$ has exactly one limit."}
+{"_id": "1444", "title": "Metric Space is Hausdorff", "text": "Let $M = \\struct {A, d}$ be a metric space. Then $M$ is a Hausdorff space."}
+{"_id": "1445", "title": "Convergent Real Sequence has Unique Limit", "text": "Let $\\sequence {s_n}$ be a real sequence. Then $\\sequence {s_n}$ can have at most one limit."}
+{"_id": "1446", "title": "Distinct Points in Metric Space have Disjoint Open Balls", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $x, y \\in M: x \\ne y$. Then there exist disjoint open $\\epsilon$-balls $B_\\epsilon \\left({x}\\right)$ and $B_\\epsilon \\left({y}\\right)$ containing $x$ and $y$ respectively."}
+{"_id": "1447", "title": "Sequentially Compact Metric Space is Lindelöf", "text": "Let $M$ be a metric space. Let $M$ be sequentially compact. Then $M$ is also a Lindelöf space. That is, from every open cover of $M$, it is possible to extract a countable subcover."}
+{"_id": "1448", "title": "Sequentially Compact Metric Space is Compact", "text": "A sequentially compact metric space is compact."}
+{"_id": "1450", "title": "Compact Subspace of Metric Space is Bounded", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $C$ be a subspace of $M$. If $C$ is compact, then it is bounded."}
+{"_id": "1451", "title": "Compact Subspace of Hausdorff Space is Closed", "text": "Let $H = \\struct {A, \\tau}$ be a Hausdorff space. Let $C$ be a compact subspace of $H$. Then $C$ is closed in $H$."}
+{"_id": "1452", "title": "Continuous Image of Compact Space is Compact", "text": "Let $T_1$ and $T_2$ be topological spaces. Let $f: T_1 \\to T_2$ be a continuous mapping. If $T_1$ is compact then so is its image $f \\sqbrk {T_1}$ under $f$. That is, compactness is a continuous invariant."}
+{"_id": "1453", "title": "Closed Subspace of Compact Space is Compact", "text": "A closed subspace of a compact space is compact. That is, the property of being compact is weakly hereditary."}
+{"_id": "1454", "title": "Closure of Real Interval is Closed Real Interval", "text": "Let $I$ be a non-empty real interval such that one of these holds: : $I = \\left({a \\,.\\,.\\, b}\\right)$ : $I = \\left[{a \\,.\\,.\\, b}\\right)$ : $I = \\left({a \\,.\\,.\\, b}\\right]$ : $I = \\left[{a \\,.\\,.\\, b}\\right]$ Let $I^-$ denote the closure of $I$. Then $I^-$ is the closed real interval $\\left[{a \\,.\\,.\\, b}\\right]$."}
+{"_id": "1455", "title": "Topological Product of Compact Spaces", "text": "Let $T_1$ and $T_2$ be topological spaces. Let $T_1 \\times T_2$ be the product space of $T_1$ and $T_2$. Then $T_1 \\times T_2$ is compact {{iff}} both $T_1$ and $T_2$ are compact."}
+{"_id": "1456", "title": "Heine-Cantor Theorem", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $M_1$ be compact. Let $f: A_1 \\to A_2$ be a continuous mapping. Then $f$ is uniformly continuous."}
+{"_id": "1457", "title": "Continuous Function on Closed Interval is Uniformly Continuous", "text": "Let $\\closedint a b$ be a closed real interval. Let $f: \\closedint a b \\to \\R$ be a continuous function. Then $f$ is uniformly continuous on $\\closedint a b$."}
+{"_id": "1458", "title": "Sine of Complement equals Cosine", "text": ":$\\sin \\left({\\dfrac \\pi 2 - \\theta}\\right) = \\cos \\theta$"}
+{"_id": "1459", "title": "Cosine of Complement equals Sine", "text": ":$\\map \\cos {\\dfrac \\pi 2 - \\theta} = \\sin \\theta$"}
+{"_id": "1460", "title": "Continuous Bijection from Compact to Hausdorff is Homeomorphism", "text": "Let $T_1$ be a compact space. Let $T_2$ be a Hausdorff space. Let $f: T_1 \\to T_2$ be a continuous bijection. Then $f$ is a homeomorphism."}
+{"_id": "1462", "title": "Indirect Proof", "text": "Let $P$ be a proposition whose truth value is to be proved (either true or false). There are two aspects to this:"}
+{"_id": "1463", "title": "Continuous Mapping of Separation", "text": "Let $T$ and $T'$ be topological spaces. Let $A \\mid B$ be a separation of $T$. Let $f: T \\to T'$ be a mapping such that the restrictions $f \\restriction_A$ and $f \\restriction_B$ are both continuous. Then $f$ is continuous on the whole of $T$."}
+{"_id": "1464", "title": "Continuity from Union of Restrictions", "text": "Let $T_1$ and $T_2$ be topological spaces. Then $f: T_1 \\to T_2$ is continuous if either: : $\\displaystyle T_1 = \\bigcup_{i \\mathop = 1}^n V_i$ where: :: each $V_i$ is closed in $T_1$, and  :: the restriction $f \\restriction_{V_i}$ is continuous for each $V_i$ or: : $\\displaystyle T_1 = \\bigcup_{i \\mathop \\in I} U_i$ where: :: $I$ is any (possibly infinite) indexing set; :: each $U_i$ is open in $T_1$, and :: the restriction $f \\restriction_{U_i}$ is continuous for each $U_i$."}
+{"_id": "1465", "title": "Fermat's Last Theorem", "text": ":$\\forall a, b, c, n \\in \\Z_{>0}, \\; n > 2$, the equation $a^n + b^n = c^n$ has no solutions."}
+{"_id": "1466", "title": "Subset of Real Numbers is Interval iff Connected", "text": "Let the real number line $\\R$ be considered as a topological space. Let $S$ be a subspace of $\\R$. Then $S$ is connected {{iff}} $S$ is an interval of $\\R$. That is, the only subspaces of $\\R$ that are connected are intervals."}
+{"_id": "1467", "title": "Continuous Image of Connected Space is Connected", "text": "Let $T_1$ and $T_2$ be topological spaces. Let $S_1 \\subseteq T_1$ be connected. Let $f: T_1 \\to T_2$ be a continuous mapping. Then the image $f \\left({S_1}\\right)$ is connected."}
+{"_id": "1468", "title": "Union of Connected Sets with Non-Empty Intersections is Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $I$ be an indexing set. Let $\\AA = \\family {A_\\alpha}_{\\alpha \\mathop \\in I}$ be an indexed family of subsets of $S$, all connected in $T$. Let $\\AA$ be such that no two of its elements are disjoint: :$\\forall B, C \\in \\AA: B \\cap C \\ne \\O$ Then $\\displaystyle \\bigcup \\AA$ is itself connected."}
+{"_id": "1469", "title": "Finite Product Space is Connected iff Factors are Connected/Basis for the Induction", "text": "Let $T_1$ and $T_2$ be topological spaces. Then the product space $T_1 \\times T_2$ is connected {{iff}} $T_1$ and $T_2$ are connected."}
+{"_id": "1470", "title": "Set between Connected Set and Closure is Connected", "text": "Let $T$ be a topological space. Let $H$ be a connected set of $T$. Let $H \\subseteq K \\subseteq H^-$, where $H^-$ denotes the closure of $H$.  Then $K$ is connected."}
+{"_id": "1471", "title": "Path-Connected Space is Connected", "text": "Let $T$ be a topological space which is path-connected. Then $T$ is connected."}
+{"_id": "1472", "title": "Hausdorff Paradox", "text": "There is a disjoint decomposition of the sphere $\\mathbb S^2$ into four sets $A, B, C, D$ such that $A, B, C, B \\cup C$ are all congruent and $D$ is countable."}
+{"_id": "1473", "title": "Closed Topologist's Sine Curve is not Path-Connected", "text": "Let $G$ be the graph of the function $y = \\sin \\left({\\dfrac 1 x}\\right)$ for $x > 0$. Let $J$ be the line segment joining the points $\\left({0, -1}\\right)$ and $\\left({0, 1}\\right)$ in $\\R^2$. Then while $G \\cup J$ is connected, it is '''not''' path-connected."}
+{"_id": "1474", "title": "Sum of Two Sides of Triangle Greater than Third Side", "text": "Given a triangle $ABC$, the sum of the lengths of any two sides of the triangle is greater than the length of the third side. {{:Euclid:Proposition/I/20}}"}
+{"_id": "1475", "title": "Lines Through Endpoints of One Side of Triangle to Point Inside Triangle is Less than Sum of Other Sides", "text": "Given a triangle and a point inside it, the sum of the lengths of the line segments from the endpoints of one side of the triangle to the point is less than the sum of the other two sides of the triangle. {{:Euclid:Proposition/I/21}}"}
+{"_id": "1476", "title": "Construction of Triangle from Given Lengths", "text": "Given three straight lines such that the sum of the lengths of any two of the lines is greater than the length of the third line, it is possible to construct a triangle having the lengths of these lines as its side lengths. {{:Euclid:Proposition/I/22}}"}
+{"_id": "1478", "title": "Connected Open Subset of Euclidean Space is Path-Connected", "text": "Let $\\R^n$ be a Euclidean $n$-space. Let $U$ be a connected open subset of $\\R^n$. Then $U$ is path-connected."}
+{"_id": "1479", "title": "Joining Paths makes Another Path", "text": "Let $T$ be a topological space. Let $I \\subseteq \\R$ be the closed real interval $\\closedint 0 1$. Let $f, g: I \\to T$ be paths in $T$ from $a$ to $b$ and from $b$ to $c$ respectively. Let $h: I \\to T$ be the mapping given by: $\\map h x = \\begin{cases} \\map f {2x} & : x \\in \\closedint 0 {\\dfrac 1 2} \\\\ \\map g {2x - 1} & : x \\in \\closedint {\\dfrac 1 2} 1 \\end{cases}$ Then $h$ is a path in $T$."}
+{"_id": "1480", "title": "Hinge Theorem", "text": "If two triangles have two pairs of sides which are the same length, the triangle with the larger included angle also has the larger third side. {{:Euclid:Proposition/I/24}}"}
+{"_id": "1481", "title": "Equivalence of Definitions of Component", "text": "{{TFAE|def = Component (Topology)|view = Component|context = Topology (Mathematical Branch)|contextview = Topology}} Let $T = \\struct {S, \\tau}$ be a topological space. Let $x \\in T$."}
+{"_id": "1483", "title": "Sequentially Compact Metric Subspace is Sequentially Compact in Itself iff Closed", "text": "Let $M$ be a metric space. Let $C \\subseteq M$ be a subspace of $M$ which is sequentially compact in $M$. Then $C$ is sequentially compact in itself {{iff}} $C$ is closed in $M$."}
+{"_id": "1485", "title": "Real Number Line is Complete Metric Space", "text": "The real number line $\\R$ with the usual (Euclidean) metric forms a complete metric space."}
+{"_id": "1486", "title": "Series Law for Extremal Length", "text": "Let $X$ be a Riemann surface. Let $\\Gamma_1$, $\\Gamma_2$ and $\\Gamma$ be families of rectifiable curves (or, more generally, families of unions of rectifiable curves) on $X$. Let every $\\gamma \\in \\Gamma$ contain a $\\gamma_1 \\in \\Gamma_1$ and a $\\gamma_2 \\in \\Gamma_2$ such that $\\gamma_1 \\cap \\gamma_2 = \\varnothing$. Then the extremal lengths of $\\Gamma_1$, $\\Gamma_2$ and $\\Gamma$ satisfy: :$\\lambda \\left({\\Gamma}\\right) \\ge \\lambda \\left({\\Gamma_1}\\right) + \\lambda \\left({\\Gamma_2}\\right)$"}
+{"_id": "1487", "title": "Extremal Length of Union", "text": "Let $X$ be a Riemann surface. Let $\\Gamma_1$ and $\\Gamma_2$ be families of rectifiable curves (or, more generally, families of unions of rectifiable curves) on $X$.  Then the extremal length of their union satisfies: :$\\dfrac 1 {\\lambda \\left({\\Gamma_1 \\cup \\Gamma_2}\\right)} \\le \\dfrac 1 {\\lambda \\left({\\Gamma_1}\\right)} + \\dfrac 1 {\\lambda \\left({\\Gamma_2}\\right)}$ Suppose that additionally $\\Gamma_1$ and $\\Gamma_2$ are disjoint in the following sense: there exist disjoint Borel subsets: :$A_1, A_2 \\subseteq X$ such that $\\displaystyle \\bigcup \\Gamma_1 \\subset A_1$ and $\\displaystyle \\bigcup \\Gamma_2 \\subset A_2$ Then :$\\dfrac 1 {\\lambda \\left({\\Gamma_1 \\cup \\Gamma_2}\\right)} = \\dfrac 1 {\\lambda \\left({\\Gamma_1}\\right)} + \\dfrac 1 {\\lambda \\left({\\Gamma_2}\\right)}$"}
+{"_id": "1489", "title": "Comparison Principle for Extremal Length", "text": "Let $X$ be a Riemann surface. Let $\\Gamma_1$ and $\\Gamma_2$ be families of rectifiable curves (or, more generally, families of unions of rectifiable curves) on $X$. Let every element of $\\Gamma_1$ contain some element of $\\Gamma_2$. Then the extremal lengths of $\\Gamma_1$ and $\\Gamma_2$ are related by: :$\\lambda \\left({\\Gamma_1}\\right) \\ge \\lambda \\left({\\Gamma_2}\\right)$ More precisely, for every conformal metric $\\rho$ as in the definition of extremal length, we have: :$L \\left({\\Gamma_1, \\rho}\\right) \\ge L \\left({\\Gamma_2, \\rho}\\right)$"}
+{"_id": "1490", "title": "Reverse Triangle Inequality", "text": "Let $M = \\struct {X, d}$ be a metric space. Then: :$\\forall x, y, z \\in X: \\size {\\map d {x, z} - \\map d {y, z} } \\le \\map d {x, y}$ === Normed Division Ring === {{:Reverse Triangle Inequality/Normed Division Ring}} === Normed Vector Space === {{:Reverse Triangle Inequality/Normed Vector Space}} === Real and Complex Numbers === {{:Reverse Triangle Inequality/Real and Complex Fields}}"}
+{"_id": "1491", "title": "Inverse Completion of Integral Domain Exists", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Then an inverse completion of $\\struct {D, \\circ}$ can be constructed."}
+{"_id": "1492", "title": "Zero of Inverse Completion of Integral Domain", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose zero is $0_D$. Let $\\struct {K, \\circ}$ be the inverse completion of $\\struct {D, \\circ}$ as defined in Inverse Completion of Integral Domain Exists. Let $x \\in K: x = \\dfrac p q$ such that $p = 0_D$. Then $x$ is equal to the zero of $K$. That is, ''any'' element of $K$ of the form $\\dfrac {0_D} q$ acts as the zero of $K$."}
+{"_id": "1493", "title": "Inverse Completion Less Zero of Integral Domain is Closed", "text": "Let $\\left({D, +, \\circ}\\right)$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $\\left({K, \\circ}\\right)$ be the inverse completion of $\\left({D, \\circ}\\right)$. Then $\\left({K^*, \\circ}\\right)$ is closed, where $K^* = K \\setminus \\left\\{{0_K}\\right\\}$."}
+{"_id": "1494", "title": "Composition of Mappings is Associative", "text": "The composition of mappings is an associative binary operation: :$\\paren {f_3 \\circ f_2} \\circ f_1 = f_3 \\circ \\paren {f_2 \\circ f_1}$ where $f_1, f_2, f_3$ are arbitrary mappings which fulfil the conditions for the relevant compositions to be defined."}
+{"_id": "1495", "title": "Group Abelian iff Cross Cancellation Property", "text": "Let $G$ be a group. Then the following are equivalent: :$(1): \\quad G$ is abelian :$(2): \\quad G$ has the cross cancellation property"}
+{"_id": "1496", "title": "Group Abelian iff Middle Cancellation Property", "text": "Let $G$ be a group. Then the following are equivalent: :$(1): \\quad G$ is abelian :$(2): \\quad G$ satisfies the middle cancellation property"}
+{"_id": "1497", "title": "Rational Numbers form Ordered Field", "text": "The set of rational numbers $\\Q$ forms an ordered field under addition and multiplication: $\\struct {\\Q, +, \\times, \\le}$."}
+{"_id": "1498", "title": "Rational Numbers are Close Packed", "text": "Let $a, b \\in \\Q$ such that $a < b$. Then $\\exists c \\in \\Q: a < c < b$. That is, the set of rational numbers is close packed."}
+{"_id": "1499", "title": "Real Addition is Closed", "text": "The set of real numbers $\\R$ is closed under addition: :$\\forall x, y \\in \\R: x + y \\in \\R$"}
+{"_id": "1501", "title": "Real Multiplication is Well-Defined", "text": "The operation of multiplication on the set of real numbers $\\R$ is well-defined."}
+{"_id": "1502", "title": "Real Multiplication is Closed", "text": "The operation of multiplication on the set of real numbers $\\R$ is closed: :$\\forall x, y \\in \\R: x \\times y \\in \\R$"}
+{"_id": "1503", "title": "Real Addition is Associative", "text": "The operation of addition on the set of real numbers $\\R$ is associative: :$\\forall x, y, z \\in \\R: x + \\paren {y + z} = \\paren {x + y} + z$"}
+{"_id": "1504", "title": "Real Multiplication is Associative", "text": "The operation of multiplication on the set of real numbers $\\R$ is associative: :$\\forall x, y, z \\in \\R: x \\times \\paren {y \\times z} = \\paren {x \\times y} \\times z$"}
+{"_id": "1505", "title": "Real Addition is Commutative", "text": "The operation of addition on the set of real numbers $\\R$ is commutative: :$\\forall x, y \\in \\R: x + y = y + x$"}
+{"_id": "1506", "title": "Real Multiplication is Commutative", "text": "The operation of multiplication on the set of real numbers $\\R$ is commutative: :$\\forall x, y \\in \\R: x \\times y = y \\times x$"}
+{"_id": "1507", "title": "Equidecomposable Nested Sets", "text": "Let $A, B, C$ be sets such that $A$ and $C$ are equidecomposable and $A \\subseteq B \\subseteq C$. Then $B$ and $C$ are equidecomposable."}
+{"_id": "1508", "title": "Rational Numbers form Metric Space", "text": "Let $\\Q$ be the set of all rational numbers. Let $d: \\Q \\times \\Q \\to \\R$ be defined as: :$\\map d {x_1, x_2} = \\size {x_1 - x_2}$ where $\\size x$ is the absolute value of $x$. Then $d$ is a metric on $\\Q$ and so $\\struct {\\Q, d}$ is a metric space."}
+{"_id": "1509", "title": "Rational Numbers form Vector Space", "text": "Let $\\Q$ be the set of rational numbers. Then the $\\Q$-module $\\Q^n$ is a vector space. It follows directly, by setting $n = 1$, that the $\\Q$-module $\\Q$ itself can also be regarded as a vector space."}
+{"_id": "1510", "title": "Five Color Theorem", "text": "A planar graph $G$ can be assigned a proper vertex $k$-coloring such that $k \\le 5$."}
+{"_id": "1511", "title": "Limit of (Cosine (X) - 1) over X", "text": ":$\\displaystyle \\lim_{x \\mathop \\to 0} \\frac {\\map \\cos x - 1} x = 0$"}
+{"_id": "1512", "title": "Real Multiplication Distributes over Addition", "text": "The operation of multiplication on the set of real numbers $\\R$ is distributive over the operation of addition: :$\\forall x, y, z \\in \\R:$ ::$x \\times \\paren {y + z} = x \\times y + x \\times z$ ::$\\paren {y + z} \\times x = y \\times x + z \\times x$"}
+{"_id": "1513", "title": "Cardano's Formula", "text": "Let $P$ be the cubic equation: :$a x^3 + b x^2 + c x + d = 0$ with $a \\ne 0$ Then $P$ has solutions: :$x_1 = S + T - \\dfrac b {3 a}$ :$x_2 = - \\dfrac {S + T} 2 - \\dfrac b {3 a} + \\dfrac {i \\sqrt 3} 2 \\paren {S - T}$ :$x_3 = - \\dfrac {S + T} 2 - \\dfrac b {3 a} - \\dfrac {i \\sqrt 3} 2 \\paren {S - T}$ where: :$S = \\sqrt [3] {R + \\sqrt {Q^3 + R^2} }$ :$T = \\sqrt [3] {R - \\sqrt {Q^3 + R^2} }$ where: :$Q = \\dfrac {3 a c - b^2} {9 a^2}$ :$R = \\dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$"}
+{"_id": "1514", "title": "Ferrari's Method", "text": "Let $P$ be the quartic equation: :$a x^4 + b x^3 + c x^2 + d x + e = 0$ such that $a \\ne 0$. Then $P$ has solutions: :$x = \\dfrac {-p \\pm \\sqrt {p^2 - 8 q} } 4$ where: :$p = \\dfrac b a \\pm \\sqrt {\\dfrac {b^2} {a^2} - \\dfrac {4 c} a + 4 y_1}$ :$q = y_1 \\mp \\sqrt {y_1^2 - \\dfrac {4 e} a}$ where $y_1$ is a real solution to the cubic: :$y^3 - \\dfrac c a y^2 + \\paren {\\dfrac {b d} {a^2} - \\dfrac {4 e} a} y + \\paren {\\dfrac {4 c e} {a^2} - \\dfrac {b^2 e} {a^3} - \\dfrac {d^2} {a^2} } = 0$ '''Ferrari's method''' is a technique for solving this quartic."}
+{"_id": "1515", "title": "Properties of Binomial Coefficients", "text": "This page gathers together some of the simpler and more common identities concerning binomial coefficients."}
+{"_id": "1516", "title": "Prime Number Theorem", "text": "The prime-counting function $\\pi \\left({n}\\right)$, that is, the number of primes less than $n$, satisfies: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map \\pi n \\frac {\\map \\ln n} n = 1$ or equivalently:  :$\\map \\pi n \\sim \\dfrac n {\\map \\ln n}$ where $\\sim$ denotes asymptotic equivalence."}
+{"_id": "1517", "title": "Induced Structure Inverse", "text": "Let $\\struct {G, \\oplus}$ be a group whose identity is $e_G$. Let $S$ be a set. Let $\\struct {G^S, \\oplus}$ be the structure on $G^S$ induced by $\\oplus$. Let $f \\in G^S$. Let $f^* \\in G^S$ be defined as follows: :$\\forall f \\in G^S: \\forall x \\in S: \\map {f^*} x = \\paren {\\map f x}^{-1}$ Then $f^*$ is the inverse of $f$ for the pointwise operation induced on $G^S$ by $\\oplus$."}
+{"_id": "1518", "title": "Quaternion Group is Hamiltonian", "text": "The quaternion group $Q$ is Hamiltonian."}
+{"_id": "1519", "title": "Finite Subset of Metric Space has no Limit Points", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $X \\subseteq A$ such that $X$ is finite. Then $X$ has no limit points."}
+{"_id": "1520", "title": "Point in Finite Metric Space is Isolated", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $X \\subseteq A$ such that $X$ is finite. Let $x \\in X$. Then $x$ is isolated in $X$."}
+{"_id": "1521", "title": "Point in Finite Hausdorff Space is Isolated", "text": "Let $T = \\left({S, \\tau}\\right)$ be a Hausdorff space. Let $X \\subseteq S$ such that $X$ is finite. Let $x \\in X$. Then $x$ is isolated in $X$."}
+{"_id": "1522", "title": "Subsequence of Sequence in Metric Space with Limit", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $\\sequence {x_n}$ be a sequence in $M$. Let $x$ be a limit point of $S = \\set {x_n: n \\in \\N}$, the set of members of $\\sequence {x_n}$. Then $\\sequence {x_n}$ has a subsequence which converges to $x$."}
+{"_id": "1523", "title": "Primes of form Power Less One", "text": "Let $m, n \\in \\N_{>0}$ be natural numbers. Let $m^n - 1$ be prime. Then $m = 2$ and $n$ is prime."}
+{"_id": "1524", "title": "Lebesgue's Number Lemma", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $M$ be sequentially compact. Then there exists a Lebesgue number for every open cover of $M$."}
+{"_id": "1527", "title": "Second Subsequence Rule", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $\\left \\langle {x_n} \\right \\rangle$ be a sequence in $M$. Suppose $\\left \\langle {x_n} \\right \\rangle$ has a subsequence which is unbounded. Then $\\left \\langle {x_n} \\right \\rangle$ is divergent."}
+{"_id": "1529", "title": "Open Set Less One Point is Open", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $U \\subseteq M$ be an open set of $M$. Let $\\alpha \\in U$. Then $U \\setminus \\left\\{{\\alpha}\\right\\}$ is open in $M$."}
+{"_id": "1531", "title": "Finite Subspace of Dense-in-itself Metric Space is not Open", "text": "Let $M = \\struct {A, d}$ be a metric space that is dense-in-itself. Let $U$ be a finite subset of $A$. Then $U$ is not an open set of $M$."}
+{"_id": "1532", "title": "Region Less One Point is Region", "text": "Let $M = \\left({A, d}\\right)$ be a dense-in-itself metric space. Let $R \\subseteq M$ be a region of $M$. Let $x \\in R$. Then $R \\setminus \\left\\{ {x}\\right\\}$ is also a region of $M$."}
+{"_id": "1533", "title": "Theorem of Even Perfect Numbers", "text": "Let $a \\in \\N$ be an even perfect number. Then $a$ is in the form: :$2^{n - 1} \\paren {2^n - 1}$ where $2^n - 1$ is prime. Similarly, if $2^n - 1$ is prime, then $2^{n - 1} \\paren {2^n - 1}$ is perfect."}
+{"_id": "1535", "title": "Lucas-Lehmer Test", "text": "Let $q$ be an odd prime. Let $\\sequence {L_n}_{n \\mathop \\in \\N}$ be the recursive sequence in $\\Z / \\paren {2^q - 1} \\Z$ defined by: :$L_0 = 4, L_{n + 1} = L_n^2 - 2 \\pmod {2^q - 1}$ Then $2^q - 1$ is prime {{iff}} $L_{q - 2} = 0 \\pmod {2^q - 1}$."}
+{"_id": "1536", "title": "Sum Over Divisors Equals Sum Over Quotients", "text": "Let $n$ be a positive integer. Let $f: \\Z_{>0} \\to \\Z_{>0}$ be a mapping on the positive integers. Let $\\displaystyle \\sum_{d \\mathop \\divides n} \\map f d$ be the sum of $\\map f d$ over the divisors of $n$. Then: :$\\displaystyle \\sum_{d \\mathop \\divides n} \\map f d = \\sum_{d \\mathop \\divides n} \\map f {\\frac n d}$."}
+{"_id": "1537", "title": "Sum of Reciprocals of Divisors equals Abundancy Index", "text": "Let $n$ be a positive integer. Let $\\map \\sigma n$ denote the sigma function of $n$. Then: :$\\displaystyle \\sum_{d \\mathop \\divides n} \\frac 1 d = \\frac {\\map \\sigma n} n$ where $\\dfrac {\\map \\sigma n} n$ is the abundancy index of $n$."}
+{"_id": "1538", "title": "Sigma Function of Prime Number", "text": "Let $n$ be a positive integer. Let $\\map \\sigma n$ be the sigma function of $n$. Then $\\map \\sigma n = n + 1$ {{iff}} $n$ is prime."}
+{"_id": "1539", "title": "Sigma Function of Power of Prime", "text": "Let $n = p^k$ be the power of a prime number $p$. Let $\\map \\sigma n$ be the sigma function of $n$. That is, let $\\map \\sigma n$ be the sum of all positive divisors of $n$. Then: :$\\map \\sigma n = \\dfrac {p^{k + 1} - 1} {p - 1}$"}
+{"_id": "1540", "title": "Tau of Power of Prime", "text": "Let $n = p^k$ be the power of a prime number $p$. Let $\\tau \\left({n}\\right)$ be the $\\tau$ function of $n$. That is, let $\\tau \\left({n}\\right)$ be the number of positive divisors of $n$. Then $\\tau \\left({n}\\right) = k + 1$."}
+{"_id": "1541", "title": "Sum Over Divisors of Multiplicative Function", "text": "Let $f: \\Z_{>0} \\to \\Z_{>0}$ be a multiplicative function. Let $n \\in \\Z_{>0}$. Let $\\displaystyle \\sum_{d \\mathop \\divides n} \\map f d$ be the sum over the divisors of $n$. Then $\\displaystyle \\map F n = \\sum_{d \\mathop \\divides n} \\map f d$ is also a multiplicative function."}
+{"_id": "1542", "title": "Divisors of Product of Coprime Integers", "text": "Let $a \\divides b c$, where $b \\perp c$. Then $\\tuple {r, s}$ satisfying: :$a = r s$, where $r \\divides b$ and $s \\divides c$ is unique up to absolute value with: :$\\size r = \\gcd \\set {a, b}$ :$\\size s = \\gcd \\set {a, c}$"}
+{"_id": "1543", "title": "Unity Function is Completely Multiplicative", "text": "Let $f_1: \\Z_{> 0} \\to \\Z_{> 0}$ be the constant function: :$\\forall n \\in \\Z_{> 0}: f_1 \\left({n}\\right) = 1$ Then $f_1$ is completely multiplicative."}
+{"_id": "1544", "title": "Identity Function is Completely Multiplicative", "text": "Let $I_{\\Z_{>0}}: \\Z_{>0} \\to \\Z_{>0}$ be the identity function: :$\\forall n \\in \\Z_{>0}: I_{\\Z_{>0}} \\left({n}\\right) = n$ Then $I_{\\Z_{>0}}$ is completely multiplicative."}
+{"_id": "1545", "title": "Tau Function is Multiplicative", "text": "The divisor counting function: :$\\displaystyle \\tau: \\Z_{>0} \\to \\Z_{>0}: \\map \\tau n = \\sum_{d \\mathop \\divides n} 1$ is multiplicative."}
+{"_id": "1546", "title": "Sigma Function is Multiplicative", "text": "The sigma function: :$\\displaystyle \\sigma: \\Z_{>0} \\to \\Z_{>0}: \\sigma \\left({n}\\right) = \\sum_{d \\mathop \\backslash n} d$ is multiplicative."}
+{"_id": "1547", "title": "Primes of form Power of Two plus One", "text": "Let $n \\in \\N$ be a natural number. Let $2^n + 1$ be prime. Then $n = 2^k$ for some natural number $k$."}
+{"_id": "1548", "title": "Integer has Multiplicative Order Modulo n iff Coprime to n", "text": "Let $a$ and $n$ be integers. Let the multiplicative order of $a$ modulo $n$ exist. Then $a \\perp n$, that is, $a$ and $n$ are coprime."}
+{"_id": "1549", "title": "Integer to Power of Multiple of Order", "text": "Let $a$ and $n$ be integers. Let $a \\perp n$, that is, let $a$ and $b$ be coprime. Let $c \\in \\Z_{>0}$ be the multiplicative order of $a$ modulo $n$. Then $a^k \\equiv 1 \\pmod n$ {{iff}} $k$ is a multiple of $c$."}
+{"_id": "1550", "title": "Sum of Infinite Geometric Sequence", "text": "Let $S$ be a standard number field, that is $\\Q$, $\\R$ or $\\C$. Let $z \\in S$. Let $\\size z < 1$, where $\\size z$ denotes: :the absolute value of $z$, for real and rational $z$ :the complex modulus of $z$ for complex $z$. Then $\\displaystyle \\sum_{n \\mathop = 0}^\\infty z^n$ converges absolutely to $\\dfrac 1 {1 - z}$."}
+{"_id": "1551", "title": "Square of Riemann Zeta Function", "text": ":$\\displaystyle \\map {\\zeta^2} z = \\sum_{k \\mathop = 1}^\\infty \\frac {\\map d k} {k^z}$ where: :$\\zeta$ is the Riemann zeta function :$d$ is the divisor function."}
+{"_id": "1552", "title": "Derivative of Riemann Zeta Function", "text": "The derivative of the Riemann zeta function is: :$\\displaystyle \\map {\\zeta'} z = \\frac {\\d \\zeta} {\\d z} = -\\sum_{n \\mathop = 2}^\\infty \\frac {\\map \\ln n} {n^z}$"}
+{"_id": "1553", "title": "Order of Divisor Function", "text": "For all $x \\ge 1$: :$\\displaystyle \\sum_{n \\mathop \\le x} \\map d n = x \\log x + \\paren {2 \\gamma - 1} x + \\map \\OO {\\sqrt x}$ where $\\gamma$ is the Euler-Mascheroni constant and $\\map d n$ is the divisor function. {{explain|Which particular divisor function? Suggest that it might be $\\tau$ function that is being referred to.}}"}
+{"_id": "1554", "title": "Order of Möbius Function", "text": "Let $\\mu$ denote the Möbius function . Then: :$\\displaystyle \\sum_{n \\mathop \\le N} \\map \\mu n = \\map o N$ where $o$ denotes little-o notation."}
+{"_id": "1555", "title": "Ingham's Theorem on Convergent Dirichlet Series", "text": "Let $\\left\\vert{a_n}\\right\\vert \\le 1$ {{explain|What exactly is $a_n$ in this context?}} For a complex number $z \\in \\C$, let $\\Re \\left({z}\\right)$ denote the real part of $z$.  Form the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n n^{-z}$ which converges to an analytic function $F \\left({z}\\right)$ for $\\Re \\left({z}\\right) > 1$. {{explain|We have \"$\\Re \\left({z}\\right) > 1$\" used here and below to mean $\\left\\{ {z \\in \\C: \\Re \\left({z}\\right) > 1}\\right\\}$? In which case, rather than just call it \"$\\Re \\left({z}\\right) > 1$\", which is unwieldy and suboptimal, it might be better and clearer to give it a single-symbol identifier.}} Let $F \\left({z}\\right)$ be analytic throughout $\\Re \\left({z}\\right) \\ge 1$. Then $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n n^{-z}$ converges throughout $\\Re \\left({z}\\right) \\ge 1$."}
+{"_id": "1556", "title": "Alexander Polynomial is a Knot Invariant", "text": "The Alexander polynomial of an elementary knot $K$ is invariant under Reidemeister moves."}
+{"_id": "1557", "title": "Product Form of Sum on Completely Multiplicative Function", "text": "Let $f$ be a completely multiplicative arithmetic function. Let the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\map f n$ be absolutely convergent. Then: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\map f n = \\prod_p \\frac 1 {1 - \\map f p}$ where the infinite product ranges over the primes."}
+{"_id": "1559", "title": "Euler Phi Function of Product with Prime", "text": "Let $p$ be prime and $n \\in \\Z: n \\ge 1$. Then: :$\\phi \\left({p n}\\right) = \\begin{cases} \\left({p - 1}\\right) \\phi \\left({n}\\right) & : p \\nmid n \\\\ p \\phi \\left({n}\\right) & : p \\mathrel \\backslash n  \\end{cases}$ where $\\phi \\left({n}\\right)$ denotes the Euler $\\phi$ function of $n$. Thus for all $n \\ge 1$ and for any prime $p$, we have that $\\phi \\left({n}\\right)$ divides $\\phi \\left({p n}\\right)$."}
+{"_id": "1560", "title": "Euler Phi Function is Even for Argument greater than 2", "text": "Let $n \\in \\Z: n \\ge 1$. Let $\\map \\phi n$ be the Euler $\\phi$ function of $n$. Then $\\map \\phi n$ is even {{iff}} $n > 2$."}
+{"_id": "1561", "title": "Cauchy-Goursat Theorem", "text": "Let $D$ be a simply connected open subset of the complex plane $\\C$. Let $\\partial D$ denote the closed contour bounding $D$. Let $f: D \\to \\C$ be holomorphic everywhere in $D$. Then: :$\\displaystyle \\oint_{\\partial D} \\map f z \\rd z = 0$"}
+{"_id": "1562", "title": "Sigma Function of Integer", "text": "Let $n$ be an integer such that $n \\ge 2$. Let $\\map \\sigma n$ be the sigma function of $n$. That is, let $\\map \\sigma n$ be the sum of all positive divisors of $n$. Let the prime decomposition of $n$ be: :$\\displaystyle n = \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i} =  p_1^{k_1} p_2^{k_2} \\cdots p_r^{k_r}$ Then: :$\\displaystyle \\map \\sigma n = \\prod_{1 \\mathop \\le i \\mathop \\le r} \\frac {p_i^{k_i + 1} - 1} {p_i - 1}$"}
+{"_id": "1563", "title": "Product of Sigma and Euler Phi Functions", "text": "Let $n$ be an integer such that $n \\ge 2$. Let the prime decomposition of $n$ be: :$n = p_1^{k_1} p_2^{k_2} \\ldots p_r^{k_r}$ Let $\\sigma \\left({n}\\right)$ be the sigma function of $n$. Let $\\phi \\left({n}\\right)$ be the Euler phi function of $n$. Then: :$\\displaystyle \\sigma \\left({n}\\right) \\phi \\left({n}\\right) = n^2 \\prod_{1 \\mathop \\le i \\mathop \\le r} \\left({1 - \\frac 1 {p_i^{k_i + 1}}}\\right)$"}
+{"_id": "1565", "title": "Order Modulo n of Power of Integer", "text": "Let $a$ have multiplicative order $c$ modulo $n$. Then for any $k \\ge 1$, $a^k$ has multiplicative order $\\dfrac c {\\gcd \\left\\{{c, k}\\right\\}}$ modulo $n$."}
+{"_id": "1566", "title": "Integers with Primitive Roots", "text": "Let $n \\in \\Z: n > 1$. Then $n$ has a primitive root {{iff}} one of the following holds: : $n = 2$ : $n = 4$ : $n = p^k$ : $n = 2 p^k$ where $p > 2$ is prime and $k \\ge 1$."}
+{"_id": "1567", "title": "Tau Function of Square-Free Integer is Power of 2", "text": "Let $n$ be a square-free integer. Let $\\tau: \\Z \\to \\Z$ be the $\\tau$ function. Then $\\map \\tau n = 2^r$ for some $r \\ge 1$. The converse is not true in general. That is, if $\\map \\tau n = 2^r$ for some $r \\ge 1$, it is not necessarily the case that $n$ is square-free."}
+{"_id": "1568", "title": "Tau Function Odd Iff Argument is Square", "text": "Let $\\tau: \\Z \\to \\Z$ be the $\\tau$ function. Then $\\tau \\left({n}\\right)$ is odd {{iff}} $n$ is square."}
+{"_id": "1569", "title": "Sigma Function Odd iff Argument is Square or Twice Square", "text": "Let $\\sigma: \\Z \\to \\Z$ be the sigma function. Then $\\sigma \\left({n}\\right)$ is odd {{iff}} $n$ is either square or twice a square."}
+{"_id": "1572", "title": "Cancellability of Congruences", "text": "Let $a, b, c, n \\in \\Z$ be integers. Then: :$c a \\equiv c b \\pmod n \\iff a \\equiv b \\pmod {n / d}$ where $d = \\gcd \\set {c, n}$."}
+{"_id": "1573", "title": "Solution of Linear Diophantine Equation", "text": "The linear Diophantine equation: :$a x + b y  = c$ has solutions {{iff}}: :$\\gcd \\set {a, b} \\divides c$ where $\\divides$ denotes divisibility. If this condition holds with $\\gcd \\set {a, b} > 1$ then division by $\\gcd \\set {a, b}$ reduces the equation to: :$a' x + b' y = c'$ where $\\gcd \\set {a', b'} = 1$. If $x_0, y_0$ is one solution of the latter equation, then the general solution is: :$\\forall k \\in \\Z: x = x_0 + b' k, y = y_0 - a' k$ or: :$\\forall k \\in \\Z: x = x_0 + \\dfrac b d k, y = y_0 - \\dfrac a d k$"}
+{"_id": "1574", "title": "GCD with Prime", "text": "Let $p$ be a prime number. Then: :$\\forall n \\in \\Z: \\gcd \\set {n, p} = \\begin{cases} p & : p \\divides n \\\\ 1 & : p \\nmid n \\end{cases}$"}
+{"_id": "1575", "title": "Prime Divides Power", "text": "Let $p$ be a prime number. Let $a, n \\in \\Z$ be integers. Then $p$ divides $a^n$ {{iff}} $p^n$ divides $a^n$."}
+{"_id": "1576", "title": "Existence of Laurent Series", "text": "Let $z_0 \\in \\C$ be a complex number. Let $R \\in \\R_{>0}$ be a real number. Let $\\map {B'} {z_0, R}$ be the open punctured disk at $z_0$ of radius $R$. Let $f: \\map {B'} {z_0, R} \\to \\C$ be holomorphic. Then there exists a sequence $\\sequence {a_n}_{n \\mathop \\in \\Z}$ such that: :$\\map f z = \\displaystyle \\sum_{n = -\\infty}^\\infty a_n \\paren {z - z_0}^n$ for all $z \\in B'(z_0, R)$."}
+{"_id": "1577", "title": "Congruence by Divisor of Modulus", "text": "Let $z \\in \\R$ be a real number. Let $a, b \\in \\R$ such that $a$ is congruent modulo $z$ to $b$, that is: :$a \\equiv b \\pmod z$ Let $m \\in \\R$ such that $z$ is an integer multiple of $m$: :$\\exists k \\in \\Z: z = k m$ Then: : $a \\equiv b \\pmod m$"}
+{"_id": "1579", "title": "Congruence of Quotient", "text": "Let $a, b \\in \\Z$ and $n \\in \\N$. Let $a$ be congruent to $b$ modulo $n$, i.e. $a \\equiv b \\pmod n$. Let $d \\in \\Z: d > 0$ such that $d$ is a common divisor of $a, b$ and $n$. Then: : $\\displaystyle \\frac a d \\equiv \\frac b d \\pmod {n / d}$"}
+{"_id": "1580", "title": "Divisibility by 11", "text": "Let $N \\in \\N$ be expressed as: :$N = a_0 + a_1 10 + a_2 10^2 + \\cdots + a_n 10^n$ Then $N$ is divisible by $11$ {{iff}} $\\displaystyle \\sum_{r \\mathop = 0}^n \\paren {-1}^r a_r$ is  divisible by $11$. That is, a divisibility test for $11$ is achieved by alternately adding and subtracting the digits and taking the result modulo $11$."}
+{"_id": "1581", "title": "Solutions of Polynomial Congruences", "text": "Let $P \\left({x}\\right)$ be an integral polynomial. Let $a \\equiv b \\pmod n$. Then $P \\left({a}\\right) \\equiv P \\left({b}\\right) \\pmod n$. In particular, $a$ is a solution to the polynomial congruence $P \\left({x}\\right) \\equiv 0 \\pmod n$ iff $b$ is also."}
+{"_id": "1582", "title": "Solution of Linear Congruence", "text": "Let $a x \\equiv b \\pmod n$ be a linear congruence. The following results hold:"}
+{"_id": "1583", "title": "Chinese Remainder Theorem/General Result", "text": "Let $n_1, n_2, \\ldots, n_r$ be positive integers such that $n_i \\perp n_j$ for all $i \\ne j$ (that is, $\\gcd \\set {n_i, n_j} = 1$). Then the system of linear congruences: :$x \\equiv b_1 \\pmod {n_1}$ :$x \\equiv b_2 \\pmod {n_2}$ :$\\ldots$ :$x \\equiv b_r \\pmod {n_r}$ has a simultaneous solution which is unique modulo $\\displaystyle \\prod_{i \\mathop = 1}^r n_i$."}
+{"_id": "1585", "title": "Second Supplement to Law of Quadratic Reciprocity", "text": ":$\\paren {\\dfrac 2 p} = \\paren {-1}^{\\paren {p^2 - 1} / 8} = \\begin{cases} +1 & : p \\equiv \\pm 1 \\pmod 8 \\\\ -1 & : p \\equiv \\pm 3 \\pmod 8 \\end{cases}$"}
+{"_id": "1586", "title": "First Supplement to Law of Quadratic Reciprocity", "text": ":$\\paren {\\dfrac {-1} p} = \\paren {-1}^{\\paren {p - 1} / 2} = \\begin{cases} +1 & : p \\equiv 1 \\pmod 4 \\\\ -1 & : p \\equiv 3 \\pmod 4 \\end{cases}$"}
+{"_id": "1587", "title": "Manipulation of Absolutely Convergent Series", "text": "Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be a real or complex series that is absolutely convergent."}
+{"_id": "1588", "title": "Product of Sums", "text": "Let $\\displaystyle \\sum_{n \\mathop \\in A} a_n$ and $\\displaystyle \\sum_{n \\mathop \\in B} b_n$ be absolutely convergent sequences. Then: :$\\displaystyle \\left({ \\sum_{i \\mathop \\in A} a_i }\\right) \\left({ \\sum_{j \\mathop \\in B} b_j }\\right) = \\sum_{\\left({i, j}\\right) \\mathop \\in A \\times B} a_i b_j$."}
+{"_id": "1589", "title": "Green's Theorem", "text": "Let $\\Gamma$ be a positively oriented piecewise smooth simple closed curve in  $\\R^2$. Let $U = \\map {\\operatorname {Int} } \\Gamma$, that is, the interior of $\\Gamma$. Let $A$ and $B$ be functions of $\\tuple {x, y}$ defined on an open region containing $U$ and have continuous partial derivatives in such a set. Then: :$\\displaystyle \\oint_\\Gamma \\paren {A \\rd x + B \\rd y} = \\iint_U \\paren {\\frac {\\partial B} {\\partial x} - \\frac {\\partial A} {\\partial y} } \\rd x \\rd y$"}
+{"_id": "1590", "title": "Gauss-Ostrogradsky Theorem", "text": "Let $U$ be a subset of $\\R^3$ which is compact and has a piecewise smooth boundary $\\partial U$. Let $\\mathbf F: \\R^3 \\to \\R^3$ be a smooth vector function defined on a neighborhood of $U$. Then: :$\\displaystyle \\iiint \\limits_U \\paren {\\nabla \\cdot \\mathbf F} \\rd V = \\iint \\limits_{\\partial U} \\mathbf F \\cdot \\mathbf n \\rd S$ where $\\mathbf n$ is the normal to $\\partial U$."}
+{"_id": "1591", "title": "Implications of Stokes' Theorem", "text": "Stokes' Theorem implies all of the following results: * Classical Stokes' Theorem * Green's Theorem * Gauss's Theorem * Fundamental Theorem of Calculus * Cauchy's Residue Theorem"}
+{"_id": "1592", "title": "Divergence Test", "text": "Let $\\sequence {a_n}$ be a sequence in $\\R$. If $\\displaystyle \\lim_{k \\mathop \\to \\infty} a_k \\ne 0$, then $\\displaystyle \\sum_{i \\mathop = 1}^\\infty a_n$ diverges."}
+{"_id": "1593", "title": "Eisenstein's Lemma", "text": "Let $p$ be an odd prime. Let $a \\in \\Z$ be an odd integer such that $p \\nmid a$. Let $\\paren {\\dfrac a p}$ be defined as the Legendre symbol. Then: :$\\paren {\\dfrac a p} = \\paren {-1}^{\\map \\alpha {a, p} }$ where: :$\\displaystyle \\map \\alpha {a, p} = \\sum_{k \\mathop = 1}^{\\frac {p - 1} 2} \\floor {\\frac {k a} p}$ :$\\floor {\\dfrac {k a} p}$ is the floor function of $\\dfrac {k a} p$."}
+{"_id": "1594", "title": "Law of Quadratic Reciprocity", "text": "Let $p$ and $q$ be distinct odd primes. Then: :$\\paren {\\dfrac p q} \\paren {\\dfrac q p} = \\paren {-1}^{\\dfrac {\\paren {p - 1} \\paren {q - 1} } 4}$ where $\\paren {\\dfrac p q}$ and $\\paren {\\dfrac q p}$ are defined as the Legendre symbol. An alternative formulation is: $\\paren {\\dfrac p q} = \\begin{cases} \\quad \\paren {\\dfrac q p} & : p \\equiv 1 \\lor q \\equiv 1 \\pmod 4 \\\\ -\\paren {\\dfrac q p} & : p \\equiv q \\equiv 3 \\pmod 4 \\end{cases}$ The fact that these formulations are equivalent is immediate. This fact is known as the '''Law of Quadratic Reciprocity''', or '''LQR''' for short."}
+{"_id": "1595", "title": "Rational Numbers and SFCFs are Equivalent", "text": "Every simple finite continued fraction has a rational value. Conversely, every rational number can be expressed as a simple finite continued fraction."}
+{"_id": "1596", "title": "Properties of Convergents of Continued Fractions", "text": "Let $n \\in \\N \\cup \\set \\infty$ be an extended natural number. Let $\\sqbrk {a_0, a_1, a_2, \\ldots}$ be a continued fraction in $\\R$ of length $n$. Let $p_0, p_1, p_2, \\ldots$ and $q_0, q_1, q_2, \\ldots$ be its numerators and denominators. Let $C_0, C_1, C_2, \\ldots$ be the convergents of $\\sqbrk {a_0, a_1, a_2, \\ldots}$. Then the following results apply:"}
+{"_id": "1597", "title": "Manipulation of Exterior Derivative", "text": "For the exterior derivative, the following statements are true: :$\\map \\d {a b} = a \\rd b + \\paren {\\d a} b$ :$\\map \\d {a \\wedge b} = \\d a \\wedge b - a \\wedge \\d b$ where $a \\wedge b$ is the wedge product."}
+{"_id": "1599", "title": "General Stokes' Theorem", "text": "Let $\\omega$ be a smooth $\\paren {n - 1}$-form with compact support on a smooth $n$-dimensional oriented manifold $X$. Let the boundary of $X$ be $\\partial X$. Then: :$\\displaystyle \\int_{\\partial X} \\omega = \\int_X \\rd \\omega$ where $\\d \\omega$ is the exterior derivative of $\\omega$."}
+{"_id": "1600", "title": "Kelvin-Stokes Theorem", "text": "Let $S$ be some orientable smooth surface with boundary in $\\R^3$. Let $\\mathbf F:\\R^3 \\to \\R^3$ be a vector-valued function with Euclidean coordinate expression: :$F = f_1 \\mathbf i + f_2 \\mathbf j + f_3 \\mathbf k$ where $f_i: \\R^3 \\to \\R$. Then: :$\\displaystyle \\oint_{\\partial S} f_1 \\rd x + f_2 \\rd y + f_3 \\rd z = \\iint_S \\paren {\\nabla \\times \\mathbf F} \\cdot \\mathbf n \\rd A$ where $\\mathbf n$ is the unit normal to $S$ and $\\d A$ is the area element on the surface."}
+{"_id": "1601", "title": "Even Convergent of Simple Continued Fraction is Strictly Smaller than Odd Convergent", "text": "Every even convergent is strictly smaller than every odd convergent."}
+{"_id": "1602", "title": "Denominators of Simple Continued Fraction are Strictly Increasing", "text": "Let $N \\in \\N \\cup \\set \\infty$ be an extended natural number. Let $\\sqbrk {a_0, a_1, a_2, \\ldots}$ be a simple continued fraction in $\\R$ of length $N$. Let $q_0, q_1, q_2, \\ldots$ be its denominators. Then with the possible exception of $q_0 = q_1$, the sequence $\\sequence {q_n}$ is strictly increasing."}
+{"_id": "1603", "title": "Simple Infinite Continued Fraction Converges to Irrational Number", "text": "The value of any simple infinite continued fraction in $\\R$ is irrational."}
+{"_id": "1604", "title": "Uniqueness of Analytic Continuation", "text": "Let $U \\subset V \\subset \\C$ be open subsets of the complex plane. Let $V$ be connected. Suppose: :$(1): \\quad F_1, F_2$ are functions defined on $V$ :$(2): \\quad f$ is a function defined on $U$. Let $F_1$ and $F_2$ be analytic continuations of $f$ to $V$. Then $F_1 = F_2$."}
+{"_id": "1605", "title": "Zeroes of Analytic Function are Isolated", "text": "Let $U \\subset \\C$ be some open set and let $f$ be an analytic function defined on $U$. Then either $f$ is a constant function, or the set $\\set {z \\in U: \\map f z = 0}$ is totally disconnected."}
+{"_id": "1606", "title": "Irrational Number is Limit of Unique Simple Infinite Continued Fraction", "text": "Let $x$ be an irrational number. Then the continued fraction expansion of $x$ is the unique simple infinite continued fraction that converges to $x$."}
+{"_id": "1607", "title": "Analytic Continuation of Riemann Zeta Function", "text": "The Riemann zeta function is meromorphic on $\\C$."}
+{"_id": "1608", "title": "Accuracy of Convergents of Continued Fraction Expansion of Irrational Number", "text": "Let $x$ be an irrational number. Let $(a_0, a_1, \\ldots)$ be its continued fraction expansion. Let $\\left \\langle {C_n}\\right \\rangle_{n \\geq 0}$ be its sequence of convergents. Let $p_0, p_1, p_2, \\ldots$ and $q_0, q_1, q_2, \\ldots$ be its numerators and denominators. Then: :$\\forall k \\ge 1: \\left\\vert{x - \\dfrac {p_{k + 1} } {q_{k + 1} } }\\right\\vert < \\dfrac 1 {q_{k + 1} q_{k + 2} } \\le \\dfrac 1 {2 q_k q_{k + 1} } < \\left\\vert{x - \\dfrac {p_k} {q_k} }\\right\\vert$ Thus: :The {{LHS}} of the inequality gives an indication of how close each convergent gets to its true value. :The {{RHS}} gives a bound that limits its accuracy."}
+{"_id": "1609", "title": "Trivial Zeroes of Riemann Zeta Function are Even Negative Integers", "text": "Let $\\rho = \\sigma + i t$ be a zero of the Riemann zeta function not contained in the critical strip: :$0 \\le \\map \\Re s \\le 1$ Then: :$s \\in \\set {-2, -4, -6, \\ldots}$ These are called the '''trivial zeros''' of $\\zeta$."}
+{"_id": "1610", "title": "Poles of Gamma Function", "text": "The gamma function $\\Gamma: \\C \\to \\C$ is analytic throughout the complex plane except at $\\set {0, -1, -2, -3, \\dotsc}$ where it has simple poles."}
+{"_id": "1611", "title": "Zeroes of Gamma Function", "text": "The Gamma function is never equal to $0$."}
+{"_id": "1612", "title": "Gamma Difference Equation", "text": ":$\\map \\Gamma {z + 1} = z \\, \\map \\Gamma z$"}
+{"_id": "1614", "title": "Convergents are Best Approximations", "text": "Let $x$ be an irrational number. Let $(p_n)_{n\\geq0}$ and $(q_n)_{n\\geq0}$ be the numerators and denominators of its continued fraction expansion. Let $\\dfrac {p_n} {q_n}$ be the $n$th convergent. Let $\\dfrac a b$ be any rational number such that $0 < b < q_{n+1}$. Then: :$\\forall n > 1: \\left\\vert{q_n x - p_n}\\right\\vert \\le \\left\\vert{b x - a}\\right\\vert$ The equality holds only if $a = p_n$ and $b = q_n$."}
+{"_id": "1615", "title": "Condition for Rational to be a Convergent", "text": "Let $x$ be an irrational number. Let the rational number $\\dfrac a b$ satisfy the inequality: :$\\left\\vert{x - \\dfrac a b}\\right\\vert < \\dfrac 1 {2 b^2}$ Then $\\dfrac a b$ is a convergent of $x$."}
+{"_id": "1616", "title": "Quadratic Irrational is Root of Quadratic Equation", "text": "Let $x$ be a quadratic irrational. Then $x$ is a solution to a quadratic equation with rational coefficients."}
+{"_id": "1617", "title": "Irrational Number has Periodic Continued Fraction iff Quadratic", "text": "Let $x$ be an irrational number. Then $x$ is a quadratic irrational {{iff}} its continued fraction expansion is periodic."}
+{"_id": "1618", "title": "Continued Fraction Expansion of Irrational Square Root", "text": "Let $n \\in \\Z$ such that $n$ is not a square. Then the continued fraction expansion of $\\sqrt n$ is of the form: :$\\sqbrk {a_1 \\sequence {b_1, b_2, \\ldots, b_{m - 1}, b_m, b_{m - 1}, \\ldots, b_2, b_1, 2 a_1} }$ or :$\\sqbrk {a_1 \\sequence {b_1, b_2, \\ldots, b_{m - 1}, b_m, b_m, b_{m - 1}, \\ldots, b_2, b_1, 2 a_1} }$ where $m \\in \\Z: m \\ge 0$. That is, it has the form as follows: :It is periodic :It starts with an integer $a_1$ :Its cycle starts with a palindromic section, either: :::$b_1, b_2, \\ldots, b_{m - 1}, b_m, b_{m - 1}, \\ldots, b_2, b_1$ ::or: :::$b_1, b_2, \\ldots, b_{m - 1}, b_m, b_m, b_{m - 1}, \\ldots, b_2, b_1$ ::which may be of length zero :Its cycle ends with twice the first partial quotient."}
+{"_id": "1619", "title": "Solution of Pell's Equation is a Convergent", "text": "Let $x = a, y = b$ be a positive solution to Pell's Equation $x^2 - n y^2 = 1$. Then $\\dfrac a b$ is a convergent of $\\sqrt n$."}
+{"_id": "1620", "title": "Convergence of P-Series", "text": "Let $p \\in \\C$ be a complex number."}
+{"_id": "1621", "title": "Square Modulo 4", "text": "Let $x \\in \\Z$ be an integer. Then $x$ is: :even {{iff}} $x^2 \\equiv 0 \\pmod 4$ :odd {{iff}} $x^2 \\equiv 1 \\pmod 4$"}
+{"_id": "1622", "title": "Parity of Smaller Elements of Primitive Pythagorean Triple", "text": "Let $\\left({x, y, z}\\right)$ be a Pythagorean triple, that is, integers such that $x^2 + y^2 = z^2$. Then $x$ and $y$ are of opposite parity."}
+{"_id": "1623", "title": "Dirichlet Series is Analytic", "text": "Let $(a_n)$ be sequence of complex numbers. Let :$\\displaystyle f \\left({z}\\right) = \\sum_{n \\mathop = 1}^\\infty \\frac {a_n} {n^z}$ be the associated Dirichlet Series, which is defined at the points where the series converges. Then $f$ is analytic in every open set such that the sum converges in the set."}
+{"_id": "1624", "title": "Consecutive Integers are Coprime", "text": "$\\forall h \\in \\Z$, $h$ and $h + 1$ have only two common factors: $1$ and $-1$. That is, consecutive integers are always coprime."}
+{"_id": "1625", "title": "Elements of Primitive Pythagorean Triple are Pairwise Coprime", "text": "Let $\\tuple {x, y, z}$ be a primitive Pythagorean triple. Then: :$x \\perp y$ :$y \\perp z$ :$x \\perp z$ That is, all elements of $\\tuple {x, y, z}$ are pairwise coprime."}
+{"_id": "1628", "title": "Odd Square Modulo 8", "text": "Let $x \\in \\Z$ be an odd square. Then $x \\equiv 1 \\pmod 8$."}
+{"_id": "1629", "title": "Sum of Consecutive Triangular Numbers is Square", "text": "The sum of two consecutive triangular numbers is a square number."}
+{"_id": "1630", "title": "If n is Triangular then so is 9n + 1", "text": "Let $n$ be a triangular number. Then $9 n + 1$ is also triangular."}
+{"_id": "1631", "title": "Sum of Arithmetic Sequence", "text": "Let $\\sequence {a_k}$ be an arithmetic sequence defined as: :$a_k = a + k d$ for $n = 0, 1, 2, \\ldots, n - 1$ Then its closed-form expression is: {{begin-eqn}} {{eqn | l = \\sum_{k \\mathop = 0}^{n - 1} \\paren {a + k d}       | r = n \\paren {a + \\frac {n - 1} 2 d}       | c =  }} {{eqn | r = \\frac {n \\paren {a + l} } 2       | c = where $l$ is the last term of $\\sequence {a_k}$ }} {{end-eqn}}"}
+{"_id": "1632", "title": "Odd Square is Eight Triangles Plus One", "text": "Let $n \\in \\Z$ be an odd integer. Then $n$ is square {{iff}} $n = 8 m + 1$ where $m$ is triangular."}
+{"_id": "1633", "title": "Triangular Number Modulo 3 and 9", "text": "Let $n$ be a triangular number. Then one of the following two conditions applies: : $n \\equiv 0 \\pmod 3$ : $n \\equiv 1 \\pmod 9$"}
+{"_id": "1634", "title": "Closed Form for Polygonal Numbers", "text": "Let $\\map P {k, n}$ be the $n$th $k$-gonal number. The closed-form expression for $\\map P {k, n}$ is given by: {{begin-eqn}} {{eqn | l = \\map P {k, n}       | r = \\frac n 2 \\paren {\\paren {k - 2} n - k + 4} }} {{eqn | r = \\frac {k - 2} 2 \\paren {n^2 - n} + n }} {{end-eqn}}"}
+{"_id": "1635", "title": "Difference Between Adjacent Polygonal Numbers is Triangular Number", "text": "Let $\\map P {k, n}$ be the $n$th $k$-gonal number. Then: :$\\map P {k + 1, n} - \\map P {k, n} = T_{n - 1}$ where $T_n$ is the $n$th triangular number."}
+{"_id": "1636", "title": "Method of Infinite Descent", "text": "Let $P \\left({n_\\alpha}\\right)$ be a propositional function depending on $n_\\alpha \\in \\N$. Let $P \\left({n_\\alpha}\\right) \\implies P \\left({n_\\beta}\\right)$ such that $0 < n_\\beta < n_\\alpha$. Then we may deduce that $P \\left({n}\\right)$ is false for all $n \\in \\N$. That is, suppose that by assuming the truth of $P \\left({n_\\alpha}\\right)$ for ''any'' natural number $n_\\alpha$, we may deduce that there always exists ''some'' number $n_\\beta$ strictly less than $n_\\alpha$ for which $P \\left({n_\\beta}\\right)$ is also true, then $P \\left({n_\\alpha}\\right)$ can not be true after all. This technique is known as the '''method of infinite descent'''. The process of deducing the truth of $P \\left({n_\\beta}\\right)$ from $P \\left({n_\\alpha}\\right)$ such that $0 < n_\\beta < n_\\alpha$ is known as the '''descent step'''."}
+{"_id": "1637", "title": "Brahmagupta-Fibonacci Identity", "text": "Let $a, b, c, d$ be numbers. Then: :$\\paren {a^2 + b^2} \\paren {c^2 + d^2} = \\paren {a c + b d}^2 + \\paren {a d - b c}^2$"}
+{"_id": "1638", "title": "Fermat's Two Squares Theorem", "text": "Let $p$ be a prime number. Then $p$ can be expressed as the sum of two squares {{iff}} either: : $p = 2$ or: : $p \\equiv 1 \\pmod 4$ The expression of a prime of the form $4 k + 1$ as the sum of two squares is unique except for the order of the two summands."}
+{"_id": "1639", "title": "Integer as Sum of Two Squares", "text": "Let $n$ be a positive integer. Then: : $n$ can be expressed as the sum of two squares {{iff}}: :each of its prime divisors of the form $4 k + 3$ (if any) occur to an even power."}
+{"_id": "1640", "title": "Square Modulo 8", "text": "Let $x \\in \\Z$ be an integer. : If $x$ is even then: ::: $x^2 \\equiv 0 \\pmod 8$ or $x^2 \\equiv 4 \\pmod 8$ : If $x$ is odd then: ::: $x^2 \\equiv 1 \\pmod 8$"}
+{"_id": "1641", "title": "Integer as Sum of Three Squares", "text": "Let $r$ be a positive integer. Then $r$ can be expressed as the sum of three squares {{iff}} it is not of the form: :$4^n \\paren {8 m + 7}$ for some $m, n \\in \\Z_{\\ge 0}$."}
+{"_id": "1642", "title": "Product of Sums of Four Squares", "text": "Let $a, b, c, d, w, x, y, z$ be numbers. Then: {{begin-eqn}} {{eqn | o =        | r = \\left({a^2 + b^2 + c^2 + d^2}\\right) \\left({w^2 + x^2 + y^2 + z^2}\\right)       | c =  }} {{eqn | l = =       | o =        | r = \\left({a w + b x + c y + d z}\\right)^2       | c =  }} {{eqn | o = +       | r = \\left({a x - b w + c z - d y}\\right)^2       | c =  }} {{eqn | o = +       | r = \\left({a y - b z - c w + d x}\\right)^2       | c =  }} {{eqn | o = +       | r = \\left({a z + b y - c x - d w}\\right)^2       | c =  }} {{end-eqn}}"}
+{"_id": "1643", "title": "Lagrange's Four Square Theorem", "text": "Every positive integer can be expressed as a sum of four squares."}
+{"_id": "1644", "title": "Hilbert-Waring Theorem", "text": "For each $k \\in \\Z: k \\ge 2$, there exists a positive integer $\\map g k$ such that every positive integer can be expressed as a sum of at most $\\map g k$ $k$th powers."}
+{"_id": "1645", "title": "Necessary Condition for Existence of BIBD", "text": "Let there exist be a BIBD with parameters $v, b, r, k, \\lambda$. Then the following are true: : $(1): \\quad b k = r v$ : $(2): \\quad \\lambda \\paren {v - 1} = r \\paren {k - 1}$ : $(3): \\quad b \\dbinom k 2 = \\lambda \\dbinom v 2$ : $(4): \\quad k < v$ : $(5): \\quad r > \\lambda$ All of $v, b, r, k, \\lambda$ are integers. Some sources prefer to report $(3)$ as: :$b = \\dfrac {\\dbinom v 2} {\\dbinom k 2} \\lambda$ which is less appealing visually, and typographically horrendous."}
+{"_id": "1646", "title": "Mapping from Singleton is Injection", "text": "Let $f: S \\to T$ be a mapping. Let $S$ be a singleton. Then $f$ is an injection."}
+{"_id": "1647", "title": "Mapping to Singleton is Surjection", "text": "Let $S$ be a non-empty set. Let $f: S \\to T$ be a mapping. Let $T$ be a singleton. Then $f$ is a surjection."}
+{"_id": "1648", "title": "Generating Function for Constant Sequence", "text": "Let $\\sequence {a_n}$ be the sequence defined as: : $\\forall n \\in \\N: a_n = r$ for some $r \\in \\R$. Then the generating function for $\\sequence {a_n}$ is given as: :$\\map G z = \\dfrac r {1 - z}$ for $\\size z < 1$"}
+{"_id": "1649", "title": "Area of Circle", "text": "The area $A$ of a circle is given by: : $A = \\pi r^2$ where $r$ is the radius of the circle."}
+{"_id": "1651", "title": "Distance Formula", "text": "The distance $d$ between two points $A = \\tuple {x_1, y_1}$ and $B = \\tuple {x_2, y_2}$ on a Cartesian plane is: :$d = \\sqrt {\\paren {x_1 - x_2}^2 + \\paren {y_1 - y_2}^2}$"}
+{"_id": "1652", "title": "Equation of Circle", "text": "The equation of a circle with radius $R$ and center $\\tuple {a, b}$ can be expressed in the following forms:"}
+{"_id": "1653", "title": "Null URM Program Computes Identity Function", "text": "The null URM program computes the '''identity function''' $I_\\N: \\N \\to \\N$, defined as: :$\\forall n \\in \\N: \\map {I_\\N} n = n$"}
+{"_id": "1654", "title": "Composition of One-Variable URM Computable Functions", "text": "Let $f: \\N \\to \\N$ and $g: \\N \\to \\N$ be URM computable functions of one variable. Let $f \\circ g$ be the composition of $f$ and $g$. Then $f \\circ g: \\N \\to \\N$ is a URM computable function."}
+{"_id": "1655", "title": "Concatenation of URM Programs is Associative", "text": "Let $P, Q, R$ be one-variable URM programs. Then the concatenated URM programs $P * \\left({Q * R}\\right)$ and $\\left({P * Q}\\right) * R$ are the same."}
+{"_id": "1656", "title": "Primitive of Cosine Function", "text": ":$\\ds \\int \\cos x \\rd x = \\sin x + C$"}
+{"_id": "1657", "title": "Primitive of Sine Function", "text": ":$\\ds \\int \\sin x \\rd x = -\\cos x + C$"}
+{"_id": "1658", "title": "Function Obtained by Substitution from URM Computable Functions", "text": "Let the functions $f: \\N^t \\to \\N, g_1: \\N^k \\to \\N, g_2: \\N^k \\to \\N, \\ldots, g_t: \\N^k \\to \\N$ all be URM computable functions. Let $h: \\N^k \\to \\N$ be defined from $f, g_1, g_2, \\ldots, g_t$ by substitution. Then $h$ is also URM computable."}
+{"_id": "1659", "title": "Normalized URM Program", "text": "Let $P$ be a URM program. Let $l = \\map \\lambda P$ be the number of basic instructions in $P$. Let $u = \\map \\rho P$ be the number of registers used by $P$. Then $P$ can be modified as follows: :Every <tt>Jump</tt> of the form $\\map J {m, n, q}$ where $q > l$ may be replaced by $\\map J {m, n, l + 1}$ :If $u > 0$, a Clear Registers Program $\\map Z {2, u}$ can be appended to the end of $P$ at lines $l + 1$ to $l + u - 1$. The new program $P'$ that results from the above modifications produces exactly the same output as $P$ for each input. Note now though that $\\map \\lambda {P'} = l + u - 1$. Such a program as $P'$ is called a '''normalized URM program'''. The point of doing this is so as to make programs easier to concatenate. Once the above have been done, each program has a well-defined exit line which can be used as the start line of the program that immediately follows it."}
+{"_id": "1660", "title": "Block Copy Program", "text": "Let $k, m, n \\in \\N$ be natural numbers such that: * $k \\ge 1$; * $\\left|{m - n}\\right| \\ge k$. The URM program defined as: {| |- ! align=\"right\" | Line !! ! align=\"left\" | Command |- | align=\"right\" | $1$ || | align=\"left\" | $C \\left({m, n}\\right)$ |- | align=\"right\" | $2$ || | align=\"left\" | $C \\left({m+1, n+1}\\right)$ |- | align=\"right\" | $\\vdots$ || | align=\"left\" | $\\vdots$ |- | align=\"right\" | $k$ || | align=\"left\" | $C \\left({m+k-1, n+k-1}\\right)$ |} is called a '''block copy program'''. It is abbreviated $C \\left({m, n, k}\\right)$. It has the effect of copying the contents of registers $R_m, R_{m+1}, \\ldots, R_{m+k-1}$ into the registers $R_n, R_{n+1}, \\ldots, R_{n+k-1}$ respectively. It has length defined as $\\lambda \\left({C \\left({m, n, k}\\right)}\\right) = k$."}
+{"_id": "1661", "title": "Function Obtained by Primitive Recursion from URM Computable Functions", "text": "Let the functions $f: \\N^k \\to \\N$ and $g: \\N^{k+2} \\to \\N$ be URM computable functions. Let $h: \\N^{k+1} \\to \\N$ be obtained from $f$ and $g$ by primitive recursion. Then $h$ is also URM computable."}
+{"_id": "1662", "title": "Double Angle Formulas", "text": "=== Double Angle Formula for Sine === {{:Double Angle Formulas/Sine}} === Double Angle Formula for Cosine === {{:Double Angle Formulas/Cosine}} === Double Angle Formula for Tangent === {{:Double Angle Formulas/Tangent}}"}
+{"_id": "1663", "title": "Half Angle Formulas", "text": "=== Half Angle Formula for Sine === {{:Half Angle Formulas/Sine}} === Half Angle Formula for Cosine === {{:Half Angle Formulas/Cosine}} === Half Angle Formula for Tangent === {{:Half Angle Formulas/Tangent}} === Half Angle Formula for Tangent: Corollary 1 === {{:Half Angle Formulas/Tangent/Corollary 1}} === Half Angle Formula for Tangent: Corollary 2 === {{:Half Angle Formulas/Tangent/Corollary 2}} === Half Angle Formula for Tangent: Corollary 3 === {{:Half Angle Formulas/Tangent/Corollary 3}} === One Plus Tangent Half Angle over One Minus Tangent Half Angle === {{:One Plus Tangent Half Angle over One Minus Tangent Half Angle}}"}
+{"_id": "1664", "title": "Primitive Recursive Function is URM Computable", "text": "Every primitive recursive function is URM computable."}
+{"_id": "1665", "title": "Constant Function is Primitive Recursive", "text": "The constant function $f_c: \\N \\to \\N$, defined as: :$\\map {f_c} n = c$ where $c \\in \\N$ is primitive recursive."}
+{"_id": "1666", "title": "Heron's Formula", "text": "Let $\\triangle ABC$ be a triangle with sides $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively. Then the area $A$ of $\\triangle ABC$ is given by: :$A = \\sqrt {s \\paren {s - a} \\paren {s - b} \\paren {s - c}}$ where $s = \\dfrac {a + b + c} 2$ is the semiperimeter of $\\triangle ABC$."}
+{"_id": "1667", "title": "Addition is Primitive Recursive", "text": "The function $\\Add: \\N^2 \\to \\N$, defined as: :$\\map \\Add {n, m} = n + m$ is primitive recursive."}
+{"_id": "1668", "title": "Exact Form of Prime-Counting Function", "text": "Let:  :$\\displaystyle \\map \\Pi x = \\map \\Li x - \\sum_\\rho \\map \\Li {x^\\rho} - \\map \\ln 2 + \\int_x^\\infty \\frac {\\d t} {t \\paren {t^2 - 1} \\, \\map \\ln t}$  where: :$\\map \\Li x$ is the offset logarithmic integral :the sum $\\displaystyle \\sum_\\rho$ is taken over all $0 < \\rho \\in \\R$ such that the zeta function $\\map \\zeta {\\alpha + i \\rho} = 0$ for some $\\alpha \\in \\R$. Then the prime-counting function is precisely: :$\\displaystyle \\map \\pi x = \\sum_{n \\mathop = 1}^\\infty \\paren {\\frac {\\map \\mu n} n \\map \\Pi {x^{1/n} } }$ where $\\map \\mu n$ denotes the Möbius function."}
+{"_id": "1669", "title": "Multiplication is Primitive Recursive", "text": "The function $\\operatorname{mult}: \\N^2 \\to \\N$, defined as: :$\\map \\Mult {n, m} = n \\times m$ is primitive recursive."}
+{"_id": "1672", "title": "Exponentiation is Primitive Recursive", "text": "The function $\\exp: \\N^2 \\to \\N$, defined as: :$\\exp \\left({n, m}\\right) = n^m$ is primitive recursive."}
+{"_id": "1673", "title": "Bézout's Lemma", "text": "Let $a, b \\in \\Z$ such that $a$ and $b$ are not both zero. Let $\\gcd \\set {a, b}$ be the greatest common divisor of $a$ and $b$. Then: :$\\exists x, y \\in \\Z: a x + b y = \\gcd \\set {a, b}$ That is, $\\gcd \\set {a, b}$ is an integer combination (or linear combination) of $a$ and $b$. Furthermore, $\\gcd \\set {a, b}$ is the smallest positive integer combination of $a$ and $b$."}
+{"_id": "1674", "title": "Predecessor Function is Primitive Recursive", "text": "The '''predecessor function''' $\\operatorname{pred}: \\N \\to \\N$ defined as: :$\\operatorname{pred} \\left({n}\\right) = \\begin{cases} 0 & : n = 0 \\\\ n-1 & : n > 0 \\end{cases}$ is primitive recursive."}
+{"_id": "1675", "title": "Cut-Off Subtraction is Primitive Recursive", "text": "The '''cut-off subtraction''' function, defined as: :$\\forall \\tuple {n, m} \\in \\N^2: n \\mathop {\\dot -} m = \\begin{cases} 0 & : n < m \\\\ n - m & : n \\ge m \\end{cases}$ is primitive recursive."}
+{"_id": "1676", "title": "Maximum Function is Primitive Recursive", "text": "The maximum function $\\max: \\N^2 \\to \\N$, defined as: :$\\max \\left({n, m}\\right) = \\begin{cases} m: & n \\le m \\\\ n: & m \\le n \\end{cases}$ is primitive recursive."}
+{"_id": "1677", "title": "Sum of Maximum and Minimum", "text": "For all numbers $a, b$ where $a, b$ in $\\N, \\Z, \\Q$ or $\\R$: :$a + b = \\max \\left({a, b}\\right) + \\min \\left({a, b}\\right)$"}
+{"_id": "1679", "title": "Absolute Difference Function is Primitive Recursive", "text": "The '''absolute difference''' function $\\operatorname {adf}: \\N^2 \\to \\N$, defined as: :$\\map {\\operatorname {adf} } {n, m} = \\size {n - m}$ where $\\size a$ is defined as the absolute value of $a$, is primitive recursive."}
+{"_id": "1680", "title": "Primitive Recursive Set is URM Computable", "text": "Every primitive recursive set is URM computable."}
+{"_id": "1681", "title": "Set Containing Only Zero is Primitive Recursive", "text": "The subset $\\left\\{{0}\\right\\} \\subset \\N$ is primitive recursive."}
+{"_id": "1682", "title": "Set of Non-Zero Natural Numbers is Primitive Recursive", "text": "Let $\\N^*$ be defined as $\\N^* = \\N \\setminus \\left\\{{0}\\right\\}$. The subset $\\N^* \\subset \\N$ is primitive recursive."}
+{"_id": "1683", "title": "Signum Function is Primitive Recursive", "text": "Let $\\operatorname{sgn}: \\N \\to \\N$ be defined as the signum function. Then: : $\\operatorname{sgn}$ is primitive recursive. : $\\overline {\\operatorname{sgn}}$ is primitive recursive."}
+{"_id": "1685", "title": "Complement of Primitive Recursive Set", "text": "Let $S \\subseteq \\N$ be primitive recursive. Then its relative complement $\\N \\setminus S$ of $S$ in $\\N$ is primitive recursive."}
+{"_id": "1686", "title": "Intersection of Primitive Recursive Sets", "text": "Let $A, B \\subseteq \\N$ be subsets of the set of natural numbers $\\N$. Let $A$ and $B$ both be primitive recursive. Then $A \\cap B$, the intersection of $A$ and $B$, is primitive recursive."}
+{"_id": "1687", "title": "Union of Primitive Recursive Sets", "text": "Let $A, B \\subseteq \\N$ be subsets of the set of natural numbers $\\N$. Let $A$ and $B$ both be primitive recursive. Then $A \\cup B$, the union of $A$ and $B$, is primitive recursive."}
+{"_id": "1688", "title": "Set of Even Numbers is Primitive Recursive", "text": "Let $E \\subseteq \\N$ be the set of all even natural numbers. Then $E$ is primitive recursive."}
+{"_id": "1689", "title": "Primitive Recursive Relation is URM Computable", "text": "Every primitive recursive relation is URM computable."}
+{"_id": "1690", "title": "Equality Relation is Primitive Recursive", "text": "The relation $\\operatorname{eq} \\subseteq \\N^2$, defined as: :$\\map {\\operatorname {eq} } {n, m} \\iff n = m$ is primitive recursive."}
+{"_id": "1692", "title": "Set Operations on Primitive Recursive Relations", "text": "Let $\\RR_1 \\subseteq N^k$ and $\\RR_2 \\subseteq N^k$ be $k$-ary relations on $N^k$. Let $\\RR_1$ and $\\RR_2$ be primitive recursive. Then the following are all primitive recursive relations: :$\\TT = \\neg \\RR_1$ :$\\UU = \\RR_1 \\land \\RR_2$ :$\\VV = \\RR_1 \\lor \\RR_2$"}
+{"_id": "1693", "title": "Ordering Relations are Primitive Recursive", "text": "The ordering relations on $\\N^2$: * $n < m$ * $n \\le m$ * $n \\ge m$ * $n > m$ are all primitive recursive."}
+{"_id": "1694", "title": "Permutation of Variables of Primitive Recursive Function", "text": "Let $f: \\N^k \\to \\N$ be a primitive recursive function. Let $\\sigma$ be a permutation of $\\left({1, 2, \\ldots, k}\\right)$. Then the function $h: \\N^k \\to \\N$ defined as: :$h \\left({n_1, n_2, \\ldots, n_k}\\right) = f \\left({n_{\\sigma \\left({1}\\right)}, n_{\\sigma \\left({2}\\right)}, \\ldots, n_{\\sigma \\left({k}\\right)}}\\right)$ is also primitive recursive."}
+{"_id": "1695", "title": "Definition by Cases is Primitive Recursive", "text": "Let $\\mathcal R_1, \\mathcal R_2, \\ldots, \\mathcal R_k$ be primitive recursive relations on $\\N^l$ such that: :$\\forall i, j \\in \\set{1, 2, \\ldots, k}: \\mathcal R_i \\implies \\lnot \\mathcal R_j$, that is, all relations are mutually exclusive :$\\forall \\tuple {n_1, n_2, \\ldots, n_l} \\in \\N^l: \\exists i \\in \\set {1, 2, \\ldots, k}: \\map {\\mathcal R_i} {n_1, n_2, \\ldots, n_l}$, that is, the set of relations is exhaustive. Let $\\forall i \\in \\set {1, 2, \\ldots, k}: g_i: \\N^l \\to \\N$  be primitive recursive functions. Then the function $f: \\N^l \\to \\N$ defined as: :$\\map f {n_1, n_2, \\ldots, n_l} = \\begin{cases} \\map {g_1} {n_1, n_2, \\ldots, n_l} & : \\map {\\mathcal R_1} {n_1, n_2, \\ldots, n_l} \\\\ \\map {g_2} {n_1, n_2, \\ldots, n_l} & : \\map {\\mathcal R_2} {n_1, n_2, \\ldots, n_l} \\\\ \\quad \\vdots & \\quad \\vdots \\\\ \\map {g_k} {n_1, n_2, \\ldots, n_l} & : \\map {\\mathcal R_k} {n_1, n_2, \\ldots, n_l} \\end{cases}$ is primitive recursive."}
+{"_id": "1697", "title": "Domain of Injection to Countable Set is Countable", "text": "Let $X$ be a set, and let $Y$ be a countable set. Let $f: X \\to Y$ be an injection. Then $X$ is also countable."}
+{"_id": "1698", "title": "Unique Code for URM Instruction", "text": "Each basic instruction $I$ in a URM Program can be identified with a unique '''code number''' $\\beta \\left({I}\\right)$. We also define the following sets: * $\\operatorname{Zinstr}$ is the set of codes of all the <tt>Zero</tt> instructions * $\\operatorname{Sinstr}$ is the set of codes of all the <tt>Successor</tt> instructions * $\\operatorname{Cinstr}$ is the set of codes of all the <tt>Copy</tt> instructions * $\\operatorname{Jinstr}$ is the set of codes of all the <tt>Jump</tt> instructions. Then we define $\\operatorname{Instr}$ to be the set of codes of all basic URM instructions. That is: :$\\operatorname{Instr} = \\operatorname{Zinstr} \\cup \\operatorname{Sinstr} \\cup \\operatorname{Cinstr} \\cup \\operatorname{Jinstr}$."}
+{"_id": "1699", "title": "URM Instructions are Countably Infinite", "text": "The set $\\Bbb I$ of all basic URM instructions is countably infinite."}
+{"_id": "1700", "title": "URM Programs are Countably Infinite", "text": "The set $\\mathbf P$ of all URM programs is countably infinite."}
+{"_id": "1701", "title": "Unique Code for URM Program", "text": "Any URM program can be assigned a unique '''code number'''."}
+{"_id": "1702", "title": "URM Computable Functions of One Variable is Countably Infinite", "text": "The set $\\mathbf U$ of all URM computable functions of $1$ variable is countably infinite."}
+{"_id": "1703", "title": "Natural Number Functions are Uncountable", "text": "The set of all natural number one-variable functions $\\set {f: \\N \\to \\N}$ is uncountably infinite."}
+{"_id": "1704", "title": "Not All Natural Number Functions are Primitive Recursive", "text": "Not all functions $f: \\N \\to \\N$ are primitive recursive."}
+{"_id": "1705", "title": "Bounded Summation is Primitive Recursive", "text": "Let the function $f: \\N^{k+1} \\to \\N$ be primitive recursive. Then so is the function $g: \\N^{k+1} \\to \\N$ defined as: :$\\displaystyle g \\left({n_1, n_2, \\ldots, n_k, z}\\right) = \\begin{cases} 0 & : z = 0 \\\\ \\sum_{y \\mathop = 1}^z f \\left({n_1, n_2, \\ldots, n_k, y}\\right) & : z > 0 \\end{cases}$"}
+{"_id": "1706", "title": "Bounded Product is Primitive Recursive", "text": "Let the function $f: \\N^{k+1} \\to \\N$ be primitive recursive. Then so is the function $g: \\N^{k+1} \\to \\N$ defined as: :$\\displaystyle g \\left({n_1, n_2, \\ldots, n_k, z}\\right) = \\begin{cases} 1 & : z = 0 \\\\ \\prod_{y=1}^z f \\left({n_1, n_2, \\ldots, n_k, y}\\right) & : z > 0 \\end{cases}$"}
+{"_id": "1707", "title": "Divisor Relation is Primitive Recursive", "text": "The divisor relation $m \\divides n$ in $\\N^2$ is primitive recursive."}
+{"_id": "1708", "title": "Divisor Counting Function is Primitive Recursive", "text": "The divisor counting ($\\tau$) function is primitive recursive."}
+{"_id": "1709", "title": "Set of Prime Numbers is Primitive Recursive", "text": "The set $\\Bbb P$ of prime numbers is primitive recursive."}
+{"_id": "1710", "title": "Prime Enumeration Function is Primitive Recursive", "text": "Let the function $p: \\N \\to \\N$ be the prime enumeration function, defined as: :$p \\left({n}\\right) = \\begin{cases} 1 & : n = 0 \\\\ \\mbox{the } n \\mbox{th prime number} & : n > 0 \\end{cases}$ Then $p$ is primitive recursive."}
+{"_id": "1712", "title": "Length Function is Primitive Recursive", "text": "Let $n \\in \\N$. Let $\\map \\len n$ denote the length of $n$. Then the function $\\len: \\N \\to \\N$ is primitive recursive."}
+{"_id": "1713", "title": "Prime Exponent Function is Primitive Recursive", "text": "Let $n \\in \\N$ be a natural number. Let $\\left({n, j}\\right): \\N^2 \\to \\N$ be defined as: :$\\left({n, j}\\right) = \\left({n}\\right)_j$ where $\\left({n}\\right)_j$ is the prime exponent function. Then $\\left({n, j}\\right)$ is primitive recursive."}
+{"_id": "1714", "title": "Set of Codes for URM Instructions is Primitive Recursive", "text": "The set $\\operatorname{Instr}$ of codes of all basic URM instructions is primitive recursive."}
+{"_id": "1715", "title": "Set of Sequence Codes is Primitive Recursive", "text": "Let $\\operatorname{Seq}$ be the set of all code numbers of finite sequences in $\\N$. Then $\\operatorname{Seq}$ is primitive recursive."}
+{"_id": "1716", "title": "Set of Codes for URM Programs is Primitive Recursive", "text": "Let $\\operatorname{Prog}$ be the set of all code numbers of URM programs. Then $\\operatorname{Prog}$ is a primitive recursive set."}
+{"_id": "1717", "title": "Upper Bounds for Prime Numbers", "text": "Let $p: \\N \\to \\N$ be the prime enumeration function. Then $\\forall n \\in \\N$, the value of $\\map p n$ is bounded above. In particular:"}
+{"_id": "1718", "title": "Minimization on Relation Equivalent to Minimization on Function", "text": "Let $\\mathcal R$ be a $k+1$-ary relation on $\\N^{k+1}$. Then the function $g: \\N^{k+1} \\to \\N$ defined as: :$g \\left({n_1, n_2, \\ldots, n_k, z}\\right) = \\mu y \\ \\mathcal R \\left({n_1, n_2, \\ldots, n_k, y}\\right)$ where $\\mu y \\ \\mathcal R \\left({n_1, n_2, \\ldots, n_k, y}\\right)$ is the minimization operation on $\\mathcal R$ is equivalent to minimization on a total function."}
+{"_id": "1720", "title": "Function Obtained by Minimization from URM Computable Functions", "text": "Let the function $f: \\N^{k+1} \\to \\N$ be a URM computable function. Let $g: \\N^k \\to \\N$ be the function obtained by minimization from $f$ thus: :$g \\left({n_1, n_2, \\ldots, n_k}\\right) \\approx \\mu y \\left({f \\left({n_1, n_2, \\ldots, n_k}\\right) = 0}\\right)$. Then $g$ is also URM computable."}
+{"_id": "1721", "title": "Function Obtained by Minimization from URM Computable Relations", "text": "Let $\\mathcal R$ be a URM computable $k+1$-ary relation on $\\N^{k+1}$. Let the function $f: \\N^{k+1} \\to \\N$ be a URM computable function. Let $g: \\N^k \\to \\N$ be the function obtained by minimization from $f$ thus: :$g \\left({n_1, n_2, \\ldots, n_k}\\right) \\approx \\mu y \\mathcal R \\left({n_1, n_2, \\ldots, n_k, y}\\right)$ Then $g$ is also URM computable."}
+{"_id": "1722", "title": "Recursive Function is URM Computable", "text": "Every recursive function is URM computable."}
+{"_id": "1723", "title": "Unique Code for State of URM Program", "text": "Every state of a URM program can be assigned a unique '''code number'''. This code number is called the '''state code''' (or '''situation code''')."}
+{"_id": "1724", "title": "State Code Function is Primitive Recursive", "text": "Let $k \\in \\N^*$. Let $e = \\gamma \\left({P}\\right)$ be the code number of a URM program $P$. Let $\\left({n_1, n_2, \\ldots, n_k}\\right)$ be the input of $P$. Let $S_k: \\N^{k+2} \\to \\N$ be the function defined as: :$S_k \\left({e, n_1, n_2, \\ldots, n_k, t}\\right)$ is the state code for $P$ at stage $t$ of computation of $P$. If $e$ does not code a URM Program then $S_k = 0$. Also, if $P$ terminates at stage $t_0$, then we put: :$\\forall t > t_0: S_k \\left({e, n_1, n_2, \\ldots, n_k, t}\\right) = S_k \\left({e, n_1, n_2, \\ldots, n_k, t_0}\\right)$. Then for all $k \\ge 1$, the function $S_k$ is primitive recursive."}
+{"_id": "1725", "title": "URM Computable Function is Recursive", "text": "Every URM computable function is recursive."}
+{"_id": "1726", "title": "Kleene's Normal Form Theorem", "text": "For each integer $k \\ge 1$, there exists: * a primitive recursive $k+1$-ary relation $T_k$; * a primitive recursive function $U: \\N \\to \\N$ such that a partial function $f: \\N^k \\to \\N$ is recursive iff, for some $e \\in \\N$ and all $\\left({n_1, n_2, \\ldots, n_k}\\right) \\in \\N^k$: :$f \\left({n_1, n_2, \\ldots, n_k}\\right) \\approx U \\left({\\mu z \\ T_k \\left({e, n_1, n_2, \\ldots, n_k, z}\\right)}\\right)$"}
+{"_id": "1727", "title": "Recursive Function uses One Minimization", "text": "Every recursive function can be obtained from the basic primitive recursive functions using: * substitution * primitive recursion * at most one minimization on a function."}
+{"_id": "1728", "title": "Universal URM Computable Functions", "text": "For each integer $k \\ge 1$, there exists a URM computable function: :$\\Phi_k: \\N^{k+1} \\to \\N$ such that for each URM computable function $f: \\N^k \\to \\N$ there exists a natural number $e$ such that: :$\\forall \\left({n_1, n_2, \\ldots, n_k}\\right) \\in \\N^k: f \\left({n_1, n_2, \\ldots, n_k}\\right) \\approx \\Phi_k \\left({e, n_1, n_2, \\ldots, n_k}\\right)$. This function $\\Phi_k$ is '''universal for URM computable functions of $k$ variables'''."}
+{"_id": "1729", "title": "Universal URM Programs", "text": "For each integer $k \\ge 1$, there exists a URM program $P_k$ such that: For each URM program $P$ there exists a natural number $e$ such that: For all $\\left({n_1, n_2, \\ldots, n_k}\\right) \\in \\N^k$, the computation using the program $P_k$ with input $\\left({e, n_1, n_2, \\ldots, n_k}\\right)$ has the same output as the computation using the program $P$ with input $\\left({n_1, n_2, \\ldots, n_k}\\right)$. This function $P_k$ is a '''universal program for URM computations with $k$ inputs'''."}
+{"_id": "1730", "title": "Combination of Recursive Functions", "text": "Let $f: \\N^k \\to \\N$ and $g: \\N^k \\to \\N$ be recursive functions (not necessarily total), where $k \\ge 1$. Let $\\mathcal R$ be a $k$-ary relation such that: : if $\\mathcal R \\left({n_1, n_2, \\ldots, n_k}\\right)$ holds, then $f \\left({n_1, n_2, \\ldots, n_k}\\right)$ is defined : if $\\mathcal R \\left({n_1, n_2, \\ldots, n_k}\\right)$ does not hold, then $g \\left({n_1, n_2, \\ldots, n_k}\\right)$ is defined. Let $h: \\N^k \\to \\N$ be the function defined as: :$h \\left({n_1, n_2, \\ldots, n_k}\\right) = \\begin{cases} f \\left({n_1, n_2, \\ldots, n_k}\\right) & : \\text{if } \\mathcal R \\left({n_1, n_2, \\ldots, n_k}\\right) \\text { holds} \\\\ g \\left({n_1, n_2, \\ldots, n_k}\\right) & : \\text{otherwise} \\end{cases}$ so that $h$ is total. Then $h$ is recursive."}
+{"_id": "1731", "title": "Not All URM Computable Functions are Primitive Recursive", "text": "There exist URM computable functions which are not primitive recursive."}
+{"_id": "1732", "title": "Cantor's Diagonal Argument", "text": "Let $S$ be a set such that $\\card S > 1$, that is, such that $S$ is not a singleton. Let $\\mathbb F$ be the set of all mappings from the natural numbers $\\N$ to $S$: :$\\mathbb F = \\set {f: \\N \\to S}$ Then $\\mathbb F$ is uncountably infinite."}
+{"_id": "1733", "title": "Infinite if Injection from Natural Numbers", "text": "Let $S$ be a set. Let there exist an injection $\\phi: \\N \\to S$ from the natural numbers to $S$. Then $S$ is infinite."}
+{"_id": "1734", "title": "Halting Problem is Not Algorithmically Decidable", "text": "Let $H: \\N^2 \\to \\N$ be the function given by: :$\\map H {m, n} = 1$ if $m$ codes a URM program which halts with input $n$ :$\\map H {m, n} = 0$ otherwise. Then $H$ is not recursive."}
+{"_id": "1735", "title": "Set of Total Functions is Not Recursive", "text": "The set $\\operatorname{Tot}$ of natural numbers which code URM programs which compute total functions of one variable is not recursive."}
+{"_id": "1736", "title": "Infinitely Many Programs for URM Computable Function", "text": "Let $g: \\N^k \\to \\N$ be a URM computable function. Then there is an infinite number of URM programs which compute $g$."}
+{"_id": "1737", "title": "Bounds of GCD for Sum and Difference Congruent Squares", "text": "Let $x, y, n$ be integers. Let: :$x \\not \\equiv \\pm y \\pmod n$ and: :$x^2 \\equiv y^2 \\pmod n$ where $a \\equiv b \\pmod n$ denotes that $a$ is congruent to $b$ modulo $n$. Then: :$1 < \\gcd \\set {x - y, n} < n$ and: :$1 < \\gcd \\set {x + y, n} < n$ where $\\gcd \\set {a, b}$ is the GCD of $a$ and $b$."}
+{"_id": "1739", "title": "Triangle Angle-Side-Angle Equality", "text": "If two triangles have: :two angles equal to two angles, respectively :the sides between the two angles equal then the remaining angles are equal, and the remaining sides equal the respective sides. That is to say, if two pairs of angles and the included sides are equal, then the triangles are congruent."}
+{"_id": "1740", "title": "Triangle Side-Angle-Angle Equality", "text": "If two triangles have: :two angles equal to two angles, respectively :the sides opposite one pair of equal angles equal then the remaining angles are equal, and the remaining sides equal the respective sides. That is to say, if two pairs of angles and a pair of opposite sides are equal, then the triangles are congruent."}
+{"_id": "1741", "title": "Equal Alternate Angles implies Parallel Lines", "text": "Given two infinite straight lines which are cut by a transversal, if the alternate angles are equal, then the lines are parallel. {{:Euclid:Proposition/I/27}}"}
+{"_id": "1744", "title": "Parallelism is Transitive Relation", "text": "Parallelism between straight lines is a transitive relation. {{:Euclid:Proposition/I/30}}"}
+{"_id": "1745", "title": "Sum of Sequence of Products of Consecutive Integers", "text": "{{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 1}^n j \\paren {j + 1}       | r = 1 \\times 2 + 2 \\times 3 + \\dotsb + n \\paren {n + 1}       | c =  }} {{eqn | r = \\frac {n \\paren {n + 1} \\paren {n + 2} } 3       | c =  }} {{end-eqn}}"}
+{"_id": "1746", "title": "Sum of Sequence of Products of Consecutive Reciprocals", "text": ":$\\displaystyle \\sum_{j \\mathop = 1}^n \\frac 1 {j \\paren {j +1} } = \\frac n {n + 1}$"}
+{"_id": "1747", "title": "Sum of Sequence of Fibonacci Numbers", "text": ":$\\displaystyle \\forall n \\in \\Z_{\\ge 0}: \\sum_{j \\mathop = 0}^n F_j = F_{n + 2} - 1$"}
+{"_id": "1749", "title": "Sum of Sequence of Even Index Fibonacci Numbers", "text": "{{begin-eqn}} {{eqn | lo= \\forall n \\ge 1:       | l = \\sum_{j \\mathop = 1}^n F_{2 j}       | r = F_2 + F_4 + F_6 + \\cdots + F_{2 n}       | c =  }} {{eqn | r = F_{2 n + 1} - 1       | c =  }} {{end-eqn}}"}
+{"_id": "1750", "title": "Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers", "text": ":$\\ds \\sum_{j \\mathop = 1}^{2 n - 1} F_j F_{j + 1} = {F_{2 n} }^2$"}
+{"_id": "1751", "title": "Lucas Number as Sum of Fibonacci Numbers", "text": "Let $L_k$ be the $k$th Lucas number, defined as: :$L_n = \\begin{cases} 2 & : n = 0 \\\\ 1 & : n = 1 \\\\ L_{n - 1} + L_{n - 2} & : \\text{otherwise} \\end{cases}$ Then: :$L_n = F_{n - 1} + F_{n + 1}$"}
+{"_id": "1752", "title": "Sum of Sequence of Product of Lucas Numbers with Powers of 2", "text": "Let $L_k$ be the $k$th Lucas number. Let $F_k$ be the $k$th Fibonacci number. Then: :$\\displaystyle \\forall n \\in \\N_{>0}: \\sum_{j \\mathop = 1}^n 2^{j - 1} L_j = 2^n F_{n + 1} - 1$ That is: :$2^0 L_1 + 2^1 L_2 + 2^2 L_3 + \\cdots + 2^{n - 1} L^n = 2^n F_{n + 1} - 1$"}
+{"_id": "1753", "title": "Sum of Odd Positive Powers", "text": "Let $n \\in \\N$ be an odd positive integer. Let $x, y \\in \\Z_{>0}$ be (strictly) positive integers. Then $x + y$ is a divisor of $x^n + y^n$."}
+{"_id": "1754", "title": "Condition for Well-Foundedness", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then $\\struct {S, \\preceq}$ is well-founded {{iff}} there is no infinite sequence $\\sequence {a_n}$ of elements of $S$ such that: :$\\forall n \\in \\N: a_{n + 1} \\prec a_n$ That is, {{iff}} there is no infinite sequence $\\sequence {a_n}$ such that $a_0 \\succ a_1 \\succ a_2 \\succ \\cdots$."}
+{"_id": "1755", "title": "Subset Relation on Power Set is Partial Ordering", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Let $\\struct {\\powerset S, \\subseteq}$ be the relational structure defined on $\\powerset S$ by the relation $\\subseteq$. Then $\\struct {\\powerset S, \\subseteq}$ is an ordered set. The ordering $\\subseteq$ is partial {{iff}} $S$ is neither empty nor a singleton; otherwise it is total."}
+{"_id": "1756", "title": "Ordering is Equivalent to Subset Relation", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then there exists a set $\\mathbb S$ of subsets of $S$ such that: :$\\struct {S, \\preceq} \\cong \\struct {\\mathbb S, \\subseteq}$ where: :$\\struct {\\mathbb S, \\subseteq}$ is the relational structure consisting of $\\mathbb S$ and the subset relation :$\\cong$ denotes order isomorphism. Hence any ordering on a set can be modelled uniquely by a set of subsets of that set under the subset relation."}
+{"_id": "1757", "title": "Subset Relation is Ordering", "text": "Let $\\mathbb S$ be a set of sets. Then $\\subseteq$ is an ordering on $\\mathbb S$. In other words, let $\\struct {\\mathbb S, \\subseteq}$ be the relational structure defined on $\\mathbb S$ by the relation $\\subseteq$. Then $\\struct {\\mathbb S, \\subseteq}$ is an ordered set."}
+{"_id": "1758", "title": "Principle of Mathematical Induction/Well-Ordered Set", "text": "Let $\\struct {S, \\preceq}$ be a well-ordered set. Let $T \\subseteq S$ be a subset of $S$ such that: :$\\forall s \\in S: \\paren {\\forall t \\in S: t \\prec s \\implies t \\in T} \\implies s \\in T$ Then $T = S$."}
+{"_id": "1759", "title": "Order Isomorphism from Woset onto Subset", "text": "Let $\\struct {S, \\preceq}$ be a woset. Let $T \\subseteq S$. Let $f: S \\to T$ be an order isomorphism. Then $\\forall x \\in S: x \\preceq \\map f x$."}
+{"_id": "1760", "title": "Order Isomorphism between Wosets is Unique", "text": "Let $\\struct {S_1, \\preceq_1}$ and $\\struct {S_2, \\preceq_2}$ be wosets. Let $\\struct {S_1, \\preceq_1} \\cong \\struct {S_2, \\preceq_2}$, that is, let $\\struct {S_1, \\preceq_1}$ and $\\struct {S_2, \\preceq_2}$ be order isomorphic. Then there is exactly one mapping $f: S_1 \\to S_2$ such that $f$ is an order isomorphism."}
+{"_id": "1761", "title": "No Isomorphism from Woset to Initial Segment", "text": "Let $\\struct {S, \\preceq}$ be a woset. Let $a \\in S$, and let $S_a$ be the initial segment of $S$ determined by $a$. Then there is no order isomorphism between $S$ and $S_a$."}
+{"_id": "1762", "title": "Woset is Isomorphic to Set of its Initial Segments", "text": "Let $\\struct {S, \\preceq}$ be a well-ordered set. Let: :$A = \\set {a^\\prec: a \\in S}$ where $a^\\prec$ is the strict lower closure of $S$ determined by $a$. Then: :$\\struct {S, \\preceq} \\cong \\struct {A, \\subseteq}$ where $\\cong$ denotes order isomorphism."}
+{"_id": "1764", "title": "Ordering on Ordinal is Subset Relation", "text": "Let $\\struct {S, \\prec}$ be an ordinal. Then $\\forall x, y \\in S:$ :$x \\in y \\iff x \\prec y \\iff S_x \\subsetneqq S_y \\iff x \\subsetneqq y$ where $S_x$ and $S_y$ are the initial segments of $S$ determined by $x$ and $y$ respectively. Thus there is no need to specify what the ordering on an ordinal is -- it is always the subset relation."}
+{"_id": "1765", "title": "Initial Segment of Ordinal is Ordinal", "text": "Let $S$ be an ordinal, and suppose that $a \\in S$. Then the initial segment $S_a = a$ of $S$ determined by $a$ is also an ordinal. In other words, every element of an ordinal is also an ordinal."}
+{"_id": "1766", "title": "Ordinal Subset of Ordinal is Initial Segment", "text": "Let $S$ be an ordinal. Let $T \\subset S$ also be an ordinal. Then $\\exists a \\in S: T = S_a$, where $S_a$ is the initial segment of $S$ determined by $a$. That is, $T = S_a = a \\in S$."}
+{"_id": "1767", "title": "Intersection of Two Ordinals is Ordinal", "text": "Let $S, T$ be ordinals. Then $S \\cap T$ is an ordinal."}
+{"_id": "1768", "title": "Relation between Two Ordinals", "text": "Let $S$ and $T$ be ordinals. Then either $S \\subseteq T$ or $T \\subseteq S$."}
+{"_id": "1769", "title": "Isomorphic Ordinals are Equal", "text": "Let $A$ and $B$ be ordinals that are order isomorphic. Then $A = B$."}
+{"_id": "1770", "title": "Condition for Woset to be Isomorphic to Ordinal", "text": "Let $\\struct {S, \\preceq}$ be a woset. Let $\\struct {S, \\preceq}$ be such that $\\forall a \\in S$, the initial segment $S_a$ of $S$ determined by $a$ is order isomorphic to some ordinal. Then $\\struct {S, \\preceq}$ itself is order isomorphic to an ordinal."}
+{"_id": "1771", "title": "Woset is Isomorphic to Unique Ordinal", "text": "Every woset is order isomorphic to a unique ordinal."}
+{"_id": "1772", "title": "Product of the Incidence Matrix of a BIBD with its Transpose", "text": "For any BIBD with parameters $v,k,\\lambda$, let $A$ be its block incidence matrix. Then: : $A^\\intercal \\cdot A = \\left({a_{ij}}\\right) = \\left({r - \\lambda}\\right) I_v + \\lambda J_v$ where: * $A$ is $v \\times b$ * $A^\\intercal$ is the transpose of $A$ * $J_v$ is the all $v\\times{v}$ $ 1$'s matrix * $I_v$ is the $v\\times{v}$ identity matrix. That is: : $A^\\intercal \\cdot A = \\begin{bmatrix} r & \\lambda & \\cdots & \\lambda \\\\ \\lambda & r & \\cdots & \\lambda \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\lambda & \\lambda & \\cdots & r \\\\ \\end{bmatrix}$"}
+{"_id": "1773", "title": "Fisher's Inequality", "text": "For any BIBD $\\struct {v, k, \\lambda}$, the number of blocks $b$ must be greater then or equal to the number of points $v$: :$ b \\ge v$"}
+{"_id": "1774", "title": "Self-Distributive Law for Conditional", "text": "The following is known as the Self-Distributive Law:"}
+{"_id": "1775", "title": "Disjunction of Conditional and Converse", "text": ": $\\vdash \\left({p \\implies q}\\right) \\lor \\left({q \\implies p}\\right)$"}
+{"_id": "1776", "title": "Characteristics of Eulerian Graph", "text": "A finite (undirected) graph is Eulerian {{iff}} it is connected and each vertex is even. Note that the definition of graph here includes: * Simple graph * Loop-graph * Multigraph * Loop-multigraph but does not include directed graph."}
+{"_id": "1778", "title": "De Morgan's Laws", "text": "'''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:"}
+{"_id": "1779", "title": "Four Color Theorem", "text": "Any planar graph $G$ can be assigned a proper vertex $k$-coloring such that $k \\le 4$."}
+{"_id": "1781", "title": "Dirac's Theorem", "text": "If a connected graph $G$ has $n \\ge 3$ vertices and the degree of each vertex is at least $\\dfrac n 2$, then $G$ is Hamiltonian."}
+{"_id": "1784", "title": "Functionally Complete Logical Connectives", "text": "These sets of logical connectives are functionally complete:"}
+{"_id": "1785", "title": "Properties of NAND", "text": "Let $\\uparrow$ signify the NAND operation. The following results hold:"}
+{"_id": "1786", "title": "Properties of NOR", "text": "Let $\\downarrow$ signify the NOR operation. The following results hold:"}
+{"_id": "1787", "title": "NAND and NOR are Functionally Complete", "text": "The NAND and NOR operators are each functionally complete. That is, NAND and NOR are Sheffer operators."}
+{"_id": "1788", "title": "Equivalence of Semantic Consequence and Logical Implication", "text": "Let $U = \\set {\\phi_1, \\phi_2, \\ldots, \\phi_m, \\ldots}$ be a countable set of propositional formulas. Let $\\psi$ be a propositional formula. Then $U \\models \\psi$ {{iff}} $U \\vdash \\psi$. That is, semantic consequence is equivalent to provable consequence."}
+{"_id": "1790", "title": "WFFs of PropLog of Length 1", "text": "The only WFFs of propositional logic of length $1$ are: * The letters of the formal grammar of propositional logic $\\mathcal L_0$ * The tautology symbol $\\top$ * The contradiction symbol $\\bot$."}
+{"_id": "1791", "title": "WFF of PropLog is Balanced", "text": "Let $\\mathbf A$ be a WFF of propositional logic. Then $\\mathbf A$ is a balanced string."}
+{"_id": "1792", "title": "Initial Part of WFF of PropLog is not WFF", "text": "Let $\\mathbf A$ be a WFF of propositional logic. Let $\\mathbf S$ be an initial part of $\\mathbf A$. Then $\\mathbf S$ is not a WFF of propositional logic."}
+{"_id": "1793", "title": "Construction of Parallel Line", "text": "Given a straight line, and a given point not on that straight line, it is possible to draw a parallel to the given straight line. {{:Euclid:Proposition/I/31}}"}
+{"_id": "1794", "title": "Sum of Angles of Triangle equals Two Right Angles", "text": "In a triangle, the sum of the three interior angles equals two right angles. {{:Euclid:Proposition/I/32}}"}
+{"_id": "1795", "title": "Equivalence of Logical Implication and Conditional", "text": ": $\\left({p \\implies q}\\right) \\dashv \\vdash \\left({p \\vdash q}\\right)$ That is, the conditional is logically equivalent to logical implication."}
+{"_id": "1797", "title": "Models for Propositional Logic", "text": "This page gathers together some useful results that can be used in the derivation of proofs by propositional tableau. Let $\\mathcal M$ be a model for propositional logic, and let $\\mathbf A$ and $\\mathbf B$ be WFFs of propositional logic. Then the following results hold. The symbol $\\models$ is used throughout to denote semantic consequence."}
+{"_id": "1798", "title": "No Boolean Interpretation Models a WFF and its Negation", "text": "Let $v$ be a boolean interpretation. Let $\\mathbf A$ be a WFF of propositional logic. Then $v$ can not model both $\\mathbf A$ and $\\neg \\mathbf A$."}
+{"_id": "1799", "title": "Extended Soundness Theorem for Propositional Tableaus and Boolean Interpretations", "text": "Tableau proofs (in terms of propositional tableaus) are a strongly sound proof system for boolean interpretations. That is, for every collection $\\mathbf H$ of WFFs of propositional logic and every WFF $\\mathbf A$: :$\\mathbf H \\vdash_{\\mathrm{PT}} \\mathbf A$ implies $\\mathbf H \\models_{\\mathrm{BI}} \\mathbf A$"}
+{"_id": "1800", "title": "Extended Completeness Theorem for Propositional Tableaus and Boolean Interpretations", "text": "Tableau proofs (in terms of propositional tableaus) are a strongly complete proof system for boolean interpretations. More precisely, for every countable collection $\\mathbf H$ of WFFs of propositional logic and every WFF $\\mathbf A$: :$\\mathbf H \\models_{\\mathrm{BI}} \\mathbf A$ implies $\\mathbf H \\vdash_{\\mathrm{PT}} \\mathbf A$"}
+{"_id": "1801", "title": "Completeness Theorem for Propositional Tableaus and Boolean Interpretations", "text": "Tableau proofs (in terms of propositional tableaus) are a complete proof system for boolean interpretations. That is, for every WFF $\\mathbf A$: :$\\models_{\\mathrm{BI}} \\mathbf A$ implies $\\vdash_{\\mathrm{PT}} \\mathbf A$"}
+{"_id": "1802", "title": "König's Tree Lemma", "text": "Let $T$ be a rooted tree with an infinite number of nodes, each with a finite number of children. Then $T$ has a branch of infinite length."}
+{"_id": "1803", "title": "Compactness Theorem for Boolean Interpretations", "text": "Let $\\mathbf H$ be a countable set of WFFs of propositional logic. Suppose $\\mathbf H$ is finitely satisfiable for boolean interpretations. That is, suppose that every finite subset $\\mathbf H' \\subseteq \\mathbf H$ is satisfiable for boolean interpretations. Then $\\mathbf H$ has a model."}
+{"_id": "1804", "title": "De Morgan's Laws (Predicate Logic)", "text": "{{:De Morgan's Laws (Predicate Logic)/Assertion of Universality}}"}
+{"_id": "1805", "title": "Initial Part of WFF of Predicate Logic is not WFF", "text": "Let $\\mathbf A$ be a WFF of predicate logic. Let $\\mathbf S$ be an initial part of $\\mathbf A$. Then $\\mathbf S$ is not a WFF of predicate logic."}
+{"_id": "1806", "title": "Unique Readability Theorem of Predicate Calculus", "text": "Each WFF of predicate logic which starts with a left bracket or a negation sign has exactly one main connective. {{Explain|probably, in the correct formulation of 'main connective', the quantifiers ought to be connectives as well}}"}
+{"_id": "1807", "title": "Quantifier has Unique Scope", "text": "Let $\\mathbf A$ be a WFF of predicate logic. Let $Q$ be an occurrence of a quantifier in $\\mathbf A$. Then there exists a unique well-formed part of $\\mathbf A$ which (omitting outermost parentheses) begins with that occurrence $Q$. This unique well-formed part of $\\mathbf A$ is called the scope of the occurrence of $Q$."}
+{"_id": "1808", "title": "Restriction of Operation Distributivity", "text": "Let $\\left({S, *, \\circ}\\right)$ be an algebraic structure. Let $T \\subseteq S$. If the operation $\\circ$ is distributive over $*$ in $\\left({S, *, \\circ}\\right)$, then it is also distributive over $*$ on a restriction $\\left({T, * \\restriction_T, \\circ \\restriction_T}\\right)$."}
+{"_id": "1810", "title": "Union of Intersections", "text": ":$\\paren {S_1 \\cap S_2} \\cup \\paren {T_1 \\cap T_2} \\subseteq S_1 \\cup T_1$"}
+{"_id": "1811", "title": "Union of Symmetric Differences", "text": "Let $R, S, T$ be sets. Then: :$\\left({R * S}\\right) \\cup \\left({S * T}\\right) = \\left({R \\cup S \\cup T}\\right) \\setminus \\left({R \\cap S \\cap T}\\right)$ where $R * S$ denotes the symmetric difference between $R$ and $S$."}
+{"_id": "1812", "title": "Difference of Unions is Subset of Union of Differences", "text": "Let $I$ be an indexing set. Let $S_\\alpha, T_\\alpha$ be sets, for all $\\alpha \\in I$. Then: :$\\displaystyle \\paren {\\bigcup_{\\alpha \\mathop \\in I} S_\\alpha} \\setminus \\paren {\\bigcup_{\\alpha \\mathop \\in I} T_\\alpha} \\subseteq \\bigcup_{\\alpha \\mathop \\in I} \\paren {S_\\alpha \\setminus T_\\alpha}$ where $S_\\alpha \\setminus T_\\alpha$ denotes set difference."}
+{"_id": "1813", "title": "Symmetric Difference of Unions is Subset of Union of Symmetric Differences", "text": "Let $I$ be an indexing set. Let $S_\\alpha, T_\\alpha$ be sets, for all $\\alpha \\in I$. Then: :$\\displaystyle \\bigcup_{\\alpha \\mathop \\in I} S_\\alpha * \\bigcup_{\\alpha \\mathop \\in I} T_\\alpha \\subseteq \\bigcup_{\\alpha \\mathop \\in I} \\paren {S_\\alpha * T_\\alpha}$ where $S * T$ is the symmetric difference between $S$ and $T$."}
+{"_id": "1814", "title": "Pseudometric Defines an Equivalence Relation", "text": "Let $X$ be a set on which there is a pseudometric $d: X \\times X \\to \\R$. For any $x, y \\in X$, let $x \\sim y$ denote that $d \\left({x, y}\\right) = 0$. Then $\\sim$ is an equivalence relation, and the equivalence classes consist of sets of elements of $X$ at zero distance from each other."}
+{"_id": "1816", "title": "First-Order Reaction", "text": "Let a substance decompose spontaneously in a '''first-order reaction'''. Let $x_0$ be a measure of the quantity of that substance at time $t = 0$. Let the quantity of the substance that remains after time $t$ be $x$. Then: :$x = x_0 e^{-k t}$ where $k$ is the rate constant."}
+{"_id": "1817", "title": "Solution to First Order ODE", "text": "Let: :$\\displaystyle \\Phi = \\frac {\\d y} {\\d x} = \\map f {x, y}$ be a first order ordinary differential eqn. Then $\\Phi$ has a general solution which can be expressed in terms of an indefinite integral of $\\map f x$: :$\\displaystyle y = \\int \\map f {x, y} \\rd x + C$ where $C$ is an arbitrary constant."}
+{"_id": "1818", "title": "Solution to First Order Initial Value Problem", "text": "Let $\\map y x$ be a solution to the first order ordinary differential equation: :$\\dfrac {\\d y} {\\d x} = \\map f {x, y}$ which is subject to an initial condition: $\\tuple {a, b}$. Then this problem is equivalent to the integral equation: :$\\displaystyle y = b + \\int_a^x \\map f {t, \\map y t} \\rd t$"}
+{"_id": "1819", "title": "Condition for Lipschitz Condition to be Satisfied", "text": "Let $f$ be a real function. Then $f$ satisfies the Lipschitz condition on a closed real interval $\\closedint a b$ if: :$\\forall y \\in \\closedint a b: \\exists A \\in \\R: \\size {\\map {\\phi'} y} \\le A$"}
+{"_id": "1820", "title": "Separation of Variables", "text": "Suppose a first order ordinary differential equation can be expressible in this form: :$\\dfrac {\\d y} {\\d x} = \\map g x \\, \\map h y$ Then the equation is said to '''have separable variables''', or '''be separable'''. Its general solution is found by solving the integration: :$\\displaystyle \\int \\frac {\\d y} {\\map h y} = \\int \\map g x \\rd x + C$"}
+{"_id": "1821", "title": "Quotient of Homogeneous Functions", "text": "Let $\\map M {x, y}$ and $\\map N {x, y}$ be homogeneous functions of the same degree. Then: :$\\dfrac {\\map M {x, y} } {\\map N {x, y} }$ is homogeneous of zero degree."}
+{"_id": "1822", "title": "Solution to Homogeneous Differential Equation", "text": "Let: :$\\map M {x, y} + \\map N {x, y} \\dfrac {\\d y} {\\d x} = 0$ be a homogeneous differential equation. It can be solved by making the substitution $z = \\dfrac y x$. Its solution is: :$\\displaystyle \\ln x = \\int \\frac {\\d z} {\\map f {1, z} - z} + C$ where: :$\\map f {x, y} = -\\dfrac {\\map M {x, y} } {\\map N {x, y} }$"}
+{"_id": "1823", "title": "Solution to Exact Differential Equation", "text": "The first order ordinary differential equation: :$F = \\map M {x, y} + \\map N {x, y} \\dfrac {\\d y} {\\d x} = 0$ is an exact differential equation {{iff}}: :$\\dfrac {\\partial M} {\\partial y} = \\dfrac {\\partial N} {\\partial x}$ The general solution of such an equation is: :$\\map f {x, y} = C$ where: :$\\dfrac {\\partial f} {\\partial x} = M$ :$\\dfrac {\\partial f} {\\partial y} = N$"}
+{"_id": "1824", "title": "Existence of Integrating Factor", "text": "Let the first order ordinary differential equation: :$(1): \\quad M \\left({x, y}\\right) + N \\left({x, y}\\right) \\dfrac {\\mathrm d y} {\\mathrm d x} = 0$ be such that $M$ and $N$ are real functions of two variables which are ''not'' homogeneous functions of the same degree. Suppose also that: :$\\dfrac {\\partial M} {\\partial y} \\ne \\dfrac {\\partial N} {\\partial x}$ that is, $(1)$ is not exact. Finally, suppose that $(1)$ has a general solution. Then it is always possible to find an integrating factor $\\mu \\left({x, y}\\right)$ such that: :$\\mu \\left({x, y}\\right) \\left({M \\left({x, y}\\right) + N \\left({x, y}\\right) \\dfrac {\\mathrm d y} {\\mathrm d x}}\\right) = 0$ is an exact differential equation. Hence it is possible to find that solution by Solution to Exact Differential Equation."}
+{"_id": "1825", "title": "Integrating Factor for First Order ODE", "text": "Let the first order ordinary differential equation: :$(1): \\quad \\map M {x, y} + \\map N {x, y} \\dfrac {\\d y} {\\d x} = 0$ be non-homogeneous and not exact. By Existence of Integrating Factor, if $(1)$ has a general solution, there exists an integrating factor $\\map \\mu {x, y}$ such that: :$\\displaystyle \\map \\mu {x, y} \\paren {\\map M {x, y} + \\map N {x, y} \\frac {\\d y} {\\d x} } = 0$ is an exact differential equation. Unfortunately, there is no systematic method of finding such a $\\map \\mu {x, y}$ for all such equations $(1)$.<ref>\"In general this is quite difficult.\" ::: -- {{BookReference|Differential Equations|1972|George F. Simmons}}: $\\S 2.9$: Integrating Factors</ref> However, there are certain types of first order ODE for which an integrating factor ''can'' be found procedurally."}
+{"_id": "1826", "title": "Solution to Linear First Order Ordinary Differential Equation", "text": "A linear first order ordinary differential equation in the form: :$\\dfrac {\\d y} {\\d x} + \\map P x y = \\map Q x$ has the general solution: :$\\ds y = e^{-\\int P \\rd x} \\paren {\\int Q e^{\\int P \\rd x} \\rd x + C}$"}
+{"_id": "1827", "title": "Extremal Length of Composition", "text": "Let $\\Gamma_1$ and $\\Gamma_2$ be families of (unions of) rectifiable curves on a Riemann surface $X$. Let $\\Gamma_1$ and $\\Gamma_2$ be disjoint in the sense that there exist disjoint Borel sets $E_1 \\subseteq X$ and $E_2 \\subseteq X$ with $\\bigcup \\Gamma_1 \\subset E_1$ and $\\bigcup \\Gamma_2 \\subset E_2$. The extremal length of the family: :$\\Gamma := \\left\\{{\\gamma_1 \\cup \\gamma_2:\\ \\gamma_1 \\in \\Gamma_1 \\text{ and }\\gamma_2 \\in \\Gamma_2}\\right\\}$ satisfies: :$\\lambda \\left({\\Gamma}\\right) = \\lambda \\left({\\Gamma_1}\\right) + \\lambda \\left({\\Gamma_2}\\right)$"}
+{"_id": "1828", "title": "Squares with All Odd Digits", "text": "The only squares whose digits<ref>That is, when written in base 10 notation.</ref> are all odd are $1$ and $9$."}
+{"_id": "1829", "title": "Parity of Integer equals Parity of its Square", "text": "Let $p \\in \\Z$ be an integer. Then $p$ is even {{iff}} $p^2$ is even."}
+{"_id": "1831", "title": "Weierstrass M-Test", "text": "Let $f_n$ be a sequence of real functions defined on a domain $D \\subseteq \\R$. Let $\\displaystyle \\sup_{x \\mathop \\in D} \\size {\\map {f_n} x} \\le M_n$ for each integer $n$ and some constants $M_n$ Let $\\displaystyle \\sum_{i \\mathop = 1}^\\infty M_i < \\infty$. Then $\\displaystyle \\sum_{i \\mathop = 1}^\\infty f_i$ converges uniformly on $D$."}
+{"_id": "1832", "title": "Measurable Sets form Sigma-Algebra", "text": "Let $\\mu^*$ be an outer measure on a set $X$. Then the set $\\map {\\mathfrak M} {\\mu^*}$ of all $\\mu^*$-measurable subsets of $X$ is a $\\sigma$-algebra."}
+{"_id": "1834", "title": "Markov's Inequality", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $f$ be an $A$-measurable function where $A \\in \\Sigma$. Then: :$\\displaystyle \\mu \\left({\\left\\{ {x \\in A: \\left\\vert{f \\left({x}\\right)}\\right\\vert \\ge t}\\right\\} }\\right) \\le \\frac 1 t \\int_A \\left\\vert{f}\\right\\vert \\mathrm d \\mu$ for any positive $t \\in \\R$."}
+{"_id": "1835", "title": "Limit Superior includes Limit Inferior", "text": "Let $\\set {E_n : n \\in \\N}$ be a sequence of sets. Then: :$\\ds \\liminf_{n \\mathop \\to \\infty} E_n \\subseteq \\limsup_{n \\mathop \\to \\infty} E_n$"}
+{"_id": "1836", "title": "Construction of Outer Measure", "text": "{{questionable|The sense of this seems to have got lost at around the time of the edit of 25th March 2012. Presumably the intention was to extend this definition from the set of mappings on the powerset of $X$ to an arbitrary subset of them -- but the definition of $\\mu^*$ seems to be inconsistent with this. Either a more rigorous explanation of exactly what all the concepts are here, or the theorem has been mis-stated.}} Let $X$ be a set. Let $\\powerset X$ be the power set of $X$. Let $\\AA$ be a subset of $\\powerset X$ which contains the empty set. Let $\\overline \\R_{\\ge 0}$ denote the set of positive extended real numbers. Let $\\gamma: \\AA \\to \\overline \\R_{\\ge 0}$ be a mapping such that $\\map \\gamma \\O = 0$. Let $\\mu^*: \\AA \\to \\overline \\R_{\\ge 0}$ be the mapping defined as: :$\\displaystyle \\forall S \\in \\powerset X: \\map {\\mu^*} S = \\inf \\set {\\sum_{n \\mathop = 0}^\\infty \\map \\gamma {A_n}: \\forall n \\in \\N : A_n \\in \\AA, \\ S \\subseteq \\bigcup_{n \\mathop = 0}^\\infty A_n}$ Then $\\mu^*$ is an outer measure on $X$. The infimum of the empty set is the greatest element, $+\\infty$."}
+{"_id": "1837", "title": "Additive and Countably Subadditive Function is Countably Additive", "text": "Let $\\Sigma$ be a $\\sigma$-algebra over a set $X$. Let $f: \\Sigma \\to \\overline \\R_{\\ge 0}$ be an additive and countably subadditive function, where $\\overline \\R_{\\ge 0}$ denotes the set of positive extended real numbers. Then $f$ is countably additive."}
+{"_id": "1839", "title": "Banach Fixed-Point Theorem", "text": "Let $\\struct {M, d}$ be a complete metric space. Let $f: M \\to M$ be a contraction. That is, there exists $q \\in \\hointr 0 1$ such that for all $x, y \\in M$: :$\\map d {\\map f x, \\map f y} \\le q \\, \\map d {x, y}$ Then there exists a unique fixed point of $f$."}
+{"_id": "1841", "title": "Measure of Empty Set is Zero", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Then $\\map \\mu \\O = 0$. That is, $\\O$ is a $\\mu$-null set."}
+{"_id": "1842", "title": "Convergence a.u. Implies Convergence a.e.", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f_n: D \\to \\R$ be a sequence of $\\Sigma$-measurable functions for $D \\in \\Sigma$. Let $f_n$ converge a.u. to a function $f$ on $D$. Then $f_n$ converges a.e. to $f$."}
+{"_id": "1844", "title": "Existence of Solution to System of First Order ODEs", "text": "Consider the system of initial value problems: :$\\begin{cases} \\dfrac {\\mathrm d y} {\\mathrm d x} = f \\left({x, y, z}\\right) & : y \\left({x_0}\\right) = y_0 \\\\ & \\\\ \\dfrac {\\mathrm d z} {\\mathrm d x} = g \\left({x, y, z}\\right) & : z \\left({x_0}\\right) = z_0 \\\\ \\end{cases}$ where $f \\left({x, y, z}\\right)$ and $g \\left({x, y, z}\\right)$ are continuous real functions in some region of space $x y z$ that contains the point $\\left({x_0, y_0, z_0}\\right)$. Then this system of equations has a unique solution which exists on some interval $\\left|{x - x_0}\\right| \\le h$."}
+{"_id": "1846", "title": "Little Bézout Theorem", "text": "Let $\\map {P_n} x$ be a polynomial of degree $n$ in $x$. Let $a$ be a constant. Then the remainder of $\\map {P_n} x$ when divided by $x - a$ is equal to $\\map {P_n} a$."}
+{"_id": "1847", "title": "Thales' Theorem", "text": "Let $A$ and $B$ be two points on opposite ends of the diameter of a circle. Let $C$ be another point on the circle such that $C \\ne A, B$. Then the lines $AC$ and $BC$ are perpendicular to each other. :400px"}
+{"_id": "1848", "title": "Edge of Polyhedron has no Curvature", "text": "The edge of a polyhedron has zero curvature."}
+{"_id": "1849", "title": "Properties of Dot Product", "text": "Let $\\mathbf u, \\mathbf v, \\mathbf w$ be vectors in the vector space $\\R^n$. Let $c$ be a real scalar. The dot product has the following properties:"}
+{"_id": "1850", "title": "Dot Product is Inner Product", "text": "The dot product is an inner product."}
+{"_id": "1851", "title": "Factor Matrix in the Inner Product", "text": ":$\\left \\langle {A \\mathbf u,\\mathbf v} \\right \\rangle = \\left \\langle {\\mathbf u, A^T\\mathbf v} \\right \\rangle$ where $\\mathbf u$ and $\\mathbf v$ are both $1 \\times n$ column vectors."}
+{"_id": "1852", "title": "Empty Set Disjoint with Itself", "text": "The empty set is disjoint with itself: :$\\varnothing \\cap \\varnothing = \\varnothing$"}
+{"_id": "1856", "title": "Pi is Irrational", "text": "Pi ($\\pi$) is irrational."}
+{"_id": "1858", "title": "Properties of Hadamard Product", "text": "Let $\\map {\\MM_S} {m, n}$ be a $m \\times n$ matrix space over $S$ over an algebraic structure $\\struct {S, \\cdot}$. Let $\\mathbf A, \\mathbf B \\in \\map {\\MM_S} {m, n}$. Let $\\mathbf A \\circ \\mathbf B$ be defined as the Hadamard product of $\\mathbf A$ and $\\mathbf B$. The operation $\\circ$ of Hadamard product satisfies the following properties: :$\\circ$ is closed on $\\map {\\MM_S} {m, n}$ {{iff}} $\\cdot$ is closed on $\\struct {S, \\cdot}$ :$\\circ$ is associative on $\\map {\\MM_S} {m, n}$ {{iff}} $\\cdot$ is associative on $\\struct {S, \\cdot}$ :$\\circ$ is commutative on $\\map {\\MM_S} {m, n}$ {{iff}} $\\cdot$ is commutative on $\\struct {S, \\cdot}$."}
+{"_id": "1860", "title": "Graph Isomorphism is Equivalence Relation", "text": "Graph isomorphism is an equivalence relation."}
+{"_id": "1861", "title": "Relation Isomorphism is Equivalence Relation", "text": "Relation isomorphism is an equivalence relation."}
+{"_id": "1862", "title": "Vertex Condition for Isomorphic Graphs", "text": "Let $G_1$ and $G_2$ be isomorphic graphs. Then the degrees of the vertices of $G_1$ are exactly the same as the degrees of the vertices of $G_2$."}
+{"_id": "1863", "title": "Pigeonhole Principle", "text": "Let $S$ be a finite set whose cardinality is $n$. Let $S_1, S_2, \\ldots, S_k$ be a partition of $S$ into $k$ subsets. Then: :at least one subset $S_i$ of $S$ contains at least $\\ceiling {\\dfrac n k}$ elements where $\\ceiling {\\, \\cdot \\,}$ denotes the ceiling function."}
+{"_id": "1865", "title": "Graph Connectedness is Equivalence Relation", "text": "Let $G = \\struct {V, E}$ be a graph. Let $\\to$ denote the relation '''is connected to''' on the set $V$. Then $\\to$ is an equivalence relation."}
+{"_id": "1866", "title": "Condition for Edge to be Bridge", "text": "Let $G = \\struct {V, E}$ be a connected graph. Let $e \\in E$ be an edge of $G$. Then $e$ is a bridge {{iff}} $e$ does not lie on any circuit of $G$."}
+{"_id": "1867", "title": "Characteristics of Traversable Graph", "text": "A finite loop-multigraph is traversable {{iff}} it is connected and no more than two vertices are odd. Any Eulerian trail which is not an Eulerian circuit must start and end at an odd vertex."}
+{"_id": "1870", "title": "Number of Edges of Regular Graph", "text": "An $r$-regular graph of order $n$ is of size $\\dfrac {n r} 2$."}
+{"_id": "1871", "title": "Graph is Bipartite iff No Odd Cycles", "text": "Let $G$ be an undirected graph. Then $G$ is bipartite {{iff}} it has no odd cycles."}
+{"_id": "1872", "title": "Ore's Theorem", "text": "Let $G = \\struct {V, E}$ be a simple graph of order $n \\ge 3$. Let $G$ be an Ore graph, that is: :For each pair of non-adjacent vertices $u, v \\in V$: :: $(1): \\quad \\deg u + \\deg v \\ge n$ Then $G$ is Hamiltonian."}
+{"_id": "1873", "title": "Path in Tree is Unique", "text": "Let $T$ be a graph. Then $T$ is a tree {{iff}} there is exactly one path between any two vertices."}
+{"_id": "1874", "title": "Size of Tree is One Less than Order", "text": "Let $T$ be a connected simple graph of order $n$. Then $T$ is a tree {{Iff}} the size of $T$ is $n-1$."}
+{"_id": "1875", "title": "Connected Subgraph of Tree is Tree", "text": "Let $T$ be a tree. Let $S$ be a subgraph of $T$ such that $S$ is connected. Then $S$ is also a tree."}
+{"_id": "1876", "title": "Finite Tree has Leaf Nodes", "text": "Every non-edgeless finite tree has at least two leaf nodes."}
+{"_id": "1877", "title": "Size of Cycle Graph equals Order", "text": "The size of a cycle graph equals its order."}
+{"_id": "1878", "title": "Path Graph from Cycle Graph", "text": "Removing one edge from a cycle graph leaves a path graph."}
+{"_id": "1879", "title": "Cycle in Balanced Signed Graph", "text": "Let $G$ be a balanced signed graph. Let $C$ be a cycle in $G$. Then $C$ has an even number of negative edges."}
+{"_id": "1880", "title": "Graph with Even Vertices Partitions into Cycles", "text": "Let $G = \\struct {V, E}$ be a graph whose vertices are all even. Then its edge set $E$ can be partitioned into cycles, no two of which share an edge. The converse also holds: a graph which can be partitioned into cycles must have all its vertices even."}
+{"_id": "1881", "title": "Condition for Bipartite Graph to be Hamiltonian", "text": "Let $G = \\struct {A \\mid B, E}$ be a bipartite graph. Let $G$ be Hamiltonian. Then $\\card A = \\card B$. That is, there is the same number of vertices in $A$ as there are in $B$."}
+{"_id": "1882", "title": "Complete Hamiltonian Bipartite Graph", "text": "Let $K_{m, n}$ be a complete bipartite graph. Then $K_{m, n}$ is Hamiltonian iff $m = n > 1$."}
+{"_id": "1883", "title": "Condition for Complete Bipartite Graph to be Semi-Hamiltonian", "text": "Let $K_{m, n}$ be a complete bipartite graph. Then $K_{m, n}$ is semi-Hamiltonian {{iff}} either: :$m = n = 1$ or: :$m = n + 1$ (or $n = m + 1$)."}
+{"_id": "1884", "title": "Equivalent Definitions for Finite Tree", "text": "Let $T$ be a finite tree of order $n$. The following statements are equivalent: : $(1): \\quad T$ is connected and has no circuits. : $(2): \\quad T$ has $n-1$ edges and has no circuits. : $(3): \\quad T$ is connected and has $n-1$ edges. : $(4): \\quad T$ is connected, and the removal of any one edge renders $T$ disconnected. : $(5): \\quad$ Any two vertices of $T$ are connected by exactly one path. : $(6): \\quad T$ has no circuits, but adding one edge creates a cycle."}
+{"_id": "1887", "title": "Number of Edges in Forest", "text": "Let $F = \\struct {V, E}$ be a forest with $n$ nodes and $m$ components. Then $F$ contains $n - m$ edges."}
+{"_id": "1888", "title": "Cayley's Formula", "text": "The number of distinct labeled trees with $n$ nodes is $n^{n - 2}$."}
+{"_id": "1889", "title": "Bijection between Prüfer Sequences and Labeled Trees", "text": "There is a one-to-one correspondence between Prüfer sequences and labeled trees. That is, every labeled tree has a unique Prüfer sequence that defines it, and every Prüfer sequence defines just one labeled tree."}
+{"_id": "1890", "title": "Tree has Center or Bicenter", "text": "Every tree has either: : $(1): \\quad$ Exactly one center or: : $(2): \\quad$ Exactly one bicenter, but never both. That is, every tree is either central or bicentral."}
+{"_id": "1892", "title": "Prim's Algorithm produces Minimum Spanning Tree", "text": "Prim's Algorithm always produces a minimum spanning tree."}
+{"_id": "1893", "title": "Number of Hamilton Cycles in Complete Graph", "text": "For all $n \\ge 3$, the number of distinct Hamilton cycles in the complete graph $K_n$ is $\\dfrac {\\left({n-1}\\right)!} 2$."}
+{"_id": "1895", "title": "Reflexive Euclidean Relation is Equivalence", "text": "A relation is an equivalence {{iff}} it is either left-Euclidean or right-Euclidean, and also reflexive."}
+{"_id": "1896", "title": "Reflexive Relation is Serial", "text": "Every reflexive relation is also a serial relation."}
+{"_id": "1897", "title": "Complement of Complete Bipartite Graph", "text": "Let $K_{p, q}$ be a complete bipartite graph. The complement of $K_{p, q}$ consists of a disconnected graph with two components: :The complete graph $K_p$ :The complete graph $K_q$."}
+{"_id": "1898", "title": "Complement of Strict Total Ordering", "text": "Let $\\left({S, \\prec}\\right)$ be a relational structure such that $\\prec$ is a strict total ordering. Then the complement of $\\prec$ is a weak total ordering."}
+{"_id": "1900", "title": "Mahler's Inequality", "text": "The geometric mean of the termwise sum of two finite sequences of positive real numbers is never less than the sum of their two separate geometric means: :$\\displaystyle \\prod_{k \\mathop = 1}^n \\paren {x_k + y_k}^{1/n} \\ge \\prod_{k \\mathop = 1}^n x_k^{1/n} + \\prod_{k \\mathop = 1}^n y_k^{1/n}$ where $x_k, y_k > 0$ for all $k$."}
+{"_id": "1901", "title": "Equivalence of Definitions of Transitive Closure (Relation Theory)/Intersection is Smallest", "text": "Let $\\RR$ be a relation on a set $S$. Then the intersection of all transitive relations on $S$ that contain $\\RR$ is the smallest transitive relation on $S$ that contains $\\RR$."}
+{"_id": "1902", "title": "Intersection of Transitive Relations is Transitive", "text": "The intersection of two transitive relations is also a transitive relation."}
+{"_id": "1903", "title": "Intersection of Reflexive Relations is Reflexive", "text": "The intersection of two reflexive relations is also a reflexive relation."}
+{"_id": "1904", "title": "Intersection of Symmetric Relations is Symmetric", "text": "The intersection of two symmetric relations is also a symmetric relation."}
+{"_id": "1905", "title": "Union of Reflexive Relations is Reflexive", "text": "The union of two reflexive relations is also a reflexive relation."}
+{"_id": "1906", "title": "Union of Symmetric Relations is Symmetric", "text": "The union of two symmetric relations is also a symmetric relation."}
+{"_id": "1907", "title": "Union of Transitive Relations Not Always Transitive", "text": "The union of transitive relations is not necessarily itself transitive."}
+{"_id": "1908", "title": "Set System Closed under Intersection is Commutative Semigroup", "text": "Let $\\mathcal S$ be a system of sets. Let $\\mathcal S$ be such that: :$\\forall A, B \\in \\mathcal S: A \\cap B \\in \\mathcal S$ Then $\\left({\\mathcal S, \\cap}\\right)$ is a commutative semigroup."}
+{"_id": "1909", "title": "Set System Closed under Union is Commutative Semigroup", "text": "Let $\\mathcal S$ be a system of sets. Let $\\mathcal S$ be such that: :$\\forall A, B \\in \\mathcal S: A \\cup B \\in \\mathcal S$ Then $\\struct {\\mathcal S, \\cup}$ is a commutative semigroup."}
+{"_id": "1910", "title": "Set System Closed under Symmetric Difference is Abelian Group", "text": "Let $\\SS$ be a system of sets. Let $\\SS$ be such that: :$\\forall A, B \\in \\SS: A * B \\in \\SS$ where $A * B$ denotes the symmetric difference between $A$ and $B$. Then $\\struct {\\SS, *}$ is an abelian group."}
+{"_id": "1911", "title": "Ring of Sets is Commutative Ring", "text": "A ring of sets $\\left({\\mathcal R, *, \\cap}\\right)$ is a commutative ring whose zero is $\\varnothing$."}
+{"_id": "1912", "title": "Union as Symmetric Difference with Intersection", "text": "Let $A$ and $B$ be sets. Then: :$A \\cup B = \\left({A * B}\\right) * \\left({A \\cap B}\\right)$ where: : $A \\cup B$ denotes set union : $A \\cap B$ denotes set intersection : $A * B$ denotes set symmetric difference"}
+{"_id": "1913", "title": "Set Difference as Symmetric Difference with Intersection", "text": ":$S \\setminus T = S * \\paren {S \\cap T}$ where: :$S \\setminus T$ denotes set difference :$S * T$ denotes set symmetric difference :$S \\cap T$ denotes set intersection."}
+{"_id": "1914", "title": "Unit of System of Sets is Unique", "text": "The unit of a system of sets, if it exists, is unique. If $U$ is the unit of a system of sets $\\mathcal S$, then $\\forall A \\in \\mathcal S: A \\subseteq U$."}
+{"_id": "1916", "title": "Power Set is Algebra of Sets", "text": "Let $S$ be a set. Let $\\mathcal P \\left({S}\\right)$ be the power set of $S$. Then $\\mathcal P \\left({S}\\right)$ is an algebra of sets where $S$ is the unit."}
+{"_id": "1917", "title": "Power Set is Closed under Intersection", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Then: :$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$"}
+{"_id": "1918", "title": "Power Set is Closed under Symmetric Difference", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Then: :$\\forall A, B \\in \\powerset S: A * B \\in \\powerset S$ where $A * B$ is the symmetric difference between $A$ and $B$."}
+{"_id": "1919", "title": "Intersection of Rings of Sets", "text": "Let $\\mathcal R_k$ be a ring of sets, where $k$ is an element of an arbitrary set of indices. Then their intersection $\\displaystyle \\mathcal R = \\bigcap_k \\mathcal R_k$ is itself a ring of sets."}
+{"_id": "1920", "title": "Minimal Ring Generated by System of Sets", "text": "Let $\\SS$ be a non-empty system of sets. Then there is a unique ring of sets $\\map \\RR \\SS$ which: :$(1): \\quad$ contains $\\SS$ :$(2): \\quad$ is contained by every ring of sets which also contains $\\SS$. This ring of sets $\\map \\RR \\SS$ is called the '''minimal ring generated by $\\SS$'''."}
+{"_id": "1921", "title": "Menelaus's Theorem", "text": "Let $ABC$ be a triangle. Let points $D, E, F$ lie on lines $BC, AC, AB$ respectively (produced if necessary). Then $D, E$ and $F$ are collinear {{iff}}: : $\\dfrac {AF} {FB} \\cdot \\dfrac {BD} {DC} \\cdot \\dfrac {CE} {EA} = -1$ In the above, the line segments $AF, BD, EA$ are determined to have negative length if they lie outside the line segments $AB, BC, CA$."}
+{"_id": "1922", "title": "Lines Joining Equal and Parallel Straight Lines are Parallel", "text": "The straight lines joining equal and parallel straight lines at their endpoints, in the same direction, are themselves equal and parallel. {{:Euclid:Proposition/I/33}}"}
+{"_id": "1923", "title": "Opposite Sides and Angles of Parallelogram are Equal", "text": "The opposite sides and angles of a parallelogram are equal to one another, and either of its diameters bisects its area. {{:Euclid:Proposition/I/34}}"}
+{"_id": "1928", "title": "Equal Sized Triangles on Same Base have Same Height", "text": "Triangles of equal area which are on the same base, and on the same side of it, are also in the same parallels. {{:Euclid:Proposition/I/39}}"}
+{"_id": "1930", "title": "Parallelogram on Same Base as Triangle has Twice its Area", "text": "A parallelogram on the same base as a triangle, and in the same parallels, has twice the area of the triangle. {{:Euclid:Proposition/I/41}}"}
+{"_id": "1931", "title": "Construction of Parallelogram equal to Triangle in Given Angle", "text": "A parallelogram can be constructed in a given angle the same size as any given triangle. {{:Euclid:Proposition/I/42}}"}
+{"_id": "1934", "title": "Construction of Parallelogram in Given Angle equal to Given Polygon", "text": "A parallelogram can be constructed in a given angle the same size as any given polygon. {{:Euclid:Proposition/I/45}}"}
+{"_id": "1939", "title": "Square of Sum", "text": ":$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$"}
+{"_id": "1940", "title": "Square of Sum less Square", "text": ":$\\forall x, y \\in \\R: \\paren {2x + y} y = \\paren {x + y}^2 - x^2$"}
+{"_id": "1941", "title": "Square of Difference", "text": ":$\\forall x, y \\in \\R: \\paren {x - y}^2 = x^2 - 2 x y + y^2$"}
+{"_id": "1942", "title": "Square of Sum with Double", "text": ":$\\forall a, b \\in \\R: \\paren {a + 2 b}^2 = a^2 + 4 a b + 4 b^2$"}
+{"_id": "1947", "title": "Construction of Square equal to Given Polygon", "text": "A square can be constructed the same size as any given polygon. {{:Euclid:Proposition/II/14}}"}
+{"_id": "1948", "title": "Finding Center of Circle", "text": "For any given circle, it is possible to find its center. {{:Euclid:Proposition/III/1}}"}
+{"_id": "1950", "title": "Perpendicular Bisector of Chord Passes Through Center", "text": "The perpendicular bisector of any chord of any given circle must pass through the center of that circle. {{:Euclid:Proposition/III/1/Porism}}"}
+{"_id": "1951", "title": "Conditions for Diameter to be Perpendicular Bisector", "text": "If in a circle a diameter bisects a chord (which is itself not a diameter), then it cuts it at right angles, and if it cuts it at right angles then it bisects it. {{:Euclid:Proposition/III/3}}"}
+{"_id": "1952", "title": "Chords do not Bisect Each Other", "text": "If in a circle two chords (which are not diameters) cut one another, then they do not bisect one another. {{:Euclid:Proposition/III/4}}"}
+{"_id": "1955", "title": "Concentric Circles do not Intersect", "text": "If two circles are concentric, they share no points on their circumferences. Alternatively, this can be worded: :If two concentric circles share one point on their circumferences, then they share them all (that is, they are the same circle)."}
+{"_id": "1958", "title": "Baire Category Theorem", "text": "Let $M = \\struct {A, d}$ be a complete metric space. Then $M = \\struct {A, d}$ is also a Baire space."}
+{"_id": "1959", "title": "Baire Characterisation Theorem", "text": "A real-valued function $f$ defined on a Banach space $X$ is a Baire-$1$ function {{iff}}: : for every non-empty closed subset $K$ of $X$, the restriction of $f$ to $K$ has a point of continuity relative to the topology of $K$."}
+{"_id": "1960", "title": "Ramsey's Theorem", "text": "In any coloring of the edges of a sufficiently large complete graph, one will find monochromatic complete subgraphs. For 2 colors, Ramsey's theorem states that for any pair of positive integers $\\tuple {r, s}$, there exists a least positive integer $\\map R {r, s}$ such that for any complete graph on $\\map R {r, s}$ vertices, whose edges are colored red or blue, there exists either a complete subgraph on $r$ vertices which is entirely red, or a complete subgraph on $s$ vertices which is entirely blue. More generally, for any given number of colors $c$, and any given integers $n_1, \\ldots, n_c$, there is a number $\\map R {n_1, \\ldots, n_c}$ such that: :if the edges of a complete graph of order $\\map R {n_1, \\ldots, n_c}$ are colored with $c$ different colours, then for some $i$ between $1$ and $c$, it must contain a complete subgraph of order $n_i$ whose edges are all color $i$. This number $\\map R {n_1, \\ldots, n_c}$ is called the Ramsey number for $n_1, \\ldots, n_c$. The special case above has $c = 2$ (and $n_1 = r$ and $n_2 = s$). Here $\\map R {r, s}$ signifies an integer that depends on both $r$ and $s$. It is understood to represent the smallest integer for which the theorem holds."}
+{"_id": "1961", "title": "Schur's Theorem (Ramsey Theory)", "text": "Let $r$ be a positive integer. Then there exists a positive integer $S$ such that: :for every partition of the integers $\\set {1, \\ldots, S}$ into $r$ parts, one of the parts contains integers $x$, $y$ and $z$ such that: ::$x + y = z$"}
+{"_id": "1963", "title": "Brahmagupta's Formula", "text": "The area of a cyclic quadrilateral with sides of lengths $a, b, c, d$ is: :$\\sqrt {\\paren {s - a} \\paren {s - b} \\paren {s - c} \\paren {s - d} }$ where $s$ is the semiperimeter: :$s = \\dfrac {a + b + c + d} 2$"}
+{"_id": "1967", "title": "Euler-Binet Formula", "text": "The Fibonacci numbers have a closed-form solution: :$F_n = \\dfrac {\\phi^n - \\paren {1 - \\phi}^n} {\\sqrt 5} = \\dfrac {\\phi^n - \\paren {-1 / \\phi}^n} {\\sqrt 5} = \\dfrac {\\phi^n - \\paren {-1}^n \\phi^{-n} } {\\sqrt 5}$ where $\\phi$ is the golden mean. Putting $\\hat \\phi = 1 - \\phi = -\\dfrac 1 \\phi$ this can be written: :$F_n = \\dfrac {\\phi^n - \\hat \\phi^n} {\\sqrt 5}$"}
+{"_id": "1968", "title": "Hölder's Inequality for Sums", "text": "Let $p, q \\in \\R_{>0}$ be strictly positive real numbers such that: :$\\dfrac 1 p + \\dfrac 1 q = 1$ Let: :$\\mathbf x = \\sequence {x_n} \\in \\ell^p$ :$\\mathbf y = \\sequence {y_n} \\in \\ell^q$ where $\\ell^p$ denotes the $p$-sequence space. Let $\\norm {\\mathbf x}_p$ denote the $p$-norm of $\\mathbf x$. Then $\\mathbf x \\mathbf y = \\sequence {x_n y_n} \\in \\ell^1$, and: :$\\norm {\\mathbf x \\mathbf y}_1 \\le \\norm {\\mathbf x}_p \\norm {\\mathbf y}_q$"}
+{"_id": "1969", "title": "Binet Form", "text": "Let $m \\in \\R$. Define: {{begin-eqn}} {{eqn | l = \\Delta       | r = \\sqrt {m^2 + 4} }} {{eqn | l = \\alpha       | r = \\frac {m + \\Delta} 2 }} {{eqn | l = \\beta       | r = \\frac {m - \\Delta} 2 }} {{end-eqn}}"}
+{"_id": "1970", "title": "Binet-Cauchy Identity", "text": ":$\\displaystyle \\left({\\sum_{i \\mathop = 1}^n a_i c_i}\\right) \\left({\\sum_{j \\mathop = 1}^n b_j d_j}\\right) = \\left({\\sum_{i \\mathop = 1}^n a_i d_i}\\right) \\left({\\sum_{j \\mathop = 1}^n b_j c_j}\\right) + \\sum_{1 \\mathop \\le i \\mathop < j \\mathop \\le n} \\left({a_i b_j - a_j b_i}\\right) \\left({c_i d_j - c_j d_i}\\right)$ where all of the $a, b, c, d$ are elements of a commutative ring. Thus the identity holds for $\\Z, \\Q, \\R, \\C$."}
+{"_id": "1971", "title": "Nicomachus's Theorem", "text": "{{begin-eqn}} {{eqn | l = 1^3       | r = 1 }} {{eqn | l = 2^3       | r = 3 + 5 }} {{eqn | l = 3^3       | r = 7 + 9 + 11 }} {{eqn | l = 4^3       | r = 13 + 15 + 17 + 19 }} {{eqn | l = \\vdots       | o =  }} {{end-eqn}} In general: :$\\forall n \\in \\N_{>0}: n^3 = \\left({n^2 - n + 1}\\right) + \\left({n^2 - n + 3}\\right) + \\ldots + \\left({n^2 + n - 1}\\right)$ In particular, the first term for $\\left({n + 1}\\right)^3$ is $2$ greater than the last term for $n^3$."}
+{"_id": "1972", "title": "Conservation of Angular Momentum", "text": "Newton's Laws of Motion imply the conservation of angular momentum in systems of masses in which no external force is acting."}
+{"_id": "1975", "title": "Determinant of Combinatorial Matrix", "text": "Let $C_n$ be the combinatorial matrix of order $n$ given by: :$C_n = \\begin{bmatrix} x + y & y & \\cdots & y \\\\ y & x + y & \\cdots & y \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ y & y & \\cdots & x + y \\end{bmatrix}$ Then the determinant of $C_n$ is given by: :$\\det \\left({C_n}\\right) = x^{n-1} \\left ({x + n y}\\right)$"}
+{"_id": "1976", "title": "Power Set is Closed under Union", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Then: :$\\forall A, B \\in \\powerset S: A \\cup B \\in \\powerset S$"}
+{"_id": "1977", "title": "Matrix Product with Adjugate Matrix", "text": "Let $R$ be a commutative ring with unity. Let $\\mathbf A \\in R^{n \\times n}$ be a square matrix of order $n$. Let $\\adj {\\mathbf A}$ be its adjugate matrix. Then: {{begin-eqn}} {{eqn | l = \\mathbf A \\cdot \\adj {\\mathbf A}       | r = \\map \\det {\\mathbf A} \\cdot \\mathbf I_n }} {{eqn | l = \\adj {\\mathbf A} \\cdot \\mathbf A       | r = \\map \\det {\\mathbf A} \\cdot \\mathbf I_n }} {{end-eqn}} where $\\map \\det {\\mathbf A}$ is the determinant of $\\mathbf A$."}
+{"_id": "1978", "title": "Square of Ones Matrix", "text": "Let $\\mathbf J = \\sqbrk 1_n$ be a square ones matrix of order $n$. Then $\\mathbf J^2 = n \\mathbf J$. That is: :$\\begin{bmatrix} 1 & 1 & \\cdots & 1 \\\\ 1 & 1 & \\cdots & 1 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 1 & 1 & \\cdots & 1 \\end{bmatrix}^2 = \\begin{bmatrix} n & n & \\cdots & n \\\\ n & n & \\cdots & n \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ n & n & \\cdots & n \\end{bmatrix}$"}
+{"_id": "1979", "title": "Inverse of Combinatorial Matrix", "text": "Let $C_n$ be the combinatorial matrix of order $n$ given by: :$C_n = \\begin{bmatrix} x + y & y & \\cdots & y \\\\ y & x + y & \\cdots & y \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ y & y & \\cdots & x + y \\end{bmatrix}$ Then its inverse $C_n^{-1} = \\sqbrk b_n$ can be specified as: :$b_{i j} = \\dfrac {-y + \\delta_{i j} \\paren {x + n y} } {x \\paren {x + n y} }$ where $\\delta_{i j}$ is the Kronecker delta."}
+{"_id": "1980", "title": "Inverse of Vandermonde Matrix", "text": "Let $V_n$ be the Vandermonde matrix of order $n$ given by: :$V_n = \\begin {bmatrix}   x_1 & x_2 & \\cdots & x_n \\\\   x_1^2 & x_2^2 & \\cdots & x_n^2 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\   x_1^n & x_2^n & \\cdots & x_n^n \\end{bmatrix}$ Then its inverse $V_n^{-1} = \\sqbrk b_n$ can be specified as: :$b_{i j} = \\begin{cases} \\paren {-1}^{j - 1} \\paren {\\dfrac {\\displaystyle \\sum_{\\substack {1 \\mathop \\le m_1 \\mathop < \\ldots \\mathop < m_{n - j} \\mathop \\le n \\\\ m_1, \\ldots, m_{n - j} \\mathop \\ne i} } x_{m_1} \\cdots x_{m_{n - j} } } {x_i \\displaystyle \\prod_{\\substack {1 \\mathop \\le m \\mathop \\le n \\\\ m \\mathop \\ne i} } \\paren {x_m - x_i} } } & : 1 \\le  j < n \\\\ \\qquad \\qquad \\qquad \\dfrac 1 {x_i \\displaystyle \\prod_{\\substack {1 \\mathop \\le m \\mathop \\le n \\\\ m \\mathop \\ne i} } \\paren {x_i - x_m} } & : j = n \\end{cases}$"}
+{"_id": "1981", "title": "Floor Function is Idempotent", "text": ":$\\floor {\\floor x} = \\floor x$"}
+{"_id": "1982", "title": "Relation between Floor and Ceiling", "text": "Let $x \\in \\R$ be a real number. Let $\\left \\lfloor {x}\\right \\rfloor$ be the floor of $x$, and $\\left \\lceil {x}\\right \\rceil$ be the ceiling of $x$. Then the following results apply:"}
+{"_id": "1983", "title": "Range of Values of Floor Function", "text": "=== Number less than Integer iff Floor less than Integer === {{:Number less than Integer iff Floor less than Integer}} === Number not less than Integer iff Floor not less than Integer === {{:Number not less than Integer iff Floor not less than Integer}} === Integer equals Floor iff between Number and One Less === {{:Integer equals Floor iff between Number and One Less}} === Integer equals Floor iff Number between Integer and One More === {{:Integer equals Floor iff Number between Integer and One More}}"}
+{"_id": "1984", "title": "Range of Values of Ceiling Function", "text": "=== Number greater than Integer iff Ceiling greater than Integer === {{:Number greater than Integer iff Ceiling greater than Integer}} === Number not greater than Integer iff Ceiling not greater than Integer === {{:Number not greater than Integer iff Ceiling not greater than Integer}} === Integer equals Ceiling iff between Number and One More === {{:Integer equals Ceiling iff between Number and One More}} === Integer equals Ceiling iff Number between Integer and One Less === {{:Integer equals Ceiling iff Number between Integer and One Less}}"}
+{"_id": "1985", "title": "Steiner Inellipse is Unique", "text": "For every triangle, there is one and only one Steiner inellipse."}
+{"_id": "1987", "title": "Kronecker's Lemma", "text": "Let $\\sequence {x_n}$ be an infinite sequence of real numbers such that: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty x_n = s$  exists and is finite. Then for $0 < b_1 \\le b_2 \\le b_3 \\le \\ldots$ and $b_n \\to \\infty$:  :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\frac 1 {b_n} \\sum_{k \\mathop = 1}^n b_k x_k = 0$"}
+{"_id": "1989", "title": "Number of Permutations", "text": "Let $S$ be a set of $n$ elements. Let $r \\in \\N: r \\le n$. Then the number of $r$-permutations of $S$ is: :${}^r P_n = \\dfrac {n!} {\\paren {n - r}!}$ When $r = n$, this becomes: :${}^n P_n = \\dfrac {n!} {\\paren {n - n}!} = n!$ Using the falling factorial symbol, this can also be expressed: :${}^r P_n = n^{\\underline r}$"}
+{"_id": "1990", "title": "Construction of Permutations", "text": "The ${}^n P_n$ permutations of $n$ objects can be generated algorithmically. By Number of Permutations, that number is given by: :${}^n P_n = n!$ where $n!$ denotes the factorial of $n$. This will be demonstrated to hold."}
+{"_id": "1991", "title": "De Polignac's Formula", "text": "Let $n!$ be the factorial of $n$. Let $p$ be a prime number. Then $p^\\mu$ is a divisor of $n!$, and $p^{\\mu + 1}$ is not, where: :$\\displaystyle \\mu = \\sum_{k \\mathop > 0} \\floor {\\frac n {p^k} }$ where $\\floor {\\, \\cdot \\,}$ denotes the floor function."}
+{"_id": "1992", "title": "Sum of Binomial Coefficients over Lower Index", "text": ":$\\displaystyle \\sum_{i \\mathop = 0}^n \\binom n i = 2^n$"}
+{"_id": "1993", "title": "Alternating Sum and Difference of Binomial Coefficients for Given n", "text": ":$\\displaystyle \\forall n \\in \\Z: \\sum_{i \\mathop = 0}^n \\left({-1}\\right)^i \\binom n i = \\delta_{n 0}$"}
+{"_id": "1994", "title": "Symmetry Rule for Binomial Coefficients", "text": "Let $n \\in \\Z_{>0}, k \\in \\Z$. Then: :$\\dbinom n k = \\dbinom n {n - k}$"}
+{"_id": "1995", "title": "Factors of Binomial Coefficient", "text": "For all $r \\in \\R, k \\in \\Z$: :$k \\dbinom r k = r \\dbinom {r - 1} {k - 1}$ where $\\dbinom r k$ is a binomial coefficient. Hence: :$\\dbinom r k = \\dfrac r k \\dbinom {r - 1} {k - 1}$ (if $k \\ne 0$) and: :$\\dfrac 1 r \\dbinom r k = \\dfrac 1 k \\dbinom {r - 1} {k - 1}$ (if $k \\ne 0$ and $r \\ne 0$)"}
+{"_id": "1996", "title": "Sum of r+k Choose k up to n", "text": ":$\\displaystyle \\forall n \\in \\Z: n \\ge 0: \\sum_{k \\mathop = 0}^n \\binom {r + k} k = \\binom {r + n + 1} n$"}
+{"_id": "1997", "title": "Negated Upper Index of Binomial Coefficient", "text": ":$\\dbinom r k = \\paren {-1}^k \\dbinom {k - r - 1} k$"}
+{"_id": "1998", "title": "Alternating Sum and Difference of r Choose k up to n", "text": ":$\\displaystyle \\sum_{k \\mathop \\le n} \\paren {-1}^k \\binom r k = \\paren {-1}^n \\binom {r - 1} n$"}
+{"_id": "1999", "title": "Product of r Choose m with m Choose k", "text": ":$\\dbinom r m \\dbinom m k = \\dbinom r k \\dbinom {r - k} {m - k}$"}
+{"_id": "2000", "title": "Chu-Vandermonde Identity", "text": ":$\\displaystyle \\sum_k \\binom r k \\binom s {n - k} = \\binom {r + s} n$"}
+{"_id": "2001", "title": "Lucas' Theorem", "text": "Let $p$ be a prime number. Let $n, k \\in \\Z_{\\ge 0}$. Then: :$\\dbinom n k \\equiv \\dbinom {\\floor {n / p} } {\\floor {k / p} } \\dbinom {n \\bmod p} {k \\bmod p} \\pmod p$ where: :$\\dbinom n k$ denotes a binomial coefficient :$n \\bmod p$ denotes the modulo operation :$\\floor \\cdot$ denotes the floor function."}
+{"_id": "2003", "title": "Congruence Modulo Real Number is Equivalence Relation", "text": "For all $z \\in \\R$, congruence modulo $z$ is an equivalence relation."}
+{"_id": "2005", "title": "Filter Basis Generates Filter", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Let $\\BB \\subset \\powerset S$. Then: :$\\FF = \\set {V \\subseteq S: \\exists U \\in \\BB: U \\subseteq V}$ is a filter on $S$ {{iff}}: :$(1): \\quad \\forall V_1, V_2 \\in \\BB: \\exists U \\in \\BB: U \\subseteq V_1 \\cap V_2$ :$(2): \\quad \\O \\notin \\BB, \\BB \\ne \\O$ That is, $\\BB$ is a filter basis of the filter $\\FF$, which is generated by $\\BB$."}
+{"_id": "2006", "title": "Limit Point iff Superfilter Converges", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $\\mathcal F$ be a filter on $S$. Let $x \\in S$. Then $x$ is a limit point of $\\mathcal F$ {{iff}} there exists a superfilter $\\mathcal F'$ of $\\mathcal F$ on $S$ which converges to $x$."}
+{"_id": "2007", "title": "Zorn's Lemma", "text": "Let $\\struct {X, \\preceq}, X \\ne \\O$ be a non-empty ordered set such that every non-empty chain in $X$ has an upper bound in $X$. Then $X$ has at least one maximal element."}
+{"_id": "2008", "title": "Ultrafilter Lemma", "text": "Let $S$ be a set. Every filter on $S$ is contained in an ultrafilter on $S$."}
+{"_id": "2010", "title": "Generated Topology is a Topology", "text": "Let $X$ be a set. Let $\\mathcal S \\subseteq \\mathcal P \\left({X}\\right)$, where $\\mathcal P \\left({X}\\right)$ is the power set of $X$. Let $\\mathcal T_\\mathcal S$ be the generated topology for $\\mathcal S$. Then $\\mathcal T_\\mathcal S$ is a topology on $X$."}
+{"_id": "2011", "title": "Equivalent Definitions of Synthetic Basis", "text": "Let $X$ be a set. Then for a subset $\\BB \\subseteq \\powerset X$ of the power set of $X$, the following are equivalent: :$(1): \\quad$ $\\displaystyle \\forall A, B \\in \\BB: \\exists \\AA \\subseteq \\BB: A \\cap B = \\bigcup \\AA$. :$(2): \\quad$ $\\displaystyle \\forall A, B \\in \\BB: \\forall x \\in A \\cap B: \\exists W \\in \\BB: x \\in W \\subseteq A \\cap B$."}
+{"_id": "2013", "title": "Product Distributes over Modulo Operation", "text": "Let $x, y, z \\in \\R$ be real numbers. Let $x \\bmod y$ denote the modulo operation. Then: :$z \\left({x \\bmod y}\\right) = \\left({z x}\\right) \\bmod \\left({z y}\\right)$"}
+{"_id": "2014", "title": "Congruence by Product of Moduli", "text": "Let $a, b, m \\in \\Z$. Let $a \\equiv b \\pmod m$ denote that $a$ is congruent to $b$ modulo $m$. Then $\\forall n \\in \\Z, n \\ne 0$: : $a \\equiv b \\pmod m \\iff a n \\equiv b n \\pmod {m n}$"}
+{"_id": "2015", "title": "Definite Integral of Partial Derivative", "text": "Let $\\map f {x, y}$ and $\\map {\\dfrac {\\partial f} {\\partial x} } {x, y}$ be continuous functions of $x$ and $y$ on $D = \\closedint {x_1} {x_2} \\times \\closedint a b$. Then: :$\\displaystyle \\frac \\d {\\d x} \\int_a^b \\map f {x, y} \\rd y = \\int_a^b \\map {\\frac {\\partial f} {\\partial x} } {x, y} \\rd y$ for $x \\in \\closedint {x_1} {x_2}$."}
+{"_id": "2016", "title": "Law of Inverses (Modulo Arithmetic)", "text": "Let $m, n \\in \\Z$. Then: :$\\exists n' \\in \\Z: n n' \\equiv d \\pmod m$ where $d = \\gcd \\set {m, n}$."}
+{"_id": "2017", "title": "Common Factor Cancelling in Congruence", "text": "Let $a, b, x, y, m \\in \\Z$. Let: :$a x \\equiv b y \\pmod m$ and $a \\equiv b \\pmod m$ where $a \\equiv b \\pmod m$ denotes that $a$ is congruent modulo $m$ to $b$. Then: :$x \\equiv y \\pmod {m / d}$ where $d = \\gcd \\set {a, m}$."}
+{"_id": "2018", "title": "Chinese Remainder Theorem", "text": "Let $a, b, r, s \\in \\Z$. Let $r \\perp s$ (that is, let $r$ and $s$ be coprime). Then: :$a \\equiv b \\pmod {r s}$ {{iff}} $a \\equiv b \\pmod r$ and $a \\equiv b \\pmod s$ where $a \\equiv b \\pmod r$ denotes that $a$ is congruent modulo $r$ to $b$."}
+{"_id": "2019", "title": "Sum of Floors not greater than Floor of Sum", "text": "Let $\\left \\lfloor {x} \\right \\rfloor$ be the floor function. Then: :$\\left \\lfloor {x} \\right \\rfloor + \\left \\lfloor {y} \\right \\rfloor \\le \\left \\lfloor {x + y} \\right \\rfloor$ The equality holds: :$\\left \\lfloor {x} \\right \\rfloor + \\left \\lfloor {y} \\right \\rfloor = \\left \\lfloor {x + y} \\right \\rfloor$ {{iff}}: :$x \\bmod 1 + y \\bmod 1 < 1$ where $x \\bmod 1$ denotes the modulo operation."}
+{"_id": "2020", "title": "Sum of Ceilings not less than Ceiling of Sum", "text": "Let $\\left \\lceil {x} \\right \\rceil$ be the ceiling function. Then: :$\\left \\lceil {x} \\right \\rceil + \\left \\lceil {y} \\right \\rceil \\ge \\left \\lceil {x + y} \\right \\rceil$ The equality holds: :$\\left \\lceil {x} \\right \\rceil + \\left \\lceil {y} \\right \\rceil = \\left \\lceil {x + y} \\right \\rceil$ {{iff}} either: :$x \\in \\Z$ or $y \\in \\Z$ or: :$x \\bmod 1 + y \\bmod 1 > 1$ where $x \\bmod 1$ denotes the modulo operation."}
+{"_id": "2021", "title": "Integer to Power of p-1 over 2 Modulo p", "text": "Let $a \\in \\Z$. Let $p$ be an odd prime. Let $b = a^{\\frac {\\paren {p - 1} } 2}$. Then one of the following cases holds: :$b \\bmod p = 0$ which happens exactly when $a \\equiv 0 \\pmod p$, or: :$b \\bmod p = 1$ or: :$b \\bmod p = p - 1$ where: :$b \\bmod p$ denotes the modulo operation :$x \\equiv y \\pmod p$ denotes that $x$ is congruent modulo $p$ to $y$."}
+{"_id": "2022", "title": "Image Filter is Filter", "text": "Let $X, Y$ be sets. Let $\\mathcal P \\left({X}\\right)$ and $\\mathcal P \\left({Y}\\right)$ be the power sets of $X$ and $Y$ respectively. Let $f: X \\to Y$ be a mapping. Let $\\mathcal F \\subset \\mathcal P \\left({X}\\right)$ be a filter on $X$. Then the image filter of $\\mathcal F$ with respect to $f$: :$f \\left[{\\mathcal F}\\right] := \\left\\{ {U \\subseteq Y: f^{-1} \\left[{U}\\right] \\in \\mathcal F}\\right\\}$ is a filter on $Y$."}
+{"_id": "2024", "title": "Filter on Product Space Converges to Point iff Projections Converge to Projections of Point", "text": "Let $\\family{X_i}_{i \\mathop \\in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set. Let $\\displaystyle X := \\prod_{i \\mathop \\in I} X_i$ be the corresponding product space. Let $\\FF$ be a filter on $X$. Let $x \\in X$. Let $\\pr_i: X \\to X_i$ denote the projection from $X$ onto $X_i$. Then $\\FF$ converges to $x$ {{iff}}: :for all $i \\in I$ the image filter $\\map {\\pr_i} \\FF$ converges to $x_i := \\map {\\pr_i} x$."}
+{"_id": "2025", "title": "Image of Ultrafilter is Ultrafilter", "text": "Let $X, Y$ be two sets, $f: X \\to Y$ a mapping and $\\mathcal F$ an ultrafilter on $X$. Then the image filter $f \\left({\\mathcal F}\\right)$ is an ultrafilter on $Y$."}
+{"_id": "2027", "title": "Schur's Inequality", "text": "Let $x, y, z \\in \\R_{\\ge 0}$ be positive real numbers. Let $t \\in \\R, t > 0$ be a (strictly) positive real number. Then: :$x^t \\paren {x - y} \\paren {x - z} + y^t \\paren {y - z} \\paren {y - x} + z^t \\paren {z - x} \\paren {z - y} \\ge 0$ The equality holds {{iff}} either: : $x = y = z$ : Two of them are equal and the other is zero. When $t$ is a positive even integer, the inequality holds for ''all'' real numbers $x, y, z$."}
+{"_id": "2028", "title": "Wallis's Product", "text": "{{begin-eqn}} {{eqn | l = \\prod_{n \\mathop = 1}^\\infty \\frac {2 n} {2 n - 1} \\cdot \\frac {2 n} {2 n + 1}       | r = \\frac 2 1 \\cdot \\frac 2 3 \\cdot \\frac 4 3 \\cdot \\frac 4 5 \\cdot \\frac 6 5 \\cdot \\frac 6 7 \\cdot \\frac 8 7 \\cdot \\frac 8 9 \\cdots       | c =  }} {{eqn | r = \\frac \\pi 2       | c =  }} {{end-eqn}}"}
+{"_id": "2030", "title": "Number of Regions in Plane Defined by Given Number of Lines", "text": "The maximum number $L_n$ of regions in the plane that can be defined by $n$ straight lines in the plane is: :$L_n = \\dfrac {n \\paren {n + 1} } 2 + 1$ {{OEIS|A000124}}"}
+{"_id": "2031", "title": "Forward Difference of Falling Factorial", "text": "Let $f: \\R \\to \\R$ be a real function. Let $\\Delta$ denote the forward difference operator. Let $x^{\\underline m}$ be the $m$th falling factorial of $x$ Then: :$\\map \\Delta {x^{\\underline m} } = m x^{\\underline {m - 1} }$"}
+{"_id": "2032", "title": "Chen's Theorem", "text": "Every sufficiently large even integer is the sum of either: :$(1): \\quad$ two primes, or  :$(2): \\quad$ a prime and a semiprime."}
+{"_id": "2033", "title": "Binomial Coefficient involving Prime", "text": "Let $p$ be a prime number. Let $\\dbinom n p$ be a binomial coefficient. Then: :$\\dbinom n p \\equiv \\floor {\\dfrac n p} \\pmod p$ where: :$\\floor {\\dfrac n p}$ denotes the floor function."}
+{"_id": "2034", "title": "Binomial Coefficient of Prime Minus One Modulo Prime", "text": "Let $p$ be a prime number. Then: :$0 \\le k \\le p - 1 \\implies \\dbinom {p - 1} k \\equiv \\left({-1}\\right)^k \\pmod p$ where $\\dbinom {p - 1} k$ denotes a binomial coefficient."}
+{"_id": "2036", "title": "Number of Primes is Infinite", "text": "The number of primes is infinite."}
+{"_id": "2039", "title": "Quotient and Remainder to Number Base", "text": "Let $n \\in \\Z: n > 0$ be an integer. Let $n$ be expressed in base $b$: :$\\displaystyle n = \\sum_{j \\mathop = 0}^m {r_j b^j}$ i.e. :$n = \\left[{r_m r_{m-1} \\ldots r_2 r_1 r_0}\\right]_b$ Then: : $\\displaystyle \\left \\lfloor {\\frac n b} \\right \\rfloor = \\left[{r_m r_{m-1} \\ldots r_2 r_1}\\right]_b$ : $n \\bmod b = r_0$ where: : $\\left \\lfloor {.} \\right \\rfloor$ denotes the floor function; : $n \\bmod b$ denotes the modulo operation."}
+{"_id": "2040", "title": "Factorial Divisible by Prime Power", "text": "Let $n \\in \\Z: n \\ge 1$. Let $p$ be a prime number. Let $n$ be expressed in base $p$ notation: :$\\displaystyle n = \\sum_{j \\mathop = 0}^m r_j p^j$ where $0 \\le r_j < p$. Let $n!$ be the factorial of $n$. Let $p^\\mu$ be the largest power of $p$ which divides $n!$, that is: :$p^\\mu \\divides n!$ :$p^{\\mu + 1} \\nmid n!$ Then: :$\\mu = \\dfrac {n - \\map {s_p} n} {p - 1}$ where $\\map {s_p} n$ is the digit sum of $n$."}
+{"_id": "2041", "title": "Wilson's Theorem/Corollary 2", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $p$ be a prime factor of $n!$ with multiplicity $\\mu$. Let $n$ be expressed in a base $p$ representation as: {{begin-eqn}} {{eqn | l = n       | r = \\sum_{j \\mathop = 0}^m a_j p^j       | c = where $0 \\le a_j < p$ }} {{eqn | r = a_0 + a_1 p + a_2 p^2 + \\cdots + a_m p^m       | c = for some $m > 0$ }} {{end-eqn}} Then: :$\\dfrac {n!} {p^\\mu} \\equiv \\paren {-1}^\\mu a_0! a_1! \\dotsb a_m! \\pmod p$"}
+{"_id": "2042", "title": "Negative Number is Congruent to Modulus minus Number", "text": ":$\\forall m, n \\in \\Z: -m \\equiv n - m \\pmod n$ where $\\bmod n$ denotes congruence modulo $n$."}
+{"_id": "2043", "title": "Gaussian Integers form Subring of Complex Numbers", "text": "The ring of Gaussian integers: :$\\struct {\\Z \\sqbrk i, +, \\times}$ forms a subring of the set of complex numbers $\\C$."}
+{"_id": "2044", "title": "Gaussian Rationals form Subfield of Complex Numbers", "text": "The set of Gaussian rationals $\\Q \\left[{i}\\right]$, under the operations of complex addition and complex multiplication, forms a subfield of the set of complex numbers $\\C$."}
+{"_id": "2045", "title": "Complex Conjugation is Automorphism", "text": "Consider the field of complex numbers $\\C$. The operation of complex conjugation: :$\\forall z \\in \\C: z \\mapsto \\overline z$ is a field automorphism."}
+{"_id": "2046", "title": "Tschirnhaus Transformation", "text": "Let $P_n \\left({x}\\right) = 0$ be a polynomial equation of order $n$: :$a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0 = 0$ Then the substitution: $y = x + \\dfrac {a_{n-1}} {n a_n}$ converts $P_n$ into a depressed polynomial: :$b_n y^n + b_{n-1} y^{n-1} + \\cdots + b_1 y + b_0 = 0$ where $b_{n-1} = 0$. Such a substitution is called a '''Tschirnhaus transformation'''."}
+{"_id": "2047", "title": "Elementary Properties of Probability Measure", "text": "Let $\\EE$ be an experiment with probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. The probability measure $\\Pr$ of $\\EE$ has the following properties:"}
+{"_id": "2048", "title": "Elementary Properties of Event Space", "text": "Let $\\mathcal E$ be an experiment with a probability space $\\left({\\Omega, \\Sigma, \\Pr}\\right)$. The event space $\\Sigma$ of $\\mathcal E$ has the following properties:"}
+{"_id": "2049", "title": "Ring of Sets is Semiring of Sets", "text": "Let $\\mathcal R$ be a ring of sets. Then $\\mathcal R$ is also a semiring of sets."}
+{"_id": "2050", "title": "Set of All Real Intervals is Semiring of Sets", "text": "Let $\\mathbb S$ be the set of all real intervals. Then $\\mathbb S$ is a semiring of sets, but is '''not''' a ring of sets."}
+{"_id": "2051", "title": "Ring of Sets Generated by Semiring", "text": "Let $\\mathcal S$ be a semiring of sets. Let $\\mathcal R \\left({\\mathcal S}\\right)$ be the minimal ring generated by $\\mathcal S$. Let $\\mathcal L$ be the system of sets $A$ with the finite expansions: :$\\displaystyle A = \\bigcup_{k \\mathop = 1}^n A_k$ with respect to the sets $A_k \\in \\mathcal S$. Then $\\mathcal L = \\mathcal R \\left({\\mathcal S}\\right)$."}
+{"_id": "2052", "title": "Ring of Sets Closed under Finite Intersection", "text": "Let $\\mathcal R$ be a ring of sets. Let $A_1, A_2, \\ldots, A_n \\in \\mathcal R$. Then: :$\\displaystyle \\bigcap_{j \\mathop = 1}^n A_j \\in \\mathcal R$"}
+{"_id": "2053", "title": "Sigma-Algebra is Delta-Algebra", "text": "A $\\sigma$-algebra is also a $\\delta$-algebra."}
+{"_id": "2054", "title": "Addition Law of Probability", "text": ":$\\map \\Pr {A \\cup B} = \\map \\Pr A + \\map \\Pr B - \\map \\Pr {A \\cap B}$"}
+{"_id": "2055", "title": "Cardinality is Additive Function", "text": "Let $S$ be a finite set. Let $\\mathcal P \\left({S}\\right)$ be the power set of $S$. The function $C: \\mathcal P \\left({S}\\right) \\to \\R$, where $C$ is defined as the cardinality of a set, is an additive function."}
+{"_id": "2056", "title": "Finite Union of Sets in Additive Function", "text": "Let $\\mathcal A$ be an algebra of sets. Let $f: \\mathcal A \\to \\overline {\\R}$ be an additive function. Let $A_1, A_2, \\ldots, A_n$ be any finite collection of pairwise disjoint elements of $\\mathcal A$. Then: :$\\displaystyle f \\left({\\bigcup_{i \\mathop = 1}^n A_i}\\right) = \\sum_{i \\mathop = 1}^n f \\left({A_i}\\right)$ That is, for any collection of pairwise disjoint elements of $\\mathcal A$, $f$ of their union equals the sum of $f$ of the individual elements."}
+{"_id": "2057", "title": "Measure is Finitely Additive Function", "text": "Let $\\Sigma$ be a $\\sigma$-algebra on a set $X$. Let $\\mu: \\Sigma \\to \\overline {\\R}$ be a measure on $\\Sigma$. Then $\\mu$ is finitely additive."}
+{"_id": "2058", "title": "Finite Union of Sets in Subadditive Function", "text": "Let $\\mathcal A$ be an algebra of sets. Let $f: \\mathcal A \\to \\overline {\\R}$ be a subadditive function. Let $A_1, A_2, \\ldots, A_n$ be any finite collection of elements of $\\mathcal A$. Then: :$\\displaystyle f \\left({\\bigcup_{i \\mathop = 1}^n A_i}\\right) \\le \\sum_{i \\mathop = 1}^n f \\left({A_i}\\right)$ That is, for any finite collection of elements of $\\mathcal A$, $f$ of their union is less than or equal to the sum of $f$ of the individual elements."}
+{"_id": "2059", "title": "Additive Function on Empty Set is Zero", "text": "Let $\\mathcal A$ be an algebra of sets. Let $f: \\mathcal A \\to \\overline {\\R}$ be an additive function on $\\mathcal A$. Then $f \\left({\\varnothing}\\right) = 0$."}
+{"_id": "2060", "title": "Additive Function is Strongly Additive", "text": "Let $\\SS$ be an algebra of sets. Let $f: \\SS \\to \\overline \\R$ be an additive function on $\\SS$. Then $f$ is also strongly additive. That is: :$\\forall A, B \\in \\SS: \\map f {A \\cup B} + \\map f {A \\cap B} = \\map f A + \\map f B$"}
+{"_id": "2061", "title": "Set Difference is Disjoint with Reverse", "text": ":$\\paren {S \\setminus T} \\cap \\paren {T \\setminus S} = \\O$"}
+{"_id": "2062", "title": "Measure is Monotone", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Then $\\mu$ is monotone, that is: :$\\forall E, F \\in \\Sigma: E \\subseteq F \\implies \\map \\mu E \\le \\map \\mu F$"}
+{"_id": "2063", "title": "Countably Additive Function also Finitely Additive", "text": "Let $\\mathcal A$ be a $\\sigma$-algebra. Let $f: \\mathcal A \\to \\overline \\R$ be a function, where $\\overline \\R$ denotes the set of extended real numbers. Let $f$ be a countably additive function: :$\\displaystyle f \\left({\\bigcup_{i \\mathop \\in \\N} A_i}\\right) = \\sum_{i \\mathop \\in \\N} f \\left({A_i}\\right)$ such that there exists at least one $A \\in \\mathcal A$ such that $f \\left({A}\\right)$ is a finite number. Then $f$ is a finitely additive function."}
+{"_id": "2064", "title": "Countably Additive Function of Null Set", "text": "Let $\\mathcal A$ be a $\\sigma$-algebra. Let $f: \\mathcal A \\to \\overline \\R$ be a function, where $\\overline \\R$ denotes the extended set of real numbers. Let $f$ be a countably additive function: :$\\displaystyle f \\left({\\bigcup_{i \\mathop \\in \\N} A_i}\\right) = \\sum_{i \\mathop \\in \\N} f \\left({A_i}\\right)$ such that there exists at least one $A \\in \\mathcal A$ where $f \\left({A}\\right)$ is a finite number. Then: : $f \\left({\\varnothing}\\right) = 0$"}
+{"_id": "2065", "title": "Non-Negative Additive Function is Monotone", "text": "Let $\\SS$ be an algebra of sets. Let $f: \\SS \\to \\overline \\R$ be an additive function, that is: :$\\forall A, B \\in \\SS: A \\cap B = \\O \\implies \\map f {A \\cup B} = \\map f A + \\map f B$ If $\\forall A \\in \\SS: \\map f A \\ge 0$, then $f$ is monotone, that is: :$A \\subseteq B \\implies \\map f A \\le \\map f B$"}
+{"_id": "2066", "title": "Power Set is Sigma-Algebra", "text": "The power set of a set is a sigma-algebra."}
+{"_id": "2068", "title": "Probability Measure on Equiprobable Outcomes", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be an equiprobability space. Let $\\card \\Omega = n$. Then: :$\\forall \\omega \\in \\Omega: \\map \\Pr \\omega = \\dfrac 1 n$ :$\\forall A \\subseteq \\Omega: \\map \\Pr A = \\dfrac {\\card A} n$."}
+{"_id": "2069", "title": "Conditional Probability Defines Probability Space", "text": "Let $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ be a measure space. Let $B \\in \\Sigma$ such that $\\Pr \\left({B}\\right) > 0$. Let $Q: \\Sigma \\to \\R$ be the real-valued function defined as: :$Q \\left({A}\\right) = \\Pr \\left({A \\mid B}\\right)$ where: :$\\Pr \\left({A \\mid B}\\right) = \\dfrac {\\Pr \\left({A \\cap B}\\right)}{\\Pr \\left({B}\\right)}$ is the conditional probability of $A$ given $B$. Then $\\left({\\Omega, \\Sigma, Q}\\right)$ is a probability space."}
+{"_id": "2070", "title": "Bayes' Theorem", "text": "Let $\\Pr$ be a probability measure on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\map \\Pr {A \\mid B}$ denote the conditional probability of $A$ given $B$. Let $\\map \\Pr A > 0$ and $\\map \\Pr B > 0$. Then: :$\\map \\Pr {B \\mid A} = \\dfrac {\\map \\Pr {A \\mid B} \\, \\map \\Pr B} {\\map \\Pr A}$"}
+{"_id": "2072", "title": "Event Independence is Symmetric", "text": "Let $A$ and $B$ be events in a probability space. Let $A$ be independent of $B$. Then $B$ is independent of $A$. That is, '''is independent of''' is a symmetric relation."}
+{"_id": "2074", "title": "Set Difference and Intersection form Partition", "text": "Let $S$ and $T$ be sets such that: :$S \\setminus T \\ne \\O$ :$S \\cap T \\ne \\O$ where $S \\setminus T$ denotes set difference and $S \\cap T$ denotes set intersection. Then $S \\setminus T$ and $S \\cap T$ form a partition of $S$."}
+{"_id": "2075", "title": "Independent Events are Independent of Complement", "text": "Let $A$ and $B$ be events in a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Then $A$ and $B$ are independent {{iff}} $A$ and $\\Omega \\setminus B$ are independent."}
+{"_id": "2076", "title": "Probability of Independent Events Not Happening", "text": "Let $\\EE = \\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $A_1, A_2, \\ldots, A_m \\in \\Sigma$ be independent events in the event space of $\\EE$. Then the probability of none of $A_1$ to $A_m$ occurring is: :$\\displaystyle \\prod_{i \\mathop = 1}^m \\paren {1 - \\map \\Pr {A_i} }$"}
+{"_id": "2077", "title": "Probability of Occurrence of At Least One Independent Event", "text": "Let $\\EE = \\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $A_1, A_2, \\ldots, A_m \\in \\Sigma$ be independent events in the event space of $\\EE$. Then the probability of at least one of $A_1$ to $A_m$ occurring is: :$\\displaystyle 1 - \\prod_{i \\mathop = 1}^m \\paren {1 - \\map \\Pr {A_i} }$"}
+{"_id": "2078", "title": "Total Probability Theorem", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $\\set {B_1, B_2, \\ldots}$ be a partition of $\\Omega$ such that $\\forall i: \\map \\Pr {B_i} > 0$. Then: :$\\displaystyle \\forall A \\in \\Sigma: \\map \\Pr A = \\sum_i \\map \\Pr {A \\mid B_i} \\, \\map \\Pr {B_i}$"}
+{"_id": "2079", "title": "Probability of Limit of Sequence of Events", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space."}
+{"_id": "2080", "title": "Boole's Inequality", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $A_1, A_2, \\ldots, A_n$ be events in $\\Sigma$. Then: :$\\displaystyle \\map \\Pr {\\bigcup_{i \\mathop = 1}^n A_i} \\le \\sum_{i \\mathop = 1}^n \\map \\Pr {A_i}$"}
+{"_id": "2081", "title": "Additive Nowhere Negative Function is Subadditive", "text": "Let $\\AA$ be an algebra of sets. Let $f: \\AA \\to \\overline {\\R}$ be an additive function such that: :$\\forall A \\in \\AA: \\map f A \\ge 0$ Then $f$ is subadditive."}
+{"_id": "2082", "title": "Probability Measure is Subadditive", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Then $\\Pr$ is a subadditive function."}
+{"_id": "2083", "title": "Inclusion-Exclusion Principle", "text": "Let $\\SS$ be an algebra of sets. Let $A_1, A_2, \\ldots, A_n$ be finite sets. Let $f: \\SS \\to \\R$ be an additive function. Then: {{begin-eqn}} {{eqn | l = \\map f {\\bigcup_{i \\mathop = 1}^n A_i}       | r = \\sum_{i \\mathop = 1}^n \\map f {A_i}       | c =  }} {{eqn | o =        | ro= -       | r = \\sum_{1 \\mathop \\le i \\mathop < j \\mathop \\le n} \\map f {A_i \\cap A_j}       | c =  }} {{eqn | o =        | ro= +       | r = \\sum_{1 \\mathop \\le i \\mathop < j \\mathop < k \\mathop \\le n} \\map f {A_i \\cap A_j \\cap A_k}       | c =  }} {{eqn | o =        | ro= \\cdots       | c =  }} {{eqn | o =        | ro= +       | r = \\paren {-1}^{n - 1} \\map f {\\bigcap_{i \\mathop = 1}^n A_i}       | c =  }} {{end-eqn}}"}
+{"_id": "2084", "title": "Existence of Probability Space and Discrete Random Variable", "text": "Let $I$ be some indexing set. Let $S = \\left\\{{s_i: i \\in I}\\right\\} \\subset \\R$ be a countable set of real numbers. Let $\\left\\{{\\pi_i: i \\in I}\\right\\} \\subset \\R$ be a countable set of real numbers which satisfies: :$\\displaystyle \\forall i \\in I: \\pi_i \\ge 0, \\sum_{i \\mathop \\in I} \\pi_i = 1$ Then there exists a probability space $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ and a discrete random variable $X$ on $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ such that the probability mass function $p_X$ of $X$ is given by: {{begin-eqn}} {{eqn | l=p_X \\left({s_i}\\right)       | r=\\pi_i       | c=if $i \\in I$ }} {{eqn | l=p_X \\left({s}\\right)       | r=0       | c=if $s \\notin S$ }} {{end-eqn}}"}
+{"_id": "2085", "title": "Sum and Product of Discrete Random Variables", "text": "Let $X$ and $Y$ be discrete random variables on the probability space $\\left({\\Omega, \\Sigma, \\Pr}\\right)$."}
+{"_id": "2086", "title": "Countable Function on Power Set of Sample Space is Discrete Random Variable", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space such that $\\Sigma$ is the power set of $\\Omega$. Let $f: \\Omega \\to \\R$ be a function such that $\\Img f$ is countable. Then $f$ is a discrete random variable on $\\struct {\\Omega, \\Sigma, \\Pr}$."}
+{"_id": "2087", "title": "Characteristic Function on Event is Discrete Random Variable", "text": "Let $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ be a probability space. Let $E \\in \\Sigma$ be any event of $\\left({\\Omega, \\Sigma, \\Pr}\\right)$. Let $\\chi_E: \\Omega \\to \\left\\{{0, 1}\\right\\}$ be the characteristic function of $E$. Then $\\chi_E$ is a discrete random variable on $\\left({\\Omega, \\Sigma, \\Pr}\\right)$."}
+{"_id": "2088", "title": "Poisson Distribution Gives Rise to Probability Mass Function", "text": "Let $X$ be a discrete random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $X$ have the poisson distribution with parameter $\\lambda$ (where $\\lambda > 0$). Then $X$ gives rise to a probability mass function."}
+{"_id": "2089", "title": "Geometric Distribution Gives Rise to Probability Mass Function", "text": "Let $X$ be a discrete random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $X$ have the geometric distribution with parameter $p$ (where $0 < p < 1$). Then $X$ gives rise to a probability mass function."}
+{"_id": "2090", "title": "Negative Binomial Distribution Gives Rise to Probability Mass Function/Second Form", "text": "Let $X$ have the negative binomial distribution (second form) with parameters $n$ and $p$ ($0 < p < 1$). Then $X$ gives rise to a probability mass function."}
+{"_id": "2092", "title": "Bernoulli Process as Binomial Distribution", "text": "Let $\\left \\langle{X_i}\\right \\rangle$ be a finite Bernoulli process of length $n$ such that each of the $X_i$ in the sequence is a Bernoulli trial with parameter $p$. Then the number of successes in $\\left \\langle{X_i}\\right \\rangle$ is modelled by a binomial distribution with parameters $n$ and $p$. Hence it can be seen that: :$X \\sim \\operatorname{B} \\left({1, p}\\right)$ is the same thing as $X \\sim \\operatorname{Bern} \\left({p}\\right)$"}
+{"_id": "2093", "title": "Binomial Distribution Approximated by Poisson Distribution", "text": "Let $X$ be a discrete random variable which has the binomial distribution with parameters $n$ and $p$. Suppose $n$ is \"very large\" and $p$ is \"very small\", but $np$ of a \"reasonable size\". Then $X$ can be approximated by a Poisson distribution with parameter $\\lambda$ where $\\lambda = np$."}
+{"_id": "2094", "title": "Bernoulli Process as Geometric Distribution", "text": "Let $\\left \\langle{X_i}\\right \\rangle$ be a Bernoulli process with parameter $p$. Let $\\mathcal E$ be the experiment which consists of performing the Bernoulli trial $X_i$ until a failure occurs, and then stop. Let $k$ be the number of successes before a failure is encountered. Then $k$ is modelled by a geometric distribution with parameter $p$."}
+{"_id": "2095", "title": "Function of Discrete Random Variable", "text": "Let $X$ be a discrete random variable on the probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $g: \\R \\to \\R$ be any real function. Then $Y = g \\sqbrk X$, defined as: :$\\forall \\omega \\in \\Omega: \\map Y \\omega = g \\sqbrk {\\map X \\omega}$ is also a discrete random variable."}
+{"_id": "2096", "title": "Expectation of Function of Discrete Random Variable", "text": "Let $X$ be a discrete random variable. Let $\\expect X$ be the expectation of $X$. Let $g: \\R \\to \\R$ be a real function. Then: :$\\displaystyle \\expect {g \\sqbrk X} = \\sum_{x \\mathop \\in \\Omega_X} \\map g x \\, \\map \\Pr {X = x}$ whenever the sum is absolutely convergent."}
+{"_id": "2097", "title": "Probability Mass Function of Function of Discrete Random Variable", "text": "Let $X$ be a discrete random variable. Let $Y = g \\left({X}\\right)$, where $g: \\R \\to \\R$ is a real function. Then the probability mass function of $Y$ is given by: :$\\displaystyle p_Y \\left({y}\\right) = \\sum_{x \\mathop \\in g^{-1} \\left({y}\\right)} \\Pr \\left({X = x}\\right)$"}
+{"_id": "2098", "title": "Variance as Expectation of Square minus Square of Expectation", "text": "Let $X$ be a random variable. Then the variance of $X$ can be expressed as: :$\\var X = \\expect {X^2} - \\paren {\\expect X}^2$ That is, it is the expectation of the square of $X$ minus the square of the expectation of $X$."}
+{"_id": "2099", "title": "Derivative of Geometric Sequence", "text": "Let $x \\in \\R: \\size x < 1$. Then: :$\\displaystyle \\sum_{n \\mathop \\ge 1} n x^{n - 1} = \\frac 1 {\\paren {1 - x}^2}$"}
+{"_id": "2100", "title": "Expectation of Shifted Geometric Distribution", "text": "Let $X$ be a discrete random variable with the shifted geometric distribution with parameter $p$. Then the expectation of $X$ is given by: :$\\expect X = \\dfrac 1 p$"}
+{"_id": "2101", "title": "Expectation of Bernoulli Distribution", "text": "Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$. Then the expectation of $X$ is given by: :$\\expect X = p$"}
+{"_id": "2102", "title": "Expectation of Binomial Distribution", "text": "Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$ for some $n \\in \\N$ and $0 \\le p \\le 1$. Then the expectation of $X$ is given by: :$\\expect X = n p$"}
+{"_id": "2103", "title": "Expectation of Poisson Distribution", "text": "Let $X$ be a discrete random variable with the Poisson distribution with parameter $\\lambda$. Then the expectation of $X$ is given by: :$\\expect X = \\lambda$"}
+{"_id": "2104", "title": "Variance of Bernoulli Distribution", "text": "Let $X$ be a discrete random variable with the Bernoulli distribution with parameter $p$: :$X \\sim \\Bernoulli p$ Then the variance of $X$ is given by: :$\\var X = p \\paren {1 - p}$"}
+{"_id": "2105", "title": "Variance of Poisson Distribution", "text": "Let $X$ be a discrete random variable with the Poisson distribution with parameter $\\lambda$. Then the variance of $X$ is given by: :$\\var X = \\lambda$"}
+{"_id": "2106", "title": "Variance of Binomial Distribution", "text": "Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$. Then the variance of $X$ is given by: :$\\var X = n p \\paren {1 - p}$"}
+{"_id": "2107", "title": "Sum of Expectations of Independent Trials", "text": "Let $\\EE_1, \\EE_2, \\ldots, \\EE_n$ be a sequence of experiments whose outcomes are independent of each other. Let $X_1, X_2, \\ldots, X_n$ be discrete random variables on $\\EE_1, \\EE_2, \\ldots, \\EE_n$ respectively. Let $\\expect {X_j}$ denote the expectation of $X_j$ for $j \\in \\set {1, 2, \\ldots, n}$. Then we have, whenever both sides are defined: :$\\displaystyle \\expect {\\sum_{j \\mathop = 1}^n X_j} = \\sum_{j \\mathop = 1}^n \\expect {X_j}$ That is, the sum of the expectations equals the expectation of the sum."}
+{"_id": "2108", "title": "Sum of Variances of Independent Trials", "text": "Let $\\EE_1, \\EE_2, \\ldots, \\EE_n$ be a sequence of experiments whose outcomes are independent of each other. Let $X_1, X_2, \\ldots, X_n$ be discrete random variables on $\\EE_1, \\EE_2, \\ldots, \\EE_n$ respectively. Let $\\var {X_j}$ be the variance of $X_j$ for $j \\in \\set {1, 2, \\ldots, n}$. Then: :$\\displaystyle \\var {\\sum_{j \\mathop = 1}^n X_j} = \\sum_{j \\mathop = 1}^n \\var {X_j}$ That is, the sum of the variances equals the variance of the sum."}
+{"_id": "2109", "title": "Variance of Shifted Geometric Distribution", "text": "Let $X$ be a discrete random variable with the shifted geometric distribution with parameter $p$. Then the variance of $X$ is given by: :$\\var X = \\dfrac {1 - p} {p^2}$"}
+{"_id": "2110", "title": "Expectation of Geometric Distribution", "text": "Let $X$ be a discrete random variable with the geometric distribution with parameter $p$. Then the expectation of $X$ is given by: :$E \\left({X}\\right) = \\dfrac p {1-p}$"}
+{"_id": "2111", "title": "Variance of Geometric Distribution", "text": "Let $X$ be a discrete random variable with the geometric distribution with parameter $p$. Then the variance of $X$ is given by: :$\\operatorname{var} \\left({X}\\right) = \\dfrac p {\\left({1 - p}\\right)^2}$"}
+{"_id": "2112", "title": "Bernoulli Process as Negative Binomial Distribution", "text": "Let $\\sequence {X_i}$ be a Bernoulli process with parameter $p$."}
+{"_id": "2113", "title": "Properties of Cumulative Distribution Function", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $X$ be a random variable on $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\map F X$ be the cumulative distribution function of $X$: :$\\forall x \\in \\R: \\map F x = \\map \\Pr {X \\le x}$ Then the following conditions apply to $\\map F X$:"}
+{"_id": "2114", "title": "Discrete Random Variable is Random Variable", "text": "Let $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ be a probability space. Let $X$ be a discrete random variable on $\\left({\\Omega, \\Sigma, \\Pr}\\right)$. Then $X$ fulfils the condition: :$\\forall x \\in \\R: \\left\\{{\\omega \\in \\Omega: X \\left({\\omega}\\right) \\le x}\\right\\} \\in \\Sigma$ That is, $X$ fulfils the condition for it to be a random variable."}
+{"_id": "2115", "title": "Expectation of Discrete Uniform Distribution", "text": "Let $X$ be a discrete random variable with the discrete uniform distribution with parameter $n$. Then the expectation of $X$ is given by: :$\\expect X = \\dfrac {n + 1} 2$"}
+{"_id": "2116", "title": "Variance of Discrete Uniform Distribution", "text": "Let $X$ be a discrete random variable with the discrete uniform distribution with parameter $p$. Then the variance of $X$ is given by: :$\\var X = \\dfrac {n^2 - 1} {12}$"}
+{"_id": "2117", "title": "Probability Generating Function of Degenerate Distribution", "text": "Let $X$ be the degenerate distribution: :$\\forall x \\in \\N: p_X \\left({x}\\right) = \\begin{cases} 1 & : x = k \\\\ 0 & : x \\ne k \\end{cases}$ where $k \\in \\N$. Then the p.g.f. of $X$ is: :$\\Pi_X \\left({s}\\right) = s^k$"}
+{"_id": "2118", "title": "Probability Generating Function of Bernoulli Distribution", "text": "Let $X$ be a discrete random variable with the Bernoulli distribution with parameter $p$. Then the p.g.f. of $X$ is: :$\\Pi_X \\left({s}\\right) = q + ps$ where $q = 1 - p$."}
+{"_id": "2119", "title": "Probability Generating Function of Binomial Distribution", "text": "Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$. Then the p.g.f. of $X$ is: :$\\Pi_X \\left({s}\\right) = \\left({q + ps}\\right)^n$ where $q = 1 - p$."}
+{"_id": "2120", "title": "Probability Generating Function of Shifted Geometric Distribution", "text": "Let $X$ be a discrete random variable with the shifted geometric distribution with parameter $p$. Then the p.g.f. of $X$ is: :$\\map {\\Pi_X} s = \\dfrac {p s} {1 - q s}$ where $q = 1 - p$."}
+{"_id": "2121", "title": "Probability Generating Function of Geometric Distribution", "text": "Let $X$ be a discrete random variable with the geometric distribution with parameter $p$. Then the p.g.f. of $X$ is: :$\\map {\\Pi_X} s = \\dfrac q {1 - p s}$ where $q = 1 - p$."}
+{"_id": "2122", "title": "Probability Generating Function of Poisson Distribution", "text": "Let $X$ be a discrete random variable with the Poisson distribution with parameter $\\lambda$. Then the p.g.f. of $X$ is: :$\\Pi_X \\left({s}\\right) = e^{-\\lambda \\left({1-s}\\right)}$"}
+{"_id": "2123", "title": "Expectation of Discrete Random Variable from PGF", "text": "Let $X$ be a discrete random variable whose probability generating function is $\\map {\\Pi_X} s$. Then the expectation of $X$ is the value of the first derivative of $\\map {\\Pi_X} s$ {{WRT|Differentiation}} $s$ at $s = 1$. That is: :$\\expect X = \\map {\\Pi'_X} 1$"}
+{"_id": "2124", "title": "Variance of Discrete Random Variable from PGF", "text": "Let $X$ be a discrete random variable whose probability generating function is $\\Pi_X \\left({s}\\right)$. Then the variance of $X$ can be easily obtained from the value of the second derivative of $\\Pi_X \\left({s}\\right)$ WRT $s$ at $x=1$: :$\\operatorname{var} \\left({X}\\right) = \\Pi''_X \\left({1}\\right) + \\mu - \\mu^2$ where $\\mu = E \\left({X}\\right)$ is the expectation of $X$."}
+{"_id": "2125", "title": "Properties of Probability Generating Function", "text": "Let $X$ be a discrete random variable whose probability generating function is $\\Pi_X \\left({s}\\right)$. Then $\\Pi_X \\left({s}\\right)$ has the following properties:"}
+{"_id": "2126", "title": "Nth Derivative of Mth Power", "text": "Let $m \\in \\Z$ be an integer such that $m \\ge 0$. The $n$th derivative of $x^m$ {{WRT|Differentiation}} $x$ is: :$\\dfrac {\\d^n} {\\d x^n} x^m = \\begin{cases} m^{\\underline n} x^{m - n} & : n \\le m \\\\ 0 & : n > m \\end{cases}$ where $m^{\\underline n}$ denotes the falling factorial."}
+{"_id": "2127", "title": "Derivative of Complex Power Series", "text": "Let $\\xi \\in \\C$ be a complex number. Let $\\sequence {a_n}$ be a sequence in $\\C$. Let $\\displaystyle f \\paren z = \\sum_{n \\mathop = 0}^\\infty a_n \\paren {z - \\xi}^n$ be a power series in a complex variable $z \\in \\C$ about $\\xi$. Let $R$ be the radius of convergence of the series defining $f \\paren z$. Let $\\cmod {z - \\xi} < R$. Then $f$ is complex-differentiable and its derivative is: :$\\displaystyle f' \\paren z = \\sum_{n \\mathop = 1}^\\infty n a_n \\paren {z - \\xi}^{n - 1}$"}
+{"_id": "2128", "title": "Probability Generating Function of Discrete Uniform Distribution", "text": "Let $X$ be a discrete random variable with the discrete uniform distribution with parameter $n$. Then the p.g.f. of $X$ is: :$\\displaystyle \\map {\\Pi_X} s = \\frac {s \\paren {1 - s^n} } {n \\paren {1 - s} }$"}
+{"_id": "2129", "title": "Derivatives of PGF of Bernoulli Distribution", "text": "Let $X$ be a discrete random variable with the Bernoulli distribution with parameter $p$. Then the derivatives of the PGF of $X$ w.r.t. $s$ are: :$\\dfrac {\\mathrm d^k} {\\mathrm d s^k} \\Pi_X \\left({s}\\right) = \\begin{cases} p & : k = 1 \\\\ 0 & : k > 1 \\end{cases}$"}
+{"_id": "2130", "title": "Derivatives of PGF of Binomial Distribution", "text": "Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$. Then the derivatives of the PGF of $X$ w.r.t. $s$ are: :$\\dfrac {\\mathrm d^k} {\\mathrm ds^k} \\Pi_X \\left({s}\\right) = \\begin{cases} n^{\\underline k} p^k \\left({q + ps}\\right)^{n-k} & : k \\le n \\\\ 0 & : k > n \\end{cases}$ where: : $n^{\\underline k}$ is the falling factorial : $q = 1 - p$"}
+{"_id": "2131", "title": "Higher Derivatives of Exponential Function", "text": "Let $\\exp x$ be the exponential function. Then: :$\\map {\\dfrac {\\d^n} {\\d x^n} } {\\exp x} = \\exp x$"}
+{"_id": "2132", "title": "Derivatives of PGF of Poisson Distribution", "text": "Let $X$ be a discrete random variable with the Poisson distribution with parameter $\\lambda$. Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are: :$\\dfrac {d^k} {\\d s^k} \\, \\map {\\Pi_X} s = \\lambda^k e^{- \\lambda \\paren {1 - s} }$"}
+{"_id": "2133", "title": "Derivatives of Function of a x + b", "text": "Let $f$ be a real function which is differentiable on $\\R$. Let $a, b \\in \\R$ be constants. Then: : $\\dfrac {\\d^n} {\\d x^n} \\left({f \\left({a x + b}\\right)}\\right) = a^n \\dfrac {\\d^n} {\\d z^n} \\left({f \\left({z}\\right)}\\right)$ where $z = a x + b$."}
+{"_id": "2134", "title": "Nth Derivative of Reciprocal of Mth Power", "text": "Let $m \\in \\Z$ be an integer such that $m > 0$. The $n$th derivative of $\\dfrac 1 {x^m}$ {{WRT|Differentiation}} $x$ is: :$\\dfrac {\\d^n} {\\d x^n} \\dfrac 1 {x^m} = \\dfrac {\\paren {-1}^n m^{\\overline n}} {x^{m + n}}$ where $m^{\\overline n}$ denotes the rising factorial."}
+{"_id": "2135", "title": "Derivatives of PGF of Geometric Distribution", "text": "Let $X$ be a discrete random variable with the geometric distribution with parameter $p$. Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are: :$\\dfrac {\\d^n} {\\d s^n} \\map {\\Pi_X} s = \\dfrac {q p^n n!} {\\paren {1 - p s}^{n + 1} }$ where $q = 1 - p$."}
+{"_id": "2136", "title": "Derivatives of PGF of Shifted Geometric Distribution", "text": "Let $X$ be a discrete random variable with the shifted geometric distribution with parameter $p$. Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are: :$\\dfrac {\\mathrm d^n} {\\mathrm d s^n} \\Pi_X \\left({s}\\right) = \\dfrac {p q^{n - 1} \\left({n - 1}\\right)!} {\\left({1 - q s}\\right)^{n + 1} }$ where $q = 1 - p$."}
+{"_id": "2140", "title": "Total Expectation Theorem", "text": "Let $\\mathcal E = \\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $x$ be a discrete random variable on $\\mathcal E$. Let $\\set {B_1 \\mid B_2 \\mid \\cdots}$ be a partition of $\\omega$ such that $\\map \\Pr {B_i} > 0$ for each $i$. Then: :$\\displaystyle \\expect X = \\sum_i \\expect {X \\mid B_i} \\, \\map \\Pr {B_i}$ whenever this sum converges absolutely. In the above: :$\\expect X$ denotes the expectation of $X$ :$\\expect {X \\mid B_i}$ denotes the conditional expectation of $X$ given $B_i$."}
+{"_id": "2141", "title": "Function of Two Discrete Random Variables", "text": "Let $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ be a probability space. Let $X$ and $Y$ be discrete random variables on $\\left({\\Omega, \\Sigma, \\Pr}\\right)$. Let $g: \\R^2 \\to \\R$ be a real-valued function. Then $Z = g \\left({X, Y}\\right)$, defined as: :$\\forall \\omega \\in \\Omega: Z \\left({\\omega}\\right) = g \\left({X \\left({\\omega}\\right), Y \\left({\\omega}\\right)}\\right)$ is also a discrete random variable."}
+{"_id": "2142", "title": "Expectation of Function of Joint Probability Mass Distribution", "text": "Let $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ be a probability space. Let $X$ and $Y$ be discrete random variables on $\\left({\\Omega, \\Sigma, \\Pr}\\right)$. Let $E \\left({X}\\right)$ be the expectation of $X$. Let $g: \\R^2 \\to \\R$ be a real-valued function Let $p_{X, Y}$ be the joint probability mass function of $X$ and $Y$. :$\\displaystyle E \\left({g \\left({X, Y}\\right)}\\right) = \\sum_{x \\mathop \\in \\Omega_X} \\sum_{y \\mathop \\in \\Omega_Y} g \\left({x, y}\\right) p_{X, Y} \\left({x, y}\\right)$ whenever the sum is absolutely convergent."}
+{"_id": "2143", "title": "Linearity of Expectation Function", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $X$ and $Y$ be random variables on $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\expect X$ denote the expectation of $X$. Then: :$\\forall \\alpha, \\beta \\in \\R: \\expect {\\alpha X + \\beta Y} = \\alpha \\, \\expect X + \\beta \\, \\expect Y$"}
+{"_id": "2145", "title": "Condition for Independence from Product of Expectations", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $X$ and $Y$ be discrete random variables on $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\expect X$ denote the expectation of $X$. Then $X$ and $Y$ are independent {{iff}}: :$\\expect {\\map g x \\map h y} = \\expect {\\map g x} \\expect {\\map h y}$ for all functions $g, h: \\R \\to \\R$ for which the latter two expectations exist."}
+{"_id": "2146", "title": "PGF of Sum of Independent Discrete Random Variables", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $X$ and $Y$ be independent discrete random variables on $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $Z$ be a discrete random variable such that $Z = X + Y$. Then: :$\\map {\\Pi_Z} s = \\map {\\Pi_X} s \\, \\map {\\Pi_Y} s$ where $\\map {\\Pi_Z} s$ is the probability generating function of $Z$."}
+{"_id": "2147", "title": "Brahmagupta Theorem", "text": "If a cyclic quadrilateral has diagonals which are perpendicular, then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. Specifically: Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ are perpendicular, crossing at $M$. Let $EF$ be a line passing through $M$ and crossing opposite sides $BC$ and $AD$ of $ABCD$. Then $EF$ is perpendicular to $BC$ {{iff}} $F$ is the midpoint of $AD$."}
+{"_id": "2149", "title": "Clavius's Law", "text": "If, from the negation of a proposition $p,$ we can derive $p$, we may conclude $p$:"}
+{"_id": "2150", "title": "Viète's Formulas", "text": "Let $P$ be a polynomial of degree $n$ with real or complex coefficients: {{begin-eqn}} {{eqn | l = \\map P x        | r = \\sum_{i \\mathop = 0}^n a_i x^i }} {{eqn | r = a_n x^n + a_{n - 1} x^{n - 1} + \\dotsb + a_1 x + a_0 }} {{end-eqn}} where $a_n \\ne 0$. Let $z_1, \\ldots, z_k$ be real or complex roots of $P$, not assumed distinct. Let $P$ be expressible in the form: :$\\displaystyle \\map P x = a_n \\prod_{k \\mathop = 1}^n \\paren {x - z_k}$ Then: {{begin-eqn}} {{eqn | l = \\paren {-1}^k \\dfrac {a_{n - k} } {a_n}       | r = e_k \\paren {\\set {z_1, \\ldots, z_n} }       | c = that is, the elementary symmetric function on $\\set {z_1, \\ldots, z_n}$ }} {{eqn | r = \\sum_{1 \\mathop \\le i_1 \\mathop < \\dotsb \\mathop < i_k \\mathop \\le n} z_{i_1} \\dotsm z_{i_k}        | c = for $k = 1, 2, \\ldots, n$. }} {{end-eqn}}"}
+{"_id": "2151", "title": "Inscribed Angle Theorem", "text": "An inscribed angle is equal to half the angle that is subtended by that arc. :300px Thus, in the figure above: :$\\angle ABC = \\frac 1 2 \\angle ADC$ {{EuclidSaid}} :''In a circle the angle at the center is double of the angle at the circumference, when the angles have the same circumference as base.'' {{EuclidPropRef|III|20}}"}
+{"_id": "2153", "title": "Existence and Uniqueness Theorem for 1st Order IVPs", "text": "Let $x' = \\map f {t, x}$, $\\map x {t_0} = x_0$ be an explicit ODE of dimension $n$. Let there exist an open ball $V = \\sqbrk {t_0 - \\ell_0, t_0 + \\ell_0} \\times \\map {\\overline B} {x_0, \\epsilon}$ of $\\tuple {t_0, x_0}$ in phase space $\\R \\times \\R^n$ such that $f$ is Lipschitz continuous on $V$. {{explain|Notation needs to be explained: $\\sqbrk {t_0 - \\ell_0, t_0 + \\ell_0}$ looks as though it should be an interval (and so needs to be written in Wirth interval notation $\\closedint {t_0 - \\ell_0} {t_0 + \\ell_0}$ so as to abide by house style rules), and $\\tuple {t_0, x_0}$ is probably an ordered pair. It's not clear enough. The immediate confusion arises because as $\\closedint {t_0 - \\ell_0} {t_0 + \\ell_0}$ is a closed interval it is counter-intuitive for it to be one of the factors of an open ball expressed as a Cartesian product.}} Then there exists $\\ell < \\ell_0$ such that there exists a unique solution $\\map x t$ defined for $t \\in \\closedint {t_0 - \\ell_0} {t_0 + \\ell_0}$. {{explain|what is an \"IVP\"?}}"}
+{"_id": "2155", "title": "Maximum Modulus Principle", "text": "A non-constant analytic function $f$ in an open connected set $D$ does not have any interior maximum points. That is, for each $z \\in D$ and $\\delta > 0$, there exists some $\\omega \\in B_\\delta \\left({z}\\right) \\cap D$, such that: :$\\left\\vert{f \\left({\\omega}\\right)}\\right\\vert > \\left\\vert{f \\left({z}\\right)}\\right\\vert$."}
+{"_id": "2156", "title": "Euler Polyhedron Formula", "text": "For any convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces:  :$V - E + F = 2$"}
+{"_id": "2158", "title": "Line Joining Centers of Two Circles Touching Internally", "text": "Let two circles touch internally. Then the straight line joining their centers passes through the point where they touch. {{:Euclid:Proposition/III/11}}"}
+{"_id": "2159", "title": "Properties of Matrix Exponential", "text": "In the following: : $\\mathrm A$ and $\\mathrm B$ are constant square matrices : $P$ is a nonsingular matrix : $t, s \\in \\R$ The matrix exponential $e^{\\mathrm A t}$ has the following properties:"}
+{"_id": "2162", "title": "Lyapunov's Stability Theorem", "text": "Let $V$ be a Lyapunov function on an open set containing an equilibrium point $x_0$. {{Disambiguate|Definition:Open Set}} Then $x_0$ is stable. If $V$ is strict, then $x_0$ is asymptotically stable. {{explain|strict}}"}
+{"_id": "2163", "title": "Bendixson-Dulac Theorem", "text": "Suppose there exists a continuously differentiable function $\\alpha \\left({x, y}\\right)$ on a simply connected domain. {{Explain|What ''is'' the domain? Reals, complex, or what?}} Suppose that: :$\\nabla \\cdot \\left({\\alpha F}\\right)$  is either always positive or always negative. Then the two-dimensional autonomous system: :$ \\left({x, y}\\right)' = F \\left({x, y}\\right)$ does not have a periodic solution."}
+{"_id": "2164", "title": "General Vector Solution of Fundamental Matrix", "text": "Let $\\map \\Phi t$ be a fundamental matrix of the system $x' = \\map A t x$. Then: :$\\map \\Phi t c$ is a general solution of $x' = \\map A t x$."}
+{"_id": "2165", "title": "General Fundamental Matrix", "text": "Let $\\map \\Phi t$ be a fundamental matrix of the system $x' = \\map A t x$. Then: :$\\map \\Phi t C$ is a general fundamental matrix of $x' = \\map A t x$, where $C$ is ''any'' nonsingular matrix."}
+{"_id": "2167", "title": "Condition for Composite Mapping on Left", "text": "Let $A, B, C$ be sets. Suppose that $C$ is non-empty. Let $f: A \\to B$ and $g: A \\to C$ be mappings. Let $\\RR: B \\to C$ be a relation such that $g = \\RR \\circ f$ is the composite of $f$ and $\\RR$. Then $\\RR$ may be a mapping {{iff}}: :$\\forall x, y \\in A: \\map f x = \\map f y \\implies \\map g x = \\map g y$ That is: :$\\forall x, y \\in A: \\map f x = \\map f y \\implies \\map g x = \\map g y$ {{iff}}: :$\\exists h: B \\to C$ such that $h$ is a mapping and $h \\circ f = g$"}
+{"_id": "2168", "title": "Condition for Composite Mapping on Right", "text": "Let $A, B, C$ be sets. Let $f: B \\to A$ and $g: C \\to A$ be mappings. Let $\\mathcal R: C \\to B$ be a relation such that $g = f \\circ \\mathcal R$ is the composite of $\\mathcal R$ and $f$. Then $\\mathcal R$ may be a mapping {{iff}}: :$\\Img g \\subseteq \\Img f$ That is: :$\\Img g \\subseteq \\Img f$ {{iff}}: :$\\exists h: C \\to B$ such that $h$ is a mapping and $f \\circ h = g$"}
+{"_id": "2169", "title": "Set Union Preserves Subsets", "text": "Let $A, B, S, T$ be sets. Then: :$A \\subseteq B, \\ S \\subseteq T \\implies A \\cup S \\subseteq B \\cup T$"}
+{"_id": "2170", "title": "Set Complement inverts Subsets", "text": ":$S \\subseteq T \\iff \\map \\complement T \\subseteq \\map \\complement S$"}
+{"_id": "2171", "title": "Set Difference with Superset is Empty Set", "text": ":$S \\subseteq T \\iff S \\setminus T = \\O$"}
+{"_id": "2172", "title": "Intersection with Subset is Subset", "text": ":$S \\subseteq T \\iff S \\cap T = S$"}
+{"_id": "2173", "title": "Union with Superset is Superset", "text": ":$S \\subseteq T \\iff S \\cup T = T$"}
+{"_id": "2174", "title": "Intersection with Complement is Empty iff Subset", "text": ":$S \\subseteq T \\iff S \\cap \\map \\complement T = \\O$"}
+{"_id": "2175", "title": "Complement Union with Superset is Universe", "text": ":$S \\subseteq T \\iff \\map \\complement S \\cup T = \\mathbb U$"}
+{"_id": "2176", "title": "Set Difference is not Associative", "text": "Let $R, S, T$ be sets. The expression: :$\\paren {R \\setminus S} \\setminus T = R \\setminus \\paren {S \\setminus T}$ holds exactly when $R \\cap T = \\O$. Here $R \\setminus S$ denotes set difference. Thus, set difference is not associative."}
+{"_id": "2177", "title": "Cardinality of Set Union", "text": "=== Union of 2 Sets === {{:Cardinality of Set Union/2 Sets}} === Union of 3 Sets === {{:Cardinality of Set Union/3 Sets}}"}
+{"_id": "2179", "title": "Group Isomorphism Preserves Identity", "text": "Let $\\phi: \\struct {G, \\circ} \\to \\struct {H, *}$ be a group isomorphism. Let: :$e_G$ be the identity of $\\struct {G, \\circ}$ :$e_H$ be the identity of $\\struct {H, *}$. Then: :$\\map \\phi {e_G} = e_H$"}
+{"_id": "2180", "title": "Strictly Positive Rational Numbers under Multiplication form Countably Infinite Abelian Group", "text": "Let $\\Q_{> 0}$ be the set of strictly positive rational numbers, i.e. $\\Q_{> 0} = \\set {x \\in \\Q: x > 0}$. The structure $\\struct {\\Q_{> 0}, \\times}$ is a countably infinite abelian group."}
+{"_id": "2181", "title": "Multinomial Theorem", "text": "Let $x_1, x_2, \\ldots, x_k \\in F$, where $F$ is a field. Then: :$\\displaystyle \\paren {x_1 + x_2 + \\cdots + x_m}^n = \\sum_{k_1 \\mathop + k_2 \\mathop + \\mathop \\cdots \\mathop + k_m \\mathop = n} \\binom n {k_1, k_2, \\ldots, k_m} {x_1}^{k_1} {x_2}^{k_2} \\cdots {x_m}^{k_m}$ where: :$m \\in \\Z_{> 0}$ is a positive integer :$n \\in \\Z_{\\ge 0}$ is a non-negative integer :$\\dbinom n {k_1, k_2, \\ldots, k_m} = \\dfrac {n!} {k_1! \\, k_2! \\, \\cdots k_m!}$ denotes a multinomial coefficient. The sum is taken for all non-negative integers $k_1, k_2, \\ldots, k_m$ such that $k_1 + k_2 + \\cdots + k_m = n$, and with the understanding that wherever $0^0$ may appear  it shall be considered to have a value of $1$. The '''multinomial theorem''' is a generalization of the Binomial Theorem."}
+{"_id": "2182", "title": "Mapping is Constant iff Increasing and Decreasing", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be ordered sets. Let $\\phi: \\left({S, \\preceq_1}\\right) \\to \\left({T, \\preceq_2}\\right)$ be a mapping. Then $\\phi$ is a constant mapping {{iff}} $\\phi$ is both increasing and decreasing."}
+{"_id": "2183", "title": "Equivalence of Definitions of Euler's Number", "text": "{{TFAE|def = Euler's Number}} === Limit of Series === {{:Definition:Euler's Number/Limit of Series}} === Limit of Sequence === {{:Definition:Euler's Number/Limit of Sequence}} === Base of Logarithm === {{:Definition:Euler's Number/Base of Logarithm}} === Exponential Function === {{:Definition:Euler's Number/Exponential Function}}"}
+{"_id": "2184", "title": "Powers Drown Logarithms", "text": "Let $r \\in \\R_{>0}$ be a (strictly) positive real number. Then: :$\\displaystyle \\lim_{x \\mathop \\to \\infty} x^{-r} \\ln x = 0$"}
+{"_id": "2185", "title": "Equivalence of Well-Ordering Principle and Induction", "text": "The Well-Ordering Principle, the Principle of Finite Induction and the Principle of Complete Finite Induction are logically equivalent. That is: :Principle of Finite Induction: Given a subset $S \\subseteq \\N$ of the natural numbers which has these properties: :: $0 \\in S$ :: $n \\in S \\implies n + 1 \\in S$ :then $S = \\N$. {{iff}}: :Principle of Complete Finite Induction: Given a subset $S \\subseteq \\N$ of the natural numbers which has these properties: :: $0 \\in S$ :: $\\set {0, 1, \\ldots, n} \\subseteq S \\implies n + 1 \\in S$ :then $S = \\N$. {{iff}}: :Well-Ordering Principle: Every non-empty subset of $\\N$ has a minimal element."}
+{"_id": "2186", "title": "Lower Bound for Subset", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $L$ be a lower bound for $S$. Let $\\left({T, \\preceq}\\right)$ be a subset of $\\left({S, \\preceq}\\right)$. Then $L$ is a lower bound for $T$."}
+{"_id": "2187", "title": "Upper Bound for Subset", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $U$ be an upper bound for $S$. Let $\\left({T, \\preceq}\\right)$ be a subset of $\\left({S, \\preceq}\\right)$. Then $U$ is an upper bound for $T$."}
+{"_id": "2188", "title": "Common Divisor Divides GCD", "text": "Let $a, b \\in \\Z$ such that not both of $a$ and $b$ are zero. Let $c$ be any common divisor of $a$ and $b$. That is, let $c \\in \\Z: c \\divides a, c \\divides b$. Then: :$c \\divides \\gcd \\set {a, b}$ where $\\gcd \\set {a, b}$ is the greatest common divisor of $a$ and $b$."}
+{"_id": "2189", "title": "Square Modulo 5", "text": "Let $x \\in \\Z$ be an integer. Then one of the following holds: {{begin-eqn}} {{eqn | l = x^2       | o = \\equiv       | r = 0 \\pmod 5       | c =  }} {{eqn | l = x^2       | o = \\equiv       | r = 1 \\pmod 5       | c =  }} {{eqn | l = x^2       | o = \\equiv       | r = 4 \\pmod 5       | c =  }} {{end-eqn}}"}
+{"_id": "2190", "title": "Square Modulo 3", "text": "Let $x \\in \\Z$ be an integer. Then one of the following holds: {{begin-eqn}} {{eqn | l = x^2       | o = \\equiv       | r = 0 \\pmod 3       | c =  }} {{eqn | l = x^2       | o = \\equiv       | r = 1 \\pmod 3       | c =  }} {{end-eqn}}"}
+{"_id": "2191", "title": "Additive Group of Integers is Countably Infinite Abelian Group", "text": "The set of integers under addition $\\struct {\\Z, +}$ forms a countably infinite abelian group."}
+{"_id": "2192", "title": "Division Laws for Groups", "text": "Let $G$ be a group. Let $a, b, x \\in G$. Then: : $(1): \\quad a x = b \\iff x = a^{-1} b$ : $(2): \\quad x a = b \\iff x = b a^{-1}$"}
+{"_id": "2193", "title": "Center is Intersection of Centralizers", "text": "The center of a group is the intersection of all the centralizers of the elements of that group: :$\\displaystyle \\map Z G = \\bigcap_{g \\mathop \\in G} \\map {C_G} g$"}
+{"_id": "2194", "title": "Equivalent Statements for Congruence Modulo Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$."}
+{"_id": "2195", "title": "Congruence Modulo Subgroup is Equivalence Relation", "text": "Let $G$ be a group, and let $H$ be a subgroup of $G$. Let $x, y \\in G$."}
+{"_id": "2196", "title": "Group Homomorphism Preserves Identity", "text": "Let $\\struct {G, \\circ}$ and $\\struct {H, *}$ be groups. Let $\\phi: \\struct {G, \\circ} \\to \\struct {H, *}$ be a group homomorphism. Let: :$e_G$ be the identity of $G$ :$e_H$ be the identity of $H$. Then: :$\\map \\phi {e_G} = e_H$"}
+{"_id": "2197", "title": "Group Homomorphism Preserves Inverses", "text": "Let $\\struct {G, \\circ}$ and $\\struct {H, *}$ be groups. Let $\\phi: \\struct {G, \\circ} \\to\\struct {H, *}$ be a group homomorphism. Let: :$e_G$ be the identity of $G$ :$e_H$ be the identity of $H$ Then: :$\\forall x \\in G: \\map \\phi {x^{-1} } = \\paren {\\map \\phi x}^{-1}$"}
+{"_id": "2198", "title": "Subset Product with Normal Subgroup is Subgroup", "text": "Let $G$ be a group whose identity is $e$. Let: :$(1): \\quad H$ be a subgroup of $G$ :$(2): \\quad N$ be a normal subgroup of $G$. Let $H N$ denote subset product. Then $H N$ and $N H$ are both subgroups of $G$."}
+{"_id": "2199", "title": "Coset by Identity", "text": "Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$."}
+{"_id": "2200", "title": "Element of Group is in its own Coset", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $x \\in G$."}
+{"_id": "2201", "title": "Element in Coset iff Product with Inverse in Subgroup", "text": "==== Element in Left Coset iff Product with Inverse in Subgroup ==== {{:Element in Left Coset iff Product with Inverse in Subgroup}} ==== Element in Right Coset iff Product with Inverse in Subgroup ==== {{:Element in Right Coset iff Product with Inverse in Subgroup}}"}
+{"_id": "2202", "title": "Cosets are Equal iff Product with Inverse in Subgroup", "text": "==== Left Cosets are Equal iff Product with Inverse in Subgroup ==== {{:Left Cosets are Equal iff Product with Inverse in Subgroup}} ==== Right Cosets are Equal iff Product with Inverse in Subgroup ==== {{:Right Cosets are Equal iff Product with Inverse in Subgroup}}"}
+{"_id": "2203", "title": "Coset Equals Subgroup iff Element in Subgroup", "text": "==== Left Coset Equals Subgroup iff Element in Subgroup ==== {{:Left Coset Equals Subgroup iff Element in Subgroup}} ==== Right Coset Equals Subgroup iff Element in Subgroup ==== {{:Right Coset Equals Subgroup iff Element in Subgroup}}"}
+{"_id": "2204", "title": "Elements in Same Coset iff Product with Inverse in Subgroup", "text": "==== Elements in Same Left Coset iff Product with Inverse in Subgroup ==== {{:Elements in Same Left Coset iff Product with Inverse in Subgroup}} ==== Elements in Same Right Coset iff Product with Inverse in Subgroup ==== {{:Elements in Same Right Coset iff Product with Inverse in Subgroup}}"}
+{"_id": "2205", "title": "Regular Representation on Subgroup is Bijection to Coset", "text": "* The mapping $\\lambda_x: H \\to x H$, where $\\lambda_x$ is the left regular representation of $H$ with respect to $x$, is a bijection from $H$ to $x H$. * The mapping $\\rho_x: H \\to H x$, where $\\rho_x$ is the right regular representation of $H$ with respect to $x$, is a bijection from $H$ to $H x$."}
+{"_id": "2206", "title": "Ring Homomorphism Preserves Zero", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring homomorphism. Let: :$0_{R_1}$ be the zero of $R_1$ :$0_{R_2}$ be the zero of $R_2$. Then: :$\\map \\phi {0_{R_1} } = 0_{R_2}$"}
+{"_id": "2207", "title": "Image of Set Difference under Mapping", "text": "Let $f: S \\to T$ be a mapping. The image of the set difference of two subsets of $S$ is a subset of the set difference of the images. That is: Let $S_1$ and $S_2$ be subsets of $S$. Then: :$f \\sqbrk {S_1} \\setminus f \\sqbrk {S_2} \\subseteq f \\sqbrk {S_1 \\setminus S_2}$ where $\\setminus$ denotes set difference."}
+{"_id": "2208", "title": "Convergence by Multiple of Error Term", "text": "Let $\\sequence {s_n}$ be a real sequence. Suppose that $\\exists \\epsilon \\in \\R, \\epsilon > 0$ such that: :$\\exists N \\in \\N: \\forall n \\ge N: \\size {s_n - l} < K \\epsilon$ for any $K \\in \\R, K > 0$, independent of both $\\epsilon$ and $N$. Then $\\sequence {s_n}$ converges to $l$."}
+{"_id": "2209", "title": "Cauchy Sequence Converges on Real Number Line", "text": "Let $\\sequence {a_n}$ be a Cauchy sequence in $\\R$. Then $\\sequence {a_n}$ is convergent. In other words, $\\struct {\\R, \\size {\\,\\cdot\\,}}$ is a Banach space."}
+{"_id": "2210", "title": "Cauchy's Convergence Criterion", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Then $\\sequence {x_n}$ is convergent {{iff}} $\\sequence {x_n}$ is a Cauchy sequence."}
+{"_id": "2211", "title": "Wedderburn's Theorem", "text": "Every finite division ring $D$ is a field."}
+{"_id": "2214", "title": "Preimages All Exist iff Surjection", "text": "Let $f: S \\to T$ be a mapping. Let $f^{-1}$ be the inverse of $f$. Let $f^{-1} \\paren t$ be the preimage of $t \\in T$. Then $f^{-1} \\paren t$ is empty for no $t \\in T$ {{iff}} $f$ is a surjection."}
+{"_id": "2215", "title": "Non-Successor Element of Peano Structure is Unique", "text": "Let $\\struct {P, s, 0}$ be a Peano structure. Then: :$P \\setminus s \\sqbrk P$ is a singleton where: :$\\setminus$ denotes set difference :$s \\sqbrk P$ denotes the image of the mapping $s$. It follows that the non-successor element $0$ is the only element of $P$ with this property."}
+{"_id": "2216", "title": "Equivalence of Formulations of Peano's Axioms", "text": "Let $P$ be a set. Let $s: P \\to P$ be a mapping. Let $0 \\in P$ be a distinguished element. {{TFAE|def = Peano's Axioms|view = Peano's Axioms}}"}
+{"_id": "2217", "title": "Product of Row Sum Unity Matrices", "text": "Let $\\mathbf A = \\left[{a}\\right]_{m n}$ be an $m \\times n$ matrix. Let $\\mathbf B = \\left[{b}\\right]_{n p}$ be an $n \\times p$ matrix. Let the row sum of $\\mathbf A$ and $\\mathbf B$ be equal to $1$. Then the row sum of their (conventional) product is also $1$."}
+{"_id": "2219", "title": "Restriction of Injection is Injection", "text": "Let $f: S \\to T$ be an injection. Let $X \\subseteq S$ be a subset of $S$. Let $f \\sqbrk X$ denote the image of $X$ under $f$. Let $Y \\subseteq T$ be a subset of $T$ such that $f \\sqbrk X \\subseteq Y$. The restriction $f \\restriction_{X \\times Y}$ of $f$ to $X \\times Y$ is an injection from $X$ to $Y$."}
+{"_id": "2221", "title": "Absolute Value is Many-to-One", "text": "Let $f: \\R \\to \\R$ be the absolute value function: :$\\forall x \\in \\R:  \\map f x = \\begin{cases} x & : x \\ge 0 \\\\ -x & : x < 0 \\end{cases}$ Then $f$ is a many-to-one relation."}
+{"_id": "2222", "title": "Laws of Logarithms", "text": "Let $x, y, b \\in \\R_{>0}$ be (strictly) positive real numbers. Let $a \\in \\R$ be any real number such that $a > 0$ and $a \\ne 1$. Let $\\log_a$ denote the logarithm to base $a$. Then:"}
+{"_id": "2223", "title": "Subset of Preimage under Relation is Preimage of Subset", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation. Let $X \\subseteq S, Y \\subseteq T$. Then: :$X \\subseteq \\mathcal R^{-1} \\sqbrk Y \\iff \\mathcal R \\sqbrk X \\subseteq Y$ In the language of direct image mappings, this can be written: :$X \\subseteq \\map {\\mathcal R^\\gets} Y \\iff \\map {\\mathcal R^\\to} X \\subseteq Y$"}
+{"_id": "2224", "title": "Composition of Direct Image Mappings of Relations", "text": "Let $A, B, C$ be non-empty sets. Let $\\mathcal R_1 \\subseteq A \\times B, \\mathcal R_2 \\subseteq B \\times C$ be relations. Let: :${\\mathcal R_1}^\\to: \\powerset A \\to \\powerset B$ and :${\\mathcal R_2}^\\to: \\powerset B \\to \\powerset C$ be the direct image mappings of $\\mathcal R_1$ and $\\mathcal R_2$. Then: :$\\paren {\\mathcal R_2 \\circ \\mathcal R_1}^\\to = {\\mathcal R_2}^\\to \\circ {\\mathcal R_1}^\\to$"}
+{"_id": "2225", "title": "Subset equals Preimage of Image iff Mapping is Injection", "text": "Let $f: S \\to T$ be a mapping. Let $f^{-1}$ denote the inverse of $f$. Then: :$\\forall A \\subseteq S: A = \\paren {f^{-1} \\circ f} \\sqbrk A$ {{iff}} $f$ is an injection where: :$f \\sqbrk A$ denotes the image of $A$ under $f$ :$f^{-1} \\circ f$ denotes the composition of $f^{-1}$ and $f$. This can be expressed in the language and notation of direct image mappings and inverse image mappings as: :$\\forall A \\in \\powerset S: A = \\map {\\paren {f^\\gets \\circ f^\\to} } A$ {{iff}} $f$ is an injection"}
+{"_id": "2226", "title": "Subset equals Image of Preimage iff Mapping is Surjection", "text": "Let $f: S \\to T$ be a mapping. Let $f^\\to: \\powerset S \\to \\powerset T$ be the direct image mapping of $f$. Similarly, let $f^\\gets: \\powerset T \\to \\powerset S$ be the inverse image mapping of $f$. Then: :$\\forall B \\in \\powerset T: B = \\map {\\paren {f^\\to \\circ f^\\gets} } B$ {{iff}} $f$ is a surjection."}
+{"_id": "2227", "title": "Inverse of Injection is Many-to-One Relation", "text": "Let $f: S \\to T$ be an injection. Let $f^{-1}: T \\to S$ be the inverse relation of $f$. Then $f^{-1}$ is many-to-one."}
+{"_id": "2228", "title": "Bijection iff Left and Right Cancellable", "text": "Let $f$ be a mapping. Then $f$ is a bijection {{iff}} $f$ is both left cancellable and right cancellable."}
+{"_id": "2229", "title": "Relation Induced by Mapping is Equivalence Relation", "text": "Let $f: S \\to T$ be a mapping. Let $\\RR_f \\subseteq S \\times S$ be the relation induced by $f$: :$\\tuple {s_1, s_2} \\in \\RR_f \\iff \\map f {s_1} = \\map f {s_2}$ Then $\\RR_f$ is an equivalence relation."}
+{"_id": "2230", "title": "Canonical Injection is Monomorphism", "text": "Let $\\struct {S_1, \\circ_1}$ and $\\struct {S_2, \\circ_2}$ be algebraic structures with identities $e_1, e_2$ respectively. The canonical injections: :$\\inj_1: \\struct {S_1, \\circ_1} \\to \\struct {S_1, \\circ_1} \\times \\struct {S_2, \\circ_2}: \\forall x \\in S_1: \\map {\\inj_1} x = \\tuple {x, e_2}$ :$\\inj_2: \\struct {S_2, \\circ_2} \\to \\struct {S_1, \\circ_1} \\times \\struct {S_2, \\circ_2}: \\forall x \\in S_2: \\map {\\inj_2} x = \\tuple {e_1, x}$ are monomorphisms."}
+{"_id": "2231", "title": "Order Embedding is Injection", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be ordered sets. Let $\\phi: S \\to T$ be an order embedding. That is: : $\\forall x, y \\in S: x \\preceq_1 y \\iff \\phi \\left({x}\\right) \\preceq_2 \\phi \\left({y}\\right)$ Then $\\phi$ is an injection."}
+{"_id": "2232", "title": "Order Isomorphism is Surjective Order Embedding", "text": "Let $\\struct {S, \\preceq_1}$ and $\\struct {T, \\preceq_2}$ be ordered sets. Let $f: S \\to T$ be a mapping. Then $f$ is an order isomorphism {{iff}}: :$(1): \\quad f$ is a surjection :$(2): \\quad \\forall x, y \\in S: x \\preceq_1 y \\iff \\map f x \\preceq_2 \\map f y$ That is, {{iff}} $f$ is an order embedding which is also a surjection."}
+{"_id": "2233", "title": "Infimum of Power Set", "text": "Let $S$ be a set. Let $\\mathcal P \\left({S}\\right)$ be the power set of $S$. Let $\\left({\\mathcal P \\left({S}\\right), \\subseteq}\\right)$ be the relational structure defined on $\\mathcal P \\left({S}\\right)$ by the relation $\\subseteq$. (From Subset Relation on Power Set is Partial Ordering, this is an ordered set.) Then the infimum of $\\left({\\mathcal P \\left({S}\\right), \\subseteq}\\right)$ is the empty set $\\varnothing$."}
+{"_id": "2234", "title": "Supremum of Power Set", "text": "Let $S$ be a set. Let $\\mathcal P \\left({S}\\right)$ be the power set of $S$. Let $\\left({\\mathcal P \\left({S}\\right), \\subseteq}\\right)$ be the relational structure defined on $\\mathcal P \\left({S}\\right)$ by the relation $\\subseteq$. (From Subset Relation on Power Set is Partial Ordering, this is an ordered set.) Then the supremum of $\\left({\\mathcal P \\left({S}\\right), \\subseteq}\\right)$ is the set $S$."}
+{"_id": "2235", "title": "Diagonal Relation is Ordering and Equivalence", "text": "Let $\\left({S, \\Delta_S}\\right)$ be a relational structure where $\\Delta_S$ is the diagonal relation, defined as: :$\\forall x, y \\in S: \\paren {x, y} \\in \\Delta_S \\iff x = y$ Then $\\Delta_S$ is the '''only''' relation on $S$ which is both an equivalence and an ordering."}
+{"_id": "2236", "title": "Cantor-Bernstein-Schröder Theorem/Proof 3", "text": "Let $S$ and $T$ be sets, such that: :$\\exists f: S \\to T$ such that $f$ is an injection :$\\exists g: T \\to S$ such that $g$ is an injection. Then there exists a bijection from $S$ to $T$."}
+{"_id": "2238", "title": "Cantor-Bernstein-Schröder Theorem/Proof 2", "text": "Let $S$ and $T$ be sets, such that: :$S \\preccurlyeq T$, that is: $T$ dominates $S$ :$T \\preccurlyeq S$, that is: $S$ dominates $T$. Then: :$S \\sim T$ that is, $S$ is equivalent to $T$."}
+{"_id": "2240", "title": "Huygens-Steiner Theorem", "text": "Let $B$ be a body of mass $M$. Let $I_0$ be the moment of inertia of $B$ about some axis $A$ through the centre of mass of $B$. Let $I$ the moment of inertia of $B$ about another axis $A'$ parallel to $A$. Then $I_0$ and $I$ are related by: :$I = I_0 + M l^2$ where $l$ is the perpendicular distance between $A$ and $A'$."}
+{"_id": "2241", "title": "Equivalence of Definitions of Well-Ordering/Definition 1 implies Definition 2", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set such that: : $\\forall T \\subseteq S: \\exists a \\in T: \\forall x \\in T: a \\preceq x$ That is, such that $\\preceq$ is a well-ordering by definition 1. Then $\\preceq$ is a total ordering."}
+{"_id": "2242", "title": "Choice Function Exists for Well-Orderable Union of Sets", "text": "Let $\\mathbb S$ be a set of sets such that: :$\\forall S \\in \\mathbb S: S \\ne \\varnothing$ that is, none of the sets in $\\mathbb S$ may be empty. Let the union $\\bigcup \\mathbb S$ be well-orderable. Then there exists a choice function $f: \\mathbb S \\to \\bigcup \\mathbb S$ defined as: :$\\forall S \\in \\mathbb S: \\exists x \\in S: f \\left({S}\\right) = x$ Thus, if every member of $\\mathbb S$ is a well-ordered, then we can create a choice function $f$ defined as: :$\\forall S \\in \\mathbb S: f \\left({S}\\right) = \\inf \\left({S}\\right)$ True, we may be making infinitely many choices, but we have a rule for doing so."}
+{"_id": "2243", "title": "Principle of Finite Choice", "text": "Let $I$ be a non-empty finite indexing set. Let $\\left\\langle{S_i}\\right\\rangle_{i \\mathop \\in I}$ be an $I$-indexed family of non-empty sets. Then there exists an $I$-indexed family $\\left\\langle{x_i}\\right\\rangle_{i \\mathop \\in I}$ such that: :$\\forall i \\in I: x_i \\in S_i$ That is, there exists a mapping: :$\\displaystyle f: I \\to \\bigcup_{i \\mathop \\in I} S_i$ such that: :$\\forall i \\in I: f \\left({i}\\right) \\in S_i$"}
+{"_id": "2244", "title": "Choice Function Exists for Set of Well-Ordered Sets", "text": "Let $\\mathbb S$ be a set of sets such that: :$\\forall S \\in \\mathbb S: S \\ne \\varnothing$ that is, none of the sets in $\\mathbb S$ may be empty. Let every element of $\\mathbb S$ be well-ordered. Then there exists a choice function $f: \\mathbb S \\to \\bigcup \\mathbb S$ satisfying: :$\\forall S \\in \\mathbb S: \\exists x \\in S: f \\left({S}\\right) = x$"}
+{"_id": "2245", "title": "Identity of Power Set with Union", "text": "Let $S$ be a set and let $\\powerset S$ be its power set. Consider the algebraic structure $\\struct {\\powerset S, \\cup}$, where $\\cup$ denotes set union. Then the empty set $\\O$ serves as the identity for $\\struct {\\powerset S, \\cup}$."}
+{"_id": "2246", "title": "Identity of Power Set with Intersection", "text": "Let $S$ be a set and let $\\powerset S$ be its power set. Consider the algebraic structure $\\struct {\\powerset S, \\cap}$, where $\\cap$ denotes set intersection. Then $S$ serves as the identity for $\\struct {\\powerset S, \\cap}$."}
+{"_id": "2247", "title": "Commutation with Inverse in Monoid", "text": "Let $x, y \\in S$ such that $y$ is invertible. Then $x$ commutes with $y$ {{iff}} $x$ commutes with $y^{-1}$."}
+{"_id": "2248", "title": "Commutation of Inverses in Monoid", "text": "Let $x, y \\in S$ such that $x$ and $y$ are both invertible. Then $x$ commutes with $y$ {{iff}} $x^{-1}$ commutes with $y^{-1}$."}
+{"_id": "2249", "title": "Inverse of Commuting Pair", "text": "Let $x, y \\in S$ such that $x$ and $y$ are both invertible. Then $x$ commutes with $y$ {{iff}}: : $\\struct {x \\circ y}^{-1} = x^{-1} \\circ y^{-1}$"}
+{"_id": "2250", "title": "Conjugate of Commuting Elements", "text": "Let $x, y \\in S$ such that $x$ and $y$ are both invertible. Then $x \\circ y \\circ x^{-1} = y$ {{iff}} $x$ and $y$ commute."}
+{"_id": "2251", "title": "Product of Commuting Elements with Inverses", "text": "Let $x, y \\in S$ such that $x$ and $y$ are both invertible. Then: :$x \\circ y \\circ x^{-1} \\circ y^{-1} = e_S = x^{-1} \\circ y^{-1} \\circ x \\circ y$ {{iff}} $x$ and $y$ commute."}
+{"_id": "2253", "title": "Exists Element Not in Set", "text": "Let $S$ be a set. Then $\\exists x: x \\notin S$. That is, for any set, there exists some element which is not in that set."}
+{"_id": "2254", "title": "Rule of Material Implication", "text": "==== Formulation 1 ==== {{:Rule of Material Implication/Formulation 1}} ==== Formulation 2 ==== {{:Rule of Material Implication/Formulation 2}}"}
+{"_id": "2255", "title": "De Morgan's Laws (Set Theory)/Proof by Induction", "text": "Let $\\mathbb T = \\left\\{{T_i: i \\mathop \\in I}\\right\\}$, where each $T_i$ is a set and $I$ is some finite indexing set. Then:"}
+{"_id": "2256", "title": "Union equals Intersection iff Sets are Equal", "text": "Let $S$ and $T$ be sets. Then: :$\\paren {S \\cup T = S} \\land \\paren {S \\cap T = S} \\iff S = T$ where: :$S \\cup T$ denotes set union :$S \\cap T$ denotes set intersection."}
+{"_id": "2257", "title": "Intersection with Set Difference is Set Difference with Intersection", "text": ": $\\left({R \\setminus S}\\right) \\cap T = \\left({R \\cap T}\\right) \\setminus S$"}
+{"_id": "2258", "title": "Set Difference is Right Distributive over Set Intersection", "text": ":$\\paren {R \\cap S} \\setminus T = \\paren {R \\setminus T} \\cap \\paren {S \\setminus T}$"}
+{"_id": "2259", "title": "Set Intersection Distributes over Set Difference", "text": ": $\\left({R \\setminus S}\\right) \\cap T = \\left({R \\cap T}\\right) \\setminus \\left({S \\cap T}\\right)$ : $R \\cap \\left({S \\setminus T}\\right) = \\left({R \\cap S}\\right) \\setminus \\left({R \\cap T}\\right)$"}
+{"_id": "2260", "title": "Set Difference with Intersection is Difference", "text": ":$S \\setminus \\paren {S \\cap T} = S \\setminus T$"}
+{"_id": "2261", "title": "Set Difference of Intersection with Set is Empty Set", "text": ":$\\left({S \\cap T}\\right) \\setminus S = \\varnothing$ :$\\left({S \\cap T}\\right) \\setminus T = \\varnothing$"}
+{"_id": "2262", "title": "Set Difference Intersection with First Set is Set Difference", "text": ":$\\left({S \\setminus T}\\right) \\cap S = S \\setminus T$"}
+{"_id": "2263", "title": "Set Difference Intersection with Second Set is Empty Set", "text": ":$\\paren {S \\setminus T} \\cap T = \\O$"}
+{"_id": "2264", "title": "Set Difference with Union is Set Difference", "text": ":$\\left({S \\cup T}\\right) \\setminus T = S \\setminus T$"}
+{"_id": "2265", "title": "Set Difference Union First Set is First Set", "text": ":$\\paren {S \\setminus T} \\cup S = S$"}
+{"_id": "2266", "title": "Set Difference Union Second Set is Union", "text": ":$\\left({S \\setminus T}\\right) \\cup T = S \\cup T$"}
+{"_id": "2267", "title": "Set Difference with Union", "text": ":$R \\setminus \\paren {S \\cup T} = \\paren {R \\cup T} \\setminus \\paren {S \\cup T} = \\paren {R \\setminus S} \\setminus T = \\paren {R \\setminus T} \\setminus S$"}
+{"_id": "2268", "title": "Set Difference is Right Distributive over Union", "text": ":$\\paren {R \\cup S} \\setminus T = \\paren {R \\setminus T} \\cup \\paren {S \\setminus T}$"}
+{"_id": "2269", "title": "Set Difference with Set Difference is Union of Set Difference with Intersection", "text": ":$R \\setminus \\paren {S \\setminus T} = \\paren {R \\setminus S} \\cup \\paren {R \\cap T}$"}
+{"_id": "2270", "title": "Set Difference is Subset of Union of Differences", "text": ": $R \\setminus S \\subseteq \\left({R \\setminus T}\\right) \\cup \\left({T \\setminus S}\\right)$"}
+{"_id": "2271", "title": "Ordered Sum of Tosets is Totally Ordered Set", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be tosets. Let $S + T = \\left({S \\cup T, \\preceq}\\right)$ be the ordered sum of $S$ and $T$. Then $\\left({S \\cup T, \\preceq}\\right)$ is itself a toset."}
+{"_id": "2272", "title": "Ordered Product of Tosets is Totally Ordered Set", "text": "Let $\\struct {S_1, \\preceq_1}$ and $\\struct {S_2, \\preceq_2}$ be tosets. Let $S_1 \\cdot S_2 = \\struct {S_1 \\times S_2, \\preceq}$ be the ordered product of $S_1$ and $S_2$. Then $\\struct {S_1 \\times S_2, \\preceq}$ is itself a toset."}
+{"_id": "2273", "title": "Restriction of Mapping is Mapping", "text": "Let $f: S \\to T$ be a mapping. Let $X \\subseteq S$. Let $f \\restriction_X$ be the restriction of $f$ to $X$. Then $f \\restriction_X: X \\to T$ is a mapping: : whose domain is $X$ : whose preimage is $X$."}
+{"_id": "2276", "title": "Hausdorff Maximal Principle", "text": "Let $\\struct {\\PP, \\preceq}$ be a partially ordered set. Then there exists a maximal chain in $\\PP$."}
+{"_id": "2277", "title": "Logical Consequence with Union", "text": "Let $U$ be a set of propositional formulas. Let $P$ be a propositional formula. Let $U \\models P$ denote that $P$ is a semantic consequence of $U$. Then: : $U \\models P$ iff: : $U \\cup P \\models P$"}
+{"_id": "2278", "title": "Logical Consequence Union Negation", "text": "Let $U$ be a set of propositional formulas. Let $P$ be a propositional formula. Let $U \\models P$ denote that $U$ is a semantic consequence $P$. Then: : $U \\models P$ iff: : $U \\cup \\left\\{{\\neg P}\\right\\}$ has no models."}
+{"_id": "2279", "title": "Union of Singleton", "text": "Consider the set of sets $\\mathbb S$ such that $\\mathbb S$ consists of just one set $S$. Then the union of $\\mathbb S$ is $S$: :$\\displaystyle \\mathbb S = \\set S \\implies \\bigcup \\mathbb S = S$"}
+{"_id": "2280", "title": "Intersection of Singleton", "text": "Consider the set of sets $\\mathbb S$ such that $\\mathbb S$ consists of just one set $S$. Then the intersection of $\\mathbb S$ is $S$: :$\\displaystyle \\mathbb S = \\set S \\implies \\bigcap \\mathbb S = S$"}
+{"_id": "2281", "title": "Union of Empty Set", "text": "Consider the set of sets $\\mathbb S$ such that $\\mathbb S$ is the empty set $\\O$. Then the union of $\\mathbb S$ is $\\O$: :$\\mathbb S = \\O \\implies \\displaystyle \\bigcup \\mathbb S = \\O$"}
+{"_id": "2282", "title": "Intersection of Empty Set", "text": "Consider the set of sets $\\mathbb S$ such that $\\mathbb S$ is the empty set $\\O$. Then the intersection of $\\mathbb S$ is $\\mathbb U$: :$\\mathbb S = \\O \\implies \\displaystyle \\bigcap \\mathbb S = \\mathbb U$ where $\\mathbb U$ is the universe. A paradoxical result."}
+{"_id": "2283", "title": "Inverse of Identity Mapping", "text": "Let $S$ be a set. Let $I_S: S \\to S$ be the identity mapping on $S$. Then the inverse of $I_S$ is itself: :$\\paren {I_S}^{-1} = I_S$"}
+{"_id": "2284", "title": "Factoring Mapping into Quotient and Injection", "text": "Let $f: S \\to T$ be a mapping. Then $f$ can be uniquely '''factored into''' a quotient mapping, followed by an injection. Thus: :$f = h \\circ q_{\\RR_f}$ where: :$q_{\\RR_f}: S \\to S / \\RR_f: \\map {q_{\\RR_f} } s = \\eqclass s {\\RR_f}$ :$h: S / \\RR_f \\to T: \\map h {\\eqclass s {\\RR_f} } = \\map f s$ :$\\eqclass s {\\RR_f}$ denotes the equivalence class of $s$ with respect to the equivalence relation $\\RR$ induced on $S$ by $f$. This can be illustrated using a commutative diagram as follows: ::$\\begin{xy}\\xymatrix@L+2mu@+1em{  S \\ar[r]^*{q_{\\RR_f} }      \\ar@{-->}[rd]_*{f = h \\circ q_{\\RR_f} } &  S / \\RR_f \\ar[d]^*{h} \\\\ &  T }\\end{xy}$"}
+{"_id": "2285", "title": "Subsets in Increasing Union", "text": "Let $S_0, S_1, S_2, \\ldots, S_i, \\ldots$ be a nested sequence of sets, that is: :$S_0 \\subseteq S_1 \\subseteq S_2 \\subseteq \\ldots \\subseteq S_i \\subseteq \\ldots$ Let $S$ be the increasing union of $S_0, S_1, S_2, \\ldots, S_i, \\ldots$: :$\\displaystyle S = \\bigcup_{i \\mathop \\in \\N} S_i$ Then: :$\\forall s \\in S: \\exists k \\in \\N: \\forall j \\ge k: s \\in S_j$"}
+{"_id": "2286", "title": "Mapping on Increasing Union", "text": "Let $S_0, S_1, S_2, \\ldots, S_i, \\ldots$ be sets such that: :$S_0 \\subseteq S_1 \\subseteq S_2 \\subseteq \\ldots \\subseteq S_i \\subseteq \\ldots$ that is, each set is contained in the next as a subset. Let $S$ be the increasing union of $S_0, S_1, S_2, \\ldots, S_i, \\ldots$: :$\\displaystyle S = \\bigcup_{i \\mathop \\in \\N} S_i$ For each $i \\in \\N$, let $f_i: S_i \\to T$ be a mapping such that: :$\\forall j < i: f_i \\restriction_{S_j} = f_j$ where $f_i \\restriction_{S_j}$ denotes the restriction of $f_i$ to $S_j$. Then there is a '''unique''' mapping: :$f: S \\to T$ which extends each $f_i$ to $S$."}
+{"_id": "2287", "title": "Domain of Injection Not Larger than Codomain", "text": "Let $f: S \\rightarrowtail T$ be an injection. Then: :$\\left|{S}\\right| \\le \\left|{T}\\right|$ where $\\left|{S}\\right|$ denotes the cardinality of $S$."}
+{"_id": "2288", "title": "Futurama Theorem", "text": "Let $A_{n - 2} \\subset A_n$ be a subgroup of the alternating group on $n$ letters $A_n$. For any element $x \\in A_{n - 2}$, let $x = x_1 x_2 \\dots x_k$, where $x_i \\in H$ is a transposition. {{explain|What is $H$?}} Then there exists $y$ which can be represented as a series of transpositions $y_1 y_2 \\dots y_j \\in A_n$ such that: :$(1): \\quad y x = z$, where $z$ contains no transpositions from $H$ :$(2): \\quad y_a \\ne x_b$ for all $a, b$."}
+{"_id": "2289", "title": "Pointwise Addition on Continuous Real-Valued Functions forms Group", "text": "Let $C$ be the set of all continuous real functions on the set of real numbers $\\R$. Let $f, g \\in C$. Let $f + g$ be the pointwise sum of $f$ and $g$: :$\\forall x \\in R: \\map {\\paren {f + g} } x = \\map f x + \\map g x$ Then $\\struct {C, +}$, the algebraic structure on $C$ induced by $+$, forms a group."}
+{"_id": "2291", "title": "General Intersection Property of Topological Space", "text": "Let $\\struct {S, \\tau}$ be a topological space. Let $S_1, S_2, \\ldots, S_n$ be open sets of $\\struct {S, \\tau}$. Then: :$\\displaystyle \\bigcap_{i \\mathop = 1}^n S_i$ is also an open set of $\\struct {S, \\tau}$. That is, the intersection of any finite number of open sets of a topology is also in $\\tau$. Conversely, if the intersection of any finite number of open sets of a topology is also in $\\tau$, then: :$(1): \\quad$ The intersection of any two elements of $\\tau$ is an element of $\\tau$ :$(2): \\quad S$ is itself an element of $\\tau$."}
+{"_id": "2292", "title": "Empty Set is Element of Topology", "text": "Let $\\struct {X, \\tau}$ be a topological space. Then $\\O$ is an open set of $\\struct {X, \\tau}$."}
+{"_id": "2293", "title": "Coarseness Relation on Topologies is Partial Ordering", "text": "Let $S$ be a set. Let $\\mathbb T$ be the set of all topologies on $S$. Let $\\le$ be the relation on $\\mathbb T$ defined as: :$\\forall \\tau_1, \\tau_2 \\in \\mathbb T: \\tau_1 \\le \\tau_2$ {{iff}} $\\tau_1$ is coarser than $\\tau_2$. Then $\\le$ is a partial ordering on $\\mathbb T$."}
+{"_id": "2294", "title": "Topology Defined by Closed Sets", "text": "Let $X$ be any set and let $\\tau$ be a collection of subsets of $X$. Then $\\tau$ is a topology on $X$ {{iff}}: :$(1): \\quad$ Any intersection of arbitrarily many closed sets of $X$ under $\\tau$ is a closed set of $X$ under $\\tau$ :$(2): \\quad$ The union of any finite number of closed sets of $X$ under $\\tau$ is a closed set of $X$ under $\\tau$ :$(3): \\quad X$ and $\\O$ are both closed sets of $X$ under $\\tau$."}
+{"_id": "2295", "title": "Complement of F-Sigma Set is G-Delta Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $X$ be an $F_\\sigma$ set of $T$. Then its complement $S \\setminus X$ is a $G_\\delta$ set of $S$."}
+{"_id": "2298", "title": "Set is Open iff Neighborhood of all its Points", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $V \\subseteq S$ be a subset of $T$. Then: :$V$ is an open set of $T$ {{iff}}: :$V$ is a neighborhood of all the points in $V$."}
+{"_id": "2299", "title": "Relationship between Limit Point Types", "text": "Let $T = \\struct {X, \\tau}$ be a topological space. Let $A \\subseteq X$. Let: :$C$ be the set of condensation points of $A$ :$W$ be the set of $\\omega$-accumulation points of $A$ :$L$ be the set of limit points of $A$ :$D$ be the set of adherent points of $A$. Then: :$C \\subseteq W \\subseteq L \\subseteq D$ That is: :Every condensation point is an $\\omega$-accumulation point :Every $\\omega$-accumulation point is a limit point :Every limit point is an adherent point. In general, the inclusions do not hold in the other direction."}
+{"_id": "2300", "title": "Limit Point of Sequence is Accumulation Point", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A \\subseteq S$. Let $\\sequence {x_n}$ be a sequence in $A$. Let $\\alpha$ be a limit point of $\\sequence {x_n}$. Then $\\alpha$ is also an accumulation point of $\\sequence {x_n}$."}
+{"_id": "2302", "title": "Filter on Set is Proper Filter", "text": "Let $S$ be a set. Let $\\mathcal P \\left({S}\\right)$ denote the power set of $S$. Let $\\left({\\mathcal P \\left({S}\\right), \\subseteq}\\right)$ be the poset defined on $\\mathcal P \\left({S}\\right)$ by the subset relation. Let $\\mathcal F$ be a filter on $S$. Then $\\mathcal F$ is a proper filter on $\\left({\\mathcal P \\left({S}\\right), \\subseteq}\\right)$."}
+{"_id": "2303", "title": "Peirce's Law is Equivalent to Law of Excluded Middle", "text": "Peirce's Law: :$\\paren {p \\implies q} \\implies p \\vdash p$ is logically equivalent to the Law of Excluded Middle: :$\\vdash p \\lor \\neg p$ That is, Peirce's Law holds {{iff}} the Law of Excluded Middle holds."}
+{"_id": "2306", "title": "Implication is Left Distributive over Conjunction", "text": "The implication operator is left distributive over the conjunction operator:"}
+{"_id": "2307", "title": "Existence of Negation Normal Form of Statement", "text": "Any propositional formula can be expressed in negation normal form (NNF)."}
+{"_id": "2308", "title": "Existence of Conjunctive Normal Form of Statement", "text": "Any propositional formula can be expressed in conjunctive normal form (CNF)."}
+{"_id": "2309", "title": "Existence of Disjunctive Normal Form of Statement", "text": "Any propositional formula can be expressed in disjunctive normal form (DNF)."}
+{"_id": "2310", "title": "Existence of Base-N Representation", "text": "{{refactor|lemmata and sub proofs}} {{tidy|Could be improved in presentational style}} Given a number $x \\in \\hointr 0 1$, there exists a representation of that number in a base-$p$ positional system. Specifically, there exists a sequence $\\sequence {a_n}$ such that: :$0 \\le a_n < p$, and :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\frac {a_n} {p^n}$ converges to $x$. Unless $\\sequence {a_n}$ terminates (that is $a_n = 0$ for all sufficiently large $n$), then this representation is unique. If $\\sequence {a_n}$ does terminate, then there is exactly one other sequence which satisfies the criteria of the theorem."}
+{"_id": "2311", "title": "Symmetry Group of Equilateral Triangle is Group", "text": "The symmetry group of the equilateral triangle is a group."}
+{"_id": "2314", "title": "Hilbert's Basis Theorem", "text": "Let $A$ be a Noetherian ring. Let $A \\sqbrk x$ be the ring of polynomial forms over $A$ in the single indeterminate $x$. Then $A \\sqbrk x$ is also a Noetherian ring."}
+{"_id": "2317", "title": "Existence of Cyclic Group of Order n", "text": "Let $n \\in \\Z_{>0}$. Then there exists a cyclic group of order $n$ which is unique up to isomorphism."}
+{"_id": "2318", "title": "Quotient Group is Group", "text": "Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Then the quotient group $G / N$ is indeed a group."}
+{"_id": "2319", "title": "Principal Ideal is Ideal", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity. Let $a \\in R$. Let $\\ideal a$ be the principal ideal of $R$ generated by $a$. Then $\\ideal a$ is an ideal of $R$. Also, if $J$ is an ideal of $R$ and $a \\in J$, then $\\ideal a \\subseteq J$. That is, $\\ideal a$ is the smallest ideal of $R$ to which $a$ belongs. {{refactor|There are a number of results that this takes in, originally taken from Whitelaw: :$(1): \\quad \\forall a \\in R: \\ideal a$ is an ideal of $R$ :$(2): \\quad \\forall a \\in R: a \\in \\ideal a$ :$(3): \\quad \\forall a \\in R:$ if $J$ is an ideal of $R$, and $a \\in J$, then $\\ideal a \\subseteq J$. :::That is, $\\ideal a$ is the smallest ideal of $R$ containing $a$. }}"}
+{"_id": "2320", "title": "Bézout's Theorem", "text": "Let $X$ and $Y$ be two plane projective curves defined over a field $F$ that do not have a common component. (This condition is true if both $X$ and $Y$ are defined by different irreducible polynomials. In particular, it holds for a pair of \"generic\" curves.) {{refactor|Link to a proof of the above, and (preferably) move this statement to the proof itself.}} Then the total number of intersection points of $X$ and $Y$ with coordinates in an algebraically closed field $E$ which contains $F$, counted with their multiplicities, is equal to the product of the degrees of $X$ and $Y$."}
+{"_id": "2321", "title": "Whitney Immersion Theorem", "text": "Let $m > 1$ be a natural number. Every smooth $m$-dimensional manifold can be immersed in Euclidean $\\left({2m-1}\\right)$-space."}
+{"_id": "2322", "title": "Pasch's Theorem", "text": "Let $a, b, c, d$ be points on a line. Let $\\left({a, b, c}\\right)$ denote that $b$ lies between $a$ and $c$. Then $\\left({a, b, c}\\right)$ and $\\left({b, c, d}\\right)$ together imply that $\\left({a, b, d}\\right)$. That is, if: : $b$ is between $a$ and $c$ and: : $c$ is between $b$ and $d$ then: : $b$ is between $a$ and $d$."}
+{"_id": "2323", "title": "Forward Difference of Power", "text": ":$\\Delta c^x = \\paren {c - 1} c^x$ where $\\Delta$ denotes the forward difference operator."}
+{"_id": "2324", "title": "Forward Difference of Harmonic Number Function", "text": "Let $H_x$ denote the harmonic number function. Then: :$\\Delta H_x = \\dfrac 1 {x + 1}$ where $\\Delta H_x$ denotes the forward difference operator."}
+{"_id": "2325", "title": "Equality of Ordered Tuples", "text": "Let $a = \\tuple {a_1, a_2, \\ldots, a_n}$ and $b = \\tuple {b_1, b_2, \\ldots, b_n}$ be ordered tuples. Then: :$a = b \\iff \\forall i: 1 \\le i \\le n: a_i = b_i$ That is, for two ordered tuples to be equal, all the corresponding elements have to be equal."}
+{"_id": "2327", "title": "Zero of Power Set with Intersection", "text": "Let $S$ be a set and let $\\powerset S$ be its power set. Consider the algebraic structure $\\struct {\\powerset S, \\cap}$, where $\\cap$ denotes set intersection. Then the empty set $\\O$ serves as the zero element for $\\struct {\\powerset S, \\cap}$."}
+{"_id": "2328", "title": "Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group", "text": "Let $\\R_{>0}$ be the set of strictly positive real numbers: :$\\R_{>0} = \\set {x \\in \\R: x > 0}$ The structure $\\struct {\\R_{>0}, \\times}$ is an uncountable abelian group."}
+{"_id": "2330", "title": "Law of Mass Action", "text": "Let $\\AA$ and $\\BB$ be two chemical substances in a solution $C$ which are involved in a second-order reaction. Let $x$ grams of $\\CC$ contain $a x$ grams of $\\AA$ and $b x$ grams of $\\BB$, where $a + b = 1$. Let there be $a A$ grams of $\\AA$ and $b B$ grams of $\\BB$ at time $t = t_0$, at which time $x = 0$. Then: :$x = \\begin{cases} \\dfrac {k A^2 a b t} {k A a b t + 1} & : A = B \\\\ & \\\\ \\dfrac {A B e^{-k \\paren {A - B} a b t}} {A - B e^{-k \\paren {A - B} a b t}} & : A \\ne B \\end{cases}$ This is known as the '''law of mass action'''"}
+{"_id": "2331", "title": "Motion of Pendulum", "text": "Consider a pendulum consisting of a bob whose mass is $m$, at the end of a rod of negligible mass of length $a$. Let the bob be pulled to one side so that the rod is at an angle $\\alpha$ from the vertical and then released. Let $T$ be the time period of the pendulum, that is, the time through which the bob takes to travel from one end of its path to the other, and back again. Then: :$T = 2 \\sqrt {\\dfrac a g} \\map K k$ where: :$k = \\map \\sin {\\dfrac \\alpha 2}$ :$\\map K k$ is the complete elliptic integral of the first kind."}
+{"_id": "2332", "title": "Body under Constant Acceleration", "text": "=== $(1):$ Velocity after Time === {{:Body under Constant Acceleration/Velocity after Time}} === $(2):$ Distance after Time === {{:Body under Constant Acceleration/Distance after Time}} === $(3):$ Velocity after Distance === {{:Body under Constant Acceleration/Velocity after Distance}} where: :$\\mathbf u$ is the velocity at time $t = 0$ :$\\mathbf v$ is the velocity at time $t$ :$\\mathbf s$ is the displacement of $B$ from its initial position at time $t$ :$\\cdot$ denotes the dot product."}
+{"_id": "2333", "title": "One-Parameter Family of Curves for First Order ODE", "text": "Every one-parameter family of curves is the general solution of some first order ordinary differential equation. Conversely, every first order ordinary differential equation has as its general solution some one-parameter family of curves."}
+{"_id": "2334", "title": "Orthogonal Trajectories of One-Parameter Family of Curves", "text": "Every one-parameter family of curves has a unique family of orthogonal trajectories."}
+{"_id": "2335", "title": "Brachistochrone is Cycloid", "text": "The shape of the brachistochrone is a cycloid."}
+{"_id": "2336", "title": "Equation of Cycloid", "text": "Consider a circle of radius $a$ rolling without slipping along the x-axis of a cartesian plane. Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis. Consider the cycloid traced out by the point $P$. Let $\\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The point $P = \\tuple {x, y}$ is described by the equations: :$x = a \\paren {\\theta - \\sin \\theta}$ :$y = a \\paren {1 - \\cos \\theta}$"}
+{"_id": "2337", "title": "Length of Arc of Cycloid", "text": "Let $C$ be a cycloid generated by the equations: :$x = a \\paren {\\theta - \\sin \\theta}$ :$y = a \\paren {1 - \\cos \\theta}$ Then the length of one arc of the cycloid is $8 a$."}
+{"_id": "2339", "title": "Goldbach's Weak Conjecture", "text": "Every odd integer greater than $7$ is the sum of three odd primes."}
+{"_id": "2340", "title": "Goldbach Conjecture implies Weak Goldbach Conjecture", "text": "The Goldbach Conjecture: : Every even integer greater than $2$ is the sum of two primes implies Goldbach's Weak Conjecture: : Every odd integer greater than $7$ is the sum of three odd primes."}
+{"_id": "2350", "title": "Unique Factorization Theorem", "text": "Let $\\struct {D, +, \\times}$ be a Euclidean domain. Then $\\struct {D, +, \\times}$ is a unique factorization domain."}
+{"_id": "2353", "title": "Integral of Power", "text": ":$\\displaystyle \\forall n \\in \\R_{\\ne -1}: \\int_0^b x^n \\rd x = \\frac {b^{n + 1} } {n + 1}$"}
+{"_id": "2354", "title": "Volume of Sphere", "text": "The volume $V$ of a sphere of radius $r$ is given by: :$V = \\dfrac {4 \\pi r^3} 3$"}
+{"_id": "2355", "title": "Einstein's Mass-Energy Equation", "text": "The energy imparted to a body to cause that body to move causes the body to increase in mass by a value $M$ as given by the equation: :$E = M c^2$ where $c$ is the speed of light."}
+{"_id": "2356", "title": "Dimension of Gravitational Constant", "text": "The dimension of the gravitational constant $G$ is $M^{-1} L^3 T^{-2}$."}
+{"_id": "2357", "title": "Dimension of Spring Force Constant", "text": "The dimension of a spring force constant is $\\mathsf {M T}^{-2}$."}
+{"_id": "2358", "title": "Vieta's Formula for Pi", "text": ":$\\pi = 2 \\times \\dfrac 2 {\\sqrt 2} \\times \\dfrac 2 {\\sqrt {2 + \\sqrt 2} } \\times \\dfrac 2 {\\sqrt {2 + \\sqrt {2 + \\sqrt 2} } } \\times \\dfrac 2 {\\sqrt {2 + \\sqrt {2 + \\sqrt {2 + \\sqrt 2 } } } } \\times \\cdots$"}
+{"_id": "2362", "title": "Derivative of Arc Length", "text": "Let $C$ be a curve in the cartesian plane described by the equation $y = \\map f x$. Let $s$ be the length along the arc of the curve from some reference point $P$. Then the derivative of $s$ with respect to $x$ is given by: :$\\dfrac {\\d s} {\\d x} = \\sqrt {1 + \\paren {\\dfrac {\\d y} {\\d x} }^2}$"}
+{"_id": "2363", "title": "Duality Principle for Sets", "text": "Any identity in set theory which uses any or all of the operations: :Set intersection $\\cap$ :Set union $\\cup$ :Empty set $\\O$ :Universal set $\\mathbb U$ and none other, remains valid if: :$\\cap$ and $\\cup$ are exchanged throughout :$\\O$ and $\\mathbb U$ are exchanged throughout."}
+{"_id": "2364", "title": "Cartesian Product is not Associative", "text": "Let $A, B, C$ be non-empty sets. Then: :$A \\times \\paren {B \\times C} \\ne \\paren {A \\times B} \\times C$ where $A \\times B$ is the cartesian product of $A$ and $B$."}
+{"_id": "2365", "title": "Rational Addition is Commutative", "text": "The operation of addition on the set of rational numbers $\\Q$ is commutative: :$\\forall x, y \\in \\Q: x + y = y + x$"}
+{"_id": "2366", "title": "Rational Addition is Associative", "text": "The operation of addition on the set of rational numbers $\\Q$ is associative: :$\\forall x, y, z \\in \\Q: x + \\paren {y + z} = \\paren {x + y} + z$"}
+{"_id": "2367", "title": "Subtraction on Numbers is Anticommutative", "text": "The operation of subtraction on the numbers is anticommutative. That is: :$a - b = b - a \\iff a = b$"}
+{"_id": "2368", "title": "Commutative Law of Addition", "text": ":$\\forall x, y \\in \\mathbb F: x + y = y + x$ That is, the operation of addition on the standard number sets is commutative."}
+{"_id": "2369", "title": "Integer Addition is Commutative", "text": "The operation of addition on the set of integers $\\Z$ is commutative: :$\\forall x, y \\in \\Z: x + y = y + x$"}
+{"_id": "2370", "title": "Integer Addition is Associative", "text": "The operation of addition on the set of integers $\\Z$ is associative: :$\\forall x, y, z \\in \\Z: x + \\paren {y + z} = \\paren {x + y} + z$"}
+{"_id": "2371", "title": "Natural Number Addition is Commutative", "text": "The operation of addition on the set of natural numbers $\\N$ is commutative: :$\\forall m, n \\in \\N: m + n = n + m$"}
+{"_id": "2372", "title": "Associative Law of Addition", "text": ":$\\forall x, y, z \\in \\mathbb F: x + \\left({y + z}\\right) = \\left({x + y}\\right) + z$ That is, the operation of addition on the standard number sets is associative."}
+{"_id": "2373", "title": "Natural Number Addition is Associative", "text": "The operation of addition on the set of natural numbers $\\N$ is associative: :$\\forall x, y, z \\in \\N: x + \\paren {y + z} = \\paren {x + y} + z$"}
+{"_id": "2374", "title": "Rational Multiplication is Commutative", "text": "The operation of multiplication on the set of rational numbers $\\Q$ is commutative: :$\\forall x, y \\in \\Q: x \\times y = y \\times x$"}
+{"_id": "2375", "title": "Rational Multiplication is Associative", "text": "The operation of multiplication on the set of rational numbers $\\Q$ is associative: :$\\forall x, y, z \\in \\Q: x \\times \\paren {y \\times z} = \\paren {x \\times y} \\times z$"}
+{"_id": "2376", "title": "Natural Number Multiplication is Commutative", "text": "The operation of multiplication on the set of natural numbers $\\N$ is commutative: :$\\forall x, y \\in \\N: x \\times y = y \\times x$"}
+{"_id": "2377", "title": "Natural Number Multiplication is Associative", "text": "The operation of multiplication on the set of natural numbers $\\N$ is associative: :$\\forall x, y, z \\in \\N: \\paren {x \\times y} \\times z = x \\times \\paren {y \\times z}$"}
+{"_id": "2378", "title": "Natural Number Multiplication Distributes over Addition", "text": "The operation of multiplication is distributive over addition on the set of natural numbers $\\N$: :$\\forall x, y, z \\in \\N:$ ::$\\paren {x + y} \\times z = \\paren {x \\times z} + \\paren {y \\times z}$ ::$z \\times \\paren {x + y} = \\paren {z \\times x} + \\paren {z \\times y}$"}
+{"_id": "2379", "title": "Multiplication of Numbers Distributes over Addition", "text": "On all the number systems: :natural numbers $\\N$ :integers $\\Z$ :rational numbers $\\Q$ :real numbers $\\R$ :complex numbers $\\C$ the operation of multiplication is distributive over addition: :$m \\paren {n + p} = m n + m p$ :$\\paren {m + n} p = m p + n p$"}
+{"_id": "2380", "title": "Rational Multiplication Distributes over Addition", "text": "The operation of multiplication on the set of rational numbers $\\Q$ is distributive over addition: :$\\forall x, y, z \\in \\Q: x \\times \\paren {y + z} = \\paren {x \\times y} + \\paren {x \\times z}$ :$\\forall x, y, z \\in \\Q: \\paren {y + z} \\times x = \\paren {y \\times x} + \\paren {z \\times x}$"}
+{"_id": "2381", "title": "Associative Law of Multiplication", "text": ":$\\forall x, y, z \\in \\mathbb F: x \\times \\paren {y \\times z} = \\paren {x \\times y} \\times z$ That is, the operation of multiplication on the standard number sets is associative."}
+{"_id": "2382", "title": "Commutative Law of Multiplication", "text": ":$\\forall x, y \\in \\mathbb F: x + y = y + x$ That is, the operation of multiplication on the standard number sets is commutative."}
+{"_id": "2383", "title": "Subtraction on Numbers is Not Associative", "text": "The operation of subtraction on the numbers is not associative. That is, in general: :$a - \\paren {b - c} \\ne \\paren {a - b} - c$"}
+{"_id": "2384", "title": "Identity Element of Multiplication on Numbers", "text": "On all the number systems: * natural numbers $\\N$ * integers $\\Z$ * rational numbers $\\Q$ * real numbers $\\R$ * complex numbers $\\C$ the identity element of multiplication is one ($1$)."}
+{"_id": "2385", "title": "Identity Element of Addition on Numbers", "text": "On all the number systems: * natural numbers $\\N$ * integers $\\Z$ * rational numbers $\\Q$ * real numbers $\\R$ * complex numbers $\\C$ the identity element of addition is zero ($0$)."}
+{"_id": "2386", "title": "Zero Element is Unique", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure that has a zero element $z \\in S$. Then $z$ is unique."}
+{"_id": "2387", "title": "Group with Zero Element is Trivial", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\struct {G, \\circ}$ have a zero element. Then $\\struct {G, \\circ}$ is the trivial group."}
+{"_id": "2388", "title": "Count of Binary Operations on Set", "text": "Let $S$ be a set whose cardinality is $n$. The number $N$ of different binary operations that can be applied to $S$ is given by: :$N = n^{\\paren {n^2} }$"}
+{"_id": "2389", "title": "Count of Commutative Binary Operations on Set", "text": "Let $S$ be a set whose cardinality is $n$. The number $N$ of possible different commutative binary operations that can be applied to $S$ is given by: :$N = n^{\\frac {n \\paren {n + 1} } 2}$"}
+{"_id": "2390", "title": "Count of Binary Operations with Fixed Identity", "text": "Let $S$ be a set whose cardinality is $n$. Let $x \\in S$. The number $N$ of possible different binary operations such that $x$ is an identity element that can be applied to $S$ is given by: :$N = n^{\\left({\\left({n-1}\\right)^2}\\right)}$"}
+{"_id": "2391", "title": "Count of Commutative Binary Operations with Fixed Identity", "text": "Let $S$ be a set whose cardinality is $n$. Let $x \\in S$. The number $N$ of possible different commutative binary operations such that $x$ is an identity element that can be applied to $S$ is given by: :$N = n^{\\frac {n \\left({n-1}\\right)}2}$"}
+{"_id": "2392", "title": "Count of Binary Operations with Identity", "text": "Let $S$ be a set whose cardinality is $n$. The number $N$ of possible different binary operations which have an identity element that can be applied to $S$ is given by: :$N = n^{\\paren {n - 1}^2 + 1}$"}
+{"_id": "2395", "title": "Power of Element in Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $\\struct {H, \\circ}$ be a subgroup of $\\struct {G, \\circ}$. Let $x \\in H$. Then: :$\\forall n \\in \\Z: x^n \\in H$"}
+{"_id": "2396", "title": "Element to Power of Remainder", "text": ":$\\forall n \\in \\Z: n = q k + r: 0 \\le r < k \\iff a^n = a^r$"}
+{"_id": "2397", "title": "Element to Power of Multiple of Order is Identity", "text": ":$\\forall n \\in \\Z: k \\divides n \\iff a^n = e$"}
+{"_id": "2398", "title": "List of Elements in Finite Cyclic Group", "text": ":$\\set {a^0, a^1, a^2, \\ldots, a^{k - 1} }$ is a complete repetition-free list of the elements of $\\gen a$"}
+{"_id": "2400", "title": "Rising Sum of Binomial Coefficients", "text": ":$\\displaystyle \\sum_{j \\mathop = 0}^m \\binom {n + j} n = \\binom {n + m + 1} {n + 1} = \\binom {n + m + 1} m$"}
+{"_id": "2401", "title": "Sum of Even Index Binomial Coefficients", "text": ":$\\displaystyle \\sum_{i \\mathop \\ge 0} \\binom n {2 i} = 2^{n - 1}$"}
+{"_id": "2402", "title": "Sum of Odd Index Binomial Coefficients", "text": ":$\\displaystyle \\sum_{i \\mathop \\ge 0} \\binom n {2 i + 1} = 2^{n - 1}$"}
+{"_id": "2404", "title": "Increasing Alternating Sum of Binomial Coefficients", "text": ":$\\displaystyle \\sum_{j \\mathop = 0}^n \\left({-1}\\right)^{n + 1} j \\binom n j = 0$"}
+{"_id": "2405", "title": "Sum of Squares of Binomial Coefficients", "text": ":$\\displaystyle \\sum_{i \\mathop = 0}^n \\binom n i^2 = \\binom {2 n} n$"}
+{"_id": "2406", "title": "Cauchy-Bunyakovsky-Schwarz Inequality/Inner Product Spaces", "text": "Let $\\mathbb K$ be a subfield of $\\C$. Let $V$ be a semi-inner product space over $\\mathbb K$. Let $x, y$ be vectors in $V$. Then: :$\\size {\\innerprod x y}^2 \\le \\innerprod x x \\innerprod y y$"}
+{"_id": "2407", "title": "Cauchy-Bunyakovsky-Schwarz Inequality/Definite Integrals", "text": "Let $f$ and $g$ be real functions which are continuous on the closed interval $\\closedint a b$. Then: :$\\displaystyle \\paren {\\int_a^b \\map f t \\, \\map g t \\rd t}^2 \\le \\int_a^b \\paren {\\map f t}^2 \\rd t \\int_a^b \\paren {\\map g t}^2 \\rd t$"}
+{"_id": "2408", "title": "Cauchy-Schwarz Inequality/Complex Numbers", "text": ":$\\displaystyle \\paren {\\sum \\cmod {w_i}^2} \\paren {\\sum \\cmod {z_i}^2} \\ge \\cmod {\\sum w_i z_i}^2$ where all of $w_i, z_i \\in \\C$."}
+{"_id": "2409", "title": "Ceiling of Floor is Floor", "text": ":$\\left \\lceil {\\left \\lfloor {x}\\right \\rfloor}\\right \\rceil = \\left \\lfloor {x}\\right \\rfloor$"}
+{"_id": "2410", "title": "Floor of Ceiling is Ceiling", "text": ": $\\left \\lfloor {\\left \\lceil {x}\\right \\rceil}\\right \\rfloor = \\left \\lceil {x}\\right \\rceil$"}
+{"_id": "2412", "title": "Ceiling of Negative equals Negative of Floor", "text": ":$\\ceiling {-x} = -\\floor x$"}
+{"_id": "2413", "title": "Floor of Negative equals Negative of Ceiling", "text": ":$\\left \\lfloor {-x}\\right \\rfloor = - \\left \\lceil {x}\\right \\rceil$"}
+{"_id": "2414", "title": "Integer is Coprime to 1", "text": "Every integer is coprime to $1$. That is: :$\\forall n \\in \\Z: n \\perp 1$"}
+{"_id": "2415", "title": "Integers Coprime to Zero", "text": "The only integers which are coprime to zero are $1$ and $-1$. That is: :$n \\in \\Z: n \\perp 0 \\iff n \\in \\set {1, -1}$ In particular, note that two integers which are coprime to each other cannot both be $0$."}
+{"_id": "2416", "title": "Sum of Binomial Coefficients over Upper Index", "text": "{{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 0}^n \\binom j m       | r = \\binom {n + 1} {m + 1}       | c =  }} {{eqn | r = \\dbinom 0 m + \\dbinom 1 m + \\dbinom 2 m + \\cdots + \\dbinom n m = \\dbinom {n + 1} {m + 1}       | c =  }} {{end-eqn}}"}
+{"_id": "2417", "title": "Divisibility of Product of Consecutive Integers", "text": "The product of $n$ consecutive positive integers is divisible by the product of the ''first'' $n$ consecutive positive integers. That is: :$\\displaystyle \\forall m, n \\in \\Z_{>0}: \\exists r \\in \\Z: \\prod_{k \\mathop = 1}^n \\paren {m + k} = r \\prod_{k \\mathop = 1}^n k$"}
+{"_id": "2418", "title": "Equal Numbers are Congruent", "text": ":$\\forall x, y, z \\in \\R: x = y \\implies x \\equiv y \\pmod z$ where $x \\equiv y \\pmod z$ denotes congruence modulo $z$."}
+{"_id": "2419", "title": "Complex Roots of Unity in Exponential Form", "text": "Let $n \\in \\Z$ be an integer such that $n > 0$. Let $z \\in \\C$ be a complex number such that $z^n = 1$. Then: :$U_n = \\set {e^{2 i k \\pi / n}: k \\in \\N_n}$ where $U_n$ is the set of $n$th roots of unity. That is: :$z \\in \\set {1, e^{2 i \\pi / n}, e^{4 i \\pi / n}, \\ldots, e^{2 \\paren {n - 1} i \\pi / n} }$ Thus for every integer $n$, the number of $n$th roots of unity is $n$. Setting $\\omega := e^{2 i \\pi / n}$, $U_n$ can then be written as: :$U_n = \\set {1, \\omega, \\omega^2, \\ldots, \\omega^{n - 1} }$"}
+{"_id": "2420", "title": "Roots of Unity under Multiplication form Cyclic Group", "text": "Let $n \\in \\Z$ be an integer such that $n > 0$. The $n$th complex roots of unity under the operation of multiplication form the cyclic group which is isomorphic to $C_n$."}
+{"_id": "2421", "title": "Inverse Mapping is Unique", "text": "Let $f: S \\to T$ be a mapping. If $f$ has an inverse mapping, then that inverse mapping is unique. That is, if: :$f$ and $g$ are inverse mappings of each other and :$f$ and $h$ are inverse mappings of each other then $g = h$."}
+{"_id": "2422", "title": "Special Linear Group is Subgroup of General Linear Group", "text": "Let $K$ be a field whose zero is $0_K$ and unity is $1_K$. Let $\\SL {n, K}$ be the special linear group of order $n$ over $K$. Then $\\SL {n, K}$ is a subgroup of the general linear group $\\GL {n, K}$."}
+{"_id": "2423", "title": "Determinant of Unit Matrix", "text": "Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. The determinant of the unit matrix of order $n$ over $R$ is equal to $1_R$."}
+{"_id": "2424", "title": "Determinant of Inverse Matrix", "text": "Let $K$ be a field whose zero is $0_K$ and whose unity is $1_K$. Let $\\mathbf A$ be an invertible matrix of order $n$ over $K$. Then the determinant of its inverse is given by: :$\\map \\det {\\mathbf A^{-1} } = \\dfrac {1_K} {\\map \\det {\\mathbf A} }$"}
+{"_id": "2425", "title": "Transposition is of Odd Parity", "text": "Let $S_n$ denote the set of permutations on $n$ letters. Let $\\pi \\in S_n$ be a transposition. Then $\\pi$ is of odd parity."}
+{"_id": "2427", "title": "Subset Product Action is Group Action", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $\\powerset G$ be the power set of $\\struct {G, \\circ}$. For any $S \\in \\powerset G$ and for any $g \\in G$, the subset product action: :$\\forall g \\in G: \\forall S \\in \\powerset G: g * S = g \\circ S$ is a group action."}
+{"_id": "2428", "title": "Subgroup of Elements whose Order Divides Integer", "text": "Let $A$ be an abelian group. Let $k \\in \\Z$ and $B$ be a set of the form: :$\\left\\{{x \\in A : x^k = e}\\right\\}$ Then $B$ is a subgroup of $A$."}
+{"_id": "2429", "title": "Abelian Group Factored by Prime", "text": "Let $G$ be a finite abelian group. Let $p$ be a prime. Factor the order of $G$ as: :$\\order G = m p^n$ such that $p$ does not divide $m$. Then: :$G = H \\times K$ where: :$H = \\set {x \\in G : x^{p^n} = e}$ and: :$K = \\set {x \\in G : x^m = e}$ Furthermore: :$\\order H = p^n$"}
+{"_id": "2430", "title": "Abelian Group of Prime-power Order is Product of Cyclic Groups", "text": "Let $G$ be an abelian group of prime-power order. Let $a$ be an element of maximal order in $G$. Then $G$ can be written in the form $\\gen a \\times K$ for some $K \\le G$."}
+{"_id": "2431", "title": "Power of Product in Abelian Group", "text": "Let $\\struct {G, \\circ}$ be an abelian group. Then: :$\\forall x, y \\in G: \\forall k \\in \\Z: \\paren {x \\circ y}^k = x^k \\circ y^k$"}
+{"_id": "2438", "title": "Identity Element of Natural Number Addition is Zero", "text": "The identity element for the natural numbers under addition is zero ($0$): :$\\forall n \\in \\N: 0 + n = n$"}
+{"_id": "2439", "title": "Simpson's Formulas", "text": "=== Cosine by Cosine === {{:Simpson's Formulas/Cosine by Cosine}} === Sine by Sine === {{:Simpson's Formulas/Sine by Sine}} === Sine by Cosine === {{:Simpson's Formulas/Sine by Cosine}} === Cosine by Sine === {{:Simpson's Formulas/Cosine by Sine}}"}
+{"_id": "2440", "title": "Prosthaphaeresis Formulas", "text": "=== Sine plus Sine === {{:Prosthaphaeresis Formulas/Sine plus Sine}} === Sine minus Sine === {{:Prosthaphaeresis Formulas/Sine minus Sine}} === Cosine plus Cosine === {{:Prosthaphaeresis Formulas/Cosine plus Cosine}} === Cosine minus Cosine === {{:Prosthaphaeresis Formulas/Cosine minus Cosine}}"}
+{"_id": "2441", "title": "Sum of Tangent and Cotangent", "text": ":$\\tan x + \\cot x = \\sec x \\csc x$"}
+{"_id": "2442", "title": "Tangent times Tangent plus Cotangent", "text": ":$\\tan x \\paren {\\tan x + \\cot x} = \\sec^2 x$"}
+{"_id": "2443", "title": "Minor Trigonometric Identities", "text": "=== Sum of Tangent and Cotangent === {{:Sum of Tangent and Cotangent}} === Tangent times Tangent plus Cotangent === {{:Tangent times Tangent plus Cotangent}} === Secant Minus Cosine === {{:Secant Minus Cosine}} === Square of Tangent Minus Square of Sine === {{:Square of Tangent Minus Square of Sine}} === Difference of Fourth Powers of Cosine and Sine === {{:Difference of Fourth Powers of Cosine and Sine}} === Cosecant Minus Sine === {{:Cosecant Minus Sine}} === Cotangent Minus Tangent === {{:Cotangent Minus Tangent}} === Sum of Cosecant and Cotangent === {{:Sum of Cosecant and Cotangent}} === Sum of Squares of Secant and Cosecant === {{:Sum of Squares of Secant and Cosecant}} === Difference of Fourth Powers of Secant and Tangent === {{:Difference of Fourth Powers of Secant and Tangent}} === Reciprocal of One Plus Sine === {{:Reciprocal of One Plus Sine}} === Reciprocal of One Minus Sine === {{:Reciprocal of One Minus Sine}} === Sum of Reciprocals of One Plus and Minus Sine === {{:Sum of Reciprocals of One Plus and Minus Sine}} === Difference of Reciprocals of One Plus and Minus Sine === {{:Difference  of Reciprocals of One Plus and Minus Sine}} === Reciprocal of One Plus Cosine === {{:Reciprocal of One Plus Cosine}} === Reciprocal of One Minus Cosine === {{:Reciprocal of One Minus Cosine}} === Sum of Secant and Tangent === {{:Sum of Secant and Tangent}} === Cosine over Sum of Secant and Tangent === {{:Cosine over Sum of Secant and Tangent}} === Secant Plus One over Secant Squared === {{:Secant Plus One over Secant Squared}} === Sine Plus Cosine times Tangent Plus Cotangent === {{:Sine Plus Cosine times Tangent Plus Cotangent}} === Tangent over Secant Plus One === {{:Tangent over Secant Plus One}} === Squares of Linear Combination of Sine and Cosine === {{:Squares of Linear Combination of Sine and Cosine}} === Reciprocal of One Minus Secant === {{:Reciprocal of One Minus Secant}} === Reciprocal of One Plus Cosecant === {{:Reciprocal of One Plus Cosecant}}"}
+{"_id": "2444", "title": "Secant Minus Cosine", "text": ":$\\sec x - \\cos x = \\sin x \\tan x$"}
+{"_id": "2445", "title": "Square of Tangent Minus Square of Sine", "text": ":$\\tan^2 x - \\sin^2 x = \\tan^2 x \\ \\sin^2 x$"}
+{"_id": "2446", "title": "Difference of Fourth Powers of Cosine and Sine", "text": ":$\\sin^4 x - \\cos^4 x = \\sin^2 x - \\cos^2 x$"}
+{"_id": "2447", "title": "Cosecant Minus Sine", "text": ":$\\csc x - \\sin x = \\cos x \\ \\cot x$"}
+{"_id": "2448", "title": "Sum of Squares of Secant and Cosecant", "text": ":$\\sec^2 x + \\csc^2 x = \\sec^2 x \\csc^2 x$"}
+{"_id": "2449", "title": "Difference of Fourth Powers of Secant and Tangent", "text": ":$\\sec^4 x - \\tan^4 x = \\sec^2 x + \\tan^2 x$"}
+{"_id": "2450", "title": "Sum of Reciprocals of One Plus and Minus Sine", "text": ":$\\dfrac 1 {1 - \\sin x} + \\dfrac 1 {1 + \\sin x} = 2 \\sec^2 x$"}
+{"_id": "2451", "title": "Difference of Reciprocals of One Plus and Minus Sine", "text": ":$\\displaystyle \\frac 1 {1 - \\sin x} - \\frac 1 {1 + \\sin x} = 2 \\tan x \\ \\sec x$"}
+{"_id": "2452", "title": "Sum of Secant and Tangent", "text": ":$\\sec x + \\tan x = \\dfrac {1 + \\sin x} {\\cos x}$"}
+{"_id": "2453", "title": "Cosine over Sum of Secant and Tangent", "text": ":$\\dfrac {\\cos x} {\\sec x + \\tan x} = 1 - \\sin x$"}
+{"_id": "2454", "title": "Secant Plus One over Secant Squared", "text": ":$\\displaystyle \\frac {\\sec x + 1} {\\sec^2 x} = \\frac {\\sin^2 x} {\\sec x - 1}$"}
+{"_id": "2455", "title": "Sine Plus Cosine times Tangent Plus Cotangent", "text": ":$\\paren {\\sin x + \\cos x} \\paren {\\tan x + \\cot x} = \\sec x + \\csc x$"}
+{"_id": "2456", "title": "Tangent over Secant Plus One", "text": ":$\\displaystyle \\frac {\\tan x} {\\sec x + 1} = \\frac {\\sec x - 1} {\\tan x}$"}
+{"_id": "2457", "title": "Squares of Linear Combination of Sine and Cosine", "text": ":$\\left({a \\cos x + b \\sin x}\\right)^2 + \\left({b \\cos x - a \\sin x}\\right)^2 = a^2 + b^2$"}
+{"_id": "2458", "title": "Reciprocal of One Minus Secant", "text": ":$\\dfrac {\\sin^2 x + 2 \\cos x - 1} {\\sin^2 x + 3 \\cos x - 3} = \\dfrac 1 {1 - \\sec x}$"}
+{"_id": "2459", "title": "Reciprocal of One Plus Cosecant", "text": ":$\\dfrac {\\cos^2 x + 3 \\sin x - 1} {\\cos^2 x + 2 \\sin x + 2} = \\dfrac 1 {1 + \\csc x}$"}
+{"_id": "2460", "title": "Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE/General Result", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive integer. The ordinary differential equation: :$a_n x^n \\, \\map {f^{\\paren n} } x + \\dotsb + a_1 x \\, \\map {f'} c + a_0 \\, \\map f x = 0$ can be transformed to a linear differential equation with constant coefficients by substitution $x = e^t$."}
+{"_id": "2461", "title": "Condition for Composite Relation with Inverse to be Identity", "text": "Let $\\RR \\subseteq S \\times T$ be a relation on $S \\times T$. Let $\\RR \\circ \\RR^{-1}$ be the composite of $\\RR$ with its inverse. Let $I_T$ be the identity mapping on $T$. Then: :$\\RR \\circ \\RR^{-1} = I_T$ {{iff}}: :$\\RR$ is many-to-one and: :$\\RR$ is right-total."}
+{"_id": "2462", "title": "Set of All Relations is a Monoid", "text": "The set of all relations $\\Bbb E = \\left\\{{\\mathcal R: \\mathcal R \\subseteq S \\times S}\\right\\}$ on a set $S$ forms a monoid in which the operation is composition of relations. The only invertible elements of $\\Bbb E$ are permutations. If $\\mathcal R$ is invertible, its inverse is $\\mathcal R^{-1}$."}
+{"_id": "2463", "title": "Inverses of Right-Total and Left-Total Relations", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation on $S \\times T$. Let $\\mathcal R^{-1} \\subseteq T \\times S$ be the inverse of $\\mathcal R$. Then:"}
+{"_id": "2464", "title": "Inverse Relation is Left and Right Inverse iff Bijection", "text": "Let $\\RR \\subseteq S \\times T$ be a relation on a cartesian product $S \\times T$. Let: :$I_S$ be the identity mapping on $S$ :$I_T$ be the identity mapping on $T$. Let $\\RR^{-1}$ be the inverse relation of $\\RR$. Then $\\RR$ is a bijection {{iff}}: :$\\RR^{-1} \\circ \\RR = I_S$ and :$\\RR \\circ \\RR^{-1} = I_T$ where $\\circ$ denotes composition of relations."}
+{"_id": "2465", "title": "Integer Multiples Closed under Addition", "text": "Let $n \\Z$ be the set of integer multiples of $n$. Then the algebraic structure $\\struct {n \\Z, +}$ is closed under addition."}
+{"_id": "2466", "title": "Integer Multiples Closed under Multiplication", "text": "Let $n \\Z$ be the set of integer multiples of $n$. Then the algebraic structure $\\struct {n \\Z, \\times}$ is closed under multiplication."}
+{"_id": "2467", "title": "Non-Zero Integers Closed under Multiplication", "text": "The set of non-zero integers is closed under multiplication."}
+{"_id": "2469", "title": "Two-Step Subgroup Test using Subset Product", "text": "Let $G$ be a group. Let $\\O\\subset H \\subseteq G$ be a non-empty subset of $G$. Then $H$ is a subgroup of $G$ {{iff}}: :$H H \\subseteq H$ :$H^{-1} \\subseteq H$ where: :$H H$ is the product of $H$ with itself :$H^{-1}$ is the inverse of $H$."}
+{"_id": "2470", "title": "One-Step Subgroup Test using Subset Product", "text": "Let $G$ be a group. Let $\\varnothing \\subset H \\subseteq G$ be a non-empty subset of $G$. Then $H$ is a subgroup of $G$ {{iff}}: :$H H^{-1} \\subseteq H$ where: :$H^{-1}$ is the inverse of $H$ :$H H ^{-1}$ is the product of $H$ with $H^{-1}$."}
+{"_id": "2471", "title": "Product of Subset with Union", "text": "Let $\\left({G, \\circ}\\right)$ be an algebraic structure. Let $X, Y, Z \\subseteq G$. Then: : $X \\circ \\left({Y \\cup Z}\\right) = \\left({X \\circ Y}\\right) \\cup \\left({X \\circ Z}\\right)$ : $\\left({Y \\cup Z}\\right) \\circ X = \\left({Y \\circ X}\\right) \\cup \\left({Z \\circ X}\\right)$ where $X \\circ Y$ denotes the subset product of $X$ and $Y$."}
+{"_id": "2473", "title": "Hilbert's Nullstellensatz", "text": "Let $k$ be an algebraically closed field. Let $n \\geq 0$ be an natural number. Let $k \\left[{x_1,\\ldots, x_n}\\right]$ be the polynomial ring in $n$ variables over $k$. Then for every ideal $J \\subseteq k \\left[{x_1,\\ldots, x_n}\\right]$, the associated ideal of its zero-locus equals its radical: :$I \\left({Z \\left({J}\\right)}\\right) = \\sqrt J$"}
+{"_id": "2474", "title": "Subset of Domain is Subset of Preimage of Image", "text": "Let $f: S \\to T$ be a mapping. Then: :$A \\subseteq S \\implies A \\subseteq \\paren {f^{-1} \\circ f} \\sqbrk A$ where: :$f \\sqbrk A$ denotes the image of $A$ under $f$ :$f^{-1} \\sqbrk A$ denotes the preimage of $A$ under $f$ :$f^{-1} \\circ f$ denotes composition of $f^{-1}$ and $f$. This can be expressed in the language and notation of direct image mappings and inverse image mappings as: :$\\forall A \\in \\powerset S: A \\subseteq \\map {\\paren {f^\\gets \\circ f^\\to} } A$"}
+{"_id": "2475", "title": "Clairaut's Differential Equation", "text": "'''Clairaut's differential equation''' is a first order ordinary differential equation which can be put into the form: :$y = x y' + \\map f {y'}$ Its general solution is: :$y = C x + \\map f C$ where $C$ is a constant."}
+{"_id": "2476", "title": "Noether Normalization Lemma", "text": "Let $k$ be a field. Let $A$ be a non-trivial finitely generated $k$-algebra. {{explain|the above link is for Definition:Non-Trivial Ring -- we need to define a Definition:Non-Trivial Algebra}} Then there exists $n \\in \\N$ and a finite injective morphism of $k$-algebra: :$k \\sqbrk {x_1, \\dotsc, x_n} \\to A$ {{DefinitionWanted|Instead of using Definition:Finite Ring Homomorphism, another page is to be generated defining the module-theory version.}}"}
+{"_id": "2477", "title": "Image of Preimage under Mapping", "text": "Let $f: S \\to T$ be a mapping. Then: :$B \\subseteq T \\implies \\paren {f \\circ f^{-1} } \\sqbrk B = B \\cap f \\sqbrk S$"}
+{"_id": "2478", "title": "Isomorphism of External Direct Products", "text": "Let: :$\\struct {S_1 \\times S_2, \\circ}$ be the external direct product of two algebraic structures $\\struct {S_1, \\circ_1}$ and $\\struct {S_2, \\circ_2}$ :$\\struct {T_1 \\times T_2, *}$ be the external direct product of two algebraic structures $\\struct {T_1, *_1}$ and $\\struct {T_2, *_2}$ :$\\phi_1$ be an isomorphism from $\\struct {S_1, \\circ_1}$ onto $\\struct {T_1, *_1}$ :$\\phi_2$ be an isomorphism from $\\struct {S_2, \\circ_2}$ onto $\\struct {T_2, *_2}$. Then the mapping $\\phi_1 \\times \\phi_2: \\struct {S_1 \\times S_2, \\circ} \\to \\struct {T_1 \\times T_2, *}$ defined as: :$\\map {\\paren {\\phi_1 \\times \\phi_2} } {x, y} = \\tuple {\\map {\\phi_1} x, \\map {\\phi_2} y}$ is an isomorphism from $\\struct {S_1 \\times S_2, \\circ}$ to $\\struct {T_1 \\times T_2, *}$."}
+{"_id": "2481", "title": "Dual of Lattice Ordering is Lattice Ordering", "text": "Let $\\struct {S, \\preccurlyeq}$ be a lattice. Let $\\preccurlyeq$ be the lattice ordering on $\\struct {S, \\preccurlyeq}$. Then its dual ordering $\\succcurlyeq$ is also a lattice ordering."}
+{"_id": "2482", "title": "Order Isomorphism on Totally Ordered Set preserves Total Ordering", "text": "Let $\\left({S, \\preccurlyeq_1}\\right)$ and $\\left({T, \\preccurlyeq_2}\\right)$ be ordered sets. Let $\\phi: \\left({S, \\preccurlyeq_1}\\right) \\to \\left({T, \\preccurlyeq_2}\\right)$ be an order isomorphism. Then $\\left({S, \\preccurlyeq_1}\\right)$ is a totally ordered set {{iff}} $\\left({T, \\preccurlyeq_2}\\right)$ is also a totally ordered set."}
+{"_id": "2483", "title": "Order Isomorphism on Well-Ordered Set preserves Well-Ordering", "text": "Let $\\struct {S_1, \\preccurlyeq_1}$ and $\\struct {S_2, \\preccurlyeq_2}$ be ordered sets. Let $\\phi: \\struct {S_1, \\preccurlyeq_1} \\to \\struct {S_2, \\preccurlyeq_2}$ be an order isomorphism. Then $\\struct {S_1, \\preccurlyeq_1}$ is a well-ordered set {{iff}} $\\struct {S_2, \\preccurlyeq_2}$ is also a well-ordered set."}
+{"_id": "2484", "title": "Order Isomorphism on Lattice preserves Lattice Structure", "text": "Let $\\left({S, \\preccurlyeq_1}\\right)$ and $\\left({T, \\preccurlyeq_2}\\right)$ be ordered sets. Let $\\phi: \\left({S, \\preccurlyeq_1}\\right) \\to \\left({T, \\preccurlyeq_2}\\right)$ be an order isomorphism. Then $\\left({S, \\preccurlyeq_1}\\right)$ is a lattice iff $\\left({T, \\preccurlyeq_2}\\right)$ is also a lattice."}
+{"_id": "2485", "title": "Set of Subgroups forms Complete Lattice", "text": "Let $\\struct {G, \\circ}$ be a group, and let $\\mathbb G$ be the set of all subgroups of $G$. Then $\\struct {\\mathbb G, \\subseteq}$ is a complete lattice."}
+{"_id": "2486", "title": "Infimum and Supremum of Subgroups", "text": "Let $\\left({G, \\circ}\\right)$ be a group. Let $\\mathbb G$ be the set of all subgroups of $G$. Let $\\left({\\mathbb G, \\subseteq}\\right)$ be the complete lattice formed by $\\mathbb G$ and $\\subseteq$. Let $H, K \\in \\mathbb G$."}
+{"_id": "2487", "title": "Index Laws/Sum of Indices/Monoid", "text": "Let $\\left({S, \\circ}\\right)$ be a monoid whose identity element is $e$. For $a \\in S$, let $\\circ^n a = a^n$ be defined as the $n$th power of $a$: :$a^n = \\begin{cases} e & : n = 0 \\\\ a^x \\circ a & : n = x + 1 \\end{cases}$ That is: :$a^n = \\underbrace{a \\circ a \\circ \\cdots \\circ a}_{n \\text{ copies of } a} = \\circ^n \\left({a}\\right)$ while: :$a^0 = e$ Then: :$\\forall m, n \\in \\N: a^{n + m} = a^n \\circ a^m$"}
+{"_id": "2488", "title": "Power of Product of Commuting Elements in Semigroup equals Product of Powers", "text": ":$\\forall n \\in \\N_{>0}: \\map {\\circ^n} {a \\circ b} = \\paren {\\circ^n a} \\circ \\paren {\\circ^n b}$"}
+{"_id": "2490", "title": "Sum of Sequences of Fifth and Seventh Powers", "text": ":$\\displaystyle \\sum_{i \\mathop = 1}^n i^5 + \\sum_{i \\mathop = 1}^n i^7 = 2 \\paren {\\sum_{i \\mathop = 1}^n i}^4$"}
+{"_id": "2491", "title": "Cardinality of Set of Strictly Increasing Mappings", "text": "Let $\\struct {S, \\preceq}$ and $\\struct {T, \\preccurlyeq}$ be tosets. Let the cardinality of $S$ and $T$ be: :$\\card S = m, \\card T = n$ Then the number of strictly increasing mappings from $S$ to $T$ is: :$\\dbinom n m = \\dfrac {n!} {m! \\ \\paren {n - m}!}$. where $\\dbinom n m$ is a binomial coefficient."}
+{"_id": "2493", "title": "Construction of Inverse Completion/Equivalence Relation", "text": "The cross-relation $\\boxtimes$ is an equivalence relation on $\\left({S \\times C, \\oplus}\\right)$."}
+{"_id": "2495", "title": "Construction of Inverse Completion/Properties of Quotient Structure", "text": "=== Identity of Quotient Structure === {{:Construction of Inverse Completion/Identity of Quotient Structure}} === Invertible Elements in Quotient Structure === {{:Construction of Inverse Completion/Invertible Elements in Quotient Structure}} === Generator for Quotient Structure === {{:Construction of Inverse Completion/Generator for Quotient Structure}} === Quotient Structure is Inverse Completion === {{:Construction of Inverse Completion/Quotient Structure is Inverse Completion}}"}
+{"_id": "2496", "title": "Integers form Commutative Ring", "text": "The set of integers $\\Z$ forms a commutative ring under addition and multiplication."}
+{"_id": "2497", "title": "Symmetric Group is Group", "text": "Let $S$ be a set. Let $\\map \\Gamma S$ denote the set of all permutations on $S$. Then $\\struct {\\map \\Gamma S, \\circ}$, the symmetric group on $S$, forms a group."}
+{"_id": "2498", "title": "Quotient Ring Defined by Ring Itself is Null Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Let $\\struct {R / R, +, \\circ}$ be the quotient ring defined by $R$. Then $\\struct {R / R, +, \\circ}$ is a null ring."}
+{"_id": "2499", "title": "Set of Subrings forms Complete Lattice", "text": "Let $\\struct {K, +, \\circ}$ be a ring. Let $\\mathbb K$ be the set of all subrings of $K$. Then $\\struct {\\mathbb K, \\subseteq}$ is a complete lattice."}
+{"_id": "2500", "title": "Set of Ideals forms Complete Lattice", "text": "Let $\\struct {K, +, \\circ}$ be a ring. Let $\\mathbb K$ be the set of all ideals of $K$. Then $\\struct {\\mathbb K, \\subseteq}$ is a complete lattice."}
+{"_id": "2501", "title": "Subset Product of Abelian Subgroups", "text": "Let $\\left({G, \\circ}\\right)$ be an abelian group. Let $H_1$ and $H_2$ be subgroups of $G$. Then $H_1 \\circ H_2$ is a subgroup of $G$."}
+{"_id": "2502", "title": "Sum of Ideals is Ideal", "text": "Let $J_1$ and $J_2$ be ideals of a ring $\\struct{R, +, \\circ}$. Then: : $J = J_1 + J_2$ is an ideal of $R$ where $J_1 + J_2$ is as defined in subset product."}
+{"_id": "2503", "title": "Principal Ideal from Element in Center of Ring", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $b \\in R$ be in the center of $R$. Then: :$\\left({b}\\right) = R \\circ b = \\left\\{{x \\circ b: x \\in R}\\right\\}$ where $\\left({b}\\right)$ is the principal ideal generated by $b$."}
+{"_id": "2504", "title": "External Direct Sum of Rings is Ring", "text": "Let $\\left({R_1, +_1, \\circ_1}\\right), \\left({R_2, +_2, \\circ_2}\\right), \\ldots, \\left({R_n, +_n, \\circ_n}\\right)$ be rings. Then their (external) direct product: : $\\displaystyle \\left({R, +, \\circ}\\right) = \\prod_{k \\mathop = 1}^n \\left({R_k, +_k, \\circ_k}\\right)$ is a ring."}
+{"_id": "2505", "title": "External Direct Product Distributivity", "text": "Let $\\left({R_1, +_1, \\circ_1}\\right), \\left({R_2, +_2, \\circ_2}\\right), \\ldots, \\left({R_n, +_n, \\circ_n}\\right)$ be ringoids. Let their external direct product be $\\displaystyle \\left({R, +, \\circ}\\right) = \\prod_{k \\mathop = 1}^n \\left({R_k, +_k, \\circ_k}\\right)$. Then the operation $\\circ$ is distributive over $+$."}
+{"_id": "2506", "title": "Ideal of External Direct Sum of Rings", "text": "Let $\\left({R_1, +_1, \\circ_1}\\right), \\left({R_2, +_2, \\circ_2}\\right), \\ldots, \\left({R_n, +_n, \\circ_n}\\right)$ be rings. Let  :$\\displaystyle \\left({R, +, \\circ}\\right) = \\prod_{k \\mathop = 1}^n \\left({R_k, +_k, \\circ_k}\\right)$  be their direct product. For each $k \\in \\left[{1 \\,.\\,.\\, n}\\right]$, let: :$R\\,'_k = \\left\\{{\\left({x_1, \\ldots, x_n}\\right) \\in R: \\forall j \\ne k: x_j = 0}\\right\\}$ Then: : $\\forall k \\in \\left[{1 \\,.\\,.\\, n}\\right]: R\\,'_k$ is an ideal of $R$."}
+{"_id": "2507", "title": "Canonical Injection from Ideal of External Direct Sum of Rings", "text": "Let $\\struct {R_1, +_1, \\circ_1}, \\struct {R_2, +_2, \\circ_2}, \\dotsc, \\struct {R_n, +_n, \\circ_n}$ be rings. Let $\\displaystyle \\struct {R, +, \\circ} = \\prod_{k \\mathop = 1}^n \\struct {R_k, +_k, \\circ_k}$ be their external direct product. For each $k \\in \\closedint 1 n$, let: :$R'_k = \\set {\\tuple {x_1, \\dotsc, x_n} \\in R: \\forall j \\ne k: x_j = 0}$ Let $\\inj_k: R_k \\to R$ be the canonical injection on the $k$th coordinate from $R_k$ into $\\struct {R, +, \\circ}$. Let $\\pr_k: R \\to R'_k$ be the projection on the $k$th coordinate of $\\struct {R, +, \\circ}$ onto $R'_k$. Then: :$\\inj_k: R_k \\to R'_k$ is an isomorphism :Its inverse is the restriction of $\\pr_k$ to $R'_k$."}
+{"_id": "2509", "title": "Conditions for Internal Ring Direct Sum", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\sequence {\\struct {S_k, +, \\circ} }$ be a sequence of subrings of $R$. Then $R$ is the ring direct sum of $\\sequence {S_k}_{1 \\mathop \\le k \\mathop \\le n}$ {{iff}}: :$(1): \\quad R = S_1 + S_2 + \\cdots + S_n$ :$(2): \\quad \\sequence {\\struct {S_k, +} }_{1 \\mathop \\le k \\mathop \\le n}$ is a sequence of independent subgroups of $\\struct {R, +}$ :$(3): \\quad \\forall k \\in \\closedint 1 n: S_k$ is an ideal of $R$."}
+{"_id": "2510", "title": "Structure Induced by Ring Operations is Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $S$ be a set. Then $\\struct {R^S, +', \\circ'}$ is a ring, where $+'$ and $\\circ'$ are the pointwise operations induced on $R^S$ by $+$ and $\\circ$."}
+{"_id": "2511", "title": "Division Subring Test", "text": "Let $\\struct {K, +, \\circ}$ be a division ring, and let $L$ be a subset of $K$. Then $\\struct {L, +, \\circ}$ is a division subring of $\\struct {K, +, \\circ}$ {{iff}} these all hold: :$(1) \\quad L^* \\ne \\O$ :$(2) \\quad \\forall x, y \\in L: x + \\paren {-y} \\in L$ :$(3) \\quad \\forall x, y \\in L: x \\circ y \\in L$ :$(4) \\quad x \\in L^* \\implies x^{-1} \\in L^*$"}
+{"_id": "2514", "title": "Homomorphism from Integers into Ring with Unity", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let the characteristic of $R$ be $p$. For any $a \\in R$, we define the mapping $g_a: \\Z \\to R$ from the integers into $R$ as: :$\\forall n \\in \\Z: \\map {g_a} n = n \\cdot a$ Then $g_a$ is a group homomorphism from $\\struct {\\Z, +}$ to $\\struct {R, +}$. Also: :$\\ideal p \\subseteq \\map \\ker {g_a}$ where: :$\\map \\ker {g_a}$ is the kernel of $g_a$; :$\\ideal p$ is the principal ideal of $\\Z$ generated by $p$. Also: :$p \\divides n \\implies n \\cdot a = 0$ where $p \\divides n$ denotes that $p$ is a divisor of $n$."}
+{"_id": "2515", "title": "Properties of Prime Subfield", "text": "Let $F$ be a field. Let $K$ be the prime subfield of $F$. Then $K$ is isomorphic to either: :$\\Q$, the field of rational numbers, or :$\\Z_p$, the Ring of Integers Modulo $p$, where $p$ is prime."}
+{"_id": "2516", "title": "Subgroup of Additive Group Modulo m is Ideal of Ring", "text": "Let $m \\in \\Z: m > 1$. Let $\\struct {\\Z_m, +_m}$ be the additive group of integers modulo $m$. Then every subgroup of $\\struct {\\Z_m, +_m}$ is an ideal of the ring of integers modulo $m$ $\\struct {\\Z_m, +_m, \\times_m}$."}
+{"_id": "2517", "title": "Bijection from Divisors to Subgroups of Cyclic Group", "text": "Let $G$ be a cyclic group of order $n$ generated by $a$. Let $S = \\left\\{{m \\in \\Z_{>0}: m \\divides n}\\right\\}$ be the set of all divisors of $n$. Let $T$ be the set of all subgroups of $G$ Let $\\phi: S \\to T$ be the mapping defined as: :$\\phi: m \\to \\left\\langle{a^{n/m}}\\right\\rangle$ where  $\\left\\langle{a^{n/m}}\\right\\rangle$ is the subgroup generated by $a^{n/m}$. Then $\\phi$ is a bijection."}
+{"_id": "2518", "title": "Generator of Additive Group Modulo m iff Unit of Ring", "text": "Let $m \\in \\Z: m > 1$. Let $\\struct {\\Z_m, +_m}$ denote the additive group of integers modulo $m$. Let $\\struct {\\Z_m, +_m, \\times_m}$ be the ring of integers modulo $m$. Let $a \\in \\Z_m$. Then: :$a$ is a generator of $\\struct {\\Z_m, +_m}$  {{iff}} :$a$ is a unit of $\\struct {\\Z_m, +_m, \\times_m}$"}
+{"_id": "2519", "title": "Subspace of Real Differentiable Functions", "text": "Let $\\mathbb J$ be an open interval of the real number line $\\R$. Let $\\map \\DD {\\mathbb J}$ be the set of all differentiable real functions on $\\mathbb J$. Then $\\struct {\\map \\DD {\\mathbb J}, +, \\times}_\\R$ is a subspace of the $\\R$-vector space $\\struct {\\R^{\\mathbb J}, +, \\times}_\\R$."}
+{"_id": "2521", "title": "Subspace of Riemann Integrable Functions", "text": "Let $\\mathbb J = \\set {x \\in \\R: a \\le x \\le b}$ be a closed interval of the real number line $\\R$. Let $\\map \\RR {\\mathbb J}$ be the set of all Riemann integrable functions on $\\mathbb J$. Then $\\struct {\\map \\RR {\\mathbb J}, +, \\times}_\\R$ is a subspace of the $\\R$-vector space $\\struct {\\R^{\\mathbb J}, +, \\times}_\\R$."}
+{"_id": "2522", "title": "Basis of Vector Space of Polynomial Functions", "text": "Let $B$ be the set of all the identity functions $I^n$ on $\\R^n$ where $n \\in \\N^*$. Then $B$ is a basis of the $\\R$-vector space $\\map P \\R$ of all polynomial functions on $\\R$."}
+{"_id": "2523", "title": "Ordered Basis for Coordinate Plane", "text": "Let $a_1, a_2 \\in \\R^2$ such that $\\left\\{{a_1, a_2}\\right\\}$ forms a linearly independent set. Then $\\left({a_1, a_2}\\right)$ is an ordered basis for the $\\R$-vector space $\\R^2$. Hence the points on the plane can be uniquely identified by means of linear combinations of $a_1$ and $a_2$."}
+{"_id": "2524", "title": "Differentiation on Polynomials is Linear Operator", "text": "Let $P \\left({\\R}\\right)$ be the vector space of all polynomial functions on the real number line $\\R$. Then the differentiation operator $D$ on $P \\left({\\R}\\right)$ is a linear operator."}
+{"_id": "2526", "title": "Noether's Theorem (Calculus of Variations)", "text": "Let $y_i$, $F$, $\\Psi_i$, $\\Phi$ be real functions. Let $x, \\epsilon \\in \\R$. Let $\\mathbf y = \\sequence {y_i}_{1 \\mathop \\le i \\mathop \\le n}$ and $\\mathbf \\Psi = \\sequence{\\Psi_i}_{1 \\mathop \\le i \\mathop \\le n}$ be vectors. Let  :$\\Phi = \\map \\Phi {x, \\mathbf y, \\mathbf y'; \\epsilon}, \\quad \\Psi_i = \\map {\\Psi_i} {x, \\mathbf y, \\mathbf y'; \\epsilon}$  such that: :$\\map \\Phi {x, \\mathbf y, \\mathbf y'; 0} = x, \\quad \\map {\\Psi_i} {x, \\mathbf y, \\mathbf y'; 0} = y_i$ where $x, \\mathbf y, \\mathbf y', \\epsilon$ are variables. Let: :$\\ds J \\sqbrk {\\mathbf y} = \\int_{x_0}^{x_1} \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ be a functional. Let: :$X = \\map \\Phi {x, \\mathbf y, \\mathbf y'; \\epsilon}, \\quad \\mathbf Y = \\map {\\mathbf \\Psi} {x, \\mathbf y, \\mathbf y'; \\epsilon}$ Suppose, $J \\sqbrk {\\mathbf y}$ is invariant under transformations $x \\rightarrow X$ and $\\mathbf y \\rightarrow \\mathbf Y$ for arbitrary $x_0$ and $x_1$. Then: :$\\nabla_{\\mathbf y'} F \\cdot \\boldsymbol \\psi + \\paren {F - \\mathbf y' \\cdot \\nabla_{\\mathbf y'} F} \\phi = C$ where $C$ is a constant and: :$\\map {\\boldsymbol \\psi} {x, \\mathbf y, \\mathbf y'} = \\dfrac {\\partial \\map {\\mathbf \\Psi} {x, \\mathbf y, \\mathbf y'; \\epsilon} } {\\partial \\epsilon} \\Bigg\\rvert_{\\epsilon \\mathop = 0}$ :$\\map \\phi {x, \\mathbf y, \\mathbf y'} = \\dfrac {\\partial \\map \\Phi {x, \\mathbf y, \\mathbf y'; \\epsilon} } {\\partial \\epsilon} \\Bigg\\rvert_{\\epsilon \\mathop = 0}$"}
+{"_id": "2528", "title": "Equivalence of Definitions of Supremum of Real-Valued Function", "text": "Let $S \\subseteq \\R$ be a subset of the real numbers. Let $f: S \\to \\R$ be a real function on $S$. {{TFAE|def = Supremum of Real-Valued Function}}"}
+{"_id": "2530", "title": "Supremum does not Precede Infimum", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $T \\subseteq S$ admit both a supremum $M$ and an infimum $m$. Then $m \\preceq M$."}
+{"_id": "2531", "title": "Integer Multiples form Commutative Ring", "text": "Let $n \\Z$ be the set of integer multiples of $n$. Then $\\struct {n \\Z, +, \\times}$ is a commutative ring. Unless $n = 1$, $\\struct {n \\Z, +, \\times}$ is not a ring with unity."}
+{"_id": "2532", "title": "Quaternions Defined by Matrices", "text": "Let $\\mathbf 1, \\mathbf i, \\mathbf j, \\mathbf k$ denote the following four elements of the matrix space $\\map {\\mathcal M_\\C} 2$: :$\\mathbf 1 = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} \\qquad \\mathbf i = \\begin{bmatrix} i & 0 \\\\ 0 & -i \\end{bmatrix} \\qquad \\mathbf j = \\begin{bmatrix} 0 & 1 \\\\ -1 & 0 \\end{bmatrix} \\qquad \\mathbf k = \\begin{bmatrix} 0 & i \\\\ i & 0 \\end{bmatrix} $ where $\\C$ is the set of complex numbers. Then $\\mathbf 1, \\mathbf i, \\mathbf j, \\mathbf k$ are related to each other in the following way: {{begin-eqn}} {{eqn | l = \\mathbf i \\mathbf j = - \\mathbf j \\mathbf i       | r = \\mathbf k }} {{eqn | l = \\mathbf j \\mathbf k = - \\mathbf k \\mathbf j       | r = \\mathbf i }} {{eqn | l = \\mathbf k \\mathbf i = - \\mathbf i \\mathbf k       | r = \\mathbf j }} {{eqn | l = \\mathbf i^2 = \\mathbf j^2 = \\mathbf k^2 = \\mathbf i \\mathbf j \\mathbf k       | r = -\\mathbf 1 }} {{end-eqn}}"}
+{"_id": "2533", "title": "Matrix Form of Quaternion", "text": "Let $\\mathbf x$ be a quaternion such that: : $\\mathbf x = a \\mathbf 1 + b \\mathbf i + c \\mathbf j + d \\mathbf k$ When the quaternion basis is expressed in the form of matrices: :$\\mathbf 1 = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} \\qquad \\mathbf i = \\begin{bmatrix} i & 0 \\\\ 0 & -i \\end{bmatrix} \\qquad \\mathbf j = \\begin{bmatrix} 0 & 1 \\\\ -1 & 0 \\end{bmatrix} \\qquad \\mathbf k = \\begin{bmatrix} 0 & i \\\\ i & 0 \\end{bmatrix}$ the general quaternion $\\mathbf x$ has the form: :$\\mathbf x = \\begin{bmatrix} a + bi & c + di \\\\ -c + di & a - bi \\end{bmatrix}$ or: :$\\mathbf x = \\begin{bmatrix} w & z \\\\ -\\overline z & \\overline w \\end{bmatrix}$ where : :$w$ and $z$ are complex numbers :$\\overline z$ is the complex conjugate of $z$."}
+{"_id": "2534", "title": "Quaternion Multiplication", "text": "Let $\\mathbf x_1 = a_1 \\mathbf 1 + b_1 \\mathbf i + c_1 \\mathbf j + d_1 \\mathbf k$ and $\\mathbf x_2 = a_2 \\mathbf 1 + b_2 \\mathbf i + c_2 \\mathbf j + d_2 \\mathbf k$ be quaternions. Then their '''product''' is given by: {{begin-eqn}} {{eqn | ll=\\mathbf x_1 \\mathbf x_2       | l==       | o=       | r=\\left({a_1 a_2 - b_1 b_2 - c_1 c_2 - d_1 d_2}\\right) \\mathbf 1       | c= }} {{eqn | o=+       | r=\\left({a_1 b_2 + b_1 a_2 + c_1 d_2 - d_1 c_2}\\right) \\mathbf i       | c= }} {{eqn | o=+       | r=\\left({a_1 c_2 - b_1 d_2 + c_1 a_2 + d_1 b_2}\\right) \\mathbf j       | c= }} {{eqn | o=+       | r=\\left({a_1 d_2 + b_1 c_2 - c_1 b_2 + d_1 a_2}\\right) \\mathbf k       | c= }} {{end-eqn}}"}
+{"_id": "2535", "title": "Quaternion Addition forms Abelian Group", "text": "Let $\\mathbb H$ be the set of quaternions. Then $\\left({\\mathbb H, +}\\right)$, where $+$ denotes quaternion addition, is an abelian group."}
+{"_id": "2536", "title": "Ring of Quaternions is Ring", "text": "The set $\\mathbb H$ of quaternions forms a ring under the operations of addition and multiplication."}
+{"_id": "2537", "title": "Quaternions Subring of Complex Matrix Space", "text": "The Ring of Quaternions is a subring of the matrix space $\\mathcal M_\\C \\left({2}\\right)$."}
+{"_id": "2538", "title": "Product of Quaternion with Conjugate", "text": "Let $\\mathbf x = a \\mathbf 1 + b \\mathbf i + c \\mathbf j + d \\mathbf k$ be a quaternion. Let $\\overline {\\mathbf x}$ be the conjugate of $\\mathbf x$. Then their product is given by: :$\\mathbf x \\overline {\\mathbf x} = \\paren {a^2 + b^2 + c^2 + d^2} \\mathbf 1 = \\overline {\\mathbf x} \\mathbf x$"}
+{"_id": "2539", "title": "Multiplicative Inverse of Quaternion", "text": "Let $\\mathbf x = a \\mathbf 1 + b \\mathbf i + c \\mathbf j + d \\mathbf k$ be a quaternion such that $\\mathbf x \\ne \\mathbf 0$. Then $\\mathbf x$ has an inverse $\\mathbf x^{-1}$ under the operation of quaternion multiplication: :$\\mathbf x^{-1} = \\lambda \\overline {\\mathbf x}$ where: :$\\lambda = \\dfrac 1 {a^2 + b^2 + c^2 + d^2} \\mathbf 1$"}
+{"_id": "2540", "title": "Multiplicative Identity for Quaternions", "text": "In the set of quaternions $\\mathbb H$, the element: :$\\mathbf 1 + 0 \\mathbf i + 0 \\mathbf j + 0 \\mathbf k$ serves as the identity element for quaternion multiplication. This element is written $\\mathbf 1$."}
+{"_id": "2541", "title": "Quaternions form Skew Field", "text": "The set $\\H$ of quaternions forms a skew field under the operations of addition and multiplication."}
+{"_id": "2542", "title": "Cancellation Law of Ring Product of Integral Domain", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose zero is $0_D$. Let $D^*$ denote $D \\setminus \\set {0_D}$, that is, $D$ without its zero. Let $a \\in D^*$. Then: :$\\forall x, y \\in D: a \\circ x = a \\circ y \\implies x = y$ That is, all elements of $D^*$ are cancellable for the ring product."}
+{"_id": "2543", "title": "Gaussian Integers form Integral Domain", "text": "The ring of Gaussian integers $\\struct {\\Z \\sqbrk i, +, \\times}$ is an integral domain."}
+{"_id": "2544", "title": "Set Intersection Not Cancellable", "text": "Let $S$ be a set and let $\\mathcal P \\left({S}\\right)$ be the power set of $S$. Let $S_1, S_2, T \\in \\mathcal P \\left({S}\\right)$. Suppose that $S_1 \\cap T = S_2 \\cap T$. Then it is not necessarily the case that $S_1 = S_2$."}
+{"_id": "2545", "title": "Sum of Triangular Matrices", "text": "Let $\\mathbf A = \\left[{a}\\right]_{n}, \\mathbf B = \\left[{b}\\right]_{n}$ be square matrices of order $n$. Let $\\mathbf C = \\mathbf A + \\mathbf B$ be the matrix entrywise sum of $\\mathbf A$ and $\\mathbf B$. If $\\mathbf A$ and $\\mathbf B$ are upper triangular matrices, then so is $\\mathbf C$. If $\\mathbf A$ and $\\mathbf B$ are lower triangular matrices, then so is $\\mathbf C$."}
+{"_id": "2546", "title": "Negative of Triangular Matrix", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $-\\mathbf A$ be the negative of $\\mathbf A$. If $\\mathbf A$ is an upper triangular matrix, then so is $-\\mathbf A$. If $\\mathbf A$ is a lower triangular matrix, then so is $-\\mathbf A$."}
+{"_id": "2548", "title": "Natural Numbers form Subsemiring of Integers", "text": "The semiring of natural numbers $\\struct {\\N, +, \\times}$ forms a subsemiring of the ring of integers $\\struct {\\Z, +, \\times}$."}
+{"_id": "2549", "title": "Integers form Subdomain of Rationals", "text": "The integral domain of integers $\\left({\\Z, +, \\times}\\right)$ forms a subdomain of the field of rational numbers."}
+{"_id": "2550", "title": "Real Numbers form Subfield of Complex Numbers", "text": "The field of real numbers $\\struct {\\R, +, \\times}$ forms a subfield of the field of complex numbers $\\struct {\\C, +, \\times}$."}
+{"_id": "2551", "title": "Sum of All Ring Products is Additive Subgroup", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring. Let $\\left({S, +}\\right)$ and $\\left({T, +}\\right)$ be additive subgroups of $\\left({R, +, \\circ}\\right)$. Let $S + T$ be defined as subset product. Let $S T$ be defined as: :$\\displaystyle S T = \\left\\{{\\sum_{i \\mathop = 1}^n s_i \\circ t_i: s_1 \\in S, t_i \\in T, i \\in \\left[{1 \\,.\\,.\\ n}\\right]}\\right\\}$ Then both $S + T$ and $S T$ are additive subgroups of $\\left({R, +, \\circ}\\right)$."}
+{"_id": "2552", "title": "Sum of All Ring Products is Closed under Addition", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring. Let $\\left({S, +}\\right)$ and $\\left({T, +}\\right)$ be additive subgroups of $\\left({R, +, \\circ}\\right)$. Let $S T$ be defined as: :$\\displaystyle S T = \\left\\{{\\sum_{i \\mathop = 1}^n s_i \\circ t_i: s_1 \\in S, t_i \\in T, i \\in \\left[{1 \\,.\\,.\\ n}\\right]}\\right\\}$ Then $\\left({S T, +}\\right)$ is a closed subset of $\\left({R, +}\\right)$."}
+{"_id": "2555", "title": "Complex Numbers form Subfield of Quaternions", "text": "The field of complex numbers $\\left({\\C, +, \\times}\\right)$ is isomorphic to the subfields of the quaternions $\\left({\\mathbb H, +, \\times}\\right)$ whose underlying subsets are: :$(1): \\quad \\mathbb H_\\mathbf i = \\left\\{{a \\mathbf 1 + b \\mathbf i + c \\mathbf j + d \\mathbf k \\in \\mathbb H: c = d = 0}\\right\\}$ :$(2): \\quad \\mathbb H_\\mathbf j = \\left\\{{a \\mathbf 1 + b \\mathbf i + c \\mathbf j + d \\mathbf k \\in \\mathbb H: b = d = 0}\\right\\}$ :$(3): \\quad \\mathbb H_\\mathbf k = \\left\\{{a \\mathbf 1 + b \\mathbf i + c \\mathbf j + d \\mathbf k \\in \\mathbb H: b = c = 0}\\right\\}$ That is: :$(1): \\quad \\mathbb H_\\mathbf i = \\left\\{{a \\mathbf 1 + b \\mathbf i \\in \\mathbb H}\\right\\}$ :$(2): \\quad \\mathbb H_\\mathbf j = \\left\\{{a \\mathbf 1 + c \\mathbf j \\in \\mathbb H}\\right\\}$ :$(3): \\quad \\mathbb H_\\mathbf k = \\left\\{{a \\mathbf 1 + d \\mathbf k \\in \\mathbb H}\\right\\}$"}
+{"_id": "2556", "title": "Ring Homomorphism whose Kernel contains Ideal", "text": "Let $R$ be a ring. Let $J$ be an ideal of $R$. Let $\\nu: R \\to R / J$ be the quotient epimorphism. Let $\\phi: R \\to S$ be a ring homomorphism such that: :$J \\subseteq \\map \\ker \\phi$ where $\\map \\ker \\phi$ is the kernel of $\\phi$. Then there exists a unique ring homomorphism $\\psi: R / J \\to S$ such that: :$\\phi = \\psi \\circ \\nu$ where $\\circ$ denotes composition of mappings. :300px Also: :$\\map \\ker \\psi = \\map \\ker \\phi / J$"}
+{"_id": "2557", "title": "Decay Equation", "text": "The first order ordinary differential equation: :$\\dfrac {\\d y} {\\d x} = k \\paren {y_a - y}$ where $k \\in \\R: k > 0$ has the general solution: :$y = y_a + C e^{-k x}$ where $C$ is an arbitrary constant. If $y = y_0$ at $x = 0$, then: :$y = y_a + \\paren {y_0 - y_a} e^{-k x}$ This differential equation is known as the '''decay equation'''."}
+{"_id": "2558", "title": "Relation on Empty Set is Equivalence", "text": "Let $S = \\varnothing$, that is, the empty set. Let $\\mathcal R \\subseteq S \\times S$ be a relation on $S$. Then $\\mathcal R$ is the null relation and is an equivalence relation."}
+{"_id": "2560", "title": "Increasing Union of Subrings is Subring", "text": "Let $R$ be a ring. Let $S_0 \\subseteq S_1 \\subseteq S_2 \\subseteq \\ldots \\subseteq S_i \\subseteq \\ldots$ be subrings of $R$. Then the increasing union $S$: :$\\displaystyle S = \\bigcup_{i \\mathop \\in \\N} S_i$ is a subring of $R$."}
+{"_id": "2561", "title": "Subrings of Integers are Sets of Integer Multiples", "text": "Let $\\struct {\\Z, +, \\times}$ be the integral domain of integers. The subrings of $\\struct {\\Z, +, \\times}$ are the rings of integer multiples: :$\\struct {n \\Z, +, \\times}$ where $n \\in \\Z: n \\ge 0$. There are no other subrings of $\\struct {\\Z, +, \\times}$ but these."}
+{"_id": "2562", "title": "Subring of Integers is Ideal", "text": "Every subring of $\\struct {\\Z, +, \\times}$ is an ideal of the ring $\\struct {\\Z, +, \\times}$."}
+{"_id": "2563", "title": "Polynomial Ring of Sequences is Ring", "text": "Let $R$ be a ring. Let $P \\left[{R}\\right]$ be the polynomial ring over sequences in $R$. Then $P \\left[{R}\\right]$ is itself a ring."}
+{"_id": "2566", "title": "Integer Addition is Closed", "text": "The set of integers is closed under addition: :$\\forall a, b \\in \\Z: a + b \\in \\Z$"}
+{"_id": "2567", "title": "Integer Addition Identity is Zero", "text": "The identity of integer addition is $0$: :$\\exists 0 \\in \\Z: \\forall a \\in \\Z: a + 0 = a = 0 + a$"}
+{"_id": "2568", "title": "Inverses for Integer Addition", "text": "Each element $x$ of the set of integers $\\Z$ has an inverse element $-x$ under the operation of integer addition: :$\\forall x \\in \\Z: \\exists -x \\in \\Z: x + \\paren {-x} = 0 = \\paren {-x} + x$"}
+{"_id": "2569", "title": "Rational Numbers form Ring", "text": "The set of rational numbers $\\Q$ forms a ring under addition and multiplication: $\\struct {\\Q, +, \\times}$."}
+{"_id": "2570", "title": "Real Numbers form Ring", "text": "The set of real numbers $\\R$ forms a ring under addition and multiplication: $\\struct {\\R, +, \\times}$."}
+{"_id": "2571", "title": "Complex Numbers form Ring", "text": "The set of complex numbers $\\C$ forms a ring under addition and multiplication: $\\struct {\\C, +, \\times}$."}
+{"_id": "2572", "title": "Complex Addition Identity is Zero", "text": "Let $\\C$ be the set of complex numbers. The identity element of $\\struct {\\C, +}$ is the complex number $0 + 0 i$."}
+{"_id": "2573", "title": "Rational Addition Identity is Zero", "text": "The identity of rational number addition is $0$: :$\\exists 0 \\in \\Q: \\forall a \\in \\Q: a + 0 = a = 0 + a$"}
+{"_id": "2574", "title": "Inverse for Complex Addition", "text": "Let $z = x + i y \\in \\C$ be a complex number. Let $-z = -x - i y \\in \\C$ be the negative of $z$. Then $-z$ is the inverse element of $z$ under the operation of complex addition: :$\\forall z \\in \\C: \\exists -z \\in \\C: z + \\paren {-z} = 0 = \\paren {-z} + z$"}
+{"_id": "2575", "title": "Equal Elements of Field of Quotients", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose zero is $0_D$. Let $\\struct {K, +, \\circ}$ be the field of quotients of $\\struct {D, +, \\circ}$. Let $x = \\dfrac p q \\in K$. Then: :$\\forall k \\in D^*: x = \\dfrac {p \\circ k} {q \\circ k}$ where: :$D^* := D \\setminus \\set {0_D}$ that is, $D$ with its zero removed."}
+{"_id": "2576", "title": "Rational Addition is Closed", "text": "The operation of addition on the set of rational numbers $\\Q$ is well-defined and closed: :$\\forall x, y \\in \\Q: x + y \\in \\Q$"}
+{"_id": "2578", "title": "Rational Multiplication is Closed", "text": "The operation of multiplication on the set of rational numbers $\\Q$ is well-defined and closed: :$\\forall x, y \\in \\Q: x \\times y \\in \\Q$"}
+{"_id": "2579", "title": "Rational Multiplication Identity is One", "text": "The identity of rational number multiplication is $1$: :$\\exists 1 \\in \\Q: \\forall a \\in \\Q: a \\times 1 = a = 1 \\times a$"}
+{"_id": "2580", "title": "Inverses for Rational Multiplication", "text": "Each element $x$ of the set of non-zero rational numbers $\\Q^*$ has an inverse element $\\dfrac 1 x$ under the operation of rational number multiplication: :$\\displaystyle \\forall x \\in \\Q^*: \\exists \\frac 1 x \\in \\Q^*: x \\times \\frac 1 x = 1 = \\frac 1 x \\times x$"}
+{"_id": "2581", "title": "Real Addition Identity is Zero", "text": "The identity of real number addition is $0$: :$\\exists 0 \\in \\R: \\forall x \\in \\R: x + 0 = x = 0 + x$"}
+{"_id": "2582", "title": "Inverses for Real Addition", "text": "Each element $x$ of the set of real numbers $\\R$ has an inverse element $-x$ under the operation of real number addition: :$\\forall x \\in \\R: \\exists -x \\in \\R: x + \\paren {-x} = 0 = \\paren {-x} + x$"}
+{"_id": "2583", "title": "Real Multiplication Identity is One", "text": "The identity element of real number multiplication is the real number $1$: :$\\exists 1 \\in \\R: \\forall a \\in \\R_{\\ne 0}: a \\times 1 = a = 1 \\times a$"}
+{"_id": "2584", "title": "Inverses for Real Multiplication", "text": "Each element $x$ of the set of non-zero real numbers $\\R_{\\ne 0}$ has an inverse element $\\dfrac 1 x$ under the operation of real number multiplication: :$\\forall x \\in \\R_{\\ne 0}: \\exists \\dfrac 1 x \\in \\R_{\\ne 0}: x \\times \\dfrac 1 x = 1 = \\dfrac 1 x \\times x$"}
+{"_id": "2585", "title": "Complex Multiplication Identity is One", "text": "Let $\\C_{\\ne 0}$ be the set of complex numbers without zero. The identity element of $\\struct {\\C_{\\ne 0}, \\times}$ is the complex number $1 + 0 i$."}
+{"_id": "2586", "title": "Inverse for Complex Multiplication", "text": "Each element $z = x + i y$ of the set of non-zero complex numbers $\\C_{\\ne 0}$ has an inverse element $\\dfrac 1 z$ under the operation of complex multiplication: :$\\forall z \\in \\C_{\\ne 0}: \\exists \\dfrac 1 z \\in \\C_{\\ne 0}: z \\times \\dfrac 1 z = 1 + 0 i = \\dfrac 1 z \\times z$ This inverse can be expressed as: :$\\dfrac 1 z = \\dfrac {x - i y} {x^2 + y^2} = \\dfrac {\\overline z} {z \\overline z}$ where $\\overline z$ is the complex conjugate of $z$."}
+{"_id": "2587", "title": "Additive Group of Integers is Normal Subgroup of Complex", "text": "Let $\\struct {\\Z, +}$ be the additive group of integers. Let $\\struct {\\C, +}$ be the additive group of complex numbers. Then $\\struct {\\Z, +}$ is a normal subgroup of $\\struct {\\C, +}$."}
+{"_id": "2590", "title": "Rational Numbers form Subfield of Complex Numbers", "text": "Let $\\struct {\\Q, +, \\times}$ denote the field of rational numbers. Let $\\struct {\\C, +, \\times}$ denote the field of complex numbers. $\\struct {\\Q, +, \\times}$ is a subfield of $\\struct {\\C, +, \\times}$."}
+{"_id": "2591", "title": "Subfield of Subfield is Subfield", "text": "Let $R$ be a ring with unity. Let $K_1, K_2$ be fields, such that: : $K_1$ is a subfield of $R$ : $K_2$ is a subfield of $K_1$ Then $K_2$ is a subfield of $R$."}
+{"_id": "2593", "title": "Characterisation of Jacobson Radical", "text": "Let $A$ be a commutative ring with unity. Let $A^\\times$ be the group of units of $A$. Let $\\map {\\operatorname {Jac} } A$ be the Jacobson radical of $A$. Then: :$\\map {\\operatorname {Jac} } A = \\set {a \\in A: 1_A - a x \\in A^\\times \\text{ for all } x \\in A}$ where $1_A$ is the unity of $A$."}
+{"_id": "2594", "title": "Prime Ideal iff Quotient Ring is Integral Domain", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $J$ be an ideal of $R$. Then $J$ is a prime ideal of $R$ {{iff}} the quotient ring $R / J$ is an integral domain."}
+{"_id": "2595", "title": "Maximal Ideal of Commutative and Unitary Ring is Prime Ideal", "text": "Let $R$ be a commutative ring with unity. Let $M$ be a maximal ideal of $R$. Then $M$ is a prime ideal of $R$."}
+{"_id": "2596", "title": "Radical Ideal iff Quotient Ring is Reduced", "text": "Let $\\left({R, +, \\circ}\\right)$ be a commutative ring with unity. Let $J$ be an ideal of $R$. Then $J$ is a radical ideal {{iff}} the quotient ring $R / J$ is a reduced ring."}
+{"_id": "2597", "title": "Properties of Degree", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose zero is $0_R$. Let $R \\sqbrk X$ denote the ring of polynomial forms over $R$ in the indeterminate $X$. For $f \\in R \\sqbrk X$ let $\\map \\deg f$ denote the degree of $f$. Then the following hold:"}
+{"_id": "2599", "title": "Set of Monomials is Closed Under Multiplication", "text": "Let $M$ be the set of all monomials on the set $\\set {X_j: j \\in J}$, with multiplication $\\circ$ defined by: :$\\ds \\paren {\\prod_{j \\mathop \\in J} X_j^{k_j} } \\circ \\paren {\\prod_{j \\mathop \\in J} X_j^{k_j'} } = \\paren {\\prod_{j \\mathop \\in J} X_j^{k_j + k_j'} }$ Then $M$ is closed under $\\circ$."}
+{"_id": "2603", "title": "Polynomial Addition is Commutative", "text": "Addition of polynomials is commutative."}
+{"_id": "2604", "title": "Multiplication of Polynomials is Associative", "text": "Multiplication of polynomials is associative."}
+{"_id": "2605", "title": "Polynomials Contain Multiplicative Identity", "text": "The set of polynomials has a multiplicative identity."}
+{"_id": "2606", "title": "Multiplication of Polynomials is Commutative", "text": "Multiplication of polynomials is commutative."}
+{"_id": "2607", "title": "Multiplication of Polynomials Distributes over Addition", "text": "Multiplication of polynomials is left- and right- distributive over addition."}
+{"_id": "2608", "title": "Distributivity is Preserved in Induced Structure", "text": "Let $\\struct {T, \\oplus, \\otimes}$ be an algebraic structure, and let $S$ be a set. Let $T^S$ denote the set of all mappings from $S$ to $T$. Let $\\struct {T^S, \\oplus}$ be the structure on $T^S$ induced by $\\oplus$. Let $\\struct {T^S, \\otimes}$ be the structure on $T^S$ induced by $\\otimes$. If $\\otimes$ is distributive over $\\oplus$, then the pointwise operation induced on $T^S$ by $\\otimes$ is distributive over the operation induced by $\\oplus$."}
+{"_id": "2609", "title": "Dirichlet Series Absolute Convergence Lemma", "text": "Let $\\displaystyle \\map f s = \\sum_{n \\mathop = 1}^\\infty \\frac {a_n} {n^s}$ be a Dirichlet series. Suppose that $f$ converges absolutely at $s_0 = \\sigma_0 + i t_0 \\in \\C$. Then $f$ converges absolutely at all points $s = \\sigma + i t \\in \\C$ with $\\sigma \\ge \\sigma_0$."}
+{"_id": "2610", "title": "Existence of Abscissa of Absolute Convergence", "text": "Let $\\displaystyle f \\left({s}\\right) = \\sum_{n \\mathop = 1}^\\infty a_n n^{-s}$ be a Dirichlet series. Let the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\left\\vert { a_n n^{-s} } \\right\\vert$ not converge for all $s \\in \\C$, or diverge for all $s \\in \\C$. Then there exists a real number $\\sigma_a$ such that $f \\left({s}\\right)$ converges absolutely for all $s = \\sigma + it$ with $\\sigma > \\sigma_a$, and does not converge absolutely for all $s$ with $\\sigma < \\sigma_a$. We call $\\sigma_a$ the '''abscissa of absolute convergence''' of the Dirichlet series."}
+{"_id": "2611", "title": "Dirichlet Series Convergence Lemma", "text": "Let $\\displaystyle f \\left({s}\\right) = \\sum_{n \\mathop = 1}^\\infty \\frac {a_n} {n^s}$ be a Dirichlet series. Let $f \\left({s}\\right)$ converge at $s_0 = \\sigma_0 + i t_0$. Then $f \\left({s}\\right)$ converge for all $s = \\sigma + i t$ where $\\sigma > \\sigma_0$."}
+{"_id": "2612", "title": "Existence of Abscissa of Convergence/General", "text": "Let $s = \\sigma + i t$ Let $\\displaystyle \\map f s = \\sum_{n \\mathop = 1}^\\infty a_n e^{-\\lambda_n s}$ be a general Dirichlet series. Then there exists a extended real number, $\\sigma_0$, such that :$(1): \\quad$ For $\\sigma < \\sigma_0$, $\\map f s$ diverges :$(2): \\quad$ For $\\sigma > \\sigma_0$, $\\map f s$ converges."}
+{"_id": "2613", "title": "Difference of Abscissae of Convergence", "text": "Let $\\displaystyle \\map f s = \\sum_{n \\mathop = 1}^\\infty a_n n^{-s}$ be a Dirichlet series. Suppose that $\\map f s$ has finite Abscissa of Convergence $\\sigma_c$. Then the Abscissa of Absolute Convergence $\\sigma_a$ is finite, and: :$0 \\le \\sigma_a - \\sigma_c \\le 1$"}
+{"_id": "2614", "title": "Completion Theorem (Metric Space)", "text": "Let $M = \\struct {A, d}$ be a metric space. Then there exists a completion $\\tilde M = \\struct {\\tilde A, \\tilde d}$ of $\\struct {A, d}$. Moreover, this completion is unique up to isometry. That is, if $\\struct {\\hat A, \\hat d}$ is another completion of $\\struct {A, d}$, then there is a bijection $\\tau: \\tilde A \\leftrightarrow \\hat A$ such that: :$(1): \\quad \\tau$ restricts to the identity on $x$: :::: $\\forall x \\in A: \\map \\tau x = x$ :$(2): \\quad \\tau$ preserves metrics: :::: $\\forall x_1, x_2 \\in A : \\map {\\hat d} {\\map \\tau {x_1}, \\map \\tau {x_2} } = \\map {\\tilde d} {x_1, x_2}$"}
+{"_id": "2615", "title": "Completeness Criterion (Metric Spaces)", "text": "Let $M = \\struct {S, d}$ be a metric space. Let $A \\subseteq S$ be a dense subset. Suppose that every Cauchy sequence in $A$ converges in $M$. Then $M$ is complete."}
+{"_id": "2616", "title": "Natural Number Addition is Closed", "text": "The operation of addition on the set of natural numbers $\\N$ is closed: :$\\forall x, y \\in \\N: x + y \\in \\N$"}
+{"_id": "2617", "title": "Properties of Ordered Field", "text": "Let $\\struct {F, +, \\cdot}$ be an ordered field with unity $1$, zero $0$. Denote the strict order by $<$ and the weak order by $\\le$. Let $\\Char F$ denote the characteristic of $F$. Then the following hold for all $x, y, z \\in k$: :$(1): \\quad x < 0 \\iff -x > 0$ :$(2): \\quad x > y \\iff x-y > 0$ :$(3): \\quad x < y \\iff -x > -y$ :$(4): \\quad \\paren {z < 0} \\land \\paren {x < y} \\implies x z > y z$ :$(5): \\quad x \\ne 0 \\implies x^2 > 0$ :$(6): \\quad 1 > 0$ :$(7): \\quad \\Char k = 0$ :$(8): \\quad x > y > 0 \\iff y^{-1} > x^{-1} > 0$"}
+{"_id": "2618", "title": "Characterisation of Ordered Fields", "text": "Let $\\struct {k, +, \\cdot}$ be a field with unity $1$ and zero $0$. Then the following are equivalent: :$(1): \\quad$ There exists a total ordering $\\le$ on $k$ such that $\\struct {k, \\le}$ is an ordered field :$(2): \\quad$ $-1$ cannot be written as a sum of squares of elements of $k$ :$(3): \\quad$ $0$ cannot be written as a non-empty sum of squares of non-zero elements of $k$"}
+{"_id": "2619", "title": "Hahn-Banach Theorem", "text": "Let $E$ be a vector space over $\\R$. Let $p: E \\to \\R$ be a Minkowski functional. Let $G \\subseteq E$ be a linear subspace of $E$. Let $f: G \\to \\R$ be a linear functional such that: :$\\forall x \\in G: \\map f x \\le \\map p x$ Then there exists a linear functional $\\tilde f$ defined on the whole space $E$ which extends $f$. That is: :$\\forall x \\in G: \\map {\\tilde f} x = \\map f x$ such that: :$\\forall x \\in E: \\map {\\tilde f} x \\le \\map p x$"}
+{"_id": "2621", "title": "Hardy-Littlewood Circle Method", "text": "Let $\\AA$ be a subset of the non-negative integers. Let: :$\\displaystyle \\map T s = \\sum_{a \\mathop \\in \\AA} s^a$ be the generating function for $\\AA$. For $N \\in \\N$, let $\\map {r_{\\AA, \\ell} } N$ be the number of solutions $\\tuple {x_1, \\ldots, x_\\ell} \\in \\AA^\\ell$ to the equation: :$x_1 + \\cdots + x_\\ell = N$ Then: :$\\displaystyle \\forall \\rho \\in \\openint 0 1: \\map {r_{\\AA, \\ell} } N = \\oint_{\\size s \\mathop = \\rho} \\frac {\\map T s^\\ell} {s^{N + 1} } \\rd s$"}
+{"_id": "2622", "title": "Vinogradov Circle Method", "text": "$\\newcommand{\\A}{\\mathcal A}$Let $\\A$ be a subset of the non-negative integers. For $\\alpha \\in \\R$, let: :$e \\left({\\alpha}\\right) := \\exp \\left({2 \\pi i \\alpha} \\right)$ Let: :$\\displaystyle T_N \\left({s}\\right) = \\sum_{\\substack {a \\mathop \\in \\A \\\\ a \\mathop \\le N}} s^a$ be the truncated generating function for $\\A$. {{explain|\"truncated\" generating function}} Let: :$V_N \\left({\\alpha}\\right) := T_N \\left({e \\left({\\alpha}\\right)}\\right)$ Let $r_{\\A, \\ell} \\left({N}\\right)$ be the number of solutions $\\left({x_1, \\dotsc, x_\\ell}\\right) \\in \\A^\\ell$ to the equation: :$(1): \\quad \\displaystyle x_1 + \\cdots + x_l = N$ Then: :$\\displaystyle r_{\\A, \\ell} \\left({N}\\right) = \\int_0^1 V_N \\left({\\alpha}\\right)^\\ell e \\left({-N \\alpha}\\right) \\, \\mathrm d \\alpha$"}
+{"_id": "2623", "title": "Euler Product", "text": "Let $a_n : \\N \\to \\C$ be an arithmetic function. Let $\\displaystyle f \\left({s}\\right) = \\sum_{n \\mathop \\in \\N} a_n n^{-s}$ be its Dirichlet series. Let $\\sigma_a$ be its abscissa of absolute convergence. Then for $\\Re \\left({s}\\right) > \\sigma_a$: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n n^{-s} = \\prod_p \\frac 1 {1 - a_p p^{-s} }$ where $p$ ranges over the primes. This representation for $f$ is called an '''Euler product''' for the Dirichlet series. {{stub|Completely multiplicative hypothesis not mentioned. Needs also the statement: $\\displaystyle f \\left({s}\\right) {{=}} \\prod_p \\left\\{{\\sum_{k \\mathop \\ge 1} a_{p^k} p^{-k s}}\\right\\}$ or however it goes for multiplicative functions which are not completely multiplicative}}"}
+{"_id": "2624", "title": "Correspondence between Linear Group Actions and Linear Representations", "text": "Let $\\left({k, +, \\cdot}\\right)$ be a field. Let $V$ be a vector space over $k$ of finite dimension. Let $\\left({G, *}\\right)$ be a finite group. There is a one-to-one correspondence between linear group actions of $G$ on $V$ and linear representations of $G$ in $V$, as follows: Let $\\phi : G \\times V \\to V$ be a group action. Let $\\rho : G \\to \\operatorname{Sym}(V)$ be a permutation representation of $G$ on $V$. The following are equivalent: :$(1): \\quad$ $\\rho$ is the permutation representation associated to $\\phi$ :$(2): \\quad$ $\\phi$ is the group action associated to $\\rho$ If this is the case, the following are equivalent: :$(1): \\quad$ $\\rho$ is a linear representation :$(2): \\quad$ $\\phi$ is a linear group action"}
+{"_id": "2625", "title": "Set Difference with Disjoint Set", "text": "Let $S, T$ be sets. Then: :$S \\cap T = \\O \\iff S \\setminus T = S$ where: :$S \\cap T$ denotes set intersection :$\\O$ denotes the empty set :$S \\setminus T$ denotes set difference."}
+{"_id": "2626", "title": "Integer Subtraction is Closed", "text": "The set of integers is closed under subtraction: :$\\forall a, b \\in \\Z: a - b \\in \\Z$"}
+{"_id": "2627", "title": "Rational Subtraction is Closed", "text": "The set of rational numbers is closed under subtraction: :$\\forall a, b \\in \\Q: a - b \\in \\Q$"}
+{"_id": "2628", "title": "Real Subtraction is Closed", "text": "The set of real numbers is closed under subtraction: :$\\forall a, b \\in \\R: a - b \\in \\R$"}
+{"_id": "2629", "title": "Complex Subtraction is Closed", "text": "The set of complex numbers is closed under subtraction: :$\\forall a, b \\in \\C: a - b \\in \\C$"}
+{"_id": "2630", "title": "Positive Real Numbers Closed under Division", "text": "The set $\\R_{>0}$ of strictly positive real numbers is closed under division: :$\\forall a, b \\in \\R_{>0}: a \\div b \\in \\R_{>0}$"}
+{"_id": "2631", "title": "Rational Numbers form Integral Domain", "text": "The set of rational numbers $\\Q$ forms an integral domain under addition and multiplication: $\\struct {\\Q, +, \\times}$."}
+{"_id": "2632", "title": "Real Numbers form Integral Domain", "text": "The set of real numbers $\\R$ forms an integral domain under addition and multiplication: $\\struct {\\R, +, \\times}$."}
+{"_id": "2633", "title": "Properties of Content", "text": "Let $f, g \\in \\Q \\sqbrk X$ and $q \\in \\Q$. The content of a polynomial satisfies the following:"}
+{"_id": "2634", "title": "Complex Numbers form Integral Domain", "text": "The set of complex numbers $\\C$ forms an integral domain under addition and multiplication: $\\struct {\\C, +, \\times}$."}
+{"_id": "2635", "title": "Root of Polynomial iff Divisible by Minimal Polynomial", "text": "Let $K$ be a field. Let $L / K$ be a field extension of $K$. Let $\\alpha \\in L$ be algebraic over $K$. Then there is a unique monic polynomial $\\mu_\\alpha \\in K \\left[{X}\\right]$ of least degree such that $\\mu_\\alpha \\left({\\alpha}\\right) = 0$. Moreover $f \\in K \\left[{X}\\right]$ is such that $f \\left({\\alpha}\\right) = 0$ {{iff}} $\\mu_\\alpha$ divides $f$. {{explain|Link to an appropriate page defining divisibility among polynomials.}}"}
+{"_id": "2636", "title": "Gauss's Lemma on Unique Factorization Domains", "text": "Let $R$ be a unique factorization domain. Then the ring of polynomials $R \\sqbrk X$ is also a unique factorization domain."}
+{"_id": "2638", "title": "Gröbner Basis", "text": "Let $F$ be a finite set of polynomials. {{explain|Explain what the underlying field of the polynomials is. Real numbers, abstract rings, general fields? Link to the appropriate definition for the case.}} Let $SP \\left({f_1, f_2}\\right)$ be the $S$-polynomial of $f_1$ and $f_2$. Let $g$ be a polynomial. Let $RF \\left({g}\\right)$ be the reduced form of $g$. Then $F$ is a Gröbner basis {{iff}}: :$\\forall f_1, f_2 \\in F \\left({RF \\left({F, SP \\left({f_1, f_2}\\right)}\\right) = 0}\\right)$ {{explain|The meaning of the above needs to be made clear. Presumably it should be interpreted something like: $\\forall f_1, f_2 \\in F: RF \\left({F, SP \\left({f_1, f_2}\\right)}\\right) {{=}} 0$. But further to this: $RF$ as defined takes a single argument, a polynomial (still to be defined upon what domain). As this statement stands, $RF$ seems to take two parameters: $F$ and $SP$. Clarification is needed.}}"}
+{"_id": "2639", "title": "Numbers of Type Integer a plus b root 2 form Subdomain of Reals", "text": "Let $\\Z \\sqbrk {\\sqrt 2}$ denote the set: :$\\Z \\sqbrk {\\sqrt 2} := \\set {a + b \\sqrt 2: a, b \\in \\Z}$ That is, all numbers of the form $a + b \\sqrt 2$ where $a$ and $b$ are integers. Then the algebraic structure: :$\\struct {\\Z \\sqbrk {\\sqrt 2}, +, \\times}$ where $+$ and $\\times$ are conventional addition and multiplication on real numbers, form an integral subdomain of the real numbers $\\R$."}
+{"_id": "2640", "title": "Sufficient Conditions for Uncountability", "text": "Let $X$ be a set. The following are equivalent: :$(1): \\quad X$ contains an uncountable subset :$(2): \\quad X$ is uncountable :$(3): \\quad $ Every sequence of distinct points $\\sequence {x_n}_{n \\mathop \\in \\N}$ in $X$ omits at least one $x \\in X$ :$(4): \\quad $ There is no surjection $\\N \\twoheadrightarrow X$ :$(5): \\quad X$ is infinite and there is no bijection $X \\leftrightarrow \\N$ Assuming the Continuum Hypothesis holds, we also have the equivalent uncountability condition: :$(6): \\quad $There exist extended real numbers $a < b$ and a surjection $X \\to \\closedint a b$"}
+{"_id": "2642", "title": "Square of Non-Zero Element of Ordered Integral Domain is Strictly Positive", "text": "Let $\\struct {D, +, \\times, \\le}$ be an ordered integral domain whose zero is $0_D$. Then: :$\\forall x \\in D: x \\ne 0_D \\iff \\map P {x \\times x}$ where $\\map P {x \\times x}$ denotes that $x \\times x$ has the (strict) positivity property. That is, the square of any element of an ordered integral domain is (strictly) positive {{iff}} that element is non-zero."}
+{"_id": "2643", "title": "Simultaneous Equation With Two Unknowns", "text": "A pair of simultaneous linear equations of the form: {{begin-eqn}} {{eqn | l=a x + b y       | r=c }} {{eqn | l=d x + e y       | r=f }} {{end-eqn}} where $ ae \\ne b d$, has as its only solution: {{begin-eqn}} {{eqn | l=x       | r= \\frac {c e - b f} {a e - b d} }} {{eqn | l=y       | r=\\frac {a f - c d} {a e - b d} }} {{end-eqn}}"}
+{"_id": "2644", "title": "Open Sets in Real Number Line", "text": "Every non-empty open set $I \\subseteq \\R$ can be expressed as a countable union of pairwise disjoint open intervals. If: :$\\displaystyle I = \\bigcup_{n \\mathop \\in \\N} J_n$ :$\\displaystyle I = \\bigcup_{n \\mathop \\in \\N} K_n$ are two such expressions, then there exists a permutation $\\sigma$ of $\\N$ such that: :$\\forall n \\in \\N: J_n = K_{\\map \\sigma n}$"}
+{"_id": "2646", "title": "Unity of Ordered Integral Domain is Strictly Positive", "text": "Let $\\struct {D, +, \\times \\le}$ be an ordered integral domain whose unity is $1_D$. Then: :$\\map P {1_D}$ where $P$ is the (strict) positivity property."}
+{"_id": "2647", "title": "Integers form Ordered Integral Domain", "text": "The integers $\\Z$ form an ordered integral domain under addition and multiplication."}
+{"_id": "2648", "title": "Rational Numbers form Ordered Integral Domain", "text": "The rational numbers $\\Q$ form an ordered integral domain under addition and multiplication."}
+{"_id": "2649", "title": "Real Numbers form Ordered Integral Domain", "text": "The set of real numbers $\\R$ forms an ordered integral domain under addition and multiplication: $\\struct {\\R, +, \\times, \\le}$."}
+{"_id": "2651", "title": "General Positivity Rule in Ordered Integral Domain", "text": "Let $\\struct {D, +, \\times}$ be an ordered integral domain, whose (strict) positivity property is denoted $P$. Let $S \\subset D$ be a subset of $D$ such that: :$\\forall s \\in S: \\map P x$ Then the following are true: :$\\displaystyle \\forall n \\in \\N_{>0}: \\forall s_i \\in S: \\map P {\\sum_{i \\mathop = 1}^n s_i}$ :$\\displaystyle \\forall n \\in \\N_{>0}: \\forall s_i \\in S: \\map P {\\prod_{i \\mathop = 1}^n s_i}$ where: :$\\displaystyle \\sum_{i \\mathop = 1}^n s_i = s_1 + s_2 + \\cdots + s_n$ :$\\displaystyle \\prod_{i \\mathop = 1}^n s_i = s_1 \\times s_2 \\times \\cdots \\times s_n$ That is, the ring sum and ring product of any number of elements in $D$ which have the (strict) positivity property also have the (strict) positivity property."}
+{"_id": "2652", "title": "Finite Integral Domain cannot be Ordered", "text": "Let $\\struct {D, +, \\times}$ be an integral domain whose unity is $1_D$ and whose order is finite. Then $D$ cannot be an ordered integral domain."}
+{"_id": "2653", "title": "Ring of Integers Modulo m cannot be Ordered Integral Domain", "text": "Let $m \\in \\Z: m \\ge 2$. Let $\\struct {\\Z_m, +, \\times}$ be the ring of integers modulo $m$. Then $\\struct {\\Z_m, +, \\times}$ cannot be an ordered integral domain."}
+{"_id": "2654", "title": "Strict Positivity Property induces Total Ordering", "text": "Let $\\struct {D, +, \\times}$ be an integral domain whose zero is $0_D$. Let $D$ be endowed with a (strict) positivity property $P: D \\to \\set {\\mathrm T, \\mathrm F}$. Then there exists a total ordering $\\le$ on $\\struct {D, +, \\times}$ induced by $P$ which is compatible with the ring structure of $\\struct {D, +, \\times}$."}
+{"_id": "2655", "title": "Relation Induced by Strict Positivity Property is Transitive", "text": "Let $\\struct {D, +, \\times}$ be an ordered integral domain where $P$ is the (strict) positivity property. Let the relation $<$ be defined on $D$ as: :$\\forall a, b \\in D: a < b \\iff \\map P {-a + b}$ Then $<$ is a transitive relation."}
+{"_id": "2656", "title": "Relation Induced by Strict Positivity Property is Compatible with Addition", "text": "Let $\\struct {D, +, \\times}$ be an ordered integral domain where $P$ is the (strict) positivity property. Let the relation $<$ be defined on $D$ as: :$\\forall a, b \\in D: a < b \\iff \\map P {-a + b}$ Then $<$ is compatible with $+$, that is: :$\\forall x, y, z \\in D: x < y \\implies \\paren {x + z} < \\paren {y + z}$ :$\\forall x, y, z \\in D: x < y \\implies \\paren {z + x} < \\paren {z + y}$"}
+{"_id": "2657", "title": "Relation Induced by Strict Positivity Property is Compatible with Multiplication", "text": "Let $\\struct {D, +, \\times}$ be an ordered integral domain where $P$ is the (strict) positivity property. Let the relation $<$ be defined on $D$ as: :$\\forall a, b \\in D: a < b \\iff \\map P {-a + b}$ Then $<$ is compatible with $\\times$ in the following sense: :$\\forall x, y, z \\in D: x < y, \\map P z \\implies \\paren {z \\times x} < \\paren {z \\times y}$ :$\\forall x, y, z \\in D: x < y, \\map P z \\implies \\paren {x \\times z} < \\paren {y \\times z}$"}
+{"_id": "2658", "title": "Relation Induced by Strict Positivity Property is Trichotomy", "text": "Let $\\struct {D, +, \\times}$ be an ordered integral domain where $P$ is the (strict) positivity property. Let the relation $<$ be defined on $D$ as: :$\\forall a, b \\in D: a < b \\iff \\map P {-a + b}$ Then $\\forall a, b \\in D:$ exactly one of the following conditions applies: :$a < b$ :$a = b$ :$a > b$ That is, $<$ is a trichotomy."}
+{"_id": "2659", "title": "Relation Induced by Strict Positivity Property is Asymmetric and Antireflexive", "text": "Let $\\struct {D, +, \\times}$ be an ordered integral domain where $P$ is the (strict) positivity property. Let the relation $<$ be defined on $D$ as: :$\\forall a, b \\in D: a < b \\iff \\map P {-a + b}$ Then $<$ is asymmetric and antireflexive."}
+{"_id": "2660", "title": "Properties of Strict Negativity", "text": "Let $\\struct {D, +, \\times}$ be an ordered integral domain, whose (strict) positivity property is denoted $P$. Let $\\le$ be the total ordering induced by $P$, and let $<$ be its strict total ordering counterpart. Let $N$ be the (strict) negativity property on $D$: :$\\forall a \\in D: \\map N a \\iff \\map P {-a}$ Then the following properties apply for all $a, b \\in D$:   === Strict Negativity is equivalent to Strictly Preceding Zero === {{:Strict Negativity is equivalent to Strictly Preceding Zero}}"}
+{"_id": "2662", "title": "Sum of Absolute Values on Ordered Integral Domain", "text": "Let $\\struct {D, +, \\times, \\le}$ be an ordered integral domain. For all $a \\in D$, let $\\size a$ denote the absolute value of $a$. Then: :$\\size {a + b} \\le \\size a + \\size b$"}
+{"_id": "2663", "title": "One Succeeds Zero in Well-Ordered Integral Domain", "text": "Let $\\struct {D, +, \\times, \\le}$ be a well-ordered integral domain. Let $0$ and $1$ be the zero and unity respectively of $D$. Then $0$ is the immediate predecessor of $1$: :$0 < 1$ :$\\neg \\exists a \\in D: 0 < a < 1$"}
+{"_id": "2664", "title": "Square of Element Less than Unity in Ordered Integral Domain", "text": "Let $\\struct {D, +, \\times, \\le}$ be an ordered integral domain. Let $x \\in D$ such that $0 < x < 1$. Then: : $0 < x \\times x < x$"}
+{"_id": "2665", "title": "Principle of Mathematical Induction/Well-Ordered Integral Domain", "text": "Let $\\struct {D, +, \\times, \\le}$ be a well-ordered integral domain whose zero is $0_D$. Let the unity of $D$ be $1_D$. Let $S \\subseteq D$ be such that: :$1_D \\in S$ :$a \\in S \\implies a + 1_D \\in S$ Then: :$D_{> 0_D} \\subseteq S$ where $D_{> 0_D}$ denotes all the elements $d \\in D$ such that $\\map P d$. That is, $D_{> 0_D}$ is the set of all (strictly) positive elements of $D$."}
+{"_id": "2666", "title": "Multiple of Divisor in Integral Domain Divides Multiple", "text": "Let $\\struct {D, +, \\times}$ be an integral domain. Let $a, b, c \\in D$. Let $a \\divides b$, where $\\divides$ denotes divisibility. Then $a \\times c$ is a divisor of $b \\times c$."}
+{"_id": "2667", "title": "Integer Divisor Results/One Divides all Integers", "text": "{{begin-eqn}} {{eqn | l = 1       | o = \\divides       | r = n }} {{eqn | l = -1       | o = \\divides       | r = n }} {{end-eqn}}"}
+{"_id": "2668", "title": "Integer Divisor Results/Integer Divides Itself", "text": ":$n \\divides n$"}
+{"_id": "2669", "title": "Integer Divisor Results/Integer Divides its Negative", "text": "{{begin-eqn}} {{eqn | l = n       | o = \\divides       | r = -n }} {{eqn | l = -n       | o = \\divides       | r = n }} {{end-eqn}}"}
+{"_id": "2670", "title": "Integer Divisor Results/Integer Divides its Absolute Value", "text": "{{begin-eqn}} {{eqn | l = n       | o = \\divides       | r = \\size n }} {{eqn | l = \\size n       | o = \\divides       | r = n }} {{end-eqn}} where: :$\\size n$ is the absolute value of $n$ :$\\divides$ denotes divisibility."}
+{"_id": "2671", "title": "Integer Divisor Results/Integer Divides Zero", "text": ":$n \\divides 0$"}
+{"_id": "2672", "title": "Integer Divisor Results/Divisors of Negative Values", "text": ": $m \\mathrel \\backslash n \\iff -m \\divides n \\iff m \\divides -n \\iff -m \\divides -n$"}
+{"_id": "2673", "title": "Divisors of One", "text": "The only divisors of $1$ are $1$ and $-1$. That is: :$a \\divides 1 \\iff a = \\pm 1$"}
+{"_id": "2674", "title": "Divisor Relation is Antisymmetric", "text": "''Divides'' is a antisymmetric relation on $\\Z_{>0}$, the set of positive integers. That is: :$\\forall a, b \\in \\Z_{>0}: a \\divides b \\land b \\divides a \\implies a = b$"}
+{"_id": "2675", "title": "Real Function is Continuous at Point iff Oscillation is Zero", "text": "Let $f: D \\to \\R$ be a real function where $D \\subseteq \\R$. Let $x$ be a point in $D$. Let $N_x$ be the set of open subset neighborhoods of $x$. Let $\\map {\\omega_f} x$ be the oscillation of $f$ at $x$: :$\\map {\\omega_f} x = \\displaystyle \\inf \\set {\\map {\\omega_f} I: I \\in N_x}$ where: :$\\map {\\omega_f} I = \\displaystyle \\sup \\set {\\size {\\map f y - \\map f z}: y, z \\in I \\cap D}$ Then $\\map {\\omega_f} x = 0$ {{iff}} $f$ is continuous at $x$."}
+{"_id": "2676", "title": "Numbers of Type Integer a plus b root 2 are Not a Field", "text": "Let $\\Z \\sqbrk {\\sqrt 2}$ denote the set: :$\\Z \\sqbrk {\\sqrt 2} := \\set {a + b \\sqrt 2: a, b \\in \\Z}$ that is, all (real) numbers of the form $a + b \\sqrt 2$ where $a$ and $b$ are integers. Then the algebraic structure: :$\\struct {\\Z \\sqbrk {\\sqrt 2}, +, \\times}$ where $+$ and $\\times$ are addition and multiplication on real numbers, is '''not''' a field."}
+{"_id": "2677", "title": "Numbers of Type Rational a plus b root 2 form Field", "text": "Let $\\Q \\sqbrk {\\sqrt 2}$ denote the set: :$\\Q \\sqbrk {\\sqrt 2} := \\set {a + b \\sqrt 2: a, b \\in \\Q}$ that is, all numbers of the form $a + b \\sqrt 2$ where $a$ and $b$ are rational numbers. Then the algebraic structure: :$\\struct {\\Q \\sqbrk {\\sqrt 2}, +, \\times}$ where $+$ and $\\times$ are conventional addition and multiplication on real numbers, is a field."}
+{"_id": "2678", "title": "Characteristic of Ring of Integers Modulo Prime", "text": "Let $\\struct {\\Z_p, +, \\times}$ be the ring of integers modulo $p$, where $p$ is a prime number. The characteristic of $\\struct {\\Z_p, +, \\times}$ is $p$."}
+{"_id": "2679", "title": "Gamma Function Extends Factorial", "text": ":$\\forall n \\in \\N: \\map \\Gamma {n + 1} = n!$"}
+{"_id": "2680", "title": "Uniform Limit of Analytic Functions is Analytic", "text": "Let $U$ be an open subset of $\\C$. Let $\\left\\{ {f_n}\\right\\}_{n \\mathop \\in \\N}$ be a sequence of analytic functions $f_n : U \\to \\C$. Let $\\left\\{ {f_n}\\right\\}$ converge locally uniformly to $f$ on $U$. Then $f$ is analytic."}
+{"_id": "2681", "title": "Modulus of Complex Integral", "text": "Let $\\left[{a \\,.\\,.\\, b}\\right]$ be a closed real interval. Let $f : \\left[{a \\,.\\,.\\, b}\\right] \\to \\C$ be a continuous complex function. Then: :$\\displaystyle \\left|{\\int_a^b f \\left({t}\\right) \\rd t}\\right| \\le \\int_a^b \\left|{f \\left({t}\\right)}\\right| \\rd t$ where the first integral is a complex Riemann integral, and the second integral is a definite real integral."}
+{"_id": "2682", "title": "Triangle Inequality for Contour Integrals", "text": "Let $C$ be a contour. Let $f: \\Img C \\to \\C$ be a continuous complex function, where $\\Img C$ denotes the image of $C$. Then: :$\\displaystyle \\size {\\int_C \\map f z \\rd z} \\le \\max_{z \\mathop \\in \\Img C} \\size {\\map f z} \\map L C$ where $\\map L C$ denotes the length of $C$."}
+{"_id": "2683", "title": "Jensen's Formula", "text": "Let $S$ be an open subset of the complex plane containing the closed disk: :$D_r = \\set {z \\in \\C : \\cmod z \\le r}$ of radius $r$ about $0$. Let $f: S \\to \\C$ be holomorphic on $S$.  Let $f$ have no zeroes on the circle $\\cmod z = r$. Let $\\map f 0 \\ne 0$. Let $\\rho_1, \\ldots, \\rho_n$ be the zeroes of $f$ in $D_r$, counted with multiplicity. Then: :$(1): \\quad \\displaystyle \\frac 1 {2 \\pi} \\int_0^{2 \\pi} \\ln \\cmod {\\map f {r e^{i \\theta} } } \\rd \\theta = \\ln \\cmod {\\map f 0} + \\sum_{k \\mathop = 1}^n \\paren {\\ln r - \\ln \\size {\\rho_k} }$"}
+{"_id": "2684", "title": "Residue Theorem", "text": "Let $U$ be a simply connected open subset of the complex plane $\\C$. Let $a_1, a_2, \\dots, a_n$ be finitely many points of $U$. Let $f: U \\to \\C$ be analytic in $U \\setminus \\set {a_1, a_2, \\dots, a_n}$.  Let $L = \\partial U$ be oriented counterclockwise. Then: :$\\displaystyle \\oint_L \\map f z \\rd z = 2 \\pi i \\sum_{k \\mathop = 1}^n \\Res f {a_k}$"}
+{"_id": "2685", "title": "Cauchy's Integral Formula", "text": "Let $D = \\set {z \\in \\C: \\cmod z \\le r}$ be the closed disk of radius $r$ in $\\C$. Let $f: U \\to \\C$ be holomorphic on some open set containing $D$. Then for each $a$ in the interior of $D$: :$\\displaystyle \\map f a = \\frac 1 {2 \\pi i} \\int_{\\partial D} \\frac {\\map f z} {\\paren {z - a} } \\rd z$ where $\\partial D$ is the boundary of $D$, and is traversed anticlockwise."}
+{"_id": "2686", "title": "Cauchy's Integral Formula/General Result", "text": "Let $n \\in \\N$ be a natural number. Then for each $a$ in the interior of $D$: :$\\displaystyle f^{\\paren n} \\paren a = \\dfrac {n!} {2 \\pi i} \\int_{\\partial D} \\frac {\\map f z} {\\paren {z - a}^{n + 1} } \\rd z$ where $\\partial D$ is the boundary of $D$, and is traversed anticlockwise."}
+{"_id": "2687", "title": "Characteristic of Subfield of Complex Numbers is Zero", "text": "The characteristic of any subfield of the field of complex numbers is $0$."}
+{"_id": "2688", "title": "Characteristic times Ring Element is Ring Zero", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity. Let the zero of $R$ be $0_R$ and the unity of $R$ be $1_R$. Let the characteristic of $R$ be $n$. Then: :$\\forall a \\in R: n \\cdot a = 0_R$"}
+{"_id": "2690", "title": "Field of Rational Functions is Field", "text": "Let $K$ be a field. Let $K \\sqbrk x$ be the integral domain of polynomial forms on $K$. Let $\\map K x$ be the field of rational functions on $K$. Then $\\map K x$ forms a field. If the characteristic of $K$ is $p$, then the characteristic of $\\map K x$ is non-zero."}
+{"_id": "2691", "title": "Hadamard Factorization Theorem", "text": "Let $f: \\C \\to \\C$ be an entire function of finite order $\\omega$. Let $0$ be a zero of $f$ of multiplicity $m\\geq0$. Let $\\left\\langle{a_n}\\right\\rangle$ be the sequence of nonzero zeroes of $f$, repeated according to multiplicity. Then $f$ has finite rank $p\\leq\\omega$ and there exists a polynomial $g$ of degree at most $\\omega$ such that: :$\\displaystyle f \\left({z}\\right) = z^m e^{g(z)} \\prod_{n \\mathop = 1}^\\infty E_p\\left( \\frac z{a_n} \\right)$ where $E_p$ denotes the $p$th Weierstrass elementary factor."}
+{"_id": "2693", "title": "Properties of Gamma Function", "text": "The gamma function $\\Gamma \\left({z}\\right)$ has the following properties:"}
+{"_id": "2694", "title": "Poisson Summation Formula", "text": "Let $f: \\R \\to \\C$ be a Schwarz function. Let $\\hat f$ be its Fourier transform. Then: :$\\displaystyle \\sum_{n \\mathop \\in \\Z} \\map f n = \\sum_{m \\mathop \\in \\Z} \\map {\\hat f} m$"}
+{"_id": "2695", "title": "Harmonic Properties of Schwarz Functions", "text": "Let $f, g : \\R \\to \\C$ be Schwarz functions. Let $\\hat f$, $\\hat g$ be the Fourier transforms of $f$ and $g$ respectively. Then: :$(1): \\quad \\hat f$, $\\hat g$ are Schwarz functions. :$(2): \\quad \\map {\\widehat {\\paren {\\hat f} } } x = \\map f {-x}$ for all $x \\in \\R$. :$(3): \\quad$ If $f * g$ is the convolution of $f$ and $g$, then: {{disambiguate|Definition:Convolution}} ::::$\\widehat {f * g} = \\hat f \\hat g$"}
+{"_id": "2696", "title": "Euler's Reflection Formula", "text": ":$\\forall z \\notin \\Z: \\map \\Gamma z \\, \\map \\Gamma {1 - z} = \\dfrac \\pi {\\map \\sin {\\pi z} }$"}
+{"_id": "2697", "title": "Legendre's Duplication Formula", "text": ":$\\forall z \\notin \\set {-\\dfrac n 2: n \\in \\N}: \\map \\Gamma z \\, \\map \\Gamma {z + \\dfrac 1 2} = 2^{1 - 2 z} \\sqrt \\pi \\, \\map \\Gamma {2 z}$ where $\\N$ denotes the natural numbers."}
+{"_id": "2698", "title": "Approximation to Reciprocal times Derivative of Gamma Function", "text": "For all $z \\in \\C$ such that $\\cmod {\\map \\arg z} < \\pi - \\epsilon, \\cmod z > 1$: :$\\dfrac {\\map {\\Gamma'} z} {\\map \\Gamma z} = \\ln z + \\map {\\mathcal O_\\epsilon} {z^{-1} }$ where: :$\\map {\\mathcal O} {z^{-1} }$ denotes big-O notation :the implied constant depends on $\\epsilon$."}
+{"_id": "2699", "title": "Complex Conjugate of Gamma Function", "text": ":$\\forall z \\in \\C \\setminus \\set {0, -1, -2, \\ldots}: \\map \\Gamma {\\overline z} = \\overline {\\map \\Gamma z}$"}
+{"_id": "2701", "title": "Intersection of Integer Ideals is Lowest Common Multiple", "text": "Let $\\ideal m$ and $\\ideal n$ be ideals of the integers $\\Z$. Let $\\ideal k$ be the intersection of $\\ideal m$ and $\\ideal n$. Then $k = \\lcm \\set {m, n}$."}
+{"_id": "2702", "title": "LCM iff Divides All Common Multiples", "text": "Let $a, b \\in \\Z$ such that $a b \\ne 0$. Let $m \\in \\Z: d > 0$. Then $m = \\lcm \\set {a, b}$ {{iff}}: : $(1): \\quad a \\divides m \\land b \\divides m$ : $(2): \\quad a \\divides n \\land b \\divides n \\implies m \\divides n$ That is, in the set of positive integers, $m$ is the LCM of $a$ and $b$ {{iff}} $m$ is a common multiple of $a$ and $b$, and $m$ also divides any other common multiple of $a$ and $b$."}
+{"_id": "2703", "title": "LCM Divides Common Multiple", "text": "Let $a, b \\in \\Z$ such that $a b \\ne 0$. Let $n$ be any common multiple of $a$ and $b$. That is, let $n \\in \\Z: a \\divides n, b \\divides n$. Then: :$\\lcm \\set {a, b} \\divides n$ where $\\lcm \\set {a, b}$ is the lowest common multiple of $a$ and $b$. {{:Euclid:Proposition/VII/35}}"}
+{"_id": "2704", "title": "Sum of Integer Ideals is Greatest Common Divisor", "text": "Let $\\ideal m$ and $\\ideal n$ be ideals of the integers $\\Z$. Let $\\ideal d = \\ideal m + \\ideal n$. Then $d = \\gcd \\set {m, n}$."}
+{"_id": "2705", "title": "Quotient Ring of Integers with Principal Ideal", "text": "Let $\\struct {\\Z, +, \\times}$ be the integral domain of integers. Let $n \\in \\Z$. Let $\\ideal n$ be the principal ideal of $\\struct {\\Z, +, \\times}$ generated by $n$. The quotient ring $\\struct {\\Z, +, \\times} / \\ideal n$ is isomorphic to $\\struct {\\Z_n, +_n, \\times_n}$, the ring of integers modulo $n$. Note the special cases where $n = 0$ or $1$: :Quotient Ring of Integers and Zero :Quotient Ring of Integers and Principal Ideal from Unity"}
+{"_id": "2706", "title": "Quotient Ring by Null Ideal", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Let $\\struct {\\set {0_R}, +, \\circ}$ be the null ideal of $\\struct {R, +, \\circ}$. Let $\\struct {R / \\set {0_R}, +, \\circ}$ be the quotient ring of $R$ defined by $\\set {0_R}$. Then $\\struct {R / \\set {0_R}, +, \\circ}$ is isomorphic to $\\struct {R, +, \\circ}$."}
+{"_id": "2707", "title": "Ring Homomorphism Preserves Negatives", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring homomorphism. Then: :$\\forall x \\in R_1: \\map \\phi {-x} = -\\paren {\\map \\phi x}$"}
+{"_id": "2708", "title": "Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function", "text": "Let $\\zeta$ be the Riemann zeta function. Let $s \\in \\C$ be a complex number with real part $\\sigma>1$. Then: :$\\displaystyle \\pi^{-s / 2} \\Gamma \\left({\\frac s 2}\\right) \\zeta \\left({s}\\right) = - \\frac 1 {s \\left({1 - s}\\right)} + \\int_1^\\infty \\left({x^{s / 2 - 1} + x^{- \\left({s + 1}\\right) / 2} }\\right) \\omega \\left({x}\\right) \\rd x$ where: : $\\Gamma$ is the gamma function : $\\displaystyle \\omega \\left({x}\\right) = \\sum_{n \\mathop = 1}^\\infty e^{- \\pi n^2 x}$"}
+{"_id": "2710", "title": "Mapping on Integers a plus b root 2 to Conjugate is Automorphism", "text": "Let $\\Z \\sqbrk {\\sqrt 2}$ denote the set: :$\\Z \\sqbrk {\\sqrt 2} := \\set {a + b \\sqrt 2: a, b \\in \\Z}$ that is, all numbers of the form $a + b \\sqrt 2$ where $a$ and $b$ are integers. Then the mapping $\\phi: \\Z \\sqbrk {\\sqrt 2} \\to \\Z \\sqbrk {\\sqrt 2}$ defined as: :$\\forall x = a + b \\sqrt 2 \\in \\Z \\sqbrk {\\sqrt 2}: \\map \\phi {a + b \\sqrt 2} = a - b \\sqrt 2$ is a ring automorphism."}
+{"_id": "2711", "title": "Ring Epimorphism from Integers to Integers Modulo m", "text": "Let $\\struct {\\Z, +, \\times}$ be the ring of integers. Let $\\struct {\\Z_m, +_m, \\times_m}$ be the ring of integers modulo $m$. Let $\\phi: \\struct {\\Z, +, \\times} \\to \\struct {\\Z_m, +_m, \\times_m}$ be the mapping defined as: :$\\forall x \\in \\Z: \\map \\phi x = \\eqclass x m$ where $\\eqclass x m$ is the residue class modulo $m$. Then $\\phi$ is a ring epimorphism, but specifically not a ring monomorphism. The image of $\\phi$ is $\\struct {\\Z_m, +_m, \\times_m}$. The kernel of $\\phi$ is $m \\Z$, the set of integer multiples of $m$."}
+{"_id": "2712", "title": "Monotone Additive Function is Linear", "text": "Let $f: \\R \\to \\R$ be a monotone real function which is additive, that is: :$\\forall x, y \\in \\R: \\map f {x + y} = \\map f x + \\map f y$ Then: :$\\exists a \\in \\R: \\forall x \\in \\R: \\map f x = a x$"}
+{"_id": "2713", "title": "Units of Ring of Polynomial Forms over Field", "text": "Let $\\struct {F, +, \\circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $F \\sqbrk X$ be the ring of polynomial forms in an indeterminate $X$ over $F$. Then the units of $F \\sqbrk X$ are all the elements of $F \\sqbrk X$ whose degree is $0$."}
+{"_id": "2714", "title": "Units of Numbers of Type Integer a plus b root 2", "text": "Let $\\Z \\sqbrk {\\sqrt 2}$ denote the set: :$\\Z \\sqbrk {\\sqrt 2} := \\set {a + b \\sqrt 2: a, b \\in \\Z}$ that is, all numbers of the form $a + b \\sqrt 2$ where $a$ and $b$ are integers. Let $\\struct {\\Z \\sqbrk {\\sqrt 2}, +, \\times}$ be the integral domain where $+$ and $\\times$ are conventional addition and multiplication on real numbers. Then numbers of the form $a + b \\sqrt 2$ such that $a^2 - 2 b^2 = \\pm 1$ are all units of $\\struct {\\Z \\sqbrk {\\sqrt 2}, +, \\times}$."}
+{"_id": "2715", "title": "Units of Gaussian Integers", "text": "Let $\\struct {\\Z \\sqbrk i, +, \\times}$ be the ring of Gaussian integers. The set of units of $\\struct {\\Z \\sqbrk i, +, \\times}$ is $\\set {1, i, -1, -i}$."}
+{"_id": "2716", "title": "Integers are Euclidean Domain", "text": "The integers $\\Z$ with the mapping $\\nu: \\Z \\to \\Z$ defined as: :$\\forall x \\in \\Z: \\map \\nu x = \\size x$ form a Euclidean domain."}
+{"_id": "2717", "title": "Gaussian Integers form Euclidean Domain", "text": "Let $\\struct {\\Z \\sqbrk i, +, \\times}$ be the integral domain of Gaussian Integers. Let $\\nu: \\Z \\sqbrk i \\to \\R$ be the real-valued function defined as:  :$\\forall a \\in \\Z \\sqbrk i: \\map \\nu a = \\cmod a^2$  where $\\cmod a$ is the (complex) modulus of $a$. Then $\\nu$ is a Euclidean valuation on $\\Z \\sqbrk i$. Hence $\\struct {\\Z \\sqbrk i, +, \\times}$ with $\\nu: \\Z \\sqbrk i \\to \\Z$ forms a Euclidean domain."}
+{"_id": "2718", "title": "Degree of Product of Polynomials over Ring", "text": ":$\\forall f, g \\in R \\left[{X}\\right]: \\deg \\left({f g}\\right) \\le \\deg \\left({f}\\right) + \\deg \\left({g}\\right)$"}
+{"_id": "2719", "title": "Degree of Sum of Polynomials", "text": ":$\\forall f, g \\in R \\sqbrk X: \\map \\deg {f + g} \\le \\max \\set {\\map \\deg f, \\map \\deg g}$"}
+{"_id": "2720", "title": "Degree of Product of Polynomials over Integral Domain not Less than Degree of Factors", "text": ":$\\forall f, g \\in R \\sqbrk X: \\map \\deg {f g} \\ge \\map \\deg f$"}
+{"_id": "2721", "title": "Polynomial Forms over Field is Euclidean Domain", "text": "Let $\\struct {F, +, \\circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $X$ be transcendental in $F$. Let $F \\sqbrk X$ be the ring of polynomial forms in $X$ over $F$. Then $F \\sqbrk X$ is a Euclidean domain."}
+{"_id": "2722", "title": "Euclidean Domain is Principal Ideal Domain", "text": "A Euclidean domain is a principal ideal domain."}
+{"_id": "2724", "title": "Elements of Euclidean Domain have Greatest Common Divisor", "text": "Let $\\struct {D, +, \\times}$ be a Euclidean domain. Then any two elements $a, b \\in D$ have a greatest common divisor $d$ such that: :$d \\divides a \\land d \\divides b$ :$x \\divides a \\land x \\divides b \\implies x \\divides d$ and $d$ is written $\\gcd \\set {a, b}$. For any $a, b \\in D$: :$\\exists s, t \\in D: s a + t b = d$ Any two greatest common divisors of any $a, b$ are associates."}
+{"_id": "2725", "title": "Associates in Ring of Polynomial Forms over Field", "text": "Let $F \\sqbrk X$ be the ring of polynomial forms over the field $F$. Let $\\map d X$ and $\\map {d'} X$ be polynomial forms in $F \\sqbrk X$. Then $\\map d X$ is an associate of $\\map {d'} X$ {{iff}} $\\map d X = c \\cdot \\map {d'} X$ for some $c \\in F, c \\ne 0_F$. Hence any two polynomials in $F \\sqbrk X$ have a unique monic GCD."}
+{"_id": "2726", "title": "Zero Element Generates Null Ideal", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. For $r \\in R$, let $\\ideal r$ denote the ideal generated by $r$. Then $\\ideal {0_R}$ is the null ideal."}
+{"_id": "2727", "title": "Irreducible Elements of Ring of Integers", "text": "Let $\\struct {\\Z, +, \\times}$ be the ring of integers. The irreducible elements of $\\struct {\\Z, +, \\times}$ are the prime numbers and their negatives."}
+{"_id": "2728", "title": "Euclid's Lemma for Euclidean Domains", "text": "Let $\\struct {D, +, \\times}$ be a Euclidean domain whose unity is $1$. Let $a, b, c \\in D$. Let $a \\divides b \\times c$, where $\\divides$ denotes divisibility. Let $a \\perp b$, where $\\perp$ denotes relative primeness. Then $a \\divides c$."}
+{"_id": "2729", "title": "Rational Polynomial is Content Times Primitive Polynomial", "text": "Let $\\Q \\sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\\map f X \\in \\Q \\sqbrk X$. Then: :$\\map f X = \\cont f \\, \\map {f^*} X$ where: :$\\cont f$ is the content of $\\map f X$ :$\\map {f^*} X$ is a primitive polynomial. For a given polynomial $\\map f X$, both $\\cont f$ and $\\map {f^*} X$ are unique."}
+{"_id": "2730", "title": "Product of Rational Polynomials", "text": "Let $\\Q \\sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\\map f X, \\map g X \\in \\Q \\sqbrk X$. Using Rational Polynomial is Content Times Primitive Polynomial, let these be expressed as: :$\\map f X = \\cont f \\cdot \\map {f^*} X$ :$\\map g X = \\cont g \\cdot \\map {g^*} X$ where: :$\\cont f, \\cont g$ are the content of $f$ and $g$ respectively :$f^*, g^*$ are primitive. Let $\\map h X = \\map f X \\, \\map g X$ be the product of $f$ and $g$. Then: :$\\map {h^*} X = \\map {f^*} X \\, \\map {g^*} X$"}
+{"_id": "2731", "title": "Parallelogram Law", "text": "Let $\\mathbf a$ and $\\mathbf b$ be vector quantities. Consider a parallelogram, two of whose adjacent sides represent $\\mathbf a$ and $\\mathbf b$ (in magnitude and direction). :300px Then the diagonal of the parallelogram through that common point represents the magnitude and direction of $\\mathbf a + \\mathbf b$, the sum of $\\mathbf a$ and $\\mathbf b$."}
+{"_id": "2732", "title": "Equivalence of Definitions of Perfect Set", "text": "{{TFAE|def = Perfect Set}} Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be a subset of $S$."}
+{"_id": "2733", "title": "Complement of Interior equals Closure of Complement", "text": "Let $T$ be a topological space. Let $H \\subseteq T$. Let $H^-$ denote the closure of $H$ and $H^\\circ$ denote the interior of $H$. Let $\\map \\complement H$ be the complement of $H$ in $T$: :$\\map \\complement H = T \\setminus H$ Then: :$\\map \\complement {H^\\circ} = \\paren {\\map \\complement H}^-$ and similarly: :$\\paren {\\map \\complement H}^\\circ = \\map \\complement {H^-}$ These can alternatively be written: :$T \\setminus H^\\circ = \\paren {T \\setminus H}^-$ :$\\paren {T \\setminus H}^\\circ = T \\setminus H^-$ which, it can be argued, is easier to follow."}
+{"_id": "2734", "title": "Interior of Finite Intersection equals Intersection of Interiors", "text": "Let $T$ be a topological space. Let $n \\in \\N$. Let: :$\\forall i \\in \\set {1, 2, \\dotsc, n}: H_i \\subseteq T$ Then: :$\\displaystyle \\left({\\bigcap_{i \\mathop = 1}^n H_i}\\right)^\\circ = \\bigcap_{i \\mathop = 1}^n H_i^\\circ$ where $H_i^\\circ$ denotes the interior of $H_i$."}
+{"_id": "2736", "title": "Finite Union of Regular Closed Sets is Regular Closed", "text": "Let $T$ be a topological space. Let $n \\in \\N$. Suppose that: :$\\forall i \\in \\set {1, 2, \\dotsc, n}: H_i \\subseteq T$ where all the $H_i$ are regular closed in $T$. That is: :$\\forall i \\in \\set {1, 2, \\dotsc, n}: H_i = H_i^{\\circ -}$ where $H_i^{\\circ -}$ denotes the closure of the interior of $H_i$ Then $\\displaystyle \\bigcup_{i \\mathop = 1}^n H_i$ is regular closed in $T$. That is: :$\\displaystyle \\bigcup_{i \\mathop = 1}^n H_i = \\paren {\\bigcup_{i \\mathop = 1}^n H_i}^{\\circ -}$"}
+{"_id": "2737", "title": "Boundary is Intersection of Closure with Closure of Complement", "text": "Let $T$ be a topological space. Let $X \\subseteq T$. Let $\\partial X$ denote the boundary of $X$, defined as: :$\\partial X = X^- \\setminus X^\\circ$ Let $\\overline X = T \\setminus X$ denote the complement of $X$ in $T$. Let $X^-$ denote the closure of $X$. Then: :$\\partial X = X^- \\cap \\paren {\\overline X}^-$"}
+{"_id": "2738", "title": "Set is Closed iff it Contains its Boundary", "text": "Let $T$ be a topological space, and let $H \\subseteq T$. Then $H$ is closed in $T$ {{iff}}: :$\\partial H \\subseteq H$ where $\\partial H$ is the boundary of $H$."}
+{"_id": "2739", "title": "Set is Open iff Disjoint from Boundary", "text": "Let $T$ be a topological space, and let $H \\subseteq T$. Then $H$ is open in $T$ {{iff}}: :$\\partial H \\cap H = \\O$ where $\\partial H$ is the boundary of $H$."}
+{"_id": "2740", "title": "Set is Clopen iff Boundary is Empty", "text": "Let $T$ be a topological space, and let $H \\subseteq T$. Then $H$ is both closed and open in $T$ {{iff}}: :$\\partial H = \\O$ where $\\partial H$ is the boundary of $H$."}
+{"_id": "2741", "title": "Boundary of Set is Closed", "text": "Let $T$ be a topological space, and let $H \\subseteq T$. Let $\\partial H$ denote the boundary of $H$. Then $\\partial H$ is closed in $T$."}
+{"_id": "2742", "title": "Boundary of Boundary is Contained in Boundary", "text": "Let $T$ be a topological space. Let $H \\subseteq T$. Then: :$\\map \\partial {\\partial H} \\subseteq \\partial H$ where $\\partial H$ is the boundary of $H$. That is, the boundary of the boundary of $H$ is contained in the boundary of $H$."}
+{"_id": "2743", "title": "Equivalence of Definitions of Exterior", "text": "{{TFAE|def = Exterior (Topology)|view = Exterior|context = Topology (Mathematical Branch)|contextview = Topology}} Let $T$ be a topological space. Let $H \\subseteq T$."}
+{"_id": "2744", "title": "Interior is Subset of Exterior of Exterior", "text": "Let $T$ be a topological space. Let $H \\subseteq T$. Let $H^e$ denote the exterior of $H$, and let $H^\\circ$ denote the interior of $H$. Then: :$H^\\circ \\subseteq \\paren {H^e}^e$"}
+{"_id": "2745", "title": "Interior is Subset of Interior of Closure", "text": "Let $T$ be a topological space. Let $H \\subseteq T$. Let $H^\\circ$ denote the interior of $H$. Let $H^-$ denote the closure of $H$. Then: :$H^\\circ \\subseteq \\left({H^-}\\right)^\\circ$"}
+{"_id": "2747", "title": "Closure of Union contains Union of Closures", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\mathbb H$ be a set of subsets of $S$. That is, let $\\mathbb H \\subseteq \\powerset S$ where $\\powerset S$ denotes the power set of $S$. Then the union of the closures of the elements of $\\mathbb H$ is a subset of the closure of the union of $\\mathbb H$: :$\\displaystyle \\bigcup_{H \\mathop \\in \\mathbb H} \\map \\cl H \\subseteq \\map \\cl {\\bigcup_{H \\mathop \\in \\mathbb H} H}$"}
+{"_id": "2748", "title": "Set is Subset of its Topological Closure", "text": "Let $T$ be a topological space. Let $H \\subseteq T$. Let $H^-$ be the closure of $H$ in $T$. Then: : $H \\subseteq H^-$"}
+{"_id": "2749", "title": "Set Closure as Intersection of Closed Sets", "text": "Let $T$ be a topological space. Let $H \\subseteq T$. Let the closure of $H$ (in $T$) be defined as: :$H^- := H \\cup H'$ where $H'$ is the derived set of $H$. Let $\\mathbb K$ be defined as: :$\\mathbb K := \\left\\{{K \\supseteq H: K}\\right.$ is closed in $\\left.{T}\\right\\}$ That is, let $\\mathbb K$ be the set of all closed sets of $T$ which contain $H$. Then the closure of $H$ (in $T$) can be defined as: :$\\displaystyle H^- := \\bigcap \\mathbb K$ that is, as the intersection of all the closed sets of $T$ which contain $H$."}
+{"_id": "2750", "title": "Set Closure is Smallest Closed Set/Topology", "text": "Let $T$ be a topological space. Let $H \\subseteq T$. Let $H^-$ denote the closure of $H$ in $T$. Then $H^-$ is the smallest superset of $H$ that is closed in $T$."}
+{"_id": "2751", "title": "Equivalence of Definitions of Interior (Topology)", "text": "{{TFAE|def = Interior (Topology)|view = interior|context = Topology (Mathematical Branch)|contextview = topology}} Let $\\struct {T, \\tau}$ be a topological space. Let $H \\subseteq T$."}
+{"_id": "2752", "title": "Intersection of Interiors contains Interior of Intersection", "text": "Let $T$ be a topological space. Let $\\mathbb H$ be a set of subsets of $T$. That is, let $\\mathbb H \\subseteq \\powerset T$ where $\\powerset T$ is the power set of $T$. Then the interior of the intersection of $\\mathbb H$ is a subset of the intersection of the interiors of the elements of $\\mathbb H$. :$\\displaystyle \\paren {\\bigcap_{H \\mathop \\in \\mathbb H} H}^\\circ \\subseteq \\bigcap_{H \\mathop \\in \\mathbb H} H^\\circ$"}
+{"_id": "2754", "title": "Exterior of Intersection contains Union of Exteriors", "text": "Let $T$ be a topological space. Let $\\mathbb H$ be a set of subsets of $T$. That is, let $\\mathbb H \\subseteq \\powerset T$ where $\\powerset T$ is the power set of $T$. Then: :$\\displaystyle \\bigcup_{H \\mathop \\in \\mathbb H} H^e \\subseteq \\paren {\\bigcap_{H \\mathop \\in \\mathbb H} H}^e$ where $H^e$ denotes the exterior of $H$."}
+{"_id": "2755", "title": "Separated Sets are Disjoint", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A, B \\subseteq S$ such that $A$ and $B$ are separated in $T$. Then $A$ and $B$ are disjoint: :$A \\cap B  = \\O$"}
+{"_id": "2756", "title": "Second-Countable Space is First-Countable", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is second-countable. Then $T$ is also first-countable."}
+{"_id": "2757", "title": "Second-Countability is Hereditary", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is second-countable. Let $T_H = \\struct {H, \\tau_H}$, where $\\O \\subset H \\subseteq S$, be a subspace of $T$. Then $T_H$ is second-countable."}
+{"_id": "2759", "title": "Basis for Topological Subspace", "text": "Let $T = \\struct {A, \\tau}$ be a topological space. Let $\\O \\subseteq H \\subseteq A$ and so let $T_H = \\struct {H, \\tau_H}$ be a subspace of $T$. Let $\\BB$ be a (synthetic) basis for $T$. Let $\\BB_H$ be defined as: :$B_H = \\set {U \\cap H: U \\in \\BB}$ Then $\\BB_H$ is a (synthetic) basis for $H$."}
+{"_id": "2761", "title": "Continuity Defined by Closure", "text": "Let $T_1 = \\struct {X_1, \\tau_1}$ and $T_2 = \\struct {X_2, \\tau_2}$ be topological spaces. Let $f: T_1 \\to T_2$ be a mapping. Then $f$ is continuous {{iff}}: :$\\forall H \\subseteq X_1: f \\sqbrk {H^-} \\subseteq \\paren {f \\sqbrk H}^-$ where $H^-$ denotes the closure of $H$ in $T_1$. That is, {{iff}} the image of the closure is a subset of the closure of the image."}
+{"_id": "2762", "title": "Valuation Ring is Local", "text": "Let $R$ be a valuation ring. Then $R$ is a local ring."}
+{"_id": "2763", "title": "Bijection is Open iff Inverse is Continuous", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ and $T_2 = \\struct {S_2, \\tau_2}$ be topological spaces. Let $f: T_1 \\to T_2$ be a bijection. Then $f$ is open {{iff}} $f^{-1}$ is continuous."}
+{"_id": "2764", "title": "Bijection is Open iff Closed", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ and $T_2 = \\struct {S_2, \\tau_2}$ be topological spaces. Let $f: T_1 \\to T_2$ be a bijection. Then $f$ is open {{iff}} $f$ is closed."}
+{"_id": "2765", "title": "T2 Space is T1 Space", "text": "Let $\\struct {S, \\tau}$ be a $T_2$ (Hausdorff) space. Then $\\struct {S, \\tau}$ is also a $T_1$ (Fréchet) space."}
+{"_id": "2766", "title": "T1 Space is T0 Space", "text": "Let $\\struct {S, \\tau}$ be a Fréchet ($T_1$) space. Then $\\struct {S, \\tau}$ is also a Kolmogorov ($T_0$) space."}
+{"_id": "2767", "title": "T5 Space is T4 Space", "text": "Let $\\struct {S, \\tau}$ be a $T_5$ space. Then $\\struct {S, \\tau}$ is also a $T_4$ space."}
+{"_id": "2768", "title": "Completely Normal Space is Normal Space", "text": "Let $\\struct {S, \\tau}$ be a completely normal space. Then $\\struct {S, \\tau}$ is also a normal space."}
+{"_id": "2769", "title": "Regular Space is T2 Space", "text": "Let $\\struct {S, \\tau}$ be a regular space. Then $\\struct {S, \\tau}$ is also a $T_2$ (Hausdorff) space."}
+{"_id": "2770", "title": "Normal Space is T3 Space", "text": "Let $\\struct {S, \\tau}$ be a normal space. Then $\\struct {S, \\tau}$ is also a $T_3$ space."}
+{"_id": "2771", "title": "Completely Hausdorff Space is Hausdorff Space", "text": "Let $\\struct {S, \\tau}$ be a $T_{2 \\frac 1 2}$ (completely Hausdorff) space. Then $\\struct {S, \\tau}$ is also a $T_2$ (Hausdorff) space."}
+{"_id": "2772", "title": "Regular Space is Completely Hausdorff Space", "text": "Let $\\struct {S, \\tau}$ be a regular space. Then $\\struct {S, \\tau}$ is also a $T_{2 \\frac 1 2}$ (completely Hausdorff) space."}
+{"_id": "2773", "title": "Equivalence of Definitions of T0 Space", "text": "{{TFAE|def = T0 Space|view = $T_0$ (Kolmogorov) space}} Let $T = \\struct {S, \\tau}$ be a topological space."}
+{"_id": "2774", "title": "Equivalence of Definitions of T1 Space", "text": "{{TFAE|def = T1 Space|view = $T_1$ (Fréchet) space}} Let $T = \\struct {S, \\tau}$ be a topological space."}
+{"_id": "2775", "title": "T3 1/2 Space is T3 Space", "text": "Let $T$ be a $T_{3 \\frac 1 2}$ space. Then $T$ is also a $T_3$ space."}
+{"_id": "2776", "title": "Tychonoff Space is Regular, T2 and T1", "text": "Let $\\struct {S, \\tau}$ be a Tychonoff space. Then $\\struct {S, \\tau}$ is also: :a regular space :a $T_2$ (Hausdorff) space :a $T_1$ (Fréchet) space."}
+{"_id": "2777", "title": "Normal Space is Tychonoff Space", "text": "Let $\\struct {S, \\tau}$ be a normal space. Then $\\struct {S, \\tau}$ is also a Tychonoff (completely regular) space."}
+{"_id": "2778", "title": "T4 and T3 Space is T 3 1/2", "text": "Let $T = \\struct {S, \\tau}$ be: :a $T_4$ space and also: :a $T_3$ space. Then $T$ is also a $T_{3 \\frac 1 2}$ space."}
+{"_id": "2779", "title": "Separation Axioms Preserved under Homeomorphism", "text": "The separation axioms are preserved under homeomorphism. Let $T_A = \\left({S_A, \\tau_A}\\right), T_B = \\left({S_B, \\tau_B}\\right)$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a homeomorphism."}
+{"_id": "2780", "title": "T0 Space is Preserved under Homeomorphism", "text": "If $T_A$ is a $T_0$ (Kolmogorov) space, then so is $T_B$."}
+{"_id": "2781", "title": "T1 Space is Preserved under Homeomorphism", "text": "If $T_A$ is a $T_1$ (Fréchet) space, then so is $T_B$."}
+{"_id": "2782", "title": "T2 Space is Preserved under Homeomorphism", "text": "If $T_A$ is a $T_2$ (Hausdorff) space, then so is $T_B$."}
+{"_id": "2783", "title": "Completely Hausdorff Space is Preserved under Homeomorphism", "text": "If $T_A$ is a $T_{2 \\frac 1 2}$ (completely Hausdorff) space, then so is $T_B$."}
+{"_id": "2784", "title": "Regular Space is Preserved under Homeomorphism", "text": "If $T_A$ is a regular space, then so is $T_B$."}
+{"_id": "2785", "title": "T3 Space is Preserved under Homeomorphism", "text": "If $T_A$ is a $T_3$ space, then so is $T_B$."}
+{"_id": "2786", "title": "T3 1/2 Space is Preserved under Homeomorphism", "text": "If $T_A$ is a $T_{3 \\frac 1 2}$ space, then so is $T_B$."}
+{"_id": "2789", "title": "T4 Space is Preserved under Homeomorphism", "text": "If $T_A$ is a $T_4$ space, then so is $T_B$."}
+{"_id": "2791", "title": "T5 Space is Preserved under Homeomorphism", "text": "If $T_A$ is a $T_5$ space, then so is $T_B$."}
+{"_id": "2792", "title": "Inverse of Homeomorphism is Homeomorphism", "text": "Let $T, T'$ be topological spaces. Let $f: T \\to T'$ be a homeomorphism. Then $f^{-1}: T' \\to T$ is also a homeomorphism."}
+{"_id": "2793", "title": "T0 Space is Preserved under Closed Bijection", "text": "Let $T_A = \\struct {S_A, \\tau_A}$ and $T_B = \\struct {S_B, \\tau_B}$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a closed bijection. If $T_A$ is a $T_0$ (Kolmogorov) space, then so is $T_B$."}
+{"_id": "2794", "title": "Completely Hausdorff Space is Preserved under Closed Bijection", "text": "Let $T_A = \\struct {S_A, \\tau_A}$ and $T_B = \\struct {S_B, \\tau_B}$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a closed bijection. If $T_A$ is a $T_{2 \\frac 1 2}$ (completely Hausdorff) space, then so is $T_B$."}
+{"_id": "2795", "title": "T1 Space is Preserved under Closed Bijection", "text": "Let $T_A = \\struct {S_A, \\tau_A}$ and $T_B = \\struct {S_B, \\tau_B}$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a closed bijection. If $T_A$ is a $T_1$ (Fréchet) space, then so is $T_B$."}
+{"_id": "2796", "title": "T2 Space is Preserved under Closed Bijection", "text": "Let $T_A = \\struct {S_A, \\tau_A}$ and $T_B = \\struct {S_B, \\tau_B}$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a closed bijection. If $T_A$ is a $T_2$ (Hausdorff) space, then so is $T_B$."}
+{"_id": "2797", "title": "Separation Properties Preserved by Expansion", "text": "These separation properties are preserved under expansion: :$T_0$ (Kolmogorov) Space :$T_1$ (Fréchet) Space :$T_2$ (Hausdorff) Space :$T_{2 \\frac 1 2}$ (Completely Hausdorff) Space"}
+{"_id": "2798", "title": "Identity Mapping to Expansion is Closed", "text": "Let $S$ be a set on which $\\tau_1$ and $\\tau_2$ are topologies such that: :$\\tau_1 \\subseteq \\tau_2$ that is, such that $\\tau_2$ is an expansion of $\\tau_1$. Let $I_X: \\struct {S, \\tau_1} \\to \\struct {S, \\tau_2}$ be the identity mapping from $\\struct {S, \\tau_1}$ to $\\struct {S, \\tau_2}$. Then $I_S$ is closed."}
+{"_id": "2799", "title": "Separation Properties Preserved under Topological Product", "text": "Let $\\mathbb S = \\family {\\struct {S_i, \\tau_i} }_{i \\mathop \\in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\\displaystyle T = \\struct {S, \\tau} = \\prod_{i \\mathop \\in I} \\struct{S_i, \\tau_i}$ be the product space of $\\mathbb S$. Then $T$ has one of the following properties {{iff}} each of $\\struct {S_i, \\tau_i}$ has the same property: :$T_0$ (Kolmogorov) Property :$T_1$ (Fréchet) Property :$T_2$ (Hausdorff) Property :$T_{2 \\frac 1 2}$ (Completely Hausdorff) Property :$T_3$ Property :$T_{3 \\frac 1 2}$ Property If $T = \\struct {S, \\tau}$ has one of the following properties then each of $\\struct {S_i, \\tau_i}$ has the same property: :$T_4$ Property :$T_5$ Property but the converse does not necessarily hold."}
+{"_id": "2800", "title": "Separation Properties Preserved in Subspace", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T_H$ be a subspace of $T$. If $T$ has one of the following properties then $T_H$ has the same property: :$T_0$ (Kolmogorov) Property :$T_1$ (Fréchet) Property :$T_2$ (Hausdorff) Property :$T_{2 \\frac 1 2}$ (Completely Hausdorff) Property :$T_3$ Property :$T_{3 \\frac 1 2}$ Property :$T_5$ Property That is, the above properties are all hereditary."}
+{"_id": "2801", "title": "T4 Property Preserved in Closed Subspace", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T_K$ be a subspace of $T$ such that $K$ is closed in $T$. If $T$ is a $T_4$ space then $T_K$ is also a $T_4$ space. That is, the property of being a $T_4$ space is weakly hereditary."}
+{"_id": "2802", "title": "Completely Normal iff Every Subspace is Normal", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then $T$ is a completely normal space {{iff}} every subspace of $T$ is normal."}
+{"_id": "2803", "title": "Tychonoff Space is Urysohn Space", "text": "Let $\\struct {S, \\tau}$ be a Tychonoff space. Then $\\struct {S, \\tau}$ is also an Urysohn space."}
+{"_id": "2804", "title": "Urysohn Space is Completely Hausdorff Space", "text": "Let $\\struct {S, \\tau}$ be an Urysohn space. Then $\\struct {S, \\tau}$ is also a $T_{2 \\frac 1 2}$ (completely Hausdorff) space."}
+{"_id": "2805", "title": "Perfectly Normal Space is Completely Normal Space", "text": "Let $T = \\struct {S, \\tau}$ be a perfectly normal space. Then $T$ is also a completely normal space."}
+{"_id": "2806", "title": "T3 Space is Semiregular", "text": "Let $T = \\struct {S, \\tau}$ be a $T_3$ space. Then $T$ is a semiregular space."}
+{"_id": "2807", "title": "Regular Space is Semiregular Space", "text": "Let $\\struct {S, \\tau}$ be a regular space. Then $\\struct {S, \\tau}$ is also a semiregular space."}
+{"_id": "2810", "title": "Compact Space is Sigma-Compact", "text": "Every compact space is $\\sigma$-compact."}
+{"_id": "2811", "title": "Sigma-Compact Space is Lindelöf", "text": "Every $\\sigma$-compact space is a Lindelöf space."}
+{"_id": "2814", "title": "Sequentially Compact Space is Countably Compact", "text": "A sequentially compact topological space is also countably compact."}
+{"_id": "2815", "title": "Countably Compact Space is Weakly Countably Compact", "text": "Every countably compact space is a weakly countably compact space."}
+{"_id": "2816", "title": "T1 Space is Weakly Countably Compact iff Countably Compact", "text": "Let $T = \\struct {S, \\tau}$ be a $T_1$ (Fréchet) topological space. Then $T$ is weakly countably compact {{iff}} $T$ is countably compact."}
+{"_id": "2817", "title": "Countably Compact Space is Pseudocompact", "text": "Let $T = \\struct {S, \\tau}$ be a countably compact space. Then $T$ is a pseudocompact space."}
+{"_id": "2819", "title": "Countably Compact Lindelöf Space is Compact", "text": "Let $T = \\struct {S, \\tau}$ be a Lindelöf space which is also countably compact. Then $T$ is compact."}
+{"_id": "2820", "title": "Compact Space is Weakly Locally Compact", "text": "Let $T = \\left({S, \\tau}\\right)$ be a compact space. Then $T$ is a weakly locally compact."}
+{"_id": "2821", "title": "Completed Riemann Zeta Function has Order One", "text": "The completed Riemann zeta function $\\xi$ has order at most $1$."}
+{"_id": "2822", "title": "Product Equation for Riemann Zeta Function", "text": "There exists a constant $B$ such that :$\\displaystyle \\frac{\\zeta'(s)}{\\zeta(s)} = B - \\frac 1{s-1} +\\frac12 \\log \\pi  - \\frac12 \\frac{\\Gamma'(s/2+1)}{\\Gamma(s/2+1)} + \\sum_\\rho \\left( \\frac 1{s-\\rho} + \\frac1\\rho \\right)$ where $\\zeta$ is the Riemann zeta function, $\\rho$ runs over the non-trivial zeros of $\\zeta$, and $\\Gamma$ is the gamma function."}
+{"_id": "2823", "title": "Zeros of Functions of Finite Order", "text": "Let $\\map f z$ be an entire function which satisfies: :$\\map f 0 \\ne 0$ :$\\cmod {\\map f z} \\ll \\map \\exp {\\map \\alpha {\\cmod z} }$ for all $z \\in \\C$ and some function $\\alpha$, where $\\ll$ is the order notation. For $T \\ge 1$, let: :$\\map N T = \\# \\set {\\rho \\in \\C: \\map f r = 0, \\ \\cmod \\rho < T}$ where $\\#$ denotes the cardinality of a set. Then: :$\\map N T \\ll \\map \\alpha {2 T}$"}
+{"_id": "2824", "title": "Strongly Locally Compact Space is Weakly Locally Compact", "text": "Let $T = \\struct {S, \\tau}$ be a strongly locally compact space. Then $T$ is weakly locally compact."}
+{"_id": "2825", "title": "Weakly Locally Compact Hausdorff Space is Strongly Locally Compact", "text": "Let $T = \\struct {S, \\tau}$ be a $T_2$ (Hausdorff) space. Let $T$ be weakly locally compact. Then $T$ is strongly locally compact."}
+{"_id": "2826", "title": "Weakly Sigma-Locally Compact iff Weakly Locally Compact and Lindelöf", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. {{TFAE}} :$(1): \\quad T$ is weakly $\\sigma$-locally compact :$(2): \\quad T$ is weakly locally compact and Lindelöf"}
+{"_id": "2827", "title": "Second-Countable Space is Lindelöf", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is second-countable. Then $T$ is also a Lindelöf space."}
+{"_id": "2828", "title": "Separable Space satisfies Countable Chain Condition", "text": "Let $T = \\struct {S, \\tau}$ be a separable space. Then $T$ satisfies the countable chain condition."}
+{"_id": "2829", "title": "Logarithmic Derivative of Riemann Zeta Function", "text": "Let $\\zeta$ be the Riemann zeta function: :$\\displaystyle \\forall s \\in \\C: \\map \\Re s > 1: \\map \\zeta s = \\sum_{n \\mathop \\ge 1} n^{-s}$ Then for all $s$ with $\\map \\Re s > 1$: :$\\displaystyle -\\frac {\\map {\\zeta'} s} {\\map \\zeta s} = \\sum_{n \\mathop \\ge 1} \\map \\Lambda n n^{-s}$ where $\\Lambda$ is von Mangoldt's function."}
+{"_id": "2830", "title": "Riemann Zeta Has No Zeros With Real Part One", "text": "Let $\\zeta$ be the Riemann zeta function. Then for all $t \\in \\R$, $\\zeta(1 + it) \\neq 0$"}
+{"_id": "2831", "title": "Compact Space is Countably Compact", "text": "Let $T = \\struct {S, \\tau}$ be a compact space. Then $T$ is countably compact."}
+{"_id": "2832", "title": "Second-Countable Space is Compact iff Countably Compact", "text": "Let $T = \\struct {S, \\tau}$ be a second-countable space. Then $T$ is compact {{iff}} $T$ is countably compact."}
+{"_id": "2833", "title": "Unsymmetric Functional Equation for Riemann Zeta Function", "text": "Let $\\zeta$ be the Riemann zeta function. Let $\\Gamma$ be the gamma function. Then for all $s \\in \\C$, :$\\displaystyle \\map \\zeta {1 - s} = 2^{1 - s} \\pi^{-s} \\, \\map \\cos {\\frac {\\pi s} 2} \\map \\Gamma s \\, \\map \\zeta s$"}
+{"_id": "2834", "title": "Countably Compact First-Countable Space is Sequentially Compact", "text": "A countably compact first-countable topological space is also sequentially compact."}
+{"_id": "2835", "title": "Residue at Simple Pole", "text": "Let $f: \\C \\to \\C$ be a function meromorphic on some region, $D$, containing $a$.  Let $f$ have a simple pole at $a$.  Then the residue of $f$ at $a$ is given by: :$\\displaystyle \\operatorname{Res} \\left({f, a}\\right) = \\lim_{z \\mathop \\to a} \\left({z - a}\\right) f \\left({z}\\right)$"}
+{"_id": "2836", "title": "Poles of Riemann Zeta Function", "text": "Let $\\zeta$ be the Riemann zeta function. Then $\\zeta$ has a simple pole at $s = 1$ with residue $1$, and no other poles."}
+{"_id": "2837", "title": "Properties of Dirichlet Convolution", "text": "Let $f, g, h$ be arithmetic functions. Let $*$ denote Dirichlet convolution. Let $\\iota$ be the identity arithmetic function. Then the following properties hold:"}
+{"_id": "2838", "title": "Ring of Arithmetic Functions is Ring with Unity", "text": "Let $\\AA$ be the set of all arithmetic functions. Let $*$ denote Dirichlet convolution, and $+$ the pointwise sum of functions. The ring of arithmetic functions $\\struct {\\AA, +, *}$ is a commutative ring with unity."}
+{"_id": "2839", "title": "Units of Ring of Arithmetic Functions", "text": "Let $f$ be an arithmetic function. Then $f$ is a unit in the ring of arithmetic functions {{iff}}: :$\\map f 1 \\ne 0$"}
+{"_id": "2840", "title": "Möbius Inversion Formula", "text": "Let $f$ and $g$ be arithmetic functions. Then: :$(1): \\quad \\displaystyle \\map f n = \\sum_{d \\mathop \\divides n} \\map g d$ {{iff}}: :$(2): \\quad \\displaystyle \\map g n = \\sum_{d \\mathop \\divides n} \\map f d \\, \\map \\mu {\\frac n d}$ where: : $d \\divides n$ denotes that $d$ is a divisor of $n$ : $\\mu$ is the Möbius function."}
+{"_id": "2841", "title": "Fully Normal Space is Normal Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a fully normal space. Then $T$ is a normal space."}
+{"_id": "2842", "title": "Paracompact Space is Countably Paracompact", "text": "Let $T = \\struct {S, \\tau}$ be a paracompact space. Then $T$ is a countably paracompact space."}
+{"_id": "2843", "title": "Metacompact Space is Countably Metacompact", "text": "Let $T = \\struct {S, \\tau}$ be a metacompact space. Then $T$ is a countably metacompact space."}
+{"_id": "2844", "title": "Fully T4 Space is T4 Space", "text": "Let $T = \\struct {S, \\tau}$ be a fully $T_4$ space. Then $T$ is a $T_4$ space."}
+{"_id": "2845", "title": "Fully Normal Space is Paracompact", "text": "Let $T = \\struct {S, \\tau}$ be a fully normal space. Then $T$ is paracompact."}
+{"_id": "2846", "title": "Subcover is Refinement of Cover", "text": "Let $S$ be a set. Let $\\mathcal C$ be a cover for $S$. Let $\\mathcal V$ be a subcover of $\\mathcal C$. Then $\\mathcal V$ is a refinement of $\\mathcal C$."}
+{"_id": "2847", "title": "Compact Space is Paracompact", "text": "Let $T = \\struct {S, \\tau}$ be a compact space. Then $T$ is paracompact."}
+{"_id": "2848", "title": "Paracompact Space is Metacompact", "text": "Let $T = \\struct {S, \\tau}$ be a paracompact space. Then $T$ is also metacompact."}
+{"_id": "2849", "title": "Countably Paracompact Space is Countably Metacompact", "text": "Let $T = \\struct {S, \\tau}$ be a countably paracompact space. Then $T$ is countably metacompact."}
+{"_id": "2850", "title": "Countably Compact Space is Countably Paracompact", "text": "Let $T = \\struct {S, \\tau}$ be a countably compact space. Then $T$ is countably paracompact."}
+{"_id": "2851", "title": "Metacompact Countably Compact Space is Compact", "text": "Let $T = \\struct {S, \\tau}$ be a countably compact space which is also metacompact. Then $T$ is compact."}
+{"_id": "2853", "title": "Countably Paracompact Lindelöf Space is Paracompact", "text": "Let $T = \\struct {S, \\tau}$ be a Lindelöf space which is also countably paracompact. Then $T$ is paracompact."}
+{"_id": "2855", "title": "Convergence of Dirichlet Series with Bounded Partial Sums", "text": "Let $\\left\\langle{a_n}\\right\\rangle_{n \\mathop \\in \\N}$ be a sequence in $\\C$. Suppose that there exists $B > 0$ such that for all $n, m \\in \\N$: :$\\displaystyle \\left\\vert{\\sum_{k \\mathop = m}^n a_n}\\right\\vert \\le B$ Then the Dirichlet series: : $\\displaystyle f \\left({s}\\right) = \\sum_{n \\mathop \\ge 1} a_n n^{-s}$ converges locally uniformly to an analytic function on $\\Re \\left({s}\\right) > 0$."}
+{"_id": "2856", "title": "Convergence of Dirichlet Series with Bounded Coefficients", "text": "Let $\\left\\langle{a_n}\\right\\rangle_{n \\mathop \\in \\N}$ be a bounded sequence in $\\C$. Then the Dirichlet series: :$\\displaystyle f \\left({s}\\right) = \\sum_{n \\mathop \\ge 1} a_n n^{-s}$ converges absolutely and locally uniformly to an analytic function on $\\Re \\left({s}\\right) > 1$."}
+{"_id": "2857", "title": "Dirichlet L-Function from Trivial Character", "text": "Let $\\chi_0$ be the trivial Dirichlet character modulo $q$. {{explain|Trivial character we got, Dirichlet character we got, we still need a page for trivial Dirichlet character. There exists on another page a link to Definition:Dirichlet Character/Trivial Character which ought to be straightforward to construct.}} Let $\\zeta$ be the Riemann zeta function. Then: :$\\displaystyle \\map L {s, \\chi_0} = \\map \\zeta s \\cdot \\prod_{p \\mathop \\divides q} \\paren {1 - p^{-s} }$ where $\\divides$ denotes divisibility."}
+{"_id": "2858", "title": "Separable Metacompact Space is Lindelöf", "text": "Let $T = \\left({S, \\tau}\\right)$ be a separable topological space which is also metacompact. Then $T$ is a Lindelöf space."}
+{"_id": "2859", "title": "Functional Equation for Dirichlet L-Functions", "text": "Let $\\chi$ be a primitive Dirichlet character to the modulus $q \\geq 1$. Let $\\map \\Lambda {s, \\chi}$ be the completed $L$-function for $\\chi$. Let $\\map \\tau \\chi$ denote the Gaussian sum. Then for all $s \\in \\C$: :$\\map \\Lambda {s, \\chi} = i^{-\\kappa} \\dfrac {\\map \\tau \\chi} {\\sqrt q} \\map \\Lambda {1 - s, \\overline \\chi}$ where $\\kappa = \\dfrac 1 2 \\paren {1 - \\map \\chi {-1} }$."}
+{"_id": "2860", "title": "Orthogonality Relations for Characters", "text": "Let $G$ be a finite abelian group with identity $e$. Let $G^*$ be the dual group of characters $\\chi : G \\to \\C_{\\ne 0}$. Let $\\chi_0$ be the trivial character on $G$. Let $\\psi: G \\to \\C_{\\ne 0}$ be any character. Let $y \\in G$ be arbitrary. Then: :$\\displaystyle \\sum_{x \\mathop \\in G} \\psi \\left({x}\\right) = \\begin{cases} \\left\\lvert{G}\\right\\rvert & : \\psi = \\chi_0 \\\\ 0 & : \\psi \\ne \\chi_0 \\end{cases}$ and: :$\\displaystyle \\sum_{\\chi \\mathop \\in G^*} \\chi \\left({y}\\right) = \\begin{cases} \\left\\lvert{G^*}\\right\\rvert & : y = e \\\\ 0 & : y \\ne e \\end{cases}$"}
+{"_id": "2861", "title": "Analytic Continuation of Dirichlet L-Function", "text": "Let $\\chi : G := \\left({\\Z / q \\Z}\\right)^\\times \\to \\C^\\times$ be a Dirichlet character modulo $q$. {{explain|$\\C^\\times$}} Let $L \\left({s, \\chi}\\right)$ be the Dirichlet $L$-function for $\\chi$. Let $\\chi$ be the trivial character. Then $L \\left({s, \\chi}\\right)$ has an analytic continuation to $\\C$ except for a simple pole at $s = 1$. Let $\\chi$ be non-trivial. Then $L \\left({s, \\chi}\\right)$ is analytic on $\\Re \\left({s}\\right) > 0$."}
+{"_id": "2862", "title": "L-Function does not Vanish at One", "text": "Let $\\psi$ be a non-trivial Dirichlet charater modulo $q$. Let $\\map L {s, \\chi}$ be the Dirichlet $L$-function associated to $\\chi$. Then $\\map L {1, \\chi} \\ne 0$."}
+{"_id": "2863", "title": "Logarithm of Dirichlet L-Functions", "text": "Let $\\chi$ be a Dirichlet character modulo $q$. The Dirichlet series :$\\map f s = \\displaystyle \\sum_{n \\mathop \\ge 1} \\sum_p \\frac {\\map \\chi p^n} {n p^{n s} }$ converges absolutely to an analytic function, where $p$ ranges over the primes. Moreover, $\\map f s$ defines a branch of $\\ln \\map L {s, \\chi}$"}
+{"_id": "2864", "title": "Dirichlet's Theorem on Arithmetic Sequences", "text": "Let $a, q$ be coprime integers. Let $\\PP_{a, q}$ be the set of primes $p$ such that $p \\equiv a \\pmod q$. Then $\\PP_{a, q}$ has Dirichlet density: :$\\map \\phi q^{-1}$ where $\\phi$ is Euler's phi function. In particular, $\\PP_{a, q}$ is infinite."}
+{"_id": "2865", "title": "Discrete Fourier Transform on Abelian Group", "text": "Let $G$ be a finite abelian group. Let $G^*$ be the dual group of characters $G \\to \\C^\\times$. Let $\\eta: G \\to \\C$ be a mapping from $G$ to the set of complex numbers. Then for all $x \\in G$: :$\\displaystyle \\eta \\left({x}\\right) = \\frac 1 {\\phi \\left({q}\\right)} \\sum_{\\chi \\mathop \\in G^*} \\langle \\eta, \\chi \\rangle_G \\chi(x)$ where: :$\\displaystyle \\langle \\eta, \\chi \\rangle_G = \\sum_{x \\mathop \\in G} \\eta \\left({x}\\right) \\overline{\\chi} \\left({x}\\right)$"}
+{"_id": "2866", "title": "Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods", "text": "Let $T = \\struct {S, \\tau}$ be a Hausdorff space. Let $V_1$ and $V_2$ be compact sets in $T$. Then $V_1$ and $V_2$ have disjoint neighborhoods."}
+{"_id": "2867", "title": "Compact Subsets of T3 Spaces", "text": "Let $T = \\struct {S, \\tau}$ be a $T_3$ space. Let $A \\subseteq S$ be compact in $T$. Then for each $U \\in \\tau$ such that $A \\subseteq U$: :$\\exists V \\in \\tau: A \\subseteq V \\subseteq V^- \\subseteq U$ where $V^-$ denotes the closure of $V$."}
+{"_id": "2868", "title": "Compact Hausdorff Space is T4", "text": "Let $T = \\struct {S, \\tau}$ be a compact Hausdorff space. Then $T$ is a $T_4$ space."}
+{"_id": "2871", "title": "Maximum Cardinality of Separable Hausdorff Space", "text": "Let $T = \\struct {S, \\tau}$ be a Hausdorff space which is separable. Then $S$ can have a cardinality no greater than $2^{2^{\\aleph_0} }$."}
+{"_id": "2872", "title": "Cluster Point of Ultrafilter is Unique", "text": "Let $S$ be a set. Let $\\FF$ be an ultrafilter on $S$. Let $x \\in S$ be a cluster point of $\\FF$. Then there is no point $y \\in S: y \\ne x$ such that $y$ is also a cluster point of $\\FF$."}
+{"_id": "2873", "title": "Sequence of Implications of Separation Axioms", "text": "Let $P_1$ and $P_2$ be separation axioms and let: :$P_1 \\implies P_2$ mean: :If a topological space $T$ satsifies separation axiom $P_1$, then $T$ also satisfies separation axiom $P_2$. Then the following sequence of separation axioms holds: {| |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | Perfectly Normal || | align=\"center\" | $\\implies$ || | align=\"center\" | Perfectly $T_4$ || | align=\"center\" | $\\implies$ || | align=\"center\" | $T_4$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | Completely Normal || | align=\"center\" | $\\implies$ || | align=\"center\" | $T_5$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | Normal || | align=\"center\" | $\\implies$ || | align=\"center\" | $T_4$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | || |- | align=\"center\" | $T_{3 \\frac 1 2}$ || | align=\"center\" | $\\impliedby$ || | align=\"center\" | Completely Regular (Tychonoff) || | align=\"center\" | $\\implies$ || | align=\"center\" | Urysohn || |- | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | $T_3$ || | align=\"center\" | $\\impliedby$ || | align=\"center\" | Regular || | align=\"center\" | $\\implies$ || | align=\"center\" | $T_{2 \\frac 1 2}$ (Completely Hausdorff) || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | Semiregular || | align=\"center\" | $\\implies$ || | align=\"center\" | $T_2$ (Hausdorff) || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $T_1$ (Fréchet) || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $T_0$ (Kolmogorov) || |}"}
+{"_id": "2874", "title": "Sequence of Implications of Global Compactness Properties", "text": "Let $P_1$ and $P_2$ be compactness properties and let: :$P_1 \\implies P_2$ mean: :If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | Sequentially Compact || | align=\"center\" | || | align=\"center\" | || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | || |- | align=\"center\" | Compact || | align=\"center\" | $\\implies$ || | align=\"center\" | Countably Compact || | align=\"center\" | $\\implies$ || | align=\"center\" | Pseudocompact || |- | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | || |- | align=\"center\" | $\\sigma$-Compact || | align=\"center\" | || | align=\"center\" | Weakly Countably Compact || | align=\"center\" | || | align=\"center\" | || |- | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || |- | align=\"center\" | Lindelöf Space || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || |}"}
+{"_id": "2875", "title": "Sequence of Implications of Local Compactness Properties", "text": "Let $P_1$ and $P_2$ be compactness properties and let: :$P_1 \\implies P_2$ mean: :If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align=\"center\" | Compact || | align=\"center\" | $\\implies$ || | align=\"center\" | Strongly Locally Compact || |- | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | Weakly $\\sigma$-Locally Compact || | align=\"center\" | $\\implies$ || | align=\"center\" | Weakly Locally Compact || | align=\"center\" | $\\Longleftarrow$ || | align=\"center\" | Locally Compact || |- | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | || |- | align=\"center\" | $\\sigma$-Compact || | align=\"center\" | || | align=\"center\" | || |- | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | || |- | align=\"center\" | Lindelöf Space || | align=\"center\" | || | align=\"center\" | || |}"}
+{"_id": "2876", "title": "Equivalence of Definitions of Integral Dependence", "text": "Let $A$ be an extension of a commutative ring with unity $R$. For $x \\in A$, the following are equivalent: {{begin-eqn}} {{eqn | n = 1       | o =       | r = \\) $x$ is integral over $R$ \\( }} {{eqn | n = 2       | o =       | r = \\) The $R$-module $R \\left[{x}\\right]$ is finitely generated \\( }} {{eqn | n = 3       | o =       | r = \\) There exists a subring $B$ of $A$ such that $x \\in B$, $R \\subseteq B$ and $B$ is a finitely generated $R$-module \\( }} {{eqn | n = 4       | o =       | r = \\) There exists a subring $B$ of $A$ such that $x B \\subseteq B$ and $B$ is finitely generated over $R$ \\( }} {{eqn | n = 5       | o =       | r = \\) There exists a faithful $R \\left[{x}\\right]$-module $B$ that is finitely generated as an $R$-module \\( }} {{end-eqn}}"}
+{"_id": "2877", "title": "Transitivity of Finite Generation", "text": "Let $A \\subseteq B \\subseteq C$ be rings. Suppose $B$ is a finitely generated $A$-module, and $C$ is a finitely generated $B$-module. Then $C$ is a finitely generated $A$-module."}
+{"_id": "2878", "title": "Integral Closure is Subring", "text": "Let $A$ be an extension of a commutative ring with unity $\\struct {R, +, \\circ}$. Let $C$ be the integral closure of $R$ in $A$. Then $C$ is a subring of $A$."}
+{"_id": "2879", "title": "Localization of Ring is Unique", "text": "Let $A$ be a commutative ring with unity. Let $S \\subseteq A$ be a multiplicatively closed subset. Let $\\left({A_S, \\iota}\\right)$ and $\\left({\\tilde A_S, \\tilde \\iota}\\right)$ both satisfy the definition of the localization of $A$ at $S$. Then there is a canonical isomorphism $\\phi: A_S \\to \\tilde A_S$."}
+{"_id": "2880", "title": "Spectrum of Ring is Nonempty", "text": "Let $A$ be a non-trivial commutative ring with unity. Then its prime spectrum is non-empty: :$\\Spec A \\ne \\O$"}
+{"_id": "2881", "title": "Compact Space is Strongly Locally Compact", "text": "Let $T = \\struct {S, \\tau}$ be a compact space. Then $T$ is a strongly locally compact space."}
+{"_id": "2882", "title": "Compact Space is Weakly Sigma-Locally Compact", "text": "Let $T = \\struct {S, \\tau}$ be a compact space. Then $T$ is a weakly $\\sigma$-locally compact space."}
+{"_id": "2883", "title": "Sequence of Implications of Paracompactness Properties", "text": "Let $P_1$ and $P_2$ be paracompactness properties and let: :$P_1 \\implies P_2$ mean: :If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | Compact || | align=\"center\" | $\\implies$ || | align=\"center\" | Countably Compact || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | Fully Normal || | align=\"center\" | $\\implies$ || | align=\"center\" | Paracompact || | align=\"center\" | $\\implies$ || | align=\"center\" | Countably Paracompact || |- | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | Fully $T_4$ || | align=\"center\" | || | align=\"center\" | Metacompact || | align=\"center\" | $\\implies$ || | align=\"center\" | Countably Metacompact || |- | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || |- | align=\"center\" | $T_4$ || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || |}"}
+{"_id": "2884", "title": "Compactness Properties in Hausdorff Spaces", "text": "Let $P_1$ and $P_2$ be compactness properties and let: :$P_1 \\implies P_2$ mean: :If a $T_2$ (Hausdorff) space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align=\"center\" | Fully $T_4$ || | align=\"center\" | $\\iff$ || | align=\"center\" | Paracompact || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | Weakly $\\sigma$-Locally Compact || | align=\"center\" | $\\implies$ || | align=\"center\" | $T_4$ || |- | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | Weakly Locally Compact || | align=\"center\" | $\\implies$ || | align=\"center\" | $T_{3 \\frac 1 2}$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | First-Countable and Countably Compact|| | align=\"center\" | $\\implies$ || | align=\"center\" | $T_3$ || |}"}
+{"_id": "2885", "title": "Zero Locus of Set is Zero Locus of Generated Ideal", "text": "Let $k$ be a field. Let $n\\geq1$ be a natural number. Let $A = k \\left[{X_1, \\ldots, X_n}\\right]$ be the ring of polynomial functions in $n$ variables over $k$. Let $T \\subseteq A$ be a subset, and $V \\left({T}\\right)$ the zero locus of $T$. Let $J = \\left({T}\\right)$ be the ideal generated by $T$. Then: : $V \\left({T}\\right) = V \\left({J}\\right)$"}
+{"_id": "2886", "title": "Compactness Properties in T3 Spaces", "text": "Let $P_1$ and $P_2$ be compactness properties and let: :$P_1 \\implies P_2$ mean: :If a $T_3$ space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align=\"center\" | Second-Countable || | align=\"center\" | $\\implies$ || | align=\"center\" | Lindelöf || |- | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | $\\Downarrow$|| | align=\"center\" | || | align=\"center\" | Fully $T_4$ $\\iff$ Paracompact || |- | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | $T_5$ || | align=\"center\" | $\\implies$ || | align=\"center\" | $T_4$ || |- |}"}
+{"_id": "2887", "title": "Finite Product of Sigma-Compact Spaces is Sigma-Compact", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\set {\\struct {S_i, \\tau_i}: 1 \\le i \\le n}$ be a finite set of topological spaces. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{i \\mathop = 1}^n \\struct {S_i, \\tau_i}$ be the product space of $\\set {\\struct {S_i, \\tau_i}: 1 \\le i \\le n}$. Let each of $\\struct {S_i, \\tau_i}$ be $\\sigma$-compact. Then $\\struct {S, \\tau}$ is also $\\sigma$-compact."}
+{"_id": "2888", "title": "Countable Product of Sequentially Compact Spaces is Sequentially Compact", "text": "Let $I$ be an indexing set with countable cardinality. Let $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be a family of topological spaces indexed by $I$. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let each of $\\struct {S_\\alpha, \\tau_\\alpha}$ be sequentially compact. Then $\\struct {S, \\tau}$ is also sequentially compact."}
+{"_id": "2889", "title": "Finite Product of Weakly Locally Compact Spaces is Weakly Locally Compact", "text": "Let $n \\in \\Z_{\\ge 0}$ be a (strictly) positive integer. Let $\\set {\\struct {S_i, \\tau_i}: 1 \\le i \\le n}$ be a finite set of topological spaces. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{i \\mathop = 1}^n \\struct {S_i, \\tau_i}$ be the product space of $\\set {\\struct {S_i, \\tau_i}: 1 \\le i \\le n}$. Let each of $\\struct {S_i, \\tau_i}$ be weakly locally compact. Then $\\struct {S, \\tau}$ is also weakly locally compact."}
+{"_id": "2891", "title": "Countable Product of Second-Countable Spaces is Second-Countable", "text": "Let $I$ be an indexing set with countable cardinality. Let $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be a family of topological spaces indexed by $I$. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let each of $\\struct {S_\\alpha, \\tau_\\alpha}$ be second-countable. Then $\\struct {S, \\tau}$ is also second-countable."}
+{"_id": "2892", "title": "Countable Product of Separable Spaces is Separable", "text": "Let $I$ be an indexing set with countable cardinality. Let $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be a family of topological spaces indexed by $I$. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let each of $\\struct {S_\\alpha, \\tau_\\alpha}$ be separable. Then $\\struct {S, \\tau}$ is also separable."}
+{"_id": "2893", "title": "Compactness Properties Preserved under Continuous Surjection", "text": "Let $T_A = \\strict {S_A, \\tau_A}$ and $T_B = \\strict {S_B, \\tau_B}$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a continuous surjection. If $T_A$ has one of the following properties, then $T_B$ has the same property: :Compact Space :$\\sigma$-Compact Space :Countable Compact Space :Sequential Compact Space :Lindelöf Space"}
+{"_id": "2894", "title": "Projection from Product Topology is Open", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ and $T_2 = \\struct {S_2, \\tau_2}$ be topological spaces. Let $T = \\struct {T_1 \\times T_2, \\tau}$ be the product space of $T_1$ and $T_2$, where $\\tau$ is the Tychonoff topology on $S$. Let $\\pr_1: T \\to T_1$ and $\\pr_2: T \\to T_2$ be the first and second projections from $T$ onto its factors. Then both $\\pr_1$ and $\\pr_2$ are open."}
+{"_id": "2895", "title": "Weak Local Compactness is Preserved under Open Continuous Surjection", "text": "Let $T_A = \\struct {S_A, \\tau_A}$ and $T_B = \\struct {S_B, \\tau_B}$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a continuous mapping which is also an open mapping and a surjection. If $T_A$ is weakly locally compact, then $T_B$ is also weakly locally compact."}
+{"_id": "2896", "title": "Compactness Properties Preserved under Projection Mapping", "text": "Let $I$ be an indexing set with countable cardinality. Let $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be a family of topological spaces indexed by $I$. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let $\\pr_\\alpha: \\struct {S, \\tau} \\to \\struct {S_\\alpha, \\tau_\\alpha}$ be the projection on the $\\alpha$ coordinate. Then $\\pr_\\alpha$ preserves the following compactness properties. That is, if $\\struct {S, \\tau}$ has one of the following properties, then each of $\\struct {S_\\alpha, \\tau_\\alpha}$ has the same property: :Compact Space :$\\sigma$-Compact Space :Countably Compact Space :Sequentially Compact Space :Lindelöf Space :Locally Compact Space :Weakly Locally Compact Space :Paracompact Space"}
+{"_id": "2897", "title": "Countability Properties Preserved under Projection Mapping", "text": "Let $\\sequence{\\struct{S_\\alpha, \\tau_\\alpha}}$ be a sequence of topological spaces. Let $\\displaystyle \\struct{S, \\tau} = \\prod \\struct{S_\\alpha, \\tau_\\alpha}$ be the product space of $\\sequence {\\struct{S_\\alpha, \\tau_\\alpha}}$. Let $\\pr_\\alpha: \\struct{S, \\tau} \\to \\struct{S_\\alpha, \\tau_\\alpha}$ be the projection on the $\\alpha$ coordinate. Then $\\pr_\\alpha$ preserves the following countability properties. That is, if $\\struct{S, \\tau}$ has one of the following properties, then each of $\\struct{S_\\alpha, \\tau_\\alpha}$ has the same property. :Separability :First-Countability :Second-Countability"}
+{"_id": "2898", "title": "Countability Axioms Preserved under Open Continuous Surjection", "text": "Let $T_A = \\left({S_A, \\tau_A}\\right)$ and $T_B = \\left({S_B, \\tau_B}\\right)$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a surjective open mapping which is also continuous. If $T_A$ has one of the following properties, then $T_B$ has the same property: :First-Countability :Second-Countability"}
+{"_id": "2899", "title": "Connectedness of Points is Equivalence Relation", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $a \\sim b $ denote the relation: :$a \\sim b \\iff a$ is connected to $b$ where $a, b \\in S$. Then $\\sim$ is an equivalence relation."}
+{"_id": "2901", "title": "Surgery for Rings", "text": "Let $R$ and $S$ be commutative rings with unity, and $\\phi: R \\to S$ a ring monomorphism. Then there is a ring $T$ isomorphic to $S$ that contains $R$ as a subring."}
+{"_id": "2902", "title": "Clopen Set contains Components of All its Points", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be both closed and open in $T$. Then $H$ contains the components of all of its points."}
+{"_id": "2903", "title": "Connectedness Between Two Points is an Equivalence Relation", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $a \\sim b $ denote the relation: :$a \\sim b \\iff \\exists S \\subseteq T: S$ is connected between the two points  $a$ and $b$ where $a, b \\in X$. Then $\\sim$ is an equivalence relation."}
+{"_id": "2904", "title": "Quasicomponent is Intersection of Clopen Sets", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $p \\in S$. Then the quasicomponent containing $p$ equals the intersection of all sets which are both open and closed which contain $p$."}
+{"_id": "2906", "title": "Localization of Ring Exists", "text": "Let $A$ be a commutative ring with unity. Let $S \\subseteq A$ be a multiplicatively closed subset with $0 \\notin S$. Then there exists a localization $\\struct {A_S, \\iota}$ of $A$ at $S$."}
+{"_id": "2907", "title": "Transitivity of Integrality", "text": "Let $A \\subseteq B \\subseteq C$ be extensions of commutative rings with unity. Suppose that $C$ is integral over $B$, and $B$ is integral over $A$. Then $C$ is integral over $A$."}
+{"_id": "2909", "title": "Universal Property for Field of Quotients", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain. Let $\\struct {F, \\oplus, \\cdot}$ be a field of quotients of $D$. Then $F$ satisfies the following universal property: There exists a (ring) homomorphism $\\iota : D \\to F$ such that: ::for every field $\\tilde F$ and :and: ::for every (ring) homomorphism $\\phi: D \\to \\tilde F$ :there exists a unique field homomorphism $\\psi: F \\to \\tilde F$ satisfying: :::$\\psi \\iota = \\phi$ That is, the following diagram commutes: :Universal Property Namely we may take: :$\\psi: a / b \\mapsto \\map \\phi a \\map \\phi b^{-1}$"}
+{"_id": "2910", "title": "Constant Mapping is Continuous", "text": "Let $T_A = \\left({A, \\tau_A}\\right)$ and $T_B = \\left({B, \\tau_B}\\right)$ be topological spaces. Let $b \\in B$ be any point in $B$. Let $f_b: A \\to B$ be the constant mapping defined by: :$\\forall x \\in A: f_b \\left({x}\\right) = b$ Then $f_b$ is continuous."}
+{"_id": "2911", "title": "Path-Connectedness is Equivalence Relation", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $a \\sim b $ denote the relation: :$a \\sim b \\iff a$ is path-connected to $b$ where $a, b \\in S$. Then $\\sim$ is an equivalence relation."}
+{"_id": "2912", "title": "Arc-Connectedness is Equivalence Relation", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $a \\sim b $ denote the relation: :$a \\sim b \\iff a$ is arc-connected to $b$ where $a, b \\in S$. Then $\\sim$ is an equivalence relation."}
+{"_id": "2913", "title": "Joining Arcs makes Another Arc", "text": "Let $T$ be a topological space. Let $\\mathbb I \\subseteq \\R$ be the closed unit interval $\\left[{0 \\,.\\,.\\, 1}\\right]$. Let $a,b,c$ be three distinct points of $T$. Let $f, g: \\mathbb I \\to T$ be arcs in $T$ from $a$ to $b$ and from $b$ to $c$ respectively. Let $h: \\mathbb I \\to T$ be the mapping given by: :$h \\left({x}\\right) = \\begin{cases} f \\left({2 x}\\right) & : x \\in \\left[{0 \\,.\\,.\\, \\dfrac 1 2}\\right] \\\\ g \\left({2 x - 1}\\right) & : x \\in \\left[{\\dfrac 1 2 \\,.\\,.\\, 1}\\right] \\end{cases}$ Then either: :$h$ is an arc in $T$ or :There exists some restriction of $h$ which, possibly after reparametrisation, is an arc in $T$."}
+{"_id": "2914", "title": "Arc in Topological Space is Path", "text": "Let $T = \\struct {X, \\tau}$ be a topological space. Let $f$ be an arc in $T$. Then $f$ is a path in $T$."}
+{"_id": "2915", "title": "Vector Times Magnitude Same Length As Magnitude Times Vector", "text": "Let $\\mathbf u$ and $\\mathbf v$ be two vectors in the vector space $\\struct {G, +_G, \\circ}_K$ over a division ring $\\struct {K, +_K, \\times}$ with subfield $\\R$ such that $\\R \\subseteq \\map Z K$ with $\\map Z K$ the center of $K$ Let $\\norm {\\mathbf u}$ and $\\norm {\\mathbf v}$ be the lengths of $\\mathbf u$ and $\\mathbf v$ respectively. Then: :$\\norm {\\paren {\\norm {\\mathbf v} \\circ \\mathbf u} } = \\norm {\\paren {\\norm {\\mathbf u} \\circ \\mathbf v} }$"}
+{"_id": "2917", "title": "Increasing Sequence in Ordered Set Terminates iff Maximal Element", "text": "Let $\\left({P, \\le}\\right)$ be an ordered set. {{TFAE}} : $(1): \\quad$ Every increasing sequence $x_1 \\leq x_2 \\leq x_3 \\leq \\cdots$ with $x_i \\in P$ eventually terminates: there is $n \\in \\N$ such that $x_n = x_{n+1} = \\cdots$. : $(2): \\quad$ Every non-empty subset of $P$ has a maximal element."}
+{"_id": "2919", "title": "Characterisation of UFDs", "text": "Let $A$ be an integral domain. {{TFAE}} : $(1): \\quad A$ is a unique factorisation domain  : $(2): \\quad A$ is a GCD domain satisfying the ascending chain condition on principal ideals. : $(3): \\quad A$ satisfies the ascending chain condition on principal ideals and every irreducible element of $A$ is a prime element of $A$."}
+{"_id": "2921", "title": "Ultraconnected Space is Path-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is ultraconnected. Then $T$ is path-connected."}
+{"_id": "2922", "title": "Continuous Real-Valued Function on Irreducible Space is Constant", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is irreducible. Let $f: S \\to \\R$ be a continuous real-valued function. Then $f$ is constant, that is: :$\\exists a \\in \\R: \\forall x \\in S: \\map f x = a$"}
+{"_id": "2923", "title": "Irreducible Space is Pseudocompact", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is irreducible. Then $T$ is pseudocompact."}
+{"_id": "2925", "title": "Arc-Connected Space is Path-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is arc-connected. Then $T$ is path-connected."}
+{"_id": "2926", "title": "Irreducible Space is Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is irreducible. Then $T$ is connected."}
+{"_id": "2927", "title": "Non-Trivial Ultraconnected Space is not T1", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is ultraconnected. If $S$ has more than one element, then $T$ is not a $T_1$ (Fréchet) space. That is, if $T$ is a $T_1$ (Fréchet) space with more than one element, it is not ultraconnected."}
+{"_id": "2929", "title": "Dot Product of Vector with Itself", "text": ":$\\mathbf u \\cdot \\mathbf u = \\norm {\\mathbf u}^2$"}
+{"_id": "2930", "title": "Cosine Formula for Dot Product", "text": "Let $\\mathbf v,\\mathbf w$ be two non-zero vectors in $\\R^n$. The dot product of $\\mathbf v$ and $\\mathbf w$ can be calculated by: :$\\mathbf v \\cdot \\mathbf w = \\norm {\\mathbf v} \\norm {\\mathbf w} \\cos \\theta$ where: :$\\norm {\\, \\cdot \\,}$ denotes vector length and :$\\theta$ is the angle between $\\mathbf v$ and $\\mathbf w$."}
+{"_id": "2931", "title": "Quasicomponents and Components are Equal in Locally Connected Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is locally connected. Then $A \\subseteq S$ is a component of $T$ {{iff}} $A \\subseteq S$ is a quasicomponent of $T$."}
+{"_id": "2932", "title": "Irreducible Space is Locally Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is irreducible. Then $T$ is locally connected."}
+{"_id": "2933", "title": "Quasicomponents and Path Components are Equal in Locally Path-Connected Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is locally path-connected. Then $A \\subseteq S$ is a path component of $T$ {{iff}} $A \\subseteq S$ is a quasicomponent of $T$."}
+{"_id": "2934", "title": "Quasicomponents and Arc Components are Equal in Locally Arc-Connected Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is locally arc-connected. Then $A \\subseteq S$ is an arc component of $T$ {{iff}} $A$ is a quasicomponent of $T$."}
+{"_id": "2936", "title": "Locally Path-Connected Space is Locally Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is locally path-connected. Then $T$ is also locally connected."}
+{"_id": "2938", "title": "UFD is GCD Domain", "text": "Let $A$ be a unique factorisation domain. Then $A$ is a GCD domain."}
+{"_id": "2941", "title": "Totally Disconnected Space is T1", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is totally disconnected. Then $T$ is a $T_1$ (Fréchet) space."}
+{"_id": "2942", "title": "Determinant of Linear Operator is Well Defined", "text": "Let $V$ be a nontrivial finite dimensional vector space over a field $K$. Let $A: V \\to V$ be a linear operator of $V$. Then the determinant $\\det A$ of $A$ is well defined, that is, does not depend on the choice of a basis of $V$."}
+{"_id": "2943", "title": "Localization Preserves Integral Closure", "text": "Let $A \\subseteq B$ be an extension of commutative rings with unity. Let $C$ be the integral closure of $A$ in $B$. Let $S \\subseteq A$ be a multiplicatively closed subset. Then $C_S$ is the integral closure of $A_S$ in $B_S$, where subscript $S$ indicates the localization at $S$."}
+{"_id": "2944", "title": "Integrally Closed is Local Property", "text": "Let $A$ be an integral domain. For a prime ideal $\\mathfrak p$ of $A$, let $A_{\\mathfrak p}$ denote the localization at $S = A \\divides \\mathfrak p$. Then the following are equivalent: :$(1): \\quad A$ is integrally closed :$(2): \\quad A_{\\mathfrak p}$ is integrally closed for all prime ideals $\\mathfrak p$. :$(3): \\quad A_{\\mathfrak m}$ is integrally closed for all maximal ideals $\\mathfrak m$."}
+{"_id": "2945", "title": "Totally Disconnected and Locally Connected Space is Discrete", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is both totally disconnected and locally connected. Then $T$ is the discrete space on $S$."}
+{"_id": "2948", "title": "Totally Separated Space is Completely Hausdorff and Urysohn", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is totally separated. Then $T$ is completely Hausdorff and Urysohn."}
+{"_id": "2949", "title": "Identity Morphism is Unique", "text": "Let $\\mathbf C$ be a category, and $X$ and object of $\\mathbf C$. Then the identity morphism $\\operatorname{id}_X : X \\to X$ is unique."}
+{"_id": "2951", "title": "Category of Sets is Category", "text": "Let $\\mathbf{Set}$ be the category of sets. Then $\\mathbf{Set}$ is a metacategory."}
+{"_id": "2953", "title": "Projection from Product Category", "text": "Let $\\mathcal C$ and $\\mathcal D$ be categories. Let $\\mathcal C \\times \\mathcal D$ be the product category. The ''projections'': :$\\pi_{\\mathcal C} : \\mathcal C \\times \\mathcal D \\to \\mathcal C : (f,g) \\mapsto f$ :$\\pi_{\\mathcal D} : \\mathcal C \\times \\mathcal D \\to \\mathcal D : (f,g) \\mapsto g$ where $(f,g) \\in \\operatorname{ob}(\\mathcal C \\times \\mathcal D)$ or $(f,g)\\in \\operatorname{mor}(\\mathcal C \\times \\mathcal D)$ are functors. Moreover, $\\pi_{\\mathcal C}$ and $\\pi_{\\mathcal D}$ satisfy the following universal property: :''For any category $\\mathcal E$ and any functors $F : \\mathcal E \\to \\mathcal C$, $G : \\mathcal E \\to \\mathcal D$, there exists a unique functor $H : \\mathcal E \\to \\mathcal C \\times \\mathcal D$ such that $F = \\pi_{\\mathcal C}H$ and $G = \\pi_{\\mathcal D}H$'' That is, the following diagram commutes: :380px {{improve|XyJax?}}"}
+{"_id": "2955", "title": "Extremally Disconnected Space is Totally Separated", "text": "Let $T = \\struct {S, \\tau}$ be an extremally disconnected topological space. Then $T$ is totally separated."}
+{"_id": "2956", "title": "Zero Dimensional Space is T3", "text": "Let $T = \\struct {S, \\tau}$ be a zero dimensional topological space. Then $T$ is a $T_3$ space."}
+{"_id": "2957", "title": "Zero Dimensional T0 Space is Totally Separated", "text": "Let $T = \\struct {S, \\tau}$ be a zero dimensional topological space which is also a $T_0$ (Kolmogorov) space. Then $T$ is totally separated."}
+{"_id": "2958", "title": "Scattered T1 Space is Totally Disconnected", "text": "Let $T = \\struct {S, \\tau}$ be a scattered topological space which is also a $T_1$ (Fréchet) space. Then $T$ is totally disconnected."}
+{"_id": "2959", "title": "Non-Trivial Connected Set in T1 Space is Dense-in-itself", "text": "Let $T = \\struct {S, \\tau}$ be a $T_1$ (Fréchet) topological space. Let $H \\subseteq S$ be connected in $T$. If $H$ has more than one element, then $H$ is dense-in-itself."}
+{"_id": "2960", "title": "Sequence of Implications of Disconnectedness Properties", "text": "Let $P_1$ and $P_2$ be disconnectedness properties and let: :$P_1 \\implies P_2$ mean: :If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align=\"center\" | Regular|| | align=\"center\" | $\\impliedby$ || | align=\"center\" | Zero Dimensional and $T_0$|| | align=\"center\" | $\\impliedby$ || | align=\"center\" | Discrete Space || | align=\"center\" | $\\implies$ || | align=\"center\" | $\\implies$ || | align=\"center\" | $\\implies$ || | align=\"center\" | Scattered and $T_1$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | Extremally Disconnected || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\implies$ || | align=\"center\" | $\\implies$ || | align=\"center\" | Totally Separated || | align=\"center\" | $\\implies$ || | align=\"center\" | Urysohn || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $T_1$ || | align=\"center\" | $\\impliedby$ || | align=\"center\" | Totally Disconnected || | align=\"center\" | $\\impliedby$ || | align=\"center\" | $\\impliedby$ || | align=\"center\" | $\\impliedby$ || | align=\"center\" | $\\impliedby$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | || | align=\"center\" | Totally Pathwise Disconnected || |}"}
+{"_id": "2962", "title": "Set with Dispersion Point is Biconnected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be a connected set in $T$. Let $p \\in H$ be a dispersion point of $H$. Then $H$ is biconnected."}
+{"_id": "2963", "title": "Totally Disconnected Space is Punctiform", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is totally disconnected. Then $T$ is punctiform."}
+{"_id": "2964", "title": "Distance in Pseudometric is Non-Negative", "text": "Let $X$ be a set on which a pseudometric $d: X \\times X \\to \\R$ has been imposed. Then: :$\\forall x, y \\in X: \\map d {x, y} \\ge 0$"}
+{"_id": "2965", "title": "Metric Defines Norm iff it Preserves Linear Structure", "text": "Let $\\struct {k, \\norm{\\,\\cdot\\,}_k}$ be a valued field. Let $V$ be a vector space over the valued field $\\struct {k, \\norm{\\,\\cdot\\,}_k}$. Let $d : V \\times V \\to k$ be a metric on $V$. Then the function $\\norm{ v } := d(v,0)$ is a norm on $V$ if and only if for all $x,y,z \\in V$, $\\lambda \\in k$: :$(1): \\quad d(x+z,y+z) = d(x,y)$ (homogeneity or translation invariance) :$(2): \\quad d(\\lambda x, \\lambda y) = \\norm{\\lambda}_k d(x,y)$ (the enlargement property)"}
+{"_id": "2966", "title": "Set of all Sets", "text": "Forming the set $\\SS$ of all sets leads to a contradiction."}
+{"_id": "2967", "title": "Real Numbers form Perfect Set", "text": "Consider the set of real numbers $\\R$ as a (complete) metric space with the usual (Euclidean) metric. Then $\\R$ forms a perfect set."}
+{"_id": "2968", "title": "Closed Interval in Reals is Uncountable", "text": "Let $a, b$ be extended real numbers such that $a < b$. Then the closed interval $\\set {x \\in \\R : a \\le x \\le b} \\subseteq \\R$ is uncountable."}
+{"_id": "2969", "title": "Metric Space is T5", "text": "Let $M = \\struct {A, d}$ be a metric space. Then $M$ is a $T_5$ space."}
+{"_id": "2970", "title": "Metric Space is Perfectly T4", "text": "Let $M = \\struct {A, d}$ be a metric space. Then $M$ is a perfectly $T_4$ space."}
+{"_id": "2971", "title": "Metric Space fulfils all Separation Axioms", "text": "Let $M = \\struct {A, d}$ be a metric space. Then $M$, considered as a topological space, fulfils all separation axioms: :$M$ is a $T_0$ (Kolmogorov) space :$M$ is a $T_1$ (Fréchet) space :$M$ is a $T_2$ (Hausdorff) space :$M$ is a semiregular space :$M$ is a $T_{2 \\frac 1 2}$ (completely Hausdorff) space  :$M$ is a $T_3$ space  :$M$ is a regular space :$M$ is an Urysohn space :$M$ is a $T_{3 \\frac 1 2}$ space :$M$ is a Tychonoff (completely regular) space :$M$ is a $T_4$ space :$M$ is a normal space :$M$ is a $T_5$ space  :$M$ is a completely normal space :$M$ is a perfectly $T_4$ space :$M$ is a perfectly normal space"}
+{"_id": "2972", "title": "Metric Space is Fully T4", "text": "Let $M = \\struct {A, d}$ be a metric space. Then $M$ is a fully $T_4$ space."}
+{"_id": "2973", "title": "Metric Space is Fully Normal", "text": "Let $M = \\struct {A, d}$ be a metric space. Then $M$ is a fully normal space."}
+{"_id": "2974", "title": "Metric Space is Paracompact", "text": "Let $M = \\struct {A, d}$ be a metric space. Then $M$ is a paracompact space."}
+{"_id": "2975", "title": "Metric Space is First-Countable", "text": "Let $M = \\struct {A, d}$ be a metric space. Then $M$ is first-countable."}
+{"_id": "2976", "title": "Metric Space is Separable iff Second-Countable", "text": "A metric space is separable {{iff}} it is second-countable."}
+{"_id": "2977", "title": "Metric Space is Lindelöf iff Second-Countable", "text": "Let $M = \\struct {A, d}$ be a metric space. Then $M$ is Lindelöf {{iff}} $M$ is second-countable."}
+{"_id": "2978", "title": "Metric Space is Countably Compact iff Sequentially Compact", "text": "Let $M$ be a metric space. Then $M$ is countably compact {{iff}} $M$ is sequentially compact."}
+{"_id": "2979", "title": "Metric Space is Weakly Countably Compact iff Countably Compact", "text": "Let $M = \\struct {A, d}$ be a metric space. Then $M$ is weakly countably compact {{iff}} $M$ is countably compact."}
+{"_id": "2980", "title": "Countably Compact Metric Space is Compact", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $M$ be countably compact. Then $M$ is compact."}
+{"_id": "2981", "title": "Metric Space is Weakly Locally Compact iff Strongly Locally Compact", "text": "Let $M = \\struct {A, d}$ be a metric space. Then $M$ is weakly locally compact {{iff}} $M$ is strongly locally compact."}
+{"_id": "2982", "title": "Sequence of Implications of Metric Space Compactness Properties", "text": "Let $P_1$ and $P_2$ be compactness properties and let: :$P_1 \\implies P_2$ mean: :If a metric space $M$ satsifies property $P_1$, then $M$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align=\"center\" | Sequentially Compact || | align=\"center\" | $\\implies$ || | align=\"center\" | Weakly $\\sigma$-Locally Compact || | align=\"center\" | $\\implies$ || | align=\"center\" | Weakly Locally Compact || |- | align=\"center\" | $\\Big\\Updownarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Updownarrow$ || |- | align=\"center\" | Countably Compact || | align=\"center\" | || | align=\"center\" | $\\sigma$-Compact || | align=\"center\" | || | align=\"center\" | Strongly Locally Compact || |- | align=\"center\" | $\\Big\\Updownarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | Compact || | align=\"center\" | || | align=\"center\" | Separable || |- | align=\"center\" | $\\Big\\Updownarrow$ || | align=\"center\" | || | align=\"center\" | $\\Big\\Updownarrow$ || |- | align=\"center\" | Weakly Countably Compact || | align=\"center\" | || | align=\"center\" | Lindelöf || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Updownarrow$ || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | Second-Countable || |}"}
+{"_id": "2983", "title": "Extremally Disconnected Metric Space is Discrete", "text": "Let $M = \\struct {A, d}$ be a metric space which is extremally disconnected. Then $M$ is the discrete topology."}
+{"_id": "2985", "title": "Indicator is Well-Defined", "text": "Let $G$ be a finite group, and $a \\in G$. Let $H$ be a subgroup of $G$. Then the indicator of $a$ in $H$ is well defined."}
+{"_id": "2986", "title": "Compact Metric Space is Totally Bounded", "text": "Let $M = \\struct {A, d}$ be a metric space which is compact. Then $M$ is totally bounded."}
+{"_id": "2987", "title": "Subgroup Generated by Subgroup and Element", "text": "Let $G$ be a finite abelian group. Let $H$ be a proper subgroup of $G$. Let $a \\in G \\setminus H$. Let $n$ be the indicator of $a$ in $H$. Then: :$K = \\left\\{{x a^k: x \\in H, \\ 0 \\le k < n}\\right\\}$ is a subgroup of $G$ such that $H \\subseteq K$, and each element of $K$ has a unique representation in this form. Moreover: :$K = \\left\\langle{H, a}\\right\\rangle$ where $\\left\\langle{H, a}\\right\\rangle$ denotes the subgroup generated by $\\left\\{{H, a}\\right\\}$, and: :$\\left\\vert{K}\\right\\vert = n \\left\\vert{H}\\right\\vert$ where $\\left\\vert{K}\\right\\vert$ denotes the order of $K$."}
+{"_id": "2989", "title": "Number of Characters on Finite Abelian Group", "text": "Let $G$ be a finite abelian group. Then the number of characters $G \\to \\C^\\times$ is $\\left\\vert{G}\\right\\vert$."}
+{"_id": "2990", "title": "Sequentially Compact Metric Space is Complete", "text": "Let $M = \\struct {A, d}$ be a metric space which is sequentially compact. Then $M$ is complete."}
+{"_id": "2991", "title": "Topological Completeness is Topological Property", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ and $T_2 = \\struct {S_2, \\tau_2}$ be topological spaces which are homeomorphic. Let $T_1$ be topologically complete. Then $T_2$ is also topologically complete."}
+{"_id": "2992", "title": "Topological Completeness is Weakly Hereditary", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is topologically complete. Let $V \\subseteq S$ be a closed subspace of $T$. Then $V$ is also topologically complete. That is, topological completeness is weakly hereditary."}
+{"_id": "2993", "title": "Metric Space Compact iff Complete in All Equivalent Metrics", "text": "Let $M_1 = \\struct {A, d_1}$ be a metric space. Then $M_1$ is compact {{iff}} $M_2 = \\struct {A, d_2}$ is a complete metric space whenever $d_2$ is equivalent to $d_1$."}
+{"_id": "2995", "title": "Urysohn's Metrization Theorem", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is regular and second-countable. Then $T$ is metrizable."}
+{"_id": "2996", "title": "Metrizable iff Regular and has Sigma-Locally Finite Basis", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then $T$ is metrizable {{iff}} $T$ is regular and has a $\\sigma$-locally finite basis."}
+{"_id": "2997", "title": "Uniformity iff Quasiuniformity has Symmetric Basis", "text": "Let $S$ be a set. Let $\\UU$ be a quasiuniformity on $S$. Then $\\UU$ is a uniformity {{iff}} $\\UU$ has a symmetric filter basis."}
+{"_id": "2998", "title": "Quasiuniformity Induces Topology", "text": "Let $\\UU$ be a quasiuniformity on a set $S$. Let $\\tau \\subseteq \\powerset S$ be a subset of the power set of $S$, created from $\\UU$ by: :$\\tau := \\set {\\map u x: u \\in \\UU, x \\in S}$ where: :$\\forall x \\in S: \\map u x = \\set {y: \\tuple {x, y} \\in u}$ That is, where $\\map u x$ is the image of $x$ under $u$, where $u$ is considered as a relation on $S$. Then $\\tau$ is a topology on $S$. That is, the quasiuniform space $\\struct {\\struct {S, \\UU}, \\tau}$ is also the topological space $\\struct {S, \\tau}$."}
+{"_id": "2999", "title": "Topological Space is Quasiuniformizable", "text": "Every topological space is quasiuniformizable."}
+{"_id": "3000", "title": "T3 1/2 Space is Uniformizable", "text": "Let $T$ be a topological space. Then $T$ is uniformizable {{iff}} $T$ is a $T_{3 \\frac 1 2}$ space."}
+{"_id": "3001", "title": "Pseudometric Space generates Uniformity", "text": "Let $P = \\struct {A, d}$ be a pseudometric space. Let $\\UU$ be the set of sets defined as: :$\\UU := \\set {u \\in A \\times A: \\exists \\epsilon \\in \\R_{>0}: u_\\epsilon \\subseteq u}$ where: :$\\R_{>0}$ is the set of strictly positive real numbers :$u_\\epsilon$ is defined as: ::$u_\\epsilon := \\set {\\tuple {x, y}: \\map d {x, y} < \\epsilon}$ Then $\\UU$ is a uniformity on $A$ which generates a uniform space with the same topology as the topology induced by $d$."}
+{"_id": "3002", "title": "Definition:Pseudometrizable Uniformity", "text": "Let $P = \\struct {A, d}$ be a pseudometric space. Let $\\UU$ be the uniformity on $A$ defined as: :$\\UU := \\set {u_\\epsilon: \\epsilon \\in \\R_{>0} }$ where: :$\\R_{>0}$ is the set of strictly positive real numbers :$u_\\epsilon$ is defined as: ::$u_\\epsilon := \\set {\\tuple {x, y}: \\map d {x, y} < \\epsilon}$ Then $\\UU$ is defined as '''pseudometrizable'''."}
+{"_id": "3003", "title": "Metric Space generates Uniformity", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $\\UU$ be the set of sets defined as: :$\\UU := \\set {u \\in A \\times A: \\exists \\epsilon \\in \\R_{>0}: u_\\epsilon \\subseteq u}$ where: :$\\R_{>0}$ is the set of strictly positive real numbers :$u_\\epsilon$ is defined as: ::$u_\\epsilon := \\set {\\tuple {x, y}: \\map d {x, y} < \\epsilon}$ Then $\\UU$ is a uniformity on $A$ which generates a uniform space with the same topology as the topology induced by $d$."}
+{"_id": "3005", "title": "Discrete Topology is Topology", "text": ":$\\tau$ is a topology on $S$."}
+{"_id": "3006", "title": "Discrete Topology is Finest Topology", "text": ":$\\tau$ is the finest topology on $S$."}
+{"_id": "3007", "title": "Set in Discrete Topology is Clopen", "text": ":$\\forall U \\subseteq S: U$ is both closed and open in $\\struct {S, \\tau}$."}
+{"_id": "3008", "title": "Topological Space is Discrete iff All Points are Isolated", "text": ":$\\tau$ is the discrete topology on $S$ {{iff}} all points in $S$ are isolated points of $T$."}
+{"_id": "3009", "title": "Point in Discrete Space is Adherent Point", "text": "Let $T = \\struct {S, \\tau}$ be a discrete topological space. Let $U \\subseteq S$. Then $x$ is an adherent point of $U$ {{iff}} $x \\in U$."}
+{"_id": "3010", "title": "Interior Equals Closure of Subset of Discrete Space", "text": "Let $A \\subseteq S$. Then: :$A = A^\\circ = A^-$ where: :$A^\\circ$ is the interior of $A$ :$A^-$ is the closure of $A$."}
+{"_id": "3012", "title": "Mapping from Discrete Space is Continuous", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ be the discrete topological space on $S_1$. Let $T_2 = \\struct {S_2, \\tau_2}$ be any other topological space. Let $\\phi: S_1 \\to S_2$ be a mapping. Then $\\phi$ is continuous."}
+{"_id": "3013", "title": "Exponential Function is Superfunction", "text": "The function $f : \\C \\to \\C$, defined as :$f \\left({ z }\\right) = c^z$ is a superfunction for any complex number $c$."}
+{"_id": "3014", "title": "Standard Discrete Metric induces Discrete Topology", "text": "Let $M = \\struct {A, d}$ be the (standard) discrete metric space on $A$. Then $d$ induces the discrete topology on $A$. Thus the discrete topology is metrizable."}
+{"_id": "3015", "title": "Discrete Space satisfies all Separation Properties", "text": "Let $T = \\struct {S, \\powerset S}$ be the discrete topological space on $S$. Then $T$ fulfils all separation axioms: :$T$ is a $T_0$ (Kolmogorov) space :$T$ is a $T_1$ (Fréchet) space :$T$ is a $T_2$ (Hausdorff) space :$T$ is a semiregular space :$T$ is a $T_{2 \\frac 1 2}$ (completely Hausdorff) space  :$T$ is a $T_3$ space  :$T$ is a regular space :$T$ is an Urysohn space :$T$ is a $T_{3 \\frac 1 2}$ space :$T$ is a Tychonoff (completely regular) space :$T$ is a $T_4$ space :$T$ is a normal space :$T$ is a $T_5$ space  :$T$ is a completely normal space :$T$ is a perfectly $T_4$ space :$T$ is a perfectly normal space"}
+{"_id": "3016", "title": "Point in Discrete Space is Neighborhood", "text": "Let $T = \\struct {S, \\tau}$ be a discrete topological space. Let $x \\in S$. Then $\\set x$ is a neighborhood of $x$ in $T$."}
+{"_id": "3017", "title": "Discrete Space is Strongly Locally Compact", "text": "Let $T = \\struct {S, \\tau}$ be a discrete topological space. Then $T$ is strongly locally compact."}
+{"_id": "3018", "title": "Discrete Space is First-Countable", "text": "Let $T = \\struct {S, \\tau}$ be a discrete topological space. Then $T$ is first-countable."}
+{"_id": "3019", "title": "Discrete Space has Open Locally Finite Cover", "text": "Let $T = \\struct {S, \\tau}$ be a discrete topological space. Consider the set $\\CC$ of all singleton subsets of $S$: :$\\CC := \\set {\\set x: x \\in S}$ Then $\\CC$ is an open cover of $T$ which is locally finite. This cover is the finest cover on $S$. That is, if $\\VV$ is a cover of $T$, then $\\CC$ is a refinement of $\\VV$."}
+{"_id": "3020", "title": "Discrete Space is Paracompact", "text": "Let $T = \\struct {S, \\tau}$ be a discrete topological space. Then $T$ is paracompact."}
+{"_id": "3021", "title": "Singleton Set in Discrete Space is Compact", "text": "Let $T = \\struct {S, \\tau}$ be a topological space where $\\tau$ is the discrete topology on $S$. Let $x \\in S$. Then $\\set x$ is compact."}
+{"_id": "3022", "title": "Countable Discrete Space is Sigma-Compact", "text": "Let $T = \\struct {S, \\tau}$ be a countable discrete topological space. Then $T$ is $\\sigma$-compact."}
+{"_id": "3023", "title": "Countable Discrete Space is Lindelöf", "text": "Let $T = \\struct {S, \\tau}$ be a countable discrete topological space. Then $T$ is a Lindelöf space."}
+{"_id": "3024", "title": "Basis for Discrete Topology", "text": "Let $S$ be a set. Let $\\tau$ be the discrete topology on $S$. Let $\\BB$ be the set of all singleton subsets of $S$: :$\\BB := \\set {\\set x: x \\in S}$. Then $\\BB$ is a basis for $T$."}
+{"_id": "3025", "title": "Countable Discrete Space is Second-Countable", "text": "Let $T = \\struct {S, \\tau}$ be a countable discrete topological space. Then $T$ is second-countable."}
+{"_id": "3026", "title": "Countable Discrete Space is Separable", "text": "Let $T = \\struct {S, \\tau}$ be a countable discrete topological space. Then $T$ is separable."}
+{"_id": "3027", "title": "Uncountable Discrete Space is not Separable", "text": "Let $T = \\struct {S, \\tau}$ be an uncountable discrete topological space. Then $T$ is not separable."}
+{"_id": "3028", "title": "Uncountable Discrete Space is not Second-Countable", "text": "Let $T = \\struct {S, \\tau}$ be an uncountable discrete topological space. Then $T$ is not second-countable."}
+{"_id": "3029", "title": "Uncountable Discrete Space is not Lindelöf", "text": "Let $T = \\struct {S, \\tau}$ be an uncountable discrete topological space. Then $T$ is not a Lindelöf space."}
+{"_id": "3030", "title": "Uncountable Discrete Space is not Sigma-Compact", "text": "Let $T = \\struct {S, \\tau}$ be an uncountable discrete topological space. Then $T$ is not $\\sigma$-compact."}
+{"_id": "3031", "title": "Discrete Space is Compact iff Finite", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space where $\\tau$ is the discrete topology on $S$. Then $T$ is compact {{iff}} $S$ is a finite set, thereby making $\\tau$ the finite discrete topology on $S$."}
+{"_id": "3032", "title": "Finite Space is Sequentially Compact", "text": "Let $T = \\struct {S, \\tau}$ be a topological space where $S$ is a finite set. Then $T$ is sequentially compact."}
+{"_id": "3033", "title": "Discrete Space is Fully Normal", "text": "Let $T = \\struct {S, \\tau}$ be a discrete topological space. Then $T$ is fully normal."}
+{"_id": "3034", "title": "Discrete Space is Fully T4", "text": "Let $T = \\struct {S, \\tau}$ be a discrete topological space. Then $T$ is fully $T_4$."}
+{"_id": "3035", "title": "Finite Topological Space is Compact", "text": "Let $T = \\struct {S, \\tau}$ be a topological space where $S$ is a finite set. Then $T$ is compact."}
+{"_id": "3036", "title": "Finite Space is Second-Countable", "text": "Let $T = \\struct {S, \\tau}$ be a topological space where $S$ is a finite set. Then $T$ is a second-countable space."}
+{"_id": "3037", "title": "Finite Discrete Space satisfies all Compactness Properties", "text": "Let $T = \\struct {S, \\tau}$ be a finite discrete topological space. Then $T$ satisfies the following compactness properties: :$T$ is compact. :$T$ is Sequentially Compact. :$T$ is Countably Compact. :$T$ is Weakly Countably Compact. :$T$ is a Lindelöf Space :$T$ is Pseudocompact. :$T$ is $\\sigma$-Compact. :$T$ is Locally Compact. :$T$ is Weakly Locally Compact. :$T$ is Strongly Locally Compact. :$T$ is $\\sigma$-Locally Compact. :$T$ is Weakly $\\sigma$-Locally Compact. :$T$ is Fully Normal. :$T$ is Fully $T_4$. :$T$ is Paracompact. :$T$ is Countably Paracompact. :$T$ is Metacompact. :$T$ is Countably Metacompact."}
+{"_id": "3038", "title": "Finite Space Satisfies All Compactness Properties", "text": "Let $T = \\struct {S, \\tau}$ be a topological space where $S$ is a finite set. Then $T$ satisfies the following compactness properties: : $T$ is compact. : $T$ is sequentially compact. : $T$ is countably compact. : $T$ is weakly countably compact. : $T$ is a Lindelöf space. : $T$ is pseudocompact. : $T$ is $\\sigma$-compact. : $T$ is strongly locally compact. : $T$ is $\\sigma$-locally compact. : $T$ is weakly $\\sigma$-locally compact. : $T$ is locally compact. : $T$ is weakly locally compact. : $T$ is paracompact. : $T$ is countably paracompact. : $T$ is metacompact. : $T$ is countably metacompact."}
+{"_id": "3041", "title": "Discrete Space is not Dense-In-Itself", "text": "Let $T = \\struct {S, \\tau}$ be a discrete topological space. Then $T$ is not dense-in-itself."}
+{"_id": "3042", "title": "Non-Trivial Discrete Space is not Connected", "text": "Let $T = \\struct {S, \\tau}$ be a non-trivial discrete topological space. Then $T$ is not connected. Thus also:"}
+{"_id": "3043", "title": "Discrete Space is Locally Path-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a discrete topological space. Then $T$ is locally path-connected."}
+{"_id": "3044", "title": "Point is Path-Connected to Itself", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $a \\in S$. Then $a$ is path-connected to itself."}
+{"_id": "3045", "title": "Discrete Uniformity is Uniformity", "text": "Let $S$ be a set. Let $\\UU$ be the discrete uniformity on $S$ Then $\\UU$ is indeed a uniformity."}
+{"_id": "3047", "title": "Indiscrete Topology is Topology", "text": ":$\\tau$ is a topology on $S$."}
+{"_id": "3048", "title": "Indiscrete Topology is Coarsest Topology", "text": ":$\\tau$ is the coarsest topology on $S$."}
+{"_id": "3049", "title": "Open and Closed Sets in Indiscrete Topology", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Let $H \\subseteq S$."}
+{"_id": "3050", "title": "Subset of Indiscrete Space is Compact and Sequentially Compact", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Let $H \\subseteq S$."}
+{"_id": "3051", "title": "Limit Points of Indiscrete Space", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space consisting of at least two points. Let $H$ be a subset of $T$ such that $H \\ne \\O$. Then every point of $T$ is a limit point of $H$."}
+{"_id": "3052", "title": "Sequence in Indiscrete Space converges to Every Point", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Let $\\sequence {s_n}$ be a sequence in $T$. Then $\\sequence {s_n}$ converges to every point of $S$."}
+{"_id": "3055", "title": "Empty Set is Nowhere Dense", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then the empty set $\\O$ is nowhere dense in $T$."}
+{"_id": "3056", "title": "Interior of Open Set", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $U \\subseteq T$ be open in $T$. Then: : $U^\\circ = U \\iff U \\in \\tau$ where $U^\\circ$ is the interior of $U$. That is, a subset of $S$ is open in $T$ {{iff}} it equals its interior."}
+{"_id": "3057", "title": "Interior of Subset of Indiscrete Space", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Let $H \\subset S$ (that is, let $H$ be a proper subset of $T$). Then: :$H^\\circ = H^{\\circ -} = H^{\\circ - \\circ} = \\O$ where: :$H^\\circ$ denotes the interior of $H$ :$H^-$ denotes the closure of $H$."}
+{"_id": "3058", "title": "Closure of Subset of Indiscrete Space", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Let $\\O \\subsetneq H \\subseteq S$ (that is, let $H$ be a non-empty subset of $T$). Then: :$H^- = H^{- \\circ} = H^{- \\circ -} = S$ where: :$H^\\circ$ denotes the interior of $H$ :$H^-$ denotes the closure of $H$."}
+{"_id": "3059", "title": "Boundary of Subset of Indiscrete Space", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Let $\\O \\subsetneq H \\subsetneq S$ (that is, let $H$ be a non-empty proper subset of $T$). Then: :$\\partial H = S$ where $\\partial H$ denotes the boundary of $H$. If $H = \\O$ or $H = S$ then $\\partial H = \\O$."}
+{"_id": "3060", "title": "Boundary of Boundary of Subset of Indiscrete Space", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Let $H \\subseteq S$. Then: :$\\map \\partial {\\partial H} = \\O$ where $\\partial H$ denotes the boundary of $H$."}
+{"_id": "3062", "title": "Indiscrete Space is Separable", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space such that $S$ has more than one element. Then $T$ is separable."}
+{"_id": "3064", "title": "Mapping to Indiscrete Space is Continuous", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ be any topological space. Let $T_2 = \\struct {S_2, \\tau_2}$ be the indiscrete topological space on $S_2$. Let $\\phi: S_1 \\to S_2$ be a mapping. Then $\\phi$ is continuous."}
+{"_id": "3065", "title": "Indiscrete Space is Path-Connected", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Then $T$ is path-connected."}
+{"_id": "3066", "title": "Indiscrete Space is Connected", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Then $T$ is connected."}
+{"_id": "3067", "title": "Indiscrete Space is Arc-Connected iff Uncountable", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Then $T$ is arc-connected {{iff}} $S$ is an uncountable set."}
+{"_id": "3068", "title": "Indiscrete Space is Irreducible", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Then $T$ is irreducible."}
+{"_id": "3069", "title": "Indiscrete Space is Ultraconnected", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Then $T$ is ultraconnected."}
+{"_id": "3070", "title": "Indiscrete Non-Singleton Space is not T0", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space which has more than one element. Then $T$ is not a $T_0$ (Kolmogorov) space."}
+{"_id": "3071", "title": "Indiscrete Space is T5", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Then $T$ is a $T_5$ space."}
+{"_id": "3073", "title": "Indiscrete Space is T3", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space . Then $T$ is a $T_3$ space."}
+{"_id": "3074", "title": "Indiscrete Space is Pseudometrizable", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Then $T$ is pseudometrizable."}
+{"_id": "3075", "title": "Partition Topology is Topology", "text": "Let $S$ be a set. Let $\\PP$ be a partition of $S$. Let $\\tau$ be the set of subsets of $S$ defined as: :$a \\in \\tau \\iff a$ is the union of sets of $\\PP$ Then $\\tau$ is a topology on $S$."}
+{"_id": "3076", "title": "Basis for Partition Topology", "text": "Let $S$ be a set. Let $\\PP$ be a partition of $S$. Let $\\tau$ be the partition topology on $S$ defined as: :$a \\in \\tau \\iff a$ is the union of sets of $\\mathcal P$ Then $\\PP$ forms a basis of $\\tau$."}
+{"_id": "3077", "title": "Open Set in Partition Topology is also Closed", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then $T$ is a partition space {{iff}}: :$\\forall U \\subseteq S: U \\in \\tau \\iff \\relcomp S U \\in \\tau$ That is, a topological space is a partition space {{iff}} all open sets are closed, and all closed sets are also open."}
+{"_id": "3079", "title": "Quotient Topology of Partition Topology is Discrete Space", "text": "Let $\\PP$ be a partition of a set $S$. Let $T = \\struct {S, \\tau}$ be the partition space formed from $\\PP$. Let $S / \\PP$ be the quotient set of $S$ by $\\PP$. Then the quotient topology $\\tau_{S / \\PP}$ is a discrete topology."}
+{"_id": "3081", "title": "Partition of Singletons yields Discrete Topology", "text": "Let $S$ be a set which is non-empty. Let $\\PP$ be the (trivial) partition of singletons on $S$: :$\\PP = \\set {\\set x: x \\in S}$ Then the partition topology on $\\PP$ is the discrete topology."}
+{"_id": "3082", "title": "Partition Topology is not T0", "text": "Let $S$ be a set and let $\\PP$ be a partition on $S$ which is not the (trivial) partition of singletons. Let $T = \\struct {S, \\tau}$ be the partition space whose basis is $\\PP$. Then $T$ is not a $T_0$ (Kolmogorov) space."}
+{"_id": "3083", "title": "Partition Topology is T5", "text": "Let $S$ be a set and let $\\PP$ be a partition on $S$ which is not the (trivial) partition of singletons. Let $T = \\struct {S, \\tau}$ be the partition space whose basis is $\\PP$. Then $T$ is a $T_5$ space."}
+{"_id": "3084", "title": "Partition Topology is T3 1/2", "text": "Let $S$ be a set and let $\\PP$ be a partition on $S$. Let $T = \\struct {S, \\tau}$ be the partition space whose basis is $\\PP$. Then $T$ is a $T_{3 \\frac 1 2}$ space."}
+{"_id": "3085", "title": "Odd-Even Topology is Second-Countable", "text": "Let $T = \\struct {\\Z_{>0}, \\tau}$ be a topological space where $\\tau$ is the odd-even topology on the strictly positive integers $\\Z_{>0}$. Then $T$ is second-countable."}
+{"_id": "3086", "title": "Odd-Even Topology is Weakly Countably Compact", "text": "Let $T = \\struct {\\Z_{>0}, \\tau}$ be a topological space where $\\tau$ is the odd-even topology on the strictly positive integers $\\Z_{>0}$. Then $T$ is weakly countably compact."}
+{"_id": "3087", "title": "Odd-Even Topology is not Countably Compact", "text": "Let $T = \\struct {\\Z_{>0}, \\tau}$ be a topological space where $\\tau$ is the odd-even topology on the strictly positive integers $\\Z_{>0}$. Then $T$ is not countably compact."}
+{"_id": "3088", "title": "Countable Discrete Space is not Weakly Countably Compact", "text": "Let $T = \\struct {S, \\tau}$ be a countable discrete space. Then $T$ is not weakly countably compact."}
+{"_id": "3089", "title": "Weak Countable Compactness is not Preserved under Continuous Maps", "text": "Let $T_A = \\struct {S_A, \\tau_A}$ be a topological space which is weakly countably compact. Let $T_B = \\struct {S_B, \\tau_B}$ be another topological space. Let $\\phi: T_A \\to T_B$ be a continuous mapping. Then $T_B$ is not necessarily weakly countably compact."}
+{"_id": "3090", "title": "Deleted Integer Topology is not Countably Compact", "text": "Let $S = \\R_{\\ge 0} \\setminus \\Z$, and let $\\tau$ be the deleted integer topology on $S$. Then the topological space $T = \\struct {S, \\tau}$ is not countably compact."}
+{"_id": "3091", "title": "Deleted Integer Topology is Second-Countable", "text": "Let $S = \\R_{\\ge 0} \\setminus \\Z$. Let $\\tau$ be the deleted integer topology on $S$. Then the topological space $T = \\struct {S, \\tau}$ is second-countable."}
+{"_id": "3092", "title": "Deleted Integer Topology is Weakly Countably Compact", "text": "Let $S = \\R_{\\ge 0} \\setminus \\Z$. Let $\\tau$ be the deleted integer topology on $S$. Then the topological space $T = \\struct {S, \\tau}$ is weakly countably compact."}
+{"_id": "3093", "title": "Pseudometric induces Topology", "text": "Let $S \\ne \\O$ be a non-empty set. Consider a pseudometric space $\\struct {S, d}$ where $d: S \\times S \\to \\R_{\\ge 0}$ is a pseudometric. Then $\\struct {S, d}$ gives rise to a topological space $\\struct {S, \\tau_d}$ whose topology $\\tau_d$ is '''defined''' (or '''induced''') by $d$."}
+{"_id": "3094", "title": "Open Sets in Pseudometric Space", "text": "Let $P = \\left({A, d}\\right)$ be a pseudometric space. Then $\\varnothing$ and $A$ are both open in $P$."}
+{"_id": "3095", "title": "Partition Space is Pseudometrizable", "text": "Let $T = \\struct {S, \\tau}$ be a partition space. Then $T$ is pseudometrizable."}
+{"_id": "3096", "title": "Double Pointed Topology is not T0", "text": "Let $T_1 = \\struct {S, \\tau_S}$ be a topological space. Let $D = \\struct {A, \\set {\\O, A} }$ be the indiscrete topology on an arbitrary doubleton $A = \\set {a, b}$. Let $T = \\struct {T_1 \\times D, \\tau}$ be the double pointed topological space on $T_1$. Then $T$ is not a $T_0$ (Kolmogorov) space."}
+{"_id": "3097", "title": "Double Pointed Discrete Real Number Space is Weakly Countably Compact", "text": "Let $T_\\R = \\struct {\\R, \\tau_\\R}$ be the (uncountable) discrete space on the set of real numbers. Let $T_D = \\struct {D, \\tau_D}$ be the indiscrete topology on the doubleton $D = \\set {a, b}$. Let $T = T_\\R \\times T_D$ be the double pointed (uncountable) discrete space which is the product space of $T_\\R$ and $T_D$. Then $T$ is weakly countably compact."}
+{"_id": "3098", "title": "Double Pointed Discrete Real Number Space is not Lindelöf", "text": "Let $T_\\R = \\struct {\\R, \\tau_\\R}$ be the (uncountable) discrete space on the set of real numbers. Let $T_D = \\struct {D, \\tau_D}$ be the indiscrete topology on the doubleton $D = \\set {a, b}$. Let $T = T_\\R \\times T_D$ be the double pointed (uncountable) discrete space which is the product space of $T_\\R$ and $T_D$. Then $T$ is '''not''' a Lindelöf space."}
+{"_id": "3099", "title": "Particular Point Topology is Topology", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Then $\\tau_p$ is a topology on $S$, and $T$ is a topological space."}
+{"_id": "3101", "title": "Limit Points in Particular Point Space", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Let $x \\in S$ such that $x \\ne p$. Then $x$ is a limit point of $p$."}
+{"_id": "3102", "title": "Closure of Open Set of Particular Point Space", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Let $U \\in \\tau_p$ be open in $T$ such that $U \\ne \\O$. Then: :$U^- = S$ where $U^-$ denotes the closure of $U$."}
+{"_id": "3103", "title": "Interior of Closed Set of Particular Point Space", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Let $V \\subseteq S$ be closed in $T$ such that $V \\ne S$. Then: :$V^\\circ = \\O$ where $V^\\circ$ denotes the interior of $V$."}
+{"_id": "3104", "title": "Point in Particular Point Space is not Omega-Accumulation Point", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Let $x \\in S$ such that $x \\ne p$. Let $H \\subseteq S$ such that $p \\in H$. Then $x$ is not an $\\omega$-accumulation point of $H$."}
+{"_id": "3105", "title": "Particular Point Space is T0", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Then $T$ is a $T_0$ (Kolmogorov) space."}
+{"_id": "3106", "title": "Non-Trivial Particular Point Topology is not T1", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space such that $S$ is not a singleton. Then $T$ is not a $T_1$ (Fréchet) space."}
+{"_id": "3107", "title": "Particular Point Topology with Three Points is not T4", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space such that $S$ is not a singleton or a doubleton. That is, such that $S$ has more than two distinct elements. Then $T$ is not a $T_4$ space."}
+{"_id": "3109", "title": "Compact Space in Particular Point Space", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Then $\\set p$ is compact in $T$."}
+{"_id": "3110", "title": "Finite Multiplicative Subgroup of Field is Cyclic", "text": "Let $\\left({F, +, \\times}\\right)$ be a field. Let $\\left({F^*, \\times}\\right)$ denote the multiplicative group of $F$. Let $C$ be a finite subgroup of $\\left({F^*, \\times}\\right)$. Then $C$ is cyclic."}
+{"_id": "3111", "title": "Induced Homomorphism of Polynomial Forms", "text": "Let $R$ and $S$ be commutative rings with unity. Let $\\phi : R \\to S$ be a ring homomorphism. Let $R \\left[{X}\\right]$ and $S \\left[{X}\\right]$ be the rings of polynomial forms over $R$ and $S$ respectively in the indeterminate $X$. Then the map $\\overline{\\phi}: R \\left[{X}\\right] \\to S \\left[{X}\\right]$ given by: :$\\overline{\\phi} \\left({a_0 + a_1 X + \\cdots + a_n X^n}\\right) = \\phi \\left({a_0}\\right) + \\phi \\left({a_1}\\right) X + \\cdots + \\phi \\left({a_n}\\right) X^n$ is a ring homomorphism."}
+{"_id": "3112", "title": "Structure of Simple Algebraic Field Extension", "text": "Let $F / K$ be a field extension. Let $\\alpha \\in F$ be algebraic over $K$. Let $\\mu_\\alpha$ be the minimal polynomial of $\\alpha$ over $K$. Let $K \\left[{\\alpha}\\right]$ (resp. $K \\left({\\alpha}\\right)$) be the subring (resp. subfield) of $F$ generated by $K \\cup \\left\\{ {\\alpha}\\right\\}$. Then: :$K \\left[{\\alpha}\\right] = K \\left({\\alpha}\\right) \\simeq K \\left[{X}\\right] / \\left\\langle{\\mu_\\alpha}\\right\\rangle$ where $\\left\\langle{\\mu_\\alpha}\\right\\rangle$ is the ideal of the ring of polynomial functions generated by $\\mu_\\alpha$. {{explain|Link to Generator needs refining}} Moreover: :$n := \\left[{K \\left({\\alpha}\\right) : K}\\right] = \\deg \\mu_\\alpha$ and: :$1, \\alpha, \\dotsc, \\alpha^{n - 1}$ is a basis of $K \\left({\\alpha}\\right)$ over $K$."}
+{"_id": "3113", "title": "Subset of Particular Point Space is either Open or Closed", "text": "Let $T = \\left({S, \\tau_p}\\right)$ be a particular point space. Let $H \\subseteq S$ be any subset of $T$. Then $H$ is either open or closed in $T$. The only sets which are both closed and open in $T$ are $S$ and $\\varnothing$."}
+{"_id": "3114", "title": "Particular Point Space is Weakly Locally Compact", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Then $T$ is weakly locally compact."}
+{"_id": "3116", "title": "Infinite Particular Point Space is not Compact", "text": "Let $T = \\struct {S, \\tau_p}$ be an infinite particular point space. Then $T$ is not compact."}
+{"_id": "3117", "title": "Uncountable Particular Point Space is not Lindelöf", "text": "Let $T = \\struct {S, \\tau_p}$ be an uncountable particular point space. Then $T$ is not a Lindelöf space."}
+{"_id": "3119", "title": "Particular Point Space is Separable", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Then $T$ is separable."}
+{"_id": "3120", "title": "Field Adjoined Set", "text": "Let $F$ be a field. Let $S \\subseteq F$ be a subset of $F$. Let $K \\le F$ be a subfield of $F$. The subring $K \\sqbrk S$ of $F$ generated by $K \\cup S$ is the set of all finite linear combinations of powers of elements of $S$ with coefficients in $K$. The subfield $\\map K S$ of $F$ generated by $K \\cup S$ is the set of all $x y^{-1} \\in F$ with $a, b \\in K \\sqbrk S$, $b \\ne 0$. $\\map K S$ is isomorphic to the field of quotients $Q$ of $K \\sqbrk S$."}
+{"_id": "3121", "title": "Separability in Uncountable Particular Point Space", "text": "Let $T = \\struct {S, \\tau_p}$ be an uncountable particular point space. Let $H = S \\setminus \\set p$ where $\\setminus$ denotes set difference. Then $H$ is not separable."}
+{"_id": "3122", "title": "Uncountable Particular Point Space is not Second-Countable", "text": "Let $T = \\struct {S, \\tau_p}$ be an uncountable particular point space. Then $T$ is not second-countable."}
+{"_id": "3123", "title": "Particular Point Space is First-Countable", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Then $T$ is first-countable."}
+{"_id": "3125", "title": "Homeomorphic Non-Comparable Particular Point Topologies", "text": "Let $S$ be a set with at least two elements. Let $p, q \\in S: p \\ne q$. Let $\\tau_p$ and $\\tau_q$ be the particular point topologies on $S$ by $p$ and $q$ respectively. Then the topological spaces $T_p = \\struct {S, \\tau_p}$ and $T_q = \\struct {S, \\tau_q}$ are homeomorphic. However, $\\tau_p$ and $\\tau_q$ are not comparable."}
+{"_id": "3126", "title": "Universal Property for Simple Field Extensions", "text": "Let $F / K$ be a field extension. Let $\\alpha \\in F$ be an algebraic over $K$. Let $\\mu_\\alpha$ be the minimal polynomial over $\\alpha$ over $K$. Let $\\psi : K \\left({\\alpha}\\right) \\to L$ be a homomorphism. Let $\\phi = \\psi \\restriction_K$. Let $\\overline \\phi: K \\left[{X}\\right] \\to L \\left[{X}\\right]$ be the Induced Homomorphism of Polynomial Forms. Then $\\psi\\left({\\alpha}\\right)$ is a root of $\\overline \\phi \\left({\\mu_\\alpha}\\right)$ in $L$. Conversely let $L$ be a field. Let $\\phi: K \\to L$ be a homomorphism. Let $\\beta$ be a root of $\\overline \\phi  \\left({\\mu_\\alpha}\\right)$ in $L$. Then there exists a unique field homomorphism $\\psi : K \\left({\\alpha}\\right) \\to L$ extending $\\phi$ that sends $\\alpha$ to $\\beta$."}
+{"_id": "3127", "title": "Finite Field Extension is Algebraic", "text": "Let $L / K$ be a finite field extension. Then $L / K$ is algebraic."}
+{"_id": "3128", "title": "Finitely Generated Algebraic Extension is Finite", "text": "Let $L/K$ be a field extension and $\\alpha_1,\\ldots,\\alpha_n \\in L$ algebraic over $K$. Then $K \\left({\\alpha_1, \\ldots, \\alpha_n}\\right) / K$ is a finite field extension."}
+{"_id": "3130", "title": "Maximal Algebraic Extension is Subfield", "text": "Let $L/K$ be a extension of fields. Let $K^a$ be the maximal algebraic extension of $K$ contained in $L$. Then $K^a$ is a subfield of $L$."}
+{"_id": "3132", "title": "Closed Set in Particular Point Space has no Limit Points", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Let $H \\subsetneq S$ be closed in $T$. Then $H$ has no limit points."}
+{"_id": "3133", "title": "Particular Point Space is Scattered", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Then $T$ is scattered."}
+{"_id": "3135", "title": "Particular Point Space is Irreducible", "text": "Let $T = \\left({S, \\tau_p}\\right)$ be a particular point space. Then $T$ is irreducible."}
+{"_id": "3136", "title": "Particular Point Space is not Ultraconnected", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space with at least three points. Then $T$ is not ultraconnected."}
+{"_id": "3137", "title": "Dispersion Point in Particular Point Space", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Then $p$ is dispersion point of $T$."}
+{"_id": "3138", "title": "Infinite Particular Point Space is not Weakly Countably Compact", "text": "Let $T = \\struct {S, \\tau_p}$ be an infinite particular point space. Then $T$ is not weakly countably compact."}
+{"_id": "3139", "title": "Fundamental Theorem of Symmetric Polynomials", "text": "Let $K$ be a field. Let $f$ a symmetric polynomial over $K$. Then $f$ can be written uniquely as a polynomial in the elementary symmetric polynomials."}
+{"_id": "3140", "title": "Particular Point Space is Pseudocompact", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Then $T$ is pseudocompact."}
+{"_id": "3141", "title": "Particular Point Space is Path-Connected", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Then $T$ is path-connected."}
+{"_id": "3142", "title": "Particular Point Space is not Arc-Connected", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Then $T$ is not arc-connected."}
+{"_id": "3143", "title": "Basis for Particular Point Space", "text": "Let $T = \\left({S, \\tau_p}\\right)$ be a particular point space. Consider the set $\\mathcal B$ defined as: :$\\mathcal B = \\left\\{{\\left\\{{x, p}\\right\\}: x \\in S}\\right\\} \\cup \\left\\{{p}\\right\\}$ Then $B$ is a basis for $S$."}
+{"_id": "3145", "title": "Particular Point Space is Non-Meager", "text": "Let $T = \\left({S, \\tau_p}\\right)$ be a particular point space. Then $T$ is non-meager."}
+{"_id": "3146", "title": "Infinite Particular Point Space is not Countably Metacompact", "text": "Let $T = \\struct {S, \\tau_p}$ be an infinite particular point space. Then $T$ is not countably metacompact."}
+{"_id": "3147", "title": "Exportation and Self-Conditional", "text": ":$p \\land q \\implies r \\dashv \\vdash \\paren {p \\implies q} \\implies \\paren {p \\implies r}$"}
+{"_id": "3148", "title": "Łoś-Vaught Test", "text": "Let $T$ be a satisfiable $\\mathcal L$-theory with no finite models. Let $T$ be $\\kappa$-categorical for some infinite cardinal $\\kappa \\ge \\left|{\\mathcal L}\\right|$. Then $T$ is complete."}
+{"_id": "3149", "title": "Sierpiński Space is Irreducible", "text": "Let $T = \\struct {\\set {0, 1}, \\tau_0}$ be a Sierpiński space. Then $T$ is irreducible."}
+{"_id": "3151", "title": "Sierpiński Space is Path-Connected", "text": "Let $T = \\struct {\\set {0, 1}, \\tau_0}$ be a Sierpiński space. Then $T$ is path-connected."}
+{"_id": "3152", "title": "Sierpiński Space is not Arc-Connected", "text": "Let $T = \\struct {\\set {0, 1}, \\tau_0}$ be a Sierpiński space. Then $T$ is not arc-connected."}
+{"_id": "3153", "title": "Sierpiński Space is T5", "text": "Let $T = \\struct {\\set {0, 1}, \\tau_0}$ be a Sierpiński space. Then $T$ is a $T_5$ space."}
+{"_id": "3154", "title": "Sierpiński Space is T4", "text": "Let $T = \\struct {\\set {0, 1}, \\tau_0}$ be a Sierpiński space. Then $T$ is a $T_4$ space."}
+{"_id": "3155", "title": "Upward Löwenheim-Skolem Theorem", "text": "{{Disambiguate|Definition:Model|I suspect model of a first-order theory $\\LL$, which is more specific than what is linked to now}} Let $T$ be an $\\LL$-theory with an infinite model. Then for each infinite cardinal $\\kappa \\ge \\card \\LL$, there exists a model of $T$ with cardinality $\\kappa$."}
+{"_id": "3156", "title": "Compactness Theorem", "text": "Let $\\LL$ be the language of predicate logic. Let $T$ be a set of $\\LL$-sentences. Then $T$ is satisfiable {{iff}} $T$ is finitely satisfiable."}
+{"_id": "3157", "title": "Łoś's Theorem", "text": "Let $\\LL$ be a language. Let $I$ be an infinite set. Let $\\UU$ be an ultrafilter on $I$. Let $\\map \\phi {v_1, \\ldots, v_n}$ be an $\\LL$-formula. Let $\\MM$ be the ultraproduct: :$\\displaystyle \\paren {\\prod_{i \\mathop \\in I} \\MM_i} / \\UU$ where each $\\MM_i$ is an $\\LL$-structure. Then, for all $m_1 = \\paren {m_{1, i} }_\\UU, \\dots, m_n = \\paren {m_{n, i} }_\\UU$ in $\\MM$: :$\\MM \\models \\map \\phi {m_1, \\ldots, m_n}$ {{iff}}: :the set $\\set {i \\in I: \\MM_i \\models \\map \\phi {m_{1, i}, \\ldots, m_{n, i} } }$ is in $\\UU$. In particular, for all $\\LL$-sentences $\\phi$, we have that: :$\\MM \\models \\phi$ {{iff}} $\\set {i \\in I: \\MM_i \\models \\phi}$ is in $\\UU$."}
+{"_id": "3158", "title": "Closed Extension Topology is Topology", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\tau^*_p$ be the closed extension topology of $\\tau$. Then $\\tau^*_p$ is a topology on $S^*_p = S \\cup \\set p$."}
+{"_id": "3160", "title": "Quantifier Free Formula is Preserved by Superstructure", "text": "Let $\\MM, \\NN$ be $\\LL$-structures such that  $\\MM$ is a substructure of $\\NN$. {{Disambiguate|Definition:Structure}} Let $\\map \\phi {\\bar x}$ be a quantifier-free $\\LL$-formula, and let $\\bar a \\in\\MM$. Then $\\MM \\models \\map \\phi {\\bar a}$ {{iff}} $\\NN \\models \\map \\phi {\\bar a}$."}
+{"_id": "3161", "title": "Algebraically Closed Field is Infinite", "text": "Let $F$ be an algebraically closed field. Then $F$ is infinite."}
+{"_id": "3162", "title": "Field of Uncountable Cardinality K has Transcendence Degree K", "text": "Let $F$ be a field of uncountable cardinality $\\kappa$. Then $F$ has transcendence degree $\\kappa$ over its prime field."}
+{"_id": "3163", "title": "Theory of Algebraically Closed Fields of Characteristic p is Complete", "text": "Let $p$ be either $0$ or a prime number. Let $ACF_p$ be the theory of algebraically closed fields of characteristic $p$ in the language $\\mathcal L_r = \\left\\{ {0, 1, +, -, \\cdot}\\right\\}$ for rings, where: :$0, 1$ are constants and: :$+, -, \\cdot$ are binary functions.  Then $ACF_p$ is complete."}
+{"_id": "3164", "title": "Lefschetz Principle (First-Order)", "text": "Let $\\phi$ be a sentence in the language $\\LL_r = \\set {0, 1, +, -, \\cdot}$ for rings, where $0, 1$ are constants and $+, -, \\cdot$ are binary functions. {{TFAE}} :$(1): \\quad \\phi$ is true in every algebraically closed field of characteristic $0$. :$(2): \\quad \\phi$ is true in some algebraically closed field of characteristic $0$. :$(3): \\quad \\phi$ is true in algebraically closed fields of characteristic $p$ for arbitrarily large primes $p$. :$(4): \\quad \\phi$ is true in algebraically closed fields of characteristic $p$ for sufficiently large primes $p$. Note in particular that since $\\C$ is an algebraically closed field of characteristic $0$, these are equivalent to $\\phi$ being true in the field $\\C$."}
+{"_id": "3165", "title": "CNF Satisfiability Problem is NP-Complete", "text": "The conjunctive normal form boolean satisfiability problem (CNF SAT) is NP-complete."}
+{"_id": "3166", "title": "Closed Sets of Closed Extension Topology", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_p = \\struct {S^*_p, \\tau^*_p}$ be the closed extension space of $T$. Then the closed sets of $T^*_p$ (apart from $S^*_p$) are the closed sets of $T$. This explains why $\\tau^*_p$ is called the closed extension topology of $\\tau$."}
+{"_id": "3167", "title": "Particular Point Topology is Closed Extension Topology of Discrete Topology", "text": "Let $S$ be a set and let $p \\in S$. Let $\\tau_p$ be the particular point topology on $S$. Let $T = \\struct {S \\setminus \\set p, \\vartheta}$ be the discrete topological space on $S \\setminus \\set p$. Then $T^* = \\struct {S, \\tau_p}$ is a closed extension space of $T$."}
+{"_id": "3168", "title": "Closed Extension Topology is not T1", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_p = \\struct {S^*_p, \\tau^*_p}$ be the closed extension space of $T$. Then $T^*_p$ is not a $T_1$ (Fréchet) space."}
+{"_id": "3169", "title": "Algebraic Numbers are Countable", "text": "The set $\\Bbb A$ of algebraic numbers is countable."}
+{"_id": "3170", "title": "Limit Points in Closed Extension Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_p = \\struct {S^*_p, \\tau^*_p}$ be the closed extension space of $T$. Let $x \\in S$. Then $x$ is a limit point of $p$."}
+{"_id": "3171", "title": "Closure of Open Set of Closed Extension Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_p = \\struct {S^*_p, \\tau^*_p}$ be the closed extension space of $T$. Then $U^- = S^*_p$ where $U^-$ denotes the closure of $U$ in $T^*_p$."}
+{"_id": "3172", "title": "Closed Extension Space is Irreducible", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_p = \\struct {S^*_p, \\tau^*_p}$ be the closed extension space of $T$. Then $T^*_p$ is irreducible."}
+{"_id": "3173", "title": "Polynomial over Field has Finitely Many Roots", "text": "Let $F$ be a field. Let $F \\left[{x}\\right]$ be the ring of polynomial functions in the indeterminate $x$. If $p \\in F \\left[{x}\\right]$ be non-null, then $p$ has finitely many roots in $F$."}
+{"_id": "3174", "title": "Set of Polynomials over Infinite Set has Same Cardinality", "text": "Let $S$ be a set of infinite cardinality $\\kappa$. Let $S \\left[{x}\\right]$ be the set of polynomial forms over $S$ in the indeterminate $x$. Then $S \\left[{x}\\right]$ has cardinality $\\kappa$."}
+{"_id": "3175", "title": "Excluded Point Topology is Topology", "text": "Let $T = \\left({S, \\tau_{\\bar p}}\\right)$ be an excluded point space. Then $\\tau_{\\bar p}$ is a topology on $S$, and $T$ is a topological space."}
+{"_id": "3177", "title": "Excluded Point Topology is Open Extension Topology of Discrete Topology", "text": "Let $S$ be a set and let $p \\in S$. Let $\\tau_{\\bar p}$ be the excluded point topology on $S$. Let $T = \\struct {S \\setminus \\set p, \\tau_D}$ be the discrete topological space on $S \\setminus \\set p$. Then $T^* = \\struct {S, \\tau_{\\bar p} }$ is an open extension space of $T$."}
+{"_id": "3178", "title": "Product of Countable Discrete Space with Sierpiński Space is Paracompact", "text": "Let $T_X = \\struct {S, \\tau}$ be a countable discrete space. Let $T_Y = \\struct {\\set {0, 1}, \\tau_0}$ be a Sierpiński space. Let $T_X \\times T_Y$ be the product space of $T_X$ and $T_Y$. Then $T_X \\times T_Y$ is paracompact."}
+{"_id": "3179", "title": "Reduction of Explicit ODE to First Order System", "text": "Let $\\map {x^{\\paren n} } t = \\map F {t, x, x', \\ldots, x^{\\paren {n - 1} } }$, $\\map x {t_0} = x_0$ be an explicit ODE with $x \\in \\R^m$. Let there exist $I \\subseteq \\R$ such that there exists a unique particular solution: :$x: I \\to \\R^m$ to this ODE. Then there exists a system of first order ODEs: :$y' = \\map {\\tilde F} {t, y}$ with $y = \\tuple {y_1, \\ldots, y_{m n} }^T \\in \\R^{m n}$ such that: :$\\tuple {\\map {y_1} t, \\ldots, \\map {y_m} t} = \\map x t$ for all $t \\in I$ and $\\map y {t_0} = x_0$."}
+{"_id": "3180", "title": "Open Continuous Image of Paracompact Space is not always Countably Metacompact", "text": "Let $T_A = \\struct {X_A, \\tau_A}$ be a topological space which is paracompact. Let $T_B = \\struct {X_B, \\tau_B}$ be another topological space. Let $\\phi: T_A \\to T_B$ be a mapping which is both continuous and open. Then it is not necessarily even the case that $T_B$ is countably metacompact, let alone paracompact."}
+{"_id": "3181", "title": "Fixed Point Formulation of Explicit ODE", "text": "Let $x' = \\map f {t, x}$ with $\\map x {t_0} = x_0$ be an explicit ODE of dimension $n$. For $a, b \\in \\R$, let $\\mathcal X = \\map {\\mathcal C} {\\closedint a b; \\R^n}$ be the space of continuous functions on the closed interval $\\closedint a b$. Let $T: \\mathcal X \\to \\mathcal X$ be the map defined by: :$\\displaystyle \\map {\\paren {T x} } t = x_0 + \\int_{t_0}^t \\map f {s, \\map x s} \\rd s$ {{explain|Definition of $\\paren {T x}$}} Then a fixed point of $T$ in $\\mathcal X$ is a solution to the above ODE."}
+{"_id": "3182", "title": "Cardinality of Infinite Union of Infinite Sets", "text": "Let $\\kappa$ be an infinite cardinal. Let $X_i$ be sets of cardinality at most $\\kappa$ indexed by a set $I$ of cardinality at most $\\kappa$. Then their union $\\displaystyle \\bigcup_{i \\mathop \\in I} X_i$ has cardinality at most $\\kappa$. Furthermore, if at least one of the $X_i$ is size $\\kappa$, then the union has cardinality $\\kappa$."}
+{"_id": "3183", "title": "Well-Ordering Theorem", "text": "Every set is well-orderable."}
+{"_id": "3184", "title": "Well-Ordering Theorem is Equivalent to Axiom of Choice", "text": "The Well-Ordering Theorem holds {{iff}} the Axiom of Choice holds. That is, every set can be well-ordered {{iff}} every collection of sets has a choice function."}
+{"_id": "3185", "title": "Tychonoff Topology is Coarsest Topology such that Projections are Continuous", "text": "Let $\\mathbb X = \\family {\\struct {X_i, \\tau_i} }_{i \\mathop \\in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $X$ be the cartesian product of $\\mathbb X$: :$\\displaystyle X := \\prod_{i \\mathop \\in I} X_i$ Let $\\tau$ be the Tychonoff topology on $X$. For each $i \\in I$, let $\\pr_i : X \\to X_i$ be the corresponding projection which maps each ordered tuple in $X$ to the corresponding element in $X_i$: :$\\forall x \\in X: \\map {\\pr_i} x = x_i$ Then $\\tau$ is the coarsest topology on $X$ such that all the $\\pr_i$ are continuous."}
+{"_id": "3187", "title": "Condition for Closed Extension Space to be T0 Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_p = \\struct {S^*_p, \\tau^*_p}$ be the closed extension space of $T$. Then $T^*_p$ is a $T_0$ (Kolmogorov) space {{iff}} $T$ is."}
+{"_id": "3188", "title": "Excluded Point Space is T0", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be an excluded point space. Then $T$ is a $T_0$ (Kolmogorov) space."}
+{"_id": "3189", "title": "Open Extension Topology is not T1", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_{\\bar p} = \\struct {S^*_p, \\tau^*_{\\bar p} }$ be the open extension space of $T$. Then $T^*_{\\bar p}$ is not a $T_1$ (Fréchet) space."}
+{"_id": "3190", "title": "Extreme Value Theorem", "text": "Let $X$ be a compact metric space and $Y$ a normed vector space. Let $f: X \\to Y$ be a continuous mapping. Then $f$ is bounded, and there exist $x, y \\in X$ such that: :$\\forall z \\in X: \\norm {\\map f x} \\le \\norm {\\map f z} \\le \\norm {\\map f y}$ where $\\norm {\\map f x}$ denotes the norm of $\\map f x$. Moreover, $\\norm f$ attains its minimum and maximum."}
+{"_id": "3191", "title": "Continuous Functions on Compact Space form Banach Space", "text": "Let $X$ be a compact Hausdorff space. Let $Y$ be a Banach space. Let $\\CC = \\CC \\struct {X; Y}$ be the set of all continuous mappings $X \\to Y$. {{explain|Work out what convention ProofWiki has already evolved for $\\CC \\struct {X; Y}$.}} Let $\\norm {\\,\\cdot\\,}_\\infty$ be the supremum norm on $\\CC$. Then $\\struct {\\CC, \\norm {\\,\\cdot\\,} }$ is a Banach space."}
+{"_id": "3192", "title": "Supremum Norm is Norm", "text": "Let $S$ be a set. Let $\\struct {X, \\norm {\\, \\cdot \\,} }$ be a normed vector space over $K \\in \\set {\\R, \\C}$. Let $\\BB$ be the set of bounded mappings $S \\to X$. Let $\\norm {\\, \\cdot \\,}_\\infty$ be the supremum norm on $\\BB$. Then $\\norm {\\, \\cdot \\,}_\\infty$ is a norm on $\\BB$. {{MissingLinks|Add a link that establishes that $\\BB$ is a vector space}}"}
+{"_id": "3193", "title": "One-Step Vector Subspace Test", "text": "Let $V$ be a vector space over a division ring $K$. Let $U \\subseteq V$ be a non-empty subset of $V$ such that: :$\\forall u, v \\in U: \\forall \\lambda \\in K: u + \\lambda v \\in U$ Then $U$ is a subspace of $V$."}
+{"_id": "3194", "title": "Ax-Grothendieck Theorem", "text": "Let $f: \\C^n \\to \\C^n$ be a polynomial map. Let $f$ be injective. Then $f$ is surjective."}
+{"_id": "3195", "title": "Condition for Open Extension Space to be T0 Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_{\\bar p} = \\struct {S^*_p, \\tau^*_{\\bar p} }$ be the open extension space of $T$. Then $T^*_{\\bar p}$ is a $T_0$ (Kolmogorov) space {{iff}} $T$ is."}
+{"_id": "3196", "title": "Non-Trivial Excluded Point Topology is not T1", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be a excluded point space such that $S$ is not a singleton. Then $T$ is not a $T_1$ (Fréchet) space."}
+{"_id": "3197", "title": "Limit Points in Open Extension Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_{\\bar p} = \\struct {S^*_p, \\tau^*_{\\bar p} }$ be the open extension space of $T$. Let $x \\in S$. Then $p$ is a limit point of $x$."}
+{"_id": "3198", "title": "Limit Points in Excluded Point Space", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be an excluded point space. Let $x \\in S$ such that $x \\ne p$. Then $p$ is the only limit point of $x$. Similarly, let $U \\subseteq S$. Then $p$ is the only limit point of $U$."}
+{"_id": "3199", "title": "Excluded Point Space is T5", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be an excluded point space. Then $T$ is a $T_5$ space."}
+{"_id": "3200", "title": "Excluded Point Space is Compact", "text": "Let $T = \\left({S, \\tau_{\\bar p}}\\right)$ be an excluded point space. Then $T$ is a compact space."}
+{"_id": "3201", "title": "Open Extension Space is Compact", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_{\\bar p} = \\struct {S^*_p, \\tau^*_{\\bar p} }$ be the open extension space of $T$. Then $T^*_{\\bar p}$ is a compact space."}
+{"_id": "3203", "title": "Open Extension Space is Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_{\\bar p} = \\struct {S^*_p, \\tau^*_{\\bar p} }$ be the open extension space of $T$. Then $T^*_{\\bar p}$ is a connected space."}
+{"_id": "3204", "title": "Excluded Point Space is Connected", "text": "Let $T = \\left({S, \\tau_{\\bar p}}\\right)$ be an excluded point space. Then $T^*_{\\bar p}$ is a connected space."}
+{"_id": "3205", "title": "Open Extension Space is Ultraconnected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_{\\bar p} = \\struct {S^*_p, \\tau^*_{\\bar p} }$ be the open extension space of $T$. Then $T^*_{\\bar p}$ is ultraconnected."}
+{"_id": "3206", "title": "Excluded Point Space is Ultraconnected", "text": "Let $T = \\left({S, \\tau_{\\bar p}}\\right)$ be an excluded point space. Then $T$ is ultraconnected."}
+{"_id": "3207", "title": "Excluded Point Space is not Irreducible", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be an excluded point space with at least three points. Then $T^*_{\\bar p}$ is not irreducible."}
+{"_id": "3208", "title": "Excluded Point Space is Path-Connected", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be an excluded point space. Then $T^*_{\\bar p}$ is path-connected."}
+{"_id": "3209", "title": "Open Extension Space is Path-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_{\\bar p} = \\struct {S^*_p, \\tau^*_{\\bar p} }$ be the open extension space of $T$. Then $T^*_{\\bar p}$ is path-connected."}
+{"_id": "3210", "title": "Excluded Point Space is not Arc-Connected", "text": "Let $S$ be a set with at least two distinct elements. Let $T = \\struct {S, \\tau_{\\bar p} }$ be an excluded point space. Then $T$ is not arc-connected."}
+{"_id": "3211", "title": "Basis for Excluded Point Space", "text": "Let $T = \\left({S, \\tau_{\\bar p}}\\right)$ be an excluded point space. Consider the set $\\mathcal B$ defined as: :$\\mathcal B = \\left\\{{\\left\\{{x}\\right\\}: x \\in S \\setminus \\left\\{{p}\\right\\}}\\right\\} \\cup \\left\\{{S}\\right\\}$ Then $B$ is a basis for $S$."}
+{"_id": "3214", "title": "Partition Topology is Zero Dimensional", "text": "Let $T = \\struct {S, \\tau}$ be a partition space. Then $T$ is zero dimensional."}
+{"_id": "3215", "title": "Discrete Space is Zero Dimensional", "text": "Let $T = \\struct {S, \\tau}$ be a discrete space. Then $T$ is zero dimensional."}
+{"_id": "3216", "title": "Discrete Space is Scattered", "text": "Let $T = \\struct {S, \\tau}$ be a topological space where $\\tau$ is the discrete topology on $S$. Then $T$ is a scattered space."}
+{"_id": "3217", "title": "Discrete Space is Extremally Disconnected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space where $\\tau$ is the discrete topology on $S$. Then $T$ is extremally disconnected."}
+{"_id": "3218", "title": "Totally Separated Space is Totally Disconnected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is totally separated. Then $T$ is totally disconnected."}
+{"_id": "3219", "title": "Open Extension Topology is not Perfectly T4", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_{\\bar p} = \\struct {S^*_p, \\tau^*_{\\bar p} }$ be the open extension space of $T$. Then $T^*_{\\bar p}$ is not a perfectly $T_4$ space."}
+{"_id": "3220", "title": "Excluded Point Space is not Perfectly T4", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be an excluded point space. Then $T$ is not a perfectly $T_4$ space."}
+{"_id": "3221", "title": "Type is Realized in some Elementary Extension", "text": "Let $\\mathcal M$ be an $\\mathcal L$-structure. Let $A$ be a subset of the universe of $\\mathcal M$. Let $p$ be an $n$-type over $A$. There exists an elementary extension of $\\mathcal M$ which realizes $p$."}
+{"_id": "3222", "title": "Subset of Excluded Point Space is not Dense-in-itself", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be a excluded point space such that $S$ is not a singleton. Let $H \\subseteq S$. Then $H$ is not dense-in-itself."}
+{"_id": "3223", "title": "Excluded Point Space is Scattered", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be an excluded point space. Then $T$ is a scattered space."}
+{"_id": "3224", "title": "Dispersion Point of Excluded Point Space", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be an excluded point space. Then $p$ is a dispersion point of $T$."}
+{"_id": "3225", "title": "Excluded Point Space is First-Countable", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be an excluded point space. Then $T$ is first-countable."}
+{"_id": "3227", "title": "Countable Excluded Point Space is Second-Countable", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be a countable excluded point space. Then $T$ is a second-countable space."}
+{"_id": "3228", "title": "Uncountable Excluded Point Space is not Second-Countable", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be an uncountable excluded point space. Then $T$ is not second-countable."}
+{"_id": "3229", "title": "Fractional Sobolev Embedding Theorem", "text": "Let $S'$ denote the space of tempered distributions. Let $\\mathcal F : S' \\to S'$ denote the Fourier transform. For each $s \\in \\R$ and $p \\in [1, \\infty]$ let $W^{s,p} \\left({\\R^n}\\right) = \\left\\{{u \\in S': \\langle \\xi \\rangle^s \\hat{u} \\in L^p \\left({\\R^n}\\right)}\\right\\}$ where $\\displaystyle \\left \\langle{\\xi}\\right \\rangle = \\left({1 + \\left|{\\xi}\\right|^2}\\right)^\\frac 1 2$. Then : 1. If $s > t$ then $W^{s,p} \\left({\\R^n}\\right)$ embeds continuously into $W^{t,q}\\left({\\R^n}\\right)$ where $q$ is given by $\\displaystyle \\frac 1 q = \\frac 1 p - \\frac {s-t} n$. {{stub}} {{namedfor|Sergei Lvovich Sobolev}} Category:Fourier Analysis 33i3o3ouq6duaqzehxwbx7ff7dp5v5c"}
+{"_id": "3230", "title": "Countable Excluded Point Space is Separable", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be a countable excluded point space. Then $T$ is a separable space."}
+{"_id": "3231", "title": "Uncountable Excluded Point Space is not Separable", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be an uncountable excluded point space. Then $T$ is not separable."}
+{"_id": "3232", "title": "Excluded Set Topology is Topology", "text": "Let $T = \\struct {S, \\tau_{\\bar H} }$ be an excluded set space. Then $\\tau_{\\bar H}$ is a topology on $S$, and $T$ is a topological space."}
+{"_id": "3234", "title": "Either-Or Topology is Topology", "text": "Let $T = \\struct {S, \\tau}$ be the either-or space. Then $\\tau$ is a topology on $T$."}
+{"_id": "3235", "title": "Closed Sets of Either-Or Topology", "text": "Let $T = \\struct {S, \\tau}$ be the either-or space. Then the closed sets of $T$ are: :$\\O$ :$S$ :$\\set {-1}$ :$\\set 1$ :$\\set {-1, 1}$ :Any subset of $\\closedint {-1} 1$ containing $\\set 0$ as a subset."}
+{"_id": "3237", "title": "Type Space is Compact", "text": "Let $\\mathcal M$ be an $\\mathcal L$-structure, and let $A$ be a subset of the universe of  $\\mathcal M$. The type space $S_n^{\\mathcal M}(A)$ of $n$-types over $A$ is compact."}
+{"_id": "3240", "title": "Omitting Types Theorem", "text": "Let $\\mathcal L$ be the language of predicate logic with a countable signature. Let $T$ be an $\\mathcal L$-theory. Let $\\left\\{{p_i: i \\in \\N}\\right\\}$ be a countable set of non-isolated $n$-types of $T$. There is a countable $\\mathcal L$-structure $\\mathcal M$ such that $\\mathcal M \\models T$ and $\\mathcal M$ omits each $p_i$."}
+{"_id": "3241", "title": "Vinogradov's Theorem", "text": "Let $\\Lambda$ be the von Mangoldt function. For $N \\in \\Z$, let: :$\\displaystyle \\map R N = \\sum_{n_1 + n_2 + n_3 \\mathop = N} \\map \\Lambda {n_1} \\, \\map \\Lambda {n_2} \\, \\map \\Lambda {n_3}$ be a weighted count of the number of representations of $N$ as a sum of three prime powers. Let $\\SS$ be the arithmetic function: :$\\displaystyle \\map \\SS N = \\prod_{p \\mathop \\nmid N} \\paren {1 + \\frac 1 {\\paren {p - 1}^3} } \\prod_{p \\mathop \\divides N} \\paren {1 - \\frac 1 {\\paren {p - 1}^2} }$ where: :$p$ ranges over the primes :$p \\nmid N$ denotes that $p$ is not a divisor of $N$ :$p \\divides N$ denotes that $p$ is a divisor of $N$. Then for any $A > 0$ and sufficiently large odd integers $N$: :$\\map R N = \\dfrac 1 2 \\map \\SS N N^2 + \\map \\OO {\\dfrac {N^2} {\\paren {\\log N}^A} }$ where $\\OO$ denotes big-O notation."}
+{"_id": "3243", "title": "Vinogradov's Theorem/Major Arcs", "text": "Let $B \\in \\R_{>0}$. Then: :$\\displaystyle \\int_\\mathcal M \\map F \\alpha^3 \\map e {-N \\alpha} \\rd \\alpha = \\frac {N^2} 2 \\map {\\mathcal S} N + \\map {\\mathcal O} {\\frac {N^2} {\\paren {\\ln N}^{B/2} } }$ where the implied constant depends only on $B$."}
+{"_id": "3244", "title": "Exponential Dominates Polynomial", "text": "Let $\\exp$ denote the real exponential function. For any fixed $k \\in \\N$ and $\\alpha > 0$ there exists $N \\in \\N$ such that $x^k < \\exp \\left({\\alpha x}\\right)$ for all real $x > N$. {{expand|it seems that the proof would apply to $x^k, x \\in \\R$ without any changes to the proof. Changes made as suggested Nov. 2016.}}"}
+{"_id": "3245", "title": "Power Dominates Logarithm", "text": "Let $\\epsilon \\in \\R_{>0}$. Let $B \\in \\N$ be arbitrary. Then there exists $N \\in \\N$ such that: :$\\forall n > N: \\left({\\ln n}\\right)^B < n^\\epsilon$"}
+{"_id": "3247", "title": "Condition for Open Extension Space to be T5 Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_{\\bar p} = \\struct {S^*_p, \\tau^*_{\\bar p} }$ be the open extension space of $T$. Then $T^*_{\\bar p}$ is a $T_5$ space {{iff}} $T$ is a $T_5$ space."}
+{"_id": "3248", "title": "Open Extension Topology is T4", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_{\\bar p} = \\struct {S^*_p, \\tau^*_{\\bar p} }$ be the open extension space of $T$. Then $T^*_{\\bar p}$ is a $T_4$ space."}
+{"_id": "3250", "title": "Open Extension Topology is not T3", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_{\\bar p} = \\struct {S^*_p, \\tau^*_{\\bar p} }$ be the open extension space of $T$. Then $T^*_{\\bar p}$ is not a $T_3$ space."}
+{"_id": "3251", "title": "Excluded Point Topology is not T3", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be a excluded point space. Then $T$ is not a $T_3$ space."}
+{"_id": "3252", "title": "Condition for Open Extension Space to be Separable", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_{\\bar p} = \\struct {S^*_p, \\tau^*_{\\bar p} }$ be the open extension space of $T$. Then $T^*_{\\bar p}$ is a separable space {{iff}} $T$ is."}
+{"_id": "3253", "title": "Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension", "text": "Let $T$ be a finitely satisfiable $\\mathcal L$-theory. Then there exists a finitely satisfiable $\\mathcal L$-theory $T'$ which contains $T$ as a subset such that: :for all $\\mathcal L$-sentences $\\phi$, either $\\phi \\in T'$ or $\\neg \\phi \\in T'$."}
+{"_id": "3255", "title": "Condition for Open Extension Space to be Second-Countable", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_{\\bar p} = \\struct {S^*_p, \\tau^*_{\\bar p} }$ be the open extension space of $T$. Then $T^*_{\\bar p}$ is a second-countable space {{iff}} $T$ is."}
+{"_id": "3258", "title": "Limit Points of Either-Or Topology", "text": "Let $T = \\struct {S, \\tau}$ be the either-or space. Let $H \\subseteq S$ be any subset of $S$. Then no element of $S$ can be a limit point of $H$ except $0$."}
+{"_id": "3259", "title": "Either-Or Topology is T5", "text": "Let $T = \\struct {S, \\tau}$ be the either-or space. Then $T$ is a $T_5$ space."}
+{"_id": "3260", "title": "Either-Or Topology is Lindelöf", "text": "Let $T = \\struct {S, \\tau}$ be the either-or space. Then $T$ is a Lindelöf space."}
+{"_id": "3261", "title": "Subspace of Either-Or Space less Zero is not Lindelöf", "text": "Let $T = \\struct {S, \\tau}$ be the either-or space. Let $H = S \\setminus \\set 0$ be the set $S$ without zero. Then the topological subspace $T_H = \\struct {H, \\tau_H}$ is not a Lindelöf space."}
+{"_id": "3265", "title": "Basis for Either-Or Topology", "text": "Let $T = \\left({S, \\tau}\\right)$ be the either-or space. Let $\\mathcal B$ be the set: :$\\mathcal B := \\left\\{{\\left\\{{x}\\right\\}: x \\in S, x \\ne 0}\\right\\} \\cup \\left\\{{ \\left({-1 \\,.\\,.\\, 1}\\right) }\\right\\}$ ... that is, the set of all singleton subsets of $S$ less $\\left\\{{0}\\right\\}$ and including the open real interval $\\left({-1 \\,.\\,.\\, 1}\\right)$. Then $\\mathcal B$ is a basis for $T$."}
+{"_id": "3266", "title": "Either-Or Topology is Locally Path-Connected", "text": "Let $T = \\struct {S, \\tau}$ be the either-or space. Then $T$ is a locally path-connected space."}
+{"_id": "3270", "title": "Limit Points of Infinite Subset of Finite Complement Space", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement space. Let $H \\subseteq S$ be an infinite subset of $S$. Then every point of $S$ is a limit point of $H$."}
+{"_id": "3271", "title": "Finite Complement Topology is Separable", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is a separable space."}
+{"_id": "3272", "title": "Closure of Infinite Subset of Finite Complement Space", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement space. Let $H \\subseteq S$ be an infinite subset of $S$. Then $H^- = S$ where $H^-$ is the closure of $S$."}
+{"_id": "3273", "title": "Subspace of Finite Complement Topology is Compact", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on an infinite set $S$. Then every topological subspace of $T$, including $T$ itself, is a compact space."}
+{"_id": "3275", "title": "Kluyver's Formula for Ramanujan's Sum", "text": "Let $q \\in \\N_{>0}$. Let $n \\in \\N$. Let $\\map {c_q} n$ be Ramanujan's sum. Let $\\mu$ denote the Möbius function. Then: :$\\displaystyle \\map {c_q} n = \\sum_{d \\mathop \\divides \\gcd \\set {q, n} } d \\map \\mu {\\frac q d}$ where $\\divides$ denotes divisibility."}
+{"_id": "3276", "title": "Condition for Complex Root of Unity to be Primitive", "text": "Let $n, k \\in \\N$. Then $\\alpha_k = \\map \\exp {\\dfrac {2 \\pi i k} n}$ is a primitive $n$th root of unity {{iff}} $\\gcd \\set {n, k} = 1$."}
+{"_id": "3277", "title": "Relation of Boubaker Polynomials to Chebyshev Polynomials", "text": "The Boubaker polynomials are related to Chebyshev polynomials through the equations: : $B_n \\left({2x}\\right) = \\dfrac {4x}n \\dfrac{\\mathrm d}{\\mathrm d x} T_n \\left({x}\\right) - 2 T_n \\left({x}\\right)$ : $B_n \\left({2x}\\right) = -2 T_n \\left({x}\\right) + 4 x U_{n-1} \\left({x}\\right)$ where: : $T_n$ denotes the Chebyshev polynomials of the first kind : $U_n$ denotes the Chebyshev polynomials of the second kind."}
+{"_id": "3279", "title": "Relation of Boubaker Polynomials to Fermat Polynomials", "text": "The Boubaker polynomials are related to Fermat polynomials by: : $B_n \\left({x}\\right) = \\dfrac 1 {\\left({\\sqrt 2}\\right)^n} F_n \\left({\\dfrac {2 \\sqrt 2 x} 3}\\right) + \\dfrac 3 {\\left({\\sqrt 2}\\right)^{n-2}} F_{n-2} \\left({\\dfrac {2 \\sqrt 2 x} 3}\\right): \\quad n = 0, 1, 2, \\ldots$"}
+{"_id": "3280", "title": "F-Sigma and G-Delta Subsets of Uncountable Finite Complement Space", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on an uncountable set $S$. Then countably infinite subsets of $S$ are $F_\\sigma$ sets and are neither open nor closed sets. Their relative complements in $S$ are $G_\\delta$ sets, and are also neither open nor closed sets."}
+{"_id": "3281", "title": "Skolem's Paradox", "text": "Let $\\mathcal L$ be a countable first-order language. Let $T$ be an $\\mathcal L$-theory which axiomatizes some version of set theory (for example, ZFC). There is a countable model of $T$."}
+{"_id": "3282", "title": "Uncountable Finite Complement Space is not First-Countable", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on an uncountable set $S$. Then $T$ is not first-countable."}
+{"_id": "3283", "title": "Separable Space need not be First-Countable", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is separable. Then $T$ does not necessarily have to be first-countable."}
+{"_id": "3285", "title": "Asymptotic Growth of Euler Phi Function", "text": "Let $\\phi$ be the Euler $\\phi$ function. For any $\\epsilon > 0$ and sufficiently large $n$: :$n^{1 - \\epsilon} < \\phi \\left({n}\\right) < n$"}
+{"_id": "3286", "title": "Finite Complement Space is T1", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on any set $S$ which contains at least two points. Then $T$ is a $T_1$ (Fréchet) space."}
+{"_id": "3287", "title": "Finite Complement Space is Irreducible", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is irreducible."}
+{"_id": "3288", "title": "Infinite Subset of Finite Complement Space Intersects Open Sets", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on an infinite set $S$. Let $H \\subseteq S$ be an infinite subset of $S$. Then the intersection of $H$ with any non-empty open set of $T$ is infinite."}
+{"_id": "3289", "title": "Irreducible Hausdorff Space is Singleton", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space which is irreducible and Hausdorff. Then $T$ contains only one point."}
+{"_id": "3290", "title": "Finite Complement Space is not T2", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is not a $T_2$ (Hausdorff) space."}
+{"_id": "3291", "title": "Finite Complement Space is not T3, T4 or T5", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is not a $T_3$ space, $T_4$ space or $T_5$ space."}
+{"_id": "3292", "title": "Double Pointed Finite Complement Topology is Compact", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on an infinite set $S$. Let $T \\times D$ be the double pointed topology on $T$. Then $T \\times D$ is compact."}
+{"_id": "3293", "title": "Double Pointed Finite Complement Topology fulfils no Separation Axioms", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on an infinite set $S$. Let $T \\times D$ be the double pointed topology on $T$. Then $T \\times D$ is not a $T_0$ space, $T_1$ space, $T_2$ space, $T_3$ space, $T_4$ space or $T_5$ space."}
+{"_id": "3294", "title": "Vaughan's Identity", "text": "Let $\\Lambda$ be von Mangoldt's function. Let $\\mu$ be the Möbius function. Then for $y, z \\ge 1$ and $n > z$: :$\\displaystyle \\map \\Lambda n = \\sum_{\\substack {d \\mathop \\divides n \\\\ d \\mathop \\le y}} \\map \\mu d \\, \\map \\ln {\\frac n d} - \\mathop {\\sum \\sum}_{\\substack {d c \\mathop \\divides n \\\\ d \\mathop \\le y, \\, c \\mathop \\le z}} \\map \\mu d \\, \\map \\Lambda c + \\mathop {\\sum \\sum}_{\\substack {d c \\mathop \\divides n \\\\ d \\mathop > y, \\, c \\mathop > z}} \\map \\mu d \\, \\map \\Lambda c$ where $\\divides$ denotes divisibility."}
+{"_id": "3295", "title": "Sum Over Divisors of von Mangoldt is Logarithm", "text": "Let $\\Lambda$ be von Mangoldt's function. Then for $n \\geq 1$ :$\\displaystyle \\sum_{d | n} \\Lambda(d) = \\log n$"}
+{"_id": "3296", "title": "Multiplicative Function that Converges to Zero on Prime Powers", "text": "Let $f$ be a multiplicative function such that: :$\\displaystyle \\lim_{p^k \\mathop \\to \\infty} \\map f {p^k} = 0$ where $p^k$ runs though ''all'' prime powers. Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map f n = 0$ where $n$ runs through the integers."}
+{"_id": "3297", "title": "Finite T1 Space is Discrete", "text": "Let $S$ be a finite set. Let $T = \\left({S, \\tau}\\right)$ be a $T_1$ (Fréchet) space. Then $\\tau$ is the discrete topology on $S$."}
+{"_id": "3298", "title": "Finite Complement Topology is Minimal T1 Topology", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement space. Let $\\tau'$ be a topology on $S$ such that $T' = \\struct {S, \\tau'}$ is a $T_1$ (Fréchet) space. Then $\\tau$ is comparable with $\\tau'$ such that $\\tau$ is coarser than $\\tau'$. That is, of all the topologies on $S$ fulfilling the $T_1$ separation axiom, the finite complement space is the smallest. Thus the finite complement topology is known as the '''minimal $T_1$ topology''' on any given set."}
+{"_id": "3299", "title": "Finite Complement Space is Connected", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is a connected space."}
+{"_id": "3300", "title": "Finite Complement Space is Locally Connected", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is a locally connected space."}
+{"_id": "3301", "title": "Mapping Induces Partition on Domain", "text": "Let $f: S \\to T$ be a mapping. Let $F$ be defined as: :$F = \\set {\\map {f^{-1} } x: x \\in T}$ where $\\map {f^{-1} } x$ is the preimage of $x$. Then $F$ is a partition of $S$."}
+{"_id": "3302", "title": "Countable Finite Complement Space is not Path-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a countable finite complement topology. Then $T$ is not path-connected."}
+{"_id": "3303", "title": "Countable Finite Complement Space is not Locally Path-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a countable finite complement topology. Then $T$ is not locally path-connected."}
+{"_id": "3304", "title": "Arc-Connectedness in Uncountable Finite Complement Space", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on a uncountable set $S$. If the Continuum Hypothesis is accepted as true, then: :$T$ is arc-connected :$T$ is locally arc-connected."}
+{"_id": "3305", "title": "Countable Complement Topology is Topology", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement space. Then $\\tau$ is a topology on $T$."}
+{"_id": "3306", "title": "Countable Complement Topology is Expansion of Finite Complement Topology", "text": "Let $T = \\struct {S, \\tau}$ be the countable complement topology on an infinite set $S$. Let $T' = \\struct {S, \\tau'}$ be the finite complement topology on the same infinite set $S$. Then $\\tau$ is an expansion of $\\tau'$."}
+{"_id": "3307", "title": "Countable Complement Space is T1", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology. Then $T$ is a $T_1$ (Fréchet) space."}
+{"_id": "3308", "title": "Uncountable Subset of Countable Complement Space Intersects Open Sets", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Let $H \\subseteq S$ be an uncountable subset of $S$. Then the intersection of $H$ with any non-empty open set of $T$ is uncountable."}
+{"_id": "3309", "title": "Countable Complement Space is not T2", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not a $T_2$ (Hausdorff) space."}
+{"_id": "3310", "title": "Countable Complement Space is Irreducible", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is an irreducible space."}
+{"_id": "3311", "title": "Countable Complement Space is not T3, T4 or T5", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not a $T_3$ space, $T_4$ space or $T_5$ space."}
+{"_id": "3313", "title": "Closed Unit Interval is not Countably Infinite Union of Disjoint Closed Sets", "text": "Let $\\mathbb I = \\closedint 0 1$ be the closed unit interval. Then $\\mathbb I$ cannot be expressed as the union of a countably infinite set of pairwise disjoint closed sets."}
+{"_id": "3314", "title": "Compact Sets in Countable Complement Space", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Then the compact sets of $T$ are exactly the finite subsets of $S$."}
+{"_id": "3315", "title": "Countable Complement Space is not Sigma-Compact", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not a $\\sigma$-compact space."}
+{"_id": "3317", "title": "Countable Complement Space is Lindelöf", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is a Lindelöf space."}
+{"_id": "3318", "title": "Countable Complement Space is not First-Countable", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not a first-countable space."}
+{"_id": "3320", "title": "Limit Points of Countable Complement Space", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement space. Let $H \\subseteq S$ be an uncountable subset of $S$. Then every point of $S$ is a limit point of $H$. Let $H \\subseteq S$ be an countable or finite subset of $S$. Then $H$ contains all its limit points."}
+{"_id": "3321", "title": "Countable Complement Space is not Separable", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not a separable space."}
+{"_id": "3322", "title": "Countable Complement Space Satisfies Countable Chain Condition", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ satisfies the countable chain condition."}
+{"_id": "3324", "title": "Countable Complement Space is Locally Connected", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is a locally connected space."}
+{"_id": "3327", "title": "Countable Complement Space is not Weakly Countably Compact", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not weakly countably compact."}
+{"_id": "3328", "title": "Saturated Implies Universal", "text": "Let $\\kappa$ be an infinite cardinal. Let $\\mathcal{M}$ be a model of the $\\mathcal{L}$-theory $T$. If $\\mathcal{M}$ is $\\kappa$-saturated, then it is $\\kappa^+$-universal, where $\\kappa^+$ is the successor cardinal of $\\kappa$."}
+{"_id": "3329", "title": "Double Pointed Countable Complement Topology fulfils no Separation Axioms", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Let $T \\times D$ be the double pointed topology on $T$. Then $T \\times D$ is not a $T_0$ space, $T_1$ space, $T_2$ space, $T_3$ space, $T_4$ space or $T_5$ space."}
+{"_id": "3330", "title": "Double Pointed Countable Complement Topology is Weakly Countably Compact", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Let $T \\times D$ be the double pointed topology on $T$. Then $T \\times D$ is weakly countably compact."}
+{"_id": "3331", "title": "Open Extension of Double Pointed Countable Complement Topology is T4 Space", "text": "Let $T = \\struct {S, \\tau_S}$ be a countable complement topology on an uncountable set $S$. Let $D = \\struct {\\set {0, 1}, \\tau_D}$ be the indiscrete topology on two points. Let $T \\times D$ be the double pointed topology on $T$. Let $\\paren {T \\times D}^*_{\\bar p}$ be the open extension topology on $S \\times \\set {0, 1} \\cup \\set p$ where $p \\notin S \\times \\set {0, 1}$. Then $\\paren {T \\times D}^*_{\\bar p}$ is a $T_4$ space, and no other separation axioms are fulfilled. That is, $\\paren {T \\times D}^*_{\\bar p}$ is not a $T_0$ space, $T_1$ space, $T_2$ space, $T_3$ space or $T_5$ space."}
+{"_id": "3332", "title": "Saturated Models of same Cardinality are Isomorphic", "text": "Let $T$ be an $\\LL$-theory. Let $\\kappa$ be an infinite cardinal. If $\\MM$ and $\\NN$ are saturated models of $T$ and the cardinality of the universes of $\\MM$ and $\\NN$ are both $\\kappa$, then $\\MM$ and $\\NN$ are isomorphic."}
+{"_id": "3334", "title": "Infinite Ramsey's Theorem", "text": "Let $k, n \\in \\N$. For any set $S$, let $S^{\\left({n}\\right)}$ denote the set $\\left\\{ {\\left\\{ {s_1, \\ldots, s_n}\\right\\}: \\text{each } s_i \\in S}\\right\\}$ of cardinality $n$ subsets of $S$. Let $X$ be an infinite set. Then: :for every partition $P$ of $X^{\\left({n}\\right)}$ into $k$ many components :there is an infinite subset $Y \\subseteq X$ such that: :each member of $Y^{\\left({n}\\right)}$ is in the same component of $P$."}
+{"_id": "3335", "title": "Compact Complement Topology is T1", "text": "Let $T = \\struct {S, \\tau}$ be a compact complement space. Then $T$ is a $T_1$ (Fréchet) space."}
+{"_id": "3336", "title": "Compact Complement Topology is Irreducible", "text": "Let $T = \\struct {\\R, \\tau}$ be the compact complement topology on $\\R$. Then $T$ is an irreducible space."}
+{"_id": "3338", "title": "Weierstrass-Casorati Theorem", "text": "Let $f$ be a holomorphic function defined on the open ball $\\map B {a, r} \\setminus \\set a$. Let $f$ have an essential singularity at $a$. Then: :$\\forall s < r: f \\sqbrk {\\map B {a, s} \\setminus \\set a}$ is a dense subset of $\\C$. {{Disambiguate|Definition:Dense}}"}
+{"_id": "3340", "title": "Compact Complement Topology is Locally Connected", "text": "Let $T = \\struct {\\R, \\tau}$ be the compact complement topology on $\\R$. Then $T$ is a locally connected space."}
+{"_id": "3342", "title": "Compact Complement Topology is Compact", "text": "Let $T = \\struct {\\R, \\tau}$ be the compact complement topology on $\\R$. Then $T$ is a compact space."}
+{"_id": "3343", "title": "Countable Local Basis in Compact Complement Topology", "text": "Let $T = \\struct {\\R, \\tau}$ be the compact complement topology on $\\R$. Let $p \\in \\R$. Then sets of the form: :$\\openint {-\\infty} {-n} \\cup \\openint {p - \\dfrac 1 n} {p + \\dfrac 1 n} \\cup \\openint n \\infty$ form a countable local basis for $p$."}
+{"_id": "3344", "title": "Compact Complement Topology is First-Countable", "text": "Let $T = \\struct {\\R, \\tau}$ be the compact complement topology on $\\R$. Then $T$ is a first-countable space."}
+{"_id": "3345", "title": "Compact Complement Topology is Second-Countable", "text": "Let $T = \\struct {\\R, \\tau}$ be the compact complement topology on $\\R$. Then $T$ is a second-countable space."}
+{"_id": "3347", "title": "Compact Complement Topology is Coarser than Euclidean Topology", "text": "Let $T = \\struct {\\R, \\tau}$ be the compact complement topology on $\\R$. Then $\\tau$ is coarser than the usual (Euclidean) topology in $\\R$."}
+{"_id": "3349", "title": "Fort Topology is Topology", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space. Then $\\tau_p$ is a topology on $T$."}
+{"_id": "3350", "title": "Fort Space is Excluded Point Space with Finite Complement Space", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on an infinite set $S$. Then $\\tau_p$ is the minimal topology that is generated by the excluded point topology and the finite complement topology."}
+{"_id": "3351", "title": "Fort Space is T1", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a $T_1$ (Fréchet) space."}
+{"_id": "3352", "title": "Fort Space is T5", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a $T_5$ space."}
+{"_id": "3353", "title": "Fort Space is Completely Normal", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a completely normal space. Consequently, $T$ satisfies all weaker separation axioms."}
+{"_id": "3354", "title": "Exchange Principle", "text": "Let $D$ be a strongly minimal set in $\\mathcal M$. Let $A$ be a subset of $D$. Let $b, c \\in D$. If $b$ is algebraic over $A \\cup \\left\\{ {c}\\right\\}$ but not over $A$, then $c$ is algebraic over $A \\cup \\left\\{ {b}\\right\\}$."}
+{"_id": "3355", "title": "Uncountable Fort Space is not Perfectly Normal", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on an uncountable set $S$. Then $T$ is not a perfectly normal space."}
+{"_id": "3356", "title": "Countable Fort Space is Perfectly Normal", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on a countably infinite set $S$. Then $T$ is a perfectly normal space."}
+{"_id": "3357", "title": "Fort Space is Compact", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a compact space."}
+{"_id": "3359", "title": "Boubaker's Theorem/Proof of Uniqueness", "text": "Let $\\left({R, +, \\circ}\\right)$ be a commutative ring. Let $\\left({D, +, \\circ}\\right)$ be an integral subdomain of $R$ whose zero is $0_D$ and whose unity is $1_D$. Let $X \\in R$ be transcendental over $D$. Let $D \\left[{X}\\right]$ be the ring of polynomial forms in $X$ over $D$. Consider the following properties: {{begin-eqn}} {{eqn | n = 1       | l = \\sum_{k \\mathop = 1}^N {p_n \\left({0}\\right)}       | r = -2N }} {{eqn | n = 2       | l = \\sum_{k \\mathop = 1}^N {p_n \\left({\\alpha_k}\\right)}       | r = 0 }} {{eqn | n = 3       | l = \\left.{\\sum_{k \\mathop = 1}^N \\frac {\\mathrm d p_x \\left({x}\\right)} {\\mathrm d x} }\\right\\vert_{x \\mathop = 0}       | r = 0 }} {{eqn | n = 4       | l = \\left.{\\sum_{k \\mathop = 1}^N \\frac {\\mathrm d {p_n}^2 \\left({x}\\right)} {\\mathrm d x^2} }\\right\\vert_{x \\mathop = 0}       | r = \\frac 8 3 N \\left({N^2 - 1}\\right) }} {{end-eqn}} where, for a given positive integer $n$, $p_n \\in D \\left[{X}\\right]$ is a non-null polynomial such that $p_n$ has $N$ roots $\\alpha_k$ in $F$. Then the subsequence $\\left \\langle {B_{4 n} \\left({x}\\right)}\\right \\rangle$ of the Boubaker polynomials is the unique polynomial sequence of $D \\left[{X}\\right]$ which verifies simultaneously the four properties $(1) - (4)$."}
+{"_id": "3361", "title": "Generating Function for Boubaker Polynomials", "text": "The Boubaker polynomials, defined as:    :$B_n \\left({x}\\right) = \\begin{cases} 1 & : n = 0 \\\\ x & : n = 1 \\\\ x^2 + 2 & : n = 2 \\\\ x B_{n-1} \\left({x}\\right) - B_{n-2} \\left({x}\\right) & : n > 2 \\end{cases}$ have as an ordinary generating function: : $\\displaystyle f_{B_n, \\operatorname{ORD}} \\left({x, t}\\right) = \\sum_{n \\mathop = 0}^{\\infty} B_n \\left({x}\\right) t^n = \\frac {1 + 3 t^2} {1 + t \\left({t - x}\\right)}$"}
+{"_id": "3362", "title": "Exponential Generating Function for Boubaker Polynomials", "text": "The Boubaker polynomials, defined as:            :$B_n \\left({x}\\right) = \\begin{cases} 1 & : n = 0 \\\\ x & : n = 1 \\\\ x^2+2 & : n = 2 \\\\ x B_{n-1} \\left({x}\\right) - B_{n-2} \\left({x}\\right) & : n > 2 \\end{cases}$ have as an exponential generating function: : $\\displaystyle f_{B_n, \\operatorname{EXP}} \\left({x, t}\\right) = \\sum_{n=0}^\\infty B_n \\left({x}\\right) \\frac{t^n}{n!} = 4 x e^{t \\frac x 2} \\frac {\\sin \\left({t \\sqrt {1 - \\left({\\frac x 2}\\right)^2}}\\right)} {\\sqrt {1 - \\left({\\frac x 2}\\right)^2}} - 2 e^{t \\frac x 2} \\cos \\left({t \\sqrt {1 - \\left({\\frac x 2}\\right)^2}}\\right) - 3$ {{Proofread}}"}
+{"_id": "3363", "title": "Clopen Points in Fort Space", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on an infinite set $S$. Let $q \\in S: q \\ne p$. Then $\\set q$ is both open and closed in $T$. $\\set p$ itself, on the other hand, is closed but not open."}
+{"_id": "3364", "title": "Fort Space is Totally Separated", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a totally separated space."}
+{"_id": "3366", "title": "Fort Space is Zero Dimensional", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a zero dimensional space."}
+{"_id": "3367", "title": "Formula and its Negation Cannot Both Cause Forking", "text": "Let $T$ be a complete $\\mathcal{L}$-theory. Let $\\mathfrak{C}$ be a monster model for $T$. Let $A\\subseteq B$ be subsets of the universe of $\\mathfrak{C}$. Let $\\pi(\\bar x)$ be an $n$-type over $B$. If $\\pi$ does not fork over $A$, then for any formula $\\phi(\\bar x, \\bar b)$, either $\\pi \\cup \\{\\phi\\}$ or $\\pi \\cup \\{\\neg\\phi\\}$ does not fork over $A$."}
+{"_id": "3369", "title": "Forking is Local", "text": "Let $T$ be a complete $\\mathcal{L}$-theory. Let $\\mathfrak{C}$ be a monster model for $T$. Let $A\\subseteq B$ be subsets of the universe of $\\mathfrak{C}$. Let $\\pi(\\bar x)$ be an $n$-type over $B$. $\\pi$ forks over $A$ if and only if a finite subset of $\\pi$ forks over $A$."}
+{"_id": "3370", "title": "Subset of Finite Set is Finite", "text": "Let $X$ be a finite set. If $Y$ is a subset of $X$, then $Y$ is also finite."}
+{"_id": "3371", "title": "Statements Equivalent to Non-Dividing Type", "text": "Let $T$ be a complete $\\mathcal L$-theory. Let $\\mathfrak C$ be a monster model for $T$. Let $A$ be a subset of the universe of $\\mathfrak C$. The following are equivalent: :$(1): \\quad$ $\\operatorname{tp}(\\bar c / A,\\bar{b})$ does not divide over $A$. :$(2): \\quad$ For every $\\{\\bar b_i : i \\in I\\}$ containing $\\bar b$ which is order indiscernible over $A$, there is some $\\bar c'$ with $\\operatorname{tp}(\\bar c' / A,\\bar b) = \\operatorname{tp}(\\bar c / A,\\bar b)$ such that $\\{\\bar b_i : i \\in I\\}$ is order indiscernible over $A,\\bar c'$. :$(3): \\quad$ For every $\\{\\bar b_i : i \\in I\\}$ containing $\\bar b$ which is order indiscernible  over $A$, there is some $A,\\bar b$-automorphism $f$ such that $\\{f(\\bar b_i) : i \\in I\\}$ is order indiscernible over $A,\\bar c$."}
+{"_id": "3372", "title": "Fortissimo Topology is Topology", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fortissimo space. Then $\\tau_p$ is a topology on $T$."}
+{"_id": "3374", "title": "Fortissimo Space is T1", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fortissimo space. Then $T$ is a $T_1$ (Fréchet) space."}
+{"_id": "3375", "title": "Fortissimo Space is Excluded Point Space with Countable Complement Space", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fortissimo space. Then $\\tau_p$ is the minimal topology that is generated by the excluded point topology and the countable complement topology."}
+{"_id": "3376", "title": "Fortissimo Space is T5", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fortissimo space on an infinite set $S$. Then $T$ is a $T_5$ space."}
+{"_id": "3377", "title": "Fortissimo Space is Lindelöf", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fortissimo space. Then $T$ is a Lindelöf space."}
+{"_id": "3378", "title": "Fortissimo Space is not Sigma-Compact", "text": "Let $T = \\struct {S, \\tau}$ be a Fortissimo space. Then $T$ is not a $\\sigma$-compact space."}
+{"_id": "3379", "title": "Fortissimo Space is not Separable", "text": "Let $T = \\struct {S, \\tau_p}$ be the fortissimo space on an uncountable set $S$. Then $T$ is not a separable space."}
+{"_id": "3380", "title": "Fortissimo Space is not First-Countable", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fortissimo space. Then $T$ is not a first-countable space."}
+{"_id": "3381", "title": "Modified Fort Topology is Topology", "text": "Let $T = \\struct {S, \\tau_{a, b} }$  be a modified Fort space. Then $\\tau_{a, b}$ is a topology on $S$."}
+{"_id": "3382", "title": "Subset of Naturals is Finite iff Bounded", "text": "Let $X$ be a subset of the natural numbers $\\N$. Then $X$ is finite {{iff}} it is bounded."}
+{"_id": "3384", "title": "Arens-Fort Space is Expansion of Countable Fort Space", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space, where $S = \\Z_{\\ge 0} \\times \\Z_{\\ge 0}$. Let $T_p = \\struct {S, \\tau_p}$ be the Fort space on $S$ where $p = \\left({0, 0}\\right)$. Then $\\tau$ is an expansion of $\\tau_p$. Furthermore, $S$ is countably infinite, so $T_p$ is a countable Fort space."}
+{"_id": "3385", "title": "Arens-Fort Space is T1", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is a $T_1$ (Fréchet) space."}
+{"_id": "3386", "title": "Arens-Fort Space is Completely Hausdorff", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is a $T_{2 \\frac 1 2}$ (completely Hausdorff) space."}
+{"_id": "3387", "title": "Arens-Fort Space is T5", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is a $T_5$ space."}
+{"_id": "3388", "title": "Arens-Fort Space is Completely Normal", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is a completely normal space. Consequently, $T$ satisfies all weaker separation axioms."}
+{"_id": "3389", "title": "Arens-Fort Space is not First-Countable", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is not a first-countable space."}
+{"_id": "3390", "title": "Countable Space is Separable", "text": "Let $T = \\struct {S, \\tau}$ be a topological space where $S$ is a countable set. Then $T$ is a separable space."}
+{"_id": "3391", "title": "Arens-Fort Space is Separable", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is a separable space."}
+{"_id": "3392", "title": "Arens-Fort Space is Sigma-Compact", "text": "Let $T = \\left({S, \\tau}\\right)$ be the Arens-Fort space. Then $T$ is a $\\sigma$-compact space."}
+{"_id": "3393", "title": "Arens-Fort Space is Lindelöf", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is a Lindelöf space."}
+{"_id": "3394", "title": "Arens-Fort Space is not Weakly Locally Compact", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is not a weakly locally compact space."}
+{"_id": "3395", "title": "Arens-Fort Space is not Compact", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is not a compact space."}
+{"_id": "3396", "title": "Arens-Fort Space is not Countably Compact", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is not a countably compact space."}
+{"_id": "3397", "title": "Lindelöf T3 Space is Paracompact", "text": "Let $T = \\struct {S, \\tau}$ be a $T_3$ space which is also Lindelöf. Then $T$ is paracompact."}
+{"_id": "3401", "title": "Arens-Fort Space is not Connected", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is not a connected space."}
+{"_id": "3402", "title": "Arens-Fort Space is not Locally Connected", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is not a locally connected space."}
+{"_id": "3403", "title": "Clopen Points in Arens-Fort Space", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Let $q \\in S: q \\ne \\tuple {0, 0}$. Then $\\set q$ is both open and closed in $T$. $\\set {\\tuple {0, 0} }$ itself is closed, but not open."}
+{"_id": "3404", "title": "Neighborhood of Origin of Arens-Fort Space is Closed", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Every neighborhood of $\\tuple {0, 0}$ is closed in $T$."}
+{"_id": "3405", "title": "Arens-Fort Space is Zero Dimensional", "text": "Let $T = \\struct {S, \\tau}$ denote the Arens-Fort space. Then $T$ is zero dimensional."}
+{"_id": "3406", "title": "Arens-Fort Space is Totally Separated", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is totally separated."}
+{"_id": "3407", "title": "Isolated Points in Arens-Fort Space", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Let $q \\in S: q \\ne \\tuple {0, 0}$. Then $q$ is an isolated point of $T$."}
+{"_id": "3408", "title": "Point in Topological Space is Open iff Isolated", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $x \\in S$. Then $\\left\\{{x}\\right\\}$ is open in $T$ {{iff}} $x$ is an isolated point of $T$."}
+{"_id": "3409", "title": "Arens-Fort Space is Scattered", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is a scattered space."}
+{"_id": "3410", "title": "Fortissimo Space is not Metrizable", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fortissimo space. Then $\\tau_p$ is not a metrizable topology."}
+{"_id": "3411", "title": "Fortissimo Space is Paracompact", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fortissimo space. Then $T$ is a paracompact space."}
+{"_id": "3412", "title": "Clopen Points in Modified Fort Space", "text": "Let $T = \\struct {S, \\tau_{a, b} }$  be a modified Fort space. Then all points in $S \\setminus \\set {a, b}$ are both open and closed in $T$. $a$ and $b$ themselves are not open in $T$, but they are closed in $T$."}
+{"_id": "3413", "title": "Modified Fort Space is Compact", "text": "Let $T = \\struct {S, \\tau_{a, b} }$ be a modified Fort space. Then $T$ is a compact space."}
+{"_id": "3414", "title": "Modified Fort Space is T1", "text": "Let $T = \\struct {S, \\tau_{a, b} }$ be a modified Fort space. Then $T$ is a $T_1$ (Fréchet) space."}
+{"_id": "3415", "title": "Modified Fort Space is not T2", "text": "Let $T = \\struct {S, \\tau_{a, b} }$ be a modified Fort space. Then $T$ is not a $T_2$ (Hausdorff) space."}
+{"_id": "3416", "title": "Modified Fort Space is not T3, T4 or T5", "text": "Let $T = \\struct {S, \\tau_{a, b} }$ be a modified Fort space. Then $T$ is not a $T_3$ space, $T_4$ space or $T_5$ space."}
+{"_id": "3417", "title": "Basis for Euclidean Topology on Real Number Line", "text": "Let $\\R$ be the set of real numbers. Let $\\BB$ be the set of subsets of $\\R$ defined as: :$\\BB = \\set {\\openint a b: a, b \\in \\R}$ That is, $\\BB$ is the set of all open real intervals of $\\R$: :$\\openint a b := \\set {x \\in \\R: a < x < b}$ Then $\\BB$ forms a basis for the Euclidean topology on $\\R$."}
+{"_id": "3420", "title": "Real Number Line satisfies all Separation Axioms", "text": "Let $\\struct {\\R, \\tau_d}$ be the the real number line with the usual (Euclidean) topology. Then $\\struct {\\R, \\tau_d}$ fulfils all separation axioms: :$\\struct {\\R, \\tau_d}$ is a $T_0$ (Kolmogorov) space :$\\struct {\\R, \\tau_d}$ is a $T_1$ (Fréchet) space :$\\struct {\\R, \\tau_d}$ is a $T_2$ (Hausdorff) space :$\\struct {\\R, \\tau_d}$ is a semiregular space :$\\struct {\\R, \\tau_d}$ is a $T_{2 \\frac 1 2}$ (completely Hausdorff) space  :$\\struct {\\R, \\tau_d}$ is a $T_3$ space  :$\\struct {\\R, \\tau_d}$ is a regular space :$\\struct {\\R, \\tau_d}$ is an Urysohn space :$\\struct {\\R, \\tau_d}$ is a $T_{3 \\frac 1 2}$ space :$\\struct {\\R, \\tau_d}$ is a Tychonoff (completely regular) space :$\\struct {\\R, \\tau_d}$ is a $T_4$ space :$\\struct {\\R, \\tau_d}$ is a normal space :$\\struct {\\R, \\tau_d}$ is a $T_5$ space  :$\\struct {\\R, \\tau_d}$ is a completely normal space :$\\struct {\\R, \\tau_d}$ is a perfectly $T_4$ space :$\\struct {\\R, \\tau_d}$ is a perfectly normal space"}
+{"_id": "3421", "title": "Elementary Amalgamation Theorem", "text": "Let $\\MM$ and $\\NN$ be $\\LL$-structures. Let $B$ be a subset of the universe of $\\MM$ such that there is a partial elementary embedding $f: B \\to \\NN$. There is an elementary extension $\\AA$ of $\\MM$ and an elementary embedding $g: \\NN \\to \\AA$ such that $\\map g {\\map f b} = b$ for all $b \\in B$. $\\array { & & \\AA& & \\\\ &\\preceq & & \\nwarrow \\exists g & & \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!: g f = \\operatorname {id}_B \\\\ \\MM & & & &\\!\\!\\!\\!\\!\\!\\!\\! \\NN \\\\ & \\supseteq & & \\nearrow f \\\\ & & B& & \\\\ }$"}
+{"_id": "3423", "title": "Baire Space iff Open Sets are Non-Meager", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then $T$ is a Baire space {{iff}} every non-empty open set of $T$ is non-meager in $T$."}
+{"_id": "3424", "title": "Extension Realizing All Types", "text": "Let $\\MM$ be an $\\LL$-structure. Let $M$ be its universe. There is an elementary extension $\\NN$ of $\\MM$ such that every type over $M$ (relative to $\\MM$) is realized in $\\NN$."}
+{"_id": "3425", "title": "Baire Space is Non-Meager", "text": "Let $T = \\struct {S, \\tau}$ be a Baire space (in the context of topology). Then $T$ is non-meager in $T$."}
+{"_id": "3427", "title": "Separation Axioms on Double Pointed Topology/T3 Axiom", "text": "Let $T_1 = \\struct {S, \\tau_S}$ be a topological space. Let $D = \\struct {A, \\set {\\O, A} }$ be the indiscrete topology on an arbitrary doubleton $A = \\set {a, b}$. Let $T = \\struct {T_1 \\times D, \\tau}$ be the double pointed topological space on $T_1$. Then $T \\times D$ is a $T_3$ space {{iff}} $T$ is also a $T_3$ space."}
+{"_id": "3428", "title": "Separation Axioms on Double Pointed Topology/T4 Axiom", "text": "Let $T_1 = \\struct {S, \\tau_S}$ be a topological space. Let $D = \\struct {A, \\set {\\O, A} }$ be the indiscrete topology on an arbitrary doubleton $A = \\set {a, b}$. Let $T = \\struct {T_1 \\times D, \\tau}$ be the double pointed topological space on $T_1$. Then $T \\times D$ is a $T_4$ space {{iff}} $T$ is also a $T_4$ space."}
+{"_id": "3429", "title": "Separation Axioms on Double Pointed Topology/T5 Axiom", "text": "Let $T_1 = \\struct {S, \\tau_S}$ be a topological space. Let $D = \\struct {A, \\set {\\O, A} }$ be the indiscrete topology on an arbitrary doubleton $A = \\set {a, b}$. Let $T = \\struct {T_1 \\times D, \\tau}$ be the double pointed topological space on $T_1$. Then $T \\times D$ is a $T_5$ space {{iff}} $T$ is also a $T_5$ space."}
+{"_id": "3430", "title": "Separation Axioms on Double Pointed Topology", "text": "Let $T = \\struct {S, \\tau_S}$ be a topological space. Let $D = \\struct {\\set {a, b}, \\tau_D}$ be the indiscrete topology on two points. Let $T \\times D$ be the double pointed topology on $T$. Then: :$T \\times D$ is not a $T_0$ (Kolmogorov) space, a $T_1$ (Fréchet) space, a $T_2$ (Hausdorff) space or a $T_{2 \\frac 1 2}$ (completely Hausdorff) space. :$T \\times D$ is a $T_3$ space, a $T_{3 \\frac 1 2}$ space, a $T_4$ space or a $T_5$ space {{iff}} $T$ is."}
+{"_id": "3431", "title": "Separation Axioms on Double Pointed Topology/T3.5 Axiom", "text": "Let $T_1 = \\struct {S, \\tau_S}$ be a topological space. Let $D = \\struct {A, \\set {\\O, A} }$ be the indiscrete topology on an arbitrary doubleton $A = \\set {a, b}$. Let $T = \\struct {T_1 \\times D, \\tau}$ be the double pointed topological space on $T_1$. Then $T \\times D$ is a $T_{3 \\frac 1 2}$ space {{iff}} $T$ is also a $T_{3 \\frac 1 2}$ space."}
+{"_id": "3432", "title": "Square of Vandermonde Matrix", "text": "The square of the Vandermonde matrix of order $n$: : $\\mathbf V = \\begin{bmatrix}   x_1 & x_2 & \\cdots & x_n \\\\   x_1^2 & x_2^2 & \\cdots & x_n^2 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\   x_1^n & x_2^n & \\cdots & x_n^n \\end{bmatrix}$ is symmetrical in $x_1, \\ldots, x_n$. {{questionable|The case $n {{=}} 2$ left me clueless to what could possibly be intended here; only $\\mathbf {V V}^T$ is trivially seen symmetric in the $x_n$, but this can hardly be called a square}}"}
+{"_id": "3433", "title": "Open and Closed Sets in Multiple Pointed Topology", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A$ be a finite set whose cardinality is greater than $1$. Let $D = \\left({A, \\left\\{{\\varnothing, A}\\right\\}}\\right)$ be the indiscrete space on $A$. Let $T \\times D$ be a multiple pointed topological space generated from $T$ and $D$. Let $H \\subseteq T$. Then: : $H \\times A$ is open in $T \\times D$ {{iff}} $H$ is open in $T$ : $H \\times A$ is closed in $T \\times D$ {{iff}} $H$ is closed in $T$ : $H \\times \\varnothing$ is both open and closed in $T \\times D$ Let $\\varnothing \\subset H_A \\subset A$, that is, let $H_A$ be a proper subset of $A$. Then $H \\times H_A$ is neither open nor closed in $T \\times D$."}
+{"_id": "3434", "title": "Basis for Box Topology", "text": "Let $\\left \\langle {\\left({S_i, \\tau_i}\\right)} \\right \\rangle_{i \\mathop \\in I}$ be an $I$-indexed family of topological spaces. Let $S$ be the cartesian product of $\\left \\langle {S_i} \\right \\rangle_{i \\mathop \\in I}$. That is: :$\\displaystyle S := \\prod_{i \\mathop \\in I} S_i$ Define: :$\\displaystyle \\mathcal B := \\left\\{{\\prod_{i \\mathop \\in I} U_i: \\forall i \\in I: U_i \\in \\tau_i}\\right\\}$ Then $\\mathcal B$ is a synthetic basis on $S$."}
+{"_id": "3435", "title": "Space with Open Point is Non-Meager", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $x \\in S$ be an open point. Then $T$ is a non-meager space."}
+{"_id": "3437", "title": "Continuous Mapping on Finite Union of Closed Sets", "text": "Let $T = \\struct {X, \\tau}$ and $S = \\struct {Y,\\sigma}$ be topological spaces. For all $i \\in \\set {1, 2, \\ldots, n}$, let $C_i$ be closed in $T$. Let $f: X \\to Y$ be a mapping such that the restriction $f \\restriction_{C_i}$ is continuous for all $i$. Then $f$ is continuous on $C = \\displaystyle \\bigcup_{i \\mathop = 1}^n C_i$, that is, $f \\restriction_C$ is continuous. If $\\family {C_i}$ is infinite, the result does not necessarily hold."}
+{"_id": "3438", "title": "Equivalence of Definitions of T2 Space", "text": "{{TFAE|def = T2 Space|view = $T_2$ (Hausdorff) space}} Let $T = \\struct {S, \\tau}$ be a topological space."}
+{"_id": "3439", "title": "Equivalence of Definitions of T3 Space", "text": "{{TFAE|def = T3 Space|view = $T_3$ space}} Let $T = \\struct {S, \\tau}$ be a topological space."}
+{"_id": "3440", "title": "Big Implies Saturated", "text": "Let $\\mathcal M$ be an $\\mathcal L$-structure. Let $\\kappa$ be a cardinal. If $\\mathcal M$ is $\\kappa$-big, then it is $\\kappa$-saturated."}
+{"_id": "3446", "title": "Limit Points in T1 Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which satisfies the $T_1$ (Fréchet) axiom. Let $H \\subset S$ be any subset of $S$. Let $x \\in H$. Then $x$ is a limit point of $H$ {{iff}} every neighborhood of $x$ contains infinitely many points of $H$."}
+{"_id": "3447", "title": "Singleton Point is Isolated", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $x \\in S$. Then $x$ is an isolated point of the singleton set $\\left\\{{x}\\right\\}$, but not necessarily an isolated point of $T$."}
+{"_id": "3450", "title": "Tietze Extension Theorem", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is normal. Let $A \\subseteq S$ be a closed set in $T$. Let $f: A \\to \\R$ be a continuous mapping from $A \\subseteq S$ to the real number line under the usual (Euclidean) topology. Then there exists a continuous extension $g: S \\to \\R$, i.e. such that: :$\\forall s \\in A: \\map f s = \\map g s$"}
+{"_id": "3451", "title": "Hermitian Operators have Real Eigenvalues", "text": "Hermitian operations have real eigenvalues."}
+{"_id": "3453", "title": "Hausdorff Space iff Diagonal Set on Product is Closed", "text": "Let $T = \\struct{S, \\tau}$ be a topological space. Let $\\Delta_S$ be the diagonal set on $S$: :$\\Delta_S = \\set{\\tuple{x, x} \\in S \\times S: x \\in S}$ where $S \\times S$ is the Cartesian product of $S$ with itself. Let $T^2 = \\struct{S \\times S, \\TT}$ be the product space with Tychonoff topology $\\TT$. Then $T$ is a $T_2$ (Hausdorff) space iff $\\Delta_S$ is closed in $T^2$."}
+{"_id": "3454", "title": "T0 Property is Hereditary", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is a $T_0$ (Kolmogorov) space. Let $T_H = \\struct {H, \\tau_H}$, where $\\O \\subset H \\subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_0$ (Kolmogorov) space."}
+{"_id": "3455", "title": "T1 Property is Hereditary", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is a $T_1$ (Fréchet) space. Let $T_H = \\struct {H, \\tau_H}$, where $\\varnothing \\subset H \\subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_1$ (Fréchet) space."}
+{"_id": "3456", "title": "T2 Property is Hereditary", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is a $T_2$ (Hausdorff) space. Let $T_H = \\struct {H, \\tau_H}$, where $\\O \\subset H \\subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_2$ (Hausdorff) space. That is, the property of being a $T_2$ (Hausdorff) space is hereditary."}
+{"_id": "3457", "title": "Completely Hausdorff Property is Hereditary", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is a $T_{2 \\frac 1 2}$ (completely Hausdorff) space. Let $T_H = \\struct {H, \\tau_H}$, where $\\O \\subset H \\subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_{2 \\frac 1 2}$ (completely Hausdorff) space."}
+{"_id": "3458", "title": "Closure in Subspace", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T_H = \\struct {H, \\tau_H}$, where $\\O \\subset H \\subseteq S$, be a subspace of $T$. Let $U \\subseteq H$ and let $\\map {\\cl_T} U$ be the closure of $U$ in $T$. Then $\\map {\\cl_T} U \\cap H$ is the closure of $U$ in $T_H$. That is: :$\\map {\\cl_T} U \\cap H = \\map {\\cl_H} U$ where $\\map {\\cl_H} U$ denotes the closure of $U$ in $T_H$."}
+{"_id": "3459", "title": "T3 Property is Hereditary", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is a $T_3$ space. Let $T_H = \\struct {H, \\tau_H}$, where $\\O \\subset H \\subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_3$ space."}
+{"_id": "3460", "title": "T3 1/2 Property is Hereditary", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is a $T_{3 \\frac 1 2}$ space. Let $T_H = \\struct {H, \\tau_H}$, where $\\O \\subset H \\subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_{3 \\frac 1 2}$ space."}
+{"_id": "3461", "title": "T5 Property is Hereditary", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is a $T_5$ space. Let $T_H = \\struct {H, \\tau_H}$, where $\\O \\subset H \\subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_5$ space."}
+{"_id": "3462", "title": "T5 Space iff Every Subspace is T4", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then $T$ is a $T_5$ space {{iff}} every subspace of $T$ is a $T_4$ space."}
+{"_id": "3463", "title": "Product Space is T0 iff Factor Spaces are T0", "text": "Let $T_A = \\struct {S_A, \\tau_A}$ and $T_B = \\struct {S_B, \\tau_B}$ be topological spaces. Let $T = T_A \\times T_B$ be the product space formed from $T_A$ and $T_B$. Then $T$ is a $T_0$ (Kolmogorov) space {{iff}} $T_A$ and $T_B$ are themselves both $T_0$ (Kolmogorov) spaces."}
+{"_id": "3464", "title": "Product Space is T1 iff Factor Spaces are T1", "text": "Let $\\mathbb S = \\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be an indexed family of topological spaces for $\\alpha$ in some indexing set $I$. Let $\\displaystyle T = \\struct {S, \\tau} = \\displaystyle \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\mathbb S$. Then $T$ is a $T_1$ (Fréchet) space {{iff}} each of $\\struct {S_\\alpha, \\tau_\\alpha}$ is a $T_1$ (Fréchet) space."}
+{"_id": "3465", "title": "Gödel's Incompleteness Theorems/First", "text": "Let $T$ be the set of theorems of some recursive set of sentences in the language of arithmetic such that $T$ contains minimal arithmetic. $T$ cannot be both consistent and complete."}
+{"_id": "3466", "title": "Undecidability Theorem", "text": "Let $T$ be the set of theorems of some consistent theory in the language of arithmetic which contains minimal arithmetic $Q$. $T$ is not recursive."}
+{"_id": "3467", "title": "Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic", "text": "Let $T$ be the set of theorems of some consistent theory in the language of arithmetic which contains minimal arithmetic. The set of Gödel numbers of the theorems of $T$ is not definable in $T$."}
+{"_id": "3468", "title": "Diagonal Lemma", "text": "Let $T$ be the set of theorems of some theory in the language of arithmetic which contains minimal arithmetic. For any formula $\\map B y$ in the language of arithmetic, there is a sentence $G$ such that :$T \\vdash G \\leftrightarrow \\map B {\\hat G}$ where $\\hat G$ is the Gödel number of $G$ (more accurately, it is the term in the language of arithmetic obtained by applying the function symbol $s$ to $0$ this many times). {{Disambiguate|Definition:Logical Formula}}"}
+{"_id": "3469", "title": "Tarski's Undefinability Theorem", "text": "Let $\\ZZ$ be the standard structure $\\struct {\\Z, +, \\cdot, s, <, 0}$ for the language of arithmetic. Let $\\operatorname {Th}_\\ZZ$ be the sentences which are true in $\\ZZ$. Let $\\Theta$ be the set of Gödel numbers of those sentences in $\\operatorname {Th}_\\ZZ$. $\\Theta$ is not definable in $\\operatorname {Th}_\\ZZ$."}
+{"_id": "3470", "title": "Equivalence of Definitions of Limit Point", "text": "{{TFAE|def = Limit Point of Set|view = limit point}} That is, let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$. Let $H^{\\complement}$ denote the relative complement of $H$ in $S$. Then the following conditions are equivalent for any point $x \\in S$: :$(1): \\quad$ Every open neighborhood $U$ of $x$ satisfies $H \\cap \\paren {U \\setminus \\set x} \\ne \\O$. :$(2): \\quad x$ belongs to the closure of $H$ but is not an isolated point of $H$. :$(3): \\quad x$ is an adherent point of $H$ but is not an isolated point of $H$. :$(4): \\quad H^{\\complement} \\cup \\set x$ is ''not'' a neighborhood of $x$."}
+{"_id": "3471", "title": "Denseness Preserved in Coarser Topology", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T' = \\struct {S, \\tau'}$ be another topological space on $S$ such that $\\tau'$ is coarser than $\\tau$. Let $H \\subseteq S$ be everywhere dense in $T$. Then $H$ is also everywhere dense in $T'$."}
+{"_id": "3472", "title": "Rationals are Dense in Compact Complement Topology", "text": "Let $T = \\struct {\\R, \\tau^*}$ be the compact complement topology on $\\R$. Let $\\Q$ be the set of rational numbers. Then $\\Q$ is everywhere dense in $T$."}
+{"_id": "3473", "title": "Compact Complement Topology is Separable", "text": "Let $T = \\struct {\\R, \\tau}$ be the compact complement topology on $\\R$. Then $T$ is a separable space."}
+{"_id": "3474", "title": "Either-Or Topology is Locally Connected", "text": "Let $T = \\struct {S, \\tau}$ be the either-or space. Then $T$ is a locally connected space."}
+{"_id": "3475", "title": "Either-Or Topology is not Locally Arc-Connected", "text": "Let $T = \\struct {S, \\tau}$ be the either-or space. Then $T$ is not a locally arc-connected space."}
+{"_id": "3477", "title": "Fort Space is Scattered", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a scattered space."}
+{"_id": "3478", "title": "Dense-in-itself Subset of T1 Space is Infinite", "text": "Let $T = \\struct {S, \\tau_p}$ be a topological space which is $T_1$ (Fréchet). Let $H \\subseteq T$ be dense-in-itself. Then $H$ is infinite."}
+{"_id": "3480", "title": "Equal Chords in Circle", "text": "In a circle, equal chords are equally distant from the center, and chords that are equally distant from the center are equal in length. {{:Euclid:Proposition/III/14}}"}
+{"_id": "3481", "title": "Relative Lengths of Chords of Circles", "text": "Of chords in a circle, the diameter is the greatest, and of the rest the nearer to the center is always greater than the more remote. {{:Euclid:Proposition/III/15}}"}
+{"_id": "3483", "title": "Construction of Tangent from Point to Circle", "text": "From a given point outside a given circle, it is possible to draw a tangent to that circle. {{:Euclid:Proposition/III/17}}"}
+{"_id": "3492", "title": "Construction of Circle from Segment", "text": "{{:Euclid:Proposition/III/25}}"}
+{"_id": "3493", "title": "Equal Angles in Equal Circles", "text": "In equal circles, equal angles stand on equal arcs, whether at the center or at the circumference of those circles. {{:Euclid:Proposition/III/26}}"}
+{"_id": "3494", "title": "Angles on Equal Arcs are Equal", "text": "In equal circles, angles standing on equal arcs are equal to one another, whether at the center or at the circumference of those circles. {{:Euclid:Proposition/III/27}}"}
+{"_id": "3497", "title": "Bisection of Arc", "text": "It is possible to bisect an arc of a circle. {{:Euclid:Proposition/III/30}}"}
+{"_id": "3498", "title": "Relative Sizes of Angles in Segments", "text": "In a circle: : the angle in a semicircle is right : the angle in a segment greater than a semicircle is acute : the angle in a segment less than a semicircle is obtuse. Further: : the angle of a segment greater than a semicircle is obtuse : the angle of a segment less than a semicircle is acute. {{:Euclid:Proposition/III/31}}"}
+{"_id": "3499", "title": "Angles made by Chord with Tangent", "text": "Let $EF$ be a tangent to a circle $ABCD$, touching it at $B$. Let $BD$ be a chord of $ABCD$. Then: : the angle in segment $BCD$ equals $\\angle DBE$ and:  : the angle in segment $BAD$ equals $\\angle DBF$. {{:Euclid:Proposition/III/32}}"}
+{"_id": "3500", "title": "Construction of Segment on Given Line Admitting Given Angle", "text": "On any given line segment, it is possible to describe a segment of a circle which admits an angle equal to any given rectilineal angle. {{:Euclid:Proposition/III/33}}"}
+{"_id": "3503", "title": "Tangent Secant Theorem", "text": "Let $D$ be a point outside a circle $ABC$. Let $DB$ be tangent to the circle $ABC$. Let $DA$ be a straight line which cuts the circle $ABC$ at $A$ and $C$. Then $DB^2 = AD \\cdot DC$. {{:Euclid:Proposition/III/36}}"}
+{"_id": "3504", "title": "Identity Element of Natural Number Multiplication is One", "text": "Let $1$ be the element one of $\\N$. Then $1$ is the identity element of multiplication: :$\\forall n \\in \\N: n \\times 1 = n = 1 \\times n$"}
+{"_id": "3505", "title": "Secant Secant Theorem", "text": "Let $C$ be a point external to a circle $ABED$. Let $CA$ and $CB$ be straight lines which cut the circle at $D$ and $E$ respectively. Then: : $CA \\cdot CD = CB \\cdot CE$"}
+{"_id": "3506", "title": "Converse of Tangent Secant Theorem", "text": "Let $D$ be a point outside a circle $ABC$. Let $DA$ be a straight line which cuts the circle $ABC$ at $A$ and $C$. Let $DB$ intersect the circle at $B$ such that $DB^2 = AD \\cdot DC$. Then $DB$ is tangent to the circle $ABC$. {{:Euclid:Proposition/III/37}}"}
+{"_id": "3507", "title": "Fitting Chord Into Circle", "text": "Into a given circle, it is possible to fit a chord equal to a given line segment which is not greater than the diameter of the circle. {{:Euclid:Proposition/IV/1}}"}
+{"_id": "3516", "title": "Construction of Isosceles Triangle whose Base Angle is Twice Apex", "text": "It is possible to construct an isosceles triangle such that each of the angles at the base is twice that at the apex. {{:Euclid:Proposition/IV/10}}"}
+{"_id": "3517", "title": "Inscribing Regular Pentagon in Circle", "text": "In a given circle, it is possible to inscribe a regular pentagon. {{:Euclid:Proposition/IV/11}}"}
+{"_id": "3519", "title": "Inscribing Circle in Regular Pentagon", "text": "In any given regular pentagon it is possible to inscribe a circle. {{:Euclid:Proposition/IV/13}}"}
+{"_id": "3521", "title": "Inscribing Regular Hexagon in Circle", "text": "In a given circle, it is possible to inscribe a regular hexagon. {{:Euclid:Proposition/IV/15}}"}
+{"_id": "3523", "title": "Multiplication of Numbers is Left Distributive over Addition", "text": "{{:Euclid:Proposition/V/1}} That is, if $ma, mb, mc$ etc. be any equimultiples of $a, b, c$ etc., then: :$m a + m b + m c + \\cdots = m \\paren {a + b + c + \\cdots }$"}
+{"_id": "3524", "title": "Multiplication of Numbers is Right Distributive over Addition", "text": "{{:Euclid:Proposition/V/2}} That is: :$ma + na + pa + \\cdots = \\paren {m + n + p + \\cdots} a$"}
+{"_id": "3526", "title": "Multiples of Terms in Equal Ratios", "text": "Let $a, b, c, d$ be quantities. Let $a : b = c : d$ where $a : b$ denotes the ratio between $a$ and $b$. Then for any numbers $m$ and $n$: :$m a : n b = m c : n d$ {{:Euclid:Proposition/V/4}}"}
+{"_id": "3529", "title": "Ratios of Equal Magnitudes", "text": "{{:Euclid:Proposition/V/7}} That is: :$a = b \\implies a : c = b : c$ :$a = b \\implies c : a = c : b$"}
+{"_id": "3530", "title": "Relative Sizes of Ratios on Unequal Magnitudes", "text": "{{:Euclid:Proposition/V/8}} That is: :$a > b \\implies a : c > b : c$ :$a > b \\implies c : a < c : b$"}
+{"_id": "3531", "title": "Magnitudes with Same Ratios are Equal", "text": "{{:Euclid:Proposition/V/9}} That is: :$a : c = b : c \\implies a = b$ :$c : a = c : b \\implies a = b$"}
+{"_id": "3532", "title": "Relative Sizes of Magnitudes on Unequal Ratios", "text": "{{:Euclid:Proposition/V/10}} That is: :$a : c > b : c \\implies a > b$ :$c : b > c : a \\implies b < a$"}
+{"_id": "3533", "title": "Equality of Ratios is Transitive", "text": "{{:Euclid:Proposition/V/11}} That is: :$A : B = C : D, C : D = E : F \\implies A : B = E : F$"}
+{"_id": "3534", "title": "Sum of Components of Equal Ratios", "text": "{{:Euclid:Proposition/V/12}} That is: :$a_1 : b_1 = a_2 : b_2 = a_3 : b_3 = \\cdots \\implies \\left({a_1 + a_2 + a_3 + \\cdots}\\right) : \\left({b_1 + b_2 + b_3 + \\cdots}\\right)$"}
+{"_id": "3535", "title": "Relative Sizes of Proportional Magnitudes", "text": "{{:Euclid:Proposition/V/13}} That is: :$a : b = c : d, c : d > e : f \\implies a : b > e : f$"}
+{"_id": "3536", "title": "Relative Sizes of Components of Ratios", "text": "{{:Euclid:Proposition/V/14}} That is, if $a : b = c : d$ then: :$a > c \\implies b > d$ :$a = c \\implies b = d$ :$a < c \\implies b < d$"}
+{"_id": "3537", "title": "Ratio Equals its Multiples", "text": "{{:Euclid:Proposition/V/15}} That is: :$a : b \\implies ma = mb$"}
+{"_id": "3538", "title": "Proportional Magnitudes are Proportional Alternately", "text": "{{:Euclid:Proposition/V/16}} That is: :$a : b = c : d \\implies a : c = b : d$"}
+{"_id": "3539", "title": "Magnitudes Proportional Compounded are Proportional Separated", "text": "{{:Euclid:Proposition/V/17}} That is: :$a : b = c : d \\implies \\left({a - b}\\right) : b = \\left({c - d}\\right) : d$"}
+{"_id": "3540", "title": "Magnitudes Proportional Separated are Proportional Compounded", "text": "{{:Euclid:Proposition/V/18}} That is: :$a : b = c : d \\implies \\left({a + b}\\right) : b = \\left({c + d}\\right) : d$"}
+{"_id": "3541", "title": "Proportional Magnitudes have Proportional Remainders", "text": "{{:Euclid:Proposition/V/19}} That is: :$a : b = c : d \\implies \\left({a - c}\\right) : \\left({b - d}\\right) = a : b$ where $a : b$ denotes the ratio of $a$ to $b$."}
+{"_id": "3542", "title": "Relative Sizes of Successive Ratios", "text": "{{:Euclid:Proposition/V/20}} That is, let: :$a : b = d : e$ :$b : c = e : f$ Then: :$a > c \\implies d > f$ :$a = c \\implies d = f$ :$a < c \\implies d < f$"}
+{"_id": "3543", "title": "Relative Sizes of Elements in Perturbed Proportion", "text": "{{:Euclid:Proposition/V/21}} That is, let: :$a : b = e : f$ :$b : c = d : e$ Then: :$a > c \\implies d > f$ :$a = c \\implies d = f$ :$a < c \\implies d < f$"}
+{"_id": "3544", "title": "Equality of Ratios Ex Aequali", "text": "{{:Euclid:Proposition/V/22}} That is, if: :$a : b = d : e$ :$b : c = e : f$ then: :$a : c = d : f$"}
+{"_id": "3545", "title": "Equality of Ratios in Perturbed Proportion", "text": "{{:Euclid:Proposition/V/23}} That is, if: :$a : b = e : f$ :$b : c = d : e$ then: :$a : c = d : f$"}
+{"_id": "3547", "title": "Sum of Antecedent and Consequent of Proportion", "text": "{{:Euclid:Proposition/V/25}} That is, if $a : b = c : d$ and $a$ is the greatest and $d$ is the least, then: :$a + d > b + c$"}
+{"_id": "3548", "title": "Areas of Triangles and Parallelograms Proportional to Base", "text": "{{:Euclid:Proposition/VI/1}} Let $ABC$ and $ACD$ be triangles. Let $EC, CF$ be parallelograms under the same height. Then: :$AC : CD = \\triangle ABC : \\triangle ACD = \\Box EC : \\Box CF$ where: :$AC : CD$ denotes the ratio of the length of $AC$ to that of $CD$ :$\\triangle ABC : \\triangle ACD$ denotes the ratio of the area of $\\triangle ABC$ to that of $\\triangle ACD$ :$\\Box EC : \\Box CF$ denotes the ratio of the area of parallelogram $EC$ to that of parallelogram $CF$."}
+{"_id": "3550", "title": "Angle Bisector Theorem", "text": "Let $\\triangle ABC$ be a triangle. Let $D$ lie on the base $BC$ of $\\triangle ABC$. Then the following are equivalent: :$(1): \\quad AD$ is the angle bisector of $\\angle BAC$ :$(2): \\quad BD : DC = AB : AC$ where $BD : DC$ denotes the ratio between the lengths $BD$ and $DC$. {{:Euclid:Proposition/VI/3}}"}
+{"_id": "3551", "title": "Equiangular Triangles are Similar", "text": "Let two triangles have the same corresponding angles. Then their corresponding sides are proportional. Thus, by definition, such triangles are similar. {{:Euclid:Proposition/VI/4}}"}
+{"_id": "3553", "title": "Triangles with One Equal Angle and Two Sides Proportional are Similar", "text": "Let two triangles have two corresponding sides which are proportional. Let the angles adjacent to both of these sides be equal. Then all of their corresponding angles are equal. Thus, by definition, such triangles are similar. {{:Euclid:Proposition/VI/6}}"}
+{"_id": "3554", "title": "Triangles with One Equal Angle and Two Other Sides Proportional are Similar", "text": "Let two triangles be such that one of the angles of one triangle equals one of the angles of the other. Let two corresponding sides which are adjacent to one of the other angles, be proportional. Let the third angle in both triangles be either both acute or both not acute. Then all of the corresponding angles of these triangles are equal. Thus, by definition, such triangles are similar. {{:Euclid:Proposition/VI/7}}"}
+{"_id": "3558", "title": "Zeckendorf's Theorem", "text": "Every positive integer has a unique Zeckendorf representation. That is: Let $n$ be a positive integer. Then there exists a unique increasing sequence of integers $\\left\\langle{c_i}\\right\\rangle$ such that: :$\\forall i \\in \\N: c_i \\ge 2$ :$c_{i + 1} > c_i + 1$ :$\\displaystyle n = \\sum_{i \\mathop = 0}^k F_{c_i}$ where $F_m$ is the $m$th Fibonacci number. For any given $n$, such a $\\left\\langle{c_i}\\right\\rangle$ is unique."}
+{"_id": "3559", "title": "Sum of Non-Consecutive Fibonacci Numbers", "text": "Let $S$ be a non-empty set of distinct non-consecutive Fibonacci numbers not containing $F_0$ or $F_1$. Let the largest element of $S$ be $F_j$. Then: :$\\displaystyle \\sum_{F_i \\mathop \\in S} F_i < F_{j + 1}$ That is, the sum of all the elements of $S$ is strictly less than the next largest Fibonacci number. That is, given some increasing sequence $\\left\\langle {c_i}\\right\\rangle$ satisfying $c_i \\ge 2$ and $c_{i + 1} \\ge c_i + 1$: :$\\displaystyle F_{c_k + 1} > \\sum_{i \\mathop = 0}^k F_{c_i}$"}
+{"_id": "3560", "title": "Construction of Third Proportional Straight Line", "text": "Given any two straight lines of given length $a$ and $b$, it is possible to construct a third straight line of length $c$ such that $a : b = b : c$. {{:Euclid:Proposition/VI/11}}"}
+{"_id": "3561", "title": "Construction of Fourth Proportional Straight Line", "text": "Given three straight lines of lengths $a, b, c$, it is possible to construct a fourth straight line of length $d$ such that $a : b = c : d$. {{:Euclid:Proposition/VI/12}}"}
+{"_id": "3562", "title": "Construction of Mean Proportional", "text": "Given any two straight lines of length $a$ and $b$ it is possible to find a straight line of length $c$ such that $a : c = c : b$. {{:Euclid:Proposition/VI/13}}"}
+{"_id": "3563", "title": "Sides of Equal and Equiangular Parallelograms are Reciprocally Proportional", "text": "{{:Euclid:Proposition/VI/14}} Note: in the above, ''equal'' is to be taken to mean ''of equal area''."}
+{"_id": "3564", "title": "Sides of Equiangular Triangles are Reciprocally Proportional", "text": "{{:Euclid:Proposition/VI/15}} Note: in the above, ''equal'' is to be taken to mean ''of equal area''."}
+{"_id": "3565", "title": "Rectangles Contained by Proportional Straight Lines", "text": "{{:Euclid:Proposition/VI/16}} Note: in the above, ''equal'' is to be taken to mean ''of equal area''."}
+{"_id": "3567", "title": "Construction of Similar Polygon", "text": "On any given straight line it is possible to construct a polygon similar to any given polygon. {{:Euclid:Proposition/VI/18}}"}
+{"_id": "3568", "title": "Triangles with Two Equal Angles are Similar", "text": "Two triangles which have two corresponding angles which are equal are similar."}
+{"_id": "3571", "title": "Ratio of Areas of Similar Triangles", "text": "{{:Euclid:Proposition/VI/19}} That is, the ratio of the areas of the similar triangles is the square of the ratio of the corresponding sides."}
+{"_id": "3574", "title": "Similarity of Polygons is Equivalence Relation", "text": "Let $A, B, C$ be polygons. If $A$ and $B$ are both similar to $C$, then $A$ is similar to $B$. {{:Euclid:Proposition/VI/21}} It is also worth noting that: :$A$ is similar to $A$, and so similarity between polygons is reflexive. :If $A$ is similar to $B$, then $B$ is similar to $A$, and so similarity between polygons is symmetric. Hence the relation of similarity between polygons is an equivalence relation."}
+{"_id": "3588", "title": "Integral Multiple of an Algebraic Number", "text": "Let $K$ be a number field and $\\alpha \\in K$. Then there exists a positive $n \\in \\Z$ such that $n \\alpha \\in \\mathcal O_K$. In this context, $ \\mathcal O_K$ denotes the algebraic integers in $K$."}
+{"_id": "3590", "title": "Schönemann-Eisenstein Theorem", "text": "Let $\\map f x = a_d x^d + a_{d - 1} x^{d - 1} + \\dotsb + a_0 \\in \\Z \\sqbrk x$ be a polynomial over the ring of integers $\\Z$. Let $p$ be a prime such that: :$(1): \\quad p \\divides a_i \\iff i \\ne d$ :$(2): \\quad p^2 \\nmid a_0$ where $p \\divides a_i$ signifies that $p$ is a divisor of $a_i$. Then $f$ is irreducible in $\\Q \\sqbrk x$."}
+{"_id": "3593", "title": "Isolated Point of Closure of Subset is Isolated Point of Subset", "text": "Let $\\struct {T, \\tau}$ be a topological space. Let $H \\subseteq T$ be a subset of $T$. Let $\\map \\cl H$ denote the closure of $H$. Let $x \\in \\map \\cl H$ be an isolated point of $\\map \\cl H$. Then $x$ is also an isolated point of $H$."}
+{"_id": "3595", "title": "Disjoint Open Sets remain Disjoint with one Closure", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A, B \\in \\tau$ such that $A \\cap B = \\varnothing$. Then: : $A^- \\cap B = \\varnothing$ where $A^-$ denotes the closure of $A$."}
+{"_id": "3596", "title": "Gauss's Lemma on Primitive Rational Polynomials", "text": "Let $\\Q$ be the field of rational numbers. Let $\\Q \\sqbrk X$ be the ring of polynomials over $\\Q$ in one indeterminate $X$. Let $\\map f X, \\map g X \\in \\Q \\sqbrk X$ be primitive polynomials. Then their product $f g$ is also a primitive polynomial."}
+{"_id": "3598", "title": "Preimage of Cover is Cover", "text": "Let $\\phi: S \\to T$ be a mapping between the sets $S$ and $T$. Let $\\mathcal U$ be a cover of $T$. Then the set: :$\\left\\{{\\phi^{-1} \\left({U}\\right): U \\in \\mathcal U}\\right\\}$ is a cover of $S$."}
+{"_id": "3599", "title": "Compactness is Preserved under Continuous Surjection", "text": "Let $T_A = \\struct {S_A, \\tau_A}$ and $T_B = \\struct {S_B, \\tau_B}$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a continuous surjection. If $T_A$ is compact, then $T_B$ is also compact."}
+{"_id": "3600", "title": "Sigma-Compactness is Preserved under Continuous Surjection", "text": "Let $T_A = \\struct {S_A, \\tau_A}$ and $T_B = \\struct {S_B, \\tau_B}$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a continuous surjection. If $T_A$ is $\\sigma$-compact, then $T_B$ is also $\\sigma$-compact."}
+{"_id": "3601", "title": "Countable Compactness is Preserved under Continuous Surjection", "text": "Let $T_A = \\struct {S_A, \\tau_A}$ and $T_B = \\struct {S_B, \\tau_B}$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a continuous surjection. If $T_A$ is countably compact, then $T_B$ is also countably compact."}
+{"_id": "3602", "title": "Sequential Compactness is Preserved under Continuous Surjection", "text": "Let $T_A = \\struct {S_A, \\tau_A}$ and $T_B = \\struct {S_B, \\tau_B}$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a continuous surjection. If $T_A$ is sequentially compact, then $T_B$ is also sequentially compact."}
+{"_id": "3603", "title": "Lindelöf Property is Preserved under Continuous Surjection", "text": "Let $T_A = \\struct {S_A, \\tau_A}$ and $T_B = \\struct {S_B, \\tau_B}$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a continuous surjection. If $T_A$ is a Lindelöf space, then $T_B$ is also a Lindelöf space."}
+{"_id": "3604", "title": "Multiples of Divisors obey Distributive Law", "text": "{{:Euclid:Proposition/VII/6}} In modern algebraic language: :$a = \\dfrac m n b, c = \\dfrac m n d \\implies a + c = \\dfrac m n \\left({b + d}\\right)$"}
+{"_id": "3605", "title": "First-Countability is Preserved under Open Continuous Surjection", "text": "Let $T_A = \\struct {S_A, \\tau_A}$ and $T_B = \\struct {S_B, \\tau_B}$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a surjective open mapping which is also continuous. If $T_A$ is first-countable, then $T_B$ is also first-countable."}
+{"_id": "3606", "title": "Second-Countability is Preserved under Open Continuous Surjection", "text": "Let $T_A = \\struct {S_A, \\tau_A}$ and $T_B = \\struct {S_B, \\tau_B}$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a surjective open mapping which is also continuous. If $T_A$ is second-countable, then $T_B$ is also second-countable."}
+{"_id": "3607", "title": "Arens-Fort Space is Countable", "text": "Let $T = \\left({S, \\tau}\\right)$ be the Arens-Fort space. Then $S$ is countably infinite."}
+{"_id": "3608", "title": "Connected iff no Proper Clopen Sets", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then $T$ is connected if and only if there exists no proper subset of $S$ which is clopen in $T$."}
+{"_id": "3609", "title": "Element of Principal Ideal Domain is Finite Product of Irreducible Elements", "text": "Let $R$ be a principal ideal domain. Let $p \\in R$ such that $p \\ne 0$ and $p$ is not a unit. Then there exist irreducible elements $p_1, \\ldots, p_n$ such that $p = p_1 \\cdots p_n$."}
+{"_id": "3610", "title": "Principal Ideal Domain fulfills Ascending Chain Condition", "text": "Let $R$ be a principal ideal domain. Then $R$ fulfills the ascending chain condition."}
+{"_id": "3611", "title": "Subspace of Product Space Homeomorphic to Factor Space", "text": "Let $\\family {\\struct {X_i, \\tau_i} }_{i \\mathop \\in I}$ be a family of topological spaces where $I$ is an arbitrary index set. Let $\\displaystyle \\struct {X, \\tau} = \\prod_{i \\mathop \\in I} \\struct {X_i, \\tau_i}$ be the product space of $\\family {\\struct {X_i, \\tau_i} }_{i \\mathop \\in I}$. Suppose that $X$ is non-empty. Then for each $i \\in I$ there is a subspace $Y_i \\subseteq X$ which is homeomorphic to $\\struct {X_i, \\tau_i}$. Specifically, for any $z \\in X$, let: :$Y_i = \\set {x \\in X: \\forall j \\in I \\setminus \\set i: x_j = z_j}$ and let $\\upsilon_i$ be the subspace topology of $Y_i$ relative to $\\tau$. Then $\\struct {Y_i, \\upsilon_i}$ is homeomorphic to $\\struct {X_i, \\tau_i}$, where the homeomorphism is the restriction of the projection $\\pr_i$ to $Y_i$."}
+{"_id": "3612", "title": "Closed Set of Countable Fort Space is G-Delta", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on a countably infinite set $S$. Let $H \\subseteq S$ be closed in $T$. Then $H$ is a $G_\\delta$ set."}
+{"_id": "3613", "title": "Fundamental Theorem of Galois Theory", "text": "Let $L / K$ be a finite Galois extension. Let $\\operatorname{Gal} \\left({L / K}\\right)$ denote the Galois group of the extension $L / K$. Let $H$ denote a subgroup of $\\operatorname{Gal} \\left({L / K}\\right)$ and $F$ denote an intermediate field. The mappings: :$H \\mapsto L_H$, and :$F \\mapsto \\operatorname{Gal} \\left({L / F}\\right)$ are inclusion-reversing and inverses. Moreover, these maps induce a bijection between the normal subgroups of $\\operatorname{Gal} \\left({L / K}\\right)$ and the normal, intermediate extensions of $L / K$."}
+{"_id": "3614", "title": "Ring of Algebraic Integers", "text": "Let $K / \\Q$ be a number field. Let $\\Z \\sqbrk x$ denote the polynomial ring in one variable over $\\Z$. Let $\\mathbb A$ denote the set of all elements of $K / \\Q$ which are a root of some monic polynomial $P \\in \\Z \\sqbrk x$. That is, let $\\mathbb A$ denote the algebraic integers over $K$. Then $\\mathbb A$ is a ring, called the '''Ring of Algebraic Integers'''."}
+{"_id": "3615", "title": "Subtraction of Divisors obeys Distributive Law", "text": "{{:Euclid:Proposition/VII/7}} In modern algebraic language: :$a = \\dfrac 1 n b, c = \\dfrac 1 n d \\implies a - c = \\dfrac 1 n \\paren {b - d}$"}
+{"_id": "3616", "title": "Minimal Polynomial is Unique", "text": "Let $L / K$ be a field extension and $\\alpha \\in L$ be algebraic over $K$. Then the minimal polynomial of $\\alpha$ over $K$ is unique."}
+{"_id": "3617", "title": "Minimal Polynomial is Irreducible", "text": "Let $L / K$ be a field extension. Let $\\alpha \\in L$ be algebraic over $K$. Then the minimal polynomial in $\\alpha$ over $K$ is unique and irreducible."}
+{"_id": "3618", "title": "Equivalence of Definitions of Normal Extension", "text": "Let $L / K$ be an algebraic field extension. {{TFAE|def = Normal Extension}}"}
+{"_id": "3619", "title": "Abstract Model of Algebraic Extensions", "text": "Let $K$ be a field and $\\alpha \\in \\overline{K}$ be an element of the algebraic closure of $K$ which is algebraic over $K$. Let $m_\\alpha$ be the minimal polynomial of $\\alpha$ over $K$. Then: :$K[\\alpha] \\cong K[x]/ \\langle m_\\alpha\\rangle$"}
+{"_id": "3620", "title": "Union of Local Bases is Basis", "text": "Let $T = \\left({X, \\tau}\\right)$ be a topological space. For each $x \\in X$, let $\\mathcal B_x$ be a local basis at $x$ which consists entirely of open sets. Then $\\displaystyle \\mathcal B = \\bigcup_{x \\mathop \\in X} \\mathcal B_x$ is a basis for the topology $\\tau$."}
+{"_id": "3621", "title": "Subtraction of Multiples of Divisors obeys Distributive Law", "text": "{{:Euclid:Proposition/VII/8}} In modern algebraic language: :$a = \\dfrac m n b, c = \\dfrac m n d \\implies a - c = \\dfrac m n \\paren {b - d}$"}
+{"_id": "3624", "title": "Proportional Numbers have Proportional Differences", "text": "{{:Euclid:Proposition/VII/11}} That is: :$a : b = c : d \\implies \\left({a - c}\\right) : \\left({b - d}\\right) = a : b$ where $a : b$ denotes the ratio of $a$ to $b$."}
+{"_id": "3627", "title": "Galois Group is Group", "text": "Let $L / K$ be a normal extension. Let $\\Gal {L / K}$ be the Galois group of $L / K$. Then $\\Gal {L / K}$ forms a group under the operation of composition of mappings."}
+{"_id": "3633", "title": "Relation of Ratios to Products", "text": "{{:Euclid:Proposition/VII/19}} That is: :$a : b = c : d \\iff ad = bc$"}
+{"_id": "3634", "title": "Ratios of Fractions in Lowest Terms", "text": "Let $a, b, c, d \\in \\Z_{>0}$ be positive integers. Let $\\dfrac a b$ be in canonical form. Let $\\dfrac a b = \\dfrac c d$. Then: :$a \\divides c$ and: :$b \\divides d$ where $\\divides$ denotes divisibility. {{:Euclid:Proposition/VII/20}}"}
+{"_id": "3636", "title": "Gelfond-Schneider Theorem", "text": "Let $\\alpha$ and $\\beta$ be algebraic numbers (possibly complex) such that $\\alpha \\notin \\set {0, 1}$. Let $\\beta$ be irrational. Then any value of $\\alpha^\\beta$ is transcendental."}
+{"_id": "3642", "title": "Associative Algebra has Multiplicative Inverses iff Unitary Division Algebra", "text": "Let $\\struct {A_R, \\oplus}$ be an associative algebra over the ring $A_R$. Then: :$\\struct {A_R, \\oplus}$ has a unique multiplicative inverse for every non-zero $a \\in A_R$ {{iff}}: :$\\struct {A_R, \\oplus}$ is a unitary division algebra."}
+{"_id": "3644", "title": "Commutator on Algebra is Alternating Bilinear Mapping", "text": "Let $\\left({A_R, \\oplus}\\right)$ be an algebra over a ring. Then the commutator on $\\left({A_R, \\oplus}\\right)$ is an alternating bilinear mapping: :$\\forall a, b \\in A_R: \\left[{a, b}\\right] = -\\left[{b, a}\\right]$"}
+{"_id": "3647", "title": "Axiom of Subsets Equivalents", "text": "The Axiom of Specification states that: :$\\forall z: \\forall P \\left({y}\\right): \\exists x: \\forall y: \\left({y \\in x \\iff \\left({y \\in z \\land P \\left({y}\\right)}\\right)}\\right)$ We will prove that this statement is equivalent to the following statements: :$\\forall z: \\forall A: \\left({\\left({z \\cap A}\\right) \\in U}\\right)$ :$\\forall z: \\forall A: \\left({A \\subseteq z \\implies A \\in U}\\right)$ In the above statements, the universe is $U$."}
+{"_id": "3649", "title": "Artin's Theorem on Alternative Algebras", "text": "Let $A = \\struct {A_R, \\oplus}$ be an algebra over the ring $R$ such that $A$ is ''not'' a boolean algebra. Then $A$ is alternative {{iff}}: :$\\forall a, b \\in A: \\paren {a \\oplus a} \\oplus b = a \\oplus \\paren {a \\oplus b}$ :$\\forall a, b \\in A: \\paren {b \\oplus a} \\oplus a = b \\oplus \\paren {a \\oplus a}$"}
+{"_id": "3650", "title": "Real Numbers form Algebra", "text": "The set of real numbers $\\R$ forms an algebra over the field of real numbers. This algebra is: :$(1): \\quad$ An associative algebra. :$(2): \\quad$ A commutative algebra. :$(3): \\quad$ A normed division algebra. :$(4): \\quad$ A nicely normed $*$-algebra whose $*$ operator is the identity mapping. :$(5): \\quad$ A real $*$-algebra."}
+{"_id": "3651", "title": "Real Numbers form Vector Space", "text": "The set of real numbers $\\R$, with the operations of addition and multiplication, forms a vector space."}
+{"_id": "3652", "title": "Burali-Forti Paradox", "text": "The existence of the set of all ordinals leads to a contradiction."}
+{"_id": "3653", "title": "Complex Numbers form Algebra", "text": "The set of complex numbers $\\C$ forms an algebra over the field of real numbers. This algebra is: :$(1): \\quad$ An associative algebra. :$(2): \\quad$ A commutative algebra. :$(3): \\quad$ A normed division algebra. :$(4): \\quad$ A nicely normed $*$-algebra. However, $\\C$ is not a real algebra."}
+{"_id": "3654", "title": "Equality implies Substitution", "text": "Let $P \\left({x}\\right)$ denote a Well-Formed Formula which contains $x$ as a free variable. Then the following are tautologies: :$\\forall x: \\left({P \\left({ x }\\right) \\iff \\exists y: \\left({y = x \\land P \\left({y}\\right)}\\right)}\\right)$ :$\\forall x: \\left({P \\left({ x }\\right) \\iff \\forall y: \\left({y = x \\implies P \\left({y}\\right)}\\right)}\\right)$ Note that when $y$ is substituted for $x$ in either formula, it is false in general; compare Confusion of Bound Variables."}
+{"_id": "3655", "title": "Fundamental Law of Universal Class", "text": ":$\\forall x: x \\in \\Bbb U$ where: : $\\Bbb U$ denotes the universal class : $x$ denotes a set."}
+{"_id": "3658", "title": "Cayley-Dickson Construction forms Star-Algebra", "text": "Let $A = \\left({A_F, \\oplus}\\right)$ be a $*$-algebra. Let $A' = \\left({A'_F, \\oplus'}\\right) = \\left({A, \\oplus}\\right)^2$ be the algebra formed from $A$ by the Cayley-Dickson construction. Then $A'$ is also a $*$-algebra."}
+{"_id": "3659", "title": "Empty Set is Small", "text": ":$\\O \\in U$ where $U$ is the universal class."}
+{"_id": "3660", "title": "Algebra from Cayley-Dickson Construction Never Real", "text": "Let $A$ be a $*$-algebra. Let $A'$ be constructed from $A$ using the Cayley-Dickson construction. Then $A'$ is not a real algebra."}
+{"_id": "3661", "title": "Cayley-Dickson Construction from Real Algebra is Commutative", "text": "Let $A = \\left({A_F, \\oplus}\\right)$ be a $*$-algebra. Let $A' = \\left({A_F, \\oplus'}\\right)$ be constructed from $A$ using the Cayley-Dickson construction. Then $A$ is a real algebra {{iff}} $A'$ is a commutative algebra."}
+{"_id": "3662", "title": "Cayley-Dickson Construction from Commutative Associative Algebra is Associative", "text": "Let $A = \\left({A_F, \\oplus}\\right)$ be a $*$-algebra. Let $A' = \\left({A_F, \\oplus'}\\right)$ be constructed from $A$ using the Cayley-Dickson construction. Then: : $A'$ is an associative algebra {{iff}}: : $A$ is both a commutative algebra and an associative algebra."}
+{"_id": "3663", "title": "Nicely Normed Cayley-Dickson Construction from Associative Algebra is Alternative", "text": "Let $A = \\left({A_F, \\oplus}\\right)$ be a $*$-algebra. Let $A' = \\left({A_F, \\oplus'}\\right)$ be constructed from $A$ using the Cayley-Dickson construction. Then $A'$ is a nicely normed alternative algebra {{iff}} $A$ is a nicely normed associative algebra."}
+{"_id": "3664", "title": "Element of Universe", "text": "Let $A$ be a class, which may be either a set or a proper class. Then: :$\\forall A: \\paren {A \\in U \\iff \\exists x: x = A}$ where $U$ is the universal class."}
+{"_id": "3666", "title": "Conjunction with Tautology", "text": ": $p \\land \\top \\dashv \\vdash p$"}
+{"_id": "3667", "title": "Disjunction with Tautology", "text": ": $p \\lor \\top \\dashv \\vdash \\top$"}
+{"_id": "3668", "title": "Conjunction with Contradiction", "text": ":$p \\land \\bot \\dashv \\vdash \\bot$"}
+{"_id": "3669", "title": "Disjunction with Contradiction", "text": ": $p \\lor \\bot \\dashv \\vdash p$"}
+{"_id": "3670", "title": "Contradiction is Negation of Tautology", "text": "A contradiction implies and is implied by the negation of a tautology: :$\\bot \\dashv \\vdash \\neg \\top$ That is, a falsehood can not be true, and a non-truth is a falsehood."}
+{"_id": "3671", "title": "Tautology is Negation of Contradiction", "text": "A tautology implies and is implied by the negation of a contradiction: :$\\top \\dashv \\vdash \\neg \\bot$ That is, a truth can not be false, and a non-falsehood must be a truth."}
+{"_id": "3672", "title": "Cayley-Dickson Construction from Nicely Normed Algebra is Nicely Normed", "text": "Let $A = \\left({A_F, \\oplus}\\right)$ be a $*$-algebra. Let $A' = \\left({A_F, \\oplus'}\\right)$ be constructed from $A$ using the Cayley-Dickson construction. Then $A'$ is a nicely normed algebra {{iff}} $A$ is also a nicely normed algebra."}
+{"_id": "3673", "title": "Quaternions form Algebra", "text": "The set of quaternions $\\Bbb H$ forms an algebra over the field of real numbers. This algebra is: :$(1): \\quad$ An associative algebra, but '''not''' a commutative algebra. :$(2): \\quad$ A normed division algebra. :$(3): \\quad$ A nicely normed $*$-algebra."}
+{"_id": "3674", "title": "Octonions form Algebra", "text": "The set of octonions $\\Bbb O$ forms an algebra over the field of real numbers. This algebra is: :$(1): \\quad$ An alternative algebra, but '''not''' an associative algebra. :$(2): \\quad$ A normed division algebra. :$(3): \\quad$ A nicely normed $*$-algebra."}
+{"_id": "3675", "title": "Division Algebra has No Zero Divisors", "text": "Let $A = \\left({A_F, \\oplus}\\right)$ be an algebra over a field $F$. Then $A$ is a division algebra {{iff}} it has no zero divisors. That is: :$\\forall a, b \\in A_F: a \\oplus b = \\mathbf 0_A \\implies a = \\mathbf 0_A \\lor b = \\mathbf 0_A$ If the product of two elements of $A$ is zero, then at least one of those elements must itself be zero. Some sources use this as the definition of a division algebra and from it deduce: :$\\forall a, b \\in A_F, b \\ne \\mathbf 0_A: \\exists_1 x \\in A_F, y \\in A_F: a = b \\oplus x, a = y \\oplus b$"}
+{"_id": "3676", "title": "Normed Division Algebra is Division Algebra", "text": "Let $A = \\struct {A_F, \\oplus}$ be a normed divison algebra over a field $F$. Let the unit of $A$ be $1_A$, and the zero of $A$ be $0_A$. Then $A$ is a unitary division algebra. Also: :$\\norm {1_A} = 1$ where $\\norm {1_A}$ denotes the norm of $1_A$."}
+{"_id": "3677", "title": "Zermelo's Theorem (Set Theory)", "text": "Every set of cardinals is well-ordered with respect to $\\le$."}
+{"_id": "3678", "title": "Cardinals form Equivalence Classes", "text": "Let $\\map \\Card S$ denote the cardinal of the set $S$. Then $\\map \\Card S$ induces an equivalence class which contains all sets which have the same cardinality as $S$."}
+{"_id": "3681", "title": "Integers form Commutative Ring with Unity", "text": "The integers $\\struct {\\Z, +, \\times}$ form a commutative ring with unity under addition and multiplication."}
+{"_id": "3682", "title": "Ring of Square Matrices over Ring is Ring", "text": "Let $R$ be a ring. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\struct {\\map {\\MM_R} n, +, \\times}$ denote the ring of square matrices of order $n$ over $R$. Then $\\struct {\\map {\\MM_R} n, +, \\times}$ is a ring."}
+{"_id": "3683", "title": "Ring of Square Matrices over Ring with Unity", "text": "Let $R$ be a ring with unity. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\struct {\\map {\\MM_R} n, +, \\times}$ denote the ring of square matrices of order $n$ over $R$. Then $\\struct {\\map {\\MM_R} n, +, \\times}$ is a ring with unity."}
+{"_id": "3684", "title": "Ring of Square Matrices over Field is Ring with Unity", "text": "Let $F$ be a field. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\struct {\\map {\\MM_F} n, +, \\times}$ denote the ring of square matrices of order $n$ over $F$. Then $\\struct {\\map {\\MM_F} n, +, \\times}$ is a ring with unity, but is not a commutative ring."}
+{"_id": "3686", "title": "Matrix Multiplication is not Commutative/Order 2 Square Matrices", "text": "Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\\map {\\MM_R} 2$ denote the $2 \\times 2$ matrix space over $R$. The operation of (conventional) matrix multiplication is not commutative over $\\map {\\MM_R} 2$."}
+{"_id": "3687", "title": "Ring Zero is Unique", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Then the ring zero of $R$ is unique."}
+{"_id": "3688", "title": "Ring Negative is Unique", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $a \\in R$. Then the ring negative $-a$ of $a$ is unique."}
+{"_id": "3689", "title": "Negative of Ring Negative", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $a \\in R$ and let $-a$ be the ring negative of $a$. Then: :$-\\paren {-a} = a$"}
+{"_id": "3690", "title": "Even Perfect Number is Triangular", "text": "All perfect numbers which are even are triangular."}
+{"_id": "3691", "title": "Field has no Proper Zero Divisors", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Then $\\struct {F, +, \\times}$ has no proper zero divisors. That is: :$a \\times b = 0_F \\implies a = 0_F \\lor b = 0_F$"}
+{"_id": "3693", "title": "Multiplicative Inverse in Field is Unique", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0_F$. Let $a \\in F$ such that $a \\ne 0_F$. Then the multiplicative inverse $a^{-1}$ of $a$ is unique."}
+{"_id": "3694", "title": "Inverse of Multiplicative Inverse", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0_F$. Let $a \\in F$ such that $a \\ne 0_F$. Let $a^{-1}$ be the multiplicative inverse of $a$. Then $\\paren {a^{-1} }^{-1} = a$."}
+{"_id": "3695", "title": "Integral Multiple Distributes over Ring Addition", "text": "Let $\\struct {R, +, \\times}$ be a ring, or a field. Let $a, b \\in R$ and $m, n \\in \\Z$. Then: :$(1): \\quad \\paren {m + n} \\cdot a = \\paren {m \\cdot a} + \\paren {n \\cdot a}$ :$(2): \\quad m \\cdot \\paren {a + b} = \\paren {m \\cdot a} + \\paren {m \\cdot b}$ where $m \\cdot a$ is as defined in integral multiple."}
+{"_id": "3696", "title": "Integral Multiple of Integral Multiple", "text": "Let $\\struct {F, +, \\times}$ be a field. Let $a \\in F$ and $m, n \\in \\Z$. Then: :$\\paren {m n} \\cdot a = m \\cdot \\paren {n \\cdot a}$ where $n \\cdot a$ is as defined in integral multiple."}
+{"_id": "3697", "title": "Product of Integral Multiples", "text": "Let $\\struct {F, +, \\times}$ be a field. Let $a, b \\in F$ and $m, n \\in \\Z$. Then: :$\\paren {m \\cdot a} \\times \\paren {n \\cdot b} = \\paren {m n} \\cdot \\paren {a \\times b}$ where $m \\cdot a$ is as defined in Integral Multiple."}
+{"_id": "3699", "title": "Characteristic of Field by Annihilator", "text": "Let $\\struct {F, +, \\times}$ be a field. Then of the following two cases, exactly one applies:"}
+{"_id": "3700", "title": "Annihilator of Ring Always Contains Zero", "text": "Let $\\struct {R, +, \\times}$ be a ring. Let $\\map {\\mathrm {Ann} } R$ be the annihilator of $R$. Then $0 \\in \\map {\\mathrm {Ann} } R$."}
+{"_id": "3701", "title": "Non-Trivial Annihilator Contains Positive Integer", "text": "Let $\\left({R, +, \\times}\\right)$ be a ring with unity. Let $A = \\operatorname{Ann} \\left({R}\\right)$ be the annihilator of $R$. Let $a \\in A$ such that $a \\ne 0$. Then $A$ contains at least one strictly positive integer."}
+{"_id": "3702", "title": "Area of Sector", "text": "Let $\\CC = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$. Let $BAC$ be the sector of $\\CC$ whose angle between $AB$ and $AC$ is $\\theta$. :300px Then the area $\\AA$ of sector $BAC$ is given by: :$\\AA = \\dfrac {r^2 \\theta} 2$ where: :$r = AB$ is the length of the radius of the circle :$\\theta$ is measured in radians."}
+{"_id": "3704", "title": "Rational Numbers form Prime Field", "text": "The field of rational numbers $\\struct {\\Q, +, \\times}$ is a prime field. That is, the only subset of $\\Q$ which sustains both addition and multiplication are $\\Q$ and $\\set 0$, and vacuously $\\O$."}
+{"_id": "3705", "title": "Field of Integers Modulo Prime is Prime Field", "text": "Let $p$ be a prime number. Let $\\struct {\\Z_p, +, \\times}$ be the field of integers modulo $p$. Then $\\struct {\\Z_p, +, \\times}$ is a prime field."}
+{"_id": "3706", "title": "Field has Prime Subfield", "text": "Let $\\struct {F, +, \\times}$ be a field. Then $F$ has a subfield which is a prime field."}
+{"_id": "3707", "title": "Constant Mapping to Identity is Homomorphism/Rings", "text": "Let $\\struct {R_1, +_1, \\circ_1}$ and $\\struct {R_2, +_2, \\circ_2}$ be rings with zeroes $0_1$ and $0_2$ respectively. Let $\\zeta$ be the zero homomorphism from $R_1$ to $R_2$, that is: :$\\forall x \\in R_1: \\map \\zeta x = 0_2$ Then $\\zeta$ is a ring homomorphism whose image is $\\set {0_2}$ and whose kernel is $R_1$."}
+{"_id": "3708", "title": "Constant Mapping to Identity is Homomorphism/Groups", "text": "Let $\\struct {G_1, \\circ_1}$ and $\\struct {G_2, \\circ_2}$ be groups with identities $e_1$ and $e_2$ respectively. Let $\\phi_e: \\struct {G_1, \\circ_1} \\to \\struct {G_2, \\circ_2}$ be the constant mapping defined as: :$\\forall x \\in G_1: \\map {\\phi_e} x = e_2$ Then $\\phi_e$ is a group homomorphism whose image is $\\set {e_2}$ and whose kernel is $G_1$."}
+{"_id": "3709", "title": "Identity Mapping is Automorphism/Groups", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Then $I_G: \\struct {G, \\circ} \\to \\struct {G, \\circ}$ is a group automorphism. Its kernel is $\\set e$."}
+{"_id": "3710", "title": "Identity Mapping is Automorphism/Rings", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0$. Then $I_R: \\struct {R, +, \\circ} \\to \\struct {R, +, \\circ}$ is a ring automorphism. Its kernel is $\\set 0$."}
+{"_id": "3712", "title": "Field Homomorphism Preserves Unity", "text": "Let $\\phi: \\struct {F_1, +_1, \\times_1} \\to \\struct {F_2, +_2, \\times_2}$ be a field homomorphism. Let: :$1_{F_1}$ be the unity of $F_1$ :$1_{F_2}$ be the unity of $F_2$. Then: :$\\map \\phi {1_{F_1} } = 1_{F_2}$"}
+{"_id": "3714", "title": "Ring Homomorphism from Field is Monomorphism or Zero Homomorphism", "text": "Let $\\struct {F, +_F, \\circ}$ be a field whose zero is $0_F$. Let $\\struct {S, +_S, *}$ be a ring whose zero is $0_S$. Let $\\phi: F \\to S$ be a ring homomorphism. Then either: :$(1): \\quad \\phi$ is a monomorphism (that is, $\\phi$ is injective) or :$(2): \\quad \\phi$ is the zero homomorphism (that is, $\\forall a \\in F: \\map \\phi a = 0_S$)."}
+{"_id": "3715", "title": "Zero and Unity of Subfield", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0$ and whose unity is $1$. Let $\\struct {K, +, \\times}$ be a subfield of $F$."}
+{"_id": "3716", "title": "Galois Field has Non-Zero Characteristic", "text": "Let $\\GF$ be a Galois field. Then the characteristic of $\\GF$ is non-zero."}
+{"_id": "3717", "title": "Power to Characteristic of Field is Monomorphism", "text": "Let $F$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let the characteristic of $F$ be $p$ where $p \\ne 0$. Let $\\phi: F \\to F$ be the mapping on $F$ defined as: :$\\forall x \\in F: \\map \\phi x = x^p$ Then $\\phi$ is a (field) monomorphism."}
+{"_id": "3719", "title": "Scalar Product with Identity", "text": ":$\\lambda \\circ e = 0_R \\circ x = e$"}
+{"_id": "3720", "title": "Scalar Product with Inverse", "text": ":$\\lambda \\circ \\struct {-x} = \\struct {-\\lambda} \\circ x = -\\struct {\\lambda \\circ x}$"}
+{"_id": "3721", "title": "Scalar Product with Sum", "text": ":$\\ds \\lambda \\circ \\paren {\\sum_{k \\mathop = 1}^m x_k} = \\sum_{k \\mathop = 1}^m \\paren {\\lambda \\circ x_k}$"}
+{"_id": "3722", "title": "Product with Sum of Scalar", "text": ":$\\ds \\paren {\\sum_{k \\mathop = 1}^m \\lambda_k} \\circ x = \\sum_{k \\mathop = 1}^m \\paren {\\lambda_k \\circ x}$"}
+{"_id": "3723", "title": "Scalar Product with Product", "text": ":$\\lambda \\circ \\paren {n \\cdot x} = n \\cdot \\paren {\\lambda \\circ x} = \\paren {n \\cdot \\lambda} \\circ x$"}
+{"_id": "3725", "title": "Expression of Vector as Linear Combination from Basis is Unique", "text": "Let $V$ be a vector space of dimension $n$. Let $\\mathcal B = \\set {\\mathbf x_1, \\mathbf x_2, \\ldots, \\mathbf x_n}$ be a basis for $V$. Let $\\mathbf x \\in V$ be any vector of $V$. Then $\\mathbf x$ can be expressed as a unique linear combination of elements of $\\mathcal B$. {{explain|The definition we have of linear combination doesn't really work for this. We probably need to expand that definition.}}"}
+{"_id": "3726", "title": "Homogeneous Linear Equations with More Unknowns than Equations", "text": "Let $\\alpha_{ij}$ be elements of a field $F$, where $1 \\le i \\le m, 1 \\le j \\le n$. Let $n > m$. Then there exist $x_1, x_2, \\ldots, x_n \\in F$ not all zero, such that: :$\\displaystyle \\forall i: 1 \\le i \\le m: \\sum_{j \\mathop = 1}^n \\alpha_{ij} x_j = 0$ Alternatively, this can be expressed as: If $n > m$, the following system of homogeneous linear equations: {{begin-eqn}} {{eqn | l = 0       | r = \\alpha_{11} x_1 + \\alpha_{12} x_2 + \\cdots + \\alpha_{1n} x_n }} {{eqn | l = 0       | r = \\alpha_{21} x_1 + \\alpha_{22} x_2 + \\cdots + \\alpha_{2n} x_n }} {{eqn | o = \\cdots }} {{eqn | l = 0       | r = \\alpha_{m1} x_1 + \\alpha_{m2} x_2 + \\cdots + \\alpha_{mn} x_n }} {{end-eqn}} has at least one solution such that not all of $x_1, \\ldots, x_n$ is zero."}
+{"_id": "3727", "title": "Generator of Vector Space Contains Basis", "text": ":$G$ contains a basis for $E$."}
+{"_id": "3728", "title": "Cardinality of Linearly Independent Set is No Greater than Dimension", "text": ":$H$ has at most $n$ elements."}
+{"_id": "3729", "title": "Cardinality of Generator of Vector Space is not Less than Dimension", "text": "Let $V$ be a vector space over a field $F$. Let $\\BB$ be a generator for $V$ containing $m$ elements. Then: :$\\map {\\dim_F} V \\le m$ where $\\map {\\dim_F} V$ is the dimension of $V$."}
+{"_id": "3730", "title": "Trichotomy Law for Real Numbers", "text": "The real numbers obey the Trichotomy Law. That is, $\\forall a, b \\in \\R$, exactly one of the following holds: {{begin-axiom}} {{axiom | n = 1         | lc= $a$ is greater than $b$:         | m = a > b }} {{axiom | n = 2         | lc= $a$ is equal to $b$:         | m = a = b }} {{axiom | n = 3         | lc= $a$ is less than $b$:         | m = a < b }} {{end-axiom}}"}
+{"_id": "3731", "title": "Real Number Ordering is Compatible with Multiplication", "text": "=== Positive Factor === {{:Real Number Ordering is Compatible with Multiplication/Positive Factor}} === Negative Factor === {{:Real Number Ordering is Compatible with Multiplication/Negative Factor}}"}
+{"_id": "3732", "title": "Real Number Ordering is Compatible with Addition", "text": ":$\\forall a, b, c \\in \\R: a < b \\implies a + c < b + c$"}
+{"_id": "3733", "title": "Real Number Inequalities can be Added", "text": "Let $a, b, c, d \\in \\R$ such that $a > b$ and $c > d$. Then: :$a + c > b + d$"}
+{"_id": "3734", "title": "Positive Real Number Inequalities can be Multiplied", "text": "Let $a, b, c, d \\in \\R$ such that $a > b$ and $c > d$. Let $b > 0$ and $d > 0$. Then $a c > b d$. If $b < 0$ or $d < 0$ the inequality does not hold."}
+{"_id": "3735", "title": "Real Ordering Incompatible with Subtraction", "text": "Let $a, b, c, d \\in R$ be real numbers such that $a > b$ and $c > d$. Then it does not necessarily hold that: :$a - c > b - d$"}
+{"_id": "3736", "title": "Real Ordering Incompatible with Division", "text": "Let $a, b, c, d \\in \\R$ be real numbers such that $a > b$ and $c > d$. Then it does not necessarily hold that: :$\\dfrac a c > \\dfrac b d$"}
+{"_id": "3737", "title": "Positive Real Number Inequalities can be Multiplied/Disproof for Negative Parameters", "text": "Let $a, b, c, d \\in \\R$ such that $a > b$ and $c > d$. Let $b > 0$ and $d > 0$. From Positive Real Number Inequalities can be Multiplied, $a c > b d$ holds. However, if $b < 0$ or $d < 0$ the inequality does ''not'' hold."}
+{"_id": "3738", "title": "Reverse Triangle Inequality/Real and Complex Fields/Corollary", "text": "Let $x$ and $y$ be elements of either the real numbers $\\R$ or the complex numbers $\\C$. Then: :$\\size {x - y} \\ge \\size x - \\size y$ where $\\size x$ denotes either the absolute value of a real number or the complex modulus of a complex number."}
+{"_id": "3743", "title": "Combination Theorem for Sequences/Real/Sum Rule", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {x_n + y_n} = l + m$"}
+{"_id": "3744", "title": "Combination Theorem for Sequences/Real/Multiple Rule", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {\\lambda x_n} = \\lambda l$"}
+{"_id": "3745", "title": "Combination Theorem for Sequences/Real/Combined Sum Rule", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {\\lambda x_n + \\mu y_n} = \\lambda l + \\mu m$"}
+{"_id": "3746", "title": "Combination Theorem for Sequences/Real/Product Rule", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {x_n y_n} = l m$"}
+{"_id": "3747", "title": "Combination Theorem for Sequences/Real/Quotient Rule", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\frac {x_n} {y_n} = \\frac l m$ provided that $m \\ne 0$."}
+{"_id": "3748", "title": "Squeeze Theorem/Sequences/Real Numbers", "text": "Let $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$. Let $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$ Suppose that: :$\\forall n \\in \\N: y_n \\le x_n \\le z_n$ Then: :$x_n \\to l$ as $n \\to \\infty$ that is: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ Thus, if $\\sequence {x_n}$ is always between two other sequences that both converge to the same limit, $\\sequence {x_n} $ is said to be '''sandwiched''' or '''squeezed''' between those two sequences and itself must therefore converge to that same limit."}
+{"_id": "3749", "title": "Squeeze Theorem/Sequences/Complex Numbers", "text": "Let $\\left \\langle {a_n} \\right \\rangle$ be a sequence in $\\R$ which is null, that is: :$a_n \\to 0$ as $n \\to \\infty$ Let $\\left \\langle {z_n} \\right \\rangle$ be a sequence in $\\C$. Suppose $\\left \\langle {a_n} \\right \\rangle$ dominates $\\left \\langle {z_n} \\right \\rangle$. That is, suppose that: : $\\forall n \\in \\N: \\left|{z_n}\\right| \\le a_n$ Then $\\left \\langle {z_n} \\right \\rangle$ is a null sequence."}
+{"_id": "3750", "title": "Squeeze Theorem/Sequences", "text": "There are two versions of this result: * one for sequences in the set of complex numbers $\\C$, and more generally for sequences in a metric space. * one for sequences in the set of real numbers $\\R$, and more generally in linearly ordered spaces (which is stronger). {{explain|stronger in what sense?}} === Sequences of Real Numbers === {{:Squeeze Theorem/Sequences/Real Numbers}} === Sequences of Complex Numbers === {{:Squeeze Theorem/Sequences/Complex Numbers}} === Sequences in a Linearly Ordered Space === {{:Squeeze Theorem/Sequences/Linearly Ordered Space}} === Sequences in a Metric Space === {{:Squeeze Theorem/Sequences/Metric Spaces}}"}
+{"_id": "3751", "title": "Squeeze Theorem/Functions", "text": "Let $a$ be a point on an open real interval $I$. Let $f$, $g$ and $h$ be real functions defined at all points of $I$ except for possibly at point $a$. Suppose that: :$\\forall x \\ne a \\in I: \\map g x \\le \\map f x \\le \\map h x$ :$\\displaystyle \\lim_{x \\mathop \\to a} \\ \\map g x = \\lim_{x \\mathop \\to a} \\ \\map h x = L$ Then: :$\\displaystyle \\lim_{x \\mathop \\to a} \\ \\map f x = L$"}
+{"_id": "3752", "title": "Weierstrass's Theorem", "text": "There exists a real function $f: \\closedint 0 1 \\to \\closedint 0 1$ such that: :$(1): \\quad f$ is continuous :$(2): \\quad f$ is nowhere differentiable."}
+{"_id": "3753", "title": "Combination Theorem for Continuous Functions", "text": "Let $X$ be one of the standard number fields $\\Q, \\R, \\C$. Let $f$ and $g$ be functions which are continuous on an open subset $S \\subseteq X$. Let $\\lambda, \\mu \\in X$ be arbitrary numbers in $X$. Then the following results hold: === Sum Rule === {{:Combination Theorem for Continuous Functions/Sum Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Functions/Multiple Rule}} === Combined Sum Rule === {{:Combination Theorem for Continuous Functions/Combined Sum Rule}} === Product Rule === {{:Combination Theorem for Continuous Functions/Product Rule}} === Quotient Rule === {{:Combination Theorem for Continuous Functions/Quotient Rule}}"}
+{"_id": "3754", "title": "Combination Theorem for Limits of Functions/Sum Rule", "text": ":$\\displaystyle \\lim_{x \\mathop \\to c} \\paren {\\map f x + \\map g x} = l + m$"}
+{"_id": "3755", "title": "Combination Theorem for Limits of Functions/Multiple Rule", "text": ":$\\displaystyle \\lim_{x \\mathop \\to c} \\ \\paren {\\lambda f \\paren x} = \\lambda l$"}
+{"_id": "3756", "title": "Combination Theorem for Limits of Functions/Combined Sum Rule", "text": ":$\\displaystyle \\lim_{x \\mathop \\to c} \\left({\\lambda f \\left({x}\\right) + \\mu g \\left({x}\\right)}\\right) = \\lambda l + \\mu m$"}
+{"_id": "3757", "title": "Combination Theorem for Limits of Functions/Product Rule", "text": ":$\\displaystyle \\lim_{x \\mathop \\to c} \\ \\paren {\\map f x \\map  g x} = l m$"}
+{"_id": "3758", "title": "Combination Theorem for Limits of Functions/Quotient Rule", "text": ":$\\displaystyle \\lim_{x \\mathop \\to c} \\frac {\\map f x} {\\map g x} = \\frac l m$ provided that $m \\ne 0$."}
+{"_id": "3759", "title": "Combination Theorem for Continuous Functions/Sum Rule", "text": ":$f + g$ is continuous on $S$."}
+{"_id": "3760", "title": "Combination Theorem for Continuous Functions/Product Rule", "text": ":$f g$ is continuous on $S$"}
+{"_id": "3761", "title": "Combination Theorem for Continuous Functions/Multiple Rule", "text": ":$\\lambda f$ is continuous on $S$."}
+{"_id": "3762", "title": "Combination Theorem for Continuous Functions/Combined Sum Rule", "text": ":$\\lambda f + \\mu g$ is continuous on $S$."}
+{"_id": "3763", "title": "Combination Theorem for Continuous Functions/Quotient Rule", "text": ":$\\dfrac f g$ is continuous on $S \\setminus \\set {x \\in S: \\map g x = 0}$ that is, on all the points $x$ of $S$ where $\\map g x \\ne 0$."}
+{"_id": "3764", "title": "Brouwer's Fixed Point Theorem/One-Dimensional Version", "text": "Let $f: \\closedint a b \\to \\closedint a b$ be a real function which is continuous on the closed interval $\\closedint a b$. Then: :$\\exists \\xi \\in \\closedint a b: \\map f \\xi = \\xi$ That is, a continuous real function from a closed real interval to itself fixes some point of that interval."}
+{"_id": "3765", "title": "Brouwer's Fixed Point Theorem/Smooth Mapping", "text": "A smooth mapping $f$ of the closed unit ball $B^n \\subset \\R^n$ into itself has a fixed point: :$\\forall f \\in \\map {C^\\infty} {B^n \\to B^n}: \\exists x \\in B^n: \\map f x = x$"}
+{"_id": "3766", "title": "Power Rule for Derivatives/Natural Number Index", "text": "Let $n \\in \\N$. Let $f: \\R \\to \\R$ be the real function defined as $f \\left({x}\\right) = x^n$. Then: :$f' \\left({x}\\right) = n x^{n-1}$ everywhere that $f \\left({x}\\right) = x^n$ is defined. When $x = 0$ and $n = 0$, $f^{\\prime} \\left({x}\\right)$ is undefined."}
+{"_id": "3767", "title": "Power Rule for Derivatives/Integer Index", "text": "Let $n \\in \\Z$. Let $f: \\R \\to \\R$ be the real function defined as $\\map f x = x^n$. Then: :$\\map {f'} x = n x^{n - 1}$ everywhere that $\\map f x = x^n$ is defined. When $x = 0$ and $n = 0$, $\\map {f'} x$ is undefined."}
+{"_id": "3768", "title": "Power Rule for Derivatives/Fractional Index", "text": "Let $n \\in \\N_{>0}$. Let $f: \\R \\to \\R$ be the real function defined as $\\map f x = x^{1 / n}$. Then: :$\\map {f'} x = n x^{n - 1}$ everywhere that $\\map f x = x^n$ is defined. When $x = 0$ and $n = 0$, $\\map {f'} x$ is undefined."}
+{"_id": "3769", "title": "Power Rule for Derivatives/Rational Index", "text": "Let $n \\in \\Q$. Let $f: \\R \\to \\R$ be the real function defined as $f \\left({x}\\right) = x^n$. Then: :$\\map {f'} x = n x^{n-1}$ everywhere that $\\map f x = x^n$ is defined. When $x = 0$ and $n = 0$, $\\map {f'} x$ is undefined."}
+{"_id": "3770", "title": "Power Rule for Derivatives/Real Number Index", "text": "Let $n \\in \\R$. Let $f: \\R \\to \\R$ be the real function defined as $\\map f x = x^n$. Then: :$\\map {f'} x = n x^{n-1}$ everywhere that $\\map f x = x^n$ is defined. When $x = 0$ and $n = 0$, $\\map {f'} x$ is undefined."}
+{"_id": "3771", "title": "Well-Ordering Minimal Elements are Unique", "text": "Let $\\left({S,\\preceq}\\right)$ be a well-ordered set. Then every non-empty subset of $S$ has a unique minimal element."}
+{"_id": "3773", "title": "Fundamental Theorem of Calculus/First Part", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$. Let $F$ be a real function which is defined on $\\closedint a b$ by: :$\\displaystyle \\map F x = \\int_a^x \\map f t \\rd t$ Then $F$ is a primitive of $f$ on $\\closedint a b$."}
+{"_id": "3774", "title": "Fundamental Theorem of Calculus/Second Part", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$. Then: :$(1): \\quad f$ has a primitive on $\\closedint a b$ :$(2): \\quad$ If $F$ is any primitive of $f$ on $\\closedint a b$, then: :::$\\displaystyle \\int_a^b \\map f t \\rd t = \\map F b - \\map F a = \\bigintlimits {\\map F t} a b$"}
+{"_id": "3775", "title": "Natural Logarithm of 1 is 0", "text": ":$\\ln 1 = 0$"}
+{"_id": "3776", "title": "Derivative of Natural Logarithm Function", "text": "Let $\\ln x$ be the natural logarithm function. Then: :$\\map {\\dfrac \\d {\\d x} } {\\ln x} = \\dfrac 1 x$"}
+{"_id": "3777", "title": "Derivative of General Logarithm Function", "text": "Let $a \\in \\R_{>0}$ such that $a \\ne 1$ Let $\\log_a x$ be the logarithm function to base $a$. Then: :$\\map {\\dfrac \\d {\\d x} } {\\log_a x} = \\dfrac {\\log_a e} x$"}
+{"_id": "3778", "title": "Logarithm is Strictly Increasing", "text": ":$\\ln x: x > 0$ is strictly increasing."}
+{"_id": "3779", "title": "Logarithm Tends to Infinity", "text": ":$\\ln x \\to +\\infty$ as $x \\to +\\infty$"}
+{"_id": "3780", "title": "Logarithm Tends to Negative Infinity", "text": ":$\\ln x \\to -\\infty$ as $x \\to 0^+$"}
+{"_id": "3781", "title": "Second Derivative of Natural Logarithm Function", "text": "Let $\\ln x$ be the natural logarithm function. Then: :$\\map {\\dfrac {\\d^2} {\\d x^2} } {\\ln x} = -\\dfrac 1 {x^2}$"}
+{"_id": "3782", "title": "Exponential of Zero and One", "text": "=== Exponential of Zero === {{:Exponential of Zero}} === Exponential of One === {{:Exponential of One}}"}
+{"_id": "3783", "title": "Exponential is Strictly Increasing", "text": ":The function $\\map f x = \\exp x$ is strictly increasing."}
+{"_id": "3784", "title": "Exponential Tends to Zero and Infinity", "text": ":$\\exp x \\to +\\infty$ as $x \\to +\\infty$ :$\\exp x \\to 0$ as $x \\to -\\infty$ Thus the exponential function has domain $\\R$ and image $\\openint 0 \\to$."}
+{"_id": "3785", "title": "Exponential of Natural Logarithm", "text": ": $\\forall x > 0: \\exp \\left({\\ln x}\\right) = x$ : $\\forall x \\in \\R: \\ln \\left({\\exp x}\\right) = x$"}
+{"_id": "3786", "title": "Exponent Combination Laws/Product of Powers", "text": ":$a^x a^y = a^{x + y}$"}
+{"_id": "3787", "title": "Exponent Combination Laws/Power of Product", "text": ":$\\paren {a b}^x = a^x b^x$"}
+{"_id": "3788", "title": "Exponent Combination Laws/Negative Power", "text": ": $a^{-x} = \\dfrac 1 {a^x}$"}
+{"_id": "3789", "title": "Exponent Combination Laws/Power of Power", "text": "Let $x, y \\in \\R$ be real numbers. Let $a^x$ be defined as $a$ to the power of $x$. Then: :$\\paren {a^x}^y = a^{x y}$"}
+{"_id": "3790", "title": "Exponent Combination Laws/Quotient of Powers", "text": ":$\\dfrac{a^x} {a^y} = a^{x - y}$"}
+{"_id": "3791", "title": "Exponent Combination Laws/Power of Quotient", "text": ":$\\paren {\\dfrac a b}^x = \\dfrac {a^x} {b^x}$"}
+{"_id": "3792", "title": "Exponent Combination Laws/Negative Power of Quotient", "text": ": $\\left({\\dfrac a b}\\right)^{-x} = \\left({\\dfrac b a}\\right)^x$"}
+{"_id": "3793", "title": "Binomial Theorem/General Binomial Theorem", "text": "Let $\\alpha \\in \\R$ be a real number. Let $x \\in \\R$ be a real number such that $\\size x < 1$. Then: {{begin-eqn}} {{eqn | l = \\paren {1 + x}^\\alpha       | r = \\sum_{n \\mathop = 0}^\\infty \\frac {\\alpha^{\\underline n} } {n!} x^n       | c =  }} {{eqn | r = \\sum_{n \\mathop = 0}^\\infty \\dbinom \\alpha n x^n       | c =  }} {{eqn | r = \\sum_{n \\mathop = 0}^\\infty \\frac 1 {n!} \\paren {\\prod_{k \\mathop = 0}^{n - 1} \\paren {\\alpha - k} } x^n       | c =  }} {{eqn | r = 1 + \\alpha x + \\dfrac {\\alpha \\paren {\\alpha - 1} } {2!} x^2 + \\dfrac {\\alpha \\paren {\\alpha - 1} \\paren {\\alpha - 2} } {3!} x^3 + \\cdots       | c =  }} {{end-eqn}} where: :$\\alpha^{\\underline n}$ denotes the falling factorial :$\\dbinom \\alpha n$ denotes a binomial coefficient."}
+{"_id": "3794", "title": "Real Sine Function is Continuous", "text": ":$\\sin x$ is continuous on $\\R$."}
+{"_id": "3795", "title": "Sine Function is Absolutely Convergent", "text": ":$\\sin x$ is absolutely convergent for all $x \\in \\R$."}
+{"_id": "3796", "title": "Sine of Zero is Zero", "text": ":$\\sin 0 = 0$"}
+{"_id": "3797", "title": "Sine Function is Odd", "text": ":$\\map \\sin {-z} = -\\sin z$ That is, the sine function is odd."}
+{"_id": "3798", "title": "Cosine Function is Continuous", "text": ":$\\cos x$ is continuous on $\\R$."}
+{"_id": "3799", "title": "Cosine Function is Absolutely Convergent", "text": ":$\\cos x$ is absolutely convergent for all $x \\in \\R$."}
+{"_id": "3800", "title": "Cosine of Zero is One", "text": ":$\\cos 0 = 1$"}
+{"_id": "3801", "title": "Cosine of Integer Multiple of Pi", "text": ":$\\forall n \\in \\Z: \\cos n \\pi = \\paren {-1}^n$"}
+{"_id": "3802", "title": "Cosine Function is Even", "text": ":$\\map \\cos {-z} = \\cos z$ That is, the cosine function is even."}
+{"_id": "3803", "title": "Image of Interval by Derivative", "text": "Let $f$ be a real function that is everywhere differentiable. Let $I \\subseteq \\Dom f$ be a real interval. Then: :$f' \\sqbrk I$ is a real interval where $f'$ denotes the derivative of $f$"}
+{"_id": "3804", "title": "Intermediate Value Theorem for Derivatives", "text": "Let $I$ be an open interval. Let $f : I \\to \\R$ be everywhere differentiable. Then $f'$ satisfies the Intermediate Value Property."}
+{"_id": "3805", "title": "Sine of Half-Integer Multiple of Pi", "text": ":$\\forall n \\in \\Z: \\map \\sin {n + \\dfrac 1 2} \\pi = \\paren {-1}^n$"}
+{"_id": "3806", "title": "Zeroes of Sine and Cosine", "text": ":$(1): \\quad \\forall n \\in \\Z: x = \\paren {n + \\dfrac 1 2} \\pi \\implies \\cos x = 0$ :$(2): \\quad \\forall n \\in \\Z: x = n \\pi \\implies \\sin x = 0$"}
+{"_id": "3807", "title": "Sine of Integer Multiple of Pi", "text": ":$\\forall n \\in \\Z: \\sin n \\pi = 0$"}
+{"_id": "3808", "title": "Cosine of Half-Integer Multiple of Pi", "text": ":$\\forall n \\in \\Z: \\map \\cos {n + \\dfrac 1 2} \\pi = 0$"}
+{"_id": "3809", "title": "Rational Square Root of Integer is Integer", "text": "Let $n$ be an integer. Suppose that $\\sqrt n$ is a rational number. Then $\\sqrt n$ is an integer."}
+{"_id": "3810", "title": "Definite Integral of Even Function", "text": "Let $f$ be an even function with a primitive on the closedinterval $\\closedint {-a} a$, where $a > 0$. Then: :$\\displaystyle \\int_{-a}^a \\map f x \\rd x = 2 \\int_0^a \\map f x \\rd x$"}
+{"_id": "3811", "title": "Definite Integral of Odd Function", "text": "Let $f$ be an odd function with a primitive on the open interval $\\closedint {-a} a$, where $a > 0$. Then: :$\\displaystyle \\int_{-a}^a \\map f x \\rd x = 0$"}
+{"_id": "3812", "title": "Union of Equivalence Classes is Whole Set", "text": "Let $\\RR \\subseteq S \\times S$ be an equivalence on a set $S$. Then the set of $\\RR$-classes constitutes the whole of $S$."}
+{"_id": "3813", "title": "Ratio of Consecutive Fibonacci Numbers", "text": "For $n \\in \\N$, let $f_n$ be the $n$th Fibonacci number. Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\frac {f_{n + 1} } {f_n} = \\phi$ where $\\phi = \\dfrac {1 + \\sqrt 5} 2$ is the golden mean."}
+{"_id": "3814", "title": "Cardinality of Image of Injection", "text": "Let $f: S \\rightarrowtail T$ be an injection. Let $A \\subseteq S$ be a finite subset of $S$. Then: :$\\card {f \\paren A} = \\card A$ where $\\card A$ denotes the cardinality of $A$."}
+{"_id": "3815", "title": "Composition of Mappings is not Commutative", "text": "The composition of mappings is '''not''' in general a commutative binary operation: :$f_2 \\circ f_1 \\ne f_1 \\circ f_2$"}
+{"_id": "3816", "title": "Intersection of Injective Image with Relative Complement", "text": "Let $f: S \\to T$ be a mapping. Then $f$ is an injection {{iff}}: :$\\forall A \\subseteq S: f \\sqbrk A \\cap f \\sqbrk {\\relcomp S A} = \\O$"}
+{"_id": "3817", "title": "Injection to Image is Bijection", "text": "Let $f: S \\rightarrowtail T$ be an injection. Let $X \\subseteq T$ be the image of $f$. Then the restriction $f {\\restriction_{S \\times X}}: S \\to X$ of $f$ to the image of $f$ is a bijection of $S$ onto $X$."}
+{"_id": "3819", "title": "Cardinality of Set of Induced Equivalence Classes of Surjection", "text": "Let $f: S \\to T$ be a mapping. Let $\\mathcal R_f \\subseteq S \\times S$ be the relation induced by $f$: :$\\tuple {s_1, s_2} \\in \\mathcal R_f \\iff \\map f {s_1} = \\map f {s_2}$ Let $f$ be a surjection. Then there are $\\card T$ different $\\mathcal R_f$-classes."}
+{"_id": "3820", "title": "Power of Elements is Subgroup", "text": "Let $G$ be an abelian group whose identity is $e$. Then for any $n \\in \\Z$, the set $G^n = \\set {x^n: x \\in G}$ is a subgroup of $G$. Moreover, if: :$(1): \\quad G$ is finite and: :$(2): \\quad n \\ne 1$ is a divisor of the order of $G$ then $G^n$ is a proper subgroup of $G$."}
+{"_id": "3821", "title": "Index Laws/Sum of Indices/Semigroup", "text": "Let $\\struct {S, \\circ}$ be a semigroup. For $a \\in S$, let $\\circ^n a = a^n$ be defined as the $n$th power of $a$: :$a^n = \\begin{cases} a & : n = 1 \\\\ a^x \\circ a & : n = x + 1 \\end{cases}$ That is: : $a^n = \\underbrace {a \\circ a \\circ \\cdots \\circ a}_{n \\text{ copies of } a} = \\circ^n \\paren a$ Then: :$\\forall m, n \\in \\N_{>0}: a^{n + m} = a^n \\circ a^m$"}
+{"_id": "3822", "title": "Index Laws/Product of Indices/Semigroup", "text": "Let $\\struct {S, \\circ}$ be a semigroup. For $a \\in S$, let $\\circ^n a = a^n$ be the $n$th power of $a$. Then: :$\\forall m, n \\in \\N_{>0}: a^{n m} = \\paren {a^n}^m = \\paren {a^m}^n$"}
+{"_id": "3823", "title": "Semigroup is Subsemigroup of Itself", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Then $\\struct {S, \\circ}$ is a subsemigroup of itself."}
+{"_id": "3824", "title": "Identity of Monoid is Unique", "text": "Let $\\struct {S, \\circ}$ be a monoid. Then $S$ has a unique identity."}
+{"_id": "3825", "title": "Identity of Group is Unique", "text": "Let $\\struct {G, \\circ}$ be a group which has an identity element $e \\in G$. Then $e$ is unique."}
+{"_id": "3826", "title": "Unity of Integral Domain is Unique", "text": "Let $\\struct {D, +, \\times}$ be an integral domain. Then the unity of $\\struct {D, +, \\times}$ is unique."}
+{"_id": "3827", "title": "Zero of Integral Domain is Unique", "text": "Let $\\struct {D, +, \\times}$ be an integral domain. Then the zero of $\\struct {D, +, \\times}$ is unique."}
+{"_id": "3828", "title": "Negative in Integral Domain is Unique", "text": "Let $\\struct {D, +, \\times}$ be an integral domain. Let $a \\in R$. Then the negative $-a$ of $a$ is unique."}
+{"_id": "3829", "title": "Inverse in Group is Unique", "text": "Let $\\struct {G, \\circ}$ be a group. Then every element $x \\in G$ has exactly one inverse: :$\\forall x \\in G: \\exists_1 x^{-1} \\in G: x \\circ x^{-1} = e^{-1} = x \\circ x$ where $e$ is the identity element of $\\struct {G, \\circ}$."}
+{"_id": "3830", "title": "Product Inverse in Ring is Unique", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity. Let $x \\in R$ be a unit of $R$. Then the product inverse $x^{-1}$ of $x$ is unique."}
+{"_id": "3831", "title": "Units of Gaussian Integers form Group", "text": "Let $U_\\C$ be the set of units of the Gaussian integers: :$U_\\C = \\set {1, i, -1, -i}$ where $i$ is the imaginary unit: $i = \\sqrt {-1}$. Let $\\struct {U_\\C, \\times}$ be the algebraic structure formed by $U_\\C$ under the operation of complex multiplication. Then $\\struct {U_\\C, \\times}$ forms a cyclic group under complex multiplication."}
+{"_id": "3833", "title": "Permutation Group is Subgroup of Symmetric Group", "text": "Let $S$ be a set. Let $\\struct {\\map \\Gamma S, \\circ}$ be the symmetric group on $S$, where $\\circ$ denotes the composition operation. Let $\\struct {H, \\circ}$ be a set of permutations of $S$ which forms a group under $\\circ$. Then $\\struct {H, \\circ}$ is a subgroup of $\\struct {\\map \\Gamma S, \\circ}$."}
+{"_id": "3834", "title": "Units of Field of Complex Numbers form Group", "text": ":$\\C^\\times = \\C \\setminus \\set 0$ where $\\C^\\times$ denotes the group of units of $\\C$."}
+{"_id": "3835", "title": "Determinant of Block Diagonal Matrix", "text": "Let $\\mathbf A$ be a block diagonal matrix of order $n$. Let $\\mathbf A_1,\\ldots,\\mathbf{A}_k$ be the square matrices on the diagonal, i.e.: :$\\displaystyle \\mathbf A = \\begin{bmatrix} \\mathbf A_1 & 0 & \\cdots & 0 \\\\ 0 & \\mathbf A_2 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & \\mathbf A_k  \\end{bmatrix}$ Then the determinant $\\det \\left({\\mathbf A}\\right)$ of $\\mathbf A$ satisfies: :$\\displaystyle \\det \\left({\\mathbf A}\\right) = \\prod_{i \\mathop = 1}^k \\det \\left({\\mathbf A_i}\\right)$"}
+{"_id": "3837", "title": "Coset Space forms Partition", "text": "=== Left Coset Space forms Partition === {{:Left Coset Space forms Partition}} === Right Coset Space forms Partition === {{:Right Coset Space forms Partition}}"}
+{"_id": "3838", "title": "Element of Group is in Unique Coset of Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $x \\in G$."}
+{"_id": "3839", "title": "General Linear Group to Determinant is Homomorphism", "text": "Let $\\GL {n, \\R}$ be the general linear group over the field of real numbers. Let $\\struct {\\R_{\\ne 0}, \\times}$ denote the multiplicative group of real numbers. Let $\\det: \\GL {n, \\R} \\to \\struct {\\R_{\\ne 0}, \\times}$ be the mapping: :$\\mathbf A \\mapsto \\map \\det {\\mathbf A}$ where $\\map \\det {\\mathbf A}$ is the determinant of $\\mathbf A$. Then $\\det$ is a group homomorphism."}
+{"_id": "3840", "title": "Logarithm on Positive Real Numbers is Group Isomorphism", "text": "Let $\\struct {\\R_{>0}, \\times}$ be the multiplicative group of positive real numbers. Let $\\struct {\\R, +}$ be the additive group of real numbers. Let $b$ be any real number such that $b > 1$. Let $\\log_b: \\struct {\\R_{>0}, \\times} \\to \\struct {\\R, +}$ be the mapping: :$x \\mapsto \\map {\\log_b} x$ where $\\log_b$ is the logarithm to base $b$. Then $\\log_b$ is a group isomorphism."}
+{"_id": "3841", "title": "Kernel is Trivial iff Monomorphism/Group", "text": "Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be a group homomorphism. Let $\\map \\ker \\phi$ be the kernel of $\\phi$. Then $\\phi$ is a group monomorphism {{iff}} $\\map \\ker \\phi$ is trivial."}
+{"_id": "3842", "title": "Kernel is Trivial iff Monomorphism/Ring", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring homomorphism. Let $\\map \\ker \\phi$ be the kernel of $\\phi$. Then $\\phi$ is a ring monomorphism {{iff}} $\\map \\ker \\phi = 0_{R_1}$."}
+{"_id": "3844", "title": "First Isomorphism Theorem/Groups", "text": "Let $\\phi: G_1 \\to G_2$ be a group homomorphism. Let $\\map \\ker \\phi$ be the kernel of $\\phi$. Then: :$\\Img \\phi \\cong G_1 / \\map \\ker \\phi$ where $\\cong$ denotes group isomorphism."}
+{"_id": "3845", "title": "First Isomorphism Theorem/Rings", "text": "Let $\\phi: R \\to S$ be a ring homomorphism. Let $\\map \\ker \\phi$ be the kernel of $\\phi$. Then: :$\\Img \\phi \\cong R / \\map \\ker \\phi$ where $\\cong$ denotes ring isomorphism."}
+{"_id": "3846", "title": "Second Isomorphism Theorem/Groups", "text": "Let $G$ be a group, and let: :$(1): \\quad H$ be a subgroup of $G$ :$(2): \\quad N$ be a normal subgroup of $G$. Then: :$\\displaystyle \\frac H {H \\cap N} \\cong \\frac {H N} N$ where $\\cong$ denotes group isomorphism."}
+{"_id": "3847", "title": "Second Isomorphism Theorem/Rings", "text": "Let $R$ be a ring, and let: :$S$ be a subring of $R$ :$J$ be an ideal of $R$. Then: :$(1): \\quad S + J$ is a subring of $R$ :$(2): \\quad J$ is an ideal of $S + J$ :$(3): \\quad S \\cap J$ is an ideal of $S$ :$(4): \\quad \\dfrac S {S \\cap J} \\cong \\dfrac {S + J} J$ where $\\cong$ denotes group isomorphism."}
+{"_id": "3848", "title": "Third Isomorphism Theorem/Groups", "text": "Let $G$ be a group, and let: : $H, N$ be normal subgroups of $G$ : $N$ be a subset of $H$. Then: :$(1): \\quad H / N$ is a normal subgroup of $G / N$ :::where $H / N$ denotes the quotient group of $H$ by $N$ :$(2): \\quad \\dfrac {G / N} {H / N} \\cong \\dfrac G H$ :::where $\\cong$ denotes group isomorphism."}
+{"_id": "3849", "title": "Third Isomorphism Theorem/Rings", "text": "Let $R$ be a ring, and let: :$J, K$ be ideals of $R$ :$J$ be a subset of $K$. Then: :$(1): \\quad K / J$ is an ideal of $R / J$ :where $K / J$ denotes the quotient ring of $K$ by $J$ :$(2): \\quad \\dfrac {R / J} {K / J} \\cong \\dfrac R K$ :where $\\cong$ denotes ring isomorphism."}
+{"_id": "3850", "title": "Continuous Mapping on Union of Open Sets", "text": "Let $T = \\left({X, \\tau}\\right)$ and $S = \\left({Y, \\sigma}\\right)$ be topological spaces. Let $I$ be an indexing set. For all $i \\in I$, let $C_i$ be open in $T$. Let $f: X \\to Y$ be a mapping such that the restriction $f \\restriction_{C_i}$ is continuous for all $i \\in I$. Then $f$ is continuous on $C = \\displaystyle \\bigcup_{i \\mathop \\in I} C_i$, i.e., $f \\restriction_C$ is continuous."}
+{"_id": "3851", "title": "Inertia Principle", "text": "Let $\\left \\langle {a_n}\\right \\rangle$ be a sequence in $\\R$. Let $a_n \\to l$ as $n \\to \\infty$, and let $c < l$ where $c \\in \\R$. Then $\\exists N \\in \\N$ such that: : $\\forall n \\in \\N, n \\ge N: c < a_n$"}
+{"_id": "3852", "title": "Composite of Homomorphisms is Homomorphism/Algebraic Structure", "text": "Let: : $\\left({S_1, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right)$ : $\\left({S_2, *_1, *_2, \\ldots, *_n}\\right)$ : $\\left({S_3, \\oplus_1, \\oplus_2, \\ldots, \\oplus_n}\\right)$ be algebraic structures. Let: : $\\phi: \\left({S_1, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right) \\to \\left({S_2, *_1, *_2, \\ldots, *_n}\\right)$ : $\\psi: \\left({S_2, *_1, *_2, \\ldots, *_n}\\right) \\to \\left({S_3, \\oplus_1, \\oplus_2, \\ldots, \\oplus_n}\\right)$ be homomorphisms. Then the composite of $\\phi$ and $\\psi$ is also a homomorphism."}
+{"_id": "3853", "title": "Composite of Homomorphisms is Homomorphism/R-Algebraic Structure", "text": "Let: : $\\left({S_1, *_1}\\right)_R$ : $\\left({S_2, *_2}\\right)_R$ : $\\left({S_3, *_3}\\right)_R$ be $R$-algebraic structures with the same number of operations. Let: : $\\phi: \\left({S_1, *_1}\\right)_R \\to \\left({S_2, *_2}\\right)_R$ : $\\psi: \\left({S_2, *_2}\\right)_R \\to \\left({S_3, *_3}\\right)_R$ be homomorphisms. Then the composite of $\\phi$ and $\\psi$ is also a homomorphism."}
+{"_id": "3854", "title": "Subgroup is Subgroup of Normalizer", "text": "Let $G$ be a group. A subgroup $H \\le G$ is a subgroup of its normalizer: :$H \\le G \\implies H \\le \\map {N_G} H$"}
+{"_id": "3855", "title": "Inclusion Mappings to Topological Sum from Components", "text": "Let $\\struct {X, \\tau_1}$ and $\\struct {Y, \\tau_2}$ be topological spaces. Let $\\struct {Z, \\tau_3}$ be the topological sum of $X$ and $Y$ where $\\tau_3$ is the topology generated by $\\tau_1$ and $\\tau_2$. Then $\\tau_3$ is the finest topology on $Z$ in which the inclusion mappings from $\\struct {X, \\tau_1}$ and $\\struct {Y, \\tau_2}$ to $\\struct {Z, \\tau_3}$ are continuous."}
+{"_id": "3856", "title": "Principal Ultrafilter is All Sets Containing Cluster Point", "text": "Let $S$ be a set. Let $\\powerset S$ denote the power set of $S$. Let $\\FF \\subset \\powerset S$ be a principal ultrafilter on $S$. Let its cluster point be $x$. Then $\\FF$ is the set of all subsets of $S$ which contain $x$."}
+{"_id": "3857", "title": "Equivalence of Definitions of T4 Space", "text": "{{TFAE|def = T4 Space|view = $T_4$ space}} Let $T = \\struct {S, \\tau}$ be a topological space."}
+{"_id": "3858", "title": "Equivalence of Definitions of T5 Space", "text": "{{TFAE|def = T5 Space|view = $T_5$ space}} Let $T = \\struct {S, \\tau}$ be a topological space."}
+{"_id": "3859", "title": "Product Space is T2 iff Factor Spaces are T2", "text": "Let $\\mathbb S = \\family{\\struct{S_\\alpha, \\tau_\\alpha}}_{\\alpha \\mathop \\in I}$ be an indexed family of topological spaces for $\\alpha$ in some indexing set $I$. Let $\\displaystyle T = \\struct{S, \\tau} = \\displaystyle \\prod_{\\alpha \\mathop \\in I} \\struct{S_\\alpha, \\tau_\\alpha}$ be the product space of $\\mathbb S$. Then $T$ is a $T_2$ (Hausdorff) space {{iff}} each of $\\struct{S_\\alpha, \\tau_\\alpha}$ is a $T_2$ (Hausdorff) space."}
+{"_id": "3860", "title": "Product Space is Completely Hausdorff iff Factor Spaces are Completely Hausdorff", "text": "Let $\\mathbb S = \\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be an indexed family of topological spaces for $\\alpha$ in some indexing set $I$. Let $\\displaystyle T = \\struct{S, \\tau} = \\displaystyle \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\mathbb S$. Then $T$ is a $T_{2 \\frac 1 2}$ (completely Hausdorff) space {{iff}} each of $\\struct{S_\\alpha, \\tau_\\alpha}$ is a $T_{2 \\frac 1 2}$ (completely Hausdorff) space."}
+{"_id": "3861", "title": "Product Space is T3 iff Factor Spaces are T3", "text": "Let $\\mathbb S = \\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be an indexed family of non-empty topological spaces for $\\alpha$ in some indexing set $I$. Let $\\displaystyle T = \\struct{S, \\tau} = \\displaystyle \\prod_{\\alpha \\mathop \\in I} \\struct{S_\\alpha, \\tau_\\alpha}$ be the product space of $\\mathbb S$. Then $T$ is a $T_3$ space {{iff}} each of $\\struct {S_\\alpha, \\tau_\\alpha}$ is a $T_3$ space."}
+{"_id": "3862", "title": "Product Space is T3 1/2 iff Factor Spaces are T3 1/2", "text": "Let $\\mathbb S = \\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be an indexed family of topological spaces for $\\alpha$ in some indexing set $I$ with $S_\\alpha \\ne \\O$ for every $\\alpha \\in I$. Let $\\displaystyle T = \\struct {S, \\tau} = \\displaystyle \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\mathbb S$. Then $T$ is a $T_{3 \\frac 1 2}$ space {{iff}} each of $\\struct {S_\\alpha, \\tau_\\alpha}$ is a $T_{3 \\frac 1 2}$ space."}
+{"_id": "3863", "title": "Factor Spaces are T4 if Product Space is T4", "text": "Let $\\mathbb S = \\family {\\struct{S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be an indexed family of non-empty topological spaces for $\\alpha$ in some indexing set $I$. Let $\\displaystyle T = \\struct {S, \\tau} = \\displaystyle \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\mathbb S$. Let $T$ be a $T_4$ space. Then each of $\\struct {S_\\alpha, \\tau_\\alpha}$ is a $T_4$ space."}
+{"_id": "3864", "title": "Factor Spaces are T5 if Product Space is T5", "text": "Let $\\mathbb S = \\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be an indexed family of topological spaces for $\\alpha$ in some indexing set $I$. Let $\\displaystyle T = \\struct {S, \\tau} = \\displaystyle \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\mathbb S$. Let $T$ be a $T_5$ space. Then each of $\\struct {S_\\alpha, \\tau_\\alpha}$ is a $T_5$ space."}
+{"_id": "3865", "title": "T4 Property is not Hereditary", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is a $T_4$ space. Let $T_H = \\struct {H, \\tau_H}$, where $\\O \\subset H \\subseteq S$, be a subspace of $T$. Then it does not necessarily follow that $T_H$ is a $T_4$ space."}
+{"_id": "3866", "title": "Derivative of Arcsecant Function", "text": ":$\\dfrac {\\map \\d {\\arcsec x} } {\\d x} = \\dfrac 1 {\\size x \\sqrt {x^2 - 1} } = \\begin {cases} \\dfrac {+1} {x \\sqrt {x^2 - 1} } & : 0 < \\arcsec x < \\dfrac \\pi 2 \\ (\\text {that is: $x > 1$}) \\\\ \\dfrac {-1} {x \\sqrt {x^2 - 1} } & : \\dfrac \\pi 2 < \\arcsec x < \\pi \\ (\\text {that is: $x < -1$})  \\\\ \\end{cases}$"}
+{"_id": "3867", "title": "Paracompactness is Preserved under Projections", "text": "Let $I$ be an indexing set. Let $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be a family of topological spaces indexed by $I$. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let $\\pr_\\alpha: \\struct {S, \\tau} \\to \\struct {S_\\alpha, \\tau_\\alpha}$ be the projection on the $\\alpha$ coordinate. If $\\struct {S, \\tau}$ is paracompact, then each of $\\struct {S_\\alpha, \\tau_\\alpha}$ is also paracompact."}
+{"_id": "3868", "title": "Connected Space is Connected Between Two Points", "text": "Let $T$ be a topological space which is connected. Then $T$ is connected between two points."}
+{"_id": "3869", "title": "Equality is Reflexive", "text": ":$\\forall a: a = a$"}
+{"_id": "3872", "title": "Closed Set in Metric Space is G-Delta", "text": "Let $\\left({X, d}\\right)$ be a metric space, and let $F \\subset X$ be a closed set.  Then $F$ is a $G_\\delta$ set of X."}
+{"_id": "3873", "title": "Totally Disconnected Space is Totally Pathwise Disconnected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is totally disconnected. Then $T$ is a totally pathwise disconnected space."}
+{"_id": "3874", "title": "Second-Countable T3 Space is T5", "text": "Let $T = \\struct {S, \\tau}$ be a $T_3$ space which is also second-countable. Then $T$ is a $T_5$ space."}
+{"_id": "3875", "title": "T3 Space is Fully T4 iff Paracompact", "text": "Let $T = \\struct {S, \\tau}$ be a $T_3$ space. Then $T$ is fully $T_4$ {{iff}} $T$ is paracompact."}
+{"_id": "3876", "title": "Mean Value Theorem for Integrals", "text": "Let $f$ be a continuous real function on the closed interval $\\closedint a b$. Then there exists a real number $k \\in \\closedint a b$ such that: :$\\displaystyle \\int_a^b \\map f x \\rd x = \\map f k \\paren {b - a}$"}
+{"_id": "3878", "title": "Cover of Doubletons of Infinite Particular Point Space has no Locally Finite Refinement", "text": "Let $T = \\struct {S, \\tau_p}$ be an infinite particular point space. Let $\\CC$ be the open cover of $T$ defined as: :$\\CC = \\set {\\set {x, p}: x \\in S, x \\ne p}$ Then $\\CC$ has no open refinement which is locally finite."}
+{"_id": "3879", "title": "Path-Connected iff Path-Connected to Point", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $U \\subseteq S$ be a non-empty subset of $T$. Then $U$ is path-connected {{iff}}: :$\\exists p \\in U: \\forall q \\in U: \\exists f: \\left[{0 \\,.\\,.\\, 1}\\right] \\to U: f$ continuous, $f \\left({0}\\right) = p$ and $f \\left({1}\\right) = q$"}
+{"_id": "3880", "title": "Uncountable Fort Space is not Separable", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on an uncountable set $S$. Then $T$ is not a separable space."}
+{"_id": "3881", "title": "Countable Fort Space is Separable", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on a countably infinite set $S$. Then $T$ is a separable space."}
+{"_id": "3882", "title": "Countable Fort Space is Second-Countable", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on a countably infinite set $S$. Then $T$ is a second-countable space."}
+{"_id": "3883", "title": "Fort Space is Regular", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space. Then $T$ is a regular space."}
+{"_id": "3884", "title": "Countable Fort Space is Metrizable", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on a countably infinite set $S$. Then $T$ is a metrizable space."}
+{"_id": "3888", "title": "Fortissimo Space is not Pseudocompact", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fortissimo space. Then $T$ is not a pseudocompact space."}
+{"_id": "3891", "title": "Double Pointed Fortissimo Space is not Sigma-Compact", "text": "Let $T = \\struct {S, \\tau}$ be a Fortissimo space. Let $T \\times D$ be the double pointed topology on $T$. Then $T \\times D$ is not $\\sigma$-compact."}
+{"_id": "3893", "title": "Neighborhood of Origin in Arens-Fort Space is not Compact", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Let $N_p$ be a neighborhood of the point $p = \\tuple {0, 0}$. Then $N_p$ is not compact."}
+{"_id": "3895", "title": "Arens-Fort Space is not Extremally Disconnected", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is not extremally disconnected."}
+{"_id": "3896", "title": "Modified Fort Space is Totally Disconnected", "text": "Let $T = \\struct {S, \\tau_{a, b} }$ be a modified Fort space. Then $T$ is totally disconnected."}
+{"_id": "3899", "title": "Sets in Modified Fort Space are Disconnected", "text": "Let $T = \\struct {S, \\tau_{a, b}}$ be a modified Fort space. Let $H$ be a subset of $S$ with more than one point. Then $H$ is disconnected."}
+{"_id": "3901", "title": "Modified Fort Space is not Totally Separated", "text": "Let $T = \\struct {S, \\tau_{a, b} }$ be a modified Fort space. Then $T$ is not totally separated."}
+{"_id": "3903", "title": "Modified Fort Space is Scattered", "text": "Let $T = \\struct {S, \\tau_{a, b} }$ be a modified Fort space. Then $T$ is scattered."}
+{"_id": "3904", "title": "Isolated Points in Subsets of Modified Fort Space", "text": "Let $T = \\struct {S, \\tau_{a, b} }$ be a modified Fort space. Let $H \\subseteq S$ contain more than two points. Then $H$ contains an isolated point."}
+{"_id": "3905", "title": "Modified Fort Space is Sequentially Compact", "text": "Let $T = \\struct {S, \\tau_{a, b} }$ be a modified Fort space. Then $T$ is sequentially compact."}
+{"_id": "3909", "title": "Tangent Function is Odd", "text": ":$\\map \\tan {-x} = -\\tan x$ That is, the tangent function is odd."}
+{"_id": "3910", "title": "Tangent of Sum", "text": ":$\\map \\tan {a + b} = \\dfrac {\\tan a + \\tan b} {1 - \\tan a \\tan b}$"}
+{"_id": "3912", "title": "Natural Logarithm Function is Differentiable", "text": "The (real) natural logarithm function is differentiable."}
+{"_id": "3913", "title": "Countable Basis of Real Number Line", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\\BB$ be the set of subsets of $\\R$ defined as: :$\\BB = \\set {\\openint a b: a, b \\in \\Q,\\ a < b}$ That is, $\\BB$ is the set of open intervals of $\\R$ whose endpoints are rational numbers. Then $\\BB$ forms a countable basis of $\\struct {\\R, \\tau_d}$"}
+{"_id": "3914", "title": "Second-Countable Space is Separable", "text": "Let $T = \\struct {S, \\tau}$ be a second-countable topological space. Then $T$ is also a separable space."}
+{"_id": "3915", "title": "Real Number Line is Second-Countable", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\\struct {\\R, \\tau_d}$ is second-countable."}
+{"_id": "3916", "title": "Real Number Line is not Countably Compact", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\\struct {\\R, \\tau_d}$ is not countably compact."}
+{"_id": "3918", "title": "Real Number Line is Sigma-Compact", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\\struct {\\R, \\tau_d}$ is $\\sigma$- compact."}
+{"_id": "3919", "title": "Closed Subset of Real Number Line is G-Delta", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $H \\subseteq \\R$ be a closed subset of $\\R$. Then $H$ is a $G_\\delta$ set."}
+{"_id": "3920", "title": "Real Number Line is Paracompact", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\\struct {\\R, \\tau_d}$ is paracompact."}
+{"_id": "3921", "title": "Real Number Line with Off-Center Distance Function is Quasimetric Space", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\\tau_d$ can be given by a quasimetric defined as: :$\\map d {x, y} = \\begin {cases} y - x & : y \\ge x \\\\ 2 \\paren {x - y} & : y < x \\end {cases}$ Thus $\\struct {\\R, \\tau_d}$ is a quasimetric space."}
+{"_id": "3922", "title": "Sub-basis for Uniformity on Real Number Line", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $a, b \\in \\R$ such that $a < b$. Let $S_{a b}$ be the set of subsets of $\\R$ defined as: :$S_{a b} = \\set {\\tuple {x, y}: x, y < b \\text{ or } x, y > a}$ Then $S_{ab}$ is a basis for a uniformity $U$ which generates the usual topology on $\\R$. Note that $U$ is clearly not the usual metric uniformity. {{MissingLinks|Lots of assumed definitions here which might not be properly specified.}}"}
+{"_id": "3923", "title": "NAND with Equal Arguments", "text": ":$p \\uparrow p \\dashv \\vdash \\neg p$ That is, the NAND of a proposition with itself corresponds to the negation operation."}
+{"_id": "3924", "title": "NAND is Commutative", "text": ":$p \\uparrow q \\dashv \\vdash q \\uparrow p$"}
+{"_id": "3925", "title": "NAND is not Associative", "text": ":$p \\uparrow \\left({q \\uparrow r}\\right) \\not \\vdash \\left({p \\uparrow q}\\right) \\uparrow r$"}
+{"_id": "3926", "title": "NOR with Equal Arguments", "text": ":$p \\downarrow p \\dashv \\vdash \\neg p$ That is, the NOR of a proposition with itself corresponds to the negation operator."}
+{"_id": "3927", "title": "NOR is Commutative", "text": ":$p \\downarrow q \\dashv \\vdash q \\downarrow p$"}
+{"_id": "3928", "title": "NOR is not Associative", "text": ":$p \\downarrow \\left({q \\downarrow r}\\right) \\not \\vdash \\left({p \\downarrow q}\\right) \\downarrow r$"}
+{"_id": "3929", "title": "Representation of Ternary Expansions", "text": "Let $x \\in \\R$ be a real number. Let $x$ be represented in base $3$ notation. While it may be possible for $x$ to have two different such representations, for example: :$\\dfrac 1 3 = 0.100000 \\ldots_3 = 0.022222 \\ldots_3$ it is not possible for $x$ be written in more than one way without using the digit $1$."}
+{"_id": "3930", "title": "Equivalence of Definitions of Cantor Set", "text": "{{TFAE|def = Cantor Set|view = the Cantor Set $\\CC$}}:"}
+{"_id": "3932", "title": "Cantor Set is Closed in Real Number Space", "text": "Let $\\CC$ be the Cantor set. Let $\\struct {\\R, \\tau_d}$ be the real number space $\\R$ under the Euclidean topology $\\tau_d$. Then $\\CC$ is a closed subset of $\\struct {\\R, \\tau_d}$."}
+{"_id": "3933", "title": "Word Metric is Metric", "text": "Let $\\left({G, \\circ}\\right)$ be a group. Let $S$ be a generating set for $G$ which is closed under inverses (that is, $x^{-1} \\in S \\iff x\\in S$). Let $d_S$ be the associated word metric. Then $d_S$ is a metric on $G$."}
+{"_id": "3934", "title": "Cantor Space is Compact", "text": "Let $\\CC$ be the Cantor set. Let $\\struct {\\R, \\tau_d}$ be the real number space $\\R$ under the Euclidean topology $\\tau_d$. Then $\\CC$ is a compact subset of $\\struct {\\R, \\tau_d}$."}
+{"_id": "3935", "title": "Cantor Space is Complete Metric Space", "text": "Let $T = \\struct {\\CC, \\tau_d}$ be the Cantor space. Then $T$ is a complete metric space."}
+{"_id": "3936", "title": "Cantor Space satisfies all Separation Axioms", "text": "Let $T = \\struct {\\CC, \\tau_d}$ be the Cantor space. Then $T$ satisfies all the separation axioms."}
+{"_id": "3937", "title": "Cantor Space is Second-Countable", "text": "Let $T = \\struct {\\CC, \\tau_d}$ be the Cantor space. Then $T$ is a second-countable space."}
+{"_id": "3938", "title": "Cantor Space is Dense-in-itself", "text": "Let $T = \\struct {\\CC, \\tau_d}$ be the Cantor space. Then $T$ is dense-in-itself."}
+{"_id": "3939", "title": "Cantor Space is not Scattered", "text": "Let $T = \\struct {\\CC, \\tau_d}$ be the Cantor space. Then $T$ is not scattered."}
+{"_id": "3940", "title": "Cantor Space is Perfect", "text": "Let $T = \\struct {\\CC, \\tau_d}$ be the Cantor space. Then $\\CC$ is a perfect set of the real number space $\\R$ under the usual (Euclidean) topology $\\tau_d$."}
+{"_id": "3941", "title": "Cantor Space is Nowhere Dense", "text": "Let $T = \\left({\\mathcal C, \\tau_d}\\right)$ be the Cantor space. Then $T$ is nowhere dense in $\\left[{0 \\,.\\,.\\, 1}\\right]$."}
+{"_id": "3942", "title": "Cantor Space is Meager in Closed Unit Interval", "text": "Let $T = \\struct {\\CC, \\tau_d}$ be the Cantor space. Then $T$ is meager in $\\closedint 0 1$."}
+{"_id": "3944", "title": "Cantor Set is Uncountable", "text": "The Cantor set $\\mathcal C$ is uncountable."}
+{"_id": "3945", "title": "Equality is Symmetric", "text": ":$\\forall a, b: a = b \\implies b = a$"}
+{"_id": "3947", "title": "Biconditional is Commutative/Formulation 1/Proof 2", "text": ": $p \\iff q \\dashv \\vdash q \\iff p$"}
+{"_id": "3948", "title": "Derivative of Hyperbolic Cosine Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\cosh u} = \\sinh u \\dfrac {\\d u} {\\d x}$"}
+{"_id": "3949", "title": "Cantor Space is Totally Separated", "text": "Let $T = \\struct {\\CC, \\tau_d}$ be the Cantor space. Then $T$ is totally separated."}
+{"_id": "3950", "title": "Cantor Space is not Extremally Disconnected", "text": "Let $T = \\struct {\\CC, \\tau_d}$ be the Cantor space. Then $T$ is not extremally disconnected."}
+{"_id": "3951", "title": "Cantor Space as Countably Infinite Product", "text": "Let $A_n = \\struct {\\set {0, 2}, \\tau_n}$ be the discrete space of the two points $0$ and $2$. Let $\\displaystyle A = \\prod_{n \\mathop = 1}^\\infty A_n$. Let $\\struct {A, \\tau}$ be the product space where $\\tau$ is the Tychonoff topology on $A$. Then $A$ is homeomorphic to the Cantor space."}
+{"_id": "3952", "title": "Cantor Space is not Locally Connected", "text": "Let $T = \\struct {\\CC, \\tau_d}$ be the Cantor space. Then $T$ is not locally connected."}
+{"_id": "3953", "title": "Local Connectedness is not Preserved under Infinite Product", "text": "The property of local connectedness is not preserved under the operation of forming an infinite product space."}
+{"_id": "3955", "title": "Difference of Squares of Hyperbolic Cosine and Sine", "text": ":$\\cosh^2 x - \\sinh^2 x = 1$ where $\\cosh$ and $\\sinh$ are hyperbolic cosine and hyperbolic sine."}
+{"_id": "3956", "title": "Derivative of Arccosecant Function", "text": ":$\\dfrac {\\map \\d {\\arccsc x} } {\\d x} = \\dfrac {-1} {\\size x \\sqrt {x^2 - 1} } = \\begin {cases} \\dfrac {-1} {x \\sqrt {x^2 - 1} } & : 0 < \\arccsc x < \\dfrac \\pi 2 \\ (\\text {that is: $x > 1$}) \\\\ \\dfrac {+1} {x \\sqrt {x^2 - 1} } & : -\\dfrac \\pi 2 < \\arccsc x < 0 \\ (\\text {that is: $x < -1$})  \\\\ \\end{cases}$"}
+{"_id": "3958", "title": "Power of Sum Modulo Prime", "text": "Let $p$ be a prime number. Then: :$\\paren {a + b}^p \\equiv a^p + b^p \\pmod p$"}
+{"_id": "3961", "title": "Volume of Right Circular Cone", "text": "The volume $V$ of a right circular cone is given by: :$V = \\dfrac 1 3 \\pi r^2 h$ where: : $r$ is the radius of the base : $h$ is the height of the cone, that is, the distance between the apex and the center of the base."}
+{"_id": "3962", "title": "Intersection of Topologies is Topology", "text": "Let $\\left({\\tau_i}\\right)_{i \\in I}$ be an arbitrary indexed set of topologies for a set $S$. Then $\\tau := \\displaystyle \\bigcap_{i \\mathop \\in I} {\\tau_i}$ is also a topology for $S$."}
+{"_id": "3963", "title": "Power Set is Closed under Set Difference", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Then: :$\\forall A, B \\in \\powerset S: A \\setminus B \\in \\powerset S$ where $A \\setminus B$ denotes the set difference of $A$ and $B$."}
+{"_id": "3964", "title": "Powers of Semigroup Element Commute", "text": "Let $\\struct {S, \\odot}$ be a semigroup. Let $a \\in S$. Let $m, n \\in \\Z_{>0}$. Then: :$\\forall m, n \\in \\Z_{>0}: a^n \\odot a^m = a^m \\odot a^n$"}
+{"_id": "3965", "title": "Equivalence of Definitions of Derivative", "text": "{{TFAE|def = Derivative of Real Function at Point}} Let $I$ be an open real interval. Let $f: I \\to \\R$ be a real function defined on $I$. Let $\\xi \\in I$ be a point in $I$."}
+{"_id": "3966", "title": "Derivative of x to the x", "text": ":$\\dfrac \\d {\\d x} x^x = x^x \\paren {\\ln x + 1}$"}
+{"_id": "3967", "title": "Commutativity of Powers in Semigroup", "text": ":$\\forall m, n \\in \\N_{>0}: a \\circ b = b \\circ a \\implies a^m \\circ b^n = b^n \\circ a^m$ but it is not necessarily the case that: :$\\forall m, n \\in \\N_{>0}: a^m \\circ b^n = b^n \\circ a^m \\implies a \\circ b = b \\circ a$"}
+{"_id": "3968", "title": "Power of Product of Commutative Elements in Semigroup", "text": ":$\\forall n \\in \\N_{>1}: \\paren {x \\circ y}^n = x^n \\circ y^n \\iff x \\circ y = y \\circ x$"}
+{"_id": "3971", "title": "Sum of Arcsecant and Arccosecant", "text": "Let $x \\in \\R$ be a real number such that $\\left|{x}\\right| \\ge 1$. Then: : $\\operatorname{arcsec} x + \\operatorname{arccsc} x = \\dfrac \\pi 2$ where $\\operatorname{arcsec}$ and $\\operatorname{arccsc}$ denote arcsecant and arccosecant respectively."}
+{"_id": "3972", "title": "Equation of Straight Line Tangent to Circle", "text": "Let $\\tuple {a, b}$ be the center of a circle $\\mathcal C$. Let $P_n = \\tuple {x_n, y_n}$ be any point on $\\mathcal C$. The equation of a non-vertical tangent line $\\mathcal T$ to $\\mathcal C$ is given by: :$y - y_n = \\dfrac {a - x_n} {y_n - b} \\paren {x - x_n}$ The equations of the vertical tangent lines to $\\mathcal C$ are: :$x = r - a$ for $P = \\tuple {r - a, b}$ :$x = a - r$ for $P = \\tuple {a - r, b}$"}
+{"_id": "3973", "title": "Index Laws for Monoids/Negative Index", "text": ":$\\forall n \\in \\Z: \\paren {a^n}^{-1} = a^{-n} = \\paren {a^{-1} }^n$"}
+{"_id": "3974", "title": "Infinite Limit Theorem", "text": "Let $f$ be a real function of $x$ of the form  :$\\map f x = \\dfrac {\\map g x} {\\map h x}$ Further, let $g$ and $h$ be continuous on some open interval $\\mathbb I$, where $c$ is a constant in $\\mathbb I$. If: :$(1): \\quad \\map g c \\ne 0$ :$(2): \\quad \\map h c = 0$ :$(3): \\quad \\forall x \\in \\mathbb I: x \\ne c \\implies \\map h x \\ne 0$ then the limit of $f$ as $x \\to c$ will not exist, and: :$\\displaystyle \\lim_{x \\mathop \\to c^+} \\map f x = +\\infty$ or $-\\infty$ :$\\displaystyle \\lim_{x \\mathop \\to c^-} \\map f x = +\\infty$ or $-\\infty$"}
+{"_id": "3975", "title": "Index Laws for Monoids/Product of Indices", "text": ":$\\forall m, n \\in \\Z: a^{n m} = \\paren {a^m}^n = \\paren {a^n}^m$"}
+{"_id": "3976", "title": "Index Laws for Monoids/Sum of Indices", "text": ":$\\forall m, n \\in \\Z: a^{n + m} = a^n \\circ a^m$"}
+{"_id": "3977", "title": "Powers of Group Elements/Negative Index", "text": ":$\\forall n \\in \\Z: \\paren {g^n}^{-1} = g^{-n} = \\paren {g^{-1} }^n$"}
+{"_id": "3978", "title": "Powers of Group Elements/Sum of Indices", "text": ":$\\forall m, n \\in \\Z: g^m \\circ g^n = g^{m + n}$"}
+{"_id": "3979", "title": "Powers of Group Elements/Product of Indices", "text": ":$\\forall m, n \\in \\Z: \\paren {g^m}^n = g^{m n} = \\paren {g^n}^m$"}
+{"_id": "3980", "title": "Group Element Commutes with Inverse", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $x \\in G$. Then: :$x \\circ x^{-1} = x^{-1} \\circ x$ That is, $x$ commutes with its inverse $x^{-1}$."}
+{"_id": "3981", "title": "Commutativity of Powers in Monoid", "text": ":$\\forall m, n \\in \\Z: a^m \\circ b^n = b^n \\circ a^m$"}
+{"_id": "3982", "title": "Power of Product of Commutative Elements in Monoid", "text": ":$\\forall n \\in \\Z: \\paren {a \\circ b}^n = a^n \\circ b^n$"}
+{"_id": "3983", "title": "Commutativity of Powers in Group", "text": ": $\\forall m, n \\in \\Z: a^m \\circ b^n = b^n \\circ a^m$"}
+{"_id": "3984", "title": "Power of Product of Commutative Elements in Group", "text": ":$a \\circ b = b \\circ a \\iff \\forall n \\in \\Z: \\paren {a \\circ b}^n = a^n \\circ b^n$"}
+{"_id": "3985", "title": "Primitive of Tangent Function/Cosine Form", "text": ":$\\displaystyle \\int \\tan x \\rd x = -\\ln \\size {\\cos x} + C$ where $\\cos x \\ne 0$."}
+{"_id": "3986", "title": "Primitive of Reciprocal", "text": ":$\\ds \\int \\frac {\\d x} x = \\ln \\size x + C$ for $x \\ne 0$."}
+{"_id": "3987", "title": "Powers of Commuting Elements of Semigroup Commute", "text": ":$\\forall m, n \\in \\N_{>0}: \\paren {\\circ^m a} \\circ \\paren {\\circ^n b} = \\paren {\\circ^n b} \\circ \\paren {\\circ^m a}$"}
+{"_id": "3988", "title": "Right Operation All Elements Right Zeroes", "text": "Let $\\struct {S, \\rightarrow}$ be an algebraic structure in which the operation $\\rightarrow$ is the right operation. Then no matter what $S$ is, $\\struct {S, \\rightarrow}$ is a semigroup all of whose elements are right zeroes. Thus it can be seen that any right zero in a semigroup is not necessarily unique."}
+{"_id": "3989", "title": "Left Operation All Elements Left Zeroes", "text": "Let $\\struct {S, \\leftarrow}$ be an algebraic structure in which the operation $\\leftarrow$ is the left operation. Then no matter what $S$ is, $\\struct {S, \\leftarrow}$ is a semigroup all of whose elements are left zeroes. Thus it can be seen that any left zero in a semigroup is not necessarily unique."}
+{"_id": "3990", "title": "More than One Right Zero then No Left Zero", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. If $\\struct {S, \\circ}$ has more than one left zero, then it has no right zero. Likewise, if $\\struct {S, \\circ}$ has more than one right zero, then it has no left zero."}
+{"_id": "3991", "title": "Left and Right Zero are the Same", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $z_L \\in S$ be a left zero, and $z_R \\in S$ be a right zero. Then $z_L = z_R$, that is, both the left and right zero are the same, and are therefore a zero $z$. Furthermore, $z$ is the ''only'' left and right zero for $\\circ$."}
+{"_id": "3992", "title": "Primitive of Cotangent Function", "text": ":$\\displaystyle \\int \\cot x \\rd x = \\ln \\size {\\sin x} + C$ where $\\sin x \\ne 0$."}
+{"_id": "3993", "title": "Inner Limit in Hausdorff Space by Open Neighborhoods", "text": "Let $\\left \\langle{C_n}\\right \\rangle_{n \\in \\N}$ be a sequence of sets in a Hausdorff topological space $\\left({\\mathcal X, \\tau}\\right)$. Let $x \\in \\mathcal X$. Let $\\mho \\left({x}\\right) := \\left\\{ {V \\in \\tau:\\ x \\in V}\\right\\}$ denote the set of open neighborhoods of $x$. Let $\\mathcal N_\\infty$ denote the set of cofinite subsets of $\\N$: : $\\mathcal N_\\infty := \\left\\{ {N \\subset \\N: \\N \\setminus N \\text{ is finite} }\\right\\}$ Then the inner limit of $\\left \\langle{C_n}\\right \\rangle_{n \\in \\N}$ is: :$\\displaystyle \\liminf_n C_n = \\left\\{ {x \\in \\mathcal X: \\forall V \\in \\mho \\left({x}\\right): \\exists N \\in \\mathcal N_\\infty: \\forall n \\in N: C_n \\cap V \\ne \\varnothing}\\right\\}$ or equivalently: : $\\displaystyle \\liminf_n C_n = \\left\\{{x \\in \\mathcal X: \\forall V \\in \\mho \\left({x}\\right): \\exists N_0 \\in \\N: \\forall n \\ge N_0: C_n \\cap V \\ne \\varnothing}\\right\\}$"}
+{"_id": "3995", "title": "Stabilizer of Subspace stabilizes Orthogonal Complement", "text": "Let $H$ be a finite-dimensional real or complex Hilbert space (that is, inner product space). Let $t: H \\to H$ be a normal operator on $H$. Let $t$ stabilize a subspace $V$. Then $t$ also stabilizes its orthogonal complement $V^\\perp$."}
+{"_id": "3996", "title": "Primitive of Secant Function/Secant plus Tangent Form", "text": ":$\\ds \\int \\sec x \\rd x = \\ln \\size {\\sec x + \\tan x} + C$ where $\\sec x + \\tan x \\ne 0$."}
+{"_id": "3997", "title": "Primitive of Cosecant Function/Cosecant plus Cotangent Form", "text": ":$\\ds \\int \\csc x \\rd x = -\\ln \\size {\\csc x + \\cot x} + C$ where $\\csc x + \\cot x \\ne 0$."}
+{"_id": "3999", "title": "Subset of Natural Numbers is Cofinal iff Infinite", "text": "Consider the ordered set $\\left({\\N, \\le}\\right)$, where $\\le$ is the usual ordering on the natural numbers. Let $S \\subseteq \\N$. Then $S$ is cofinal {{iff}} it is infinite."}
+{"_id": "4000", "title": "Inner Limit in Hausdorff Space by Set Closures", "text": "Let $\\struct {\\XX, \\tau}$ be a Hausdorff space. Let $\\sequence {C_n}_{n \\mathop \\in \\N}$ be a sequence of sets in $\\XX$. Then: :$\\displaystyle \\liminf_n C_n = \\bigcap_{N \\mathop \\in \\NN_\\infty^\\#} \\map \\cl {\\bigcup_{n \\mathop \\in N} C_n}$ where: :$\\cl$ denotes set closure :$\\NN_\\infty^\\#$ denotes the set of cofinal subsets of $\\N$."}
+{"_id": "4002", "title": "Ordinal is Transitive", "text": "Every ordinal (by Definition 3) is a transitive set."}
+{"_id": "4003", "title": "Foundational Relation has no Relational Loops", "text": "Let $\\prec$ be a foundational relation on $A$ and let $x_1, x_2, \\ldots, x_n \\in A$. Then: : $\\neg \\left({x_1 \\prec x_2 \\land x_3 \\prec x_4 \\cdots \\land x_n \\prec x_1}\\right)$ That is, there are no relational loops within $A$."}
+{"_id": "4004", "title": "Epsilon is Foundational", "text": "Let $\\Epsilon$ denote the epsilon relation. Then $\\Epsilon$ is a foundational relation on every class $A$."}
+{"_id": "4005", "title": "Inner Limit in Normed Spaces by Open Balls", "text": "Let $\\left \\langle{C_n}\\right \\rangle_{n \\in \\N}$ be a sequence of sets in a normed vector space $\\left({\\mathcal X, \\left\\Vert{\\cdot}\\right\\Vert}\\right)$. Then the inner limit of $\\left \\langle{C_n}\\right \\rangle_{n \\in \\N}$ is: : $\\displaystyle \\liminf_n \\ C_n = \\left\\{{x: \\forall \\epsilon > 0: \\exists N \\in \\mathcal N_\\infty: \\forall n \\in N: x \\in C_n + \\epsilon B}\\right\\}$ where $B$ denotes the open unit ball of the space. {{explain|What are $N$ and $\\mathcal N_\\infty$ in this context? Also, what point is at the center of $B$? And what does $C_n + \\epsilon B$ mean? For the latter one suspects $\\cup$, but this needs to be checked. This page might need to be rewritten from a new perspective, as the original author was touchy about symbols used and departed ProofWiki in a rage when his notation was changed.}}"}
+{"_id": "4006", "title": "Inner Limit is Closed Set", "text": "Let $\\struct {S, \\tau}$ be a Hausdorff topological space. Let $\\sequence {C_n}_{n \\mathop \\in \\N}$ be a sequence of sets in $S$. Then the inner limit $\\liminf_n C_n$ is a closed set."}
+{"_id": "4007", "title": "Local Basis of Topological Vector Space", "text": "Let $\\struct {\\XX, \\tau}$ be a topological vector space. Let $0_\\XX$ denote the zero vector of $\\XX$. Then there exists a local basis $\\BB$ of $0_\\XX$ with the following properties: :$(1): \\quad \\forall W \\in \\BB: \\exists V \\in \\BB$ such that $V + V \\subseteq W$ (where the addition $V + V$ is meant in the sense of the Minkowski sum) :$(2): \\quad$ Every $W \\in \\BB$ is star-shaped (balanced) :$(3): \\quad$ Every $W \\in \\BB$ is absorbent. :$(4): \\quad \\displaystyle \\bigcap_\\BB W = \\set {0_\\XX}$."}
+{"_id": "4008", "title": "No Membership Loops", "text": "For any proper classes or sets $A_1, A_2, \\ldots, A_n$: :$\\neg \\paren {A_1 \\in A_2 \\land A_2 \\in A_3 \\land \\cdots \\land A_n \\in A_1}$"}
+{"_id": "4009", "title": "Composite of Isomorphisms is Isomorphism/Algebraic Structure", "text": "Let: : $\\left({S_1, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right)$ : $\\left({S_2, *_1, *_2, \\ldots, *_n}\\right)$ : $\\left({S_3, \\oplus_1, \\oplus_2, \\ldots, \\oplus_n}\\right)$ be algebraic structures. Let: : $\\phi: \\left({S_1, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right) \\to \\left({S_2, *_1, *_2, \\ldots, *_n}\\right)$ : $\\psi: \\left({S_2, *_1, *_2, \\ldots, *_n}\\right) \\to \\left({S_3, \\oplus_1, \\oplus_2, \\ldots, \\oplus_n}\\right)$ be isomorphisms. Then the composite of $\\phi$ and $\\psi$ is also an isomorphism."}
+{"_id": "4010", "title": "Composite of Isomorphisms is Isomorphism/R-Algebraic Structure", "text": "Let: : $\\left({S_1, \\ast_1}\\right)_R$ : $\\left({S_2, \\ast_2}\\right)_R$ : $\\left({S_3, \\ast_3}\\right)_R$ be $R$-algebraic structures with the same number of operations. Let: : $\\phi: \\left({S_1, \\ast_1}\\right)_R \\to \\left({S_2, \\ast_2}\\right)_R$ : $\\psi: \\left({S_2, \\ast_2}\\right)_R \\to \\left({S_3, \\ast_3}\\right)_R$ be isomorphisms. Then the composite of $\\phi$ and $\\psi$ is also an isomorphism."}
+{"_id": "4011", "title": "Element of Transitive Class", "text": "Let $B$ be a transitive class. Then: :$A \\in B \\implies A \\subsetneq B$ where $\\subsetneq$ denotes a proper subset)."}
+{"_id": "4014", "title": "Restriction of Foundational Relation is Foundational", "text": "Let $S$ be a set. Let $\\RR \\subseteq S \\times S$ be a foundational relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\RR {\\restriction_T} \\subseteq T \\times T$ be the restriction of $\\RR$ to $T$. Then $\\RR {\\restriction_T}$ is a foundational relation on $T$."}
+{"_id": "4015", "title": "Restriction of Strict Well-Ordering is Strict Well-Ordering", "text": "Let $R$ be a strict well-ordering of $A$. Let $B \\subseteq A$. Then $R$ is a strict well-ordering of $B$."}
+{"_id": "4016", "title": "Alternative Definition of Ordinal", "text": "A set $S$ is an ordinal {{iff}} $S$ is transitive and is strictly well-ordered by the $\\in$-relation."}
+{"_id": "4017", "title": "Ordinal Class is Ordinal", "text": "The Ordinal Class $\\On$ is an ordinal."}
+{"_id": "4018", "title": "Transitive Set is Proper Subset of Ordinal iff Element of Ordinal", "text": "Let $A$ be an ordinal. Let $B$ be a transitive set. Then: :$B \\subsetneq A \\iff B \\in A$"}
+{"_id": "4021", "title": "Ordinal Membership is Trichotomy", "text": "Let $A$ and $B$ be ordinals. Then: :$\\left({A = B}\\right) \\lor \\left({A \\in B}\\right) \\lor \\left({B \\in A}\\right)$ where $\\lor$ denotes logical or."}
+{"_id": "4022", "title": "Second Derivative of Concave Real Function is Non-Positive", "text": "Let $f$ be a real function which is twice differentiable on the open interval $\\openint a b$. Then $f$ is concave on $\\openint a b$ {{iff}} its second derivative $f'' \\le 0$ on $\\openint a b$."}
+{"_id": "4023", "title": "Ordinal is Member of Ordinal Class", "text": "Let $A$ be an ordinal. Then: :$A \\in \\On \\lor A = \\On$ where $\\On$ denote the class of ordinals ."}
+{"_id": "4025", "title": "De Morgan's Laws (Set Theory)/Set Difference", "text": "{{:De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection}}"}
+{"_id": "4027", "title": "De Morgan's Laws (Set Theory)/Set Difference/General Case", "text": "Let $S$ and $T$ be sets. Let $\\mathcal P \\left({T}\\right)$ be the power set of $T$. Let $\\mathbb T \\subseteq \\mathcal P \\left({T}\\right)$. Then: ==== Difference with Intersection ==== {{:De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection}} ==== Difference with Union ==== {{:De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Union}}"}
+{"_id": "4029", "title": "De Morgan's Laws (Set Theory)/Set Complement", "text": "{{:De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection}}"}
+{"_id": "4033", "title": "Complement of Preimage equals Preimage of Complement", "text": "Let $f: S \\to T$ be a mapping. Let $T_1$ be a subset of $T$. Then: :$\\relcomp S {f^{-1} \\sqbrk {T_1} } = f^{-1} \\sqbrk {\\relcomp T {T_1} }$ where: :$\\complement_S$ (in this context) denotes relative complement :$f^{-1} \\sqbrk {T_1}$ denotes preimage."}
+{"_id": "4034", "title": "One-to-Many Image of Set Difference/Corollary 1", "text": "Let $\\RR \\subseteq S \\times T$ be a relation which is one-to-many. Let $A \\subseteq B \\subseteq S$. Then: :$\\relcomp {\\RR \\sqbrk B} {\\RR \\sqbrk A} = \\RR \\sqbrk {\\relcomp B A}$ where $\\complement$ (in this context) denotes relative complement."}
+{"_id": "4035", "title": "One-to-Many Image of Set Difference/Corollary 2", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation which is one-to-many. Let $A$ be a subset of $S$. Then: :$\\relcomp {\\Img {\\mathcal R} } {\\mathcal R \\sqbrk A} = \\mathcal R \\sqbrk {\\relcomp S A}$ where $\\complement$ (in this context) denotes relative complement. In the language of direct image mappings this can be presented as: :$\\forall A \\in \\powerset S: \\map {\\paren {\\complement_{\\Img {\\mathcal R} } \\circ \\mathcal R^\\to} } A = \\map {\\paren {\\mathcal R^\\to \\circ \\complement_S} } A$"}
+{"_id": "4036", "title": "Integer Power Function is Bijective iff Index is Odd", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Let $f_n: \\R \\to \\R$ be the real function defined as: :$\\map {f_n} x = x^n$ Then $f_n$ is a bijection {{iff}} $n$ is odd."}
+{"_id": "4037", "title": "Set of Infinite Sequences is Uncountable", "text": "Let $S$ be a set which contains more than one element. Let $S^\\infty$ denote the set of all sequences of elements of $S$. Then $S^\\infty$ is uncountable."}
+{"_id": "4038", "title": "Union of Topologies is not necessarily Topology", "text": "Let $\\left({\\tau_i}\\right)_{i \\in I}$ be an arbitrary indexed set of topologies for a set $S$. Then $\\tau := \\displaystyle \\bigcup_{i \\mathop \\in I} {\\tau_i}$ is not necessarily also a topology for $S$."}
+{"_id": "4039", "title": "Niven's Theorem", "text": "Consider the angles $\\theta$ in the range $0 \\le \\theta \\le \\dfrac \\pi 2$. The only values of $\\theta$ such that both $\\dfrac \\theta \\pi$ and $\\sin \\theta$ are rational are: :$\\theta = 0: \\sin \\theta = 0$ :$\\theta = \\dfrac \\pi 6: \\sin \\theta = \\dfrac 1 2$ :$\\theta = \\dfrac \\pi 2: \\sin \\theta = 1$"}
+{"_id": "4048", "title": "Sorgenfrey Line is Topology", "text": "The Sorgenfrey Line is a topological space."}
+{"_id": "4049", "title": "Sorgenfrey Line is Hausdorff", "text": "Let $T = \\struct {\\R, \\tau}$ be the Sorgenfrey line. Then $T$ is Hausdorff."}
+{"_id": "4051", "title": "Sorgenfrey Line is Perfectly Normal", "text": "Let $T = \\struct {\\R, \\tau}$ be the Sorgenfrey line. Then $T$ is perfectly normal."}
+{"_id": "4052", "title": "Sorgenfrey Line is Expansion of Real Line", "text": "Let $\\R = \\struct {\\R, d}$ be the metric space defined in Real Number Line is Metric Space. Let $T = \\struct {\\R, \\tau}$ be the Sorgenfrey line. Then $T$ is an expansion of $\\R$ as a topological space."}
+{"_id": "4053", "title": "Sorgenfrey Line satisfies all Separation Axioms", "text": "Let $T = \\left({\\R, \\tau}\\right)$ be the Sorgenfrey line. Then $T$ satisfies all separation axioms."}
+{"_id": "4054", "title": "Argument of Product equals Sum of Arguments", "text": "Let $z_1, z_2 \\in \\C$ be complex numbers. Let $\\arg$ be the argument operator. Then: :$\\map \\arg {z_1 z_2} = \\map \\arg {z_1} + \\map \\arg {z_2} + 2 k \\pi$ where $k$ can be $0$, $1$ or $-1$."}
+{"_id": "4055", "title": "Convergence of Sequence in Discrete Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a discrete topological space. Let $H = \\left \\langle{x_n}\\right \\rangle_{n \\in \\N}$ be a sequence in $S$. Then $H$ converges in $T$ to a limit iff: :$\\exists k \\in \\N: \\forall m \\in \\N: m > k: x_m = x_k$ That is, iff the sequence reaches some value of $S$ and \"stays there\"."}
+{"_id": "4056", "title": "Discrete Subspace of Fortissimo Space", "text": "Let $T = \\left({S, \\tau_p}\\right)$ be a Fortissimo space. Let $T' = \\left({S \\setminus \\left\\{{p}\\right\\}, \\tau_p}\\right)$ be the topological subspace induced on $T$ by the subset $S \\setminus \\left\\{{p}\\right\\}$. Then $T'$ is a discrete topological space."}
+{"_id": "4057", "title": "Power Reduction Formulas", "text": "=== Square of Sine === {{:Power Reduction Formulas/Sine Squared}} === Square of Cosine === {{:Power Reduction Formulas/Cosine Squared}} === Square of Tangent === {{:Power Reduction Formulas/Tangent Squared}} === Cube of Sine === {{:Power Reduction Formulas/Sine Cubed}} === Cube of Cosine === {{:Power Reduction Formulas/Cosine Cubed}} === Fourth Power of Sine === {{:Power Reduction Formulas/Sine to 4th}} === Fourth Power of Cosine === {{:Power Reduction Formulas/Cosine to 4th}} === Fifth Power of Sine === {{:Power Reduction Formulas/Sine to 5th}} === Fifth Power of Cosine === {{:Power Reduction Formulas/Cosine to 5th}}"}
+{"_id": "4058", "title": "Countable Space is Sigma-Compact", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space, where $S$ is a countable set. Then $T$ is $\\sigma$-compact."}
+{"_id": "4059", "title": "Closed Sets of Fortissimo Space", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fortissimo space. Then $H \\subseteq S$ is closed in $T$ {{iff}}: :$p \\in H$ or :$H$ is countable or both."}
+{"_id": "4060", "title": "Union of Topologies on Singleton or Doubleton is Topology", "text": "Let $S$ be a set containing either exactly one or exactly two elements. Let $\\family {\\tau_i}_{i \\mathop \\in I}$ be an arbitrary non-empty indexed set of topologies for a set $S$. Then $\\tau := \\displaystyle \\bigcup_{i \\mathop \\in I} {\\tau_i}$ is also a topology for $S$."}
+{"_id": "4061", "title": "Meager Sets in Arens-Fort Space", "text": "Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Let $A \\subseteq S$. Then $A$ is meager {{iff}} $A = \\set {\\tuple {0, 0} }$."}
+{"_id": "4062", "title": "Right and Left Regular Representations in Topological Group are Homeomorphisms", "text": "Let $\\left({G, \\cdot, \\tau}\\right)$ be a topological group. Let $g \\in G$ be an element of $G$. Then the left and right regular representations with respect to $g$: :$L_g: \\left({G, \\tau}\\right) \\to \\left({G, \\tau}\\right)$ and: :$R_g: \\left({G, \\tau}\\right) \\to \\left({G, \\tau}\\right)$ are homeomorphisms."}
+{"_id": "4063", "title": "Center of Group is Subgroup", "text": "Let $G$ be a group. The center $\\map Z G$ of $G$ is a subgroup of $G$."}
+{"_id": "4064", "title": "Center of Group is Abelian Subgroup", "text": "The center $Z \\left({G}\\right)$ of any group $G$ is a subgroup of $G$ which is abelian."}
+{"_id": "4065", "title": "Dihedral Group/Group Presentation", "text": "The dihedral group $D_n$ has the group presentation: :$D_n = \\gen {\\alpha, \\beta: \\alpha^n = \\beta^2 = e, \\beta \\alpha \\beta = \\alpha^{−1} }$ That is, the dihedral group $D_n$ is generated by two elements $\\alpha$ and $\\beta$ such that: :$(1): \\quad \\alpha^n = e$ :$(2): \\quad \\beta^2 = e$ :$(3): \\quad \\beta \\alpha = \\alpha^{n - 1} \\beta$"}
+{"_id": "4066", "title": "Intersection with Subgroup Product of Superset", "text": "Let $X, Y, Z$ be subgroups of a group $\\struct {G, \\circ}$. Let $Y \\subseteq X$. Then: :$X \\cap \\paren {Y \\circ Z} = Y \\circ \\paren {X \\cap Z}$ where $Y \\circ Z$ denotes subset product."}
+{"_id": "4067", "title": "Properties of Semi-Inner Product", "text": "Let $V$ be a vector space over $\\Bbb F \\in \\left\\{{\\R, \\C}\\right\\}$. Let $\\left \\langle{\\cdot, \\cdot}\\right \\rangle$ be a semi-inner product on $V$. Denote, for $x \\in V$, $\\left\\Vert{x}\\right\\Vert := \\left\\langle{x, x}\\right\\rangle^{1 / 2}$. Then, $\\forall x, y \\in V, a \\in \\Bbb F$: :$(1): \\quad \\left\\Vert{x + y}\\right\\Vert \\le \\left\\Vert{x}\\right\\Vert + \\left\\Vert{y}\\right\\Vert$ :$(2): \\quad \\left\\Vert{a x}\\right\\Vert = \\left|{a}\\right| \\left\\Vert{x}\\right\\Vert$"}
+{"_id": "4068", "title": "Inner Product Norm is Norm", "text": "Let $V$ be an inner product space over a subfield $\\Bbb F$ of $\\C$. Let $\\norm {\\, \\cdot \\,}$ denote the inner product norm on $V$. Then $\\norm {\\, \\cdot \\,}$ is a norm on $V$."}
+{"_id": "4069", "title": "Metric Induced by Norm is Metric", "text": "Let $V$ be a normed vector space. Let $\\norm{\\,\\cdot\\,}$ denote its norm. Let $d$ be the metric induced by $\\norm{\\,\\cdot\\,}$. Then $d$ is a metric."}
+{"_id": "4070", "title": "Product of Sums of Four Squares/Corollary", "text": "Let $a_1, a_2, \\ldots, a_n, b_1, b_2, \\ldots, b_n, c_1, c_2, \\ldots, c_n, d_1, d_2, \\ldots, d_n$ be integers. Then: : $\\displaystyle \\exists w, x, y, z \\in \\Z: \\prod_{j \\mathop = 1}^n \\left({a_j^2 + b_j^2 + c_j^2 + d_j^2}\\right) = w^2 + x^2 + y^2 + z^2$ That is, the product of any number of sums of four squares is also a sum of four squares."}
+{"_id": "4071", "title": "Subgroup Subset of Subgroup Product", "text": "Let $\\left({G, \\circ}\\right)$ be a group whose identity is $e$. Let $H$ and $K$ be subgroups of $G$. Then: :$H \\subseteq H \\circ K \\supseteq K$ where $H \\circ K$ denotes the subset product of $H$ and $K$."}
+{"_id": "4072", "title": "Product Space is Path-connected iff Factor Spaces are Path-connected", "text": "Let $\\SS = \\family {\\struct{S_i, \\tau_i}}_{i \\in I}$ be an indexed family of topological spaces for $i$ in some indexing set $I$ such that $\\forall i \\in I: S_i \\ne \\varnothing$. Let $\\displaystyle T = \\struct{S, \\tau} = \\prod_{i \\mathop \\in I} \\struct{S_i, \\tau_i}$ be the product space of $\\SS$. Then $T$ is a path-connected space {{iff}} each of $\\struct{S_i, \\tau_i}$ is a path-connected space."}
+{"_id": "4073", "title": "Normal Subgroup of Subset Product of Subgroups", "text": "Let $G$ be a group whose identity is $e$. Let: : $H$ be a subgroup of $G$ : $N$ be a normal subgroup of $G$. Then: : $N \\lhd N H$ where: : $\\lhd$ denotes normal subgroup : $N H$ denotes subset product."}
+{"_id": "4075", "title": "Non-Abelian Order 8 Group with One Order 2 Element is Quaternion Group", "text": "Let $G$ be a group with the following properties: :$(1): \\quad G$ is non-abelian. :$(2): \\quad G$ is of order $8$. :$(3): \\quad G$ has precisely one element of order $2$. Then $G$ is isomorphic to the quaternion group $Q$."}
+{"_id": "4077", "title": "Subset Product of Normal Subgroups with Trivial Intersection", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $H, K$ be normal subgroups of $G$. Let $H \\cap K = e$. Then $H K$ is isomorphic to $H \\times K$ where: : $H K$ denotes the subset product of $H$ and $K$ : $H \\times K$ denotes the direct product of $H$ and $K$."}
+{"_id": "4080", "title": "Completion Theorem (Inner Product Space)", "text": "Let $V$ be an inner product space over a subfield $\\Bbb F$ of $\\C$. Let $\\left\\langle{\\cdot, \\cdot}\\right\\rangle_V$ be the inner product on $V$. Let $d: V \\times V \\to \\R_{\\ge 0}$ be the metric induced by the inner product norm. Let $H$ be the completion of $V$ with respect to $d$. Then $\\left\\langle{\\cdot, \\cdot}\\right\\rangle_V$ can be extended to an inner product on $H$. By definition, $H$ will be a Hilbert space. Therefore, the theorem can alternatively be stated as: :Any inner product space may be completed to a Hilbert space."}
+{"_id": "4081", "title": "Identity Mapping is Group Endomorphism", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Then $I_G: \\struct {G, \\circ} \\to \\struct {G, \\circ}$ is a group endomorphism."}
+{"_id": "4082", "title": "Pythagoras's Theorem (Hilbert Space)", "text": "Let $H$ be a Hilbert space with inner product norm $\\norm {\\, \\cdot \\,}$. Let $f_1, \\ldots, f_n \\in H$ be pairwise orthogonal. Then: :$\\displaystyle \\norm {\\sum_{i \\mathop = 1}^n f_i}^2 = \\sum_{i \\mathop = 1}^n \\norm {f_i}^2$"}
+{"_id": "4083", "title": "Primitive of Reciprocal of x squared plus a squared/Arctangent Form", "text": ":$\\ds \\int \\frac {\\d x} {x^2 + a^2} = \\frac 1 a \\arctan \\frac x a + C$"}
+{"_id": "4084", "title": "Endomorphism from Integers to Multiples", "text": "Let $\\struct {\\Z, +}$ be the additive group of integers. Let $\\phi: \\struct {\\Z, +} \\to \\struct {\\Z, +}$ be a mapping. Then $\\phi$ is a group endomorphism {{iff}}: :$\\exists k \\in \\Z: \\forall n \\in \\Z: \\map \\phi n = k n$"}
+{"_id": "4085", "title": "Finite Cyclic Group is Isomorphic to Integers under Modulo Addition", "text": "Let $\\struct {G, \\circ}$ be a finite group whose identity element is $e$. Then $\\struct {G, \\circ}$ is cyclic of order $n$ {{iff}} $\\struct {G, \\circ}$ is isomorphic with the additive group of integers modulo $n$ $\\struct {\\Z_n, +_n}$."}
+{"_id": "4086", "title": "Parallelogram Law (Hilbert Space)", "text": "Let $H$ be a Hilbert space with associated norm $\\norm {\\, \\cdot \\,}$. Let $f, g \\in H$ be arbitrary. Then: :$\\norm {f + g}^2 + \\norm {f - g}^2 = 2 \\paren {\\norm f^2 + \\norm g^2}$"}
+{"_id": "4087", "title": "Unique Point of Minimal Distance", "text": "Let $H$ be a Hilbert space, and let $h \\in H$. Let $K \\subseteq H$ be a closed, convex, non-empty subset of $H$. Then there is a unique point $k_0 \\in K$ such that: :$\\norm {h - k_0} = \\map d {h, K}$ where $d$ denotes distance to a set. {{refactor|Own page, and prove it}} Furthermore, if $K$ is a linear subspace, this point is characterised by: :$\\norm {h - k_0} = \\map d {h, K} \\iff \\paren {h - k_0} \\perp K$ where $\\perp$ signifies orthogonality."}
+{"_id": "4088", "title": "Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form", "text": ":$\\displaystyle \\int \\frac 1 {\\sqrt {a^2 - x^2} } \\rd x = \\arcsin \\frac x a + C$"}
+{"_id": "4089", "title": "Inner Automorphism is Automorphism", "text": "Let $G$ be a group. Let $x \\in G$. Let $\\kappa_x$ be the inner automorphism of $x$ in $G$. Then $\\kappa_x$ is an automorphism of $G$."}
+{"_id": "4090", "title": "Order of Automorphism Group", "text": "Let $G$ be a finite group whose order is greater than $2$. Let $\\Aut G$ be the automorphism group of $G$. Then the order of $\\Aut G$ is greater than $1$."}
+{"_id": "4092", "title": "Properties of Orthogonal Projection", "text": "Let $H$ be a Hilbert space. Let $K$ be a closed linear subspace of $H$. Let $P_K$ denote the orthogonal projection on $K$. Then $P_K$ has the following properties: :$(1):\\qquad P_K$ is a linear transformation on $H$. :$(2):\\qquad \\forall h\\in H: \\left\\|{P_K(h)}\\right\\| \\le \\left\\|{h}\\right\\|$ :$(3):\\qquad P_K \\circ P_K = P_K$ :$(4):\\qquad \\ker P_K = K^{\\perp}$ and $\\operatorname{ran} P_K = K$ {{wtd|I am not entirely happy with this setup of the page, with four results stated. Strictly speaking, each should have its own page. But I am not up to thinking of a name for each of the results separately. Probably I will put up four transcluded proof pages. Lastly, note that these four results are one theorem $I.2.7$ in Conway. --LF<br/><br/>Four transcluded subpages is a technique that's been used before for this sort of linked set of results. The fact they're all part of the same proof in Conway is incidental. -- pm}}"}
+{"_id": "4093", "title": "Quotient Theorem for Group Homomorphisms", "text": "Let $\\phi: G \\to G'$ be a (group) homomorphism between two groups $G$ and $G'$. Then $\\phi$ can be decomposed into the form: :$\\phi = \\alpha \\beta \\gamma$ where: :$\\alpha: \\Img \\phi \\to G'$ is a monomorphism :$\\beta: G / \\map \\ker \\phi \\to \\Img \\phi$ is an isomorphism :$\\gamma: G \\to G / \\map \\ker \\phi$ is an epimorphism."}
+{"_id": "4094", "title": "Inclusion Mapping is Monomorphism", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure. Let $\\left({T, \\circ}\\right)$ be an algebraic substructure of $S$. Let $\\iota: T \\to S$ be the inclusion mapping from $T$ to $S$. Then $\\iota$ is a monomorphism."}
+{"_id": "4097", "title": "Schreier-Zassenhaus Theorem", "text": "Let $G$ be a finite group. Let $\\HH_1$ and $\\HH_2$ be two normal series for $G$. Then $\\HH_1$ and $\\HH_2$ have refinements of equal length whose factors are isomorphic."}
+{"_id": "4099", "title": "Jordan-Hölder Theorem", "text": "Let $G$ be a finite group. Let $\\HH_1$ and $\\HH_2$ be two composition series for $G$. Then: :$\\HH_1$ and $\\HH_2$ have the same length :Corresponding factors of $\\HH_1$ and $\\HH_2$ are isomorphic."}
+{"_id": "4100", "title": "Double Orthocomplement is Closed Linear Span", "text": "Let $\\HH$ be a Hilbert space. Let $A \\subseteq \\HH$ be a subset of $\\HH$. Then the following identity holds: :$\\paren {A^\\perp}^\\perp = \\vee A$ Here $A^\\perp$ denotes orthocomplementation, and $\\vee A$ denotes the closed linear span."}
+{"_id": "4101", "title": "Orthocomplement is Closed Linear Subspace", "text": "Let $H$ be a Hilbert space. Let $A \\subseteq H$ be a subset of $H$. Then the orthocomplement $A^\\perp$ of $A$ is a closed linear subspace of $H$."}
+{"_id": "4102", "title": "Linear Subspace Dense iff Zero Orthocomplement", "text": "Let $H$ be a Hilbert space. Let $K$ be a linear subspace of $H$. Then $K$ is everywhere dense {{iff}} $K^\\perp = \\left({0}\\right)$, where $K^\\perp$ is the orthocomplement of $K$, and $\\left({0}\\right)$ denotes the zero subspace."}
+{"_id": "4103", "title": "Finite Group has Composition Series", "text": "Let $G$ be a finite group. Then $G$ has a composition series."}
+{"_id": "4104", "title": "Length of Subgroup Plus Length of Quotient Group", "text": "Let $G$ be a finite group. Let $H$ be a normal subgroup of $G$. Then: :$\\map l G = \\map l H + \\map l {G / H}$ where: :$\\map l G$ denotes the length of $G$ :$G / H$ denotes the quotient group of $G$ by $H$."}
+{"_id": "4105", "title": "Length of Abelian Group", "text": "Let $G$ be an abelian group whose order is $n$. Let $n$ have the prime decomposition: :$\\displaystyle n = \\prod_{i \\mathop = 1}^r p_i^{k_i} = p_1^{k_1} p_2^{k_2} \\cdots p_r^{k_r}$ where $p_1 < p_2 < \\cdots < p_r$ are distinct primes and $k_1, k_2, \\ldots, k_r$ are positive integers. Then the length of $G$ is given by: :$\\displaystyle \\map l G = \\sum_{i \\mathop = 1}^r k_i = k_1 + k_2 + \\cdots + k_r$"}
+{"_id": "4106", "title": "Condition for Composition Series", "text": "Let $G$ be a finite group. Then: :a normal series $\\HH$ for $G$ is a composition series for $G$ {{iff}}: :every factor group of $\\HH$ is a simple group."}
+{"_id": "4107", "title": "Factors of Composition Series for Prime Power Group", "text": "Let $G$ be a group such that $\\order G = p^n$ where $p$ is a prime number. Then $G$ has a composition series in which each factor group is cyclic of order $p$."}
+{"_id": "4108", "title": "Continuity of Linear Functionals", "text": "Let $H$ be a Hilbert space, and let $L$ be a linear functional on $H$. Then the following four statements are equivalent: :$(1):\\quad L$ is continuous :$(2):\\quad L$ is continuous at $\\mathbf{0}_H$ :$(3):\\quad L$ is continuous at some point :$(4):\\quad \\exists c > 0: \\forall h \\in H: \\left|{Lh}\\right| \\le c \\left\\|{h}\\right\\|$"}
+{"_id": "4109", "title": "Equivalence of Definitions of Closed Linear Span", "text": "Let $H$ be a Hilbert space over $\\Bbb F \\in \\set {\\R, \\C}$, and let $A \\subseteq H$ be a subset. {{TFAE|def = Closed Linear Span|view = closed linear span of $A$}} :$(1): \\quad \\displaystyle \\vee A = \\bigcap \\Bbb M$, where $\\Bbb M$ consists of all closed linear subspaces $M$ of $H$ with $A \\subseteq M$ :$(2): \\quad \\displaystyle \\vee A$ is the smallest closed linear subspace $M$ of $H$ with $A \\subseteq M$ :$(3): \\quad \\displaystyle \\vee A = \\map \\cl {\\set {\\sum_{k \\mathop = 1}^n \\alpha_k f_k: n \\in \\N_{\\ge 1}, \\alpha_i \\in \\Bbb F, f_i \\in A} }$, where $\\cl$ denotes closure"}
+{"_id": "4110", "title": "Equivalence of Definitions of Norm of Linear Functional", "text": "Let $H$ be a Hilbert space, and let $L$ be a bounded linear functional on $H$. {{explain|We can weaken the assumption that $H$ be a Hilbert space by saying that it is just a normed space, I think. It is not strictly necessary to consider a Hilbert space.}} Define the following norms of $L$: :$(1): \\quad \\left\\|{L}\\right\\|_1 = \\sup \\left\\{{\\left|{Lh}\\right|: \\left\\|{h}\\right\\| \\le 1}\\right\\}$ :$(2): \\quad \\left\\|{L}\\right\\|_2 = \\sup \\left\\{{\\left|{Lh}\\right|: \\left\\|{h}\\right\\| = 1}\\right\\}$ :$(3): \\quad \\left\\|{L}\\right\\|_3 = \\displaystyle \\sup \\left\\{{\\dfrac {\\left|{Lh}\\right|} {\\left\\|{h}\\right\\|}: h \\in H\\setminus \\left\\{\\mathbf 0 \\right\\}}\\right\\}$ :$(4): \\quad \\left\\|{L}\\right\\|_4 = \\inf \\left\\{{c > 0: \\forall h \\in H: \\left|{Lh}\\right| \\le c \\left\\|{h}\\right\\|}\\right\\}$ Then: : $\\left\\|{L}\\right\\|_1 = \\left\\|{L}\\right\\|_2 = \\left\\|{L}\\right\\|_3 = \\left\\|{L}\\right\\|_4$"}
+{"_id": "4111", "title": "Riesz Representation Theorem (Hilbert Spaces)", "text": "Let $H$ be a Hilbert space. Let $L$ be a bounded linear functional on $H$. Then there is a unique $h_0 \\in H$ such that: :$\\forall h \\in H: L h = \\innerprod h {h_0}$"}
+{"_id": "4112", "title": "Orthonormal Subset Extends to Basis", "text": "Let $H$ be a Hilbert space, and let $S$ be an orthonormal subset of $H$. Then there exists a basis for $H$ that contains $S$ as a subset."}
+{"_id": "4113", "title": "Gram-Schmidt Orthogonalization", "text": "Let $H$ be a Hilbert space. Let $S = \\set {h_n: n \\in \\N}$ be a linearly independent subset of $H$.  Then there exists an orthonormal subset $E = \\set {e_n: n \\in \\N}$ of $H$ such that: :$\\forall k \\in \\N: \\operatorname{span} \\set {h_n: 0 \\le n \\le k} = \\operatorname{span} \\set {e_n: 0 \\le n \\le k}$ where $\\operatorname{span}$ denotes linear span."}
+{"_id": "4114", "title": "Orthogonal Projection onto Closed Linear Span", "text": "Let $H$ be a Hilbert space, and let $E = \\left\\{{e_1, \\ldots, e_n}\\right\\}$ be an orthonormal subset of $H$. Let $M = \\vee E$, the closed linear span of $E$.  Then the orthogonal projection $P$ onto $M$ satisfies, $\\forall h \\in H$: :$Ph = \\displaystyle \\sum_{k=1}^n \\left\\langle{h, e_k}\\right\\rangle e_k$"}
+{"_id": "4115", "title": "Bessel's Inequality", "text": "Let $H$ be a Hilbert space, and let $E = \\left\\{{e_n: n \\in \\N}\\right\\}$ be a countably infinite orthonormal subset of $H$. Then, for all $h \\in H$, one has the inequality: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\left|{\\left\\langle{h, e_n}\\right\\rangle}\\right|^2 \\le \\left\\|{h}\\right\\|^2$"}
+{"_id": "4116", "title": "Between two Rational Numbers exists Irrational Number", "text": "Let $a, b \\in \\Q$ where $a < b$.  Then: :$\\exists \\xi \\in \\R \\setminus \\Q: a < \\xi < b$"}
+{"_id": "4117", "title": "Abelian Group is Simple iff Prime", "text": "Let $G$ be a non-trivial abelian group. Then $G$ is simple {{iff}} $G$ is a prime group."}
+{"_id": "4118", "title": "Cyclic Group is Simple iff Prime", "text": "Let $G$ be a cyclic group. Then $G$ is simple {{iff}} $G$ is a prime group."}
+{"_id": "4119", "title": "Factors of Solvable Group are Prime", "text": "Let $G$ be a solvable group. Let $\\mathcal H$ be a composition series of $G$. Then all factor groups of $\\mathcal H$ must be prime."}
+{"_id": "4121", "title": "Singleton is Convex Set", "text": "Let $V$ be a vector space over $\\R$ or $\\C$, and let $v \\in V$. Then the singleton $S = \\left\\{{v}\\right\\}$ is a convex set."}
+{"_id": "4124", "title": "Group is Solvable iff Normal Subgroup and Quotient are Solvable", "text": "Let $G$ be a finite group. Let $H$ be a normal subgroup of $G$. Then $G$ is solvable {{iff}}: :$(1): \\quad H$ is solvable and :$(2): \\quad G / H$ is solvable where $G / H$ is the quotient group of $G$ by $H$."}
+{"_id": "4126", "title": "Intersection of Convex Sets is Convex Set (Vector Spaces)", "text": "Let $V$ be a vector space over $\\R$ or $\\C$. Let $\\mathcal C$ be a family of convex subsets of $V$. Then the intersection $\\displaystyle \\bigcap \\mathcal C$ is also a convex subset of $V$."}
+{"_id": "4127", "title": "Prime Power Group is Solvable", "text": "Let $G$ be a group whose order is $p^n$ where $p$ is a prime number and $n$ is a positive integer. Then $G$ is solvable."}
+{"_id": "4128", "title": "Subgroup of Solvable Group is Solvable", "text": "Let $G$ be a solvable group. Let $H$ be a subgroup of $G$. Then $H$ is solvable."}
+{"_id": "4129", "title": "Group with Normal Series with Solvable Factor Groups is Solvable", "text": "Let $G$ be a solvable group. Let $\\HH$ be a normal series for $G$. Let all the factor groups of $\\HH$ be solvable. Then $G$ is solvable."}
+{"_id": "4130", "title": "Group with Order Less than 60 is Solvable", "text": "Every group whose order is less than $60$ is solvable."}
+{"_id": "4131", "title": "Direct Product of Solvable Groups is Solvable", "text": "Let $G$ and $H$ be groups which are solvable. Then their (external) direct product $G \\times H$ is also solvable."}
+{"_id": "4132", "title": "Group with One Sylow Subgroup per Prime Divisor is Solvable", "text": "Let $G$ be a group of order $n$. Suppose that, for each prime number $p$ which divides $n$, $G$ has exactly one $p$-Sylow subgroup. Then $G$ is solvable."}
+{"_id": "4133", "title": "Permutation Group as Effective Transformation Group", "text": "Let $*$ be a group action of $G$ on $X$. Then $G$ is a permutation group {{iff}} $G$ acts acts effectively on $X$."}
+{"_id": "4135", "title": "Characterization of Bases (Hilbert Spaces)", "text": "Let $H$ be a Hilbert space, and let $E$ be an orthonormal subset of $H$. Then the following six statements are equivalent: :$(1): \\qquad E$ is a basis for $H$ :$(2): \\qquad h \\in H, h \\perp E \\implies h = \\mathbf 0$, where $\\perp$ denotes orthogonality :$(3): \\qquad \\vee E = H$, where $\\vee E$ denotes the closed linear span of $E$ :$(4): \\qquad \\forall h \\in H: h = \\displaystyle \\sum \\left\\{{\\left\\langle{h, e}\\right\\rangle e: e \\in E}\\right\\}$ :$(5): \\qquad \\forall g, h \\in H: \\left\\langle{g, h}\\right\\rangle = \\displaystyle \\sum \\left\\{{\\left\\langle{g, e}\\right\\rangle\\left\\langle{e, h}\\right\\rangle: e \\in E}\\right\\}$ :$(6): \\qquad \\forall h \\in H: \\left\\Vert{h}\\right\\Vert^2 = \\displaystyle \\sum \\left\\{{\\left\\vert{\\left\\langle{h, e}\\right\\rangle}\\right\\vert^2: e \\in E}\\right\\}$ In the last three statements, $\\displaystyle \\sum$ denotes a generalized sum. Statement $(6)$ is commonly known as '''Parseval's identity'''."}
+{"_id": "4136", "title": "Square Number Less than One", "text": "Let $x$ be a real number such that $x^2 < 1$. Then: : $x \\in \\left({-1 \\,.\\,.\\, 1}\\right)$ where $\\left({-1 \\,.\\,.\\, 1}\\right)$ is the open interval $\\left\\{{x \\in \\R: -1 < x < 1}\\right\\}$."}
+{"_id": "4138", "title": "Subgroups of Symmetric Group Isomorphic to Product of Subgroups", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $k \\in \\closedint 1 n$. Then there are $\\dbinom n k$ subgroups of $S_n$ which are isomorphic to $S_k \\times S_{n - k}$, where $\\dbinom n k$ denotes the binomial coefficient. All of these $\\dbinom n k$ subgroups are conjugate."}
+{"_id": "4139", "title": "Uniform Limit Theorem", "text": "Let $\\left({M, d_M}\\right)$ and $\\left({N, d_N}\\right)$ be metric spaces. Let $\\left \\langle{f_n}\\right \\rangle$ be a sequence of mappings from $M$ to $N$ such that: :$(1): \\quad \\forall n \\in \\N: f_n$ is continuous at every point of $M$ :$(2): \\quad  \\langle f_n\\rangle$ converges uniformly to $f$ Then: : $f$ is continuous at every point of $M$."}
+{"_id": "4140", "title": "Equality is Transitive", "text": ":$\\forall a, b, c: \\paren {a = b} \\land \\paren {b = c} \\implies a = c$"}
+{"_id": "4141", "title": "Axiom:Axioms of Equality", "text": "The '''axioms of equality''' are strictly speaking not axiomatic at all, as they can be deduced from still more basic axioms, in particular Leibniz's law: :$x = y \\dashv \\vdash P \\left({x}\\right) \\iff P \\left({y}\\right)$ where $P \\left({x}\\right)$ and $P \\left({y}\\right)$ are propositional functions on the elements $x$ and $y$ of the universe of discourse."}
+{"_id": "4142", "title": "Axiom of Pairing from Infinity and Replacement", "text": "The Axiom of Pairing is a consequence of: :the Axiom of Infinity and :the Axiom of Replacement."}
+{"_id": "4143", "title": "Axiom of Pairing from Powers and Replacement", "text": "The Axiom of Pairing is a consequence of: :the Axiom of the Empty Set and :the Axiom of Powers and also :the Axiom of Replacement."}
+{"_id": "4144", "title": "Axiom of Pairing from Axiom of Subsets", "text": "Let it be supposed that there exists a set which contains at least two elements. Then the Axiom of Pairing is a consequence of the Axiom of Specification."}
+{"_id": "4145", "title": "Existence of Singleton Set", "text": "Let $a$ be a set. Then the singleton set $\\set a$ may be constructed such that: :$a \\in \\set a$"}
+{"_id": "4147", "title": "Intersection of Unions with Complements is Subset of Union", "text": "Let $R, S, T$ be sets. Then: :$\\left({R \\cup S}\\right) \\cap \\left({\\overline R \\cup T}\\right) \\subseteq S \\cup T$"}
+{"_id": "4148", "title": "Conjunction implies Disjunction of Conjunctions with Complements", "text": ":$p \\land q \\vdash \\left({p \\land r}\\right) \\lor \\left({q \\land \\neg r}\\right)$"}
+{"_id": "4149", "title": "Conjunction of Disjunctions with Complements implies Disjunction", "text": ":$\\paren {p \\lor r} \\land \\paren {q \\lor \\neg r} \\vdash p \\lor q$"}
+{"_id": "4150", "title": "Intersection of Elements of Power Set", "text": "Let $S$ be a set. Let: : $\\displaystyle \\mathbb S = \\bigcap_{X \\mathop \\in \\mathcal P \\left({S}\\right)} X$ where $\\mathcal P \\left({S}\\right)$ is the power set of $S$. Then $\\mathbb S = \\varnothing$."}
+{"_id": "4151", "title": "Power Set of Subset", "text": "Let $S \\subseteq T$ where $S$ and $T$ are both sets. Then: :$\\powerset S \\subseteq \\powerset T$ where $\\powerset S$ denotes the power set of $S$."}
+{"_id": "4153", "title": "Equivalence of Definitions of Ordered Pair", "text": "The following definitions of an ordered pair are equivalent:"}
+{"_id": "4154", "title": "Subset of Cartesian Product", "text": "Let $S$ be a set of ordered pairs. Then $S$ is the subset of the cartesian product of two sets."}
+{"_id": "4157", "title": "Continuously Differentiable Curve has Finite Arc Length", "text": "Let $y = \\map f x$ be a real function which is continuous on the closed interval $\\closedint a b$ and continuously differentiable on the open interval $\\openint a b$. The definite integral:  :$s = \\displaystyle \\int_{x \\mathop = a}^{x \\mathop = b} \\sqrt {1 + \\paren {\\frac {\\d y} {\\d x} }^2} \\rd x$ exists, and is called the '''arc length''' of $f$ between $a$ and $b$."}
+{"_id": "4158", "title": "Arc Length for Parametric Equations", "text": "Let $x = \\map f t$ and $y = \\map g t$ be real functions of a parameter $t$.  Let these equations describe a curve $\\mathcal C$ that is continuous for all $t \\in \\closedint a b$ and continuously differentiable for all $t \\in \\openint a b$. Suppose that the graph of the curve does not intersect itself for any $t \\in \\openint a b$. Then the arc length of $\\mathcal C$ between $a$ and $b$ is given by: :$s = \\displaystyle \\int_a^b \\sqrt {\\paren {\\frac {\\d x} {\\d t} }^2 + \\paren {\\frac {\\d y} {\\d t} }^2} \\rd t$ for $\\dfrac {\\d x} {\\d t} \\ne 0$."}
+{"_id": "4159", "title": "Inverse Element of Injection", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be an injection. Then: :$\\map {f^{-1} } y = x \\iff \\map f x = y$"}
+{"_id": "4160", "title": "Hilbert Space Separable iff Countable Dimension", "text": "Let $H$ be a Hilbert space. Then $H$ is a separable space iff it has countable dimension."}
+{"_id": "4161", "title": "Net Convergence Equivalent to Absolute Convergence", "text": "Let $V$ be a Banach space. Let $\\left({v_n}\\right)_{n \\in \\N}$ be a sequence of elements in $V$. Then the following two statements are equivalent: :$(1): \\qquad \\displaystyle \\sum_{n=1}^\\infty \\left\\Vert{v_n}\\right\\Vert$ converges (absolute convergence) :$(2): \\qquad \\displaystyle \\sum \\left\\{{v_n: n \\in \\N}\\right\\}$ converges (generalised or net convergence)"}
+{"_id": "4162", "title": "Intersection of Infinite Successor Sets", "text": "Let $\\mathbb S$ be a non-empty indexed family of infinite successor sets. Then $\\bigcap \\mathbb S$ is an infinite successor set."}
+{"_id": "4163", "title": "Minimal Infinite Successor Set Exists", "text": "There exists a minimal infinite successor set $\\omega$ that is a subset of every other infinite successor set."}
+{"_id": "4164", "title": "Minimal Infinite Successor Set forms Peano Structure", "text": "Let $\\omega$ be the minimal infinite successor set. Let $\\cdot^+: \\omega \\to \\omega$ be the mapping assigning to a set its successor set: :$n^+ := n \\cup \\left\\{{n}\\right\\}$ Let $\\varnothing \\in \\omega$ be the empty set. Then $\\left({\\omega, \\cdot^+, \\varnothing}\\right)$ is a Peano structure."}
+{"_id": "4165", "title": "Integral of Logarithm", "text": ":$\\displaystyle \\int \\ln x \\ \\mathrm d x = x \\ln x - x + C$ :$\\displaystyle \\int \\log_a x \\ \\mathrm d x =  \\frac 1 {\\ln a}\\left({x \\ln x - x}\\right) + C$ for $x > 0$."}
+{"_id": "4166", "title": "Finite Ordinal is not Subset of one of its Elements", "text": "Let $n$ be a finite ordinal. Then: :$\\nexists x \\in n: n \\subseteq x$ that is, $n$ is not a subset of one of its elements."}
+{"_id": "4167", "title": "Element of Minimal Infinite Successor Set is Transitive Set", "text": "Let $\\omega$ be the minimal infinite successor set. Let $n \\in \\omega$. Then $x \\in n \\implies x \\subseteq n$. That is, every element of $n$ is also a subset of it. In other words, each element of $\\omega$ is a transitive set."}
+{"_id": "4168", "title": "Primitive of Exponential Function", "text": ":$\\displaystyle \\int e^x \\rd x = e^x + C$ where $C$ is an arbitrary constant."}
+{"_id": "4170", "title": "Natural Number Addition Commutes with Zero", "text": "Let $\\N$ be the natural numbers. Then: :$\\forall n \\in \\N: 0 + n = n = n + 0$"}
+{"_id": "4171", "title": "Natural Number Addition Commutativity with Successor", "text": "Let $\\N$ be the natural numbers. Then: :$\\forall m, n \\in \\N: m^+ + n = \\paren {m + n}^+$"}
+{"_id": "4172", "title": "Natural Numbers are Comparable", "text": "Let $\\N$ be the natural numbers. Let $m, n \\in \\N$. Then $m$ and $n$ are comparable by the ordering relation $\\le$. That is, either: :$(1): \\quad m \\le n$ or: :$(2): \\quad n \\le m$ or possibly both."}
+{"_id": "4177", "title": "Zero is Zero Element for Natural Number Multiplication", "text": "Let $\\N$ be the natural numbers. Then $0$ is a zero element for multiplication: :$\\forall n \\in \\N: 0 \\times n = 0 = n \\times 0$"}
+{"_id": "4178", "title": "Hilbert Space Isomorphism is Equivalence Relation", "text": "Hilbert space isomorphism is an equivalence relation."}
+{"_id": "4179", "title": "Element of Finite Ordinal iff Subset", "text": "Let $m, n$ be distinct finite ordinals. Then: :$m \\in n \\iff m \\subseteq n$"}
+{"_id": "4180", "title": "Ordering on Natural Numbers is Compatible with Addition", "text": "Let $m, n, k \\in \\N$ where $\\N$ is the set of natural numbers. Then: :$m < n \\iff m + k < n + k$"}
+{"_id": "4181", "title": "Hilbert Space Isomorphism is Bijection", "text": "Let $H, K$ be Hilbert spaces. Denote by $\\left\\langle{\\cdot, \\cdot}\\right\\rangle_H$ and $\\left\\langle{\\cdot, \\cdot}\\right\\rangle_K$ their respective inner products. Let $U: H \\to K$ be an isomorphism. Then $U$ is a bijection."}
+{"_id": "4182", "title": "Hilbert Spaces Isomorphic iff Same Dimension", "text": "Two Hilbert spaces are isomorphic iff they have the same dimension."}
+{"_id": "4183", "title": "Surjection that Preserves Inner Product is Linear", "text": "Let $H, K$ be Hilbert spaces, and denote by $\\left\\langle{\\cdot, \\cdot}\\right\\rangle_H$ and $\\left\\langle{\\cdot, \\cdot}\\right\\rangle_K$ their respective inner products. Let $U: H \\to K$ be a surjection such that: :$\\forall g,h \\in H: \\left\\langle{g, h}\\right\\rangle_H = \\left\\langle{Ug, Uh}\\right\\rangle_K$ Then $U$ is a linear map, and hence an isomorphism."}
+{"_id": "4184", "title": "Linear Operator on General Logarithm", "text": "Let $\\phi: \\R^\\R \\to \\R^\\R, y \\mapsto \\phi\\left({y}\\right)$ be a linear operator on the space of functions from $\\R\\to\\R$. Let $y$ be a real function such that $\\forall x \\in \\R$, $y\\left({x}\\right) > \\mathbf{0}\\left({x}\\right) = 0$. Let $\\log_a y$ be the logarithm of $y$ to base $a$. Then: :$\\phi \\left({\\log_a y}\\right) = \\dfrac 1 {\\ln a} \\left({\\phi \\left({\\ln y}\\right)}\\right)$ where $\\ln$ is the natural logarithm."}
+{"_id": "4185", "title": "Ordering on Natural Numbers is Compatible with Multiplication", "text": "Let $m, n, k \\in \\N$ where $\\N$ is the set of natural numbers. Let $k \\ne 0$. Then: :$m < n \\iff m \\times k < n \\times k$"}
+{"_id": "4186", "title": "Natural Number Ordering is Transitive", "text": "Let $m, n, k \\in \\N$ where $\\N$ is the set of natural numbers. Let $<$ be the relation defined on $\\N$ such that: :$m < n \\iff m \\in n$ where $\\N$ is defined as the minimal infinite successor set $\\omega$. Then: :$k < m, m < n \\implies k < n$ That is: $<$ is a transitive relation."}
+{"_id": "4187", "title": "Proper Subset of Finite Ordinal is Equivalent to Smaller Ordinal", "text": "Let $n$ be a finite ordinal. Let $x \\subsetneq n$. Then for some finite ordinal $m < n$: :$m \\sim x$ where $m \\sim x$ denotes that $m$ is (set) equivalent to $x$. That is, every proper subset of a finite ordinal $n$ is equivalent to some finite ordinal smaller than $n$."}
+{"_id": "4188", "title": "Hilbert Space Direct Sum is Hilbert Space", "text": "Let $\\left({H_i}\\right)_{i \\in I}$ be a $I$-indexed family of Hilbert spaces over $\\Bbb F \\in \\left\\{{\\R, \\C}\\right\\}$. Let $H = \\displaystyle \\bigoplus_{i \\mathop \\in I} H_i$ be their Hilbert space direct sum. Then $H$ is a Hilbert space."}
+{"_id": "4189", "title": "Morera's Theorem", "text": "Let $D$ be a simply connected domain in $\\C$. Let $f: D \\to \\C$ be a continuous function. Let $f$ be such that: :$\\displaystyle \\int_\\gamma \\map f z \\rd z = 0$ for every simple closed contour $\\gamma$ in $D$ Then $f$ is analytic on $D$."}
+{"_id": "4190", "title": "Relation is Antisymmetric iff Intersection with Inverse is Coreflexive", "text": "Let $\\RR$ be a relation on $S$. Then: :$\\RR$ is antisymmetric {{iff}}: :$\\RR \\cap \\RR^{-1}$ is coreflexive where $\\RR^{-1}$ is the inverse of $\\RR$."}
+{"_id": "4191", "title": "Min Operation on Toset is Semigroup", "text": "Let $\\left({S, \\preceq}\\right)$ be a totally ordered set. Let $\\min \\left({x, y}\\right)$ denote the min operation on $x, y \\in S$. Then $\\left({S, \\min}\\right)$ is a semigroup."}
+{"_id": "4192", "title": "Max Semigroup is Commutative", "text": "Let $\\left({S, \\preceq}\\right)$ be a totally ordered set.  Then the semigroup $\\left({S, \\max}\\right)$ is commutative."}
+{"_id": "4193", "title": "Relation is Connected iff Union with Inverse and Diagonal is Trivial Relation", "text": "Let $\\mathcal R$ be a relation on $S$. Then $\\mathcal R$ is a connected relation {{iff}}: :$\\mathcal R \\cup \\mathcal R^{-1} \\cup \\Delta_S = S \\times S$ where $\\mathcal R^{-1}$ is the inverse of $\\mathcal R$ and $\\Delta_S$ is the diagonal relation."}
+{"_id": "4195", "title": "Absolutely Convergent Generalized Sum Converges", "text": "Let $V$ be a Banach space. Let $\\norm {\\, \\cdot \\,}$ denote the norm on $V$. Let $d$ denote the corresponding induced metric. Let $\\family {v_i}_{i \\mathop \\in I}$ be an indexed subset of $V$ such that the generalized sum $\\displaystyle \\sum_{i \\mathop \\in I} \\set {v_i}$ converges absolutely. Then the generalized sum $\\displaystyle \\sum \\set {v_i: i \\in I}$ converges."}
+{"_id": "4196", "title": "Generalized Sum is Monotone", "text": "Let $\\left({a_i}\\right)_{i \\in I}$ be an $I$-indexed family of positive real numbers. That is, let $a_i \\in \\R_{\\ge 0}$ for all $i \\in I$. Then, for every finite subset $F$ of $I$: :$\\displaystyle \\sum_{i \\mathop \\in F} a_i \\le \\sum \\left\\{{a_i : i \\in I}\\right\\}$ provided the generalized sum on the right converges."}
+{"_id": "4197", "title": "Convergent Generalized Sum of Positive Reals has Countably Many Non-Zero Terms", "text": "Let $\\left({a_i}\\right)_{i \\mathop \\in I}$ be an $I$-indexed family of positive real numbers. That is, let $a_i \\in \\R_{\\ge 0}$ for all $i \\in I$. Suppose that $\\displaystyle \\sum \\left\\{ {a_i: i \\in I}\\right\\}$ converges. Then the set $I_{>0} := \\left\\{ {i \\in I: a_i > 0}\\right\\}$ is countable."}
+{"_id": "4200", "title": "Finite Subsets form Directed Set", "text": "Let $I$ be a set. Denote with $\\mathcal F$ the set of finite subsets of $I$. Let $\\subseteq$ be the subset relation on $\\mathcal F$. Then $\\struct {\\mathcal F, \\subseteq}$ is a directed set."}
+{"_id": "4201", "title": "Strictly Precedes is Strict Ordering", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $\\prec$ be the relation on $S$ defined as: :$a \\prec b \\iff (a \\ne b) \\land (a \\preceq b)$ That is, $a \\prec b$ {{iff}} $a$ strictly precedes $b$. Then: : $a \\preceq b \\iff (a = b) \\lor (a \\prec b)$ and $\\prec$ is a strict ordering on $S$."}
+{"_id": "4202", "title": "Smallest Element is Unique", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. If $S$ has a smallest element, then it can have only one. That is, if $a$ and $b$ are both smallest elements of $S$, then $a = b$."}
+{"_id": "4203", "title": "Greatest Element is Unique", "text": "Let $\\struct {S, \\preceq}$ be a ordered set. If $S$ has a greatest element, then it can have only one. That is, if $a$ and $b$ are both greatest elements of $S$, then $a = b$."}
+{"_id": "4204", "title": "Smallest Element is Lower Bound", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T \\subseteq S$. Let $T$ have a smallest element $m \\in T$. Then $m$ is a lower bound of $T$. It follows by definition that $T$ is bounded below."}
+{"_id": "4205", "title": "Greatest Element is Upper Bound", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $T \\subseteq S$. Let $T$ have a greatest element $M \\in T$. Then $M$ is an upper bound of $T$. It follows by definition that $T$ is bounded above."}
+{"_id": "4206", "title": "Intersection of Subset with Lower Bounds", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $T \\subseteq S$. Let $T_*$ be the set of all lower bounds of $T$ in $S$. Then $T_* \\cap T \\ne \\O$ {{iff}}: :$T$ has a smallest element $m$ and :$T_* \\cap T$ is a singleton such that $T_* \\cap T = \\set m$"}
+{"_id": "4208", "title": "Nesbitt's Inequality", "text": "Let $a$, $b$ and $c$ be positive real numbers. Then: :$\\dfrac a {b + c} + \\dfrac b {a + c} + \\dfrac c {a + b} \\ge \\dfrac 3 2$"}
+{"_id": "4209", "title": "Greatest Element is Maximal", "text": "Let $\\struct {S, \\preceq}$ be an ordered set which has a greatest element. Let $M$ be the greatest element of $\\struct {S, \\preceq}$. Then $M$ is a maximal element."}
+{"_id": "4210", "title": "Smallest Element is Minimal", "text": "Let $\\struct {S, \\preceq}$ be an ordered set which has a smallest element. Let $m$ be the smallest element of $\\struct {S, \\preceq}$. Then $m$ is a minimal element."}
+{"_id": "4211", "title": "Minimal Element in Toset is Unique and Smallest", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. Let $m$ be a minimal element of $\\struct {S, \\preceq}$. Then: :$(1): \\quad m$ is the smallest element of $\\struct {S, \\preceq}$ :$(2): \\quad m$ is the only minimal element of $\\struct {S, \\preceq}$."}
+{"_id": "4212", "title": "Maximal Element in Toset is Unique and Greatest", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. Let $M$ be a maximal element of $\\struct {S, \\preceq}$. Then: :$(1): \\quad M$ is the greatest element of $\\struct {S, \\preceq}$. :$(2): \\quad M$ is the only maximal element of $\\struct {S, \\preceq}$."}
+{"_id": "4213", "title": "Whitney Embedding Theorem", "text": "Every smooth $m$-dimensional manifold admits a smooth embedding into Euclidean space $\\R^{2m+1}$."}
+{"_id": "4214", "title": "Max Semigroup is Idempotent", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. Then the semigroup $\\struct {S, \\max}$ is an idempotent semigroup."}
+{"_id": "4215", "title": "Tarski's Geometry is Complete", "text": "Tarski's geometry is both consistent and complete."}
+{"_id": "4216", "title": "Continuity of Linear Transformations", "text": "Let $H, K$ be Hilbert spaces, and let $A: H \\to K$ be a linear transformation. Then the following four statements are equivalent: :$(1): \\quad A$ is continuous :$(2): \\quad A$ is continuous at $\\mathbf 0_H$ :$(3): \\quad A$ is continuous at some point :$(4): \\quad \\exists c > 0: \\forall h \\in H: \\norm {\\map A h}_K \\le c \\norm h_H$"}
+{"_id": "4217", "title": "Equivalence of Definitions of Norm of Linear Transformation", "text": "Let $H, K$ be Hilbert spaces, and let $A: H \\to K$ be a bounded linear transformation. {{TFAE|def = Norm on Bounded Linear Transformation}} :$(1): \\qquad \\left\\Vert{A}\\right\\Vert = \\sup \\left\\{{\\left\\Vert{Ah}\\right\\Vert_K: \\left\\Vert{h}\\right\\Vert_H \\le 1}\\right\\}$ :$(2): \\qquad \\left\\Vert{A}\\right\\Vert = \\sup \\left\\{{\\dfrac {\\left\\Vert{Ah}\\right\\Vert_K} {\\left\\Vert{h}\\right\\Vert_H}: h \\in H, h \\ne \\mathbf{0}_H}\\right\\}$ :$(3): \\qquad \\left\\Vert{A}\\right\\Vert = \\sup \\left\\{{\\left\\Vert{Ah}\\right\\Vert_K: \\left\\Vert{h}\\right\\Vert_H \\le 1}\\right\\}$ :$(4): \\qquad \\left\\Vert{A}\\right\\Vert = \\inf \\left\\{{c > 0: \\forall h \\in H: \\left\\Vert{Ah}\\right\\Vert_K \\le c \\left\\Vert{h}\\right\\Vert_H}\\right\\}$"}
+{"_id": "4218", "title": "Max Semigroup on Toset is Semilattice", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set.  Then the max semigroup $\\struct {S, \\max}$ is a semilattice."}
+{"_id": "4219", "title": "Max yields Supremum of Parameters", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set.  Let $x, y \\in S$. Then: :$\\max \\set {x, y} = \\sup \\set {x, y}$ where: :$\\max$ denotes the max operation :$\\sup$ denotes the supremum."}
+{"_id": "4220", "title": "Mappings to Vector Space form Vector Space", "text": "Let $X$ be a non-empty set. Let $V$ be a vector space over a field (or division ring) $K$. Let $V^X$ denote the set of all mappings from $X$ to $V$. Let $+$ denote pointwise addition on $V^X$. Let $\\circ$ denote pointwise ($K$)-scalar multiplication on $V^X$. Then $\\left({V^X, +, \\circ}\\right)_K$ is a vector space over $K$."}
+{"_id": "4221", "title": "Mappings to Algebraic Structure form Similar Algebraic Structure", "text": "Let $X$ be a nonempty set. Let $G$ be a magma with respect to the binary operations $\\circ_1, \\ldots, \\circ_n$ on $G$. Let $G^X$ be the set of all mappings from $X$ to $G$. Denote also by $\\circ_1, \\ldots, \\circ_n$ the binary operations defined on $G^X$ by pointwise addition. {{wtd|Transcluded pages for group, monoid, abelian group, and so on, and so forth}}"}
+{"_id": "4222", "title": "Mappings to R-Algebraic Structure form Similar R-Algebraic Structure", "text": "Let $X$ be a nonempty set, and let $R$ be a ring. Let $\\left({G, \\circ}\\right)_R$ be an $R$-algebraic structure. Let $G^X$ be the set of all mappings from $X$ to $G$. Denote also by $\\circ$ the binary operation defined on $G^X$ by pointwise ($R$)-scalar multiplication. {{wtd|Transcluded pages for module, vector space, and so on, and so forth}}"}
+{"_id": "4223", "title": "Space of Bounded Linear Transformations is Banach Space", "text": "Let $H, K$ be Hilbert spaces. Let $\\map B {H, K}$ denote the space of bounded linear transformations from $H$ to $K$. Let $\\Bbb F \\in \\set {\\R, \\C}$ denote the ground field of $K$. Now $\\map B {H, K} \\subseteq K^H$, the set of mappings from $H$ to $K$. Therefore, $\\map B {H, K}$ can be endowed with pointwise addition ($+$) and ($\\Bbb F$)-scalar multiplication ($\\circ$). Let $\\norm{\\,\\cdot\\,}$ denote the norm on bounded linear transformations. Then $\\norm{\\,\\cdot\\,}$ is a norm on $\\map B {H, K}$. Furthermore, $B \\left({H, K}\\right)$ is a Banach space with respect to this norm."}
+{"_id": "4224", "title": "Equivalence of Formulations of Pasch's Axiom", "text": "The two forms of Pasch's Axiom in Tarski's Geometry are consistent. That is, the expressions: :$(1): \\quad \\forall a, b, c, p, q: \\exists x: \\mathsf B a p c \\land \\mathsf B b q c \\implies \\mathsf B p x b \\land \\mathsf B q x a$ and: :$(2): \\quad \\forall a, b, c, p, q: \\exists x: \\mathsf B a p c \\land \\mathsf B q c b \\implies \\mathsf B a x q \\land \\mathsf B b p x$ are logically equivalent."}
+{"_id": "4225", "title": "Square-Summable Indexed Sets Closed Under Addition", "text": "Let $\\family {a_i}_{i \\mathop \\in I}, \\family {b_i}_{i \\mathop \\in I}$ be $I$-indexed families of real numbers. Let: :$\\displaystyle \\sum \\set {a_i^2: i \\in I} < \\infty$ :$\\displaystyle \\sum \\set {b_i^2: i \\in I} < \\infty$ where $\\displaystyle \\sum$ denotes the generalized sums. Then: :$\\displaystyle \\sum \\set {\\paren {a_i + b_i}^2: i \\in I} < \\infty$"}
+{"_id": "4226", "title": "Generalized Sum Preserves Inequality", "text": "Let $\\left({a_i}\\right)_{i \\in I}, \\left({b_i}\\right)_{i \\in I}$ be $I$-indexed families of positive real numbers. That is, let $a_i, b_i \\in \\R_{\\ge 0}$ for all $i \\in I$. Suppose that for all $i \\in I$, $a_i \\le b_i$. Furthermore, suppose that $\\displaystyle \\sum \\left\\{{ b_i: i \\in I }\\right\\}$ converges. Then: : $\\displaystyle \\sum \\left\\{{ a_i: i \\in I }\\right\\} \\le \\sum \\left\\{{ b_i: i \\in I }\\right\\}$ In particular, $\\displaystyle \\sum \\left\\{{ a_i: i \\in I }\\right\\}$ converges."}
+{"_id": "4227", "title": "Generalized Sum is Linear", "text": "Let $\\left({z_i}\\right)_{i \\in I}, \\left({w_i}\\right)_{i \\in I}$ be $I$-indexed families of complex numbers. That is, let $z_i, w_i \\in \\C$ for all $i \\in I$. Suppose that $\\displaystyle \\sum \\left\\{{ z_i: i \\in I }\\right\\}, \\sum \\left\\{{ w_i: i \\in I }\\right\\}$ converge to $z, w \\in \\C$, respectively. Then: :$(1): \\quad \\displaystyle \\sum \\left\\{{ z_i + w_i: i \\in I }\\right\\}$ converges to $z+w$ :$(2): \\quad \\forall \\lambda \\in \\C: \\displaystyle \\sum \\left\\{{ \\lambda z_i: i \\in I }\\right\\}$ converges to $\\lambda z$"}
+{"_id": "4228", "title": "Min yields Infimum of Parameters", "text": "Let $\\left({S, \\preceq}\\right)$ be a totally ordered set.  Let $x, y \\in S$. Then: : $\\min \\left({x, y}\\right) = \\inf \\left({\\left\\{{x, y}\\right\\}}\\right)$ where: :$\\min$ denotes the min operation :$\\inf$ denotes the infimum."}
+{"_id": "4229", "title": "Convergence of Generalized Sum of Complex Numbers", "text": "Let $\\left({z_i}\\right)_{i \\mathop \\in I}$ be an $I$-indexed family of complex numbers. That is, let $z_i \\in \\C$ for all $i \\in I$. Denote by $\\operatorname{Re} z_i$, resp. $\\operatorname{Im} z_i$ the families of real, resp. imaginary parts of the family $z_i$. Then the following are equivalent: :$(1): \\quad \\displaystyle \\sum \\left\\{{z_i : i \\in I}\\right\\}$ converges to $z \\in \\C$ :$(2): \\quad \\displaystyle \\sum \\left\\{{\\operatorname{Re} z_i : i \\in I}\\right\\}, \\sum \\left\\{{\\operatorname{Im} z_i : i \\in I}\\right\\}$ converge to $\\operatorname{Re} z, \\operatorname{Im} z \\in \\R$, respectively"}
+{"_id": "4231", "title": "Lower and Upper Bounds for Sequences/Corollary", "text": "Let $\\sequence {x_n}, \\sequence {y_n}$ be sequences in $\\R$. Let $x_n \\to l, y_n \\to m$ as $n \\to \\infty$. Suppose that for all $n \\in \\N$, $x_n \\le y_n$. Then: :$l \\le m$ that is: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n \\le \\lim_{n \\mathop \\to \\infty} y_n$ This is often phrased as: ''limits preserve inequalities''."}
+{"_id": "4232", "title": "Bounded Linear Transformation Induces Bounded Sesquilinear Form", "text": "Let $H, K$ be Hilbert spaces over $\\Bbb F \\in \\left\\{{\\R, \\C}\\right\\}$. Let $A \\in B \\left({H, K}\\right), B \\in B \\left({K, H}\\right)$ be bounded linear transformations. Let $u, v: H \\times K \\to \\Bbb F$ be defined by: :$u \\left({h, k}\\right) := \\left\\langle{Ah, k}\\right\\rangle_K$ :$v \\left({h, k}\\right) := \\left\\langle{h, Bk}\\right\\rangle_H$ Then $u$ and $v$ are bounded sesquilinear forms."}
+{"_id": "4233", "title": "Classification of Bounded Sesquilinear Forms", "text": "Let $H, K$ be Hilbert spaces over $\\Bbb F \\in \\left\\{{\\R, \\C}\\right\\}$. Let $u: H \\times K \\to \\Bbb F$ be a bounded sesquilinear form with bound $M$. Then there exist unique bounded linear transformations $A \\in B \\left({H, K}\\right), B \\in B \\left({K, H}\\right)$ such that: :$\\forall h \\in H, k \\in K: u \\left({h, k}\\right) = \\left\\langle{Ah, k}\\right\\rangle_K = \\left\\langle{h, Bk}\\right\\rangle_H$ Furthermore, $\\left\\Vert{A}\\right\\Vert, \\left\\Vert{B}\\right\\Vert \\le M$."}
+{"_id": "4234", "title": "Isomorphism iff Inverse Equals Adjoint", "text": "Let $H, K$ be Hilbert spaces. Let $U \\in B \\struct {H, K}$ be a bounded linear transformation. Then the following are equivalent: :$(1): \\quad U$ is an isomorphism :$(2): \\quad U$ is invertible and $U^{-1} = U^*$, where $U^*$ denotes the adjoint of $U$."}
+{"_id": "4235", "title": "Limit of Sequence is Limit of Real Function", "text": "Let $\\sequence {a_n}$ be a real sequence. Let $f: x \\mapsto \\map f x$ be a real function. Suppose the limit: :$\\displaystyle \\lim_{x \\mathop \\to +\\infty} \\map f x$ exists. If for every $n$ in the domain of $\\sequence {a_n}$: :$\\map f n = a_n$ then: :$\\displaystyle \\lim_{n \\mathop \\to +\\infty} \\ a_n = \\displaystyle \\lim_{x \\mathop \\to +\\infty} \\map f x$"}
+{"_id": "4237", "title": "Equality of Cartesian Products", "text": "Let $A, B, C, D$ be nonempty sets. Then: : $A \\times B = C \\times D \\iff A = C \\land B = D$ where $\\times$ denotes cartesian product."}
+{"_id": "4238", "title": "Adjoining is Linear", "text": "Let $H, K$ be Hilbert spaces over $\\Bbb F \\in \\left\\{{\\R, \\C}\\right\\}$. Let $A, B \\in B \\left({H, K}\\right)$ be bounded linear transformations. Then the operation of adjoining $^*$ satisfies, for all $\\lambda \\in \\Bbb F$: :$(1): \\qquad \\left({\\lambda A}\\right)^* = \\overline \\lambda A^*$ :$(2): \\qquad \\left({A + B}\\right)^* = A^* + B^*$ That is, $^*: B \\left({H, K}\\right) \\to B \\left({K, H}\\right)$ is a linear transformation."}
+{"_id": "4239", "title": "Adjoint of Composition", "text": "Let $H, K, L$ be Hilbert spaces. Let $A \\in \\map B {K, L}, B \\in \\map B {H, K}$ be bounded linear transformations. Then $\\paren {A B}^* = B^* A^*$, where $^*$ denotes adjoining."}
+{"_id": "4240", "title": "Double Adjoint is Itself", "text": "Let $H, K$ be Hilbert spaces. Let $A \\in B \\left({H, K}\\right)$ be a bounded linear transformation. Then $A^{**} := \\left({A^*}\\right)^* = A$."}
+{"_id": "4242", "title": "Identity of Points", "text": "Two points share the same position {{iff}} they are the same points."}
+{"_id": "4243", "title": "Successor of Omega", "text": ":$\\omega + 1 = \\set {0, 1, 2, ...; \\omega}$ where $\\omega$ is the minimal infinite successor set and $\\omega + 1$ is the successor of $\\omega$. Note the use of the semicolon; this is the notation for multipart infinite sets."}
+{"_id": "4244", "title": "Equivalence of Versions of Axiom of Choice", "text": "The following formulations of the Axiom of Choice are equivalent:"}
+{"_id": "4245", "title": "Norm of Adjoint", "text": "Let $H, K$ be Hilbert spaces. Let $A \\in \\map B {H, K}$ be a bounded linear transformation. Then the norm of $A$ satisfies: :$\\norm A^2 = \\norm {A^*}^2 = \\norm {A^* A}$ where $A^*$ denotes the adjoint of $A$."}
+{"_id": "4246", "title": "Operator Self-Adjoint iff Inner Product Real", "text": "Let $H$ be a Hilbert space over $\\C$. Let $A \\in \\map B H$ be a bounded linear operator. Then $A$ is self-adjoint {{iff}}: :$\\forall h \\in H: \\innerprod {A h} h_H \\mathop \\in \\R$"}
+{"_id": "4247", "title": "Norm of Self-Adjoint Operator", "text": "Let $H$ be a Hilbert space. Let $A \\in B \\left({H}\\right)$ be a self-adjoint operator. Then the norm of $A$ satisfies: :$\\left\\Vert{A}\\right\\Vert = \\sup \\left\\{{ \\left\\vert{ \\left\\langle{Ah, h}\\right\\rangle_H }\\right\\vert: h \\in H, \\left\\Vert{h}\\right\\Vert_H = 1 }\\right\\}$"}
+{"_id": "4248", "title": "Operator Zero iff Inner Product Zero", "text": "Let $H$ be a Hilbert space over $\\C$. Let $A \\in B \\left({H}\\right)$ be a bounded linear operator. Suppose that: :$\\forall h \\in H: \\left\\langle{Ah, h}\\right\\rangle_H = 0$ Then $A$ is the zero operator."}
+{"_id": "4249", "title": "Linear Operator is Sum of Real and Imaginary Parts", "text": "Let $H$ be a Hilbert space over $\\C$. Let $A \\in B \\left({H}\\right)$ be a bounded linear operator. Let $B$ and $C$ be the real and imaginary parts of $A$, respectively. Then $A = B + iC$."}
+{"_id": "4250", "title": "Characterization of Normal Operators", "text": "Let $H$ be a Hilbert space. Let $A \\in B \\left({H}\\right)$ be a bounded linear operator. Then the following are equivalent: :$(1): \\qquad AA^* = A^*A$, i.e., $A$ is normal :$(2): \\qquad \\forall h \\in H: \\left\\Vert{Ah}\\right\\Vert_H = \\left\\Vert{A^*h}\\right\\Vert_H$ If $H$ is a Hilbert space over $\\C$, these are also equivalent to: :$(3): \\qquad \\operatorname{Re} \\left({A}\\right) \\operatorname{Im} \\left({A}\\right) = \\operatorname{Im} \\left({A}\\right) \\operatorname{Re} \\left({A}\\right)$, i.e., the real and imaginary parts of $A$ commute"}
+{"_id": "4251", "title": "Isometry iff Adjoint is Left-Inverse", "text": "Let $H, K$ be Hilbert spaces. Let $A \\in B \\left({H, K}\\right)$ be a bounded linear transformation. Then $A$ is an isometry iff: :$A^*A = I_H$ where $A^*$ denotes the adjoint of $A$, and $I_H$ the identity operator on $H$."}
+{"_id": "4252", "title": "Characterization of Unitary Operators", "text": "Let $H$ be a Hilbert space. Let $A \\in \\map B H$ be a bounded linear operator. Then the following are equivalent: :$(1): \\quad A$ is a unitary operator :$(2): \\quad A^* A = A A^* = I$, where $A^*$ denotes the adjoint of $A$, and $I$ denotes the identity operator :$(3): \\quad A$ is a normal isometry"}
+{"_id": "4253", "title": "Kernel of Linear Transformation is Orthocomplement of Range of Adjoint", "text": "Let $H, K$ be Hilbert spaces. Let $A \\in B \\struct {H, K}$ be a bounded linear transformation. Then $\\ker A = \\paren {\\Img {A^*} }^\\perp$, where: :$A^*$ denotes the adjoint of $A$ :$\\ker A$ is the kernel of $A$ :$\\Img {A^*}$ is the image of $A^*$ :$\\perp$ signifies orthocomplementation"}
+{"_id": "4254", "title": "Relation Contains Mapping is Equivalent to AoC", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation on $S \\times T$. Then: :there exists a mapping $f \\subseteq \\RR$ whose domain is the same as the preimage of $\\RR$ {{iff}} :the axiom of choice holds."}
+{"_id": "4255", "title": "Existence of Set with Singleton Intersections with Disjoint Collection", "text": "Let $\\mathcal C$ be a set of sets all of which are pairwise disjoint. Then: : there exists a set $A$ such that $\\forall S \\in \\mathcal C: A \\cap S$ is a singleton {{iff}} : the axiom of choice holds."}
+{"_id": "4259", "title": "Complementary Idempotent is Idempotent", "text": "Let $H$ be a Hilbert space. Let $A$ be an idempotent operator. Then the complementary idempotent $I - A$ is also idempotent."}
+{"_id": "4260", "title": "Range of Idempotent is Kernel of Complementary Idempotent", "text": "Let $H$ be a Hilbert space. Let $A$ be an idempotent operator. Then $\\Rng A = \\map \\ker {I - A}$."}
+{"_id": "4261", "title": "Range and Kernel of Idempotent are Algebraically Complementary", "text": "Let $H$ be a Hilbert space. Let $A$ be an idempotent operator. Then $\\ker A$ and $\\Rng A$ are algebraically complementary, that is: :$\\ker A \\cap \\Rng A = \\left({0}\\right)$, the zero subspace :$\\ker A + \\Rng A = H$, where $+$ signifies setwise addition. {{explain|Determine whether range means image or codomain, and replace appropriately}}"}
+{"_id": "4262", "title": "Natural Logarithm Function is Continuous", "text": "The natural logarithm function is continuous."}
+{"_id": "4263", "title": "Characterization of Projections", "text": "Let $H$ be a Hilbert space. Let $A \\in B \\left({H}\\right)$ be an idempotent operator. Then the following are equivalent: :$(1): \\qquad A$ is a projection :$(2): \\qquad A$ is the orthogonal projection onto $\\operatorname{ran} A$ :$(3): \\qquad \\left\\Vert{A}\\right\\Vert = 1$, where $\\left\\Vert{\\cdot}\\right\\Vert$ is the norm on bounded linear operators. :$(4): \\qquad A$ is self-adjoint :$(5): \\qquad A$ is normal :$(6): \\qquad \\forall h \\in H: \\left\\langle{Ah, h}\\right\\rangle_H \\ge 0$"}
+{"_id": "4264", "title": "Set of Linear Subspaces is Closed under Intersection", "text": "Let $\\struct {V, +, \\circ}_K$ be a $K$-vector space. Let $\\family {M_i}_{i \\mathop \\in I}$ be an $I$-indexed family of subspaces of $V$. Then $M := \\ds \\bigcap_{i \\mathop \\in I} M_i$ is also a subspace of $V$."}
+{"_id": "4265", "title": "Closed Linear Subspaces Closed under Intersection", "text": "Let $V$ be a topological vector space. Let $\\family {M_i}_{i \\mathop \\in I}$ be an $I$-indexed family of closed linear subspaces of $V$. Then $M := \\ds \\bigcap_{i \\mathop \\in I} M_i$ is also a closed linear subspace of $V$."}
+{"_id": "4266", "title": "Orthogonal Difference is Closed Linear Subspace", "text": "Let $H$ be a Hilbert space. Let $M, N$ be closed linear subspaces of $H$. Then the orthogonal difference $M \\ominus N$ is also a closed linear subspace of $H$."}
+{"_id": "4267", "title": "Closed Linear Subspaces Closed under Setwise Addition", "text": "Let $H$ be a Hilbert space. Let $M, N$ be closed linear subspaces of $H$. Then $M + N$ is also a closed linear subspace of $H$, where $+$ denotes setwise addition."}
+{"_id": "4268", "title": "Linear Subspaces Closed under Setwise Addition", "text": "Let $V$ be a $K$-vector space. Let $M, N$ be linear subspaces of $V$. Then $L := M + N$ is also a linear subspace of $V$, where $+$ denotes setwise addition."}
+{"_id": "4269", "title": "Complementary Projection is Projection", "text": "Let $H$ be a Hilbert space. Let $A$ be a projection. Then the complementary projection $I - A$ is also a projection."}
+{"_id": "4270", "title": "Direct Sum of Subspace and Orthocomplement", "text": "Let $H$ be a Hilbert space. Let $M$ be a closed linear subspace of $H$. Denote by $M^\\perp$ its orthocomplement. Then the direct sum $M \\oplus M^\\perp$ is isomorphic to $H$."}
+{"_id": "4271", "title": "Characterization of Invariant Subspaces", "text": "Let $H$ be a Hilbert space. Let $A \\in B \\left({H}\\right)$ be a bounded linear operator. Let $\\begin{pmatrix}  W & X \\\\  Y & Z \\end{pmatrix}$ be the matrix notation for $A$ with respect to $M$. Let $M$ be a closed linear subspace of $H$; denote by $P$ the orthogonal projection on $M$. Then the following three statements are equivalent: :$(1): \\qquad M$ is an invariant subspace for $A$ :$(2): \\qquad PAP = AP$ :$(3): \\qquad Y = 0$"}
+{"_id": "4272", "title": "Characterization of Reducing Subspaces", "text": "Let $\\HH$ be a Hilbert space. Let $A \\in \\map B \\HH$ be a bounded linear operator. Let $M$ be a closed linear subspace of $\\HH$. Let $P$ denote the orthogonal projection on $M$. Let $\\begin{pmatrix}  W & X \\\\  Y & Z \\end{pmatrix}$ be the matrix notation for $A$ with respect to $M$. Then the following four statements are equivalent: :$(1): \\quad M$ is a reducing subspace for $A$ :$(2): \\quad P A = A P$ :$(3): \\quad X = Y = 0$ :$(4): \\quad M$ is an invariant subspace for both $A$ and its adjoint $A^*$"}
+{"_id": "4273", "title": "Subset of Toset is Toset", "text": "Let $\\left({S, \\preceq}\\right)$ be a totally ordered set. Let $T \\subseteq S$. Then $\\left({T, \\preceq \\restriction_T}\\right)$ is also a totally ordered set. In the above, $\\preceq \\restriction_T$ denotes the restriction of $\\preceq$ to $T$."}
+{"_id": "4274", "title": "Bretschneider's Formula", "text": "Let $ABCD$ be a general quadrilateral. Then the area $\\mathcal A$ of $ABCD$ is given by: :$\\mathcal A = \\sqrt{\\left({s - a}\\right) \\left({s - b}\\right) \\left({s - c}\\right) \\left({s - d}\\right) - a b c d \\cos^2 \\left({\\dfrac {\\alpha + \\gamma} 2}\\right)}$ where: : $a, b, c, d$ are the lengths of the sides of the quadrilateral : $s = \\dfrac {a + b + c + d} 2$ is the semiperimeter : $\\alpha$ and $\\gamma$ are opposite angles."}
+{"_id": "4275", "title": "Transfinite Induction/Principle 1", "text": "Let $\\On$ denote the class of all ordinals. Let $A$ denote a class. Suppose that: :For all elements $x$ of $\\On$, if $x$ is a subset of $A$, then $x$ is an element of $A$. Then $\\On \\subseteq A$."}
+{"_id": "4276", "title": "Limit of Functions that Agree", "text": "Let $f$ and $g$ be real functions. Let $f$ and $g$ agree for all $x$ in a deleted neighborhood of $c$. Let the limit: :$\\displaystyle \\lim_{x \\to c} \\ f \\left({x}\\right)$ exist. Then the limit: :$\\displaystyle \\lim_{x \\to c} \\ g \\left({x}\\right)$ also exists, and: :$\\displaystyle \\lim_{x \\to c} \\ f \\left({x}\\right) =  \\lim_{x \\to c} \\ g \\left({x}\\right)$"}
+{"_id": "4277", "title": "Ordering of Reciprocals", "text": "Let $x, y \\in \\R$ be real numbers such that $x, y \\in \\openint 0 \\to$ or $x, y \\in \\openint \\gets 0$ Then: :$x \\le y \\iff \\dfrac 1 y \\le \\dfrac 1 x$"}
+{"_id": "4278", "title": "Comparison Test for Divergence", "text": "Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n$ be a divergent series of positive real numbers. Let $\\sequence {a_n}$ be a sequence in $\\R$. Let: :$\\forall n \\in \\N_{>0}: b_n \\le a_n$ Then the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ diverges."}
+{"_id": "4280", "title": "Limit at Infinity of Real Identity Function", "text": "Let $I_\\R: \\R \\to \\R$ be the identity function on $\\R$. Then: :$(1): \\quad \\displaystyle \\lim_{x \\mathop \\to +\\infty} \\map {I_\\R} x = +\\infty$ :$(2): \\quad \\displaystyle \\lim_{x \\mathop \\to -\\infty} \\map {I_\\R} x = -\\infty$"}
+{"_id": "4281", "title": "Push Theorem", "text": "Let $f$ be a real function which is continuous on the open interval $\\openint a \\to$, $a \\in \\R$, such that: :$\\displaystyle \\lim_{x \\mathop \\to +\\infty} \\map f x = +\\infty$ Let $g$ be a real function defined on some open interval $\\openint b \\to$ such that, for sufficiently large $x$: :$\\map g x > \\map f x$ Then: :$\\displaystyle \\lim_{x \\mathop \\to +\\infty} \\map g x = +\\infty$"}
+{"_id": "4284", "title": "Orthocomplement Reverses Subset", "text": "Let $H$ be a Hilbert space. Let $A, B$ be subsets of $H$, and let $A \\subseteq B$. Then $B^\\perp \\subseteq A^\\perp$, where $\\perp$ signifies orthocomplementation."}
+{"_id": "4285", "title": "Limit at Infinity of x^n", "text": "Let $x \\mapsto x^n$, $n \\in \\R$ be a real function which is continuous on the open interval $\\openint 1 {+\\infty}$. Let $n > 0$. Then $x^n \\to +\\infty$ as $x \\to +\\infty$."}
+{"_id": "4286", "title": "Monotonicity of Real Sequences", "text": "Let $\\sequence {a_n}: \\mathbb D \\to \\R$ be a real sequence, where $\\mathbb D$ is a subset of $\\N$. Let $\\Bbb X$ be a real interval such that $\\Bbb D \\subseteq \\Bbb X$. Let $f: \\Bbb X \\to \\R, x \\mapsto \\map f x$ be a differentiable real function. Suppose that for every $n \\in \\mathbb D$: :$\\map f n = a_n$ Then: :If $\\forall x \\in \\Bbb X: D_x \\map f x \\ge 0$, $\\sequence {a_n}$ is increasing :If $\\forall x \\in \\Bbb X: D_x \\map f x >  0$, $\\sequence {a_n}$ is strictly increasing :If $\\forall x \\in \\Bbb X: D_x \\map f x \\le 0$, $\\sequence {a_n}$ is decreasing :If $\\forall x \\in \\Bbb X: D_x \\map f x < 0$, $\\sequence {a_n}$ is strictly decreasing where $D_x$ denotes differentiation {{WRT|Differentiation}} $x$."}
+{"_id": "4287", "title": "Hausdorff Maximal Principle is equivalent to Axiom of Choice", "text": "Every ordered set has a maximal chain {{iff}} the axiom of choice holds."}
+{"_id": "4290", "title": "Young's Inequality for Products", "text": "Let $p, q \\in \\R_{> 0}$ be strictly positive real numbers such that: : $\\dfrac 1 p + \\dfrac 1 q = 1$ Then, for any $a, b \\in \\R_{\\ge 0}$: : $a b \\le \\dfrac {a^p} p + \\dfrac{b^q} q$ Equality occurs {{iff}}: :$b = a^{p-1}$."}
+{"_id": "4292", "title": "Restriction of Monotone Function is Monotone", "text": "The restriction of a monotone mapping is monotone."}
+{"_id": "4293", "title": "Taylor's Theorem/One Variable/Statement of Theorem", "text": "Let $f$ be a real function which is: :of differentiability class $C^n$ on the closed interval $\\closedint a x$ and: :at least $n + 1$ times differentiable on the open interval $\\openint a x$. Then: {{begin-eqn}} {{eqn | l = \\map f x       | r = \\frac 1 {0!} \\map f a       | c = }} {{eqn | o =       | ro= +       | r = \\frac 1 {1!} \\paren {x - a} \\map {f'} a       | c =  }} {{eqn | o =       | ro= +       | r = \\frac 1 {2!} \\paren {x - a}^2 \\map {f''} a       | c =  }} {{eqn | o =       | ro= +       | r = \\cdots       | c =  }} {{eqn | o =       | ro= +       | r = \\frac 1 {n!} \\paren {x - a}^n \\map {f^{\\paren n} } a       | c =  }} {{eqn | o =       | ro= +       | r = R_n       | c =  }} {{end-eqn}} where $R_n$ (sometimes denoted $E_n$) is known as the '''error term''' or '''remainder''', and can be presented in one of $2$ forms: ;Lagrange Form :$R_n = \\dfrac {\\map {f^{\\paren {n + 1} } } \\xi} {\\paren {n + 1}!} \\paren {x - a}^{n + 1}$ for some $\\xi \\in \\openint a x$. ;Cauchy Form :$R_n = \\dfrac {\\map {f^{\\paren {n + 1} } } \\xi \\paren {x - \\xi}^n} {n!} \\paren {x - a}$ for some $\\xi \\in \\openint a x$."}
+{"_id": "4294", "title": "Elements of Minimal Infinite Successor Set are Well-Ordered", "text": "Let $\\omega$ be the minimal infinite successor set. Let $a \\in \\omega$. Then $a$ is well-ordered by $\\subseteq$."}
+{"_id": "4295", "title": "Finite Lexicographic Order on Well-Ordered Sets is Well-Ordering", "text": "Let $S$ be a set which is well-ordered by $\\preceq$. Let $\\preccurlyeq$ be the lexicographic order on the set $T_n$ of all ordered $n$-tuples of $S$: :$\\left({x_1, x_2, \\ldots, x_n}\\right) \\prec \\left({y_1, y_2, \\ldots, y_n}\\right)$ {{iff}}: ::$\\exists k: 1 \\le k \\le n$ such that $\\forall 1 \\le j < k: x_j = y_j$ but $x_k \\prec y_k$ in $S$. {{explain|Disentangle this.}} Then for a given $n \\in \\N_{>0}$, $\\preccurlyeq$ is a well-ordering on $T_n$."}
+{"_id": "4296", "title": "Infinite Lexicographic Order on Well-Ordered Sets is not Well-Ordering", "text": "Let $\\struct {S, \\preceq}$ be a well-ordered set. Let $\\preccurlyeq$ be the lexicographic order on the set $T$ of all ordered tuples of $S$: :$\\tuple {x_1, x_2, \\ldots, x_m} \\prec \\tuple {y_1, y_2, \\ldots, y_n}$ {{iff}}: ::$\\exists k: 1 \\le k \\le \\map \\min {m, n}$ such that $\\forall 1 \\le j < k: x_j = y_j$ but $x_k \\prec y_k$ in $S$ :or: ::$m < n$ and $\\forall 1 \\le j < m: x_j = y_j$. Then $\\preccurlyeq$ is '''not''' a well-ordering on $T$."}
+{"_id": "4297", "title": "Minimal Infinite Successor Set is Well-Ordered", "text": "Let $\\omega$ be the minimal infinite successor set. Then $\\omega$ is well-ordered by $\\subseteq$."}
+{"_id": "4298", "title": "Transfinite Induction", "text": "=== Principle 1 === {{:Transfinite Induction/Principle 1}} === Schema 1 === {{:Transfinite Induction/Schema 1}} === Principle 2 === {{:Transfinite Induction/Principle 2}} === Schema 2 === {{:Transfinite Induction/Schema 2}}"}
+{"_id": "4299", "title": "Transfinite Recursion/Theorem 1", "text": "Let $G$ be a mapping. {{explain|What are the domain and range of $G$?}} Let $K$ be a class of mappings $f$ that satisfy: :the domain of $f$ is some ordinal $y$ :$\\forall x \\in y: f \\left({x}\\right) = G \\left({f {\\restriction_x} }\\right)$ where $f {\\restriction_x}$ denotes the restriction of $f$ to $x$. Let $F = \\bigcup K$, the union of $K$. Then: :$(1): \\quad F$ is a mapping with domain $\\operatorname{On}$ :$(2): \\quad \\forall x \\in \\operatorname{On}: F \\left({x}\\right) = G \\left({F {\\restriction_x} }\\right)$ :$(3): \\quad F$ is unique. That is, if another mapping $A$ has the above two properties, then $A = F$."}
+{"_id": "4300", "title": "Squeeze Theorem for Absolutely Convergent Series", "text": "Let $\\displaystyle \\sum \\size {a_n}$ be an absolutely convergent series in $\\R$. Suppose that: :$\\displaystyle -\\sum \\size {a_n} = \\sum \\size {a_n}$ Then $\\displaystyle \\sum a_n$ equals the above two series."}
+{"_id": "4301", "title": "N Choose k is not greater than n^k", "text": ":$\\forall n \\in \\Z, k \\in \\Z: 1 \\le k \\le n: \\dbinom n k < n^k$  where $\\dbinom n k$ is a binomial coefficient. Equality holds when $k = 0$ and $k = 1$."}
+{"_id": "4302", "title": "Absolutely Convergent Series is Convergent/Real Numbers", "text": "Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be an absolutely convergent series in $\\R$. Then $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ is convergent."}
+{"_id": "4303", "title": "Definition:Natural Numbers/Zermelo Construction", "text": "The natural numbers $\\N = \\set {0, 1, 2, 3, \\ldots}$ can be defined as a series of subsets: :$0 := \\O = \\set {}$ :$1 := \\set 0 = \\set \\O$ :$2 := \\set 1 = \\set {\\set \\O}$ :$3 := \\set 2 = \\set {\\set {\\set \\O} }$ ::::$\\vdots$ Thus the natural number $n$ consists of $\\O$ enclosed in $n$ pairs of braces."}
+{"_id": "4304", "title": "Set is Element of its Power Set", "text": "A set is an element of its power set: :$S \\in \\powerset S$"}
+{"_id": "4305", "title": "Power Set of Empty Set", "text": "The power set of the empty set $\\O$ is the set $\\set \\O$."}
+{"_id": "4306", "title": "Symmetric Difference is Subset of Union", "text": "The symmetric difference of two sets is a subset of their union: :$S * T \\subseteq S \\cup T$"}
+{"_id": "4309", "title": "Sum of Projections/General Case", "text": "Let $H$ be a Hilbert space. Let $\\left({M_i}\\right)_{i \\in I}$ be an $I$-indexed set of closed linear subspaces of $H$. Let $M_i$ and $M_j$ be orthogonal whenever $i \\ne j$. Denote, for each $i \\in I$, by $P_i$ the orthogonal projection onto $M_i$. Denote by $P$ the orthogonal projection onto the closed linear span $\\vee \\left\\{{M_i: i \\in I}\\right\\}$ of the $M_i$. Then for all $h \\in H$, $\\displaystyle \\sum \\left\\{{P_i h: i \\in I}\\right\\} = Ph$, where $\\displaystyle \\sum$ denotes a generalized sum."}
+{"_id": "4310", "title": "Product of Projections", "text": "Let $H$ be a Hilbert space. Let $P, Q$ be projections. Then the following are equivalent: :$(1): \\quad PQ$ is a projection :$(2): \\quad PQ = QP$ :$(3): \\quad P + Q - PQ$ is a projection {{MissingLinks|Provide proper linking to the def of addition and multiplication of operators}}"}
+{"_id": "4311", "title": "Difference of Projections", "text": "Let $H$ be a Hilbert space. Let $P, Q$ be projections. Then the following are equivalent: :$(1): \\quad P - Q$ is a projection :$(2): \\quad P Q = Q$ :$(3): \\quad Q P = Q$ {{MissingLinks|Provide proper linking to the def of addition of operators}}"}
+{"_id": "4313", "title": "Union of Countable Sets of Sets", "text": "Let $\\mathcal A$ and $\\mathcal B$ be countable sets of sets. Then: :$\\left\\{{A \\cup B: A \\in \\mathcal A, B \\in \\mathcal B}\\right\\}$ is also countable."}
+{"_id": "4314", "title": "Set of Finite Subsets of Countable Set is Countable", "text": "Let $A$ be a countable set. Then the set of finite subsets of $A$ is countable."}
+{"_id": "4315", "title": "Cantor's Theorem (Strong Version)", "text": "Let $S$ be a set. Let $\\map {\\PP^n} S$ be defined recursively by: :$\\map {\\PP^n} S = \\begin{cases} S & : n = 0 \\\\ \\powerset {\\map {\\PP^{n - 1} } S} & : n > 0 \\end{cases}$ where $\\powerset S$ denotes the power set of $S$. Then $S$ is not equivalent to $\\map {\\PP^n} S$ for any $n > 0$."}
+{"_id": "4316", "title": "Ordering Induced by Injection is Ordering", "text": "Let $\\left({T, \\le}\\right)$ be an ordered set, and let $S$ be a set. Let $f: S \\to T$ be an injection. Then $\\le_f$, the ordering induced by $f$, is an ordering."}
+{"_id": "4318", "title": "Injection Induces Total Ordering", "text": "Let $\\left({T, \\le}\\right)$ be a totally ordered set, and let $S$ be a set. Let $f: S \\to T$ be an injection. Then $\\le_f$, the ordering induced by $f$, is a total ordering."}
+{"_id": "4319", "title": "Injection Induces Well-Ordering", "text": "Let $\\left({T, \\le}\\right)$ be a well-ordered set. Let $S$ be a set. Let $f: S \\to T$ be an injection. Then $\\le_f$, the ordering induced by $f$, is a well-ordering."}
+{"_id": "4323", "title": "Hartogs' Lemma (Set Theory)", "text": "Let $S$ be a set. Then there exists an ordinal $\\alpha$ such that there is no injection from $\\alpha$ to $S$."}
+{"_id": "4324", "title": "Bourbaki-Witt Fixed Point Theorem", "text": "Let $\\struct {X, \\le}$ be a non-empty chain complete poset (that is, an ordered set in which every chain has a supremum). Let $f: X \\to X$ be an inflationary mapping, that is, so that $\\map f x \\ge x$. Then for every $x \\in X$ there exists $y \\in X$ where $y \\ge x$ such that $\\map f y = y$."}
+{"_id": "4329", "title": "Convergent Sequence with Finite Number of Terms Deleted is Convergent", "text": "Let $\\left({X, d}\\right)$ be a metric space. Let $\\left\\langle{x_k}\\right\\rangle$ be a sequence in $X$. Let $\\left\\langle{x_k}\\right\\rangle$ be convergent. Let a finite number of terms be deleted from $\\left \\langle {x_k} \\right \\rangle$. Then the resulting subsequence is convergent."}
+{"_id": "4330", "title": "Cauchy Condensation Test", "text": "Let $\\sequence {a_n}: n \\mapsto \\map a n$ be a decreasing sequence of strictly positive terms in $\\R$ which converges with a limit of zero. That is, for every $n$ in the domain of $\\sequence {a_n}$: $a_n > 0$, $a_n \\ge a_{n + 1}$, and $a_n \\to 0$ as $n \\to +\\infty$. Then the series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$  converges {{iff}} the condensed series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty 2^n \\map a {2^n}$ converges."}
+{"_id": "4331", "title": "Between Two Sets Exists Injection or Surjection", "text": "Let $S$ and $T$ be sets. Then either or both of the following cases hold: :$(1):$ There exists a mapping $f: S \\to T$ such that $f$ is an injection :$(2):$ There exists a mapping $f: S \\to T$ such that $f$ is a surjection."}
+{"_id": "4332", "title": "Injection from Set to Power Set", "text": "For every set $S$, there exists an injection from $S$ to its power set $\\mathcal P \\left({S}\\right)$."}
+{"_id": "4333", "title": "Minimum Degree is at Least Connectivity", "text": "Let $G = \\left({V, E}\\right)$ be a simple graph. Then: :$\\delta \\left({G}\\right) \\geq \\kappa \\left({G}\\right)$ That is, the minimum degree of $G$ is at least its connectivity."}
+{"_id": "4334", "title": "Homeomorphism Relation is Equivalence", "text": "Let $T_1$ and $T_2$ be topological spaces. Let $T_1 \\sim T_2$ denote that $T_1$ and $T_2$ are homeomorphic. The relation $\\sim$ is an equivalence relation."}
+{"_id": "4335", "title": "Identity Mapping is Homeomorphism", "text": "Let $T$ be a topological space. The identity mapping $I_T: T \\to T$ defined as: :$\\forall x \\in T: I_T \\left({x}\\right) = x$ is a homeomorphism."}
+{"_id": "4336", "title": "Composite of Homeomorphisms is Homeomorphism", "text": "Let $T_1, T_2, T_3$ be topological spaces.  Let $f: T_1 \\to T_2$ and $g: T_2 \\to T_3$ be homeomorphisms. Then $g \\circ f: T_1 \\to T_3$ is also a homeomorphism."}
+{"_id": "4337", "title": "Identity Mapping is Continuous", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. The identity mapping $I_S: S \\to S$ defined as: :$\\forall x \\in S: \\map {I_S} x = x$ is a continuous mapping."}
+{"_id": "4341", "title": "Kuratowski's Lemma", "text": "Let $\\struct {S, \\preceq}, S \\ne \\O$ be a non-empty ordered set.  Then every chain in $S$ is the subset of some maximal chain."}
+{"_id": "4342", "title": "Included Set Topology is Topology", "text": "Let $T = \\left({S, \\tau_H}\\right)$ be an included set space. Then $\\tau_H$ is a topology on $S$, and $T$ is a topological space."}
+{"_id": "4343", "title": "Space of Compact Linear Transformations is Banach Space", "text": "Let $H, K$ be Hilbert spaces, and let $B_0 \\left({H, K}\\right)$ be the space of compact linear transformations from $H$ to $K$. Let $\\Bbb F \\in \\left\\{{\\R, \\C}\\right\\}$ be the ground field of $K$. Now $B_0 \\left({H, K}\\right) \\subseteq K^H$, the set of mappings from $H$ to $K$. Therefore, $B_0 \\left({H, K}\\right)$ can be endowed with pointwise addition ($+$) and ($\\Bbb F$)-scalar multiplication ($\\circ$). Let $\\left\\Vert{\\cdot}\\right\\Vert$ denote the norm on bounded linear operators. Then $\\left\\Vert{\\cdot}\\right\\Vert$ is a norm on $B_0 \\left({H, K}\\right)$. Furthermore, $B_0 \\left({H, K}\\right)$ is a Banach space with respect to this norm."}
+{"_id": "4344", "title": "Compact Linear Transformation is Bounded", "text": "Let $H, K$ be Hilbert spaces. Let $T \\in B_0 \\left({H, K}\\right)$ be a compact linear transformation. Then $T$ is also a bounded linear transformation. That is, $B_0 \\left({H, K}\\right) \\subseteq B \\left({H, K}\\right)$."}
+{"_id": "4345", "title": "Compact Linear Transformations Composed with Bounded Linear Operator", "text": "Let $H, K$ be Hilbert spaces. Let $T \\in B_0 \\left({H, K}\\right)$ be a compact linear transformation. Let $A \\in B \\left({H}\\right), B \\in B \\left({K}\\right)$ be bounded linear operators. Then the compositions $TA$ and $BT$ are also compact linear transformations."}
+{"_id": "4347", "title": "Linear Transformation Compact iff Adjoint Compact", "text": "Let $H, K$ be Hilbert spaces. Let $T: H \\to K$ be a linear transformation. Then $T$ is compact iff its adjoint $T^*$ is."}
+{"_id": "4348", "title": "Finite Rank Operators Dense in Compact Linear Transformations", "text": "Let $H, K$ be Hilbert spaces. Then: :$\\map {B_{00} } {H, K}$ is everywhere dense in $\\map {B_0} {H, K}$ where: :$\\map {B_{00} } {H, K}$ is the space of continuous finite rank operators from $H$ to $K$ :$\\map {B_0} {H, K}$ is the space of compact linear transformations from $H$ to $K$. That is, for every $T \\in \\map {B_0} {H, K}$, there is a sequence $\\sequence {T_n}_{n \\mathop \\in \\N}$ in $\\map {B_{00} } {H, K}$ such that: :$\\ds \\lim_{n \\mathop \\to \\infty} \\norm {T_n - T} = 0$ where $\\norm {\\, \\cdot \\,}$ denotes the norm on bounded linear transformations."}
+{"_id": "4349", "title": "Inverse of Matrix Product", "text": "Let $\\mathbf {A, B}$ be square matrices of order $n$ Let $\\mathbf I$ be the $n \\times n$ unit matrix. Let $\\mathbf A$ and $\\mathbf B$ be invertible. Then the matrix product $\\mathbf {AB}$ is also invertible, and: :$\\paren {\\mathbf A \\mathbf B}^{-1} = \\mathbf B^{-1} \\mathbf A^{-1}$"}
+{"_id": "4350", "title": "Closure of Range of Compact Linear Transformation is Separable", "text": "Let $H, K$ be Hilbert spaces. Let $T \\in \\map {B_0} {H, K}$ be a compact linear transformation. Then $\\map \\cl {\\Rng T}$ is separable."}
+{"_id": "4351", "title": "Elementary Row Operations as Matrix Multiplications", "text": "Let $e$ be an elementary row operation. Let $\\mathbf E$ be the elementary row matrix of order $m$ defined as: :$\\mathbf E = e \\paren {\\mathbf I}$ where $\\mathbf I$ is the unit matrix. Then for every $m \\times n$ matrix $\\mathbf A$: :$e \\paren {\\mathbf A} = \\mathbf {E A}$ where $\\mathbf {E A}$ denotes the conventional matrix product."}
+{"_id": "4353", "title": "Included Set Topology on Finite Intersection", "text": "Let $T = \\struct {S, \\tau}$ be a topological space on a set $S$. Let $A_1, A_2, \\ldots, A_n$ be a finite set of subsets of $S$: :$\\forall i \\in \\closedint 1 n: A_i \\subseteq S$ Let $\\forall i \\in \\closedint 1 n: \\map T {A_i} = \\struct {S, \\tau_{A_i} }$ be the included set spaces on $S$ by $A_i$. Let: :$\\forall i \\in \\closedint 1 n: \\map T {A_i} \\le T$ where $\\map T {A_i} \\le T$ denotes that $\\map T {A_i}$ is coarser than $T$. Then: :$\\displaystyle \\map T {\\bigcap A_i} \\le T$ where $\\displaystyle \\map T {\\bigcap A_i}$ is the included set space on $S$ by $\\displaystyle \\bigcap_{i \\mathop = 1}^n A_i$."}
+{"_id": "4355", "title": "Included Set Topology on Union", "text": "Let $T = \\struct {S, \\tau}$ be a topological space on a set $S$. Let $\\family {A_i}_{i \\mathop \\in I}$ be a family of subsets of $S$ indexed by the indexing set $I$: :$\\forall i \\in I: A_i \\subseteq S$ Let $\\forall i \\in I: \\map T {A_i} = \\struct {S, \\tau_{A_i} }$ be the included set spaces on $S$ by $A_i$. Let: :$\\forall i \\in I: \\map T {A_i} \\ge T$ where $\\map T {A_i} \\ge T$ denotes that $\\map T {A_i}$ is finer than $T$. Then: :$\\map T {\\bigcup A_i} \\ge T$ where $\\map T {\\bigcup A_i}$ is the included set space on $S$ by $\\displaystyle \\bigcup_{i \\mathop \\in I} A_i$."}
+{"_id": "4356", "title": "Kuratowski's Free Set Theorem", "text": "Let $n \\in \\Z_{>0}$ be a positive integer. Let $X$ be an infinite set. Let $\\left[{X}\\right]^n$ denote the set of $n$-element subsets of $X$ whose cardinality is $n$. Let $\\left[{X}\\right]^{< \\aleph_0}$ denote the set of finitesubsets of $X$. Then: : the cardinality of $X$ is greater than or equal to $\\aleph_n$ {{iff}} : for every mapping $f: \\left[{X}\\right]^n \\to \\left[{X}\\right]^{<\\aleph_0}$ there exists an $(n + 1)$-element free subset of $X$ with respect to $f$. That is, a subset $Y$ of $X$ with $n+1$ elements such that $y \\notin f \\left({Y \\setminus \\left\\{{y}\\right\\} }\\right)$ for all $y \\in Y$."}
+{"_id": "4357", "title": "Diagonalizable Operator Bounded iff Value Set Bounded", "text": "Let $H$ be a Hilbert space. Let $A: H \\to H$ be a diagonalizable operator. Let $\\left({\\alpha_e}\\right)_{e \\in E}$ be the value set of $A$, with respect to a suitable basis $E$ for $H$. Then $A$ is bounded iff $\\left({\\alpha_e}\\right)_{e \\in E}$ is, i.e., iff: :$\\exists M \\in \\R: \\forall e \\in E: \\left\\vert{\\alpha_e}\\right\\vert \\le M$"}
+{"_id": "4358", "title": "Diagonalizable Operator Compact iff Value Set Converges to Zero", "text": "Let $H$ be a Hilbert space of countable dimension. Let $A: H \\to H$ be a diagonalizable operator. Let $\\left({\\alpha_n}\\right)_{n \\in \\N}$ be the value set of $A$, with respect to a suitable basis $E = \\left({e_n}\\right)_{n \\in \\N}$ for $H$. Then $A$ is compact iff: :$\\displaystyle \\lim_{n \\to \\infty} \\alpha_n = 0$"}
+{"_id": "4359", "title": "Nonzero Eigenvalue of Compact Operator has Finite Dimensional Eigenspace", "text": "Let $H$ be a Hilbert space. Let $T \\in B_0 \\left({H}\\right)$ be a compact operator. Let $\\lambda \\in \\sigma_p \\left({T}\\right), \\lambda \\ne 0$ be a nonzero eigenvalue of $T$. Then the eigenspace for $\\lambda$ has finite dimension."}
+{"_id": "4360", "title": "Condition for Nonzero Eigenvalue of Compact Operator", "text": "Let $H$ be a Hilbert space over $\\Bbb F \\in \\set {\\R, \\C}$. Let $T \\in \\map {B_0} H$ be a compact operator. Let $\\lambda \\in \\Bbb F, \\lambda \\ne 0$ be a nonzero scalar. Suppose that the following holds: :$\\inf \\set {\\norm {\\paren {T - \\lambda I} h}_H: \\norm h_H = 1} = 0$ Then $\\lambda \\in \\map {\\sigma_p} T$, that is, $\\lambda$ is an eigenvalue for $T$."}
+{"_id": "4361", "title": "Finite Rank Operator is Compact", "text": "Let $H, K$ be Hilbert spaces. Let $T \\in \\map {B_{00} } {H, K}$ be a bounded finite rank operator. Then $T \\in \\map {B_0} {H, K}$, that is, $T$ is compact."}
+{"_id": "4362", "title": "Adjoint of Finite Rank Operator", "text": "Let $H, K$ be Hilbert spaces. Let $T \\in \\map {B_{00} } {H, K}$ be a bounded finite rank operator. Then: :$T^* \\in \\map {B_{00} } {K, H}$ that is, the adjoint of $T$ is also a bounded finite rank operator."}
+{"_id": "4363", "title": "Compact Idempotent is of Finite Rank", "text": "Let $H$ be a Hilbert space. Let $T \\in \\map {B_0} H$ be a compact linear operator, and let $T$ be idempotent. Then: :$T \\in \\map {B_{00} } H$ that is, $T$ is a bounded finite rank operator."}
+{"_id": "4364", "title": "Characterization of Finite Rank Operators", "text": "Let $H$ be a Hilbert space. Let $T \\in B_{00} \\left({H}\\right)$ be a bounded finite rank operator. Let $n = \\operatorname{dim} \\left({\\operatorname{ran} T}\\right)$ be the rank of $T$. Then there are orthonormal vectors $e_1, \\ldots, e_n$ and vectors $g_1, \\ldots, g_n$ of $H$ such that: :$\\forall h \\in H: Th = \\displaystyle \\sum_{i=1}^n \\left\\langle{h, e_i}\\right\\rangle_H g_i$"}
+{"_id": "4365", "title": "Compact Operator on Hilbert Space Direct Sum", "text": "Let $\\sequence {H_n}_{n \\mathop \\in \\N}$ be a sequence of Hilbert spaces. Denote by $H = \\displaystyle \\bigoplus_{n \\mathop = 1}^\\infty H_n$ their Hilbert space direct sum. For each $n \\in \\N$, let $T_n \\in \\map B {H_n}$ be a bounded linear operator. Suppose that one has $\\displaystyle \\sup_{n \\mathop \\in \\N} \\norm {T_n} < \\infty$, where $\\norm {\\, \\cdot \\, }$ signifies the norm on bounded linear operators. Define $T \\in \\map B H$ by:  :$\\forall h = \\sequence {h_n}_{n \\mathop \\in \\N}: T h = \\sequence {T_n h_n}_{n \\mathop \\in \\N} \\in H$ (That $T$ is indeed bounded follows from Bounded Linear Operator on Hilbert Space Direct Sum.) Then $T$ is compact {{iff}} the following conditions hold: : For each $n \\in \\N$, $T_n$ is compact : $\\displaystyle \\lim_{n \\mathop \\to \\infty} \\norm {T_n} = 0$"}
+{"_id": "4366", "title": "Bounded Linear Operator on Hilbert Space Direct Sum", "text": "Let $\\left({H_n}\\right)_{n \\in \\N}$ be a sequence of Hilbert spaces. Denote by $H = \\displaystyle \\bigoplus_{n \\mathop = 1}^\\infty H_n$ their Hilbert space direct sum. For each $n \\in \\N$, let $T_n \\in B \\left({H_n}\\right)$ be a bounded linear operator. Suppose that one has $\\displaystyle \\sup_{n \\mathop \\in \\N} \\, \\left\\Vert{T_n}\\right\\Vert < \\infty$, where $\\left\\Vert{\\cdot}\\right\\Vert$ signifies the norm on bounded linear operators. Define $T: H \\to H$ by:  :$\\forall h = \\left({h_n}\\right)_{n \\in \\N}: T h = \\left({T_n h_n}\\right)_{n \\in \\N} \\in H$ Then $T \\in B \\left({H}\\right)$ is a bounded linear operator."}
+{"_id": "4367", "title": "Compact Self-Adjoint Operator has Countable Point Spectrum", "text": "Let $H$ be a Hilbert space. Let $T \\in B_0 \\left({H}\\right)$ be a compact, self-adjoint operator. Then its point spectrum $\\sigma_p \\left({T}\\right)$ is countable."}
+{"_id": "4368", "title": "Spectral Theorem for Compact Self-Adjoint Operators", "text": "Let $H$ be a Hilbert space. Let $T \\in B_0 \\left({H}\\right)$ be a compact, self-adjoint operator. Then there exists a (possibly finite) sequence $\\left({\\lambda_n}\\right)$ of distinct nonzero eigenvalues of $T$ such that: * $P_n P_m = P_m P_n = 0$ if $n \\ne m$ * $\\displaystyle \\lim_{k \\to \\infty} \\left\\Vert{T - \\sum_{n=1}^k \\lambda_n P_n}\\right\\Vert = 0$, that is, $T = \\displaystyle \\sum_{n=1}^\\infty \\lambda_n P_n$ where $P_n$ is the orthogonal projection onto the eigenspace of $\\lambda_n$, and $\\left\\Vert{\\cdot}\\right\\Vert$ denotes the norm on bounded linear operators. {{refactor|Split corollaries to subpages}} === Corollary 1 === There exists a (possibly finite) sequence $\\left({\\mu_n}\\right)$ of real numbers and a basis $E = \\left({e_n}\\right)$ for $\\left({\\operatorname{ker} T}\\right)^\\perp$ such that: :$\\forall h \\in H: Th = \\displaystyle \\sum_{n=1}^\\infty \\left\\langle{h, e_n}\\right\\rangle_H \\mu_n e_n$"}
+{"_id": "4369", "title": "Eigenspace for Normal Operator is Reducing Subspace", "text": "Let $H$ be a Hilbert space over $\\Bbb F \\in \\left\\{{\\R, \\C}\\right\\}$. Let $A \\in B \\left({H}\\right)$ be a normal operator. Let $\\lambda \\in \\Bbb F$. Then $\\ker \\left({A - \\lambda}\\right)$ is a reducing subspace for $A$. Here $\\ker$ denotes kernel."}
+{"_id": "4371", "title": "Eigenvalues of Self-Adjoint Operator are Real", "text": "Let $H$ be a Hilbert space. Let $A \\in B \\left({H}\\right)$ be a self-adjoint operator. Then all eigenvalues of $A$ are real."}
+{"_id": "4372", "title": "Operator Diagonalizable iff Basis of Eigenvectors", "text": "Let $H$ be a Hilbert space. Let $A:H \\to H$ be a linear operator on $H$. Then $A$ is diagonalizable iff there exists a basis $E$ of $H$, consisting of eigenvectors for $A$."}
+{"_id": "4373", "title": "Existence of Hartogs Number", "text": "Let $S$ be a set. Then $S$ has a Hartogs number."}
+{"_id": "4375", "title": "Left and Right Inverse Mappings Implies Bijection", "text": "Let $f: S \\to T$ be a mapping. Let $f$ be such that: : $\\exists g_1: T \\to S: g_1 \\circ f = I_S$ : $\\exists g_2: T \\to S: f \\circ g_2 = I_T$ where both $g_1$ and $g_2$ are mappings. Then $f$ is a bijection."}
+{"_id": "4378", "title": "Left and Right Inverses of Mapping are Inverse Mapping", "text": "Let $f: S \\to T$ be a mapping such that: :$(1): \\quad \\exists g_1: T \\to S: g_1 \\circ f = I_S$ :$(2): \\quad \\exists g_2: T \\to S: f \\circ g_2 = I_T$ Then: : $g_1 = g_2 = f^{-1}$ where $f^{-1}$ is the inverse of $f$."}
+{"_id": "4381", "title": "Operator Commuting with Diagonalizable Operator", "text": "Let $H$ be a Hilbert space. Let $A = \\displaystyle \\sum_{i \\mathop \\in I} \\alpha_i P_i$ be a diagonalizable operator on $H$. Let $B \\in \\map B H$ be a bounded linear operator. Then the following are equivalent: :$(1): \\quad A B = B A$ :$(2): \\quad$ For all $i \\in I$, $\\Rng {P_i}$ is a reducing subspace for $B$ where $\\Rng {P_i}$ denotes range."}
+{"_id": "4385", "title": "Composition of Mappings is Composition of Relations", "text": "Let $S_1, S_2, S_3$ be sets. Let $f_1: S_1 \\to S_2$ and $f_2: S_2 \\to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$. Let $f_2 \\circ f_1$ be the composite of $f_1$ and $f_2$. Let $f_1$ and $f_2$ be considered as relations on $S_1 \\times S_2$ and $S_2 \\times S_3$ respectively. Then $f_2 \\circ f_1$ defined as the composition of relations coincides with the definition of $f_2 \\circ f_1$ as the composition of mappings."}
+{"_id": "4386", "title": "Null Sequence in Exponential Sequence", "text": "Let $\\sequence {a_n}_{n \\mathop \\in \\N} \\in \\C$ be a sequence of complex numbers such that: :$\\displaystyle \\lim_{n \\mathop \\to +\\infty}a_n = 0$ Then: :$\\displaystyle \\lim_{n \\mathop \\to +\\infty} \\paren {1 + \\dfrac {a_n} n}^n = 1$"}
+{"_id": "4387", "title": "Identity Mapping is Right Identity", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Then: :$f \\circ I_S = f$ where $I_S$ is the identity mapping on $S$, and $\\circ$ signifies composition of mappings."}
+{"_id": "4388", "title": "Diagonal Relation is Left Identity", "text": "Let $\\RR \\subseteq S \\times T$ be a relation on $S \\times T$. Then: :$\\Delta_T \\circ \\RR = \\RR$ where $\\Delta_T$ is the diagonal relation on $T$, and $\\circ$ signifies composition of relations."}
+{"_id": "4389", "title": "Isomorphism Preserves Associativity", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be algebraic structures. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be an isomorphism. Then $\\circ$ is associative {{iff}} $*$ is associative."}
+{"_id": "4390", "title": "Isomorphism Preserves Commutativity", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be algebraic structures. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be an isomorphism. Then $\\circ$ is commutative {{iff}} $*$ is commutative."}
+{"_id": "4391", "title": "Isomorphism Preserves Inverses", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be algebraic structures. Let $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ be an isomorphism. Let $\\left({S, \\circ}\\right)$ have an identity $e_S$. Then $x^{-1}$ is an inverse of $x$ for $\\circ$ iff $\\phi \\left({x^{-1}}\\right)$ is an inverse of $\\phi \\left({x}\\right)$ for $*$. That is, iff: : $\\phi \\left({x^{-1}}\\right) = \\left({\\phi \\left({x}\\right)}\\right)^{-1}$"}
+{"_id": "4392", "title": "Isomorphism Preserves Identity", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be algebraic structures. Let $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ be an isomorphism. Then $\\circ$ has an identity $e_S$ {{iff}} $\\phi \\left({e_S}\\right)$ is the identity for $*$."}
+{"_id": "4398", "title": "Non-Zero Natural Numbers under Multiplication form Commutative Semigroup", "text": "Let $\\N_{>0}$ be the set of natural numbers without zero, that is, $\\N_{>0} = \\N \\setminus \\set 0$. The structure $\\struct {\\N_{>0}, \\times}$ forms an infinite commutative semigroup."}
+{"_id": "4399", "title": "Isomorphism Preserves Groups", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be algebraic structures. Let $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ be an isomorphism. If $\\left({S, \\circ}\\right)$ is a group, then so is $\\left({T, *}\\right)$."}
+{"_id": "4400", "title": "Isomorphism Preserves Semigroups", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be algebraic structures. Let $\\phi: S \\to T$ be an isomorphism. If $\\struct {S, \\circ}$ is a semigroup, then so is $\\struct {T, *}$."}
+{"_id": "4401", "title": "Left Cancellable iff Left Regular Representation Injective", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Then $a \\in S$ is left cancellable {{iff}} the left regular representation $\\map {\\lambda_a} x$ is injective."}
+{"_id": "4402", "title": "Right Cancellable iff Right Regular Representation Injective", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Then $a \\in S$ is right cancellable {{iff}} the right regular representation $\\map {\\rho_a} x$ is injective."}
+{"_id": "4403", "title": "Isomorphism Preserves Cancellability", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be algebraic structures. Let $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ be an isomorphism. Then: : $a \\in S$ is cancellable in $\\left({S, \\circ}\\right)$ iff $\\phi \\left({a}\\right) \\in T$ is cancellable in $\\left({T, *}\\right)$."}
+{"_id": "4404", "title": "Isomorphism Preserves Left Cancellability", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be algebraic structures. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be an isomorphism. Then: :$a \\in S$ is left cancellable in $\\struct {S, \\circ}$ {{iff}}: :$\\map \\phi a \\in T$ is left cancellable in $\\struct {T, *}$."}
+{"_id": "4405", "title": "Isomorphism Preserves Right Cancellability", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be algebraic structures. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be an isomorphism. Then: :$a \\in S$ is right cancellable in $\\struct {S, \\circ}$ {{iff}} $\\map \\phi a \\in T$ is right cancellable in $\\struct {T, *}$."}
+{"_id": "4406", "title": "Differentiation of Power Series", "text": "Let $\\xi \\in \\R$ be a real number. Let $\\left \\langle {a_n} \\right \\rangle$ be a sequence in $\\R$. Let $\\displaystyle \\sum_{m \\mathop \\ge 0} a_m \\left({x - \\xi}\\right)^m$ be the power series in $x$ about the point $\\xi$. Then within the interval of convergence: :$\\displaystyle \\frac {\\mathrm d^n} {\\mathrm d x^n} \\sum_{m \\mathop \\ge 0} a_m \\left({x - \\xi}\\right)^m = \\sum_{m \\mathop \\ge n} a_m m^{\\underline n} \\left({x - \\xi}\\right)^{m - n}$ where $m^{\\underline n}$ denotes the falling factorial."}
+{"_id": "4407", "title": "General Associativity Theorem/Formulation 1", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Let $\\sequence {a_k}_{p + 1 \\mathop \\le k \\mathop \\le p + n}$ be a sequence of elements of $S$. Let $\\sequence {r_k}_{0 \\mathop \\le k \\mathop \\le s}$ be a strictly increasing sequence of natural numbers such that $r_0 = p$ and $r_s = p+n$. Suppose: :$\\displaystyle \\forall k \\in \\closedint 1 s: b_k = \\prod_{j \\mathop = r_{k - 1} \\mathop + 1}^{r_k} {a_j}$ Then: :$\\displaystyle \\forall n \\in \\N_{>0}: \\prod_{k \\mathop = 1}^s {b_k} = \\prod_{k \\mathop = p \\mathop + 1}^{p \\mathop + n} {a_k}$ That is: :$\\displaystyle \\forall n \\in \\N_{>0}: \\prod_{k \\mathop = 1}^s \\paren {a_{r_{k - 1} + 1} \\circ a_{r_{k - 1} + 2} \\circ \\ldots \\circ a_{r_k} } = a_{p + 1} \\circ \\ldots \\circ a_{p + n}$"}
+{"_id": "4408", "title": "General Associativity Theorem/Formulation 2", "text": "Let $n \\in \\N_{>0}$ and let $a_1, \\ldots, a_n$ be elements of a set $S$. Let $\\circ$ be an associative operation on $S$. Let the set $\\map {P_n} {a_1, a_2, \\ldots, a_n}$ be defined inductively by: :$\\map {P_1} {a_1} = \\set {a_1}$ :$\\map {P_2} {a_1, a_2} = \\set {a_1 \\circ a_2}$ :$\\map {P_n} {a_1, a_2, \\ldots, a_n} = \\set {x \\circ y: x \\in \\map {P_r} {a_1, a_2, \\ldots, a_r} \\land y \\in \\map {P_s} {a_{r + 1}, a_{r + 2}, \\ldots, a_{r + s} }, n = r + s}$ Then $\\map {P_n} {a_1, a_2, \\ldots, a_n}$ consists of a unique entity which we can denote $a_1 \\circ a_2 \\circ \\ldots \\circ a_n$."}
+{"_id": "4411", "title": "Monoid of Self-Inverse Elements is Abelian Group", "text": "Let $\\left({S, \\circ}\\right)$ be a monoid such that: :$\\forall x \\in S: x \\circ x = e$ where $e$ is the identity element of $\\left({S, \\circ}\\right)$. Then $\\left({S, \\circ}\\right)$ is an abelian group."}
+{"_id": "4413", "title": "Product is Right Identity Therefore Right Cancellable", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroup. Let $e_R \\in S$  be a right identity of $S$. Let $a \\in S$ such that: :$\\exists b \\in S: a \\circ b = e_R$ Then $a$ is right cancellable in $\\left({S, \\circ}\\right)$."}
+{"_id": "4415", "title": "Integer Multiples Greater than Positive Integer Closed under Multiplication", "text": "Let $n \\Z$ be the set of integer multiples of $n$. Let $p \\in \\Z: p \\ge 0$ be a positive integer. Let $S \\subseteq n \\Z$  be defined as: :$S := \\left\\{{x \\in n \\Z: x > p}\\right\\}$ that is, the set of integer multiples of $n$ greater than $p$. Then the algebraic structure $\\struct {S, \\times}$ is closed under multiplication."}
+{"_id": "4418", "title": "Non-Zero Complex Numbers Closed under Multiplication", "text": "The set of non-zero complex numbers is closed under multiplication."}
+{"_id": "4419", "title": "Non-Zero Modulo Numbers Closed under Multiplication then Modulo is Prime", "text": "Let $\\left({\\Z_m, +_m, \\times_m}\\right)$ be the ring of integers modulo $m$ for $m > 1$. Let $\\Z'_m$ be the set of non-zero integers modulo $m$. Let $\\left({\\Z_m, \\times_m}\\right)$ be closed under modulo multiplication. Then $m$ is prime."}
+{"_id": "4420", "title": "Magma is Submagma of Itself", "text": "Let $\\left({S, \\circ}\\right)$ be a magma. Then $\\left({S, \\circ}\\right)$ is a submagma of itself."}
+{"_id": "4421", "title": "Empty Set is Submagma of Magma", "text": "Let $\\left({S, \\circ}\\right)$ be a magma. Then: : $\\left({\\varnothing, \\circ}\\right)$ is a submagma of $\\left({S, \\circ}\\right)$ where $\\varnothing$ is the empty set."}
+{"_id": "4422", "title": "Max Operation on Toset is Semigroup", "text": "Let $\\left({S, \\preceq}\\right)$ be a totally ordered set. Let $\\max \\left({x, y}\\right)$ denote the max operation on $x, y \\in S$. Then $\\left({S, \\max}\\right)$ is a semigroup."}
+{"_id": "4424", "title": "Min Semigroup is Commutative", "text": "Let $\\left({S, \\preceq}\\right)$ be a totally ordered set.  Then the semigroup $\\left({S, \\min}\\right)$ is commutative."}
+{"_id": "4425", "title": "Min Semigroup is Idempotent", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. Then the semigroup $\\struct {S, \\min}$ is an idempotent semigroup."}
+{"_id": "4426", "title": "Max Operation on Woset is Monoid", "text": "Let $\\struct {S, \\preceq}$ be a well-ordered set. Let $\\max \\set {x, y}$ denote the max operation on $x, y \\in S$. Then $\\struct {S, \\max}$ is a monoid. Its identity element is the smallest element of $S$."}
+{"_id": "4427", "title": "Max Operation on Natural Numbers forms Monoid", "text": "Let $\\left({\\N, \\max}\\right)$ denote the algebraic structure formed from the natural numbers $\\N$ and the max operation. Then $\\left({\\N, \\max}\\right)$ is a monoid. Its identity element is the zero."}
+{"_id": "4428", "title": "Complex Modulus Function is Continuous", "text": "Let $z_0 \\in \\C$ be a complex number. Then the complex modulus function is continuous at $z_0$."}
+{"_id": "4430", "title": "Euclidean Space is Banach Space", "text": "Let $m$ be a positive integer. Then the Euclidean space $\\R^m$, along with the Euclidean norm, forms a Banach space over $\\R$."}
+{"_id": "4431", "title": "Complex Plane is Complete Metric Space", "text": "The complex plane, along with the metric induced by the norm given by the complex modulus, forms a complete metric space."}
+{"_id": "4432", "title": "Euclidean Space is Complete Metric Space", "text": "Let $m$ be a positive integer. Then the Euclidean space $\\R^m$, along with the Euclidean metric, forms a complete metric space."}
+{"_id": "4433", "title": "Complex Plane is Banach Space", "text": "The complex plane, along with the complex modulus, forms a Banach space over $\\C$."}
+{"_id": "4434", "title": "Rational Number Space is not Complete Metric Space", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean metric $d$. Then $\\struct {\\Q, \\tau_d}$ is not a complete metric space."}
+{"_id": "4442", "title": "Limit of Subsequence equals Limit of Sequence/Real Numbers", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let $l \\in \\R$ such that: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ Let $\\sequence {x_{n_r} }$ be a subsequence of $\\sequence {x_n}$. Then: :$\\displaystyle \\lim_{r \\mathop \\to \\infty} x_{n_r} = l$"}
+{"_id": "4445", "title": "Distance Function of Metric Space is Continuous", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $\\tau_A$ be the topology on $A$ induced by $d$. Let $\\struct {A \\times A, \\tau}$ be the product space of $\\struct {A, \\tau_A}$ and itself. Then the distance function $d: A \\times A \\to \\R$ is a continuous mapping."}
+{"_id": "4447", "title": "Sequence Converges to Within Half Limit/Complex Numbers", "text": "Let $\\sequence {z_n}$ be a sequence in $\\C$. Let $\\sequence {z_n}$ be convergent to the limit $l$. That is, let $\\displaystyle \\lim_{n \\mathop \\to \\infty} z_n = l$ where $l \\ne 0$. Then: : $\\exists N: \\forall n > N: \\cmod {z_n} > \\dfrac {\\cmod l} 2$"}
+{"_id": "4448", "title": "Number to Reciprocal Power is Decreasing", "text": "The real sequence $\\sequence {n^{1/n} }$ is decreasing for $n \\ge 3$."}
+{"_id": "4453", "title": "Inverse of Diagonal Matrix", "text": "Let: :$\\mathbf D = \\begin{bmatrix} a_{11} & 0 & \\cdots & 0 \\\\ 0 & a_{22} & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & a_{nn} \\\\ \\end{bmatrix}$ be an $n \\times n$ diagonal matrix. Then its inverse is given by: : $\\mathbf D^{-1} = \\begin{bmatrix} \\dfrac 1 {a_{11}} & 0 & \\cdots & 0 \\\\ 0 & \\dfrac 1 {a_{22}} & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & \\dfrac 1 {a_{nn}} \\\\ \\end{bmatrix}$ provided that none of the diagonal elements are zero. If any of the diagonal elements are zero, $\\mathbf D$ is not invertible.  {{expand|If any diagonal element isn't a unit}}"}
+{"_id": "4454", "title": "Subspace of Complete Metric Space is Closed iff Complete", "text": "Let $\\struct {M, d}$ be a complete metric space. Let $\\struct {S, d}$ be a subspace of $\\struct {M, d}$. Then $S$ is closed {{iff}} $S$ is complete."}
+{"_id": "4455", "title": "Set of Chains is Chain Complete for Inclusion", "text": "Let $\\left({P, \\leq}\\right)$ be an ordered set. Let $\\operatorname{ch} \\left({P}\\right)$ be the set of chains in $P$. Let $\\mathcal C$ be a chain in $\\operatorname{ch} \\left({P}\\right)$ with respect to the subset ordering. Then the union $\\bigcup \\mathcal C$ of $\\mathcal C$ is also a chain in $P$."}
+{"_id": "4456", "title": "Jensen's Inequality (Real Analysis)", "text": "Let $I$ be a real interval. Let $\\phi: I \\to \\R$ be a convex function. Let $x_1, x_2, \\ldots, x_n \\in I$. Let $\\lambda_1, \\lambda_2, \\ldots, \\lambda_n \\ge 0$ be real numbers, at least one of which is non-zero. Then: :$\\displaystyle \\map \\phi {\\frac {\\sum_{k \\mathop = 1}^n \\lambda_k x_k} {\\sum_{k \\mathop = 1}^n \\lambda_k} } \\le \\frac {\\sum_{k \\mathop = 1}^n \\lambda_k \\map \\phi {x_k} } {\\sum_{k \\mathop = 1}^n \\lambda_k}$ For $\\phi$ strictly convex, equality holds {{iff}} $x_1 = x_2 = \\cdots = x_n$."}
+{"_id": "4458", "title": "Real Linear Subspace Contains Zero Vector", "text": "Let $\\mathbb W \\subseteq \\R^n$ such that $\\mathbb W$ is a linear subspace of $\\R^n$. Then $\\mathbb W$ contains the zero vector: :$\\mathbf 0_{n \\times 1} = \\begin{bmatrix} 0 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{bmatrix} \\in \\mathbb W$"}
+{"_id": "4459", "title": "Zero Subspace is Subspace", "text": "Let $V$ be a vector space over $K$ with zero vector $\\mathbf 0$. The zero subspace $\\left\\{{\\mathbf 0}\\right\\}$ is a subspace of $V$."}
+{"_id": "4460", "title": "Cartesian Product of Natural Numbers with Itself is Countable", "text": "The Cartesian product $\\N \\times \\N$ of the set of natural numbers $\\N$ with itself is countable."}
+{"_id": "4463", "title": "Minkowski's Inequality for Sums", "text": "Let $a_1, a_2, \\ldots, a_n, b_1, b_2, \\ldots, b_n \\in \\R_{\\ge 0}$ be non-negative real numbers. Let $p \\in \\R$, $p \\ne 0$ be a real number. If $p < 0$, then we require that $a_1, a_2, \\ldots, a_n, b_1, b_2, \\ldots, b_n$ be strictly positive. If $p > 1$, then: :$\\displaystyle \\left({\\sum_{k \\mathop = 1}^n \\left({a_k + b_k}\\right)^p}\\right)^{1/p} \\le \\left({\\sum_{k \\mathop = 1}^n a_k^p}\\right)^{1/p} + \\left({\\sum_{k \\mathop = 1}^n b_k^p}\\right)^{1/p}$ If $p < 1$, $p \\ne 0$, then: :$\\displaystyle \\left({\\sum_{k \\mathop = 1}^n \\left({a_k + b_k}\\right)^p}\\right)^{1/p} \\ge \\left({\\sum_{k \\mathop = 1}^n a_k^p}\\right)^{1/p} + \\left({\\sum_{k \\mathop = 1}^n b_k^p}\\right)^{1/p}$"}
+{"_id": "4464", "title": "Minkowski's Inequality for Integrals", "text": "Let $f, g$ be integrable functions in $X \\subseteq \\R^n$ with respect to the volume element $dV$. {{explain|Riemann integrable or Lebesgue integrable?}} :$(1):\\quad$ Let $p > 1$. Then: :: $\\displaystyle \\paren {\\int_X \\size {f + g}^p \\rd V}^{1/p} \\le \\paren {\\int_X \\size f^p \\rd V}^{1/p} + \\paren {\\int_X \\size g^p \\rd V}^{1/p}$ :$(2):\\quad$ Let $p < 1, p \\ne 0$. Then: :: $\\displaystyle \\paren {\\int_X \\size {f + g}^p \\rd V}^{1/p} \\ge \\paren {\\int_X \\size f^p \\rd V}^{1/p} + \\paren {\\int_X \\size g^p \\rd V}^{1/p}$"}
+{"_id": "4465", "title": "Characteristic Function Determined by 1-Fiber", "text": "Let $A \\subseteq S$. Let $f:S \\to \\left\\{{0, 1}\\right\\}$ be a mapping. Denote by $\\chi_A$ the characteristic function on $A$. Then the following are equivalent: :$(1):\\quad f = \\chi_A$ :$(2):\\quad \\forall s \\in S: f \\left({s}\\right) = 1 \\iff s \\in A$ Using the notion of a fiber, $(2)$ may also be expressed as: :$(2'):\\quad f^{-1} \\left({1}\\right) = A$"}
+{"_id": "4466", "title": "Characteristic Function of Intersection", "text": "Let $A, B \\subseteq S$. Let $\\chi_{A \\cap B}$ be the characteristic function of their intersection $A \\cap B$."}
+{"_id": "4467", "title": "Characteristic Function of Union", "text": "Let $A, B \\subseteq S$. Let $\\chi_{A \\mathop \\cup B}$ be the characteristic function of their union $A \\cup B$."}
+{"_id": "4468", "title": "Characteristic Function of Set Difference", "text": "Let $A, B \\subseteq S$. Then: :$\\chi_{A \\mathop \\setminus B} = \\chi_A - \\chi_{A \\cap B}$  where: :$A \\setminus B$ denotes set difference :$\\chi$ denotes characteristic function."}
+{"_id": "4469", "title": "Characteristic Function of Symmetric Difference", "text": "Let $A, B \\subseteq S$. Then: :$\\chi_{A * B} = \\chi_A + \\chi_B - 2 \\chi_{A \\cap B}$  where: : $\\chi$ denotes characteristic function : $*$ denotes symmetric difference."}
+{"_id": "4470", "title": "Vector Subspace of Real Vector Space", "text": "Let $\\R^n$ be a real vector space. Let $\\mathbb W \\subseteq \\R^n$. Then $\\mathbb W$ is a linear subspace of $\\R^n$ {{iff}}: :$(1): \\quad \\mathbf 0 \\in \\mathbb W$, where $\\mathbf 0$ is the zero vector with $n$ entries :$(2): \\quad \\mathbb W$ is closed under vector addition :$(3): \\quad \\mathbb W$ is closed under scalar multiplication."}
+{"_id": "4471", "title": "Reverse Young's Inequality for Products", "text": "Let $p, q \\in \\R_{> 0}$ be strictly positive real numbers satisfying: :$\\displaystyle \\frac 1 p - \\frac 1 q = 1$ Let $a \\in \\R_{\\ge 0}$ be a positive real number. Let $b \\in \\R_{> 0}$ be a strictly positive real number. Then: :$\\displaystyle a b \\ge \\frac {a^p} p - \\frac {b^{-q}} q$"}
+{"_id": "4472", "title": "Reverse Hölder's Inequality for Sums", "text": "Let $p, q \\in \\R_{>0}$ be strictly positive real numbers such that $\\dfrac 1 p - \\dfrac 1 q = 1$. Suppose that the sequences $\\mathbf x = \\sequence {x_n}$ and $\\mathbf y = \\sequence {y_n}$ in $\\C$ (or $\\R$) are such that the series :$\\displaystyle \\paren {\\sum_{n \\mathop = 1}^\\infty \\size {x_n}^p}^{1/p}$ and :$\\displaystyle \\paren {\\sum_{n \\mathop = 1}^\\infty \\size {y_n}^{-q} }^{-1/q}$ are convergent. Denote these values by $\\norm {\\mathbf x}_p$ and $\\norm {\\mathbf y}_{-q}$, respectively. Here, the notation $\\norm {\\mathbf x}_p$ does not denote a norm, but is instead just a convenient notation similar to that of the $p$-norm, which is only defined when $p \\ge 1$. Let $\\norm {\\mathbf {x y} }_1$ denote the $1$-norm of $\\mathbf {x y}$, if $\\mathbf {x y}$ is in the Lebesgue space $\\ell^1$. Then $\\norm {\\mathbf {x y} }_1 \\ge \\norm {\\mathbf x}_p \\norm {\\mathbf y}_{-q}$."}
+{"_id": "4473", "title": "Real Vector Space is Vector Space", "text": "Let $\\R$ be the set of real numbers. Then the real vector space $\\R^n$ is a vector space."}
+{"_id": "4475", "title": "Sigma-Algebra Contains Empty Set", "text": "Let $X$ be a set, and let $\\Sigma$ be a $\\sigma$-algebra on $X$. Then $\\varnothing \\in \\Sigma$."}
+{"_id": "4476", "title": "Sigma-Algebra Closed under Union", "text": "Let $X$ be a set, and let $\\Sigma$ be a $\\sigma$-algebra on $X$. Let $A, B \\in \\Sigma$ be measurable sets. Then $A \\cup B \\in \\Sigma$, where $\\cup$ denotes set union."}
+{"_id": "4477", "title": "Sigma-Algebra Closed under Countable Intersection", "text": "Let $X$ be a set, and let $\\Sigma$ be a $\\sigma$-algebra on $X$. Suppose that $\\left({E_n}\\right)_{n \\in \\N} \\in \\Sigma$ is a collection of measurable sets. Then $\\displaystyle \\bigcap_{n \\mathop \\in \\N} E_n \\in \\Sigma$, where $\\displaystyle \\bigcap$ denotes set intersection."}
+{"_id": "4478", "title": "Sigma-Algebra of Countable Sets", "text": "Let $X$ be a set. Let $\\Sigma$ be the collection of countable and co-countable subsets of $X$. Then $\\Sigma$ is a $\\sigma$-algebra."}
+{"_id": "4479", "title": "Trace Sigma-Algebra is Sigma-Algebra", "text": "Let $X$ be a set, and let $\\Sigma$ be a $\\sigma$-algebra on $X$. Let $E \\subseteq X$ be a subset of $X$. Then the trace $\\sigma$-algebra $\\Sigma_E$ is a $\\sigma$-algebra on $E$."}
+{"_id": "4480", "title": "Pre-Image Sigma-Algebra on Domain is Sigma-Algebra", "text": "Let $X, X'$ be sets, and let $f: X \\to X'$ be a mapping. Let $\\Sigma'$ be a $\\sigma$-algebra on $X'$. Then $f^{-1} \\left({\\Sigma'}\\right)$, the pre-image $\\sigma$-algebra on the domain of $f$, is a $\\sigma$-algebra on $X$."}
+{"_id": "4481", "title": "Intersection of Sigma-Algebras", "text": "Let $X$ be a set. Let $\\family {\\Sigma_i}_{i \\mathop \\in I}$ be a indexed set of $\\sigma$-algebras on $X$. Then $\\Sigma := \\displaystyle \\bigcap_{i \\mathop \\in I} \\Sigma_i$ is also a $\\sigma$-algebra on $X$. Here, $\\displaystyle \\bigcap$ denotes set intersection."}
+{"_id": "4482", "title": "Generated Sigma-Algebra Preserves Subset", "text": "Let $X$ be a set. Let $\\mathcal F, \\mathcal G \\subseteq \\mathcal P \\left({X}\\right)$ be collections of subsets of $X$. Suppose that $\\mathcal F \\subseteq \\mathcal G$. Then $\\sigma \\left({\\mathcal F}\\right) \\subseteq \\sigma \\left({\\mathcal G}\\right)$, where $\\sigma \\left({\\mathcal G}\\right)$ denotes the $\\sigma$-algebra generated by $\\mathcal G$"}
+{"_id": "4483", "title": "Characterization of Euclidean Borel Sigma-Algebra", "text": "Let $\\mathcal{O}^n$, $\\mathcal{C}^n$ and $\\mathcal{K}^n$ be the collections of open, closed and compact subsets of the Euclidean space $\\left({\\R^n, \\tau}\\right)$, respectively. Let $\\mathcal{J}_{ho}^n$ be the collection of half-open rectangles in $\\R^n$. Let $\\mathcal{J}^n_{ho, \\text{rat}}$ be the collection of half-open rectangles in $\\R^n$ with rational endpoints. Then the Borel $\\sigma$-algebra $\\mathcal B \\left({\\R^n}\\right)$ satisfies: :$\\mathcal B \\left({\\R^n}\\right) = \\sigma \\left({\\mathcal{O}^n}\\right) = \\sigma \\left({\\mathcal{C}^n}\\right) = \\sigma \\left({\\mathcal{K}^n}\\right) = \\sigma \\left({\\mathcal{J}_{ho}^n}\\right) = \\sigma \\left({\\mathcal{J}^n_{ho, \\text{rat}}}\\right)$ where $\\sigma$ denotes generated $\\sigma$-algebra."}
+{"_id": "4484", "title": "Generated Sigma-Algebra Preserves Finiteness", "text": "Let $X$ be a set, and let $A_1, \\ldots, A_n \\subseteq X$ be subsets of $X$. Then $\\sigma \\left({\\left\\{{A_1, \\ldots, A_n}\\right\\}}\\right)$ is a finite set, where $\\sigma$ denotes generated $\\sigma$-algebra."}
+{"_id": "4485", "title": "Set of Invertible Mappings forms Symmetric Group", "text": "Let $S$ be a set. Let $\\struct {S^S, \\circ}$ be the monoid of all the mappings from $S$ to itself. Let $\\mathcal G$ be the set of all invertible mappings from $S$ to $S$. Then $\\struct {\\mathcal G, \\circ}$ is the symmetric group on $S$."}
+{"_id": "4486", "title": "Group of Units of Field", "text": "Let $k$ be a field. Then $k^\\times = k \\setminus \\set 0$"}
+{"_id": "4487", "title": "Automorphism Group is Subgroup of Symmetric Group", "text": "Let $\\struct {S, *}$ be an algebraic structure. Let $\\Aut S$ be the automorphism group of $\\struct {S, *}$. Then $\\Aut S$ is a subgroup of the symmetric group  $\\struct {\\Gamma \\paren S, \\circ}$ on $S$."}
+{"_id": "4488", "title": "Differentiation of Vector-Valued Function Componentwise", "text": "Let: :$\\mathbf r: t \\mapsto \\left\\langle{r_1 \\left({t}\\right), r_2 \\left({t}\\right), \\ldots, r_n \\left({t}\\right)}\\right\\rangle$  be a differentiable vector-valued function. The derivative of a vector-valued function can be calculated by differentiating each of its component functions: :$\\dfrac {\\d \\mathbf r \\left({t}\\right)} {\\d t} = \\left\\langle{D_t r_1 \\left({t}\\right), D_t r_2 \\left({t}\\right), \\ldots, D_t r_n \\left({t}\\right)}\\right\\rangle$"}
+{"_id": "4489", "title": "Derivative of Dot Product of Vector-Valued Functions", "text": "Let: : $\\mathbf r: x \\mapsto \\left\\langle{r_1 \\left({x}\\right), r_2 \\left({x}\\right), \\ldots, r_n \\left({x}\\right)}\\right\\rangle$ : $\\mathbf q: x \\mapsto \\left\\langle{q_1 \\left({x}\\right), q_2 \\left({x}\\right), \\ldots, q_n \\left({x}\\right)}\\right\\rangle$ be differentiable vector-valued functions. The derivative of their dot product is given by: :$\\dfrac \\d {\\d x} \\left({\\mathbf r \\left({x}\\right) \\cdot \\mathbf q \\left({x}\\right)}\\right) = \\mathbf r' \\left({x}\\right) \\cdot \\mathbf q \\left({x}\\right) + \\mathbf r \\left({x}\\right) \\cdot \\mathbf q' \\left({x}\\right)$"}
+{"_id": "4490", "title": "Borel Sigma-Algebra of Subset is Trace Sigma-Algebra", "text": "Let $\\left({X, \\tau}\\right)$ be a topological space, and let $A \\subseteq X$ be a subset of $X$. Let $\\tau_A$ be the subspace topology on $A$. Then the following equality of $\\sigma$-algebras on $A$ holds: :$\\mathcal B \\left({A, \\tau_A}\\right) = \\mathcal B \\left({X, \\tau}\\right)_A$ where $\\mathcal B$ signifies Borel $\\sigma$-algebra, and $\\mathcal B \\left({X, \\tau}\\right)_A$ signifies trace $\\sigma$-algebra."}
+{"_id": "4491", "title": "Derivative of Vector Cross Product of Vector-Valued Functions", "text": "Let: :$\\mathbf r: t \\mapsto \\begin {bmatrix} x \\\\ y \\\\ z \\end{bmatrix}$ :$\\mathbf q: t \\mapsto \\begin {bmatrix} \\chi \\\\ \\gamma \\\\ \\zeta \\end{bmatrix}$ be differentiable vector-valued functions, where: :$x, y, z, \\chi, \\gamma, \\zeta$ are (images of) differentiable real functions. The derivative of the vector cross product of $\\mathbf r$ and $\\mathbf q$ is given by: :$D_t \\left({\\mathbf r \\left({t}\\right) \\times \\mathbf q \\left({t}\\right)}\\right) = \\mathbf r' \\left({x}\\right) \\times \\mathbf q \\left({x}\\right) + \\mathbf r \\left({x}\\right) \\times \\mathbf q'\\left({x}\\right)$"}
+{"_id": "4492", "title": "Existence and Uniqueness of Monotone Class Generated by Collection of Subsets", "text": "Let $X$ be a set. Let $\\mathcal G \\subseteq \\mathcal P \\left({X}\\right)$ be a collection of subsets of $X$. Then $\\mathfrak m \\left({\\mathcal G}\\right)$, the monotone class generated by $\\mathcal G$, exists and is unique."}
+{"_id": "4499", "title": "Derivative of Product of Real Function and Vector-Valued Function", "text": "Let: :$\\mathbf r:x \\mapsto \\mathbf z$ be a differentiable vector-valued function, where: :$\\mathbf{z} = \\begin{bmatrix} z_1 \\\\ z_2 \\\\ \\vdots \\\\ z_n \\end{bmatrix}$ such that: :$z_1, z_2, \\cdots, z_n$ are (images of) differentiable real functions. Let: :$f: x \\mapsto y$ be a differentiable real function. Then: :$D_x \\left({y \\, \\mathbf z}\\right) = \\dfrac {\\d y} {\\d x} \\mathbf z + y \\dfrac {\\d \\mathbf z} {\\d x}$"}
+{"_id": "4504", "title": "Generated Sigma-Algebra by Generated Monotone Class", "text": "Let $X$ be a set, and let $\\mathcal G \\subseteq \\mathcal P \\left({X}\\right)$ be a nonempty collection of subsets of $X$. Suppose that $\\mathcal G$ satisfies the following condition: :$(1):\\quad A \\in \\mathcal G \\implies \\complement_X \\left({A}\\right) \\in \\mathcal G$ that is, $\\mathcal G$ is closed under complement in $X$. Then: :$\\mathfrak m \\left({\\mathcal G}\\right) = \\sigma \\left({\\mathcal G}\\right)$ where $\\mathfrak m$ denotes generated monotone class, and $\\sigma$ denotes generated $\\sigma$-algebra."}
+{"_id": "4507", "title": "Heine-Borel Theorem/Metric Space", "text": "A metric space is compact {{iff}} it is both complete and totally bounded."}
+{"_id": "4511", "title": "Borel Sigma-Algebra on Euclidean Space by Monotone Class", "text": "Let $\\left({\\R^n, \\tau}\\right)$ be the $n$-dimensional Euclidean space. Then: :$\\mathcal B \\left({\\R^n, \\tau}\\right) = \\mathfrak m \\left({\\tau}\\right)$ where $\\mathcal B$ denotes Borel $\\sigma$-algebra, and $\\mathfrak m$ denotes generated monotone class."}
+{"_id": "4512", "title": "Measure is Strongly Additive", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Then $\\mu$ is strongly additive, that is: : $\\forall E, F \\in \\Sigma: \\mu \\left({E \\cap F}\\right) + \\mu \\left({E \\cup F}\\right) = \\mu \\left({E}\\right) + \\mu \\left({F}\\right)$"}
+{"_id": "4513", "title": "Measure is Subadditive", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Then $\\mu$ is subadditive, that is: :$\\forall E, F \\in \\Sigma: \\mu \\left({E \\cup F}\\right) \\le \\mu \\left({E}\\right) + \\mu \\left({F}\\right)$"}
+{"_id": "4514", "title": "Representation of Degree One is Irreducible", "text": "Let $\\left({G, \\cdot}\\right)$ be a finite group. Let $\\rho: G \\to \\operatorname{GL} \\left({V}\\right)$ be a linear representation of $G$ on $V$ of degree $1$.  Then $\\rho$ is an  irreducible linear representation."}
+{"_id": "4515", "title": "Irreducible Representations of Abelian Group", "text": "Let $\\left({G, \\cdot}\\right)$ be a finite abelian group. Let $V$ be a non-null vector space over an algebraically closed field $k$. Let $\\rho: G \\to \\operatorname{GL} \\left({V}\\right)$ be a linear representation. Then $\\rho$ is irreducible iff $\\dim \\left({V}\\right) = 1$, where, $\\dim$ denotes dimension."}
+{"_id": "4516", "title": "Schur's Lemma (Representation Theory)", "text": "Let $\\left({G, \\cdot}\\right)$ be a finite group. Let $V$ and $V'$ be two irreducible $G$-modules. Let $f: V \\to V'$ be a homomorphism of $G$-modules. Then either: :$f \\left({v}\\right) = 0$ for all $v \\in V$ or: : $f$ is an isomorphism."}
+{"_id": "4518", "title": "Kernel is G-Module", "text": "Let $\\struct {G, \\cdot}$ be a group. Let $f: \\struct {V, \\phi} \\to \\struct {V', \\mu}$ be a homomorphism of $G$-modules. Then its kernel $\\map \\ker f$ is a $G$-submodule of $V$."}
+{"_id": "4519", "title": "Image is G-Module", "text": "Let $\\left({G, \\cdot}\\right)$ be a group and let $f: \\left({V, \\phi}\\right) \\to \\left({V', \\mu}\\right)$ be a homomorphism of $G$-modules. Then $\\operatorname{Im} \\left({f}\\right)$ is a $G$-submodule of $V'$."}
+{"_id": "4520", "title": "Set Intersection Preserves Subsets/Families of Sets", "text": "Let $I$ be an indexing set. Let $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$ and $\\family {B_\\alpha}_{\\alpha \\mathop \\in I}$ be indexed families of subsets of a set $S$. Let: :$\\forall \\beta \\in I: A_\\beta \\subseteq B_\\beta$ Then: :$\\displaystyle \\bigcap_{\\alpha \\mathop \\in I} A_\\alpha \\subseteq \\bigcap_{\\alpha \\mathop \\in I} B_\\alpha$"}
+{"_id": "4521", "title": "Set Intersection Preserves Subsets", "text": "Let $A, B, S, T$ be sets. Then: :$A \\subseteq B, \\ S \\subseteq T \\implies A \\cap S \\subseteq B \\cap T$"}
+{"_id": "4522", "title": "Null Space Contains Zero Vector", "text": "Let: :$\\map {\\mathrm N} {\\mathbf A} = \\set {\\mathbf x \\in \\R^n: \\mathbf A \\mathbf x = \\mathbf 0}$ be the null space of $\\mathbf A$, where: :$ \\mathbf A_{m \\times n} = \\begin {bmatrix} a_{1 1} & a_{1 2} & \\cdots & a_{1 n} \\\\ a_{2 1} & a_{2 2} & \\cdots & a_{2 n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ a_{m 1} & a_{m 2} & \\cdots & a_{m n} \\\\ \\end{bmatrix}$  is a matrix in the matrix space $\\map {\\MM_\\R} {m, n}$. Then the null space of $\\mathbf A$ contains the zero vector: :$\\mathbf 0 \\in \\map {\\mathrm N} {\\mathbf A}$ where: :$\\mathbf 0 = \\mathbf 0_{m \\times 1} = \\begin {bmatrix} 0 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end {bmatrix}$"}
+{"_id": "4524", "title": "Commutative Linear Transformation is G-Module Homomorphism", "text": "Let $\\rho: G \\to \\operatorname{GL} \\left({V}\\right)$ be a representation. Let $f: V \\to V$ be a linear mapping. Let: :$\\forall g \\in G: \\rho \\left({g}\\right) \\circ f = f \\circ \\rho \\left({g}\\right)$ Then $f: V \\to V$ is a $G$-module homomorphism. {{explain|between which $G$-modules? (probably $\\left({G, \\rho}\\right) \\to \\left({G, \\rho}\\right)$, which makes it rather tautologous)}}"}
+{"_id": "4525", "title": "Characterization of Measures", "text": "Let $\\struct {X, \\Sigma}$ be a measurable space. Denote $\\overline \\R_{\\ge 0}$ for the set of positive extended real numbers. A mapping $\\mu: \\Sigma \\to \\overline \\R_{\\ge 0}$ is a measure {{iff}}: :$(1):\\quad \\map \\mu \\O = 0$ :$(2):\\quad \\mu$ is finitely additive :$(3):\\quad$ For every increasing sequence $\\sequence {E_n}_{n \\mathop \\in \\N}$ in $\\Sigma$, if $E_n \\uparrow E$, then: ::::$\\map \\mu E = \\ds \\lim_{n \\mathop \\to \\infty} \\map \\mu {E_n}$ where $E_n \\uparrow E$ denotes limit of increasing sequence of sets. Alternatively, and equivalently, $(3)$ may be replaced by either of: :$(3'):\\quad$ For every decreasing sequence $\\sequence {E_n}_{n \\mathop \\in \\N}$ in $\\Sigma$ for which $\\map \\mu {E_1}$ is finite, if $E_n \\downarrow E$, then: ::::$\\map \\mu E = \\ds \\lim_{n \\mathop \\to \\infty} \\map \\mu {E_n}$ :$(3''):\\quad$ For every decreasing sequence $\\sequence {E_n}_{n \\mathop \\in \\N}$ in $\\Sigma$ for which $\\map \\mu {E_1}$ is finite, if $E_n \\downarrow \\O$, then: ::::$\\ds \\lim_{n \\mathop \\to \\infty} \\map \\mu {E_n} = 0$ where $E_n \\downarrow E$ denotes limit of decreasing sequence of sets."}
+{"_id": "4526", "title": "Measure is Countably Subadditive", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Then $\\mu$ is a countably subadditive function."}
+{"_id": "4527", "title": "Dirac Measure is Measure", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $x \\in X$, and let $\\delta_x$ be the Dirac measure at $x$. Then $\\delta_x$ is a measure."}
+{"_id": "4528", "title": "Dirac Measure is Probability Measure", "text": "Let $\\left({X, \\mathcal A}\\right)$ be a measurable space. Let $x \\in X$, and let $\\delta_x$ be the Dirac measure at $x$. Then $\\delta_x$ is a probability measure."}
+{"_id": "4529", "title": "Definition:Co-Countable Measure", "text": "Let $X$ be an uncountable set. Let $\\Sigma$ be the $\\sigma$-algebra of countable sets on $X$. Then the '''co-countable measure (on $X$)''' is the measure defined by: :$\\mu: \\Sigma \\to \\overline{\\R}, \\ \\mu \\left({E}\\right) := \\begin{cases} 0 & : \\text{if $E$ is countable}\\\\ 1 & : \\text{if $E$ is co-countable}\\end{cases}$ where: : $\\overline{\\R}$ denotes the extended real numbers : $E$ is co-countable iff $X \\setminus E$ is countable."}
+{"_id": "4530", "title": "Co-Countable Measure is Measure", "text": "Let $X$ be an uncountable set. Let $\\Sigma$ be the $\\sigma$-algebra of countable sets on $X$. Then the co-countable measure $\\mu$ on $X$ is a measure."}
+{"_id": "4532", "title": "Counting Measure is Measure", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Then the counting measure $\\left\\vert{\\cdot}\\right\\vert$ on $\\left({X, \\Sigma}\\right)$ is a measure."}
+{"_id": "4533", "title": "Trivial Vector Space iff Zero Dimension", "text": "Let $V$ be a vector space. Then $V = \\left\\{{\\mathbf 0}\\right\\}$ iff $\\dim \\left({V}\\right) = 0$, where $\\dim$ signifies dimension."}
+{"_id": "4534", "title": "G-Submodule Test", "text": "Let $\\left({V, \\phi}\\right)$ be a $G$-module over a field $k$. Let $W$ be a vector subspace of $V$. Then $\\left({W, \\phi_W}\\right)$, where $\\phi_W: G \\times W \\to W$ is the restriction of $\\phi$ to $G \\times W$, is a $G$-submodule of $V$ iff $\\phi \\left({G, W}\\right) \\subseteq W$."}
+{"_id": "4535", "title": "Composite of Continuous Mappings is Continuous/Point", "text": "Let $T_1, T_2, T_3$ be topological spaces. Let the mapping $f : T_1 \\to T_2$ be continuous at $x$. Let the mapping $g : T_2 \\to T_3$ be continuous at $x$ at $\\map f x$. Then the composite mapping $g \\circ f : T_1 \\to T_3$ is continuous at $x$ at $x$."}
+{"_id": "4537", "title": "Null Space Closed under Vector Addition", "text": "Let: :$\\map {\\mathrm N} {\\mathbf A} = \\set {\\mathbf x \\in \\R^n : \\mathbf A \\mathbf x = \\mathbf 0}$ be the null space of $\\mathbf A$, where: :$\\mathbf A_{m \\times n} = \\begin {bmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} \\\\ \\end{bmatrix}$, $\\mathbf x_{n \\times 1} = \\begin{bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{bmatrix}$ and $\\mathbf 0_{m \\times 1} = \\begin {bmatrix} 0 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end {bmatrix}$  are matrices :the column matrix $\\mathbf x_{n \\times 1}$ is interpreted as a vector in $\\R^n$. Then $\\map {\\mathrm N} {\\mathbf A}$ is closed under vector addition: :$\\forall \\mathbf v, \\mathbf w \\in \\map {\\mathrm N} {\\mathbf A}: \\mathbf v + \\mathbf w \\in \\map {\\mathrm N} {\\mathbf A}$"}
+{"_id": "4538", "title": "Null Space Closed under Scalar Multiplication", "text": "Let: :$\\map {\\mathrm N} {\\mathbf A} = \\set {\\mathbf x \\in \\R^n : \\mathbf {A x} = \\mathbf 0}$ be the null space of $\\mathbf A$, where: :$ \\mathbf A_{m \\times n} = \\begin{bmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} \\\\ \\end {bmatrix}$,  $\\mathbf x_{n \\times 1} = \\begin {bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end {bmatrix}$, $\\mathbf 0_{m \\times 1} = \\begin {bmatrix} 0 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end {bmatrix}$ are matrices where each column is an element of a real vector space. Then $\\map {\\mathrm N} {\\mathbf A}$ is closed under scalar multiplication: :$\\forall \\mathbf v \\in \\map {\\mathrm N} {\\mathbf A} ,\\forall \\lambda \\in \\R: \\lambda \\mathbf v \\in \\map {\\mathrm N} {\\mathbf A}$"}
+{"_id": "4539", "title": "Null Space is Subspace", "text": "Let: :$\\operatorname{N} \\left({\\mathbf A}\\right) = \\left\\{{\\mathbf x \\in \\R^n: \\mathbf {Ax} = \\mathbf 0}\\right\\}$ be the null space of $\\mathbf A$, where: :$ \\mathbf A_{m \\times n} = \\begin{bmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} \\\\ \\end{bmatrix}$,  $\\mathbf x_{n \\times 1} = \\begin{bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{bmatrix}$, $\\mathbf 0_{m \\times 1} = \\begin{bmatrix} 0 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{bmatrix}$ are matrices. Then $\\operatorname{N} \\left({\\mathbf A}\\right)$ is a linear subspace of $\\R^n$."}
+{"_id": "4540", "title": "Norm is Continuous", "text": "Let $\\struct {V, \\norm {\\,\\cdot\\,} }$ be a normed vector space. Then the mapping $x \\mapsto \\norm x$ is continuous. Here, the metric used is the metric $d$ induced by $\\norm {\\,\\cdot\\,}$."}
+{"_id": "4542", "title": "P-adic Valuation is Valuation", "text": "The $p$-adic valuation $\\nu_p: \\Q \\to \\Z \\cup \\left\\{{+\\infty}\\right\\}$ is a valuation on $\\Q$."}
+{"_id": "4543", "title": "P-adic Norm not Complete on Rational Numbers", "text": "Let $\\norm {\\,\\cdot\\,}_p$ be the $p$-adic norm on the rationals $\\Q$ for some prime $p$. Then: :the valued field $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$ is not complete. That is, there exists a Cauchy sequence in $\\struct {\\Q, \\norm{\\,\\cdot\\,}_p}$ which does not converge to a limit in $\\Q$."}
+{"_id": "4544", "title": "P-adic Valuation on Integers is Valuation", "text": "Let $\\nu_p^\\Z: \\Z \\to \\Z \\cup \\set {+\\infty}$ be the $p$-adic valuation restricted to the integers. Then $\\nu_p^\\Z$ is a valuation."}
+{"_id": "4546", "title": "Infinite Measure is Measure", "text": "Let $\\struct {X, \\Sigma}$ be a measurable space. Then the infinite measure $\\mu$ on $\\struct {X, \\Sigma}$ is a measure."}
+{"_id": "4547", "title": "Linear Combination of Measures", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $\\mu, \\nu$ be measures on $\\left({X, \\Sigma}\\right)$. Then for all positive real numbers $a, b \\in \\R_{\\ge 0}$, the pointwise sum: :$a \\mu + b \\nu: \\Sigma \\to \\overline \\R, \\ \\left({a \\mu + b \\nu}\\right) \\left({E}\\right) := a \\mu \\left({E}\\right) + b \\nu \\left({E}\\right)$ is also a measure on $\\left({X, \\Sigma}\\right)$."}
+{"_id": "4548", "title": "Series of Measures is Measure", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $\\left({\\mu_n}\\right)_{n \\mathop \\in \\N}$ be a sequence of measures on $\\left({X, \\Sigma}\\right)$. Let $\\left({a_n}\\right)_{n \\mathop \\in \\N} \\subseteq \\R_{\\ge 0}$ be a sequence of positive real numbers. Then the series of measures $\\mu: \\Sigma \\to \\overline{\\R}$, defined by: :$\\displaystyle \\mu \\left({E}\\right) := \\sum_{n \\mathop \\in \\N} a_n \\mu_n \\left({E}\\right)$ is also a measure on $\\left({X, \\Sigma}\\right)$."}
+{"_id": "4550", "title": "Quotient Set Determined by Relation Induced by Partition is That Partition", "text": "Let $S$ be a set. Let $\\mathcal P$ be a partition of $S$. Let $\\mathcal R$ be the relation induced by $\\mathcal P$. Then the quotient set $S / \\mathcal R$ of $S$ is $\\mathcal P$ itself."}
+{"_id": "4551", "title": "Null Sets Closed under Subset", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $N \\in \\Sigma$ be a $\\mu$-null set, and let $M \\in \\Sigma$ be a subset of $N$. Then $M$ is also a $\\mu$-null set."}
+{"_id": "4552", "title": "Null Sets Closed under Countable Union", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $\\left({N_n}\\right)_{n \\in \\N}$ be a sequence of $\\mu$-null sets. Then $N := \\displaystyle \\bigcup_{n \\mathop \\in \\N} N_n$ is also a $\\mu$-null set."}
+{"_id": "4553", "title": "Absolute Value induces Equivalence Compatible with Integer Multiplication", "text": "Let $\\Z$ be the set of integers. Let $\\RR$ be the relation on $\\Z$ defined as: :$\\forall x, y \\in \\Z: \\struct {x, y} \\in \\RR \\iff \\size x = \\size y$ where $\\size x$ denotes the absolute value of $x$. Then $\\RR$ is a congruence relation for integer multiplication."}
+{"_id": "4554", "title": "Completion Theorem (Measure Spaces)", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Then there exists a completion $\\left({X, \\Sigma^*, \\bar \\mu}\\right)$ of $\\left({X, \\Sigma, \\mu}\\right)$."}
+{"_id": "4555", "title": "Restricted Measure is Measure", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $\\Sigma'$ be a sub-$\\sigma$-algebra of $\\Sigma$. Then the restricted measure $\\mu \\restriction_{\\Sigma'}$ is a measure on the measurable space $\\left({X, \\Sigma'}\\right)$."}
+{"_id": "4556", "title": "Restricting Measure Preserves Finiteness", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $\\mu$ be a finite measure. Let $\\Sigma'$ be a sub-$\\sigma$-algebra of $\\Sigma$. Then the restricted measure $\\mu \\restriction_{\\Sigma'}$ is also a finite measure."}
+{"_id": "4560", "title": "Central Subgroup is Normal", "text": "Let $G$ be a group. Let $H$ be a central subgroup of $G$. Then $H$ is a normal subgroup of $G$."}
+{"_id": "4562", "title": "Angle Between Non-Zero Vectors Always Defined", "text": "The angle between two non-zero vectors in $\\R^n$ is always defined."}
+{"_id": "4563", "title": "Dynkin System Contains Empty Set", "text": "Let $X$ be a set, and let $\\mathcal D$ be a Dynkin system on $X$. Then the empty set $\\varnothing$ is an element of $\\mathcal D$."}
+{"_id": "4564", "title": "Dynkin System Closed under Disjoint Union", "text": "Let $X$ be a set, and let $\\mathcal D$ be a Dynkin system on $X$. Let $D, E \\in \\mathcal D$ be disjoint. Then the union $D \\cup E$ is also an element of $\\mathcal D$."}
+{"_id": "4565", "title": "Sigma-Algebra is Dynkin System", "text": "Let $X$ be a set, and let $\\Sigma$ be a $\\sigma$-algebra on $X$. Then $\\Sigma$ is a Dynkin system on $X$."}
+{"_id": "4566", "title": "Existence and Uniqueness of Dynkin System Generated by Collection of Subsets", "text": "Let $X$ be a set. Let $\\mathcal G \\subseteq \\mathcal P \\left({X}\\right)$ be a collection of subsets of $X$. Then $\\delta \\left({\\mathcal G}\\right)$, the Dynkin system generated by $\\mathcal G$, exists and is unique."}
+{"_id": "4567", "title": "Generated Sigma-Algebra Contains Generated Dynkin System", "text": "Let $X$ be a set. Let $\\mathcal G \\subseteq \\mathcal P \\left({X}\\right)$ be a collection of subsets of $X$. Then $\\delta \\left({\\mathcal G}\\right) \\subseteq \\sigma \\left({\\mathcal G}\\right)$. Here $\\delta$ denotes generated Dynkin system, and $\\sigma$ denotes generated $\\sigma$-algebra."}
+{"_id": "4568", "title": "Dynkin System Closed under Intersections is Sigma-Algebra", "text": "Let $X$ be a set, and let $\\DD$ be a Dynkin system on $X$. Suppose that $\\DD$ satisfies the following condition: :$(1):\\quad \\forall D, E \\in \\DD: D \\cap E \\in \\DD$ That is, $\\DD$ is closed under intersection. Then $\\DD$ is a $\\sigma$-algebra."}
+{"_id": "4569", "title": "Dynkin System with Generator Closed under Intersection is Sigma-Algebra", "text": "Let $X$ be a set. Let $\\GG \\subseteq \\powerset X$ be a collection of subsets of $X$. Suppose that $\\GG$ satisfies the following condition: :$(1):\\quad \\forall G, H \\in \\GG: G \\cap H \\in \\GG$ That is, $\\GG$ is closed under intersection. Then: :$\\map \\delta \\GG = \\map \\sigma \\GG$ where $\\delta$ denotes generated Dynkin system, and $\\sigma$ denotes generated $\\sigma$-algebra."}
+{"_id": "4570", "title": "Carathéodory's Theorem (Measure Theory)", "text": "Let $X$ be a set. Let $\\SS \\subseteq \\powerset X$ be a semi-ring of subsets of $X$. Let $\\mu: \\SS \\to \\overline \\R$ be a pre-measure on $\\SS$. Let $\\map \\sigma \\SS$ be the $\\sigma$-algebra generated by $\\SS$. Then $\\mu$ extends to a measure $\\mu^*$ on $\\map \\sigma \\SS$. {{Refactor|extract corollary}}"}
+{"_id": "4574", "title": "Congruence Modulo Normal Subgroup is Congruence Relation", "text": "Let $\\struct {G, \\circ}$ be a group. Let $N$ be a normal subgroup of $G$. Then congruence modulo $N$ is a congruence relation for the group operation $\\circ$."}
+{"_id": "4575", "title": "Intersection Measure is Measure", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $F \\in \\Sigma$ be a measurable set. Then the intersection measure $\\mu_F$ is a measure on the measurable space $\\left({X, \\Sigma}\\right)$."}
+{"_id": "4576", "title": "Uniqueness of Measures/Proof 1", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $\\mathcal G \\subseteq \\mathcal P \\left({X}\\right)$ be a generator for $\\Sigma$; i.e., $\\Sigma = \\sigma \\left({\\mathcal G}\\right)$. Suppose that $\\mathcal G$ satisfies the following conditions: :$(1):\\quad \\forall G, H \\in \\mathcal G: G \\cap H \\in \\mathcal G$ :$(2):\\quad$ There exists an exhausting sequence $\\left({G_n}\\right)_{n \\in \\N} \\uparrow X$ in $\\mathcal G$ Let $\\mu, \\nu$ be measures on $\\left({X, \\Sigma}\\right)$, and suppose that: :$(3):\\quad \\forall G \\in \\mathcal G: \\mu \\left({G}\\right) = \\nu \\left({G}\\right)$ :$(4):\\quad \\forall n \\in \\N: \\mu \\left({G_n}\\right)$ is finite Then $\\mu = \\nu$. Alternatively, by Countable Cover induces Exhausting Sequence, the exhausting sequence in $(2)$ may be replaced by a countable $\\mathcal G$-cover $\\left({G_n}\\right)_{n \\in \\N}$, still subject to $(4)$."}
+{"_id": "4577", "title": "Lebesgue Measure Invariant under Translations", "text": "Let $\\lambda^n$ be the $n$-dimensional Lebesgue measure on $\\R^n$ equipped with the Borel $\\sigma$-algebra $\\mathcal B \\left({\\R^n}\\right)$. Let $\\mathbf x \\in \\R^n$. Then $\\lambda^n$ is translation-invariant; i.e., for all $B \\in \\mathcal B \\left({\\R^n}\\right)$, have: :$\\lambda^n \\left({\\mathbf x + B}\\right) = \\lambda^n \\left({B}\\right)$ where $\\mathbf x + B$ is the set $\\left\\{{\\mathbf x + \\mathbf b: \\mathbf b \\in B}\\right\\}$."}
+{"_id": "4578", "title": "Translation-Invariant Measure on Euclidean Space is Multiple of Lebesgue Measure", "text": "Let $\\mu$ be a measure on $\\R^n$ equipped with the Borel $\\sigma$-algebra $\\mathcal B \\left({\\R^n}\\right)$. Suppose that $\\mu$ is translation-invariant. Also, suppose that $\\kappa := \\mu \\left({\\left[{0 \\,.\\,.\\, 1}\\right)^n }\\right) < +\\infty$. Then $\\mu = \\kappa \\lambda^n$, where $\\lambda^n$ is the $n$-dimensional Lebesgue measure."}
+{"_id": "4579", "title": "Dynkin System Closed under Set Difference with Subset", "text": "Let $X$ be a set, and let $\\mathcal D$ be a Dynkin system on $X$. Let $D, E \\in \\mathcal D$ and suppose that $E \\subseteq D$. Then the set difference $D \\setminus E$ is also an element of $\\mathcal D$."}
+{"_id": "4585", "title": "Coset Product of Normal Subgroup is Consistent with Subset Product Definition", "text": "Let $\\struct {G, \\circ}$ be a group. Let $N$ be a normal subgroup of $G$. Let $a, b \\in G$. Let $a \\circ N$ and $b \\circ N$ be the left cosets of $a$ and $b$ by $N$. Then the coset product: :$\\paren {a \\circ N} \\circ \\paren {b \\circ N} = \\paren {a \\circ b} \\circ N$ is consistent with the definition of the coset product as the subset product of $a \\circ N$ and $b \\circ  N$: :$\\paren {a \\circ N} \\paren {b \\circ N} = \\set {x \\circ y: x \\in a \\circ N, y \\in b \\circ N}$"}
+{"_id": "4587", "title": "Congruence Relation iff Compatible with Operation", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\RR$ be an equivalence relation on $S$. Then $\\RR$ is a congruence relation for $\\circ$ {{iff}}: {{begin-eqn}} {{eqn | ll= \\forall x, y, z \\in S:       | l = x \\mathrel \\RR y       | o = \\implies       | r = \\paren {x \\circ z} \\mathrel \\RR \\paren {y \\circ z} }} {{eqn | l = x \\mathrel \\RR y       | o = \\implies       | r = \\paren {z \\circ x} \\mathrel \\RR \\paren {z \\circ y} }} {{end-eqn}} That is, {{iff}} $\\RR$ is compatible with $\\circ$."}
+{"_id": "4589", "title": "Outer Measure of Limit of Increasing Sequence of Sets", "text": "Let $\\mu^*$ be an outer measure on a set $X$. Let $\\left\\langle{S_n}\\right\\rangle$ be an increasing sequence of $\\mu^*$-measurable sets, and let $S_n \\uparrow S$ (as $n \\to \\infty$). Then for any subset $A \\subseteq X$: : $\\displaystyle \\mu^* \\left({A \\cap S}\\right) = \\lim_{n \\to \\infty} \\mu^* \\left({A \\cap S_n}\\right)$"}
+{"_id": "4590", "title": "Half-Open Rectangles form Semiring of Sets", "text": "The half-open $n$-rectangles $\\JJ_{ho}^n$ form a semiring of sets."}
+{"_id": "4591", "title": "Lebesgue Pre-Measure is Pre-Measure", "text": "The Lebesgue pre-measure $\\lambda^n$ on the half-open $n$-rectangles $\\mathcal{J}_{ho}^n$ is a pre-measure."}
+{"_id": "4592", "title": "Existence and Uniqueness of Lebesgue Measure", "text": "Let $\\lambda^n$ be the Lebesgue pre-measure on the half-open $n$-rectangles $\\mathcal{J}_{ho}^n$. Then Lebesgue measure, the extension of $\\lambda^n$ to the Borel $\\sigma$-algebra $\\mathcal B \\left({\\R^n}\\right)$, exists and is unique."}
+{"_id": "4593", "title": "Lebesgue Measure is Diffuse", "text": "Let $\\lambda^n$ be Lebesgue measure on $\\R^n$. Then $\\lambda^n$ is a diffuse measure."}
+{"_id": "4595", "title": "Half-Open Rectangles Closed under Intersection", "text": "Let $\\lefthalf-open $n$-rectangles. Then $\\lefthalf-open $n$-rectangle."}
+{"_id": "4597", "title": "Equivalence of Definitions of Null Set in Euclidean Space", "text": "Let $\\lambda^n$ be $n$-dimensional Lebesgue measure on $\\R^n$. Let $E \\subseteq \\R^n$. Then the following are equivalent: :$(1):\\quad \\exists B \\in \\mathcal B \\left({\\R^n}\\right): E \\subseteq B, \\lambda^n \\left({B}\\right) = 0$ :$(2):\\quad$ For every $\\epsilon > 0$, there exists a countable cover $\\left({J_i}\\right)_{i \\mathop \\in \\N}$ of $E$ by open $n$-rectangles, such that: ::$\\displaystyle \\sum_{i \\mathop = 1}^\\infty \\operatorname{vol} \\left({J_i}\\right) \\le \\epsilon$ {{explain|link to some definition (might not exist) for $\\operatorname{vol}$}}"}
+{"_id": "4606", "title": "Homomorphism to Group Preserves Inverses", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\struct {T, *}$ be a group. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be a homomorphism. Let $\\struct {S, \\circ}$ have an identity $e_S$. Let $x^{-1}$ be an inverse of $x$ for $\\circ$. Then $\\map \\phi {x^{-1} }$ is an inverse of $\\map \\phi x$ for $*$."}
+{"_id": "4607", "title": "Equivalence Induced by Epimorphism is Congruence Relation", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be algebraic structures. Let $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ be an epimorphism. Let $\\mathcal R_\\phi$ be the equivalence induced by $\\phi$. Then the induced equivalence $\\mathcal R_\\phi$ is a congruence relation for $\\circ$."}
+{"_id": "4608", "title": "Unique Isomorphism from Quotient Mapping to Epimorphism Domain", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be algebraic structures. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be an epimorphism. Let $\\mathcal R_\\phi$ be the equivalence induced by $\\phi$. Let $S / \\mathcal R_\\phi$ be the quotient of $S$ by $\\mathcal R_\\phi$. Let $q_{\\mathcal R_\\phi}: S \\to S / \\mathcal R_\\phi$ be the quotient mapping induced by $\\mathcal R_\\phi$. Let $\\struct {S / \\mathcal R_\\phi}, {\\circ_{\\mathcal R_\\phi} }$ be the quotient structure defined by $\\mathcal R_\\phi$. Then there is one and only one isomorphism: :$\\psi: \\struct {S / \\mathcal R_\\phi}, {\\circ_{\\mathcal R_\\phi} } \\to \\struct {T, *}$ which satisfies: :$\\psi \\bullet q_{\\mathcal R_\\phi} = \\phi$ where, in order not to cause notational confusion, $\\bullet$ is used as the symbol to denote composition of mappings."}
+{"_id": "4609", "title": "Identity is in Kernel of Group Homomorphism", "text": "Let $G$ and $H$ be groups. Let $e_G$ and $e_H$ be the identity elements of $G$ and $H$ respectively. Let $\\phi: G \\to H$ be a (group) homomorphism from $G$ to $H$. Then: :$e_G \\in \\map \\ker \\phi$ where $\\map \\ker \\phi$ is the kernel of $\\phi$."}
+{"_id": "4613", "title": "Union is Dominated by Disjoint Union", "text": "Let $I$ be an indexing set. For all $i \\in I$, let $S_i$ be a set. Then: :$\\displaystyle \\bigcup_{i \\mathop \\in I} S_i \\preccurlyeq \\bigsqcup_{i \\mathop \\in I} S_i$ where $\\preccurlyeq$ denotes domination, $\\bigcup$ denotes union, and $\\bigsqcup$ denotes disjoint union."}
+{"_id": "4614", "title": "Disjoint Union Preserves Domination", "text": "Let $I$ be an indexing set. For all $i \\in I$, let $A_i$ and $B_i$ be sets such that $A_i \\preccurlyeq B_i$. Here, $\\preccurlyeq$ denotes domination. Then: :$\\displaystyle \\bigsqcup_{i \\mathop \\in I} A_i \\preccurlyeq \\bigsqcup_{i \\mathop \\in I} B_i$ where $\\bigsqcup$ denotes disjoint union."}
+{"_id": "4615", "title": "Epimorphism from Real Numbers to Circle Group", "text": "Let $\\struct {K, \\times}$ be the circle group, that is: :$K = \\set {z \\in \\C: \\cmod z = 1}$ under complex multiplication. Let $f: \\R \\to K$ be the mapping from the real numbers to $K$ defined as: :$\\forall x \\in \\R: \\map f x = \\cos x + i \\sin x$ Then $f: \\struct {\\R, +} \\to \\struct {K, \\times}$ is a group epimorphism. Its kernel is: :$\\map \\ker f = \\set {2 \\pi n: n \\in \\Z}$"}
+{"_id": "4617", "title": "Increasing Sequence of Sets induces Partition on Limit", "text": "Let $\\sequence {S_n}_{n \\mathop \\in \\N} \\uparrow S$ be an increasing sequence of sets with limit $S$. Define $T_1 = S_1$, and, for $n \\in \\N$, $T_{n + 1} = S_{n + 1} \\setminus S_n$, where $\\setminus$ denotes set difference. Then $\\sequence {T_n}_{n \\mathop \\in \\N}$ is a countable partition of $S$."}
+{"_id": "4618", "title": "Power Function on Complex Numbers is Epimorphism", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive integer. Let $\\struct {\\C_{\\ne 0}, \\times}$ be the multiplicative group of complex numbers. Let $f_n: \\C_{\\ne 0} \\to \\C_{\\ne 0}$ be the mapping from the set of complex numbers less zero to itself defined as: :$\\forall z \\in \\C_{\\ne 0}: \\map {f_n} z = z^n$ Then $f_n: \\struct {\\C_{\\ne 0}, \\times} \\to \\struct {\\C_{\\ne 0}, \\times}$ is a group epimorphism. The kernel of $f_n$ is the set of complex $n$th roots of unity."}
+{"_id": "4619", "title": "Real Part as Mapping is Endomorphism for Complex Addition", "text": "Let $\\struct {\\C, +}$ be the additive group of complex numbers. Let $\\struct {\\R, +}$ be the additive group of real numbers. Let $f: \\C \\to \\R$ be the mapping from the complex numbers to the real numbers defined as: :$\\forall z \\in \\C: \\map f z = \\map \\Re z$ where $\\map \\Re z$ denotes the real part of $z$. Then $f: \\struct {\\C, +} \\to \\struct {\\R, +}$ is a group epimorphism. Its kernel is the set: :$\\map \\ker f = \\set {i x: x \\in \\R}$ of wholly imaginary numbers."}
+{"_id": "4620", "title": "Equivalence of Definitions of Semiring of Sets", "text": "{{TFAE|def = Semiring of Sets}} A collection $\\SS$ of subsets of a set $X$ is a semiring (of sets) {{iff}}: :$(1):\\quad \\O \\in \\SS$ :$(2):\\quad A, B \\in \\SS \\implies A \\cap B \\in \\SS$; that is, $\\SS$ is $\\cap$-stable :$(3):\\quad$ If $A, A_1 \\in \\SS$ such that $A_1 \\subseteq A$, then there exists a finite sequence $A_2, A_3, \\ldots, A_n \\in \\SS$ such that: ::$(3a):\\quad \\displaystyle A = \\bigcup_{k \\mathop = 1}^n A_k$ ::$(3b):\\quad$ The $A_k$ are pairwise disjoint We prove that criterion $(3)$ can be replaced by: :$(3'):\\quad$ If $A, B \\in \\SS$, then there exist finite sequence of pairwise disjoint sets $A_1, A_2, \\ldots, A_n \\in \\SS$ such that $\\displaystyle A \\setminus B = \\bigcup_{k \\mathop = 1}^n A_k$."}
+{"_id": "4621", "title": "Imaginary Part as Mapping is Endomorphism for Complex Addition", "text": "Let $\\struct {\\C, +}$ be the additive group of complex numbers. Let $\\struct {\\R, +}$ be the additive group of real numbers. Let $f: \\C \\to \\R$ be the mapping from the complex numbers to the real numbers defined as: :$\\forall z \\in \\C: \\map f z = \\map \\Im z$ where $\\map \\Im z$ denotes the imaginary part of $z$. Then $f: \\struct {\\C, +} \\to \\struct {\\R, +}$ is a group epimorphism. Its kernel is the set: :$\\map \\ker f = \\R$ of (wholly) real numbers."}
+{"_id": "4622", "title": "Reduced Echelon Matrix is Unique", "text": "Every $m \\times n$ matrix is row equivalent to exactly one $m \\times n$ reduced echelon matrix. That is, the reduced echelon form of a matrix is unique."}
+{"_id": "4623", "title": "Measure of Set Difference with Subset", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $S, T \\in \\Sigma$ be such that $S \\subseteq T$, and suppose that $\\mu \\paren S < +\\infty$. Then: :$\\mu \\paren {T \\setminus S} = \\mu \\paren T - \\mu \\paren S$ where $T \\setminus S$ denotes set difference."}
+{"_id": "4624", "title": "Subset of Codomain is Superset of Image of Preimage", "text": "Let $f: S \\to T$ be a mapping. Then: :$B \\subseteq T \\implies \\paren {f \\circ f^{-1} } \\sqbrk B \\subseteq B$ where: :$f \\sqbrk B$ denotes the image of $B$ under $f$ :$f^{-1}$ denotes the inverse of $f$ :$f \\circ f^{-1}$ denotes composition of $f$ and $f^{-1}$. This can be expressed in the language and notation of direct image mappings and inverse image mappings as: :$\\forall B \\in \\powerset T: \\map {\\paren {f^\\to \\circ f^\\gets} } B \\subseteq B$"}
+{"_id": "4625", "title": "Induced Outer Measure Restricted to Semiring is Pre-Measure", "text": "Let $\\mathcal S$ be a semiring over a set $X$. Let $\\mu: \\mathcal S \\to \\overline \\R_{\\ge 0}$ be a pre-measure on $\\mathcal S$, where $\\overline \\R_{\\ge 0}$ denotes the set of positive extended real numbers. Let $\\mu^*: \\mathcal P \\left({X}\\right) \\to \\overline \\R_{\\ge 0}$ be the outer measure induced by $\\mu$. Then: :$\\displaystyle \\mu^*\\restriction_{\\mathcal S} \\, = \\mu$ where $\\restriction$ denotes restriction."}
+{"_id": "4626", "title": "Ordering on Extended Real Numbers is Ordering", "text": "Denote with $\\le$ the usual ordering on the extended real numbers $\\overline \\R$. Then $\\le$ is an ordering, and so $\\overline \\R$ is an ordered set."}
+{"_id": "4627", "title": "Ordering on Extended Real Numbers is Total Ordering", "text": "Let $\\le$ denote the ordering on the extended real numbers $\\overline \\R$. Then $\\le$ is a total ordering, and so $\\overline \\R$ is a toset."}
+{"_id": "4628", "title": "Extended Real Number Space is Compact", "text": "The extended real number space is compact."}
+{"_id": "4629", "title": "Euclidean Space is Subspace of Extended Real Number Space", "text": "Let $\\struct {\\overline \\R, \\tau}$ be the extended real number space. Then $\\tau {\\restriction_\\R}$, the subspace topology on $\\R$, is the Euclidean topology. That is, Euclidean $1$-space is a subspace of the extended real number space."}
+{"_id": "4631", "title": "Lagrange's Formula", "text": "Let: :$\\mathbf a = \\begin{bmatrix} a_x \\\\ a_y \\\\ a_z \\end{bmatrix}$, $\\mathbf b = \\begin{bmatrix} b_x \\\\ b_y \\\\ b_z \\end{bmatrix}$, $\\mathbf c = \\begin{bmatrix} c_x \\\\ c_y \\\\ c_z \\end{bmatrix}$ be vectors in a vector space of $3$ dimensions. Then: :$\\mathbf a \\times \\paren {\\mathbf b \\times \\mathbf c} = \\paren {\\mathbf a \\cdot \\mathbf c} \\mathbf b - \\paren {\\mathbf a \\cdot \\mathbf b} \\mathbf c$"}
+{"_id": "4633", "title": "External Direct Product of Groups is Group/Finite Product", "text": "The external direct product of a finite sequence of groups is itself a group."}
+{"_id": "4634", "title": "External Direct Product Inverses/General Result", "text": "Let $\\displaystyle \\left({S, \\circ}\\right) = \\prod_{k \\mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\\left({S_1, \\circ_1}\\right), \\left({S_2, \\circ_2}\\right), \\ldots, \\left({S_n, \\circ_n}\\right)$. Let $\\left({x_1, x_2, \\ldots, x_n}\\right) \\in S$. Let $y_k$ be an inverse of $x_k$ in $\\left({S_k, \\circ_k}\\right)$ for each of $k \\in \\N^*_n$. Then $\\left({y_1, y_2, \\ldots, y_n}\\right)$ is the inverse of $\\left({x_1, x_2, \\ldots, x_n}\\right) \\in S$ in $\\left({S, \\circ}\\right)$."}
+{"_id": "4635", "title": "External Direct Product Identity/General Result", "text": "Let $\\displaystyle \\left({S, \\circ}\\right) = \\prod_{k \\mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\\left({S_1, \\circ_1}\\right), \\left({S_2, \\circ_2}\\right), \\ldots, \\left({S_n, \\circ_n}\\right)$. Let $e_1, e_2, \\ldots, e_n$ be the identity elements of $\\left({S_1, \\circ_1}\\right), \\left({S_2, \\circ_2}\\right), \\ldots, \\left({S_n, \\circ_n}\\right)$ respectively. Then $\\left({e_1, e_2, \\ldots, e_n}\\right)$ is the identity element of $\\left({S, \\circ}\\right)$."}
+{"_id": "4636", "title": "External Direct Product Commutativity/General Result", "text": "Let $\\displaystyle \\left({S, \\circ}\\right) = \\prod_{k \\mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\\left({S_1, \\circ_1}\\right), \\left({S_2, \\circ_2}\\right), \\ldots, \\left({S_n, \\circ_n}\\right)$. If $\\circ_1, \\ldots, \\circ_n$ are all commutative, then so is $\\circ$."}
+{"_id": "4637", "title": "External Direct Product Associativity/General Result", "text": "Let $\\displaystyle \\left({S, \\circ}\\right) = \\prod_{k \\mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\\left({S_1, \\circ_1}\\right), \\left({S_2, \\circ_2}\\right), \\ldots, \\left({S_n, \\circ_n}\\right)$. If $\\circ_1, \\ldots, \\circ_n$ are all associative, then so is $\\circ$."}
+{"_id": "4638", "title": "Vector Cross Product is not Associative", "text": "The vector cross product is ''not'' associative. That is, in general: :$\\mathbf a \\times \\left({\\mathbf b \\times \\mathbf c}\\right) \\ne \\left({\\mathbf a \\times \\mathbf b}\\right) \\times \\mathbf c$ for $\\mathbf {a}, \\mathbf {b}, \\mathbf {c} \\in \\R^3$."}
+{"_id": "4639", "title": "Vector Cross Product is Anticommutative", "text": "The vector cross product is anticommutative: :$\\forall \\mathbf a, \\mathbf b \\in \\R^3: \\mathbf a \\times \\mathbf b = -\\left({\\mathbf b \\times \\mathbf a}\\right)$"}
+{"_id": "4641", "title": "Canonical Injection is Monomorphism/General Result", "text": "Let $\\struct {S_1, \\circ_1}, \\struct {S_2, \\circ_2}, \\dotsc, \\struct {S_j, \\circ_j}, \\dotsc, \\struct {S_n, \\circ_n}$ be algebraic structures with identities $e_1, e_2, \\dotsc, e_j, \\dotsc, e_n$ respectively. Then the canonical injection: :$\\displaystyle \\inj_j: \\struct {S_j, \\circ_j} \\to \\prod_{i \\mathop = 1}^n \\struct {S_i, \\circ_i}$ defined as: :$\\map {\\inj_j} x = \\tuple {e_1, e_2, \\dotsc, e_{j - 1}, x, e_{j + 1}, \\dotsc, e_n}$ is a monomorphism."}
+{"_id": "4642", "title": "Orthogonality of Solutions to the Sturm-Liouville Equation with Distinct Eigenvalues", "text": "Let $f \\left({x}\\right)$ and $g \\left({x}\\right)$ be solutions of the Sturm-Liouville equation: :$(1): \\quad -\\dfrac {\\mathrm d} {\\mathrm d x} \\left({p \\left({x}\\right) \\dfrac {\\mathrm d y} {\\mathrm d x}}\\right) + q \\left({x}\\right) y = \\lambda w \\left({x}\\right) y$ where $y$ is a function of the free variable $x$. The functions $p \\left({x}\\right)$, $q \\left({x}\\right)$ and $w \\left({x}\\right)$ are specified. In the simplest cases they are continuous on the closed interval $\\left[{a \\,.\\,.\\, b}\\right]$. In addition: :$(1a): \\quad p \\left({x}\\right) > 0$ has a continuous derivative :$(1b): \\quad w \\left({x}\\right) > 0$ :$(1c): \\quad y$ is typically required to satisfy some boundary conditions at $a$ and $b$. Assume that the Sturm-Liouville problem is regular, that is, $p \\left({x}\\right)^{-1} > 0$, $q \\left({x}\\right)$, and $w \\left({x}\\right) > 0$ are real-valued integrable functions over the closed interval  $\\left[{a \\,.\\,.\\, b}\\right]$, with ''separated boundary conditions'' of the form:  :$(2): \\quad y \\left({a}\\right) \\cos \\alpha - p \\left({a}\\right) y' \\left({a}\\right)\\sin \\alpha = 0$ :$(3): \\quad y \\left({b}\\right) \\cos \\beta - p \\left({b}\\right) y' \\left({b}\\right)\\sin \\beta = 0$ where $\\alpha, \\beta \\in \\left[{0 \\,.\\,.\\, \\pi}\\right)$. Then: :$\\displaystyle \\left\\langle{f, g}\\right\\rangle = \\int_a^b \\overline {f \\left({x}\\right)} q \\left({x}\\right) w \\left({x}\\right) \\, \\mathrm d x = 0$ where $f \\left({x}\\right)$ and $g \\left({x}\\right)$ are solutions to the Sturm-Liouville equation corresponding to distinct eigenvalues and $w \\left({x}\\right)$ is the \"weight\" or \"density\" function."}
+{"_id": "4643", "title": "Row Equivalent Matrix for Homogeneous System has same Solutions", "text": "Let $\\mathbf A$ be a matrix in the matrix space $\\map {\\MM_\\R} {m, n}$ such that: :$\\mathbf A \\mathbf x = \\mathbf 0$ represents a homogeneous system of linear equations. Let $\\mathbf H$ be row equivalent to $\\mathbf A$. Then the solution set of $\\mathbf H \\mathbf x = \\mathbf 0$ equals the solution set of $\\mathbf A \\mathbf x = \\mathbf 0$. That is: :$\\mathbf A \\sim \\mathbf H \\implies \\set {\\mathbf x: \\mathbf A \\mathbf x = \\mathbf 0} = \\set {\\mathbf x: \\mathbf H \\mathbf x = \\mathbf 0}$ where $\\sim$ represents row equivalence."}
+{"_id": "4644", "title": "Null Space of Reduced Echelon Form", "text": "Let $\\mathbf A$ be a matrix in the matrix space $\\map {\\MM_\\R} {m, n}$ such that: :$\\mathbf A \\mathbf x = \\mathbf 0$ represents a homogeneous system of linear equations. The null space of $\\mathbf A$ is the same as that of the null space of the reduced row echelon form of $\\mathbf A$: :$\\map {\\mathrm N} {\\mathbf A} = \\map {\\mathrm N} {\\map {\\mathrm {rref} } {\\mathbf A} }$"}
+{"_id": "4645", "title": "Null Space Contains Only Zero Vector iff Columns are Independent", "text": "Let: {{begin-eqn}} {{eqn | l = \\mathbf A_{m \\times n}       | r = \\begin{bmatrix} \\mathbf a_1 & \\mathbf a_2 & \\cdots & \\mathbf a_n \\end{bmatrix} }} {{end-eqn}} be a matrix where: :$\\forall i: 1 \\le i \\le n: \\mathbf a_i = \\begin{bmatrix} a_{1i} \\\\ a_{2i} \\\\ \\vdots \\\\ a_{mi} \\end{bmatrix} \\in \\R^m$ are vectors. Then: :$\\set {\\mathbf a_1, \\mathbf a_2, \\cdots, \\mathbf a_n}$ is a linearly independent set  {{iff}}: :$\\map {\\mathrm N} {\\mathbf A} = \\set {\\mathbf 0_{n \\times 1} }$ where $\\map {\\mathrm N} {\\mathbf A}$ is the null space of $\\mathbf A$."}
+{"_id": "4646", "title": "Axiom of Dependent Choice Implies Axiom of Countable Choice", "text": "The axiom of dependent choice implies the axiom of countable choice."}
+{"_id": "4647", "title": "Axiom of Choice Implies Axiom of Dependent Choice", "text": "The axiom of choice implies the axiom of dependent choice."}
+{"_id": "4648", "title": "Infimum of Empty Set is Greatest Element", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Suppose that $\\map \\inf \\O$, the infimum of the empty set, exists in $S$. Then $\\forall s \\in S: s \\preceq \\map \\inf \\O$. That is, $\\map \\inf \\O$ is the greatest element of $S$."}
+{"_id": "4649", "title": "Supremum of Empty Set is Smallest Element", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then: :the supremum of the empty set exists {{iff}} the smallest element of $S$ exists in which case: :$\\map \\sup \\O$ is the smallest element of $S$"}
+{"_id": "4650", "title": "Countable Union of Countable Sets is Countable", "text": "Let the axiom of countable choice be accepted. Then it can be proved that a countable union of countable sets is countable."}
+{"_id": "4657", "title": "Extended Real Addition is Commutative", "text": "Extended real addition $+_{\\overline{\\R}}$ is commutative. That is, for all $x, y \\in \\overline{\\R}$: :$x +_{\\overline{\\R}} y = y +_{\\overline{\\R}} x$ whenever at least one of the sides is defined."}
+{"_id": "4658", "title": "Extended Real Addition is Associative", "text": "Extended real addition $+_{\\overline{\\R}}$ is commutative. That is, for all $x, y, z \\in \\overline{\\R}$: :$(1):\\quad x +_{\\overline{\\R}} \\left({y +_{\\overline{\\R}} z}\\right) = \\left({x +_{\\overline{\\R}} y}\\right) +_{\\overline{\\R}} z$ whenever at least one of the sides is defined."}
+{"_id": "4659", "title": "Extended Real Multiplication is Commutative", "text": "Extended real multiplication $\\times_{\\overline \\R}$ is commutative. That is, for all $x, y \\in \\overline \\R$: :$x \\times_{\\overline \\R} y = y \\times_{\\overline \\R} x$"}
+{"_id": "4660", "title": "Extended Real Multiplication is Associative", "text": "Extended real multiplication $\\cdot_{\\overline \\R}$ is commutative. That is, for all $x, y, z \\in \\overline \\R$: :$(1): \\quad x \\cdot_{\\overline \\R} \\left({y \\cdot_{\\overline \\R} z}\\right) = \\left({x \\cdot_{\\overline \\R} y}\\right) \\cdot_{\\overline \\R} z$"}
+{"_id": "4661", "title": "Extended Real Numbers under Multiplication form Monoid", "text": "Denote with $\\overline \\R$ the extended real numbers. Denote with $\\cdot_{\\overline \\R}$ the extended real multiplication. The algebraic structure $\\struct {\\overline \\R, \\cdot_{\\overline \\R} }$ is a monoid."}
+{"_id": "4662", "title": "Extended Real Numbers under Multiplication form Commutative Monoid", "text": "Denote with $\\overline \\R$ the extended real numbers. Denote with $\\cdot_{\\overline \\R}$ the extended real multiplication. The algebraic structure $\\left({\\overline \\R, \\cdot_{\\overline \\R}}\\right)$ is a commutative monoid."}
+{"_id": "4665", "title": "Translation in Euclidean Space is Measurable Mapping", "text": "Let $\\mathcal B$ be the Borel $\\sigma$-algebra on $\\R^n$. Let $\\mathbf x \\in \\R^n$, and denote with $\\tau_{\\mathbf x}: \\R^n \\to \\R^n$ translation by $\\mathbf x$. Then $\\tau_{\\mathbf x}$ is $\\mathcal B \\, / \\, \\mathcal B$-measurable."}
+{"_id": "4667", "title": "Internal Group Direct Product Commutativity/General Result", "text": "Let $\\struct {G, \\circ}$ be the internal group direct product of $H_1, H_2, \\ldots, H_n$. Let $h_i$ and $h_j$ be elements of $H_i$ and $H_j$ respectively, $i \\ne j$. Then $h_i \\circ h_j = h_j \\circ h_i$."}
+{"_id": "4669", "title": "Mapping Measurable iff Measurable on Generator", "text": "Let $\\left({X, \\Sigma}\\right)$ and $\\left({X', \\Sigma'}\\right)$ be measurable spaces. Suppose that $\\Sigma'$ is generated by $\\mathcal{G}'$. Then a mapping $f: X \\to X'$ is $\\Sigma \\, / \\, \\Sigma'$-measurable iff: :$\\forall G' \\in \\mathcal{G}': f^{-1} \\left({G'}\\right) \\in \\Sigma$ That is, iff the preimage of every generator under $f$ is a measurable set."}
+{"_id": "4670", "title": "Internal Direct Product Theorem/General Result", "text": "Let $G$ be a group whose identity is $e$. Let $\\sequence {H_k}_{1 \\mathop \\le k \\mathop \\le n}$ be a sequence of subgroups of $G$. Then $G$ is the internal group direct product of $\\sequence {H_k}_{1 \\mathop \\le k \\mathop \\le n}$ {{iff}}: :$(1): \\quad G = H_1 H_2 \\cdots H_n$ :$(2): \\quad \\sequence {H_k}_{1 \\mathop \\le k \\mathop \\le n}$ is a sequence of independent subgroups :$(3): \\quad \\forall k \\in \\set {1, 2, \\ldots, n}: H_k \\lhd G$ where $H_k \\lhd G$ denotes that $H_k$ is a normal subgroup of $G$."}
+{"_id": "4671", "title": "Structure Induced by Group Operation is Group", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $S$ be a set. Let $\\struct {G^S, \\oplus}$ be the structure on $G^S$ induced by $\\circ$. Then $\\struct {G^S, \\oplus}$ is a group."}
+{"_id": "4672", "title": "Continuous Mapping is Measurable", "text": "Let $\\left({X, \\tau}\\right)$ and $\\left({X', \\tau'}\\right)$ be topological spaces. Denote with $\\mathcal B \\left({X, \\tau}\\right)$ and $\\mathcal B \\left({X', \\tau'}\\right)$ their respective Borel $\\sigma$-algebras. Let $f: X \\to X'$ be a continuous mapping. Then $f$ is $\\mathcal B \\left({X, \\tau}\\right) \\, / \\, \\mathcal B \\left({X', \\tau'}\\right)$-measurable."}
+{"_id": "4673", "title": "Composition of Measurable Mappings is Measurable", "text": "Let $\\struct {X_1, \\Sigma_1}$, $\\struct {X_2, \\Sigma_2}$ and $\\struct {X_3, \\Sigma_3}$ be measurable spaces. Let $f: X_1 \\to X_2$ be a $\\Sigma_1 \\, / \\, \\Sigma_2$-measurable mapping. Let $g: X_2 \\to X_3$ be a $\\Sigma_2 \\, / \\, \\Sigma_3$-measurable mapping. Then their composition $g \\circ f: X_1 \\to X_3$ is $\\Sigma_1 \\, / \\, \\Sigma_3$-measurable."}
+{"_id": "4674", "title": "Restriction of Equivalence Relation is Equivalence", "text": "Let $S$ be a set. Let $\\mathcal R \\subseteq S \\times S$ be an equivalence relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\mathcal R {\\restriction_T} \\subseteq T \\times T$ be the restriction of $\\mathcal R$ to $T$. Then $\\mathcal R {\\restriction_T}$ is an equivalence relation on $T$."}
+{"_id": "4675", "title": "Restriction of Reflexive Relation is Reflexive", "text": "Let $S$ be a set. Let $\\mathcal R \\subseteq S \\times S$ be a reflexive relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\mathcal R {\\restriction_T} \\subseteq T \\times T$ be the restriction of $\\mathcal R$ to $T$. Then $\\mathcal R {\\restriction_T}$ is a reflexive relation on $T$."}
+{"_id": "4676", "title": "Restriction of Symmetric Relation is Symmetric", "text": "Let $S$ be a set. Let $\\mathcal R \\subseteq S \\times S$ be a symmetric relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\mathcal R {\\restriction_T} \\subseteq T \\times T$ be the restriction of $\\mathcal R$ to $T$. Then $\\mathcal R {\\restriction_T}$ is a symmetric relation on $T$."}
+{"_id": "4677", "title": "Restriction of Transitive Relation is Transitive", "text": "Let $S$ be a set. Let $\\RR \\subseteq S \\times S$ be a transitive relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\RR {\\restriction_T} \\subseteq T \\times T$ be the restriction of $\\RR$ to $T$. Then $\\RR {\\restriction_T}$ is a transitive relation on $T$."}
+{"_id": "4678", "title": "Restriction of Asymmetric Relation is Asymmetric", "text": "Let $S$ be a set. Let $\\mathcal R \\subseteq S \\times S$ be a asymmetric relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\mathcal R \\restriction_T \\ \\subseteq T \\times T$ be the restriction of $\\mathcal R$ to $T$. Then $\\mathcal R \\restriction_T$ is a asymmetric relation on $T$."}
+{"_id": "4679", "title": "Restriction of Antisymmetric Relation is Antisymmetric", "text": "Let $S$ be a set. Let $\\RR \\subseteq S \\times S$ be an antisymmetric relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\RR {\\restriction_T} \\subseteq T \\times T$ be the restriction of $\\RR$ to $T$. Then $\\RR {\\restriction_T}$ is an antisymmetric relation on $T$."}
+{"_id": "4680", "title": "Restriction of Antireflexive Relation is Antireflexive", "text": "Let $S$ be a set. Let $\\mathcal R \\subseteq S \\times S$ be an antireflexive relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\mathcal R \\restriction_T \\ \\subseteq T \\times T$ be the restriction of $\\mathcal R$ to $T$. Then $\\mathcal R \\restriction_T$ is an antireflexive relation on $T$."}
+{"_id": "4681", "title": "Restriction of Antitransitive Relation is Antitransitive", "text": "Let $S$ be a set. Let $\\mathcal R \\subseteq S \\times S$ be an antitransitive relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\mathcal R \\restriction_T \\ \\subseteq T \\times T$ be the restriction of $\\mathcal R$ to $T$. Then $\\mathcal R \\restriction_T$ is an antitransitive relation on $T$."}
+{"_id": "4682", "title": "Relation is Total iff Union with Inverse is Trivial Relation", "text": "Let $\\mathcal R$ be a relation on $S$. Then $\\mathcal R$ is a total relation {{iff}}: :$\\mathcal R \\cup \\mathcal R^{-1} = S \\times S$ where: : $\\mathcal R^{-1}$ is the inverse of $\\mathcal R$. : $S \\times S$ is the trivial relation on $S$."}
+{"_id": "4683", "title": "Restriction of Connected Relation is Connected", "text": "Let $S$ be a set. Let $\\mathcal R \\subseteq S \\times S$ be a connected relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\mathcal R \\restriction_T \\ \\subseteq T \\times T$ be the restriction of $\\mathcal R$ to $T$. Then $\\mathcal R \\restriction_T$ is a connected relation on $T$."}
+{"_id": "4684", "title": "Restriction of Ordering is Ordering", "text": "Let $S$ be a set. Let $\\preceq$ be an ordering on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\preceq \\restriction_T$ be the restriction of $\\preceq$ to $T$. Then $\\preceq \\restriction_T$ is an ordering on $T$."}
+{"_id": "4686", "title": "Restriction of Serial Relation is Not Necessarily Serial", "text": "Let $S$ be a set. Let $\\mathcal R \\subseteq S \\times S$ be a serial relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\mathcal R \\restriction_T \\ \\subseteq T \\times T$ be the restriction of $\\mathcal R$ to $T$. Then $\\mathcal R \\restriction_T$ is not necessarily a serial relation on $T$."}
+{"_id": "4687", "title": "Restriction of Non-Reflexive Relation is Not Necessarily Non-Reflexive", "text": "Let $S$ be a set. Let $\\RR \\subseteq S \\times S$ be a non-reflexive relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\RR {\\restriction_T} \\subseteq T \\times T$ be the restriction of $\\RR$ to $T$. Then $\\RR {\\restriction_T}$ is not necessarily a non-reflexive relation on $T$."}
+{"_id": "4688", "title": "Restriction of Non-Symmetric Relation is Not Necessarily Non-Symmetric", "text": "Let $S$ be a set. Let $\\RR \\subseteq S \\times S$ be a non-symmetric relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\RR {\\restriction_T} \\ \\subseteq T \\times T$ be the restriction of $\\RR$ to $T$. Then $\\RR {\\restriction_T}$ is not necessarily a non-symmetric relation on $T$."}
+{"_id": "4689", "title": "Restriction of Non-Transitive Relation is Not Necessarily Non-Transitive", "text": "Let $S$ be a set. Let $\\mathcal R \\subseteq S \\times S$ be a non-transitive relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\mathcal R \\restriction_T \\ \\subseteq T \\times T$ be the restriction of $\\mathcal R$ to $T$. Then $\\mathcal R \\restriction_T$ is not necessarily a non-transitive relation on $T$."}
+{"_id": "4690", "title": "Existence and Uniqueness of Sigma-Algebra Generated by Collection of Mappings", "text": "Let $\\left({X_i, \\Sigma_i}\\right)$ be measurable spaces, with $i \\in I$ for some index set $I$. Let $X$ be a set, and let, for $i \\in I$, $f_i: X \\to X_i$ be a mapping. Then $\\sigma \\left({f_i: i \\in I}\\right)$, the $\\sigma$-algebra generated by $\\left({f_i}\\right)_{i \\in I}$, exists and is unique."}
+{"_id": "4691", "title": "Characterization of Sigma-Algebra Generated by Collection of Mappings", "text": "Let $\\left({X_i, \\Sigma_i}\\right)$ be measurable spaces, with $i \\in I$ for some index set $I$. Let $X$ be a set, and let, for $i \\in I$, $f_i: X \\to X_i$ be a mapping. Then: :$\\sigma \\left({f_i: i \\in I}\\right) = \\sigma \\left({\\displaystyle \\bigcup_{i \\mathop \\in I} f_i^{-1} \\left({\\Sigma_i}\\right)}\\right)$ where: :$\\sigma \\left({f_i: i \\in I}\\right)$ is the $\\sigma$-algebra generated by $\\left({f_i}\\right)_{i \\in I}$ :$\\sigma \\left({\\displaystyle \\bigcup_{i \\mathop \\in I} f_i^{-1} \\left({\\Sigma_i}\\right)}\\right)$ is the $\\sigma$-algebra generated by $\\displaystyle \\bigcup_{i \\mathop \\in I} f_i^{-1} \\left({\\Sigma_i}\\right)$ :$f_i^{-1} \\left({\\Sigma_i}\\right)$ denotes the pre-image $\\sigma$-algebra on $X$ by $f$"}
+{"_id": "4692", "title": "Restriction of Non-Connected Relation is Not Necessarily Non-Connected", "text": "Let $S$ be a set. Let $\\RR \\subseteq S \\times S$ be a non-connected relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\RR {\\restriction_T} \\subseteq T \\times T$ be the restriction of $\\RR$ to $T$. Then $\\RR {\\restriction_T}$ is not necessarily a non-connected relation on $T$."}
+{"_id": "4693", "title": "Duals of Isomorphic Ordered Sets are Isomorphic", "text": "Let $\\struct {S, \\preccurlyeq_1}$ and $\\struct {T, \\preccurlyeq_2}$ be ordered sets. Let $\\struct {S, \\succcurlyeq_1}$ and $\\struct {T, \\succcurlyeq_2}$ be the dual ordered sets of $\\struct {S, \\preccurlyeq_1}$ and $\\struct {T, \\preccurlyeq_2}$ respectively. Let $f: \\struct {S, \\preccurlyeq_1} \\to \\struct {T, \\preccurlyeq_2}$ be an order isomorphism. Then $f: \\struct {S, \\succcurlyeq_1} \\to {T, \\succcurlyeq_2} $ is also an order isomorphism."}
+{"_id": "4694", "title": "Pushforward Measure is Measure", "text": "Let $\\struct {X, \\Sigma}$ and $\\struct {X', \\Sigma'}$ be measurable spaces. Let $\\mu$ be a measure on $\\struct {X, \\Sigma}$. Let $f: X \\to X'$ be a $\\Sigma \\, / \\, \\Sigma'$-measurable mapping. Then the pushforward measure $f_* \\mu: \\Sigma' \\to \\overline \\R$ is a measure."}
+{"_id": "4695", "title": "Lebesgue Measure Invariant under Orthogonal Group", "text": "Let $M \\in \\map {\\mathrm O} {n, \\R}$ be an orthogonal matrix. Let $\\lambda^n$ be $n$-dimensional Lebesgue measure. Then the pushforward measure $M_* \\lambda^n$ equals $\\lambda^n$."}
+{"_id": "4696", "title": "Pushforward of Lebesgue Measure under General Linear Group", "text": "Let $M \\in \\GL {n, \\R}$ be an invertible matrix. Let $\\lambda^n$ be $n$-dimensional Lebesgue measure. Then the pushforward measure $M_* \\lambda^n$ satisfies: :$M_* \\lambda^n = \\size {\\det M^{-1} } \\cdot \\lambda^n$"}
+{"_id": "4697", "title": "Pre-Image Sigma-Algebra on Domain is Generated by Mapping", "text": "Let $X, X'$ be sets, and let $f: X \\to X'$ be a mapping. Let $\\Sigma'$ be a $\\sigma$-algebra on $X'$. Let $f: X \\to X'$ be a mapping. Then: :$\\sigma \\left({f}\\right) = f^{-1} \\left({\\Sigma'}\\right)$ where :$\\sigma \\left({f}\\right)$ denotes the $\\sigma$-algebra generated by $f$ :$f^{-1} \\left({\\Sigma'}\\right)$ denotes the pre-image $\\sigma$-algebra under $f$"}
+{"_id": "4698", "title": "Mapping between Euclidean Spaces Measurable iff Components Measurable", "text": "Let $\\R^n$ and $\\R^m$ be Euclidean spaces. Denote by $\\mathcal{B}^n$ and $\\mathcal{B}^m$ their respective Borel $\\sigma$-algebras. Denote with $\\mathcal B$ the Borel $\\sigma$-algebra on $\\R$. Let $f: \\R^n \\to \\R^m$ be a mapping, and write: :$f \\left({\\mathbf x}\\right) = \\begin{bmatrix}f_1 \\left({\\mathbf x}\\right) \\\\ \\vdots \\\\ f_m \\left({\\mathbf x}\\right)\\end{bmatrix}$ with, for $1 \\le i \\le m$, $f_i: \\R^n \\to \\R$. Then $f$ is $\\mathcal{B}^n \\, / \\, \\mathcal{B}^m$-measurable iff: :$\\forall i:f_i: \\R^n \\to \\R$ is $\\mathcal{B}^n \\, / \\, \\mathcal B$-measurable"}
+{"_id": "4699", "title": "Pre-Image Sigma-Algebra of Generated Sigma-Algebra", "text": "Let $f: X \\to Y$ be a mapping. Let $\\mathcal G \\subseteq \\mathcal P \\left({Y}\\right)$ be a collection of subsets of $Y$. Then the following equality of $\\sigma$-algebras on $X$ holds: :$f^{-1} \\left({\\sigma \\left({\\mathcal G}\\right)}\\right) = \\sigma \\left({f^{-1} \\left({\\mathcal G}\\right)}\\right)$ where :$\\sigma$ denotes a generated $\\sigma$-algebra :$f^{-1} \\left({\\sigma \\left({\\mathcal G}\\right)}\\right)$ denotes the pre-image $\\sigma$-algebra :$f^{-1} \\left({\\mathcal G}\\right)$ is the preimage of $\\mathcal G$ under $f$"}
+{"_id": "4700", "title": "Stieltjes Function of Measure is Stieltjes Function", "text": "Let $\\mu$ be a measure on $\\R$ with the Borel $\\sigma$-algebra $\\mathcal B \\left({\\R}\\right)$. Suppose that for every $n \\in \\N$: :$\\mu \\left[{-n \\,.\\,.\\, n}\\right] < +\\infty$ Then $F_\\mu: \\R \\to \\overline \\R$, the Stieltjes function of $\\mu$, is a Stieltjes function."}
+{"_id": "4701", "title": "Characterization of Differentiability", "text": "{{refactor|one-dimensional case deserves its own page, perhaps}} Let $\\mathbb X$ be an open rectangle of $\\R^n$.  Let $f: \\mathbb X \\to \\R, \\mathbf x \\mapsto \\map f {\\mathbf x}$ be a real-valued function. Let $\\mathbf x = \\begin {bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end {bmatrix} \\in \\R^n$. Let $\\map {\\Delta f} {\\mathbf x} = \\map f {\\mathbf x + \\Delta \\mathbf x} - \\map f {\\mathbf x}$ . Let $\\dfrac {\\partial f} {\\partial x_j}$ be the partial derivative of $f$ {{WRT|Differentiation}} $x_j$. Then $f$ is differentiable {{iff}} there exists some $\\map {\\Delta f} {\\mathbf x}$ such that: :$\\map {\\Delta  f} {\\mathbf x} = \\displaystyle \\sum_{i \\mathop = 1}^n \\frac {\\partial \\map f {\\mathbf x} } {\\partial x_i} \\Delta x_i + \\sum_{i \\mathop = 1}^n \\varepsilon_i \\Delta x_i$ where $\\forall i: 1 \\le i \\le n: \\varepsilon_i \\to 0$ as $\\Delta x_i \\to 0$."}
+{"_id": "4702", "title": "Chain Rule for Real-Valued Functions", "text": "Let $f: \\R^n \\to \\R, \\mathbf x \\mapsto z$ be a differentiable real-valued function. Let $\\mathbf x = \\begin{bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{bmatrix} \\in \\R^n$. Further, let every element $x_i: 1 \\le i \\le n$ represent an implicitly defined differentiable real function of $t$. Then $z$ is itself differentiable {{WRT|Differentiation}} $t$ and: {{begin-eqn}} {{eqn | l = \\frac {\\d z} {\\d t}       | r = \\sum_{k \\mathop = 1}^n \\frac {\\partial z }{\\partial x_i} \\frac {\\d x_i} {\\d t} }} {{end-eqn}} where $\\dfrac {\\partial z} {\\partial x_i}$ is the partial derivative of $z$ {{WRT|Differentiation}} $x_i$."}
+{"_id": "4704", "title": "Pre-Measure of Finite Stieltjes Function is Pre-Measure", "text": "Let $\\mathcal J_{ho}$ denote the collection of half-open intervals in $\\R$. Let $f: \\R \\to \\R$ be a finite Stieltjes function. Then the pre-measure of $f$, $\\mu_f: \\mathcal{J}_{ho} \\to \\overline \\R_{\\ge 0}$ is a pre-measure. Here, $\\overline \\R_{\\ge 0}$ denotes the set of positive extended real numbers."}
+{"_id": "4705", "title": "Pre-Measure of Finite Stieltjes Function Extends to Unique Measure", "text": "Let $\\mathcal{J}_{ho}$ denote the collection of half-open intervals in $\\R$. Let $f: \\R \\to \\R$ be a finite Stieltjes function. Then the pre-measure of $f$, $\\mu_f$, extends uniquely to a measure $\\mu$ on $\\mathcal B \\left({\\R}\\right)$, the Borel $\\sigma$-algebra on $\\R$. This unique measure $\\mu$ is the measure of $f$."}
+{"_id": "4706", "title": "Stieltjes Function of Measure of Finite Stieltjes Function", "text": "Let $f: \\R \\to \\R$ be a finite Stieltjes function. Let $\\mu_f$ be the measure of $f$. Let $f_{\\mu_f}$ be the Stieltjes function of $\\mu_f$. Then $f_{\\mu_f} = f$."}
+{"_id": "4707", "title": "Measure of Stieltjes Function of Measure", "text": "Let $\\mu$ be a measure on $\\mathcal B \\left({\\R}\\right)$, the Borel $\\sigma$-algebra on $\\R$. Suppose that for all $n \\in \\N$, $\\mu$ satisfies: :$\\mu \\left({ \\left[{-n \\,.\\,.\\, n}\\right) \\ }\\right) < +\\infty$ Let $f_\\mu$ be the Stieltjes function of $\\mu$. Let $\\mu_{f_\\mu}$ be the measure of $f_\\mu$. Then $\\mu_{f_\\mu} = \\mu$."}
+{"_id": "4710", "title": "Characterization of Measurable Functions", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $f: X \\to \\overline{\\R}$ be an extended real-valued function. Then the following are all equivalent: {{begin-eqn}} {{eqn | n=1       | o=       | r=f\\) is measurable \\( }} {{eqn | n=2       | o=       | r=\\forall \\alpha \\in \\R: \\left\\{ {x \\in X: f \\left({x}\\right) \\le \\alpha}\\right\\} \\in \\Sigma }} {{eqn | n=2'       | o=       | r=\\forall \\alpha \\in \\Q: \\left\\{ {x \\in X: f \\left({x}\\right) \\le \\alpha}\\right\\} \\in \\Sigma }} {{eqn | n=3       | o=       | r=\\forall \\alpha \\in \\R: \\left\\{ {x \\in X: f \\left({x}\\right) < \\alpha}\\right\\} \\in \\Sigma }} {{eqn | n=3'       | o=       | r=\\forall \\alpha \\in \\Q: \\left\\{ {x \\in X: f \\left({x}\\right) < \\alpha}\\right\\} \\in \\Sigma }} {{eqn | n=4       | o=       | r=\\forall \\alpha \\in \\R: \\left\\{ {x \\in X: f \\left({x}\\right) \\ge \\alpha}\\right\\} \\in \\Sigma }} {{eqn | n=4'       | o=       | r=\\forall \\alpha \\in \\Q: \\left\\{ {x \\in X: f \\left({x}\\right) \\ge \\alpha}\\right\\} \\in \\Sigma }} {{eqn | n=5       | o=       | r=\\forall \\alpha \\in \\R: \\left\\{ {x \\in X: f \\left({x}\\right) > \\alpha}\\right\\} \\in \\Sigma }} {{eqn | n=5'       | o=       | r=\\forall \\alpha \\in \\Q: \\left\\{ {x \\in X: f \\left({x}\\right) > \\alpha}\\right\\} \\in \\Sigma }} {{end-eqn}}"}
+{"_id": "4711", "title": "Characterization of Extended Real Sigma-Algebra", "text": "Let $\\mathcal B \\left({\\R}\\right)$ be the Borel $\\sigma$-algebra on $\\R$. Let $\\overline{\\mathcal B}$ be the extended real $\\sigma$-algebra. Define $\\mathcal S := \\mathcal P \\left({\\left\\{{+\\infty, -\\infty}\\right\\}}\\right)$, where $\\mathcal P$ denotes power set. Then: :$\\overline{\\mathcal B} = \\left\\{{B \\cup S: B \\in \\mathcal B \\left({\\R}\\right), S \\in \\mathcal S}\\right\\}$"}
+{"_id": "4712", "title": "Extended Real Sigma-Algebra Induces Borel Sigma-Algebra on Reals", "text": "Let $\\overline \\BB$ be the extended real $\\sigma$-algebra. Let $\\map \\BB \\R$ be the Borel $\\sigma$-algebra on $\\R$. Then: :$\\overline \\BB_\\R = \\map \\BB \\R$ where $\\overline \\BB_\\R$ denotes a trace $\\sigma$-algebra."}
+{"_id": "4713", "title": "Generators for Extended Real Sigma-Algebra", "text": "Let $\\overline{\\mathcal B}$ be the extended real $\\sigma$-algebra. Then $\\overline{\\mathcal B}$ is generated by each of the following collections of extended real intervals: {{begin-eqn}} {{eqn | n = 1       | o =        | r = \\left\\{ {\\ \\left[{a \\,.\\,.\\, +\\infty}\\right]: a \\in \\R}\\right\\} }} {{eqn | n = 1'       | o =        | r = \\left\\{ {\\ \\left[{a \\,.\\,.\\, +\\infty}\\right]: a \\in \\Q}\\right\\} }} {{eqn | n = 2       | o =        | r = \\left\\{ {\\ \\left({b \\,.\\,.\\, +\\infty}\\right]: b \\in \\R}\\right\\} }} {{eqn | n = 2'       | o =        | r = \\left\\{ {\\ \\left({b \\,.\\,.\\, +\\infty}\\right]: b \\in \\Q}\\right\\} }} {{eqn | n = 3       | o =        | r = \\left\\{ {\\ \\left[{-\\infty \\,.\\,.\\, c}\\right): c \\in \\R}\\right\\} }} {{eqn | n = 3'       | o =        | r = \\left\\{ {\\ \\left[{-\\infty \\,.\\,.\\, c}\\right): c \\in \\Q}\\right\\} }} {{eqn | n = 4       | o =        | r = \\left\\{ {\\ \\left[{-\\infty \\,.\\,.\\, d}\\right]: d \\in \\R}\\right\\} }} {{eqn | n = 4'       | o =        | r = \\left\\{ {\\ \\left[{-\\infty \\,.\\,.\\, d}\\right]: d \\in \\Q}\\right\\} }} {{end-eqn}}"}
+{"_id": "4714", "title": "Characteristic Function Measurable iff Set Measurable", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $E \\subseteq X$. Then the following are equivalent: :$(1): \\quad E \\in \\Sigma$; i.e., $E$ is a $\\Sigma$-measurable set :$(2): \\quad \\chi_E: X \\to \\left\\{{0, 1}\\right\\}$, the characteristic function of $E$, is $\\Sigma$-measurable"}
+{"_id": "4715", "title": "Simple Function is Measurable", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $f: X \\to \\R$ be a simple function. Then $f$ is $\\Sigma$-measurable."}
+{"_id": "4716", "title": "Identity Mapping is Relation Isomorphism", "text": "Let $\\struct {S, \\RR}$ be a relational structure. Then the identity mapping $I_S: S \\to S$ is a relation isomorphism from $\\struct {S, \\RR}$ to itself."}
+{"_id": "4717", "title": "Inverse of Relation Isomorphism is Relation Isomorphism", "text": "Let $\\struct {S, \\RR_1}$ and $\\struct {T, \\RR_2}$ be relational structures. Let $\\phi: \\struct {S, \\RR_1} \\to \\struct {T, \\RR_2}$ be a bijection. Then: :$\\phi: \\struct {S, \\RR_1} \\to \\struct {T, \\RR_2}$ is a relation isomorphism {{iff}}: :$\\phi^{-1}: \\struct {T, \\RR_2} \\to \\struct {S, \\RR_1}$ is also a relation isomorphism."}
+{"_id": "4718", "title": "Measurable Function is Simple Function iff Finite Image Set", "text": "Let $\\struct {X, \\Sigma}$ be a measurable space. Let $f: X \\to \\R$ be a measurable function. Then $f$ is a simple function {{iff}} its image is finite: :$\\card {\\Img f} < \\infty$"}
+{"_id": "4719", "title": "Composite of Relation Isomorphisms is Relation Isomorphism", "text": "Let $\\struct {S_1, \\RR_1}$, $\\struct {S_2, \\RR_2}$ and $\\struct {S_3, \\RR_3}$ be relational structures. Let: :$\\phi: \\struct {S_1, \\RR_1} \\to \\struct {S_2, \\RR_2}$ and: :$\\psi: \\struct {S_2, \\RR_2} \\to \\struct {S_3, \\RR_3}$ be relation isomorphisms. Then $\\psi \\circ \\phi: \\struct {S_1, \\RR_1} \\to \\struct {S_3, \\RR_3}$ is also a relation isomorphism."}
+{"_id": "4720", "title": "Pointwise Sum of Simple Functions is Simple Function", "text": "Let $\\struct {X, \\Sigma}$ be a measurable space. Let $f, g : X \\to \\R$ be simple functions. Then $f + g: X \\to \\R, \\map {\\paren {f + g} } x := \\map f x + \\map g x$ is also a simple function."}
+{"_id": "4721", "title": "Pointwise Product of Simple Functions is Simple Function", "text": "Let $\\struct {X, \\Sigma}$ be a measurable space. Let $f, g : X \\to \\R$ be simple functions. Then $f \\cdot g: X \\to \\R, \\map {\\paren {f \\cdot g} } x := \\map f x \\cdot \\map g x$ is also a simple function."}
+{"_id": "4722", "title": "Positive Part of Simple Function is Simple Function", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $f: X \\to \\R$ be a simple function. Then $f^+: X \\to \\R$, the positive part of $f$, is also a simple function."}
+{"_id": "4723", "title": "Negative Part of Simple Function is Simple Function", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $f: X \\to \\R$ be a simple function. Then $f^-: X \\to \\R$, the negative part of $f$ is also a simple function."}
+{"_id": "4724", "title": "Difference of Positive and Negative Parts", "text": "Let $X$ be a set, and let $f: X \\to \\overline{\\R}$ be an extended real-valued function. Let $f^+$, $f^-: X \\to \\overline{\\R}$ be the positive and negative parts of $f$, respectively. Then $f = f^+ - f^-$."}
+{"_id": "4725", "title": "Sum of Positive and Negative Parts", "text": "Let $X$ be a set, and let $f: X \\to \\overline{\\R}$ be an extended real-valued function. Let $f^+, f^-: X \\to \\overline{\\R}$ be the positive and negative parts of $f$, respectively. Then $\\left\\vert{f}\\right\\vert = f^+ + f^-$, where $\\left\\vert{f}\\right\\vert$ is the absolute value of $f$."}
+{"_id": "4726", "title": "Absolute Value of Simple Function is Simple Function", "text": "Let $\\struct {X, \\Sigma}$ be a measurable space. Let $f: X \\to \\R$ be a simple function. Then $\\size f: X \\to \\R$, the absolute value of $f$, is also a simple function."}
+{"_id": "4727", "title": "Measurable Function Pointwise Limit of Simple Functions", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $f: X \\to \\overline{\\R}$ be a $\\Sigma$-measurable function. Then there exists a sequence $\\left({f_n}\\right)_{n \\in \\N} \\in \\mathcal E \\left({\\Sigma}\\right)$ of simple functions, such that: :$\\forall x \\in X: f \\left({x}\\right) = \\displaystyle \\lim_{n \\to \\infty} f_n \\left({x}\\right)$ That is, such that $f = \\displaystyle \\lim_{n \\to \\infty} f_n$ pointwise. The sequence $\\left({f_n}\\right)_{n \\in \\N}$ may furthermore be taken to satisfy: :$\\forall n \\in \\N: \\left\\vert{f_n}\\right\\vert \\le \\left\\vert{f}\\right\\vert$ where $\\left\\vert{f}\\right\\vert$ denotes the absolute value of $f$."}
+{"_id": "4728", "title": "Finite Cartesian Product of Non-Empty Sets is Non-Empty", "text": "Let $S_1, S_2, \\ldots, S_n$ be non-non-empty sets. Then their cartesian product $S_1 \\times S_2 \\times \\cdots \\times S_n$ is non-empty."}
+{"_id": "4729", "title": "Equivalent Conditions for Dedekind-Infinite Set", "text": "For a set $S$, the following conditions are equivalent: :$(1): \\quad$ $S$ is Dedekind-infinite. :$(2): \\quad$ $S$ has a countably infinite subset. The above equivalence can be proven in Zermelo-Fraenkel set theory. If the axiom of countable choice is accepted, then it can be proven that the following condition is also equivalent to the above two: :$(3): \\quad$ $S$ is infinite."}
+{"_id": "4730", "title": "Relation Isomorphism Preserves Reflexivity", "text": "Let $\\struct {S, \\mathcal R_1}$ and $\\struct {T, \\mathcal R_2}$ be relational structures. Let $\\struct {S, \\mathcal R_1}$ and $\\struct {T, \\mathcal R_2}$ be (relationally) isomorphic. Then $\\mathcal R_1$ is a reflexive relation {{iff}} $\\mathcal R_2$ is also a reflexive relation."}
+{"_id": "4731", "title": "Preimage of Subset under Composite Mapping", "text": "Let $S_1, S_2, S_3$ be sets. Let $f: S_1 \\to S_2$ and $g: S_2 \\to S_3$ be mappings. Denote with $g \\circ f: S_1 \\to S_3$ the composition of $g$ and $f$. Let $S_3' \\subseteq S_3$ be a subset of $S_3$. Then: :$\\paren {g \\circ f}^{-1} \\sqbrk {S_3'} = \\paren {f^{-1} \\circ g^{-1} } \\sqbrk {S_3'}$ where $g^{-1} \\sqbrk {S_3'}$ denotes the preimage of $S_3'$ under $g$."}
+{"_id": "4732", "title": "Infinite Set has Countably Infinite Subset/Proof 4", "text": "If the axiom of countable choice is accepted, then it can be proven that every infinite set has a countably infinite subset."}
+{"_id": "4733", "title": "Set is Infinite iff exist Subsets of all Finite Cardinalities", "text": "A set $S$ is infinite {{iff}} for all $n \\in \\N$, there exists a subset of $S$ whose cardinality is $n$."}
+{"_id": "4734", "title": "Relation Isomorphism Preserves Symmetry", "text": "Let $\\left({S, \\mathcal R_1}\\right)$ and $\\left({T, \\mathcal R_2}\\right)$ be relational structures. Let $\\left({S, \\mathcal R_1}\\right)$ and $\\left({T, \\mathcal R_2}\\right)$ be (relationally) isomorphic. Then $\\mathcal R_1$ is a symmetric relation {{iff}} $\\mathcal R_2$ is also a symmetric relation."}
+{"_id": "4735", "title": "Set Difference with Proper Subset", "text": "Let $S$ be a set. Let $T \\subsetneq S$ be a proper subset of $S$.   Let $S \\setminus T$ denote the set difference between $S$ and $T$. Then: :$S \\setminus T \\ne \\O$ where $\\O$ denotes the empty set."}
+{"_id": "4736", "title": "Order Isomorphism iff Strictly Increasing Surjection", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be totally ordered sets. A mapping $\\phi: \\left({S, \\preceq_1}\\right) \\to \\left({T, \\preceq_2}\\right)$ is an order isomorphism iff: : $(1): \\quad \\phi$ is a surjection : $(2): \\quad \\forall x, y \\in S: x \\mathop{\\prec_1} y \\implies \\phi \\left({x}\\right) \\mathop{\\prec_2} \\phi \\left({y}\\right)$"}
+{"_id": "4737", "title": "Relation Isomorphism Preserves Transitivity", "text": "Let $\\struct {S, \\RR_1}$ and $\\struct {T, \\RR_2}$ be relational structures. Let $\\struct {S, \\RR_1}$ and $\\struct {T, \\RR_2}$ be (relationally) isomorphic. Then $\\RR_1$ is a transitive relation {{iff}} $\\RR_2$ is a transitive relation."}
+{"_id": "4740", "title": "Pointwise Supremum of Measurable Functions is Measurable", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space, and let $I$ be a countable set. Let $\\left({f_i}\\right)_{i \\in I}$, $f_i: X \\to \\overline \\R$ be an $I$-indexed collection of $\\Sigma$-measurable functions. Then the pointwise supremum $\\displaystyle \\sup_{i \\mathop \\in I} f_i: X \\to \\overline \\R$ is also $\\Sigma$-measurable."}
+{"_id": "4741", "title": "Pointwise Infimum of Measurable Functions is Measurable", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space, and let $I$ be a countable set. Let $\\left({f_i}\\right)_{i \\in I}$, $f_i: X \\to \\overline{\\R}$ be an $I$-indexed collection of $\\Sigma$-measurable functions. Then the pointwise infimum $\\displaystyle \\inf_{i \\mathop \\in I} f_i: X \\to \\overline{\\R}$ is also $\\Sigma$-measurable."}
+{"_id": "4742", "title": "Pointwise Upper Limit of Measurable Functions is Measurable", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $\\left({f_n}\\right)_{n \\in \\N}$, $f_n: X \\to \\overline{\\R}$ be a sequence of $\\Sigma$-measurable functions. Then the pointwise upper limit $\\displaystyle \\limsup_{n \\to \\infty} f_n: X \\to \\overline{\\R}$ is also $\\Sigma$-measurable."}
+{"_id": "4743", "title": "Pointwise Lower Limit of Measurable Functions is Measurable", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $\\left({f_n}\\right)_{n \\mathop \\in \\N}$, $f_n: X \\to \\overline \\R$ be a sequence of $\\Sigma$-measurable functions. Then the pointwise lower limit: :$\\displaystyle \\liminf_{n \\mathop \\to \\infty} f_n: X \\to \\overline \\R$ is also $\\Sigma$-measurable."}
+{"_id": "4744", "title": "Pointwise Limit of Measurable Functions is Measurable", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $\\left({f_n}\\right)_{n \\in \\N}$, $f_n: X \\to \\overline{\\R}$ be a sequence of $\\Sigma$-measurable functions. Then the pointwise limit $\\displaystyle \\lim_{n \\to \\infty} f_n: X \\to \\overline{\\R}$ is also $\\Sigma$-measurable."}
+{"_id": "4745", "title": "Pointwise Sum of Measurable Functions is Measurable", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $f, g: X \\to \\overline{\\R}$ be $\\Sigma$-measurable functions. Assume that the pointwise sum $f + g: X \\to \\overline{\\R}$ is well-defined. Then $f + g$ is a $\\Sigma$-measurable function."}
+{"_id": "4746", "title": "Pointwise Difference of Measurable Functions is Measurable", "text": "Let $\\struct {X, \\Sigma}$ be a measurable space. Let $f, g: X \\to \\overline \\R$ be $\\Sigma$-measurable functions. Assume that the pointwise difference $f - g: X \\to \\overline \\R$ is well-defined. Then $f - g$ is a $\\Sigma$-measurable function."}
+{"_id": "4747", "title": "Pointwise Product of Measurable Functions is Measurable", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $f, g: X \\to \\overline{\\R}$ be $\\Sigma$-measurable functions. Then the pointwise product $f \\cdot g: X \\to \\overline{\\R}$ is also $\\Sigma$-measurable."}
+{"_id": "4748", "title": "Pointwise Maximum of Measurable Functions is Measurable", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $f, g: X \\to \\overline{\\R}$ be $\\Sigma$-measurable functions. Then the pointwise maximum $\\max \\left({f, g}\\right): X \\to \\overline{\\R}$ is also $\\Sigma$-measurable."}
+{"_id": "4750", "title": "Function Measurable iff Positive and Negative Parts Measurable", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $f: X \\to \\overline{\\R}$ be an extended real-valued function. Let $f^+, f^-: X \\to \\overline{\\R}$ be the positive and negative parts of $f$. Then $f$ is $\\Sigma$-measurable iff both $f^+$ and $f^-$ are $\\Sigma$-measurable."}
+{"_id": "4752", "title": "Factorization Lemma/Extended Real-Valued Function", "text": "Then an extended real-valued function $g: X \\to \\overline{\\R}$ is $\\sigma \\left({f}\\right)$-measurable {{iff}}: :There exists a $\\Sigma$-measurable mapping $\\tilde g: Y \\to \\overline{\\R}$ such that $g = \\tilde g \\circ f$ where: :$\\sigma \\left({f}\\right)$ denotes the $\\sigma$-algebra generated by $f$"}
+{"_id": "4753", "title": "Factorization Lemma", "text": "Let $X$ be a set, and $\\left({Y, \\Sigma}\\right)$ be a measurable space. Let $f: X \\to Y$ be a mapping."}
+{"_id": "4754", "title": "Piecewise Combination of Measurable Mappings is Measurable/Binary Case", "text": "Let $f, g: X \\to X'$ be $\\Sigma \\, / \\, \\Sigma'$-measurable mappings. Let $E \\in \\Sigma$ be a measurable set. Define $h: X \\to X'$ by: :$\\forall x \\in X: \\map h x := \\begin{cases} \\map f x & : \\text {if $x \\in E$} \\\\ \\map g x & : \\text {if $x \\notin E$} \\end{cases}$ Then $h$ is also a $\\Sigma \\, / \\, \\Sigma'$-measurable mapping."}
+{"_id": "4755", "title": "Piecewise Combination of Measurable Mappings is Measurable/General Case", "text": "Let $\\left({E_n}\\right)_{n \\in \\N} \\in \\Sigma, \\displaystyle \\bigcup_{n \\mathop \\in \\N} E_n = X$ be a countable cover of $X$ by $\\Sigma$-measurable sets. For each $n \\in \\N$, let $f_n: E_n \\to X'$ be a $\\Sigma_{E_n} \\, / \\, \\Sigma'$-measurable mapping. Here, $\\Sigma_{E_n}$ is the trace $\\sigma$-algebra of $E_n$ in $\\Sigma$. Suppose that for every $m, n \\in \\N$, $f_m$ and $f_n$ satisfy: :$(1): \\quad f_m \\restriction_{E_m \\cap E_n} = f_n \\restriction_{E_m \\cap E_n}$ that is, $f_m$ and $f_n$ coincide whenever both are defined; here $\\restriction$ denotes restriction. Define $f: X \\to X'$ by: :$\\displaystyle \\forall n \\in \\N, x \\in E_n: f \\left({x}\\right) := f_n \\left({x}\\right)$ Then $f$ is a $\\Sigma \\, / \\, \\Sigma'$-measurable mapping."}
+{"_id": "4756", "title": "Piecewise Combination of Measurable Mappings is Measurable", "text": "Let $\\struct {X, \\Sigma}$ and $\\struct {X', \\Sigma'}$ be measurable spaces."}
+{"_id": "4757", "title": "Function Simple iff Positive and Negative Parts Simple", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $g: X \\to \\overline{\\R}$ be an extended real-valued function. Then $g$ is a simple function {{iff}} its positive part $g^+$ and negative part $g^-$ are simple functions."}
+{"_id": "4758", "title": "Bounded Measurable Function Uniform Limit of Simple Functions", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $f: X \\to \\overline{\\R}$ be a bounded $\\Sigma$-measurable function. Then there exists a sequence $\\left({f_n}\\right)_{n \\in \\N} \\in \\mathcal E \\left({\\Sigma}\\right)$ of simple functions, such that: :$\\forall \\epsilon > 0: \\exists n \\in \\N: \\forall x \\in X: \\left\\vert{f \\left({x}\\right) - f_n \\left({x}\\right)}\\right\\vert < \\epsilon$ That is, such that $f = \\displaystyle \\lim_{n \\to \\infty} f_n$ uniformly. The sequence $\\left({f_n}\\right)_{n \\in \\N}$ may furthermore be taken to satisfy: :$\\forall n \\in \\N: \\left\\vert{f_n}\\right\\vert \\le \\left\\vert{f}\\right\\vert$ where $\\left\\vert{f}\\right\\vert$ denotes the absolute value of $f$."}
+{"_id": "4760", "title": "Integral of Characteristic Function", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $E \\in \\Sigma$ be a measurable set, and let $\\chi_E: X \\to \\R$ be its characteristic function. Then $I_\\mu \\left({\\chi_E}\\right) = \\mu \\left({E}\\right)$, where $I_\\mu \\left({\\chi_E}\\right)$ is the $\\mu$-integral of $\\chi_E$."}
+{"_id": "4761", "title": "Integral of Positive Simple Function is Positive Homogeneous", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $f: X \\to \\R, f \\in \\mathcal E^+$ be a positive simple function. Let $\\lambda \\in \\R_{\\ge 0}$ be a positive real number. Then $I_\\mu \\left({\\lambda \\cdot f}\\right) = \\lambda \\cdot I_\\mu \\left({f}\\right)$, where: :$\\lambda \\cdot f$ is the pointwise $\\lambda$-multiple of $f$ :$I_\\mu$ denotes $\\mu$-integration This can be summarized by saying that $I_\\mu$ is positive homogeneous."}
+{"_id": "4762", "title": "Integral of Positive Simple Function is Additive", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $f,g: X \\to \\R$, $f,g \\in \\mathcal{E}^+$ be positive simple functions. Then $I_\\mu \\left({f + g}\\right) = I_\\mu \\left({f}\\right) + I_\\mu \\left({g}\\right)$, where: :$f + g$ is the pointwise sum of $f$ and $g$ :$I_\\mu$ denotes $\\mu$-integration This can be summarized by saying that $I_\\mu$ is additive."}
+{"_id": "4763", "title": "Strict Ordering Preserved under Product with Invertible Element", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be an ordered semigroup. Let $z \\in S$ be invertible. Suppose that either $x \\circ z \\prec y \\circ z$ or $z \\circ x \\prec z \\circ y$. Then $x \\prec y$."}
+{"_id": "4764", "title": "Strict Ordering Preserved under Cancellability in Totally Ordered Semigroup", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a totally ordered semigroup. If either: : $x \\circ z \\prec y \\circ z$ or : $z \\circ x \\prec z \\circ y$ then $x \\prec y$."}
+{"_id": "4765", "title": "Relation Compatibility in Totally Ordered Semigroup", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be an ordered semigroup such that: :$(1): \\quad$ All the elements of $\\left({S, \\circ, \\preceq}\\right)$ are cancellable for $\\circ$ :$(2): \\quad \\preceq$ is a total ordering. Then: :$\\forall x, y, z \\in S: x \\circ z \\preceq y \\circ z \\iff x \\preceq y$"}
+{"_id": "4766", "title": "Integral of Positive Measurable Function Extends Integral of Positive Simple Function", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $f: X \\to \\R, f \\in \\mathcal{E}^+$ be a positive simple function. Then $\\displaystyle \\int f \\, \\mathrm d\\mu = I_\\mu \\left({f}\\right)$, where: :$\\displaystyle \\int \\cdot \\, \\mathrm d\\mu$ denotes the $\\mu$-integral of positive measurable functions :$I_\\mu$ denotes the $\\mu$-integral of positive simple functions That is, $\\displaystyle \\int \\cdot \\, \\mathrm d\\mu \\restriction_{\\mathcal{E}^+} = I_\\mu$, using the notion of restriction, $\\restriction$."}
+{"_id": "4767", "title": "Beppo Levi's Theorem", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $\\sequence {f_n}_{n \\mathop \\in \\N} \\in \\MM_{\\overline \\R}^+$ be an increasing sequence of positive $\\Sigma$-measurable functions. Let $\\displaystyle \\sup_{n \\mathop \\in \\N} f_n: X \\to \\overline \\R$ be the pointwise supremum of $\\sequence {f_n}_{n \\mathop \\in \\N}$, where $\\overline \\R$ denotes the extended real numbers. Then: :$\\displaystyle \\int \\sup_{n \\mathop \\in \\N} f_n \\rd \\mu = \\sup_{n \\mathop \\in \\N} \\int f_n \\rd \\mu$ where the supremum on the {{RHS}} is in the ordering on $\\overline \\R$."}
+{"_id": "4768", "title": "Ordered Semigroup Isomorphism is Surjective Monomorphism", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ and $\\left({T, *, \\preccurlyeq}\\right)$ be ordered semigroups. Let $\\phi: \\left({S, \\circ, \\preceq}\\right) \\to \\left({T, *, \\preccurlyeq}\\right)$ be a mapping. Then $\\phi$ is an ordered semigroup isomorphism iff: :$(1): \\quad \\phi$ is an ordered semigroup monomorphism :$(2): \\quad \\phi$ is a surjection."}
+{"_id": "4769", "title": "Ordered Semigroup Monomorphism into Image is Isomorphism", "text": "Let $\\struct {S, \\circ, \\preceq}$ and $\\struct {T, *, \\preccurlyeq}$ be ordered semigroups. Let $\\phi: \\struct {S, \\circ, \\preceq} \\to \\struct {T, *, \\preccurlyeq}$ be an ordered semigroup monomorphism. Let $S'$ be the image of $\\phi$. Then $\\phi$ is an ordered semigroup isomorphism from $\\struct {S, \\circ, \\preceq}$ into $\\struct {S', * {\\restriction_{S'} }, \\preccurlyeq \\restriction_{S'} }$. Here: :$* {\\restriction_{S'}}$ denotes the restriction of $*$ to $S' \\times S'$ :$\\preccurlyeq \\restriction_{S'}$ denotes the restriction of $\\preccurlyeq$ to $S' \\times S'$."}
+{"_id": "4770", "title": "Order Completion Unique up to Isomorphism", "text": "Let $\\left({S, \\preceq_S}\\right)$ be an ordered set. Suppose that both $\\left({T, \\preceq_T}\\right)$ and $\\left({T', \\preceq_{T'}}\\right)$ are order completions for $\\left({S, \\preceq_S}\\right)$. Then there exists a unique order isomorphism $\\psi: T \\to T'$. In particular, $\\left({T, \\preceq_T}\\right)$ and $\\left({T', \\preceq_{T'}}\\right)$ are isomorphic."}
+{"_id": "4771", "title": "Intersection of Strict Upper Closures in Toset", "text": "Let $\\left({S, \\preceq}\\right)$ be a totally ordered set. Let $a, b \\in S$. Then: :$a^\\succ \\cap b^\\succ = \\left({\\max \\left({a, b}\\right)}\\right)^\\succ$ where: :$a^\\succ$ denotes strict upper closure of $a$ :$\\max$ denotes the max operation."}
+{"_id": "4772", "title": "Intersection of Weak Upper Closures in Toset", "text": "Let $\\left({S, \\preccurlyeq}\\right)$ be a totally ordered set. Let $a, b \\in S$. Then: :$a^\\succcurlyeq \\cap b^\\succcurlyeq = \\left({\\max \\left({a, b}\\right)}\\right)^\\succcurlyeq$ where: : $a^\\succcurlyeq$ denotes weak upper closure of $a$ : $\\max$ denotes the max operation."}
+{"_id": "4773", "title": "Intersection of Weak Lower Closures in Toset", "text": "Let $\\struct {S, \\preccurlyeq}$ be a totally ordered set. Let $a, b \\in S$. Then: :$a^\\preccurlyeq \\cap b^\\preccurlyeq = \\paren {\\min \\set {a, b} }^\\preccurlyeq$ where: :$a^\\preccurlyeq$ denotes the weak lower closure of $a$ :$\\min$ denotes the min operation."}
+{"_id": "4775", "title": "Naturally Ordered Semigroup Exists", "text": "There exists a Naturally Ordered Semigroup."}
+{"_id": "4776", "title": "Strict Upper Closure in Restricted Ordering", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T \\subseteq S$ be a subset of $S$, and let $\\preceq \\restriction_T$ be the restricted ordering on $T$. Then for all $t \\in T$: :$t^{\\succ T} = T \\cap t^{\\succ S}$ where: : $t^{\\succ T}$ is the strict upper closure of $t$ in $\\left({T, \\preceq \\restriction_T}\\right)$ : $t^{\\succ S}$ is the strict upper closure of $t$ in $\\left({S, \\preceq}\\right)$."}
+{"_id": "4777", "title": "Weak Upper Closure in Restricted Ordering", "text": "Let $\\left({S, \\preccurlyeq}\\right)$ be an ordered set. Let $T \\subseteq S$ be a subset of $S$. Let $\\preccurlyeq \\restriction_T$ be the restricted ordering on $T$. Then for all $t \\in T$: :$t^{\\succcurlyeq T} = T \\cap t^{\\succcurlyeq S}$ where: : $t^{\\succcurlyeq T}$ is the weak upper closure of $t$ in $\\left({T, \\preccurlyeq \\restriction_T}\\right)$ : $t^{\\succcurlyeq S}$ is the weak upper closure of $t$ in $\\left({S, \\preccurlyeq}\\right)$."}
+{"_id": "4778", "title": "Strict Lower Closure in Restricted Ordering", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T \\subseteq S$ be a subset of $S$, and let $\\preceq \\restriction_T$ be the restricted ordering on $T$. Then for all $t \\in T$: :$t^{\\prec T} = T \\cap t^{\\prec S}$ where: : $t^{\\prec T}$ is the strict lower closure of $t$ in $\\left({T, \\preceq \\restriction_T}\\right)$ : $t^{\\prec S}$ is the strict lower closure of $t$ in $\\left({S, \\preceq}\\right)$."}
+{"_id": "4779", "title": "Weak Lower Closure in Restricted Ordering", "text": "Let $\\left({S, \\preccurlyeq}\\right)$ be an ordered set. Let $T \\subseteq S$ be a subset of $S$. Let $\\preccurlyeq \\restriction_T$ be the restricted ordering on $T$. Then for all $t \\in T$: :$t^{\\preccurlyeq T} = T \\cap t^{\\preccurlyeq S}$ where: :$t^{\\preccurlyeq T}$ is the weak lower closure of $t$ in $\\left({T, \\preccurlyeq \\restriction_T}\\right)$ :$t^{\\preccurlyeq S}$ is the weak lower closure of $t$ in $\\left({S, \\preccurlyeq}\\right)$."}
+{"_id": "4781", "title": "Natural Numbers under Multiplication form Ordered Commutative Semigroup", "text": "Let $\\N$ be the natural numbers. Let $\\times$ be multiplication. Let $\\le$ be the ordering on $\\N$. Then $\\left({\\N, \\times, \\le}\\right)$ is an ordered commutative semigroup."}
+{"_id": "4782", "title": "Invertible Elements under Natural Number Multiplication", "text": "Let $\\N$ be the natural numbers. Let $\\times$ denote multiplication. Then the only invertible element of $\\N$ for $\\times$ is $1$."}
+{"_id": "4783", "title": "Graph containing Closed Walk of Odd Length also contains Odd Cycle", "text": "Let $G$ be a graph. {{explain|This proof works for a simple graph, but the theorem may hold for loop graphs and/or multigraphs. Clarification needed as to what applies.}} Let $G$ have a closed walk of odd length. Then $G$ has an odd cycle."}
+{"_id": "4784", "title": "Homomorphism of Powers/Naturally Ordered Semigroup", "text": "Let $\\struct {S, \\circ, \\preceq}$ be a naturally ordered semigroup. For a given $a \\in T_1$, let $\\map {\\odot^n} a$ be the $n$th power of $a$ in $T_1$. For a given $a \\in T_2$, let $\\map {\\oplus^n} a$ be the $n$th power of $a$ in $T_2$. Then: :$\\forall a \\in T_1: \\forall n \\in \\struct {S^*, \\circ, \\preceq}: \\map \\phi {\\map {\\odot^n} a} = \\map {\\oplus^n} {\\map \\phi a}$ where $S^* = S \\setminus \\set 0$."}
+{"_id": "4785", "title": "Homomorphism of Powers/Natural Numbers", "text": "Let $n \\in \\N$. Let $\\odot^n$ and $\\oplus^n$ be the $n$th powers of $\\odot$ and $\\oplus$, respectively. Then: :$\\forall a \\in T_1: \\forall n \\in \\N: \\map \\phi {\\map {\\odot^n} a} = \\map {\\oplus^n} {\\map \\phi a}$"}
+{"_id": "4786", "title": "Homomorphism of Powers/Integers", "text": "Let $\\struct {T_1, \\odot}$ and $\\struct {T_2, \\oplus}$ be monoids. Let $\\phi: \\struct {T_1, \\odot} \\to \\struct {T_2, \\oplus}$ be a (semigroup) homomorphism. Let $a$ be an invertible element of $T_1$. Let $n \\in \\Z$. Let $\\odot^n$ and $\\oplus^n$ be as defined as in Index Laws for Monoids. Then: :$\\forall n \\in \\Z: \\map \\phi {\\map {\\odot^n} a} = \\map {\\oplus^n} {\\map \\phi a}$"}
+{"_id": "4787", "title": "Right Operation is Left Distributive over All Operations", "text": "Let $\\struct {S, \\circ, \\rightarrow}$ be an algebraic structure where: :$\\rightarrow$ is the right operation :$\\circ$ is any arbitrary binary operation. Then $\\rightarrow$ is left distributive over $\\circ$."}
+{"_id": "4788", "title": "Left Operation is Right Distributive over All Operations", "text": "Let $\\struct {S, \\circ, \\leftarrow}$ be an algebraic structure where: :$\\leftarrow$ is the left operation :$\\circ$ is any arbitrary binary operation. Then $\\leftarrow$ is right distributive over $\\circ$."}
+{"_id": "4789", "title": "Right Operation is Distributive over Idempotent Operation", "text": "Let $\\struct {S, \\circ, \\rightarrow}$ be an algebraic structure where: :$\\rightarrow$ is the right operation :$\\circ$ is any arbitrary binary operation. Then: :$\\rightarrow$ is distributive over $\\circ$ {{iff}} :$\\circ$ is idempotent."}
+{"_id": "4790", "title": "Left Operation is Distributive over Idempotent Operation", "text": "Let $\\struct {S, \\circ, \\leftarrow}$ be an algebraic structure where: :$\\leftarrow$ is the left operation :$\\circ$ is any arbitrary binary operation. Then: :$\\leftarrow$ is distributive over $\\circ$ {{iff}} :$\\circ$ is idempotent."}
+{"_id": "4791", "title": "Integral of Positive Measurable Function as Limit of Integrals of Positive Simple Functions", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $f: X \\to \\overline{\\R} \\in \\mathcal{M}_{\\overline{\\R}}^+$ be a positive $\\Sigma$-measurable function. Let $\\left({f_n}\\right)_{n \\in \\N} \\in \\mathcal{E}^+$, $f_n: X \\to \\R$ be a sequence of positive simple functions such that: :$\\displaystyle \\lim_{n \\to \\infty} f_n = f$ where $\\lim$ denotes a pointwise limit. Then: :$\\displaystyle \\int f \\, \\mathrm d\\mu = \\lim_{n \\to \\infty} \\int f_n \\, \\mathrm d\\mu$ where the integral signs denote $\\mu$-integration."}
+{"_id": "4792", "title": "Integral of Characteristic Function/Corollary", "text": ":$\\displaystyle \\int \\chi_E \\, \\mathrm d\\mu = \\mu \\left({E}\\right)$ where the integral sign denotes the $\\mu$-integral of $\\chi_E$."}
+{"_id": "4793", "title": "Integral of Positive Measurable Function is Positive Homogeneous", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f: X \\to \\R, f \\in \\mathcal M_{\\overline \\R}^+$ be a positive measurable function. Let $\\lambda \\in \\R_{\\ge 0}$ be a positive real number. Then: :$\\displaystyle \\int \\lambda f \\rd \\mu = \\lambda \\int f \\rd \\mu$ where: :$\\lambda f$ is the pointwise $\\lambda$-multiple of $f$ :The integral sign denotes $\\mu$-integration This can be summarized by saying that $\\displaystyle \\int \\cdot \\rd \\mu$ is positive homogeneous."}
+{"_id": "4794", "title": "Linear Transformation as Matrix Product", "text": "Let $T: \\R^n \\to \\R^m, \\mathbf x \\mapsto \\map T {\\mathbf x}$ be a linear transformation. Then: :$\\map T {\\mathbf x} = \\mathbf A_T \\mathbf x$ where $\\mathbf A_T$ is the $m \\times n$ matrix defined as: :$\\mathbf A_T = \\begin {bmatrix} \\map T {\\mathbf e_1} & \\map T {\\mathbf e_2} & \\cdots & \\map T {\\mathbf e_n} \\end {bmatrix}$ where $\\tuple {\\mathbf e_1, \\mathbf e_2, \\cdots, \\mathbf e_n}$ is the standard ordered basis of $\\R^n$."}
+{"_id": "4795", "title": "Integral of Positive Measurable Function is Additive", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $f,g: X \\to \\overline{\\R}$, $f,g \\in \\mathcal{M}_{\\overline{\\R}}^+$ be positive measurable functions. Then: :$\\displaystyle \\int f + g \\, \\mathrm d\\mu = \\displaystyle \\int f \\, \\mathrm d\\mu + \\displaystyle \\int g \\, \\mathrm d\\mu$ where: :$f + g$ is the pointwise sum of $f$ and $g$ :The integral sign denotes $\\mu$-integration This can be summarized by saying that $\\displaystyle \\int \\cdot \\, \\mathrm d\\mu$ is additive."}
+{"_id": "4796", "title": "Integral of Positive Simple Function is Increasing", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $f, g: X \\to \\R$, $f, g \\in \\mathcal{E}^+$ be positive simple functions. Suppose that: : $f \\le g$ where $\\le$ denotes pointwise inequality. Then: :$I_\\mu \\left({f}\\right) \\le I_\\mu \\left({g}\\right)$ where $I_\\mu$ denotes $\\mu$-integration This can be summarized by saying that $I_\\mu$ is an increasing mapping."}
+{"_id": "4797", "title": "Integral of Positive Measurable Function is Monotone", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $f,g: X \\to \\overline{\\R}$, $f,g \\in \\mathcal{M}_{\\overline{\\R}}^+$ be positive measurable functions. Suppose that $f \\le g$, where $\\le$ denotes pointwise inequality. Then: :$\\displaystyle \\int f \\, \\mathrm d\\mu \\le \\int g \\, \\mathrm d\\mu$ where the integral sign denotes $\\mu$-integration. This can be summarized by saying that $\\displaystyle \\int \\cdot \\, \\mathrm d\\mu$ is monotone."}
+{"_id": "4798", "title": "Matrix Multiplication is Homogeneous of Degree 1", "text": "Let $\\mathbf A$ be an $m \\times n$ matrix and $\\mathbf B$ be an $n \\times p$ matrix such that the columns of $\\mathbf A$ and $\\mathbf B$ are members of $\\R^m$ and $\\R^n$, respectively. Let $\\lambda \\in \\mathbb F \\in \\set {\\R, \\C}$ be a scalar. Then: :$\\mathbf A \\paren {\\lambda \\mathbf B} = \\lambda \\paren {\\mathbf A \\mathbf B}$"}
+{"_id": "4799", "title": "Series of Positive Measurable Functions is Positive Measurable Function", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $\\left({f_n}\\right)_{n \\in \\N} \\in \\mathcal{M}_{\\overline{\\R}}^+$, $f_n: X \\to \\overline{\\R}$ be a sequence of positive measurable functions. Let $\\displaystyle \\sum_{n \\mathop \\in \\N} f_n: X \\to \\overline{\\R}$ be the pointwise series of the $f_n$. Then $\\displaystyle \\sum_{n \\mathop \\in \\N} f_n$ is also a positive measurable function."}
+{"_id": "4800", "title": "Integral of Series of Positive Measurable Functions", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $\\left({f_n}\\right)_{n \\in \\N} \\in \\mathcal{M}_{\\overline{\\R}}^+$, $f_n: X \\to \\overline{\\R}$ be a sequence of positive measurable functions. Let $\\displaystyle \\sum_{n \\mathop \\in \\N} f_n: X \\to \\overline{\\R}$ be the pointwise series of the $f_n$. Then: :$\\displaystyle \\int \\sum_{n \\mathop \\in \\N} f_n \\, \\mathrm d \\mu = \\sum_{n \\mathop \\in \\N} \\int f_n \\, \\mathrm d \\mu$ where the integral sign denotes $\\mu$-integration."}
+{"_id": "4802", "title": "Integral with respect to Discrete Measure", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $\\displaystyle \\mu = \\sum_{n \\mathop \\in \\N} \\lambda_n \\delta_{x_n}$ be a discrete measure on $\\left({X, \\Sigma}\\right)$. Let $f \\in \\mathcal{M}_{\\overline{\\R}}^+, f: X \\to \\overline{\\R}$ be a positive measurable function. Then: :$\\displaystyle \\int f \\, \\mathrm d\\mu = \\sum_{n \\mathop \\in \\N} \\lambda_n f \\left({x_n}\\right)$ where the integral sign denotes $\\mu$-integration."}
+{"_id": "4803", "title": "Matrix Product as Linear Transformation", "text": "Let: :$ \\mathbf A_{m \\times n} = \\begin{bmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} \\\\ \\end{bmatrix}$ :$\\mathbf x_{n \\times 1} = \\begin{bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{bmatrix}$ :$\\mathbf y_{n \\times 1} = \\begin{bmatrix} y_1 \\\\ y_2 \\\\ \\vdots \\\\ y_n \\end{bmatrix}$ be matrices where each column is an element of a real vector space. Let $T$ be the mapping: :$T: \\R^m \\to \\R^n, \\mathbf x \\mapsto \\mathbf A \\mathbf x$ Then $T$ is a linear transformation."}
+{"_id": "4804", "title": "Fatou's Lemma for Integrals", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space."}
+{"_id": "4805", "title": "Linear Transformation Maps Zero Vector to Zero Vector", "text": "Let $\\mathbf V$ be a vector space, with zero $\\mathbf 0$. Likewise let $\\mathbf V\\,'$ be another vector space, with zero $\\mathbf 0'$. Let $T: \\mathbf V \\to \\mathbf V\\,'$ be a linear transformation. Then: :$T: \\mathbf 0 \\mapsto \\mathbf 0'$"}
+{"_id": "4806", "title": "Infimum of Product", "text": "Let $\\left({G, \\circ, \\preceq}\\right)$ be an ordered group. Suppose that subsets $A$ and $B$ of $G$ admit infima in $G$. Then: :$\\inf \\left({A \\circ_{\\mathcal P} B}\\right) = \\inf A \\circ \\inf B$ where $\\circ_{\\mathcal P}$ denotes subset product."}
+{"_id": "4807", "title": "Supremum of Product", "text": "Let $\\struct {G, \\circ, \\preceq}$ be an ordered group. Suppose that subsets $A$ and $B$ of $G$ admit suprema in $G$. Then: :$\\sup \\paren {A \\circ_{\\PP} B} = \\sup A \\circ \\sup B$ where $\\circ_{\\PP}$ denotes subset product."}
+{"_id": "4808", "title": "Integral with respect to Series of Measures", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $\\displaystyle \\mu := \\sum_{n \\mathop \\in \\N} \\lambda_n \\mu_n$ be a series of measures on $\\left({X, \\Sigma}\\right)$. Then for all positive measurable functions $f: X \\to \\overline \\R, f \\in \\mathcal M_{\\overline{\\R}}^+$: :$\\displaystyle \\int f \\, \\mathrm d \\mu = \\sum_{n \\mathop \\in \\N} \\int f \\, \\mathrm d \\mu_n$ where the integral signs denote integration with respect to a measure."}
+{"_id": "4809", "title": "Reverse Fatou's Lemma", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space."}
+{"_id": "4810", "title": "Characteristic Function of Limit Inferior of Sequence of Sets", "text": "Let $\\left({E_n}\\right)_{n \\in \\N}$ be a sequence of sets. Let $E := \\displaystyle \\liminf_{n \\mathop \\to \\infty} \\, E_n$ be the limit inferior of the $E_n$. Then: :$\\displaystyle \\chi_E = \\liminf_{n \\mathop \\to \\infty} \\, \\chi_{E_n}$ where: :$\\chi$ denotes characteristic function :$\\displaystyle \\liminf_{n \\to \\infty} \\, \\chi_{E_n}$ is the pointwise limit inferior of the $\\chi_{E_n}$"}
+{"_id": "4811", "title": "Characteristic Function of Limit Superior of Sequence of Sets", "text": "Let $\\left({E_n}\\right)_{n \\in \\N}$ be a sequence of sets. Let $E := \\displaystyle \\limsup_{n \\mathop \\to \\infty} \\, E_n$ be the limit superior of the $E_n$. Then: :$\\displaystyle \\chi_E = \\limsup_{n \\to \\infty} \\, \\chi_{E_n}$ where: :$\\chi$ denotes characteristic function :$\\displaystyle \\liminf_{n \\to \\infty} \\, \\chi_{E_n}$ is the pointwise limit superior of the $\\chi_{E_n}$"}
+{"_id": "4812", "title": "Fatou's Lemma for Measures", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $\\sequence {E_n}_{n \\mathop \\in \\N} \\in \\Sigma$ be a sequence of $\\Sigma$-measurable sets. Then: :$\\displaystyle \\map \\mu {\\liminf_{n \\mathop \\to \\infty} E_n} \\le \\liminf_{n \\mathop \\to \\infty} \\map \\mu {E_n}$ where: :$\\displaystyle \\liminf_{n \\mathop \\to \\infty} E_n$ is the limit inferior of the $E_n$ :the {{RHS}} limit inferior is taken in the extended real numbers $\\overline \\R$."}
+{"_id": "4813", "title": "Kernel Transformation of Measure is Measure", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $N: X \\times \\Sigma \\to \\overline{\\R}_{\\ge0}$ be a kernel. Then $\\mu N: X \\to \\overline{\\R}$, the kernel transformation of $\\mu$, is a measure."}
+{"_id": "4814", "title": "Kernel Transformation of Positive Measurable Function is Positive Measurable Function", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $N: X \\times \\Sigma \\to \\overline{\\R}_{\\ge0}$ be a kernel. Let $f: X \\to \\overline{\\R}$ be a positive measurable function. Then $N f: X \\to \\overline{\\R}$, the transformation of $f$ by $N$, is also a positive measurable function."}
+{"_id": "4815", "title": "Integral with respect to Kernel Transformation of Measure", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $N: X \\times \\Sigma \\to \\overline \\R_{\\ge 0}$ be a kernel. Let $f: X \\to \\overline \\R$ be a positive measurable function. Then: :$\\displaystyle \\int f \\map \\rd {\\mu N} = \\int N f \\rd \\mu$ where: :The integral sign denotes integration with respect to a measure :$\\mu N$ is the transformation of $\\mu$ by $N$ :$N f$ is the transformation of $f$ by $N$ Writing $\\map \\mu f$ in place of $\\displaystyle \\int f \\rd \\mu$, the theorem statement can be conveniently expressed as: :$\\map {\\mu N} f = \\map \\mu {N f}$"}
+{"_id": "4816", "title": "Canonical Injection is Injection", "text": "Let $\\struct {S_1, \\circ_1}$ and $\\struct {S_2, \\circ_2}$ be algebraic structures with identities $e_1, e_2$ respectively. The canonical injections: :$\\inj_1: \\struct {S_1, \\circ_1} \\to \\struct {S_1, \\circ_1} \\times \\struct {S_2, \\circ_2}: \\forall x \\in S_1: \\map {\\inj_1} x = \\tuple {x, e_2}$ :$\\inj_2: \\struct {S_2, \\circ_2} \\to \\struct {S_1, \\circ_1} \\times \\struct {S_2, \\circ_2}: \\forall x \\in S_2: \\map {\\inj_2} x = \\tuple {e_1, x}$ are injections."}
+{"_id": "4817", "title": "Canonical Injection is Injection/General Result", "text": "Let $\\struct {S_1, \\circ_1}, \\struct {S_2, \\circ_2}, \\dotsc, \\struct {S_j, \\circ_j}, \\dotsc, \\struct {S_n, \\circ_n}$ be algebraic structures with identities $e_1, e_2, \\ldots, e_j, \\ldots, e_n$ respectively. The canonical injection: :$\\displaystyle \\inj_j: \\struct {S_j, \\circ_j} \\to \\prod_{i \\mathop = 1}^n \\struct {S_i, \\circ_i}$ defined as: :$\\map {\\inj_j} x = \\tuple {e_1, e_2, \\dotsc, e_{j - 1}, x, e_{j + 1}, \\dotsc, e_n}$ is an injection."}
+{"_id": "4819", "title": "Finite Union of Countable Sets is Countable", "text": "The union of a finite number of countable sets is countable."}
+{"_id": "4822", "title": "Particular Values of Binomial Coefficients", "text": "=== Binomial Coefficient $\\dbinom 0 0$ === {{:Zero Choose Zero}} === Binomial Coefficient $\\dbinom 0 n$ === {{:Zero Choose n}} === Binomial Coefficient $\\dbinom 1 n$ === {{:One Choose n}} === N Choose Negative Number is Zero === {{:N Choose Negative Number is Zero}} === Binomial Coefficient with Zero === {{:Binomial Coefficient with Zero}} === Binomial Coefficient with One === {{:Binomial Coefficient with One}} === Binomial Coefficient with Self === {{:Binomial Coefficient with Self}} === Binomial Coefficient with Self minus One === {{:Binomial Coefficient with Self minus One}} === Binomial Coefficient with Two === {{:Binomial Coefficient with Two}}"}
+{"_id": "4823", "title": "Binomial Coefficient with Zero", "text": ":$\\forall r \\in \\R: \\dbinom r 0 = 1$"}
+{"_id": "4824", "title": "Binomial Coefficient with One", "text": ":$\\forall r \\in \\R: \\dbinom r 1 = r$"}
+{"_id": "4825", "title": "Binomial Coefficient with Self", "text": ":$\\forall n \\in \\Z: \\dbinom n n = \\sqbrk {n \\ge 0}$ where $\\sqbrk {n \\ge 0}$ denotes Iverson's convention. That is: :$\\forall n \\in \\Z_{\\ge 0}: \\dbinom n n = 1$ :$\\forall n \\in \\Z_{< 0}: \\dbinom n n = 0$"}
+{"_id": "4826", "title": "Binomial Coefficient with Two", "text": ":$\\forall r \\in \\R: \\dbinom r 2 = \\dfrac {r \\left({r - 1}\\right)} 2$"}
+{"_id": "4834", "title": "Binomial Theorem/Integral Index", "text": "Let $X$ be one of the set of numbers $\\N, \\Z, \\Q, \\R, \\C$. Let $x, y \\in X$. Then: {{begin-eqn}} {{eqn | ll= \\forall n \\in \\Z_{\\ge 0}:       | l = \\paren {x + y}^n       | r = \\sum_{k \\mathop = 0}^n \\binom n k x^{n - k} y^k       | c =  }} {{eqn | r = x^n + \\binom n 1 x^{n - 1} y + \\binom n 2 x^{n - 2} y^2 + \\binom n 3 x^{n - 3} y^3 + \\cdots       | c =  }} {{eqn | r = x^n + n x^{n - 1} y + \\frac {n \\paren {n - 1} } {2!} x^{n - 2} y^2 + \\frac {n \\paren {n - 1} \\paren {n - 3} } {3!} x^{n - 3} y^3 + \\cdots       | c =  }} {{end-eqn}} where $\\dbinom n k$ is $n$ choose $k$."}
+{"_id": "4835", "title": "Binomial Theorem/Ring Theory", "text": "Let $\\left({R, +, \\odot}\\right)$ be a ringoid such that $\\left({R, \\odot}\\right)$ is a commutative semigroup. Let $n \\in \\Z: n \\ge 2$. Then: :$\\displaystyle \\forall x, y \\in R: \\odot^n \\left({x + y}\\right) = \\odot^n x + \\sum_{k \\mathop = 1}^{n-1} \\binom n k \\left({\\odot^{n-k} x}\\right) \\odot \\left({\\odot^k y}\\right) + \\odot^n y$ where $\\dbinom n k = \\dfrac {n!} {k! \\ \\left({n - k}\\right)!}$ (see Binomial Coefficient). If $\\left({R, \\odot}\\right)$ has an identity element $e$, then: :$\\displaystyle \\forall x, y \\in R: \\odot^n \\left({x + y}\\right) = \\sum_{k \\mathop = 0}^n \\binom n k \\left({\\odot^{n - k} x}\\right) \\odot \\left({\\odot^k y}\\right)$"}
+{"_id": "4836", "title": "Derivative of Absolute Value Function", "text": "Let $\\size x$ be the absolute value of $x$ for real $x$. Then: :$\\dfrac \\d {\\d x} \\size x = \\dfrac x {\\size x}$ for $x \\ne 0$. At $x = 0$, $\\size x$ is not differentiable."}
+{"_id": "4837", "title": "Inverse Element in Inverse Completion of Commutative Monoid", "text": "Let $\\left({S, \\circ}\\right)$ be a commutative monoid. Let $\\left ({C, \\circ}\\right) \\subseteq \\left({S, \\circ}\\right)$ be the subsemigroup of cancellable elements of $\\left({S, \\circ}\\right)$. Let $\\left({T, \\circ'}\\right)$ be an inverse completion of $\\left({S, \\circ}\\right)$. Then the inverse of an element of $S$ which is invertible for $\\circ$ is also its inverse for $\\circ'$."}
+{"_id": "4838", "title": "Construction of Inverse Completion/Congruence Relation", "text": "The cross-relation $\\boxtimes$ is a congruence relation on $\\left({S \\times C, \\oplus}\\right)$. === Members of Equivalence Classes === {{:Construction of Inverse Completion/Equivalence Relation/Members of Equivalence Classes}} === Equivalence Class of Equal Elements === {{:Construction of Inverse Completion/Equivalence Relation/Equivalence Class of Equal Elements}}"}
+{"_id": "4839", "title": "Construction of Inverse Completion/Equivalence Relation/Members of Equivalence Classes", "text": "$\\forall x, y \\in S, a, b \\in C:$ :$(1): \\quad \\tuple {x \\circ a, a} \\boxtimes \\tuple {y \\circ b, b} \\iff x = y$ :$(2): \\quad \\eqclass {\\tuple {x \\circ a, y \\circ a} } \\boxtimes = \\eqclass {\\tuple {x, y} } \\boxtimes$ where $\\eqclass {\\tuple {x, y} } \\boxtimes$ is the equivalence class of $\\tuple {x, y}$ under $\\boxtimes$."}
+{"_id": "4840", "title": "Construction of Inverse Completion/Equivalence Relation/Equivalence Class of Equal Elements", "text": ":$\\forall c, d \\in C: \\left({c, c}\\right) \\boxtimes \\left({d, d}\\right)$"}
+{"_id": "4841", "title": "Construction of Inverse Completion/Quotient Structure is Commutative Semigroup", "text": ":$\\left({T', \\oplus'}\\right)$ is a commutative semigroup."}
+{"_id": "4842", "title": "Construction of Inverse Completion/Quotient Mapping is Injective", "text": "Let the mapping $\\psi: S \\to T'$ be defined as: :$\\forall x \\in S: \\psi \\left({x}\\right) = \\left[\\!\\left[{\\left({x \\circ a, a}\\right)}\\right]\\!\\right]_\\boxtimes$ Then $\\psi: S \\to T'$ is an injection, and does not depend on the particular element $a$ chosen."}
+{"_id": "4843", "title": "Construction of Inverse Completion/Quotient Mapping is Monomorphism", "text": "The mapping $\\psi: S \\to T'$ is a monomorphism."}
+{"_id": "4844", "title": "Construction of Inverse Completion/Quotient Mapping to Image is Isomorphism", "text": "Let $S'$ be the image $\\psi \\left({S}\\right)$ of $S$. Then $\\psi$ is an isomorphism from $S$ onto $S'$."}
+{"_id": "4845", "title": "Construction of Inverse Completion/Quotient Mapping/Image of Cancellable Elements", "text": "The set $C'$ of cancellable elements of the semigroup $S'$ is $\\psi \\left[{C}\\right]$."}
+{"_id": "4846", "title": "Construction of Inverse Completion/Image of Quotient Mapping is Subsemigroup", "text": "Let $S'$ be the image $\\psi \\left({S}\\right)$ of $S$. Then $\\left({S', \\oplus'}\\right)$ is a subsemigroup of $\\left({T', \\oplus'}\\right)$."}
+{"_id": "4847", "title": "Construction of Inverse Completion/Identity of Quotient Structure", "text": "Let $c \\in C$ be arbitrary. Then: :$\\eqclass {\\tuple {c, c} } \\boxtimes$ is the identity of $T'$."}
+{"_id": "4848", "title": "Construction of Inverse Completion/Invertible Elements in Quotient Structure", "text": "Every cancellable element of $S'$ is invertible in $T'$."}
+{"_id": "4849", "title": "Construction of Inverse Completion/Generator for Quotient Structure", "text": "$T' = S' \\cup \\left({C'}\\right)^{-1}$ is a generator for the semigroup $T'$."}
+{"_id": "4850", "title": "Construction of Inverse Completion/Quotient Structure is Inverse Completion", "text": "$T'$ is an inverse completion of its subsemigroup $S'$."}
+{"_id": "4851", "title": "Ring of Integers Modulo Composite is not Integral Domain", "text": "Let $m \\in \\Z: m \\ge 2$. Let $\\struct {\\Z_m, +, \\times}$ be the ring of integers modulo $m$. Let $m$ be a composite number. Then $\\struct {\\Z_m, +, \\times}$ is not an integral domain."}
+{"_id": "4852", "title": "Ring Zero is not Cancellable", "text": "Let $\\struct {R, +, \\circ}$ be a ring which is not null. Let $0$ be the ring zero of $R$. Then $0$ is not a cancellable element for the ring product $\\circ$."}
+{"_id": "4853", "title": "Congruence Relation on Ring induces Ideal", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring. Let $\\mathcal E$ be a congruence relation on $R$. Let $J = \\left[\\!\\left[{0_R}\\right]\\!\\right]_\\mathcal E$ be the equivalence class of $0_R$ under $\\mathcal E$. Then $J$ is an ideal of $R$."}
+{"_id": "4854", "title": "Ideal induces Congruence Relation on Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $J$ be an ideal of $R$ Then $J$ induces a congruence relation $\\EE_J$ on $R$ such that $\\struct {R / J, +, \\circ}$ is a quotient ring."}
+{"_id": "4855", "title": "Vector Space has Basis between Linearly Independent Set and Spanning Set", "text": "Let $V$ be a vector space over a field $F$. Let $L$ be a linearly independent subset of $V$. Let $S$ be a set that spans $V$. Suppose that $L \\subseteq S$. Then $V$ has a basis $B$ such that $L \\subseteq B \\subseteq S$."}
+{"_id": "4858", "title": "Null Ring is Trivial Ring", "text": "Let $R$ be the null ring. Then $R$ is a trivial ring and therefore a commutative ring."}
+{"_id": "4859", "title": "Null Ring is Ring", "text": "Let $R$ be the null ring. That is, let: : $R := \\left({\\left\\{{0_R}\\right\\}, +, \\circ}\\right)$ where ring addition and the ring product are defined as: * $0_R + 0_R = 0_R$ * $0_R \\circ 0_R = 0_R$ Then $R$ is a ring."}
+{"_id": "4860", "title": "Trivial Group is Cyclic Group", "text": "The trivial group is a cyclic group and therefore abelian."}
+{"_id": "4861", "title": "Trivial Ring is Commutative Ring", "text": "Let $\\struct {R, +, \\circ}$ be a trivial ring. Then $\\struct {R, +, \\circ}$ is a commutative ring."}
+{"_id": "4862", "title": "Congruence Relation and Ideal are Equivalent", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring. Let $\\mathcal E$ be an equivalence relation on $R$ compatible with both $\\circ$ and $+$, i.e. a congruence relation on $R$. Let $J = \\left[\\!\\left[{0_R}\\right]\\!\\right]_\\mathcal E$ be the equivalence class of $0_R$ under $\\mathcal E$. Then: : $(1a): \\quad J = \\left[\\!\\left[{0_R}\\right]\\!\\right]_\\mathcal E$ is an ideal of $R$ : $(2a): \\quad$ The equivalence defined by the quotient ring $R / J$ is $\\mathcal E$ itself. Similarly, let $J$ be an ideal of $R$. Then: : $(1b): \\quad J$ induces a congruence relation $\\mathcal E_J$ on $R$ : $(2b): \\quad$ The ideal of $R$ defined by $\\mathcal E_J$ is $J$ itself."}
+{"_id": "4863", "title": "Quotient Ring is Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $J$ be an ideal of $R$. Let $\\struct {R / J, +, \\circ}$ be the quotient ring of $R$ by $J$. Then $R / J$ is also a ring."}
+{"_id": "4864", "title": "Ideal is Additive Normal Subgroup", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring. Let $J$ be an ideal of $R$. Then $\\left({J, +}\\right)$ is a normal subgroup of $\\left({R, +}\\right)$."}
+{"_id": "4867", "title": "Characterization of Integrable Functions", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f: X \\to \\overline \\R, f \\in \\MM_{\\overline \\R}$ be a $\\Sigma$-measurable function. Then the following are equivalent: :$(1): \\quad f \\in \\map {\\LL_{\\overline \\R} } \\mu$, that is, $f$ is $\\mu$-integrable. :$(2): \\quad$ The positive and negative parts $f^+$ and $f^-$ are $\\mu$-integrable. :$(3): \\quad$ The absolute value $\\size f$ of $f$ is $\\mu$-integrable. :$(4): \\quad$ There exists an $\\mu$-integrable function $g: X \\to \\overline \\R$ such that $\\size f \\le g$ pointwise."}
+{"_id": "4868", "title": "Quotient Ring is Ring/Quotient Ring Addition is Well-Defined", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let $\\struct {R / J, +, \\circ}$ be the quotient ring of $R$ by $J$. Then $+$ is well-defined on $R / J$, that is: :$x_1 + J = x_2 + J, y_1 + J = y_2 + J \\implies \\paren {x_1 + y_1} + J = \\paren {x_2 + y_2} + J$"}
+{"_id": "4869", "title": "Implicitly Defined Real-Valued Function", "text": "Let $F: \\struct {\\mathbf X' \\subseteq \\R^{n + 1} } \\to \\struct {\\mathbb I' \\subseteq \\R}$ have continuous partial derivatives.  {{explain|Can the language of this be brought into line with existing definitions of implicit functions?}} Let $\\tuple {\\mathbf x, z}$ denote an element of $\\R^{n + 1}$, where $\\mathbf x \\in \\R^n$ and $z \\in \\R$. Suppose $\\exists \\tuple {\\mathbf x_0, z_0} \\in \\mathbf X'$ such that: :$\\map F {\\mathbf x_0, z_0} = 0$ :$\\dfrac \\partial {\\partial z} \\map F {\\mathbf x_0, z_0} \\ne 0$ Then there exists a unique mapping of the form: :$g: \\mathbf X \\to \\mathbb I$ where $\\mathbf X \\subseteq \\R^n$ contains $\\mathbf x_0$ and $\\mathbb I$ is an open real interval containing $z_0$, such that: :$\\forall \\mathbf x \\in \\mathbf X, z \\in \\mathbb I: \\map F {\\mathbf x, z} = 0 \\iff z = \\map g {\\mathbf x}$ and $g$ itself has continuous partial derivatives. {{proofread}}"}
+{"_id": "4873", "title": "Characteristic Function of Subset", "text": "Let $A \\subseteq B \\subseteq S$. Then: :$\\forall s \\in S: \\chi_A \\left({s}\\right) \\le \\chi_B \\left({s}\\right)$ where $\\chi$ denotes characteristic function."}
+{"_id": "4874", "title": "Characteristic Function Determined by 0-Fiber", "text": "Let $A \\subseteq S$. Let $f: S \\to \\set {0, 1}$ be a mapping. Denote by $\\chi_A$ the characteristic function on $A$. Then the following are equivalent: :$(1): \\quad f = \\chi_A$ :$(2): \\quad \\forall s \\in S: \\map f s = 0 \\iff s \\notin A$ Using the notion of a fiber, $(2)$ may also be expressed as: :$(2'):\\quad \\map {f^{-1} } 0 = S \\setminus A$"}
+{"_id": "4875", "title": "Ideal induced by Congruence Relation defines that Congruence", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring. Let $\\mathcal E$ be a congruence relation on $R$. Let $J = \\left[\\!\\left[{0_R}\\right]\\!\\right]_\\mathcal E$ be the ideal induced by $\\mathcal E$. Then the equivalence defined by the coset space $\\left({R, +}\\right) / \\left({J, +}\\right)$ is $\\mathcal E$ itself."}
+{"_id": "4876", "title": "Rational Numbers Null Set under Lebesgue Measure", "text": "Let $\\lambda$ be $1$-dimensional Lebesgue measure on $\\R$. Let $\\Q$ be the set of rational numbers. Then $\\lambda \\left({\\Q}\\right) = 0$, i.e. $\\Q$ is a $\\lambda$-null set."}
+{"_id": "4877", "title": "Cartesian Product of Intersections/General Case", "text": ":$\\displaystyle \\paren{ \\prod_{i \\mathop \\in I} S_i } \\cap \\paren{ \\prod_{i \\mathop \\in I} T_i } = \\prod_{i \\mathop \\in I} \\paren{S_i \\cap T_i}$"}
+{"_id": "4878", "title": "Cartesian Product of Semirings of Sets", "text": "Let $\\SS$ and $\\TT$ be semirings of sets. Then $\\SS \\times \\TT$ is also a semiring of sets. Here, $\\times$ denotes Cartesian product."}
+{"_id": "4879", "title": "Congruence Relation on Group induces Normal Subgroup", "text": "Let $\\left({G, \\circ}\\right)$ be a group whose identity is $e$. Let $\\mathcal R$ be a congruence relation for $\\circ$. Let $H = \\left[\\!\\left[{e}\\right]\\!\\right]_\\mathcal R$, where $\\left[\\!\\left[{e}\\right]\\!\\right]_\\mathcal R$ is the equivalence class of $e$ under $\\mathcal R$. Then: : $\\left({H, \\circ \\restriction_H}\\right)$ is a normal subgroup of $G$ where $\\circ \\restriction_H$ denotes the restriction of $\\circ$ to $H$."}
+{"_id": "4880", "title": "Normal Subgroup induced by Congruence Relation defines that Congruence", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $\\mathcal R$ be a congruence relation for $\\circ$. Let $\\eqclass e {\\mathcal R}$ be the equivalence class of $e$ under $\\mathcal R$. Let $N = \\eqclass e {\\mathcal R}$ be the normal subgroup induced by $\\mathcal R$. {{explain|Check the above because it does not look right}} Then $\\mathcal R$ is the equivalence relation $\\mathcal R_N$ defined by $N$."}
+{"_id": "4881", "title": "Quotient Structure on Group defined by Congruence equals Quotient Group", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $\\mathcal R$ be a congruence relation for $\\circ$. Let $\\struct {G / \\mathcal R, \\circ_\\mathcal R}$ be the quotient structure defined by $\\mathcal R$. Let $N = \\eqclass e {\\mathcal R}$ be the normal subgroup induced by $\\mathcal R$. Let $\\struct {G / N, \\circ_N}$ be the quotient group of $G$ by $N$. Then $\\struct {G / \\mathcal R, \\circ_\\mathcal R}$ is the subgroup $\\struct {G / N, \\circ_N}$ of the semigroup $\\struct {\\powerset G, \\circ_\\mathcal P}$."}
+{"_id": "4882", "title": "Vector Scaled by Zero is Zero Vector", "text": "Let $F$ be a field whose zero is $0_F$. Let $\\struct {\\mathbf V, +, \\circ}_F$ be a vector space over $F$, as defined by the vector space axioms. Then: :$\\forall \\mathbf v \\in \\mathbf V: 0_F \\circ \\mathbf v = \\bszero$"}
+{"_id": "4883", "title": "Zero Vector Scaled is Zero Vector", "text": "Let $\\struct {\\mathbf V, +, \\circ}_F$ be a vector space over a field $F$, as defined by the vector space axioms. Then: :$\\forall \\lambda \\in \\mathbb F: \\lambda \\circ \\bszero = \\bszero$ where $\\bszero \\in \\mathbf V$ is the zero vector."}
+{"_id": "4884", "title": "Vector Product is Zero only if Factor is Zero", "text": "Let $F$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $\\struct {\\mathbf V, +, \\circ}_F$ be a vector space over $F$, as defined by the vector space axioms. Then: :$\\forall \\lambda \\in F: \\forall \\mathbf v \\in \\mathbf V: \\lambda \\circ \\mathbf v = \\bszero \\implies \\paren {\\lambda = 0_F \\lor \\mathbf v = \\mathbf 0}$ where $\\bszero \\in \\mathbf V$ is the zero vector."}
+{"_id": "4886", "title": "Null Ring is Ideal", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Then the null ring $\\struct {\\set {0_R}, +, \\circ}$ is an ideal of $R$."}
+{"_id": "4887", "title": "Ring Epimorphism Preserves Subrings", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring epimorphism. Let $S$ be a subring of $R_1$. Then $\\phi \\sqbrk S$ is a subring of $R_2$."}
+{"_id": "4888", "title": "Preimage of Image of Ideal under Ring Homomorphism", "text": "Let $\\phi: \\left({R_1, +_1, \\circ_1}\\right) \\to \\left({R_2, +_2, \\circ_2}\\right)$ be a ring homomorphism. Let $K = \\ker \\left({\\phi}\\right)$, where $\\ker \\left({\\phi}\\right)$ is the kernel of $\\phi$.  Let $J$ be an ideal of $R_1$. Then: :$\\phi^{-1} \\left({\\phi \\left({J}\\right)}\\right) = J + K$"}
+{"_id": "4889", "title": "Image of Preimage of Ideal under Ring Epimorphism", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring epimorphism. Let $S_2$ be an ideal of $R_2$. Then: :$\\phi \\sqbrk {\\phi^{-1} \\sqbrk {S_2} } = S_2$"}
+{"_id": "4890", "title": "Vectors are Right Cancellable", "text": "Let $\\struct {\\mathbf V, +, \\circ}_{\\mathbb F}$ be a vector space over $\\mathbb F$, as defined by the vector space axioms. Then every $\\mathbf v \\in \\struct {\\mathbf V, +}$ is right cancellable: :$\\forall \\mathbf a, \\mathbf b, \\mathbf c \\in \\mathbf V: \\mathbf a + \\mathbf c = \\mathbf b + \\mathbf c \\implies \\mathbf a = \\mathbf b$"}
+{"_id": "4891", "title": "Vectors are Left Cancellable", "text": "Let $\\left({\\mathbf V, +, \\circ}\\right)_{\\mathbb F}$ be a vector space over $\\mathbb F$, as defined by the vector space axioms. Then every $\\mathbf v \\in \\left({\\mathbf V, +}\\right)$ is left cancellable: :$\\forall \\mathbf a, \\mathbf b, \\mathbf c \\in \\mathbf V: \\mathbf c + \\mathbf a = \\mathbf c + \\mathbf b \\implies \\mathbf a = \\mathbf b$"}
+{"_id": "4893", "title": "Additive Inverse in Vector Space is Unique", "text": "Let $\\struct {\\mathbf V, +, \\circ}_F$ be a vector space over a field $F$, as defined by the vector space axioms. Then for every $\\mathbf v \\in \\mathbf V$, the additive inverse of $\\mathbf v$ is unique: :$\\forall \\mathbf v \\in \\mathbf V: \\exists! \\paren {-\\mathbf v} \\in \\mathbf V: \\mathbf v + \\paren {-\\mathbf v} = \\mathbf 0$"}
+{"_id": "4894", "title": "Vector Inverse is Negative Vector", "text": "Let $F$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $\\struct {\\mathbf V, +, \\circ}_F$ be a vector space over $F$, as defined by the vector space axioms. Then: :$\\forall \\mathbf v \\in \\mathbf V: -\\mathbf v = -1_F \\circ \\mathbf v$"}
+{"_id": "4895", "title": "Non-Trivial Commutative Division Ring is Field", "text": "Let $\\struct {R, +, \\circ}$ be a non-trivial division ring such that $\\circ$ is commutative. Then $\\struct {R, +, \\circ}$ is a field. Similarly, let $\\struct {F, +, \\circ}$ be a field. Then $\\struct {F, +, \\circ}$ is a non-trivial  division ring such that $\\circ$ is commutative."}
+{"_id": "4896", "title": "Division Ring has No Proper Zero Divisors", "text": "Let $\\left({R, +, \\circ}\\right)$ be a division ring. Then $\\left({R, +, \\circ}\\right)$ has no proper zero divisors."}
+{"_id": "4903", "title": "Linear Transformation is Injective iff Kernel Contains Only Zero", "text": "Let $\\mathbf V, \\mathbf V'$ be vector spaces, with respective zeroes $\\mathbf 0, \\mathbf 0'$. Let $T: \\mathbf V \\to \\mathbf V'$ be a linear transformation. Then: :$T$ is injective {{iff}} $\\map \\ker T = \\set {\\mathbf 0}$ where: :$\\mathbf 0$ is the zero of the domain of $T$ :$\\map \\ker T$ is the kernel of $T$."}
+{"_id": "4904", "title": "Diagonal Relation on Ring is Ordering Compatible with Ring Structure", "text": "Let $\\struct {R, +, \\circ, \\preceq}$ be a ring whose zero is $0_R$. Then the diagonal relation $\\Delta_R$ on $R$ is an ordering compatible with the ring structure of $R$."}
+{"_id": "4905", "title": "Nagata-Smirnov Metrization Theorem", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Then $T$ is metrizable {{iff}} $T$ is regular and has a basis that is countably locally finite."}
+{"_id": "4906", "title": "Smirnov Metrization Theorem", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Then $T$ is metrizable {{iff}} it is paracompact and locally metrizable."}
+{"_id": "4907", "title": "Identity Mapping is Automorphism/Semigroups", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroup. Then $I_S: \\left({S, \\circ}\\right) \\to \\left({S, \\circ}\\right)$ is a semigroup automorphism."}
+{"_id": "4909", "title": "Inverse of Reflexive Relation is Reflexive", "text": "Let $\\mathcal R$ be a relation on a set $S$. If $\\mathcal R$ is reflexive, then so is $\\mathcal R^{-1}$."}
+{"_id": "4910", "title": "Inverse of Antireflexive Relation is Antireflexive", "text": "Let $\\mathcal R$ be a relation on a set $S$. If $\\mathcal R$ is antireflexive, then so is $\\mathcal R^{-1}$."}
+{"_id": "4911", "title": "Inverse of Non-Reflexive Relation is Non-Reflexive", "text": "Let $\\mathcal R$ be a relation on a set $S$. If $\\mathcal R$ is non-reflexive, then so is $\\mathcal R^{-1}$."}
+{"_id": "4912", "title": "Inverse of Symmetric Relation is Symmetric", "text": "Let $\\mathcal R$ be a relation on a set $S$. If $\\mathcal R$ is symmetric, then so is $\\mathcal R^{-1}$."}
+{"_id": "4913", "title": "Inverse of Asymmetric Relation is Asymmetric", "text": "Let $\\mathcal R$ be a relation on a set $S$. If $\\mathcal R$ is asymmetric, then so is $\\mathcal R^{-1}$."}
+{"_id": "4914", "title": "Inverse of Antisymmetric Relation is Antisymmetric", "text": "Let $\\RR$ be a relation on a set $S$. If $\\RR$ is antisymmetric, then so is $\\RR^{-1}$."}
+{"_id": "4915", "title": "Inverse of Non-Symmetric Relation is Non-Symmetric", "text": "Let $\\mathcal R$ be a relation on a set $S$. If $\\mathcal R$ is non-symmetric, then so is $\\mathcal R^{-1}$."}
+{"_id": "4916", "title": "Inverse of Transitive Relation is Transitive", "text": "Let $\\mathcal R$ be a relation on a set $S$. Let $\\mathcal R$ be transitive. Then its inverse $\\mathcal R^{-1}$ is also transitive."}
+{"_id": "4917", "title": "Inverse of Antitransitive Relation is Antitransitive", "text": "Let $\\RR$ be a relation on a set $S$. If $\\RR$ is antitransitive, then so is $\\RR^{-1}$."}
+{"_id": "4918", "title": "Inverse of Non-Transitive Relation is Non-Transitive", "text": "Let $\\mathcal R$ be a relation on a set $S$. If $\\mathcal R$ is non-transitive, then so is $\\mathcal R^{-1}$."}
+{"_id": "4919", "title": "Inverse of Ordered Ring Isomorphism is Ordered Ring Isomorphism", "text": "Let $\\left({S, +, \\circ, \\preceq}\\right)$ and $\\left({T, \\oplus, *, \\preccurlyeq}\\right)$ be ordered rings. Let $\\phi: S \\to T$ be an ordered ring isomorphism. Then $\\phi^{-1}: T \\to S$ is also an ordered ring isomorphism."}
+{"_id": "4920", "title": "Feit-Thompson Theorem", "text": "All finite groups of odd order are solvable. That is, every non-abelian group finite simple group is of even order."}
+{"_id": "4921", "title": "Orthogonal Group is Subgroup of General Linear Group", "text": "Let $k$ be a field. Let $\\operatorname O \\left({n, k}\\right)$ be the $n$th orthogonal group on $k$. Let $\\operatorname{GL} \\left({n, k}\\right)$ be the $n$th general linear group on $k$. Then $\\operatorname O \\left({n, k}\\right)$ is a subgroup of $\\operatorname{GL} \\left({n, k}\\right)$."}
+{"_id": "4922", "title": "Orthogonal Group is Group", "text": "Let $k$ be a field. The $n$th orthogonal group on $k$ is a group."}
+{"_id": "4923", "title": "Composite of Ordered Ring Isomorphisms is Ordered Ring Isomorphism", "text": "Let $\\left({S_1, +_1, \\circ_1, \\preccurlyeq_1}\\right), \\left({S_2, +_2, \\circ_2, \\preccurlyeq_2}\\right), \\left({S_3, +_3, \\circ_3, \\preccurlyeq_3}\\right)$ be ordered rings. Let $\\phi: S_1 \\to S_2$ and $\\psi: S_2 \\to S_3$ be ordered ring isomorphisms. Then the composite mapping $\\psi \\circ \\phi: S_1 \\to S_3$ is also an ordered ring isomorphism."}
+{"_id": "4924", "title": "Composite of Ordered Ring Monomorphisms is Ordered Ring Monomorphism", "text": "Let $\\left({S_1, +_1, \\circ_1, \\preccurlyeq_1}\\right), \\left({S_2, +_2, \\circ_2, \\preccurlyeq_2}\\right), \\left({S_3, +_3, \\circ_3, \\preccurlyeq_3}\\right)$ be ordered rings. Let $\\phi: S_1 \\to S_2$ and $\\psi: S_2 \\to S_3$ be ordered ring monomorphisms. Then the composite mapping $\\psi \\circ \\phi: S_1 \\to S_3$ is also an ordered ring monomorphism."}
+{"_id": "4927", "title": "Rescaling is Linear Transformation", "text": "Let $\\left({R, +, \\cdot}\\right)$ be a commutative ring. Let $\\left({V, +, \\circ}\\right)_R$ be an $R$-module. Then for any $r \\in R$, the rescaling: :$m_r: V \\to V, v \\mapsto r \\circ v$ is a linear transformation."}
+{"_id": "4928", "title": "Determinant of Rescaling Matrix", "text": "Let $R$ be a commutative ring. Let $r \\in R$. Let $r \\, \\mathbf I_n$ be the square matrix of order $n$ defined by: :$\\sqbrk {r \\, \\mathbf I_n}_{i j} = \\begin{cases} r & : i = j \\\\ 0 & : i \\ne j \\end{cases}$ Then: :$\\map \\det {r \\, \\mathbf I_n} = r^n$ where $\\det$ denotes determinant."}
+{"_id": "4929", "title": "Inverse of Rescaling Matrix", "text": "Let $R$ be a commutative ring with unity. Let $r \\in R$ be a unit in $R$. Let $r \\, \\mathbf I_n$ be the $n \\times n$ rescaling matrix of $r$. Then $\\left({r \\, \\mathbf I_n}\\right)^{-1} = r^{-1} \\, \\mathbf I_n$."}
+{"_id": "4930", "title": "Powers of Ring Elements/General Result", "text": ":$\\forall m, n \\in \\Z: \\forall x \\in R: \\paren {m \\cdot x} \\circ \\paren {n \\cdot x} = \\paren {m n} \\cdot \\paren {x \\circ x}$."}
+{"_id": "4933", "title": "Characteristic of Division Ring is Zero or Prime", "text": "Let $\\struct {D, +, \\circ}$ be a division ring. Let $\\map {\\operatorname {Char} } D$ be the characteristic of $D$. Then $\\map {\\operatorname {Char} } D$ is either $0$ or a prime number."}
+{"_id": "4937", "title": "Magdy's Exchanger", "text": "'''Magdy's Exchanger''' is a simple method to get a semi-algebraic formula for inverse trigonometric functions using the help of inverse hyperbolic functions that has a real logarthmic formula hence we will have an example to get the inverse sine function. First: :$\\displaystyle \\arcsin x  = \\int \\frac 1 {\\sqrt{1 - x^2}} \\, \\mathrm d x$ Using Integration by Parts: :$\\displaystyle \\int \\frac 1 {\\sqrt{1-x^2}} \\,\\mathrm d x = -\\frac {\\sqrt {1-x^2}} x - \\int \\frac{\\sqrt{1-x^2}} {x^2} \\, \\mathrm d x$ The problem is that the integration on the {{RHS}} will inverse itself if we solve it by parts. Then the result will be: :$0 = 0$ Magdy's method will come into effect now. As we know the inverse hyperbolic function that is most similar to the inverse sine function is the inverse sinh function. The idea is to get the last integration in the {{RHS}} from the inverse sinh function. So we use parts again on the derivative of inverse sinh as follows: :$\\displaystyle \\operatorname{arcsinh} x = \\int \\frac 1 {\\sqrt{1 + x^2}} \\, \\mathrm d x$ {{begin-eqn}} {{eqn | l = \\operatorname{arcsinh} x       | r = \\int \\frac 1 {\\sqrt{1 + x^2} } \\, \\mathrm d x       | c = Primitive of $\\dfrac 1 {\\sqrt {x^2 + a^2} }$ }} {{eqn | r = \\frac {\\sqrt {1 + x^2} } x + \\int \\frac {\\sqrt {1 + x^2} } {x^2} \\,\\mathrm d x       | c = Integration by Parts }} {{eqn | r = \\ln \\left({x + \\sqrt{x^2 + 1} }\\right)       | c = Definition of Inverse Sinh Function }} {{end-eqn}} By Primitive of Root of $\\dfrac {\\sqrt {x^2 + a^2} } {x^2}$: :$\\displaystyle \\int \\frac {\\sqrt {1 + x^2} } {x^2} \\, \\mathrm d x = \\ln \\left({x + \\sqrt{x^2 + 1} }\\right) - \\frac {\\sqrt {1 + x^2} } x$ Taking the derivativeof both sides: :$\\displaystyle \\frac{\\sqrt{1 + x^2} } {x^2} = \\frac {1 + x^2 + x \\sqrt{x^2 + 1} } {x^3 + x^2 \\sqrt {x^2 + 1} } = A$ Squaring both sides: :$\\dfrac {1 + x^2} {x^4} = A^2$ Now the most important step. As we want to have $1 - x^2$ in the numerator we will have the fraction on the {{LHS}}} as: :$\\dfrac {1 - x^2 + 2 x^2} {x^4}$ which does not affect the value of the fraction. Now the {{LHS}} is: :$\\dfrac {1 - x^2} {x^4} + \\dfrac 2 {x^2}$ then the wanted fraction will be alone on the {{LHS}}: :$\\dfrac {1 - x^2} {x^4} = A^2 - \\dfrac 2 {x^2}$ Taking the root of both sides: :$\\dfrac {\\sqrt{1 - x^2} } {x^2} = \\sqrt {A^2 - \\dfrac 2 {x^2} }$ then integrating both sides: :$\\displaystyle \\int \\frac {\\sqrt{1 - x^2} } {x^2} \\, \\mathrm d x = \\int \\sqrt{A^2 - \\frac 2 {x^2}} \\, \\mathrm d x$ Now the {{RHS}} is complicated but evaluates as follows: :$\\displaystyle \\int \\sqrt{A^2 - \\frac 2 {x^2} } \\, \\mathrm d x = \\frac {x^2 \\sqrt {\\frac 1 {x^4} - \\frac 1 {x^2} } \\ln \\left({2 \\sqrt {x^2 - 1} + 2 x}\\right)} {\\sqrt {x^2 - 1} } - x \\sqrt {\\frac 1 {x^4} - \\frac 1 {x^2} } + C$ Substituting the last result in the first equation of the inverse sine we get: :$\\displaystyle \\arcsin x = - \\frac {\\sqrt {1 - x^2} } x - \\frac {x^2 \\sqrt {\\frac 1 {x^4} - \\frac 1 {x^2} } \\ln \\left({2 \\sqrt {x^2 - 1} + 2 x}\\right)} {\\sqrt {x^2 - 1}} +  x \\sqrt {\\frac 1 {x^4} - \\frac 1 {x^2} } + C$ Hence we get a semi-algebraic formula for the inverse sine function. {{qed}}"}
+{"_id": "4940", "title": "Sigma-Algebra Generated by Complements of Generators", "text": "Let $\\Sigma$ be a $\\sigma$-algebra on a set $X$. Let $\\mathcal G$ be a generator for $\\Sigma$. Then: :$\\mathcal{G}' := \\left\\{{X \\setminus G: G \\in \\mathcal G}\\right\\}$ the set of relative complements of $\\mathcal G$, is also a generator for $\\Sigma$."}
+{"_id": "4950", "title": "Primary Decomposition Theorem", "text": "Let $K$ be a field. Let $V$ be a vector space over $K$. Let $T: V \\to V$ be a linear operator on $V$. Let $\\map p x \\in K \\sqbrk x$ be a polynomial such that: :$\\map \\deg p \\ge 1$ :$\\map p T = 0$ where $0$ is the zero operator on $V$. {{explain|Link to definition of $K \\sqbrk x$}} Let $\\map {p_1} x, \\map {p_2} x, \\ldots, \\map {p_r} x$ be distinct irreducible monic polynomials. Let $c \\in K \\setminus \\set 0$ and $a_1, a_2, \\ldots, a_r, r \\in \\Z_{\\ge 1}$ be constants. We have that: :$\\map p x = c \\map {p_1} x^{a_1} \\map {p_2} x^{a_2} \\dotsm \\map {p_r} x^{a_r}$ The primary decomposition theorem then states the following : :$(1): \\quad \\map \\ker {\\map {p_i} T^{a_i} }$ is a $T$-invariant subspace of $V$ for all $i = 1, 2, \\dotsc, r$ :$(2): \\quad \\displaystyle V = \\bigoplus_{i \\mathop = 1}^r \\map \\ker {\\map {p_i} T^{a_i} }$ {{explain|Link to definition of $\\bigoplus$ in this context.}} {{improve|Rather than dragging the unwieldy $\\map \\ker {\\map {p_i} T^{a_i} }$ all around the page, suggest that a symbol e.g. $\\kappa_i$ be used for it instead. There are many more places where the exposition is repetitive and could benefit from being broken into more modular units. And there are still places where the logical flow is compromised by being wrapped backwards upon itself with \"because\" and \"indeed\" and nested \"if-thens\", although I have done my best to clean out most of these.}}"}
+{"_id": "4961", "title": "Sigma-Algebra Extended by Single Set", "text": "Let $\\Sigma$ be a $\\sigma$-algebra on a set $X$. Let $S \\subseteq X$ be a subset of $X$. For subsets $T \\subseteq X$ of $X$, denote $T^c$ for the set difference $X \\setminus T$. Then: :$\\sigma \\left({\\Sigma \\cup \\left\\{{S}\\right\\}}\\right) = \\left\\{{\\left({E_1 \\cap S}\\right) \\cup \\left({E_2 \\cap S^c}\\right): E_1, E_2 \\in \\Sigma}\\right\\}$ where $\\sigma$ denotes generated $\\sigma$-algebra."}
+{"_id": "4966", "title": "Subring Module/Special Case", "text": "Let $S$ be a subring of the ring $\\struct {R, +, \\circ}$. Let $\\circ_S$ be the restriction of $\\circ$ to $S \\times R$. Then $\\struct {R, +, \\circ_S}_S$ is an $S$-module. If $\\struct {R, +, \\circ}$ has a unity, $1_R$, and $1_R \\in S$, then $\\struct {R, +, \\circ_S}_S$ is a unitary $S$-module."}
+{"_id": "4967", "title": "Trivial Module is Module", "text": "Let $\\struct {G, +_G}$ be an abelian group whose identity is $e_G$. Let $\\struct {R, +_R, \\circ_R}$ be a ring. Let $\\struct {G, +_G, \\circ}_R$ be the trivial $R$-module, such that: :$\\forall \\lambda \\in R: \\forall x \\in G: \\lambda \\circ x = e_G$ Then $\\struct {G, +_G, \\circ}_R$ is a module."}
+{"_id": "4969", "title": "Null Module is Module", "text": "Let $\\left({R, +_R, \\circ_R}\\right)$ be a ring. Let $G$ be the trivial group. Let $\\left({G, +_G, \\circ}\\right)_R$ be the null module. Then $\\left({G, +_G, \\circ}\\right)_R$ is a module."}
+{"_id": "4972", "title": "Submodule of Module of Polynomial Functions", "text": "Let $K$ be a commutative ring with unity. Let $P \\left({K}\\right)$ be the set of all polynomial functions on $K$. Consider the set $P_m \\left({K}\\right)$ of all the polynomial functions: :$\\displaystyle \\sum_{k \\mathop = 0}^{m-1} \\alpha_k {I_K}^k$ for some $m \\in \\N^*$ where: :$\\left \\langle {\\alpha_k} \\right \\rangle_{k \\in \\left[{0 \\,.\\,.\\, m-1}\\right]}$ is any sequence of $m$ terms of $K$. Then $P_m \\left({K}\\right)$ is a submodule of $P \\left({K}\\right)$."}
+{"_id": "4973", "title": "Condition on Equality of Generated Sigma-Algebras", "text": "Let $X$ be a set, and let $\\mathcal G$, $\\mathcal H$ be sets of subsets of $X$. Suppose that: :$\\mathcal G \\subseteq \\mathcal H \\subseteq \\sigma \\left({\\mathcal G}\\right)$ where $\\sigma$ denotes generated $\\sigma$-algebra. Then: :$\\sigma \\left({\\mathcal G}\\right) = \\sigma \\left({\\mathcal H}\\right)$"}
+{"_id": "4974", "title": "Sigma-Algebra is Monotone Class", "text": "Let $\\Sigma$ be a $\\sigma$-algebra on a set $X$. Then $\\Sigma$ is also a monotone class."}
+{"_id": "4976", "title": "Trace Sigma-Algebra of Generated Sigma-Algebra", "text": "Let $X$ be a Set, and let $\\GG \\subseteq \\powerset X$ be a collection of subsets of $X$. Let $A \\subseteq X$ be a subset of $X$. Then the following equality holds: :$A \\cap \\map \\sigma \\GG = \\map \\sigma {A \\cap \\GG}$ where :$\\map \\sigma \\GG$ denotes the smallest $\\sigma$-algebra on $X$ that contains $\\GG$ :$\\map \\sigma {A \\cap \\GG}$ denotes the smallest $\\sigma$-algebra on $A$ that contains ${A \\cap \\GG}$ :$A \\cap \\map \\sigma \\GG$ denotes the trace $\\sigma$-algebra on $A$ :$A \\cap \\GG$ is a shorthand for $\\set {A \\cap G: G \\in \\GG}$"}
+{"_id": "4981", "title": "Intersection is Subset/General Result", "text": "Let $S$ be a set. Let $\\mathcal P \\left({S}\\right)$ be the power set of $S$. Let $\\mathbb S \\subseteq \\mathcal P \\left({S}\\right)$. Then: : $\\displaystyle \\forall T \\in \\mathbb S: \\bigcap \\mathbb S \\subseteq T$ === Family of Sets === In the context of a family of sets, the result can be presented as follows: {{:Intersection is Subset/Family of Sets}}"}
+{"_id": "4982", "title": "Union of Relative Complements of Nested Subsets", "text": "Let $R \\subseteq S \\subseteq T$ be sets with the indicated inclusions. Then: :$\\complement_T \\left({S}\\right) \\cup \\complement_S \\left({R}\\right) = \\complement_T \\left({R}\\right)$ where $\\complement$ denotes relative complement. Phrased via Set Difference as Intersection with Relative Complement: :$\\left({T \\setminus S}\\right) \\cup \\left({S \\setminus R}\\right) = T \\setminus R$ where $\\setminus$ denotes set difference."}
+{"_id": "4983", "title": "Homomorphic Image of R-Module is R-Module", "text": "Let $\\left({R, +_R, \\times_R}\\right)$ be a ring. Let $\\left({G, +_G, \\circ_G}\\right)_R$ be an $R$-module. Let $\\left({H, +_H, \\circ_H}\\right)_R$ be an $R$-algebraic structure. Let $\\phi: G \\to H$ be a homomorphism. Then the homomorphic image of $\\phi$ is an $R$-module."}
+{"_id": "4986", "title": "Evaluation Linear Transformation is Linear Transformation", "text": "Let $R$ be a commutative ring. Let $G$ be an $R$-module. Let $G^*$ be the algebraic dual of $G$. Let $G^{**}$ be the algebraic dual of $G^*$. Let the mapping $J: G \\to G^{**}$ be the evaluation linear transformation from $G$ into $G^{**}$. For each $x \\in G$,  $x^\\wedge: G^* \\to R$ is defined as: :$\\forall t' \\in G^*: \\map {x^\\wedge} {t'} = \\map {t'} x$ Let the mapping $J: G \\to G^{**}$ be the evaluation linear transformation from $G$ into $G^{**}$ defined as: :$\\forall x \\in G: \\map J x = x^\\wedge$ where for each $x \\in G$, $x^\\wedge: G^* \\to R$ is defined as: :$\\forall t' \\in G^*: \\map {x^\\wedge} {t'} = \\map {t'} x$   Then: :$(1): \\quad x^\\wedge \\in G^{**}$ :$(2): \\quad J$ is a linear transformation."}
+{"_id": "4987", "title": "Evaluation Isomorphism is Isomorphism", "text": "Let $R$ be a commutative ring. Let $G$ be a unitary $R$-module whose dimension is finite. Then the evaluation linear transformation $J: G \\to G^{**}$ is an isomorphism."}
+{"_id": "4990", "title": "Zero Matrix is Identity for Hadamard Product", "text": "Let $\\struct {S, \\cdot}$ be a monoid whose identity is $e$. Let $\\map {\\MM_S} {m, n}$ be an $m \\times n$ matrix space over $S$. Let $\\mathbf e = \\sqbrk e_{m n}$ be the zero matrix of $\\map {\\MM_S} {m, n}$. Then $\\mathbf e$ is the identity element for Hadamard product."}
+{"_id": "4991", "title": "Sum of Ideals is Ideal/General Result", "text": "Let $J_1, J_2, \\ldots, J_n$ be ideals of a ring $\\struct {R, +, \\circ}$. Then: : $J = J_1 + J_2 + \\cdots + J_n$ is an ideal of $R$. where $J_1 + J_2 + \\cdots + J_n$ is as defined in subset product."}
+{"_id": "4998", "title": "Integer is Expressible as Product of Primes", "text": "Let $n$ be an integer such that $n > 1$. Then $n$ can be expressed as the product of one or more primes."}
+{"_id": "4999", "title": "Expression for Integer as Product of Primes is Unique", "text": "Let $n$ be an integer such that $n > 1$. Then the expression for $n$ as the product of one or more primes is unique up to the order in which they appear."}
+{"_id": "5005", "title": "Modulo Addition is Closed/Integers", "text": "Let $m \\in \\Z$ be an integer. Then addition modulo $m$ on the set of integers modulo $m$ is closed: :$\\forall \\eqclass x m, \\eqclass y m \\in \\Z_m: \\eqclass x m +_m \\eqclass y m \\in \\Z_m$."}
+{"_id": "5023", "title": "Set is Subset of Union/General Result", "text": "Let $S$ be a set. Let $\\mathcal P \\left({S}\\right)$ be the power set of $S$. Let $\\mathbb S \\subseteq \\mathcal P \\left({S}\\right)$. Then: : $\\forall T \\in \\mathbb S: T \\subseteq \\bigcup \\mathbb S$"}
+{"_id": "5026", "title": "Intersection Distributes over Union/General Result", "text": "Let $S$ and $T$ be sets. Let $\\powerset T$ be the power set of $T$. Let $\\mathbb T$ be a subset of $\\powerset T$. Then: :$\\displaystyle S \\cap \\bigcup \\mathbb T = \\bigcup_{X \\mathop \\in \\mathbb T} \\paren {S \\cap X}$"}
+{"_id": "5027", "title": "Union Distributes over Intersection/General Result", "text": "Let $S$ and $T$ be sets. Let $\\powerset T$ be the power set of $T$. Let $\\mathbb T$ be a subset of $\\powerset T$. Then: :$\\displaystyle S \\cup \\bigcap \\mathbb T = \\bigcap_{X \\mathop \\in \\mathbb T} \\paren {S \\cup X}$"}
+{"_id": "5028", "title": "Scalar Multiple of Simple Function is Simple Function", "text": "Let $\\struct {X, \\Sigma}$ be a measurable space. Let $f: X \\to \\R$ be a simple function, and let $\\lambda \\in \\R$. Then the pointwise scalar multiple $\\lambda f: X \\to \\R$ of $f$ is also a simple function."}
+{"_id": "5029", "title": "Image of Composite Mapping", "text": "Let $f: S \\to T$ and $g: R \\to S$ be mappings. Then: :$\\operatorname{Im} \\left({f \\circ g}\\right) = f \\left[{\\operatorname{Im} \\left({g}\\right)}\\right]$ where: : $f \\circ g$ is the composition of $g$ and $f$ : $\\operatorname{Im}$ denotes image : $f \\left[{\\cdot}\\right]$ denotes taking image of a subset under $f$."}
+{"_id": "5030", "title": "Scalar Multiple of Integrable Function is Integrable Function", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f: X \\to \\overline \\R$ be a $\\mu$-integrable function, and let $\\lambda \\in \\R$. Then $\\lambda f: X \\to \\overline \\R$, the pointwise $\\lambda$-multiple of $f$, is also $\\mu$-integrable. That is, the space of integrable functions $\\LL^1_{\\overline \\R}$ is closed under pointwise $\\R$-scalar multiplication."}
+{"_id": "5037", "title": "Left Distributive Law for Natural Numbers", "text": "The operation of multiplication is left distributive over addition on the set of natural numbers $\\N$: :$\\forall x, y, n \\in \\N_{> 0}: n \\times \\paren {x + y} = \\paren {n \\times x} + \\paren {n \\times y}$"}
+{"_id": "5038", "title": "Right Distributive Law for Natural Numbers", "text": "The operation of multiplication is right distributive over addition on the set of natural numbers $\\N_{> 0}$: :$\\forall x, y, n \\in \\N_{> 0}: \\paren {x + y} \\times n = \\paren {x \\times n} + \\paren {y \\times n}$"}
+{"_id": "5039", "title": "Standard Machinery", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $\\mathcal{L}^1_{\\overline \\R} \\left({\\mu}\\right)$ be the space of $\\mu$-integrable functions. Let $P \\left({f_1, \\ldots, f_n}\\right)$ be a proposition, where the variables $f_i$ denote $\\mu$-measurable functions $f_i: X \\to \\overline{\\R}$. Let every occurrence of an $f_i$ be of the form: :$\\displaystyle \\int \\Phi \\left({f_i}\\right) \\, \\mathrm d\\mu$ for a suitable index set $I$ and multilinear mapping $\\Phi: \\mathcal{L}^1_{\\overline \\R} \\left({\\mu}\\right)^I \\to \\mathcal{L}^1_{\\overline \\R} \\left({\\mu}\\right)$. Denote with $\\chi \\left({\\Sigma}\\right)$ the set of characteristic functions of elements of $\\Sigma$, i.e.: :$\\chi \\left({\\Sigma}\\right) := \\left\\{{\\chi_E: X \\to \\R: E \\in \\Sigma}\\right\\}$ Then the following are equivalent: :$(A): \\quad \\forall f_1, \\ldots, f_n \\in \\chi \\left({\\Sigma}\\right): P \\left({f_1, \\ldots, f_n}\\right)$ :$(B): \\quad \\forall f_1, \\ldots, f_n \\in \\mathcal{L}^1_{\\overline \\R} \\left({\\mu}\\right): P \\left({f_1, \\ldots, f_n}\\right)$ that is, it suffices to verify $P$ holds for characteristic functions. {{WIP|Not completely valid as stated; need to restrict to $E$ with $\\mu(E)<+\\infty$ and impose more conditions on $P$}}"}
+{"_id": "5043", "title": "Third Principle of Mathematical Induction", "text": "Let $\\map P n$ be a propositional function depending on $n \\in \\N$. If: :$(1): \\quad \\map P n$ is true for all $n \\le d$ for some $d \\in \\N$ :$(2): \\quad \\forall m \\in \\N: \\paren {\\forall k \\in \\N, m \\le k < m + d: \\map P k} \\implies \\map P {m + d}$ then $\\map P n$ is true for all $n \\in \\N$."}
+{"_id": "5046", "title": "Cross-Relation is Equivalence Relation", "text": "{{Cross-Relation Context Definition}} Then $\\boxtimes$ is an equivalence relation on $\\struct {S_1 \\times S_2, \\oplus}$."}
+{"_id": "5047", "title": "Cross-Relation on Natural Numbers is Equivalence Relation", "text": "Let $\\struct {\\N, +}$ be the semigroup of natural numbers under addition. Let $\\struct {\\N \\times \\N, \\oplus}$ be the (external) direct product of $\\struct {\\N, +}$ with itself, where $\\oplus$ is the operation on $\\N \\times \\N$ induced by $+$ on $\\N$. The relation $\\boxtimes$ defined on $\\N \\times \\N$ by: :$\\tuple {x_1, y_1} \\boxtimes \\tuple {x_2, y_2} \\iff x_1 + y_2 = x_2 + y_1$ is an equivalence relation on $\\struct {\\N \\times \\N, \\oplus}$."}
+{"_id": "5048", "title": "Equivalence Classes of Cross-Relation on Natural Numbers", "text": "Let $\\struct {\\N, +}$ be the semigroup of natural numbers under addition. Let $\\struct {\\N \\times \\N, \\oplus}$ be the (external) direct product of $\\struct {\\N, +}$ with itself, where $\\oplus$ is the operation on $\\N \\times \\N$ induced by $+$ on $\\N$. Let $\\boxtimes$ be the cross-relation defined on $\\N \\times \\N$ by: :$\\tuple {x_1, y_1} \\boxtimes \\tuple {x_2, y_2} \\iff x_1 + y_2 = x_2 + y_1$ Then $\\eqclass {\\tuple {x, y} } \\boxtimes$ is the equivalence class of $\\tuple {x, y}$ under $\\boxtimes$, where: :$\\eqclass {\\tuple {x_1, y_1} } \\boxtimes = \\eqclass {\\tuple {x_2, y_2} } \\boxtimes \\iff \\tuple {x_1, y_1} \\boxtimes \\tuple {x_2, y_2}$ The equivalence class $\\eqclass {\\tuple {x, y} } \\boxtimes$ is more often denoted more simply as $\\eqclass {x, y} {}$."}
+{"_id": "5052", "title": "Addition of Cross-Relation Equivalence Classes on Natural Numbers is Cancellable", "text": "Let $\\struct {\\N, +}$ be the semigroup of natural numbers under addition. Let $\\struct {\\N \\times \\N, \\oplus}$ be the (external) direct product of $\\struct {\\N, +}$ with itself, where $\\oplus$ is the operation on $\\N \\times \\N$ induced by $+$ on $\\N$. Let $\\boxtimes$ be the cross-relation defined on $\\N \\times \\N$ by: :$\\tuple {x_1, y_1} \\boxtimes \\tuple {x_2, y_2} \\iff x_1 + y_2 = x_2 + y_1$ Let $\\eqclass {x, y} {}$ denote the equivalence class of $\\tuple {x, y}$ under $\\boxtimes$. {{WIP|Introduce the language of the Definition:Quotient Set.}} The operation $\\oplus$ on these equivalence classes is cancellable, in the sense that: {{begin-eqn}} {{eqn | l = \\eqclass {a, b} {} \\oplus \\eqclass {c_1, d_1} {}       | r = \\eqclass {a, b} {} \\oplus \\eqclass {c_2, d_2} {}       | c =  }} {{eqn | ll= \\leadsto       | l = \\eqclass {c_1, d_1} {}       | r = \\eqclass {c_2, d_2} {}       | c =  }} {{end-eqn}} and similarly: {{begin-eqn}} {{eqn | l = \\eqclass {c_1, d_1} {} \\oplus \\eqclass {a, b} {}       | r = \\eqclass {c_2, d_2} {} \\oplus \\eqclass {a, b} {}       | c =  }} {{eqn | ll= \\leadsto       | l = \\eqclass {c_1, d_1} {}       | r = \\eqclass {c_2, d_2} {}       | c =  }} {{end-eqn}}"}
+{"_id": "5053", "title": "Cross-Relation Equivalence Classes on Natural Numbers are Cancellable for Addition", "text": "Let $\\struct {\\N, +}$ be the semigroup of natural numbers under addition. Let $\\struct {\\N \\times \\N, \\oplus}$ be the (external) direct product of $\\struct {\\N, +}$ with itself, where $\\oplus$ is the operation on $\\N \\times \\N$ induced by $+$ on $\\N$. Let $\\boxtimes$ be the cross-relation defined on $\\N \\times \\N$ by: :$\\tuple {x_1, y_1} \\boxtimes \\tuple {x_2, y_2} \\iff x_1 + y_2 = x_2 + y_1$ Let $\\eqclass {x, y} {}$ denote the equivalence class of $\\tuple {x, y}$ under $\\boxtimes$. {{wtd|Introduce the language of the Definition:Quotient Set.}} The operation $\\oplus$ on these equivalence classes is cancellable, in the sense that: {{begin-eqn}} {{eqn | l = \\eqclass {a_1, b_1} {}       | r = \\eqclass {a_2, b_2} {}       | c =  }} {{eqn | l = \\eqclass {c_1, d_1} {}       | r = \\eqclass {c_2, d_2} {}       | c =  }} {{eqn | ll= \\leadsto       | l = \\eqclass {a_1, b_1} {} \\oplus \\eqclass {c_1, d_1} {}       | r = \\eqclass {a_2, b_2} {} \\oplus \\eqclass {c_2, d_2} {}       | c =  }} {{end-eqn}}"}
+{"_id": "5054", "title": "Cross-Relation is Congruence Relation", "text": "{{Cross-Relation Context Definition}} The cross-relation $\\boxtimes$ is a congruence relation on $\\struct {S_1 \\times S_2, \\oplus}$."}
+{"_id": "5055", "title": "Elements of Cross-Relation Equivalence Class", "text": "{{Cross-Relation Context Definition}} Let $\\left[\\!\\left[{\\left({x, y}\\right)}\\right]\\!\\right]_\\boxtimes$ be the $\\boxtimes$-equivalence class of $\\left({x, y}\\right)$, where $\\left({x, y}\\right) \\in S_1 \\times S_2$. Then: $\\forall x, y \\in S_1, a, b \\in S_2:$ : $(1): \\quad \\left[\\!\\left[{\\left({x \\circ a, a}\\right)}\\right]\\!\\right]_\\boxtimes = \\left[\\!\\left[{\\left({y \\circ b, b}\\right)}\\right]\\!\\right]_\\boxtimes \\iff x = y$ : $(2): \\quad \\left[\\!\\left[{\\left({x \\circ a, y \\circ a}\\right)}\\right]\\!\\right]_\\boxtimes = \\left[\\!\\left[{\\left({x, y}\\right)}\\right]\\!\\right]_\\boxtimes$"}
+{"_id": "5056", "title": "Equivalence Class of Equal Elements of Cross-Relation", "text": "{{Cross-Relation Context Definition}} Then: :$\\forall c, d \\in S_1 \\cap S_2: \\tuple {c, c} \\boxtimes \\tuple {d, d}$"}
+{"_id": "5057", "title": "Length of Concatenation", "text": "Let $S$ and $T$ be words, and let $ST$ be their concatenation. Then: :$\\left\\vert{ST}\\right\\vert = \\left\\vert{S}\\right\\vert + \\left\\vert{T}\\right\\vert$ where $\\left\\vert{S}\\right\\vert$ denotes the length of $S$."}
+{"_id": "5061", "title": "Relation is Connected and Reflexive iff Total", "text": "Let $S$ be a set. Let $\\mathcal R$ be a relation on $S$. Then: :$\\mathcal R$ is both a connected relation and a reflexive relation {{iff}}: :$\\mathcal R$ is a total relation."}
+{"_id": "5062", "title": "Total Ordering is Total Relation", "text": "Let $S$ be a set. Let $\\mathcal R \\subseteq S \\times S$ be a total ordering. Then $\\mathcal R$ is a total relation."}
+{"_id": "5066", "title": "Natural Number Addition Commutativity with Successor/Proof 2", "text": "Let $\\N$ be the natural numbers. Then: :$\\forall m, n \\in \\N_{> 0}: \\left({m + 1}\\right) + n = \\left({m + n}\\right) + 1$"}
+{"_id": "5067", "title": "Natural Number Commutes with 1 under Addition", "text": "Let $n \\in \\N_{> 0}$ be a natural number. Then $n$ commutes with $1$ under the operation of addition: :$\\forall n \\in \\N_{> 0}: n + 1 = 1 + n$"}
+{"_id": "5068", "title": "Natural Number Addition is Commutative/Proof 3", "text": "The operation of addition on the set of natural numbers $\\N_{> 0}$ is commutative: :$\\forall x, y \\in \\N_{> 0}: x + y = y + x$"}
+{"_id": "5070", "title": "Ordering on Natural Numbers is Trichotomy", "text": "Let $\\N$ be the natural numbers. Let $<$ be the (strict) ordering on $\\N$. Then exactly one of the following is true: :$(1): \\quad a = b$ :$(2): \\quad a > b$ :$(3): \\quad a < b$ That is, $<$ is a trichotomy on $\\N$."}
+{"_id": "5071", "title": "Ordering on 1-Based Natural Numbers is Transitive", "text": "Let $\\N_{> 0}$ be the 1-based natural numbers. Let $<$ be the strict ordering on $\\N_{>0}$. Then $<$ is a transitive relation."}
+{"_id": "5072", "title": "Trichotomy is Antireflexive", "text": "Let $\\RR$ be a trichotomy. Then $\\RR$ is an antireflexive relation."}
+{"_id": "5073", "title": "Ordering on 1-Based Natural Numbers is Total Ordering", "text": "Let $\\N_{> 0}$ be the $1$-based natural numbers. Let $<$ be the strict ordering on $\\N_{>0}$. Then $<$ is a (strict) total ordering."}
+{"_id": "5074", "title": "Left Cancellable Commutative Operation is Right Cancellable", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\circ$ be left cancellable and also commutative. Then $\\circ$ is also right cancellable."}
+{"_id": "5075", "title": "Right Cancellable Commutative Operation is Left Cancellable", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\circ$ be right cancellable and also commutative. Then $\\circ$ is also left cancellable."}
+{"_id": "5076", "title": "Natural Number Addition is Cancellable", "text": "Let $\\N$ be the natural numbers. Let $+$ be addition on $\\N$. Then: :$\\forall a, b, c \\in \\N: a + c = b + c \\implies a = b$ :$\\forall a, b, c \\in \\N: a + b = a + c \\implies b = c$ That is, $+$ is cancellable on $\\N$."}
+{"_id": "5077", "title": "Uniqueness of Measures", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $\\mathcal G \\subseteq \\mathcal P \\left({X}\\right)$ be a generator for $\\Sigma$; i.e., $\\Sigma = \\sigma \\left({\\mathcal G}\\right)$. Suppose that $\\mathcal G$ satisfies the following conditions: :$(1):\\quad \\forall G, H \\in \\mathcal G: G \\cap H \\in \\mathcal G$ :$(2):\\quad$ There exists an exhausting sequence $\\left({G_n}\\right)_{n \\in \\N} \\uparrow X$ in $\\mathcal G$ Let $\\mu, \\nu$ be measures on $\\left({X, \\Sigma}\\right)$, and suppose that: :$(3):\\quad \\forall G \\in \\mathcal G: \\mu \\left({G}\\right) = \\nu \\left({G}\\right)$ :$(4):\\quad \\forall n \\in \\N: \\mu \\left({G_n}\\right)$ is finite Then $\\mu = \\nu$. Alternatively, by Countable Cover induces Exhausting Sequence, the exhausting sequence in $(2)$ may be replaced by a countable $\\mathcal G$-cover $\\left({G_n}\\right)_{n \\in \\N}$, still subject to $(4)$."}
+{"_id": "5078", "title": "Uniqueness of Measures/Proof 2", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $\\mathcal G \\subseteq \\mathcal P \\left({X}\\right)$ be a generator for $\\Sigma$; i.e., $\\Sigma = \\sigma \\left({\\mathcal G}\\right)$. Suppose that $\\mathcal G$ satisfies the following conditions: :$(1):\\quad \\forall G, H \\in \\mathcal G: G \\cap H \\in \\mathcal G$ :$(2):\\quad$ There exists an exhausting sequence $\\left({G_n}\\right)_{n \\in \\N} \\uparrow X$ in $\\mathcal G$ Let $\\mu, \\nu$ be measures on $\\left({X, \\Sigma}\\right)$, and suppose that: :$(3):\\quad \\forall G \\in \\mathcal G: \\mu \\left({G}\\right) = \\nu \\left({G}\\right)$ :$(4):\\quad \\forall n \\in \\N: \\mu \\left({G_n}\\right)$ is finite Then $\\mu = \\nu$. Alternatively, by Countable Cover induces Exhausting Sequence, the exhausting sequence in $(2)$ may be replaced by a countable $\\mathcal G$-cover $\\left({G_n}\\right)_{n \\in \\N}$, still subject to $(4)$."}
+{"_id": "5079", "title": "Natural Number Multiplication is Cancellable", "text": "Let $\\N$ be the natural numbers. Let $\\times$ be multiplication on $\\N$. Then: :$\\forall x, y, z \\in \\N_{>0}: x \\times z = y \\times z \\implies x = y$ :$\\forall x, y, z \\in \\N_{>0}: x \\times y = x \\times z \\implies y = z$ That is, $\\times$ is cancellable on $\\N_{>0}$."}
+{"_id": "5080", "title": "Natural Number Addition is Cancellable for Ordering", "text": "Let $\\N$ be the natural numbers. Let $<$ be the strict ordering on $\\N$. Let $+$ be addition on $\\N$. Then: :$\\forall a, b, c \\in \\N_{>0}: a + c < b + c \\implies a < b$ :$\\forall a, b, c \\in \\N_{>0}: a + b < a + c \\implies b < c$ That is, $+$ is cancellable on $\\N$ for $<$."}
+{"_id": "5081", "title": "Natural Number Multiplication is Cancellable for Ordering", "text": "Let $\\N$ be the natural numbers. Let $\\times$ be multiplication on $\\N$. Let $<$ be the strict ordering on $\\N$. Then: :$\\forall a, b, c \\in \\N: a \\times c < b \\times c \\implies a < b$ :$\\forall a, b, c \\in \\N: a \\times b < a \\times c \\implies b < c$ That is, $\\times$ is cancellable on $\\N$ for $<$."}
+{"_id": "5082", "title": "Successor to Natural Number", "text": "Let $\\N_{> 0}$ be the 1-based natural numbers: :$\\N_{> 0} = \\left\\{{1, 2, 3, \\ldots}\\right\\}$ Let $<$ be the ordering on $\\N_{> 0}$: :$\\forall a, b \\in \\N_{>0}: a < b \\iff \\exists c \\in \\N_{>0}: a + c = b$ Let $a \\in \\N_{>0}$. Then there exists no natural number $n$ such that $a < n < a + 1$."}
+{"_id": "5083", "title": "Natural Number is Not Equal to Successor", "text": "Let $\\N_{> 0}$ be the 1-based natural numbers: :$\\N_{> 0} = \\left\\{{1, 2, 3, \\ldots}\\right\\}$ Then: :$\\forall n \\in \\N_{> 0}: n \\ne n + 1$"}
+{"_id": "5084", "title": "Biconditional is Commutative", "text": "==== Formulation 1 ==== {{:Biconditional is Commutative/Formulation 1}} ==== Formulation 2 ==== {{:Biconditional is Commutative/Formulation 2}}"}
+{"_id": "5085", "title": "Biconditional is Associative", "text": "==== Formulation 1 ==== {{:Biconditional is Associative/Formulation 1}} ==== Formulation 2 ==== {{:Biconditional is Associative/Formulation 2}}"}
+{"_id": "5086", "title": "Biconditional is Reflexive", "text": ": $\\vdash p \\iff p$"}
+{"_id": "5087", "title": "Biconditional is Transitive", "text": "The biconditional operator is transitive:"}
+{"_id": "5088", "title": "Ordinals are Totally Ordered", "text": "The ordinals are totally ordered."}
+{"_id": "5089", "title": "Successor Set of Transitive Set is Transitive", "text": "Let $S$ be a transitive set. Then its successor set $S\\,^+ = S \\cup \\left\\{{S}\\right\\}$ is also transitive."}
+{"_id": "5090", "title": "Successor Set of Ordinal is Ordinal", "text": "Let $S$ be an ordinal. Then its successor set $S^+ = S \\cup \\set S$ is also an ordinal."}
+{"_id": "5091", "title": "Ordinals are Well-Ordered", "text": "The ordinals are well-ordered."}
+{"_id": "5093", "title": "Set Intersection Preserves Subsets/Corollary", "text": "Let $A, B, S$ be sets. Then: :$A \\subseteq B \\implies A \\cap S \\subseteq B \\cap S$"}
+{"_id": "5094", "title": "Set Intersection Preserves Subsets/Families of Sets/Corollary", "text": "Let $I$ be an indexing set. Let $\\family {B_\\alpha}_{\\alpha \\mathop \\in I}$ be an indexed family of subsets of a set $S$. Let $A$ be a set such that $A \\subseteq B_\\alpha$ for all $\\alpha \\in I$. Then: :$\\displaystyle A \\subseteq \\bigcap_{\\alpha \\mathop \\in I} B_\\alpha$"}
+{"_id": "5095", "title": "Intersection is Largest Subset/General Result", "text": "Let $T$ be a set. Let $\\powerset T$ be the power set of $T$. Let $\\mathbb T$ be a subset of $\\powerset T$. Then: :$\\paren {\\forall X \\in \\mathbb T: S \\subseteq X} \\iff S \\subseteq \\bigcap \\mathbb T$ === Family of Sets === In the context of a family of sets, the result can be presented as follows: {{:Intersection is Largest Subset/Family of Sets}}"}
+{"_id": "5096", "title": "Union is Smallest Superset/General Result", "text": "Let $S$ and $T$ be sets. Let $\\powerset S$ denote the power set of $S$. Let $\\mathbb S$ be a subset of $\\powerset S$. Then: :$\\displaystyle \\paren {\\forall X \\in \\mathbb S: X \\subseteq T} \\iff \\bigcup \\mathbb S \\subseteq T$ === Family of Sets === In the context of a family of sets, the result can be presented as follows: {{:Union is Smallest Superset/Family of Sets}}"}
+{"_id": "5097", "title": "Mapping is Involution iff Bijective and Symmetric", "text": "Let $S$ be a set. Let $f: S \\to S$ be a mapping on $S$. Then $f$ is an involution {{iff}} $f$ is both a bijection and a symmetric relation."}
+{"_id": "5098", "title": "De Morgan's Laws (Set Theory)/Relative Complement/General Case", "text": "Let $S$ be a set. Let $T$ be a subset of $S$. Let $\\mathcal P \\left({T}\\right)$ be the power set  of $T$. Let $\\mathbb T \\subseteq \\mathcal P \\left({T}\\right)$.  Then: ==== Complement of Intersection ==== {{:De Morgan's Laws (Set Theory)/Relative Complement/General Case/Complement of Intersection}} ==== Complement of Union ==== {{:De Morgan's Laws (Set Theory)/Relative Complement/General Case/Complement of Union}}"}
+{"_id": "5099", "title": "Absorption Laws (Set Theory)", "text": "These two results together are known as the '''Absorption Laws''', corresponding to the equivalent results in logic."}
+{"_id": "5104", "title": "Cardinality of Set Union/Corollary", "text": "Let $S_1, S_2, \\ldots, S_n$ be finite sets which are pairwise disjoint. Then: :$\\displaystyle \\card {\\bigcup_{i \\mathop = 1}^n S_i} = \\sum_{i \\mathop = 1}^n \\card {S_i}$ Specifically: : $\\card {S_1 \\cup S_2} = \\card {S_1} + \\card {S_2}$"}
+{"_id": "5110", "title": "Image of Intersection under Relation/General Result", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation. Let $\\powerset S$ be the power set of $S$. Let $\\mathbb S \\subseteq \\powerset S$. Then: :$\\displaystyle \\RR \\sqbrk {\\bigcap \\mathbb S} \\subseteq \\bigcap_{X \\mathop \\in \\mathbb S} \\RR \\sqbrk X$"}
+{"_id": "5111", "title": "Image of Intersection under One-to-Many Relation/General Result", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation. Let $\\powerset S$ be the power set of $S$. Then: :$\\displaystyle \\forall \\mathbb S \\subseteq \\powerset S: \\RR \\sqbrk {\\bigcap \\mathbb S} = \\bigcap_{X \\mathop \\in \\mathbb S} \\RR \\sqbrk X$  {{iff}} $\\RR$ is one-to-many."}
+{"_id": "5112", "title": "Image of Union under Relation/General Result", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation. Let $\\powerset S$ be the power set of $S$. Let $\\mathbb S \\subseteq \\powerset S$. Then: :$\\displaystyle \\RR \\sqbrk {\\bigcup \\mathbb S} = \\bigcup_{X \\mathop \\in \\mathbb S} \\RR \\sqbrk X$"}
+{"_id": "5114", "title": "Preimage of Union under Mapping/General Result", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let $\\powerset T$ be the power set of $T$. Let $\\mathbb T \\subseteq \\powerset T$. Then: :$\\displaystyle f^{-1} \\sqbrk {\\bigcup \\mathbb T} = \\bigcup_{X \\mathop \\in \\mathbb T} f^{-1} \\sqbrk X$"}
+{"_id": "5115", "title": "Preimage of Union under Relation/General Result", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation. Let $\\powerset T$ be the power set of $T$. Let $\\mathbb T \\subseteq \\powerset T$. Then: :$\\displaystyle \\RR^{-1} \\sqbrk {\\bigcup \\mathbb T} = \\bigcup_{X \\mathop \\in \\mathbb T} \\RR^{-1} \\sqbrk X$ where $\\RR^{-1} \\sqbrk X$ denotes the preimage of $X$ under $\\RR$."}
+{"_id": "5116", "title": "Image of Intersection under Mapping/General Result", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let $\\powerset S$ be the power set of $S$. Let $\\mathbb S \\subseteq \\powerset S$. Then: :$\\displaystyle f \\sqbrk {\\bigcap \\mathbb S} \\subseteq \\bigcap_{X \\mathop \\in \\mathbb S} f \\sqbrk X$"}
+{"_id": "5117", "title": "Preimage of Intersection under Mapping/General Result", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let $\\mathcal P \\left({T}\\right)$ be the power set of $T$. Let $\\mathbb T \\subseteq \\mathcal P \\left({T}\\right)$. Then: :$\\displaystyle f^{-1} \\left[{\\bigcap \\mathbb T}\\right] = \\bigcap_{X \\mathop \\in \\mathbb T} f^{-1} \\left[{X}\\right]$"}
+{"_id": "5118", "title": "Bijective Relation has Left and Right Inverse", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation on a cartesian product $S \\times T$. Let: :$I_S$ be the identity mapping on $S$ :$I_T$ be the identity mapping on $T$. Let $\\mathcal R^{-1}$ be the inverse relation of $\\mathcal R$. Let $\\mathcal R$ be a bijection. Then: :$\\mathcal R^{-1} \\circ \\mathcal R = I_S$ and :$\\mathcal R \\circ \\mathcal R^{-1} = I_T$ where $\\circ$ denotes composition of relations."}
+{"_id": "5119", "title": "Left and Right Inverse Relations Implies Bijection", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation on a cartesian product $S \\times T$. Let: :$I_S$ be the identity mapping on $S$ :$I_T$ be the identity mapping on $T$. Let $\\mathcal R^{-1}$ be the inverse relation of $\\mathcal R$. Let $\\mathcal R$ be such that: :$\\mathcal R^{-1} \\circ \\mathcal R = I_S$ and :$\\mathcal R \\circ \\mathcal R^{-1} = I_T$ where $\\circ$ denotes composition of relations. Then $\\mathcal R$ is a bijection."}
+{"_id": "5120", "title": "Bijection has Left and Right Inverse", "text": "Let $f: S \\to T$ be a bijection. Let: : $I_S$ be the identity mapping on $S$ : $I_T$ be the identity mapping on $T$. Let $f^{-1}$ be the inverse of $f$. Then: : $f^{-1} \\circ f = I_S$ and: : $f \\circ f^{-1} = I_T$ where $\\circ$ denotes composition of mappings."}
+{"_id": "5124", "title": "Symmetry in Space Implies Conservation of Momentum", "text": "The total derivative of the action $S_{12}$ from states $1$ to $2$ with regard to position is equal to the difference in momentum from states $1$ to $2$: :$\\dfrac {\\d S_{1 2} } {\\d x} = p_2 - p_1$ {{MissingLinks|Although we do have a page Definition:State, it refers to a concept in game theory and not physics.}}"}
+{"_id": "5126", "title": "Mappings in Product of Sets are Surjections", "text": "Let $S$ and $T$ be sets. Let $\\struct {P, \\phi_1, \\phi_2}$ be a product of $S$ and $T$. Then $\\phi_1$ and $\\phi_1$ are surjections."}
+{"_id": "5127", "title": "Cartesian Product is Set Product", "text": "Let $S$ and $T$ be sets. Let $S \\times T$ be the Cartesian product of $S$ and $T$. Let $\\pr_1: S \\times T \\to S$ and $\\pr_2: S \\times T \\to T$ be the first and second projections respectively on $S \\times T$. Then $\\struct {S \\times T, \\pr_1, \\pr_2}$ is a set product."}
+{"_id": "5131", "title": "Positive Rational Number as Power of Number with Power of Itself", "text": "Every positive rational number can be written either as: : $a^{a^a}$ for some irrational number $a$ or as: : $n^{n^n}$ for some natural number $n$."}
+{"_id": "5133", "title": "Rational Number as Power of Number with Itself", "text": "Every rational number in the interval $\\openint {\\paren {\\dfrac 1 e}^{\\frac 1 e} }{+\\infty}$ can be written either as: : $a^a$ for some irrational number $a$ or as: : $n^n$ for some natural number $n$."}
+{"_id": "5134", "title": "Integral of Integrable Function is Homogeneous", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f: X \\to \\overline \\R$ be a $\\mu$-integrable function. Let $\\lambda \\in \\R$. Then: :$\\displaystyle \\int \\lambda f \\rd \\mu = \\lambda \\int f \\rd \\mu$ where $\\lambda f$ is the pointwise $\\lambda$-multiple of $f$."}
+{"_id": "5135", "title": "Pointwise Sum of Integrable Functions is Integrable Function", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f, g: X \\to \\overline \\R$ be $\\mu$-integrable functions. Suppose that their pointwise sum $f + g$ is well-defined. Then $f + g$ is also a $\\mu$-integrable function. That is, the space of $\\mu$-integrable functions $\\LL^1_{\\overline \\R}$ is closed under pointwise addition."}
+{"_id": "5136", "title": "Integral of Integrable Function is Additive", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f, g: X \\to \\overline \\R$ be $\\mu$-integrable functions. Suppose that their pointwise sum $f + g$ is well-defined. Then: :$\\displaystyle \\int f + g \\rd \\mu = \\int f \\rd \\mu + \\int g \\rd \\mu$"}
+{"_id": "5137", "title": "Pointwise Minimum of Integrable Functions is Integrable Function", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f, g: X \\to \\overline \\R$ be $\\mu$-integrable functions. Then $\\map \\min {f, g}$, the pointwise minimum of $f$ and $g$, is also a $\\mu$-integrable function. That is, the space of $\\mu$-integrable functions $\\LL^1_{\\overline \\R}$ is closed under pointwise minimum."}
+{"_id": "5138", "title": "Pointwise Maximum of Integrable Functions is Integrable Function", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f, g: X \\to \\overline \\R$ be $\\mu$-integrable functions. Then $\\map \\max {f, g}$, the pointwise maximum of $f$ and $g$, is also a $\\mu$-integrable function. That is, the space of $\\mu$-integrable functions $\\LL^1_{\\overline \\R}$ is closed under pointwise maximum."}
+{"_id": "5139", "title": "Integral of Integrable Function is Monotone", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f, g: X \\to \\overline \\R$ be $\\mu$-integrable functions. Suppose that $f \\le g$ pointwise. Then: :$\\displaystyle \\int f \\rd \\mu \\le \\int g \\rd \\mu$"}
+{"_id": "5140", "title": "Triangle Inequality for Integrals", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f: X \\to \\overline \\R$ be a $\\mu$-integrable function. Then: :$\\displaystyle \\size {\\int_X f \\rd \\mu} \\le \\int_X \\size f \\rd \\mu$"}
+{"_id": "5141", "title": "Space of Integrable Functions is Vector Space", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $\\mathcal{L}^1 \\left({\\mu}\\right)$ be the space of real-valued $\\mu$-integrable functions. Then $\\mathcal{L}^1 \\left({\\mu}\\right)$, endowed with pointwise $\\R$-scalar multiplication and pointwise addition, forms a vector space over $\\R$."}
+{"_id": "5142", "title": "Measure with Density is Measure", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $f: X \\to \\overline{\\R}_{\\ge 0}$ be a positive $\\mu$-measurable function. Then the $f \\mu$, the measure with density $f$ with respect to $\\mu$ is a measure."}
+{"_id": "5143", "title": "Integrable Function Zero A.E. iff Absolute Value has Zero Integral", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f: X \\to \\overline \\R$ be a $\\mu$-integrable function. Then the following are equivalent: :$f = 0$ almost everywhere :$\\displaystyle \\int \\size f \\rd \\mu = 0$"}
+{"_id": "5144", "title": "Integral of Integrable Function over Null Set", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f: X \\to \\overline \\R$ be a $\\mu$-integrable function. Let $N$ be a $\\mu$-null set. Then: :$\\displaystyle \\int_N f \\rd \\mu = 0$ where $\\displaystyle \\int_N$ signifies an integral over $N$."}
+{"_id": "5146", "title": "Integrable Function is A.E. Real-Valued", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f: X \\to \\overline \\R$ be a $\\mu$-integrable function. Then $\\map f x \\in \\R$ for almost all $x \\in X$."}
+{"_id": "5147", "title": "Integrable Functions with Equal Integrals on Sub-Sigma-Algebra are A.E. Equal", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $\\GG$ be a sub-$\\sigma$-algebra of $\\Sigma$. Let $f, g: X \\to \\overline \\R$ be $\\GG$-integrable functions. Suppose that, for all $G \\in \\GG$: :$\\displaystyle \\int_G f \\rd \\mu = \\int_G g \\rd \\mu$ Then $f = g$ $\\mu$-almost everywhere."}
+{"_id": "5148", "title": "Measurable Functions with Equal Integrals on Sub-Sigma-Algebra are A.E. Equal", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $\\GG$ be a sub-$\\sigma$-algebra of $\\Sigma$. Suppose that $\\mu \\restriction_\\GG$, the restriction of $\\mu$ to $\\GG$, is $\\sigma$-finite. Let $f, g: X \\to \\overline \\R$ be $\\GG$-measurable functions. Suppose that, for all $G \\in \\GG$: :$\\displaystyle \\int_G f \\rd \\mu = \\int_G g \\rd \\mu$ Then $f = g$ $\\mu$-almost everywhere."}
+{"_id": "5149", "title": "Intersection is Largest Subset/Family of Sets", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of sets indexed by $I$. Then for all sets $X$: :$\\ds \\paren {\\forall i \\in I: X \\subseteq S_i} \\iff X \\subseteq \\bigcap_{i \\mathop \\in I} S_i$ where $\\ds \\bigcap_{i \\mathop \\in I} S_i$ is the intersection of $\\family {S_i}$."}
+{"_id": "5150", "title": "Countable Set is Null Set under Lebesgue Measure", "text": "Let $S \\subseteq \\R$ be a countable set. Then $\\lambda \\left({S}\\right) = 0$, where $\\lambda$ is Lebesgue measure. That is, $S$ is a $\\lambda$-null set."}
+{"_id": "5151", "title": "Intersection is Subset/Family of Sets", "text": "Let $\\family {S_\\alpha}_{\\alpha \\mathop \\in I}$ be a family of sets indexed by $I$. Then: :$\\displaystyle \\forall \\beta \\in I: \\bigcap_{\\alpha \\mathop \\in I} S_\\alpha \\subseteq S_\\beta$ where $\\displaystyle \\bigcap_{\\alpha \\mathop \\in I} S_\\alpha$ is the intersection of $\\family {S_\\alpha}_{\\alpha \\mathop \\in I}$."}
+{"_id": "5152", "title": "Union is Smallest Superset/Family of Sets", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of sets indexed by $I$. Then for all sets $X$: :$\\ds \\paren {\\forall i \\in I: S_i \\subseteq X} \\iff \\bigcup_{i \\mathop \\in I} S_i \\subseteq X$ where $\\ds \\bigcup_{i \\mathop \\in I} S_i$ is the union of $\\family {S_i}$."}
+{"_id": "5153", "title": "Fatou's Lemma for Integrals/Integrable Functions", "text": "Let $\\sequence {f_n}_{n mathop \\in \\N} \\in \\LL^1$, $f_n: X \\to \\R$ be a sequence of $\\mu$-integrable functions. Let $\\displaystyle \\liminf_{n mathop \\to \\infty} f_n: X \\to \\overline \\R$ be the pointwise limit inferior of the $f_n$. Suppose that there exists an $\\mu$-integrable $f: X \\to \\R$ such that for all $n \\in \\N$, $f \\le f_n$ pointwise. Then: :$\\displaystyle \\int \\liminf_{n \\mathop \\to \\infty} f_n \\rd \\mu \\le \\liminf_{n \\mathop \\to \\infty} \\int f_n \\rd \\mu$ where: :the integral sign denotes $\\mu$-integration :the {{RHS}} limit inferior is taken in the extended real numbers $\\overline \\R$."}
+{"_id": "5154", "title": "Fatou's Lemma for Integrals/Positive Measurable Functions", "text": "Let $\\sequence {f_n}_{n \\mathop \\in \\N} \\in \\MM_{\\overline \\R}^+$, $f_n: X \\to \\overline \\R$ be a sequence of positive measurable functions. Let $\\displaystyle \\liminf_{n \\mathop \\to \\infty} f_n: X \\to \\overline \\R$ be the pointwise limit inferior of the $f_n$. Then: :$\\displaystyle \\int \\liminf_{n \\mathop \\to \\infty} f_n \\rd \\mu \\le \\liminf_{n \\mathop \\to \\infty} \\int f_n \\rd \\mu$ where: :the integral sign denotes $\\mu$-integration :the {{RHS}} limit inferior is taken in the extended real numbers $\\overline \\R$."}
+{"_id": "5155", "title": "Reverse Fatou's Lemma/Integrable Functions", "text": "Let $\\sequence {f_n}_{n \\mathop \\in \\N} \\in \\LL^1$, $f_n: X \\to \\R$ be a sequence of $\\mu$-integrable functions. Let $\\displaystyle \\limsup_{n \\mathop \\to \\infty} f_n: X \\to \\overline \\R$ be the pointwise limit superior of the $f_n$. Suppose that there exists an $\\mu$-integrable $f: X \\to \\R$ such that for all $n \\in \\N$, $f_n \\le f$ pointwise. Then: :$\\displaystyle \\limsup_{n \\mathop \\to \\infty} \\int f_n \\rd \\mu \\le \\int \\limsup_{n \\mathop \\to \\infty} f_n \\rd \\mu$ where: :the integral sign denotes $\\mu$-integration :the {{RHS}} limit inferior is taken in the extended real numbers $\\overline \\R$."}
+{"_id": "5156", "title": "Set is Subset of Union/Family of Sets", "text": "Let $\\family {S_\\alpha}_{\\alpha \\mathop \\in I}$ be a family of sets indexed by $I$. Then: :$\\displaystyle \\forall \\beta \\in I: S_\\beta \\subseteq \\bigcup_{\\alpha \\mathop \\in I} S_\\alpha$ where $\\displaystyle \\bigcup_{\\alpha \\mathop \\in I} S_\\alpha$ is the union of $\\family {S_\\alpha}$."}
+{"_id": "5157", "title": "Reverse Fatou's Lemma/Positive Measurable Functions", "text": "Let $\\sequence {f_n}_{n \\mathop \\in \\N} \\in \\MM_{\\overline \\R}^+$, $f_n: X \\to \\overline \\R$ be a sequence of positive measurable functions. Suppose that there exists a positive measurable function $f: X \\to \\overline \\R$ such that: :$\\displaystyle \\int f \\rd \\mu < +\\infty$ :$\\forall n \\in \\N: f_n \\le f$ where $\\le$ signifies a pointwise inequality. Let $\\displaystyle \\limsup_{n \\mathop \\to \\infty} f_n: X \\to \\overline \\R$ be the pointwise limit superior of the $f_n$. Then: :$\\displaystyle \\limsup_{n \\mathop \\to \\infty} \\int f_n \\rd \\mu \\le \\int \\limsup_{n \\mathop \\to \\infty} f_n \\rd \\mu$ where: :the integral sign denotes $\\mu$-integration :the {{LHS}} limit superior is taken in the extended real numbers $\\overline \\R$."}
+{"_id": "5158", "title": "Monotone Convergence Theorem (Measure Theory)", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $\\sequence {u_n}_{n \\mathop \\in \\N} \\in \\map {\\LL^1} \\mu$, $u_n: X \\to \\R$ be a increasing sequence of $\\mu$-integrable functions. Let $\\displaystyle \\sup_{n \\mathop \\in \\N} u_n: X \\to \\overline \\R$ be the pointwise supremum of the $u_n$. Then $\\displaystyle \\sup_{n \\mathop \\in \\N} u_n$ is $\\mu$-integrable {{iff}}: :$\\displaystyle \\sup_{n \\mathop \\in \\N} \\int u_n \\rd \\mu < +\\infty$ and, in that case: :$\\displaystyle \\int \\sup_{n \\mathop \\in \\N} u_n \\rd \\mu = \\sup_{n \\mathop \\in \\N} \\int u_n \\rd \\mu$"}
+{"_id": "5159", "title": "Intersection is Associative/Family of Sets", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ and $\\family {I_\\lambda}_{\\lambda \\mathop \\in \\Lambda}$ be indexed families of sets. Let $\\displaystyle I = \\bigcap_{\\lambda \\mathop \\in \\Lambda} I_\\lambda$. Then: :$\\displaystyle \\bigcap_{i \\mathop \\in I} S_i = \\bigcap_{\\lambda \\mathop \\in \\Lambda} \\paren {\\bigcap_{i \\mathop \\in I_\\lambda} S_i}$"}
+{"_id": "5160", "title": "Union is Associative/Family of Sets", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ and $\\family {I_\\lambda}_{\\lambda \\mathop \\in \\Lambda}$ be indexed families of sets. Let $\\displaystyle I = \\bigcup_{\\lambda \\mathop \\in \\Lambda} I_\\lambda$ denote the union of  $\\family {I_\\lambda}_{\\lambda \\mathop \\in \\Lambda}$. Then: :$\\displaystyle \\bigcup_{i \\mathop \\in I} S_i = \\bigcup_{\\lambda \\mathop \\in \\Lambda} \\paren {\\bigcup_{i \\mathop \\in I_\\lambda} S_i}$"}
+{"_id": "5161", "title": "Lebesgue's Dominated Convergence Theorem", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $\\sequence {f_n}_{n \\mathop \\in \\N} \\in \\map {\\LL^1} \\mu$, $f_n: X \\to \\R$ be a sequence of $\\mu$-integrable functions. Suppose that for some $\\mu$-integrable $g: X \\to \\R$, it holds that: :$\\forall n \\in \\N: \\size {f_n} \\le g$ pointwise Suppose that the pointwise limit $f := \\displaystyle \\lim_{n \\mathop \\to \\infty} f_n$ exists almost everywhere. Then $f$ is $\\mu$-integrable, and: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\int \\size {f_n - f} \\rd \\mu = 0$ :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\int f_n \\rd \\mu = \\int \\lim_{n \\mathop \\to \\infty} f_n \\rd \\mu$"}
+{"_id": "5162", "title": "Continuity under Integral Sign", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $U$ be a non-empty open set of a metric space. Let $f: U \\times X \\to \\R$ be a mapping satisfying: :$(1): \\quad$ For all $\\lambda \\in U$, the mapping $x \\mapsto \\map f {\\lambda, x}$ is $\\mu$-integrable :$(2): \\quad$ For $\\mu$-almost all $x \\in X$, the mapping $\\lambda \\mapsto \\map f {\\lambda, x}$ is continuous :$(3): \\quad$ There exists a $\\mu$-integrable $g: X \\to \\R$ such that: ::::$\\forall \\tuple {\\lambda, x} \\in U \\times X: \\size {\\map f {\\lambda, x} } \\le \\map g x$ Then the mapping $h: U \\to \\R$ defined by: :$\\displaystyle \\map h \\lambda := \\int \\map f {\\lambda, x} \\map {\\rd \\mu} x$ is continuous."}
+{"_id": "5163", "title": "Differentiability under Integral Sign", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $\\openint a b$ be a non-empty open interval. Let $f: \\openint a b \\times X \\to \\R$ be a mapping satisfying: :$(1): \\quad$ For all $t \\in \\openint a b$, the mapping $x \\mapsto f \\left({t, x}\\right)$ is $\\mu$-integrable :$(2): \\quad$ For all $x \\in X$, the mapping $t \\mapsto \\map f {t, x}$ is differentiable :$(3): \\quad$ There exists a $\\mu$-integrable $g: X \\to \\R$ such that: ::::$\\displaystyle \\forall \\tuple {t, x} \\in \\openint a b \\times X: \\size {\\frac \\partial {\\partial t} \\map f {t, x} } \\le \\map g x$ Then the mapping $h: \\openint a b \\to \\R$ defined by: :$\\displaystyle \\map h t := \\int \\map f {t, x} \\map {\\rd \\mu} x$ is differentiable, and its derivative is: :$\\displaystyle D_t \\map h t = \\int \\frac \\partial {\\partial t} \\map f {t, x} \\map {\\rd \\mu} x$"}
+{"_id": "5164", "title": "Riemann-Lebesgue Theorem", "text": "Let $f: \\closedint a b \\to \\R$ be a bounded mapping. Let $\\mu$ be a one-dimensional Lebesgue measure. {{explain|Link to definition of one-dimensional in this specific context}} Then $f$ is Darboux integrable {{iff}} the set of all discontinuities of $f$ is a $\\mu$-null set."}
+{"_id": "5165", "title": "Gamma Function is Continuous on Positive Reals", "text": "Let $\\Gamma: \\R_{>0} \\to \\R$ be the Gamma function, restricted to the strictly positive real numbers. Then $\\Gamma$ is continuous."}
+{"_id": "5166", "title": "Gamma Function is Smooth on Positive Reals", "text": "Let $\\Gamma: \\R_{>0} \\to \\R$ be the Gamma function, restricted to the strictly positive real numbers. Then $\\Gamma$ is smooth. {{expand|I'm quite sure it's analytic on all of its domain}}"}
+{"_id": "5167", "title": "Log of Gamma Function is Convex on Positive Reals", "text": "Let $\\Gamma: \\R_{>0} \\to \\R$ be the Gamma function, restricted to the strictly positive real numbers. Let $\\ln$ denote the natural logarithm function. Then the composite mapping $\\ln \\circ \\operatorname \\Gamma$ is a convex function."}
+{"_id": "5168", "title": "Pratt's Lemma", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let: :$\\sequence {g_n}_{n \\mathop \\in \\N}$ :$\\sequence {G_n}_{n \\mathop \\in \\N}$ :$\\sequence {f_n}_{n \\mathop \\in \\N}$ be sequences of $\\mu$-integrable functions. Let the pointwise limits: :$\\displaystyle f := \\lim_{n \\mathop \\to \\infty} f_n$ :$\\displaystyle g := \\lim_{n \\mathop \\to \\infty} g_n$ :$\\displaystyle G := \\lim_{n \\mathop \\to \\infty} G_n$ exist. Let $g$ and $G$ be $\\mu$-integrable. Suppose that, for all $x \\in X$ and $n \\in \\N$: :$\\map {g_n} x \\le \\map {f_n} x \\le \\map {G_n} x$ Finally, suppose the following hold: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\int g_n \\rd \\mu = \\int g \\rd \\mu$ :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\int G_n \\rd \\mu = \\int G \\rd \\mu$ Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\int f_n \\rd \\mu = \\int f \\rd \\mu$ and the latter is finite."}
+{"_id": "5169", "title": "Hölder's Inequality", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $p, q \\in \\R$ such that $\\dfrac 1 p + \\dfrac 1 q = 1$. Let $f \\in \\map {\\LL^p} \\mu, f: X \\to \\R$, and $g \\in \\map {\\LL^q} \\mu, g: X \\to \\R$, where $\\LL$ denotes Lebesgue space. Then their pointwise product $f g$ is $\\mu$-integrable, that is: :$f g \\in \\map {\\LL^1} \\mu$ and: :$\\size {f g}_1 = \\displaystyle \\int \\size {f g} \\rd \\mu \\le \\norm f_p \\cdot \\norm g_q$ where the $\\norm {\\, \\cdot \\,}_p$ signify $p$-seminorms."}
+{"_id": "5170", "title": "Hölder's Inequality/Equality", "text": ":$\\displaystyle \\int \\size {f g} \\rd \\mu = \\norm f_p \\cdot \\norm g_q$ holds {{iff}}, for almost all $x \\in X$: :$\\dfrac {\\size {\\map f x}^p} {\\norm f_p^p} = \\dfrac {\\size {\\map g x}^q} {\\norm g_q^q}$"}
+{"_id": "5171", "title": "Condition for Membership of Equivalence Class", "text": "Let $\\RR$ be an equivalence relation on a set $S$. Let $\\eqclass x \\RR$ denote the $\\RR$-equivalence class of $x$. Then: :$\\forall y \\in S: y \\in \\eqclass x \\RR \\iff \\tuple {x, y} \\in \\RR$"}
+{"_id": "5174", "title": "Cartesian Product is Set Product/Family of Sets", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of sets. For all $j \\in I$, let $\\pr_i: \\displaystyle \\prod_{j \\mathop \\in I} \\family {S_j} \\to S_i$ be the $i$th projection from $\\displaystyle \\prod_{j \\mathop \\in I} \\family {S_j}$ to $S_i$. Then $\\struct {\\displaystyle \\prod_{j \\mathop \\in I} \\family {S_j}, \\family {\\pr_i}_{i \\mathop \\in I} }$ is a set product."}
+{"_id": "5176", "title": "Preimage of Intersection under Relation/General Result", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R \\subseteq S \\times T$ be a relation. Let $\\mathcal P \\left({T}\\right)$ be the power set of $T$. Let $\\mathbb T \\subseteq \\mathcal P \\left({T}\\right)$. Then: :$\\displaystyle \\mathcal R^{-1} \\left[{\\bigcap \\mathbb T}\\right] \\subseteq \\bigcap_{X \\mathop \\in \\mathbb T} \\mathcal R^{-1} \\left[{X}\\right]$"}
+{"_id": "5177", "title": "Image of Union under Relation/Family of Sets", "text": "Let $S$ and $T$ be sets. Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of subsets of $S$. Let $\\RR \\subseteq S \\times T$ be a relation. Then: :$\\displaystyle \\RR \\sqbrk {\\bigcup_{i \\mathop \\in I} S_i} = \\bigcup_{i \\mathop \\in I} \\RR \\sqbrk {S_i}$ where $\\displaystyle \\bigcup_{i \\mathop \\in I} S_i$ denotes the union of $\\family {S_i}_{i \\mathop \\in I}$."}
+{"_id": "5178", "title": "Image of Intersection under Relation/Family of Sets", "text": "Let $S$ and $T$ be sets. Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of subsets of $S$. Let $\\RR \\subseteq S \\times T$ be a relation. Then: :$\\ds \\RR \\sqbrk {\\bigcap_{i \\mathop \\in I} S_i} \\subseteq \\bigcap_{i \\mathop \\in I} \\RR \\sqbrk {S_i}$ where $\\ds \\bigcap_{i \\mathop \\in I} S_i$ denotes the intersection of $\\family {S_i}_{i \\mathop \\in I}$."}
+{"_id": "5179", "title": "Preimage of Union under Relation/Family of Sets", "text": "Let $S$ and $T$ be sets. Let $\\family {T_i}_{i \\mathop \\in I}$ be a family of subsets of $T$. Let $\\RR \\subseteq S \\times T$ be a relation. Then: :$\\displaystyle \\RR^{-1} \\sqbrk {\\bigcup_{i \\mathop \\in I} T_i} = \\bigcup_{i \\mathop \\in I} \\RR^{-1} \\sqbrk {T_i}$ where: :$\\displaystyle \\bigcup_{i \\mathop \\in I} T_i$ denotes the union of $\\family {T_i}_{i \\mathop \\in I}$ :$\\RR^{-1} \\left[{T_i}\\right]$ denotes the preimage of $T_i$ under $\\RR$."}
+{"_id": "5182", "title": "Image of Intersection under Mapping/Family of Sets", "text": "Let $S$ and $T$ be sets. Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of subsets of $S$. Let $f: S \\to T$ be a mapping. Then: :$\\ds f \\sqbrk {\\bigcap_{i \\mathop \\in I} S_i} \\subseteq \\bigcap_{i \\mathop \\in I} f \\sqbrk {S_i}$ where $\\ds \\bigcap_{i \\mathop \\in I} S_i$ denotes the intersection of $\\family {S_i}_{i \\mathop \\in I}$."}
+{"_id": "5183", "title": "Image of Union under Mapping/Family of Sets", "text": "Let $S$ and $T$ be sets. Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of subsets of $S$. Let $f: S \\to T$ be a mapping. Then: :$\\ds f \\sqbrk {\\bigcup_{i \\mathop \\in I} S_i} = \\bigcup_{i \\mathop \\in I} f \\sqbrk {S_i}$ where $\\ds \\bigcup_{i \\mathop \\in I} S_i$ denotes the union of $\\family {S_i}_{i \\mathop \\in I}$."}
+{"_id": "5184", "title": "Minkowski's Inequality/Lebesgue Spaces", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $p \\in \\R$, $p \\ge 1$. Let $f, g: X \\to \\R$ be $p$-integrable, that is, elements of Lebesgue $p$-space $\\map {\\mathcal L^p} \\mu$. Then their pointwise sum $f + g: X \\to \\R$ is also $p$-integrable, and: :$\\norm {f + g}_p \\le \\norm f_p + \\norm g_p$ where $\\norm {\\, \\cdot \\, }_p$ denotes the $p$-seminorm."}
+{"_id": "5185", "title": "Lebesgue Space is Vector Space", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $\\mathcal L^p \\left({\\mu}\\right)$ be Lebesgue $p$-space for $\\mu$. Then $\\mathcal L^p \\left({\\mu}\\right)$ is a vector subspace of $\\mathcal M \\left({\\Sigma}\\right)$, the space of $\\Sigma$-measurable functions. In particular, it is a vector space."}
+{"_id": "5186", "title": "Triangle Inequality for Series/Lebesgue Spaces", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $p \\in \\R$, $p \\ge 1$. Let $\\sequence {f_n}_{n \\mathop \\in \\N} \\in \\map {\\LL^p} {\\mu}$ be a sequence of $p$-integrable functions, that is, a sequence in Lebesgue $p$-space. Suppose that for all $n \\in \\N$, $f_n \\ge 0$ holds pointwise. {{expand|This condition could possibly be weakened/altered, using Lebesgue's Dominated Convergence Theorem in place of Beppo Levi's Theorem}} Then: :$\\displaystyle \\norm {\\sum_{n \\mathop = 1}^\\infty f_n}_p \\le \\sum_{n \\mathop = 1}^\\infty \\norm {f_n}_p$ where $\\norm {\\, \\cdot \\,}_p$ denotes the $p$-seminorm."}
+{"_id": "5187", "title": "Riesz-Fischer Theorem", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $p \\in \\R$, $p \\ge 1$. The Lebesgue $p$-space $\\map {\\LL^p} \\mu$, endowed with the $p$-norm, is a complete metric space."}
+{"_id": "5188", "title": "Pointwise Convergent Bounded Sequence in Lebesgue Space Converges in Norm", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $p \\in \\R_{\\ge 1}$. Let $\\sequence {f_n}_{n \\mathop \\in \\N}, f_n: X \\to \\R$ be a sequence in Lebesgue $p$-space $\\LL^p \\left({\\mu}\\right)$. Suppose that the pointwise limit $f := \\displaystyle \\lim_{n \\mathop \\to \\infty} f_n$ exists $\\mu$-almost everywhere. Suppose that for some $g \\in \\map {\\LL^p} \\mu$, the pointwise inequality $\\size {f_n} \\le g$ holds for all $n \\in \\N$. Then $f \\in \\map {\\LL^p} \\mu$, and: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\norm {f - f_n}_p = 0$ where $\\norm {\\, \\cdot \\,}_p$ denotes the $p$-seminorm."}
+{"_id": "5189", "title": "Riesz's Convergence Theorem", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $p \\in \\R$, $p \\ge 1$. Let $\\sequence {f_n}_{n \\mathop \\in \\N}, f_n: X \\to \\R$ be a sequence in Lebesgue $p$-space $\\map {\\LL^p} \\mu$. Suppose that the pointwise limit $f := \\displaystyle \\lim_{n \\mathop \\to \\infty} f_n$ exists $\\mu$-almost everywhere, and that $f \\in \\map {\\LL^p} \\mu$. Then the following are equivalent: :$(1): \\quad \\displaystyle \\lim_{n \\mathop \\to \\infty} \\norm {f - f_n}_p = 0$ :$(2): \\quad \\displaystyle \\lim_{n \\mathop \\to \\infty} \\norm {f_n}_p = \\norm f_p$ where $\\norm {\\, \\cdot \\,}_p$ denotes the $p$-seminorm."}
+{"_id": "5190", "title": "Preimage of Union under Mapping/Family of Sets", "text": "Let $S$ and $T$ be sets. Let $\\family {T_i}_{i \\mathop \\in I}$ be a family of subsets of $T$. Let $f: S \\to T$ be a relation. Then: :$\\displaystyle f^{-1} \\sqbrk {\\bigcup_{i \\mathop \\in I} T_i} = \\bigcup_{i \\mathop \\in I} f^{-1} \\sqbrk {T_i}$ where: :$\\displaystyle \\bigcup_{i \\mathop \\in I} T_i$ denotes the union of $\\family {T_i}_{i \\mathop \\in I}$ :$f^{-1} \\sqbrk {T_i}$ denotes the preimage of $T_i$ under $f$."}
+{"_id": "5191", "title": "Image of Intersection under One-to-Many Relation/Family of Sets", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation. Then $\\RR$ is a one-to-many relation {{iff}}: :$\\ds \\RR \\sqbrk {\\bigcap_{i \\mathop \\in I} S_i} = \\bigcap_{i \\mathop \\in I} \\RR \\sqbrk {S_i}$  where $\\family {S_i}_{i \\mathop \\in I}$ is ''any'' family of subsets of $S$."}
+{"_id": "5192", "title": "Preimage of Intersection under Mapping/Family of Sets", "text": "Let $S$ and $T$ be sets. Let $\\family {T_i}_{i \\mathop \\in I}$ be a family of subsets of $T$. Let $f: S \\to T$ be a mapping. Then: :$\\displaystyle f^{-1} \\sqbrk {\\bigcap_{i \\mathop \\in I} T_i} = \\bigcap_{i \\mathop \\in I} f^{-1} \\sqbrk {T_i}$ where: :$\\displaystyle \\bigcap_{i \\mathop \\in I} T_i$ denotes the intersection of $\\family {T_i}_{i \\mathop \\in I}$. :$f^{-1} \\sqbrk {T_i}$ denotes the preimage of $T_i$ under $f$."}
+{"_id": "5193", "title": "De Morgan's Laws (Set Theory)/Set Complement/General Case", "text": "Let $\\mathbb T$ be a set of sets, all of which are subsets of a universe $\\mathbb U$. Then: ==== Complement of Intersection ==== {{:De Morgan's Laws (Set Theory)/Set Complement/General Case/Complement of Intersection}} ==== Complement of Union ==== {{:De Morgan's Laws (Set Theory)/Set Complement/General Case/Complement of Union}}"}
+{"_id": "5194", "title": "De Morgan's Laws (Set Theory)/Relative Complement/Family of Sets", "text": "Let $S$ be a set. Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of subsets of $S$. Then: ==== Complement of Intersection ==== {{:De Morgan's Laws (Set Theory)/Relative Complement/Family of Sets/Complement of Intersection}} ==== Complement of Union ==== {{:De Morgan's Laws (Set Theory)/Relative Complement/Family of Sets/Complement of Union}}"}
+{"_id": "5195", "title": "Simple P-Integrable Functions Dense in Lebesgue Space", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space, and let $p \\in \\R$, $p \\ge 1$. Let $\\mathcal L^p \\left({\\mu}\\right)$ be Lebesgue $p$-space for $\\mu$. Let $\\mathcal E \\left({\\Sigma}\\right) \\cap \\mathcal L^p \\left({\\mu}\\right)$ be the space of $\\Sigma$-simple, $p$-integrable functions. Then $\\mathcal E \\left({\\Sigma}\\right) \\cap \\mathcal L^p \\left({\\mu}\\right)$ is everywhere dense in $\\mathcal L^p \\left({\\mu}\\right)$."}
+{"_id": "5196", "title": "De Morgan's Laws (Set Theory)/Set Difference/Family of Sets", "text": "Let $S$ and $T$ be sets. Let $\\family {T_i}_{i \\mathop \\in I}$ be a family of subsets of $T$. Then: ==== Difference with Intersection ==== {{:De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Intersection}} ==== Difference with Union ==== {{:De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Union}}"}
+{"_id": "5197", "title": "De Morgan's Laws (Set Theory)/Set Complement/Family of Sets", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of sets, all of which are subsets of a universe $\\Bbb U$. Then: ==== Complement of Intersection ==== {{:De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection}} ==== Complement of Union ==== {{:De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Union}}"}
+{"_id": "5198", "title": "Jensen's Inequality (Measure Theory)", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f: X \\to \\R$ be a $\\mu$-integrable function such that $f \\ge 0$ pointwise."}
+{"_id": "5199", "title": "Jensen's Inequality (Measure Theory)/Convex Functions", "text": "Let $V: \\hointr 0 \\infty \\to \\hointr 0 \\infty$ be a convex function. Then for all positive measurable functions $g: X \\to \\R$, $g \\in \\map {\\mathcal M^+} \\Sigma$: :$\\map V {\\dfrac {\\int g \\cdot f \\rd \\mu} {\\int f \\rd \\mu} } \\le \\dfrac {\\int \\paren {V \\circ g} \\cdot f \\rd \\mu} {\\int f \\rd \\mu}$ where $\\circ$ denotes composition, and $\\cdot$ denotes pointwise multiplication."}
+{"_id": "5200", "title": "Jensen's Inequality (Measure Theory)/Concave Functions", "text": "Let $\\Lambda: \\hointr 0 \\infty \\to \\hointr 0 \\infty$ be a concave function. Then for all positive measurable functions $g: X \\to \\R$, $g \\in \\map {\\MM^+} \\Sigma$: :$\\dfrac {\\int \\paren {\\Lambda \\circ g} \\cdot f \\rd \\mu} {\\int f \\rd \\mu} \\le \\map \\Lambda {\\dfrac {\\int g \\cdot f \\rd \\mu} {\\int f \\rd \\mu} }$ where $\\circ$ denotes composition, and $\\cdot$ denotes pointwise multiplication."}
+{"_id": "5202", "title": "Power Set is Lattice", "text": "Let $S$ be a set. Let $\\left({\\mathcal P \\left({S}\\right), \\subseteq}\\right)$ be the relational structure defined on $\\mathcal P \\left({S}\\right)$ by the subset relation $\\subseteq$. Then $\\left({\\mathcal P \\left({S}\\right), \\subseteq}\\right)$ is a lattice."}
+{"_id": "5203", "title": "Divisor Relation induces Lattice", "text": "Let $\\struct {\\Z_{> 0}, \\divides}$ be the ordered set comprising: :The set of positive integers $\\Z_{> 0}$ :The divisor relation $\\divides$ defined as: ::$a \\divides b := \\exists k \\in \\Z_{> 0}: b = ka$ Then $\\struct {\\Z_{> 0}, \\divides}$ is a lattice."}
+{"_id": "5204", "title": "Generator for Product Sigma-Algebra", "text": "Let $\\struct {X, \\Sigma_1}$ and $\\struct {Y, \\Sigma_2}$ be measurable spaces. Let $\\GG_1$ and $\\GG_2$ be generators for $\\Sigma_1$ and $\\Sigma_2$, respectively. Then $\\GG_1 \\times \\GG_2$ is a generator for the product $\\sigma$-algebra $\\Sigma_1 \\otimes \\Sigma_2$."}
+{"_id": "5205", "title": "Uniqueness of Product Measures", "text": "Let $\\struct {X, \\Sigma_1, \\mu}$ and $\\struct {Y, \\Sigma_2, \\nu}$ be measure spaces. Let $\\GG_1$ and $\\GG_2$ be generators for $\\Sigma_1$ and $\\Sigma_2$, respectively. Suppose that $\\GG_1$ and $\\GG_2$ are closed under intersection. Suppose further that there are exhausting sequences $\\sequence {G_{1, n} }_{n \\mathop \\in \\N}$ and $\\sequence {G_{2, n} }_{n \\mathop \\in \\N}$ in $\\GG_1$ and $\\GG_2$, respectively, such that: :$\\forall n \\in \\N: \\map \\mu {G_{1, n} } < \\infty$ :$\\forall n \\in \\N: \\map \\nu {G_{2, n} } < \\infty$ Then there is at most one measure $\\rho$ on the product space $\\struct {X \\times Y, \\Sigma_1 \\otimes \\Sigma_2}$ such that: :$\\forall G_1 \\in \\GG_1, G_2 \\in \\GG_2: \\map \\rho {G_1 \\times G_2} = \\map \\mu {G_1} \\, \\map \\nu {G_2}$ That is, there can be at most one product measure on $\\struct {X \\times Y, \\Sigma_1 \\otimes \\Sigma_2}$."}
+{"_id": "5206", "title": "Existence of Product Measures", "text": "Let $\\left({X, \\Sigma_1, \\mu}\\right)$ and $\\left({Y, \\Sigma_2, \\nu}\\right)$ be $\\sigma$-finite measure spaces. Then there exists a $\\sigma$-finite measure $\\rho$ on the product space $\\left({X \\times Y, \\Sigma_1 \\otimes \\Sigma_2}\\right)$ such that: :$\\forall E_1 \\in \\Sigma_1, E_2 \\in \\Sigma_2: \\rho \\left({E_1 \\times E_2}\\right) = \\mu \\left({E_1}\\right) \\nu \\left({E_2}\\right)$ This $\\rho$ is called product measure and is also denoted $\\mu \\times \\nu$."}
+{"_id": "5207", "title": "Tonelli's Theorem", "text": "Let $\\struct {X, \\Sigma_1, \\mu}$ and $\\struct {Y, \\Sigma_2, \\nu}$ be $\\sigma$-finite measure spaces. Let $\\struct {X \\times Y, \\Sigma_1 \\otimes \\Sigma_2, \\mu \\times \\nu}$ be the product measure space of $\\struct {X, \\Sigma_1, \\mu}$ and $\\struct {Y, \\Sigma_2, \\nu}$. Let $f: X \\times Y \\to \\overline \\R_{\\ge 0}$ be a positive $\\Sigma_1 \\otimes \\Sigma_2$-measurable function. Then: :$\\displaystyle \\int_{X \\times Y} f \\map \\rd {\\mu \\times \\nu} = \\int_Y \\int_X \\map f {x, y} \\map {\\d \\mu} x \\map {\\d \\nu} y = \\int_X \\int_Y \\map f {x, y} \\map {\\d \\nu} y \\map {\\d \\mu} x$"}
+{"_id": "5208", "title": "Condition for Power Set to be Totally Ordered", "text": "Let $\\mathcal P \\left({S}\\right)$ be the power set of a set $S$. Let $\\left({\\mathcal P \\left({S}\\right), \\subseteq}\\right)$ be the set $\\mathcal P \\left({S}\\right)$ ordered by $\\subseteq$. Then $\\left({\\mathcal P \\left({S}\\right), \\subseteq}\\right)$ is totally ordered iff $S$ is either the empty set or a singleton."}
+{"_id": "5209", "title": "Fubini's Theorem", "text": "Let $\\struct {X, \\Sigma_1, \\mu}$ and $\\struct {Y, \\Sigma_2, \\nu}$ be $\\sigma$-finite measure spaces. Let $\\struct {X \\times Y, \\Sigma_1 \\otimes \\Sigma_2, \\mu \\times \\nu}$ be the product measure space of $\\struct {X, \\Sigma_1, \\mu}$ and $\\struct {Y, \\Sigma_2, \\nu}$. Let $f: X \\times Y \\to \\R$ be a $\\Sigma_1 \\otimes \\Sigma_2$-measurable function. Suppose that: :$\\displaystyle \\int_{X \\times Y} \\size f \\map {\\rd} {\\mu \\times \\nu} < \\infty$ Then $f$ is $\\mu \\times \\nu$-integrable, and: :$\\displaystyle \\int_{X \\times Y} f \\map {\\rd} {\\mu \\times \\nu} = \\int_Y \\int_X \\map f {x, y} \\map {\\rd \\mu} x \\map {\\rd \\nu} y = \\int_X \\int_Y \\map f {x, y} \\map {\\rd \\nu} y \\map {\\rd \\mu} x$ {{expand|Implement a version of this for real functions}}"}
+{"_id": "5210", "title": "Product Sigma-Algebra Generated by Projections", "text": "Let $\\struct {X, \\Sigma_1}$ and $\\struct {Y, \\Sigma_2}$ be measurable spaces. Let $\\Sigma_1 \\otimes \\Sigma_2$ be the product $\\sigma$-algebra on $X \\times Y$. Let $\\pr_1: X \\times Y \\to X$ and $\\pr_2: X \\times Y \\to Y$ be the first and second projections, respectively. Then: :$\\Sigma_1 \\otimes \\Sigma_2 = \\map \\sigma {\\pr_1, \\pr_2}$ where $\\sigma$ denotes generated $\\sigma$-algebra."}
+{"_id": "5211", "title": "Measurable Mappings to Product Measurable Space", "text": "Let $\\left({X, \\Sigma}\\right)$, $\\left({X_1, \\Sigma_1}\\right)$ and $\\left({X_2, \\Sigma_2}\\right)$ be measurable spaces. Let $\\Sigma_1 \\otimes \\Sigma_2$ be the product $\\sigma$-algebra on $X_1 \\times X_2$. Let $\\operatorname{pr}_1: X_1 \\times X_2 \\to X_1$ and $\\operatorname{pr}_2: X_1 \\times X_2 \\to X_2$ be the first and second projections, respectively. A mapping $f: X \\to X_1 \\times X_2$ is $\\Sigma \\, / \\, \\Sigma_1 \\otimes \\Sigma_2$-measurable {{iff}}: :$\\operatorname{pr}_i \\circ f: X \\to X_i$ is $\\Sigma \\, / \\, \\Sigma_i$-measurable, for $i = 1, 2$"}
+{"_id": "5212", "title": "Measurable Mappings from Product Measurable Space", "text": "Let $\\left({X, \\Sigma}\\right)$, $\\left({X_1, \\Sigma_1}\\right)$ and $\\left({X_2, \\Sigma_2}\\right)$ be measurable spaces. Let $\\Sigma_1 \\otimes \\Sigma_2$ be the product $\\sigma$-algebra on $X_1 \\times X_2$. Let $f: X_1 \\times X_2 \\to X$ be a $\\Sigma_1 \\otimes \\Sigma_2 \\, / \\, \\Sigma$-measurable mapping. Then: :$\\forall x_1 \\in X_1: f \\left({x_1, \\cdot}\\right): X_2 \\to X$ is $\\Sigma_2 \\, / \\, \\Sigma$-measurable :$\\forall x_2 \\in X_2: f \\left({\\cdot, x_2}\\right): X_1 \\to X$ is $\\Sigma_1 \\, / \\, \\Sigma$-measurable"}
+{"_id": "5213", "title": "Integral of Survival Function", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a $\\sigma$-finite measure space. Let $f: X \\to \\R_{\\ge 0}$ be a positive $\\Sigma$-measurable function. Let $F \\left({f}\\right): \\R \\to \\R$ be the survival function of $f$. Then: :$\\displaystyle \\int f \\, \\mathrm d \\mu = \\int_{\\left({0 \\,.\\,.\\, \\infty}\\right)} F \\left({f}\\right) \\, \\mathrm d \\lambda$ where $\\lambda$ is Lebesgue measure."}
+{"_id": "5214", "title": "Cardinal Zero is Less than Cardinal One", "text": "The zero cardinal $0$ is less than one: :$0 < 1$"}
+{"_id": "5217", "title": "Idempotent Non-Trivial Quasigroup is Not a Loop", "text": "Let $\\left({S, \\circ}\\right)$ be an idempotent quasigroup whose underlying set $S$ comprises more than one element. Then $\\left({S, \\circ}\\right)$ is not an algebra loop, that is, it has no identity element."}
+{"_id": "5218", "title": "B-Algebra is Right Cancellable", "text": "Let $\\struct {X, \\circ}$ be a $B$-algebra. Then $\\circ$ is right-cancellable for $X$. That is: :$\\forall x, y, z \\in X: x \\circ z = y \\circ z \\implies x = y$"}
+{"_id": "5219", "title": "Right Regular Representation of 0 is Bijection in B-Algebra", "text": "Let $\\left({X, \\circ}\\right)$ be a $B$-algebra. Then the right regular representation of $\\left({X, \\circ}\\right)$ with respect to $0$ is a bijection."}
+{"_id": "5220", "title": "Product of Cardinals is Commutative", "text": "Let $\\mathbf a$ and $\\mathbf b$ be cardinals. Then: : $\\mathbf a \\mathbf b = \\mathbf b \\mathbf a$ where $\\mathbf a \\mathbf b$ denotes the product of $\\mathbf a$ and $\\mathbf b$."}
+{"_id": "5222", "title": "B-Algebra Identity: xy=x(0(0y))", "text": "Let $\\left({X, \\circ}\\right)$ be a $B$-algebra. Then: :$\\forall x,y \\in X: x \\circ y = x \\circ\\left({0 \\circ\\left({ 0 \\circ y}\\right)}\\right)$"}
+{"_id": "5223", "title": "B-Algebra Identity: x (y z) = (x (0 z)) y", "text": "Let $\\left({X, \\circ}\\right)$ be a $B$-algebra. Then: :$\\forall x,y,z \\in X: x \\circ \\left({y \\circ z}\\right) = \\left({x \\circ \\left({0 \\circ z}\\right)}\\right) \\circ y$"}
+{"_id": "5224", "title": "B-Algebra Identity: xy = 0 iff x = y", "text": "Let $\\left({X, \\circ}\\right)$ be a $B$-algebra. Then: :$\\forall x,y \\in X: x \\circ y = 0 \\iff x = y$"}
+{"_id": "5225", "title": "First Power of Element in B-Algebra", "text": "Let $\\left({X, \\circ}\\right)$ be a $B$-algebra. Then: :$\\forall x \\in X: x^1 = x$ where $x^k$ for $k \\in \\N$ denotes the $k$th power of the element $x$."}
+{"_id": "5226", "title": "B-Algebra Power Law", "text": "Let $\\left({X, \\circ}\\right)$ be a B-algebra. Let $n, m \\in \\N$ such that $n \\ge m$. Then: :$\\forall x \\in X: x^n \\circ x^m = x^{n-m}$ where $x^k$ for $k \\in \\N$ denotes the $k$th power of the element $x$."}
+{"_id": "5227", "title": "B-Algebra is Left Cancellable", "text": "Let $\\struct {X, \\circ}$ be a $B$-algebra. Then $\\circ$ is a left cancellable operation."}
+{"_id": "5228", "title": "B-Algebra is Quasigroup", "text": "Let $\\left({X, \\circ}\\right)$ be a $B$-algebra. Then $\\left({X, \\circ}\\right)$ is a quasigroup."}
+{"_id": "5229", "title": "Quasigroup is not necessarily B-Algebra", "text": "Let $\\struct {S, \\circ}$ be a quasigroup. Then $\\struct {S, \\circ}$ is not necessarily a $B$-algebra."}
+{"_id": "5230", "title": "Group is Quasigroup", "text": "Let $\\left({G, \\circ}\\right)$ be a group. Then $\\left({G, \\circ}\\right)$ is a quasigroup."}
+{"_id": "5231", "title": "B-Algebra Power Law with Zero", "text": ":$\\forall x \\in X: n, m \\in \\N_{>0} \\implies x^m \\circ x^n = 0 \\circ x^{n-m}$"}
+{"_id": "5232", "title": "Product of Powers in B-Algebra", "text": "Let $\\left({X, \\circ}\\right)$ be a $B$-algebra. Let $x \\in X$ and $m, n \\in \\N$. Then: :$x^m \\circ x^n = \\begin{cases} x^{m-n} & : m \\ge n \\\\ 0 \\circ x^{n-m} & : n > m \\end{cases}$"}
+{"_id": "5233", "title": "Superset of Co-Countable Set", "text": "Every superset of a co-countable set is co-countable."}
+{"_id": "5234", "title": "Group Induces B-Algebra", "text": "Let $\\left({G, \\circ}\\right)$ be a group whose identity element is $e$. Let $*$ be the binary operation on $G$ defined as: :$\\forall a, b \\in G: a * b = a \\circ b^{-1}$ where $b^{-1}$ is the inverse element of $b$ under the operation $\\circ$. Then the algebraic structure $\\left({G, *}\\right)$ is a $B$-algebra."}
+{"_id": "5236", "title": "Identity of Cardinal Product is One", "text": "Let $\\mathbf a$ be a cardinal. Then: : $\\mathbf 1 \\mathbf a = \\mathbf a$ where $\\mathbf 1 \\mathbf a$ denotes the product of the (cardinal) one and $\\mathbf a$. That is, $\\mathbf 1$ is the identity element of the product operation on cardinals."}
+{"_id": "5237", "title": "Zero of Cardinal Product is Zero", "text": "Let $\\mathbf a$ be a cardinal. Then: : $\\mathbf 0 \\mathbf a = \\mathbf 0$ where $\\mathbf 0 \\mathbf a$ denotes the product of the (cardinal) zero and $\\mathbf a$. That is, $\\mathbf 0$ is the zero element of the product operation on cardinals."}
+{"_id": "5238", "title": "Sum of Cardinals is Commutative", "text": "Let $\\mathbf a$ and $\\mathbf b$ be cardinals. Then: :$\\mathbf a + \\mathbf b = \\mathbf b + \\mathbf a$ where $\\mathbf a + \\mathbf b$ denotes the sum of $\\mathbf a$ and $\\mathbf b$."}
+{"_id": "5240", "title": "B-Algebra is Commutative iff x(xy)=y", "text": "Let $\\left({X, \\circ}\\right)$ be a $B$-algebra. Then $\\left({X, \\circ}\\right)$ is commutative {{iff}}: :$\\forall x, y \\in X: x \\circ \\left({x \\circ y}\\right) = y$"}
+{"_id": "5241", "title": "Identity of Cardinal Sum is Zero", "text": "Let $\\mathbf a$ be a cardinal. Then: : $\\mathbf a + \\mathbf 0 = \\mathbf a$ where $\\mathbf a + \\mathbf 0$ denotes the sum of the zero cardinal and $\\mathbf a$. That is, $\\mathbf 0$ is the identity element of the sum operation on cardinals."}
+{"_id": "5242", "title": "Commutative B-Algebra Implies (zy)(zx)=xy", "text": "Let $\\struct {B, \\circ}$ be a commutative $B$-algebra.  Then: :$\\forall x, y, z \\in X: \\paren {z \\circ y} \\circ \\paren {z \\circ x} = x \\circ y$"}
+{"_id": "5243", "title": "Cardinal Product Distributes over Cardinal Sum", "text": "Let $\\mathbf a$, $\\mathbf b$ and $\\mathbf c$ be cardinals. Then: : $\\mathbf a \\left({\\mathbf b + \\mathbf c}\\right) = \\mathbf a \\mathbf b + \\mathbf a \\mathbf c$ where: : $\\mathbf a + \\mathbf b$ denotes the sum of $\\mathbf a$ and $\\mathbf b$. : $\\mathbf a \\mathbf b$ denotes the product of $\\mathbf a$ and $\\mathbf b$."}
+{"_id": "5244", "title": "Ordering of Cardinals Compatible with Cardinal Product", "text": "Let $\\mathbf a$, $\\mathbf b$ and $\\mathbf c$ be cardinals. Then: : $\\mathbf a \\le \\mathbf b \\implies \\mathbf a \\mathbf c \\le \\mathbf b \\mathbf c$ where $\\mathbf a \\mathbf c$ denotes the product of $\\mathbf a$ and $\\mathbf c$."}
+{"_id": "5245", "title": "Ordering of Cardinals Compatible with Cardinal Sum", "text": "Let $\\mathbf a$, $\\mathbf b$ and $\\mathbf c$ be cardinals. Then: :$\\mathbf a \\le \\mathbf b \\implies \\mathbf a + \\mathbf c \\le \\mathbf b + \\mathbf c$ where $\\mathbf a \\mathbf c$ denotes the sum of $\\mathbf a$ and $\\mathbf c$."}
+{"_id": "5246", "title": "Group is B-Algebra Iff All Elements Self-Inverse", "text": "Let $\\left({G, \\circ}\\right)$ be a group whose identity element is $e$. Then $\\left({G, \\circ}\\right)$ is also a $B$-algebra {{iff}}: :$\\forall g \\in G: g = g^{-1}$ That is, {{iff}} all elements of $G$ are self-inverse."}
+{"_id": "5247", "title": "Condition for Existence of Cardinal Sum", "text": "Let $\\mathbf a$ and $\\mathbf b$ be cardinals. Then: : $\\mathbf a \\le \\mathbf b \\iff \\exists \\mathbf c: \\mathbf a + \\mathbf c = \\mathbf b$ where $\\mathbf c$ is also a cardinal."}
+{"_id": "5248", "title": "Cardinal One is Cancellable for Cardinal Sum", "text": "Let $\\mathbf a$ and $\\mathbf b$ be cardinals. Then: :$\\mathbf a + \\mathbf 1 = \\mathbf b + \\mathbf 1 \\implies \\mathbf a = \\mathbf b$ where $\\mathbf 1$ is (cardinal) one."}
+{"_id": "5250", "title": "Mapping from Finite Set to Itself is Injection iff Surjection", "text": "Let $S$ be a finite set. Let $f: S \\to S$ be a mapping. Then $f$ is injective {{iff}} $f$ is surjective."}
+{"_id": "5251", "title": "Finite Cardinals form Infinite Set", "text": "The finite cardinals form a set which is infinite."}
+{"_id": "5253", "title": "Increasing Function has Countable Discontinuities", "text": "Let $f: \\R \\to \\R$ be an increasing real function. Then $f$ has at most countably many discontinuities."}
+{"_id": "5254", "title": "Integral of Increasing Function Composed with Measurable Function", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a $\\sigma$-finite measure space. Let $f: X \\to \\R_{\\ge 0}$ be a positive measurable function. Let $\\phi: \\R_{\\ge 0} \\to \\R_{\\ge 0}$ be a continuously differentiable, increasing function such that $\\phi \\left({0}\\right) = 0$. Then: :$\\displaystyle \\int \\phi \\circ f \\rd \\mu = \\int_0^\\infty \\phi' \\left({t}\\right) F \\left({t}\\right) \\rd t$ where: : $F$ is the survival function of $f$ : $\\displaystyle \\int_0^\\infty$ denotes an improper integral."}
+{"_id": "5255", "title": "Minkowski's Inequality for Double Integrals", "text": "Let $\\struct {X, \\Sigma, \\mu}$ and $\\struct {Y, \\Sigma', \\nu}$ be $\\sigma$-finite measure spaces. Let $\\struct {X \\times Y, \\Sigma \\otimes \\Sigma', \\mu \\times \\nu}$ be their product measure space. Let $f: X \\times Y \\to \\overline \\R$ be a $\\Sigma \\otimes \\Sigma'$-measurable function. Then, for all $p \\in \\R$ with $p \\ge 1$: :$\\displaystyle \\paren {\\int_X \\paren {\\int_Y \\size {\\map f {x, y} } \\map {\\rd \\nu} y}^p \\map {\\rd \\mu} x}^{1/p} \\le \\int_Y \\paren {\\int_X \\size {\\map f {x, y} }^p \\map {\\rd \\mu} x}^{1/p} \\map {\\rd \\nu} y$"}
+{"_id": "5256", "title": "Integral with respect to Pushforward Measure", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $\\struct {X', \\Sigma'}$ be a measurable space. Let $T: X \\to X'$ be a $\\Sigma \\, / \\, \\Sigma'$-measurable mapping. Let $f: X' \\to \\overline \\R$ be a $\\map T \\mu$-integrable function, where $\\map T \\mu$ denotes the pushforward measure of $\\mu$ under $T$. Then $f \\circ T: X \\to \\overline \\R$ is $\\mu$-integrable, and: :$\\displaystyle \\int_{X'} f \\rd \\map T \\mu = \\int_X f \\circ T \\rd \\mu$"}
+{"_id": "5257", "title": "Integrable Function under Pushforward Measure", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $\\struct {X', \\Sigma'}$ be a measurable space. Let $T: X \\to X'$ be a $\\Sigma \\, / \\, \\Sigma'$-measurable mapping. Let $f: X' \\to \\overline \\R$ be a mapping. Then the following are equivalent: :$(1): \\quad f$ is $\\map T \\mu$-integrable :$(2): \\quad f \\circ T$ is $\\mu$-integrable where $\\map T \\mu$ is the pushforward measure of $\\mu$ under $T$."}
+{"_id": "5259", "title": "0 in B-Algebra is Left Cancellable Element", "text": "Let $\\left({X, \\circ}\\right)$ be a $B$-Algebra. Then: :$\\forall x, y \\in X: 0 \\circ x = 0 \\circ y \\implies x = y$"}
+{"_id": "5260", "title": "B-Algebra Identity: 0(0x)=x", "text": "Let $\\struct {X, \\circ}$ be a $B$-algebra.  Then: :$\\forall x \\in X: 0 \\circ \\paren {0 \\circ x} = x$"}
+{"_id": "5261", "title": "B-Algebra Induces Group", "text": "Let $\\struct {X, \\circ}$ be a $B$-algebra. Let $*$ be the binary operation on $X$ defined as: :$\\forall a, b \\in X: a * b := a \\circ \\paren {0 \\circ b}$ Then the algebraic structure $\\struct {X, *}$ is a group such that: :$\\forall x \\in X: 0 \\circ x$ is the inverse element of $x$ under $*$. That is: :$\\forall a, b \\in X: a * b^{-1} := a \\circ b$"}
+{"_id": "5269", "title": "Convolution of Measurable Functions is Bilinear", "text": "Let $\\BB^n$ be the Borel $\\sigma$-algebra on $\\R^n$. Let $f, f', g, g': \\R^n \\to \\R$ be $\\BB^n$-measurable functions. Then for all $\\lambda \\in \\R$: :$\\paren {\\lambda f + f'} * g = \\lambda \\paren {f * g} + f' * g$ :$f * \\paren {\\lambda g + g'} = \\lambda \\paren {f * g} + f * g'$ provided the convolutions in these expressions exist. That is, convolution $*$ is a bilinear operation."}
+{"_id": "5270", "title": "Convolution of Measurable Function and Measure is Bilinear", "text": "Let $\\mu$ and $\\nu$ be measures on the Borel $\\sigma$-algebra $\\mathcal B^n$ on $\\R^n$. Let $f, f': \\R^n \\to \\R$ be $\\mathcal B^n$-measurable functions. Then for all $\\lambda \\in \\R$: :$\\left({\\lambda f + f'}\\right) * \\mu = \\lambda \\left({f * \\mu}\\right) + f' * \\mu$ :$f * \\left({\\lambda \\mu + \\nu}\\right) = \\lambda \\left({f * \\mu}\\right) + f * \\nu$ provided the convolutions in these expressions exist. That is, convolution $*$ is a bilinear operation."}
+{"_id": "5271", "title": "Convolution of Measures is Bilinear", "text": "Let $\\mu, \\mu', \\nu$ and $\\nu'$ be measures on the Borel $\\sigma$-algebra $\\mathcal B^n$ on $\\R^n$. Then for all $\\lambda \\in \\R$: :$\\left({\\lambda \\mu + \\mu'}\\right) * \\nu = \\lambda \\left({\\mu * \\nu}\\right) + \\mu' * \\nu$ :$\\mu * \\left({\\lambda \\nu + \\nu'}\\right) = \\lambda \\left({\\mu * \\nu}\\right) + \\mu * \\nu'$ where $*$ denotes convolution. That is, convolution is a bilinear operation."}
+{"_id": "5272", "title": "Convolution of Measures as Pushforward Measure", "text": "Let $\\mu$ and $\\nu$ be measures on the Borel $\\sigma$-algebra $\\BB^n$ on $\\R^n$. Let $\\alpha: \\R^n \\times \\R^n \\to \\R^n, \\ \\map \\alpha {\\mathbf x, \\mathbf y} = \\mathbf x + \\mathbf y$ be vector addition on $\\R^n$. Then we have the following equality of measures on $\\BB^n$: :$\\mu * \\nu = \\map {\\alpha_*} {\\mu \\times \\nu}$ where $\\mu * \\nu$ is the convolution of $\\mu$ and $\\nu$, and $\\map {\\alpha_*} {\\mu \\times \\nu}$ is the pushforward of the product measure $\\mu \\times \\nu$ under $\\alpha$."}
+{"_id": "5274", "title": "Continuous Functions with Compact Support Dense in Lebesgue P-Space", "text": "Let $C_c \\left({\\R^n}\\right)$ be the space of continuous functions with compact support on $\\R^n$. Let $p \\in \\R$, $p \\ge 1$, and let $\\mathcal L^p \\left({\\lambda^n}\\right)$ be Lebesgue $p$-space for Lebesgue measure $\\lambda^n$. Then $C_c \\left({\\R^n}\\right)$ is everywhere dense in $\\mathcal L^p \\left({\\lambda^n}\\right)$ with respect to the $p$-seminorm $\\left\\Vert{\\, \\cdot \\,}\\right\\Vert_p$."}
+{"_id": "5275", "title": "Convolution of Integrable Function with Bounded Function", "text": "Let $f: \\R^n \\to \\R$ be a Lebesgue integrable function. Let $g: \\R^n \\to \\R$ be an essentially bounded function under Lebesgue measure $\\lambda^n$. Then the convolution $f * g$ of $f$ and $g$ is bounded and continuous. In particular, $f * g$ is again essentially bounded."}
+{"_id": "5276", "title": "F-Sigma Sets Closed under Union", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $F, F'$ be $F_\\sigma$ sets of $T$. Then their union $F \\cup F'$ is also an $F_\\sigma$ set of $T$."}
+{"_id": "5277", "title": "F-Sigma Sets Closed under Intersection", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $F, F'$ be $F_\\sigma$ sets of $T$. Then their intersection $F \\cap F'$ is also a $F_\\sigma$ set of $T$."}
+{"_id": "5278", "title": "G-Delta Sets Closed under Union", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $G, G'$ be $G_\\delta$ sets of $T$. Then their union $G \\cup G'$ is also a $G_\\delta$ set of $T$."}
+{"_id": "5279", "title": "G-Delta Sets Closed under Intersection", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $G, G'$ be $G_\\delta$ sets of $T$. Then their intersection $G \\cap G'$ is also a $G_\\delta$ set of $T$."}
+{"_id": "5286", "title": "Limit Point is Limit of Convergent Sequence", "text": "Let $M = \\left({X, d}\\right)$ be a metric space. Let $E \\subseteq X$ be a subset of $X$. Let $p$ be a limit point of $E$. Then there exists a sequence $\\left\\langle{x_n}\\right\\rangle \\subseteq E$ which converges to a limit: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = p$ where $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n$ is the limit of the sequence $\\left\\langle{x_n}\\right\\rangle$."}
+{"_id": "5287", "title": "F-Sigma Sets form Lattice", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $\\mathcal F$ be the collection of all $F_\\sigma$ sets of $T$. Then $\\left({\\mathcal F, \\subseteq}\\right)$ is a lattice, where $\\subseteq$ denotes the subset relation."}
+{"_id": "5288", "title": "G-Delta Sets form Lattice", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $\\mathcal G$ be the collection of all $G_\\delta$ sets of $T$. Then $\\left({\\mathcal G, \\subseteq}\\right)$ is a lattice, where $\\subseteq$ denotes the subset relation."}
+{"_id": "5289", "title": "Bounded Sequence in Euclidean Space has Convergent Subsequence", "text": "Let $\\left\\langle{x_i}\\right\\rangle_{i \\in \\N}$ be a bounded sequence in the Euclidean space $\\R^n$. Then some subsequence of $\\left\\langle{x_i}\\right\\rangle_{i \\in \\N}$ converges to a limit."}
+{"_id": "5290", "title": "Exponential Growth Equation/Special Case", "text": "All solutions of the differential equation $y' = y$ take the form $y = C e^x$."}
+{"_id": "5292", "title": "Existence-Uniqueness Theorem for First-Order Differential Equation", "text": "Let $P$ and $Q$ be continuous functions on some open interval $I \\subseteq \\R$. Let $a \\in I$. Let $b \\in \\R$. There is a unique function $f(x)=y$ on $I$ that satisfies the differential equation  :$y' + P(x)y = Q(x)$ along with the initial condition  :$f(a)=b$ This function is :$\\displaystyle f(x) = be^{-A(x)} + e^{-A(x)}\\int_a^x Q(t) e^{A(t)} dt$ where $\\displaystyle A(x) = \\int_a^x P(t)dt$"}
+{"_id": "5293", "title": "Proper Well-Ordering determines Smallest Elements", "text": "Let $S$ be a class. Let $\\preceq$ be a proper well-ordering on $S$. Let $B$ be a nonempty subclass of $S$. Then $B$ has a $\\preceq$-smallest element."}
+{"_id": "5295", "title": "Subset of Ordinals has Minimal Element", "text": "Let $A$ be an ordinal (we shall allow $A$ to be a proper class). Let $B$ be a nonempty subset of $A$. Then $B$ has an $\\Epsilon$-minimal element. {{explain|$\\Epsilon$}} That is: : $\\exists x \\in B: B \\cap x = \\varnothing$"}
+{"_id": "5296", "title": "Union of Subset of Ordinals is Ordinal", "text": "{{explain|If $A$ is a proper class, then its union may be the class of all ordinals, which we do not consider an ordinal.}} Let $A$ be a class of ordinals. That is, $A \\subseteq \\On$, where $\\On$ denotes the ordinal class. Then $\\bigcup A$ is an ordinal."}
+{"_id": "5297", "title": "Union of Ordinals is Least Upper Bound", "text": "Let $A \\subset \\operatorname{On}$. That is, let $A$ be a class of ordinals (every member of $A$ is an ordinal). Then $\\bigcup A$, the union of $A$, is the least upper bound of $A$: :$\\displaystyle \\forall x \\in A: x \\le A$ :$\\displaystyle \\forall y \\in A: y \\le x \\implies \\bigcup A \\le x$"}
+{"_id": "5298", "title": "Ordinal is Less than Successor", "text": "Let $x$ be an ordinal. Let $x^+$ denote the successor of $x$. Then: :$x \\in x^+$ :$x \\subset x^+$"}
+{"_id": "5299", "title": "No Natural Number between Number and Successor", "text": "Let $\\N$ be the natural numbers. Let $n \\in \\N$. Then no natural number $m$ exists strictly between $n$ and its successor: :$\\neg \\exists m \\in \\N: \\paren {n < m < n^+}$ That is: :If $m \\le n \\le m^+$, then $m = n$ or $m = n^+$."}
+{"_id": "5300", "title": "No Largest Ordinal", "text": "Let $a$ be a set of ordinals. Then: :$\\forall x \\in a: x \\prec \\left({\\bigcup a}\\right)^+$"}
+{"_id": "5301", "title": "Minimal Infinite Successor Set is Ordinal", "text": "Let $\\omega$ denote the minimal infinite successor set. Then $\\omega$ is an ordinal."}
+{"_id": "5304", "title": "No Infinitely Descending Membership Chains", "text": "Let $\\omega$ denote the minimal infinite successor set. Let $F$ be a function whose domain is $\\omega$. Then: :$\\exists n \\in \\omega: \\map F {n^+} \\notin \\map F n$"}
+{"_id": "5307", "title": "Transfinite Recursion/Uniqueness of Transfinite Recursion", "text": "Let $f$ be a mapping with a domain $y$ where $y$ is an ordinal. Let $f$ satisfy the condition that: : $\\forall x \\in y: f \\left({x}\\right) = G \\left({f \\restriction x}\\right)$ where $f \\restriction x$ denotes the restriction of $f$ to $x$. {{explain|What is $G$?}} Let $g$ be a mapping with a domain $z$ where $z$ is an ordinal. Let $g$ satisfy the condition that: : $\\forall x \\in z: g \\left({x}\\right) = G \\left({g \\restriction x}\\right)$ Let $y \\subseteq z$. Then: : $\\forall x \\in y: f \\left({x}\\right) = g \\left({x}\\right)$"}
+{"_id": "5308", "title": "Transfinite Induction/Schema 1", "text": "Let $P \\left({x}\\right)$ be a property  Suppose that: : If $P \\left({x}\\right)$ holds for all ordinals $x$ less than $y$, then $P \\left({y}\\right)$ also holds. Then $P \\left({x}\\right)$ holds for all ordinals $x$."}
+{"_id": "5309", "title": "Transfinite Induction/Principle 2", "text": "Let $A$ be a class satisfying the following conditions: * $\\varnothing \\in A$ * $\\forall x \\in A: x^+ \\in A$ * If $y$ is a limit ordinal, then $\\left({ \\forall x < y: x \\in A }\\right) \\implies y \\in A$ where $x^+$ denotes the successor of $x$. Then $\\operatorname{On} \\subseteq A$."}
+{"_id": "5310", "title": "Transfinite Induction/Schema 2", "text": "Let $\\phi \\left({x}\\right)$ be a property satisfying the following conditions: :$(1): \\quad \\phi \\left({\\varnothing}\\right)$ is true :$(2): \\quad$ If $x$ is an ordinal, then $\\phi \\left({x}\\right) \\implies \\phi \\left({x^+}\\right)$ :$(3): \\quad$ If $y$ is a limit ordinal, then $\\left({\\forall x < y: \\phi \\left({x}\\right)}\\right) \\implies \\phi \\left({y}\\right)$ where $x^+$ denotes the successor of $x$. Then, $\\phi \\left({x}\\right)$ is true for all ordinals $x$."}
+{"_id": "5311", "title": "Transfinite Recursion/Theorem 2", "text": "Let $\\Dom x$ denote the domain of $x$. Let $\\Img x$ denote the image of the mapping $x$. {{explain|We infer that $x$ is a mapping, but what is its context?}} Let $G$ be a class of ordered pairs $\\tuple {x, y}$ satisfying at least one of the following conditions: :$(1): \\quad x = \\O$ and $y = a$ {{explain|What is $a$?}} :$(2): \\quad \\exists \\beta: \\Dom x = \\beta^+$ and $y = \\map H {\\map x {\\bigcup \\Dom x} }$ {{explain|What is $H$?}} :$(3): \\quad \\Dom x$ is a limit ordinal and $y = \\bigcup \\Rng x$. {{explain|Is this invoking well-founded recursion?}} Let $\\map F \\alpha = \\map G {F \\restriction \\alpha}$ for all ordinals $\\alpha$. Then: :$F$ is a mapping and the domain of $F$ is the ordinals, $\\On$. :$\\map F \\O = a$ :$\\map F {\\beta^+} = \\map H {\\map F \\beta}$ :For limit ordinals $\\beta$, $\\displaystyle \\map F \\beta = \\bigcup_{\\gamma \\mathop \\in \\beta} \\map F \\gamma$ :$F$ is unique. ::That is, if there is another function $A$ satisfying the above three properties, then $A = F$."}
+{"_id": "5312", "title": "Transfinite Recursion/Corollary", "text": "Let $x$ be an ordinal. Let $G$ be a mapping There exists a unique mapping $f$ that satisfies the following properties: : The domain of $f$ is $x$ : $\\forall y \\in x: f \\left({y}\\right) = G \\left({f \\restriction y}\\right)$"}
+{"_id": "5313", "title": "Inverse of Product/Monoid/General Result", "text": "Let $\\left({S, \\circ}\\right)$ be a monoid whose identity is $e$. Let $a_1, a_2, \\ldots, a_n \\in S$ be invertible for $\\circ$, with inverses $a_1^{-1}, a_2^{-1}, \\ldots, a_n^{-1}$. Then $a_1 \\circ a_2 \\circ \\cdots \\circ a_n$ is invertible for $\\circ$, and: :$\\forall n \\in \\N_{> 0}: \\left({a_1 \\circ a_2 \\circ \\cdots \\circ a_n}\\right)^{-1} = a_n^{-1} \\circ \\cdots \\circ a_2^{-1} \\circ a_1^{-1}$"}
+{"_id": "5314", "title": "Inverse of Group Product", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $a, b \\in G$, with inverses $a^{-1}, b^{-1}$. Then: :$\\paren {a \\circ b}^{-1} = b^{-1} \\circ a^{-1}$"}
+{"_id": "5315", "title": "Inverse of Group Product/General Result", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $a_1, a_2, \\ldots, a_n \\in G$, with inverses $a_1^{-1}, a_2^{-1}, \\ldots, a_n^{-1}$. Then: : $\\paren {a_1 \\circ a_2 \\circ \\cdots \\circ a_n}^{-1} = a_n^{-1} \\circ \\cdots \\circ a_2^{-1} \\circ a_1^{-1}$"}
+{"_id": "5316", "title": "Well-Ordered Transitive Subset is Equal or Equal to Initial Segment", "text": "Let $\\left({\\prec, A}\\right)$ be a well-ordered set. For every $x \\in A$, let every $\\prec$-initial segment $A_x$ be a set. Let $B$ be a subclass of $A$ such that :$\\forall x \\in A: \\forall y \\in B: \\left({x \\prec y \\implies x \\in B}\\right)$. That is, $B$ must be $\\prec$-transitive. Then: :$A = B$ or: :$\\exists x \\in A: B = A_x$"}
+{"_id": "5317", "title": "Inverse of Inverse/General Algebraic Structure", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure with an identity element $e$. Let $x \\in S$ be invertible, and let $y$ be an inverse of $x$. Then $x$ is also an inverse of $y$."}
+{"_id": "5318", "title": "Inverse of Group Inverse", "text": "Let $\\struct {G, \\circ}$ be a group. Let $g \\in G$, with inverse $g^{-1}$. Then: :$\\paren {g^{-1} }^{-1} = g$"}
+{"_id": "5319", "title": "Condition for Injective Mapping on Ordinals", "text": "Let $F$ be a mapping satisfying the following properties: :$(1): \\quad$ The domain of $F$ is $\\On$, the ordinal class :$(2): \\quad$ For all ordinals $x$, $\\map F x = \\map G {F \\restriction x}$ :$(3): \\quad$ For all ordinals $x$, $\\map G {F \\restriction x} \\in \\paren {A \\setminus \\Img x}$ where $\\Img x$ is the image of $x$ under $F$. Let $\\Img F$ denote the image of $F$. Then the following properties hold: :$(1): \\quad \\Img F \\subseteq A$ :$(2): \\quad F$ is injective :$(3): \\quad A$ is a proper class. Note that only the third property of $F$ is the most important. For any function $G$, a function $F$ can be constructed satisfying the first two using transfinite recursion. {{explain|Exactly which of the above statements is the \"third\" is unclear. This statement needs to be made precise. It also needs to be explained. Also, it needs to be put in a separate section as it is peripheral to the statement of the theorem.}}"}
+{"_id": "5320", "title": "Maximal Injective Mapping from Ordinals to a Set", "text": "Let $F$ be a mapping satisfying the following properties: {{explain|What is $G$?}} : The domain of $F$ is $\\operatorname{On}$, the ordinal class : For all ordinals $x$, $F \\left({x}\\right) = G \\left({F \\restriction x}\\right)$. : For all ordinals $x$, if $(A \\setminus \\operatorname{Im} \\left({x}\\right) ) \\ne \\varnothing$, then $G  \\left({F \\restriction x}\\right) \\in (A \\setminus \\operatorname{Im} \\left({x}\\right))$ where $\\operatorname{Im} \\left({x}\\right)$ is the image of the subset $x$ under $F$. : $A$ is a set. Then there exists an ordinal $y$ satisfying the following properties: : $\\forall x \\in y: \\left({A \\setminus \\operatorname{Im} \\left({x}\\right)}\\right) \\ne \\varnothing$ : $\\operatorname {Im} \\left({y}\\right) = A$ : $F \\restriction y$ is an injective mapping. Note that the first third and fourth properties of $F$ are the most important.  For any mapping $G$, a mapping $F$ can be constructed satisfying the first two properties using transfinite recursion."}
+{"_id": "5321", "title": "Power of Element/Semigroup", "text": "Let $\\struct {S, \\oplus}$ be a magma. Let $a \\in S$. Let $n \\in \\N_{>0}$. Let $\\tuple {a_1, a_2, \\ldots, a_n}$ be the ordered $n$-tuple defined by $a_k = a$ for each $k \\in \\N_n$. Then: :$\\displaystyle \\bigoplus_{k \\mathop = 1}^n a_k = \\oplus^n a$ where: :$\\displaystyle \\bigoplus_{k \\mathop = 1}^n a_k$ is the composite of $\\tuple {a_1, a_2, \\ldots, a_n}$ for $\\oplus$ :$\\oplus^n a$ is the $n$th power of $a$ under $\\oplus$."}
+{"_id": "5322", "title": "Order Isomorphism between Ordinals and Proper Class", "text": "Let $\\left({A, \\prec}\\right)$ be a strict well-ordering. Let $A$ be a proper class. Let the initial segment of $x$ be a set for every $x \\in A$. Then we may make the following definitions: Set $G$ equal to the collection of ordered pairs $\\left({x, y}\\right)$ such that: :$y \\in \\left({A \\setminus \\operatorname{Im} \\left({x}\\right)}\\right)$ :$\\left({A \\setminus \\operatorname{Im}\\left({x}\\right)}\\right) \\cap A_y = \\varnothing$ Use transfinite recursion to construct a mapping $F$ such that: : The domain of $F$ is $\\operatorname{On}$ : For all ordinals $x$, $F \\left({x}\\right) = G \\left({F {\\restriction_x} }\\right)$ Then $F: \\operatorname{On} \\to A$ is an order isomorphism between $\\left({\\operatorname{On}, \\in}\\right)$ and $\\left({A, \\prec}\\right)$."}
+{"_id": "5325", "title": "Strict Well-Ordering Isomorphic to Unique Ordinal under Unique Mapping", "text": "Let $S$ be a set. Let $\\left({S, \\prec}\\right)$ be a strict well-ordering. Then there exists a unique ordinal $x$ and unique mapping $f$ such that $f: x \\to S$ is an order isomorphism."}
+{"_id": "5326", "title": "Unique Isomorphism between Ordinal Subset and Unique Ordinal", "text": "Let $\\operatorname{On}$ be the class of ordinals. Let $S \\subset \\operatorname{On}$ where $S$ is a set. Then there exists a unique mapping $\\phi$ and a unique ordinal $x$ such that $\\phi : x \\to S$ is an order isomorphism."}
+{"_id": "5327", "title": "Lexicographic Order forms Well-Ordering on Ordered Pairs of Ordinals", "text": "The Lexicographic Order $\\operatorname{Le}$ is a strict well-ordering on $\\left({\\operatorname{On} \\times \\operatorname{On}}\\right)$."}
+{"_id": "5329", "title": "Canonical Order Well-Orders Ordered Pairs of Ordinals", "text": "The canonical order, $R_0$ strictly well-orders the ordered pairs of ordinal numbers."}
+{"_id": "5332", "title": "Initial Segment of Canonical Order is Set", "text": "Let $R_0$ denote the canonical ordering of $\\left({\\operatorname{On} \\times \\operatorname{On} }\\right)$. Then, for all $\\left({x, y}\\right) \\in \\left({\\operatorname{On} \\times \\operatorname{On} }\\right)$, the $R_0$-initial segment is a set. {{explain|The $R_0$-initial segment of what?}}"}
+{"_id": "5333", "title": "Ordinal Addition is Closed", "text": "Let $\\operatorname{On}$ be the class of all ordinals. Then: :$\\forall x, y \\in \\operatorname{On}: \\left({x + y}\\right) \\in \\operatorname{On}$ That is: the sum $x+y$ is an ordinal."}
+{"_id": "5334", "title": "Union of Ordinals is Ordinal", "text": "Let $y$ be a set. Let $F \\left({z}\\right)$ be a mapping such that: :$\\displaystyle \\forall z \\in y: F \\left({z}\\right) \\in \\operatorname{On}$ Then: :$\\displaystyle \\bigcup_{z \\mathop \\in y} F \\left({z}\\right) \\in \\operatorname{On}$"}
+{"_id": "5335", "title": "Ordinal Addition by Zero", "text": "Let $x$ be an ordinal. Let $\\varnothing$ be the zero ordinal. Then: :$x + \\varnothing = x = \\varnothing + x$ where $+$ denotes ordinal addition."}
+{"_id": "5336", "title": "Union of Limit Ordinal", "text": "Let $x$ be a limit ordinal. Then: :$\\displaystyle x = \\bigcup x$ :$\\displaystyle x = \\bigcup_{y \\mathop \\in x} y$"}
+{"_id": "5337", "title": "Substitutivity of Equality", "text": "Let $x$ and $y$ be sets. Let $\\map P x$ be a well-formed formula of the language of set theory. Let $\\map P y$ be the same proposition $\\map P x$ with some (not necessarily all) free instances of $x$ replaced with free instances of $y$. Let $=$ denote set equality. :$x = y \\implies \\paren {\\map P x \\iff \\map P y}$"}
+{"_id": "5338", "title": "Membership is Left Compatible with Ordinal Addition", "text": "Let $x$, $y$, and $z$ be ordinals. Let $<$ denote membership $\\in$, since $\\in$ is a strict well-ordering on the ordinals. Then: :$x < y \\implies \\paren {z + x} < \\paren {z + y}$"}
+{"_id": "5339", "title": "Ordinal Addition is Left Cancellable", "text": "Let $x, y, z$ be ordinals. Then: :$\\left({z + x}\\right) = \\left({z + y}\\right) \\implies x = y$ That is, ordinal addition is left cancellable."}
+{"_id": "5340", "title": "Supremum Inequality for Ordinals", "text": "Let $A \\subseteq \\operatorname{On}$ and $B \\subseteq \\operatorname{On}$ be ordinals. Then: :$\\displaystyle \\forall x \\in A: \\exists y \\in B: x \\le y \\implies \\bigcup A \\le \\bigcup B$"}
+{"_id": "5341", "title": "Subset is Right Compatible with Ordinal Addition", "text": "Let $x, y, z$ be ordinals. Then: :$x \\le y \\implies \\left({x + z}\\right) \\le \\left({y + z}\\right)$"}
+{"_id": "5342", "title": "Ordinal Subtraction when Possible is Unique", "text": "Let $x$ and $y$ be ordinals such that $x \\le y$. Then there exists a unique ordinal $z$ such that $\\paren {x + z} = y$. That is: :$x \\le y \\implies \\exists! z \\in \\On: \\paren {x + z} = y$"}
+{"_id": "5343", "title": "Equality of Successors", "text": "Let $x$ and $y$ be ordinals. Let $x^+$ denote the successor set of $x$. Then, $x = y \\iff x^+ = y^+$"}
+{"_id": "5345", "title": "Finite Ordinal Plus Transfinite Ordinal", "text": "Let $n$ be a finite ordinal. Let $x$ be a transfinite ordinal. Then: : $n + x = x$"}
+{"_id": "5346", "title": "Class is Transitive iff Union is Subset", "text": "A class $A$ is transitive {{iff}}: :$\\displaystyle \\bigcup A \\subseteq A$"}
+{"_id": "5347", "title": "Union of Successor Ordinal", "text": "Let $x$ be an ordinal. Let $x^+$ denote the successor of $x$. Then: :$\\displaystyle \\map \\bigcup {x^+} = x$"}
+{"_id": "5348", "title": "Union Distributes over Union", "text": "Set union is distributive over itself: :$\\forall A, B, C: \\paren {A \\cup B} \\cup \\paren {A \\cup C} = A \\cup B \\cup C = \\paren {A \\cup C} \\cup \\paren {B \\cup C}$ where $A, B, C$ are sets."}
+{"_id": "5349", "title": "Subset of Empty Set", "text": "Let $A$ be a class. Then: : $A$ is a subset of the empty set $\\O$ {{iff}}; : $A$ is equal to the empty set: :$\\displaystyle A \\subseteq \\O \\iff A = \\O$"}
+{"_id": "5350", "title": "Ordinal is Subset of Successor", "text": "Let $x$ and $y$ be ordinals. Let $x^+$ denote the successor of $x$. Then: : $x \\subseteq y^+ \\iff \\left({x \\subseteq y \\lor x = y^+}\\right)$"}
+{"_id": "5351", "title": "Indexed Union Equality", "text": "Let $A$, $B_x$, and $C_x$ be classes. Then: :$\\displaystyle \\forall x \\in A: B_x = C_x \\implies \\bigcup_{x \\mathop \\in A} B_x = \\bigcup_{x \\mathop \\in A} C_x$ {{MissingLinks|$\\bigcup_{x \\mathop \\in A} B_x$ and $\\bigcup_{x \\mathop \\in A} C_x$}}"}
+{"_id": "5352", "title": "Indexed Union Subset", "text": "Let $A$, $B_x$ and $C_x$ be classes. {{explain|It can be inferred from the context, but the meaning of the subscript on $B_x$ and $C_x$ needs to be explained.}} Then: :$\\displaystyle \\forall x \\in A: B_x \\subseteq C_x \\implies \\bigcup_{x \\mathop \\in A} B_x \\subseteq \\bigcup_{x \\mathop \\in A} C_x$ {{MissingLinks|$\\bigcup_{x \\mathop \\in A} B_x$ and $\\bigcup_{x \\mathop \\in A} C_x$}}"}
+{"_id": "5353", "title": "Ordinal is Less than Sum", "text": "Let $x$ and $y$ be ordinals. Then: :$x \\le \\left({x + y}\\right)$ :$x \\le \\left({y + x}\\right)$"}
+{"_id": "5354", "title": "Limit Ordinals Preserved Under Ordinal Addition", "text": "Let $x$ and $y$ be ordinals such that $x$ is a limit ordinal. Then $\\left({y + x}\\right)$ is a limit ordinal. That is, letting $K_{II}$ denote the class of all limit ordinals: :$\\forall x \\in K_{II}: \\left({y + x}\\right) \\in K_{II}$"}
+{"_id": "5355", "title": "Successor in Limit Ordinal", "text": "Suppose that $x$ is a limit ordinal and that $y \\in x$. Then $y^+ \\in x$ where $y^+$ denotes the successor set of $y$: :$\\forall y \\in x: y^+ \\in x$"}
+{"_id": "5356", "title": "Successor is Less than Successor", "text": "Let $x$ and $y$ be ordinals and let $x^+$ denote the successor set of $x$. Then, $x \\in y \\iff x^+ \\in y^+$."}
+{"_id": "5357", "title": "Ordinal Addition is Associative", "text": "Ordinal addition is associative, i.e.: :$\\left({x + y}\\right) + z = x + \\left({y + z}\\right)$ holds for all ordinals $x$, $y$ and $z$."}
+{"_id": "5360", "title": "Derivative of Composite Function/Corollary", "text": ":$\\dfrac {\\d y} {\\d x} = \\dfrac {\\paren {\\dfrac {\\d y} {\\d u} } } {\\paren {\\dfrac {\\d x} {\\d u} } }$ for $\\dfrac {\\d x} {\\d u} \\ne 0$."}
+{"_id": "5365", "title": "Product of Subset with Intersection/Corollary", "text": "Let $\\struct {G, \\circ}$ be a group. Let $X, Y, Z \\subseteq G$ such that $X$ is a singleton. Then: :$X \\circ \\paren {Y \\cap Z} = \\paren {X \\circ Y} \\cap \\paren {X \\circ Z}$ :$\\paren {Y \\cap Z} \\circ X = \\paren {Y \\circ X} \\cap \\paren {Z \\circ X}$ where $X \\circ Y$ denotes the subset product of $X$ and $Y$."}
+{"_id": "5368", "title": "Single Instruction URM Programs/Zero Function", "text": "The zero function $\\Zero: \\N \\to \\N$, defined as: :$\\forall n \\in \\N: \\map \\Zero n = 0$"}
+{"_id": "5369", "title": "Single Instruction URM Programs/Successor Function", "text": "The successor function $\\Succ: \\N \\to \\N$, defined as: :$\\forall n \\in \\N: \\map \\Succ n = n + 1$"}
+{"_id": "5370", "title": "Single Instruction URM Programs/Projection Function", "text": "The projection functions $\\pr_j^k: \\N^k \\to \\N$, defined as: :$\\forall j \\in \\closedint 1 k: \\forall \\tuple {n_1, n_2, \\ldots, n_k} \\in \\N^k: \\map {\\pr_j^k} {n_1, n_2, \\ldots, n_k} = n_j$"}
+{"_id": "5373", "title": "Antiassociative Operation has no Idempotent Elements", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\circ$ be antiassociative on $S$. Then no element of $S$ is idempotent under $ \\circ$. That is: :$\\forall x \\in S: x \\circ x \\ne x$"}
+{"_id": "5374", "title": "Antiassociative Operation is not Commutative", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure. Let $\\circ$ be antiassociative on $S$. Then $\\circ$ is not commutative on $S$."}
+{"_id": "5375", "title": "Antiassociative Structure of Finite Order", "text": "Let $n \\in \\N$ such that $n > 2$. Then there exists an algebraic structure $\\left({S, \\circ}\\right)$ of order $n$ such that $\\circ$ is antiassociative on $S$."}
+{"_id": "5376", "title": "Example:Antiassociative Structure", "text": "Let $\\left({\\R_{>0}, \\circ}\\right)$ be an algebraic structure where $\\R_{>0}$ denotes the strictly positive real numbers. Define: :$\\forall x, y \\in \\R_{>0}: x \\circ y = x y + y$ Then $\\circ$ is antiassociative on $\\R_{>0}$: :$\\forall x, y, z \\in \\R_{>0}: \\left({x \\circ y}\\right) \\circ z \\ne x \\circ \\left({y \\circ z}\\right)$"}
+{"_id": "5377", "title": "Reciprocal Sequence is Strictly Decreasing", "text": "The reciprocal sequence: :$\\sequence {\\operatorname {recip} }: \\N_{>0} \\to \\R$: $n \\mapsto \\dfrac 1 n$ is strictly decreasing."}
+{"_id": "5382", "title": "Two-Step Vector Subspace Test", "text": "Let $V$ be a vector space over a division ring $K$. Let $U \\subseteq V$ be a non-empty subset of $V$ such that: :$(1): \\qquad \\forall u \\in U, \\lambda \\in K: \\lambda u \\in U$ :$(2): \\qquad \\forall u, v \\in U: u + v \\in U$ Then $U$ is a subspace of $V$."}
+{"_id": "5385", "title": "Reciprocal Function is Strictly Decreasing", "text": "The reciprocal function: :$\\operatorname{recip}: \\R \\setminus \\set 0 \\to \\R$, $x \\mapsto \\dfrac 1 x$ is strictly decreasing: :on the open interval $\\openint 0 \\to$ :on the open interval $\\openint \\gets 0$"}
+{"_id": "5386", "title": "Coefficients of Polynomial Product", "text": "Let $J$ be a set. Let $p_1, \\ldots p_n$ be polynomial forms in the indeterminates $\\set {X_j : j \\in J}$ over a commutative ring $R$. Suppose that for each $i$ with $1 \\le i \\le n$, we have, for appropriate $a_{i, k} \\in R$: :$p_i = \\displaystyle \\sum_{k \\mathop \\in Z} a_{i, k} X^k$ where $Z$ comprises the multiindices of natural numbers over $J$. Then: :$\\displaystyle \\prod_{i \\mathop = 1}^n p_i = \\displaystyle \\sum_{k \\mathop \\in Z} b_k X^k$ where: :$\\displaystyle b_k := \\sum_{k_1 + \\cdots + k_n \\mathop = k} \\paren {\\prod_{i \\mathop = 1}^n a_{i, k_i} }$"}
+{"_id": "5389", "title": "Subset Product of Subgroups/Necessary Condition", "text": "Let $\\left({G, \\circ}\\right)$ be a group. Let $H, K$ be subgroups of $G$. Let $H \\circ K$ be a subgroup of $G$. Then $H$ and $K$ are permutable. That is: :$H \\circ K = K \\circ H$ where $H \\circ K$ denotes subset product."}
+{"_id": "5390", "title": "Subset Product of Subgroups/Sufficient Condition", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H, K$ be subgroups of $G$. Let $H$ and $K$ be permutable subgroups of $G$. That is, suppose: :$H \\circ K = K \\circ H$ where $H \\circ K$ denotes subset product. Then $H \\circ K$ is a subgroup of $G$."}
+{"_id": "5395", "title": "Characteristics of Eulerian Graph/Necessary Condition", "text": "Let $G$ be a finite (undirected) graph. Let $G$ be Eulerian. Then $G$ is connected and each vertex of $G$ is even. Note that the definition of graph here includes: * Simple graph * Loop-graph * Multigraph * Loop-multigraph but does not include directed graph."}
+{"_id": "5396", "title": "Characteristics of Eulerian Graph/Sufficient Condition/Proof 1", "text": "Let $G$ be a finite (undirected) graph which is connected Let each vertex of $G$ be even. Then $G$ is an Eulerian graph. Note that the definition of graph here includes: * Simple graph * Loop-graph * Multigraph * Loop-multigraph but does not include directed graph."}
+{"_id": "5397", "title": "Characteristics of Eulerian Graph/Sufficient Condition/Proof 2", "text": "Let $G$ be a finite (undirected) graph which is connected. Let each vertex of $G$ be even. Then $G$ is an Eulerian graph. Note that the definition of graph here includes: * Simple graph * Loop-graph * Multigraph * Loop-multigraph but does not include directed graph."}
+{"_id": "5398", "title": "Characteristics of Eulerian Graph/Sufficient Condition", "text": "Let $G$ be a finite (undirected) graph which is connected Let each vertex of $G$ be even. Then $G$ is an Eulerian graph. Note that the definition of graph here includes: * Simple graph * Loop-graph * Multigraph * Loop-multigraph but does not include directed graph."}
+{"_id": "5399", "title": "Characterization of Pre-Measures", "text": "Let $X$ be a set, and let $\\mathcal S \\subseteq \\mathcal P \\left({X}\\right)$ be a collection of subsets of $X$. Let $\\varnothing \\in \\mathcal S$. Denote $\\overline \\R_{\\ge 0}$ for the set of positive extended real numbers. A mapping $\\mu: \\mathcal S \\to \\overline \\R_{\\ge 0}$ is a pre-measure {{iff}}: :$(1):\\quad \\mu \\left({\\varnothing}\\right) = 0$ :$(2):\\quad \\mu$ is finitely additive :$(3):\\quad$ For every increasing sequence $\\left({E_n}\\right)_{n \\in \\N}$ in $\\mathcal S$, if $E_n \\uparrow E$ for some $E \\in \\mathcal S$, then: ::$\\mu \\left({E}\\right) = \\displaystyle \\lim_{n \\to \\infty} \\mu \\left({E_n}\\right)$ where $E_n \\uparrow E$ denotes limit of increasing sequence of sets. Alternatively, and equivalently, $(3)$ may be replaced by either of: :$(3'):\\quad$ For every decreasing sequence $\\left({E_n}\\right)_{n \\in \\N}$ in $\\mathcal S$ for which $\\mu \\left({E_1}\\right)$ is finite, if $E_n \\downarrow E$ for some $E \\in \\mathcal S$, then: ::$\\mu \\left({E}\\right) = \\displaystyle \\lim_{n \\to \\infty} \\mu \\left({E_n}\\right)$ :$(3''):\\quad$ For every decreasing sequence $\\left({E_n}\\right)_{n \\in \\N}$ in $\\mathcal S$ for which $\\mu \\left({E_1}\\right)$ is finite, if $E_n \\downarrow \\varnothing$, then: ::$\\displaystyle \\lim_{n \\to \\infty} \\mu \\left({E_n}\\right) = 0$ where $E_n \\downarrow E$ denotes limit of decreasing sequence of sets."}
+{"_id": "5400", "title": "Power Set is Magma of Sets", "text": "Let $X$ be a set. Let $\\family {\\phi_i}: i \\in I$ be an indexed family of mappings. Then $\\powerset X$, the power set of $X$, is a magma of sets for $\\family {\\phi_i}: i \\in I$ on $X$."}
+{"_id": "5401", "title": "Existence and Uniqueness of Magma of Sets Generated by Collection of Subsets", "text": "Let $X$ be a set, and let $\\Phi := \\left\\{{\\phi_i: i \\in I}\\right\\}$ be a collection of partial mappings with codomain $\\mathcal P \\left({X}\\right)$, the power set of $X$. Let $\\mathcal G \\subseteq \\mathcal P \\left({X}\\right)$ be a collection of subsets of $X$. Then the magma of sets generated by $\\mathcal G$ exists and is unique."}
+{"_id": "5402", "title": "Intersection of Magmas of Sets is Magma of Sets", "text": "Let $X$ be a set. Let $\\Phi := \\set {\\phi_i: i \\in I}$ be a collection of partial mappings with codomain $\\powerset X$, the power set of $X$. Let $\\mathcal S_j$ be a magma of sets for $\\Phi$, for each $j \\in J$, for some index set $J$. Then: :$\\mathcal S := \\displaystyle \\bigcap_{j \\mathop \\in J} \\mathcal S_j$ is also a magma of sets for $\\Phi$."}
+{"_id": "5403", "title": "Sigma-Algebra as Magma of Sets", "text": "The concept of $\\sigma$-algebra is an instance of a magma of sets."}
+{"_id": "5404", "title": "Ordinal Multiplication is Closed", "text": "Let $x$ and $y$ be ordinals. Let $\\operatorname{On}$ denote the ordinal class. :$\\left({x \\cdot y}\\right) \\in \\operatorname{On}$"}
+{"_id": "5405", "title": "Natural Number Multiplication is Closed", "text": "Let $m$ and $n$ be natural numbers. Then: :$m \\times n \\in \\N$ where $\\times$ denotes natural number multiplication."}
+{"_id": "5406", "title": "Ordinal Multiplication by Zero", "text": "Let $x$ be an ordinal. {{begin-eqn}} {{eqn | l = \\left({x \\cdot \\varnothing}\\right)       | r = \\varnothing       | c =  }} {{eqn | l = \\left({\\varnothing \\cdot x}\\right)       | r = \\varnothing       | c =  }} {{end-eqn}}"}
+{"_id": "5407", "title": "Ordinal Multiplication by One", "text": "Let $x$ be an ordinal. Let $1$ denote the successor of $\\varnothing$. {{begin-eqn}} {{eqn|l = \\left({x \\cdot 1}\\right)      |r = x      |c =  }} {{eqn|l = \\left({1 \\cdot x}\\right)      |r = x      |c =  }} {{end-eqn}}"}
+{"_id": "5408", "title": "Ordinal Addition by One", "text": "Let $x$ be an ordinal. Let $x^+$ denote the successor of $x$. Let $1$ denote (ordinal) one, the successor of the zero ordinal $\\O$. Then: :$x + 1 = x^+$ where $+$ denotes ordinal addition."}
+{"_id": "5411", "title": "Closed Set Measurable in Borel Sigma-Algebra", "text": "Let $\\left({S, \\tau}\\right)$ be a topological space, and let $\\mathcal B \\left({\\tau}\\right)$ be the associated Borel $\\sigma$-algebra. Let $C$ be a closed set in $S$. Then $C$ is $\\mathcal B \\left({\\tau}\\right)$-measurable."}
+{"_id": "5412", "title": "Euler Phi Function of Product with Prime/Corollary", "text": ":$d \\divides n \\implies \\map \\phi d \\divides \\map \\phi n$ where $d \\divides n$ denotes that $d$ is a divisor of $n$."}
+{"_id": "5416", "title": "Commutative B-Algebra Induces Abelian Group", "text": "Let $\\left({X, \\circ }\\right)$ be a commutative $B$-algebra. Let $*$ be the binary operation on $X$ defined as: :$\\forall a, b \\in X: a * b := a \\circ \\left({0 \\circ b}\\right)$ Then the algebraic structure $\\left({X, *}\\right)$ is an abelian group such that: :$\\forall x \\in X: 0 \\circ x$ is the inverse element of $x$ under $*$. That is: :$\\forall a, b \\in X: a * b^{-1} := a \\circ b$"}
+{"_id": "5417", "title": "Membership is Left Compatible with Ordinal Multiplication", "text": "Let $x$, $y$, and $z$ be ordinals. Then: :$\\left({ x < y \\land z > 0 }\\right) \\iff \\left({ z \\cdot x }\\right) < \\left({ z \\cdot y }\\right)$"}
+{"_id": "5418", "title": "Ordinal Multiplication is Left Cancellable", "text": "Let $x, y, z$ be ordinals. Let $z \\ne 0$. Then: :$\\displaystyle \\left({z \\cdot x}\\right) = \\left({z \\cdot y}\\right) \\implies x = y$ That is, ordinal multiplication is left cancellable."}
+{"_id": "5419", "title": "Subset is Right Compatible with Ordinal Multiplication", "text": "Let $x, y, z$ be ordinals. Then: :$x \\le y \\implies \\left({x \\cdot z}\\right) \\le \\left({y \\cdot z}\\right)$"}
+{"_id": "5420", "title": "Ordinals have No Zero Divisors", "text": "Let $x$ and $y$ be ordinals. Then: :$\\paren {x \\cdot y} = 0 \\iff \\paren {x = 0 \\lor y = 0}$"}
+{"_id": "5423", "title": "Absolute Value of Simple Function is Simple Function/Proof 2", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $f: X \\to \\R$ be a simple function. Then $\\left\\vert{f}\\right\\vert: X \\to \\R$, the absolute value of $f$, is also a simple function."}
+{"_id": "5424", "title": "Absolute Value of Simple Function is Simple Function/Proof 1", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $f: X \\to \\R$ be a simple function. Then $\\left\\vert{f}\\right\\vert: X \\to \\R$, the absolute value of $f$, is also a simple function."}
+{"_id": "5427", "title": "Convergence in Norm Implies Convergence in Measure", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space, and let $p \\in \\R, p \\ge 1$. Let $\\sequence {f_n}_{n \\mathop \\in \\N}, f_n : X \\to \\R$ be a sequence of $p$-integrable functions. Also, let $f: X \\to \\R$ be a $p$-integrable function. Suppose that $f_n$ converges in norm to $f$ (in the $p$-norm). Then $f_n$ converges in measure to $f$ (in $\\mu$). That is: :$\\displaystyle \\operatorname {\\mathcal L^{\\textit p}-\\!\\lim\\,} \\limits_{n \\mathop \\to \\infty} f_n = f \\implies \\operatorname {\\mu-\\!\\lim\\,} \\limits_{n \\mathop \\to \\infty} f_n = f$"}
+{"_id": "5428", "title": "Pointwise Convergence Implies Convergence in Measure", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $\\sequence {f_n}_{n \\mathop \\in \\N}, f_n : X \\to \\R$ be a sequence of measurable functions. Also, let $f: X \\to \\R$ be a measurable function. Suppose that $f$ is the pointwise limit of the $f_n$ $\\mu$-almost everywhere. Then $f_n$ converges in measure to $f$ (in $\\mu$). That is: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map {f_n} x \\stackrel {a.e.} {=} \\map f x \\implies \\operatorname {\\mu-\\!\\lim\\,} \\limits_{n \\mathop \\to \\infty} f_n = f$"}
+{"_id": "5429", "title": "Convergence in Sigma-Finite Measure", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a $\\sigma$-finite measure space. Let $\\left({f_n}\\right)_{n \\in \\N}, f_n: X \\to \\R$ be a sequence of measurable functions. Also, let $f, g: X \\to \\R$ be measurable functions. Suppose that $f_n$ converges in measure to both $f$ and $g$ (in $\\mu$). Then $f$ and $g$ are equal $\\mu$-almost everywhere."}
+{"_id": "5430", "title": "Vitali's Theorem", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a $\\sigma$-finite measure space, and let $p \\in \\R, p \\ge 1$. Let $\\left({f_n}\\right)_{n \\in \\N}, f_n: X \\to \\R$ be a sequence of $p$-integrable functions. Also, let $f: X \\to \\R$ be a measurable function. Suppose that $\\displaystyle \\operatorname{\\mu-\\!\\lim\\,} \\limits_{n \\to \\infty} f_n = f$, i.e. $f_n$ converges in measure to $f$. Then the following are equivalent: :$(1): \\quad \\displaystyle \\lim_{n \\to \\infty} \\left\\Vert{f_n - f}\\right\\Vert_p = 0$, where $\\left\\Vert{\\cdot}\\right\\Vert_p$ is the $p$-seminorm (i.e. $\\displaystyle \\operatorname{\\mathcal L^{\\textit p}-\\!\\lim\\,} \\limits_{n \\to \\infty} f_n = f$) :$(2): \\left({\\left\\vert{f_n}\\right\\vert^p}\\right)_{n \\in \\N}$ is a uniformly integrable collection :$(3): \\displaystyle \\lim_{n \\to \\infty} \\int \\left\\vert{f_n}\\right\\vert^p \\, \\mathrm d \\mu = \\int \\left\\vert{f}\\right\\vert^p \\, \\mathrm d \\mu$ If $\\left({X, \\Sigma, \\mu}\\right)$ is not $\\sigma$-finite, $(1)$ and $(3)$ must be replaced by, respectively: :$(1'): \\quad \\left({f_n}\\right)_{n \\in \\N}$ converges in $\\mathcal L^p$ :$(3'): \\quad \\displaystyle \\lim_{n \\to \\infty} \\int \\left\\vert{f_n}\\right\\vert^p \\, \\mathrm d \\mu$ exists in $\\R$"}
+{"_id": "5432", "title": "Trivial Subgroup is Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Then the trivial subgroup $\\struct {\\set e, \\circ}$ is indeed a subgroup of $\\struct {G, \\circ}$."}
+{"_id": "5433", "title": "Trivial Subgroup is Normal", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Then the trivial subgroup $\\struct {\\set e, \\circ}$ of $G$ is a normal subgroup in $G$."}
+{"_id": "5434", "title": "Group is Normal in Itself", "text": "Let $\\struct {G, \\circ}$ be a group. Then $\\struct {G, \\circ}$ is a normal subgroup of itself."}
+{"_id": "5435", "title": "Relativisation is Standard Model", "text": "Let $P$ be a well-formed formula. Let $A$ be a finite set such that $x \\in A$ {{iff}} $x$ is a free variable in $P$. Then: :$\\displaystyle A \\subseteq B \\implies \\left({B \\models P \\iff P^B}\\right)$ {{explain|Definition of $P^B$}}"}
+{"_id": "5436", "title": "Induction on Well-Formed Formulas", "text": "Let $\\mathcal F$ be a formal language with a bottom-up grammar. Let $\\Phi$ be a proposition about the well-formed formulas of $\\mathcal F$. Suppose that $\\Phi$ is true for all letters of $\\mathcal F$. Suppose further that every rule of formation preserves $\\Phi$, i.e. when fed well-formed formulas satisfying $\\Phi$, it yields new well-formed formulas satisfying $\\Phi$. Then all well-formed formulas of $\\mathcal F$ satisfy $\\Phi$."}
+{"_id": "5437", "title": "Substitution of Elements", "text": "Let $a$, $b$, and $x$ be sets. :$a = b \\implies \\left({a \\in x \\iff b \\in x}\\right)$"}
+{"_id": "5438", "title": "Limit Ordinals Preserved Under Ordinal Multiplication", "text": "Let $x$ and $y$ be ordinals. Let $x$ be non-empty. Let $y$ be a limit ordinal. It follows that the ordinal product $\\left({x \\times y}\\right)$ is a limit ordinal."}
+{"_id": "5439", "title": "Ordinal is Less than Ordinal times Limit", "text": "Let $y$ be a limit ordinal. Let $x$ and $z$ be ordinals. Then: :$z < \\left({x \\times y}\\right) \\iff \\exists w < y: z < \\left({x \\times w}\\right)$"}
+{"_id": "5440", "title": "Epsilon Induction", "text": "Let $A$ be a class. Let $\\Bbb U$ denote the universe. :$\\left({\\forall x: \\left({x \\subseteq A \\implies x \\in A}\\right)}\\right) \\implies A = \\Bbb U$"}
+{"_id": "5441", "title": "Class Equality is Reflexive", "text": "Let $A$ be a class. Then: :$A = A$ where $=$ denotes class equality."}
+{"_id": "5443", "title": "Class Equality is Transitive", "text": "Let $A$, $B$, and $C$ be classes. Let $=$ denote class equality. Then :$\\left({ A = B \\land B = C }\\right) \\implies B = A$"}
+{"_id": "5444", "title": "Substitutivity of Class Equality", "text": "Let $A$ and $B$ be classes. Let $P \\left({A}\\right)$ be a well-formed formula of the language of set theory. Let $P \\left({B}\\right)$ be the same proposition $P \\left({A}\\right)$ with all instances of $A$ replaced with instances of $B$. Let $=$ denote class equality. :$A = B \\implies \\left({ P \\left({A}\\right) \\iff P \\left({B}\\right) }\\right)$"}
+{"_id": "5445", "title": "Class Equal to All its Elements", "text": "Let $A$ be a class. Then: :$A = \\left\\{{x : x \\in A }\\right\\}$"}
+{"_id": "5447", "title": "Class Member of Class Builder", "text": "Let $A$ be a class. Let $x$ be a set. Let $P \\left({x}\\right)$ be a well-formed formula in the language of set theory. Let $P \\left({A}\\right)$ denote the formula $P\\left({x}\\right)$ with all free instances of $x$ replaced with instances of $A$. Let $\\left\\{{x: P \\left({x}\\right)}\\right\\}$ be a class specified using class builder notation. Then: :$A \\in \\left\\{{x : P \\left({x}\\right)}\\right\\} \\iff \\left({\\exists x: x = A \\land P \\left({A}\\right)}\\right)$"}
+{"_id": "5448", "title": "Class Equality Extension of Set Equality", "text": "Let $=_1$ denote set equality. Let $=_2$ denote class equality. Let $x$ and $y$ be sets. Then $x =_1 y$ iff $x =_2 y$."}
+{"_id": "5449", "title": "B-Algebra Induced by Group Induced by B-Algebra", "text": "Let $\\left({S, *}\\right)$ be a $B$-algebra. Let $\\left({S, \\circ}\\right)$ be the group described on $B$-Algebra Induces Group. Let $\\left({S, *'}\\right)$ be the $B$-algebra described on Group Induces $B$-Algebra. Then $\\left({S, *'}\\right) = \\left({S, *}\\right)$."}
+{"_id": "5453", "title": "Union of Doubleton", "text": "Let $\\set {x, y}$ be a doubleton. Then: :$\\displaystyle \\bigcup \\set {x, y} = x \\cup y$"}
+{"_id": "5454", "title": "Union of Small Classes is Small", "text": "Let $x$ and $y$ be small classes. Then $x \\cup y$ is also small."}
+{"_id": "5459", "title": "Set Difference is Set", "text": "Let $x$ be a small class. Let $A$ be a class. Let $\\map \\MM B$ denote that $B$ is small. Then:  :$\\map \\MM {x \\setminus A}$"}
+{"_id": "5460", "title": "Nonempty Class has Members", "text": "Let $A$ be a class. Then: :$A \\ne \\O \\iff \\exists x: x \\in A$"}
+{"_id": "5461", "title": "Class is Not Element of Itself", "text": "Let $A$ be a class. Then $A$ is not an element of itself: :$A \\notin A$"}
+{"_id": "5462", "title": "Category of Relations is Category", "text": "Let $\\mathbf{Rel}$ be the category of relations. Then $\\mathbf{Rel}$ is a metacategory."}
+{"_id": "5464", "title": "Identity Functor is Functor", "text": "Let $\\mathbf C$ be a metacategory. Let $\\operatorname{id}_{\\mathbf C}: \\mathbf C \\to \\mathbf C$ be the identity functor on $\\mathbf C$. Then $\\operatorname{id}_{\\mathbf C}$ is a functor."}
+{"_id": "5465", "title": "Category of Categories is Category", "text": "Let $\\mathbf{Cat}$ be the category of categories. Then $\\mathbf{Cat}$ is a metacategory."}
+{"_id": "5466", "title": "Identity Functor is Left Identity", "text": "Let $\\mathbf C$ and $\\mathbf D$ be metacategories. Let $F: \\mathbf C \\to \\mathbf D$ be a functor, and let $\\operatorname{id}_{\\mathbf D}$ be the identity functor on $\\mathbf D$. Then the composite functor $\\operatorname{id}_{\\mathbf D} F$ satisfies: :$\\operatorname{id}_{\\mathbf D} F = F$"}
+{"_id": "5467", "title": "Identity Functor is Right Identity", "text": "Let $\\mathbf C$ and $\\mathbf D$ be metacategories. Let $F: \\mathbf C \\to \\mathbf D$ be a functor, and let $\\operatorname{id}_{\\mathbf C}$ be the identity functor on $\\mathbf C$. Then the composite functor $F \\operatorname{id}_{\\mathbf C}$ satisfies: :$F \\operatorname{id}_{\\mathbf C} = F$"}
+{"_id": "5468", "title": "Composition of Functors is Associative", "text": "Let $\\mathbf A$, $\\mathbf B$, $\\mathbf C$ and $\\mathbf D$ be metacategories. Let $F: \\mathbf A \\to \\mathbf B$, $G: \\mathbf B \\to \\mathbf C$ and $H: \\mathbf C \\to \\mathbf D$ be functors. Composition of functors is associative: :$H \\paren {G F} = \\paren {H G} F$"}
+{"_id": "5469", "title": "Abel-Ruffini Theorem", "text": "There is no general algebraic solution for determining all the roots of a polynomial of degree $5$ or higher."}
+{"_id": "5470", "title": "Cartesian Product is Small", "text": "Let $a$ and $b$ be small classes. Then their Cartesian product $\\left({a \\times b}\\right)$ is small: :$\\mathscr M \\left({a \\times b}\\right)$"}
+{"_id": "5471", "title": "Preorder Category is Category", "text": "Let $\\struct {S, \\precsim}$ be a preordered set. Let $\\mathbf S$ be its associated preorder category. Then $\\mathbf S$ is a category."}
+{"_id": "5472", "title": "Category Induces Preorder", "text": "Let $\\mathbf S$ be a category with set of objects $S$. Then the binary relation $\\precsim$ defined by: :$\\forall a, b \\in S: a \\precsim b \\iff \\exists f: a \\to b$ is a preorder on $S$."}
+{"_id": "5473", "title": "Preorder Induced by Preorder Category", "text": "Let $\\left({S, \\precsim}\\right)$ be a preordered set. Let $\\mathbf S$ be its associated preorder category. Let $\\precsim'$ be the preorder induced by $\\mathbf S$ as on Category Induces Preorder. Then $\\precsim'$ is the same as $\\precsim$."}
+{"_id": "5474", "title": "Functor between Order Categories", "text": "Let $\\struct {S, \\preceq}$ and $\\struct {T, \\preceq'}$ be ordered sets. Let $\\mathbf S$ and $\\mathbf T$ be their associated order categories, respectively. Let $F: \\mathbf S \\to \\mathbf T$ be a functor. Then its object functor $F: S \\to T$ is a monotone mapping."}
+{"_id": "5475", "title": "Discrete Category on Set is Discrete Category", "text": "Let $S$ be a set. Let $\\mathbf{Dis} \\left({S}\\right)$ be the discrete category on $S$. Then $\\mathbf{Dis} \\left({S}\\right)$ determines a unique (up to isomorphism discrete category $\\mathbf{Dis} \\left({S}\\right)$ whose objects precisely comprise $S$. {{expand|That is, $\\mathbf{Dis}: \\mathbf{Set} \\to \\mathbf{Cat}$ is a functor (with left inverse $\\mathbf{ob}$) }}"}
+{"_id": "5477", "title": "Image of Small Class under Mapping is Small", "text": "Let $A$ be a mapping. Let $a$ be a small class. Then, the image of $a$ under $A$ is small. {{improve|$A$ being a class variable, I strongly suggest a different letter to denote the mapping}}"}
+{"_id": "5478", "title": "Inverse of Small Relation is Small", "text": "Let $a$ be a small class. Let $a$ also be a relation. Then the inverse relation of $a$ is small."}
+{"_id": "5479", "title": "Functor between Monoid Categories", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be monoids. Let $\\mathbf S$ and $\\mathbf T$ be the associated monoid categories. Let $F: \\mathbf S \\to \\mathbf T$ be a functor. Then the morphism functor $F_1$ of $F$ is a monoid homomorphism."}
+{"_id": "5483", "title": "Cayley's Theorem (Category Theory)", "text": "Let $\\mathbf C$ be a small category. Denote with $\\mathbf{Set}$ the category of sets. Then there exists a category $\\mathbf D$, subject to: :$(1): \\quad $ The objects of $\\mathbf D$ are sets. :$(2): \\quad $ The morphisms of $\\mathbf D$ are mappings. :$(3): \\quad \\mathbf C \\cong \\mathbf D$, i.e. $\\mathbf C$ and $\\mathbf D$ are isomorphic. That is, $\\mathbf C$ is isomorphic to a subcategory of $\\mathbf{Set}$."}
+{"_id": "5484", "title": "Domain of Small Relation is Small", "text": "Let $a$ be a small class. Let $a$ also be a relation. Then the domain of $a$ is small."}
+{"_id": "5485", "title": "Range of Small Relation is Small", "text": "Let $a$ be a small class. Let $a$ also be a relation. Then the range of $a$ is small."}
+{"_id": "5486", "title": "Definition:Opposite Group", "text": "Let $\\struct {G, \\circ}$ be a group. We define a new operation $*$ on $G$ by: :$\\forall a, b \\in G: a * b = b \\circ a$ The algebraic structure $\\struct {G, *}$ is called the '''opposite group''' to $G$."}
+{"_id": "5487", "title": "Center of Opposite Group", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\struct {G, *}$ be the opposite group to $G$. Let $\\map Z {G, \\circ}$ and $\\map Z {G, *}$ be the centers of $\\struct {G, \\circ}$ and $\\struct {G, *}$, respectively. Then: :$\\map Z {G, \\circ} = \\map Z {G, *}$"}
+{"_id": "5488", "title": "Opposite Group of Opposite Group", "text": "Let $\\left({G, \\circ}\\right)$ be a group. Let $\\left({G, *}\\right)$ be the opposite group to $\\left({G, \\circ}\\right)$. Let $\\left({G, \\circ'}\\right)$ be the opposite group to $\\left({G, *}\\right)$. Then: :$\\left({G, \\circ}\\right) = \\left({G, \\circ'}\\right)$"}
+{"_id": "5489", "title": "Slice Category is Category", "text": "Let $\\mathbf C$ be a metacategory. Let $C \\in \\mathbf C_0$ be a object of $\\mathbf C$. Let $\\mathbf C / C$ be the slice category of $\\mathbf C$ over $C$. Then $\\mathbf C / C$ is a metacategory."}
+{"_id": "5490", "title": "Monoid Category is Category", "text": "Let $\\left({S, \\circ}\\right)$ be a monoid with identity $e_S$. Let $\\mathbf S$ be the associated monoid category. Then $\\mathbf S$ is a category."}
+{"_id": "5491", "title": "Dual Category is Category", "text": "Let $\\mathbf C$ be a metacategory. Let $\\mathbf C^{\\text{op}}$ be its dual category. Then $\\mathbf C^{\\text{op}}$ is also a metacategory."}
+{"_id": "5492", "title": "Signum Function on Integers is Extension of Signum on Natural Numbers", "text": "Let $\\sgn_\\Z: \\Z \\to \\set {-1, 0, 1}$ be the signum function on the integers. Let $\\sgn_\\N: \\N \\to \\set {0, 1}$ be the signum function on the natural numbers. Then $\\sgn_\\Z: \\Z \\to \\Z$ is an extension of $\\sgn_\\N: \\N \\to \\N$."}
+{"_id": "5494", "title": "Cartesian Product with Proper Class is Proper Class", "text": "Let $A$ be a proper class. Let $B$ be a class which is not empty. Then the Cartesian product $\\paren {A \\times B}$ is a proper class."}
+{"_id": "5495", "title": "Uniqueness Condition for Relation Value", "text": "Let $\\mathcal R$ be a relation. Let $\\left({x, y}\\right) \\in \\mathcal R$. Let: :$\\exists ! y: \\left({x, y}\\right) \\in \\mathcal R$ Then: :$\\mathcal R \\left({x}\\right) = y$ where $\\mathcal R \\left({x}\\right)$ denotes the image of $\\mathcal R$ at $x$. If $y$ is not unique, then: :$\\mathcal R \\left({x}\\right) = \\varnothing$"}
+{"_id": "5496", "title": "Value of Relation is Small", "text": "The value of a relation is always a small class."}
+{"_id": "5497", "title": "Mapping whose Domain is Small Class is Small", "text": "Let $F$ be a mapping. Let the domain of $F$ be a small class. Then, $F$ is a small class."}
+{"_id": "5498", "title": "Restriction of Mapping to Small Class is Small", "text": "Let $F$ be a mapping. Let $A$ be a small class. Then the restriction $F {\\restriction_A}$ is a small class."}
+{"_id": "5499", "title": "Preimage of Singleton", "text": "Let $\\RR$ be a relation. Let $\\map {\\RR^{-1} } t$ denote the preimage of $t$ under $\\RR$. Let $\\RR^{-1} \\sqbrk {\\set t}$ denote the preimage of $\\set t$ under $\\RR$. Then: :$\\RR^{-1} \\sqbrk {\\set t} = \\map {\\RR^{-1} } t$"}
+{"_id": "5500", "title": "Order Isomorphism Preserves Minimal Elements", "text": "Let $A_1$ and $A_2$ be classes. Let $\\prec_1$ and $\\prec_2$ be relations. Let $\\phi : A_1 \\to A_2$ create an order isomorphism between $\\left({ A_1 , \\prec_1 }\\right)$ and $\\left({ A_2 , \\prec_2 }\\right)$. Let $B \\subseteq A_1$. Then $\\phi$ maps the $\\prec_1$-minimal elements (for strict orderings) of $B$ to the $\\prec_2$-minimal element of $\\phi \\left({B}\\right)$."}
+{"_id": "5501", "title": "Order Isomorphism Preserves Initial Segments", "text": "Let $A_1$ and $A_2$ be classes. Let $\\prec_1$ and $\\prec_2$ be strict orderings. Let $\\phi : A_1 \\to A_2$ create an order isomorphism between $\\left({A_1, \\prec_1}\\right)$ and $\\left({A_2, \\prec_2}\\right)$. Suppose $x \\in A_1$. Then $\\phi$ maps the $\\prec_1$-initial segment of $x$ to the $\\prec_2$-initial segment of $\\phi \\left({x}\\right)$."}
+{"_id": "5503", "title": "Alternating Group is Set of Even Permutations", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $A_n$ be the alternating group on $n$ letters. Then $A_n$ consists of the set of even permutations of $S_n$."}
+{"_id": "5504", "title": "Induced Relation Generates Order Isomorphism", "text": "Let $\\left({ A_1 , \\preceq_1 }\\right)$ be an ordered set. Let $\\phi : A_1 \\to A_2$ be a bijection. Let: : $\\preceq_2 \\mathop{:=} \\left\\{ \\left({\\phi\\left({x}\\right), \\phi\\left({y}\\right) }\\right): x \\in A_1 \\land y \\in A_1 \\land x \\mathop{\\preceq_1} y \\right\\}$ Then $\\phi : \\left({ A_1 , \\preceq_1 }\\right) \\to \\left({ A_2 , \\preceq_2 }\\right)$ is an order isomorphism."}
+{"_id": "5505", "title": "Ordinal Multiplication is Left Distributive", "text": "Let $x$, $y$, and $z$ be ordinals. Let $\\times$ denote ordinal multiplication. Let $+$ denote ordinal addition. Then: :$x \\times \\left({ y + z }\\right) = \\left({ x \\times y }\\right) + \\left({ x \\times z }\\right)$"}
+{"_id": "5506", "title": "Ordinal Multiplication is Associative", "text": "Let $x, y, z$ be ordinals. Let $\\times$ denote ordinal multiplication. Then: :$x \\times \\left({y \\times z}\\right) = \\left({x \\times y}\\right) \\times z$"}
+{"_id": "5507", "title": "Division Theorem for Ordinals", "text": "Let $x$ and $y$ be ordinals. Let $0$ denote the zero ordinal. Suppose $y \\ne 0$. Then there exist unique ordinals $z$ and $w$ such that: :$x = \\paren {y \\times z} + w$ and $w < y$."}
+{"_id": "5508", "title": "Finite Ordinal Times Ordinal", "text": "Let $m$ and $n$ be finite ordinals. Let $m \\ne 0$, where $0$ is the zero ordinal. Let $x$ be a limit ordinal. Then: :$m \\times \\left({ x + n }\\right) = x + \\left({ m \\times n }\\right)$ {{expand|Via Cantor normal form, all ordinals are of the form $x+n$}}"}
+{"_id": "5509", "title": "Factorization of Limit Ordinals", "text": "Let $x$ be a limit ordinal. Then: :$x = ( \\omega \\times y )$ for some $y \\in \\operatorname{On}$ where $\\omega$ is the minimal infinite successor set."}
+{"_id": "5510", "title": "Ordinal Exponentiation is Closed", "text": "Let $x$ and $y$ be ordinals. Then: :$x^y \\in \\operatorname{On}$ That is, ordinal exponentiation is closed."}
+{"_id": "5511", "title": "Exponent Base of One", "text": "Let $x$ be an ordinal. Then: :$1^x = 1$"}
+{"_id": "5512", "title": "Exponent Not Equal to Zero", "text": "Let $x$ and $y$ be ordinals. Let $x \\ne 0$. Then: :$x^y \\ne 0$"}
+{"_id": "5513", "title": "Membership is Left Compatible with Ordinal Exponentiation", "text": "Let $x$, $y$, and $z$ be ordinals. Suppose $1 < z$. Then: :$x < y \\iff z^x < z^y$"}
+{"_id": "5514", "title": "Conjugacy Action on Group Elements is Group Action", "text": "The conjugacy action on $G$: : $\\forall g, h \\in G: g * h = g \\circ h \\circ g^{-1}$ is a  group action on itself."}
+{"_id": "5515", "title": "Conjugacy Action on Subgroups is Group Action", "text": "Let $X$ be the set of all subgroups of $G$. For any $H \\le G$ and for any $g \\in G$, the conjugacy action: :$g * H := g \\circ H \\circ g^{-1}$ is a group action."}
+{"_id": "5516", "title": "Slice Category of Order Category", "text": "Let $\\mathbf P$ be a order category, and denote its ordering by $\\preceq$. Let $p \\in \\mathbf P_0$ be an object of $\\mathbf P$. Then: :$\\mathbf P \\mathbin / p \\cong p^\\preceq$ where: :$\\mathbf P \\mathbin / p$ is the slice of $\\mathbf P$ over $p$ :$p^\\preceq$ is the order category defined by the weak lower closure of $p$."}
+{"_id": "5517", "title": "Category of Pointed Sets as Coslice Category", "text": "Let $\\mathbf{Set}_*$ be the category of pointed sets. Let $\\mathbf{Set}$ be the category of sets. Let $1 := \\left\\{{*}\\right\\}$ be any singleton. Then: :$\\mathbf{Set}_* \\cong 1 \\mathbin / \\mathbf{Set}$ where $1 \\mathbin / \\mathbf{Set}$ denotes the coslice of $\\mathbf{Set}$ under $1$ and $\\cong$ signifies isomorphic categories."}
+{"_id": "5518", "title": "Subset is Right Compatible with Ordinal Exponentiation", "text": "Let $x, y, z$ be ordinals. Then: :$x \\le y \\implies x^z \\le y^z$"}
+{"_id": "5519", "title": "Condition for Membership is Right Compatible with Ordinal Exponentiation", "text": "Let $x, y, z$ be ordinals. Let $z$ be the successor of some ordinal $w$. Then: :$x < y \\iff x^z < y^z$"}
+{"_id": "5520", "title": "Lower Bound for Ordinal Exponentiation", "text": "Let $x$ and $y$ be ordinals. Let $x$ be greater than $1$, where $1$ denotes the successor of the zero ordinal. Then: :$y \\le x^y$"}
+{"_id": "5521", "title": "Unique Ordinal Exponentiation Inequality", "text": "Let $x$ and $y$ be ordinals. Let $x > 1$ and $y > 0$. Then there exists a unique ordinal $z$ such that: :$x^z ≤ y$ and $y < x^{z^+}$"}
+{"_id": "5522", "title": "Limit Ordinals Closed under Ordinal Exponentiation", "text": "Let $x$ and $y$ be ordinals. Let $y$ be a limit ordinal. Let $x^y$ denote ordinal exponentiation. Then: :If $x > 1$, then $x^y$ is a limit ordinal. :If $x \\ne \\varnothing$, then $y^x$ is a limit ordinal."}
+{"_id": "5523", "title": "Successor of Element of Ordinal is Subset", "text": "Let $x$ and $y$ be ordinals. Then: :$x \\in y \\iff x^+ \\subseteq y$"}
+{"_id": "5524", "title": "Ordinal is Less than Ordinal to Limit Power", "text": "Let $x$, $y$, and $z$ be ordinals. Let $z$ be a limit ordinal. Then: :$x < y^z \\iff \\exists w \\in z: x < y^w$"}
+{"_id": "5525", "title": "Ordinal Sum of Powers", "text": "Let $x$, $y$, and $z$ be ordinals. Then: :$x^y \\times x^z = x^{y + z}$"}
+{"_id": "5526", "title": "Kleene Closure is Monoid", "text": "Let $S$ be a set, and let $S^*$ be its Kleene closure. Let $*$ denote concatenation of ordered tuples. Then $\\struct {S^*, *}$ is a monoid."}
+{"_id": "5529", "title": "Ordinal Power of Power", "text": "Let $x$, $y$, and $z$ be ordinals. Then: :$\\left({x^y}\\right)^z = x^{y \\mathop \\times z}$"}
+{"_id": "5530", "title": "Upper Bound of Ordinal Sum", "text": "Let $x$ and $y$ be ordinals. Suppose $x > 1$. Let $\\langle a_n \\rangle$ be a finite sequence of ordinals such that: :$a_n < x$ for all $n$ Let $\\left\\langle{b_n}\\right\\rangle$ be a strictly decreasing finite sequence of ordinals such that: :$b_n < y$ for all $n$ Then: :$\\displaystyle \\sum_{i \\mathop = 1}^n x^{b_i} a_i < x^y$"}
+{"_id": "5531", "title": "General Associative Law for Ordinal Sum", "text": "Let $x$ be a finite ordinal. Let $\\left\\langle{a_i}\\right\\rangle$ be a sequence of ordinals. Then: :$\\displaystyle \\sum_{i \\mathop = 1}^{x + 1} a_i = a_1 + \\sum_{i \\mathop = 1}^x a_{i + 1}$"}
+{"_id": "5532", "title": "Power Sets of Equinumerous Sets are Equinumerous", "text": "Let $x$ and $y$ be sets such that $x \\sim y$. Then: :$\\powerset x \\sim \\powerset y$"}
+{"_id": "5533", "title": "General Associative Law for Ordinal Sum/Proof 1", "text": "Let $x$ be a finite ordinal. Let $\\left\\langle{a_i}\\right\\rangle$ be a sequence of ordinals. Then: :$\\displaystyle \\sum_{i \\mathop = 1}^{x + 1} a_i = a_1 + \\sum_{i \\mathop = 1}^x a_{i + 1}$"}
+{"_id": "5534", "title": "General Associative Law for Ordinal Sum/Proof 2", "text": "Let $x$ be a finite ordinal. Let $\\left\\langle{a_i}\\right\\rangle$ be a sequence of ordinals. Then: :$\\displaystyle \\sum_{i \\mathop = 1}^{x + 1} a_i = a_1 + \\sum_{i \\mathop = 1}^x a_{i + 1}$"}
+{"_id": "5537", "title": "Basis Representation Theorem for Ordinals", "text": "Let $x$ and $y$ be ordinals. Let $x > 1$ and $y > 0$. Then there exist unique finite sequences of ordinals: :$\\sequence {a_i}, \\sequence {b_i}$ both of unique length $n$ such that: :$(1): \\quad \\sequence {a_i}$ is a strictly decreasing sequence for $1 \\le i \\le n$ :$(2): \\quad 0 < b_i < x$ for all $1 \\le i \\le n$ :$(3): \\quad \\ds y = \\sum_{i \\mathop = 1}^n x^{a_i} b_i$"}
+{"_id": "5539", "title": "Ordinal Multiplication via Cantor Normal Form/Limit Base", "text": "Let $x$ and $y$ be ordinals. Let $x$ be a limit ordinal. Let $y > 0$. Let $\\left\\langle{a_i}\\right\\rangle$ be a sequence of ordinals that is strictly decreasing on $1 \\le i \\le n$. Let $\\left\\langle{b_i}\\right\\rangle$ be a sequence of finite ordinals. Then: :$\\displaystyle \\sum_{i \\mathop = 1}^n \\left({x^{a_i} b_i}\\right) \\times x^y = x^{a_1 \\mathop + y}$"}
+{"_id": "5540", "title": "Ordinal Exponentiation of Terms", "text": "Let $x, y, z$ be ordinals. Let $n$ be a finite ordinal. Let $x$ be a limit ordinal. Let $y, z, n$ all be greater than $0$. Then: :$\\left({x^y \\times n}\\right)^z = x^{y \\mathop \\times z} \\times n$ if $z$ is not a limit ordinal :$\\left({x^y \\times n }\\right)^z = x^{y \\mathop \\times z}$ if $z$ is a limit ordinal"}
+{"_id": "5541", "title": "Prime Triplet is Unique", "text": "Let $n \\in \\Z$ be an integer such that $n > 3$. Then $n$, $n + 2$, $n + 4$ cannot all be prime. That is, the only prime triplet is $\\set {3, 5, 7}$."}
+{"_id": "5543", "title": "Ordinal Exponentiation via Cantor Normal Form/Limit Exponents", "text": "Let $x$ and $y$ be ordinals. Let $x$ and $y$ be limit ordinals. Let $\\sequence {a_i}$ be a sequence of ordinals that is strictly decreasing on $1 \\le i \\le n$. Let $\\sequence {b_i}$ be a sequence of natural numbers. Then: :$\\displaystyle \\paren {\\sum_{i \\mathop = 1}^n x^{a_i} \\times b_i}^y = x^{a_1 \\mathop \\times y}$"}
+{"_id": "5545", "title": "Inverse Relation Functor is Contravariant Functor", "text": "Let $\\mathbf{Rel}$ be the category of relations. Let $C: \\mathbf{Rel} \\to \\mathbf{Rel}$ be the inverse relation functor. Then $C$ is a contravariant functor."}
+{"_id": "5546", "title": "Injection iff Monomorphism in Category of Sets", "text": "Let $\\mathbf{Set}$ be the category of sets. Let $f: X \\to Y$ be a morphism in $\\mathbf{Set}$, i.e. a mapping. Then $f$ is an injection iff it is a monomorphism."}
+{"_id": "5548", "title": "Morphism in Preorder Category is Monic", "text": "Let $\\mathbf P$ be a preorder category. Let $f \\in \\mathbf P_1$ be a morphism. Then $f$ is monic."}
+{"_id": "5550", "title": "Surjection iff Epimorphism in Category of Sets", "text": "Let $\\mathbf{Set}$ be the category of sets. Let $f: X \\to Y$ be a morphism in $\\mathbf{Set}$, i.e. a mapping. Then $f$ is a surjection iff it is an epimorphism."}
+{"_id": "5551", "title": "Inclusion of Natural Numbers in Integers is Epimorphism", "text": "Let $\\mathbf{Mon}$ be the category of monoids. Let $\\left({\\N, +}\\right)$ denote the monoid of natural numbers as on Natural Numbers under Addition form Commutative Monoid. Let $\\left({\\Z, +}\\right)$ denote the monoid of integers as on additive group of integers. Denote with $\\iota: \\N \\to \\Z$ the inclusion mapping. Then $\\iota: \\N \\to \\Z$ is an epimorphism in $\\mathbf{Mon}$."}
+{"_id": "5552", "title": "Isomorphism (Category Theory) is Monic", "text": "Let $\\mathbf C$ be a metacategory. Let $f: C \\to D$ be an isomorphism. Then $f: C \\rightarrowtail D$ is monic."}
+{"_id": "5555", "title": "Stabilizer of Element under Conjugacy Action is Centralizer", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $*$ be the conjugacy action on $G$ defined by the rule: : $\\forall g, h \\in G: g * h = g \\circ h \\circ g^{-1}$ Let $x \\in G$. Then the stabilizer of $x$ under this conjugacy action is: :$\\Stab x = \\map {C_G} x$ where $\\map {C_G} x$ is the centralizer of $x$ in $G$."}
+{"_id": "5556", "title": "Stabilizer of Subset Product Action on Power Set", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $\\powerset G$ be the power set of $\\struct {G, \\circ}$. Let $*: G \\times \\powerset G \\to \\powerset G$ be the subset product action on $\\powerset G$ defined as: :$\\forall g \\in G: \\forall S \\in \\powerset G: g * S = g \\circ S$ where $g \\circ S$ is the subset product $\\set g \\circ S$. Then the stabilizer of $S$ in $\\powerset G$ is the set: :$\\Stab S = S$"}
+{"_id": "5557", "title": "Stabilizer of Coset Action on Set of Subgroups", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $\\powerset G$ denote the power set of $G$. Let $\\HH \\subseteq \\powerset G$ denote the set of subgroups of $G$. Let $*$ be the subset product action on $\\HH \\subseteq \\powerset G$ defined as: :$\\forall g \\in G: \\forall H \\in \\HH: g * H = g \\circ H$ where $g \\circ H$ is the (left) coset of $g$ by $H$. Then the stabilizer of $H$ in $\\powerset G$ is $H$ itself: :$\\Stab H = H$"}
+{"_id": "5558", "title": "Isomorphism (Category Theory) is Epic", "text": "Let $\\mathbf C$ be a metacategory. Let $f: C \\to D$ be an isomorphism. Then $f: C \\twoheadrightarrow D$ is epic."}
+{"_id": "5559", "title": "Split Epimorphism is Epic", "text": "Let $\\mathbf C$ be a metacategory. Let $f: C \\to D$ be a split epimorphism. Then $f: C \\twoheadrightarrow D$ is epic."}
+{"_id": "5560", "title": "Split Monomorphism is Monic", "text": "Let $\\mathbf C$ be a metacategory. Let $f: C \\to D$ be a split monomorphism. Then $f: C \\rightarrowtail D$ is monic."}
+{"_id": "5562", "title": "Relational Closure Exists for Set-Like Relation", "text": "Let $A$ be a class. Let $\\prec$ be a relation on $A$. Furthermore, let ${\\prec^{-1}}\\left({a}\\right)$ be a small class for each $a \\in A$. Let $S$ be a small class that is a subset of the class $A$. Let $G$ be a mapping such that: :$G \\left({ x }\\right) = A \\cap \\left({ {\\prec^{-1}} \\left({ x }\\right) }\\right)$ Let $F$ be defined using the Principle of Recursive Definition: :$F \\left({0}\\right) = S$ :$F \\left({n^+}\\right) = F \\left({n}\\right) \\cup G \\left({F \\left({n}\\right)}\\right)$ Let $\\displaystyle T = \\bigcup_{n \\mathop \\in \\omega} F\\left({n}\\right)$. Then: :$(1): \\quad T$ is a set and satisfies: :::::$\\forall x \\in A: \\forall y \\in T: \\left({ x \\prec y \\implies x \\in T }\\right)$ :::::In other words, $T$ is $\\prec$-transitive. :$(2): \\quad S \\subseteq T$ :$(3): \\quad$ If $R$ is $\\prec$-transitive and $S \\subseteq R$, then $T \\subseteq R$. That is, given any set $S$, there is an explicit construction for its relational closure."}
+{"_id": "5563", "title": "Stabilizer of Conjugacy Action on Subgroup is Normalizer", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $X$ be the set of all subgroups of $G$. Let $*$ be the conjugacy action on $H$: :$\\forall g \\in G, H \\in X: g * H = g \\circ H \\circ g^{-1}$ Then the stabilizer of $H$ in $\\powerset G$ is given by: :$\\Stab H = \\map {N_G} H$ where $\\map {N_G} H$ is the normalizer of $H$ in $G$."}
+{"_id": "5564", "title": "Well-Founded Proper Relational Structure Determines Minimal Elements", "text": "Let $A$ and $B$ be classes. Let $\\left({ A, \\prec }\\right)$ be a proper relational structure. Let $\\prec$ be a foundational relation. Suppose $B \\subset A$ and $B \\ne \\varnothing$. Then $B$ has a $\\prec$-minimal element."}
+{"_id": "5565", "title": "Initial Object is Unique", "text": "Let $\\mathbf C$ be a metacategory. Let $0$ and $0'$ be two initial objects of $\\mathbf C$. Then there is a unique isomorphism $u: 0 \\to 0'$. Hence, initial objects are unique up to unique isomorphism."}
+{"_id": "5566", "title": "Terminal Object is Unique", "text": "Let $\\mathbf C$ be a metacategory. Let $1$ and $1'$ be two terminal objects of $\\mathbf C$. Then there is a unique isomorphism $u: 1 \\to 1'$. Hence, terminal objects are unique up to unique isomorphism."}
+{"_id": "5567", "title": "Empty Set is Initial Object", "text": "Let $\\mathbf{Set}$ be the category of sets. Then the empty set $\\O$ is an initial object of $\\mathbf{Set}$."}
+{"_id": "5568", "title": "Singleton is Terminal Object of Category of Sets", "text": "Let $\\mathbf{Set}$ be the category of sets. Let $S = \\left\\{{x}\\right\\}$ be any singleton set. Then $S$ is a terminal object of $\\mathbf{Set}$."}
+{"_id": "5569", "title": "Zero (Category) is Initial Object", "text": "Let $\\mathbf{Cat}$ be the category of categories. Let $\\mathbf 0$ be the zero category. Then $\\mathbf 0$ is an initial object of $\\mathbf{Cat}$."}
+{"_id": "5570", "title": "One (Category) is Terminal Object", "text": "Let $\\mathbf{Cat}$ be the category of categories. Let $\\mathbf 1$ be the category one. Then $\\mathbf 1$ is a terminal object of $\\mathbf{Cat}$."}
+{"_id": "5571", "title": "Empty Mapping is Mapping", "text": "For each set $T$, the empty mapping, where the domain is the empty set, is a mapping."}
+{"_id": "5572", "title": "Empty Mapping is Unique", "text": "For each set $T$ there exists exactly one empty mapping $f: \\O \\to T$ whose domain is the empty set."}
+{"_id": "5573", "title": "Null Relation is Mapping iff Domain is Empty Set", "text": "Let $S$ and $T$ be sets. The null relation $\\RR = \\O \\subseteq S \\times T$ is a mapping {{iff}} $S = \\O$."}
+{"_id": "5574", "title": "Orbit of Element of Group Acting on Itself is Group", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $*$ be the group action of $\\struct {G, \\circ}$ on itself by the rule: :$\\forall g, h \\in G: g * h = g \\circ h$ Then the orbit of an element $x \\in G$ is given by: :$\\Orb x = G$"}
+{"_id": "5575", "title": "Orbit of Element under Conjugacy Action is Conjugacy Class", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $*$ be the conjugacy group action on $G$: : $\\forall g, h \\in G: g * h = g \\circ h \\circ g^{-1}$ Let $x \\in G$. Then the orbit of $x$ under this group action is: :$\\Orb x = C_x$ where $C_x$ is the conjugacy class of $x$."}
+{"_id": "5576", "title": "Axiom of Foundation (Strong Form)", "text": "Let $B$ be a class. Suppose $B$ is non-empty. Then $B$ has a $\\in$-minimal element."}
+{"_id": "5578", "title": "Trivial Group is Terminal Object of Category of Groups", "text": "Let $\\mathbf{Grp}$ be the category of groups. Let $\\left\\{{e}\\right\\}$ be the trivial group. Then $\\left\\{{e}\\right\\}$ is a terminal object of $\\mathbf{Grp}$."}
+{"_id": "5580", "title": "Orbit of Conjugacy Action on Subgroup is Set of Conjugate Subgroups", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $X$ be the set of all subgroups of $G$. Let $*$ be the conjugacy action on $H$ defined as: : $\\forall g \\in G, H \\in X: g * H = g \\circ H \\circ g^{-1}$ Then the orbit $\\Orb H$ of $H$ in $\\powerset G$ is the set of subgroups of $G$ conjugate to $H$."}
+{"_id": "5582", "title": "Closure for Finite Collection of Relations and Operations", "text": "Let $\\RR_1, \\RR_2, \\ldots \\RR_n$ be relations. Let $\\SS_1, \\SS_2, \\ldots \\SS_m$ be operations. Let $T$ be a small class. Let the image of $\\RR_i$ over any small class $x$ be small classes for $1 \\le i \\le n$. Let the image of $\\SS_i$ over any Cartesian product $x \\times x$ be small classes for $1 \\le i \\le m$. Then there exists a small class $X$ such that: :$(1): \\quad$ $T \\subseteq X$ :$(2): \\quad$ Each $\\RR_i$ is closed with respect to $X$.  Each $\\SS_i$ is closed with respect to $X \\times X$. :$(3): \\quad$ $X$ is the smallest small class satisfying conditions $(1)$ and $(2)$ above. If $Y$ satisfies conditions $(1)$ and $(2)$, then $X \\subseteq Y$."}
+{"_id": "5583", "title": "Well-Founded Recursion", "text": "Let $A$ be a class. Let $\\mathcal R$ be a relation that is foundational on $A$. Let every $\\mathcal R$-initial segment of any element $x$ of $A$ be a small class. Let $K$ be a class of mappings $f$ that satisfy: :$(1): \\quad$ The domain of $f$ is some subset $y \\subseteq A$ such that $y$ is transitive with respect to $\\mathcal R$. :$(2): \\quad \\forall x \\in y: f \\left({x}\\right) = G \\left({F \\restriction \\mathcal R^{-1} x}\\right)$ where $F \\restriction R^{-1} x$ denotes the restriction of $F$ to the $\\mathcal R$-initial segment of $x$. {{explain|What is $G$? And should $F$ not be introduced before it is used, instead of after?}} Let $F = \\bigcup K$, the union of $K$. Then: : $F$ is a mapping with domain $A$ : $\\forall x \\in A: F \\left({x}\\right) = G \\left({F \\restriction \\mathcal R^{-1} x}\\right)$ : $F$ is unique, in the sense that if another mapping $A$ has the above two properties, then $A = F$."}
+{"_id": "5584", "title": "Von Neumann Hierarchy is Supertransitive", "text": "Let $V$ denote the Von Neumann Hierarchy. Let $x$ be an ordinal. Then $\\map V x$ is supertransitive."}
+{"_id": "5585", "title": "Von Neumann Hierarchy Comparison", "text": "Let $x$ and $y$ be ordinals such that $x < y$. Then: :$\\map V x \\in \\map V y$ :$\\map V x \\subset \\map V y$ {{explain|$\\map V x$ etc.}}"}
+{"_id": "5586", "title": "Every Set in Von Neumann Universe", "text": "Let $S$ be a small class. Then $S$ is well-founded. {{explain|The connection between \"well-founded\" and \"in von Neumann universe\".}}"}
+{"_id": "5588", "title": "Ordinal Equal to Rank", "text": "Let $x$ be an ordinal. Let $S$ be a small class. Let $V \\left({ x }\\right)$ denote the von Neumann hierarchy on the ordinal $x$. Then $x$ equals the rank of $S$ iff $S \\in V \\left({x+1}\\right) \\land S \\notin V \\left({x}\\right)$."}
+{"_id": "5590", "title": "Tau of Prime Number", "text": "Let $p \\in \\Z_{> 0}$. Then $p$ is a prime number {{iff}}: :$\\map \\tau p = 2$ where $\\map \\tau p$ denotes the tau function of $p$."}
+{"_id": "5591", "title": "Join is Commutative", "text": "Let $\\left({S, \\vee, \\preceq}\\right)$ be a join semilattice. Then $\\vee$ is commutative."}
+{"_id": "5592", "title": "Meet is Commutative", "text": "Let $\\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Then $\\wedge$ is commutative."}
+{"_id": "5593", "title": "Join Succeeds Operands", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a, b \\in S$ admit a join $a \\vee b \\in S$. Then: :$a \\preceq a \\vee b$ :$b \\preceq a \\vee b$ i.e., $a \\vee b$ succeeds its operands $a$ and $b$."}
+{"_id": "5594", "title": "Meet Precedes Operands", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a, b \\in S$ admit a meet $a \\wedge b \\in S$. Then: :$a \\wedge b \\preceq a$ :$a \\wedge b \\preceq b$ i.e., $a \\wedge b$ precedes its operands $a$ and $b$."}
+{"_id": "5595", "title": "Meet is Associative", "text": "Let $\\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Then $\\wedge$ is associative."}
+{"_id": "5597", "title": "Poset Elements Equal iff Equal Weak Upper Closure", "text": "Let $\\left({S, \\preccurlyeq}\\right)$ be an ordered set. Let $s, t \\in S$. Then $s = t$ {{iff}}: :$s^\\succcurlyeq = t^\\succcurlyeq$ where $s^\\succcurlyeq$ denotes weak upper closure of $s$. That is, {{iff}}, for all $r \\in S$: :$s \\preccurlyeq r \\iff t \\preccurlyeq r$"}
+{"_id": "5599", "title": "Join is Associative", "text": "Let $\\struct {S, \\vee, \\preceq}$ be a join semilattice. Then $\\vee$ is associative."}
+{"_id": "5601", "title": "Prime iff Equal to Product", "text": "Let $p \\in \\Z$ be an integer such that $p \\ne 0$ and $p \\ne \\pm 1$. Then $p$ is prime {{iff}}: :$\\forall a, b \\in \\Z: p = ab \\implies p = \\pm a \\lor p = \\pm b$"}
+{"_id": "5603", "title": "Multiple of Divisor Divides Multiple", "text": "Let $a, b, c \\in \\Z$. Let: :$a \\divides b$ where $\\divides$ denotes divisibility. Then: :$a c \\divides b c$"}
+{"_id": "5606", "title": "Divisor Relation is Transitive", "text": "The divisibility relation is a transitive relation on $\\Z$, the set of integers. That is: :$\\forall x, y, z \\in \\Z: x \\divides y \\land y \\divides z \\implies x \\divides z$"}
+{"_id": "5615", "title": "Polynomial Forms over Field form Integral Domain/Formulation 1", "text": "Let $\\struct {F, +, \\circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $X$ be transcendental in $F$. Let $F \\sqbrk X$ be the ring of polynomial forms in $X$ over $F$. Then $F \\sqbrk X$ is an integral domain."}
+{"_id": "5616", "title": "Sum of Indices of Real Number/Positive Integers", "text": "Let $n, m \\in \\Z_{\\ge 0}$ be positive integers. Let $r^n$ be defined as $r$ to the power of $n$. Then: : $r^{n + m} = r^n \\times r^m$"}
+{"_id": "5617", "title": "Product of Indices of Real Number/Positive Integers", "text": "Let $n, m \\in \\Z_{\\ge 0}$ be positive integers. Let $r^n$ be defined as $r$ to the power of $n$. Then: :$\\paren {r^n}^m = r^{n m}$"}
+{"_id": "5619", "title": "Smallest Element is Initial Object", "text": "Let $\\mathbf P$ be an order category. Suppose the objects $\\mathbf P_0$ of $\\mathbf P$, considered as an ordered set, have a smallest element $p$. Then $p$ is an initial object of $\\mathbf P$."}
+{"_id": "5620", "title": "Ordinal is Subset of Rank of Small Class iff Not in Von Neumann Hierarchy", "text": "Let $x$ be an ordinal. Let $S$ be a small class. Let $V \\left({ x }\\right)$ denote the von Neumann hierarchy on the ordinal $x$. Then $x$ is a subset of the rank of $S$ {{iff}} $S \\notin V \\left({x}\\right)$."}
+{"_id": "5621", "title": "Membership Rank Inequality", "text": "Let $S$ and $T$ be sets. Let $\\operatorname{rank} \\left({ S }\\right)$ denote the rank of $S$. Then: :$S \\in T \\implies \\operatorname{rank} \\left({S}\\right) < \\operatorname{rank} \\left({ T }\\right)$"}
+{"_id": "5622", "title": "Greatest Element is Terminal Object", "text": "Let $\\mathbf P$ be an order category. Let $p$ be the greatest element of the objects $\\mathbf P_0$ of $\\mathbf P$, considered as a ordered set. Then $p$ is a terminal object of $\\mathbf P$."}
+{"_id": "5625", "title": "Rank of Set Determined by Members", "text": "Let $S$ be a set. Let $\\operatorname{rank} \\left({ S }\\right)$ denote the rank of $S$. Then: :$\\operatorname{rank} \\left({ S }\\right) = \\bigcap \\left\\{ x \\in \\operatorname{On} : \\forall y \\in S: \\operatorname{rank} \\left({ y }\\right) < x \\right\\}$"}
+{"_id": "5627", "title": "Bounded Rank implies Small Class", "text": "Let $S$ be a class. Suppose the rank, denoted $\\operatorname{rank} \\left({x}\\right)$, of each $x \\in S$ is bounded above by some ordinal $y$. {{MissingLinks|rank}} Then $S$ is a small class."}
+{"_id": "5628", "title": "Count of Subsets with Even Cardinality", "text": "Let $S$ be a set whose cardinality is $n$. Then the number of subsets of $S$ whose cardinality is even is $2^{n-1}$."}
+{"_id": "5630", "title": "Count of Subsets with Odd Cardinality", "text": "Let $S$ be a set whose cardinality is $n$. Then the number of subsets of $S$ whose cardinality is odd is $2^{n-1}$."}
+{"_id": "5634", "title": "Well-Founded Relation is Strongly Well-Founded", "text": "Let $A$ be a class. Let $\\prec$ be a foundational relation on $A$. Let $B$ be a nonempty class such that $B \\subseteq A$. Then $B$ has a $\\prec$-minimal element."}
+{"_id": "5635", "title": "Category of Ordered Sets has Enough Constants", "text": "Let $\\mathbf{OrdSet}$ be the category of ordered sets. Then $\\mathbf{OrdSet}$ has enough constants."}
+{"_id": "5638", "title": "Cardinal Number is Ordinal", "text": "Let $S$ be a set such that $S \\sim x$ for some ordinal $x$. Let $\\left|{S}\\right|$ denote the cardinal number of $S$. Then: :$\\left|{S}\\right| \\in \\operatorname{On}$ where $\\operatorname{On}$ denotes the class of ordinals."}
+{"_id": "5641", "title": "Condition for Set Equivalent to Cardinal Number", "text": "Let $S$ be a set. Let $\\card S$ denote the cardinal number of $S$. That is, let $\\card S$ be the intersection of all ordinals equivalent to $S$. {{refactor|This following statement needs to go somewhere else as it's a non-sequitur in this context.}} Note that in the absence of the Axiom of Choice, $\\card S$ may be the class of all sets. Then the following are equivalent: : $(1): \\quad S \\sim \\card S$ : $(2): \\quad \\exists x \\in \\On: S \\sim x$ : $(3): \\quad \\exists x \\in \\On: \\exists y: \\paren {y \\subseteq x \\land S \\sim y}$ where $A \\sim B$ means that there is a bijection from $A$ onto $B$, and the quantification over $y$ is unbounded (so $y$ may be any set)."}
+{"_id": "5642", "title": "Cardinal Number Equivalence or Equal to Universe", "text": "Let $S$ be a set. Let $\\left\\vert{S}\\right\\vert$ denote the cardinal number of $S$. Let $\\mathbb U$ denote the universal class. Then: :$S \\sim \\left\\vert{S}\\right\\vert \\lor \\left\\vert{S}\\right\\vert = \\mathbb U$"}
+{"_id": "5643", "title": "Ordinal Number Equivalent to Cardinal Number", "text": "Let $x$ be an ordinal. Let $\\card x$ denote the cardinal number of $x$. Then: :$x \\sim \\card x$ where $\\sim$ denotes set equivalence."}
+{"_id": "5644", "title": "Cardinal Number Less than Ordinal", "text": "Let $S$ be a set. Let $\\left|{S}\\right|$ denote the cardinal number of $S$. Let $x$ be an ordinal such that $S \\sim x$. Then: :$\\left|{S}\\right| \\le x$"}
+{"_id": "5645", "title": "Equivalent Sets have Equal Cardinal Numbers", "text": "Let $S$ and $T$ be sets. Let $\\card S$ denote the cardinal number of $S$. Then: :$S \\sim T \\implies \\card S = \\card T$"}
+{"_id": "5646", "title": "Condition for Set Union Equivalent to Associated Cardinal Number", "text": "Let $S$ and $T$ be sets. Let $\\left|{S}\\right|$ denote the cardinal number of $S$. Let $\\sim$ denote set equivalence. Then: :$S \\cup T \\sim \\left|{S \\cup T}\\right| \\iff S \\sim \\left|{S}\\right| \\land T \\sim \\left|{T}\\right|$"}
+{"_id": "5648", "title": "Cardinal of Cardinal Equal to Cardinal", "text": "Let $S$ be a set such that $S$ is equivalent to its cardinal. If the axiom of choice holds, then this condition holds for any set. Then: :$\\left|{ \\left({ \\left|{S}\\right| }\\right) }\\right| = \\left|{S}\\right|$ where $\\left|{S}\\right|$ denotes the cardinal number of $S$."}
+{"_id": "5649", "title": "Cardinal of Finite Ordinal", "text": "Let $n$ be a finite ordinal. Let $\\left|{n}\\right|$ denote the cardinal number of $n$. Then: :$\\left|{n}\\right| = n$"}
+{"_id": "5650", "title": "Finite Ordinal is equal to Natural Number", "text": "Let $n$ be an element of the minimal infinite successor set. Let $x$ be an ordinal. Then: :$n \\sim x \\implies n = x$"}
+{"_id": "5651", "title": "Product (Category Theory) is Unique", "text": "Let $\\mathbf C$ be a metacategory. Let $A$ and $B$ be objects of $\\mathbf C$. Let $A \\times B$ and $A \\times' B$ both be products of $A$ and $B$. Then there is a unique isomorphism $u: A \\times B \\to A \\times' B$. That is, products are unique up to unique isomorphism. {{expand|cover extended theorem (for general def)}}"}
+{"_id": "5653", "title": "Subset implies Cardinal Inequality", "text": "Let $S$ and $T$ be sets such that $S \\subseteq T$. Furthermore, let: :$T \\sim \\card T$ where $\\card T$ denotes the cardinality of $T$. Then: :$\\card S \\le \\card T$"}
+{"_id": "5656", "title": "Subset of Ordinal implies Cardinal Inequality", "text": "Let $S$ be a set. Let $x$ be an ordinal such that $S \\subseteq x$. Then: :$\\left|{S}\\right| \\le \\left|{x}\\right|$ where $\\left|{S}\\right|$ denotes the cardinality of $S$."}
+{"_id": "5657", "title": "Set Less than Cardinal Product", "text": "Let $S$ and $T$ be sets. Let $T$ be nonempty. Suppose that $S \\times T \\sim \\left|{ S \\times T }\\right|$. Then: :$\\left|{ S }\\right| \\le \\left|{ S \\times T }\\right|$"}
+{"_id": "5658", "title": "Injection implies Cardinal Inequality", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be an injection. Let $\\card T$ denote the cardinal number of $T$. Let: :$T \\sim \\card T$ where $\\sim$ denotes set equivalence Then: :$\\card S \\le \\card T$"}
+{"_id": "5659", "title": "Set is Equivalent to Image under Injection", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be an injection. Then the image of $S$ under $f$ is equivalent to $S$."}
+{"_id": "5660", "title": "Cardinality of Image of Mapping not greater than Cardinality of Domain", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let $\\card S$ denote the cardinal number of $S$. Let $S \\sim \\card S$. Then: :$\\card {\\Img f} \\le \\card S$"}
+{"_id": "5661", "title": "Injection iff Cardinal Inequality", "text": "Let $\\card T$ denote the cardinal number of $T$. Let $S$ and $T$ be sets such that $S \\sim \\card S$ and $T \\sim \\card T$. Then: :$\\card S \\le \\card T $ {{iff}} there exists an injection $f: S \\to T$."}
+{"_id": "5662", "title": "Surjection iff Cardinal Inequality", "text": "Let $S$ and $T$ be sets such that $S \\sim \\card S$ and $T \\sim \\card T$. Furthermore, let $S$ be non-empty. Then: :$0 < \\card T \\le \\card S$ {{iff}} there exists a surjection $f: S \\to T$."}
+{"_id": "5667", "title": "Right Identity while exists Right Inverse for All is Identity", "text": "Let $\\struct  {S, \\circ}$ be a semigroup with a right identity $e_R$ such that: :$\\forall x \\in S: \\exists x_R: x \\circ x_R = e_R$ That is, every element of $S$ has a right inverse with respect to the right identity. Then $e_R$ is also a left identity, that is, is an identity."}
+{"_id": "5668", "title": "Left Identity while exists Left Inverse for All is Identity", "text": "Let $\\struct {S, \\circ}$ be a semigroup with a left identity $e_L$ such that: :$\\forall x \\in S: \\exists x_L: x_L \\circ x = e_L$ That is, every element of $S$ has a left inverse with respect to the left identity. Then $e_L$ is also a right identity, that is, is an identity."}
+{"_id": "5669", "title": "Right Inverse for All is Left Inverse", "text": "Let $\\struct {S, \\circ}$ be a semigroup with a right identity $e_R$ such that: :$\\forall x \\in S: \\exists x_R: x \\circ x_R = e_R$ That is, every element of $S$ has a right inverse with respect to the right identity. Then $x_R \\circ x = e_R$, that is, $x_R$ is also a left inverse with respect to the right identity."}
+{"_id": "5676", "title": "Exponential on Complex Plane is Group Homomorphism", "text": "Let $\\struct {\\C, +}$ be the additive group of complex numbers. Let $\\struct {\\C_{\\ne 0}, \\times}$ be the multiplicative group of complex numbers. Let $\\exp: \\struct {\\C, +} \\to \\struct {\\C_{\\ne 0}, \\times}$ be the mapping: :$x \\mapsto \\map \\exp x$ where $\\exp$ is the complex exponential function. Then $\\exp$ is a group homomorphism."}
+{"_id": "5679", "title": "Group Direct Product is Product in Category of Groups", "text": "Let $\\mathbf{Grp}$ be the category of groups. Let $G$ and $H$ be groups, and let $G \\times H$ be their direct product. Then $G \\times H$ is a binary product of $G$ and $H$ in $\\mathbf{Grp}$."}
+{"_id": "5680", "title": "Product Category is Product in Category of Categories", "text": "Let $\\mathbf{Cat}$ be the category of categories. Let $\\mathbf C$ and $\\mathbf D$ be small categories, and let $\\mathbf C \\times \\mathbf D$ be their product category. Then $\\mathbf C \\times \\mathbf D$ is a binary product of $\\mathbf C$ and $\\mathbf D$ in $\\mathbf{Cat}$."}
+{"_id": "5681", "title": "Infimum is Product in Order Category", "text": "Let $\\mathbf P$ be an order category whose ordering is $\\preceq$. Let $p,q \\in \\mathbf P_0$, and suppose that they have some infimum $r = \\inf \\left\\{{p, q}\\right\\}$. Then $r$ is a binary product of $p$ and $q$ in $\\mathbf P$."}
+{"_id": "5688", "title": "Cardinal of Union Less than Cardinal of Cartesian Product", "text": "Let $S$ and $T$ be sets that are equivalent to their cardinal numbers. Let $\\card S$ denote the cardinal number of $S$. Let $\\card S > 1$ and $\\card T > 1$. Then: :$\\card {S \\cup T} \\le \\card {S \\times T}$"}
+{"_id": "5689", "title": "Cardinal Product Equinumerous to Ordinal Product", "text": "Let $S$ and $T$ be sets that are equivalent to their cardinal numbers. Let $\\left|{ S }\\right|$ denote the cardinal number of $S$. Let $\\cdot$ denote ordinal multiplication and let $\\times$ denote the Cartesian product. Then: :$S \\times T \\sim \\left|{ S }\\right| \\cdot \\left|{ T }\\right|$"}
+{"_id": "5690", "title": "Product of Finite Sets is Finite", "text": "Let $S$ and $T$ be finite sets. Then $S \\times T$ is a finite set."}
+{"_id": "5691", "title": "Union of Finite Sets is Finite", "text": "Let $S$ and $T$ be finite sets. Then $S \\cup T$ is a finite set."}
+{"_id": "5692", "title": "Ordinal is Finite iff Natural Number", "text": "Let $x$ be an ordinal. Then $x$ is a finite set {{iff}} $x$ is an element of the minimal infinite successor set."}
+{"_id": "5695", "title": "Cardinal Inequality implies Ordinal Inequality", "text": "Let $T$ be a set. Let $\\left|{T}\\right|$ denote the cardinal number of $T$. Let $x$ be an ordinal. Then: :$x < \\left|{T}\\right| \\iff \\left|{x}\\right| < \\left|{T}\\right|$"}
+{"_id": "5696", "title": "Cardinal Number Plus One Less than Cardinal Product", "text": "Let $x$ be an ordinal such that $x > 1$. Then: :$\\left|{ x + 1 }\\right| \\le \\left|{ x \\times x }\\right|$ Where $\\times$ denotes the Cartesian product."}
+{"_id": "5697", "title": "Non-Finite Cardinal is equal to Cardinal Product", "text": "Let $\\omega$ denote the minimal infinite successor set. Let $x$ be an ordinal such that $x \\ge \\omega$. Then: :$\\left|{x}\\right| = \\left|{ x\\times x}\\right|$ where $\\times$ denotes the Cartesian product."}
+{"_id": "5701", "title": "Cartesian Product Preserves Cardinality", "text": "Let $R$, $S$, and $T$ be sets. Suppose that $S$ is equivalent to $T$. Then: :$R \\times S \\sim R \\times T$ :$S \\times R \\sim T \\times R$"}
+{"_id": "5702", "title": "Product of Composite Morphisms", "text": "Let $\\mathbf C$ be a metacategory. Let $f \\times f': A \\times A' \\to B \\times B'$ and $g \\times g': B \\times B' \\to C \\times C'$ be two composable products of morphisms in $\\mathbf C$. Then: :$\\left({g \\circ f}\\right) \\times \\left({g' \\circ f'}\\right) = \\left({g \\times g'}\\right) \\circ \\left({f \\times f'}\\right)$ where $\\times$ signifies product of morphisms."}
+{"_id": "5703", "title": "Cardinal Product Equal to Maximum", "text": "Let $S$ and $T$ be sets that are equinumerous to their cardinal number. Let $\\left|{S}\\right|$ denote the cardinal number of $S$. Suppose $S$ is infinite. Suppose $T > 0$. Then: :$\\left|{S \\times T}\\right| = \\max\\left({\\left|{S}\\right|, \\left|{T}\\right| }\\right)$"}
+{"_id": "5704", "title": "Cardinal of Union Equal to Maximum", "text": "Let $S$ and $T$ be sets that are equinumerous to their cardinal number. Let $\\left|{ S }\\right|$ denote the cardinal number of $S$. Suppose $S$ is infinite. Then: :$\\left|{ S \\cup T }\\right| = \\max\\left({ \\left|{ S }\\right| , \\left|{ T }\\right| }\\right)$"}
+{"_id": "5705", "title": "Cardinal Class is Subset of Ordinal Class", "text": "Let $\\mathcal N$ denote the class of all cardinal numbers. Let $\\operatorname{On}$ denote the class of all ordinals. Then: :$\\mathcal N \\subseteq \\operatorname{On}$"}
+{"_id": "5706", "title": "Cardinal of Cardinal Equal to Cardinal/Corollary", "text": "Let $\\mathcal N$ denote the class of all cardinal numbers. Let $x$ be an ordinal. Then: :$x \\in \\mathcal N \\iff x = \\left|{ x }\\right|$"}
+{"_id": "5708", "title": "Cardinal Equal to Collection of All Dominated Ordinals", "text": "Let $S$ be a set. Let $\\preccurlyeq$ denote the dominance relation. Let $\\On$ denote the class of all ordinals. Let $x = \\set {y \\in \\On: y \\preccurlyeq S}$ Then: :$(1): \\quad x$ is an element of the class of cardinals. :$(2): \\quad$ There is no injection $f$ such that $f : x \\to S$"}
+{"_id": "5709", "title": "Cardinal Class is Proper Class", "text": "The class $\\mathcal N$ of all cardinal numbers is a proper class."}
+{"_id": "5710", "title": "Infinite Cardinal Class is Proper Class", "text": "The infinite cardinal class $\\mathcal N’$ is a proper class."}
+{"_id": "5711", "title": "Ordinal in Aleph iff Cardinal in Aleph", "text": "Let $x$ and $y$ be ordinals. Then: :$x \\in \\aleph_y \\iff \\left|{ x }\\right| \\in \\aleph_y$ where $\\aleph$ denotes the aleph mapping."}
+{"_id": "5712", "title": "Aleph Product is Aleph", "text": "Let $x$ be an ordinal. Then: :$\\left|{\\aleph_x \\times \\aleph_x}\\right| = \\aleph_x$ where $\\aleph$ denotes the aleph mapping."}
+{"_id": "5713", "title": "Aleph is Infinite Cardinal", "text": "Let $x$ be an ordinal. Then $\\aleph_x$ is an infinite cardinal where $\\aleph$ denotes the aleph mapping."}
+{"_id": "5714", "title": "Surjection from Aleph to Ordinal", "text": "Let $x$ and $y$ be ordinals. Suppose that: :$0 < y < \\aleph_{x+1}$ Then there is a surjection: :$f : \\aleph_x \\to y$"}
+{"_id": "5715", "title": "Ordinal Less than Successor Aleph", "text": "Let $x$ and $y$ be ordinals. Then: :$y < \\aleph_{x+1} \\iff y < \\aleph_x \\lor y \\sim \\aleph_x$"}
+{"_id": "5716", "title": "Aleph is Infinite", "text": "Let $x$ be an ordinal. :$\\aleph_x \\ge \\omega$ where: :$\\aleph$ denotes the aleph mapping :$\\omega$ denotes the minimal infinite successor set."}
+{"_id": "5717", "title": "Aleph-Null", "text": "Let $\\omega$ denote the minimal infinite successor set. :$\\omega = \\aleph_0$ where $\\aleph$ denotes the aleph mapping."}
+{"_id": "5718", "title": "Minimal Infinite Successor Set is Infinite Cardinal", "text": "$\\omega$, the minimal infinite successor set, is an element of the infinite cardinal class $\\mathcal N’$."}
+{"_id": "5719", "title": "Set of All Mappings is Small Class", "text": "Let $S$ and $T$ be small classes. It follows that the set of all mappings $S^T$ is a small class."}
+{"_id": "5721", "title": "Covariant Hom Functor is Functor", "text": "Let $\\mathbf{Set}$ be the category of sets. Let $\\mathbf C$ be a locally small category. Let $C \\in \\mathbf C_0$ be an object of $\\mathbf C$. Let $\\hom_\\mathbf C \\paren {C, \\cdot}: \\mathbf C \\to \\mathbf{Set}$ be the covariant representable functor based at $C$. Then $\\hom_\\mathbf C \\paren {C, \\cdot}$ is a functor."}
+{"_id": "5722", "title": "Inverse Morphism is Unique", "text": "Let $\\mathbf C$ be a metacategory. Let $f: C \\to D$ be an isomorphism of $\\mathbf C$. Then $f$ admits a unique inverse morphism $g: D \\to C$."}
+{"_id": "5723", "title": "Cofinal Ordinal Relation is Reflexive", "text": "Let $x$ be an ordinal. Then $x$ is cofinal to itself. That is: :$\\operatorname{cof} \\left({x, x}\\right)$"}
+{"_id": "5724", "title": "Cofinal Ordinal Relation is Transitive", "text": "Let $x$, $y$, and $z$ be ordinals. Let $\\operatorname{cof}$ denote the cofinal relation. Then: :$\\operatorname{cof} \\left({ x,y }\\right) \\land \\operatorname{cof} \\left({ y,z }\\right) \\implies \\operatorname{cof} \\left({ x,z }\\right)$"}
+{"_id": "5725", "title": "Cofinal to Zero iff Ordinal is Zero", "text": "Let $x$ be an ordinal. Let $\\operatorname{cof}$ denote the cofinal relation. Let $0$ denote the zero ordinal. Then the following are equivalent: :$\\operatorname{cof} \\left({ x,0 }\\right)$ :$\\operatorname{cof} \\left({ 0,x }\\right)$ :$x = 0$"}
+{"_id": "5726", "title": "Condition for Cofinal Nonlimit Ordinals", "text": "Let $x$ and $y$ be nonlimit ordinals. Let $\\operatorname{cof}$ denote the cofinal relation. Let $\\le$ denote the subset relation. {{explain|This statement could be worded a little more carefully. It is, from examining the link, clear that the subset relation and the ordering relation are the same thing, but it grates to see $\\le$ being defined as the subset relation. I'm not saying it ''is'' wrong, because it's not wrong, but it ''looks'' uncomfortable.}} Furthermore, let $x$ and $y$ satisfy the condition: :$0 < x \\le y$ Then: :$\\map {\\operatorname{cof}} { y,x }$"}
+{"_id": "5728", "title": "Cofinal Limit Ordinals", "text": "Let $x$ and $y$ be ordinals. Let $\\operatorname{cof}$ denote the cofinal relation. Let $K_{II}$ denote the class of all limit ordinals. Then: :$\\operatorname{cof} \\left({ x,y }\\right) \\implies \\left({ x \\in K_{II} \\iff y \\in K_{II} }\\right)$"}
+{"_id": "5734", "title": "Matrix is Invertible iff Determinant has Multiplicative Inverse", "text": "Let $R$ be a commutative ring with unity. Let $\\mathbf A \\in R^{n \\times n}$ be a square matrix of order $n$. Then $\\mathbf A$ is invertible {{iff}} its determinant is invertible in $R$. If $R$ is one of the standard number fields $\\Q$, $\\R$ or $\\C$, this translates into: :$\\mathbf A$ is invertible {{iff}} its determinant is non-zero."}
+{"_id": "5738", "title": "Prime Groups of Same Order are Isomorphic", "text": "Two prime groups of the same order are isomorphic to each other."}
+{"_id": "5739", "title": "Infinite Cyclic Group is Unique up to Isomorphism", "text": "All infinite cyclic groups are isomorphic. That is, up to isomorphism, there is only one infinite cyclic group."}
+{"_id": "5740", "title": "Intersection of Subgroups is Subgroup/General Result", "text": "Let $\\mathbb S$ be a set of subgroups of $\\struct {G, \\circ}$, where $\\mathbb S \\ne \\O$. Then the intersection $\\displaystyle \\bigcap \\mathbb S$ of the elements of $\\mathbb S$ is itself a subgroup of $G$. Also, $\\displaystyle \\bigcap \\mathbb S$ is the largest subgroup of $\\struct {G, \\circ}$ contained in each element of $\\mathbb S$."}
+{"_id": "5741", "title": "Union of Subgroups/Corollary 2", "text": "Let $H \\vee K$ be the join of $H$ and $K$. Then $H \\vee K = H \\cup K$ {{iff}} $H \\subseteq K$ or $K \\subseteq H$."}
+{"_id": "5742", "title": "Existence of Unique Subgroup Generated by Subset/Singleton Generator", "text": "Let $a \\in G$. Then $H = \\gen a = \\set {a^n: n \\in \\Z}$ is the unique smallest subgroup of $G$ such that $a \\in H$. That is: : $K \\le G: a \\in K \\implies H \\subseteq K$"}
+{"_id": "5744", "title": "Join of Subgroups is Group Generated by Union", "text": "Let $G$ be a group. Let $H$ and $K$ be subgroups of $G$. Let $S$ be the set of words of $H \\cup K$. Then $S$ is a subgroup of $K$ such that: :$S = \\gen {H \\cup K} = H \\vee K$ where $H \\vee K$ denotes the join of $H$ and $K$."}
+{"_id": "5745", "title": "Order of Subgroup Product/Corollary", "text": ":$\\size {H \\vee K} \\ge \\dfrac {\\order H \\order K} {\\order {H \\cap K} }$ or :$\\dfrac {\\size {H \\vee K} } {\\order H} \\ge \\dfrac {\\order K} {\\order {H \\cap K} }$ where $H \\vee K$ denotes join and $\\order H$ denotes the order of $H$."}
+{"_id": "5747", "title": "Homomorphic Image of Group Element is Coset", "text": "Let $\\phi: \\left({G, \\circ}\\right) \\to \\left({H, *}\\right)$ be a group homomorphism. Let $\\ker \\left({\\phi}\\right)$ be the kernel of $\\phi$. Let $h \\in H$. Then $\\operatorname{Im}^{-1} \\left({h}\\right)$ is either the empty set or a coset of $\\ker \\left({\\phi}\\right)$."}
+{"_id": "5748", "title": "Characterization of Metacategory via Equations", "text": "Let $\\mathbf C_0$ and $\\mathbf C_1$ be collections of objects. Let $\\operatorname{cod}$ and $\\operatorname{dom}$ assign to every element of $\\mathbf C_1$ an element of $\\mathbf C_0$. Let $\\operatorname{id}$ assign to every element of $\\mathbf C_0$ an element of $\\mathbf C_1$. Denote with $\\mathbf C_2$ the collection of pairs $\\left({f, g}\\right)$ of elements of $\\mathbf C_1$ satisfying: :$\\operatorname{dom} g = \\operatorname{cod} f$ Let $\\circ$ assign to every such pair an element of $\\mathbf C_1$. Then $\\mathbf C_0, \\mathbf C_1, \\operatorname{cod}, \\operatorname{dom}, \\operatorname{id}$ and $\\circ$ together determine a metacategory $\\mathbf C$ iff the following seven axioms are satisfied: {{begin-eqn}} {{eqn|l = \\operatorname{dom} \\operatorname{id}_A = A      |o = \\qquad      |r = \\operatorname{cod} \\operatorname{id}_A = A }} {{eqn|l = f \\circ \\operatorname{id}_{\\operatorname{dom} f} = f      |o =       |r = \\operatorname{id}_{\\operatorname{cod} f} \\circ f = f }} {{eqn|l = \\operatorname{dom} \\left({g \\circ f}\\right) = \\operatorname{dom} f      |o =       |r = \\operatorname{cod} \\left({g \\circ f}\\right) = \\operatorname{cod} g }} {{eqn|l = h \\circ \\left({g \\circ f}\\right)      |r = \\left({h \\circ g}\\right) \\circ f }} {{end-eqn}} with $A$ and $f,g,h$ arbitrary elements of $\\mathbf C_0$ and $\\mathbf C_1$, respectively. Further, in the last two lines it is presumed that all compositions are defined. Hence it follows that: * $\\mathbf C_0$ and $\\mathbf C_1$ represent the collections of objects and morphisms of $\\mathbf C$ * $\\operatorname{dom}$ and $\\operatorname{cod}$ represent the domain and codomain of a morphism of $\\mathbf C$ * $\\operatorname{id}$ represents the identity morphisms of $\\mathbf C$ * $\\mathbf C_2$ represents the collection of composable morphisms of $\\mathbf C$ * $\\circ$ represents the composition of morphisms in $\\mathbf C$"}
+{"_id": "5751", "title": "Kernel of Group Homomorphism Corresponds with Normal Subgroup of Domain", "text": "Let $\\struct {G, \\circ}$ and $\\struct {H, *}$ be groups. Let $\\phi: \\struct {G, \\circ} \\to \\struct {H, *}$ be a group homomorphism. Let $\\map \\ker \\phi$ be the kernel of $\\phi$. Then there exists $N \\lhd G$, a normal subgroup of $G$ such that: :$N = \\map \\ker \\phi$ Conversely, let $N \\lhd G$ be normal subgroup of $G$. Then there exists $\\phi: \\struct {G, \\circ} \\to \\struct {H, *}$, a group homomorphism, whose kernel $\\map \\ker \\phi$ is such that: :$\\map \\ker \\phi = N$"}
+{"_id": "5752", "title": "Inner Automorphism Maps Normal Subgroup to Itself", "text": "Let $G$ be a group. Let $x \\in G$. Let $\\kappa_x$ be the inner automorphism of $x$ in $G$. Let $N$ be a normal subgroup of $G$. Then: :$\\kappa_x \\sqbrk N = N$"}
+{"_id": "5753", "title": "Group Epimorphism Preserves Normal Subgroups", "text": "Let $\\left({G, \\circ}\\right)$ and $\\left({H, *}\\right)$ be groups. Let $\\phi: G \\to H$ be a group epimorphism. Let $N \\lhd G$, where $\\lhd$ denotes that $N$ is a normal subgroup of $G$. Then $\\phi \\left({N}\\right) \\lhd H$. That is, the image under $\\phi$ of a normal subgroup is itself normal."}
+{"_id": "5754", "title": "Group Epimorphism Induces Bijection between Subgroups", "text": "Let $G_1$ and $G_2$ be groups whose identities are $e_{G_1}$ and $e_{G_2}$ respectively. Let $\\phi: G_1 \\to G_2$ be a group epimorphism. Let $K := \\ker \\left({\\phi}\\right)$ be the kernel of $\\phi$. Let $\\mathbb H_1 = \\left\\{{H \\subseteq G_1: H \\le G_1, K \\subseteq H}\\right\\}$ be the set of subgroups of $G_1$ which contain $K$. Let $\\mathbb H_2 = \\left\\{{H \\subseteq G_2: H \\le G_2}\\right\\}$ be the set of subgroups of $G_2$. Then there exists a bijection $Q: \\mathbb H_1 \\leftrightarrow \\mathbb H_2$ such that: :$\\forall N \\lhd G_1: Q \\left({N}\\right) \\lhd G_2$ :$\\forall N \\lhd G_2: Q^{-1} \\left({N}\\right) \\lhd G_1$ where $N \\lhd G_1$ denotes that $N$ is a normal subgroup of $G_1$. That is, normal subgroups map bijectively to normal subgroups under $Q$."}
+{"_id": "5756", "title": "Homomorphic Image of Quotient Group under Epimorphism", "text": "Let $G_1$ and $G_2$ be groups whose identities are $e_{G_1}$ and $e_{G_2}$ respectively. Let $\\phi: G_1 \\to G_2$ be a group epimorphism. Let $K := \\ker \\left({\\phi}\\right)$ be the kernel of $\\phi$. Let $N$ be a normal subgroup of $G_1$ such that $K \\subseteq N$. Then: :$\\dfrac {G_1} N \\cong \\dfrac {G_2}{\\phi \\left({N}\\right)}$ where $\\dfrac {G_1} N$ denotes the quotient group of $G_1$ by $N$."}
+{"_id": "5758", "title": "Cauchy's Group Theorem", "text": "Let $G$ be a finite group whose identity is $e$. Let $p$ be a prime number which divides order of $G$. Then $G$ has a subgroup of order $p$."}
+{"_id": "5767", "title": "Connected Subspace of Linearly Ordered Space", "text": "Let $\\struct {S, \\preceq, \\tau}$ be a linearly ordered space. Let $Y \\subseteq S$. Then $Y$ is connected in $\\struct {S, \\tau}$ {{iff}} both of the following hold: :$(1): \\quad Y$ is convex in $S$ :$(2): \\quad \\struct {Y, \\preceq \\restriction_Y}$ is a linear continuum, where $\\restriction$ denotes restriction."}
+{"_id": "5768", "title": "Abnormal Subgroup is Self-Normalizing Subgroup", "text": "Let $G$ be a group. Let $H$ be an abnormal subgroup of $G$. Then $H$ is a self-normalizing subgroup of $G$."}
+{"_id": "5770", "title": "Duality Principle (Category Theory)", "text": "In the study of metacategories and categories, the following two '''duality principles''' are very useful."}
+{"_id": "5771", "title": "Duality Principle (Category Theory)/Conceptual Duality", "text": "Let $\\Sigma$ be a statement about metacategories, be it in natural language or otherwise. Suppose that $\\Sigma$ holds for all metacategories. Then so does its dual statement $\\Sigma^*$."}
+{"_id": "5772", "title": "Product of Complex Number with Conjugate", "text": "Let $z = a + i b \\in \\C$ be a complex number. Let $\\overline z$ denote the complex conjugate of $z$. Then: :$z \\overline z = a^2 + b^2 = \\cmod z^2$ and thus is wholly real."}
+{"_id": "5779", "title": "Compact Subspace of Linearly Ordered Space/Lemma", "text": "Let $\\left({X, \\preceq, \\tau}\\right)$ be a linearly ordered space. Let $Y \\subseteq X$ be a non-empty subset of $X$. Let $Y$ be a compact subspace of $\\left({X, \\tau}\\right)$. Then $\\left({Y, \\preceq \\restriction_Y}\\right)$ is a complete lattice, where $\\restriction$ denotes restriction."}
+{"_id": "5781", "title": "Inclusion Mapping on Subgroup is Homomorphism", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\struct {H, \\circ_{\\restriction H} }$ be a subgroup of $G$. Let $i: H \\to G$ be the inclusion mapping from $H$ to $G$. Then $i$ is a group homomorphism."}
+{"_id": "5782", "title": "Mapping to Identity is Unique Constant Homomorphism", "text": "Let $\\struct {G, \\circ}$ and $\\struct {H, *}$ be groups whose identities are $e_G$ and $e_H$ respectively. Then there exists a unique constant mapping from $G$ to $H$ which is a homomorphism: :$\\phi_{e_H}: G \\to H: \\forall g \\in G: \\map {\\phi_{e_H} } g = e_H$"}
+{"_id": "5783", "title": "Subgroup is Superset of Conjugate iff Normal", "text": "Let $\\struct {G, \\circ}$ be a group. Let $N$ be a subgroup of $G$. Then $N$ is normal in $G$ (by definition 1) {{iff}}: : $\\forall g \\in G: g \\circ N \\circ g^{-1} \\subseteq N$ : $\\forall g \\in G: g^{-1} \\circ N \\circ g \\subseteq N$"}
+{"_id": "5784", "title": "Union Distributes over Union/General Result", "text": "Let $\\left\\langle{\\mathbb S_i}\\right\\rangle_{i \\in I}$ be an $I$-indexed family of sets of sets. Then: :$\\displaystyle \\bigcup_{i \\mathop \\in I} \\bigcup \\mathbb S_i = \\bigcup \\bigcup_{i \\mathop \\in I} \\mathbb S_i$"}
+{"_id": "5785", "title": "Subgroup is Subset of Conjugate iff Normal", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $N$ be a subgroup of $G$. Then $N$ is normal in $G$ (by definition 1) {{iff}}: :$\\forall g \\in G: N \\subseteq g \\circ N \\circ g^{-1}$ :$\\forall g \\in G: N \\subseteq g^{-1} \\circ N \\circ g$"}
+{"_id": "5786", "title": "Existence and Uniqueness of Generated Topology", "text": "Let $X$ be a set. Let $\\SS \\subseteq \\powerset X$ be a subset of the power set of $X$. Then there exists a unique topology $\\map \\tau \\SS$ on $X$ such that: :$(1): \\quad \\SS \\subseteq \\map \\tau \\SS$. :$(2): \\quad$ For any topology $\\TT$ on $X$, the implication $\\SS \\subseteq \\TT \\implies \\map \\tau \\SS \\subseteq \\TT$ holds."}
+{"_id": "5787", "title": "Equivalent Conditions for Cover by Collection of Subsets", "text": "Let $X$ be a set. Then the following conditions are equivalent for a subset $\\mathcal C \\subseteq \\mathcal P \\left({X}\\right)$ of the power set of $X$: :$\\left({1}\\right): \\quad$ $\\mathcal C$ is a cover for $X$. :$\\left({2}\\right): \\quad$ $\\displaystyle X = \\bigcup \\mathcal C$. :$\\left({3}\\right): \\quad$ $\\displaystyle \\exists \\mathcal S \\subseteq \\mathcal C: X = \\bigcup \\mathcal S$."}
+{"_id": "5788", "title": "Subgroup equals Conjugate iff Normal", "text": ":$\\forall g \\in G: g \\circ N \\circ g^{-1} = N$ :$\\forall g \\in G: g^{-1} \\circ N \\circ g = N$"}
+{"_id": "5791", "title": "Subgroup is Normal iff Contains Conjugate Elements", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $N$ be a subgroup of $G$. Then $N$ is normal in $G$ {{iff}}: :$\\forall g \\in G: \\paren {n \\in N \\iff g \\circ n \\circ g^{-1} \\in N}$ :$\\forall g \\in G: \\paren {n \\in N \\iff g^{-1} \\circ n \\circ g \\in N}$"}
+{"_id": "5792", "title": "Subgroup is Normal iff Left Cosets are Right Cosets", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $N$ be a subgroup of $G$. Then $N$ is normal in $G$ (by definition 1) {{iff}}: : Every right coset of $N$ in $G$ is a left coset or equivalently: : The right coset space of $N$ in $G$ equals its left coset space."}
+{"_id": "5793", "title": "Symmetric Group has Non-Normal Subgroup", "text": "Let $S_n$ be the (full) symmetric group on $n$ elements, where $n \\ge 3$. Then $S_n$ contains at least one subgroup which is not normal."}
+{"_id": "5797", "title": "Inclusion Mapping on Subgroup is Monomorphism", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\struct {H, \\circ {\\restriction_H} }$ be a subgroup of $G$. Let $i: H \\to G$ be the inclusion mapping from $H$ to $G$. Then $i$ is a group monomorphism."}
+{"_id": "5798", "title": "Inverse of Group Isomorphism is Isomorphism", "text": "Let $\\struct {G, \\circ}$ and $\\struct {H, *}$ be groups. Let $\\phi: \\struct {G, \\circ} \\to \\struct {H, *}$ be a mapping. Then $\\phi$ is an isomorphism {{iff}} $\\phi^{-1}: \\struct {H, *} \\to \\struct {G, \\circ}$ is also an isomorphism."}
+{"_id": "5804", "title": "Composite of Group Homomorphisms is Homomorphism", "text": "Let: :$\\struct {G_1, \\circ}$ :$\\struct {G_2, *}$ :$\\struct {G_3, \\oplus}$ be groups. Let: :$\\phi: \\struct {G_1, \\circ} \\to \\struct {G_2, *}$ :$\\psi: \\struct {G_2, *} \\to \\struct {G_3, \\oplus}$ be homomorphisms. Then the composite of $\\phi$ and $\\psi$ is also a homomorphism."}
+{"_id": "5805", "title": "Composite of Group Monomorphisms is Monomorphism", "text": "Let: :$\\struct {G_1, \\circ}$ :$\\struct {G_2, *}$ :$\\struct {G_3, \\oplus}$ be groups. Let: :$\\phi: \\struct {G_1, \\circ} \\to \\struct {G_2, *}$ :$\\psi: \\struct {G_2, *} \\to \\struct {G_3, \\oplus}$ be monomorphisms. Then the composite of $\\phi$ and $\\psi$ is also a monomorphism."}
+{"_id": "5806", "title": "Composite of Group Epimorphisms is Epimorphism", "text": "Let: :$\\struct {G_1, \\circ}$ :$\\struct {G_2, *}$ :$\\struct {G_3, \\oplus}$ be groups. Let: :$\\phi: \\struct {G_1, \\circ} \\to \\struct {G_2, *}$ :$\\psi: \\struct {G_2, *} \\to \\struct {G_3, \\oplus}$ be (group) epimorphisms. Then the composite of $\\phi$ and $\\psi$ is also a (group) epimorphism."}
+{"_id": "5807", "title": "Composite of Group Isomorphisms is Isomorphism", "text": "Let $\\struct {G_1, \\circ}$, $\\struct {G_2, *}$ and $\\struct {G_3, \\oplus}$ be groups. Let $\\phi: \\struct {G_1, \\circ} \\to \\struct {G_2, *}$ and $\\psi: \\struct {G_2, *} \\to \\struct {G_3, \\oplus}$ be group isomorphisms. Then the composite of $\\psi$ with $\\phi$ is also a group isomorphism."}
+{"_id": "5808", "title": "Sum of Deviations from Mean", "text": "Let $x_1, x_2, \\ldots, x_n$ be real data about some quantitative variable. Let $\\overline x$ be the arithmetic mean of the above data. Then:  :$\\displaystyle \\sum_{i \\mathop = 1}^n \\left({x_i - \\overline x}\\right) = 0$"}
+{"_id": "5809", "title": "Third Isomorphism Theorem/Groups/Corollary 2", "text": "Let $G$ and $H$ be groups whose identities are $e_G$ and $e_H$ respectively. Let $\\phi: G \\to H$ be a group homomorphism. Let $K$ be the kernel of $\\phi$. Let $N$ be a normal subgroup of $G$. Let $q: G \\to \\dfrac G N$ be the quotient epimorphism from $G$ to the quotient group $\\dfrac G N$. Then $N \\subseteq K$ {{iff}} there exists a group homomorphism $\\psi: \\dfrac G N \\to H$ such that: :$\\phi = \\psi \\circ q$ and: :$\\dfrac G K \\cong \\dfrac {G / N} {K / N}$"}
+{"_id": "5812", "title": "Identity of Group Direct Product", "text": "Let $\\struct {G \\times H, \\circ}$ be the group direct product of the two groups $\\struct {G, \\circ_1}$ and $\\struct {H, \\circ_2}$. Let $e_G$ be the identity for $\\struct {G, \\circ_1}$. Let $e_H$ be the identity for $\\struct {H, \\circ_2}$. Then $\\tuple {e_G, e_H}$ is the identity for $\\struct {G \\times H, \\circ}$."}
+{"_id": "5815", "title": "Inverses in Group Direct Product", "text": "Let $\\struct {G \\times H, \\circ}$ be the group direct product of the two groups $\\struct {G, \\circ_1}$ and $\\struct {H, \\circ_2}$. Let $g^{-1}$ be an inverse of $g \\in \\struct {G, \\circ_1}$. Let $h^{-1}$ be an inverse of $h \\in \\struct {H, \\circ_2}$. Then $\\tuple {g^{-1}, h^{-1} }$ is the inverse of $\\tuple {g, h} \\in \\struct {G \\times H, \\circ}$."}
+{"_id": "5818", "title": "Associativity of Operation in Group Direct Product", "text": "Let $\\struct {G \\times H, \\circ}$ be the group direct product of the two groups $\\struct {G, \\circ_1}$ and $\\struct {H, \\circ_2}$. Then the operation $\\circ$ in $\\struct {G \\times T, \\circ}$ is associative."}
+{"_id": "5819", "title": "Anticommutativity of External Direct Product", "text": "Let $\\left({S, \\circ_1}\\right)$ and $\\left({T, \\circ_2}\\right)$ be algebraic structures where $S$ and $T$ both have at least two distinct elements. Let $\\left({S \\times T, \\circ}\\right)$ be their external direct product. Then $\\left({S \\times T, \\circ}\\right)$ is anticommutative iff at least one of $\\left({S, \\circ_1}\\right)$ and $\\left({T, \\circ_2}\\right)$ is."}
+{"_id": "5822", "title": "Projection on Group Direct Product is Epimorphism", "text": "Let $\\struct {G_1, \\circ_1}$ and $\\struct {G_2, \\circ_2}$ be groups. Let $\\struct {G, \\circ}$ be the group direct product of $\\struct {G_1, \\circ_1}$ and $\\struct {G_2, \\circ_2}$. Then: :$\\pr_1$ is an epimorphism from $\\struct {G, \\circ}$ to $\\struct {G_1, \\circ_1}$ :$\\pr_2$ is an epimorphism from $\\struct {G, \\circ}$ to $\\struct {G_2, \\circ_2}$ where $\\pr_1$ and $\\pr_2$ are the first and second projection respectively of $\\struct {G, \\circ}$."}
+{"_id": "5823", "title": "Canonical Injection on Group Direct Product is Monomorphism", "text": "Let $\\struct {G_1, \\circ_1}$ and $\\struct {G_2, \\circ_2}$ be groups with identities $e_1, e_2$ respectively. Let $\\struct {G_1 \\times G_2, \\circ}$ be the group direct product of $\\struct {G_1, \\circ_1}$ and $\\struct {G_2, \\circ_2}$ Then the canonical injections: :$\\inj_1: \\struct {G_1, \\circ_1} \\to \\struct {G_1, \\circ_1} \\times \\struct {G_2, \\circ_2}: \\forall x \\in G_1: \\map {\\inj_1} x = \\tuple {x, e_2}$ :$\\inj_2: \\struct {G_2, \\circ_2} \\to \\struct {G_1, \\circ_1} \\times \\struct {G_2, \\circ_2}: \\forall x \\in G_2: \\map {\\inj_2} x = \\tuple {e_1, x}$ are group monomorphisms."}
+{"_id": "5826", "title": "Canonical Injection is Right Inverse of Projection", "text": "Let $\\struct {G_1, \\circ_1}$ and $\\struct {G_2, \\circ_2}$ be groups with identity elements $e_1$ and $e_2$ respectively. Let $\\struct {G_1 \\times G_2, \\circ}$ be the group direct product of $\\struct {G_1, \\circ_1}$ and $\\struct {G_2, \\circ_2}$ Let: :$\\pr_1: \\struct {G_1 \\times G_2, \\circ} \\to \\struct {G_1, \\circ_1}$ be the first projection from $\\struct {G_1 \\times G_2, \\circ}$ to $\\struct {G_1, \\circ_1}$ :$\\pr_2: \\struct {G_1 \\times G_2, \\circ} \\to \\struct {G_2, \\circ_2}$ be the second projection from $\\struct {G_1 \\times G_2, \\circ}$ to $\\struct {G_2, \\circ_2}$. Let: :$\\inj_1: \\struct {G_1, \\circ_1} \\to \\struct {G_1 \\times G_2, \\circ}$ be the canonical injection from $\\struct {G_1, \\circ_1}$ to $\\struct {G_1 \\times G_2, \\circ}$ :$\\inj_2: \\struct {G_2, \\circ_2} \\to \\struct {G_1 \\times G_2, \\circ}$ be the canonical injection from $\\struct {G_2, \\circ_2}$ to $\\struct {G_1 \\times G_2, \\circ}$. Then: :$(1): \\quad \\pr_1 \\circ \\inj_1 = I_{G_1}$ :$(2): \\quad \\pr_2 \\circ \\inj_2 = I_{G_2}$ where $I_{G_1}$ and $I_{G_2}$ are the identity mappings on $G_1$ and $G_2$ respectively."}
+{"_id": "5827", "title": "Image of Canonical Injection is Kernel of Projection", "text": "Let $\\struct {G_1, \\circ_1}$ and $\\struct {G_2, \\circ_2}$ be groups with identity elements $e_1$ and $e_2$ respectively. Let $\\struct {G_1 \\times G_2, \\circ}$ be the group direct product of $\\struct {G_1, \\circ_1}$ and $\\struct {G_2, \\circ_2}$ Let: :$\\pr_1: \\struct {G_1 \\times G_2, \\circ} \\to \\struct {G_1, \\circ_1}$ be the first projection from $\\struct {G_1 \\times G_2, \\circ}$ to $\\struct {G_1, \\circ_1}$ :$\\pr_2: \\struct {G_1 \\times G_2, \\circ} \\to \\struct {G_2, \\circ_2}$ be the second projection from $\\struct {G_1 \\times G_2, \\circ}$ to $\\struct {G_2, \\circ_2}$. Let: :$\\inj_1: \\struct {G_1, \\circ_1} \\to \\struct {G_1 \\times G_2, \\circ}$ be the canonical injection from $\\struct {G_1, \\circ_1}$ to $\\struct {G_1 \\times G_2, \\circ}$ :$\\inj_2: \\struct {G_2, \\circ_2} \\to \\struct {G_1 \\times G_2, \\circ}$ be the canonical injection from $\\struct {G_2, \\circ_2}$ to $\\struct {G_1 \\times G_2, \\circ}$. Then: :$(1): \\quad \\Img {\\inj_1} = \\map \\ker {\\pr_2}$ :$(2): \\quad \\Img {\\inj_2} = \\map \\ker {\\pr_1}$ That is: :the image of the (first) canonical injection is the kernel of the second projection :the image of the (second) canonical injection is the kernel of the first projection."}
+{"_id": "5829", "title": "Definition:Direct Product of Group Homomorphisms", "text": "Let $G, H_1$ and $H_2$ be groups. Let $f_1: G \\to H_1$ and $f_2: G \\to H_2$ be group homomorphisms. Then $f_1 \\times f_2: G \\to H_1 \\times H_2$, defined as: :$\\forall g \\in G: \\map{\\paren {f_1 \\times f_2} } g = \\tuple {\\map {f_1} g, \\map {f_2} g}$ is called the '''direct product of $f_1$ and $f_2$'''."}
+{"_id": "5830", "title": "Direct Product of Group Homomorphisms is Homomorphism", "text": "Let $\\struct {G, \\circ}, \\struct {H_1, *_1}$ and $\\struct {H_2, *_2}$ be groups. Let $\\struct {H_1 \\times H_2, *}$ be the group direct product of $H_1$ and $H_2$. Let $f_1: G \\to H_1$ and $f_2: G \\to H_2$ be group homomorphisms. Let $f_1 \\times f_2: g \\to H_1 \\times H_2$ be the direct product of $f_1$ and $f_2$. Then $f_1 \\times f_2$ is a group homomorphism."}
+{"_id": "5831", "title": "Projections on Direct Product of Group Homomorphisms", "text": "Let $G, H_1$ and $H_2$ be groups. Let $H_1 \\times H_2$ be the group direct product of $H_1$ and $H_2$. Let $f_1: G \\to H_1$ and $f_2: G \\to H_2$ be group homomorphisms. Let $f_1 \\times f_2: g \\to H_1 \\times H_2$ be the direct product of $f_1$ and $f_2$. Let: :$\\pr_1: H_1 \\times H_2 \\to H_1$ be the first projection from $H_1 \\times H_2$ to $H_1$ :$\\pr_2: H_1 \\times H_2 \\to H_2$ be the second projection from $H_1 \\times H_2$ to $H_2$. Then: :$(1) \\quad \\pr_1 \\circ \\paren {f_1 \\times f_2} = f_1$ :$(2) \\quad \\pr_2 \\circ \\paren {f_1 \\times f_2} = f_2$ where $\\circ$ is the operation of composition of mappings."}
+{"_id": "5844", "title": "Pointwise Addition is Associative", "text": "Let $S$ be a non-empty set. Let $\\mathbb F$ be one of the standard number sets: $\\Z, \\Q, \\R$ or $\\C$. Let $f, g, h: S \\to \\mathbb F$ be functions. Let $f + g: S \\to \\mathbb F$ denote the pointwise sum of $f$ and $g$. Then: :$\\paren {f + g} + h = f + \\paren {g + h}$ That is, pointwise addition is associative."}
+{"_id": "5845", "title": "Pointwise Addition on Complex-Valued Functions is Associative", "text": "Let $f, g, h: S \\to \\C$ be complex-valued functions. Let $f + g: S \\to \\C$ denote the pointwise sum of $f$ and $g$. Then: :$\\paren {f + g} + h = f + \\paren {g + h}$"}
+{"_id": "5846", "title": "Pointwise Addition is Commutative", "text": "Let $S$ be a non-empty set. Let $\\mathbb F$ be one of the standard number sets: $\\Z, \\Q, \\R$ or $\\C$. Let $f, g: S \\to \\mathbb F$ be functions. Let $f + g: S \\to \\mathbb F$ denote the pointwise sum of $f$ and $g$. Then: :$f + g = g + f$ </onlyinclude> That is, pointwise addition is commutative."}
+{"_id": "5847", "title": "Pointwise Multiplication is Associative", "text": "Let $S$ be a non-empty set. Let $\\mathbb F$ be one of the standard number sets: $\\Z, \\Q, \\R$ or $\\C$. Let $f, g, h: S \\to \\mathbb F$ be functions. Let $f \\times g: S \\to \\mathbb F$ denote the pointwise product of $f$ and $g$. Then: :$\\paren {f \\times g} \\times h = f \\times \\paren {g \\times h}$ That is, pointwise multiplication is associative."}
+{"_id": "5848", "title": "Pointwise Multiplication is Commutative", "text": "Let $S$ be a non-empty set. Let $\\mathbb F$ be one of the standard number sets: $\\Z, \\Q, \\R$ or $\\C$. Let $f, g, h: S \\to \\mathbb F$ be functions. Let $f \\times g: S \\to \\mathbb F$ denote the pointwise product of $f$ and $g$. Then: :$f \\times g = g \\times f$ That is, pointwise multiplication is commutative."}
+{"_id": "5853", "title": "Pointwise Multiplication on Real-Valued Functions is Associative", "text": "Let $f, g, h: S \\to \\R$ be real-valued functions. Let $f \\times g: S \\to \\R$ denote the pointwise product of $f$ and $g$. Then: :$\\left({f \\times g}\\right) \\times h = f \\times \\left({g \\times h}\\right)$"}
+{"_id": "5854", "title": "Pointwise Multiplication on Rational-Valued Functions is Associative", "text": "Let $f, g, h: S \\to \\Q$ be rational-valued functions. Let $f \\times g: S \\to \\Q$ denote the pointwise product of $f$ and $g$. Then: :$\\paren {f \\times g} \\times h = f \\times \\paren {g \\times h}$"}
+{"_id": "5855", "title": "Set Equation: Union", "text": "Let $A$ and $B$ be sets. Consider the set equation: :$A \\cup X = B$ The solution set of this is: :$\\varnothing$ if $A \\nsubseteq B$ :$\\left\\{ {\\left({B \\setminus A}\\right) \\cup Y: Y \\subseteq A}\\right\\}$ otherwise."}
+{"_id": "5856", "title": "Set Equation: Intersection", "text": "Let $A$ and $B$ be sets. Consider the set equation: :$A \\cap X = B$ The solution set of this is: :$\\O$ if $B \\nsubseteq A$ :$\\set {B \\cup Y: A \\nsubseteq Y}$ otherwise."}
+{"_id": "5859", "title": "Non-Abelian Group has Order Greater than 4", "text": "Let $\\struct {G, \\circ}$ be a non-abelian group. Then the order of $\\struct {G, \\circ}$ is greater than $4$."}
+{"_id": "5860", "title": "Bayes' Theorem/General Result", "text": "Let $\\set {B_1, B_2, \\ldots}$ be a  partition of the event space $\\Sigma$. Then, for any $B_i$ in the partition: :$\\map \\Pr {B_i \\mid A} = \\dfrac {\\map \\Pr {A \\mid B_i} \\map \\Pr {B_i} } {\\map \\Pr A} = \\dfrac {\\map \\Pr {A \\mid B_i} \\map \\Pr {B_i} } {\\sum_j \\map \\Pr {A \\mid B_j} \\map \\Pr {B_j} }$ where $\\ds \\sum_j$ denotes the sum over $j$."}
+{"_id": "5861", "title": "Topological Sum is Coproduct in Category of Topological Spaces", "text": "Let $\\mathbf{Top}$ be the category of topological spaces. Let $X$ and $Y$ be topological spaces, and let $X \\sqcup Y$ be their topological sum. Then $X \\sqcup Y$ is the coproduct of $X$ and $Y$ in $\\mathbf{Top}$."}
+{"_id": "5862", "title": "Coproduct of Ordered Sets", "text": "Let $\\mathbf{OrdSet}$ be the category of ordered sets. Let $\\left({P, \\preceq_1}\\right)$ and $\\left({Q, \\preceq_2}\\right)$ be ordered sets. Let $P \\sqcup Q$ be the disjoint union of $P$ and $Q$. Let $\\preceq$ be the ordering on $P \\sqcup Q$ defined by: :$\\left({x, i}\\right) \\preceq \\left({y, j}\\right)$ {{iff}} $i = j$ and $x \\preceq_i y$ where $i = 1$ or $i = 2$ depending on whether $x,y \\in P$ or $x,y \\in Q$. Then $\\left({P \\sqcup Q, \\preceq}\\right)$ is the coproduct of $P$ and $Q$ in $\\mathbf{OrdSet}$."}
+{"_id": "5863", "title": "Supremum is Coproduct in Order Category", "text": "Let $\\mathbf P$ be an order category with ordering $\\preceq$. Let $p, q \\in P_0$, and suppose they have some supremum $r = \\sup \\left\\{{p, q}\\right\\}$. Then $r$ is the coproduct of $p$ and $q$ in $\\mathbf P$."}
+{"_id": "5865", "title": "Real Function is Continuous at Isolated Point", "text": "Let $A \\subseteq \\R$ be any subset of the real numbers. Let $f: A \\to \\R$ be a real function. Let $x \\in A$ be an isolated point of $A$. Then $f$ is continuous at $x$."}
+{"_id": "5866", "title": "Metric Space Continuity by Epsilon-Delta", "text": "Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \\to A_2$ be a mapping from $A_1$ to $A_2$. Let $a \\in A_1$ be a point in $A_1$. Then the following definitions of continuity of $f$ at $a$ with respect to $d_1$ and $d_2$ are equivalent:"}
+{"_id": "5867", "title": "Equalizer is Monomorphism", "text": "Let $\\mathbf C$ be a metacategory. Let $e: E \\to C$ be the equalizer of two morphisms $f, g: C \\to D$. Then $e$ is a monomorphism."}
+{"_id": "5868", "title": "Metric Space Continuity by Open Ball", "text": "Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \\to A_2$ be a mapping from $A_1$ to $A_2$. Let $a \\in A_1$ be a point in $A_1$. Then the following definitions of continuity of $f$ at $a$ with respect to $d_1$ and $d_2$ are equivalent:"}
+{"_id": "5869", "title": "Mapping from Standard Discrete Metric on Real Number Line is Continuous", "text": "Let $\\R$ be the real number line. Let $\\left({\\R, d_1}\\right)$ be the metric space such that $d_1$ be the Euclidean metric on $\\R$. Let $\\left({\\R, d_2}\\right)$ be the metric space such that $d_2$ be the standard discrete metric on $\\R$. Let $f: \\left({\\R, d_2}\\right) \\to \\left({\\R, d_1}\\right)$ be a real function. Then $f$ is $\\left({d_2, d_1}\\right)$-continuous on $\\R$."}
+{"_id": "5870", "title": "Properties of Affine Spaces", "text": "Let $\\mathcal E$ be an affine space with difference space $V$. Let $0$ denote the zero element of $V$. Then the following hold for all $p,q,r \\in \\mathcal E$ and all $u,v \\in V$: : $(1): \\quad p - p = 0$ : $(2): \\quad p + 0 = p$ : $(3): \\quad p + u = p + v \\iff u = v$ : $(4): \\quad q - p = r - p \\iff q = r$"}
+{"_id": "5871", "title": "P-Product Metric on Real Vector Space is Metric/Proof 1", "text": "Let $\\R^n$ be an $n$-dimensional real vector space. Let $p \\in \\R_{\\ge 1}$. Let $d_p: \\R^n \\times \\R^n \\to \\R$ be the $p$-product metric on $\\R^n$: : $\\displaystyle d_p \\left({x, y}\\right) :=  \\left({\\sum_{i \\mathop = 1}^n \\left \\vert {x_i - y_i} \\right \\vert^p}\\right)^{\\frac 1 p}$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({y_1, y_2, \\ldots, y_n}\\right) \\in \\R^n$. Then $d_p$ is a metric."}
+{"_id": "5872", "title": "P-Product Metric on Real Vector Space is Metric/Proof 2", "text": "Let $\\R^n$ be an $n$-dimensional real vector space. Let $p \\in \\R_{\\ge 1}$. Let $d_p: \\R^n \\times \\R^n \\to \\R$ be the $p$-product metric on $\\R^n$: : $\\displaystyle d_p \\left({x, y}\\right) :=  \\left({\\sum_{i \\mathop = 1}^n \\left \\vert {x_i - y_i} \\right \\vert^p}\\right)^{\\frac 1 p}$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({y_1, y_2, \\ldots, y_n}\\right) \\in \\R^n$. Then $d_p$ is a metric."}
+{"_id": "5873", "title": "Surjection from Natural Numbers iff Right Inverse", "text": "Let $S$ be a set. Let $f: \\N \\to S$ be a mapping, where $\\N$ denotes the set of natural numbers. Then $f$ is a surjection {{iff}} $f$ admits a right inverse."}
+{"_id": "5874", "title": "Sum of Squared Deviations from Mean", "text": "Let $x_1, x_2, \\ldots, x_n$ be real data about some quantitative variable. Let $\\overline x$ be the arithmetic mean of the above data. Then:  :$\\displaystyle \\sum_{i \\mathop = 1}^n \\paren {x_i - \\overline x}^2 = \\sum_{i \\mathop = 1}^n \\paren {x_i^2 - \\overline x^2}$"}
+{"_id": "5875", "title": "Surjection from Natural Numbers iff Countable", "text": "Let $S$ be a non-empty set. Then $S$ is countable {{iff}} there exists a surjection $f: \\N \\to S$."}
+{"_id": "5876", "title": "Summation is Linear", "text": "Let $\\left({x_1, \\ldots, x_n}\\right)$ and $\\left({y_1, \\ldots, y_n}\\right)$ be finite sequences of numbers of equal length. Let $\\lambda$ be a number. Then:"}
+{"_id": "5877", "title": "Sum of Identical Terms", "text": "Let $x$ be a number, and let $n \\in \\N$ be a natural number. Suppose that $n \\ge 1$. Then: :$\\displaystyle \\sum_{i \\mathop = 1}^n x = n x$"}
+{"_id": "5880", "title": "Synthetic Sub-Basis and Analytic Sub-Basis are Compatible", "text": "Let $\\left({X, \\tau}\\right)$ be a topological space. Let $\\mathcal S \\subseteq \\mathcal P \\left({X}\\right)$, where $\\mathcal P \\left({X}\\right)$ denotes the power set of $X$. Then $\\mathcal S$ is an analytic sub-basis for $\\tau$ {{iff}} $\\tau$ is the topology on $X$ generated by the synthetic sub-basis $\\mathcal S$."}
+{"_id": "5881", "title": "Analytic Basis is Analytic Sub-Basis", "text": "Let $\\left({X, \\tau}\\right)$ be a topological space. Let $\\mathcal B \\subseteq \\tau$ be an analytic basis for $\\tau$. Then $\\mathcal B$ is an analytic sub-basis for $\\tau$."}
+{"_id": "5882", "title": "Continuity Test using Sub-Basis", "text": "Let $\\struct {X_1, \\tau_1}$ and $\\struct {X_2, \\tau_2}$ be topological spaces. Let $f: X_1 \\to X_2$ be a mapping. Let $\\SS$ be an analytic sub-basis for $\\tau_2$. Suppose that: :$\\forall S \\in \\SS: f^{-1} \\sqbrk S \\in \\tau_1$ where $f^{-1} \\sqbrk S$ denotes the preimage of $S$ under $f$. Then $f$ is continuous."}
+{"_id": "5883", "title": "Lagrange Interpolation Formula", "text": "Let $\\tuple {x_0, \\ldots, x_n}$ and $\\tuple {a_0, \\ldots, a_n}$ be ordered tuples of real numbers such that $x_i \\ne x_j$ for $i \\ne j$. Then there exists a unique polynomial $P \\in \\R \\sqbrk X$ of degree at most $n$ such that: :$\\map P {x_i} = a_i$ for all $i \\in \\set {0, 1, \\ldots, n}$ Moreover $P$ is given by the formula: :$\\displaystyle \\map P X = \\sum_{j \\mathop = 0}^n a_i \\map {L_j} X$ where $\\map {L_j} X$ is the $j$th Lagrange basis polynomial associated to the $x_i$."}
+{"_id": "5884", "title": "Product Formula for Sine", "text": ":$\\displaystyle \\map \\sin {n z} = 2^{n - 1} \\prod_{k \\mathop = 0}^{n - 1} \\map \\sin {z + \\frac{k \\pi} n}$"}
+{"_id": "5885", "title": "Final Topology is Topology", "text": "Let $X$ be a set. Let $I$ be an indexing set. Let $\\family {\\struct {Y_i, \\tau_i} }_{i \\mathop \\in I}$ be an $I$-indexed family of topological spaces. Let $\\family {f_i: Y_i \\to X}_{i \\mathop \\in I}$ be an $I$-indexed family of mappings. Let $\\tau$ be the final topology on $X$ with respect to $\\family {f_i}_{i \\mathop \\in I}$. Then $\\tau$ is a topology on $X$."}
+{"_id": "5888", "title": "Subspace Topology is Initial Topology with respect to Inclusion Mapping", "text": "Let $\\struct {X, \\tau}$ be a topological space. Let $Y$ be a non-empty subset of $X$. Let $\\iota: Y \\to X$ be the inclusion mapping. Let $\\tau_Y$ be the initial topology on $Y$ with respect to $\\iota$. Then $\\struct {Y, \\tau_Y}$ is a topological subspace of $\\struct {X, \\tau}$. That is: :$\\tau_Y = \\set {U \\cap Y: U \\in \\tau}$"}
+{"_id": "5889", "title": "Topological Subspace is Topological Space", "text": "Let $\\left({X, \\tau}\\right)$ be a topological space. Let $H \\subseteq X$ be a non-empty subset of $X$. Let $\\tau_H = \\left\\{{U \\cap H: U \\in \\tau}\\right\\}$ be the subspace topology on $H$. Then the topological subspace $\\left({H, \\tau_H}\\right)$ is a topological space."}
+{"_id": "5891", "title": "Initial Topology with respect to Mapping equals Set of Preimages", "text": "Let $X$ be a set. Let $\\struct {Y, \\tau_Y}$ be a topological space. Let $f: X \\to Y$ be a mapping. Let $\\tau_X$ be the initial topology on $X$ with respect to $f$. Then: :$\\tau_X = \\set {f^{-1} \\sqbrk U: U \\in \\tau_Y}$"}
+{"_id": "5900", "title": "Sommerfeld-Watson Transform", "text": "Let $f \\left({z}\\right)$ be a mapping with isolated poles. {{explain|Explain the context in which this theorem is placed. For example, what is the domain and range of $f$? One supposes $\\C$ but it needs to be made clear.}} Let $f$ go to zero faster than $\\dfrac 1 {\\left|{z}\\right|}$ as $\\left|{z}\\right| \\to \\infty$. {{explain|\"faster\"}} Let $C$ be a contour that is deformed such that all poles of $f \\left({z}\\right)$ are contained in $C$. Then: :$\\displaystyle \\sum \\limits_{n \\mathop = -\\infty}^\\infty \\left({-1}\\right)^n f \\left({n}\\right) = \\frac 1 {2 i} \\oint_C \\frac {f \\left({z}\\right)} {\\sin \\pi z} \\, \\mathrm d z$"}
+{"_id": "5901", "title": "Linear Transformation of Arithmetic Mean", "text": "Let $D = \\set {x_0, x_1, x_2, \\ldots, x_n}$ be a set of real data describing a quantitative variable. Let $\\overline x$ be the arithmetic mean of the data in $D$. Let $T: \\R \\to \\R$ be a linear transformation such that: :$\\forall i \\in \\set {0, 1, \\ldots, n}: \\map T {x_i} = \\lambda x_i + \\gamma$ Let $T \\sqbrk D$ be the image of $D$ under $T$. Then the arithmetic mean of the data in $T \\sqbrk D$ is given by: :$\\map T {\\overline x} = \\lambda \\overline x + \\gamma$"}
+{"_id": "5905", "title": "Category of Pointed Sets is Category", "text": "Let $\\mathbf{Set}_*$ be the category of pointed sets. Then $\\mathbf{Set}_*$ is a metacategory."}
+{"_id": "5906", "title": "Closed Set in Topological Subspace/Corollary", "text": "Let subspace $T'$ be closed in $T$. Then $V \\subseteq T'$ is closed in $T'$ {{iff}} $V$ is closed in $T$."}
+{"_id": "5907", "title": "Unit Interval is Path-Connected in Real Numbers", "text": "Let $\\R$ be the real number line with the usual (Euclidean} metric. The closed unit interval $\\mathbf I = \\closedint 0 1$ is a path-connected metric subspace of $\\R$."}
+{"_id": "5908", "title": "Subset of Real Numbers is Path-Connected iff Interval", "text": "Let $\\R$ be the real number line considered as an Euclidean space. Let $S \\subseteq \\R$ be a subset of $\\R$. Then $S$ is a path-connected metric subspace of $\\R$ {{iff}} $S$ is a real interval."}
+{"_id": "5909", "title": "Size of Linearly Independent Subset is at Most Size of Finite Generator", "text": "Let $R$ be a division ring. Let $V$ be an $R$-vector space. Let $F \\subseteq V$ be a finite generator of $V$ over $R$. Let $L \\subseteq V$ be linearly independent over $R$. Then: :$\\size L \\le \\size F$"}
+{"_id": "5910", "title": "Intermediate Value Theorem/Corollary", "text": "Let $0 \\in \\R$ lie between $f \\left({a}\\right)$ and $f \\left({b}\\right)$. That is, either: : $f \\left({a}\\right) < 0 < f \\left({b}\\right)$ or: : $f \\left({b}\\right) < 0 < f \\left({a}\\right)$ Then $f$ has a root in $\\left({a \\,.\\,.\\, b}\\right)$."}
+{"_id": "5914", "title": "Euclidean Space is Path-Connected", "text": "Let $\\R^n$ be the $n$-dimensional Euclidean space for $n \\in \\N$ a natural number. Then $\\R^n$ is path-connected."}
+{"_id": "5915", "title": "Coequalizer is Epimorphism", "text": "Let $\\mathbf C$ be a metacategory. Let $q: D \\to Q$ be the coequalizer of two morphisms $f, g: C \\to D$. Then $q$ is an epimorphism."}
+{"_id": "5919", "title": "Euclidean Space without Origin is Path-Connected", "text": "Let $n \\in \\Z: n \\ge 2$. Let $\\R^n$ be the $n$-dimensional Euclidean space. Let $\\R^n \\setminus \\set {\\mathbf 0}$ be $\\R^n$ with the origin removed. Then $\\R^n \\setminus \\set {\\mathbf 0}$ is path-connected."}
+{"_id": "5921", "title": "Composite of Continuous Mappings is Continuous/Corollary", "text": "Let $T_1, T_2, T_3$ each be one of: :metric spaces :the complex plane :the real number line Let $f: T_1 \\to T_2$ and $g: T_2 \\to T_3$ be continuous mappings. Then the composite mapping $g \\circ f: T_1 \\to T_3$ is continuous."}
+{"_id": "5922", "title": "Binomial Theorem/Multiindex", "text": "Let $\\alpha$ be a multiindex, indexed by $\\left\\{{1, \\ldots, n}\\right\\}$ such that $\\alpha_j \\ge 0$ for $j = 1, \\ldots, n$. Let $x = \\left({x_1, \\ldots, x_n}\\right)$ and $y = \\left({y_1, \\ldots, y_n}\\right)$ be ordered tuples of real numbers. Then: :$\\displaystyle \\left({x + y}\\right)^\\alpha = \\sum_{0 \\mathop \\le \\beta \\mathop \\le \\alpha} {\\alpha \\choose \\beta} x^\\beta y^{\\alpha - \\beta}$ where $\\displaystyle {n \\choose k}$ is a binomial coefficient."}
+{"_id": "5924", "title": "Leibniz's Rule/One Variable", "text": "Let $f$ and $g$ be real functions defined on the open interval $I$. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $x \\in I$ be a point in $I$ at which both $f$ and $g$ are $n$ times differentiable. Then: :$\\displaystyle \\paren {\\map f x \\, \\map g x}^{\\paren n} = \\sum_{k \\mathop = 0}^n \\binom n k \\map {f^{\\paren k} } x \\, \\map {g^{\\paren {n - k} } } x$ where $\\paren n$ denotes the order of the derivative."}
+{"_id": "5928", "title": "Open Sets of Double Pointed Topology", "text": "Let $\\struct {S, \\tau_S}$ be a topological space. Let $D$ be a doubleton endowed with the indiscrete topology. Let $\\struct {S \\times D, \\tau}$ be the double pointed topology on $S$. Then $X \\subseteq S \\times D$ is open in $\\tau$ {{iff}} for some $U \\in \\tau$: :$X = U \\times D$"}
+{"_id": "5929", "title": "Open Sets of Double Pointed Topology/Corollary", "text": "A subset $X \\subseteq S \\times D$ is closed in $\\tau$ {{iff}} for some closed set $C$ of $\\tau$: :$X = C \\times D$"}
+{"_id": "5931", "title": "Open Set Less One Point is Open/Corollary", "text": "Let $S = \\left\\{{\\alpha_1, \\alpha_2, \\ldots, \\alpha_n}\\right\\} \\subseteq U$ be a finite set of points in $U$. Then $U \\setminus S$ is open in $M$."}
+{"_id": "5932", "title": "Closed Real Interval is Neighborhood Except at Endpoints", "text": "Let $\\R$ be the real number line considered as an Euclidean space. Let $\\left[{a \\,.\\,.\\, b}\\right] \\subset \\R$ be a closed interval of $\\R$. Then $\\left[{a \\,.\\,.\\, b}\\right]$ is a neighborhood of all of its points except $a$ and $b$."}
+{"_id": "5934", "title": "Open Real Interval is Open Set", "text": "Let $\\R$ be the real number line considered as an Euclidean space. Let $\\openint a b \\subset \\R$ be an open interval of $\\R$. Then $\\openint a b$ is an open set of $\\R$."}
+{"_id": "5935", "title": "Closed Real Interval is not Open Set", "text": "Let $\\R$ be the real number line considered as an Euclidean space. Let $\\left[{a \\,.\\,.\\, b}\\right] \\subset \\R$ be a closed interval of $\\R$. Then $\\left[{a \\,.\\,.\\, b}\\right]$ is not an open set of $\\R$."}
+{"_id": "5936", "title": "Open Set may not be Open Ball", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $U \\subseteq M$ be an open set of $M$. Then it is not necessarily the case that $U$ is an open ball of some $x \\in A$."}
+{"_id": "5938", "title": "Closed Real Interval is Closed Set", "text": "Let $\\R$ be the real number line considered as an Euclidean space. Let $\\closedint a b \\subset \\R$ be a closed interval of $\\R$. Then $\\closedint a b$ is a closed set of $\\R$."}
+{"_id": "5939", "title": "Kernel of Linear Transformation is Null Space of Matrix Representation", "text": "Let $V$ and $W$ be finite dimensional vector spaces. Let $\\phi: V \\to W$ be a linear transformation from $V$ to $W$. Let $\\left({e_1, \\ldots, e_n}\\right)$ and $\\left({f_1, \\ldots, f_m}\\right)$ be ordered bases of $V$ and $W$ respectively. Let $A$ be the matrix of $\\phi$ in these bases. Define $f: V \\to \\R^n$ by: :$\\displaystyle \\sum_{i \\mathop = 1}^n a_i e_i \\mapsto \\left({a_1, \\ldots, a_n}\\right)$ and $g : W \\to \\R^m$ by: :$\\displaystyle \\sum_{i \\mathop = 1}^m b_i f_i \\mapsto \\left({b_1, \\ldots, b_m}\\right)$ Let $N \\left({A}\\right) = \\left\\{ {x \\in \\R^n: A x = 0}\\right\\}$ be the null space of $A$. Let $\\ker \\phi = \\left\\{ {x \\in V: \\phi x = 0}\\right\\}$ be the kernel of $\\phi$. Then: :$f \\left[{\\ker \\phi}\\right] = N \\left({A}\\right)$ and :$f^{-1} \\left[{N \\left({A}\\right)}\\right] = \\ker \\phi$ where $f \\left[{X}\\right]$ denotes the image set of a subset $X$ of the domain of $f$."}
+{"_id": "5940", "title": "Half-Open Real Interval is neither Open nor Closed", "text": "Let $\\R$ be the real number line considered as an Euclidean space. Let $\\left[{a \\,.\\,.\\, b}\\right) \\subset \\R$ be a half-open interval of $\\R$. Then $\\left[{a \\,.\\,.\\, b}\\right)$ is neither an open set nor a closed set of $\\R$. Similarly, the half-open interval $\\left({a \\,.\\,.\\, b}\\right] \\subset \\R$ is neither an open set nor a closed set of $\\R$."}
+{"_id": "5941", "title": "Compact Sets of Double Pointed Topology", "text": "Let $\\struct {S, \\tau_S}$ be a topological space. Let $D$ be a doubleton endowed with the indiscrete topology. Let $\\struct {S \\times D, \\tau}$ be the double pointed topology on $S$. Then $X \\subseteq S \\times D$ is compact in $\\tau$ {{iff}} for some compact set $C$ of $\\tau_S$: :$\\map {\\pr_1} X = C$ where $\\pr_1$ denotes the first projection on $S \\times D$."}
+{"_id": "5942", "title": "Closure of Subset of Double Pointed Topological Space", "text": "Let $\\struct {S, \\tau_S}$ be a topological space. Let $D$ be a doubleton endowed with the indiscrete topology. Let $\\struct {S \\times D, \\tau}$ be the double pointed topology on $S$. Let $X \\subseteq S \\times D$ be a subset of $S \\times D$. Then the closure of $X$ in $\\tau$ is: :$\\map \\cl X = \\map \\cl {\\map {\\pr_1} X} \\times D$ where $\\pr_1$ denotes the first projection on $S \\times D$."}
+{"_id": "5943", "title": "Interior of Subset of Double Pointed Topological Space", "text": "Let $\\struct {S, \\tau_S}$ be a topological space. Let $D$ be a doubleton endowed with the indiscrete topology. Let $\\struct {S \\times D, \\tau}$ be the double pointed topology on $S$. Let $X \\subseteq S \\times D$ be a subset of $S \\times D$. Define $A \\subseteq S$ by: :$A := \\set {s \\in S: \\paren {\\forall d \\in D: \\tuple {s, d} \\in X} }$ Then the interior of $X$ in $\\tau$ is: :$X^\\circ = A^\\circ \\times D$"}
+{"_id": "5946", "title": "Projection of Complement Contains Complement of Projection", "text": "Let $S$ and $T$ be non-empty sets. Let $X \\subseteq S \\times T$ be a subset of the Cartesian product $S \\times T$. Denote with $\\operatorname{pr}_1, \\operatorname{pr}_2$ and $\\complement$ the first and second projections, and the complement operation, respectively. Then: {{begin-eqn}} {{eqn|l = \\complement \\left({\\operatorname{pr}_1 \\left({X}\\right)}\\right)       |o = \\subseteq      |r = \\operatorname{pr}_1 \\left({\\complement \\left({X}\\right)}\\right) }} {{eqn|l = \\complement \\left({\\operatorname{pr}_2 \\left({X}\\right)}\\right)       |o = \\subseteq      |r = \\operatorname{pr}_2 \\left({\\complement \\left({X}\\right)}\\right) }} {{end-eqn}}"}
+{"_id": "5947", "title": "Interior of Subset", "text": "Let $\\left({S, \\tau}\\right)$ be a topological space. Let $X$ and $Y$ be subsets of $S$, and suppose that $X \\subseteq Y$. Then: :$X^\\circ \\subseteq Y^\\circ$ where $X^\\circ$ denotes the interior of $X$."}
+{"_id": "5948", "title": "Real Interval is Bounded in Real Numbers", "text": "Let $\\R$ be the real number line considered as an Euclidean space. Let $a, b \\in \\R$. Let $\\mathcal I$ be one of the following real intervals: {{begin-eqn}} {{eqn | l = \\openint a b       | o = :=       | r = \\set {x \\in \\R: a < x < b}       | c = Open Real Interval }} {{eqn | l = \\hointr a b       | o = :=       | r = \\set {x \\in \\R: a \\le x < b}       | c = Half-Open (to the right) Real Interval }} {{eqn | l = \\hointl a b       | o = :=       | r = \\set {x \\in \\R: a < x \\le b}       | c = Half-Open (to the left) Real Interval }} {{eqn | l = \\closedint a b       | o = :=       | r = \\set {x \\in \\R: a \\le x \\le b}       | c = Closed Real Interval }} {{end-eqn}} where $b \\ge a$. Then $\\mathcal I$ is bounded in $\\R$."}
+{"_id": "5949", "title": "Set of Integers is not Bounded", "text": "Let $\\R$ be the real number line considered as an Euclidean space. The set $\\Z$ of integers is not bounded in $\\R$."}
+{"_id": "5950", "title": "Closed Real Interval is Compact", "text": "Let $\\R$ be the real number line considered as an Euclidean space. Let $I = \\closedint a b$ be a closed real interval. Then $I$ is compact."}
+{"_id": "5952", "title": "Open Real Interval is not Closed Set", "text": "Let $\\R$ be the real number line with the usual (Euclidean) metric. Let $I = \\openint a b$ be an open real interval. Then $I$ is not a closed set of $\\R$."}
+{"_id": "5953", "title": "Open Real Interval is not Compact", "text": "Let $\\R$ be the real number line considered as an Euclidean space. Let $I = \\openint a b$ be an open real interval. Then $I$ is not compact."}
+{"_id": "5954", "title": "Set of Integers is not Compact", "text": "Let $\\R$ be the real number line considered as an Euclidean space. Let $\\Z$ be the set of integers. Then $\\Z$ is not compact."}
+{"_id": "5955", "title": "Heine-Borel Theorem/Euclidean Space", "text": "Let $n \\in \\N_{> 0}$. Let $C$ be a subspace of the Euclidean space $\\R^n$. Then $C$ is closed and bounded {{iff}} it is compact."}
+{"_id": "5956", "title": "Continuous Function on Compact Space is Bounded", "text": "Let $\\struct {X, \\tau}$ be a topological space. Let $\\struct {Y, \\norm {\\, \\cdot \\, } }$ be a normed vector space. Let $f: X \\to Y$ be continuous. Then $f$ is bounded."}
+{"_id": "5959", "title": "Hausdorff Space is Hereditarily Compact iff Finite", "text": "Let $\\left({S, \\tau}\\right)$ be a Hausdorff space. Then $\\left({S, \\tau}\\right)$ is hereditarily compact {{iff}} $S$ is finite."}
+{"_id": "5961", "title": "Complete and Totally Bounded Metric Space is Sequentially Compact", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $M$ be complete and totally bounded. Then $M$ is sequentially compact."}
+{"_id": "5966", "title": "Closed Bounded Subset of Real Numbers is Compact", "text": "Let $\\R$ be the real number line considered as an Euclidean space. Let $S \\subseteq \\R$ be a closed and bounded subspace of $\\R$. Then $S$ is compact in $\\R$."}
+{"_id": "5971", "title": "Non-Closed Set of Real Numbers is not Compact", "text": "Let $\\R$ be the set of real numbers considered as an Euclidean space. Let $S \\subseteq \\R$ be non-closed in $\\R$. Then $S$ is not a compact subspace of $\\R$."}
+{"_id": "5972", "title": "Compact Subspace of Real Numbers is Closed and Bounded", "text": "Let $\\R$ be the real number line considered as a Euclidean space. Let $S \\subseteq \\R$ be compact subspace of $\\R$. Then $S$ is closed and bounded in $\\R$."}
+{"_id": "5973", "title": "Category of Subobjects is Category", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ be an object of $\\mathbf C$. Let $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$ be the category of subobjects of $C$. Then $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$ is a metacategory."}
+{"_id": "5974", "title": "Category of Subobjects is Preorder Category", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ be an object of $\\mathbf C$. Let $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$ be the category of subobjects of $C$. Then $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$ is a preorder category."}
+{"_id": "5976", "title": "Integral of Power/Fermat's Proof", "text": ":$\\displaystyle \\forall n \\in \\Q_{>0}: \\int_0^b x^n \\rd x = \\frac {b^{n + 1} } {n + 1}$"}
+{"_id": "5977", "title": "Composition of Mapping with Inclusion is Restriction", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let $A \\subseteq S$ be a subset of the domain of $S$. Let $i_A: A \\to S$ be the inclusion mapping from $A$ to $S$. Then: :$f \\circ i_A = f \\restriction_A$ where $f \\restriction_A$ denotes the restriction of $f$ to $A$."}
+{"_id": "5978", "title": "Equivalence of Subobjects is Equivalence", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ be an object of $\\mathbf C$. Let $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$ be the category of subobjects of $C$. The relation $\\sim$ on $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$ defined by: :$m \\sim m'$ iff $m$ and $m'$ are equivalent is an equivalence. {{improve|That's not a relation since we can't know we're dealing with sets; cover this}}"}
+{"_id": "5982", "title": "Morphism Class Equivalence is Equivalence", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ be an object of $\\mathbf C$. Let $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$ be the category of subobjects of $C$. Let $\\sim$ denote morphism class equivalence on the morphisms of $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$. Then $\\sim$ is an equivalence."}
+{"_id": "5983", "title": "Subobject Class in Category of Sets", "text": "Let $\\mathbf{Set}$ be the category of sets. Let $S$ be a set. Let $\\overline{\\mathbf{Sub}}_{\\mathbf{Set}} \\left({S}\\right)$ be the category of subobject classes of $S$. Let $\\mathcal P \\left({S}\\right)$ be the order category on the power set of $S$ induced by Subset Relation on Power Set is Partial Ordering. Then $\\overline{\\mathbf{Sub}}_{\\mathbf{Set}} \\left({S}\\right) \\cong \\mathcal P \\left({S}\\right)$."}
+{"_id": "5984", "title": "Category of Subobject Classes is Category", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ be an object of $\\mathbf C$. Let $\\map {\\overline {\\mathbf {Sub} }_{\\mathbf C} } C$ be the category of subobject classes of $C$. Then $\\map {\\overline {\\mathbf {Sub} }_{\\mathbf C} } C$ is a metacategory."}
+{"_id": "5985", "title": "Category of Subobject Classes is Order Category", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ be an object of $\\mathbf C$. Let $\\overline{\\mathbf{Sub}}_{\\mathbf C} \\left({C}\\right)$ be the category of subobject classes of $C$. Then $\\overline{\\mathbf{Sub}}_{\\mathbf C} \\left({C}\\right)$ is an order category."}
+{"_id": "5987", "title": "Reverse Triangle Inequality/Real and Complex Fields", "text": "Let $x$ and $y$ be elements of either the real numbers $\\R$ or the complex numbers $\\C$. Then: :$\\cmod {x - y} \\ge \\size {\\cmod x - \\cmod y}$"}
+{"_id": "5990", "title": "Triangle Inequality/Real Numbers", "text": "Let $x, y \\in \\R$ be real numbers. Let $\\size x$ denote the absolute value of $x$. Then: :$\\size {x + y} \\le \\size x + \\size y$"}
+{"_id": "5991", "title": "Triangle Inequality/Complex Numbers", "text": "Let $z_1, z_2 \\in \\C$ be complex numbers. Let $\\cmod z$ denote the modulus of $z$. Then: :$\\cmod {z_1 + z_2} \\le \\cmod {z_1} + \\cmod {z_2}$"}
+{"_id": "5992", "title": "Triangle Inequality/Vectors in Euclidean Space", "text": "Let $\\mathbf x, \\mathbf y$ be vectors in $\\R^n$. Let $\\norm {\\, \\cdot \\,}$ denote vector length. Then: :$\\norm {\\mathbf x + \\mathbf y} \\le \\norm {\\mathbf x} + \\norm {\\mathbf y}$ If the two vectors are scalar multiples where said scalar is non-negative, an equality holds: :$\\exists \\lambda \\in \\R, \\lambda \\ge 0: \\mathbf x = \\lambda \\mathbf y \\iff \\norm {\\mathbf x + \\mathbf y} = \\norm {\\mathbf x} + \\norm {\\mathbf y}$"}
+{"_id": "5996", "title": "Reverse Triangle Inequality/Real and Complex Fields/Corollary/Proof 2", "text": "Let $x$ and $y$ be elements of either the real numbers $\\R$ or the complex numbers $\\C$. Then: :$\\cmod {x - y} \\ge \\cmod x - \\cmod y$ where $\\cmod x$ denotes either the absolute value of a real number or the complex modulus of a complex number."}
+{"_id": "5997", "title": "Negative of Complex Modulus", "text": "Let $z \\in \\C$ be a complex number. Then: :$-\\cmod z \\le \\cmod z$ where $\\cmod z$ denotes the complex modulus of $z$. The equality holds {{iff}} $z = 0$."}
+{"_id": "5999", "title": "Limit of Function by Convergent Sequences/Corollary", "text": "Let $\\openint a b$ be an open real interval. Let $f: \\openint a b \\to \\R$ be a real function. Let $l \\in \\R$. Then: :$(1): \\quad \\displaystyle \\lim_{x \\mathop \\to a^+} \\map f x = l \\iff \\forall \\sequence {x_n} \\subseteq \\openint a b: \\lim_{n \\mathop \\to \\infty} x_n = a \\implies \\lim_{n \\mathop \\to \\infty} \\map f {x_n} = l$ :$(2): \\quad \\displaystyle \\lim_{x \\mathop \\to b^-} \\map f x = l \\iff \\forall \\sequence {x_n} \\subseteq \\openint a b: \\lim_{n \\mathop \\to \\infty} x_n = b \\implies \\lim_{n \\mathop \\to \\infty} \\map f {x_n} = l$ where: :$\\displaystyle \\lim_{x \\mathop \\to a^+} \\map f x$ denotes the limit of $f$ from the right :$\\displaystyle \\lim_{x \\mathop \\to b^-} \\map f x$ denotes the limit of $f$ from the left."}
+{"_id": "6003", "title": "Half-Open Real Interval is not Open Set", "text": "Let $\\R$ be the real number line considered as an Euclidean space. Let $\\hointr a b \\subset \\R$ be a half-open interval of $\\R$. Then $\\hointr a b$ is not an open set of $\\R$. Similarly, the half-open interval $\\hointl a b \\subset \\R$ is not an open set of $\\R$."}
+{"_id": "6004", "title": "Empty Set is Open in Metric Space", "text": "Let $M = \\struct {A, d}$ be a metric space. Then the empty set $\\varnothing$ is an open set of $M$."}
+{"_id": "6005", "title": "Metric Space is Open in Itself", "text": "Let $M = \\struct {A, d}$ be a metric space. Then the set $A$ is an open set of $M$."}
+{"_id": "6006", "title": "Subset of Standard Discrete Metric Space is Open", "text": "Let $M = \\left({A, d}\\right)$ be a standard discrete metric space. Let $S \\subseteq A$ be a subset of $A$. Then $S$ is an open set of $M$."}
+{"_id": "6007", "title": "Image of Open Set under Continuous Mapping in Metric Space may not be Open", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f: A_1 \\to A_2$ be a $\\left({d_1, d_2}\\right)$-continuous mapping from $A_1$ to $A_2$. Let $U \\subseteq A_1$ be an open set of $M_1$. Then it is not necessarily the case that $f \\left({U}\\right)$ is an open set of $M_2$."}
+{"_id": "6009", "title": "Equivalence of Definitions of Adherent Point", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A \\subseteq S$. {{TFAE|def = Adherent Point|view = adherent point of $A$}}"}
+{"_id": "6011", "title": "Metric Space is Compact iff Countably Compact", "text": "A metric space is compact {{iff}} it is countably compact."}
+{"_id": "6012", "title": "Countably Compact Metric Space is Sequentially Compact", "text": "Let $M$ be a countably compact metric space. Then $M$ is sequentially compact."}
+{"_id": "6015", "title": "Countably Infinite Set in Countably Compact Space has Omega-Accumulation Point", "text": "Let $\\struct {X, \\tau}$ be a countably compact topological space. Let $A \\subseteq X$ be countably infinite. Then $A$ has an $\\omega$-accumulation point in $X$."}
+{"_id": "6018", "title": "First-Countable Space is Sequentially Compact iff Countably Compact", "text": "A first-countable topological space is sequentially compact {{iff}} it is countably compact."}
+{"_id": "6019", "title": "Accumulation Point of Infinite Sequence in First-Countable Space is Subsequential Limit", "text": "Let $\\struct {X, \\tau}$ be a first-countable topological space. Let $\\sequence {x_n}_{n \\mathop \\in \\N}$ be an infinite sequence in $X$. Let $x$ be an accumulation point of $\\sequence {x_n}$. Then $x$ is a subsequential limit of $\\sequence {x_n}$."}
+{"_id": "6020", "title": "Sequentially Compact Metric Space is Totally Bounded", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $M$ be sequentially compact. Then $M$ is totally bounded."}
+{"_id": "6021", "title": "Separable Metric Space is Second-Countable", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $M$ be separable. Then $M$ is second-countable."}
+{"_id": "6022", "title": "Lipschitz Equivalence is Equivalence Relation", "text": "Let $A$ be a set. Let $\\mathcal D$ be the set of all metrics on $A$. Let $\\sim$ be the relation on $\\mathcal D$ defined as: :$\\forall d_1, d_2 \\in \\mathcal D: d_1 \\sim d_2 \\iff d_1$ is Lipschitz equivalent to $d_2$ Then $\\sim$ is an equivalence relation."}
+{"_id": "6024", "title": "Sequentially Compact Metric Space is Separable", "text": "A sequentially compact metric space is separable."}
+{"_id": "6025", "title": "Sequentially Compact Metric Space is Second-Countable", "text": "A sequentially compact metric space is second-countable."}
+{"_id": "6027", "title": "Lipschitz Equivalent Metrics are Topologically Equivalent", "text": "Let $M_1 = \\struct {A, d_1}$ and $M_2 = \\struct {A, d_2}$ be metric spaces on the same underlying set $A$. Let $d_1$ and $d_2$ be Lipschitz equivalent. Then $d_1$ and $d_2$ are topologically equivalent."}
+{"_id": "6030", "title": "Compact Metric Space is Complete", "text": "A compact metric space is complete."}
+{"_id": "6031", "title": "Pullback as Equalizer", "text": "Let $\\mathbf C$ be a metacategory with binary products. Suppose that the following diagram commutes (possibly except for the square): ::$\\begin{xy}\\xymatrix@+1em{  P   \\ar@/^/[rrd]^*+{p_1}   \\ar@/_/[ddr]_*+{p_2}   \\ar[rd]^*+{e} \\\\ &  A \\times B   \\ar[r]_*+{\\pi_1}   \\ar[d]^*+{\\pi_2} &  A   \\ar[d]^*+{f} \\\\ &  B   \\ar[r]_*+{g} &  C }\\end{xy}$ where $A \\times B$ is the binary product of $A$ and $B$, and $\\pi_1, \\pi_2$ are the associated projections. Then the border of the diagram, i.e.: ::$\\begin{xy}\\xymatrix{  P   \\ar[r]^*+{p_1}   \\ar[d]_*+{p_2} &  A   \\ar[d]^*+{f} \\\\  B   \\ar[r]_*+{g} &  C }\\end{xy}$ is a pullback of $f$ and $g$ iff $e: P \\to A \\times B$ is the equalizer of $f \\circ \\pi_1$ and $g \\circ \\pi_2$."}
+{"_id": "6032", "title": "Minkowski's Inequality for Sums/Index 2", "text": "Let $a_1, a_2, \\ldots, a_n, b_1, b_2, \\ldots, b_n \\ge 0$ be non-negative real numbers. Then: :$\\displaystyle \\paren {\\sum_{k \\mathop = 1}^n \\paren {a_k + b_k}^2}^{1/2} \\le \\paren {\\sum_{k \\mathop = 1}^n a_k^2}^{1/2} + \\paren {\\sum_{k \\mathop = 1}^n b_k^2}^{1/2}$"}
+{"_id": "6033", "title": "Banach-Steinhaus Theorem", "text": "Let $X$ be a Banach space. Let $Y$ be a normed vector space with norm $\\norm {\\,\\cdot\\,}_Y$. Let $\\family {T_\\alpha: X \\to Y}_{\\alpha \\mathop \\in A}$ be an $A$-indexed family of bounded linear transformations from $X$ to $Y$. Suppose that: :$\\displaystyle \\forall x \\in X: \\sup_{\\alpha \\mathop \\in A} \\norm {T_\\alpha x}_Y < \\infty$ Then: :$\\displaystyle \\sup_{\\alpha \\mathop \\in A} \\norm {T_\\alpha} < \\infty$ where $\\norm {T_\\alpha}$ denotes the norm of the linear transformation $T_\\alpha$. {{explain|We may need a definitive explanation for the intuitively understood $< \\infty$.}}"}
+{"_id": "6034", "title": "Minkowski's Inequality for Sums/Index Greater than 1", "text": "Let $a_1, a_2, \\ldots, a_n, b_1, b_2, \\ldots, b_n \\in \\R_{\\ge 0}$ be non-negative real numbers. Let $p \\in \\R$ be a real number such that $p > 1$. Then: :$\\displaystyle \\left({\\sum_{k \\mathop = 1}^n \\left({a_k + b_k}\\right)^p}\\right)^{1/p} \\le \\left({\\sum_{k \\mathop = 1}^n a_k^p}\\right)^{1/p} + \\left({\\sum_{k \\mathop = 1}^n b_k^p}\\right)^{1/p}$"}
+{"_id": "6035", "title": "Minkowski's Inequality for Sums/Index Less than 1", "text": "Let $a_1, a_2, \\ldots, a_n, b_1, b_2, \\ldots, b_n \\in \\R_{\\ge 0}$ be non-negative real numbers. Let $p \\in \\R$ be a real number. If $p < 0$, then we require that $a_1, a_2, \\ldots, a_n, b_1, b_2, \\ldots, b_n$ be strictly positive. If $p < 1$, $p \\ne 0$, then: :$\\displaystyle \\left({\\sum_{k \\mathop = 1}^n \\left({a_k + b_k}\\right)^p}\\right)^{1/p} \\ge \\left({\\sum_{k \\mathop = 1}^n a_k^p}\\right)^{1/p} + \\left({\\sum_{k \\mathop = 1}^n b_k^p}\\right)^{1/p}$"}
+{"_id": "6036", "title": "Monotone Real Function is Darboux Integrable", "text": "Let $\\closedint a b$ be a closed real interval, where $a < b$. Let $f: \\closedint a b \\to \\R$ be a monotone real function. Then $f$ is Darboux integrable over $\\closedint a b$."}
+{"_id": "6037", "title": "Topologies Not Always Comparable by Coarseness", "text": "Let $S$ be a set with at least $2$ elements. Let $\\mathbb T$ be the set of all topologies on $S$. For two topologies $\\tau_a, \\tau_b \\in \\mathbb T$, let $\\tau_a \\le \\tau_b$ denote that $\\tau_a$ is coarser than $\\tau_b$. Then there exist $\\tau_1, \\tau_2 \\in \\mathbb T$ such that neither: :$\\tau_1 \\le \\tau_2$ nor: :$\\tau_2 \\le \\tau_1$ That is, there are always topologies on $S$ which are non-comparable."}
+{"_id": "6053", "title": "Pullback of Subset Inclusion", "text": "Denote with $\\mathbf{Set}$ the category of sets. Let $A, B$ be sets, and let $f: A \\to B$ be a mapping. Let $V \\subseteq B$ be a subset of $B$. Denote with $i: V \\to B$ the inclusion mapping. Let $f^{-1} \\left({V}\\right) \\subseteq A$ be the preimage of $V$ under $f$. Denote with $j: f^{-1} \\left({V}\\right) \\to A$ the inclusion mapping. Denote with $\\bar f = f \\restriction_{f^{-1} \\left({V}\\right)}$ the restriction of $f$ to $f^{-1} \\left({V}\\right)$. Then: ::$\\begin{xy}\\xymatrix{  f^{-1} \\left({V}\\right)   \\ar[r]^*+{\\bar f}   \\ar[d]_*+{j} &  V   \\ar[d]^*+{i} \\\\  A   \\ar[r]_*+{f} &  B }\\end{xy}$ is a pullback diagram in $\\mathbf{Set}$."}
+{"_id": "6054", "title": "Poincaré Plane is Abstract Geometry", "text": "The Poincaré plane $\\struct {\\H, L_H}$ is an abstract geometry."}
+{"_id": "6055", "title": "Pullback Lemma", "text": "Let $\\mathbf C$ be a metacategory. Suppose that the following is a commutative diagram in $\\mathbf C$: ::$\\begin{xy}\\xymatrix@+0.5em{  F   \\ar[d]_*+{h''}   \\ar[r]^*+{f'} &  E   \\ar[d]^*+{h'}   \\ar[r]^*+{g'} &  D   \\ar[d]^*+{h} \\\\  A   \\ar[r]_*+{f} &  B   \\ar[r]_*+{g} &  C }\\end{xy}$ Suppose furthermore that the right square is a pullback. Then the left square is a pullback iff the outer rectangle is."}
+{"_id": "6057", "title": "Carathéodory's Theorem (Convex Analysis)", "text": "Let $E \\subset \\R^\\ell$. Let $\\mathbb x \\in \\map {\\operatorname{co} } E$. {{explain|Definition of $\\map {\\operatorname{co} } E$}} Then $\\mathbf x$ is a convex combination of affinely independent points of $E$.  In particular, $\\mathbf x $ is a convex combination of at most $\\ell + 1$ points of $E$."}
+{"_id": "6058", "title": "Limit of Sine of X over X/Corollary", "text": ":$\\displaystyle \\lim_{x \\mathop \\to 0} \\frac x {\\sin x} = 1$"}
+{"_id": "6064", "title": "Dicyclic Group is Group/Corollary", "text": "The quaternion group $Q_4$ is a non-abelian group."}
+{"_id": "6065", "title": "Integral of Constant/Definite", "text": ":$\\displaystyle \\int_a^b c \\ \\ \\mathrm dx = c \\left({b-a}\\right)$."}
+{"_id": "6066", "title": "Primitive of Constant", "text": ":$\\ds \\int c \\rd x = c x + C$ where $C$ is an arbitrary constant."}
+{"_id": "6067", "title": "Unbounded Set of Real Numbers is not Compact", "text": "Let $\\R$ be the set of real numbers considered as a Euclidean space. Let $S \\subseteq \\R$ be unbounded in $\\R$. Then $S$ is not a compact subspace of $\\R$."}
+{"_id": "6069", "title": "Continuous Mapping is Sequentially Continuous", "text": "Let $X$ and $Y$ be topological spaces. Let $x \\in X$. Let $f: X \\to Y$ be a mapping that is continuous at $x$. Then $f$ is sequentially continuous at $x$."}
+{"_id": "6070", "title": "Sequential Continuity is Equivalent to Continuity in Metric Space", "text": "Let $\\left({X, d}\\right)$ and $\\left({Y, e}\\right)$ be metric spaces. Let $f: X \\to Y$ be a mapping. Let $x \\in X$. Then $f$ is continuous at $x$ {{iff}} $f$ is sequentially continuous at $x$."}
+{"_id": "6071", "title": "Pullback of Commutative Triangle", "text": "Let $\\mathbf C$ be a metacategory. Suppose that the following is a commutative diagram in $\\mathbf C$: ::$\\begin{xy}\\xymatrix@+1em@L+2px{  A'   \\ar[rr]^*{h_\\alpha}   \\ar[dd]_*{\\alpha'} & &  A   \\ar[rd]^*{\\gamma}   \\ar[dd]^(.4)*{\\alpha} \\\\ &  B'   \\ar[ld]^*{\\beta'}   \\ar[rr] |{\\hole} ^(.3)*{h_\\beta} & &  B   \\ar[ld]^*{\\beta} \\\\  C'   \\ar[rr]_*{h} & &  C }\\end{xy}$ and that the two squares in it are pullback diagrams. Then there is a unique morphism $\\gamma': A' \\to B'$ making the following commute: ::$\\begin{xy}\\xymatrix@+1em@L+2px{  A'   \\ar[rr]^*{h_\\alpha}   \\ar[dd]_*{\\alpha'}   \\ar@{-->}[rd]^*{\\gamma'} & &  A   \\ar[rd]^*{\\gamma}   \\ar[dd]^(.4)*{\\alpha} \\\\ &  B'   \\ar[ld]^*{\\beta'}   \\ar[rr] |{\\hole} ^(.3)*{h_\\beta} & &  B   \\ar[ld]^*{\\beta} \\\\  C'   \\ar[rr]_*{h} & &  C }\\end{xy}$ Furthermore, $\\gamma'$ makes the following a pullback: ::$\\begin{xy}\\xymatrix@+.5em@L+2px{  A'   \\ar[r]^*{h_\\alpha}   \\ar[d]_*{\\gamma'} &  A   \\ar[d]^*{\\gamma} \\\\  B'   \\ar[r]_*{h_\\beta} &  B }\\end{xy}$"}
+{"_id": "6072", "title": "Associative Commutative Idempotent Operation is Distributive over Itself", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure, such that: :$(1): \\quad \\circ$ is associative :$(2): \\quad \\circ$ is commutative :$(3): \\quad \\circ$ is idempotent. Then $\\circ$ is distributive over itself. That is: :$\\forall a, b, c \\in S: \\left({a \\circ b}\\right) \\circ \\left({a \\circ c}\\right) = a \\circ b \\circ c = \\left({a \\circ c}\\right) \\circ \\left({b \\circ c}\\right)$"}
+{"_id": "6073", "title": "Intersection Distributes over Intersection", "text": "Set intersection is distributive over itself: :$\\forall A, B, C: \\paren {A \\cap B} \\cap \\paren {A \\cap C} = A \\cap B \\cap C = \\paren {A \\cap C} \\cap \\paren {B \\cap C}$ where $A, B, C$ are sets."}
+{"_id": "6074", "title": "Inclusion Mapping is Restriction of Identity", "text": "Let $T$ be a set. Let $S \\subseteq T$ be a subset of $T$. Let $i_S: S \\to T$ be the inclusion mapping on $S$. Then $i_S$ is the restriction of the identity mapping $I_T: T \\to T$ on $T$."}
+{"_id": "6075", "title": "Preimage of Subset under Inclusion Mapping", "text": "Let $S$ be a set. Let $H \\subseteq S$ be a subset of $S$. Let $i_H: H \\to S$ be the inclusion mapping on $H$. Let $T \\subseteq S$. Then: :$i_H^{-1} \\left({T}\\right) = T \\cap H$ where $i_H^{-1} \\left({T}\\right)$ is the preimage of $T$ under $i_H$."}
+{"_id": "6076", "title": "Inclusion Mapping is Continuous", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T_H = \\struct {H, \\tau_H}$ be a topological subspace of $T$ where $H \\subseteq S$. Let $i_H: H \\to S$ be the inclusion mapping on $H$. Then $i_H$ is a $\\struct {\\tau_H, \\tau}$-continuous mapping."}
+{"_id": "6077", "title": "Continuity of Composite with Inclusion/Mapping on Inclusion", "text": "If $f$ is $\\tuple {\\tau, \\tau'}$-continuous, then $f \\circ i$ is $\\tuple {\\tau_H, \\tau'}$-continuous"}
+{"_id": "6078", "title": "Continuity of Composite with Inclusion/Inclusion on Mapping", "text": "$g$ is $\\left({\\tau', \\tau_H}\\right)$-continuous iff $i \\circ g$ is $\\left({\\tau', \\tau}\\right)$-continuous."}
+{"_id": "6079", "title": "Continuity of Composite with Inclusion/Uniqueness of Induced Topology", "text": "The induced topology $\\tau_H$ is the ''only'' topology on $H$ satisfying Continuity of Composite with Inclusion: Inclusion on Mapping for all possible $g$."}
+{"_id": "6080", "title": "Products of Open Sets form Local Basis in Product Space", "text": "Let $T_1 = \\struct{A_1, \\tau_1}$ and $T_2 = \\struct{A_2, \\tau_2}$ be topological spaces. Let $\\struct{T, \\tau} = T_1 \\times T_2$ be the product space of $T_1$ and $T_2$. Let $\\tuple{x, y} \\in A_1 \\times A_2$. Let $W \\in \\tau$ be an open set of $T$ such that $\\tuple{x, y} \\in W$. Then: :$\\exists U_1 \\in \\tau_1, U_2 \\in \\tau_2: \\tuple{x, y} \\in U_1 \\times U_2 \\subseteq W$ That is, products of open sets from $T_1$ and $T_2$ form a local basis at $\\tuple{x, y}$."}
+{"_id": "6081", "title": "Open Real Intervals are Homeomorphic", "text": "Consider the real numbers $\\R$ as a metric space under the Euclidean metric. Let $I_1 := \\left({a \\,.\\,.\\, b}\\right)$ and $I_2 := \\left({c \\,.\\,.\\, d}\\right)$ be non-empty open real intervals. Then $I_1$ and $I_2$ are homeomorphic."}
+{"_id": "6082", "title": "Half-Open Real Interval is not Closed in Real Number Line", "text": "Let $\\R$ be the real number line considered as an Euclidean space. Let $\\left[{a \\,.\\,.\\, b}\\right) \\subset \\R$ be a half-open interval of $\\R$. Then $\\left[{a \\,.\\,.\\, b}\\right)$ is not a closed set of $\\R$. Similarly, the half-open interval $\\left({a \\,.\\,.\\, b}\\right] \\subset \\R$ is not a closed set of $\\R$."}
+{"_id": "6084", "title": "Underlying Set of Topological Space is Clopen", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Then the underlying set $S$ of $T$ is both open and closed in $T$."}
+{"_id": "6085", "title": "Empty Set is Closed in Topological Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then $\\O$ is closed in $T$."}
+{"_id": "6086", "title": "Basis induces Local Basis", "text": "Let $\\left({S, \\tau}\\right)$ be a topological space. Let $\\mathcal B$ be an analytic basis for $\\tau$. Let $x \\in S$, and denote with $\\mathcal B_x$ the set $\\left\\{{B \\in \\mathcal B: x \\in B}\\right\\}$. Then $\\mathcal B_x$ is a local basis at $x$."}
+{"_id": "6089", "title": "Supremum of Bounded Above Set of Reals is in Closure", "text": "Let $\\R$ be the real number line under the Euclidean metric. Let $H \\subseteq \\R$ be a bounded above subset of $\\R$ such that $H \\ne \\varnothing$. Let $u = \\map \\sup H$ be the supremum of $H$. Then: :$u \\in \\map \\cl H$ where $\\map \\cl H$ denotes the closure of $H$ in $\\R$."}
+{"_id": "6090", "title": "Topological Space is Open Neighborhood of Subset", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $H \\subseteq S$ be a subset of $S$. Then $S$ is an open neighborhood of $H$."}
+{"_id": "6091", "title": "Open Superset is Open Neighborhood", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $U \\in \\tau$ be an open set of $T$. Let $H \\subseteq U$. Then $U$ is an open neighborhood of $H$."}
+{"_id": "6092", "title": "Category has Products and Equalizers iff Pullbacks and Terminal Object", "text": "Let $\\mathbf C$ be a metacategory. Then the following are equivalent: :$(1): \\mathbf C$ has all finite products and equalizers. :$(2): \\mathbf C$ has all pullbacks and a terminal object."}
+{"_id": "6093", "title": "Infimum of Bounded Below Set of Reals is in Closure", "text": "Let $\\R$ be the real number line with the  usual (Euclidean} metric. Let $H \\subseteq \\R$ be a bounded below subset of $\\R$ such that $H \\ne \\O$. Let $l = \\map \\inf H$ be the infimum of $H$. Then: :$l \\in \\map \\cl H$ where $\\map \\cl H$ denotes the closure of $H$ in $\\R$."}
+{"_id": "6098", "title": "Nowhere Dense iff Complement of Closure is Everywhere Dense/Corollary", "text": "Let $H$ be a closed set of $T$. Then $H$ is nowhere dense in $T$ {{iff}} $S \\setminus H$ is everywhere dense in $T$."}
+{"_id": "6099", "title": "Equivalence of Definitions of Topology Generated by Synthetic Basis", "text": "Let $S$ be a set. Let $\\BB$ be a synthetic basis on $S$. {{TFAE|def = Topology Generated by Synthetic Basis}}"}
+{"_id": "6100", "title": "Union from Synthetic Basis is Topology", "text": "Let $\\BB$ be a synthetic basis on a set $X$. Let $\\displaystyle \\tau = \\set {\\bigcup \\AA: \\AA \\subseteq \\BB}$. Then $\\tau$ is a topology on $X$."}
+{"_id": "6102", "title": "Product as Limit", "text": "Let $\\mathbf C$ be a metacategory. Let $C_1, C_2$ be objects of $\\mathbf C$. Let their binary product $C_1 \\times C_2$ exist in $\\mathbf C$. Then $C_1 \\times C_2$ is the limit of the diagram $D: 2 \\to \\mathbf C$ defined by: :$D_0 := C_1, D_1 := C_2$ where $2$ is the discrete category with two objects $0, 1$."}
+{"_id": "6103", "title": "Equalizer as Limit", "text": "Let $\\mathbf C$ be a metacategory. Let $f_1, f_2: C_1 \\to C_2$ be morphisms of $\\mathbf C$. Let their equalizer $e: E \\to C_1$ exist in $\\mathbf C$. Then $\\left({E, e}\\right)$ is the limit of the diagram $D: \\mathbf J \\to \\mathbf C$ defined by: ::$\\begin{xy}\\xymatrix@+1em@L+3px{  \\mathbf{J}: &  \\ast   \\ar[r]<2pt>   \\ar[r]<-2pt> &  \\star }\\end{xy}$ ::$\\begin{xy}\\xymatrix@+1em@L+3px{  D: &  C_1   \\ar[r]<2pt>^*+{f_1}   \\ar[r]<-2pt>_*+{f_2} &  C_2 }\\end{xy}$"}
+{"_id": "6104", "title": "Terminal Object as Limit", "text": "Let $\\mathbf C$ be a metacategory. Let $\\mathbf C$ have a terminal object $1$. Then $1$ is the limit of the unique diagram $D: \\mathbf 0 \\to \\mathbf C$, where $\\mathbf 0$ is the zero category."}
+{"_id": "6105", "title": "Pullback as Limit", "text": "Let $\\mathbf C$ be a metacategory. Let $f_1: A \\to C$ and $f_2: B \\to C$ be morphisms of $\\mathbf C$. Let their pullback: ::$\\begin{xy}\\xymatrix@+1em@L+3px{  P   \\ar[r]^*+{p_2}   \\ar[d]_*+{p_1} &  A   \\ar[d]^*+{f_1} \\\\  B   \\ar[r]_*+{f_2} &  C }\\end{xy}$ exist in $\\mathbf C$. Then $\\left({P, p_1, p_2}\\right)$ is the limit of the diagram $D: \\mathbf J \\to \\mathbf C$ defined by: ::$\\begin{xy}\\xymatrix@+1em@L+3px{  \\save[]+<0em,-2em>*{\\mathbf{J}:} \\restore & &  \\cdot   \\ar[d] \\\\ &  \\cdot   \\ar[r] &  \\cdot }\\end{xy}$ ::$\\begin{xy}\\xymatrix@+1em@L+3px{  \\save[]+<0em,-2em>*{D:} \\restore & &  A   \\ar[d]^*+{f_1} \\\\ &  B   \\ar[r]_*+{f_2} &  C }\\end{xy}$"}
+{"_id": "6106", "title": "Identification Topology is Topology", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \\to S_2$ be a mapping. Let $\\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\\struct {S_1, \\tau_1}$. Then $\\tau_2$ is a topology on $S_2$."}
+{"_id": "6107", "title": "Identification Mapping is Continuous", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \\to S_2$ be a mapping. Let $\\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\\struct {S_1, \\tau_1}$. Then the identification mapping $f$ is continuous."}
+{"_id": "6108", "title": "Existence and Uniqueness of Identification Topology", "text": "Let $T_1 = \\left({S_1, \\tau_1}\\right)$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \\to S_2$ be a mapping. Let $\\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\\left({S_1, \\tau_1}\\right)$. Then the identification topology on $S_2$ with respect to $f$ and $\\left({S_1, \\tau_1}\\right)$ always exists and is unique."}
+{"_id": "6110", "title": "Identification Topology is Finest Topology for Mapping to be Continuous", "text": "Let $T_1 := \\struct {S_1, \\tau_1}$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \\to S_2$ be a mapping. Let $\\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\\struct {S_1, \\tau_1}$. Let $T_2 := \\struct {S_2, \\tau_2}$ be the resulting topological space. Then $\\tau_2$ is the finest topology on $S_2$ such that $f: T_1 \\to T_2$ is continuous."}
+{"_id": "6111", "title": "Identification Topology equals Quotient Topology on Induced Equivalence", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \\to S_2$ be a mapping. Let $\\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\\struct {S_1, \\tau_1}$: :$\\tau_2 = \\set {V \\in \\powerset {S_2}: f^{-1} \\sqbrk V \\in \\tau_1}$ Let $\\RR_f \\subseteq S_1 \\times S_1$ be the equivalence on $S_1$ induced by $f$: :$\\tuple {s_1, s_2} \\in \\RR_f \\iff \\map f {s_1} = \\map f {s_2}$ Let $\\tau_{\\RR_f}$ be the quotient topology on $S / \\RR_f$ by $q_{\\RR_f}$: :$\\tau_{\\RR_f} := \\set {U \\subseteq S / \\RR_f: q_{\\RR_f}^{-1} \\sqbrk U \\in \\tau_1}$ Then: :$\\tau_2$ is homeomorphic to $\\tau_{\\RR_f}$."}
+{"_id": "6112", "title": "Continuity of Composite with Identification Mapping", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ and $T_2 = \\struct {S_2, \\tau_2}$ be topological spaces. Let $f_1: T_1 \\to T_2$ be a mapping. Let $S_3$ be a set. Let $p: S_1 \\to S_3$ be a mapping. Let $\\tau_3$ be the identification topology on $S_3$ with respect to $p$ and $T_1$. Let $T_3 = \\struct {S_3, \\tau_3}$ be the resulting topological space. Let $f_2: S_3 \\to S_2$ be a mapping such that: :$f_1 = f_2 \\circ p$ where $f_2 \\circ p$ is the composition of $f_2$ with $p$. Then $f_1$ is a continuous mapping {{iff}} $f_2$ is a continuous mapping."}
+{"_id": "6113", "title": "Young's Inequality for Increasing Functions", "text": "Let $a_0$ and $b_0$ be strictly positive real numbers. Let $f: \\closedint 0 {a_0} \\to \\closedint 0 {b_0}$ be a strictly increasing bijection. Let $a$ and $b$ be real numbers such that $0 \\le a \\le a_0$ and $0 \\le b \\le b_0$. Then: :$\\displaystyle ab \\le \\int_0^a \\map f u \\rd u + \\int_0^b \\map {f^{-1}} v \\rd v$ where $\\displaystyle \\int$ denotes the definite integral."}
+{"_id": "6114", "title": "Set is Closed iff Equals Topological Closure", "text": "Let $T$ be a topological space. Let $H \\subseteq T$. Then $H$ is closed in $T$ {{iff}}: :$H = \\map \\cl H$"}
+{"_id": "6116", "title": "Equivalence of Definitions of Compact Topological Subspace", "text": "{{TFAE|def = Compact Subspace}} Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $T_H = \\left({H, \\tau_H}\\right)$ be a topological subspace of $T$, where $H \\subseteq S$."}
+{"_id": "6117", "title": "Covariant Hom Functor is Continuous", "text": "Let $\\mathbf{Set}$ be the category of sets. Let $\\mathbf C$ be a locally small category. Let $C$ be an object of $\\mathbf C$, and let $\\hom \\paren {C, \\cdot}: \\mathbf C \\to \\mathbf{Set}$ be the covariant hom functor based at $C$. Then $\\hom \\paren {C, \\cdot}$ is a continuous functor."}
+{"_id": "6118", "title": "Contravariant Hom Functor maps Colimits to Limits", "text": "Let $\\mathbf{Set}$ be the category of sets. Let $\\mathbf C$ be a locally small category. Let $C$ be an object of $\\mathbf C$. Let $\\map \\hom {\\cdot, C}: \\mathbf C \\to \\mathbf{Set}$ be the contravariant hom functor based at $C$. Then $\\map \\hom {\\cdot, C}$ maps every colimit to a limit, in that: :$\\map \\hom { {\\varinjlim \\,}_j \\,D_j, C} \\cong {\\varprojlim \\,}_j \\, \\map \\hom {D_j, C}$ for every diagram $D: \\mathbf J \\to \\mathbf C$."}
+{"_id": "6119", "title": "Composite Mapping is Mapping", "text": "Let $S_1, S_2, S_3$ be sets. Let $f: S_1 \\to S_2$ and $g: S_2 \\to S_3$ be mappings. Then the composite mapping $g \\circ f$ is also a mapping."}
+{"_id": "6120", "title": "Normal Space is Regular Space", "text": "Let $\\struct {S, \\tau}$ be a normal space. Then $\\struct {S, \\tau}$ is also a regular space."}
+{"_id": "6121", "title": "Product of Hausdorff Factor Spaces is Hausdorff/General Result", "text": "Let $\\SS = \\family {\\struct {S_\\alpha, \\tau_\\alpha} }$ be an indexed family of topological spaces for $\\alpha$ in some indexing set $I$. Let $\\displaystyle T = \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\SS$. Let each of $\\struct {S_\\alpha, \\tau_\\alpha}$ for $\\alpha \\in I$ be $T_2$ (Hausdorff) spaces. Then $T$ is a $T_2$ (Hausdorff) space."}
+{"_id": "6122", "title": "Product of Hausdorff Factor Spaces is Hausdorff", "text": "Let $T_\\alpha = \\struct {S_\\alpha, \\tau_\\alpha}$ and $T_\\beta = \\struct {S_\\beta, \\tau_\\beta}$ be topological spaces. Let $T = T_\\alpha \\times T_\\beta$ be the product space of $T_\\alpha$ and $T_\\beta$ Let $T_\\alpha$ and $T_\\beta$ both be $T_2$ (Hausdorff) spaces. Then $T$ is also a $T_2$ (Hausdorff) space."}
+{"_id": "6123", "title": "Domain of Continuous Injection to Hausdorff Space is Hausdorff", "text": "Let $T_\\alpha = \\struct {S_\\alpha, \\tau_\\alpha}$ and $T_\\beta = \\struct {S_\\beta, \\tau_\\beta}$ be topological spaces. Let $f: S_\\alpha \\to S_\\beta$ be a continuous mapping which is an injection. If $T_\\beta$ is a $T_2$ (Hausdorff) space, then $T_\\alpha$ is also a $T_2$ (Hausdorff) space."}
+{"_id": "6126", "title": "Quotient Space of Hausdorff Space is not necessarily Hausdorff", "text": "Let $T = \\struct {S, \\tau}$ be a Hausdorff space. Let $\\RR \\subseteq S \\times S$ be an equivalence relation on $S$. Let $T_\\RR := \\struct {S / \\RR, \\tau_\\RR}$ be the quotient space of $S$ by $\\RR$. Then $T_\\RR$ is not necessarily also a Hausdorff space."}
+{"_id": "6132", "title": "Topological Product of Compact Spaces/Finite Product", "text": "Let $T_1, T_2, \\ldots, T_n$ be topological spaces. Let $\\displaystyle \\prod_{i \\mathop = 1}^n T_i$ be the product space of $T_1, T_2, \\ldots, T_n$. Then $\\displaystyle \\prod_{i \\mathop = 1}^n T_i$ is compact {{iff}} all of $T_1, T_2, \\ldots, T_n$ are compact."}
+{"_id": "6133", "title": "Uniform Continuity on Metric Space does not imply Compactness", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f: A_1 \\to A_2$ be a uniformly continuous mapping on $A_1$. Then $M_1$ does not necessarily have to be a compact metric space."}
+{"_id": "6134", "title": "Uniformly Continuous Function is Continuous/Metric Space", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_1, d_1}\\right)$ be metric spaces. Let the mapping $f: M_1 \\to M_2$ be uniformly continuous on $M_1$. Then $f$ is continuous on $M_1$."}
+{"_id": "6137", "title": "Uniformly Continuous Function is Continuous/Real Function", "text": "Let $I$ be an interval of $\\R$. Let $f: I \\to \\R$ be a uniformly continuous real function on $I$. Then $f$ is continuous on $I$."}
+{"_id": "6139", "title": "Category of Finite Sets is Cartesian Closed", "text": "Let $\\mathbf{Finset}$ be the category of finite sets. Then $\\mathbf{Finset}$ is Cartesian closed."}
+{"_id": "6140", "title": "Components of Separation are Clopen", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A \\mid B$ be a separation of $T$. Then both $A$ and $B$ are clopen in $T$."}
+{"_id": "6141", "title": "Equivalence of Definitions of Connected Topological Space/No Separation iff No Union of Closed Sets", "text": "{{TFAE|def = Connected Topological Space}} Let $T = \\left({S, \\tau}\\right)$ be a topological space."}
+{"_id": "6143", "title": "Complements of Components of Two-Component Partition form Partition", "text": "Let $S$ be a set with at least two elements. Let $A, B \\subseteq S$. Let $\\complement_S$ denote the complement relative to $S$. $A \\mid B$ is a partition of $S$ {{iff}} $\\relcomp S A \\mid \\relcomp S B$ is a partition of $S$."}
+{"_id": "6144", "title": "Sigma-Algebra Closed under Finite Intersection", "text": "Let $A_1, \\ldots, A_n \\in \\Sigma$. Then $\\displaystyle \\bigcap_{k \\mathop = 1}^n A_k \\in \\Sigma$."}
+{"_id": "6145", "title": "Equivalence of Definitions of Connected Topological Space/No Union of Closed Sets implies No Subsets with Empty Boundary", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $T$ have no two disjoint non-empty closed sets whose union is $S$. Then the only subsets of $S$ whose boundary is empty are $S$ and $\\varnothing$."}
+{"_id": "6146", "title": "Equivalence of Definitions of Connected Topological Space/No Subsets with Empty Boundary implies No Clopen Sets", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $T$ be such that the only subsets of $S$ whose boundary is empty are $S$ and $\\varnothing$. Then the only clopen sets of $T$ are $S$ and $\\varnothing$."}
+{"_id": "6147", "title": "Equivalence of Definitions of Connected Topological Space/No Clopen Sets implies No Union of Separated Sets", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let the only clopen sets of $T$ be $S$ and $\\O$. Then there are no two non-empty separated sets of $T$ whose union is $S$."}
+{"_id": "6148", "title": "Equivalence of Definitions of Connected Topological Space/No Union of Separated Sets implies No Continuous Surjection to Discrete Two-Point Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T$ be such that there are no two non-empty separated sets whose union is $S$. Then there exists no continuous surjection from $T$ onto a discrete two-point space."}
+{"_id": "6149", "title": "Equivalence of Definitions of Connected Topological Space/No Continuous Surjection to Discrete Two-Point Space implies No Separation", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $T$ be such that there exists no continuous surjection from $T$ onto a discrete two-point space. Then there exist no open sets $A, B \\in \\tau$ such that $A, B \\ne \\varnothing$, $A \\cup B = S$ and $A \\cap B = \\varnothing$."}
+{"_id": "6150", "title": "Existence of Vector Space Bases implies Axiom of Choice", "text": "The supposition that every vector space has a basis, along with the Zermelo-Fraenkel axioms, implies that the axiom of choice holds."}
+{"_id": "6152", "title": "Category of Posets is Cartesian Closed", "text": "Let $\\mathbf{OrdSet}$ be the category of ordered sets. Then $\\mathbf{OrdSet}$ is Cartesian closed."}
+{"_id": "6153", "title": "Definition:Exponentiation Functor", "text": "Let $\\mathbf C$ be a Cartesian closed metacategory. Let $A$ be an object of $\\mathbf C$. Then '''exponentiation by $A$''', denoted $\\left({-}\\right)^A: \\mathbf C \\to \\mathbf C$, is the functor defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = C^A := C^A        |rc= $C^A$ is the exponential of $C$ by $A$ }} {{axiom|lc= Morphism functor:        |m = f^A := \\widetilde{\\left({f \\circ \\epsilon}\\right)}        |rc= $f: B \\to C$ is a morphism of $\\mathbf C$ }} {{end-axiom}} Here $\\epsilon: B^A \\times A \\to B$ denotes the evaluation morphism, and $\\widetilde{\\left({f \\circ \\epsilon}\\right)}: B^A \\to C^A$ is the exponential transpose of $f \\circ \\epsilon$. That it is in fact a functor is shown on Exponentiation Functor is Functor."}
+{"_id": "6154", "title": "Exponentiation Functor is Functor", "text": "Let $\\mathbf C$ be a Cartesian closed metacategory. Let $A$ be an object of $\\mathbf C$. Let $\\left({-}\\right)^A: \\mathbf C \\to \\mathbf C$ be the exponentiation functor. Then $\\left({-}\\right)^A$ is a functor."}
+{"_id": "6155", "title": "Cosine Exponential Formulation", "text": ":$\\cos z = \\dfrac {\\map \\exp {i z} + \\map \\exp {-i z} } 2$"}
+{"_id": "6157", "title": "Reciprocal of i", "text": "Let $i$ be the imaginary unit such that $i^2 = -1$. Then $\\dfrac 1 i = -i$."}
+{"_id": "6160", "title": "Squeeze Theorem/Functions/Proof 3", "text": "Let $a$ be a point on an open real interval $I$. Also let $f$, $g$ and $h$ be real functions defined at all points of $I$ except for possibly at point $a$. Suppose that: : $\\forall x \\ne a \\in I: g \\left({x}\\right) \\le f \\left({x}\\right) \\le h \\left({x}\\right)$ : $\\displaystyle \\lim_{x \\mathop \\to a} \\ g \\left({x}\\right) = \\lim_{x \\mathop \\to a} \\ h \\left({x}\\right) = L$ Then: : $\\displaystyle \\lim_{x \\mathop \\to a} \\ f \\left({x}\\right) = L$"}
+{"_id": "6167", "title": "Rational Numbers are not Connected", "text": "The set of rational numbers $\\Q$ is not a connected topological space."}
+{"_id": "6168", "title": "Closure of Connected Set is Connected", "text": "Let $T$ be a topological space. Let $H$ be a connected set of $T$. Let $H^-$ denote the closure of $H$ in $T$.  Then $H^-$ is connected in $T$."}
+{"_id": "6173", "title": "Closed Topologist's Sine Curve is Connected", "text": "Let $G$ be the graph of the function $y = \\map \\sin {\\dfrac 1 x}$ for $x > 0$. Let $J$ be the line segment joining the points $\\tuple {0, -1}$ and $\\tuple {0, 1}$ in $\\R^2$. Then $G \\cup J$ is connected."}
+{"_id": "6174", "title": "Separated Sets are Clopen in Union", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A$ and $B$ be separated sets in $T$. Let $H = A \\cup B$ be given the subspace topology. Then $A$ and $B$ are each both open and closed in $H$."}
+{"_id": "6175", "title": "Compatibility of Atlases is Equivalence Relation", "text": "Let $M$ be a topological space. Let $d$ and $k$ be natural numbers. Let $\\mathcal A$ denote the set of all $d$-dimensional atlases of class $\\mathcal C^k$ on $M$. Define a relation $\\sim$ on $\\mathcal A$ by putting, for any two $\\mathcal C^k$-atlases $\\mathcal F$ and $\\mathcal G$: :$\\mathcal F \\sim \\mathcal G$ {{iff}} $\\mathcal F$ and $\\mathcal G$ are $C^k$-compatible. Then $\\sim$ is an equivalence relation on $\\mathcal A$."}
+{"_id": "6176", "title": "Addition of Real and Imaginary Parts", "text": "Let $z_0, z_1 \\in \\C$ be two complex numbers. Then: :$\\map \\Re {z_0 + z_1} = \\map \\Re {z_0} + \\map \\Re {z_1}$ and: :$\\map \\Im {z_0 + z_1} = \\map \\Im {z_0} + \\map \\Im {z_1}$ Here, $\\map \\Re {z_0}$ denotes the real part of $z_0$, and $\\map \\Im {z_0}$ denotes the imaginary part of $z_0$."}
+{"_id": "6177", "title": "Multiplication of Real and Imaginary Parts", "text": "Let $w, z \\in \\C$ be complex numbers. $(1)$ If $w$ is wholly real, then: :$\\map \\Re {w z} = w \\, \\map \\Re z$ and: :$\\map \\Im {w z} = w \\, \\map \\Im z$ $(2)$ If $w$ is wholly imaginary, then: :$\\map \\Re {w z} = -\\map \\Im w \\, \\map \\Im z$ and: :$\\map \\Im {w z} = \\map \\Im w \\, \\map \\Re z$ Here, $\\map \\Re z$ denotes the real part of $z$, and $\\map \\Im z$ denotes the imaginary part of $z$."}
+{"_id": "6178", "title": "Rational Numbers are Totally Disconnected", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\Q, \\tau_d}$ is a totally disconnected space."}
+{"_id": "6182", "title": "Rational Numbers are not Discrete Space", "text": "The set of rational numbers $\\Q$ does not form a discrete space."}
+{"_id": "6183", "title": "Rational Numbers are not Extremally Disconnected", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\Q, \\tau_d}$ does not form an extremally disconnected space."}
+{"_id": "6185", "title": "Real and Imaginary Part Projections are Continuous", "text": "Define the real-valued functions $x, y: \\C \\to \\R$ by: :$\\forall z \\in \\C: \\map x z = \\map \\Re z$ :$\\forall z \\in \\C: \\map y z = \\map \\Im z$ Equip $\\R$ with the usual Euclidean metric. Equip $\\C$ with the usual Euclidean metric. {{refactor|Reconsider the above link so that it goes to an actual definition page.}} Then both $x$ and $y$ are continuous functions."}
+{"_id": "6186", "title": "Continuous Complex Function is Complex Riemann Integrable", "text": "Let $\\closedint a b$ be a closed real interval. Let $f: \\closedint a b \\to \\C$ be a continuous complex function. Then $f$ is complex Riemann integrable over $\\closedint a b$."}
+{"_id": "6187", "title": "Compact Subspace of Metric Space is Sequentially Compact in Itself", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $C \\subseteq M$ be a subspace of $M$ such that $C$ is compact. Then $C$ is sequentially compact in itself."}
+{"_id": "6191", "title": "Sum of Complex Integrals on Adjacent Intervals", "text": "Let $\\left[{a \\,.\\,.\\, b}\\right]$ be a closed real interval. Let $f: \\left[{a \\,.\\,.\\, b}\\right] \\to \\C$ be a continuous complex function. Let $c \\in \\left[{a \\,.\\,.\\, b}\\right]$. Then: :$\\displaystyle \\int_a^c f \\left({t}\\right) \\ \\mathrm dt + \\int_c^b f \\left({t}\\right) \\ \\mathrm dt = \\int_a^b f \\left({t}\\right) \\ \\mathrm dt$"}
+{"_id": "6192", "title": "Equivalence of Definitions of Metric Space Continuity at Point", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f: A_1 \\to A_2$ be a mapping from $A_1$ to $A_2$. Let $a \\in A_1$ be a point in $A_1$. {{TFAE|def = Continuous at Point of Metric Space|continuity at a point}}"}
+{"_id": "6194", "title": "Ordering Principle", "text": "Let $S$ be a set. Then there exists a total ordering on $S$."}
+{"_id": "6195", "title": "Number Smaller than Lebesgue Number is also Lebesgue Number", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $\\epsilon \\in \\R_{>0}$ be a Lebesgue number for $M$. Let $\\epsilon' \\in \\R_{>0}: \\epsilon' < \\epsilon$. Then $\\epsilon'$ is also a Lebesgue number for $M$."}
+{"_id": "6196", "title": "Open Cover may not have Lebesgue Number", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $\\mathcal C$ be an open cover of $M$. Then it may not necessarily be the case that $\\mathcal C$ has a Lebesgue number."}
+{"_id": "6197", "title": "Burnside's Theorem", "text": "Let $G$ be a finite group. Let the order of $G$ be $p^m q^n$ where: : $p, q$ are prime : $m, n \\in \\N$ Then $G$ is solvable."}
+{"_id": "6198", "title": "Constant Function is Uniformly Continuous/Metric Space", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f_c: A_1 \\to A_2$ be the constant mapping from $A_1$ to $A_2$: :$\\exists c \\in A_2: \\forall a \\in A_1: f_c \\left({a}\\right) = c$ That is, every point in $A_1$ maps to the same point $c$ in $A_2$. Then $f_c$ is uniformly continuous throughout $A_1$ with respect to $d_1$ and $d_2$."}
+{"_id": "6199", "title": "Constant Function is Uniformly Continuous/Real Function", "text": "Let $f_c: \\R \\to \\R$ be the constant mapping: :$\\exists c \\in \\R: \\forall a \\in \\R: \\map {f_c} a = c$ Then $f_c$ is uniformly continuous on $\\R$."}
+{"_id": "6200", "title": "Linear Combination of Complex Integrals", "text": "Let $\\closedint a b$ be a closed real interval. Let $f, g: \\closedint a b \\to \\C$ be complex Riemann integrable functions over $\\closedint a b$. Let $\\lambda, \\mu \\in \\C$ be complex constants. Then: :$\\displaystyle \\int_a^b \\paren {\\lambda \\map f t + \\mu \\map g t} \\rd t = \\lambda \\int_a^b \\map f t \\rd t + \\mu \\int_a^b \\map g t \\rd t$"}
+{"_id": "6201", "title": "Modulus Larger than Real Part and Imaginary Part", "text": "Let $z \\in \\C$ be a complex number. Let $\\operatorname{Re} \\left({z}\\right)$ denote the real part of $z$, and $\\operatorname{Im} \\left({z}\\right) $ the imaginary part of $z$. Then:"}
+{"_id": "6202", "title": "Pointwise Difference of Simple Functions is Simple Function", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $f,g : X \\to \\R$ be simple functions. Then $f - g: X \\to \\R, \\left({f - g}\\right) \\left({x}\\right) := f \\left({x}\\right) - g \\left({x}\\right)$ is also a simple function."}
+{"_id": "6203", "title": "Modulus and Argument of Complex Exponential", "text": "Let $z \\in \\C$ be a complex number. Let $\\hointr a {a + 2 \\pi}$ be a half open interval of length $2 \\pi$. Let $r \\in \\hointr 0 {+\\infty}$ and $\\theta \\in \\hointr a {a + 2 \\pi}$. Then: :$r = \\cmod z$ and $\\theta = \\map \\arg z$ {{iff}}: :$z = r e^{i \\theta}$ where: :$\\cmod z$ denotes the modulus of $z$ :$\\map \\arg z$ denotes the argument of $z$ :$x \\mapsto e^x$ is the complex exponential function. If $z = 0$ or $r = 0$, then $\\theta$ may be any number in $\\hointr a {a + 2 \\pi}$."}
+{"_id": "6204", "title": "Bases of Vector Space have Equal Cardinality", "text": "Let $k$ be a division ring. Let $V$ be a vector space over $k$. Let $X$ and $Y$ be bases of $V$. Then $X$ and $Y$ are equinumerous."}
+{"_id": "6205", "title": "Equivalence of Definitions of Symmetric Difference/(1) iff (2)", "text": "Let $S$ and $T$ be sets. {{TFAENocat|def = Symmetric Difference|view = symmetric difference $S * T$ between $S$ and $T$}}"}
+{"_id": "6206", "title": "Equivalence of Definitions of Symmetric Difference/(1) iff (3)", "text": "Let $S$ and $T$ be sets. {{TFAENocat|def = Symmetric Difference|view = symmetric difference $S * T$ between $S$ and $T$}}"}
+{"_id": "6207", "title": "Equivalence of Definitions of Symmetric Difference/(2) iff (4)", "text": "Let $S$ and $T$ be sets. {{TFAENocat|def = Symmetric Difference|view = symmetric difference $S * T$ between $S$ and $T$}}"}
+{"_id": "6208", "title": "Subset of Countable Set is Countable", "text": "A subset of a countable set is countable."}
+{"_id": "6209", "title": "Sine Exponential Formulation", "text": ":$\\sin z = \\dfrac {\\map \\exp {i z} - \\map \\exp {-i z} } {2 i}$"}
+{"_id": "6210", "title": "Sine and Cosine Exponential Formulation", "text": "For any complex number $x$:"}
+{"_id": "6211", "title": "Sine of Sum", "text": ":$\\map \\sin {a + b} = \\sin a \\cos b + \\cos a \\sin b$"}
+{"_id": "6215", "title": "Hyperbolic Sine Function is Odd", "text": ":$\\map \\sinh {-x} = -\\sinh x$"}
+{"_id": "6216", "title": "Exponential of Sum/Complex Numbers", "text": "Let $z_1, z_2 \\in \\C$ be complex numbers. Let $\\exp z$ be the exponential of $z$. Then: :$\\map \\exp {z_1 + z_2} = \\paren {\\exp z_1} \\paren {\\exp z_2}$"}
+{"_id": "6217", "title": "Hyperbolic Cosine Function is Even", "text": ":$\\map \\cosh {-x} = \\cosh x$"}
+{"_id": "6218", "title": "Hyperbolic Tangent Function is Odd", "text": "Let $\\tanh: \\C \\to \\C$ be the hyperbolic tangent function on the set of complex numbers. Then $\\tanh$ is odd: :$\\map \\tanh {-x} = -\\tanh x$"}
+{"_id": "6219", "title": "Hyperbolic Sine in terms of Sine", "text": "Let $z \\in \\C$ be a complex number. Then: :$i \\sinh z = \\map \\sin {i z}$ where: :$\\sin$ denotes the complex sine :$\\sinh$ denotes the hyperbolic sine :$i$ is the imaginary unit: $i^2 = -1$."}
+{"_id": "6220", "title": "Hyperbolic Cosine in terms of Cosine", "text": "Let $z \\in \\C$ be a complex number. Then: :$\\cosh z = \\map \\cos {i z}$ where: :$\\cos$ denotes the complex cosine :$\\cosh$ denotes the hyperbolic cosine :$i$ is the imaginary unit: $i^2 = -1$."}
+{"_id": "6221", "title": "Hyperbolic Tangent in terms of Tangent", "text": "Let $z \\in \\C$ be a complex number. Then: :$i \\tanh z = \\map \\tan {i z}$ where: :$\\tan$ denotes the tangent function :$\\tanh$ denotes the hyperbolic tangent :$i$ is the imaginary unit: $i^2 = -1$."}
+{"_id": "6222", "title": "Order-Extension Principle", "text": "Let $S$ be a set. Let $\\preceq$ be an ordering on $S$. Then there exists a total ordering $\\le$ on $S$ such that: :$\\forall a, b \\in S: \\left({a \\preceq b \\implies a \\le b}\\right)$"}
+{"_id": "6223", "title": "Union of Nest of Orderings is Ordering", "text": "Let $S$ be a set. Let $C$ be a nonempty nest of orderings on $S$. Then $\\bigcup C$ is an ordering on $S$."}
+{"_id": "6224", "title": "Ordering can be Expanded to compare Additional Pair", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a$ and $b$ be non-comparable elements of $S$. That is, let: :$a \\not\\preceq b$ and: :$b \\not\\preceq a$ Let ${\\preceq'} = {\\preceq} \\cup \\left\\{ {\\left({a, b}\\right)} \\right\\}$. Let $\\preceq'^+$ be the transitive closure of $\\preceq'$. Then: :$\\preceq'^+$ is an ordering. $\\preceq'^+$ can be defined by letting $p \\preceq'^+ q$ {{iff}}: :$p \\preceq q$ or :$p \\preceq a$ and $b \\preceq q$."}
+{"_id": "6226", "title": "Contour Integral is Well-Defined", "text": "Let $C$ be a contour defined by a finite sequence $C_1, \\ldots, C_n$ of directed smooth curves in the complex plane$\\C$. Let $C_i$ be parameterized by the smooth path $\\gamma_i: \\left[{a_i \\,.\\,.\\, b_i}\\right] \\to \\C$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. Let $f: \\operatorname{Im} \\left({C}\\right) \\to \\C$ be a continuous complex function, where $\\operatorname{Im} \\left({C}\\right)$ denotes the image of $C$. Suppose that $\\sigma_i: \\left[{c_i \\,.\\,.\\, d_i}\\right] \\to \\C$ is a reparameterization of $C_i$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. Then: :$\\displaystyle \\sum_{i \\mathop = 1}^n \\int_{a_i}^{b_i} f \\left({\\gamma_i \\left({t}\\right) }\\right) \\gamma_i' \\left({t}\\right) \\rd t = \\sum_{i \\mathop = 1}^n \\int_{c_i}^{d_i} f \\left({\\sigma_i \\left({t}\\right) }\\right) \\sigma_i' \\left({t}\\right) \\rd t$ and all complex Riemann integrals in the equation are defined."}
+{"_id": "6227", "title": "Boolean Prime Ideal Theorem", "text": "Let $\\left({S, \\le}\\right)$ be a Boolean algebra. Let $I$ be an ideal in $S$. Let $F$ be a filter on $S$. Let $I \\cap F = \\varnothing$. Then there exists a prime ideal $P$ in $S$ such that: : $I \\subseteq P$ and: : $P \\cap F = \\varnothing$"}
+{"_id": "6233", "title": "Tangent Exponential Formulation", "text": "Let $z$ be a complex number. Let $\\tan z$ denote the tangent function and $i$ denote the imaginary unit: $i^2 = -1$. Then:"}
+{"_id": "6234", "title": "Arcsine Logarithmic Formulation", "text": "For any real number $x: -1 \\le x \\le 1$: :$\\arcsin x = \\dfrac 1 i \\map \\ln {i x + \\sqrt {1 - x^2} }$ where $\\arcsin x$ is the arcsine and $i^2 = -1$."}
+{"_id": "6235", "title": "Arccosine Logarithmic Formulation", "text": "For any real number $x: -1 \\le x \\le 1$: :$\\arccos x = \\dfrac 1 i \\map \\ln {x + \\sqrt {x^2 - 1} }$ where $\\arccos x$ is the arccosine and $i^2 = -1$."}
+{"_id": "6236", "title": "Arctangent Logarithmic Formulation", "text": "For any real number $x$: :$\\arctan x = \\dfrac 1 2 i \\, \\map \\ln {\\dfrac {1 - i x} {1 + i x} }$ where $\\arctan x$ is the arctangent and $i^2 = -1$."}
+{"_id": "6239", "title": "Two Ring is Boolean Ring", "text": "Let $2$ be the two ring. Then $2$ is a Boolean ring."}
+{"_id": "6240", "title": "Idempotent Ring is Commutative", "text": "Let $\\struct {R, +, \\circ}$ be an idempotent ring. Denote with $0_R$ the zero of $R$. Then $\\struct {R, +, \\circ}$ is a commutative ring."}
+{"_id": "6241", "title": "Idempotent Ring has Characteristic Two", "text": "Let $\\struct {R, +, \\circ}$ be an idempotent non-null ring. Denote with $0_R$ the zero of $R$. Then $\\struct {R, +, \\circ}$ has characteristic $2$."}
+{"_id": "6242", "title": "Two-Valued Functions form Boolean Ring", "text": "Let $S$ be a set, and let $2$ be the two ring. Let $2^S$ be the set of all $2$-valued functions on $S$. Denote with $+$ and $\\cdot$ the pointwise operations induced on $2^S$ by $+_2$ and $\\times_2$, respectively. Then $\\struct {2^S, +, \\cdot}$ is a Boolean ring."}
+{"_id": "6245", "title": "Complex Integration by Substitution", "text": "Let $\\left[{a \\,.\\,.\\, b}\\right]$ be a closed real interval. Let $\\phi: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$ be a real function which has a derivative on $\\left[{a \\,.\\,.\\, b}\\right]$. Let $f: A \\to \\C$ be a continuous complex function, where $A$ is a subset of the image of $\\phi$. If $\\phi \\left({a}\\right) \\le \\phi \\left({b}\\right)$, then: :$\\displaystyle \\int_{\\phi \\left({a}\\right)}^{\\phi \\left({b}\\right)} f \\left({t}\\right) \\, \\mathrm d t = \\int_a^b f \\left({\\phi \\left({u}\\right)}\\right) \\phi' \\left({u}\\right) \\, \\mathrm d u$ If $\\phi \\left({a}\\right) > \\phi \\left({b}\\right)$, then: :$\\displaystyle \\int_{\\phi \\left({b}\\right)}^{\\phi \\left({a}\\right)} f \\left({t}\\right) \\, \\mathrm  dt = -\\int_a^b f \\left({\\phi \\left({u}\\right)}\\right) \\phi' \\left({u}\\right) ,\\ \\mathrm d u$"}
+{"_id": "6246", "title": "Continuous Injection of Interval is Strictly Monotone", "text": "Let $I$ be a real interval. Let $f: I \\to \\R$ be an injective continuous real function. Then $f$ is strictly monotone."}
+{"_id": "6247", "title": "Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice", "text": "Let $L$ be a distributive lattice. {{explain|Full notational specification needed}} Let $F$ be a filter in $L$. Let $M$ be an ideal in $L$ which is disjoint from $F$ such that no ideal in $L$ larger than $M$ is disjoint from $F$. Then $M$ is a prime ideal."}
+{"_id": "6248", "title": "Prime Ideal in Lattice", "text": "Let $\\left({L, \\le}\\right)$ be a lattice. Let $I$ be an ideal in $L$. Then $I$ is a prime ideal {{iff}}: :$\\forall a, b \\in L: a \\wedge b \\in I \\implies a \\in I \\text{ or } b \\in I$ where $a \\wedge b$ denotes $\\min \\left\\{{a, b}\\right\\}$, the meet of $a$ and $b$."}
+{"_id": "6249", "title": "Inverse of Increasing Bijection need not be Increasing", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be ordered sets. Let $\\phi: S \\to T$ be a bijection which is increasing. Then $\\phi^{-1}: T \\to S$ is not necessarily also increasing."}
+{"_id": "6252", "title": "Order Isomorphic Sets are Equivalent", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be ordered sets. Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be order isomorphic. Then $S$ and $T$ are equivalent."}
+{"_id": "6253", "title": "Linear Combination of Contour Integrals", "text": "Let $C$ be a contour in $\\C$. Let $f, g: \\Img C \\to \\C$ be continuous complex functions, where $\\Img C$ denotes the image of $C$. Let $\\lambda, \\mu \\in \\C$ be complex constants. Then: :$\\displaystyle \\int_C \\paren {\\lambda \\map f z + \\mu \\map g z} \\rd z = \\lambda \\int_C \\map f z \\rd z + \\mu \\int_C \\map g z \\rd z$"}
+{"_id": "6254", "title": "P-Product Metric Induces Product Topology", "text": "Let $M_A = \\struct{A, d_A}$ and $M_B = \\struct{B, d_B}$ be metric spaces. Let $\\tau_A$ and $\\tau_B$ be the topologies on $A$ and $B$ induced by $d_A$ and $d_B$, respectively. Let $p \\ge 1$ be an extended real number. Let $M = \\struct{A \\times B, d}$ be the $p$-product of $M_A$ and $M_B$. We have that $M$ is a metric space. Let $\\tau$ be the topology on $A \\times B$ induced by $d$. Then $\\struct{A \\times B, \\tau}$ is the product space of $\\struct{A, \\tau_A}$ and $\\struct{B, \\tau_B}$."}
+{"_id": "6255", "title": "P-Product Metrics are Lipschitz Equivalent", "text": "Let $M_A = \\left({A, d_A}\\right)$ and $M_B = \\left({B, d_B}\\right)$ be metric spaces. Let $\\tau_A$ and $\\tau_B$ be the topologies on $A$ and $B$ induced by $d_A$ and $d_B$, respectively. For all extended real numbers $p \\ge 1$, let $M_p = \\left({A \\times B, d_p}\\right)$ be the $p$-product of $M_A$ and $M_B$. Then all the metrics $\\left\\{{d_p: p \\ge 1}\\right\\}$ are Lipschitz equivalent."}
+{"_id": "6256", "title": "Ordered Sum of Tosets is Totally Ordered Set/General Result", "text": "Let $S_1, S_2, \\ldots, S_n$ all be tosets. Let $T_n$ be the ordered sum of $S_1, S_2, \\ldots, S_n$: :$\\forall n \\in \\N_{>0}: T_n = \\begin{cases} S_1 & : n = 1 \\\\ T_{n-1} + S_n & : n > 1 \\end{cases}$ Then $T_n$ is a toset."}
+{"_id": "6257", "title": "Supremum of Suprema", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $\\mathbb T \\subseteq \\powerset S$, where $\\powerset S$ is the power set of $S$. Suppose all $T \\in \\mathbb T$ admit a supremum $\\sup T$ in $S$. Then: :$\\sup \\bigcup \\mathbb T = \\sup {\\set {\\sup T: T \\in \\mathbb T} }$ if one of these two quantities exists (in $S$)."}
+{"_id": "6258", "title": "Ordered Product of Tosets is Totally Ordered Set/General Result", "text": "Let $S_1, S_2, \\ldots, S_n$ all be tosets. Let $T_n$ be the ordered product of $S_1, S_2, \\ldots, S_n$: :$\\forall n \\in \\N_{>0}: T_n = \\begin{cases} S_1 & : n = 1 \\\\ T_{n-1} \\cdot S_n & : n > 1 \\end{cases}$ Then $T_n$ is a toset."}
+{"_id": "6259", "title": "Derivative of Complex Composite Function", "text": "Let $f: D \\to \\C$ be a complex-differentiable function, where $D \\subseteq \\C$ is an open set. Let $g: \\operatorname{Im} \\left({f}\\right) \\to \\C$ be a complex-differentiable function, where $\\operatorname{Im} \\left({f}\\right)$ denotes the image of $f$. Define $h = f \\circ g: D \\to C$ as the composite of $f$ and $g$. Then $h$ is complex-differentiable, and its derivative is defined as: :$\\forall z \\in D: h' \\left({z}\\right) =  f' \\left({g \\left({z}\\right)}\\right) g' \\left({z}\\right)$"}
+{"_id": "6260", "title": "Ring of Idempotents is Idempotent Ring", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring. Let $\\struct {A, \\oplus, \\circ}$ be its ring of idempotents. Then $\\struct {A, \\oplus, \\circ}$ is an idempotent ring."}
+{"_id": "6261", "title": "Kelley's Theorem", "text": "Let $\\struct {D, \\preceq}$ be a directed set, Let $S$ be a non-empty set. Let $n: D \\to S$ be a net in $S$. Then $n$ has a universal subnet."}
+{"_id": "6263", "title": "Contour Integral is Independent of Parameterization", "text": "Let $C$ be a contour defined by a finite sequence $C_1, \\ldots, C_n$ of directed smooth curves. Let $C_i$ be parameterized by the smooth path $\\gamma_i: \\left[{a_i \\,.\\,.\\, b_i}\\right] \\to \\C$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. Let $f: \\operatorname{Im} \\left({C}\\right) \\to \\C$ be a continuous complex function, where $\\operatorname{Im} \\left({C}\\right)$ denotes the image of $C$. Suppose that $\\sigma_i: \\left[{c_i \\,.\\,.\\, d_i}\\right] \\to \\C$ is a reparameterization of $C_i$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. Then: :$\\displaystyle \\int_C f \\left({z}\\right) \\rd z = \\sum_{i \\mathop = 1}^n \\int_{a_i}^{b_i} f \\left({\\gamma_i \\left({t}\\right) }\\right) \\gamma_i' \\left({t}\\right) \\rd t = \\sum_{i \\mathop = 1}^n \\int_{c_i}^{d_i} f \\left({\\sigma_i \\left({t}\\right) }\\right) \\sigma_i' \\left({t}\\right) \\rd t$"}
+{"_id": "6264", "title": "Unity is Unity in Ring of Idempotents", "text": "Let $\\left({R, +, \\circ}\\right)$ be a commutative and unitary ring whose unity is $1_R$. Let $\\left({A, \\oplus, \\circ}\\right)$ be the ring of idempotents of $R$. Then $1_R$ is also a unity for $\\left({A, \\oplus, \\circ}\\right)$."}
+{"_id": "6265", "title": "Ring of Idempotents of Commutative and Unitary Ring is Boolean Ring", "text": "Let $\\struct {R, +, \\circ}$ be a commutative and unitary ring. Let $\\struct {A, \\oplus, \\circ}$ be its ring of idempotents. Then $\\struct {A, \\oplus, \\circ}$ is a Boolean ring."}
+{"_id": "6266", "title": "Definition:Boolean Group", "text": "Let $\\struct {G, \\circ}$ be a group. Then $\\struct {G, \\circ}$ is a '''Boolean group''' {{iff}} all its elements, other than the identity, have order $2$."}
+{"_id": "6267", "title": "Boolean Ring has Proper Zero Divisor", "text": "Let $\\left({R, +, \\circ}\\right)$ be a Boolean ring whose zero is $0_R$. Suppose that $R$ has more than two elements. Then $R$ has a proper zero divisor."}
+{"_id": "6268", "title": "Continuous Real Function on Closed Interval is Bijective iff Strictly Monotone", "text": "Let $\\closedint a b$ and $\\closedint c d$ be closed real intervals. Let $f: \\closedint c d \\to \\closedint a b$ be a continuous real function. Let $\\map f c, \\map f d \\in \\set {a, b}$. Then $f$ is bijective {{iff}} $f$ is strictly monotone."}
+{"_id": "6269", "title": "Concatenation of Contours is Contour", "text": "Let $C$ and $D$ be contours. That is, $C$ is a finite sequence of directed smooth curves $C_1, \\ldots, C_n$. Let $C_i$ be parameterized by the smooth path $\\gamma_i: \\left[{a_i\\,.\\,.\\,b_i}\\right] \\to \\C$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. Similarly, $D$ is a finite sequence of directed smooth curves $D_1, \\ldots, D_m$. Let $D_i$ be parameterized by the smooth path $\\sigma_i: \\left[{c_i\\,.\\,.\\,d_i}\\right] \\to \\C$ for all $i \\in \\left\\{ {1, \\ldots, m}\\right\\}$. Suppose $\\gamma_n \\left({b_n}\\right) = \\sigma_1 \\left({c_1}\\right)$. Then the finite sequence: :$C_1, \\ldots, C_n, D_1, \\ldots, D_m$ defines a contour."}
+{"_id": "6270", "title": "Contour Integral of Concatenation of Contours", "text": "Let $C$ and $D$ be contours in $\\C$. Suppose that the end point of $C$ is equal to the start point of $D$, so the concatenation $C \\cup D$ is defined. Let $f: \\operatorname{Im} \\left({C \\cup D}\\right) \\to \\C$ be a continuous complex function, where $\\operatorname{Im} \\left({C \\cup D}\\right)$ denotes the image of $C \\cup D$. Then: :$\\displaystyle \\int_{C \\cup D} f \\left({z}\\right) \\ \\mathrm dz = \\int_C f \\left({z}\\right) \\rd z + \\int_D f \\left({z}\\right) \\rd z$"}
+{"_id": "6271", "title": "Regular Expression is Accepted by Finite State Machine", "text": "Let $R$ be a regular expression. Then there exists a finite state machine $F$ such that its accepted language $L \\left({F}\\right)$ is exactly $L \\left({R}\\right)$, the language defined by $R$."}
+{"_id": "6272", "title": "Directed Smooth Curve Relation is Equivalence", "text": "Let $\\sim$ denote a relation on the set of all smooth paths: $\\left\\{ {\\gamma: I \\to \\C : \\text{$I$ is a closed real interval, $\\gamma$ is a smooth path} }\\right\\}$. Let $\\gamma : \\left[{ a \\,.\\,.\\, b }\\right] \\to \\C$ and $\\sigma : \\left[{c \\,.\\,.\\, d}\\right] \\to \\C$ be smooth paths. Define $\\sim$ as follows:  :$\\gamma \\sim \\sigma$ {{iff}} there exists a bijective differentiable strictly increasing real function $\\phi: \\left[{c \\,.\\,.\\, d}\\right] \\to \\left[{a \\,.\\,.\\, b}\\right]$ such that $\\sigma = \\gamma \\circ \\phi$. Then $\\sim$ is an equivalence relation on the set of all smooth paths."}
+{"_id": "6273", "title": "Max Operation Representation on Real Numbers", "text": "Let $x, y \\in \\R$. Then: :$\\max \\left\\{{x, y}\\right\\} = \\frac 1 2 \\left({x + y + \\left\\vert{x - y}\\right\\vert}\\right)$ where $\\operatorname{max}$ denotes the max operation."}
+{"_id": "6274", "title": "Reparameterization of Directed Smooth Curve Preserves Image", "text": "Let $\\left[{a \\,.\\,.\\, b}\\right]$ and $\\left[{c\\,.\\,.\\, d}\\right]$ be closed real intervals. Let $\\gamma : \\left[{a \\,.\\,.\\, b}\\right] \\to \\C$ be a smooth path. Let $C$ be a directed smooth curve with parameterization $\\gamma$. Suppose that $\\sigma : \\left[{c \\,.\\,.\\, d}\\right] \\to \\C$ is a reparameterization of $C$. Then $\\operatorname{Im} \\left({\\gamma}\\right) = \\operatorname{Im} \\left({\\sigma}\\right)$, where $\\operatorname{Im} \\left({\\gamma}\\right)$ denotes the image of $\\gamma$."}
+{"_id": "6275", "title": "Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints", "text": "Let $\\R^n$ be a real cartesian space of $n$ dimensions. Let $\\left[{a \\,.\\,.\\, b}\\right]$ and $\\left[{c \\,.\\,.\\, d}\\right]$ be closed real intervals. Let $\\gamma: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R^n$ be a smooth path in $\\R^n$. Let $C$ be a directed smooth curve with parameterization $\\gamma$. Suppose that $\\sigma: \\left[{c \\,.\\,.\\, d}\\right] \\to \\C$ is a reparameterization of $C$. Then the start points and end points of $\\gamma$ and $\\sigma$ are identical: :$\\gamma \\left({a}\\right) = \\sigma \\left({c}\\right)$ :$\\gamma \\left({b}\\right) = \\sigma \\left({d}\\right)$"}
+{"_id": "6276", "title": "Product of Ring Negatives", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Then: :$\\forall x, y \\in \\struct {R, +, \\circ}: \\paren {-x} \\circ \\paren {-y} = x \\circ y$ where $\\paren {-x}$ denotes the negative of $x$."}
+{"_id": "6277", "title": "Distance Function for Distinct Elements in Metric Space is Strictly Positive", "text": "Let $A$ be a set. Let $d: A \\times A \\to \\R$ be a real-valued function on $A$ with the following properties: {{begin-axiom}} {{axiom|n = M1'        |q = \\forall x, y \\in A        |m = d \\left({x, y}\\right) = 0 \\iff x = y }} {{axiom|n = M2        |q = \\forall x, y, z \\in A        |m = d \\left({x, y}\\right) + d \\left({y, z}\\right) \\ge d \\left({x, z}\\right) }} {{axiom|n = M3        |q = \\forall x, y \\in A        |m = d \\left({x, y}\\right) = d \\left({y, x}\\right) }} {{end-axiom}} which can be considered as being an alternative formulation of the metric space axioms. Then: :$\\forall x, y \\in A: x \\ne y \\implies d \\left({x, y}\\right) > 0$ which is metric space axiom $(M4)$. Thus $d$ is a distance function, so making $M := \\left({A, d}\\right)$ a metric space."}
+{"_id": "6280", "title": "Transpose of Upper Triangular Matrix is Lower Triangular", "text": "The transpose of an upper triangular matrix is a lower triangular matrix."}
+{"_id": "6281", "title": "Derivative iff Right and Left Derivative", "text": "Let $f$ be a real function. Then $f$ is differentiable {{iff}} it has both right- and left-hand derivatives which agree."}
+{"_id": "6282", "title": "Composition of Ring Homomorphisms is Ring Homomorphism", "text": "Let: :$\\struct {R_1, +_1, \\circ_1}$ :$\\struct {R_2, +_2, \\circ_2}$ :$\\struct {R_3, +_3, \\circ_3}$ be rings. Let: :$\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ :$\\psi: \\struct {R_2, +_2, \\circ_2} \\to \\struct {R_3, +_3, \\circ_3}$ be homomorphisms. Then the composite of $\\phi$ and $\\psi$ is also a homomorphism."}
+{"_id": "6285", "title": "Composition of Ring Epimorphisms is Ring Epimorphism", "text": "Let: :$\\struct {R_1, +_1, \\circ_1}$ :$\\struct {R_2, +_2, \\circ_2}$ :$\\struct {R_3, +_3, \\circ_3}$ be rings. Let: :$\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ :$\\psi: \\struct {R_2, +_2, \\circ_2} \\to \\struct {R_3, +_3, \\circ_3}$ be (ring) epimorphisms. Then the composite of $\\phi$ and $\\psi$ is also a (ring) epimorphism."}
+{"_id": "6287", "title": "Composition of Ring Isomorphisms is Ring Isomorphism", "text": "Let: * $\\left({R_1, +_1, \\circ_1}\\right)$ * $\\left({R_2, +_2, \\circ_2}\\right)$ * $\\left({R_3, +_3, \\circ_3}\\right)$ be rings. Let: * $\\phi: \\left({R_1, +_1, \\circ_1}\\right) \\to \\left({R_2, +_2, \\circ_2}\\right)$ * $\\psi: \\left({R_2, +_2, \\circ_2}\\right) \\to \\left({R_3, +_3, \\circ_3}\\right)$ be (ring) isomorphisms. Then the composite of $\\phi$ and $\\psi$ is also a (ring) isomorphism."}
+{"_id": "6288", "title": "Composition of Ring Endomorphisms is Ring Endomorphism", "text": "Let $R$ be a set. Let: : $\\left({R, +_1, \\circ_1}\\right)$ : $\\left({R, +_2, \\circ_2}\\right)$ : $\\left({R, +_3, \\circ_3}\\right)$ be rings. Let: : $\\phi: \\left({R, +_1, \\circ_1}\\right) \\to \\left({R, +_2, \\circ_2}\\right)$ : $\\psi: \\left({R, +_2, \\circ_2}\\right) \\to \\left({R, +_3, \\circ_3}\\right)$ be (ring) endomorphisms. Then the composite of $\\phi$ and $\\psi$ is also a (ring) endomorphism."}
+{"_id": "6289", "title": "Composition of Ring Automorphisms is Ring Automorphism", "text": "Let $R$ be a set. Let: * $\\left({R, +_1, \\circ_1}\\right)$ * $\\left({R, +_2, \\circ_2}\\right)$ * $\\left({R, +_3, \\circ_3}\\right)$ be rings. Let: * $\\phi: \\left({R, +_1, \\circ_1}\\right) \\to \\left({R, +_2, \\circ_2}\\right)$ * $\\psi: \\left({R, +_2, \\circ_2}\\right) \\to \\left({R, +_3, \\circ_3}\\right)$ be (ring) automorphisms. Then the composite of $\\phi$ and $\\psi$ is also a (ring) automorphism."}
+{"_id": "6290", "title": "Set has Rank", "text": "Let $S$ be a set. Then $S$ has a rank."}
+{"_id": "6292", "title": "Quotient Epimorphism is Epimorphism/Group", "text": "Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $G / N$ be the quotient group of $G$ by $N$. Let $q_N: G \\to G / N$ be the quotient epimorphism from $G$ to $G / N$: :$\\forall x \\in G: \\map {q_N} x = x N$ Then $q_N$ is a group epimorphism whose kernel is $N$."}
+{"_id": "6293", "title": "Strictly Increasing Mapping on Well-Ordered Class", "text": "Let $ \\left({S, \\prec}\\right)$ be a strictly well-ordered class. Let $\\left({T, <}\\right)$ be a strictly ordered class. Let $f$ be a mapping from $S$ to $T$. For each $i \\in S$ such that $i$ is not maximal in $S$, let: : $f \\left({i}\\right) < f \\left({\\operatorname{succ} \\left({i}\\right)}\\right)$ where $\\operatorname{succ} \\left({i}\\right)$ is the immediate successor element of $i$. Let: :$\\forall i, j \\in S: i \\preceq j \\implies f \\left({i}\\right) \\le f \\left({j}\\right)$ Then for each $i, j \\in S$ such that $i \\prec j$: :$f \\left({i}\\right) < f \\left({j}\\right)$ {{MissingLinks|In the above, link to definitions of increasing mapping and strictly increasing mapping, in order to correlate with page title.}}"}
+{"_id": "6296", "title": "Non-Maximal Element of Well-Ordered Class has Immediate Successor", "text": "Let $\\left({C, \\le}\\right)$ be a well-ordered class. Let $x \\in C$. Suppose that $x$ is maximal in $C$. {{mistake|maximal?}} Then $x$ has an immediate successor in $C$."}
+{"_id": "6297", "title": "Set Contained in Smallest Transitive Set", "text": "Let $S$ be a set. Then there exists a transitive set $G$ such that: :$S \\subseteq G$ and: :if $Q$ is any transitive set such that $S \\subseteq Q$, then $G \\subseteq Q$."}
+{"_id": "6298", "title": "Transitive Set Contained in Von Neumann Hierarchy Level", "text": "Let $G$ be a transitive set. Then for some ordinal $i$, $G \\subseteq V_i$."}
+{"_id": "6299", "title": "External Direct Product Closure/General Result", "text": "Let $\\displaystyle \\struct {S, \\circ} = \\prod_{k \\mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\\struct {S_1, \\circ_1}, \\struct {S_2, \\circ_2}, \\ldots, \\struct {S_n, \\circ_n}$. Let $\\struct {S_1, \\circ_1}, \\struct {S_2, \\circ_2}, \\ldots, \\struct {S_n, \\circ_n}$ all be closed algebraic structures. Then $\\struct {S, \\circ}$ is also a closed algebraic structure."}
+{"_id": "6300", "title": "External Direct Product of Abelian Groups is Abelian Group/General Result", "text": "The external direct product of a finite sequence of abelian groups is itself an abelian group."}
+{"_id": "6301", "title": "Homomorphism of External Direct Products/General Result", "text": "Let $n \\in \\N_{>0}$. Let: : $\\displaystyle \\left({\\mathcal S_n, \\circledcirc_n}\\right) := \\prod_{k \\mathop = 1}^n S_k = \\left({S_1, \\circ_1}\\right) \\times \\left({S_2, \\circ_2}\\right) \\times \\cdots \\times \\left({S_n, \\circ_n}\\right)$ : $\\displaystyle \\left({\\mathcal T_n, \\circledast_n}\\right) := \\prod_{k \\mathop = 1}^n T_k = \\left({T_1, \\ast_1}\\right) \\times \\left({T_2, \\ast_2}\\right) \\times \\cdots \\times \\left({T_n, \\ast_n}\\right)$ be external direct products of algebraic structures. Let $\\Phi_n: \\left({\\mathcal S_n, \\circledcirc_n}\\right) \\to \\left({\\mathcal T_n, \\circledast_n}\\right)$ be the mapping defined as: :$\\Phi_n: \\left({s_1, \\ldots, s_n}\\right) := \\begin{cases} \\phi_1 \\left({s_1}\\right) & : n = 1 \\\\ \\left({\\phi_1 \\left({s_1}\\right), \\phi_2 \\left({s_2}\\right)}\\right) & : n = 2 \\\\ \\left({\\Phi_n \\left({s_1, \\ldots, s_{n - 1} }\\right), \\phi_n \\left({s_n}\\right)}\\right) & : n > 2 \\\\ \\end{cases}$ That is: :$\\Phi_n: \\left({s_1, \\ldots, s_n}\\right) := \\left({\\phi_1 \\left({s_1}\\right), \\phi_2 \\left({s_2}\\right), \\ldots, \\phi_n \\left({s_n}\\right)}\\right)$ Let $\\phi_k: \\left({S_k, \\circ_k}\\right) \\to \\left({T_k, \\ast_k}\\right)$ be a homomorphism for each $k \\in \\left\\{{1, 2, \\ldots, n}\\right\\}$. Then $\\Phi_n$ is a homomorphism from $\\left({\\mathcal S_n, \\circledcirc_n}\\right)$ to $\\left({\\mathcal T_n, \\circledast_n}\\right)$."}
+{"_id": "6302", "title": "Cesàro Mean", "text": "Let $\\sequence {a_n}$ be a sequence of complex numbers. Suppose that $\\sequence {a_n}$ converges to $\\ell$ in $\\C$: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} a_n = \\ell$ Then also: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\frac {a_1 + \\dotsb + a_n} n = \\ell$"}
+{"_id": "6303", "title": "Characterization of Minimal Element", "text": "Let $C$ be a class. Let $\\prec$ be a relation on $C$. Let $B$ be a subclass of $C$. Let $x \\in B$. Let $S_x = \\left\\{ {y \\in C: y \\prec x \\text{ and } y \\ne x}\\right\\}$ be the initial segment of $x$ in $C$. Then $x$ is a minimal element of $B$ {{iff}} $B \\cap S_x = \\varnothing$."}
+{"_id": "6304", "title": "Non-Empty Class has Element of Least Rank", "text": "Let $C$ be a class. Let $C \\ne \\varnothing$. Then $C$ has an element of least rank. That is: :$\\exists x \\in C: \\forall y \\in C: \\operatorname{rank} \\left({x}\\right) \\le \\operatorname {rank}\\left({y}\\right)$ where $\\operatorname{rank}\\left({x}\\right)$ is the rank of $x$."}
+{"_id": "6305", "title": "Reversed Directed Smooth Curve is Directed Smooth Curve", "text": "Let $C$ be a directed smooth curve in $\\C$. Let $C$ be parameterized by the smooth path $\\gamma: \\left[{a \\,.\\,.\\, b}\\right] \\to \\C$. Define $\\psi: \\left[{a \\,.\\,.\\, b}\\right] \\to \\left[{a \\,.\\,.\\, b}\\right]$ by $\\psi \\left({t}\\right) = a + b - t$. Define $\\rho: \\left[{a \\,.\\,.\\, b}\\right] \\to \\C$ by $\\rho = \\gamma \\circ \\psi$. Then $\\rho$ is a smooth path which parameterizes a directed smooth curve $-C$. The directed smooth curve $-C$ is independent of the parameterization $\\gamma$."}
+{"_id": "6306", "title": "Reversed Contour is Contour", "text": "Let $\\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ be a contour in $\\R^n$ that is defined as a concatenation of a finite sequence $C_1, \\ldots, C_n$ of directed smooth curves in $\\R^n$. Then the finite sequence of reversed directed smooth curves: :$-C_n, -C_{n - 1}, \\ldots, -C_1$ defines a contour that is independent of the parameterization of $C_1, \\ldots, C_n$."}
+{"_id": "6307", "title": "Contour Integral along Reversed Contour", "text": "Let $C$ be a contour. Let $f: \\Img C \\to \\C$ be a continuous complex functions, where $\\Img C$ denotes the image of $C$. Then the contour integral of $f$ along the reversed contour $-C$ is: :$\\displaystyle \\int_{-C} \\map f z \\rd z = -\\int_C \\map f z \\rd z$"}
+{"_id": "6308", "title": "Well-Founded Relation is Strongly Well-Founded/Lemma", "text": "Let $A$ be a non-empty class. Let $\\prec$ be a foundational relation on $A$. Then $A$ has a $\\prec$-minimal element."}
+{"_id": "6315", "title": "Alternative Differentiability Condition", "text": "Let $\\mathbb K$ be either $\\R$ or $\\C$. Let $f: D \\to \\mathbb K$ be a continuous mapping, where $D \\subseteq \\mathbb K$ is an open set. Let $z \\in \\mathbb K$. Then $f$ is differentiable at $z$ {{iff}} there exist $\\alpha \\in \\mathbb K$ and $r \\in \\R_{>0}$ such that for all $h \\in \\map {B_r} 0 \\setminus \\set 0$: :$\\map f {z + h} = \\map f z + h \\paren {\\alpha + \\map \\epsilon h}$ where $\\map {B_r} 0$ denotes an open ball of $0$, and $\\epsilon: \\map {B_r} 0 \\setminus \\set 0 \\to \\mathbb K$ is a continuous mapping with $\\displaystyle \\lim_{h \\mathop \\to 0} \\map \\epsilon h = 0$. If the conditions are true, then $\\alpha = \\map {f'} z$."}
+{"_id": "6316", "title": "Von Neumann Hierarchy is Cumulative", "text": "For any two ordinals $x$ and $y$, If $x < y$ then $V(x) \\subsetneqq V(y)$."}
+{"_id": "6328", "title": "Odd Number Theorem/Corollary", "text": "A recurrence relation for the square numbers is: :$S_n = S_{n - 1} + 2 n - 1$"}
+{"_id": "6331", "title": "Effect of Sequence of Elementary Row Operations on Determinant", "text": "Let $\\hat o_1, \\ldots, \\hat o_m$ be a finite sequence of elementary row operations.  Here, $\\hat o_i$ denotes an elementary row operation on a square matrix of order $n$ over a commutative ring with unity $\\struct {R, +, \\circ}$. Here, $i \\in \\set {1, \\ldots, m}$. Then there exists $c \\in R$ such that for all square matrices of order $n$ $\\mathbf A$ over $R$: :$\\map \\det {\\mathbf A} = c \\map \\det {\\mathbf A'}$ where $\\mathbf A'$ is the square matrix of order $n$ that results from applying the elementary row operations $\\hat o_1, \\ldots, \\hat o_m$ on $\\mathbf A$."}
+{"_id": "6334", "title": "Alternative Differentiability Condition/Proof 1", "text": "Let $\\mathbb K$ be either $\\R$ or $\\C$. Let $f: D \\to \\mathbb K$ be a continuous mapping, where $D \\subseteq \\mathbb K$ is an open set. Let $z \\in \\mathbb K$. Then: : $f$ is differentiable at $z$ {{iff}}: : there exist $\\alpha \\in \\mathbb K$ and $r \\in \\R_{>0}$ such that for all $h \\in B_r \\left({0}\\right) \\setminus \\left\\{ {0}\\right\\}$: ::$f \\left({z + h}\\right) = f \\left({z}\\right) + h \\left({\\alpha + \\epsilon \\left({h}\\right) }\\right)$ where: : $B_r \\left({0}\\right)$ denotes an open ball of $0$ : $\\epsilon: B_r \\left({0}\\right) \\setminus \\left\\{ {0}\\right\\} \\to \\mathbb K$ is a continuous mapping with $\\displaystyle \\lim_{h \\to 0} \\ \\epsilon \\left({h}\\right) = 0$. If the conditions are true, then $\\alpha = f' \\left({z}\\right)$."}
+{"_id": "6335", "title": "Alternative Differentiability Condition/Proof 2", "text": "Let $\\mathbb K$ be either $\\R$ or $\\C$. Let $f: D \\to \\mathbb K$ be a continuous mapping, where $D \\subseteq \\mathbb K$ is an open set. Let $z \\in \\mathbb K$. Then $f$ is differentiable at $z$ {{iff}} there exist $\\alpha \\in \\mathbb K$ and $r \\in \\R_{>0}$ such that for all $h \\in B_r \\left({0}\\right) \\setminus \\left\\{ {0}\\right\\}$: :$f \\left({z + h}\\right) = f \\left({z}\\right) + h \\left({\\alpha + \\epsilon \\left({h}\\right) }\\right)$ where $B_r \\left({0}\\right)$ denotes an open ball of $0$, and $\\epsilon: B_r \\left({0}\\right) \\setminus \\left\\{ {0}\\right\\} \\to \\mathbb K$ is a continuous mapping with $\\displaystyle \\lim_{h \\to 0} \\ \\epsilon \\left({h}\\right) = 0$. If the conditions are true, then $\\alpha = f' \\left({z}\\right)$."}
+{"_id": "6336", "title": "Every Filter has Limit Point implies Every Ultrafilter Converges", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let each filter on $S$ have a limit point in $S$. Then each ultrafilter on $S$ converges to a point in $S$."}
+{"_id": "6337", "title": "Every Ultrafilter Converges implies Every Filter has Limit Point", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let every ultrafilter on $S$ be convergent. Then every filter on $S$ has a limit point."}
+{"_id": "6338", "title": "Left Distributive and Commutative implies Distributive", "text": "Let $\\struct {S, \\circ, *}$ be an algebraic structure. Let the operation $\\circ$ be left distributive over the operation $*$. Let $\\circ$ be commutative. Then $\\circ$ is distributive over $*$."}
+{"_id": "6339", "title": "Filter on Product Space Converges iff Projections Converge", "text": "Let $\\family{X_i}_{i \\mathop \\in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set. Let $\\displaystyle X := \\prod_{i \\mathop \\in I} X_i$ be the corresponding product space. Let $\\pr_i: X \\to X_i$ denote the projection from $X$ onto $X_i$. Let $\\FF \\subset \\powerset X$ be a filter on $X$. Then $\\FF$ converges {{iff}} for all $i \\in I$ the image filter $\\map {\\pr_i} \\FF$ converges."}
+{"_id": "6340", "title": "Tychonoff's Theorem for Hausdorff Spaces", "text": "Let $\\family {X_i}_{i \\mathop \\in I}$ be an indexed family of non-empty Hausdorff spaces, where $I$ is an arbitrary index set. Let $\\displaystyle X = \\prod_{i \\mathop \\in I} X_i$ be the corresponding product space. Then $X$ is compact {{iff}} each $X_i$ is compact. == Proof == First assume that $X$ is compact.  From Projection from Product Topology is Continuous, the projections $\\pr_i : X \\to X_i$ are continuous. From Continuous Image of Compact Space is Compact, it follows that the $X_i$ are compact. Assume now that each $X_i$ is compact. By Equivalent Definitions of Compactness it is enough to show that every ultrafilter on $X$ converges. Thus let $\\FF$ be an ultrafilter on $X$. From Image of Ultrafilter is Ultrafilter, for each $i \\in I$, the image filter $\\map {\\pr_i} \\FF$ is an ultrafilter on $X_i$. Each $X_i$ is compact by assumption. So by Equivalent Definitions of Compactness, each $\\map {\\pr_i} \\FF$ converges. From Filter on Product of Hausdorff Spaces Converges iff Projections Converge, $\\FF$ converges. {{explain|Find the appropriate link for \"converges\" and explain why this proves the result.}} {{qed}}"}
+{"_id": "6341", "title": "Filter on Product of Hausdorff Spaces Converges iff Projections Converge", "text": "Let $\\family{X_i}_{i \\mathop \\in I}$ be an indexed family of non-empty Hausdorff spaces where $I$ is an arbitrary index set. Let $\\displaystyle X := \\prod_{i \\mathop \\in I} X_i$ be the corresponding product space. Let $\\pr_i: X \\to X_i$ denote the projection from $X$ onto $X_i$. Let $\\FF \\subset \\powerset X$ be a filter on $X$. Then $\\FF$ converges {{iff}} for each $i \\in I$ the image filter $\\map {\\pr_i} \\FF$ converges."}
+{"_id": "6342", "title": "Tychonoff's Theorem/General Case", "text": "Let $\\family {X_i}_{i \\mathop \\in I}$ be an indexed family of non-empty topological spaces, where $I$ is an arbitrary index set. Let $\\displaystyle X = \\prod_{i \\mathop \\in I} X_i$ be the corresponding product space. Then $X$ is compact {{iff}} each $X_i$ is compact."}
+{"_id": "6343", "title": "Positive Real has Real Square Root", "text": "Let $x \\in \\R_{>0}$ be a (strictly) positive real number. Then: :$\\exists y \\in \\R: x = y^2$"}
+{"_id": "6344", "title": "Join is Idempotent", "text": "Let $\\struct {S, \\vee, \\preceq}$ be a join semilattice. Then $\\vee$ is idempotent."}
+{"_id": "6346", "title": "Supremum of Singleton", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then for all $a \\in S$: :$\\sup \\set a = a$ where $\\sup$ denotes supremum."}
+{"_id": "6347", "title": "Infimum of Singleton", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then for all $a \\in S$: :$\\inf \\set a = a$ where $\\inf$ denotes infimum."}
+{"_id": "6348", "title": "Meet is Idempotent", "text": "Let $\\struct {S, \\wedge, \\preceq}$ be a meet semilattice. Then $\\wedge$ is idempotent."}
+{"_id": "6349", "title": "GCD from Prime Decomposition", "text": "Let $a, b \\in \\Z$. From Expression for Integers as Powers of Same Primes, let: {{begin-eqn}} {{eqn | l = a       | r = p_1^{k_1} p_2^{k_2} \\ldots p_r^{k_r} }} {{eqn | l = b       | r = p_1^{l_1} p_2^{l_2} \\ldots p_r^{l_r} }} {{eqn | ll= \\forall i \\in \\set {1, 2, \\dotsc, r}:       | l = p_i       | o = \\divides       | r = a       | c =  }} {{eqn | lo= \\lor       | l = p_i       | o = \\divides       | r = b }} {{end-eqn}} That is, the primes given in these prime decompositions may be divisors of ''either'' of the numbers $a$ or $b$. Then: :$\\gcd \\set {a, b} = p_1^{\\min \\set {k_1, l_1} } p_2^{\\min \\set {k_2, l_2} } \\ldots p_r^{\\min \\set {k_r, l_r} }$ where $\\gcd \\set {a, b}$ denotes the greatest common divisor of $a$ and $b$."}
+{"_id": "6350", "title": "LCM from Prime Decomposition", "text": "Let $a, b \\in \\Z$. From Expression for Integers as Powers of Same Primes, let: {{begin-eqn}} {{eqn | l = a       | r = p_1^{k_1} p_2^{k_2} \\ldots p_r^{k_r} }} {{eqn | l = b       | r = p_1^{l_1} p_2^{l_2} \\ldots p_r^{l_r} }} {{eqn | ll= \\forall i \\in \\set {1, 2, \\dotsc, r}:       | l = p_i       | o = \\divides       | r = a       | c =  }} {{eqn | lo= \\lor       | l = p_i       | o = \\divides       | r = b }} {{end-eqn}} That is, the primes given in these prime decompositions may be divisors of ''either'' of the numbers $a$ or $b$. Then: :$\\lcm \\set {a, b} = p_1^{\\max \\set {k_1, l_1} } p_2^{\\max \\set {k_2, l_2} } \\ldots p_r^{\\max \\set {k_r, l_r} }$ where $\\lcm \\set {a, b}$ denotes the lowest common multiple of $a$ and $b$."}
+{"_id": "6351", "title": "Product of Positive Element and Element Greater than One", "text": "Let $\\left({R, +, \\circ, \\le}\\right)$ be an ordered ring with unity $1_R$ and zero $0_R$. Let $x,y \\in R$. Suppose that $x > 0_R$ and $y > 1_R$. Then $x \\circ y > x$ and $y \\circ x > x$."}
+{"_id": "6352", "title": "Strictly Positive Power of Strictly Positive Element Greater than One Succeeds Element", "text": "Let $\\left({R, +, \\circ, \\le}\\right)$ be an ordered ring with unity. Let $x \\in R$ with $x > 1$ and $x > 0$. Let $n \\in \\N_{>0}$. Then: : $\\circ^n x \\ge x$"}
+{"_id": "6353", "title": "Directed Set has Strict Successors iff Unbounded Above", "text": "Let $\\left({S, \\le}\\right)$ be a directed set. Then every element of $S$ has a strict successor in $S$ iff $S$ has no upper bound in $S$."}
+{"_id": "6354", "title": "Complex Numbers are Uncountable", "text": "The set of complex numbers $\\C$ is uncountably infinite."}
+{"_id": "6355", "title": "Infimum of Infima", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $\\mathbb T$ be a collection of subsets of $S$. Suppose all $T \\in \\mathbb T$ admit an infimum $\\inf T$ in $S$. Then: :$\\inf \\bigcup \\mathbb T = \\inf \\left\\{{\\inf T: T \\in \\mathbb T}\\right\\}$ as soon as one of these two quantities exists."}
+{"_id": "6358", "title": "Join Semilattice is Ordered Structure", "text": "Let $\\left({S, \\vee, \\preceq}\\right)$ be a join semilattice. Then $\\left({S, \\vee, \\preceq}\\right)$ is an ordered structure. That is, $\\preceq$ is compatible with $\\vee$."}
+{"_id": "6359", "title": "Meet Semilattice is Ordered Structure", "text": "Let $\\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Then $\\left({S, \\wedge, \\preceq}\\right)$ is an ordered structure."}
+{"_id": "6360", "title": "Join Semilattice is Semilattice", "text": "Let $\\struct {S, \\vee, \\preceq}$ be a join semilattice. Then $\\struct {S, \\vee}$ is a semilattice."}
+{"_id": "6361", "title": "Meet Semilattice is Semilattice", "text": "Let $\\struct {S, \\wedge, \\preceq}$ be a meet semilattice. Then $\\struct {S, \\wedge}$ is a semilattice."}
+{"_id": "6362", "title": "Semilattice Induces Ordering", "text": "Let $\\struct {S, \\circ}$ be a semilattice. Let $\\preceq$ be the relation on $S$ defined by, for all $a, b \\in S$: :$a \\preceq b$ {{iff}} $a \\circ b = b$ Then $\\preceq$ is an ordering."}
+{"_id": "6363", "title": "Elementary Row Operations Commute with Matrix Multiplication", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring. Let $\\mathbf A = \\sqbrk a_{m n}$ be an $m \\times n$ matrix over $R$. Let $\\mathbf B = \\sqbrk b_{n p}$ be an $n \\times p$ matrix over $R$. Let $\\hat o_1, \\ldots, \\hat o_{\\hat n}$ be a finite sequence of elementary row operations that can be performed on a matrix over $R$ with $m$ rows. Let $\\mathbf A'$ denote the $m \\times n$-matrix that results from using $\\hat o_1, \\ldots, \\hat o_{\\hat n}$ on $\\mathbf A$. Let $\\mathbf C = \\mathbf A \\mathbf B$ be the matrix product of $\\mathbf A$ and $\\mathbf B$. Let $\\mathbf C'$ denote the $m \\times p$-matrix that results from using $\\hat o_1, \\ldots, \\hat o_{\\hat n}$ on $\\mathbf C$. Then: :$\\mathbf C' = \\mathbf A' \\mathbf B$"}
+{"_id": "6364", "title": "Hypothetical Syllogism/Formulation 1", "text": ":$p \\implies q, q \\implies r \\vdash p \\implies r$"}
+{"_id": "6368", "title": "Hypothetical Syllogism/Formulation 2/Proof 1", "text": ": $p \\implies q, q \\implies r, p \\vdash r$"}
+{"_id": "6370", "title": "Hypothetical Syllogism/Formulation 2/Proof 3", "text": ": $p \\implies q, q \\implies r, p \\vdash r$"}
+{"_id": "6371", "title": "Union of Relations Compatible with Operation is Compatible", "text": "Let $\\left({S, \\circ}\\right)$ be a closed algebraic structure. Let $\\mathcal F$ be a family of relations on $S$. Let each element of $\\mathcal F$ be compatible with $\\circ$. Let $\\mathcal Q = \\bigcup \\mathcal F$. Then $Q$ is a relation compatible with $\\circ$."}
+{"_id": "6372", "title": "Complement of Relation Compatible with Group is Compatible", "text": "Let $\\left({G, \\circ}\\right)$ be a group. Let $\\mathcal R$ be a relation on $G$. Let $\\mathcal R$ be compatible with $\\circ$. Let $\\mathcal Q = \\complement_{G \\times G} \\mathcal R$, so that: :$\\forall a, b \\in G: a \\mathop {\\mathcal Q} b \\iff \\neg \\left({a \\mathop {\\mathcal R} b}\\right)$ Then $\\mathcal Q$ is a relation compatible with $\\circ$."}
+{"_id": "6373", "title": "Relation Compatible with Group Operation is Strongly Compatible", "text": "Let $\\left({S,\\circ}\\right)$ be a group. Let $\\mathcal R$ be an endorelation on $S$ compatible with $\\circ$. Let $x, y, z \\in S$. Then $\\mathcal R$ is strongly compatible with $\\circ$. That is, the following equivalences hold: :$x \\mathrel{\\mathcal R} y \\iff x \\circ z \\mathrel{\\mathcal R} y \\circ z$ :$x \\mathrel{\\mathcal R} y \\iff z \\circ x \\mathrel{\\mathcal R} z \\circ y$"}
+{"_id": "6374", "title": "Intersection of Relations Compatible with Operation is Compatible", "text": "Let $\\struct {S, \\circ}$ be a closed algebraic structure. Let $\\mathscr F$ be a indexed family of relations on $S$. Suppose that each element of $\\mathscr F$ is compatible with $\\circ$. Let $\\QQ = \\bigcap \\mathscr F$ be the intersection of $\\mathscr F$. Then $\\QQ$ is a relation compatible with $\\circ$."}
+{"_id": "6375", "title": "Inverse of Relation Compatible with Operation is Compatible", "text": "Let $\\left({S, \\circ}\\right)$ be a closed algebraic structure. Let $\\mathcal R$ be a relation on $S$ which is compatible with $\\circ$. Let $\\mathcal Q$ be the inverse relation of $\\mathcal R$. Then $\\mathcal Q$ is compatible with $\\circ$."}
+{"_id": "6376", "title": "Properties of Ordered Group/OG1", "text": "{{begin-axiom}} {{axiom|n = \\operatorname{OG}1.1        |m = x \\preceq y \\iff x \\circ z \\preceq y \\circ z }} {{axiom|n = \\operatorname{OG}1.2        |m = x \\preceq y \\iff z \\circ x \\preceq z \\circ y }} {{axiom|n = \\operatorname{OG}1.1'        |m = x \\prec y \\iff x \\circ z \\prec y \\circ z }} {{axiom|n = \\operatorname{OG}1.2'        |m = x \\prec y \\iff z \\circ x \\prec z \\circ y }} {{end-axiom}}"}
+{"_id": "6377", "title": "Relations Compatible with Operation Form Complete Distributive Lattice", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure. Let $C$ be the set of relations on $S$ compatible with $\\circ$. Then $\\left({C, \\cap,\\cup,\\subseteq}\\right)$ is a complete distributive lattice."}
+{"_id": "6378", "title": "Relations Compatible with Group Form Complete Boolean Algebra", "text": "Let $\\left({S,\\circ}\\right)$ be a group. Let $C$ be the set of relations on $S$ which are compatible with $\\circ$. Then $\\left({C, \\cap, \\cup, \\subseteq}\\right)$ is a complete Boolean lattice."}
+{"_id": "6379", "title": "Duality Principle (Order Theory)", "text": "Let $\\Sigma$ be a statement about ordered sets (whether in natural or a formal language). Let $\\Sigma^*$ be the dual statement of $\\Sigma$."}
+{"_id": "6380", "title": "Hypothetical Syllogism/Formulation 3", "text": ":$\\vdash \\paren {\\paren {p \\implies q} \\land \\paren {q \\implies r} } \\implies \\paren {p \\implies r}$"}
+{"_id": "6383", "title": "Duality Principle (Order Theory)/Global Duality", "text": "The following are equivalent: :$(1): \\quad \\Sigma$ is true for all ordered sets :$(2): \\quad \\Sigma^*$ is true for all ordered sets"}
+{"_id": "6384", "title": "Duality Principle (Order Theory)/Local Duality", "text": "Let $\\struct {S, \\preceq}$ be an ordered set, and let $\\struct {S, \\succeq}$ be its dual. Then the following are equivalent: :$(1): \\quad \\Sigma$ is true for $\\struct {S, \\preceq}$ :$(2): \\quad \\Sigma^*$ is true for $\\struct {S, \\succeq}$"}
+{"_id": "6387", "title": "Diagonal Complement Relation Compatible with Group Operation", "text": "Let $\\left({G, \\circ}\\right)$ be a group. Let $\\Delta_G$ be the diagonal relation on $G$. Then $\\Delta_G^c = \\complement_{G \\times G} \\Delta_G$ is a relation compatible with $\\circ$. {{explain|Find another way to write the above, it's ugly}} In other words, $\\ne$ is a relation compatible with $\\circ$."}
+{"_id": "6388", "title": "Reflexive Closure of Relation Compatible with Operation is Compatible", "text": "Let $\\left({S, \\circ}\\right)$ be a magma. Let $\\mathcal R$ be a relation on $S$ which is compatible with $\\circ$. Let $\\mathcal R^=$ be the reflexive closure of $\\prec$. That is, $\\mathcal R^=$ is defined as the union of $\\mathcal R$ with the diagonal relation for $S$. Then $\\mathcal R^=$ is compatible with $\\circ$."}
+{"_id": "6389", "title": "Reflexive Reduction of Relation Compatible with Group Operation is Compatible", "text": "Let $\\struct {S, \\circ}$ be a group. Let $\\RR$ be a relation on $S$ which is compatible with $\\circ$. Let $\\RR^\\ne$ be the reflexive reduction of $\\RR$. Then $\\RR^\\ne$ is compatible with $\\circ$."}
+{"_id": "6392", "title": "Double Negation/Double Negation Introduction/Sequent Form/Formulation 2", "text": ": $\\vdash p \\implies \\neg \\neg p$"}
+{"_id": "6394", "title": "Dual of Dual Ordering", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $\\left({S, \\succeq}\\right)$ be its dual. Then the dual of $\\left({S, \\succeq}\\right)$ is again $\\left({S, \\preceq}\\right)$."}
+{"_id": "6395", "title": "Dual of Dual Statement (Order Theory)", "text": "Let $\\Sigma$ be a statement about ordered sets. Let $\\Sigma^*$ be its dual statement. Then $\\Sigma$ is also the dual statement of $\\Sigma^*$."}
+{"_id": "6396", "title": "Double Negation/Double Negation Elimination/Sequent Form/Formulation 2", "text": ": $\\vdash \\neg \\neg p \\implies p$"}
+{"_id": "6397", "title": "Double Negation/Double Negation Elimination", "text": "The rule of '''double negation elimination''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Double Negation/Double Negation Elimination/Proof Rule}} === Sequent Form === {{:Double Negation/Double Negation Elimination/Sequent Form}}"}
+{"_id": "6398", "title": "Double Negation/Formulation 1", "text": ":$p \\dashv \\vdash \\neg \\neg p$"}
+{"_id": "6402", "title": "Double Negation/Formulation 1/Proof 2", "text": ":$p \\dashv \\vdash \\neg \\neg p$"}
+{"_id": "6406", "title": "Rule of Transposition/Formulation 1", "text": "A statement and its contrapositive have the same truth value: :$p \\implies q \\dashv \\vdash \\neg q \\implies \\neg p$ Its abbreviation in a tableau proof is $\\textrm {TP}$."}
+{"_id": "6407", "title": "Rule of Transposition/Formulation 1/Proof 1", "text": ":$p \\implies q \\dashv \\vdash \\neg q \\implies \\neg p$"}
+{"_id": "6408", "title": "Rule of Transposition/Formulation 1/Proof 2", "text": ":$p \\implies q \\dashv \\vdash \\neg q \\implies \\neg p$"}
+{"_id": "6409", "title": "Positive Infinity is Maximal", "text": "Let $\\struct {\\overline \\R, \\le}$ be the extended real numbers with the usual ordering. Then $+\\infty$ is a maximal element of $\\overline \\R$."}
+{"_id": "6410", "title": "Negative Infinity is Minimal", "text": "Let $\\left({\\overline \\R, \\le}\\right)$ be the extended real numbers with the usual ordering. Then $-\\infty$ is a minimal element of $\\overline \\R$."}
+{"_id": "6413", "title": "Inversion Mapping Reverses Ordering in Ordered Group", "text": "{{begin-eqn}} {{eqn | n = \\text{OG} 3       | l = x \\preceq y       | o = \\iff       | r = y^{-1} \\preceq x^{-1} }} {{eqn | n = \\text{OG}3'       | l = x \\prec y       | o = \\iff       | r = y^{-1} \\prec x^{-1} }} {{end-eqn}}"}
+{"_id": "6415", "title": "Operating on Transitive Relationships Compatible with Operation", "text": "Let $\\left({S, \\circ}\\right)$ be a magma. Let $\\mathcal R$ be a transitive relation on $S$ which is compatible with $\\circ$. Let $\\mathcal R^=$ be the reflexive closure of $\\mathcal R$. Let $x, y, z, w \\in S$. Then the following implications hold: $(1)\\quad$ If $x \\mathrel{\\mathcal R} y$ and $z \\mathrel{\\mathcal R} w$, then $x \\circ z \\mathrel{\\mathcal R} y \\circ w$. $(2)\\quad$ If $x \\mathrel{\\mathcal R} y$ and $z \\mathrel{\\mathcal R^=} w$, then $x \\circ z \\mathrel{\\mathcal R} y \\circ w$. $(3)\\quad$ If $x \\mathrel{\\mathcal R^=} y$ and $z \\mathrel{\\mathcal R} w$, then $x \\circ z \\mathrel{\\mathcal R} y \\circ w$. $(4)\\quad$ If $x \\mathrel{\\mathcal R^=} y$ and $z \\mathrel{\\mathcal R^=} w$, then $x \\circ z \\mathrel{\\mathcal R^=} y \\circ w$."}
+{"_id": "6418", "title": "Rule of Transposition/Formulation 2/Proof 1", "text": ": $\\vdash \\left({p \\implies q}\\right) \\iff \\left({\\neg q \\implies \\neg p}\\right)$"}
+{"_id": "6419", "title": "Rule of Transposition/Formulation 2/Forward Implication/Proof", "text": ":$\\vdash \\paren {p \\implies q} \\implies \\paren {\\neg q \\implies \\neg p}$"}
+{"_id": "6420", "title": "Rule of Transposition/Formulation 2/Reverse Implication/Proof", "text": ": $\\vdash \\left({\\neg q \\implies \\neg p}\\right) \\implies \\left({p \\implies q}\\right)$"}
+{"_id": "6421", "title": "Complex Modulus is Norm", "text": "The complex modulus is a norm on the set of complex numbers $\\C$."}
+{"_id": "6422", "title": "Set Difference of Relations Compatible with Group Operation is Compatible", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\RR, \\QQ$ be relations on $G$ which are compatible with $\\circ$. Then the difference $\\RR \\setminus \\QQ$ is compatible with $\\circ$."}
+{"_id": "6423", "title": "Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroup. Let $\\mathcal R$ be a transitive relation on $S$ which is compatible with $\\circ$. Let $x, y \\in S$ such that $x \\mathrel{\\mathcal R} y$. Let $n \\in \\N_{>0}$ be a strictly positive integer. Then: : $x^n \\mathrel{\\mathcal R} y^n$ where $x^n$ is the $n$th power of $x$."}
+{"_id": "6424", "title": "Reflexive Closure of Transitive Antisymmetric Relation is Ordering", "text": "Let $S$ be a set. Let $\\prec$ be a transitive, antisymmetric relation on $S$. Let $\\preceq$ be the reflexive closure of $\\prec$. Then $\\preceq$ is an ordering on $S$."}
+{"_id": "6427", "title": "Reflexive Closure of Transitive Relation is Transitive", "text": "Let $S$ be a set. Let $\\mathcal R$ be a transitive relation. Let $\\mathcal R^=$ be the reflexive closure of $\\mathcal R$. Then $\\mathcal R^=$ is also transitive."}
+{"_id": "6429", "title": "Positive Infinity is Greatest Element", "text": "Let $\\left({\\overline \\R, \\le}\\right)$ be the extended real numbers with their usual ordering. Then $+\\infty$ is the greatest element of $\\overline \\R$."}
+{"_id": "6430", "title": "Negative Infinity is Smallest Element", "text": "Let $\\left({\\overline \\R, \\le}\\right)$ be the extended real numbers with their usual ordering. Then $-\\infty$ is the smallest element of $\\overline \\R$."}
+{"_id": "6431", "title": "Properties of Relation Compatible with Group Operation/CRG1", "text": ":$x \\mathrel {\\mathcal R} y \\iff x \\circ z \\mathrel {\\mathcal R} y \\circ z$ :$x \\mathrel {\\mathcal R} y \\iff z \\circ x \\mathrel {\\mathcal R} z \\circ y$"}
+{"_id": "6432", "title": "Properties of Relation Compatible with Group Operation/CRG2", "text": ":$(1): \\quad x \\mathrel {\\mathcal R} y \\iff e \\mathrel {\\mathcal R} y \\circ x^{-1}$ :$(2): \\quad x \\mathrel {\\mathcal R} y \\iff e \\mathrel {\\mathcal R} x^{-1} \\circ y$ :$(3): \\quad x \\mathrel {\\mathcal R} y \\iff x \\circ y^{-1} \\mathrel {\\mathcal R} e$ :$(4): \\quad x \\mathrel {\\mathcal R} y \\iff y^{-1} \\circ x \\mathrel {\\mathcal R} e$"}
+{"_id": "6433", "title": "Properties of Relation Compatible with Group Operation/CRG4", "text": ":$x \\mathrel{\\mathcal R} e \\iff e \\mathrel{\\mathcal R} x^{-1}$ :$e \\mathrel{\\mathcal R} x \\iff x^{-1} \\mathrel{\\mathcal R} e$"}
+{"_id": "6434", "title": "Dual Pairs (Order Theory)", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a, b \\in S$, and let $T \\subseteq S$. Then the following phrases about, and concepts pertaining to $\\left({S, \\preceq}\\right)$ are dual to one another: :{| style=\"text-align:center\" | $b \\preceq a$ | width = \"20px\" | | $a \\preceq b$ |- | $a$ succeeds $b$ | | $a$ precedes $b$ |- | $a$ strictly succeeds $b$ | | $a$ strictly precedes $b$ |- | $a$ is an upper bound for $T$ | | $a$ is a lower bound for $T$ |- | $a$ is a supremum for $T$ | | $a$ is an infimum for $T$ |- | $a$ is a maximal element of $T$ | | $a$ is a minimal element of $T$ |- | $a$ is the greatest element | | $a$ is the smallest element |- | the weak lower closure $a^\\preceq$ of $a$ | | the weak upper closure $a^\\succeq$ of $a$ |- | the strict lower closure $a^\\prec$ of $a$ | | the strict upper closure $a^\\succ$ of $a$ |- | the strict lower closure $T^\\prec$ of $T$ | | the strict upper closure $T^\\succ$ of $T$ |- | the join $a \\vee b$ of $a$ and $b$ | | the meet $a \\wedge b$ of $a$ and $b$ |- | $T$ is a lower set in $S$ | | $T$ is an upper set in $S$ |}"}
+{"_id": "6435", "title": "Conditional is not Left Self-Distributive/Formulation 1", "text": "While this holds: :$\\paren {p \\implies q} \\implies r \\vdash \\paren {p \\implies r} \\implies \\paren {q \\implies r}$ its converse does not: :$\\paren {p \\implies r} \\implies \\paren {q \\implies r} \\not \\vdash \\paren {p \\implies q} \\implies r$"}
+{"_id": "6438", "title": "Cauchy-Riemann Equations", "text": "Let $D \\subseteq \\C$ be an open subset of the set of complex numbers $\\C$. Let $f: D \\to \\C$ be a complex function on $D$. Let $u, v: \\set {\\tuple {x, y} \\in \\R^2: x + i y = z \\in D} \\to \\R$ be two real-valued functions defined as: :$\\map u {x, y} = \\map \\Re {\\map f z}$ :$\\map v {x, y} = \\map \\Im {\\map f z}$ where: :$\\map \\Re {\\map f z}$ denotes the real part of $\\map f z$ :$\\map \\Im {\\map f z}$ denotes the imaginary part of $\\map f z$. Then $f$ is complex-differentiable in $D$ {{iff}}: :$u$ and $v$ are differentiable in their entire domain and: :The following two equations, known as the '''Cauchy-Riemann equations''', hold for the continuous partial derivatives of $u$ and $v$: :$(1): \\quad \\dfrac {\\partial u} {\\partial x} = \\dfrac {\\partial v} {\\partial y}$ :$(2): \\quad \\dfrac {\\partial u} {\\partial y} = -\\dfrac {\\partial v} {\\partial x}$ If the conditions are true, then for all $z \\in D$: :$\\map {f'} z = \\map {\\dfrac {\\partial f} {\\partial x} } z = -i \\map {\\dfrac {\\partial f} {\\partial y} } z$"}
+{"_id": "6440", "title": "Cauchy-Riemann Equations/Necessary Condition", "text": "Let $D \\subseteq \\C$ be an open subset of the set of complex numbers $\\C$. Let $f: D \\to \\C$ be a complex function on $D$. Let $u, v: \\left\\{ {\\left({x, y}\\right) \\in \\R^2: x + i y = z \\in D }\\right\\} \\to \\R$ be two real-valued functions defined as: :$u \\left({x, y}\\right) = \\operatorname{Re} \\left({f \\left({z}\\right) }\\right)$ :$v \\left({x, y}\\right) = \\operatorname{Im} \\left({f \\left({z}\\right) }\\right)$ where: : $\\operatorname{Re} \\left({f \\left({z}\\right)}\\right)$ denotes the real part of $f \\left({z}\\right)$ : $\\operatorname{Im} \\left({f \\left({z}\\right)}\\right) $ denotes the imaginary part of $f \\left({z}\\right)$. Then $f$ is complex-differentiable in $D$ {{iff}}: : $u$ and $v$ are differentiable in their entire domain and: : The following two equations, known as the '''Cauchy-Riemann equations''', hold for the partial derivatives of $u$ and $v$: :$(1): \\quad \\dfrac{\\partial u}{\\partial x} = \\dfrac{\\partial v}{\\partial y}$ :$(2): \\quad \\dfrac{\\partial u}{\\partial y} = - \\dfrac{\\partial v}{\\partial x}$ If the conditions are true, then for all $z \\in D$: :$f' \\left({z}\\right) = \\dfrac{\\partial f}{\\partial x} \\left({z}\\right) = -i \\dfrac{\\partial f}{\\partial y} \\left({z}\\right)$"}
+{"_id": "6442", "title": "Cauchy-Riemann Equations/Sufficient Condition", "text": "Let $D \\subseteq \\C$ be an open subset of the set of complex numbers $\\C$. Let $f: D \\to \\C$ be a complex function on $D$. Let $u, v: \\left\\{ {\\left({x, y}\\right) \\in \\R^2: x + i y = z \\in D }\\right\\} \\to \\R$ be two real-valued functions defined as: :$u \\left({x, y}\\right) = \\operatorname{Re} \\left({f \\left({z}\\right) }\\right)$ :$v \\left({x, y}\\right) = \\operatorname{Im} \\left({f \\left({z}\\right) }\\right)$ where: : $\\operatorname{Re} \\left({f \\left({z}\\right)}\\right)$ denotes the real part of $f \\left({z}\\right)$ : $\\operatorname{Im} \\left({f \\left({z}\\right)}\\right) $ denotes the imaginary part of $f \\left({z}\\right)$. Let: :$u$ and $v$ be differentiable in their entire domain and: :The following two equations, known as the '''Cauchy-Riemann equations''', hold for the continuous partial derivatives of $u$ and $v$: ::$(1): \\quad \\dfrac{\\partial u}{\\partial x} = \\dfrac{\\partial v}{\\partial y}$ ::$(2): \\quad \\dfrac{\\partial u}{\\partial y} = - \\dfrac{\\partial v}{\\partial x}$ Then: :$f$ is complex-differentiable in $D$ and: :for all $z \\in D$: ::$f' \\left({z}\\right) = \\dfrac {\\partial f} {\\partial x} \\left({z}\\right) = -i \\dfrac{\\partial f}{\\partial y} \\left({z}\\right)$"}
+{"_id": "6444", "title": "Self-Distributive Law for Conditional/Formulation 1/Proof 2", "text": ": $p \\implies \\left({q \\implies r}\\right) \\dashv \\vdash \\left({p \\implies q}\\right) \\implies \\left({p \\implies r}\\right)$"}
+{"_id": "6445", "title": "Cauchy-Riemann Equations/Expression of Derivative", "text": "Let $D \\subseteq \\C$ be an open subset of the set of complex numbers $\\C$. Let $f: D \\to \\C$ be a complex function on $D$. Let $u, v: \\left\\{ {\\left({x, y}\\right) \\in \\R^2: x + i y = z \\in D }\\right\\} \\to \\R$ be two real-valued functions defined as: :$u \\left({x, y}\\right) = \\operatorname{Re} \\left({f \\left({z}\\right) }\\right)$ :$v \\left({x, y}\\right) = \\operatorname{Im} \\left({f \\left({z}\\right) }\\right)$ where: : $\\operatorname{Re} \\left({f \\left({z}\\right)}\\right)$ denotes the real part of $f \\left({z}\\right)$ : $\\operatorname{Im} \\left({f \\left({z}\\right)}\\right) $ denotes the imaginary part of $f \\left({z}\\right)$. Then $f$ is complex-differentiable in $D$ {{iff}}: : $u$ and $v$ are differentiable in their entire domain and: : The following two equations, known as the '''Cauchy-Riemann equations''', hold for the partial derivatives of $u$ and $v$: :$(1): \\quad \\dfrac {\\partial u} {\\partial x} = \\dfrac {\\partial v} {\\partial y}$ :$(2): \\quad \\dfrac {\\partial u} {\\partial y} = - \\dfrac {\\partial v} {\\partial x}$ If the conditions are true, then for all $z \\in D$: :$f' \\left({z}\\right) = \\dfrac{\\partial f}{\\partial x} \\left({z}\\right) = -i \\dfrac{\\partial f}{\\partial y} \\left({z}\\right)$"}
+{"_id": "6446", "title": "Self-Distributive Law for Conditional/Forward Implication/Formulation 1/Proof", "text": ": $p \\implies \\left({q \\implies r}\\right) \\vdash \\left({p \\implies q}\\right) \\implies \\left({p \\implies r}\\right)$"}
+{"_id": "6450", "title": "Succeed is Dual to Precede", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a, b \\in S$. The following are dual statements: :$a$ succeeds $b$ :$a$ precedes $b$"}
+{"_id": "6451", "title": "Strictly Succeed is Dual to Strictly Precede", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $a, b \\in S$. The following are dual statements: :$a$ strictly succeeds $b$ :$a$ strictly precedes $b$"}
+{"_id": "6452", "title": "Upper Bound is Dual to Lower Bound", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a \\in S$ and $T \\subseteq S$. The following are dual statements: :$a$ is an upper bound for $T$ :$a$ is a lower bound for $T$"}
+{"_id": "6453", "title": "Supremum is Dual to Infimum", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a \\in S$ and $T \\subseteq S$. The following are dual statements: :$a$ is a supremum for $T$ :$a$ is an infimum for $T$"}
+{"_id": "6454", "title": "Maximal Element is Dual to Minimal Element", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $T \\subseteq S$, and $a \\in T$. The following are dual statements: :$a$ is a maximal element of $T$ :$a$ is a minimal element of $T$"}
+{"_id": "6455", "title": "Greatest Element is Dual to Smallest Element", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $a \\in S$. The following are dual statements: :$a$ is the greatest element of $S$ :$a$ is the smallest element of $S$"}
+{"_id": "6456", "title": "Lower Closure is Dual to Upper Closure", "text": "Let $\\left({S, \\preccurlyeq}\\right)$ be an ordered set. Let $a, b \\in S$. Let $T \\subseteq S$ The following are pairs of dual statements: :$b \\in a^\\preccurlyeq$, the lower closure of $a$ :$b \\in a^\\succcurlyeq$, the upper closure of $a$ :$b \\in T^\\preccurlyeq$, the lower closure of $T$ :$b \\in T^\\succcurlyeq$, the upper closure of $T$"}
+{"_id": "6457", "title": "Strict Lower Closure is Dual to Strict Upper Closure", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a, b \\in S$. The following are dual statements: :$b \\in a^\\prec$, the strict lower closure of $a$ :$b \\in a^\\succ$, the strict upper closure of $a$"}
+{"_id": "6458", "title": "Join is Dual to Meet", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a, b, c \\in S$. The following are dual statements: :$c = a \\vee b$, the join of $a$ and $b$ :$c = a \\wedge b$ the meet of $a$ and $b$"}
+{"_id": "6459", "title": "Self-Distributive Law for Conditional/Forward Implication/Formulation 1", "text": ": $p \\implies \\left({q \\implies r}\\right) \\vdash \\left({p \\implies q}\\right) \\implies \\left({p \\implies r}\\right)$"}
+{"_id": "6460", "title": "Self-Distributive Law for Conditional/Reverse Implication/Formulation 1", "text": ":$\\paren {p \\implies q} \\implies \\paren {p \\implies r} \\vdash p \\implies \\paren {q \\implies r}$"}
+{"_id": "6466", "title": "Square Matrix with Duplicate Columns has Zero Determinant", "text": "If two columns of a square matrix over a commutative ring $\\struct {R, +, \\circ}$ are identical, then its determinant is zero."}
+{"_id": "6467", "title": "Properties of Relation Compatible with Group Operation/CRG3", "text": ":$x \\mathrel{\\mathcal R} y \\iff y^{-1} \\mathrel{\\mathcal R} x^{-1}$"}
+{"_id": "6471", "title": "Ordering Induced by Join Semilattice", "text": "Let $\\struct {S, \\vee, \\preceq}$ be a join semilattice. By Join Semilattice is Semilattice, $\\struct {S, \\vee}$ is a semilattice. By Semilattice Induces Ordering, $\\struct {S, \\vee}$ induces an ordering $\\preceq'$ on $S$, by: :$a \\preceq' b$ {{iff}} $a \\vee b = b$ for all $a, b \\in S$. The ordering $\\preceq'$ coincides with the original ordering $\\preceq$."}
+{"_id": "6472", "title": "Combination Theorem for Complex Derivatives", "text": "Let $D$ be an open subset of the set of complex numbers $\\C$. Let $f, g: D \\to \\C$ be complex-differentiable functions on $D$ Let $z \\in D$. Let $w, c \\in \\C$ be arbitrary complex numbers. Then the following results hold:"}
+{"_id": "6473", "title": "Combination Theorem for Complex Derivatives/Sum Rule", "text": ":$\\left({f + g}\\right)' \\left({z}\\right) = f' \\left({z}\\right) + g' \\left({z}\\right)$"}
+{"_id": "6474", "title": "Combination Theorem for Complex Derivatives/Multiple Rule", "text": ":$\\left({w f}\\right)' \\left({z}\\right) = w f' \\left({z}\\right)$"}
+{"_id": "6475", "title": "Join Absorbs Meet", "text": "Let $\\struct {S, \\wedge, \\preceq}$ be a meet semilattice. Let $\\vee$ denote join. Then $\\vee$ absorbs $\\wedge$. That is, for all $a, b \\in S$: :$a \\vee \\paren {a \\wedge b} = a$"}
+{"_id": "6476", "title": "Meet Absorbs Join", "text": "Let $\\left({S, \\vee, \\preceq}\\right)$ be a join semilattice. Let $\\wedge$ denote meet. Then $\\wedge$ absorbs $\\vee$. That is, for all $a, b \\in S$: :$a \\wedge \\left({a \\vee b}\\right) = a$"}
+{"_id": "6479", "title": "Rule of Exportation/Reverse Implication/Formulation 2/Proof 1", "text": ":$\\vdash \\paren {p \\implies \\paren {q \\implies r} } \\implies \\paren {\\paren {p \\land q} \\implies r}$"}
+{"_id": "6482", "title": "Rule of Exportation/Forward Implication/Formulation 1", "text": ": $\\left ({p \\land q}\\right) \\implies r \\vdash p \\implies \\left ({q \\implies r}\\right)$"}
+{"_id": "6483", "title": "Rule of Exportation/Reverse Implication/Formulation 1", "text": ":$p \\implies \\paren {q \\implies r} \\vdash \\paren {p \\land q} \\implies r$"}
+{"_id": "6485", "title": "Rule of Commutation/Conjunction/Formulation 1/Proof 2", "text": ":$p \\land q \\dashv \\vdash q \\land p$"}
+{"_id": "6486", "title": "Rule of Commutation/Conjunction/Formulation 1", "text": ":$p \\land q \\dashv \\vdash q \\land p$"}
+{"_id": "6489", "title": "Rule of Commutation/Disjunction/Formulation 1", "text": ":$p \\lor q \\dashv \\vdash q \\lor p$"}
+{"_id": "6490", "title": "Rule of Commutation/Conjunction/Formulation 2", "text": ":$\\vdash \\paren {p \\land q} \\iff \\paren {q \\land p}$"}
+{"_id": "6491", "title": "Rule of Commutation/Disjunction/Formulation 2", "text": ":$\\vdash \\paren {p \\lor q} \\iff \\paren {q \\lor p}$"}
+{"_id": "6492", "title": "Combination Theorem for Complex Derivatives/Product Rule", "text": ":$\\left({f g}\\right)' \\left({z}\\right) = f' \\left({z}\\right) g \\left({z}\\right) + f \\left({z}\\right) g' \\left({z}\\right)$"}
+{"_id": "6493", "title": "Dedekind Completion is Unique up to Isomorphism", "text": "Let $S$ be an ordered set. Let $\\struct {X, f}$ and $\\struct {Y, g}$ be Dedekind completions of $S$. Then there exists a unique order isomorphism $\\phi: X \\to Y$ such that $\\phi \\circ f = g$."}
+{"_id": "6494", "title": "Combination Theorem for Complex Derivatives/Combined Sum Rule", "text": "Let $\\dfrac \\d {\\d z} \\left({c f + w g}\\right)$ denote the derivative of $c f + w g$. Then: :$\\dfrac \\d {\\d z} \\left({c f + w g}\\right) \\left({z}\\right) = c \\dfrac \\d {\\d z} f \\left({z}\\right) + w \\dfrac \\d {\\d z} g \\left({z}\\right)$"}
+{"_id": "6496", "title": "Reflexive Reduction of Transitive Antisymmetric Relation is Strict Ordering", "text": "Let $S$ be a set. Let $\\mathcal R$ be a transitive, antisymmetric relation on $S$. Let $\\mathcal R^\\ne$ denote the reflexive reduction of $\\mathcal R$. Then $\\mathcal R^\\ne$ is a strict ordering."}
+{"_id": "6497", "title": "Reflexive Reduction of Antisymmetric Relation is Asymmetric", "text": "Let $S$ be a set. Let $\\mathcal R$ be an antisymmetric relation on $S$. Let $\\mathcal R^\\ne$ be the reflexive reduction of $\\mathcal R$. Then $\\mathcal R^\\ne$ is asymmetric."}
+{"_id": "6498", "title": "Properties of Ordered Group", "text": "Let $\\left({G, \\circ, \\preceq}\\right)$ be an ordered group with identity $e$. Let $x,y,z,w \\in G$. Then the following hold:"}
+{"_id": "6504", "title": "Reductio ad Absurdum/Variant 1", "text": ": $\\neg p \\implies \\bot \\vdash p$"}
+{"_id": "6507", "title": "Reductio ad Absurdum/Variant 2", "text": ":$\\neg p \\implies \\paren {q \\land \\neg q} \\vdash p$"}
+{"_id": "6508", "title": "Proof by Contradiction/Variant 1", "text": ":$\\left({p \\vdash \\left({q \\land \\neg q}\\right)}\\right) \\vdash \\neg p$"}
+{"_id": "6509", "title": "Proof by Contradiction/Variant 2", "text": "==== Formulation 1 ==== {{:Proof by Contradiction/Variant 2/Formulation 1}} ==== Formulation 2 ==== {{:Proof by Contradiction/Variant 2/Formulation 2}}"}
+{"_id": "6510", "title": "Proof by Contradiction/Variant 3", "text": "==== Formulation 1 ==== {{:Proof by Contradiction/Variant 3/Formulation 1}} ==== Formulation 2 ==== {{:Proof by Contradiction/Variant 3/Formulation 2}}"}
+{"_id": "6511", "title": "Equivalence of Definitions of Topological Group", "text": "Let $\\left({G, \\odot}\\right)$ be a group. On its underlying set $G$, let $\\left({G, \\tau}\\right)$ be a topological space. {{TFAE|def = Topological Group}}"}
+{"_id": "6512", "title": "Implication is Left Distributive over Conjunction/Forward Implication/Formulation 1/Proof", "text": ": $p \\implies \\left({q \\land r}\\right) \\vdash \\left({p \\implies q}\\right) \\land \\left({p \\implies r}\\right)$"}
+{"_id": "6513", "title": "Implication is Left Distributive over Conjunction/Reverse Implication/Formulation 1/Proof", "text": ": $\\left({p \\implies q}\\right) \\land \\left({p \\implies r}\\right) \\vdash p \\implies \\left({q \\land r}\\right)$"}
+{"_id": "6514", "title": "Implication is Left Distributive over Conjunction/Forward Implication/Formulation 1", "text": ": $p \\implies \\left({q \\land r}\\right) \\vdash \\left({p \\implies q}\\right) \\land \\left({p \\implies r}\\right)$"}
+{"_id": "6515", "title": "Implication is Left Distributive over Conjunction/Reverse Implication/Formulation 1", "text": ": $\\left({p \\implies q}\\right) \\land \\left({p \\implies r}\\right) \\vdash p \\implies \\left({q \\land r}\\right)$"}
+{"_id": "6516", "title": "Implication is Left Distributive over Conjunction/Formulation 1/Proof 1", "text": ": $p \\implies \\left({q \\land r}\\right) \\dashv \\vdash \\left({p \\implies q}\\right) \\land \\left({p \\implies r}\\right)$"}
+{"_id": "6518", "title": "Implication is Left Distributive over Conjunction/Forward Implication/Formulation 2/Proof", "text": ": $\\vdash \\left({p \\implies \\left({q \\land r}\\right)}\\right) \\implies \\left({\\left({p \\implies q}\\right) \\land \\left({p \\implies r}\\right)}\\right)$"}
+{"_id": "6519", "title": "Implication is Left Distributive over Conjunction/Reverse Implication/Formulation 2/Proof", "text": ": $\\vdash \\left({\\left({p \\implies q}\\right) \\land \\left({p \\implies r}\\right)}\\right) \\implies \\left({p \\implies \\left({q \\land r}\\right)}\\right)$"}
+{"_id": "6522", "title": "Implication is Left Distributive over Conjunction/Formulation 2/Proof 1", "text": ": $\\vdash \\left({p \\implies \\left({q \\land r}\\right)}\\right) \\iff \\left({\\left({p \\implies q}\\right) \\land \\left({p \\implies r}\\right)}\\right)$"}
+{"_id": "6524", "title": "Criterion for Ring with Unity to be Topological Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity. Let $\\tau$ be a topology over $R$. Suppose that $+$ and $\\circ$ are $\\tau$-continuous mappings. Then $\\struct {R, +, \\circ, \\tau}$ is a topological ring."}
+{"_id": "6525", "title": "Union of Relations is Relation", "text": "Let $S$ and $T$ be sets. Let $\\mathcal F$ be a family of relations from $S$ to $T$. Let $\\displaystyle \\mathcal R = \\bigcup \\mathcal F$, the union of all the elements of $\\mathcal F$.   Then $\\mathcal R$ is a relation from $S$ to $T$. {{expand|Binary case}}"}
+{"_id": "6526", "title": "Combination Theorem for Complex Derivatives/Quotient Rule", "text": "For all $z \\in D$ with $\\map g z \\ne 0$: :$\\map {\\paren {\\dfrac f g}'} z = \\dfrac {\\map {f'} z \\map g z - \\map f z \\map {g'} z} {\\paren {\\map g z}^2}$"}
+{"_id": "6527", "title": "Complex-Differentiable Function is Continuous", "text": "Let $f: D \\to \\C$ be a complex function, where $D \\subseteq \\C$ is an open set. Suppose that $f$ is complex-differentiable at $z \\in D$. Then $f$ is continuous at $z$."}
+{"_id": "6529", "title": "Proof by Cases/Formulation 1/Forward Implication/Proof 1", "text": ": $\\left({p \\implies r}\\right) \\land \\left({q \\implies r}\\right) \\vdash \\left({p \\lor q}\\right) \\implies r$"}
+{"_id": "6532", "title": "Closed Ball in Euclidean Space is Compact", "text": "Let $x \\in \\R_n$ be a point in the Euclidean space $\\R^n$. Let $\\epsilon \\in \\R_{>0}$. Then the closed $\\epsilon$-ball $\\map {B_\\epsilon^-} x$ is compact."}
+{"_id": "6533", "title": "Isometric Image of Cauchy Sequence is Cauchy Sequence", "text": "Let $\\struct {S_1, d_1}$ and $\\struct {S_2, d_2}$ be metric spaces. Let $f: S_1 \\to S_2$ be an isometry. Let $\\sequence {x_n}$ be a Cauchy sequence in $S_1$. Let $\\sequence {y_n} = \\sequence {\\map f {x_n} }$ be the image of $\\sequence {x_n}$ under $f$. Then $\\sequence {y_n}$ is a Cauchy sequence."}
+{"_id": "6534", "title": "Union of Subsets is Subset", "text": "Let $S_1$, $S_2$, and $T$ be sets. Let $S_1$ and $S_2$ both be subsets of $T$. Then: :$S_1 \\cup S_2 \\subseteq T$ That is: :$\\paren {S_1 \\subseteq T} \\land \\paren {S_2 \\subseteq T} \\implies \\paren {S_1 \\cup S_2} \\subseteq T$"}
+{"_id": "6537", "title": "Proof by Cases/Formulation 1", "text": ": $\\left({p \\implies r}\\right) \\land \\left({q \\implies r}\\right) \\dashv \\vdash \\left({p \\lor q}\\right) \\implies r$"}
+{"_id": "6538", "title": "Distance between Closed Sets in Euclidean Space", "text": "Let $S, T \\subseteq \\R^n$ be closed, non-empty subsets of the real Euclidean space $R^n$. Suppose that $S$ is bounded, and $S$ and $T$ are disjoint. Then there exists $x \\in S$ and $y \\in T$ such that: :$\\map d {x, y} = \\map d {S, T} > 0$ where: :$d$ denotes the Euclidean metric :$\\map d {S, T}$ is the distance between $S$ and $T$."}
+{"_id": "6539", "title": "Complement in Distributive Lattice is Unique", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded distributive lattice. Then every $a \\in S$ admits at most one complement."}
+{"_id": "6540", "title": "Equivalence of Definitions of Top", "text": "{{TFAE|def = Top (Lattice Theory)|view = $\\top$ (top)|context = Lattice Theory}} Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a lattice."}
+{"_id": "6541", "title": "Equivalence of Definitions of Bottom", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a lattice. Let $\\bot$ be a bottom of $\\left({S, \\vee, \\wedge, \\preceq}\\right)$. {{TFAE|def = Bottom (Lattice Theory)|view = Bottom|context = Lattice Theory}}"}
+{"_id": "6542", "title": "Top is Unique", "text": "Let $\\struct {S, \\vee, \\wedge, \\preceq}$ be a lattice. Then $S$ has at most one top."}
+{"_id": "6544", "title": "Isometry is Homeomorphism of Induced Topologies", "text": "Let $\\struct {S_1, d_1}$ and $\\struct {S_2, d_2}$ be metric spaces or pseudometric spaces. Let $f: S_1 \\to S_2$ be an isometry from $\\struct {S_1, d_1}$ to $\\struct {S_2, d_2}$. Let $\\tau_1$ and $\\tau_2$ be the topologies induced on $S_1$ and $S_2$ by the metrics $d_1$ and $d_2$, respectively. Then $f$ is a homeomorphism from $\\struct {S_1, \\tau_1}$ to $\\struct {S_2, \\tau_2}$."}
+{"_id": "6545", "title": "Equivalence of Definitions of Distributive Lattice", "text": "{{TFAE|def = Distributive Lattice}} Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a lattice. {{begin-axiom}} {{axiom | n = 1         | q = \\forall x, y, z \\in S         | m = x \\wedge \\left({y \\vee z}\\right) = \\left({x \\wedge y}\\right) \\vee \\left({x \\wedge z}\\right) }} {{axiom | n = 1'         | q = \\forall x, y, z \\in S         | m = \\left({x \\vee y}\\right) \\wedge z = \\left({x \\wedge z}\\right) \\vee \\left({y \\wedge z}\\right) }} {{axiom | n = 2         | q = \\forall x, y, z \\in S         | m = x \\vee \\left({y \\wedge z}\\right) = \\left({x \\vee y}\\right) \\wedge \\left({x \\vee z}\\right) }} {{axiom | n = 2'         | q = \\forall x, y, z \\in S         | m = \\left({x \\wedge y}\\right) \\vee z = \\left({x \\vee z}\\right) \\wedge \\left({x \\vee y}\\right) }} {{end-axiom}}"}
+{"_id": "6546", "title": "Distance-Preserving Surjection is Isometry of Metric Spaces", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $\\phi: M_1 \\to M_2$ be a surjective distance-preserving mapping. That is: :$\\forall a, b \\in M_1: d_1 \\tuple {a, b} = d_2 \\tuple {\\map \\phi a, \\map \\phi b}$ Then $\\phi$ is an isometry."}
+{"_id": "6548", "title": "Existence of Positive Root of Positive Real Number", "text": "Let $x \\in \\R$ be a real number such that $x > 0$. Let $n \\in \\Z$ be an integer such that $n \\ne 0$. Then there exists a $y \\in \\R: y \\ge 0$ such that $y^n = x$."}
+{"_id": "6549", "title": "Proof by Cases/Formulation 2", "text": ":$\\vdash \\paren {\\paren {p \\implies r} \\land \\paren {q \\implies r} } \\iff \\paren {\\paren {p \\lor q} \\implies r}$"}
+{"_id": "6550", "title": "Proof by Cases/Formulation 2/Forward Implication", "text": ": $\\vdash \\left({\\left({p \\implies r}\\right) \\land \\left({q \\implies r}\\right)}\\right) \\implies \\left({\\left({p \\lor q}\\right) \\implies r}\\right)$"}
+{"_id": "6551", "title": "Proof by Cases/Formulation 2/Reverse Implication", "text": ": $\\vdash \\left({\\left({p \\lor q}\\right) \\implies r}\\right) \\implies \\left({\\left({p \\implies r}\\right) \\land \\left({q \\implies r}\\right)}\\right)$"}
+{"_id": "6552", "title": "Set is Subset of Union/Set of Sets", "text": "Let $\\mathbb S$ be a set of sets. Then: : $\\displaystyle \\forall T \\in \\mathbb S: T \\subseteq \\bigcup \\mathbb S$"}
+{"_id": "6553", "title": "Proof by Cases/Formulation 3", "text": ": $\\vdash \\left({\\left({p \\lor q}\\right) \\land \\left({p \\implies r}\\right) \\land \\left({q \\implies r}\\right)}\\right) \\implies r$"}
+{"_id": "6554", "title": "Properties of Relation Compatible with Group Operation", "text": "Let $\\left({G,\\circ}\\right)$ be a group with identity element $e$. Let $\\mathcal R$ be a endorelation on $G$ which is compatible with $\\circ$. Let $x,y,z \\in G$. Then the following hold:"}
+{"_id": "6555", "title": "Existence of Positive Root of Positive Real Number/Positive Exponent", "text": "Let $x \\in \\R$ be a real number such that $x > 0$. Let $n \\in \\Z$ be an integer such that $n > 0$. Then there exists a $y \\in \\R: y \\ge 0$ such that $y^n = x$."}
+{"_id": "6556", "title": "Existence of Positive Root of Positive Real Number/Negative Exponent", "text": "Let $x \\in \\R$ be a real number such that $x > 0$. Let $n \\in \\Z$ be an integer such that $n < 0$. Then there exists a $y \\in \\R: y \\ge 0$ such that $y^n = x$."}
+{"_id": "6557", "title": "Power of Ring Negative", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $n \\in \\N_{>0}$ be a strictly positive integer. Let $x \\in R$. Then: :If $n$ is even: :::$\\map {\\circ^n} {-x} = \\map {\\circ^n} x$ :If $n$ is odd: :::$\\map {\\circ^n} {-x} = -\\map {\\circ^n} x$"}
+{"_id": "6558", "title": "Paving Lemma", "text": "Let $S$ be an open subset of the Euclidean space $\\R^m$ or the set of complex numbers $\\C$. Let $\\gamma: \\closedint a b \\to S$ be a path in $S$. Then there exists $K \\in \\R_{>0}$ such that: :For all $\\epsilon \\in \\openint 0 K$, there exists a normal subdivision $\\set {x_0, x_1, \\ldots, x_{n-1}, x_n}$ of the closed interval $\\closedint a b$ such that: ::$\\displaystyle \\bigcup_{i \\mathop = 0}^n \\map {B_\\epsilon} {\\map \\gamma {x_i} } \\subseteq S$ :and for all $i \\in \\set {0, 1, \\ldots, n - 1}$: ::$\\map \\gamma {\\closedint {x_i} {x_{i + 1} } } \\subseteq \\map {B_\\epsilon} {\\map \\gamma {x_i} }$ Here, $\\map {B_\\epsilon} {\\map \\gamma {x_i} }$ denotes the open ball of $\\map \\gamma {x_i}$ with radius $\\epsilon$."}
+{"_id": "6559", "title": "Proof by Cases/Formulation 1/Reverse Implication", "text": ": $\\left({p \\lor q}\\right) \\implies r \\vdash \\left({p \\implies r}\\right) \\land \\left({q \\implies r}\\right)$"}
+{"_id": "6560", "title": "Praeclarum Theorema/Formulation 1/Proof 2", "text": ":$\\paren {p \\implies q} \\land \\paren {r \\implies s} \\vdash \\paren {p \\land r} \\implies \\paren {q \\land s}$"}
+{"_id": "6561", "title": "Praeclarum Theorema/Formulation 1", "text": ": $\\left({p \\implies q}\\right) \\land \\left({r \\implies s}\\right) \\vdash \\left({p \\land r}\\right) \\implies \\left({q \\land s}\\right)$"}
+{"_id": "6562", "title": "Praeclarum Theorema/Formulation 2", "text": ": $\\vdash \\left({\\left({p \\implies q}\\right) \\land \\left({r \\implies s}\\right)}\\right) \\implies \\left({\\left({p \\land r}\\right) \\implies \\left({q \\land s}\\right)}\\right)$"}
+{"_id": "6563", "title": "Strictly Positive Integer Power Function Strictly Succeeds Each Element", "text": "Let $\\struct {R, +, \\circ, \\le}$ be an ordered ring with unity. Let $\\struct {R, \\le}$ be a directed set with no upper bound. Let $n \\in \\N_{>0}$. Let $f: R \\to R$ be defined by: :$\\forall x \\in R: \\map f x = \\circ^n x$ Then the image of $f$ has elements strictly succeeding each elements of $R$."}
+{"_id": "6565", "title": "Constructive Dilemma/Formulation 1/Proof 2", "text": ": $p \\implies q, r \\implies s \\vdash p \\lor r \\implies q \\lor s$"}
+{"_id": "6566", "title": "Constructive Dilemma/Formulation 1", "text": ": $p \\implies q, r \\implies s \\vdash p \\lor r \\implies q \\lor s$"}
+{"_id": "6567", "title": "Constructive Dilemma/Formulation 2", "text": ": $\\vdash \\paren {\\paren {p \\lor r} \\land \\paren {p \\implies q} \\land \\paren {r \\implies s} } \\implies \\paren {q \\lor s}$"}
+{"_id": "6570", "title": "Clavius's Law/Formulation 1", "text": ":$\\neg p \\implies p \\vdash p$"}
+{"_id": "6572", "title": "Equivalent Matrices may not be Similar", "text": "If two square matrices of order $n > 1$ over a ring with unity $R$ are equivalent, they are not necessarily similar."}
+{"_id": "6573", "title": "Odd Power Function is Strictly Increasing/General Result", "text": "Let $\\struct {R, +, \\circ, \\le}$ be a totally ordered ring. Let $n$ be an odd positive integer. Let $f: R \\to R$ be the mapping defined by: :$\\map f x = \\map {\\circ^n} x$ Then $f$ is strictly increasing on $R$."}
+{"_id": "6574", "title": "Number of Bijective Restrictions", "text": "Let $f: S \\to T$ be a surjection. Let $B$ be the set of all bijective restrictions of $f$. Then the cardinality of $B$ is: :$\\displaystyle \\card {\\prod_{i \\mathop \\in I} \\family {S / \\mathcal R_f}_i}$ where $S / \\mathcal R_f$ denotes the quotient set of the induced equivalence of $f$ indexed by $I$."}
+{"_id": "6575", "title": "Number of Injective Restrictions", "text": "Let $f: S \\to T$ be a mapping. Let $Q$ be the set of all injective restrictions of $f$. Then the cardinality of $Q$ is: :$\\displaystyle \\card {\\prod_{i \\mathop \\in I} \\prod_{j \\mathop \\in J_i} \\family {\\family {\\powerset {S / \\mathcal R_f} }_i}_j}$ where: :$\\mathcal P$ denotes power set :$S / \\mathcal R_f$ denotes quotient set of the induced equivalence of $f$."}
+{"_id": "6576", "title": "Box Topology on Finite Product Space is Tychonoff Topology", "text": "Let $n \\in \\N$. For all $k \\in \\set {1, \\ldots, n}$, let $T_k = \\struct {X_k, \\tau_k}$ be topological spaces. Let $\\displaystyle X = \\prod_{k \\mathop = 1}^n X_k$ be the cartesian product of $X_1, \\ldots, X_n$. Then the box topology and the Tychonoff topology on $X$ are identical."}
+{"_id": "6577", "title": "Strictly Positive Integer Power Function is Unbounded Above/General Case", "text": "Let $\\struct {R, +, \\circ, \\le}$ be a totally ordered ring with unity. Suppose that $R$ has no upper bound. Let $n \\in \\N_{>0}$. Let $f: R \\to R$ be defined by: :$\\map f x = \\circ^n x$ Then the image of $f$ is unbounded above in $R$."}
+{"_id": "6578", "title": "Strictly Positive Integer Power Function is Unbounded Above", "text": "Let $\\R$ be the real numbers with the usual ordering. Let $n \\in \\N_{>0}$. Let $f: \\R \\to \\R$ be defined by: :$\\map f x = x^n$ Then $f$ is unbounded above."}
+{"_id": "6579", "title": "Ring Product preserves Inequalities on Positive Elements", "text": "Let $\\struct {R, +, \\circ, \\le}$ be an ordered ring. Let $x, y, z, w \\in R$. Let $0 < x < y$ and $0 < z < w$. Then: :$0 < z \\circ x < w \\circ y$"}
+{"_id": "6580", "title": "Power Function is Strictly Increasing on Positive Elements", "text": "Let $\\struct {R, +, \\circ, \\le}$ be an ordered ring. Let $x, y \\in R$. Let $n \\in \\N_{>0}$ be a strictly positive integer. Let $0 < x < y$. Then: :$0 < \\map {\\circ^n} x < \\map {\\circ^n} y$"}
+{"_id": "6581", "title": "Biconditional is Associative/Formulation 1", "text": ": $p \\iff \\paren {q \\iff r} \\dashv \\vdash \\paren {p \\iff q} \\iff r$"}
+{"_id": "6582", "title": "Biconditional is Associative/Formulation 2", "text": ":$\\vdash \\paren {p \\iff \\paren {q \\iff r} } \\iff \\paren {\\paren {p \\iff q} \\iff r}$"}
+{"_id": "6583", "title": "Open Ball is Convex Set", "text": "Let $V$ be a normed vector space with norm $\\left\\Vert{\\cdot}\\right\\Vert$ over $\\R$ or $\\C$. An open ball in the metric induced by $\\left\\Vert{\\cdot}\\right\\Vert$ is a convex set."}
+{"_id": "6584", "title": "Pseudometric Space is Metric Space iff Kolmogorov", "text": "Let $M = \\struct {S, d}$ be a pseudometric space. Let $T = \\struct {S, \\tau}$ be the topological space over $S$ induced by $d$. Then $M$ is a metric space {{iff}} $T$ is a $T_0$ (Kolmogorov} space."}
+{"_id": "6585", "title": "Sequence Converges to Point Relative to Metric iff it Converges Relative to Induced Topology", "text": "Let $M = \\struct {S, d}$ be a metric space or a pseudometric space. Let $T = \\struct {S, \\tau}$ be the topological space induced by $d$. Let $\\sequence {x_n}$ be a infinite sequence in $S$. Let $l \\in S$. Then $\\sequence {x_n}$ converges to $l$ relative to $d$ {{iff}} $\\sequence {x_n}$ converges to $l$ relative to $\\tau$."}
+{"_id": "6586", "title": "Isometry Preserves Sequence Convergence", "text": "Let $M_1 = \\struct {S_1, d_1}$ and $M_2 = \\struct {S_2, d_2}$ both be metric spaces or pseudometric spaces. Let $\\phi: S_1 \\to S_2$ be an isometry. Let $\\sequence {x_n}$ be an infinite sequence in $S_1$. Suppose that $\\sequence {x_n}$ converges to a point $p \\in S_1$. Then $\\sequence {\\map \\phi {x_n}}$ converges to $\\map \\phi p$."}
+{"_id": "6587", "title": "Connected Domain is Connected by Staircase Contours", "text": "Let $D \\subseteq \\C$ be an open set. Then $D$ is a connected domain {{iff}}: :for all $z, w \\in \\C$, there exists a staircase contour in $D$ with start point $z$ and end point $w$."}
+{"_id": "6588", "title": "Dedekind Completeness is Self-Dual", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then $\\struct {S, \\preceq}$ is Dedekind complete {{iff}} every non-empty subset of $S$ that is bounded below admits an infimum in $S$. That is, an ordered set is Dedekind complete {{iff}} its dual is Dedekind complete."}
+{"_id": "6591", "title": "Biconditional is Transitive/Formulation 1", "text": ":$p \\iff q, q \\iff r \\vdash p \\iff r$"}
+{"_id": "6592", "title": "Biconditional is Transitive/Formulation 2", "text": ": $\\vdash \\left({\\left({p \\iff q}\\right) \\land \\left({q \\iff r}\\right)}\\right) \\implies \\left({p \\iff r}\\right)$"}
+{"_id": "6594", "title": "Biconditional as Disjunction of Conjunctions/Formulation 1/Reverse Implication", "text": ": $\\left({p \\land q}\\right) \\lor \\left({\\neg p \\land \\neg q}\\right) \\vdash p \\iff q$"}
+{"_id": "6596", "title": "Biconditional as Disjunction of Conjunctions/Formulation 1", "text": ": $p \\iff q \\dashv \\vdash \\paren {p \\land q} \\lor \\paren {\\neg p \\land \\neg q}$"}
+{"_id": "6597", "title": "Biconditional as Disjunction of Conjunctions/Formulation 2", "text": ":$\\vdash \\paren {p \\iff q} \\iff \\paren {\\paren {p \\land q} \\lor \\paren {\\neg p \\land \\neg q} }$"}
+{"_id": "6598", "title": "Biconditional as Disjunction of Conjunctions", "text": "==== Formulation 1 ==== {{:Biconditional as Disjunction of Conjunctions/Formulation 1}} ==== Formulation 2 ==== {{:Biconditional as Disjunction of Conjunctions/Formulation 2}}"}
+{"_id": "6600", "title": "Biconditional Equivalent to Biconditional of Negations/Formulation 1/Reverse Implication", "text": ":$\\neg p \\iff \\neg q \\vdash p \\iff q$"}
+{"_id": "6601", "title": "Biconditional Equivalent to Biconditional of Negations/Formulation 1", "text": ": $p \\iff q \\dashv \\vdash \\neg p \\iff \\neg q$"}
+{"_id": "6602", "title": "Biconditional Equivalent to Biconditional of Negations/Formulation 2", "text": ": $\\vdash \\left({p \\iff q}\\right) \\iff \\left({\\neg p \\iff \\neg q}\\right)$"}
+{"_id": "6603", "title": "Biconditional Equivalent to Biconditional of Negations", "text": "==== Formulation 1 ==== {{:Biconditional Equivalent to Biconditional of Negations/Formulation 1}} ==== Formulation 2 ==== {{:Biconditional Equivalent to Biconditional of Negations/Formulation 2}}"}
+{"_id": "6604", "title": "Biconditional iff Disjunction implies Conjunction/Formulation 1/Forward Implication", "text": ": $p \\iff q \\vdash \\left({p \\lor q}\\right) \\implies \\left({p \\land q}\\right)$"}
+{"_id": "6605", "title": "Biconditional iff Disjunction implies Conjunction/Formulation 1/Reverse Implication", "text": ": $\\left({p \\lor q}\\right) \\implies \\left({p \\land q}\\right) \\vdash p \\iff q$"}
+{"_id": "6607", "title": "Biconditional iff Disjunction implies Conjunction/Formulation 1", "text": ":$p \\iff q \\dashv \\vdash \\paren {p \\lor q} \\implies \\paren {p \\land q}$"}
+{"_id": "6608", "title": "Biconditional iff Disjunction implies Conjunction/Formulation 2", "text": ": $\\vdash \\left({p \\iff q}\\right) \\iff \\left({\\left({p \\lor q}\\right) \\implies \\left({p \\land q}\\right)}\\right)$"}
+{"_id": "6609", "title": "Biconditional iff Disjunction implies Conjunction", "text": "==== Formulation 1 ==== {{:Biconditional iff Disjunction implies Conjunction/Formulation 1}} ==== Formulation 2 ==== {{:Biconditional iff Disjunction implies Conjunction/Formulation 2}}"}
+{"_id": "6611", "title": "Biconditional is Transitive/Formulation 1/Proof 1", "text": ": $p \\iff q, q \\iff r \\vdash p \\iff r$"}
+{"_id": "6612", "title": "Order of Squares in Totally Ordered Ring without Proper Zero Divisors", "text": "Let $\\struct {R, +, \\circ, \\le}$ be a totally ordered ring without proper zero divisors whose zero is $0_R$. Let $x, y \\in R$ be positive, that is, $0_R \\le x, y$. Then $x \\le y \\iff x \\circ x \\le y \\circ y$. That is, the square mapping is an order embedding of $\\struct {R_{\\ge 0}, \\le}$ into itself. When $R$ is one of the standard sets of numbers $\\Z, \\Q, \\R$, then this translates into: :If $x, y$ are positive, then $x \\le y \\iff x^2 \\le y^2$."}
+{"_id": "6613", "title": "Order of Squares in Ordered Field", "text": "Let $\\struct {R, +, \\circ, \\le}$ be an ordered field whose zero is $0_R$ and whose unity is $1_R$. Suppose that $\\forall a \\in R: 0 < a \\implies 0 < a^{-1}$. Let $x, y \\in \\struct {R, +, \\circ, \\le}$ such that $0_R \\le x, y$. Then $x \\le y \\iff x \\circ x \\le y \\circ y$. That is, the square function is an order embedding of $\\struct {R_{\\ge 0}, \\le}$ into itself. When $R$ is one of the standard fields of numbers $\\Q$ and $\\R$, then this translates into: :If $x, y$ are positive then $x \\le y \\iff x^2 \\le y^2$."}
+{"_id": "6614", "title": "Law of Identity/Formulation 1/Proof 1", "text": ": $p \\vdash p$"}
+{"_id": "6616", "title": "Law of Identity/Formulation 1/Proof 2", "text": ": $p \\vdash p$"}
+{"_id": "6619", "title": "Law of Identity/Formulation 2", "text": "Every proposition entails itself: :$\\vdash p \\implies p$"}
+{"_id": "6620", "title": "Vandermonde Determinant/Alternative Formulations", "text": "The '''Vandermonde determinant of order $n$''' is the determinant defined as follows: :$V_n = \\begin{vmatrix}      1 &    x_1 &  x_1^2 & \\cdots & x_1^{n - 2} & x_1^{n - 1} \\\\      1 &    x_2 &  x_2^2 & \\cdots & x_2^{n - 2} & x_2^{n - 1} \\\\ \\vdots & \\vdots & \\vdots & \\ddots &      \\vdots &      \\vdots \\\\      1 &    x_n &  x_n^2 & \\cdots & x_n^{n - 2} & x_n^{n - 1} \\end{vmatrix}$ Its value is given by: :$\\displaystyle V_n = \\prod_{1 \\mathop \\le i \\mathop < j \\mathop \\le n} \\paren {x_j - x_i}$"}
+{"_id": "6623", "title": "Neighborhood Condition for Coarser Topology", "text": "Let $S$ be a set. Let $\\tau_1$ and $\\tau_2$ be two topologies on $S$. Suppose that for all $z \\in S$ and for all open neighborhoods $N_z$ of $z$ with respect to $\\tau_1$, there exists $U \\in \\tau_2$ such that $U \\subseteq N_z$. Then $\\tau_1$ is coarser than $\\tau_2$."}
+{"_id": "6627", "title": "Absorption Laws (Logic)/Conjunction Absorbs Disjunction/Proof 2", "text": ":$p \\land \\left({p \\lor q}\\right) \\dashv \\vdash p$"}
+{"_id": "6630", "title": "Absorption Laws (Logic)/Conjunction Absorbs Disjunction", "text": ":$p \\land \\paren {p \\lor q} \\dashv \\vdash p$"}
+{"_id": "6632", "title": "Absorption Laws (Logic)/Disjunction Absorbs Conjunction/Proof 2", "text": ":$p \\lor \\left ({p \\land q}\\right) \\dashv \\vdash p$"}
+{"_id": "6635", "title": "Absorption Laws (Logic)/Disjunction Absorbs Conjunction", "text": ":$p \\lor \\left ({p \\land q}\\right) \\dashv \\vdash p$"}
+{"_id": "6636", "title": "Extended Transitivity", "text": "Let $S$ be a set. Let $\\RR$ be a transitive relation on $S$. Let $\\RR^=$ be the reflexive closure of $\\RR$. Let $a, b, c \\in S$. Then: {{begin-eqn}} {{eqn | n = 1       | l = \\paren {a \\mathrel \\RR b} \\land \\paren {b \\mathrel \\RR c}       | o = \\implies       | r = a \\mathrel \\RR c }} {{eqn | n = 2       | l = \\paren {a \\mathrel \\RR b} \\land \\paren {b \\mathrel {\\RR^=} c}       | o = \\implies       | r = a \\mathrel \\RR c }} {{eqn | n = 3       | l = \\paren {a \\mathrel {\\RR^=} b} \\land \\paren {b \\mathrel \\RR c}       | o = \\implies       | r = a \\mathrel \\RR c }} {{eqn | n = 4       | l = \\paren {a \\mathrel {\\RR^=} b} \\land \\paren {b \\mathrel {\\RR^=} c}       | o = \\implies       | r = a \\mathrel {\\RR^=} c }} {{end-eqn}}"}
+{"_id": "6637", "title": "Rule of Idempotence/Conjunction", "text": "{{:Rule of Idempotence/Conjunction/Formulation 1}}"}
+{"_id": "6638", "title": "Rule of Idempotence/Disjunction", "text": "{{:Rule of Idempotence/Disjunction/Formulation 1}}"}
+{"_id": "6639", "title": "Rule of Idempotence/Disjunction/Formulation 1/Proof", "text": ": $p \\dashv \\vdash p \\lor p$"}
+{"_id": "6652", "title": "Ordering Cycle implies Equality", "text": "Let $\\left({S,\\preceq}\\right)$ be an ordered set. Let $x_1$, $x_2$, and $x_3$ be elements of $S$. Suppose that {{begin-eqn}} {{eqn|l = x_1      |o = \\preceq      |r = x_2 }} {{eqn|l = x_2      |o = \\preceq      |r = x_3 }} {{eqn|l = x_3      |o = \\preceq      |r = x_1 }} {{end-eqn}} Then $x_1 = x_2 = x_3$."}
+{"_id": "6653", "title": "Euclidean Topology is Tychonoff Topology", "text": "Let $T_1 = \\left({\\R, \\tau_1}\\right)$ be the topological space such that $\\tau_1$ is the Euclidean topology on $\\R$. Let $T_n = \\left({\\R^n, \\tau_n}\\right)$ be the topological space such that $\\tau_n$ is the Tychonoff topology on the cartesian product $\\displaystyle \\R_n = \\prod_{i \\mathop = 1}^n \\R$. Then the Euclidean topology on $\\R^n$ and the Tychonoff topology on $\\R^n$ are the same."}
+{"_id": "6654", "title": "Modus Ponendo Tollens/Variant/Formulation 1/Proof", "text": ":$\\neg \\paren {p \\land q} \\dashv \\vdash p \\implies \\neg q$"}
+{"_id": "6655", "title": "Modus Ponendo Tollens/Variant/Formulation 1/Forward Implication", "text": ": $\\neg \\left({p \\land q}\\right) \\vdash p \\implies \\neg q$"}
+{"_id": "6656", "title": "Modus Ponendo Tollens/Variant/Formulation 1/Reverse Implication", "text": ":$p \\implies \\neg q \\vdash \\neg \\paren {p \\land q}$"}
+{"_id": "6657", "title": "Modus Ponendo Tollens/Variant/Formulation 1", "text": ": $\\neg \\left({p \\land q}\\right) \\dashv \\vdash p \\implies \\neg q$"}
+{"_id": "6658", "title": "Modus Ponendo Tollens/Variant/Formulation 2", "text": ": $\\vdash \\left({\\neg \\left({p \\land q}\\right)}\\right) \\iff \\left({p \\implies \\neg q}\\right)$"}
+{"_id": "6660", "title": "Implication Equivalent to Negation of Conjunction with Negative/Formulation 1/Reverse Implication", "text": ":$\\neg \\left({p \\land \\neg q}\\right) \\vdash p \\implies q$"}
+{"_id": "6661", "title": "Implication Equivalent to Negation of Conjunction with Negative/Formulation 1/Forward Implication", "text": ":$p \\implies q \\vdash \\neg \\left({p \\land \\neg q}\\right)$"}
+{"_id": "6662", "title": "Implication Equivalent to Negation of Conjunction with Negative/Formulation 1", "text": ":$p \\implies q \\dashv \\vdash \\neg \\paren {p \\land \\neg q}$"}
+{"_id": "6663", "title": "Implication Equivalent to Negation of Conjunction with Negative", "text": "==== Formulation 1 ==== {{:Implication Equivalent to Negation of Conjunction with Negative/Formulation 1}} ==== Formulation 2 ==== {{:Implication Equivalent to Negation of Conjunction with Negative/Formulation 2}}"}
+{"_id": "6664", "title": "Zero Derivative implies Constant Complex Function", "text": "Let $D \\subseteq \\C$ be a connected domain of $\\C$. Let $f: D \\to \\C$ be a complex-differentiable function. For all $z \\in D$, let $\\map {f'} z = 0$. Then $f$ is constant on $D$."}
+{"_id": "6670", "title": "Relation between P-Product Metric and Chebyshev Distance on Real Vector Space", "text": "For $n \\in \\N$, let $\\R^n$ be a Euclidean space. Let $p \\in \\R_{\\ge 1}$. Let $d_p$ be the $p$-product metric on $\\R^n$.  Let $d_\\infty$ be the Chebyshev distance on $\\R^n$. Then :$\\forall x, y \\in \\R^n: \\map {d_\\infty} {x, y} \\le \\map {d_p} {x, y} \\le n^{1/p} \\map {d_\\infty} {x, y}$"}
+{"_id": "6671", "title": "P-Product Metrics on Real Vector Space are Topologically Equivalent/Inequality for General Case", "text": "For $n \\in \\N$, let $\\R^n$ be a real vector Space. Let $r, t \\in \\R_{\\ge 1}$. Let $d_r$ and $d_t$ be $p$-product metrics on $\\R^n$. Then $d_r$ and $d_t$ are topologically equivalent."}
+{"_id": "6675", "title": "De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 1/Forward Implication", "text": ": $\\neg p \\lor \\neg q \\vdash \\neg \\left({p \\land q}\\right)$"}
+{"_id": "6676", "title": "De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 1/Reverse Implication", "text": ": $\\neg \\left({p \\land q}\\right) \\vdash \\neg p \\lor \\neg q$"}
+{"_id": "6677", "title": "De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 1", "text": ":$\\neg p \\lor \\neg q \\dashv \\vdash \\neg \\paren {p \\land q}$"}
+{"_id": "6678", "title": "De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 2", "text": ":$\\vdash \\paren {\\neg p \\lor \\neg q} \\iff \\paren {\\neg \\paren {p \\land q} }$"}
+{"_id": "6680", "title": "De Morgan's Laws (Logic)/Conjunction of Negations/Formulation 1/Forward Implication", "text": ":$\\neg p \\land \\neg q \\vdash \\neg \\paren {p \\lor q}$"}
+{"_id": "6681", "title": "De Morgan's Laws (Logic)/Conjunction of Negations/Formulation 1/Reverse Implication", "text": ":$\\neg \\paren {p \\lor q} \\vdash \\neg p \\land \\neg q$"}
+{"_id": "6684", "title": "De Morgan's Laws (Logic)/Conjunction of Negations", "text": "{{:De Morgan's Laws (Logic)/Conjunction of Negations/Formulation 1}}"}
+{"_id": "6685", "title": "De Morgan's Laws (Logic)/Conjunction", "text": "{{:De Morgan's Laws (Logic)/Conjunction/Formulation 1}}"}
+{"_id": "6686", "title": "De Morgan's Laws (Logic)/Conjunction/Formulation 1/Forward Implication", "text": ": $p \\land q \\vdash \\neg \\left({\\neg p \\lor \\neg q}\\right)$"}
+{"_id": "6687", "title": "De Morgan's Laws (Logic)/Conjunction/Formulation 1/Reverse Implication", "text": ": $\\neg \\left({\\neg p \\lor \\neg q}\\right) \\vdash p \\land q$"}
+{"_id": "6688", "title": "De Morgan's Laws (Logic)/Conjunction/Formulation 1/Proof", "text": ": $p \\land q \\dashv \\vdash \\neg \\left({\\neg p \\lor \\neg q}\\right)$"}
+{"_id": "6691", "title": "De Morgan's Laws (Logic)/Disjunction", "text": "{{:De Morgan's Laws (Logic)/Disjunction/Formulation 1}}"}
+{"_id": "6698", "title": "Rule of Distribution/Disjunction Distributes over Conjunction", "text": "=== Disjunction is Left Distributive over Conjunction === {{:Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive}} === Disjunction is Right Distributive over Conjunction === {{:Rule of Distribution/Disjunction Distributes over Conjunction/Right Distributive}}"}
+{"_id": "6699", "title": "Rule of Distribution/Conjunction Distributes over Disjunction", "text": "=== Conjunction is Left Distributive over Disjunction === {{:Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive}} === Conjunction is Right Distributive over Disjunction === {{:Rule of Distribution/Conjunction Distributes over Disjunction/Right Distributive}}"}
+{"_id": "6700", "title": "Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1/Proof", "text": ":$p \\land \\left({q \\lor r}\\right) \\dashv \\vdash \\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)$"}
+{"_id": "6701", "title": "Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 2", "text": ":$\\vdash \\paren {p \\land \\paren {q \\lor r} } \\iff \\paren {\\paren {p \\land q} \\lor \\paren {p \\land r} }$"}
+{"_id": "6702", "title": "Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive", "text": "==== Formulation 1 ==== {{:Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1}} ==== Formulation 2 ==== {{:Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 2}}"}
+{"_id": "6703", "title": "Rule of Distribution/Conjunction Distributes over Disjunction/Right Distributive", "text": "==== Formulation 1 ==== {{:Rule of Distribution/Conjunction Distributes over Disjunction/Right Distributive/Formulation 1}} ==== Formulation 2 ==== {{:Rule of Distribution/Conjunction Distributes over Disjunction/Right Distributive/Formulation 2}}"}
+{"_id": "6704", "title": "Rule of Distribution/Conjunction Distributes over Disjunction/Right Distributive/Formulation 1", "text": ":$\\paren {q \\lor r} \\land p \\dashv \\vdash \\paren {q \\land p} \\lor \\paren {r \\land p}$"}
+{"_id": "6705", "title": "Rule of Distribution/Conjunction Distributes over Disjunction/Right Distributive/Formulation 2", "text": ":$\\vdash \\paren {\\paren {q \\lor r} \\land p} \\iff \\paren {\\paren {q \\land p} \\lor \\paren {r \\land p} }$"}
+{"_id": "6711", "title": "Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive", "text": "==== Formulation 1 ==== {{:Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 1}} ==== Formulation 2 ==== {{:Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 2}}"}
+{"_id": "6712", "title": "Rule of Distribution/Disjunction Distributes over Conjunction/Right Distributive", "text": "==== Formulation 1 ==== {{:Rule of Distribution/Disjunction Distributes over Conjunction/Right Distributive/Formulation 1}} ==== Formulation 2 ==== {{:Rule of Distribution/Disjunction Distributes over Conjunction/Right Distributive/Formulation 2}}"}
+{"_id": "6715", "title": "Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 2", "text": ":$\\vdash \\paren {p \\lor \\paren {q \\land r} } \\iff \\paren {\\paren {p \\lor q} \\land \\paren {p \\lor r} }$"}
+{"_id": "6716", "title": "Rule of Distribution/Disjunction Distributes over Conjunction/Right Distributive/Formulation 1", "text": ":$\\paren {q \\land r} \\lor p \\dashv \\vdash \\paren {q \\lor p} \\land \\paren {r \\lor p}$"}
+{"_id": "6717", "title": "Rule of Distribution/Disjunction Distributes over Conjunction/Right Distributive/Formulation 2", "text": ":$\\vdash \\paren {\\paren {q \\land r} \\lor p} \\iff \\paren {\\paren {q \\lor p} \\land \\paren {r \\lor p} }$"}
+{"_id": "6719", "title": "Union of Subsets is Subset/Set of Sets", "text": "Let $T$ be a set. Let $\\mathbb S$ be a set of sets. Suppose that for each $S \\in \\mathbb S$, $S \\subseteq T$. Then: :$\\displaystyle \\bigcup \\mathbb S \\subseteq T$"}
+{"_id": "6720", "title": "Union is Smallest Superset/Set of Sets", "text": "Let $T$ be a set. Let $\\mathbb S$ be a set of sets. Then: :$\\displaystyle \\paren {\\forall X \\in \\mathbb S: X \\subseteq T} \\iff \\bigcup \\mathbb S \\subseteq T$"}
+{"_id": "6721", "title": "Intersection of Relations is Relation", "text": "Let $S$ and $T$ be sets. Let $\\mathcal F$ be a family of relations from $S$ to $T$. Let $\\displaystyle \\mathcal R = \\bigcap \\mathcal F$, the intersection of all the elements of $\\mathcal F$.   Then $\\mathcal R$ is a relation from $S$ to $T$. {{Expand|Binary case}}"}
+{"_id": "6722", "title": "Intersection of Subsets is Subset/Set of Sets", "text": "Let $T$ be a set. Let $\\mathbb S$ be a non-empty set of sets. Suppose that for each $S \\in \\mathbb S$: :$S \\subseteq T$ Then: :$\\bigcap \\mathbb S \\subseteq T$"}
+{"_id": "6723", "title": "Equivalence of Definitions of Initial Topology", "text": "Let $X$ be a set. Let $I$ be an indexing set. Let $\\family {\\struct{Y_i, \\tau_i}}_{i \\mathop \\in I}$ be an indexed family of topological spaces indexed by $I$. Let $\\family {f_i: X \\to Y_i}_{i \\mathop \\in I}$ be an indexed family of mappings indexed by $I$. {{TFAE|def = Initial Topology}}"}
+{"_id": "6728", "title": "Radius of Convergence from Limit of Sequence/Complex Case", "text": "Let $\\xi \\in \\C$ be a complex number. Let $\\displaystyle S \\paren z = \\sum_{n \\mathop = 0}^\\infty a_n \\paren {z - \\xi}^n$ be a (complex) power series about $\\xi$. Let the sequence $\\sequence {\\cmod {\\dfrac {a_{n + 1} } {a_n} } }_{n \\mathop \\in \\N}$ converges. Then $R$ is given by: :$\\displaystyle \\dfrac 1 R = \\lim_{n \\mathop \\to \\infty} \\cmod {\\dfrac {a_{n + 1} } {a_n} }$ If: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\cmod {\\dfrac {a_{n + 1} } {a_n} } = 0$ then the radius of convergence is infinite, and $S \\paren z$ is absolutely convergent for all $z \\in \\C$."}
+{"_id": "6729", "title": "Radius of Convergence from Limit of Sequence/Real Case", "text": "Let $\\xi \\in \\R$ be a real number. Let $\\displaystyle S \\paren x = \\sum_{n \\mathop = 0}^\\infty a_n \\paren {x - \\xi}^n$ be a power series about $\\xi$. Then the radius of convergence $R$ of $S \\paren x$ is given by: : $\\displaystyle \\frac 1 R = \\lim_{n \\mathop \\to \\infty} \\size {\\frac {a_{n + 1} } {a_n} }$ if this limit exists and is nonzero. If : $\\displaystyle \\lim_{n \\mathop \\to \\infty} \\size {\\frac {a_{n + 1} } {a_n} } = 0$ then the radius of convergence is infinite and therefore the interval of convergence is $\\R$."}
+{"_id": "6732", "title": "Existence of Radius of Convergence of Complex Power Series", "text": "Let $\\xi \\in \\C$. Let $\\displaystyle S \\paren z = \\sum_{n \\mathop = 0}^\\infty a_n \\paren {z - \\xi}^n $ be a (complex) power series about $\\xi$. Then there exists a radius of convergence $R \\in \\overline \\R$ of $S \\paren z$."}
+{"_id": "6733", "title": "Conjunction Equivalent to Negation of Implication of Negative/Formulation 1", "text": ":$p \\land q \\dashv \\vdash \\neg \\left({p \\implies \\neg q}\\right)$"}
+{"_id": "6736", "title": "Conjunction Equivalent to Negation of Implication of Negative/Formulation 2", "text": ":$\\vdash \\paren {p \\land q} \\iff \\paren {\\neg \\paren {p \\implies \\neg q} }$"}
+{"_id": "6738", "title": "Conjunction Equivalent to Negation of Implication of Negative/Formulation 1/Forward Implication", "text": ":$p \\land q \\vdash \\neg \\left({p \\implies \\neg q}\\right)$"}
+{"_id": "6739", "title": "Conjunction Equivalent to Negation of Implication of Negative/Formulation 1/Reverse Implication", "text": ":$\\neg \\left({p \\implies \\neg q}\\right) \\vdash p \\land q$"}
+{"_id": "6740", "title": "Rule of Material Implication/Formulation 1/Forward Implication", "text": ":$p \\implies q \\vdash \\neg p \\lor q$"}
+{"_id": "6741", "title": "Rule of Material Implication/Formulation 1", "text": ":$p \\implies q \\dashv \\vdash \\neg p \\lor q$"}
+{"_id": "6742", "title": "Rule of Material Implication/Formulation 1/Reverse Implication", "text": ":$\\neg p \\lor q \\vdash p \\implies q$"}
+{"_id": "6743", "title": "Rule of Material Implication/Formulation 1/Proof", "text": ":$p \\implies q \\dashv \\vdash \\neg p \\lor q$"}
+{"_id": "6744", "title": "Rule of Material Implication/Formulation 2", "text": ":$\\vdash \\paren {p \\implies q} \\iff \\paren {\\neg p \\lor q}$"}
+{"_id": "6746", "title": "Modus Tollendo Ponens/Variant/Formulation 1", "text": ":$p \\lor q \\dashv \\vdash \\neg p \\implies q$"}
+{"_id": "6747", "title": "Modus Tollendo Ponens/Sequent Form", "text": "==== Case 1 ==== {{:Modus Tollendo Ponens/Sequent Form/Case 1}} ==== Case 2 ==== {{:Modus Tollendo Ponens/Sequent Form/Case 2}}"}
+{"_id": "6748", "title": "Modus Tollendo Ponens/Variant/Formulation 1/Proof", "text": ":$p \\lor q \\dashv \\vdash \\neg p \\implies q$"}
+{"_id": "6751", "title": "Radius of Convergence of Power Series over Factorial/Complex Case", "text": "Let $\\xi \\in \\C$ be a complex number. Let $\\displaystyle \\map f z = \\sum_{n \\mathop = 0}^\\infty \\dfrac {\\paren {z - \\xi}^n} {n!}$. Then $\\map f z$ converges absolutely for all $z \\in \\C$. That is, the radius of convergence of the power series $\\displaystyle \\sum_{n \\mathop = 0}^\\infty \\frac {\\paren {z - \\xi}^n} {n!}$ is infinite."}
+{"_id": "6752", "title": "Radius of Convergence of Power Series over Factorial/Real Case", "text": "Let $\\xi \\in \\R$ be a real number. Let $\\displaystyle f \\left({x}\\right) = \\sum_{n \\mathop = 0}^\\infty \\frac {\\left({x - \\xi}\\right)^n} {n!}$. Then $f \\left({x}\\right)$ converges for all $x \\in \\R$. That is, the interval of convergence of the power series $\\displaystyle \\sum_{n \\mathop = 0}^\\infty \\frac {\\left({x - \\xi}\\right)^n} {n!}$ is $\\R$."}
+{"_id": "6753", "title": "Sum of Logarithms/Natural Logarithm", "text": "Let $x, y \\in \\R$ be strictly positive real numbers. Then: :$\\ln x + \\ln y = \\map \\ln {x y}$ where $\\ln$ denotes the natural logarithm."}
+{"_id": "6756", "title": "Sum of Logarithms/General Logarithm", "text": "Let $x, y, b \\in \\R$ be strictly positive real numbers such that $b > 1$. Then: :$\\log_b x + \\log_b y = \\map {\\log_b} {x y}$ where $\\log_b$ denotes the logarithm to base $b$."}
+{"_id": "6757", "title": "Rule of Simplification/Sequent Form/Formulation 1/Proof", "text": ":$(1): \\quad p \\land q \\vdash p$ :$(2): \\quad p \\land q \\vdash q$"}
+{"_id": "6760", "title": "Rule of Simplification/Sequent Form", "text": "The Rule of Simplification can be symbolised by the sequents:"}
+{"_id": "6761", "title": "Rule of Conjunction/Sequent Form", "text": "The Rule of Conjunction can be symbolised in sequent form as follows:"}
+{"_id": "6764", "title": "Radius of Convergence of Derivative of Complex Power Series", "text": "Let $\\xi \\in \\C$. For all $z \\in \\C$, define the power series: : $\\displaystyle S \\paren z = \\sum_{n \\mathop = 0}^\\infty a_n \\paren {z - \\xi}^n$ and: : $\\displaystyle S' \\paren z = \\sum_{n \\mathop = 1}^\\infty n a_n \\paren {z - \\xi}^{n - 1}$ Let $R$ be the radius of convergence of $S \\paren z$, and let $R'$ be the radius of convergence of $S' \\paren z$. Then $R =R'$."}
+{"_id": "6765", "title": "Rule of Addition/Sequent Form", "text": "The Rule of Addition can be symbolised by the sequents: :$(1): \\quad p \\vdash p \\lor q$ :$(2): \\quad q \\vdash p \\lor q$"}
+{"_id": "6766", "title": "Rule of Addition/Sequent Form/Proof 2", "text": ":$(1): \\quad p \\vdash p \\lor q$ :$(2): \\quad q \\vdash p \\lor q$"}
+{"_id": "6767", "title": "Rule of Addition/Sequent Form/Proof 1", "text": "The Rule of Addition can be symbolised by the sequents: :$(1): \\quad p \\vdash p \\lor q$ :$(2): \\quad q \\vdash p \\lor q$"}
+{"_id": "6770", "title": "Rule of Addition/Sequent Form/Form 2/Proof 1", "text": ":$q \\vdash p \\lor q$"}
+{"_id": "6772", "title": "Proof by Cases/Sequent Form", "text": "Proof by Cases can be symbolised by the sequent: :$p \\lor q, \\paren {p \\vdash r}, \\paren {q \\vdash r} \\vdash r$"}
+{"_id": "6776", "title": "Modus Ponendo Ponens/Sequent Form", "text": ":$p \\implies q, p \\vdash q$"}
+{"_id": "6780", "title": "Logarithm of Power/Natural Logarithm", "text": "Let $x \\in \\R$ be a strictly positive real number. Let $a \\in \\R$ be a real number such that $a > 1$. Let $r \\in \\R$ be any real number. Let $\\ln x$ be the natural logarithm of $x$. Then: :$\\map \\ln {x^r} = r \\ln x$"}
+{"_id": "6781", "title": "Logarithm of Power/General Logarithm", "text": "Let $x \\in \\R$ be a strictly positive real number. Let $a \\in \\R$ be a real number such that $a > 1$. Let $r \\in \\R$ be any real number. Let $\\log_a x$ be the logarithm to the base $a$ of $x$. Then: :$\\map {\\log_a} {x^r} = r \\log_a x$"}
+{"_id": "6783", "title": "Rule of Implication/Sequent Form", "text": "The Rule of Implication can be symbolised by the sequent: :$\\left({p \\vdash q}\\right) \\vdash p \\implies q$"}
+{"_id": "6784", "title": "Rule of Implication/Sequent Form/Proof 2", "text": ":$\\left({p \\vdash q}\\right) \\vdash p \\implies q$"}
+{"_id": "6785", "title": "Sum of Absolutely Convergent Series", "text": "Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ and $\\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n$ be two real or complex series that are absolutely convergent. Then the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\paren {a_n + b_n}$ is absolutely convergent, and: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\paren {a_n + b_n} = \\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n + \\sum_{n \\mathop = 1}^\\infty b_n$"}
+{"_id": "6786", "title": "Rule of Implication/Sequent Form/Proof 1", "text": ":$\\left({p \\vdash q}\\right) \\vdash p \\implies q$"}
+{"_id": "6787", "title": "Rule of Simplification/Sequent Form/Formulation 1", "text": ":$(1): \\quad p \\land q \\vdash p$ :$(2): \\quad p \\land q \\vdash q$"}
+{"_id": "6789", "title": "Rule of Simplification/Sequent Form/Formulation 2", "text": ":$(1): \\quad \\vdash p \\land q \\implies p$ :$(2): \\quad \\vdash p \\land q \\implies q$"}
+{"_id": "6790", "title": "Rule of Simplification/Sequent Form/Formulation 2/Proof 1/Form 1", "text": ":$\\vdash p \\land q \\implies p$"}
+{"_id": "6791", "title": "Rule of Simplification/Sequent Form/Formulation 2/Proof 1/Form 2", "text": ":$\\vdash p \\land q \\implies q$"}
+{"_id": "6792", "title": "Complement of Complement in Uniquely Complemented Lattice", "text": "Let $\\left({S, \\wedge, \\vee, \\preceq}\\right)$ be a uniquely complemented lattice. For each $x \\in S$, let $\\neg x$ be the complement of $x$. Then for each $x \\in S$: :$\\neg \\neg x = x$"}
+{"_id": "6795", "title": "Complement of Bottom/Boolean Algebra", "text": "Let $\\left({S, \\vee, \\wedge, \\neg}\\right)$ be a Boolean algebra. Then $\\neg \\bot = \\top$."}
+{"_id": "6796", "title": "Complement of Bottom/Bounded Lattice", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded lattice. Then the bottom $\\bot$ has a unique complement, namely $\\top$, top."}
+{"_id": "6798", "title": "Complement of Top/Boolean Algebra", "text": "Let $\\struct {S, \\vee, \\wedge, \\neg}$ be a Boolean algebra. Then $\\neg \\top = \\bot$."}
+{"_id": "6800", "title": "Duality Principle (Boolean Algebras)", "text": "Let $\\struct {S, \\vee, \\wedge}$ be a Boolean algebra. Then any theorem in $\\struct {S, \\vee, \\wedge}$ remains valid if both $\\vee$ and $\\wedge$ are interchanged, and also $\\bot$ and $\\top$ are interchanged throughout the whole theorem."}
+{"_id": "6801", "title": "Identities of Boolean Algebra also Zeroes", "text": "Let $\\struct {S, \\vee, \\wedge, \\neg}$ be a Boolean algebra, defined as in Definition 1. Let the identity for $\\vee$ be $\\bot$ and the identity for $\\wedge$ be $\\top$. Then: :$(1): \\quad \\forall x \\in S: x \\vee \\top = \\top$ :$(2): \\quad \\forall x \\in S: x \\wedge \\bot = \\bot$ That is, $\\bot$ is a zero element for $\\wedge$, and $\\top$ is a zero element for $\\vee$."}
+{"_id": "6802", "title": "Complement of Complement (Boolean Algebras)", "text": "Let $\\left({S, \\vee, \\wedge, \\neg}\\right)$ be a Boolean algebra. Then for all $a \\in S$: :$\\neg (\\neg a) = a$"}
+{"_id": "6803", "title": "De Morgan's Laws (Boolean Algebras)", "text": ":$\\neg \\paren {a \\vee b} = \\neg a \\wedge \\neg b$ :$\\neg \\paren {a \\wedge b} = \\neg a \\vee \\neg b$"}
+{"_id": "6804", "title": "Complement in Boolean Algebra is Unique", "text": "Let $\\left({S, \\vee, \\wedge}\\right)$ be a Boolean algebra. Then for all $a \\in S$, there is a unique $b \\in S$ such that: :$a \\wedge b = \\bot, a \\vee b = \\top$ i.e., a valid choice for $\\neg a$ as in axiom $(BA \\ 4)$ for Boolean algebras."}
+{"_id": "6805", "title": "Cancellation of Join in Boolean Algebra", "text": "Let $\\left({S, \\vee, \\wedge, \\neg}\\right)$ be a Boolean algebra. Let $a, b, c \\in S$, and suppose that: {{begin-eqn}} {{eqn | l = a \\vee c       | r = b \\vee c }} {{eqn | l = a \\vee \\neg c       | r = b \\vee \\neg c }} {{end-eqn}} Then $a = b$."}
+{"_id": "6806", "title": "Cancellation of Meet in Boolean Algebra", "text": "Let $\\struct {S, \\vee, \\wedge, \\neg}$ be a Boolean algebra. Let $a, b, c \\in S$. Let: {{begin-eqn}} {{eqn | l = a \\wedge c       | r = b \\wedge c }} {{eqn | l = a \\wedge \\neg c       | r = b \\wedge \\neg c }} {{end-eqn}} Then: : $a = b$"}
+{"_id": "6807", "title": "Operations of Boolean Algebra are Associative", "text": "Let $\\struct {S, \\vee, \\wedge, \\neg}$ be a Boolean algebra, defined as in Definition 1. Then: :$\\forall a, b, c \\in S: \\paren {a \\wedge b} \\wedge c = a \\wedge \\paren {b \\wedge c}$ :$\\forall a, b, c \\in S: \\paren {a \\vee b} \\vee c = a \\vee \\paren {b \\vee c}$ That is, both $\\vee$ and $\\wedge$ are associative operations."}
+{"_id": "6808", "title": "Absorption Laws (Boolean Algebras)", "text": "Let $\\struct {S, \\vee, \\wedge}$ be a Boolean algebra, defined as in Definition 1. Then for all $a, b \\in S$: :$a = a \\vee \\paren {a \\wedge b}$ :$a = a \\wedge \\paren {a \\vee b}$ That is, $\\vee$ absorbs $\\wedge$, and $\\wedge$ absorbs $\\vee$."}
+{"_id": "6811", "title": "Difference of Absolutely Convergent Series", "text": "Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ and $\\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n$ be two real or complex series that are absolutely convergent. Then the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\paren {a_n - b_n}$ is absolutely convergent, and: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\paren {a_n - b_n} = \\sum_{n \\mathop = 1}^\\infty a_n - \\sum_{n \\mathop = 1}^\\infty b_n$"}
+{"_id": "6814", "title": "True Statement is implied by Every Statement/Formulation 1", "text": ":$p \\vdash q \\implies p$"}
+{"_id": "6815", "title": "True Statement is implied by Every Statement/Formulation 2", "text": ":$\\vdash q \\implies \\paren {p \\implies q}$"}
+{"_id": "6816", "title": "True Statement is implied by Every Statement", "text": "==== Formulation 1 ==== {{:True Statement is implied by Every Statement/Formulation 1}} ==== Formulation 2 ==== {{:True Statement is implied by Every Statement/Formulation 2}}"}
+{"_id": "6817", "title": "False Statement implies Every Statement", "text": "==== Formulation 1 ==== {{:False Statement implies Every Statement/Formulation 1}} ==== Formulation 2 ==== {{:False Statement implies Every Statement/Formulation 2}}"}
+{"_id": "6818", "title": "False Statement implies Every Statement/Formulation 1", "text": ": $\\neg p \\vdash p \\implies q$"}
+{"_id": "6820", "title": "False Statement implies Every Statement/Formulation 1/Proof 2", "text": ":$\\neg p \\vdash p \\implies q$"}
+{"_id": "6821", "title": "False Statement implies Every Statement/Formulation 2", "text": ":$\\vdash \\neg p \\implies \\paren {p \\implies q}$"}
+{"_id": "6825", "title": "Modus Tollendo Ponens/Sequent Form/Case 1", "text": ":$p \\lor q, \\neg p \\vdash q$"}
+{"_id": "6826", "title": "Modus Tollendo Ponens/Sequent Form/Case 2", "text": ":$p \\lor q, \\neg q \\vdash p$"}
+{"_id": "6827", "title": "Two is Boolean Algebra", "text": "Let $\\mathbf 2$ denote two. Then $\\mathbf 2$ is a Boolean algebra."}
+{"_id": "6828", "title": "Two-Valued Functions form Boolean Algebra", "text": "Let $\\mathbf 2$ be the Boolean algebra two, and let $X$ be a set. Let $\\mathbf 2^X$ be the set of all mappings $p: X \\to \\mathbf 2$. Define the operations $\\vee$, $\\wedge$ and $\\neg$ on $\\mathbf 2^X$ in pointwise fashion thus: :$\\vee: \\mathbf 2^X \\times \\mathbf 2^X \\to \\mathbf 2^X, \\left({p \\vee q}\\right) (x) := p (x) \\vee q (x)$ :$\\wedge: \\mathbf 2^X \\times \\mathbf 2^X \\to \\mathbf 2^X, \\left({p \\wedge q}\\right) (x) := p (x) \\wedge q (x)$ :$\\neg: \\mathbf 2^X \\to \\mathbf 2^X, \\left({\\neg p}\\right) (x) := \\neg p (x)$ Furthermore, write $\\bot$ and $\\top$ for the constant mappings with these values, viz: :$\\bot: X \\to \\mathbf 2, \\bot (x) := \\bot$ :$\\top: X \\to \\mathbf 2, \\top (x) := \\top$ Then $\\left({\\mathbf 2^X, \\vee, \\wedge, \\neg}\\right)$ is a Boolean algebra, with $\\bot$ and $\\top$ as identities for $\\vee$ and $\\wedge$, respectively."}
+{"_id": "6829", "title": "Power Set is Boolean Ring", "text": "Let $S$ be a set, and let $\\mathcal P \\left({S}\\right)$ be its power set. Denote with $*$ and $\\cap$ symmetric difference and intersection, respectively. Then $\\left({S, *, \\cap}\\right)$ is a Boolean ring."}
+{"_id": "6830", "title": "Power Set and Two-Valued Functions are Isomorphic Boolean Rings", "text": "Let $S$ be a set. Let $\\mathbf 2$ be the Boolean ring two. Let $\\mathcal P \\left({S}\\right)$ be the power set of $S$; by Power Set is Boolean Ring, it is a Boolean ring. Let $\\mathbf 2^S$ be the set of all mappings $f: S \\to \\mathbf 2$; by Two-Valued Functions form Boolean Ring, it is also a Boolean ring. Let $\\chi_{\\left({\\cdot}\\right)}: \\mathcal P \\left({S}\\right) \\to \\mathbf 2^S$ be the characteristic function operation. Then $\\chi_{\\left({\\cdot}\\right)}$ is a ring isomorphism."}
+{"_id": "6831", "title": "Peirce's Law/Formulation 1", "text": ": $\\left({p \\implies q}\\right) \\implies p \\vdash p$"}
+{"_id": "6833", "title": "Peirce's Law/Formulation 2/Proof 2", "text": ": $\\vdash \\left({\\left({p \\implies q}\\right) \\implies p}\\right) \\implies p$"}
+{"_id": "6835", "title": "Peirce's Law/Strong Form/Formulation 1", "text": ":$\\paren {\\paren {p \\implies q} \\implies p} \\dashv \\vdash p$"}
+{"_id": "6838", "title": "Conjunction with Negative Equivalent to Negation of Implication", "text": "==== Formulation 1 ==== {{:Conjunction with Negative Equivalent to Negation of Implication/Formulation 1}} ==== Formulation 2 ==== {{:Conjunction with Negative Equivalent to Negation of Implication/Formulation 2}}"}
+{"_id": "6839", "title": "Conjunction with Negative Equivalent to Negation of Implication/Formulation 1", "text": ":$p \\land \\neg q \\dashv \\vdash \\neg \\left({p \\implies q}\\right)$"}
+{"_id": "6840", "title": "Conjunction with Negative Equivalent to Negation of Implication/Formulation 1/Forward Implication", "text": ":$p \\land \\neg q \\vdash \\neg \\paren {p \\implies q}$"}
+{"_id": "6841", "title": "Conjunction with Negative Equivalent to Negation of Implication/Formulation 1/Reverse Implication", "text": ":$\\neg \\left({p \\implies q}\\right) \\vdash p \\land \\neg q$"}
+{"_id": "6842", "title": "Conjunction with Negative Equivalent to Negation of Implication/Formulation 1/Proof", "text": ":$p \\land \\neg q \\dashv \\vdash \\neg \\left({p \\implies q}\\right)$"}
+{"_id": "6843", "title": "Conjunction with Negative Equivalent to Negation of Implication/Formulation 2", "text": ":$\\vdash \\paren {p \\land \\neg q} \\iff \\paren {\\neg \\paren {p \\implies q} }$"}
+{"_id": "6844", "title": "Reflexive Circular Relation is Equivalence", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a reflexive and circular relation in $S$. Then $\\mathcal R$ is an equivalence relation."}
+{"_id": "6845", "title": "Inverse of Algebraic Structure Isomorphism is Isomorphism/General Result", "text": "Let $\\phi: \\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right)$ and $\\left({T, *_1, *_2, \\ldots, *_n}\\right)$ be algebraic structures. Let $\\phi: \\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right) \\to \\left({T, *_1, *_2, \\ldots, *_n}\\right)$ be a mapping. Then: : $\\phi: \\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right) \\to \\left({T, *_1, *_2, \\ldots, *_n}\\right)$ is an isomorphism {{iff}}: : $\\phi^{-1}: \\left({T, *_1, *_2, \\ldots, *_n}\\right) \\to \\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right)$ is also an isomorphism."}
+{"_id": "6846", "title": "Fundamental Theorem of Calculus for Complex Riemann Integrals", "text": "Let $\\closedint a b$ be a closed real interval. Let $F, f: \\closedint a b \\to \\C$ be complex functions. Suppose that $F$ is a primitive of $f$. Then the complex Riemann integral of $f$ satisfies: :$\\displaystyle \\int_a^b \\map f t \\rd t = \\map F b - \\map F a$"}
+{"_id": "6847", "title": "Fundamental Theorem of Calculus for Contour Integrals", "text": "Let $F, f: D \\to \\C$ be complex functions, where $D$ is a connected domain. Let $C$ be a contour that is a concatenation of the directed smooth curves $C_1, \\ldots, C_n$. Let $C_k$ be parameterized by the smooth path $\\gamma_k: \\closedint {a_k} {b_k} \\to D$ for all $k \\in \\set {1, \\ldots, n}$. Suppose that $F$ is a primitive of $f$. If $C$ has start point $z$ and end point $w$, then: :$\\displaystyle \\int_C \\map f z \\rd z = \\map F w - \\map F z$ If $C$ is a closed contour, then: :$\\displaystyle \\oint_C \\map f z \\rd z = 0$"}
+{"_id": "6848", "title": "Power Series is Taylor Series", "text": "Let $\\displaystyle f \\left({z}\\right) = \\sum_{n \\mathop = 0}^\\infty a_n \\left({z - \\xi}\\right)^n$ be a complex power series about $\\xi \\in \\C$. Let $R$ be the radius of convergence of $f$. Then, $f$ is of differentiability class $C^\\infty$. For all $n \\in \\N$: :$a_n = \\dfrac{f^{\\left({n}\\right) } \\left({\\xi}\\right) }{ n! }$ Hence, $f$ is equal to its Taylor series expansion about $\\xi$: :$\\displaystyle \\forall z \\in \\C, \\left\\vert{z - \\xi}\\right\\vert < R: \\quad f \\left({z}\\right) = \\sum_{n \\mathop = 0}^\\infty \\dfrac{\\left({z - \\xi}\\right)^n}{n!} f^{\\left({n}\\right) } \\left({\\xi}\\right)$"}
+{"_id": "6850", "title": "Image of Complex Exponential Function", "text": "The image of the complex exponential function is $\\C \\setminus \\left\\{ {0}\\right\\}$."}
+{"_id": "6851", "title": "Reciprocal of Complex Exponential", "text": ":$\\dfrac 1 {\\map \\exp z} = \\map \\exp {-z}$"}
+{"_id": "6852", "title": "Power Function Strictly Preserves Ordering in Ordered Group", "text": "Let $n \\in \\N_{>0}$ be a strictly positive integer."}
+{"_id": "6853", "title": "Period of Complex Exponential Function", "text": ":$\\map \\exp {z + 2 k \\pi i} = \\map \\exp z$"}
+{"_id": "6854", "title": "Zero Staircase Integral Condition for Primitive", "text": "Let $f: D \\to \\C$ be a continuous complex function, where $D$ is a connected domain. Let $z_0 \\in D$. Suppose that $\\displaystyle \\oint_C \\map f z \\rd z = 0$ for all closed staircase contours $C$ in $D$. Then $f$ has a primitive $F: D \\to \\C$ defined by: :$\\displaystyle \\map F w = \\int_{C_w} \\map f z \\rd z$ where $C_w$ is any staircase contour in $D$ with start point $z_0$ and end point $w$."}
+{"_id": "6855", "title": "Derivative of Complex Polynomial", "text": "Let $a_n \\in \\C$ for $n \\in \\left\\{ {0, 1, \\ldots, N}\\right\\}$, where $N \\in \\N$. Let $f: \\C \\to \\C$ be a complex polynomial defined by $\\displaystyle f \\left({z}\\right) = \\sum_{n \\mathop = 0}^N a_n z^n$. Then $f$ is complex differentiable and its derivative is: :$\\displaystyle f' \\left({z}\\right) = \\sum_{n \\mathop = 1}^N n a_n z^{n-1}$"}
+{"_id": "6856", "title": "Nested Sphere Theorem", "text": "Let $M = \\left({A, d}\\right)$ be a complete metric space. Let $\\left\\langle{S_n}\\right\\rangle$ be a sequence of closed balls in $M$ defined by: : $S_n = B^-_{\\rho_n} \\left({x_n}\\right)$ where $\\rho_n \\to 0$ as $n \\to \\infty$ and: : $S_1 \\supseteq S_2 \\supseteq \\cdots \\supseteq S_n \\supseteq \\cdots$ Then there exists $x \\in A$ such that: :$\\displaystyle \\bigcap_{n \\mathop = 1}^\\infty S_n = \\left\\{{x}\\right\\}$"}
+{"_id": "6858", "title": "Principle of Dilemma/Formulation 1/Forward Implication", "text": ": $\\left({p \\implies q}\\right) \\land \\left({\\neg p \\implies q}\\right) \\vdash q$"}
+{"_id": "6861", "title": "Principle of Dilemma/Formulation 1/Forward Implication/Proof 3", "text": ": $\\left({p \\implies q}\\right) \\land \\left({\\neg p \\implies q}\\right) \\vdash q$"}
+{"_id": "6863", "title": "Principle of Dilemma/Formulation 1/Reverse Implication", "text": ": $q \\vdash \\left({p \\implies q}\\right) \\land \\left({\\neg p \\implies q}\\right)$"}
+{"_id": "6866", "title": "Principle of Dilemma/Formulation 2/Reverse Implication", "text": ":$\\vdash q \\implies \\paren {\\paren {p \\implies q} \\land \\paren {\\neg p \\implies q} }$"}
+{"_id": "6867", "title": "Properties of Complex Exponential Function", "text": "Let $z \\in \\C$ be a complex number. Let $\\exp z$ be the exponential of $z$. Then:"}
+{"_id": "6868", "title": "Open Domain is Connected iff it is Path-Connected", "text": "Let $D \\subseteq \\C$ be a open subset of the set of complex numbers. Then $D$ is connected {{iff}} $D$ is path-connected."}
+{"_id": "6869", "title": "Proof by Contradiction/Variant 2/Formulation 1", "text": ":$p \\implies q, p \\implies \\neg q \\vdash \\neg p$"}
+{"_id": "6870", "title": "Proof by Contradiction/Variant 2/Formulation 2", "text": ":$\\vdash \\paren {\\paren {p \\implies q} \\land \\paren {p \\implies \\neg q} } \\implies \\neg p$"}
+{"_id": "6871", "title": "Proof by Contradiction/Variant 3/Formulation 1", "text": ":$p \\implies \\neg p \\vdash \\neg p$"}
+{"_id": "6872", "title": "Proof by Contradiction/Variant 3/Formulation 2", "text": ":$\\vdash \\paren {p \\implies \\neg p} \\implies \\neg p$"}
+{"_id": "6873", "title": "Primitive of Function on Connected Domain", "text": "Let $f: D \\to \\C$ be a continuous complex function, where $D$ is a connected domain. Then the following three conditions are equivalent: :$(1): \\quad$ $f$ has a primitive. :$(2): \\quad$ For any two contours $C_1, C_2$ in $D$ with identical start points $z_1 \\in D$ and end points $z_2 \\in D$, we have: ::$\\displaystyle \\int_{C_1} \\map f z \\rd z = \\int_{C_2} \\map f z \\rd z$ :$(3): \\quad$ For all closed contours $C$ in $D$, we have: ::$\\displaystyle \\oint_C \\map f z \\rd z = 0$ If the conditions hold, we can choose any $z_0 \\in D$ and define a primitive $F: D \\to \\C$ of $f$ by: :$\\displaystyle \\map F w = \\int_{C_w} \\map f z \\rd z$ where $C_w$ is any contour in $D$ with start point $z_0$ and end point $w$."}
+{"_id": "6874", "title": "Mind the Gap", "text": "Let $\\left({S, \\preccurlyeq}\\right)$ be a totally ordered set. Let $a, b \\in S$ with $a \\prec b$. Suppose that: :$\\left\\{ {x \\in S: a \\prec x \\prec b}\\right\\} = \\varnothing$ Then: : $a^\\succ = b^\\succcurlyeq$ and : $b^\\prec = a^\\preccurlyeq$ where: : $a^\\succ$ is the strict upper closure of $a$ : $b^\\succcurlyeq$ is the weak upper closure of $b$ : $b^\\prec$ is the strict lower closure of $b$ : $a^\\preccurlyeq$ is the weak lower closure of $a$."}
+{"_id": "6875", "title": "Proof by Contradiction/Sequent Form", "text": ":$\\paren {p \\vdash \\bot} \\vdash \\neg p$"}
+{"_id": "6876", "title": "Left or Right Inverse of Matrix is Inverse", "text": "Let $\\mathbf A, \\mathbf B$ be square matrices of order $n$ over a commutative ring with unity $\\left({R, +, \\circ}\\right)$. Suppose that: : $\\mathbf A \\mathbf B = \\mathbf I_n$ where $\\mathbf I_n$ is the unit matrix of order $n$. Then $\\mathbf A$ and $\\mathbf B$ are invertible matrices, and furthermore: :$\\mathbf B = \\mathbf A^{-1}$ where $\\mathbf A^{-1}$ is the inverse of $\\mathbf A$."}
+{"_id": "6877", "title": "Banach-Alaoglu Theorem", "text": "Let $X$ be a separable normed space. Then the closed unit sphere in its dual $X^*$ is weak* sequentially compact."}
+{"_id": "6878", "title": "Empty Product is Terminal Object", "text": "Let $\\mathbf C$ be a metacategory. Suppose $\\mathbf C$ admits a product $\\displaystyle \\prod \\O$ for the empty set. Then $\\displaystyle \\prod \\O$ is a terminal object of $\\mathbf C$."}
+{"_id": "6879", "title": "Unary Product for Object is Itself", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ be an object of $\\mathbf C$. Then $\\displaystyle \\prod \\set C = C$, where $\\displaystyle \\prod$ denotes product."}
+{"_id": "6884", "title": "Vector Cross Product is Orthogonal to Factors", "text": "Let $\\mathbf a$ and $\\mathbf b$ be vectors in the Euclidean space $\\R^3$. Let $\\mathbf a \\times \\mathbf b$ denote the vector cross product. Then: :$(1): \\quad$ $\\mathbf a$ and $\\mathbf a \\times \\mathbf b$ are orthogonal. :$(2): \\quad$ $\\mathbf b$ and $\\mathbf a \\times \\mathbf b$ are orthogonal."}
+{"_id": "6885", "title": "Squeeze Theorem/Sequences/Metric Spaces", "text": "Let $M = \\left({S, d}\\right)$ be a metric space or pseudometric space. Let $\\left\\langle{x_n}\\right\\rangle$ be a sequence in $S$. Let $p \\in S$.  Let $\\left\\langle{r_n}\\right\\rangle$ be a sequence in $\\R_{\\ge 0}$. Let $\\left\\langle{r_n}\\right\\rangle$ converge to $0$. For each $n$, let $d \\left({p, x_n}\\right) \\le r_n$. Then $\\left\\langle{x_n}\\right\\rangle$ converges to $p$."}
+{"_id": "6886", "title": "Squeeze Theorem/Sequences/Linearly Ordered Space", "text": "Let $\\left({S, \\le, \\tau}\\right)$ be a linearly ordered space. Let $\\left\\langle{x_n}\\right\\rangle$, $\\left\\langle{y_n}\\right\\rangle$, and $\\left\\langle{z_n}\\right\\rangle$ be sequences in $S$. Let $p \\in S$. Let $\\left\\langle{x_n}\\right\\rangle$ and $\\left\\langle{z_n}\\right\\rangle$ both converge to $p$ For each $n$, let $x_n \\le y_n \\le z_n$. Then $\\left\\langle{y_n}\\right\\rangle$ converges to $p$."}
+{"_id": "6888", "title": "Non-Zero Vectors Orthogonal iff Perpendicular", "text": "Let $\\mathbf u$, $\\mathbf v$ be non-zero vectors in the Euclidean space $\\R^n$. Then $\\mathbf u$ and $\\mathbf v$ are orthogonal {{iff}} they are perpendicular."}
+{"_id": "6892", "title": "Exponential Function is Continuous/Complex", "text": ":$\\forall z_0 \\in \\C: \\displaystyle \\lim_{z \\mathop \\to z_0} \\exp z = \\exp z_0$"}
+{"_id": "6894", "title": "Order Topology on Convex Subset is Subspace Topology", "text": "Let $\\left({S, \\preceq,\\tau}\\right)$ be a linearly ordered space. Let $A \\subseteq S$ be a convex set in $S$. Let $\\upsilon$ be the order topology on $A$. Let $\\tau'$ be the $\\tau$-relative subspace topology on $A$. Then $\\upsilon = \\tau'$."}
+{"_id": "6896", "title": "Exponential Function is Continuous/Real Numbers", "text": "The real exponential function is continuous. That is: :$\\forall x_0 \\in \\R: \\displaystyle \\lim_{x \\mathop \\to x_0} \\ \\exp x = \\exp x_0$"}
+{"_id": "6897", "title": "Edge of Tree is Bridge", "text": "Let $T$ be a tree. Let $e$ be an edge of $T$. Then $e$ is a bridge of $T$."}
+{"_id": "6898", "title": "Sub-Basis for Topological Subspace", "text": "Let $\\left({X,\\tau}\\right)$ be a topological space. Let $K$ be a sub-basis for $\\tau$. Let $\\left({S, \\tau'}\\right)$ be a subspace of $\\left({X, \\tau}\\right)$. Let $K' = \\left\\{{ U \\cap S: U \\in K }\\right\\}$. That is, $K'$ consists of the $\\tau'$-open sets in $S$ corresponding to elements of $K$. Then $K'$ is a sub-basis for $\\tau'$."}
+{"_id": "6899", "title": "Modus Ponendo Ponens/Variant 1", "text": ":  $p \\vdash \\left({p \\implies q}\\right) \\implies q$"}
+{"_id": "6901", "title": "Modus Ponendo Ponens/Variant 1/Proof 2", "text": ":$p \\vdash \\left({p \\implies q}\\right) \\implies q$"}
+{"_id": "6902", "title": "Modus Ponendo Ponens/Variant 2", "text": ":  $\\vdash p \\implies \\left({\\left({p \\implies q}\\right) \\implies q}\\right)$"}
+{"_id": "6904", "title": "Modus Ponendo Ponens/Variant 2/Proof 2", "text": ":$\\vdash p \\implies \\paren {\\paren {p \\implies q} \\implies q}$"}
+{"_id": "6908", "title": "Rule of Commutation/Conjunction", "text": "Conjunction is commutative: === Formulation 1 === {{:Rule of Commutation/Conjunction/Formulation 1}} === Formulation 2 === {{:Rule of Commutation/Conjunction/Formulation 2}}"}
+{"_id": "6909", "title": "Rule of Commutation/Disjunction", "text": "Disjunction is commutative: === Formulation 1 === {{:Rule of Commutation/Disjunction/Formulation 1}} === Formulation 2 === {{:Rule of Commutation/Disjunction/Formulation 2}}"}
+{"_id": "6910", "title": "Factor Principles/Conjunction on Right/Formulation 1", "text": ": $p \\implies q \\vdash \\left({p \\land r}\\right) \\implies \\left ({q \\land r}\\right)$"}
+{"_id": "6911", "title": "Factor Principles/Conjunction on Right/Formulation 2", "text": ": $\\vdash \\left({p \\implies q}\\right) \\implies \\left({\\left({p \\land r}\\right) \\implies \\left ({q \\land r}\\right)}\\right)$"}
+{"_id": "6912", "title": "Interior of Convex Angle is Convex Set", "text": "Let $\\mathbf v, \\mathbf w$ be two non-zero vectors in $\\R^2$, and let $p$ be a point in $\\R^2$. Suppose that the angle between $\\mathbf v$ and $\\mathbf w$ is a convex angle. Then the set  :$U = \\left\\{ {p + st \\mathbf v + \\left({1-s}\\right) t \\mathbf w : s \\in \\left({0\\,.\\,.\\,1}\\right) , t \\in \\R_{>0} }\\right\\}$ is a convex set. {{expand|It'd be really nice to have a picture of $U$ to support intuition and connect with the page title}}"}
+{"_id": "6913", "title": "Factor Principles/Conjunction on Right", "text": "==== Formulation 1 ==== {{:Factor Principles/Conjunction on Right/Formulation 1}} ==== Formulation 2 ==== {{:Factor Principles/Conjunction on Right/Formulation 2}}"}
+{"_id": "6914", "title": "Factor Principles/Conjunction on Left", "text": "==== Formulation 1 ==== {{:Factor Principles/Conjunction on Left/Formulation 1}} ==== Formulation 2 ==== {{:Factor Principles/Conjunction on Left/Formulation 2}}"}
+{"_id": "6915", "title": "Factor Principles/Conjunction on Left/Formulation 1", "text": ": $p \\implies q \\vdash \\left({r \\land p}\\right) \\implies \\left ({r \\land q}\\right)$"}
+{"_id": "6916", "title": "Factor Principles/Conjunction on Left/Formulation 2", "text": ":$\\vdash \\paren {p \\implies q} \\implies \\paren {\\paren {r \\land p} \\implies \\paren {r \\land q} }$"}
+{"_id": "6917", "title": "Triangle is Convex Set", "text": "The interior of a triangle embedded in $\\R^2$ is a convex set."}
+{"_id": "6920", "title": "Subset Relation is Compatible with Subset Product", "text": "Let $\\left({S,\\circ}\\right)$ be a magma. Let $\\mathcal P \\left({S}\\right)$ be the power set of $S$. Let $\\circ_\\mathcal P$ be the operation induced on $\\mathcal P \\left({S}\\right)$ by $\\circ$. Then the subset relation $\\subseteq$ is compatible with $\\circ_\\mathcal P$."}
+{"_id": "6923", "title": "Subset Relation is Compatible with Subset Product/Corollary 1", "text": "Let $A, B, C, D \\in \\powerset S$. Let $A \\subseteq B$ and $C \\subseteq D$. Then: :$A \\circ_\\PP C \\subseteq B \\circ_\\PP D$"}
+{"_id": "6924", "title": "Subset Relation is Compatible with Subset Product/Corollary 2", "text": "Let $A,B \\in \\mathcal P \\left({S}\\right)$, the power set of $S$. {{improve|No need to use power set. $A \\subseteq B \\subseteq S$ sufficient.}} Let $A \\subseteq B$. Let $x \\in S$. Then: :$x \\circ A \\subseteq x \\circ B$ :$A \\circ x \\subseteq B \\circ x$"}
+{"_id": "6926", "title": "Equivalence of Definitions of Normal Subset/1 iff 2", "text": "Let $\\left({G, \\circ}\\right)$ be a group. Let $S$ be a subset of $G$. Then Normal Subset/Definition 1 is equivalent to Normal Subset/Definition 2. That is, the following three statements are equivalent: :$(1)\\quad \\forall g \\in G: g \\circ S = S \\circ g$ :$(2)\\quad \\forall g \\in G: g \\circ S \\circ g^{-1} = S$ :$(3)\\quad \\forall g \\in G: g^{-1} \\circ S \\circ g = S$"}
+{"_id": "6927", "title": "Subset Product with Identity", "text": "Let $\\left({S, \\circ}\\right)$ be a magma. Let $\\left({S, \\circ}\\right)$ have an identity element $e$. Then $e \\circ S = S \\circ e = S$, where $\\circ$ is understood to be the subset product with singleton."}
+{"_id": "6928", "title": "Equivalence of Definitions of Normal Subset/2 implies 3", "text": "Let $\\left({G, \\circ}\\right)$ be a group. Let $S \\subseteq G$. Let $S$ be a normal subset of $G$ by Definition 2. Then $S$ is a normal subset of $G$ by Definition 3. That is, if: :$\\forall g \\in G: g \\circ S \\circ g^{-1} = S$ or: :$\\forall g \\in G: g^{-1} \\circ S \\circ g = S$ then: :$\\forall g \\in G: g \\circ S \\circ g^{-1} \\subseteq S$ and: :$\\forall g \\in G: g^{-1} \\circ S \\circ g \\subseteq S$"}
+{"_id": "6929", "title": "Superset Relation is Compatible with Subset Product", "text": "Let $\\left({S,\\circ}\\right)$ be a magma. Let $\\circ_{\\mathcal P}$ be the subset product on $\\mathcal P \\left({S}\\right)$, the power set of $S$. Then the superset relation $\\supseteq$ is compatible with $\\circ_{\\mathcal P}$."}
+{"_id": "6930", "title": "Equivalence of Definitions of Normal Subset/3 iff 4", "text": "Let $\\left({G,\\circ}\\right)$ be a group. Let $S \\subseteq G$. Then: : $S$ is a normal subset of $G$ by Definition 3 iff: : $S$ is a normal subset of $G$ by Definition 4. That is, the following conditions are equivalent: : $(1)\\quad \\forall g \\in G: g \\circ S \\circ g^{-1} \\subseteq S$ : $(2)\\quad \\forall g \\in G: g^{-1} \\circ S \\circ g \\subseteq S$ : $(3)\\quad \\forall g \\in G: S \\subseteq g \\circ S \\circ g^{-1}$ : $(4)\\quad \\forall g \\in G: S \\subseteq g^{-1} \\circ S \\circ g$"}
+{"_id": "6931", "title": "Equivalence of Definitions of Normal Subset/3 and 4 imply 2", "text": "Let $\\left({G,\\circ}\\right)$ be a group. Let $S \\subseteq G$. Let $S$ be a normal subset of $G$ by Definition 3 and Definition 4. Then $S$ is a normal subset of $G$ by Definition 2."}
+{"_id": "6932", "title": "Equivalence of Definitions of Normal Subset/3 iff 5", "text": "Let $\\struct {G, \\circ}$ be a group. Let $S \\subseteq G$. Then: : $S$ is a normal subset of $G$ by Definition 3 {{iff}}: : $S$ is a normal subset of $G$ by Definition 5."}
+{"_id": "6933", "title": "Equivalence of Definitions of Normal Subgroup", "text": "Let $G$ be a group. Let $N$ be a subgroup of $G$. {{TFAE|def = Normal Subgroup}}"}
+{"_id": "6934", "title": "Subgroup is Normal iff Normal Subset", "text": "Let $\\left({G, \\circ}\\right)$ be a group. Let $N$ be a subgroup of $G$. Then $N$ is normal in $G$ (by definition 1) {{iff}} it is a normal subset of $G$."}
+{"_id": "6935", "title": "Rule of Explosion/Sequent Form", "text": ":$\\bot \\vdash \\phi$"}
+{"_id": "6936", "title": "Law of Excluded Middle/Sequent Form", "text": "The Law of Excluded Middle can be symbolised by the sequent: :$\\vdash p \\lor \\neg p$"}
+{"_id": "6937", "title": "Power Set of Monoid under Induced Operation is Monoid", "text": "Let $\\left({G, \\circ}\\right)$ be a monoid with identity $e$. Let $\\left({\\mathcal P \\left({G}\\right), \\circ_\\mathcal P}\\right)$ be the algebraic structure consisting of the power set of $G$ and the operation induced on $\\mathcal P \\left({S}\\right)$ by $\\circ$. Then $\\left({\\mathcal P \\left({G}\\right), \\circ_\\mathcal P}\\right)$ is a monoid with identity $\\left\\{{e}\\right\\}$."}
+{"_id": "6938", "title": "Principle of Non-Contradiction/Sequent Form/Formulation 2", "text": ":$\\vdash \\neg \\paren {p \\land \\neg p}$"}
+{"_id": "6941", "title": "Power Set of Semigroup under Induced Operation is Semigroup", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroup. Let $\\left({\\mathcal P \\left({S}\\right), \\circ_\\mathcal P}\\right)$ be the algebraic structure consisting of the power set of $S$ and the operation induced on $\\mathcal P \\left({S}\\right)$ by $\\circ$. Then $\\left({\\mathcal P \\left({S}\\right), \\circ_\\mathcal P}\\right)$ is a semigroup."}
+{"_id": "6942", "title": "Power Set of Group under Induced Operation is Monoid", "text": "Let $\\left({G, \\circ}\\right)$ be a group with identity $e$. Let $\\left({\\mathcal P \\left({G}\\right), \\circ_\\mathcal P}\\right)$ be the algebraic structure consisting of the power set of $G$ and the operation induced on $\\mathcal P \\left({G}\\right)$ by $\\circ$. Then $\\left({\\mathcal P \\left({G}\\right), \\circ_\\mathcal P}\\right)$ is a monoid with identity $\\left\\{{e}\\right\\}$."}
+{"_id": "6944", "title": "Power Set of Magma under Induced Operation is Magma", "text": "Let $\\left({S, \\circ}\\right)$ be a magma. Let $\\left({\\mathcal P \\left({S}\\right), \\circ_\\mathcal P}\\right)$ be the algebraic structure consisting of the power set of $S$ and the operation induced on $\\mathcal P \\left({S}\\right)$ by $\\circ$. Then $\\left({\\mathcal P \\left({S}\\right), \\circ_\\mathcal P}\\right)$ is a magma."}
+{"_id": "6945", "title": "Element in Left Coset iff Product with Inverse in Subgroup", "text": "Let $y H$ denote the left coset of $H$ by $y$. Then: :$x \\in y H \\iff x^{-1} y \\in H$"}
+{"_id": "6946", "title": "Element in Right Coset iff Product with Inverse in Subgroup", "text": "Let $H \\circ y$ denote the right coset of $H$ by $y$. Then: :$x \\in H y \\iff x y^{-1} \\in H$"}
+{"_id": "6949", "title": "Inverse of Inverse of Subset of Group", "text": "Let $\\struct {G, \\circ}$ be a group. Let $X \\subseteq G$. Then: : $\\paren {X^{-1} }^{-1} = X$. where $X^{-1}$ denotes the inverse of $X$."}
+{"_id": "6953", "title": "Exclusive Or is Commutative", "text": ": $p \\oplus q \\dashv \\vdash q \\oplus p$"}
+{"_id": "6954", "title": "Exclusive Or is Associative", "text": "Exclusive or is associative: : $p \\oplus \\left({q \\oplus r}\\right) \\dashv \\vdash \\left({p \\oplus q}\\right) \\oplus r$"}
+{"_id": "6955", "title": "Exclusive Or with Itself", "text": "Exclusive or destroys copies of itself: : $p \\oplus p \\dashv \\vdash \\bot$"}
+{"_id": "6956", "title": "Non-Equivalence as Disjunction of Conjunctions/Formulation 1/Proof 1", "text": ": $\\neg \\left ({p \\iff q}\\right) \\dashv \\vdash \\left({\\neg p \\land q}\\right) \\lor \\left({p \\land \\neg q}\\right)$"}
+{"_id": "6957", "title": "Non-Equivalence as Disjunction of Conjunctions/Formulation 1/Proof 2", "text": ": $\\neg \\left ({p \\iff q}\\right) \\dashv \\vdash \\left({\\neg p \\land q}\\right) \\lor \\left({p \\land \\neg q}\\right)$"}
+{"_id": "6958", "title": "Non-Equivalence as Disjunction of Conjunctions/Formulation 1", "text": ": $\\neg \\left ({p \\iff q}\\right) \\dashv \\vdash \\left({\\neg p \\land q}\\right) \\lor \\left({p \\land \\neg q}\\right)$"}
+{"_id": "6959", "title": "Exclusive Or is Negation of Biconditional", "text": "Exclusive or is equivalent to the negation of the biconditional: :$p \\oplus q \\dashv \\vdash \\neg \\paren {p \\iff q}$"}
+{"_id": "6960", "title": "Exclusive Or as Disjunction of Conjunctions", "text": ":$p \\oplus q \\dashv \\vdash \\paren {\\neg p \\land q} \\lor \\paren {p \\land \\neg q}$"}
+{"_id": "6962", "title": "Non-Equivalence as Disjunction of Negated Implications/Proof 1", "text": ": $\\neg \\left ({p \\iff q}\\right) \\dashv \\vdash \\neg \\left({p \\implies q}\\right) \\lor \\neg \\left({q \\implies p}\\right)$"}
+{"_id": "6963", "title": "Non-Equivalence as Disjunction of Negated Implications/Proof 2", "text": ": $\\neg \\left ({p \\iff q}\\right) \\dashv \\vdash \\neg \\left({p \\implies q}\\right) \\lor \\neg \\left({q \\implies p}\\right)$"}
+{"_id": "6964", "title": "Non-Equivalence as Disjunction of Negated Implications", "text": ": $\\neg \\left ({p \\iff q}\\right) \\dashv \\vdash \\neg \\left({p \\implies q}\\right) \\lor \\neg \\left({q \\implies p}\\right)$"}
+{"_id": "6965", "title": "Conjunction of Disjunction with Negation is Conjunction with Negation", "text": ":$\\left({p \\lor q}\\right) \\land \\neg q \\dashv \\vdash p \\land \\neg q$"}
+{"_id": "6966", "title": "Non-Equivalence as Conjunction of Disjunction with Negation of Conjunction/Proof 1", "text": ": $\\neg \\left ({p \\iff q}\\right) \\dashv \\vdash \\left({p \\lor q} \\right) \\land \\neg \\left({p \\land q}\\right)$"}
+{"_id": "6967", "title": "Non-Equivalence as Conjunction of Disjunction with Negation of Conjunction/Proof 2", "text": ": $\\neg \\left ({p \\iff q}\\right) \\dashv \\vdash \\left({p \\lor q} \\right) \\land \\neg \\left({p \\land q}\\right)$"}
+{"_id": "6968", "title": "Non-Equivalence as Conjunction of Disjunction with Negation of Conjunction", "text": ": $\\neg \\left ({p \\iff q}\\right) \\dashv \\vdash \\left({p \\lor q} \\right) \\land \\neg \\left({p \\land q}\\right)$ That is, negation of the biconditional means the same thing as '''either-or but not both''', that is, exclusive or."}
+{"_id": "6975", "title": "Inversion Mapping is Involution", "text": "Let $G$ be a group, and let $\\iota: G \\to G$ be the inversion mapping. Then $\\iota$ is an involution. That is: :$\\forall g \\in G: \\map \\iota {\\map \\iota g} = g$"}
+{"_id": "6977", "title": "Rule of Association/Conjunction", "text": "Conjunction is associative: === Formulation 1 === {{:Rule of Association/Conjunction/Formulation 1}} === Formulation 2 === {{:Rule of Association/Conjunction/Formulation 2}}"}
+{"_id": "6978", "title": "Rule of Association/Disjunction", "text": "Disjunction is associative: === Formulation 1 === {{:Rule of Association/Disjunction/Formulation 1}} === Formulation 2 === {{:Rule of Association/Disjunction/Formulation 2}}"}
+{"_id": "6979", "title": "Rule of Association/Conjunction/Formulation 1", "text": ":$p \\land \\left({q \\land r}\\right) \\dashv \\vdash \\left({p \\land q}\\right) \\land r$"}
+{"_id": "6980", "title": "Rule of Association/Disjunction/Formulation 1", "text": ":$p \\lor \\paren {q \\lor r} \\dashv \\vdash \\paren {p \\lor q} \\lor r$"}
+{"_id": "6984", "title": "Rule of Association/Disjunction/Formulation 1/Proof 2", "text": ":$p \\lor \\left({q \\lor r}\\right) \\dashv \\vdash \\left({p \\lor q}\\right) \\lor r$"}
+{"_id": "6985", "title": "Rule of Association/Conjunction/Formulation 2", "text": ":$\\vdash \\paren {p \\land \\paren {q \\land r} } \\iff \\paren {\\paren {p \\land q} \\land r}$"}
+{"_id": "6986", "title": "Rule of Association/Disjunction/Formulation 2", "text": ":$\\vdash \\paren {p \\lor \\paren {q \\lor r} } \\iff \\paren {\\paren {p \\lor q} \\lor r}$"}
+{"_id": "6987", "title": "Principle of Commutation/Forward Implication/Formulation 1", "text": ": $p \\implies \\left({q \\implies r}\\right) \\vdash q \\implies \\left({p \\implies r}\\right)$"}
+{"_id": "6990", "title": "Principle of Commutation/Reverse Implication/Formulation 1", "text": ":$q \\implies \\paren {p \\implies r} \\vdash p \\implies \\paren {q \\implies r}$"}
+{"_id": "6991", "title": "Principle of Commutation/Reverse Implication/Formulation 1/Proof", "text": ": $q \\implies \\left({p \\implies r}\\right) \\vdash p \\implies \\left({q \\implies r}\\right)$"}
+{"_id": "6992", "title": "Principle of Commutation/Formulation 1/Proof 1", "text": ": $p \\implies \\left({q \\implies r}\\right) \\dashv \\vdash q \\implies \\left({p \\implies r}\\right)$"}
+{"_id": "6993", "title": "Principle of Commutation/Formulation 1/Proof 2", "text": ": $p \\implies \\left({q \\implies r}\\right) \\dashv \\vdash q \\implies \\left({p \\implies r}\\right)$"}
+{"_id": "6994", "title": "Principle of Commutation/Formulation 2", "text": ":$\\vdash \\paren {p \\implies \\paren {q \\implies r} } \\iff \\paren {q \\implies \\paren {p \\implies r} }$"}
+{"_id": "6999", "title": "Linearly Ordered Space is Connected iff Linear Continuum", "text": "Let $T = \\struct {S, \\preceq, \\tau}$ be a linearly ordered space. Then $S$ is a connected space {{iff}} it is a linear continuum."}
+{"_id": "7000", "title": "Rule of Transposition/Variant 1/Formulation 2/Forward Implication", "text": ": $\\vdash \\left({p \\implies \\neg q}\\right) \\implies \\left({q \\implies \\neg p}\\right)$"}
+{"_id": "7002", "title": "Boundary of Polygon is Jordan Curve", "text": "Let $P$ be a polygon embedded in $\\R^2$. Then there exists a Jordan curve $\\gamma: \\closedint 0 1 \\to \\R^2$ such that the image of $\\gamma$ is equal to the boundary $\\partial P$ of $P$."}
+{"_id": "7003", "title": "Rule of Explosion/Variant 1", "text": ":$\\vdash p \\implies \\paren {\\neg p \\implies q}$"}
+{"_id": "7005", "title": "Negation of Conditional implies Antecedent", "text": ":$\\vdash \\neg \\paren {p \\implies q} \\implies p$"}
+{"_id": "7006", "title": "Negation of Conditional implies Negation of Consequent", "text": ":$\\vdash \\neg \\left({p \\implies q}\\right) \\implies \\neg q$"}
+{"_id": "7009", "title": "Compactness from Basis", "text": "Let $\\struct {X, \\tau}$ be a topological space. Let $B$ be a basis for $\\tau$. Then the following propositions are equivalent: {{begin-axiom}} {{axiom | n = 1         | t = $\\struct {X, \\tau}$ is compact. }} {{axiom | n = 2         | t = Every open cover of $X$ by elements of $B$ has a finite subcover. }} {{end-axiom}}"}
+{"_id": "7014", "title": "Implication Equivalent to Negation of Conjunction with Negative/Formulation 2/Reverse Implication", "text": ":$\\vdash \\left({\\neg \\left({p \\land \\neg q}\\right)}\\right) \\implies \\left({p \\implies q}\\right)$"}
+{"_id": "7015", "title": "Biconditional Elimination", "text": "The rule of '''biconditional  elimination''' is a valid deduction sequent in propositional logic."}
+{"_id": "7017", "title": "Biconditional Introduction/Sequent Form", "text": ":$p \\implies q, q \\implies p \\vdash p \\iff q$"}
+{"_id": "7020", "title": "Biconditional Elimination/Sequent Form/Proof 1/Form 1", "text": ":$p \\iff q \\vdash p \\implies q$"}
+{"_id": "7021", "title": "Biconditional Elimination/Sequent Form/Proof 1/Form 2", "text": ":$p \\iff q \\vdash q \\implies p$"}
+{"_id": "7022", "title": "Biconditional Elimination/Sequent Form/Proof 1", "text": ":$(1): \\quad p \\iff q \\vdash p \\implies q$ :$(2): \\quad p \\iff q \\vdash q \\implies p$"}
+{"_id": "7023", "title": "Biconditional Elimination/Sequent Form/Proof 2", "text": ":$(1): \\quad p \\iff q \\vdash p \\implies q$ :$(2): \\quad p \\iff q \\vdash q \\implies p$"}
+{"_id": "7024", "title": "Biconditional is Commutative/Formulation 1/Proof 1", "text": ": $p \\iff q \\dashv \\vdash q \\iff p$"}
+{"_id": "7025", "title": "Rule of Material Equivalence/Formulation 1", "text": ":$p \\iff q \\dashv \\vdash \\paren {p \\implies q} \\land \\paren {q \\implies p}$"}
+{"_id": "7026", "title": "Equivalences are Interderivable/Proof 1", "text": "If two propositional formulas are interderivable, they are equivalent: : $\\left ({p \\dashv \\vdash q}\\right) \\dashv \\vdash \\left ({p \\iff q}\\right)$"}
+{"_id": "7027", "title": "Equivalences are Interderivable/Forward Implication", "text": ": $\\left ({p \\dashv \\vdash q}\\right) \\vdash \\left ({p \\iff q}\\right)$"}
+{"_id": "7029", "title": "Compact Subspace of Linearly Ordered Space/Reverse Implication", "text": "Let $\\struct {X, \\preceq, \\tau}$ be a linearly ordered space. Let $Y \\subseteq X$ be a non-empty subset of $X$. Let the following hold: :$(1): \\quad$ For every non-empty $S \\subseteq Y$, $S$ has a supremum and an infimum in $X$. :$(2): \\quad$ For every non-empty $S \\subseteq Y$: $\\sup S, \\inf S \\in Y$. Then $Y$ is a compact subspace of $\\struct {X, \\tau}$."}
+{"_id": "7030", "title": "Modus Ponendo Tollens/Sequent Form", "text": "==== Case 1 ==== {{:Modus Ponendo Tollens/Sequent Form/Case 1}} ==== Case 2 ==== {{:Modus Ponendo Tollens/Sequent Form/Case 2}}"}
+{"_id": "7031", "title": "Modus Ponendo Tollens/Sequent Form/Case 1", "text": ":$\\neg \\paren {p \\land q}, p \\vdash \\neg q$"}
+{"_id": "7032", "title": "Modus Ponendo Tollens/Sequent Form/Case 2", "text": ":$\\neg \\left({p \\land q}\\right), q \\vdash \\neg p$"}
+{"_id": "7033", "title": "Union of Subsets is Subset/Subset of Power Set", "text": "Let $S$ and $T$ be sets. Let $\\powerset S$ be the power set of $S$. Let $\\mathbb S$ be a subset of $\\powerset S$. Then: :$\\displaystyle \\paren {\\forall X \\in \\mathbb S: X \\subseteq T} \\implies \\bigcup \\mathbb S \\subseteq T$"}
+{"_id": "7035", "title": "Conjunction with Negative Equivalent to Negation of Implication/Formulation 2/Forward Implication", "text": ":$\\vdash \\paren {p \\land \\neg q} \\implies \\paren {\\neg \\paren {p \\implies q} }$"}
+{"_id": "7042", "title": "Neighborhood Sub-Basis Criterion for Filter Convergence", "text": "Let $\\left({X, \\tau}\\right)$ be a topological space. Let $\\mathcal F$ be a filter on $X$. Let $p \\in X$. Then $\\mathcal F$ converges to $p$ iff $\\mathcal F$ contains as a subset a neighborhood sub-basis at $x$."}
+{"_id": "7045", "title": "Rule of Material Implication/Formulation 2/Forward Implication", "text": ": $\\vdash \\left({p \\implies q}\\right) \\implies \\left({\\neg p \\lor q}\\right)$"}
+{"_id": "7047", "title": "Implication is Left Distributive over Conjunction/Formulation 1/Proof 2", "text": ":$p \\implies \\left({q \\land r}\\right) \\dashv \\vdash \\left({p \\implies q}\\right) \\land \\left({p \\implies r}\\right)$"}
+{"_id": "7049", "title": "Compact Subspace of Linearly Ordered Space", "text": "Let $\\struct {X, \\preceq, \\tau}$ be a linearly ordered space. Let $Y \\subseteq X$ be a non-empty subset of $X$. Then $Y$ is a compact subspace of $\\struct {X, \\tau}$ {{iff}} both of the following hold: :$(1): \\quad$ For every non-empty $S \\subseteq Y$, $S$ has a supremum and an infimum in $X$. :$(2): \\quad$ For every non-empty $S \\subseteq Y$: $\\sup S, \\inf S \\in Y$."}
+{"_id": "7050", "title": "Order Topology is Hausdorff", "text": "Let $\\struct {X, \\preceq, \\tau}$ be a linearly ordered space. Then $\\struct {X, \\tau}$ is a Hausdorff space."}
+{"_id": "7051", "title": "Subset of Linearly Ordered Space which is Order-Complete and Closed but not Compact", "text": "Let $X = \\left[{0 \\,.\\,.\\, 1}\\right) \\cup \\left({2 \\,.\\,.\\, 3}\\right) \\cup \\left\\{{4}\\right\\}$. Let $\\preceq$ be the ordering on $X$ induced by the usual ordering of the real numbers. Let $\\tau$ be the $\\preceq$ order topology on $X$. Let $Y = \\left[{0 \\,.\\,.\\, 1}\\right) \\cup \\left\\{{4}\\right\\}$. Let $\\tau'$ be the $\\tau$-relative subspace topology on $Y$. Then: :$\\left({Y, \\preceq}\\right)$ is a complete lattice :$Y$ is closed in $X$ but: :$\\left({Y, \\tau'}\\right)$ is not compact."}
+{"_id": "7052", "title": "Rule of Explosion/Variant 2", "text": ":$\\vdash \\paren {p \\land \\neg p} \\implies q$"}
+{"_id": "7054", "title": "Jordan Curve Theorem", "text": "Let $\\gamma: \\closedint 0 1 \\to \\R^2$ be a Jordan curve. Let $\\Img \\gamma$ denote the image of $\\gamma$. Then $\\R^2 \\setminus \\Img \\gamma$ is a union of two disjoint connected components. Both components are open in $\\R^2$, and both components have $\\Img \\gamma$ as their boundary. One component is bounded, and is called the interior of $\\gamma$. The other component is unbounded, and is called the exterior of $\\gamma$."}
+{"_id": "7055", "title": "Heine-Borel Theorem/Dedekind Complete Space", "text": "Let $T = \\left({X, \\preceq, \\tau}\\right)$ be a Dedekind-complete linearly ordered space. Let $Y$ be a non-empty subset of $X$. Then $Y$ is compact {{iff}} $Y$ is closed and bounded in $T$."}
+{"_id": "7056", "title": "Heine-Borel iff Dedekind Complete", "text": "Let $\\struct {X, \\preceq, \\tau}$ be a linearly ordered space. Then $X$ is Dedekind complete {{iff}} every closed, bounded subset of $X$ is compact."}
+{"_id": "7057", "title": "Jordan Polygon Theorem", "text": "Let $P$ be a polygon embedded in $\\R^2$. Denote the boundary of $P$ as $\\partial P$. Then, $\\R^2 \\setminus \\partial P$ is a union of two connected components. Both components are open in $\\R^2$. One component is bounded, and is called the interior of $P$. The other component is unbounded, and is called the exterior of $P$."}
+{"_id": "7058", "title": "Jordan Polygon Interior and Exterior Criterion", "text": "Let $P$ be a polygon embedded in $\\R^2$. Let $q \\in \\R^2 \\setminus \\partial P$, where $\\partial P$ denotes the boundary of $P$. Let $\\mathbf v \\in \\R^2 \\setminus \\left\\{ {\\mathbf 0}\\right\\}$ be a non-zero vector, and let $\\mathcal L = \\left\\{ {q + s \\mathbf v: s \\in \\R_{\\ge 0} }\\right\\}$ be a ray with start point $q$. Let $N \\left({q}\\right) \\in \\N$ be the number of crossings between $\\mathcal L$ and $\\partial P$. Then: :$(1): \\quad$ $q \\in \\operatorname{Int} \\left({P}\\right)$, iff $N \\left({q}\\right) \\equiv 1 \\pmod 2$ :$(2): \\quad$ $q \\in \\operatorname{Ext} \\left({P}\\right)$, iff $N \\left({q}\\right) \\equiv 0 \\pmod 2$ Here, $\\operatorname{Int} \\left({P}\\right)$ and $\\operatorname{Ext} \\left({P}\\right)$ denote the interior and exterior of $\\partial P$, when $\\partial P$ is considered as a Jordan curve."}
+{"_id": "7059", "title": "Closed Set in Linearly Ordered Space", "text": "Let $\\struct {X, \\preceq, \\tau}$ be a linearly ordered space. Let $C$ be a subset of $X$. Then $C$ is closed in $X$ {{iff}} for all non-empty subsets $S$ of $C$: :If $s \\in X$ is a supremum or infimum of $S$ in $X$, then $s \\in C$."}
+{"_id": "7064", "title": "Factor Principles/Disjunction on Left/Formulation 2", "text": ": $\\vdash \\left({p \\implies q}\\right) \\implies \\left({\\left({r \\lor p}\\right) \\implies \\left ({r \\lor q}\\right)}\\right)$"}
+{"_id": "7065", "title": "Factor Principles/Disjunction on Right/Formulation 2", "text": ": $\\vdash \\left({p \\implies q}\\right) \\implies \\left({\\left({p \\lor r}\\right) \\implies \\left ({q \\lor r}\\right)}\\right)$"}
+{"_id": "7066", "title": "Factor Principles/Disjunction on Right/Formulation 1", "text": ":$p \\implies q \\vdash \\paren {p \\lor r} \\implies \\paren {q \\lor r}$"}
+{"_id": "7067", "title": "Factor Principles/Disjunction on Left/Formulation 1", "text": ": $p \\implies q \\vdash \\left({r \\lor p}\\right) \\implies \\left ({r \\lor q}\\right)$"}
+{"_id": "7068", "title": "Disjunction of Implications", "text": ":$\\vdash \\paren {p \\implies q} \\lor \\paren {q \\implies r}$"}
+{"_id": "7073", "title": "Principle of Composition/Formulation 1/Forward Implication", "text": ":$\\paren {p \\implies r} \\lor \\paren {q \\implies r} \\vdash \\paren {p \\land q} \\implies r$"}
+{"_id": "7074", "title": "Principle of Composition/Formulation 1/Reverse Implication", "text": ":$\\paren {p \\land q} \\implies r \\vdash \\paren {p \\implies r} \\lor \\paren {q \\implies r}$"}
+{"_id": "7075", "title": "Principle of Composition/Formulation 2", "text": ":$\\left({\\left({p \\implies r}\\right) \\lor  \\left({q \\implies r}\\right)}\\right) \\iff \\left({\\left({p \\land q}\\right) \\implies r}\\right)$"}
+{"_id": "7076", "title": "Inversion Mapping on Topological Group is Homeomorphism", "text": "Let $T = \\struct {G, \\circ, \\tau}$ be a topological group. Let $\\phi: G \\to G$ be the inversion mapping of $T$. Then $\\phi$ is a homeomorphism."}
+{"_id": "7077", "title": "De Morgan's Laws (Logic)/Conjunction of Negations/Formulation 2/Forward Implication", "text": ": $\\left({\\neg p \\land \\neg q}\\right) \\implies \\left({\\neg \\left({p \\lor q}\\right)}\\right)$"}
+{"_id": "7078", "title": "De Morgan's Laws (Logic)/Conjunction of Negations/Formulation 2/Reverse Implication", "text": ": $\\left({\\neg \\left({p \\lor q}\\right)}\\right) \\implies \\left({\\neg p \\land \\neg q}\\right)$"}
+{"_id": "7080", "title": "Equivalence of Definitions of Generalized Ordered Space/Definition 2 implies Definition 1", "text": "Let $\\struct {S, \\preceq, \\tau}$ be a generalized ordered space by Definition 2: {{:Definition:Generalized Ordered Space/Definition 2}} Then $\\struct {S, \\preceq, \\tau}$ is a generalized ordered space by Definition 1: {{:Definition:Generalized Ordered Space/Definition 1}}"}
+{"_id": "7081", "title": "Inverse Image of Convex Set under Monotone Mapping is Convex", "text": "Let $\\left({X, \\le}\\right)$ and $\\left({Y, \\preceq}\\right)$ be ordered sets. Let $f: X \\to Y$ be a monotone mapping. Let $C$ be a convex subset of $Y$. Then $f^{-1} \\left[{C}\\right]$ is convex in $X$."}
+{"_id": "7082", "title": "Conjunction with Law of Excluded Middle", "text": ":$\\vdash p \\iff \\paren {p \\land q} \\lor \\paren {p \\land \\neg q}$"}
+{"_id": "7083", "title": "Proof by Cases with Contradiction", "text": ":$\\vdash p \\iff \\left({p \\lor q}\\right) \\land \\left({p \\lor \\neg q}\\right)$"}
+{"_id": "7084", "title": "Ray is Convex", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $I$ be a ray, either open or closed. Then $I$ is convex in $S$."}
+{"_id": "7085", "title": "Intersection of Convex Sets is Convex Set (Order Theory)", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $\\mathcal C$ be a set of convex sets in $S$. Then $\\displaystyle \\bigcap \\mathcal C$ is convex."}
+{"_id": "7086", "title": "Interval of Ordered Set is Convex", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. Let $I$ be an interval: be it open, closed, or half-open in $S$. Then $I$ is convex in $S$."}
+{"_id": "7087", "title": "Upper and Lower Closures are Convex", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a \\in S$. Then $a^\\succeq$, $a^\\succ$, $a^\\preceq$, and $a^\\prec$ are convex in $S$."}
+{"_id": "7088", "title": "Transitive Closure of Symmetric Relation is Symmetric", "text": "Let $S$ be a set. Let $\\mathcal R$ be a symmetric relation on $S$. Let $\\mathcal T$ be the transitive closure of $\\mathcal R$. The $\\mathcal T$ is symmetric."}
+{"_id": "7089", "title": "Transitive Closure of Reflexive Relation is Reflexive", "text": "Let $S$ be a set. Let $\\mathcal R$ be a reflexive relation on $S$. Let $\\mathcal T$ be the transitive closure of $\\mathcal R$. Then $\\mathcal T$ is reflexive."}
+{"_id": "7090", "title": "Transitive Closure of Reflexive Symmetric Relation is Equivalence", "text": "Let $S$ be a set. Let $\\mathcal R$ be a symmetric and reflexive relation on $S$. Then the transitive closure of $\\mathcal R$ is an equivalence relation."}
+{"_id": "7091", "title": "Union of Overlapping Convex Sets in Toset is Convex", "text": "Let $\\left({ S, \\preceq }\\right)$ be a totally ordered set. Let $U$ and $V$ be convex sets in $S$. Let $U \\cap V \\ne \\varnothing$. Then $U \\cup V$ is also convex."}
+{"_id": "7092", "title": "Rule of Material Equivalence", "text": "==== Formulation 1 ==== {{:Rule of Material Equivalence/Formulation 1}} ==== Formulation 2 ==== {{:Rule of Material Equivalence/Formulation 2}}"}
+{"_id": "7093", "title": "Union of Overlapping Convex Sets in Toset is Convex/Infinite Union", "text": "Let $\\left({S, \\preceq}\\right)$ be a totally ordered set. Let $\\mathcal A$ be a set of convex subsets of $S$. For any $P, Q \\in \\mathcal A$, let there be elements $C_0, \\dotsc, C_n \\in \\mathcal A$ such that: :$C_0 = P$ :$C_n = Q$ :For $k = 0, \\dots, n-1: C_k \\cap C_{k+1} \\ne \\varnothing$ Then $\\bigcup \\mathcal A$ is convex in $S$."}
+{"_id": "7094", "title": "Rule of Material Equivalence/Formulation 2", "text": ":$\\vdash \\left({p \\iff q}\\right) \\iff \\left({\\left({p \\implies q}\\right) \\land \\left({q \\implies p}\\right)}\\right)$"}
+{"_id": "7095", "title": "Non-Equivalence as Disjunction of Conjunctions", "text": "==== Formulation 1 ==== {{:Non-Equivalence as Disjunction of Conjunctions/Formulation 1}} ==== Formulation 2 ==== {{:Non-Equivalence as Disjunction of Conjunctions/Formulation 2}}"}
+{"_id": "7096", "title": "Continuous Involution is Homeomorphism", "text": "Let $\\struct {S, \\tau}$ be a topological space. Let $f: S \\to S$ be a continuous involution. Then $f$ is a homeomorphism."}
+{"_id": "7097", "title": "Involution is Permutation", "text": "Let $S$ be a set. Let $f: S \\to S$ be an involution. Then $f$ is a permutation."}
+{"_id": "7099", "title": "Non-Equivalence as Disjunction of Conjunctions/Formulation 1/Forward Implication/Proof", "text": ": $\\neg \\left ({p \\iff q}\\right) \\vdash \\left({\\neg p \\land q}\\right) \\lor \\left({p \\land \\neg q}\\right)$"}
+{"_id": "7100", "title": "Non-Equivalence as Disjunction of Conjunctions/Formulation 1/Reverse Implication/Proof", "text": ": $\\left({\\neg p \\land q}\\right) \\lor \\left({p \\land \\neg q}\\right) \\vdash \\neg \\left ({p \\iff q}\\right)$"}
+{"_id": "7102", "title": "Non-Equivalence as Disjunction of Conjunctions/Formulation 2", "text": ":$\\vdash \\paren {\\neg \\paren {p \\iff q} } \\iff \\paren {\\paren {\\neg p \\land q} \\lor \\paren {p \\land \\neg q} }$"}
+{"_id": "7103", "title": "Non-Equivalence as Equivalence with Negation/Formulation 1/Forward Implication", "text": ":$\\neg \\paren {p \\iff q} \\vdash \\paren {p \\iff \\neg q}$"}
+{"_id": "7105", "title": "Non-Equivalence as Equivalence with Negation/Formulation 1", "text": ":$\\neg \\paren {p \\iff q} \\dashv \\vdash \\paren {p \\iff \\neg q}$"}
+{"_id": "7109", "title": "Sign of Function Matches Sign of Definite Integral", "text": "Let $f$ be a real function continuous on some closed interval $\\closedint a b$, where $a < b$. Then: :If $\\forall x \\in \\closedint a b: \\map f x \\ge 0$ then $\\displaystyle \\int_a^b \\map f x \\rd x \\ge 0$ :If $\\forall x \\in \\closedint a b: \\map f x > 0$ then $\\displaystyle \\int_a^b \\map f x \\rd x > 0$ :If $\\forall x \\in \\closedint a b: \\map f x \\le 0$ then $\\displaystyle \\int_a^b \\map f x \\rd x \\le 0$ :If $\\forall x \\in \\closedint a b: \\map f x < 0$ then $\\displaystyle \\int_a^b \\map f x \\rd x < 0$"}
+{"_id": "7110", "title": "Non-Equivalence as Equivalence with Negation/Formulation 2", "text": ": $\\vdash \\neg \\left ({p \\iff q}\\right) \\iff \\left({p \\iff \\neg q}\\right)$"}
+{"_id": "7113", "title": "Symmetric Closure of Relation Compatible with Operation is Compatible", "text": "Let $\\struct {S, \\circ}$ be a magma. Let $\\RR$ be a relation compatible with $\\circ$. Let $\\RR^\\leftrightarrow$ be the symmetric closure of $\\RR$. Then $\\RR^\\leftrightarrow$ is compatible with $\\circ$."}
+{"_id": "7114", "title": "Singleton is Convex Set (Order Theory)", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $x \\in S$. Then the singleton $\\left\\{{x}\\right\\}$ is convex."}
+{"_id": "7115", "title": "Rule of Transposition/Variant 1/Formulation 2", "text": ":$\\vdash \\paren {p \\implies \\neg q} \\iff \\paren {q \\implies \\neg p}$"}
+{"_id": "7116", "title": "Rule of Transposition/Variant 1", "text": "==== Formulation 1 ==== {{:Rule of Transposition/Variant 1/Formulation 1}} ==== Formulation 2 ==== {{:Rule of Transposition/Variant 1/Formulation 2}}"}
+{"_id": "7117", "title": "Rule of Transposition/Variant 1/Formulation 2/Reverse Implication", "text": ": $\\vdash \\left({q \\implies \\neg p}\\right) \\implies \\left({p \\implies \\neg q}\\right)$"}
+{"_id": "7119", "title": "Rule of Transposition/Variant 1/Formulation 1", "text": ":$p \\implies \\neg q \\dashv \\vdash q \\implies \\neg p$"}
+{"_id": "7124", "title": "Rule of Transposition/Variant 2", "text": "==== Formulation 1 ==== {{:Rule of Transposition/Variant 2/Formulation 1}} ==== Formulation 2 ==== {{:Rule of Transposition/Variant 2/Formulation 2}}"}
+{"_id": "7125", "title": "Rule of Transposition/Variant 2/Formulation 1", "text": ": $\\neg p \\implies q \\dashv \\vdash \\neg q \\implies p$"}
+{"_id": "7130", "title": "Rule of Transposition/Variant 2/Formulation 1/Proof 2", "text": ":$\\neg p \\implies q \\dashv \\vdash \\neg q \\implies p$"}
+{"_id": "7134", "title": "Rule of Transposition/Variant 2/Formulation 2", "text": ": $\\vdash \\left({\\neg p \\implies q}\\right) \\iff \\left({\\neg q \\implies p}\\right)$"}
+{"_id": "7135", "title": "Boundary of Polygon is Topological Boundary", "text": "Let $P$ be a polygon embedded in $\\R^2$. Denote the boundary of $P$ as $\\partial P$. Let $\\Int P$ and $\\Ext P$ denote the interior and exterior of $\\partial P$, where $\\partial P$ is considered as a Jordan curve. Then the topological boundary of $\\Int P$ is equal to $\\partial P$, and the topological boundary of $\\Ext P$ is equal to $\\partial P$."}
+{"_id": "7136", "title": "Jordan Curve and Jordan Arc form Two Jordan Curves", "text": "Let $\\closedint a b$ denote the closed real interval between $a \\in \\R, b \\in \\R: a \\le b$. Let $\\gamma: \\closedint a b \\to \\R^2$ be a Jordan curve. Let the interior of $\\gamma$ be denoted $\\Int \\gamma$. Let the image of $\\gamma$ be denoted $\\Img \\gamma$. Let $\\sigma: \\closedint c d \\to \\R^2$ be a Jordan arc such that: :$\\map \\sigma c \\ne \\map \\sigma d$ :$\\map \\sigma c, \\map \\sigma d \\in \\Img \\gamma$ and: :$\\forall t \\in \\openint c d: \\map \\sigma t \\in \\Int \\gamma$ Let $t_1 = \\map {\\gamma^{-1} } {\\map \\sigma c}$. Let $t_2 = \\map {\\gamma^{-1} } {\\map \\sigma d}$. Let $t_1 < t_2$. Define: :$-\\sigma: \\closedint c d \\to \\Img \\sigma$ by $-\\map \\sigma t = \\map \\sigma {c + d - t}$ Let $*$ denote concatenation of paths. Let $\\gamma \\restriction_{\\closedint a {t_1} }$ denote the restriction of $\\gamma$ to $\\closedint a {t_1}$. Define: :$\\gamma_1 = \\gamma {\\closedint a {t_1} } * \\sigma * \\gamma {\\restriction_{\\closedint {t_2} b} }$ Define: :$\\gamma_2 = \\gamma {\\closedint a {t_1} } * \\paren {-\\sigma}$ Then $\\gamma_1$ and $\\gamma_2$ are Jordan curves such that: :$\\Int {\\gamma_1} \\subseteq \\Int \\gamma$ and: :$\\Int {\\gamma_2} \\subseteq \\Int \\gamma$"}
+{"_id": "7137", "title": "Restriction of Total Ordering is Total Ordering", "text": "Let $\\struct {S, \\preceq}$ be a total ordering. Let $T \\subseteq S$. Let $\\preceq \\restriction_T$ be the restriction of $\\preceq$ to $T$. Then $\\preceq \\restriction_T$ is a total ordering of $T$."}
+{"_id": "7138", "title": "Ordering Cycle implies Equality/General Case", "text": "Let $\\left({S,\\preceq}\\right)$ be an ordered set. Let $x_0, x_1, \\dots, x_n \\in S$. Suppose that for $k = 0, 1, \\dots, n - 1: x_k \\preceq x_{k+1}$. Suppose also that $x_n \\preceq x_0$. Then $x_0 = x_1 = \\dots = x_n$."}
+{"_id": "7139", "title": "Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T \\subseteq S$ be a finite, non-empty subset of $S$. Then $T$ has a maximal element and a minimal element."}
+{"_id": "7140", "title": "Star Convex Set is Path-Connected", "text": "Let $A$ be a star convex subset of a vector space $V$ over $\\R$ or $\\C$. Then $A$ is path-connected."}
+{"_id": "7143", "title": "Minimal Element of Chain is Smallest Element", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $C$ be a chain in $S$. Let $m$ be a minimal element of $C$. Then $m$ is the smallest element of $C$."}
+{"_id": "7144", "title": "Maximal Element of Chain is Greatest Element", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $C$ be a chain in $S$. Let $m$ be a maximal element of $C$. Then $m$ is the greatest element of $C$."}
+{"_id": "7148", "title": "Modus Ponendo Ponens/Variant 3", "text": ":$\\vdash \\paren {\\paren {p \\implies q} \\land p} \\implies q$"}
+{"_id": "7152", "title": "Rule of Material Equivalence/Formulation 1/Proof 2", "text": ":$p \\iff q \\dashv \\vdash \\left({p \\implies q}\\right) \\land \\left({q \\implies p}\\right)$"}
+{"_id": "7153", "title": "Rule of Material Equivalence/Formulation 2/Proof 1", "text": ":$\\vdash \\left({p \\iff q}\\right) \\iff \\left({\\left({p \\implies q}\\right) \\land \\left({q \\implies p}\\right)}\\right)$"}
+{"_id": "7155", "title": "Rule of Material Equivalence/Formulation 2/Proof 2", "text": ":$\\vdash \\left({p \\iff q}\\right) \\iff \\left({\\left({p \\implies q}\\right) \\land \\left({q \\implies p}\\right)}\\right)$"}
+{"_id": "7160", "title": "Modus Tollendo Ponens/Variant/Formulation 2/Proof 2", "text": ": $\\vdash \\left({p \\lor q}\\right) \\iff \\left({\\neg p \\implies q}\\right)$"}
+{"_id": "7161", "title": "Rule of Transposition/Formulation 2/Proof 2", "text": ": $\\vdash \\left({p \\implies q}\\right) \\iff \\left({\\neg q \\implies \\neg p}\\right)$"}
+{"_id": "7166", "title": "Convex Set Characterization (Order Theory)", "text": "Let $\\left({S, \\preceq, \\tau}\\right)$ be an ordered set. Let $C \\subseteq S$. {{TFAE}} {{begin-axiom}} {{axiom | n = 1         | t = $C$ is convex. }} {{axiom | n = 2         | t = $C$ is the intersection of an upper set with a lower set. }} {{axiom | n = 3         | t = $C$ is the intersection of its upper closure with its lower closure. }} {{end-axiom}}"}
+{"_id": "7168", "title": "Constructive Dilemma/Formulation 3", "text": ":$\\paren {p \\implies q} \\land \\paren {r \\implies s}, p \\lor r \\vdash q \\lor s$"}
+{"_id": "7169", "title": "Upper Closure is Upper Set", "text": "Let $(S, \\preceq, \\tau)$ be an ordered set. Let $T$ be a subset of $S$. Let $U$ be the upper closure of $T$. Then $U$ is an upper set."}
+{"_id": "7172", "title": "Destructive Dilemma/Formulation 1/Proof 3", "text": ":$p \\implies q, r \\implies s \\vdash \\neg q \\lor \\neg s \\implies \\neg p \\lor \\neg r$"}
+{"_id": "7174", "title": "Open Ray is Open in GO-Space/Definition 1", "text": "Let $\\struct {S, \\preceq, \\tau}$ be a generalized ordered space. Let $p \\in S$. Then: :$p^\\prec$ and $p^\\succ$ are $\\tau$-open where: :$p^\\prec$ is the strict lower closure of $p$ :$p^\\succ$ is the strict upper closure of $p$."}
+{"_id": "7175", "title": "Upper and Lower Closures of Open Set in GO-Space are Open", "text": "Let $\\left({X, \\preceq, \\tau}\\right)$ be a Generalized Ordered Space/Definition 1. Let $A$ be open in $X$. Then the upper and lower closures of $A$ are open."}
+{"_id": "7176", "title": "Equivalence of Definitions of Generalized Ordered Space/Definition 1 implies Definition 3", "text": "Let $\\struct {S, \\preceq, \\tau}$ be a generalized ordered space by Definition 1: {{:Definition:Generalized Ordered Space/Definition 1}} Then $\\struct {S, \\preceq, \\tau}$ is a generalized ordered space by Definition 3: {{:Definition:Generalized Ordered Space/Definition 3}}"}
+{"_id": "7177", "title": "Upper Set is Convex", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T \\subseteq S$ be an upper set. Then $T$ is convex in $S$."}
+{"_id": "7178", "title": "Lower Set is Convex", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T \\subseteq S$ be a lower set. Then $T$ is convex in $S$."}
+{"_id": "7180", "title": "GO-Space Embeds Densely into Linearly Ordered Space", "text": "Let $\\left({Y, \\preceq, \\tau}\\right)$ be a generalized ordered space (GO-space) by Definition 3. That is: : let $\\left({Y, \\tau}\\right)$ be a Hausdorff space and: : let $\\tau$ have a sub-basis consisting of upper sets and lower sets relative to $\\preceq$. Then $\\left({Y, \\preceq, \\tau}\\right)$ is a GO-space by Definition 2. That is, there is a linearly ordered space $\\left({X, \\preceq', \\tau'}\\right)$ and a mapping from $Y$ to $X$ which is a order embedding and a topological embedding."}
+{"_id": "7181", "title": "Upper Set with no Minimal Element", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $U \\subseteq S$. Then: :$U$ is an upper set in $S$ with no minimal element {{iff}}: :$\\displaystyle U = \\bigcup \\set {u^\\succ: u \\in U}$ where $u^\\succ$ is the strict upper closure of $u$."}
+{"_id": "7182", "title": "Lower Sets in Totally Ordered Set form Nest", "text": "Let $\\left({S, \\preceq}\\right)$ be a totally ordered set. Let $\\mathcal L$ be a set of lower sets in $S$. Then $\\mathcal L$ is a nest. That is, $\\mathcal L$ is totally ordered by $\\subseteq$."}
+{"_id": "7183", "title": "Exclusive Or is Self-Inverse", "text": ":$\\paren {p \\oplus q} \\oplus q \\dashv \\vdash p$ where $\\oplus$ denotes the exclusive or operator."}
+{"_id": "7184", "title": "Conjunction has no Inverse", "text": "Let $\\land$ denote the conjunction operation of propositional logic. Then there exists no binary logical connective $\\circ$ such that: :$(1): \\quad \\forall p, q \\in \\left\\{{T, F}\\right\\}: \\left({p \\land q}\\right) \\circ q = p$"}
+{"_id": "7185", "title": "Disjunction has no Inverse", "text": "Let $\\lor$ denote the disjunction operation of propositional logic. Then there exists no binary logical connective $\\circ$ such that: :$(1): \\quad \\forall p, q \\in \\left\\{{T, F}\\right\\}: \\left({p \\lor q}\\right) \\circ q = p$"}
+{"_id": "7186", "title": "Binary Logical Connectives with Inverse", "text": "Let $\\circ$ be a binary logical connective. Then there exists another binary logical connective $*$ such that: :$\\forall p, q \\in \\set {\\F, \\T}: \\paren {p \\circ q} * q \\dashv \\vdash p \\dashv \\vdash q * \\paren {p \\circ q}$ {{iff}} $\\circ$ is either: :$(1): \\quad$ the exclusive or operator or: :$(2): \\quad$ the biconditional operator. That is, the only truth functions that have an inverse operation are the exclusive or and the biconditional."}
+{"_id": "7187", "title": "Biconditional is Self-Inverse", "text": ":$\\paren {p \\iff q} \\iff q \\dashv \\vdash p$ where $\\iff$ denotes the biconditional operator."}
+{"_id": "7188", "title": "Finite Chain is Order-Isomorphic to Finite Ordinal", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $C$ be a finite chain in $S$. Then for some finite ordinal $\\mathbf n$: :$\\left({C, {\\preceq \\restriction_C} }\\right)$ is order-isomorphic to $\\mathbf n$. That is: :$\\left({C, {\\preceq \\restriction_C} }\\right)$ is order-isomorphic to $\\N_n$ where $\\N_n$ is the initial segment of $\\N$ determined by $n$: :$\\N_n = \\left\\{ {k \\in \\N: k < n}\\right\\} = \\left\\{ {0, 1, \\ldots, n - 1}\\right\\}$"}
+{"_id": "7189", "title": "Complete Linearly Ordered Space is Compact", "text": "Let $\\left({X, \\preceq, \\tau}\\right)$ be a linearly ordered space. Let $\\left({X, \\preceq}\\right)$ be a complete lattice. Then $\\left({X, \\tau}\\right)$ is compact."}
+{"_id": "7190", "title": "Condition for Well-Foundedness/Reverse Implication", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Suppose that there is no infinite sequence $\\left \\langle {a_n}\\right \\rangle$ of elements of $S$ such that $\\forall n \\in \\N: a_{n+1} \\prec a_n$. Then $\\left({S, \\preceq}\\right)$ is well-founded."}
+{"_id": "7191", "title": "Condition for Well-Foundedness/Reverse Implication/Proof 1", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Suppose that there is no infinite sequence $\\sequence {a_n}$ of elements of $S$ such that $\\forall n \\in \\N: a_{n + 1} \\prec a_n$. Then $\\struct {S, \\preceq}$ is well-founded."}
+{"_id": "7193", "title": "Inversion Mapping on Ordered Group is Dual Order-Isomorphism", "text": "Let $\\struct {G, \\circ, \\preceq}$ be an ordered group. Let $\\iota: G \\to G$ be the inversion mapping, defined by $\\map \\phi x = x^{-1}$. Then $\\iota$ is a dual order-isomorphism."}
+{"_id": "7195", "title": "Properties of Ordered Group/OG2/Proof 2", "text": "Let $\\left({G, \\circ, \\preceq}\\right)$ be an ordered group with identity $e$. Let $\\prec$ be the reflexive reduction of $\\preceq$. Let $x, y \\in G$. Then the following equivalences hold: :$(\\operatorname{OG}2.1):\\quad x \\preceq y \\iff e \\preceq y \\circ x^{-1}$ :$(\\operatorname{OG}2.2):\\quad x \\preceq y \\iff e \\preceq x^{-1} \\circ y$ :$(\\operatorname{OG}2.3):\\quad x \\preceq y \\iff x \\circ y^{-1} \\preceq e$ :$(\\operatorname{OG}2.4):\\quad x \\preceq y \\iff y^{-1} \\circ x \\preceq e$ :$(\\operatorname{OG}2.1'):\\quad x \\prec y \\iff e \\prec y \\circ x^{-1}$ :$(\\operatorname{OG}2.2'):\\quad x \\prec y \\iff e \\prec x^{-1} \\circ y$ :$(\\operatorname{OG}2.3'):\\quad x \\prec y \\iff x \\circ y^{-1} \\prec e$ :$(\\operatorname{OG}2.4'):\\quad x \\prec y \\iff y^{-1} \\circ x \\prec e$"}
+{"_id": "7196", "title": "Properties of Ordered Group/OG4/Proof 1", "text": "Let $\\struct {G, \\circ, \\preceq}$ be an ordered group with identity $e$. Let $x \\in G$. Then the following equivalences hold: {{begin-axiom}} {{axiom | n = \\operatorname {OG} 4.1         | m = x \\preceq e \\iff e \\preceq x^{-1} }} {{axiom | n = \\operatorname {OG} 4.2         | m = e \\preceq x \\iff x^{-1} \\preceq e }} {{axiom | n = \\operatorname {OG} 4.1'         | m = x \\prec e \\iff e \\prec x^{-1} }} {{axiom | n = \\operatorname {OG} 4.2'         | m = e \\prec x \\iff x^{-1} \\prec e }} {{end-axiom}}"}
+{"_id": "7197", "title": "Properties of Ordered Group/OG4/Proof 2", "text": "Let $\\left({G, \\circ, \\preceq}\\right)$ be an ordered group with identity $e$. Let $x \\in G$. Then the following equivalences hold: {{begin-axiom}} {{axiom|n = \\operatorname{OG}4.1        |m = x \\preceq e \\iff e \\preceq x^{-1} }} {{axiom|n = \\operatorname{OG}4.2        |m = e \\preceq x \\iff x^{-1} \\preceq e }} {{axiom|n = \\operatorname{OG}4.1'        |m = x \\prec e \\iff e \\prec x^{-1} }} {{axiom|n = \\operatorname{OG}4.2'        |m = e \\prec x \\iff x^{-1} \\prec e }} {{end-axiom}}"}
+{"_id": "7198", "title": "Properties of Ordered Group/OG5/Proof 1", "text": "Let $\\left({G, \\circ, \\preceq}\\right)$ be an ordered group with identity $e$. Let $\\prec$ be the reflexive reduction of $\\preceq$. Let $x \\in G$. Let $n \\in \\N_{>0}$ be a strictly positive integer. Then the following hold: :$x \\preceq e \\implies x^n \\preceq e$ :$e \\preceq x \\implies e \\preceq x^n$ :$x \\prec e \\implies x^n \\prec e$ :$e \\prec x \\implies e \\prec x^n$"}
+{"_id": "7199", "title": "Properties of Ordered Group/OG5/Proof 2", "text": "Let $\\left({G, \\circ, \\preceq}\\right)$ be an ordered group with identity $e$. Let $\\prec$ be the reflexive reduction of $\\preceq$. Let $x \\in G$. Let $n \\in \\N_{>0}$ be a strictly positive integer. Then the following hold: :$x \\preceq e \\implies x^n \\preceq e$ :$e \\preceq x \\implies e \\preceq x^n$ :$x \\prec e \\implies x^n \\prec e$ :$e \\prec x \\implies e \\prec x^n$"}
+{"_id": "7200", "title": "Mapping from Totally Ordered Set is Dual Order Embedding iff Strictly Decreasing", "text": "Let $\\left({S, \\preceq_1}\\right)$ be a totally ordered set. Let $\\left({T, \\preceq_2}\\right)$ be an ordered set. Let $\\phi: S \\to T$ be a mapping. Then $\\phi$ is a dual order embedding {{iff}} $\\phi$ is strictly decreasing. That is: :$\\forall x, y \\in S: x \\preceq_1 y \\iff \\phi \\left({y}\\right) \\preceq_2 \\phi \\left({x}\\right)$ {{iff}} :$\\forall x, y \\in S: x \\prec_1 y \\implies \\phi \\left({y}\\right) \\prec_2 \\phi \\left({x}\\right)$"}
+{"_id": "7203", "title": "Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing/Reverse Implication", "text": "Let $\\struct {S, \\preceq_1}$ be a totally ordered set and let $\\struct {T, \\preceq_2}$ be an ordered set. Let $\\phi: S \\to T$ be a strictly increasing mapping. Then $\\phi$ is an order embedding. <section notitle=\"true\" name=\"proofs\"> == Proof 1 == {{:Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing/Reverse Implication/Proof 1}} == Proof 2 == {{:Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing/Reverse Implication/Proof 2}} </section> Category:Order Embeddings Category:Total Orderings ip0v3n07lm6pfuiq8dh8hmlp58or3cs"}
+{"_id": "7204", "title": "Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing/Reverse Implication/Proof 2", "text": "Let $\\struct {S, \\preceq_1}$ be a totally ordered set and let $\\struct {T, \\preceq_2}$ be an ordered set. Let $\\phi: S \\to T$ be a strictly increasing mapping. Then $\\phi$ is an order embedding."}
+{"_id": "7205", "title": "Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing/Reverse Implication/Proof 1", "text": "Let $\\struct {S, \\preceq_1}$ be a totally ordered set. Let $\\struct {T, \\preceq_2}$ be an ordered set. Let $\\phi: S \\to T$ be a strictly increasing mapping. Then $\\phi$ is an order embedding ."}
+{"_id": "7206", "title": "Foundational Relation is Antireflexive", "text": "Let $\\mathcal R$ be a foundational relation on a set or class $A$. Then $\\mathcal R$ is antireflexive."}
+{"_id": "7207", "title": "Foundational Relation is Asymmetric", "text": "Let $\\struct {S, \\RR}$ be a relational structure, where $S$ is a set or a proper class. Let $\\RR$ be a foundational relation. Then $\\RR$ is asymmetric."}
+{"_id": "7209", "title": "Upper Set with no Smallest Element is Open in GO-Space", "text": "Let $\\struct {S, \\preceq, \\tau}$ be a generalized ordered space. Let $U$ be an upper set in $S$ with no smallest element. Then $U$ is open in $\\struct {S, \\preceq, \\tau}$."}
+{"_id": "7210", "title": "Lower Set with no Maximal Element", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $L \\subseteq S$. Then: : $L$ is a lower set in $S$ with no maximal element {{iff}}: : $\\displaystyle L = \\bigcup \\left\\{{l^\\prec: l \\in L }\\right\\}$ where $l^\\prec$ is the strict lower closure of $l$."}
+{"_id": "7211", "title": "Lower Set with no Greatest Element is Open in GO-Space", "text": "Let $\\struct {S, \\preceq, \\tau}$ be a generalized ordered space. Let $L$ be a lower set in $S$ with no greatest element. Then $L$ is open in $\\struct {S, \\preceq, \\tau}$."}
+{"_id": "7212", "title": "Lower Set is Dual to Upper Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T \\subseteq S$. The following are dual statements: :$T$ is a lower set in $S$ :$T$ is an upper set in $S$"}
+{"_id": "7214", "title": "Open Ray is Dual to Open Ray", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. Let $R$ be an open ray in $\\struct {S, \\preceq}$. Then $R$ is an open ray in $\\struct {S, \\succeq}$, where $\\succeq$ is the dual ordering of $\\preceq$."}
+{"_id": "7215", "title": "Topologies on Set form Complete Lattice", "text": "Let $X$ be a non-empty set. Let $\\mathcal L$ be the set of topologies on $X$. Then $\\left({\\mathcal L, \\subseteq}\\right)$ is a complete lattice."}
+{"_id": "7216", "title": "Complement of Lower Set is Upper Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $L$ be a lower set. Then $S \\setminus L$ is an upper set."}
+{"_id": "7217", "title": "GO-Space Embeds as Closed Subspace of Linearly Ordered Space", "text": "Let $(X, \\preceq_X, \\tau_X)$ be a generalized ordered space. Then there is a linearly ordered space $(Y, \\preceq_Y, \\tau_Y)$ and a mapping $\\phi: X \\to Y$ such that $\\phi$ is a topological embedding and an order embedding, and $\\phi(X)$ is closed in $Y$."}
+{"_id": "7218", "title": "Union of Total Ordering with Lower Sets is Total Ordering", "text": "Let $\\left({Y, \\preceq}\\right)$ be a totally ordered set. Let $X$ be the disjoint union of $Y$ with the set of lower sets of $Y$. Define a relation $\\preceq'$ on $X$ extending $\\preceq$ by letting: :$y_1 \\preceq' y_2 \\iff y_1 \\preceq y_2$ :$y \\preceq' L \\iff y \\in L$ :$L_1 \\preceq' L_2 \\iff L_1 \\subseteq L_2$ :$L \\preceq' y \\iff y \\in Y \\setminus L$ Then $\\preceq'$ is a total ordering."}
+{"_id": "7219", "title": "Lower Closure is Lower Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T$ be a subset of $S$. Let $L$ be the lower closure of $T$. Then $L$ is a lower set."}
+{"_id": "7221", "title": "Ordered Set is Lower Set in Itself", "text": "Let $(S, \\preceq)$ be an ordered set. Then $S$ is a lower set in $S$."}
+{"_id": "7222", "title": "Ordered Set is Convex in Itself", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then $S$ is a convex set in $S$."}
+{"_id": "7223", "title": "Equivalence of Definitions of Generalized Ordered Space/Definition 3 implies Definition 1", "text": "Let $\\left({S, \\preceq, \\tau}\\right)$ be a generalized ordered space by Definition 3: {{:Definition:Generalized Ordered Space/Definition 3}} Then $\\left({S, \\preceq, \\tau}\\right)$ is a generalized ordered space by Definition 1: {{:Definition:Generalized Ordered Space/Definition 1}}"}
+{"_id": "7224", "title": "Strict Upper Closure is Upper Set", "text": "Let $(S, \\preceq)$ be an ordered set. Let $p \\in S$. Then $p^\\succ$, the strict upper closure of $p$, is an upper set."}
+{"_id": "7225", "title": "Strict Lower Closure is Lower Set", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $p \\in S$. Then $p^\\prec$, the strict lower closure of $p$, is a lower set."}
+{"_id": "7226", "title": "Topology Discrete iff All Singletons Open", "text": "Let $(X, \\tau)$ be a topological space. Then $\\tau$ is the discrete topology on $X$ iff: : For all $x \\in X$: $\\{ x \\} \\in \\tau$ That is, iff every singleton of $X$ is $\\tau$-open."}
+{"_id": "7229", "title": "Characteristic Function of Universe", "text": "Let $S$ be a set. Let $\\chi_S: S \\to \\left\\{ {0, 1}\\right\\}$ be its characteristic function (in itself). Then: :$\\chi_S = f_1$ where $f_1: S \\to \\left\\{ {0, 1}\\right\\}$ is the constant mapping with value $1$."}
+{"_id": "7230", "title": "Faltings' Theorem", "text": "Let $C$ be a curve over $\\Q$ of genus $g > 1$. Then $C$ has only finitely many rational points."}
+{"_id": "7231", "title": "Supremum of Lower Closure of Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T \\subseteq S$. Let $L = T^\\preceq$ be the lower closure of $T$ in $S$. Let $s \\in S$ Then $s$ is the supremum of $T$ {{iff}} it is the supremum of $L$."}
+{"_id": "7232", "title": "Upper Closure is Smallest Containing Upper Set", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $T \\subseteq S$. Let $U = T^\\succeq$ be the upper closure of $T$. Then $U$ is the smallest upper set containing $T$ as a subset."}
+{"_id": "7233", "title": "Upper Closure is Closure Operator", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T^\\succeq$ be the upper closure of $T$ for each $T \\subseteq S$. Then $\\cdot^\\succeq$ is a closure operator."}
+{"_id": "7234", "title": "Equivalence of Definitions of Upper Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $U \\subseteq S$. {{TFAE|def = Upper Set}}"}
+{"_id": "7235", "title": "Topological Closure is Closure Operator", "text": "The topological closure operator is a closure operator."}
+{"_id": "7236", "title": "Reflexive Closure is Closure Operator", "text": "Let $S$ be a set. Let $R$ be the set of all endorelations on $S$. Then the reflexive closure operator on $R$ is a closure operator."}
+{"_id": "7237", "title": "Set Closure is Smallest Closed Set/Closure Operator", "text": "Let $S$ be a set. Let $\\cl: \\powerset S \\to \\powerset S$ be a closure operator. Let $T \\subseteq S$. Then $\\map \\cl T$ is the smallest closed set (with respect to $\\cl$) containing $T$ as a subset."}
+{"_id": "7238", "title": "Equivalence of Definitions of Lower Set", "text": "{{TFAE|def = Lower Set}} Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $U \\subseteq S$. Then the following are equivalent: {{begin-axiom}} {{axiom | n = 1         | m = \\forall u \\in U: \\forall s \\in S: s \\preceq u \\implies s \\in U }} {{axiom | n = 2         | m = U^\\preceq \\subseteq U }} {{axiom | n = 3         | m = U^\\preceq = U }} {{end-axiom}} where $U^\\preceq$ is the lower closure of $U$."}
+{"_id": "7239", "title": "Zero and One are the only Consecutive Perfect Squares/Proof 1", "text": "{{:Zero and One are the only Consecutive Perfect Squares}} <section title=\"Proof\" name=\"proof\"> Let $x$ and $h$ be integers such that $x^2 + 1 = \\paren {x - h}^2$ {{begin-eqn}} {{eqn | l = x^2 + 1       | r = \\left({x - h}\\right)^2 }} {{eqn | l = 1       | r = -2 x h + h^2 }} {{eqn | l = 2 x h       | r = h^2 - 1 }} {{eqn | l = 2 x h       | r = \\paren {h - 1} \\paren {h + 1} }} {{end-eqn}} We have that Consecutive Integers are Coprime. However, both sides must have the same unique prime factorization by the Fundamental Theorem of Arithmetic Therefore $h$ cannot have any prime factors since they cannot be shared by $\\paren {h - 1} \\paren {h + 1}$. This leaves $h = -1$, $h = 0$, or $h = 1$ as the only possibilities since they are the only integers with no prime factors. If $h = -1$ then $h + 1 = 0$, so $2 x h = 0$. It follows that $x = 0$. If $h = 1$ then $h - 1 = 0$, so $2 x h = 0$. It follows that $x = 0$. If $h = 0$, then $2 x \\cdot 0 = \\paren {-1} \\paren 1$, which is a contradiction. Therefore the only pairs of consecutive perfect squares are: :$0^2 = 0$ and $\\paren {0 + \\paren {-1} }^2 = \\paren {-1}^2 = 1$ and: :$0^2 = 0$ and $\\paren {0 + 1}^2 = 1^2 = 1$ {{qed}} </section> Category:Zero and One are the only Consecutive Perfect Squares htz0vf4du4yfpb36uyojnh9tye85uv2"}
+{"_id": "7240", "title": "Zero and One are the only Consecutive Perfect Squares/Proof 2", "text": "{{:Zero and One are the only Consecutive Perfect Squares}} <section title=\"Proof\" name=\"proof\"> Suppose that $k, l \\in \\Z$ are such that their squares are consecutive, i.e.: :$l^2 - k^2 = 1$ Then we can factor the left-hand side as: :$l^2 - k^2 = \\left({l + k}\\right) \\left({l - k}\\right)$ By Invertible Integers under Multiplication, it follows that: :$l + k = \\pm 1 = l - k$ Therefore, it must be that: :$\\left({l + k}\\right) - \\left({l - k}\\right) = 0$ That is, $2 k = 0$, from which we conclude $k = 0$. So if $n$ and $n + 1$ are squares, then necessarily $n = 0$. The result follows. {{qed}} </section> Category:Zero and One are the only Consecutive Perfect Squares k8g8dzfo5fx677zrb5d7aaqbvi1ouhp"}
+{"_id": "7241", "title": "Convergent Series of Natural Numbers", "text": "Let $\\left({a_n}\\right)_{n \\in \\N}$ be a sequence of natural numbers. Then the following are equivalent: $(1): \\quad \\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ converges $(2): \\quad \\exists N \\in \\N: \\forall n \\ge N: a_n = 0$ That is, $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ converges {{iff}} only finitely many of the $a_n$ are non-zero."}
+{"_id": "7242", "title": "Closure Operator from Closed Sets", "text": "Let $S$ be a set. Let $\\CC$ be a set of subsets of $S$. Let $\\CC$ be closed under arbitrary intersections: :$\\forall \\KK \\in \\powerset \\CC: \\bigcap \\KK \\in \\CC$ where $\\bigcap \\O$ is taken to be $S$. Define $\\cl: \\powerset S \\to \\CC$ by letting: :$\\map \\cl T = \\bigcap \\set {C \\in \\CC: T \\subseteq C}$ Then $\\cl$ is a closure operator whose closed sets are the elements of $\\CC$."}
+{"_id": "7243", "title": "Intersection is Decreasing", "text": "Let $U$ be a set. Let $\\mathcal F$ and $\\mathcal G$ be sets of subsets of $U$. Then $\\mathcal F \\subseteq \\mathcal G \\implies \\bigcap \\mathcal G \\subseteq \\bigcap \\mathcal F$, where by convention $\\bigcap \\varnothing = U$. That is, $\\bigcap$ is a decreasing mapping from $(\\mathcal P(\\mathcal P(U)), \\subseteq)$ to $(\\mathcal P(U), \\subseteq)$, where $\\mathcal P(U)$ is the power set of $U$."}
+{"_id": "7245", "title": "Open Ray is Open in GO-Space", "text": "Let $\\left({S, \\preceq, \\tau}\\right)$ be a generalized ordered space. Let $p \\in S$. Then: : $p^\\prec$ and $p^\\succ$ are $\\tau$-open where: : $p^\\prec$ is the strict lower closure of $p$ : $p^\\succ$ is the strict upper closure of $p$."}
+{"_id": "7247", "title": "Union is Increasing", "text": "Let $U$ be a set. Let $\\mathcal F$ and $\\mathcal G$ be sets of subsets of $U$. Then $\\mathcal F \\subseteq \\mathcal G \\implies \\bigcup \\mathcal F \\subseteq \\bigcup \\mathcal G$. That is, $\\bigcup$ is an increasing mapping from $(\\mathcal P(\\mathcal P(U)), \\subseteq)$ to $(\\mathcal P(U), \\subseteq)$, where $\\mathcal P(U)$ is the power set of $U$."}
+{"_id": "7248", "title": "Intersection is Idempotent/Indexed Family", "text": "Let $\\family {F_i}_{i \\mathop \\in I}$ be a non-empty indexed family of sets. Suppose that all the sets in the $\\family {F_i}_{i \\mathop \\in I}$ are the same. That is, suppose that for some set $S$: :$\\forall i \\in I: F_i = S$ Then: :$\\displaystyle \\bigcap_{i \\mathop \\in I} F_i = S$ where $\\displaystyle \\bigcap_{i \\mathop \\in I} F_i$ is the intersection of $\\family {F_i}_{i \\mathop \\in I}$."}
+{"_id": "7256", "title": "Factor Principles/Conjunction on Left/Formulation 1/Proof 2", "text": ":$p \\implies q \\vdash \\paren {r \\land p} \\implies \\paren {r \\land q}$"}
+{"_id": "7257", "title": "Factor Principles/Conjunction on Left/Formulation 1/Proof 1", "text": ": $p \\implies q \\vdash \\paren {r \\land p} \\implies \\paren {r \\land q}$"}
+{"_id": "7263", "title": "Constructive Dilemma for Join Semilattices", "text": "Let $\\struct {S, \\vee, \\preceq}$ be a join semilattice. Let $a, b, c, d \\in S$. Let $a \\preceq b$. Let $c \\preceq d$. Then $\\paren {a \\vee c} \\preceq \\paren {b \\vee d}$."}
+{"_id": "7264", "title": "Praeclarum Theorema for Meet Semilattices", "text": "Let $(S, \\wedge, \\preceq)$ be a meet semilattice. Let $a, b, c, d \\in S$. Let $a \\preceq b$. Let $c \\preceq d$. Then $(a \\wedge c) \\preceq (b \\wedge d)$."}
+{"_id": "7265", "title": "Supremum is Increasing relative to Product Ordering", "text": "Let $(S, \\preceq)$ be an ordered set. Let $I$ be a set. Let $f, g: I \\to S$. Let $f \\left[{I}\\right]$ denote the image of $I$ under $f$. Let: :$\\forall i \\in I: f \\left({i}\\right) \\preceq g \\left({i}\\right)$ That is, let $f \\preceq g$ in the product ordering. Let $f \\left[{I}\\right]$ and $g \\left[{I}\\right]$ admit suprema. Then: : $\\sup f \\left[{I}\\right] \\preceq \\sup g \\left[{I}\\right]$"}
+{"_id": "7266", "title": "Reflexive Closure of Strict Ordering is Ordering", "text": "Let $S$ be a set. Let $\\prec$ be a strict ordering on $S$. Let $\\preceq$ be the reflexive closure of $\\prec$. Then $\\preceq$ is an ordering."}
+{"_id": "7267", "title": "Reflexive Closure is Reflexive", "text": "Let $\\mathcal R$ be a relation on a set $S$. Then $\\mathcal R^=$, the reflexive closure of $\\mathcal R$, is reflexive."}
+{"_id": "7269", "title": "Reflexive Closure is Inflationary", "text": "Let $S$ be a set. Let $R$ denote the set of all endorelations on $S$. Then the reflexive closure operator is an inflationary mapping on $R$."}
+{"_id": "7272", "title": "Reflexive Closure is Order Preserving", "text": "Let $S$ be a set. Let $R$ denote the set of all endorelations on $S$. Then the reflexive closure operator is an order preserving mapping on $R$. That is: :$\\forall \\RR, \\SS \\in R: \\RR \\subseteq \\SS \\implies \\mathcal R^= \\subseteq \\SS^=$ where $\\RR^=$ and $\\SS^=$ denote the reflexive closure of $\\RR$ and $\\SS$ respectively."}
+{"_id": "7273", "title": "Reflexive Closure is Idempotent", "text": "Let $S$ be a set. Let $R$ denote the set of all endorelations on $S$. Then the reflexive closure operator is an idempotent mapping on $R$. That is: :$\\forall \\RR \\in R: \\RR^= = \\paren {\\RR^=}^=$ where $\\RR^=$ denotes the reflexive closure of $\\RR$."}
+{"_id": "7276", "title": "Transitive Closure Always Exists (Relation Theory)/Proof 1", "text": "Let $\\mathcal R$ be a relation on a set $S$. Then the transitive closure $\\mathcal R^+$ of $\\mathcal R$ always exists."}
+{"_id": "7277", "title": "Transitive Closure Always Exists (Relation Theory)/Proof 2", "text": "Let $\\mathcal R$ be a relation on a set $S$. Then the transitive closure $\\mathcal R^+$ of $\\mathcal R$ always exists."}
+{"_id": "7278", "title": "Equivalence of Definitions of Transitive Closure (Relation Theory)/Union of Compositions is Smallest", "text": "Let $\\RR$ be a relation on a set $S$. Let: :$\\RR^n := \\begin{cases} \\RR & : n = 0 \\\\ \\RR^{n - 1} \\circ \\RR & : n > 0 \\end{cases}$ where $\\circ$ denotes composition of relations. {{explain|Really? I would have thought $\\RR^1 {{=}} \\RR$, not $\\RR^0 {{=}} \\RR$. If anything, the diagonal relation $\\Delta_S$ should be $\\RR^0$.}} Finally, let: :$\\displaystyle \\RR^+ = \\bigcup_{i \\mathop \\in \\N} \\RR^i$ Then $\\RR^+$ is the smallest transitive relation on $S$ that contains $\\RR$."}
+{"_id": "7279", "title": "Equivalence of Definitions of Transitive Closure (Relation Theory)/Finite Chain Equivalent to Union of Compositions", "text": "The finite chain and union of compositions definitions of '''transitive closure''' are equivalent."}
+{"_id": "7280", "title": "Chasles' Relation", "text": "Let $\\mathcal E$ be an affine space. Let $p, q, r \\in \\mathcal E$ be points. Then: :$\\vec{p q} = \\vec{p r} + \\vec{r q}$"}
+{"_id": "7281", "title": "Affine Coordinates are Well-Defined", "text": "Let $\\mathcal E$ be an affine space with difference space $V$ over a field $k$. Let $\\mathcal R = \\left({p_0, e_1, \\ldots, e_n}\\right)$ be an affine frame in $\\mathcal E$. Define a mapping $\\Theta_{\\mathcal R} : k^n \\to \\mathcal E$ by: :$\\displaystyle \\Theta_\\mathcal R \\left({\\lambda_1, \\ldots, \\lambda_n}\\right) = p_0 + \\sum_{i \\mathop = 1}^n \\lambda_i e_i$ Then $\\Theta_\\mathcal R$ is a bijection."}
+{"_id": "7283", "title": "Transitive Chaining", "text": "Let $\\mathcal R$ be a transitive relation on a set $S$. Let $n \\in \\N$ be a natural number. Let $n \\ge 2$. Let $\\langle x_k \\rangle_{k \\in \\left\\{ {1, 2, \\dots, n}\\right\\} }$ be a sequence of $n$ terms. For each $k \\in \\left\\{ {1, 2, \\dots, n-1}\\right\\}$, let $x_k \\mathrel {\\mathcal R} x_{k+1}$. That is, let $x_1 \\mathrel {\\mathcal R} x_2$, $x_2 \\mathrel {\\mathcal R} x_3, \\dotsc, x_n-1 \\mathrel {\\mathcal R} x_n$. Then $x_1 \\mathrel {\\mathcal R} x_n$"}
+{"_id": "7286", "title": "Szpilrajn Extension Theorem", "text": "Let $\\struct {S, \\prec}$ be a strictly ordered set. {{Disambiguate|Definition:Strictly Ordered Set}} Then there is a strict total ordering on $S$ of which $\\prec$ is a subset."}
+{"_id": "7287", "title": "Strict Ordering can be Expanded to Compare Additional Pair", "text": "Let $\\left({S, \\prec}\\right)$ be an ordered set. Let $a$ and $b$ be distinct, $\\prec$-incomparable elements of $S$. That is, let: :$a \\not \\prec b$ and $b \\not \\prec a$. Let ${\\prec'} = {\\prec} \\cup \\left\\{ {\\left({a, b}\\right)} \\right\\}$. Define a relation $\\prec'^+$ by letting $p \\prec'^+ q$ {{iff}}: : $p \\prec q$ or: : $p \\preceq a$ and $b \\preceq q$ where $\\preceq$ is the reflexive closure of $\\prec$. Then: : $\\prec'^+$ is a strict ordering : $\\prec^+$ is the transitive closure of $\\prec'$."}
+{"_id": "7289", "title": "Characterization of Interior of Triangle", "text": "Let $\\triangle$ be a triangle embedded in $\\R^2$. Denote the vertices of $\\triangle$ as $A_1, A_2, A_3$. For $i \\in \\set {1, 2, 3}$, put $j = i \\bmod 3 + 1$, $k = \\paren {i + 1} \\bmod 3 + 1$, and: :$U_i = \\set {A_i + s t \\paren {A_j - A_i} + \\paren {1 - s} t \\paren {A_k - A_i} : s \\in \\openint 0 1, t \\in \\R_{>0} }$ Then: :$\\displaystyle \\Int \\triangle = \\bigcap_{i \\mathop = 1}^3 U_i$ where $\\Int \\triangle$ denotes the interior of the boundary of $\\triangle$."}
+{"_id": "7290", "title": "Equivalence of Definitions of Reflexive Transitive Closure", "text": "Let $\\mathcal R$ be a relation on a set $S$. {{TFAE|def = Reflexive Transitive Closure}}"}
+{"_id": "7291", "title": "Intersection of Relation with Inverse is Symmetric Relation", "text": "Let $\\mathcal R$ be a relation on a set $S$. Then $\\mathcal R \\cap \\mathcal R^{-1}$, the intersection of $\\mathcal R$ with its inverse, is symmetric."}
+{"_id": "7292", "title": "Intersection of Closed Sets is Closed/Closure Operator", "text": "Let $S$ be a set. Let $f: \\powerset S \\to \\powerset S$ be a closure operator on $S$. Let $\\CC$ be the set of all subsets of $S$ that are closed with respect to $f$. Let $\\AA \\subseteq \\CC$. Then $\\bigcap \\AA \\in \\CC$."}
+{"_id": "7293", "title": "Closure is Closed/Power Set", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Let $\\cl: \\powerset S \\to \\powerset S$ be a closure operator. Let $T \\subseteq S$. Then $\\map \\cl T$ is a closed set with respect to $\\cl$."}
+{"_id": "7294", "title": "Relation Intersection Inverse is Greatest Symmetric Subset of Relation", "text": "Let $\\RR$ be a relation on a set $S$. Let $\\powerset \\RR$ be the power set of $\\RR$. By definition, $\\powerset \\RR$ is the set of all relations on $S$ that are subsets of $\\RR$. Then the greatest element of $\\powerset \\RR$ that is symmetric is: :$\\RR \\cap \\RR^{-1}$"}
+{"_id": "7295", "title": "Composition of Compatible Closure Operators", "text": "Let $S$ be a set. Let $f, g: \\mathcal P \\left({S}\\right) \\to \\mathcal P \\left({S}\\right)$ be closure operators on $S$. Let $\\mathcal C_f$ and $\\mathcal C_g$ be the sets of closed sets of $S$ with respect to $f$ and $g$ respectively. For each subset $T$ of $S$, let the following hold: : $(1): \\quad$ If $T$ is closed with respect to $g$, then $f \\left({T}\\right)$ is closed with respect to $g$. :: That is, if $T \\in \\mathcal C_g$ then $f \\left({T}\\right) \\in \\mathcal C_g$. : $(2): \\quad$ If $T$ is closed with respect to $f$, then $g \\left({T}\\right)$ is closed with respect to $f$. :: That is, if $T \\in \\mathcal C_f$ then $g \\left({T}\\right) \\in \\mathcal C_f$. Let $\\mathcal C_h = \\mathcal C_f \\cap \\mathcal C_g$. Then: : $\\mathcal C_h$ induces a closure operator $h$ on $S$ : $f \\circ g = g \\circ f = h$, where $\\circ$ represents composition of mappings."}
+{"_id": "7297", "title": "Closure Operator from Closed Elements", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $C \\subseteq S$. Suppose that $C$ is a subset of $S$ with the property that every element of $S$ has a smallest successor in $C$. Let $\\cl: S \\to S$ be defined as follows: For $x \\in S$: :$\\map \\cl x = \\map \\min {C \\cap x^\\succeq}$ where $x^\\succeq$ is the upper closure of $x$. That is, let $\\map \\cl x$ be the smallest successor of $x$ in $C$. Then: :$\\cl$ is a closure operator on $S$ :The closed elements of $\\cl$ are precisely the elements of $C$."}
+{"_id": "7298", "title": "Product of Affine Spaces is Affine Space", "text": "Let $\\mathcal E, \\mathcal F$ be affine spaces. Let $\\mathcal G = \\mathcal E \\times \\mathcal F$ be the product of $\\mathcal E$ and $\\mathcal F$. Then $\\mathcal G$ is an affine space."}
+{"_id": "7299", "title": "Intersection of Complete Meet Subsemilattices invokes Closure Operator", "text": "Let $\\struct {S, \\preccurlyeq}$ be an ordered set. Let $f_i$ be a closure operator on $S$ for each $i \\in I$. Let $C_i = \\map {f_i} S$ be the set of closed elements with respect to $f_i$ for each $i \\in I$. Suppose that for each $i \\in I$, $C_i$ is a '''complete meet subsemilattice''' of $S$ in the following sense: :For each $D \\subseteq C_i$, $D$ has an infimum in $S$ such that $\\inf D \\in C_i$. Then $C = \\displaystyle \\bigcap_{i \\mathop \\in I} C_i$ induces a closure operator on $S$."}
+{"_id": "7301", "title": "Reflexive Reduction of Ordering is Strict Ordering", "text": "Let $\\mathcal R$ be an ordering on a set $S$. Let $\\mathcal R^\\ne$ be the reflexive reduction of $\\mathcal R$. Then $\\mathcal R^\\ne$ is a strict ordering on $S$."}
+{"_id": "7304", "title": "Vectorialization of Affine Space is Vector Space", "text": "Let $\\mathcal E$ be an affine space over a field $k$ with difference space $E$. Let $\\mathcal R = \\tuple {p_0, e_1, \\ldots, e_n}$ be an affine frame in $\\mathcal E$. Let $\\struct {\\mathcal E, +, \\cdot}$ be the vectorialization of $\\mathcal E$. Then $\\struct {\\mathcal E, +, \\cdot}$ is a vector space."}
+{"_id": "7305", "title": "Reflexive Reduction is Antireflexive", "text": "Let $\\mathcal R$ be a relation on a set $S$. Let $\\mathcal R^\\ne$, be the reflexive reduction of $\\mathcal R$. Then $\\mathcal R^\\ne$ is antireflexive."}
+{"_id": "7307", "title": "Subband iff Idempotent under Induced Operation", "text": "Let $\\left({S, \\circ}\\right)$ be a band. Let $\\left({\\mathcal P \\left({S}\\right), \\circ_\\mathcal P}\\right)$ be the algebraic structure consisting of the power set of $S$ and the operation induced on $\\mathcal P \\left({S}\\right)$ by $\\circ$. Let $X \\in \\mathcal P \\left({S}\\right)$. Then $X$ is idempotent {{iff}} $\\left({X, \\circ}\\right)$ is a subband of $\\left({S, \\circ}\\right)$."}
+{"_id": "7308", "title": "Restriction of Idempotent Operation is Idempotent", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure. Let $T \\subseteq S$. Let the operation $\\circ$ be idempotent. Then $\\circ$ is also idempotent upon restriction  to $\\left({T, \\circ \\restriction_T}\\right)$."}
+{"_id": "7310", "title": "Composition of Commuting Idempotent Mappings is Idempotent", "text": "Let $S$ be a set. Let $f, g: S \\to S$ be idempotent mappings from $S$ to $S$. Let: :$f \\circ g = g \\circ f$ where $\\circ$ denotes composition. Then $f \\circ g$ is idempotent."}
+{"_id": "7311", "title": "Compositions of Closure Operators are both Closure Operators iff Operators Commute", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $f$ and $g$ be closure operators on $S$. Then the following are equivalent: :$(1): \\quad f \\circ g$ and $g \\circ f$ are both closure operators. :$(2): \\quad f$ and $g$ commute (that is, $f \\circ g = g \\circ f$). :$(3): \\quad \\operatorname{img}\\left({f \\circ g}\\right) = \\operatorname{img}\\left({g \\circ f}\\right)$ where $\\operatorname{img}$ represents the image of a mapping."}
+{"_id": "7312", "title": "Composition of Increasing Mappings is Increasing", "text": "Let $\\struct {S, \\preceq_S}$, $\\struct {T, \\preceq_T}$ and $\\struct {U, \\preceq_U}$ be ordered sets. Let $g: S \\to T$ and $f: T \\to U$ be increasing mappings. Then their composition $f \\circ g: S \\to U$ is also increasing."}
+{"_id": "7313", "title": "Composition of Inflationary Mappings is Inflationary", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $f, g: S \\to S$ be inflationary mappings. Then $f \\circ g$, the composition of $f$ and $g$, is also inflationary."}
+{"_id": "7315", "title": "Fixed Point of Idempotent Mapping", "text": "Let $S$ be a set. Let $f: S \\to S$ be an idempotent mapping. Let $f \\left[{S}\\right]$ be the image of $S$ under $f$. Let $x \\in S$. Then $x$ is a fixed point of $f$ {{iff}} $x \\in f \\left[{S}\\right]$."}
+{"_id": "7316", "title": "Symmetric Closure of Ordering may not be Transitive", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $\\preceq^\\leftrightarrow$ be the symmetric closure of $\\preceq$. Then it is not necessarily the case that $\\preceq^\\leftrightarrow$ is transitive."}
+{"_id": "7317", "title": "Composition of Idempotent Mappings", "text": "Let $S$ be a set. Let $f, g: S \\to S$ be idempotent mappings. Suppose that $f \\circ g$ and $g \\circ f$ have the same images. That is, suppose that $f \\sqbrk {g \\sqbrk S} = g \\sqbrk {f \\sqbrk S}$. Then $f \\circ g$ and $g \\circ f$ are idempotent."}
+{"_id": "7318", "title": "Composition of Inflationary and Idempotent Mappings", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $f$ and $g$ be inflationary and idempotent mappings on $S$. Then the following are equivalent: :$(1): \\quad f \\circ g$ and $g \\circ f$ are both idempotent :$(2): \\quad f$ and $g$ commute (that is, $f \\circ g = g \\circ f$) :$(3): \\quad \\Img {f \\circ g} = \\Img {g \\circ f}$ where: :$\\circ$ represents composition :$\\Img f$ represents the image of a mapping $f$."}
+{"_id": "7319", "title": "Closure is Smallest Closed Successor", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $f: S \\to S$ be a closure operator on $S$. Let $x \\in S$. Then $\\map f x$ is the smallest closed element that succeeds $x$."}
+{"_id": "7320", "title": "Closed Elements Uniquely Determine Closure Operator", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $f, g: S \\to S$ be closure operators on $S$. Suppose that $f$ and $g$ have the same closed elements. Then $f = g$."}
+{"_id": "7322", "title": "Square of Number Always Exists", "text": "Let $x$ be a number. Then its square $x^2$ is guaranteed to exist."}
+{"_id": "7323", "title": "Schröder Rule", "text": "Let $A$, $B$ and $C$ be relations on a set $S$. Then the following are equivalent statements: :$(1): \\quad A \\circ B \\subseteq C$ :$(2): \\quad A^{-1} \\circ \\overline C \\subseteq \\overline B$ :$(3): \\quad \\overline C \\circ B^{-1} \\subseteq \\overline A$ where: : $\\circ$ denotes relation composition : $A^{-1}$ denotes the inverse of $A$ : $\\overline A$ denotes the complement of $A$."}
+{"_id": "7324", "title": "Equivalence of Definitions of Dual Relation", "text": "{{TFAE|def = Dual Relation}} Let $\\RR \\subseteq S \\times T$ be a relation."}
+{"_id": "7325", "title": "Trivial Gradation is Gradation", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring. Let $\\left({M, e,\\cdot}\\right)$ be a monoid. Let :$\\displaystyle R = \\bigoplus_{m \\mathop \\in M} R_m$ be the trivial $M$-gradation on $R$. This is a gradation on $R$."}
+{"_id": "7326", "title": "Fixed Point of Mappings is Fixed Point of Composition", "text": "Let $S$ be a set. Let $f, g: S \\to S$ be mappings. Let $x \\in S$ be a fixed point of both $f$ and $g$. Then $x$ is also a fixed point of $f \\circ g$, the composition of $f$ and $g$."}
+{"_id": "7327", "title": "Fixed Point of Mappings is Fixed Point of Composition/General Result", "text": "Let $S$ be a set. Let $n \\in \\N$ be a strictly positive integer. Let $\\N_n$ be the initial segment of $n$ in $\\N$. That is, let $\\N_n = \\left\\{{0, 1, \\dots, n-1}\\right\\}$. For each $i \\in \\N_n$, let $f_i: S \\to S$ be a mapping. Let $x \\in S$ be a fixed point of $f_i$ for each $i \\in \\N_n$. Let $g = f_0 \\circ f_1 \\circ \\dots \\circ f_{n-1}$ be the composition of all the $f_i$s. Then $x$ is a fixed point of $g$."}
+{"_id": "7328", "title": "Polynomial has Integer Coefficients iff Content is Integer", "text": "$f$ has integer coefficients {{iff}} $\\cont f$ is an integer."}
+{"_id": "7330", "title": "Content of Scalar Multiple", "text": ":$\\cont {q f} = q \\cont f$"}
+{"_id": "7331", "title": "Fixed Point of Composition of Inflationary Mappings", "text": "Let $\\left({S, \\preceq}\\right)$ be a ordered set. Let $f, g: S \\to S$ be inflationary mappings. Let $x \\in S$. Then: : $x$ is a fixed point of $f \\circ g$ {{iff}}: : $x$ is a fixed point of both $f$ and $g$."}
+{"_id": "7332", "title": "Group of Units is Group", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity. Then the set of units of $\\struct {R, +, \\circ}$ forms a group under $\\circ$. Hence the justification for referring to the group of units of $\\struct {R, +, \\circ}$."}
+{"_id": "7333", "title": "Principal Ideal of Principal Ideal Domain is of Irreducible Element iff Maximal/Forward Implication", "text": "Let $\\struct {D, +, \\circ}$ be a principal ideal domain. Let $p$ be an irreducible element of $D$. Let $\\ideal p$ be the principal ideal of $D$ generated by $p$. Then $\\ideal p$ is a maximal ideal of $D$."}
+{"_id": "7334", "title": "Principal Ideal of Principal Ideal Domain is of Irreducible Element iff Maximal/Reverse Implication", "text": "Let $\\left({D, +, \\circ}\\right)$ be a principal ideal domain. Let $\\left({p}\\right)$ be the principal ideal of $D$ generated by $p$. Let $\\left({p}\\right)$ be a maximal ideal of $D$. Then $p$ is irreducible."}
+{"_id": "7335", "title": "Subring of Polynomials over Integral Domain Contains that Domain", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring. Let $\\struct {D, +, \\circ}$ be an integral subdomain of $R$. Let $x \\in R$. Let $D \\sqbrk x$ denote the ring of polynomials in $x$ over $D$. Then $D \\sqbrk x$ contains $D$ as a subring and $x$ as an element."}
+{"_id": "7336", "title": "Subring of Polynomials over Integral Domain is Smallest Subring containing Element and Domain", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring. Let $\\struct {D, +, \\circ}$ be an integral subdomain of $R$. Let $x \\in R$. Let $D \\sqbrk x$ denote the ring of polynomials in $x$ over $D$. Then $D \\sqbrk x$ is the smallest subring of $R$ which contains $D$ as a subring and $x$ as an element."}
+{"_id": "7337", "title": "Cantor-Bernstein-Schröder Theorem/Lemma", "text": "Let $S$ be a set. Let $T \\subseteq S$. Suppose that $f: S \\to T$ is an injection. Then there is a bijection $g: S \\to T$."}
+{"_id": "7338", "title": "Cantor-Bernstein-Schröder Theorem/Proof 5", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ and $g: T \\to S$ be injections. Then there exists a bijection $\\phi: S \\to T$."}
+{"_id": "7339", "title": "Quotient Space of Real Line may be Indiscrete", "text": "Let $\\struct {\\R, \\tau}$ denote the real number line with the usual (Euclidean) topology. Let $\\Q$ be the rational numbers. Let $\\mathbb I$ be the irrational numbers. Then $\\set {\\Q, \\mathbb I}$ is a partition of $\\R$. Let $\\sim$ be the equivalence relation induced on $\\R$ by $\\set {\\Q, \\mathbb I}$. Let $T_\\sim := \\struct {\\R / {\\sim}, \\tau_\\sim} $ be the quotient space of $\\R$ by $\\sim$. Then $T_\\sim$ is an indiscrete space."}
+{"_id": "7340", "title": "Quotient Space of Real Line may not be Kolmogorov", "text": "Let $\\struct {\\R, \\tau}$ denote the real number line with the usual (Euclidean) topology. Then there exists an equivalence relation $\\sim$ on $\\R$ such that the quotient space $\\struct {\\R / {\\sim}, \\tau_\\sim}$ is not Kolmogorov."}
+{"_id": "7341", "title": "Quotient Space of Real Line may be Kolmogorov but not Fréchet", "text": "Let $\\struct {\\R, \\tau}$ denote the real number line with the usual (Euclidean) topology. Define an equivalence relation $\\sim$ by letting $x \\sim y$ {{iff}} either: :$x = y$ or: :$x, y \\in \\Q$ Let $\\struct {\\R / {\\sim}, \\tau_\\sim}$ be the quotient space of $\\R$ by $\\sim$. Then $\\struct {\\R / {\\sim}, \\tau_\\sim}$ is a Kolmogorov space but not a Fréchet space."}
+{"_id": "7342", "title": "Unique Representation in Polynomial Forms/General Result", "text": "Let $f$ be a polynomial form in the indeterminates $\\set {X_j: j \\in J}$ such that $f: \\mathbf X^k \\mapsto a_k$. For $r \\in \\R$, $\\mathbf X^k \\in M$, let $r \\mathbf X^k$ denote the polynomial form that takes the value $r$ on $\\mathbf X^k$ and zero on all other monomials. Let $Z$ denote the set of all multiindices indexed by $J$. Then the sum representation: :$\\ds \\hat f = \\sum_{k \\mathop \\in Z} a_k \\mathbf X^k$ has only finitely many non-zero terms. Moreover it is everywhere equal to $f$, and is the unique such sum."}
+{"_id": "7344", "title": "Law of Excluded Middle for Two Variables", "text": ":$\\vdash (p \\land q) \\lor (\\lnot p \\land q) \\lor (p \\land \\lnot q) \\lor (\\lnot p \\land \\lnot q)$"}
+{"_id": "7348", "title": "Ring of Polynomial Forms is Commutative Ring with Unity", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $A = R \\sqbrk {\\set {X_j: j \\in J} }$ be the set of all polynomial forms over $R$ in the indeterminates $\\set {X_j: j \\in J}$. Then $\\struct {A, +, \\circ}$ is a commutative ring with unity."}
+{"_id": "7351", "title": "Idempotent Elements form Subsemigroup of Commutative Semigroup", "text": "Let $\\struct {S, \\circ}$ be a semigroup such that $\\circ$ is commutative. Let $I$ be the set of all elements of $S$ that are idempotent under $\\circ$. That is: :$I = \\set {x \\in S: x \\circ x = x}$ Then $\\struct {I, \\circ}$ is a subsemigroup of $\\struct {S, \\circ}$."}
+{"_id": "7352", "title": "Product of Commuting Idempotent Elements is Idempotent", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Let $a, b \\in S$ be idempotent elements of $S$. Let $a$ and $b$ commute: :$a \\circ b = b \\circ a$ Then $a \\circ b$ is idempotent."}
+{"_id": "7353", "title": "Set of All Self-Maps is Semigroup", "text": "Let $S$ be a set. Let $S^S$ be the set of all mappings from $S$ to itself. Let the operation $\\circ$ represent composition of mappings. Then the algebraic structure $\\struct {S^S, \\circ}$ is a semigroup."}
+{"_id": "7354", "title": "Integers form Subdomain of Reals", "text": "The integral domain of integers $\\struct {\\Z, +, \\times}$ forms a subdomain of the field of real numbers."}
+{"_id": "7355", "title": "Identity Element is Idempotent", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $e \\in S$ be an identity with respect to $\\circ$. Then $e$ is idempotent under $\\circ$."}
+{"_id": "7356", "title": "Identity of Algebraic Structure is Preserved in Substructure", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure with identity $e$. Let $\\struct {T, \\circ}$ be a algebraic substructure of $\\struct {S, \\circ}$. That is, let $T \\subseteq S$. Let $e \\in T$. Then $e$ is an identity of $\\struct {T, \\circ}$."}
+{"_id": "7357", "title": "Idempotent Elements form Submonoid of Commutative Monoid", "text": "Let $\\left({S, \\circ}\\right)$ be a commutative monoid. Let $e \\in S$ be the identity element of $\\left({S, \\circ}\\right)$. Let $I$ be the set of all elements of $S$ that are idempotent under $\\circ$. That is: :$I = \\left\\{{x \\in S: x \\circ x = x}\\right\\}$ Then $\\left({I, \\circ}\\right)$ is a submonoid of $\\left({S, \\circ}\\right)$ with identity $e$."}
+{"_id": "7358", "title": "Inverse Image under Order Embedding of Strict Upper Closure of Image of Point", "text": "Let $\\struct {S, \\preceq}$ and $\\struct {T, \\preceq'}$ be ordered sets. Let $\\phi: S \\to T$ be an order embedding of $\\struct {S, \\preceq}$ into $\\struct {T, \\preceq'}$ Let $p \\in S$. Then: :$\\map {\\phi^{-1} } {\\map \\phi p^{\\succ'} } = p^\\succ$ where $\\cdot^\\succ$ and $\\cdot^{\\succ'}$ represent strict upper closure with respect to $\\preceq$ and $\\preceq'$, respectively."}
+{"_id": "7360", "title": "Equivalence of Definitions of Order Embedding/Definition 1 implies Definition 3", "text": "Let $\\struct {S, \\preceq_1}$ and $\\struct {T, \\preceq_2}$ be ordered sets. Let $\\phi: S \\to T$ be a mapping. Let $\\phi: S \\to T$ be an order embedding by Definition 1: {{:Definition:Order Embedding/Definition 1}} Then $\\phi: S \\to T$ is an order embedding by Definition 3: {{:Definition:Order Embedding/Definition 3}}"}
+{"_id": "7361", "title": "Equivalence of Definitions of Order Embedding/Definition 3 implies Definition 1", "text": "Let $\\struct {S, \\preceq_1}$ and $\\struct {T, \\preceq_2}$ be ordered sets. Let $\\phi: S \\to T$ be a mapping. Let $\\phi: S \\to T$ be an order embedding by Definition 3: {{:Definition:Order Embedding/Definition 3}} Then $\\phi: S \\to T$ is an order embedding by Definition 1: {{:Definition:Order Embedding/Definition 1}}"}
+{"_id": "7363", "title": "Path as Parameterization of Contour", "text": "Let $\\left[{a \\,.\\,.\\, b}\\right]$ be a closed real interval. Let $\\gamma: \\left[{a \\,.\\,.\\, b}\\right] \\to \\C$ be a path. Let there exist $n \\in \\N$ and a subdivision $\\left\\{{a_0, a_1, \\ldots, a_n}\\right\\}$ of $\\left[{a \\,.\\,.\\, b}\\right]$ such that: : $\\gamma {\\restriction_{ \\left[{a_{k - 1} \\,.\\,.\\, a_k}\\right] } }$ is a smooth path for all $k \\in \\left\\{ {1, \\ldots, n}\\right\\}$ where $\\gamma {\\restriction_{\\left[{a_{k - 1} \\,.\\,.\\, a_k}\\right]} }$ denotes the restriction of $\\gamma$ to $\\left[{a_{k - 1} \\,.\\,.\\, a_k}\\right]$. Then there exists a contour $C$ with parameterization $\\gamma$ and these properties: :$(1): \\quad$ If $\\gamma$ is a closed path, then $C$ is a closed contour. :$(2): \\quad$ If $\\gamma$ is a Jordan arc, then $C$ is a simple contour. :$(3): \\quad$ If $\\gamma$ is a Jordan curve, then $C$ is a simple closed contour."}
+{"_id": "7364", "title": "Kernel of Induced Homomorphism of Polynomial Forms", "text": "Let $R$ and $S$ be commutative rings with unity. Let $\\phi: R \\to S$ be a ring homomorphism. Let $K = \\ker \\phi$. Let $R \\left[{X}\\right]$ and $S \\left[{X}\\right]$ be the rings of polynomial forms over $R$ and $S$ respectively in the indeterminate $X$. Let $\\bar\\phi: R \\left[{X}\\right] \\to S \\left[{X}\\right]$ be the induced morphism of polynomial rings. Then the kernel of $\\bar\\phi$ is: :$\\ker \\bar\\phi = \\left\\{{ a_0 + a_1 X + \\cdots + a_n X^n \\in R \\left[{X}\\right] : \\phi \\left({a_i}\\right) = 0 \\text{ for } i = 0, \\ldots, n }\\right\\}$ Or, more concisely: :$\\ker \\bar\\phi = \\left({\\ker \\phi}\\right) \\left[{X}\\right]$"}
+{"_id": "7365", "title": "Boundary of Polygon as Contour", "text": "Let $P$ be a polygon embedded in the complex plane $\\C$. Denote the boundary of $P$ as $\\partial P$. Then there exists a simple closed contour $C$ such that: : $\\operatorname{Im} \\left({C}\\right) = \\partial P$ where $\\operatorname{Im} \\left({C}\\right)$ denotes the image of $C$."}
+{"_id": "7366", "title": "Zero Simple Staircase Integral Condition for Primitive", "text": "Let $f: D \\to \\C$ be a continuous complex function, where $D$ is a connected domain. Let $\\displaystyle \\oint_C \\map f z \\rd z = 0$ for all simple closed staircase contours $C$ in $D$. Then $f$ has a primitive $F: D \\to \\C$."}
+{"_id": "7367", "title": "Preordering induces Equivalence Relation", "text": "Let $\\struct {S, \\precsim}$ be a preordered set. Define a relation $\\sim$ on $S$ by letting $x \\sim y$ {{iff}} $x \\precsim y$ and $y \\precsim x$. Then $\\sim$ is an equivalence relation."}
+{"_id": "7368", "title": "Content of Rational Polynomial is Multiplicative", "text": "Let $h \\in \\Q \\sqbrk X$ be a polynomial with rational coefficients. Let $\\cont h$ denote the content of $h$. Then for any polynomials $f, g \\in \\Q \\sqbrk X$ with rational coefficients: :$\\cont {f g} = \\cont f \\cont g$"}
+{"_id": "7369", "title": "Gauss's Lemma on Irreducible Polynomials", "text": "Let $\\Z$ be the ring of integers. Let $\\Z \\sqbrk X$ be the ring of polynomials over $\\Z$. Let $h \\in \\Z \\sqbrk X$ be a polynomial. {{TFAE}} :$(1): \\quad h$ is irreducible in $\\Q \\sqbrk X$ and primitive :$(2): \\quad h$ is irreducible in $\\Z \\sqbrk X$."}
+{"_id": "7370", "title": "Antisymmetric Quotient of Preordered Set is Ordered Set", "text": "Let $\\struct {S, \\precsim}$ be a preordered set. Let $\\sim$ be the equivalence relation on $S$ induced by $\\precsim$. Let $\\struct {S / {\\sim}, \\preceq}$ be the antisymmetric quotient of $\\struct {S, \\precsim}$. Then: :$\\struct {S / {\\sim}, \\preceq}$ is an ordered set. :$\\forall P, Q \\in S / {\\sim}: \\paren {P \\preceq Q} \\land \\paren {p \\in P} \\land \\paren {q \\in Q} \\implies p \\precsim q$ This second statement means that we could just as well have defined $\\preceq$ by letting $P \\preceq Q$ iff: :$\\forall p \\in P: \\forall q \\in Q: p \\precsim q$"}
+{"_id": "7371", "title": "Ordering on Partition Determines Preordering", "text": "Let $S$ be a set. Let $\\PP$ be a partition of $S$. Let $\\phi: S \\to \\PP$ be the quotient mapping. Let $\\preceq$ be a ordering of $\\PP$. Define a relation $\\precsim$ on $S$ by letting $p \\precsim q$ {{iff}}: :$\\map \\phi p \\preceq \\map \\phi q$ Then: :$\\precsim$ is a preordering on $S$. :$\\precsim$ is the only preordering on $S$ that induces the $\\preceq$ ordering on $\\PP$."}
+{"_id": "7372", "title": "Units of Ring of Polynomial Forms over Commutative Ring", "text": "Let $\\struct {R, +, \\circ}$ be a non-null commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $R \\sqbrk X$ be the ring of polynomial forms in an indeterminate $X$ over $R$. Let $\\map P X = a_0 + a_1 X + \\cdots + a_n X^n \\in R \\sqbrk X$. Then: :$\\map P X$ is a unit of $R \\sqbrk X$ {{iff}}: :$a_0$ is a unit of $R$ Also, for $i = 1, \\ldots, n$, $a_i$ is nilpotent in $R$."}
+{"_id": "7374", "title": "Polynomials Closed under Addition/Polynomials over Integral Domain", "text": "Let $\\left({R, +, \\circ}\\right)$ be a commutative ring with unity. Let $\\left({D, +, \\circ}\\right)$ be an integral subdomain of $R$. Then $\\forall x \\in R$, the set $D \\left[{x}\\right]$ of polynomials in $x$ over $D$ is closed under the operation $+$."}
+{"_id": "7375", "title": "Polynomials Closed under Addition/Polynomials over Ring", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring. Let $\\left({S, +, \\circ}\\right)$ be a subring of $R$. Then $\\forall x \\in R$, the set $S \\left[{x}\\right]$ of polynomials in $x$ over $S$ is closed under the operation $+$."}
+{"_id": "7381", "title": "Nilpotent Element is Zero Divisor", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Suppose further that $R$ is not the null ring. Let $x \\in R$ be a nilpotent element of $R$. Then $x$ is a zero divisor in $R$."}
+{"_id": "7383", "title": "Units of Ring of Polynomial Forms over Integral Domain", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain. Let $D \\sqbrk X$ be the ring of polynomial forms in an indeterminate $X$ over $D$. Then the group of units of $D \\sqbrk X$ is precisely the group of elements of $D \\sqbrk X$ of degree zero that are units of $D$."}
+{"_id": "7384", "title": "Kernel of Magma Homomorphism is Submagma", "text": "Let $\\left({S, *}\\right)$ and $\\left({T, \\circ}\\right)$ be algebraic structures. Let $\\left({T, \\circ}\\right)$ have an identity $e$. Let $\\phi: S \\to T$ be a magma homomorphism. Then the kernel of $\\phi$ is a submagma of $\\left({S, *}\\right)$. That is: :$\\left({\\phi^{-1} \\left({e}\\right), *}\\right)$ is a submagma of $\\left({S, *}\\right)$ where $\\phi^{-1} \\left({e}\\right)$ denote the preimage of $e$."}
+{"_id": "7385", "title": "Preimage of Zero of Homomorphism is Submagma", "text": "Let $\\struct {S, *}$ be a magma. Let $\\struct {T, \\circ}$ be a magma with a zero element $0$. Let $\\phi: S \\to T$ be a magma homomorphism. Then $\\struct {\\phi^{-1} \\sqbrk 0, *}$ is a submagma of $\\struct {S, *}$."}
+{"_id": "7386", "title": "Polynomial over Field is Reducible iff Scalar Multiple is Reducible", "text": "Let $K$ be a field. Let $K \\left[{X}\\right]$ be the ring of polynomial forms over $K$. Let $P \\in K \\left[{X}\\right]$. Let $\\lambda \\in K \\setminus \\left\\{{0}\\right\\}$. Then $P$ is irreducible in $K \\left[{X}\\right]$ iff $\\lambda P$ is also irreducible in $K \\left[{X}\\right]$. {{expand|Investigate whether this result also holds where $K$ is a general ring.}}"}
+{"_id": "7387", "title": "Conjunction of Disjunctions Consequence", "text": ":$\\left({p \\lor q}\\right) \\land \\left({r \\lor s}\\right) \\vdash p \\lor r \\lor \\left({q \\land s}\\right)$"}
+{"_id": "7390", "title": "Existence of Ring of Polynomial Forms in Transcendental over Integral Domain", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $\\struct {D, +, \\circ}$ be an integral subdomain of $R$ whose zero is $0_D$. Let $X \\in R$ be transcendental over $D$ Then the ring of polynomials $D \\sqbrk X$ in $X$ over $D$ exists."}
+{"_id": "7395", "title": "Eisenstein Integers form Integral Domain", "text": "The ring of Eisenstein integers $\\struct {\\Z \\sqbrk \\omega, +, \\times}$ is an integral domain."}
+{"_id": "7396", "title": "Eisenstein Integers form Subring of Complex Numbers", "text": "The set of Eisenstein integers $\\Z \\sqbrk \\omega$, under the operations of complex addition and complex multiplication, forms a subring of the set of complex numbers $\\C$."}
+{"_id": "7399", "title": "Polynomial Forms over Field form Integral Domain/Formulation 2", "text": "Let $\\struct {F, +, \\circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $\\GF$ be the set of all polynomials over $\\struct {F, +, \\circ}$ defined as sequences. Let polynomial addition and polynomial multiplication be defined as: :$\\forall f = \\sequence {a_k} = \\tuple {a_0, a_1, a_2, \\ldots}, g = \\sequence {b_k} = \\tuple {b_0, b_1, b_2, \\ldots} \\in \\GF$: ::$f \\oplus g := \\tuple {a_0 + b_0, a_1 + b_1, a_2 + b_2, \\ldots}$ ::$f \\otimes g := \\tuple {c_0, c_1, c_2, \\ldots}$ where $\\displaystyle c_i = \\sum_{j \\mathop + k \\mathop = i} a_j \\circ b_k$ Then $\\struct {\\GF, \\oplus, \\otimes}$ is an integral domain."}
+{"_id": "7400", "title": "Maximal Spectrum of Ring is Nonempty", "text": "Let $A$ be a non-trivial commutative ring with unity. Then its maximal spectrum is non-empty: :$\\operatorname {Max} \\Spec A \\ne \\O$"}
+{"_id": "7404", "title": "Knaster-Tarski Lemma", "text": "Let $\\left({L, \\preceq}\\right)$ be a complete lattice. Let $f: L \\to L$ be an increasing mapping. Then $f$ has a least fixed point and a greatest fixed point."}
+{"_id": "7405", "title": "Knaster-Tarski Theorem", "text": "Let $\\left({L, \\preceq}\\right)$ be a complete lattice. Let $f: L \\to L$ be an increasing mapping. Let $F$ be the set (or class) of fixed points of $f$. Then $\\left({F, \\preceq}\\right)$ is a complete lattice."}
+{"_id": "7406", "title": "Degree of Product of Polynomials over Ring/Corollary 2", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose zero is $0_D$. Let $D \\sqbrk X$ be the ring of polynomials over $D$ in the indeterminate $X$. For $f \\in D \\sqbrk X$ let $\\map \\deg f$ denote the degree of $f$. Then: :$\\forall f, g \\in D \\sqbrk X: \\map \\deg {f g} = \\map \\deg f + \\map \\deg g$"}
+{"_id": "7407", "title": "Knaster-Tarski Lemma/Power Set", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Let $f: \\powerset S \\to \\powerset S$ be a $\\subseteq$-increasing mapping. That is, suppose that for all $T, U \\in \\powerset S$: :$T \\subseteq U \\implies \\map f T \\subseteq \\map f U$ Then $f$ has a greatest fixed point and a least fixed point."}
+{"_id": "7408", "title": "Cantor-Bernstein-Schröder Theorem/Proof 6", "text": "Let $A$ and $B$ be sets. Let $f: A \\to B$ and $g: B \\to A$ be injections. Then there is a bijection $h: A \\to B$; so that $A$ and $B$ are equivalent. Furthermore: : For all $x \\in A$ and $y \\in B$, if $y = h \\left({x}\\right)$ then either $y = f \\left({x}\\right)$ or $x = g \\left({y}\\right)$."}
+{"_id": "7409", "title": "Ring of Polynomial Forms over Integral Domain is Integral Domain", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose zero is $0_D$. Let $\\struct {D \\sqbrk X, \\oplus, \\odot}$ be the ring of polynomial forms over $D$ in the indeterminate $X$. Then $\\struct {D \\sqbrk X, \\oplus, \\odot}$ is an integral domain."}
+{"_id": "7410", "title": "Union of One-to-Many Relations with Disjoint Images is One-to-Many", "text": "Let $S_1, S_2, T_1, T_2$ be sets or classes. Let $\\mathcal R_1$ be a one-to-many relation on $S_1 \\times T_1$. Let $\\mathcal R_2$ be a one-to-many relation on $S_2 \\times T_2$. Suppose that the images of $\\mathcal R_1$ and $\\mathcal R_2$ are disjoint. Then $\\mathcal R_1 \\cup \\mathcal R_2$ is a one-to-many relation on $(S_1 \\cup S_2) \\times (T_1 \\cup T_2)$."}
+{"_id": "7412", "title": "Union of Many-to-One Relations with Disjoint Domains is Many-to-One", "text": "Let $S_1, S_2, T_1, T_2$ be sets or classes. Let $\\RR_1$ be a many-to-one relation on $S_1 \\times T_1$. Let $\\RR_2$ be a many-to-one relation on $S_2 \\times T_2$. Suppose that the domains of $\\RR_1$ and $\\RR_2$ are disjoint. Then $\\RR_1 \\cup \\RR_2$ is a many-to-one relation on $\\paren {S_1 \\cup S_2} \\times \\paren {T_1 \\cup T_2}$."}
+{"_id": "7413", "title": "Nth Root of Integer is Integer or Irrational", "text": "Let $n$ be a natural number. Let $x$ be an integer. If the $n$th root of $x$ is not an integer, it must be irrational."}
+{"_id": "7414", "title": "Union of Bijections with Disjoint Domains and Codomains is Bijection", "text": "Let $A$, $B$, $C$, and $D$ be sets or classes. Let $A \\cap B = C \\cap D = \\varnothing$. Let $f: A \\to C$ and $g: B \\to D$ be bijections. Then $f \\cup g: A \\cup B \\to C \\cup D$ is also a bijection."}
+{"_id": "7415", "title": "Interval in Complete Lattice is Complete Lattice", "text": "Let $\\left({L, \\preceq}\\right)$ be a complete lattice. Let $a, b \\in L$ with $a \\preceq b$. Let $\\left[{a \\,.\\,.\\, b}\\right]$ be the closed interval between $a$ and $b$. {{explain|Demonstrate that for each $a, b \\in L$ that $\\left[{a \\,.\\,.\\, b}\\right]$ exists and is unique.}} Then $\\left[{a \\,.\\,.\\, b}\\right]$ is also a complete lattice under $\\preceq$."}
+{"_id": "7417", "title": "Set Difference with Subset is Superset of Set Difference", "text": "Let $A, B, S$ be sets or classes. Suppose that $A \\subseteq B$. Then $S \\setminus B \\subseteq S \\setminus A$, where $\\setminus$ represents set difference."}
+{"_id": "7420", "title": "Knaster-Tarski Lemma/Corollary", "text": "Let $\\struct {L, \\preceq}$ be a complete lattice. Let $f: L \\to L$ be an increasing mapping. Then $f$ has a fixed point"}
+{"_id": "7421", "title": "Knaster-Tarski Lemma/Corollary/Power Set", "text": "Let $S$ be a set. Let $\\mathcal P \\left({S}\\right)$ be the power set of $S$. Let $f: \\mathcal P \\left({S}\\right) \\to \\mathcal P \\left({S}\\right)$ be a $\\subseteq$-increasing mapping. That is, suppose that for all $T, U \\in \\mathcal P \\left({S}\\right)$: : $T \\subseteq U \\implies f \\left({T}\\right) \\subseteq f\\left({U}\\right)$ Then $f$ has a fixed point."}
+{"_id": "7430", "title": "Alternative Definition of Ordinal in Well-Founded Theory", "text": "A set $S$ is an ordinal {{iff}} $S$ is transitive and $\\forall x, y \\in S: \\left({x \\in y \\lor x = y \\lor y \\in x}\\right)$."}
+{"_id": "7431", "title": "Foundational Relation is Antireflexive/Corollary", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Suppose that $S$ is non-empty. Then $\\preceq$ is not a foundational relation."}
+{"_id": "7432", "title": "Reflexive Reduction of Well-Founded Ordering is Foundational Relation", "text": "Let $S$ be a set. Let $\\preceq$ be a well-founded ordering of $S$. Let $\\prec$ be the reflexive reduction of $\\preceq$. Then $\\prec$ is a foundational relation."}
+{"_id": "7433", "title": "Epsilon Relation is Proper", "text": "Let $\\mathbb U$ be the universal class. Let $\\Epsilon$ be the epsilon relation. Then $\\left({\\mathbb U, \\Epsilon}\\right)$ is a proper relational structure."}
+{"_id": "7436", "title": "Relationship between Transitive Closure Definitions", "text": "Let $x$ be a set. Let $a$ be the smallest set such that $x \\in a$ and $a$ is transitive. Let $b$ be the smallest set such that $x \\subseteq b$ and $b$ is transitive. Then $a = b \\cup \\set x$."}
+{"_id": "7438", "title": "Ordinal is not Element of Itself", "text": "Let $x$ be an ordinal. Then $x \\notin x$."}
+{"_id": "7439", "title": "Set is Element of Successor", "text": "Let $x$ be a set. Let $x^+$ be the successor of $x$. Then $x \\in x^+$."}
+{"_id": "7440", "title": "Element of Ordinal is Ordinal", "text": "Let $n$ be an ordinal. Let $m \\in n$. Then $m$ is also an ordinal."}
+{"_id": "7441", "title": "Modulus of Exponential of Imaginary Number is One", "text": "Let $\\cmod z$ denote the modulus of a complex number $z$. Let $e^z$ be the complex exponential of $z$. Let $x$ be wholly real. Then: :$\\cmod {e^{i x} } = 1$"}
+{"_id": "7442", "title": "Absolute Value of Power", "text": "Let $x$, $y$ be real numbers. Let $x^y$, $x$ to the power of $y$, be real. Then: :$\\size {x^y} = \\size x^y$"}
+{"_id": "7443", "title": "Count of Rows of Truth Table", "text": "Let $P$ be a WFF of propositional logic. Suppose $\\mathcal P$ is of finite size such that it contains $n$ different letters. Then a truth table constructed to express $P$ will contain $2^n$ rows."}
+{"_id": "7445", "title": "Denumerable Class is Set", "text": "Let $A$ be a class. Let $\\N$ be the natural numbers. Suppose that $F: \\N \\to A$ is a bijection. Then $A$ is a set."}
+{"_id": "7448", "title": "Relative Complement inverts Subsets", "text": "Let $S$ be a set. Let $A \\subseteq S, B \\subseteq S$ be subsets of $S$. Then: :$A \\subseteq B \\iff \\relcomp S B \\subseteq \\relcomp S A$ where $\\complement_S$ denotes the complement relative to $S$."}
+{"_id": "7452", "title": "Union of Subsets is Subset/Family of Sets", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of sets indexed by $I$. Then for all sets $X$: :$\\displaystyle \\paren {\\forall i \\in I: S_i \\subseteq X} \\implies \\bigcup_{i \\mathop \\in I} S_i \\subseteq X$ where $\\displaystyle \\bigcup_{i \\mathop \\in I} S_i$ is the union of $\\family {S_i}$."}
+{"_id": "7453", "title": "Union Distributes over Union/Sets of Sets", "text": "Let $A$ and $B$ denote sets of sets. Then: :$\\displaystyle \\bigcup \\left({A \\cup B}\\right) = \\left({\\bigcup A}\\right) \\cup \\left({\\bigcup B}\\right)$ where $\\displaystyle \\bigcup A$ denotes the union of $A$."}
+{"_id": "7454", "title": "Union Distributes over Union/Families of Sets", "text": "Let $I$ be an indexing set. Let $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$ and $\\family {B_\\alpha}_{\\alpha \\mathop \\in I}$ be indexed families of subsets of a set $S$. Then: :$\\displaystyle \\map {\\bigcup_{\\alpha \\mathop \\in I} } {A_\\alpha \\cup B_\\alpha} = \\paren {\\bigcup_{\\alpha \\mathop \\in I} A_\\alpha} \\cup \\paren {\\bigcup_{\\alpha \\mathop \\in I} B_\\alpha}$ where $\\displaystyle \\bigcup_{\\alpha \\mathop \\in I} A_\\alpha$ denotes the union of $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$."}
+{"_id": "7455", "title": "Intersection Distributes over Intersection/Sets of Sets", "text": "Let $A$ and $B$ denote sets of sets. Then: :$\\displaystyle \\bigcap \\paren {A \\cap B} = \\paren {\\bigcap A} \\cap \\paren {\\bigcap B}$ where $\\displaystyle \\bigcap A$ denotes the intersection of $A$."}
+{"_id": "7456", "title": "Intersection Distributes over Intersection/Families of Sets", "text": "Let $I$ be an indexing set. Let $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$ and $\\family {B_\\alpha}_{\\alpha \\mathop \\in I}$ be indexed families of subsets of a set $S$. Then: :$\\displaystyle \\map {\\bigcap_{\\alpha \\mathop \\in I} } {A_\\alpha \\cap B_\\alpha} = \\paren {\\bigcap_{\\alpha \\mathop \\in I} A_\\alpha} \\cap \\paren {\\bigcap_{\\alpha \\mathop \\in I} B_\\alpha}$ where $\\displaystyle \\bigcap_{\\alpha \\mathop \\in I} A_i$ denotes the intersection of $\\family {A_\\alpha}$."}
+{"_id": "7457", "title": "Intersection Distributes over Intersection/General Result", "text": "Let $\\left\\langle{\\mathbb S_i}\\right\\rangle_{i \\in I}$ be an $I$-indexed family of sets of sets. Then: :$\\displaystyle \\bigcap_{i \\mathop \\in I} \\bigcap \\mathbb S_i = \\bigcap \\bigcap_{i \\mathop \\in I} \\mathbb S_i$"}
+{"_id": "7458", "title": "Intersection Distributes over Union/Family of Sets", "text": "Let $I$ be an indexing set. Let $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$ be a indexed family of subsets of a set $S$. Let $B \\subseteq S$. Then: :$\\displaystyle \\map {\\bigcup_{\\alpha \\mathop \\in I} } {A_\\alpha \\cap B} = \\paren {\\bigcup_{\\alpha \\mathop \\in I} A_\\alpha} \\cap B$ where $\\displaystyle \\bigcup_{\\alpha \\mathop \\in I} A_\\alpha$ denotes the union of $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$."}
+{"_id": "7459", "title": "Union Distributes over Intersection/Family of Sets", "text": "Let $I$ be an indexing set. Let $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$ be an indexed family of subsets of a set $S$. Let $B \\subseteq S$. Then: :$\\displaystyle \\map {\\bigcap_{\\alpha \\mathop \\in I} } {A_\\alpha \\cup B} = \\paren {\\bigcap_{\\alpha \\mathop \\in I} A_\\alpha} \\cup B$ where $\\displaystyle \\bigcap_{\\alpha \\mathop \\in I} A_\\alpha$ denotes the intersection of $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$."}
+{"_id": "7463", "title": "Equivalence of Definitions of Transitive Closure (Relation Theory)/Finite Chain is Smallest", "text": "Let $S$ be a set or class. Let $\\RR$ be a relation on $S$. Let $\\RR^+$ be the transitive closure of $\\RR$ by the finite chain definition. That is, for $x, y \\in S$ let $x \\mathrel {\\RR^+} y$ {{iff}} for some natural number $n > 0$ there exist $s_0, s_1, \\dots, s_n \\in S$ such that $s_0 = x$, $s_n = y$, and: :$\\forall k \\in \\N_n: s_k \\mathrel \\RR s_{k+1}$ Then $\\RR^+$ is transitive and if $\\QQ$ is a transitive relation on $S$ such that $\\RR \\subseteq \\QQ$ then $\\RR \\subseteq \\QQ$."}
+{"_id": "7464", "title": "Order-Preserving Bijection on Wosets is Order Isomorphism", "text": "Let $\\struct {S, \\preceq_1}$ and $\\struct {T, \\preceq_2}$ be well-ordered sets. Let $\\phi: S \\to T$ be a bijection such that $\\phi: S \\to T$ is order-preserving: :$\\forall x, y \\in S: x \\preceq_1 y \\implies \\map \\phi x \\preceq_2 \\map \\phi y$ Then: :$\\forall x, y \\in S: \\map \\phi x \\preceq_2 \\map \\phi y \\implies x \\preceq_1 y$ That is, $\\phi: S \\to T$ is an order isomorphism."}
+{"_id": "7467", "title": "Inverse Image of Set under Set-Like Relation is Set", "text": "Let $A$ be a class. Let $\\RR$ be a set-like endorelation on $A$. Let $B \\subseteq A$ be a set. Then $\\map {\\RR^{-1} } B$, the inverse image of $B$ under $\\RR$, is also a set."}
+{"_id": "7470", "title": "Reciprocal of Holomorphic Function", "text": "Let $f: \\C \\to \\C$ be a complex function. Let $U \\subseteq \\C$ be an open set such that $f$ has no zeros in $U$. Suppose further that $f$ is holomorphic in $U$. Then the complex function :$\\dfrac 1 {f_{\\restriction U} } : U \\to \\C$ is holomorphic."}
+{"_id": "7471", "title": "Transitive Closure of Set-Like Relation is Set-Like", "text": "Let $A$ be a class. Let $\\RR$ be a set-like endorelation on $A$. Let $\\RR^+$ be the transitive closure of $\\RR$. Then $\\RR^+$ is also a set-like relation."}
+{"_id": "7473", "title": "Minimal WRT Restriction", "text": "Let $A$ be a set or class. Let $\\mathcal R$ be a relation on $A$. Let $B$ be a subset or subclass of $A$. Let $\\mathcal R'$ be the restriction of $\\mathcal R$ to $B$. Let $m \\in B$. Then: :$m$ is $\\mathcal R$-minimal in $B$ {{iff}}: :$m$ is $\\mathcal R'$-minimal in $B$."}
+{"_id": "7474", "title": "Intersection of Ordinals is Ordinal", "text": "Let $A$ be a non-empty class of ordinals. Then $\\bigcap A$ is an ordinal."}
+{"_id": "7475", "title": "Meet with Complement is Bottom", "text": "Let $\\struct {S, \\vee, \\wedge, \\neg}$ be a Boolean algebra, defined as in Definition 2. Then: :$\\exists \\bot \\in S: \\forall a \\in S: a \\wedge \\neg a = \\bot$ where $\\wedge$ denotes the meet operation in $S$. This element $\\bot$ is unique for any given $S$, and is named '''bottom'''."}
+{"_id": "7476", "title": "Join with Complement is Top", "text": "Let $\\struct {S, \\vee, \\wedge, \\neg}$ be a Boolean algebra, defined as in Definition 2. Then: :$\\exists \\top \\in S: \\forall a \\in S: a \\vee \\neg a = \\top$ where $\\wedge$ denotes the meet operation in $S$. This element $\\top$ is unique for any given $S$, and is named '''top'''."}
+{"_id": "7479", "title": "Ordering is Equivalent to Subset Relation/Lemma", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then: : $\\forall a_1, a_2 \\in S: \\paren ({a_1 \\preceq a_2 \\implies {a_1}^\\preceq \\subseteq {a_2}^\\preceq}$ where ${a_1}^\\preceq$ denotes the lower closure of $a_1$."}
+{"_id": "7480", "title": "Smallest Element WRT Restricted Ordering", "text": "Let $S$ be a set or class. Let $\\preceq$ be an ordering on $S$. Let $T$ be a subset or subclass of $S$. Let $\\preceq'$ be the restriction of $\\preceq$ to $T$. Let $m \\in T$. Then $m$ is the $\\preceq$-smallest element of $T$ iff $m$ is the $\\preceq'$-smallest element of $T$."}
+{"_id": "7482", "title": "Restriction of Well-Founded Ordering", "text": "Let $S$ be a set or class. Let $T$ be a subset or subclass of $S$. Let $\\preceq$ be a well-founded ordering of $A$. Let $\\preceq'$ be the restriction of $\\preceq$ to $T$. Then $\\preceq'$ is a well-founded ordering of $T$."}
+{"_id": "7483", "title": "Restriction of Well-Ordering is Well-Ordering", "text": "Let $S$ be a set or class. Let $\\preceq$ be a well-ordering of $S$. Let $T$ be a subset or subclass of $S$. Let $\\preceq'$ be the restriction of $\\preceq$ to $T$. Then $\\preceq'$ well-orders $T$."}
+{"_id": "7485", "title": "Ordering is Equivalent to Subset Relation/Proof 2", "text": "{{:Ordering is Equivalent to Subset Relation}} Specifically: Let :$\\mathbb S := \\set {a^\\preceq: a \\in S}$ where $a^\\preceq$ is the lower closure of $a$. That is: :$a^\\preceq := \\set {b \\in S: b \\preceq a}$ Let the mapping $\\phi: S \\to \\mathbb S$ be defined as: :$\\map \\phi a = a^\\preceq$ Then $\\phi$ is an order isomorphism from $\\struct {S, \\preceq}$ to $\\struct {\\mathbb S, \\subseteq}$."}
+{"_id": "7486", "title": "Event Space contains Sample Space", "text": ":$\\Omega \\in \\Sigma$"}
+{"_id": "7487", "title": "Event Space contains Empty Set", "text": ":$\\O \\in \\Sigma$"}
+{"_id": "7489", "title": "Power Set of Sample Space is Event Space", "text": "Let $\\EE$ be an experiment whose sample space is $\\Omega$. Let $\\powerset \\Omega$ be the power set of $\\Omega$. Then $\\powerset \\Omega$ is an event space of $\\EE$."}
+{"_id": "7491", "title": "Event Space from Single Subset of Sample Space", "text": "Let $\\EE$ be an experiment whose sample space is $\\Omega$. Let $\\O \\subsetneqq A \\subsetneqq \\Omega$. Then $\\Sigma := \\set {\\O, A, \\Omega \\setminus A, \\Omega}$ is an event space of $\\EE$."}
+{"_id": "7492", "title": "Intersection of Events is Event", "text": ":$A, B \\in \\Sigma \\implies A \\cap B \\in \\Sigma$"}
+{"_id": "7493", "title": "Set Difference of Events is Event", "text": ":$A, B \\in \\Sigma \\implies A \\setminus B \\in \\Sigma$"}
+{"_id": "7495", "title": "Symmetric Difference of Events is Event", "text": ":$A, B \\in \\Sigma \\implies A \\ast B \\in \\Sigma$"}
+{"_id": "7496", "title": "Characterization of Affine Transformations", "text": "Let $\\mathcal E$ and $\\mathcal F$ be affine spaces over a field $k$. Let $\\mathcal L: \\mathcal E \\to \\mathcal F$ be a mapping. Then $\\mathcal L$ is an affine transformation {{iff}} for all points $p, q \\in \\mathcal E$ and all $\\lambda \\in k$: :$\\mathcal L \\left({\\lambda p + \\left({1 - \\lambda}\\right) q}\\right) = \\lambda \\mathcal L \\left({p}\\right) + \\left({1 - \\lambda}\\right) \\mathcal L \\left({q}\\right)$ where $\\lambda p + \\left({1 - \\lambda}\\right) q$ and $\\lambda \\mathcal L \\left({p}\\right) + \\left({1 - \\lambda}\\right) \\mathcal L \\left({q}\\right)$ denote barycenters."}
+{"_id": "7497", "title": "Probability of Empty Event is Zero", "text": ":$\\map \\Pr \\O = 0$"}
+{"_id": "7499", "title": "Ordinal Membership is Asymmetric", "text": "Let $m$ and $n$ be ordinals. Then it is not the case that $m \\in n$ and $n \\in m$."}
+{"_id": "7500", "title": "Intersection of Ordinals is Smallest", "text": "Let $A$ be a non-empty set or class of ordinals. Let $m = \\bigcap A$ be the intersection of all the elements of $A$. Then $m$ is the smallest element of $A$."}
+{"_id": "7501", "title": "Ordinal Class is Strongly Well-Ordered by Subset", "text": "Let $\\On$ be the class of all ordinals. Then the restriction of the subset relation, $\\subseteq$, to $\\On$ is a strong well-ordering. That is: :$\\subseteq$ is an ordering on $\\On$. :If $A$ is a non-empty subclass of $\\On$, then $A$ has a $\\subseteq$-smallest element."}
+{"_id": "7504", "title": "Subset is Compatible with Ordinal Successor", "text": "Let $x$ and $y$ be ordinals and let $x^+$ denote the successor set of $x$. Let $x \\in y$. Then: :$x^+ \\in y^+$"}
+{"_id": "7505", "title": "Successor is Less than Successor/Sufficient Condition/Proof 1", "text": "Let $x$ and $y$ be ordinals and let $x^+$ denote the successor set of $x$. Let $x^+ \\in y^+$. Then: : $x \\in y$"}
+{"_id": "7506", "title": "Successor is Less than Successor/Sufficient Condition/Proof 2", "text": "Let $x$ and $y$ be ordinals and let $x^+$ denote the successor set of $x$. Let $x^+ \\in y^+$. Then: : $x \\in y$"}
+{"_id": "7507", "title": "Successor is Less than Successor/Sufficient Condition", "text": "== Proof 1 == {{:Successor is Less than Successor/Sufficient Condition/Proof 1}}"}
+{"_id": "7508", "title": "Probability of Event not Occurring", "text": ":$\\forall A \\in \\Sigma: \\map \\Pr {\\Omega \\setminus A} = 1 - \\map \\Pr A$"}
+{"_id": "7509", "title": "Delta-Algebra is Sigma-Algebra", "text": "Every $\\delta$-algebra is a $\\sigma$-algebra."}
+{"_id": "7510", "title": "Countable Intersection of Events is Event", "text": ":$\\quad A_1, A_2, \\ldots \\in \\Sigma \\implies \\ds \\bigcap_{i \\mathop = 1}^\\infty A_i \\in \\Sigma$"}
+{"_id": "7513", "title": "Subset Relation is Ordering/General Result", "text": "Let $\\mathbb S$ be a set of sets or class. Then $\\subseteq$ is an ordering on $\\mathbb S$. In other words, let $\\left({\\mathbb S, \\subseteq}\\right)$ be the relational structure defined on $\\mathbb S$ by the relation $\\subseteq$. Then $\\left({\\mathbb S, \\subseteq}\\right)$ is an ordered set."}
+{"_id": "7514", "title": "Noetherian Domain is Factorization Domain", "text": "Let $R$ be a noetherian integral domain. Then $R$ is a factorization domain."}
+{"_id": "7515", "title": "Probability Measure is Monotone", "text": "Let $A, B \\in \\Sigma$ such that $A \\subseteq B$. Then: :$\\map \\Pr A \\le \\map \\Pr B$"}
+{"_id": "7516", "title": "Transfinite Induction/Principle 1/Proof 2", "text": "Let $\\On$ denote the class of all ordinals. Let $A$ denote a class. Suppose that: :For each element $x$ of $\\On$, if $\\forall y \\in \\On: \\paren {y < x \\implies y \\in A}$ then $x$ is an element of $A$. Then $\\On \\subseteq A$."}
+{"_id": "7517", "title": "Immediate Successor is Unique in Toset", "text": "Let $(S, \\preceq)$ be a totally ordered set. Let $x, y \\in S$. Suppose that $y$ is an immediate successor of $x$. Then $y$ is the ''only'' immediate successor of $x$."}
+{"_id": "7518", "title": "Mapping from Set to Ordinal Class is Bounded Above", "text": "Let $x$ be a set. Let $\\operatorname{On}$ be the class of all ordinals. Let $f: x \\to \\operatorname{On}$ be a mapping. Then $f$ has an upper bound."}
+{"_id": "7519", "title": "Equivalence of Definitions of Independent Events", "text": "Let $\\EE$ be an experiment with probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $A, B \\in \\Sigma$ be events of $\\EE$ such that $\\map \\Pr A > 0$ and $\\map \\Pr B > 0$. {{TFAE|def = Independent Events}}"}
+{"_id": "7520", "title": "Union of Subset of Ordinals is Ordinal/Corollary", "text": "Let $y$ be a set. Let $\\On$ be the class of all ordinals. Let $F: y \\to \\On$ be a mapping. Then: :$\\displaystyle \\bigcup \\map F y \\in \\On$ where $\\map F y$ is the image of $y$ under $F$."}
+{"_id": "7521", "title": "Independent Events are Independent of Complement/General Result", "text": "Let $A_1, A_2, \\ldots, A_m$ be events in a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Then $A_1, A_2, \\ldots, A_m$ are independent {{iff}} $\\Omega \\setminus A_1, \\Omega \\setminus A_2, \\ldots, \\Omega \\setminus A_m$ are also independent."}
+{"_id": "7522", "title": "Probability of Independent Events Not Happening/Corollary", "text": "Let $A$ be an event in an event space of an experiment $\\EE$ whose probability space is $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\map \\Pr A = p$. Suppose that the nature of $\\EE$ is that its outcome is independent of previous trials of $\\EE$. Then the probability that $A$ does not occur during the course of $m$ trials of $\\EE$ is $\\paren {1 - p}^m$."}
+{"_id": "7523", "title": "Probability of Limit of Sequence of Events/Increasing", "text": "Let $\\left \\langle{A_n}\\right \\rangle_{n \\mathop \\in \\N}$ be an increasing sequence of events. Let $\\displaystyle A = \\bigcup_{i \\mathop \\in \\N} A_i$ be the limit of $\\left \\langle{A_n}\\right \\rangle_{n \\mathop \\in \\N}$. Then: :$\\displaystyle \\Pr \\left({A}\\right) = \\lim_{n \\to \\infty} \\Pr \\left({A_n}\\right)$"}
+{"_id": "7524", "title": "Probability of Limit of Sequence of Events/Decreasing", "text": "Let $\\left \\langle{B_n}\\right \\rangle_{n \\mathop \\in \\N}$ be a decreasing sequence of events. Let $\\displaystyle B = \\bigcap_{i \\mathop \\in \\N} B_i$ be the limit of $\\left \\langle{B_n}\\right \\rangle_{n \\mathop \\in \\N}$. Then: :$\\displaystyle \\Pr \\left({B}\\right) = \\lim_{n \\to \\infty} \\Pr \\left({B_n}\\right)$"}
+{"_id": "7525", "title": "Sum of Discrete Random Variables", "text": "Let $U: \\Omega \\to \\R$ be defined as: :$\\forall \\omega \\in \\Omega: \\map U \\omega = \\map X \\omega + \\map Y \\omega$ Then $U$ is also a discrete random variable on $\\struct {\\Omega, \\Sigma, \\Pr}$."}
+{"_id": "7526", "title": "Product of Discrete Random Variables", "text": "Let $V: \\Omega \\to \\R$ be defined as: :$\\forall \\omega \\in \\Omega: V \\left({\\omega}\\right) = X \\left({\\omega}\\right) Y \\left({\\omega}\\right)$ Then $V$ is also a discrete random variable on $\\left({\\Omega, \\Sigma, \\Pr}\\right)$."}
+{"_id": "7527", "title": "Bernoulli Process as Geometric Distribution/Shifted", "text": "Let $\\left \\langle{Y_i}\\right \\rangle$ be a Bernoulli process with parameter $p$. Let $\\mathcal E$ be the experiment which consists of performing the Bernoulli trial $Y_i$ as many times as it takes to achieve a success, and then stop. Let $k$ be the number of Bernoulli trials to achieve a success. Then $k$ is modelled by a shifted geometric distribution with parameter $p$."}
+{"_id": "7528", "title": "Geometric Distribution Gives Rise to Probability Mass Function/Shifted", "text": "Let $Y$ be a discrete random variable on a probability space $\\left({\\Omega, \\Sigma, \\Pr}\\right)$. Let $Y$ have the shifted geometric distribution with parameter $p$ (where $0 < p < 1$). Then $Y$ gives rise to a probability mass function."}
+{"_id": "7529", "title": "Negative Binomial Distribution Gives Rise to Probability Mass Function", "text": "Let $X$ be a discrete random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$."}
+{"_id": "7530", "title": "Negative Binomial Distribution Gives Rise to Probability Mass Function/First Form", "text": "Let $X$ have the negative binomial distribution (first form) with parameters $n$ and $p$ ($0 < p < 1$). Then $X$ gives rise to a probability mass function."}
+{"_id": "7532", "title": "Bernoulli Process as Negative Binomial Distribution/First Form", "text": "Let $\\EE$ be the experiment which consists of performing the Bernoulli trial $X_i$ until a total of $n$ failures have been encountered. Let $X$ be the discrete random variable defining the number of successes before $n$ failures have been encountered. Then $X$ is modeled by a negative binomial distribution of the first form."}
+{"_id": "7533", "title": "Bernoulli Process as Negative Binomial Distribution/Second Form", "text": "Let $\\EE$ be the experiment which consists of performing the Bernoulli trial $X_i$ as many times as it takes to achieve a total of $n$ successes, and then stops. Let $Y$ be the discrete random variable defining the number of trials before $n$ successes have been achieved. Then $X$ is modeled by a negative binomial distribution of the second form."}
+{"_id": "7534", "title": "Negative Binomial Distribution as Generalized Geometric Distribution/Second Form", "text": "The second form of the negative binomial distribution is a generalization of the shifted geometric distribution: Let $\\sequence {Y_i}$ be a Bernoulli process with parameter $p$. Let $\\FF$ be the experiment which consists of: :Perform the Bernoulli trial $Y_i$ as many times as it takes to achieve $n$ successes, and then stop. Let $k$ be the number of Bernoulli trials that need to be taken in order to achieve up to (and including) the $n$th success. Let $\\FF'$ be the experiment which consists of: :Perform the Bernoulli trial $Y_i$ until '''one''' success is achieved, and then stop. Then $k$ is modelled by the experiment: :Perform experiment $\\FF'$ until $n$ failures occur, and then stop."}
+{"_id": "7536", "title": "Derivative of Geometric Sequence/Corollary", "text": ":$\\displaystyle \\sum_{n \\mathop \\ge 1} n \\paren {n + 1} x^{n - 1} = \\frac 2 {\\paren {1 - x}^3}$"}
+{"_id": "7539", "title": "Equivalence of Definitions of Countable Set", "text": "Let $S$ be a set. {{TFAE|def = Countable Set}}"}
+{"_id": "7540", "title": "Infinite Set of Natural Numbers is Countably Infinite", "text": "Let $\\N$ be the set of natural numbers. Let $S$ be an infinite subset of $\\N$. Then $S$ is countably infinite. That is, there is a bijection $f: \\N \\to S$."}
+{"_id": "7542", "title": "Generating Function for Natural Numbers", "text": "Let $\\sequence {a_n}$ be the sequence defined as: : $\\forall n \\in \\N_{> 0}: a_n = n - 1$ That is: :$\\sequence {a_n} = 0, 1, 2, 3, 4, \\ldots$ Then the generating function for $\\sequence {a_n}$ is given as: :$G \\paren z = \\dfrac 1 {\\paren {1 - z}^2}$"}
+{"_id": "7543", "title": "Injection has Surjective Left Inverse Mapping", "text": "Let $S$ and $T$ be sets such that $S \\ne \\O$. Let $f: S \\to T$ be a injection. Then there exists a surjection $g: T \\to S$ such that: :$g \\circ f = I_S$"}
+{"_id": "7546", "title": "Many-to-One Relation Extends to Mapping", "text": "Let $S$ and $T$ be sets. Let $T$ be non-empty. Let $\\mathcal R \\subset S \\times T$ be a many-to-one relation. Then there exists a mapping $f: S \\to T$ such that $\\mathcal R \\subseteq f$."}
+{"_id": "7551", "title": "Probability Generating Function as Expectation", "text": "Let $X$ be a discrete random variable whose codomain, $\\Omega_X$, is a subset of the natural numbers $\\N$. Let $p_X$ be the probability mass function for $X$. Let $\\Pi_X \\left({s}\\right)$ be the probability generating function for $X$. Then: :$\\Pi_X \\left({s}\\right) = E \\left({s^X}\\right)$ where $E \\left({s^X}\\right)$ denotes the expectation of $s^X$."}
+{"_id": "7553", "title": "Probability Generating Function of Negative Binomial Distribution/First Form", "text": "Let $X$ be a discrete random variable with the negative binomial distribution (first form) with parameters $n$ and $p$. Then the p.g.f. of $X$ is: :$\\Pi_X \\left({s}\\right) = \\left({\\dfrac q {1 - ps}}\\right)^n$ where $q = 1 - p$."}
+{"_id": "7554", "title": "Probability Generating Function of Negative Binomial Distribution/Second Form", "text": "Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$. Then the p.g.f. of $X$ is: :$\\displaystyle \\Pi_X \\left({s}\\right) = \\left({\\frac {ps} {1 - qs}}\\right)^n$ where $q = 1 - p$."}
+{"_id": "7556", "title": "Probability Generating Function of Scalar Multiple of Random Variable", "text": "Let $X$ be a discrete random variable whose probability generating function is $\\Pi_X \\left({s}\\right)$. Let $k \\in \\Z_{\\ge 0}$ be a positive integer. Let $Y$ be a discrete random variable such that $Y = m X$. Then :$\\Pi_Y \\left({s}\\right) = \\Pi_X \\left({s^m}\\right)$. where $\\Pi_Y \\left({s}\\right)$ is the probability generating function of $Y$."}
+{"_id": "7558", "title": "Derivatives of Probability Generating Function at One", "text": "Let $X$ be a discrete random variable whose probability generating function is $\\Pi_X \\left({s}\\right)$. Then the $n$th derivative of $\\Pi_X \\left({s}\\right)$ at $s = 1$ is given by: :$\\dfrac {\\mathrm d^n} {\\mathrm d s^n} \\Pi_X \\left({1}\\right) = E \\left({X \\left({X - 1}\\right) \\cdots \\left({X - n + 1}\\right)}\\right)$ for $n = 1, 2, \\ldots$"}
+{"_id": "7559", "title": "Differentiation of Power Series/Corollary", "text": "The value of $\\displaystyle \\frac {\\mathrm d^n}{\\mathrm d x^n} \\sum_{m \\mathop \\ge 0} a_m \\left({x - \\xi}\\right)^m$ at $x = \\xi$ is: :$\\displaystyle \\left.{\\frac {\\mathrm d^n}{\\mathrm d x^n} \\sum_{m \\mathop \\ge 0} a_m \\left({x - \\xi}\\right)^m}\\right|_{x = \\xi} = a_n n!$"}
+{"_id": "7562", "title": "Non-Equivalence of Proposition and Negation/Formulation 1", "text": ":$p \\implies \\neg p, \\neg p \\implies p \\vdash \\bot$"}
+{"_id": "7567", "title": "Non-Equivalence of Proposition and Negation/Formulation 2", "text": ":$\\vdash \\neg \\left({p \\iff \\neg p}\\right)$"}
+{"_id": "7568", "title": "No Injection from Power Set to Set/Lemma", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Then there does not exist a set $B$ such that there is an injection from $B$ into $S$ and a surjection from $B$ onto $\\powerset S$."}
+{"_id": "7569", "title": "No Injection from Power Set to Set", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Then there is no injection from $\\powerset S$ into $S$."}
+{"_id": "7571", "title": "Inverse Image Mapping of Injection is Surjection", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a injection. Let $f^\\gets: \\powerset T \\to \\powerset S$ be the inverse image mapping of $f$. Then $f^\\gets$ is a surjection."}
+{"_id": "7573", "title": "Expectation of Negative Binomial Distribution/Second Form", "text": "Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$. Then the expectation of $X$ is given by: :$E \\left({X}\\right) = \\dfrac n p$"}
+{"_id": "7574", "title": "Derivatives of PGF of Negative Binomial Distribution/Second Form", "text": "Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$. Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are: :$\\dfrac {\\mathrm d^k} {\\mathrm d s^k} \\Pi_X \\left({s}\\right) = ...$"}
+{"_id": "7575", "title": "First Derivative of PGF of Negative Binomial Distribution/Second Form", "text": "Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$. Then the first derivative of the PGF of $X$ {{WRT|Differentiation}} $s$ is: :$\\dfrac \\d {\\d s} \\map {\\Pi_X} s = n p \\paren {\\dfrac {\\paren {p s}^{n - 1} } {\\paren {1 - q s}^{n + 1} } }$"}
+{"_id": "7578", "title": "Nth Derivative of Reciprocal of Mth Power/Corollary", "text": "The $n$th derivative of $\\dfrac 1 x$ {{WRT|Differentiation}} $x$ is: :$\\dfrac {\\d^n} {\\d x^n} \\dfrac 1 x = \\dfrac {\\paren {-1}^n n!} {x^{n + 1} }$ where $n!$ denotes $n$ factorial."}
+{"_id": "7581", "title": "First Derivative of PGF of Negative Binomial Distribution/First Form", "text": "Let $X$ be a discrete random variable with the negative binomial distribution (first form) with parameters $n$ and $p$. Then the first derivative of the PGF of $X$ w.r.t. $s$ is: :$\\dfrac \\d {\\d s} \\map {\\Pi_X} s = \\dfrac {n p} q \\paren {\\dfrac q {1 - p s} }^{n + 1}$"}
+{"_id": "7583", "title": "Expectation of Negative Binomial Distribution/First Form", "text": "Let $X$ be a discrete random variable with the negative binomial distribution (first form) with parameters $n$ and $p$. Then the expectation of $X$ is given by: :$E \\left({X}\\right) = \\dfrac {n p} q$"}
+{"_id": "7584", "title": "Closure Condition for Hausdorff Space", "text": "Let $\\struct {X, \\tau}$ be a topological space. Then $\\struct {X, \\tau}$ is a Hausdorff space {{iff}}: :For all $x, y \\in X$ such that $x \\ne y$, there exists an open set $U$ such that $x \\in U$ and $y \\notin U^-$, where $U^-$ is the closure of $U$."}
+{"_id": "7586", "title": "Second Derivative of PGF of Negative Binomial Distribution/Second Form", "text": "Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$. Then the second derivative of the PGF of $X$ w.r.t. $s$ is: :$\\dfrac {\\mathrm d^2} {\\mathrm d s^2} \\Pi_X \\left({s}\\right) = \\left({\\dfrac {ps} {1 - qs} }\\right)^{n+2} \\left({\\dfrac {n \\left({n-1}\\right) + 2 n q s} {\\left({p s^2}\\right)^2} }\\right)$"}
+{"_id": "7588", "title": "Derivatives of PGF of Negative Binomial Distribution/First Form", "text": "Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$. Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are: :$\\dfrac {\\mathrm d^k} {\\mathrm d s^k} \\Pi_X \\left({s}\\right) = \\dfrac {n^{\\overline k} p^k} {q^k} \\left({\\dfrac q {1 - p s} }\\right)^{n + k}$ where: : $n^{\\overline k}$ is the rising factorial: $n^{\\overline k} = n \\left({n + 1}\\right) \\left({n + 2}\\right) \\cdots \\left({n + k - 1}\\right)$ : $q = 1 - p$"}
+{"_id": "7590", "title": "Variance of Negative Binomial Distribution/Second Form", "text": "Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$. Then the variance of $X$ is given by: :$\\displaystyle \\operatorname{var} \\left({X}\\right) = \\frac {n q} {p^2}$"}
+{"_id": "7591", "title": "PGF of Sum of Independent Discrete Random Variables/General Result", "text": "Let: :$Z = X_1 + X_2 + \\cdots + X_n$ where each of $X_1, X_2, \\ldots, X_n$ are independent discrete random variables with PGFs $\\map {\\Pi_{X_1} } s, \\map {\\Pi_{X_2} } s, \\ldots, \\map {\\Pi_{X_n} } s$. Then: :$\\displaystyle \\map {\\Pi_Z} s = \\prod_{j \\mathop = 1}^n \\map {\\Pi_{X_j} } s$"}
+{"_id": "7592", "title": "Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods/Lemma", "text": "Let $\\left({S, \\tau}\\right)$ be a Hausdorff space. Let $C$ be a compact subspace of $S$. Let $x \\in S \\setminus C$. Then there exist open sets $U$ and $V$ such that $x \\in U$, $C \\subseteq V$, and $U \\cap V = \\varnothing$."}
+{"_id": "7593", "title": "Sum of Independent Binomial Random Variables", "text": "Let $X$ and $Y$ be discrete random variables with a binomial distribution: :$X \\sim \\operatorname{B} \\left({m, p}\\right)$ and :$Y \\sim \\operatorname{B} \\left({n, p}\\right)$ Let $X$ and $Y$ be independent. Then their sum $Z = X + Y$ is distributed as: :$Z \\sim \\operatorname{B} \\left({\\left({m + n}\\right), p}\\right)$"}
+{"_id": "7594", "title": "Field has Algebraic Closure", "text": "Every field has an algebraic closure."}
+{"_id": "7595", "title": "Point Finite Set of Open Sets in Separable Space is Countable", "text": "Let $\\struct {X, \\tau}$ be a separable space. Let $\\mathcal F$ be a point finite set of open sets of $X$. Then $\\mathcal F$ is countable."}
+{"_id": "7596", "title": "Countable Union of Finite Sets is Countable", "text": "The following statements are equivalent in $\\mathrm{ZF}^-$: {{explain|$\\mathrm{ZF}^-$}} :The Axiom of Countable Choice for Finite Sets holds. :The union of any countable set of finite sets is countable."}
+{"_id": "7597", "title": "Reverse Triangle Inequality/Normed Vector Space", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,} }$ be a normed vector space. Then: :$\\forall x, y \\in X: \\norm {x - y} \\ge \\size {\\norm x - \\norm y}$"}
+{"_id": "7601", "title": "Equivalence of Forms of Axiom of Countable Choice", "text": "The following forms of the Axiom of Countable Choice are equivalent in $\\mathrm{ZF}^-$: {{explain|$\\mathrm{ZF}^-$}}"}
+{"_id": "7602", "title": "GCD from Generator of Ideal", "text": "Let $m, n \\in \\Z$, with either $m \\ne 0$ or $n \\ne 0$. Let $I = \\ideal {m, n}$ be the ideal generated by $m$ and $n$. Let $d$ be a non-negative generator for the principal ideal $I$. Then: :$\\gcd \\set {m, n} =  d$ where $\\gcd \\set {m, n}$ denotes the greatest common divisor of $m$ and $n$."}
+{"_id": "7603", "title": "Finite Union of Finite Sets is Finite", "text": "Let $S$ be a finite set of finite sets. Then the union of $S$ is finite."}
+{"_id": "7604", "title": "Open Set Characterization of Denseness", "text": "Let $\\left({X, \\tau}\\right)$ be a topological space. Let $S \\subseteq X$. Then $S$ is (everywhere) dense in $X$ {{iff}} every non-empty ($\\tau$-)open set contains an element of $S$."}
+{"_id": "7605", "title": "Union of Set of Sets when a Set Intersects All", "text": "Let $F$ be a set of sets. Let $S$ be a set or class. Suppose that: :$\\forall A \\in F: A \\cap S \\ne \\O$ Then: :$\\ds F = \\bigcup_{x \\mathop \\in S} \\set {A \\in F: x \\in A}$"}
+{"_id": "7607", "title": "Order of Group of Units of Integers Modulo m", "text": "Let $n \\in \\Z_{\\ge 0}$ be an integer. Let $\\struct {\\Z / n \\Z, +, \\cdot}$ be the ring of integers modulo $n$. Let $U = \\struct {\\paren {\\Z / n \\Z}^\\times, \\cdot}$ denote the group of units of this ring. Then: :$\\order U = \\map \\phi n$ where $\\phi$ denotes the Euler $\\phi$-function."}
+{"_id": "7608", "title": "Cyclicity Condition for Units of Ring of Integers Modulo m", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Let $\\struct {\\Z / n \\Z, +, \\times}$ be the ring of integers modulo $n$. Let $U = \\struct {\\paren {\\Z / n \\Z}^\\times, \\times}$ denote the group of units of $\\struct {\\Z / n \\Z, +, \\times}$. Then $U$ is cyclic {{iff}} either: :$n = p^\\alpha$ or: :$n = 2 p^\\alpha$ where $p \\ge 3$ is prime and $\\alpha \\ge 0$."}
+{"_id": "7609", "title": "Chinese Remainder Theorem/Corollary", "text": "Let $n_1, n_2, \\ldots, n_r$ be positive integers such that $n_i \\perp n_j$ for all $i \\ne j$ (that is, $\\gcd \\left\\{{n_i, n_j}\\right\\} = 1$). Let $N = n_1 \\cdots n_r$. For an integer $k$, let $\\Z / k \\Z$ denote the ring of integers modulo $k$. Then we have a ring isomorphism: :$\\Z / N \\Z \\simeq \\Z / n_1 \\Z \\times \\cdots \\times \\Z / n_r \\Z$"}
+{"_id": "7611", "title": "Minkowski's Theorem", "text": "Let $L$ be a lattice in $\\R^n$. Let $d$ be the covolume of $L$. Let $\\mu$ be a translation invariant measure on $\\R^n$ Let $S$ be a convex subset of $\\R^n$ that is symmetric about the origin, i.e. such that: :$\\forall p \\in S : -p \\in S$  Let the volume of $S$ be greater than $2^n d$. Then $S$ contains a non-zero point of $L$."}
+{"_id": "7612", "title": "Component of Finite Union in Ultrafilter", "text": "Let $S$ be a non-empty set. Let $\\mathcal U$ be a ultrafilter on $S$. Let $\\left\\{{Y_1, \\dots, Y_n}\\right\\}$ be a pairwise disjoint set of subsets of $S$ such that $S = Y_1 \\cup \\cdots \\cup Y_n$. Then there is a unique $k \\in \\left\\{{1, \\dots, n}\\right\\}$ such that $Y_k \\in \\mathcal U$."}
+{"_id": "7613", "title": "Restricted Tukey-Teichmüller Theorem/Weak Form", "text": "Let $X$ be a set. Let $\\mathcal A$ be a non-empty set of subsets of $X$. Let $'$ be a unary operation on $X$. Let $\\mathcal A$ have finite character. For all $A \\in \\mathcal A$ and all $x \\in X$, let either: :$A \\cup \\set x \\in \\mathcal A$ or: :$A \\cup \\set {x'} \\in \\mathcal A$ Then there exists a $B \\in \\mathcal A$ such that for all $x \\in X$, either $x \\in B$ or $x' \\in B$."}
+{"_id": "7614", "title": "Restricted Tukey-Teichmüller Theorem/Strong Form", "text": "Let $X$ be a set. Let $\\mathcal A$ be a non-empty set of subsets of $X$. Let $'$ be a unary operation on $X$. Let $\\mathcal A$ have finite character. For all $A \\in \\mathcal A$ and all $x \\in X$, let either: : $A \\cup \\left\\{ {x}\\right\\} \\in \\mathcal A$ or: :$A \\cup \\left\\{ {x'}\\right\\} \\in \\mathcal A$ Then for each $A \\in \\mathcal A$ there exists a $C \\in \\mathcal A$ such that: :$A \\subseteq C$ and: :for all $x \\in X$, either $x \\in C$ or $x' \\in C$."}
+{"_id": "7616", "title": "Boolean Prime Ideal Theorem/Extension Lemma", "text": "Let $\\struct {B, \\vee, \\wedge, \\neg, \\bot, \\top}$ be a Boolean algebra. Let $J \\subseteq B$ have the finite join property. Let $z \\in B$. Then either $J \\vee z$ or $J \\vee \\neg z$ also has the finite join property."}
+{"_id": "7617", "title": "Finite Character for Sets of Mappings", "text": "Let $S$ and $T$ be sets. Let $\\mathcal F$ be a set of mappings from subsets of $S$ to $T$. That is, let $\\mathcal F$ be a set of partial mappings from $S$ to $T$. Then the following are equivalent: {{begin-axiom}} {{axiom | n = 1         | t = $\\mathcal F$ has finite character in the sense of Definition:Finite Character/Mappings. }} {{axiom | n = 2         | t = $\\mathcal F$ has finite character as a set of subsets of $S \\times T$ in the sense of Definition:Finite Character. }} {{end-axiom}}"}
+{"_id": "7618", "title": "Cowen-Engeler Lemma", "text": "Let $X$ and $Y$ be sets. Let $M$ be a set of mappings from subsets of $X$ to $Y$. Thus each element of $M$ is a partial mapping from $X$ to $Y$. Define a mapping $\\Phi: X \\to \\powerset Y$ thus: :$\\map \\Phi x = \\set {\\map f x: f \\in M \\text{ and } x \\in \\Dom f}$ That is: :$\\map \\Phi x = \\set {y \\in Y: \\exists f \\in M: \\tuple {x, y} \\in f}$ Suppose that the following hold: {{begin-axiom}} {{axiom | n = 1         | t = $\\map \\Phi x$ is finite for each $x \\in X$. }} {{axiom | n = 2         | t = For each finite subset $F$ of $X$, there exists an element $f \\in M$ such that the domain of $f$ is $F$. }} {{axiom | n = 3         | t = $M$ has finite character. }} {{axiom | t = That is, a mapping $f$ from some subset of $X$ to $Y$ is an element of $M$ {{iff}} its restriction to each finite subset of $\\Dom f$ is in $M$. }} {{end-axiom}} Then there exists an element of $M$ with domain $X$."}
+{"_id": "7619", "title": "Ultrafilter Lemma/Corollary", "text": "Let $S$ be a non-empty set. Let $\\mathcal A$ be a set of subsets of $S$. Suppose that $\\mathcal A$ has the finite intersection property. Then there is an ultrafilter $\\mathcal U$ on $S$ such that $\\mathcal A \\subseteq \\mathcal U$."}
+{"_id": "7621", "title": "Order-Extension Principle/Strict", "text": "Let $S$ be a set. Let $\\prec$ be a strict ordering on $S$. Then there exists a strict total ordering $<$ on $S$ such that: :$\\forall a, b \\in S: a \\prec b \\implies a < b$"}
+{"_id": "7622", "title": "Order-Extension Principle/Strict/Finite Set", "text": "Let $T$ be a finite set. Let $\\prec$ be a strict ordering on $T$. Then there exists a strict total ordering $<$ on $T$ such that: :$\\forall a, b \\in T: \\left({a \\prec b \\implies a < b}\\right)$"}
+{"_id": "7623", "title": "Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension/Lemma", "text": "Let $T$ be a finitely satisfiable $\\mathcal L$-theory. Let $\\phi$ be an $\\mathcal L$-sentence. Then either $T \\cup \\left\\{ {\\phi}\\right\\}$ or $T \\cup \\left\\{ {\\neg \\phi}\\right\\}$ is finitely satisfiable."}
+{"_id": "7624", "title": "Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension/Proof 2", "text": "Let $T$ be a finitely satisfiable $\\mathcal L$-theory. There is a finitely satisfiable $\\mathcal L$-theory $T'$ which contains $T$ as a subset such that for all $\\mathcal L$-sentences $\\phi$, either $\\phi \\in T'$ or $\\neg\\phi \\in T'$. {{explain|Does this actually mean the same as the statement of the theorem in Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension? If so, replace it. If not, then it is a different proof altogether. If the two statements are equivalent, then this needs to be demonstrated.}}"}
+{"_id": "7626", "title": "Reflexive Closure of Strict Total Ordering is Total Ordering", "text": "Let $S$ be a set. Let $\\prec$ be a strict total ordering of $S$. Let $\\preceq$ be the reflexive closure of $\\prec$. Then $\\preceq$ is an total ordering of $S$."}
+{"_id": "7634", "title": "Node of Rooted Tree is on Branch", "text": "Let $T$ be a rooted tree with root node $r_T$. Let $t$ be a node of $T$. Then there exists a branch $\\Gamma$ of $T$ such that $t \\in \\Gamma$."}
+{"_id": "7635", "title": "Leaf of Rooted Tree is on One Branch", "text": "Let $T$ be a rooted tree with root node $r_T$. Let $t$ be a leaf node of $T$. Then there exists a unique branch $\\Gamma$ of $T$ such that $t \\in \\Gamma$."}
+{"_id": "7636", "title": "Node of Rooted Tree with Multiple Children is on Multiple Branches", "text": "Let $T$ be a rooted tree with root node $r_T$. Let $t$ be a node of $T$. Let $t$ have more than one child nodes. Then $t$ is on more than one branch of $T$."}
+{"_id": "7637", "title": "Branch of Finite Tree is Finite", "text": "Let $T$ be a finite rooted tree with root node $r_T$. Let $\\Gamma$ be a branch of $T$. Then $\\Gamma$ is a finite branch."}
+{"_id": "7638", "title": "Same Dimensional Vector Spaces are Isomorphic", "text": "Let $K$ be a division ring. Let $V$, $W$ be finite dimensional $K$-vector spaces. Suppose that $\\dim_K V = \\dim_K W$. Then: :$V \\cong W$ That is, $V$ and $W$ are isomorphic."}
+{"_id": "7639", "title": "Five Lemma", "text": "Let $A$ be a commutative ring with unity. Let: ::$\\begin{xy}\\xymatrix@L+2mu@+1em{  M_1 \\ar[r]^*{\\alpha_1}      \\ar[d]^*{\\phi_1} &  M_2 \\ar[r]^*{\\alpha_2}      \\ar[d]^*{\\phi_2} &  M_3 \\ar[r]^*{\\alpha_3}      \\ar[d]^*{\\phi_3} &  M_4 \\ar[r]^*{\\alpha_4}      \\ar[d]^*{\\phi_4} &  M_5 \\ar[d]^*{\\phi_5} \\\\  N_1 \\ar[r]_*{\\beta_1} &  N_2 \\ar[r]_*{\\beta_2} &  N_3 \\ar[r]_*{\\beta_3} &  N_4 \\ar[r]_*{\\beta_4} &  N_5 }\\end{xy}$ be a commutative diagram of $A$-modules. Suppose that the rows are exact. Then: :If $\\phi_2$ and $\\phi_4$ are surjective and $\\phi_5$ is injective then $\\phi_3$ is surjective. :If $\\phi_2$ and $\\phi_4$ are injective and $\\phi_1$ is surjective then $\\phi_3$ is injective."}
+{"_id": "7640", "title": "Snake Lemma", "text": "Let $A$ be a commutative ring with unity. Let: ::$\\begin{xy}\\xymatrix@L+2mu@+1em{ &  M_1 \\ar[r]_*{\\alpha_1}      \\ar[d]^*{\\phi_1} &  M_2 \\ar[r]_*{\\alpha_2}      \\ar[d]^*{\\phi_2} &  M_3 \\ar[d]^*{\\phi_3}      \\ar[r] &   0 \\\\  0 \\ar[r] &  N_1 \\ar[r]_*{\\beta_1} &  N_2 \\ar[r]_*{\\beta_2} &  N_3 & }\\end{xy}$ be a commutative diagram of $A$-modules. Suppose that the rows are exact. Then we have a commutative diagram: ::$\\begin{xy}\\xymatrix@L+2mu@+1em{ &  \\ker \\phi_1 \\ar[r]_*{\\tilde\\alpha_1}      \\ar[d]^*{\\iota_1} &  \\ker \\phi_2 \\ar[r]_*{\\tilde\\alpha_2}      \\ar[d]^*{\\iota_2} &  \\ker \\phi_3      \\ar[d]^*{\\iota_3} & \\\\ &  M_1 \\ar[r]_*{\\alpha_1}      \\ar[d]^*{\\phi_1} &  M_2 \\ar[r]_*{\\alpha_2}      \\ar[d]^*{\\phi_2} &  M_3 \\ar[d]^*{\\phi_3}      \\ar[r] &   0 \\\\  0 \\ar[r] &  N_1 \\ar[r]_*{\\beta_1}      \\ar[d]^*{\\pi_1} &  N_2 \\ar[r]_*{\\beta_2}      \\ar[d]^*{\\pi_2} &  N_3 \\ar[d]^*{\\pi_3} \\\\ &  \\operatorname{coker} \\phi_1 \\ar[r]_*{\\bar\\beta_1} &  \\operatorname{coker} \\phi_2 \\ar[r]_*{\\bar\\beta_2} &  \\operatorname{coker} \\phi_3 & }\\end{xy}$ where: *For $i=1,2,3$, $\\iota_i$ is the inclusion mapping *For $i=1,2,3$, $\\pi_i$ is the canonical epimorphism *For $i = 1,2$, $\\tilde\\alpha_i = \\left({ \\alpha_i }\\right)_{| \\ker \\phi_i}$ *For $i = 1,2$, $\\bar\\beta_i$ is defined by: ::$\\forall\\; n_i + \\operatorname{im}\\phi_i \\in \\operatorname{coker}\\phi_i : \\bar\\beta_i\\left({ n_i + \\operatorname{im}\\phi_i }\\right) = \\beta_i\\left({ n_i }\\right) + \\operatorname{im} \\phi_{i+1}$ Moreover there exists a morphism $\\delta : \\ker\\phi_3 \\to \\operatorname{coker}\\phi_1$ such that we have an exact sequence: ::$\\begin{xy}\\xymatrix@L+2mu@+1em{  \\ker \\phi_1 \\ar[r]_*{\\tilde\\alpha_1} &  \\ker \\phi_2 \\ar[r]_*{\\tilde\\alpha_2} &  \\ker\\phi_3 \\ar[r]_*{\\delta} &   \\operatorname{coker}\\phi_1 \\ar[r]_*{\\bar\\beta_1} &  \\operatorname{coker}\\phi_2 \\ar[r]_*{\\bar\\beta_2} &  \\operatorname{coker}\\phi_3 }\\end{xy}$"}
+{"_id": "7641", "title": "Hall's Marriage Theorem/Finite Set", "text": "Let $\\SS = \\sequence {S_k}_{k \\in I}$ be a finite indexed family of finite sets. For each $F \\subseteq I$, let $\\displaystyle Y_F = \\bigcup_{k \\mathop \\in F} S_k$. Let $Y = Y_I$. Then the following are equivalent: :$(1): \\quad \\SS$ satisfies the '''marriage condition''': for each subset $F$ of $I$, $\\card F \\le \\card {Y_F}$. :$(2): \\quad$ There exists an injection $f: I \\to Y$ such that $\\forall k \\in I: \\map f k \\in S_k$."}
+{"_id": "7642", "title": "Connecting Homomorphism is Functorial", "text": "Let $A$ be a commutative ring with unity. Let: ::$\\begin{xy}\\xymatrix{ &&& M_1 \\ar@{->}[rr]      \\ar@{->}[dl]      \\ar@{->}[dd]|!{[d];[d]}\\hole && M_2 %      \\ar@{->}[rr]      \\ar@{->}[dl]      \\ar@{->}[dd]|!{[d];[d]}\\hole && M_3 \\ar@{->}[dl]       \\ar@{->}[dd]|!{[d];[d]}\\hole       \\ar@{->}[rr] && 0 \\\\ && M_1' \\ar@{->}[rr]    \\ar@{->}[dd] && M_2' \\ar@{->}[rr]      \\ar@{->}[dd] && M_3' \\ar@{->}[dd]        \\ar@{->}[rr] && 0 \\\\ & 0 \\ar@{->}[rr]|!{[r];[r]}\\hole && N_1 \\ar@{->}[rr]|!{[r];[r]}\\hole      \\ar@{->}[dl]_{F’} && N_2 \\ar@{->}[rr]|!{[r];[r]}\\hole      \\ar@{->}[dl] && N_3 \\ar@{->}[dl]_{F’’} \\\\ % 0 \\ar@{->}[rr] && N_1' \\ar@{->}^(.65){e’}[rr] && N_2' \\ar@{->}[rr] && N_3' }\\end{xy}$ be a commutative diagram of $A$-modules. Suppose that the rows are exact. {{finish}}"}
+{"_id": "7645", "title": "Law of Cosines/Proof 3", "text": "=== Lemma: Right Triangle === {{:Law of Cosines/Right Triangle}} === Acute Triangle === {{:Law of Cosines/Proof 3/Acute Triangle}} === Obtuse Triangle === {{:Law of Cosines/Proof 3/Obtuse Triangle}}"}
+{"_id": "7646", "title": "Hall's Marriage Theorem/General Set", "text": "Let $\\SS = \\sequence {S_k}_{k \\in I}$ be an indexed family of finite sets. For each $F \\subseteq I$, let $\\displaystyle Y_F = \\bigcup_{k \\mathop \\in F} S_k$. Let $Y = Y_I$. Then the following are equivalent: :$(1): \\quad \\SS$ satisfies the '''marriage condition''': for each finite subset $F$ of $I$, $\\card F \\le \\card {Y_F}$. :$(2): \\quad$ There exists an injection $f: I \\to Y$ such that $\\forall k \\in I: \\map f k \\in S_k$."}
+{"_id": "7648", "title": "Branch of Finite Propositional Tableau is Finite", "text": "Let $T$ be a finite propositional tableau. Let $\\Gamma$ be a branch of $T$. Then $\\Gamma$ is a finite branch."}
+{"_id": "7649", "title": "Tableau Confutation is Finished", "text": "Let $T$ be a tableau confutation. Then $T$ is a finished tableau."}
+{"_id": "7650", "title": "Expression of Vector as Linear Combination from Basis is Unique/General Result", "text": "Let $V$ be a vector space over a division ring $R$. Let $B$ be a basis for $V$. Let $x \\in V$. Then there is a unique finite subset $C$ of $R \\times B$ such that: :$\\displaystyle x = \\sum_{\\left({r, v}\\right) \\mathop \\in C} r \\cdot v$ :$\\forall \\left({r, v}\\right) \\in C: r \\ne 0_R$"}
+{"_id": "7651", "title": "Axiom:Axiom of Choice for Finite Sets/Proof from Ordering Principle", "text": "Suppose that the Ordering Principle holds. Let $\\SS$ be a non-empty set of finite non-empty sets. Then there exists a choice function for $\\SS$."}
+{"_id": "7652", "title": "Axiom:Axiom of Choice for Finite Sets/Proof from Hall's Marriage Theorem", "text": "Suppose that Hall's Marriage Theorem holds for all sets of finite sets. Let $\\SS$ be a non-empty set of finite, non-empty sets. Then there exists a choice function for $\\SS$."}
+{"_id": "7653", "title": "Dependent Choice for Finite Sets", "text": "Let $\\RR$ be a binary relation on a non-empty set $S$. For each $a \\in S$, let $C_a = \\set {b \\in S: a \\mathrel \\RR b }$ Suppose that: :For all $a \\in S$, $C_a$ is a non-empty finite set. Let $s \\in S$. Then there exists a sequence $\\sequence {x_n}_{n \\mathop \\in \\N}$ in $S$ such that: :$x_0 = s$ :$\\forall n \\in \\N: x_n \\mathrel \\RR x_{n+1}$"}
+{"_id": "7654", "title": "Dependent Choice (Fixed First Element)", "text": "Let $\\mathcal R$ be a binary relation on a non-empty set $S$. Suppose that: :$\\forall a \\in S: \\exists b \\in S: a \\mathrel{\\mathcal R} b$ that is, that $\\mathcal R$ is a left-total relation (specifically a ''serial relation''). Let $s \\in S$. Then there exists a sequence $\\left\\langle{x_n}\\right\\rangle_{n \\in \\N}$ in $S$ such that: :$x_0 = s$ :$\\forall n \\in \\N: x_n \\mathrel{\\mathcal R} x_{n+1}$"}
+{"_id": "7657", "title": "Locally Finite Connected Graph is Countable", "text": "Let $G = \\struct {V, E}$ be a graph which is connected and locally finite. Then $G$ has countably many vertices and countably many edges."}
+{"_id": "7658", "title": "Law of Cosines/Right Triangle", "text": "Let $\\triangle ABC$ be a triangle whose sides $a, b, c$ are such that: : $a$ is opposite $A$ : $b$ is opposite $B$ : $c$ is opposite $C$. Let $\\triangle ABC$ be a right triangle such that $\\angle A$ is right. Then: :$c^2 = a^2 + b^2 - 2 a b \\cos C$"}
+{"_id": "7659", "title": "Minimal Infinite Successor Set is Minimal", "text": "The minimal infinite successor set $\\omega$ is a subset of every infinite successor set."}
+{"_id": "7660", "title": "Law of Cosines/Proof 3/Acute Triangle", "text": "Let $\\triangle ABC$ be a triangle whose sides $a, b, c$ are such that: : $a$ is opposite $A$ : $b$ is opposite $B$ : $c$ is opposite $C$. Let $\\triangle ABC$ be an acute triangle. Then: :$c^2 = a^2 + b^2 - 2a b \\cos C$"}
+{"_id": "7661", "title": "Law of Cosines/Proof 3/Obtuse Triangle", "text": "Let $\\triangle ABC$ be a triangle whose sides $a, b, c$ are such that: : $a$ is opposite $A$ : $b$ is opposite $B$ : $c$ is opposite $C$. Let $\\triangle ABC$ be an obtuse triangle such that $A$ is obtuse Then: :$c^2 = a^2 + b^2 - 2a b \\cos C$"}
+{"_id": "7662", "title": "König's Lemma/Countable", "text": "Let $G = \\struct {V, E}$ be a graph with countably infinitely many vertices which is connected and is locally finite. Then every vertex of $G$ lies on a path of infinite length."}
+{"_id": "7664", "title": "Schröder Rule/Proof 2", "text": "Let $A$, $B$ and $C$ be relations on a set $S$. Then the following are equivalent statements: :$(1): \\quad A \\circ B \\subseteq C$ :$(2): \\quad A^{-1} \\circ \\overline{C} \\subseteq \\overline{B}$ :$(3): \\quad \\overline{C} \\circ B^{-1} \\subseteq \\overline{A}$ where: : $\\circ$ denotes relation composition : $A^{-1}$ denotes the inverse of $A$ : $\\overline{A}$ denotes the complement of $A$."}
+{"_id": "7665", "title": "Equivalence of Definitions of Logical Consistence", "text": "Let $\\mathbf H$ be a countable set (either finite or infinite) of WFFs of propositional logic. The following statements are logically equivalent: :$(1): \\quad$ $\\mathbf H$ has a model. :$(2): \\quad$ $\\mathbf H$ is consistent for the proof system of propositional tableaus. :$(3): \\quad$ $\\mathbf H$ has no tableau confutation."}
+{"_id": "7666", "title": "Paths from Vertex of Minimal Length form Tree", "text": "Let $G = \\left({V, E}\\right)$ be a simple graph. Let $r \\in V$ be a vertex in $G$. Let $P$ be the set of finite open paths beginning at $r$ that are of minimal length to reach their endpoints. {{refactor|Extract the below sentence and write it as a definition of a \"minimal length path\".}} That is, an open path $p$ beginning at $r$ and ending at some vertex $s$ is in $P$ {{iff}} there is no path $q$ beginning at $r$ and ending at $s$ such that the length of $q$ is less than the length of $p$. Let $p, q \\in P$. Let $\\left\\{{p, q}\\right\\} \\in E'$ {{iff}} either: :$q$ is formed by extending $p$ with one edge and one vertex of $G$ or: :$p$ is formed by extending $q$ with one edge and one vertex of $G$. Then $T = \\left({P, E'}\\right)$ is a tree."}
+{"_id": "7667", "title": "Finite Sequences in Set Form Acyclic Graph", "text": "Let $S$ be a set. Let $V$ be the set of finite sequences in $S$. Let $E$ be the set of unordered pairs $\\{p, q\\}$ of elements of $V$ such that either: : $q$ is formed by extending $p$ by one element or : $p$ is formed by extending $q$ by one element. That is: : $| \\operatorname{Dom}(p) * \\operatorname{Dom}(q) | = 1$, where $*$ is symmetric difference and : $p \\restriction D = q \\restriction D$, where $D = \\operatorname{Dom}(p) \\cap \\operatorname{Dom}(q)$ Then $T = (V, E)$ is an acyclic graph."}
+{"_id": "7669", "title": "Connected Vertices are Connected by Path", "text": "Let $G = \\struct {V, E}$ be a simple graph. Let $x, y \\in V$. Let there exist a walk $w: \\N_n \\to V$ from $x$ to $y$. Then there exists a subsequence $z_n$ of $w$ which is a path from $x$ to $y$."}
+{"_id": "7676", "title": "Negation as Implication of Bottom", "text": "$p \\implies \\bot \\dashv\\vdash \\neg p$"}
+{"_id": "7677", "title": "Clavius's Law/Formulation 1/Proof 2", "text": "From Peirce's Law: :$\\left({p \\implies q}\\right) \\implies p \\vdash p$ follows Clavius's Law: :$\\neg p \\implies p \\vdash p$"}
+{"_id": "7678", "title": "Clavius's Law implies Law of Excluded Middle", "text": "From Clavius's Law: :$\\neg p \\implies p \\vdash p$ follows the Law of Excluded Middle: :$\\vdash p \\lor \\neg p$"}
+{"_id": "7680", "title": "Parity of Integer equals Parity of its Square/Even", "text": "Let $p \\in \\Z$ be an integer. Let $p$ be even. Then $p^2$ is also even."}
+{"_id": "7681", "title": "Parity of Integer equals Parity of its Square/Odd", "text": "Let $p \\in \\Z$ be an integer. Let $p$ be odd. Then $p^2$ is also odd."}
+{"_id": "7683", "title": "Negation of Excluded Middle is False/Form 1", "text": "$\\neg (p \\lor \\neg p) \\vdash \\bot$"}
+{"_id": "7684", "title": "Negation of Excluded Middle is False/Form 2", "text": "$\\vdash \\neg \\neg (p \\lor \\neg p)$"}
+{"_id": "7685", "title": "Double Negation Elimination implies Law of Excluded Middle/Proof 2", "text": "Let the Law of Double Negation Elimination be supposed to hold: :$\\neg \\neg p \\vdash p$ Then the Law of Excluded Middle likewise holds: :$\\vdash p \\lor \\neg p$"}
+{"_id": "7686", "title": "Injection iff Left Cancellable/Necessary Condition", "text": "Let $f: Y \\to Z$ be an injection. Then $f$ is left cancellable."}
+{"_id": "7687", "title": "Injection iff Left Cancellable/Sufficient Condition", "text": "Let $f: Y \\to Z$ be a mapping which is left cancellable. Then $f$ is an injection."}
+{"_id": "7688", "title": "Peirce's Law implies Law of Excluded Middle", "text": "From Peirce's Law: :$\\left({p \\implies q}\\right) \\implies p \\vdash p$ follows the Law of Excluded Middle: :$\\vdash p \\lor \\neg p$"}
+{"_id": "7689", "title": "Pseudocomplemented Lattice is Bounded", "text": "Let $(L, \\wedge, \\vee, \\preceq)$ be a pseudocomplemented lattice. Then $(L, \\wedge, \\vee, \\preceq)$ is a bounded lattice."}
+{"_id": "7690", "title": "Topology forms Complete Lattice", "text": "Let $\\struct {X, \\tau}$ be a topological space. Then $\\struct {\\tau, \\subseteq}$ is a complete lattice."}
+{"_id": "7691", "title": "Law of Excluded Middle implies Peirce's Law", "text": "From the Law of Excluded Middle follows Peirce's Law: :$\\left({p \\lor \\neg p}\\right) \\vdash \\left({\\left({p \\implies q}\\right) \\implies p}\\right) \\implies p$"}
+{"_id": "7693", "title": "Inverse of Composite Bijection/Proof 2", "text": "Let $f$ and $g$ be bijections. Then: :$\\paren {g \\circ f}^{-1} = f^{-1} \\circ g^{-1}$ and $f^{-1} \\circ g^{-1}$ is itself a bijection."}
+{"_id": "7695", "title": "Sierpiński's Theorem", "text": "Let $\\left({S, \\tau}\\right)$ be a compact connected Hausdorff space. Let $\\left\\{{F_n: n \\in \\N}\\right\\}$ be a pairwise disjoint closed cover of $S$. {{explain|In context it's obvious, but worth mentioning that $F_n$ is a finite cover as well?}} Then $F_n = S$ for some $n \\in \\N$."}
+{"_id": "7696", "title": "Sierpiński's Theorem/Lemma 1", "text": "Let $\\struct {S, \\tau}$ be a compact connected Hausdorff space. Let $A$ be a closed, non-empty proper subset of $S$. Let $C$ be a component of $A$. Then: :$C \\cap \\partial A \\ne \\O$ where $\\partial A$ denotes the boundary of $A$."}
+{"_id": "7697", "title": "Quasicomponent of Compact Hausdorff Space is Connected", "text": "Let $\\left({X, \\tau}\\right)$ be a compact Hausdorff space. Let $C$ be a quasicomponent of $\\left({X, \\tau}\\right)$. Then $C$ is connected."}
+{"_id": "7700", "title": "Quasicomponents and Components are Equal in Compact Hausdorff Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a compact Hausdorff space. Then for each $A \\subseteq S$: $A$ is a component of $S$ {{iff}} $A$ is a quasicomponent of $S$."}
+{"_id": "7701", "title": "Derived Set in T1 Space is Closed", "text": "Let $\\struct {X, \\tau}$ be a $T_1$ space. Let $S \\subseteq X$. Let $S'$ be the derived set of $S$. Then $S'$ is closed."}
+{"_id": "7707", "title": "Multiplicative Inverse in Monoid of Integers Modulo m", "text": "Let $\\struct {\\Z_m, \\times_m}$ be the multiplicative monoid of integers modulo $m$. Then: :$\\eqclass k m \\in \\Z_m$ has an inverse in $\\struct {\\Z_m, \\times_m}$ {{iff}}: :$k \\perp m$"}
+{"_id": "7708", "title": "Element Commutes with Square in Group", "text": "Let $\\left({G, \\circ}\\right)$ be a group. Let $x \\in G$. Then $x$ commutes with $x \\circ x$."}
+{"_id": "7711", "title": "Element Commutes with Square in Semigroup", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroup. Let $x \\in S$. Then $x$ commutes with $x \\circ x$."}
+{"_id": "7714", "title": "Subsets of Disjoint Sets are Disjoint", "text": "Let $S$ and $T$ be disjoint sets. Let $S' \\subseteq S$ and $T' \\subseteq T$. Then $S'$ and $T'$ are disjoint."}
+{"_id": "7718", "title": "Network with Positive Integer Mapping is Multigraph", "text": "Let $N = \\struct {V, E, w}$ be a network whose weights are all strictly positive integers. Then $N$ can be represented as a multigraph. Conversely, any multigraph can be expressed as a network whose weights are all strictly positive integers."}
+{"_id": "7719", "title": "Complete Graph is Regular", "text": "Let $K_p$ be the complete graph of order $p$. Then $K_p$ is $p-1$-regular."}
+{"_id": "7720", "title": "No Simple Graph is Perfect", "text": "Let $G$ be a simple graph whose order is $2$ or greater. Then $G$ is not perfect."}
+{"_id": "7721", "title": "Tukey-Teichmüller Lemma", "text": "Let $\\FF$ be a non-empty set of finite character. Then $\\FF$ has an element which is maximal with respect to the set inclusion relation."}
+{"_id": "7724", "title": "Circuit has Three Edges or More", "text": "Let $G$ be a simple graph. Let $C$ be a circuit in $G$. Then $C$ has at least $3$ edges."}
+{"_id": "7728", "title": "Dirac's Theorem/Proof 1", "text": "If a connected graph $G$ has $n \\ge 3$ vertices and the degree of each vertex is at least $\\dfrac n 2$, then $G$ is Hamiltonian."}
+{"_id": "7729", "title": "Dirac's Theorem/Proof 2", "text": "If a connected graph $G$ has $n \\ge 3$ vertices and the degree of each vertex is at least $\\dfrac n 2$, then $G$ is Hamiltonian."}
+{"_id": "7730", "title": "Graph of Cube is Hamiltonian", "text": "The graph of the cube is Hamiltonian."}
+{"_id": "7731", "title": "Graph of Icosahedron is Hamiltonian", "text": "The graph of the icosahedron is Hamiltonian."}
+{"_id": "7733", "title": "Path in Tree is Unique/Necessary Condition", "text": "Let $T$ be a tree. Then there is exactly one path between any two vertices."}
+{"_id": "7734", "title": "Path in Tree is Unique/Sufficient Condition", "text": "Let $T$ be a graph. Let $T$ be such that between any two vertices there is exactly one path. Then $T$ is a tree."}
+{"_id": "7737", "title": "Size of Tree is One Less than Order/Necessary Condition", "text": "Let $T$ be a tree of order $n$. Then the size of $T$ is $n-1$."}
+{"_id": "7738", "title": "Size of Tree is One Less than Order/Sufficient Condition", "text": "Let $T$ be a connected simple graph of order $n$. Let the size of $T$ be $n-1$. Then $T$ is a tree."}
+{"_id": "7739", "title": "Size of Tree is One Less than Order/Necessary Condition/Induction Step/Proof 1", "text": "Let the following hold: :A tree of order $k$ is of size $k - 1$. Then this holds: :A tree of order $k + 1$  is of size $k$."}
+{"_id": "7740", "title": "Size of Tree is One Less than Order/Necessary Condition/Induction Step/Proof 2", "text": "Let the following hold: :For all $j \\le k$, a tree of order $j$ is of size $j-1$. Then this holds: :A tree of order $k+1$  is of size $k$."}
+{"_id": "7743", "title": "Kruskal's Algorithm produces Minimum Spanning Tree", "text": "Kruskal's Algorithm produces a minimum spanning tree."}
+{"_id": "7744", "title": "Regular Graph is Tree iff Complete Graph of Order 2", "text": "Let $G$ be a non-edgeless regular graph. Then $G$ is a tree {{iff}} $G$ is $K_2$, the complete graph of order $2$."}
+{"_id": "7746", "title": "Perimeter of Circle", "text": "The perimeter $C$ of a circle with radius $r$ is given by: : $C = 2 \\pi r$"}
+{"_id": "7748", "title": "Euler's Criterion", "text": "Let $p$ be an odd prime. Let $a \\not \\equiv 0 \\pmod p$. Then: {{begin-eqn}} {{eqn | l = a^{\\frac {p-1} 2}       | o = \\equiv       | r = 1       | rr= \\pmod p       | c = {{iff}} $a$ is a quadratic residue of $p$ }} {{eqn | l=a ^{\\frac {p-1} 2}       | o = \\equiv       | r = -1       | rr= \\pmod p       | c = {{iff}} $a$ is a quadratic non-residue of $p$. }} {{end-eqn}}"}
+{"_id": "7755", "title": "Value of Degree in Radians", "text": "The value of a degree in radians is given by: :$1 \\degrees = \\dfrac {\\pi} {180} \\radians \\approx 0.01745 \\ 32925 \\ 19943 \\ 29576 \\ 92 \\ldots \\radians$ {{OEIS|A019685}}"}
+{"_id": "7764", "title": "Difference of Two Squares/Geometric Proof", "text": ":$\\forall x, y \\in \\R: x^2 - y^2 = \\paren {x + y} \\paren {x - y}$"}
+{"_id": "7776", "title": "Area of Parallelogram/Square", "text": "The area of a square equals the product of one of its bases and the associated altitude."}
+{"_id": "7777", "title": "Area of Parallelogram/Rectangle", "text": "The area of a rectangle equals the product of one of its bases and the associated altitude."}
+{"_id": "7778", "title": "Area of Parallelogram/Parallelogram", "text": "Let $ABCD$ be a parallelogram whose adjacent sides are of length $a$ and $b$ enclosing an angle $\\theta$. The area of $ABCD$ equals the product of one of its bases and the associated altitude: {{begin-eqn}} {{eqn | l = \\map \\Area {ABCD}       | r = b h       | c =  }} {{eqn | r = a b \\sin \\theta       | c =  }} {{end-eqn}} where: :$b$ is the side of $ABCD$ which has been chosen to be the base :$h$ is the altitude of $ABCD$ from $b$."}
+{"_id": "7779", "title": "Perimeter of Rectangle", "text": "Let $ABCD$ be a rectangle whose side lengths are $a$ and $b$. The perimeter of $ABCD$ is $2 a + 2 b$."}
+{"_id": "7780", "title": "Perimeter of Parallelogram", "text": "Let $ABCD$ be a parallelogram whose side lengths are $a$ and $b$. The perimeter of $ABCD$ is $2 a + 2 b$."}
+{"_id": "7783", "title": "Perimeter of Triangle", "text": "Let $ABC$ be a triangle. Then the perimeter $P$ of $ABC$ is given by: :$P = a + b + c$ where $a, b, c$ are the lengths of the sides of $ABC$."}
+{"_id": "7784", "title": "Area of Trapezoid", "text": ":410px Let $ABCD$ be a trapezoid: :whose parallel sides are of lengths $a$ and $b$ and :whose height is $h$. Then the area of $ABCD$ is given by: :$\\Box ABCD = \\dfrac {h \\paren {a + b} } 2$"}
+{"_id": "7785", "title": "Perimeter of Trapezoid", "text": ":400px Let $ABCD$ be a trapezoid: :whose parallel sides are of lengths $a$ and $b$ :whose height is $h$. and :whose non-parallel sides are at angles $\\theta$ and $\\phi$ with the parallels. The perimeter $P$ of $ABCD$ is given by: :$P = a + b + h \\paren {\\csc \\theta + \\csc \\phi}$ where $\\csc$ denotes cosecant."}
+{"_id": "7787", "title": "Perimeter of Regular Polygon", "text": "Let $P$ be a regular $n$-sided polygon whose side length is $b$. Then the perimeter $L$ of $P$ is given by: :$L = n b$"}
+{"_id": "7788", "title": "Commensurability is Transitive", "text": "Let $a$, $b$, $c$ be three real numbers. Let $a$ and $b$ be commensurable, and $b$ and $c$ be commensurable. Then $a$ and $c$ are commensurable."}
+{"_id": "7791", "title": "Arc Length of Sector", "text": "Let $\\mathcal C = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$. Let $BAC$ be the sector of $\\mathcal C$ whose angle between $AB$ and $AC$ is $\\theta$. :300px Then the length $s$ of arc $BC$ is given by: :$\\mathcal s = r \\theta$ where: :$r = AB$ is the length of the radius of the circle :$\\theta$ is measured in radians."}
+{"_id": "7792", "title": "Length of Inradius of Triangle", "text": "Let $\\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$. Then the length of the inradius $r$ of $\\triangle ABC$ is given by: :$r = \\dfrac {\\sqrt {s \\paren {s - a} \\paren {s - b} \\paren {s - c} } } s$ where $s = \\dfrac {a + b + c} 2$ is the semiperimeter of $\\triangle ABC$."}
+{"_id": "7793", "title": "Length of Circumradius of Triangle", "text": "Let $\\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$. Then the length of the circumradius $R$ of $\\triangle ABC$ is given by: :$R = \\dfrac {abc} {4 \\sqrt {s \\paren {s - a} \\paren {s - b} \\paren {s - c} } }$ where $s = \\dfrac {a + b + c} 2$ is the semiperimeter of $\\triangle ABC$."}
+{"_id": "7794", "title": "Area of Triangle in Terms of Inradius", "text": "Let $\\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$. Then the area $\\mathcal A$ of $\\triangle ABC$ is given by: :$\\mathcal A = r s$ where: :$r$ is the inradius of $\\triangle ABC$ :$s = \\dfrac {a + b + c} 2$ is the semiperimeter of $\\triangle ABC$."}
+{"_id": "7795", "title": "Area of Regular Polygon by Circumradius", "text": "Let $P$ be a regular $n$-gon. Let $C$ be a circumcircle of $P$. Let the radius of $C$ be $r$. Then the area $\\mathcal A$ of $P$ is given by: :$\\mathcal A = \\dfrac 1 2 n r^2 \\sin \\dfrac {2 \\pi} n$"}
+{"_id": "7798", "title": "Perimeter of Regular Polygon by Inradius", "text": "Let $P$ be a regular $n$-gon. Let $C$ be an incircle of $P$. Let the radius of $C$ be $r$. Then the perimeter $\\mathcal P$ of $P$ is given by: :$\\mathcal P = 2 n r \\tan \\dfrac \\pi n$"}
+{"_id": "7799", "title": "Image of Interval by Continuous Function is Interval/Proof 1", "text": "Let $I$ be a real interval. Let $f: I \\to \\R$ be a continuous real function. Then the image of $f$ is a real interval."}
+{"_id": "7800", "title": "Image of Interval by Continuous Function is Interval/Proof 2", "text": "Let $I$ be a real interval. Let $f: I \\to \\R$ be a continuous real function. Then the image of $f$ is a real interval."}
+{"_id": "7801", "title": "Area of Isosceles Triangle", "text": "Let $\\triangle ABC$ be an isosceles triangle whose apex is $A$. Let $\\theta$ be the angle of the apex $A$. Let $r$ be the length of a leg of $\\triangle ABC$. Then the area $\\mathcal A$ of $\\triangle ABC$ is given by: :$\\mathcal A = \\dfrac 1 2 r^2 \\sin \\theta$"}
+{"_id": "7802", "title": "Area of Segment of Circle", "text": "Let $C$ be a circle of radius $r$. Let $S$ be a segment of $C$ such that its base subtends an angle of $\\theta$ at the center of the circle. Then the area $\\AA$ of $S$ is given by: :$\\AA = \\dfrac 1 2 r^2 \\paren {\\theta - \\sin \\theta}$ where $\\theta$ is measured in radians."}
+{"_id": "7805", "title": "Mills' Theorem", "text": "There exists a real number $A$ such that $\\left\\lfloor{A^{3^n} }\\right\\rfloor$ is a prime number for all $n \\in \\N_{>0}$, where: :$\\left\\lfloor{x}\\right\\rfloor$ denotes the floor function of $x$ :$\\N$ denotes the set of all natural numbers."}
+{"_id": "7806", "title": "Equation of Ellipse in Reduced Form", "text": "Let $K$ be an ellipse aligned in a cartesian plane in reduced form. Let: :the major axis of $K$ have length $2 a$ :the minor axis of $K$ have length $2 b$."}
+{"_id": "7807", "title": "Area of Ellipse", "text": "Let $K$ be an ellipse whose major axis is of length $2 a$ and whose minor axis is of length $2 b$. The area $\\mathcal A$ of $K$ is given by: : $\\mathcal A = \\pi a b$"}
+{"_id": "7808", "title": "Equation of Ellipse in Reduced Form/Cartesian Frame", "text": "The equation of $K$ is: :$\\dfrac {x^2} {a^2} + \\dfrac {y^2} {b^2} = 1$"}
+{"_id": "7809", "title": "Equation of Ellipse in Reduced Form/Cartesian Frame/Parametric Form", "text": "The equation of $K$ in parametric form is: :$x = a \\cos \\theta, y = b \\sin \\theta$"}
+{"_id": "7810", "title": "Perimeter of Ellipse", "text": "Let $K$ be an ellipse whose major axis is of length $2 a$ and whose minor axis is of length $2 b$. The perimeter $\\PP$ of $K$ is given by: :$\\ds \\PP = 4 a \\int_0^{\\pi / 2} \\sqrt {1 - k^2 \\sin^2 \\theta} \\rd \\theta$ where: :$k = \\dfrac {\\sqrt {a^2 - b^2} } a$ The definite integral: :$\\displaystyle \\PP = \\int_0^{\\pi / 2} \\sqrt{1 - k^2 \\sin^2 \\theta} \\rd \\theta$ is the complete elliptic integral of the second kind."}
+{"_id": "7812", "title": "Distance of Point from Origin in Cartesian Coordinates", "text": "Let $P = \\tuple {x, y}$ be a point in the cartesian plane. Then $P$ is at a distance of $\\sqrt {x^2 + y^2}$ from the origin."}
+{"_id": "7813", "title": "Composite of Monomorphisms is Monomorphism", "text": "Let: :$\\struct {S_1, \\circ_1, \\circ_2, \\ldots, \\circ_n}$ :$\\struct {S_2, *_1, *_2, \\ldots, *_n}$ :$\\struct {S_3, \\oplus_1, \\oplus_2, \\ldots, \\oplus_n}$ be algebraic structures. Let: :$\\phi: \\struct {S_1, \\circ_1, \\circ_2, \\ldots, \\circ_n} \\to \\struct {S_2, *_1, *_2, \\ldots, *_n}$ :$\\psi: \\struct {S_2, *_1, *_2, \\ldots, *_n} \\to \\struct {S_3, \\oplus_1, \\oplus_2, \\ldots, \\oplus_n}$ be monomorphisms. Then the composite of $\\phi$ and $\\psi$ is also a monomorphism."}
+{"_id": "7815", "title": "Composite of Automorphisms is Automorphism", "text": "Let $\\struct {S, \\circ_1, \\circ_2, \\ldots, \\circ_n}$ be an algebraic structure. Let: :$\\phi: \\struct {S, \\circ_1, \\circ_2, \\ldots, \\circ_n} \\to \\struct {S, \\circ_1, \\circ_2, \\ldots, \\circ_n}$ :$\\psi: \\struct {S, \\circ_1, \\circ_2, \\ldots, \\circ_n} \\to \\struct {S, \\circ_1, \\circ_2, \\ldots, \\circ_n}$ be automorphisms. Then the composite of $\\phi$ and $\\psi$ is also an automorphism."}
+{"_id": "7816", "title": "Composite of Endomorphisms is Endomorphism", "text": "Let $\\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right)$ be an algebraic structure. Let: : $\\phi: \\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right) \\to \\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right)$ : $\\psi: \\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right) \\to \\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right)$ be endomorphisms. Then the composite of $\\phi$ and $\\psi$ is also an endomorphism."}
+{"_id": "7817", "title": "Sine of Angle in Cartesian Plane", "text": "Let $P = \\tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\\sin \\theta = \\dfrac y r$ where $\\sin$ denotes the sine of $\\theta$."}
+{"_id": "7820", "title": "Equivalence of Definitions of Sine of Angle", "text": "Let $\\theta$ be an angle. {{TFAE|def = Sine of Angle|view = sine of $\\theta$}}"}
+{"_id": "7824", "title": "Cardinality of Subset Relation on Power Set of Finite Set", "text": "Let $S$ be a set such that: :$\\card S = n$ where $\\card S$ denotes the cardinality of $S$. From Subset Relation on Power Set is Partial Ordering we have that $\\struct {\\powerset S, \\subseteq}$ is an ordered set. The cardinality of $\\subseteq$ as a relation is $3^n$."}
+{"_id": "7825", "title": "Quotient Structure of Semigroup is Semigroup", "text": "Let $\\RR$ be a congruence relation on a semigroup $\\struct {S, \\circ}$. Then the quotient structure $\\struct {S / \\RR, \\circ_\\RR}$ is a semigroup."}
+{"_id": "7826", "title": "Quotient Structure of Monoid is Monoid", "text": "Let $\\RR$ be a congruence relation on a monoid $\\struct {S, \\circ}$ with an identity $e$. Then the quotient structure $\\struct {S / \\RR, \\circ_\\RR}$ is a monoid."}
+{"_id": "7827", "title": "Quotient Structure of Group is Group", "text": "Let $\\RR$ be a congruence relation on a group $\\struct {G, \\circ}$. Then the quotient structure $\\struct {G / \\RR, \\circ_\\RR}$ is a group."}
+{"_id": "7828", "title": "Quotient Structure of Abelian Group is Abelian Group", "text": "Let $\\RR$ be a congruence relation on an abelian group $\\struct {G, \\circ}$. Then the quotient structure $\\struct {G / \\RR, \\circ_\\RR}$ is an abelian group."}
+{"_id": "7829", "title": "Quotient Structure is Similar to Structure", "text": "Let $\\RR$ be a congruence relation on a algebraic structure $\\struct {G, \\circ}$. {{MissingLinks|\"similar\" structure?}} Then the quotient structure $\\struct {G / \\RR, \\circ_\\RR}$ is a similar structure to $\\struct {G, \\circ}$."}
+{"_id": "7830", "title": "Shape of Secant Function", "text": "The nature of the secant function on the set of real numbers $\\R$ is as follows: :$(1): \\quad \\sec x$ is continuous and strictly increasing on the intervals $\\hointr 0 {\\dfrac \\pi 2}$ and $\\hointl {\\dfrac \\pi 2} \\pi$ :$(2): \\quad \\sec x$ is continuous and strictly decreasing on the intervals $\\hointr {-\\pi} {-\\dfrac \\pi 2}$ and $\\hointl {-\\dfrac \\pi 2} 0$ :$(3): \\quad \\sec x \\to + \\infty$ as $x \\to -\\dfrac \\pi 2^+$ :$(4): \\quad \\sec x \\to + \\infty$ as $x \\to \\dfrac \\pi 2^-$ :$(5): \\quad \\sec x \\to - \\infty$ as $x \\to \\dfrac \\pi 2^+$ :$(6): \\quad \\sec x \\to - \\infty$ as $x \\to \\dfrac {3 \\pi} 2^-$"}
+{"_id": "7831", "title": "Shape of Cosecant Function", "text": "The nature of the cosecant function on the set of real numbers $\\R$ is as follows: :$(1): \\quad$ strictly decreasing on the intervals $\\hointr {-\\dfrac \\pi 2} 0$ and $\\hointl 0 {\\dfrac \\pi 2}$ :$(2): \\quad$ strictly increasing on the intervals $\\hointr {\\dfrac \\pi 2} \\pi$ and $\\hointl \\pi {\\dfrac {3 \\pi} 2}$ :$(3): \\quad$ $\\csc x \\to +\\infty$ as $x \\to 0^+$ :$(4): \\quad$ $\\csc x \\to +\\infty$ as $x \\to \\pi^-$ :$(5): \\quad$ $\\csc x \\to -\\infty$ as $x \\to \\pi^+$ :$(6): \\quad$ $\\csc x \\to -\\infty$ as $x \\to 2 \\pi^-$"}
+{"_id": "7833", "title": "Integer Multiples under Multiplication form Semigroup", "text": "Let $n \\Z$ be the set of integer multiples of $n$. Then $\\struct {n \\Z, \\times}$ is a semigroup. If $\\size n > 1$ then $\\struct {n \\Z, \\times}$ has no identity."}
+{"_id": "7834", "title": "Order of External Direct Product", "text": "Let $\\struct {S, \\circ_1}$ and $\\struct {T, \\circ_2}$ be algebraic structures. Then the order of $\\struct {S \\times T, \\circ}$ is $\\card S \\times \\card T$."}
+{"_id": "7835", "title": "Congruence (Number Theory) is Congruence Relation", "text": "Congruence modulo $m$ is a congruence relation on $\\left({\\Z, +}\\right)$."}
+{"_id": "7839", "title": "Equivalence of Definitions of Cotangent of Angle", "text": "Let $\\theta$ be an angle. {{TFAE|def = Cotangent of Angle|cotangent}}"}
+{"_id": "7841", "title": "Equivalence of Definitions of Cosecant of Angle", "text": "Let $\\theta$ be an angle. {{TFAE|def = Cosecant of Angle|view = cosecant}}"}
+{"_id": "7842", "title": "Cosine of Angle in Cartesian Plane", "text": "Let $P = \\tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\\cos \\theta = \\dfrac x r$ where $\\cos$ denotes the cosine of $\\theta$."}
+{"_id": "7843", "title": "Tangent of Angle in Cartesian Plane", "text": "Let $P = \\tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\\tan \\theta = \\dfrac y x$ where $\\tan$ denotes the tangent of $\\theta$."}
+{"_id": "7844", "title": "Cotangent of Angle in Cartesian Plane", "text": "Let $P = \\tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\\cot \\theta = \\dfrac x y$ where $\\cot$ denotes the cotangent of $\\theta$."}
+{"_id": "7845", "title": "Secant of Angle in Cartesian Plane", "text": "Let $P = \\tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\\sec \\theta = \\dfrac r x$ where $\\sec$ denotes the secant of $\\theta$."}
+{"_id": "7846", "title": "Cosecant of Angle in Cartesian Plane", "text": "Let $P = \\tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\\csc \\theta = \\dfrac r x$ where $\\csc$ denotes the secant of $\\theta$."}
+{"_id": "7847", "title": "Tangent is Sine divided by Cosine", "text": "Let $\\theta$ be an angle such that $\\cos \\theta \\ne 0$. Then: :$\\tan \\theta = \\dfrac {\\sin \\theta} {\\cos \\theta}$ where $\\tan$, $\\sin$ and $\\cos$ mean tangent, sine and cosine respectively."}
+{"_id": "7848", "title": "Cotangent is Cosine divided by Sine", "text": "Let $\\theta$ be an angle such that $\\sin \\theta \\ne 0$. Then: :$\\cot \\theta = \\dfrac {\\cos \\theta} {\\sin \\theta}$ where $\\cot$, $\\sin$ and $\\cos$ mean cotangent, sine and cosine respectively."}
+{"_id": "7849", "title": "Cotangent is Reciprocal of Tangent", "text": ":$\\cot \\theta = \\dfrac 1 {\\tan \\theta}$"}
+{"_id": "7850", "title": "Secant is Reciprocal of Cosine", "text": ":$\\sec \\theta = \\dfrac 1 {\\cos \\theta}$"}
+{"_id": "7851", "title": "Cosecant is Reciprocal of Sine", "text": ":$\\csc \\theta = \\dfrac 1 {\\sin \\theta}$"}
+{"_id": "7852", "title": "Particular Values of Sine Function", "text": "The following values of the sine function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Sine of Zero === {{:Sine of Zero is Zero}} === Sine of 15 Degrees === {{:Sine of 15 Degrees}} === Sine of 30 Degrees === {{:Sine of 30 Degrees}} === Sine of 45 Degrees === {{:Sine of 45 Degrees}} === Sine of 60 Degrees === {{:Sine of 60 Degrees}} === Sine of 75 Degrees === {{:Sine of 75 Degrees}} === Sine of Right Angle === {{:Sine of Right Angle}} === Sine of 105 Degrees === {{:Sine of 105 Degrees}} === Sine of 120 Degrees === {{:Sine of 120 Degrees}} === Sine of 135 Degrees === {{:Sine of 135 Degrees}} === Sine of 150 Degrees === {{:Sine of 150 Degrees}} === Sine of 165 Degrees === {{:Sine of 165 Degrees}} === Sine of Straight Angle === {{:Sine of Straight Angle}} === Sine of 195 Degrees === {{:Sine of 195 Degrees}} === Sine of 210 Degrees === {{:Sine of 210 Degrees}} === Sine of 225 Degrees === {{:Sine of 225 Degrees}} === Sine of 240 Degrees === {{:Sine of 240 Degrees}} === Sine of 255 Degrees === {{:Sine of 255 Degrees}} === Sine of Three Right Angles === {{:Sine of Three Right Angles}} === Sine of 285 Degrees === {{:Sine of 285 Degrees}} === Sine of 300 Degrees === {{:Sine of 300 Degrees}} === Sine of 315 Degrees === {{:Sine of 315 Degrees}} === Sine of 330 Degrees === {{:Sine of 330 Degrees}} === Sine of 345 Degrees === {{:Sine of 345 Degrees}} === Sine of Full Angle === {{:Sine of Full Angle}}"}
+{"_id": "7853", "title": "Positive-Term Generalized Sum Converges iff Supremum", "text": "Let $\\struct {G, \\circ, \\le}$ be an abelian totally ordered group, considered under the order topology. Let $\\set {x_i: i \\in I}$ be an indexed set of positive elements of $G$. {{explain|What actually does \"positive\" mean in this context?}} Then: :the generalized sum $\\displaystyle \\sum \\set {x_i: i \\in I}$ converges to a point $x \\in G$ {{iff}} :$x$ is the supremum of: ::$P := \\displaystyle \\set {\\sum_{i \\mathop \\in F} x_i: \\text{$F \\subseteq I$ and $F$ is finite} }$"}
+{"_id": "7856", "title": "Half Angle Formulas/Sine", "text": "{{begin-eqn}} {{eqn | l = \\sin \\frac \\theta 2       | r = +\\sqrt {\\frac {1 - \\cos \\theta} 2}       | c = for $\\dfrac \\theta 2$ in quadrant $\\text I$ or quadrant $\\text {II}$ }} {{eqn | l = \\sin \\frac \\theta 2       | r = -\\sqrt {\\dfrac {1 - \\cos \\theta} 2}       | c = for $\\dfrac \\theta 2$ in quadrant $\\text {III}$ or quadrant $\\text {IV}$ }} {{end-eqn}}"}
+{"_id": "7859", "title": "Half Angle Formulas/Cosine", "text": "{{begin-eqn}} {{eqn | l = \\cos \\frac \\theta 2       | r = +\\sqrt {\\frac {1 + \\cos \\theta} 2}       | c = for $\\dfrac \\theta 2$ in quadrant $\\text I$ or quadrant $\\text {IV}$ }} {{eqn | l = \\cos \\frac \\theta 2       | r = -\\sqrt {\\frac {1 + \\cos \\theta} 2}       | c = for $\\dfrac \\theta 2$ in quadrant $\\text {II}$ or quadrant $\\text {III}$ }} {{end-eqn}}"}
+{"_id": "7860", "title": "Half Angle Formulas/Tangent", "text": "{{begin-eqn}} {{eqn | l = \\tan \\frac \\theta 2       | r = +\\sqrt {\\dfrac {1 - \\cos \\theta} {1 + \\cos \\theta} }       | c = for $\\dfrac \\theta 2$ in quadrant $\\text I$ or quadrant $\\text {III}$ }} {{eqn | l = \\tan \\frac \\theta 2       | r = -\\sqrt {\\dfrac {1 - \\cos \\theta} {1 + \\cos \\theta} }       | c = for $\\dfrac \\theta 2$ in quadrant $\\text {II}$ or quadrant $\\text {IV}$ }} {{end-eqn}} where $\\tan$ denotes tangent and $\\cos$ denotes cosine. When $\\theta = \\paren {2 k + 1} \\pi$, $\\tan \\dfrac \\theta 2$ is undefined."}
+{"_id": "7861", "title": "Kernel of Normal Operator is Kernel of Adjoint", "text": "Let $H$ be a Hilbert space. Let $A \\in \\map B H$ be a normal operator. Then: :$\\ker A = \\ker A^*$ where: :$\\ker$ denotes kernel :$A^*$ denotes the adjoint of $A$."}
+{"_id": "7862", "title": "Half Angle Formulas/Tangent/Corollary 1", "text": ":$\\tan \\dfrac \\theta 2 = \\dfrac {\\sin \\theta} {1 + \\cos \\theta}$"}
+{"_id": "7863", "title": "Half Angle Formulas/Tangent/Corollary 2", "text": ":$\\tan \\dfrac \\theta 2 = \\dfrac {1 - \\cos \\theta} {\\sin \\theta}$"}
+{"_id": "7864", "title": "Half Angle Formulas/Tangent/Corollary 3", "text": ":$\\tan \\dfrac \\theta 2 = \\csc \\theta - \\cot \\theta$"}
+{"_id": "7865", "title": "Bisection of Angle in Cartesian Plane", "text": "Let $\\theta$ be the angular coordinate of a point $P$ in a polar coordinate plane. Let $QOR$ be a straight line that bisects the angle $\\theta$. Then the angular coordinates of $Q$ and $R$ are $\\dfrac \\theta 2$ and $\\pi + \\dfrac \\theta 2$."}
+{"_id": "7866", "title": "Sine of 15 Degrees", "text": ":$\\sin 15 \\degrees = \\sin \\dfrac \\pi {12} = \\dfrac {\\sqrt 6 - \\sqrt 2} 4$"}
+{"_id": "7867", "title": "Sine of 30 Degrees", "text": ":$\\sin 30 \\degrees = \\sin \\dfrac \\pi 6 = \\dfrac 1 2$"}
+{"_id": "7869", "title": "Sine of 45 Degrees", "text": ":$\\sin 45 \\degrees = \\sin \\dfrac \\pi 4 = \\dfrac {\\sqrt 2} 2$"}
+{"_id": "7871", "title": "Sine of 60 Degrees", "text": ":$\\sin 60 \\degrees = \\sin \\dfrac \\pi 3 = \\dfrac {\\sqrt 3} 2$"}
+{"_id": "7874", "title": "Sine of 75 Degrees", "text": ":$\\sin 75 \\degrees = \\sin \\dfrac {5 \\pi} {12} = \\dfrac {\\sqrt 6 + \\sqrt 2} 4$"}
+{"_id": "7875", "title": "Sine of Right Angle", "text": ":$\\sin 90^\\circ = \\sin \\dfrac \\pi 2 = 1$"}
+{"_id": "7876", "title": "Sine of Angle plus Right Angle", "text": ": $\\sin \\paren {x + \\dfrac \\pi 2} = \\cos x$"}
+{"_id": "7877", "title": "Sine of 105 Degrees", "text": ":$\\sin 105^\\circ = \\sin \\dfrac {7 \\pi} {12} = \\dfrac {\\sqrt 6 + \\sqrt 2} 4$"}
+{"_id": "7878", "title": "Sine of 120 Degrees", "text": ":$\\sin 120 \\degrees = \\sin \\dfrac {2 \\pi} 3 = \\dfrac {\\sqrt 3} 2$"}
+{"_id": "7879", "title": "Sine of 135 Degrees", "text": ":$\\sin 135 \\degrees = \\sin \\dfrac {3 \\pi} 4 = \\dfrac {\\sqrt 2} 2$"}
+{"_id": "7880", "title": "Sine of 150 Degrees", "text": ":$\\sin 150 \\degrees = \\sin \\dfrac {5 \\pi} 6 = \\dfrac 1 2$"}
+{"_id": "7881", "title": "Sine of 165 Degrees", "text": ":$\\sin 165 \\degrees = \\sin \\dfrac {11 \\pi} {12} = \\dfrac {\\sqrt 6 - \\sqrt 2} 4$"}
+{"_id": "7882", "title": "Sine of Straight Angle", "text": ":$\\sin 180 \\degrees = \\sin \\pi = 0$"}
+{"_id": "7889", "title": "Sine of Supplementary Angle", "text": ":$\\sin \\paren {\\pi - \\theta} = \\sin \\theta$ where $\\sin$ denotes sine. That is, the sine of an angle equals its supplement."}
+{"_id": "7890", "title": "Cosine of Supplementary Angle", "text": ":$\\map \\cos {\\pi - \\theta} = -\\cos \\theta$ where $\\cos$ denotes cosine. That is, the cosine of an angle is the negative of its supplement."}
+{"_id": "7891", "title": "Sine of Conjugate Angle", "text": ":$\\map \\sin {2 \\pi - \\theta} = -\\sin \\theta$ where $\\sin$ denotes sine. That is, the sine of an angle is the negative of its conjugate."}
+{"_id": "7892", "title": "Cosine of Conjugate Angle", "text": ":$\\map \\cos {2 \\pi - \\theta} = \\cos \\theta$ where $\\cos$ denotes cosine. That is, the cosine of an angle equals its conjugate."}
+{"_id": "7893", "title": "Sine of 195 Degrees", "text": ":$\\sin 195 \\degrees = \\sin \\dfrac {13 \\pi} {12} = -\\dfrac {\\sqrt 6 - \\sqrt 2} 4$"}
+{"_id": "7894", "title": "Sine of 210 Degrees", "text": ":$\\sin 210 \\degrees = \\sin \\dfrac {7 \\pi} 6 = -\\dfrac 1 2$"}
+{"_id": "7895", "title": "Sine of 225 Degrees", "text": ":$\\sin 225 \\degrees = \\sin \\dfrac {5 \\pi} 4 = -\\dfrac {\\sqrt 2} 2$"}
+{"_id": "7896", "title": "Sine of 240 Degrees", "text": ":$\\sin 240 \\degrees = \\sin \\dfrac {4 \\pi} 3 = -\\dfrac {\\sqrt 3} 2$"}
+{"_id": "7897", "title": "Sine of 255 Degrees", "text": ":$\\sin 255^\\circ = \\sin \\dfrac {17 \\pi} {12} = - \\dfrac {\\sqrt 6 + \\sqrt 2} 4$"}
+{"_id": "7898", "title": "Sine of Three Right Angles", "text": ":$\\sin 270 \\degrees = \\sin \\dfrac {3 \\pi} 2 = -1$"}
+{"_id": "7899", "title": "Sine of 285 Degrees", "text": ":$\\sin 285^\\circ = \\sin \\dfrac {19 \\pi} {12} = - \\dfrac {\\sqrt 6 + \\sqrt 2} 4$"}
+{"_id": "7900", "title": "Sine of 300 Degrees", "text": ":$\\sin 300 \\degrees = \\sin \\dfrac {5 \\pi} 3 = -\\dfrac {\\sqrt 3} 2$"}
+{"_id": "7901", "title": "Sine of 315 Degrees", "text": ":$\\sin 315 \\degrees = \\sin \\dfrac {7 \\pi} 4 = -\\dfrac {\\sqrt 2} 2$"}
+{"_id": "7902", "title": "Sine of 330 Degrees", "text": ":$\\sin 330 \\degrees = \\sin \\dfrac {11 \\pi} 6 = -\\dfrac 1 2$"}
+{"_id": "7903", "title": "Sine of 345 Degrees", "text": ":$\\sin 345^\\circ = \\sin \\dfrac {23 \\pi} {12} = - \\dfrac {\\sqrt 6 - \\sqrt 2} 4$"}
+{"_id": "7904", "title": "Sine of Full Angle", "text": ":$\\sin 360^\\circ = \\sin 2 \\pi = 0$"}
+{"_id": "7905", "title": "Particular Values of Cosine Function", "text": "The following values of the cosine function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Cosine of Zero === {{:Cosine of Zero is One}} === Cosine of 15 Degrees === {{:Cosine of 15 Degrees}} === Cosine of 30 Degrees === {{:Cosine of 30 Degrees}} === Cosine of 45 Degrees === {{:Cosine of 45 Degrees}} === Cosine of 60 Degrees === {{:Cosine of 60 Degrees}} === Cosine of 75 Degrees === {{:Cosine of 75 Degrees}} === Cosine of Right Angle === {{:Cosine of Right Angle}} === Cosine of 105 Degrees === {{:Cosine of 105 Degrees}} === Cosine of 120 Degrees === {{:Cosine of 120 Degrees}} === Cosine of 135 Degrees === {{:Cosine of 135 Degrees}} === Cosine of 150 Degrees === {{:Cosine of 150 Degrees}} === Cosine of 165 Degrees === {{:Cosine of 165 Degrees}} === Cosine of Straight Angle === {{:Cosine of Straight Angle}} === Cosine of 195 Degrees === {{:Cosine of 195 Degrees}} === Cosine of 210 Degrees === {{:Cosine of 210 Degrees}} === Cosine of 225 Degrees === {{:Cosine of 225 Degrees}} === Cosine of 240 Degrees === {{:Cosine of 240 Degrees}} === Cosine of 255 Degrees === {{:Cosine of 255 Degrees}} === Cosine of Three Right Angles === {{:Cosine of Three Right Angles}} === Cosine of 285 Degrees === {{:Cosine of 285 Degrees}} === Cosine of 300 Degrees === {{:Cosine of 300 Degrees}} === Cosine of 315 Degrees === {{:Cosine of 315 Degrees}} === Cosine of 330 Degrees === {{:Cosine of 330 Degrees}} === Cosine of 345 Degrees === {{:Cosine of 345 Degrees}} === Cosine of Full Angle === {{:Cosine of Full Angle}}"}
+{"_id": "7906", "title": "Cosine of 15 Degrees", "text": ":$\\cos 15 \\degrees = \\cos \\dfrac \\pi {12} = \\dfrac {\\sqrt 6 + \\sqrt 2} 4$"}
+{"_id": "7909", "title": "Cosine of 30 Degrees", "text": ":$\\cos 30 \\degrees = \\cos \\dfrac \\pi 6 = \\dfrac {\\sqrt 3} 2$"}
+{"_id": "7910", "title": "Cosine of 45 Degrees", "text": ":$\\cos 45 \\degrees = \\cos \\dfrac \\pi 4 = \\dfrac {\\sqrt 2} 2$"}
+{"_id": "7911", "title": "Cosine of 60 Degrees", "text": ":$\\cos 60 \\degrees = \\cos \\dfrac \\pi 3 = \\dfrac 1 2$"}
+{"_id": "7912", "title": "Cosine of 75 Degrees", "text": ":$\\cos 75^\\circ = \\cos \\dfrac {5 \\pi}{12} = \\dfrac {\\sqrt 6 - \\sqrt 2} 4$"}
+{"_id": "7913", "title": "Cosine of Right Angle", "text": ":$\\cos 90^\\circ = \\cos \\dfrac \\pi 2 = 0$"}
+{"_id": "7914", "title": "Cosine of 105 Degrees", "text": ":$\\cos 105 \\degrees = \\cos \\dfrac {7 \\pi} {12} = - \\dfrac {\\sqrt 6 - \\sqrt 2} 4$"}
+{"_id": "7915", "title": "Cosine of Angle plus Right Angle", "text": ":$\\map \\cos {x + \\dfrac \\pi 2} = -\\sin x$"}
+{"_id": "7916", "title": "Cosine of 120 Degrees", "text": ":$\\cos 120 \\degrees = \\cos \\dfrac {2 \\pi} 3 = -\\dfrac 1 2$"}
+{"_id": "7917", "title": "Cosine of 135 Degrees", "text": ":$\\cos 135 \\degrees = \\cos \\dfrac {3 \\pi} 4 = -\\dfrac {\\sqrt 2} 2$"}
+{"_id": "7918", "title": "Cosine of 150 Degrees", "text": ":$\\cos 150 \\degrees = \\cos \\dfrac {5 \\pi} 6 = -\\dfrac {\\sqrt 3} 2$"}
+{"_id": "7919", "title": "Cosine of 165 Degrees", "text": ":$\\cos 165 \\degrees = \\cos \\dfrac {11 \\pi} {12} = - \\dfrac {\\sqrt 6 + \\sqrt 2} 4$"}
+{"_id": "7920", "title": "Cosine of Straight Angle", "text": ":$\\cos 180^\\circ = \\cos \\pi = - 1$"}
+{"_id": "7921", "title": "Cosine of 195 Degrees", "text": ":$\\cos 195 \\degrees = \\cos \\dfrac {13 \\pi} {12} = - \\dfrac {\\sqrt 6 + \\sqrt 2} 4$"}
+{"_id": "7922", "title": "Cosine of 210 Degrees", "text": ":$\\cos 210 \\degrees = \\cos \\dfrac {7 \\pi} 6 = -\\dfrac {\\sqrt 3} 2$"}
+{"_id": "7923", "title": "Cosine of 225 Degrees", "text": ":$\\cos 225 \\degrees = \\cos \\dfrac {5 \\pi} 4 = -\\dfrac {\\sqrt 2} 2$"}
+{"_id": "7924", "title": "Cosine of 240 Degrees", "text": ":$\\cos 240 \\degrees = \\cos \\dfrac {4 \\pi} 3 = -\\dfrac 1 2$"}
+{"_id": "7925", "title": "Cosine of 255 Degrees", "text": ":$\\cos 255^\\circ = \\cos \\dfrac {17 \\pi} {12} = - \\dfrac {\\sqrt 6 - \\sqrt 2} 4$"}
+{"_id": "7926", "title": "Cosine of Three Right Angles", "text": ":$\\cos 270^\\circ = \\cos \\dfrac {3 \\pi} 2 = 0$"}
+{"_id": "7927", "title": "Cosine of 285 Degrees", "text": ":$\\cos 285^\\circ = \\cos \\dfrac {19 \\pi} {12} = \\dfrac {\\sqrt 6 - \\sqrt 2} 4$"}
+{"_id": "7928", "title": "Cosine of 300 Degrees", "text": ":$\\cos 300 \\degrees = \\cos \\dfrac {5 \\pi} 3 = \\dfrac 1 2$"}
+{"_id": "7929", "title": "Cosine of 315 Degrees", "text": ":$\\cos 315 \\degrees = \\cos \\dfrac {7 \\pi} 4 = \\dfrac {\\sqrt 2} 2$"}
+{"_id": "7930", "title": "Cosine of 330 Degrees", "text": ":$\\cos 330 \\degrees = \\cos \\dfrac {11 \\pi} 6 = \\dfrac {\\sqrt 3} 2$"}
+{"_id": "7931", "title": "Cosine of 345 Degrees", "text": ":$\\cos 345 \\degrees = \\cos \\dfrac {23 \\pi} {12} = \\dfrac {\\sqrt 6 + \\sqrt 2} 4$"}
+{"_id": "7932", "title": "Cosine of Full Angle", "text": ":$\\cos 360 \\degrees = \\cos 2 \\pi = 1$"}
+{"_id": "7933", "title": "Particular Values of Tangent Function", "text": "The following values of the tangent function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Tangent of Zero === {{:Tangent of Zero}} === Tangent of 15 Degrees === {{:Tangent of 15 Degrees}} === Tangent of 22.5 Degrees === {{:Tangent of 22.5 Degrees}} === Tangent of 30 Degrees === {{:Tangent of 30 Degrees}} === Tangent of 45 Degrees === {{:Tangent of 45 Degrees}} === Tangent of 60 Degrees === {{:Tangent of 60 Degrees}} === Tangent of 67.5 Degrees === {{:Tangent of 67.5 Degrees}} === Tangent of 75 Degrees === {{:Tangent of 75 Degrees}} === Tangent of Right Angle === {{:Tangent of Right Angle}} === Tangent of 105 Degrees === {{:Tangent of 105 Degrees}} === Tangent of 120 Degrees === {{:Tangent of 120 Degrees}} === Tangent of 135 Degrees === {{:Tangent of 135 Degrees}} === Tangent of 150 Degrees === {{:Tangent of 150 Degrees}} === Tangent of 165 Degrees === {{:Tangent of 165 Degrees}} === Tangent of Straight Angle === {{:Tangent of Straight Angle}} === Tangent of 195 Degrees === {{:Tangent of 195 Degrees}} === Tangent of 210 Degrees === {{:Tangent of 210 Degrees}} === Tangent of 225 Degrees === {{:Tangent of 225 Degrees}} === Tangent of 240 Degrees === {{:Tangent of 240 Degrees}} === Tangent of 255 Degrees === {{:Tangent of 255 Degrees}} === Tangent of Three Right Angles === {{:Tangent of Three Right Angles}} === Tangent of 285 Degrees === {{:Tangent of 285 Degrees}} === Tangent of 300 Degrees === {{:Tangent of 300 Degrees}} === Tangent of 315 Degrees === {{:Tangent of 315 Degrees}} === Tangent of 330 Degrees === {{:Tangent of 330 Degrees}} === Tangent of 345 Degrees === {{:Tangent of 345 Degrees}} === Tangent of Full Angle === {{:Tangent of Full Angle}}"}
+{"_id": "7934", "title": "Tangent of Zero", "text": ":$\\tan 0 = 0$"}
+{"_id": "7935", "title": "Tangent of 15 Degrees", "text": ":$\\tan 15^\\circ = \\tan \\dfrac {\\pi} {12} = 2 - \\sqrt 3$"}
+{"_id": "7939", "title": "Tangent of 30 Degrees", "text": ":$\\tan 30^\\circ = \\tan \\dfrac \\pi 6 = \\dfrac {\\sqrt 3} 3$"}
+{"_id": "7940", "title": "Particular Values of Cotangent Function", "text": "The following values of the cotangent function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Cotangent of Zero === {{:Cotangent of Zero}} === Cotangent of 15 Degrees === {{:Cotangent of 15 Degrees}} === Cotangent of 30 Degrees === {{:Cotangent of 30 Degrees}} === Cotangent of 45 Degrees === {{:Cotangent of 45 Degrees}} === Cotangent of 60 Degrees === {{:Cotangent of 60 Degrees}} === Cotangent of 75 Degrees === {{:Cotangent of 75 Degrees}} === Cotangent of Right Angle === {{:Cotangent of Right Angle}} === Cotangent of 105 Degrees === {{:Cotangent of 105 Degrees}} === Cotangent of 120 Degrees === {{:Cotangent of 120 Degrees}} === Cotangent of 135 Degrees === {{:Cotangent of 135 Degrees}} === Cotangent of 150 Degrees === {{:Cotangent of 150 Degrees}} === Cotangent of 165 Degrees === {{:Cotangent of 165 Degrees}} === Cotangent of Straight Angle === {{:Cotangent of Straight Angle}} === Cotangent of 195 Degrees === {{:Cotangent of 195 Degrees}} === Cotangent of 210 Degrees === {{:Cotangent of 210 Degrees}} === Cotangent of 225 Degrees === {{:Cotangent of 225 Degrees}} === Cotangent of 240 Degrees === {{:Cotangent of 240 Degrees}} === Cotangent of 255 Degrees === {{:Cotangent of 255 Degrees}} === Cotangent of Three Right Angles === {{:Cotangent of Three Right Angles}} === Cotangent of 285 Degrees === {{:Cotangent of 285 Degrees}} === Cotangent of 300 Degrees === {{:Cotangent of 300 Degrees}} === Cotangent of 315 Degrees === {{:Cotangent of 315 Degrees}} === Cotangent of 330 Degrees === {{:Cotangent of 330 Degrees}} === Cotangent of 345 Degrees === {{:Cotangent of 345 Degrees}} === Cotangent of Full Angle === {{:Cotangent of Full Angle}}"}
+{"_id": "7941", "title": "Particular Values of Secant Function", "text": "The following values of the secant function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Secant of Zero === {{:Secant of Zero}} === Secant of $15 \\degrees$ === {{:Secant of 15 Degrees}} === Secant of $30 \\degrees$ === {{:Secant of 30 Degrees}} === Secant of $45 \\degrees$ === {{:Secant of 45 Degrees}} === Secant of $60 \\degrees$ === {{:Secant of 60 Degrees}} === Secant of $75 \\degrees$ === {{:Secant of 75 Degrees}} === Secant of Right Angle === {{:Secant of Right Angle}} === Secant of $105 \\degrees$ === {{:Secant of 105 Degrees}} === Secant of $120 \\degrees$ === {{:Secant of 120 Degrees}} === Secant of $135 \\degrees$ === {{:Secant of 135 Degrees}} === Secant of $150 \\degrees$ === {{:Secant of 150 Degrees}} === Secant of $165 \\degrees$ === {{:Secant of 165 Degrees}} === Secant of Straight Angle === {{:Secant of Straight Angle}} === Secant of $195 \\degrees$ === {{:Secant of 195 Degrees}} === Secant of $210 \\degrees$ === {{:Secant of 210 Degrees}} === Secant of $225 \\degrees$ === {{:Secant of 225 Degrees}} === Secant of $240 \\degrees$ === {{:Secant of 240 Degrees}} === Secant of $255 \\degrees$ === {{:Secant of 255 Degrees}} === Secant of Three Right Angles === {{:Secant of Three Right Angles}} === Secant of $285 \\degrees$ === {{:Secant of 285 Degrees}} === Secant of $300 \\degrees$ === {{:Secant of 300 Degrees}} === Secant of $315 \\degrees$ === {{:Secant of 315 Degrees}} === Secant of $330 \\degrees$ === {{:Secant of 330 Degrees}} === Secant of $345 \\degrees$ === {{:Secant of 345 Degrees}} === Secant of Full Angle === {{:Secant of Full Angle}}"}
+{"_id": "7942", "title": "Particular Values of Cosecant Function", "text": "The following values of the cosecant function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Cosecant of Zero === {{:Cosecant of Zero}} === Cosecant of 15 Degrees === {{:Cosecant of 15 Degrees}} === Cosecant of 30 Degrees === {{:Cosecant of 30 Degrees}} === Cosecant of 45 Degrees === {{:Cosecant of 45 Degrees}} === Cosecant of 60 Degrees === {{:Cosecant of 60 Degrees}} === Cosecant of 75 Degrees === {{:Cosecant of 75 Degrees}} === Cosecant of Right Angle === {{:Cosecant of Right Angle}} === Cosecant of 105 Degrees === {{:Cosecant of 105 Degrees}} === Cosecant of 120 Degrees === {{:Cosecant of 120 Degrees}} === Cosecant of 135 Degrees === {{:Cosecant of 135 Degrees}} === Cosecant of 150 Degrees === {{:Cosecant of 150 Degrees}} === Cosecant of 165 Degrees === {{:Cosecant of 165 Degrees}} === Cosecant of Straight Angle === {{:Cosecant of Straight Angle}} === Cosecant of 195 Degrees === {{:Cosecant of 195 Degrees}} === Cosecant of 210 Degrees === {{:Cosecant of 210 Degrees}} === Cosecant of 225 Degrees === {{:Cosecant of 225 Degrees}} === Cosecant of 240 Degrees === {{:Cosecant of 240 Degrees}} === Cosecant of 255 Degrees === {{:Cosecant of 255 Degrees}} === Cosecant of Three Right Angles === {{:Cosecant of Three Right Angles}} === Cosecant of 285 Degrees === {{:Cosecant of 285 Degrees}} === Cosecant of 300 Degrees === {{:Cosecant of 300 Degrees}} === Cosecant of 315 Degrees === {{:Cosecant of 315 Degrees}} === Cosecant of 330 Degrees === {{:Cosecant of 330 Degrees}} === Cosecant of 345 Degrees === {{:Cosecant of 345 Degrees}} === Cosecant of Full Angle === {{:Cosecant of Full Angle}}"}
+{"_id": "7943", "title": "Tangent of 45 Degrees", "text": ":$\\tan 45 \\degrees = \\tan \\dfrac \\pi 4 = 1$"}
+{"_id": "7944", "title": "Tangent of 60 Degrees", "text": ":$\\tan 60^\\circ = \\tan \\dfrac \\pi 3 = \\sqrt 3$"}
+{"_id": "7945", "title": "Tangent of 75 Degrees", "text": ":$\\tan 75^\\circ = \\tan \\dfrac {5 \\pi} {12} = 2 + \\sqrt 3$"}
+{"_id": "7946", "title": "Tangent of Complement equals Cotangent", "text": ":$\\map \\tan {\\dfrac \\pi 2 - \\theta} = \\cot \\theta$ for $\\theta \\ne n \\pi$ where $\\tan$ and $\\cot$ are tangent and cotangent respectively. That is, the cotangent of an angle is the tangent of its complement. This relation is defined wherever $\\sin \\theta \\ne 0$."}
+{"_id": "7947", "title": "Cotangent of Complement equals Tangent", "text": ":$\\cot \\left({\\dfrac \\pi 2 - \\theta}\\right) = \\tan \\theta$ for $\\theta \\ne \\left({2 n + 1}\\right) \\dfrac \\pi 2$ where $\\cot$ and $\\tan$ are cotangent and tangent respectively. That is, the tangent of an angle is the cotangent of its complement. This relation is defined wherever $\\cos \\theta \\ne 0$."}
+{"_id": "7948", "title": "Tangent of Angle plus Right Angle", "text": ": $\\tan \\left({x + \\dfrac \\pi 2}\\right) = -\\cot x$"}
+{"_id": "7949", "title": "Cotangent of Angle plus Right Angle", "text": ": $\\cot \\left({x + \\dfrac \\pi 2}\\right) = -\\tan x$"}
+{"_id": "7950", "title": "Tangent of Supplementary Angle", "text": ":$\\map \\tan {\\pi - \\theta} = -\\tan \\theta$ where $\\tan$ denotes tangent. That is, the tangent of an angle is the negative of its supplement."}
+{"_id": "7951", "title": "Cotangent of Supplementary Angle", "text": ":$\\map \\cot {\\pi - \\theta} = -\\cot \\theta$ where $\\cot$ denotes tangent. That is, the cotangent of an angle is the negative of its supplement."}
+{"_id": "7952", "title": "Tangent of Conjugate Angle", "text": ":$\\map \\tan {2 \\pi - \\theta} = -\\tan \\theta$ where $\\tan$ denotes tangent. That is, the tangent of an angle is the negative of its conjugate."}
+{"_id": "7953", "title": "Cotangent of Conjugate Angle", "text": ":$\\map \\cot {2 \\pi - \\theta} = -\\cot \\theta$ where $\\cot$ denotes cotangent. That is, the cotangent of an angle is the negative of its conjugate."}
+{"_id": "7954", "title": "Tangent of Right Angle", "text": ":$\\tan 90^\\circ = \\tan \\dfrac \\pi 2$ is undefined"}
+{"_id": "7955", "title": "Tangent of 105 Degrees", "text": ":$\\tan 105^\\circ = \\tan \\dfrac {7 \\pi} {12} = - \\left({2 + \\sqrt 3}\\right)$"}
+{"_id": "7956", "title": "Tangent of 120 Degrees", "text": ":$\\tan 120 \\degrees = \\tan \\dfrac {2 \\pi} 3 = -\\sqrt 3$"}
+{"_id": "7957", "title": "Tangent of 135 Degrees", "text": ":$\\tan 135^\\circ = \\tan \\dfrac {3 \\pi} 4 = - 1$"}
+{"_id": "7958", "title": "Tangent of 150 Degrees", "text": ":$\\tan 150^\\circ = \\tan \\dfrac {5 \\pi} 6 = - \\dfrac {\\sqrt 3} 3$"}
+{"_id": "7959", "title": "Tangent of 165 Degrees", "text": ":$\\tan 165^\\circ = \\tan \\dfrac {11 \\pi} {12} = - \\left({2 - \\sqrt 3}\\right)$"}
+{"_id": "7960", "title": "Tangent of Straight Angle", "text": ":$\\tan 180^\\circ = \\tan \\pi = 0$"}
+{"_id": "7961", "title": "Tangent of 195 Degrees", "text": ":$\\tan 195^\\circ = \\tan \\dfrac {13 \\pi} {12} = 2 - \\sqrt 3$"}
+{"_id": "7962", "title": "Tangent of 210 Degrees", "text": ":$\\tan 210 \\degrees = \\tan \\dfrac {7 \\pi} 6 = \\dfrac {\\sqrt 3} 3$"}
+{"_id": "7963", "title": "Tangent of 225 Degrees", "text": ":$\\tan 225 \\degrees = \\tan \\dfrac {5 \\pi} 4 = 1$"}
+{"_id": "7964", "title": "Tangent of 240 Degrees", "text": ":$\\tan 240 \\degrees = \\tan \\dfrac {4 \\pi} 3 = \\sqrt 3$"}
+{"_id": "7965", "title": "Tangent of 255 Degrees", "text": ":$\\tan 255^\\circ = \\tan \\dfrac {17 \\pi} {12} = 2 + \\sqrt 3$"}
+{"_id": "7966", "title": "Tangent of Three Right Angles", "text": ":$\\tan 270^\\circ = \\tan \\dfrac {3 \\pi} 2$ is undefined"}
+{"_id": "7967", "title": "Tangent of 285 Degrees", "text": ":$\\tan 285^\\circ = \\tan \\dfrac {19 \\pi} {12} = -\\left({2 + \\sqrt 3}\\right)$"}
+{"_id": "7968", "title": "Tangent of 300 Degrees", "text": ":$\\tan 300^\\circ = \\tan \\dfrac {5 \\pi} 3 = -\\sqrt 3$"}
+{"_id": "7969", "title": "Tangent of 315 Degrees", "text": ":$\\tan 315^\\circ = \\tan \\dfrac {7 \\pi} 4 = -1$"}
+{"_id": "7970", "title": "Tangent of 330 Degrees", "text": ":$\\tan 330^\\circ = \\tan \\dfrac {11 \\pi} 6 = -\\dfrac {\\sqrt 3} 3$"}
+{"_id": "7971", "title": "Tangent of 345 Degrees", "text": ":$\\tan 345^\\circ = \\tan \\dfrac {23 \\pi} {12} = -\\left({2 - \\sqrt 3}\\right)$"}
+{"_id": "7972", "title": "Tangent of Full Angle", "text": ":$\\tan 360^\\circ = \\tan 2 \\pi = 0$"}
+{"_id": "7973", "title": "Cotangent of Zero", "text": ":$\\cot 0$ is undefined"}
+{"_id": "7974", "title": "Cotangent of 30 Degrees", "text": ":$\\cot 30^\\circ = \\cot \\dfrac {\\pi} 6 = \\sqrt 3$"}
+{"_id": "7975", "title": "Cotangent of 15 Degrees", "text": ":$\\cot 15^\\circ = \\cot \\dfrac {\\pi} {12} = 2 + \\sqrt 3$"}
+{"_id": "7976", "title": "Cotangent of 45 Degrees", "text": ":$\\cot 45 \\degrees = \\cot \\dfrac \\pi 4 = 1$"}
+{"_id": "7977", "title": "Cotangent of 60 Degrees", "text": ":$\\cot 60^\\circ = \\cot \\dfrac {\\pi} 3 = \\dfrac {\\sqrt 3} 3$"}
+{"_id": "7978", "title": "Cotangent of 75 Degrees", "text": ":$\\cot 75 \\degrees = \\cot \\dfrac {5 \\pi} {12} = 2 - \\sqrt 3$"}
+{"_id": "7979", "title": "Cotangent of Right Angle", "text": ":$\\cot 90^\\circ = \\cot \\dfrac \\pi 2 = 0$"}
+{"_id": "7980", "title": "Cotangent of 105 Degrees", "text": ":$\\cot 105^\\circ = \\cot \\dfrac {7 \\pi} {12} = -\\left({2 - \\sqrt 3}\\right)$"}
+{"_id": "7981", "title": "Cotangent of 120 Degrees", "text": ":$\\cot 120 \\degrees = \\cot \\dfrac {2 \\pi} 3 = -\\dfrac {\\sqrt 3} 3$"}
+{"_id": "7982", "title": "Cotangent of 135 Degrees", "text": ":$\\cot 135 \\degrees = \\cot \\dfrac {3 \\pi} 4 = -1$"}
+{"_id": "7983", "title": "Cotangent of 150 Degrees", "text": ":$\\cot 150 \\degrees = \\cot \\dfrac {5 \\pi} 6 = -\\sqrt 3$"}
+{"_id": "7984", "title": "Cotangent of 165 Degrees", "text": ":$\\cot 165 \\degrees = \\cot \\dfrac {11 \\pi} {12} = -\\paren {2 + \\sqrt 3}$"}
+{"_id": "7985", "title": "Cotangent of Straight Angle", "text": ":$\\cot 180^\\circ = \\cot \\pi$ is undefined"}
+{"_id": "7986", "title": "Cotangent of 195 Degrees", "text": ":$\\cot 195 \\degrees = \\cot \\dfrac {13 \\pi} {12} = 2 + \\sqrt 3$"}
+{"_id": "7987", "title": "Cotangent of 210 Degrees", "text": ":$\\cot 210^\\circ = \\cot \\dfrac {7 \\pi} 6 = \\sqrt 3$"}
+{"_id": "7988", "title": "Cotangent of 225 Degrees", "text": ":$\\cot 225^\\circ = \\cot \\dfrac {5 \\pi} 4 = 1$"}
+{"_id": "7989", "title": "Cotangent of 240 Degrees", "text": ":$\\cot 240^\\circ = \\cot \\dfrac {4 \\pi} 3 = \\dfrac {\\sqrt 3} 3$"}
+{"_id": "7990", "title": "Cotangent of 255 Degrees", "text": ":$\\cot 255 \\degrees = \\cot \\dfrac {17 \\pi} {12} = 2 - \\sqrt 3$"}
+{"_id": "7991", "title": "Cotangent of Three Right Angles", "text": ":$\\cot 270^\\circ = \\cot \\dfrac {3 \\pi} 2 = 0$"}
+{"_id": "7992", "title": "Cotangent of 285 Degrees", "text": ":$\\cot 285 \\degrees = \\cot \\dfrac {19 \\pi} {12} = -\\paren {2 - \\sqrt 3}$"}
+{"_id": "7993", "title": "Cotangent of 300 Degrees", "text": ":$\\cot 300 \\degrees = \\cot \\dfrac {5 \\pi} 3 = - \\dfrac {\\sqrt 3} 3$"}
+{"_id": "7994", "title": "Cotangent of 315 Degrees", "text": ":$\\cot 315^\\circ = \\cot \\dfrac {7 \\pi} 4 = -1$"}
+{"_id": "7995", "title": "Cotangent of 330 Degrees", "text": ":$\\cot 330^\\circ = \\cot \\dfrac {11 \\pi} 6 = -\\sqrt 3$"}
+{"_id": "7996", "title": "Cotangent of 345 Degrees", "text": ":$\\cot 345 \\degrees = \\cot \\dfrac {23 \\pi} {12} = -\\paren {2 + \\sqrt 3}$"}
+{"_id": "7997", "title": "Cotangent of Full Angle", "text": ":$\\cot 360^\\circ = \\cot 2 \\pi$ is undefined"}
+{"_id": "7998", "title": "Secant of Zero", "text": ":$\\sec 0 = 0$"}
+{"_id": "7999", "title": "Secant of 15 Degrees", "text": ":$\\sec 15 \\degrees = \\sec \\dfrac \\pi {12} = \\sqrt 6 - \\sqrt 2$"}
+{"_id": "8000", "title": "Secant of 30 Degrees", "text": ":$\\sec 30 \\degrees = \\sec \\dfrac \\pi 6 = \\dfrac {2 \\sqrt 3} 3$"}
+{"_id": "8001", "title": "Secant of 45 Degrees", "text": ":$\\sec 45 \\degrees = \\sec \\dfrac \\pi 4 = \\sqrt 2$"}
+{"_id": "8002", "title": "Secant of 60 Degrees", "text": ":$\\sec 60 \\degrees = \\sec \\dfrac \\pi 3 = 2$"}
+{"_id": "8003", "title": "Secant of 75 Degrees", "text": ":$\\sec 75 \\degrees = \\sec \\dfrac {5 \\pi} {12} = \\sqrt 6 + \\sqrt 2$"}
+{"_id": "8004", "title": "Secant of Right Angle", "text": ":$\\sec 90 \\degrees = \\sec \\dfrac \\pi 2$ is undefined"}
+{"_id": "8005", "title": "Secant of Angle plus Right Angle", "text": ": $\\sec \\left({x + \\dfrac \\pi 2}\\right) = -\\csc x$"}
+{"_id": "8006", "title": "Cosecant of Angle plus Right Angle", "text": ": $\\csc \\left({x + \\dfrac \\pi 2}\\right) = \\sec x$"}
+{"_id": "8010", "title": "Cosecant of Supplementary Angle", "text": ":$\\csc \\left({\\pi - \\theta}\\right) = \\csc \\theta$ where $\\csc$ denotes cosecant. That is, the cosecant of an angle equals its supplement."}
+{"_id": "8011", "title": "Secant of Conjugate Angle", "text": ":$\\map \\sec {2 \\pi - \\theta} = \\sec \\theta$ where $\\sec$ denotes secant. That is, the secant of an angle equals its conjugate."}
+{"_id": "8012", "title": "Cosecant of Conjugate Angle", "text": ":$\\map \\csc {2 \\pi - \\theta} = -\\csc \\theta$ where $\\csc$ denotes cosecant. That is, the cosecant of an angle is the negative of its conjugate."}
+{"_id": "8013", "title": "Secant of 105 Degrees", "text": ":$\\sec 105 \\degrees = \\sec \\dfrac {7 \\pi} {12} = -\\paren {\\sqrt 6 + \\sqrt 2}$"}
+{"_id": "8014", "title": "Secant of 120 Degrees", "text": ":$\\sec 120 \\degrees = \\sec \\dfrac {2 \\pi} 3 = -2$"}
+{"_id": "8015", "title": "Secant of 135 Degrees", "text": ":$\\sec 135 \\degrees = \\sec \\dfrac {3 \\pi} 4 = -\\sqrt 2$"}
+{"_id": "8016", "title": "Secant of 150 Degrees", "text": ":$\\sec 150 \\degrees = \\sec \\dfrac {5 \\pi} 6 = -\\dfrac {2 \\sqrt 3} 3$"}
+{"_id": "8017", "title": "Secant of 165 Degrees", "text": ":$\\sec 165 \\degrees = \\sec \\dfrac {11 \\pi} {12} = -\\paren {\\sqrt 6 - \\sqrt 2}$"}
+{"_id": "8018", "title": "Secant of Straight Angle", "text": ":$\\sec 180 \\degrees = \\sec \\pi = -1$"}
+{"_id": "8019", "title": "Secant of 195 Degrees", "text": ":$\\sec 195 \\degrees = \\sec \\dfrac {13 \\pi} {12} = -\\paren {\\sqrt 6 - \\sqrt 2}$"}
+{"_id": "8020", "title": "Secant of 210 Degrees", "text": ":$\\sec 210 \\degrees = \\sec \\dfrac {7 \\pi} 6 = -2 \\dfrac {\\sqrt 3} 3$"}
+{"_id": "8021", "title": "Secant of 225 Degrees", "text": ":$\\sec 225 \\degrees = \\sec \\dfrac {5 \\pi} 4 = -\\sqrt 2$"}
+{"_id": "8022", "title": "Secant of 240 Degrees", "text": ":$\\sec 240 \\degrees = \\sec \\dfrac {4 \\pi} 3 = - 2$"}
+{"_id": "8023", "title": "Secant of 255 Degrees", "text": ":$\\sec 255 \\degrees = \\sec \\dfrac {17 \\pi} {12} = -\\paren {\\sqrt 6 + \\sqrt 2}$"}
+{"_id": "8024", "title": "Secant of Three Right Angles", "text": ":$\\sec 270 \\degrees = \\sec \\dfrac {3 \\pi} 2$ is undefined"}
+{"_id": "8025", "title": "Secant of 285 Degrees", "text": ":$\\sec 285 \\degrees = \\sec \\dfrac {19 \\pi} {12} = \\sqrt 6 + \\sqrt 2$"}
+{"_id": "8026", "title": "Secant of 300 Degrees", "text": ":$\\sec 300 \\degrees = \\sec \\dfrac {5 \\pi} 3 = 2$"}
+{"_id": "8027", "title": "Secant of 315 Degrees", "text": ":$\\sec 315 \\degrees = \\sec \\dfrac {7 \\pi} 4 = \\sqrt 2$"}
+{"_id": "8028", "title": "Secant of 330 Degrees", "text": ":$\\sec 330 \\degrees = \\sec \\dfrac {11 \\pi} 6 = 2 \\dfrac {\\sqrt 3} 3$"}
+{"_id": "8029", "title": "Secant of 345 Degrees", "text": ":$\\sec 345 \\degrees = \\sec \\dfrac {23 \\pi} {12} = \\sqrt 6 - \\sqrt 2$"}
+{"_id": "8030", "title": "Secant of Full Angle", "text": ":$\\sec 360 \\degrees = \\sec 2 \\pi = 1$"}
+{"_id": "8031", "title": "Cosecant of Zero", "text": ":$\\csc 0$ is undefined"}
+{"_id": "8032", "title": "Cosecant of 15 Degrees", "text": ":$\\csc 15^\\circ = \\csc \\dfrac \\pi {12} = \\sqrt 6 + \\sqrt 2$"}
+{"_id": "8033", "title": "Cosecant of 30 Degrees", "text": ":$\\csc 30^\\circ = \\csc \\dfrac \\pi 6 = 2$"}
+{"_id": "8034", "title": "Cosecant of 45 Degrees", "text": ":$\\csc 45^\\circ = \\csc \\dfrac \\pi 4 = \\sqrt 2$"}
+{"_id": "8035", "title": "Cosecant of 60 Degrees", "text": ":$\\csc 60^\\circ = \\csc \\dfrac \\pi 3 = \\dfrac {2 \\sqrt 3} 3$"}
+{"_id": "8036", "title": "Cosecant of 75 Degrees", "text": ":$\\csc 75 \\degrees = \\csc \\dfrac {5 \\pi} {12} = \\sqrt 6 - \\sqrt 2$"}
+{"_id": "8037", "title": "Cosecant of Right Angle", "text": ":$\\csc 90^\\circ = \\csc \\dfrac \\pi 2 = 1$"}
+{"_id": "8038", "title": "Cosecant of 105 Degrees", "text": ":$\\csc 105 \\degrees = \\csc \\dfrac {7 \\pi} {12} = \\sqrt 6 - \\sqrt 2$"}
+{"_id": "8039", "title": "Cosecant of 120 Degrees", "text": ":$\\csc 120 \\degrees = \\csc \\dfrac {2 \\pi} 3 = \\dfrac {2 \\sqrt 3} 3$"}
+{"_id": "8040", "title": "Cosecant of 135 Degrees", "text": ":$\\csc 135 \\degrees = \\csc \\dfrac {3 \\pi} 4 = \\sqrt 2$"}
+{"_id": "8041", "title": "Cosecant of 150 Degrees", "text": ":$\\csc 150 \\degrees = \\csc \\dfrac {5 \\pi} 6 = \\sqrt 2$"}
+{"_id": "8042", "title": "Cosecant of 165 Degrees", "text": ":$\\csc 165 \\degrees = \\csc \\dfrac {11 \\pi} {12} = \\sqrt 6 + \\sqrt 2$"}
+{"_id": "8043", "title": "Cosecant of Straight Angle", "text": ":$\\csc 180^\\circ = \\csc \\pi$ is undefined"}
+{"_id": "8044", "title": "Cosecant of 195 Degrees", "text": ":$\\csc 195^\\circ = \\csc \\dfrac {13 \\pi} {12} = - \\left({\\sqrt 6 + \\sqrt 2}\\right)$"}
+{"_id": "8045", "title": "Cosecant of 210 Degrees", "text": ":$\\csc 210^\\circ = \\csc \\dfrac {7 \\pi} 6 = -2$"}
+{"_id": "8046", "title": "Cosecant of 225 Degrees", "text": ":$\\csc 225 \\degrees = \\csc \\dfrac {5 \\pi} 4 = -\\sqrt 2$"}
+{"_id": "8047", "title": "Cosecant of 240 Degrees", "text": ":$\\csc 240 \\degrees = \\csc \\dfrac {4 \\pi} 3 = -\\dfrac {2 \\sqrt 3} 3$"}
+{"_id": "8048", "title": "Cosecant of 255 Degrees", "text": ":$\\csc 255^\\circ = \\csc \\dfrac {17 \\pi} {12} = -\\left({\\sqrt 6 - \\sqrt 2}\\right)$"}
+{"_id": "8049", "title": "Cosecant of Three Right Angles", "text": ":$\\csc 270^\\circ = \\csc \\dfrac {3 \\pi} 2 = -1$"}
+{"_id": "8050", "title": "Cosecant of 285 Degrees", "text": ":$\\csc 285^\\circ = \\csc \\dfrac {19 \\pi} {12} = -\\left({\\sqrt 6 - \\sqrt 2}\\right)$"}
+{"_id": "8051", "title": "Cosecant of 300 Degrees", "text": ":$\\csc 300^\\circ = \\csc \\dfrac {5 \\pi} 3 = -\\dfrac {2 \\sqrt 3} 3$"}
+{"_id": "8052", "title": "Cosecant of 315 Degrees", "text": ":$\\csc 315^\\circ = \\csc \\dfrac {7 \\pi} 4 = -\\sqrt 2$"}
+{"_id": "8053", "title": "Cosecant of 330 Degrees", "text": ":$\\csc 330 \\degrees = \\csc \\dfrac {11 \\pi} 6 = -2$"}
+{"_id": "8054", "title": "Cosecant of 345 Degrees", "text": ":$\\csc 345 \\degrees = \\csc \\dfrac {23 \\pi} {12} = -\\paren {\\sqrt 6 + \\sqrt 2}$"}
+{"_id": "8055", "title": "Cosecant of Full Angle", "text": ":$\\csc 360 \\degrees = \\csc 2 \\pi$ is undefined"}
+{"_id": "8056", "title": "Shape of Sine Function/Graph", "text": ":800px"}
+{"_id": "8057", "title": "Shape of Cosine Function/Graph", "text": ":800px"}
+{"_id": "8058", "title": "Shape of Tangent Function/Graph", "text": ":800px"}
+{"_id": "8059", "title": "Shape of Cotangent Function/Graph", "text": ":800px"}
+{"_id": "8060", "title": "Shape of Secant Function/Graph", "text": ":800px"}
+{"_id": "8061", "title": "Cosecant Function is Odd", "text": ":$\\map \\csc {-x} = -\\csc x$ That is, the cosecant function is odd."}
+{"_id": "8062", "title": "Secant Function is Even", "text": ":$\\map \\sec {-x} = \\sec x$ That is, the secant function is even."}
+{"_id": "8063", "title": "Cotangent Function is Odd", "text": ":$\\map \\cot {-x} = -\\cot x$ That is, the cotangent function is odd."}
+{"_id": "8064", "title": "Cotangent of Sum", "text": ":$\\cot \\left({a + b}\\right) = \\dfrac {\\cot a \\cot b - 1} {\\cot b + \\cot a}$ where $\\cot $ is cotangent. === Corollary === {{:Cotangent of Sum/Corollary}}"}
+{"_id": "8065", "title": "Half-Integer is Half Odd Integer", "text": "Let $r$ be a number. Then $r$ is a half-integer {{iff}} $r = \\dfrac n 2$ where $n$ is an odd integer."}
+{"_id": "8071", "title": "Sine of Angle plus Straight Angle", "text": ":$\\map \\sin {x + \\pi} = -\\sin x$"}
+{"_id": "8072", "title": "Cosine of Angle plus Straight Angle", "text": ":$\\map \\cos {x + \\pi} = -\\cos x$"}
+{"_id": "8073", "title": "Sine of Angle plus Full Angle", "text": ":$\\map \\sin {x + 2 \\pi} = \\sin x$"}
+{"_id": "8074", "title": "Cosine of Angle plus Full Angle", "text": ": $\\cos \\paren {x + 2 \\pi} = \\cos x$"}
+{"_id": "8075", "title": "Tangent of Angle plus Full Angle", "text": ": $\\tan \\left({x + 2 \\pi}\\right) = \\tan x$"}
+{"_id": "8076", "title": "Tangent of Angle plus Straight Angle", "text": ": $\\tan \\left({x + \\pi}\\right) = \\tan x$"}
+{"_id": "8077", "title": "Cotangent of Angle plus Straight Angle", "text": ":$\\map \\cot {x + \\pi} = \\cot x$"}
+{"_id": "8078", "title": "Secant of Angle plus Straight Angle", "text": ":$\\map \\sec {x + \\pi} = -\\sec x$"}
+{"_id": "8079", "title": "Cosecant of Angle plus Straight Angle", "text": ":$\\map \\csc {x + \\pi} = -\\csc x$"}
+{"_id": "8080", "title": "Cotangent of Angle plus Full Angle", "text": ": $\\cot \\left({x + 2 \\pi}\\right) = \\cot x$"}
+{"_id": "8081", "title": "Secant of Angle plus Full Angle", "text": ": $\\sec \\left({x + 2 \\pi}\\right) = \\sec x$"}
+{"_id": "8082", "title": "Cosecant of Angle plus Full Angle", "text": ": $\\csc \\left({x + 2 \\pi}\\right) = \\csc x$"}
+{"_id": "8087", "title": "Sine of Angle plus Three Right Angles", "text": ":$\\map \\sin {x + \\dfrac {3 \\pi} 2} = -\\cos x$"}
+{"_id": "8088", "title": "Cosine of Angle plus Three Right Angles", "text": ":$\\map \\cos {x + \\dfrac {3 \\pi} 2} = \\sin x$"}
+{"_id": "8089", "title": "Tangent of Angle plus Three Right Angles", "text": ":$\\map \\tan {x + \\dfrac {3 \\pi} 2} = -\\cot x$"}
+{"_id": "8090", "title": "Cotangent of Angle plus Three Right Angles", "text": ":$\\map \\cot {x + \\dfrac {3 \\pi} 2} = -\\tan x$"}
+{"_id": "8093", "title": "Sine of Three Right Angles less Angle", "text": ":$\\map \\sin {\\dfrac {3 \\pi} 2 - \\theta} = -\\cos \\theta$ where $\\sin$ and $\\cos$ are sine and cosine respectively."}
+{"_id": "8094", "title": "Cosine of Three Right Angles less Angle", "text": ":$\\cos \\left({\\dfrac {3 \\pi} 2 - \\theta}\\right) = - \\sin \\theta$ where $\\cos$ and $\\sin$ are cosine and sine respectively."}
+{"_id": "8096", "title": "Cosecant of Three Right Angles less Angle", "text": ":$\\map \\csc {\\dfrac {3 \\pi} 2 - \\theta} = -\\sec \\theta$ where $\\csc$ and $\\sec$ are cosecant and secant respectively."}
+{"_id": "8099", "title": "Double Angle Formulas/Sine", "text": ":$\\sin 2 \\theta = 2 \\sin \\theta \\cos \\theta$"}
+{"_id": "8102", "title": "Double Angle Formulas/Cosine", "text": ":$\\cos 2 \\theta = \\cos^2 \\theta - \\sin^2 \\theta$"}
+{"_id": "8105", "title": "Double Angle Formulas/Tangent", "text": ":$\\map \\tan {2 \\theta} = \\dfrac {2 \\tan \\theta} {1 - \\tan^2 \\theta}$"}
+{"_id": "8106", "title": "Triple Angle Formulas", "text": "=== Triple Angle Formula for Sine === {{:Triple Angle Formulas/Sine}} === Triple Angle Formula for Cosine === {{:Triple Angle Formulas/Cosine}} === Triple Angle Formula for Tangent === {{:Triple Angle Formulas/Tangent}}"}
+{"_id": "8107", "title": "Triple Angle Formulas/Sine", "text": ":$\\sin 3 \\theta = 3 \\sin \\theta - 4 \\sin^3 \\theta$"}
+{"_id": "8108", "title": "Triple Angle Formulas/Cosine", "text": ":$\\cos 3 \\theta = 4 \\cos^3 \\theta - 3 \\cos \\theta$"}
+{"_id": "8109", "title": "Triple Angle Formulas/Tangent", "text": ":$\\map \\tan {3 \\theta} = \\dfrac {3 \\tan \\theta - \\tan^3 \\theta} {1 - 3 \\tan^2 \\theta}$"}
+{"_id": "8110", "title": "Quadruple Angle Formulas", "text": "=== Quadruple Angle Formula for Sine === {{:Quadruple Angle Formulas/Sine}} === Quadruple Angle Formula for Cosine === {{:Quadruple Angle Formulas/Cosine}} === Quadruple Angle Formula for Tangent === {{:Quadruple Angle Formulas/Tangent}}"}
+{"_id": "8111", "title": "Quadruple Angle Formulas/Sine", "text": ":$\\sin 4 \\theta = 4 \\sin \\theta \\cos \\theta - 8 \\sin^3 \\theta \\cos \\theta$"}
+{"_id": "8112", "title": "Quadruple Angle Formulas/Cosine", "text": ":$\\cos 4 \\theta = 8 \\cos^4 \\theta - 8 \\cos^2 \\theta + 1$"}
+{"_id": "8113", "title": "Quadruple Angle Formulas/Tangent", "text": ":$\\tan 4 \\theta = \\dfrac {4 \\tan \\theta - 4 \\tan^3 \\theta} {1 - 6 \\tan^2 \\theta + \\tan^4 \\theta}$"}
+{"_id": "8114", "title": "Quintuple Angle Formulas/Sine", "text": ":$\\sin 5 \\theta = 5 \\sin \\theta - 20 \\sin^3 \\theta + 16 \\sin^5 \\theta$"}
+{"_id": "8115", "title": "Quintuple Angle Formulas/Cosine", "text": ":$\\cos 5 \\theta = 16 \\cos^5 \\theta - 20 \\cos^3 \\theta + 5 \\cos \\theta$"}
+{"_id": "8116", "title": "Quintuple Angle Formulas/Tangent", "text": ":$\\tan 5 \\theta = \\dfrac {\\tan^5 \\theta - 10 \\tan^3 \\theta + 5 \\tan \\theta} {1 - 10 \\tan^2 \\theta + 5 \\tan^4 \\theta}$"}
+{"_id": "8117", "title": "Power Reduction Formulas/Sine Squared", "text": ":$\\sin^2 x = \\dfrac {1 - \\cos 2 x} 2$"}
+{"_id": "8118", "title": "Power Reduction Formulas/Cosine Squared", "text": ":$\\cos^2 x = \\dfrac {1 + \\cos 2 x} 2$"}
+{"_id": "8119", "title": "Power Reduction Formulas/Tangent Squared", "text": ":$\\tan^2x = \\dfrac {1 - \\cos2x} {1 + \\cos2x}$"}
+{"_id": "8120", "title": "Power Reduction Formulas/Sine Cubed", "text": ":$\\sin^3 x = \\dfrac {3 \\sin x - \\sin 3 x} 4$"}
+{"_id": "8121", "title": "Power Reduction Formulas/Cosine Cubed", "text": ":$\\cos^3 x = \\dfrac {3 \\cos x + \\cos 3 x} 4$"}
+{"_id": "8122", "title": "Power Reduction Formulas/Sine to 4th", "text": ":$\\sin^4 x = \\dfrac {3 - 4 \\cos 2 x + \\cos 4 x} 8$"}
+{"_id": "8123", "title": "Power Reduction Formulas/Cosine to 4th", "text": ":$\\cos^4 x = \\dfrac {3 + 4 \\cos 2 x + \\cos 4 x} 8$"}
+{"_id": "8124", "title": "Power Reduction Formulas/Sine to 5th", "text": ":$\\sin^5 x = \\dfrac {10 \\sin x - 5 \\sin 3 x + \\sin 5 x} {16}$"}
+{"_id": "8125", "title": "Power Reduction Formulas/Cosine to 5th", "text": ":$\\cos^5 x = \\dfrac {10 \\cos x + 5 \\cos 3 x + \\cos 5 x} {16}$"}
+{"_id": "8126", "title": "Orthocomplement of Subset of Orthocomplement is Superset", "text": "Let $H$ be a Hilbert space. Let $A, B \\subseteq H$ be subsets of $H$ such that $B \\subseteq A^\\perp$, where $A^\\perp$ is the orthocomplement of $A$. Then $A \\subseteq B^\\perp$."}
+{"_id": "8127", "title": "Simpson's Formulas/Cosine by Cosine", "text": ":$\\cos \\alpha \\cos \\beta = \\dfrac {\\map \\cos {\\alpha - \\beta} + \\map \\cos {\\alpha + \\beta} } 2$"}
+{"_id": "8128", "title": "Simpson's Formulas/Sine by Sine", "text": ":$\\sin \\alpha \\sin \\beta = \\dfrac {\\map \\cos {\\alpha - \\beta} - \\map \\cos {\\alpha + \\beta} } 2$"}
+{"_id": "8129", "title": "Simpson's Formulas/Cosine by Sine", "text": ":$\\cos \\alpha \\sin \\beta = \\dfrac {\\map \\sin {\\alpha + \\beta} - \\map \\sin {\\alpha - \\beta} } 2$"}
+{"_id": "8130", "title": "Simpson's Formulas/Sine by Cosine", "text": ":$\\sin \\alpha \\cos \\beta = \\dfrac {\\map \\sin {\\alpha + \\beta} + \\map \\sin {\\alpha - \\beta} } 2$"}
+{"_id": "8131", "title": "Prosthaphaeresis Formulas/Sine plus Sine", "text": ":$\\sin \\alpha + \\sin \\beta = 2 \\map \\sin {\\dfrac {\\alpha + \\beta} 2} \\map \\cos {\\dfrac {\\alpha - \\beta} 2}$"}
+{"_id": "8132", "title": "Prosthaphaeresis Formulas/Sine minus Sine", "text": ":$\\sin \\alpha - \\sin \\beta = 2 \\map \\cos {\\dfrac {\\alpha + \\beta} 2} \\map \\sin {\\dfrac {\\alpha - \\beta} 2}$"}
+{"_id": "8133", "title": "Prosthaphaeresis Formulas/Cosine plus Cosine", "text": ":$\\cos \\alpha + \\cos \\beta = 2 \\, \\map \\cos {\\dfrac {\\alpha + \\beta} 2} \\, \\map \\cos {\\dfrac {\\alpha - \\beta} 2}$"}
+{"_id": "8134", "title": "Prosthaphaeresis Formulas/Cosine minus Cosine", "text": ":$\\cos \\alpha - \\cos \\beta = -2 \\, \\map \\sin {\\dfrac {\\alpha + \\beta} 2} \\, \\map \\sin {\\dfrac {\\alpha - \\beta} 2}$"}
+{"_id": "8135", "title": "Sine of Integer Multiple of Argument/Formulation 1", "text": "{{begin-eqn}} {{eqn | l = \\sin n \\theta       | r = \\sin \\theta \\paren {\\paren {2 \\cos \\theta}^{n - 1} - \\dbinom {n - 2} 1 \\paren {2 \\cos \\theta}^{n - 3} + \\dbinom {n - 3} 2 \\paren {2 \\cos \\theta}^{n - 5} - \\cdots}       | c =  }} {{eqn | r = \\sin \\theta \\paren {\\sum_{k \\mathop \\ge 0} \\paren {-1}^k \\binom {n - \\paren {k + 1} } k \\paren {2 \\cos \\theta}^{n - \\paren {2 k + 1} } }       | c =  }} {{end-eqn}}"}
+{"_id": "8140", "title": "Cosine to Power of Odd Integer", "text": ":$\\displaystyle \\cos^{2n+1} \\theta = \\frac 1 {2^{2n}} \\sum_{k \\mathop = 0}^n \\binom {2n+1} k \\cos \\left({2n - 2k + 1}\\right) \\theta$"}
+{"_id": "8143", "title": "Sum of Arctangent and Arccotangent", "text": "Let $x \\in \\R$ be a real number. Then: : $\\arctan x + \\operatorname{arccot} x = \\dfrac \\pi 2$ where $\\arctan$ and $\\operatorname{arccot}$ denote arctangent and arccotangent respectively."}
+{"_id": "8144", "title": "Arcsine of Reciprocal equals Arccosecant", "text": ":$\\map \\arcsin {\\dfrac 1 x} = \\arccsc x$"}
+{"_id": "8145", "title": "Arccosine of Reciprocal equals Arcsecant", "text": ":$\\map \\arccos {\\dfrac 1 x} = \\arcsec x$"}
+{"_id": "8146", "title": "Arctangent of Reciprocal equals Arccotangent", "text": ":$\\map \\arctan {\\dfrac 1 x} = \\arccot x$"}
+{"_id": "8147", "title": "Inverse Sine is Odd Function", "text": "Everywhere that the function is defined: :$\\map \\arcsin {-x} = -\\arcsin x$"}
+{"_id": "8148", "title": "Arccosine of Negative Argument", "text": "Everywhere that the function is defined: :$\\map \\arccos {-x} = \\pi - \\arccos x$"}
+{"_id": "8149", "title": "Inverse Tangent is Odd Function", "text": "Everywhere that the function is defined: :$\\map \\arctan {-x} = -\\arctan x$"}
+{"_id": "8150", "title": "Arccotangent of Negative Argument", "text": "Everywhere that the function is defined: :$\\map \\arccot {-x} = \\pi - \\arccot x$"}
+{"_id": "8151", "title": "Arcsecant of Negative Argument", "text": "Everywhere that the function is defined: :$\\map \\arcsec {-x} = \\pi - \\arcsec x$"}
+{"_id": "8152", "title": "Inverse Cosecant is Odd Function", "text": "Everywhere that the function is defined: :$\\map \\arccsc {-x} = -\\arccsc x$"}
+{"_id": "8156", "title": "Sine of Angle of Triangle by Semiperimeter", "text": "Let $\\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$. Then: : $\\sin A = \\dfrac 2 {b c} \\sqrt {s \\paren {s - a} \\paren {s - b} \\paren {s - c} }$ where $\\sin$ denotes sine and $s$ is the semiperimeter: $s = \\dfrac {a + b + c} 2$."}
+{"_id": "8157", "title": "Functionally Complete Logical Connectives/Negation, Conjunction, Disjunction and Implication", "text": ": $\\left\\{{\\neg, \\land, \\lor, \\implies}\\right\\}$: Not, And, Or and Implies"}
+{"_id": "8158", "title": "Functionally Complete Logical Connectives/Negation and Disjunction", "text": ":$\\set {\\neg, \\lor}$: Not and Or"}
+{"_id": "8159", "title": "Functionally Complete Logical Connectives/Negation and Conjunction", "text": ":$\\set {\\neg, \\land}$: Not and And"}
+{"_id": "8160", "title": "Functionally Complete Logical Connectives/Negation and Conditional", "text": ":$\\set {\\neg, \\implies}$: Not and Implies"}
+{"_id": "8161", "title": "Functionally Complete Logical Connectives/Conjunction, Negation and Disjunction", "text": ":$\\set {\\neg, \\land, \\lor}$: Not, And and Or"}
+{"_id": "8162", "title": "Functionally Complete Logical Connectives/NAND", "text": ":$\\set {\\uparrow}$: NAND"}
+{"_id": "8163", "title": "Functionally Complete Logical Connectives/NOR", "text": ":$\\set {\\downarrow}$: NOR"}
+{"_id": "8165", "title": "NOR with Equal Arguments/Proof 1", "text": "Let $\\downarrow$ signify the NOR operation. Then for any proposition $p$: :$p \\downarrow p \\dashv \\vdash \\neg p$ That is, the NOR of a proposition with itself corresponds to the negation operator."}
+{"_id": "8166", "title": "NOR with Equal Arguments/Proof 2", "text": "Let $\\downarrow$ signify the NOR operation. Then for any proposition $p$: :$p \\downarrow p \\dashv \\vdash \\neg p$ That is, the NOR of a proposition with itself corresponds to the negation operator."}
+{"_id": "8167", "title": "NAND with Equal Arguments/Proof 1", "text": "Let $\\uparrow$ signify the NAND operation. Then, for any proposition $p$: :$p \\uparrow p \\dashv \\vdash \\neg p$ That is, the NAND of a proposition with itself corresponds to the negation operation."}
+{"_id": "8168", "title": "NAND with Equal Arguments/Proof 2", "text": "Let $\\uparrow$ signify the NAND operation. Then, for any proposition $p$: :$p \\uparrow p \\dashv \\vdash \\neg p$ That is, the NAND of a proposition with itself corresponds to the negation operation."}
+{"_id": "8169", "title": "Conditional and Converse are not Equivalent", "text": "A conditional statement: :$p \\implies q$ is not logically equivalent to its converse: :$q \\implies p$"}
+{"_id": "8172", "title": "NOR is Commutative/Proof 1", "text": "Let $\\downarrow$ signify the NOR operation. Then, for any two propositions $p$ and $q$: :$p \\downarrow q \\dashv \\vdash q \\downarrow p$ That is, NOR is commutative."}
+{"_id": "8173", "title": "NOR is Commutative/Proof 2", "text": "Let $\\downarrow$ signify the NOR operation. Then, for any two propositions $p$ and $q$: :$p \\downarrow q \\dashv \\vdash q \\downarrow p$ That is, NOR is commutative."}
+{"_id": "8176", "title": "NOR is not Associative/Proof 1", "text": "Let $\\downarrow$ signify the NOR operation. Then there exist propositions $p,q,r$ such that: :$p \\downarrow \\left({q \\downarrow r}\\right) \\not \\vdash \\left({p \\downarrow q}\\right) \\downarrow r$ That is, NOR is not associative."}
+{"_id": "8177", "title": "NOR is not Associative/Proof 2", "text": "Let $\\downarrow$ signify the NOR operation. Then there exist propositions $p, q, r$ such that: :$p \\downarrow \\paren {q \\downarrow r} \\not \\vdash \\paren {p \\downarrow q} \\downarrow r$ That is, NOR is not associative."}
+{"_id": "8178", "title": "De Morgan's Laws (Predicate Logic)/Assertion of Universality", "text": ":$\\forall x: \\map P x \\dashv \\vdash \\neg \\exists x: \\neg \\map P x$ ::''If everything '''is''', there exists nothing that '''is not'''.''"}
+{"_id": "8179", "title": "De Morgan's Laws (Predicate Logic)/Denial of Existence", "text": ":$\\forall x: \\neg \\map P x \\dashv \\vdash \\neg \\exists x: \\map P x$ ::''If everything '''is not''', there exists nothing that '''is'''.''"}
+{"_id": "8180", "title": "De Morgan's Laws (Predicate Logic)/Denial of Universality", "text": ":$\\neg \\forall x: \\map P x \\dashv \\vdash \\exists x: \\neg \\map P x$ ::''If not everything '''is''', there exists something that '''is not'''.''"}
+{"_id": "8181", "title": "De Morgan's Laws (Predicate Logic)/Assertion of Existence", "text": ":$\\neg \\forall x: \\neg \\map P x \\dashv \\vdash \\exists x: \\map P x$ ::''If not everything '''is not''', there exists something that '''is'''.''"}
+{"_id": "8182", "title": "Conjunction in terms of NAND", "text": ":$p \\land q \\dashv \\vdash \\paren {p \\uparrow q} \\uparrow \\paren {p \\uparrow q}$ where $\\land$ denotes logical conjunction and $\\uparrow$ denotes logical NAND."}
+{"_id": "8183", "title": "Disjunction in terms of NOR", "text": ":$p \\lor q \\dashv \\vdash \\paren {p \\downarrow q} \\downarrow \\paren {p \\downarrow q}$ where $\\lor$ denotes logical disjunction and $\\downarrow$ denotes logical NOR."}
+{"_id": "8184", "title": "Disjunction in terms of NAND", "text": ":$p \\lor q \\dashv \\vdash \\paren {p \\uparrow p} \\uparrow \\paren {q \\uparrow q}$ where $\\lor$ denotes logical disjunction and $\\uparrow$ denotes logical NAND."}
+{"_id": "8185", "title": "Implication in terms of NAND", "text": ":$p \\implies q \\dashv \\vdash p \\uparrow \\paren {q \\uparrow q}$"}
+{"_id": "8195", "title": "Mapping is Injection and Surjection iff Inverse is Mapping", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Then: : $f: S \\to T$ can be defined as a bijection in the sense that: ::$(1): \\quad f$ is an injection ::$(2): \\quad f$ is a surjection {{iff}}: :the inverse $f^{-1}$ of $f$ is itself a mapping."}
+{"_id": "8197", "title": "Condition for Composite Mapping to be Identity", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ and $g: T \\to S$ be mappings such that: : $g \\circ f = I_S$ where $I_S$ is the identity mapping on $S$. Then $f$ is an injection and $g$ is a surjection."}
+{"_id": "8199", "title": "Eigenvalues of G-Representation are Roots of Unity", "text": "Let $G$ be a finite group. Let $\\left({K, +, \\cdot}\\right)$ be a field. Let $V$ be a $G$-module over $K$ (i.e. $V$ is a $K \\left[{G}\\right]$-module). Then $\\forall g \\in G$, the eigenvalues of the action by the vector $g \\in K \\left[{G}\\right]$ on $V$ are roots of unity."}
+{"_id": "8200", "title": "Character of Representations over C are Algebraic Integers", "text": "Let $G$ be a finite group. Let $\\chi$ be the character of any [[Definition:G-Module|$\\C \\left[{G}\\right]$-module]] $\\left({V, \\rho}\\right)$. Then for all $g \\in G$, it follows that $\\chi \\left({g}\\right)$ is an algebraic integer."}
+{"_id": "8201", "title": "Injection from Finite Set to Itself is Surjection", "text": "Let $S$ be a finite set. Let $f: S \\to S$ be an injection. Then $f$ is also a surjection."}
+{"_id": "8202", "title": "Codomain of Bijection is Domain of Inverse", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a bijection. Let $f^{-1}: T \\to S$ be the inverse of $f$. Then the domain of $f^{-1}$ equals the codomain of $f$."}
+{"_id": "8203", "title": "Domain of Bijection is Codomain of Inverse", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a bijection. Let $f^{-1}: T \\to S$ be the inverse of $f$. Then the codomain of $f^{-1}$ equals the domain of $f$."}
+{"_id": "8204", "title": "Conditions for Uniqueness of Left Inverse Mapping", "text": "Let $S$ and $T$ be sets such that $S \\ne \\O$. Let $f: S \\to T$ be an injection. Then a left inverse mapping of $f$ is in general not unique. Uniqueness occurs under either of two circumstances: :$(1): \\quad S$ is a singleton :$(2): \\quad f$ is a bijection."}
+{"_id": "8205", "title": "Surjection iff Right Inverse/Non-Uniqueness", "text": "A right inverse of $f$ is in general not unique. Uniqueness occurs {{iff}} $f$ is a bijection."}
+{"_id": "8206", "title": "Mapping reflects Preordering", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let ${\\precsim} \\subseteq T \\times T$ be a preordering on $T$. Let $\\RR$ be the relation defined on $S$ by the rule: :$x \\mathrel \\RR y \\iff \\map f x \\precsim \\map f y$ Then $\\RR$ is a preordering on $S$."}
+{"_id": "8209", "title": "Bernoulli's Inequality", "text": "Let $x \\in \\R$ be a real number such that $x > -1$. Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Then: :$\\paren {1 + x}^n \\ge 1 + n x$"}
+{"_id": "8212", "title": "Multiple of Ring Product", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring. Let $x, y \\in \\left({R, +, \\circ}\\right)$. Then: :$\\forall n \\in \\Z_{> 0}: \\left({n \\cdot x} \\right) \\circ y = n \\cdot \\left({x \\circ y}\\right) = x \\circ \\left({n \\cdot y}\\right)$ where $n \\cdot x$ denotes the $n$th multiple of $x$."}
+{"_id": "8216", "title": "Ordering is Preserved on Integers by Addition", "text": "The usual ordering on the integers is preserved by the operation of addition: :$\\forall a, b, c, d, \\in \\Z: a \\le b, c \\le d \\implies a + c \\le b + d$"}
+{"_id": "8219", "title": "Power Set of Finite Set is Finite", "text": "Let $S$ be a finite set. Then the power set of $S$ is likewise finite."}
+{"_id": "8222", "title": "Common Divisor Divides Difference", "text": "Let $c$ be a common divisor of two integers $a$ and $b$. That is: :$a, b, c \\in \\Z: c \\divides a \\land c \\divides b$ Then: :$c \\divides \\paren {a - b}$"}
+{"_id": "8226", "title": "Divisor Divides Multiple", "text": "Let $a, b$ be integers. Let: :$a \\divides b$ where $\\divides$ denotes divisibility. Then: :$\\forall c \\in \\Z: a \\divides b c$"}
+{"_id": "8229", "title": "One is not Prime", "text": "The integer $1$ (one) is not a prime number."}
+{"_id": "8230", "title": "Initial Part of String is Substring", "text": "Let $S$ be a string, and let $T$ be an initial part of $S$. Then $T$ is a substring of $S$."}
+{"_id": "8231", "title": "Ring Zero is Idempotent", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring whose ring zero is $0_R$. Then $0_R$ is an idempotent element of $R$ under the ring product $\\circ$: :$0_R \\circ 0_R = 0_R$"}
+{"_id": "8233", "title": "Prime iff Coprime to all Smaller Positive Integers", "text": "Let $p$ be a prime number Then: :$\\forall x \\in \\Z, 0 < x < p: x \\perp p$ That is, $p$ is relatively prime to all smaller (strictly) positive integers."}
+{"_id": "8234", "title": "Division Theorem/Positive Divisor/Positive Dividend", "text": "<onlyinclude> For every pair of integers $a, b$ where $a \\ge 0$ and $b > 0$, there exist unique integers $q, r$ such that $a = q b + r$ and $0 \\le r < b$: :$\\forall a, b \\in \\Z, a \\ge 0, b > 0: \\exists! q, r \\in \\Z: a = q b + r, 0 \\le r < b$ <onlyinclude> In the above equation: * $a$ is the '''dividend''' * $b$ is the '''divisor''' * $q$ is the '''quotient''' * $r$ is the '''principal remainder''', or, more usually, just the '''remainder'''."}
+{"_id": "8235", "title": "Division Theorem/Positive Divisor", "text": "<onlyinclude> For every pair of integers $a, b$ where $b > 0$, there exist unique integers $q, r$ such that $a = q b + r$ and $0 \\le r < b$: :$\\forall a, b \\in \\Z, b > 0: \\exists! q, r \\in \\Z: a = q b + r, 0 \\le r < b$ <onlyinclude> In the above equation: :$a$ is the '''dividend''' :$b$ is the '''divisor''' :$q$ is the '''quotient''' :$r$ is the '''principal remainder''', or, more usually, just the '''remainder'''."}
+{"_id": "8239", "title": "Division Theorem/Positive Divisor/Positive Dividend/Existence", "text": "For every pair of integers $a, b$ where $a \\ge 0$ and $b > 0$, there exist integers $q, r$ such that $a = q b + r$ and $0 \\le r < b$: :$\\forall a, b \\in \\Z, a \\ge 0, b > 0: \\exists q, r \\in \\Z: a = q b + r, 0 \\le r < b$"}
+{"_id": "8242", "title": "Division Theorem/Positive Divisor/Positive Dividend/Uniqueness", "text": "For every pair of integers $a, b$ where $a \\ge 0$ and $b > 0$, the integers $q, r$ such that $a = q b + r$ and $0 \\le r < b$ are unique: :$\\forall a, b \\in \\Z, a \\ge 0, b > 0: \\exists! q, r \\in \\Z: a = q b + r, 0 \\le r < b$"}
+{"_id": "8244", "title": "Division Theorem/Positive Divisor/Existence/Proof 2", "text": "For every pair of integers $a, b$ where $b > 0$, there exist integers $q, r$ such that $a = q b + r$ and $0 \\le r < b$: :$\\forall a, b \\in \\Z, b > 0: \\exists q, r \\in \\Z: a = q b + r, 0 \\le r < b$"}
+{"_id": "8245", "title": "Division Theorem/Positive Divisor/Existence", "text": "For every pair of integers $a, b$ where $b > 0$, there exist integers $q, r$ such that $a = q b + r$ and $0 \\le r < b$: :$\\forall a, b \\in \\Z, b > 0: \\exists q, r \\in \\Z: a = q b + r, 0 \\le r < b$"}
+{"_id": "8246", "title": "Division Theorem/Positive Divisor/Uniqueness/Proof 1", "text": "For every pair of integers $a, b$ where $b > 0$, the integers $q, r$ such that $a = q b + r$ and $0 \\le r < b$ are unique: :$\\forall a, b \\in \\Z, b > 0: \\exists! q, r \\in \\Z: a = q b + r, 0 \\le r < b$"}
+{"_id": "8247", "title": "Division Theorem/Positive Divisor/Uniqueness/Proof 2", "text": "For every pair of integers $a, b$ where $b > 0$, the integers $q, r$ such that $a = q b + r$ and $0 \\le r < b$ are unique: :$\\forall a, b \\in \\Z, b > 0: \\exists! q, r \\in \\Z: a = q b + r, 0 \\le r < b$"}
+{"_id": "8248", "title": "Division Theorem/Positive Divisor/Uniqueness", "text": "For every pair of integers $a, b$ where $b > 0$, the integers $q, r$ such that $a = q b + r$ and $0 \\le r < b$ are unique: :$\\forall a, b \\in \\Z, b > 0: \\exists! q, r \\in \\Z: a = q b + r, 0 \\le r < b$"}
+{"_id": "8249", "title": "Division Theorem/Positive Divisor/Existence/Proof 3", "text": "For every pair of integers $a, b$ where $b > 0$, there exist integers $q, r$ such that $a = q b + r$ and $0 \\le r < b$: :$\\forall a, b \\in \\Z, b > 0: \\exists q, r \\in \\Z: a = q b + r, 0 \\le r < b$"}
+{"_id": "8250", "title": "Division Theorem/Positive Divisor/Uniqueness/Proof 3", "text": "For every pair of integers $a, b$ where $b > 0$, the integers $q, r$ such that $a = q b + r$ and $0 \\le r < b$ are unique: :$\\forall a, b \\in \\Z, b > 0: \\exists! q, r \\in \\Z: a = q b + r, 0 \\le r < b$"}
+{"_id": "8252", "title": "Integer Coprime to all Factors is Coprime to Whole", "text": "Let $a, b \\in \\Z$. Let $\\displaystyle b = \\prod_{j \\mathop = 1}^r b_j$ Let $a$ be coprime to each of $b_1, \\ldots, b_r$. Then $a$ is coprime to $b$."}
+{"_id": "8253", "title": "All Factors Divide Integer then Whole Divides Integer", "text": "Let $S = \\set {a_1, a_2, \\ldots, a_r} \\subseteq \\Z$ be a finite subset of the integers. Let $S$ be pairwise coprime. Let: :$\\forall j \\in \\set {1, 2, \\ldots, r}: a_r \\divides b$ where $\\divides$ denotes divisibility. Then: :$\\displaystyle \\prod_{j \\mathop = 1}^r a_j \\divides b$"}
+{"_id": "8254", "title": "Piecewise Continuous Function with One-Sided Limits is Bounded", "text": "Let $f$ be a real function defined on a closed interval $\\closedint a b$. Let $f$ be a piecewise continuous function with one-sided limits. Then $f$ is a bounded piecewise continuous function."}
+{"_id": "8255", "title": "Upper Bounds for Prime Numbers/Result 1", "text": ":$\\forall n \\in \\N: \\map p n \\le 2^{2^{n - 1} }$"}
+{"_id": "8256", "title": "Upper Bounds for Prime Numbers/Result 2", "text": ":$\\forall n \\in \\N: \\map p n \\le \\paren {p \\paren {n - 1} }^{n - 1} + 1$"}
+{"_id": "8258", "title": "Product of Integers of form 4n + 1", "text": "Let $m, n \\in \\Z$ such that both $m$ and $n$ are of the form $4 k + 1$ where $k \\in \\Z$. Then $m n$ is also of the form $4 k + 1$."}
+{"_id": "8259", "title": "Infinite Number of Primes of form 4n - 1", "text": "There are infinitely many prime numbers of the form $4 n - 1$."}
+{"_id": "8262", "title": "Integer Coprime to Modulus iff Linear Congruence to 1 exists", "text": "Let $a, m \\in \\Z$. The linear congruence: :$a x \\equiv 1 \\pmod m$ has a solution $x$ {{iff}} $a$ and $m$ are coprime."}
+{"_id": "8263", "title": "Integer Coprime to Modulus iff Linear Congruence to 1 exists/Corollary", "text": "Let $p$ be a prime number. The linear congruence: :$a x \\equiv 1 \\pmod p$ has a solution $x$ {{iff}} $a \\not \\equiv 0 \\pmod p$."}
+{"_id": "8264", "title": "Solution of Linear Congruence/Existence", "text": "$a x \\equiv b \\pmod n$ has at least one solution {{iff}}: : $\\gcd \\set {a, n} \\divides b$ that is, {{iff}} $\\gcd \\set {a, n}$ is a divisor of $b$."}
+{"_id": "8265", "title": "Solution of Linear Congruence/Number of Solutions", "text": "Let $\\gcd \\set {a, n} = d$. Then $a x \\equiv b \\pmod n$ has $d$ solutions which are given by the unique solution modulo $\\dfrac n d$ of the congruence: : $\\dfrac a d x \\equiv \\dfrac b d \\paren {\\bmod \\dfrac n d}$"}
+{"_id": "8266", "title": "Solution of Linear Congruence/Unique iff Coprime to Modulus", "text": "If $\\gcd \\set {a, n} = 1$, then $a x \\equiv b \\pmod n$ has a unique solution."}
+{"_id": "8267", "title": "Ring Direct Product of Modulo Integers is Isomorphic to Ring Modulo Product iff Coprime", "text": "Let $m, n \\in \\Z_{>1}$. Let $\\struct {\\Z_m, +_m, \\times_m}$ and $\\struct {\\Z_n, +_n, \\times_n}$ be the rings of integers modulo $m$ and $n$ respectively. Let $\\struct {\\Z_m \\times \\Z_n}$ be the direct product of $\\Z_m$ and $\\Z_n$. Let $\\struct {\\Z_{m n}, +_{m n}, \\times_{m n} }$ be the ring of integers modulo $mn$. Then $\\struct {\\Z_m \\times \\Z_n}$ is isomorphic to $\\struct {\\Z_{m n}, +_{m n}, \\times_{m n} }$ {{iff}} $m$ and $n$ are coprime."}
+{"_id": "8269", "title": "Trivial Norm on Division Ring is Non-Archimedean", "text": "Let $\\struct {R, +, \\circ}$ be a division ring whose ring zero is $0_R$. Then the trivial norm $\\norm {\\, \\cdot \\,}: R \\to \\R_{\\ge 0}$, which is given by: :$\\norm x = \\begin{cases}   0 & : x = 0_R \\\\   1 & : \\text{ otherwise} \\end{cases}$ is non-archimedean: :$\\norm {x + y} \\le \\max \\set {\\norm x, \\norm y}$"}
+{"_id": "8270", "title": "Ring of Integers Modulo m is Ring", "text": "For all $m \\in \\N: m \\ge 2$, the ring of integers modulo $m$: :$\\struct {\\Z_m, +_m, \\times_m}$ is a commutative ring with unity $\\eqclass 1 m$. The zero of $\\struct {\\Z_m, +_m, \\times_m}$ is $\\eqclass 0 m$."}
+{"_id": "8271", "title": "Piecewise Continuous Function with One-Sided Limits is Uniformly Continuous on Each Piece", "text": "Let $f$ be a real function defined on a closed interval $\\left[{a \\,.\\,.\\, b}\\right]$. Let $f$ be piecewise continuous with one-sided limits: {{:Definition:Piecewise Continuous Function with One-Sided Limits}} Then: :for all $i \\in \\left\\{ {1, 2, \\ldots, n}\\right\\}$, $f$ is uniformly continuous on $\\left({x_{i − 1} \\,.\\,.\\, x_i}\\right)$ ."}
+{"_id": "8272", "title": "Principle of Definition by Structural Induction", "text": "Let $\\LL$ be a formal language. Let the formal grammar of $\\LL$ be a bottom-up grammar with unique parsability. A definition $\\map D \\phi$ (in the metalanguage of $\\LL$) for all well-formed formulas $\\phi$ of $\\LL$ is uniquely specified by: :$(1): \\quad$ A definition $\\map D a$ for each letter $a$ of $\\LL$ :$(2): \\quad$ For each rule of formation for $\\LL$, a definition $\\map D \\phi$ of the resultant WFF $\\phi$ in terms of the constituent WFFs' definitions $\\map D {\\phi_1}, \\ldots, \\map D {\\phi_n}$."}
+{"_id": "8273", "title": "Principle of Structural Induction", "text": "Let $\\LL$ be a formal language. Let the formal grammar of $\\LL$ be a bottom-up grammar. Let $\\map P \\phi$ be a statement (in the metalanguage of $\\LL$) about well-formed formulas $\\phi$ of $\\LL$. Then $P$ is true for all WFFs of $\\LL$ {{iff}} both: :$\\map P a$ is true for all letters $a$ of $\\LL$, and, for each rule of formation of $\\LL$, if $\\phi$ is a WFF resulting from WFFs $\\phi_1, \\ldots, \\phi_n$ by applying that rule, then: :$\\map P \\phi$ is true only if $\\map P {\\phi_1}, \\ldots, \\map P {\\phi_n}$ are all true."}
+{"_id": "8274", "title": "Integers under Multiplication form Monoid", "text": "The set of integers under multiplication $\\struct {\\Z, \\times}$ is a monoid."}
+{"_id": "8276", "title": "Integers under Multiplication form Semigroup", "text": "The set of integers under multiplication $\\struct {\\Z, \\times}$ is a semigroup."}
+{"_id": "8279", "title": "Canonical Form of Rational Number is Unique", "text": "The canonical form of a rational number is unique."}
+{"_id": "8280", "title": "Existence of Canonical Form of Rational Number", "text": "Let $r \\in \\Q$. Then: :$\\exists p \\in \\Z, q \\in \\Z_{>0}: r = \\dfrac p q, p \\perp q$ That is, every rational number can be expressed in its canonical form."}
+{"_id": "8281", "title": "Existence of Dyadic Rational between two Rationals", "text": "Let $a$ and $b$ be rational numbers such that $a < b$. Then there exist integers $m$ and $r$ such that: :$a < \\dfrac m {2^r} < b$ That is, there exists a dyadic rational between any pair of rational numbers."}
+{"_id": "8282", "title": "Field with 4 Elements has only Order 2 Elements", "text": "Let $\\struct {\\GF, +, \\times}$ be a field which has exactly $4$ elements. Then: :$\\forall a \\in \\GF: a + a = 0_\\GF$ where $0_\\GF$ is the zero of $\\GF$."}
+{"_id": "8283", "title": "Count of Truth Functions", "text": "There are $2^{\\paren {2^k} }$ distinct truth functions on $k$ variables."}
+{"_id": "8284", "title": "Unary Truth Functions", "text": "There are $4$ distinct unary truth functions: :$(1): \\quad$ The constant function $\\map f p = \\F$ :$(2): \\quad$ The constant function $\\map f p = \\T$ :$(3): \\quad$ The identity function $\\map f p = p$ :$(4): \\quad$ The logical not function $\\map f p = \\neg p$"}
+{"_id": "8285", "title": "Binary Truth Functions", "text": "There are $16$ distinct binary truth functions: * Two constant functions: ** $\\map {f_\\F} {p, q} = \\F$ ** $\\map {f_\\T} {p, q} = \\T$ * Two projections: ** $\\map {\\pr_1} {p, q} = p$ ** $\\map {\\pr_2} {p, q} = q$ * Two negated projections: ** $\\map {\\overline {\\pr_1} } {p, q} = \\neg p$ ** $\\map {\\overline {\\pr_2} } {p, q} = \\neg q$ * The conjunction: $p \\land q$ * The disjunction: $p \\lor q$ * Two conditionals: ** $p \\implies q$ ** $q \\implies p$ * The biconditional: $p \\iff q$ * The exclusive or: $\\map \\neg {p \\iff q}$ * Two negated conditionals: ** $\\map \\neg {p \\implies q}$ ** $\\map \\neg {q \\implies p}$ * The NAND: $p \\uparrow q$ * The NOR: $p \\downarrow q$"}
+{"_id": "8287", "title": "Inverse Hyperbolic Sine Logarithmic Formulation", "text": "{{finish|The true story is more complicated than this. The inverse hyperbolic sine is a multifunction. So is $\\ln$ in the complex plane. And $\\sqrt {z^2 + 1}$ has two values, pos and neg, so you need to justify which one is taken. The best context to put this page is probably into Definition:Inverse Hyperbolic Sine/Arcsine, where an explanation needs to be made as to why the principal value is taken thus.}} For any complex number $z \\in \\C$: :$\\operatorname {arsinh} z = \\map \\ln {z + \\sqrt {z^2 + 1} }$ where $\\operatorname {arsinh} z$ is the inverse hyperbolic sine."}
+{"_id": "8288", "title": "Value of Cauchy Determinant", "text": "Let $D_n$ be a Cauchy determinant of order $n$: :$\\begin{vmatrix} \\dfrac 1 {x_1 + y_1} & \\dfrac 1 {x_1 + y_2} & \\cdots & \\dfrac 1 {x_1 + y_n} \\\\ \\dfrac 1 {x_2 + y_1} & \\dfrac 1 {x_2 + y_2} & \\cdots & \\dfrac 1 {x_2 + y_n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\dfrac 1 {x_n + y_1} & \\dfrac 1 {x_n + y_2} & \\cdots & \\dfrac 1 {x_n + y_n} \\\\ \\end{vmatrix}$ Then the value of $D_n$ is given by: :$D_n = \\dfrac {\\displaystyle \\prod_{1 \\mathop \\le i \\mathop < j \\mathop \\le n} \\left({x_j - x_i}\\right) \\left({y_j - y_i}\\right)} {\\displaystyle \\prod_{1 \\mathop \\le i, \\, j \\mathop \\le n} \\left({x_i + y_j}\\right)}$ If $D_n$ is given by: :$\\begin{vmatrix} \\dfrac 1 {x_1 - y_1} & \\dfrac 1 {x_1 - y_2} & \\cdots & \\dfrac 1 {x_1 - y_n} \\\\ \\dfrac 1 {x_2 - y_1} & \\dfrac 1 {x_2 - y_2} & \\cdots & \\dfrac 1 {x_2 - y_n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\dfrac 1 {x_n - y_1} & \\dfrac 1 {x_n - y_2} & \\cdots & \\dfrac 1 {x_n - y_n} \\\\ \\end{vmatrix}$ then its determinant is given by: :$D_n = \\dfrac {\\displaystyle \\prod_{1 \\mathop \\le i \\mathop < j \\mathop \\le n} \\left({x_j - x_i}\\right) \\left({y_i - y_j}\\right)} {\\displaystyle \\prod_{1 \\mathop \\le i, \\, j \\mathop \\le n} \\left({x_i - y_j}\\right)}$"}
+{"_id": "8289", "title": "Integers under Addition form Abelian Group", "text": "The set of integers under addition $\\struct {\\Z, +}$ forms an abelian group."}
+{"_id": "8292", "title": "Piecewise Continuous Function with One-Sided Limits is Darboux Integrable", "text": "Let $f$ be a real function defined on a closed interval $\\closedint a b$. Let $f$ be piecewise continuous with one-sided limits on $\\closedint a b$. Then $f$ is Darboux integrable on $\\closedint a b$."}
+{"_id": "8294", "title": "Schröder Rule/Proof 1", "text": "Let $A$, $B$ and $C$ be relations on a set $S$. Then the following are equivalent statements: :$(1): \\quad A \\circ B \\subseteq C$ :$(2): \\quad A^{-1} \\circ \\overline{C} \\subseteq \\overline{B}$ :$(3): \\quad \\overline{C} \\circ B^{-1} \\subseteq \\overline{A}$ where: : $\\circ$ denotes relation composition : $A^{-1}$ denotes the inverse of $A$ : $\\overline{A}$ denotes the complement of $A$."}
+{"_id": "8295", "title": "Condition for Well-Foundedness/Forward Implication", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $\\struct {S, \\preceq}$ be well-founded. Then there is no infinite sequence $\\sequence {a_n}$ of elements of $S$ such that $\\forall n \\in \\N: a_{n + 1} \\prec a_n$."}
+{"_id": "8296", "title": "Extendability Theorem for Function Continuous on Open Interval", "text": "Let $f$ be a continuous real function that is defined on an open interval $\\openint a b$. Let $g$ be a real function that satisfies: :$g$ is defined on $\\closedint a b$ :$g$ is continuous on $\\closedint a b$ :$g = f$ on $\\openint a b$. Then $g$ exists {{iff}} $\\displaystyle \\lim_{x \\mathop \\to a^+} \\map f x$ and $\\displaystyle \\lim_{x \\mathop \\to b^-} \\map f x$ exist."}
+{"_id": "8297", "title": "Definition:Trivial Ordering", "text": "The '''trivial ordering''' is an ordering $\\mathcal R$ defined on a set $S$ by: :$\\forall a, b \\in S: a \\mathrel{\\mathcal R} b \\iff a = b$ That is, there is no ordering defined on any two distinct elements of the set $S$."}
+{"_id": "8300", "title": "Integrability Theorem for Functions Continuous on Open Intervals", "text": "Let $f$ be a real function defined on a closed interval $\\closedint a b$ such that $a < b$. Let $f$ be continuous on $\\openint a b$. Let the one-sided limits $\\displaystyle \\lim_{x \\mathop \\to a^+} \\map f x$ and $\\displaystyle \\lim_{x \\mathop \\to b^-} \\map f x$ exist. Then $f$ is Darboux integrable on $\\closedint a b$."}
+{"_id": "8301", "title": "Set of Rational Numbers whose Numerator Divisible by p is Closed under Multiplication", "text": "Let $p$ be a prime number. Let $A_p$ be the set of all rational numbers which, when expressed in canonical form has a numerator which is divisible by $p$. Then $A_p$ is closed under rational multiplication."}
+{"_id": "8302", "title": "Integer Less One divides Power Less One", "text": "Let $q, n \\in \\Z_{>0}$. Then: :$\\paren {q - 1} \\divides \\paren {q^n - 1}$ where $\\divides$ denotes divisibility."}
+{"_id": "8303", "title": "Number Plus One divides Power Plus One iff Odd", "text": "Let $q, n \\in \\Z_{>0}$. Then: :$\\paren {q + 1} \\divides \\paren {q^n + 1}$ {{iff}} $n$ is odd. In the above, $\\divides$ denotes divisibility."}
+{"_id": "8304", "title": "Mapping Bounded on Union iff Bounded on Each Component/Real-Valued Function", "text": "Let $f$ be a real-valued function. Then: : $f$ is bounded on the union of a finite number of sets within the domain of $f$ {{iff}}: :$f$ is bounded on each of the sets."}
+{"_id": "8305", "title": "Parity of Integer equals Parity of Positive Power", "text": "Let $p \\in \\Z$ be an integer. Let $n \\in \\Z_{>0}$ be a strictly positive integer. Then $p$ is even {{iff}} $p^n$ is even. That is, the parity of an integer equals the parity of all its (strictly) positive powers."}
+{"_id": "8306", "title": "Two divides Power Plus One iff Odd", "text": "Let $q, n \\in \\Z_{>0}$. Then: :$2 \\divides \\paren {q^n + 1}$ {{iff}} $q$ is odd. In the above, $\\divides$ denotes divisibility."}
+{"_id": "8307", "title": "Primes of form Power plus One", "text": "Let $q, n \\in \\Z_{>0}$ such that $q > 1$. Then $q^n + 1$ is prime only if: :$(1): \\quad q$ is even and :$(2): \\quad n$ is of the form $2^m$ for some positive integer $m$."}
+{"_id": "8310", "title": "Fermat Number whose Index is Sum of Integers", "text": "Let $F_n = 2^{\\left({2^n}\\right)} + 1$ be the $n$th Fermat number. Let $k \\in \\Z_{>0}$. Then: :$F_{n + k} - 1 = \\left({F_n - 1}\\right)^{2^k}$"}
+{"_id": "8311", "title": "Goldbach's Theorem", "text": "Let $F_m$ and $F_n$ be Fermat numbers such that $m \\ne n$. Then $F_m$ and $F_n$ are coprime."}
+{"_id": "8312", "title": "Number of Boolean Interpretations for Finite Set of Variables", "text": "Let $\\mathcal P_0$ be the vocabulary of language of propositional logic. Let $S \\subseteq \\mathcal P_0$ be a finite set of $n$ letters from $\\mathcal P_0$. Then there are $2^n$ different partial boolean interpretations for $S$."}
+{"_id": "8316", "title": "Rational Numbers under Addition form Monoid", "text": "The set of rational numbers under addition $\\struct {\\Q, +}$ forms a monoid."}
+{"_id": "8317", "title": "Real Numbers form Field", "text": "The set of real numbers $\\R$ forms a field under addition and multiplication: $\\struct {\\R, +, \\times}$."}
+{"_id": "8318", "title": "Rational Numbers form Field", "text": "Consider the algebraic structure $\\struct {\\Q, +, \\times}$, where: :$\\Q$ is the set of all rational numbers :$+$ is the operation of rational addition :$\\times$ is the operation of rational multiplication. Then $\\struct {\\Q, +, \\times}$ forms a field."}
+{"_id": "8320", "title": "Integers Modulo m under Addition form Abelian Group", "text": "Let $\\Z_m$ is the set of integers modulo $m$ Let $+_m$ be the operation of addition modulo $m$. Then the structure $\\struct {\\Z_m, +_m}$ is an abelian group."}
+{"_id": "8322", "title": "Rational Numbers under Multiplication form Monoid", "text": "The set of rational numbers under multiplication $\\struct {\\Q, \\times}$ forms a monoid."}
+{"_id": "8323", "title": "Real Numbers under Multiplication form Monoid", "text": "The set of real numbers under multiplication $\\struct {\\R, \\times}$ forms a monoid."}
+{"_id": "8324", "title": "Complex Numbers under Multiplication form Monoid", "text": "The set of complex numbers under multiplication $\\struct {\\C, \\times}$ forms a monoid."}
+{"_id": "8326", "title": "Non-Zero Natural Numbers under Addition do not form Monoid", "text": "Let $\\N_{>0}$ be the set of natural numbers without zero, i.e. $\\N_{>0} = \\N \\setminus \\left\\{{0}\\right\\}$. The structure $\\left({\\N_{>0}, +}\\right)$ does ''not'' form a monoid."}
+{"_id": "8327", "title": "Natural Numbers Bounded Below under Addition form Commutative Semigroup", "text": "Let $m \\in \\N$ where $\\N$ is the set of natural numbers. Let $M \\subseteq \\N$ be defined as: :$M := \\set {x \\in \\N: x \\ge m}$ That is, $M$ is the set of all natural numbers greater than or equal to $m$. Then the algebraic structure $\\struct {M, +}$ is a commutative semigroup."}
+{"_id": "8328", "title": "Positive Real Numbers under Max Operation form Monoid", "text": "Let $\\R_{\\ge 0}$ be the set of positive (that is, non-negative) real numbers. Let $\\max: \\R_{\\ge 0}^2 \\to \\R_{\\ge 0}$ be the max operation on $\\R_{\\ge 0}$. Then $\\left({\\R_{\\ge 0}, \\max}\\right)$ is a monoid whose identity is $0$."}
+{"_id": "8329", "title": "Free Commutative Monoid on One Element is Isomorphic to Positive Integers under Addition", "text": "Let $X = \\left\\{{x}\\right\\}$ be a singleton. Let $M$ be the free commutative monoid on $X$. Then $M$ is isomorphic to the additive monoid of natural numbers."}
+{"_id": "8335", "title": "Finite Monoid with Right Cancellable Operation is Group", "text": "Let $\\struct {S, \\circ}$ be a finite monoid. Let $\\circ$ be a right cancellable operation. Then $\\struct {S, \\circ}$ is a group."}
+{"_id": "8337", "title": "Multiplicative Group of Field is Abelian Group", "text": "Let $\\struct {F, +, \\times}$ be a field. Let $F^* := F \\setminus \\set 0$ be the set $F$ less its zero. The algebraic structure $\\struct {F^*, \\times}$ is an abelian group."}
+{"_id": "8340", "title": "One and Minus One form Subgroup of Multiplicative Group of Rational Numbers", "text": "Let $\\left({\\Q_{\\ne 0}, \\times}\\right)$ be the multiplicative group of rational numbers. Let $S \\subseteq \\Q$ where $S = \\left\\{{1, -1}\\right\\}$. Then $\\left({S, \\times}\\right)$ is a subgroup of $\\left({\\Q_{\\ne 0}, \\times}\\right)$."}
+{"_id": "8342", "title": "Symmetric Group is not Abelian", "text": "Let $S_n$ be the symmetric group of order $n$ where $n \\ge 3$. Then $S_n$ is not abelian."}
+{"_id": "8345", "title": "Möbius Function is Multiplicative/Corollary", "text": "Let $\\gcd \\set {m, n} > 1$. Then: :$\\map \\mu {m n} = 0$"}
+{"_id": "8348", "title": "Identity of Affine Group of One Dimension", "text": "The $1$-dimensional affine group on $\\R$ $\\operatorname{Af}_1 \\left({\\R}\\right)$ has $f_{1, 0}$ as an identity element."}
+{"_id": "8349", "title": "Inverse in Affine Group of One Dimension", "text": "Let $\\map {\\operatorname {Af}_1} \\R$ denote the $1$-dimensional affine group on $\\R$. Let $f_{a b} \\in \\map {\\operatorname {Af}_1} \\R$. Let $c = \\dfrac 1 a$ and $d = \\dfrac {-b} a$. Then $f_{c d} \\in \\map {\\operatorname {Af}_1} \\R$ is the inverse of $f_{a b}$."}
+{"_id": "8352", "title": "Commutation with Group Elements implies Commuation with Product with Inverse", "text": "Let $G$ be a group. Let $a, b, c \\in G$ such that $a$ commutes with both $b$ and $c$. Then $a$ commutes with $b c^{-1}$."}
+{"_id": "8356", "title": "Centralizer of Subset is Intersection of Centralizers of Elements", "text": "Let $\\struct {G, \\circ}$ be a group. Let $S \\subseteq G$. Let $\\map {C_G} S$ be the centralizer of $S$ in $G$. Then: :$\\ds \\map {C_G} S = \\bigcap_{x \\mathop \\in S} \\map {C_G} x$ where $\\map {C_G} z$ is the centralizer of $x$ in $G$."}
+{"_id": "8358", "title": "Group Action of Symmetric Group", "text": "Let $\\N_n$ denote the set $\\set {1, 2, \\ldots, n}$. Let $\\struct {S_n, \\circ}$ denote the symmetric group on $\\N_n$. The mapping $*: S_n \\times \\N_n \\to \\N_n$ defined as: :$\\forall \\pi \\in S_n, \\forall n \\in \\N_n: \\pi * n = \\map \\pi n$ is a group action."}
+{"_id": "8360", "title": "Trivial Group Action is Group Action", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $S$ be a set. Let $*: G \\times S \\to S$ be the trivial group action: :$\\forall \\tuple {g, s} \\in G \\times S: g * s = s$ Then $*$ is indeed a group action."}
+{"_id": "8362", "title": "Right Regular Representation by Inverse is Group Action", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $*: G \\times G \\to G$ be the operation: :$\\forall g, h \\in G: g * h = \\map {\\rho_{g^{-1} } } h$ where $\\rho_g$ is the right regular representation of $G$ with respect to $g$. Then $*$ is a group action."}
+{"_id": "8363", "title": "Left Regular Representation is Group Action", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $*: G \\times G \\to G$ be the operation: :$\\forall g, h \\in G: g * h = \\map {\\lambda_g} h$ where $\\lambda_g$ is the left regular representation of $G$ with respect to $g$. Then $*$ is a group action."}
+{"_id": "8364", "title": "Right Regular Representation by Inverse is Transitive Group Action", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $*: G \\times G \\to G$ be the group action: :$\\forall g, h \\in G: g * h = \\map {\\rho_{g^{-1} } } h$ where $\\rho_g$ is the right regular representation of $G$ with respect to $g$. Then $*$ is a transitive group action."}
+{"_id": "8366", "title": "Conjugacy Action is not Transitive", "text": "Let $\\struct {G, \\circ}$ be a non-trivial group whose identity is $e$. Let $*: G \\times G \\to G$ be the conjugacy group action: : $\\forall g, h \\in G: g * h = g \\circ h \\circ g^{-1}$ Then $*$ is ''not'' a transitive group action."}
+{"_id": "8369", "title": "Group Action on Subgroup by Right Regular Representation", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $*: H \\times G \\to G$ be the operation defined as: :$\\forall \\tuple {h, g} \\in H \\times G: h * g = \\map {\\rho_{h^{-1} } } g$ where $\\map {\\rho_{h^{-1} } } g$ is the right regular representation of $g$ by $h^{-1}$. Then $*$ is a group action."}
+{"_id": "8371", "title": "Orbit of Group Action on Subgroup by Right Regular Representation is Right Coset", "text": "Let $G$ be a group. Let $H$ be a proper subgroup of $G$. Let $*: H \\times G \\to G$ be the group action defined as: :$\\forall \\tuple {h, g} \\in H \\times G: h * g = \\map {\\rho_{h^{-1} } } g$ where $\\map {\\rho_{h^{-1} } } g$ is the right regular representation of $g$ by $h^{-1}$. Let $x \\in G$. Then the orbit of $x$ under $*$ is given by: :$\\forall x \\in G: \\Orb x = H x$ where $H x$ is the right coset of $H$ by $x$."}
+{"_id": "8372", "title": "Left Coset by Identity", "text": "Then: : $e H = H$ where $e H$ is the left coset of $H$ by $e$."}
+{"_id": "8373", "title": "Right Coset by Identity", "text": "Then: : $H = H e$ where $H e$ is the right coset of $H$ by $e$."}
+{"_id": "8374", "title": "Inversion Mapping is Permutation", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\iota: G \\to G$ be the inversion mapping on $G$. Then $\\iota$ is a permutation on $G$."}
+{"_id": "8375", "title": "Inversion Mapping is Mapping", "text": "Let $\\left({G, \\circ}\\right)$ be a group. Let $\\iota: G \\to G$ be the inversion mapping on $G$. Then $\\iota$ is indeed a mapping."}
+{"_id": "8378", "title": "Cartesian Product of Group Actions", "text": "Let $\\struct {G, \\circ}$ be a group. Let $S$ and $T$ be sets. Let $*_S: G \\times S \\to S$ and $*_T: G \\times T \\to T$ be group actions. Then the operation $*: G \\times \\paren {S \\times T} \\to S \\times T$ defined as: :$\\forall \\tuple {g, \\tuple {s, t} } \\in G \\times \\paren {S \\times T}: g * \\tuple {s, t} = \\tuple {g *_S s, g *_T t}$ is a group action."}
+{"_id": "8380", "title": "Index in Subgroup", "text": "Let $G$ be a group. Let $H, K$ be subgroups of finite index of $G$. Then: :$\\index H {H \\cap K} \\le \\index G K$ where $\\index G K$ denotes the index of $K$ in $G$. Equality happens {{iff}} $G = H K$."}
+{"_id": "8381", "title": "Finite Cyclic Group has Euler Phi Generators", "text": "Let $C_n$ be a (finite) cyclic group of order $n$. Then $C_n$ has $\\map \\phi n$ generators, where $\\map \\phi n$ denotes the Euler $\\phi$ function."}
+{"_id": "8395", "title": "Extendability Theorem for Derivatives Continuous on Open Intervals", "text": "Let $f$ be a continuous real function defined on an interval $\\closedint a b$ where $a < b$. Then $f$ is continuously differentiable on $\\closedint a b$ {{iff}}: :$f$ is continuously differentiable on $\\openint a b$ and: :$\\displaystyle \\lim_{x \\mathop \\to a^+} \\map {f'} x$ and $\\displaystyle \\lim_{x \\mathop \\to b^-} \\map {f'} x$ exist."}
+{"_id": "8411", "title": "Universal Affirmative and Universal Negative are Contrary iff First Predicate is not Vacuous", "text": "Consider the categorical statements: :$\\mathbf A: \\quad$ The universal affirmative: $\\forall x: \\map S x \\implies \\map P x$ :$\\mathbf E: \\quad$ The universal negative: $\\forall x: \\map S x \\implies \\neg \\map P x$ Then: :$\\mathbf A$ and $\\mathbf E$ are contrary {{iff}}: :$\\exists x: \\map S x$ Using the symbology of predicate logic: :$\\exists x: \\map S x \\iff \\neg \\paren {\\paren {\\forall x: \\map S x \\implies \\map P x} \\land \\paren {\\forall x: \\map S x \\implies \\neg \\map P x} }$"}
+{"_id": "8412", "title": "Particular Affirmative and Particular Negative are Subcontrary iff First Predicate is not Vacuous", "text": "Consider the categorical statements: :$\\mathbf I: \\quad$ The particular affirmative: $\\exists x: \\map S x \\land \\map P x$ :$\\mathbf O: \\quad$ The particular negative: $\\exists x: \\map S x \\land \\neg \\map P x$ Then: :$\\mathbf I$ and $\\mathbf O$ are subcontrary {{iff}}: :$\\exists x: \\map S x$ Using the symbology of predicate logic: :$\\exists x: \\map S x \\iff \\paren {\\paren {\\exists x: \\map S x \\land \\map P x} \\lor \\paren {\\exists x: \\map S x \\land \\neg \\map P x} }$"}
+{"_id": "8413", "title": "Universal Affirmative and Particular Negative are Contradictory", "text": "Consider the categorical statements: :$\\mathbf A: \\quad$ The universal affirmative: $\\forall x: \\map S x \\implies \\map P x$ :$\\mathbf O: \\quad$ The particular negative: $\\exists x: \\map S x \\land \\neg \\map P x$ Then $\\mathbf A$ and $\\mathbf O$ are contradictory. Using the symbology of predicate logic: :$\\neg \\paren {\\paren {\\forall x: \\map S x \\implies \\map P x} \\iff \\paren {\\exists x: \\map S x \\land \\neg \\map P x} }$"}
+{"_id": "8414", "title": "Particular Affirmative and Universal Negative are Contradictory", "text": "Consider the categorical statements: :$\\mathbf I: \\quad$ The particular affirmative: $\\exists x: \\map S x \\land \\map P x$ :$\\mathbf E: \\quad$ The universal negative: $\\forall x: \\map S x \\implies \\neg \\map P x$ Then $\\mathbf I$ and $\\mathbf E$ are contradictory. Using the symbology of predicate logic: :$\\neg \\paren {\\paren {\\exists x: \\map S x \\land \\map P x} \\iff \\paren {\\forall x: \\map S x \\implies \\neg \\map P x} }$"}
+{"_id": "8415", "title": "Universal Affirmative implies Particular Affirmative iff First Predicate is not Vacuous", "text": "Consider the categorical statements: :$\\map {\\mathbf A} {S, P}: \\quad$ The universal affirmative: $\\forall x: \\map S x \\implies \\map P x$ :$\\map {\\mathbf I} {S, P}: \\quad$ The particular affirmative: $\\exists x: \\map S x \\land \\map P x$ Then: :$\\map {\\mathbf A} {S, P} \\implies \\map {\\mathbf I} {S, P}$ {{iff}}: :$\\exists x: \\map S x$ Using the symbology of predicate logic: :$\\exists x: \\map S x \\iff \\paren {\\paren {\\forall x: \\map S x \\implies \\map P x} \\implies \\paren {\\exists x: \\map S x \\land \\map P x} }$"}
+{"_id": "8417", "title": "Socrates is Mortal", "text": ":$(1): \\quad$ ''All humans are mortal.'' :$(2): \\quad$ ''{{AuthorRef|Socrates}} is human.'' :$(3): \\quad$ ''Therefore {{AuthorRef|Socrates}} is mortal.''"}
+{"_id": "8418", "title": "Right-Hand Differentiable Function is Right-Continuous", "text": "Let $f$ be a real function defined on an interval $I$. Let $a$ be a point in $I$ where $f$ is right-hand differentiable. Then $f$ is right-continuous at $a$."}
+{"_id": "8419", "title": "Left-Hand Differentiable Function is Left-Continuous", "text": "Let $f$ be a real function defined on an interval $I$. Let $a$ be a point in $I$ where $f$ is left-hand differentiable. Then $f$ is left-continuous at $a$."}
+{"_id": "8420", "title": "Left-Hand and Right-Hand Differentiable Function is Continuous", "text": "Let $f$ be a real function defined on an interval $I$. Let $a$ be a point in $I$ where $f$ is left- and right-hand differentiable. Then $f$ is continuous at $a$."}
+{"_id": "8421", "title": "Universal Affirmative and Negative are both False iff Particular Affirmative and Negative are both True", "text": "Consider the categorical statements: {{begin-axiom}} {{axiom | q = \\mathbf A \\left({S, P}\\right)         | lc= The universal affirmative:         | ml= \\forall x: S \\left({x}\\right)         | mo= \\implies         | mr= P \\left({x}\\right) }} {{axiom | q = \\mathbf E \\left({S, P}\\right)         | lc= The universal negative:         | ml= \\forall x: S \\left({x}\\right)         | mo= \\implies         | mr= \\neg P \\left({x}\\right) }} {{axiom | q = \\mathbf I \\left({S, P}\\right)         | lc= The particular affirmative:         | ml= \\exists x: S \\left({x}\\right)         | mo= \\land          | mr= P \\left({x}\\right) }} {{axiom | q = \\mathbf O \\left({S, P}\\right)         | lc= The particular negative:         | ml= \\exists x: S \\left({x}\\right)         | mo= \\land          | mr= \\neg P \\left({x}\\right) }} {{end-axiom}} Then: :$\\mathbf A \\left({S, P}\\right)$ and $\\mathbf E \\left({S, P}\\right)$ are both false {{iff}}: :$\\mathbf I \\left({S, P}\\right)$ and $\\mathbf O \\left({S, P}\\right)$ are both true."}
+{"_id": "8423", "title": "Law of Simple Conversion of I", "text": "Consider the particular affirmative categorical statement ''Some $S$ is $P$'': :$\\map {\\mathbf I} {S, P}: \\exists x: \\map S x \\land  \\map P x$ Then ''Some $P$ is $S$'': :$\\map {\\mathbf I} {P, S}$"}
+{"_id": "8424", "title": "Law of Simple Conversion of E", "text": "Consider the universal negative categorical statement ''No $S$ is $P$'': :$\\mathbf E \\left({S, P}\\right): \\forall x: S \\left({x}\\right) \\implies \\neg P \\left({x}\\right)$ Then ''No $P$ is $S$'': :$\\mathbf E \\left({P, S}\\right)$"}
+{"_id": "8425", "title": "Conversion per Accidens", "text": "Consider the categorical statements: :$\\map {\\mathbf A} {S, P}: \\quad$ The universal affirmative: $\\forall x: \\map S x \\implies \\map P x$ :$\\map {\\mathbf I} {P, S}: \\quad$ The particular affirmative: $\\exists x: \\map P x \\land \\map S x$ Then: :$\\map {\\mathbf A} {S, P} \\implies \\map {\\mathbf I} {P, S}$ {{iff}}: :$\\exists x: \\map S x$ Using the symbology of predicate logic: :$\\exists x: \\map S x \\iff \\paren {\\paren {\\forall x: \\map S x \\implies \\map P x} \\implies \\paren {\\exists x: \\map P x \\land \\map S x} }$ This law has the traditional name '''conversion per accidens of $\\mathbf A$'''. Thus the $\\mathbf A$ form '''converts per accidens''' to the $\\mathbf I$ form."}
+{"_id": "8426", "title": "Number of Standard Instances of Categorical Syllogism", "text": "There are $256$ distinct standard instances of the categorical syllogism."}
+{"_id": "8436", "title": "No Valid Categorical Syllogism contains two Negative Premises", "text": "No categorical syllogism of which both premises are negative categorical statements is valid."}
+{"_id": "8437", "title": "Valid Patterns of Categorical Syllogism", "text": "The following categorical syllogisms are valid: :$\\begin{array}{rl} \\text{I} & AAA \\\\ \\text{I} & AII \\\\ \\text{I} & EAE \\\\ \\text{I} & EIO \\\\ * \\text{I} & AAI \\\\ * \\text{I} & EAO \\\\ \\end{array} \\qquad \\begin{array}{rl} \\text{II} & EAE \\\\ \\text{II} & AEE \\\\ \\text{II} & AOO \\\\ \\text{II} & EIO \\\\ * \\text{II} & EAO \\\\ * \\text{II} & AEO \\\\ \\end{array} \\qquad \\begin{array}{rl} \\dagger \\text{III} & AAI \\\\ \\text{III} & AII \\\\ \\text{III} & IAI \\\\ \\dagger \\text{III} & EAO \\\\ \\text{III} & EIO \\\\ \\text{III} & OAO \\\\ \\end{array} \\qquad \\begin{array}{rl} \\S \\text{IV} & AAI \\\\ \\text{IV} & AEE \\\\ \\dagger \\text{IV} & EAO \\\\ \\text{IV} & EIO \\\\ \\text{IV} & IAI \\\\ * \\text{IV} & AEO \\\\ \\end{array}$ In the above: :$\\text{I}, \\text{II}, \\text{III}, \\text{IV}$ denote the four figures of the categorical syllogisms :$A, E, I, O$ denote the universal affirmative, universal negative, particular affirmative and particular negative respectively: see Shorthand for Categorical Syllogism :Syllogisms marked $*$ require the assumption that $\\exists x: \\map S x$, that is, that there exists an object fulfilling the secondary predicate :Syllogisms marked $\\dagger$ require the assumption that $\\exists x: \\map M x$, that is, that there exists an object fulfilling the middle predicate :Syllogisms marked $\\S$ require the assumption that $\\exists x: \\map P x$, that is, that there exists an object fulfilling the primary predicate"}
+{"_id": "8439", "title": "Distributed Term of Conclusion of Valid Categorical Syllogism is Distributed in Premise", "text": "Let $T$ be a term of the conclusion $C$ of a valid categorical syllogism $Q$. Let $T$ be distributed in $C$. Then $T$ is also distributed in whichever premise of $Q$ in which it appears."}
+{"_id": "8440", "title": "Middle Term of Valid Categorical Syllogism is Distributed at least Once", "text": "Let $M$ be the middle term of a valid categorical syllogism $Q$. Then $M$ is distributed in at least one of the premises of $Q$ in which it appears."}
+{"_id": "8442", "title": "Conclusion of Valid Categorical Syllogism is Negative iff one Premise is Negative", "text": "The conclusion of a valid categorical syllogism is negative {{iff}} one of the premises is also negative."}
+{"_id": "8443", "title": "No Valid Categorical Syllogism contains two Particular Premises", "text": "Let $Q$ be a valid categorical syllogism. Then at least one of the premises of $Q$ is universal."}
+{"_id": "8444", "title": "No Valid Categorical Syllogism with Particular Premise has Universal Conclusion", "text": "Let $Q$ be a valid categorical syllogism. Let one of the premises of $Q$ be particular. Then the conclusion of $Q$ is also particular."}
+{"_id": "8445", "title": "Elimination of all but 48 Categorical Syllogisms as Invalid", "text": "Of the $256$ different types of categorical syllogism, all but $48$ can immediately be identified as invalid by consideration of the Rules of Quantity and the Rules of Quality."}
+{"_id": "8446", "title": "Conjunction implies Disjunction", "text": ":$\\vdash \\paren {p \\land q} \\implies \\paren {p \\lor q}$"}
+{"_id": "8450", "title": "Angle Bisector Vector", "text": "Let $\\mathbf u$ and $\\mathbf v$ be vectors of non-zero length. Let $\\left\\Vert{\\mathbf u}\\right\\Vert$ and $\\left\\Vert{\\mathbf v}\\right\\Vert$ be their respective lengths. Then $\\left\\Vert{\\mathbf u}\\right\\Vert \\mathbf v + \\left\\Vert{\\mathbf v}\\right\\Vert \\mathbf u$ is the angle bisector of $\\mathbf u$ and $\\mathbf v$."}
+{"_id": "8452", "title": "Injection from Finite Set to Itself is Surjection/Corollary", "text": "Let $S$ be a finite set. Let $f: S \\to S$ be an injection. Then $f$ is a permutation."}
+{"_id": "8453", "title": "Surjection from Finite Set to Itself is Permutation", "text": "Let $S$ be a finite set. Let $f: S \\to S$ be an surjection. Then $f$ is a permutation."}
+{"_id": "8454", "title": "Order of Power of Group Element", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $g \\in G$ be an element of $G$ such that: :$\\order g = n$ where $\\order g$ denotes the order of $g$. Then: :$\\forall m \\in \\Z: \\order {g^m} = \\dfrac n {\\gcd \\set {m, n} }$ where $\\gcd \\set {m, n}$ denotes the greatest common divisor of $m$ and $n$."}
+{"_id": "8455", "title": "Existence of Group of Finite Order", "text": "Let $n \\in \\Z_{>0}$. Then there exists at least one group whose order is $n$."}
+{"_id": "8456", "title": "Vitali Theorem", "text": "There exists a set of real numbers which is not Lebesgue measurable."}
+{"_id": "8457", "title": "Finite Number of Groups of Given Finite Order", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then there exists a finite number of types of group of order $n$."}
+{"_id": "8458", "title": "Nu of Prime Number is 1", "text": "Let $p$ be a prime number. Then: :$\\map \\nu p = 1$ where $\\nu$ denotes the $\\nu$ function: the number of types of group of a given order."}
+{"_id": "8459", "title": "Image of Subset under Relation equals Union of Images of Elements", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation on $S \\times T$. Let $X \\subseteq S$ be a subset of $S$. Then: :$\\displaystyle \\RR \\sqbrk X = \\bigcup_{x \\mathop \\in X} \\map \\RR x$ where: :$\\RR \\sqbrk X$ is the image of the subset $X$ under $\\RR$ :$\\map \\RR x$ is the image of the element $x$ under $\\RR$."}
+{"_id": "8460", "title": "Preimage of Subset under Relation equals Union of Preimages of Elements", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation on $S \\times T$. Let $\\RR^{-1} \\subseteq T \\times S$ be the inverse relation to $\\RR$ Let $Y \\subseteq T$ be a subset of $T$. Then: :$\\RR^{-1} \\sqbrk Y = \\displaystyle \\bigcup_{y \\mathop \\in Y} \\map {\\RR^{-1} } y$ where: :$\\RR^{-1} \\sqbrk Y$ is the preimage of the subset $Y$ under $\\RR$ :$\\RR^{-1} \\sqbrk Y$ is the preimage of the element $y$ under $\\RR$."}
+{"_id": "8461", "title": "Preimage of Subset under Mapping equals Union of Preimages of Elements", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping from $S$ to $T$. Let $f^{-1} \\subseteq T \\times S$ be the inverse of $f$, defined as: :$f^{-1} = \\set {\\tuple {t, s}: \\map f s = t}$ Let $Y \\subseteq T$ be a subset of $T$. Then: :$\\displaystyle f^{-1} \\sqbrk Y = \\bigcup_{y \\mathop \\in Y} \\map {f^{-1} } y$ where: :$f^{-1} \\sqbrk Y$ is the preimage of the subset $Y$ under $f$ :$\\map {f^{-1} } y$ is the preimage of the element $y$ under $f$."}
+{"_id": "8463", "title": "Image of Domain of Mapping is Image Set", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. The image of $S$ is the image set of $f$: :$f \\sqbrk S = \\Img f$"}
+{"_id": "8464", "title": "Image of Singleton under Mapping", "text": "Let $f: S \\to T$ be a mapping. Then the image of an element of $S$ is equal to the image of a singleton containing that element, the singleton being a subset of $S$: :$\\forall s \\in S: \\set {\\map f s} = f \\sqbrk {\\set s}$"}
+{"_id": "8465", "title": "Image of Subset under Mapping equals Union of Images of Elements", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping from $S$ to $T$. Let $X \\subseteq S$ be a subset of $S$. Then: :$\\displaystyle f \\sqbrk X = \\bigcup_{x \\mathop \\in X} \\map f x$ where: :$f \\sqbrk X$ is the image of the subset $X$ under $f$ :$\\map f x$ is the image of the element $x$ under $f$."}
+{"_id": "8466", "title": "Mapping Images are Disjoint only if Domains are Disjoint", "text": "Let $S$ and $T$ be sets. Let: :$f \\sqbrk S \\cap f \\sqbrk T = \\O$ where $f \\sqbrk S$ denotes the image set of $S$. Then: :$S \\cap T = \\O$"}
+{"_id": "8467", "title": "Image of Relation is Domain of Inverse Relation", "text": "Let $\\RR \\subseteq S \\times T$ be a relation. Let $\\RR^{-1} \\subseteq T \\times S$ be the inverse of $\\RR$. Then: :$\\Img \\RR = \\Dom {\\RR^{-1} }$ That is, the image of a relation is the domain of its inverse."}
+{"_id": "8468", "title": "Domain of Relation is Image of Inverse Relation", "text": "Let $\\RR \\subseteq S \\times T$ be a relation. Let $\\RR^{-1} \\subseteq T \\times S$ be the inverse of $\\RR$. Then: :$\\Dom \\RR = \\Img {\\RR^{-1} }$ That is, the domain of a relation is the image of its inverse."}
+{"_id": "8471", "title": "Image of Set Difference under Injection", "text": "Let $f: S \\to T$ be a mapping. Let $S_1$ and $S_2$ be subsets of $S$. Let $S_1 \\setminus S_2$ denote the set difference between $S_1$ and $S_2$. Then: :$\\forall S_1, S_2 \\subseteq S: f \\left[{S_1}\\right] \\setminus f \\left[{S_2}\\right] = f \\left[{S_1 \\setminus S_2}\\right]$ {{iff}} $f$ is an injection."}
+{"_id": "8472", "title": "De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Union", "text": ":$\\displaystyle \\map \\complement {\\bigcup_{i \\mathop \\in I} S_i} = \\bigcap_{i \\mathop \\in I} \\map \\complement {S_i}$"}
+{"_id": "8473", "title": "De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection", "text": ":$\\displaystyle \\map \\complement {\\bigcap_{i \\mathop \\in I} S_i} = \\bigcup_{i \\mathop \\in I} \\map \\complement {S_i}$"}
+{"_id": "8475", "title": "Closure of Intersection and Symmetric Difference imply Closure of Set Difference", "text": "Let $\\mathcal R$ be a system of sets such that for all $A, B \\in \\mathcal R$: : $(1): \\quad A \\cap B \\in \\mathcal R$ : $(2): \\quad A * B \\in \\mathcal R$ where $\\cap$ denotes set intersection and $*$ denotes set symmetric difference. Then: :$\\forall A, B \\in \\mathcal R: A \\setminus B \\in \\mathcal R$ where $\\setminus$ denotes set difference."}
+{"_id": "8476", "title": "Closure of Intersection and Symmetric Difference imply Closure of Union", "text": "Let $\\mathcal R$ be a system of sets such that for all $A, B \\in \\mathcal R$: : $(1): \\quad A \\cap B \\in \\mathcal R$ : $(2): \\quad A * B \\in \\mathcal R$ where $\\cap$ denotes set intersection and $*$ denotes set symmetric difference. Then: :$\\forall A, B \\in \\mathcal R: A \\cup B \\in \\mathcal R$ where $\\cup$ denotes set union."}
+{"_id": "8484", "title": "Complement of Limit Inferior is Limit Superior of Complements", "text": "Let $\\left\\{{E_n : n \\in \\N}\\right\\}$ be a sequence of sets. Then: :$\\displaystyle \\complement \\left({\\liminf_{n \\mathop \\to \\infty} \\ E_n}\\right) = \\limsup_{n \\mathop \\to \\infty} \\ \\complement \\left({E_n}\\right)$ where $\\liminf$ and $\\limsup$ denote the limit inferior and limit superior, respectively."}
+{"_id": "8486", "title": "Sigma-Ring is Closed under Countable Intersections", "text": "Let $\\mathcal R$ be a $\\sigma$-ring. Let $\\left \\langle{A_n}\\right \\rangle_{n \\mathop \\in \\N} \\in \\mathcal R$ be a sequence of sets in $\\mathcal R$. Then: :$\\displaystyle \\bigcap_{n \\mathop = 1}^\\infty A_n \\in \\mathcal R$"}
+{"_id": "8487", "title": "Pappus's Hexagon Theorem", "text": "Let $A, B, C$ be a set of collinear points. Let $a, b, c$ be another set of collinear points. Let $X, Y, Z$ be the points of intersection of each of the straight lines $Ab$ and $aB$, $Ac$ and $aC$, and $Bc$ and $bC$. Then $X, Y, Z$ are collinear points."}
+{"_id": "8489", "title": "Two Planes have Line in Common", "text": "Two distinct planes have exactly one (straight) line in common."}
+{"_id": "8491", "title": "Principle of Duality in the Plane", "text": "Let $P$ be a theorem of projective geometry proven using the propositions of incidence. Let $Q$ be the statement created from $P$ by interchanging: :$(1): \\quad$ the terms '''point''' and '''(straight) line''' :$(2): \\quad$ the terms '''collinear''' (of points) and '''concurrent''' (of lines) :$(3): \\quad$ the terms '''lie on''' and '''intersect at''' and so on. Then $Q$ is also a theorem of projective geometry."}
+{"_id": "8492", "title": "Principle of Duality in Space", "text": "Let $P$ be a theorem of projective geometry proven using the propositions of incidence. Let $Q$ be the statement created from $P$ by interchanging: :$(1) \\quad$ the terms '''point''' and '''plane''' :$(2) \\quad$ the terms '''lie on''' and '''intersect at''' and so on. Then $Q$ is also a theorem of projective geometry."}
+{"_id": "8493", "title": "Desargues' Theorem", "text": "Let $\\triangle ABC$ and $\\triangle A'B'C'$ be triangles lying in the same or different planes. Let the lines $AA'$, $BB'$ and $CC'$ intersect in the point $O$. Then $BC$ meets $B'C'$ in $L$, $CA$ meets $C'A'$ in $M$ and $AB$ meets $A'B'$ in $N$, where $L, M, N$ are collinear."}
+{"_id": "8495", "title": "Domain of Composite Mapping", "text": "Let $S_1, S_2, S_3$ be sets. Let $f_1: S_1 \\to S_2$ and $f_2: S_2 \\to S_3$ be mappings. Let $f_2 \\circ f_1: S_1 \\to S_3$ be the composite mapping of $f_1$ and $f_2$. Then: :$\\Dom {f_1} = \\Dom {f_2 \\circ f_1}$ where $\\Dom {f_1}$ denotes the domain of $f_1$."}
+{"_id": "8497", "title": "Union of Functions Theorem", "text": "Let $X$ be a set. Let $\\set {X_i: i \\in \\N}$ be an exhausting sequence of sets on $X$. For each $i \\in \\N$, let $g_i: X_i \\to Y$ be a mapping such that: :$g_{i + 1} \\restriction X_i = g_i$ where $g_{i + 1} \\restriction X_i$ denotes the restriction of $g_{i + 1}$ to $g_i$. Then: :$\\displaystyle \\bigcup \\set {g_i: i \\in \\N}$ is a mapping from $X$ to $Y$."}
+{"_id": "8498", "title": "Inductive Definition of Sequence", "text": "Let $X$ be a set. Let $h \\in \\N$. Let $a_i \\in X$ for all $i \\in \\set {1, 2, \\ldots, h}$. Let $S$ be the set of all finite sequences whose codomains are in $X$. Let $G: S \\to X$ be a mapping. Then there is a unique sequence $f$ whose codomain is in $X$ such that: :$f_i = \\begin{cases} a_i & : i \\in \\set {1, 2, \\ldots, h} \\\\ \\map G {f_1, f_2, \\ldots, f_{i - 1} } & : i \\ge h + 1 \\end{cases}$"}
+{"_id": "8499", "title": "Model of Root of Propositional Tableau is Model of Branch", "text": "Let $\\left({T, \\mathbf H, \\Phi}\\right)$ be a propositional tableau. Let $v: \\mathcal L_0 \\to \\left\\{{T, F}\\right\\}$ be a boolean interpretation such that: :$v \\models_{\\mathrm{BI}} \\mathbf H$ that is, such that $v$ is a model for the root $\\mathbf H$ of $T$. Then there exists a branch $\\Gamma$ of $T$ such that: :$v \\models_{\\mathrm{BI}} \\Phi \\left[{\\Gamma}\\right]$ where $\\Phi \\left[{\\Gamma}\\right]$ denotes the image of $\\Gamma$ under $\\Phi$."}
+{"_id": "8500", "title": "Antireflexive and Transitive Relation is Antisymmetric", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation which is not null. Let $\\mathcal R$ be antireflexive and transitive. Then $\\mathcal R$ is also antisymmetric."}
+{"_id": "8501", "title": "Tableau Confutation implies Unsatisfiable", "text": "Let $\\mathbf H$ be a collection of WFFs of propositional logic. Suppose there exists a tableau confutation of $\\mathbf H$. Then $\\mathbf H$ is unsatisfiable for boolean interpretations."}
+{"_id": "8502", "title": "Finished Branch Lemma", "text": "Let $\\Gamma$ be a finished branch of a propositional tableau $\\left({T, \\mathbf H, \\Phi}\\right)$. Let $v$ be a boolean interpretation such that: :$v \\models_{\\mathrm{BI}} \\mathbf A$ for every basic WFF $\\mathbf A$ that occurs along $\\Gamma$. Then: :$v \\models_{\\mathrm{BI}} \\Phi \\left[{\\Gamma}\\right]$ where $\\Phi \\left[{\\Gamma}\\right]$ is the image of $\\Gamma$ under $\\Phi$."}
+{"_id": "8503", "title": "Diagonal Relation is Serial", "text": "Let $S$ be a set. Let $\\Delta_S$ be the diagonal relation on $S$. Then $\\Delta_S$ is a serial relation."}
+{"_id": "8504", "title": "Serial Relation is not Null", "text": "Let $S$ be a set such that $S \\ne \\varnothing$. Let $\\mathcal R$ be a serial relation on $S$. Then $\\mathcal R$ is not a null relation."}
+{"_id": "8505", "title": "Transitive Relation is Antireflexive iff Asymmetric", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation which is not null. Let $\\mathcal R$ be transitive. Then $\\mathcal R$ is antireflexive iff $\\mathcal R$ is asymmetric."}
+{"_id": "8506", "title": "Symmetric and Antisymmetric Relation is Transitive", "text": "Let $S$ be a set. Let $\\RR \\subseteq S \\times S$ be a relation in $S$ which is both symmetric and antisymmetric. Then $\\RR$ is transitive."}
+{"_id": "8509", "title": "Relation is Reflexive and Coreflexive iff Diagonal", "text": "Let $S$ be a set. Let $\\mathcal R \\subseteq S \\times S$ be a relation on $S$. Then $\\mathcal R$ is reflexive and coreflexive iff: : $\\mathcal R = \\Delta_S$ where $\\Delta_S$ is the diagonal relation."}
+{"_id": "8510", "title": "Symmetric Preordering is Equivalence Relation", "text": "Let $\\RR \\subseteq S \\times S$ be a preordering on a set $S$. Let $\\RR$ also be symmetric. Then $\\RR$ is an equivalence relation on $S$."}
+{"_id": "8511", "title": "Antisymmetric Preordering is Ordering", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a preordering on a set $S$. Let $\\mathcal R$ also be antisymmetric. Then $\\mathcal R$ is an ordering on $S$."}
+{"_id": "8514", "title": "Relation is Antisymmetric and Reflexive iff Intersection with Inverse equals Diagonal Relation", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation on a set $S$. Then $\\mathcal R$ is both antisymmetric and reflexive iff: :$\\mathcal R \\cap \\mathcal R^{-1} = \\Delta_S$ where $\\Delta_S$ denotes the diagonal relation."}
+{"_id": "8515", "title": "Reflexive and Transitive Relation is Idempotent", "text": "Let $\\RR \\subseteq S \\times S$ be a relation on a set $S$. Let $\\RR$ be both reflexive and  transitive. Then $\\RR$ is idempotent, in the sense that: :$\\RR \\circ \\RR = \\RR$ where $\\circ$ denotes composition of relations."}
+{"_id": "8517", "title": "Equivalence of Definitions of Ordering/Proof 2", "text": "The following definitions of ordering are equivalent:"}
+{"_id": "8518", "title": "Ordered Tuple/Equality", "text": "Let: :$(1): \\quad \\sequence {a_m} = \\tuple {a_1, a_2, \\ldots, a_m}$ and :$(2): \\quad \\sequence {b_n} = \\tuple {b_1, b_2, \\ldots, b_n}$ be ordered tuples for some $m, n \\in \\N_{>0}$. Then: :$\\sequence {a_m} = \\sequence {b_n} \\iff n = m \\land \\forall j \\in \\N^*_n: a_j = b_j$"}
+{"_id": "8520", "title": "Image of Domain of Relation is Image Set", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R \\subseteq S \\times T$ be a relation. The image of the domain of $\\mathcal R$ is the image set of $\\mathcal R$: :$\\mathcal R \\left [{\\operatorname{Dom} \\left({\\mathcal R}\\right)}\\right] = \\operatorname{Im} \\left ({\\mathcal R}\\right)$ where $\\operatorname{Im} \\left ({\\mathcal R}\\right)$ is the image of $\\mathcal R$."}
+{"_id": "8521", "title": "Condition for Relation to be Transitive and Antitransitive", "text": "Let $S$ be a set. Let $\\RR \\subseteq S \\times S$ be a relation in $S$. Then: : $\\RR$ is both transitive and antitransitive {{iff}}: : $\\neg \\paren {\\exists x, y, z \\in S: x \\mathrel {\\RR} y \\land y \\mathrel {\\RR} z}$"}
+{"_id": "8522", "title": "Transitive and Antitransitive Relation is Asymmetric", "text": "Let $S$ be a set. Let $\\mathcal R \\subseteq S \\times S$ be a relation in $S$. Let $\\mathcal R$ be both transitive and antitransitive. Then $\\mathcal R$ is asymmetric."}
+{"_id": "8523", "title": "Valid Syllogism in Figure I needs Affirmative Minor Premise and Universal Major Premise", "text": "Let $Q$ be a valid categorical syllogism in Figure $\\text I$. Then it is a necessary condition that: :The major premise of $Q$ be a universal categorical statement and :The minor premise of $Q$ be an affirmative categorical statement."}
+{"_id": "8524", "title": "Valid Syllogism in Figure II needs Negative Conclusion and Universal Major Premise", "text": "Let $Q$ be a valid categorical syllogism in Figure $\\text{II}$. Then it is a necessary condition that: :The major premise of $Q$ be a universal categorical statement and :The conclusion of $Q$ be a negative categorical statement."}
+{"_id": "8525", "title": "Valid Syllogism in Figure III needs Particular Conclusion and if Negative then Negative Major Premise", "text": "Let $Q$ be a valid categorical syllogism in Figure $\\text {III}$. Then it is a necessary condition that: :The conclusion of $Q$ be a particular categorical statement and: :If the conclusion of $Q$ be a negative categorical statement, then so is the major premise of $Q$."}
+{"_id": "8526", "title": "Valid Syllogisms in Figure IV", "text": "Let $Q$ be a valid categorical syllogism in Figure $\\text {IV}$. Then it is a necessary condition that: :$(1): \\quad$ Either: :: the major premise of $Q$ be a negative categorical statement :or: :: the minor premise of $Q$ be a universal categorical statement :or both. :$(2): \\quad$ If the conclusion of $Q$ be a negative categorical statement, then the major premise of $Q$ be a universal categorical statement. :$(3): \\quad$ If the conclusion of $Q$ be a universal categorical statement, then the minor premise of $Q$ be a negative categorical statement."}
+{"_id": "8527", "title": "Elimination of all but 24 Categorical Syllogisms as Invalid", "text": "Of the $256$ different types of categorical syllogism, all but $24$ can be identified as invalid. These are the $24$ patterns which may still be valid: :$\\begin{array}{rl} \\text{I} & AAA \\\\ \\text{I} & AII \\\\ \\text{I} & EAE \\\\ \\text{I} & EIO \\\\ \\text{I} & AAI \\\\ \\text{I} & EAO \\\\ \\end{array} \\qquad \\begin{array}{rl} \\text{II} & EAE \\\\ \\text{II} & AEE \\\\ \\text{II} & AOO \\\\ \\text{II} & EIO \\\\ \\text{II} & EAO \\\\ \\text{II} & AEO \\\\ \\end{array} \\qquad \\begin{array}{rl} \\text{III} & AAI \\\\ \\text{III} & AII \\\\ \\text{III} & IAI \\\\ \\text{III} & EAO \\\\ \\text{III} & EIO \\\\ \\text{III} & OAO \\\\ \\end{array} \\qquad \\begin{array}{rl} \\text{IV} & AAI \\\\ \\text{IV} & AEE \\\\ \\text{IV} & EAO \\\\ \\text{IV} & EIO \\\\ \\text{IV} & IAI \\\\ \\text{IV} & AEO \\\\ \\end{array}$"}
+{"_id": "8528", "title": "Extension of Contradictory Branch is Contradictory", "text": "Let $T$ be a propositional tableau. Let $\\Gamma$ be a contradictory branch of $T$. Let $\\Gamma'$ be an extension of $\\Gamma$. Then $\\Gamma'$ is also contradictory."}
+{"_id": "8529", "title": "Finished Propositional Tableau has Finished Branch or is Confutation", "text": "Let $\\struct {T, \\mathbf H, \\Phi}$ be a finished propositional tableau. Then one of the following holds: :$T$ has a finished branch :$T$ is a tableau confutation."}
+{"_id": "8530", "title": "Definition:P-Series", "text": "Let $p \\in \\C$ be a complex number. The series defined as: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\frac 1 {n^p} = 1 + \\frac 1 {2^p} + \\frac 1 {3^p} + \\frac 1 {4^p} + \\dotsb$ is known as a '''$p$-series'''."}
+{"_id": "8531", "title": "Alternating Harmonic Series is Conditionally Convergent", "text": "The alternating harmonic series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\frac {\\paren {-1}^\\paren {n - 1} } n = 1 - \\frac 1 2 + \\frac 1 3 - \\frac 1 4 + \\cdots$ is conditionally convergent."}
+{"_id": "8532", "title": "Manipulation of Absolutely Convergent Series/Permutation", "text": "If $\\pi: \\N \\to \\N$ is a permutation of $N$, then: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n = \\sum_{n \\mathop = 1}^\\infty a_{\\map \\pi n}$"}
+{"_id": "8533", "title": "Manipulation of Absolutely Convergent Series/Characteristic Function", "text": "Let $A \\subseteq \\N$. Then: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n \\map {\\chi_A} n = \\sum_{n \\mathop \\in A} a_n$ where $\\chi_A$ is the characteristic function of $A$."}
+{"_id": "8534", "title": "Manipulation of Absolutely Convergent Series/Scale Factor", "text": "Let $c \\in \\R$, or $c \\in \\C$. Then: :$\\displaystyle c \\sum_{n \\mathop = 1}^\\infty a_n = \\sum_{n \\mathop = 1}^\\infty c a_n$"}
+{"_id": "8535", "title": "Cauchy Product of Absolutely Convergent Series", "text": "Let $\\displaystyle \\sum_{n \\mathop = 0}^\\infty a_n$ and $\\displaystyle \\sum_{n \\mathop = 0}^\\infty b_n$ be two real series that are absolutely convergent. Then the Cauchy product of $\\displaystyle \\sum_{n \\mathop = 0}^\\infty a_n$ and $\\displaystyle \\sum_{n \\mathop = 0}^\\infty b_n$ is absolutely convergent."}
+{"_id": "8536", "title": "Binomial Coefficient with Self minus One", "text": ":$\\forall n \\in \\N_{>0}: \\dbinom n {n - 1} = n$"}
+{"_id": "8537", "title": "Rising Sum of Binomial Coefficients/Marginal Cases", "text": "=== $n = 0$ === When $n = 0$ we have: {{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 0}^m \\binom j 0       | r = \\binom 0 0 + \\binom 1 0 + \\binom 2 0 + \\cdots \\binom m 0       | c =  }} {{eqn | r = 1 + 1 + \\cdots + 1       | c = from $0$ to $m$ }} {{eqn | r = m + 1       | c = as there are $m+1$ of them }} {{eqn | r = \\binom {m+1} 1       | c =  }} {{end-eqn}} So the theorem holds for $n = 0$. {{qed}} === $n = 1$ === When $n = 1$ we have: {{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 0}^m \\binom {1 + j} 1       | r = \\binom 1 1 + \\binom 2 1 + \\binom 3 1 + \\cdots \\binom {m + 1} 1       | c =  }} {{eqn | r = 1 + 2 + \\cdots + \\paren {m + 1}       | c =  }} {{eqn | r = \\frac {\\paren {m + 1} \\paren {m + 2} } 2       | c = Closed Form for Triangular Numbers }} {{eqn | r = \\binom {m + 2} 2       | c = {{Defof|Binomial Coefficient}} }} {{end-eqn}} So the theorem holds for $n = 1$. {{qed}}"}
+{"_id": "8538", "title": "Uniformly Convergent Series of Continuous Functions is Continuous", "text": "Let $\\sequence {f_n}$ be a sequence of real functions. Let each of $\\sequence {f_n}$ be continuous on the interval $\\hointr a b$. {{explain|Investigation needed as to whether there is a mistake in {{BookReference|Special Functions of Mathematics for Engineers|1992|Larry C. Andrews|ed = 2nd|edpage = Second Edition}} -- should it actually be a closed interval?}} Let the series: :$\\displaystyle \\map f x := \\sum_{n \\mathop = 1}^\\infty \\map {f_n} x$ be uniformly convergent for all $x \\in \\closedint a b$. Then $f$ is continuous on $\\hointr a b$."}
+{"_id": "8539", "title": "Definite Integral of Uniformly Convergent Series of Continuous Functions", "text": "Let $\\sequence {f_n}$ be a sequence of real functions. Let each of $\\sequence {f_n}$ be continuous on the interval $\\hointr a b$. {{explain|Investigation needed as to whether there is a mistake in {{BookReference|Special Functions of Mathematics for Engineers|1992|Larry C. Andrews|ed = 2nd|edpage = Second Edition}} -- should it actually be a closed interval?}} Let the series: :$\\displaystyle \\map f x := \\sum_{n \\mathop = 1}^\\infty \\map {f_n} x$ be uniformly convergent for all $x \\in \\closedint a b$. Then: :$\\displaystyle \\int_a^b \\map f x \\rd x = \\sum_{n \\mathop = 1}^\\infty \\int_a^b \\map {f_n} x \\rd x$"}
+{"_id": "8540", "title": "Derivative of Uniformly Convergent Series of Continuously Differentiable Functions", "text": "Let $\\sequence {f_n}$ be a sequence of real functions. Let each of $\\sequence {f_n}$ be continuously differentiable on the interval $\\closedint a b$. Let the series: :$\\displaystyle \\map f x := \\sum_{n \\mathop = 1}^\\infty \\map {f_n} x$ be pointwise convergent for all $x \\in \\closedint a b$. Let the series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\frac \\d {\\d x} \\map {f_n} x$ be uniformly convergent for all $x \\in \\closedint a b$. Then: :$\\displaystyle \\frac \\d {\\d x} \\map f x := \\sum_{n \\mathop = 1}^\\infty \\frac \\d {\\d x} \\map {f_n} x$"}
+{"_id": "8541", "title": "Power Series Converges Uniformly within Radius of Convergence", "text": "Let $\\displaystyle S := \\sum_{n \\mathop = 0}^\\infty a_n \\paren {x - \\xi}^n$ be a power series about a point $\\xi$. Let $R$ be the radius of convergence of $S$. Let $\\rho \\in \\R$ such that $0 \\le \\rho < R$. Then $S$ is uniformly convergent on $D = \\set {x: \\size {x - \\xi} \\le \\rho}$."}
+{"_id": "8542", "title": "Power Series Converges to Continuous Function", "text": "Let $\\displaystyle \\map f x := \\sum_{n \\mathop = 0}^\\infty a_n \\paren {x - \\xi}^n$ be a power series about a point $\\xi$. Let $R$ be the radius of convergence of $S$. Then $\\map f x$ is a continuous function on $\\set {x: \\size {x - \\xi} < R}$."}
+{"_id": "8543", "title": "Power Series is Termwise Integrable within Radius of Convergence", "text": "Let $\\displaystyle \\map f x := \\sum_{n \\mathop = 0}^\\infty a_n \\paren {x - \\xi}^n$ be a power series about a point $\\xi$. Let $R$ be the radius of convergence of $S$. Then: :$\\displaystyle \\int_a^b \\map f x \\rd x = \\sum_{n \\mathop = 0}^\\infty \\int_a^b a_n x^n \\rd x = \\sum_{n \\mathop = 0}^\\infty a_n \\frac {x^{n + 1} } {n + 1}$"}
+{"_id": "8544", "title": "Power Series is Termwise Differentiable within Radius of Convergence", "text": "Let $\\displaystyle \\map f x := \\sum_{n \\mathop = 0}^\\infty a_n \\paren {x - \\xi}^n$ be a power series about a point $\\xi$. Let $R$ be the radius of convergence of $S$. Then: :$\\displaystyle \\frac \\d {\\d x} \\map f x = \\sum_{n \\mathop = 0}^\\infty \\frac \\d {\\d x} a_n x^n = \\sum_{n \\mathop = 1}^\\infty  n a_n x^{n - 1}$"}
+{"_id": "8545", "title": "Power Series Expansion for Logarithm of 1 + x", "text": "The Newton-Mercator series defines the  natural logarithm function as a power series expansion: {{begin-eqn}} {{eqn | l = \\map \\ln {1 + x}       | r = \\sum_{n \\mathop = 1}^\\infty \\paren {-1}^{n - 1} \\frac {x^n} n }} {{eqn | r = x - \\frac {x^2} 2 + \\frac {x^3} 3 - \\frac {x^4} 4 + \\cdots }} {{end-eqn}} valid for all $x \\in \\R$ such that $-1 < x \\le 1$."}
+{"_id": "8548", "title": "Newton-Mercator Series/Examples/2", "text": "The Newton-Mercator Series for $x = 1$ converges to the natural logarithm of $2$: {{begin-eqn}} {{eqn | l = \\sum_{n \\mathop = 1}^\\infty \\frac {\\paren {-1}^\\paren {n - 1} } n       | r = 1 - \\frac 1 2 + \\frac 1 3 - \\frac 1 4 + \\dotsb       | c =  }} {{eqn | r = \\ln 2       | c =  }} {{end-eqn}} This real number is known as Mercator's constant."}
+{"_id": "8551", "title": "Power Series Expansion for Sine Function", "text": "The sine function has the power series expansion: {{begin-eqn}} {{eqn | l = \\sin x       | r = \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {x^{2 n + 1} } {\\paren {2 n + 1}!}       | c =  }} {{eqn | r = x - \\frac {x^3} {3!} + \\frac {x^5} {5!} - \\frac {x^7} {7!} + \\cdots       | c =  }} {{end-eqn}} valid for all $x \\in \\R$."}
+{"_id": "8552", "title": "Power Series Expansion for Cosine Function", "text": "The cosine function has the power series expansion: {{begin-eqn}} {{eqn | l = \\cos x       | r = \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {x^{2 n} } {\\paren {2 n}!}       | c =  }} {{eqn | r = 1 - \\frac {x^2} {2!} + \\frac {x^4} {4!} - \\frac {x^6} {6!} + \\cdots       | c =  }} {{end-eqn}} valid for all $x \\in \\R$."}
+{"_id": "8553", "title": "Spacing Limit Theorem", "text": "Let $X_{\\left({i}\\right)}$ be the $i$th ordered statistic of $N$ samples from a continuous random distribution with density function $f_X \\left({x}\\right)$. Then the spacing between the ordered statistics given $X_{\\left({i}\\right)}$ converges in distribution to exponential for sufficiently large sampling according to: :$N \\left({X_{\\left({i + 1}\\right)} - X_{\\left({i}\\right)} }\\right) \\xrightarrow D \\exp \\left({ \\dfrac 1 { f \\left({ X_{\\left({i}\\right)} }\\right)} }\\right)$ as $N \\to \\infty$ for $i = 1, 2, 3, \\dotsc, N-1$."}
+{"_id": "8561", "title": "Union of Closures of Singleton Rationals is Rational Space", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the usual (Euclidean) topology $\\tau_d$. Let $B_\\alpha$ denote the singleton containing the rational number $\\alpha$. Then the union of the closures in the set of real numbers $\\R$ of all $B_\\alpha$ is $\\Q$: :$\\displaystyle \\bigcup_{\\alpha \\mathop \\in \\Q} \\map \\cl {B_\\alpha} = \\Q$"}
+{"_id": "8562", "title": "Closure of Union of Singleton Rationals is Real Number Line", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the usual (Euclidean) topology $\\tau_d$. Let $B_\\alpha$ be the singleton containing the rational number $\\alpha$. Then the closure in the set of real numbers $\\R$ of the union of all $B_\\alpha$ is $\\R$ itself: :$\\displaystyle \\map \\cl {\\bigcup_{\\alpha \\mathop \\in \\Q} B_\\alpha} = \\R$"}
+{"_id": "8563", "title": "Closure of Rational Numbers is Real Numbers", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the same topology. Then: :$\\Q^- = \\R$ where $\\Q^-$ denotes the closure of $\\Q$."}
+{"_id": "8565", "title": "Real Number is Closed in Real Number Line", "text": "Let $\\struct {\\R, \\tau}$ be the real number line with the usual (Euclidean) topology. Let $\\alpha \\in \\R$ be a real number. Then $\\set \\alpha$ is closed in $\\struct {\\R, \\tau}$."}
+{"_id": "8566", "title": "Intersection of Exteriors of Singleton Rationals is Irrationals", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the usual (Euclidean) topology $\\tau_d$. Let $B_\\alpha$ be the singleton containing the rational number $\\alpha$. Then: :$\\displaystyle \\bigcap_{\\alpha \\mathop \\in \\Q} B_\\alpha^e = \\R \\setminus \\Q$ where $B_\\alpha^e$ denotes the exterior of $B_\\alpha$ in $\\R$."}
+{"_id": "8567", "title": "Rational Numbers form F-Sigma Set in Reals", "text": "Let $\\Q$ be the set of rational numbers. Let $\\struct {\\R, \\tau}$ be the real number line with the usual (Euclidean) topology. Then $\\Q$ is a $F_\\sigma$ set in $\\R$."}
+{"_id": "8568", "title": "Set of Rational Numbers is not Closed in Reals", "text": "Let $\\Q$ be the set of rational numbers. Let $\\struct {\\R, \\tau}$ denote the real number line with the usual (Euclidean) topology. Then $\\Q$ is not closed in $\\R$."}
+{"_id": "8569", "title": "Set of Rational Numbers is not G-Delta Set in Reals", "text": "Let $\\Q$ be the set of rational numbers. Let $\\struct {\\R, \\tau}$ denote the real number line with the usual (Euclidean) topology. Then $\\Q$ is not a $G_\\delta$ set in $\\R$."}
+{"_id": "8571", "title": "Eigenvalue of Matrix Powers", "text": "Let $A$ be a square matrix. Let $\\lambda$ be an eigenvalue of $A$ and $\\mathbf v$ be the corresponding eigenvector. Then: :$A^n \\mathbf v = \\lambda^n \\mathbf v$ holds for each positive integer $n$."}
+{"_id": "8572", "title": "Closure of Intersection of Rationals and Irrationals is Empty Set", "text": "Let $\\struct {\\R, \\tau}$ denote the real number line with the usual (Euclidean) topology. Let $\\Q$ be the set of rational numbers. Then: :$\\paren {\\Q \\cap \\paren {\\R \\setminus \\Q} }^- = \\O$ where: :$\\R \\setminus \\Q$ denotes the set of irrational numbers :$\\paren {\\Q \\cap \\paren {\\R \\setminus \\Q} }^-$ denotes the closure of $\\Q \\cap \\paren {\\R \\setminus \\Q}$."}
+{"_id": "8573", "title": "Intersection of Closures of Rationals and Irrationals is Reals", "text": "Let $\\struct {\\R, \\tau}$ be the real number line with the usual (Euclidean) topology. Let $\\Q$ be the set of rational numbers. Then: :$\\Q^- \\cap \\paren {\\R \\setminus \\Q}^- = \\R$ where: :$\\R \\setminus \\Q$ denotes the set of irrational numbers :$\\Q^-$ denotes the closure of $\\Q$."}
+{"_id": "8574", "title": "Irrational Numbers form Metric Space", "text": "Let $\\mathbb I = \\R \\setminus \\Q$ be the set of all irrational numbers. Let $d: \\mathbb I \\times \\mathbb I \\to \\R$ be defined as: :$\\map d {x_1, x_2} = \\size {x_1 - x_2}$ where $\\size x$ is the absolute value of $x$. Then $d$ is a metric on $\\mathbb I$ and so $\\struct {\\mathbb I, d}$ is a metric space."}
+{"_id": "8575", "title": "Rational Number Space is Completely Normal", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\Q, \\tau_d}$ is a completely normal space."}
+{"_id": "8576", "title": "Irrational Number Space is Completely Normal", "text": "Let $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\R \\setminus \\Q, \\tau_d}$ is a completely normal space."}
+{"_id": "8577", "title": "Rational Number Space is Paracompact", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\Q, \\tau_d}$ is paracompact."}
+{"_id": "8579", "title": "Euclidean Plus Metric is Metric", "text": "Let $\\R$ be the set of real numbers. Let $d: \\R \\times \\R \\to \\R$ be the Euclidean plus metric: :$\\map d {x, y} := \\size {x - y} + \\displaystyle \\sum_{i \\mathop = 1}^\\infty 2^{-i} \\map \\inf {1, \\size {\\max_{j \\mathop \\le i} \\frac 1 {\\size {x - r_j}} - \\max_{j \\mathop \\le i} \\frac 1 {\\size {y - r_j} } } }$ Then $d$ is indeed a metric."}
+{"_id": "8580", "title": "Open Ball in Euclidean Plus Metric is Subset of Equivalent Ball in Euclidean Metric", "text": "Let $\\R$ be the set of real numbers. Let $d: \\R \\times \\R \\to \\R$ be the Euclidean plus metric: :$\\map d {x, y} := \\size {x - y} + \\displaystyle \\sum_{i \\mathop = 1}^\\infty 2^{-i} \\map \\inf {1, \\size {\\max_{j \\mathop \\le i} \\frac 1 {\\size {x - r_j} } - \\max_{j \\mathop \\le i} \\frac 1 {\\size {y - r_j} } } }$ Let $d': \\R \\times \\R \\to \\R$ be the Euclidean metric. Let $\\epsilon \\in \\R_{>0}$ be a (strictly) positive real number. Let $\\map {B_\\epsilon} {p; d}$ be an open $\\epsilon$-ball of $p$ in $\\R$ on $d$. Let $\\map {B'_\\epsilon} {p; d'}$ be an open $\\epsilon$-ball of $p$ in $\\R$ on $d'$. Then: :$\\map {B_\\epsilon} {p; d} \\subseteq \\map {B'_\\epsilon} {p; d'}$"}
+{"_id": "8581", "title": "Irrational Number Space is Complete Metric Space", "text": "Let $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\R \\setminus \\Q, \\tau_d}$ is a complete metric space."}
+{"_id": "8583", "title": "Rational Number Space is Meager", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\Q, \\tau_d}$ is meager."}
+{"_id": "8584", "title": "Rational Number Space is Separable", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\Q, \\tau_d}$ is separable."}
+{"_id": "8585", "title": "Underlying Set of Topological Space is Everywhere Dense", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then the underlying set $S$ of $T$ is everywhere dense in $T$."}
+{"_id": "8586", "title": "Irrational Number Space is Separable", "text": "Let $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\R \\setminus \\Q, \\tau_d}$ is separable."}
+{"_id": "8588", "title": "Irrational Number Space is Second-Countable", "text": "Let $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\R \\setminus \\Q, \\tau_d}$ is second-countable."}
+{"_id": "8589", "title": "Compact Set of Rational Numbers is Nowhere Dense", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Let $S \\subseteq \\Q$ be a compact set of $\\Q$. Then $S$ is nowhere dense in $\\Q$."}
+{"_id": "8590", "title": "Compact Set of Irrational Numbers is Nowhere Dense", "text": "Let $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$. Let $S \\subseteq \\R \\setminus \\Q$ be a compact set of $\\R \\setminus \\Q$. Then $S$ is nowhere dense in $\\R \\setminus \\Q$."}
+{"_id": "8591", "title": "Rational Number Space is not Locally Compact Hausdorff Space", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\Q, \\tau_d}$ is not a locally compact Hausdorff Space."}
+{"_id": "8592", "title": "Irrational Number Space is not Locally Compact Hausdorff Space", "text": "Let $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\R \\setminus \\Q, \\tau_d}$ is not a locally compact Hausdorff Space."}
+{"_id": "8596", "title": "Rationals plus Irrational are Everywhere Dense in Irrationals", "text": "Let $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$. Let $x \\in \\R \\setminus \\Q$ be an arbitrary irrational number. Let $S_x$ be the set defined as: :$S_x := \\set {x + q: q \\in \\Q}$ Then $S_x$ is everywhere dense in $\\struct {\\R \\setminus \\Q, \\tau_d}$."}
+{"_id": "8597", "title": "Rational Number Space is Totally Separated", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\Q, \\tau_d}$ is totally separated."}
+{"_id": "8598", "title": "Irrational Number Space is Totally Separated", "text": "Let $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\R \\setminus \\Q, \\tau_d}$ is totally separated."}
+{"_id": "8600", "title": "Rational Number Space is Dense-in-itself", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\Q, \\tau_d}$ is dense-in-itself."}
+{"_id": "8601", "title": "Irrational Number Space is Dense-in-itself", "text": "Let $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\R \\setminus \\Q, \\tau_d}$ is dense-in-itself."}
+{"_id": "8602", "title": "Irrational Number Space is not Scattered", "text": "Let $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\R \\setminus \\Q, \\tau_d}$ is not scattered."}
+{"_id": "8603", "title": "Rational Number Space is Zero Dimensional", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\Q, \\tau_d}$ is zero dimensional."}
+{"_id": "8604", "title": "Irrational Number Space is Zero Dimensional", "text": "Let $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\R \\setminus \\Q, \\tau_d}$ is zero dimensional."}
+{"_id": "8605", "title": "Open Unit Interval on Rational Number Space is Bounded but not Compact", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Then: :$\\openint 0 1 \\cap \\Q$ is totally bounded but not compact where $\\openint 0 1$ is the open unit interval."}
+{"_id": "8607", "title": "Closure of Integer Reciprocal Space", "text": "Let $A \\subseteq \\R$ be the set of all points on $\\R$ defined as: :$A := \\set {\\dfrac 1 n: n \\in \\Z_{>0} }$ Let $\\struct {A, \\tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Then: :$A^- = A \\cup \\set 0$ where $A^-$ denotes the closure of $A$ in $\\R$."}
+{"_id": "8608", "title": "Zero is Limit Point of Integer Reciprocal Space", "text": "Let $A \\subseteq \\R$ be the set of all points on $\\R$ defined as: :$A := \\set {\\dfrac 1 n : n \\in \\Z_{>0} }$ Let $\\struct {A, \\tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Then $0$ is the only limit point of $A$ in $\\R$."}
+{"_id": "8610", "title": "Zero is Omega-Accumulation Point of Integer Reciprocal Space Union with Closed Interval", "text": "Let $A \\subseteq \\R$ be the set of all points on $\\R$ defined as: :$A := \\set {\\dfrac 1 n : n \\in \\Z_{>0} }$ Let $\\struct {A, \\tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Let $B$ be the uncountable set: :$B := A \\cup \\closedint 2 3$ where $\\closedint 2 3$ is a closed interval of $\\R$. $2$ and $3$ are to all intents arbitrary, but convenient. Then $0$ is an $\\omega$-accumulation point of $B$ in $\\R$."}
+{"_id": "8611", "title": "Zero is not Condensation Point of Integer Reciprocal Space Union with Closed Interval", "text": "Let $A \\subseteq \\R$ be the set of all points on $\\R$ defined as: :$A := \\set {\\dfrac 1 n : n \\in \\Z_{>0} }$ Let $\\struct {A, \\tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Let $B$ be the uncountable set: :$B := A \\cup \\closedint 2 3$ where $\\closedint 2 3$ is a closed interval of $\\R$. $2$ and $3$ are to all intents arbitrary, but convenient. Then $0$ is not a condensation point of $B$ in $\\R$."}
+{"_id": "8612", "title": "Integer Reciprocal Space contains Cauchy Sequence with no Limit Point", "text": "Let $A \\subseteq \\R$ be the set of all points on $\\R$ defined as: :$A := \\set {\\dfrac 1 n : n \\in \\Z_{>0} }$ Let $\\struct {A, \\tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Then $A$ has a Cauchy sequence which has no limit point in $A$."}
+{"_id": "8613", "title": "Integer Reciprocal Space with Zero is not Locally Connected", "text": "Let $A \\subseteq \\R$ be the set of all points on $\\R$ defined as: :$A := \\set 0 \\cup \\set {\\dfrac 1 n : n \\in \\Z_{>0} }$ Let $\\struct {A, \\tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology. Then $A$ is not locally connected."}
+{"_id": "8614", "title": "Discrete Space is Locally Connected", "text": "Let $T = \\struct {S, \\tau}$ be a discrete topological space. Then $T$ is locally connected."}
+{"_id": "8616", "title": "Integer Reciprocal Space with Zero is Totally Separated", "text": "Let $A \\subseteq \\R$ be the set of all points on $\\R$ defined as: :$A := \\set 0 \\cup \\set {\\dfrac 1 n : n \\in \\Z_{>0} }$ Let $\\struct {A, \\tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology. Then $A$ is totally separated."}
+{"_id": "8618", "title": "Quasicomponents of Integer Reciprocal Space with Zero are Single Points", "text": "Let $A \\subseteq \\R$ be the set of all points on $\\R$ defined as: :$A := \\set 0 \\cup \\set {\\dfrac 1 n : n \\in \\Z_{>0} }$ Let $\\struct {A, \\tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology. Then the quasicomponents of $A$ are singletons."}
+{"_id": "8621", "title": "Interior of Union of Adjacent Open Intervals", "text": "Let $a, b, c \\in R$ where $a < b < c$. Let $A$ be the union of the two adjacent open intervals: :$A := \\openint a b \\cup \\openint b c$ Then: :$A = A^\\circ$ where $A^\\circ$ is the interior of $A$."}
+{"_id": "8622", "title": "Interior of Closure of Interior of Union of Adjacent Open Intervals", "text": "Let $a, b, c \\in R$ where $a < b < c$. Let $A$ be the union of the two adjacent open intervals: :$A := \\openint a b \\cup \\openint b c$ Then: :$A^{\\circ - \\circ} = A^{- \\circ} = \\openint a c$ where: :$A^\\circ$ is the interior of $A$ :$A^-$ is the closure of $A$."}
+{"_id": "8623", "title": "Closure of Interior of Closure of Union of Adjacent Open Intervals", "text": "Let $a, b, c \\in R$ where $a < b < c$. Let $A$ be the union of the two adjacent open intervals: :$A := \\openint a b \\cup \\openint b c$ Then: :$A^{- \\circ -} = A^{\\circ -} = A^- = \\closedint a c$ where: :$A^\\circ$ is the interior of $A$ :$A^-$ is the closure of $A$."}
+{"_id": "8624", "title": "Exterior of Exterior of Union of Adjacent Open Intervals", "text": "Let $A$ be the union of the two adjacent open intervals: :$A := \\openint a b \\cup \\openint b c$ Then: :$A^{ee} = \\openint a c$ where $A^e$ is the exterior of $A$."}
+{"_id": "8626", "title": "Open Real Interval is Regular Open", "text": "Let $\\struct {\\R, \\tau}$ denote the real number line with the usual (Euclidean) topology. Let $\\openint a b$ be an open interval of $\\R$. Then $\\openint a b$ is regular open in $\\struct {\\R, \\tau_d}$."}
+{"_id": "8627", "title": "Closure of Open Real Interval is Closed Real Interval", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology.   Let $\\openint a b$ be an open interval of $\\R$. Then the closure of $\\openint a b$ is the closed interval $\\closedint a b$."}
+{"_id": "8628", "title": "Limit Points of Open Real Interval", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line under the usual (Euclidean) topology.   Let $\\openint a b$ be an open interval of $\\R$. Then the limit points of $\\openint a b$ are: :all the points in $\\openint a b$ as well as: :the points $a$ and $b$."}
+{"_id": "8629", "title": "Interior of Closed Real Interval is Open Real Interval", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology.   Let $\\closedint a b$ be a closed interval of $\\R$. Then the interior of $\\closedint a b$ is the open interval $\\left({a, b}\\right)$."}
+{"_id": "8630", "title": "Union of Regular Open Sets is not necessarily Regular Open", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $U$ and $V$ be regular open sets of $T$. Then $U \\cup V$ is not also necessarily a regular open set of $T$."}
+{"_id": "8631", "title": "Closure of Intersection may not equal Intersection of Closures", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H_1$ and $H_2$ be subsets of $S$. Let ${H_1}^-$ and ${H_2}^-$ denote the closures of $H_1$ and $H_2$ respectively. Then it is not necessarily the case that: :$\\paren {H_1 \\cap H_2}^- = {H_1}^- \\cap {H_2}^-$"}
+{"_id": "8632", "title": "Closure of Empty Set is Empty Set", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Then the closure of the empty set $\\varnothing$ in $T$ is $\\varnothing$."}
+{"_id": "8633", "title": "Closed Real Interval is Regular Closed", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\\closedint a b$ be a closed interval of $\\R$. Then $\\closedint a b$ is regular closed in $\\struct {\\R, \\tau_d}$."}
+{"_id": "8634", "title": "Closed Real Interval is Closed in Real Number Line", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\\closedint a b$ be a closed interval of $\\R$. Then $\\closedint a b$ is closed (in the topological sense) in $\\struct {\\R, \\tau_d}$."}
+{"_id": "8635", "title": "Intersection of Regular Closed Sets is not necessarily Regular Closed", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $U$ and $V$ be regular closed sets of $T$. Then $U \\cap V$ is not also necessarily a regular closed set of $T$."}
+{"_id": "8636", "title": "Interior of Union is not necessarily Union of Interiors", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H_1$ and $H_2$ be subsets of $S$. Let ${H_1}^\\circ$ and ${H_2}^\\circ$ denote the interiors of $H_1$ and $H_2$ respectively. Then it is not necessarily the case that: :$\\left({H_1 \\cup H_2}\\right)^\\circ = {H_1}^\\circ \\cup {H_2}^\\circ$"}
+{"_id": "8639", "title": "Expectation of Exponential Distribution", "text": "Let $X$ be a continuous random variable of the exponential distribution with parameter $\\beta$ for some $\\beta \\in \\R_{> 0}$ Then the expectation of $X$ is given by: :$\\expect X = \\beta$"}
+{"_id": "8643", "title": "Von Mangoldt Equivalence", "text": "For $n \\in \\N_{>0}$, let $\\Lambda \\left({n}\\right)$ be the von Mangoldt function. Then: :$\\displaystyle \\lim_{N \\to \\infty} \\frac 1 N \\sum_{n \\mathop = 1}^N \\Lambda \\left({n}\\right) = 1$ is logically equivalent to the Prime Number Theorem."}
+{"_id": "8644", "title": "Zeta Equivalence to Prime Number Theorem", "text": "Let $\\map \\zeta z$ be the Riemann $\\zeta$ function. The Prime Number Theorem is logically equivalent to the statement that the average of the first $N$ coefficients of $\\dfrac {\\zeta'} {\\zeta}$ tend to $-1$ as $N$ goes to infinity. {{explain|What does $z$ range over, and what does it mean by \"first $N$ coefficients\" of $\\dfrac {\\zeta'} {\\zeta}$?}}"}
+{"_id": "8647", "title": "Substitution Rule for Matrices", "text": "Let $\\mathbf A$ be a square matrix of order $n$. Then: :$(1): \\quad \\ds \\sum_{j \\mathop = 1}^n \\delta_{i j} a_{j k} = a_{i k}$ :$(2): \\quad \\ds \\sum_{j \\mathop = 1}^n \\delta_{i j} a_{k j} = a_{k i}$ where: :$\\delta_{i j}$ is the Kronecker delta :$a_{j k}$ is element $\\tuple {j, k}$ of $\\mathbf A$."}
+{"_id": "8652", "title": "Matrix is Invertible iff Determinant has Multiplicative Inverse/Necessary Condition", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $\\mathbf A \\in R^{n \\times n}$ be an invertible square matrix of order $n$. Let $\\mathbf B = \\mathbf A^{-1}$ be the inverse of $\\mathbf A$. Let $\\map \\det {\\mathbf A}$ be the determinant of $\\mathbf A$. Then: :$\\map \\det {\\mathbf B} = \\dfrac 1 {\\map \\det {\\mathbf A} }$"}
+{"_id": "8653", "title": "Product of Orthogonal Matrix with Transpose is Identity", "text": "Let $\\mathbf Q$ be an orthogonal matrix. Then: :$\\mathbf Q \\mathbf Q^\\intercal = \\mathbf I = \\mathbf Q^\\intercal \\mathbf Q$ where: :$\\mathbf Q^\\intercal$ is the transpose of $\\mathbf Q$ :$\\mathbf I$ is a unit (identity) matrix"}
+{"_id": "8654", "title": "Determinant of Orthogonal Matrix is Plus or Minus One", "text": "Let $\\mathbf Q$ be an orthogonal matrix. Then: :$\\det \\mathbf Q = \\pm 1$ where $\\det \\mathbf Q$ is the determinant of $\\mathbf Q$."}
+{"_id": "8655", "title": "Product of Proper Orthogonal Matrices is Proper Orthogonal Matrix", "text": "Let $\\mathbf P$ and $\\mathbf Q$ be proper orthogonal matrices. Let $\\mathbf P \\mathbf Q$ be the (conventional) matrix product of $\\mathbf P$ and $\\mathbf Q$. Then $\\mathbf P \\mathbf Q$ is a proper orthogonal matrix."}
+{"_id": "8656", "title": "Product of Orthogonal Matrices is Orthogonal Matrix", "text": "Let $\\mathbf P$ and $\\mathbf Q$ be orthogonal matrices. Let $\\mathbf P \\mathbf Q$ be the (conventional) matrix product of $\\mathbf P$ and $\\mathbf Q$. Then $\\mathbf P \\mathbf Q$ is an orthogonal matrix."}
+{"_id": "8658", "title": "Internal Group Direct Product is Injective/General Result", "text": "Let $G$ be a group whose identity is $e$. Let $\\sequence {H_n}$ be a sequence of subgroups of $G$. Let $\\displaystyle \\phi_n: \\prod_{j \\mathop = 1}^n H_j \\to G$ be a mapping defined by: :$\\displaystyle \\map {\\phi_n} {h_1, h_2, \\ldots, h_n} = \\prod_{j \\mathop = 1}^n h_j$ Then $\\phi_n$ is injective {{iff}}: :$\\forall i, j \\in \\set {1, 2, \\ldots, n}: i \\ne j \\implies H_i \\cap H_j = \\set e$ That is, {{iff}} $\\sequence {H_n}$ is a sequence of independent subgroups."}
+{"_id": "8659", "title": "Trace of Matrix Product", "text": "Let $\\mathbf A$ and $\\mathbf B$ be square matrices of order $n$. Let $\\mathbf A \\mathbf B$ be the (conventional) matrix product of $\\mathbf A$ and $\\mathbf B$. Then: :$\\ds \\map \\tr {\\mathbf A \\mathbf B} = \\sum_{i \\mathop = 1}^n \\sum_{j \\mathop = 1}^n a_{i j} b_{j i}$ where $\\map \\tr {\\mathbf A \\mathbf B}$ denotes the trace of $\\mathbf A \\mathbf B$. Using the Einstein summation convention, this can be expressed as: :$\\map \\tr {\\mathbf A \\mathbf B} = a_{i j} b_{j i}$"}
+{"_id": "8660", "title": "Trace of Matrix Product/General Result", "text": "Let $\\mathbf A_1, \\mathbf A_2, \\ldots, \\mathbf A_m$ be square matrices of order $n$. Let $\\mathbf A_1 \\mathbf A_2 \\cdots \\mathbf A_m$ be the (conventional) matrix product of $\\mathbf A_1, \\mathbf A_2, \\ldots, \\mathbf A_m$. Then: :$(1): \\quad \\ds \\map \\tr {\\mathbf A_1 \\mathbf A_2 \\cdots \\mathbf A_m} = \\map {a_1} {i_1, i_2} \\map {a_2} {i_2, i_3} \\cdots \\map {a_{m - 1} } {i_{m - 1}, i_m} \\map {a_m} {i_m, i_1}$ where: :$\\map {a_1} {i_1, i_2}$ (for example) denotes the element of $\\mathbf A_1$ whose indices are $i_1$ and $i_2$ :$\\map \\tr {\\mathbf A_1 \\mathbf A_2 \\cdots \\mathbf A_m}$ denotes the trace of $\\mathbf A_1 \\mathbf A_2 \\cdots \\mathbf A_m$. In $(1)$, the Einstein summation convention is used, with the implicit understanding that a summation is performed over each of the indices $i_1$ to $i_m$."}
+{"_id": "8661", "title": "Product of Finite Sequence of Matrices", "text": "Let $\\mathbf A_1, \\mathbf A_2, \\ldots, \\mathbf A_n$ be matrices. Let the order of $\\mathbf A_j$ be $d_j \\times d_{j + 1}$. Let $\\displaystyle \\mathbf C := \\prod_{j \\mathop = 1}^n \\mathbf A_j = \\mathbf A_1 \\mathbf A_2 \\cdots \\mathbf A_n$ be the (conventional) matrix product of $\\mathbf A_1, \\mathbf A_2, \\ldots, \\mathbf A_n$. Then: :$(1): \\quad \\displaystyle \\map c {i_1, i_{n + 1} } = \\sum_{i_n \\mathop = 1}^{d_n} \\dotsm \\sum_{i_3 \\mathop = 1}^{d_3} \\sum_{i_2 \\mathop = 1}^{d_2} \\map {a_1} {i_1, i_2} \\map {a_2} {i_2, i_3} \\dotsm \\map {a_{n - 1} } {i_{n - 1}, i_n} \\map {a_n} {i_n, i_{n + 1} }$ where: :$\\map {a_1} {i_1, i_2}$ (for example) denotes the element of $\\mathbf A_1$ whose indices are $i_1$ and $i_2$ :the order of $\\mathbf C$ is $d_1 \\times d_{n+1}$."}
+{"_id": "8662", "title": "Characteristic of Galois Field is Prime", "text": "Let $\\GF$ be a Galois field.  Then the characteristic of $\\GF$ is a prime number."}
+{"_id": "8663", "title": "Finite Ring with No Proper Zero Divisors is Field", "text": "Let $\\struct {R, +, \\circ}$ be a finite non-null ring with no proper zero divisors. Then $R$ is a field."}
+{"_id": "8664", "title": "Equivalence of Definitions of Dot Product", "text": "{{TFAE|def = Dot Product}} Let $\\mathbf a$ and $\\mathbf b$ be vectors in the real Euclidean space $\\R^n$."}
+{"_id": "8665", "title": "Determinant of Kronecker Delta Elements", "text": "Let $\\lambda$ and $\\pi$ be permutations on $\\left\\{{1, 2, 3}\\right\\}$. Let: : $\\left({i, j, k}\\right) = \\left({\\lambda \\left({1}\\right), \\lambda \\left({2}\\right), \\lambda \\left({3}\\right)}\\right)$ : $\\left({r, s, t}\\right) = \\left({\\pi \\left({1}\\right), \\pi \\left({2}\\right), \\pi \\left({3}\\right)}\\right)$ Then: :$\\begin{vmatrix}   \\delta_{ir} & \\delta_{is} & \\delta_{it} \\\\   \\delta_{jr} & \\delta_{js} & \\delta_{jt} \\\\   \\delta_{kr} & \\delta_{ks} & \\delta_{kt} \\end{vmatrix} = \\operatorname{sgn} \\left({i, j, k}\\right) \\operatorname{sgn} \\left({r, s, t}\\right)$ where: : $\\delta_{ir}$ denotes the Kronecker delta : $\\begin{vmatrix} \\cdot \\end{vmatrix}$ denotes a determinant : $\\operatorname{sgn} \\left({i, j, k}\\right)$ is the sign of the permutation $\\left({i, j, k}\\right)$ of the set $\\left\\{{1, 2, 3}\\right\\}$."}
+{"_id": "8666", "title": "Row Equivalent Matrix for Homogeneous System has same Solutions/Corollary", "text": ":$\\set {\\mathbf x: \\mathbf A \\mathbf x = \\mathbf 0} = \\set {\\mathbf x: \\map {\\mathrm {ref} } {\\mathbf A} \\mathbf x = \\mathbf 0}$ where $\\map {\\mathrm {ref} } {\\mathbf A}$ is the reduced echelon form of $\\mathbf A$."}
+{"_id": "8669", "title": "Continuous Mapping is Sequentially Continuous/Corollary", "text": "Let $f$ be continuous (everywhere) on $X$. Then $f$ is sequentially continuous on $X$."}
+{"_id": "8670", "title": "Product of Complex Numbers in Polar Form", "text": "Let $z_1 := \\polar {r_1, \\theta_1}$ and $z_2 := \\polar {r_2, \\theta_2}$ be complex numbers expressed in polar form. Then: :$z_1 z_2 = r_1 r_2 \\paren {\\map \\cos {\\theta_1 + \\theta_2} + i \\map \\sin {\\theta_1 + \\theta_2} }$"}
+{"_id": "8672", "title": "Subtraction of Complex Numbers", "text": "Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers. The subtraction operation on $z_1$ and $z_2$ is: :$z_1 - z_2 = \\paren {a_1 - a_2} + i \\paren {b_1 - b_2}$"}
+{"_id": "8673", "title": "Division of Complex Numbers", "text": "Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers such that $z_2 \\ne 0$. The operation of division is performed on $z_1$ by $z_2$ as follows: :$\\dfrac {z_1} {z_2} = \\dfrac {a_1 a_2 + b_1 b_2} {a_2^2 + b_2^2} + i \\dfrac {a_2 b_1 - a_1 b_2} {a_2^2 + b_2^2}$"}
+{"_id": "8674", "title": "De Moivre's Formula/Proof 1", "text": "Let $z \\in \\C$ be a complex number expressed in complex form: :$z = r \\paren {\\cos x + i \\sin x}$ Then: :$\\forall \\omega \\in \\C: \\paren {r \\paren {\\cos x + i \\sin x} }^\\omega = r^\\omega \\map \\cos {\\omega x} + i \\, \\map \\sin {\\omega x}$"}
+{"_id": "8676", "title": "Division of Complex Numbers in Polar Form", "text": "Let $z_1 := \\polar {r_1, \\theta_1}$ and $z_2 := \\polar {r_2, \\theta_2}$ be complex numbers expressed in polar form, such that $z_2 \\ne 0$. Then: :$\\dfrac {z_1} {z_2} = \\dfrac {r_1} {r_2} \\paren {\\map \\cos {\\theta_1 - \\theta_2} + i \\map \\sin {\\theta_1 - \\theta_2} }$ or: :$\\dfrac {z_1} {z_2} = \\dfrac {r_1} {r_2} \\map \\cis {\\theta_1 - \\theta_2}$"}
+{"_id": "8677", "title": "Roots of Complex Number", "text": "Let $z := \\polar {r, \\theta}$ be a complex number expressed in polar form, such that $z \\ne 0$. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then the $n$th roots of $z$ are given by: :$z^{1 / n} = \\set {r^{1 / n} \\paren {\\map \\cos {\\dfrac {\\theta + 2 \\pi k} n} + i \\, \\map \\sin {\\dfrac {\\theta + 2 \\pi k} n} }: k \\in \\set {0, 1, 2, \\ldots, n - 1} }$ There are $n$ distinct such $n$th roots."}
+{"_id": "8680", "title": "Zeroth Power of Real Number equals One", "text": "Let $a \\in \\R_{>0}$ be a (strictly) positive real number. Let $a^x$ be defined as $a$ to the power of $x$. Then: :$a^0 = 1$"}
+{"_id": "8681", "title": "Exponential of One", "text": ": $\\exp 1 = e$ where $e$ is Euler's number: $e = 2.718281828\\ldots$"}
+{"_id": "8684", "title": "Exponential of Zero", "text": ": $\\exp 0 = 1$"}
+{"_id": "8685", "title": "Difference of Logarithms", "text": ":$\\log_b x - \\log_b y = \\map {\\log_b} {\\dfrac x y}$"}
+{"_id": "8688", "title": "Change of Base of Logarithm/Base 10 to Base e", "text": "==== Form 1 ==== {{:Change of Base of Logarithm/Base 10 to Base e/Form 1}} ==== Form 2 ==== {{:Change of Base of Logarithm/Base 10 to Base e/Form 2}}"}
+{"_id": "8689", "title": "Change of Base of Logarithm/Base e to Base 10", "text": "==== Form 1 ==== {{:Change of Base of Logarithm/Base e to Base 10/Form 1}} ==== Form 2 ==== {{:Change of Base of Logarithm/Base e to Base 10/Form 2}}"}
+{"_id": "8690", "title": "Power of Identity is Identity", "text": "Let $\\struct {M, \\circ}$ be a monoid whose identity element is $e$. Then: :$\\forall n \\in \\Z: e^n = e$"}
+{"_id": "8694", "title": "Tangent Exponential Formulation/Formulation 2", "text": ":$\\tan z = \\dfrac {e^{i z} - e^{-i z} } {i \\paren {e^{i z} + e^{-i z} } }$"}
+{"_id": "8698", "title": "Cotangent Exponential Formulation", "text": ":$\\cot z = i \\dfrac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }$"}
+{"_id": "8699", "title": "Cosecant Exponential Formulation", "text": "Let $z$ be a complex number. Let $\\csc z$ denote the cosecant function and $i$ denote the imaginary unit: $i^2 = -1$. Then: :$\\csc z = \\dfrac {2 i} {e^{i z} - e^{-i z} }$"}
+{"_id": "8700", "title": "Secant Exponential Formulation", "text": "Let $z$ be a complex number. Let $\\sec z$ denote the secant function and $i$ denote the imaginary unit: $i^2 = -1$. Then: :$\\sec z = \\dfrac 2 {e^{i z} + e^{-i z} }$"}
+{"_id": "8701", "title": "Periodicity of Complex Exponential Function", "text": "For all $k \\in \\Z$: :$e^{i \\paren {\\theta + 2 k \\pi} } = e^{i \\theta}$"}
+{"_id": "8702", "title": "Product of Complex Numbers in Exponential Form", "text": "Let $z_1 := r_1 e^{i \\theta_1}$ and $z_2 := r_2 e^{i \\theta_2}$ be complex numbers expressed in exponential form. Then: :$z_1 z_2 = r_1 r_2 e^{i \\paren {\\theta_1 + \\theta_2} }$"}
+{"_id": "8703", "title": "Division of Complex Numbers in Exponential Form", "text": "Let $z_1 := r_1 e^{i \\theta_1}$ and $z_2 := r_2 e^{i \\theta_2}$ be complex numbers expressed in exponential form. Then: :$\\dfrac {z_1} {z_2} = \\dfrac {r_1} {r_2} e^{i \\paren {\\theta_1 - \\theta_2} }$"}
+{"_id": "8704", "title": "De Moivre's Formula/Exponential Form", "text": ":$\\forall \\omega \\in \\C: \\paren {r e^{i \\theta} }^\\omega = r^\\omega e^{i \\omega \\theta}$"}
+{"_id": "8706", "title": "Z/(m)-Module Associated with Ring of Characteristic m", "text": "Let $\\left({R,+,*}\\right)$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let the characteristic of $R$ be $m$. Let $\\left({\\Z_m, +_m, \\times_m}\\right)$ be the ring of integers modulo $m$. Let $\\circ$ be the mapping from $\\Z_m \\times R$ to $R$ defined as: :$\\forall \\left[\\!\\left[a\\right]\\!\\right]_m \\in \\Z_m: \\forall x \\in R: \\left[\\!\\left[a\\right]\\!\\right]_m \\circ x = a \\cdot x$ where $\\left[\\!\\left[a\\right]\\!\\right]_m$ is the residue class of $a$ modulo $m$ and $a \\cdot x$ is the $a$th power of $x$. Then $\\left({R, +, \\circ}\\right)_{\\Z_m}$ is a unitary $\\Z_m$-module."}
+{"_id": "8707", "title": "Sum of Squares of Hyperbolic Secant and Tangent", "text": ":$\\sech^2 x + \\tanh^2 x = 1$ where $\\sech$ and $\\tanh$ are hyperbolic secant and hyperbolic tangent."}
+{"_id": "8708", "title": "Difference of Squares of Hyperbolic Cotangent and Cosecant", "text": ":$\\coth^2 x - \\csch^2 x = 1$ where $\\coth$ and $\\csch$ are hyperbolic cotangent and hyperbolic cosecant."}
+{"_id": "8709", "title": "Hyperbolic Cosecant Function is Odd", "text": "Let $\\csch: \\C \\to \\C$ be the hyperbolic cosecant function on the set of complex numbers. Then $\\csch$ is odd: :$\\map \\csch {-x} = -\\csch x$"}
+{"_id": "8710", "title": "Hyperbolic Secant Function is Even", "text": ":$\\map \\sech {-x} = \\sech x$"}
+{"_id": "8711", "title": "Hyperbolic Cotangent Function is Odd", "text": "Let $\\coth: \\C \\to \\C$ be the hyperbolic cotangent function on the set of complex numbers. Then $\\coth$ is odd: :$\\map \\coth {-x} = -\\coth x$"}
+{"_id": "8712", "title": "Hyperbolic Sine of Sum", "text": ":$\\map \\sinh {a + b} = \\sinh a \\cosh b + \\cosh a \\sinh b$"}
+{"_id": "8713", "title": "Hyperbolic Cosine of Sum", "text": ":$\\map \\cosh {a + b} = \\cosh a \\cosh b + \\sinh a \\sinh b$"}
+{"_id": "8715", "title": "Hyperbolic Tangent of Sum", "text": ":$\\map \\tanh {a + b} = \\dfrac {\\tanh a + \\tanh b} {1 + \\tanh a \\tanh b}$"}
+{"_id": "8716", "title": "Hyperbolic Cotangent of Sum", "text": ":$\\map \\coth {a + b} = \\dfrac {\\coth a \\coth b + 1} {\\coth b + \\coth a}$"}
+{"_id": "8717", "title": "Double Angle Formulas/Hyperbolic Sine", "text": ":$\\sinh 2 x = 2 \\sinh x \\cosh x$"}
+{"_id": "8718", "title": "Double Angle Formulas/Hyperbolic Cosine", "text": ":$\\cosh 2 x = \\cosh^2 x + \\sinh^2 x$"}
+{"_id": "8719", "title": "Double Angle Formulas/Hyperbolic Tangent", "text": ": $\\tanh 2 x = \\dfrac {2 \\tanh x} {1 + \\tanh^2 x}$"}
+{"_id": "8722", "title": "Half Angle Formulas/Hyperbolic Sine", "text": "{{begin-eqn}} {{eqn | l = \\sinh \\frac x 2       | r = +\\sqrt {\\frac {\\cosh x - 1} 2}       | c = for $x \\ge 0$ }} {{eqn | l = \\sinh \\frac x 2       | r = -\\sqrt {\\dfrac {\\cosh x - 1} 2}       | c = for $x \\le 0$ }} {{end-eqn}}"}
+{"_id": "8723", "title": "Half Angle Formulas/Hyperbolic Cosine", "text": ":$\\cosh \\dfrac x 2 = +\\sqrt {\\dfrac {\\cosh x + 1} 2}$"}
+{"_id": "8724", "title": "Half Angle Formulas/Hyperbolic Tangent", "text": "{{begin-eqn}} {{eqn | l = \\tanh \\frac x 2       | r = +\\sqrt {\\frac {\\cosh x - 1} {\\cosh x + 1} }       | c = for $x \\ge 1$ }} {{eqn | l = \\tanh \\frac x 2       | r = -\\sqrt {\\frac {\\cosh x - 1} {\\cosh x + 1} }       | c = for $x \\le 1$ }} {{end-eqn}}"}
+{"_id": "8726", "title": "Half Angle Formulas/Hyperbolic Tangent/Corollary 2", "text": "For $x \\ne 0$: :$\\tanh \\dfrac x 2 = \\dfrac {\\cosh x - 1} {\\sinh x}$"}
+{"_id": "8727", "title": "Triple Angle Formulas/Hyperbolic Sine", "text": ":$\\sinh 3 x = 3 \\sinh x + 4 \\sinh^3 x$"}
+{"_id": "8728", "title": "Triple Angle Formulas/Hyperbolic Cosine", "text": ": $\\cosh 3 x = 4 \\cosh^3 x - 3 \\cosh x$"}
+{"_id": "8729", "title": "Triple Angle Formulas/Hyperbolic Tangent", "text": ": $\\tanh \\left({3 x}\\right) = \\dfrac {3 \\tanh x + \\tanh^3 x} {1 + 3 \\tanh^2 x}$"}
+{"_id": "8730", "title": "Quadruple Angle Formulas/Hyperbolic Sine", "text": ": $\\sinh \\left({4 x}\\right) = 8 \\sinh^3 x \\cosh x + 4 \\sinh x \\cosh x$"}
+{"_id": "8731", "title": "Quadruple Angle Formulas/Hyperbolic Cosine", "text": ": $\\cosh \\left({4 x}\\right) = 8 \\cosh^4 x - 8 \\cosh^2 x + 1$"}
+{"_id": "8732", "title": "Quadruple Angle Formulas/Hyperbolic Tangent", "text": ": $\\tanh \\left({4 x}\\right) = \\dfrac {4 \\tanh x + 4 \\tanh^3 x} {1 + 6 \\tanh^2 x + \\tanh^4 x}$"}
+{"_id": "8733", "title": "Power Reduction Formulas/Hyperbolic Sine Squared", "text": ":$\\sinh^2 x = \\dfrac {\\cosh 2 x - 1} 2$"}
+{"_id": "8734", "title": "Power Reduction Formulas/Hyperbolic Cosine Squared", "text": ":$\\cosh^2 x = \\dfrac {\\cosh 2 x + 1} 2$"}
+{"_id": "8737", "title": "Power Reduction Formulas/Hyperbolic Sine to 4th", "text": ":$\\sinh^4 x = \\dfrac {3 - 4 \\cosh 2 x + \\cosh 4 x} 8$"}
+{"_id": "8739", "title": "Prosthaphaeresis Formulas/Hyperbolic Sine plus Hyperbolic Sine", "text": ":$\\sinh x + \\sinh y = 2 \\sinh \\left({\\dfrac {x + y} 2}\\right) \\cosh \\left({\\dfrac {x - y} 2}\\right)$"}
+{"_id": "8740", "title": "Prosthaphaeresis Formulas/Hyperbolic Sine minus Hyperbolic Sine", "text": ":$\\sinh x - \\sinh y = 2 \\map \\cosh {\\dfrac {x + y} 2} \\map \\sinh {\\dfrac {x - y} 2}$"}
+{"_id": "8741", "title": "Prosthaphaeresis Formulas/Hyperbolic Cosine plus Hyperbolic Cosine", "text": ":$\\cosh x + \\cosh y = 2 \\map \\cosh {\\dfrac {x + y} 2} \\map \\cosh {\\dfrac {x - y} 2}$"}
+{"_id": "8743", "title": "Simpson's Formulas/Hyperbolic Sine by Hyperbolic Sine", "text": ":$\\sinh x \\sinh y = \\dfrac {\\map \\cosh {x + y} - \\map \\cosh {x - y} } 2$"}
+{"_id": "8744", "title": "Simpson's Formulas/Hyperbolic Cosine by Hyperbolic Cosine", "text": ":$\\cosh x \\cosh y = \\dfrac {\\cosh \\paren {x + y} + \\cosh \\paren {x - y} } 2$"}
+{"_id": "8745", "title": "Simpson's Formulas/Hyperbolic Sine by Hyperbolic Cosine", "text": ":$\\sinh x \\cosh y = \\dfrac {\\sinh \\paren {x + y} + \\sinh \\paren {x - y} } 2$"}
+{"_id": "8747", "title": "Graph of Hyperbolic Cosine Function", "text": ":600px"}
+{"_id": "8752", "title": "Inverse Hyperbolic Sine of Reciprocal equals Inverse Hyperbolic Cosecant", "text": "Everywhere that the function is defined: :$\\map {\\sinh^{-1} } {\\dfrac 1 x} = \\csch^{-1} x$ where $\\sinh^{-1}$ and $\\csch^{-1}$ denote inverse hyperbolic sine and inverse hyperbolic cosecant respectively."}
+{"_id": "8753", "title": "Inverse Hyperbolic Cosine of Reciprocal equals Inverse Hyperbolic Secant", "text": "Everywhere that the function is defined: :$\\map {\\cosh^{-1} } {\\dfrac 1 x} = \\sech^{-1} x$ where $\\sinh^{-1}$ and $\\csch^{-1}$ denote inverse hyperbolic cosine and inverse hyperbolic secant respectively."}
+{"_id": "8754", "title": "Inverse Hyperbolic Tangent of Reciprocal equals Inverse Hyperbolic Cotangent", "text": "Everywhere that the function is defined: :$\\map {\\tanh^{-1} } {\\dfrac 1 x} = \\coth^{-1} x$ where $\\tanh^{-1}$ and $\\coth^{-1}$ denote inverse hyperbolic tangent and inverse hyperbolic cotangent respectively."}
+{"_id": "8755", "title": "Inverse Hyperbolic Sine is Odd Function", "text": "Let $x \\in \\R$. Then: :$\\map {\\sinh^{-1} } {-x} = -\\sinh^{-1} x$ where $\\map {\\sinh^{-1} } {-x}$ denotes the inverse hyperbolic sine function."}
+{"_id": "8756", "title": "Inverse Hyperbolic Tangent is Odd Function", "text": ":$\\map {\\tanh^{-1} } {-x} = -\\tanh^{-1} x$"}
+{"_id": "8758", "title": "Inverse Hyperbolic Cosecant is Odd Function", "text": "Let $x \\in \\R$. Then: :$\\map {\\csch^{-1} } {-x} = -\\csch^{-1} x$ where $\\map {\\csch^{-1} } {-x}$ denotes the inverse hyperbolic cosecant function."}
+{"_id": "8765", "title": "Hyperbolic Cosecant in terms of Cosecant", "text": "Let $z \\in \\C$ be a complex number. Then: :$i \\csch z = -\\csc \\paren {i z}$ where: : $\\csc$ denotes the cosecant function : $\\csch$ denotes the hyperbolic cosecant : $i$ is the imaginary unit: $i^2 = -1$."}
+{"_id": "8766", "title": "Hyperbolic Secant in terms of Secant", "text": "Let $z \\in \\C$ be a complex number. Then: :$\\sech z = \\sec \\paren {i z}$ where: : $\\sec$ denotes the secant function : $\\sech$ denotes the hyperbolic secant : $i$ is the imaginary unit: $i^2 = -1$."}
+{"_id": "8767", "title": "Hyperbolic Cotangent in terms of Cotangent", "text": "Let $z \\in \\C$ be a complex number. Then: :$\\coth z = -\\cot \\paren {i z}$ where: : $\\cot$ denotes the cotangent function : $\\coth$ denotes the hyperbolic cotangent : $i$ is the imaginary unit: $i^2 = -1$."}
+{"_id": "8768", "title": "Sine in terms of Hyperbolic Sine", "text": "Let $z \\in \\C$ be a complex number. Then: :$i \\sin z = \\map \\sinh {i z}$ where: :$\\sin$ denotes the complex sine :$\\sinh$ denotes the hyperbolic sine :$i$ is the imaginary unit: $i^2 = -1$."}
+{"_id": "8769", "title": "Cosine in terms of Hyperbolic Cosine", "text": "Let $z \\in \\C$ be a complex number. Then: :$\\cos z = \\map \\cosh {i z}$ where: :$\\cos$ denotes the complex cosine :$\\cosh$ denotes the hyperbolic cosine :$i$ is the imaginary unit: $i^2 = -1$."}
+{"_id": "8770", "title": "Tangent in terms of Hyperbolic Tangent", "text": "Let $z \\in \\C$ be a complex number. Then: :$i \\tan z = \\map \\tanh {i z}$ where: :$\\tan$ denotes the tangent function :$\\tanh$ denotes the hyperbolic tangent :$i$ is the imaginary unit: $i^2 = -1$."}
+{"_id": "8771", "title": "Cosecant in terms of Hyperbolic Cosecant", "text": "Let $z \\in \\C$ be a complex number. Then: :$i \\csc = -\\csch \\paren {i z}$ where: : $\\csc$ denotes the cosecant function : $\\csch$ denotes the hyperbolic cosecant : $i$ is the imaginary unit: $i^2 = -1$."}
+{"_id": "8772", "title": "Secant in terms of Hyperbolic Secant", "text": "Let $z \\in \\C$ be a complex number. Then: :$\\sec z = \\map \\sech {i z}$ where: :$\\sec$ denotes the secant function :$\\sech$ denotes the  hyperbolic secant :$i$ is the imaginary unit: $i^2 = -1$."}
+{"_id": "8773", "title": "Cotangent in terms of Hyperbolic Cotangent", "text": "Let $z \\in \\C$ be a complex number. Then: :$i \\cot z = -\\coth \\paren {i z}$ where: : $\\cot$ denotes the cotangent function : $\\coth$ denotes the hyperbolic cotangent : $i$ is the imaginary unit: $i^2 = -1$."}
+{"_id": "8774", "title": "General Periodicity Property", "text": "Let $f: X \\to X$ be a periodic function, where $X$ is either the set of real numbers $\\R$ or the set of complex numbers $\\C$. Let $L$ be a periodic element of $f$. Then: :$\\forall n \\in \\Z: \\forall x \\in X: \\map f x = \\map f {x + n L}$ That is, after every distance $L$, the function $f$ repeats itself."}
+{"_id": "8775", "title": "Periodicity of Hyperbolic Sine", "text": "Let $k \\in \\Z$. Then: :$\\map \\sinh {x + 2 k \\pi i} = \\sinh x$"}
+{"_id": "8776", "title": "Periodicity of Hyperbolic Cosine", "text": "Let $k \\in \\Z$. Then: :$\\cosh \\left({x + 2 k \\pi i}\\right) = \\cosh x$"}
+{"_id": "8777", "title": "Periodicity of Hyperbolic Tangent", "text": "Let $k \\in \\Z$. Then: :$\\tanh \\left({x + 2 k \\pi i}\\right) = \\tanh x$"}
+{"_id": "8778", "title": "Periodicity of Hyperbolic Cosecant", "text": "Let $k \\in \\Z$. Then: :$\\operatorname{csch} \\left({x + 2 k \\pi i}\\right) = \\operatorname{csch} x$"}
+{"_id": "8779", "title": "Periodicity of Hyperbolic Secant", "text": "Let $k \\in \\Z$. Then: :$\\map \\sech {x + 2 k \\pi i} = \\sech x$"}
+{"_id": "8780", "title": "Periodicity of Hyperbolic Cotangent", "text": "Let $k \\in \\Z$. Then: :$\\map \\coth {x + 2 k \\pi i} = \\coth x$"}
+{"_id": "8781", "title": "Inverse Sine of Imaginary Number", "text": ":$\\map {\\sin^{-1} } {i x} = i \\sinh^{-1} x$"}
+{"_id": "8782", "title": "Inverse Hyperbolic Sine of Imaginary Number", "text": ":$\\sinh^{-1} \\left({i x}\\right) = i \\sin^{-1} x$"}
+{"_id": "8786", "title": "Inverse Hyperbolic Tangent of Imaginary Number", "text": ":$\\map {\\tanh^{-1} } {i x} = i \\tan^{-1} x$"}
+{"_id": "8788", "title": "Inverse Hyperbolic Cotangent of Imaginary Number", "text": ":$\\map {\\coth^{-1} } {i x} = i \\cot^{-1} x$"}
+{"_id": "8791", "title": "Inverse Cosecant of Imaginary Number", "text": ":$\\map {\\csc^{-1} } {i x} = i \\csch^{-1} x$"}
+{"_id": "8792", "title": "Inverse Hyperbolic Cosecant of Imaginary Number", "text": ":$\\map {\\csch^{-1} } {i x} = -i \\csc^{-1} x$"}
+{"_id": "8793", "title": "Arccosecant of Reciprocal equals Arcsine", "text": ":$\\map \\arccsc {\\dfrac 1 x} = \\arcsin x$"}
+{"_id": "8794", "title": "Arcsecant of Reciprocal equals Arccosine", "text": ":$\\map \\arcsec {\\dfrac 1 x} = \\arccos x$"}
+{"_id": "8795", "title": "Arccotangent of Reciprocal equals Arctangent", "text": ":$\\map \\arccot {\\dfrac 1 x} = \\arctan x$"}
+{"_id": "8796", "title": "Solution to Quadratic Equation/Real Coefficients", "text": "Let $a, b, c \\in \\R$. The quadratic equation $a x^2 + b x + c = 0$ has: :Two real solutions if $b^2 - 4 a c > 0$ :One real solution if $b^2 - 4 a c = 0$ :Two complex solutions if $b^2 - 4 a c < 0$, and those two solutions are complex conjugates."}
+{"_id": "8797", "title": "Sum of Roots of Quadratic Equation", "text": "Let $P$ be the quadratic equation $a x^2 + b x + c = 0$. Let $\\alpha$ and $\\beta$ be the roots of $P$. Then: :$\\alpha + \\beta = -\\dfrac b a$"}
+{"_id": "8798", "title": "Product of Roots of Quadratic Equation", "text": "Let $P$ be the quadratic equation $a x^2 + b x + c = 0$. Let $\\alpha$ and $\\beta$ be the roots of $P$. Then: :$\\alpha \\beta = \\dfrac c a$"}
+{"_id": "8799", "title": "Definition:Discriminant of Polynomial/Cubic Equation", "text": "Let: : $Q = \\dfrac {3 a c - b^2} {9 a^2}$ : $R = \\dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$ The '''discriminant''' of the cubic equation is given by: : $D := Q^3 + R^2$"}
+{"_id": "8802", "title": "Ring is Module over Itself", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Then $\\struct {R, +, \\circ}_R$ is an $R$-module.  If $\\struct {R, +, \\circ}$ has a unity, then $\\struct {R, +, \\circ}_R$ is unitary."}
+{"_id": "8805", "title": "Rational Division is Closed", "text": "The set of rational numbers less zero is closed under division: :$\\forall a, b \\in \\Q_{\\ne 0}: a / b \\in \\Q_{\\ne 0}$"}
+{"_id": "8806", "title": "Complex Modulus of Product of Complex Numbers/General Result", "text": "Let $z_1, z_2, \\ldots, z_n \\in \\C$ be complex numbers. Let $\\cmod z$ be the modulus of $z$. Then: : $\\cmod {z_1 z_2 \\cdots z_n} = \\cmod {z_1} \\cdot \\cmod {z_2} \\cdots \\cmod {z_n}$"}
+{"_id": "8808", "title": "Complex Modulus of Reciprocal of Complex Number", "text": "Let $z \\in \\C$ be a complex number such that $z \\ne 0$. Let $\\cmod z$ denote the complex modulus of $z$. Then: :$\\cmod {\\dfrac 1 z} = \\dfrac 1 {\\cmod z}$"}
+{"_id": "8810", "title": "Triangle Inequality/Complex Numbers/Corollary 1", "text": "Let $z_1, z_2 \\in \\C$ be complex numbers. Let $\\cmod z$ be the modulus of $z$. Then: : $\\cmod {z_1 + z_2} \\ge \\cmod {z_1} - \\cmod {z_2}$"}
+{"_id": "8811", "title": "Complex Modulus of Additive Inverse", "text": "Let $z \\in \\C$ be a complex number. Let $-z$ be the negative of $z$: :$z + \\paren {-z} = 0$ Then: :$\\cmod z = \\cmod {\\paren {-z} }$ where $\\cmod z$ denotes the modulus of $z$."}
+{"_id": "8812", "title": "Even Power of Negative Real Number", "text": "Let $x \\in \\R$ be a real number. Let $n \\in \\Z$ be an even integer. Then: :$\\paren {-x}^n = x^n$"}
+{"_id": "8813", "title": "Complex Addition is Closed/Proof 1", "text": "The set of complex numbers $\\C$ is closed under addition: :$\\forall z, w \\in \\C: z + w \\in \\C$"}
+{"_id": "8814", "title": "Complex Addition is Closed/Proof 2", "text": "The set of complex numbers $\\C$ is closed under addition: :$\\forall z, w \\in \\C: z + w \\in \\C$"}
+{"_id": "8815", "title": "Cardinality of Cartesian Product/General Result", "text": "Let $\\displaystyle \\prod_{k \\mathop = 1}^n S_k$ be the cartesian product of a (finite) sequence of sets $\\sequence {S_n}$. Then: :$\\displaystyle \\card {\\prod_{k \\mathop = 1}^n S_k} = \\prod_{k \\mathop = 1}^n \\card {S_k}$"}
+{"_id": "8816", "title": "Cardinality of Cartesian Product/General Result/Corollary", "text": "Let $S^n$ be a cartesian space. Then: :$\\card {S^n} = \\card S^n$"}
+{"_id": "8819", "title": "Product of Complex Numbers in Polar Form/General Result", "text": "Let $z_1, z_2, \\ldots, z_n \\in \\C$ be complex numbers. Let $z_j = \\polar {r_j, \\theta_j}$ be $z_j$ expressed in polar form for each $j \\in \\set {1, 2, \\ldots, n}$. Then: :$z_1 z_2 \\cdots z_n = r_1 r_2 \\cdots r_n \\paren {\\map \\cos {\\theta_1 + \\theta_2 + \\cdots + \\theta_n} + i \\map \\sin {\\theta_1 + \\theta_2 + \\cdots + \\theta_n} }$"}
+{"_id": "8822", "title": "Complex Roots of Unity are Vertices of Regular Polygon Inscribed in Circle", "text": "Let $n \\in \\Z$ be an integer such that $n \\ge 3$. Let $z \\in \\C$ be a complex number such that $z^n = 1$. Let $U_n = \\set {e^{2 i k \\pi / n}: k \\in \\N_n}$ be the set of $n$th roots of unity. Let $U_n$ be plotted on the complex plane. Then the elements of $U_n$ are located at the vertices of a regular $n$-sided polygon $P$, such that: :$(1):\\quad$ $P$ is circumscribed by a unit circle whose center is at $\\tuple {0, 0}$ :$(2):\\quad$ one of those vertices is at $\\tuple {1, 0}$."}
+{"_id": "8823", "title": "Equation of Unit Circle in Complex Plane", "text": "Consider the unit circle $C$ whose center is at $\\tuple {0, 0}$ on the complex plane. Its equation is given by: :$\\cmod z = 1$ where $\\cmod z$ denotes the complex modulus of $z$."}
+{"_id": "8824", "title": "Equation of Unit Circle", "text": "Let the unit circle have its center at the origin of the Cartesian plane. Its equation is given by: :$x^2 + y^2 = 1$ {{expand|Present it in polar coordinates as well}}"}
+{"_id": "8827", "title": "Tautological Consequent", "text": ":$p \\implies \\top \\dashv \\vdash \\top$"}
+{"_id": "8830", "title": "Tautological Antecedent", "text": ":$\\top \\implies p \\dashv \\vdash p$"}
+{"_id": "8833", "title": "Contradictory Consequent", "text": ":$p \\implies \\bot \\dashv \\vdash \\neg p$"}
+{"_id": "8836", "title": "Contradictory Antecedent", "text": ":$\\bot \\implies p \\dashv \\vdash \\top$"}
+{"_id": "8839", "title": "Biconditional with Tautology", "text": ":$p \\iff \\top \\dashv \\vdash p$"}
+{"_id": "8848", "title": "Unique Representation of Complex Number in Spherical Form", "text": "Let $\\mathcal P$ be the complex plane. Let $\\mathbb S$ be the unit sphere which is tangent to $\\mathcal P$ at $\\tuple {0, 0}$ (that is, where $z = 0$). Let the diameter of $\\mathbb S$ perpendicular to $\\mathcal P$ through $\\tuple {0, 0}$ be $NS$ where $S$ is the point $\\tuple {0, 0}$. Let the point $N$ be referred to as the '''north pole''' of $\\mathbb S$ and $S$ be referred to as the '''south pole''' of $\\mathbb S$. :900px Let $A$ be a point on $P$. Let the line $NA$ be constructed. Then $NA$ passes through exactly one point $A'$ on the surface of $\\mathbb S$ apart from $N$. Similarly, let $A'$ be a point on the surface of $\\mathbb S$ apart from $N$. Let the line $NA'$ be constructed. Then $NA'$ passes through exactly one point $A$ on $P$."}
+{"_id": "8849", "title": "Exclusive Or with Contradiction", "text": ":$p \\oplus \\bot \\dashv \\vdash p$"}
+{"_id": "8857", "title": "Product of Complex Number with Conjugate by Dot and Cross Product", "text": "Let $z_1$ and $z_2$ be complex numbers. Then: :$\\overline {z_1} z_2 = \\paren {z_1 \\circ z_2} + i \\paren {z_1 \\times z_2}$ where: :$\\overline {z_1}$ denotes the complex conjugate of $z_1$ :$z_1 \\circ z_2$ denotes the complex dot product of $z_1$ with $z_2$ :$z_1 \\times z_2$ denotes the complex cross product of $z_1$ with $z_2$."}
+{"_id": "8858", "title": "Product of Complex Number with Conjugate in Exponential Form", "text": "Let $z_1$ and $z_2$ be complex numbers. Then: :$\\overline {z_1} z_2 = \\cmod {z_1} \\, \\cmod {z_2} e^{i \\theta}$ where: :$\\overline {z_1}$ denotes the complex conjugate of $z_1$ :$\\cmod {z_1}$ denotes the complex modulus of $z_1$ :$\\theta$ denotes the angle from $z_1$ to $z_2$, measured in the positive direction."}
+{"_id": "8859", "title": "Complex Numbers are Perpendicular iff Dot Product is Zero", "text": "Let $z_1$ and $z_2$ be complex numbers in vector form such that $z_1 \\ne 0$ and $z_2 \\ne 0$. Then $z_1$ and $z_2$ are perpendicular {{iff}}: :$z_1 \\circ z_2 = 0$ where $z_1 \\circ z_2$ denotes the complex dot product of $z_1$ with $z_2$."}
+{"_id": "8860", "title": "Complex Numbers are Parallel iff Cross Product is Zero", "text": "Let $z_1$ and $z_2$ be complex numbers in vector form such that $z_1 \\ne 0$ and $z_2 \\ne 0$. Then $z_1$ and $z_2$ are parallel {{iff}}: :$z_1 \\times z_2 = 0$ where $z_1 \\times z_2$ denotes the complex cross product of $z_1$ with $z_2$."}
+{"_id": "8861", "title": "Tautology iff Negation is Unsatisfiable", "text": "Let $\\mathbf A$ be a WFF of propositional logic. Then $\\mathbf A$ is a tautology iff its negation $\\neg \\mathbf A$ is unsatisfiable."}
+{"_id": "8862", "title": "Satisfiable iff Negation is Falsifiable", "text": "Let $\\mathbf A$ be a WFF of propositional logic. Then $\\mathbf A$ is satisfiable iff its negation $\\neg \\mathbf A$ is falsifiable."}
+{"_id": "8864", "title": "Subset of Satisfiable Set is Satisfiable", "text": "Let $\\mathcal L$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\mathcal L$. Let $\\mathcal F$ be an $\\mathscr M$-satisfiable set of formulas from $\\mathcal L$. Let $\\mathcal F'$ be a subset of $\\mathcal F$. Then $\\mathcal F'$ is also $\\mathscr M$-satisfiable."}
+{"_id": "8865", "title": "Satisfiable Set Union Tautology is Satisfiable", "text": "Let $\\mathcal L$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\mathcal L$. Let $\\mathcal F$ be an $\\mathscr M$-satisfiable set of formulas from $\\mathcal L$. Let $\\phi$ be a tautology for $\\mathscr M$. Then $\\mathcal F \\cup \\left\\{{\\phi}\\right\\}$ is also $\\mathscr M$-satisfiable."}
+{"_id": "8866", "title": "Unsatisfiable Set Union Formula is Unsatisfiable", "text": "Let $\\LL$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\LL$. Let $\\FF$ be an $\\mathscr M$-unsatisfiable set of formulas from $\\LL$. Let $\\phi$ be a logical formula. Then $\\FF \\cup \\set \\phi$ is also $\\mathscr M$-unsatisfiable."}
+{"_id": "8867", "title": "Superset of Unsatisfiable Set is Unsatisfiable", "text": "Let $\\LL$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\LL$. Let $\\FF$ be an $\\mathscr M$-unsatisfiable set of formulas from $\\LL$. Let $\\FF'$ be a superset of $\\FF$. Then $\\FF'$ is also $\\mathscr M$-unsatisfiable."}
+{"_id": "8868", "title": "Unsatisfiable Set minus Tautology is Unsatisfiable", "text": "Let $\\mathcal L$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\mathcal L$. Let $\\mathcal F$ be an $\\mathscr M$-unsatisfiable set of formulas from $\\mathcal L$. Let $\\phi \\in \\mathcal F$ be a tautology. Then $\\mathcal F \\setminus \\set {\\phi}$ is also $\\mathscr M$-unsatisfiable."}
+{"_id": "8873", "title": "Semantic Consequence of Set Union Formula", "text": "Let $\\mathcal L$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\mathcal L$. Let $\\mathcal F$ be a set of logical formulas from $\\mathcal L$. Let $\\phi$ be an $\\mathscr M$-semantic consequence of $\\mathcal F$. Let $\\psi$ be another logical formula. Then: :$\\mathcal F \\cup \\left\\{{\\psi}\\right\\} \\models_{\\mathscr M} \\phi$ that is, $\\phi$ is also a semantic consequence of $\\mathcal F \\cup \\left\\{{\\psi}\\right\\}$."}
+{"_id": "8874", "title": "Semantic Consequence of Superset", "text": "Let $\\LL$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\LL$. Let $\\FF$ be a set of logical formulas from $\\LL$. Let $\\phi$ be an $\\mathscr M$-semantic consequence of $\\FF$. Let $\\FF'$ be another set of logical formulas. Then: :$\\FF \\cup \\FF' \\models_{\\mathscr M} \\phi$ that is, $\\phi$ is also a semantic consequence of $\\FF \\cup \\FF'$."}
+{"_id": "8875", "title": "Semantic Consequence of Set minus Tautology", "text": "Let $\\mathcal L$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\mathcal L$. Let $\\mathcal F$ be a set of logical formulas from $\\mathcal L$. Let $\\phi$ be an $\\mathscr M$-semantic consequence of $\\mathcal F$. Let $\\psi \\in \\mathcal F$ be a tautology. Then: :$\\mathcal F \\setminus \\left\\{{\\psi}\\right\\} \\models_{\\mathscr M} \\phi$ that is, $\\phi$ is also a semantic consequence of $\\mathcal F \\setminus \\left\\{{\\psi}\\right\\}$."}
+{"_id": "8878", "title": "Set of Literals Satisfiable iff No Complementary Pairs", "text": "Let $S$ be a set of literals. Then $S$ is satisfiable {{iff}} it contains no complementary pairs."}
+{"_id": "8879", "title": "Equivalence of Definitions of Exterior Point (Complex Analysis)", "text": "{{TFAE|def = Exterior Point (Complex Analysis)|view = exterior point|context = Complex Analysis}} Let $S \\subseteq \\C$ be a subset of the complex plane. Let $z_0 \\in \\C$."}
+{"_id": "8889", "title": "Quintuple Angle Formulas/Sine/Corollary", "text": "For all $\\theta$ such that $\\theta \\ne 0, \\pm \\pi, \\pm 2 \\pi \\ldots$ :$\\dfrac {\\sin 5 \\theta} {\\sin \\theta} = 16 \\cos^4 \\theta - 12 \\cos^2 \\theta + 1$ where $\\sin$ denotes sine and $\\cos$ denotes cosine."}
+{"_id": "8891", "title": "Complex Multiplication as Geometrical Transformation", "text": "Let $z_1 = \\left\\langle{r_1, \\theta_1}\\right\\rangle$ and $z_2 = \\left\\langle{r_2, \\theta_2}\\right\\rangle$ be complex numbers expressed in polar form. Let $z_1$ and $z_2$ be represented on the complex plane $\\C$ in vector form. Let $z = z_1 z_2$ be the product of $z_1$ and $z_2$. Then $z$ can be interpreted as the result of: : rotating $z_1$ about the origin of $\\C$ by $\\theta_2$ in the positive direction : multiplying the modulus of $z_1$ by $r_2$."}
+{"_id": "8892", "title": "Argument of Quotient equals Difference of Arguments", "text": "Let $z_1$ and $z_2$ be complex numbers. Then: :$\\map \\arg {\\dfrac {z_1} {z_2} } = \\map \\arg {z_1} - \\map \\arg {z_1} + 2 k \\pi$ where: :$\\arg$ denotes the argument of a complex number :$k$ can be $0$, $1$ or $-1$."}
+{"_id": "8894", "title": "Cardinality of Finite Vector Space", "text": "Let $V$ be a $K$-vector space. Let $K$ be finite. Let the dimension of $V$ be finite. Then: :$\\size V = \\size K^{\\map \\dim V}$"}
+{"_id": "8895", "title": "Sum of Roots of Polynomial", "text": "Let $P$ be the polynomial equation: : $a_n z^n + a_{n - 1} z^{n - 1} + \\cdots + a_1 z + a_0 = 0$ such that $a_n \\ne 0$. The sum of the roots of $P$ is $-\\dfrac {a_{n - 1} } {a_n}$."}
+{"_id": "8896", "title": "Product of Roots of Polynomial", "text": "Let $P$ be the polynomial equation: :$a_n z^n + a_{n - 1} z^{n - 1} + \\cdots + a_1 z + a_0 = 0$ such that $a_n \\ne 0$. The product of the roots of $P$ is $\\dfrac {\\paren {-1}^n a_0} {a_n}$."}
+{"_id": "8899", "title": "Sum of Sines of Fractions of Pi", "text": "Let $n \\in \\Z$ such that $n > 1$. Then: :$\\displaystyle \\sum_{k \\mathop = 1}^{n - 1} \\sin \\frac {2 k \\pi} n = 0$"}
+{"_id": "8900", "title": "Area of Parallelogram in Complex Plane", "text": "Let $z_1$ and $z_2$ be complex numbers expressed as vectors. Let $ABCD$ be the parallelogram formed by letting $AD = z_1$ and $AB = z_2$. Then the area $\\AA$ of $ABCD$ is given by: :$\\AA = z_1 \\times z_2$ where $z_1 \\times z_2$ denotes the cross product of $z_1$ and $z_2$."}
+{"_id": "8901", "title": "Area of Triangle in Determinant Form", "text": "Let $A = \\tuple {x_1, y_1}, B = \\tuple {x_2, y_2}, C = \\tuple {x_3, y_3}$ be points in the Cartesian plane. The area $\\mathcal A$ of the triangle whose vertices are at $A$, $B$ and $C$ is given by: :$\\mathcal A = \\dfrac 1 2 \\size {\\paren {\\begin{vmatrix} x_1 & y_1 & 1 \\\\ x_2 & y_2 & 1 \\\\ x_3 & y_3 & 1 \\\\ \\end{vmatrix} } }$"}
+{"_id": "8902", "title": "Equation of Circle in Complex Plane/Formulation 2", "text": "Let $\\C$ be the complex plane. Let $C$ be a circle in $\\C$. Then $C$ may be written as: :$\\alpha z \\overline z + \\beta z + \\overline \\beta \\overline z + \\gamma = 0$ where: :$\\alpha \\in \\R_{\\ne 0}$ is real and non-zero :$\\gamma \\in \\R$ is real :$\\beta \\in \\C$ is complex such that $\\cmod \\beta^2 > \\alpha \\gamma$. The curve $C$ is a straight line {{iff}} $\\alpha = 0$ and $\\beta \\ne 0$."}
+{"_id": "8903", "title": "Equation of Line in Complex Plane/Formulation 1", "text": "Let $\\C$ be the complex plane. Let $L$ be a straight line in $\\C$. Then $L$ may be written as: :$\\beta z + \\overline \\beta \\overline z  + \\gamma = 0$ where $\\gamma$ is real and $\\beta$ may be complex."}
+{"_id": "8904", "title": "Vertices of Equilateral Triangle in Complex Plane", "text": "Let $z_1$, $z_2$ and $z_3$ be complex numbers. Then: : $z_1$, $z_2$ and $z_3$ represent on the complex plane the vertices of an equilateral triangle {{iff}}: :${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$"}
+{"_id": "8905", "title": "Product of Sines of Fractions of Pi", "text": "Let $m \\in \\Z$ such that $m > 1$. Then: :$\\displaystyle \\prod_{k \\mathop = 1}^{m - 1} \\sin \\frac {k \\pi} m = \\frac m {2^{m - 1} }$"}
+{"_id": "8906", "title": "Product of Complex Conjugates/General Result", "text": "Let $z_1, z_2, \\ldots, z_n \\in \\C$ be complex numbers. Let $\\overline z$ be the complex conjugate of the complex number $z$. Then: :$\\displaystyle \\overline {\\prod_{j \\mathop = 1}^n z_j} = \\prod_{j \\mathop = 1}^n \\overline {z_j}$"}
+{"_id": "8907", "title": "Quotient of Complex Conjugates", "text": "Let $z_1, z_2 \\in \\C$ be complex numbers. Let $\\overline z$ be the complex conjugate of the complex number $z$. Then: :$\\overline {\\paren {\\dfrac {z_1} {z_2} } } = \\dfrac {\\paren {\\overline {z_1} } } {\\paren {\\overline {z_2} } }$ for $z_2 \\ne 0$."}
+{"_id": "8908", "title": "Condition for Points in Complex Plane to form Parallelogram", "text": "Let $A = z_1$, $B = z_2$, $C = z_3$ and $D = z_4$ represent on the complex plane the vertices of a quadrilateral. Then $ABCD$ is a parallelogram {{iff}}: :$z_1 - z_2 + z_3 - z_4 = 0$"}
+{"_id": "8909", "title": "Quadrilateral with Bisecting Diagonals is Parallelogram", "text": "Let $ABCD$ be a quadrilateral. Let the diagonals of $ABCD$ bisect each other. Then $ABCD$ is a parallelogram."}
+{"_id": "8910", "title": "Medians of Triangle Meet at Point", "text": "Let $\\triangle ABC$ be a triangle. Then the medians of $\\triangle ABC$ meet at a single point. This point is called the centroid of $\\triangle ABC$. {{expand|State where that point is, that is $1/3$ the way along each median}}"}
+{"_id": "8911", "title": "Midpoints of Sides of Quadrilateral form Parallelogram", "text": "Let $\\Box ABCD$ be a quadrilateral. Let $E, F, G, H$ be the midpoints of $AB, BC, CD, DA$ respectively. Then $\\Box EFGH$ is a parallelogram."}
+{"_id": "8912", "title": "Line from Bisector of Side of Parallelogram to Vertex Trisects Diagonal", "text": "Let $ABCD$ be a parallelogram. Let $E$ be the midpoint of $AD$. Then the point at which the line $BE$ meets $AC$ trisects $AC$."}
+{"_id": "8914", "title": "Sum of Complex Numbers in Exponential Form", "text": "Let $z_1 = r_1 e^{i \\theta_1}$ and $z_2 = r_2 e^{i \\theta_2}$ be complex numbers expressed in exponential form. Let $z_3 = r_3 e^{i \\theta_3} = z_1 + z_2$. Then: :$r_3 = \\sqrt {r_1^2 + r_2^2 + 2 r_1 r_2 \\cos \\left({\\theta_1 - \\theta_2}\\right)}$ :$\\theta_3 = \\map \\arctan {\\dfrac {r_1 \\sin \\theta_1 + r_2 \\sin \\theta_2} {r_1 \\cos \\theta_1 + r_2 \\cos \\theta_2} }$"}
+{"_id": "8923", "title": "De Moivre's Formula/Positive Integer Index", "text": "Let $z \\in \\C$ be a complex number expressed in polar form: :$z = r \\paren {\\cos x + i \\sin x}$ Then: :$\\forall n \\in \\Z_{>0}: \\paren {r \\paren {\\cos x + i \\sin x} }^n = r^n \\paren {\\map \\cos {n x} + i \\map \\sin {n x} }$"}
+{"_id": "8924", "title": "De Moivre's Formula/Integer Index", "text": "Let $z \\in \\C$ be a complex number expressed in complex form: :$z = r \\paren {\\cos x + i \\sin x}$ Then: :$\\forall n \\in \\Z: \\paren {r \\paren {\\cos x + i \\sin x} }^n = r^n \\paren {\\map \\cos {n x} + i \\, \\map \\sin {n x} }$"}
+{"_id": "8925", "title": "De Moivre's Formula/Negative Integer Index", "text": "Let $z \\in \\C$ be a complex number expressed in complex form: :$z = r \\left({\\cos x + i \\sin x}\\right)$ Then: :$\\forall n \\in \\Z_{\\le 0}: \\paren {r \\paren {\\cos x + i \\sin x} }^n = r^n \\paren {\\map \\cos {n x} + i \\, \\map \\sin {n x} }$"}
+{"_id": "8926", "title": "De Moivre's Formula/Rational Index", "text": "Let $z \\in \\C$ be a complex number expressed in complex form: :$z = r \\paren {\\cos x + i \\sin x}$ Then: :$\\forall p \\in \\Q: \\paren {r \\paren {\\cos x + i \\sin x} }^p = r^p \\paren {\\map \\cos {p x} + i \\, \\map \\sin {p x} }$"}
+{"_id": "8929", "title": "Dot Product Operator is Commutative", "text": ":$\\mathbf u \\cdot \\mathbf v = \\mathbf v \\cdot \\mathbf u$"}
+{"_id": "8932", "title": "Dot Product with Self is Non-Negative", "text": ":$\\mathbf u \\cdot \\mathbf u \\ge 0$"}
+{"_id": "8935", "title": "Dot Product with Self is Zero iff Zero Vector", "text": ":$\\mathbf u \\cdot \\mathbf u = 0 \\iff \\mathbf u = \\mathbf 0$"}
+{"_id": "8936", "title": "Vector Cross Product is Anticommutative/Complex", "text": ":$\\forall z_1, z_2 \\in \\C: z_1 \\times z_2 = -\\paren {z_2 \\times z_1}$"}
+{"_id": "8937", "title": "Polar Form of Complex Conjugate", "text": "Let $z := r \\left({\\cos \\theta + i \\sin \\theta}\\right) \\in \\C$ be a complex number expressed in polar form. Then: :$\\overline z = r \\left({\\cos \\theta - i \\sin \\theta}\\right)$ where $\\overline z$ denotes the complex conjugate of $z$."}
+{"_id": "8938", "title": "Exponential Form of Complex Conjugate", "text": "Let $z := r e^{i \\theta} \\in \\C$ be a complex number expressed in exponential form. Then: :$\\overline z = r e^{-i \\theta}$ where $\\overline z$ denotes the complex conjugate of $z$."}
+{"_id": "8939", "title": "Complex Dot Product in Exponential Form", "text": "Let $z_1 := r_1 e^{i \\theta_1}, z_2 := r_2 e^{i \\theta_2} \\in \\C$ be complex numbers expressed in exponential form. Then: :$z_1 \\circ z_2 = r_1 r_2 \\, \\map \\cos {\\theta_2 - \\theta_1}$ where $z_1 \\circ z_2$ denotes the dot product of $z_1$ and $z_2$."}
+{"_id": "8940", "title": "Complex Cross Product in Exponential Form", "text": "Let $z_1 := r_1 e^{i \\theta_1}, z_2 := r_2 e^{i \\theta_2} \\in \\C$ be complex numbers expressed in exponential form. Then: :$z_1 \\times z_2 = r_1 r_2 \\map \\sin {\\theta_2 - \\theta_1}$ where $z_1 \\times z_2$ denotes the dot product of $z_1$ and $z_2$."}
+{"_id": "8943", "title": "Dot Product Distributes over Addition", "text": ":$\\paren {\\mathbf u + \\mathbf v} \\cdot \\mathbf w = \\mathbf u \\cdot \\mathbf w + \\mathbf v \\cdot \\mathbf w$"}
+{"_id": "8945", "title": "Dot Product Operator is Bilinear", "text": ":$\\paren {c \\mathbf u + \\mathbf v} \\cdot \\mathbf w = c \\paren {\\mathbf u \\cdot \\mathbf w} + \\paren {\\mathbf v \\cdot \\mathbf w}$"}
+{"_id": "8949", "title": "Dot Product Associates with Scalar Multiplication", "text": ":$\\paren {c \\mathbf u} \\cdot \\mathbf v = c \\paren {\\mathbf u \\cdot \\mathbf v}$"}
+{"_id": "8950", "title": "Semantic Tableau Algorithm Terminates", "text": "Let $\\mathbf A$ be a WFF of propositional logic. Then the Semantic Tableau Algorithm for $\\mathbf A$ terminates. Each leaf node of the resulting semantic tableau is marked."}
+{"_id": "8952", "title": "Complement of Closed Set in Complex Plane is Open", "text": "Let $S \\subseteq \\C$ be a closed subset of the complex plane $\\C$. Then the complement of $S$ in $\\C$ is open."}
+{"_id": "8956", "title": "Condition for Quartic with Real Coefficients to have Wholly Imaginary Root", "text": "Let $Q$ be the quartic equation: :$(1): \\quad z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4 = 0$ such that all of $a_1, a_2, a_3, a_4$ are real numbers. Then $Q$ has a root which is wholly imaginary {{iff}}: :$\\text {(a)}: \\quad a_3^2 + a_1^2 a_4 = a_1 a_2 a_3$ :$\\text {(b)}: \\quad a_1 a_3 > 0$"}
+{"_id": "8957", "title": "Soundness and Completeness of Semantic Tableaus", "text": "Let $\\mathbf A$ be a WFF of propositional logic. Let $T$ be a completed semantic tableau for $\\mathbf A$. Then $\\mathbf A$ is unsatisfiable {{iff}} $T$ is closed."}
+{"_id": "8958", "title": "Semantic Tableau Algorithm is Decision Procedure for Tautologies", "text": "The Semantic Tableau Algorithm is a decision procedure for tautologies."}
+{"_id": "8959", "title": "Soundness Theorem for Semantic Tableaus", "text": "Let $\\mathbf A$ be a WFF of propositional logic. Let $T$ be a completed tableau for $\\mathbf A$. Suppose that $T$ is closed. Then $\\mathbf A$ is unsatisfiable for boolean interpretations."}
+{"_id": "8960", "title": "Completeness Theorem for Semantic Tableaus", "text": "Let $\\mathbf A$ be a WFF of propositional logic. Let $\\mathbf A$ be unsatisfiable for boolean interpretations. Then every completed tableau for $\\mathbf A$ is closed."}
+{"_id": "8963", "title": "Cosine to Power of Even Integer/Proof 2", "text": "Let $n \\in \\Z$ be an even integer. Then: :$\\displaystyle \\cos^n \\theta = \\frac 1 {2^{n - 1} } \\sum_{k \\mathop = 0}^{n / 2} \\paren {\\binom n k \\cos \\paren {n - 2 k} \\theta}$ That is: :$\\cos^n \\theta = \\dfrac 1 {2^{n - 1} } \\paren {\\cos n \\theta + n \\cos \\paren {n - 2} \\theta + \\dfrac {n \\paren {n - 1} } 2 \\cos \\paren {n - 4} \\theta + \\cdots + \\dfrac {n!} {\\paren {\\paren {\\frac n 2}!}^2} \\cos \\theta}$"}
+{"_id": "8965", "title": "Conjugate of Real Polynomial is Polynomial in Conjugate", "text": "Let $\\map P z$ be a polynomial in a complex number $z$. Let the coefficients of $P$ all be real. Then: :$\\overline {\\map P z} = \\map P {\\overline z}$ where $\\overline z$ denotes the complex conjugate of $z$."}
+{"_id": "8967", "title": "Equation for Line through Two Points in Complex Plane", "text": "Let $z_1, z_2 \\in \\C$ be complex numbers. Let $L$ be a straight line through $z_1$ and $z_2$ in the complex plane. === Formulation 1 === {{:Equation for Line through Two Points in Complex Plane/Formulation 1}} === Parametric Form $1$ === {{:Equation for Line through Two Points in Complex Plane/Parametric Form 1}} === Parametric Form $2$ === {{:Equation for Line through Two Points in Complex Plane/Parametric Form 2}} === Symmetric Form === {{:Equation for Line through Two Points in Complex Plane/Symmetric Form}}"}
+{"_id": "8968", "title": "Absolute Value of Components of Complex Number no greater than Root 2 of Modulus", "text": "Let $z = x + i y \\in \\C$ be a complex number. Then: :$\\size x + \\size y \\le \\sqrt 2 \\cmod z$ where: :$\\size x$ and $\\size y$ denote the absolute value of $x$ and $y$ :$\\cmod z$ denotes the complex modulus of $z$."}
+{"_id": "8969", "title": "Exclusive Or with Factor of Exclusive Or", "text": ":$\\left({p \\oplus q}\\right) \\oplus q \\dashv \\vdash p$"}
+{"_id": "8970", "title": "Biconditional with Factor of Biconditional", "text": ":$\\paren {p \\iff q} \\iff q \\dashv \\vdash p$"}
+{"_id": "8972", "title": "Conditional iff Biconditional of Consequent with Disjunction", "text": ":$p \\implies q \\dashv \\vdash q \\iff \\left({p \\lor q}\\right)$"}
+{"_id": "8974", "title": "Functionally Incomplete Logical Connectives", "text": "These sets of logical connectives are ''not'' functionally complete."}
+{"_id": "8975", "title": "Functionally Incomplete Logical Connectives/Conjunction and Disjunction", "text": ":$\\set {\\land, \\lor}$: And and Or"}
+{"_id": "8976", "title": "Functionally Incomplete Logical Connectives/Negation and Biconditional", "text": ":$\\set {\\neg, \\iff}$: Not and Iff"}
+{"_id": "8977", "title": "Theory of Set of Formulas is Theory", "text": "Let $\\mathcal L$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\mathcal L$. Let $\\mathcal F$ be a set of $\\mathcal L$-formulas. Let $T \\left({\\mathcal F}\\right)$ be the $\\mathcal L$-theory of $\\mathcal F$. Then $T \\left({\\mathcal F}\\right)$ is a theory."}
+{"_id": "8978", "title": "Semantic Consequence is Transitive", "text": "Let $\\mathcal L$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\mathcal L$. Let $\\mathcal F, \\mathcal G$ and $\\mathcal H$ be sets of $\\mathcal L$-formulas. Suppose that: :$\\mathcal F \\models_{\\mathscr M} \\mathcal G$ :$\\mathcal G \\models_{\\mathscr M} \\mathcal H$ Then $\\mathcal F \\models_{\\mathscr M} \\mathcal H$."}
+{"_id": "8979", "title": "Lemniscate of Bernoulli as Locus in Complex Plane", "text": "The locus of $z$ on the complex plane such that: :$\\cmod {z - a} \\cmod {z - a} = a^2$ is a lemniscate of Bernoulli."}
+{"_id": "8980", "title": "Sum of Cosines of Arithmetic Sequence of Angles", "text": "Let $\\alpha \\in \\R$ be a real number such that $\\alpha \\ne 2 \\pi k$ for $k \\in \\Z$. Then:"}
+{"_id": "8981", "title": "Sum of Sines of Arithmetic Sequence of Angles", "text": "Let $\\alpha \\in \\R$ be a real number such that $\\alpha \\ne 2 \\pi k$ for $k \\in \\Z$. Then:"}
+{"_id": "8982", "title": "Product of Tangent and Cotangent", "text": ":$\\tan \\theta \\cot \\theta = 1$"}
+{"_id": "8983", "title": "Product of Cotangents of Fractions of Pi", "text": "Let $m \\in \\Z$ such that $m > 1$. Then: :$\\displaystyle \\prod_{k \\mathop = 1}^{m - 1} \\cot \\frac {k \\pi} {2 m} = 1$"}
+{"_id": "8986", "title": "Uncountable Set less Countable Set is Uncountable", "text": "Let $S$ be an uncountable set. Let $T \\subseteq S$ be a countable subset of $S$. Then: :$S \\setminus T$ is uncountable where $\\setminus$ denotes set difference."}
+{"_id": "8987", "title": "Irrational Numbers are Uncountably Infinite", "text": "The set $\\R \\setminus \\Q$ of irrational numbers is uncountable."}
+{"_id": "8988", "title": "Product of Diagonals from Point of Regular Polygon", "text": "Let $A_0, A_1, \\ldots, A_{n - 1}$ be the vertices of a regular $n$-gon $P = A_0 A_1 \\cdots A_{n - 1}$ which is circumscribed by a unit circle. Then: :$\\displaystyle \\prod_{k \\mathop = 2}^{n - 2} A_0 A_k = \\frac {n \\csc^2 \\frac \\pi n} 4$ where $A_0 A_k$ is the length of the line joining $A_0$ to $A_k$."}
+{"_id": "8992", "title": "Condition on Conjugate from Real Product of Complex Numbers", "text": "Let $z_1, z_2 \\in \\C$ be complex numbers such that $z_1 z_2 \\in \\R_{\\ne 0}$. Then: :$\\exists p \\in \\R: z_1 = p \\overline {z_2}$ where $\\overline {z_2}$ denotes the complex conjugate of $z_2$."}
+{"_id": "8993", "title": "Exponential of Sum", "text": "=== Real Numbers === {{:Exponential of Sum/Real Numbers}} === Complex Numbers === {{:Exponential of Sum/Complex Numbers}} == Corollary == {{:Exponential of Sum/Corollary}}"}
+{"_id": "9000", "title": "Cayley-Menger Determinant", "text": "Let $S$ be a $j$-simplex in $\\R^n$. Let the vertices of $S$ be $v_1, v_2, \\ldots, v_{j+1}$. Let $B = \\left[{\\beta_{ij} }\\right]$ denote the $\\left({j + 1}\\right) \\times \\left({j + 1}\\right)$ matrix given by: :$\\beta_{ij} = \\left\\vert{v_i - v_j}\\right\\vert_2^2$ where $\\left\\vert{v_i - v_j}\\right\\vert_2$ is the vector 2-norm of the vector $v_i - v_j$. Then the content $V_j$ is given by: {{begin-eqn}} {{eqn | l = V_j^2 \\left({S}\\right)         | r = \\frac {\\left({-1}\\right)^{j+1} } {2^j \\left({j!}\\right)^2} \\det C  }} {{end-eqn}} where $C$ is the $\\left({j + 2}\\right) \\times \\left({j + 2}\\right)$ matrix obtained from $B$ by bordering $B$ with a top row $\\left({0, 1, \\ldots, 1}\\right)$ and a left column $\\left({0, 1, \\ldots, 1}\\right)^\\intercal$."}
+{"_id": "9002", "title": "Rule of Commutation/Disjunction/Formulation 2/Forward Implication", "text": ":$\\vdash \\left({p \\lor q}\\right) \\implies \\left({q \\lor p}\\right)$"}
+{"_id": "9003", "title": "Provable by Gentzen Proof System iff Negation has Closed Tableau", "text": "Let $\\mathscr G$ be instance 1 of a Gentzen proof system."}
+{"_id": "9004", "title": "Provable by Gentzen Proof System iff Negation has Closed Tableau/Formula", "text": "Let $\\mathbf A$ be a WFF of propositional logic. Then $\\mathbf A$ is a $\\mathscr G$-theorem iff: :$\\neg \\mathbf A$ has a closed semantic tableau where $\\neg \\mathbf A$ is the negation of $\\mathbf A$."}
+{"_id": "9005", "title": "Provable by Gentzen Proof System iff Negation has Closed Tableau/Set of Formulas", "text": "Let $U$ be a set of WFFs of propositional logic. Then $U$ is a $\\mathscr G$-theorem iff: :$\\bar U$ has a closed semantic tableau where $\\bar U$ is the set comprising the logical complements of all WFFs in $U$."}
+{"_id": "9009", "title": "Domain of Real Natural Logarithm", "text": "Let $\\ln$ be the natural logarithm function on the real numbers. Then the domain of $\\ln$ is the set of strictly positive real numbers: :$\\Dom \\ln = \\R_{>0}$"}
+{"_id": "9010", "title": "Image of Real Natural Logarithm", "text": "Let $\\ln$ be the natural logarithm function on the real numbers. Then the image of $\\ln$ is the set of real numbers: :$\\Img \\ln = \\R$"}
+{"_id": "9015", "title": "Soundness and Completeness of Gentzen Proof System", "text": "Let $\\mathscr G$ be instance 1 of a Gentzen proof system. Let $\\mathrm{BI}$ be the formal semantics of boolean interpretations. Then $\\mathscr G$ is a sound and complete proof system for $\\mathrm{BI}$."}
+{"_id": "9017", "title": "Definition:Deduction Rule", "text": "Let $\\mathcal L$ be the language of propositional logic. The '''Deduction Rule''' is the rule of inference: :From $U, \\mathbf A \\vdash \\mathbf B$, one may infer $U \\vdash \\mathbf A \\implies \\mathbf B$ where: :$\\mathbf A, \\mathbf B$ are WFFs of propositional logic :$U$ is a set of WFFs For a given proof system, the '''Deduction Rule''' can be either a basic rule of inference, or a derived rule."}
+{"_id": "9018", "title": "Deduction Theorem", "text": "Let $\\mathscr H$ be instance 1 of a Hilbert proof system. Then the deduction rule: ::$\\dfrac{U,\\mathbf A \\vdash \\mathbf B}{U \\vdash \\mathbf A \\implies \\mathbf B}$ is a derived rule for $\\mathscr H$."}
+{"_id": "9025", "title": "Soundness Theorem for Hilbert Proof System", "text": "Let $\\mathscr H$ be instance 1 of a Hilbert proof system. Let $\\mathrm{BI}$ be the formal semantics of boolean interpretations. Then $\\mathscr H$ is a sound proof system for $\\mathrm{BI}$: :Every $\\mathscr H$-theorem is a tautology."}
+{"_id": "9026", "title": "Equivalence of Definitions of Dominate (Set Theory)", "text": "Let $S, T$ be sets. {{TFAE|def = Dominate (Set Theory)|view = Dominate|context = Set Theory}}"}
+{"_id": "9027", "title": "Element of Toset has at most One Immediate Successor", "text": "Let $\\struct {S, \\preceq}$ be a toset. Let $a \\in S$. Then $a$ has at most one immediate successor."}
+{"_id": "9028", "title": "Element of Toset has at most One Immediate Predecessor", "text": "Let $\\struct {S, \\preceq}$ be a toset. Let $a \\in S$. Then $a$ has at most one immediate predecessor."}
+{"_id": "9033", "title": "Equivalence of Definitions of Complex Inverse Tangent Function", "text": "{{TFAE|def = Complex Inverse Tangent}} Let $S$ be the subset of the complex plane: :$S = \\C \\setminus \\set {0 + i, 0 - i}$"}
+{"_id": "9034", "title": "Equivalence of Definitions of Complex Inverse Cotangent Function", "text": "{{TFAE|def = Complex Inverse Cotangent}} Let $S$ be the subset of the complex plane: :$S = \\C \\setminus \\left\\{{0 + i, 0 - i}\\right\\}$"}
+{"_id": "9042", "title": "Square Root of Complex Number in Cartesian Form", "text": "Let $z \\in \\C$ be a complex number. Let $z = x + i y$ where $x, y \\in \\R$ are real numbers. Let $z$ not be wholly real, that is, such that $y \\ne 0$. Then the square root of $z$ is given by: :$z^{1/2} = \\pm \\paren {a + i b}$ where: {{begin-eqn}} {{eqn | l = a       | r = \\sqrt {\\frac {x + \\sqrt {x^2 + y^2} } 2}       | c =  }} {{eqn | l = b       | r = \\frac y {\\cmod y} \\sqrt {\\frac {-x + \\sqrt {x^2 + y^2} } 2}       | c =  }} {{end-eqn}}"}
+{"_id": "9043", "title": "Zero is Identity in Naturally Ordered Semigroup", "text": "Let $\\struct {S, \\circ, \\preceq}$ be a naturally ordered semigroup. Let $0$ be the zero of $\\struct {S, \\circ, \\preceq}$. Then $0$ is the identity for $\\circ$. That is: :$\\forall n \\in S: n \\circ 0 = n = 0 \\circ n$"}
+{"_id": "9044", "title": "Reduction Formula for Integral of Power of Sine", "text": "Let $n \\in \\Z_{> 0}$ be a (strictly) positive integer. Then: :$\\displaystyle \\int \\sin^n x \\rd x = \\dfrac {n - 1} n \\int \\sin^{n - 2} x \\rd x - \\dfrac {\\sin^{n - 1} x \\cos x} n$ is a reduction formula for $\\displaystyle \\int \\sin^n x \\rd x$."}
+{"_id": "9045", "title": "Reduction Formula for Definite Integral of Power of Sine", "text": "Let $n \\in \\Z_{> 0}$ be a positive integer. Let $I_n$ be defined as: :$\\displaystyle I_n = \\int_0^{\\frac \\pi 2} \\sin^n x \\rd x$ Then $\\left\\langle{I_n}\\right\\rangle$ is a decreasing sequence of real numbers which satisfies: :$n I_n = \\left({n - 1}\\right) I_{n - 2}$ Thus: :$I_n = \\dfrac {n - 1} n  I_{n - 2}$ is a reduction formula for $I_n$."}
+{"_id": "9046", "title": "Stirling's Formula/Proof 2/Lemma 4", "text": "Let $I_n$ be defined as: :$\\displaystyle I_n = \\int_0^{\\frac \\pi 2} \\sin^n x \\rd x$ Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\frac {I_{2 n} } {I_{2 n + 1} } = 1$"}
+{"_id": "9049", "title": "Definite Integral from 0 to Half Pi of Even Power of Sine x", "text": ":$\\displaystyle \\int_0^{\\frac \\pi 2} \\sin^{2 n} x \\rd x = \\dfrac {\\paren {2 n}!} {\\paren {2^n n!}^2} \\dfrac \\pi 2$"}
+{"_id": "9050", "title": "Definite Integral from 0 to Half Pi of Odd Power of Sine x", "text": ":$\\displaystyle \\int_0^{\\frac \\pi 2} \\sin^{2 n + 1} x \\rd x = \\dfrac {\\paren {2^n n!}^2} {\\paren {2 n + 1}!}$"}
+{"_id": "9051", "title": "Zero Complement is Not Empty", "text": "Let $\\struct {S, \\circ, \\preceq}$ be a naturally ordered semigroup. Let $S^*$ be the zero complement of $S$. Then $S^*$ is not empty."}
+{"_id": "9052", "title": "Stirling's Formula/Proof 2/Lemma 5", "text": ":$\\dfrac {n!} {n^n \\sqrt n e^{-n} } \\to \\sqrt {2 \\pi}$ as $n \\to \\infty$"}
+{"_id": "9053", "title": "Limit of Error in Stirling's Formula", "text": "Consider Stirling's Formula: :$n! \\sim \\sqrt {2 \\pi n} \\left({\\dfrac n e}\\right)^n$ The ratio of $n!$ to its approximation $\\sqrt {2 \\pi n} \\left({\\dfrac n e}\\right)^n$ is bounded as follows: :$e^{1 / \\left({12 n + 1}\\right)} \\le \\dfrac {n!} {\\sqrt {2 \\pi n} n^n e^{-n} } \\le e^{1 / 12 n}$"}
+{"_id": "9054", "title": "Zero Strictly Precedes One", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a naturally ordered semigroup. Let $0$ be the zero of $S$. Let $1$ be the one of $S$. Then: :$0 \\prec 1$"}
+{"_id": "9055", "title": "Stirling's Formula/Proof 2/Lemma 2", "text": "The sequence $\\left \\langle {d_n} \\right \\rangle$ defined as: : $\\displaystyle d_n = \\ln \\left({n!}\\right) - \\left({n + \\frac 1 2}\\right) \\ln n + n$ is decreasing."}
+{"_id": "9058", "title": "Power Series Expansion for Real Arctangent Function", "text": "The arctangent function has a Taylor series expansion: :$\\arctan x = \\begin {cases} \\displaystyle \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {x^{2 n + 1} } {2 n + 1} & : -1 \\le x \\le 1 \\\\ \\displaystyle \\frac \\pi 2 - \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac 1 {\\paren {2 n + 1} x^{2 n + 1} } & : x \\ge 1 \\\\ \\displaystyle -\\frac \\pi 2 - \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac 1 {\\paren {2 n + 1} x^{2 n + 1} } & : x \\le -1 \\end {cases}$ That is: :$\\arctan x = \\begin {cases} \\displaystyle x - \\frac {x^3} 3 + \\frac {x^5} 5 - \\frac {x^7} 7 + \\cdots & : -1 \\le x \\le 1 \\\\ \\displaystyle \\frac \\pi 2 - \\frac 1 x + \\frac 1 {3 x^3} - \\frac 1 {5 x^5} + \\cdots & : x \\ge 1 \\\\ \\displaystyle -\\frac \\pi 2 - \\frac 1 x + \\frac 1 {3 x^3} - \\frac 1 {5 x^5} + \\cdots & : x \\le -1 \\end {cases}$"}
+{"_id": "9059", "title": "Ordering in terms of Addition", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a naturally ordered semigroup. Then $\\forall m, n \\in S$:  :$m \\preceq n \\iff \\exists p \\in S: m \\circ p = n$"}
+{"_id": "9060", "title": "Difference in Naturally Ordered Semigroup is Unique", "text": "Let $\\struct {S, \\circ, \\preceq}$ be a naturally ordered semigroup. Let $n, m \\in S$ such that $m \\preceq n$. Then there exists a unique difference $n - m$ of $m$ and $n$."}
+{"_id": "9061", "title": "Equivalence of Definitions of Beta Function", "text": "{{TFAE|def = Beta Function}} For $\\map \\Re x, \\map \\Re y > 0$:"}
+{"_id": "9062", "title": "Ordering of Naturally Ordered Semigroup is Strongly Compatible", "text": ":$\\forall m, n, p \\in S: m \\preceq n \\iff m \\circ p \\preceq n \\circ p$"}
+{"_id": "9063", "title": "Strict Ordering of Naturally Ordered Semigroup is Strongly Compatible", "text": ":$\\forall m, n, p \\in S: m \\prec n \\iff m \\circ p \\prec n \\circ p$"}
+{"_id": "9065", "title": "Sum with One is Immediate Successor in Naturally Ordered Semigroup", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a naturally ordered semigroup. Let $1$ be the one of $S$. Let $n \\in S$. Then $n \\circ 1$ is the immediate successor of $n$. That is, for all $m \\in S$: :$n \\prec m \\iff n \\circ 1 \\preceq m$"}
+{"_id": "9066", "title": "Riemann Zeta Function at Even Integers", "text": "The Riemann $\\zeta$ function can be calculated for even integers as follows: {{begin-eqn}} {{eqn | l = \\map \\zeta {2 n}       | r = \\paren {-1}^{n + 1} \\dfrac {B_{2 n} 2^{2 n - 1} \\pi^{2 n} } {\\paren {2 n}!}       | c =  }} {{eqn | r = \\frac 1 {1^{2 n} } + \\frac 1 {2^{2 n} } + \\frac 1 {3^{2 n} } + \\frac 1 {4^{2 n} } + \\cdots       | c =  }} {{end-eqn}} where: :$B_n$ are the Bernoulli numbers :$n$ is a positive integer."}
+{"_id": "9067", "title": "Tangent Inequality", "text": ":$x < \\tan x$ for all $x$ in the interval $\\left({0 \\,.\\,.\\, \\dfrac {\\pi} 2}\\right)$."}
+{"_id": "9068", "title": "Integral Representation of Riemann Zeta Function in terms of Gamma Function", "text": "For $\\Re \\paren s > 1$, the Riemann Zeta function is given by: :$\\displaystyle \\map \\zeta s = \\frac 1 {\\map \\Gamma s} \\int_0^\\infty \\frac {t^{s - 1}} {e^t - 1} \\rd t$ where $\\Gamma$ is the Gamma function."}
+{"_id": "9071", "title": "Gamma Function is Unique Extension of Factorial", "text": "Let $f: \\R_{>0} \\to \\R$ be a real function which is positive and continuous. Let $\\ln \\mathop \\circ f$ be convex on $\\R_{>0}$. Let $f$ satisfy the conditions: :$f \\left({x + 1}\\right) = \\begin{cases} 1 & : x = 0 \\\\ x f \\left({x}\\right) & : x > 0 \\end{cases}$ Then: : $\\forall x \\in \\R_{>0}: f \\left({x}\\right) = \\Gamma \\left({x}\\right)$ where $\\Gamma \\left({x}\\right)$ is the Gamma function."}
+{"_id": "9072", "title": "Equivalence of Definitions of Convex Real Function", "text": "Let $f$ be a real function which is defined on a real interval $I$. {{TFAE|def = Convex Real Function}}"}
+{"_id": "9074", "title": "Power Series Expansion for Real Arcsine Function", "text": "The (real) arcsine function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \\arcsin x       | r = \\sum_{n \\mathop = 0}^\\infty \\frac {\\paren {2 n}!} {2^{2 n} \\paren {n!}^2} \\frac {x^{2 n + 1} } {2 n + 1}       | c =  }} {{eqn | r = x + \\frac 1 2 \\frac {x^3} 3 + \\frac {1 \\times 3} {2 \\times 4} \\frac {x^5} 5 + \\frac {1 \\times 3 \\times 5} {2 \\times 4 \\times 6} \\frac {x^7} 7 + \\cdots       | c =  }} {{end-eqn}} which converges for $-1 \\le x \\le 1$."}
+{"_id": "9075", "title": "Power Series Expansion for Real Arccosine Function", "text": "The arccosine function has a Taylor Series expansion: {{begin-eqn}} {{eqn | l = \\arccos x       | r = \\frac \\pi 2 - \\sum_{n \\mathop = 0}^\\infty \\frac {\\paren {2 n}!} {2^{2 n} \\paren {n!}^2} \\frac {x^{2 n + 1} } {2 n + 1}       | c =  }} {{eqn | r = \\frac \\pi 2 - \\paren {x + \\frac 1 2 \\frac {x^3} 3 + \\frac {1 \\times 3} {2 \\times 4} \\frac {x^5} 5 + \\frac {1 \\times 3 \\times 5} {2 \\times 4 \\times 6} \\frac {x^7} 7 + \\cdots}       | c =  }} {{end-eqn}} which converges for $-1 \\le x \\le 1$."}
+{"_id": "9077", "title": "Real Function is Concave iff its Negative is Convex", "text": "Let $f$ be a real function. Let $I \\subseteq \\R$ be an interval of $\\R$. Then $f$ is concave on $I$ {{iff}} $-f$ is convex on $\\R$."}
+{"_id": "9080", "title": "Concave Real Function is Left-Hand and Right-Hand Differentiable", "text": "Let $f$ be a real function which is concave on the open interval $\\left({a \\,.\\,.\\, b}\\right)$. Then the left-hand derivative $f'_- \\left({x}\\right)$ and right-hand derivative $f'_+ \\left({x}\\right)$ both exist for all $x \\in \\left({a \\,.\\,.\\, b}\\right)$."}
+{"_id": "9082", "title": "Absolute Value Function is Convex", "text": "Let $f: \\R \\to \\R$ be the absolute value function on the real numbers. Then $f$ is convex."}
+{"_id": "9083", "title": "Real Function is Concave iff Derivative is Decreasing", "text": "Let $f$ be a real function which is differentiable on the open interval $\\openint a b$. Then $f$ is concave on $\\openint a b$ {{iff}} its derivative $f'$ is decreasing on $\\openint a b$. Thus the intuitive result that a concave function \"gets less steep\"."}
+{"_id": "9084", "title": "Real Function is Strictly Convex iff Derivative is Strictly Increasing", "text": "Let $f$ be a real function which is differentiable on the open interval $\\openint a b$. Then $f$ is strictly convex on $\\openint a b$ {{iff}} its derivative $f'$ is strictly increasing on $\\openint a b$."}
+{"_id": "9085", "title": "Real Function is Strictly Concave iff Derivative is Strictly Decreasing", "text": "Let $f$ be a real function which is differentiable on the open interval $\\openint a b$. Then $f$ is strictly concave on $\\openint a b$ {{iff}} its derivative $f'$ is strictly decreasing on $\\openint a b$."}
+{"_id": "9086", "title": "Inverse of Strictly Decreasing Convex Real Function is Convex", "text": "Let $f$ be a real function which is convex on the open interval $I$. Let $J = f \\left[{I}\\right]$. If $f$ be strictly decreasing on $I$, then $f^{-1}$ is convex on $J$."}
+{"_id": "9087", "title": "Inverse of Strictly Increasing Concave Real Function is Convex", "text": "Let $f$ be a real function which is concave on the open interval $I$. Let $J = f \\sqbrk I$. If $f$ is strictly increasing on $I$, then $f^{-1}$ is convex on $J$."}
+{"_id": "9088", "title": "Inverse of Strictly Decreasing Concave Real Function is Concave", "text": "Let $f$ be a real function which is concave on the open interval $I$. Let $J = f \\left[{I}\\right]$. If $f$ be strictly decreasing on $I$, then $f^{-1}$ is concave on $J$."}
+{"_id": "9089", "title": "Mean Value of Concave Real Function", "text": "Let $f$ be a real function which is continuous on the closed interval $\\left[{a \\,.\\,.\\, b}\\right]$ and differentiable on the open interval $\\left({a \\,.\\,.\\, b}\\right)$. Let $f$ be concave on $\\left({a \\,.\\,.\\, b}\\right)$. Then: : $\\forall \\xi \\in \\left({a \\,.\\,.\\, b}\\right): f \\left({x}\\right) - f \\left({\\xi}\\right) \\le f^{\\prime} \\left({\\xi}\\right) \\left({x - \\xi}\\right)$"}
+{"_id": "9090", "title": "Differentiable Bounded Concave Real Function is Constant", "text": "Let $f$ be a real function which is: :$(1): \\quad$ Differentiable on $\\R$ :$(2): \\quad$ Bounded on $\\R$ :$(3): \\quad$ Concave on $\\R$. Then $f$ is constant."}
+{"_id": "9091", "title": "Second Derivative of Strictly Concave Real Function is Strictly Negative", "text": "Let $f$ be a real function which is twice differentiable on the open interval $\\left({a \\,.\\,.\\, b}\\right)$. Then $f$ is strictly concave on $\\left({a \\,.\\,.\\, b}\\right)$ {{iff}} its second derivative $f'' < 0$ on $\\left({a \\,.\\,.\\, b}\\right)$."}
+{"_id": "9092", "title": "Second Derivative of Strictly Convex Real Function is Strictly Positive", "text": "Let $f$ be a real function which is twice differentiable on the open interval $\\openint a b$. Then $f$ is strictly convex on $\\openint a b$ {{iff}} its second derivative $f'' > 0$ on $\\openint a b$."}
+{"_id": "9094", "title": "Inverse of Strictly Decreasing Strictly Concave Real Function is Strictly Concave", "text": "Let $f$ be a real function which is strictly concave on the open interval $I$. Let $J = f \\left[{I}\\right]$. If $f$ be strictly decreasing on $I$, then $f^{-1}$ is strictly concave on $J$."}
+{"_id": "9095", "title": "Inverse of Strictly Increasing Strictly Convex Real Function is Strictly Concave", "text": "Let $f$ be a real function which is strictly convex on the open interval $I$. Let $J = f \\sqbrk I$. If $f$ be strictly increasing on $I$, then $f^{-1}$ is strictly concave on $J$."}
+{"_id": "9096", "title": "Inverse of Strictly Increasing Strictly Concave Real Function is Strictly Convex", "text": "Let $f$ be a real function which is strictly concave on the open interval $I$. Let $J = f \\left[{I}\\right]$. If $f$ be strictly increasing on $I$, then $f^{-1}$ is strictly convex on $J$."}
+{"_id": "9097", "title": "Gamma Function as Integral of Natural Logarithm", "text": "Let $x \\in \\R_{>0}$ be a strictly positive real number. Then: :$\\displaystyle \\map \\Gamma x = \\int_{\\to 0}^1 \\paren {\\ln \\frac 1 t}^{x - 1} \\rd t$ where $\\Gamma$ denotes the Gamma function."}
+{"_id": "9098", "title": "Reversal of Limits of Definite Integral", "text": "Let $a \\le b$. Then: :$\\displaystyle \\int_a^b \\map f x \\rd x = - \\int_b^a \\map f x \\rd x$"}
+{"_id": "9100", "title": "Comparison Test for Improper Integral", "text": "Let $I = \\openint a b$ be an open real interval. Let $\\phi$ be a real function which is continuous on $I$ and also non-negative on $I$. Let $f$ be a real function which is continuous on $I$. Let $f$ satisfy: :$\\forall x \\in I: \\size {\\map f x} \\le \\map \\phi x$ If the improper integral of $\\phi$ over $I$ exists, then so does that of $f$."}
+{"_id": "9101", "title": "Beta Function is Continuous and Positive on Positive Reals", "text": "Let $x, y \\in \\R$ be real numbers. Let $\\map \\Beta {x, y}$ be the Beta function: :$\\ds \\map \\Beta {x, y} := \\int_{\\mathop \\to 0}^{\\mathop \\to 1} t^{x - 1} \\paren {1 - t}^{y - 1} \\rd t$ Let $y \\in \\R_{>0}$ be given. Then $\\map \\Beta {x, y}$ is a positive and continuous function of $x$ on $\\R_{>0}$."}
+{"_id": "9102", "title": "Logarithm of Beta Function is Convex on Positive Reals", "text": "Let $x, y \\in \\R$ be real numbers. Let $\\Beta \\left({x, y}\\right)$ be the Beta function: :$\\displaystyle \\Beta \\left({x, y}\\right) := \\int_{\\mathop \\to 0}^{\\mathop \\to 1} t^{x - 1} \\left({1 - t}\\right)^{y - 1} \\ \\mathrm d t$ Let $y \\in \\R_{>0}$ be given. Then $\\ln \\left({\\Beta \\left({x, y}\\right)}\\right)$ is a convex function of $x$ on $\\R_{>0}$."}
+{"_id": "9103", "title": "Gamma Function of One Half", "text": ":$\\map \\Gamma {\\dfrac 1 2} = \\sqrt \\pi$ Its decimal expansion starts: :$\\map \\Gamma {\\dfrac 1 2} = 1 \\cdotp 77245 \\, 38509 \\, 05516 \\, 02729 \\, 81674 \\, 83341 \\, 14518 \\, 27975 \\ldots$"}
+{"_id": "9110", "title": "Sum of Reciprocals of Primes is Divergent/Proof 2", "text": "The series: :$\\displaystyle \\sum_{p \\mathop \\in \\Bbb P} \\frac 1 p$ where: :$\\Bbb P$ is the set of all prime numbers is divergent."}
+{"_id": "9112", "title": "Sum of Reciprocals of Primes is Divergent/Lemma", "text": "Let $C \\in \\R_{>0}$ be a (strictly) positive real number. Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {\\map \\ln {\\ln n} - C} = + \\infty$"}
+{"_id": "9113", "title": "Strictly Increasing Infinite Sequence of Integers is Cofinal in Natural Numbers", "text": "Let $S = \\left\\langle{x_n}\\right\\rangle$ be an infinite sequence of integers which is strictly increasing.   Then $S$ is a cofinal subset of $\\left({\\Z, \\le}\\right)$ where $\\le$ is the usual ordering on the integers."}
+{"_id": "9114", "title": "Product of Even and Odd Functions", "text": "Let $\\mathcal O$ be an odd real function defined on some symmetric set $S$. Let $\\mathcal E$ be an even real function defined on some symmetric set $S'$. Let $\\mathcal O\\mathcal E$ be their pointwise product, defined on the intersection of the domains of $\\mathcal O$ and $\\mathcal E$. Then $\\mathcal O\\mathcal E$ is odd.  That is: :$\\forall x \\in S \\cap S': \\left({\\mathcal O\\mathcal E}\\right)\\left({-x}\\right) = - \\left({\\mathcal O\\mathcal E}\\right)\\left({x}\\right)$."}
+{"_id": "9115", "title": "Millin Series", "text": "The '''Millin series''' is the series defined as: :$\\displaystyle \\sum_{n \\mathop = 0}^\\infty \\frac 1 {F_{2^n}} = \\frac {7 - \\sqrt 5} 2$"}
+{"_id": "9117", "title": "Odd Bernoulli Numbers Vanish", "text": "Let $B_n$ denote the $n$th Bernoulli Number. Then: :$B_{2n + 1} = 0$ for $n \\ge 1$."}
+{"_id": "9118", "title": "Gauss Multiplication Formula", "text": "Let $\\Gamma$ denote the Gamma Function. Then: :$\\displaystyle \\forall z \\notin \\set {-\\frac m n: m \\in \\N}: \\prod_{k \\mathop = 0}^{n - 1} \\map \\Gamma {z + \\frac k n} = \\paren {2 \\pi}^{\\paren {n - 1} / 2} n^{1/2 - n z} \\map \\Gamma {n z}$ where $\\N$ denotes the natural numbers."}
+{"_id": "9119", "title": "Dirichlet Integral", "text": ":$\\displaystyle \\int_0^\\infty \\frac {\\sin x} x \\rd x = \\frac \\pi 2$"}
+{"_id": "9120", "title": "Riemann Zeta Function and Prime Counting Function", "text": "For $\\map \\Re s > 1$: :$\\displaystyle \\log \\map \\zeta s = s \\int_0^{\\mathop \\to \\infty} \\frac {\\map \\pi x} {x \\paren {x^s - 1} } \\rd x$ where: :$\\zeta$ denotes the Riemann Zeta Function :$\\pi$ denotes the Prime-Counting Function."}
+{"_id": "9121", "title": "Faulhaber's Formula", "text": "Let $n$ and $p$ be positive integers. Then: :$\\displaystyle \\sum_{k \\mathop = 1}^n k^p = \\frac 1 {p + 1} \\sum_{i \\mathop = 0}^p \\paren {-1}^i \\binom {p + 1} i B_i n^{p + 1 - i}$ where $B_n$ denotes the $n$th Bernoulli number."}
+{"_id": "9122", "title": "Complex Exponential Tends to Zero", "text": "Let $\\exp z$ be the complex exponential function. Then: :$\\displaystyle \\lim_{\\map \\Re z \\mathop \\to +\\infty} e^{-z} = 0$ where $\\map \\Re z$ denotes the real part of $z$."}
+{"_id": "9123", "title": "Modulus of Exponential is Exponential of Real Part", "text": "Let $z \\in \\C$ be a complex number. Let $\\exp z$ denote the complex exponential function. Let $\\cmod {\\, \\cdot \\,}$ denote the complex modulus Then: :$\\cmod {\\exp z} = \\map \\exp {\\Re z}$ where $\\Re z$ denotes the real part of $z$."}
+{"_id": "9124", "title": "Laplace Transform of Exponential/Real Argument", "text": "Let $\\laptrans f$ denote the Laplace transform of a function $f$. Let $e^x$ be the real exponential. Then: :$\\map {\\laptrans {e^{a t} } } s = \\dfrac 1 {s - a}$ where $a \\in \\R$ is constant, and $\\map \\Re s > \\map \\Re a$."}
+{"_id": "9125", "title": "Laplace Transform of Cosine", "text": "Let $\\cos$ be the real cosine function. Let $\\laptrans f$ denote the Laplace transform of the real function $f$. Then: :$\\laptrans {\\cos a t} = \\dfrac s {s^2 + a^2}$ where $a \\in \\R_{>0}$ is constant, and $\\map \\Re s > a$."}
+{"_id": "9126", "title": "Laplace Transform of Sine", "text": "Let $\\sin$ denote the real sine function. Let $\\laptrans f$ denote the Laplace transform of a real function $f$. Then: :$\\laptrans {\\sin at} = \\dfrac a {s^2 + a^2}$ where $a \\in \\R_{>0}$ is constant, and $\\map \\Re s > 0$."}
+{"_id": "9127", "title": "Sum Rule for Derivatives/General Result", "text": "Let $f_1 \\left({x}\\right), f_2 \\left({x}\\right), \\ldots, f_n \\left({x}\\right)$ be real functions all differentiable. Then for all $n \\in \\N_{>0}$: : $\\displaystyle D_x \\left({\\sum_{i \\mathop = 1}^n f_i \\left({x}\\right)}\\right) = \\sum_{i \\mathop = 1}^n D_x \\left({f_i \\left({x}\\right)}\\right)$"}
+{"_id": "9128", "title": "Product Rule for Derivatives/General Result", "text": "Let $f_1 \\left({x}\\right), f_2 \\left({x}\\right), \\ldots, f_n \\left({x}\\right)$ be real functions differentiable on the open interval $I$. then: :$\\forall x \\in I: \\displaystyle D_x \\left({\\prod_{i \\mathop = 1}^n f_i \\left({x}\\right)}\\right) = \\sum_{i \\mathop = 1}^n \\left({D_x \\left({f_i \\left({x}\\right)}\\right) \\prod_{j \\mathop \\ne i} f_j \\left({x}\\right)}\\right)$"}
+{"_id": "9129", "title": "Derivative of Arccotangent Function", "text": ":$\\dfrac {\\map \\d {\\arccot x} } {\\d x} = \\dfrac {-1} {1 + x^2}$"}
+{"_id": "9131", "title": "Asymptotic Formula for Bernoulli Numbers", "text": "The Bernoulli numbers with even index can be approximated by the asymptotic formula: :$B_{2 n} \\sim \\paren {-1}^{n + 1} 4 \\sqrt {\\pi n} \\paren {\\dfrac n {\\pi e} }^{2 n}$ where: :$B_n$ denotes the $n$th Bernoulli number :$\\sim$ denotes asymptotically equal."}
+{"_id": "9132", "title": "Cotangent Minus Tangent", "text": ":$\\cot x - \\tan x = 2 \\cot 2 x$"}
+{"_id": "9133", "title": "Derivative of Hyperbolic Cotangent Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\coth u} = -\\csch^2 u \\dfrac {\\d u} {\\d x}$"}
+{"_id": "9134", "title": "Derivative of Hyperbolic Cosecant Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\csch u} = -\\csch u \\coth u \\dfrac {\\d u} {\\d x}$"}
+{"_id": "9135", "title": "Derivative of Inverse Hyperbolic Sine", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\sinh^{-1} x} = \\dfrac 1 {\\sqrt {x^2 + 1} }$"}
+{"_id": "9136", "title": "Derivative of Inverse Hyperbolic Cosine", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\cosh^{-1} x} = \\dfrac 1 {\\sqrt {x^2 - 1} }$"}
+{"_id": "9137", "title": "Derivative of Inverse Hyperbolic Tangent", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\tanh^{-1} x} = \\dfrac 1 {1 - x^2}$"}
+{"_id": "9138", "title": "Derivative of Inverse Hyperbolic Cotangent", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\coth^{-1} x} = \\dfrac {-1} {x^2 - 1}$"}
+{"_id": "9139", "title": "Derivative of Inverse Hyperbolic Secant", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\sech^{-1} x} = \\dfrac {-1} {x \\sqrt{1 - x^2} }$"}
+{"_id": "9140", "title": "Derivative of Inverse Hyperbolic Cosecant", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\csch^{-1} x} = \\dfrac {-1} {\\size x \\sqrt {1 + x^2} }$"}
+{"_id": "9141", "title": "Sum of Cosecant and Cotangent", "text": ":$\\displaystyle \\csc x + \\cot x = \\cot {\\frac x 2}$"}
+{"_id": "9143", "title": "Leibniz's Rule/One Variable/Third Derivative", "text": "Let $f$ and $g$ be real functions defined on the open interval $I$. Let $x \\in I$ be a point in $I$ at which both $f$ and $g$ are thrice differentiable. Then: :$\\paren {\\map f x \\, \\map g x}''' = \\map f x \\, \\map {g'''} x + 3 \\map {f'} x \\, \\map {g''} x + 3 \\map {f''} x \\, \\map {g'} x + \\map {f'''} x \\, \\map g x$"}
+{"_id": "9144", "title": "Power Series Expansion for Tangent Function", "text": "The tangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \\tan x       | r = \\sum_{n \\mathop = 1}^\\infty \\frac {\\paren {-1}^{n - 1} 2^{2 n} \\paren {2^{2 n} - 1} B_{2 n} \\, x^{2 n - 1} } {\\paren {2 n}!}       | c =  }} {{eqn | r = x + \\frac {x^3} 3 + \\frac {2 x^5} {15} + \\frac {17 x^7} {315} + \\cdots       | c =  }} {{end-eqn}} where $B_{2 n}$ denotes the Bernoulli numbers. This converges for $\\size x < \\dfrac \\pi 2$."}
+{"_id": "9145", "title": "Partial Differentiation Operator is Commutative for Continuous Functions", "text": "Let $\\map f {x, y}$ be a function of the two independent variables $x$ and $y$. Let $\\map f {x, y}$ be continuous. Let the partial deriviatives of $f$ also be continuous. Then: :$\\dfrac {\\partial^2 f} {\\partial x \\partial y} = \\dfrac {\\partial^2 f} {\\partial y \\partial x}$"}
+{"_id": "9146", "title": "Primitive of Constant Multiple of Function", "text": "Let $f$ be a real function which is integrable. Let $c$ be a constant. Then: :$\\displaystyle \\int c \\map f x \\rd x = c \\int \\map f x \\rd x$"}
+{"_id": "9147", "title": "Primitive of Pointwise Sum of Functions", "text": "Let $f_1, f_2, \\ldots, f_n$ be real functions which are integrable. Then: :$\\ds \\int \\map {\\paren {f_1 \\pm f_2 \\pm \\, \\cdots \\pm f_n} } x \\rd x = \\int \\map {f_1} x \\rd x \\pm \\int \\map {f_2} x \\rd x \\pm \\, \\cdots \\pm \\int \\map {f_n} x \\rd x$"}
+{"_id": "9148", "title": "Hurwitz's Theorem (Number Theory)", "text": "Let $\\xi$ be an irrational number. Then there are infinitely many relatively prime integers $m, n \\in \\Z$ such that: :$\\left|{\\xi - \\dfrac p q}\\right| < \\dfrac 1 {\\sqrt 5 \\, q^2}$"}
+{"_id": "9149", "title": "Primitive of Function of Constant Multiple", "text": "Let $f$ be a real function which is integrable. Let $c$ be a constant. Then: :$\\displaystyle \\int \\map f {c x} \\rd x = \\frac 1 c \\int \\map f u \\d u$ where $u = c x$."}
+{"_id": "9150", "title": "Primitive of Composite Function", "text": "Let $f$ and $g$ be a real functions which are integrable. Let the composite function $g \\circ f$ also be integrable. Then: {{begin-eqn}} {{eqn | l = \\int \\map {\\paren {g \\circ f} } x \\rd x       | r = \\int \\map g u \\frac {\\d x} {\\d u} \\rd u       | c =  }} {{eqn | r = \\int \\frac {\\map g u} {\\map {f'} x} \\rd u       | c =  }} {{end-eqn}} where $u = \\map f x$."}
+{"_id": "9151", "title": "Weierstrass Product Inequality", "text": "For $n \\ge 1$: :$\\displaystyle \\prod_{i \\mathop = 1}^n \\left({1 - a_i}\\right) \\ge 1 - \\sum_{i \\mathop = 1}^n a_i$ where all of $a_i$ are in the closed interval $\\left[{0 \\,.\\,.\\,1 }\\right]$."}
+{"_id": "9154", "title": "Primitive of Power", "text": "Let $n \\in \\R: n \\ne -1$. Then: :$\\ds \\int x^n \\rd x = \\frac {x^{n + 1} } {n + 1} + C$ where $C$ is an arbitrary constant. That is: :$\\dfrac {x^{n + 1} } {n + 1}$ is a primitive of $x^n$."}
+{"_id": "9155", "title": "Sine Inequality", "text": ":$\\size {\\sin x} \\le \\size x$ for all $x \\in \\R$."}
+{"_id": "9158", "title": "Jordan's Inequality", "text": ":$\\dfrac 2 {\\pi} x \\le \\sin x \\le x$ for all $x$ in the interval $\\left[{0 \\,.\\,.\\, \\dfrac {\\pi} 2}\\right]$"}
+{"_id": "9159", "title": "Cosine Inequality", "text": ":$1 - \\dfrac {x^2} 2 \\le \\cos x$ for all $x \\in \\R$."}
+{"_id": "9162", "title": "Laplace Transform of Constant Mapping", "text": "Let $a \\in \\R$ be a real constant. Let $f_a: \\R \\to \\R$ or $\\C$ be the constant mapping, defined as: :$\\forall t \\in \\R: \\map {f_a} t = a$ Let $\\laptrans {f_a}$ be the Laplace transform of $f_a$. Then: :$\\laptrans {\\map {f_a} t} = \\dfrac a s$ for $\\map \\Re s > a$."}
+{"_id": "9164", "title": "Laplace Transform of Positive Integer Power", "text": "Let $\\laptrans f$ denote the Laplace transform of a function $f$. Let $t^n: \\R \\to \\R$ be $t$ to the $n$th power for some $n \\in \\N_{\\ge 0}$. Then: :$\\laptrans {t^n} = \\dfrac {n!} { s^{n + 1} }$ for $\\map \\Re s > 0$."}
+{"_id": "9165", "title": "Primitive of Function under its Derivative", "text": "Let $f$ be a real function which is integrable. Then: :$\\ds \\int \\frac {\\map {f'} x} {\\map f x} \\rd x = \\ln \\size {\\map f x} + C$ where $C$ is an arbitrary constant."}
+{"_id": "9166", "title": "Primitive of Exponential Function/General Result", "text": "Let $a \\in \\R_{>0}$ be a constant such that $a \\ne 1$. Then: :$\\ds \\int a^x \\rd x = \\frac {a^x} {\\ln a} + C$ where $C$ is an arbitrary constant."}
+{"_id": "9167", "title": "Primitive of Tangent Function/Secant Form", "text": ":$\\ds \\int \\tan x \\rd x = \\ln \\size {\\sec x} + C$ where $\\sec x$ is defined."}
+{"_id": "9168", "title": "Primitive of Secant Function/Tangent plus Angle Form", "text": ":$\\ds \\int \\sec x \\rd x = \\ln \\size {\\map \\tan {\\frac x 2 + \\frac \\pi 4} } + C$"}
+{"_id": "9169", "title": "Exponential of x not less than 1+x", "text": ":$e^x \\ge 1 + x$ for all $x \\in \\R$."}
+{"_id": "9170", "title": "Primitive of Cosecant Function/Cosecant minus Cotangent Form", "text": ":$\\ds \\int \\csc x \\rd x = \\ln \\size {\\csc x - \\cot x} + C$ where $\\csc x - \\cot x \\ne 0$."}
+{"_id": "9171", "title": "Primitive of Cosecant Function/Tangent Form", "text": ":$\\ds \\int \\csc x \\rd x = \\ln \\size {\\tan \\frac x 2} + C$ where $\\tan \\dfrac x 2 \\ne 0$."}
+{"_id": "9172", "title": "Primitive of Square of Secant Function", "text": ":$\\ds \\int \\sec^2 x \\rd x = \\tan x + C$ where $C$ is an arbitrary constant."}
+{"_id": "9173", "title": "Primitive of Square of Cosecant Function", "text": ":$\\ds \\int \\csc^2 x \\rd x = -\\cot x + C$ where $C$ is an arbitrary constant."}
+{"_id": "9174", "title": "Primitive of Square of Tangent Function", "text": ":$\\displaystyle \\int \\tan^2 x \\rd x = \\tan x - x + C$"}
+{"_id": "9175", "title": "Primitive of Square of Cotangent Function", "text": ":$\\displaystyle \\int \\cot^2 x \\rd x = -\\cot x - x + C$ where $C$ is an arbitrary constant."}
+{"_id": "9176", "title": "Primitive of Square of Sine Function", "text": ":$\\ds \\int \\sin^2 x \\rd x = \\frac x 2 - \\frac {\\sin 2 x} 4 + C$ where $C$ is an arbitrary constant."}
+{"_id": "9177", "title": "Primitive of Square of Cosine Function", "text": ":$\\ds \\int \\cos^2 x \\rd x = \\frac x 2 + \\frac {\\sin 2 x} 4 + C$"}
+{"_id": "9178", "title": "Primitive of Product of Secant and Tangent", "text": ":$\\ds \\int \\sec x \\tan x \\rd x = \\sec x + C$ where $C$ is an arbitrary constant."}
+{"_id": "9180", "title": "Primitive of Hyperbolic Sine Function", "text": ":$\\ds \\int \\sinh x \\rd x = \\cosh x + C$"}
+{"_id": "9181", "title": "Primitive of Hyperbolic Cosine Function", "text": ":$\\displaystyle \\int \\cosh x \\rd x = \\sinh x + C$"}
+{"_id": "9182", "title": "Primitive of Hyperbolic Tangent Function", "text": ":$\\ds \\int \\tanh x \\rd x = \\map \\ln {\\cosh x} + C$"}
+{"_id": "9183", "title": "Primitive of Hyperbolic Cotangent Function", "text": ":$\\ds \\int \\coth x \\rd x = \\ln \\size {\\sinh x} + C$ where $\\sinh x \\ne 0$."}
+{"_id": "9184", "title": "Limit of Tan X over X", "text": ":$\\displaystyle \\lim_{x \\mathop \\to 0} \\frac {\\tan x} x = 1$"}
+{"_id": "9191", "title": "Logarithm of Reciprocal", "text": ":$\\map {\\log_b} {\\dfrac 1 x} = -\\log_b x$"}
+{"_id": "9193", "title": "Primitive of Hyperbolic Secant Function/Arcsine Form", "text": ":$\\ds \\int \\sech x \\rd x = \\map \\arcsin {\\tanh x} + C$"}
+{"_id": "9194", "title": "Primitive of Hyperbolic Secant Function/Arctangent of Exponential Form", "text": ":$\\ds \\int \\sech x \\rd x = 2 \\map \\arctan {e^x} + C$"}
+{"_id": "9198", "title": "Primitive of Hyperbolic Cosecant Function/Hyperbolic Tangent Form", "text": ":$\\ds \\int \\csch x \\rd x = \\ln \\size {\\tanh \\frac x 2} + C$ where $\\tanh \\dfrac x 2 \\ne 0$."}
+{"_id": "9201", "title": "Primitive of Square of Hyperbolic Secant Function", "text": ":$\\ds \\int \\sech^2 x \\rd x = \\tanh x + C$ where $C$ is an arbitrary constant."}
+{"_id": "9202", "title": "Primitive of Square of Hyperbolic Cosecant Function", "text": ":$\\ds \\int \\csch^2 x \\rd x = -\\coth x + C$ where $C$ is an arbitrary constant."}
+{"_id": "9203", "title": "Primitive of Square of Hyperbolic Tangent Function", "text": ":$\\displaystyle \\int \\tanh^2 x \\rd x = x - \\tanh x + C$ where $C$ is an arbitrary constant."}
+{"_id": "9204", "title": "Primitive of Square of Hyperbolic Cotangent Function", "text": ":$\\displaystyle \\int \\coth^2 x \\rd x = x - \\coth x + C$ where $C$ is an arbitrary constant."}
+{"_id": "9205", "title": "Heaviside Step Function is Piecewise Continuous", "text": "Let $c \\ge 0$ be a constant real number. The Heaviside step function: :$\\mu_c \\left({t}\\right) = \\begin{cases} 1 & : t > c \\\\ 0 & : t < c \\end{cases}$ is piecewise continuous for any interval of the form: :$\\left[{c - M \\,.\\,.\\, c + M}\\right]$ where $M > 0$ is some arbitrarily large constant."}
+{"_id": "9206", "title": "Euler Formula for Sine Function/Complex Numbers/Proof 1/Lemma 1", "text": "The function: :$\\dfrac {\\sinh x} x$  is increasing for positive real $x$."}
+{"_id": "9207", "title": "Hyperbolic Tangent Less than X", "text": ":$\\tanh x \\le x$ for $x \\ge 0$."}
+{"_id": "9208", "title": "Primitive of Square of Hyperbolic Sine Function", "text": ":$\\displaystyle \\int \\sinh^2 x \\rd x = \\frac {\\sinh 2 x} 4 - \\frac x 2 + C$ where $C$ is an arbitrary constant."}
+{"_id": "9209", "title": "Primitive of Square of Hyperbolic Cosine Function", "text": ":$\\displaystyle \\int \\cosh^2 x \\rd x = \\frac {\\sinh 2 x} 4 + \\frac x 2 + C$ where $C$ is an arbitrary constant."}
+{"_id": "9211", "title": "Primitive of Product of Hyperbolic Cosecant and Cotangent", "text": ":$\\ds \\int \\csch x \\coth x \\rd x = -\\csch x + C$ where $C$ is an arbitrary constant."}
+{"_id": "9212", "title": "Primitive of Reciprocal of x squared minus a squared", "text": "thumbright600px$\\color {blue} {\\dfrac 1 {x^2 - a^2} } \\qquad \\color {green} {-\\dfrac 1 a \\tanh^{-1} \\dfrac x a} \\qquad \\color {red} {-\\dfrac 1 a \\coth^{-1} \\dfrac x a}$ Let $a \\in \\R_{>0}$ be a strictly positive real constant."}
+{"_id": "9216", "title": "Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1", "text": ":$\\displaystyle \\int \\frac {\\d x} {a^2 - x^2} = \\begin {cases} \\dfrac 1 {2 a} \\map \\ln {\\dfrac {a + x} {a - x} } + C & : \\size x < a\\\\ & \\\\  \\dfrac 1 {2 a} \\map \\ln {\\dfrac {x + a} {x - a} } + C & : \\size x > a \\\\ & \\\\ \\text {undefined} & : \\size x = a \\end {cases}$"}
+{"_id": "9217", "title": "Primitive of Reciprocal of a squared minus x squared", "text": "thumbright600px$\\color {blue} {\\dfrac 1 {a^2 - x^2} } \\qquad \\color {red} {\\dfrac 1 a \\tanh^{-1} \\dfrac x a} \\qquad \\color {green} {\\dfrac 1 a \\coth^{-1} \\dfrac x a}$ Let $a \\in \\R_{>0}$ be a strictly positive real constant."}
+{"_id": "9219", "title": "Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form", "text": ":$\\ds \\int \\frac {\\d x} {\\sqrt {x^2 + a^2} } = \\map \\ln {x + \\sqrt {x^2 + a^2} } + C$"}
+{"_id": "9220", "title": "Limit at Infinity of Polynomial over Complex Exponential", "text": "Let $n \\in \\N$. Let $\\map {P_n} x$ be a real polynomial, of degree $n$. Let $e^z$ be the complex exponential function, where $z = x + i y$. Let $a \\in \\R_{>0}$. Then: :$\\displaystyle \\lim_{x \\mathop \\to +\\infty} \\frac {\\map {P_n} x} {e^{a z} } = 0$"}
+{"_id": "9221", "title": "Primitives of Trigonometric Functions", "text": "This page gathers together primitives of trigonometric functions. In the below, $C$ is an arbitrary constant throughout."}
+{"_id": "9222", "title": "Primitives of Hyperbolic Functions", "text": "This page gathers together primitives of hyperbolic functions. In the below, $C$ is an arbitrary constant throughout."}
+{"_id": "9223", "title": "Derivatives of Trigonometric Functions", "text": "This page gathers together derivatives of trigonometric functions."}
+{"_id": "9224", "title": "Derivatives of Inverse Trigonometric Functions", "text": "This page gathers together derivatives of inverse trigonometric functions."}
+{"_id": "9225", "title": "Derivatives of Hyperbolic Functions", "text": "This page gathers together derivatives of hyperbolic functions. Let $u$ be a differentiable real function of $x$."}
+{"_id": "9226", "title": "Derivatives of Inverse Hyperbolic Functions", "text": "This page gathers together derivatives of inverse hyperbolic functions."}
+{"_id": "9227", "title": "Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form", "text": ":$\\ds \\int \\frac {\\d x} {\\sqrt {x^2 - a^2} } = \\ln \\size {x + \\sqrt {x^2 - a^2} } + C$"}
+{"_id": "9229", "title": "Primitive of Reciprocal of x by Root of x squared minus a squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\sqrt {x^2 - a^2} } = \\frac 1 a \\arcsec \\size {\\frac x a} + C$ for $\\size x > a$."}
+{"_id": "9231", "title": "Primitive of Reciprocal of x by Root of x squared plus a squared/Logarithm Form", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\sqrt {x^2 + a^2} } = -\\frac 1 a \\map \\ln {\\frac {a + \\sqrt {x^2 + a^2} } x} + C$"}
+{"_id": "9233", "title": "Primitive of Reciprocal of x by Root of a squared minus x squared/Logarithm Form", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\sqrt {a^2 - x^2} } = -\\frac 1 a \\map \\ln {\\frac {a + \\sqrt {a^2 - x^2} } x} + C$"}
+{"_id": "9234", "title": "Generalized Integration by Parts", "text": "Let $\\map f x, \\map g x$ be real functions which are integrable and at least $n$ times differentiable. Then: {{begin-eqn}} {{eqn | l = \\int f^{\\paren n} g \\rd x       | r = \\sum_{j \\mathop = 0}^{n - 1} \\paren {-1}^j f^{\\paren {n - j - 1} } g^{\\paren j} + \\paren {-1}^n \\int f g^{\\paren n} \\rd x       | c =  }} {{eqn | r = f^{\\paren {n - 1} } g - f^{\\paren {n - 2} } g' + f^{\\paren {n - 3} } g'' - \\cdots + \\paren {-1}^n \\int f g^{\\paren n} \\rd x       | c =  }} {{end-eqn}} where $f^{\\paren n}$ denotes the $n$th derivative of $f$."}
+{"_id": "9235", "title": "Primitive of Function of a x + b", "text": ":$\\displaystyle \\int \\map F {a x + b} \\rd x = \\frac 1 a \\int \\map F u \\rd u$ where $u = a x + b$."}
+{"_id": "9236", "title": "Primitive of Function of Root of a x + b", "text": ":$\\displaystyle \\int F \\left({\\sqrt {a x + b}}\\right) \\ \\mathrm d x = \\frac 2 a \\int u \\ F \\left({u}\\right) \\ \\mathrm d u$ where $u = \\sqrt {a x + b}$."}
+{"_id": "9237", "title": "Derivative of Nth Root", "text": "Let $n \\in \\N_{>0}$. Let $f: \\R \\to \\R$ be the real function defined as $f \\left({x}\\right) = \\sqrt [n] x$. Then: :$f' \\left({x}\\right) = \\dfrac 1 {n \\left({\\sqrt [n] x}\\right)^{n-1} }$ everywhere that $f \\left({x}\\right) = \\sqrt [n] x$ is defined."}
+{"_id": "9239", "title": "Primitive of Function of Root of a squared minus x squared", "text": ":$\\displaystyle \\int F \\left({\\sqrt {a^2 - x^2}}\\right) \\ \\mathrm d x = a \\int \\cos u \\ F \\left({a \\cos u}\\right) \\ \\mathrm d u$ where $x = a \\sin u$."}
+{"_id": "9241", "title": "Primitive of Function of Root of x squared minus a squared", "text": ":$\\displaystyle \\int F \\left({\\sqrt {x^2 - a^2}}\\right) \\ \\mathrm d x = a \\int \\sec u \\tan u \\ F \\left({a \\tan u}\\right) \\ \\mathrm d u$ where $x = a \\sec u$."}
+{"_id": "9242", "title": "Primitive of Function of Exponential Function", "text": ":$\\displaystyle \\int \\map F {e^{a x} } \\rd x = \\frac 1 a \\int \\frac {\\map F u} u \\rd u$ where $u = e^{a x}$."}
+{"_id": "9244", "title": "Primitive of Function of Arcsine", "text": ":$\\displaystyle \\int F \\left({\\arcsin \\frac x a}\\right) \\ \\mathrm d x = a \\int F \\left({u}\\right) \\cos u \\ \\mathrm d u$ where $u = \\arcsin \\dfrac x a$."}
+{"_id": "9245", "title": "Primitive of Function of Arccosine", "text": ":$\\displaystyle \\int F \\paren {\\arccos \\frac x a} \\rd x = -a \\int F \\paren u \\sin u \\rd u$ where $u = \\arccos \\dfrac x a$."}
+{"_id": "9246", "title": "Primitive of Function of Arctangent", "text": ":$\\displaystyle \\int F \\left({\\arctan \\frac x a}\\right) \\ \\mathrm d x = a \\int F \\left({u}\\right) \\sec^2 u \\ \\mathrm d u$ where $u = \\arctan \\dfrac x a$."}
+{"_id": "9247", "title": "Primitive of Function of Arccotangent", "text": ":$\\displaystyle \\int F \\left({\\operatorname{arccot} \\frac x a}\\right) \\ \\mathrm d x = -a \\int F \\left({u}\\right) \\csc^2 u \\ \\mathrm d u$ where $u = \\operatorname{arccot} \\dfrac x a$."}
+{"_id": "9248", "title": "Primitive of Function of Arcsecant", "text": ":$\\displaystyle \\int \\map F {\\arcsec \\frac x a} \\rd x = a \\int \\map F u \\sec u \\tan u \\rd u$ where $u = \\arcsec \\dfrac x a$."}
+{"_id": "9251", "title": "Primitive of Reciprocal of a x + b", "text": ":$\\displaystyle \\int \\frac {\\d x} {a x + b} = \\frac 1 a \\ln \\size {a x + b} + C$"}
+{"_id": "9252", "title": "Primitive of x over a x + b", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {a x + b} = \\frac x a - \\frac b {a^2} \\ln \\size {a x + b} + C$"}
+{"_id": "9253", "title": "Laplace Transform of Exponential times Function", "text": ":$\\laptrans {e^{a t} \\map f t} = \\map F {s - a}$ everywhere that $\\laptrans f$ exists, for $\\map \\Re s > a$"}
+{"_id": "9254", "title": "Laplace Transform of Derivative", "text": "Let $f'$ be piecewise continuous with one-sided limits on said intervals. Then $\\laptrans f$ exists for $\\map \\Re s > a$, and: :$\\laptrans {\\map {f'} t} = s \\laptrans {\\map f t} - \\map f 0$"}
+{"_id": "9255", "title": "Primitive of x squared over a x + b", "text": ":$\\displaystyle \\int \\frac {x^2 \\rd x} {a x + b} = \\frac {\\paren {a x + b}^2} {2 a^3} - \\frac {2 b \\paren {a x + b} } {a^3} + \\frac {b^2} {a^3} \\ln \\size {a x + b} + C$"}
+{"_id": "9256", "title": "Primitive of x cubed over a x + b", "text": ":$\\displaystyle \\int \\frac {x^3 \\rd x} {a x + b} = \\frac {\\paren {a x + b}^3} {3 a^4} - \\frac {3 b \\paren {a x + b}^2} {2 a^4} - \\frac {3 b^2 \\paren {a x + b} } {a^4} + \\frac {b^3} {a^4} \\ln \\size {a x + b} + C$"}
+{"_id": "9257", "title": "Primitive of Reciprocal of x by a x + b", "text": ":$\\displaystyle \\int \\frac {\\rd x} {x \\paren {a x + b} } = \\frac 1 b \\ln \\size {\\frac x {a x + b} } + C$"}
+{"_id": "9258", "title": "Primitive of Reciprocal of x squared by a x + b", "text": ":$\\ds \\int \\frac {\\d x} {x^2 \\paren {a x + b} } = -\\frac 1 {b x} + \\frac a {b^2} \\ln \\size {\\frac {a x + b} x} + C$"}
+{"_id": "9259", "title": "Primitive of Reciprocal of x cubed by a x + b", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^3 \\left({a x + b}\\right)} = \\frac {2 a x - b} {2 b^2 x^2} + \\frac {a^2} {b^3} \\ln \\left\\vert{\\frac x {a x + b} }\\right\\vert + C$"}
+{"_id": "9260", "title": "Primitives of Rational Functions involving a x + b", "text": "This page gathers together the primitives of some rational functions involving $a x + b$."}
+{"_id": "9261", "title": "Primitive of Reciprocal of a x + b squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\paren {a x + b}^2} = -\\frac 1 {a \\paren {a x + b} } + C$"}
+{"_id": "9262", "title": "Primitive of x over a x + b squared", "text": ":$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\left({a x + b}\\right)^2} = \\frac b {a^2 \\left({a x + b}\\right)} + \\frac 1 {a^2} \\ln \\left\\vert{a x + b}\\right\\vert + C$"}
+{"_id": "9263", "title": "Primitive of x squared over a x + b squared", "text": ":$\\displaystyle \\int \\frac {x^2 \\rd x} {\\paren {a x + b}^2} = \\frac {a x + b} {a^3} - \\frac {b^2} {a^3 \\paren {a x + b} } - \\frac {2 b} {a^3} \\ln \\size {a x + b} + C$"}
+{"_id": "9264", "title": "Primitive of x cubed over a x + b squared", "text": ":$\\displaystyle \\int \\frac {x^3 \\ \\mathrm d x} {\\left({a x + b}\\right)^2} = \\frac {\\left({a x + b}\\right)^2} {2 a^4} - \\frac {3 b \\left({a x + b}\\right)} {a^4} + \\frac {b^3} {a^4 \\left({a x + b}\\right)} + \\frac {3 b^2} {a^4} \\ln \\left\\vert{a x + b}\\right\\vert + C$"}
+{"_id": "9265", "title": "Primitive of Reciprocal of x by a x + b squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\paren {a x + b}^2} = \\frac 1 {b \\paren {a x + b} } + \\frac 1 {b^2} \\ln \\size {\\frac x {a x + b} } + C$"}
+{"_id": "9266", "title": "Primitive of Reciprocal of x squared by a x + b squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^2 \\left({a x + b}\\right)^2} = \\frac {-a} {b^2 \\left({a x + b}\\right)} - \\frac 1 {b^2 x} + \\frac {2 a} {b^3} \\ln \\left\\vert{\\frac {a x + b} x}\\right\\vert + C$"}
+{"_id": "9267", "title": "Primitive of Reciprocal of x cubed by a x + b squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^3 \\left({a x + b}\\right)^2} = - \\frac {\\left({a x + b}\\right)^2} {2 b^4 x^2} + \\frac {3 a \\left({a x + b}\\right)} {b^4 x} - \\frac {a^3 x} {b^4 \\left({a x + b}\\right)} + \\frac {3 a^2} {b^4} \\ln \\left\\vert{\\frac x {a x + b} }\\right\\vert + C$"}
+{"_id": "9268", "title": "Primitives of Rational Functions involving a x + b squared", "text": "This page gathers together the primitives of some rational functions involving $\\left({a x + b}\\right)^2$."}
+{"_id": "9269", "title": "Primitive of Reciprocal of a x + b cubed", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\paren {a x + b}^3} = -\\frac 1 {2 a \\paren {a x + b}^2} + C$"}
+{"_id": "9270", "title": "Primitive of x over a x + b cubed", "text": ":$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\left({a x + b}\\right)^3} = \\frac {-1} {a^2 \\left({a x + b}\\right)} + \\frac b {2 a^2 \\left({a x + b}\\right)^2} + C$"}
+{"_id": "9273", "title": "Primitive of Reciprocal of x by a x + b cubed", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x \\left({a x + b}\\right)^3} = \\frac {a^2 x^2} {2 b^3 \\left({a x + b}\\right)^2} - \\frac {2 a x} {b^3 \\left({a x + b}\\right)} + \\frac 1 {b^3} \\ln \\left\\vert{\\frac x {a x + b} }\\right\\vert + C$"}
+{"_id": "9274", "title": "Primitive of Reciprocal of x squared by a x + b cubed", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^2 \\left({a x + b}\\right)^3} = \\frac {-a} {2 b^2 \\left({a x + b}\\right)^2} - \\frac {2 a} {b^3 \\left({a x + b}\\right)} - \\frac 1 {b^3 x} + \\frac {3 a} {b^4} \\ln \\left\\vert{\\frac {a x + b} x}\\right\\vert + C$"}
+{"_id": "9275", "title": "Primitive of Reciprocal of x cubed by a x + b cubed", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^3 \\paren {a x + b}^3} = \\frac {a^4 x^2} {2 b^5 \\paren {a x + b}^2} - \\frac {4 a^3 x} {b^5 \\paren {a x + b} } - \\frac {\\paren {a x + b}^2} {2 b^5 x^2} + \\frac {4 a} {b^4 x} + \\frac {6 a^2} {b^5} \\ln \\size {\\frac x {a x + b} } + C$"}
+{"_id": "9276", "title": "Laplace Transform of Second Derivative", "text": "Let $f: \\R \\to \\R$ or $\\R \\to \\C$ be a continuous function on any interval of the form $0 \\le t \\le a$. Let $f$ be twice differentiable. Let $f'$ be continuous and $f''$ piecewise continuous with one-sided limits on said intervals. Let $f$ and $f'$ be of exponential order. Let $\\laptrans f$ denote the Laplace transform of $f$. Then $\\laptrans {f''}$ exists for $\\map \\Re s > a$, and: :$\\laptrans {\\map {f''} t} = s^2 \\laptrans {\\map f t} - s \\, \\map f 0 - \\map {f'} 0$"}
+{"_id": "9277", "title": "Euler's Number is Transcendental", "text": "Euler's Number $e$ is transcendental."}
+{"_id": "9279", "title": "Laplace Transform of Hyperbolic Cosine", "text": "Let $\\cosh t$ be the hyperbolic cosine, where $t$ is real. Let $\\laptrans f$ denote the Laplace transform of the real function $f$. Then: :$\\laptrans {\\cosh a t} = \\dfrac s {s^2 - a^2}$ where $a \\in \\R_{>0}$ is constant, and $\\map \\Re s > a$."}
+{"_id": "9280", "title": "Laplace Transform of Hyperbolic Sine", "text": "Let $\\sinh t$ be the hyperbolic sine, where $t$ is real. Let $\\laptrans f$ denote the Laplace transform of the real function $f$. Then: :$\\laptrans {\\sinh a t} = \\dfrac a {s^2 - a^2}$ where $a \\in \\R_{>0}$ is constant, and $\\map \\Re s > a$."}
+{"_id": "9281", "title": "Primitives of Rational Functions involving a x + b cubed", "text": "This page gathers together the primitives of some rational functions involving $\\left({a x + b}\\right)^3$."}
+{"_id": "9282", "title": "Primitive of Power of a x + b", "text": ":$\\displaystyle \\int \\paren {a x + b}^n \\rd x = \\frac {\\paren {a x + b}^{n + 1} } {\\paren {n + 1} a} + C$"}
+{"_id": "9283", "title": "Primitive of x by Power of a x + b", "text": ":$\\displaystyle \\int x \\left({a x + b}\\right)^n \\ \\mathrm d x = \\frac {\\left({a x + b}\\right)^{n + 2} } {\\left({n + 2}\\right) a^2} - \\frac {b \\left({a x + b}\\right)^{n + 1} } {\\left({n + 1}\\right) a^2} + C$"}
+{"_id": "9284", "title": "Primitive of x squared by Power of a x + b", "text": ":$\\displaystyle \\int x^2 \\left({a x + b}\\right)^n \\ \\mathrm d x = \\frac {\\left({a x + b}\\right)^{n + 3} } {\\left({n + 3}\\right) a^3} - \\frac {2 b \\left({a x + b}\\right)^{n + 2} } {\\left({n + 2}\\right) a^3} + \\frac {b^2 \\left({a x + b}\\right)^{n + 1} } {\\left({n + 1}\\right) a^3} + C$"}
+{"_id": "9285", "title": "Reduction Formula for Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b", "text": ":$\\displaystyle \\int x^m \\left({a x + b}\\right)^n \\rd x = \\frac {x^{m+1} \\left({a x + b}\\right)^n} {m + n + 1} + \\frac {n b} {m + n + 1} \\int x^m \\left({a x + b}\\right)^{n - 1} \\rd x$"}
+{"_id": "9290", "title": "Primitive of x over a x + b cubed/Proof 1", "text": ":$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\left({a x + b}\\right)^3} = \\frac {-1} {a^2 \\left({a x + b}\\right)} + \\frac b {2 a^2 \\left({a x + b}\\right)^2} + C$"}
+{"_id": "9291", "title": "Primitive of x over a x + b cubed/Proof 2", "text": ":$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\left({a x + b}\\right)^3} = \\frac {-1} {a^2 \\left({a x + b}\\right)} + \\frac b {2 a^2 \\left({a x + b}\\right)^2} + C$"}
+{"_id": "9292", "title": "Reduction Formula for Primitive of Power of x by Power of a x + b", "text": "=== Decrement of Power of $a x + b$ === {{:Reduction Formula for Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b}} === Decrement of Power of $x$ === {{:Reduction Formula for Primitive of Power of x by Power of a x + b/Decrement of Power of x}} === Increment of Power of $a x + b$ === {{:Reduction Formula for Primitive of Power of x by Power of a x + b/Increment of Power of a x + b}} === Increment of Power of $x$ === {{:Reduction Formula for Primitive of Power of x by Power of a x + b/Increment of Power of x}}"}
+{"_id": "9293", "title": "Distance from Subset to Element", "text": "Let $\\struct {M, d}$ be a metric space. Let $S \\subseteq M$ be a subset of $M$. Let $s \\in S$. Then: :$\\map d {s, S} = 0$ where $\\map d {s, S}$ denotes the distance between $s$ and $S$."}
+{"_id": "9294", "title": "Reduction Formula for Primitive of Power of x by Power of a x + b/Decrement of Power of x", "text": ":$\\displaystyle \\int x^m \\left({a x + b}\\right)^n \\rd x = \\frac {x^m \\left({a x + b}\\right)^{n + 1} } {\\left({m + n + 1}\\right) a} - \\frac {m b} {\\left({m + n + 1}\\right) a} \\int x^{m - 1} \\left({a x + b}\\right)^n \\rd x$"}
+{"_id": "9295", "title": "Distance from Subset to Supremum", "text": "Let $S \\subseteq \\R$ be a subset of the real numbers. Suppose that the supremum $\\sup S$ of $S$ exists. Then: :$\\map d {\\sup S, S} = 0$ where $\\map d {\\sup S, S}$ is the distance between $\\sup S$ and $S$."}
+{"_id": "9296", "title": "Distance between Element and Subset is Nonnegative", "text": "Let $\\struct {M, d}$ be a metric space. Let $x \\in M$ and $S \\subseteq M$. Then: :$\\map d {x, S} \\ge 0$ where $\\map d {x, S}$ is the distance between $x$ and $S$."}
+{"_id": "9297", "title": "Distance from Subset to Infimum", "text": "Let $S \\subseteq \\R$ be a subset of the real numbers. Suppose that the infimum $\\inf S$ of $S$ exists. Then: :$\\map d {\\inf S, S} = 0$ where $\\map d {\\inf S, S}$ is the distance between $\\inf S$ and $S$."}
+{"_id": "9298", "title": "Reduction Formula for Primitive of Power of x by Power of a x + b/Increment of Power of a x + b", "text": ":$\\displaystyle \\int x^m \\paren {a x + b}^n \\rd x = \\frac {-x^{m + 1} \\paren {a x + b}^{n + 1} } {\\paren {n + 1} b} + \\frac {m + n + 2} {\\paren {n + 1} b} \\int x^m \\paren {a x + b}^{n + 1} \\rd x$"}
+{"_id": "9299", "title": "Reduction Formula for Primitive of Power of x by Power of a x + b/Increment of Power of x", "text": ":$\\displaystyle \\int x^m \\left({a x + b}\\right)^n \\rd x = \\frac {x^{m+1} \\left({a x + b}\\right)^{n + 1} } {\\left({m + 1}\\right) b} - \\frac {\\left({m + n + 2}\\right) a} {\\left({m + 1}\\right) b} \\int x^{m + 1} \\left({a x + b}\\right)^n \\rd x$"}
+{"_id": "9300", "title": "Primitive of Reciprocal of Root of a x + b", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\sqrt{a x + b} } = \\frac {2 \\sqrt {a x + b} } a + C$"}
+{"_id": "9303", "title": "Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form", "text": ":$\\ds \\int \\frac {\\d x} {x^2 - a^2} = -\\frac 1 a \\coth^{-1} {\\frac x a} + C$ where $\\size x > a$."}
+{"_id": "9304", "title": "Primitive of Reciprocal of x by Root of a x + b", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\sqrt{a x + b} } = \\begin {cases} \\dfrac 1 {\\sqrt b} \\map \\ln {\\dfrac {\\sqrt {a x + b} - \\sqrt b} {\\sqrt {a x + b} + \\sqrt b} } + C & : b > 0 \\\\ \\dfrac 2 {\\sqrt {-b} } \\arctan \\sqrt {\\dfrac {a x + b} {-b} } + C & : b < 0 \\end {cases}$"}
+{"_id": "9305", "title": "Primitive of Reciprocal of x squared by Root of a x + b", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^2 \\sqrt{a x + b} } = -\\frac {\\sqrt{a x + b} } {b x} - \\frac a {2 b} \\int \\frac {\\mathrm d x} {x \\sqrt{a x + b} }$"}
+{"_id": "9306", "title": "Primitive of Root of a x + b", "text": ":$\\displaystyle \\int \\sqrt{a x + b} \\ \\mathrm d x = \\frac {2 \\sqrt{\\left({a x + b}\\right)^3} } {3 a}$"}
+{"_id": "9307", "title": "Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form", "text": ":$\\ds \\int \\frac {\\d x} {a^2 - x^2} = \\frac 1 a \\tanh^{-1} \\frac x a + C$ where $\\size x < a$."}
+{"_id": "9308", "title": "Primitive of Reciprocal of Root of x squared plus a squared/Inverse Hyperbolic Sine Form", "text": ":$\\ds \\int \\frac {\\d x} {\\sqrt {x^2 + a^2} } = \\sinh^{-1} {\\frac x a} + C$"}
+{"_id": "9309", "title": "Primitive of Reciprocal of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\sqrt {x^2 - a^2} } = \\dfrac {\\size x} x \\cosh^{-1} {\\size {\\frac x a} } + C$ for $x > a$."}
+{"_id": "9310", "title": "Primitive of Reciprocal of x by Root of x squared plus a squared/Inverse Hyperbolic Cosecant Form", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\sqrt {x^2 + a^2} } = -\\frac 1 a \\csch^{-1} {\\frac x a} + C$"}
+{"_id": "9311", "title": "Primitive of Reciprocal of x by Root of a squared minus x squared/Inverse Hyperbolic Secant Form", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\sqrt {a^2 - x^2} } = -\\frac 1 a \\sech^{-1} {\\frac x a} + C$"}
+{"_id": "9316", "title": "Primitive of x by Root of a x + b", "text": ":$\\displaystyle \\int x \\sqrt{a x + b} \\ \\mathrm d x = \\frac {2 \\left({3 a x - 2 b}\\right) } {15 a^2} \\sqrt{\\left({a x + b}\\right)^3}$"}
+{"_id": "9317", "title": "Laplace Transform of Heaviside Step Function times Function", "text": "Let $\\map f t: \\R \\to \\R$ or $\\R \\to \\C$ be a function of exponential order $a$ for some constant $a \\in \\R$. Let $f$ be piecewise continuous with one-sided limits on any closed interval of the form $\\closedint 0 b$ where $b > 0$. Let $\\map {u_c} t$ be the Heaviside step function. Let $\\laptrans {\\map f t} = \\map F s$ denote the Laplace transform of $f$. Then: :$\\laptrans {\\map {u_c} t \\, \\map f {t - c} } = e^{-s c} \\map F s$ for $\\map \\Re s > a$."}
+{"_id": "9318", "title": "Primitive of x squared by Root of a x + b", "text": ":$\\displaystyle \\int x^2 \\sqrt{a x + b} \\ \\mathrm d x = \\frac {2 \\left({15 a^2 x^2 - 12 a b x + 8 b^2}\\right) } {105 a^3} \\sqrt{\\left({a x + b}\\right)^3} + C$"}
+{"_id": "9320", "title": "Primitive of Root of a x + b over x", "text": ":$\\displaystyle \\int \\frac {\\sqrt {a x + b} } x \\rd x = 2 \\sqrt {a x + b} + b \\int \\frac {\\d x} {x \\sqrt{a x + b} }$"}
+{"_id": "9322", "title": "Primitive of Root of a x + b over x squared", "text": ":$\\displaystyle \\int \\frac {\\sqrt {a x + b} } {x^2} \\rd x = -\\frac {\\sqrt {a x + b} } x + \\frac a 2 \\int \\frac {\\d x} {x \\sqrt {a x + b} }$"}
+{"_id": "9323", "title": "Primitive of Power of x over Root of a x + b", "text": ":$\\ds \\int \\frac {x^m} {\\sqrt{a x + b} } \\rd x = \\frac {2 x^m \\sqrt{a x + b} } {\\paren {2 m + 1} a} - \\frac {2 m b} {\\paren {2 m + 1} a} \\int \\frac {x^{m - 1} } {\\sqrt{a x + b} } \\rd x$"}
+{"_id": "9324", "title": "Primitive of Reciprocal of Power of x by Root of a x + b", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^m \\sqrt {a x + b} } = -\\frac {\\sqrt {a x + b} } {\\paren {m - 1} b x^{m - 1} } - \\frac {\\paren {2 m - 3} a} {\\paren {2 m - 2} b} \\int \\frac {\\d x} {x^{m - 1} \\sqrt {a x + b} }$"}
+{"_id": "9325", "title": "Primitive of Root of a x + b over Power of x", "text": "=== Formulation 1 === {{:Primitive of Root of a x + b over Power of x/Formulation 1}} === Formulation 2 === {{:Primitive of Root of a x + b over Power of x/Formulation 2}}"}
+{"_id": "9327", "title": "Primitive of Power of Root of a x + b over x", "text": ":$\\displaystyle \\int \\frac {\\left({\\sqrt{a x + b} }\\right)^m} x \\ \\mathrm d x = \\frac {2 \\left({\\sqrt{a x + b} }\\right)^m } m + b \\int \\frac {\\left({\\sqrt{a x + b} }\\right)^{m - 2} } x \\ \\mathrm d x$"}
+{"_id": "9328", "title": "Primitive of Reciprocal of x by Power of Root of a x + b", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x \\left({\\sqrt{a x + b} }\\right)^m} = \\frac 2 {\\left({m - 2}\\right) b \\left({\\sqrt{a x + b} }\\right)^{m - 2} }  + \\frac 1 b \\int \\frac {\\mathrm d x} {x \\left({\\sqrt{a x + b} }\\right)^{m - 2} }$"}
+{"_id": "9329", "title": "Primitive of Reciprocal of a x + b by p x + q", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({a x + b}\\right) \\left({p x + q}\\right)} = \\frac 1 {b p - a q} \\ln \\left\\vert{\\frac {p x + q} {a x + b} }\\right\\vert + C$"}
+{"_id": "9330", "title": "Primitive of x over a x + b by p x + q", "text": ":$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\left({a x + b}\\right) \\left({p x + q}\\right)} = \\frac 1 {b p - a q} \\left({\\frac b a \\ln \\left\\vert{a x + b}\\right\\vert - \\frac q p \\ln \\left\\vert{p x + q}\\right\\vert}\\right) + C$"}
+{"_id": "9331", "title": "Primitive of Reciprocal of a x + b squared by p x + q", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({a x + b}\\right)^2 \\left({p x + q}\\right)} = \\frac 1 {b p - a q} \\left({\\frac 1 {a x + b} + \\frac p {b p - a q} \\ln \\left\\vert{\\frac {p x + q} {a x + b} }\\right\\vert}\\right) + C$"}
+{"_id": "9332", "title": "Primitive of x over a x + b squared by p x + q", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {\\paren {a x + b}^2 \\paren {p x + q} } = \\frac 1 {b p - a q} \\paren {\\frac q {b p - a q} \\ln \\size {\\frac {a x + b} {p x + q} } - \\frac b {a \\paren {a x + b} } } + C$"}
+{"_id": "9333", "title": "Primitive of x squared over a x + b squared by p x + q", "text": ":$\\displaystyle \\int \\frac {x^2 \\ \\mathrm d x} {\\left({a x + b}\\right)^2 \\left({p x + q}\\right)} = \\frac {b^2} {\\left({b p - a q}\\right) a^2 \\left({a x + b}\\right)} + \\frac 1 {\\left({b p - a q}\\right)^2} \\left({\\frac {q^2} p \\ln \\left\\vert{p x + q}\\right\\vert + \\frac {b \\left({b p - 2 a q}\\right)} {a^2} \\ln \\left\\vert{a x + b}\\right\\vert}\\right) + C$"}
+{"_id": "9334", "title": "Friedrichs' Inequality", "text": "Let $G \\subset \\R^n$ be bounded domain. Then for any $u \\in \\map {W^{2, 1}_0} G$: :$\\norm u_{\\map {L^2} G} \\le \\map {\\operatorname {diam} } G \\norm {\\nabla u}_{\\map {L^2} G}$ where: :$\\map {\\operatorname {diam} } G := \\sup \\limits_{x, y \\mathop \\in G} \\size {x - y}$"}
+{"_id": "9335", "title": "Primitive of Reciprocal of Power of a x + b by Power of p x + q", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({a x + b}\\right)^m \\left({p x + q}\\right)^n} = \\frac {-1} {\\left({n - 1}\\right) \\left({b p - a q}\\right)} \\left({\\frac 1 {\\left({a x + b}\\right)^{m-1} \\left({p x + q}\\right)^{n-1} } + a \\left({m + n - 2}\\right) \\int \\frac {\\mathrm d x} {\\left({a x + b}\\right)^m \\left({p x + q}\\right)^{n-1} } }\\right)$"}
+{"_id": "9337", "title": "Reduction Formula for Primitive of Power of a x + b by Power of p x + q/Decrement of Power", "text": ":$\\displaystyle \\int \\left({a x + b}\\right)^m \\left({p x + q}\\right)^n \\ \\mathrm d x = \\frac {\\left({a x + b}\\right)^{m+1} \\left({p x + q}\\right)^n} {\\left({m + n + 1}\\right) a} - \\frac {n \\left({b p - a q}\\right)} {\\left({m + n + 1}\\right) a} \\int \\left({a x + b}\\right)^m \\left({p x + q}\\right)^{n-1} \\ \\mathrm d x$"}
+{"_id": "9338", "title": "Reduction Formula for Primitive of Power of a x + b by Power of p x + q/Increment of Power", "text": ":$\\displaystyle \\int \\left({a x + b}\\right)^m \\left({p x + q}\\right)^n \\ \\mathrm d x = \\frac 1 {\\left({n + 1}\\right) \\left({b p - a q}\\right)} \\left({\\left({a x + b}\\right)^{m+1} \\left({p x + q}\\right)^{n+1} - a \\left({m + n + 2}\\right) \\int \\left({a x + b}\\right)^m \\left({p x + q}\\right)^{n+1} \\ \\mathrm d x}\\right)$"}
+{"_id": "9339", "title": "Primitive of a x + b over p x + q", "text": ":$\\displaystyle \\int \\frac {a x + b} {p x + q} \\ \\mathrm d x  = \\frac {a x} p + \\frac {b p - a q} {p^2} \\ln \\left\\vert{p x + q}\\right\\vert + C$"}
+{"_id": "9340", "title": "Reduction Formula for Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b/Proof 1", "text": ":$\\displaystyle \\int x^m \\left({a x + b}\\right)^n \\rd x = \\frac {x^{m+1} \\left({a x + b}\\right)^n} {m + n + 1} + \\frac {n b} {m + n + 1} \\int x^m \\left({a x + b}\\right)^{n - 1} \\rd x$"}
+{"_id": "9348", "title": "Primitive of Power of a x + b over Power of p x + q/Formulation 1", "text": ":$\\displaystyle \\int \\frac {\\left({a x + b}\\right)^m} {\\left({p x + q}\\right)^n} \\rd x = \\frac {-1} {\\left({n - 1}\\right) \\left({b p - a q}\\right)} \\left({\\frac {\\left({a x + b}\\right)^{m + 1} } {\\left({p x + q}\\right)^{n - 1} } + \\left({n - m - 2}\\right) a \\int \\frac {\\left({a x + b}\\right)^m} { \\left({p x + q}\\right)^{n - 1} } \\rd x}\\right)$"}
+{"_id": "9349", "title": "Primitive of Power of a x + b over Power of p x + q", "text": "=== Formulation 1 === {{:Primitive of Power of a x + b over Power of p x + q/Formulation 1}} === Formulation 2 === {{:Primitive of Power of a x + b over Power of p x + q/Formulation 2}} === Formulation 3 === {{:Primitive of Power of a x + b over Power of p x + q/Formulation 3}}"}
+{"_id": "9350", "title": "Primitive of Power of a x + b over Power of p x + q/Formulation 2", "text": ":$\\displaystyle \\int \\frac {\\paren {a x + b}^m} {\\paren {p x + q}^n} \\rd x = \\frac {-1} {\\paren {n - m - 1} p} \\paren {\\frac {\\paren {a x + b}^m} {\\paren {p x + q}^{n - 1} } + m \\paren {b p - a q} \\int \\frac {\\paren {a x + b}^{m - 1} } {\\paren {p x + q}^n} \\rd x}$"}
+{"_id": "9351", "title": "Primitive of Power of a x + b over Power of p x + q/Formulation 3", "text": ":$\\displaystyle \\int \\frac {\\paren {a x + b}^m} {\\paren {p x + q}^n} \\rd x = \\frac {-1} {\\paren {n - 1} p} \\paren {\\frac {\\paren {a x + b}^m} {\\paren {p x + q}^{n - 1} } - m a \\int \\frac {\\paren {a x + b}^{m - 1} } {\\paren {p x + q}^{n - 1}} \\rd x}$"}
+{"_id": "9353", "title": "Primitive of Reciprocal of p x + q by Root of a x + b", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\paren {p x + q} \\sqrt {a x + b} } = \\begin {cases} \\dfrac 1 {\\sqrt {b p - a q} \\sqrt p} \\ln \\size {\\dfrac {\\sqrt {p \\paren {a x + b} } - \\sqrt {b p - a q} } {\\sqrt {p \\paren {a x + b} } + \\sqrt {b p - a q} } } & : b p - a q > 0 \\\\ \\dfrac 2 {\\sqrt {a q - b p} \\sqrt p} \\arctan \\sqrt {\\dfrac {p \\paren {a x + b} } {a q - b p} } & : b p - a q < 0 \\\\ \\end {cases}$"}
+{"_id": "9354", "title": "Primitive of Root of a x + b over p x + q", "text": ":$\\displaystyle \\int \\frac {\\sqrt{a x + b} } {p x + q} \\rd x = \\begin{cases} \\dfrac {2 \\sqrt{a x + b} } p + \\dfrac {\\sqrt {b p - a q} } {p \\sqrt p} \\ln \\size {\\dfrac {\\sqrt {p \\paren {a x + b} } - \\sqrt {b p - a q} } {\\sqrt {p \\paren {a x + b} } + \\sqrt {b p - a q} } } & : b p - a q > 0 \\\\ \\dfrac {2 \\sqrt{a x + b} } p - \\dfrac {\\sqrt {a q - b p} } {p \\sqrt p} \\arctan \\sqrt {\\dfrac {p \\paren {a x + b} } {a q - b p} } & : b p - a q < 0 \\\\ \\end{cases}$"}
+{"_id": "9355", "title": "Primitive of Power of p x + q by Root of a x + b", "text": ":$\\displaystyle \\int \\paren {p x + q}^n \\sqrt {a x + b} \\rd x = \\frac {2 \\paren {p x + q}^{n + 1} \\sqrt {a x + b} } {\\paren {2 n + 3} p} + \\frac {b p - a q} {\\paren {2 n + 3} p} \\int \\frac {\\paren {p x + q}^n} {\\sqrt{a x + b} } \\rd x$"}
+{"_id": "9356", "title": "Primitive of Reciprocal of Power of p x + q by Root of a x + b", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({p x + q}\\right)^n \\sqrt{a x + b} } = \\frac {\\sqrt{a x + b} } {\\left({n - 1}\\right) \\left({a q - b p}\\right) \\left({p x + q}\\right)^{n-1} } + \\frac {\\left({2 n - 3}\\right) a} {2 \\left({n - 1}\\right)  \\left({a q - b p}\\right)} \\int \\frac {\\mathrm d x} {\\left({p x + q}\\right)^{n-1} \\sqrt{a x + b} }$"}
+{"_id": "9357", "title": "Primitive of Power of p x + q over Root of a x + b", "text": ":$\\displaystyle \\int \\frac {\\left({p x + q}\\right)^n} {\\sqrt{a x + b} } \\ \\mathrm d x = \\frac {2 \\left({p x + q}\\right)^n \\sqrt{a x + b} } {\\left({2 n + 1}\\right) a} + \\frac {2 n \\left({a q - b p}\\right)} {\\left({2 n + 1}\\right) a} \\int \\frac {\\left({p x + q}\\right)^{n-1} } {\\sqrt{a x + b} } \\ \\mathrm d x$"}
+{"_id": "9359", "title": "Primitive of Reciprocal of Root of a x + b by Root of p x + q", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\sqrt {\\paren {a x + b} \\paren {p x + q} } } = \\begin {cases} \\dfrac 2 {\\sqrt {a p} } \\map \\ln {\\sqrt {p \\paren {a x + b} } + \\sqrt {a \\paren {p x + q} } } + C & : \\dfrac {b p - a q} p > 0 \\\\ \\dfrac 2 {\\sqrt {a p} } \\sinh^{-1} \\sqrt {\\dfrac {p \\paren {a x + b} } {b p - a q} } + C & : \\dfrac {b p - a q} p < 0 \\\\ \\end {cases}$"}
+{"_id": "9360", "title": "Derivative of Laplace Transform", "text": "Let $f: \\R \\to \\R$ or $\\R \\to \\C$ be a continuous function, differentiable on any closed interval $\\closedint 0 a$. Let $\\laptrans f = F$ denote the Laplace transform of $f$. Then, everywhere that $\\dfrac \\d {\\d s} \\laptrans f$ exists: :$\\dfrac \\d {\\d s} \\laptrans {\\map f t} = -\\laptrans {t \\, \\map f t}$"}
+{"_id": "9361", "title": "Primitive of x over Root of a x + b by Root of p x + q", "text": ":$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\sqrt{\\left({a x + b}\\right) \\left({p x + q}\\right)} } = \\frac {\\sqrt{\\left({a x + b}\\right) \\left({p x + q}\\right)} } {a p} - \\frac {b p + a q} {2 a p} \\int \\frac {\\mathrm d x} {\\sqrt{\\left({a x + b}\\right) \\left({p x + q}\\right)} }$"}
+{"_id": "9362", "title": "Higher Order Derivatives of Laplace Transform", "text": ":$\\dfrac {\\d^n} {\\d s^n} \\laptrans {\\map f t} = \\paren {-1}^n \\laptrans {t^n \\, \\map f t}$"}
+{"_id": "9363", "title": "Primitive of Root of a x + b by Root of p x + q", "text": ":$\\displaystyle \\int \\sqrt {\\paren {a x + b} \\paren {p x + q} } \\rd x = \\frac {2 a p x + b p + a q} {4 a p} \\sqrt {\\paren {a x + b} \\paren {p x + q} } - \\frac {\\paren {b p - a q}^2} {8 a p} \\int \\frac {\\d x} {\\sqrt {\\paren {a x + b} \\paren {p x + q} } }$"}
+{"_id": "9364", "title": "Primitive of Root of p x + q over Root of a x + b", "text": ":$\\displaystyle \\int \\sqrt {\\frac {p x + q} {a x + b} } \\rd x = \\frac {\\sqrt {\\paren {a x + b} \\paren {p x + q} } } a + \\frac {a q - b p} {2 a} \\int \\frac {\\d x} {\\sqrt {\\paren {a x + b} \\paren {p x + q} } }$"}
+{"_id": "9365", "title": "Primitive of Reciprocal of p x + q by Root of a x + b by Root of p x + q", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\paren {p x + q} \\sqrt {\\paren {a x + b} \\paren {p x + q} } } = \\frac {2 \\sqrt{a x + b} } {\\paren {a q - b p} \\sqrt {p x + q} } + C$"}
+{"_id": "9366", "title": "Primitive of x over x squared plus a squared", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {x^2 + a^2} = \\frac 1 2 \\map \\ln {x^2 + a^2} + C$"}
+{"_id": "9367", "title": "Primitive of x squared over x squared plus a squared", "text": ":$\\displaystyle \\int \\frac {x^2 \\rd x} {x^2 + a^2} = x - a \\arctan {\\frac x a} + C$"}
+{"_id": "9368", "title": "Primitive of x cubed over x squared plus a squared", "text": ":$\\displaystyle \\int \\frac {x^3 \\rd x} {x^2 + a^2} = \\frac {x^2} 2 - \\frac {a^2} 2 \\map \\ln {x^2 + a^2} + C$"}
+{"_id": "9369", "title": "Primitive of Reciprocal of x by x squared plus a squared", "text": ":$\\ds \\int \\frac {\\rd x} {x \\paren {x^2 + a^2} } = \\frac 1 {2 a^2} \\map \\ln {\\frac {x^2} {x^2 + a^2} } + C$"}
+{"_id": "9370", "title": "Primitive of Reciprocal of x squared by x squared plus a squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^2 \\paren {x^2 + a^2} } = -\\frac 1 {a^2 x} - \\frac 1 {a^3} \\arctan \\frac x a + C$"}
+{"_id": "9371", "title": "Primitive of Reciprocal of x cubed by x squared plus a squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^3 \\left({x^2 + a^2}\\right)} = -\\frac 1 {2 a^2 x^2} - \\frac 1 {2 a^4} \\ln \\left({\\frac {x^2 + a^2} {x^2} }\\right) + C$"}
+{"_id": "9372", "title": "Primitive of Reciprocal of x squared plus a squared squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\paren {x^2 + a^2}^2} = \\frac x {2 a^2 \\paren {x^2 + a^2} } + \\frac 1 {2 a^3} \\arctan \\frac x a + C$"}
+{"_id": "9373", "title": "Primitive of x over x squared plus a squared squared", "text": ":$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\left({x^2 + a^2}\\right)^2} = -\\frac 1 {2 \\left({x^2 + a^2}\\right)} + C$"}
+{"_id": "9374", "title": "Primitive of x squared over x squared plus a squared squared", "text": ":$\\displaystyle \\int \\frac {x^2 \\ \\mathrm d x} {\\left({x^2 + a^2}\\right)^2} = \\frac {-x} {2 \\left({x^2 + a^2}\\right)} + \\frac 1 {2 a} \\arctan \\frac x a + C$"}
+{"_id": "9375", "title": "Primitive of x cubed over x squared plus a squared squared", "text": ":$\\displaystyle \\int \\frac {x^3 \\ \\mathrm d x} {\\left({x^2 + a^2}\\right)^2} = \\frac {a^2} {2 \\left({x^2 + a^2}\\right)} + \\frac 1 2 \\ln \\left({x^2 + a^2}\\right) + C$"}
+{"_id": "9376", "title": "Primitive of Reciprocal of x by x squared plus a squared squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\paren {x^2 + a^2}^2} = \\frac 1 {2 a^2 \\paren {x^2 + a^2} } + \\frac 1 {2 a^4} \\map \\ln {\\frac {x^2} {x^2 + a^2} } + C$"}
+{"_id": "9377", "title": "Primitive of Reciprocal of x squared by x squared plus a squared squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^2 \\paren {x^2 + a^2}^2} = -\\frac 1 {a^4 x} - \\frac x {2 a^4 \\paren {x^2 + a^2} } - \\frac 3 {2 a^5} \\arctan \\frac x a + C$"}
+{"_id": "9378", "title": "Primitive of Reciprocal of x cubed by x squared plus a squared squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^3 \\paren {x^2 + a^2}^2} = -\\frac 1 {2 a^4 x^2} - \\frac 1 {2 a^4 \\paren {x^2 + a^2} } - \\frac 1 {a^6} \\map \\ln {\\frac {x^2} {x^2 + a^2} } + C$"}
+{"_id": "9379", "title": "Primitive of Reciprocal of Power of x squared plus a squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({x^2 + a^2}\\right)^n} = \\frac x {2 \\left({n - 1}\\right) a^2 \\left({x^2 + a^2}\\right)^{n - 1} } + \\frac {2 n - 3} {\\left({2 n - 2}\\right) a^2} \\int \\frac {\\mathrm d x} {\\left({x^2 + a^2}\\right)^{n - 1} }$"}
+{"_id": "9381", "title": "Primitive of Reciprocal of x by Power of x squared plus a squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x \\left({x^2 + a^2}\\right)^n} = \\frac 1 {2 \\left({n - 1}\\right) a^2 \\left({x^2 + a^2}\\right)^{n - 1} } + \\frac 1 {a^2} \\int \\frac {\\mathrm d x} {x \\left({x^2 + a^2}\\right)^{n - 1} }$"}
+{"_id": "9382", "title": "Primitive of Reciprocal of Power of x by Power of x squared plus a squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^m \\paren {x^2 + a^2}^n} = \\frac 1 {a^2} \\int \\frac {\\d x} {x^m \\paren {x^2 + a^2}^{n - 1} } - \\frac 1 {a^2} \\int \\frac {\\d x} {x^{m - 2} \\paren {x^2 + a^2}^n}$"}
+{"_id": "9383", "title": "Primitive of x over x squared minus a squared", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {x^2 - a^2} = \\frac 1 2 \\map \\ln {x^2 - a^2} + C$ for $x^2 > a^2$."}
+{"_id": "9385", "title": "Primitive of x cubed over x squared minus a squared", "text": ":$\\displaystyle \\int \\frac {x^3 \\ \\mathrm d x} {x^2 - a^2} = \\frac {x^2} 2 + \\frac {a^2} 2 \\ln \\left({x^2 - a^2}\\right) + C$ for $x^2 > a^2$."}
+{"_id": "9386", "title": "Primitive of Reciprocal of x by x squared minus a squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\paren {x^2 - a^2} } = \\frac 1 {2 a^2} \\, \\map \\ln {\\frac {x^2 - a^2} {x^2} } + C$ for $x^2 > a^2$."}
+{"_id": "9387", "title": "Primitive of Reciprocal of x squared by x squared minus a squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^2 \\left({x^2 - a^2}\\right)} = \\frac 1 {a^2 x} + \\frac 1 {2 a^3} \\ln \\left({\\frac {x - a} {x + a} }\\right) + C$ for $x^2 > a^2$."}
+{"_id": "9388", "title": "Primitive of Reciprocal of x cubed by x squared minus a squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^3 \\left({x^2 - a^2}\\right)} = \\frac 1 {2 a^2 x^2} + \\frac 1 {2 a^4} \\ln \\left({\\frac {x^2 - a^2} {x^2} }\\right) + C$ for $x^2 > a^2$."}
+{"_id": "9389", "title": "Primitive of Reciprocal of x squared minus a squared squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\left({x^2 - a^2}\\right)^2} = \\frac {-x} {2 a^2 \\left({x^2 - a^2}\\right)} + \\frac 1 {4 a^3} \\ln \\left({\\frac {x + a} {x - a} }\\right) + C$ for $x^2 > a^2$."}
+{"_id": "9391", "title": "Primitive of x over x squared minus a squared squared", "text": ":$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\left({x^2 - a^2}\\right)^2} = \\frac {-1} {2 \\left({x^2 - a^2}\\right)} + C$ for $x^2 > a^2$."}
+{"_id": "9392", "title": "Primitive of x squared over x squared minus a squared squared", "text": ":$\\displaystyle \\int \\frac {x^2 \\ \\mathrm d x} {\\left({x^2 - a^2}\\right)^2} = \\frac {-x} {2 \\left({x^2 - a^2}\\right)} + \\frac 1 {4 a} \\ln \\left({\\frac {x - a} {x + a} }\\right) + C$ for $x^2 > a^2$."}
+{"_id": "9395", "title": "Primitive of x cubed over x squared minus a squared squared", "text": ":$\\displaystyle \\int \\frac {x^3 \\ \\mathrm d x} {\\left({x^2 - a^2}\\right)^2} = \\frac {-a^2} {2 \\left({x^2 - a^2}\\right)} + \\frac 1 2 \\ln \\left({x^2 - a^2}\\right) + C$ for $x^2 > a^2$."}
+{"_id": "9398", "title": "Primitive of Reciprocal of x by x squared minus a squared squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\paren {x^2 - a^2}^2} = \\frac {-1} {2 a^2 \\left({x^2 - a^2}\\right)} + \\frac 1 {2 a^4} \\ln \\left({\\frac {x^2} {x^2 - a^2} }\\right) + C$ for $x^2 > a^2$."}
+{"_id": "9399", "title": "Primitive of Reciprocal of x squared by x squared minus a squared squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^2 \\left({x^2 - a^2}\\right)^2} = \\frac {-1} {a^4 x} - \\frac x {2 a^4 \\left({x^2 - a^2}\\right)} + \\frac 3 {4 a^5} \\ln \\left({\\frac {x + a} {x - a} }\\right) + C$ for $x^2 > a^2$."}
+{"_id": "9400", "title": "Primitive of Reciprocal of x cubed by x squared minus a squared squared", "text": ":$\\ds \\int \\frac {\\d x} {x^3 \\paren {x^2 - a^2}^2} = \\frac {-1} {2 a^4 x^2} - \\frac 1 {2 a^4 \\paren {x^2 - a^2} } + \\frac 1 {a^6} \\map \\ln {\\frac {x^2} {x^2 - a^2} } + C$ for $x^2 > a^2$."}
+{"_id": "9401", "title": "Primitive of Reciprocal of Power of x squared minus a squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({x^2 - a^2}\\right)^n} = \\frac {-x} {2 \\left({n - 1}\\right) a^2 \\left({x^2 - a^2}\\right)^{n - 1} } - \\frac {2 n - 3} {\\left({2 n - 2}\\right) a^2} \\int \\frac {\\mathrm d x} {\\left({x^2 - a^2}\\right)^{n - 1} }$ for $x^2 > a^2$."}
+{"_id": "9402", "title": "Primitive of x over Power of x squared minus a squared", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {\\paren {x^2 - a^2}^n} = \\frac {-1} {2 \\paren {n - 1} \\paren {x^2 - a^2}^{n - 1} }$ for $x^2 > a^2$."}
+{"_id": "9404", "title": "Primitive of Power of x over Power of x squared minus a squared", "text": ":$\\displaystyle \\int \\frac {x^m \\rd x} {\\paren {x^2 - a^2}^n} = \\int \\frac {x^{m - 2} \\rd x} {\\paren {x^2 - a^2}^{n - 1} } + a^2 \\int \\frac {x^{m - 2} \\rd x} {\\paren {x^2 - a^2}^n}$ for $x^2 > a^2$."}
+{"_id": "9406", "title": "Sign of Quotient of Factors of Difference of Squares", "text": "Let $a, b \\in \\R$ such that $a \\ne b$. Then :$\\map \\sgn {a^2 - b^2} = \\map \\sgn {\\dfrac {a + b} {a - b} } = \\map \\sgn {\\dfrac {a - b} {a + b} }$ where $\\sgn$ denotes the signum of a real number."}
+{"_id": "9407", "title": "Signum Function is Completely Multiplicative", "text": "The signum function on the set of real numbers is a completely multiplicative function: :$\\forall x, y \\in \\R: \\map \\sgn {x y} = \\map \\sgn x \\map \\sgn y$"}
+{"_id": "9409", "title": "Absolute Value Function is Completely Multiplicative", "text": "The absolute value function on the real numbers $\\R$ is completely multiplicative: :$\\forall x, y \\in \\R: \\left\\vert{x y}\\right\\vert = \\left\\vert{x}\\right\\vert \\, \\left\\vert{y}\\right\\vert$ where $\\left \\vert{a}\\right \\vert$ denotes the absolute value of $a$."}
+{"_id": "9411", "title": "Signum Function of Reciprocal", "text": "Let $x \\in \\R$ such that $x \\ne 0$. Then: :$\\map \\sgn x = \\map \\sgn {\\dfrac 1 x}$ where $\\map \\sgn x$ denotes the signum of $x$."}
+{"_id": "9412", "title": "Primitive of x over a squared minus x squared", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {a^2 - x^2} = -\\frac 1 2 \\, \\map \\ln {a^2 - x^2} + C$ for $x^2 < a^2$."}
+{"_id": "9413", "title": "Linear Combination of Laplace Transforms", "text": "Then: :$\\laptrans {\\lambda \\, \\map f t + \\mu \\, \\map g t} = \\lambda \\laptrans {\\map f t} + \\mu \\laptrans {\\map g t}$ everywhere all the above expressions are defined."}
+{"_id": "9415", "title": "Primitive of x cubed over a squared minus x squared", "text": ":$\\displaystyle \\int \\frac {x^3 \\ \\mathrm d x} {a^2 - x^2} = -\\frac {x^2} 2 - \\frac {a^2} 2 \\ln \\left({a^2 - x^2}\\right) + C$ for $x^2 < a^2$."}
+{"_id": "9416", "title": "Primitive of Reciprocal of x squared by a squared minus x squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^2 \\paren {a^2 - x^2} } = \\frac {-1} {a^2 x} + \\frac 1 {2 a^3} \\map \\ln {\\frac {a + x} {a - x} } + C$ for $x^2 < a^2$."}
+{"_id": "9417", "title": "Primitive of x over a squared minus x squared squared", "text": ":$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\left({a^2 - x^2}\\right)^2} = \\frac 1 {2 \\left({a^2 - x^2}\\right)} + C$ for $x^2 < a^2$."}
+{"_id": "9418", "title": "Primitive of x squared over a squared minus x squared squared", "text": ":$\\displaystyle \\int \\frac {x^2 \\rd x} {\\paren {a^2 - x^2}^2} = \\frac x {2 \\paren {a^2 - x^2} } - \\frac 1 {4 a} \\map \\ln {\\frac {a + x} {a - x} } + C$ for $x^2 < a^2$."}
+{"_id": "9419", "title": "Primitive of x cubed over a squared minus x squared squared", "text": ":$\\displaystyle \\int \\frac {x^3 \\rd x} {\\paren {a^2 - x^2}^2} = \\frac {a^2} {2 \\paren {a^2 - x^2} } + \\frac 1 2 \\map \\ln {a^2 - x^2} + C$ for $x^2 < a^2$."}
+{"_id": "9420", "title": "Primitive of Reciprocal of x by a squared minus x squared squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\paren {a^2 - x^2}^2} = \\frac 1 {2 a^2 \\paren {a^2 - x^2} } + \\frac 1 {2 a^4} \\map \\ln {\\frac {x^2} {a^2 - x^2} } + C$ for $x^2 < a^2$."}
+{"_id": "9421", "title": "Primitive of Reciprocal of x squared by a squared minus x squared squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^2 \\paren {a^2 - x^2}^2} = \\frac {-1} {a^4 x} + \\frac x {2 a^4 \\paren {a^2 - x^2} } + \\frac 3 {4 a^5} \\map \\ln {\\frac {a + x} {a - x} } + C$ for $x^2 < a^2$."}
+{"_id": "9422", "title": "Primitive of Reciprocal of Power of a squared minus x squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({a^2 - x^2}\\right)^n} = \\frac x {2 \\left({n - 1}\\right) a^2 \\left({a^2 - x^2}\\right)^{n - 1} } + \\frac {2 n - 3} {\\left({2 n - 2}\\right) a^2} \\int \\frac {\\mathrm d x} {\\left({a^2 - x^2}\\right)^{n - 1} }$ for $x^2 > a^2$."}
+{"_id": "9425", "title": "Primitive of Power of x over Power of a squared minus x squared", "text": ":$\\displaystyle \\int \\frac {x^m \\rd x} {\\paren {a^2 - x^2}^n} = a^2 \\int \\frac {x^{m - 2} \\rd x} {\\paren {a^2 - x^2}^n} - \\int \\frac {x^{m - 2} \\rd x} {\\paren {a^2 - x^2}^{n - 1} }$ for $x^2 < a^2$."}
+{"_id": "9426", "title": "Primitive of Reciprocal of Power of x by Power of a squared minus x squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^m \\left({a^2 - x^2}\\right)^n} = \\frac 1 {a^2} \\int \\frac {\\mathrm d x} {x^m \\left({a^2 - x^2}\\right)^{n-1} } + \\frac 1 {a^2} \\int \\frac {\\mathrm d x} {x^{m-2} \\left({a^2 - x^2}\\right)^n}$ for $x^2 < a^2$."}
+{"_id": "9427", "title": "Primitive of x over Root of x squared plus a squared", "text": ":$\\ds \\int \\frac {x \\rd x} {\\sqrt {x^2 + a^2} } = \\sqrt {x^2 + a^2} + C$"}
+{"_id": "9429", "title": "Primitive of x cubed over Root of x squared plus a squared", "text": ":$\\displaystyle \\int \\frac {x^2 \\rd x} {\\sqrt {x^2 + a^2} } = \\frac {\\paren {\\sqrt {x^2 + a^2} }^3} 3 - a^2 \\sqrt {x^2 + a^2} + C$"}
+{"_id": "9430", "title": "Primitive of Reciprocal of x squared by Root of x squared plus a squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^2 \\sqrt {x^2 + a^2} } = \\frac {-\\sqrt {x^2 + a^2} } {a^2 x} + C$"}
+{"_id": "9431", "title": "Primitive of Reciprocal of x cubed by Root of x squared plus a squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^3 \\sqrt {x^2 + a^2} } = \\frac {- \\sqrt {x^2 + a^2} } {2 a^2 x^2} + \\frac 1 {2 a^3} \\ln \\left({\\frac {a + \\sqrt {x^2 + a^2} } x}\\right) + C$"}
+{"_id": "9433", "title": "Primitive of x by Root of x squared plus a squared", "text": ":$\\displaystyle \\int x \\sqrt {x^2 + a^2} \\rd x = \\frac {\\paren {\\sqrt {x^2 + a^2} }^3} 3 + C$"}
+{"_id": "9436", "title": "Primitive of Root of x squared plus a squared over x", "text": ":$\\displaystyle \\int \\frac {\\sqrt {x^2 + a^2} } x \\ \\mathrm d x = \\sqrt {x^2 + a^2} - a \\ln \\left({\\frac {a + \\sqrt {x^2 + a^2} } a}\\right) + C$"}
+{"_id": "9437", "title": "Primitive of Root of x squared plus a squared over x squared", "text": ":$\\displaystyle \\int \\frac {\\sqrt {x^2 + a^2} } {x^2} \\rd x = \\frac {-\\sqrt {x^2 + a^2} } x + \\map \\ln {x + \\sqrt {x^2 + a^2} } + C$"}
+{"_id": "9438", "title": "Primitive of Root of x squared plus a squared over x cubed", "text": ":$\\displaystyle \\int \\frac {\\sqrt {x^2 + a^2} } {x^3} \\ \\mathrm d x = \\frac {-\\sqrt {x^2 + a^2} } {2 x^2} - \\frac 1 {2 a} \\ln \\left({\\frac {a + \\sqrt {x^2 + a^2} } x}\\right) + C$"}
+{"_id": "9439", "title": "Primitive of Reciprocal of Root of x squared plus a squared cubed", "text": ":$\\ds \\int \\frac {\\d x} {\\paren {\\sqrt {x^2 + a^2} }^3} = \\frac x {a^2 \\sqrt {x^2 + a^2} } + C$"}
+{"_id": "9440", "title": "Primitive of x over Root of x squared plus a squared cubed", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {\\paren {\\sqrt {x^2 + a^2} }^3} = \\frac {-1} {\\sqrt {x^2 + a^2} } + C$"}
+{"_id": "9443", "title": "Primitive of Reciprocal of x by Root of x squared plus a squared cubed", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\paren {\\sqrt {x^2 + a^2} }^3} = \\frac 1 {a^2 \\sqrt {x^2 + a^2} } - \\frac 1 {a^3} \\map \\ln {\\frac {a + \\sqrt {x^2 + a^2} } x} + C$"}
+{"_id": "9445", "title": "Primitive of Reciprocal of x cubed by Root of x squared plus a squared cubed", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^3 \\left({\\sqrt {x^2 + a^2} }\\right)^3} = \\frac {-1} {2 a^2 x^2 \\sqrt {x^2 + a^2} } - \\frac 3 {2 a^4 \\sqrt {x^2 + a^2} } + \\frac 3 {2 a^5} \\ln \\left({\\frac {a + \\sqrt {x^2 + a^2} } x}\\right) + C$"}
+{"_id": "9446", "title": "Primitive of Root of x squared plus a squared cubed", "text": ":$\\ds \\int \\paren {\\sqrt {x^2 + a^2} }^3 \\rd x = \\frac {x \\paren {\\sqrt {x^2 + a^2} }^3} 4 + \\frac {3 a^2 x \\sqrt {x^2 + a^2} } 8 + \\frac {3 a^4} 8 \\map \\ln {x + \\sqrt {x^2 + a^2} } + C$"}
+{"_id": "9447", "title": "Primitive of x by Root of x squared plus a squared cubed", "text": ":$\\displaystyle \\int x \\left({\\sqrt {x^2 + a^2} }\\right)^3 \\ \\mathrm d x = \\frac {\\left({\\sqrt {x^2 + a^2} }\\right)^5} 5 + C$"}
+{"_id": "9448", "title": "Primitive of x squared by Root of x squared plus a squared cubed", "text": ":$\\displaystyle \\int x^2 \\left({\\sqrt {x^2 + a^2} }\\right)^3 \\ \\mathrm d x = \\frac {x \\left({\\sqrt {x^2 + a^2} }\\right)^5} 6 - \\frac {a^2 x \\left({\\sqrt {x^2 + a^2} }\\right)^3} {24} - \\frac {a^4 x \\sqrt {x^2 + a^2} } {16} - \\frac {a^6} {16} \\ln \\left({x + \\sqrt {x^2 + a^2} }\\right) + C$"}
+{"_id": "9449", "title": "Primitive of x cubed by Root of x squared plus a squared cubed", "text": ":$\\displaystyle \\int x^3 \\paren {\\sqrt {x^2 + a^2} }^3 \\rd x = \\frac {\\paren {\\sqrt {x^2 + a^2} }^7} 7 - \\frac {a^2 \\paren {\\sqrt {x^2 + a^2} }^5} 5 + C$"}
+{"_id": "9453", "title": "Primitive of x over Root of x squared minus a squared", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {\\sqrt {x^2 - a^2} } = \\sqrt {x^2 - a^2} + C$"}
+{"_id": "9454", "title": "Primitive of x squared over Root of x squared minus a squared", "text": "{{:Primitive of x squared over Root of x squared minus a squared/Inverse Hyperbolic Cosine Form}}"}
+{"_id": "9455", "title": "Primitive of x cubed over Root of x squared minus a squared", "text": ":$\\displaystyle \\int \\frac {x^3 \\rd x} {\\sqrt {x^2 - a^2} } = \\frac {\\paren {\\sqrt {x^2 - a^2} }^3} 3 + a^2 \\sqrt {x^2 - a^2} + C$"}
+{"_id": "9456", "title": "Primitive of Reciprocal of x squared by Root of x squared minus a squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^2 \\sqrt {x^2 - a^2} } = \\frac {\\sqrt {x^2 - a^2} } {a^2 x} + C$"}
+{"_id": "9457", "title": "Primitive of Reciprocal of x cubed by Root of x squared minus a squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^3 \\sqrt {x^2 - a^2} } = \\frac {\\sqrt {x^2 - a^2} } {2 a^2 x^2} + \\frac 1 {2 a^3} \\arcsec \\size {\\frac x a} + C$"}
+{"_id": "9458", "title": "Primitive of Root of x squared minus a squared", "text": "{{:Primitive of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form}}"}
+{"_id": "9459", "title": "Primitive of x by Root of x squared minus a squared", "text": ":$\\displaystyle \\int x \\sqrt {x^2 - a^2} \\rd x = \\frac {\\paren {\\sqrt {x^2 - a^2} }^3} 3 + C$"}
+{"_id": "9460", "title": "Primitive of x squared by Root of x squared minus a squared", "text": ":$\\displaystyle \\int x^2 \\sqrt {x^2 - a^2} \\rd x = \\frac {x \\paren {\\sqrt {x^2 - a^2} }^3} 4 + \\frac {a^2 x \\sqrt {x^2 - a^2} } 8 - \\frac {a^4} 8 \\ln \\size {x + \\sqrt {x^2 - a^2} } + C$"}
+{"_id": "9462", "title": "Primitive of Root of x squared minus a squared over x", "text": ":$\\displaystyle \\int \\frac {\\sqrt {x^2 - a^2} } x \\rd x = \\sqrt {x^2 - a^2} - \\frac 1 {2 a} \\arcsec \\size {\\frac x a} + C$"}
+{"_id": "9464", "title": "Primitive of Root of x squared minus a squared over x cubed", "text": ":$\\displaystyle \\int \\frac {\\sqrt {x^2 - a^2} } {x^3} \\rd x = \\frac {-\\sqrt {x^2 - a^2} } {2 x^2} + \\frac 1 {2 a} \\arcsec \\size {\\frac x a} + C$"}
+{"_id": "9465", "title": "Primitive of Reciprocal of Root of x squared minus a squared cubed", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\paren {\\sqrt {x^2 - a^2} }^3} = \\frac {-x} {a^2 \\sqrt {x^2 - a^2} } + C$"}
+{"_id": "9466", "title": "Primitive of x over Root of x squared minus a squared cubed", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {\\paren {\\sqrt {x^2 - a^2} }^3} = \\frac {-1} {\\sqrt {x^2 - a^2} } + C$"}
+{"_id": "9467", "title": "Primitive of x squared over Root of x squared minus a squared cubed", "text": ":$\\displaystyle \\int \\frac {x^2 \\rd x} {\\paren {\\sqrt {x^2 - a^2} }^3} = \\frac {-x} {\\sqrt {x^2 - a^2} } + \\ln \\size {x + \\sqrt {x^2 - a^2} } + C$"}
+{"_id": "9468", "title": "Primitive of x cubed over Root of x squared minus a squared cubed", "text": ":$\\displaystyle \\int \\frac {x^3 \\rd x} {\\paren {\\sqrt {x^2 - a^2} }^3} = \\sqrt {x^2 - a^2} - \\frac {a^2} {\\sqrt {x^2 - a^2} } + C$"}
+{"_id": "9469", "title": "Primitive of Reciprocal of x by Root of x squared minus a squared cubed", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\paren {\\sqrt {x^2 - a^2} }^3} = \\frac {-1} {a^2 \\sqrt {x^2 - a^2} } - \\frac 1 {a^3} \\arcsec \\size {\\frac x a} + C$"}
+{"_id": "9471", "title": "Primitive of Reciprocal of x cubed by Root of x squared minus a squared cubed", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^3 \\paren {\\sqrt {x^2 - a^2} }^3} = \\frac 1 {2 a^2 x^2 \\sqrt {x^2 - a^2} } - \\frac 3 {2 a^4 \\sqrt {x^2 - a^2} } + \\frac 3 {2 a^5} \\arcsec \\size {\\frac x a} + C$"}
+{"_id": "9472", "title": "Primitive of Root of x squared minus a squared cubed", "text": ":$\\displaystyle \\int \\paren {\\sqrt {x^2 - a^2} }^3 \\rd x = \\frac {x \\paren {\\sqrt {x^2 - a^2} }^3} 4 - \\frac {3 a^2 x \\sqrt {x^2 - a^2} } 8 + \\frac {3 a^4} 8 \\ln \\size {x + \\sqrt {x^2 - a^2} } + C$"}
+{"_id": "9473", "title": "Primitive of x by Root of x squared minus a squared cubed", "text": ":$\\displaystyle \\int x \\paren {\\sqrt {x^2 - a^2} }^3 \\rd x = \\frac {\\paren {\\sqrt {x^2 - a^2} }^5} 5 + C$"}
+{"_id": "9475", "title": "Primitive of x cubed by Root of x squared minus a squared cubed", "text": ":$\\displaystyle \\int x^3 \\paren {\\sqrt {x^2 - a^2} }^3 \\rd x = \\frac {\\paren {\\sqrt {x^2 - a^2} }^7} 7 + \\frac {a^2 \\paren {\\sqrt {x^2 - a^2} }^5} 5 + C$"}
+{"_id": "9477", "title": "Primitive of Root of x squared minus a squared cubed over x squared", "text": ":$\\displaystyle \\int \\frac {\\paren {\\sqrt {x^2 - a^2} }^3} {x^2} \\rd x = \\frac {-\\paren {\\sqrt {x^2 - a^2} }^3} x + \\frac{3 x \\sqrt {x^2 - a^2} } 2 - \\frac {3 a^2} 2 \\ln \\size {x + \\sqrt {x^2 - a^2} } + C$"}
+{"_id": "9478", "title": "Primitive of Root of x squared minus a squared cubed over x cubed", "text": ":$\\displaystyle \\int \\frac {\\paren {\\sqrt {x^2 - a^2} }^3} {x^3} \\rd x = \\frac {-\\paren {\\sqrt {x^2 - a^2} }^3} {2 x^2} + \\frac {3 \\sqrt {x^2 - a^2} } 2 - \\frac {3 a} 2 \\arcsec \\size {\\frac x a} + C$"}
+{"_id": "9479", "title": "Primitive of x over Root of a squared minus x squared", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {\\sqrt {a^2 - x^2} } = -\\sqrt {a^2 - x^2} + C$"}
+{"_id": "9480", "title": "Primitive of x cubed over Root of a squared minus x squared", "text": ":$\\displaystyle \\int \\frac {x^3 \\rd x} {\\sqrt {a^2 - x^2} } = \\frac {\\left({\\sqrt {a^2 - x^2} }\\right)^3} 3 - a^2 \\sqrt {a^2 - x^2} + C$"}
+{"_id": "9481", "title": "Primitive of Reciprocal of x squared by Root of a squared minus x squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^2 \\sqrt {a^2 - x^2} } = \\frac {-\\sqrt {a^2 - x^2} } {a^2 x} + C$"}
+{"_id": "9482", "title": "Primitive of Reciprocal of x cubed by Root of a squared minus x squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^3 \\sqrt {a^2 - x^2} } = \\frac {-\\sqrt {a^2 - x^2} } {2 a^2 x^2} - \\frac 1 {2 a^3} \\map \\ln {\\frac {a + \\sqrt {a^2 - x^2} } x} + C$"}
+{"_id": "9483", "title": "Primitive of Root of a squared minus x squared", "text": ":$\\displaystyle \\int \\sqrt {a^2 - x^2} \\rd x = \\frac {x \\sqrt {a^2 - x^2} } 2 + \\frac {a^2} 2 \\arcsin \\frac x a + C$"}
+{"_id": "9484", "title": "Primitive of x by Root of a squared minus x squared", "text": ":$\\displaystyle \\int x \\sqrt {a^2 - x^2} \\ \\mathrm d x = \\frac {-\\left({\\sqrt {a^2 - x^2} }\\right)^3} 3 + C$"}
+{"_id": "9486", "title": "Primitive of x cubed by Root of a squared minus x squared", "text": ":$\\displaystyle \\int x^3 \\sqrt {a^2 - x^2} \\ \\mathrm d x = \\frac {\\left({\\sqrt {a^2 - x^2} }\\right)^5} 5 - \\frac {a^2 \\left({\\sqrt {a^2 - x^2} }\\right)^3} 3 + C$"}
+{"_id": "9487", "title": "Primitive of Root of a squared minus x squared over x", "text": ":$\\displaystyle \\int \\frac {\\sqrt {a^2 - x^2} } x \\rd x = \\sqrt {a^2 - x^2} - a \\, \\map \\ln {\\frac {a + \\sqrt {a^2 - x^2} } x} + C$"}
+{"_id": "9488", "title": "Primitive of Root of a squared minus x squared over x squared", "text": ":$\\displaystyle \\int \\frac {\\sqrt {a^2 - x^2} } {x^2} \\rd x = \\frac {-\\sqrt {a^2 - x^2} } x - \\arcsin \\frac x a + C$"}
+{"_id": "9489", "title": "Primitive of Reciprocal of Root of a squared minus x squared cubed", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({\\sqrt {a^2 - x^2} }\\right)^3} = \\frac x {a^2 \\sqrt {a^2 - x^2} } + C$"}
+{"_id": "9490", "title": "Primitive of x over Root of a squared minus x squared cubed", "text": ":$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\left({\\sqrt {a^2 - x^2} }\\right)^3} = \\frac 1 {\\sqrt {a^2 - x^2} } + C$"}
+{"_id": "9491", "title": "Primitive of x squared over Root of a squared minus x squared cubed", "text": ":$\\displaystyle \\int \\frac {x^2 \\rd x} {\\paren {\\sqrt {a^2 - x^2} }^3} = \\frac x {\\sqrt {a^2 - x^2} } - \\arcsin \\frac x a + C$"}
+{"_id": "9492", "title": "Primitive of Reciprocal of x by Root of a squared minus x squared cubed", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x \\left({\\sqrt {a^2 - x^2} }\\right)^3} = \\frac 1 {a^2 \\sqrt {a^2 - x^2} } - \\frac 1 {a^3} \\ln \\left({\\frac {a + \\sqrt {a^2 - x^2} } x}\\right) + C$"}
+{"_id": "9493", "title": "Primitive of Reciprocal of x squared by Root of a squared minus x squared cubed", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^2 \\left({\\sqrt {a^2 - x^2} }\\right)^3} = \\frac {-\\sqrt {a^2 - x^2} } {a^4 x} + \\frac x {a^4 \\sqrt {a^2 - x^2} } + C$"}
+{"_id": "9495", "title": "Primitive of Root of a squared minus x squared cubed", "text": ":$\\displaystyle \\int \\paren {\\sqrt {a^2 - x^2} }^3 \\rd x = \\frac {x \\paren {\\sqrt {a^2 - x^2} }^3} 4 + \\frac {3 a^2 x \\sqrt {a^2 - x^2} } 8 + \\frac {3 a^4} 8 \\arcsin \\frac x a + C$"}
+{"_id": "9496", "title": "Primitive of x by Root of a squared minus x squared cubed", "text": ":$\\displaystyle \\int x \\left({\\sqrt {a^2 - x^2} }\\right)^3 \\ \\mathrm d x = \\frac {-\\left({\\sqrt {a^2 - x^2} }\\right)^5} 5 + C$"}
+{"_id": "9497", "title": "Primitive of x squared by Root of a squared minus x squared cubed", "text": ":$\\displaystyle \\int x^2 \\paren {\\sqrt {a^2 - x^2} }^3 \\rd x = \\frac {-x \\paren {\\sqrt {a^2 - x^2} }^5} 6 + \\frac {a^2 x \\paren {\\sqrt {a^2 - x^2} }^3} {24} + \\frac {a^4 x \\sqrt {a^2 - x^2} } {16} + \\frac {a^6} {16} \\arcsin \\frac x a + C$"}
+{"_id": "9499", "title": "Primitive of Reciprocal of a x squared plus b x plus c", "text": ":$\\displaystyle \\int \\frac {\\d x} {a x^2 + b x + c} = \\begin{cases} \\dfrac 2 {\\sqrt {4 a c - b^2} } \\, \\map \\arctan {\\dfrac {2 a x + b} {\\sqrt {4 a c - b^2} } } + C & : b^2 - 4 a c < 0 \\\\ \\dfrac 1 {\\sqrt {b^2 - 4 a c} } \\ln \\size {\\dfrac {2 a x + b - \\sqrt {b^2 - 4 a c} } {2 a x + b + \\sqrt {b^2 - 4 a c} } } + C & : b^2 - 4 a c > 0 \\\\ \\dfrac {-2} {2 a x + b} + C & : b^2 = 4 a c \\end{cases}$"}
+{"_id": "9503", "title": "Primitive of Reciprocal of a x squared plus b x plus c/a equal to 0", "text": ":$\\displaystyle \\int \\frac {\\d x} {a x^2 + b x + c} = \\frac 1 b \\ln \\size {b x + c} + C$ when $a = 0$."}
+{"_id": "9504", "title": "Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0/Proof 2", "text": "Let $a \\in \\R_{\\ne 0}$. Let $b = 0$. Then: :$\\displaystyle \\int \\frac {\\mathrm d x} {a x^2 + b x + c} = \\begin{cases} \\dfrac 1 {\\sqrt {a c} } \\arctan \\left({x \\sqrt {\\dfrac a c} }\\right) + C & : a c > 0 \\\\ \\dfrac 1 {2 \\sqrt {-a c} } \\ln \\left\\vert{\\dfrac {a x - \\sqrt {-a c} } {a x + \\sqrt {-a c} } }\\right\\vert + C & : a c < 0 \\\\ \\dfrac {-1} {a x} + C & : c = 0 \\end{cases}$"}
+{"_id": "9505", "title": "Primitive of x over a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {a x^2 + b x + c} = \\frac 1 {2 a} \\ln \\left\\vert{a x^2 + b x + c}\\right\\vert - \\frac b {2 a} \\int \\frac {\\mathrm d x} {a x^2 + b x + c}$"}
+{"_id": "9507", "title": "Primitive of x squared over a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {x^2 \\rd x} {a x^2 + b x + c} = \\frac x a - \\frac b {2 a^2} \\ln \\size {a x^2 + b x + c} - \\frac {b^2 - 2 a c} {2 a^2} \\int \\frac {\\d x} {a x^2 + b x + c}$"}
+{"_id": "9511", "title": "Primitive of Reciprocal of a x squared plus b x plus c/Negative Discriminant", "text": "Let $a \\in \\R_{\\ne 0}$. Let $b^2 - 4 a c < 0$. Then: :$\\displaystyle \\int \\frac {\\d x} {a x^2 + b x + c} = \\frac 2 {\\sqrt {4 a c - b^2} } \\map \\arctan {\\dfrac {2 a x + b} {\\sqrt {4 a c - b^2} } } + C$"}
+{"_id": "9512", "title": "Primitive of Reciprocal of a x squared plus b x plus c/Positive Discriminant", "text": "Let $a \\in \\R_{\\ne 0}$. Let $b^2 - 4 a c > 0$. Then: :$\\displaystyle \\int \\frac {\\d x} {a x^2 + b x + c} = \\frac 1 {\\sqrt {b^2 - 4 a c} } \\ln \\size {\\dfrac {2 a x + b - \\sqrt {b^2 - 4 a c} } {2 a x + b + \\sqrt {b^2 - 4 a c} } } + C$"}
+{"_id": "9513", "title": "Primitive of Reciprocal of a x squared plus b x plus c/Zero Discriminant", "text": "Let $a \\in \\R_{\\ne 0}$. Let $b^2 - 4 a c = 0$. Then: :$\\displaystyle \\int \\frac {\\d x} {a x^2 + b x + c} = \\frac {-2} {2 a x + b} + C$"}
+{"_id": "9514", "title": "Primitive of Reciprocal of square of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({a x^2 + b x + c}\\right)^2} = \\frac {2 a x + b} {\\left({4 a c - b^2}\\right) \\left({a x^2 + b x + c}\\right)} + \\frac {2 a} {4 a c - b^2} \\int \\frac {\\mathrm d x} {a x^2 + b x + c}$"}
+{"_id": "9515", "title": "Completing the Square", "text": "Let $a, b, c, x$ be real numbers with $a \\ne 0$. Then: :$a x^2 + b x + c = \\dfrac {\\paren {2 a x + b}^2 + 4 a c - b^2} {4 a}$ This process is known as '''completing the square'''."}
+{"_id": "9516", "title": "Primitive of x over square of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\left({a x^2 + b x + c}\\right)^2} = \\frac {- \\left({b x + 2 c}\\right)} {\\left({4 a c - b^2}\\right) \\left({a x^2 + b x + c}\\right)} - \\frac b {4 a c - b^2} \\int \\frac {\\mathrm d x} {a x^2 + b x + c}$"}
+{"_id": "9517", "title": "Primitive of x squared over square of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {x^2 \\ \\mathrm d x} {\\left({a x^2 + b x + c}\\right)^2} = \\frac {\\left({b^2 - 2 a c}\\right) x + b c} {a \\left({4 a c - b^2}\\right) \\left({a x^2 + b x + c}\\right)} + \\frac {2 c} {4 a c - b^2} \\int \\frac {\\mathrm d x} {a x^2 + b x + c}$"}
+{"_id": "9518", "title": "Primitive of Power of x over Power of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: {{begin-eqn}} {{eqn | l = \\int \\frac {x^m \\rd x} {\\paren {a x^2 + b x + c}^n}       | r = \\frac {x^{m - 1} } {\\paren {2 n - m - 1} a \\paren {a x^2 + b x + c}^{n - 1} }       | c =  }} {{eqn | o =       | ro= +       | r = \\frac {\\paren {m - 1} c} {\\paren {2 n - m - 1} a} \\int \\frac {x^{m - 2} \\rd x} {\\paren {a x^2 + b x + c}^n}       | c =  }} {{eqn | o =       | ro= -       | r = \\frac {\\paren {n - m} b} {\\paren {2 n - m - 1} a} \\int \\frac {x^{m - 1} \\rd x} {\\paren {a x^2 + b x + c}^n}       | c =  }} {{end-eqn}}"}
+{"_id": "9519", "title": "Primitive of Reciprocal of x by square of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {\\d x} {x \\paren {a x^2 + b x + c}^2} = \\frac 1 {2 c \\paren {a x^2 + b x + c} } - \\frac b {2 c} \\int \\frac {\\d x} {\\paren {a x^2 + b x + c}^2} + \\frac 1 c \\int \\frac {\\d x} {x \\paren {a x^2 + b x + c} }$"}
+{"_id": "9520", "title": "Primitive of Reciprocal of x squared by square of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {\\d x} {x^2 \\paren {a x^2 + b x + c}^2} = \\frac {-1} {c x \\paren {a x^2 + b x + c} } - \\frac {3 a} c \\int \\frac {\\d x} {\\paren {a x^2 + b x + c}^2} - \\frac {2 b} c \\int \\frac {\\d x} {x \\paren {a x^2 + b x + c}^2}$"}
+{"_id": "9521", "title": "Primitive of Reciprocal of Power of x by Power of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: {{begin-eqn}} {{eqn | l = \\int \\frac {\\d x} {x^m \\paren {a x^2 + b x + c}^n}       | r = \\frac {-1} {\\paren {m - 1} c x^{m - 1} \\paren {a x^2 + b x + c}^{n - 1} } }} {{eqn | o =       | ro= -       | r = \\frac {\\paren {m + 2 n - 3} a} {\\paren {m - 1} c} \\int \\frac {\\d x} {x^{m - 2} \\paren {a x^2 + b x + c}^n} }} {{eqn | o =       | ro= -       | r = \\frac {\\paren {m - n + 2} b} {\\paren {m - 1} c} \\int \\frac {\\d x} {x^{m - 1} \\paren {a x^2 + b x + c}^n} }} {{end-eqn}}"}
+{"_id": "9522", "title": "Ring of Sets Closed under Finite Union", "text": "Let $\\mathcal R$ be a ring of sets. Let $A_1, A_2, \\ldots, A_n \\in \\mathcal R$. Then: :$\\displaystyle \\bigcup_{j \\mathop = 1}^n A_j \\in \\mathcal R$"}
+{"_id": "9523", "title": "Sigma-Ring contains Limit Superior of Sequence of Sets", "text": "Let $\\mathcal R$ be a $\\sigma$-ring. Let $\\left \\langle{A_n}\\right \\rangle_{n \\mathop \\in \\N} \\in \\mathcal R$ be a sequence of sets in $\\mathcal R$. Then: :$\\displaystyle \\limsup_{n \\to \\infty} \\ A_n \\in \\mathcal R$"}
+{"_id": "9524", "title": "Gamma Function for Non-Negative Integer Argument", "text": "The Gamma function satisfies: :$\\map \\Gamma z = \\dfrac {\\map \\Gamma {z + 1} } z$ for any $z$ which is not a nonpositive integer."}
+{"_id": "9525", "title": "Gamma Function of Positive Half-Integer", "text": "{{begin-eqn}} {{eqn | l = \\Gamma \\left({m + \\frac 1 2}\\right)       | r = \\frac {\\left({2 m}\\right)!} {2^{2 m} m!} \\sqrt \\pi       | c =  }} {{eqn | r = \\frac {1 \\times 3 \\times 5 \\times \\cdots \\times \\left({2 m - 1}\\right)} {2^m} \\sqrt \\pi       | c =  }} {{end-eqn}} where: : $m + \\dfrac 1 2$ is a half-integer such that $m > 0$ : $\\Gamma$ denotes the Gamma function."}
+{"_id": "9526", "title": "Gamma Function of Negative Half-Integer", "text": "{{begin-eqn}} {{eqn | l = \\map \\Gamma {-m + \\frac 1 2}       | r = \\frac {\\paren {-1}^m 2^{2 m} m!} {\\paren {2 m}!} \\sqrt \\pi       | c =  }} {{eqn | r = \\frac {\\paren {-1}^m 2^m} {1 \\times 3 \\times 5 \\times \\cdots \\times \\paren {2 m - 1} } \\sqrt \\pi       | c =  }} {{end-eqn}} where: :$-m + \\dfrac 1 2$ is a half-integer such that $m > 0$ :$\\Gamma$ denotes the Gamma function."}
+{"_id": "9527", "title": "Reciprocal times Derivative of Gamma Function", "text": "Let $\\Gamma$ denote the Gamma function. Then: :$\\displaystyle \\dfrac {\\Gamma\\,' \\left({z}\\right)} {\\Gamma \\left({z}\\right)} = -\\gamma + \\sum_{n \\mathop = 1}^\\infty \\left({\\frac 1 n - \\frac 1 {z + n - 1} }\\right)$ where: : $\\Gamma\\,' \\left({z}\\right)$ denotes the derivative of the Gamma function : $\\gamma$ denotes the Euler-Mascheroni constant."}
+{"_id": "9528", "title": "Derivative of Gamma Function at 1", "text": "Let $\\Gamma$ denote the Gamma function. Then: :$\\map {\\Gamma'} 1 = -\\gamma$ where: :$\\map {\\Gamma'} 1$ denotes the derivative of the Gamma function evaluated at $1$ :$\\gamma$ denotes the Euler-Mascheroni constant."}
+{"_id": "9530", "title": "Stirling's Formula for Gamma Function", "text": "Let $\\Gamma$ denote the Gamma function. Then: :$\\Gamma \\left({x + 1}\\right) = \\sqrt {2 \\pi x} \\, x^x e^{-x} \\left({1 + \\dfrac 1 {12 x} + \\dfrac 1 {288 x^2} - \\dfrac {139} {51 \\, 480 x^3} + \\cdots}\\right)$"}
+{"_id": "9532", "title": "Beta Function as Integral of Power of t over Power of t plus 1", "text": ":$\\displaystyle \\map \\Beta {x, y} = \\int_{\\mathop \\to 0}^{\\mathop \\to \\infty} \\frac {t^{x - 1} } {\\paren {1 + t}^{x + y} } \\rd t$ where $\\Beta$ denotes the Beta function."}
+{"_id": "9533", "title": "Beta Function as Integral of Power of t by Power of 1 minus t over Power of r plus t", "text": ":$\\displaystyle \\Beta \\left({x, y}\\right) := r^y \\left({r + 1}\\right)^x \\int_{\\mathop \\to 0}^{\\mathop \\to 1} \\frac {t^{x - 1} \\left({1 - t}\\right)^{y - 1} } {\\left({r + t}\\right)^{x + y} } \\rd t$ where $\\Beta$ denotes the Beta function."}
+{"_id": "9534", "title": "Separation of Variables/General Result", "text": "Suppose a first order ordinary differential equation can be expressible in this form: :$\\map {g_1} x \\map {h_1} y + \\map {g_2} x \\map {h_2} y \\dfrac {\\d y} {\\d x} = 0$ Then the equation is said to '''have separable variables''', or '''be separable'''. Its general solution is found by solving the integration: :$\\displaystyle \\int \\frac {\\map {g_1} x} {\\map {g_2} x} \\rd x + \\int \\frac {\\map {h_2} y} {\\map {h_1} y} \\rd y = C$"}
+{"_id": "9535", "title": "Sum of Arithmetic-Geometric Sequence", "text": "Let $\\sequence {a_k}$ be an arithmetic-geometric sequence defined as: :$a_k = \\paren {a + k d} r^k$ for $k = 0, 1, 2, \\ldots, n - 1$ Then its closed-form expression is: :$\\displaystyle \\sum_{k \\mathop = 0}^{n - 1} \\paren {a + k d} r^k = \\frac {a \\paren {1 - r^n} } {1 - r} + \\frac {r d \\paren {1 - n r^{n - 1} + \\paren {n - 1} r^n} } {\\paren {1 - r}^2}$"}
+{"_id": "9539", "title": "Reduction Formula for Integral of Power of Cosine", "text": "Let $n \\in \\Z_{> 0}$ be a (strictly) positive integer. Then: :$\\displaystyle \\int \\cos^n x \\rd x = \\dfrac {\\cos^{n - 1} x \\sin x} n + \\dfrac {n - 1} n \\int \\cos^{n - 2} x \\rd x$ is a reduction formula for $\\displaystyle \\int \\cos^n x \\rd x$."}
+{"_id": "9540", "title": "Primitives of Rational Functions involving Power of a x + b", "text": "This page gathers together the primitives of some rational functions involving a general power of $a x + b$."}
+{"_id": "9541", "title": "Primitives of Functions involving Root of a x + b", "text": "This page gathers together the primitives of some rational functions involving $\\sqrt {a x + b}$."}
+{"_id": "9542", "title": "Primitives of Functions involving Power of Root of a x + b", "text": "This page gathers together the primitives of some rational functions involving $\\sqrt {a x + b}$."}
+{"_id": "9543", "title": "Primitives of Functions involving a x + b and p x + q", "text": "This page gathers together the primitives of some rational functions involving $a x + b$ and $p x + q$."}
+{"_id": "9544", "title": "Primitives of Functions involving Root of a x + b and p x + q", "text": "This page gathers together the primitives of some functions involving $\\sqrt{a x + b}$ and $p x + q$."}
+{"_id": "9545", "title": "Primitives of Functions involving Root of a x + b and Root of p x + q", "text": "This page gathers together the primitives of some functions involving $\\sqrt{a x + b}$ and $\\sqrt{p x + q}$."}
+{"_id": "9546", "title": "Primitives involving x squared plus a squared", "text": "This page gathers together the primitives of some expressions involving $x^2 + a^2$."}
+{"_id": "9547", "title": "Primitives involving x squared plus a squared squared", "text": "This page gathers together the primitives of some expressions involving $\\left({x^2 + a^2}\\right)^2$."}
+{"_id": "9548", "title": "Primitives involving Power of x squared plus a squared", "text": "This page gathers together the primitives of some expressions involving $\\left({x^2 + a^2}\\right)^n$."}
+{"_id": "9549", "title": "Primitives involving x squared minus a squared", "text": "This page gathers together the primitives of some expressions involving $x^2 - a^2$."}
+{"_id": "9550", "title": "Primitives involving x squared minus a squared squared", "text": "This page gathers together the primitives of some expressions involving $\\left({x^2 - a^2}\\right)^2$."}
+{"_id": "9551", "title": "Primitives involving Power of x squared minus a squared", "text": "This page gathers together the primitives of some expressions involving $\\left({x^2 - a^2}\\right)^n$."}
+{"_id": "9552", "title": "Primitives involving a squared minus x squared", "text": "This page gathers together the primitives of some expressions involving $a^2 - x^2$."}
+{"_id": "9553", "title": "Primitives involving a squared minus x squared squared", "text": "This page gathers together the primitives of some expressions involving $\\left({a^2 - x^2}\\right)^2$."}
+{"_id": "9554", "title": "Primitives involving Power of a squared minus x squared", "text": "This page gathers together the primitives of some expressions involving $\\left({x^2 - a^2}\\right)^n$."}
+{"_id": "9555", "title": "Primitives involving Root of x squared plus a squared", "text": "This page gathers together the primitives of some expressions involving $\\sqrt{x^2 + a^2}$."}
+{"_id": "9556", "title": "Primitives involving Root of x squared plus a squared cubed", "text": "This page gathers together the primitives of some expressions involving $\\paren {\\sqrt {x^2 + a^2} }^3$."}
+{"_id": "9557", "title": "Primitives involving Root of x squared minus a squared", "text": "This page gathers together the primitives of some expressions involving $\\sqrt{x^2 - a^2}$."}
+{"_id": "9558", "title": "Primitives involving Root of x squared minus a squared cubed", "text": "This page gathers together the primitives of some expressions involving $\\left({\\sqrt{x^2 - a^2} }\\right)^3$."}
+{"_id": "9559", "title": "Primitives involving Root of a squared minus x squared", "text": "This page gathers together the primitives of some expressions involving $\\sqrt {a^2 - x^2}$."}
+{"_id": "9560", "title": "Primitives involving Root of a squared minus x squared cubed", "text": "This page gathers together the primitives of some expressions involving $\\left({\\sqrt{a^2 - x^2} }\\right)^3$."}
+{"_id": "9561", "title": "Primitives involving a x squared plus b x plus c", "text": "This page gathers together the primitives of some expressions involving $a x^2 + b x + c$."}
+{"_id": "9564", "title": "Primitive of Reciprocal of Root of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {\\d x} {\\sqrt {a x^2 + b x + c} } = \\begin{cases} \\dfrac 1 {\\sqrt a} \\map \\ln {2 \\sqrt a \\sqrt {a x^2 + b x + c} + 2 a x + b} + C & : b^2 - 4 a c > 0 \\\\ \\dfrac 1 {\\sqrt a} \\map {\\sinh^{-1} } {\\dfrac {2 a x + b} {\\sqrt {4 a c - b^2} } } + C & : b^2 - 4 a c < 0 \\\\ \\dfrac 1 {\\sqrt a} \\ln \\size {2 a x + b} + C & : b^2 - 4 a c = 0 \\end{cases}$"}
+{"_id": "9565", "title": "Primitive of Reciprocal of Root of a x squared plus b x plus c/Negative Discriminant", "text": "Let $a \\in \\R_{\\ne 0}$. Let $b^2 - 4 a c < 0$. Then: :$\\displaystyle \\int \\frac {\\d x} {\\sqrt {a x^2 + b x + c} } = \\frac 1 {\\sqrt a} \\sinh^{-1} \\paren {\\dfrac {2 a x + b} {\\sqrt {4 a c - b^2} } } + C$"}
+{"_id": "9566", "title": "Primitive of Reciprocal of Root of a x squared plus b x plus c/Positive Discriminant", "text": "Let $a \\in \\R_{\\ne 0}$. Let $b^2 - 4 a c > 0$. Then: :$\\displaystyle \\int \\frac {\\d x} {\\sqrt {a x^2 + b x + c} } = \\frac 1 {\\sqrt a} \\map \\ln {2 \\sqrt a \\sqrt {a x^2 + b x + c} + 2 a x + b} + C$"}
+{"_id": "9567", "title": "Primitive of Reciprocal of Root of a x squared plus b x plus c/Zero Discriminant", "text": "Let $a \\in \\R_{\\ne 0}$. Let $b^2 - 4 a c = 0$. Then: :$\\displaystyle \\int \\frac {\\mathrm d x} {\\sqrt {a x^2 + b x + c} } = \\frac 1 {\\sqrt a} \\ln \\left\\vert{2 a x + b}\\right\\vert + C$"}
+{"_id": "9569", "title": "Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0/Proof 1", "text": "Let $a \\in \\R_{\\ne 0}$. Let $b = 0$. Then: :$\\displaystyle \\int \\frac {\\d x} {a x^2 + b x + c} = \\begin {cases} \\dfrac 1 {\\sqrt {a c} } \\map \\arctan {x \\sqrt {\\dfrac a c} } + C & : a c > 0 \\\\ \\dfrac 1 {2 \\sqrt {-a c} } \\ln \\size {\\dfrac {a x - \\sqrt {-a c} } {a x + \\sqrt {-a c} } } + C & : a c < 0 \\\\ \\dfrac {-1} {a x} + C & : c = 0 \\end {cases}$"}
+{"_id": "9583", "title": "Primitive of Reciprocal of x by Root of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {\\d x} {x \\sqrt {a x^2 + b x + c} } = \\begin{cases} \\dfrac {-1} {\\sqrt c} \\ln \\left({\\dfrac {2 \\sqrt c \\sqrt {a x^2 + b x + c} + b x + 2 c} x}\\right) & : b^2 - 4 a c > 0 \\\\ \\dfrac {-1} {\\sqrt c} \\sinh^{-1} \\left({\\dfrac {b x + 2 c} {\\left\\vert{x}\\right\\vert \\sqrt {4 a c - b^2} } }\\right) & : b^2 - 4 a c < 0 \\\\ \\dfrac {-1} {\\sqrt c} \\ln \\left\\vert{\\dfrac {2 c} x + b}\\right\\vert + C & : b^2 - 4 a c = 0 \\end{cases}$"}
+{"_id": "9584", "title": "Primitive of Reciprocal of x squared by Root of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {\\d x} {x^2 \\sqrt {a x^2 + b x + c} } = -\\frac {\\sqrt {a x^2 + b x + c} } {c x} - \\frac b {2 c} \\int \\frac {\\d x} {x \\sqrt {a x^2 + b x + c} }$"}
+{"_id": "9585", "title": "Primitive of Root of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\sqrt {a x^2 + b x + c} \\rd x = \\frac {\\paren {2 a x + b} \\sqrt {a x^2 + b x + c} } {4 a} + \\frac {4 a c - b^2} {8 a} \\int \\frac {\\d x} {\\sqrt {a x^2 + b x + c} }$"}
+{"_id": "9586", "title": "Primitive of x by Root of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int x \\sqrt {a x^2 + b x + c} \\ \\mathrm d x = \\frac {\\left({\\sqrt {a x^2 + b x + c} }\\right)^3} {3 a} - \\frac {b \\left({2 a x + b}\\right) \\sqrt {a x^2 + b x + c} } {8 a^2} - \\frac {b \\left({4 a c - b^2}\\right)} {16 a^2} \\int \\frac {\\mathrm d x} {\\sqrt {a x^2 + b x + c} }$"}
+{"_id": "9587", "title": "Primitive of x squared by Root of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int x^2 \\sqrt {a x^2 + b x + c} \\rd x = \\frac {6 a x - 5 b} {24 a^2} \\paren {\\sqrt {a x^2 + b x + c} }^3 + \\frac {5 b^2 - 4 a c} {16 a^2} \\int \\sqrt {a x^2 + b x + c} \\rd x$"}
+{"_id": "9588", "title": "Primitive of Root of a x squared plus b x plus c over x", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {\\sqrt {a x^2 + b x + c} } x \\rd x = \\sqrt {a x^2 + b x + c} + \\frac b 2 \\int \\frac {\\d x} {\\sqrt {a x^2 + b x + c} } + c \\int \\frac {\\d x} {x \\sqrt {a x^2 + b x + c} }$"}
+{"_id": "9589", "title": "Primitive of Root of a x squared plus b x plus c over x squared", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {\\sqrt {a x^2 + b x + c} } {x^2} \\ \\mathrm d x = \\frac {-\\sqrt {a x^2 + b x + c} } x + a \\int \\frac {\\mathrm d x} {\\sqrt {a x^2 + b x + c} } + \\frac b 2 \\int \\frac {\\mathrm d x} {x \\sqrt {a x^2 + b x + c} }$"}
+{"_id": "9590", "title": "Primitive of Reciprocal of Cube of Root of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({\\sqrt {a x^2 + b x + c} }\\right)^3} = \\frac {2 \\left({2 a x + b}\\right)} {\\left({4 a c - b^2}\\right) \\sqrt {a x^2 + b x + c} } + C$"}
+{"_id": "9591", "title": "Primitive of x over Cube of Root of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\left({\\sqrt {a x^2 + b x + c} }\\right)^3} = \\frac {2 \\left({b x + 2 c}\\right)} {\\left({b^2 - 4 a c}\\right) \\sqrt {a x^2 + b x + c} }$"}
+{"_id": "9593", "title": "Primitive of Reciprocal of x by Cube of Root of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {\\mathrm d x} {x \\left({\\sqrt {a x^2 + b x + c} }\\right)^3} = \\frac 1 {c \\sqrt {a x^2 + b x + c} } + \\frac 1 c \\int \\frac {\\mathrm d x} {x \\sqrt {a x^2 + b x + c} } - \\frac b {2 c} \\int \\frac {\\mathrm d x} {\\left({\\sqrt {a x^2 + b x + c} }\\right)^3}$"}
+{"_id": "9596", "title": "Primitive of x by Half Integer Power of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int x \\left({a x^2 + b x + c}\\right)^{n + \\frac 1 2} \\ \\mathrm d x = \\frac {\\left({a x^2 + b x + c}\\right)^{n + \\frac 3 2} } {a \\left({2 n + 3}\\right)} - \\frac b {2 a} \\int \\left({a x^2 + b x + c}\\right)^{n + \\frac 1 2} \\ \\mathrm d x$"}
+{"_id": "9599", "title": "Primitive of x over x cubed plus a cubed", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {x^3 + a^3} = \\frac 1 {6 a} \\map \\ln {\\frac {x^2 - a x + a^2} {\\paren {x + a}^2} } + \\frac 1 {a \\sqrt 3} \\arctan \\frac {2 x - a} {a \\sqrt 3}$"}
+{"_id": "9600", "title": "Primitive of Reciprocal of x cubed plus a cubed", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^3 + a^3} = \\frac 1 {6 a^2} \\ln \\size {\\frac {\\paren {x + a}^2} {x^2 - a x + a^2} } + \\frac 1 {a^2 \\sqrt 3} \\arctan \\frac {2 x - a} {a \\sqrt 3}$"}
+{"_id": "9602", "title": "Primitive of Reciprocal of x by x cubed plus a cubed", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x \\left({x^3 + a^3}\\right)} = \\frac 1 {3 a^3} \\ln \\left\\vert{\\frac {x^3} {x^3 + a^3} }\\right\\vert + C$"}
+{"_id": "9604", "title": "Primitive of Reciprocal of x cubed plus a cubed squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({x^3 + a^3}\\right)^2} = \\frac x {3 a^3 \\left({x^3 + a^3}\\right)} + \\frac 1 {9 a^5} \\ln \\left({\\frac {\\left({x + a}\\right)^2} {x^2 - a x + a^2} }\\right) + \\frac 2 {3 a^5 \\sqrt 3} \\arctan \\frac {2 x - a} {a \\sqrt 3}$"}
+{"_id": "9606", "title": "Primitive of x squared over x cubed plus a cubed squared", "text": ":$\\displaystyle \\int \\frac {x^2 \\ \\mathrm d x} {\\left({x^3 + a^3}\\right)^2} = \\frac {-1} {3 \\left({x^3 + a^3}\\right)} + C$"}
+{"_id": "9607", "title": "Primitive of Reciprocal of x by x cubed plus a cubed squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x \\left({x^3 + a^3}\\right)^2} = \\frac 1 {3 a^3 \\left({x^3 + a^3}\\right)} + \\frac 1 {3 a^6} \\ln \\left\\vert{\\frac {x^3} {x^3 + a^3} }\\right\\vert$"}
+{"_id": "9609", "title": "Primitive of Power of x over x cubed plus a cubed", "text": ":$\\displaystyle \\int \\frac {x^m \\rd x} {x^3 + a^3} = \\frac {x^{m - 2} } {m - 2} - a^3 \\int \\frac {x^{m - 3} \\rd x} {x^3 + a^3}$"}
+{"_id": "9610", "title": "Primitive of Reciprocal of x fourth plus a fourth", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^4 + a^4} = \\frac 1 {4 a^3 \\sqrt 2} \\map \\ln {\\frac {x^2 + a x \\sqrt 2 + a^2} {x^2 - a x \\sqrt 2 + a^2} } - \\frac 1 {2 a^3 \\sqrt 2} \\paren {\\map \\arctan {1 - \\frac {x \\sqrt 2} a} - \\map \\arctan {1 + \\frac {x \\sqrt 2} a} }$"}
+{"_id": "9611", "title": "Primitive of x over x fourth plus a fourth", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {x^4 + a^4} = \\frac 1 {2 a^2} \\arctan \\frac {x^2} {a^2}$"}
+{"_id": "9612", "title": "Primitive of x squared over x fourth plus a fourth", "text": ":$\\displaystyle \\int \\frac {x^2 \\rd x} {x^4 + a^4} = \\frac 1 {4 a \\sqrt 2} \\map \\ln {\\frac {x^2 - a x \\sqrt 2 + a^2} {x^2 + a x \\sqrt 2 + a^2} } - \\frac 1 {2 a \\sqrt 2} \\paren {\\map \\arctan {1 - \\frac {x \\sqrt 2} a} - \\map \\arctan {1 + \\frac {x \\sqrt 2} a} }$"}
+{"_id": "9613", "title": "Primitive of x cubed over x fourth plus a fourth", "text": ":$\\displaystyle \\int \\frac {x^3 \\ \\mathrm d x} {x^4 + a^4} = \\frac {\\ln \\left({x^4 + a^4}\\right)} 4$"}
+{"_id": "9614", "title": "Primitive of Reciprocal of x by x fourth plus a fourth", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x \\left({x^4 + a^4}\\right)} = \\frac 1 {4 a^4} \\ln \\left({\\frac {x^4} {x^4 + a^4} }\\right)$"}
+{"_id": "9616", "title": "Primitive of Reciprocal of x cubed by x fourth plus a fourth", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^3 \\left({x^4 + a^4}\\right)} = \\frac {-1} {2 a^4 x^2} - \\frac 1 {2 a^6} \\arctan \\frac {x^2} {a^2}$"}
+{"_id": "9618", "title": "Primitive of x over x fourth minus a fourth", "text": ":$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {x^4 - a^4} = \\frac 1 {4 a^2} \\ln \\left\\vert{\\frac {x^2 - a^2} {x^2 + a^2} }\\right\\vert + C$"}
+{"_id": "9619", "title": "Primitive of x squared over x fourth minus a fourth", "text": ":$\\displaystyle \\int \\frac {x^2 \\rd x} {x^4 - a^4} = \\frac 1 {4 a} \\ln \\size {\\frac {x - a} {x + a} } + \\frac 1 {2 a} \\arctan \\frac x a + C$"}
+{"_id": "9620", "title": "Primitive of x cubed over x fourth minus a fourth", "text": ":$\\displaystyle \\int \\frac {x^3 \\ \\mathrm d x} {x^4 - a^4} = \\frac {\\ln \\left\\vert{x^4 - a^4}\\right\\vert} 4 + C$"}
+{"_id": "9622", "title": "Primitive of Reciprocal of x squared by x fourth minus a fourth", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^2 \\left({x^4 - a^4}\\right)} = \\frac 1 {a^4 x} + \\frac 1 {4 a^5} \\ln \\left\\vert{\\frac {x - a} {x + a} }\\right\\vert + \\frac 1 {2 a^5} \\arctan \\frac x a + C$"}
+{"_id": "9623", "title": "Primitive of Reciprocal of x cubed by x fourth minus a fourth", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^3 \\left({x^4 - a^4}\\right)} = \\frac 1 {2 a^4 x^2} + \\frac 1 {4 a^6} \\ln \\left\\vert{\\frac {x^2 - a^2} {x^2 + a^2} }\\right\\vert + C$"}
+{"_id": "9624", "title": "Primitive of Reciprocal of x by Power of x plus Power of a", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x \\left({x^n + a^n}\\right)} = \\frac 1 {n a^n} \\ln \\left\\vert{\\frac {x^n} {x^n + a^n} }\\right\\vert + C$"}
+{"_id": "9628", "title": "Absolute Value of Even Power", "text": "Let $x \\in \\R$ be a real number. Let $n \\in \\Z$ be an integer. Then: :$\\left\\vert{x^{2 n} }\\right\\vert = x^{2 n}$"}
+{"_id": "9629", "title": "Primitive of Power of x less one over Power of x plus Power of a", "text": ":$\\displaystyle \\int \\frac {x^{n - 1} \\ \\mathrm d x} {x^n + a^n} = \\frac 1 n \\ln \\left\\vert{x^n + a^n}\\right\\vert + C$"}
+{"_id": "9630", "title": "Primitive of x cubed over x fourth plus a fourth/Proof 1", "text": ":$\\displaystyle \\int \\frac {x^3 \\ \\mathrm d x} {x^4 + a^4} = \\frac {\\ln \\left({x^4 + a^4}\\right)} 4$"}
+{"_id": "9631", "title": "Primitive of x cubed over x fourth plus a fourth/Proof 2", "text": ":$\\displaystyle \\int \\frac {x^3 \\ \\mathrm d x} {x^4 + a^4} = \\frac {\\ln \\left({x^4 + a^4}\\right)} 4$"}
+{"_id": "9636", "title": "Primitive of Reciprocal of Power of x by Power of Power of x plus Power of a", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^m \\ \\left({x^n + a^n}\\right)^r} = \\frac 1 {a^n} \\int \\frac {\\d x} {x^m \\left({x^n + a^n}\\right)^{r - 1} } - \\frac 1 {a^n} \\int \\frac {\\d x} {x^{m - n} \\left({x^n + a^n}\\right)^r}$"}
+{"_id": "9637", "title": "Primitive of Reciprocal of x by Root of Power of x plus Power of a", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\sqrt {x^n + a^n} } = \\frac 1 {n \\sqrt {a^n} } \\ln \\size {\\frac {\\sqrt {x^n + a^n} - \\sqrt {a^n} } {\\sqrt {x^n + a^n} + \\sqrt {a^n} } } + C$"}
+{"_id": "9638", "title": "Primitive of Reciprocal of x by Power of x minus Power of a", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\paren {x^n - a^n} } = \\frac 1 {n a^n} \\ln \\size {\\frac {x^n - a^n} {x^n} } + C$"}
+{"_id": "9639", "title": "Primitive of Power of x less one over Power of x minus Power of a", "text": ":$\\displaystyle \\int \\frac {x^{n - 1} \\rd x} {x^n - a^n} = \\frac 1 n \\ln \\size {x^n - a^n} + C$"}
+{"_id": "9645", "title": "Primitive of x by Sine of a x", "text": ":$\\displaystyle \\int x \\sin a x \\rd x = \\frac {\\sin a x} {a^2} - \\frac {x \\cos a x} a + C$"}
+{"_id": "9646", "title": "Primitive of x squared by Sine of a x", "text": ":$\\displaystyle \\int x^2 \\sin a x \\rd x = \\frac {2 x \\sin a x} {a^2} + \\paren {\\frac 2 {a^3} - \\frac {x^2} a} \\cos a x + C$"}
+{"_id": "9649", "title": "Primitive of Sine of a x over x squared", "text": ":$\\displaystyle \\int \\frac {\\sin a x \\ \\mathrm d x} {x^2} = -\\frac {\\sin a x} x + a \\int \\frac {\\cos a x \\ \\mathrm d x} x$"}
+{"_id": "9653", "title": "Primitive of x over Sine of a x", "text": "{{begin-eqn}} {{eqn | l = \\int \\frac {x \\rd x} {\\sin a x}       | r = \\frac 1 {a^2} \\sum_{n \\mathop = 0}^\\infty \\frac {\\paren {-1}^{n - 1} 2 \\paren {2^{2 n - 1} - 1} B_{2 n} \\paren {a x}^{2 n + 1} } {\\paren {2 n + 1}!} + C       | c =  }} {{eqn | r = \\frac 1 {a^2} \\paren {a x + \\frac {\\paren {a x}^3} {18} + \\frac {7 \\paren {a x}^5} {1800} \\cdots} + C       | c =  }} {{end-eqn}}"}
+{"_id": "9654", "title": "Primitive of Square of Sine of a x", "text": ":$\\displaystyle \\int \\sin^2 a x \\rd x = \\frac x 2 - \\frac {\\sin 2 a x} {4 a} + C$"}
+{"_id": "9655", "title": "Primitive of x by Square of Sine of a x", "text": ":$\\ds \\int x \\sin^2 a x \\rd x = \\frac {x^2} 4 - \\frac {x \\sin 2 a x} {4 a} - \\frac {\\cos 2 a x} {8 a^2} + C$"}
+{"_id": "9656", "title": "Primitive of Cube of Sine of a x", "text": ":$\\displaystyle \\int \\sin^3 a x \\rd x = -\\frac {\\cos a x} a + \\frac {\\cos^3 a x} {3 a} + C$"}
+{"_id": "9658", "title": "Primitive of Square of Cosecant of a x", "text": ":$\\displaystyle \\int \\csc^2 a x \\ \\mathrm d x = -\\frac {\\cot a x} a + C$"}
+{"_id": "9659", "title": "Primitive of Reciprocal of Square of Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\sin^2 a x} = \\frac {-\\cot a x} a + C$"}
+{"_id": "9660", "title": "Primitive of Reciprocal of Cube of Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\sin^3 a x} = \\frac {-\\cos a x} {2 a \\sin^2 a x} + \\frac 1 {2 a} \\ln \\left\\vert{\\tan \\frac {a x} 2}\\right\\vert + C$"}
+{"_id": "9661", "title": "Primitive of Sine of p x by Sine of q x", "text": "For $p \\ne q$: :$\\displaystyle \\int \\sin p x \\sin q x \\rd x = \\frac {\\map \\sin {p - q} x} {2 \\paren {p - q} } - \\frac {\\map \\sin {p + q} x} {2 \\paren {p + q} } + C$"}
+{"_id": "9662", "title": "Primitive of Reciprocal of 1 minus Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {1 - \\sin a x} = \\frac 1 a \\tan \\left({\\frac \\pi 4 + \\frac {a x} 2}\\right) + C$"}
+{"_id": "9663", "title": "Primitive of x over 1 minus Sine of a x", "text": ":$\\ds \\int \\frac {x \\rd x} {1 - \\sin a x} = \\frac x a \\map \\tan {\\frac \\pi 4 + \\frac {a x} 2} + \\frac 2 {a^2} \\ln \\size {\\map \\sin {\\frac \\pi 4 - \\frac {a x} 2} } + C$"}
+{"_id": "9664", "title": "Primitive of Reciprocal of 1 plus Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {1 + \\sin a x} = -\\frac 1 a \\map \\tan {\\frac \\pi 4 - \\frac {a x} 2} + C$"}
+{"_id": "9665", "title": "Primitive of x over 1 plus Sine of a x", "text": ":$\\ds \\int \\frac {x \\rd x} {1 + \\sin a x} = -\\frac x a \\map \\tan {\\frac \\pi 4 - \\frac {a x} 2} + \\frac 2 {a^2} \\ln \\size {\\map \\sin {\\frac \\pi 4 + \\frac {a x} 2} } + C$"}
+{"_id": "9667", "title": "Reciprocal of One Minus Sine", "text": ":$\\dfrac 1 {1 - \\sin x} = \\dfrac 1 2 \\map {\\sec^2} {\\dfrac \\pi 4 + \\dfrac x 2}$"}
+{"_id": "9668", "title": "Reciprocal of One Plus Sine", "text": ":$\\dfrac 1 {1 + \\sin x} = \\dfrac 1 2 \\map {\\sec^2} {\\dfrac \\pi 4 - \\dfrac x 2}$"}
+{"_id": "9671", "title": "Power Series Expansion for Cosecant Function", "text": "The cosecant function has a Laurent series expansion: {{begin-eqn}} {{eqn | l = \\csc x       | r = \\sum_{n \\mathop = 0}^\\infty \\dfrac {\\paren {-1}^{n - 1} 2 \\paren {2^{2 n - 1} - 1} B_{2 n} \\,  x^{2 n - 1} } {\\paren {2 n}!}       | c =  }} {{eqn | r = \\frac 1 x + \\frac x 6 + \\frac {7 x^3} {360} + \\frac {31 x^5} {15 \\, 120} + \\cdots       | c =  }} {{end-eqn}} where $B_n$ denotes the Bernoulli numbers. This converges for $0 < \\size x < \\pi$."}
+{"_id": "9672", "title": "Periodic Function plus Constant", "text": "Let $f: \\mathbb F \\to \\mathbb F$ be a function, where $\\mathbb F \\in \\left\\{{\\R, \\C}\\right\\}$. Let $k \\in \\mathbb F$ be constant. Then $f$ is periodic with period $L$ {{iff}} $f + k$ is periodic with period $L$."}
+{"_id": "9674", "title": "Sum of Two Odd Powers/Examples/Sum of Two Cubes", "text": ":$x^3 + y^3 = \\paren {x + y} \\paren {x^2 - x y + y^2}$"}
+{"_id": "9675", "title": "Primitive of Periodic Function", "text": "Let $f: \\R \\to \\R$ be a real function. Let $F$ be a primitive of $f$ that is bounded on all of $\\R$. Let $f$ be periodic with period $L$. Then $F$ is also periodic with period $L$."}
+{"_id": "9676", "title": "Primitive of Reciprocal of x squared by x cubed plus a cubed/Lemma", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x^2 \\left({x^3 + a^3}\\right)} = \\frac {-1} {a^3 x} - \\frac 1 {a^3} \\int \\frac {x \\ \\mathrm d x} {\\left({x^3 + a^3}\\right)}$"}
+{"_id": "9677", "title": "Sum of Two Fourth Powers", "text": ":$x^4 + y^4 = \\paren {x^2 + \\sqrt 2 x y + y^2} \\paren {x^2 - \\sqrt 2 x y + y^2}$"}
+{"_id": "9680", "title": "One Plus Tangent Half Angle over One Minus Tangent Half Angle", "text": ":$\\dfrac {1 + \\tan \\frac x 2} {1 - \\tan \\frac x 2} = \\sec x + \\tan x$"}
+{"_id": "9683", "title": "Primitive of Power of Cosecant of a x", "text": ":$\\displaystyle \\int \\csc^n a x \\ \\mathrm d x = \\frac{-\\csc^{n - 2} a x \\cot a x} {a \\left({n - 1}\\right)} + \\frac {n - 2} {n - 1} \\int \\csc^{n - 2} a x \\ \\mathrm d x$"}
+{"_id": "9684", "title": "Primitive of Cosecant of a x/Tangent Form", "text": ":$\\ds \\int \\csc a x \\rd x = \\frac 1 a \\ln \\size {\\tan \\frac {a x} 2} + C$ where $\\tan \\dfrac {a x} 2 \\ne 0$."}
+{"_id": "9687", "title": "Primitive of Reciprocal of Square of 1 plus Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({1 + \\sin a x}\\right)^2} = \\frac {-1} {2a} \\tan \\left({\\frac \\pi 4 - \\frac {a x} 2}\\right) - \\frac 1 {6 a} \\tan^3 \\left({\\frac \\pi 4 - \\frac {a x} 2}\\right) + C$"}
+{"_id": "9688", "title": "Primitive of Reciprocal of p plus q by Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {p + q \\sin a x} = \\begin{cases}  \\displaystyle \\frac 2 {a \\sqrt {p^2 - q^2} } \\map \\arctan {\\frac {p \\tan \\dfrac {a x} 2 + q} {\\sqrt {p^2 - q^2} } } + C & : q^2 - p^2 < 0 \\\\ \\displaystyle \\frac 1 {a \\sqrt {q^2 - p^2} } \\ln \\size {\\frac {p \\tan \\dfrac {a x} 2 + q - \\sqrt {p^2 - q^2} } {p \\tan \\dfrac {a x} 2 + q + \\sqrt {p^2 - q^2} } } + C & : q^2 - p^2 > 0 \\\\ \\end{cases}$"}
+{"_id": "9689", "title": "Primitive of Reciprocal of square of p plus q by Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\paren {p + q \\sin a x}^2} = \\frac {q \\cos a x} {a \\paren {p^2 - q^2} \\paren {p + q \\sin a x} } + \\frac p {p^2 - q^2} \\int \\frac {\\d x} {p + q \\sin a x}$"}
+{"_id": "9690", "title": "Primitive of Reciprocal of p squared plus square of q by Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {p^2 + q^2 \\sin^2 a x} = \\frac 1 {a p \\sqrt {p^2 + q^2} } \\arctan \\frac {\\sqrt {p^2 + q^2} \\tan a x} p + C$"}
+{"_id": "9693", "title": "Primitive of Sine of a x over Power of x", "text": ":$\\displaystyle \\int \\frac {\\sin a x} {x^n} \\rd x = \\frac {-\\sin a x} {\\paren {n - 1} x^{n - 1} } + \\frac a {n - 1} \\int \\frac {\\cos a x} {x^{n - 1} } \\rd x$"}
+{"_id": "9694", "title": "Primitive of Reciprocal of Power of Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\sin^n a x} = \\frac {- \\cos a x} {a \\left({n - 1}\\right) \\sin^{n - 1} a x} + \\frac {n - 2} {n - 1} \\int \\frac {\\mathrm d x} {\\sin^{n - 2} a x}$"}
+{"_id": "9695", "title": "Primitive of x over Power of Sine of a x", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {\\sin^n a x} = \\frac {-x \\cos a x} {a \\paren {n - 1} \\sin^{n - 1} a x} - \\frac 1 {a^2 \\paren {n - 1} \\paren {n - 2} \\sin^{n - 2} a x} + \\frac {n - 2} {n - 1} \\int \\frac {x \\rd x} {\\sin^{n - 2} a x}$"}
+{"_id": "9696", "title": "Primitive of x by Cosine of a x", "text": ":$\\displaystyle \\int x \\cos a x \\rd x = \\frac {\\cos a x} {a^2} + \\frac {x \\sin a x} a + C$"}
+{"_id": "9697", "title": "Primitive of x squared by Cosine of a x", "text": ":$\\displaystyle \\int x^2 \\cos a x \\rd x = \\frac {2 x \\cos a x} {a^2} + \\paren {\\frac {x^2} a - \\frac 2 {a^3} } \\sin a x + C$"}
+{"_id": "9698", "title": "Primitive of x cubed by Cosine of a x", "text": ":$\\displaystyle \\int x^3 \\map \\cos {a x} \\rd x = \\paren {\\frac {3 x^2} {a^2} - \\frac 6 {a^4} } \\cos a x + \\paren {\\frac {x^3} a - \\frac {6 x} {a^3} } \\sin a x + C$"}
+{"_id": "9702", "title": "Primitive of Reciprocal of Cosine of a x/Logarithm of Tangent Form", "text": ":$\\ds \\int \\frac {\\d x} {\\cos a x} = \\frac 1 a \\ln \\size {\\map \\tan {\\frac \\pi 4 + \\frac {a x} 2} } + C$"}
+{"_id": "9704", "title": "Primitive of Square of Cosine of a x", "text": ":$\\displaystyle \\int \\cos^2 a x \\, \\mathrm d x = \\frac x 2 + \\frac {\\sin 2 a x} {4 a} + C$"}
+{"_id": "9705", "title": "Primitive of x by Square of Cosine of a x", "text": ":$\\displaystyle \\int x \\cos^2 a x \\ \\mathrm d x = \\frac {x^2} 4 + \\frac {x \\sin 2 a x} {4 a} + \\frac {\\cos 2 a x} {8 a^2} + C$"}
+{"_id": "9707", "title": "Primitive of Fourth Power of Cosine of a x", "text": ":$\\displaystyle \\int \\cos^4 a x \\ \\mathrm d x = \\frac {3 x} 8 + \\frac {\\sin 2 a x} {4 a} + \\frac {\\sin 4 a x} {32 a} + C$"}
+{"_id": "9708", "title": "Primitive of Reciprocal of Square of Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\cos^2 a x} = \\frac {\\tan a x} a + C$"}
+{"_id": "9709", "title": "Primitive of Square of Secant of a x", "text": ":$\\displaystyle \\int \\sec^2 a x \\ \\mathrm d x = \\frac {\\tan a x} a + C$"}
+{"_id": "9710", "title": "Primitive of Secant of a x/Tangent Form", "text": ":$\\ds \\int \\sec a x \\rd x = \\frac 1 a \\ln \\size {\\map \\tan {\\frac \\pi 4 + \\frac {a x} 2} } + C$ where $\\map \\tan {\\dfrac \\pi 4 + \\dfrac {a x} 2} \\ne 0$."}
+{"_id": "9711", "title": "Primitive of Secant of a x/Secant plus Tangent Form", "text": ":$\\ds \\int \\sec a x \\rd x = \\frac 1 a \\ln \\size {\\sec a x + \\tan a x} + C$ where $\\sec a x + \\tan a x \\ne 0$."}
+{"_id": "9713", "title": "Primitive of Power of Secant of a x", "text": ":$\\displaystyle \\int \\sec^n a x \\rd x = \\frac {\\sec^{n - 2} a x \\tan a x} {a \\paren {n - 1} } + \\frac {n - 2} {n - 1} \\int \\sec^{n - 2} a x \\rd x$"}
+{"_id": "9714", "title": "Primitive of Reciprocal of Cube of Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\cos^3 a x} = \\frac {\\sin a x} {2 a \\cos^2 a x} + \\frac 1 {2 a} \\ln \\size {\\map \\tan {\\frac \\pi 4 + \\frac {a x} 2} } + C$"}
+{"_id": "9715", "title": "Primitive of Cosine of a x by Cosine of p x", "text": ":$\\displaystyle \\int \\cos a x \\cos p x \\rd x = \\frac {\\map \\sin {\\paren {a - p} x} } {2 \\paren {a - p} } + \\frac {\\map \\sin {\\paren {a + p} x} } {2 \\paren {a + p} } + C$"}
+{"_id": "9716", "title": "Primitive of Reciprocal of 1 minus Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {1 - \\cos a x} = \\frac {-1} a \\cot \\frac {a x} 2 + C$"}
+{"_id": "9717", "title": "Primitive of x over 1 minus Cosine of a x", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {1 - \\cos a x} = \\frac {-x} a \\cot \\frac {a x} 2 + \\frac 2 {a^2} \\ln \\size {\\sin \\frac {a x} 2} + C$"}
+{"_id": "9718", "title": "Primitive of Reciprocal of 1 plus Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {1 + \\cos a x} = \\frac 1 a \\tan \\frac {a x} 2 + C$"}
+{"_id": "9722", "title": "Primitive of Reciprocal of p plus q by Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {p + q \\cos a x} = \\begin{cases}  \\displaystyle \\frac 2 {a \\sqrt {p^2 - q^2} } \\arctan \\left({\\sqrt {\\frac {p - q} {p + q} } \\tan \\dfrac {a x} 2}\\right) + C & : p^2 > q^2 \\\\ \\displaystyle \\frac 1 {a \\sqrt {q^2 - p^2} } \\ln \\left\\vert{\\frac {\\tan \\dfrac {a x} 2 + \\sqrt {\\dfrac {q + p} {q - p} } } {\\tan \\dfrac {a x} 2 - \\sqrt {\\dfrac {q + p} {q - p} } } }\\right\\vert + C & : p^2 < q^2 \\\\ \\end{cases}$"}
+{"_id": "9726", "title": "Primitive of Power of x by Cosine of a x", "text": ":$\\displaystyle \\int x^m \\cos a x \\rd x = \\frac {x^m \\sin a x} a + \\frac {m x^{m - 1} \\cos a x} {a^2} - \\frac {m \\paren {m - 1} } {a^2} \\int x^{m - 2} \\cos a x \\rd x$"}
+{"_id": "9727", "title": "Primitive of Reciprocal of Power of Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\cos^n a x} = \\frac {\\sin a x} {a \\left({n - 1}\\right) \\cos^{n - 1} a x} + \\frac {n - 2} {n - 1} \\int \\frac {\\mathrm d x} {\\cos^{n - 2} a x}$"}
+{"_id": "9728", "title": "Primitive of x over Power of Cosine of a x", "text": ":$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\cos^n a x} = \\frac {x \\sin a x} {a \\left({n - 1}\\right) \\cos^{n - 1} a x} - \\frac 1 {a^2 \\left({n - 1}\\right) \\left({n - 2}\\right) \\cos^{n - 2} a x} + \\frac {n - 2} {n - 1} \\int \\frac {\\mathrm d x} {\\cos^{n - 2} a x} + C$"}
+{"_id": "9729", "title": "Primitive of Sine of a x by Cosine of a x", "text": ":$\\displaystyle \\int \\sin a x \\cos a x \\rd x = \\frac {\\sin^2 a x} {2 a} + C$"}
+{"_id": "9730", "title": "Primitive of Sine of p x by Cosine of q x", "text": ":$\\displaystyle \\int \\sin p x \\cos q x \\rd x = \\frac {-\\cos \\left({p - q}\\right) x} {2 \\left({p - q}\\right)} - \\frac {\\cos \\left({p + q}\\right) x} {2 \\left({p + q}\\right)} + C$ for $p, q \\in \\R: p \\ne q$"}
+{"_id": "9731", "title": "Primitive of Power of Sine of a x by Cosine of a x", "text": ":$\\displaystyle \\int \\sin^n a x \\cos a x \\rd x = \\frac {\\sin^{n + 1} a x} {\\paren {n + 1} a} + C$"}
+{"_id": "9732", "title": "Primitive of Power of Cosine of a x by Sine of a x", "text": ":$\\displaystyle \\int \\cos^n a x \\sin a x \\rd x = \\frac {-\\cos^{n + 1} a x} {\\paren {n + 1} a} + C$"}
+{"_id": "9733", "title": "Primitive of Sine of a x squared by Cosine of a x squared", "text": ":$\\displaystyle \\int \\sin^2 a x \\cos^2 a x \\rd x = \\frac x 8 - \\frac {\\sin 4 a x} {32 a} + C$"}
+{"_id": "9734", "title": "Primitive of Reciprocal of Sine of a x by Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\sin a x \\cos a x} = \\frac 1 a \\ln \\left\\vert{\\tan a x}\\right\\vert + C$"}
+{"_id": "9735", "title": "Primitive of Square of Secant of a x over Tangent of a x", "text": ":$\\displaystyle \\int \\frac {\\sec^2 a x \\ \\mathrm d x} {\\tan a x} = \\frac 1 a \\ln \\left\\vert{\\tan a x}\\right\\vert + C$"}
+{"_id": "9736", "title": "Primitive of Reciprocal of Square of Sine of a x by Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\sin^2 a x \\cos a x} = \\frac 1 a \\ln \\size {\\map \\tan {\\frac \\pi 4 + \\frac {a x} 2} } - \\frac 1 {a \\sin a x} + C$"}
+{"_id": "9737", "title": "Primitive of Power of Cosecant of a x by Cotangent of a x", "text": ":$\\displaystyle \\int \\csc^n a x \\cot a x \\ \\mathrm d x = \\frac {-\\csc^n a x} {n a} + C$"}
+{"_id": "9740", "title": "Primitive of Square of Sine of a x over Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\sin^2 a x \\rd x} {\\cos a x} = \\frac {-\\sin a x} a + \\frac 1 a \\ln \\size {\\map \\tan {\\frac \\pi 4 + \\frac {a x} 2} } + C$"}
+{"_id": "9741", "title": "Primitive of Square of Cosine of a x over Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\cos^2 a x \\ \\mathrm d x} {\\sin a x} = \\frac {\\cos a x} a + \\frac 1 a \\ln \\left\\vert{\\tan \\frac {a x} 2}\\right\\vert + C$"}
+{"_id": "9742", "title": "Primitive of Reciprocal of Cosine of a x by 1 plus Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\cos a x \\left({1 + \\sin a x}\\right)} = \\frac {-1} {2 a \\left({1 + \\sin a x}\\right)} + \\frac 1 {2 a} \\ln \\left\\vert{\\tan \\left({\\frac {a x} 2 + \\frac \\pi 4}\\right)}\\right\\vert + C$"}
+{"_id": "9743", "title": "Primitive of Reciprocal of Cosine of a x by 1 minus Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\cos a x \\paren {1 - \\sin a x} } = \\frac 1 {2 a \\paren {1 - \\sin a x} } + \\frac 1 {2 a} \\ln \\size {\\map \\tan {\\frac {a x} 2 + \\frac \\pi 4} } + C$"}
+{"_id": "9744", "title": "Primitive of Reciprocal of Sine of a x by 1 plus Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\sin a x \\paren {1 + \\cos a x} } = \\frac 1 {2 a \\paren {1 + \\cos a x} } + \\frac 1 {2 a} \\ln \\size {\\tan \\frac {a x} 2} + C$"}
+{"_id": "9746", "title": "Primitive of Reciprocal of Sine of a x plus Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\sin a x + \\cos a x} = \\frac 1 {a \\sqrt 2} \\ln \\size {\\map \\tan {\\frac {a x} 2 + \\frac \\pi 8} } + C$"}
+{"_id": "9747", "title": "Primitive of Reciprocal of Sine of a x minus Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\sin a x - \\cos a x} = \\frac 1 {a \\sqrt 2} \\ln \\left\\vert{\\tan \\left({\\frac {a x} 2 - \\frac \\pi 8}\\right)}\\right\\vert + C$"}
+{"_id": "9748", "title": "Primitive of Sine of a x over Sine of a x plus Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\sin a x \\ \\mathrm d x} {\\sin a x + \\cos a x} = \\frac x 2 - \\frac 1 {2 a} \\ln \\left\\vert{\\sin a x + \\cos a x}\\right\\vert + C$"}
+{"_id": "9749", "title": "Primitive of Sine of a x over Sine of a x minus Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\sin a x \\rd x} {\\sin a x - \\cos a x} = \\frac x 2 + \\frac 1 {2 a} \\ln \\size {\\sin a x - \\cos a x} + C$"}
+{"_id": "9750", "title": "Primitive of Cosine of a x over Sine of a x plus Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\cos a x \\rd x} {\\sin a x + \\cos a x} = \\frac x 2 + \\frac 1 {2 a} \\ln \\size {\\sin a x + \\cos a x} + C$"}
+{"_id": "9751", "title": "Primitive of Cosine of a x over Sine of a x minus Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\cos a x \\ \\mathrm d x} {\\sin a x - \\cos a x} = \\frac {-x} 2 + \\frac 1 {2 a} \\ln \\left\\vert{\\sin a x - \\cos a x}\\right\\vert + C$"}
+{"_id": "9752", "title": "Primitive of Sine of a x over p plus q of Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\sin a x \\rd x} {p + q \\cos a x} = \\frac {-1} {a q} \\ln \\size {p + q \\cos a x} + C$"}
+{"_id": "9754", "title": "Primitive of Sine of a x over Power of p plus q of Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\sin a x \\ \\mathrm d x} {\\left({p + q \\cos a x}\\right)^n} = \\frac 1 {a q \\left({n - 1}\\right) \\left({p + q \\cos a x}\\right)^{n - 1} } + C$"}
+{"_id": "9756", "title": "Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {p \\sin a x + q \\cos a x} = \\frac 1 {a \\sqrt {p^2 + q^2} } \\ln \\tan \\left\\vert{\\frac {a x + \\arctan \\dfrac q p} 2}\\right\\vert + C$"}
+{"_id": "9757", "title": "Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x plus r", "text": ":$\\ds \\int \\frac {\\d x} {p \\sin a x + q \\cos a x + r} = \\begin{cases} \\ds \\frac 2 {a \\sqrt {r^2 - p^2 - q^2} } \\map \\arctan {\\frac {p + \\paren {r - q} \\tan \\dfrac {a x} 2} {\\sqrt {r^2 - p^2 - q^2} } } + C & : p^2 + q^2 < r^2 \\\\ \\ds \\frac 1 {a \\sqrt {p^2 + q^2 - r^2} } \\ln \\size {\\frac {p - \\sqrt {p^2 + q^2 - r^2} + \\paren {r - q} \\tan \\dfrac {a x} 2} {p + \\sqrt {p^2 + q^2 - r^2} + \\paren {r - q} \\tan \\dfrac {a x} 2} } + C & : p^2 + q^2 > r^2 \\end{cases}$"}
+{"_id": "9758", "title": "Primitive of Reciprocal of p by Sine of a x plus q by 1 plus Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {p \\sin a x + q \\left({1 + \\cos a x}\\right)} = \\frac 1 {a p} \\ln \\left\\vert{q + p \\tan \\frac {a x} 2}\\right\\vert + C$"}
+{"_id": "9761", "title": "Primitive of Reciprocal of p squared by square of Sine of a x plus q squared by square of Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {p^2 \\sin^2 a x + q^2 \\cos^2 a x} = \\frac 1 {a p q} \\map \\arctan {\\frac {p \\tan a x} q} + C$"}
+{"_id": "9762", "title": "Primitive of Reciprocal of p squared by square of Sine of a x minus q squared by square of Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {p^2 \\sin^2 a x - q^2 \\cos^2 a x} = \\frac 1 {2 a p q} \\ln \\size {\\frac {p \\tan a x - q} {p \\tan a x + q} } + C$"}
+{"_id": "9763", "title": "Primitive of Power of Sine of a x by Power of Cosine of a x/Reduction of Power of Sine", "text": ":$\\displaystyle \\int \\sin^m a x \\cos^n a x \\rd x = \\frac {-\\sin^{m - 1} a x \\cos^{n + 1} a x} {a \\paren {m + n} } + \\frac {m - 1} {m + n} \\int \\sin^{m - 2} a x \\cos^n a x \\rd x + C$"}
+{"_id": "9764", "title": "Primitive of Power of Sine of a x by Power of Cosine of a x/Reduction of Power of Cosine", "text": ":$\\displaystyle \\int \\sin^m a x \\cos^n a x \\rd x = \\frac {\\sin^{m + 1} a x \\cos^{n - 1} a x} {a \\paren {m + n} } + \\frac {n - 1} {m + n} \\int \\sin^m a x \\cos^{n - 2} a x \\rd x + C$"}
+{"_id": "9766", "title": "Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Both Powers", "text": ":$\\displaystyle \\int \\frac {\\sin^m a x} {\\cos^n a x} \\rd x = \\frac {\\sin^{m - 1} a x} {a \\paren {n - 1} \\cos^{n - 1} a x} - \\frac {m - 1} {n - 1} \\int \\frac {\\sin^{m - 2} a x} {\\cos^{n - 2} a x} \\rd x + C$"}
+{"_id": "9767", "title": "Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Power of Cosine", "text": ":$\\displaystyle \\int \\frac {\\sin^m a x} {\\cos^n a x} \\ \\mathrm d x = \\frac {\\sin^{m + 1} a x} {a \\left({n - 1}\\right) \\cos^{n - 1} a x} - \\frac {m - n + 2} {n - 1} \\int \\frac {\\sin^m a x} {\\cos^{n - 2} a x} \\ \\mathrm d x + C$"}
+{"_id": "9770", "title": "Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Both Powers", "text": ":$\\displaystyle \\int \\frac {\\cos^m a x} {\\sin^n a x} \\rd x = \\frac {-\\cos^{m - 1} a x} {a \\paren {n - 1} \\sin^{n - 1} a x} - \\frac {m - 1} {n - 1} \\int \\frac {\\cos^{m - 2} a x} {\\sin^{n - 2} a x} \\rd x + C$"}
+{"_id": "9771", "title": "Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Power of Sine", "text": ":$\\displaystyle \\int \\frac {\\cos^m a x} {\\sin^n a x} \\rd x = \\frac {-\\cos^{m + 1} a x} {a \\paren {n - 1} \\sin^{n - 1} a x} - \\frac {m - n + 2} {n - 1} \\int \\frac {\\cos^m a x} {\\sin^{n - 2} a x} \\rd x + C$"}
+{"_id": "9774", "title": "Primitive of Reciprocal of Power of Cosine of a x by Power of Sine of a x/Reduction of Power of Cosine", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\sin^m a x \\cos^n a x} = \\frac 1 {a \\left({n - 1}\\right) \\sin^{m - 1} a x \\cos^{n - 1} a x} + \\frac {m + n - 2} {n - 1} \\int \\frac {\\mathrm d x} {\\sin^m a x \\cos^{n - 2} a x}$"}
+{"_id": "9777", "title": "Primitive of Tangent of a x/Cosine Form", "text": ":$\\ds \\int \\tan a x \\rd x = \\frac {-\\ln \\size {\\cos a x} } a + C$"}
+{"_id": "9780", "title": "Primitive of Square of Tangent of a x", "text": ":$\\displaystyle \\int \\tan^2 a x \\ \\mathrm d x = \\frac {\\tan a x} a - x + C$"}
+{"_id": "9781", "title": "Primitive of Cube of Tangent of a x", "text": ":$\\displaystyle \\int \\tan^3 a x \\rd x = \\frac {\\tan^2 a x} {2 a} + \\frac 1 a \\ln \\size {\\cos a x} + C$"}
+{"_id": "9782", "title": "Primitive of Power of Tangent of a x by Square of Secant of a x", "text": ":$\\displaystyle \\int \\tan^n a x \\sec^2 a x \\rd x = \\frac {\\tan^{n + 1} a x} {\\paren {n + 1} a} + C$"}
+{"_id": "9783", "title": "Primitive of Cotangent of a x", "text": ":$\\displaystyle \\int \\cot a x \\rd x = \\frac {\\ln \\size {\\sin a x} } a + C$"}
+{"_id": "9785", "title": "Primitive of x by Tangent of a x", "text": ":$\\displaystyle \\int x \\tan a x \\rd x = \\frac 1 {a^2} \\paren {\\frac {\\paren {a x} ^ 3} 3 + \\frac {\\paren {a x}^5} {15} + \\frac {2 \\paren {a x}^7} {105} + \\cdots + \\frac {\\paren {-1}^{n - 1} 2^{2 n} \\paren {2^{2 n} - 1} B_{2 n} \\paren {a x}^{2 n + 1} } {\\paren {2 n + 1}!} + \\cdots} + C$ where $B_{2 n}$ denotes the $2 n$th Bernoulli number."}
+{"_id": "9786", "title": "Primitive of Tangent of a x over x", "text": ":$\\displaystyle \\int \\frac {\\tan a x} x \\ \\mathrm d x = a x + \\frac {\\paren {a x}^3} 9 + \\frac {2 \\paren {a x}^5} {75} + \\cdots + \\frac {\\paren {-1}^{n - 1} 2^{2 n} \\paren {2^{2 n} - 1} B_{2 n} \\paren {a x}^{2 n - 1} } {\\paren {2 n - 1} \\paren {2 n}!} + \\cdots + C$ where $B_n$ denotes the $n$th Bernoulli number."}
+{"_id": "9788", "title": "Primitive of Reciprocal of p plus q by Tangent of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {p + q \\tan a x} = \\frac {p x} {p^2 + q^2} + \\frac q {a \\paren {p^2 + q^2} } \\ln \\size {q \\sin a x + p \\cos a x} + C$"}
+{"_id": "9790", "title": "Primitive of Square of Cotangent of a x", "text": ":$\\displaystyle \\int \\cot^2 a x \\rd x = \\frac {-\\cot a x} a - x + C$"}
+{"_id": "9791", "title": "Primitive of Cube of Cotangent of a x", "text": ":$\\displaystyle \\int \\cot^3 a x \\rd x = \\frac {-\\cot^2 a x} {2 a} - \\frac 1 a \\ln \\size {\\sin a x} + C$"}
+{"_id": "9792", "title": "Primitive of Power of Cotangent of a x by Square of Cosecant of a x", "text": ":$\\displaystyle \\int \\cot^n a x \\csc^2 a x \\ \\mathrm d x = \\frac {-\\cot^{n + 1} a x} {\\left({n + 1}\\right) a} + C$"}
+{"_id": "9793", "title": "Primitive of Square of Cosecant of a x over Cotangent of a x", "text": ":$\\displaystyle \\int \\frac {\\csc^2 a x \\ \\mathrm d x} {\\cot a x} = \\frac {-\\ln \\left\\vert{\\cot a x}\\right\\vert} a + C$"}
+{"_id": "9796", "title": "Primitive of Cotangent of a x over x", "text": ":$\\displaystyle \\int \\frac {\\cot a x} x \\ \\mathrm d x = \\frac {-1} a x - \\frac {a x} 3 - \\frac {\\paren {a x}^3} {135} - \\cdots - \\frac {\\paren {-1}^{n - 1} 2^{2 n} B_{2 n} \\paren {a x}^{2 n - 1} } {\\paren {2 n - 1} \\paren {2 n}!} - \\cdots + C$ where $B_n$ denotes the $n$th Bernoulli number."}
+{"_id": "9797", "title": "Primitive of x by Square of Cotangent of a x", "text": ":$\\displaystyle \\int x \\cot^2 a x \\ \\mathrm d x = \\frac {-x \\cot a x} a + \\frac 1 {a^2} \\ln \\left\\vert{\\sin a x}\\right\\vert - \\frac {x^2} 2 + C$"}
+{"_id": "9799", "title": "Primitive of Power of Cotangent of a x", "text": ":$\\displaystyle \\int \\cot^n a x \\ \\mathrm d x = \\frac {-\\cot^{n - 1} a x} {\\left({n - 1}\\right) a} - \\int \\cot^{n - 2} a x \\ \\mathrm d x + C$"}
+{"_id": "9800", "title": "Primitive of Cube of Secant of a x", "text": ":$\\displaystyle \\int \\sec^3 a x \\rd x = \\frac 1 {2 a} \\paren {\\sec a x \\tan a x + \\ln \\size {\\sec a x + \\tan a x} } + C$"}
+{"_id": "9802", "title": "Primitive of x by Secant of a x", "text": "{{begin-eqn}} {{eqn | l = \\int x \\sec a x \\rd x       | r = \\frac 1 {a^2} \\paren {\\frac {\\paren {a x}^2} 2 - \\frac {\\paren {a x}^4} 8 + \\frac {5 \\paren {a x}^6} {144} - \\cdots + \\frac {\\paren {-1}^n E_{2 n} \\paren {a x}^{2 n + 2} } {\\paren {2 n + 2} \\paren {2 n}!} + \\cdots} + C       | c =  }} {{eqn | r = \\frac 1 {a^2} \\sum_{n \\mathop = 0}^\\infty \\frac {\\paren {-1}^n E_{2 n} \\paren {a x}^{2 n + 2} } {\\paren {2 n + 2} \\paren {2 n}!} + C       | c =  }} {{end-eqn}} where $E_{2 n}$ is the $2 n$th Euler number."}
+{"_id": "9803", "title": "Primitive of Secant of a x over x", "text": ":$\\displaystyle \\int \\frac {\\sec a x} x \\ \\mathrm d x = \\ln \\size x + \\frac {\\paren {a x}^2} 4 + \\frac {5 \\paren {a x}^4} {96} + \\frac {61 \\paren {a x}^6} {4320} + \\cdots + \\frac {\\paren {-1}^n E_n \\paren {a x}^{2 n} } {\\paren {2 n} \\paren {2 n}!} + \\cdots + C$ where $E_n$ is the $n$th Euler number."}
+{"_id": "9804", "title": "Primitive of x by Square of Secant of a x", "text": ":$\\displaystyle \\int x \\sec^2 a x \\ \\mathrm d x = \\frac {x \\tan a x} a + \\frac  1 {a^2} \\ln \\left\\vert{\\cos a x}\\right\\vert + C$"}
+{"_id": "9805", "title": "Primitive of Reciprocal of q plus p by Secant of a x", "text": ":$\\ds \\int \\frac {\\d x} {q + p \\sec a x} = \\frac x q - \\frac p q \\int \\frac {\\d x} {p + q \\cos a x} + C$"}
+{"_id": "9808", "title": "Primitive of x by Cosecant of a x", "text": ":$\\displaystyle \\int x \\csc a x \\rd x = \\frac 1 {a^2} \\paren {a x + \\frac {\\paren {a x}^3} {18} + \\frac {7 \\paren {a x}^5} {1800} + \\cdots + \\frac {\\paren {-1}^{n - 1} 2 \\paren {2^{2 n - 1} - 1} B_n \\paren {a x}^{2 n + 1} } {\\paren {2 n + 1}!} + \\cdots} + C$ where $B_{2 n}$ is the $2 n$th Bernoulli number."}
+{"_id": "9809", "title": "Primitive of Cosecant of a x over x", "text": ":$\\displaystyle \\int \\frac {\\csc a x} x \\rd x = \\frac {-1} {a x} + \\frac {a x} 6 + \\frac {7 \\paren {a x}^3} {1080} + \\cdots + \\frac {\\paren {-1}^{n - 1} 2 \\paren {2^{2 n - 1} - 1} B_{2 n} \\paren {a x}^{2 n - 1} } {\\paren {2 n - 1} \\paren {2 n}!} + \\cdots + C$ where $B_n$ is the $n$th Bernoulli number."}
+{"_id": "9810", "title": "Primitive of x by Square of Cosecant of a x", "text": ":$\\displaystyle \\int x \\csc^2 a x \\ \\mathrm d x = \\frac {-x \\cot a x} a + \\frac  1 {a^2} \\ln \\left\\vert{\\sin a x}\\right\\vert + C$"}
+{"_id": "9811", "title": "Primitive of Reciprocal of q plus p by Cosecant of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {q + p \\csc a x} = \\frac x q - \\frac p q \\int \\frac {\\mathrm d x} {p + q \\sin a x} + C$"}
+{"_id": "9812", "title": "Primitive of Arcsine of x over a", "text": ":$\\displaystyle \\int \\arcsin \\frac x a \\rd x = x \\arcsin \\frac x a + \\sqrt {a^2 - x^2} + C$"}
+{"_id": "9818", "title": "Primitive of x squared by Arcsine of x over a", "text": ":$\\displaystyle \\int x^2 \\arcsin \\frac x a \\rd x = \\frac {x^3} 3 \\arcsin \\frac x a + \\frac {\\paren {x^2 + 2 a^2} \\sqrt {a^2 - x^2} } 9 + C$"}
+{"_id": "9821", "title": "Primitive of Arccosine of x over a", "text": ":$\\displaystyle \\int \\arccos \\frac x a \\rd x = x \\arccos \\frac x a - \\sqrt {a^2 - x^2} + C$"}
+{"_id": "9825", "title": "Primitive of x by Arccosine of x over a/Proof 1", "text": ":$\\displaystyle \\int x \\arccos \\frac x a \\ \\mathrm d x = \\left({\\frac {x^2} 2 - \\frac {a^2} 4}\\right) \\arccos \\frac x a - \\frac {x \\sqrt {a^2 - x^2} } 4 + C$"}
+{"_id": "9826", "title": "Primitive of x by Arccosine of x over a/Proof 2", "text": ":$\\displaystyle \\int x \\arccos \\frac x a \\rd x = \\paren {\\frac {x^2} 2 - \\frac {a^2} 4} \\arccos \\frac x a - \\frac {x \\sqrt {a^2 - x^2} } 4 + C$"}
+{"_id": "9827", "title": "Primitive of x squared by Arccosine of x over a", "text": ":$\\displaystyle \\int x^2 \\arccos \\frac x a \\rd x = \\frac {x^3} 3 \\arccos \\frac x a - \\frac {\\paren {x^2 + 2 a^2} \\sqrt {a^2 - x^2} } 9 + C$"}
+{"_id": "9829", "title": "Primitive of Arccosine of x over a over x squared", "text": ":$\\displaystyle \\int \\frac {\\arccos \\frac x a \\ \\mathrm d x} {x^2} = \\frac {-\\arccos \\frac x a} x + \\frac 1 a \\ln \\left({\\frac {a + \\sqrt {a^2 - x^2} } x}\\right) + C$"}
+{"_id": "9830", "title": "Primitive of Square of Arccosine of x over a", "text": ":$\\displaystyle \\int \\paren {\\arccos \\frac x a}^2 \\rd x = x \\paren {\\arccos \\frac x a}^2 - 2 x - 2 \\sqrt {a^2 - x^2} \\arccos \\frac x a + C$"}
+{"_id": "9831", "title": "Primitive of Arctangent of x over a", "text": ":$\\displaystyle \\int \\arctan \\frac x a \\rd x = x \\arctan \\frac x a - \\frac a 2 \\ln \\paren {x^2 + a^2} + C$"}
+{"_id": "9834", "title": "Primitive of x by Arctangent of x over a", "text": ":$\\displaystyle \\int x \\arctan \\frac x a \\ \\mathrm d x = \\frac {x^2 + a^2} 2 \\arctan \\frac x a - \\frac {a x} 2 + C$"}
+{"_id": "9835", "title": "Primitive of x by Arccotangent of x over a", "text": ":$\\displaystyle \\int x \\arccot \\frac x a \\rd x = \\frac {x^2 + a^2} 2 \\arccot \\frac x a + \\frac {a x} 2 + C$"}
+{"_id": "9836", "title": "Primitive of x squared by Arctangent of x over a", "text": ":$\\displaystyle \\int x^2 \\arctan \\frac x a \\ \\mathrm d x = \\frac {x^3} 3 \\arctan \\frac x a - \\frac {a x^2} 6 + \\frac {a^3} 6 \\ln \\left({x^2 + a^2}\\right) + C$"}
+{"_id": "9838", "title": "Primitive of Arctangent of x over a over x", "text": "{{begin-eqn}} {{eqn \t| l = \\int \\frac 1 x \\arctan \\paren {\\frac x a} \\rd x       \t| r = \\sum_{k \\mathop = 0}^\\infty \\paren {-1}^k \\frac {x^{2 k + 1} } {\\paren {2 k + 1}^2 a^{2 k + 1} }  }} {{eqn \t| r = \\frac x a - \\frac {x^3} {3^2 a^3} + \\frac {x^5} {5^2 a^5} - \\frac {x^7} {7^2 a^7} + \\cdots + C }} {{end-eqn}}"}
+{"_id": "9841", "title": "Primitive of Arccotangent of x over a over x squared", "text": ":$\\displaystyle \\int \\frac {\\arccot \\frac x a \\rd x} {x^2} = \\frac {-\\arccot \\frac x a} x + \\frac 1 {2 a} \\map \\ln {\\frac {x^2 + a^2} {x^2} } + C$"}
+{"_id": "9842", "title": "Primitive of Arccotangent of x over a", "text": ":$\\displaystyle \\int \\operatorname{arccot} \\frac x a \\ \\mathrm d x = x \\operatorname{arccot} \\frac x a + \\frac a 2 \\ln \\left({x^2 + a^2}\\right) + C$"}
+{"_id": "9843", "title": "Primitive of Arccotangent of x over a/Proof 1", "text": ":$\\displaystyle \\int \\arccot \\frac x a \\rd x = x \\arccot \\frac x a + \\frac a 2 \\, \\map \\ln {x^2 + a^2} + C$"}
+{"_id": "9844", "title": "Primitive of Arccotangent of x over a/Proof 2", "text": ":$\\displaystyle \\int \\operatorname{arccot} \\frac x a \\ \\mathrm d x = x \\operatorname{arccot} \\frac x a + \\frac a 2 \\ln \\left({x^2 + a^2}\\right) + C$"}
+{"_id": "9847", "title": "Primitive of x by Arcsecant of x over a", "text": ":$\\displaystyle \\int x \\operatorname{arcsec} \\frac x a \\ \\mathrm d x = \\begin{cases} \\displaystyle \\frac {x^2} 2 \\operatorname{arcsec} \\frac x a - \\frac {a \\sqrt{x^2 - a^2} } 2 + C & : 0 < \\operatorname{arcsec} \\dfrac x a < \\dfrac \\pi 2 \\\\ \\displaystyle \\frac {x^2} 2 \\operatorname{arcsec} \\frac x a + \\frac {a \\sqrt{x^2 - a^2} } 2 + C & : \\dfrac \\pi 2 < \\operatorname{arcsec} \\dfrac x a < \\pi \\\\ \\end{cases}$"}
+{"_id": "9848", "title": "Primitive of x by Arccosecant of x over a", "text": ":$\\displaystyle \\int x \\arccsc \\frac x a \\rd x = \\begin{cases} \\displaystyle \\frac {x^2} 2 \\arccsc \\frac x a + \\frac {a \\sqrt {x^2 - a^2} } 2 + C & : 0 < \\arccsc \\dfrac x a < \\dfrac \\pi 2 \\\\ \\displaystyle \\frac {x^2} 2 \\arccsc \\frac x a - \\frac {a \\sqrt {x^2 - a^2} } 2 + C & : -\\dfrac \\pi 2 < \\arccsc \\dfrac x a < 0 \\\\ \\end{cases}$"}
+{"_id": "9849", "title": "Primitive of x squared by Arcsecant of x over a", "text": ":$\\displaystyle \\int x^2 \\operatorname{arcsec} \\frac x a \\ \\mathrm d x = \\begin{cases} \\displaystyle \\frac {x^3} 3 \\operatorname{arcsec} \\frac x a - \\frac {a x \\sqrt{x^2 - a^2} } 6 - \\frac {a^3} 6 \\ln \\left({x + \\sqrt {x^2 - a^2} }\\right) + C & : 0 < \\operatorname{arcsec} \\dfrac x a < \\dfrac \\pi 2 \\\\ \\displaystyle \\frac {x^3} 3 \\operatorname{arcsec} \\frac x a + \\frac {a x \\sqrt{x^2 - a^2} } 6 + \\frac {a^3} 6 \\ln \\left({x + \\sqrt {x^2 - a^2} }\\right) + C & : \\dfrac \\pi 2 < \\operatorname{arcsec} \\dfrac x a < \\pi \\\\ \\end{cases}$"}
+{"_id": "9853", "title": "Primitive of Arccosecant of x over a over x squared", "text": ":$\\displaystyle \\int \\frac {\\arccsc \\frac x a} {x^2} \\rd x = \\begin{cases} \\displaystyle \\frac {-\\arccsc \\frac x a} x - \\frac {\\sqrt{x^2 - a^2} } {a x} + C & : 0 < \\arccsc \\dfrac x a < \\dfrac \\pi 2 \\\\ \\displaystyle \\frac {-\\arccsc \\frac x a} x + \\frac {\\sqrt{x^2 - a^2} } {a x} + C & : -\\dfrac \\pi 2 < \\arccsc \\dfrac x a < 0 \\\\ \\end{cases}$"}
+{"_id": "9856", "title": "Primitive of Power of x by Arccosecant of x over a", "text": ":$\\displaystyle \\int x^m \\arccsc \\frac x a \\rd x = \\begin{cases} \\displaystyle \\frac {x^{m + 1} } {m + 1} \\arccsc \\frac x a + \\frac a {m + 1} \\int \\frac {x^m \\rd x} {\\sqrt {x^2 - a^2} } + C & : 0 < \\arccsc \\dfrac x a < \\dfrac \\pi 2 \\\\ \\displaystyle \\frac {x^{m + 1} } {m + 1} \\arccsc \\frac x a - \\frac a {m + 1} \\int \\frac {x^m \\rd x} {\\sqrt {x^2 - a^2} } + C & : -\\dfrac \\pi 2 < \\arccsc \\dfrac x a < 0 \\\\ \\end{cases}$"}
+{"_id": "9857", "title": "Primitive of Exponential of a x", "text": ":$\\displaystyle \\int e^{a x} \\rd x = \\frac {e^{a x} } a + C$"}
+{"_id": "9858", "title": "Primitive of x by Exponential of a x", "text": ":$\\displaystyle \\int x e^{a x} \\rd x = \\frac {e^{a x} } a \\paren {x - \\frac 1 a} + C$"}
+{"_id": "9859", "title": "Primitive of x squared by Exponential of a x", "text": ":$\\displaystyle \\int x^2 e^{a x} \\rd x = \\frac {e^{a x} } a \\paren {x^2 - \\frac {2 x} a + \\frac 2 {a^2} } + C$"}
+{"_id": "9860", "title": "Primitive of Power of x by Exponential of a x", "text": "Let $n$ be a positive integer. Let $a$ be a non-zero real number. Then: {{begin-eqn}} {{eqn | l = \\int x^n e^{a x} \\rd x       | r = \\frac {e^{a x} } a \\paren {x^n - \\dfrac {n x^{n - 1} } a + \\dfrac {n \\paren {n - 1} x^{n - 2} } {a^2} - \\dfrac {n \\paren {n - 1} \\paren {n - 2} x^{n - 3} } {a^3} + \\cdots + \\dfrac {\\paren {-1}^n n!} {a^n} } + C       | c =  }} {{eqn | r = \\frac {e^{a x} } a \\sum_{k \\mathop = 0}^n \\paren {\\paren {-1}^k \\frac {n^{\\underline k} x^{n - k} } {a^k} } + C       | c =  }} {{end-eqn}} where $n^{\\underline k}$ denotes the $k$th falling factorial power of $n$."}
+{"_id": "9861", "title": "Primitive of Power of x by Exponential of a x/Lemma", "text": "Let $n$ be a positive integer. Then: :$\\displaystyle \\int x^n e^{a x} \\rd x = \\frac {x^n e^{a x} } a - \\frac n a \\int x^{n - 1} e^{a x} \\rd x + C$"}
+{"_id": "9862", "title": "Primitive of Exponential of a x over x", "text": ":$\\displaystyle \\int \\frac {e^{a x} \\rd x} x = \\ln \\size x + \\sum_{k \\mathop \\ge 1} \\frac {\\paren {a x}^k} {k \\times k!} + C$"}
+{"_id": "9863", "title": "Primitive of Exponential of a x over Power of x", "text": ":$\\displaystyle \\int \\frac {e^{a x} \\rd x} {x^n} = \\frac {-e^{a x} } {\\paren {n - 1} x^{n - 1} } + \\frac a {n - 1} \\int \\frac {e^{a x} \\rd x} {x^{n - 1} } + C$ where $n \\ne 1$."}
+{"_id": "9864", "title": "Primitive of Reciprocal of p plus q by Exponential of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {p + q e^{a x} } = \\frac x p - \\frac 1 {a p} \\ln \\size {p + q e^{a x} } + C$"}
+{"_id": "9866", "title": "Primitive of Reciprocal of p by Exponential of a x plus q by Exponential of -a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {p e^{a x} + q e^{-a x} } = \\begin{cases} \\displaystyle \\frac 1 {a \\sqrt {p q} } \\map \\arctan {\\sqrt {\\frac p q} e^{a x} } & : \\sqrt {p q} > 0 \\\\ \\displaystyle \\frac 1 {2 a \\sqrt {-p q} } \\ln \\size {\\frac {e^{a x} - \\sqrt {-\\frac q p} } {e^{a x} + \\sqrt {-\\frac q p} } } & : \\sqrt {p q} < 0 \\\\ \\end{cases}$"}
+{"_id": "9867", "title": "Primitive of Exponential of a x by Sine of b x", "text": ":$\\displaystyle \\int e^{a x} \\sin b x \\rd x = \\frac {e^{a x} \\paren {a \\sin b x - b \\cos b x} } {a^2 + b^2} + C$"}
+{"_id": "9868", "title": "Primitive of Exponential of a x by Cosine of b x", "text": ":$\\displaystyle \\int e^{a x} \\cos b x \\rd x = \\frac {e^{a x} \\paren {a \\cos b x + b \\sin b x}} {a^2 + b^2} + C$"}
+{"_id": "9869", "title": "Primitive of x by Exponential of a x by Sine of b x", "text": ":$\\displaystyle \\int x e^{a x} \\sin b x \\ \\mathrm d x = \\frac {x e^{a x} \\left({a \\sin b x - b \\cos bx}\\right)} {a^2 + b^2} - \\frac {e^{a x} \\left({\\left({a^2 - b^2}\\right) \\sin b x - 2 a b \\cos bx}\\right)} {\\left({a^2 + b^2}\\right)^2} + C$"}
+{"_id": "9873", "title": "Primitive of Exponential of a x by Logarithm of x", "text": ":$\\displaystyle \\int e^{a x} \\ln x \\rd x = \\frac {e^{a x} \\ln x} a - \\frac 1 a \\int \\frac {e^{a x} } x \\rd x + C$"}
+{"_id": "9874", "title": "Primitive of Exponential of a x by Power of Sine of b x", "text": ":$\\displaystyle \\int e^{a x} \\sin^n b x \\ \\mathrm d x = \\frac {e^{a x} \\sin^{n - 1} b x} {a^2 + n^2 b^2} \\left({a \\sin b x - n b \\cos b x}\\right) + \\frac {n \\left({n - 1}\\right) b^2} {a^2 + n^2 b^2} \\int e^{a x} \\sin^{n - 2} b x \\ \\mathrm d x + C$"}
+{"_id": "9875", "title": "Primitive of Exponential of a x by Power of Cosine of b x", "text": ":$\\displaystyle \\int e^{a x} \\cos^n b x \\rd x = \\frac {e^{a x} \\cos^{n - 1} b x} {a^2 + n^2 b^2} \\paren {a \\cos b x + n b \\sin b x} + \\frac {n \\paren {n - 1} b^2} {a^2 + n^2 b^2} \\int e^{a x} \\cos^{n - 2} b x \\rd x + C$"}
+{"_id": "9876", "title": "Primitive of Logarithm of x", "text": ":$\\ds \\int \\ln x \\rd x = x \\ln x - x + C$"}
+{"_id": "9877", "title": "Primitive of x by Logarithm of x", "text": ":$\\displaystyle \\int x \\ln x \\rd x = \\frac {x^2} 2 \\left({\\ln x - \\frac 1 2}\\right) + C$"}
+{"_id": "9878", "title": "Primitive of Power of x by Logarithm of x", "text": ":$\\displaystyle \\int x^m \\ln x \\rd x = \\frac {x^{m + 1} } {m + 1} \\left({\\ln x - \\frac 1 {m + 1} }\\right) + C$ where $m \\ne -1$."}
+{"_id": "9881", "title": "Primitive of Logarithm of x over x", "text": ":$\\displaystyle \\int \\frac {\\ln x} x \\ \\mathrm d x = \\frac {\\ln^2 x} 2 + C$"}
+{"_id": "9883", "title": "Primitive of Square of Logarithm of x", "text": ":$\\displaystyle \\int \\ln^2 x \\ \\mathrm d x = x \\ln^2 x - 2 x \\ln x + 2 x + C$"}
+{"_id": "9884", "title": "Primitive of Power of Logarithm of x over x", "text": ":$\\displaystyle \\int \\frac {\\ln^n x} x \\ \\mathrm d x = \\frac {\\ln^{n + 1} x} {n + 1} + C$"}
+{"_id": "9885", "title": "Primitive of Reciprocal of x by Logarithm of x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x \\ln x} = \\ln \\left({\\ln x}\\right) + C$"}
+{"_id": "9887", "title": "Primitive of Power of x over Logarithm of x", "text": ":$\\displaystyle \\int \\frac {x^m \\rd x} {\\ln x} = \\map \\ln {\\ln x} + \\paren {m + 1} \\ln x + \\sum_{k \\mathop \\ge 2}^n \\frac {\\paren {m + 1}^k \\paren {\\ln x}^k} {k \\times k!} + C$"}
+{"_id": "9890", "title": "Primitive of Logarithm of x squared plus a squared", "text": ":$\\displaystyle \\int \\map \\ln {x^2 + a^2} \\rd x = x \\map \\ln {x^2 + a^2} - 2 x + 2 a \\arctan \\frac x a + C$"}
+{"_id": "9892", "title": "Primitive of Power of x by Logarithm of x squared plus a squared", "text": ":$\\displaystyle \\int x^m \\map \\ln {x^2 + a^2} \\rd x = \\frac {x^{m + 1} \\map \\ln {x^2 + a^2} } {m + 1} - \\frac 2 {m + 1} \\int \\frac {x^{m + 2} } {x^2 + a^2} \\rd x$"}
+{"_id": "9893", "title": "Primitive of Power of x by Logarithm of x squared minus a squared", "text": ":$\\displaystyle \\int x^m \\, \\map \\ln {x^2 - a^2} \\rd x = \\frac {x^{m + 1} \\, \\map \\ln {x^2 - a^2} } {m + 1} -  \\frac 2 {m + 1} \\int \\frac {x^{m + 2} } {x^2 - a^2} \\rd x$"}
+{"_id": "9894", "title": "Primitive of Hyperbolic Sine of a x", "text": ":$\\displaystyle \\int \\sinh a x \\ \\mathrm d x = \\frac {\\cosh a x} a + C$"}
+{"_id": "9895", "title": "Primitive of Hyperbolic Cosine of a x", "text": ":$\\displaystyle \\int \\cosh a x \\ \\mathrm d x = \\frac {\\sinh a x} a + C$"}
+{"_id": "9896", "title": "Primitive of Hyperbolic Tangent of a x", "text": ":$\\displaystyle \\int \\tanh a x \\rd x = \\frac {\\ln \\size {\\cosh a x} } a + C$"}
+{"_id": "9897", "title": "Primitive of Hyperbolic Cotangent of a x", "text": ":$\\displaystyle \\int \\coth a x \\ \\mathrm d x = \\frac {\\ln \\left\\vert{\\sinh a x}\\right\\vert} a + C$"}
+{"_id": "9899", "title": "Primitive of Hyperbolic Cosecant of a x", "text": ":$\\displaystyle \\int \\csch a x \\rd x = \\frac 1 a \\ln \\size {\\tanh \\frac {a x} 2} + C$"}
+{"_id": "9900", "title": "Primitive of x by Hyperbolic Sine of a x", "text": ":$\\displaystyle \\int x \\sinh a x \\ \\mathrm d x = \\frac {x \\cosh a x} a - \\frac {\\sinh a x} {a^2} + C$"}
+{"_id": "9901", "title": "Primitive of x by Hyperbolic Cosine of a x", "text": ":$\\displaystyle \\int x \\cosh a x \\rd x = \\frac {x \\sinh a x} a - \\frac {\\cosh a x} {a^2} + C$"}
+{"_id": "9903", "title": "Primitive of x squared by Hyperbolic Cosine of a x", "text": ":$\\displaystyle \\int x^2 \\cosh a x \\ \\mathrm d x = \\left({\\frac {x^2} a + \\frac 2 {a^3} }\\right) \\sinh a x - \\frac {2 x \\cosh a x} {a^2} + C$"}
+{"_id": "9904", "title": "Primitive of Hyperbolic Cosine of a x over x", "text": "{{begin-eqn}} {{eqn | l = \\int \\frac {\\cosh a x \\ \\mathrm d x} x       | r = \\ln \\left\\vert{x}\\right\\vert + \\sum_{k \\mathop \\ge 1} \\frac { \\left({a x}\\right)^{2 k} } {\\left({2 k}\\right) \\left({2 k}\\right)!} + C }} {{eqn | r = \\ln \\left\\vert{x}\\right\\vert + \\frac {\\left({a x}\\right)^2} {2 \\times 2!} + \\frac {\\left({a x}\\right)^4} {4 \\times 4!} + \\frac {\\left({a x}\\right)^6} {6 \\times 6!} + \\cdots + C }} {{end-eqn}}"}
+{"_id": "9905", "title": "Primitive of Hyperbolic Tangent of a x over x", "text": "{{begin-eqn}} {{eqn | l = \\int \\frac {\\tanh a x \\ \\mathrm d x} x       | r = \\sum_{k \\mathop \\ge 1} \\frac {\\left({-1}\\right)^{k - 1} 2^{2 k} \\left({2^{2 k} - 1}\\right) B_k \\left({a x}\\right)^{2 k - 1} } {\\left({2 k - 1}\\right) \\left({2 k}\\right)!} + C }} {{eqn | r = a x - \\frac {\\left({a x}\\right)^3} 9 + \\frac {2 \\left({a x}\\right)^5} {75} - \\cdots + C }} {{end-eqn}} where $B_k$ denotes the $k$th Bernoulli number."}
+{"_id": "9906", "title": "Primitive of Hyperbolic Cotangent of a x over x", "text": "{{begin-eqn}} {{eqn | l = \\int \\frac {\\coth a x \\ \\mathrm d x} x       | r = \\sum_{k \\mathop \\ge 0} \\frac {\\left({-1}\\right)^k 2^{2 k} B_k \\left({a x}\\right)^{2 k - 1} } {\\left({2 k - 1}\\right) \\left({2 k}\\right)!} + C }} {{eqn | r = -\\frac 1 {a x} + \\frac {a x} 3 - \\frac {\\left({a x}\\right)^3} {135} + \\cdots + C }} {{end-eqn}} where $B_k$ denotes the $k$th Bernoulli number."}
+{"_id": "9907", "title": "Primitive of Hyperbolic Secant of a x over x", "text": "{{begin-eqn}} {{eqn | l = \\int \\frac {\\operatorname{sech} a x \\ \\mathrm d x} x       | r = \\ln \\left\\vert{x}\\right\\vert + \\sum_{k \\mathop \\ge 1} \\frac {\\left({-1}\\right)^k E_k \\left({a x}\\right)^{2 k} } {\\left({2 k}\\right) \\left({2 k}\\right)!} + C }} {{eqn | r = \\ln \\left\\vert{x}\\right\\vert - \\frac {\\left({a x}\\right)^2} 4 + \\frac {\\left({a x}\\right)^4} {96} - \\frac {\\left({a x}\\right)^6} {4320} + \\cdots + C }} {{end-eqn}} where $E_k$ denotes the $k$th Euler number."}
+{"_id": "9908", "title": "Primitive of Hyperbolic Cosecant of a x over x", "text": "{{begin-eqn}} {{eqn | l = \\int \\frac {\\operatorname{csch} a x \\ \\mathrm d x} x       | r = \\sum_{k \\mathop \\ge 0} \\frac {\\left({-1}\\right)^k 2 \\left({2^{2 k - 1} - 1}\\right) B_k \\left({a x}\\right)^{2 k - 1} } {\\left({2 k - 1}\\right) \\left({2 k}\\right)!} + C }} {{eqn | r = - \\frac 1 {a x} - \\frac {a x} 6 + \\frac {7 \\left({a x}\\right)^3} {1080} + \\cdots + C }} {{end-eqn}} where $E_k$ denotes the $k$th Bernoulli number."}
+{"_id": "9909", "title": "Derivative of Hyperbolic Sine of a x", "text": ":$\\map {D_x} {\\sinh a x} = a \\cosh a x$"}
+{"_id": "9910", "title": "Derivative of Hyperbolic Cosine of a x", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\cosh a x} = a \\sinh a x$"}
+{"_id": "9911", "title": "Primitive of Hyperbolic Sine of a x over x squared", "text": ":$\\displaystyle \\int \\frac {\\sinh a x \\ \\mathrm d x} {x^2} = -\\frac {\\sinh a x} x + a \\int \\frac {\\cosh a x \\ \\mathrm d x} x$"}
+{"_id": "9913", "title": "Primitive of Reciprocal of Hyperbolic Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\sinh a x} = \\frac 1 a \\ln \\left\\vert {\\tanh \\frac {a x} 2} \\right\\vert + C$"}
+{"_id": "9915", "title": "Primitive of Reciprocal of Hyperbolic Tangent of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\tanh a x} = \\frac {\\ln \\left\\vert {\\sinh a x} \\right\\vert} a + C$"}
+{"_id": "9916", "title": "Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent", "text": ":$\\coth x = \\dfrac 1 {\\tanh x}$ where $\\tanh$ and $\\coth$ denote hyperbolic tangent and hyperbolic cotangent respectively."}
+{"_id": "9917", "title": "Primitive of Reciprocal of Hyperbolic Cotangent of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\coth a x} = \\frac {\\ln \\left\\vert {\\cosh a x} \\right\\vert} a + C$"}
+{"_id": "9919", "title": "Primitive of Reciprocal of Hyperbolic Cosecant of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\operatorname{csch} a x} = \\frac {\\cosh a x} a + C$"}
+{"_id": "9922", "title": "Primitive of Square of Hyperbolic Sine of a x", "text": ":$\\displaystyle \\int \\sinh^2 a x \\rd x = \\dfrac {\\sinh a x \\cosh a x} {2 a} - \\frac x 2 + C$"}
+{"_id": "9923", "title": "Primitive of Square of Hyperbolic Cosine of a x", "text": ":$\\displaystyle \\int \\cosh^2 a x \\rd x = \\frac {\\sinh a x \\cosh a x} {2 a} + \\frac x 2 + C$"}
+{"_id": "9924", "title": "Primitive of Square of Hyperbolic Tangent of a x", "text": ":$\\displaystyle \\int \\tanh^2 a x \\rd x = x - \\frac {\\tanh a x} a + C$"}
+{"_id": "9925", "title": "Primitive of Square of Hyperbolic Cotangent of a x", "text": ":$\\displaystyle \\int \\coth^2 a x \\ \\mathrm d x = x - \\frac {\\coth a x} a + C$"}
+{"_id": "9926", "title": "Primitive of Square of Hyperbolic Secant of a x", "text": ":$\\ds \\int \\sech^2 a x \\rd x = \\frac {\\tanh a x} a + C$"}
+{"_id": "9927", "title": "Primitive of Square of Hyperbolic Cosecant of a x", "text": ":$\\displaystyle \\int \\operatorname{csch}^2 a x \\ \\mathrm d x = \\frac {-\\coth a x} a + C$"}
+{"_id": "9928", "title": "Primitive of Square of Hyperbolic Sine of a x/Corollary", "text": ":$\\displaystyle \\int \\sinh^2 a x \\rd x = \\frac {\\sinh 2 a x} {4 a} - \\frac x 2 + C$"}
+{"_id": "9929", "title": "Primitive of x by Square of Hyperbolic Sine of a x", "text": ":$\\ds \\int x \\sinh^2 a x \\rd x = \\dfrac {x \\sinh 2 a x} {4 a} - \\frac {\\cosh 2 a x} {8 a^2} - \\frac {x^2} 4 + C$"}
+{"_id": "9930", "title": "Primitive of Square of Hyperbolic Cosine of a x/Corollary", "text": ":$\\displaystyle \\int \\cosh^2 a x \\rd x = \\frac {\\sinh 2 a x} {4 a} + \\frac x 2 + C$"}
+{"_id": "9931", "title": "Primitive of x by Square of Hyperbolic Cosine of a x", "text": ":$\\displaystyle \\int x \\cosh^2 a x \\ \\mathrm d x = \\frac {x \\sinh 2 a x} {4 a} - \\frac {\\cosh 2 a x} {8 a^2} + \\frac {x^2} 4 + C$"}
+{"_id": "9932", "title": "Primitive of x by Square of Hyperbolic Tangent of a x", "text": ":$\\displaystyle \\int x \\tanh^2 a x \\ \\mathrm d x = \\frac {x^2} 2 - \\frac {x \\tanh a x} a + \\frac 1 {a^2} \\ln \\left\\vert{\\cosh a x}\\right\\vert + C$"}
+{"_id": "9933", "title": "Primitive of x by Square of Hyperbolic Cotangent of a x", "text": ":$\\displaystyle \\int x \\coth^2 a x \\ \\mathrm d x = \\frac {x^2} 2 - \\frac {x \\coth a x} a + \\frac 1 {a^2} \\ln \\left\\vert{\\sinh a x}\\right\\vert + C$"}
+{"_id": "9934", "title": "Primitive of x by Square of Hyperbolic Secant of a x", "text": ":$\\displaystyle \\int x \\operatorname{sech}^2 a x \\ \\mathrm d x = \\frac {x \\tanh a x} a - \\frac 1 {a^2} \\ln \\left\\vert{\\cosh a x}\\right\\vert + C$"}
+{"_id": "9938", "title": "Primitive of Hyperbolic Sine of a x by Hyperbolic Sine of p x", "text": ":$\\displaystyle \\int \\sinh a x \\sinh p x \\rd x = \\frac {\\map \\sinh {a + p} x} {2 \\paren {a + p} } - \\frac {\\map \\sinh {a - p} x} {2 \\paren {a - p} } + C$"}
+{"_id": "9939", "title": "Primitive of Hyperbolic Cosine of a x by Hyperbolic Cosine of p x", "text": ":$\\displaystyle \\int \\cosh a x \\cosh p x \\rd x = \\frac {\\map \\sinh {a + p} x} {2 \\paren {a + p} } + \\frac {\\map \\sinh {a - p} x} {2 \\paren {a - p} } + C$"}
+{"_id": "9940", "title": "Primitive of Hyperbolic Sine of a x by Sine of p x", "text": ":$\\displaystyle \\int \\sinh a x \\sin p x \\ \\mathrm d x = \\frac {a \\cosh a x \\sin p x - p \\sinh a x \\cos p x} {a^2 + p^2} + C$"}
+{"_id": "9942", "title": "Primitive of Hyperbolic Cosine of a x by Sine of p x", "text": ":$\\displaystyle \\int \\cosh a x \\sin p x \\ \\mathrm d x = \\frac {a \\sinh a x \\sin p x - p \\cosh a x \\cos p x} {a^2 + p^2} + C$"}
+{"_id": "9943", "title": "Primitive of Hyperbolic Cosine of a x by Cosine of p x", "text": ":$\\displaystyle \\int \\cosh a x \\cos p x \\ \\mathrm d x = \\frac {a \\sinh a x \\cos p x + p \\cosh a x \\sin p x} {a^2 + p^2} + C$"}
+{"_id": "9944", "title": "Primitive of Reciprocal of p plus q by Hyperbolic Sine of a x", "text": ":$\\ds \\int \\frac {\\d x} {p + q \\sinh a x}  = \\frac 1 {a \\sqrt{p^2 + q^2} } \\ln \\size {\\frac {q e^{a x} + p - \\sqrt {p^2 + q^2} } {q e^{a x} + p + \\sqrt {p^2 + q^2} } } + C$"}
+{"_id": "9948", "title": "Primitive of Reciprocal of q plus p by Hyperbolic Secant of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {q + p \\operatorname{sech} a x} = \\frac x q - \\frac p q \\int \\frac {\\mathrm d x} {p + q \\cosh a x} + C$"}
+{"_id": "9949", "title": "Primitive of Reciprocal of q plus p by Hyperbolic Cosecant of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {q + p \\csch a x} = \\frac x q - \\frac p q \\int \\frac {\\d x} {p + q \\sinh a x} + C$"}
+{"_id": "9957", "title": "Primitive of Power of x by Hyperbolic Cosine of a x", "text": ":$\\ds \\int x^m \\cosh a x \\rd x = \\frac {x^m \\sinh a x} a - \\frac m a \\int x^{m - 1} \\sinh a x \\rd x + C$"}
+{"_id": "9958", "title": "Primitive of Power of Hyperbolic Sine of a x", "text": ":$\\displaystyle \\int \\sinh^n a x \\rd x = \\frac {\\sinh^{n - 1} a x \\cosh a x} {a n} - \\frac {n - 1} n \\int \\sinh^{n - 2} a x \\rd x + C$"}
+{"_id": "9959", "title": "Primitive of Power of Hyperbolic Cosine of a x", "text": ":$\\displaystyle \\int \\cosh^n a x \\rd x = \\frac {\\cosh^{n - 1} a x \\sinh a x} {a n} + \\frac {n - 1} n \\int \\cosh^{n - 2} a x \\rd x + C$"}
+{"_id": "9960", "title": "Primitive of Power of Hyperbolic Tangent of a x by Square of Hyperbolic Secant of a x", "text": ":$\\displaystyle \\int \\tanh^n a x \\operatorname{sech}^2 a x \\ \\mathrm d x = \\frac {\\tanh^{n + 1} a x} {\\left({n + 1}\\right) a} + C$"}
+{"_id": "9961", "title": "Derivative of Hyperbolic Tangent of a x", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\tanh a x} = a \\sech^2 a x$"}
+{"_id": "9962", "title": "Derivative of Hyperbolic Cotangent of a x", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\coth a x} = -a \\csch^2 a x$"}
+{"_id": "9963", "title": "Primitive of Power of Hyperbolic Cotangent of a x by Square of Hyperbolic Cosecant of a x", "text": ":$\\ds \\int \\coth^n a x \\csch^2 a x \\rd x = \\frac {-\\coth^{n + 1} a x} {\\paren {n + 1} a} + C$"}
+{"_id": "9964", "title": "Primitive of Power of Hyperbolic Cotangent of a x", "text": ":$\\ds \\int \\coth^n a x \\rd x = \\frac {-\\coth^{n - 1} a x} {a \\paren {n - 1} } + \\int \\coth^{n - 2} a x \\rd x + C$"}
+{"_id": "9965", "title": "Derivative of Hyperbolic Secant of a x", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\sech a x} = -a \\sech a x \\tanh a x$"}
+{"_id": "9966", "title": "Derivative of Hyperbolic Cosecant of a x", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\csch a x} = -a \\csch a x \\coth a x$"}
+{"_id": "9967", "title": "Primitive of Power of Hyperbolic Secant of a x", "text": ":$\\displaystyle \\int \\sech^n a x \\rd x = \\frac {\\sech^{n - 2} a x \\tanh a x} {a \\paren {n - 1} } + \\frac {n - 2} {n - 1} \\int \\sech^{n - 2} a x \\rd x + C$"}
+{"_id": "9968", "title": "Primitive of Power of Hyperbolic Cosecant of a x", "text": ":$\\displaystyle \\int \\csch^n a x \\rd x = \\frac {-\\csch^{n - 2} a x \\coth a x} {a \\paren {n - 1} } - \\frac {n - 2} {n - 1} \\int\\csch^{n - 2} a x \\rd x + C$"}
+{"_id": "9971", "title": "Primitive of x over Power of Hyperbolic Sine of a x", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {\\sinh^n a x} = \\frac {-x \\cosh a x} {a \\paren {n - 1} \\sinh^{n - 1} a x} - \\frac 1 {a^2 \\paren {n - 1} \\paren {n - 2} \\sinh^{n - 2} a x} - \\frac {n - 2} {n - 1} \\int \\frac {x \\rd x} {\\sinh^{n - 2} a x} + C$"}
+{"_id": "9972", "title": "Primitive of x over Power of Hyperbolic Cosine of a x", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {\\cosh^n a x} = \\frac {x \\sinh a x} {a \\paren {n - 1} \\cosh^{n - 1} a x} + \\frac 1 {a^2 \\paren {n - 1} \\paren {n - 2} \\cosh^{n - 2} a x} + \\frac {n - 2} {n - 1} \\int \\frac {x \\rd x} {\\cosh^{n - 2} a x} + C$"}
+{"_id": "9973", "title": "Primitive of Reciprocal of Hyperbolic Cosine of a x plus 1", "text": ":$\\ds \\int \\frac {\\d x} {\\cosh a x + 1} = \\frac 1 a \\tanh \\frac {a x} 2 + C$"}
+{"_id": "9976", "title": "Primitive of Reciprocal of Square of Hyperbolic Cosine of a x minus 1", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\paren {\\cosh a x - 1}^2} = \\frac 1 {2 a} \\coth \\frac {a x} 2 - \\frac 1 {6 a} \\coth^3 \\frac {a x} 2 + C$"}
+{"_id": "9977", "title": "Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x", "text": ":$\\displaystyle \\int \\sinh a x \\cosh a x \\rd x = \\frac {\\sinh^2 a x} {2 a} + C$"}
+{"_id": "9981", "title": "Primitive of Power of Hyperbolic Sine of a x by Hyperbolic Cosine of a x", "text": ":$\\displaystyle \\int \\sinh^n a x \\cosh a x \\rd x = \\frac {\\sinh^{n + 1} a x} {\\paren {n + 1} a} + C$ for $n \\ne -1$."}
+{"_id": "9984", "title": "Primitive of Power of Hyperbolic Cosine of a x by Hyperbolic Sine of a x", "text": ":$\\displaystyle \\int \\cosh^n a x \\sinh a x \\rd x = \\frac {\\cosh^{n + 1} a x} {\\paren {n + 1} a} + C$ for $n \\ne -1$."}
+{"_id": "9986", "title": "Primitive of Square of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x", "text": ":$\\displaystyle \\int \\sinh^2 a x \\cosh^2 a x \\rd x = \\frac {\\sinh 4 a x} {32 a} - \\frac x 8 + C$"}
+{"_id": "9987", "title": "Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\sinh a x \\cosh a x} = \\frac 1 a \\ln \\left\\vert{\\tanh a x}\\right\\vert + C$"}
+{"_id": "9988", "title": "Primitive of Square of Hyperbolic Secant of a x over Hyperbolic Tangent of a x", "text": ":$\\displaystyle \\int \\frac {\\operatorname{sech}^2 a x \\ \\mathrm d x} {\\tanh a x} = \\frac 1 a \\ln \\left\\vert{\\tanh a x}\\right\\vert + C$"}
+{"_id": "9989", "title": "Primitive of Reciprocal of Square of Hyperbolic Sine of a x by Hyperbolic Cosine of a x", "text": ":$\\ds \\int \\frac {\\d x} {\\sinh^2 a x \\cosh a x} = -\\frac 1 a \\map \\arctan {\\sinh a x} - \\frac {\\csch a x} a + C$"}
+{"_id": "9991", "title": "Primitive of Power of Hyperbolic Secant of a x by Hyperbolic Tangent of a x", "text": ":$\\displaystyle \\int \\sech^n a x \\tanh a x \\rd x = \\frac {-\\sech^n a x} {n a} + C$"}
+{"_id": "9992", "title": "Primitive of Power of Hyperbolic Cosecant of a x by Hyperbolic Cotangent of a x", "text": ":$\\displaystyle \\int \\csch^n a x \\coth a x \\rd x = \\frac {-\\csch^n a x} {n a} + C$"}
+{"_id": "9994", "title": "Primitive of Square of Hyperbolic Sine of a x over Hyperbolic Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\sinh^2 a x \\ \\mathrm d x} {\\cosh a x} = \\frac {\\sinh a x} a - \\frac 1 a \\arctan \\left({\\sinh a x}\\right) + C$"}
+{"_id": "9995", "title": "Primitive of Reciprocal of Hyperbolic Cosine of a x by 1 plus Hyperbolic Sine of a x", "text": ":$\\ds \\int \\frac {\\rd x} {\\cosh a x \\paren {1 + \\sinh a x} } = \\frac 1 {2 a} \\ln \\size {\\frac {1 + \\sinh a x} {\\cosh a x} } + \\frac 1 a \\map \\arctan {e^{a x} } + C$"}
+{"_id": "9997", "title": "Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x minus 1", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\sinh a x \\paren {\\cosh a x - 1} } = \\frac {-1} {2 a} \\ln \\size {\\tanh \\frac {a x} 2} - \\frac 1 {2 a \\paren {\\cosh a x - 1} } + C$"}
+{"_id": "9998", "title": "Primitive of Cube of Hyperbolic Tangent of a x", "text": ":$\\displaystyle \\int \\tanh^3 a x \\ \\mathrm d x = \\frac {\\ln \\left\\vert{\\cosh a x}\\right\\vert} a - \\frac {\\tanh^2 a x} {2 a} + C$"}
+{"_id": "9999", "title": "Primitive of Cube of Hyperbolic Tangent of a x/Proof 1", "text": ":$\\displaystyle \\int \\tanh^3 a x \\ \\mathrm d x = \\frac {\\ln \\left\\vert{\\cosh a x}\\right\\vert} a - \\frac {\\tanh^2 a x} {2 a} + C$"}
+{"_id": "10000", "title": "Primitive of Cube of Hyperbolic Tangent of a x/Proof 2", "text": ":$\\displaystyle \\int \\tanh^3 a x \\ \\mathrm d x = \\frac {\\ln \\left\\vert{\\cosh a x}\\right\\vert} a - \\frac {\\tanh^2 a x} {2 a} + C$"}
+{"_id": "10002", "title": "Primitive of Cube of Hyperbolic Cotangent of a x/Proof 1", "text": ":$\\displaystyle \\int \\coth^3 a x \\ \\mathrm d x = \\frac {\\ln \\left\\vert{\\sinh a x}\\right\\vert} a - \\frac {\\coth^2 a x} {2 a} + C$"}
+{"_id": "10003", "title": "Primitive of Cube of Hyperbolic Cotangent of a x/Proof 2", "text": ":$\\displaystyle \\int \\coth^3 a x \\ \\mathrm d x = \\frac {\\ln \\left\\vert{\\sinh a x}\\right\\vert} a - \\frac {\\coth^2 a x} {2 a} + C$"}
+{"_id": "10004", "title": "Primitive of x by Hyperbolic Tangent of a x", "text": ":$\\displaystyle \\int x \\tanh a x \\rd x = \\frac 1 {a^2} \\paren {\\frac {\\paren {a x}^3} 3 - \\frac {\\paren {a x}^5} {15} + \\frac {2 \\paren {a x}^7} {105} + \\cdots + \\frac { 2^{2 n} \\paren {2^{2 n} - 1} B_{2 n} \\paren {a x}^{2 n + 1} } {\\paren {2 n + 1}!} + \\cdots} + C$"}
+{"_id": "10005", "title": "Primitive of x by Hyperbolic Cotangent of a x", "text": ":$\\displaystyle \\int x \\coth a x \\rd x = \\frac 1 {a^2} \\paren {a x + \\frac {\\paren {a x}^3} 9 - \\frac {\\paren {a x}^5} {225} + \\cdots + \\frac {2^{2 n} B_{2 n} \\paren {a x}^{2 n + 1} } {\\paren {2 n + 1}!} + \\cdots} + C$"}
+{"_id": "10006", "title": "Primitive of Square of Hyperbolic Cosecant of a x over Hyperbolic Cotangent of a x", "text": ":$\\displaystyle \\int \\frac {\\csch^2 a x \\rd x} {\\coth a x} = \\frac {-1} a \\ln \\size {\\coth a x} + C$"}
+{"_id": "10007", "title": "Primitive of Cube of Hyperbolic Secant of a x", "text": ":$\\displaystyle \\int \\sech^3 a x \\rd x = \\frac {\\sech a x \\tanh a x} {2 a} + \\frac 1 {2 a} \\map \\arctan {\\sinh a x} + C$"}
+{"_id": "10008", "title": "Primitive of Cube of Hyperbolic Cosecant of a x", "text": ":$\\ds \\int \\csch^3 a x \\rd x = \\frac {-\\csch a x \\coth a x} {2 a} - \\frac 1 {2 a} \\ln \\size {\\tanh a x} + C$"}
+{"_id": "10009", "title": "Primitive of Hyperbolic Secant Function/Arctangent of Hyperbolic Sine Form", "text": ":$\\ds \\int \\sech x \\rd x =  \\map \\arctan {\\sinh x} + C$"}
+{"_id": "10010", "title": "Primitive of Hyperbolic Secant of a x/Arcsine Form", "text": ":$\\displaystyle \\int \\operatorname{sech} a x \\ \\mathrm d x = \\frac {\\arcsin \\left({\\tanh a x}\\right)} a + C$"}
+{"_id": "10011", "title": "Primitive of Hyperbolic Secant of a x/Arctangent of Exponential Form", "text": ":$\\ds \\int \\sech a x \\rd x = \\frac {2 \\map \\arctan {e^{a x} } } a + C$"}
+{"_id": "10012", "title": "Primitive of Hyperbolic Secant of a x/Arctangent of Hyperbolic Sine Form", "text": ":$\\displaystyle \\int \\operatorname{sech} a x \\ \\mathrm d x = \\frac {\\arctan \\left({\\sinh a x}\\right)} a + C$"}
+{"_id": "10014", "title": "Primitive of x by Hyperbolic Cosecant of a x", "text": ":$\\displaystyle \\int x \\csch a x \\rd x = \\frac 1 {a^2} \\sum_{n \\mathop = 0}^\\infty \\frac {2 \\paren {1 - 2^{2 n - 1} } B_{2 n} \\paren {a x}^{2 n + 1} } {\\paren {2 n + 1}!} + C$"}
+{"_id": "10015", "title": "Derivative of Inverse Hyperbolic Sine of x over a", "text": ":$\\dfrac {\\map \\d {\\map {\\sinh^{-1} } {\\frac x a} } } {\\d x} = \\dfrac 1 {\\sqrt {x^2 + a^2}}$"}
+{"_id": "10016", "title": "Primitive of Inverse Hyperbolic Sine of x over a", "text": ":$\\displaystyle \\int \\sinh^{-1} \\frac x a \\rd x = x \\sinh^{-1} \\frac x a - \\sqrt {x^2 + a^2} + C$"}
+{"_id": "10017", "title": "Derivative of Inverse Hyperbolic Cosine of x over a", "text": ":$\\dfrac {\\map \\d {\\map {\\cosh^{-1} } {\\frac x a} } } {\\d x} = \\dfrac 1 {\\sqrt {x^2 - a^2} }$ where $x > a$."}
+{"_id": "10018", "title": "Derivative of Inverse Hyperbolic Tangent of x over a", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\map {\\tanh^{-1} } {\\dfrac x a} } = \\dfrac a {a^2 - x^2}$ where $-a < x < a$."}
+{"_id": "10019", "title": "Derivative of Inverse Hyperbolic Cotangent of x over a", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\map {\\coth^{-1} } {\\dfrac x a} } = \\dfrac {-a} {x^2 - a^2}$ where $x^2 > a^2$."}
+{"_id": "10020", "title": "Derivative of Inverse Hyperbolic Secant of x over a", "text": ":$\\dfrac {\\mathrm d \\left({\\operatorname{sech}^{-1} \\left({\\frac x a}\\right)}\\right)} {\\mathrm d x} = \\dfrac {-a} {x \\sqrt{a^2 - x^2} }$ where $0 < x < a$."}
+{"_id": "10021", "title": "Derivative of Inverse Hyperbolic Cosecant of x over a", "text": ":$\\dfrac {\\map \\d {\\csch^{-1} } {\\frac x a} } {\\d x} = \\dfrac {-a} {\\size x \\sqrt {a^2 + x^2} }$ where $x \\ne 0$."}
+{"_id": "10023", "title": "Primitive of Inverse Hyperbolic Tangent of x over a", "text": ":$\\displaystyle \\int \\tanh^{-1} \\frac x a \\rd x = x \\tanh^{-1} \\dfrac x a + \\frac {a \\ln \\left({a^2 - x^2}\\right)} 2 + C$"}
+{"_id": "10024", "title": "Primitive of Inverse Hyperbolic Cotangent of x over a", "text": ":$\\displaystyle \\int \\coth^{-1} \\frac x a \\ \\mathrm d x = x \\coth^{-1} \\dfrac x a + \\frac {a \\ln \\left({x^2 - a^2}\\right)} 2 + C$"}
+{"_id": "10025", "title": "Primitive of Inverse Hyperbolic Secant of x over a", "text": ":$\\displaystyle \\int \\sech^{-1} \\frac x a \\rd x = \\begin{cases} x \\sech^{-1} \\dfrac x a + a \\arcsin \\dfrac x a + C & : \\sech^{-1} \\dfrac x a > 0 \\\\ x \\sech^{-1} \\dfrac x a - a \\arcsin \\dfrac x a + C & : \\sech^{-1} \\dfrac x a < 0 \\end{cases}$"}
+{"_id": "10029", "title": "Inverse Hyperbolic Cosine of x over a in Logarithm Form", "text": ":$\\cosh^{-1} \\dfrac x a = \\map \\ln {x + \\sqrt {x^2 - a^2} } - \\ln a$"}
+{"_id": "10030", "title": "Inverse Hyperbolic Tangent of x over a in Logarithm Form", "text": ":$\\tanh^{-1} \\dfrac x a = \\dfrac 1 2 \\map \\ln {\\dfrac {a + x} {a - x} }$"}
+{"_id": "10031", "title": "Inverse Hyperbolic Cotangent of x over a in Logarithm Form", "text": ":$\\coth^{-1} \\dfrac x a = \\dfrac 1 2 \\map \\ln {\\dfrac {x + a} {x - a} }$"}
+{"_id": "10032", "title": "Inverse Hyperbolic Secant of x over a in Logarithm Form", "text": ":$\\operatorname{sech}^{-1} \\dfrac x a = \\ln \\left({\\dfrac {a + \\sqrt{a^2 - x^2} } x}\\right)$"}
+{"_id": "10033", "title": "Inverse Hyperbolic Cosecant of x over a in Logarithm Form", "text": ":$\\csch^{-1} \\dfrac x a = \\map \\ln {\\dfrac {a + \\sqrt {a^2 + x^2} } x}$"}
+{"_id": "10035", "title": "Primitive of Root of x squared plus a squared/Inverse Hyperbolic Sine Form", "text": ":$\\displaystyle \\int \\sqrt {x^2 + a^2} \\rd x = \\frac {x \\sqrt {x^2 + a^2} } 2 + \\frac {a^2} 2 \\sinh^{-1} \\frac x a + C$"}
+{"_id": "10036", "title": "Primitive of x squared over Root of x squared plus a squared/Inverse Hyperbolic Sine Form", "text": ":$\\displaystyle \\int \\frac {x^2 \\rd x} {\\sqrt {x^2 + a^2} } = \\frac {x \\sqrt {x^2 + a^2} } 2 - \\frac {a^2} 2 \\sinh^{-1} \\frac x a + C$"}
+{"_id": "10038", "title": "Primitive of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form", "text": ":$\\displaystyle \\int \\sqrt {x^2 - a^2} \\rd x = \\frac {x \\sqrt {x^2 - a^2} } 2 - \\frac {a^2} 2 \\cosh^{-1} \\frac x a + C$"}
+{"_id": "10039", "title": "Primitive of Root of x squared minus a squared/Logarithm Form", "text": ":$\\displaystyle \\int \\sqrt {x^2 - a^2} \\rd x = \\frac {x \\sqrt {x^2 - a^2} } 2 - \\frac {a^2} 2 \\ln \\size {x + \\sqrt {x^2 - a^2} } + C$"}
+{"_id": "10040", "title": "Primitive of x squared over Root of x squared minus a squared/Inverse Hyperbolic Cosine Form", "text": ":$\\displaystyle \\int \\frac {x^2 \\rd x} {\\sqrt {x^2 - a^2} } = \\frac {x \\sqrt {x^2 - a^2} } 2 + \\frac {a^2} 2 \\cosh^{-1} \\frac x a + C$ for $x > a$."}
+{"_id": "10042", "title": "Primitive of x squared over x squared minus a squared/Inverse Hyperbolic Cotangent Form", "text": ":$\\displaystyle \\int \\frac {x^2 \\ \\mathrm d x} {x^2 - a^2} = x - a \\coth^{-1} \\frac x a + C$ for $x^2 > a^2$."}
+{"_id": "10043", "title": "Primitive of x squared over a squared minus x squared/Logarithm Form", "text": ":$\\displaystyle \\int \\frac {x^2 \\rd x} {a^2 - x^2} = -x + \\frac a 2 \\map \\ln {\\frac {a + x} {a - x} } + C$ for $x^2 < a^2$."}
+{"_id": "10044", "title": "Primitive of x squared over a squared minus x squared/Inverse Hyperbolic Tangent Form", "text": ":$\\displaystyle \\int \\frac {x^2 \\rd x} {a^2 - x^2} = -x + a \\tanh^{-1} \\frac x a + C$ for $x^2 < a^2$."}
+{"_id": "10045", "title": "Primitive of x by Inverse Hyperbolic Tangent of x over a", "text": ":$\\displaystyle \\int x \\tanh^{-1} \\frac x a \\rd x = \\frac {a x} 2 + \\frac {x^2 - a^2} 2 \\tanh^{-1} \\frac x a + C$"}
+{"_id": "10047", "title": "Primitive of x squared by Inverse Hyperbolic Sine of x over a", "text": ":$\\displaystyle \\int x^2 \\sinh^{-1} \\frac x a \\rd x = \\frac {x^3} 3 \\sinh^{-1} \\frac x a + \\frac {\\paren {2 a^2 - x^2} \\sqrt {x^2 + a^2} } 9 + C$"}
+{"_id": "10049", "title": "Primitive of x squared by Inverse Hyperbolic Tangent of x over a", "text": ":$\\displaystyle \\int x^2 \\tanh^{-1} \\frac x a \\ \\mathrm d x = \\frac {a x^2} 6 + \\frac {x^3} 3 \\tanh^{-1} \\frac x a + \\frac {a^3} 6 \\ln \\left({a^2 - x^2}\\right) + C$"}
+{"_id": "10057", "title": "Primitive of Inverse Hyperbolic Sine of x over a over x squared", "text": ":$\\displaystyle \\int \\frac {\\sinh^{-1} \\dfrac x a \\ \\mathrm d x} {x^2} = \\frac {-\\sinh^{-1} \\dfrac x a} x - \\frac 1 a \\ln \\left({\\frac {a + \\sqrt {x^2 + a^2} } x}\\right)$"}
+{"_id": "10058", "title": "Primitive of Inverse Hyperbolic Cosine of x over a over x squared", "text": ":$\\displaystyle \\int \\frac {\\cosh^{-1} \\dfrac x a \\ \\mathrm d x} {x^2} = \\begin{cases} \\displaystyle \\frac {-\\cosh^{-1} \\dfrac x a} x - \\frac 1 a \\ln \\left({\\frac {a + \\sqrt {x^2 + a^2} } x}\\right) & : \\cosh^{-1} \\dfrac x a > 0 \\\\ \\displaystyle \\frac {-\\cosh^{-1} \\dfrac x a} x + \\frac 1 a \\ln \\left({\\frac {a + \\sqrt {x^2 + a^2} } x}\\right) & : \\cosh^{-1} \\dfrac x a < 0 \\\\ \\end{cases}$"}
+{"_id": "10059", "title": "Primitive of Inverse Hyperbolic Tangent of x over a over x squared", "text": ":$\\ds \\int \\frac {\\tanh^{-1} \\dfrac x a \\rd x} {x^2} = \\frac {-\\tanh^{-1} \\dfrac x a} x + \\frac 1 {2 a} \\map \\ln {\\frac {x^2} {a^2 - x^2} } + C$"}
+{"_id": "10060", "title": "Primitive of Inverse Hyperbolic Cotangent of x over a over x squared", "text": ":$\\displaystyle \\int \\frac {\\coth^{-1} \\dfrac x a \\rd x} {x^2} = \\frac {-\\coth^{-1} \\dfrac x a} x + \\frac 1 {2 a} \\map \\ln {\\frac {x^2} {x^2 - a^2} } + C$"}
+{"_id": "10061", "title": "Primitive of Power of x by Inverse Hyperbolic Sine of x over a", "text": ":$\\displaystyle \\int x^m \\sinh^{-1} \\frac x a \\ \\mathrm d x = \\frac {x^{m + 1} } {m + 1} \\sinh^{-1} \\frac x a- \\frac 1 {m + 1}  \\int \\frac {x^{m + 1} } {\\sqrt {x^2 + a^2} } \\ \\mathrm d x + C$"}
+{"_id": "10062", "title": "Primitive of Power of x by Inverse Hyperbolic Cotangent of x over a", "text": ":$\\displaystyle \\int x^m \\coth^{-1} \\frac x a \\rd x = \\frac {x^{m + 1} } {m + 1} \\coth^{-1} \\frac x a - \\frac a {m + 1} \\int \\frac {x^{m + 1} } {a^2 - x^2} \\rd x + C$"}
+{"_id": "10064", "title": "Primitive of Power of x by Inverse Hyperbolic Cosecant of x over a", "text": ":$\\displaystyle \\int x^m \\operatorname{csch}^{-1} \\frac x a \\rd x = \\begin{cases} \\displaystyle \\frac {x^{m + 1} } {m + 1} \\operatorname{csch}^{-1} \\frac x a + \\frac 1 {m + 1} \\int \\frac {x^m} {\\sqrt {x^2 + a^2} } \\rd x + C & : x > 0 \\\\ \\displaystyle \\frac {x^{m + 1} } {m + 1} \\operatorname{csch}^{-1} \\frac x a - \\frac 1 {m + 1} \\int \\frac {x^m} {\\sqrt {x^2 + a^2} } \\rd x + C & : x < 0 \\\\ \\end{cases}$"}
+{"_id": "10065", "title": "Hyperbolic Tangent Half-Angle Substitution", "text": ":$\\displaystyle \\int \\map F {\\sinh x, \\cosh x} \\rd x = 2 \\int \\map F {\\frac {2 u} {1 - u^2}, \\frac {1 + u^2} {1 - u^2} } \\frac {\\d u} {1 - u^2}$ where $u = \\tanh \\dfrac x 2$."}
+{"_id": "10069", "title": "Reciprocal of One Plus Cosine", "text": ":$\\dfrac 1 {1 + \\cos x} = \\dfrac 1 2 \\sec^2 \\dfrac x 2$"}
+{"_id": "10070", "title": "Reciprocal of One Minus Cosine", "text": ":$\\dfrac 1 {1 - \\cos x} = \\dfrac 1 2 \\csc^2 \\left({\\dfrac x 2}\\right)$"}
+{"_id": "10073", "title": "Irrationality of Logarithm", "text": "Let $a, b \\in \\N_{>0}$ such that both $\\nexists m, n \\in \\N_{>0}: a^m = b^n$. Then $\\log_b a$ is irrational."}
+{"_id": "10076", "title": "Tangent of 22.5 Degrees", "text": ":$\\tan 22.5^\\circ = \\tan \\dfrac \\pi 8 = \\sqrt 2 - 1$"}
+{"_id": "10077", "title": "Tangent of 67.5 Degrees", "text": ":$\\tan 67.5 \\degrees = \\tan \\dfrac {3 \\pi} 8 = \\sqrt 2 + 1$"}
+{"_id": "10078", "title": "Sine of x plus Cosine of x/Cosine Form", "text": ":$\\sin x + \\cos x = \\sqrt 2 \\, \\map \\cos {x - \\dfrac \\pi 4}$"}
+{"_id": "10080", "title": "Sine of x minus Cosine of x/Sine Form", "text": ":$\\sin x - \\cos x = \\sqrt 2 \\sin \\left({x - \\dfrac \\pi 4}\\right)$"}
+{"_id": "10082", "title": "Sine of x minus Cosine of x", "text": "=== Sine Form === {{:Sine of x minus Cosine of x/Sine Form}} === Cosine Form === {{:Sine of x minus Cosine of x/Cosine Form}}"}
+{"_id": "10083", "title": "Sine of x minus Cosine of x/Cosine Form", "text": ":$\\sin x - \\cos x = \\sqrt 2 \\, \\map \\cos {x - \\dfrac {3 \\pi} 4}$"}
+{"_id": "10084", "title": "Multiple of Sine plus Multiple of Cosine", "text": "=== Cosine Form === {{:Multiple of Sine plus Multiple of Cosine/Cosine Form}} === Sine Form === {{:Multiple of Sine plus Multiple of Cosine/Sine Form}}"}
+{"_id": "10085", "title": "Sum Formulas for Sine and Cosine", "text": "=== Sine of x plus Cosine of x: Sine Form === {{:Sine of x plus Cosine of x/Sine Form}} === Sine of x plus Cosine of x: Cosine Form === {{:Sine of x plus Cosine of x/Cosine Form}} === Sine of x minus Cosine of x: Sine Form === {{:Sine of x minus Cosine of x/Sine Form}} === Sine of x minus Cosine of x: Cosine Form === {{:Sine of x minus Cosine of x/Cosine Form}} === Cosine of x minus Sine of x: Sine Form === {{:Cosine of x minus Sine of x/Sine Form}} === Cosine of x minus Sine of x: Cosine Form === {{:Cosine of x minus Sine of x/Cosine Form}} == Multiple of Sine plus Multiple of Cosine == {{:Multiple of Sine plus Multiple of Cosine}}"}
+{"_id": "10086", "title": "Multiple of Sine plus Multiple of Cosine/Cosine Form", "text": ":$p \\sin x + q \\cos x = \\sqrt {p^2 + q^2} \\map \\cos {x + \\arctan \\dfrac {-p} q}$"}
+{"_id": "10087", "title": "Multiple of Sine plus Multiple of Cosine/Sine Form", "text": ":$p \\sin x + q \\cos x = \\sqrt {p^2 + q^2} \\sin \\left({x + \\arctan \\dfrac q p}\\right)$"}
+{"_id": "10090", "title": "Primitive of Cosine of a x over Sine of a x plus phi", "text": ":$\\displaystyle \\int \\frac {\\cos a x \\rd x} {\\map \\sin {a x + \\phi} } = \\frac {\\ln \\size {\\map \\sin {a x + \\phi} } } {a \\cos \\phi} + \\tan \\phi \\int \\frac {\\sin a x \\rd x} {\\map \\sin {a x + \\phi} } + C$"}
+{"_id": "10091", "title": "Primitive of Sine of a x over Sine of a x plus phi", "text": ":$\\displaystyle \\int \\frac {\\sin a x \\ \\mathrm d x} {\\sin \\left({a x + \\phi}\\right)} = \\frac x {\\cos \\phi} - \\tan \\phi \\int \\frac {\\cos a x \\ \\mathrm d x} {\\sin \\left({a x + \\phi}\\right)} + C$"}
+{"_id": "10092", "title": "Reciprocal of One Plus Hyperbolic Cosine", "text": ":$\\dfrac 1 {1 + \\cosh x} = \\dfrac 1 2 \\operatorname{sech}^2 \\dfrac x 2$"}
+{"_id": "10093", "title": "Reciprocal of Hyperbolic Cosine Minus One", "text": ":$\\dfrac 1 {\\cosh x - 1} = \\dfrac 1 2 \\csch^2 \\dfrac x 2$"}
+{"_id": "10097", "title": "Triangle Right-Angle-Hypotenuse-Side Equality", "text": "If two triangles have: : one right angle each : the sides opposite to the right angle equal : another two respective sides equal they will also have: : their third sides equal : the remaining two angles equal to their respective remaining angles."}
+{"_id": "10098", "title": "Internal Angles of Regular Polygon", "text": "The size $A$ of each internal angle of a regular $n$-gon is given by: :$A = \\dfrac {\\paren {n - 2} 180 \\degrees} n$"}
+{"_id": "10099", "title": "Sum of Internal Angles of Polygon", "text": "The sum $S$ of all internal angles of a polygon with $n$ sides is given by the formula $S = \\paren {n - 2} 180 \\degrees$."}
+{"_id": "10100", "title": "Five Platonic Solids", "text": "There exist exactly five platonic solids: :$\\paren 1: \\quad$ the regular tetrahedron :$\\paren 2: \\quad$ the cube :$\\paren 3: \\quad$ the regular octahedron :$\\paren 4: \\quad$ the regular dodecahedron :$\\paren 5: \\quad$ the regular icosahedron. {{:Euclid:Proposition/XIII/18/Endnote}}"}
+{"_id": "10102", "title": "Geometric Sequence with Coprime Extremes is in Lowest Terms", "text": "Let $G_n = \\sequence {a_0, a_1, \\ldots, a_n}$ be a geometric sequence of integers. Let: :$a_0 \\perp a_n$ where $\\perp$ denotes coprimality. Then $G_n$ is in its lowest terms."}
+{"_id": "10103", "title": "Measurements of Common Angles/Straight Angle", "text": "The measurement of a straight angle is $\\dfrac{360^\\circ} 2 = 180^\\circ$ or $\\dfrac {2 \\pi} 2 = \\pi$."}
+{"_id": "10104", "title": "Measurements of Common Angles/Right Angle", "text": "The measurement of a right angle is $\\dfrac {180 \\degrees} 2 = 90 \\degrees$ or $\\dfrac \\pi 2$."}
+{"_id": "10105", "title": "Measurements of Common Angles/Full Angle", "text": "A full angle is equal to one full rotation."}
+{"_id": "10106", "title": "Measurements of Common Angles/Acute Angle", "text": "An acute angle measures $\\theta$, where: :$0 \\degrees < \\theta < 90 \\degrees$ or: :$0 < \\theta < \\dfrac \\pi 2$"}
+{"_id": "10107", "title": "Measurements of Common Angles/Obtuse Angle", "text": "An obtuse angle measures $\\theta$, where: : $90^\\circ < \\theta < 180^\\circ$ or: : $\\dfrac \\pi 2 < \\theta < \\pi$"}
+{"_id": "10108", "title": "Measurements of Common Angles/Reflex Angle", "text": "A reflex angle measures $\\theta$, where: : $180^\\circ < \\theta < 360^\\circ$ or: : $\\pi < \\theta < 2 \\pi$"}
+{"_id": "10111", "title": "Supplementary Interior Angles implies Parallel Lines", "text": "Given two infinite straight lines which are cut by a transversal, if the interior angles on the same side of the transversal are supplementary, then the lines are parallel."}
+{"_id": "10112", "title": "Equal Corresponding Angles implies Parallel Lines", "text": "Given two infinite straight lines which are cut by a transversal, if the corresponding angles are equal, then the lines are parallel."}
+{"_id": "10113", "title": "Equivalence of Definitions of Tangent Vector", "text": "Let $M$ be a smooth manifold. Let $m \\in M$ be a point.  Let $V$ be an open neighborhood of $m$.  Let $C^\\infty \\left({V, \\R}\\right)$ be defined as the set of all smooth mappings $f: V \\to \\R$. {{TFAE|def = Tangent Vector}}"}
+{"_id": "10114", "title": "Parallelism implies Equal Alternate Angles", "text": "Given two infinite straight lines which are cut by a transversal, if the lines are parallel, then the alternate angles are equal."}
+{"_id": "10115", "title": "Parallelism implies Equal Corresponding Angles", "text": "Given two infinite straight lines which are cut by a transversal, if the lines are parallel, then the corresponding angles are equal."}
+{"_id": "10116", "title": "Proportion is Reflexive", "text": "Proportion is a reflexive relation. That is, every real variable is proportional to itself: :$\\forall x \\in \\R: x \\propto x$"}
+{"_id": "10117", "title": "Proportion is Symmetric", "text": "Proportion is a symmetric relation. That is: :$\\forall x, y \\in \\R: x \\propto y \\implies y \\propto x$"}
+{"_id": "10118", "title": "Proportion is Transitive", "text": "Proportion is a transitive relation. That is: :$\\forall x,y,z \\in \\R: x \\propto y \\land y \\propto z \\implies x \\propto z$"}
+{"_id": "10119", "title": "Proportion is Equivalence Relation", "text": "Proportion is an equivalence relation."}
+{"_id": "10120", "title": "Parallelism implies Supplementary Interior Angles", "text": "Given two infinite straight lines which are cut by a transversal, if the lines are parallel, then the interior angles on the same side of the transversal are supplementary."}
+{"_id": "10122", "title": "Sine of 18 Degrees", "text": ":$\\sin 18 \\degrees = \\sin \\dfrac \\pi {10} = \\dfrac {\\sqrt 5 - 1} 4$ where $\\sin$ denotes the sine function."}
+{"_id": "10123", "title": "Cosine of 72 Degrees", "text": ":$\\cos 72 \\degrees = \\cos \\dfrac {2 \\pi} 5 = \\dfrac {\\sqrt 5 - 1} 4$"}
+{"_id": "10124", "title": "Sine of 72 Degrees", "text": ":$\\sin 72 \\degrees = \\sin \\dfrac {2 \\pi} 5 = \\dfrac {\\sqrt{10 + 2 \\sqrt 5} } 4$ where $\\sin$ denotes the sine function."}
+{"_id": "10125", "title": "Cosine of 18 Degrees", "text": ":$\\cos 18^\\circ = \\cos \\dfrac \\pi {10} = \\dfrac {\\sqrt{10 + 2 \\sqrt 5}} 4$ where $\\cos$ denotes the cosine function."}
+{"_id": "10127", "title": "Sine of 3 Degrees", "text": ":$\\sin 3^\\circ = \\sin \\dfrac \\pi {60} = \\dfrac {\\sqrt{30} + \\sqrt{10} - \\sqrt 6 - \\sqrt 2 - 2 \\sqrt {15 + 3 \\sqrt 5} + 2 \\sqrt {5 + \\sqrt 5} } {16}$ where $\\sin$ denotes the sine function."}
+{"_id": "10128", "title": "Cosine of 3 Degrees", "text": ":$\\cos 3^\\circ = \\cos \\dfrac {\\pi} {60} = \\dfrac{ \\sqrt{30} - \\sqrt{10} - \\sqrt 6 + \\sqrt 2 + 2 \\sqrt{15 + 3 \\sqrt 5} + 2 \\sqrt{5 + \\sqrt 5} } {16}$ where $\\cos$ denotes the cosine function."}
+{"_id": "10129", "title": "Square Root of Sum as Sum of Square Roots", "text": "Let $a, b \\in \\R, a \\ge b$. Then: :$\\sqrt {a + b} = \\sqrt {\\dfrac a 2 + \\dfrac {\\sqrt {a^2 - b^2}} 2} + \\sqrt {\\dfrac a 2 - \\dfrac {\\sqrt {a^2 - b^2}} 2}$"}
+{"_id": "10130", "title": "Sine of 1 Degree", "text": ":$\\sin 1^\\circ = \\sin \\dfrac {\\pi} {180} = $ where $\\sin$ denotes the sine function."}
+{"_id": "10131", "title": "Rectangle is Parallelogram", "text": "Let $ABCD$ be a rectangle. Then $ABCD$ is a parallelogram."}
+{"_id": "10135", "title": "Line at Right Angles to Diameter of Circle/Porism", "text": "{{:Euclid:Proposition/III/16/Porism}}"}
+{"_id": "10141", "title": "Characteristic Function of Union/Variant 2", "text": ":$\\chi_{A \\mathop \\cup B} = \\chi_A + \\chi_B - \\chi_{A \\mathop \\cap B}$"}
+{"_id": "10143", "title": "Characteristic Function of Intersection/Variant 1", "text": ":$\\chi_{A \\cap B} = \\chi_A \\chi_B$"}
+{"_id": "10148", "title": "Sum of Square Roots as Square Root of Sum", "text": ":$\\sqrt a + \\sqrt b = \\sqrt {a + b + \\sqrt {4 a b} }$"}
+{"_id": "10160", "title": "Divisor of One of Coprime Numbers is Coprime to Other", "text": "Let $a, b \\in \\N$ be numbers such that $a$ and $b$ are coprime: :$a \\perp b$ Let $c > 1$ be a divisor of $a$: :$c \\divides a$ Then $c$ and $b$ are coprime: :$c \\perp b$ {{:Euclid:Proposition/VII/23}}"}
+{"_id": "10163", "title": "Integer Coprime to all Factors is Coprime to Whole/Proof 1", "text": "Let $a, b \\in \\Z$. Let $\\displaystyle b = \\prod_{j \\mathop = 1}^r b_j$ Let $a$ be coprime to each of $b_1, \\ldots, b_r$. Then $a$ is coprime to $b$."}
+{"_id": "10164", "title": "Integer Coprime to all Factors is Coprime to Whole/Proof 2", "text": "Let $a, b, c \\in \\Z$. Let $c$ be coprime to each of $a$ and $b$. Then $c$ is coprime to $a b$. {{:Euclid:Proposition/VII/24}}"}
+{"_id": "10165", "title": "Square of Coprime Number is Coprime", "text": "Let $a$ and $b$ be coprime integers: :$a, b \\in \\Z: a \\perp b$ Then: :$a^2 \\perp b$ {{:Euclid:Proposition/VII/25}}"}
+{"_id": "10166", "title": "Product of Coprime Pairs is Coprime", "text": "Let $a, b, c, d$ be integers. Let:  :$a \\perp c, b \\perp c, a \\perp d, b \\perp d$ where $a \\perp c$ denotes that $a$ and $c$ are coprime. Then: :$a b \\perp c d$ {{:Euclid:Proposition/VII/26}}"}
+{"_id": "10167", "title": "Powers of Coprime Numbers are Coprime", "text": "Let $a, b$ be coprime integers: :$a \\perp b$ Then: :$\\forall n \\in \\N_{>0}: a^n \\perp b^n$ {{:Euclid:Proposition/VII/27}}"}
+{"_id": "10168", "title": "Numbers are Coprime iff Sum is Coprime to Both", "text": "Let $a, b$ be integers. Then: :$a \\perp b \\iff a \\perp \\paren {a + b}$ where $a \\perp b$ denotes that $a$ and $b$ are coprime. {{:Euclid:Proposition/VII/28}}"}
+{"_id": "10172", "title": "Natural Number is Prime or has Prime Factor", "text": "Let $a$ be a natural number greater than $1$. Then either: :$a$ is a prime number or: :there exists a prime number $p \\ne a$ such that $p \\divides a$ where $\\divides$ denotes '''is a divisor of'''. {{:Euclid:Proposition/VII/32}}"}
+{"_id": "10177", "title": "Negative of Real Function that Increases Without Bound", "text": "Let $f: \\R \\to \\R$ be a real function. Then: :$(1): \\quad \\displaystyle \\lim_{x \\mathop \\to +\\infty} \\map f x = +\\infty \\implies \\lim_{x \\mathop \\to +\\infty} -\\map f x = -\\infty$ :$(2): \\quad \\displaystyle \\lim_{x \\mathop \\to -\\infty} \\map f x = +\\infty \\implies \\lim_{x \\mathop \\to -\\infty} -\\map f x = -\\infty$"}
+{"_id": "10178", "title": "Integer Divided by Divisor is Integer", "text": "Let $a, b \\in \\N$. Then: :$b \\divides a \\implies \\dfrac 1 b \\times a \\in \\N$ where $\\divides$ denotes divisibilty. {{:Euclid:Proposition/VII/37}}"}
+{"_id": "10179", "title": "Divisor is Reciprocal of Divisor of Integer", "text": "Let $a, b, c \\in \\Z_{>0}$. Then: :$b = \\dfrac 1 c \\times a \\implies c \\divides a$ where $\\divides$ denotes divisibilty. {{:Euclid:Proposition/VII/38}}"}
+{"_id": "10181", "title": "Negative of Real Function that Decreases Without Bound", "text": "Let $f: \\R \\to \\R$ be a real function. Then: :$(1): \\quad \\displaystyle \\lim_{x \\mathop \\to +\\infty} \\map f x = -\\infty \\implies \\lim_{x \\mathop \\to +\\infty} -\\map f x = +\\infty$ :$(2): \\quad \\displaystyle \\lim_{x \\mathop \\to -\\infty} \\map f x = -\\infty \\implies \\lim_{x \\mathop \\to -\\infty} -\\map f x = +\\infty$"}
+{"_id": "10182", "title": "Construction of Geometric Sequence in Lowest Terms", "text": "It is possible to find a geometric sequence of integers $G_n$ of length $n + 1$ with a given common ratio such that $G_n$ is in its lowest terms. {{:Euclid:Proposition/VIII/2}}"}
+{"_id": "10183", "title": "Geometric Sequence in Lowest Terms has Coprime Extremes", "text": "A geometric sequence of integers in lowest terms has extremes which are coprime. {{:Euclid:Proposition/VIII/3}}"}
+{"_id": "10187", "title": "First Element of Geometric Sequence not dividing Second", "text": "Let $P = \\sequence {a_j}_{0 \\mathop \\le j \\mathop \\le n}$ be a geometric sequence of integers of length $n$. Let $a_0$ not be a divisor of $a_1$. Then: :$\\forall j, k \\in \\set {0, 1, \\ldots, n}, j \\ne k: a_j \\nmid a_k$ That is, if the initial term of $P$ does not divide the second, no term of $P$ divides any other term of $P$. {{:Euclid:Proposition/VIII/6}}"}
+{"_id": "10188", "title": "First Element of Geometric Sequence that divides Last also divides Second", "text": "Let $P = \\sequence {a_j}_{0 \\mathop \\le j \\mathop \\le n}$ be a geometric sequence of integers of length $n$. Let $a_0$ be a divisor of $a_n$. Then $a_0$ is a divisor of $a_2$. {{:Euclid:Proposition/VIII/7}}"}
+{"_id": "10189", "title": "Geometric Sequences in Proportion have Same Number of Elements", "text": "Let $P = \\sequence {a_j}_{0 \\mathop \\le j \\mathop \\le n}$ be a geometric sequence of integers of length $n$. Let $r$ be the common ratio of $P$. Let $Q = \\sequence {b_j}_{0 \\mathop \\le j \\mathop \\le m}$ be a geometric sequence of integers of length $m$. Let $r$ be the common ratio of $Q$. Let $b_0$ and $b_m$ be such that $\\dfrac {b_0} {b_m} = \\dfrac {a_0} {a_n}$. Then $m = n$. {{:Euclid:Proposition/VIII/8}}"}
+{"_id": "10190", "title": "Elements of Geometric Sequence between Coprime Numbers", "text": "Let $P = \\sequence {a_j}_{0 \\mathop \\le j \\mathop \\le n}$ be a geometric sequence of integers of length $n + 1$. Let $a_0$ be coprime to $a_n$. Then there exist geometric sequences of integers $Q_1$ and $Q_2$ of length $n + 1$ such that: :the initial term of both $Q_1$ and $Q_2$ is $1$ :the final term of $Q_1$ is $a_0$ :the final term of $Q_2$ is $a_n$. {{:Euclid:Proposition/VIII/9}}"}
+{"_id": "10191", "title": "Product of Geometric Sequences from One", "text": "Let $Q_1 = \\sequence {a_j}_{0 \\mathop \\le j \\mathop \\le n}$ and $Q_2 = \\sequence {b_j}_{0 \\mathop \\le j \\mathop \\le n}$ be geometric sequences of integers of length $n + 1$. Let $a_0 = b_0 = 1$. Then the sequence $P = \\sequence {p_j}_{0 \\mathop \\le j \\mathop \\le n}$ defined as: :$\\forall j \\in \\set {0, \\ldots, n}: p_j = b^j a^{n - j}$ is a geometric sequence. {{:Euclid:Proposition/VIII/10}}"}
+{"_id": "10194", "title": "Powers of Elements of Geometric Sequence are in Geometric Sequence", "text": "Let $P = \\sequence {a_j}_{0 \\mathop \\le j \\mathop \\le n}$ be a geometric sequence of integers. Then the sequence $Q = \\sequence {b_j}_{0 \\mathop \\le j \\mathop \\le n}$ defined as: :$\\forall j \\in \\set {0, 1, \\ldots, n}: b_j = a_j^m$ where $m \\in \\Z_{>0}$, is a geometric sequence. {{:Euclid:Proposition/VIII/13}}"}
+{"_id": "10195", "title": "Number divides Number iff Square divides Square", "text": "Let $a, b \\in \\Z$. Then: :$a^2 \\divides b^2 \\iff a \\divides b$ where $\\divides$ denotes integer divisibility. {{:Euclid:Proposition/VIII/14}}"}
+{"_id": "10196", "title": "Number divides Number iff Cube divides Cube", "text": "Let $a, b \\in \\Z$. Then: :$a^3 \\divides b^3 \\iff a \\divides b$ where $\\divides$ denotes integer divisibility. {{:Euclid:Proposition/VIII/15}}"}
+{"_id": "10198", "title": "Number does not divide Number iff Cube does not divide Cube", "text": "Let $a, b \\in \\Z$ be integers. Then: :$a \\nmid b \\iff a^3 \\nmid b^3$ where $a \\nmid b$ denotes that $a$ is not a divisor of $b$. {{:Euclid:Proposition/VIII/17}}"}
+{"_id": "10204", "title": "If First of Four Numbers in Geometric Sequence is Cube then Fourth is Cube", "text": "Let $P = \\tuple {a, b, c, d}$ be a geometric sequence of integers. Let $a$ be a cube number. Then $d$ is also a cube number. {{:Euclid:Proposition/VIII/23}}"}
+{"_id": "10205", "title": "If Ratio of Square to Number is as between Two Squares then Number is Square", "text": "Let $a, b, c, d \\in \\Z$ be integers such that: :$\\dfrac a b = \\dfrac {c^2} {d^2}$ Let $a$ be a square number. Then $b$ is also a square number. {{:Euclid:Proposition/VIII/24}}"}
+{"_id": "10211", "title": "Square of Cube Number is Cube", "text": "Let $a \\in \\N$ be a natural number. Let $a$ be a cube number. Then $a^2$ is also a cube number. {{:Euclid:Proposition/IX/3}}"}
+{"_id": "10212", "title": "Cube Number multiplied by Cube Number is Cube", "text": "Let $a, b \\in \\N$ be natural numbers. Let $a$ and $b$ be cube numbers. Then $a b$ is also a cube number. {{:Euclid:Proposition/IX/4}}"}
+{"_id": "10214", "title": "Number Squared making Cube is itself Cube", "text": "Let $a \\in \\Z$ be an integer. Let $a^2$ be a cube number. Then $a$ is a cube number. {{:Euclid:Proposition/IX/6}}"}
+{"_id": "10215", "title": "Product of Composite Number with Number is Solid Number", "text": "Let $a, b \\in \\Z$ be positive integers. Let $a$ be a composite number. Then $a b$ is a solid number. {{:Euclid:Proposition/IX/7}}"}
+{"_id": "10216", "title": "Square Numbers are Similar Plane Numbers", "text": "Let $a$ and $b$ be square numbers. Then $a$ and $b$ are similar plane numbers."}
+{"_id": "10218", "title": "Elements of Geometric Sequence from One where First Element is Power of Number", "text": "Let $G_n = \\sequence {a_n}_{0 \\mathop \\le i \\mathop \\le n}$ be a geometric sequence of integers. Let $a_0 = 1$. Let $m \\in \\Z_{> 0}$. Let $a_1$ be the $m$th power of an integer. Then all the terms of $G_n$ are $m$th powers of  integers. {{:Euclid:Proposition/IX/9}}"}
+{"_id": "10225", "title": "Sum of Pair of Elements of Geometric Sequence with Three Elements in Lowest Terms is Coprime to other Element", "text": "Let $P = \\tuple {a, b, c}$ be a geometric sequence of integers in its lowest terms. Then $\\paren {a + b}$, $\\paren {b + c}$ and $\\paren {a + c}$ are all coprime to each of $a$, $b$ and $c$. {{:Euclid:Proposition/IX/15}}"}
+{"_id": "10226", "title": "Two Coprime Integers have no Third Integer Proportional", "text": "Let $a, b \\in \\Z_{>0}$ be integers such that $a$ and $b$ are coprime. Then there is no integer $c \\in \\Z$ such that: :$\\dfrac a b = \\dfrac b c$ {{:Euclid:Proposition/IX/16}}"}
+{"_id": "10227", "title": "Last Element of Geometric Sequence with Coprime Extremes has no Integer Proportional as First to Second", "text": "Let $G_n = \\sequence {a_j}_{0 \\mathop \\le j \\mathop \\le n}$ be a geometric sequence of integers such that $a_0 \\ne 1$. Let $a_0 \\perp a_n$, where $\\perp$ denotes coprimality. Then there does not exist an integer $b$ such that: :$\\dfrac {a_0} {a_1} = \\dfrac {a_n} b$ {{:Euclid:Proposition/IX/17}}"}
+{"_id": "10228", "title": "Condition for Existence of Third Number Proportional to Two Numbers", "text": "Let $a, b, c \\in \\Z$ be integers. Let $\\tuple {a, b, c}$ be a geometric sequence. In order for this to be possible, both of these conditions must be true: :$(1): \\quad a$ and $b$ cannot be coprime :$(2): \\quad a \\divides b^2$ where $\\divides$ denotes divisibility. {{:Euclid:Proposition/IX/18}}"}
+{"_id": "10238", "title": "Odd Divisor of Even Number also divides its Half", "text": "Let $a, b \\in \\Z$ be integers. Let $a$ be odd and $b$ be even. Let: :$a \\divides b$ where $\\divides$ denotes divisibility. Then: :$a \\divides \\dfrac b 2$  {{:Euclid:Proposition/IX/30}}"}
+{"_id": "10239", "title": "Odd Number Coprime to Number is also Coprime to its Double", "text": "Let $a, b \\in \\Z$ be integers. Let $a$ be odd. Let: : $a \\perp b$ where $\\perp$ denotes coprimality. Then: : $a \\perp 2 b$ {{:Euclid:Proposition/IX/31}}"}
+{"_id": "10242", "title": "Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd", "text": "Let $a \\in \\Z$ be an integer such that: :$(1): \\quad a$ is not a power of $2$ :$(2): \\quad \\dfrac a 2$ is an even integer. Then $a$ is both even-times even and even-times odd. {{:Euclid:Proposition/IX/34}}"}
+{"_id": "10243", "title": "Scaled Real Function that Increases Without Bound", "text": "Let $f: \\R \\to \\R$ be a real function. Let $\\lambda \\in \\R_{\\ne 0}$ be a nonzero constant. Then: For $\\lambda > 0$: :$\\displaystyle \\lim_{x \\mathop \\to +\\infty} \\map f x = +\\infty \\implies \\lim_{x \\mathop \\to +\\infty} \\lambda \\map f x = +\\infty$ :$\\displaystyle \\lim_{x \\mathop \\to -\\infty} \\map f x = +\\infty \\implies \\lim_{x \\mathop \\to -\\infty} \\lambda \\map f x = + \\infty$ For $\\lambda < 0$: :$\\displaystyle \\lim_{x \\mathop \\to +\\infty} \\map f x = +\\infty \\implies \\lim_{x \\mathop \\to +\\infty} \\lambda \\map f x = -\\infty$ :$\\displaystyle \\lim_{x \\mathop \\to -\\infty} \\map f x = +\\infty \\implies \\lim_{x \\mathop \\to -\\infty} \\lambda \\map f x = -\\infty$"}
+{"_id": "10244", "title": "Scaled Real Function that Decreases Without Bound", "text": "Let $f: \\R \\to \\R$ be a real function. Let $\\lambda \\in \\R_{\\ne 0}$ be a nonzero constant. Then: For $\\lambda > 0$: :$\\displaystyle \\lim_{x \\mathop \\to +\\infty} \\map f x = -\\infty \\implies \\lim_{x \\mathop \\to +\\infty} \\lambda \\map f x = -\\infty$ :$\\displaystyle \\lim_{x \\mathop \\to -\\infty} \\map f x = -\\infty \\implies \\lim_{x \\mathop \\to -\\infty} \\lambda \\map f x = -\\infty$ For $\\lambda < 0$: :$\\displaystyle \\lim_{x \\mathop \\to +\\infty} \\map f x = -\\infty \\implies \\lim_{x \\mathop \\to +\\infty} \\lambda \\map f x = +\\infty$ :$\\displaystyle \\lim_{x \\mathop \\to -\\infty} \\map f x = -\\infty \\implies \\lim_{x \\mathop \\to -\\infty} \\lambda \\map f x = +\\infty$"}
+{"_id": "10245", "title": "Existence of Fraction of Number Smaller than Given", "text": "Let $a, b \\in \\R_{>0}$ be two (strictly) positive real numbers, such that $a > b$. Let the sequence $\\left\\langle{a_n}\\right\\rangle$ be defined recursively as: :$a_i = \\begin{cases} a & : i = 1 \\\\ a_{i - 1} - c: \\dfrac {a_{i - 1} } 2 < c < a_{i - 1} & : i > 1 \\end{cases}$ Then: : $\\exists n \\in \\N_{>0}: a_n < b$ {{:Euclid:Proposition/X/1}}"}
+{"_id": "10249", "title": "Restriction of Continuous Mapping is Continuous/Topological Spaces", "text": "Let $T_1 = \\left({S_1, \\tau_1}\\right)$ and $T_2 = \\left({S_2, \\tau_2}\\right)$ be topological spaces. Let $M_1 \\subseteq S_1$ be a subset of $S_1$. Let $f: S_1 \\to S_2$ be a mapping which is continuous. Let $M_2 \\subseteq S_2$ be a subset of $S_2$ such that $f\\left[{M_1}\\right] \\subseteq M_2$. Let $f {\\restriction_{M_1 \\times M_2}}: M_1 \\to M_2$ be the restriction of $f$ to $M_1 \\times M_2$. Then $f {\\restriction_{M_1 \\times M_2}}$ is continuous, where $M_1$ and $M_2$ are equipped with the respective subspace topologies."}
+{"_id": "10250", "title": "Restriction of Continuous Mapping is Continuous/Metric Spaces", "text": "Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces. Let $S \\subseteq M_1$ be a subset of $M_1$. Let $f: M_1 \\to M_2$ be a mapping which is continuous at a point $\\alpha \\in S$. Let $f \\restriction_S = g: S \\to M_2$ be the restriction of $f$ to $S$. Then $g$ is continuous at $\\alpha$."}
+{"_id": "10251", "title": "Restriction of Inverse is Inverse of Restriction", "text": "Let $S_1$ and $S_2$ be sets. Let $f: S_1 \\to S_2$ be a bijection. Let $S \\subseteq S_1$ be a subset of $S_1$. Let $f^{-1}$ be the inverse of $f$. Let $f {\\restriction_{S \\times f \\left[{S}\\right]}}$ be the restriction of $f$ to $S \\times f\\left[{S}\\right]$. Let $f^{-1} {\\restriction_{f \\left[{S}\\right] \\times S}}$ be the restriction of $f^{-1}$ to $f \\left[{S}\\right] \\times S$. Then: : $f {\\restriction_{S \\times f \\left[{S}\\right]}}$ is a bijection and: : ${\\left({f {\\restriction_{S \\times f \\left[{S}\\right]}} }\\right)}^{-1} = f^{-1} {\\restriction_{f \\left[{S}\\right] \\times S}}$"}
+{"_id": "10253", "title": "Hölder's Inequality/General", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. For $i = 1, \\ldots, n$ let $p_i \\in \\R$ such that: :$\\displaystyle \\sum_{i \\mathop = 1}^n \\frac 1 {p_i} = 1$ Let $f_i \\in \\map {\\LL^{p_i} } \\mu, f_i: X \\to \\R$, where $\\LL$ denotes Lebesgue space. Then their pointwise product $\\displaystyle \\prod_{i \\mathop = 1}^n f_i$ is integrable, that is: :$\\displaystyle \\prod_{i \\mathop = 1}^n f_i \\in \\map {\\LL^1} \\mu$ and: :$\\displaystyle \\norm {\\prod_{i \\mathop = 1}^n f_i}_1 = \\int \\size {\\prod_{i \\mathop = 1}^n f_i} \\rd \\mu \\le \\prod_{i \\mathop = 1}^n \\norm {f_i}_{p_i}$ where the various instances of $\\norm {\\, \\cdot \\,}$ signify $p$-seminorms."}
+{"_id": "10254", "title": "Theorem of Even Perfect Numbers/Sufficient Condition", "text": "Let $n \\in \\N$ be such that $2^n - 1$ is prime. Then $2^{n - 1} \\paren {2^n - 1}$ is perfect. {{:Euclid:Proposition/IX/36}}"}
+{"_id": "10255", "title": "Theorem of Even Perfect Numbers/Necessary Condition", "text": "Let $a \\in \\N$ be an even perfect number. Then $a$ is in the form: :$2^{n - 1} \\paren {2^n - 1}$ where $2^n - 1$ is prime."}
+{"_id": "10269", "title": "Condition for Commensurability of Roots of Quadratic Equation/Lemma", "text": "{{:Euclid:Proposition/X/17/Lemma}}"}
+{"_id": "10271", "title": "Condition for Incommensurability of Roots of Quadratic Equation", "text": "Consider the quadratic equation: :$(1): \\quad a x - x^2 = \\dfrac {b^2} 4$ Then $x$ and $a - x$ are incommensurable {{iff}} $\\sqrt {a^2 - b^2}$ and $a$ are incommensurable. {{:Euclid:Proposition/X/18}}"}
+{"_id": "10272", "title": "Product of Rationally Expressible Numbers is Rational/Lemma", "text": "{{:Euclid:Proposition/X/19/Lemma}}"}
+{"_id": "10273", "title": "Product of Rationally Expressible Numbers is Rational", "text": "Let $a, b \\in \\set {x \\in \\R_{>0} : x^2 \\in \\Q}$ be the lengths of rational line segments. Furthermore, let $\\dfrac a b \\in \\Q$. Then, $a b \\in \\Q$. {{:Euclid:Proposition/X/19}}"}
+{"_id": "10333", "title": "Construction of Apotome is Unique", "text": "Let $D$ be the domain $\\left\\{{x \\in \\R_{>0} : x^2 \\in \\Q}\\right\\}$, the rationally expressible numbers. Let $a, b \\in D$ be two rationally expressible numbers such that $a - b$ is an apotome. Then, there exists only one $x \\in D$ such that $a - b + x$ and $a$ are commensurable in square only. {{:Euclid:Proposition/X/79}}"}
+{"_id": "10370", "title": "Lower Bound of Natural Logarithm", "text": ":$\\forall x \\in \\R_{>0}: 1 - \\dfrac 1 x \\le \\ln x$ where $\\ln x$ denotes the natural logarithm of $x$."}
+{"_id": "10371", "title": "Bounds of Natural Logarithm", "text": "Let $\\ln y$ be the natural logarithm of $y$ where $y \\in \\R_{>0}$. Then $\\ln$ satisfies the compound inequality: :$\\displaystyle 1 - \\frac 1 y \\le \\ln y \\le y - 1$"}
+{"_id": "10396", "title": "Construction of Solid Angle from Three Plane Angles any Two of which are Greater than Other Angle/Lemma", "text": "{{:Euclid:Proposition/XI/23/Lemma}}"}
+{"_id": "10424", "title": "Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights", "text": "{{:Euclid:Proposition/XII/9}}"}
+{"_id": "10433", "title": "Construction of Polyhedron in Outer of Concentric Spheres/Porism", "text": "{{:Euclid:Proposition/XII/17/Porism}}"}
+{"_id": "10461", "title": "Definition:Elevated/Point", "text": "Let $P$ be a plane. An '''elevated point (relative to $P$)''' is a point in space which does not lie on $P$."}
+{"_id": "10462", "title": "Definition:Elevated/Line", "text": "Let $P$ be a plane. An '''elevated line (relative to $P$)''' is a line in space which does not lie entirely in $P$."}
+{"_id": "10465", "title": "Common Ratio in Rational Geometric Sequence is Rational", "text": "Let $\\sequence {a_k}$ be a geometric sequence whose terms are rational. Then the common ratio of $\\sequence {a_k}$ is rational."}
+{"_id": "10466", "title": "Common Ratio in Integer Geometric Sequence is Rational", "text": "Let $\\sequence {a_k}$ be a geometric sequence whose terms are all integers. Then the common ratio of $\\sequence {a_k}$ is rational."}
+{"_id": "10473", "title": "Form of Geometric Sequence of Integers", "text": "Let $P = \\sequence {a_j}_{0 \\mathop \\le j \\mathop \\le n}$ be a geometric sequence of length $n + 1$ consisting of integers only. Then the $j$th term of $P$ is given by: :$a_j = k p^{n - j} q^j$ where: : the common ratio of $P$ expressed in canonical form is $\\dfrac q p$ : $k$ is an integer."}
+{"_id": "10476", "title": "Geometric Sequence of Integers with Integer Common Ratio", "text": "Let $P = \\sequence {a_j}_{1 \\mathop \\le j \\mathop \\le n}$ be a geometric sequence of length $n$ consisting entirely of integers. Let $r$ be the common ratio of $P$. Then $r$ is an integer {{iff}}: :$\\forall i, j \\in \\set {1, 2, \\ldots, n}, i \\le j: a_i \\divides a_j$ That is, terms of $P$ divide later terms of $P$ {{iff}} $r$ is an integer."}
+{"_id": "10477", "title": "Integers are Coprime iff Powers are Coprime", "text": "Let $a, b \\in \\Z$ be integers. Then: :$a \\perp b \\iff \\forall n \\in \\N: a^n \\perp b^n$ That is, two integers are coprime {{iff}} all their positive integer powers are coprime."}
+{"_id": "10479", "title": "Integer Addition is Cancellable", "text": "The operation of addition on the set of integers $\\Z$ is cancellable: :$\\forall x, y, z \\in \\Z: x + z = y + z \\implies x = y$"}
+{"_id": "10481", "title": "Complement Relative to Subset is Subset of Complement Relative to Superset", "text": "Let $A, B, C$ be sets such that $A \\subseteq B \\subseteq C$. Then: :$\\relcomp B A \\subseteq \\relcomp C A$"}
+{"_id": "10482", "title": "Complement of Relative Complement is Union with Complement", "text": "Let $A, B, C$ be sets such that $A \\subseteq B \\subseteq C$. Then: :$\\relcomp C {\\relcomp B A} = A \\cup \\relcomp C B$"}
+{"_id": "10483", "title": "Set Union Preserves Subsets/Families of Sets", "text": "Let $I$ be an indexing set. Let $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$ and $\\family {B_\\alpha}_{\\alpha \\mathop \\in I}$ be indexed families of subsets of a set $S$. Let: :$\\forall \\beta \\in I: A_\\beta \\subseteq B_\\beta$ Then: :$\\displaystyle \\bigcup_{\\alpha \\mathop \\in I} A_\\alpha \\subseteq \\bigcup_{\\alpha \\mathop \\in I} B_\\alpha$"}
+{"_id": "10484", "title": "Set Union Preserves Subsets/General Result", "text": "Let $\\mathbb S, \\mathbb T$ be sets of sets. Suppose that for each $S \\in \\mathbb S$ there exists a $T \\in \\mathbb T$ such that $S \\subseteq T$. Then $\\bigcup \\mathbb S \\subseteq \\bigcup \\mathbb T$."}
+{"_id": "10485", "title": "Intersection of Family is Subset of Intersection of Subset of Family", "text": "Let $I$ be an indexing set. Let $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$ be an indexed family of subsets of a set $S$. Let $J \\subseteq I$. Then: :$\\displaystyle \\bigcap_{\\alpha \\mathop \\in I} A_\\alpha \\subseteq \\bigcap_{\\alpha \\mathop \\in J} A_\\alpha$ where $\\displaystyle \\bigcap_{\\alpha \\mathop \\in I} A_\\alpha$ denotes the intersection of $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$."}
+{"_id": "10486", "title": "Union of Subset of Family is Subset of Union of Family", "text": "Let $I$ be an indexing set. Let $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$ be an indexed family of subsets of a set $S$. Let $J \\subseteq I$ Then: :$\\displaystyle \\bigcup_{\\alpha \\mathop \\in J} A_\\alpha \\subseteq \\bigcup_{\\alpha \\mathop \\in I} A_\\alpha$ where $\\displaystyle \\bigcup_{\\alpha \\mathop \\in I} A_\\alpha$ denotes the union of $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$."}
+{"_id": "10487", "title": "Intersection of Open Intervals of Positive Reals is Empty", "text": "Let $\\R_{> 0}$ be the set of strictly positive real numbers. For all $x \\in \\R_{> 0}$, let $A_x$ be the open real interval $\\openint 0 x$. Then: :$\\displaystyle \\bigcap_{x \\mathop \\in \\R_{> 0} } A_x = \\O$"}
+{"_id": "10490", "title": "Union of Closed Intervals of Positive Reals is Set of Positive Reals", "text": "Let $\\R_{> 0}$ be the set of strictly positive real numbers. For all $x \\in \\R_{> 0}$, let $B_x$ be the closed real interval $\\closedint 0 x$. Then: :$\\displaystyle \\bigcup_{x \\mathop \\in \\R_{> 0} } B_x = \\R_{\\ge 0}$"}
+{"_id": "10492", "title": "Subset of Cartesian Product not necessarily Cartesian Product of Subsets", "text": "Let $A$ and $B$ be sets. Let $A$ and $B$ both have at least two distinct elements. Then there exists $W \\subseteq A \\times B$ such that $W$ is not the cartesian product of a subset of $A$ and a subset of $B$."}
+{"_id": "10493", "title": "Subset equals Preimage of Image implies Injection", "text": "Let $f: S \\to T$ be a mapping. Let $f^\\to: \\powerset S \\to \\powerset T$ be the direct image mapping of $f$. Similarly, let $f^\\gets: \\powerset T \\to \\powerset S$ be the inverse image mapping of $f$. Let: :$\\forall A \\in \\powerset S: A = \\map {\\paren {f^\\gets \\circ f^\\to} } A$ Then $f$ is an injection."}
+{"_id": "10494", "title": "Preimage of Image of Subset under Injection equals Subset", "text": "Let $f: S \\to T$ be an injection. Then: :$\\forall A \\subseteq S: A = \\paren {f^{-1} \\circ f} \\sqbrk A$ where: :$f \\sqbrk A$ denotes the image of $A$ under $f$ :$f^{-1}$ denotes the inverse of $f$ :$f^{-1} \\circ f$ denotes composition of $f^{-1}$ and $f$."}
+{"_id": "10495", "title": "Subset equals Image of Preimage implies Surjection", "text": "Let $f: S \\to T$ be a mapping. Let: :$\\forall B \\subseteq T: B = \\paren {f \\circ f^{-1} } \\sqbrk B$ where $f \\sqbrk B$ denotes the image of $B$ under $f$. Then $f$ is a surjection."}
+{"_id": "10496", "title": "Image of Preimage of Subset under Surjection equals Subset", "text": "Let $f: S \\to T$ be a surjection. Then: :$\\forall B \\subseteq T: B = \\left({f \\circ f^{-1} }\\right) \\sqbrk B$ where: :$f \\sqbrk B$ denotes the image of $B$ under $f$ :$f^{-1}$ denotes the inverse of $f$ :$f \\circ f^{-1}$ denotes composition of $f$ and $f^{-1}$."}
+{"_id": "10503", "title": "Intersection of Image with Subset of Codomain", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let $A \\subseteq S$ and $B \\subseteq T$. Then: :$f \\sqbrk {A \\cap f^{-1} \\sqbrk B} = f \\sqbrk A \\cap B$"}
+{"_id": "10504", "title": "Projection is Injection iff Factor is Singleton", "text": "Let $S_1, S_2, \\ldots, S_n$ be non-empty sets. Let $\\displaystyle S = \\prod_{i \\mathop = 1}^n S_i$ be the cartesian product of $S_1, S_2, \\ldots, S_n$. Let $\\pr_j: S \\to S_j$ be the $j$th projection on $S$. Then $\\pr_j$ is an injection {{iff}} $S_k$ is a singleton for all $k \\in \\set {1, 2, \\dotsc, n}$ where $k \\ne j$."}
+{"_id": "10505", "title": "Preimage of Element under Projection", "text": "Let $A$ and $B$ be sets. Let $A \\times B$ be the cartesian product of $A$ and $B$. Let $\\pr_1: A \\times B \\to A$ be the first projection of $A \\times B$. Let $a \\in A$. Then: :$\\pr_1^{-1} \\sqbrk {\\set a} = \\set {\\tuple {a, b}: b \\in B}$ that is: :$\\pr_1^{-1} \\sqbrk {\\set a} = \\set a \\times B$"}
+{"_id": "10508", "title": "Divisors of Factorial", "text": "Let $n \\in \\N_{>0}$. Then all natural numbers less than or equal to $n$ are divisors of $n!$: :$\\forall k \\in \\left\\{{1, 2, \\ldots, n}\\right\\}: n! \\equiv 0 \\pmod k$"}
+{"_id": "10510", "title": "Positive Difference Relation on Reals is Transitive", "text": "Let $P \\subseteq \\R$ be a subset of the real numbers such that: :$(1): \\quad 1 \\in P$ :$(2): \\quad a, b \\in P \\implies a + b \\in P$ :$(3): \\quad$ For all $x \\in \\R$, exactly one of these is true: :::$x \\in P$ :::$x = 0$ :::$-x \\in P$ Let $Q \\subseteq \\R \\times \\R$ be the relation on $\\R$ defined as: :$Q = \\set {\\tuple {a, b} \\in \\R:  a - b \\in P}$ Then $Q$ is a transitive relation."}
+{"_id": "10513", "title": "Composition of Repeated Compositions of Injections", "text": "Let $S$ be a set. Let $f: S \\to S$ be an injection. Let the sequence of mappings: :$f^0, f^1, f^2, \\ldots, f^n, \\ldots$ be defined as: :$\\forall n \\in \\N: f^n \\left({x}\\right) = \\begin{cases} x & : n = 0 \\\\ f \\left({x}\\right) & : n = 1 \\\\ f \\left({f^{n-1} \\left({x}\\right)}\\right) & : n > 1 \\end{cases}$ Then: :$\\forall m, n \\in \\Z_{\\ge 0}: f^n \\circ f^m = f^{m + n}$ where $f^n \\circ f^m$ denotes composition of mappings."}
+{"_id": "10514", "title": "Derivative Function on Set of Functions induces Equivalence Relation", "text": "Let $X$ be the set of real functions $f: \\R \\to \\R$ which possess continuous derivatives. Let $\\mathcal R \\subseteq X \\times X$ be the relation on $X$ defined as: :$\\mathcal R = \\set {\\tuple {f, g} \\in X \\times X: D f = D g}$ where $D f$ denotes the first derivative of $f$. Then $\\mathcal R$ is an equivalence relation."}
+{"_id": "10515", "title": "Factorial Divides Product of Successive Numbers", "text": "Let $m, n \\in \\N_{\\ge 1}$ be natural numbers Let $m^{\\overline n}$ be $m$ to the power of $n$ rising. Then: :$m^{\\overline n} \\equiv 0 \\bmod n!$ That is, the factorial of $n$ divides the product of $n$ successive numbers."}
+{"_id": "10516", "title": "Binomial Coefficient is Integer", "text": "Let $\\dbinom n k$ be a binomial coefficient. Then $\\dbinom n k$ is an integer."}
+{"_id": "10518", "title": "Set of Mappings which map to Same Element induces Equivalence Relation", "text": "Let $X$ and $Y$ be sets. Let $E$ be the set of all mappings from $X$ to $Y$. Let $b \\in X$. Let $\\mathcal R \\subseteq E \\times E$ be the relation on $E$ defined as: :$\\mathcal R := \\set {\\tuple {f, g} \\in \\mathcal R: \\map f b = \\map g b}$ Then $\\mathcal R$ is an equivalence relation."}
+{"_id": "10520", "title": "Equivalence of Definitions of Inverse Mapping", "text": "Let $S$ and $T$ be sets. {{TFAE|def = Inverse Mapping}}"}
+{"_id": "10521", "title": "Inverse Mapping is Bijection", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ and $g: T \\to S$ be inverse mappings of each other. Then $f$ and $g$ are bijections."}
+{"_id": "10522", "title": "Mapping is Injection and Surjection iff Inverse is Mapping/Proof 2", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Then: : $f: S \\to T$ can be defined as a bijection in the sense that: ::$(1): \\quad f$ is an injection ::$(2): \\quad f$ is a surjection. {{iff}}: :the inverse $f^{-1}$ of $f$ is such that: ::for each $y \\in T$, the preimage $\\map {f^{-1} } y$ has exactly one element. :That is, such that $f^{-1} \\subseteq T \\times S$ is itself a mapping."}
+{"_id": "10523", "title": "Mapping is Extension iff Composite with Inclusion", "text": "Let $S$ and $T$ be sets. Let $A \\subseteq S$. Let $f: S \\to T$ and $g: A \\to T$ be mappings. Then $f$ is an extension of $g$ {{iff}}: :$f = g \\circ i_A$ where $i_A$ is the inclusion mapping on $A$. This can be illustrated using a commutative diagram as follows: ::$\\begin {xy} \\xymatrix@L + 2mu@ + 1em {  A \\ar[r]^*{i_A}      \\ar@{-->}[rd]_*{f = g \\circ i_A} &  S \\ar[d]^*{g} \\\\ &  T } \\end {xy}$"}
+{"_id": "10524", "title": "Cardinality of Set of Restrictions of Mapping", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let the cardinality of $S$ be $n$. Let $F$ be the set of restrictions of $f$ to a subset of $S$. Then there are $2^n$ elements of $F$."}
+{"_id": "10525", "title": "Projection is Surjection/Family of Sets", "text": "Let $\\family {S_\\alpha}_{\\alpha \\mathop \\in I}$ be a family of sets. Let $\\displaystyle \\prod_{\\alpha \\mathop \\in I} S_\\alpha$ be the Cartesian product of $\\family {S_\\alpha}_{\\alpha \\mathop \\in I}$. Let each of $S_\\alpha$ be non-empty. For each $\\beta \\in I$, let $\\displaystyle \\pr_\\beta: \\prod_{\\alpha \\mathop \\in I} S_\\alpha \\to S_\\beta$ be the $\\beta$th projection on $\\displaystyle S = \\prod_{\\alpha \\mathop \\in I} S_\\alpha$. Then $\\pr_\\beta$ is a surjection."}
+{"_id": "10527", "title": "Chebyshev Distance on Real Vector Space is Metric", "text": "The Chebyshev distance on $\\R^n$: : $\\displaystyle \\forall x, y \\in \\R^n: d_\\infty \\left({x, y}\\right):= \\max_{i \\mathop = 1}^n {\\left\\vert{x_i - y_i}\\right\\vert}$ is a metric."}
+{"_id": "10528", "title": "Chebyshev Distance is Metric", "text": "Let $M_1 = \\struct {A_1, d_1}, M_2 = \\struct {A_2, d_2}, \\ldots, M_n = \\struct {A_n, d_n}$ be metric spaces. Let $\\displaystyle \\mathcal A = \\prod_{i \\mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \\ldots, A_n$. Let $d_\\infty: \\mathcal A \\times \\mathcal A \\to \\R$ be the Chebyshev distance on $\\mathcal A$: :$\\displaystyle \\map {d_\\infty} {x, y} = \\max_{i \\mathop = 1}^n \\set {\\map {d_i} {x_i, y_i} }$ where $x = \\tuple {x_1, x_2, \\ldots, x_n}, y = \\tuple {y_1, y_2, \\ldots, y_n} \\in \\mathcal A$. Then $d_\\infty$ is a metric,"}
+{"_id": "10534", "title": "Set of Successive Numbers contains Unique Multiple", "text": "Let $m \\in \\Z_{\\ge 1}$. Then $\\set {m, m + 1, \\ldots, m + n - 1}$ contains a unique integer that is a multiple of $n$. That is, in any set containing $n$ successive integers, $n$ divides exactly one of those integers."}
+{"_id": "10535", "title": "Taxicab Metric is Topologically Equivalent to Chebyshev Distance on Real Vector Space", "text": "For $n \\in \\N$, let $\\R^n$ be a real vector space. Let $d_1$ be the taxicab metric on $\\R^n$. Let $d_\\infty$ be the Chebyshev distance on $\\R^n$. Then :$\\forall x, y \\in \\R^n: d_\\infty \\left({x, y}\\right) \\le d_1 \\left({x, y}\\right) \\le n \\cdot d_\\infty \\left({x, y}\\right)$ It follows that $d_1$ and $d_\\infty$ are Lipschitz equivalent."}
+{"_id": "10537", "title": "Euclidean Metric is Metric", "text": "Let $M_{1'} = \\left({A_{1'}, d_{1'}}\\right), M_{2'} = \\left({A_{2'}, d_{2'}}\\right), \\ldots, M_{n'} = \\left({A_{n'}, d_{n'}}\\right)$ be metric spaces. Let $\\displaystyle \\mathcal A = \\prod_{i \\mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \\ldots, A_{n'}$. The Euclidean metric on $\\mathcal A$ is a metric."}
+{"_id": "10538", "title": "Euclidean Space is Normed Space", "text": "The Euclidean norm is a norm on the Euclidean space $\\R^n$."}
+{"_id": "10539", "title": "Number of Multiples less than Given Number", "text": "Let $m, n \\in \\N_{\\ge 1}$. The number of multiples of $m$ not greater than $n$ is given by: :$q = \\floor {\\dfrac n m}$ where $\\floor {\\cdot}$ denotes the floor function"}
+{"_id": "10542", "title": "Restriction of Non-Continuous Mapping on Metric Space to Subspace may be Continuous", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be a metric spaces. Let $f: A_1 \\to A_2$ be a mapping. Let $Y \\subseteq A_1$. Let $f {\\restriction_Y}: Y \\to A_2$ be the restriction of $f$ to $Y$. Let $f {\\restriction_Y}$ be $\\left({d_Y, d_2}\\right)$-continuous. Then it is not necessarily the case that $f$ is also $\\left({d_1, d_2}\\right)$-continuous."}
+{"_id": "10543", "title": "Supremum Metric on Bounded Real Functions on Closed Interval is Metric", "text": "Let $\\closedint a b \\subseteq \\R$ be a closed real interval. Let $A$ be the set of all bounded real functions $f: \\closedint a b \\to \\R$. Let $d: A \\times A \\to \\R$ be the supremum metric on $A$. Then $d$ is a metric."}
+{"_id": "10544", "title": "Supremum Metric on Continuous Real Functions is Subspace of Bounded", "text": "Let $\\left[{a \\,.\\,.\\, b}\\right] \\subseteq \\R$ be a closed real interval. Let $\\mathscr C \\left[{a \\,.\\,.\\, b}\\right]$ be the set of all continuous functions $f: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$. Let $\\mathscr B \\left({\\left[{a \\,.\\,.\\, b}\\right], \\R}\\right)$ be the set of all bounded real functions $f: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$. Let $d$ be the supremum metric on $\\mathscr B \\left({\\left[{a \\,.\\,.\\, b}\\right], \\R}\\right)$. Then $\\left({\\mathscr C \\left[{a \\,.\\,.\\, b}\\right], d_{\\mathscr C} }\\right)$ is a subspace of $\\left({\\mathscr B \\left({\\left[{a \\,.\\,.\\, b}\\right], \\R}\\right), d}\\right)$."}
+{"_id": "10545", "title": "Zero Definite Integral of Nowhere Negative Function implies Zero Function", "text": "Let $\\left[{a \\,.\\,.\\, b}\\right] \\subseteq \\R$ be a closed real interval. Let $h: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$ be a continuous real function such that: :$\\forall x \\in \\left[{a \\,.\\,.\\, b}\\right]: h \\left({x}\\right) \\ge 0$ Let: :$\\displaystyle \\int_a^b h \\left({x}\\right) \\rd x = 0$ Then: :$\\forall x \\in \\left[{a \\,.\\,.\\, b}\\right]: h \\left({x}\\right) = 0$"}
+{"_id": "10546", "title": "L2 Metric on Closed Real Interval is Metric", "text": "Let $S$ be the set of all real functions which are continuous on the closed interval $\\closedint a b$. Let $d: S \\times S \\to \\R$ be the $L^2$ metric on $\\closedint a b$: :$\\displaystyle \\forall f, g \\in S: \\map {d_2} {f, g} := \\paren {\\int_a^b \\paren {\\map f t - \\map g t}^2 \\rd t}^{\\frac 1 2}$ Then $d_2$ is a metric."}
+{"_id": "10547", "title": "Element in Bounded Metric Space has Bound", "text": "Let $M = \\left({X, d}\\right)$ be a metric space. Let $M' = \\left({Y, d_Y}\\right)$ be a subspace of $M$. Let $M'$ be bounded in $M$. Then: :$\\forall a' \\in X: \\exists K' \\in \\R: \\forall x \\in Y: d \\left({x, a}\\right) \\le K'$ That is, if there is one element of $X$ which satisfies the condition for $Y$ to be bounded in $M$, they ''all'' do."}
+{"_id": "10549", "title": "Convergence of Square of Linear Combination of Sequences whose Squares Converge", "text": "Let $\\sequence {x_i}$ and $\\sequence {y_i}$ be real sequences such that the series $\\ds \\sum_{i \\mathop \\ge 0} x_i^2$ and $\\ds \\sum_{i \\mathop \\ge 0} y_i^2$ are convergent. Let $\\lambda, \\mu \\in \\R$ be real numbers. Then $\\ds \\sum_{i \\mathop \\ge 0} \\paren {\\lambda x_i + \\mu y_i}^2$ is convergent."}
+{"_id": "10550", "title": "Hilbert Sequence Space is Metric Space", "text": "Let $A$ be the set of all real sequences $\\left\\langle{x_i}\\right\\rangle$ such that the series $\\displaystyle \\sum_{i \\mathop \\ge 0} x_i^2$ is convergent. Let $\\ell^2 = \\left({A, d_2}\\right)$ be the Hilbert sequence space on $\\R$. Then $\\ell^2$ is a metric space."}
+{"_id": "10551", "title": "Supremum Metric on Bounded Real-Valued Functions is Metric", "text": "Let $X$ be a set. Let $A$ be the set of all bounded real-valued functions $f: X \\to \\R$. Let $d: A \\times A \\to \\R$ be the supremum metric on $A$. Then $d$ is a metric."}
+{"_id": "10552", "title": "Supremum Metric on Bounded Real Sequences is Metric", "text": "Let $A$ be the set of all bounded real sequences. Let $d: A \\times A \\to \\R$ be the supremum metric on $A$. Then $d$ is a metric."}
+{"_id": "10553", "title": "Supremum Metric is Metric", "text": "Let $S$ be a set. Let $M = \\left({A', d'}\\right)$ be a metric space. Let $A$ be the set of all bounded mappings $f: S \\to M$. Let $d: A \\times A \\to \\R$ be the supremum metric on $A$. Then $d$ is a metric."}
+{"_id": "10560", "title": "Supremum Metric on Bounded Continuous Mappings is Metric", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $A$ be the set of all continuous mappings $f: M_1 \\to M_2$ which are also bounded. Let $d: A \\times A \\to \\R$ be the supremum metric on $A$. Then $d$ is a metric."}
+{"_id": "10561", "title": "Harmonic Number is not Integer", "text": "Let $H_n$ be the $n$th harmonic number. Then $H_n$ is not an integer for $n \\ge 2$. That is, the only harmonic numbers that are integers are $H_0$ and $H_1$."}
+{"_id": "10562", "title": "Supremum Metric on Differentiability Class is Metric", "text": "Let $\\left[{a \\,.\\,.\\, b}\\right] \\subseteq \\R$ be a closed real interval. Let $r \\in \\N$ be a natural number. Let $A := \\mathscr D^r \\left[{a \\,.\\,.\\, b}\\right]$ be the set of all continuous functions $f: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$ which are of differentiability class $r$. Let $d: A \\times A \\to \\R$ be the supremum metric on $A$. Then $d$ is a metric."}
+{"_id": "10564", "title": "Subspace of Metric Space is Metric Space", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $H \\subseteq A$. Let $d_H: H \\times H \\to \\R$ be the restriction $d \\restriction_{H \\times H}$ of $d$ to $H$. Let $\\left({H, d_H}\\right)$ be a metric subspace of $\\left({A, d}\\right)$. Then $d_H$ is a metric on $H$."}
+{"_id": "10567", "title": "Lp Metric on Closed Real Interval is Metric", "text": "Let $S$ be the set of all real functions which are continuous on the closed interval $\\left[{a \\,.\\,.\\, b}\\right]$. Let $p \\in \\R_{\\ge 1}$. Let $d_p: S \\times S \\to \\R$ be the $L^p$ metric on $\\left[{a \\,.\\,.\\, b}\\right]$: :$\\displaystyle \\forall f, g \\in S: d \\left({f, g}\\right) := \\left({\\int_a^b \\left\\vert{f \\left({t}\\right) - g \\left({t}\\right)}\\right\\vert^p \\ \\mathrm d t}\\right)^{\\frac 1 p}$ Then $d_p$ is a metric."}
+{"_id": "10568", "title": "P-Sequence Space of Real Sequences is Metric Space", "text": "Let $A$ be the set of all real sequences $\\left\\langle{x_i}\\right\\rangle$ such that the series $\\displaystyle \\sum_{i \\mathop \\ge 0} x_i^2$ is convergent. Let $d_p$ be the $p$-sequence metric on $\\R$. Then $\\ell^p := \\left({A, d_p}\\right)$ is a metric space."}
+{"_id": "10571", "title": "Peano Structure is Unique", "text": "Let $\\left({P, s, 0}\\right)$ and $\\left({P', s', 0'}\\right)$ be Peano structures. Then there is a unique bijection $f: P \\to P'$ such that: {{begin-eqn}} {{eqn|l = f \\left({0}\\right)      |r = 0' }} {{eqn|l = \\forall n \\in P: f \\left({s \\left({n}\\right)}\\right)      |r = s' \\left({f \\left({n}\\right)}\\right) }} {{end-eqn}}"}
+{"_id": "10573", "title": "Constant Function is Continuous/Metric Space/Proof 2", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f_c: A_1 \\to A_2$ be the constant mapping from $A_1$ to $A_2$: :$\\exists c \\in A_2: \\forall a \\in A_1: f_c \\left({a}\\right) = c$ That is, every point in $A_1$ maps to the same point $c$ in $A_2$. Then $f_c$ is continuous throughout $A_1$ with respect to $d_1$ and $d_2$."}
+{"_id": "10574", "title": "Constant Function is Continuous/Metric Space/Proof 1", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f_c: A_1 \\to A_2$ be the constant mapping from $A_1$ to $A_2$: :$\\exists c \\in A_2: \\forall a \\in A_1: f_c \\left({a}\\right) = c$ That is, every point in $A_1$ maps to the same point $c$ in $A_2$. Then $f_c$ is continuous throughout $A_1$ with respect to $d_1$ and $d_2$."}
+{"_id": "10575", "title": "Constant Function is Continuous/Metric Space", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f_c: A_1 \\to A_2$ be the constant mapping from $A_1$ to $A_2$: :$\\exists c \\in A_2: \\forall a \\in A_1: f_c \\left({a}\\right) = c$ That is, every point in $A_1$ maps to the same point $c$ in $A_2$. Then $f_c$ is continuous throughout $A_1$ with respect to $d_1$ and $d_2$."}
+{"_id": "10576", "title": "Constant Function is Continuous/Real Function", "text": "Let $f_c: \\R \\to \\R$ be the constant mapping: :$\\exists c \\in \\R: \\forall a \\in \\R: \\map {f_c} a = c$ Then $f_c$ is continuous on $\\R$."}
+{"_id": "10577", "title": "Identity Mapping is Continuous/Metric Space", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. The identity mapping $I_A: A \\to A$ defined as: :$\\forall x \\in A: I_A \\left({x}\\right) = x$ is a continuous mapping."}
+{"_id": "10578", "title": "Identity Mapping on Real Vector Space from Chebyshev to Euclidean Metric is Continuous", "text": "Let $\\R^n$ be an $n$-dimensional real vector space. Let $d_2$ be the Euclidean metric on $\\R^n$. Let $d_\\infty$ be the Chebyshev distance on $\\R^n$. Let $I: \\R^n \\to \\R^n$ be the identity mapping from $\\R^n$ to itself. Then the mapping: :$I: \\struct {\\R^n, d_\\infty} \\to \\struct {\\R^n, d_2}$ is $\\tuple {d_\\infty, d_2}$-continuous."}
+{"_id": "10579", "title": "Identity Mapping on Real Vector Space from Euclidean to Chebyshev Distance is Continuous", "text": "Let $\\R^n$ be an $n$-dimensional real vector space. Let $d_2$ be the Euclidean metric on $\\R^n$. Let $d_\\infty$ be the Chebyshev distance on $\\R^n$. Let $I: \\R^n \\to \\R^n$ be the identity mapping from $\\R^n$ to itself. Then the mapping: :$I: \\left({\\R^n, d_2}\\right) \\to \\left({\\R^n, d_\\infty}\\right)$ is $\\left({d_2, d_\\infty}\\right)$-continuous."}
+{"_id": "10582", "title": "Composite of Continuous Mappings at Point between Metric Spaces is Continuous at Point", "text": "Let $M_1 = \\left({X_1, d_1}\\right), M_2 = \\left({X_2, d_2}\\right), M_3 = \\left({X_3, d_3}\\right)$ be metric spaces. Let $f: M_1 \\to M_2$ be continuous at $a \\in X_1$. Let $g: M_2 \\to M_3$ be continuous at $f \\left({a}\\right) \\in X_2$. Then their composite $g \\circ f: M_1 \\to M_3$ is continuous at $a \\in X_1$."}
+{"_id": "10583", "title": "Composite of Continuous Mappings between Metric Spaces is Continuous", "text": "Let $M_1 = \\left({X_1, d_1}\\right), M_2 = \\left({X_2, d_2}\\right), M_3 = \\left({X_3, d_3}\\right)$ be metric spaces. Let $f: M_1 \\to M_2$ and $g: M_2 \\to M_3$ be continuous mappings. Then their composite $g \\circ f: M_1 \\to M_3$ is continuous."}
+{"_id": "10587", "title": "Mapping from L1 Space to Real Number Space is Continuous", "text": "Let $\\left({\\R, d}\\right)$ be the real number line under the usual metric $d$. Let $X$ be the set of continuous real functions $f: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$. Let $d_1$ be the $L^1$ metric on $X$. Let $I: X \\to \\R$ be the real-valued function defined as: :$\\displaystyle \\forall f \\in X: I \\left({f}\\right) := \\int_a^b f \\left({t}\\right) \\ \\mathop d t$ Then the mapping: :$I : \\left({X, d_1}\\right) \\to \\left({\\R, d}\\right)$ is continuous."}
+{"_id": "10588", "title": "Cartesian Product under Chebyshev Distance of Continuous Mappings between Metric Spaces is Continuous", "text": "Let $n \\in \\N_{>0}$. Let $M_1 = \\left({A_1, d_1}\\right), M_2 = \\left({A_2, d_2}\\right), \\ldots, M_n = \\left({A_n, d_n}\\right)$ be metric spaces. Let $N_1 = \\left({B_1, d'_1}\\right), N_2 = \\left({B_2, d'_2}\\right), \\ldots, N_n = \\left({B_n, d'_n}\\right)$ be metric spaces. Let $f_i: M_i \\to N_i$ be continuous mappings for all $i \\in \\left\\{{1, 2, \\ldots, n}\\right\\}$. Let $\\displaystyle \\mathcal M = \\prod_{i \\mathop = 1}^n M_i$ be the cartesian product of $A_1, A_2, \\ldots, A_n$. Let $\\displaystyle \\mathcal N = \\prod_{i \\mathop = 1}^n N_i$ be the cartesian product of $B_1, B_2, \\ldots, B_n$. Let $d_\\infty$ be the Chebyshev distance on $\\displaystyle \\mathcal A = \\prod_{i \\mathop = 1}^n A_i$, and $\\displaystyle \\mathcal B = \\prod_{i \\mathop = 1}^n B_i$, defined as: : $\\displaystyle d_\\infty \\left({x, y}\\right) = \\max_{i \\mathop = 1}^n \\left\\{ {d_i \\left({x_i, y_i}\\right)}\\right\\}$ : $\\displaystyle d_\\infty \\left({x, y}\\right) = \\max_{i \\mathop = 1}^n \\left\\{ {d'_i \\left({x_i, y_i}\\right)}\\right\\}$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({y_1, y_2, \\ldots, y_n}\\right) \\in \\mathcal A$ or $\\mathcal B$. Let $F: M \\to N$ be the mapping defined as: :$\\forall x \\in \\mathcal A: F \\left({x_1, x_2, \\ldots, x_n}\\right) = \\left({f \\left({x_1}\\right), \\left({x_2}\\right), \\ldots,  \\left({x_n}\\right)}\\right)$ Then $F$ is continuous."}
+{"_id": "10590", "title": "Addition of Coordinates on Euclidean Plane is Continuous Function", "text": "Let $\\struct {\\R^2, d_2}$ be the real number plane with the usual (Euclidean) metric. Let $f: \\R^2 \\to \\R$ be the real-valued function defined as: :$\\forall \\tuple {x_1, x_2} \\in \\R^2: \\map f {x_1, x_2} = x_1 + x_2$ Then $f$ is continuous."}
+{"_id": "10591", "title": "Metric Space Continuity by Inverse of Mapping between Open Balls", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f: A_1 \\to A_2$ be a mapping from $A_1$ to $A_2$. Let $a \\in A_1$ be a point in $A_1$. $f$ is continuous at $a$ with respect to the metrics $d_1$ and $d_2$ iff: :$\\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: B_\\delta \\left({a; d_1}\\right) \\subseteq f^{-1} \\left[{B_\\epsilon \\left({f \\left({a}\\right); d_2}\\right)}\\right]$ where $B_\\epsilon \\left({f \\left({a}\\right); d_2}\\right)$ denotes the open $\\epsilon$-ball of $f \\left({a}\\right)$ with respect to the metric $d_2$, and similarly for $B_\\delta \\left({a; d_1}\\right)$."}
+{"_id": "10592", "title": "Open Ball is Neighborhood of all Points Inside", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $a \\in A$. Let $B_\\epsilon \\left({a}\\right)$ be an open $\\epsilon$-ball of $a$ in $M$. Let $x \\in B_\\epsilon \\left({a}\\right)$. Then $B_\\epsilon \\left({a}\\right)$ is a neighborhoods of $x$ in $M$."}
+{"_id": "10593", "title": "Metric Space Continuity by Neighborhood", "text": "Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \\to A_2$ be a mapping from $A_1$ to $A_2$. Let $a \\in A_1$ be a point in $A_1$. Then the following definitions of continuity of $f$ at $a$ with respect to $d_1$ and $d_2$ are equivalent:"}
+{"_id": "10594", "title": "Sequence Characterization of Open Sets", "text": "Let $\\left({X, d}\\right)$ be a metric space. Let $G \\subseteq X$. Then the following are equivalent: :$(1): \\quad G \\subseteq X$ is an open set of $\\left({X, d}\\right)$ :$(2): \\quad \\forall x \\in G: \\forall \\left\\langle{x_n}\\right\\rangle \\in X: x_n \\to x: \\exists n_0 \\in \\N: \\forall n \\ge n_0: \\left\\langle{x_n}\\right\\rangle \\in G$"}
+{"_id": "10595", "title": "Metric Space Continuity by Inverse of Mapping between Neighborhoods", "text": "Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \\to A_2$ be a mapping from $A_1$ to $A_2$. Let $a \\in A_1$ be a point in $A_1$. $f$ is continuous at $a$ with respect to the metrics $d_1$ and $d_2$ {{iff}}: :for each neighborhood $N$ of $\\map f a$ in $M_2$, $f^{-1} \\sqbrk N$ is a neighborhood of $a$."}
+{"_id": "10596", "title": "Point in Metric Space has Neighborhood", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $a \\in A$ be a point in $M$. Then there exists some neighborhood of $a$ in $M$."}
+{"_id": "10597", "title": "Point in Metric Space is Element of its Neighborhood", "text": "Let $N$ be a neighborhood of $a$ in $M$. Then $a \\in N$."}
+{"_id": "10598", "title": "Superset of Neighborhood in Metric Space is Neighborhood", "text": "Let $N$ be a neighborhood of $a$ in $M$. Let $N \\subseteq N' \\subseteq A$. Then $N'$ is a neighborhood of $a$ in $M$."}
+{"_id": "10599", "title": "Intersection of Neighborhoods in Metric Space is Neighborhood", "text": "Let $N, N'$ be neighborhoods of $a$ in $M$. Then $N \\cap N'$ is a neighborhood of $a$ in $M$."}
+{"_id": "10600", "title": "Neighborhood in Metric Space has Subset Neighborhood", "text": "Let $N$ be a neighborhood of $a$ in $M$. Then there exists a neighborhood $N'$ of $a$ such that: :$(1): \\quad N' \\subseteq N$ :$(2): \\quad N'$ is a neighborhood of each of its points."}
+{"_id": "10601", "title": "Basic Properties of Neighborhood in Metric Space", "text": "This page gathers together the basic properties of a neighborhood in a metric space."}
+{"_id": "10602", "title": "Basis for Element of Real Number Line", "text": "Let $M = \\struct {\\R, d}$ denote the real number line with the usual (Euclidean) metric. Let $a \\in \\R$ be a point in $M$. Then the set of all open intervals containing $a$ is a basis for the neighborhood system of $a$."}
+{"_id": "10603", "title": "Open Ball in Real Number Line is Open Interval", "text": "Let $\\struct {\\R, d}$ denote the real number line $\\R$ with the usual (Euclidean) metric $d$. Let $x \\in \\R$ be a point in $\\R$. Let $\\map {B_\\epsilon} x$ be the open $\\epsilon$-ball at $x$. Then $\\map {B_\\epsilon} x$ is the open interval $\\openint {x - \\epsilon} {x + \\epsilon}$."}
+{"_id": "10604", "title": "Open Real Interval is Open Ball", "text": "Let $\\R$ denote the real number line with the usual (Euclidean) metric. Let $I := \\openint a b \\subseteq \\R$ be an open real interval. Then $I$ is the open $\\epsilon$-ball $\\map {B_\\epsilon} \\alpha$ of some $\\alpha \\in \\R$."}
+{"_id": "10605", "title": "Neighborhoods in Standard Discrete Metric Space", "text": "Let $M = \\left({A, d}\\right)$ be a metric space where $d$ is the standard discrete metric. Let $a \\in A$. Then $\\left\\{ {a}\\right\\}$ is a neighborhood of $a$ which forms a basis for the system of neighborhoods of $a$."}
+{"_id": "10606", "title": "Subset of Standard Discrete Metric Space is Neighborhood of Each Point", "text": "Let $M = \\left({A, d}\\right)$ be a metric space where $d$ is the standard discrete metric. Let $S \\subseteq A$. Let $a \\in S$. Then $S$ is a neighborhood of $a$. That is, every subset of $A$ is a neighborhood of each of its points."}
+{"_id": "10609", "title": "Continuity of Heaviside Step Function", "text": "Let $\\mu_c: \\R \\to \\R$ be the Heaviside step function: :$\\mu_c \\left({x}\\right) = \\begin{cases} 0 & : x < c \\\\ 1 & : x > c \\\\ \\text{arbitrary} & : x = c \\end{cases}$ Then $\\mu_c$ is continuous at every point of $\\R$ except at $c$."}
+{"_id": "10610", "title": "Metric Space Continuity by Neighborhood Basis", "text": "Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \\to A_2$ be a mapping from $A_1$ to $A_2$. Let $a \\in A_1$ be a point in $A_1$. Let $\\BB_{\\map f a}$ be a basis for the neighborhood system at $\\map f a$. $f$ is continuous at $a$ with respect to the metrics $d_1$ and $d_2$ {{iff}}: :for each neighborhood $N$ in $\\BB_{\\map f a}$, $f^{-1} \\sqbrk N$ is a neighborhood of $a$."}
+{"_id": "10611", "title": "Closed Intervals form Neighborhood Basis in Real Number Line", "text": "Let $\\R$ be the real number line with the usual (Euclidean) metric. Let $a \\in \\R$ be a point in $\\R$. Let $\\BB_a$ be defined as: :$\\BB_a := \\set {\\closedint {a - \\epsilon} {a + \\epsilon}: \\epsilon \\in \\R_{>0} }$ that is, the set of all closed intervals of $\\R$ with $a$ as a midpoint. Then $\\BB_a$ is a basis for the neighborhood system of $a$."}
+{"_id": "10613", "title": "Open Reciprocal-N Balls form Neighborhood Basis in Real Number Line", "text": "Let $\\R$ denote the real number line with the usual (Euclidean) metric. Let $a \\in \\R$ be a point in $\\R$. Let $\\BB_a$ be defined as: :$\\BB_a := \\set {\\map {B_\\epsilon} a: \\epsilon \\in \\set {\\dfrac 1 n: n \\in \\N} }$ that is, the set of all open $\\epsilon$-balls of $a$ for $\\epsilon$ which are reciprocals of integers. Then $\\BB_a$ is a basis for the neighborhood system of $a$."}
+{"_id": "10614", "title": "Subset of Open Reciprocal-N Balls forms Neighborhood Basis in Real Number Line", "text": "Let $\\R$ denote the real number line with the usual (Euclidean) metric. Let $a \\in \\R$ be a point in $\\R$. Let $k \\in \\Z$ be some fixed integer. Let $\\BB_a$ be defined as: :$\\BB_a := \\set {\\map {B_\\epsilon} a: \\epsilon \\in \\set {\\dfrac 1 n: n \\in \\N, n > k} }$ that is, the set of all open $\\epsilon$-balls of $a$ for $\\epsilon$ which are reciprocals of integers greater than $k$. Then the $\\BB_a$ is a basis for the neighborhood system of $a$."}
+{"_id": "10615", "title": "Neighborhood Basis in Real Number Line is Infinite", "text": "Let $\\R$ be the real number line with the usual (Euclidean) metric. Let $a \\in \\R$ be a point in $\\R$. Let $\\BB_a$ be a basis for the neighborhood system of $a$. Then $\\BB_a$ is an infinite set."}
+{"_id": "10616", "title": "Distinct Points in Metric Space have Disjoint Neighborhoods", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $x, y \\in M: x \\ne y$. Then there exist neighborhoods $N_x$ and $N_y$ of $x$ and $y$ respectively such that $N_x \\cap N_y = \\varnothing$, that is, that are disjoint."}
+{"_id": "10617", "title": "Open Ball in Cartesian Product under Chebyshev Distance", "text": "Let $M_1 = \\struct {A_1, d_1}, M_2 = \\struct {A_2, d_2}, \\ldots, M_n = \\struct {A_n, d_n}$ be metric spaces. Let $\\displaystyle \\AA = \\prod_{i \\mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \\ldots, A_n$. Let $d_\\infty: \\AA \\times \\AA \\to \\R$ be the Chebyshev distance on $\\AA$: :$\\displaystyle \\map {d_\\infty} {x, y} = \\max_{i \\mathop = 1}^n \\set {\\map {d_i} {x_i, y_i} }$ where $x = \\tuple {x_1, x_2, \\ldots, x_n}, y = \\tuple {y_1, y_2, \\ldots, y_n} \\in \\AA$. Let $a = \\tuple {a_1, a_2, \\ldots, a_n} \\in \\AA$. Let $\\epsilon \\in \\R_{>0}$. Let $\\map {B_\\epsilon} {a; d_\\infty}$ be the open $\\epsilon$-ball of $a$ in $M = \\struct {\\AA, d_\\infty}$. Then: :$\\displaystyle \\map {B_\\epsilon} {a; d_\\infty} = \\prod_{i \\mathop = 1}^n \\map {B_\\epsilon} {a_i; d_i}$"}
+{"_id": "10619", "title": "Neighborhood Basis in Cartesian Product under Chebyshev Distance", "text": "Let $M_1 = \\left({A_1, d_1}\\right), M_2 = \\left({A_2, d_2}\\right), \\ldots, M_n = \\left({A_n, d_n}\\right)$ be metric spaces. Let $\\displaystyle \\mathcal A = \\prod_{i \\mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \\ldots, A_n$. Let $d_\\infty: \\mathcal A \\times \\mathcal A \\to \\R$ be the Chebyshev distance on $\\mathcal A$: : $\\displaystyle d_\\infty \\left({x, y}\\right) = \\max_{i \\mathop = 1}^n \\left\\{ {d_i \\left({x_i, y_i}\\right)}\\right\\}$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({y_1, y_2, \\ldots, y_n}\\right) \\in \\mathcal A$. Let $a = \\left({a_1, a_2, \\ldots, a_n}\\right) \\in \\mathcal A$. For all $i \\in \\left\\{ {1, 2, \\ldots, n}\\right\\}$, let $\\mathcal B_{a_i}$ be a basis for the system of neighborhoods for $a_i$ in $M_i$. Let $\\mathcal B_a$ be the cartesian product of $\\mathcal B_{a_1}, \\mathcal B_{a_2}, \\ldots, \\mathcal B_{a_n}$: :$\\displaystyle \\mathcal B_a = \\prod_{i \\mathop = 1}^n \\mathcal B_{a_i}$ Then $\\mathcal B_a$ is a basis for the system of neighborhoods for $a$ in $M$."}
+{"_id": "10620", "title": "Projection from Cartesian Product under Chebyshev Distance is Continuous", "text": "Let $M_1 = \\struct {A_1, d_1}, M_2 = \\struct {A_2, d_2}, \\ldots, M_n = \\struct {A_n, d_n}$ be metric spaces. Let $\\displaystyle \\AA = \\prod_{i \\mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \\ldots, A_n$. Let $d_\\infty: \\AA \\times \\AA \\to \\R$ be the Chebyshev distance on $\\AA$: :$\\displaystyle \\map {d_\\infty} {x, y} = \\max_{i \\mathop = 1}^n \\set {\\map {d_i} {x_i, y_i} }$ where $x = \\tuple {x_1, x_2, \\ldots, x_n}, y = \\tuple {y_1, y_2, \\ldots, y_n} \\in \\AA$. For all $i \\in \\set {1, 2, \\ldots, n}$, let $\\pr_i: \\AA \\to A_i$ be the $i$th projection on $\\AA$: :$\\forall a \\in \\AA: \\map {\\pr_i} a = a_i$ where $a = \\tuple {a_1, a_2, \\ldots, a_n} \\in \\AA$. Then for all $i \\in \\set {1, 2, \\ldots, n}$, $p_i$ is continuous on $\\AA$."}
+{"_id": "10623", "title": "Limit of Sequence in Metric Space in Neighborhood", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $\\left\\langle{a_n}\\right\\rangle$ be a sequence in $A$. Then $\\displaystyle \\lim_{n \\mathop \\to \\infty} a_n = a$ iff for each neighborhood $V$ of $a$: :$\\exists N \\in \\N: n > N \\implies a_n \\in V$"}
+{"_id": "10624", "title": "Square of Real Number is Non-Negative", "text": "Let $x \\in \\R$. Then: : $0 \\le x^2$"}
+{"_id": "10627", "title": "Real Number between Zero and One is Greater than Square", "text": "Let $x \\in \\R$. Let $0 < x < 1$. Then: : $0 < x^2 < x$"}
+{"_id": "10628", "title": "Real Number Greater than One is Less than Square", "text": "Let $x \\in \\R$. Let $x > 1$. Then: : $x^2 > x$"}
+{"_id": "10634", "title": "Power of Real Number greater than One is Unbounded Above", "text": "Let $x \\in \\R$ be a real number such that $x > 1$. Let set $S = \\set {x^n: n \\in \\N}$. Then $S$ is unbounded above."}
+{"_id": "10635", "title": "Power of Real Number between Zero and One is Bounded", "text": "Let $x \\in \\R$ be a real number. Let $0 < x < 1$. Let set $S = \\set {x^n: n \\in \\N}$. Then: :$\\inf S = 0$ and: :$\\sup S = 1$ where $\\inf S$ and $\\sup S$ are the infimum and supremum of $S$ respectively."}
+{"_id": "10636", "title": "Limit of Function by Convergent Sequences/Real Number Line", "text": "Let $f$ be a real function defined on an open interval $\\openint a b$, except possibly at the point $c \\in \\openint a b$. Then $\\displaystyle \\lim_{x \\mathop \\to c} \\map f x = l$ {{iff}}: :for each sequence $\\sequence {x_n}$ of points of $\\openint a b$ such that $\\forall n \\in \\N_{>0}: x_n \\ne c$ and $\\displaystyle \\lim_{n \\to \\mathop \\infty} x_n = c$ it is true that: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map f {x_n} = l$"}
+{"_id": "10637", "title": "Continuity of Mapping between Metric Spaces by Convergent Sequence", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f: A_1 \\to A_2$ be a mapping. Then $f$ is continuous at $a \\in X$ iff: : whenever $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = a$ for a sequence $\\left \\langle {x_n} \\right \\rangle$ of points of $A_1$   it is true that: : $\\displaystyle \\lim_{n \\mathop \\to \\infty} f \\left({x_n}\\right) = f \\left({a}\\right)$"}
+{"_id": "10639", "title": "Infimum of Subset of Real Numbers is Arbitrarily Close", "text": "Let $A \\subseteq \\R$ be a subset of the real numbers. Let $b$ be an infimum of $A$. Let $\\epsilon \\in \\R_{>0}$. Then: :$\\exists x \\in A: x - b < \\epsilon$"}
+{"_id": "10640", "title": "Existence of Sequence in Set of Real Numbers whose Limit is Infimum", "text": "Let $A \\subseteq \\R$ be a non-empty subset of the real numbers. Let $b$ be an infimum of $A$. Then there exists a sequence $\\sequence {a_n}$ in $\\R$ such that: :$(1): \\quad \\forall n \\in \\N: a_n \\in A$ :$(2): \\quad \\displaystyle \\lim_{n \\mathop \\to \\infty} a_n = b$"}
+{"_id": "10641", "title": "Existence of Sequence in Subset of Metric Space whose Limit is Infimum", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $a \\in A$. Let $S \\subseteq A$ be a non-empty subset of $A$. Then there exists a sequence $\\sequence {a_n}$ of points of $S$ such that: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map d {a, a_n} = \\map d {a, S}$"}
+{"_id": "10642", "title": "Limit of Sequence in Product of Metric Spaces under Chebyshev Distance", "text": "Let $M_1 = \\struct {A_1, d_1}, M_2 = \\struct {A_2, d_2}, \\ldots, M_k = \\struct {A_n, d_k}$ be metric spaces. Let $\\displaystyle \\AA = \\prod_{i \\mathop = 1}^k A_i$ be the cartesian product of $A_1, A_2, \\ldots, A_k$. Let $d_\\infty: \\AA \\times \\AA \\to \\R$ be the Chebyshev distance on $\\AA$: :$\\displaystyle \\map {d_\\infty} {x, y} = \\max_{i \\mathop = 1}^k \\set {\\map {d_i} {x_i, y_i} }$ where $x = \\tuple {x_1, x_2, \\ldots, x_k}, y = \\tuple {y_1, y_2, \\ldots, y_k} \\in \\AA$. Let $\\sequence {a_n}$ be a sequence of points of $\\AA$: :$a_n = \\tuple {a_{n 1}, a_{n 2}, \\ldots, a_{n k} } \\in \\AA$ Let $c = \\tuple {c_1, c_2, \\ldots, c_k} \\in \\AA$. Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} a_n = c \\iff \\forall i \\in \\set {1, 2, \\ldots, k}: \\lim_{n \\mathop \\to \\infty} a_{n i} = c_i$ where $\\displaystyle \\lim_{n \\mathop \\to \\infty}$ denotes a limit."}
+{"_id": "10643", "title": "Recurrence Formula for Sum of Sequence of Fibonacci Numbers", "text": "Let $g_n$ be the sum of the first $n$ Fibonacci numbers. Then: :$\\forall n \\ge 2: g_n = g_{n - 1} + g_{n - 2} + 1$"}
+{"_id": "10644", "title": "Limit of Subsequence equals Limit of Sequence/Metric Space", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $\\left \\langle {x_n} \\right \\rangle$ be a sequence in $A$. Let $l \\in A$ such that: : $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ Let $\\left \\langle {x_{n_r}} \\right \\rangle$ be a subsequence of $\\left \\langle {x_n} \\right \\rangle$. Then: : $\\displaystyle \\lim_{r \\mathop \\to \\infty} x_{n_r} = l$"}
+{"_id": "10647", "title": "Convergent Real Sequence is Bounded", "text": "Let $\\left \\langle {x_n} \\right \\rangle$ be a sequence in $\\R$. Let $l \\in A$ such that $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$. Then $\\left \\langle {x_n} \\right \\rangle$ is bounded."}
+{"_id": "10648", "title": "Monotone Convergence Theorem (Real Analysis)/Decreasing Sequence", "text": "Let $\\sequence {x_n}$ be decreasing and bounded below. Then $\\sequence {x_n}$ converges to its infimum."}
+{"_id": "10649", "title": "Monotone Convergence Theorem (Real Analysis)/Increasing Sequence", "text": "Let $\\sequence {x_n}$ be increasing and bounded above. Then $\\sequence {x_n}$ converges to its supremum."}
+{"_id": "10652", "title": "Triangle Inequality for Distance from Element to Set", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $S \\subseteq A$ be a non-empty subset of $A$. Let $x, y \\in A$. Then: :$\\map d {x, S} \\le \\map d {x, y} + \\map d {y, S}$ where $\\map d {x, S}$ is the distance from $x$ to $S$."}
+{"_id": "10653", "title": "Open Ball in Euclidean Plane is Interior of Circle", "text": "Let $\\R^2$ be the real number plane with the usual (Euclidean) metric. Let $x = \\tuple {x_1, x_2} \\in \\R^2$ be a point in $\\R^2$. Let $\\map {B_\\epsilon} x$ be the open $\\epsilon$-ball at $x$. Then $\\map {B_\\epsilon} x$ is the interior of the circle whose center is $x$ and whose radius is $\\epsilon$."}
+{"_id": "10654", "title": "Open Ball in Euclidean 3-Space is Interior of Sphere", "text": "Let $\\R^3$ be the real Euclidean $3$-space considered as a metric space under the usual metric. Let $x = \\tuple {x_1, x_2, x_3} \\in \\R^3$ be a point in $\\R^3$. Let $\\map {B_\\epsilon} x$ be the open $\\epsilon$-ball at $x$. Then $\\map {B_\\epsilon} x$ is the interior of the sphere whose center is $x$ and whose radius is $\\epsilon$."}
+{"_id": "10656", "title": "Open Ball in Standard Discrete Metric Space", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $d$ be the standard discrete metric on $M$. Let $a \\in A$. Let $\\map {B_\\epsilon} {a; d}$ be an open $\\epsilon$-ball of $a$ in $M$. Then: :$\\map {B_\\epsilon} {a; d} = \\begin{cases} \\set a & : \\epsilon \\le 1 \\\\ A & : \\epsilon > 1 \\end{cases}$"}
+{"_id": "10657", "title": "Distance from Point to Subset is Continuous Function", "text": "Let $M = \\struct {X, d}$ be a metric space. Let $A \\subseteq X$ be a non-empty subset of $X$. Let $f: X \\to \\R$ be the function defined as: :$\\forall x \\in X: \\map f x = \\map d {x, A}$ where $\\map d {x, A}$ denotes the distance from $x$ to $A$. Then $f$ is continuous."}
+{"_id": "10659", "title": "Mapping from Cartesian Product under Chebyshev Distance to Real Number Line is Continuous", "text": "Let $M = \\struct {A, d'}$ be a metric space. Let $\\displaystyle \\AA = A \\times A$ be the cartesian product of $A$ with itself. Let $d_\\infty: \\AA \\times \\AA \\to \\R$ be the Chebyshev distance on $\\AA$: : $\\displaystyle \\map {d_\\infty} {x, y} = \\max \\set {\\map {d'} {x_1, y_1}, \\map {d'} {x_2, y_2} }$ where $x = \\tuple {x_1, x_2}, y = \\tuple {y_1, y_2} \\in \\AA$. Then the mapping: :$d': \\struct {A \\times A, d_\\infty} \\to \\struct {\\R, d}$ where $d$ is the usual metric, is continuous."}
+{"_id": "10662", "title": "Empty Set is Closed in Metric Space", "text": "Let $M = \\struct {A, d}$ be a metric space. Then the empty set $\\varnothing$ is closed in $M$."}
+{"_id": "10663", "title": "Metric Space is Closed in Itself", "text": "Let $M = \\struct {A, d}$ be a metric space. Then $A$ is closed in $M$."}
+{"_id": "10664", "title": "Metric Space is Open and Closed in Itself", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Then $A$ is both open and closed in $M$."}
+{"_id": "10665", "title": "Subset of Metric Space contains Limits of Sequences iff Closed", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $H \\subseteq A$. Then $H$ is closed in $M$ iff: :for each sequence $\\left\\langle{a_n}\\right\\rangle$ of points of $H$ that converges to a point $a \\in A$, it follows that $a \\in H$."}
+{"_id": "10666", "title": "Subset of Metric Space is Closed iff contains all Zero Distance Points", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $H \\subseteq A$. Then $H$ is closed in $M$ {{iff}}: :$\\forall x \\in A: \\map d {x, H} = 0 \\implies x \\in H$ where $\\map d {x, H}$ denotes the distance between $x$ and $H$."}
+{"_id": "10667", "title": "Continuity of Mapping between Metric Spaces by Closed Sets", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f: A_1 \\to A_2$ be a mapping. Then $f$ is continuous iff: :for every $V \\subseteq A_2$ which is closed in $M_2$, $f^{-1} \\left({V}\\right)$ is closed in $M_1$."}
+{"_id": "10668", "title": "Metric Space defined by Closed Sets", "text": "Let $M = \\struct {A, d}$ be a metric space. Then: {{begin-axiom}} {{axiom | n = C1         | lc= $A$ is closed in $M$ }} {{axiom | n = C2         | lc= $\\O$ is closed in $M$ }} {{axiom | n = C3         | lc= The union of a finite number of closed sets of $M$ is a closed set of $M$ }} {{axiom | n = C4         | lc= The intersection of arbitrarily many closed sets of $M$ is a closed set of $M$ }} {{end-axiom}}"}
+{"_id": "10669", "title": "Infinite Union of Closed Sets of Metric Space may not be Closed", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $V_1, V_2, V_3, \\ldots$ be an infinite set of closed sets of $M$. Then it is not necessarily the case that $\\displaystyle \\bigcup_{n \\mathop \\in \\N} V_n$ is itself a closed set of $M$."}
+{"_id": "10670", "title": "Open Sets of Cartesian Product of Metric Spaces under Chebyshev Distance", "text": "Let $M_1 = \\left({A_1, d_1}\\right), M_2 = \\left({A_2, d_2}\\right), \\ldots, M_n = \\left({A_n, d_n}\\right)$ be metric spaces. Let $\\displaystyle \\mathcal A = \\prod_{i \\mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \\ldots, A_n$. Let $d_\\infty: \\mathcal A \\times \\mathcal A \\to \\R$ be the Chebyshev distance on $\\mathcal A$: : $\\displaystyle d_\\infty \\left({x, y}\\right) = \\max_{i \\mathop = 1}^n \\left\\{ {d_i \\left({x_i, y_i}\\right)}\\right\\}$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({y_1, y_2, \\ldots, y_n}\\right) \\in \\mathcal A$. For $i \\in \\left\\{ {1, 2, \\ldots, n}\\right\\}$, let $U_i$ be open in $M_i$. Then $\\displaystyle \\prod_{i \\mathop = 1}^n U_i$ is open in $M = \\left({\\mathcal A, d_\\infty}\\right)$."}
+{"_id": "10671", "title": "Basis for Product of Metric Spaces under Chebyshev Distance", "text": "Let $M_1 = \\left({A_1, d_1}\\right), M_2 = \\left({A_2, d_2}\\right), \\ldots, M_n = \\left({A_n, d_n}\\right)$ be metric spaces. Let $\\displaystyle \\mathcal A = \\prod_{i \\mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \\ldots, A_n$. Let $d_\\infty: \\mathcal A \\times \\mathcal A \\to \\R$ be the Chebyshev distance on $\\mathcal A$: : $\\displaystyle d_\\infty \\left({x, y}\\right) = \\max_{i \\mathop = 1}^n \\left\\{ {d_i \\left({x_i, y_i}\\right)}\\right\\}$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({y_1, y_2, \\ldots, y_n}\\right) \\in \\mathcal A$. For $i \\in \\left\\{ {1, 2, \\ldots, n}\\right\\}$, let $U_i$ be open in $M_i$. Then $\\left\\{ {\\displaystyle \\prod_{i \\mathop = 1}^n U_i}\\right\\}$ is a basis for the open sets of $M$."}
+{"_id": "10672", "title": "Open Balls form Basis for Open Sets of Metric Space", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $\\mathcal B$ be the set of all open balls of $M$. Then $\\mathcal B$ is a basis for the open sets of $M$."}
+{"_id": "10673", "title": "Graph of Continuous Mapping between Metric Spaces is Closed in Chebyshev Product", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $\\displaystyle \\mathcal A = A_1 \\times A_2$ be the cartesian product of $A_1$ and $A_2$. Let $d_\\infty: \\mathcal A \\times \\mathcal A \\to \\R$ be the Chebyshev distance on $\\mathcal A$: : $\\displaystyle d_\\infty \\left({x, y}\\right) = \\max \\left\\{ {d_1 \\left({x_1, y_1}\\right), d_2 \\left({x_2, y_2}\\right)}\\right\\}$ where $x = \\left({x_1, x_2}\\right), y = \\left({y_1, y_2}\\right) \\in \\mathcal A$. Let $\\Gamma_f$ be the graph of $f$. Then $\\Gamma_f$ is a closed set of $\\left({\\mathcal A, d_\\infty}\\right)$."}
+{"_id": "10677", "title": "Point of Metric Space is Isolated iff not Limit Point", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $H \\subseteq A$ be a subset of $A$. Let $H'$ be the set of limit points of $H$. Let $H^i$ be the set of isolated points of $H$. Then: :$A' \\cap A^i = \\O$"}
+{"_id": "10679", "title": "Subset of Metric Space is Subset of its Closure", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $H \\subseteq A$ be a subset of $A$. Then: :$H \\subseteq H^-$ where $H^-$ denotes the closure of $H$."}
+{"_id": "10680", "title": "Point in Closure of Subset of Metric Space iff Limit of Sequence", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $H \\subseteq A$ be a subset of $A$. Let $H^-$ denote the closure of $H$. Let $a \\in A$. Then $a \\in H^-$ {{iff}} there exists a sequence $\\sequence {x_n}$ of points of $H$ which converges to the limit $a$."}
+{"_id": "10681", "title": "Closure of Subset of Closed Set of Metric Space is Subset", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $F$ be a closed set of $M$. Let $H \\subseteq F$ be a subset of $F$. Let $H^-$ denote the closure of $H$. Then $H^- \\subseteq F$."}
+{"_id": "10682", "title": "Closure of Subset of Metric Space is Intersection of Closed Supersets", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $H \\subseteq A$ be a subset of $A$. Let $H^-$ denote the closure of $H$. Then $H^-$ is the intersection of all closed sets of $M$ of which $H$ is a subset."}
+{"_id": "10683", "title": "Rational Numbers form Metric Subspace of Real Numbers under Euclidean Metric", "text": "Let $\\struct {\\Q, d_\\Q}$ be the set of rational numbers under the function $d_\\Q: \\Q \\times \\Q \\to \\R$ defined as: :$\\forall x, y \\in \\Q: \\map d {x, y} = \\size {x - y}$ Let $\\struct {\\R, d}$ denote the real number line with the usual (Euclidean) metric. Then $\\struct {\\Q, d_\\Q}$ is a metric subspace of $\\struct {\\R, d}$, where:"}
+{"_id": "10685", "title": "Euclidean Metric on Real Number Line is Metric", "text": "The Euclidean metric on the real number line $\\R$ is a metric."}
+{"_id": "10687", "title": "Definition:Unit n-Cube", "text": "Let $n \\in \\N$. The '''unit $n$-cube''' $I^n$ is the Cartesian product of $n$ instances of the closed real interval $\\left\\{ {x \\in \\R: 0 \\le x \\le 1}\\right\\}$: :$I^n = \\left[{0 \\,.\\,.\\, 1}\\right]^n$"}
+{"_id": "10688", "title": "Unit n-Cube under Chebyshev Distance is Subspace of Real Vector Space", "text": "Let $n \\in \\N$. Let $I^n$ denote the unit $n$-cube: :$I^n = \\left[{0 \\,.\\,.\\, 1}\\right]^n$ that is, the Cartesian product of $n$ instances of the closed real interval $\\left\\{ {x \\in \\R: 0 \\le x \\le 1}\\right\\}$. Let $d_c: I^n \\times I^n \\to \\R$ be defined as: :$\\displaystyle d_c \\left({x, y}\\right) = \\max_{i \\mathop = 1}^n \\left\\{ {\\left\\lvert{x_i - y_i}\\right\\rvert}\\right\\}$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({x_1, x_2, \\ldots, x_n}\\right) \\in I^n$. Then $\\left({I_n, d_c}\\right)$ is a metric subspace of $\\left({\\R^n, d_\\infty}\\right)$, where $d_\\infty$ is the Chebyshev distance on the real vector space $\\R_n$."}
+{"_id": "10689", "title": "Unit n-Sphere under Euclidean Metric is Metric Subspace of Euclidean Real Vector Space", "text": "Let $\\Bbb S^n$ be the unit $n$-sphere. Let $d_S: \\Bbb S^n \\times \\Bbb S^n \\to \\R$ be the real-valued function defined as: :$\\displaystyle \\forall x, y \\in \\Bbb S^n: d_S \\left({x, y}\\right) = \\sqrt {\\sum_{i \\mathop = 1}^{n + 1} \\left({x_i - y_i}\\right)^2}$ where $x = \\left({x_1, x_2, \\ldots, x_{n + 1} }\\right), y = \\left({y_1, y_2, \\ldots, y_{n + 1} }\\right)$. Then $\\left({\\Bbb S^n, d_S}\\right)$ is a metric subspace of $\\left({\\R^{n + 1}, d}\\right)$, where $d$ is the Euclidean metric on the real vector space $\\R^{n + 1}$."}
+{"_id": "10690", "title": "Vector Subspace of Real Vector Space under Chebyshev Metric is Metric Subspace", "text": "Let $n \\in \\N$. Let $A$ be the set of all ordered $n+1$-tuples $\\tuple {x_1, x_2, \\ldots, x_{n + 1} }$ of real numbers such that $x_{n + 1} = 0$. Let $d: A \\times A \\to \\R$ be the function defined as: :$\\displaystyle \\forall x, y \\in A: \\map d {x, y} = \\max_{i \\mathop = 1}^n \\set {\\size {x_i - y_i} }$ where $x = \\tuple {x_1, x_2, \\ldots, x_{n + 1} }, y = \\tuple {y_1, y_2, \\ldots, y_{n + 1} }$. Then $\\struct {A, d}$ is a metric subspace of $\\struct {\\R^{n + 1}, d_\\infty}$ where $d_\\infty$ is the Chebyshev distance on the real vector space $\\R^{n + 1}$."}
+{"_id": "10692", "title": "Bertrand-Chebyshev Theorem", "text": "For all $n \\in \\N_{>0}$, there exists a prime number $p$ with $n < p \\le 2 n$."}
+{"_id": "10693", "title": "Sequence of Binomial Coefficients is Strictly Increasing to Half Upper Index", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive integer. Let $\\dbinom n k$ be the binomial coefficient of $n$ over $k$ for a positive integer $k \\in \\Z_{\\ge 0}$. Let $S_n = \\left\\langle{x_k}\\right\\rangle$ be the sequence defined as: :$x_k = \\dbinom n k$ Then $S_n$ is strictly increasing exactly where $0 \\le k < \\dfrac n 2$."}
+{"_id": "10697", "title": "Isometry between Metric Spaces is Continuous", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $\\phi: M_1 \\to M_2$ be an isometry. Then $\\phi: M_1 \\to M_2$ is a continuous mapping."}
+{"_id": "10699", "title": "Isometric Metric Spaces are Homeomorphic", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $M_1$ and $M_2$ be isometric. Then $M_1$ and $M_2$ are homeomorphic."}
+{"_id": "10700", "title": "Homeomorphic Metric Spaces are not necessarily Isometric", "text": "Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces. Let $M_1$ and $M_2$ be homeomorphic. Then it is not necessarily the case that $M_1$ and $M_2$ are isometric."}
+{"_id": "10701", "title": "Identity Mapping between Metrics separated by Scale Factor is Continuous", "text": "Let $M_1 = \\struct {A, d_1}$ and $M_2 = \\struct {A, d_2}$ be metric spaces on the same underlying set $A$. Let $d_1$ and $d_2$ be such that: :$\\forall x, y \\in A: \\map {d_2} {x, y} \\le K \\map {d_2} {x, y}$ Let $I_A: A \\to A$ be the identity mapping on $A$. Then $I_A$ is continuous from $M_1$ to $M_2$."}
+{"_id": "10704", "title": "Metric Spaces on Topologically Equivalent Metrics on same Underlying Set are Homeomorphic", "text": "Let $M_1 = \\left({A, d_1}\\right)$ and $M_2 = \\left({A, d_2}\\right)$ be metric spaces on the same underlying set $A$. Let $d_1$ and $d_2$ be topologically equivalent. Then $M_1$ and $M_2$ are homeomorphic."}
+{"_id": "10705", "title": "Real Vector Space under Chebyshev Distance is Homeomorphic to that under Euclidean Metric", "text": "Let $\\R^n$ be an $n$-dimensional real vector space. Let $d_\\infty: \\R^n \\times \\R^n \\to \\R$ be the Chebyshev distance on $\\R^n$. Let $d_2: \\R^n \\times \\R^n \\to \\R$ be the Euclidean metric on $\\R^n$. Let $M_1 = \\left({\\R^n, d_\\infty}\\right)$ and $M_2 = \\left({\\R^n, d_2}\\right)$ be the corresponding metric spaces. Then $M_1$ and $M_2$ are homeomorphic."}
+{"_id": "10706", "title": "Equivalence of Definitions of Homeomorphic Metric Spaces", "text": "{{TFAE|def = Homeomorphic Metric Spaces}} Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f: A_1 \\to A_2$ be a bijection."}
+{"_id": "10707", "title": "Liouville's Theorem (Number Theory)", "text": "Let $x$ be an irrational number that is algebraic of degree $n$. Then there exists a constant $c > 0$ (which can depend on $x$) such that: :$\\size {x - \\dfrac p q} \\ge \\dfrac c {q^n}$ for every pair $p, q \\in \\Z$ with $q \\ne 0$."}
+{"_id": "10708", "title": "Liouville Numbers are Irrational", "text": "Liouville numbers are irrational."}
+{"_id": "10709", "title": "Cardinality of Power Set of Natural Numbers Equals Cardinality of Real Numbers", "text": "The cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers."}
+{"_id": "10710", "title": "Liouville's Constant is Transcendental", "text": "Liouville's constant: {{begin-eqn}} {{eqn | l = \\sum_{n \\mathop \\ge 1} \\dfrac 1 {10^{n!} }       | r = \\frac 1 {10^1} + \\frac 1 {10^2} + \\frac 1 {10^6} +  \\frac 1 {10^{24} } + \\cdots       | c =  }} {{eqn | r = 0.11000 \\, 10000 \\, 00000 \\, 00000 \\, 00010 \\, 00 \\ldots       | c =  }} {{end-eqn}} is transcendental."}
+{"_id": "10711", "title": "Isometry of Metric Spaces is Equivalence Relation", "text": "Let $M_1$ and $M_2$ be metric spaces. Let $M_1 \\sim M_2$ denote that $M_1$ and $M_2$ are isometric. The relation $\\sim$ is an equivalence relation."}
+{"_id": "10712", "title": "Homeomorphism of Metric Spaces is Equivalence Relation", "text": "Let $M_1$ and $M_2$ be metric spaces. Let $M_1 \\sim M_2$ denote that $M_1$ and $M_2$ are homeomorphic. The relation $\\sim$ is an equivalence relation."}
+{"_id": "10713", "title": "Isometry of Metric Spaces is Homeomorphism", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f: M_1 \\to M_2$ be an isometry. Then $f$ is a homeomorphism from $M_1$ to $M_2$."}
+{"_id": "10714", "title": "Equivalence Class of Isometries is Subset of Equivalence Class of Homeomorphisms", "text": "Let $\\mathcal C_I$ be an equivalence class of isometries on the set of metric spaces. Then $\\mathcal C_I$ is a subset of an equivalence class of homeomorphisms on the set of metric spaces."}
+{"_id": "10715", "title": "Equivalence of Local Uniform Convergence and Compact Convergence", "text": "Let $U \\subseteq \\C$ be an open set of the complex plane. Let $f_n: U \\to \\C$ be a sequence of functions which converges pointwise to $f: U \\to \\C$. Then $f_n$ converges to $f$ uniformly on all compact subsets of $U$ {{iff}} $f_n$ converges locally uniformly on $U$."}
+{"_id": "10716", "title": "Frattini's Argument", "text": "Let $\\struct {G, \\circ}$ be a group. Let $K$ be a finite normal subgroup of $G$, and $p$ a prime which divides the order of $K$. Let $P$ be a Sylow $p$-subgroup of $K$, and $\\map {N_G} P$ the normalizer of $P$ in $G$. Then: :$G = \\map {N_G} P \\circ K = K \\circ \\map {N_G} P$"}
+{"_id": "10718", "title": "Closure of Pointwise Operation on Algebraic Structure", "text": "Let $S$ be a set such that $S \\ne \\varnothing$. Let $\\struct {T, \\circ}$ be an algebraic structure. Let $T^S$ be the set of all mappings from $S$ to $T$. Let $f, g \\in T^S$, that is, let $f: S \\to T$ and $g: S \\to T$ be mappings. Let $\\oplus: T^S \\to T^S$ be the pointwise operation on $T^S$ induced by $\\circ$. Then $\\oplus$ is closed on $T^S$ {{iff}} $\\struct {T, \\circ}$ is closed."}
+{"_id": "10719", "title": "Pointwise Operation on Distributive Structure is Distributive", "text": "Let $S$ be a set. Let $\\left({T, +, \\circ}\\right)$ be an algebraic structure with two operations $+$ and $\\circ$. Let $T^S$ be the set of all mappings from $S$ to $T$. Let $\\left({T^S, +', \\circ'}\\right)$ be the structure induced on $T^S$ by $+$ and $\\circ$. Let $\\circ$ be distributive over $+$ in $T$. Then $\\circ'$ is distributive over $+'$ in $T$."}
+{"_id": "10720", "title": "Identity Mapping on Metric Space is Homeomorphism", "text": "Let $M = \\struct {A, d}$ be a metric space. The identity mapping $I_A: M \\to M$ defined as: :$\\forall x \\in A: \\map {I_A} x = x$ is a homeomorphism."}
+{"_id": "10721", "title": "Inverse of Homeomorphism between Metric Spaces is Homeomorphism", "text": "Let $M = \\left({A_1, d_1}\\right), M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f: M_1 \\to M_2$ be a homeomorphism. Then $f^{-1}: M_2 \\to M_1$ is also a homeomorphism."}
+{"_id": "10722", "title": "Composite of Homeomorphisms between Metric Spaces is Homeomorphism", "text": "Let $M_1, M_2, M_3$ be metric spaces.  Let $f: M_1 \\to M_2$ and $g: M_2 \\to M_3$ be homeomorphisms. Then $g \\circ f: M_1 \\to M_3$ is also a homeomorphism."}
+{"_id": "10725", "title": "Metrizable Space is Hausdorff", "text": "Let $T$ be a metrizable topological space. Then $T$ is a $T_2$ (Hausdorff) space."}
+{"_id": "10727", "title": "Underlying Set of Topological Space is Closed", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then the underlying set $S$ of $T$ is closed in $T$."}
+{"_id": "10728", "title": "Basic Properties of Neighborhood in Topological Space", "text": "This page gathers together the basic properties of a neighborhood of a point in a topological space."}
+{"_id": "10729", "title": "Point in Topological Space has Neighborhood", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $x \\in S$. Then there exists in $T$ at least one neighborhood of $x$."}
+{"_id": "10730", "title": "Point in Topological Space is Element of its Neighborhood", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $x \\in S$. Let $N$ be a neighborhood of $x$ in $T$. Then $a \\in N$."}
+{"_id": "10732", "title": "Intersection of Neighborhoods in Topological Space is Neighborhood", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $x \\in S$. Let $M, N$ be a neighborhoods of $x$ in $T$. Then $M \\cap N$ is a neighborhood of $x$ in $T$."}
+{"_id": "10733", "title": "Neighborhood in Topological Space has Subset Neighborhood", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $x \\in S$. Let $N$ be a neighborhood of $x$ in $T$. Then there exists a neighborhood $N'$ of $x$ such that: :$(1): \\quad N' \\subseteq N$ :$(2): \\quad N'$ is a neighborhood of each of its points."}
+{"_id": "10734", "title": "Empty Set is Open in Neighborhood Space", "text": "Let $\\struct {S, \\mathcal N}$ be a neighborhood space. Then the empty set $\\varnothing$ is an open set of $\\struct {S, \\mathcal N}$."}
+{"_id": "10735", "title": "Whole Space is Open in Neighborhood Space", "text": "Let $\\struct {S, \\NN}$ be a neighborhood space. Then $S$ itself is an open set of $\\struct {S, \\NN}$."}
+{"_id": "10736", "title": "Intersection of two Open Sets of Neighborhood Space is Open", "text": "Let $\\left({S, \\mathcal N}\\right)$ be a neighborhood space. Let $U$ and $V$ be open sets of $\\left({S, \\mathcal N}\\right)$. Then $U \\cap V$ is an open set of $\\left({S, \\mathcal N}\\right)$."}
+{"_id": "10737", "title": "Union of Open Sets of Neighborhood Space is Open", "text": "Let $S$ be a neighborhood space. Let $I$ be an indexing set. Let $\\family {U_\\alpha}_{\\alpha \\mathop \\in I}$ be a family of open sets of $\\struct {S, \\NN}$ indexed by $I$. Then their union $\\ds \\bigcup_{\\alpha \\mathop \\in I} U_i$ is an open set of $\\struct {S, \\NN}$."}
+{"_id": "10738", "title": "Neighborhood Space is Topological Space", "text": "Let $\\left({S, \\mathcal N}\\right)$ be a neighborhood space. Let $\\tau = \\left\\{{N: N \\in \\mathcal N}\\right\\}$ be the set of all open sets of $\\left({S, \\mathcal N}\\right)$. Then $\\left({S, \\tau}\\right)$ forms a topological space."}
+{"_id": "10739", "title": "Subset in Neighborhood Space is Neighborhood iff it contains Open Set", "text": "Let $\\left({S, \\mathcal N}\\right)$ be a neighborhood space. Let $x \\in S$ be a point of $S$. Let $N \\subseteq S$ be a subset of $S$. Then $N$ is a neighborhood of $x$ {{iff}} there exists an open set $U$ of $\\left({S, \\mathcal N}\\right)$ such that $x \\in U \\subseteq N$."}
+{"_id": "10740", "title": "Definition:Neighborhood Space Induced by Topological Space", "text": "Let $S$ be a set. Let $\\tau$ be a topology on $S$, thus forming the topological space $\\left({S, \\tau}\\right)$. For all $x \\in S$, let $\\mathcal N_x$ be the neighborhood filter of $x$ in $S$. Let $\\mathcal N$ be the set of subsets of $S$ such that: :$\\forall N \\in \\mathcal N: N$ is a neighborhood of each of its points Then $\\left({S, \\mathcal N}\\right)$ is the '''neighborhood space induced by $\\left({S, \\tau}\\right)$'''."}
+{"_id": "10742", "title": "Topological Space Induced by Neighborhood Space is Topological Space", "text": "Let $\\left({S, \\mathcal N}\\right)$ be a neighborhood space. Let $\\left({S, \\tau}\\right)$ be the topological space induced by $\\left({S, \\mathcal N}\\right)$. Then $\\left({S, \\tau}\\right)$ is a topological space."}
+{"_id": "10743", "title": "Topological Space induced by Neighborhood Space induced by Topological Space", "text": "Let $S$ be a set. Let $\\tau$ be a topology on $S$, thus forming the topological space $\\left({S, \\tau}\\right)$. Let $\\left({S, \\mathcal N}\\right)$ be the neighborhood space induced by $\\tau$ on $S$. Let $\\left({S, \\tau'}\\right)$ be the topological space induced by $\\mathcal N$ on $S$. Then $\\tau = \\tau'$."}
+{"_id": "10744", "title": "Correspondence between Neighborhood Space and Topological Space", "text": "Let $S$ be a set. Let $\\left({S, \\tau}\\right)$ be a topological space. Let $\\left({S, \\mathcal N}\\right)$ be the neighborhood space induced by $\\tau$ on $S$. Let $\\phi: \\left({S, \\tau}\\right) \\to \\left({S, \\mathcal N}\\right)$ be the mapping defined as: :$\\forall x \\in S: \\phi \\left({x}\\right) = x$ :$\\forall U \\in \\tau: \\phi \\left({U}\\right) = U \\in \\mathcal N$ Let $\\left({T, \\mathcal N'}\\right)$ be a neighborhood space. Let $\\left({T, \\tau'}\\right)$ be the topological space induced by $\\mathcal N$ on $S$. Let $\\psi: \\left({T, \\mathcal N'}\\right) \\to \\left({T, \\tau'}\\right)$ be the mapping defined as: :$\\forall y \\in T: \\psi \\left({y}\\right) = y$ :$\\forall V \\in \\mathcal N': \\psi \\left({V}\\right) = V \\in \\tau'$ Then: :$(1): \\quad \\phi^{-1} \\left({\\phi \\left({S, \\tau}\\right)}\\right) = \\left({S, \\tau}\\right)$ :$(2): \\quad \\psi^{-1} \\left({\\psi \\left({T, \\mathcal N'}\\right)}\\right) = \\left({T, \\mathcal N'}\\right)$"}
+{"_id": "10745", "title": "Neighborhood Space induced by Topological Space induced by Neighborhood Space", "text": "Let $\\left({S, \\mathcal N}\\right)$ be a neighborhood space. Let $\\left({S, \\tau}\\right)$ be the topological space induced by $\\mathcal N$ on $S$. Let $\\left({S, \\mathcal N'}\\right)$ be the neighborhood space induced by $\\tau$ on $S$. Then $\\mathcal N = \\mathcal N'$."}
+{"_id": "10746", "title": "Binomial Coefficient with Zero/Integer Coefficients", "text": ":$\\forall n \\in \\N: \\dbinom n 0 = 1$ where $\\dbinom n 0$ denotes a binomial coefficient."}
+{"_id": "10750", "title": "Form of Geometric Sequence of Integers in Lowest Terms", "text": "Let $G_n = \\sequence {a_j}_{0 \\mathop \\le j \\mathop \\le n}$ be a geometric sequence of length $n$ consisting of positive integers only. Let $r$ be the common ratio of $G_n$. Let the elements of $G_n$ be the smallest positive integers such that $G_n$ has common ratio $r$. Then the $j$th term of $G_n$ is given by: :$a_j = p^{n - j} q^j$ where $r = \\dfrac q p$. That is: :$G_n = \\tuple {p^n, p^{n - 1} q, p^{n - 2} q^2, \\ldots, p q^{n - 1}, q^n}$"}
+{"_id": "10751", "title": "Form of Geometric Sequence of Integers with Coprime Extremes", "text": "Let $Q_n = \\sequence {a_j}_{0 \\mathop \\le j \\mathop \\le n}$ be a geometric sequence of length $n$ consisting of positive integers only. Let $a_1$ and $a_n$ be coprime. Then the $j$th term of $Q_n$ is given by: :$a_j = q^j p^{n - j}$"}
+{"_id": "10752", "title": "Naturally Ordered Semigroup forms Peano Structure", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a naturally ordered semigroup. Let $0 \\in S$ be the zero of $S$. Let $1 \\in S$ be the one of $S$. Let $s: S \\to S$ be the mapping defined as: :$s \\left({n}\\right) := n \\circ 1$ Then $\\left({S, 0, s}\\right)$ is a Peano structure."}
+{"_id": "10753", "title": "Divisibility of Elements in Geometric Sequence of Integers", "text": "Let $Q_n = \\sequence {a_j}_{0 \\mathop \\le j \\mathop \\le n}$ be a geometric sequence of integers. Let $j \\ne k$. Then: :$\\paren {\\exists j \\in \\set {0, 1, \\ldots, n - 1}: a_j \\divides a_{j + 1} } \\iff \\paren {\\forall j, k \\in \\set {0, 1, \\ldots, n}, j < k: a_j \\divides a_k}$ where $\\divides$ denotes integer divisibility. That is: :One term of a geometric sequence of integers is the divisor of the next term {{iff}} :All terms are divisors of all later terms."}
+{"_id": "10754", "title": "Form of Geometric Sequence of Integers from One", "text": "Let $Q_n = \\sequence {a_j}_{0 \\mathop \\le j \\mathop \\le n}$ be a geometric sequence consisting of integers only. Let $a_0 = 1$. Then the $j$th term of $Q_n$ is given by: :$a_j = a^j$ where: :the common ratio of $Q_n$ is $a$ :$a = a_1$. Thus: :$Q_n = \\tuple {1, a, a^2, \\ldots, a^n}$"}
+{"_id": "10755", "title": "Divisors of Power of Prime", "text": "Let $p$ be a prime number. Let $n \\in \\Z_{> 0}$ be a (strictly) positive integer. Then the only divisors of $p^n$ are $1, p, p^2, \\ldots, p^{n - 1}, p^n$."}
+{"_id": "10759", "title": "Piecewise Continuously Differentiable Function/Definition 2 is Continuous", "text": "Let $f$ be a real function defined on a closed interval $\\closedint a b$. Let $f$ satisfy Piecewise Continuously Differentiable Function/Definition 2. Then $f$ is continuous."}
+{"_id": "10760", "title": "Homomorphism of Chain Complexes induces Homomorphism of Homology", "text": "Let $A_\\bullet$ and $B_\\bullet$ be chain complexes of abelian groups. Let $f: A_\\bullet \\to B_\\bullet$ be a homomorphism. Then for every $n$, $f$ induces a morphism $H_n \\left({A_\\bullet}\\right) \\to H_n \\left({B_\\bullet}\\right)$ of homology groups. {{explain|Domain of $n$}}"}
+{"_id": "10761", "title": "Homotopic Chain Maps Induce Equal Maps on Homology", "text": "Let $A_\\bullet$, $B_\\bullet$ be chain complexes of abelian groups. Let $f, g: A_\\bullet \\to B_\\bullet$ be chain maps which are homotopic. Then $f$ and $g$ induce equal maps on homology."}
+{"_id": "10762", "title": "Equivalence of Definitions of Piecewise Continuously Differentiable Function", "text": "{{TFAE|def = Piecewise Continuously Differentiable Function}} Let $f$ be a real function defined on a closed interval $\\closedint a b$."}
+{"_id": "10763", "title": "Bounded Function Continuous on Open Interval is Darboux Integrable", "text": "Let $f$ be a real function defined on an interval $\\closedint a b$ such that $a < b$. Let $f$ be continuous on $\\closedint a b$. Let $f$ be bounded on $\\closedint a b$. Then $f$ is Darboux integrable on $\\closedint a b$."}
+{"_id": "10764", "title": "Square Inscribed in Circle is greater than Half Area of Circle", "text": "A square inscribed in a circle has an area greater than half that of the circle."}
+{"_id": "10765", "title": "Bounded Piecewise Continuous Function is Darboux Integrable", "text": "Let $f$ be a real function defined on the closed interval $\\closedint a b$. Let $f$ be piecewise continuous and bounded on $\\closedint a b$. Then $f$ is Darboux integrable on $\\closedint a b$."}
+{"_id": "10769", "title": "Rational Numbers form Subset of Real Numbers", "text": "The set $\\Q$ of rational numbers forms a subset of the real numbers $\\R$."}
+{"_id": "10770", "title": "Real Number is not necessarily Rational Number", "text": "Let $x$ be a real number. Then it is not necessarily the case that $x$ is also a rational number."}
+{"_id": "10771", "title": "Zero is both Positive and Negative", "text": "The number $0$ (zero) is the only (real) number which is both: :a positive (real) number and :a negative (real) number."}
+{"_id": "10772", "title": "Binary Operation on Subset is Binary Operation", "text": "Let $S$ be a set. Let $\\circ$ be a binary operation on $S$. Let $T \\subseteq S$. Let $\\circ {\\restriction}_T$ be the restriction of $\\circ$ to $T$. Then $\\circ {\\restriction}_T$ is a binary operation on $T$."}
+{"_id": "10777", "title": "Matrix Entrywise Addition over Ring is Associative", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\map {\\MM_R} {m, n}$ be a $m \\times n$ matrix space over $R$. For $\\mathbf A, \\mathbf B \\in \\map {\\MM_R} {m, n}$, let $\\mathbf A + \\mathbf B$ be defined as the matrix entrywise sum of $\\mathbf A$ and $\\mathbf B$. The operation $+$ is associative on $\\map {\\MM_R} {m, n}$. That is: :$\\paren {\\mathbf A + \\mathbf B} + \\mathbf C = \\mathbf A + \\paren {\\mathbf B + \\mathbf C}$ for all $\\mathbf A$, $\\mathbf B$ and $\\mathbf C$ in $\\map {\\MM_R} {m, n}$."}
+{"_id": "10778", "title": "Final Topology with respect to Mapping", "text": "Let $\\left({X, \\tau_X}\\right)$ be a topological space. Let $Y$ be a set. Let $f: X \\to Y$ be a mapping. Let $\\tau_Y$ be the final topology on $Y$ with respect to $f$. Then: :$\\tau_Y = \\left\\{ U \\subseteq Y : f^{-1} \\left({U}\\right) \\in \\tau_X \\right\\}$ Observe that the set on the right-hand side of the equality is sometimes denoted $f\\tau$. Further, the following holds true: # $\\forall \\left(Z, \\tau_Z \\right)$ topological space $: \\forall g: \\left({Y, f\\tau}\\right) \\to \\left({Z, \\tau_Z}\\right)$ mapping $: g$ is continuous $\\iff g \\circ f$ is continuous (where $f$ is defined from $\\left({X, \\tau_X}\\right)$ to $\\left(Y, f\\tau \\right)$). # $\\forall \\tau_Y$ topology in $Y : \\forall \\left(Z, \\tau_Z \\right)$ topological space, the following are equivalent: ##$\\tau_Y = f\\tau$ ##$\\forall g: \\left({Y, \\tau_Y}\\right) \\to \\left({Z, \\tau_Z}\\right) : g$ is continuous $\\iff g \\circ f$ is continuous (where $f$ is defined from $\\left({X, \\tau_X}\\right)$ to $\\left(Y, \\tau_Y \\right)$). That is, the topology $f\\tau$ is ''characterised'' by the fact that it is the only one which makes all mappings defined from it to any topological space continuous."}
+{"_id": "10781", "title": "Composition of Symmetries is Symmetry", "text": "Let $P$ be a geometric figure. Let $S_P$ be the set of all symmetries of $P$. Let $\\circ$ denote composition of mappings. Let $\\phi$ and $\\psi$ be symmetries of $P$. Then $\\phi \\circ \\psi$ is also a symmetry of $P$."}
+{"_id": "10782", "title": "Composition of Symmetries is Associative", "text": "Let $P$ be a geometric figure. Let $S_P$ be the set of all symmetries of $P$. Let $\\circ$ denote composition of mappings. Let $\\phi, \\psi, \\chi$ be symmetries of $P$. Then: : $\\paren {\\phi \\circ \\psi} \\circ \\chi = \\phi \\circ \\paren {\\psi \\circ \\chi}$ That is, composition of symmetries is associative."}
+{"_id": "10783", "title": "Square of Modulo less One equals One", "text": "Let $m \\in \\Z$ be an integer. Let $\\Z_m$ be the set of integers modulo $m$: :$\\Z_m = \\set {\\eqclass 0 m, \\eqclass 1 m, \\ldots, \\eqclass {m - 1} m}$ Then: :$\\eqclass {m - 1} m \\times_m \\eqclass {m - 1} m = \\eqclass 1 m$ where $\\times_m$ denotes multiplication modulo $m$."}
+{"_id": "10785", "title": "Symmetry Group of Equilateral Triangle is Symmetric Group", "text": "Let $D_3$ denote the symmetry group of the equilateral triangle. Let $S_3$ denote the symmetric group on $3$ letters. Then $D_3$ is isomorphic to $S_3$."}
+{"_id": "10786", "title": "De Morgan's Laws (Set Theory)/Relative Complement/General Case/Complement of Intersection", "text": ":$\\ds \\relcomp S {\\bigcap \\mathbb T} = \\bigcup_{H \\mathop \\in \\mathbb T} \\relcomp S H$"}
+{"_id": "10787", "title": "De Morgan's Laws (Set Theory)/Relative Complement/General Case/Complement of Union", "text": ":$\\displaystyle \\relcomp S {\\bigcup \\mathbb T} = \\bigcap_{H \\mathop \\in \\mathbb T} \\relcomp S H$"}
+{"_id": "10788", "title": "De Morgan's Laws (Set Theory)/Relative Complement/Family of Sets/Complement of Union", "text": ":$\\displaystyle \\relcomp S {\\bigcup_{i \\mathop \\in I} \\mathbb S_i} = \\bigcap_{i \\mathop \\in I} \\relcomp S {S_i}$"}
+{"_id": "10789", "title": "De Morgan's Laws (Set Theory)/Relative Complement/Family of Sets/Complement of Intersection", "text": ":$\\displaystyle \\relcomp S {\\bigcap_{i \\mathop \\in I} \\mathbb S_i} = \\bigcup_{i \\mathop \\in I} \\relcomp S {S_i}$"}
+{"_id": "10790", "title": "De Morgan's Laws (Set Theory)/Relative Complement/Complement of Union", "text": ":$\\relcomp S {T_1 \\cup T_2} = \\relcomp S {T_1} \\cap \\relcomp S {T_2}$"}
+{"_id": "10791", "title": "De Morgan's Laws (Set Theory)/Relative Complement/Complement of Intersection", "text": ":$\\relcomp S {T_1 \\cap T_2} = \\relcomp S {T_1} \\cup \\relcomp S {T_2}$"}
+{"_id": "10792", "title": "De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Intersection", "text": ":$\\displaystyle S \\setminus \\bigcap_{i \\mathop \\in I} T_i = \\bigcup_{i \\mathop \\in I} \\paren {S \\setminus T_i}$ where: :$\\displaystyle \\bigcup_{i \\mathop \\in I} T_i := \\set {x: \\exists i \\in I: x \\in T_i}$ that is, the union of $\\family {T_i}_{i \\mathop \\in I}$."}
+{"_id": "10793", "title": "De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Union", "text": ":$\\displaystyle S \\setminus \\bigcup_{i \\mathop \\in I} T_i = \\bigcap_{i \\mathop \\in I} \\paren {S \\setminus T_i}$ where: :$\\displaystyle \\bigcap_{i \\mathop \\in I} T_i := \\set {x: \\forall i \\in I: x \\in T_i}$ that is, the intersection of $\\family {T_i}_{i \\mathop \\in I}$."}
+{"_id": "10794", "title": "De Morgan's Laws (Set Theory)/Set Complement/General Case/Complement of Intersection", "text": ":$\\displaystyle \\complement \\paren {\\bigcap \\mathbb T} = \\bigcup_{H \\mathop \\in \\mathbb T} \\complement \\paren H$"}
+{"_id": "10795", "title": "De Morgan's Laws (Set Theory)/Set Complement/General Case/Complement of Union", "text": ":$\\displaystyle \\complement \\paren {\\bigcup \\mathbb T} = \\bigcap_{H \\mathop \\in \\mathbb T} \\complement \\paren H$"}
+{"_id": "10796", "title": "De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection", "text": ":$\\overline {T_1 \\cap T_2} = \\overline T_1 \\cup \\overline T_2$"}
+{"_id": "10797", "title": "De Morgan's Laws (Set Theory)/Set Complement/Complement of Union", "text": ":$\\overline {T_1 \\cup T_2} = \\overline T_1 \\cap \\overline T_2$"}
+{"_id": "10798", "title": "De Morgan's Laws (Set Theory)/Proof by Induction/Difference with Intersection/Proof", "text": "Let $\\mathbb T = \\set {T_i: i \\mathop \\in I}$, where each $T_i$ is a set and $I$ is some finite indexing set. Then: :$\\displaystyle S \\setminus \\bigcap_{i \\mathop \\in I} T_i = \\bigcup_{i \\mathop \\in I} \\paren {S \\setminus T_i}$"}
+{"_id": "10802", "title": "De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection", "text": ":$\\displaystyle S \\setminus \\bigcap \\mathbb T = \\bigcup_{T' \\mathop \\in \\mathbb T} \\paren {S \\setminus T'}$ where: :$\\displaystyle \\bigcap \\mathbb T := \\set {x: \\forall T' \\in \\mathbb T: x \\in T'}$ that is, the intersection of $\\mathbb T$"}
+{"_id": "10803", "title": "De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Union", "text": ":$\\displaystyle S \\setminus \\bigcup \\mathbb T = \\bigcap_{T' \\mathop \\in \\mathbb T} \\paren {S \\setminus T'}$ where: :$\\displaystyle \\bigcup \\mathbb T := \\set {x: \\exists T' \\in \\mathbb T: x \\in T'}$ that is, the union of $\\mathbb T$."}
+{"_id": "10804", "title": "De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection", "text": ":$S \\setminus \\paren {T_1 \\cap T_2} = \\paren {S \\setminus T_1} \\cup \\paren {S \\setminus T_2}$"}
+{"_id": "10805", "title": "De Morgan's Laws (Set Theory)/Set Difference/Difference with Union", "text": ":$S \\setminus \\paren {T_1 \\cup T_2} = \\paren {S \\setminus T_1} \\cap \\paren {S \\setminus T_2}$"}
+{"_id": "10806", "title": "Isomorphism between Ring of Integers Modulo 2 and Parity Ring", "text": "The ring of integers modulo $2$ and the parity ring are isomorphic."}
+{"_id": "10807", "title": "Parity Addition is Associative", "text": "Let $R := \\struct {\\set {\\text{even}, \\text{odd} }, +, \\times}$ be the parity ring. The operation $+$ is associative: :$\\forall a, b, c \\in R: \\paren {a + b} + c = a + \\paren {b + c}$"}
+{"_id": "10810", "title": "Parity Multiplication is Commutative", "text": "Let $R := \\struct {\\set {\\text{even}, \\text{odd} }, +, \\times}$ be the parity ring. The operation $\\times$ is commutative: :$\\forall a, b \\in R: a \\times b = b \\times a$"}
+{"_id": "10819", "title": "Isomorphism between Roots of Unity under Multiplication and Integers under Modulo Addition", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\struct {R_n, \\times}$ be the complex $n$th roots of unity under complex multiplication. Let $\\struct {\\Z_n, +_n}$ be the integers modulo $n$ under modulo addition. Then $\\struct {R_n, \\times}$ and $\\struct {\\Z_n, +_n}$ are isomorphic algebraic structures."}
+{"_id": "10820", "title": "Isomorphism between Gaussian Integer Units and Integers Modulo 4 under Addition", "text": "Let $\\struct {U_\\C, \\times}$ be the group of Gaussian integer units under complex multiplication. Let $\\struct {\\Z_n, +_4}$ be the integers modulo $4$ under modulo addition. Then $\\struct {U_\\C, \\times}$ and $\\struct {\\Z_4, +_4}$ are isomorphic algebraic structures."}
+{"_id": "10823", "title": "Gaussian Integer Units are 4th Roots of Unity", "text": "The units of the ring of Gaussian integers: :$\\set {1, i, -1, -i}$ are the (complex) $4$th roots of $1$."}
+{"_id": "10827", "title": "Isomorphism by Cayley Table", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be algebraic structures whose underlying sets are both finite. Then $\\struct {S, \\circ}$ and $\\struct {T, *}$ are isomorphic {{iff}}: :a bijection $f: S \\to T$ can be found such that: ::the Cayley table of $\\struct {T, *}$ can be generated from the Cayley table of $\\struct {S, \\circ}$ by replacing each entry of $S$ with its image under $f$."}
+{"_id": "10828", "title": "Bijection between Integers and Even Integers", "text": "Let $\\Z$ be the set of integers. Let $2 \\Z$ be the set of even integers. Then there exists a bijection $f: \\Z \\to 2 \\Z$ between the two."}
+{"_id": "10829", "title": "Set of Integers under Addition is Isomorphic to Set of Even Integers under Addition", "text": "Let $\\struct {\\Z, +}$ be the algebraic structure formed by the set of integers under the operation of addition. Let $\\struct {2 \\Z, +}$ be the algebraic structure formed by the set of even integers under the operation of addition. Then $\\struct {\\Z, +}$ and $\\struct {2 \\Z, +}$ are isomorphic."}
+{"_id": "10830", "title": "Natural Numbers under Addition do not form Group", "text": "The algebraic structure $\\struct {\\N, +}$ consisting of the set of natural numbers $\\N$ under addition $+$ is not a group."}
+{"_id": "10831", "title": "Natural Numbers under Multiplication do not form Group", "text": "The algebraic structure $\\struct {\\N, \\times}$ consisting of the set of natural numbers $\\N$ under multiplication $\\times$ is not a group."}
+{"_id": "10835", "title": "Symmetry Group of Square is Group", "text": "The symmetry group of the square is a non-abelian group."}
+{"_id": "10836", "title": "Dihedral Group is Group", "text": "Let $D_n$ be the dihedral group of order $2 n$. Then $D_n$ is indeed a group."}
+{"_id": "10837", "title": "Symmetric Group is Subgroup of Monoid of Self-Maps", "text": "Let $S$ be a set. Let $S^S$ be the set of all mappings from $S$ to itself Let $\\struct {\\Gamma \\paren S, \\circ}$ denote the symmetric group on $S$. Let $\\struct {S^S, \\circ}$ be the monoid of self-maps under composition of mappings. Then $\\struct {\\Gamma \\paren S, \\circ}$ is a subgroup of $\\struct {S^S, \\circ}$."}
+{"_id": "10838", "title": "Cancellability by Cayley Table", "text": "Let $\\left({S, \\circ}\\right)$ be a finite algebraic structure. Let $\\mathcal T$ be the Cayley table for $\\left({S, \\circ}\\right)$. Let $a \\in S$ be an element of $S$. Then $a$ is cancellable for $\\circ$ {{iff}}: :$(1): \\quad$ no element of $S$ is repeated in $\\mathcal T$ in the row headed by $a$ and: :$(2): \\quad$ no element of $S$ is repeated in $\\mathcal T$ in the column headed by $a$."}
+{"_id": "10841", "title": "Piecewise Continuous Function does not necessarily have Improper Integrals", "text": "Let $f$ be a real function defined on a closed interval $\\left[{a \\,.\\,.\\, b}\\right]$, $a < b$.  Let $f$ be a piecewise continuous function: {{:Definition:Piecewise Continuous Function}} Then it is not necessarily the case that $f$ is a piecewise continuous function with improper integrals: {{:Definition:Piecewise Continuous Function with Improper Integrals}}"}
+{"_id": "10842", "title": "Right Cancellable Element is Right Cancellable in Subset", "text": "Let $\\left ({S, \\circ}\\right)$ be an algebraic structure. Let $\\left ({T, \\circ}\\right) \\subseteq \\left ({S, \\circ}\\right)$. Let $x \\in T$ be right cancellable in $S$. Then $x$ is also right cancellable in $T$."}
+{"_id": "10843", "title": "Left Cancellable Element is Left Cancellable in Subset", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\struct {T, \\circ} \\subseteq \\struct {S, \\circ}$. Let $x \\in T$ be left cancellable in $S$. Then $x$ is also left cancellable in $T$."}
+{"_id": "10844", "title": "Left Cancellable Elements of Semigroup form Subsemigroup", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Let $C_\\lambda$ be the set of left cancellable elements of $\\struct {S, \\circ}$. Then $\\struct {C_\\lambda, \\circ}$ is a subsemigroup of $\\struct {S, \\circ}$."}
+{"_id": "10845", "title": "Right Cancellable Elements of Semigroup form Subsemigroup", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Let $C_\\rho$ be the set of right cancellable elements of $\\struct {S, \\circ}$. Then $\\struct {C_\\rho, \\circ}$ is a subsemigroup of $\\struct {S, \\circ}$."}
+{"_id": "10848", "title": "Surjection iff Right Cancellable/Necessary Condition", "text": "Let $f$ be a surjection. Then $f$ is right cancellable."}
+{"_id": "10851", "title": "Surjection iff Right Cancellable/Sufficient Condition", "text": "Let $f$ be a mapping which is right cancellable. Then $f$ is a surjection."}
+{"_id": "10852", "title": "Absolute Value Function on Integers induces Equivalence Relation", "text": "Let $\\Z$ be the set of integers. Let $\\mathcal R$ be the relation on $\\Z$ defined as: :$\\forall x, y \\in \\Z: \\left({x, y}\\right) \\in \\mathcal R \\iff \\left\\vert{x}\\right\\vert = \\left\\vert{y}\\right\\vert$ where $\\left\\vert{x}\\right\\vert$ denotes the absolute value of $x$. Then $\\mathcal R$ is an equivalence relation."}
+{"_id": "10853", "title": "Absolute Value induces Equivalence not Compatible with Integer Addition", "text": "Let $\\Z$ be the set of integers. Let $\\RR$ be the relation on $\\Z$ defined as: :$\\forall x, y \\in \\Z: \\tuple {x, y} \\in \\RR \\iff \\size x = \\size y$ where $\\size x$ denotes the absolute value of $x$. Then $\\RR$ is not a congruence relation for integer addition."}
+{"_id": "10854", "title": "Equivalence Relation is Congruence for Right Operation", "text": "Every equivalence is a congruence for the right operation."}
+{"_id": "10855", "title": "Equivalence Relation is Congruence for Left Operation", "text": "Every equivalence is a congruence for the left operation."}
+{"_id": "10856", "title": "Left Coset Space forms Partition", "text": "The left coset space of $H$ forms a partition of its group $G$, and hence: {{begin-eqn}} {{eqn | l = x \\equiv^l y \\pmod H       | o = \\iff       | r = x H = y H }} {{eqn | l = \\neg \\paren {x \\equiv^l y} \\pmod H       | o = \\iff       | r = x H \\cap y H = \\O }} {{end-eqn}}"}
+{"_id": "10857", "title": "Right Coset Space forms Partition", "text": "The right coset space of $H$ forms a partition of its group $G$: {{begin-eqn}} {{eqn | l = x \\equiv^r y \\pmod H       | o = \\iff       | r = H x = H y }} {{eqn | l = \\neg \\paren {x \\equiv^r y} \\pmod H       | o = \\iff       | r = H x \\cap H y = \\O }} {{end-eqn}}"}
+{"_id": "10858", "title": "Right Cosets are Equal iff Product with Inverse in Subgroup", "text": "Let $H x$ denote the right coset of $H$ by $x$. Then: :$H x = H y \\iff x y^{-1} \\in H$"}
+{"_id": "10859", "title": "Left Cosets are Equal iff Product with Inverse in Subgroup", "text": "Let $x H$ denote the left coset of $H$ by $x$. Then: :$x H = y H \\iff x^{-1} y \\in H$"}
+{"_id": "10860", "title": "Left Congruence Class Modulo Subgroup is Left Coset", "text": "Let $\\mathcal R^l_H$ be the equivalence defined as left congruence modulo $H$. The equivalence class $\\eqclass g {\\mathcal R^l_H}$ of an element $g \\in G$ is the left coset $g H$. This is known as the '''left congruence class of $g \\bmod H$'''."}
+{"_id": "10862", "title": "Left Congruence Modulo Subgroup is Equivalence Relation", "text": "Let $x \\equiv^l y \\pmod H$ denote the relation that $x$ is left congruent modulo $H$ to $y$. Then the relation $\\equiv^l$ is an equivalence relation."}
+{"_id": "10863", "title": "Right Congruence Modulo Subgroup is Equivalence Relation", "text": "Let $x \\equiv^r y \\pmod H$ denote the relation that $x$ is right congruent modulo $H$ to $y$ Then the relation $\\equiv^r$ is an equivalence relation."}
+{"_id": "10864", "title": "Group Epimorphism is Isomorphism iff Kernel is Trivial", "text": "Let $\\struct {G, \\oplus}$ and $\\struct {H, \\odot}$ be groups. Let $\\phi: \\struct {G, \\oplus} \\to \\struct {H, \\odot}$ be a group epimorphism. Let $e_G$ and $e_H$ be the identities of $G$ and $H$ respectively. Let $K = \\map \\ker \\phi$ be the kernel of $\\phi$. Then: :the epimorphism $\\phi$ is an isomorphism {{iff}} :$K = \\set {e_G}$"}
+{"_id": "10867", "title": "Quotient Group of Quotient Group is Isomorphic to Quotient Group by Preimage under Quotient Mapping", "text": "Let $G$ be a group. Let $H \\lhd G$ where $\\lhd$ denotes that $H$ is a normal subgroup of $G$. Let $K \\lhd G/H$ and $L = q_H^{-1} \\left[{K}\\right]$, where: :$q_H: G \\to G/H$ is the quotient epimorphism from $G$ to the quotient group $G/H$ :$q_H^{-1} \\left[{K}\\right]$ is the preimage of $K$ under $q_H$. Then there exists a group isomorphism $\\phi: \\left({G / H}\\right) / K \\to G / L$ defined as: ::$\\phi \\circ q_K \\circ q_H = q_L$"}
+{"_id": "10870", "title": "Real Numbers are not Well-Ordered under Conventional Ordering", "text": "Let $\\left({\\R, \\leqslant}\\right)$ be the ordered structure consisting of the real numbers under the conventional ordering. Then $\\left({\\R, \\leqslant}\\right)$ is not a well-ordered set."}
+{"_id": "10871", "title": "Supremum is Unique", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $T$ be a non-empty subset of $S$. Then $T$ has at most one supremum in $S$."}
+{"_id": "10872", "title": "Infimum is Unique", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $T$ be a non-empty subset of $S$. Then $T$ has at most one infimum in $S$."}
+{"_id": "10873", "title": "Infimum of Subgroups in Lattice", "text": "Then: :$\\inf \\set {H, K} = H \\cap K$"}
+{"_id": "10874", "title": "Supremum of Subgroups in Lattice", "text": "Let either $H$ or $K$ be normal in $G$. Then: :$\\sup \\left\\{{H, K}\\right\\} = H \\circ K$ where $H \\circ K$ denotes subset product."}
+{"_id": "10875", "title": "Lexicographic Order on Totally Ordered Sets is Total Ordering", "text": "Let $\\struct {S_1, \\preceq_1}$ and $\\struct {S_2, \\preceq_2}$ be ordered sets. Let $\\preccurlyeq$ be the lexicographic order on $S_1 \\times S_2$''': :$\\tuple {x_1, x_2} \\preccurlyeq \\tuple {y_1, y_2} \\iff \\paren {x_1 \\prec_1 y_1} \\lor \\paren {x_1 = y_1 \\land x_2 \\preceq_2 y_2}$ Then: :$\\preccurlyeq$ is a total ordering on $S_1 \\times S_2$ {{iff}} :both $\\preceq_1$ and $\\preceq_2$ are total orderings."}
+{"_id": "10876", "title": "Lexicographic Order is Ordering", "text": "Let $\\left({S_1, \\preceq_1}\\right)$ and $\\left({S_2, \\preceq_2}\\right)$ be ordered sets. Let $\\preccurlyeq$ be the lexicographic order on $S_1 \\times S_2$''': :$\\left({x_1, x_2}\\right) \\preccurlyeq \\left({y_1, y_2}\\right) \\iff \\left({x_1 \\prec_1 y_1}\\right) \\lor \\left({x_1 = y_1 \\land x_2 \\preceq_2 y_2}\\right)$ Then $\\preccurlyeq$ is an ordering on $S_1 \\times S_2$."}
+{"_id": "10879", "title": "Powers of Commuting Elements of Monoid Commute", "text": ":$\\forall m, n \\in \\N: \\paren {\\circ^m a} \\circ \\paren {\\circ^n b} = \\paren {\\circ^n b} \\circ \\paren {\\circ^m a}$"}
+{"_id": "10880", "title": "Power of Product of Commuting Elements in Monoid equals Product of Powers", "text": ":$\\forall n \\in \\N: \\circ^n \\paren {a \\circ b} = \\paren {\\circ^n a} \\circ \\paren {\\circ^n b}$"}
+{"_id": "10882", "title": "Index Laws/Product of Indices/Monoid", "text": "Let $\\struct {S, \\circ}$ be a monoid whose identity element is $e$. For $a \\in S$, let $\\circ^n a = a^n$ be the $n$th power of $a$. Then: :$\\forall m, n \\in \\N: a^{n m} = \\paren {a^n}^m = \\paren {a^m}^n$"}
+{"_id": "10884", "title": "Cardinality of Proper Subset of Finite Set", "text": "Let $A$ and $B$ be finite sets such that $A \\subsetneqq B$. Let $\\card B = n$, where $\\card {\\, \\cdot \\,}$ denotes cardinality. Then $\\card A < n$."}
+{"_id": "10885", "title": "First-Countable Space is Hausdorff iff All Convergent Sequences have Unique Limit", "text": "Let $T = \\struct {S, \\tau}$ be a first-countable topological space. Then $T$ is Hausdorff {{iff}} all convergent sequences on $T$ have a unique limit point."}
+{"_id": "10886", "title": "Topologically Distinguishable Points are Distinct", "text": "Let $T = \\left({X, \\tau}\\right)$ be a topological space. Let $x, y \\in X$ be topologically distinguishable. Then the singleton sets $\\left\\{{x}\\right\\}$ and $\\left\\{{y}\\right\\}$ are disjoint and so: : $x \\ne y$"}
+{"_id": "10887", "title": "Restriction of Strict Total Ordering is Strict Total Ordering", "text": "Let $\\left({S, \\prec}\\right)$ be a strict total ordering. Let $T \\subseteq S$. Let $\\prec \\restriction_T$ be the restriction of $\\prec$ to $T$. Then $\\prec \\restriction_T$ is a strict total ordering of $T$."}
+{"_id": "10888", "title": "Principle of Recursive Definition", "text": "Let $\\N$ be the natural numbers. Let $T$ be a set. Let $a \\in T$. Let $g: T \\to T$ be a mapping. Then there exists exactly one mapping $f: \\N \\to T$ such that: :$\\forall x \\in \\N: \\map f x = \\begin{cases} a & : x = 0 \\\\ \\map g {\\map f n} & : x = n + 1 \\end{cases}$"}
+{"_id": "10891", "title": "Principle of Recursive Definition/General Result", "text": "Let $p \\in \\N$. Let $p^\\ge$ be the upper closure of $p$ in $\\N$: :$p^\\ge := \\set {x \\in \\N: x \\ge p} = \\set {p, p + 1, p + 2, \\ldots}$ Then there exists exactly one mapping $f: p^\\ge \\to T$ such that: :$\\forall x \\in p^\\ge: \\map f x = \\begin{cases} a & : x = p \\\\ \\map g {\\map f n} & : x = n + 1 \\end{cases}$"}
+{"_id": "10893", "title": "Real Zero is Zero Element", "text": ":$\\forall x \\in \\R: 0 \\times x = 0$"}
+{"_id": "10894", "title": "Negative of Real Zero equals Zero", "text": "Let $0$ denote the identity for addition in the real numbers $\\R$. Then: :$-0 = 0$"}
+{"_id": "10895", "title": "Negative of Negative Real Number", "text": ":$\\forall x \\in \\R: -\\paren {-x} = x$"}
+{"_id": "10896", "title": "Multiplication by Negative Real Number", "text": ":$\\forall x, y \\in \\R: x \\times \\paren {-y} = -\\paren {x \\times y} = \\paren {-x} \\times y$"}
+{"_id": "10899", "title": "Negative of Sum of Real Numbers", "text": ":$\\forall x, y \\in \\R: -\\paren {x + y} = -x - y$"}
+{"_id": "10901", "title": "Real Multiplication Identity is One/Corollary", "text": ":$\\forall x \\in \\R_{\\ne 0}: x \\times y = x \\implies y = 1$"}
+{"_id": "10902", "title": "Real Number Divided by Itself", "text": ":$\\forall x \\in \\R_{\\ne 0}: \\dfrac x x = 1$"}
+{"_id": "10906", "title": "Product of Reciprocals of Real Numbers", "text": ":$\\forall x, y \\in \\R_{\\ne 0}: \\dfrac 1 x \\times \\dfrac 1 y = \\dfrac 1 {x \\times y}$"}
+{"_id": "10907", "title": "Product of Quotients of Real Numbers", "text": ":$\\forall x, w \\in \\R, y, z \\in \\R_{\\ne 0}: \\dfrac x y \\times \\dfrac w z = \\dfrac {x \\times w} {y \\times z}$"}
+{"_id": "10908", "title": "Sum of Quotients of Real Numbers", "text": ":$\\forall x, w \\in \\R, y, z \\in \\R_{\\ne 0}: \\dfrac x y + \\dfrac w z = \\dfrac {\\paren {x \\times z} + \\paren {y \\times w} } {y \\times z}$"}
+{"_id": "10910", "title": "Reciprocal of Quotient of Real Numbers", "text": ":$\\forall x, y \\in \\R_{\\ne 0}: \\dfrac 1 {x / y} = \\dfrac y x$"}
+{"_id": "10911", "title": "Quotient of Quotients of Real Numbers", "text": ":$\\forall x \\in \\R, y, w, z \\in \\R_{\\ne 0}: \\dfrac {x / y} {w / z} = \\dfrac {x \\times z} {y \\times w}$"}
+{"_id": "10912", "title": "Product of Real Number with Quotient", "text": ":$\\forall a, x \\in \\R, y \\in \\R_{\\ne 0}: \\dfrac {a \\times x} y = a \\times \\dfrac x y$"}
+{"_id": "10913", "title": "Neighborhood in Topological Subspace", "text": "Let $\\struct {X, \\tau}$ be a topological space. Let $S \\subseteq X$ be a subset of $X$. Let $\\tau_S$ denote the subspace topology on $S$. Let $x \\in S$ be an arbitrary point of $S$. Let $E \\subseteq S$. Then: :$E$ is a neighborhood of $x$ in $\\struct {S, \\tau_S}$ {{iff}}: :$\\exists D \\subseteq X$ such that: ::$D$ is a neighborhood of $x$ in $X$ ::$E = D \\cap S$."}
+{"_id": "10914", "title": "Negative of Quotient of Real Numbers", "text": ":$\\forall x \\in \\R, y \\in \\R_{\\ne 0}: \\dfrac {-x} y = -\\dfrac x y = \\dfrac x {-y}$"}
+{"_id": "10917", "title": "Product of Strictly Positive Real Numbers is Strictly Positive", "text": ":$x, y \\in \\R_{>0} \\implies x \\times y \\in \\R_{>0}$"}
+{"_id": "10918", "title": "Real Number is Greater than Zero iff its Negative is Less than Zero", "text": ":$\\forall x \\in \\R: x > 0 \\iff \\paren {-x} < 0$"}
+{"_id": "10919", "title": "Order of Real Numbers is Dual of Order of their Negatives", "text": ":$\\forall x, y \\in \\R: x > y \\iff \\paren {-x}< \\paren {-y}$"}
+{"_id": "10920", "title": "Square of Non-Zero Real Number is Strictly Positive", "text": ":$\\forall x \\in \\R: x \\ne 0 \\implies x^2 > 0$"}
+{"_id": "10923", "title": "Real Zero is Less than Real One", "text": "The real number $0$ is less than the real number $1$: :$0 < 1$"}
+{"_id": "10924", "title": "Product of Real Numbers is Positive iff Numbers have Same Sign", "text": "The product of two real numbers is greater than $0$ {{iff}} either both are greater than $0$ or both are less than $0$. :$\\forall x, y \\in \\R: x \\times y > 0 \\iff \\paren {x, y \\in \\R_{>0} } \\lor \\paren {x, y \\in \\R_{<0} }$"}
+{"_id": "10925", "title": "Reciprocal of Strictly Positive Real Number is Strictly Positive", "text": ":$\\forall x \\in \\R: x > 0 \\implies \\dfrac 1 x > 0$"}
+{"_id": "10926", "title": "Order of Strictly Positive Real Numbers is Dual of Order of their Reciprocals", "text": ":$\\forall x, y \\in \\R: x > y > 0 \\implies \\dfrac 1 x < \\dfrac 1 y$"}
+{"_id": "10927", "title": "Mean of Unequal Real Numbers is Between them", "text": ":$\\forall x, y \\in \\R: x < y \\implies x < \\dfrac {x + y} 2 < y$"}
+{"_id": "10928", "title": "Intersection of Inductive Set as Subset of Real Numbers is Inductive Set", "text": "Let $\\AA$ be a set of inductive sets defined as subsets of real numbers. Then their intersection is an inductive set."}
+{"_id": "10931", "title": "Non-Empty Subset of Initial Segment of Natural Numbers has Greatest Element", "text": "Let $n \\in \\N_{>0}$ be a non-zero natural number. Let $\\N^*_n$ denote the Initial segment $\\set {1, 2, \\ldots, n}$ of the non-zero natural numbers. Then every non-empty subset of $\\N^*_n$ has a greatest element."}
+{"_id": "10932", "title": "Not every Non-Empty Subset of Natural Numbers has Greatest Element", "text": "Let $S \\subseteq \\N_{>0}$. Then, despite Non-Empty Subset of Initial Segment of Natural Numbers has Greatest Element, it is not necessarily the case that $S$ has a greatest element."}
+{"_id": "10935", "title": "Existence of Integral on Union of Adjacent Intervals", "text": "Let $f$ be a real function defined on a closed interval $\\closedint a b$ where $a < b$. Let $c$ be a point in $\\openint a b$. Then: :$f$ is Darboux integrable on $\\closedint a c$ and $\\closedint c b$ {{iff}}: :$f$ is Darboux integrable on $\\closedint a b$."}
+{"_id": "10936", "title": "Cartesian Product of Bijections is Bijection", "text": "Let $S_1 \\times S_2$ be the Cartesian product of two sets $S_1$ and $S_2$. Let $T_1 \\times T_2$ be the Cartesian product of two sets $T_1$ and $T_2$. Let $f_1: S_1 \\to T_1$ and $f_2: S_2 \\to T_2$ be bijections. Let $f_1 \\times f_2: S_1 \\times S_2 \\to T_1 \\times T_2$ be the Cartesian product of $f_1$ and $f_2$ defined as: :$\\forall \\left({s_1, s_2}\\right) \\in S_1 \\times S_2: f_1 \\times f_2 \\left({s_1, s_2}\\right) := \\left({f_1 \\left({s_1}\\right), f_2 \\left({s_2}\\right)}\\right)$ Then $f_1 \\times f_2$ is a bijection."}
+{"_id": "10941", "title": "First Order ODE/y' - f (y) phi' (x) over f' (y) = phi (x) phi' (x) over f' (y)", "text": "Let $\\map f y$ and $\\map \\phi x$ be known real functions of $y$ and $x$ respectively. The general solution of: :$(1): \\quad \\dfrac {\\d y} {\\d x} - \\dfrac {\\map f y} {\\map {f'} y} \\map {\\phi'} x = \\dfrac {\\map \\phi x \\, \\map {\\phi'} x} {\\map {f'} y}$ is: :$\\map f y = C e^{\\map \\phi x} - \\map \\phi x - 1$"}
+{"_id": "10942", "title": "Cardinality of Set of All Mappings/Finite Sets", "text": "Let $S$ and $T$ be finite sets. The cardinality of the set of all mappings from $S$ to $T$ (that is, the total number of mappings from $S$ to $T$) is: :$\\card {T^S} = \\card T^{\\card S}$"}
+{"_id": "10943", "title": "Cardinality of Set of All Mappings/Infinite Sets", "text": "Let $S$ and $T$ be sets such that either $S$ or $T$ is infinite. The cardinality of the set of all mappings from $S$ to $T$ (that is, the total number of mappings from $S$ to $T$) is: :$\\card {T^S} = \\card T^{\\card S}$"}
+{"_id": "10944", "title": "Cardinality of Set Union/General Case", "text": "Let $S_1, S_2, \\ldots$ be sets. Then: {{begin-eqn}}      {{eqn | l = \\card {\\bigcup_{i \\mathop = 1}^n S_i}       | r = \\sum_{i \\mathop = 1}^n \\card {S_i}       | c = }} {{eqn | o = -       | r = \\sum_{1 \\mathop \\le i \\mathop < j \\mathop \\le n} \\card {S_i \\cap S_j}       | c = }} {{eqn | o = +       | r = \\sum_{1 \\mathop \\le i \\mathop < j \\mathop < k \\mathop \\le n} \\card {S_i \\cap S_j \\cap S_k}       | c = }} {{eqn | o = \\cdots       | c = }} {{eqn | o = +       | r = \\paren {-1}^{n - 1} \\card {\\bigcap_{i \\mathop = 1}^n S_i}       | c = }} {{end-eqn}}"}
+{"_id": "10946", "title": "Structure of Inverse Completion of Commutative Semigroup", "text": "Let $\\left({S, \\circ}\\right)$ be a commutative semigroup. Let $\\left ({C, \\circ}\\right) \\subseteq \\left({S, \\circ}\\right)$ be the subsemigroup of cancellable elements of $\\left({S, \\circ}\\right)$. Let $\\left({T, \\circ'}\\right)$ be an inverse completion of $\\left({S, \\circ}\\right)$. Then: :$T = S \\circ' C^{-1}$ where: :$C^{-1}$ is the inverse of $C$ in $T$ :$S \\circ' C^{-1}$ is the subset product of $S$ with $C^{-1}$."}
+{"_id": "10947", "title": "Inverse in Monoid is Unique", "text": "Let $\\struct {S, \\circ}$ be a monoid. Then an element $x \\in S$ can have at most one inverse for $\\circ$."}
+{"_id": "10948", "title": "Inverse of Inverse/Monoid", "text": "Let $\\struct {S, \\circ}$ be a monoid. Let $x \\in S$ be invertible, and let its inverse be $x^{-1}$. Then $x^{-1}$ is also invertible, and: :$\\paren {x^{-1} }^{-1} = x$"}
+{"_id": "10949", "title": "Inverse of Product/Monoid", "text": "Let $\\left({S, \\circ}\\right)$ be a monoid whose identity is $e$. Let $a, b \\in S$ be invertible for $\\circ$, with inverses $a^{-1}, b^{-1}$. Then $a \\circ b$ is invertible for $\\circ$, and: : $\\left({a \\circ b}\\right)^{-1} = b^{-1} \\circ a^{-1}$"}
+{"_id": "10950", "title": "Right Identity Element is Idempotent", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $e_R \\in S$ be a right identity with respect to $\\circ$. Then $e_R$ is idempotent under $\\circ$."}
+{"_id": "10952", "title": "More than one Right Identity then no Left Identity", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure. If $\\left({S, \\circ}\\right)$ has more than one right identity, then it has no left identity."}
+{"_id": "10953", "title": "Inverse Completion is Commutative Monoid", "text": "Let $\\left({S, \\circ}\\right)$ be a commutative semigroup. Let $\\left ({C, \\circ}\\right) \\subseteq \\left({S, \\circ}\\right)$ be the subsemigroup of cancellable elements of $\\left({S, \\circ}\\right)$. Let $\\left({T, \\circ'}\\right)$ be an inverse completion of $\\left({S, \\circ}\\right)$. Then $\\left({T, \\circ'}\\right)$ is a commutative monoid."}
+{"_id": "10954", "title": "Inverse Completion is Commutative Semigroup", "text": "Let $\\left({S, \\circ}\\right)$ be a commutative semigroup. Let $\\left ({C, \\circ}\\right) \\subseteq \\left({S, \\circ}\\right)$ be the subsemigroup of cancellable elements of $\\left({S, \\circ}\\right)$. Let $\\left({T, \\circ'}\\right)$ be an inverse completion of $\\left({S, \\circ}\\right)$. Then $T = S \\circ' C^{-1}$, and is a commutative semigroup."}
+{"_id": "10956", "title": "Complement of G-Delta Set is F-Sigma Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $X$ be an $G_\\delta$ set of $T$. Then its complement $S \\setminus X$ is an $F_\\sigma$ set of $T$."}
+{"_id": "10957", "title": "Bounded Piecewise Continuous Function has Improper Integrals", "text": "Let $f$ be a real function defined on a closed interval $\\closedint a b$, $a < b$.  Let $f$ be piecewise continuous and bounded on $\\closedint a b$. Then $f$ is a piecewise continuous function with improper integrals."}
+{"_id": "10958", "title": "F-Sigma Set is not necessarily Closed Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $X$ be an $F_\\sigma$ set of $T$. Then it is not necessarily the case that $X$ is a closed set of $T$."}
+{"_id": "10959", "title": "Not every Closed Set is G-Delta Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $V$ be a closed set of $T$. Then it is not necessarily the case that $V$ is a $G_\\delta$ set of $T$."}
+{"_id": "10960", "title": "Open Set of Uncountable Finite Complement Topology is not F-Sigma", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on an uncountable set $S$. Let $U \\in \\tau$ be an open set of $T$. Then $U$ is not an $F_\\sigma$ set."}
+{"_id": "10961", "title": "Closed Set of Uncountable Finite Complement Topology is not G-Delta", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on an uncountable set $S$. Let $V \\in \\tau$ be a closed set of $T$. Then $V$ is not a $G_\\delta$ set."}
+{"_id": "10962", "title": "G-Delta Set is not necessarily Open Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $X$ be a $G_\\delta$ set of $T$. Then it is not necessarily the case that $X$ is a open set of $T$."}
+{"_id": "10964", "title": "Limit Point of Sequence may only be Adherent Point of Range", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A \\subseteq S$. Let $\\sequence {x_n}$ be a sequence in $A$. Let $\\alpha$ be a limit point of $\\sequence {x_n}$. Then $\\alpha$ may be only an adherent point of $A$ and not a limit point of $A$."}
+{"_id": "10965", "title": "Limit Point of Sequence in Discrete Space not always Limit Point of Open Set", "text": "Let $T = \\struct {S, \\tau}$ be a discrete topological space. Let $U \\in \\tau$ be an open set of $T$. Let $\\sequence {x_n}$ be a sequence in $U$. Let $x$ be the limit point of $\\sequence {x_n}$. Then $x$ is not always a limit point of $U$."}
+{"_id": "10966", "title": "Accumulation Point of Sequence of Distinct Terms is Omega-Accumulation Point of Range", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\sequence {x_n}$ be a sequence of distinct terms of $S$. Let $\\alpha$ be an accumulation point of $\\sequence {x_n}$. Then $\\alpha$ is also an $\\omega$-accumulation point of $\\set {x_n: n \\in \\N}$."}
+{"_id": "10967", "title": "Limit Point of Sequence is Adherent Point of Range", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $\\left \\langle {x_n} \\right \\rangle$ be a sequence in $S$. Let $\\alpha$ be a limit point of $\\left \\langle {x_n} \\right \\rangle$. Then $\\alpha$ is an adherent point of $\\left\\{ {x_n: n \\in \\N}\\right\\}$."}
+{"_id": "10968", "title": "Limit Point of Sequence of Distinct Terms is Omega-Accumulation Point of Range", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\sequence {x_n}$ be a sequence of distinct terms of $S$. Let $\\alpha$ be a limit point of $\\sequence {x_n}$. Then $\\alpha$ is also an $\\omega$-accumulation point of $\\set {x_n: n \\in \\N}$."}
+{"_id": "10969", "title": "Equivalence of Definitions of Isolated Point", "text": "{{TFAE|def = Isolated Point of Subset|view = isolated point}} Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be a subset of $S$."}
+{"_id": "10971", "title": "Equivalence of Definitions of Closed Set", "text": "{{TFAE|def = Closed Set (Topology)|view = Closed Set|context = Topology (Mathematical Branch)|contextview = topology}} Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$."}
+{"_id": "10972", "title": "Kuratowski's Closure-Complement Problem", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A \\subseteq S$ be a subset of $T$. By successive applications of the operations of complement relative to $S$ and the closure, there can be as many as $14$ distinct subsets of $S$ (including $A$ itself)."}
+{"_id": "10973", "title": "Kuratowski's Closure-Complement Problem/Complement", "text": "The complement of $A$ in $\\R$ is given by: {{begin-eqn}} {{eqn | l = A'       | r = \\left({\\gets \\,.\\,.\\, 0}\\right]       | c = {{Defof|Unbounded Closed Real Interval}} }} {{eqn | o =       | ro= \\cup       | r = \\left\\{ {1} \\right\\}       | c = {{Defof|Singleton}} }} {{eqn | o =       | ro= \\cup       | r = \\left[{2 \\,.\\,.\\, 3}\\right)       | c = {{Defof|Half-Open Real Interval}} }} {{eqn | o =       | ro= \\cup       | r = \\left({3 \\,.\\,.\\, 4}\\right]       | c = ... adjacent to Half-Open Real Interval }} {{eqn | o =       | ro= \\cup       | r = \\left({\\R \\setminus \\Q \\cap \\left[{4 \\,.\\,.\\, 5}\\right]}\\right)       | c = Irrational Numbers from $4$ to $5$ }} {{eqn | o =       | ro= \\cup       | r = \\left[{5 \\,.\\,.\\, \\to}\\right)       | c = {{Defof|Unbounded Closed Real Interval}} }} {{end-eqn}} :500px"}
+{"_id": "10974", "title": "Kuratowski's Closure-Complement Problem/Interior", "text": "The interior of $A$ in $\\R$ is given by: {{begin-eqn}} {{eqn | l = A^\\circ       | r = \\left({0 \\,.\\,.\\, 1}\\right) \\cup \\left({1 \\,.\\,.\\, 2}\\right)       | c = Union of Adjacent Open Intervals }} {{end-eqn}} :500px"}
+{"_id": "10975", "title": "Kuratowski's Closure-Complement Problem/Closure", "text": "The closure of $A$ in $\\R$ is given by: {{begin-eqn}} {{eqn | l = A^-       | r = \\closedint 0 2       | c = {{Defof|Closed Real Interval}} }} {{eqn | o =       | ro= \\cup       | r = \\set 3       | c = {{Defof|Singleton}} }} {{eqn | o =       | ro= \\cup       | r = \\closedint 4 5       | c = {{Defof|Closed Real Interval}} }} {{end-eqn}} :500px"}
+{"_id": "10976", "title": "Kuratowski's Closure-Complement Problem/Exterior", "text": "The exterior of $A$ in $\\R$ is given by: {{begin-eqn}} {{eqn | l = A^e       | r = \\openint \\gets 0       | c = {{Defof|Unbounded Open Real Interval}} }} {{eqn | o =       | ro= \\cup       | r = \\openint 2 3 \\cup \\openint 3 4       | c = {{Defof|Union of Adjacent Open Intervals}} }} {{eqn | o =       | ro= \\cup       | r = \\openint 5 \\to       | c = {{Defof|Unbounded Open Real Interval}} }} {{end-eqn}} :500px"}
+{"_id": "10977", "title": "Kuratowski's Closure-Complement Problem/Closure of Complement", "text": "The closure of the complement of $A$ in $\\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\\prime \\, -}       | r = \\hointl \\gets 0       | c = {{Defof|Unbounded Closed Real Interval}} }} {{eqn | o =       | ro= \\cup       | r = \\set 1       | c = {{Defof|Singleton}} }} {{eqn | o =       | ro= \\cup       | r = \\hointr 2 \\to       | c = {{Defof|Unbounded Closed Real Interval}} }} {{end-eqn}} :500px"}
+{"_id": "10978", "title": "Kuratowski's Closure-Complement Problem/Closure of Interior", "text": "The closure of the interior of $A$ in $\\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\\circ \\, -}       | r = \\closedint 0 2       | c = {{Defof|Closed Real Interval}} }} {{end-eqn}} :500px"}
+{"_id": "10979", "title": "Kuratowski's Closure-Complement Problem/Interior of Closure", "text": "The interior of the closure of $A$ in $\\R$ is given by: {{begin-eqn}} {{eqn | l = A^{- \\, \\circ}       | r = \\openint 0 2       | c = {{Defof|Open Real Interval}} }} {{eqn | o =       | ro= \\cup       | r = \\openint 4 5       | c = {{Defof|Open Real Interval}} }} {{end-eqn}} :500px"}
+{"_id": "10980", "title": "Kuratowski's Closure-Complement Problem/Interior of Closure of Interior", "text": "The interior of the closure of the interior of $A$ in $\\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\\circ \\, - \\, \\circ}       | r = \\left({0 \\,.\\,.\\, 2}\\right)       | c = {{Defof|Open Real Interval}} }} {{end-eqn}} :500px"}
+{"_id": "10981", "title": "Kuratowski's Closure-Complement Problem/Interior of Complement of Interior", "text": "The interior of the complement of the interior of $A$ in $\\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\\circ \\, \\prime \\, \\circ}       | r = \\openint \\gets 0       | c = {{Defof|Unbounded Open Real Interval}} }} {{eqn | o =       | ro= \\cup       | r = \\openint 2 \\to       | c = {{Defof|Unbounded Open Real Interval}} }} {{end-eqn}} :500px"}
+{"_id": "10982", "title": "Kuratowski's Closure-Complement Problem/Closure of Interior of Complement", "text": "The closure of the interior of the complement of $A$ in $\\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\\prime \\, \\circ \\, -}       | r = \\hointl \\gets 0       | c = {{Defof|Unbounded Closed Real Interval}} }} {{eqn | o =       | ro= \\cup       | r = \\closedint 2 4       | c = {{Defof|Closed Real Interval}} }} {{eqn | o =       | ro= \\cup       | r = \\hointr 5 \\to       | c = {{Defof|Unbounded Closed Real Interval}} }} {{end-eqn}} :500px"}
+{"_id": "10983", "title": "Kuratowski's Closure-Complement Problem/Closure of Interior of Closure", "text": "The closure of the interior of the closure of $A$ in $\\R$ is given by: {{begin-eqn}} {{eqn | l = A^{- \\, \\circ \\, -}       | r = \\left[{0 \\,.\\,.\\, 2}\\right]       | c = {{Defof|Closed Real Interval}} }} {{eqn | o =       | ro= \\cup       | r = \\left[{4 \\,.\\,.\\, 5}\\right]       | c = {{Defof|Closed Real Interval}} }} {{end-eqn}} :500px"}
+{"_id": "10984", "title": "Kuratowski's Closure-Complement Problem/Interior of Complement of Interior of Closure", "text": "The interior of the complement of the interior of the closure of $A$ in $\\R$ is given by: {{begin-eqn}} {{eqn | l = A^{- \\, \\circ \\, \\prime \\, \\circ}       | r = \\left({\\gets \\,.\\,.\\, 0}\\right)       | c = {{Defof|Unbounded Open Real Interval}} }} {{eqn | o =       | ro= \\cup       | r = \\left({2 \\,.\\,.\\, 4}\\right)       | c = {{Defof|Open Real Interval}} }} {{eqn | o =       | ro= \\cup       | r = \\left({5 \\,.\\,.\\, \\to}\\right)       | c = {{Defof|Unbounded Open Real Interval}} }} {{end-eqn}} :500px"}
+{"_id": "10985", "title": "Kuratowski's Closure-Complement Problem/Complement of Interior of Closure of Interior", "text": "The complement of the interior of the closure of the interior of $A$ in $\\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\\circ \\, - \\, \\circ \\, \\prime}       | r = \\left({\\gets \\,.\\,.\\, 0}\\right]       | c = {{Defof|Unbounded Closed Real Interval}} }} {{eqn | o =       | ro= \\cup       | r = \\left[{2 \\,.\\,.\\, \\to}\\right)       | c = {{Defof|Unbounded Closed Real Interval}} }} {{end-eqn}} :500px"}
+{"_id": "10986", "title": "Sum of Integrals on Adjacent Intervals for Integrable Functions", "text": "Let $f$ be a real function which is Darboux integrable on any closed interval $\\mathbb I$. Let $a, b, c \\in \\mathbb I$. Then: :$\\displaystyle \\int_a^c \\map f t \\rd t + \\int_c^b \\map f t \\rd t = \\int_a^b \\map f t \\rd t$"}
+{"_id": "10987", "title": "Closure of Irrational Numbers is Real Numbers", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the same topology. Then: :$\\paren {\\R \\setminus \\Q}^- = \\R$ where $\\paren {\\R \\setminus \\Q}-$ denotes the closure of $\\R \\setminus \\Q$."}
+{"_id": "10988", "title": "Closure of Union of Adjacent Open Intervals", "text": "Let $a, b, c \\in R$ where $a < b < c$. Let $A$ be the union of the two adjacent open intervals: :$A := \\left({a \\,.\\,.\\, b}\\right) \\cup \\left({b \\,.\\,.\\, c}\\right)$ Then: :$A^- = \\left[{a \\,.\\,.\\, c}\\right]$ where: : $A^-$ is the closure of $A$."}
+{"_id": "10989", "title": "Interior equals Complement of Closure of Complement", "text": "Let $T$ be a topological space. Let $H \\subseteq T$. Let $H^-$ denote the closure of $H$ and $H^\\circ$ denote the interior of $H$. Let $H^\\prime$ denote the complement of $H$ in $T$: :$H^\\prime = T \\setminus H$ Then: :$H^\\circ = H^{\\prime \\, - \\, \\prime}$"}
+{"_id": "10990", "title": "Closure of Half-Open Real Interval is Closed Real Interval", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line under the usual (Euclidean) topology.   Let $H_1 = \\hointl a b$ and $H_2 = \\hointr a b$ be half-open intervals of $\\R$. Then the closure of both $H_1$ and $H_2$ in $\\R$ are the closed interval $\\closedint a b$."}
+{"_id": "10993", "title": "Closure of Irrational Interval is Closed Real Interval", "text": "Let $\\struct {\\R, \\tau_d}$ be the real numbers under the usual (Euclidean) topology. Let $\\struct{\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the same topology. Let $a, b \\in \\R$ such that $a < b$. Let $\\Bbb I \\subseteq \\R$ be an interval of $\\R$ Then the closure of the set: :$\\Bbb I \\cap \\paren {\\R \\setminus \\Q}$ is the closed real interval $\\closedint a b$."}
+{"_id": "10994", "title": "Closure of Rational Interval is Closed Real Interval", "text": "Let $\\struct {\\R, \\tau_d}$ be the real numbers under the usual (Euclidean) topology. Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the same topology. Let $a, b \\in \\R$ such that $a < b$. Let $\\Bbb I \\subseteq \\R$ be an interval of $\\R$ Then the closure of the set : :$\\Bbb I \\cap \\Q$ is the closed real interval $\\closedint a b$."}
+{"_id": "10995", "title": "Interior of Singleton in Real Number Line is Empty", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $a \\in \\R$ be a real number. Then: :$\\set a^\\circ = \\O$ where $\\set a^\\circ$ denotes the interior of $\\set a$ in $\\R$."}
+{"_id": "10996", "title": "Kuratowski's Closure-Complement Problem/Proof of Maximum", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A \\subseteq S$ be a subset of $T$. By successive applications of the operations of complement relative to $S$ and the closure, there can be no more than $14$ distinct subsets of $S$ (including $A$ itself)."}
+{"_id": "10997", "title": "Closure of Complement of Closure is Regular Closed", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A \\subseteq S$ be a subset of $T$. Let $A^-$ denote the closure of $A$ in $T$. Let $A'$ denote the complement of $A$ in $S$: $A' = S \\setminus A$. Then $A^{- ' -}$ is regular closed."}
+{"_id": "10998", "title": "Topology induced by Usual Metric on Positive Integers is Discrete", "text": "Let $\\Z_{>0}$ be the set of (strictly) positive integers. Let $d: \\Z_{>0} \\times \\Z_{>0} \\to \\R$ be the usual (Euclidean) metric on $\\Z_{>0}$. Then the metric topology for $d$ is a discrete topology."}
+{"_id": "10999", "title": "Topology induced by Scaled Euclidean Metric on Positive Integers is Discrete", "text": "Let $\\Z_{>0}$ be the set of (strictly) positive integers. Let $\\delta: \\Z_{>0} \\times \\Z_{>0} \\to \\R$ be the metric on $\\Z_{>0}$ defined as: :$\\forall x, y \\in \\Z_{>0}: \\map \\delta {x, y} = \\dfrac {\\size {x - y} } {x y}$ Then the metric topology for $\\delta$ is a discrete topology."}
+{"_id": "11000", "title": "Scaled Euclidean Metric is Metric", "text": "Let $\\R_{>0}$ be the set of (strictly) positive integers. Let $\\delta: \\R_{>0} \\times \\R_{>0} \\to \\R$ be the metric on $\\R_{>0}$ defined as: :$\\forall x, y \\in \\R_{>0}: \\delta \\left({x, y}\\right) = \\dfrac {\\left\\lvert{x - y}\\right\\rvert} {x y}$ Then $\\delta$ is a metric."}
+{"_id": "11001", "title": "Topologies induced by Usual Metric and Scaled Euclidean Metric on Positive Integers are Homeomorphic", "text": "Let $\\Z_{>0}$ be the set of (strictly) positive integers. Let $d: \\Z_{>0} \\times \\Z_{>0} \\to \\R$ be the usual (Euclidean) metric on $\\Z_{>0}$. Let $\\delta: \\Z_{>0} \\times \\Z_{>0} \\to \\R$ be the metric on $\\Z_{>0}$ defined as: :$\\forall x, y \\in \\Z_{>0}: \\map \\delta {x, y} = \\dfrac {\\size {x - y} } {x y}$ Let $\\tau_d$ denote the metric topology for $d$. Let $\\tau_\\delta$ denote the metric topology for $\\delta$. Then $\\struct {\\Z_{>0}, \\tau_d}$ and $\\struct {\\Z_{>0}, \\tau_\\delta}$ are homeomorphic."}
+{"_id": "11002", "title": "Cauchy Sequence in Positive Integers under Usual Metric is eventually Constant", "text": "Let $\\Z_{>0}$ be the set of (strictly) positive integers. Let $d: \\Z_{>0} \\times \\Z_{>0} \\to \\R$ be the usual (Euclidean) metric on $\\Z_{>0}$. Let $\\sequence {x_n}$ be a Cauchy sequence in $\\struct {\\Z_{>0}, d}$. Then: :$\\exists m, n \\in \\Z_{>0}: \\forall r > n: x_r = m$ That is, $\\sequence {x_n}$ is eventually constant."}
+{"_id": "11003", "title": "Positive Integers under Usual Metric is Complete Metric Space", "text": "Let $\\Z_{>0}$ be the set of (strictly) positive integers. Let $d: \\Z_{>0} \\times \\Z_{>0} \\to \\R$ be the usual (Euclidean) metric on $\\Z_{>0}$. Then $\\struct {\\Z_{>0}, d}$ is a complete metric space."}
+{"_id": "11004", "title": "Cauchy Sequence in Positive Integers under Scaled Euclidean Metric", "text": "Let $\\Z_{>0}$ be the set of (strictly) positive integers. Let $\\delta: \\Z_{>0} \\times \\Z_{>0} \\to \\R$ be the scaled Euclidean metric on $\\Z_{>0}$ defined as: :$\\forall x, y \\in \\Z_{>0}: \\map \\delta {x, y} = \\dfrac {\\size {x - y} } {x y}$ The sequence $\\sequence {x_n}$ in $\\Z_{>0}$ defined as: :$\\forall n \\in \\N: x_n = n$ is a Cauchy sequence in $\\struct {\\Z_{>0}, \\delta}$."}
+{"_id": "11005", "title": "Positive Integers under Scaled Euclidean Metric is not Complete Metric Space", "text": "Let $\\Z_{>0}$ be the set of (strictly) positive integers. Let $\\delta: \\Z_{>0} \\times \\Z_{>0} \\to \\R$ be the scaled Euclidean metric on $\\Z_{>0}$ defined as: :$\\forall x, y \\in \\Z_{>0}: \\map \\delta {x, y} = \\dfrac {\\size {x - y} } {x y}$ Then $\\struct {\\Z_{>0}, \\delta}$ is not a complete metric space."}
+{"_id": "11006", "title": "Subset of Euclidean Plane whose Product of Coordinates are Greater Than or Equal to 1 is Closed", "text": "Let $\\struct {\\R^2, \\tau_d}$ be the real number plane with the usual (Euclidean) topology. Let $A \\subseteq R^2$ be the set of all points defined as: :$A := \\set {\\tuple {x, y} \\in \\R^2: x y \\ge 1}$ Then $A$ is a closed set in $\\struct {\\R^2, d}$."}
+{"_id": "11007", "title": "Projection on Real Euclidean Plane is Open Mapping", "text": "Let $\\struct {\\R^2, d}$ be the real number plane with the usual (Euclidean) topology. Let $\\rho: \\R^2 \\to \\R$ be the first projection on $\\R^2$ defined as: :$\\forall \\tuple{x, y} \\in \\R^2: \\map \\rho {x, y} = x$ Then $\\rho$ is an open mapping. The same applies with the second projection on $\\R^2$."}
+{"_id": "11008", "title": "Projection on Real Euclidean Plane is not Closed Mapping", "text": "Let $\\struct {\\R^2, d}$ be the real number plane with the usual (Euclidean) topology. Let $\\rho: \\R^2 \\to \\R$ be the first projection on $\\R^2$ defined as: :$\\forall \\tuple {x, y} \\in \\R^2: \\map \\rho {x, y} = x$ Then $\\rho$ is not a closed mapping. The same applies with the second projection on $\\R^2$."}
+{"_id": "11009", "title": "Complement of Set of Rational Pairs in Real Euclidean Plane is Arc-Connected", "text": "Let $\\struct {\\R^2, d}$ be the real number plane with the usual (Euclidean) topology. Let $S \\subseteq \\R^2$ be the subset of $\\R^2$ defined as: :$\\forall x, y \\in \\R^2: \\tuple {x, y} \\in S \\iff x, y \\in \\Q$ Hence let $A := \\R^2 \\setminus S$: :$\\tuple {x, y} \\in A$ {{iff}} either $x$ or $y$ or both is irrational. Then $A$ is arc-connected."}
+{"_id": "11010", "title": "Empty Set is Compact Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Then the empty set $\\varnothing$ is a compact subspace of $T$."}
+{"_id": "11011", "title": "Alexandroff Extension is Topology", "text": "Let $T = \\struct {S, \\tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \\cup \\set p$. Let $T^* = \\struct {S^*, \\tau^*}$ be the Alexandroff extension on $S$. Then $\\tau^*$ is a topology on $S^*$."}
+{"_id": "11012", "title": "Intersection of Closed Set with Compact Subspace is Compact", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be closed in $T$. Let $K \\subseteq S$ be compact in $T$. Then $H \\cap K$ is compact in $T$."}
+{"_id": "11013", "title": "Finite Union of Compact Sets is Compact", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $n \\in \\N$ be a natural number. Let $\\left\\langle{U_i}\\right\\rangle_{1 \\mathop \\le i \\mathop \\le n}$ be a finite sequence of compact subsets of $T$. Let $\\mathcal U_n := \\displaystyle \\bigcup_{i \\mathop = 1}^n U_i$ be the union of $\\left\\langle{U_i}\\right\\rangle$. Then $\\mathcal U_n$ is compact in $T$."}
+{"_id": "11015", "title": "Principle of Mathematical Induction", "text": "Let $\\map P n$ be a propositional function depending on $n \\in \\Z$. Let $n_0 \\in \\Z$ be given. Suppose that: :$(1): \\quad \\map P {n_0}$ is true :$(2): \\quad \\forall k \\in \\Z: k \\ge n_0 : \\map P k \\implies \\map P {k + 1}$ Then: :$\\map P n$ is true for all $n \\in \\Z$ such that $n \\ge n_0$."}
+{"_id": "11016", "title": "Principle of Finite Induction", "text": "Let $S \\subseteq \\Z$ be a subset of the integers. Let $n_0 \\in \\Z$ be given. Suppose that: :$(1): \\quad n_0 \\in S$ :$(2): \\quad \\forall n \\ge n_0: n \\in S \\implies n + 1 \\in S$ Then: :$\\forall n \\ge n_0: n \\in S$ That is: :$S = \\set {n \\in \\Z: n \\ge n_0}$"}
+{"_id": "11017", "title": "Principle of Mathematical Induction/Peano Structure", "text": "Let $\\struct {P, s, 0}$ be a Peano structure. Let $\\map Q n$ be a propositional function depending on $n \\in P$. Suppose that: :$(1): \\quad \\map Q 0$ is true :$(2): \\quad \\forall n \\in P: \\map Q n \\implies \\map Q {\\map s n}$ Then: :$\\forall n \\in P: \\map Q n$"}
+{"_id": "11019", "title": "Principle of Mathematical Induction/Naturally Ordered Semigroup", "text": "Let $\\struct {S, \\circ, \\preceq}$ be a naturally ordered semigroup. Let $T \\subseteq S$ such that $0 \\in T$ and $n \\in T \\implies n \\circ 1 \\in T$. Then $T = S$."}
+{"_id": "11020", "title": "Consecutive Fibonacci Numbers are Coprime", "text": "Let $F_k$ be the $k$th Fibonacci number. Then: :$\\forall n \\ge 2: \\gcd \\set {F_n, F_{n + 1} } = 1$ where $\\gcd \\set {a, b}$ denotes the greatest common divisor of $a$ and $b$. That is, a Fibonacci number and the one next to it are coprime."}
+{"_id": "11021", "title": "Divisibility of Fibonacci Number", "text": ":$\\forall m, n \\in \\Z_{> 2} : m \\divides n \\iff F_m \\divides F_n$ where $\\divides$ denotes divisibility."}
+{"_id": "11022", "title": "Fibonacci Number in terms of Smaller Fibonacci Numbers", "text": ":$\\forall m, n \\in \\Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$"}
+{"_id": "11023", "title": "GCD of Fibonacci Numbers", "text": ":$\\forall m, n \\in \\Z_{> 2}: \\gcd \\set {F_m, F_n} = F_{\\gcd \\set {m, n} }$ where $\\gcd \\set {a, b}$ denotes the greatest common divisor of $a$ and $b$."}
+{"_id": "11024", "title": "Catalan's Identity", "text": ":${F_n}^2 - F_{n - r} F_{n + r} = \\left({-1}\\right)^{n - r} {F_r}^2$"}
+{"_id": "11025", "title": "Fibonacci Number with Negative Index", "text": ":$\\forall n \\in \\Z_{> 0} : F_{-n} = \\left({-1}\\right)^{n + 1} F_n$"}
+{"_id": "11026", "title": "Fibonacci Number in terms of Larger Fibonacci Numbers", "text": ":$\\forall m, n \\in \\Z_{>0} : F_{m - n} = \\left({-1}\\right)^{n + 1} F_{m - 1} F_n + \\left({-1}\\right)^n F_m F_{n - 1}$"}
+{"_id": "11028", "title": "Vajda's Identity", "text": "==== Formulation 1 ==== {{:Vajda's Identity/Formulation 1}} ==== Formulation 2 ==== {{:Vajda's Identity/Formulation 2}}"}
+{"_id": "11031", "title": "Determinant of Matrix Product/General Case", "text": "Let $\\mathbf A_1, \\mathbf A_2, \\cdots, \\mathbf A_n$ be square matrices of order $n$, where $n > 1$. Then: :$\\map \\det {\\mathbf A_1 \\mathbf A_2 \\cdots \\mathbf A_n} = \\map \\det {\\mathbf A_1} \\map \\det {\\mathbf A_2} \\cdots \\map \\det {\\mathbf A_n}$"}
+{"_id": "11032", "title": "Opposite Sides Equal implies Parallelogram", "text": "Let $ABCD$ be a convex quadrilateral with $AB = CD$ and $BC = AD$. Then $ABCD$ is a parallelogram."}
+{"_id": "11033", "title": "Properties of Fibonacci Numbers", "text": "Let $F_n$ denote the $n$th Fibonacci number: {{:Definition:Fibonacci Number}}"}
+{"_id": "11034", "title": "Principle of Mathematical Induction/Naturally Ordered Semigroup/General Result", "text": "Let $\\struct {S, \\circ, \\preceq}$ be a naturally ordered semigroup. Let $p \\in S$. Let $T \\subseteq S$ such that: :$x \\in T \\implies p \\preceq x \\land \\paren {x \\in T \\implies x \\circ 1 \\in T}$ Then: :$S \\setminus S_p \\subseteq T$ where: :$\\setminus$ denotes set difference :$S_p$ denotes the set of all elements of $S$ preceding $p$."}
+{"_id": "11035", "title": "Principle of Mathematical Induction for Minimal Infinite Successor Set", "text": "Let $\\omega$ be the minimal infinite successor set. Let $S \\subseteq \\omega$. Suppose that: :$(1): \\quad \\varnothing \\in S$ :$(2): \\quad \\forall x: x \\in S \\implies x^+ \\in S$ where $x^+$ is the successor set of $x$. Then: :$S = \\omega$ </onlyinclude>"}
+{"_id": "11036", "title": "Principle of Mathematical Induction for Natural Numbers in Real Numbers", "text": "Let $\\struct {\\R, +, \\times, \\le}$ be the field of real numbers. Let $\\N$ be the natural numbers in $\\R$. Suppose that $A \\subseteq \\N$ is an inductive set. Then $A = \\N$."}
+{"_id": "11037", "title": "Chinese Remainder Theorem (Commutative Algebra)", "text": "Let $A$ be a commutative ring. {{explain|Presumably a Definition:Commutative and Unitary Ring is actually required here?}} Let $I_1, \\ldots, I_k$ for some $k \\ge 1$ be pairwise coprime ideals in $A$, that is: : $\\forall i \\ne j: I_i + I_j = A$ Then there is an isomorphism of rings: :$A / \\left({I_1 \\cap \\ldots \\cap I_k}\\right) \\to A / I_1 \\times \\cdots \\times A / I_k$ which is induced by the ring homomorphism $\\phi: A \\to A / I_1 \\times \\cdots \\times A / I_k$ defined as: :$\\phi \\left({x}\\right) = \\left({x + I_1, \\ldots, x + I_k}\\right)$ which passes through the quotient. {{explain|what is meant by \"passes through the quotient\"?}}"}
+{"_id": "11039", "title": "Sine of i", "text": ":$\\sin i = \\paren {\\dfrac e 2 - \\dfrac 1 {2 e} } i$"}
+{"_id": "11040", "title": "Cosine of i", "text": ":$\\cos i = \\dfrac e 2 + \\dfrac 1 {2 e}$"}
+{"_id": "11041", "title": "Tangent of i", "text": ":$\\tan i = \\left({\\dfrac {e^2 - 1} {e^2 + 1} }\\right) i$"}
+{"_id": "11042", "title": "Cosecant of i", "text": ":$\\csc i = \\left({\\dfrac {2 e} {1 - e^2} }\\right) i$"}
+{"_id": "11043", "title": "Secant of i", "text": ":$\\sec i = \\dfrac {2 e} {e^2 + 1}$"}
+{"_id": "11044", "title": "Cotangent of i", "text": ":$\\cot i = \\left({\\dfrac {1 + e^2} {1 - e^2} }\\right) i$"}
+{"_id": "11058", "title": "Sine in terms of Cosine", "text": "{{begin-eqn}} {{eqn | l = \\sin x       | r = +\\sqrt {1 - \\cos ^2 x}       | c = if there exists an integer $n$ such that $2 n \\pi < x < \\paren {2 n + 1} \\pi$ }} {{eqn | l = \\sin x       | r = -\\sqrt {1 - \\cos ^2 x}       | c = if there exists an integer $n$ such that $\\paren {2 n + 1} \\pi < x < \\paren {2 n + 2} \\pi$ }} {{end-eqn}}"}
+{"_id": "11061", "title": "Sine is Reciprocal of Cosecant", "text": ":$\\sin \\theta = \\dfrac 1 {\\csc \\theta}$"}
+{"_id": "11062", "title": "Tangent is Reciprocal of Cotangent", "text": ":$\\tan \\theta = \\dfrac 1 {\\cot \\theta}$"}
+{"_id": "11064", "title": "Sine of Complex Number", "text": "Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\\sin \\paren {a + b i} = \\sin a \\cosh b + i \\cos a \\sinh b$ where: :$\\sin$ denotes the sine function (real and complex) :$\\cos$ denotes the real cosine function :$\\sinh$ denotes the hyperbolic sine function :$\\cosh$ denotes the hyperbolic cosine function."}
+{"_id": "11065", "title": "Cosine of Complex Number", "text": "Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\\cos \\left({a + b i}\\right) = \\cos a \\cosh b - i \\sin a \\sinh b$ where: :$\\cos$ denotes the cosine function (real and complex) :$\\sin$ denotes the real sine function :$\\sinh$ denotes the hyperbolic sine function :$\\cosh$ denotes the hyperbolic cosine function"}
+{"_id": "11066", "title": "Tangent of Complex Number", "text": "Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then:"}
+{"_id": "11068", "title": "Sine in terms of Secant", "text": "{{begin-eqn}} {{eqn | l = \\sin x       | r = + \\frac {\\sqrt{\\sec ^2 x - 1} } {\\sec x}       | c = if there exists an integer $n$ such that $n \\pi < x < \\paren {n + \\dfrac 1 2} \\pi$ }} {{eqn | l = \\sin x       | r = - \\frac {\\sqrt{\\sec ^2 x - 1} } {\\sec x}       | c = if there exists an integer $n$ such that $\\paren {n + \\dfrac 1 2} \\pi < x < \\paren {n + 1} \\pi$ }} {{end-eqn}}"}
+{"_id": "11069", "title": "Sign of Sine", "text": "Let $x$ be a real number. {{begin-eqn}} {{eqn | l = \\sin x       | o = >       | r = 0       | c = if there exists an integer $n$ such that $2 n \\pi < x < \\paren {2 n + 1} \\pi$ }} {{eqn | l = \\sin x       | o = <       | r = 0       | c = if there exists an integer $n$ such that $\\paren {2 n + 1} \\pi < x < \\paren {2 n + 2} \\pi$ }} {{end-eqn}} where $\\sin$ is the real sine function."}
+{"_id": "11070", "title": "Second Principle of Mathematical Induction", "text": "Let $\\map P n$ be a propositional function depending on $n \\in \\Z$. Let $n_0 \\in \\Z$ be given. Suppose that: :$(1): \\quad \\map P {n_0}$ is true :$(2): \\quad \\forall k \\in \\Z: k \\ge n_0: \\map P {n_0} \\land \\map P {n_0 + 1} \\land \\ldots \\land \\map P {k - 1} \\land \\map P k \\implies \\map P {k + 1}$ Then: :$\\map P n$ is true for all $n \\ge n_0$. This process is called '''proof by (mathematical) induction'''."}
+{"_id": "11071", "title": "Second Principle of Finite Induction", "text": "Let $S \\subseteq \\Z$ be a subset of the integers. Let $n_0 \\in \\Z$ be given. Suppose that: :$(1): \\quad n_0 \\in S$ :$(2): \\quad \\forall n \\ge n_0: \\paren {\\forall k: n_0 \\le k \\le n \\implies k \\in S} \\implies n + 1 \\in S$ Then: :$\\forall n \\ge n_0: n \\in S$"}
+{"_id": "11072", "title": "Sign of Cosine", "text": "Let $x$ be a real number. Then: {{begin-eqn}} {{eqn | l = \\cos x       | o = >       | r = 0       | c = if there exists an integer $n$ such that $\\paren {2 n - \\dfrac 1 2} \\pi < x < \\paren {2 n + \\dfrac 1 2} \\pi$ }} {{eqn | l = \\cos x       | o = <       | r = 0       | c = if there exists an integer $n$ such that $\\paren {2 n + \\dfrac 1 2} \\pi < x < \\paren {2 n + \\dfrac 3 2} \\pi$ }} {{end-eqn}} where $\\cos$ is the real cosine function."}
+{"_id": "11073", "title": "Sign of Tangent", "text": "Let $x$ be a real number. Then: {{begin-eqn}} {{eqn | l = \\tan x       | o = >       | r = 0       | c = if there exists an integer $n$ such that $n \\pi < x < \\paren {n + \\dfrac 1 2} \\pi$ }} {{eqn | l = \\tan x       | o = <       | r = 0       | c = if there exists an integer $n$ such that $\\paren {n + \\dfrac 1 2} \\pi < x < \\paren {n + 1} \\pi$ }} {{end-eqn}} where $\\tan$ denotes the tangent function."}
+{"_id": "11076", "title": "Reciprocal of Strictly Negative Real Number is Strictly Negative", "text": ":$\\forall x \\in \\R: x < 0 \\implies \\dfrac 1 x < 0$"}
+{"_id": "11078", "title": "Sign of Secant", "text": "Let $x$ be a real number. {{begin-eqn}} {{eqn | l = \\sec x       | o = >       | r = 0       | c = if there exists an integer $n$ such that $\\paren {2 n - \\dfrac 1 2} \\pi < x < \\paren {2 n + \\dfrac 1 2} \\pi$ }} {{eqn | l = \\sec x       | o = <       | r = 0       | c = if there exists an integer $n$ such that $\\paren {2 n + \\dfrac 1 2} \\pi < x < \\paren {2 n + \\dfrac 3 2} \\pi$ }} {{end-eqn}} where $\\sec$ is the real secant function."}
+{"_id": "11082", "title": "Cosine in terms of Tangent", "text": "{{begin-eqn}} {{eqn | l = \\cos x       | r = +\\frac 1 {\\sqrt {1 + \\tan^2 x} }       | c = if there exists an integer $n$ such that $\\paren {2 n - \\dfrac 1 2} \\pi < x < \\paren {2 n + \\dfrac 1 2} \\pi$ }} {{eqn | l = \\cos x       | r = -\\frac 1 {\\sqrt {1 + \\tan^2 x} }       | c = if there exists an integer $n$ such that $\\paren {2 n + \\dfrac 1 2} \\pi < x < \\paren {2 n + \\dfrac 3 2} \\pi$ }} {{end-eqn}}"}
+{"_id": "11083", "title": "Tangent in terms of Secant", "text": "{{begin-eqn}} {{eqn | l = \\tan x       | r = +\\sqrt {\\sec^2 x - 1}       | c = if there exists an integer $n$ such that $n \\pi < x < \\paren {n + \\dfrac 1 2} \\pi$ }} {{eqn | l = \\tan x       | r = -\\sqrt {\\sec^2 x - 1}       | c = if there exists an integer $n$ such that $\\paren {n + \\dfrac 1 2} \\pi < x < \\paren {n + 1} \\pi$ }} {{end-eqn}}"}
+{"_id": "11084", "title": "Union of Left-Total Relations is Left-Total", "text": "Let $S_1, S_2, T_1, T_2$ be sets or classes. Let $\\mathcal R_1 \\subseteq S_1 \\times T_1$ and $\\mathcal R_2 \\subseteq S_2 \\times T_2$ be left-total relations. Then $\\mathcal R_1 \\cup \\mathcal R_2$ is left-total."}
+{"_id": "11085", "title": "Union of Inverse is Inverse of Union", "text": "Let for $i \\in \\left\\{1,2\\right\\}$ $\\mathcal R_i \\subseteq S_i \\times T_i$ be a relation on $S_i \\times T_i$. Let $\\mathcal R_i^{-1} \\subseteq T_i \\times S_i$ be the inverse of $\\mathcal R_i$. Then $\\mathcal R_1^{-1} \\cup \\mathcal R_2^{-1} = \\left(\\mathcal R_1 \\cup \\mathcal R_2\\right)^{-1}$"}
+{"_id": "11086", "title": "Condition for Darboux Integrability", "text": "Let $\\closedint a b$ be a closed real interval. Let $f$ be a bounded real function defined on $\\closedint a b$. Then $f$ is Darboux integrable {{iff}}: :for every $\\epsilon \\in \\R_{>0}$, there exists a finite subdivision $S$ of $\\closedint a b$ such that $\\map U S – \\map L S < \\epsilon$ where :$\\map U S$ is the upper sum of $f$ on $\\closedint a b$ with respect to $S$ :$\\map L S$ is the lower sum of $f$ on $\\closedint a b$ with respect to $S$"}
+{"_id": "11087", "title": "Open Set minus Closed Set is Open", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. For $A \\subseteq S$ denote by $\\complement_S \\left({A}\\right)$ the relative complement of $A$ in $S$. Let $U \\in \\tau$ and $\\complement_S \\left({V}\\right) \\in \\tau$. Then: :$U \\setminus V \\in \\tau$ and: :$\\complement_S \\left({V \\setminus U}\\right) \\in \\tau$"}
+{"_id": "11088", "title": "Union of Right-Total Relations is Right-Total", "text": "Let $S_1, S_2, T_1, T_2$ be sets or classes. Let $\\mathcal R_1 \\subseteq S_1 \\times T_1$ and $\\mathcal R_2 \\subseteq S_2 \\times T_2$ be right-total relations. Then $\\mathcal R_1 \\cup \\mathcal R_2$ is right-total."}
+{"_id": "11089", "title": "Primitive of x over a x + b squared by p x + q/Corollary", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {\\paren {a x + b}^2 \\paren {p x + q} } = \\frac 1 {b p - a q} \\paren {\\frac q {b p - a q} \\ln \\size {\\frac {a x + b} {p x + q} } + \\frac x {a x + b} } + C$"}
+{"_id": "11090", "title": "Arctangent of Imaginary Number", "text": "Let $x$ belong to the open real interval $\\openint {-1} 1$. Then: :$\\map {\\tan^{-1} } {i x} = \\dfrac i 2 \\map \\ln {\\dfrac {1 + x} {1 - x} }$ where $\\tan$ is the complex tangent function, $\\ln$ is the real natural logarithm, and $i$ is the imaginary unit."}
+{"_id": "11091", "title": "Equivalence of Definitions of Real Inverse Hyperbolic Tangent", "text": "Let $S$ denote the open real interval: : $S := \\left({-1 \\,.\\,.\\, 1}\\right)$ {{TFAE|def = Real Inverse Hyperbolic Tangent}}"}
+{"_id": "11092", "title": "Square Root is Strictly Increasing", "text": "The positive square root function is strictly increasing, that is: :$ \\forall x,y \\in \\R_{>0}: x < y \\implies \\sqrt x < \\sqrt y$"}
+{"_id": "11094", "title": "Minimum of Real Hyperbolic Cosine Function", "text": "Let $x$ be a real number. Then: :$\\cosh x \\ge 1$ where $\\cosh$ denotes the hyperbolic cosine function."}
+{"_id": "11096", "title": "Exponential of Real Number is Strictly Positive", "text": "Let $x$ be a real number. Let $\\exp$ denote the (real) exponential function. Then: :$\\forall x \\in \\R : \\exp x > 0$"}
+{"_id": "11098", "title": "Derivative of Power of Function", "text": "Let $\\map u x$ be a differentiable real function of $x$. Let $n$ be a real number such that $n \\ne -1$. Then: :$\\map {\\dfrac \\d {\\d x} } {\\map u x^n} = n \\map u x^{n - 1} \\map {\\dfrac \\d {\\d x} } {\\map u x}$"}
+{"_id": "11099", "title": "Natural Logarithm of e is 1", "text": ":$\\ln e = 1$"}
+{"_id": "11103", "title": "Real Inverse Hyperbolic Cosine is Strictly Increasing", "text": "The real inverse hyperbolic cosine function is strictly increasing, that is: :$\\forall x, y \\ge 1 : x < y \\implies \\cosh^{-1} x < \\cosh^{-1} y$"}
+{"_id": "11104", "title": "Laplace Transform of Constant Multiple", "text": "Let $a \\in \\C$ or $\\R$ be constant. Then: :$a \\laptrans {\\map f {a t} } = \\map F {\\dfrac s a}$"}
+{"_id": "11105", "title": "Laplace Transform of Function of t minus a", "text": "Let $g$ be the function defined as: :$\\map g t = \\begin{cases} \\map f {t - a} & : t > a \\\\ 0 & : t \\le a \\end{cases}$ Then: :$\\laptrans {\\map g t} = e^{-a s} \\map F s$"}
+{"_id": "11106", "title": "Seifert-van Kampen Theorem", "text": "The functor $\\pi_1 : \\mathbf{Top_\\bullet} \\to \\mathbf{Grp}$ preserves pushouts of inclusions."}
+{"_id": "11107", "title": "Functions of Independent Random Variables are Independent", "text": "Let $X$ and $Y$ be independent random variables on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $g$ and $h$ be real-valued functions defined on the codomains of $X$ and $Y$ respectively. Then $\\map g X$ and $\\map h Y$ are independent random variables."}
+{"_id": "11108", "title": "Multiplication Property of Characteristic Functions", "text": "Let $X$ and $Y$ be independent random variables on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\phi_X$ and $\\phi_Y$ denote the characteristic functions of $X$ and $Y$ respectively. Then: :$\\phi_{X + Y} = \\phi_X \\phi_Y$"}
+{"_id": "11109", "title": "Relationship between Limit Inferior and Lower Limit", "text": "Let $\\struct {S, \\tau}$ be a topological space. Let $f: S \\to \\R \\cup \\set {-\\infty, \\infty}$ be an extended real-valued function. Let $\\sequence {s_n}_{n \\mathop \\in \\N}$ be a convergent sequence in $S$ such that $s_n \\to \\bar s$. Then the lower limit of $f$ at $\\bar s$ is bounded above by the limit inferior of $\\sequence {\\map f {s_n} }$: :$\\displaystyle \\liminf_{s \\mathop \\to \\bar s} \\map f s \\le \\liminf_{n \\mathop \\to \\infty} \\map f {s_n}$"}
+{"_id": "11111", "title": "Sequence on Finite Product Space Converges to Point iff Projections Converge to Projections of Point", "text": "Let $N \\in \\N$. For all $k \\in \\set {1, \\ldots, N}$, let $T_k = \\struct {X_k, \\tau_k}$ be topological spaces. Let $\\displaystyle X = \\prod_{k \\mathop = 1}^N X_k$ be the cartesian product of $X_1, \\ldots, X_N$. Let $\\tau$ be the product topology on $X$. Denote by $\\pr_k : X \\to X_k$ the projection from $X$ onto $X_k$. Let $\\sequence {x_n}$ be a sequence on $X$ and let $x \\in X$. Then $\\sequence {x_n}$ converges to $x$ {{iff}}: :for all $k \\in \\set {1, \\ldots, N}$ the sequence $\\sequence {\\map {\\pr_k} {x_n} }$ converges to $\\map {\\pr_k} x$."}
+{"_id": "11112", "title": "Real Convergent Sequence is Cauchy Sequence", "text": "Every convergent real sequence in $\\R$ is a Cauchy sequence."}
+{"_id": "11113", "title": "Real Sequence is Cauchy iff Convergent", "text": "Let $\\sequence {a_n}$ be a sequence in $\\R$. Then $\\sequence {a_n}$ is a Cauchy sequence {{iff}} it is convergent."}
+{"_id": "11114", "title": "Complex Sequence is Cauchy iff Convergent", "text": "Let $\\sequence {z_n}$ be a complex sequence. Then $\\sequence {z_n}$ is a Cauchy sequence {{iff}} it is convergent."}
+{"_id": "11117", "title": "Sum of Arctangents", "text": ":$\\arctan a + \\arctan b = \\arctan \\dfrac {a + b} {1 - a b}$ where $\\arctan$ denotes the arctangent."}
+{"_id": "11118", "title": "Difference of Arctangents", "text": ":$\\arctan a - \\arctan b = \\arctan \\dfrac {a - b} {1 + a b}$ where $\\arctan$ denotes the arctangent."}
+{"_id": "11119", "title": "Sum of Arccotangents", "text": ":$\\arccot a + \\arccot b = \\arccot \\dfrac {a b - 1} {a + b}$ where $\\arccot$ denotes the arccotangent."}
+{"_id": "11120", "title": "Difference of Arccotangents", "text": ":$\\arccot a - \\arccot b = \\arccot \\dfrac {a b + 1} {a - b}$ where $\\arccot$ denotes the arccotangent."}
+{"_id": "11121", "title": "Multiple Angle Formula for Tangent", "text": ":$\\displaystyle \\tan \\left({n \\theta}\\right) = \\frac {\\displaystyle \\sum_{i \\mathop = 0}^{\\left\\lfloor{\\frac {n - 1} 2}\\right\\rfloor} \\left({-1}\\right)^i \\binom n {2 i + 1} \\tan^{2 i + 1}\\theta} {\\displaystyle \\sum_{i \\mathop = 0}^{\\left\\lfloor{\\frac n 2}\\right\\rfloor} \\left({-1}\\right)^i \\binom n {2 i} \\tan^{2 i}\\theta}$"}
+{"_id": "11129", "title": "Modulus of Complex Number equals its Distance from Origin", "text": "The modulus of a complex number equals its distance from the origin on the complex plane."}
+{"_id": "11130", "title": "Empty Set is Countable", "text": "The empty set $\\O$ is countable."}
+{"_id": "11139", "title": "Area of Isosceles Triangle in terms of Sides", "text": "Let $\\triangle ABC$ be an isosceles triangle whose apex is $A$. Let $r$ be the length of a leg of $\\triangle ABC$. Let $b$ be the length of the base of $\\triangle ABC$. Then the area $\\mathcal A$ of $\\triangle ABC$ is given by: :$\\mathcal A = \\dfrac b 4 \\sqrt{4 r^2 - b^2}$"}
+{"_id": "11141", "title": "Even Function Times Even Function is Even", "text": "Let $X \\subset \\R$ be a symmetric set of real numbers: :$\\forall x \\in X: -x \\in X$ Let $f, g: X \\to \\R$ be two even functions. Let $f \\cdot g$ denote the pointwise product of $f$ and $g$. Then $\\paren {f \\cdot g}: X \\to \\R$ is also an even function."}
+{"_id": "11142", "title": "Odd Function Times Even Function is Odd", "text": "Let $X \\subset \\R$ be a symmetric set of real numbers: :$\\forall x \\in X: -x \\in X$ Let $f: X \\to \\R$ be an odd function. Let $g: X \\to \\R$ be an even function. Let $f \\cdot g$ denote the pointwise product of $f$ and $g$. Then $\\paren {f \\cdot g}: X \\to \\R$ is an odd function."}
+{"_id": "11143", "title": "Odd Function Times Odd Function is Even", "text": "Let $S \\subset \\R$ be a symmetric set of real numbers: :$\\forall x \\in S: -x \\in X$ Let $f, g: X \\to \\R$ be two odd functions. Let $f \\cdot g$ denote the pointwise product of $f$ and $g$. Then $\\paren {f \\cdot g}: S \\to \\R$ is an even function."}
+{"_id": "11144", "title": "Mellin Transform of Exponential", "text": "Let $a$ be a complex constant and $e^t$ be the complex exponential. Let $\\MM$ be the Mellin transform. Then: :$\\map {\\MM \\set {e^{-a t} } } s = a^{-s} \\, \\map \\Gamma s$ where $\\map \\Re a, \\map \\Re s > 0$"}
+{"_id": "11145", "title": "Mellin Transform of Dirac Delta Function", "text": "Let $\\delta \\left({t}\\right)$ be the Dirac delta function. Let $c$ be a positive real number. Let $\\mathcal M$ be the Mellin transform. Then: :$\\mathcal M \\left\\{ { \\delta_c \\left({t}\\right)} \\right\\} \\left({s}\\right) = c^{s-1}$"}
+{"_id": "11148", "title": "Mellin Transform of Power Times Function", "text": "Let $t^n: \\R \\to \\R$ be $t$ to the $n$th power for some $n \\in \\N_{\\ge 0}$. Let $\\mathcal M$ be the Mellin transform. Then: :$\\mathcal M \\left\\{ {t^n f \\left({t}\\right)} \\right\\} \\left({s}\\right) = \\mathcal M \\left\\{ {f \\left({t}\\right)}\\right\\} \\left({s + n}\\right)$ given that both transforms exist."}
+{"_id": "11149", "title": "Mellin Transform of Dirac Delta Function by Function", "text": "Let $f: \\R \\to \\R$ be a function. Let $c \\in \\R_{>0}$ be a positive constant real number. Let $\\delta_c \\left({t}\\right)$ be the Dirac delta function. Let $\\mathcal M$ be the Mellin transform. Then: :$\\mathcal M \\left\\{ {\\delta_c \\left({t}\\right) f \\left({t}\\right)} \\right\\} \\left({s}\\right) = c^{s - 1} f \\left({c}\\right)$"}
+{"_id": "11152", "title": "Unity of Ring is Idempotent", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring with unity whose unity is $1_R$. Then $1_R$ is an idempotent element of $R$ under the ring product $\\circ$: :$1_R \\circ 1_R = 1_R$"}
+{"_id": "11153", "title": "Mellin Transform of Higher Order Exponential", "text": "Let $a$ be a complex constant. Let $n$ be a natural number. Let $e^t$ be the complex exponential of $t$. Let $\\MM$ be the Mellin transform. Then: :$\\map {\\MM \\set {e^{-a t^n} } } s = \\dfrac {a^{-s/n} } n \\map \\Gamma {\\dfrac s n}$ where $\\map \\Gamma z$ is the Gamma function and $\\map \\Re a$, $\\map \\Re s > 0$."}
+{"_id": "11163", "title": "Complex Numbers as Quotient Ring of Real Polynomial", "text": "Let $\\C$ be the set of complex numbers. Let $P \\left[{x}\\right]$ be the set of polynomials over real numbers, where the coefficients of the polynomials are real. Let $\\left\\langle{x^2 + 1}\\right\\rangle = \\left\\{ {Q \\left({x}\\right) \\left({x^2 + 1}\\right): Q \\left({x}\\right) \\in P \\left[{x}\\right]}\\right\\}$ be the ideal generated by $x^2 + 1$ in $P \\left[{x}\\right]$. Let $D = P \\left[{x}\\right] / \\left\\langle{x^2 + 1}\\right\\rangle$ be the quotient of $P \\left[{x}\\right]$ modulo $\\left\\langle{x^2 + 1}\\right\\rangle$. Then: :$\\left({\\C, +, \\times}\\right) \\cong \\left({D, +, \\times}\\right)$"}
+{"_id": "11164", "title": "Quaternion Modulus in Terms of Conjugate", "text": "Let $\\mathbf x = a \\mathbf 1 + b \\mathbf i + c \\mathbf j + d \\mathbf k$ be a quaternion. Let $\\left\\vert{\\mathbf x}\\right\\vert$ be the modulus of $\\mathbf x$. Let $\\overline{\\mathbf x}$ be the conjugate of $\\mathbf x$. Then: : $\\left\\vert{\\mathbf x}\\right\\vert^2 \\mathbf 1 = \\mathbf x \\overline{\\mathbf x}$"}
+{"_id": "11167", "title": "Sufficient Condition for Quaternion Multiplication to Commute", "text": "In general, quaternion multiplication does not commute. But, for $\\mathbf x,\\mathbf y \\in \\H$, $\\mathbf x \\times \\mathbf y = \\mathbf y \\times \\mathbf x$ if any one of the following conditions hold: {{begin-eqn}} {{eqn | n = 1a       | l = \\mathbf x, \\mathbf y       | o = \\in       | r = \\set {a \\mathbf 1 + b \\mathbf i + 0 \\mathbf j + 0 \\mathbf k: a, b \\in \\R} }} {{eqn | n = 1b       | l = \\mathbf x, \\mathbf y       | o = \\in       | r = \\set {a \\mathbf 1 + 0 \\mathbf i + c \\mathbf j + 0 \\mathbf k: a, c \\in \\R} }} {{eqn | n = 1c       | l = \\mathbf x, \\mathbf y       | o = \\in       | r = \\set {a \\mathbf 1 + 0 \\mathbf i + 0 \\mathbf j + d \\mathbf k: a, d \\in \\R} }} {{eqn | n = 2a       | l = \\mathbf x       | o = \\in       | r = \\set {a \\mathbf 1 + 0 \\mathbf i + 0 \\mathbf j + 0 \\mathbf k: a \\in \\R} }} {{eqn | n = 2b       | l = \\mathbf y       | o = \\in       | r = \\set {a \\mathbf 1 + 0 \\mathbf i + 0 \\mathbf j + 0 \\mathbf k: a \\in \\R} }} {{eqn | n = 3       | l = \\mathbf x       | r = \\paren {a \\mathbf 1 + 0 \\mathbf i + 0 \\mathbf j + 0 \\mathbf k} \\times \\overline {\\mathbf y}: a \\in \\R }} {{end-eqn}}"}
+{"_id": "11168", "title": "Complex Conjugation is Involution", "text": "Let $z = x + i y$ be a complex number. Let $\\overline z$ denote the complex conjugate of $z$. Then the operation of complex conjugation is an involution: :$\\overline {\\paren {\\overline z} } = z$"}
+{"_id": "11175", "title": "Quaternion Modulus of Conjugate", "text": "Let $z = a \\mathbf 1 + b \\mathbf i + c \\mathbf j + d \\mathbf k$ be a quaternion. Let $\\overline z$ be the conjugate of $z$. Let $\\cmod z$ be the quaternion modulus of $z$. Then: :$\\cmod {\\overline z} = \\cmod z$"}
+{"_id": "11176", "title": "Quaternion Conjugation is Involution", "text": "Let $\\mathbf x = a \\mathbf 1 + b \\mathbf i + c \\mathbf j + d \\mathbf k$ be a quaternion. Let $\\overline {\\mathbf x}$ denote the quaternion conjugate of $\\mathbf x$. Then the operation of quaternion conjugation is an involution: :$\\overline {\\paren {\\overline {\\mathbf x} } } = \\mathbf x$"}
+{"_id": "11179", "title": "Sum of Quaternion Conjugates", "text": "Let $\\mathbf x, \\mathbf y \\in \\mathbb H$ be quaternions. Let $\\overline {\\mathbf x}$ be the conjugate of $\\mathbf x$. Then: :$\\overline {\\mathbf x + \\mathbf y} = \\overline {\\mathbf x} + \\overline {\\mathbf y}$"}
+{"_id": "11182", "title": "Principle of Recursive Definition for Peano Structure", "text": "Let $\\struct {P, 0, s}$ be a Peano structure. Let $T$ be a set. Let $a \\in T$. Let $g: T \\to T$ be a mapping. Then there exists exactly one mapping $f: P \\to T$ such that: :$\\forall x \\in P: \\map f x = \\begin{cases} a & : x = 0 \\\\ \\map g {\\map f n} & : x = \\map s n \\end{cases}$"}
+{"_id": "11183", "title": "Principle of Recursive Definition for Minimal Infinite Successor Set", "text": "Let $\\omega$ be the minimal infinite successor set. Let $T$ be a set. Let $a \\in T$. Let $g: T \\to T$ be a mapping. Then there exists exactly one mapping $f: \\omega \\to T$ such that: :$\\forall x \\in \\omega: \\map f x = \\begin {cases} a & : x = \\O \\\\ \\map g {\\map f n} & : x = n^+ \\end {cases}$ where $n^+$ is the successor set of $n$."}
+{"_id": "11184", "title": "Ordering on 1-Based Natural Numbers is Trichotomy", "text": "Let $\\N_{> 0}$ be the $1$-based natural numbers. Let $<$ be the strict ordering on $\\N_{>0}$. Then exactly one of the following is true: :$(1): \\quad a = b$ :$(2): \\quad a > b$ :$(3): \\quad a < b$ That is, $<$ is a trichotomy on $\\N_{> 0}$."}
+{"_id": "11185", "title": "Product of Quaternion Conjugates", "text": "Let $\\mathbf x, \\mathbf y \\in \\mathbb H$ be quaternions. Let $\\overline{\\mathbf x}$ be the conjugate of $\\mathbf x$. Then: :$\\overline{\\mathbf x \\times \\mathbf y} = \\overline{\\mathbf y} \\times \\overline{\\mathbf x}$ but in general: :$\\overline{\\mathbf x \\times \\mathbf y} \\ne \\overline{\\mathbf x} \\times \\overline{\\mathbf y}$"}
+{"_id": "11186", "title": "Quaternion Modulus of Product of Quaternions", "text": "Let $\\mathbf x, \\mathbf y$ be quaternions. Let $\\left\\vert{\\mathbf x}\\right\\vert$ be the modulus of $\\mathbf x$. Then: :$\\left\\vert{\\mathbf x \\mathbf y}\\right\\vert = \\left\\vert{\\mathbf x}\\right\\vert \\left\\vert{\\mathbf y}\\right\\vert$"}
+{"_id": "11196", "title": "Octonion Conjugation is Involution", "text": "Let $x = \\tuple {a, b}: a, b \\in \\mathbb H$ be a octonion. Let $\\overline x$ be the conjugate of $x$. Then: :$\\overline \\cdot: x \\mapsto \\overline x$ is an involution. That is: :$\\overline {\\paren {\\overline x} } = x$"}
+{"_id": "11197", "title": "Ordering on 1-Based Natural Numbers is Compatible with Addition", "text": "Let $\\N_{> 0}$ be the $1$-based natural numbers. Let $+$ denote addition on $\\N_{>0}$. Let $<$ be the strict ordering on $\\N_{>0}$. Then: :$\\forall a, b, n \\in \\N_{>0}: a < b \\implies a + n < b + n$ That is, $>$ is compatible with $+$ on $\\N_{>0}$."}
+{"_id": "11198", "title": "Ordering on 1-Based Natural Numbers is Compatible with Multiplication", "text": "Let $\\N_{> 0}$ be the $1$-based natural numbers. Let $\\times$ denote multiplication on $\\N_{>0}$. Let $<$ be the strict ordering on $\\N_{>0}$. Then: :$\\forall a, b, n \\in \\N_{>0}: a < b \\implies a \\times n < b \\times n$ That is, $>$ is compatible with $\\times$ on $\\N_{>0}$."}
+{"_id": "11199", "title": "Integral between Limits is Independent of Direction", "text": "Let $f$ be a real function which is integrable on the interval $\\openint a b$. Then: :$\\ds \\int_a^b \\map f x \\rd x = \\int_a^b \\map f {a + b - x} \\rd x$"}
+{"_id": "11200", "title": "Countable Set equals Range of Sequence", "text": "Let $S$ be a set. Then $S$ is countable {{iff}} there exists a sequence $\\sequence {s_i}_{i \\mathop \\in N}$ where $N$ is a subset of $\\N$ such that $S$ equals the range of $\\sequence {s_i}_{i \\mathop \\in N}$."}
+{"_id": "11202", "title": "Initial Segment of Natural Numbers determined by Zero is Empty", "text": "Let $\\N_k$ denote the initial segment of the natural numbers determined by $k$: :$\\N_k = \\set {0, 1, 2, 3, \\ldots, k - 1}$ Then $\\N_0 = \\O$."}
+{"_id": "11203", "title": "Initial Segment of One-Based Natural Numbers determined by Zero is Empty", "text": "Let $\\N^*_k$ denote the initial segment of the one-based natural numbers determined by $k$: :$\\N^*_k = \\set {1, 2, 3, \\ldots, k - 1, k}$ Then $\\N^*_0 = \\O$."}
+{"_id": "11204", "title": "Heine-Borel Theorem/Real Line/Closed and Bounded Interval", "text": "Let $\\left[{a \\,.\\,.\\, b}\\right]$, $a < b$, be a closed and bounded real interval. Let $S$ be a set of open real sets. Let $S$ be a cover of $\\left[{a \\,.\\,.\\, b}\\right]$. Then there is a finite subset of $S$ that covers $\\left[{a \\,.\\,.\\, b}\\right]$."}
+{"_id": "11205", "title": "Addition on 1-Based Natural Numbers is Cancellable", "text": "Let $\\N_{> 0}$ be the $1$-based natural numbers. Let $+$ be addition on $\\N_{>0}$. Then: :$\\forall a, b, c \\in \\N_{>0}: a + c = b + c \\implies a = b$ :$\\forall a, b, c \\in \\N_{>0}: a + b = a + c \\implies b = c$ That is, $+$ is cancellable on $\\N_{>0}$."}
+{"_id": "11206", "title": "Addition on 1-Based Natural Numbers is Cancellable for Ordering", "text": "Let $\\N_{> 0}$ be the $1$-based natural numbers. Let $<$ be the strict ordering on $\\N_{>0}$. Let $+$ be addition on $\\N_{>0}$. Then: :$\\forall a, b, c \\in \\N_{>0}: a + c < b + c \\implies a < b$ :$\\forall a, b, c \\in \\N_{>0}: a + b < a + c \\implies b < c$ That is, $+$ is cancellable on $\\N_{>0}$ for $<$."}
+{"_id": "11207", "title": "Multiplication on 1-Based Natural Numbers is Cancellable", "text": "Let $\\N_{> 0}$ be the $1$-based natural numbers. Let $\\times$ be multiplication on $\\N_{>0}$. Then: :$\\forall a, b, c \\in \\N_{>0}: a \\times c = b \\times c \\implies a = b$ :$\\forall a, b, c \\in \\N_{>0}: a \\times b = a \\times c \\implies b = c$ That is, $\\times$ is cancellable on $\\N_{>0}$."}
+{"_id": "11208", "title": "Multiplication on 1-Based Natural Numbers is Cancellable for Ordering", "text": "Let $\\N_{> 0}$ be the $1$-based natural numbers. Let $\\times$ be multiplication on $\\N_{>0}$. Let $<$ be the strict ordering on $\\N_{>0}$. Then: :$\\forall a, b, c \\in \\N_{>0}: a \\times c < b \\times c \\implies a < b$ :$\\forall a, b, c \\in \\N_{>0}: a \\times b < a \\times c \\implies b < c$ That is, $\\times$ is cancellable on $\\N_{>0}$ for $<$."}
+{"_id": "11209", "title": "Index of Trivial Subgroup is Cardinality of Group", "text": "Let $G$ be a group whose identity element is $e$. Let $\\set e$ be the trivial subgroup of $G$. Then: :$\\index G {\\set e} = \\order G$ where: :$\\index G {\\set e}$ denotes the index of $\\set e$ in $G$ :$\\order G$ denotes the cardinality of $G$."}
+{"_id": "11210", "title": "Index is One iff Subgroup equals Group", "text": "Let $G$ be a group whose identity element is $e$. Let $H$ be a subgroup of $G$. Then: :$\\index G H = 1 \\iff G = H$ where $\\index G H$ denotes the index of $H$ in $G$."}
+{"_id": "11212", "title": "Set Equality is Equivalence Relation", "text": "Let $S$ be a set. Set equality is an equivalence relation on the power set $\\powerset S$ of $S$."}
+{"_id": "11220", "title": "Subset Relation is Antisymmetric", "text": "The relation \"is a subset of\" is antisymmetric: :$\\paren {R \\subseteq S} \\land \\paren {S \\subseteq R} \\iff R = S$"}
+{"_id": "11221", "title": "There Exists No Universal Set", "text": "There exists no set which is an absolutely universal set. That is: :$\\neg \\left({\\exists \\mathcal U: \\forall T: T \\in \\mathcal U}\\right)$ where $T$ is any arbitrary object at all. That is, a set that contains ''everything'' cannot exist."}
+{"_id": "11222", "title": "Union of Disjoint Singletons is Doubleton", "text": "Let $\\set a$ and $\\set b$ be singletons such that $a \\ne b$. Then: :$\\set a \\cup \\set b = \\set {a, b}$"}
+{"_id": "11223", "title": "Distributive Laws", "text": "== Intersection Distributes over Union == {{:Intersection Distributes over Union}}"}
+{"_id": "11224", "title": "Intersection of Empty Set/Paradoxical Implications", "text": "Although it appears counter-intuitive, the reasoning is sound. This result is therefore classed as a veridical paradox. {{AuthorRef|Paul R. Halmos}} declares, in Section $5$ of his {{BookLink|Naive Set Theory|Paul R. Halmos}} of $1960$ that: :''There is no profound problem here; it is merely a nuisance to be forced always to be making qualifications and exceptions just because some set somewhere along some construction might turn out to be empty. There is nothing to be done about this; it is just a fact of life.'' However, later in that same work (in Section $9$, in the context of indexed families of sets) he says that: :''... an empty intersection does not make sense.'' a sentiment which is repeated in the $2008$ collaboration with {{AuthorRef|Steven Givant}}, {{BookLink|Introduction to Boolean Algebras|Paul Halmos}}, p. $457$. In {{BookReference|Topology|2000|James R. Munkres|ed = 2nd|edpage = Second Edition}}, the author recognizes the result, but does not adopt it: :''If one has a given large set $X$ that is specified at the outset of the discussion to be one's \"universe of discourse,\" and one considers only subsets of $X$ throughout, it is reasonable to let $\\displaystyle \\bigcap_{A \\mathop \\in \\AA} A = X$ when $\\AA$ is empty. Not all mathematicians follow this convention, however. To avoid difficulty, we shall not define the intersection when $\\AA$ is empty.'' {{BookReference|Topology: An Introduction with Application to Topological Groups|1967|George McCarty}} accepts this result, but cautiously: :''A natural mnemonic for these extreme cases is that $\\bigcap \\SS$ \"grows larger\" as $\\SS$ \"grows smaller\", and $\\bigcup \\SS$ grows smaller as $\\SS$ grows smaller. No other convention is possible, but the case $\\SS = \\O$ will often be treated redundantly by itself in definitions and proofs, as a reminder of the null case.'' {{BookReference|Set Theory|1999|András Hajnal|author2 = Peter Hamburger}} dismiss it casually: :''As usual, we adopt the convention that in case $A = \\O$ the expression $\\bigcap A$ is defined only in case we work with the subsets of an underlying set $X$. In this case we put $\\bigcap A = X$.'' and {{BookReference|Programming, Games and Transportation Networks|1965|Claude Berge|author2 = A. Ghouila-Houri}} do not even acknowledge that there may be a problem in the first place: :''In the case where $I = \\O$, we have ::$\\displaystyle \\bigcap_{i \\mathop \\in I} A_i = X$;'' :''this moreover is the only case where $X$ plays a role; in fact, if $I$ is not empty, clearly we have: ::$\\displaystyle \\bigcap_{i \\mathop \\in I} A_i = \\set {x \\mid x \\in A_i \\text { for every } i \\in I}$."}
+{"_id": "11225", "title": "Ordered Pair/Kuratowski Formalization/Motivation", "text": "The only reason for the Kuratowski formalization of ordered pairs: :$\\tuple {a, b} = \\set {\\set a, \\set {a, b} }$ is so their existence can be justified in the strictures of the axiomatic set theory, in particular Zermelo-Fraenkel set theory (ZF). Once that has been demonstrated, there is no need to invoke it again. The fact that this formulation allows that: :$\\tuple {a, b} = \\tuple {c, d} \\iff a = c, b = d$ is its stated aim. The fact that $\\set {a, b} \\in \\tuple {a, b}$ is an unfortunate side-effect brought about by means of the definition. It would be possible to add another axiom to ZF or ZFC specifically to allow for ordered pairs to be defined, and in some systems of axiomatic set theory this is what is done."}
+{"_id": "11226", "title": "Heine-Borel Theorem/Real Line/Closed and Bounded Set", "text": "Let $F$ be a closed and bounded real set. Let $C$ be a set of open real sets. Let $C$ be a cover of $F$. Then there is a finite subset of $C$ that covers $F$."}
+{"_id": "11227", "title": "Cardinality of Set of All Mappings from Empty Set", "text": "Let $T$ be a set. Let $T^\\O$ denote the set of all mappings from $\\O$ to $S$. Then: :$\\card {T^\\O} = 1$ where $\\card {T^\\O}$ denotes the cardinality of $\\O^S$."}
+{"_id": "11228", "title": "Cardinality of Set of All Mappings to Empty Set", "text": "Let $S$ be a set. Let $\\O^S$ be the set of all mappings from $S$ to $\\O$. Then: :$\\card {\\O^S} = \\begin{cases} 1 & : S = \\O \\\\  0 & : S \\ne \\O \\end{cases}$ where $\\card {\\O^S}$ denotes the cardinality of $\\O^S$."}
+{"_id": "11229", "title": "Intersection Distributes over Union/Family of Sets/Corollary", "text": "Let $I$ and $J$ be indexing sets. Let $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$ and $\\family {B_\\beta}_{\\beta \\mathop \\in J}$ be indexed families of subsets of a set $S$. Then: :$\\displaystyle \\bigcup_{\\tuple {\\alpha, \\beta} \\mathop \\in I \\times J} \\paren {A_\\alpha \\cap B_\\beta} = \\paren {\\bigcup_{\\alpha \\mathop \\in I} A_\\alpha} \\cap \\paren {\\bigcup_{\\beta \\mathop \\in J} B_\\beta}$ where $\\displaystyle \\bigcup_{\\alpha \\mathop \\in I} A_\\alpha$ denotes the union of $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$."}
+{"_id": "11231", "title": "Cartesian Product of Unions/General Result", "text": "Let $I$ and $J$ be indexing sets. Let $\\family {A_i}_{i \\mathop \\in I}$ and $\\family {B_j}_{j \\mathop \\in J}$ be families of sets indexed by $I$ and $J$ respectively. Then: :$\\displaystyle \\paren {\\bigcup_{i \\mathop \\in I} A_i} \\times \\paren {\\bigcup_{j \\mathop \\in J} B_j} = \\bigcup_{\\tuple {i, j} \\mathop \\in I \\times J} \\paren {A_i \\times B_j}$ where: :$\\displaystyle \\bigcup_{i \\mathop \\in I} A_i$ denotes the union of $\\family {A_i}_{i \\mathop \\in I}$ and so on :$\\times$ denotes Cartesian product."}
+{"_id": "11232", "title": "Preimages All Exist iff Surjection/Corollary", "text": ":$\\forall B \\subseteq T, B \\ne \\O: f^{-1} \\sqbrk B \\ne \\O$ {{iff}}: :$f$ is a surjection where $f^{-1} \\sqbrk B$ denotes the preimage of $B \\subseteq T$."}
+{"_id": "11238", "title": "Image of Preimage under Relation is Subset", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation. Then: :$B \\subseteq T \\implies \\paren {\\mathcal R \\circ \\mathcal R^{-1} } \\sqbrk B \\subseteq B$ where: :$\\mathcal R \\sqbrk B$ denotes the image of $B$ under $\\mathcal R$ :$\\mathcal R^{-1} \\sqbrk B$ denotes the preimage of $B$ under $\\mathcal R$ :$\\mathcal R \\circ \\mathcal R^{-1}$ denotes composition of $\\mathcal R$ and $\\mathcal R^{-1}$."}
+{"_id": "11239", "title": "Inverse of Direct Image Mapping does not necessarily equal Inverse Image Mapping", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R \\subseteq S \\times T$ be a relation. Let $\\mathrel R^\\to$ be the direct image mapping of $\\mathcal R$. Let $\\mathrel R^\\gets$ be the inverse image mapping of $\\mathcal R$. Then it is not necessarily the case that: :$\\paren {\\mathrel R^\\to}^{-1} = \\mathcal R^\\gets$ where $\\paren {\\mathrel R^\\to}^{-1}$ denote the inverse of $\\mathrel R^\\to$. That is, the inverse of the direct image mapping of $\\mathcal R$ does not always equal the inverse image mapping of $\\mathcal R$."}
+{"_id": "11246", "title": "Singleton of Power Set less Empty Set is Minimal Subset", "text": "Let $S$ be a set which is non-empty. Let $\\mathcal C = \\mathcal P \\left({S}\\right) \\setminus \\varnothing$, that is, the power set of $S$ without the empty set. Let $x \\in S$. Then $\\left\\{{x}\\right\\}$ is a minimal element of the ordered structure $\\left({\\mathcal C, \\subseteq}\\right)$."}
+{"_id": "11248", "title": "Natural Numbers under Multiplication form Semigroup", "text": "Let $\\N$ be the set of natural numbers. Let $\\times$ denote the operation of multiplication on $\\N$. The structure $\\struct {\\N, \\times}$ forms a semigroup."}
+{"_id": "11251", "title": "Bernoulli's Hanging Chain Problem", "text": "Consider a uniform chain $C$ whose physical properties are as follows: :$C$ is of length $l$ :The mass per unit length of $C$ is $m$ :$C$ is of zero stiffness. Let $C$ be suspended in a vertical line from a fixed point and otherwise free to move. Let $C$ be slightly disturbed in a vertical plane from its position of stable equilibrium. Let $\\map y t$ be the horizontal displacement at time $t$ from its position of stable equilibrium of a particle of $C$ which is a vertical distance $x$ from its point of attachment. The $2$nd order ODE describing the motion of $y$ is: :$\\dfrac {\\d^2 y} {\\d t^2} = g \\paren {l - x} \\dfrac {\\d^2 y} {\\d x^2} - g \\dfrac {\\d y} {\\d x}$"}
+{"_id": "11252", "title": "Real Line Continuity by Inverse of Mapping", "text": "Let $f$ be a real function. Let the domain of $f$ be open. Let $f^{-1}$ be the inverse of $f$. Then $f$ is continuous {{iff}}: :for every open real set $O$ that overlaps with the image of $f$, the preimage $f^{-1} \\left [{O}\\right]$ is open."}
+{"_id": "11253", "title": "Trisecting the Angle/Neusis Construction", "text": "Let $\\alpha$ be an angle which is to be trisected. This can be achieved by means of a neusis construction."}
+{"_id": "11256", "title": "Squaring the Circle by Compass and Straightedge Construction is Impossible", "text": "There is no compass and straightedge construction to allow a square to be constructed whose area is equal to that of a given circle."}
+{"_id": "11257", "title": "Difference between Distances from Point on Hyperbola to Foci is Constant", "text": "Let $K$ be a hyperbola. Let $F_1$ and $F_2$ be the foci of $K$. Let $P$ be an arbitrary point on $K$. Then the distance from $P$ to $F_1$ minus the distance from $P$ to $F_2$ is constant for all $P$ on $K$."}
+{"_id": "11258", "title": "Time Taken for Body to Fall at Earth's Surface", "text": "Let an object $m$ be released above ground from a point near the Earth's surface and allowed to fall freely. Let $m$ fall a distance $s$ in time $t$. Then: :$s = \\dfrac 1 2 g t^2$ or: :$t = \\sqrt {\\dfrac {2 s} g}$ where $g$ is the Acceleration Due to Gravity at the height through which $m$ falls. It is supposed that the distance $s$ is small enough that $g$ can be considered constant throughout."}
+{"_id": "11259", "title": "Length of Chord of Circle", "text": "Let $C$ be a circle of radius $r$. Let $AB$ be a chord which joins the endpoints of the arc $ADB$. Then: :$AB = 2 r \\sin \\dfrac \\theta 2$ where $\\theta$ is the angle subtended by $AB$ at the center of $C$."}
+{"_id": "11260", "title": "Spherical Triangles with Same Angles are Congruent", "text": "Two triangles on the surface of a given sphere which have the same angles are congruent."}
+{"_id": "11261", "title": "Sum of Angles of Spherical Triangle", "text": "The sum of the angles of a spherical triangle is between $\\pi$ and $3 \\pi$ radians."}
+{"_id": "11262", "title": "Ptolemy's Theorem", "text": "Let $ABCD$ be a cyclic quadrilateral. Then: :$AB \\times CD + AD \\times BC = AC \\times BD$"}
+{"_id": "11263", "title": "Spherical Law of Sines", "text": "Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Then: :$\\dfrac {\\sin a} {\\sin A} = \\dfrac {\\sin b} {\\sin B} = \\dfrac {\\sin c} {\\sin C}$"}
+{"_id": "11264", "title": "Spherical Law of Cosines", "text": "Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Then: :$\\cos a = \\cos b \\cos c + \\sin b \\sin c \\cos A$"}
+{"_id": "11265", "title": "Spherical Law of Cosines/Angles", "text": ":$\\cos A = -\\cos B \\cos C + \\sin B \\sin C \\cos a$"}
+{"_id": "11266", "title": "Spherical Law of Tangents", "text": ":$\\dfrac {\\tan \\frac 1 2 \\paren {A + B} } {\\tan \\frac 1 2 \\paren {A - B} } = \\dfrac {\\tan \\frac 1 2 \\paren {a + b} } {\\tan \\frac 1 2 \\paren {a - b} }$"}
+{"_id": "11267", "title": "Cosine of Half Angle for Spherical Triangles", "text": ":$\\cos \\dfrac A 2 = \\sqrt {\\dfrac {\\sin s \\, \\map \\sin {s - a} } {\\sin b \\sin c} }$ where $s = \\dfrac {a + b + c} 2$."}
+{"_id": "11268", "title": "Cosine of Half Side for Spherical Triangles", "text": ":$\\cos \\dfrac a 2 = \\sqrt {\\dfrac {\\map \\cos {S - B} \\, \\map \\cos {S - C} } {\\sin B \\sin C} }$ where $S = \\dfrac {A + B + C} 2$."}
+{"_id": "11269", "title": "Napier's Rules for Right Angled Spherical Triangles", "text": ":410px Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let either angle $\\angle C$ or side $c$ be a right angle. Let the remaining parts of $\\triangle ABC$ be arranged in a circle as above: :for $\\angle C$ a right angle, the '''interior''' :for $c$ a right angle, the '''exterior''' where the symbol $\\Box$ denotes a right angle. Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Let the remaining two parts be called '''opposite parts'''."}
+{"_id": "11270", "title": "Equation of Circle/Cartesian", "text": ":$\\paren {x - a}^2 + \\paren {y - b}^2 = R^2$"}
+{"_id": "11271", "title": "Equation of Circle/Parametric", "text": ":$x = a + R \\cos t, \\ y = b + R \\sin t$"}
+{"_id": "11273", "title": "Equation of Circle/Cartesian/Corollary 1", "text": "The equation: :$A \\paren {x^2 + y^2} + B x + C y + D = 0$ is the equation of a circle with radius $R$ and center $\\tuple {a, b}$, where: :$R = \\dfrac 1 {2 A} \\sqrt {B^2 + C^2 - 4 A D}$ :$\\tuple {a, b} = \\tuple {\\dfrac {-B} {2 A}, \\dfrac {-C} {2 A} }$ provided: :$A > 0$ :$B^2 + C^2 \\ge 4 A D$"}
+{"_id": "11274", "title": "Equation of Circle/Cartesian/Corollary 2", "text": "The equation of a circle with radius $R$ whose center is at the origin expressed in Cartesian coordinates is: :$x^2 + y^2 = R^2$"}
+{"_id": "11275", "title": "Distance Formula/3 Dimensions", "text": "The distance $d$ between two points $A = \\tuple {x_1, y_1, z_1}$ and $B = \\tuple {x_2, y_2, z_2}$ in a Cartesian space of 3 dimensions is: :$d = \\sqrt {\\paren {x_1 - x_2}^2 + \\paren {y_1 - y_2}^2 + \\paren {z_1 - z_2}^2}$"}
+{"_id": "11276", "title": "Equation of Sphere/Rectangular Coordinates", "text": ":$\\paren {x - a}^2 + \\paren {y - b}^2 + \\paren {z - c}^2 = R^2$"}
+{"_id": "11277", "title": "Equation of Sphere/Rectangular Coordinates/Corollary", "text": "The equation of a sphere with radius $R$ whose center is at the origin expressed in Cartesian coordinates is: :$x^2 + y^2 + z^2 = R^2$"}
+{"_id": "11278", "title": "Equation of Conic Section", "text": "The general conic section can be expressed in Cartesian coordinates in the form: :$a x^2 + b x y + c y^2 + d x + e y + f = 0$ for some $a, b, c, d, e, f \\in \\R$."}
+{"_id": "11279", "title": "Graph of Quadratic describes Parabola", "text": "The locus of the equation defining a quadratic: :$y = a x^2 + b x + c$ describes a parabola."}
+{"_id": "11280", "title": "Graph of Quadratic describes Parabola/Corollary 1", "text": "The locus of the equation of the square function: :$y = x^2$ describes a parabola."}
+{"_id": "11282", "title": "Natural Numbers under Addition form Commutative Semigroup", "text": "The algebraic structure $\\left({\\N, +}\\right)$ consisting of the set of natural numbers $\\N$ under addition $+$ is a commutative semigroup."}
+{"_id": "11283", "title": "Gödel's Incompleteness Theorems/First/Corollary", "text": "If $T$ is both consistent and complete, it does not contain minimal arithmetic."}
+{"_id": "11284", "title": "Monomorphism that is Split Epimorphism is Split Monomorphism", "text": "Let $\\mathbf C$ be a metacategory. Let $f: C \\to D$ be a morphism in $\\mathbf C$ such that $f$ is a monomorphism and a split epimorphism. Then $f: C \\to D$ is a split monomorphism. {{explain|What are $C$ and $D$?}}"}
+{"_id": "11285", "title": "Epimorphism that is Split Monomorphism is Split Epimorphism", "text": "Let $\\mathbf C$ be a metacategory. Let $f: C \\to D$ be a epimorphism and a split monomorphism. Then $f: C \\to D$ is a split epimorphism."}
+{"_id": "11287", "title": "Westwood's Puzzle/Proof 1", "text": ":500px Take any rectangle $ABCD$ and draw the diagonal $AC$. Inscribe a circle $GFJ$ in one of the resulting triangles $\\triangle ABC$. Drop perpendiculars $IEF$ and $HEJ$ from the center of this incircle $E$ to the sides of the rectangle. Then the area of the rectangle $DHEI$ equals half the area of the rectangle $ABCD$."}
+{"_id": "11288", "title": "Westwood's Puzzle/Proof 2", "text": ":500px Take any rectangle $ABCD$ and draw the diagonal $AC$. Inscribe a circle $GFJ$ in one of the resulting triangles $\\triangle ABC$. Drop perpendiculars $IEF$ and $HEJ$ from the center of this incircle $E$ to the sides of the rectangle. Then the area of the rectangle $DHEI$ equals half the area of the rectangle $ABCD$."}
+{"_id": "11289", "title": "Vector Cross Product Operator is Bilinear", "text": "Let $\\mathbf u$, $\\mathbf v$ and $\\mathbf w$ be vectors in a vector space $\\mathbf V$ of $3$ dimensions: {{begin-eqn}} {{eqn | l = \\mathbf u       | r = u_i \\mathbf i + u_j \\mathbf j + u_k \\mathbf k }} {{eqn | l = \\mathbf v       | r = v_i \\mathbf i + v_j \\mathbf j + v_k \\mathbf k }} {{eqn | l = \\mathbf w       | r = w_i \\mathbf i + w_j \\mathbf j + w_k \\mathbf k }} {{end-eqn}} where $\\left({\\mathbf i, \\mathbf j, \\mathbf k}\\right)$ is the standard ordered basis of $\\mathbf V$. Let $c$ be a real number. Then: : $\\left({c \\mathbf u + \\mathbf v}\\right) \\times \\mathbf w = c \\left({ \\mathbf u \\times \\mathbf w}\\right) + \\mathbf v \\times \\mathbf w$"}
+{"_id": "11290", "title": "Natural Numbers have No Proper Zero Divisors", "text": "Let $\\N$ be the natural numbers. Then for all $m, n \\in \\N$: :$m \\times n = 0 \\iff m = 0 \\lor n = 0$ That is, $\\N$ has no proper zero divisors."}
+{"_id": "11291", "title": "Diagonals of Rhombus Bisect Angles", "text": "Let $OABC$ be a rhombus. Then: :$(1): \\quad OB$ bisects $\\angle AOC$ and $\\angle ABC$ :$(2): \\quad AC$ bisects $\\angle OAB$ and $\\angle OCB$ 400px"}
+{"_id": "11296", "title": "Divergent Sequence with Finite Number of Terms Deleted is Divergent", "text": "Let $\\left({X, d}\\right)$ be a metric space. Let $\\left \\langle {x_k} \\right \\rangle$ be a sequence in $X$. Let $\\left \\langle {x_k} \\right \\rangle$ be divergent. Let a finite number of terms be deleted from $\\left \\langle {x_k} \\right \\rangle$. Then the resulting subsequence is divergent."}
+{"_id": "11300", "title": "Tensor Product is Module", "text": "Let $R$ be a ring. Let $M$ be a $R$-right module. Let $N$ be a $R$-left module. Then: :$T = \\displaystyle \\bigoplus_{s \\mathop \\in M \\mathop \\times N} R s$ is a left module."}
+{"_id": "11301", "title": "Supremum of Subset of Real Numbers is Arbitrarily Close", "text": "Let $A \\subseteq \\R$ be a subset of the real numbers. Let $b$ be a supremum of $A$. Let $\\epsilon \\in \\R_{>0}$. Then: :$\\exists x \\in A: b − x < \\epsilon$"}
+{"_id": "11302", "title": "Tensor with Zero Element is Zero in Tensor", "text": "Let $R$ be a ring. Let $M$ be a right $R$-module. Let $N$ be a left $R$-module. Let $M \\otimes_R N$ denote their tensor product. Then: :$0\\otimes_R n = m \\otimes_R 0 = 0 \\otimes_R 0$ is the zero in $M \\otimes_R N$."}
+{"_id": "11303", "title": "Primitive of Arcsecant of x over a/Formulation 1", "text": ":$\\displaystyle \\int \\operatorname{arcsec} \\frac x a \\ \\mathrm d x = \\begin{cases} \\displaystyle x \\operatorname{arcsec} \\frac x a - a \\ln \\left({x + \\sqrt {x^2 - a^2} }\\right) + C & : 0 < \\operatorname{arcsec} \\dfrac x a < \\dfrac \\pi 2 \\\\ \\displaystyle x \\operatorname{arcsec} \\frac x a + a \\ln \\left({x + \\sqrt {x^2 - a^2} }\\right) + C & : \\dfrac \\pi 2 < \\operatorname{arcsec} \\dfrac x a < \\pi \\\\ \\end{cases}$"}
+{"_id": "11307", "title": "Cross Product of Vector with Itself is Zero", "text": "Let $\\mathbf x$ be a vector in a vector space of $3$ dimensions: : $\\mathbf x = x_i \\mathbf i + x_j \\mathbf j + x_k \\mathbf k$ Then: :$\\mathbf x \\times \\mathbf x = \\mathbf 0$ where $\\times$ denotes vector cross product."}
+{"_id": "11309", "title": "Reciprocal of Riemann Zeta Function", "text": "For $\\Re \\left({z}\\right) > 1$: :$\\displaystyle \\frac 1 {\\zeta \\left({z}\\right)} = \\sum_{k \\mathop = 1}^\\infty \\frac{\\mu \\left({k}\\right)} {k^z}$ where: : $\\zeta$ is the Riemann zeta function : $\\mu$ is the Möbius function."}
+{"_id": "11312", "title": "Condition for Agreement of Family of Mappings", "text": "Let $\\left({A_i}\\right)_{i \\mathop \\in I}, \\left({B_i}\\right)_{i \\mathop \\in I}$ be families of non empty sets. Let $\\left({f_i}\\right)_{i \\mathop \\in I}$ be a family of mappings such that: :$\\forall i \\in I: f_i \\in \\mathcal F \\left({A_i, B_i}\\right)$ {{explain|Clarify: what is $\\mathcal F \\left({A_i, B_i}\\right)$? From the context it can be understood as being the set of all mappings $f_i: A_i \\to B_i$ but this is just a guess. The domain and range of each of the $f_i$ could help with being explicitly defined.<br/>Note also that I have taken the liberty of exchanging the difficult-to-read $\\mathscr F$ with the more eye-friendly $\\mathcal F$.}} We have that: :$\\displaystyle \\bigcup_{i \\mathop \\in I} f_i \\in \\mathcal F \\left({\\bigcup_{i \\mathop \\in I} A_i, \\bigcup_{i \\mathop \\in I} B_i }\\right)$ {{iff}}: :$\\displaystyle \\forall i, j \\in I: \\operatorname{Dom} f_i \\cap \\operatorname{Dom} f_j \\ne \\varnothing \\implies \\left({\\forall a \\in \\left({\\operatorname{Dom} f_i \\cap \\operatorname{Dom} f_j}\\right), \\left({a, b}\\right) \\in f_i \\implies \\left({a, b}\\right) \\in f_j}\\right)$"}
+{"_id": "11313", "title": "Supremum of Set of Real Numbers is at least Supremum of Subset", "text": "Let $S$ be a set of real numbers. Let $S$ have a supremum. Let $T$ be a non-empty subset of $S$. Then $\\sup T$ exists and: :$\\sup T \\le \\sup S$"}
+{"_id": "11316", "title": "Supremum of Subset of Union Equals Supremum of Union", "text": "Let $S$ be a non-empty real set. Let $S$ have a supremum. Let $\\set {S_i: i \\in \\set {1, 2, \\ldots, n} }$, $n \\in \\N_{>0}$, be a set of non-empty subsets of $S$. Let $\\bigcup S_i = S$. Then there exists a $j$ in $\\set {1, 2, \\ldots, n}$ such that: :$\\sup S_j = \\sup S$"}
+{"_id": "11319", "title": "Condition for Ideal to be Total Ring", "text": "Let $\\left({A, +, \\circ}\\right)$ be a commutative ring with unity. Let $I$ be an ideal of $A$ such that the quotient ring $A / I$ is a field. Let $J$ be an ideal of $A$ such that $I \\subsetneq J$. Then: :$A = J$"}
+{"_id": "11320", "title": "Area between Two Non-Intersecting Chords", "text": "Let $AB$ and $CD$ be two chords of a circle whose center is at $O$ and whose radius is $r$. :400px :400px Let $\\alpha$ and $\\theta$ be respectively the measures in radians of the angles $\\angle COD$ and $\\angle AOB$. Then the area $\\mathcal A$ between the two chords is given by: : $\\mathcal A = \\dfrac {r^2} 2 \\left({\\theta - \\sin \\theta - \\alpha + \\sin \\alpha}\\right)$ if $O$ is not included in the area, and: : $\\mathcal A = r^2 \\left({\\pi - \\dfrac 1 2 \\left({\\theta - \\sin \\theta + \\alpha - \\sin \\alpha}\\right)}\\right)$ if $O$ is included in the area."}
+{"_id": "11321", "title": "Supremum of Absolute Value of Difference equals Supremum of Difference", "text": "Let $S$ be a non-empty real set. Let $\\displaystyle \\sup_{x, y \\mathop \\in S} \\paren {x - y}$ exist. Then $\\displaystyle \\sup_{x, y \\mathop \\in S} \\size {x - y}$ exists and: :$\\displaystyle \\sup_{x, y \\mathop \\in S} \\size {x - y} = \\sup_{x, y \\mathop \\in S} \\paren {x - y}$"}
+{"_id": "11322", "title": "Supremum of Sum equals Sum of Suprema", "text": "Let $A$ and $B$ be non-empty sets of real numbers. Let $A + B$ be $\\set {x + y: x \\in A, y \\in B}$. Let either $A$ and $B$ have suprema or $A + B$ have a supremum. Then all $\\sup A$, $\\sup B$, and $\\sup \\paren {A + B}$ exist and: :$\\sup \\paren {A + B} = \\sup A + \\sup B$"}
+{"_id": "11324", "title": "Edge is Minimum Weight Bridge iff in All Minimum Spanning Trees", "text": "Let $G$ be an undirected network. Let every edge of $G$ have a unique weight. Let $e$ be an edge of $G$. Then $e$ is a bridge of minimum weight in $G$ {{iff}} $e$ belongs to every minimum spanning tree of $G$."}
+{"_id": "11325", "title": "Maximum Weight Edge in all Minimum Spanning Trees is Bridge", "text": "Let $G$ be an undirected network. Let every edge of $G$ have a unique weight. Let $e$ be an edge of $G$ that belongs to every minimum spanning tree of $G$. Let $e$ have maximum weight in $G$. Then $e$ is a bridge in $G$."}
+{"_id": "11329", "title": "Scaling Property of Dirac Delta Function", "text": "Let $\\delta \\left({t}\\right)$ be the Dirac delta function. Let $a$ be a non zero constant real number. Then: :$\\delta \\left({a t}\\right) = \\dfrac {\\delta \\left({t}\\right)} {\\left \\vert a \\right \\vert}$"}
+{"_id": "11330", "title": "Supremum of Function is less than Supremum of Greater Function", "text": "Let $f$ and $g$ be real functions. Let $S$ be a subset of $\\operatorname{Dom} \\left({f}\\right) \\cap \\operatorname{Dom} \\left({g}\\right)$. Let $f \\left({x}\\right) \\le g \\left({x}\\right)$ for every $x \\in S$. Let $\\displaystyle \\sup_{x \\mathop \\in S} g \\left({x}\\right)$ exist. Then $\\displaystyle \\sup_{x \\mathop \\in S} f \\left({x}\\right)$ exists and: :$\\displaystyle \\sup_{x \\mathop \\in S} f \\left({x}\\right) \\le \\sup_{x \\mathop \\in S} g \\left({x}\\right)$."}
+{"_id": "11331", "title": "Characterization of Boundary by Open Sets", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A$ be a subset of $T$. Let $x$ be a point of $T$. Then $x \\in \\operatorname{Fr} A$ {{iff}}: :for every open set $U$ of $T$: ::if $x \\in U$ ::then $A \\cap U \\ne \\varnothing$ and $\\complement_S \\left({A}\\right) \\cap U \\ne \\varnothing$ where: :$\\complement_S \\left({A}\\right) = S \\setminus A$ denotes the complement of $A$ in $S$ :$\\operatorname{Fr} A$ denotes the boundary of $A$."}
+{"_id": "11332", "title": "Characterization of Closure by Open Sets", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A$ be a subset of $S$. Let $x$ be a point of $T$. Let $A^-$ denote the closure of $A$. Then $x \\in A^-$ {{iff}}: :for every open set $U$ of $T$: ::$x \\in U \\implies A \\cap U \\ne \\varnothing$"}
+{"_id": "11333", "title": "Summation is Linear/Sum of Summations", "text": ":$\\displaystyle \\sum_{i \\mathop = 1}^n x_i + \\sum_{i \\mathop = 1}^n y_i = \\sum_{i \\mathop = 1}^n \\paren {x_i + y_i}$"}
+{"_id": "11334", "title": "Summation is Linear/Scaling of Summations", "text": ":$\\displaystyle \\lambda \\sum_{i \\mathop = 1}^n x_i = \\sum_{i \\mathop = 1}^n \\lambda x_i$"}
+{"_id": "11335", "title": "Supremum of Absolute Value of Difference equals Difference between Supremum and Infimum", "text": "Let $f$ be a real function. Let $S$ be a subset of the domain of $f$. Let $\\displaystyle \\sup_{x \\mathop \\in S} \\set {\\map f x}$ and $\\displaystyle \\inf_{x \\mathop \\in S} \\set {\\map f x}$ exist. Then $\\displaystyle \\sup_{x, y \\mathop \\in S} \\set {\\size {\\map f x - \\map f y} }$ exists and: :$\\displaystyle \\sup_{x, y \\mathop \\in S} \\set {\\size {\\map f x - \\map f y} } = \\sup_{x \\mathop \\in S} \\set {\\map f x} - \\inf_{x \\mathop \\in S} \\set {\\map f x}$"}
+{"_id": "11336", "title": "Diaconescu-Goodman-Myhill Theorem", "text": "The axiom of choice implies the law of excluded middle."}
+{"_id": "11337", "title": "Characterization of Boundary by Basis", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\BB \\subseteq \\tau$ be a basis. Let $A$ be a subset of $T$. Let $x$ be a point of $T$. Then $x \\in \\partial A$ {{iff}}: :for every $U \\in \\BB$: ::if $x \\in U$ ::then $A \\cap U \\ne \\O$ and $\\relcomp S A \\cap U \\ne \\O$ where: :$\\relcomp S A = S \\setminus A$ denotes the complement of $A$ in $S$ :$\\partial A$ denotes the boundary of $A$ in $T$."}
+{"_id": "11338", "title": "Union of Interiors and Boundary Equals Whole Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A$ be a subset of $T$. Then: :$S = \\Int A \\cup \\partial A \\cup \\Int {A'}$ where: :$A' = S \\setminus A$ denotes the complement of $A$ relative to $S$ :$\\Int A$ denotes the interior of $A$ :$\\partial A$ denotes the boundary of $A$."}
+{"_id": "11339", "title": "Characterization of Derivative by Open Sets", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A$ be a subset of $T$. Let $x$ be a point of $T$. Then :$x \\in A'$ {{iff}}: :for every open set $U$ of $T$: ::if $x \\in U$ ::then there exists a point $y$ of $T$ such that $y \\in A \\cap U$ and $x \\ne y$ where :$A'$ denotes the derivative of $A$."}
+{"_id": "11341", "title": "Derivative is Included in Closure", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A$ be a subset of $S$. Then :$A' \\subseteq A^-$ where :$A'$ denotes the derivative of $A$ :$A^-$ denotes the closure of $A$."}
+{"_id": "11342", "title": "Closure Equals Union with Derivative", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A$ be a subset of $S$. Then: :$A^- = A \\cup A'$ where :$A'$ denotes the derivative of $A$ :$A^-$ denotes the closure of $A$."}
+{"_id": "11343", "title": "Derivative of Subset is Subset of Derivative", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A$, $B$ be subsets of $S$. Then :$A \\subseteq B \\implies A' \\subseteq B'$ where $A'$ denotes the derivative of $A$ in $T$."}
+{"_id": "11344", "title": "Derivative of Union is Union of Derivatives", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A$, $B$ be subsets of $S$. Then :$\\left({A \\cup B}\\right)' = A' \\cup B\\,'$ where :$A'$ denotes the derivative of $A$."}
+{"_id": "11345", "title": "Derivative of Derivative is Subset of Derivative in T1 Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a $T_1$ topological space. Let $A$ be a subset of $S$. Then :$A'' \\subseteq A'$ where :$A'$ denotes the derivative of $A$"}
+{"_id": "11346", "title": "Closure of Derivative is Derivative in T1 Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a $T_1$ topological space. Let $A$ be a subset of $S$. Then :$A'^- = A'$ where :$A'$ denotes the derivative of $A$ :$A^-$ denotes the closure of $A$."}
+{"_id": "11347", "title": "Union of Derivatives is Subset of Derivative of Union", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let: :$\\FF \\subseteq \\powerset S$ be a set of subsets of $S$ where $\\powerset S$ denotes the power set of $S$. Then: :$\\ds \\bigcup_{A \\mathop \\in \\FF} A' \\subseteq \\paren {\\bigcup_{A \\mathop \\in \\FF} A}'$ where $A'$ denotes the derivative of $A$."}
+{"_id": "11348", "title": "Point is Isolated iff not Accumulation Point", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $H \\subseteq S$. Let $x \\in H$. Then: :$x$ is an isolated point in $H$ {{iff}}: :$x$ is not an accumulation point of $H$"}
+{"_id": "11350", "title": "Dense-in-itself iff Subset of Derivative", "text": "Let $T$ be a topological space. Let $A \\subseteq T$. Then: :$A$ is dense-in-itself {{iff}}: :$A \\subseteq A'$ where :$A'$ denotes the derivative of $A$."}
+{"_id": "11351", "title": "Closure of Dense-in-itself is Dense-in-itself in T1 Space", "text": "Let $T$ be a $T_1$ topological space. Let $A \\subseteq T$.  Let $A$ be dense-in-itself. Then the closure $A^-$ of $A$ is also dense-in-itself."}
+{"_id": "11354", "title": "Equivalence of Definitions of Weight of Topological Space", "text": "Let $T$ be a topological space. Let $\\mathbb B$ be the set of all bases of $T$. The following definitions of the weight of $T$ are equivalent:"}
+{"_id": "11356", "title": "Difference of Two Powers/Examples/Difference of Two Cubes", "text": ":$x^3 - y^3 = \\paren {x - y} \\paren {x^2 + x y + y^2}$"}
+{"_id": "11357", "title": "Existence of Subfamily of Cardinality not greater than Weight of Space and Unions Equal", "text": "Let $T$ be a topological space. Let $\\mathcal F$ be a set of open sets of $T$. There exists a subset $\\mathcal G \\subseteq \\mathcal F$ such that: :$\\displaystyle \\bigcup \\mathcal G = \\bigcup \\mathcal F$ and: :$\\left\\vert{\\mathcal G}\\right\\vert \\leq w \\left({T}\\right)$ where: :$w \\left({T}\\right)$ denotes the weight of $T$ :$\\left\\vert{\\mathcal G}\\right\\vert$ denotes the cardinality of $\\mathcal G$."}
+{"_id": "11358", "title": "Set of Singletons is Smallest Basis of Discrete Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a discrete topological space. Let $\\mathcal B = \\left\\{{\\left\\{{x}\\right\\} : x \\in S}\\right\\}$. Then $\\mathcal B$ is the smallest basis of $T$. That is: :$\\mathcal B$ is a basis of $T$ and: :for every basis $\\mathcal C$ of $T$, $\\mathcal B \\subseteq \\mathcal C$."}
+{"_id": "11362", "title": "Cardinality of Set of Singletons", "text": "Let $S$ be a set. Let $T = \\left\\{ {\\left\\{{x}\\right\\}: x \\in S}\\right\\}$ be the set of all singletons of elements of $S$. Then: :$\\left\\vert T \\right\\vert = \\left\\vert S \\right\\vert$ where $\\left\\vert S \\right\\vert$ denotes the cardinality of $S$."}
+{"_id": "11375", "title": "Basis has Subset Basis of Cardinality equal to Weight of Space", "text": "Let $T = \\struct {X, \\tau}$ be a topological space. Let $\\BB$ be a basis of $T$. Then there exists a basis $\\BB_0$ of $T$ such that :$\\BB_0 \\subseteq \\BB$ and $\\card {\\BB_0} = \\map w T$ where: :$\\card {\\BB_0}$ denotes the cardinality of $\\BB_0$ :$\\map w T$ denotes the weight of $T$."}
+{"_id": "11376", "title": "Cardinality of Union not greater than Product", "text": "Let $\\FF$ be a set of sets. Let: :$\\size \\FF \\le \\mathbf m$ where  :$\\size \\FF$ denotes the cardinality of $\\FF$ :$\\mathbf m$ is cardinal number (possibly infinite). Let: :$\\forall A \\in \\FF: \\size A \\le \\mathbf n$ where :$\\mathbf n$ is cardinal number (possibly infinite). Then: :$\\ds \\size {\\bigcup \\FF} \\le \\size {\\mathbf m \\times \\mathbf n} = \\mathbf m \\mathbf n$"}
+{"_id": "11377", "title": "Image of Mapping of Intersections is Smallest Basis", "text": "Let $T = \\left({X, \\tau}\\right)$ be a topological space. Let $f:X \\to \\tau$ be a mapping such that :$\\forall x \\in X: \\left({x \\in f \\left({x}\\right) \\land \\forall U \\in \\tau: x \\in U \\implies f \\left({x}\\right) \\subseteq U}\\right)$. Then the image $\\operatorname{Im} \\left({f}\\right)$ is subset of every basis of $T$."}
+{"_id": "11378", "title": "Cardinality of Image of Mapping of Intersections is not greater than Weight of Space", "text": "Let $T = \\struct {X, \\tau}$ be a topological space. Let $f: X \\to \\tau$ be a mapping such that: :$\\forall x \\in X: \\paren {x \\in \\map f x \\land \\forall U \\in \\tau: x \\in U \\implies \\map f x \\subseteq U}$ Then the cardinality of the image of $f$ is no greater than the weight of $T$: $\\card {\\Img f} \\le \\map w T$."}
+{"_id": "11379", "title": "Finite Weight Space has Basis equal to Image of Mapping of Intersections", "text": "Let $T = \\struct {X, \\tau}$ be a topological space with finite weight. Then there exist a basis $\\BB$ of $T$ and a mapping $f:X \\to \\tau$ such that: :$\\BB = \\Img f$ and :$\\forall x \\in X: \\paren {x \\in \\map f x \\land \\forall U \\in \\tau: x \\in U \\implies \\map f x \\subseteq U}$ where $\\Img f$ denotes the image of $f$."}
+{"_id": "11380", "title": "Rubik's Cube has 54 Facets", "text": "Let $S$ be the set of facets of Rubik's cube. Then the cardinality of $S$ is given by: :$\\card S = 54$ That is: :A Rubik's cube has $54$ facets."}
+{"_id": "11381", "title": "Equivalence of Definitions of Symmetric Difference/(3) iff (5)", "text": "Let $S$ and $T$ be sets. {{TFAENocat|def = Symmetric Difference|view = symmetric difference $S * T$ between $S$ and $T$}}"}
+{"_id": "11382", "title": "Equivalence of Definitions of Symmetric Difference/(2) iff (5)", "text": "Let $S$ and $T$ be sets. {{TFAENocat|def = Symmetric Difference|view = symmetric difference $S * T$ between $S$ and $T$}}"}
+{"_id": "11389", "title": "Even and Odd Integers form Partition of Integers", "text": "The odd integers and even integers form a partition of the integers."}
+{"_id": "11390", "title": "Analog between Logic and Set Theory", "text": "The concepts of set theory have directly corresponding concepts in logic: :{| border = \"1\" |- ! style=\"padding: 2px 10px\" | Set Theory ! style=\"padding: 2px 10px\" | Logic |- | align=\"left\" style=\"padding: 2px 10px\"| Set: $S, T$ | align=\"left\" style=\"padding: 2px 10px\"| Statement: $p, q$ |- | align=\"left\" style=\"padding: 2px 10px\"| Union: $S \\cup T$ | align=\"left\" style=\"padding: 2px 10px\"| Disjunction: $p \\lor q$ |- | align=\"left\" style=\"padding: 2px 10px\"| Intersection: $S \\cap T$ | align=\"left\" style=\"padding: 2px 10px\"| Conjunction: $p \\land q$ |- | align=\"left\" style=\"padding: 2px 10px\"| Subset: $S \\subseteq T$ | align=\"left\" style=\"padding: 2px 10px\"| Conditional: $p \\implies q$ |- | align=\"left\" style=\"padding: 2px 10px\"| Symmetric Difference: $S * T$ | align=\"left\" style=\"padding: 2px 10px\"| Exclusive Or: $p \\oplus q$ |- | align=\"left\" style=\"padding: 2px 10px\"| Complement: $\\relcomp {} S$ | align=\"left\" style=\"padding: 2px 10px\"| Logical Not: $\\lnot p$ |- | align=\"left\" style=\"padding: 2px 10px\"| Set Equality: $S = T$ | align=\"left\" style=\"padding: 2px 10px\"| Biconditional: $p \\iff q$ |- | align=\"left\" style=\"padding: 2px 10px\"| Venn Diagram | align=\"left\" style=\"padding: 2px 10px\"| Truth Table |}"}
+{"_id": "11392", "title": "Cardinality of Set is Finite iff Set is Finite", "text": "Let $A$ be a set. $\\left\\vert{A}\\right\\vert$ is finite {{iff}} $A$ is finite where $\\left\\vert{A}\\right\\vert$ denotes the cardinality of $A$."}
+{"_id": "11394", "title": "Topology Generated by Closed Sets", "text": "Let $X$ be a set. Let $\\mathcal F$ be a set of subsets of $X$. Suppose that :$\\varnothing \\in \\mathcal F$ and :for every subsets $A$ and $B$ of $X$ if $A, B \\in \\mathcal F$, then $A \\cup B \\in \\mathcal F$ and :for every subset $\\mathcal G \\subseteq \\mathcal F$, $\\bigcap \\mathcal G \\in \\mathcal F$ and :$\\tau = \\left\\{{\\complement_X \\left(A\\right): A \\in \\mathcal F}\\right\\}$. Then: :$T = \\left( {X, \\tau} \\right)$ is topological space and :for every subset $A$ of $X$, $A$ is closed in $T$ {{iff}} $A \\in \\mathcal F$."}
+{"_id": "11395", "title": "Equivalence of Definitions of Countably Infinite Set", "text": "Let $S$ be a set. {{TFAE|def = Countably Infinite Set}}"}
+{"_id": "11396", "title": "Set of Odd Integers is Countably Infinite", "text": "Let $\\Bbb O$ be the set of odd integers. Then $\\Bbb O$ is countably infinite."}
+{"_id": "11397", "title": "Unique Readability for Polish Notation", "text": "Let $\\AA$ be an alphabet. Then Polish notation for $\\AA$ has the unique readability property."}
+{"_id": "11400", "title": "Product of Rook Matrices is Rook Matrix", "text": "Let $\\mathbf A$ and $\\mathbf B$ be rook matrices. Their product $\\mathbf {A B}$ is also a rook matrix."}
+{"_id": "11401", "title": "Topology Defined by Basis", "text": "Let $S$ be a set. Let $\\mathcal B$ be a set of subsets of $S$. Suppose that :$(B1): \\quad \\forall A_1, A_2 \\in \\mathcal B: \\forall x \\in A_1 \\cap A_2: \\exists A \\in \\mathcal B: x \\in A \\subseteq A_1 \\cap A_2$ :$(B2): \\quad \\forall x \\in X: \\exists A \\in \\mathcal B: x \\in A$ ::$\\tau = \\left\\{{\\bigcup \\mathcal G: \\mathcal G \\subseteq \\mathcal B}\\right\\}$ Then: :$T = \\left({S, \\tau}\\right)$ is a topological space :$\\mathcal B$ is a basis of $T$."}
+{"_id": "11403", "title": "Matrix is Non-Invertible iff Product with Non-Zero Vector is Zero", "text": "Let $\\mathbf A$ be a square matrix of order $n$. Then $\\mathbf A$ is non-invertible if there exists a vector $\\mathbf v$ of $n$ such that: :$\\mathbf v \\ne \\mathbf 0$ :$\\mathbf A \\mathbf v = \\mathbf 0$ where $\\mathbf 0$ is the zero vector."}
+{"_id": "11404", "title": "Equivalence of Definitions of Integer Congruence", "text": "Let $m \\in \\Z_{> 0}$. {{TFAE|def = Congruence (Number Theory)/Integers|view = congruence modulo $m$}}"}
+{"_id": "11406", "title": "Krull's Theorem", "text": "Let $R$ be a non-null ring with unity. Then $R$ has a maximal ideal."}
+{"_id": "11410", "title": "NAND as Disjunction of Negations", "text": ": $p \\uparrow q \\dashv \\vdash \\neg p \\lor \\neg q$"}
+{"_id": "11416", "title": "Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication", "text": ": $\\vdash \\left({p \\lor p}\\right) \\implies p$"}
+{"_id": "11421", "title": "Rule of Addition/Sequent Form/Formulation 1/Form 2/Proof 2", "text": ":$q \\vdash p \\lor q$"}
+{"_id": "11424", "title": "Rule of Addition/Sequent Form/Formulation 2/Form 2", "text": ":$\\vdash q \\implies \\left({p \\lor q}\\right)$"}
+{"_id": "11433", "title": "Hilbert Proof System Instance 2 is Consistent", "text": "Instance 2 of the Hilbert proof systems $\\mathscr H_2$ is consistent."}
+{"_id": "11434", "title": "Set of Local Minimum is Countable", "text": "Let $X$ be a subset of $\\R$. The set: :$\\leftset {x \\in X: x}$ is local minimum in $\\rightset X$ is countable."}
+{"_id": "11435", "title": "Set of Pairwise Disjoint Intervals is Countable", "text": "Let $X$ be a subset of $\\mathcal P \\left({\\R}\\right)$ such that: :$(1): \\quad X$ is pairwise disjoint: ::::$\\forall A,B \\in X: A \\ne B \\implies A \\cap B = \\varnothing$. :$(2): \\quad$ every element of $X$ contains an open interval: ::::$\\forall A \\in X: \\exists x, y \\in \\R: x < y \\land \\left({x \\,.\\,.\\, y}\\right) \\subseteq A$. Then $X$ is countable."}
+{"_id": "11436", "title": "Set is Countable if Cardinality equals Cardinality of Countable Set", "text": "Let $X, Y$ be sets. Let: : $\\left\\vert{X}\\right\\vert = \\left\\vert{Y}\\right\\vert$ where $\\left\\vert{X}\\right\\vert$ denotes the cardinality of $X$. If $X$ is countable then $Y$ is countable."}
+{"_id": "11437", "title": "Double Negation/Double Negation Introduction/Sequent Form", "text": "{{:Double Negation/Double Negation Introduction/Sequent Form/Formulation 1}}"}
+{"_id": "11438", "title": "Double Negation/Double Negation Elimination/Proof Rule", "text": ":If we can conclude $\\neg \\neg \\phi$, then we may infer $\\phi$."}
+{"_id": "11439", "title": "Double Negation/Double Negation Elimination/Formulation 1/Sequent Form", "text": "{{:Double Negation/Double Negation Elimination/Sequent Form/Formulation 1}}"}
+{"_id": "11440", "title": "Biconditional Elimination/Also known as", "text": "Some sources refer to the Biconditional Elimination as the rule of '''Biconditional-Conditional'''."}
+{"_id": "11442", "title": "Principle of Non-Contradiction/Explanation", "text": "The '''Principle of Non-Contradiction''' can be expressed in natural language as follows: :A statement can not be both true and not true at the same time. This means: if we have managed to deduce that a statement is both true and false, then the sequence of deductions show that the pool of assumptions upon which the sequent rests contains assumptions which are mutually contradictory. Thus it provides a means of eliminating a logical not from a sequent."}
+{"_id": "11443", "title": "Principle of Non-Contradiction/Proof Rule", "text": ":If we can conclude both $\\phi$ and $\\neg \\phi$, we may infer a contradiction."}
+{"_id": "11449", "title": "Solutions of Pythagorean Equation/Primitive", "text": "The set of all primitive Pythagorean triples is generated by: :$\\tuple {2 m n, m^2 - n^2, m^2 + n^2}$ where: :$m, n \\in \\Z_{>0}$ are (strictly) positive integers :$m \\perp n$, that is, $m$ and $n$ are coprime :$m$ and $n$ are of opposite parity :$m > n$"}
+{"_id": "11450", "title": "Solutions of Pythagorean Equation/General", "text": "Let $x, y, z$ be a solution to the Pythagorean equation. Then $x = k x', y = k y', z = k z'$, where: :$\\tuple {x', y', z'}$ is a primitive Pythagorean triple :$k \\in \\Z: k \\ge 1$"}
+{"_id": "11453", "title": "Goldbach Conjecture implies Goldbach's Marginal Conjecture", "text": "Suppose the Goldbach Conjecture holds: :Every even integer greater than $2$ is the sum of two primes. Then Goldbach's Marginal Conjecture follows: :Every integer greater than $5$ can be written as the sum of three primes."}
+{"_id": "11454", "title": "Infinite Number of Chen Primes", "text": "There exists an infinite number of Chen primes. That is, there exists an infinite number of pairs: :$\\set {p, p + 2}$ where: :$p$ is a prime :$p + 2$ is either a prime or a semiprime."}
+{"_id": "11455", "title": "Congruent Integers are of same Quadratic Character", "text": "Let $p$ be an odd prime. Let $a \\in \\Z$ be an integer such that $a \\not \\equiv 0 \\pmod p$. Let $a \\equiv b \\pmod p$. Then $a$ and $b$ have the same quadratic character."}
+{"_id": "11456", "title": "Weight of Sorgenfrey Line is Continuum", "text": "Let $T = \\struct {\\R, \\tau}$ be the Sorgenfrey line. Then $\\map w T = \\mathfrak c$ where :$\\map w T$ denotes the weight of $T$ :$\\mathfrak c$ denotes continuum, the cardinality of real numbers."}
+{"_id": "11458", "title": "Set of Subset of Reals with Cardinality less than Continuum has not Interval in Union Closure", "text": "Let $\\mathcal B$ be a set of subsets of $\\R$, the set of all real numbers. Let: :$\\left\\vert{\\mathcal B}\\right\\vert < \\mathfrak c$ where :$\\left\\vert{\\mathcal B}\\right\\vert$ denotes the cardinality of $\\mathcal B$ :$\\mathfrak c = \\left\\vert{\\R}\\right\\vert$ denotes continuum. Then: :$\\exists x, y \\in \\R: x < y \\land \\left[{x \\,.\\,.\\, y}\\right) \\notin \\left\\{{\\bigcup \\mathcal G: \\mathcal G \\subseteq \\mathcal B}\\right\\}$"}
+{"_id": "11459", "title": "Cardinality of Basis of Sorgenfrey Line not greater than Continuum", "text": "Let $T = \\struct {\\R, \\tau}$ be the Sorgenfrey line. Let :$\\BB = \\set {\\hointr x y: x, y \\in \\R \\land x < y}$ be the basis of $T$. Then $\\card \\BB \\le \\mathfrak c$ where :$\\card \\BB$ denotes the cardinality of $\\BB$ :$\\mathfrak c = \\card \\R$ denotes the continuum."}
+{"_id": "11460", "title": "Construction of Regular Prime p-Gon Exists iff p is Fermat Prime", "text": "Let $p$ be a prime number. Then there exists a compass and straightedge construction for a regular $p$-gon {{iff}} $p$ is a Fermat prime."}
+{"_id": "11461", "title": "Cardinalities form Inequality implies Difference is Nonempty", "text": "Let $X, Y$ be sets. Let :$\\left\\vert{X}\\right\\vert < \\left\\vert{Y}\\right\\vert$ where $\\left\\vert{X}\\right\\vert$ denotes the cardinality of $X$. Then: :$Y \\setminus X \\ne \\varnothing$"}
+{"_id": "11462", "title": "Set of Subsets of Reals with Cardinality less than Continuum Cardinality of Local Minimums of Union Closure less than Continuum", "text": "Let $\\BB$ be a set of subsets of $\\R$. Let: :$\\size \\BB < \\mathfrak c$ where :$\\size \\BB$ denotes the cardinality of $\\BB$ :$\\mathfrak c = \\size \\R$ denotes continuum. Let :$X = \\leftset {x \\in \\R: \\exists U \\in \\set {\\bigcup \\GG: \\GG \\subseteq \\BB}: x}$ is local minimum in $\\rightset U$ Then: :$\\size X < \\mathfrak c$"}
+{"_id": "11463", "title": "Slope of Secant", "text": "Let $f: \\R \\to \\R$ be a real function. Let the graph of $f$ be depicted on a Cartesian plane. :400px Let $AB$ be a secant of $f$ where: :$A = \\tuple {x, \\map f x}$ :$A = \\tuple {x + h, \\map f {x + h} }$ Then the slope of $AB$ is given by: :$\\dfrac {\\map f {x + h} - \\map f x} h$"}
+{"_id": "11464", "title": "Derivative of Curve at Point", "text": "Let $f: \\R \\to \\R$ be a real function. Let the graph $G$ of $f$ be depicted on a Cartesian plane. Then the derivative of $f$ at $x = \\xi$ is equal to the tangent to $G$ at $x = \\xi$."}
+{"_id": "11465", "title": "Derivative of Square Function", "text": "Let $f: \\R \\to \\R$ be the square function: :$\\forall x \\in \\R: \\map f x = x^2$ Then the derivative of $f$ is given by: :$\\map {f'} x = 2 x$"}
+{"_id": "11466", "title": "Countable iff Cardinality not greater than Aleph Zero", "text": "Let $X$ be set. $X$ is countable {{iff}}: $\\left\\vert{X}\\right\\vert \\leq \\aleph_0$ where: :$\\left\\vert{X}\\right\\vert$ denotes the cardinality of $X$ :$\\aleph_0 = \\left\\vert{\\N}\\right\\vert$ by Aleph Zero equals Cardinality of Naturals."}
+{"_id": "11467", "title": "Aleph Zero equals Cardinality of Naturals", "text": "$\\aleph_0 = \\left\\vert{\\N}\\right\\vert$ where :$\\aleph$ denotes the aleph mapping, :$\\left\\vert{\\N}\\right\\vert$ denotes the cardinality of $\\N$."}
+{"_id": "11469", "title": "Body under Constant Acceleration/Velocity after Time", "text": ": $\\mathbf v = \\mathbf u + \\mathbf a t$"}
+{"_id": "11470", "title": "Body under Constant Acceleration/Distance after Time", "text": ":$\\mathbf s = \\mathbf u t + \\dfrac {\\mathbf a t^2} 2$"}
+{"_id": "11471", "title": "Body under Constant Acceleration/Velocity after Distance", "text": ":$\\mathbf v \\cdot \\mathbf v = \\mathbf u \\cdot \\mathbf u + 2 \\mathbf a \\cdot \\mathbf s$"}
+{"_id": "11472", "title": "Body in Free Fall moves in Parabolic Path", "text": "A body in free fall above the surface of the Earth follows a path approximating to a parabola."}
+{"_id": "11475", "title": "Aleph Zero is less than Continuum", "text": "$\\aleph_0 < \\mathfrak c$ where :$\\aleph$ denotes the aleph mapping, :$\\mathfrak c$ denotes continuum, the cardinality of real numbers."}
+{"_id": "11476", "title": "Cardinality of Set less than Cardinality of Power Set", "text": "Let $X$ be a set. Then: :$\\left\\vert{X}\\right\\vert < \\left\\vert{\\mathcal P \\left({X}\\right)}\\right\\vert$ where :$\\left\\vert{X}\\right\\vert$ denotes the cardinality of $X$, :$\\mathcal P \\left({X}\\right)$ denotes the power set of $X$."}
+{"_id": "11477", "title": "Volume of Solid of Revolution", "text": "Let $f: \\R \\to \\R$ be a real function which is integrable on the interval $\\closedint a b$. Let the points be defined: :$A = \\tuple {a, \\map f a}$ :$B = \\tuple {b, \\map f b}$ :$C = \\tuple {b, 0}$ :$D = \\tuple {a, 0}$ Let the figure $ABCD$ be defined as being bounded by the straight lines $y = 0$, $x = a$, $x = b$ and the curve defined by $\\set {\\map f x: a \\le x \\le b}$. Let the solid of revolution $S$ be generated by rotating $ABCD$ around the $x$-axis (that is, $y = 0$). Then the volume $V$ of $S$ is given by: :$\\displaystyle V = \\pi \\int_a^b \\paren {\\map f x}^2 \\rd x$"}
+{"_id": "11478", "title": "Continuum equals Cardinality of Power Set of Naturals", "text": "$\\mathfrak c = \\card {\\powerset \\N}$ where :$\\powerset \\N$ denotes the power set of $\\N$ :$\\card {\\powerset \\N}$ denotes the cardinality of $\\powerset \\N$ :$\\mathfrak c = \\card \\R$ denotes the continuum."}
+{"_id": "11479", "title": "Acceleration is Second Derivative of Displacement with respect to Time", "text": "The '''acceleration''' $\\mathbf a$ of a body $M$ is the second derivative of the displacement $\\mathbf s$ of $M$ from a given point of reference with respect to time $t$: :$\\mathbf a = \\dfrac {\\d^2 \\mathbf s} {\\d t^2}$"}
+{"_id": "11480", "title": "Equation of Catenary/Formulation 2", "text": "The '''catenary''' is described by the equation: :$y = \\dfrac a 2 \\paren {e^{x / a} + e^{-x / a} } = a \\cosh \\dfrac x a$ where $a$ is a constant. The lowest point of the chain is at $\\tuple {0, a}$."}
+{"_id": "11481", "title": "Cardinality of Power Set is Invariant", "text": "Let $X, Y$ be sets. Let $\\card X = \\card Y$ where $\\card X$ denotes the cardinality of $X$. Then: :$\\card {\\powerset X} = \\card {\\powerset Y}$ where $\\powerset X$ denotes the power set of $X$."}
+{"_id": "11482", "title": "Reals are Isomorphic to Dedekind Cuts", "text": "Let $\\mathscr D$ be set of all Dedekind cuts of the totally ordered set $\\struct {\\Q, \\le}$. Define a mapping $f: \\R \\to \\mathscr D$ as: :$\\forall x \\in \\R: \\map f x = \\set {y \\in \\Q: y < x}$ Then $f$ is a bijection."}
+{"_id": "11484", "title": "Slope of Orthogonal Curves", "text": "Let $C_1$ and $C_2$ be curves in a cartesian plane. Let $C_1$ and $C_2$ intersect each other at $P$. Let the slope of $C_1$ and $C_2$ at $P$ be $m_1$ and $m_2$. Then $C_1$ and $C_2$ are orthogonal {{iff}}: :$m_1 = -\\dfrac 1 {m_2}$"}
+{"_id": "11485", "title": "Orthogonal Trajectories/Concentric Circles", "text": "Consider the one-parameter family of curves: :$(1): \\quad x^2 + y^2 = c$ Its family of orthogonal trajectories is given by the equation: :$y = c x$"}
+{"_id": "11486", "title": "Angle of Tangent to Radius in Polar Coordinates", "text": "Let $C$ be a curve embedded in a plane defined by polar coordinates. Let $P$ be the point at $\\polar {r, \\theta}$. Then the angle $\\psi$ made by the tangent to $C$ at $P$ with the radial coordinate is given by: :$\\tan \\psi = r \\dfrac {\\d \\theta} {\\d r}$"}
+{"_id": "11487", "title": "Orthogonal Trajectories/Circles Tangent to Y Axis", "text": "Consider the one-parameter family of curves: :$(1): \\quad x^2 + y^2 = 2 c x$ which describes the loci of circles tangent to the $y$-axis at the origin. Its family of orthogonal trajectories is given by the equation: :$x^2 + y^2 = 2 c y$ which describes the loci of circles tangent to the $x$-axis at the origin. :600px"}
+{"_id": "11491", "title": "Orthogonal Trajectories/Parabolas Tangent to X Axis", "text": "Consider the one-parameter family of curves of parabolas which are tangent to the $x$-axis at the origin: :$(1): \\quad y = c x^2$ Its family of orthogonal trajectories is given by the equation: :$x^2 + 2 y^2 = c$ :600px"}
+{"_id": "11493", "title": "Orthogonal Trajectories/Exponential Functions", "text": "Consider the one-parameter family of curves of graphs of the exponential function: :$(1): \\quad y = c e^x$ Its family of orthogonal trajectories is given by the equation: :$y^2 = -2 x + c$ :600px"}
+{"_id": "11494", "title": "Orthogonal Trajectories/Parabolas with Focus at Origin", "text": "Consider the one-parameter family of curves of parabolas whose focus is at the origin and whose axis is the $x$-axis: :$(1): \\quad y^2 = 4 c \\paren {x + c}$ Its family of orthogonal trajectories is given by the equation: :$y^2 = 4 c \\paren {x + c}$ :600px"}
+{"_id": "11500", "title": "Speed of Body under Free Fall from Height", "text": "Let an object $B$ be released above ground from a point near the Earth's surface and allowed to fall freely. Let $B$ fall a distance $s$. Then: :$v = \\sqrt {2 g s}$ where: :$v$ is the speed of $B$ after having fallen a distance $s$ :$g$ is the Acceleration Due to Gravity at the height through which $B$ falls. It is supposed that the distance $s$ is small enough that $g$ can be considered constant throughout."}
+{"_id": "11505", "title": "Approximate Motion of Pendulum", "text": "Consider a pendulum consisting of a bob whose mass is $m$, at the end of a rod of negligible mass of length $a$. Let the bob be pulled to one side so that the rod is at a small angle $\\alpha$ (less than about $10 \\degrees$ or $15 \\degrees$) from the vertical and then released. Let $T$ be the period of the pendulum, that is, the time through which the bob takes to travel from one end of its path to the other, and back again. Then: :$T = 2 \\pi \\sqrt {\\dfrac a g}$ where $g$ is the Acceleration Due to Gravity."}
+{"_id": "11506", "title": "Space is Separable iff Density not greater than Aleph Zero", "text": "Let $T$ be a topological space. Then: :$T$ is separable {{iff}} $d \\left({T}\\right) \\leq \\aleph_0$ where :$d \\left({T}\\right)$ denotes the density of $T$, :$\\aleph$ denotes the aleph mapping."}
+{"_id": "11508", "title": "Boundary of Empty Set is Empty", "text": "Let $T$ be a topological space. Then:  :$\\partial_T \\O = \\O$ where $\\partial_T \\O$ denotes the boundary in topology $T$ of $\\O$."}
+{"_id": "11513", "title": "Escape Velocity of Projectile fired Upwards", "text": "Let $P$ be a planet. Let $P$ have an Acceleration Due to Gravity at its surface of $g$. Let $P$ have a radius of $R$. Then the escape velocity of $P$ is given by: :$V = \\sqrt {2 g R}$"}
+{"_id": "11514", "title": "Union of Boundaries", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A, B$ be subsets of $S$. Then: :$\\partial A \\cup \\partial B = \\partial \\left({A \\cup B}\\right) \\cup \\partial \\left({A \\cap B}\\right) \\cup \\left({\\partial A \\cap \\partial B}\\right)$ where $\\partial A$ denotes the boundary of $A$."}
+{"_id": "11516", "title": "Length of Arch of Sine Function", "text": "The length of one arch of the sine function: :$y = \\sin x$ is given by: :$L = 2 \\sqrt 2 \\map E {\\dfrac {\\sqrt 2} 2}$ where $E$ denotes the incomplete elliptic integral of the second kind."}
+{"_id": "11517", "title": "Length of Lemniscate of Bernoulli", "text": "The total length of the lemniscate of Bernoulli given in polar coordinates as: :$r^2 = a^2 \\cos 2 \\theta$ is given by: {{begin-eqn}} {{eqn | l = L       | r = 4 a \\map F {\\sqrt 2, \\dfrac \\pi 4}       | c =  }} {{eqn | r = \\dfrac 1 {\\sqrt {2 \\pi} } \\paren {\\map \\Gamma {\\dfrac 1 4} }^2       | c =  }} {{end-eqn}} where $F$ denotes the incomplete elliptic integral of the first kind."}
+{"_id": "11518", "title": "Boundary of Union is Subset of Union of Boundaries", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A, B$ be subsets of $S$. Then: :$\\partial \\left({A \\cup B}\\right) \\subseteq \\partial A \\cup \\partial B$ where $\\partial A$ denotes the boundary of $A$."}
+{"_id": "11519", "title": "Boundary of Intersection is Subset of Union of Boundaries", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A, B$ be subsets of $S$. Then: :$\\partial \\paren {A \\cap B} \\subseteq \\partial A \\cup \\partial B$ where $\\partial A$ denotes the boundary of $A$."}
+{"_id": "11520", "title": "Complete Elliptic Integral of the First Kind as Power Series", "text": "The '''complete elliptic integral of the first kind''': :$\\displaystyle \\map K k = \\int_0^{\\pi / 2} \\frac {\\rd \\phi} {\\sqrt {1 - k^2 \\sin^2 \\phi} } = \\int_0^1 \\frac {\\rd v} {\\sqrt {\\paren {1 - v^2} \\paren {1 - k^2 v^2} } }$ can be expressed as the power series: {{begin-eqn}} {{eqn | l = \\map K k       | r = \\frac \\pi 2 \\sum_{i \\mathop \\ge 0} \\paren {\\prod_{j \\mathop = 1}^i \\frac {2 j - 1} {2 j} }^2 k^{2 i}       | c =  }} {{eqn | r = \\frac \\pi 2 \\paren {1 + \\paren {\\frac 1 2}^2 k^2 + \\paren {\\frac {1 \\cdot 3} {2 \\cdot 4} }^2 k^4 + \\paren {\\frac {1 \\cdot 3 \\cdot 5} {2 \\cdot 4 \\cdot 6} }^2 k^6 + \\cdots}       | c =  }} {{end-eqn}}"}
+{"_id": "11521", "title": "Complete Elliptic Integral of the Second Kind as Power Series", "text": "The '''complete elliptic integral of the second kind''': :$\\displaystyle \\map E k = \\int_0^{\\pi / 2} \\sqrt {1 - k^2 \\sin^2 \\phi} \\, \\rd \\phi = \\int_0^1 \\dfrac {\\sqrt {1 - k^2 v^2} } {\\sqrt {1 - v^2}} \\, \\rd v$ can be expressed as the power series: {{begin-eqn}} {{eqn | l = \\map E k       | r = \\frac \\pi 2 \\sum_{i \\mathop \\ge 0} \\paren {\\prod_{j \\mathop = 1}^i \\frac {2 j - 1} { 2 j} }^2 \\frac {k^{2 i} } {1 - 2 i}       | c =  }} {{eqn | r = \\frac \\pi 2 \\paren {1 - \\paren {\\frac 1 2}^2 k^2 - \\paren {\\frac {1 \\cdot 3} {2 \\cdot 4} }^2 \\frac {k^4} 3 - \\paren {\\frac {1 \\cdot 3 \\cdot 5} {2 \\cdot 4 \\cdot 6} }^2 \\frac {k^6} 5 - \\cdots}       | c =  }} {{end-eqn}}"}
+{"_id": "11523", "title": "Set is Countable iff Cardinality not greater Aleph Zero", "text": "Let $X$ be a set. Then: :$X$ is countable {{iff}} $\\size X \\le \\aleph_0$ where :$\\size X$ denotes the cardinality of $X$, :$\\aleph$ denotes the aleph mapping."}
+{"_id": "11524", "title": "Bernoulli's Theorem", "text": "Let the probability of the occurrence of an event be $p$. Let $n$ independent trials be made, with $k$ successes. Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\frac k n = p$"}
+{"_id": "11525", "title": "Time of Travel down Brachistochrone", "text": "Let a wire $AB$ be curved into the shape of a brachistochrone. Let $AB$ be embedded in a constant and uniform gravitational field where Acceleration Due to Gravity is $g$. Let a bead $P$ be released at $A$ to slide down without friction to $B$. Then the time taken for $P$ to slide from $A$ to $B$ is: :$T = \\pi \\sqrt {\\dfrac a g}$ where $a$ is the radius of the generating circle of the cycloid which forms $AB$."}
+{"_id": "11526", "title": "Time of Travel down Brachistochrone/Corollary", "text": "Let a bead $P$ be released from anywhere on the wire between $A$ and $B$ to slide down without friction to $B$. Then the time taken for $P$ to slide to $B$ is: :$T = \\pi \\sqrt{\\dfrac a g}$"}
+{"_id": "11528", "title": "Union of Set of Singletons", "text": "Let $S$ be a set. Let $T = \\set {\\set x: x \\in S}$ be the set of all singletons of elements of $S$. Then: :$\\ds \\bigcup T = S$ where $\\ds \\bigcup T$ denotes the union of $T$."}
+{"_id": "11530", "title": "T1 Space is T1/2 Space", "text": "Let $T$ be a $T_1$ topological space. Then $T$ is $T_{\\frac 1 2}$ space."}
+{"_id": "11532", "title": "Parabolas Inscribed in Shared Tangent Lines", "text": "Let the function $\\map f x = A x^2 + B x + C_1$ be a curve embedded in the Euclidean Plane. Let $\\map {y_1} x$ be the equation of the tangent line at $\\tuple {Q, \\map f Q}$ on $f$. Let $\\map {y_2} x$ be the equation of the tangent line at $\\tuple {-Q, \\map f {-Q} }$ on $f$. Then there exists another function $\\map g x$ also embedded in the Euclidean Plane defined as: :$\\map g x = -A x^2 + B x + C_2$. with: :tangent lines $\\map {y_3} x$ being the equation of the tangent line at $\\tuple {Q, \\map g Q}$ on $g$ and: :$\\map {y_4} x$ being the equation of the tangent line at $\\tuple {-Q, \\map g {-Q} }$ on $g$. so that the tangent lines $y_3$ and $y_4$ inscribe $\\map f x$ and the tangent lines $y_1$ and $y_2$ inscribe $\\map g x$."}
+{"_id": "11533", "title": "Characterization of T0 Space by Closures of Singletons", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Then :$T$ is a $T_0$ space {{iff}} ::$\\forall x, y \\in S: x \\ne y \\implies x \\notin \\left\\{{y}\\right\\}^- \\lor y \\notin \\left\\{{x}\\right\\}^-$ where $\\left\\{{y}\\right\\}^-$ denotes the closure of $\\left\\{{y}\\right\\}$."}
+{"_id": "11534", "title": "Characterization of T0 Space by Distinct Closures of Singletons", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Then :$T$ is a $T_0$ space {{iff}} ::$\\forall x, y \\in S: x \\ne y \\implies \\left\\{{x}\\right\\}^- \\ne \\left\\{{y}\\right\\}^-$ where $\\left\\{{y}\\right\\}^-$ denotes the closure of $\\left\\{{y}\\right\\}$."}
+{"_id": "11535", "title": "Characterization of T0 Space by Closed Sets", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Then :$T$ is a $T_0$ space {{iff}} ::for every points $x, y \\in S$ if $x \\ne y$ then :::there exists a closed subset $F$ of $S$ such that $x \\in F$ and $y \\notin F$ ::or :::there exists a closed subset $F$ of $S$ such that $x \\notin F$ and $y \\in F$"}
+{"_id": "11536", "title": "Focus of Ellipse from Major and Minor Axis", "text": "Let $K$ be an ellipse whose major axis is $2 a$ and whose minor axis is $2 b$. Let $c$ be the distance of the foci of $K$ from the center. Then: :$a^2 = b^2 + c^2$"}
+{"_id": "11537", "title": "Equidistance of Ellipse equals Major Axis", "text": "Let $K$ be an ellipse whose foci are $F_1$ and $F_2$. Let $P$ be an arbitrary point on $K$. Let $d$ be the constant distance such that: :$d_1 + d_2 = d$ where: : $d_1 = P F_1$ : $d_2 = P F_2$ Then $d$ is equal to the major axis of $K$."}
+{"_id": "11538", "title": "Equidistance of Hyperbola equals Transverse Axis", "text": "Let $K$ be an hyperbola whose foci are $F_1$ and $F_2$. Let $P$ be an arbitrary point on $K$. Let $d$ be the constant distance such that: :$\\left\\lvert{d_1 - d_2}\\right\\rvert = d$ where: : $d_1 = P F_1$ : $d_2 = P F_2$ Then $d$ is equal to the transverse axis of $K$."}
+{"_id": "11539", "title": "Equivalence of Definitions of Ellipse", "text": "The following definitions of an ellipse are equivalent:"}
+{"_id": "11541", "title": "Equation of Hyperbola in Reduced Form", "text": "Let $K$ be an hyperbola aligned in a cartesian plane in reduced form. Let: :the transverse axis of $K$ have length $2 a$ :the conjugate axis of $K$ have length $2 b$."}
+{"_id": "11542", "title": "Equation of Hyperbola in Reduced Form/Cartesian Frame", "text": "The equation of $K$ is: :$\\dfrac {x^2} {a^2} - \\dfrac {y^2} {b^2} = 1$"}
+{"_id": "11543", "title": "Focus of Hyperbola from Transverse and Conjugate Axis", "text": "Let $K$ be a hyperbola whose transverse axis is $2 a$ and whose conjugate axis is $2 b$. Let $c$ be the distance of the foci of $K$ from the center. Then: :$c^2 = a^2 + b^2$"}
+{"_id": "11544", "title": "Equation of Hyperbola in Reduced Form/Cartesian Frame/Parametric Form", "text": "The equation of $K$ in parametric form is: :$x = a \\cosh \\theta, y = b \\sinh \\theta$"}
+{"_id": "11545", "title": "Set of Condensation Points is Subset of Derivative", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A$ be a subset of $S$. Then: :$A^0 \\subseteq A'$ where :$A^0$ denotes the set of condensation points of $A$ :$A'$ denotes the derivative of $A$"}
+{"_id": "11546", "title": "Closure of Set of Condensation Points equals Itself", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A$ be a subset of $S$. Then: :$\\left({A^0}\\right)^- = A^0$ where :$A^0$ denotes the set of condensation points of $A$ :$A^-$ denotes the closure of $A$"}
+{"_id": "11547", "title": "First Order ODE/(x + y) dx = (x - y) dy", "text": "is a homogeneous differential equation with general solution: :$\\arctan \\dfrac y x = \\ln \\sqrt {x^2 + y^2} + C$"}
+{"_id": "11550", "title": "Set of Condensation Points is Monotone", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A, B$ be subsets of $S$. Then: :$A \\subseteq B \\implies {A^0} \\subseteq B^0$ where :$A^0$ denotes the set of condensation points of $A$"}
+{"_id": "11551", "title": "First Order ODE/x^2 y' = 3 (x^2 + y^2) arctan (y over x) + x y", "text": "is a homogeneous differential equation with solution: :$y = x \\tan C x^3$"}
+{"_id": "11552", "title": "First Order ODE/x sine (y over x) y' = y sine (y over x) + x", "text": "is a homogeneous differential equation with solution: :$\\cos \\dfrac y x + \\ln C x = 0$"}
+{"_id": "11554", "title": "First Order ODE in form y' = f (a x + b y + c)", "text": "The first order ODE: :$\\dfrac {\\mathrm d y} {\\mathrm d x} = f \\left({a x + b y + c}\\right)$ can be solved by substituting: :$z := a x + b y + c$ to obtain: :$\\displaystyle x = \\int \\frac {\\mathrm d z} {b f \\left({z}\\right) + a}$"}
+{"_id": "11556", "title": "First Order ODE/y' = sin^2 (x - y + 1)", "text": "The first order ODE: :$\\dfrac {\\d y} {\\d x} = \\map {\\sin^2} {x - y + 1}^2$ has the general solution: :$\\map \\tan {x - y + 1} = x + C$"}
+{"_id": "11557", "title": "First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f))", "text": "The first order ODE: :$\\dfrac {\\d y} {\\d x} = \\map F {\\dfrac {a x + b y + c} {d x + e y + f} }$ such that: :$ a e \\ne b d$ can be solved by substituting: :$x := z - h$ :$y := w - k$ where: :$h = \\dfrac {c e - b f} {a e - b d}$ :$k = \\dfrac {a f - c d} {a e - b d}$ to obtain: :$\\dfrac {\\d w} {\\d z} = \\map F {\\dfrac {a z + b w} {d z + e w} }$ which can be solved by the technique of Solution to Homogeneous Differential Equation."}
+{"_id": "11559", "title": "First Order ODE/(x + y + 4) over (x - y - 6)", "text": "The first order ODE: :$(1): \\quad \\dfrac {\\d y} {\\d x} = \\dfrac {x + y + 4} {x - y - 6}$ has the general solution: :$\\map \\arctan {\\dfrac {y + 5} {x - 1} } = \\ln \\sqrt {\\paren {x - 1}^2 + \\paren {y + 5}^2} + C$"}
+{"_id": "11560", "title": "First Order ODE/(x + y + 4) over (x + y - 6)", "text": "The first order ODE: :$(1): \\quad \\dfrac {\\d y} {\\d x} = \\dfrac {x + y + 4} {x + y - 6}$ has the general solution: :$y - x = 5 \\, \\map \\ln {x + y - 1} + C$"}
+{"_id": "11561", "title": "First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f)) where a e = b d/Example", "text": "=== $\\dfrac {\\d y} {\\d x} = \\dfrac {x + y + 4} {x + y - 6}$ === {{:First Order ODE/(x + y + 4) over (x + y - 6)}}"}
+{"_id": "11562", "title": "First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f))/Example", "text": "=== $\\dfrac {\\d y} {\\d x} = \\dfrac {x + y + 4} {x - y - 6}$ === {{:First Order ODE/(x + y + 4) over (x - y - 6)}}"}
+{"_id": "11564", "title": "First Order ODE/(x + (2 over y)) dy + y dx = 0", "text": "is an exact differential equation with solution: :$x y + 2 \\ln y = C$"}
+{"_id": "11565", "title": "First Order ODE/(y - x^3) dx + (x + y^3) dy = 0", "text": "is an exact differential equation with solution: :$4 x y - x^4 + y^4 = C$"}
+{"_id": "11566", "title": "First Order ODE/(y + y cosine x y) dx + (x + x cosine x y) dy = 0", "text": "is an exact differential equation with solution: :$x y + \\sin x y = C$"}
+{"_id": "11567", "title": "First Order ODE/(sine x sine y - x e^y) dy = (e^y + cosine x cosine y) dx", "text": "is an exact differential equation with solution: :$\\sin x \\cos y + x e^y = C$"}
+{"_id": "11569", "title": "First Order ODE/(2 x y^3 + y cosine x) dx + (3 x^2 y^2 + sine x) dy", "text": "is an exact differential equation with solution: :$x^2 y^3 + y \\sin x = C$"}
+{"_id": "11571", "title": "Heart Curve", "text": "The following equations define curves that are more or less heart shaped:"}
+{"_id": "11575", "title": "Limit Point of Subset is Limit Point of Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A, B$ be subset of $S$ such that :$A \\subseteq B$ Let $x$ be a point of $S$. Then: :if $x$ is limit point of $A$, then $x$ is limit point of $B$."}
+{"_id": "11580", "title": "Integrating Factor for First Order ODE/Preliminary Work", "text": "Let the first order ordinary differential equation: :$(1): \\quad \\map M {x, y} + \\map N {x, y} \\dfrac {\\d y} {\\d x} = 0$ be non-homogeneous and not exact. Let $\\map \\mu x$ be an integrating factor for $(1)$. Let: :$\\map P {x, y} := \\dfrac {\\partial M} {\\partial y} - \\dfrac {\\partial N} {\\partial x}$ Then: :$\\dfrac 1 \\mu = \\dfrac {\\map P {x, y} } {N \\dfrac {\\partial \\mu} {\\partial x} - M  \\dfrac {\\partial \\mu} {\\partial y} }$"}
+{"_id": "11581", "title": "Integrating Factor for First Order ODE/Function of One Variable", "text": "Suppose that: :$g \\left({x}\\right) = \\dfrac {\\dfrac {\\partial M} {\\partial y} - \\dfrac {\\partial N} {\\partial x} } {N \\left({x, y}\\right)}$ is a function of $x$ only. Then: :$\\mu \\left({x}\\right) = e^{\\int g \\left({x}\\right) \\mathrm d x}$ is an integrating factor for $(1)$. Similarly, suppose that: :$h \\left({y}\\right) = \\dfrac {\\dfrac {\\partial M} {\\partial y} - \\dfrac {\\partial N} {\\partial x} } {M \\left({x, y}\\right)}$ is a function of $y$ only. Then: :$\\mu \\left({y}\\right) = e^{\\int -h \\left({y}\\right) \\mathrm d y}$ is an integrating factor for $(1)$."}
+{"_id": "11582", "title": "Integrating Factor for First Order ODE/Function of Sum of Variables", "text": "Suppose that: :$g \\left({z}\\right) = \\dfrac {\\dfrac {\\partial M} {\\partial y} - \\dfrac {\\partial N} {\\partial x} } {N \\left({x, y}\\right) - M \\left({x, y}\\right)}$ is a function of $z$, where $z = x + y$. Then: :$\\mu \\left({x + y}\\right) = \\mu \\left({z}\\right) = e^{\\int g \\left({z}\\right) \\mathrm d z}$ is an integrating factor for $(1)$."}
+{"_id": "11583", "title": "Integrating Factor for First Order ODE/Function of Product of Variables", "text": ":$\\map g z = \\dfrac {\\dfrac {\\partial M} {\\partial y} - \\dfrac {\\partial N} {\\partial x}} {N y - M x}$ is a function of $z$, where $z = x y$. Then: :$\\map \\mu {x y} = \\map \\mu z = e^{\\int \\map g z \\d z}$ is an integrating factor for $(1)$."}
+{"_id": "11584", "title": "Set of Condensation Points of Union is Union of Sets of Condensation Points", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A, B$ be subsets of $S$. Then: :$\\left({A \\cup B}\\right)^0 = A^0 \\cup B^0$"}
+{"_id": "11585", "title": "Set of Condensation Points of Union is Union of Sets of Condensation Points/Lemma", "text": "Let $x$ be a point of $S$. Then: :if $x$ is condensation point of $A \\cup B$, :then $x$ is condensation point of $A$ or $x$ is condensation point of $B$."}
+{"_id": "11586", "title": "Integrating Factor for First Order ODE/Conclusion", "text": "Let the first order ordinary differential equation: :$(1): \\quad \\map M {x, y} + \\map N {x, y} \\dfrac {\\d y} {\\d x} = 0$ be non-homogeneous and not exact. Let $\\map \\mu {x, y}$be an integrating factor for $(1)$. If one of these is the case: :$\\mu$ is a function of $x$ only :$\\mu$ is a function of $y$ only :$\\mu$ is a function of $x + y$ :$\\mu$ is a function of $x y$ then: :$\\mu = e^{\\int \\map f w \\rd w}$ where $w$ depends on the nature of $\\mu$."}
+{"_id": "11587", "title": "First Order ODE/y dx + (x^2 y - x) dy = 0", "text": "The first order ODE: :$(1): \\quad y \\rd x + \\paren {x^2 y - x} \\rd y = 0$ has the general solution: :$\\dfrac {y^2} 2 - \\dfrac y x = C$"}
+{"_id": "11588", "title": "Set of Condensation Points of Countable Set is Empty", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A$ be a subset of $S$. Then: :if $A$ is countable, :then $A^0 = \\varnothing$."}
+{"_id": "11589", "title": "Set of Condensation Points of Countable Set is Empty/Lemma", "text": ":if $A$ is countable, :then there exists no point $x$ of $S$ such that $x$ is a condensation point of $A$."}
+{"_id": "11590", "title": "First Order ODE/(3 x^2 - y^2) dy - 2 x y dx = 0", "text": "The first order ODE: :$(1): \\quad \\paren {3 x^2 - y^2} \\rd y - 2 x y \\rd x = 0$ has the general solution: :$\\dfrac 1 y - \\dfrac {x^2} {y^3} = C$"}
+{"_id": "11592", "title": "First Order ODE/y dx + x dy + 3 x^3 y^4 dy", "text": "The first order ODE: :$(1): \\quad y \\rd x + x \\rd y + 3 x^3 y^4 \\rd y = 0$ has the general solution: :$-\\dfrac 1 {2 x^2 y^2} + \\dfrac {3 y^2} 2 = C$"}
+{"_id": "11593", "title": "Curved Mirror producing Parallel Rays is Paraboloid", "text": "Let $M$ be a curved mirror embedded in a real cartesian $3$- space. Let there be a source of light at the origin. Let $M$ reflect the light in a beam parallel to the $x$-axis. Then $M$ is the solid of revolution produced by rotating about the $x$-axis the parabola whose equation is: :$y^2 = 2 c x + c^2$"}
+{"_id": "11595", "title": "Quotient Rule for Differentials/Formulation 1", "text": ":$\\map \\d {\\dfrac y x} = \\dfrac {x \\rd y - y \\rd x} {x^2}$"}
+{"_id": "11598", "title": "Product Rule for Differentials", "text": ":$\\map \\d {x y} = x \\rd y + y \\rd x$"}
+{"_id": "11599", "title": "Differential of Sum of Squares", "text": ":$\\map \\d {x^2 + y^2} = 2 \\paren {x \\map \\rd x + y \\map \\rd y}$"}
+{"_id": "11600", "title": "Differential of Arctangent of Quotient", "text": ":$\\mathrm d \\left({\\arctan \\dfrac x y}\\right) = \\dfrac{y \\, \\mathrm d x - x \\, \\mathrm d y} {x^2 + y^2}$"}
+{"_id": "11601", "title": "Differential of Logarithm of Quotient", "text": ":$\\map \\d {\\ln \\dfrac x y} = \\dfrac {y \\rd x - x \\rd y} {x y}$"}
+{"_id": "11602", "title": "First Order ODE/y dx + (x^2 y - x) dy = 0/Proof 1", "text": "{{:First Order ODE/y dx + (x^2 y - x) dy = 0}} This can also be presented in the form: :$\\dfrac {\\d y} {\\d x} + \\dfrac y {x^2 y - x}$"}
+{"_id": "11604", "title": "First Order ODE/y dx - x dy = x y^3 dy", "text": "The first order ODE: :$(1): \\quad y rd x - x \\rd y = x y^3 \\rd y$ has the general solution: :$\\ln \\dfrac x y = \\dfrac {y^3} 3 + C$"}
+{"_id": "11606", "title": "First Order ODE/(x + y) dx = (x - y) dy/Proof 1", "text": "The first order ordinary differential equation: :$(1): \\quad \\paren {x + y} \\rd x = \\paren {x - y} \\rd y$ is a homogeneous differential equation with solution: :$\\arctan \\dfrac y x = \\ln \\sqrt{x^2 + y^2} + C$"}
+{"_id": "11608", "title": "First Order ODE/x dy = (y + x^2 + 9 y^2) dx", "text": "The first order ODE: :$(1): \\quad x \\rd y = \\paren {y + x^2 + 9 y^2} \\rd x$ has the general solution: :$\\map \\arctan {\\dfrac {3 y} x} = 3 x + C$"}
+{"_id": "11613", "title": "Linear First Order ODE/y' + (y over x) = 3 x", "text": "The linear first order ODE: :$\\dfrac {\\d y} {\\d x} + \\dfrac y x = 3 x$ has the general solution: :$x y = x^3 + C$ or: :$y = x^2 + \\dfrac C x$"}
+{"_id": "11614", "title": "Linear First Order ODE/x y' - 3 y = x^4", "text": "The linear first order ODE: :$(1): \\quad x \\dfrac {\\d y} {\\d x} - 3y = x^4$ has the general solution: :$y = x^4 + \\dfrac C {x^3}$"}
+{"_id": "11615", "title": "Linear First Order ODE/y' + y = 1 over (1 + exp 2 x)", "text": "The linear first order ODE: :$(1): \\quad y' + y = \\dfrac 1 {1 + e^{2 x} }$ has the general solution: :$y = e^{-x} \\map \\arctan {e^x} + C e^{-x}$"}
+{"_id": "11616", "title": "Linear First Order ODE/(1 + x^2) dy + 2 x y dx = cotangent x dx", "text": "The linear first order ODE: :$(1): \\quad \\paren {1 + x^2} \\rd y + 2 x y \\rd x = \\cot x \\rd x$ has the general solution: :$y = \\dfrac {\\map \\ln {\\sin x} } {1 + x^2} + \\dfrac C {1 + x^2}$"}
+{"_id": "11619", "title": "Linear First Order ODE/(2 y - x^3) dx = x dy", "text": "The linear first order ODE: :$(1): \\quad \\paren {2 y - x^3} \\rd x = x \\rd y$ has the general solution: :$y = -x^3 + C x^2$"}
+{"_id": "11620", "title": "Solution to Bernoulli's Equation", "text": "'''Bernoulli's equation''': :$(1): \\quad \\dfrac {\\d y} {\\d x} + \\map P x y = \\map Q x y^n$ where: :$n \\ne 0, n \\ne 1$ has the general solution: :$\\displaystyle \\frac {\\map \\mu x} {y^{n - 1} } = \\paren {1 - n} \\int \\map Q x \\map \\mu x \\rd x + C$ where: :$\\map \\mu x = e^{\\paren {1 - n} \\int \\map P x \\rd x}$"}
+{"_id": "11621", "title": "Bernoulli's Equation/x y' + y = x^4 y^3", "text": "The first order ODE: :$(1): \\quad x y' + y = x^4 y^3$ has the general solution: :$\\dfrac 1 {y^2} = - x^4 + C x^2$"}
+{"_id": "11625", "title": "Topology Defined by Neighborhood System", "text": "Let $S$ be a set. Let $\\family {\\NN_x}_{x \\mathop \\in S}$ be an indexed family where $\\NN_x$ is non-empty set of subsets of $S$. Assume that :$(\\text N 1): \\quad \\forall x \\in S, U \\in \\NN_x: x \\in U$ :$(\\text N 2): \\quad \\forall x \\in S, U \\in \\NN_x, y \\in U:\\exists V \\in \\NN_y: V \\subseteq U$ :$(\\text N 3): \\quad \\forall x \\in S, U_1, U_2 \\in \\NN_x: \\exists U \\in \\NN_x: U \\subseteq U_1 \\cap U_2$ Then $T = \\struct {S, \\tau}$ is a topological space where: :$\\tau = \\displaystyle \\set {\\bigcup \\GG: \\GG \\subseteq \\bigcup_{x \\mathop \\in S} \\NN_x}$ Moreover, $\\family {\\NN_x}_{x \\mathop \\in S}$ is a neighborhood system of $T$."}
+{"_id": "11626", "title": "Orthogonal Trajectories/x + C exp -x", "text": "Consider the one-parameter family of curves: :$(1): \\quad y = x + C e^{-x}$ Its family of orthogonal trajectories is given by the equation: :$x = y - 2 + C e^{-y}$ :600px"}
+{"_id": "11628", "title": "Solution of Second Order Differential Equation with Missing Dependent Variable", "text": "Let $\\map f {x, y', y''} = 0$ be a second order ordinary differential equation in which the dependent variable $y$ is not explicitly present. Then $f$ can be reduced to a first order ordinary differential equation, whose solution can be determined."}
+{"_id": "11630", "title": "Niemytzki Plane is Topology", "text": "Niemytzki plane is a topological space."}
+{"_id": "11631", "title": "Solution of Second Order Differential Equation with Missing Independent Variable", "text": "Let $\\map g {y, \\dfrac {\\d y} {\\d x}, \\dfrac {\\d^2 y} {\\d x^2} } = 0$ be a second order ordinary differential equation in which the independent variable $x$ is not explicitly present. Then $g$ can be reduced to a first order ordinary differential equation, whose solution can be determined."}
+{"_id": "11632", "title": "Linear Second Order ODE/y'' + k^2 y = 0", "text": "The second order ODE: :$(1): \\quad y'' + k^2 y = 0$ has the general solution: :$y = A \\, \\map \\sin {k x + B}$ or can be expressed as: :$y = C_1 \\sin k x + C_2 \\cos k x$"}
+{"_id": "11633", "title": "Second Order ODE/y y'' + (y')^2 = 0", "text": "The second order ODE: :$(1): \\quad y y'' + \\paren {y'}^2 = 0$ has the general solution: :$y^2 = C_1 x + C_2$"}
+{"_id": "11634", "title": "Second Order ODE/x y'' = y' + (y')^3", "text": "The second order ODE: :$(1): \\quad x y'' = y' + \\paren {y'}^3$ has the general solution: :$x^2 + \\paren {y - C_2}^2 = C_1^2$"}
+{"_id": "11635", "title": "Linear Second Order ODE/y'' - k^2 y = 0", "text": "The second order ODE: :$(1): \\quad y'' - k^2 y = 0$ has the general solution: :$y = C_1 e^{k x} + C_2 e^{-k x}$"}
+{"_id": "11636", "title": "Second Order ODE/x^2 y'' = 2 x y' + (y')^2", "text": "The second order ODE: :$x^2 y'' = 2 x y' + \\paren {y'}^2$ has the general solution: :$y = -\\dfrac {x^2} 2 - C_1 x - {C_1}^2 \\, \\map \\ln {x - C_1} + C_2$"}
+{"_id": "11638", "title": "Second Order ODE/y y'' = y^2 y' + (y')^2", "text": "The second order ODE: :$(1): \\quad y y'' = y^2 y' + \\paren {y'}^2$ subject to the initial conditions: :$y = -\\dfrac 1 2$ and $y' = 1$ when $x = 0$ has the particular solution: :$2 y - 3 = 8 y \\, \\map \\exp {\\dfrac {3 x} 2}$"}
+{"_id": "11639", "title": "Second Order ODE/y'' = 1 + (y')^2", "text": "The second order ODE: :$(1): \\quad y'' = 1 + \\paren {y'}^2$ has the general solution: :$y = \\map {\\ln \\sec} {x + C_1} + C_2$"}
+{"_id": "11643", "title": "Velocity of Bead on Brachistochrone", "text": "Consider a wire bent into the shape of an arc of a cycloid $C$ and inverted so that its cusps are uppermost and on the same horizontal line. Let $C$ be defined by Equation of Cycloid embedded in a cartesian plane: :$x = a \\left({\\theta - \\sin \\theta}\\right)$ :$y = -a \\left({1 - \\cos \\theta}\\right)$ Let a bead $B$ be released from some point on the wire. Let $B$ slide without friction under the influence of a constant gravitational field exerting an acceleration $g$. Let $s_0$ be the arc length along the cycloid. Let $s$ be the arc length along the cycloid at any subsequent point in time. Then the speed $v$ of $B$ relative to $C$ is defined by the equation: :$4 a v^2 = g \\left({ {s_0}^2 - s^2}\\right)$"}
+{"_id": "11644", "title": "Open Ball is Subset of Open Ball", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $x, y$ be points of $A$. Then: : $\\epsilon-\\delta \\ge \\map d {x, y} \\implies \\map {B_\\delta} y \\subseteq \\map {B_\\epsilon} x$ where $\\map {B_\\epsilon} x$ denotes the open $\\epsilon$-ball in $M = \\struct {A, d}$."}
+{"_id": "11645", "title": "Pursuit Curve of Boat in River", "text": "Consider a straight river $R$ whose parallel banks are aligned with the $y$-axis and the line $x = c$ of a cartesian plane. Let the current of $R$ have a constant and uniform speed $a$ in the negative $y$ direction. Let a boat $B$ be launched from the point $\\left({c, 0}\\right)$ and headed directly towards the origin with speed $b$ relative to the water. The path of $B$ is defined by the equation: :$c^k \\left({y + \\sqrt {x^2 + x^2} }\\right) = x^{k + 1}$"}
+{"_id": "11646", "title": "Current in Electric Circuit/L, R, C in Series", "text": "Consider the electrical circuit $K$ consisting of: : a resistance $R$ : an inductance $L$ : a capacitance $C$ in series with a source of electromotive force $E$ which is a function of time $t$. :File:CircuitRLCseries.png The electric current $I$ in $K$ is given by the linear second order ODE: :$L \\dfrac {\\mathrm d^2 I} {\\mathrm d t^2} + R \\dfrac {\\mathrm d I} {\\mathrm d t} + \\dfrac 1 C I = \\dfrac {\\mathrm d E} {\\mathrm d t}$"}
+{"_id": "11647", "title": "Electric Charge in Electric Circuit/L, R, C in Series", "text": "Consider the electrical circuit $K$ consisting of: : a resistance $R$ : an inductance $L$ : a capacitance $C$ in series with a source of electromotive force $E$ which is a function of time $t$. :File:CircuitRLCseries.png The electric charge $Q$ in $K$ is given by the linear second order ODE: :$L \\dfrac {\\mathrm d^2 Q} {\\mathrm d t^2} + R \\dfrac {\\mathrm d Q} {\\mathrm d t} + \\dfrac 1 C Q = E$"}
+{"_id": "11648", "title": "Current in Electric Circuit/L, R in Series", "text": "Consider the electrical circuit $K$ consisting of: :a resistance $R$ :an inductance $L$ in series with a source of electromotive force $E$ which is a function of time $t$. :File:CircuitRLseries.png The electric current $I$ in $K$ is given by the linear first order ODE: :$L \\dfrac {\\d I} {\\d t} + R I = E$"}
+{"_id": "11649", "title": "Current in Electric Circuit/L, R in Series/Constant EMF at t = 0", "text": "Let the electric current flowing in $K$ at time $t = 0$ be $I_0$. Let a constant EMF $E_0$ be imposed upon $K$ at time $t = 0$. The electric current $I$ in $K$ is given by the equation: :$I = \\dfrac {E_0} R + \\paren {I_0 - \\dfrac {E_0} R} e^{-R t / L}$"}
+{"_id": "11650", "title": "Current in Electric Circuit/L, R in Series/Constant EMF at t = 0/Corollary 1", "text": "Let the electric current flowing in $K$ at time $t = 0$ be $I_0$. Let a constant EMF $E_0$ be imposed upon $K$ at time $t = 0$. After a sufficiently long time, the electric current $I$ in $K$ is given by the equation: :$E_0  = R I$"}
+{"_id": "11651", "title": "Current in Electric Circuit/L, R in Series/Constant EMF at t = 0/Corollary 2", "text": "Let the electric current flowing in $K$ at time $t = 0$ be $0$. Let a constant EMF $E_0$ be imposed upon $K$ at time $t = 0$. The electric current $I$ in $K$ is given by the equation: :$I = \\dfrac {E_0} R \\paren {1 - e^{-R t / L} }$"}
+{"_id": "11652", "title": "Current in Electric Circuit/L, R in Series/Constant EMF at t = 0/Corollary 3", "text": "Let the electric current flowing in $K$ at time $t = 0$ be $I_0$. Let  EMF imposed upon $K$ be zero. The electric current $I$ in $K$ is given by the equation: :$I = I_0 e^{-R t / L}$"}
+{"_id": "11655", "title": "Current in Electric Circuit/L, R in Series/Condition for Ohm's Law", "text": "Ohm's Law is satisfied by $K$ whenever the current $I$ is at a maximum or a minimum."}
+{"_id": "11656", "title": "Current in Electric Circuit/L, R in Series/Minimum Current implies Increasing EMF", "text": "Let the current $I$ be at a minimum. Then the EMF $E$ is increasing."}
+{"_id": "11657", "title": "Current in Electric Circuit/L, R in Series/Maximum Current implies Decreasing EMF", "text": "Let the current $I$ be at a maximum. Then the EMF $E$ is decreasing."}
+{"_id": "11658", "title": "Second Order ODE/y y'' = (y')^2", "text": "The second order ODE: :$(1): \\quad y y'' = \\paren {y'}^2$ has the general solution: :$y = C_2 e^{C_1 x}$"}
+{"_id": "11659", "title": "First Order ODE/(1 - x y) y' = y^2", "text": "The first order ODE: :$(1): \\quad \\paren {1 - x y} y' = y^2$ has the general solution: :$x y = \\ln y + C$"}
+{"_id": "11660", "title": "First Order ODE/(2 x + 3 y + 1) dx + (2 y - 3 x + 5) dy = 0", "text": "The first order ODE: :$(1): \\quad \\paren {2 x + 3 y + 1} \\rd x + \\paren {2 y - 3 x + 5} \\rd y = 0$ has the general solution: :$3 \\, \\map \\arctan {\\dfrac {y + 1} {x - 1} } = \\map \\ln {\\paren {y + 1}^2 + \\paren {x - 1}^2} + C$"}
+{"_id": "11661", "title": "First Order ODE/x y' = Root of (x^2 + y^2)", "text": "is a homogeneous differential equation with solution: :$3 x^2 \\ln x = y \\sqrt {x^2 + y^2} + x^2 \\, \\map \\ln {y + \\sqrt {x^2 + y^2} } + y^2 + C x^2$"}
+{"_id": "11662", "title": "Bernoulli's Equation/y^2 dx = (x^3 - x y) dy", "text": "The first order ODE: :$(1): \\quad y^2 \\rd x = \\paren {x^3 - x y} \\rd y$ has the general solution: :$3 y = 2 x^2 + C x^2 y^2$"}
+{"_id": "11663", "title": "First Order ODE/(x^2 y^3 + y) dx = (x^3 y^2 - x) dy", "text": "The first order ODE: :$(1): \\quad \\paren {x^2 y^3 + y} \\rd x = \\paren {x^3 y^2 - x} \\rd y$ has the general solution: :$-\\dfrac 1 {2 x^2 y^2} = \\ln \\dfrac y x + C$"}
+{"_id": "11664", "title": "Second Order ODE/y y'' + (y')^2 - 2 y y' = 0", "text": "The second order ODE: :$y y'' + \\paren {y'}^2 - 2 y y' = 0$ has the general solution: :$y^2 = C_2 e^{2 x} + C_1$"}
+{"_id": "11665", "title": "Rationals are Everywhere Dense in Sorgenfrey Line", "text": "$\\Q$ is everywhere dense in the Sorgenfrey line."}
+{"_id": "11666", "title": "Linear First Order ODE/x dy + y dx = x cosine x dx", "text": "The linear first order ODE: :$x \\rd y + y \\rd x = x \\cos x \\rd x$ has the general solution: :$x y = x \\sin x + \\cos x + C$"}
+{"_id": "11667", "title": "First Order ODE/x y dy = x^2 dy + y^2 dx", "text": "The first order ODE: :$(1): \\quad x y \\rd y = x^2 \\rd y + y^2 \\rd x$ has the general solution: :$y = x \\ln y + C x$"}
+{"_id": "11669", "title": "Second Order ODE/y'' + 2 x (y')^2 = 0", "text": "The second order ODE: :$(1): \\quad y'' + 2 x \\paren {y'}^2 = 0$ has the general solution: :$C_1 \\map \\arctan {C_1 x} = y + C_2$"}
+{"_id": "11670", "title": "First Order ODE/(y over x^2) dx + (y - 1 over x) dy = 0", "text": "The first order ordinary differential equation: :$(1): \\quad \\dfrac y {x^2} \\rd x + \\paren {y - \\dfrac 1 x} \\rd y = 0$ is an exact differential equation with solution: :$\\dfrac {y^2} 2 - \\dfrac y x = C$"}
+{"_id": "11671", "title": "Focal Property of Parabola", "text": ":500px"}
+{"_id": "11672", "title": "First Order ODE/(3 x^2 over y^4 - 1 over y^2) dy - 2 x over y^3 dx = 0", "text": "The first order ODE: :$(1): \\quad \\paren {\\dfrac {3 x^2} {y^4} - \\dfrac 1 {y^2} } \\rd y - \\dfrac {2 x} {y^3} \\rd x = 0$ is an exact differential equation with general solution: :$\\dfrac 1 y - \\dfrac {x^2} {y^3} = C$"}
+{"_id": "11673", "title": "First Order ODE/(y - 1 over x) dx + (x - y) dy = 0", "text": "The first order ODE: :$(1): \\quad \\paren {y - \\dfrac 1 x} \\rd x + \\paren {x - y} \\rd y = 0$ is an exact differential equation with solution: :$x y - \\ln x - \\dfrac {y^2} 2 + C$"}
+{"_id": "11674", "title": "First Order ODE/1 over x^3 y^2 dx + (1 over x^2 y^3 + 3 y) dy = 0", "text": "The first order ODE: :$(1): \\quad \\dfrac 1 {x^3 y^2} \\rd x + \\paren {\\dfrac 1 {x^2 y^3} + 3 y} \\rd y = 0$ is an exact differential equation with solution: :$-\\dfrac 1 {2 x^2 y^2} + \\dfrac {3 y^2} 2 = C$"}
+{"_id": "11675", "title": "Linear First Order ODE/y' - (y over x) = 3 x", "text": "The linear first order ODE: :$(1): \\quad \\dfrac {\\d y} {\\d x} - \\dfrac y x = 3 x$ has the general solution: :$\\dfrac y x = 3 x + C$ or: :$y = 3 x^2 + C x$"}
+{"_id": "11676", "title": "Characterization of Closure by Basis", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $\\mathcal B \\subseteq \\tau$ be a basis. Let $A$ be a subset of $T$. Let $x$ be a point of $T$. Then $x \\in A^-$ {{iff}}: :for every $U \\in \\mathcal B$: ::if $x \\in U$ then $A \\cap U \\ne \\varnothing$ where: :$A^-$ denotes the closure of $A$"}
+{"_id": "11677", "title": "Sorgenfrey Line is Separable", "text": "The Sorgenfrey line is separable."}
+{"_id": "11678", "title": "First Order ODE/y dx + x dy = 0", "text": "The first order ODE: :$(1): \\quad y \\rd x + x \\rd y = 0$ has the general solution: :$x y = C$"}
+{"_id": "11679", "title": "First Order ODE/y dy = k dx", "text": "Let $k \\in \\R$ be a real number. The first order ODE: :$y \\, \\dfrac {\\d y} {\\d x} = k$ has the general solution: :$y^2 = 2 k x + C$"}
+{"_id": "11680", "title": "First Order ODE/x dy = (y + y^3) dx", "text": "The first order ODE: :$(1): \\quad x \\dfrac {\\d y} {\\d x} = y + y^3$ has the general solution: :$y = \\dfrac x {\\sqrt {C^2 - x^2} }$"}
+{"_id": "11681", "title": "First Order ODE/y dy = k x dx", "text": "Let $k \\in \\R$ be a real number. The first order ODE: :$y \\rd y = k x \\rd x$ has the general solution: :$y^2 = k x^2 + C$"}
+{"_id": "11682", "title": "Primitive of Reciprocal of Root of x squared plus k", "text": "Let $k \\in \\R$. Then: :$\\displaystyle \\int \\frac {\\d x} {\\sqrt {\\size {x^2 + k} } } = \\map \\ln {x + \\sqrt {\\size {x^2 + k} } } + C$"}
+{"_id": "11683", "title": "Bernoulli's Equation/x^2 dy = (2 x y + y^2) dx", "text": "The first order ODE: :$(1): \\quad x^2 \\rd y = \\paren {2 x y + y^2} \\rd x$ has the general solution: :$y = - \\dfrac {x^2} {x + C}$"}
+{"_id": "11684", "title": "Bernoulli's Equation/2 x y dx + (x^2 + 2 y) dy = 0", "text": "The first order ODE: :$(1): \\quad 2 x y \\rd x + \\paren {x^2 + 2 y} \\rd y = 0$ has the solution: :$y \\paren {x^2 + y} = C$"}
+{"_id": "11685", "title": "Linear First Order ODE/y' - (y over x) = k x", "text": "Let $k \\in \\R$ be a real number. The linear first order ODE: :$(1): \\quad \\dfrac {\\d y} {\\d x} - \\dfrac y x = k x$ has the general solution: :$\\dfrac y x = k x + C$ or: :$y = k x^2 + C x$"}
+{"_id": "11686", "title": "First Order ODE/x dy = k y dx", "text": "The first order ODE: :$(1): \\quad x \\rd y = k y \\rd x$ has the general solution: :$y = C x^k$"}
+{"_id": "11687", "title": "First Order ODE/dy = k y dx", "text": "Let $k \\in \\R$ be a real number. The first order ODE: :$\\dfrac {\\d y} {\\d x} = k y$ has the general solution: :$y = C e^{k x}$"}
+{"_id": "11688", "title": "First Order ODE/(1 over x^3 y^2 + 1 over x) dx + (1 over x^2 y^3 - 1 over y) dy = 0", "text": "The first order ordinary differential equation: :$(1): \\quad \\paren {\\dfrac 1 {x^3 y^2} + \\dfrac 1 x} \\rd x + \\paren {\\dfrac 1 {x^2 y^3} - \\dfrac 1 y} \\rd y = 0$ is an exact differential equation with solution: :$-\\dfrac 1 {2 x^2 y^2} = \\ln \\dfrac y x + C$"}
+{"_id": "11689", "title": "Linear First Order ODE/y' + (y over x) = k x^n", "text": "Let $k, n \\in \\R$ be real numbers. The linear first order ODE: :$(1): \\quad \\dfrac {\\d y} {\\d x} + \\dfrac y x = k x^n$ has the general solution: :$y = \\begin{cases} \\dfrac {k x^{n + 1} } {n + 2} + \\dfrac C x & : n \\ne -2 \\\\ & \\\\ \\dfrac {k \\ln x} x + \\dfrac C x & : n = -2 \\end{cases}$"}
+{"_id": "11692", "title": "Linear First Order ODE/(x^2 + y) dx = x dy", "text": "The linear first order ODE: :$(1): \\quad \\paren {x^2 + y} \\rd x = x \\rd y$ has the general solution: :$y = x^2 + C x$"}
+{"_id": "11693", "title": "Linear First Order ODE/x y' + y = x^2 cosine x", "text": "The linear first order ODE: :$x \\, \\dfrac {\\d y} {\\d x} + y = x^2 \\cos x$ has the general solution: :$y = 2 \\cos x + x \\sin x - \\dfrac 2 x \\sin x + \\dfrac C x$"}
+{"_id": "11696", "title": "First Order ODE/(6x + 4y + 3) dx + (3x + 2y + 2) dy = 0", "text": "The first order ODE: :$(1): \\quad \\paren {6 x + 4 y + 3} \\rd x + \\paren {3 x + 2 y + 2} \\rd y = 0$ has the general solution: :$3 x + 2 y + \\map \\ln {\\paren {3 x + 2 y}^2} + x = C$"}
+{"_id": "11697", "title": "First Order ODE/Cosine (x + y) dx = sine (x + y) dx + x sine (x + y) dy", "text": "The first order ordinary differential equation: :$\\map \\cos {x + y} \\rd x = \\map \\sin {x + y} \\rd x + x \\map \\sin {x + y} \\rd y$ is an exact differential equation with solution: :$x \\map \\cos {x + y} = C$"}
+{"_id": "11698", "title": "Linear First Order ODE/x y' + y = f (x)", "text": "The linear first order ODE: :$(1): \\quad x \\, \\dfrac {\\d y} {\\d x} + y = \\map f x$ has the general solution: :$\\displaystyle x y = \\int \\map f x \\rd x + C$"}
+{"_id": "11702", "title": "Second Order ODE/x^2 y'' + x y' = 1", "text": "The second order ODE: :$x^2 y'' + x y' = 1$ has the general solution: :$y = \\dfrac {\\paren {\\ln x}^2} 2 + C_1 \\ln x + C_2$"}
+{"_id": "11703", "title": "Sorgenfrey Line is Lindelöf", "text": "The Sorgenfrey line is Lindelöf."}
+{"_id": "11705", "title": "First Order ODE/y' ln (x - y) = 1 + ln (x - y)", "text": "The first order ordinary differential equation: :$(1): \\quad \\dfrac {\\d y} {\\d x} \\map \\ln {x - y} \\d x = 1 + \\map \\ln {x - y}$ is an exact differential equation with solution: :$\\paren {x - y} \\map \\ln {x - y} = C - y$"}
+{"_id": "11706", "title": "Linear First Order ODE/y' + 2 x y = exp -x^2", "text": "The linear first order ODE: :$\\dfrac {\\d y} {\\d x} + 2 x y = \\map \\exp {-x^2}$ has the general solution: :$y = \\paren {x + C} \\map \\exp {-x^2}$"}
+{"_id": "11708", "title": "Linear First Order ODE/(1 + x^2) y' + 2 x y = 4 x^3", "text": "The linear first order ODE: :$\\paren {1 + x^2} \\dfrac {\\d y} {\\d x} + 2 x y = 4 x^3$ has the general solution: :$y = \\dfrac {x^4} {1 + x^2} + \\dfrac C {1 + x^2}$"}
+{"_id": "11710", "title": "First Order ODE/(1 + x^2) y' + x y = 0", "text": "The first order ODE: :$\\paren {1 + x^2} y' + x y = 0$ has the general solution: :$y = \\dfrac C {\\sqrt {1 + x^2} }$"}
+{"_id": "11712", "title": "First Order ODE/(x exp y + y - x^2) dy = (2 x y - exp y - x) dx", "text": "The first order ordinary differential equation: :$(1): \\quad \\paren {x e^y + y - x^2} \\rd y = \\paren {2 x y - e^y - x} \\rd x$ is an exact differential equation with solution: :$2 x e^y + x^2 + y^2 - 2 x^2 y = C$"}
+{"_id": "11713", "title": "First Order ODE/exp x (1 + x) dx = (x exp x - y exp y) dy", "text": "The first order ordinary differential equation: :$(1): \\quad e^x \\paren {1 + x} \\rd x = \\paren {x e^x - y e^y} \\rd y$ has the solution: :$2 x e^x e^{-y} + y^2 + C$"}
+{"_id": "11714", "title": "Two Stage Radioactive Decay", "text": "Let $R_A$ be a radioactive isotope which decays into another radioactive isotope $R_B$ with the rate constant $k_1$. Let $R_B$ decay into a third element $R_C$ with the rate constant $k_2$. Let the initial quantity of $R_A$ be $x_0$. Let the amounts of $R_A$ and $R_B$ present at time $t$ be $x$ and $y$ respectively. Then: :$y = \\begin{cases} \\dfrac {k_1 x_0} {k_2 - k_1} \\left({e^{-k_1 t} - e^{-k_2 t} }\\right) & : k_1 \\ne k_2 \\\\ & \\\\ k_1 x_0 t e^{-k_1 t} & : k_1 = k_2 \\end{cases}$"}
+{"_id": "11715", "title": "General Solution of Riccati Equation from Particular Solution", "text": "Consider the Riccati equation: :$(1): \\quad y' = \\map p x + \\map q x y + \\map r x y^2$ Let $\\map {y_1} x$ be a particular solution to $(1)$. Then the general solution to $(1)$ has the form: :$\\map y x = \\map {y_1} x + \\map z x$ where $\\map z x$ is the general solution to the Bernoulli equation: :$z' - \\paren {q - 2 r y_1} z = r z^2$"}
+{"_id": "11716", "title": "Riccati Equation/y' = (y over x) + x^3 y^2 - x^5", "text": "The Riccati equation: :$(1): \\quad y' = \\dfrac y x + x^3 y^2 - x^5$ has the general solution: :$C \\exp \\dfrac {2 x^5} 5 = \\dfrac {y - x} {y + x}$"}
+{"_id": "11717", "title": "Bernoulli's Equation/y' - (1 over x + 2 x^4) y = x^3 y^2", "text": "The first order ODE: :$(1): \\quad y' - \\paren {\\dfrac 1 x + 2 x^4} y = x^3 y^2$ has the general solution: :$y = \\dfrac {2 x} {C \\, \\map \\exp {-\\dfrac {2 x^5} 5} - 1}$"}
+{"_id": "11718", "title": "Motion of Body with Variable Mass", "text": "Let $B$ be a body undergoing a force $\\mathbf F$. Let $B$ be travelling at a velocity $\\mathbf v$ at time $t$. Let mass travelling at a velocity $\\mathbf v + \\mathbf w$ be added to $B$ at a rate of $\\dfrac {\\d m} {\\d t}$. Let $m$ be the mass of $B$ at time $t$. Then the equation of motion of $B$ is given by: :$\\mathbf w \\dfrac {\\d m} {\\d t} + \\mathbf F = m \\dfrac {\\d \\mathbf v} {\\d t}$"}
+{"_id": "11719", "title": "Burnout Velocity of Upward Rocket under Constant Gravity", "text": "Let $R$ be a rocket whose structural mass is $m_1$. Let $R$ contain fuel of initial mass $m_2$. Let $R$ be fired straight up from the surface of a planet whose gravitational field exerts an Acceleration Due to Gravity of $g$, assumed constant. Let $R$ burn fuel at a constant rate $a$, producing a constant exhaust velocity $b$ relative to $R$. Let all forces on $R$ except that due to the gravitational field be neglected. Then the burnout velocity of $R$ is given by: :$v_b = b \\ln \\left({1 + \\dfrac {m_1} {m_2} }\\right) - \\dfrac {g m_2} a$"}
+{"_id": "11720", "title": "Burnout Height of Upward Rocket under Constant Gravity", "text": "Let $R$ be a rocket whose structural mass is $m_1$. Let $R$ contain fuel of initial mass $m_2$. Let $R$ be fired straight up from the surface of a planet whose gravitational field exerts an Acceleration Due to Gravity of $g$, assumed constant. Let $R$ burn fuel at a constant rate $a$, with a constant exhaust velocity $b$ relative to $R$. Let all forces on $R$ except that due to  the gravitational field be neglected. Then the burnout height of $R$ is given by: :$h_b = -\\dfrac {g m_2^2} {2 a^2} + \\dfrac {b m_2} a + \\dfrac {b m_1} a \\ln \\dfrac {m_1} {m_1 + m_2}$"}
+{"_id": "11721", "title": "Linear First Order ODE/y' = x + y", "text": "The linear first order ODE: :$(1): \\quad \\dfrac {\\d y} {\\d x} = x + y$ has the general solution: :$y = C e^x - x - 1$"}
+{"_id": "11722", "title": "Linear First Order ODE/y' = x + y/y(0) = 1", "text": "The linear first order ODE: :$(1): \\quad \\dfrac {\\d y} {\\d x} = x + y$ with initial condition: :$\\map y 0 = 1$ has the particular solution: :$y = 2 e^x - x - 1$"}
+{"_id": "11723", "title": "Existence and Uniqueness of Solution for Linear Second Order ODE with two Initial Conditions", "text": "Let $\\map P x$, $\\map Q x$ and $\\map R x$ be continuous real functions on a closed real interval $\\closedint a b$. Let $x_0$ be any point in $\\closedint a b$. Let $y_0$ and ${y_0}'$ be real numbers. Then the linear second order ordinary differential equation: :$(1): \\quad \\dfrac {\\d^2 y} {\\d x^2} + \\map P x \\dfrac {\\d y} {\\d x} + \\map Q x y = \\map R x$ has a unique particular solution $\\map y x$ on $\\closedint a b$ such that: :$\\map y {x_0} = y_0$ and: :$\\map {y'} {x_0} = {y_0}'$"}
+{"_id": "11725", "title": "General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution", "text": "Consider the nonhomogeneous linear second order ODE: :$(1): \\quad \\dfrac {\\d^2 y} {\\d x^2} + \\map P x \\dfrac {\\d y} {\\d x} + \\map Q x y = \\map R x$ Let $\\map {y_g} x$ be the general solution of the homogeneous linear second order ODE: :$(2): \\quad \\dfrac {\\d^2 y} {\\d x^2} + \\map P x \\dfrac {\\d y} {\\d x} + \\map Q x y = 0$ Let $\\map {y_p} x$ be a particular solution of $(1)$. Then $\\map {y_g} x + \\map {y_p} x$ is the general solution of $(1)$."}
+{"_id": "11726", "title": "Trivial Solution of Homogeneous Linear 2nd Order ODE", "text": "The homogeneous linear second order ODE: :$\\dfrac {\\d^2 y} {\\d x^2} + \\map P x \\dfrac {\\d y} {\\d x} + \\map Q x y = 0$ has the particular solution: :$\\map y x = 0$ that is, the zero constant function. This particular solution is referred to as the '''trivial solution'''."}
+{"_id": "11727", "title": "Linear Combination of Solutions to Homogeneous Linear 2nd Order ODE", "text": "Let $c_1$ and $c_2$ be real numbers. Let $\\map {y_1} x$ and $\\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE: :$(1): \\quad \\dfrac {\\d^2 y} {\\d x^2} + \\map P x \\dfrac {\\d y} {\\d x} + \\map Q x y = 0$ Then: :$c_1 \\, \\map {y_1} x + c_2 \\, \\map {y_2} x$ is also a particular solution to $(1)$. That is, a linear combination of particular solutions to a homogeneous linear second order ODE is also a particular solution to that ODE."}
+{"_id": "11729", "title": "Solutions of Linear 2nd Order ODE have Common Zero iff Linearly Dependent", "text": "Let $\\map {y_1} x$ and $\\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE: :$(1): \\quad \\dfrac {\\d^2 y} {\\d x^2} + \\map P x \\dfrac {\\d y} {\\d x} + \\map Q x y = 0$ on a closed interval $\\closedint a b$. Let $y_1$ and $y_2$ both have a zero for the same value of $x$ in $\\closedint a b$. Then $y_1$ and $y_2$ are constant multiples of each other. That is, $y_1$ and $y_2$ are linearly dependent."}
+{"_id": "11730", "title": "Real Function is Linearly Dependent with Zero Function", "text": "Let $f \\left({x}\\right)$ be a real function defined on a closed interval $\\left[{a \\,.\\,.\\, b}\\right]$. Let $g \\left({x}\\right)$ be the constant zero function on $\\left[{a \\,.\\,.\\, b}\\right]$: :$\\forall x \\in \\left[{a \\,.\\,.\\, b}\\right]: g \\left({x}\\right) = 0$ Then $f$ and $g$ are linearly dependent on $\\left[{a \\,.\\,.\\, b}\\right]$."}
+{"_id": "11731", "title": "Two Linearly Independent Solutions of Homogeneous Linear Second Order ODE generate General Solution", "text": "Let $\\map {y_1} x$ and $\\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE: :$(1): \\quad \\dfrac {\\d^2 y} {\\d x^2} + \\map P x \\dfrac {\\d y} {\\d x} + \\map Q x y = 0$ on a closed interval $\\closedint a b$. Let $y_1$ and $y_2$ be linearly independent. Then the general solution to $(1)$ is: :$y = C_1 \\map {y_1} x + C_2 \\map {y_2} x$ where $C_1 \\in \\R$ and $C_2 \\in \\R$ are arbitrary constants."}
+{"_id": "11732", "title": "Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE", "text": "Let $\\map {y_1} x$ and $\\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE: :$(1): \\quad \\dfrac {\\d^2 y} {\\d x^2} + \\map P x \\dfrac {\\d y} {\\d x} + \\map Q x y = 0$ on a closed interval $\\closedint a b$. Let $y_1$ and $y_2$ be linearly independent. Then their Wronskian is either never zero, or zero everywhere on $\\closedint a b$."}
+{"_id": "11733", "title": "Real Number Line is Lindelöf", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\\struct {\\R, \\tau_d}$ is Lindelöf."}
+{"_id": "11734", "title": "Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent", "text": "Let $\\map {y_1} x$ and $\\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE: :$(1): \\quad \\dfrac {\\d^2 y} {\\d x^2} + \\map P x \\dfrac {\\d y} {\\d x} + \\map Q x y = 0$ on a closed interval $\\closedint a b$. Then: :$y_1$ and $y_2$ are linearly dependent {{iff}}: :the Wronskian $\\map W {y_1, y_2}$ of $y_1$ and $y_2$ is zero everywhere on $\\closedint a b$."}
+{"_id": "11736", "title": "Linear Second Order ODE/y'' + y = 0", "text": "The second order ODE: :$(1): \\quad y'' + y = 0$ has the general solution: :$y = C_1 \\sin x + C_2 \\cos x$"}
+{"_id": "11741", "title": "Linear Second Order ODE/x^2 y'' - 2 x y' + 2 y = 0", "text": "The second order ODE: :$(1): \\quad x^2 y'' - 2 x y' + 2 y = 0$ has the general solution: :$y = C_1 x + C_2 x^2$ on any closed real interval which does not contain $0$."}
+{"_id": "11742", "title": "Linear Second Order ODE/x^2 y'' - 2 x y' + 2 y = 0/y(1) = 3, y'(1) = 5", "text": "The second order ODE: :$(1): \\quad x^2 y'' - 2 x y' + 2 y = 0$ with initial conditions: :$\\map y 1 = 3$ :$\\map {y'} 1 = 5$ has the particular solution: :$y = x + 2 x^2$"}
+{"_id": "11743", "title": "Linear Second Order ODE/y'' - 3 y' + 2 y = 0", "text": "The second order ODE: :$(1): \\quad y'' - 3 y' + 2 y = 0$ has the general solution: :$y = C_1 e^x + C_2 e^{2 x}$"}
+{"_id": "11744", "title": "Linear Second Order ODE/y'' - 3 y' + 2 y = 0/y(0) = -1, y'(0) = 1", "text": "The second order ODE: :$(1): \\quad y'' - 3 y' + 2 y = 0$ with initial conditions: :$\\map y 0 = -1$ :$\\map {y'} 0 = 1$ has the particular solution: :$y = -3 e^x + 2 e^{2 x}$"}
+{"_id": "11745", "title": "Linear Second Order ODE/y'' - 4 y' + 4 y = 0", "text": "The second order ODE: :$(1): \\quad y'' - 4 y' + 4 y = 0$ has the general solution: :$y = C_1 e^{2 x} + C_2 x e^{2 x}$"}
+{"_id": "11746", "title": "Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another", "text": "Let $\\map {y_1} x$ be a particular solution to the homogeneous linear second order ODE: :$(1): \\quad \\dfrac {\\d^2 y} {\\d x^2} + \\map P x \\dfrac {\\d y} {\\d x} + \\map Q x y = 0$ such that $y_1$ is not the trivial solution. Then there exists a standard procedure to determine another particular solution $\\map {y_2} x$ of $(1)$ such that $y_1$ and $y_2$ are linearly independent."}
+{"_id": "11747", "title": "Induced Solution to Homogeneous Linear Second Order ODE is Linearly Independent with Inducing Solution", "text": "Let $\\map {y_1} x$ be a particular solution to the homogeneous linear second order ODE: :$(1): \\quad \\dfrac {\\d^2 y} {\\d x^2} + \\map P x \\dfrac {\\d y} {\\d x} + \\map Q x y = 0$ such that $y_1$ is not the trivial solution. Let $\\map {y_2} x$ be the real function defined as: :$\\map {y_2} x = \\map v x \\, \\map {y_1} x$ where: :$\\displaystyle v = \\int \\dfrac 1 { {y_1}^2} e^{-\\int P \\rd x} \\rd x$ Then $y_2$ and $y_1$ are linearly independent."}
+{"_id": "11748", "title": "Linear Second Order ODE/x^2 y'' + x y' - y = 0", "text": "The second order ODE: :$(1): \\quad x^2 y'' + x y' - y = 0$ has the general solution: :$y = C_1 x + \\dfrac {C_2} x$"}
+{"_id": "11750", "title": "Linear Second Order ODE/y'' - y = 0", "text": "The second order ODE: :$(1): \\quad y'' - y = 0$ has the general solution: :$y = C_1 e^x + C_2 e^{-x}$"}
+{"_id": "11752", "title": "Second Order ODE/x y'' + 3 y' = 0", "text": "The second order ODE: :$(1): \\quad x y'' + 3 y' = 0$ has the general solution: :$y = C_1 + \\dfrac {C_2} {x^2}$"}
+{"_id": "11753", "title": "Linear Second Order ODE/x^2 y'' + x y' - 4 y = 0", "text": "The second order ODE: :$(1): \\quad x^2 y'' + x y' - 4 y = 0$ has the general solution: :$y = C_1 x^2 + \\dfrac {C_2} {x^2}$"}
+{"_id": "11755", "title": "Topological Subspace of Real Number Line is Lindelöf", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $W$ be a non-empty subset of $\\R$. Then $R_W$ is Lindelöf where $R_W$ denotes the topological subspace of $R$ on $W$."}
+{"_id": "11757", "title": "Bessel's Equation/x^2 y'' + x y' + (x^2 - (1 over 4)) y = 0/Particular Solution", "text": "The special case of Bessel's equation: :$(1): \\quad x^2 y'' + x y' + \\left({x^2 - \\dfrac 1 4}\\right) y = 0$ has a particular solution: :$y = \\dfrac {\\sin x} {\\sqrt x}$"}
+{"_id": "11758", "title": "Linear Second Order ODE/(x - 1) y'' - y' + y = 0", "text": "The second order ODE: :$(1): \\quad \\paren {x - 1} y'' - x y' + y = 0$ has the general solution: :$y = C_1 x + C_2 e^x$"}
+{"_id": "11759", "title": "Linear Second Order ODE/x^2 y'' + 2 x y' - 2 y = 0", "text": "The second order ODE: :$(1): \\quad x^2 y'' + 2 x y' - 2 y = 0$ has the general solution: :$y = C_1 x + \\dfrac {C_2} {x^2}$"}
+{"_id": "11761", "title": "Second Order ODE/x y'' - (2 x + 1) y' + (x + 1) y = 0", "text": "The second order ODE: :$(1): \\quad x y'' - \\paren {2 x + 1} y' + \\paren {x + 1} y = 0$ has the general solution: :$y = C_1 e^x + C_2 x^2 e^x$"}
+{"_id": "11763", "title": "Exponential Function is Solution of Constant Coefficient Homogeneous LSOODE iff Index is Root of Auxiliary Equation", "text": "Let: :$(1): \\quad y'' + p y' + q y = 0$ be a constant coefficient homogeneous linear second order ODE. Then: :$y = e^{m_1 x}$ is a solution to $(1)$ {{iff}} :$m_1$ is a root of the auxiliary equation $m^2 + p m + q = 0$"}
+{"_id": "11764", "title": "Solution of Constant Coefficient Homogeneous LSOODE", "text": "Let: :$(1): \\quad y'' + p y' + q y = 0$ be a constant coefficient homogeneous linear second order ODE. Let $m_1$ and $m_2$ be the roots of the auxiliary equation $m^2 + p m + q = 0$."}
+{"_id": "11766", "title": "Solution of Constant Coefficient Homogeneous LSOODE/Real Roots of Auxiliary Equation", "text": "{{:Solution of Constant Coefficient Homogeneous LSOODE}} Let $p^2 > 4 q$. Then $(1)$ has the general solution: :$y = C_1 e^{m_1 x} + C_2 e^{m_2 x}$"}
+{"_id": "11767", "title": "Solution of Constant Coefficient Homogeneous LSOODE/Complex Roots of Auxiliary Equation", "text": "{{:Solution of Constant Coefficient Homogeneous LSOODE}} Let $p^2 < 4 q$. Then $(1)$ has the general solution: :$y = e^{a x} \\paren {C_1 \\cos b x + C_2 \\sin b x}$ where: :$m_1 = a + i b$ :$m_2 = a - i b$"}
+{"_id": "11768", "title": "Solution of Constant Coefficient Homogeneous LSOODE/Equal Real Roots of Auxiliary Equation", "text": "{{:Solution of Constant Coefficient Homogeneous LSOODE}} Let $p^2 = 4 q$. Then $(1)$ has the general solution: :$y = C_1 e^{m_1 x} + C_2 x e^{m_1 x}$"}
+{"_id": "11769", "title": "Linear Second Order ODE/y'' + y' - 6 y = 0", "text": "The second order ODE: :$(1): \\quad y'' + y' - 6 y = 0$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{-3 x}$"}
+{"_id": "11770", "title": "Linear Second Order ODE/y'' + 2 y' + y = 0", "text": "The second order ODE: :$(1): \\quad y'' + 2 y' + y = 0$ has the general solution: :$y = C_1 e^{-x} + C_2 x e^{-x}$"}
+{"_id": "11772", "title": "Linear Second Order ODE/2 y'' - 4 y' + 8 y = 0", "text": "The second order ODE: :$(1): \\quad 2 y'' - 4 y + 8 y = 0$ has the general solution: :$y = e^x \\paren {C_1 \\cos \\sqrt 3 x + C_2 \\sin \\sqrt 3 x}$"}
+{"_id": "11775", "title": "Linear Second Order ODE/y'' - 9 y' + 20 y = 0", "text": "The second order ODE: :$(1): \\quad y'' - 9 y' + 20 y = 0$ has the general solution: :$y = C_1 e^{4 x} + C_2 e^{5 x}$"}
+{"_id": "11777", "title": "Condition for Solutions to Constant Coefficient Homogeneous LSOODE to tend to Zero", "text": "Let: :$(1): \\quad y'' + p y' + q y = 0$ be a constant coefficient homogeneous linear second order ODE. Let the general solution to $(1)$ be $\\map y {x, C_1, C_2}$. Then: :$\\displaystyle \\lim_{x \\mathop \\to \\infty} \\map y {x, C_1, C_2} = 0$ {{iff}} :$p$ and $q$ are both strictly positive."}
+{"_id": "11778", "title": "Derivative of Solution to Constant Coefficient Homogeneous LSOODE is also Solution", "text": "Let: :$(1): \\quad y'' + p y' + q y = 0$ be a constant coefficient homogeneous linear second order ODE. Let $\\map y x$ be a particular solution of $(1)$. Then its derivative $\\map {y'} x$ is also a particular solution of $(1)$."}
+{"_id": "11779", "title": "Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE", "text": "Consider the Cauchy-Euler equation: :$(1): \\quad x^2 \\dfrac {\\d^2 y} {\\d x^2} + p x \\dfrac {\\d y} {\\d x} + q y = 0$ By making the substitution: :$x = e^t$ it is possible to convert $(1)$ into a constant coefficient homogeneous linear second order ODE: :$\\dfrac {\\d^2 y} {\\d t^2} + \\paren {p - 1} \\dfrac {\\d y} {\\d t} + q y = 0$"}
+{"_id": "11782", "title": "Linear Second Order ODE/x^2 y'' + 2 x y' - 12 y = 0", "text": "The second order ODE: :$(1): \\quad x^2 y'' + 2 x y' - 12 y = 0$ has the general solution: :$y = C_1 x^3 + C_2 x^{-4}$"}
+{"_id": "11783", "title": "Cardinality of Image of Set not greater than Cardinality of Set", "text": "Let $X, Y$ be sets. Let $f:X \\to Y$ be a mapping. Let $A$ be a subset of $X$. Then $\\left\\vert{f^\\to\\left({A}\\right)}\\right\\vert \\le \\left\\vert{A}\\right\\vert$ where $\\left\\vert{A}\\right\\vert$ denotes the cardinality of $A$."}
+{"_id": "11784", "title": "Projector has Norm 1", "text": "An idempotent operator $P$ is a projector on the Hilbert Space $H$ {{iff}} $P$ has norm $1$: :$\\displaystyle \\norm P \\equiv \\sup_{x \\mathop \\in H} \\frac {\\norm P} {\\norm x} = 1$"}
+{"_id": "11785", "title": "Linear Second Order ODE/y'' + 3 y' - 10 y = 0", "text": "The second order ODE: :$(1): \\quad y'' + 3 y' - 10 y = 0$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{-5 x}$"}
+{"_id": "11787", "title": "Linear Second Order ODE/y'' + 4 y = 0", "text": "The second order ODE: :$(1): \\quad y'' + 4 y = 0$ has the general solution: :$y = C_1 \\cos 2 x + C_2 \\sin 2 x$"}
+{"_id": "11788", "title": "Linear Second Order ODE/y'' + 4 y = 3 sine x", "text": "The second order ODE: :$(1): \\quad y'' + 4 y = 3 \\sin x$ has the general solution: :$y = C_1 \\cos 2 x + C_2 \\sin 2 x + \\sin x$"}
+{"_id": "11793", "title": "Linear Second Order ODE/y'' + 10 y' + 25 y = 0", "text": "The second order ODE: :$(1): \\quad y'' + 10 y' + 25 y = 0$ has the general solution: :$y = C_1 e^{-5 x} + C_2 x e^{-5 x}$"}
+{"_id": "11794", "title": "Linear Second Order ODE/y'' + 10 y' + 25 y = 14 exp -5 x", "text": "The second order ODE: :$(1): \\quad y'' + 10 y' + 25 y = 14 e^{-5 x}$ has the general solution: :$y = C_1 \\cos 2 x + C_2 \\sin 2 x + \\sin x$"}
+{"_id": "11795", "title": "Linear Second Order ODE/y'' - 2 y' + 5 y = 0", "text": "The second order ODE: :$(1): \\quad y'' - 2 y' + 5 y = 0$ has the general solution: :$y = e^x \\paren {C_1 \\cos 2 x + C_2 \\sin 2 x}$"}
+{"_id": "11799", "title": "Linear Second Order ODE/y'' + y' - 12 y = 0", "text": "The second order ODE: :$(1): \\quad y'' + y' - 12 y = 0$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-4 x}$"}
+{"_id": "11800", "title": "Linear Second Order ODE/y'' + 2 y' + 10 y = 0", "text": "The second order ODE: :$(1): \\quad y'' + 2 y' + 10 y = 0$ has the general solution: :$y = e^{-x} \\paren {C_1 \\cos 3 x + C_2 \\sin 3 x}$"}
+{"_id": "11801", "title": "Linear Second Order ODE/y'' + 4 y' + 4 y = 0", "text": "The second order ODE: :$(1): \\quad y'' + 4 y' + 4 y = 0$ has the general solution: :$y = \\paren {C_1 + C_2 x} e^{-2 x}$"}
+{"_id": "11803", "title": "Linear Second Order ODE/y'' + y' - 2 y = 0", "text": "The second order ODE: :$(1): \\quad y'' + y' - 2 y = 0$ has the general solution: :$y = C_1 e^x + C_2 e^{-2 x}$"}
+{"_id": "11807", "title": "Linear Second Order ODE/y'' - 4 y = 0", "text": "The second order ODE: :$(1): \\quad y'' - 4 y = 0$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{-2 x}$"}
+{"_id": "11818", "title": "Linear Second Order ODE/y'' - y' - 6 y = 0", "text": "The second order ODE: :$(1): \\quad y'' - y' - 6 y = 0$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-2 x}$"}
+{"_id": "11819", "title": "Linear Second Order ODE/y'' - y' - 6 y = 20 exp -2 x", "text": "The second order ODE: :$(1): \\quad y'' - y' - 6 y = 20 e^{-2 x}$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-2 x} - 4 x e^{-2 x}$"}
+{"_id": "11822", "title": "Linear Second Order ODE/y'' - 2 y' = 0", "text": "The second order ODE: :$(1): \\quad y'' - 2 y' = 0$ has the general solution: :$y = C_1 + C_2 e^{2 x}$"}
+{"_id": "11824", "title": "Directed iff Finite Subsets have Upper Bounds", "text": "Let $\\left({S, \\precsim}\\right)$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is directed {{iff}} :for every a finite subset $A$ of $H$ ::$\\exists h \\in H: \\forall a \\in A: a \\precsim h$"}
+{"_id": "11825", "title": "Linear Second Order ODE/y'' + k^2 y = sine b x", "text": "The second order ODE: :$(1): \\quad y'' + k^2 y = \\sin b x$ has the general solution: :$y = \\begin{cases} C_1 \\sin k x + C_2 \\cos k x + \\dfrac {\\sin b x} {k^2 - b^2} & : b \\ne k \\\\ C_1 \\sin k x + C_2 \\cos k x - \\dfrac {x \\cos k x} {2 k} & : b = k  \\end{cases}$"}
+{"_id": "11826", "title": "Combination of Solutions to Non-Homogeneous LSOODE with same Homogeneous Part", "text": "Let $\\map {y_1} x$ be a particular solution of the linear second order ODE: :$(1): \\quad y'' + \\map P x y' + \\map Q x y = \\map {R_1} x$ Let $\\map {y_2} x$ be a particular solution of the linear second order ODE: :$(2): \\quad y'' + \\map P x y' + \\map Q x y = \\map {R_2} x$ Then $\\map y x = \\map {y_1} x + \\map {y_2} x$ is a particular solution of the linear second order ODE: :$(3): \\quad y'' + \\map P x y' + \\map Q x y = \\map {R_1} x + \\map {R_2} x$"}
+{"_id": "11827", "title": "Linear Second Order ODE/y'' + 4 y = 4 cosine 2 x + 6 cosine x + 8 x^2 - 4 x", "text": "The second order ODE: :$(1): \\quad y'' + 4 y = 4 \\cos 2 x + 6 \\cos x + 8 x^2 - 4 x$ has the general solution: :$y = C_1 \\sin 2 x + C_2 \\cos 2 x + x \\sin 2 x + 2 \\cos x - 1 - x + 2 x^2$"}
+{"_id": "11828", "title": "Linear Second Order ODE/y'' + 4 y = 4 cosine 2 x", "text": "The second order ODE: :$(1): \\quad y'' + 4 y = 4 \\cos 2 x$ has the general solution: :$y = C_1 \\sin 2 x + C_2 \\cos 2 x + x \\sin 2 x$"}
+{"_id": "11829", "title": "Linear Second Order ODE/y'' + 4 y = 6 cosine x", "text": "The second order ODE: :$(1): \\quad y'' + 4 y = 6 \\cos x$ has the general solution: :$y = C_1 \\sin 2 x + C_2 \\cos 2 x + 2 \\cos x$"}
+{"_id": "11830", "title": "Linear Second Order ODE/y'' + 4 y = 8 x^2 - 4 x", "text": "The second order ODE: :$(1): \\quad y'' + 4 y = 8 x^2 - 4 x$ has the general solution: :$y = C_1 \\sin 2 x + C_2 \\cos 2 x - 1 - x + 2 x^2$"}
+{"_id": "11832", "title": "Linear Second Order ODE/y'' - 2 y' + y = 0", "text": "The second order ODE: :$(1): \\quad y'' - 2 y' + y = 0$ has the general solution: :$y = C_1 e^x + C_2 x e^x$"}
+{"_id": "11833", "title": "Linear Second Order ODE/y'' - 2 y' + y = 2 x", "text": "The second order ODE: :$(1): \\quad y'' - 2 y' + y = 2 x$ has the general solution: :$y = C_1 e^x + C_2 x e^x + 2 x + 4$"}
+{"_id": "11837", "title": "Linear Second Order ODE/y'' - y' - 6 y = exp -x", "text": "The second order ODE: :$(1): \\quad y'' - y' - 6 y = e^{-x}$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-2 x} - \\dfrac {e^{-x} } 4$"}
+{"_id": "11840", "title": "Filtered iff Finite Subsets have Lower Bounds", "text": "Let $\\struct {S, \\precsim}$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is filtered {{iff}}: :for every a finite subset $A$ of $H$ ::$\\exists h \\in H: \\forall a \\in A: h \\precsim a$"}
+{"_id": "11841", "title": "Tangent of Half Angle plus Quarter Pi", "text": ":$\\map \\tan {\\dfrac x 2 + \\dfrac \\pi 4} = \\tan x + \\sec x$"}
+{"_id": "11842", "title": "Linear Second Order ODE/y'' + 4 y = tangent 2 x", "text": "The second order ODE: :$(1): \\quad y'' + 4 y = \\tan 2 x$ has the general solution: :$y = C_1 \\cos 2 x + C_2 \\sin 2 x - \\dfrac 1 4 \\cos 2 x \\, \\map \\ln {\\sec 2 x + \\tan 2 x}$"}
+{"_id": "11843", "title": "Linear Second Order ODE/y'' + 2 y' + y = exp -x log x", "text": "The second order ODE: :$(1): \\quad y'' + 2 y' + y = e^{-x} \\ln x$ has the general solution: :$y = C_1 e^{-x} + C_2 x e^{-x} - \\dfrac {x^2 e^{-x} \\ln x} 2 - \\dfrac 3 4 x^2 e^{-x}$"}
+{"_id": "11844", "title": "Linear Second Order ODE/y'' - 2 y' - 3 y = 0", "text": "The second order ODE: :$(1): \\quad y'' - 2 y' - 3 y = 0$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-x}$"}
+{"_id": "11846", "title": "Linear Second Order ODE/y'' + 2 y' + 5 y = 0", "text": "The second order ODE: :$(1): \\quad y'' + 2 y' + 5 y = 0$ has the general solution: :$y = e^{-x} \\paren {C_1 \\cos 2 x + C_2 \\sin 2 x}$"}
+{"_id": "11847", "title": "Linear Second Order ODE/y'' + 2 y' + 5 y = exp -x secant 2 x", "text": "The second order ODE: :$(1): \\quad y'' + 2 y' + 5 y = e^{-x} \\sec 2 x$ has the general solution: :$y = e^{-x} \\paren {C_1 \\cos 2 x + C_2 \\sin 2 x} + \\dfrac {x e^{-x} \\sin 2 x} 2 + \\dfrac {e^{-x} \\cos 2 x \\ln \\cos 2 x} 4$"}
+{"_id": "11848", "title": "Singleton is Directed and Filtered Subset", "text": "Let $\\struct {S, \\precsim}$ be a preordered set. Let $x$ be an element of $S$. Then $\\set x$ is directed and filtered subset of $S$."}
+{"_id": "11849", "title": "Second Order ODE/(x^2 - 1) y'' - 2 x y' + 2 y = 0", "text": "The second order ODE: :$(1): \\quad \\paren {x^2 - 1} y'' - 2 x y' + 2 y = 0$ has the general solution: :$y = C_1 x + C_2 \\paren {x^2 + 1}$"}
+{"_id": "11850", "title": "Linear Second Order ODE/(x^2 - 1) y'' - 2 x y' + 2 y = (x^2 - 1)^2", "text": "The second order ODE: :$(1): \\quad \\paren {x^2 - 1} y'' - 2 x y' + 2 y = \\paren {x^2 - 1}^2$ has the general solution: :$y = C_1 x + C_2 \\paren {x^2 + 1} + \\dfrac {x^4} 6 - \\dfrac {x^2} 2$"}
+{"_id": "11851", "title": "Directed iff Lower Closure Directed", "text": "Let $\\left({S, \\precsim}\\right)$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is directed {{iff}} $H^\\precsim$ is directed where $H^\\precsim$ denotes the lower closure of set."}
+{"_id": "11852", "title": "Filtered iff Upper Closure Filtered", "text": "Let $\\left({S, \\precsim}\\right)$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is filtered {{iff}} $H^\\succsim$ is filtered where $H^\\succsim$ denotes the upper closure of set."}
+{"_id": "11853", "title": "Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = 0", "text": "The second order ODE: :$(1): \\quad \\paren {x^2 + x} y'' + \\paren {2 - x^2} y' - \\paren {2 + x} y = 0$ has the general solution: :$y = C_1 e^x + \\dfrac {C_2} x$"}
+{"_id": "11859", "title": "Position of Cart attached to Wall by Spring/x = x0 at t = 0", "text": "Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the horizontal position of $C$ at time $t$ can be expressed as: :$x = x_0 \\cos \\alpha t$"}
+{"_id": "11864", "title": "Directed in Join Semilattice", "text": "Let $\\left({S, \\preceq}\\right)$ be a join semilattice. Let $H$ be a non-empty lower subset of $S$. Then $H$ is directed {{iff}} :$\\forall x, y \\in H: x \\vee y \\in H$"}
+{"_id": "11871", "title": "Position of Cart attached to Wall by Spring under Damping/Underdamped/x = x0 at t = 0", "text": "Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the horizontal position of $C$ at time $t$ can be expressed as: :$x = \\dfrac {x_0} \\alpha e^{-b t} \\left({\\alpha \\cos \\alpha t + b \\sin \\alpha t}\\right)$"}
+{"_id": "11872", "title": "Canonical Form of Underdamped Oscillatory System", "text": "Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \\quad \\dfrac {\\d^2 x} {\\d t^2} + 2 b \\dfrac {\\d x} {\\d t} + a^2 x = 0$ for $a, b \\in \\R_{>0}$. Let $b < a$, so as to make $S$ underdamped. Then the value of $x$ can be expressed in the form: :$x = \\dfrac {x_0 \\, a} \\alpha e^{-b t} \\, \\map \\cos {\\alpha t - \\theta}$ where: :$\\alpha = \\sqrt {a^2 - b^2}$ :$\\theta = \\map \\arctan {\\dfrac b \\alpha}$ This can be referred to as the '''canonical form''' of the solution of $(1)$."}
+{"_id": "11873", "title": "Period of Oscillation of Underdamped System is Regular", "text": "Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$\\dfrac {\\d^2 x} {\\d t^2} + 2 b \\dfrac {\\d x} {\\d t} + a^2 x = 0$ for $a, b \\in \\R_{>0}$. Let $b < a$, so as to make $S$ underdamped. Then the period of its movement is well-defined, in the sense that its zeroes are regularly spaced, and given by: :$T = \\dfrac {2 \\pi} {\\sqrt {a^2 - b^2} }$"}
+{"_id": "11874", "title": "Period of Oscillation of Underdamped Cart attached to Wall by Spring", "text": "Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the period of oscillation of $C$ can be expressed as: :$T = \\dfrac {2 \\pi} {\\sqrt {\\dfrac k m - \\dfrac {c^2} {4 m^2} } }$"}
+{"_id": "11875", "title": "Natural Frequency of Underdamped System", "text": "Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$\\dfrac {\\d^2 x} {\\d t^2} + 2 b \\dfrac {\\d x} {\\d t} + a^2 x = 0$ for $a, b \\in \\R_{>0}$. Let $b < a$, so as to make $S$ underdamped. Then the natural frequency of $S$ is given by: :$\\nu = \\dfrac {\\sqrt {a^2 - b^2} } {2 \\pi}$"}
+{"_id": "11876", "title": "Natural Frequency of Underdamped Cart attached to Wall by Spring", "text": "Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the natural frequency of $C$ can be expressed as: :$\\nu = \\dfrac 1 {2 \\pi} \\sqrt {\\dfrac k m - \\dfrac {c^2} {4 m^2} }$"}
+{"_id": "11878", "title": "Linear Second Order ODE/y'' + 2 b y' + a^2 y = 0/b less than a", "text": "The second order ODE: :$(1): \\quad y'' + 2 b y' + a^2 y = 0$ where $b^2 < a^2$ has the general solution: :$y = e^{-b x} \\paren {C_1 \\, \\map \\cos {\\sqrt {a^2 - b^2} } x + C_2 \\, \\map \\sin {\\sqrt {a^2 - b^2} } x}$"}
+{"_id": "11879", "title": "Linear Second Order ODE/y'' + 2 b y' + a^2 y = K cosine omega x/b less than a", "text": "The second order ODE: :$(1): \\quad y'' + 2 b y' + a^2 y = K \\cos \\omega x$ where $b^2 < a^2$ has the general solution: :$y = e^{-b x} \\paren {C_1 \\cos \\alpha x + C_2 \\sin \\alpha x} + \\dfrac K {\\sqrt {4 b^2 \\omega^2 + \\paren {a^2 - \\omega^2}^2} } \\map \\cos {\\omega x - \\phi}$ where: :$\\alpha = \\sqrt {a^2 - b^2}$ :$\\phi = \\map \\arctan {\\dfrac {2 b \\omega} {a^2 - \\omega^2} }$"}
+{"_id": "11882", "title": "Condition for Resonance in Forced Vibration of Underdamped System", "text": "Consider a physical system $S$ whose behaviour is defined by the second order ODE: :$(1): \\quad \\dfrac {\\d^2 y} {\\d x^2} + 2 b \\dfrac {\\d y} {\\d x} + a^2 x = K \\cos \\omega x$ where: :$K \\in \\R: k > 0$ :$a, b \\in \\R_{>0}: b < a$ Then $S$ is in resonance when: :$\\omega = \\sqrt {a^2 - 2 b^2}$ and thus the resonance frequency is: :$\\nu_R = \\dfrac {\\sqrt {a^2 - 2 b^2} } {2 \\pi}$ This resonance frequency exists only when $a^2 - 2 b^2 > 0$."}
+{"_id": "11883", "title": "Filtered in Meet Semilattice", "text": "Let $\\left({S, \\preceq}\\right)$ be a meet semilattice. Let $H$ be a non-empty upper subset of $S$. Then $H$ is filtered {{iff}} :$\\forall x, y \\in H: x \\wedge y \\in H$"}
+{"_id": "11885", "title": "Directed in Join Semilattice with Finite Suprema", "text": "Let $\\left({S, \\preceq}\\right)$ be a join semilattice. Let $H$ be a non-empty lower subset of $S$. Then $H$ is directed {{iff}} :for every non-empty finite subset $A$ of $H$, $\\sup A \\in H$"}
+{"_id": "11887", "title": "Filtered in Meet Semilattice with Finite Infima", "text": "Let $\\left({S, \\preceq}\\right)$ be a meet semilattice. Let $H$ be a non-empty lower subset of $S$. Then $H$ is filtered {{iff}} :for every non-empty finite subset $A$ of $H$, $\\inf A \\in H$"}
+{"_id": "11888", "title": "Interval between Local Maxima for Underdamped Free Vibration", "text": "Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \\quad \\dfrac {\\mathrm d^2 x} {\\mathrm d t^2} + 2 b \\dfrac {\\mathrm d x} {\\mathrm d t} + a^2 x = 0$ for $a, b \\in \\R_{>0}$. Let $b < a$, so as to make $S$ underdamped. :600px Let $T$ be the period of oscillation of $S$. Then the successive local maxima of $x$ occur for $t = 0, T, 2T, \\ldots$"}
+{"_id": "11889", "title": "Ratio of Successive Local Maxima for Underdamped Free Vibration", "text": "Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \\quad \\dfrac {\\mathrm d^2 x} {\\mathrm d t^2} + 2 b \\dfrac {\\mathrm d x} {\\mathrm d t} + a^2 x = 0$ for $a, b \\in \\R_{>0}$. Let $b < a$, so as to make $S$ underdamped. :600px Let $T$ be the period of oscillation of $S$. Let $x_1$ and $x_2$ be successive local maxima of $x$. Then: :$\\dfrac {x_1} {x_2} = e^{b T}$"}
+{"_id": "11890", "title": "Motion of Particle in Polar Coordinates", "text": "Consider a particle $p$ of mass $m$ moving in the plane under the influence of a force $\\mathbf F$. Let the position of $p$ at time $t$ be given in polar coordinates as $\\left\\langle{r, \\theta}\\right\\rangle$. Let $\\mathbf F$ be expressed as: :$\\mathbf F = F_r \\mathbf u_r + F_\\theta \\mathbf u_\\theta$ where: :$\\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$ :$\\mathbf u_\\theta$ is the unit vector in the direction of the angular coordinate of $p$ :$F_r$ and $F_\\theta$ are the magnitudes of the components of $\\mathbf F$ in the directions of $\\mathbf u_r$ and $\\mathbf u_\\theta$ respectively. Then the second order ordinary differential equations governing the motion of $m$ under the force $\\mathbf F$ are: {{begin-eqn}} {{eqn | l = F_\\theta       | r = m \\left({r \\dfrac {\\mathrm d^2 \\theta} {\\mathrm d t^2} + 2 \\dfrac {\\mathrm d r} {\\mathrm d t} \\dfrac {\\mathrm d \\theta} {\\mathrm d t} }\\right)       | c =  }} {{eqn | l = F_r       | r = m \\left({\\dfrac {\\mathrm d^2 r} {\\mathrm d t^2} - r \\left({\\dfrac {\\mathrm d \\theta} {\\mathrm d t} }\\right)^2}\\right)       | c =  }} {{end-eqn}}"}
+{"_id": "11891", "title": "Derivative of Angular Component under Central Force", "text": "Let a point mass $p$ of mass $m$ be under the influence of a central force $\\mathbf F$. Let the position of $p$ at time $t$ be given in polar coordinates as $\\polar {r, \\theta}$. Let $\\mathbf r$ be the radius vector from the origin to $p$. Then the rate of change of the angular coordinate of $p$ is inversely proportional to the square of the radial coordinate of $p$."}
+{"_id": "11892", "title": "Linear Second Order ODE/y'' + y = K", "text": "The second order ODE: :$(1): \\quad y'' + y = K$ has the general solution: :$y = C_1 \\sin x + C_2 \\cos x + K$"}
+{"_id": "11893", "title": "Velocity Vector in Polar Coordinates", "text": "Consider a particle $p$ moving in the plane. Let the position of $p$ at time $t$ be given in polar coordinates as $\\left\\langle{r, \\theta}\\right\\rangle$. Then the velocity $\\mathbf v$ of $p$ can be expressed as: :$\\mathbf v = r \\dfrac {\\mathrm d \\theta} {\\mathrm d t} \\mathbf u_\\theta + \\dfrac {\\mathrm d r} {\\mathrm d t} \\mathbf u_r$ where: :$\\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$ :$\\mathbf u_\\theta$ is the unit vector in the direction of the angular coordinate of $p$"}
+{"_id": "11894", "title": "Derivatives of Unit Vectors in Polar Coordinates", "text": "Consider a particle $p$ moving in the plane. Let the position of $p$ be given in polar coordinates as $\\left\\langle{r, \\theta}\\right\\rangle$. Let: :$\\mathbf u_r$ be the unit vector in the direction of the radial coordinate of $p$ :$\\mathbf u_\\theta$ be the unit vector in the direction of the angular coordinate of $p$ Then the derivative of $\\mathbf u_r$ and $\\mathbf u_\\theta$ {{WRT}} $\\theta$ can be expressed as: {{begin-eqn}} {{eqn | l = \\dfrac {\\mathrm d \\mathbf u_r} {\\mathrm d \\theta}       | r = \\mathbf u_\\theta }} {{eqn | l = \\dfrac {\\mathrm d \\mathbf u_\\theta} {\\mathrm d \\theta}       | r = -\\mathbf u_r }} {{end-eqn}}"}
+{"_id": "11895", "title": "Acceleration Vector in Polar Coordinates", "text": "Consider a particle $p$ moving in the plane. Let the position of $p$ at time $t$ be given in polar coordinates as $\\left\\langle{r, \\theta}\\right\\rangle$. Then the acceleration $\\mathbf a$ of $p$ can be expressed as: :$\\mathbf a = \\left({r \\dfrac {\\mathrm d^2 \\theta} {\\mathrm d t^2} + 2 \\dfrac {\\mathrm d r} {\\mathrm d t} \\dfrac {\\mathrm d \\theta} {\\mathrm d t} }\\right) \\mathbf u_\\theta + \\left({\\dfrac {\\mathrm d^2 r} {\\mathrm d t^2} - r \\left({\\dfrac {\\mathrm d \\theta} {\\mathrm d t} }\\right)^2}\\right) \\mathbf u_r$ where: :$\\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$ :$\\mathbf u_\\theta$ is the unit vector in the direction of the angular coordinate of $p$"}
+{"_id": "11897", "title": "Existence of Non-Empty Finite Suprema in Join Semilattice", "text": "Let $\\left({S, \\preceq}\\right)$ be a join semilattice. Let $A$ be a non-empty finite subset of $S$. Then $A$ admits a supremum in $\\left({S, \\preceq}\\right)$."}
+{"_id": "11899", "title": "Euler's Product form of Riemann Zeta Function", "text": "Let $s \\in \\R: s > 1$. Then: :$\\displaystyle \\sum_{k \\mathop \\in \\N_{>0} } \\dfrac 1 {k^s} = \\prod_{p \\mathop \\in \\Bbb P} \\dfrac 1 {1 - 1 / p^s}$ where $\\Bbb P$ denotes the set of all prime numbers."}
+{"_id": "11900", "title": "Variance of Exponential Distribution", "text": "Let $X$ be a continuous random variable with the exponential distribution with parameter $\\beta$. Then the variance of $X$ is: :$\\var X = \\beta^2$"}
+{"_id": "11901", "title": "Cardinality of Singleton", "text": "Let $A$ be a set. Then $\\left\\vert{A}\\right\\vert = 1$ {{iff}} $\\exists a: A = \\left\\{ {a}\\right\\}$ where $\\left\\vert{A}\\right\\vert$ denotes the cardinality of $A$."}
+{"_id": "11902", "title": "Existence of Non-Empty Finite Infima in Meet Semilattice", "text": "Let $\\struct {S, \\preceq}$ be a meet semilattice. Let $A$ be a non-empty finite subset of $S$. Then $A$ admits a infimum in $\\struct {S, \\preceq}$."}
+{"_id": "11905", "title": "Thales' Theorem/Historical Note", "text": "This result was known by the ancient Babylonians as early as $2000$ BCE. The proof of what is now known as Thales' Theorem that {{AuthorRef|Thales of Miletus}} actually used is unknown.  Whether he was aware of the Interior Angles Theorem and used it in his proof cannot be decided. The Interior Angles Theorem is traditionally ascribed to {{AuthorRef|Pythagoras of Samos|Pythagoras}}, who lived near {{AuthorRef|Thales of Miletus|Thales}}, and may have met him. Thus it is possible that {{AuthorRef|Pythagoras of Samos|Pythagoras}} learned of the Interior Angles Theorem either directly or indirectly from {{AuthorRef|Thales of Miletus|Thales}} himself."}
+{"_id": "11908", "title": "External Angle of Triangle equals Sum of other Internal Angles", "text": "The external angle of a triangle equals the sum of the other two internal angles. {{:Euclid:Proposition/I/32}}"}
+{"_id": "11912", "title": "Pappus's Centroid Theorem", "text": "=== Pappus's Centroid Theorem for Surface Area === {{:Pappus's Centroid Theorem/Surface Area}} === Pappus's Centroid Theorem for Volume === {{:Pappus's Centroid Theorem/Volume}}"}
+{"_id": "11913", "title": "Pappus's Centroid Theorem/Volume", "text": "Let $C$ be a plane figure that lies entirely on one side of a straight line $L$. Let $S$ be the solid of revolution generated by $C$ around $L$. Then the volume of $S$ is equal to the area of $C$ multiplied by the distance travelled by the centroid of $C$ around $L$ when generating $S$."}
+{"_id": "11914", "title": "Pappus's Centroid Theorem/Surface Area", "text": "Let $C$ be a plane figure that lies entirely on one side of a straight line $L$. Let $S$ be the solid of revolution generated by $C$ around $L$. Then the surface area of $S$ is equal to the perimeter length of $C$ multiplied by the distance travelled by the centroid of $C$ around $L$ when generating $S$."}
+{"_id": "11915", "title": "Pythagoras's Theorem for Parallelograms", "text": "Let $\\triangle ABC$ be a triangle. Let $ACDE$ and $BCFG$ be parallelograms constructed on the sides $AC$ and $BC$ of $\\triangle ABC$. Let $DE$ and $FG$ be produced to intersect at $H$. Let $AJ$ and $BI$ be constructed on $A$ and $B$ parallel to and equal to $HC$. Then the area of the parallelogram $ABIJ$ equals the sum of the areas of the parallelograms $ACDE$ and $BCFG$."}
+{"_id": "11918", "title": "Mapping Preserves Finite and Filtered Infima", "text": "Let $\\left({S_1, \\preceq_1}\\right)$, $\\left({S_2, \\preceq_2}\\right)$ be meet semilattices. Let $f: S_1 \\to S_2$ be a mapping. Let $f$ preserve finite infima and preserve filtered infima. Then $f$ also preserves all infima"}
+{"_id": "11920", "title": "Volume of Sphere from Surface Area", "text": "The volume $V$ of a sphere of radius $r$ is given by: :$V = \\dfrac {r A} 3$ where $A$ is the surface area of the sphere."}
+{"_id": "11922", "title": "Volume of Solid of Revolution/Historical Note", "text": "The technique of finding the Volume of Solid of Revolution by dividing up the solid of revolution into many thin disks and approximating them to cylinders was devised by {{AuthorRef|Johannes Kepler}} sometime around or after $1612$, reportedly on the occasion of his wedding in $1613$. His inspiration was in the problem of finding the volume of wine barrels accurately. He published his technique in his $1615$ work {{BookLink|Nova Stereometria Doliorum Vinariorum|Johannes Kepler}} (New Stereometry of Wine Barrels). {{AuthorRef|Gottfried Wilhelm von Leibniz}} redefined the problem by applying the techniques of integral calculus around $1680$."}
+{"_id": "11924", "title": "Factors of Mersenne Number M67", "text": "The Mersenne number $M_{67}$ has the factors: :$193 \\, 707 \\, 721$ :$761 \\, 838 \\, 257 \\, 287$"}
+{"_id": "11925", "title": "Lower Bound is Lower Bound for Subset", "text": "Let $\\struct {S, \\preceq}$ be a preordered set. Let $A, B$ be subsets of $S$ such that :$B \\subseteq A$ Let $L$ be an element of $S$. Let $L$ be a lower bound for $A$. Then $L$ is a lower bound for $B$."}
+{"_id": "11926", "title": "Upper Bound is Upper Bound for Subset", "text": "Let $\\left({S, \\preceq}\\right)$ be a preordered set. Let $A, B$ be subsets of $S$ such that :$B \\subseteq A$ Let $U$ be an upper bound for $A$. Then $U$ is an upper bound for $B$."}
+{"_id": "11927", "title": "Largest Rectangle with Given Perimeter is Square", "text": "Let $\\SS$ be the set of all rectangles with a given perimeter $L$. The element of $\\SS$ with the largest area is the square with length of side $\\dfrac L 4$."}
+{"_id": "11928", "title": "Mapping Preserves Finite and Directed Suprema", "text": "Let $\\left({S_1, \\preceq_1}\\right)$, $\\left({S_2, \\preceq_2}\\right)$ be join semilattices. Let $f: S_1 \\to S_2$ be a mapping. Let $f$ preserve finite suprema and preserve directed suprema. Then $f$ also preserves all suprema"}
+{"_id": "11929", "title": "Integral of Power/Historical Note", "text": "The conventional proof of Integral of Power of course holds for all real $n \\ne -1$, not just where $n$ is a strictly positive rational. However, the real point of this page is Fermat's proof, which demonstrates how integration was achieved before the full machinery of calculus had been thoroughly constructed. {{AuthorRef|Bonaventura Francesco Cavalieri}} had previously made progress with this problem, proving it for integral $1 \\le n \\le 9$ but the algebra for the proof of each power was more difficult than the previous one, and he found $10$ too much hard work. The clear beauty of {{AuthorRef|Pierre de Fermat|Fermat}}'s approach was that it works for ''all'' $n$, rational as well as integral."}
+{"_id": "11930", "title": "Integer as Sum of Polygonal Numbers", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then $n$ is: :$(1): \\quad$ Either triangular or the sum of $2$ or $3$ triangular numbers :$(2): \\quad$ Either square or the sum of $2$, $3$ or $4$ square numbers :$(3): \\quad$ Either pentagonal or the sum of $2$, $3$, $4$ or $5$ pentagonal numbers :and so on. That is: :for all $k \\ge 3$, $n$ is the sum of no more than $k$ polygonal numbers of order $k$."}
+{"_id": "11931", "title": "Infima Preserving Mapping on Filters is Increasing", "text": "Let $\\struct {S, \\preceq}$, $\\struct {T, \\precsim}$ be ordered sets. Let $f: S \\to T$ be a mapping. For every filter $F$ in $\\struct {S, \\preceq}$, let $f$ preserve the infimum on $F$. Then $f$ is increasing."}
+{"_id": "11932", "title": "Infimum of Upper Closure of Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T \\subseteq S$. Let $U = T^\\succeq$ be the upper closure of $T$ in $S$. Let $s \\in S$. Then $s$ is the infimum of $T$ {{iff}} it is the infimum of $U$."}
+{"_id": "11935", "title": "Cavalieri's Principle/Extension", "text": "Let two solid figures $S_1$ and $S_2$ have equal height. Let the areas of the sections made by planes parallel to their bases and at equal distances from the bases always have the same ratio. Then the volumes of $S_1$ and $S_2$ are in that same ratio."}
+{"_id": "11936", "title": "Cavalieri's Principle", "text": "Let two solid figures $S_1$ and $S_2$ have equal height. Let sections made by planes parallel to their bases and at equal distances from the bases always have equal area. Then the volumes of $S_1$ and $S_2$ are equal."}
+{"_id": "11938", "title": "Volume of Cone", "text": "Let $K$ be a cone whose base is of area $A$ and whose height is $h$. Then the volume of $K$ is given by: :$V_K = \\dfrac {A h} 3$"}
+{"_id": "11939", "title": "Volume of Cylinder", "text": "The volume $V_C$ of a cylinder whose bases are circles of radius $r$ and whose height is $h$ is given by the formula: :$V_C = \\pi r^2 h$"}
+{"_id": "11941", "title": "Slope of Tangent to Cycloid", "text": "Let $C$ be a cycloid generated by the equations: :$x = a \\paren {\\theta - \\sin \\theta}$ :$y = a \\paren {1 - \\cos \\theta}$ The slope of the tangent to $C$ at the point $\\tuple {x, y}$ is given by: :$\\dfrac {\\d y} {\\d x} = \\cot \\dfrac \\theta 2$"}
+{"_id": "11943", "title": "Upper Closure of Singleton", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $s$ be an element of $S$. Then: :$\\left\\{ {s}\\right\\}^\\succeq = s^\\succeq$ where: :$\\left\\{ {s}\\right\\}^\\succeq$ denotes the upper closure of $\\left\\{ {s}\\right\\}$ :$s^\\succeq$ denotes the upper closure of $s$"}
+{"_id": "11944", "title": "Lower Closure of Singleton", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $s$ be an element of $S$. Then: : $\\left\\{ {s}\\right\\}^\\preceq = s^\\preceq$ where: :$\\left\\{ {s}\\right\\}^\\preceq$ denotes the lower closure of $\\left\\{ {s}\\right\\}$ :$s^\\preceq$ denotes the lower closure of $s$."}
+{"_id": "11945", "title": "Upper Closure of Element is Filter", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $s$ be an element of $S$. Then: :$s^\\succeq$ is a filter in $\\struct {S, \\preceq}$ where $s^\\succeq$ denotes the upper closure of $s$."}
+{"_id": "11948", "title": "Length of Logarithmic Spiral", "text": "Consider a logarithmic spiral $S$ given by the equation: :$r = a e^{b \\theta}$ Construct a tangent to $S$ at the point $Q = \\left({a, 0}\\right)$. Let the tangent cross the $y$-axis at $P$. Then the length of $PQ$ equals the total length of $S$ from $P$ inwards to the origin."}
+{"_id": "11950", "title": "Length of Logarithmic Spiral/Historical Note", "text": "The length of a logarithmic spiral was first found by {{AuthorRef|Evangelista Torricelli}} in $1645$. This was the first time anybody had found the length of a non-straight-line curve for anything other than a circle. Before this had been done, few people could accept that this was possible to do. For example, {{AuthorRef|René Descartes}} had stated in his {{BookLink|La Géométrie|René Descartes}} in $1637$: :''Geometry should not include lines that are like strings, in that they are sometimes straight and sometimes curved, since the ratios between straight and curved lines are not known, and I believe cannot be discovered by human minds.'' {{AuthorRef|Galileo}}'s response was: :'' Who is so blind as not to see that, if there are two equal straight lines, one of which is then bent into a curve, that curve will be equal to the straight line?''"}
+{"_id": "11951", "title": "Infimum of Upper Closure of Element", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $s$ be an element of $S$. Then: :$\\map \\inf {s^\\succeq} = s$ where $s^\\succeq$ denotes the upper closure of $s$."}
+{"_id": "11952", "title": "Volume of Gabriel's Horn", "text": "Consider Gabriel's horn, the solid of revolution formed by rotating about the $x$-axis the curve: :$y = \\dfrac 1 x$ Consider the volume $V$ of the space enclosed by the planes $x = 1$, $x = a$ and the portion of Gabriel's horn where $1 \\le x \\le a$. Then: :$V = \\pi \\left({1 - \\dfrac 1 a}\\right)$"}
+{"_id": "11953", "title": "Pascal's Theorem", "text": "Let $ABCDEF$ be a hexagon whose $6$ vertices lie on a conic section and whose opposite sides are not parallel. Then the points of intersection of the opposite sides, when produced as necessary, all lie on a straight line."}
+{"_id": "11954", "title": "Lower Closure of Element is Ideal", "text": "Let $\\left({S, \\preceq}\\right)$ ne an ordered set. Let $s$ be an element of $S$. Then $s^\\preceq$ is ideal in $\\left({S, \\preceq}\\right)$ where $s^\\preceq$ denotes the lower closure of $s$."}
+{"_id": "11955", "title": "Supremum of Lower Closure of Element", "text": "Let $\\left({S. \\preceq}\\right)$ be an ordered set. Let $s$ be an element of $S$. Then: :$\\sup \\left({s^\\preceq}\\right) = s$ where $s^\\preceq$ denotes the lower closure of $s$."}
+{"_id": "11956", "title": "Upper Closure is Decreasing", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $x, y$ be elements of $S$ such that :$x \\preceq y$ then $y^\\succeq \\subseteq x^\\succeq$ where $y^\\succeq$ denotes the upper closure of $y$."}
+{"_id": "11957", "title": "Lower Closure is Increasing", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $x, y$ be elements of $S$ such that :$x \\preceq y$ then $x^\\preceq \\subseteq y^\\preceq$ where $y^\\preceq$ denotes the lower closure of $y$."}
+{"_id": "11958", "title": "Suprema Preserving Mapping on Ideals is Increasing", "text": "Let $\\left({S, \\preceq}\\right)$, $\\left({T, \\precsim}\\right)$ be ordered sets. Let $f: S \\to T$ be a mapping. For every ideal $I$ in $\\left({S, \\preceq}\\right)$, let $f$ preserve the supremum on $I$. Then $f$ is increasing."}
+{"_id": "11959", "title": "Imaginary Unit to Power of Itself", "text": ":$i^i = e^{-\\pi / 2}$ where $i$ is the imaginary unit. Its decimal expansion is: :$0 \\cdotp 20787 \\, 95763 \\, 50761 \\, 90854 \\, 6955 \\ldots$ {{OEIS|A049006}}"}
+{"_id": "11962", "title": "Imaginary Unit to Power of Itself/Complete", "text": ":$i^i = \\set {\\exp \\paren {\\dfrac {4 k + 3} 2 \\pi}: k \\in \\Z}$ where $i$ is the imaginary unit."}
+{"_id": "11963", "title": "Riemann Zeta Function of 4", "text": "The Riemann zeta function of $4$ is given by: {{begin-eqn}} {{eqn | l = \\map \\zeta 4       | r = \\dfrac 1 {1^4} +  \\dfrac 1 {2^4} +  \\dfrac 1 {3^4} +  \\dfrac 1 {4^4} + \\cdots       | c =  }} {{eqn | r = \\dfrac {\\pi^4} {90}       | c =  }} {{eqn | o = \\approx       | r = 1 \\cdotp 08232 \\, 3 \\ldots       | c =  }} {{end-eqn}}"}
+{"_id": "11964", "title": "Riemann Zeta Function of 6", "text": "The Riemann zeta function of $6$ is given by: {{begin-eqn}} {{eqn | l = \\map \\zeta 6       | r = \\dfrac 1 {1^6} +  \\dfrac 1 {2^6} +  \\dfrac 1 {3^6} +  \\dfrac 1 {4^6} + \\cdots       | c =  }} {{eqn | r = \\dfrac {\\pi^6} {945}       | c =  }} {{eqn | o = \\approx       | r = 1 \\cdotp 01734 \\, 3 \\ldots       | c =  }} {{end-eqn}}"}
+{"_id": "11965", "title": "Infima Preserving Mapping on Filters Preserves Filtered Infima", "text": "Let $\\struct {S, \\preceq}$, $\\struct {T, \\precsim}$ be ordered sets. Let $f: S \\to T$ be a mapping. For every filter $F$ in $\\struct {S, \\preceq}$, let $f$ preserve the infimum on $F$. Then $f$ preserves filtered infima."}
+{"_id": "11966", "title": "Approximate Formula for Number of Partitions", "text": "The number of partitions $\\map p n$ of a (strictly) positive integer $n$ is given by the approximation: :$\\map p n \\approx \\dfrac {e^{\\pi \\sqrt {2 n / 3} } } {4 n \\sqrt 3}$"}
+{"_id": "11967", "title": "Number of Partitions as Coefficient of Power Series", "text": "The number of partitions $\\map p n$ of a (strictly) positive integer $n$ is equal to the coefficient of $x^n$ when the expression: :$\\map f n = \\dfrac 1 {\\paren {1 - x} \\paren {1 - x^2} \\paren {1 - x^3} \\cdots}$ is expanded into a power series. That is: :$\\map f n = 1 + \\map p 1 x + \\map p 2 x^2 + \\map p 3 x^3 + \\cdots$ or: :$\\displaystyle \\sum_{n \\mathop \\in \\Z_{\\ge 0} } \\map p n q^n = \\prod_{j \\mathop \\in \\Z_{>0} } \\dfrac 1 {1 - q^j}$ where $\\map p 0 := 1$."}
+{"_id": "11968", "title": "Suprema Preserving Mapping on Ideals Preserves Directed Suprema", "text": "Let $\\left({S, \\preceq}\\right)$, $\\left({T, \\precsim}\\right)$ be ordered sets. Let $f: S \\to T$ be a mapping. Let every filter $F$ in $\\left({S, \\preceq}\\right)$, $f$ preserve the infimum on $F$. Then $f$ preserves directed suprema."}
+{"_id": "11969", "title": "Euler's Equations of Motion for Rotation of Rigid Body", "text": "Let a rigid body $B$ rotate about an axis $A$ which is fixed in relation to $B$ and parallel to the principal axis of inertia of $B$. Then the rotation of $B$ about $A$ is described by: :$\\mathbf I \\cdot \\dot {\\boldsymbol \\omega} + \\boldsymbol \\omega \\times \\left({\\mathbf I \\cdot \\boldsymbol\\omega}\\right) = \\mathbf M$ where: :$\\mathbf M$ is the applied torque applied to $B$ about $A$ :$\\mathbf I$ is the moment of inertia of $B$ with respect to $A$ :$\\boldsymbol \\omega$ is the angular velocity about $A$."}
+{"_id": "11970", "title": "Euler's Hydrodynamical Equation for Flow of Ideal Incompressible Fluid", "text": ":$\\begin{cases} \\dfrac {D \\mathbf u} {D t} & = & - \\nabla w + \\mathbf g \\\\ \\nabla \\cdot \\mathbf u & = & 0 \\end{cases}$ where: :$\\mathbf u$ denotes the flow velocity vector, with components in an $N$-dimensional space $u_1, u_2, \\dots, u_N$ :$\\dfrac D {D t}$ denotes the material derivative in time :$\\cdot$ denotes the dot product :$\\nabla$ denotes the nabla operator, used to represent the specific thermodynamic work gradient (first equation), and the flow velocity divergence (second equation) :$\\mathbf u \\cdot \\nabla$ is the convective derivative :$w$ is the thermodynamic work per unit mass, the internal source term :$\\mathbf g$ denotes body acceleration per unit mass acting on the continuum. {{stub|Absolutely no idea as to how best to present this. Leaving it as a stub for now.}} {{Namedfor|Leonhard Paul Euler|cat = Euler}}"}
+{"_id": "11972", "title": "Euler Buckling Formula", "text": ":$F = \\dfrac {\\pi^2 E I} {\\paren {K L}^2}$ where: :$F$ = maximum or critical force (vertical load on column) :$E$ = modulus of elasticity :$I$ = area moment of inertia of the cross section of the rod :$L$ = unsupported length of column :$K$ = column effective length factor, whose value depends on the conditions of end support of the column, as follows: ::For both ends pinned (hinged, free to rotate), $K = 1.0$ ::For both ends fixed, $K = 0.50$ ::For one end fixed and the other end pinned, $K \\approx 0.699$ ::For one end fixed and the other end free to move laterally, $K = 2.0$ :$K L$ is the effective length of the column"}
+{"_id": "11973", "title": "Mellin Transform of Heaviside Step Function/Lemma", "text": "Let $t \\in \\R$. Let $s \\in \\C$ with $\\map \\Re s < 0$. Then: :$\\displaystyle \\lim_{t \\mathop \\to +\\infty} t^s = 0$"}
+{"_id": "11974", "title": "Up-Complete Lower Bounded Join Semilattice is Complete", "text": "Let $\\left({S, \\preceq}\\right)$ be an up-complete lower bounded join semillattice. Then $\\left({S, \\preceq}\\right)$ is complete."}
+{"_id": "11975", "title": "Gregory Series", "text": "For $-\\dfrac \\pi 4 \\le \\theta \\le \\dfrac \\pi 4$: :$\\theta = \\tan \\theta - \\dfrac 1 3 \\tan^3 \\theta + \\dfrac 1 5 \\tan^5 \\theta - \\ldots$"}
+{"_id": "11976", "title": "Differential Form of Arc Length on Curved Surface", "text": "The first fundamental form for the element of arc length on a curved surface is given by: :$\\d s^2 = E \\rd u^2 + 2 F \\d u \\rd v + G \\rd v^2$ where $E$, $F$ and $G$ are the coefficients of the first fundamental form."}
+{"_id": "11977", "title": "Lattice is Complete iff it Admits All Suprema", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Then $\\left({S, \\preceq}\\right)$ is a complete lattice {{iff}} :$\\forall X \\subseteq S: X$ admits a supremum."}
+{"_id": "11978", "title": "Theorema Egregium", "text": "The Gaussian curvature of a surface does not change if one bends the surface without stretching it. That is, Gaussian curvature can be determined entirely by measuring angles, distances and their rates on the surface itself, without further reference to the particular way in which the surface is embedded in the ambient $3$-dimensional Euclidean space. Thus the Gaussian curvature is an intrinsic invariant of a surface."}
+{"_id": "11979", "title": "Gauss-Bonnet Theorem", "text": "Let $M$ be a compact $2$-dimensional Riemannian manifold with boundary $\\partial M$. Let $\\Kappa$ be the Gaussian curvature of $M$. Let $k_g$ be the geodesic curvature of $\\partial M$. Then : :$\\displaystyle \\int_M \\kappa \\, \\mathrm d A + \\int_{\\partial M} k_g \\, \\mathrm d s = 2 \\pi \\chi\\left({M}\\right)$ where: :$\\mathrm d A$ is the element of area of the surface :$\\mathrm d s$ is the line element along $\\partial M$ :$\\chi\\left({M}\\right)$ is the Euler characteristic of $M$."}
+{"_id": "11980", "title": "Unique Factorization Theorem for Gaussian Integers", "text": "For every Gaussian integer $n$ such that $\\left\\vert{n}\\right\\vert > 1$, $n$ can be expressed as the product of one or more Gaussian primes, uniquely up to the order in which they appear."}
+{"_id": "11981", "title": "Dirichlet's Principle (Harmonic Functions)", "text": "Let the function $\\map u x$ be the particular solution to Poisson's equation: :$\\Delta u + f = 0$ on a domain $\\Omega$ of $\\R^n$ with boundary condition: :$u = g$ on $\\partial \\Omega$ Then $u$ can be obtained as the minimizer of the Dirichlet's energy: :$\\displaystyle E \\sqbrk {\\map v x} = \\int_\\Omega \\paren {\\frac 1 2 \\cmod {\\nabla v}^2 - v f} \\rd x$ amongst all twice differentiable functions $v$ such that $v = g$ on $\\partial \\Omega$ . This result holds provided that there exists at least one function which makes the Dirichlet integral finite."}
+{"_id": "11982", "title": "Galois Connection is Expressed by Minimum", "text": "Let $\\struct {S, \\preceq}$, $\\struct {T, \\precsim}$ be ordered sets. Let $g: S \\to T$, $d: T \\to S$ be mappings. Then $\\tuple {g, d}$ is a Galois connection {{iff}}: :$g$ is an increasing mapping and ::$\\forall t \\in T: \\map d t = \\min \\set {g^{-1} \\sqbrk {t^\\succsim} }$ where :$\\min$ denotes the minimum :$g^{-1} \\sqbrk {t^\\succsim}$ denotes the image of $t^\\succsim$ under relation $g^{-1}$ :$t^\\succsim$ denotes the upper closure of $t$"}
+{"_id": "11984", "title": "Closed Form for Pentagonal Numbers", "text": "The closed-form expression for the $n$th pentagonal number is: :$P_n = \\dfrac {n \\paren {3 n - 1} } 2$"}
+{"_id": "11986", "title": "Closed Form for Hexagonal Numbers", "text": "The closed-form expression for the $n$th hexagonal number is: :$H_n = n \\paren {2 n - 1}$"}
+{"_id": "11990", "title": "Sum of Even Integers is Even", "text": "The sum of any finite number of even integers is itself even."}
+{"_id": "11991", "title": "Galois Connection is Expressed by Maximum", "text": "Let $\\struct {S, \\preceq}$, $\\struct {T, \\precsim}$ be ordered sets. Let $g: S \\to T$, $d: T \\to S$ be mappings. Then $\\tuple {g, d}$ is a Galois connection {{iff}} :$d$ is an increasing mapping and: ::$\\forall s \\in S: \\map g s = \\map \\max {d^{-1} \\sqbrk {s^\\preceq} }$ where: :$\\max$ denotes the maximum :$d^{-1} \\sqbrk {s^\\preceq}$ denotes the image of $s^\\preceq$ under relation $d^{-1}$ :$s^\\preceq$ denotes the lower closure of $t$"}
+{"_id": "11992", "title": "Compass and Straightedge Construction for Regular Heptagon does not exist", "text": "There exists no compass and straightedge construction for the regular heptagon."}
+{"_id": "11995", "title": "Integer to Rational Power is Irrational iff not Integer or Reciprocal", "text": "Let $m$ be a rational number. Let $n$ be a positive integer. Then $n^m$ is an irrational number {{iff}} $n^{\\size m}$ is not an integer."}
+{"_id": "11999", "title": "Squaring the Circle by Archimedean Spiral", "text": "Let $C$ be a circle. It is possible to construct a square of the same area as $C$ using an Archimedean spiral."}
+{"_id": "12001", "title": "Surface Area of Sphere", "text": "The surface area $A$ of a sphere whose radius $a$ is given by: :$A = 4 \\pi a^2$"}
+{"_id": "12002", "title": "Upper Bound for Number of Grains of Sand to fill Universe", "text": "The number of grains of sand that would fill the Universe, to the extent it was believed to exist in the days of {{AuthorRef|Archimedes of Syracuse|Archimedes}}, is bounded above by $10^{63}$."}
+{"_id": "12004", "title": "Lower Adjoint Preserves All Suprema", "text": "Let $\\left({S, \\preceq}\\right)$, $\\left({T, \\precsim}\\right)$ be ordered sets. Let $d: T \\to S$ be an lower adjoint of Galois connection. Then $d$ preserves all suprema."}
+{"_id": "12005", "title": "Oscillation at Point (Infimum) equals Oscillation at Point (Limit)", "text": "Let $f: D \\to \\R$ be a real function where $D \\subseteq \\R$. Let $x$ be a point in $D$. Let $N_x$ be the set of open subset neighborhoods of $x$. Let $\\map {\\omega_f} x$ be the oscillation of $f$ at $x$ as defined by: :$\\map {\\omega_f} x = \\inf \\set {\\map {\\omega_f} I: I \\in N_x}$ where $\\map {\\omega_f} I$ is the oscillation of $f$ on a real set $I$: :$\\map {\\omega_f} I = \\sup \\set {\\size {\\map f y - \\map f z}: y, z \\in I \\cap D}$ Let $\\map {\\omega^L_f} x$ be the oscillation of $f$ at $x$ as defined by: :$\\map {\\omega^L_f} x = \\displaystyle \\lim_{h \\mathop \\to 0^+} \\map {\\omega_f} {\\openint {x - h} {x + h} }$ Then: :$\\map {\\omega_f} x \\in \\R$ {{iff}} $\\map {\\omega^L_f} x \\in \\R$ and, if $\\map {\\omega_f} x$ and $\\map {\\omega^L_f} x$ exist as real numbers: :$\\map {\\omega_f} x = \\map {\\omega^L_f} x$"}
+{"_id": "12007", "title": "All Infima Preserving Mapping is Upper Adjoint of Galois Connection", "text": "Let $\\left({S, \\preceq}\\right)$ be a complete lattice. Let $\\left({T, \\precsim}\\right)$ be an ordered set. Let $g: S \\to T$ be all infima preserving mapping. Then there exists a mapping $d: T \\to S$ such that $\\left({g, d}\\right)$ is Galois connection and :$\\forall t \\in T: d\\left({t}\\right) = \\min\\left({g^{-1}\\left[{t^\\succsim}\\right]}\\right)$ where :$\\min$ denotes the minimum :$g^{-1}\\left[{t^\\succsim}\\right]$ denotes the image of $t^\\succsim$ under relation $g^{-1}$ :$t^\\succsim$ denotes the upper closure of $t$"}
+{"_id": "12008", "title": "Binomial Theorem/General Binomial Theorem/Historical Note", "text": "The General Binomial Theorem was first conceived by {{AuthorRef|Isaac Newton}} during the years $1665$ to $1667$ when he was living in his home in Woolsthorpe. He announced the result formally, in letters to [https://en.wikipedia.org/wiki/Henry_Oldenburg Henry Oldenburg] on $13$th June $1676$ and $24$th October $1676$ but did not provide a proper proof (at that time the need for the appropriate level of rigor had not been recognised). {{AuthorRef|Leonhard Paul Euler}} made an incomplete attempt in $1774$, but the full proof had to wait for {{AuthorRef|Carl Friedrich Gauss}} to provide it in $1812$. This was, in fact, the first time anything about infinite summations was proved adequately."}
+{"_id": "12009", "title": "Image of Inverse Image", "text": "Let $S, T$ be sets. Let $f: S \\to T$ be a mapping. Let $X$ be a subset of $T$. Then: :$f\\left[{f^{-1}\\left[{X}\\right]}\\right] \\subseteq X$ where: :$f^{-1}\\left[{X}\\right]$ denotes the image of $X$ under the relation $f^{-1}$."}
+{"_id": "12010", "title": "All Suprema Preserving Mapping is Lower Adjoint of Galois Connection", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $\\struct {T, \\precsim}$ be a complete lattice. Let $d: T \\to S$ be all suprema preserving mapping. Then there exists a mapping $g: S \\to T$ such that $\\left({g, d}\\right)$ is Galois connection and :$\\forall s \\in S: \\map g s = \\map \\max {d^{-1} \\sqbrk {s^\\preceq} }$ where :$\\min$ denotes the maximum :$d^{-1} \\sqbrk {s^\\preceq}$ denotes the image of $s^\\preceq$ under relation $d^{-1}$ :$s^\\preceq$ denotes the lower closure of $t$"}
+{"_id": "12012", "title": "Galois Connection Implies Order on Mappings", "text": "Let $\\struct {S, \\preceq}$, $\\struct {T, \\precsim}$ be ordered sets. Let $g: S \\to T$ and $d: T \\to S$ be mappings such that :$\\tuple {g, d}$ is Galois connection. Then $d \\circ g \\preceq I_S$ and $I_T \\precsim g \\circ d$ where :$\\preceq, \\precsim$ denote the orderings on mappings, :$I_S$ denotes the identity mapping of $S$"}
+{"_id": "12015", "title": "Primitive of exp (-x^2) has no Solution in Elementary Functions", "text": "The primitive: :$\\displaystyle \\int \\map \\exp {-x^2} \\rd x$ cannot be expressed in terms of a finite number of elementary functions."}
+{"_id": "12016", "title": "Primitive of exp x over x has no Solution in Elementary Functions", "text": "The primitive: :$\\displaystyle \\int \\frac {e^{a x} \\, \\mathrm d x} x$ cannot be expressed in terms of a finite number of elementary functions."}
+{"_id": "12017", "title": "Primitive of Sine of x over x has no Solution in Elementary Functions", "text": "The primitive: :$\\displaystyle \\int \\frac {\\sin x \\, \\mathrm d x} x$ cannot be expressed in terms of a finite number of elementary functions."}
+{"_id": "12018", "title": "Primitive of Reciprocal of Logarithm of x has no Solution in Elementary Functions", "text": "The primitive: :$\\displaystyle \\int \\frac {\\mathrm d x} {\\ln x}$ cannot be expressed in terms of a finite number of elementary functions."}
+{"_id": "12019", "title": "Pi is Transcendental", "text": "$\\pi$ (pi) is transcendental."}
+{"_id": "12021", "title": "Solution of Qunitic Equation using Elliptic Functions", "text": "A quintic equation can be solved using elliptic functions."}
+{"_id": "12022", "title": "Riemann Surface is Path-Connected", "text": "A Riemann surface is path-connected."}
+{"_id": "12023", "title": "Properties of Riemann Surface", "text": "This page gathers some results about Riemann surfaces."}
+{"_id": "12024", "title": "Radó's Theorem (Riemann Surfaces)", "text": "A Riemann surface is second countable."}
+{"_id": "12025", "title": "Conformal Isomorphism of Universal Cover of Riemann Surface", "text": "The universal cover of a Riemann surface is conformally isomorphic to either: :the Riemann sphere :the complex plane or :the unit disk."}
+{"_id": "12026", "title": "Riemann Surface is Metrizable", "text": "A Riemann surface is metrizable."}
+{"_id": "12027", "title": "Riemann Surface admits Metric of Constant Curvature", "text": "A Riemann surface admits a metric of constant curvature."}
+{"_id": "12028", "title": "Riemann Sphere is only Elliptic Riemann Surface", "text": "The Riemann sphere is the only elliptic Riemann surface (up to conformal isomorphism)."}
+{"_id": "12029", "title": "Parabolic Riemann Surface is Plane, Punctured Plane or Torus", "text": "A parabolic Riemann surface is conformally isomorphic to either: :the complex plane :the punctured complex plane $\\C \\setminus \\set 0$ or: :a torus."}
+{"_id": "12030", "title": "Riemann's Rearrangement Theorem", "text": "Let $S$ be an infinite series which is conditionally convergent. Then its terms can be arranged in a permutation so that the new series converges to any given value, or diverges."}
+{"_id": "12031", "title": "Riemann-Christoffel Tensor in Two Dimensions is Gaussian Curvature", "text": "Let $M$ be a Riemannian manifold of dimension $2$. Then the Riemann-Christoffel tensor on $M$ reduces to the Gaussian curvature on $M$."}
+{"_id": "12032", "title": "Riemannian Manifold has Zero Gaussian Curvature iff Euclidean", "text": "Let $M$ be a Riemannian manifold of dimension $2$. The Gaussian curvature on $M$ is zero {{iff}} the Riemannian metric on $\\mathcal M$ is the same as the Euclidean metric."}
+{"_id": "12033", "title": "Particle on Curved Surface under no Force moves along Geodesic", "text": "Consider a particle $P$ which is constrained to move on a curved surface $C$. Let $P$ be such that no force acts upon it. Then $P$ moves along a geodesic."}
+{"_id": "12034", "title": "Sum of Reciprocals of Powers as Euler Product", "text": "Let $\\zeta$ be the Riemann zeta function. Let $s \\in \\C$ be a complex number with real part $\\sigma > 1$. Then: :$\\displaystyle \\map \\zeta s = \\prod_{\\text {$p$ prime} } \\frac 1 {1 - p^{-s} }$ where the infinite product runs over the prime numbers."}
+{"_id": "12037", "title": "Shift Mapping is Lower Adjoint iff Appropriate Maxima Exist", "text": "Let $\\left({S, \\preceq}\\right)$ be a meet semilattice. Then the following two conditions are equivalent: $(1): \\quad \\forall x \\in S, f:S \\to S: \\left({\\forall s \\in S: f\\left({s}\\right) = x \\wedge s}\\right) \\implies f$ is lower adjoint $(2): \\quad \\forall x, t \\in S: \\max \\left\\{ {s \\in S: x \\wedge s \\preceq t}\\right\\}$ exists."}
+{"_id": "12038", "title": "Brouwerian Lattice iff Shift Mapping is Lower Adjoint", "text": "Let $\\left({S, \\preceq}\\right)$ be a lattice. Then $\\left({S, \\preceq}\\right)$ is a Brouwerian lattice {{iff}} :$\\forall x \\in S, f: S \\to S: \\left({\\forall s \\in S: f \\left({s}\\right) = x \\wedge s}\\right) \\implies f$ is a lower adjoint"}
+{"_id": "12039", "title": "Brouwerian Lattice is Distributive", "text": "Let $\\left({S, \\preceq}\\right)$ be a Brouwerian lattice. Then $\\left({S, \\preceq}\\right)$ is a distributive lattice"}
+{"_id": "12040", "title": "Definite Integral of Reciprocal of Root of a Squared minus x Squared", "text": ":$\\displaystyle \\int_0^x \\frac {\\d t} {\\sqrt{1 - t^2} } = \\arcsin x$"}
+{"_id": "12041", "title": "Meromorphic Function is Elliptic iff Doubly Periodic", "text": "Let $f: \\C \\to \\C$ be a meromorphic function.  Then $\\map f z$ is an elliptic function {{iff}} it is also doubly periodic."}
+{"_id": "12042", "title": "Entire Function is Transcendental iff Power Series Expansion is Infinite", "text": "Let $f$ be an entire function. Then $f$ is '''transcendental''' {{iff}} the power series expansion of $f$ has an infinite number of coefficients which are non-zero."}
+{"_id": "12043", "title": "Entire Function is Transcendental iff not Complex Polynomial Function", "text": "Let $f$ be an entire function. Then $f$ is '''transcendental''' {{iff}} it is not a complex polynomial function."}
+{"_id": "12044", "title": "Complex Function is Entire iff it has Everywhere Convergent Power Series", "text": "Let $f: \\C \\to \\C$ be a complex function. Then $f$ is an '''entire function''' {{iff}} $f$ can be given by an everywhere convergent power series: :$\\displaystyle \\map f z = \\sum_{n \\mathop = 0}^\\infty a_n z^n; \\quad \\lim_{n \\mathop \\to \\infty} \\sqrt [n] {\\size {a_n} } = 0$"}
+{"_id": "12045", "title": "Brouwerian Lattice is Upper Bounded", "text": "Let $\\struct {S, \\vee, \\wedge, \\preceq}$ be a Brouwerian lattice. Then $S$ is upper bounded."}
+{"_id": "12046", "title": "Extreme Value Theorem/Real Function", "text": "Let $f$ be continuous in a closed real interval $\\left[{a \\,.\\,.\\, b}\\right]$. Then: :$\\forall x \\in \\left[{a \\,.\\,.\\, b}\\right]: \\exists x_s \\in \\left[{a \\,.\\,.\\, b}\\right]: f \\left({x_s}\\right) \\le f \\left({x}\\right)$ :$\\forall x \\in \\left[{a \\,.\\,.\\, b}\\right]: \\exists x_n \\in \\left[{a \\,.\\,.\\, b}\\right]: f \\left({x_n}\\right) \\ge f \\left({x}\\right)$"}
+{"_id": "12047", "title": "Galileo's Paradox", "text": "The natural numbers $\\N$ are in one-to-one correspondence with their squares: That is, the mapping: :$\\forall n \\in \\N: f: n \\mapsto n^2$ is a bijection. Hence the set of square numbers is equinumerous to the set of natural numbers. That is, a set is equinumerous to one of its proper subsets."}
+{"_id": "12048", "title": "Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a Brouwerian lattice. Let $a$ be an element of $S$. Let $g, d: S \\to S$ be mappings such that :$\\forall s \\in S: g\\left({s}\\right) = a \\to s$ and :$\\forall s \\in S: d\\left({s}\\right) = a \\wedge s$ Then $\\left({g, d}\\right)$ is Galois connection."}
+{"_id": "12049", "title": "Inequality with Meet Operation is Equivalent to Inequality with Relative Pseudocomplement in Brouwerian Lattice", "text": "Let $\\struct {S, \\vee, \\wedge, \\preceq}$ be s Brouwerian lattice. Let $a, x, y \\in S$. Then :$a \\wedge x \\preceq y$ {{iff}} $x \\preceq a \\to y$"}
+{"_id": "12050", "title": "Weierstrass Approximation Theorem", "text": "Let $f$ be a real function which is continuous on the closed interval $\\Bbb I$. Then $f$ can be uniformly approximated on $\\Bbb I$ by a polynomial function to any given degree of accuracy."}
+{"_id": "12051", "title": "Construction of Right Angle by Stretched Rope", "text": "Let a rope be knotted at regular intervals so that there are $12$ equal segments separated by knots. Tie the rope in a loop consisting of those $12$ segments. Fix one of the knots to the ground at the point you want the right angle to be placed. Stretch the rope tightly in the direction of one of the legs of the right angle you want to create and fix the fourth knot to the ground. Stretch the rope tightly in the direction of the other leg of the right angle you want to create and fix the third knot in the ground, at the same time making sure the remaining segment of $5$ knots is also tight. The point where the $3$-knot section and the $4$-knot section meet, the rope will be bent at a right angle. :400px"}
+{"_id": "12055", "title": "Up-Complete Product", "text": "Let $\\struct {S, \\preceq_1}$, $\\struct {T, \\preceq_2}$ be non-empty ordered sets. Let $\\struct {S \\times T, \\preceq}$ be Cartesian product of $\\struct {S, \\preceq_1}$ and $\\struct {T, \\preceq_2}$."}
+{"_id": "12056", "title": "Cartesian Product of Ordered Sets is Ordered Set", "text": "Let $\\left({S_1, \\preceq_1}\\right)$, $\\left({S_2, \\preceq_2}\\right)$ be ordered sets. Let $\\left({S_1 \\times S_2, \\preceq}\\right)$ be the Cartesian product of $\\left({S_1, \\preceq_1}\\right)$ and $\\left({S_2, \\preceq_2}\\right)$. Then $\\left({S_1 \\times S_2, \\preceq}\\right)$ is also an ordered set."}
+{"_id": "12061", "title": "Motion of Rocket in Outer Space", "text": "Let $B$ be a rocket travelling in outer space. Let the velocity of $B$ at time $t$ be $\\mathbf v$. Let the mass of $B$ at time $t$ be $m$. Let the exhaust velocity of $B$ be constant at $\\mathbf b$. Then the equation of motion of $B$ is given by: :$m \\dfrac {\\mathrm d \\mathbf v} {\\mathrm d t} = - \\mathbf b \\dfrac {\\mathrm d m} {\\mathrm d t}$"}
+{"_id": "12063", "title": "Velocity of Rocket in Outer Space", "text": "Let $B$ be a rocket travelling in outer space. Let the velocity of $B$ at time $t$ be $\\mathbf v$. Let the mass of $B$ at time $t$ be $m$. Let the exhaust velocity of $B$ be constant at $\\mathbf b$. Then the velocity of $B$ at time $t$ is given by: :$\\map {\\mathbf v} t = \\map {\\mathbf v} 0 + \\mathbf b \\ln \\dfrac {\\map m 0} {\\map m t}$ where $\\map {\\mathbf v} 0$ and $\\map m 0$ are the velocity and mass of $B$ at time $t = 0$."}
+{"_id": "12064", "title": "Acceleration of Rocket in Outer Space", "text": "Let $B$ be a rocket travelling in outer space. Let the velocity of $B$ at time $t$ be $\\mathbf v$. Let the mass of $B$ at time $t$ be $m$. Let the exhaust velocity of $B$ be constant at $\\mathbf b$. Then the acceleration of $B$ at time $t$ is given by: :$\\mathbf a = \\dfrac 1 m \\left({- \\mathbf b \\dfrac {\\mathrm d m} {\\mathrm d t} }\\right)$"}
+{"_id": "12065", "title": "Up-Complete Product/Lemma 2", "text": "Let $X$ be a directed subset of $S \\times T$. Then :$\\map {\\pr_1^\\to} X$ and $\\map {\\pr_2^\\to} X$ are directed where  :$\\pr_1$ denotes the first projection on $S \\times T$ :$\\pr_2$ denotes the second projection on $S \\times T$ :$\\map {\\pr_1^\\to} X$ denotes the image of $X$ under $\\pr_1$"}
+{"_id": "12067", "title": "Sum of Cosines of Multiples of Angle", "text": "{{begin-eqn}} {{eqn | l = \\frac 1 2 + \\sum_{k \\mathop = 1}^n \\cos \\left({k x}\\right)       | r = \\frac 1 2 + \\cos x + \\cos 2 x + \\cos 3 x + \\cdots + \\cos n x       | c =  }} {{eqn| r = \\frac {\\sin \\left({\\left({2 n + 1}\\right) x / 2}\\right)} {2 \\sin \\left({x / 2}\\right)}       | c =  }} {{end-eqn}} where $x$ is not an integer multiple of $2 \\pi$."}
+{"_id": "12068", "title": "Eratosthenes' Measurement of Earth", "text": "The circumference of Earth was measured by {{AuthorRef|Eratosthenes of Cyrene}} to be of the order of $45 \\, 000 \\, \\mathrm{km}$."}
+{"_id": "12070", "title": "Sum of Reciprocals of Twin Primes", "text": "The sum of the reciprocals of all the twin primes: :$\\dfrac 1 3 + \\dfrac 1 5 + \\dfrac 1 7 + \\dfrac 1 {11} + \\dfrac 1 {13} + \\dfrac 1 {17} + \\dfrac 1 {19} + \\dfrac 1 {29} + \\dfrac 1 {31} + \\cdots$ is either finite or convergent."}
+{"_id": "12072", "title": "Prime Number Theorem in Eulerian Logarithmic Integral Form", "text": "The Prime Number Theorem is equivalent to: :$\\displaystyle \\lim_{x \\mathop \\to \\infty} \\frac {\\map \\pi x} {\\map \\Li x} = 1$ where: :$\\map \\pi x$ is the prime-counting function :$\\map \\Li x$ is the Eulerian logarithmic integral: ::$\\displaystyle \\map \\Li x := \\int_2^x \\dfrac {\\d t} {\\ln t}$"}
+{"_id": "12073", "title": "Exponential of Rational Number is Irrational", "text": "Let $r$ be a rational number such that $r \\ne 0$. Then: :$e^r$ is irrational where $e$ is Euler's number."}
+{"_id": "12074", "title": "Rational Points on Graph of Exponential Function", "text": "Consider the graph $f$ of the exponential function in the real Cartesian plane $\\R^2$: :$f := \\set {\\tuple {x, y} \\in \\R^2: y = e^x}$ The only rational point of $f$ is $\\tuple {0, 1}$."}
+{"_id": "12075", "title": "Rational Points on Graph of Logarithm Function", "text": "Consider the graph of the logarithm function in the real Cartesian plane $\\R^2$: :$f := \\set {\\tuple {x, y} \\in \\R^2: y = \\ln x}$ The only rational point of $f$ is $\\tuple {1, 0}$."}
+{"_id": "12076", "title": "Rational Points on Graph of Sine Function", "text": "Consider the graph of the sine function in the real Cartesian plane $\\R^2$: :$f := \\set {\\tuple {x, y} \\in \\R^2: y = \\sin x}$ The only rational point of $f$ is $\\tuple {0, 0}$."}
+{"_id": "12077", "title": "Rational Points on Graph of Cosine Function", "text": "Consider the graph of the cosine function in the real Cartesian plane $\\R^2$: :$f := \\left\\{ {\\left({x, y}\\right) \\in \\R^2: y = \\cos x}\\right\\}$ The only rational point of $f$ is $\\left({0, 1}\\right)$."}
+{"_id": "12079", "title": "Meet-Continuous iff Ideal Supremum is Meet Preserving", "text": "Let $\\mathscr S = \\struct {S, \\vee, \\wedge, \\preceq}$ be an up-complete lattice. Let $f: \\map {\\it Ids} {\\mathscr S} \\to S$ be a mapping such that: :$\\forall I \\in \\map {\\it Ids} {\\mathscr S}: \\map f I = \\sup_{\\mathscr S} I$ where  :$\\map {\\it Ids} {\\mathscr S}$ denotes the set of all ideals in $\\mathscr S$ Then :$\\mathscr S$ is meet-continuous {{iff}} :$f$ preserves meet as a mapping from $\\struct {\\map {\\it Ids} {\\mathscr S}, \\subseteq}$ into $\\mathscr S$"}
+{"_id": "12080", "title": "Rational Number is Algebraic", "text": "Let $r \\in \\Q$ be a rational number. Then $r$ is also an algebraic number."}
+{"_id": "12082", "title": "Square Root of 2 is Algebraic of Degree 2", "text": "The square root of $2$ is an algebraic number of degree $2$."}
+{"_id": "12083", "title": "Constructible Length with Compass and Straightedge", "text": "Let $L$ be a line segment in a Eucldiean space. Let the length of $L$ be $d$. Let $L'$ be a line segment of length $d'$ constructed from $L$ using a compass and straightedge construction. Then: :$d' = q d$ where $q$ is an algebraic number whose degree is at most $2$."}
+{"_id": "12084", "title": "Gelfond-Schneider Constant is Transcendental", "text": "The Gelfond-Schneider constant: :$2^{\\sqrt 2}$ is transcendental."}
+{"_id": "12089", "title": "Sum of Bernoulli Numbers by Binomial Coefficients Vanishes", "text": ":$\\forall n \\in \\Z_{>1}: \\displaystyle \\sum_{k \\mathop = 0}^{n - 1} \\binom n k B_k = 0$ where $B_k$ denotes the $k$th Bernoulli number."}
+{"_id": "12090", "title": "Bernoulli Numbers are Rational", "text": "The Bernoulli numbers are rational."}
+{"_id": "12091", "title": "Power Series Expansion for Tangent Function/Proof of Convergence", "text": "The radius of convergence of the Power Series Expansion for Tangent Function: :$\\displaystyle \\tan x = \\sum_{n \\mathop = 1}^\\infty \\frac {\\paren {-1}^{n - 1} 2^{2 n} \\paren {2^{2 n} - 1} B_{2 n} } {\\paren {2 n}!} x^{2 n - 1}$ where $B_{2 n}$ denotes the Bernoulli numbers, is given as: :$\\size x < \\dfrac \\pi 2$"}
+{"_id": "12094", "title": "Power Series Expansion for Tangent Function/Sequence", "text": "The Power Series Expansion for Tangent Function begins: :$\\tan x = x + \\dfrac 1 3 x^3 + \\dfrac 2 {15} x^5 + \\dfrac {17} {315} x^7 + \\dfrac {62} {2835} x^9 + \\cdots$"}
+{"_id": "12095", "title": "Partial Fractions Expansion of Cotangent", "text": "Let $x \\in \\R - \\Z$ , that is such that $x$ is a real number that is not an integer. Then: :$\\pi \\cot \\pi x = \\dfrac 1 2 + 2 x \\displaystyle \\sum_{n \\mathop = 1}^\\infty \\frac 1 {x^2 - n^2}$"}
+{"_id": "12098", "title": "Tangent to Cycloid is Vertical at Cusps", "text": "The tangent to the cycloid whose locus is given by: : $x = a \\paren {\\theta - \\sin \\theta}$ : $y = a \\paren {1 - \\cos \\theta}$ is vertical at the cusps."}
+{"_id": "12099", "title": "Tangent to Cycloid passes through Top of Generating Circle", "text": "Let $C$ be a cycloid generated by the equations: :$x = a \\paren {\\theta - \\sin \\theta}$ :$y = a \\paren {1 - \\cos \\theta}$ Then the tangent to $C$ at a point $P$ on $C$ passes through the top of the generating circle of $C$."}
+{"_id": "12100", "title": "Tangent to Cycloid", "text": "Let $C$ be a cycloid generated by the equations: : $x = a \\left({\\theta - \\sin \\theta}\\right)$ : $y = a \\left({1 - \\cos \\theta}\\right)$ Then the tangent to $C$ at a point $\\left({x, y}\\right)$ on $C$ is given by the equation: :$y - a \\left({1 - \\cos \\theta}\\right) = \\dfrac {\\sin \\theta} {1 - \\cos \\theta} \\left({x - a \\theta + a \\sin \\theta}\\right)$"}
+{"_id": "12101", "title": "Equation of Hypocycloid", "text": "Let a circle $C_1$ of radius $b$ roll without slipping around the inside of a circle $C_2$ of (larger) radius $a$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let $P$ be a point on the circumference of $C_1$. Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \\tuple {a, 0}$ on the $x$-axis. Let $H$ be the hypocycloid traced out by the point $P$. Let $\\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The point $P = \\tuple {x, y}$ is described by the equations: :$x = \\paren {a - b} \\cos \\theta + b \\map \\cos {\\paren {\\dfrac {a - b} b} \\theta}$ :$y = \\paren {a - b} \\sin \\theta - b \\map \\sin {\\paren {\\dfrac {a - b} b} \\theta}$"}
+{"_id": "12102", "title": "Meet in Set of Ideals", "text": "Let $\\mathscr S = \\left({S, \\preceq}\\right)$ be a meet semilattice. Let ${\\it Ids}\\left({\\mathscr S}\\right)$ be the set of all ideals in $\\mathscr S$. Let $I_1, I_2$ be ideals in $\\mathscr S$. Then :$I_1 \\wedge I_2 = \\left\\{ {i_1 \\wedge i_2: i_1 \\in I_1, i_2 \\in I_2}\\right\\}$ where :$I_1 \\wedge I_2$ denotes the meet in $\\left({ {\\it Ids}\\left({\\mathscr S}\\right), \\subseteq}\\right)$"}
+{"_id": "12103", "title": "Intersection of Semilattice Ideals is Ideal", "text": "Let $\\struct {S, \\preceq}$ be a meet semilattice. Let $I_1, I_2$ be ideals in $\\struct {S, \\preceq}$. Then $I_1 \\cap I_2$ is ideal in $\\struct {S, \\preceq}$"}
+{"_id": "12104", "title": "Number of Cusps of Hypocycloid from Integral Ratio of Circle Radii", "text": "Let $H$ be a hypocycloid $H$ generated by a circle $C_1$ of radius $b$ rolling within a circle $C_2$ of (larger) radius $a$. Let $a = n b$ where $n$ is an integer. Then $H$ has $n$ cusps."}
+{"_id": "12105", "title": "Number of Cusps of Hypocycloid from Rational Ratio of Circle Radii", "text": "Consider the hypocycloid $H$ generated by a circle $C_1$ of radius $b$ rolling within a circle $C_2$ of (larger) radius $a$. Let $k = \\dfrac a b$ be a rational number. Let $k$ be expressed in canonical form: :$k = \\dfrac p q$ where $p$ and $q$ are coprime. Then $H$ has $p$ cusps."}
+{"_id": "12106", "title": "Cusps of Hypocycloid from Irrational Ratio of Circle Radii", "text": "Consider the hypocycloid $H$ generated by a rotor $C_1$ of radius $b$ rolling within a stator $C_2$ of (larger) radius $a$. Let $k = \\dfrac a b$ be an irrational number. Then $H$ has an infinite number of cusps, which are evenly and densely distributed around the circumference of $C_2$."}
+{"_id": "12107", "title": "Equation of Astroid", "text": "Let $H$ be the astroid generated by the rotor $C_1$ of radius $b$ rolling without slipping around the inside of a stator $C_2$ of radius $a = 4 b$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let $P$ be a point on the circumference of $C_1$. Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \\tuple {a, 0}$ on the $x$-axis. Let $\\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. === Parametric Form === {{:Equation of Astroid/Parametric Form}} === Cartesian Form === {{:Equation of Astroid/Cartesian Form}}"}
+{"_id": "12109", "title": "Tangent to Astroid between Coordinate Axes has Constant Length", "text": "Let $C_1$ be a circle of radius $b$ roll without slipping around the inside of a circle $C_2$ of radius $a = 4 b$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let $P$ be a point on the circumference of $C_1$. Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \\tuple {a, 0}$ on the $x$-axis. Let $H$ be the astroid formed by the locus of $P$. The segment of the tangent to $H$ between the $x$-axis and the $y$-axis is constant, immaterial of the point of tangency."}
+{"_id": "12110", "title": "Pendulum Contained by Cycloid moves along Cycloidal Path", "text": "Let a pendulum with a flexible rod be suspended from a point $P$. Let the rod be contained by a pair of bodies shaped as the arcs of a cycloid such that $P$ is the cusp between those two arcs. :400px Then the bob is constrained to move such that its path traces the arc of a cycloid."}
+{"_id": "12111", "title": "Lower Closure of Meet of Lower Closures", "text": "Let $\\left({S, \\preceq}\\right)$ be a meet semilattice. Let $D_1, D_2$ be subsets of $S$. Then :$\\left\\{ {i_1 \\wedge i_2: i_1 \\in D_1^\\preceq, i_2 \\in D_2^\\preceq}\\right\\}^\\preceq = \\left\\{ {i_1 \\wedge i_2: i_1 \\in D_1, i_2 \\in D_2}\\right\\}^\\preceq$ where :$D_1^\\preceq$ denotes the lower closure of $D_1$."}
+{"_id": "12112", "title": "Equation of Cycloid in Cartesian Coordinates", "text": "Consider a circle of radius $a$ rolling without slipping along the $x$-axis of a cartesian plane. Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis. Consider the cycloid traced out by the point $P$. Let $\\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The point $P = \\tuple {x, y}$ is described by the equation: :$a \\sin^{-1} \\paren {\\dfrac {\\sqrt {2 a y - y^2} } a} = \\sqrt {2 a y - y^2} + x$"}
+{"_id": "12113", "title": "Second Derivative of Locus of Cycloid", "text": "Consider a circle of radius $a$ rolling without slipping along the x-axis of a cartesian plane. Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis. Consider the cycloid traced out by the point $P$. Let $\\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The second derivative of the locus of $P$ is given by: :$y'' = -\\dfrac a {y^2}$"}
+{"_id": "12118", "title": "Rate of Change of Cartesian Coordinates of Cycloid", "text": "Let a circle $C$ of radius $a$ roll without slipping along the x-axis of a cartesian plane at a constant speed such that the center moves with a velocity $\\mathbf v_0$ in the direction of increasing $x$. Consider a point $P$ on the circumference of this circle. Let $\\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. Then the rate of change of $x$ and $y$ can be expresssed as: {{begin-eqn}} {{eqn | l = \\frac {\\d x} {\\d t}       | r = \\mathbf v_0 \\paren {1 - \\cos \\theta}       | c =  }} {{eqn | l = \\frac {\\d y} {\\d t}       | r = \\mathbf v_0 \\sin \\theta       | c =  }} {{end-eqn}} where $\\theta$ is the angle turned by $C$ after time $t$."}
+{"_id": "12119", "title": "Maximum Rate of Change of X Coordinate of Cycloid", "text": "Let a circle $C$ of radius $a$ roll without slipping along the x-axis of a cartesian plane at a constant speed such that the center moves with a velocity $\\mathbf v_0$ in the direction of increasing $x$. Consider a point $P$ on the circumference of this circle. Let $\\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The maximum rate of change of $x$ is $2 \\mathbf v_0$, which happens when $P$ is at the top of the circle $C$."}
+{"_id": "12121", "title": "Lower Closure is Closure Operator", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Then :lower closure of set is a closure operator."}
+{"_id": "12122", "title": "Area inside Astroid", "text": "The area inside an astroid $H$ constructed within a circle of radius $a$ is given by: :$\\AA = \\dfrac {3 \\pi a^2} 8$"}
+{"_id": "12123", "title": "Length of Arc of Astroid", "text": "The total length of the arcs of an astroid constructed within a stator of radius $a$ is given by: :$\\LL = 6 a$"}
+{"_id": "12124", "title": "Area of Surface of Revolution from Astroid", "text": "Let $H$ be the astroid constructed within a circle of radius $a$. The surface of revolution formed by rotating $H$ around the $x$-axis: :400px evaluates to: :$\\mathcal S = \\dfrac {12 \\pi a^2} 5$"}
+{"_id": "12125", "title": "Tusi Couple is Diameter of Stator", "text": "A Tusi couple is a degenerate case of the hypocycloid whose form is a straight line that forms a diameter of the stator."}
+{"_id": "12126", "title": "Meet Preserves Directed Suprema", "text": "Let $\\mathscr S = \\struct {S, \\preceq}$ be an up-complete meet semilattice such that :$\\forall x \\in S$, a directed subset $D$ of $S$: $x \\preceq \\sup D \\implies x \\preceq \\sup \\set {x \\wedge y: y \\in D}$"}
+{"_id": "12127", "title": "Equation of Astroid/Parametric Form", "text": "The point $P = \\tuple {x, y}$ is described by the parametric equation: :$\\begin{cases} x & = a \\cos^3 \\theta \\\\ y & = a \\sin^3 \\theta \\end{cases}$ where $\\theta$ is the angle between the $x$-axis and the line joining the origin to the center of $C_1$."}
+{"_id": "12128", "title": "Equation of Astroid/Cartesian Form", "text": "The point $P = \\tuple {x, y}$ is described by the equation: :$x^{2/3} + y^{2/3} = a^{2/3}$"}
+{"_id": "12129", "title": "Preceding iff Meet equals Less Operand", "text": "Let $\\left({S, \\preceq}\\right)$ be a meet semilattice. Let $x, y \\in S$. Then :$x \\preceq y$ {{iff}} $x \\wedge y = x$"}
+{"_id": "12130", "title": "Simple Harmonic Motion of Point on Tusi Couple", "text": "Let $C_1$ and $C_2$ be the rotor and stator respectively of a Tusi couple $H$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let the center of $C_1$ move at a constant angular velocity $\\omega$ around the center of $C_2$. Let $P$ be the point on the circumference of $C_1$ whose locus is $H$. Let $C_1$ be initially positioned so that $P$ its point of tangency to $C_2$, located on the $x$-axis. Then $P$ moves back and forward on the $x$-axis with simple harmonic motion with period $\\dfrac {2 \\pi} \\omega$ and maximum speed $a \\omega$."}
+{"_id": "12131", "title": "Maximum Rate of Change of Y Coordinate of Astroid", "text": "Let $C_1$ and $C_2$ be the rotor and stator respectively of an astroid $H$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let the center $C$ of $C_1$ move at a constant angular velocity $\\omega$ around the center of $C_2$. Let $P$ be the point on the circumference of $C_1$ whose locus is $H$. Let $C_1$ be positioned at time $t = 0$ so that $P$ its point of tangency to $C_2$, located on the $x$-axis. Let $\\theta$ be the angle made by $OC$ to the $x$-axis at time $t$. Then the maximum rate of change of the $y$ coordinate of $P$ in the first quadrant occurs when $P$ is at the point where: :$x = a \\paren {\\dfrac 1 3}^{3/2}$ :$y = a \\paren {\\dfrac 2 3}^{3/2}$"}
+{"_id": "12132", "title": "Equation of Deltoid", "text": "The point $P = \\tuple {x, y}$ is described by the parametric equation: :$\\begin{cases} x & = 2 b \\cos \\theta + b \\cos 2 \\theta \\\\ y & = 2 b \\sin \\theta - b \\sin 2 \\theta \\end{cases}$ where $\\theta$ is the angle between the $x$-axis and the line joining the origin to the center of $C_1$."}
+{"_id": "12133", "title": "Length of Arc of Deltoid", "text": "The total length of the arcs of a deltoid constructed within a stator of radius $a$ is given by: :$\\LL = \\dfrac {16 a} 3$"}
+{"_id": "12134", "title": "Equation of Epicycloid", "text": "Let a circle $C_1$ of radius $b$ roll without slipping around the outside of a circle $C_2$ of radius $a$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let $P$ be a point on the circumference of $C_1$. Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \\tuple {a, 0}$ on the $x$-axis. Let $H$ be the epicycloid traced out by the point $P$. Let $\\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The point $P = \\tuple {x, y}$ is described by the equations: :$x = \\paren {a + b} \\cos \\theta - b \\map \\cos {\\paren {\\dfrac {a + b} b} \\theta}$ :$y = \\paren {a + b} \\sin \\theta - b \\map \\sin {\\paren {\\dfrac {a + b} b} \\theta}$"}
+{"_id": "12136", "title": "Equation of Nephroid", "text": "The point $P = \\tuple {x, y}$ is described by the parametric equation: :$\\begin{cases} x & = 3 b \\cos \\theta - b \\cos 3 \\theta \\\\ y & = 3 b \\sin \\theta - b \\sin 3 \\theta \\end{cases}$ where $\\theta$ is the angle between the $x$-axis and the line joining the origin to the center of $C_1$."}
+{"_id": "12137", "title": "Length of Arc of Nephroid", "text": "The total length of the arcs of a nephroid constructed around a stator of radius $a$ is given by: :$\\LL = 12 a$"}
+{"_id": "12138", "title": "Supremum of Meet Image of Directed Set", "text": "Let $\\struct {S, \\preceq}$ be an up-complete meet semilattice. Let $f: S \\times S \\to S$ be a mapping such that :$\\forall \\tuple {x, y} \\in S \\times S: \\map f {x, y} = x \\wedge y$ Let $D$ be directed subset of $S \\times S$ in Cartesian product $\\struct {S \\times S, \\precsim}$ of $\\struct {S, \\preceq}$ and $\\struct {S, \\preceq}$. Then: :$\\map \\sup {\\map {f^\\to} D} = \\sup \\set {x \\wedge y: x \\in \\map {\\pr_1^\\to} D, y \\in \\map {\\pr_2^\\to} D}$ where :$\\pr_1$ denotes the first projection on $S \\times S$ :$\\pr_2$ denotes the second projection on $S \\times S$ :$\\map {\\pr_1^\\to} D$ denotes the image of $D$ under $\\pr_1$"}
+{"_id": "12139", "title": "Meet is Increasing", "text": "Let $\\left({S, \\preceq}\\right)$ be a meet semilattice. Let $f: S \\times S \\to S$ be a mapping such that :$\\forall s, t \\in S: f\\left({s, t}\\right) = s \\wedge t$ Then: :$f$ is increasing as a mapping from Cartesian product $\\left({S\\times S, \\precsim}\\right)$ of $\\left({S, \\preceq}\\right)$ and $\\left({S, \\preceq}\\right)$ into $\\left({S, \\preceq}\\right)$."}
+{"_id": "12140", "title": "Meet Preserves Directed Suprema/Lemma 2", "text": "Let $x$ be an element of $S$, $D$ be a directed subset of $S$. Then: :$\\paren {\\sup D} \\wedge x = \\sup \\set {d \\wedge x: d \\in D}$"}
+{"_id": "12141", "title": "Supremum by Suprema of Directed Set in Cartesian Product", "text": "Let $\\struct {S, \\preceq}$ be an up-complete meet semilattice. Let $\\struct {S \\times S, \\precsim}$ be the Cartesian product of $\\struct {S, \\preceq}$ and $\\struct {S, \\preceq}$. Let $D$ be a directed subset of $S \\times S$. Then: :$\\sup D = \\tuple {\\map \\sup {\\map {\\pr_1^\\to} D}, \\map \\sup {\\map {\\pr_2^\\to} D} }$"}
+{"_id": "12142", "title": "Evolute of Circle is its Center", "text": "The evolute of a circle is a single point: its center."}
+{"_id": "12145", "title": "Parametric Equations for Evolute/Formulation 1", "text": "Let $C$ be a curve expressed as the locus of an equation $\\map f {x, y} = 0$. The parametric equations for the evolute of $C$ can be expressed as: :$\\begin{cases} X = x - \\dfrac {y' \\paren {1 + y'^2} } {y''} \\\\ Y = y + \\dfrac {1 + y'^2} {y''} \\end{cases}$ where: :$\\left({x, y}\\right)$ denotes the Cartesian coordinates of a general point on $C$ :$\\left({X, Y}\\right)$ denotes the Cartesian coordinates of a general point on the evolute of $C$ :$y'$ and $y''$ denote the derivative and second derivative respectively of $y$ {{WRT|Differentiation}} $x$."}
+{"_id": "12147", "title": "Parametric Equations for Evolute/Formulation 2", "text": "Let $C$ be a curve expressed as the locus of an equation $\\map f {x, y} = 0$. The parametric equations for the evolute of $C$ can be expressed as: :$\\begin{cases} X = x - \\dfrac {y' \\paren {x'^2 + y'^2} } {x' y'' - y' x''} \\\\ Y = y + \\dfrac {x' \\paren {x'^2 + y'^2} } {x' y'' - y' x''} \\end{cases}$ where: :$\\tuple {x, y}$ denotes the Cartesian coordinates of a general point on $C$ :$\\tuple {X, Y}$ denotes the Cartesian coordinates of a general point on the evolute of $C$ :$x'$ and $x''$ denote the derivative and second derivative respectively of $x$ {{WRT|Differentiation}} $t$ :$y'$ and $y''$ denote the derivative and second derivative respectively of $y$ {{WRT|Differentiation}} $t$."}
+{"_id": "12149", "title": "Normal to Curve is Tangent to Evolute", "text": "Let $C$ be a curve defined by a real function which is twice differentiable. Let the curvature of $C$ be non-constant. Let $P$ be a point on $C$. Let $Q$ be the center of curvature of $C$ at $P$. The normal to $C$ at $P$ is tangent to the evolute $E$ of $C$ at $Q$."}
+{"_id": "12150", "title": "Length of Arc of Evolute equals Difference in Radii of Curvature", "text": "Let $C$ be a curve defined by a real function which is twice differentiable. Let the curvature of $C$ be non-constant. The length of arc of the evolute $E$ of $C$ between any two points $Q_1$ and $Q_2$ of $C$ is equal to the difference between the radii of curvature at the corresponding points $P_1$ and $P_2$ of $C$."}
+{"_id": "12151", "title": "Curve is Involute of Evolute", "text": "Let $C$ be a curve defined by a real function which is twice differentiable. Let the curvature of $C$ be non-constant. Let $E$ be the evolute $C$. Then the involute of $E$ is $C$."}
+{"_id": "12152", "title": "Evolute of Cycloid is Cycloid", "text": "The evolute of a cycloid is another cycloid."}
+{"_id": "12153", "title": "Body behaves as Particle under Gravitation", "text": "A body of finite volume under the force of gravity behaves identically to a point mass under the same force."}
+{"_id": "12154", "title": "Meet of Directed Subsets is Directed", "text": "Let $\\left({S, \\preceq}\\right)$ be a meet semilattice. Let $D_1, D_2$ be directed subset of $S$. Then :$\\left\\{ {x \\wedge y: x \\in D_1, y \\in D_2}\\right\\}$ is directed subset of $S$"}
+{"_id": "12155", "title": "Image of Directed Subset under Increasing Mapping is Directed", "text": "Let $\\left({S, \\preceq}\\right)$, $\\left({T, \\precsim}\\right)$ be ordered sets. Let $f: S \\to T$ be an increasing mapping. Let $D$ be a directed subset of $S$. Then :$f^\\to \\left({D}\\right)$ is also a directed subset of $T$ where :$f^\\to \\left({D}\\right)$ denotes the image of $D$ under $f$."}
+{"_id": "12156", "title": "Mass of Sun from Gravitational Constant", "text": "Let the gravitational constant be known. Let the mean distance from the Earth to the sun be known. Then it is possible to calculate the mass of the sun."}
+{"_id": "12157", "title": "Meet is Directed Suprema Preserving implies Meet of Suprema equals Supremum of Meet of Directed Subsets", "text": "Let $\\struct {S, \\preceq}$ be an up-complete meet semilattice. Let $\\struct {S \\times S, \\precsim}$ be the Cartesian product of $\\struct {S, \\preceq}$ and $\\struct {S, \\preceq}$. Let $f: S \\times S \\to S$ be a mapping such that :$\\forall s, t \\in S: \\map f {s, t} = s \\wedge t$ and :$f$ preserves directed suprema. Let $D_1, D_2$ be directed subsets of $S$. Then :$\\paren {\\sup D_1} \\wedge \\paren {\\sup D_2} = \\sup \\set {x \\wedge y: x \\in D_1, y \\in D_2}$"}
+{"_id": "12158", "title": "Meet of Suprema equals Supremum of Meet of Ideals implies Ideal Supremum is Meet Preserving", "text": "Let $\\mathscr S = \\struct {S, \\wedge, \\preceq}$ be an up-complete meet semilattice. Let $f: \\map {\\it Ids} {\\mathscr S} \\to S$ be a mapping such that: :$\\forall I \\in \\map {\\it Ids} {\\mathscr S}: \\map f I = \\sup_{\\mathscr S} I$ where  :$\\map {\\it Ids} {\\mathscr S}$ denotes the set of all ideals in $\\mathscr S$ Let :$\\forall I_1, I_2 \\in \\map {\\it Ids} {\\mathscr S}: \\paren {\\sup I_1} \\wedge \\paren {\\sup I_2} = \\sup \\set {i \\wedge j: i \\in I_1, j \\in I_2}$ Then: $f$ preserves meet as a mapping from $\\struct {\\map {\\it Ids} {\\mathscr S}, \\subseteq}$ into $\\mathscr S$"}
+{"_id": "12161", "title": "Field Norm of Quaternion is Positive Definite", "text": "Let $\\mathbf x = a \\mathbf 1 + b \\mathbf i + c \\mathbf j + d \\mathbf k$ be a quaternion. Let $\\overline {\\mathbf x}$ be the conjugate of $\\mathbf x$. The field norm of $\\mathbf x$: :$\\map n {\\mathbf x} := \\cmod {\\mathbf x \\overline {\\mathbf x} }$ is positive definite."}
+{"_id": "12162", "title": "Field Norm of Quaternion is Multiplicative", "text": "Let $\\mathbf x$ be a quaternion. Let $\\overline {\\mathbf x}$ be the conjugate of $\\mathbf x$. The field norm of $\\mathbf x$: :$\\map n {\\mathbf x} := \\size {\\mathbf x \\overline {\\mathbf x} }$ is a multiplicative function."}
+{"_id": "12163", "title": "Equivalence of Definitions of Oscillation at Point for Real Functions", "text": "Let $X$ and $Y$ be real sets. Let $f: X \\to Y$ be a real function. Let $x \\in X$. {{TFAE|def = Oscillation/Real Space/Oscillation at Point|view = Oscillation at a Point}}"}
+{"_id": "12165", "title": "Supremum of Cartesian Product", "text": "Let $\\left({S_1, \\preceq_1}\\right)$, $\\left({S_2, \\preceq_2}\\right)$ be an ordered set. Let $\\left({S_1 \\times S_2, \\precsim}\\right)$ be the Cartesian product of $\\left({S_1, \\preceq_1}\\right)$ and $\\left({S_2, \\preceq_2}\\right)$. Let $X_1$ be a non-empty subset of $S_1$, $X_2$ be a non-empty subset of $S_2$ such that :$X_1$ and $X_2$ admit suprema. Then :$X_1 \\times X_2$ admits a supremum and :$\\sup \\left({X_1 \\times X_2}\\right) = \\left({\\sup X_1, \\sup X_2}\\right)$"}
+{"_id": "12167", "title": "Nonassociative Division Algebra with Real Scalars has Dimension of Power of 2", "text": "Let $A$ be a nonassociative division algebra with real scalars. Then the dimension of $A$ is a power of $2$."}
+{"_id": "12168", "title": "Bott-Milnor-Kervaire 1,2,4,8 Theorem", "text": "Let $A$ be a division algebra with real scalars. Then the dimension of $A$ is either: :$1$: the real numbers $\\R$ :$2$: the complex numbers $\\C$ :$4$: the quaternions $\\Bbb H$ or: :$8$: the octonions $\\Bbb O$."}
+{"_id": "12169", "title": "Montel's Theorem", "text": "Let $U \\subseteq \\C$ be an open subset of the complex numbers. Let $\\mathcal H \\left({U}\\right)$ be the space of holomorphic mappings on $U$. Then a family of mappings $\\mathcal F \\subseteq \\mathcal H \\left({U}\\right)$ is normal {{iff}} $\\mathcal F$ is locally bounded."}
+{"_id": "12170", "title": "Vitali's Convergence Theorem", "text": "Let $U$ be an  open,  connected subset of $\\C$. Let $S \\subseteq U$ contain a limit point $\\sigma$. Let $\\sequence {f_n}_{n \\mathop \\in \\N}$ be a  normal family of holomorphic mappings $f_n : U \\to \\C$. Let $\\sequence {f_n}_{n \\mathop \\in \\N}$  converge to some holomorphic mapping $f : U \\to \\C$ at $\\sigma$. Then $f_n$ converges uniformly to $f$ on all compact subsets of $U$."}
+{"_id": "12171", "title": "Identity Theorem", "text": "Let $U$ be an open connected subset of the complex plane $\\C$. Let $f$ and $g$ be complex functions whose domain is $U$. Let $S = \\left\\{{z \\in U: f \\left({z}\\right) = g \\left({z}\\right)}\\right\\}$. Let $f$ and $g$ be analytic on $U$. Let $S$ have a limit point in $U$. Then: :$\\forall z \\in U : f \\left({z}\\right) = g \\left({z}\\right)$"}
+{"_id": "12172", "title": "Exponential Sequence is Uniformly Convergent on Compact Sets", "text": "Let $\\mathcal E = \\left \\langle{E_n}\\right \\rangle$ denote the sequence of functions $E_n: \\C \\to \\C$ defined as: :$E_n \\left({z}\\right) = \\left({1 + \\dfrac z n}\\right)^n$ Let $K$ be a compact subset of $\\C$. Then $\\mathcal E$ is uniformly convergent on $K$."}
+{"_id": "12173", "title": "Uniformly Convergent Sequence Evaluated on Convergent Sequence", "text": "Let $X = \\struct {A, d}$ and $Y = \\struct {B, \\rho}$ be metric spaces. {{explain|Definition:Complete Metric Space is invoked, but it is called just \"metric space\". Analyse to determine whether it needs to be complete here or not.}} Let $K$ be a subspace of $X$. Let $\\FF = \\sequence {f_n}$ be a sequence of continuous mappings $f_n: X \\to Y$ uniformly convergent on $K$. Let $\\sequence {a_n}$ be a Convergent Sequence in $K$ with limit $a \\in K$. Then $\\sequence {\\map {f_n} {a_n} }$ is convergent such that: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map {f_n} {a_n} = \\map f a$"}
+{"_id": "12174", "title": "Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Ideals", "text": "Let $\\mathscr S = \\struct {S, \\vee, \\wedge, \\preceq}$ be an up-complete lattice. Then :$\\mathscr S$ is meet-continuous {{iff}} :for every ideals $I, J$ in $\\mathscr S$: $\\paren {\\sup I} \\wedge \\paren {\\sup J} = \\sup \\set {i \\wedge j: i \\in I, j \\in J}$"}
+{"_id": "12175", "title": "Ellipse is Mechanical Curve", "text": "The ellipse is a mechanical curve. That is, it cannot be constructed using a compass and straightedge. However, as many points as is desired can be so constructed that lie on any given ellipse."}
+{"_id": "12176", "title": "Parabola is Mechanical Curve", "text": "The parabola is a mechanical curve. That is, it cannot be constructed using a compass and straightedge. However, as many points as is desired can be so constructed that lie on any given parabola."}
+{"_id": "12177", "title": "Hyperbola is Mechanical Curve", "text": "The hyperbola is a mechanical curve. That is, it cannot be constructed using a compass and straightedge. However, as many points as is desired can be so constructed that lie on any given hyperbola."}
+{"_id": "12179", "title": "Derivative of Sequence of Holomorphic Functions", "text": "Let $U$ be an  open,  connected subset of $\\C$. Let $\\left \\langle{f_n}\\right \\rangle$ be a sequence of  holomorphic mappings $f_n: U \\to \\C$. Let $\\left \\langle{f_n}\\right \\rangle$  converge pointwise to some function $f: U \\to \\C$. Let $\\left \\langle{f_n}\\right \\rangle$  converge uniformly on  compact subsets of $U$. Then $f$ is holomorphic on $U$. Further, the sequence of  derivatives $\\left \\langle{f_n'}\\right \\rangle$ converges to $f'$ on $U$.  This convergence is uniform on compact subsets of $U$."}
+{"_id": "12180", "title": "Uniformly Convergent Sequence of Bounded Functions is Uniformly Bounded", "text": "Let $X = \\left({A, d}\\right)$ and $Y = \\left({B, \\rho}\\right)$ be metric spaces. Let $\\left \\langle{f_i}\\right \\rangle_{i \\in I}$ be a uniformly convergent sequence of mappings $f_i: X \\to Y$. $\\forall i \\in I$, let $f_i$ be bounded. Then $\\left \\langle{f_i}\\right \\rangle$ is uniformly bounded."}
+{"_id": "12181", "title": "Product of Uniformly Convergent Sequences of Bounded Functions is Uniformly Convergent", "text": "Let $X = \\struct {A, d}$ and $Y = \\struct {B, \\rho}$ be metric spaces. Let $\\sequence {f_n}$ and $\\sequence {g_n}$ be sequences of mappings from $X$ to $Y$. Let $\\sequence {f_n}$ and $\\sequence {g_n}$ be uniformly convergent on some subspace $S$ of $X$.  $\\forall n \\in \\N$, let $f_n$ and $g_n$ be bounded. Then the sequence $\\sequence {f_n g_n}$ is uniformly convergent on $S$. {{explain|for the product to be defined, we probably want $Y$ to have some more structure}}"}
+{"_id": "12182", "title": "Area between Radii and Curve in Polar Coordinates", "text": "Let $C$ be a curve expressed in polar coordinates $\\polar {r, \\theta}$ as: :$r = \\map g \\theta$ where $g$ is a real function. Let $\\theta = \\theta_a$ and $\\theta = \\theta_b$ be the two rays from the pole at angles $\\theta_a$ and $\\theta_b$ to the polar axis respectively. Then the area $\\mathcal A$ between $\\theta_a$, $\\theta_b$ and $C$ is given by: :$\\displaystyle \\mathcal A = \\int \\limits_{\\theta \\mathop = \\theta_a}^{\\theta \\mathop = \\theta_b} \\frac {\\paren {\\map g \\theta}^2 \\rd \\theta} 2$ as long as $\\paren {\\map g \\theta}^2$ is integrable."}
+{"_id": "12183", "title": "Derivative of Exponential Function/Complex", "text": "The complex exponential function is its own derivative. That is: :$\\map {D_z} {\\exp z} = \\exp z$"}
+{"_id": "12184", "title": "Exponential Sequence is Eventually Increasing", "text": "Let $\\sequence {E_n}$ be the sequence of real functions $E_n: \\R \\to \\R$ defined as: :$\\map {E_n} x = \\paren {1 + \\dfrac x n}^n$ Then, for sufficiently large $n \\in \\N$, $\\sequence {\\map {E_n} x}$ is increasing with respect to $n$. That is: :$\\forall x \\in \\R: \\forall n \\in \\N: n \\ge \\ceiling {\\size x} \\implies \\map {E_n} x \\le \\map {E_{n + 1} } x$ where $\\ceiling x$ denotes the ceiling of $x$."}
+{"_id": "12185", "title": "Euler's Number: Limit of Sequence implies Base of Logarithm", "text": "Let $e$ be  Euler's number defined by: :$\\displaystyle e := \\lim_{n \\mathop \\to \\infty} \\left({1 + \\frac 1 n}\\right) ^n$ Then $e$ is the unique solution to the equation $\\ln \\left({x}\\right) = 1$. That is: :$\\ln \\left({x}\\right) = 1 \\iff x = e$"}
+{"_id": "12186", "title": "Diophantine Equation y cubed equals x squared plus 2", "text": "The indeterminate Diophantine equation: :$y^3 = x^2 + 2$ has only one integer solution: :$x = 5, y = 3$"}
+{"_id": "12187", "title": "Continuous Extension from Dense Subset", "text": "Let $X$ be a metric space. Let $D$ be a dense subset of $X$. Let $f: D \\to \\R$ be a uniformly continuous mapping. Then there exists a unique continuous extension of $f$ to $X$."}
+{"_id": "12188", "title": "Sum of two Fourth Powers cannot be Fourth Power", "text": "$\\forall a, b, c \\in \\Z_{>0}$, the equation $a^4 + b^4 = c^4$ has no solutions."}
+{"_id": "12189", "title": "Moore-Osgood Theorem", "text": "Let $X$ and $Y$ be metric spaces. Let $S$ be a subspace of $X$. Let $c$ be a limit point of $S$. Let $\\left \\langle{f_n}\\right \\rangle$ be a sequence of mappings $f_n : X \\to Y$. Suppose that: :$(1): \\quad \\left \\langle{f_n}\\right \\rangle$ is uniformly convergent on $S$ :$(2): \\quad \\displaystyle \\forall n \\in \\N : \\lim_{x \\to c} f_n \\left({ x }\\right)$ exists Then: :$\\displaystyle \\lim_{x \\to c} \\lim_{n \\to \\infty} f_n \\left({ x }\\right) = \\lim_{n \\to \\infty} \\lim_{x \\to c} f_n \\left({ x }\\right)$"}
+{"_id": "12190", "title": "Cycloid is Best Shape for Arch", "text": "The optimal shape for an arch is a cycloid."}
+{"_id": "12191", "title": "Center of Gravity of Cycloid", "text": "Let $C$ be a cycloid. The center of gravity of an arc of $C$ is at {{stub|research needed}}"}
+{"_id": "12194", "title": "Ore Graph is Connected", "text": "Let $G = \\left({V, E}\\right)$ be an Ore graph. Then $G$ is connected."}
+{"_id": "12195", "title": "Power Function to Rational Power permits Unique Continuous Extension", "text": "Let $a \\in \\R_{> 0}$. Let $f: \\Q \\to \\R$ be the real-valued function defined as: :$\\map f q = a^q$ where $a^q$ denotes $a$ to the power of $q$. Then there exists a unique continuous extension of $f$ to $\\R$."}
+{"_id": "12201", "title": "Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Directed Subsets", "text": "Let $\\mathscr S = \\struct {S, \\vee, \\wedge, \\preceq}$ be an up-complete lattice. Then: :$\\mathscr S$ is meet-continuous {{iff}}: :for every directed subsets $D_1, D_2$ of $S$: $\\paren {\\sup D_1} \\wedge \\paren {\\sup D_2} = \\sup \\set {d_1 \\wedge d_2: d_1 \\in D_1, d_2 \\in D_2}$"}
+{"_id": "12202", "title": "Brouwerian Lattice iff Meet-Continuous and Distributive", "text": "Let $\\mathscr S = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Then :$\\mathscr S$ is a Brouwerian lattice {{iff}} :$\\mathscr S$ is meet-continuous and distributive."}
+{"_id": "12203", "title": "Power is Well-Defined/Integer", "text": "Let $x$ be a non-zero real number. Let $k$ be an integer. Then $x^k$ is well-defined."}
+{"_id": "12205", "title": "Plane Figure with Maximum Area for given Perimeter is Circle", "text": "Let $F$ be a plane figure. Let $P$ be the length of the perimeter of $F$. Let the area of $F$ be the largest of all the plane figures whose perimeters are of length $P$. Then $F$ is a circle."}
+{"_id": "12206", "title": "Real Number between Zero and One is Greater than Power/Natural Number", "text": "Let $x \\in \\R$. Let $0 < x < 1$. Let $n$ be a natural number. Then: : $0 < x^n \\le x$"}
+{"_id": "12207", "title": "Power Function is Strictly Increasing over Positive Reals/Natural Exponent", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive integer. Let $f: \\R_{>0} \\to \\R$ be the real function defined as: :$f \\left({x}\\right) = x^n$ where $x^n$ denotes $x$ to the power of $n$. Then $f$ is strictly increasing."}
+{"_id": "12208", "title": "Root is Strictly Increasing", "text": "Let $x \\in \\R_{> 0}$. Let $n \\in \\N$. Let $f: \\R_{> 0} \\to \\R$ be the real function defined as: :$\\map f x = \\sqrt [n] x$ where $\\sqrt [n] x$ denotes the $n$th root of $x$. Then $f$ is strictly increasing."}
+{"_id": "12209", "title": "Shift Mapping is Lower Adjoint iff Meet is Distributive with Supremum", "text": "Let $\\left({S, \\preceq}\\right)$ be a complete lattice. Then :$\\forall x \\in S, f: S \\to S:\\left({\\forall y \\in S: f\\left({y}\\right) = x \\wedge y}\\right) \\implies f$ is lower adjoint of Galois connection {{iff}} :$\\forall x \\in S, X \\subseteq S: x \\wedge \\sup X = \\sup \\left\\{ {x \\wedge y: y \\in X}\\right\\}$"}
+{"_id": "12210", "title": "Greatest Common Divisor is Associative", "text": "Let $a, b, c \\in \\Z$. Then: :$\\gcd \\left\\{ {a, \\gcd \\left\\{{b, c}\\right\\} }\\right\\} = \\gcd \\left\\{ {\\gcd \\left\\{{a, b}\\right\\}, c}\\right\\}$ where $\\gcd$ denotes the greatest common divisor."}
+{"_id": "12211", "title": "Lowest Common Multiple is Associative", "text": "Let $a, b, c \\in \\Z$. Then: :$\\lcm \\left\\{{a, \\lcm \\left\\{{ b , c }\\right\\} }\\right\\} = \\lcm \\left\\{{\\lcm \\left\\{{a, b}\\right\\}, c}\\right\\}$ where $\\lcm$ denotes the lowest common multiple."}
+{"_id": "12212", "title": "LCM of Coprime Integers", "text": "Let $a, b \\in \\Z_{>0}$ be coprime integers. Then: :$\\lcm \\set {a, b} = a b$ where $\\lcm$ denotes the lowest common multiple."}
+{"_id": "12213", "title": "GCD with One Fixed Argument is Multiplicative Function", "text": "Let $a, b, c \\in \\Z: b \\perp c$ where $b \\perp c$ denotes that $b$ is coprime to $c$. Then: :$\\gcd \\set {a, b} \\gcd \\set {a, c} = \\gcd \\set {a, b c}$ That is, GCD is multiplicative."}
+{"_id": "12214", "title": "Power Function on Base between Zero and One is Strictly Decreasing/Positive Integer", "text": "Let $a \\in \\R$ be a real number such that $0 < a < 1$. Let $f: \\Z_{\\ge 0} \\to \\R$ be the real-valued function defined as: :$\\map f n = a^n$ where $a^n$ denotes $a$ to the power of $n$. Then $f$ is strictly decreasing."}
+{"_id": "12215", "title": "Power Function on Base between Zero and One is Strictly Decreasing/Rational Number", "text": "Let $a \\in \\R$ be a real number such that $0 < a < 1$. Let $f: \\Q \\to \\R$ be the real-valued function defined as: :$\\map f q = a^q$ where $a^q$ denotes $a$ to the power of $q$. Then $f$ is strictly decreasing."}
+{"_id": "12216", "title": "Power Function on Base Greater than One is Strictly Increasing/Positive Integer", "text": "Let $a \\in \\R$ be a real number such that $a > 1$. Let $f: \\Z_{\\ge 0} \\to \\R$ be the real-valued function defined as: :$\\map f n = a^n$ where $a^n$ denotes $a$ to the power of $n$. Then $f$ is strictly increasing."}
+{"_id": "12217", "title": "Root of Reciprocal is Reciprocal of Root", "text": "Let $x \\in \\R_{\\ge 0}$. Let $n \\in \\N$. Let $\\sqrt [n] x$ denote the $n$th root of $x$. Then: :$\\sqrt [n] {\\dfrac 1 x} = \\dfrac 1 {\\sqrt [n] x}$"}
+{"_id": "12218", "title": "Power Function on Base Greater than One is Strictly Increasing/Rational Number", "text": "Let $a \\in \\R$ be a real number such that $a > 1$. Let $f: \\Q \\to \\R$ be the real-valued function defined as: :$\\map f q = a^q$ where $a^q$ denotes $a$ to the power of $q$. Then $f$ is strictly increasing."}
+{"_id": "12219", "title": "Existence of Square Roots of Positive Real Number", "text": "Let $r \\in \\R_{\\ge 0}$ be a positive real number. Then: :$\\exists y_1 \\in \\R_{\\ge 0}: {y_1}^2 = r$ :$\\exists y_2 \\in \\R_{\\le 0}: {y_2}^2 = r$"}
+{"_id": "12220", "title": "Inequality iff Difference is Positive", "text": "Let $x, y \\in \\R$. Then the following are equivalent: :$(1): \\quad x < y$ :$(2): \\quad y - x > 0$"}
+{"_id": "12221", "title": "Power Function on Base between Zero and One is Strictly Decreasing/Integer", "text": "Let $a \\in \\R$ be a real number such that $0 < a < 1$. Let $f: \\Z \\to \\R$ be the real-valued function defined as: :$\\map f k = a^k$ where $a^k$ denotes $a$ to the power of $k$. Then $f$ is strictly decreasing."}
+{"_id": "12222", "title": "Meet-Continuous iff Meet Preserves Directed Suprema", "text": "Let $\\mathscr S = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be an up-complete lattice. Let $\\left({S \\times S, \\precsim}\\right)$ be the Cartesian product of $\\mathscr S$ and $\\mathscr S$. Let $f: S \\times S \\to S$ be a mapping such that :$\\forall x, y \\in S: f\\left({x, y}\\right) = x \\wedge y$ Then :$\\mathscr S$ is meet-continuous {{iff}} :$f$ preserves directed suprema"}
+{"_id": "12224", "title": "Real Star-Algebra is Commutative", "text": "Let $A = \\left( A_F , \\oplus \\right)$ be a real *-algebra whose conjugation is denoted as $*$. Then: :$\\forall a,b \\in A, a \\oplus b = b \\oplus a$ That is, real *-algebra is commutative."}
+{"_id": "12225", "title": "Meet-Continuous implies Shift Mapping Preserves Directed Suprema", "text": "Let $\\mathscr S = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a meet-continuous lattice. Let $x \\in S$. Let $f: S \\to S$ be a mapping such that :$\\forall y \\in S: f\\left({y}\\right) = x \\wedge y$ Then :$f$ preserves directed suprema."}
+{"_id": "12226", "title": "Power of Strictly Positive Real Number is Strictly Positive/Positive Integer", "text": "Let $x \\in \\R_{>0}$ be a (strictly) positive real number. Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Then: :$x^n > 0$ where $x^n$ denotes the $n$th power of $x$."}
+{"_id": "12227", "title": "Power of Positive Real Number is Positive/Integer", "text": "Let $x \\in \\R_{>0}$ be a (strictly) positive real number. Let $n \\in \\Z$ be an integer. Then: :$x^n > 0$ where $x^n$ denotes the $n$th power of $x$."}
+{"_id": "12228", "title": "Power of Positive Real Number is Positive/Rational Number", "text": "Let $x \\in \\R_{>0}$ be a (strictly) positive real number. Let $q \\in \\Q$ be a rational number. Then: :$x^q > 0$ where $x^q$ denotes the $x$ to the power of $q$."}
+{"_id": "12229", "title": "Power Function is Monotone/Rational Number", "text": "Let $a \\in \\R_{>0}$. Let $f: \\Q \\to \\R$ be the real-valued function defined as: :$\\map f r = a^r$ where $a^r$ denotes $a$ to the power of $r$. Then $f$ is monotone. Further, $f$ is strictly monotone unless $a = 1$."}
+{"_id": "12230", "title": "Power Function tends to One as Power tends to Zero/Rational Number", "text": "Let $a \\in \\R_{> 0}$. Let $f: \\Q \\to \\R$ be the real-valued function defined as: :$\\map f q = a^q$ where $a^q$ denotes $a$ to the power of $q$. Then: :$\\displaystyle \\lim_{x \\mathop \\to 0} \\map f x = 1$"}
+{"_id": "12231", "title": "Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number/Lemma", "text": "Let $a \\in \\R$ be a real number such that $a > 1$. Let $r \\in \\Q_{> 0}$ be a strictly positive rational number such that $r < 1$. Then: :$1 < a^r < 1 + a r$"}
+{"_id": "12232", "title": "Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number", "text": "Let $a \\in \\R_{> 0}$ be a strictly positive real number such that $a > 1$. Let $f: \\Q \\to \\R$ be the real-valued function defined as: :$\\map f r = a^r$ where $a^r$ denotes $a$ to the power of $r$. Then: :$\\displaystyle \\lim_{r \\mathop \\to 0} \\map f r = 1$"}
+{"_id": "12233", "title": "Power Function on Base between Zero and One Tends to One as Power Tends to Zero/Rational Number", "text": "Let $a \\in \\R_{> 0}$ be a strictly positive real number such that $0 < a < 1$. Let $f: \\Q \\to \\R$ be the real-valued function defined as: :$\\map f r = a^r$ where $a^r$ denotes $a$ to the power of $r$. Then: :$\\displaystyle \\lim_{r \\mathop \\to 0} \\map f r = 1$"}
+{"_id": "12234", "title": "Euler's Number to Rational Power permits Unique Continuous Extension", "text": "Let $e$ be Euler's number. Let $f: \\Q \\to \\R$ be the real-valued function defined as: :$f \\left({q}\\right) = e^q$ where $e^q$ denotes  $e$ to the power of $q$. Then there exists a unique continuous extension of $f$ to $\\R$."}
+{"_id": "12235", "title": "Open Unit Interval is Proper Subset of Closed Unit Interval", "text": "The open unit interval: :$I_o = \\left({0 \\,.\\,.\\, 1}\\right)$ is a proper subset of the closed unit interval: :$I_c = \\left[{0 \\,.\\,.\\, 1}\\right]$"}
+{"_id": "12236", "title": "Vector Cross Product satisfies Jacobi Identity", "text": "Let $\\mathbf a, \\mathbf b, \\mathbf c$ be vectors in $3$ dimensional Euclidean space. Let $\\times$ denotes the cross product. Then: :$\\mathbf a \\times \\paren {\\mathbf b \\times \\mathbf c} + \\mathbf b \\times \\paren {\\mathbf c \\times \\mathbf a} + \\mathbf c \\times \\paren {\\mathbf a \\times \\mathbf b} = \\mathbf 0$ That is, the cross product operation satisfies the Jacobi identity."}
+{"_id": "12237", "title": "Meet-Continuous and Distributive implies Shift Mapping Preserves Finite Suprema", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a meet-continuous distributive complete lattice. Let $x \\in S$. Let $f: S \\to S$ be a mapping such that :$\\forall y \\in S: f \\left({y}\\right) = x \\wedge y$ Then :$f$ preserves finite suprema"}
+{"_id": "12238", "title": "Set Union is not Cancellable", "text": "Set union is not a cancellable operation. That is, for a given $A, B, C \\subseteq S$ for some $S$, it is not always the case that: :$A \\cup B = A \\cup C \\implies B = C$"}
+{"_id": "12240", "title": "Intersection of Real Intervals is Real Interval", "text": "Let $I_1$ and $I_2$ be real intervals. Then $I_1 \\cap I_2$ is also a real interval."}
+{"_id": "12241", "title": "Union of Real Intervals is not necessarily Real Interval", "text": "Let $I_1$ and $I_2$ be real intervals. Then $I_1 \\cup I_2$ is not necessarily a real interval."}
+{"_id": "12242", "title": "Pasting Lemma/Corollary 1", "text": "Let $A$ and $B$ be closed in $X$. Let $f : A \\to Y$ and $g : B \\to Y$ be continuous mappings that agree on $A \\cap B$. Then the mapping $f \\cup g : A \\cup B \\to Y$ is continuous."}
+{"_id": "12243", "title": "Pasting Lemma/Corollary 2", "text": "Let $\\mathcal A = \\left\\{ {A_i: i \\in I} \\right\\}$ be a set of sets that are open in $X$. Let $f: \\bigcup \\mathcal A \\to Y$ be a mapping such that: :$\\forall i \\in I : f \\restriction A_i$ is continuous Then $f$ is continuous on $\\bigcup \\mathcal A$."}
+{"_id": "12244", "title": "Pasting Lemma", "text": "Let $X$ and $Y$ be topological spaces. Let $A$ and $B$ be open in $X$. Let $f: A \\to Y$ and $g: B \\to Y$ be continuous mappings that agree on $A \\cap B$. Let $f \\cup g$ be the union of the mappings $f$ and $g$: :$\\forall x \\in A \\cup B: \\map {f \\cup g} x = \\begin {cases} \\map f x & : x \\in A \\\\ \\map g x & : x \\in B \\end {cases}$ Then the mapping $f \\cup g : A \\cup B \\to Y$ is continuous."}
+{"_id": "12247", "title": "Domain of Real Square Root Function", "text": "The domain of the real square root function is the set of positive real numbers $\\R_{\\ge 0}$: :$\\left\\{ {x \\in \\R: x \\ge 0}\\right\\}$"}
+{"_id": "12248", "title": "Image of Real Square Root Function", "text": "The image of the real square root function is the entire set of positive real numbers $\\R_{\\ge 0}$."}
+{"_id": "12249", "title": "Exponent Combination Laws/Product of Powers/Proof 1", "text": "Let $a \\in \\R_{> 0}$ be a positive real number. Let $x, y \\in \\R$ be real numbers. Let $a^x$ be defined as $a$ to the power of $x$. Then: :$a^x a^y = a^{x + y}$"}
+{"_id": "12251", "title": "Inequality of Product of Unequal Numbers", "text": "Let $a, b, c, d \\in \\R$. Then: :$0 < a < b \\land 0 < c < d \\implies 0 < a c < b d$"}
+{"_id": "12252", "title": "Negative of Absolute Value/Corollary 3", "text": "Let $y \\in \\R_{\\ge 0}$. Let $z \\in \\R$. Then: :$\\left\\vert{x - z}\\right\\vert < y \\iff z - y < x < z + y$"}
+{"_id": "12253", "title": "Exponent Combination Laws/Product of Powers/Proof 2/Lemma", "text": "Let $x_1, x_2, y_1, y_2 \\in \\R_{>0}$ be strictly positive real numbers. Let $\\epsilon \\in \\openint 0 {\\min \\set {y_1, y_2, 1} }$. Then: :$\\size {x_1 - y_1} < \\epsilon \\land \\size {x_2 - y_2} < \\epsilon \\implies \\size {x_1 x_2 - y_1 y_2} < \\epsilon \\paren {y_1 + y_2 + 1}$"}
+{"_id": "12254", "title": "Union of Unordered Tuples", "text": "Let $x_1, \\dots, x_n, x_{n+1}, \\dots, x_m$ be arbitrary. Then :$\\set {x_1, \\dots, x_n} \\cup \\set {x_{n + 1}, \\dots, x_m} = \\set {x_1, \\dots, x_n, x_{n + 1}, \\dots, x_m}$"}
+{"_id": "12255", "title": "Gauss-Lucas Theorem", "text": "Let $P$ be a (nonconstant) polynomial with complex coefficients. Then all zeros of its derivative $P'$  belong to the convex hull of the set of zeros of $P$."}
+{"_id": "12256", "title": "Way Below implies Preceding", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $x, y \\in S$ such that :$x \\ll y$ where $\\ll$ denotes element is way below second element. Then :$x \\preceq y$"}
+{"_id": "12257", "title": "Preceding and Way Below implies Way Below", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $u, x, y, z \\in S$ such that :$u \\preceq x \\ll y \\preceq z$ where $\\ll$ denotes the way below relation. Then :$u \\ll z$"}
+{"_id": "12258", "title": "Join is Way Below if Operands are Way Below", "text": "Let $\\left({S, vee, \\preceq}\\right)$ be a join semilattice. Let $x, y, z \\in S$ such that :$x \\ll z$ and $y \\ll z$ where $\\ll$ denotes the way below relation. Then :$x \\vee y \\ll z$"}
+{"_id": "12259", "title": "Way Below Relation is Transitive", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $x, y, z \\in S$ such that :$x \\ll y \\ll z$ Then :$x \\ll z$"}
+{"_id": "12260", "title": "Way Below Relation is Antisymmetric", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $x, y \\in S$ such that :$x \\ll y$ and $y \\ll x$ Then :$x = y$"}
+{"_id": "12261", "title": "Continued Fraction Expansion of Golden Mean", "text": "The golden mean has the simplest possible continued fraction expansion, namely $\\sqbrk {1, 1, 1, 1, \\ldots}$: :$\\phi = 1 + \\cfrac 1 {1 + \\cfrac 1 {1 + \\cfrac 1 {\\ddots} } }$"}
+{"_id": "12263", "title": "Fibonacci Number less than Golden Section to Power less One", "text": "For all $n \\in \\N_{> 0}$: :$F_n \\le \\phi^{n - 1}$ where: :$F_n$ is the $n$th Fibonacci number :$\\phi$ is the golden section: $\\phi = \\dfrac {1 + \\sqrt 5} 2$"}
+{"_id": "12264", "title": "Square of Golden Mean equals One plus Golden Mean", "text": ":$\\phi^2 = \\phi + 1$ where $\\phi$ denotes the golden mean."}
+{"_id": "12265", "title": "Fibonacci Number greater than Golden Section to Power less Two", "text": "For all $n \\in \\N_{\\ge 2}$: :$F_n \\ge \\phi^{n - 2}$ where: :$F_n$ is the $n$th Fibonacci number :$\\phi$ is the golden section: $\\phi = \\dfrac {1 + \\sqrt 5} 2$"}
+{"_id": "12267", "title": "Triangular Number as Alternating Sum and Difference of Squares", "text": "{{begin-eqn}} {{eqn | ll= \\forall n \\in \\N:        | l = \\frac {n \\paren {n + 1} } 2       | r = \\sum_{j \\mathop = 0}^{n - 1} \\paren {-1}^j \\paren {n - j}^2       | c =  }} {{eqn | r = n^2 - \\paren {n - 1}^2 + \\paren {n - 2}^2 - \\cdots + \\paren {-1}^{n - 1}        | c =  }} {{end-eqn}} Thus the $n$th triangular number can be expressed as the alternating sum and difference of squares: So: {{begin-eqn}} {{eqn | l = 1       | r = 1^2       | c =  }} {{eqn | l = 3       | r = 2^2 - 1^2       | c =  }} {{eqn | l = 6       | r = 3^2 - 2^2 + 1^2       | c =  }} {{eqn | l = 10       | r = 4^2 - 3^2 + 2^2 - 1^2       | c =  }} {{end-eqn}} and so on."}
+{"_id": "12269", "title": "Bernoulli's Inequality/Corollary", "text": "Let $x \\in \\R$ be a real number such that $0 < x < 1$. Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Then: :$\\left({1 - x}\\right)^n \\ge 1 - n x$"}
+{"_id": "12272", "title": "2 to the n is Greater than n Cubed when n is 10 and above", "text": ":$\\forall n \\in \\Z, n \\ge 10: 2^n > n^3$"}
+{"_id": "12274", "title": "Set of Integers can be Well-Ordered", "text": "The set of integers $\\Z$ can be well-ordered with an appropriately chosen ordering."}
+{"_id": "12276", "title": "Existence of Digital Root", "text": "Let $n \\in \\N$ be a natural number. Let $b \\in \\N$ such that $b \\ge 2$ also be a natural number. Let $n$ be expressed in base $b$. Then the digital root base $b$ exists for $n$."}
+{"_id": "12277", "title": "Golden Mean is Irrational", "text": "The golden mean $\\phi$ is irrational."}
+{"_id": "12278", "title": "Way Below iff Preceding Finite Supremum", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $x, y \\in S$. Then $x \\ll y$ {{iff}} :$\\forall X \\subseteq S: y \\preceq \\sup X \\implies \\exists A \\in {\\it Fin}\\left({X}\\right): x \\preceq \\sup A$ where :$\\ll$ denotes the way below relation, :${\\it Fin}\\left({X}\\right)$ denotes the set of all finite subsets of $X$."}
+{"_id": "12279", "title": "Exponent Combination Laws/Product of Powers/Proof 2", "text": "Let $a \\in \\R_{> 0}$ be a positive real number. Let $x, y \\in \\R$ be real numbers. Let $a^x$ be defined as $a$ to the power of $x$. Then: : $a^x a^y = a^{x + y}$ {{proofread}}"}
+{"_id": "12280", "title": "Exponent Combination Laws/Power of Power/Proof 1", "text": "Let $a \\in \\R_{>0}$ be a (strictly) positive real number. {{:Exponent Combination Laws/Power of Power}}"}
+{"_id": "12281", "title": "Exponent Combination Laws/Power of Power/Proof 2", "text": "Let $a \\in \\R_{>0}$ be a (strictly) positive real number. {{:Exponent Combination Laws/Power of Power}}"}
+{"_id": "12282", "title": "Rational Sequence Increasing to Real Number", "text": "Let $x \\in \\R$ be a real number. Then there exists some increasing rational sequence that converges to $x$."}
+{"_id": "12283", "title": "Rational Sequence Decreasing to Real Number", "text": "Let $x \\in \\R$ be a real number. Then there exists some decreasing rational sequence that converges to $x$."}
+{"_id": "12284", "title": "Power Function on Base Greater than One is Strictly Increasing/Real Number", "text": "Let $a \\in \\R$ be a real number such that $a > 1$. Let $f: \\R \\to \\R$ be the real function defined as: :$\\map f x = a^x$ where $a^x$ denotes $a$ to the power of $x$. Then $f$ is strictly increasing."}
+{"_id": "12285", "title": "Power Function on Strictly Positive Base is Continuous/Real Power", "text": "Let $f : \\R \\to \\R$ be the real function defined as: :$\\map f x = a^x$ where $a^x$ denotes $a$ to the power of $x$. Then $f$ is continuous."}
+{"_id": "12286", "title": "Power Function on Strictly Positive Base is Continuous/Rational Power", "text": "Let $f: \\Q \\to \\R$ be the real-valued function defined as: :$\\map f x = a^x$ where $a^x$ denotes $a$ to the power of $x$. Then $f$ is continuous."}
+{"_id": "12287", "title": "Defining Sequence of Natural Logarithm is Convergent", "text": "Let $x \\in \\R$ be a real number such that $x > 0$. Let $\\left\\langle{ f_n }\\right\\rangle$ be the sequence of mappings $f_n : \\R_{>0} \\to \\R$ defined as: :$f_n \\left({ x }\\right) = n \\left({ \\sqrt[n]{ x } - 1 }\\right)$ Then $\\left\\langle{ f_n }\\right\\rangle$ is pointwise convergent."}
+{"_id": "12288", "title": "Exponent Combination Laws/Positive Integers", "text": "Let $r \\in \\R_{>0}$ be (strictly) positive real numbers."}
+{"_id": "12293", "title": "Sum of Indices of Real Number/Rational Numbers", "text": "Let $x, y \\in \\Q$ be rational numbers. Let $r^x$ be defined as $r$ to the power of $n$. Then: : $r^{x + y} = r^x \\times r^y$"}
+{"_id": "12294", "title": "Exponent Combination Laws/Rational Numbers", "text": "Let $r \\in \\R_{> 0}$ be a (strictly) positive real number."}
+{"_id": "12295", "title": "Product of Indices of Real Number/Rational Numbers", "text": "Let $x, y \\in \\Q$ be rational numbers. Let $r^x$ be defined as $r$ to the power of $x$. Then: :$\\paren {r^x}^y = r^{x y}$"}
+{"_id": "12296", "title": "Sum of Indices of Real Number", "text": "Let $r \\in \\R_{> 0}$ be a (strictly) positive real number."}
+{"_id": "12297", "title": "Product of Indices of Real Number", "text": "Let $r \\in \\R_{> 0}$ be a (strictly) positive real number."}
+{"_id": "12298", "title": "Sum of Indices of Real Number/Integers", "text": "Let $n, m \\in \\Z$ be integers. Let $r^n$ be defined as $r$ to the power of $n$. Then: :$r^{n + m} = r^n \\times r^m$"}
+{"_id": "12299", "title": "Product of Indices of Real Number/Integers", "text": "Let $n, m \\in \\Z$ be positive integers. Let $r^n$ be defined as $r$ to the power of $n$. Then: :$\\paren {r^n}^m = r^{n m}$"}
+{"_id": "12300", "title": "Uniformly Convergent iff Difference Under Supremum Metric Vanishes", "text": "Let $X$ and $Y$ be metric spaces Let $\\sequence {f_n}$ be a sequence of mappings defined on $X$. Let $f: X \\to Y$ be a mapping. Let $d_S: S \\times S \\to Y$ denote the supremum metric on $S \\subseteq X$. Then $\\sequence {f_n}$ converges uniformly to $f$ on $S$ {{iff}} $\\map {d_S} {f_n, f} \\to 0$ as $n \\to \\infty$."}
+{"_id": "12301", "title": "Uniform Convergence is Hereditary", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $\\sequence {f_n}$ be a sequence of mappings defined on $A$. Let $\\sequence {f_n}$ be uniformly convergent on $S \\subseteq A$. Then $\\sequence {f_n}$ is uniformly convergent on every metric subspace of $S$. That is, uniform convergence is a hereditary property of a metric space."}
+{"_id": "12302", "title": "Real Number to Negative Power", "text": "Let $r \\in \\R_{> 0}$ be a (strictly) positive real number."}
+{"_id": "12303", "title": "Real Number to Negative Power/Integer", "text": "Let $n \\in \\Z$ be an integer. Let $r^n$ be defined as $r$ to the power of $n$. Then: :$r^{-n} = \\dfrac 1 {r^n}$"}
+{"_id": "12304", "title": "Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal", "text": "Let $\\mathscr S = \\left({S, \\preceq}\\right)$ be an up-complete ordered set. Let $x, y \\in S$. Then $x \\ll y$ {{iff}} :$\\forall I \\in {\\it Ids}\\left({\\mathscr S}\\right): y \\preceq \\sup I \\implies x \\in I$ where :$\\ll$ denotes the way below relation, :${\\it Ids}\\left({\\mathscr S}\\right)$ denotes the set of all ideals in $\\mathscr S$."}
+{"_id": "12305", "title": "Lower Closure of Directed Subset is Ideal", "text": "Let $\\mathscr S = \\left({S, \\preceq}\\right)$ be an ordered set. Let $D$ be a directed subset of $S$. Then :$D^\\preceq$ is an ideal in $\\mathscr S$ where $D^\\preceq$ denotes the lower closure of $D$."}
+{"_id": "12306", "title": "Sum of Geometric Sequence/Corollary 2", "text": ":$\\displaystyle \\sum_{j \\mathop = 0}^{n - 1} j x^j = \\frac {\\paren {n - 1} x^{n + 1} - n x^n + x} {\\paren {x - 1}^2}$"}
+{"_id": "12307", "title": "Defining Sequence of Natural Logarithm is Strictly Decreasing", "text": "Let $x \\in \\R$ be a real number such that $x > 0$. Let $\\sequence {f_n}$ be the sequence of mappings $f_n : \\R_{>0} \\to \\R$ defined as: :$\\map {f_n} x = n \\paren {\\sqrt [n] x - 1}$ Then $\\forall x \\in \\R_{>0}: \\sequence {\\map {f_n} x}$ is strictly decreasing."}
+{"_id": "12309", "title": "Logarithm Base 10 of 2 is Irrational", "text": "The common logarithm of $2$: :$\\log_{10} 2 \\approx 0.30102 \\, 99956 \\, 63981 \\, 19521 \\, 37389 \\ldots$ is irrational."}
+{"_id": "12310", "title": "Nth Root of 1 plus x not greater than 1 plus x over n", "text": "Let $x \\in \\R_{>0}$ be a (strictly) positive real number. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: :$\\sqrt [n] {1 + x} \\le 1 + \\dfrac x n$"}
+{"_id": "12312", "title": "Reciprocal of Logarithm", "text": "Let $x, y \\in \\R_{> 0}$ be (strictly) positive real numbers. Then: :$\\dfrac 1 {\\log_x y} = \\log_y x$"}
+{"_id": "12313", "title": "Discontinuity of Monotonic Function is Jump Discontinuity", "text": "Let $X$ be an open subset of $\\R$. Let $f: X \\to Y$ be a monotone real function. Then $f$ is discontinuous at a point $c \\in X$ {{iff}} $c$ is a jump discontinuity of $f$."}
+{"_id": "12314", "title": "Surjective Monotone Function is Continuous", "text": "Let $X$ be an open subset of $\\R$. Let $Y$ be a real interval. Let $f: X \\to Y$ be a surjective monotone real function. Then $f$ is continuous on $X$."}
+{"_id": "12322", "title": "Limit of Bounded Convergent Sequence is Bounded", "text": "Let $\\left \\langle {x_n} \\right \\rangle$, $\\left \\langle {a_n} \\right \\rangle$, and $\\left \\langle {b_n} \\right \\rangle$ be convergent sequences in $\\R$. Let $\\left \\langle {x_n} \\right \\rangle$, $\\left \\langle {a_n} \\right \\rangle$, and $\\left \\langle {b_n} \\right \\rangle$ converge to $x, a, b \\in \\R$, respectively. Suppose that: :$\\exists N \\in \\N : n \\geq N \\implies a_n \\leq x_n \\leq b_n$ Then: :$ a \\le x \\le b$"}
+{"_id": "12323", "title": "Logarithm of Power/Natural Logarithm/Natural Power", "text": "Let $x \\in \\R$ be a strictly positive real number. Let $n \\in \\R$ be any natural number. Let $\\ln x$ be the natural logarithm of $x$. Then: :$\\ln \\left({x^n}\\right) = n \\ln x$"}
+{"_id": "12324", "title": "Logarithm of Power/Natural Logarithm/Integer Power", "text": "Let $x \\in \\R$ be a strictly positive real number. Let $n \\in \\R$ be any integer. Let $\\ln x$ be the natural logarithm of $x$. Then: :$\\ln \\left({ x^n }\\right) = n \\ln x$"}
+{"_id": "12325", "title": "Logarithm of Power/Natural Logarithm/Rational Power", "text": "Let $x \\in \\R$ be a strictly positive real number. Let $r \\in \\R$ be any rational number. Let $\\ln x$ be the natural logarithm of $x$. Then: :$\\ln \\left({ x^r }\\right) = r \\ln x$"}
+{"_id": "12326", "title": "Dini's Theorem", "text": "Let $K \\subseteq \\R$ be compact. Let $\\sequence {f_n}$ be a sequence of continuous real functions defined on $K$. Let $\\sequence {f_n}$ converge pointwise to a continuous function $f$. Suppose that: :$\\forall x \\in K : \\sequence {\\map {f_n} x}$ is monotone. Then the convergence of $\\sequence {f_n}$ to $f$ is uniform."}
+{"_id": "12327", "title": "Way Below in Meet-Continuous Lattice", "text": "Let $\\mathscr S = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a meet-continuous bounded below lattice. Let $x, y \\in S$. Then $x \\ll y$ {{iff}} :$\\forall I \\in {\\it Ids}\\left({\\mathscr S}\\right): y = \\sup I \\implies x \\in I$ where :$\\ll$ denotes the way below relation, :${\\it Ids}\\left({\\mathscr S}\\right)$ denotes the set of all ideals in $\\mathscr S$."}
+{"_id": "12328", "title": "Approximation to Binary Logarithm from Natural and Common Logarithm", "text": "The binary logarithm $\\lg x$ can be approximated, to within $1 \\%$, by the expression: :$\\lg x \\approx \\ln x + \\log_{10} x$ That is, by the sum of the natural logarithm and common logarithm."}
+{"_id": "12331", "title": "Monotone Real Function with Everywhere Dense Image is Continuous", "text": "Let $I$ and $J$ be real intervals. Let $f: I \\to J$ be a monotone real function. Let $f \\sqbrk I$ be everywhere dense in $J$, where $f \\sqbrk I$ denotes the image of $I$ under $f$. Then $f$ is continuous on $I$."}
+{"_id": "12332", "title": "Relative Difference between Infinite Set and Finite Set is Infinite", "text": "Let $S$ be an infinite set. Let $T$ be a finite set. Then $S \\setminus T$ is an infinite set."}
+{"_id": "12333", "title": "Set Difference over Subset", "text": "Let $A$, $B$, and $S$ be sets. Let $A \\subseteq B$. Then: :$A \\setminus S \\subseteq B \\setminus S$"}
+{"_id": "12334", "title": "Subset of Empty Set iff Empty", "text": "Let $S$ be a set. Let $\\O$ denote the empty set. Then $S \\subseteq \\O$ {{iff}} $S = \\O$."}
+{"_id": "12335", "title": "Finite Subset of Metric Space is Closed", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $S \\subseteq A$ be finite. Then $S$ is closed in $M$."}
+{"_id": "12336", "title": "Monotone Real Function with Everywhere Dense Image is Continuous/Lemma", "text": ":$\\displaystyle \\openint {\\lim_{x \\mathop \\to c^-} \\map f x} {\\lim_{x \\mathop \\to c+ } \\map f x} \\cap f \\sqbrk I \\subseteq \\set {\\map f c}$"}
+{"_id": "12337", "title": "Defining Sequence of Natural Logarithm is Uniformly Convergent on Compact Sets", "text": "Let $x \\in \\R$ be a real number such that $x > 0$. Let $\\sequence {f_n}$ be the sequence of mappings $f_n : \\R_{>0} \\to \\R$ defined as: :$\\map {f_n} x = n \\paren {\\sqrt [n] x - 1}$ Let $K \\subseteq \\R_{>0}$ be compact. Then $\\sequence {f_n}$ is uniformly convergent on $K$."}
+{"_id": "12338", "title": "Multiplication of Positive Number by Real Number Greater than One", "text": "Let $x$ and $y$ be real numbers. Let $x > 1$. Let $y > 0$. Then $\\dfrac y x < y$."}
+{"_id": "12341", "title": "Set of Finite Suprema is Directed", "text": "Let $\\left({S, \\vee, \\preceq}\\right)$ be a join semilattice. Let $X$ be a non-empty subset of $S$. Then :$\\left\\{ {\\sup A: A \\in {\\it Fin}\\left({X}\\right) \\land A \\ne \\varnothing}\\right\\}$ is directed. where ${\\it Fin}\\left({X}\\right)$ denotes the set of all finite subsets of $X$."}
+{"_id": "12342", "title": "Change of Index Variable of Summation", "text": ":$\\displaystyle \\sum_{R \\left({i}\\right)} a_i = \\sum_{R \\left({j}\\right)} a_j$"}
+{"_id": "12343", "title": "Singleton is Finite", "text": "Let $x$ be arbitrary. Then $\\set x$ is a finite set."}
+{"_id": "12344", "title": "T4 Property Preserved in Closed Subspace/Corollary", "text": "If $T$ is a normal space then $T_K$ is also a normal space. That is, the property of being a normal space is weakly hereditary."}
+{"_id": "12345", "title": "Derivative of Uniformly Convergent Sequence of Differentiable Functions", "text": "Let $J$ be a bounded interval. Let $\\left\\langle{f_n}\\right\\rangle$ be a sequence of real functions $f_n: J \\to \\R$. Let each of $\\left\\langle{f_n}\\right\\rangle$ be differentiable on $J$. Let $\\left\\langle{ f_n \\left({ x_0 }\\right) }\\right\\rangle$ be convergent for some $x_0 \\in J$. Let the sequence of derivatives $\\left\\langle{f_n'}\\right\\rangle$ converge uniformly on $J$ to a function $g : J \\to \\R$. Then $\\left\\langle{f_n}\\right\\rangle$ converge uniformly on $J$ to a differentiable function $f: J \\to \\R$ and $D_x f = g$."}
+{"_id": "12346", "title": "Derivative of Natural Logarithm Function/Proof 4/Lemma", "text": "Let $\\sequence {f_n}_n$ be the sequence of real functions $f_n: \\R_{>0} \\to \\R$ defined as: :$\\map {f_n} x = n \\paren {\\sqrt [n] x - 1}$ Let $k \\in \\N$. Let $J = \\closedint {\\dfrac 1 k} k$. Then the sequence of derivatives $\\sequence { {f_n}'}_n$ converges uniformly to some real function $g: J \\to \\R$."}
+{"_id": "12348", "title": "Exponential Function is Well-Defined/Real", "text": "Let $x \\in \\R$ be a real number. Let $\\exp x$ be the exponential of $x$. Then $\\exp x$ is well-defined."}
+{"_id": "12350", "title": "Tail of Convergent Sequence", "text": "Let $\\left\\langle{a_n}\\right\\rangle$ be a real sequence. Let $N \\in \\N$ be a natural number. Let $a \\in R$ be a real number. Then: :$a_n \\to a$ {{iff}}: :$a_{n + N} \\to a$"}
+{"_id": "12354", "title": "Union of Inverses of Mappings is Inverse of Union of Mappings", "text": "Let $I$ be an indexing set. Let $\\family {f_i: i \\in I}$ be an indexed family of mappings. For each $i \\in I$, let $f^{-1}$ denote the inverse of $f$. Then the inverse of the union of $\\family {f_i: i \\in I}$ is the union of the inverses of $f_i, i \\in I$. That is: :$\\ds \\paren {\\bigcup \\family {f_i: i \\in I} }^{-1} = \\bigcup \\family {f_i^{-1}: i \\in I}$"}
+{"_id": "12355", "title": "Union of Functions Theorem/Corollary", "text": "For each $i \\in \\N$, let $g_i : X_i \\to Y$ be invertible. Then $\\displaystyle \\bigcup \\left\\{{g_i: i \\in \\N}\\right\\}$ is invertible and: :$\\displaystyle \\left({\\bigcup \\left\\{{g_i: i \\in \\N}\\right\\} }\\right)^{-1} = \\bigcup \\left\\{{g_i^{-1}: i \\in \\N}\\right\\}$"}
+{"_id": "12356", "title": "Permutation of Indices of Summation", "text": ":$\\displaystyle \\sum_{\\map R j} a_j = \\sum_{\\map R {\\map \\pi j} } a_{\\map \\pi j}$"}
+{"_id": "12357", "title": "Translation of Index Variable of Summation", "text": "Let $R: \\Z \\to \\left\\{ {\\mathrm T, \\mathrm F}\\right\\}$ be a propositional function on the set of integers. Let $\\displaystyle \\sum_{R \\left({j}\\right)} a_j$ denote a summation over $R$. Then: :$\\displaystyle \\sum_{R \\left({j}\\right)} a_j = \\sum_{R \\left({c \\mathop + j}\\right)} a_{c \\mathop + j} = \\sum_{R \\left({c \\mathop - j}\\right)} a_{c \\mathop - j}$ where $c$ is an integer constant which is not dependent upon $j$."}
+{"_id": "12358", "title": "Axiom of Approximation in Up-Complete Semilattice", "text": "Let $\\mathscr S = \\left({S, \\wedge, \\preceq}\\right)$ be an up-complete meet semilattice. Let :$\\forall x \\in S: x^\\ll$ is directed Then :$\\mathscr S$ satisfies axiom of approximation {{iff}} :$\\forall x, y \\in S: x \\npreceq y \\implies \\exists u \\in S: u \\ll x \\land u \\npreceq y$"}
+{"_id": "12359", "title": "Operand is Upper Bound of Way Below Closure", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $x \\in S$. Then :$x$ is upper bound for $x^\\ll$ where $x^\\ll$ denotes the way below closure of $x$."}
+{"_id": "12360", "title": "Exchange of Order of Summation", "text": ":$\\displaystyle \\sum_{R \\left({i}\\right)} \\sum_{S \\left({j}\\right)} a_{i j} = \\sum_{S \\left({j}\\right)} \\sum_{R \\left({i}\\right)} a_{i j}$"}
+{"_id": "12361", "title": "Sum of Summations equals Summation of Sum/Infinite Sequence", "text": "Let $R: \\Z \\to \\set {\\mathrm T, \\mathrm F}$ be a propositional function on the set of integers $\\Z$. Let $\\displaystyle \\sum_{\\map R i} x_i$ denote a summation over $R$. Let the fiber of truth of $R$ be infinite. Let $\\displaystyle \\sum_{\\map R i} b_i$ and $\\displaystyle \\sum_{\\map R i} c_i$ be convergent. Then: :$\\displaystyle \\sum_{\\map R i} \\paren {b_i + c_i} = \\sum_{\\map R i} b_i + \\sum_{\\map R i} c_i$"}
+{"_id": "12362", "title": "Exchange of Order of Summation with Dependency on Both Indices", "text": ":$\\displaystyle \\sum_{R \\left({i}\\right)} \\sum_{S \\left({i, j}\\right)} a_{i j} = \\sum_{S' \\left({j}\\right)} \\sum_{R' \\left({i, j}\\right)} a_{i j}$ where: :$S' \\left({j}\\right)$ denotes the propositional function: ::there exists an $i$ such that both $R \\left({i}\\right)$ and $S \\left({i, j}\\right)$ hold :$R' \\left({i, j}\\right)$ denotes the propositional function: ::both $R \\left({i}\\right)$ and $S \\left({i, j}\\right)$ hold."}
+{"_id": "12363", "title": "Summation of i from 1 to n of Summation of j from 1 to i", "text": ":$\\displaystyle \\sum_{i \\mathop = 1}^n \\sum_{j \\mathop = 1}^i a_{i j} = \\sum_{j \\mathop = 1}^n \\sum_{i \\mathop = j}^n a_{i j}$"}
+{"_id": "12364", "title": "Exchange of Order of Summation/Infinite Series", "text": "Let the fiber of truth of both $R$ and $S$ be infinite. Let: :$\\displaystyle \\sum_{R \\left({i}\\right)} \\sum_{S \\left({j}\\right)} \\left\\vert{a_{i j} }\\right\\vert$ exist. Then: :$\\displaystyle \\sum_{R \\left({i}\\right)} \\sum_{S \\left({j}\\right)} a_{i j} = \\sum_{S \\left({j}\\right)} \\sum_{R \\left({i}\\right)} a_{i j}$"}
+{"_id": "12365", "title": "Exchange of Order of Summation with Dependency on Both Indices/Infinite Series", "text": "Let the fiber of truth of both $R$ and $S$ be infinite. Let: :$\\displaystyle \\sum_{R \\left({i}\\right)} \\sum_{S \\left({i, j}\\right)} \\left\\vert{a_{i j} }\\right\\vert$ exist. Then: :$\\displaystyle \\sum_{R \\left({i}\\right)} \\sum_{S \\left({i, j}\\right)} a_{i j} = \\sum_{S' \\left({j}\\right)} \\sum_{R' \\left({i, j}\\right)} a_{i j}$ where: :$S' \\left({j}\\right)$ denotes the propositional function: ::there exists an $i$ such that both $R \\left({i}\\right)$ and $S \\left({i, j}\\right)$ hold :$R' \\left({i, j}\\right)$ denotes the propositional function: ::both $R \\left({i}\\right)$ and $S \\left({i, j}\\right)$ hold."}
+{"_id": "12366", "title": "Exchange of Order of Summation/Finite and Infinite Series", "text": "Let the fiber of truth of $R$ be infinite. Let the fiber of truth of $S$ be finite. For all $j$ in the fiber of truth of $S$, let $\\displaystyle \\sum_{R \\left({i}\\right)} a_{i j}$ be convergent. Then: :$\\displaystyle \\sum_{R \\left({i}\\right)} \\sum_{S \\left({j}\\right)} a_{i j} = \\sum_{S \\left({j}\\right)} \\sum_{R \\left({i}\\right)} a_{i j}$"}
+{"_id": "12367", "title": "Permutation of Indices of Summation/Infinite Series", "text": "Let the fiber of truth of $R$ be infinite. Let $\\displaystyle \\sum_{\\map R i} a_i$ be absolutely convergent. Then: :$\\displaystyle \\sum_{\\map R j} a_j = \\sum_{\\map R {\\map \\pi j} } a_{\\map \\pi j}$"}
+{"_id": "12368", "title": "Sum of Summations equals Summation of Sum", "text": "Let $R: \\Z \\to \\set {\\T, \\F}$ be a propositional function on the set of integers. Let $\\displaystyle \\sum_{\\map R i} x_i$ denote a summation over $R$. Let the fiber of truth of $R$ be finite. Then: :$\\displaystyle \\sum_{\\map R i} \\paren {b_i + c_i} = \\sum_{\\map R i} b_i + \\sum_{\\map R i} c_i$"}
+{"_id": "12371", "title": "Sum of Summations over Overlapping Domains", "text": ":$\\displaystyle \\sum_{R \\left({j}\\right)} a_j + \\sum_{S \\left({j}\\right)} a_j = \\sum_{R \\left({j}\\right) \\mathop \\lor S \\left({j}\\right)} a_j + \\sum_{R \\left({j}\\right) \\mathop \\land S \\left({j}\\right)} a_j$ where $\\lor$ and $\\land$ signify logical disjunction and logical conjunction respectively."}
+{"_id": "12373", "title": "Rules for Manipulating Summations", "text": "Let $R: \\Z \\to \\left\\{ {\\mathrm T, \\mathrm F}\\right\\}$ and $S: \\Z \\to \\left\\{ {\\mathrm T, \\mathrm F}\\right\\}$ be propositional functions on the set of integers. Let $S: \\Z \\times \\Z \\to \\left\\{ {\\mathrm T, \\mathrm F}\\right\\}$ be a propositional functions on the Cartesian product of the set of integers with itself. Let $\\displaystyle \\sum_{R \\left({i}\\right)} x_i$ denote a summation over $R$. Let $\\pi$ be a permutation on the fiber of truth of $R$."}
+{"_id": "12374", "title": "Uniqueness of Continuously Differentiable Solution to Initial Value Problem", "text": "Let $D \\subseteq \\R^2$ be a region containing $\\tuple {a, b}$. Let $f: D \\to \\R$ be real-valued mapping such that $f$ and $\\dfrac {\\partial f} {\\partial x}$ are continuous on $D$. Consider the initial value problem: :$\\dfrac {\\d x} {\\d t} = \\map f {t, x}$ :$\\map x a = b$ Suppose the initial value problem above has a solution $\\phi$ for all $x$ in some interval $J$ containing $a$. Then the solution $\\phi$ is  unique on $J$."}
+{"_id": "12379", "title": "Exhausting Sequence of Sets on the Strictly Positive Real Numbers", "text": "For each $k \\in \\N$, let $S_k = \\openint {\\dfrac 1 k} k$. Then $\\sequence {S_k}_k$ is an exhausting sequence of sets on $\\R_{>0}$."}
+{"_id": "12380", "title": "Way Below in Ordered Set of Topology", "text": "Let $\\left({S, \\tau}\\right)$ be a topological space. Let $\\left({\\tau, \\preceq}\\right)$ be an ordered set where $\\preceq \\mathop = \\subseteq\\restriction_{\\tau \\times \\tau}$ Let $x, y \\in \\tau$. Then :$x \\ll y$ in $\\left({\\tau, \\preceq}\\right)$ {{iff}} :for every set $F$ of open subsets of $S$: if $y \\subseteq \\bigcup F$, then ::there exists a finite subset $G$ of $F$: $x \\subseteq \\bigcup G$ where $\\ll$ denotes the way below relation."}
+{"_id": "12381", "title": "Continuous Inverse Theorem", "text": "Let $f$ be a real function defined on an interval $I$. Let $f$ be strictly monotone and continuous on $I$. Let $g$ be the inverse mapping to $f$. Let $J := f \\left[{I}\\right]$ be the image of $I$ under $f$. Then $g$ is strictly monotone and continuous on $J$."}
+{"_id": "12382", "title": "Logarithm is Strictly Concave", "text": ":$\\ln x: x > 0$ strictly concave."}
+{"_id": "12383", "title": "Exponential is Strictly Convex", "text": ":The function $f \\left({x}\\right) = \\exp x$ is strictly convex."}
+{"_id": "12384", "title": "Way Below in Complete Lattice", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $x, y \\in S$. Then :$x \\ll y$ {{iff}} :$\\forall X \\subseteq S: y \\preceq \\sup X \\implies \\exists A \\in {\\it Fin}\\left({X}\\right): x \\preceq \\sup A$ where :$\\ll$ denotes the way below relation, :${\\it Fin}\\left({X}\\right)$ denotes the set of all finite subsets of $X$."}
+{"_id": "12385", "title": "Continuous Midpoint-Convex Function is Convex", "text": "Let $f$ be a real function which is defined on a real interval $I$. Let $f$ be midpoint-convex and continuous on $I$. Then $f$ is convex."}
+{"_id": "12387", "title": "Power Function on Strictly Positive Base is Continuous", "text": "Let $a \\in \\R_{>0}$."}
+{"_id": "12388", "title": "Continuous Strictly Midpoint-Concave Function is Strictly Concave", "text": "Let $f$ be strictly midpoint-concave and continuous on $I$. Then $f$ is strictly concave."}
+{"_id": "12389", "title": "Continuous Midpoint-Concave Function is Concave", "text": "Let $f$ be midpoint-concave and continuous on $I$. Then $f$ is concave."}
+{"_id": "12390", "title": "Continuous Strictly Midpoint-Convex Function is Strictly Convex", "text": "Let $f$ be strictly midpoint-convex and continuous on $I$. Then $f$ is strictly convex."}
+{"_id": "12391", "title": "Power Function on Strictly Positive Base is Convex", "text": "Let $a \\in \\R_{>0}$ be a strictly positive real number. Let $f: \\R \\to \\R$ be the real function defined as: :$\\map f x = a^x$ where $a^x$ denotes $a$ to the power of $x$. Then $f$ is convex."}
+{"_id": "12392", "title": "Natural Logarithm as Derivative of Exponential at Zero", "text": "Let $\\ln: \\R_{>0}$ denote the real natural logarithm. Then: :$\\displaystyle \\forall x \\in \\R_{>0}: \\ln x = \\lim_{h \\mathop \\to 0} \\frac {x^h - 1} h$"}
+{"_id": "12395", "title": "Product of Increasing Positive Functions is Increasing", "text": "Let $f$ and $g$ be real functions defined on an interval $I$. Let $f$ and $g$ be increasing and positive on $I$. Then the product $f g$ is increasing on $I$."}
+{"_id": "12397", "title": "Derivative of Exponential Function/Proof 5/Lemma", "text": ":$\\forall x \\in \\R : n \\ge \\left\\lceil{\\left\\vert{x}\\right\\vert}\\right\\rceil \\implies \\left\\langle{\\dfrac n {n + x} \\left({1 + \\dfrac x n}\\right)^n}\\right\\rangle$ is increasing."}
+{"_id": "12399", "title": "Change of Index Variable of Product", "text": ":$\\displaystyle \\prod_{R \\left({i}\\right)} a_i = \\prod_{R \\left({j}\\right)} a_j$"}
+{"_id": "12400", "title": "Translation of Index Variable of Summation/Corollary", "text": ":$\\displaystyle \\sum_{j \\mathop = m}^n a_j = \\sum_{j \\mathop = m + c}^{n + c} a_{j - c}$ where $c$ is an integer constant which is not dependent upon $j$."}
+{"_id": "12401", "title": "Permutation of Indices of Product", "text": ":$\\displaystyle \\prod_{\\map R j} a_j = \\prod_{\\map R {\\map \\pi j} } a_{\\map \\pi j}$"}
+{"_id": "12402", "title": "Translation of Index Variable of Product", "text": ":$\\displaystyle \\prod_{R \\left({j}\\right)} a_j = \\prod_{R \\left({c \\mathop + j}\\right)} a_{c \\mathop + j} = \\prod_{R \\left({c \\mathop - j}\\right)} a_{c \\mathop - j}$ where: :$\\displaystyle \\prod_{R \\left({j}\\right)} a_j$ denotes the product over $a_j$ for all $j$ that satisfy the propositional function $R \\left({j}\\right)$ :$c$ is an integer constant which is not dependent upon $j$."}
+{"_id": "12403", "title": "Exchange of Order of Product", "text": ":$\\displaystyle \\prod_{R \\left({i}\\right)} \\prod_{S \\left({j}\\right)} a_{i j} = \\prod_{S \\left({j}\\right)} \\prod_{R \\left({i}\\right)} a_{i j}$"}
+{"_id": "12404", "title": "Product of Products over Overlapping Domains", "text": ":$\\displaystyle \\prod_{R \\left({j}\\right)} a_j \\prod_{S \\left({j}\\right)} a_j = \\left({\\prod_{R \\left({j}\\right) \\mathop \\lor S \\left({j}\\right)} a_j}\\right) \\left({\\prod_{R \\left({j}\\right) \\mathop \\land S \\left({j}\\right)} a_j}\\right)$ where $\\lor$ and $\\land$ signify logical disjunction and logical conjunction respectively."}
+{"_id": "12408", "title": "Topology is Locally Compact iff Ordered Set of Topology is Continuous", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $L = \\left({\\tau, \\preceq}\\right)$ be the ordered set where $\\preceq$ is the inclusion relation. Then :$(1): \\quad T$ is locally compact implies $L$ is continuous :$(2): \\quad T$ is $T_3$ space and $L$ is continuous implies $T$ is locally compact"}
+{"_id": "12412", "title": "Sum of Sequence of Power by Index", "text": ":$\\displaystyle \\sum_{j \\mathop = 0}^n j x^j = \\frac {n x^{n + 2} - \\paren {n + 1} x^{n + 1} + x} {\\paren {x - 1}^2}$ for $x \\ne 1$."}
+{"_id": "12415", "title": "Ordering of Series of Ordered Sequences", "text": "Let $\\sequence {a_n}$ and $\\sequence {b_n}$ be two sequences. Let $\\displaystyle \\sum_{n \\mathop = 1}^{\\infty} a_n$ and $\\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n$ be convergent series. For each $n \\in \\N$, let $a_n < b_n$. Then: :$\\displaystyle \\sum_{n \\mathop = 0}^\\infty a_n < \\displaystyle \\sum_{n \\mathop = 0}^\\infty b_n$"}
+{"_id": "12416", "title": "Reciprocal of Real Exponential", "text": ":$\\dfrac 1 {\\map \\exp x} = \\map \\exp {-x}$"}
+{"_id": "12417", "title": "Telescoping Series/Example 2", "text": "Let $\\left \\langle {b_n} \\right \\rangle$ be a sequence in $\\R$. Let $\\left \\langle {a_n} \\right \\rangle$ be a sequence whose terms are defined as: :$a_k = b_k - b_{k - 1}$ Then: :$\\displaystyle \\sum_{k \\mathop = m}^n a_k = b_n - b_{m - 1}$"}
+{"_id": "12418", "title": "Repunit Integer as Product of Base - 1 by Increasing Digit Integer", "text": "{{begin-eqn}} {{eqn | l = 9 \\times 1 + 2       | r = 11 }} {{eqn | l = 9 \\times 12 + 3       | r = 111 }} {{eqn | l = 9 \\times 123 + 4       | r = 1111 }} {{eqn | l = 9 \\times 1234 + 5       | r = 11111 }} {{eqn | l = 9 \\times 12345 + 6       | r = 11111 }} {{eqn | o = \\ldots }} {{end-eqn}} That is: :$\\displaystyle 9 \\sum_{j \\mathop = 0}^n \\left({n - j}\\right)10^j + n + 1 = \\sum_{j \\mathop = 0}^n 10^j$"}
+{"_id": "12419", "title": "Repunit Integer as Product of Base - 1 by Increasing Digit Integer/General Result", "text": ":$\\displaystyle \\paren {b - 1} \\sum_{j \\mathop = 0}^n \\paren {n - j}b^j + n + 1 = \\sum_{j \\mathop = 0}^n b^j$"}
+{"_id": "12425", "title": "Exponential of Real Number is Strictly Positive/Proof 5/Lemma", "text": ":$\\forall x \\in \\R: \\exp x \\ne 0$"}
+{"_id": "12426", "title": "Product of Products", "text": "Let $R: \\Z \\to \\set {\\mathrm T, \\mathrm F}$ be a propositional function on the set of integers. Let $\\displaystyle \\prod_{R \\paren i} x_i$ denote a product over $R$. Let the fiber of truth of $R$ be finite. Then: :$\\displaystyle \\prod_{R \\paren i} \\paren {b_i c_i} = \\paren {\\prod_{R \\paren i} b_i} \\paren {\\prod_{R \\paren i} c_i}$"}
+{"_id": "12428", "title": "Summation of General Logarithms", "text": "Let $R: \\Z \\to \\left\\{ {\\mathrm T, \\mathrm F}\\right\\}$ be a propositional function on the set of integers. Let $\\displaystyle \\prod_{R \\left({i}\\right)} a_i$ denote a product over $R$. Let the fiber of truth of $R$ be finite. Then: :$\\displaystyle \\log_b \\left({\\prod_{R \\left({i}\\right)} a_i}\\right) = \\sum_{R \\left({i}\\right)} \\log_b a_i$"}
+{"_id": "12429", "title": "Way Below Closure is Directed in Bounded Below Join Semilattice", "text": "Let $\\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let $x \\in S$. Then :$x^\\ll$ is directed."}
+{"_id": "12430", "title": "Bottom is Way Below Any Element", "text": "Let $\\left({S, \\preceq}\\right)$ be a bounded below ordered set. Let $x \\in S$. Then :$\\bot \\ll x$ where $\\bot$ denotes the smallest element in $S$."}
+{"_id": "12432", "title": "Equivalence of Definitions of Real Exponential Function/Proof 2", "text": "The following definitions of the exponential function are equivalent."}
+{"_id": "12433", "title": "Product to n of Product to Index", "text": ":$\\displaystyle \\prod_{i \\mathop = 0}^n \\prod_{j \\mathop = 0}^i a_i a_j = \\prod_{i \\mathop = 0}^n {a_i}^{n + 2}$"}
+{"_id": "12434", "title": "Bernoulli's Inequality/Corollary/General Result", "text": "For all $n \\in \\Z_{\\ge 0}$: :$\\displaystyle \\prod_{j \\mathop = 1}^n \\left({1 - a_j}\\right) \\ge 1 - \\sum_{j \\mathop = 1}^n a^j$ where $0 < a_j < 1$ for all $j$."}
+{"_id": "12435", "title": "Product of Sequence of 1 minus Reciprocal of Squares", "text": "For all $n \\in \\Z_{\\ge 1}$: :$\\displaystyle \\prod_{j \\mathop = 2}^n \\paren {1 - \\dfrac 1 {j^2} } = \\dfrac {n + 1} {2 n}$"}
+{"_id": "12438", "title": "Product of Summations", "text": ":$\\displaystyle \\prod_{j \\mathop = 1}^n \\sum_{i \\mathop = 1}^m a_{i j} = \\sum_{1 \\mathop \\le i_1, \\mathop \\ldots \\mathop , i_n \\le m} a_{i_1 1} \\cdots a_{i_n n}$"}
+{"_id": "12439", "title": "Summation of Powers over Product of Differences", "text": ":$\\displaystyle \\sum_{j \\mathop = 1}^n \\begin{pmatrix} {\\dfrac { {x_j}^r} {\\displaystyle \\prod_{\\substack {1 \\mathop \\le k \\mathop \\le n \\\\ k \\mathop \\ne j} } \\left({x_j - x_k}\\right)} } \\end{pmatrix} = \\begin{cases} 0 & : 0 \\le r < n - 1 \\\\ 1 & : r = n - 1 \\\\ \\displaystyle \\sum_{j \\mathop = 1}^n x_j & : r = n \\end{cases}$"}
+{"_id": "12441", "title": "Summation by k of Product by r of x plus k minus r over Product by r less k of k minus r", "text": ":$\\displaystyle \\sum_{k \\mathop = 1}^n \\left({\\dfrac {\\displaystyle \\prod_{\\substack {1 \\mathop \\le r \\mathop \\le n \\\\ r \\mathop \\ne m} } \\left({x + k - r}\\right)} {\\displaystyle \\prod_{\\substack {1 \\mathop \\le r \\mathop \\le n \\\\ r \\mathop \\ne k} } \\left({k - r}\\right)} }\\right) = 1$ where $1 \\le m \\le n$ and $x$ is arbitrary."}
+{"_id": "12442", "title": "Summation by k of Product by r of x plus k minus r over Product by r less k of k minus r/Example", "text": ":$\\dfrac {x \\left({x - 2}\\right) \\left({x - 3}\\right)} {\\left({-1}\\right) \\left({-2}\\right) \\left({-3}\\right)} + \\dfrac {\\left({x + 1}\\right) \\left({x - 1}\\right) \\left({x - 2}\\right)} {\\left({1}\\right) \\left({-1}\\right) \\left({-2}\\right)} + \\dfrac {\\left({x + 2}\\right) x \\left({x - 1}\\right)} {\\left({2}\\right) \\left({1}\\right) \\left({-1}\\right)} + \\dfrac {\\left({x + 3}\\right) \\left({x + 1}\\right) x} {\\left({3}\\right) \\left({2}\\right) \\left({1}\\right)} = 1$"}
+{"_id": "12443", "title": "Inverse of Cauchy Matrix", "text": "Let $C_n$ be the square Cauchy matrix of order $n$: :$C_n = \\begin{bmatrix} \\dfrac 1 {x_1 + y_1} & \\dfrac 1 {x_1 + y_2} & \\cdots & \\dfrac 1 {x_1 + y_n} \\\\ \\dfrac 1 {x_2 + y_1} & \\dfrac 1 {x_2 + y_2} & \\cdots & \\dfrac 1 {x_2 + y_n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\dfrac 1 {x_n + y_1} & \\dfrac 1 {x_n + y_2} & \\cdots & \\dfrac 1 {x_n + y_n} \\\\ \\end{bmatrix}$ Then its inverse $C_n^{-1} = \\sqbrk b_n$ can be specified as: :$\\begin{bmatrix} b_{ij} \\end{bmatrix} = \\begin{bmatrix} \\dfrac {\\displaystyle \\prod_{k \\mathop = 1}^n \\paren {x_j + y_k} \\paren {x_k + y_i} } {\\displaystyle \\paren {x_j + y_i} \\paren {\\prod_{\\substack {1 \\mathop \\le k \\mathop \\le n \\\\ k \\mathop \\ne j} } \\paren {x_j - x_k} } \\paren {\\prod_{\\substack {1 \\mathop \\le k \\mathop \\le n \\\\ k \\mathop \\ne i} } \\paren {y_i - y_k} } } \\end{bmatrix}$"}
+{"_id": "12446", "title": "Sum of Elements in Inverse of Cauchy Matrix", "text": "Let $C_n$ be the Cauchy matrix of order $n$ given by: :$C_n = \\begin{bmatrix} \\dfrac 1 {x_1 + y_1} & \\dfrac 1 {x_1 + y_2 } & \\cdots & \\dfrac 1 {x_1 + y_n} \\\\ \\dfrac 1 {x_2 + y_1} & \\dfrac 1 {x_2 + y_2 } & \\cdots & \\dfrac 1 {x_2 + y_n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\dfrac 1 {x_m + y_1} & \\dfrac 1 {x_m + y_2 } & \\cdots & \\dfrac 1 {x_m + y_n} \\\\ \\end{bmatrix}$ Let $C_n^{-1}$ be its inverse, from Inverse of Cauchy Matrix: :$b_{i j} = \\dfrac {\\displaystyle \\prod_{k \\mathop = 1}^n \\paren {x_j + y_k} \\paren {x_k + y_i} } {\\displaystyle \\paren {x_j + y_i} \\paren {\\prod_{\\substack {1 \\mathop \\le k \\mathop \\le n \\\\ k \\mathop \\ne j} } \\paren {x_j - x_k} } \\paren {\\prod_{\\substack {1 \\mathop \\le k \\mathop \\le n \\\\ k \\mathop \\ne i} } \\paren {y_i - x_k} } }$ The sum of all the elements of $C_n^{-1}$ is: :$\\displaystyle \\sum_{1 \\mathop \\le i, \\ j \\mathop \\le n} b_{i j} = \\sum_{k \\mathop = 1}^n x_k + \\sum_{k \\mathop = 1}^n y_k$"}
+{"_id": "12447", "title": "Hilbert Matrix is Cauchy Matrix", "text": "A Hilbert matrix is a special case of a Cauchy matrix."}
+{"_id": "12448", "title": "Inverse of Hilbert Matrix", "text": "Let $H_n$ be the Hilbert matrix of order $n$: :$\\begin{bmatrix} a_{i j} \\end{bmatrix} = \\begin{bmatrix} \\dfrac 1 {i + j - 1} \\end{bmatrix}$ Then its inverse $H_n^{-1} = \\sqbrk b_n$ can be specified as: :$\\begin{bmatrix} b_{i j} \\end{bmatrix} = \\begin{bmatrix} \\dfrac {\\paren {-1}^{i + j} \\paren {i + n - 1}! \\paren {j + n - 1}!} {\\paren {\\paren {i - 1}!}^2 \\paren {\\paren {j - 1}!}^2 \\paren {n - j}! \\paren {n - i}! \\paren {i + j - 1} } \\end{bmatrix}$"}
+{"_id": "12450", "title": "Sum of Elements in Inverse of Hilbert Matrix", "text": "Let $H_n$ be the Hilbert matrix of order $n$: :$\\begin{bmatrix} a_{i j} \\end{bmatrix} = \\begin{bmatrix} \\dfrac 1 {i + j - 1} \\end{bmatrix}$ Consider its inverse $H_n^{-1}$. All the elements of $H_n^{-1}$ are integers. The sum of all the elements $b_{i j}$ of $H_n^{-1}$ is: :$\\displaystyle \\sum_{1 \\mathop \\le i, \\ j \\mathop \\le n} b_{i j} = n^2$"}
+{"_id": "12451", "title": "Suprema of two Real Sets", "text": "Let $S$ and $T$ be real sets. Let $S$ and $T$ admit suprema. Then: : $\\sup S \\le \\sup T \\iff \\forall \\epsilon \\in \\R_{>0}: \\forall s \\in S: \\exists t \\in T: s < t + \\epsilon$"}
+{"_id": "12452", "title": "Sequential Continuity is Equivalent to Continuity in the Reals", "text": "Let $A \\subseteq \\R$ be a subset of the real numbers. Let $c \\in A$. Let $f : A \\to \\R$ be a real function. Then $f$ is continuous at $c$ {{iff}}: :for each sequence $\\sequence {x_n}$ in $A$ that converges to $c$, the sequence $\\sequence {\\map f {x_n} }$ converges to $\\map f c$."}
+{"_id": "12453", "title": "Rising Factorial as Quotient of Factorials", "text": ":$x^{\\overline n} = \\dfrac {\\paren {x + n - 1}!} {\\paren {x - 1}!} = \\dfrac {\\map \\Gamma {x + n} } {\\map \\Gamma x}$"}
+{"_id": "12454", "title": "Equivalent Definition for Alternating Bilinear Mapping", "text": "Let $\\left({A_R, \\oplus}\\right)$ be an algebra over a ring $R$ with the property that $\\operatorname{Char}\\left({R}\\right) \\neq 2$. Then the following definitions for alternating bilinear maps are equivalent: * $\\oplus$ is an alternating bilinear map {{iff}} for all $a \\in A_R$, $a \\oplus a = 0$. * $\\oplus$ is an alternating bilinear map {{iff}} for all $a, b \\in A_R$, $a \\oplus b + b \\oplus a = 0$."}
+{"_id": "12456", "title": "Way Below if Between is Compact Set in Ordered Set of Topology", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $L = \\left({\\tau, \\preceq}\\right)$ be an ordered set where $\\preceq \\mathop = \\subseteq\\restriction_{\\tau \\times \\tau}$ Let $x, y \\in \\tau$ such that :$\\exists H \\subseteq S: x \\subseteq H \\subseteq y \\land H$ is compact Then :$x \\ll y$"}
+{"_id": "12457", "title": "Locally Compact iff Open Neighborhood contains Compact Set", "text": "Let $T = \\struct{S, \\tau}$ be a topological space. Then :$T$ is locally compact {{iff}} :$\\forall U \\in \\tau, x \\in U: \\exists K \\subseteq S: x \\in K^\\circ \\land K \\subseteq U \\land K$ is compact where $K^\\circ$ denotes the interior of $K$."}
+{"_id": "12458", "title": "Cauchy-Binet Formula", "text": "Let $\\mathbf A$ be an $m \\times n$ matrix. Let $\\mathbf B$ be an $n \\times m$ matrix. Let $1 \\le j_1, j_2, \\ldots, j_m \\le n$. Let $\\mathbf A_{j_1 j_2 \\ldots j_m}$ denote the $m \\times m$ matrix consisting of columns $j_1, j_2, \\ldots, j_m$ of $\\mathbf A$. Let $\\mathbf B_{j_1 j_2 \\ldots j_m}$ denote the $m \\times m$ matrix consisting of rows $j_1, j_2, \\ldots, j_m$ of $\\mathbf B$. Then: :$\\displaystyle \\map \\det {\\mathbf A \\mathbf B} = \\sum_{1 \\mathop \\le j_1 \\mathop < j_2 \\mathop < \\cdots \\mathop < j_m \\le n} \\map \\det {\\mathbf A_{j_1 j_2 \\ldots j_m} } \\map \\det {\\mathbf B_{j_1 j_2 \\ldots j_m} }$ where $\\det$ denotes the determinant."}
+{"_id": "12460", "title": "Auxiliary Relation is Congruent", "text": "Let $\\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let $\\mathcal R$ be relation on $S$ satisfying conditions $(ii)$ and $(iii)$ of auxiliary relation. Then :$\\forall x, y, z, u \\in S: \\left({x, z}\\right) \\in \\mathcal R \\land \\left({y, u}\\right) \\in \\mathcal R \\implies \\left({x \\vee y, z \\vee u}\\right) \\in \\mathcal R$"}
+{"_id": "12462", "title": "Cauchy-Binet Formula/Example/m equals 1", "text": "Let $\\mathbf A = \\left[{a}\\right]_{1 n}$ be a row matrix with $n$ columns. and $\\mathbf B = \\left[{b}\\right]_{n 1}$ be a column matrix with $n$ rows. Let $\\mathbf A \\mathbf B$ be the (conventional) matrix product of $\\mathbf A$ and $\\mathbf B$. Then: :$\\displaystyle \\det \\left({\\mathbf A \\mathbf B}\\right) = \\sum_{j \\mathop = 1}^n a_j b_j$ where: :$a_j$ is element $a_{1 j}$ of $\\mathbf A$ :$b_j$ is element $b_{j 1}$ of $\\mathbf B$."}
+{"_id": "12463", "title": "Preceding is Auxiliary Relation", "text": "Let $\\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Then :$\\preceq$ is auxiliary relation."}
+{"_id": "12464", "title": "Cauchy-Binet Formula/Example/Matrix by Transpose", "text": "Let $\\mathbf A$ be an $m \\times n$ matrix. Let $\\mathbf A^\\intercal$ be the transpose $\\mathbf A$. Let $1 \\le j_1, j_2, \\ldots, j_m \\le n$. Let $\\mathbf A_{j_1 j_2 \\ldots j_m}$ denote the $m \\times m$ matrix consisting of columns $j_1, j_2, \\ldots, j_m$ of $\\mathbf A$. Let $\\mathbf A^\\intercal_{j_1 j_2 \\ldots j_m}$ denote the $m \\times m$ matrix consisting of rows $j_1, j_2, \\ldots, j_m$ of $\\mathbf A^\\intercal$. Then: :$\\displaystyle \\det \\left({\\mathbf A \\mathbf A^\\intercal}\\right) = \\sum_{1 \\mathop \\le j_1 \\mathop < j_2 \\mathop < \\cdots \\mathop < j_m \\le n} \\left({\\det \\left({\\mathbf A_{j_1 j_2 \\ldots j_m} }\\right)}\\right)^2$ where $\\det$ denotes the determinant."}
+{"_id": "12465", "title": "Preceding is Top in Ordered Set of Auxiliary Relations", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let ${\\it Aux}\\left({L}\\right)$ be the set of all auxiliary relations on $S$. Let $P = \\left({ {\\it Aux}\\left({L}\\right), \\precsim}\\right)$ be an ordered set where $\\precsim \\mathop = \\subseteq\\restriction_{{\\it Aux}\\left({L}\\right) \\times {\\it Aux}\\left({L}\\right)}$ Then :$\\preceq \\mathop = \\top_P$ where $\\top_P$ denotes the greatest element in $P$."}
+{"_id": "12466", "title": "Bottom Relation is Auxiliary Relation", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let $B = \\left\\{ {\\left({\\bot, x}\\right): x \\in S}\\right\\}$ where $\\bot$ denotes the smallest element in $L$. Then :$B$ is auxiliary relation."}
+{"_id": "12467", "title": "Bottom Relation is Bottom in Ordered Set of Auxiliary Relations", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let ${\\it Aux}\\left({L}\\right)$ be the set of all auxiliary relations on $S$. Let $P = \\left({ {\\it Aux}\\left({L}\\right), \\precsim}\\right)$ be an ordered set where $\\precsim \\mathop = \\subseteq\\restriction_{{\\it Aux}\\left({L}\\right) \\times {\\it Aux}\\left({L}\\right)}$ Let $B = \\left\\{ {\\left({\\bot_L, x}\\right): x \\in S}\\right\\}$ where $\\bot_L$ denotes smallest element in $L$. Then :$B \\mathop = \\bot_P$"}
+{"_id": "12468", "title": "Inverse of Conditional is Converse of Contrapositive", "text": "Let $p \\implies q$ be a conditional. Then the inverse of $p \\implies q$ is the converse of its contrapositive."}
+{"_id": "12469", "title": "Inverse of Conditional is Contrapositive of Converse", "text": "Let $p \\implies q$ be a conditional. Then the inverse of $p \\implies q$ is the contrapositive of its converse."}
+{"_id": "12470", "title": "Converse of Conditional is Contrapositive of Inverse", "text": "Let $p \\implies q$ be a conditional. Then the converse of $p \\implies q$ is the contrapositive of its inverse."}
+{"_id": "12471", "title": "Converse of Conditional is Inverse of Contrapositive", "text": "Let $p \\implies q$ be a conditional. Then the converse of $p \\implies q$ is the inverse of its contrapositive."}
+{"_id": "12472", "title": "Intersection of Auxiliary Relations is Auxiliary Relation", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let $\\mathcal F$ be a non-empty set of auxiliary relations on $S$. Then :$\\bigcap \\mathcal F$ is auxiliary relation."}
+{"_id": "12474", "title": "Cauchy-Binet Formula/Example/m greater than n", "text": "Let $\\mathbf A$ be an $m \\times n$ matrix. Let $\\mathbf B$ be an $n \\times m$ matrix. Let $m > n$. Then: :$\\displaystyle \\det \\left({\\mathbf A \\mathbf B}\\right) = 0$"}
+{"_id": "12477", "title": "Krattenthaler's Identity", "text": ":$\\begin{vmatrix} \\left({x + q_2}\\right) \\left({x + q_3}\\right) & \\left({x + p_1}\\right) \\left({x + q_3}\\right) & \\left({x + p_1}\\right) \\left({x + p_2}\\right) \\\\ \\left({y + q_2}\\right) \\left({y + q_3}\\right) & \\left({y + p_1}\\right) \\left({y + q_3}\\right) & \\left({y + p_1}\\right) \\left({y + p_2}\\right) \\\\ \\left({z + q_2}\\right) \\left({z + q_3}\\right) & \\left({z + p_1}\\right) \\left({z + q_3}\\right) & \\left({z + p_1}\\right) \\left({z + p_2}\\right) \\end{vmatrix} = \\left({x - y}\\right) \\left({x - z}\\right) \\left({y - z}\\right) \\left({p_1 - q_2}\\right) \\left({p_1 - q_3}\\right) \\left({p_2 - q_3}\\right)$ where $\\left\\vert{\\, \\cdot \\,}\\right\\vert$ denotes determinant."}
+{"_id": "12483", "title": "Infima of two Real Sets", "text": "Let $S$ and $T$ be sets of real numbers. Let $S$ and $T$ admit infima. Then: : $\\inf S \\ge \\inf T \\iff \\forall \\epsilon \\in \\R_{>0}: \\forall s \\in S: \\exists t \\in T: s + \\epsilon > t$"}
+{"_id": "12484", "title": "Floor Function is Integer", "text": "Let $x$ be a real number. Then the floor function of $x$ is an integer: :$\\floor x \\in \\Z$"}
+{"_id": "12485", "title": "Floor of Number plus Integer", "text": ":$\\forall n \\in \\Z: \\floor x + n = \\floor {x + n}$"}
+{"_id": "12486", "title": "Floor is between Number and One Less", "text": ":$x - 1 < \\left\\lfloor{x}\\right\\rfloor \\le x$ where $\\left\\lfloor{x}\\right\\rfloor$ is the floor of $x$."}
+{"_id": "12487", "title": "Equivalence of Definitions of Floor Function", "text": "Let $x$ be a real number. {{TFAE|def = Floor Function}}"}
+{"_id": "12488", "title": "Properties of Floor Function", "text": "This page gathers together some basic propeties of the floor function."}
+{"_id": "12489", "title": "Properties of Ceiling Function", "text": "This page gathers together some basic propeties of the ceiling function."}
+{"_id": "12490", "title": "Ceiling Function is Integer", "text": "Let $x$ be a real number. Then the ceiling function of $x$ is an integer: :$\\ceiling x \\in \\Z$"}
+{"_id": "12491", "title": "Real Number is Integer iff equals Ceiling", "text": ":$x = \\ceiling x \\iff x \\in \\Z$"}
+{"_id": "12492", "title": "Ceiling Function is Idempotent", "text": ":$\\ceiling {\\ceiling x} = \\ceiling x$"}
+{"_id": "12493", "title": "Way Below Relation is Auxiliary Relation", "text": "Lrt $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Then :$\\ll$ is auxiliary relation where $\\ll$ denotes the way below relation."}
+{"_id": "12495", "title": "Number is between Ceiling and One Less", "text": ":$\\ceiling x - 1 < x \\le \\ceiling x$"}
+{"_id": "12496", "title": "Ceiling is between Number and One More", "text": ":$x \\le \\ceiling x < x + 1$ where $\\ceiling x$ is the ceiling of $x$."}
+{"_id": "12497", "title": "Ceiling of Number plus Integer", "text": ":$\\forall n \\in \\Z: \\left \\lceil {x} \\right \\rceil + n = \\left \\lceil {x + n} \\right \\rceil$"}
+{"_id": "12498", "title": "Relation Segment of Auxiliary Relation is Ideal", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let $R$ be auxiliary relation on $S$. Let $x \\in S$. Then :$x^R$ is ideal in $L$ where $x^R$ denotes the $R$-segment of $x$."}
+{"_id": "12499", "title": "Quotient of Modulo Operation with Modulus", "text": "Let $x, y \\in \\R$ be real numbers. Let $x \\bmod y$ denote the modulo operation: :$x \\bmod y := \\begin{cases} x - y \\left \\lfloor {\\dfrac x y}\\right \\rfloor & : y \\ne 0 \\\\ x & : y = 0 \\end{cases}$ where $\\left \\lfloor {\\dfrac x y}\\right \\rfloor$ denotes the floor of $\\dfrac x y$. Let $y \\ne 0$. Then: :$0 \\le \\dfrac x y - \\left \\lfloor {\\dfrac x y}\\right \\rfloor = \\dfrac {x \\bmod y} y < 1$"}
+{"_id": "12500", "title": "Range of Modulo Operation for Positive Modulus", "text": "Let $x, y \\in \\R$ be real numbers. Let $x \\bmod y$ denote the modulo operation: :$x \\bmod y := \\begin{cases} x - y \\floor {\\dfrac x y} & : y \\ne 0 \\\\ x & : y = 0 \\end{cases}$ where $\\floor {\\dfrac x y}$ denotes the floor of $\\dfrac x y$. Let $y > 0$. Then: :$0 \\le x \\bmod y < y$"}
+{"_id": "12501", "title": "Range of Modulo Operation for Negative Modulus", "text": "Let $x, y \\in \\R$ be real numbers. Let $x \\bmod y$ denote the modulo operation: :$x \\bmod y := \\begin{cases} x - y \\left \\lfloor {\\dfrac x y}\\right \\rfloor & : y \\ne 0 \\\\ x & : y = 0 \\end{cases}$ where $\\left \\lfloor {\\dfrac x y}\\right \\rfloor$ denotes the floor of $\\dfrac x y$. Let $y < 0$. Then: :$0 \\ge x \\bmod y > y$"}
+{"_id": "12502", "title": "Zero is Integer Multiple of Zero", "text": "Zero is an integer multiple of zero."}
+{"_id": "12505", "title": "Modulo Operation/Examples/18 mod 3", "text": ":$18 \\bmod 3 = 0$"}
+{"_id": "12507", "title": "Number is Divisor iff Modulo is Zero", "text": "Let $x, y \\in \\R$ be real numbers. Let $x \\bmod y$ denote the modulo operation: :$x \\bmod y := \\begin {cases} x - y \\floor {\\dfrac x y} & : y \\ne 0 \\\\ x & : y = 0 \\end {cases}$ where $\\floor {\\dfrac x y}$ denotes the floor of $\\dfrac x y$. Then $x \\bmod y = 0$ {{iff}} $x$ is an integer multiple of $y$."}
+{"_id": "12509", "title": "Number less than Integer iff Floor less than Integer", "text": ":$\\left \\lfloor{x}\\right \\rfloor < n \\iff x < n$"}
+{"_id": "12510", "title": "Number not less than Integer iff Floor not less than Integer", "text": ":$x \\ge n \\iff \\left \\lfloor{x}\\right \\rfloor \\ge n$"}
+{"_id": "12511", "title": "Number not greater than Integer iff Ceiling not greater than Integer", "text": ":$\\left \\lceil{x}\\right \\rceil \\le n \\iff x \\le n$"}
+{"_id": "12512", "title": "Number greater than Integer iff Ceiling greater than Integer", "text": ":$\\left \\lceil{x}\\right \\rceil > n \\iff x > n$"}
+{"_id": "12513", "title": "Integer equals Floor iff between Number and One Less", "text": ":$\\floor x = n \\iff x - 1 < n \\le x$"}
+{"_id": "12514", "title": "Integer equals Floor iff Number between Integer and One More", "text": ":$\\floor x = n \\iff n \\le x < n + 1$"}
+{"_id": "12515", "title": "Integer equals Ceiling iff between Number and One More", "text": ":$\\left \\lceil{x}\\right \\rceil = n \\iff x \\le n < x + 1$"}
+{"_id": "12516", "title": "Integer equals Ceiling iff Number between Integer and One Less", "text": ":$\\ceiling x = n \\iff n - 1 < x \\le n$"}
+{"_id": "12517", "title": "Floor of Root of Floor equals Floor of Root", "text": ":$\\displaystyle \\floor {\\sqrt {\\floor x} } = \\floor {\\sqrt x}$"}
+{"_id": "12518", "title": "Ceiling of Root of Ceiling equals Ceiling of Root", "text": ":$\\displaystyle \\ceiling {\\sqrt {\\ceiling x} } = \\ceiling {\\sqrt x}$"}
+{"_id": "12519", "title": "Congruence by Divisor of Modulus/Integer Modulus", "text": "Let $r, s \\in \\Z$ be integers. Let $a, b \\in \\Z$ such that $a$ is congruent modulo $r s$ to $b$, that is: :$a \\equiv b \\pmod {r s}$ Then: :$a \\equiv b \\pmod r$ and: :$a \\equiv b \\pmod s$"}
+{"_id": "12520", "title": "Modulo Addition is Well-Defined/Real Modulus", "text": "Let $z \\in \\R$ be a real number. Let: :$a \\equiv b \\pmod z$ and: :$x \\equiv y \\pmod z$ where $a, b, x, y \\in \\R$. Then: : $a + x \\equiv b + y \\pmod z$"}
+{"_id": "12522", "title": "Congruence by Product of Moduli/Real Modulus", "text": "Let $a, b, z \\in \\R$. Let $a \\equiv b \\pmod z$ denote that $a$ is congruent to $b$ modulo $z$. Then $\\forall y \\in \\R, y \\ne 0$: :$a \\equiv b \\pmod z \\iff y a \\equiv y b \\pmod {y z}$"}
+{"_id": "12523", "title": "Segment of Auxiliary Relation is Subset of Lower Closure", "text": "Let $\\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let $R$ be auxiliary relation on $S$. Let $x \\in S$. Then :$x^R \\subseteq x^\\preceq$ where :$x^R$ denotes the $R$-segment of $x$, :$x^\\preceq$ denotes the lower closure of $x$."}
+{"_id": "12525", "title": "Preceding implies Inclusion of Segments of Auxiliary Relation", "text": "Let $\\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let $R$ be an auxiliary relation on $S$. Let $x, y \\in S$ such that :$x \\preceq y$ Then :$x^R \\subseteq y^R$ where $x^R$ denotes the $R$-segment of $x$."}
+{"_id": "12526", "title": "Characteristic Function of Square-Free Integers is Multiplicative", "text": "Let $S \\subseteq \\Z$ be the set of positive integers defined as: :$S := \\set {n \\in \\Z: \\forall k \\in \\Z_{>1}: k^2 \\nmid n}$ That is, let $S$ be the set of all square-free positive integers. Let $\\chi_S: \\N \\to \\Z$ denote the characteristic function of $S$: :$\\forall n \\in \\Z: \\map {\\chi_S} n = \\sqbrk {n \\in S}$ where $\\sqbrk {n \\in S}$ is Iverson's convention. Then $\\chi_S$ is multiplicative."}
+{"_id": "12527", "title": "Segment of Auxiliary Relation Mapping is Increasing", "text": "Let $R = \\left({S, \\preceq}\\right)$ be an ordered set. Let ${\\it Ids}\\left({R}\\right)$ be the set of all ideals in $R$. Let $L = \\left({ {\\it Ids}\\left({R}\\right), \\precsim}\\right)$ be an ordered set where $\\precsim \\mathop = \\subseteq\\restriction_{ {\\it Ids}\\left({R}\\right) \\times {\\it Ids}\\left({R}\\right)}$ Let $r$ be an auxiliary relation on $S$. Let $f: S \\to {\\it Ids}\\left({R}\\right)$ be a mapping such that :$\\forall x \\in S: f\\left({x}\\right) = x^r$ where $x^r$ denotes the $r$-segment of $x$. Then :$f$ is increasing mapping."}
+{"_id": "12529", "title": "Product of Multiplicative Functions is Multiplicative", "text": "Let $f: \\N \\to \\C$ and $g: \\N \\to \\C$ be multiplicative functions. Then their pointwise product: :$f \\times g: \\Z \\to \\Z: \\forall s \\in S: \\map {\\paren {f \\times g} } s := \\map f s \\times \\map g s$ is also multiplicative."}
+{"_id": "12530", "title": "Segment of Auxiliary Relation Mapping is Element of Increasing Mappings Satisfying Inclusion in Lower Closure", "text": "Let $R = \\left({S, \\preceq}\\right)$ be an ordered set. Let $\\mathit{Ids}\\left({R}\\right)$ be the set of all ideals in $R$. Let $L = \\left({ \\mathit{Ids}\\left({R}\\right), \\precsim}\\right)$ be an ordered set where $\\precsim \\mathop = \\subseteq\\restriction_{\\mathit{Ids}\\left({R}\\right) \\times \\mathit{Ids}\\left({R}\\right)}$. Let $r$ be an auxiliary relation on $S$. Let $M = \\left({F, \\preccurlyeq}\\right)$ be the ordered set of increasing mappings $g$ satisfying $\\forall x \\in S: g\\left({x}\\right) \\subseteq x^\\preceq$. Let $f: S \\to \\mathit{Ids}\\left({R}\\right)$ be a mapping such that: :$\\forall x \\in S: f\\left({x}\\right) = x^r$ where $x^r$ denotes the $r$-segment of $x$. Then: :$(1): \\quad f \\in F$ Let $h: S \\to \\mathit{Ids}\\left({R}\\right): x \\mapsto x^\\preceq$ Then: :$(2): \\quad h \\in F \\land f \\preccurlyeq h$ Let $k: S \\to \\mathit{Ids}\\left({R}\\right): x \\mapsto \\left\\{ {\\bot}\\right\\}$ where $\\bot$ denotes the smallest element in $L$. Then: :$(3): \\quad k \\in F$ and $k \\preccurlyeq f$"}
+{"_id": "12531", "title": "Singleton of Bottom is Ideal", "text": "Let $\\struct {S, \\preceq}$ be a bounded below ordered set. Then :$\\set \\bot$ is an ideal in $\\struct {S, \\preceq}$ where $\\bot$ denotes the smallest element in $S$."}
+{"_id": "12535", "title": "Rising Factorial in terms of Falling Factorial", "text": ":$x^{\\overline n} = \\left({x + n - 1}\\right)^{\\underline n}$"}
+{"_id": "12536", "title": "Rising Factorial in terms of Falling Factorial of Negative", "text": ":$x^{\\overline k} = \\paren {-1}^k \\paren {-x}^{\\underline k}$"}
+{"_id": "12537", "title": "Falling Factorial as Quotient of Factorials", "text": ":$x^{\\underline n} = \\dfrac {x!} {\\paren {x - n}!} = \\dfrac {\\map \\Gamma {x + 1} } {\\map \\Gamma {x - n + 1} }$"}
+{"_id": "12538", "title": "One to Integer Rising is Integer Factorial", "text": ":$1^{\\overline n} = n!$"}
+{"_id": "12539", "title": "Number to Power of One Rising is Itself", "text": ":$x^{\\overline 1} = x$"}
+{"_id": "12540", "title": "Integer to Power of Itself Falling is Factorial", "text": ":$n^{\\underline n} = n!$"}
+{"_id": "12541", "title": "Number to Power of One Falling is Itself", "text": ":$x^{\\underline 1} = x$"}
+{"_id": "12542", "title": "Number of Permutations of One Less", "text": ":${}^{n - 1} P_n = {}^n P_n$ where ${}^k P_n$ denotes the number of ordered selections of $k$ objects from $n$."}
+{"_id": "12543", "title": "Element of Increasing Mappings Satisfying Inclusion in Lower Closure is Generated by Auxiliary Relation", "text": "Let $R = \\left({S, \\preceq}\\right)$ be a bounded below join semilattice. Let $\\mathit{Ids}\\left({R}\\right)$ be the set of all ideals in $R$. Let $L = \\left({ \\mathit{Ids}\\left({R}\\right), \\precsim}\\right)$ be an ordered set where $\\precsim \\mathop = \\subseteq\\restriction_{\\mathit{Ids}\\left({R}\\right) \\times \\mathit{Ids}\\left({R}\\right)}$. Let $M = \\left({F, \\preccurlyeq}\\right)$ be the ordered set of increasing mappings $g:S \\to \\mathit{Ids}\\left({R}\\right)$ satisfying $\\forall x \\in S: g\\left({x}\\right) \\subseteq x^\\preceq$. Let $f \\in F$. Then :there exists an auxiliary relation $\\mathcal R$ on $S$ such that ::$\\forall x \\in S:f\\left({x}\\right) = x^{\\mathcal R}$ where $x^{\\mathcal R}$ denotes the $\\mathcal R$-segment of $x$."}
+{"_id": "12544", "title": "Binomial Coefficient with Two/Corollary", "text": ":$\\forall n \\in \\N: \\dbinom n 2 = T_{n - 1} = \\dfrac {n \\paren {n - 1} } 2$ where $T_n$ is the $n$th triangular number."}
+{"_id": "12546", "title": "Increasing Mappings Satisfying Inclusion in Lower Closure is Isomorphic to Auxiliary Relations", "text": "Let $R = \\left({S, \\preceq}\\right)$ be a bounded below join semilattice. Let $\\mathit{Ids}\\left({R}\\right)$ be the set of all ideals in $R$. Let $L = \\left({ \\mathit{Ids}\\left({R}\\right), \\precsim}\\right)$ be an ordered set where $\\precsim \\mathop = \\subseteq\\restriction_{\\mathit{Ids}\\left({R}\\right) \\times \\mathit{Ids}\\left({R}\\right)}$. Let $M = \\left({F, \\preccurlyeq}\\right)$ be the ordered set of increasing mappings $g:S \\to \\mathit{Ids}\\left({R}\\right)$ satisfying $\\forall x \\in S: g\\left({x}\\right) \\subseteq x^\\preceq$. Let $\\mathit{Aux}\\left({R}\\right)$ be the set of all auxiliary relations on $S$. Let $P = \\left({ \\mathit{Aux}\\left({R}\\right), \\precsim'}\\right)$ be an ordered set where $\\precsim' \\mathop = \\subseteq\\restriction_{\\mathit{Aux}\\left({R}\\right) \\times \\mathit{Aux}\\left({R}\\right)}$. Then :there exists an order isomorphism between $P$ and $M$"}
+{"_id": "12547", "title": "Binomial Theorem/Abel's Generalisation", "text": ":$\\displaystyle \\left({x + y}\\right)^n = \\sum_k \\binom n k x \\left({x - k z}\\right)^{k - 1} \\left({y + k z}\\right)^{n - k}$"}
+{"_id": "12548", "title": "Sum over k of r Choose m+k by s Choose n+k", "text": "Let $s \\in \\R, r \\in \\Z_{\\ge 0}, m, n \\in \\Z$. Then: :$\\displaystyle \\sum_k \\binom r {m + k} \\binom s {n + k} = \\binom {r + s} {r - m + n}$"}
+{"_id": "12549", "title": "Sum over k of r Choose k by s+k Choose n by -1^r-k", "text": "Let $s \\in \\R, r \\in \\Z_{\\ge 0}, n \\in \\Z$. Then: :$\\displaystyle \\sum_k \\binom r k \\binom {s + k} n \\left({-1}\\right)^{r - k} = \\binom s {n - r}$"}
+{"_id": "12551", "title": "Sum over k of r-k Choose m by s+k Choose n", "text": "Let $m, n, r, s \\in \\Z_{\\ge 0}$ such that $n \\ge s$. Then: :$\\displaystyle \\sum_{k \\mathop = 0}^r \\binom {r - k} m \\binom {s + k} n = \\binom {r + s + 1} {m + n + 1}$"}
+{"_id": "12552", "title": "Sum over k of r-tk Choose k by s-t(n-k) Choose n-k by r over r-tk", "text": "Let $r, s, t \\in \\R, n \\in \\Z$. Then: :$\\displaystyle \\sum_{k \\mathop \\ge 0} \\binom {r - t k} k \\binom {s - t \\left({n - k}\\right)} {n - k} \\frac r {r - t k} = \\binom {r + s - t n} n$"}
+{"_id": "12555", "title": "Sum over k of r Choose k by s Choose k by k", "text": ":$\\displaystyle \\sum_k \\binom r k \\binom s k k = \\binom {r + s - 1} {r - 1} s$"}
+{"_id": "12556", "title": "Relation Segment is Increasing", "text": "Let $S$ be a set. Let $\\RR, \\QQ$ be relations on $S$ such that :$\\RR \\subseteq \\QQ$ Let $x \\in S$. Then :$x^\\RR \\subseteq x^\\QQ$ where $x^\\RR$ denotes the $\\RR$-segment of $x$."}
+{"_id": "12557", "title": "Sum over k of n+k Choose 2 k by 2 k Choose k by -1^k over k+1", "text": ":$\\displaystyle \\sum_k \\binom {n + k} {2 k} \\binom {2 k} k \\frac {\\paren {-1}^k} {k + 1} = \\sqbrk {n = 0}$"}
+{"_id": "12564", "title": "Intersection of Ideals with Suprema Succeed Element equals Way Below Closure of Element", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $x \\in S$. Then :$\\displaystyle \\bigcap \\left\\{ {I \\in \\mathit{Ids}: x \\preceq \\sup I}\\right\\} = x^\\ll$ where $\\mathit{Ids}$ denotes the set of all ideals in $L$."}
+{"_id": "12565", "title": "Way Below Closure is Ideal in Bounded Below Join Semilattice", "text": "Let $L = \\struct {S, \\vee, \\preceq}$ be a bounded below join semilattice. Let $x \\in S$. Then :$x^\\ll$ is ideal in $L$."}
+{"_id": "12568", "title": "Sum over k of r Choose k by -1^r-k by Polynomial", "text": "Let $r \\in \\Z_{\\ge 0}$. Then: :$\\displaystyle \\sum_k \\binom r k \\paren {-1}^{r - k} \\map {P_r} k = r! \\, b_r$ where: :$\\map {P_r} k = b_0 + b_1 k + \\cdots + b_r k^r$ is a polynomial in $k$ of degree $r$."}
+{"_id": "12569", "title": "Sum over k of r Choose k by s-kt Choose r by -1^k", "text": "Let $r \\in \\Z_{\\ge 0}$. Then: :$\\displaystyle \\sum_k \\binom r k \\binom {s - k t} r \\left({-1}\\right)^k = t^r$ where $\\dbinom r k$ etc. are binomial coefficients."}
+{"_id": "12573", "title": "Multinomial Coefficient expressed as Product of Binomial Coefficients", "text": ":$\\dbinom {k_1 + k_2 + \\cdots + k_m} {k_1, k_2, \\ldots, k_m} = \\dbinom {k_1 + k_2} {k_1} \\dbinom {k_1 + k_2 + k_3} {k_1 + k_2} \\cdots \\dbinom {k_1 + k_2 + \\cdots + k_m} {k_1 + k_2 + \\cdots + k_{m - 1} }$ where: :$\\dbinom {k_1 + k_2 + \\cdots + k_m} {k_1, k_2, \\ldots, k_m}$ denotes a multinomial coefficient :$\\dbinom {k_1 + k_2} {k_1}$ etc. denotes binomial coefficients."}
+{"_id": "12577", "title": "Universal Closures are Semantically Equivalent", "text": "Let $\\mathbf A$ be a WFF of predicate logic. Let $\\mathbf B, \\mathbf B'$ be universal closures of $\\mathbf A$. Then $\\mathbf B$ and $\\mathbf B'$ are semantically equivalent."}
+{"_id": "12578", "title": "Meet-Continuous iff if Element Precedes Supremum of Directed Subset then Element equals Supremum of Meet of Element by Directed Subset", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be an up-complete lattice. Then :$L$ is meet-continuous {{iff}} :$\\forall x \\in S$, directed subset $D$ of $S: x \\preceq \\sup D \\implies x = \\sup \\set {x \\wedge d: d \\in D}$"}
+{"_id": "12579", "title": "Continuous Lattice is Meet-Continuous", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below continuous lattice. Then $L$ is meet-continuous."}
+{"_id": "12581", "title": "Number to Power of Zero Rising is One", "text": ":$x^{\\overline 0} = 1$"}
+{"_id": "12582", "title": "Number to Power of Zero Falling is One", "text": ":$x^{\\underline 0} = 1$"}
+{"_id": "12583", "title": "Product of Number by its Falling Factorial", "text": "Let $x^{\\underline n}$ denote the $n$th falling factorial power of $x$. Then: :$x x^{\\underline n} = x^{\\underline {n + 1} } + n x^{\\underline n}$"}
+{"_id": "12584", "title": "Product of Number by its Rising Factorial", "text": "Let $x^{\\overline n}$ denote the $n$th rising factorial power of $x$. Then: :$x x^{\\overline n} = x^{\\overline {n + 1} } - n x^{\\overline n}$"}
+{"_id": "12586", "title": "Value of Formula under Assignment Determined by Free Variables", "text": "Let $\\mathbf A$ be a WFF of predicate logic. Let $\\mathcal A$ be a structure for predicate logic. Let $\\sigma, \\sigma'$ be assignments for $\\mathbf A$ in $\\mathcal A$ such that: :For each free variable $x$ of $\\mathbf A$, $\\sigma \\left({x}\\right) = \\sigma' \\left({x}\\right)$ Then: :$\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\mathbf A}\\right) } \\left[{\\sigma}\\right] = \\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\mathbf A}\\right) } \\left[{\\sigma'}\\right]$ where $\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\mathbf A}\\right) } \\left[{\\sigma}\\right]$ is the value of $\\mathbf A$ under $\\sigma$."}
+{"_id": "12587", "title": "First Inversion Formula for Stirling Numbers", "text": "For all $m, n \\in \\Z_{\\ge 0}$: :$\\displaystyle \\sum_k \\left[{n \\atop k}\\right] \\left\\{ {k \\atop m}\\right\\} \\left({-1}\\right)^{n - k} = \\delta_{m n}$ where: :$\\displaystyle \\left[{n \\atop k}\\right]$ denotes an unsigned Stirling number of the first kind :$\\displaystyle \\left\\{ {k \\atop m}\\right\\}$ denotes a Stirling number of the second kind :$\\delta_{m n}$ denotes the Kronecker delta."}
+{"_id": "12588", "title": "Unsigned Stirling Number of the First Kind of 1", "text": ":$\\displaystyle {1 \\brack n} = \\delta_{1 n}$"}
+{"_id": "12589", "title": "Signed Stirling Number of the First Kind of 1", "text": ":$s \\left({1, n}\\right) = \\delta_{1 n}$"}
+{"_id": "12590", "title": "Stirling Number of the Second Kind of 1", "text": ":$\\displaystyle {1 \\brace n} = \\delta_{1 n}$"}
+{"_id": "12591", "title": "Second Inversion Formula for Stirling Numbers", "text": "For all $m, n \\in \\Z_{\\ge 0}$: :$\\displaystyle \\sum_k \\left\\{ {n \\atop k}\\right\\} \\left[{k \\atop m}\\right] \\left({-1}\\right)^{n - k} = \\delta_{m n}$ where: :$\\displaystyle \\left\\{ {n \\atop k}\\right\\}$ denotes a Stirling number of the second kind :$\\displaystyle \\left[{k \\atop m}\\right]$ denotes an unsigned Stirling number of the first kind :$\\delta_{m n}$ denotes the Kronecker delta."}
+{"_id": "12592", "title": "Zero Choose n", "text": ":$\\dbinom 0 n = \\delta_{0 n}$"}
+{"_id": "12593", "title": "Unsigned Stirling Number of the First Kind of 0", "text": ":$\\displaystyle \\left[{0 \\atop n}\\right] = \\delta_{0 n}$"}
+{"_id": "12594", "title": "Particular Values of Signed Stirling Numbers of the First Kind", "text": "This page gathers together some particular values of signed Stirling numbers of the first kind."}
+{"_id": "12595", "title": "Particular Values of Unsigned Stirling Numbers of the First Kind", "text": "This page gathers together some particular values of unsigned Stirling numbers of the first kind."}
+{"_id": "12597", "title": "Particular Values of Stirling Numbers of the Second Kind", "text": "This page gathers together some particular values of Stirling numbers of the second kind."}
+{"_id": "12598", "title": "Stirling Number of the Second Kind of 0", "text": ":$\\displaystyle {0 \\brace n} = \\delta_{0 n}$"}
+{"_id": "12599", "title": "Unsigned Stirling Number of the First Kind of Number with Self", "text": ":$\\displaystyle \\left[{n \\atop n}\\right] = 1$"}
+{"_id": "12600", "title": "Unsigned Stirling Number of the First Kind of Number with Greater", "text": "Let $\\displaystyle {n \\brack k}$ denote an unsigned Stirling number of the first kind. Then: :$\\displaystyle {n \\brack k} = 0$"}
+{"_id": "12601", "title": "Signed Stirling Number of the First Kind of Number with Greater", "text": "Let $\\map s {n, k}$ denote a signed Stirling number of the first kind. Then: :$\\map s {n, k} = 0$"}
+{"_id": "12603", "title": "Stirling Number of Number with Greater", "text": "Let $n, k \\in \\Z_{\\ge 0}$. Let $k > n$."}
+{"_id": "12604", "title": "Stirling Number of the Second Kind of Number with Self", "text": ":$\\displaystyle {n \\brace n} = 1$"}
+{"_id": "12605", "title": "Value of Term under Assignment Determined by Variables", "text": "Let $\\tau$ be a term of predicate logic. Let $\\mathcal A$ be a structure for predicate logic. Let $\\sigma, \\sigma'$ be assignments for $\\tau$ in $\\mathcal A$ such that: :For each variable $x$ occurring in $\\tau$, $\\sigma \\left({x}\\right) = \\sigma' \\left({x}\\right)$ Then: :$\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau}\\right) } \\left[{\\sigma}\\right] = \\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau}\\right) } \\left[{\\sigma'}\\right]$ where $\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau}\\right) } \\left[{\\sigma}\\right]$ is the value of $\\tau$ under $\\sigma$."}
+{"_id": "12606", "title": "Relation between Signed and Unsigned Stirling Numbers of the First Kind", "text": "Let $m, n \\in \\Z_{\\ge 0}$ be positive integers. Then: :$\\displaystyle {n \\brack m} = \\paren {-1}^{n + m} \\map s {n, m}$ where: :$\\displaystyle {n \\brack m}$ denotes an unsigned Stirling number of the first kind :$\\map s {n, m}$ denotes a signed Stirling number of the first kind."}
+{"_id": "12607", "title": "Unsigned Stirling Number of the First Kind of Number with Greater/Proof 1", "text": "Let $n, k \\in \\Z_{\\ge 0}$ such that $k > n$. {{:Unsigned Stirling Number of the First Kind of Number with Greater}}"}
+{"_id": "12608", "title": "Unsigned Stirling Number of the First Kind of Number with Greater/Proof 2", "text": "Let $n, k \\in \\Z_{\\ge 0}$ such that $k > n$. {{:Unsigned Stirling Number of the First Kind of Number with Greater}}"}
+{"_id": "12609", "title": "Stirling Number of the Second Kind of Number with Greater/Proof 1", "text": "Let $n, k \\in \\Z_{\\ge 0}$ such that $k > n$. {{:Stirling Number of the Second Kind of Number with Greater}}"}
+{"_id": "12610", "title": "Stirling Number of the Second Kind of Number with Greater/Proof 2", "text": "Let $n, k \\in \\Z_{\\ge 0}$ such that $k > n$. {{:Stirling Number of the Second Kind of Number with Greater}}"}
+{"_id": "12611", "title": "Signed Stirling Number of the First Kind of Number with Self", "text": ":$s \\left({n, n}\\right) = 1$"}
+{"_id": "12612", "title": "Unsigned Stirling Number of the First Kind of n with n-1", "text": ":$\\displaystyle \\left[{n \\atop n - 1}\\right] = \\binom n 2$"}
+{"_id": "12613", "title": "Stirling Number of the Second Kind of n with n-1", "text": ":$\\displaystyle {n \\brace n - 1} = \\binom n 2$"}
+{"_id": "12614", "title": "Signed Stirling Number of the First Kind of n with n-1", "text": ":$\\map s {n, n - 1} = -\\dbinom n 2$"}
+{"_id": "12615", "title": "Unsigned Stirling Number of the First Kind of n+1 with 0", "text": ":$\\displaystyle {n + 1 \\brack 0} = 0$"}
+{"_id": "12616", "title": "Stirling Number of the Second Kind of n+1 with 0", "text": ":$\\displaystyle {n + 1 \\brace 0} = 0$"}
+{"_id": "12618", "title": "Unsigned Stirling Number of the First Kind of n+1 with 1", "text": ":$\\displaystyle \\left[{n + 1 \\atop 1}\\right] = n!$"}
+{"_id": "12619", "title": "Stirling Number of the Second Kind of n+1 with 1", "text": ":$\\displaystyle {n + 1 \\brace 1} = 1$"}
+{"_id": "12620", "title": "Signed Stirling Number of the First Kind of n+1 with 1", "text": ":$s \\left({n + 1, 1}\\right) = \\left({-1}\\right)^n n!$"}
+{"_id": "12622", "title": "Sum over k of Unsigned Stirling Numbers of the First Kind of n with k by k choose m", "text": "Let $m, n \\in \\Z_{\\ge 0}$. :$\\displaystyle \\sum_k \\left[{n \\atop k}\\right] \\binom k m = \\left[{n + 1 \\atop m + 1}\\right]$ where: :$\\displaystyle \\left[{n \\atop k}\\right]$ denotes an unsigned Stirling number of the first kind :$\\dbinom k m$ denotes a binomial coefficient."}
+{"_id": "12623", "title": "Semantically Equivalent Terms are Equal", "text": "Let $\\tau_1, \\tau_2$ be terms. Suppose that they are semantically equivalent with respect to the empty set. Then $\\tau_1 = \\tau_2$."}
+{"_id": "12624", "title": "Down Mapping is Generated by Approximating Relation", "text": "Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be a bounded below meet-continuous meet semilattice. Let $\\mathit{Ids}\\left({L}\\right)$ be the set of all ideals in $L$. Let $I$ be an ideal in $L$. Let $f: S \\to \\mathit{Ids}\\left({L}\\right)$ be a mapping such that :$\\forall x \\in S: x \\preceq \\sup I \\implies f\\left({x}\\right) = \\left\\{ {x \\wedge i: i \\in I}\\right\\}$ and :$\\forall x \\in S: x \\npreceq \\sup I \\implies f\\left({x}\\right) = x^\\preceq$ where $x^\\preceq$ denotes the lower closure of $x$. Then :there exists an auxiliary approximating relation $\\mathcal R$ on $S$ such that ::$\\forall s \\in S: f\\left({s}\\right) = s^{\\mathcal R}$ where $s^{\\mathcal R}$ denotes the $\\mathcal R$-segment of $s$."}
+{"_id": "12626", "title": "Substitution Instance of Term is Term", "text": "Let $\\beta, \\tau$ be terms of predicate logic. Let $x \\in \\operatorname {VAR}$ be a variable. Let $\\map \\beta {x \\gets \\tau}$ be the substitution instance of $\\beta$ substituting $\\tau$ for $x$. Then $\\map \\beta {x \\gets \\tau}$ is a term."}
+{"_id": "12627", "title": "Substitution Theorem for Terms", "text": "Let $\\beta, \\tau$ be terms. Let $x \\in \\mathrm{VAR}$ be a variable. Let $\\beta \\left({x \\gets \\tau}\\right)$ be the substitution instance of $\\beta$ substituting $\\tau$ for $x$. Let $\\mathcal A$ be a structure for predicate logic. Let $\\sigma$ be an assignment for $\\beta$ and $\\tau$. Suppose that: :$\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau}\\right) } \\left[{\\sigma}\\right] = a$ where $\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau}\\right) } \\left[{\\sigma}\\right]$ is the value of $\\tau$ under $\\sigma$. Then: :$\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\beta \\left({x \\gets \\tau}\\right) }\\right) } \\left[{\\sigma}\\right] = \\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\beta}\\right) } \\left[{\\sigma + \\left({x / a}\\right)}\\right]$ where $\\sigma + \\left({x / a}\\right)$ is the extension of $\\sigma$ by mapping $x$ to $a$."}
+{"_id": "12629", "title": "Sum over k of Unsigned Stirling Numbers of the First Kind of n+1 with k+1 by k choose m by -1^k-m", "text": "Let $m, n \\in \\Z_{\\ge 0}$. :$\\displaystyle \\sum_k \\left[{n + 1 \\atop k + 1}\\right] \\binom k m \\left({-1}\\right)^{k - m} = \\left[{n \\atop m}\\right]$ where: :$\\displaystyle \\left[{n + 1 \\atop k + 1}\\right]$ etc. denotes an unsigned Stirling number of the first kind :$\\dbinom k m$ denotes a binomial coefficient."}
+{"_id": "12630", "title": "Substitution Theorem for Well-Formed Formulas", "text": "Let $\\mathbf A$ be a WFF of predicate logic. Let $x \\in \\mathrm{VAR}$ be a variable. Let $\\tau$ be a term of predicate logic which is freely substitutable for $x$ in $\\mathbf A$. Let $\\mathbf A \\left({x \\gets \\tau}\\right)$ be the substitution instance of $\\mathbf A$ substituting $\\tau$ for $x$. Let $\\mathcal A$ be a structure for predicate logic. Let $\\sigma$ be an assignment for $\\mathbf A$ and $\\tau$. Suppose that: :$\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau}\\right) } \\left[{\\sigma}\\right] = a$ where $\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau}\\right) } \\left[{\\sigma}\\right]$ is the value of $\\tau$ under $\\sigma$. Then: :$\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\mathbf A \\left({x \\gets \\tau}\\right) }\\right) } \\left[{\\sigma}\\right] = \\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\mathbf A}\\right) } \\left[{\\sigma + \\left({x / a}\\right)}\\right]$ where $\\sigma + \\left({x / a}\\right)$ is the extension of $\\sigma$ by mapping $x$ to $a$."}
+{"_id": "12631", "title": "Sum over k of Stirling Numbers of the Second Kind of k with m by n choose k", "text": "Let $m, n \\in \\Z_{\\ge 0}$. :$\\displaystyle \\sum_k \\left\\{ {k \\atop m}\\right\\} \\binom n k = \\left\\{ {n + 1 \\atop m + 1}\\right\\}$ where: :$\\displaystyle \\left\\{ {k \\atop m}\\right\\}$ denotes a Stirling number of the second kind :$\\dbinom n k$ denotes a binomial coefficient."}
+{"_id": "12632", "title": "Sum over k of Stirling Numbers of the Second Kind of k+1 with m+1 by n choose k by -1^k-m", "text": "Let $m, n \\in \\Z_{\\ge 0}$. :$\\displaystyle \\sum_k \\left\\{ {k + 1 \\atop m + 1}\\right\\} \\binom n k \\left({-1}\\right)^{n - k} = \\left\\{ {n \\atop m}\\right\\}$ where: :$\\displaystyle \\left\\{ {k + 1 \\atop m + 1}\\right\\}$ etc. denotes a Stirling number of the second kind :$\\dbinom n k$ denotes a binomial coefficient."}
+{"_id": "12633", "title": "Universal Instantiation/Informal Statement", "text": "Suppose we have a universal statement: :$\\forall x: \\map P x$ where $\\forall$ is the universal quantifier and $\\map P x$ is a propositional function. Then we can deduce: :$\\map P {\\mathbf a}$ where $\\mathbf a$ is any arbitrary object we care to choose in the universe of discourse. In natural language: :''Suppose $P$ is true of everything in the universe of discourse.'' :''Let $\\mathbf a$ be an element of the universe of discourse.\" :''Then $P$ is true of $\\mathbf a$.''"}
+{"_id": "12634", "title": "Universal Instantiation/Model", "text": "Let $\\map {\\mathbf A} x$ be a WFF of predicate logic. Let $\\tau$ be a term which is freely substitutable for $x$ in $\\mathbf A$. Then $\\forall x: \\map {\\mathbf A} x \\implies \\map {\\mathbf A} \\tau$ is a tautology."}
+{"_id": "12635", "title": "Existential Generalisation/Informal Statement", "text": ":$P \\left({\\mathbf a}\\right) \\vdash \\exists x: P \\left({x}\\right)$ Suppose we have the following: : We can find an arbitrary object $\\mathbf a$ in our universe of discourse which has the property $P$. Then we may infer that: : there exists in that universe ''at least one'' object $x$ which has that property $P$. This is called the '''Rule of Existential Generalisation''' and often appears in a proof with its abbreviation '''EG'''."}
+{"_id": "12636", "title": "Way Below is Approximating Relation", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below continuous lattice. Then $\\ll$ is an approximating relation on $S$."}
+{"_id": "12637", "title": "Intersection of Relation Segments of Approximating Relations equals Way Below Closure", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below meet-continuous lattice. Let $\\mathit{App}\\left({L}\\right)$ be the set of all auxiliary approximating relations on $S$. Let $x \\in S$. Then : $\\displaystyle \\bigcap \\left\\{ {x^{\\mathcal R}: \\mathcal R \\in \\mathit{App}\\left({L}\\right)}\\right\\} = x^\\ll$"}
+{"_id": "12638", "title": "Limit of Monotone Real Function/Increasing", "text": "Let $f$ be a real function which is increasing and bounded above on the open interval $\\openint a b$. Let the supremum of $f$ on $\\openint a b$ be $L$. Then: :$\\displaystyle \\lim_{x \\mathop \\to b^-} \\map f x = L$ where $\\displaystyle \\lim_{x \\mathop \\to b^-} \\map f x$ is the limit of $f$ from the left at $b$."}
+{"_id": "12639", "title": "Limit of Monotone Real Function/Decreasing", "text": "Let $f$ be a real function which is decreasing and bounded below on the open interval $\\openint a b$. Let the infimum of $f$ on $\\openint a b$ be $l$. Then: :$\\displaystyle \\lim_{x \\mathop \\to b^-} \\map f x = l$ where $\\displaystyle \\lim_{x \\mathop \\to b^-} \\map f x$ is the limit of $f$ from the left at $b$."}
+{"_id": "12640", "title": "Sum over k of m choose k by -1^m-k by k to the n", "text": "Let $m, n \\in \\Z_{\\ge 0}$. :$\\displaystyle \\sum_k \\binom m k \\paren {-1}^{m - k} k^n = m! {n \\brace m}$ where: :$\\dbinom m k$ denotes a binomial coefficient :$\\displaystyle {n \\brace m}$ etc. denotes a Stirling number of the second kind :$m!$ denotes a factorial."}
+{"_id": "12641", "title": "Sum over k of m-n choose m+k by m+n choose n+k by Stirling Number of the Second Kind of m+k with k", "text": "Let $m, n \\in \\Z_{\\ge 0}$. :$\\displaystyle \\sum_k \\binom {m - n} {m + k} \\binom {m + n} {n + k} {m + k \\brace k} = {n \\brack n - m}$ where: :$\\dbinom {m - n} {m + k}$ etc. denote binomial coefficients :$\\displaystyle {m + k \\brace k}$ denotes a Stirling number of the second kind :$\\displaystyle {n \\brack n - m}$ denotes an unsigned Stirling number of the first kind."}
+{"_id": "12642", "title": "Sum over k of m-n choose m+k by m+n choose n+k by Unsigned Stirling Number of the First Kind of m+k with k", "text": "Let $m, n \\in \\Z_{\\ge 0}$. :$\\displaystyle \\sum_k \\binom {m - n} {m + k} \\binom {m + n} {n + k} {m + k \\brack k} = {n \\brace n - m}$ where: :$\\dbinom {m - n} {m + k}$ etc. denote binomial coefficients :$\\displaystyle {m + k \\brack k}$ denotes an unsigned Stirling number of the first kind :$\\displaystyle {n \\brace n - m}$ denotes a Stirling number of the second kind."}
+{"_id": "12643", "title": "Sum over k of Stirling Number of the Second Kind of n+1 with k+1 by Unsigned Stirling Number of the First Kind of k with m by -1^k-m", "text": "Let $m, n \\in \\Z_{\\ge 0}$. :$\\displaystyle \\sum_k \\left\\{ {n + 1 \\atop k + 1}\\right\\} \\left[{k \\atop m}\\right] \\left({-1}\\right)^{k - m} = \\binom n m$ where: :$\\dbinom n m$ denotes a binomial coefficient :$\\displaystyle \\left[{k \\atop m}\\right]$ denotes an unsigned Stirling number of the first kind :$\\displaystyle \\left\\{ {n + 1 \\atop k + 1}\\right\\}$ denotes a Stirling number of the second kind."}
+{"_id": "12645", "title": "Intersection of Applications of Down Mappings at Element equals Way Below Closure of Element", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below meet-continuous lattice. Let $\\mathit{Ids}$ be the set of all ideals in $L$. Let for all $I \\in \\mathit{Ids}$: $m_I: S \\to \\mathit{Ids}$ be a mapping: :$\\forall x \\in S: x \\preceq \\sup I \\implies m_I\\left({x}\\right) = \\left\\{ {x \\wedge i: i \\in I}\\right\\}$ and :$\\forall x \\in S: x \\npreceq \\sup I \\implies m_I\\left({x}\\right) = x^\\preceq$ where $x^\\preceq$ denotes the lower closure of $x$. Let $x \\in S$. Then :$\\displaystyle \\bigcap \\left\\{ {m_I \\left({x}\\right): I \\in \\mathit{Ids} }\\right\\} = x^\\ll$ where $x^\\ll$ denotes the way below closure of $x$."}
+{"_id": "12649", "title": "Zero Choose Zero", "text": ":$\\dbinom 0 0 = 1$"}
+{"_id": "12653", "title": "Intersection of Lower Closure of Element with Ideal equals Meet of Element and Ideal", "text": "Let $\\struct {S, \\preceq}$ be a meet semilattice. Let $I$ be an ideal in $\\struct {S, \\preceq}$. Let $x \\in S$. Then: :$\\paren {x^\\preceq} \\cap I = \\set {x \\wedge i: i \\in I}$ where $x^\\preceq$ denotes the lower closure of $x$."}
+{"_id": "12655", "title": "Condition on Lower Coefficient for Binomial Coefficient to be Maximum", "text": "Let $n, k \\in \\Z_{\\ge 0}$ be positive integers. Let $\\dbinom n k$ denote the binomial coefficient of $n$ choose $k$. Then for a given value of $n$, the value of $k$ for which $\\dbinom n k$ is a maximum is: :$k_\\max = \\floor {\\dfrac n 2}$ and also: :$k_\\max = \\ceiling {\\dfrac n 2}$"}
+{"_id": "12657", "title": "N Choose Negative Number is Zero", "text": "Let $n \\in \\Z$ be an integer. Let $k \\in \\Z_{<0}$ be a (strictly) negative integer. Then: :$\\dbinom n k = 0$"}
+{"_id": "12658", "title": "Kummer's Theorem", "text": "Let $p$ be a prime number. Let $a, b \\in \\Z_{\\ge 0}$. Let: :$p^n \\divides \\dbinom {a + b} b$ but :$p^{n + 1} \\nmid \\dbinom {a + b} b$ where: :$\\divides$ denotes divisibility :$\\nmid$ denotes non-divisibility :$\\dbinom {a + b} b$ denotes a binomial coefficient. Then $n$ equals the number of carries that occur when $a$ is added to $b$ using the classical addition algorithm in base $p$."}
+{"_id": "12659", "title": "Numbers in Row of Pascal's Triangle all Odd iff Row number 2^n - 1", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Then the non-zero elements of the $n$th row of Pascal's triangle are all odd {{iff}}: :$n = 2^m - 1$ for some $m \\in \\Z_{\\ge 0}$. {{:Definition:Pascal's Triangle}} As can be seen, the entries in rows $0, 1, 3, 7$ are all odd."}
+{"_id": "12660", "title": "Sum of Sequence of Fourth Powers", "text": ":$\\displaystyle \\sum_{k \\mathop = 0}^n k^4 = \\dfrac {\\paren {n + 1} n \\paren {n + \\frac 1 2} \\paren {3 n^2 + 3 n - 1} } {15}$"}
+{"_id": "12661", "title": "Complete Lattice is Bounded", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Then :$L$ is bounded"}
+{"_id": "12662", "title": "Constant Function is Primitive Recursive/General Case", "text": "The constant function of $k$ variables: $f_c^k: \\N^k \\to \\N$, defined as: :$\\map {f_c^k} {n_1, n_2, \\ldots, n_k} = c$ where $c \\in \\N$ is primitive recursive."}
+{"_id": "12665", "title": "Value of Multiplicative Function at One", "text": "Let $f: \\N \\to \\C$ be a multiplicative function. If $f$ is not identically zero, then $\\map f 1 = 1$."}
+{"_id": "12666", "title": "Value of Multiplicative Function is Product of Values of Prime Power Factors", "text": "Let $f: \\N \\to \\C$ be a multiplicative function. Let $n = p_1^{k_1} p_2^{k_2} \\cdots p_r^{k_r}$ be the prime decomposition of $n$. Then: :$\\map f n = \\map f {p_1^{k_1} } \\, \\map f {p_2^{k_2} } \\dotsm \\map f {p_r^{k_r} }$"}
+{"_id": "12667", "title": "-1^n by -n choose k-1 equals -1^k by -k choose n-1", "text": ":$\\paren {-1}^n \\dbinom {-n} {k - 1} = \\paren {-1}^k \\dbinom {-k} {n - 1}$"}
+{"_id": "12670", "title": "One Choose n", "text": ":$\\dbinom 1 n = \\begin{cases} 1 & : n \\in \\left\\{ {0, 1}\\right\\} \\\\ 0 & : \\text {otherwise} \\end{cases}$"}
+{"_id": "12671", "title": "Summations of Products of Binomial Coefficients", "text": "This page gathers together some identities concerning summations of products of binomial coefficients. In the following, unless otherwise specified: :$k, m, n \\in \\Z$ :$r, s, t \\in \\R$."}
+{"_id": "12672", "title": "Separable Discrete Space is Countable", "text": "Let $T = \\struct {S, \\tau}$ be a discrete topological space. Let $T$ be separable. Then $S$ is countable."}
+{"_id": "12678", "title": "Sum over k of r-tk Choose k by s-t(n-k) Choose n-k by r over r-tk/Proof 1/Basis for the Induction", "text": "Let $r, s, t \\in \\R, n \\in \\Z$. Consider the equation: :$\\displaystyle (1): \\quad \\sum_{k \\mathop \\ge 0} \\binom {r - t k} k \\binom {s - t \\paren {n - k} } {n - k} \\frac r {r - t k} = \\binom {r + s - t n} n$ where $\\dbinom {r - t k} k$ etc. are binomial coefficients. Then equation $(1)$ holds for the special case where $s = n - 1 - r + n t$."}
+{"_id": "12681", "title": "Continuous Lattice iff Auxiliary Approximating Relation is Superset of Way Below Relation", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $\\mathit{App}\\left({L}\\right)$ be the set of all auxiliary approximating relation on $S$. Then :$L$ is continuous {{iff}} :$\\forall \\mathcal R \\in \\mathit{App}\\left({L}\\right): \\ll \\subseteq \\mathcal R$ and $\\ll$ is an approximating relation"}
+{"_id": "12682", "title": "Completely Additive Function is Additive", "text": "Let $f: \\N \\to \\C$ be a completely additive function. Then $f$ is also additive."}
+{"_id": "12683", "title": "Real Logarithm is Completely Additive", "text": "Let $\\log_b: \\R_{>0} \\to \\R$ be the real logarithm to base $b$. Then $\\log_b$ is completely additive."}
+{"_id": "12684", "title": "Sum over k of r-tk Choose k by s-t(n-k) Choose n-k by r over r-tk/Proof 1/Lemma", "text": "Let this hold for $\\tuple {r, s, t, n}$: :$\\displaystyle \\sum_{k \\mathop \\ge 0} \\binom {r - t k} k \\binom {s - t \\paren {n - k} } {n - k} \\frac r {r - t k} = \\binom {r + s - t n} n$ and also for $\\tuple {r, s - t, t, n - 1}$. Then it also holds for $\\tuple {r, s + 1, t, n}$."}
+{"_id": "12687", "title": "Sum over k of r-kt choose k by r over r-kt by z^k", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Let $A_n \\left({x, t}\\right)$ be the polynomial of degree $n$ defined as: :$A_n \\left({x, t}\\right) := \\dbinom {x - n t} n \\dfrac x {x - n t}$ for $x \\ne n t$. Let $z = x^{t + 1} - x^t$. Then: :$\\displaystyle \\sum_k A_k \\left({r, t}\\right) z^k = x^r$ for sufficiently small $z$."}
+{"_id": "12688", "title": "Not Preceding implies Approximating Relation and not Preceding", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $x, y \\in S$ such that :$x \\npreceq y$ Let $\\mathcal R$ be an approximating relation on $S$. Then :$\\exists u \\in S: \\left({u, x}\\right) \\in \\mathcal R \\land u \\npreceq y$"}
+{"_id": "12689", "title": "Auxiliary Approximating Relation has Quasi Interpolation Property", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $x, z \\in S$. Let $\\mathcal R$ be an auxiliary approximating relation on $S$ such that :$\\left({x, z}\\right) \\in \\mathcal R \\land x \\ne z$ Then :$\\exists y \\in S: x \\preceq y \\land \\left({y, z}\\right) \\in \\mathcal R \\land x \\ne y$"}
+{"_id": "12690", "title": "Preceding iff Join equals Larger Operand", "text": "Let $\\left({S, \\preceq}\\right)$ be a join semilattice. Let $x, y \\in S$. Then :$x \\preceq y$ {{iff}} $x \\vee y = y$"}
+{"_id": "12691", "title": "Sum over k of -1^k by n choose k by r-kt choose n by r over r-kt", "text": ":$\\displaystyle \\sum_k \\left({-1}\\right)^k \\dbinom n k \\dbinom {r - k t} n \\dfrac r {r - k t} = \\delta_{n 0}$ where $\\delta_{n 0}$ is the Kronecker delta."}
+{"_id": "12692", "title": "Sum over k of r-kt choose k by z^k", "text": "Let $n \\in \\Z_{\\ge 0}$ be a non-negative integer. Then: :$\\displaystyle \\sum_k \\dbinom {r - t k} k z^k = \\frac {x^{r + 1} } {\\left({t + 1}\\right)x - t}$ where $\\dbinom {r - t k} k$ denotes a binomial coefficient."}
+{"_id": "12698", "title": "Auxiliary Approximating Relation has Interpolation Property", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $x, z \\in S$ such that :$x \\ll z \\land x \\ne z$ Let $\\mathcal R$ be an auxiliary approximating relation on $S$. Then :$\\exists y \\in S: \\left({x, y}\\right) \\in \\mathcal R \\land \\left({y, z}\\right) \\in \\mathcal R \\land x \\ne y$"}
+{"_id": "12701", "title": "Way Below has Strong Interpolation Property", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below continuous lattice. Let $x, z \\in S$ such that :$x \\ll z \\land x \\ne z$ Then :$\\exists y \\in S: x \\ll y \\land y \\ll z \\land x \\ne y$"}
+{"_id": "12702", "title": "Way Below has Interpolation Property", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below continuous lattice. Let $x, z \\in S$ such that :$x \\ll z$ Then :$\\exists y \\in S: x \\ll y \\land y \\ll z$"}
+{"_id": "12703", "title": "Left Coset Equals Subgroup iff Element in Subgroup", "text": ":$x H = H \\iff x \\in H$"}
+{"_id": "12704", "title": "Right Coset Equals Subgroup iff Element in Subgroup", "text": ":$H x = H \\iff x \\in H$"}
+{"_id": "12705", "title": "Elements in Same Left Coset iff Product with Inverse in Subgroup", "text": ": $x, y$ are in the same left coset of $H$ {{iff}} $x^{-1} y \\in H$."}
+{"_id": "12706", "title": "Elements in Same Right Coset iff Product with Inverse in Subgroup", "text": ": $x, y$ are in the same right coset of $H$ {{iff}} $x y^{-1} \\in H$"}
+{"_id": "12708", "title": "Sum over k of m+r+s Choose k by n+r-s Choose n-k by r+k Choose m+n", "text": "Let $m, n \\in \\Z_{\\ge 0}$. Then: :$\\displaystyle \\sum_k \\binom {m - r + s} k \\binom {n + r - s} {n - k} \\binom {r + k} {m - n} = \\binom r m \\binom s n$"}
+{"_id": "12709", "title": "Sum over k of Unsigned Stirling Numbers of First Kind by x^k", "text": ":$\\displaystyle \\sum_k \\left[{n \\atop k}\\right] x^k = x^{\\overline n}$ where: :$\\displaystyle \\left[{n \\atop k}\\right]$ denotes an unsigned Stirling number of the first kind :$x^{\\overline n}$ denotes $x$ to the $n$ rising."}
+{"_id": "12710", "title": "Capelli's Sum", "text": ":$\\displaystyle \\left({x + y}\\right)^{\\overline n} = \\sum_k \\binom n k x^{\\overline k} y^{\\overline {n - k} }$ where: :$\\dbinom n k$ denotes a binomial coefficient :$x^{\\overline k}$ denotes $x$ to the $k$ rising."}
+{"_id": "12712", "title": "Way Below iff Second Operand Preceding Supremum of Directed Set There Exists Element of Directed Set First Operand Way Below Element", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below continuous lattice. Let $x, y$ be elements of $S$. Then :$x \\ll y$ {{iff}} :for every directed subset $D$ of $S$ such that $y \\preceq \\sup D$ ::there exists an element $d$ of $D$: $x \\ll d$"}
+{"_id": "12713", "title": "Continuous iff Way Below Closure is Ideal and Element Precedes Supremum", "text": "Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be an up-complete meet semilattice. Then :$L$ is continuous {{iff}} :$\\forall x \\in S: x^\\ll$ is an ideal in $L$ and $x \\preceq \\sup \\left({x^\\ll}\\right)$ and ::for every ideal $I$ in $L$: $x \\preceq \\sup I \\implies x^\\ll \\subseteq I$ where $x^\\ll$ denotes the way below closure of $x$."}
+{"_id": "12714", "title": "Characterization of Euclidean Borel Sigma-Algebra/Open equals Closed", "text": "Let $\\mathcal O^n$, $\\mathcal C^n$ and $\\mathcal K^n$ be the collections of open and closed subsets of the Euclidean space $\\left({\\R^n, \\tau}\\right)$, respectively. Then: :$\\sigma \\left({\\mathcal O^n}\\right) = \\sigma \\left({\\mathcal C^n}\\right)$ where $\\sigma$ denotes generated $\\sigma$-algebra."}
+{"_id": "12715", "title": "Characterization of Euclidean Borel Sigma-Algebra/Closed equals Compact", "text": "Let $\\mathcal C^n$ and $\\mathcal K^n$ be the collections of closed and compact subsets of the Euclidean space $\\left({\\R^n, \\tau}\\right)$, respectively. Then: :$\\sigma \\left({\\mathcal C^n}\\right) = \\sigma \\left({\\mathcal K^n}\\right)$ where $\\sigma$ denotes generated $\\sigma$-algebra."}
+{"_id": "12716", "title": "Characterization of Euclidean Borel Sigma-Algebra/Open equals Rectangle", "text": "Let $\\mathcal O^n$ be the collection of open subsets of the Euclidean space $\\left({\\R^n, \\tau}\\right)$. Let $\\mathcal J_{ho}^n$ be the collection of half-open rectangles in $\\R^n$. Then: :$\\sigma \\left({\\mathcal O^n}\\right) = \\sigma \\left({\\mathcal J_{ho}^n}\\right)$ where $\\sigma$ denotes generated $\\sigma$-algebra."}
+{"_id": "12717", "title": "Characterization of Euclidean Borel Sigma-Algebra/Rectangle equals Rational Rectangle", "text": "Let $\\mathcal J_{ho}^n$ be the collection of half-open rectangles in $\\R^n$. Let $\\mathcal J^n_{ho, \\text{rat}}$ be the collection of half-open rectangles in $\\R^n$ with rational endpoints. Then: :$\\sigma \\left({\\mathcal J_{ho}^n}\\right) = \\sigma \\left({\\mathcal J^n_{ho, \\text{rat}}}\\right)$ where $\\sigma$ denotes generated $\\sigma$-algebra."}
+{"_id": "12718", "title": "Rising Factorial as Factorial by Binomial Coefficient", "text": "Let $x \\in \\R$ be a real number. Let $n \\in \\Z_{\\ge 0}$ be a positive integer. :$x^{\\overline n} = n! \\dbinom {x + n - 1} n$ where: :$x^{\\overline n}$ denotes $x$ to the $n$ rising :$n!$ denotes the factorial of $n$ :$\\dbinom {x + n - 1} n$ denotes a binomial coefficient."}
+{"_id": "12719", "title": "Ramus's Identity", "text": "Let $k, m, n \\in \\Z_{\\ge 0}$ be positive integers such that $0 \\le k < m$. Then: {{begin-eqn}} {{eqn | l = \\sum_{j \\mathop \\ge 0} \\binom n {j m + k}       | r = \\dbinom n k + \\dbinom n {m + k} +  \\dbinom n {2 m + k} + \\cdots        | c =  }} {{eqn | r = \\dfrac 1 m \\sum_{0 \\mathop \\le j \\mathop < m} \\paren {2 \\cos \\dfrac {j \\pi} m}^n \\cos \\dfrac {j \\paren {n - 2 k} \\pi} m       | c =  }} {{end-eqn}}"}
+{"_id": "12723", "title": "Way Below Closure is Lower Set", "text": "Let $L = \\struct {S, \\vee, \\preceq}$ be an ordered set. Let $x \\in S$. Then :$x^\\ll$ is a lower set."}
+{"_id": "12724", "title": "Continuous iff For Every Element There Exists Ideal Element Precedes Supremum", "text": "Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be an up-complete meet semilattice. Then :$L$ is continuous {{iff}} :for every element $x$ of $S$ there exists ideal $I$ in $L$: ::$x \\preceq \\sup I$ and for every ideal $J$ in $L: x \\preceq \\sup J \\implies I \\subseteq J$"}
+{"_id": "12725", "title": "Beta Function of x+1 with y plus Beta Function of x with y+1", "text": ":$\\Beta \\left({x + 1, y}\\right) + \\Beta \\left({x, y + 1}\\right) = \\Beta \\left({x, y}\\right)$"}
+{"_id": "12726", "title": "Beta Function of x with y+1 by x+y over y", "text": ":$\\map \\Beta {x, y} = \\dfrac {x + y} y \\map \\Beta {x, y + 1}$"}
+{"_id": "12727", "title": "Partial Gamma Function expressed as Integral", "text": "Let $m \\in \\Z_{\\ge 1}$. Let $\\map {\\Gamma_m} x$ denote the partial Gamma function, defined as: :$\\map {\\Gamma_m} x := \\dfrac {m^x m!} {x \\paren {x + 1} \\paren {x + 2} \\cdots \\paren {x + m} }$ Then: :$\\displaystyle \\map {\\Gamma_m} x = \\int_0^m \\paren {1 - \\frac t m}^m t^{x - 1} \\rd t$ for $x > 0$."}
+{"_id": "12728", "title": "Partial Gamma Function expressed as Integral/Lemma", "text": ":$(1): \\quad \\displaystyle \\int_0^m \\paren {1 - \\frac t m}^m t^{x - 1} \\rd t = m^x \\int_0^1 \\paren {1 - t}^m t^{x - 1} \\rd t$ for $x > 0$."}
+{"_id": "12732", "title": "Integral Form of Gamma Function equivalent to Euler Form", "text": "{{TFAENocat|def = Gamma Function}} === Integral Form === {{:Definition:Integral Form of Gamma Function}} === Euler Form === {{:Definition:Euler Form of Gamma Function}}"}
+{"_id": "12734", "title": "Derivative of Composite Function/Second Derivative", "text": ":$D_x^2 w = D_u^2 w \\paren {D_x^1 u}^2 + D_u^1 w D_x^2 u$"}
+{"_id": "12735", "title": "Faà di Bruno's Formula", "text": "Let $D_x^k u$ denote the $k$th derivative of a function $u$ {{WRT|Differentiation}} $x$. Then: :$\\displaystyle D_x^n w = \\sum_{j \\mathop = 0}^n D_u^j w \\sum_{\\substack {\\sum_{p \\mathop \\ge 1} k_p \\mathop = j \\\\ \\sum_{p \\mathop \\ge 1} p k_p \\mathop = n \\\\ \\forall p \\mathop \\ge 1: k_p \\mathop \\ge 0} } n! \\prod_{m \\mathop = 1}^n \\dfrac {\\left({D_x^m u}\\right)^{k_m} } {k_m! \\left({m!}\\right)^{k_m} }$"}
+{"_id": "12737", "title": "Cardinality of Reduced Residue System", "text": "Let $n \\ge 2$. Let $\\Z'_n$ be the reduced residue system modulo $n$. Then: :$\\card {\\Z'_n} = \\map \\phi n$ where $\\map \\phi n$ is the Euler phi function."}
+{"_id": "12738", "title": "Equivalence of Definitions of Norm of Linear Functional/Corollary", "text": "For all $h \\in H$, the following inequality holds: :$\\left|{Lh}\\right| \\le \\left\\|{L}\\right\\| \\left\\|{h}\\right\\|$"}
+{"_id": "12739", "title": "Faà di Bruno's Formula/Example/0/Proof", "text": "Consider Faà di Bruno's Formula: {{:Faà di Bruno's Formula}} When $n = 0$ we have: {{:Faà di Bruno's Formula/Example/0}}"}
+{"_id": "12740", "title": "Faà di Bruno's Formula/Example/1/Proof", "text": "Consider Faà di Bruno's Formula: {{:Faà di Bruno's Formula}} When $n = 1$ we have: {{:Faà di Bruno's Formula/Example/1}}"}
+{"_id": "12741", "title": "Faà di Bruno's Formula/Example/2/Proof", "text": "Consider Faà di Bruno's Formula: {{:Faà di Bruno's Formula}} When $n = 2$ we have: {{:Faà di Bruno's Formula/Example/2}}"}
+{"_id": "12742", "title": "Faà di Bruno's Formula/Lemma 1", "text": ":$D_x \\left({\\left({D_x^m u}\\right)^{k_m} }\\right) = k_m \\left({D_x^m u}\\right)^{k_m - 1} D_x^{m + 1} u$"}
+{"_id": "12743", "title": "Faà di Bruno's Formula/Lemma 2", "text": ":$\\displaystyle D_x \\left({\\prod_{m \\mathop = 1}^r \\left({\\dfrac {\\left({D_x^m u}\\right)^{k_m} } {k_m! \\left({m!}\\right)^{k_m} } }\\right) }\\right) = \\prod_{m \\mathop = 1}^r \\left({\\dfrac {\\left({D_x^m u}\\right)^{k_m} } {k_m! \\left({m!}\\right)^{k_m} } }\\right) \\sum_{m \\mathop = 1}^r k_m \\dfrac {D_x^{m + 1} u} {D_x^m u}$"}
+{"_id": "12745", "title": "Supremum of Ideals is Increasing", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an up-complete ordered set. Let $\\mathit{Ids}\\left({L}\\right)$ be the set of all ideals in $L$. Let $P = \\left({\\mathit{Ids}\\left({L}\\right), \\precsim}\\right)$ be an ordered set where $\\mathord \\precsim = \\subseteq\\restriction_{\\mathit{Ids}\\left({L}\\right)\\times \\mathit{Ids}\\left({L}\\right)}$ Let $f: \\mathit{Ids}\\left({L}\\right) \\to S$ be a mapping such that :$\\forall I \\in \\mathit{Ids}\\left({L}\\right): f\\left({I}\\right) = \\sup I$ Then $f$ is an increasing mapping."}
+{"_id": "12747", "title": "Derivative of Composite Function/Third Derivative", "text": ":$D_x^3 w = D_u^3 w \\paren {D_x^1 u}^3 + 3 D_u^2 w D_x^2 u D_x^1 u + D_u^1 w D_x^3 u$"}
+{"_id": "12748", "title": "Number of Set Partitions by Number of Components", "text": "Let $S$ be a (finite) set whose cardinality is $n$. Let $\\map f {n, k}$ denote the number of different ways $S$ can be partitioned into $k$ components. Then: :$\\displaystyle \\map f {n, k} = {n \\brace k}$ where $\\displaystyle {n \\brace k}$ denotes a Stirling number of the second kind."}
+{"_id": "12750", "title": "Floor of Half of n+m plus Floor of Half of n-m+1", "text": "Let $n, m \\in \\Z$ be integers. :$\\floor {\\dfrac {n + m} 2} + \\floor {\\dfrac {n - m + 1} 2} = n$ where $\\floor x$ denotes the floor of $x$."}
+{"_id": "12751", "title": "Ceiling of Half of n+m plus Ceiling of Half of n-m+1", "text": "Let $n, m \\in \\Z$ be integers. :$\\left\\lceil{\\dfrac {n + m} 2}\\right\\rceil + \\left\\lceil{\\dfrac {n - m + 1} 2}\\right\\rceil = n + 1$ where $\\left\\lceil{x}\\right\\rceil$ denotes the ceiling of $x$."}
+{"_id": "12752", "title": "Floor of Non-Integer", "text": "Let $x \\in \\R$ be a real number. Let $x \\notin \\Z$. Then: :$\\left\\lfloor{x}\\right\\rfloor < x$ where $\\left\\lfloor{x}\\right\\rfloor$ denotes the floor of $x$."}
+{"_id": "12753", "title": "Logarithm of Base", "text": "Let $b \\in \\R_{>0}$ such that $b \\ne 1$. Then: :$\\log_b b = 1$ where $\\log_b$ denotes the logarithm to base $b$."}
+{"_id": "12755", "title": "Floor of x+m over n", "text": "Let $m, n \\in \\Z$ such that $n > 0$. Let $x \\in \\R$. Then: :$\\floor {\\dfrac {x + m} n} = \\floor {\\dfrac {\\floor x + m} n}$ where $\\floor x$ denotes the floor of $x$."}
+{"_id": "12756", "title": "McEliece's Theorem (Integer Functions)", "text": "Let $f: \\R \\to \\R$ be a continuous, strictly increasing real function defined on an interval $A$. Let: :$\\forall x \\in A: \\left \\lfloor{x}\\right \\rfloor \\in A, \\left \\lceil{x}\\right \\rceil \\in A$ where: :$\\left \\lfloor{x}\\right \\rfloor$ denotes the floor of $x$ :$\\left \\lceil{x}\\right \\rceil$ denotes the ceiling of $x$ Then: :$\\forall x \\in A: \\left \\lfloor{f \\left({x}\\right)}\\right \\rfloor = \\left \\lfloor{f \\left({\\left \\lfloor{x}\\right \\rfloor}\\right)}\\right \\rfloor \\iff \\left \\lceil{f \\left({x}\\right)}\\right \\rceil = \\left \\lceil{f \\left({\\left \\lceil{x}\\right \\rceil}\\right)}\\right \\rceil \\iff \\left(f \\left({x}\\right) \\in \\Z \\implies x \\in \\Z)\\right)$"}
+{"_id": "12757", "title": "Ceiling of Non-Integer", "text": "Let $x \\in \\R$ be a real number. Let $x \\notin \\Z$. Then: :$\\left \\lceil{x}\\right \\rceil > x$ where $\\left \\lceil{x}\\right \\rceil$ denotes the ceiling  of $x$."}
+{"_id": "12762", "title": "Ceiling of x+m over n", "text": "Let $m, n \\in \\Z$ such that $n > 0$. Let $x \\in \\R$. Then: :$\\ceiling {\\dfrac {x + m} n} = \\ceiling {\\dfrac {\\ceiling x + m} n}$ where $\\ceiling x$ denotes the ceiling of $x$."}
+{"_id": "12767", "title": "Summation over k of Floor of mk+x over n", "text": "Let $m, n \\in \\Z$ such that $n > 0$. Let $x \\in \\R$. Then: :$\\displaystyle \\sum_{0 \\mathop \\le k \\mathop < n} \\left \\lfloor{\\dfrac {m k + x} n}\\right \\rfloor = \\dfrac {\\left({m - 1}\\right) \\left({n - 1}\\right)} 2 + \\dfrac {d - 1} 2 + d \\left \\lfloor{\\dfrac x d}\\right \\rfloor$ where: :$\\left \\lfloor{x}\\right \\rfloor$ denotes the floor of $x$ :$d$ is the greatest common divisor of $m$ and $n$."}
+{"_id": "12768", "title": "Summation over k of Floor of x plus k over y", "text": "Let $x, y \\in \\R$ such that $y > 0$. Then: :$\\displaystyle \\sum_{0 \\mathop \\le k \\mathop < y} \\left \\lfloor{x + \\dfrac k y}\\right \\rfloor = \\left \\lfloor{x y + \\left \\lfloor{x + 1}\\right \\rfloor \\left({\\left \\lceil{y}\\right \\rceil - y}\\right)}\\right \\rfloor$"}
+{"_id": "12774", "title": "Semantic Consequence preserved in Supersignature", "text": "Let $\\mathcal L, \\mathcal L'$ be signatures for the language of predicate logic. Let $\\mathcal L'$ be a supersignature of $\\mathcal L$. Let $\\mathbf A$ be an $\\mathcal L$-sentence. Let $\\Sigma$ be a set of $\\mathcal L$-sentences. Then the following are equivalent: :$\\mathcal A \\models_{\\mathrm{PL}} \\mathbf A$ for all $\\mathcal L$-structure $\\mathcal A$ for which $\\mathcal A \\models_{\\mathrm{PL}} \\Sigma$ :$\\mathcal A' \\models_{\\mathrm{PL}} \\mathbf A$ for all $\\mathcal L'$-structure $\\mathcal A'$ for which $\\mathcal A' \\models_{\\mathrm{PL}} \\Sigma$ where $\\models_{\\mathrm{PL}}$ denotes the models relation. That is to say, the notion of semantic consequence is preserved in passing to a supersignature."}
+{"_id": "12775", "title": "Satisfiability preserved in Supersignature", "text": "Let $\\mathcal L, \\mathcal L'$ be signatures for the language of predicate logic. Let $\\mathcal L'$ be a supersignature of $\\mathcal L$. Let $\\Sigma$ be a set of $\\mathcal L$-sentences. Then the following are equivalent: :$\\mathcal A \\models_{\\mathrm{PL}} \\Sigma$ for some $\\mathcal L$-structure $\\mathcal A$ :$\\mathcal A' \\models_{\\mathrm{PL}} \\Sigma$ for some $\\mathcal L'$-structure $\\mathcal A'$ where $\\models_{\\mathrm{PL}}$ is the models relation. That is to say, the notion of satisfiability is preserved in passing to a supersignature."}
+{"_id": "12777", "title": "Subtract Half is Replicative Function", "text": "Let $f: \\R \\to \\R$ be the real function defined as: :$\\forall x \\in \\R: \\map f x = x - \\dfrac 1 2$ Then $f$ is a replicative function."}
+{"_id": "12778", "title": "Membership of Set of Integers is Replicative Function", "text": "Let $f: \\R \\to \\R$ be the real function defined as: :$\\forall x \\in \\R: \\map f x = \\sqbrk {x \\in \\Z}$ where $\\sqbrk {\\cdots}$ is Iverson's convention. Then $f$ is a replicative function."}
+{"_id": "12781", "title": "Logarithm of Absolute Value of 2 times Sine of pi x is Replicative Function", "text": "Let $f: \\R \\to \\R$ be the real function defined as: :$\\forall x \\in \\R: \\map f x = \\log \\, \\size {2 \\sin \\pi x}$ Then $f$ is a replicative function."}
+{"_id": "12784", "title": "Constant Multiple of Replicative Function is Replicative", "text": "Let $f: \\R \\to \\R$ be a real function. Let $f$ be a replicative function. Let $c \\in \\R$ be a constant. Let $g: \\R \\to \\R$ be the real function defined as: :$g \\left({x}\\right) = c f \\left({x}\\right)$ Then $g$ is also a replicative function."}
+{"_id": "12785", "title": "Replicative Function of x minus Floor of x is Replicative", "text": "Let $f: \\R \\to \\R$ be a real function. Let $f$ be a replicative function. Let $g: \\R \\to \\R$ be the real function defined as: :$g \\left({x}\\right) = f \\left({x - \\left \\lfloor{x}\\right \\rfloor}\\right)$ Then $g$ is also a replicative function."}
+{"_id": "12786", "title": "Closed Form for Sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ...", "text": "Let $a_1, a_2, a_3, \\ldots$ be the sequence: :$\\left \\langle{a_n}\\right \\rangle = 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \\ldots$ Then: :$a_n = \\left \\lceil{\\dfrac {\\sqrt {1 + 8 n} - 1} 2}\\right \\rceil$"}
+{"_id": "12787", "title": "Sum of Sequence as Summation of Difference of Adjacent Terms", "text": "Let $n \\in \\Z_{> 0}$ be a strictly positive integer. Then: :$\\displaystyle \\sum_{k \\mathop = 1}^n a_k = n a_n - \\sum_{k \\mathop = 1}^{n - 1} k \\left({a_{k + 1} - a_k}\\right)$"}
+{"_id": "12788", "title": "Continuous Lattice and Way Below implies Preceding implies Preceding", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a continuous complete lattice. Let $a, b \\in S$. Let :$\\forall c \\in S: c \\ll a \\implies c \\preceq b$ Then $a \\preceq b$"}
+{"_id": "12791", "title": "Sum over k of Floor of Log base b of k", "text": "Let $n \\in \\Z_{> 0}$ be a strictly positive integer. Let $b \\in \\Z$ such that $b \\ge 2$. Then: :$\\displaystyle \\sum_{k \\mathop = 1}^n \\left \\lfloor{\\log_b k}\\right \\rfloor = \\left({n + 1}\\right) \\left \\lfloor{\\log_b n}\\right \\rfloor - \\dfrac {b^{\\left \\lfloor{\\log_b n}\\right \\rfloor + 1} - b} {b - 1}$"}
+{"_id": "12792", "title": "Sum over k of Floor of Root k", "text": "Let $n \\in \\Z_{> 0}$ be a strictly positive integer. Let $b \\in \\Z$ such that $b \\ge 2$. Then: :$\\displaystyle \\sum_{k \\mathop = 1}^n \\left \\lfloor{\\sqrt k}\\right \\rfloor = \\left \\lfloor{\\sqrt n}\\right \\rfloor \\left({n - \\dfrac {\\left({2 \\left \\lfloor{\\sqrt n}\\right \\rfloor + 5}\\right) \\left({\\left \\lfloor{\\sqrt n}\\right \\rfloor - 1}\\right)} 6 }\\right)$"}
+{"_id": "12793", "title": "Top is Meet Irreducible", "text": "Let $\\left({S, \\wedge, \\preceq}\\right)$ be a bounded above meet semilattice. Then $\\top$ is meet irreducible where $\\top$ denotes the greatest element in $S$."}
+{"_id": "12795", "title": "Sum over k of Sum over j of Floor of n + jb^k over b^k+1", "text": "Let $n, b \\in \\Z$ such that $n \\ge 0$ and $b \\ge 2$. Then: :$\\displaystyle \\sum_{k \\mathop \\ge 0} \\sum_{1 \\mathop \\le j \\mathop < b} \\left \\lfloor{\\dfrac {n + j b^k} {b^{k + 1} } }\\right \\rfloor = n$"}
+{"_id": "12798", "title": "Equivalence of Definitions of Legendre Symbol", "text": "Let $p$ be an odd prime. Let $a \\in \\Z$. {{TFAE|def = Legendre Symbol}}"}
+{"_id": "12799", "title": "Upper Closure of Element without Element is Filter implies Element is Meet Irreducible", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a lattice. Let $x \\in S$. Let : $x^\\succeq \\setminus \\left\\{ {x}\\right\\}$ be a filter in $L$. Then $x$ is meet irreducible."}
+{"_id": "12800", "title": "Maximal Element of Complement of Filter is Meet Irreducible", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a lattice. Let $F$ be a filter in $L$. Let $p \\in S$. Let $p = \\max \\complement_S\\left({F}\\right)$. Then $p$ is meet irreducible."}
+{"_id": "12801", "title": "De Polignac's Formula/Technique", "text": "When calculating $\\mu$, the easiest way to calculate the next term is simply to divide the previous term by $p$ and discard the remainder: :$\\floor {\\dfrac n {p^{k + 1} } } = \\floor {\\floor {\\dfrac n {p^k} } / p}$"}
+{"_id": "12802", "title": "Not Preceding implies There Exists Meet Irreducible Element Not Preceding", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below continuous lattice. Let $x, y \\in S$ such that :$y \\npreceq x$ Then :$\\exists p \\in S: p$ is meet irreducible and $x \\preceq p$ and $y \\npreceq p$"}
+{"_id": "12805", "title": "Characteristics of Floor and Ceiling Function", "text": "Let $f: \\R \\to \\Z$ be an integer-valued function which satisfies both of the following: :$(1): \\quad \\map f {x + 1} = \\map f x + 1$ :$(2): \\quad \\forall n \\in \\Z_{> 0}: \\map f x = \\map f {\\dfrac {\\map f {n x} } n}$ Then either: :$\\forall x \\in \\Q: \\map f x = \\floor x$ or: :$\\forall x \\in \\Q: \\map f x = \\ceiling x$"}
+{"_id": "12807", "title": "Continuous Replicative Function", "text": "Let $f: \\R \\to \\R$ be a real function. Let $f$ be continuous on $\\R$. Let $f$ also be a replicative function. Then $f$ is of the form: :$f \\left({x}\\right) = \\left({x - \\dfrac 1 2}\\right) a$ where $a \\in \\R$."}
+{"_id": "12808", "title": "Sum over k to p over 2 of Floor of 2kq over p", "text": "Let $p \\in \\Z$ be an odd prime. Let $q \\in \\Z$ be an odd integer. Then: :$\\displaystyle \\sum_{0 \\mathop \\le k \\mathop < p / 2} \\floor {\\dfrac {2 k q} p}r \\equiv  \\sum_{0 \\mathop \\le k \\mathop < p / 2} \\floor {\\dfrac {k q} p} \\pmod 2$"}
+{"_id": "12809", "title": "Way Below implies There Exists Way Below Open Filter Subset of Way Above Closure", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below continuous lattice. Let $x, y \\in S$ such that :$x \\ll y$ where $\\ll$ denotes the way below relation. Then there exists a way below open filter in $L$: $y \\in F \\land F \\subseteq x^\\gg$ where $x^\\gg$ denotes the way above closure of $x$."}
+{"_id": "12811", "title": "Powers of Group Elements/Negative Index/Additive Notation", "text": ":$\\forall n \\in \\Z: -\\paren {n g} = \\paren {-n} g = n \\paren {-g}$"}
+{"_id": "12812", "title": "Powers of Group Elements/Product of Indices/Additive Notation", "text": ":$\\forall m, n \\in \\Z: n \\paren {m g} = \\paren {m n} g = m \\paren {n g}$"}
+{"_id": "12813", "title": "Power of Product in Abelian Group/Additive Notation", "text": ": $k \\left({x + y}\\right) = k x + k y$"}
+{"_id": "12814", "title": "Power of Idempotent Element", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $s \\in S$ be an idempotent element with respect to $\\circ$. Then: :$\\forall n \\in \\Z_{> 0}: s^n = s$ where $s^n$ is defined as: :$s^n = \\begin{cases} s & : n = 1 \\\\ s^{n - 1} \\circ s & : n > 1 \\end{cases}$"}
+{"_id": "12815", "title": "Powers of Group Element Commute", "text": "Let $\\struct {G, \\circ}$ be a group. Let $g \\in G$. Let $m, n \\in \\N_{>0}$. Then: :$\\forall m, n \\in \\N_{>0}: g^n \\circ g^m = g^m \\circ g^n$"}
+{"_id": "12818", "title": "Strictly Positive Real Numbers under Multiplication form Subgroup of Non-Zero Real Numbers", "text": "Let $\\R_{>0}$ be the set of strictly positive real numbers, that is: :$\\R_{>0} = \\set {x \\in \\R: x > 0}$ The structure $\\struct {\\R_{>0}, \\times}$ forms a subgroup of $\\struct {\\R_{\\ne 0}, \\times}$, where $\\R_{\\ne 0}$ is the set of real numbers without zero, that is: :$\\R_{\\ne 0} = \\R \\setminus \\set 0$"}
+{"_id": "12819", "title": "Strictly Positive Rational Numbers under Multiplication form Subgroup of Non-Zero Rational Numbers", "text": "Let $\\Q_{> 0}$ be the set of strictly positive rational numbers, that is $\\Q_{> 0} = \\set { x \\in \\Q: x > 0}$. The structure $\\struct {\\Q_{> 0}, \\times}$ is a subgroup of $\\struct {\\Q_{\\ne 0}, \\times}$, where $\\Q_{\\ne 0}$ is the set of rational numbers without zero: $\\Q_{\\ne 0} = \\Q \\setminus \\set 0$."}
+{"_id": "12820", "title": "Union of Upper Sets is Upper", "text": "Let $\\left({S, \\preceq}\\right)$ be a preordered set. Let $A$ be a set of subsets of $S$ such that :$\\forall X \\in A: X$ is an upper set. Then: :$\\bigcup A$ is also an upper set."}
+{"_id": "12821", "title": "Set of Rotations in Space about Fixed Point forms Infinite Group", "text": "Let $\\SS$ be a rigid body in space. Let $O$ be a fixed point in space. The set of all rotations of $\\SS$ through some line through $O$ forms an infinite group."}
+{"_id": "12822", "title": "Group of Rotations about Fixed Point is not Abelian", "text": "Let $\\SS$ be a rigid body in space. Let $O$ be a fixed point in space. Let $\\GG$ be the group of all rotations of $\\SS$ around $O$. Then $\\GG$ is not an abelian group."}
+{"_id": "12823", "title": "Union of Filtered Sets is Filtered", "text": "Let $\\left({S, \\preceq}\\right)$ be a preordered set. Let $A$ be a set of subsets of $S$ such that :$\\forall X \\in A: X$ is filtered and :$\\forall X, Y \\in A: \\exists Z \\in A: X \\cup Y \\subseteq Z$ Then: :$\\bigcup A$ is also filtered."}
+{"_id": "12824", "title": "Special Orthogonal Group is Group", "text": "Let $k$ be a field. The $n$th orthogonal group on $k$ is a group."}
+{"_id": "12825", "title": "Unit Matrix is its own Inverse", "text": "The inverse of the unit matrix $\\mathbf I_n$ of order $n$ is $\\mathbf I_n$. That is, a unit matrix it its own inverse."}
+{"_id": "12826", "title": "Way Above Closure is Subset of Upper Closure of Element", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $x \\in S$. Then $x^\\gg \\subseteq x^\\succeq$ where :$x^\\gg$ denotes the way above closure of $x$, :$x^\\succeq$ denotes the upper closure of $x$."}
+{"_id": "12827", "title": "Diagonal Matrix is Symmetric", "text": "Let $D$ be a diagonal matrix. Then $D$ is symmetric."}
+{"_id": "12828", "title": "Upper Way Below Open Subset Complement is Non Empty implies There Exists Maximal Element of Complement", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice. Let $X$ be upper way below open subset of $S$. Let $x \\in S$ such that :$x \\in \\relcomp S X$ Then :$\\exists m \\in S: x \\preceq m \\land m = \\max \\relcomp S X$"}
+{"_id": "12829", "title": "Unit Matrix is Orthogonal", "text": "The unit matrix $\\mathbf I_n$ of order $n$ is orthogonal."}
+{"_id": "12830", "title": "Unit Matrix is Proper Orthogonal", "text": "The unit matrix $\\mathbf I_n$ of order $n$ is proper orthogonal."}
+{"_id": "12831", "title": "Inverse of Orthogonal Matrix is Orthogonal", "text": "Let $\\mathbf A$ be an orthogonal matrix. Then its inverse $\\mathbf A^{-1}$ is also orthogonal."}
+{"_id": "12832", "title": "Inverse of Inverse of Matrix", "text": "Let $\\mathbf A$ be an invertible matrix. Then: :$\\paren {\\mathbf A^{-1} }^{-1} = \\mathbf A$ That is, an invertible matrix equals the inverse of its inverse."}
+{"_id": "12833", "title": "Inverse of Proper Orthogonal Matrix is Proper Orthogonal", "text": "Let $\\mathbf A$ be a proper orthogonal matrix. Then its inverse $\\mathbf A^{-1}$ is also proper orthogonal."}
+{"_id": "12834", "title": "Special Orthogonal Group is Subgroup of Orthogonal Group", "text": "Let $k$ be a field. Let $\\map {\\operatorname O} {n, k}$ be the $n$th orthogonal group on $k$. Let $\\map {\\operatorname {SO} } {n, k}$ be the $n$th special orthogonal group on $k$. Then $\\map {\\operatorname {SO} } {n, k}$ is a subgroup of $\\map {\\operatorname O} {n, k}$."}
+{"_id": "12835", "title": "Negative Matrix is Inverse for Hadamard Product", "text": "Let $\\struct {G, \\cdot}$ be a group whose identity is $e$. Let $\\map {\\MM_G} {m, n}$ be a $m \\times n$ matrix space over $\\struct {G, \\cdot}$. Let $\\mathbf A$ be an element of $\\map {\\MM_G} {m, n}$. Let $-\\mathbf A$ be the negative of $\\mathbf A$. Then $-\\mathbf A$ is the inverse for the operation $\\circ$, where $\\circ$ is the Hadamard product."}
+{"_id": "12837", "title": "Left Regular Representation wrt Left Cancellable Element on Finite Semigroup is Bijection", "text": "Let $\\struct {S, \\circ}$ be a finite semigroup. Let $a \\in S$ be left cancellable. Then the left regular representation $\\lambda_a$ of $\\struct {S, \\circ}$ with respect to $a$ is a bijection."}
+{"_id": "12838", "title": "Singleton is Chain", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $x \\in S$. Then $\\left\\{ {x}\\right\\}$ is a chain of $\\left({S, \\preceq}\\right)$."}
+{"_id": "12839", "title": "Identity Elements occupy Diagonal of Cayley Table in Inverse Row Form", "text": "Let $\\struct {G, \\circ}$ be a finite group. Let $\\CC$ be a Cayley table for $\\struct {G, \\circ}$ presented in inverse row form. Then all the entries in the main diagonal of $\\CC$ are instances of the identity element."}
+{"_id": "12841", "title": "Scaling preserves Modulo Addition", "text": "Let $m \\in \\Z_{> 0}$. Let $x, y, c \\in \\Z$. Let $x \\equiv y \\pmod m$. Then: :$c x \\equiv c y \\pmod m$"}
+{"_id": "12842", "title": "Modulo Addition is Linear", "text": "Let $m \\in \\Z_{> 0}$. Let $x_1, x_2, y_1, y_2, c_1, c_2 \\in \\Z$. Let: :$x_1 \\equiv y_1 \\pmod m$ :$x_2 \\equiv y_2 \\pmod m$ Then: :$c_1 x_1 + c_2 x_2 \\equiv c_1 y_1 + c_2 y_2 \\pmod m$"}
+{"_id": "12843", "title": "Euler Phi Function of Prime", "text": "Let $p$ be a prime number $p > 1$. Then: :$\\map \\phi p = p - 1$ where $\\phi: \\Z_{>0} \\to \\Z_{>0}$ is the Euler $\\phi$ function."}
+{"_id": "12844", "title": "Chain is Directed", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $C$ be a non-empty chain of $S$. Then $C$ is directed."}
+{"_id": "12845", "title": "Reduced Residue System Modulo Prime", "text": "Let $p$ be a prime number. The reduced residue system modulo $p$ contains $p - 1$ elements: :$Z'_p = \\set {\\eqclass 1 m, \\eqclass 2 m, \\ldots, \\eqclass {p - 1} m}$ and so can be defined as: :$Z'_p = Z_p \\setminus \\set {\\eqclass 0 m}$"}
+{"_id": "12846", "title": "Modulo Multiplication on Reduced Residue System is Closed", "text": "Let $m \\in \\Z_{> 0}$ be a (strictly) positive integer. Let $\\Z'_m$ be the reduced residue system modulo $m$: :$\\Z'_m = \\set {\\eqclass k m \\in \\Z_m: k \\perp m}$ Let $S = \\struct {\\Z'_m, \\times_m}$ be the algebraic structure consisting of $\\Z'_m$ under the modulo multiplication. Then $S$ is closed, in the sense that: :$\\forall a, b \\in \\Z'_m: a \\times_m b \\in \\Z'_m$"}
+{"_id": "12847", "title": "Modulo Multiplication on Reduced Residue System is Cancellable", "text": "Let $m \\in \\Z_{> 0}$ be a (strictly) positive integer. Let $\\Z'_m$ be the reduced residue system modulo $m$: :$\\Z'_m = \\set {\\eqclass k m \\in \\Z_m: k \\perp m}$ Let $S = \\struct {\\Z'_m, \\times_m}$ be the algebraic structure consisting of $\\Z'_m$ under modulo multiplication. Then $\\times_m$ is cancellable, in the sense that: :$\\forall a, b, c \\in \\Z'_m: a \\times_m c = b \\times_m c \\implies a = b$ and: :$\\forall a, b, c \\in \\Z'_m: c \\times_m a = c \\times_m b \\implies a = b$"}
+{"_id": "12849", "title": "Order Generating iff Every Element is Infimum", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice. Let $X$ be a subset of $S$. Then :$X$ is order generating {{iff}} :$\\forall x \\in S: \\exists Y \\subseteq X: x = \\inf Y$"}
+{"_id": "12850", "title": "Weierstrass Factorization Theorem", "text": "Let $f$ be an entire function. Let $0$ be a zero of $f$ of multiplicity $m \\ge 0$. Let the sequence $\\sequence {a_n}$ consist of the nonzero zeroes of $f$, repeated according to multiplicity."}
+{"_id": "12851", "title": "Equivalence of Axiom Schemata for Finite Group", "text": "The following axiom schemata for the definition of a finite group are logically equivalent:"}
+{"_id": "12852", "title": "Order Generating iff Every Superset Closed on Infima is Whole Space", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $X$ be a subset of $S$. Then :$X$ is order generating {{iff}} :$\\forall Y \\subseteq S: Y \\supseteq X \\land \\left({\\forall Z \\subseteq Y: \\inf Z \\in Y}\\right) \\implies Y = S$"}
+{"_id": "12853", "title": "Group Isomorphism Preserves Inverses", "text": "Let $\\phi: \\struct {G, \\circ} \\to \\struct {H, *}$ be a group isomorphism. Let: :$e_G$ be the identity of $\\struct {G, \\circ}$ :$e_H$ be the identity of $\\struct {H, *}$. Then: : $\\forall g \\in G: \\map \\phi {g^{-1} } = \\paren {\\map \\phi g}^{-1}$"}
+{"_id": "12860", "title": "Preimage of Composite Relation", "text": "Let $\\RR_1 \\subseteq S_1 \\times T_1$ and $\\RR_2 \\subseteq S_2 \\times T_2$ be relations. Let $\\RR_2 \\circ \\RR_1 \\subseteq S_1 \\times T_2$ be the composition of $\\RR_1$ and $\\RR_2$. Then the preimage of $\\RR_2 \\circ \\RR_1$ is given by: :$\\Preimg {\\RR_2 \\circ \\RR_1} = \\Preimg {\\Img {\\RR_1} \\cap \\Preimg {\\RR_2} }$"}
+{"_id": "12861", "title": "Image of Composite Relation", "text": "Let $\\RR_1 \\subseteq S_1 \\times T_1$ and $\\RR_2 \\subseteq S_2 \\times T_2$ be relations. Let $\\RR_2 \\circ \\RR_1 \\subseteq S_1 \\times T_2$ be the composition of $\\RR_1$ and $\\RR_2$. Then the image of $\\RR_2 \\circ \\RR_1$ is given by: :$\\Img {\\RR_2 \\circ \\RR_1} = \\Img {\\Img {\\RR_1} \\cap \\Preimg {\\RR_2} }$"}
+{"_id": "12862", "title": "Weierstrass Product Theorem", "text": "Let $\\sequence {a_k}$ be a sequence of non-zero complex numbers such that: :$\\cmod {a_n} \\to \\infty$ as $n \\to \\infty$ Let $\\sequence {p_n}$ be a sequence of non-negative integers for which the series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\size {\\dfrac r {a_n} }^{1 + p_n}$ converges for every $r \\in \\R_{> 0}$. Let: :$\\displaystyle \\map f z = \\prod_{n \\mathop = 1}^\\infty \\map {E_{p_n} } {\\frac z {a_n} }$ where $E_{p_n}$ are Weierstrass elementary factors. Then $f$ is entire and its zeroes are the points $a_n$, counted with multiplicity."}
+{"_id": "12868", "title": "Isomorphism between Gaussian Integer Units and Rotation Matrices Order 4", "text": "Let $\\struct {U_\\C, \\times}$ be the group of Gaussian integer units under complex multiplication. Let $\\struct {R_4, \\times}$ be the group of rotation matrices of order $4$ under modulo addition. Then $\\struct {U_\\C, \\times}$ and $\\struct {R_4, \\times}$ are isomorphic algebraic structures."}
+{"_id": "12869", "title": "Isomorphism between Gaussian Integer Units and Reduced Residue System Modulo 5 under Multiplication", "text": "Let $\\struct {U_\\C, \\times}$ be the group of Gaussian integer units under complex multiplication. Let $\\struct {\\Z'_5, \\times_5}$ be the multiplicative group of reduced residues modulo $5$. Then $\\struct {U_\\C, \\times}$ and $\\struct {\\Z'_5, \\times_5}$ are isomorphic algebraic structures."}
+{"_id": "12870", "title": "Group Generated by Reciprocal of z and 1 minus z", "text": "Let $\\struct {S, \\circ}$ denote the '''group generated by $\\dfrac 1 z$ and $1 - z$'''. Then $\\struct {S, \\circ}$ is a finite group of order $6$."}
+{"_id": "12871", "title": "Group Generated by Reciprocal of z and Minus z is Klein Four-Group", "text": "Let $K_4$ denote the Klein $4$-group. Let $S$ be the group generated by $1 / z$ and $-z$. Then $K_4$ and $S$ are isomorphic algebraic structures."}
+{"_id": "12872", "title": "Group of Reflection Matrices Order 4 is Klein Four-Group", "text": "Let $K_4$ denote the Klein $4$-group. Let $R_4$ be the Group of Reflection Matrices Order $4$. Then $K_4$ and $R_4$ are isomorphic algebraic structures."}
+{"_id": "12874", "title": "Klein Four-Group and Group of Cyclic Group of Order 4 are not Isomorphic", "text": "The Klein $4$-group $K_4$ and the cyclic group of order $4$ $C_4$ are not isomorphic."}
+{"_id": "12875", "title": "Order Generating iff Not Preceding implies There Exists Element Preceding and Not Preceding", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $X$ be a subset of $S$. Then :$X$ is order generating {{iff}} :$\\forall x, y \\in S: \\left({ y \\npreceq x \\implies \\exists p \\in X: x \\preceq p \\land y \\npreceq p}\\right)$"}
+{"_id": "12876", "title": "Permutation Induces Equivalence Relation/Corollary", "text": ":$i \\mathrel {\\mathcal R_\\pi} j$ {{iff}} $i$ and $j$ are in the same cycle of $\\pi$."}
+{"_id": "12877", "title": "Order of Group Element not less than Order of Power", "text": "Let $\\left({G, \\circ}\\right)$ be a group whose identity is $e$. Let $g \\in G$ be an element of $g$. Let $\\left\\lvert{g}\\right\\rvert$ denote the order of $g$ in $G$. Then: :$\\forall m \\in \\Z: \\left\\lvert{g^m}\\right\\rvert \\le \\left\\lvert{g}\\right\\rvert$ where $g^m$ denotes the $m$th power of $g$ in $G$."}
+{"_id": "12880", "title": "Order of Group Element equals Order of Coprime Power", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $g \\in G$ be an element of $g$. Let $\\order g$ denote the order of $g$ in $G$. Then: :$\\forall m \\in \\Z: \\order {g^m} = \\order g \\iff m \\perp \\order g$ where: :$g^m$ denotes the $m$th power of $g$ in $G$ :$\\perp$ denotes coprimality."}
+{"_id": "12883", "title": "Unique Composition of Group Element whose Order is Product of Coprime Integers", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $g \\in G$ be an element of $g$. Let: :$\\order g = m n$ where: :$\\order g$ denotes the order of $g$ in $G$ :$m$ and $n$ are coprime integers. Then $g$ can be expressed uniquely as the product of two commuting elements $a$ and $b$ of order $m$ and $n$ respectively."}
+{"_id": "12884", "title": "Order of Cyclic Group equals Order of Generator", "text": "Let $G$ be a finite cyclic group which is generated by $a \\in G$. Then: :$\\order a = \\order G$ where: :$\\order a$ denotes the order of $a$ in $G$ :$\\order G$ denotes the order of $G$."}
+{"_id": "12885", "title": "Group whose Order equals Order of Element is Cyclic", "text": "Let $G$ be a finite group of order $n$. Let $g \\in G$ have order $n$. Then $G$ is a cyclic group which is generated by $g$."}
+{"_id": "12886", "title": "Integer Powers of 2 under Multiplication form Infinite Abelian Group", "text": "Let $S$ be the set of integers defined as: :$S = \\set {2^k: k \\in \\Z}$ Then $\\struct {S, \\times}$ is an infinite abelian group."}
+{"_id": "12887", "title": "Numbers of form 1 + 2m over 1 + 2n form Infinite Abelian Group under Multiplication", "text": "Let $S$ be the set of integers defined as: :$S = \\set {\\dfrac {1 + 2 m} {1 + 2 n}: m, n \\in \\Z}$ Then $\\struct {S, \\times}$ is an infinite abelian group."}
+{"_id": "12889", "title": "Superset of Order Generating is Order Generating", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $X, Y$ be subsets of $S$ such that :$X$ is order generating and :$X \\subseteq Y$ Then $Y$ is order generating."}
+{"_id": "12890", "title": "Modulus 1 Rational Argument Complex Numbers under Multiplication form Infinite Abelian Group", "text": "Let $S$ be the set defined as: :$S = \\set {\\cos \\theta + i \\sin \\theta: \\theta \\in \\Q}$ Then the algebraic structure $\\struct {S, \\times}$ is an infinite abelian group."}
+{"_id": "12891", "title": "Inversion Mapping is Automorphism iff Group is Abelian", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\iota: G \\to G$ be the inversion mapping on $G$, defined as: :$\\forall g \\in G: \\map \\iota g = g^{-1}$ Then $\\iota$ is an automorphism {{iff}} $G$ is abelian."}
+{"_id": "12892", "title": "Even Order Group has Odd Number of Order 2 Elements", "text": "Let $G$ be a group whose identity is $e$. Let $G$ be of even order. Then $G$ has an odd number of elements of order $2$."}
+{"_id": "12894", "title": "Roots of Resolvent of Cubic", "text": "Let $P$ be the cubic equation: : $a x^3 + b x^2 + c x + d = 0$ with $a \\ne 0$ Let $\\map r P$ be the resolvent equation of $P$, given by: : $u^6 - 2 R u^3 - Q^3$ Let the roots of $P$ be $\\alpha_1, \\alpha_2, \\alpha_3$. Then the roots of $\\map r P$ can be expressed as: {{begin-eqn}} {{eqn | l = v       | r = \\frac 1 3 \\paren {\\alpha_1 + \\omega \\alpha_2 + \\omega^2 \\alpha_3} }} {{eqn | l = \\omega v       | r = \\frac 1 3 \\paren {\\alpha_3 + \\omega \\alpha_1 + \\omega^2 \\alpha_2}       | c =  }} {{eqn | l = \\omega^2 v       | r = \\frac 1 3 \\paren {\\alpha_2 + \\omega \\alpha_3 + \\omega^2 \\alpha_1}       | c =  }} {{eqn | l = u       | r = \\frac 1 3 \\paren {\\alpha_1 + \\omega \\alpha_3 + \\omega^2 \\alpha_2} }} {{eqn | l = \\omega u       | r = \\frac 1 3 \\paren {\\alpha_2 + \\omega \\alpha_1 + \\omega^2 \\alpha_3}       | c =  }} {{eqn | l = \\omega^2 u       | r = \\frac 1 3 \\paren {\\alpha_3 + \\omega \\alpha_2 + \\omega^2 \\alpha_1}       | c =  }} {{end-eqn}} where $\\omega = -\\dfrac {-1 + \\sqrt {-3} } 2$ is one of the primitive (complex) cube roots of $1$."}
+{"_id": "12895", "title": "Roots of Complex Number/Examples/Cube Roots", "text": "Let $z := \\polar {r, \\theta}$ be a complex number expressed in polar form, such that $z \\ne 0$. Then the complex cube roots of $z$ are given by: :$z^{1 / 3} = \\set {r^{1 / 3} \\paren {\\map \\cos {\\dfrac {\\theta + 2 \\pi k} 3} + i \\, \\map \\sin {\\dfrac {\\theta + 2 \\pi k} 3} }: k \\in \\set {0, 1, 2} }$ There are $3$ distinct such complex cube roots. These can also be expressed as: :$z^{1 / 3} = \\set {r^{1 / 3} e^{i \\paren {\\theta + 2 \\pi k} / 3}: k \\in \\set {0, 1, 2} }$ or: :$z^{1 / 3} = \\set {r^{1 / 3} e^{i \\theta / 3} \\omega^k: k \\in \\set {0, 1, 2} }$ where $\\omega = e^{2 i \\pi / 3} = -\\dfrac 1 2 + \\dfrac {i \\sqrt 3} 2$ is the first cube root of unity."}
+{"_id": "12896", "title": "Roots of Complex Number/Corollary", "text": "Let $z := \\polar {r, \\theta}$ be a complex number expressed in polar form, such that $z \\ne 0$. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $w$ be one of the complex $n$th roots of $z$. Then the $n$th roots of $z$ are given by: :$z^{1 / n} = \\set {w \\epsilon^k: k \\in \\set {1, 2, \\ldots, n - 1} }$ where $\\epsilon$ is a primitive $n$th root of unity."}
+{"_id": "12897", "title": "Top is Prime Element", "text": "Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be a bounded above meet semilattice. Then $\\top$ is a prime element where $\\top$ denotes the greatest element of $L$."}
+{"_id": "12898", "title": "Characterization of Prime Element in Meet Semilattice", "text": "Let $L = \\struct {S, \\wedge, \\preceq}$ be a meet semilattice. Let $p \\in S$, Then: :$p$ is prime element {{iff}}: :for all non-empty finite subsets $A$ of $S$: ::if $\\inf A \\preceq p$, then there exists element $x$ of $A$ such that $x \\preceq p$."}
+{"_id": "12900", "title": "Prime Element is Meet Irreducible", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a lattice. Let $p \\in S$. Let $p$ be a prime element of $L$. Then $p$ is meet irreducible in $L$."}
+{"_id": "12901", "title": "Prime Element iff Meet Irreducible in Distributive Lattice", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a distributive lattice. Let $p \\in S$. Then $p$ is a prime element {{iff}} $p$ is meet irreducible."}
+{"_id": "12902", "title": "Multiple Equilibrium Points all have Equal Payoffs", "text": "Let $G$ be a two-person zero-sum game. Let $G$ have more than one equilibrium point. Then every equilibrium point of $G$ has the same payoff."}
+{"_id": "12903", "title": "Definition:Information of Game", "text": "The '''information''' of a game is the knowledge each player has of the history of all the moves of all the players of the game."}
+{"_id": "12907", "title": "Multigraph with Finite Vertex Set may not be Finite", "text": "Let $G$ be a multigraph. Let the vertex set of $G$ be finite. Then it is not necessarily the case that $G$ is also finite."}
+{"_id": "12908", "title": "Modulus Larger than Real Part", "text": ":$\\cmod z \\ge \\size {\\map \\Re z}$"}
+{"_id": "12909", "title": "Modulus Larger than Imaginary Part", "text": ":$\\cmod z \\ge \\size {\\map \\Im z}$"}
+{"_id": "12910", "title": "Product Space is T0 iff Factor Spaces are T0/General Result", "text": "Let $\\SS = \\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\\displaystyle T = \\struct{S, \\tau} = \\prod \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\SS$. Then $T$ is a $T_0$ (Kolmogorov) space {{iff}} each of $\\struct{S_\\alpha, \\tau_\\alpha}$ is a $T_0$ (Kolmogorov) space."}
+{"_id": "12911", "title": "Separation Properties Preserved under Topological Product/Corollary", "text": "$T = \\struct {S, \\tau}$ has one of the following properties {{iff}} each of $\\struct {S_i, \\tau_i}$ has the same property: :Regular Property :Tychonoff (Completely Regular) Property If $T = \\struct {S, \\tau}$ has one of the following properties then each of $\\struct {S_i, \\tau_i}$ has the same property: :Normal Property :Completely Normal Property but the converse does not necessarily hold."}
+{"_id": "12913", "title": "Quotient and Remainder to Number Base/General Result", "text": ":$\\left \\lfloor {\\dfrac n {b^s}} \\right \\rfloor = \\left[{r_m r_{m-1} \\ldots r_{s+1} r_s}\\right]_b$ :$\\displaystyle n \\,\\bmod\\, {b^s} = \\sum_{j \\mathop = 0}^{s-1} {r_j b^j} = \\left[{r_{s-1} r_{s-2} \\ldots r_1 r_0}\\right]_b$"}
+{"_id": "12914", "title": "Remainder is Primitive Recursive", "text": "Let $m, n \\in \\N$ be natural numbers. Let us define the function $\\operatorname{rem}: \\N^2 \\to \\N$: :$\\map \\rem {n, m} = \\begin{cases} \\text{the remainder when } n \\text{ is divided by } m & : m \\ne 0 \\\\ 0 & : m = 0 \\end{cases}$ where the $\\text{remainder}$ is as defined in the Division Theorem: :If $n = m q + r$, where $0 \\le r < m$, then $r$ is the remainder. Then $\\rem$ is primitive recursive."}
+{"_id": "12915", "title": "Quotient is Primitive Recursive", "text": "Let $m, n \\in \\N$ be natural numbers. Let us define the function $\\operatorname {quot}: \\N^2 \\to \\N$: :$\\map {\\operatorname {quot} } {n, m} = \\begin{cases} \\text{the quotient when } n \\text{ is divided by } m & : m \\ne 0 \\\\ 0 & : m = 0 \\end{cases}$ where the $\\text {quotient}$ and $\\text {remainder}$ are as defined in the Division Theorem: :If $n = m q + r$, where $0 \\le r < m$, then $q$ is the quotient. Then $\\operatorname {quot}$ is primitive recursive."}
+{"_id": "12918", "title": "Not Every Two-Person Zero-Sum Game has Saddle Point", "text": "Not every two-person zero-sum game has a saddle point."}
+{"_id": "12920", "title": "Two-Person Zero-Sum Game with Multiple Solutions", "text": "There exists a two-person zero-sum game with more than one solution."}
+{"_id": "12921", "title": "Two-Person Zero-Sum Game with Finite Strategies has Solution", "text": "Let $G$ be a two-person zero-sum game. Let each player of $G$ have a finite set of strategies available. Then $G$ has at least one solution."}
+{"_id": "12923", "title": "Matching Pennies is Completely Mixed Game", "text": "The game of matching pennies is a completely mixed game."}
+{"_id": "12925", "title": "Value of Skew-Symmetric Game is Zero", "text": "Let $G$ be a two-person zero-sum game. Let $G$ be represented by a payoff table that is skew-symmetric. Then the value of $G$ is zero."}
+{"_id": "12926", "title": "Prime Element iff Complement of Lower Closure is Filter", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a bounded above lattice. Let $p \\in S$ such that :$p \\ne \\top$ where $\\top$ denotes the top of $L$. Then: :$p$ is a prime element {{iff}} :$\\relcomp S {p^\\preceq}$ is filter in $L$ where :$p^\\preceq$ denotes the lower closure of $p$ :$\\relcomp S {p^\\preceq}$ denotes the relative complement of $p^\\preceq$ relative to $S$."}
+{"_id": "12927", "title": "Oscillation at Point (Infimum) equals Oscillation at Point (Epsilon-Neighborhood)", "text": "Let $f: D \\to \\R$ be a real function where $D \\subseteq \\R$. Let $x$ be a point in $D$. Let $N_x$ be the set of open subset neighborhoods of $x$. Let $E_x$ be the set of $\\epsilon$-neighborhoods of $x$. Let $\\map {\\omega_f} x$ be the oscillation of $f$ at $x$ based on $N_x$: :$\\map {\\omega_f} x = \\inf \\set {\\map {\\omega_f} I: I \\in N_x}$ where $\\map {\\omega_f} I$ is the oscillation of $f$ on $I$: :$\\map {\\omega_f} I = \\sup \\set {\\size {\\map f y - \\map f z}: y, z \\in I \\cap D}$ Let $\\map {\\omega^E_f} x$ be the oscillation of $f$ at $x$ based on $E_x$: :$\\map {\\omega^E_f} x = \\inf \\set {\\map {\\omega_f} I: I \\in E_x}$ Then: :$\\map {\\omega_f} x \\in \\R$ {{iff}} $\\map {\\omega^E_f} x \\in \\R$ and, if $\\map {\\omega_f} x$ and $\\map {\\omega^E_f} x$ exist as real numbers: :$\\map {\\omega_f} x = \\map {\\omega^E_f} x$"}
+{"_id": "12929", "title": "Prime Element iff There Exists Way Below Open Filter which Complement has Maximum", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below continuous distributive lattice. Let $p \\in S$ such that :$p \\ne \\top$ where $\\top$ denotes the top of $L$. Then :$p$ is a prime element {{iff}} :there exists a way below open filter $F$ in $L$: $p = \\max \\left({ \\complement_S\\left({F}\\right) }\\right)$"}
+{"_id": "12933", "title": "Characterization of Prime Ideal", "text": "Let $L = \\struct {S, \\wedge, \\preceq}$ be a meet semilattice. Let $I$ be an proper ideal in $L$. Then :$I$ is a prime ideal {{iff}} :$\\forall x, y \\in S: \\paren {x \\wedge y \\in I \\implies x \\in I \\lor y \\in I}$"}
+{"_id": "12934", "title": "Matching Pennies is Strictly Competitive", "text": "The game of Matching Pennies is strictly competitive."}
+{"_id": "12936", "title": "Second Price Auction has Inefficient Equilibria", "text": "A second price auction has Nash equilibria which are inefficient."}
+{"_id": "12937", "title": "Expected-Utility Maximization Theorem", "text": "Let $P$ be a rational decision-maker. Let $A$ be a set of possible moves available to $P$. Then there exists a way of assigning values to a payoff function $u: A \\to \\R$ so that $P$ will always choose the move such that $u$ is a maximum."}
+{"_id": "12938", "title": "Epic Equalizer is Isomorphism", "text": "Let $\\mathbf C$ be a metacategory. Let $e: E \\to C$ be the equalizer of two morphisms $f, g: C \\to D$. Let $e$ be an epimorphism. Then $e$ is an isomorphism."}
+{"_id": "12939", "title": "Infimum of Subset of Extended Real Numbers is Arbitrarily Close", "text": "Let $A \\subseteq \\overline \\R$ be a subset of the extended real numbers. Let $b$ be an infimum (in $\\R$) of $A$. Let $\\epsilon \\in \\R_{>0}$. Then: :$\\exists x \\in \\paren {A \\cap \\R}: x - b < \\epsilon$"}
+{"_id": "12940", "title": "Oscillation on Set is an Extended Real Number", "text": "Let $f: D \\to \\R$ be a real function where $D \\subseteq \\R$. Let $x$ be a point in $D$. Let $I$ be a real set that contains (as an element) $x$. Let $\\omega_f \\left({I}\\right)$ be the oscillation of $f$ on $I$: :$\\omega_f \\left({I}\\right) = \\sup \\left\\{{\\left\\vert{f \\left({y}\\right) - f \\left({z}\\right)}\\right\\vert: y, z \\in I \\cap D}\\right\\}$ Then: :$\\omega_f \\left({I}\\right) \\in \\overline \\R_{\\ge 0}$ and: :$\\omega_f \\left({I}\\right) = \\begin{cases} \\text{a positive real number} & \\left\\{{\\left\\vert{f \\left({y}\\right) - f \\left({z}\\right)}\\right\\vert: y, z \\in I \\cap D}\\right\\} \\text{is bounded above} \\\\ \\infty & \\left\\{{\\left\\vert{f \\left({y}\\right) - f \\left({z}\\right)}\\right\\vert: y, z \\in I \\cap D}\\right\\} \\text{is not bounded above} \\end{cases}$"}
+{"_id": "12941", "title": "Oscillation on Subset", "text": "Let $f: D \\to \\R$ be a real function where $D \\subseteq \\R$. Let $x$ be a point in $D$. Let $S_x$ be a set of real sets that contain (as an element) $x$. Let $\\map {\\omega_f} I$ be the oscillation of $f$ on a set $I$ in $S_x$: :$\\map {\\omega_f} I = \\sup \\set {\\size {\\map f y - \\map f z}: y, z \\in I \\cap D}$ Let $I \\in S_x$. Let $\\map {\\omega_f} I \\in \\R$. Let $J \\in S_x$ be a subset of $I$. Then: :$\\map {\\omega_f} J \\in \\R$ :$\\map {\\omega_f} J \\le \\map {\\omega_f} I$"}
+{"_id": "12942", "title": "Generating Fraction for Lucas Numbers", "text": "The fraction: :$\\dfrac {199} {9899}$ has a decimal expansion which contains within it the start of the Lucas sequence: :$0 \\cdotp 02010 \\, 30407 \\, 11 \\ldots$"}
+{"_id": "12943", "title": "Transcendental Numbers are Uncountable", "text": "The set of transcendental real numbers is uncountable."}
+{"_id": "12944", "title": "Almost All Real Numbers are Transcendental", "text": "Almost all real numbers are transcendental."}
+{"_id": "12945", "title": "Champernowne Constant is Transcendental", "text": "The Champernowne constant: :$0 \\cdotp 12345 \\, 67891 \\, 01112 \\, 13141 \\, 51617 \\, 18192 \\, 02122 \\ldots$ is transcendental."}
+{"_id": "12946", "title": "Champernowne Constant is Normal", "text": "The Champernowne constant: :$0 \\cdotp 12345 \\, 67891 \\, 01112 \\, 13141 \\, 51617 \\, 18192 \\, 02122 \\ldots$ is normal with respect to base $10$."}
+{"_id": "12950", "title": "Anomalous Cancellation/Examples/3544 over 7531", "text": ":$\\dfrac {344} {731} = \\dfrac {3544} {7531} = \\dfrac {35544} {75531} = \\cdots$"}
+{"_id": "12951", "title": "Anomalous Cancellation/Examples/143 185 over 17 018 560", "text": ":$\\dfrac {1435} {170 \\, 560} = \\dfrac {143 \\, 185} {17 \\, 018 \\, 560} = \\dfrac {14 \\, 318 \\, 185} {1 \\, 701 \\, 818 \\, 560} = \\cdots$"}
+{"_id": "12952", "title": "Number of Digits in Power of 2", "text": "Let $n$ be a positive integer. Expressed in conventional decimal notation, the number of digits in the $n$th power of $2$: :$2^n$ is equal to: :$\\ceiling {n \\log_{10} 2}$ where $\\ceiling x$ denotes the ceiling of $x$."}
+{"_id": "12953", "title": "One Third as Quotient of Sequences of Odd Numbers", "text": ":$\\dfrac 1 3 = \\dfrac {1 + 3} {5 + 7} = \\dfrac {1 + 3 + 5} {7 + 9 + 11} = \\dfrac {1 + 3 + 5 + 7} {9 + 11 + 13 + 15} = \\cdots$"}
+{"_id": "12954", "title": "Euler's Number as Sum of Egyptian Fractions", "text": "The reciprocal of Euler's number $e$ can be approximated by the following sequence of Egyptian fractions: :$\\dfrac 1 e = \\dfrac 1 3 + \\dfrac 1 {29} + \\dfrac 1 {15 \\, 786} + \\dfrac 1 {513 \\, 429 \\, 610} + \\cdots$ {{OEIS|A006526}}"}
+{"_id": "12955", "title": "One Half as Pandigital Fraction", "text": "There are $12$ ways $\\dfrac 1 2$ can be expressed as a pandigital fraction: :$\\dfrac 1 2 = \\dfrac {6729} {13 \\, 458}$ :$\\dfrac 1 2 = \\dfrac {6792} {13 \\, 584}$ :$\\dfrac 1 2 = \\dfrac {6927} {13 \\, 854}$ :$\\dfrac 1 2 = \\dfrac {7269} {14 \\, 538}$ :$\\dfrac 1 2 = \\dfrac {7293} {14 \\, 586}$ :$\\dfrac 1 2 = \\dfrac {7329} {14 \\, 658}$ :$\\dfrac 1 2 = \\dfrac {7692} {15 \\, 384}$ :$\\dfrac 1 2 = \\dfrac {7923} {15 \\, 846}$ :$\\dfrac 1 2 = \\dfrac {7932} {15 \\, 864}$ :$\\dfrac 1 2 = \\dfrac {9267} {18 \\, 534}$ :$\\dfrac 1 2 = \\dfrac {9273} {18 \\, 546}$ :$\\dfrac 1 2 = \\dfrac {9327} {18 \\, 654}$"}
+{"_id": "12956", "title": "One Seventh as Pandigital Fraction", "text": "There are $7$ ways $\\dfrac 1 7$ can be expressed as a pandigital fraction: :$\\dfrac 1 2 = \\dfrac {2394} {16 \\, 758}$ :$\\dfrac 1 2 = \\dfrac {2637} {18 \\, 459}$ :$\\dfrac 1 2 = \\dfrac {4527} {31 \\, 689}$ :$\\dfrac 1 2 = \\dfrac {5274} {36 \\, 918}$ :$\\dfrac 1 2 = \\dfrac {5418} {37 \\, 926}$ :$\\dfrac 1 2 = \\dfrac {5976} {41 \\, 832}$ :$\\dfrac 1 2 = \\dfrac {7614} {53 \\, 298}$"}
+{"_id": "12957", "title": "One Third as Pandigital Fraction", "text": "There are $2$ ways $\\dfrac 1 3$ can be expressed as a pandigital fraction: :$\\dfrac 1 3 = \\dfrac {5823} {17469}$ :$\\dfrac 1 3 = \\dfrac {5832} {17496}$"}
+{"_id": "12958", "title": "One Quarter as Pandigital Fraction", "text": "There are $4$ ways $\\dfrac 1 4$ can be expressed as a pandigital fraction: :$\\dfrac 1 4 = \\dfrac {3942} {15768}$ :$\\dfrac 1 4 = \\dfrac {4392} {17568}$ :$\\dfrac 1 4 = \\dfrac {5796} {23184}$ :$\\dfrac 1 4 = \\dfrac {7956} {31824}$"}
+{"_id": "12959", "title": "One Fifth as Pandigital Fraction", "text": "There are $12$ ways $\\dfrac 1 5$ can be expressed as a pandigital fraction: :$\\dfrac 1 5 = \\dfrac {2697} {13485}$ :$\\dfrac 1 5 = \\dfrac {2769} {13845}$ :$\\dfrac 1 5 = \\dfrac {2937} {14685}$ :$\\dfrac 1 5 = \\dfrac {2967} {14835}$ :$\\dfrac 1 5 = \\dfrac {2973} {14865}$ :$\\dfrac 1 5 = \\dfrac {3297} {16485}$ :$\\dfrac 1 5 = \\dfrac {3729} {18645}$ :$\\dfrac 1 5 = \\dfrac {6297} {31485}$ :$\\dfrac 1 5 = \\dfrac {7629} {38145}$ :$\\dfrac 1 5 = \\dfrac {9237} {46185}$ :$\\dfrac 1 5 = \\dfrac {9627} {48135}$ :$\\dfrac 1 5 = \\dfrac {9723} {48615}$"}
+{"_id": "12960", "title": "One Sixth as Pandigital Fraction", "text": "There are $3$ ways $\\dfrac 1 6$ can be expressed as a pandigital fraction: :$\\dfrac 1 6 = \\dfrac {2943} {17658}$ :$\\dfrac 1 6 = \\dfrac {4653} {27918}$ :$\\dfrac 1 6 = \\dfrac {5697} {34182}$"}
+{"_id": "12961", "title": "One Eighth as Pandigital Fraction", "text": "There are $46$ ways $\\dfrac 1 8$ can be expressed as a pandigital fraction: :$\\dfrac 1 8 = \\dfrac {3187} {25496}$ :$\\dfrac 1 8 = \\dfrac {4589} {36712}$ :$\\dfrac 1 8 = \\dfrac {4591} {36728}$ :$\\dfrac 1 8 = \\dfrac {4689} {37512}$ :$\\dfrac 1 8 = \\dfrac {4691} {37528}$ :$\\dfrac 1 8 = \\dfrac {4769} {38152}$ :$\\dfrac 1 8 = \\dfrac {5237} {41896}$ :$\\dfrac 1 8 = \\dfrac {5371} {42968}$ :$\\dfrac 1 8 = \\dfrac {5789} {46312}$ :$\\dfrac 1 8 = \\dfrac {5791} {46328}$ :$\\dfrac 1 8 = \\dfrac {5839} {46712}$ :$\\dfrac 1 8 = \\dfrac {5892} {47136}$ :$\\dfrac 1 8 = \\dfrac {5916} {47328}$ :$\\dfrac 1 8 = \\dfrac {5921} {47368}$ :$\\dfrac 1 8 = \\dfrac {6479} {51832}$ :$\\dfrac 1 8 = \\dfrac {6741} {53928}$ :$\\dfrac 1 8 = \\dfrac {6789} {54312}$ :$\\dfrac 1 8 = \\dfrac {6791} {54328}$ :$\\dfrac 1 8 = \\dfrac {6839} {54712}$ :$\\dfrac 1 8 = \\dfrac {7123} {56984}$ :$\\dfrac 1 8 = \\dfrac {7312} {58496}$ :$\\dfrac 1 8 = \\dfrac {7364} {58912}$ :$\\dfrac 1 8 = \\dfrac {7416} {59328}$ :$\\dfrac 1 8 = \\dfrac {7421} {59368}$ :$\\dfrac 1 8 = \\dfrac {7894} {63152}$ :$\\dfrac 1 8 = \\dfrac {7941} {63528}$ :$\\dfrac 1 8 = \\dfrac {8174} {65392}$ :$\\dfrac 1 8 = \\dfrac {8179} {65432}$ :$\\dfrac 1 8 = \\dfrac {8394} {67152}$ :$\\dfrac 1 8 = \\dfrac {8419} {67352}$ :$\\dfrac 1 8 = \\dfrac {8439} {67512}$ :$\\dfrac 1 8 = \\dfrac {8932} {71456}$ :$\\dfrac 1 8 = \\dfrac {8942} {71536}$ :$\\dfrac 1 8 = \\dfrac {8953} {71624}$ :$\\dfrac 1 8 = \\dfrac {8954} {71632}$ :$\\dfrac 1 8 = \\dfrac {9156} {73248}$ :$\\dfrac 1 8 = \\dfrac {9158} {73264}$ :$\\dfrac 1 8 = \\dfrac {9182} {73456}$ :$\\dfrac 1 8 = \\dfrac {9316} {74528}$ :$\\dfrac 1 8 = \\dfrac {9321} {74568}$ :$\\dfrac 1 8 = \\dfrac {9352} {74816}$ :$\\dfrac 1 8 = \\dfrac {9416} {75328}$ :$\\dfrac 1 8 = \\dfrac {9421} {75368}$ :$\\dfrac 1 8 = \\dfrac {9523} {76184}$ :$\\dfrac 1 8 = \\dfrac {9531} {76248}$ :$\\dfrac 1 8 = \\dfrac {9541} {76328}$"}
+{"_id": "12962", "title": "One Ninth as Pandigital Fraction", "text": "There are $3$ ways $\\dfrac 1 9$ can be expressed as a pandigital fraction: :$\\dfrac 1 9 = \\dfrac {6381} {57429}$ :$\\dfrac 1 9 = \\dfrac {6471} {58239}$ :$\\dfrac 1 9 = \\dfrac {8361} {75249}$"}
+{"_id": "12963", "title": "Four Fifths as Pandigital Fraction", "text": "The fraction $\\dfrac 4 5$ can be expressed as a pandigital fraction in the following interesting way: :$\\dfrac 4 5 = \\dfrac {9876} {12 \\, 345}$"}
+{"_id": "12964", "title": "Probability of Two Random Integers having no Common Divisor", "text": "Let $a$ and $b$ be integers chosen at random. The probability that $a$ and $b$ are coprime is given by: :$\\map \\Pr {a \\perp b} = \\dfrac 1 {\\map \\zeta 2} = \\dfrac 6 {\\pi^2}$ where $\\zeta$ denotes the zeta function. The decimal expansion of $\\dfrac 1 {\\map \\zeta 2}$ starts: :$\\dfrac 1 {\\map \\zeta 2} = 0 \\cdotp 60792 \\, 71018 \\, 54026 \\, 6 \\ldots$ {{OEIS|A059956}}"}
+{"_id": "12965", "title": "Probability of Random Integer being Square-Free", "text": "Let $a$ be an integer chosen at random. The probability that $a$ is square-free is given by: :$\\map \\Pr {\\neg \\exists b \\in \\Z: b^2 \\divides a} = \\dfrac 1 {\\map \\zeta 2} = \\dfrac 6 {\\pi^2}$ where $\\zeta$ denotes the zeta function. The decimal expansion of $\\dfrac 1 {\\map \\zeta 2}$ starts: :$\\dfrac 1 {\\map \\zeta 2} = 0 \\cdotp 60792 \\, 71018 \\, 54026 \\, 6 \\ldots$ {{OEIS|A059956}}"}
+{"_id": "12966", "title": "Kepler's Conjecture", "text": "The densest packing of identical spheres in space is obtained when the spheres are arranged with their centers at the points of a face-centered cubic lattice. This obtains a density of $\\dfrac \\pi {3 \\sqrt 2} = \\dfrac \\pi {\\sqrt {18} }$: :$\\dfrac \\pi {\\sqrt {18} } = 0 \\cdotp 74048 \\ldots$"}
+{"_id": "12967", "title": "Prime Ideal is Prime Element", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a lattice. Let $I$ be an ideal in $L$. Then: :$I$ is a prime ideal {{iff}}: :$I$ is a prime element in $\\struct {\\map {\\mathit {Ids} } L, \\precsim}$ where: :$\\map {\\mathit {Ids} } L$ denotes the set of all ideals in $L$ :$\\mathord \\precsim := \\mathord \\subseteq \\restriction_{\\map {\\mathit {Ids} } L \\times \\map {\\mathit {Ids} } L}$"}
+{"_id": "12968", "title": "Probability of Three Random Integers having no Common Divisor", "text": "Let $a, b$ and $c$ be integers chosen at random. The probability that $a, b$ and $c$ have no common divisor: :$\\map \\Pr {\\map \\perp {a, b, c} } = \\dfrac 1 {\\map \\zeta 3}$ where $\\zeta$ denotes the zeta function: :$\\map \\zeta 3 = \\dfrac 1 {1^3} + \\dfrac 1 {2^3} + \\dfrac 1 {3^3} + \\dfrac 1 {4^3} + \\cdots$ The decimal expansion of $\\dfrac 1 {\\map \\zeta 3}$ starts: :$\\dfrac 1 {\\map \\zeta 3} = 0 \\cdotp 83190 \\, 73725 \\, 80707 \\ldots$ {{OEIS|A088453}}"}
+{"_id": "12969", "title": "Densest Packing of Identical Circles", "text": "The densest packing of identical circles in the plane obtains a density of $\\dfrac \\pi {2 \\sqrt 3} = \\dfrac \\pi {\\sqrt {12} }$: :$\\dfrac \\pi {2 \\sqrt 3} = 0 \\cdotp 90689 \\, 96821 \\ldots$ {{OEIS|A093766}} This happens when they are packed together in a hexagonal array, with each circle touching $6$ others."}
+{"_id": "12971", "title": "Meet is Intersection in Set of Ideals", "text": "Let $\\mathscr S = \\struct {S, \\wedge, \\preceq}$ be a meet semilattice. Let $\\map {\\mathit {Ids} } {\\mathscr S}$ be the set of all ideals in $\\mathscr S$. Let $P = \\struct {\\map {\\mathit {Ids} } {\\mathscr S}, \\precsim}$ be an ordered set where $\\mathord \\precsim = \\mathord \\subseteq \\restriction_{\\map {\\mathit {Ids} } {\\mathscr S} \\times \\map {\\mathit {Ids} } {\\mathscr S} }$ Let $I_1, I_2$ be ideals in $\\mathscr S$. Then :$I_1 \\wedge_P I_2 = I_1 \\cap I_2$"}
+{"_id": "12972", "title": "Characterization of Prime Ideal by Finite Infima", "text": "Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Let $I$ be an ideal in $L$. Then :$I$ is a prime ideal {{iff}} :for all non-empty finite subset $A$ of $S: \\left({ \\inf A \\in I \\implies \\exists a \\in A: a \\in I}\\right)$"}
+{"_id": "12973", "title": "Doubling the Cube/Archytas Curve", "text": "The Archytas curve can be used for Doubling the Cube."}
+{"_id": "12974", "title": "Doubling the Cube/Intersection of Parabolas", "text": "The problem of Doubling the Cube can be solved by finding the intersection of two parabolas."}
+{"_id": "12975", "title": "Doubling the Cube/Intersection of Parabola and Hyperbola", "text": "The problem of Doubling the Cube can be solved by finding the intersection of a parabola and a hyperbola."}
+{"_id": "12976", "title": "Unordered Pair is Finite", "text": "Let $x, y$ be arbitrary. Then $\\left\\{ {x, y}\\right\\}$ is finite."}
+{"_id": "12977", "title": "Pair is Union of Singletons", "text": "Let $x, y$ be arbitrary. {{questionable|Need to state that $x \\ne y$.}} Then $\\left\\{ {x, y}\\right\\} = \\left\\{ {x}\\right\\} \\cup \\left\\{ {y}\\right\\}$"}
+{"_id": "12978", "title": "Doubling the Cube/Conchoid of Nicomedes", "text": "The problem of Doubling the Cube can be solved by using a conchoid of Nicomedes."}
+{"_id": "12979", "title": "Doubling the Cube/Cissoid of Diocles", "text": "The problem of Doubling the Cube can be solved by using a cissoid of Diocles."}
+{"_id": "12980", "title": "Length of Diagonal of Unit Square", "text": "The length of a diagonal of a square of side length $1$ is $\\sqrt 2$) (the square root of $2$)."}
+{"_id": "12981", "title": "Sequence of Best Rational Approximations to Square Root of 2", "text": "A sequence of best rational approximations to the square root of $2$ starts: :$\\dfrac 1 1, \\dfrac 3 2, \\dfrac 7 5, \\dfrac {17} {12}, \\dfrac {41} {29}, \\dfrac {99} {70}, \\dfrac {239} {169}, \\dfrac {577} {408}, \\ldots$ where: :the numerators are half of the Pell-Lucas numbers, $\\dfrac 1 2 Q_n$ :the denominators are the Pell numbers $P_n$ starting from $\\dfrac {\\tfrac12 Q_1} {P_1}$. {{OEIS-Numerators|A001333}} {{OEIS-Denominators|A000129}}"}
+{"_id": "12982", "title": "Difference between Adjacent Convergents of Simple Continued Fraction", "text": "Then for $k \\ge 1$: :$p_k q_{k - 1} - p_{k - 1} q_k = \\paren {-1}^{k + 1}$ That is: :$C_k - C_{k - 1} = \\dfrac {p_k} {q_k} - \\dfrac {p_{k - 1} } {q_{k - 1} } = \\dfrac {\\paren {-1}^{k + 1} } {q_k q_{k - 1} }$"}
+{"_id": "12983", "title": "Difference between Adjacent Convergents But One of Simple Continued Fraction", "text": "For $k \\ge 2$: :$p_k q_{k - 2} - p_{k - 2} q_k = \\paren {-1}^k a_k$ That is: :$C_k - C_{k-2} = \\dfrac {p_k} {q_k} - \\dfrac {p_{k - 2} } {q_{k - 2} } = \\dfrac {\\paren {-1}^k a_k} {q_k q_{k - 2} }$"}
+{"_id": "12984", "title": "Convergents of Simple Continued Fraction are Rationals in Canonical Form", "text": "For all $k \\ge 1$, $\\dfrac {p_k} {q_k}$ is in canonical form: :$p_k$ and $q_k$ are coprime :$q_k > 0$."}
+{"_id": "12986", "title": "Dual of Preordered Set is Preordered Set", "text": "Let $P = \\left({S, \\preceq}\\right)$ be a preordered set. Then dual of $P$, $P^{-1} = \\left({S, \\succeq}\\right)$ is also a preordered set. {{finish|Dual Ordered Set $\\ne$ Dual Preordered Set}}"}
+{"_id": "12987", "title": "Dual Ordered Set is Ordered Set", "text": "Let $P = \\left({S, \\preceq}\\right)$ be an ordered set. Then its dual, $P^{-1} = \\left({S, \\succeq}\\right)$, is also an ordered set."}
+{"_id": "12988", "title": "Value of Finite Continued Fraction equals Numerator Divided by Denominator", "text": "Let $F$ be a field. Let $\\tuple {a_0, a_1, \\ldots, a_n}$ be a finite continued fraction of length $n \\ge 0$. Let $p_n$ and $q_n$ be its $n$th numerator and denominator. Then the value $\\sqbrk {a_0, a_1, \\ldots, a_n}$ equals $\\dfrac {p_n} {q_n}$."}
+{"_id": "12989", "title": "Simple Infinite Continued Fraction Converges", "text": "Let $C = (a_0, a_1, \\ldots)$ be a simple infinite continued fraction in $\\R$. Then $C$ converges."}
+{"_id": "12991", "title": "Continued Fraction Identities/First/Infinite", "text": "Let $\\left[{a_0, a_1, a_2, \\ldots}\\right]$ be a simple infinite continued fraction. Then: :$\\left[{a_1, a_2, a_3, \\ldots}\\right] = a_1 + \\dfrac 1 {\\left[{a_2, a_3, \\ldots}\\right]}$"}
+{"_id": "12992", "title": "Relation between Adjacent Best Rational Approximations to Root 2", "text": "Consider the Sequence of Best Rational Approximations to Square Root of 2: :$\\sequence S := \\dfrac 1 1, \\dfrac 3 2, \\dfrac 7 5, \\dfrac {17} {12}, \\dfrac {41} {29}, \\dfrac {99} {70}, \\dfrac {239} {169}, \\dfrac {577} {408}, \\ldots$ Let $\\dfrac {p_n} {q_n}$ and $\\dfrac {p_{n + 1} } {q_{n + 1} }$ be adjacent terms of $\\sequence S$. Then: :$\\dfrac {p_{n + 1} } {q_{n + 1} } = \\dfrac {p_n + 2 q_n} {p_n + q_n}$"}
+{"_id": "12993", "title": "Parity of Pell Numbers", "text": "Consider the Pell numbers $P_0, P_1, P_2, \\ldots$ :$0, 1, 2, 5, 12, 29, \\ldots$ $P_n$ has the same parity as $n$. That is: :if $n$ is odd then $P_n$ is odd :if $n$ is even then $P_n$ is even."}
+{"_id": "12994", "title": "Parity of Best Rational Approximations to Root 2", "text": "Consider the Sequence of Best Rational Approximations to Square Root of 2: :$\\sequence S := \\dfrac 1 1, \\dfrac 3 2, \\dfrac 7 5, \\dfrac {17} {12}, \\dfrac {41} {29}, \\dfrac {99} {70}, \\dfrac {239} {169}, \\dfrac {577} {408}, \\ldots$ where $S_1 := \\dfrac 1 1$. The numerators of the terms of $\\sequence S$ are all odd. For all $n$, the parity of the denominator of term $S_n$ is the same as the parity of $n$."}
+{"_id": "12996", "title": "Prime Ideal is Prime Filter in Dual Lattice", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a lattice. Let $X$ be a subset of $S$. Then :$X$ is a prime ideal in $L$ {{iff}}: :$X$ is a prime filter in $L^{-1}$ where $L^{-1} = \\struct {S, \\succeq}$ denotes the dual of $L$."}
+{"_id": "12997", "title": "Ideal is Filter in Dual Ordered Set", "text": "Let $P = \\left({S, \\preceq}\\right)$ be an ordered set. Let $X$ be a subset of $S$. Then :$X$ is ideal in $P$ {{iff}} :$X$ is filter in $P^{-1}$ where $P^{-1} = \\left({S, \\succeq}\\right)$ denotes the dual of $P$."}
+{"_id": "12998", "title": "Square Root of 2 as Sum of Egyptian Fractions", "text": "The square root of $2$ can be approximated by the following sequence of Egyptian fractions: :$\\sqrt 2 = 1 + \\dfrac 1 3 + \\dfrac 1 {253} + \\dfrac 1 {218 \\, 201} + \\dfrac 1 {61 \\, 323 \\, 543 \\, 802} + \\cdots$ {{OEIS|A006487}}"}
+{"_id": "12999", "title": "Sprague's Property of Root 2", "text": "Let $S = \\sequence {s_n}$ be the sequence of fractions defined as follows: Let the numerator of $s_n$ be: :$\\floor {n \\sqrt 2}$ where $\\floor x$ denotes the floor of $x$. Let the denominators of the terms of $S$ be the (strictly) positive integers missing from the numerators of $S$: :$S := \\dfrac 1 3, \\dfrac 2 6, \\dfrac 4 {10}, \\dfrac 5 {13}, \\dfrac 7 {17}, \\dfrac 8 {20}, \\ldots$ Then the difference between the numerator and denominator of $s_n$ is equal to $2 n$."}
+{"_id": "13000", "title": "Steiner's Calculus Problem", "text": "Let $f: \\R_{>0} \\to \\R$ be the real function defined as: :$\\forall x \\in \\R_{>0}: \\map f x = x^{1/x}$ Then $\\map f x$ reaches its maximum at $x = e$ where $e$ is Euler's number ."}
+{"_id": "13001", "title": "Ratio of Lengths of Arms of Pentagram", "text": "Consider a pentagram. :400px Let $AC$ be the length of one of the lines which span the pentagram and define it. Let $B$ be one of the points where $AC$ intersects one of the other such lines such that $AB > AC$. Then: :$\\dfrac {AC} {AB} = \\phi$ where $\\phi$ denotes the golden mean."}
+{"_id": "13002", "title": "Filter is Ideal in Dual Ordered Set", "text": "Let $P = \\left({S, \\preceq}\\right)$ be an ordered set. Let $X$ be a subset of $S$. Then :$X$ is filter in $P$ {{iff}} :$X$ is ideal in $P^{-1}$ where $P^{-1} = \\left({S, \\succeq}\\right)$ denotes the dual of $P$."}
+{"_id": "13003", "title": "Prime Filter is Prime Ideal in Dual Lattice", "text": "Let $L = \\struct {S, \\preceq}$ be a lattice. Let $X$ be a subset of $S$. Then :$X$ is a prime filter in $L$ {{iff}}: :$X$ is a prime ideal in $L^{-1}$ where $L^{-1} = \\struct {S, \\succeq}$ denotes the dual of $L$."}
+{"_id": "13004", "title": "Trefoil Knot in Paper forms Pentagon", "text": "Take a strip of paper with parallel edges. Tie a simple trefoil knot in it. Flatten the knot and pull the paper tight. The knot will be in the shape of a regular pentagon. The edges of the paper strip trace out a pentagram. :500px"}
+{"_id": "13005", "title": "Sequence of Golden Rectangles", "text": "A golden rectangle can be divided into a square and another golden rectangle. That golden rectangle can in turn be divided into a square and another golden rectangle. The sequence can be continued indefinitely. :700px"}
+{"_id": "13006", "title": "Sequence of Golden Rectangles/Equiangular Spiral", "text": "The points where the vertices of successive squares of this sequence meet can be joined together by an equiangular spiral. This equiangular spiral can be approximated by quarter circles constructed as shown. The equiangular spiral tends towards the point of intersection of the diagonals of the golden rectangles."}
+{"_id": "13007", "title": "Continued Fraction Expansion of Golden Mean/Rate of Convergence", "text": "This continued fraction expansion has the slowest rate of convergence of all simple infinite continued fractions."}
+{"_id": "13008", "title": "Continued Fraction Expansion of Golden Mean/Successive Convergents", "text": "The $n$th convergent is given by: :$C_n = \\dfrac {F_{n + 1} } {F_n}$ where $F_n$ denotes the $n$th Fibonacci number."}
+{"_id": "13009", "title": "Continued Fraction Expansion of Pi", "text": "The constant $\\pi$ (pi) has the continued fraction expansion: :$\\pi = \\sqbrk {3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, \\ldots}$"}
+{"_id": "13010", "title": "Continued Fraction Expansion of Pi/Convergents", "text": "The convergents of the continued fraction expansion to $\\pi$ (pi) are: :$3, \\dfrac {22} 7, \\dfrac {333} {106}, \\dfrac {355} {113}, \\dfrac {103993} {33102}, \\dfrac {104348} {33215}$ {{OEIS-Numerators|A002485}} {{OEIS-Denominators|A002486}} These best rational approximations are accurate to $0, 2, 4, 6, 9, 9, 9, 10, 11, 11, 12, 13, \\ldots$ decimals. {{OEIS|A114526}} === Zu Chongzhi Fraction === {{:Definition:Zu Chongzhi Fraction}}"}
+{"_id": "13011", "title": "Filter is Prime iff For Every Element Element either Negation Belongs to Filter in Boolean Lattice", "text": "Let $B = \\struct {S, \\vee, \\wedge, \\neg, \\preceq}$ be a Boolean lattice. Let $F$ be a filter in $B$. Then :$F$ is prime {{iff}} :$\\forall x \\in S: x \\in F \\lor \\left({\\neg x}\\right) \\in F$"}
+{"_id": "13012", "title": "Beatty's Theorem", "text": "Let $r, s \\in \\R \\setminus \\Q$ be an irrational number such that $r > 1$ and $s > 1$. Let $\\BB_r$ and $\\BB_s$ be the Beatty sequences on $r$ and $s$ respectively. Then $\\BB_r$ and $\\BB_s$ are complementary Beatty sequences {{iff}}: :$\\dfrac 1 r + \\dfrac 1 s = 1$"}
+{"_id": "13013", "title": "Proper and Prime iff Ultrafilter in Boolean Lattice", "text": "Let $B = \\left({S, \\vee, \\wedge, \\neg, \\preceq}\\right)$ be a Boolean lattice. Let $F$ be a filter in $B$. Then :$F$ is a proper subset of $S$ and $F$ is a prime filter in $B$ {{iff}}: :$F$ is ultrafilter on $B$"}
+{"_id": "13017", "title": "Difference between Terms of Wythoff Pair", "text": "Let $\\tuple {\\floor {k \\phi}, \\floor {k \\phi^2} }$ be a Wythoff pair. The difference between the coordinates of this Wythoff pair is $k$. That is: :$\\floor {k \\phi^2} - \\floor {k \\phi} = k$"}
+{"_id": "13018", "title": "Finite Infima Set and Upper Closure is Smallest Filter", "text": "Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Let $X$ be a non-empty subset of $S$. Then :$X \\subseteq {\\operatorname{fininfs}\\left({X}\\right)}^\\succeq$ and :for every a filter $F$ in $L$: $\\left({X \\subseteq F \\implies {\\operatorname{fininfs}\\left({X}\\right)}^\\succeq \\subseteq F}\\right)$ where : $\\operatorname{fininfs}\\left({X}\\right)$ denotes the finite infima set of $X$, : $X^\\succeq$ denotes the upper closure of $X$."}
+{"_id": "13019", "title": "Smaller Number of Wythoff Pair is Smallest Number not yet in Sequence", "text": "Consider the sequence of Wythoff pairs arranged in sequential order: :$\\tuple {0, 0}, \\tuple {1, 2}, \\tuple {3, 5}, \\tuple {4, 7}, \\tuple {6, 10}, \\tuple {8, 13}, \\ldots$ The first coordinate of each Wythoff pair is the smallest positive integer which has not yet appeared in the sequence."}
+{"_id": "13020", "title": "Sequence of Wythoff Pairs contains all Positive Integers exactly Once Each", "text": "Consider the sequence of Wythoff pairs arranged in sequential order: :$\\tuple {0, 0}, \\tuple {1, 2}, \\tuple {3, 5}, \\tuple {4, 7}, \\tuple {6, 10}, \\tuple {8, 13}, \\ldots$ Apart from the first Wythoff pair $\\tuple {0, 0}$, every positive integer appears in this sequence exactly once."}
+{"_id": "13021", "title": "Wythoff Pairs are Winning Positions in Wythoff's Game", "text": "The Wythoff pairs are the winning positions in Wythoff's game."}
+{"_id": "13023", "title": "Zeta of 2 as Product of Fractions with Prime Numerators", "text": "{{begin-eqn}} {{eqn | l = \\map \\zeta 2       | r = \\prod_p \\paren {\\frac p {p - 1} } \\paren {\\frac p {p + 1} }       | c =  }} {{eqn | r = \\dfrac 2 1 \\times \\dfrac 2 3 \\times \\dfrac 3 2 \\times \\dfrac 3 4 \\times \\dfrac 5 4 \\times \\dfrac 5 6 \\times \\dfrac 7 6 \\times \\dfrac 7 8 \\times \\dfrac {11} {10} \\times \\dfrac {11} {12} \\times \\dfrac {13} {12} \\times \\dfrac {13} {14} \\times \\cdots       | c =  }} {{end-eqn}} where: : $\\zeta$ denotes the Riemann zeta function : $\\displaystyle \\prod_p$ denotes the product over all prime numbers."}
+{"_id": "13026", "title": "Snub Cube Inscribed in Octahedron", "text": "A snub cube inscribed in an octahedron involves the Tribonacci constant. {{explain|a bit more detail could help here}}"}
+{"_id": "13027", "title": "Snub Cube Inscribed in Cube", "text": "A snub cube inscribed in a cube involves the Tribonacci constant. {{explain|a bit more detail could help here}}"}
+{"_id": "13028", "title": "Icosahedron Inscribed in Octahedron", "text": "An icosahedron inscribed in an octahedron involves the Golden Ratio. {{stub|a bit more detail could help here}}"}
+{"_id": "13029", "title": "Top in Filter", "text": "Let $\\left({S, \\preceq}\\right)$ be a bounded above ordered set. Let $F$ be a filter on $S$. Then $\\top \\in F$ where $\\top$ denotes the greatest element of $S$."}
+{"_id": "13030", "title": "Finite Infima Set and Upper Closure is Filter", "text": "Let $P = \\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Let $X$ be a non-empty subset of $S$. Then :${\\operatorname{fininfs}\\left({X}\\right)}^\\succeq$ is filter in $P$. where :$\\operatorname{fininfs}\\left({X}\\right)$ denotes the finite infima set of $X$, :$X^\\succeq$ denotes the upper closure of $X$."}
+{"_id": "13031", "title": "Brun's Theorem", "text": "The sum of the reciprocals of all the twin primes converges."}
+{"_id": "13032", "title": "Divisibility by 2", "text": "An integer $N$ expressed in decimal notation is divisible by $2$ {{iff}} the least significant digit of $N$ is divisible by $2$. That is: :$N = [a_n \\ldots a_2 a_1 a_0]_{10} = a_0 + a_1 10 + a_2 10^2 + \\cdots + a_n 10^n$ is divisible by $2$  {{iff}}: :$a_0$ is divisible by $2$."}
+{"_id": "13033", "title": "Divisibility by Power of 2", "text": "Let $r \\in \\Z_{\\ge 1}$ be a strictly positive integer. An integer $N$ expressed in decimal notation is divisible by $2^r$ {{iff}} the last $r$ digits of $N$ form an integer divisible by $2^r$. That is: :$N = [a_n \\ldots a_2 a_1 a_0]_{10} = a_0 + a_1 10 + a_2 10^2 + \\cdots + a_n 10^n$ is divisible by $2^r$  {{iff}}: :$a_0 + a_1 10 + a_2 10^2 + \\cdots + a_r 10^r$ is divisible by $2^r$."}
+{"_id": "13035", "title": "Equivalence of Definitions of Deficient Number", "text": "The following definitions of a deficient number are equivalent:"}
+{"_id": "13037", "title": "Power of Prime is Deficient", "text": "Let $n \\in \\Z_{>0}$ be a power of a prime number $p$: :$n = p^k$ for some $k \\in \\Z_{>0}$. Then $n$ is deficient."}
+{"_id": "13039", "title": "Set is Subset of Finite Infima Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $X$ be a subset of $S$. Then $X \\subseteq \\operatorname{fininfs}\\left({X}\\right)$ where $\\operatorname{fininfs}\\left({X}\\right)$ denotes the finite infima set of $X$."}
+{"_id": "13040", "title": "Set is Subset of Upper Closure", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $X$ be a subset of $S$. Then $X \\subseteq X^\\succeq$ where $X^\\succeq$ denotes the upper closure of $X$."}
+{"_id": "13041", "title": "Tamref's Last Theorem", "text": "The Diophantine equation: :$n^x + n^y = n^z$ has exactly one form of solutions in integers: :$2^x + 2^x = 2^{x + 1}$ for all $x \\in \\Z$."}
+{"_id": "13042", "title": "3^x + 4^y equals 5^z has Unique Solution", "text": "The Diophantine equation: :$3^x + 4^y = 5^z$ has exactly one solution in integers: :$3^2 + 4^2 = 5^2$"}
+{"_id": "13043", "title": "5^x + 12^y equals 13^z has Unique Solution", "text": "The Diophantine equation: :$5^x + 12^y = 13^z$ has exactly one solution in integers: :$5^2 + 12^2 = 13^2$"}
+{"_id": "13044", "title": "Upper Closure of Subset is Subset of Upper Closure", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $X, Y$ be subsets of $S$. Then :$X \\subseteq Y \\implies X^\\succeq \\subseteq Y^\\succeq$ where $X^\\succeq$ denotes the upper closure of $X$."}
+{"_id": "13045", "title": "Euler's Number as Limit of n over nth Root of n Factorial", "text": ":$\\displaystyle e = \\lim_{n \\mathop \\to \\infty} \\dfrac n {\\sqrt [n] {n!} }$ where: :$e$ denotes Euler's number :$n!$ denotes $n$ factorial."}
+{"_id": "13046", "title": "Continued Fraction Expansion of Euler's Number", "text": "The constant Euler's number $e$ has the continued fraction expansion: {{begin-eqn}} {{eqn | l = e       | r = \\sqbrk {2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \\ldots }       | c = }} {{eqn | l =        | r = \\sqbrk {1, 0, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \\ldots }       | c = }} {{end-eqn}}"}
+{"_id": "13047", "title": "Continued Fraction Expansion of Euler's Number/Convergents", "text": "The convergents of the continued fraction expansion to Euler's number $e$ are: :$2, 3, \\dfrac 8 3, \\dfrac {11} 4, \\dfrac {19} 7, \\dfrac {87} {32}, \\dfrac {106} {39}, \\dfrac {193} {71}, \\dfrac {1264} {465}, \\dfrac {1457} {536}, \\dfrac {2721} {1001}, \\ldots$ {{OEIS-Numerators|A007676}} {{OEIS-Denominators|A007677}} These best rational approximations are accurate to $0, 0, 1, 1, 2, 3, 3, 4, 5, 5, \\ldots$ decimals. {{OEIS|A114539}} The fraction $\\dfrac {878} {323}$ is exceptionally easy to remember: :$\\dfrac {878} {323} = 2 \\cdotp 71826 \\, 625 \\ldots$ although this does not occur in the above continued fraction expansion."}
+{"_id": "13048", "title": "Number of Random Fractional Reals whose Total Exceeds 1", "text": "Let real numbers be selected at random following a continuous uniform distribution from the interval $\\closedint 0 1$ until their total sum is greater than $1$. The expectation of the number of selections is Euler's number $e$."}
+{"_id": "13049", "title": "Sine of X over X as Infinite Product", "text": "Let $z$ be a non-zero Complex Number. Then: :$\\displaystyle \\frac {\\sin z} z = \\cos \\frac z 2 \\cos \\frac z 4 \\cos \\frac z 8 \\cdots = \\prod_{i \\mathop = 1}^{\\infty} \\cos \\frac z {2^i}$ where $\\sin$ denotes the sine function and $\\cos$ denotes the cosine function."}
+{"_id": "13051", "title": "Maximum of Three Mutually Perpendicular Lines in Ordinary Space", "text": "In ordinary space, there can be no more than $3$ straight lines which are pairwise perpendicular. Thus, in a configuration of $4$ straight lines in space, at least one pair will not be perpendicular to each other."}
+{"_id": "13052", "title": "Complex Logarithm of 1", "text": ":$\\ln 1 = \\set {2 k \\pi i: k \\in \\Z}$"}
+{"_id": "13053", "title": "Sum Rule for Complex Derivatives", "text": "Let $\\map f z, \\map j z, \\map k z$ be single-valued continuous complex functions in a domain $D \\subseteq \\C$, where $D$ is open. Let $f$, $j$, and $k$ be complex-differentiable at all points in $D$. Let $\\map f z = \\map j z + \\map k z$. Then: : $\\forall z \\in D: \\map {f'} z = \\map {j'} z + \\map {k'} z$"}
+{"_id": "13054", "title": "Product Rule for Complex Derivatives", "text": "Let $\\map f z, \\map j z, \\map k z$ be single-valued continuous complex functions in a domain $D \\subseteq \\C$, where $D$ is open. Let $f$, $j$, and $k$ be complex-differentiable at all points in $D$. Let $\\map f z = \\map j z \\, \\map k z$. Then: :$\\forall z \\in D: \\map {f'} z = \\map j z \\, \\map {k'} z + \\map {j'} z \\, \\map k z$"}
+{"_id": "13055", "title": "Derivative of Constant/Complex", "text": "Let $f_c \\left({z}\\right)$ be the constant function on an open domain $D \\in \\C$, where $c \\in \\C$. Then: :$\\forall z \\in D : f_c' \\left({z}\\right) = 0$"}
+{"_id": "13056", "title": "Derivative of Identity Function/Real", "text": "Let $I_\\R: \\R \\to \\R$ be the identity mapping on the real numbers $\\R$. Then: :$\\map {I_\\R'} x = 1$"}
+{"_id": "13057", "title": "Derivative of Identity Function/Complex", "text": "Let $I_\\C: \\C \\to \\C$ be the identity function. Then: :$\\map {I_\\C'} z = 1$"}
+{"_id": "13058", "title": "Chain Rule for Real-Valued Functions/Corollary", "text": "Let $\\Psi$ represent a differentiable function of $x$ and $y$. Let $y$ represent a differentiable function of $x$. Then: {{begin-eqn}} {{eqn | l = \\frac {\\d \\Psi} {\\d x}       | r = \\frac {\\partial \\Psi} {\\partial x} + \\frac {\\partial \\Psi} {\\partial y} \\frac {\\d y} {\\d x} }} {{end-eqn}}"}
+{"_id": "13059", "title": "Trisecting the Angle/Hyperbola", "text": "Let $\\alpha$ be an angle which is to be trisected. This can be achieved by means of a hyperbola."}
+{"_id": "13062", "title": "Trisecting the Angle/Quadratrix of Hippias", "text": "Let $\\alpha$ be an angle which is to be trisected. This can be achieved by means of a quadratrix of Hippias."}
+{"_id": "13063", "title": "Trisecting the Angle/Cissoid of Diocles", "text": "Let $\\alpha$ be an angle which is to be trisected. This can be achieved by means of a cissoid of Diocles."}
+{"_id": "13064", "title": "Trisecting the Angle/Conchoid of Nicomedes", "text": "Let $\\alpha$ be an angle which is to be trisected. This can be achieved by means of a conchoid of Nicomedes."}
+{"_id": "13065", "title": "Laplace Transform of Periodic Function", "text": "Let $f$ be periodic, that is: :$\\exists T \\in \\R_{\\ne 0}: \\forall x \\in \\R: \\map f x = \\map f {x + T}$ Then: :$\\laptrans {\\map f t} = \\dfrac 1 {1 - e^{-s T} } \\displaystyle \\int_0^T e^{-s t} \\map f t \\rd t$ where $\\laptrans {\\map f t}$ denotes the Laplace transform."}
+{"_id": "13068", "title": "Triangle is Medial Triangle of Larger Triangle", "text": "Let $\\triangle ABC$ be a triangle. $\\triangle ABC$ is the medial triangle of a larger triangle."}
+{"_id": "13069", "title": "Altitudes of Triangle Meet at Point", "text": "Let $\\triangle ABC$ be a triangle. The altitudes of $\\triangle ABC$ all intersect at the same point."}
+{"_id": "13070", "title": "Perpendicular Bisectors of Triangle Meet at Point", "text": "Let $\\triangle ABC$ be a triangle. The perpendicular bisectors of $AB$, $BC$ and $AC$ all intersect at the same point."}
+{"_id": "13071", "title": "Centroid of Triangle is Centroid of Medial", "text": "Let $\\triangle ABC$ be a triangle. Let $\\triangle DEF$ be the medial triangle of $\\triangle ABC$. Let $G$ be the centroid of $\\triangle ABC$. Then $G$ is also the centroid of $\\triangle DEF$."}
+{"_id": "13072", "title": "Diameters of Parallelogram Bisect each other", "text": "Let $\\Box ABCD$ be a parallelogram with diameters $AC$ and $BD$. Let $AC$ and $BD$ intersect at $E$. Then $E$ is the midpoint of both $AC$ and $BD$."}
+{"_id": "13074", "title": "Position of Centroid on Euler Line", "text": "Let $\\triangle ABC$ be a triangle which is not equilateral. Let $O$ be the circumcenter of $\\triangle ABC$. Let $G$ be the centroid of $\\triangle ABC$. Let $H$ be the orthocenter of $\\triangle ABC$. Then $G$ lies on the straight line connecting $O$ and $H$ such that: :$OG : GH = 1 : 2$ The line $OGH$ is the '''Euler line''' of $\\triangle ABC$."}
+{"_id": "13075", "title": "Perpendicular Bisector of Triangle is Altitude of Medial Triangle", "text": "Let $\\triangle ABC$ be a triangle. Let $\\triangle DEF$ be the medial triangle of $\\triangle ABC$. Let a perpendicular bisector be constructed on $AC$ at $F$ to intersect $DE$ at $P$. Then $FP$ is an altitude of $\\triangle DEF$."}
+{"_id": "13076", "title": "Orthocenter, Centroid and Circumcenter Coincide iff Triangle is Equilateral", "text": "Let $\\triangle ABC$ be a triangle. Let $O$ be the circumcenter of $\\triangle ABC$. Let $G$ be the centroid of $\\triangle ABC$. Let $H$ be the orthocenter of $\\triangle ABC$. Then $O$, $G$ and $H$ are the same points {{iff}} $\\triangle ABC$ is equilateral. If $\\triangle ABC$ is not equilateral, then $O$, $G$ and $H$ are all distinct."}
+{"_id": "13079", "title": "Altitude, Median and Perpendicular Bisector Coincide iff Triangle is Isosceles", "text": "Let $\\triangle ABC$ be a triangle. Then: ::the altitude from $AB$ to $C$ ::the median from $AB$ to $C$ ::the perpendicular bisector of $AB$ :are all the same straight line {{iff}}: :$\\triangle ABC$ is isosceles where $AB$ is the base."}
+{"_id": "13080", "title": "Three Points Describe a Circle", "text": "Let $A$, $B$ and $C$ be points which are not collinear. Then there exists exactly one circle whose circumference passes through all $3$ points $A$, $B$ and $C$."}
+{"_id": "13081", "title": "Three Regular Tessellations", "text": "There exist exactly $3$ regular tessellations of the plane."}
+{"_id": "13082", "title": "Integer is Sum of Three Triangular Numbers", "text": "Let $n$ be a positive integer. Then $n$ is the sum of $3$ triangular numbers."}
+{"_id": "13083", "title": "Fermat's Last Theorem/Cubic", "text": "The Diophantine equation $a^3 + b^3 = c^3$ has no solutions in strictly positive integers."}
+{"_id": "13084", "title": "Smallest Magic Square is of Order 3", "text": "Apart from the trivial order $1$ magic square: {{:Magic Square/Examples/Order 1}} the smallest magic square is the order $3$ magic square: {{:Magic Square/Examples/Order 3}}"}
+{"_id": "13085", "title": "Euler Phi Function of n equal to Euler Phi Function of n+3", "text": "Let $\\phi$ denote the Euler $\\phi$ function. The only solutions to the equation: :$\\phi \\left({n}\\right) = \\phi \\left({n + 3}\\right)$ less than $1 \\, 000 \\, 000$ are: :$\\phi \\left({3}\\right) = \\phi \\left({6}\\right) = 2$ :$\\phi \\left({5}\\right) = \\phi \\left({8}\\right) = 4$"}
+{"_id": "13087", "title": "Friendship Theorem", "text": "Let there be a group of $6$ people. The traditional setting is that these $6$ people are at a party. Then (at least) one of the following $2$ statements is true: :$(1): \\quad$ At least $3$ of these $6$ people have all met each other before :$(2): \\quad$ At least $3$ of these $6$ people have never met each other before. That is, either there exists a set of $3$ mutual acquaintances, or there exists a set of $3$ mutual strangers."}
+{"_id": "13088", "title": "Volume of Smallest Tetrahedron with Integer Edges and Integer Volume", "text": "The volume of the smallest tetrahedron with integer edges and integer volume is $3$. There are $2$ possible sets of edges: :$32, 33, 35, 40, 70, 76$ :$21, 32, 47, 56, 58, 76$"}
+{"_id": "13091", "title": "Barbier's Theorem", "text": "Let $K$ be a (closed) curve of constant diameter. {{explain|\"constant diameter\"}} Let the circumference of $K$ be $c$. Let the diameter of $K$ be $d$. Then: :$\\dfrac c d = \\pi$"}
+{"_id": "13093", "title": "Machin's Formula for Pi", "text": ":$\\dfrac \\pi 4 = 4 \\arctan \\dfrac 1 5 - \\arctan \\dfrac 1 {239} \\approx 0 \\cdotp 78539 \\, 81633 \\, 9744 \\ldots$"}
+{"_id": "13094", "title": "Ordinals under Addition form Ordered Semigroup", "text": "$\\left({\\operatorname{On}, +, \\le}\\right)$ forms an ordered semigroup, where: : $\\operatorname{On}$ denotes the ordinal class, and : $+$ denotes ordinal addition."}
+{"_id": "13095", "title": "Subset is Left Compatible with Ordinal Addition", "text": "Let $x, y, z$ be ordinals. Then: :$x \\le y \\implies \\paren {z + x} \\le \\paren {z + y}$"}
+{"_id": "13096", "title": "Subset is Compatible with Ordinal Addition", "text": "Let $x, y, z$ be ordinals. Then: : $(1): x \\le y \\implies \\left({z + x}\\right) \\le \\left({z + y}\\right)$ : $(2): x \\le y \\implies \\left({x + z}\\right) \\le \\left({y + z}\\right)$"}
+{"_id": "13097", "title": "Ordinals under Addition form Semigroup", "text": "$\\left({\\operatorname{On}, +}\\right)$ forms an semigroup, where: : $\\operatorname{On}$ denotes the ordinal class, and : $+$ denotes ordinal addition."}
+{"_id": "13100", "title": "Ordinals under Multiplication form Semigroup", "text": "$\\struct {\\On, \\times}$ forms an semigroup, where: : $\\On$ denotes the ordinal class, and : $\\times$ denotes ordinal multiplication."}
+{"_id": "13101", "title": "Subset is Left Compatible with Ordinal Multiplication", "text": "Let $x, y, z$ be ordinals. Then: :$x \\le y \\implies \\left({z \\cdot x}\\right) \\le \\left({z \\cdot y}\\right)$"}
+{"_id": "13102", "title": "Subset is Compatible with Ordinal Multiplication", "text": "Let $x, y, z$ be ordinals. Then: : $(1): x \\le y \\implies \\left({z \\cdot x}\\right) \\le \\left({z \\cdot y}\\right)$ : $(2): x \\le y \\implies \\left({x \\cdot z}\\right) \\le \\left({y \\cdot z}\\right)$"}
+{"_id": "13103", "title": "Ordinals under Multiplication form Ordered Semigroup", "text": "$\\struct {\\On, \\times, \\le}$ forms an ordered semigroup, where: : $\\On$ denotes the ordinal class, and : $\\times$ denotes ordinal multiplication."}
+{"_id": "13104", "title": "Ordinals under Addition form Monoid", "text": "$\\struct {\\On, +}$ forms an monoid, where: :$\\On$ denotes the ordinal class :$+$ denotes ordinal addition."}
+{"_id": "13105", "title": "Ordinals under Addition form Ordered Monoid", "text": "$\\left({\\operatorname{On}, +, \\le}\\right)$ forms an ordered monoid, where: : $\\operatorname{On}$ denotes the ordinal class, and : $+$ denotes ordinal addition."}
+{"_id": "13106", "title": "Ordinals under Multiplication form Monoid", "text": "$\\left({\\operatorname{On}, \\cdot}\\right)$ forms an monoid, where: : $\\operatorname{On}$ denotes the ordinal class, and : $\\cdot$ denotes ordinal multiplication."}
+{"_id": "13107", "title": "Ordinals under Multiplication form Ordered Monoid", "text": "$\\struct {\\On, \\cdot, \\le}$ forms an ordered monoid, where: :$\\On$ denotes the ordinal class, and :$\\cdot$ denotes ordinal multiplication."}
+{"_id": "13108", "title": "Buffon's Needle", "text": "Let a horizontal plane be divided into strips by a series of parallel lines a fixed distance apart, like floorboards. Let a needle whose length equals the distance between the parallel lines be dropped onto the plane randomly from a random height. Then the probability that the needle falls across one of the parallel lines is $\\dfrac 2 \\pi$."}
+{"_id": "13109", "title": "Pi as Sum of Odd Reciprocals Alternating in Sign in Pairs", "text": ":$\\dfrac {\\pi \\sqrt 2} 4 = 1 + \\dfrac 1 3 - \\dfrac 1 5 - \\dfrac 1 7 + \\dfrac 1 9 + \\dfrac 1 {11} - \\dfrac 1 {13} - \\dfrac 1 {15} \\cdots$"}
+{"_id": "13111", "title": "Sum of Reciprocals of Squares of Odd Integers", "text": "{{begin-eqn}} {{eqn | l = \\sum_{n \\mathop = 1}^\\infty \\frac 1 {\\paren {2 n - 1}^2}       | r = 1 + \\dfrac 1 {3^2} + \\dfrac 1 {5^2} + \\dfrac 1 {7^2} + \\dfrac 1 {9^2} + \\cdots }} {{eqn | r = \\dfrac {\\pi^2} 8 }} {{end-eqn}}"}
+{"_id": "13112", "title": "Basel Problem as Infinite Product", "text": ":$\\displaystyle \\dfrac {\\pi^2} 6 = \\prod_{p \\mathop \\in \\mathbb P} \\dfrac {p^2} {p^2 - 1}$"}
+{"_id": "13113", "title": "Convergence of Taylor Series of Function Analytic on Disk", "text": "Let $F$ be a complex function. Let $x_0$ be a point in $\\R$. Let $R$ be an extended real number greater than zero. Let $F$ be analytic at every point $z \\in \\C$ satisfying $\\cmod {z - x_0} < R$. Let $f = F {\\restriction_\\R}$ be a real function. Then: :the Taylor series of $f$ about $x_0$ converges to $f$ at every point $x \\in \\R$ satisfying $\\size {x - x_0} < R$"}
+{"_id": "13114", "title": "Taylor Series of Analytic Function has infinite Radius of Convergence", "text": "Let $F$ be a complex function. Let $F$ be analytic everywhere. Let $f = F {\\restriction_{\\R}}$ be a real function. Let $x_0$ be a point in $\\R$. Then the Taylor series of $f$ about $x_0$ converges to $f$ at every point in $\\R$."}
+{"_id": "13116", "title": "Number of Binary Digits in Power of 10", "text": "Let $n$ be a positive integer. Expressed in binary notation, the number of digits in the $n$th power of $10$: :$10^n$ is equal to: :$\\ceiling {n \\log_2 10}$ where $\\ceiling x$ denotes the ceiling of $x$."}
+{"_id": "13120", "title": "Regular Octahedron is Dual of Cube", "text": "The regular octahedron is the dual of the cube."}
+{"_id": "13121", "title": "Plane Figure with Bilateral Symmetry about Two Lines has 4 Congruent Parts", "text": "Let $F$ be a plane figure. Let $F$ have two different axes of bilateral symmetry. Then those two axes divide $F$ into $4$ congruent parts. :500px"}
+{"_id": "13122", "title": "Hyperbola can be Drawn through Four Non-Collinear Points", "text": "Let $A, B, C, D$ be points in the plane of which no $3$ are collinear. Then a hyperbola can be drawn so that it passes through all points $A, B, C, D$."}
+{"_id": "13123", "title": "Classification of Finite Simple Groups", "text": "The finite simple groups can be classified as follows:"}
+{"_id": "13124", "title": "Divisibility of n-1 Factorial by Composite n", "text": "Let $n \\in \\Z$ be composite. Then: :$n \\divides \\paren {n - 1}! \\iff n \\ne 4$ where: :$\\divides$ denotes divisibility :$n!$ denotes the factorial of $n$."}
+{"_id": "13125", "title": "Divisibility by 4", "text": "An integer $N$ expressed in decimal notation is divisible by $4$ {{iff}} the $2$ least significant digits of $N$ form a $2$-digit integer divisible by $4$. That is: :$N = \\sqbrk {a_n \\ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \\cdots + a_n 10^n$ is divisible by $4$  {{iff}}: :$10 a_1 + a_0$ is divisible by $4$."}
+{"_id": "13126", "title": "Ratio of Number to Reversal which is Multiple", "text": "Take a (strictly) positive integer $n$, written in conventional decimal notation. Let $m$ be the reversal of $n$. Let $m = k n$ where $k$ is an integer. Then $k$ is either $4$ or $9$."}
+{"_id": "13127", "title": "Smallest Pythagorean Triangle is 3-4-5", "text": "The smallest Pythagorean triangle has sides of length $3$, $4$ and $5$. :300px"}
+{"_id": "13128", "title": "Pythagorean Triangle with Sides in Arithmetic Sequence", "text": "The $3-4-5$ triangle is the only Pythagorean triangle such that: :the lengths of whose sides are in arithmetic sequence and: :the lengths of whose sides form a primitive Pythagorean triple."}
+{"_id": "13129", "title": "Injectivity of Laplace Transform", "text": "Let $f$, $g$ be functions from $\\left [{0 \\,.\\,.\\, \\to} \\right ) \\to \\mathbb F$ of a real variable $t$, where $\\mathbb F \\in \\left\\{ {\\R, \\C}\\right\\}$. Further let $f$ and $g$ be continuous everywhere on their domains. Let $f$ and $g$ both admit Laplace transforms. Suppose that the Laplace transforms $\\mathcal L \\left\\{{f}\\right\\}$ and $\\mathcal L \\left\\{{g}\\right\\}$ satisfy: :$\\forall t \\ge 0: \\mathcal L \\left\\{{f\\left({t}\\right)}\\right\\} = \\mathcal L \\left\\{{g\\left({t}\\right)}\\right\\}$ Then $f = g$ everywhere on $\\left[{0 \\,.\\,.\\, \\to}\\right)$."}
+{"_id": "13130", "title": "Pythagorean Triangle whose Area is Half Perimeter", "text": "The $3-4-5$ triangle is the only Pythagorean triangle whose area is half its perimeter."}
+{"_id": "13132", "title": "Square Modulo 5/Corollary", "text": "When written in conventional base $10$ notation, no square number ends in one of $2, 3, 7, 8$."}
+{"_id": "13133", "title": "Prime equals Plus or Minus One modulo 6", "text": "Let $p$ be a prime number greater than $3$. Then $p$ is either of the form: :$p = 6 n + 1$ or: :$p = 6 n - 1$ That is: :$p = \\pm 1 \\pmod 6$"}
+{"_id": "13134", "title": "Conic Section through Five Points", "text": "Let $A, B, C, D, E$ be distinct points in the plane such that no $3$ of them are collinear. Then it is possible to draw a conic section that passes through all $5$ points."}
+{"_id": "13136", "title": "Lamé's Theorem", "text": "Let $a, b \\in \\Z_{>0}$ be (strictly) positive integers. Let $c$ and $d$ be the number of digits in $a$ and $b$ respectively when expressed in decimal notation. Let the Euclidean Algorithm be employed to find the GCD of $a$ and $b$. Then it will take no more than $5 \\times \\min \\set {c, d}$ cycles around the Euclidean Algorithm to find $\\gcd \\set {a, b}$."}
+{"_id": "13137", "title": "Volume of Unit Hypersphere", "text": "The volume of the unit sphere in $n$-dimensional space increases as $n$ goes up to $5$, but decreases thereafter."}
+{"_id": "13138", "title": "Finite Subset Bounds Element of Finite Infima Set and Upper Closure", "text": "Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be meet semilattice. Let $F$ be filter in $L$. Let $X$ be non empty finite subset of $S$. Let $x \\in S$ such that : $x \\in \\left({\\operatorname{fininfs}\\left({F \\cup X}\\right)}\\right)^\\succeq$ where :$\\operatorname{fininfs}$ denotes the finite infima set, :$X^\\succeq$ denotes the upper closure of $X$. Then there exists $a \\in S$: $a \\in F \\land x \\succeq a \\wedge \\inf X$"}
+{"_id": "13139", "title": "Length of Reciprocal of Product of Powers of 2 and 5", "text": "Let $n \\in \\Z$ be an integer. Let $\\dfrac 1 n$, when expressed as a decimal expansion, terminate after $m$ digits. Then $n$ is of the form $2^p 5^q$, where $m$ is the greater of $p$ and $q$."}
+{"_id": "13140", "title": "Structure of Recurring Decimal", "text": "Let $\\dfrac 1 m$, when expressed as a decimal expansion, recur with a period of $p$ digits. Let $\\dfrac 1 n$, when expressed as a decimal expansion, terminate after $q$ digits. Then $\\dfrac 1 {m n}$ has a nonperiodic part of $q$ digits, and a recurring part of $p$ digits."}
+{"_id": "13141", "title": "Characteristics of Pentatope", "text": "A pentatope has $5$ cells, $10$ faces, $10$ edges and $5$ vertices."}
+{"_id": "13142", "title": "Pentatope is Self-Dual", "text": "A pentatope is self-dual."}
+{"_id": "13143", "title": "Riemann Hypothesis implies Odd Number is Sum of at most 5 Primes", "text": "Let the truth of the Riemann Hypothesis be assumed. Let $n$ be an odd integer. Then $n$ is the sum of at most $5$ primes."}
+{"_id": "13144", "title": "Numbers of Zeroes that Factorial does not end with", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Let $n!$ denote the factorial of $n$. Let $n!$ be expressed in decimal notation. Then $n!$ cannot end in the following numbers of zeroes: :$5, 11, 17, 23, 29, 30, 36, 42, \\ldots$ {{OEIS|A000966}}"}
+{"_id": "13145", "title": "Euler-Binet Formula/Historical Note", "text": "The Euler-Binet Formula, derived by {{AuthorRef|Jacques Philippe Marie Binet|Binet}} in $1843$, was already known to {{AuthorRef|Leonhard Paul Euler|Euler}}, {{AuthorRef|Abraham de Moivre|de Moivre}} and {{AuthorRef|Daniel Bernoulli}} over a century earlier. However, it was {{AuthorRef|Jacques Philippe Marie Binet|Binet}} who derived the more general Binet Form of which this is an elementary application."}
+{"_id": "13147", "title": "Cosine is of Exponential Order Zero", "text": "Let $\\cos t$ be the cosine of $t$, where $t \\in \\R$. Then $\\cos t$ is of exponential order $0$."}
+{"_id": "13148", "title": "Sine is of Exponential Order Zero", "text": "Let $\\sin t$ be the sine of $t$, where $t \\in \\R$. Then $\\sin t$ is of exponential order $0$."}
+{"_id": "13149", "title": "Scalar Multiple of Function of Exponential Order", "text": "Let $f: \\R \\to \\F$ be a function, where $\\F \\in \\set {\\R, \\C}$. Let $\\lambda$ be a complex constant. Suppose $f$ is of exponential order $a$. Then $\\lambda f$ is also of exponential order $a$."}
+{"_id": "13150", "title": "Function of Exponential Order of Scalar Multiple", "text": "Let $f: \\R \\to \\F$ be a function, where $\\F \\in \\set {\\R, \\C}$. Let $\\lambda$ be a real constant. Let $\\map f t$ be of exponential order $a$. Then the function defined by $t \\mapsto \\map f {\\lambda t}$ is of exponential order $a\\lambda$."}
+{"_id": "13151", "title": "Identity is of Exponential Order Epsilon", "text": "Let $I_\\R: t \\mapsto t$ be the identity mapping on $\\R_{\\ge 0}$. Then $I_\\R$ is of exponential order $\\epsilon$ for any  $\\epsilon > 0$ arbitrarily small in magnitude."}
+{"_id": "13152", "title": "Product of Functions of Exponential Order", "text": "Let $f, g: \\R \\to \\F$ be functions, where $\\F \\in \\set {\\R, \\C}$. Let $f$ be of exponential order $a$ and $g$ be of exponential order $b$. Then $f g: t \\mapsto \\map f t \\, \\map g t$ is of exponential order $a+b$."}
+{"_id": "13153", "title": "Sum of Functions of Exponential Order", "text": "Let $f, g: \\R \\to \\F$ be functions, where $\\F \\in \\set {\\R, \\C}$. Suppose $f$ is of exponential order $a$ and $g$ is of exponential order $b$. Then $f + g: t \\mapsto \\map f t + \\map g t$ is of exponential order $\\max \\set {a, b}$."}
+{"_id": "13154", "title": "Linear Combination of Functions of Exponential Order", "text": "Let $f, g: \\R \\to \\F$ be functions, where $\\F \\in \\set {\\R, \\C}$. Let $\\lambda, \\mu$ be complex numbers. Suppose $f$ is of exponential order $a$ and $g$ is of exponential order $b$. Then $\\map {\\paren {\\lambda f + \\mu g} } t = \\lambda \\, \\map f t + \\mu \\, \\map g t$ is of exponential order $\\max \\set {a, b}$."}
+{"_id": "13155", "title": "Constant Function is of Exponential Order Zero", "text": "Let $f_C: \\R \\to \\mathbb F: t \\mapsto C$ be a constant function, where $\\mathbb F \\in \\set {\\R, \\C}$. Then $f_C$ is of exponential order $0$."}
+{"_id": "13156", "title": "Polynomial is of Exponential Order Epsilon", "text": "Let $P: \\R \\to \\mathbb F$ be a polynomial, where $\\mathbb F \\in \\set {\\R, \\C}$. Then $P$ is of exponential order $\\epsilon$ for any $\\epsilon > 0$ arbitrarily small in magnitude."}
+{"_id": "13157", "title": "Fibonacci Number by Power of 2", "text": "{{begin-eqn}} {{eqn | ll= \\forall n \\in \\Z_{\\ge 0}:       | l = 2^{n - 1} F_n       | r = \\sum_k 5^k \\dbinom n {2 k + 1}       | c =  }} {{eqn | r = \\dbinom n 1 + 5 \\dbinom n 3 + 5^2 \\dbinom n 5 + \\cdots       | c =  }} {{end-eqn}} where: :$F_n$ denotes the $n$th Fibonacci number :$\\dbinom n {2 k + 1} \\ $ denotes a binomial coefficient."}
+{"_id": "13159", "title": "Approximation to Golden Rectangle using Fibonacci Squares", "text": "An approximation to a golden rectangle can be obtained by placing adjacent to one another squares with side lengths corresponding to consecutive Fibonacci numbers in the following manner: :800px It can also be noted, as from Sequence of Golden Rectangles, that an equiangular spiral can be approximated by constructing quarter circles as indicated."}
+{"_id": "13160", "title": "Fibonacci Number of Index 2n as Sum of Squares of Fibonacci Numbers", "text": "Let $F_n$ denote the $n$th Fibonacci number. Then: :$F_{2 n} = {F_{n + 1} }^2 - {F_{n - 1} }^2$"}
+{"_id": "13163", "title": "Euler-Binet Formula/Also known as", "text": "The '''Euler-Binet Formula''' is also known as '''{{AuthorRef|Jacques Philippe Marie Binet|Binet}}'s formula'''."}
+{"_id": "13164", "title": "If Ideal and Filter are Disjoint then There Exists Prime Ideal Including Ideal and Disjoint from Filter", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a distributive lattice. Let $I$ be an ideal in $L$. Let $F$ be a filter on $L$ such that :$I \\cap F = \\O$ Then there exists a prime ideal $P$ in $L$: $I \\subseteq P$ and $P \\cap F = \\O$"}
+{"_id": "13165", "title": "No 4 Fibonacci Numbers can be in Arithmetic Sequence", "text": "Let $a, b, c, d$ be distinct Fibonacci numbers. Then, except for the trivial case: :$a = 0, b = 1, c = 2, d = 3$ it is not possible that $a, b, c, d$ are in arithmetic sequence."}
+{"_id": "13166", "title": "Sum of Alternating Sign Reciprocals of Sequence of Pairs of Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared", "text": "{{begin-eqn}} {{eqn | l = \\sum_{k \\mathop \\ge 2} \\left({-1}\\right)^k \\dfrac 1 {F_k F_{k + 1} }       | r = \\dfrac 1 {1 \\times 2} - \\dfrac 1 {2 \\times 3} + \\dfrac 1 {3 \\times 5} - \\dfrac 1 {5 \\times 8} + \\cdots       | c =  }} {{eqn | r = \\phi^{-2}       | c =  }} {{end-eqn}} where: :$F_k$ denotes the $k$th Fibonacci number :$\\phi$ denotes the golden mean."}
+{"_id": "13167", "title": "Sum of Reciprocals of Sequence of Pairs of Even Index Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared", "text": "{{begin-eqn}} {{eqn | l = \\sum_{k \\mathop \\ge 1} \\dfrac 1 {F_{2 k} F_{2 k + 2} }       | r = \\dfrac 1 {1 \\times 3} + \\dfrac 1 {3 \\times 8} + \\dfrac 1 {8 \\times 21} + \\dfrac 1 {21 \\times 55} + \\cdots       | c =  }} {{eqn | r = \\phi^{-2}       | c =  }} {{end-eqn}} where: :$F_k$ denotes the $k$th Fibonacci number :$\\phi$ denotes the golden mean."}
+{"_id": "13168", "title": "Number of Fibonacci Numbers between n and 2n", "text": "Let $n \\in \\Z_{> 0}$ be a (strictly) positive integer. Then there exists either one or two Fibonacci numbers between $n$ and $2 n$ inclusive."}
+{"_id": "13169", "title": "Square Root is of Exponential Order Epsilon", "text": "The positive square root function: :$t \\mapsto \\sqrt t$ is of exponential order $\\epsilon$ for any  $\\epsilon > 0$ arbitrarily small in magnitude."}
+{"_id": "13170", "title": "Number of Fibonacci Numbers with Same Number of Decimal Digits", "text": "Let $n$ be an integer such that $n > 1$. When expressed in decimal notation, there are either $4$ or $5$ Fibonacci numbers with $n$ digits."}
+{"_id": "13171", "title": "Prime Number divides Infinite Number of Fibonacci Numbers", "text": "Let $p$ be a prime number. Then there exist an infinite number of Fibonacci numbers which are divisible by $p$."}
+{"_id": "13172", "title": "Prime Number divides Fibonacci Number", "text": "For $n \\in \\Z$, let $F_n$ denote the $n$th Fibonacci number. Let $p$ be a prime number. Then: :$p \\equiv \\pm 1 \\pmod 5 \\implies p \\divides F_{p - 1}$ :$p \\equiv \\pm 2 \\pmod 5 \\implies p \\divides F_{p + 1}$ where $\\divides$ denotes divisibility. Thus in all cases, except where $p = 5$ itself: :$p \\divides F_{p \\pm 1}$"}
+{"_id": "13173", "title": "Existence of Fibonacci Number Divisible by Number", "text": "Let $m \\in \\Z$ be an integer. Then in the first $m^2$ Fibonacci numbers there exists at least one Fibonacci number which is divisible by $m$."}
+{"_id": "13174", "title": "Fibonacci Prime has Prime Index except for 3", "text": "Let $F_n$ denote the $n$th Fibonacci number. Let $F_n$ be a prime number. Then, apart from $F_4 = 3$, $n$ is a prime number."}
+{"_id": "13175", "title": "Fibonacci Number with Prime Index is not necessarily Prime", "text": "Let $p \\in \\Z_{>0}$ be a prime number. Let $F_p$ be the $p$th Fibonacci number. Then $F_p$ is not itself necessarily prime."}
+{"_id": "13176", "title": "Definition:Fibonacci Prime Pair", "text": "A '''Fibonacci prime pair''' is a pair of Fibonacci numbers $F_p$ and $F_{p + 2}$ such that: :$p$ and $p + 2$ are both prime numbers :$F_p$ and $F_{p + 2}$ are both prime numbers."}
+{"_id": "13177", "title": "Complex Riemann Integral is Contour Integral", "text": "Let $f: \\R \\to \\C$ be a complex Riemann integrable function over some closed real interval $\\left[{a \\,.\\,.\\, b}\\right]$. Then: :$\\displaystyle \\int_a^b f \\left({t}\\right) \\rd t = \\int_\\mathcal C f \\left({t}\\right) \\rd t$ where: : the integral on the {{LHS}} is a complex Riemann integral : the integral on the {{RHS}} is a contour integral : $\\mathcal C$ is a straight line segment along the real axis, connecting $a$ to $b$."}
+{"_id": "13178", "title": "Fibonacci Number is not Product of Two Smaller Fibonacci Numbers", "text": "Let $m, n \\in \\Z$ be integers. Suppose $\\size m, \\size n \\ge 3$. Let $F_m$ and $F_n$ be the $m$th and $n$th Fibonacci numbers. Then $F_m \\times F_n$ is not a Fibonacci number."}
+{"_id": "13179", "title": "Sequence of Fibonacci Numbers ending in Index", "text": "Let $F_k$ denote the $k$th Fibonacci number. For all $k \\in \\Z$, let $F_k$ be expressed in decimal notation. The sequence of integers $\\sequence n$ such that $F_n$ ends in $n$ starts: :$0, 1, 5, 25, 29, 41, 49, 61, 65, 85, 89, 101, 125, 145, 149, 245, 265, 365, 385, 485, 505, 601, \\ldots$ {{OEIS|A000350}}"}
+{"_id": "13180", "title": "Transformation of P-Norm", "text": "Let $p, q \\ge 1$ be real numbers. Let $\\ell^p$ denote the $p$-sequence space. Let $\\norm {\\mathbf x}_p$ denote the $p$-norm. Let $\\mathbf x = \\sequence {x_n} \\in \\ell^{p q}$. Suppose further that $\\mathbf x^p = \\sequence { {x_n}^p} \\in \\ell^q$. Then: :$\\norm {\\mathbf x^p}_q = \\norm {\\mathbf x}_{p q}^p$"}
+{"_id": "13181", "title": "Set is Subset of Finite Suprema Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $X$ be a subset of $S$. Then $X \\subseteq \\operatorname{finsups}\\left({X}\\right)$ where $\\operatorname{finsups}\\left({X}\\right)$ denotes finite suprema set of $X$."}
+{"_id": "13182", "title": "Lower Closure of Subset is Subset of Lower Closure", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $X, Y$ be subsets of $S$. Then :$X \\subseteq Y \\implies X^\\preceq \\subseteq Y^\\preceq$ where $X^\\preceq$ is the lower closure of $X$."}
+{"_id": "13183", "title": "Finite Suprema Set and Lower Closure is Smallest Ideal", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a join semilattice. Let $X$ be a subset of $S$. Then $X \\subseteq \\operatorname{finsups}\\left({X}\\right)^\\preceq$ and :for every ideal $I$ in $L$: $X \\subseteq I \\implies \\operatorname{finsups}\\left({X}\\right)^\\preceq \\subseteq I$ where :$\\operatorname{finsups}\\left({X}\\right)$ denotes the finite suprema set of $X$, :$X^\\preceq$ denotes the lower closure of $X$."}
+{"_id": "13184", "title": "Set is Subset of Lower Closure", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $X$ be a subset of $S$. Then $X \\subseteq X^\\preceq$ where $X^\\preceq$ denotes the lower closure of $X$."}
+{"_id": "13185", "title": "Algorithm to determine whether Polynomial Diophantine Equation has Integer Solution", "text": "There is no algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution."}
+{"_id": "13186", "title": "Maximum Volume of Unit Radius Sphere in Fractional Dimensions", "text": "The volume of a unit sphere in $x$-dimensional Euclidean space for real $x$ occurs when $x$ is given as: :$x = 5 \\cdotp 25694 \\, 64048 \\, 60 \\ldots$ {{OEIS|A074455}} The corresponding volume at that dimension is given by: :$V = 5 \\cdotp 27776 \\, 80211 \\, 13400 \\, 997 \\ldots$ {{OEIS|A074454}}"}
+{"_id": "13187", "title": "Perfect Number is Sum of Successive Odd Cubes except 6", "text": "Let $n$ be an even perfect number such that $n \\ne 6$. Then: :$\\displaystyle n = \\sum_{k \\mathop = 1}^m \\paren {2 k - 1}^3 = 1^3 + 3^3 + \\cdots + \\paren {2 m - 1}^3$ for some $m \\in \\Z_{>0}$. That is, every even perfect number is the sum of the sequence of the first $r$ odd cubes, for some $r$."}
+{"_id": "13189", "title": "Sum of Sequence of Odd Cubes", "text": ":$\\displaystyle \\sum_{j \\mathop = 1}^n \\paren {2 j - 1}^3 = 1^3 + 3^3 + 5^3 + \\dotsb + \\paren {2 n − 1}^3 = n^2 \\paren {2 n^2 − 1}$"}
+{"_id": "13190", "title": "Bottom in Ideal", "text": "Let $\\left({S, \\preceq}\\right)$ be a bounded below ordered set. Let $I$ be a ideal in $S$. Then $\\bot \\in I$ where $\\bot$ denotes the smallest element of $S$."}
+{"_id": "13191", "title": "Real Power is of Exponential Order Epsilon", "text": "Let:  :$f: \\hointr 0 \\to \\to \\R: t \\mapsto t^r$ be $t$ to the power of $r$, for $r \\in \\R, r > -1$. Then $f$ is of exponential order $\\epsilon$ for any  $\\epsilon > 0$ arbitrarily small in magnitude."}
+{"_id": "13192", "title": "Finite Suprema Set and Lower Closure is Ideal", "text": "Let $P = \\left({S, \\vee, \\preceq}\\right)$ be a join semilattice. Let $X$ be a non-empty subset of $S$. Then :${\\operatorname{finsups}\\left({X}\\right)}^\\preceq$ is ideal in $P$. where :$\\operatorname{finsups}\\left({X}\\right)$ denotes the finite suprema set of $X$, :$X^\\preceq$ denotes the lower closure of $X$."}
+{"_id": "13195", "title": "Only Number which is Sum of 3 Factors is 6", "text": "The only positive integer which is the sum of exactly $3$ of its distinct coprime divisors is $6$."}
+{"_id": "13196", "title": "Consecutive Integers whose Sums of Squares of Divisors are Equal", "text": "The only two consecutive positive integers whose sums of the squares of their divisors are equal are $6$ and $7$."}
+{"_id": "13197", "title": "Local Maxima of Number of Goldbach Decompositions", "text": "Let $\\mathbb E$ be the set of even positive integers. Let $G: \\mathbb E \\to \\N$ be the mapping defined as: :$\\forall n \\in \\mathbb E: \\map G n =$ the number of Goldbach decompositions of $n$. Then $G$ has local maxima when $n$ is a multiple of $6$."}
+{"_id": "13198", "title": "Digital Root of 3 Consecutive Numbers ending in Multiple of 3", "text": "Let $n$, $n + 1$ and $n + 2$ be positive integers such that $n + 2$ is a multiple of $3$. Let $m = n + \\paren {n + 1} + \\paren {n + 2}$. Then the digital root of $m$ is $6$."}
+{"_id": "13199", "title": "Six Regular 4-Dimensional Polytopes", "text": "There exist exactly six $4$-dimensional regular polytopes: :the pentatope (also known as the $4$-simplex) :the tesseract :the $16$-cell :the $24$-cell :the $120$-cell :the $600$-cell"}
+{"_id": "13200", "title": "Characteristics of Regular 4-Dimensional Polytopes", "text": "The $4$-dimensional regular polytopes have the following characteristics: {| border=\"1\" |- ! Name ! No. of cells ! No. of faces ! No. of edges ! No. of vertices ! Dual |- | align=\"right\" | Pentatope | align=\"right\" | $5$ | align=\"right\" | $10$ | align=\"right\" | $10$ | align=\"right\" | $5$ | align=\"right\" | Self-dual |- | align=\"right\" | Tesseract | align=\"right\" | $8$ | align=\"right\" | $24$ | align=\"right\" | $32$ | align=\"right\" | $16$ | align=\"right\" | $16$-cell |- | align=\"right\" | $16$-cell | align=\"right\" | $16$ | align=\"right\" | $32$ | align=\"right\" | $24$ | align=\"right\" | $8$ | align=\"right\" | Tesseract |- | align=\"right\" | $24$-cell | align=\"right\" | $24$ | align=\"right\" | $96$ | align=\"right\" | $96$ | align=\"right\" | $24$ | align=\"right\" | Self-dual |- | align=\"right\" | $120$-cell | align=\"right\" | $120$ | align=\"right\" | $720$ | align=\"right\" | $1200$ | align=\"right\" | $600$ | align=\"right\" | $600$-cell |- | align=\"right\" | $600$-cell | align=\"right\" | $600$ | align=\"right\" | $1200$ | align=\"right\" | $720$ | align=\"right\" | $120$ | align=\"right\" | $120$-cell |}"}
+{"_id": "13201", "title": "Six Equal Circles Tangent to Equal Circle", "text": "It is possible to arrange $6$ equal circles around another circles equal in area to each of the $6$: :500px"}
+{"_id": "13202", "title": "Complex Power is of Exponential Order Epsilon", "text": "Let:  :$f: \\hointr 0 \\to \\to \\C: t \\mapsto t^\\phi$ be $t$ to the power of $\\phi$, for $\\phi \\in \\C$, defined on its principal branch. Let $\\map \\Re \\phi > -1$. Then $f$ is of exponential order $\\epsilon$ for any  $\\epsilon > 0$ arbitrarily small in magnitude."}
+{"_id": "13203", "title": "Three Regular Tessellations/Hexagons", "text": "Regular hexagons form a regular tessellation: :400px"}
+{"_id": "13204", "title": "Limiting Area of Polygon with given Perimeter", "text": "Let $\\mathcal P$ be the set of plane geometric figures with perimeter $L$. The element of $P$ with the largest area is the circle of radius $\\dfrac L {2 \\pi}$ which has area $\\dfrac {L^2} {4 \\pi}$."}
+{"_id": "13205", "title": "Brianchon's Theorem", "text": "Let tangents to $6$ points on a conic section $K$ form a hexagon $H$ to circumscribe the $K$. Then the main diagonals of $H$ meet at a single point. :400px"}
+{"_id": "13206", "title": "Brianchon's Theorem is Projective Dual to Pascal's Theorem", "text": "Brianchon's Theorem is the projective dual of Pascal's Theorem."}
+{"_id": "13207", "title": "Steiner Tree of Unit Cube", "text": "The length of the Steiner tree for the vertices of the unit cube is given approximately by: :$L = 6 \\cdotp 196 \\ldots$"}
+{"_id": "13208", "title": "Integer as Sum of Three Squares/Sequence", "text": "The sequence of positive integers that cannot be expressed as the sum of at most $3$ squares begins: :$7, 15, 23, 28, 31, 39, 47, 55, 60, \\ldots$"}
+{"_id": "13209", "title": "Finite Subset Bounds Element of Finite Suprema Set and Lower Closure", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be join semilattice. Let $I$ be ideal in $L$. Let $X$ be non empty finite subset of $S$. Let $x \\in S$ such that : $x \\in \\left({\\operatorname{finsups}\\left({F \\cup X}\\right)}\\right)^\\preceq$ where :$\\operatorname{finsups}$ denotes the finite suprema set, :$X^\\preceq$ denotes the lower closure of $X$. Then there exists $a \\in S$: $a \\in I \\land x \\preceq a \\vee \\sup X$"}
+{"_id": "13211", "title": "Divisibility of Elements of Pythagorean Triple by 7", "text": "Let $\\tuple {a, b, c}$ be a Pythagorean triple such that $a^2 + b^2 = c^2$. Then at least one of $a$, $b$, $a + b$ or $a - b$ is divisible by $7$."}
+{"_id": "13212", "title": "Smaller Elements of Pythagorean Triple not both Odd", "text": "Let $\\left({x, y, z}\\right)$ be a Pythagorean triple, i.e. integers such that $x^2 + y^2 = z^2$. Then $x$ and $y$ cannot both be odd."}
+{"_id": "13213", "title": "Integer as Sum of Seven Positive Cubes", "text": "Every sufficiently large integer can be expressed as the sum of no more than $7$ positive cubes."}
+{"_id": "13214", "title": "Maximum of Seven Colors Needed for Proper Vertex Coloring on Torus", "text": "Let $G$ be a graph embedded on the surface of a torus. $G$ can be assigned a proper vertex $k$-coloring such that $k \\le 7$."}
+{"_id": "13215", "title": "Every Element is Directed and Every Two Elements are Included in Third Element implies Union is Directed", "text": "Let $P = \\left({S, \\preceq}\\right)$ be an ordered set. Let $A$ be a set of subsets of $S$. Let :$\\forall X \\in A: X$ is directed. Let :$\\forall X, Y \\in A: \\exists Z \\in A: X \\cup Y \\subseteq Z$ Then $\\bigcup A$ is directed."}
+{"_id": "13216", "title": "Every Element is Lower implies Union is Lower", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $A$ be a set of subsets of $S$. Let :$\\forall X \\in A: X$ is a lower set. Then $\\bigcup A$ is a lower set."}
+{"_id": "13217", "title": "Rectangle Divided into Incomparable Subrectangles", "text": "Let $R$ be a rectangle. Let $R$ be divided into $n$ smaller rectangles which are pairwise incomparable. Then $n \\ge 7$. The smallest rectangle with integer sides that can be so divided into rectangles with integer sides is $13 \\times 22$. :600px"}
+{"_id": "13218", "title": "If Element Does Not Belong to Ideal then There Exists Prime Ideal Including Ideal and Excluding Element", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a distributive lattice. Let $I$ be an ideal in $L$. Let $x$ be an element of $S$. Suppose $x \\notin I$ Then there exists a prime ideal $P$ in $L$: $I \\subseteq P$ and $x \\notin P$"}
+{"_id": "13219", "title": "Definite Integral to Infinity of Exponential of -x by Logarithm of x", "text": "Let $\\ln t$ denote the natural logarithm function for real $t > 0$. Let $e^{-t}$ denote the real exponential. Then: :$\\displaystyle \\int_{0^+}^{ \\mathop \\to +\\infty} \\ln t \\,  e^{-t} \\, \\mathrm d t = - \\gamma$ where the {{LHS}} is an improper integral, and $\\gamma$ is the Euler-Mascheroni Constant."}
+{"_id": "13220", "title": "Bottom not in Proper Filter", "text": "Let $L = \\struct {S, \\preceq}$ be a bounded below preordered set. Let $F$ be a filter on $L$. Then $F$ is proper filter {{iff}} $\\bot \\notin F$ where $\\bot$ denotes the smallest element of $S$."}
+{"_id": "13221", "title": "Rectangle Divided into Differently Shaped Equal Area Subrectangles", "text": "Let $R$ be a rectangle. Let $R$ be divided into $n$ smaller rectangles which are of equal area but with different lengths of sides. Then $n \\ge 7$."}
+{"_id": "13223", "title": "If Ideal and Filter are Disjoint then There Exists Prime Filter Including Filter and Disjoint from Ideal", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a distributive lattice. Let $I$ be an ideal in $L$. Let $F$ be a filter on $L$ such that :$F \\cap I = \\O$ Then there exists a prime filter $P$ in $L$: $F \\subseteq P$ and $P \\cap I = \\O$"}
+{"_id": "13224", "title": "Determinant of Elementary Row Matrix", "text": "Let $\\mathbf E$ be an elementary row matrix. The determinant of $\\mathbf E$ is as follows:"}
+{"_id": "13225", "title": "Dual Distributive Lattice is Distributive", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a lattice. Then :$L$ is a distributive lattice {{iff}} :$L^{-1}$ is a distributive lattice where $L^{-1} = \\left({S, \\succeq}\\right)$ denotes the dual of $L$."}
+{"_id": "13226", "title": "Exchange of Rows as Sequence of Other Elementary Row Operations", "text": "Let $\\mathbf A$ be an $m \\times n$ matrix. Let $i, j \\in \\closedint 1 m: i \\ne j$ Let $r_k$ denote the $k$th row of $\\mathbf A$ for $1 \\le k \\le m$: :$r_k = \\begin {pmatrix} a_{k 1} & a_{k 2} & \\cdots & a_{k n} \\end {pmatrix}$ Let $e$ be the elementary row operation acting on $\\mathbf A$ as: {{begin-axiom}} {{axiom | n = \\text {ERO} 3         | t = Interchange rows $i$ and $j$         | m = r_i \\leftrightarrow r_j }} {{end-axiom}} Then $e$ can be expressed as a finite sequence of exactly $4$ instances of the other two elementary row operations. {{begin-axiom}} {{axiom | n = \\text {ERO} 1         | t = For some $\\lambda \\in K_{\\ne 0}$, multiply row $i$ by $\\lambda$         | m = r_i \\to \\lambda r_i }} {{axiom | n = \\text {ERO} 2         | t = For some $\\lambda \\in K$, add $\\lambda$ times row $j$ to row $i$         | m = r_i \\to r_i + \\lambda r_j }} {{end-axiom}}"}
+{"_id": "13228", "title": "Seven Different Frieze Groups", "text": "There are $7$ different symmetry groups for a frieze."}
+{"_id": "13229", "title": "Way Below iff Second Operand Preceding Supremum of Prime Ideal implies First Operand is Element of Ideal", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a distributive complete lattice. Let $x, y \\in S$. Then $x \\ll y$ {{iff}}: :for every prime ideal $P$ in $L$: $y \\preceq \\sup P \\implies x \\in P$"}
+{"_id": "13230", "title": "Lower Closure is Prime Ideal for Prime Element", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a lattice. Let $p \\in S$ be a prime element. Then $p^\\preceq$ is a prime ideal."}
+{"_id": "13231", "title": "Prime is Pseudoprime (Order Theory)", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be an up-complete lattice. Let $p \\in S$ be a prime element. Then $p$ is pseudoprime."}
+{"_id": "13233", "title": "Smallest Prime Number whose Period is of Maximum Length", "text": "$7$ is the smallest prime number the period of whose reciprocal, when expressed in decimal notation, is maximum: :$\\dfrac 1 7 = 0 \\cdotp \\dot 14285 \\dot 7$"}
+{"_id": "13234", "title": "Characterization of Pseudoprime Element by Finite Infima", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a continuous lattice. Let $p \\in S$ be a pseudoprime element. Let $A$ be a non-empty finite subset of $S$ such that :$\\inf A \\ll p$ where $\\ll$ denotes the way below relation. Then $\\exists a \\in A: a \\preceq p$"}
+{"_id": "13235", "title": "Elementary Row Operations as Matrix Multiplications/Corollary", "text": "Let $\\mathbf X$ and $\\mathbf Y$ be two $m \\times n$ matrices that differ by exactly one elementary row operation. Then there exists an elementary row matrix of order $m$ such that: :$\\mathbf {E X} = \\mathbf Y$"}
+{"_id": "13236", "title": "Cyclotomic Polynomial of Prime Index", "text": "Let $p$ be a prime number. The '''$p$th cyclotomic polynomial''' is: :$\\map {\\Phi_p} x = x^{p - 1} + x^{p - 2} + \\cdots + x + 1$"}
+{"_id": "13237", "title": "Product of Cyclotomic Polynomials", "text": "Let $n > 0$ be a (strictly) positive integer. Then: :$\\displaystyle \\prod_{d \\mathop \\backslash n} \\Phi_d \\left({x}\\right) = x^n-1$ where: :$\\Phi_d \\left({x}\\right)$ denotes the $d$th cyclotomic polynomial :the product runs over all divisors of $n$."}
+{"_id": "13238", "title": "First Cyclotomic Polynomial", "text": "The '''first cyclotomic polynomial''' is: :$\\map {\\Phi_1} x = x - 1$"}
+{"_id": "13239", "title": "Cyclotomic Polynomial has Integer Coefficients", "text": "Let $n \\in \\Z_{>0}$ be a positive integer. Then the $n$th cyclotomic polynomial $\\map {\\Phi_n} x$ has integer coefficients."}
+{"_id": "13240", "title": "Formal Derivative of Polynomials Satisfies Leibniz's Rule", "text": "Let $R$ be a commutative ring with unity. Let $R \\left[{X}\\right]$ be the polynomial ring over $R$. Let $f, g \\in R \\left[{X}\\right]$ be polynomials. Let $f'$ and $g'$ denote their formal derivatives. Then: :$\\left({f g}\\right)' = f g' + f' g$"}
+{"_id": "13241", "title": "Double Root of Polynomial is Root of Derivative", "text": "Let $R$ be a commutative ring with unity. Let $f \\in R \\left[{X}\\right]$ be a polynomial. Let $a \\in R$ be a root of $f$ with multiplicity at least $2$. Let $f'$ denote the formal derivative of $f$. Then $a$ is a root of $f'$."}
+{"_id": "13242", "title": "Fano Plane is Unique Projective Plane of Order 2", "text": "The '''Fano plane''' is the unique finite projective plane of order $2$."}
+{"_id": "13243", "title": "7 Prime Knots with 7 Crossings", "text": "There exist exactly $7$ prime knots which have exactly $7$ crossings."}
+{"_id": "13244", "title": "Prime Divisors of Cyclotomic Polynomials", "text": "Let $n \\ge 1$ be a positive integer. Let $\\map {\\Phi_n} x$ denote the $n$th cyclotomic polynomial. Let $a \\in \\Z$ be an integer such that $\\map {\\Phi_n} a \\ne 0$. Let $p$ be a prime divisor of $\\map {\\Phi_n} a$. Then $p \\equiv 1 \\pmod n$ or $p \\divides n$."}
+{"_id": "13248", "title": "Sum of Complex Indices of Real Number", "text": "Let $r \\in \\R_{> 0}$ be a (strictly) positive real number. Let $\\psi, \\tau \\in \\C$ be complex numbers. Let $r^\\lambda$ be defined as the the principal branch of a positive real number raised to a complex number. Then: :$r^{\\psi \\mathop + \\tau} = r^\\psi \\times r^\\tau$"}
+{"_id": "13249", "title": "Universal Property of Free Modules", "text": "Let $R$ be a ring. Let $M$ be a free $R$-module with basis $\\{e_i\\mid i\\in I\\}$. Let $N$ be an $R$-module. Let $\\{n_i\\mid i\\in I\\}$ be a family of elements of $N$. Then there exists a unique $R$-module homomorphism that maps $e_i$ to $n_i$ for all $i\\in I$."}
+{"_id": "13250", "title": "Odd Squares 7 Less than Nearest Power of 2", "text": "There exist exactly $3$ odd squares which are $7$ less than the nearest power of $2$: :$5^2 = 25 = 2^5 - 7$ :$11^2 = 121 = 2^7 - 7$ :$181^2 = 32 \\, 761 = 2^{15} - 7$ {{OEIS|A038198}} Note that this sequence includes $1$ and $3$, being all the squares which are $7$ less than a power of $2$. However, for $1$ and $3$, those powers ($8$ and $16$ respectively) are not the nearest power of $2$ ($1$ and $4$ respectively)."}
+{"_id": "13251", "title": "Universal Property of Direct Product of Modules", "text": "Let $R$ be a ring. Let $N$ be an $R$-module. Let $\\left({M_i}\\right)_{i \\mathop \\in I}$ be a family of $R$-modules. Let $M = \\displaystyle \\prod_{i \\mathop \\in I} M_i$ be their direct product. Let $\\left({\\psi_i}\\right)_{i \\mathop \\in I}$ be a family of $R$-module morphisms $N \\to M_i$. Then there exists a unique morphism: :$\\Psi: N \\to M$ such that: : $\\forall i: \\psi_i = \\pi_i \\circ \\Psi$ where $\\pi_i: M \\to M_i$ is the $i$th canonical projection."}
+{"_id": "13252", "title": "Universal Property of Direct Sum of Modules", "text": "Let $R$ be a ring. Let $N$ be an $R$-module. Let $\\left({M_i}\\right)_{i \\mathop \\in I}$ be a family of $R$-modules. Let $M = \\displaystyle \\bigoplus_{i \\mathop \\in I} M_i$ be their direct sum. Let $\\left({\\psi_i}\\right)_{i \\mathop \\in I}$ be a family of $R$-module morphisms $M_i \\to N$. Then there exists a unique morphism: :$\\Psi: M \\to N$ such that: : $\\forall i: \\psi_i = \\Psi \\circ \\iota_i$ where $\\iota_i: M_i \\to M$ is the $i$th canonical injection."}
+{"_id": "13255", "title": "Morphism from Ring with Unity to Module", "text": "Let $R$ be a ring with unity. Let $M$ be an $R$-module. Then for every $m \\in M$ there exists a unique $R$-module morphism: :$\\psi: R \\to M$ that sends $1$ to $m$."}
+{"_id": "13256", "title": "Universal Property of Free Module on Set", "text": "Let $R$ be a ring with unity. Let $R^{\\left({I}\\right)}$ be the free $R$-module on $I$. Let $M$ be an $R$-module. Let $\\left\\langle{m_i}\\right\\rangle_{i \\mathop \\in I}$ be a family of elements of $M$. Then there exists a unique $R$-module morphism: :$\\Psi: R^{\\left({I}\\right)}\\to M$ that sends the $i$th canonical basis element to $m_i$, for all $i\\in I$. Moreover: :$\\displaystyle \\Psi((r_i)_{i \\mathop \\in I}) = \\sum_{i \\mathop \\in I} r_i m_i$"}
+{"_id": "13257", "title": "Multiplicative Auxiliary Relation iff Images are Filtered", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below lattice. Let $\\mathcal R$ be an auxiliary relation on $S$. Then $\\mathcal R$ is multiplicative {{iff}} :for all $x \\in S$: $\\mathcal R\\left({x}\\right)$ is filtered where $\\mathcal R\\left({x}\\right)$ denotes the $\\mathcal R$-image of $x$."}
+{"_id": "13258", "title": "Characterisation of Spanning Set through Free Module Indexed by Set", "text": "Let $M$ be a unitary $R$-module. Let $S = \\left\\langle{m_i}\\right\\rangle_{i \\mathop \\in I}$ be a family of elements of $M$. Let $\\Psi: R^{\\left({I}\\right)} \\to M$ be the morphism given by Universal Property of Free Module Indexed by Set. Then $S$ is a spanning set of $M$ {{iff}} $\\Psi$ is surjective."}
+{"_id": "13259", "title": "Characterisation of Linearly Independent Set through Free Module Indexed by Set", "text": "Let $M$ be a unitary $R$-module. Let $S = \\left\\langle{m_i}\\right\\rangle_{i \\mathop \\in I}$ be a family of elements of $M$. Let $\\Psi : R^{\\left({I}\\right)} \\to M$ be the module homomorphism given by Universal Property of Free Module on Set. Then the following are equivalent: :$S$ linearly independent :$\\Psi$ is injective."}
+{"_id": "13260", "title": "Free Module is Isomorphic to Free Module on Set", "text": "Let $M$ be a unitary $R$-module. Let $\\mathcal B = \\left\\langle{b_i}\\right\\rangle_{i \\mathop \\in I}$ be a family of elements of $M$. Let $\\Psi: R^{\\left({I}\\right)} \\to M$ be the morphism given by Universal Property of Free Module on Set. Then the following are equivalent: :$\\mathcal B$ is a basis of $M$ :$\\Psi$ is an isomorphism"}
+{"_id": "13261", "title": "Direct Product of Modules is Module", "text": "Let $R$ be a ring. Let $\\left\\{\\left\\langle {M_i,+_i,\\circ_i} \\right\\rangle\\right\\}_{i \\in I}$ be a family of $R$-modules. Let $\\left\\langle{M, +, \\circ}\\right\\rangle$ be their direct product. Then $\\left\\langle{M, +, \\circ}\\right\\rangle$ is a module."}
+{"_id": "13262", "title": "Direct Product of Unitary Modules is Unitary Module", "text": "Let $R$ be a ring with unity. Let $\\left\\{ {M_i}\\right\\}_{i \\in I}$ be a family of unitary $R$-modules. Let $\\left({M, +, \\circ}\\right)$ be their direct product. Then $\\left({M, +, \\circ}\\right)$ is a unitary $R$-module."}
+{"_id": "13263", "title": "Auxiliary Relation Image of Element is Upper Set", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an ordered set. Let $R$ be an auxiliary relation on $S$. Let $x \\in S$. Then $R\\left({x}\\right)$ is an upper set where $R\\left({x}\\right)$ denotes the image of $x$ under $R$."}
+{"_id": "13264", "title": "Multiplicative Auxiliary Relation iff Congruent", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below lattice. Let $\\mathcal R$ be an auxiliary relation on $S$. Then $\\mathcal R$ is multiplicative {{iff}}: :$\\forall a, b, x, y \\in S: \\left({a, x}\\right), \\left({b, y}\\right) \\in \\mathcal R \\implies \\left({a \\wedge b, x \\wedge y}\\right) \\in \\mathcal R$ That is iff $\\mathcal R$ is a congruence relation for $\\wedge$."}
+{"_id": "13265", "title": "Cube which is One Less than a Square", "text": "$8$ is the only cube number which is $1$ less than a square: :$2^3 + 1 = 3^2$"}
+{"_id": "13266", "title": "Prime Powers Differing by One", "text": "$8$ and $9$ are the only powers of prime numbers which differ by exactly $1$: :$2^3 + 1 = 3^2$"}
+{"_id": "13267", "title": "Cubic Fibonacci Numbers", "text": "The only Fibonacci numbers which are also cubes are: :$F_1 = 1 = 1^3$ :$F_6 = 8 = 2^3 = 3 + 5$"}
+{"_id": "13268", "title": "Monoid Ring of Commutative Monoid over Commutative Ring is Commutative", "text": "Let $R$ be a commutative ring. Let $G$ be a commutative monoid. Let $R[G]$ be the monoid ring of $G$ over $R$. Then $R[G]$ is commutative."}
+{"_id": "13269", "title": "3 Non-Parallel Planes divide Space into 8", "text": "Let $3$ planes which are pairwise non-parallel be constructed in ordinary $3$-dimensional space. Then that space is divided into $8$ parts by those planes."}
+{"_id": "13270", "title": "8 Mutually Non-Attacking Queens on Chessboard", "text": "On a standard chessboard, it is possible to arrange a maximum of $8$ queens so that no queen is attacking any other queen. There are $12$ such arrangements, up to rotation and reflection."}
+{"_id": "13271", "title": "Divisibility by 8", "text": "An integer $N$ expressed in decimal notation is divisible by $8$ {{iff}} the $3$ least significant digits of $N$ form a $3$-digit integer divisible by $8$. That is: :$N = \\sqbrk {a_n \\ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \\cdots + a_n 10^n$ is divisible by $8$  {{iff}}: :$100 a_2 + 10 a_1 + a_0$ is divisible by $8$."}
+{"_id": "13274", "title": "No Perfect Magic Cube of Order Less than 5 Exists", "text": "Apart from the trivial order $1$ case, no perfect magic cube exists whose order is $4$ or less."}
+{"_id": "13275", "title": "Modulus of Positive Real Number to Complex Power is Positive Real Number to Power of Real Part", "text": "Let $z \\in \\C$ be a complex number. Let $t > 0$ be wholly real. Let $t^z$ be $t$ to the power of $z$ defined on its principal branch. Then: :$\\cmod {t^z} = t^{\\map \\Re z}$"}
+{"_id": "13276", "title": "Characterization of Pseudoprime Element when Way Below Relation is Multiplicative", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below continuous lattice such that :$\\ll$ is multiplicative relation where $\\ll$ denotes the way below relation of $L$. Let $p \\in S$. Then $p$ is pseudoprime element {{iff}} :$\\forall a, b \\in S: a \\wedge b \\ll p \\implies a \\preceq p \\lor b \\preceq p$"}
+{"_id": "13278", "title": "Eight Convex Deltahedra", "text": "There exist exactly $8$ distinct convex deltahedra: :$4$ faces: regular tetrahedron :$6$ faces: triangular bipyramid :$8$ faces: regular octahedron :$10$ faces: pentagonal bipyramid :$12$ faces: snub disphenoid (split a regular tetrahedron into two wedges and join them with a band of $8$ equilateral triangles) :$14$ faces: triaugmented triangular prism (attach $3$ square pyramids to a triangular prism) :$16$ faces: gyroelongated square bipyramid (attach $2$ square pyramids to a square antiprism) :$20$ faces: regular icosahedron."}
+{"_id": "13279", "title": "Way Below Relation is Multiplicative implies Pseudoprime Element is Prime", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below continuous lattice such that :$\\ll$ is multiplicative relation where $\\ll$ denotes the way below relation of $L$. Let $p \\in S$. Then $p$ is a pseudoprime element is a prime element."}
+{"_id": "13280", "title": "If Every Element Pseudoprime is Prime then Way Below Relation is Multiplicative", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below continuous distributive lattice. Let every element $p \\in S$: $p$ is pseudoprime $\\implies p$ is prime. Then $\\ll$ is multiplicative where $\\ll$ denotes the way below relation of $L$."}
+{"_id": "13281", "title": "Upper Closure of Element is Way Below Open Filter iff Element is Compact", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $x \\in S$. Then : $x^\\succeq$ is a way below open filter on $L$ {{iff}} :$x$ is compact"}
+{"_id": "13282", "title": "9 is Only Square which is Sum of 2 Consecutive Positive Cubes", "text": "Discounting the trivial solution: :$1^2 = 1 = 0^3 + 1^3$ $9$ is the only square number which is the sum of $2$ consecutive positive cube numbers: :$3^2 = 9 = 1^3 + 2^3$"}
+{"_id": "13283", "title": "If Compact Between then Way Below", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $x, k, y \\in S$ such that :$x \\preceq k$ and $k \\preceq y$ and $k \\in K\\left({L}\\right)$ where $K\\left({L}\\right)$ denotes the compact subset of $L$. Then $x \\ll y$ where $\\ll$ denotes the way below relation."}
+{"_id": "13284", "title": "Nine Regular Polyhedra", "text": "There exist $9$ regular polyhedra."}
+{"_id": "13285", "title": "Way Below is Congruent for Join", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a join semilattice. Then $\\ll$ is congruence relation for $\\vee$: :$\\forall a, b, x, y \\in S: a \\ll x \\land b \\ll y \\implies a \\vee b \\ll x \\vee y$ where $\\ll$ denotes the way below relation."}
+{"_id": "13286", "title": "Lifting The Exponent Lemma", "text": "Let $x, y \\in \\Z$ be distinct integers. Let $n \\geq1$ be a natural number. Let $p$ be an odd prime. Let: :$p \\mathrel \\backslash x - y$ and: :$p \\nmid x y$ where $\\backslash$ and $\\nmid$ denote divisibility and non-divisibility respectively. Then :$\\nu_p \\left({x^n - y^n}\\right) = \\nu_p \\left({x - y}\\right) + \\nu_p \\left({n}\\right)$ where $\\nu_p$ denotes $p$-adic valuation."}
+{"_id": "13289", "title": "Zsigmondy's Theorem", "text": "Let $a > b > 0$ be coprime positive integers. Let $n \\ge 1$ be a (strictly) positive integer. Then there is a prime number $p$ such that :$p$ divides $a^n - b^n$ :$p$ does not divide $a^k - b^k$ for all $k < n$ with the following exceptions: :$n = 1$ and $a - b = 1$ :$n = 2$ and $a + b$ is a power of $2$ :$n = 6$, $a = 2$, $b = 1$"}
+{"_id": "13290", "title": "Compact Subset is Join Subsemilattice", "text": "Let $L = \\struct {S, \\vee, \\preceq}$ be a bounded below join semilattice. Let $\\map K L$ be a compact subset of $L$. Then $\\map K L$ is join subsemilattice: :$\\forall x, y \\in \\map K L: x \\vee y \\in \\map K L$"}
+{"_id": "13291", "title": "Totally Ordered Ring Zero Precedes Element or its Inverse", "text": "Let $\\struct {R, +, \\circ, \\preceq}$ be an ordered ring. From the definition of ordered ring, $\\preceq$ is compatible with $+$. Let $0_R$ be the zero element of $R$. Let $x \\ne 0_R$ be a non-zero element of $R$. Let $-x$ be the ring negative of $x$. Then: :$0_R \\prec x \\lor 0_R \\prec -x$ but not both."}
+{"_id": "13293", "title": "Bottom in Compact Subset", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Then $\\bot \\in K\\left({L}\\right)$ where $\\bot$ denotes the smallest element of $L$, :$K\\left({L}\\right)$ denotes the compact subset of $L$."}
+{"_id": "13294", "title": "Only 1 by 1 Matrices Generally Commute Under Multiplication", "text": "Let $R$ be a commutative ring. Let $\\MM_{m \\times n}$ be the set of all $m \\times n$ matrices over $R$. Then under conventional matrix multiplication: :$\\mathbf {A B} = \\mathbf {B A}$ for all $\\mathbf A \\in \\MM_{m \\times n}, \\; \\mathbf B \\in \\MM_{n \\times p}$ {{iff}} $p = m = n = 1$."}
+{"_id": "13295", "title": "Composition of Relations is not Commutative", "text": "Composition of relations is, in general, not commutative: That is, it is usually the case that: :$\\mathcal R_1 \\circ \\mathcal R_2 \\ne \\mathcal R_2 \\circ \\mathcal R_1$ for relations $\\mathcal R_1$ and $\\mathcal R_2$."}
+{"_id": "13296", "title": "Compact Closure is Intersection of Lower Closure and Compact Subset", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let $x \\in S$. Then $x^{\\mathrm{compact} } = x^\\preceq \\cap K\\left({L}\\right)$ where :$x^{\\mathrm{compact} }$ denotes the compact closure of $x$, :$x^\\preceq$ denotes the lower closure of $x$, :$K\\left({L}\\right)$ denotes the compact subset of $L$."}
+{"_id": "13297", "title": "Compact Closure is Subset of Way Below Closure", "text": "Let $L = \\struct {S, \\preceq}$ be an ordered set. Let $x \\in S$. Then $x^{\\mathrm {compact} } \\subseteq x^\\ll$ where :$x^{\\mathrm {compact} }$ denotes the compact closure of $x$, :$x^\\ll$ denotes the way below closure of $x$."}
+{"_id": "13298", "title": "Algebraic iff Continuous and For Every Way Below Exists Compact Between", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a lattice. Then $L$ is algebraic {{iff}}: :$L$ is continuous and :$\\forall x, y \\in S: x \\ll y \\implies \\exists k \\in \\map K L: x \\preceq k \\preceq y$ where :$\\ll$ denotes the way below relation, :$\\map K L$ denotes the compact subset of $L$."}
+{"_id": "13299", "title": "Non-Empty Way Below Closure is Directed in Join Semilattice", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a join semilattice. Let $x \\in S$ such that :$x^\\ll \\ne \\varnothing$ where $x^\\ll$ denotes the way below closure os $x$. Then $x^\\ll$ is directed."}
+{"_id": "13300", "title": "Image under Subset of Relation is Subset of Image under Relation", "text": "Let $S$ and $T$ be sets. Let $\\RR_1 \\subseteq S \\times T$ be a relation in $S \\times T$. Let $\\RR_2 \\subseteq \\RR_1$. Let $A \\subseteq S$. Then: :$\\RR_2 \\sqbrk A \\subseteq \\RR_1 \\sqbrk A$ where $\\RR_1 \\sqbrk A$ denotes the image of $A$ under $\\RR_1$."}
+{"_id": "13302", "title": "Intersection of Transitive Relations is Transitive/General Result", "text": "Let $\\left\\{ {\\mathcal R_i: i \\mathop \\in I}\\right\\}$ be an $I$-indexed collection of transitive relations on a set $S$. Then their intersection $\\displaystyle \\bigcap_{i \\mathop \\in I} \\mathcal R_i$ is also a transitive relation on $S$."}
+{"_id": "13305", "title": "Upper Bounds are Equivalent implies Suprema are equal", "text": "Let $L = \\struct {S, \\preceq}$ be an ordered set. Let $X, Y$ be subsets of $S$. Assume that :$X$ admits a supremum and :$\\forall x \\in S: x$ is upper bound for $X \\iff x$ is upper bound for $Y$ Then $\\sup X = \\sup Y$"}
+{"_id": "13306", "title": "Inverse of Subset of Relation is Subset of Inverse", "text": "Let $S$ and $T$ be sets Let $\\RR_1 = S \\times T$ be a relation on $S \\times T$. Let $\\RR_2 \\subseteq \\RR_1$. Then: :$\\RR_2^{-1} \\subseteq \\RR_1^{-1}$ where $\\RR_1^{-1}$ denotes the inverse of $\\RR_1$."}
+{"_id": "13307", "title": "Preceding implies if Less Upper Bound then Greater Upper Bound", "text": "Let $L = \\struct {S, \\preceq}$ be an ordered set. Let $x, y \\in S$ such that :$x \\preceq y$ Let $X \\subseteq S$. Then :$x$ is upper bound for $X \\implies y$ is upper bound for $X$ and :$y$ is lower bound for $X \\implies x$ is lower bound for $X$."}
+{"_id": "13308", "title": "Three Regular Tessellations/Squares", "text": "Squares form a regular tessellation: :400px"}
+{"_id": "13309", "title": "Three Regular Tessellations/Triangles", "text": "Equilateral triangles form a regular tessellation: :400px"}
+{"_id": "13312", "title": "Arithmetic iff Way Below Relation is Multiplicative in Algebraic Lattice", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below algebraic lattice. Then $L$ is arithmetic {{iff}} :$\\ll$ is a multiplicative relation where $\\ll$ denotes the way below relation of $L$."}
+{"_id": "13313", "title": "Endomorphism Ring of Abelian Group is Ring with Unity", "text": "Let $\\struct {G, +}$ be an abelian group. Let $\\struct {\\map {\\mathrm {End} } G, +, \\circ}$ be its endomorphism ring. Then $\\struct {\\map {\\mathrm {End} } G, +, \\circ}$ is a ring with unity $I_G$, where $I_G$ is the identity mapping on $G$."}
+{"_id": "13314", "title": "Elements with Support in Ideal form Submagma of Direct Product", "text": "Let $\\left({S_i, \\circ_i}\\right)_{i \\mathop \\in I}$ be a family of magmas with identity. Let $\\displaystyle S = \\prod_{i \\mathop \\in I} S_i$ be their direct product. Let $J \\subset I$ be an ideal of $I$. {{explain|The link to ideal suggests that $I$ would need to be an ordered set, which does not appear to be the case here.}} Let $T = \\left\\{ {s \\in S: \\operatorname{supp} \\left({s}\\right) \\in J}\\right\\}$ where $\\operatorname{supp}$ denotes support. Then $T$ is a submagma of $S$."}
+{"_id": "13315", "title": "Elements of Finite Support form Submagma of Direct Product", "text": "Let $\\struct {S_i, \\circ_i}_{i \\mathop \\in I}$ be a family of magmas with identity. Let $\\displaystyle S = \\prod_{i \\mathop \\in I} S_i$ be their direct product. Let $T$ be the subset of elements of $S$ whose support is finite: :$T = \\set {s \\in S: \\map {\\operatorname {supp} } s \\text{ is finite} }$ Then $T$ is a submagma of $S$."}
+{"_id": "13316", "title": "Direct Sum of Modules is Module", "text": "Let $A$ be a commutative ring with unity. Let $\\left\\{ {M_i}\\right\\}_{i \\in I}$ be a family of $A$-modules indexed by $I$. Let $\\displaystyle M = \\bigoplus_{i \\mathop \\in I} M_i$ be their direct sum. Then $M$ is a module."}
+{"_id": "13317", "title": "Möbius Inversion Formula/Abelian Group", "text": "Let $G$ be an abelian group. Let $f, g: \\N \\to G$ be mappings. Then :$\\displaystyle \\map f n = \\prod_{d \\mathop \\divides n} \\map g d$ {{iff}}: :$\\displaystyle \\map g n = \\prod_{d \\mathop \\divides n} \\map f d^{\\mu \\paren {\\frac n d} }$"}
+{"_id": "13318", "title": "Möbius Inversion Formula for Cyclotomic Polynomials", "text": "Let $n > 0$ be a (strictly) positive integer. Let $\\Phi_n$ be the $n$th cyclotomic polynomial. Then: :$\\map {\\Phi_n} x = \\displaystyle \\prod_{d \\mathop \\divides n} \\paren {x^d - 1}^{\\map \\mu {n / d} }$ where: :the product runs over all divisors of $n$. :$\\mu$ is the Möbius function."}
+{"_id": "13319", "title": "Composite of Inverse of Mapping with Mapping", "text": "Let $f: S \\to T$ be a mapping. Then: :$f \\circ f^{-1} = I_{\\Img f}$ where: : $f \\circ f^{-1}$ is the composite of $f$ and $f^{-1}$ : $f^{-1}$ is the inverse of $f$ : $I_{\\Img f}$ is the identity mapping on the image set of $f$."}
+{"_id": "13320", "title": "Multilinear Mapping from Free Modules is Determined by Bases", "text": "Let $R$ be a commutative ring with unity. Let $M_1,\\ldots,M_n$ be free $R$-modules. Let $B_1,\\ldots,B_n$ be bases of $M_1,\\ldots,M_n$. Let $N$ be an $R$-module. Let $f:B_1\\times\\cdots\\times B_n\\to N$ be a function. Then there exists a unique multilinear map $\\phi:M_1\\times\\cdots\\times M_n\\to N$ such that $\\phi(b)=f(b)$ for all $b\\in B_1\\times\\cdots\\times B_n$."}
+{"_id": "13321", "title": "Equivalence of Definitions of Order Complete Set", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. {{TFAE|def = Order Complete Set}}"}
+{"_id": "13322", "title": "Condition for Uniqueness of Increasing Mappings between Tosets", "text": "Let $\\struct {S, \\preceq}$ and $\\struct {T, \\preccurlyeq}$ be tosets. Let $f: S \\to T$ and $g: S \\to T$ be increasing mappings from $S$ to $T$. Let $H \\subseteq S$ be a subset of $S$. Let $f$ and $g$ agree on $H$. Let $K = f \\sqbrk H$ be the image set of $H$ under $f$. Let the intersection of $K$ with every set of the form: :$\\set {y \\in T: u < y < v: u, v \\in T, u < v}$ be non-empty. Then $f = g$."}
+{"_id": "13324", "title": "Arithmetic iff Compact Subset form Lattice in Algebraic Lattice", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below algebraic lattice. Then $L$ is arithmetic {{iff}} $\\left({K\\left({L}\\right), \\precsim}\\right)$ is a lattice, where $K\\left({L}\\right)$ denotes the compact subset of $L$, :$\\mathord \\precsim = \\mathord \\preceq \\cap \\left({K\\left({L}\\right) \\times K\\left({L}\\right)}\\right)$"}
+{"_id": "13326", "title": "Four Kepler-Poinsot Polyhedra", "text": "There exist exactly four Kepler-Poinsot polyhedra: : $(1): \\quad$ the small stellated dodecahedron : $(2): \\quad$ the great stellated dodecahedron : $(3): \\quad$ the great dodecahedron : $(4): \\quad$ the great icosahedron."}
+{"_id": "13327", "title": "Dissection of Rectangle into 9 Distinct Integral Squares", "text": "Let $R$ be a rectangle. Let $R$ be divided into $n$ squares which all have different lengths of sides. Then $n \\ge 9$. The smallest rectangle with integer sides that can be so divided into squares with integer sides is $32 \\times 33$. :600px"}
+{"_id": "13328", "title": "Nine Point Circle Theorem", "text": "Let $\\triangle ABC$ be a triangle. These $9$ points: :the feet of the altitudes of $\\triangle ABC$ :the midpoints of the sides of $\\triangle ABC$ :the midpoints of the lines from the vertices of $\\triangle ABC$ to the orthocenter $H$ of $\\triangle ABC$ all lie on the circumference of a circle. The center $M$ lies on the Euler line of $\\triangle ABC$, at the midpoint between the orthocenter $H$ and the circumcenter $O$. :500px"}
+{"_id": "13329", "title": "Feuerbach's Theorem", "text": "Let $\\triangle ABC$ be a triangle. The Feuerbach circle of $\\triangle ABC$ is tangent to: :the incircle of $\\triangle ABC$ and: :the $3$ excircles of $\\triangle ABC$. :800px"}
+{"_id": "13332", "title": "Round Peg fits in Square Hole better than Square Peg fits in Round Hole", "text": "A round peg fits better in a square hole than a square peg fits in a round hole. :600px"}
+{"_id": "13333", "title": "Ratios of Sizes of Mutually Inscribed Multidimensional Cubes and Spheres", "text": "Consider: : a cube $C_n$ of $n$ dimensions inscribed within a sphere $S_n$ of $n$ dimensions : a sphere $S'_n$ of $n$ dimensions  inscribed within a cube $C'_n$ of $n$ dimensions. Let: : $A_{cn}$ be the $n$ dimensional volume of $C_n$ : $A_{sn}$ be the $n$ dimensional volume of $S_n$ : $A'_{cn}$ be the $n$ dimensional volume of $C'_n$ : $A'_{sn}$ be the $n$ dimensional volume of $S'_n$. For $n < 9$: :$\\dfrac {S_n} {C_n} > \\dfrac {C'_n} {S'_n}$ but for $n \\ge 9$: :$\\dfrac {S_n} {C_n} < \\dfrac {C'_n} {S'_n}$ That is, for dimension $n$ less than $9$, the $n$ dimensional round peg fits better into an $n$ dimensional square hole than an $n$ dimensional square peg fits into an $n$ dimensional round hole, but for $9$ and higher dimensions, the situation is reversed."}
+{"_id": "13334", "title": "Pseudoprime Element is Prime in Arithmetic Lattice", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a bounded below arithmetic lattice. Let $p \\in S$. Then if $p$ is pseudoprime element, then $p$ is prime element."}
+{"_id": "13335", "title": "Every Pseudoprime Element is Prime implies Lattice is Arithmetic", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below algebraic distributive lattice. Assume that :for every element $p$ of $S$ if $p$ is pseudoprime element, then $p$ is prime element. Then $L$ is arithmetic."}
+{"_id": "13336", "title": "General Periodicity Property/Corollary", "text": "Let $f: \\R \\to \\R$ be a real function. Then $L$ is a periodic element of $f$ {{iff}}: :$\\forall x \\in \\R: \\map f {x \\bmod L} = \\map f x$ where $x \\bmod L$ is the modulo operation."}
+{"_id": "13338", "title": "No Configurations of 7 or 8 Lines with 3 Intersection Points on each Line", "text": "There exist no configurations of $7$ or $8$ straight lines each of which has exactly $3$ points of intersection."}
+{"_id": "13339", "title": "3 Configurations of 9 Lines with 3 Intersection Points on each Line", "text": "There exist exactly $3$ essentially different configurations of $9$ straight lines each of which has exactly $3$ points of intersection. This is one: there are two others. :400px"}
+{"_id": "13340", "title": "Power of Positive Real Number is Positive/Real Number", "text": "Let $x \\in \\R_{>0}$ be a (strictly) positive real number. Let $r \\in \\R$ be a real number. Then: :$x^r > 0$ where $x^r$ denotes the $x$ to the power of $r$."}
+{"_id": "13341", "title": "Element is Finite iff Element is Compact in Lattice of Power Set", "text": "Let $X$ be a set. Let $L = \\left({\\mathcal P\\left({X}\\right), \\cup, \\cap, \\subseteq}\\right)$ be a lattice of power set. Let $x \\in \\mathcal P\\left({X}\\right)$. Then $x$ is a finite set {{iff}} $x$ is a compact element."}
+{"_id": "13342", "title": "Function with Limit at Infinity of Exponential Order Zero", "text": "Let $f: \\hointr 0 \\to \\to \\R$ be a real function. Let $f$ be continuous everywhere on their domains, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\\hointr 0 \\to$. {{explain|Establish whether it is \"finite subinterval\" that is needed here, or what we have already defined as \"Definition:Finite Subdivision\". Also get the correct instance of \"continuous\".}} Let $f$ have a (finite) limit at infinity. Then $f$ is of exponential order $0$."}
+{"_id": "13343", "title": "Bounded Function is of Exponential Order Zero", "text": "Let $f: \\hointr 0 \\to \\to \\mathbb F$ be a function, where $\\mathbb F \\in \\set {\\R, \\C}$. Let $f$ be continuous everywhere on its domain, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\\hointr 0 \\to$. {{explain|Establish whether it is \"finite subinterval\" that is needed here, or what we have already defined as \"Definition:Finite Subdivision\".}} {{explain|Pin down a specific page on which the relevant definition of \"Continuous\" can be found for this context.}} Let $f$ be bounded. Then $f$ is of exponential order $0$."}
+{"_id": "13344", "title": "Arctangent is of Exponential Order Zero", "text": "Let $\\arctan: \\R \\to \\openint {-\\dfrac \\pi 2} {\\dfrac \\pi 2}$ be the real arctangent. Then $\\arctan$ is of exponential order $0$."}
+{"_id": "13345", "title": "Arccotangent is of Exponential Order Zero", "text": "Let $\\arccot: \\R \\to \\openint 0 \\pi$ be the real arccotangent. Then $\\arccot$ is of exponential order $0$."}
+{"_id": "13346", "title": "Limit at Infinity of Sine Integral Function", "text": "Let $\\Si: \\R \\to \\R$ denote the sine integral function. Then $\\Si$ has a (finite) limit at infinity: :$\\displaystyle \\lim_{x \\mathop \\to +\\infty} \\map \\Si x = \\frac \\pi 2$"}
+{"_id": "13347", "title": "Natural Number Power is of Exponential Order Epsilon", "text": "Let $n \\in \\N$ be a natural number. Then:  :$t \\mapsto t^n$ is of exponential order $\\epsilon$ for any $\\epsilon > 0$ arbitrarily small in magnitude."}
+{"_id": "13348", "title": "Raising Exponential Order", "text": "Let $\\map f t: \\R \\to \\mathbb F$ a function, where $\\mathbb F \\in \\set {\\R, \\C}$. Let $f$ be continuous on the real interval $\\hointr 0 \\to$, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\\hointr 0 \\to$. {{explain|Establish whether it is \"finite subinterval\" that is needed here, or what we have already defined as \"Definition:Finite Subdivision\".}} {{explain|Pin down a specific page on which the relevant definition of \"Continuous\" can be found for this context.}} Let $f$ be of exponential order $a$. Let $b > a$. Then $f$ is of exponential order $b$."}
+{"_id": "13350", "title": "Factorial is not of Exponential Order", "text": "Let $\\Gamma$ denote the gamma function. Let $f \\left({t}\\right) = \\Gamma \\left({t+1}\\right) = t!$. Then: :$f$ is not of exponential order. That is, it grows faster than any exponential."}
+{"_id": "13351", "title": "X to the x is not of Exponential Order", "text": "Let $f: \\R \\to \\R$ be defined on $\\hointr 0 \\to$ with $\\map f x = x^x$. Then: :$f$ is not of exponential order. That is, it grows faster than any exponential."}
+{"_id": "13352", "title": "Empty is Bottom of Lattice of Power Set", "text": "Let $X$ be a set. Let $L = \\struct {\\powerset X, \\cup, \\cap, \\subseteq}$ be the lattice of the power set of $X$. Then: :$\\O = \\bot$ where $\\bot$ denotes the bottom of $L$."}
+{"_id": "13354", "title": "Hermite-Lindemann-Weierstrass Theorem", "text": "Let $a_1, \\cdots, a_n$ be algebraic numbers (possibly complex) that are linearly independent over the rational numbers $\\Q$. Then: :$e^{a_1}, \\cdots, e^{a_n}$ are algebraically independent. where $e$ is Euler's number."}
+{"_id": "13355", "title": "Hermite-Lindemann-Weierstrass Theorem/Weaker", "text": "Let $a$ be a non-zero algebraic number (possibly complex). Then: :$e^a$ is transcendental where $e$ is Euler's number."}
+{"_id": "13357", "title": "Schanuel's Conjecture Implies Transcendence of Pi by Euler's Number", "text": "Let Schanuel's Conjecture be true. Then $\\pi \\times e$ is transcendental."}
+{"_id": "13358", "title": "Schanuel's Conjecture Implies Transcendence of Pi plus Euler's Number", "text": "Let Schanuel's Conjecture be true. Then $\\pi + e$ is transcendental."}
+{"_id": "13359", "title": "Pascal's Mystic Hexagram", "text": "Let $ABCDEF$ be a hexagram whose $6$ vertices lie on an ellipse such that the order of vertices along the ellipse is $AECFBD$. :500px Then the points of intersection of the sides of $ABCDEF$ lie on a straight line."}
+{"_id": "13360", "title": "Transcendence of Sum or Product of Transcendentals", "text": "Let $a$ and $b$ be two transcendental numbers. Then at least one of $a + b$ and $a \\times b$ is transcendental."}
+{"_id": "13361", "title": "Schanuel's Conjecture Implies Algebraic Independence of Pi and Euler's Number over the Rationals", "text": "Let Schanuel's Conjecture be true. Then $\\pi$ (pi) and $e$ (Euler's number) are algebraically independent over the rational numbers $\\Q$."}
+{"_id": "13362", "title": "Image of Mapping from Finite Set is Finite", "text": "Let $X, Y$ be sets. Let $f: X \\to Y$ be a mapping. Let $X$ be a finite set. Then $f \\sqbrk X$ is a finite set."}
+{"_id": "13363", "title": "Magic Constant of Order 3 Magic Square", "text": "The magic constant of the order $3$ magic square is $15$."}
+{"_id": "13366", "title": "Sum of Terms of Magic Square", "text": "The total of all the entries in a magic square of order $n$ is given by: :$T_n = \\dfrac {n^2 \\paren {n^2 + 1} } 2$"}
+{"_id": "13367", "title": "Schanuel's Conjecture Implies Transcendence of 2 to the power of Euler's Number", "text": "Let Schanuel's Conjecture be true. Then $2$ to the power of Euler's number $e$: :$2^e$ is transcendental, where $e$ is Euler's number."}
+{"_id": "13368", "title": "Schanuel's Conjecture Implies Transcendence of Pi to the power of Euler's Number", "text": "Let Schanuel's Conjecture be true. Then $\\pi$(pi) to the power of Euler's number $e$: :$\\pi^e$ is transcendental."}
+{"_id": "13369", "title": "Schanuel's Conjecture Implies Transcendence of Euler's Number to the power of Euler's Number", "text": "Let Schanuel's Conjecture be true. Then Euler's number $e$ to the power of itself: :$e^e$ is transcendental."}
+{"_id": "13370", "title": "Magic Constant of Magic Square", "text": "The magic constant of a magic square of order $n$ is given by: :$S_n = \\dfrac {n \\paren {n^2 + 1} } 2$"}
+{"_id": "13371", "title": "Magic Square of Order 3 is Unique", "text": "Up to rotations and reflections, the magic square of order $3$ is unique: {{:Magic Square/Examples/Order 3}}"}
+{"_id": "13372", "title": "Wholly Real Number and Wholly Imaginary Number are Linearly Independent over the Rationals", "text": "Let $z_1$ be a non-zero wholly real number. Let $z_2$ be a non-zero wholly imaginary number. Then, $z_1$ and $z_2$ are linearly independent over the rational numbers $\\Q$, where the group is the complex numbers $\\C$."}
+{"_id": "13373", "title": "Schanuel's Conjecture Implies Algebraic Independence of Pi and Log of Pi over the Rationals", "text": "Let Schanuel's Conjecture be true. Then $\\pi$ (pi) and the logarithm of $\\pi$ (pi): :$\\ln \\pi$ are algebraically independent over the rational numbers $\\Q$."}
+{"_id": "13374", "title": "Hermite-Lindemann-Weierstrass Theorem/Weaker/Corollary", "text": "Let $a$ be a algebraic number (possibly complex) which is neither $0$ nor $1$. Then: :any value of $\\ln a$ is transcendental where $\\ln$ denotes complex natural logarithm."}
+{"_id": "13376", "title": "Sums of Squares in Lines of Order 3 Magic Square", "text": "Consider the order 3 magic square: {{:Magic Square/Examples/Order 3}} : The sums of the squares of the top and bottom rows are equal, and differ by $18$ from the sums of the squares of the middle row : The sums of the squares of the left and right columns are equal , and differ by $18$ from the sums of the squares of the middle column."}
+{"_id": "13378", "title": "Lambert W of Zero is Zero", "text": "Let $W_0$ denote principal branch of the Lambert W function. Then: :$W_0 \\left({0}\\right) = 0$"}
+{"_id": "13379", "title": "Sums of Squares of Lines of Order 3 Magic Square", "text": "Consider the order 3 magic square: {{:Magic Square/Examples/Order 3}} : The sums of the squares of the rows, when expressed as $3$-digit decimal numbers, are equal to the sums of the squares of those same rows of that same order 3 magic square when reflected in a vertical axis: :$\\begin{array}{|c|c|c|} \\hline 6 & 7 & 2 \\\\ \\hline 1 & 5 & 9 \\\\ \\hline 8 & 3 & 4 \\\\ \\hline \\end{array}$ Similarly: : The sums of the squares of the columns, when expressed as $3$-digit decimal numbers, are equal to the sums of the squares of those same columns of that same order 3 magic square when reflected in a horizontal axis: :$\\begin{array}{|c|c|c|} \\hline 4 & 3 & 8 \\\\ \\hline 9 & 5 & 1 \\\\ \\hline 2 & 7 & 6 \\\\ \\hline \\end{array}$"}
+{"_id": "13380", "title": "123456789 x 8 + 9 = 987654321", "text": "{{begin-eqn}} {{eqn | l = 1 \\times 8 + 1       | r = 9 }} {{eqn | l = 12 \\times 8 + 2       | r = 98 }} {{eqn | l = 123 \\times 8 + 3       | r = 987 }} {{eqn | l = 1234 \\times 8 + 4       | r = 9876 }} {{eqn | l = 12345 \\times 8 + 5       | r = 98765 }} {{eqn | l = 123456 \\times 8 + 6       | r = 987654 }} {{eqn | l = 1234567 \\times 8 + 7       | r = 9876543 }} {{eqn | l = 12345678 \\times 8 + 8       | r = 98765432 }} {{eqn | l = 123456789 \\times 8 + 9       | r = 987654321 }} {{end-eqn}} The above pattern is an instance of the identity: :$\\displaystyle \\left({b - 2}\\right) \\sum_{j \\mathop = 1}^n j b^{n - j} + n = \\sum_{j \\mathop = 1}^n \\left({b - j}\\right) b^{n - j}$ where $b = 10$ and $n$ goes from $1$ to $9$."}
+{"_id": "13381", "title": "123456789 x 9 + 10 = 1111111111", "text": "{{begin-eqn}} {{eqn | l = 1 \\times 9 + 2       | r = 11 }} {{eqn | l = 12 \\times 9 + 3       | r = 111 }} {{eqn | l = 123 \\times 9 + 4       | r = 1111 }} {{eqn | l = 1234 \\times 9 + 5       | r = 11111 }} {{eqn | l = 12345 \\times 9 + 6       | r = 111111 }} {{eqn | l = 123456 \\times 9 + 7       | r = 1111111 }} {{eqn | l = 1234567 \\times 9 + 8       | r = 11111111 }} {{eqn | l = 12345678 \\times 9 + 9       | r = 111111111 }} {{eqn | l = 123456789 \\times 9 + 10       | r = 1111111111 }} {{end-eqn}} The above pattern is an instance of the identity: :$\\displaystyle \\left({b - 1}\\right) \\sum_{j \\mathop = 1}^n j b^{n - j} + n + 1 = \\sum_{j \\mathop = 0}^n b^j$ where $b = 10$ and $n$ goes from $1$ to $9$."}
+{"_id": "13382", "title": "Algebraic Numbers form Field", "text": "The set of real numbers $\\R$ forms a field under addition and multiplication: $\\left({\\R, +, \\times}\\right)$. {{wtd|correct the statement}}"}
+{"_id": "13383", "title": "Linearly Independent over the Rational Numbers iff Linearly Independent over the Integers", "text": "Let $z_1, z_2, \\ldots, z_n$ be complex numbers. Then: :$z_1, z_2, \\ldots, z_n$ are linearly independent over the rational numbers $\\Q$ {{iff}}: :$z_1, z_2, \\ldots, z_n$ are linearly independent over the integers $\\Z$."}
+{"_id": "13384", "title": "Sums of Squares of Diagonals of Order 3 Magic Square", "text": "Consider the order 3 magic square: {{:Magic Square/Examples/Order 3}} The sums of the squares of the diagonals, when expressed as $3$-digit decimal numbers, are equal to the sums of the squares of those same diagonals of that same order 3 magic square when reversed. {{improve|Find a way to describe the \"diagonals\" accurately, as what is being demonstrated here does not match the description.}}"}
+{"_id": "13385", "title": "Lambert W of Non-Zero Algebraic Number is Transcendental", "text": "Let $W$ denote the (general) Lambert W function. Let $a$ be a non-zero algebraic number. Then, $W \\left({a}\\right)$ is transcendental."}
+{"_id": "13387", "title": "Way Below in Lattice of Power Set", "text": "Let $X$ be a set. Let $L = \\left({\\mathcal P\\left({X}\\right), \\cup, \\cap, \\preceq}\\right)$ be a lattice of power set of $X$ where $\\mathord\\preceq = \\mathord\\subseteq \\cap \\left({\\mathcal P\\left({X}\\right) \\times \\mathcal P\\left({X}\\right)}\\right)$ Let $x, y \\in \\mathcal P\\left({X}\\right)$. Then $x \\ll y$ {{iff}} :for every a set $Y$ of subsets of $X$ such that $y \\subseteq \\bigcup Y$ ::then there exists a finite subset $Z$ of $Y$: $x \\subseteq \\bigcup Z$ where $\\ll$ denotes the way below relation."}
+{"_id": "13388", "title": "Divisibility by 10", "text": "An integer $N$ expressed in decimal notation is divisible by $10$ {{iff}} the least significant digit of $N$ is $0$. That is: :$N = \\sqbrk {a_n \\ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \\cdots + a_n 10^n$ is divisible by $10$  {{iff}}: :$a_0 = 0$"}
+{"_id": "13390", "title": "Difference of Two Squares cannot equal 2 modulo 4", "text": "Let $n \\in \\Z_{>0}$ be of the form $4 k + 2$ for some $k \\in \\Z$. Then $n$ cannot be expressed in the form: :$n = a^2 - b^2$ for $a, b \\in \\Z$."}
+{"_id": "13391", "title": "Compact Closure is Set of Finite Subsets in Lattice of Power Set", "text": "Let $X$ be a set. Let $L = \\left({\\mathcal P\\left({X}\\right), \\cup, \\cap, \\preceq}\\right)$ be the lattice of power set of $X$ where $\\mathord\\preceq = \\mathord\\subseteq \\cap \\mathcal P\\left({X}\\right) \\times \\mathcal P\\left({X}\\right)$ Let $x \\in \\mathcal P\\left({X}\\right)$. Then $x^{\\mathrm{compact} } = \\mathit{Fin}\\left({x}\\right)$ where $\\mathit{Fin}\\left({x}\\right)$ denotes the set of all finite subsets of $x$."}
+{"_id": "13392", "title": "10 is Only Triangular Number that is Sum of Consecutive Odd Squares", "text": "$10$ is the only triangular number which is the sum of two consecutive odd squares: :$10 = 1^2 + 3^2$"}
+{"_id": "13393", "title": "10 Consecutive Integers contain Coprime Integer", "text": "Let $n \\in \\Z$ be an integer. Let $S := \\set {n, n + 1, n + 2, \\ldots, n + 9}$ be the set of $10$ consecutive integers starting from $n$. Then at least one element of $S$ is coprime to every other element of $S$."}
+{"_id": "13394", "title": "Two Fifths as Pandigital Fraction", "text": "There are $3$ ways $\\dfrac 2 5$ can be expressed as a pandigital fraction: :$\\dfrac 2 5 = \\dfrac {6894} {17235}$ :$\\dfrac 2 5 = \\dfrac {8694} {21735}$ :$\\dfrac 2 5 = \\dfrac {9486} {23715}$"}
+{"_id": "13395", "title": "Two Sevenths as Pandigital Fraction", "text": "There are $6$ ways $\\dfrac 2 7$ can be expressed as a pandigital fraction: :$\\dfrac 2 7 = \\dfrac {3654} {12789}$ :$\\dfrac 2 7 = \\dfrac {3674} {12859}$ :$\\dfrac 2 7 = \\dfrac {5342} {18697}$ :$\\dfrac 2 7 = \\dfrac {7418} {25963}$ :$\\dfrac 2 7 = \\dfrac {9786} {34251}$ :$\\dfrac 2 7 = \\dfrac {9862} {34517}$"}
+{"_id": "13396", "title": "Two Ninths as Pandigital Fraction", "text": "There are $2$ ways $\\dfrac 2 9$ can be expressed as a pandigital fraction: :$\\dfrac 2 9 = \\dfrac {3924} {17658}$ :$\\dfrac 2 9 = \\dfrac {7596} {34182}$"}
+{"_id": "13397", "title": "Closed Form for Tetrahedral Numbers", "text": "The closed-form expression for the $n$th tetrahedral number is: :$H_n = \\dfrac {n \\paren {n + 1} \\paren {n + 2} } 6$"}
+{"_id": "13398", "title": "Factorial as Product of Two Factorials", "text": "Apart from the general pattern, following directly from the definition of the factorial: :$\\paren {n!}! = n! \\paren {n! - 1}!$ the only known factorial which is the product of two factorials is: :$10! = 6! \\, 7!$"}
+{"_id": "13400", "title": "Two Thirds as Pandigital Fraction", "text": "$\\dfrac 2 3$ cannot be expressed as a pandigital fraction."}
+{"_id": "13401", "title": "Three Quarters as Pandigital Fraction", "text": "$\\dfrac 3 4$ cannot be expressed as a pandigital fraction."}
+{"_id": "13402", "title": "Three Fifths as Pandigital Fraction", "text": "$\\dfrac 3 5$ cannot be expressed as a pandigital fraction."}
+{"_id": "13403", "title": "Three Sevenths as Pandigital Fraction", "text": "$\\dfrac 3 7$ cannot be expressed as a pandigital fraction."}
+{"_id": "13404", "title": "Three Eighths as Pandigital Fraction", "text": "$\\dfrac 3 8$ cannot be expressed as a pandigital fraction."}
+{"_id": "13405", "title": "Four Sevenths as Pandigital Fraction", "text": "$\\dfrac 4 7$ cannot be expressed as a pandigital fraction."}
+{"_id": "13406", "title": "Five Sevenths as Pandigital Fraction", "text": "$\\dfrac 5 7$ cannot be expressed as a pandigital fraction."}
+{"_id": "13407", "title": "Six Sevenths as Pandigital Fraction", "text": "$\\dfrac 6 7$ cannot be expressed as a pandigital fraction."}
+{"_id": "13408", "title": "Five Sixths as Pandigital Fraction", "text": "$\\dfrac 5 6$ cannot be expressed as a pandigital fraction."}
+{"_id": "13409", "title": "Five Eighths as Pandigital Fraction", "text": "$\\dfrac 5 8$ cannot be expressed as a pandigital fraction."}
+{"_id": "13410", "title": "Seven Eighths as Pandigital Fraction", "text": "$\\dfrac 7 8$ cannot be expressed as a pandigital fraction."}
+{"_id": "13411", "title": "Four Ninths as Pandigital Fraction", "text": "$\\dfrac 4 9$ cannot be expressed as a pandigital fraction."}
+{"_id": "13412", "title": "Five Ninths as Pandigital Fraction", "text": "$\\dfrac 5 9$ cannot be expressed as a pandigital fraction."}
+{"_id": "13413", "title": "Seven Ninths as Pandigital Fraction", "text": "$\\dfrac 7 9$ cannot be expressed as a pandigital fraction."}
+{"_id": "13414", "title": "Eight Ninths as Pandigital Fraction", "text": "$\\dfrac 8 9$ cannot be expressed as a pandigital fraction."}
+{"_id": "13415", "title": "Generating Function for Triangular Numbers", "text": "Let $T_n$ denote the $n$th triangular number. Then the generating function for $\\sequence {T_n}$ is given as: :$\\displaystyle \\map G z = \\frac z {\\paren {1 - z}^3}$"}
+{"_id": "13416", "title": "Lattice of Power Set is Algebraic", "text": "Let $X$ be a set. Let $L = \\left({\\mathcal P\\left({X}\\right), \\cup, \\cap, \\preceq}\\right)$ be the lattice of power set of $X$ where $\\mathord\\preceq = \\mathord\\subseteq \\cap \\left({\\mathcal P\\left({X}\\right) \\times \\mathcal P\\left({X}\\right)}\\right)$ Then $L$ is algebraic."}
+{"_id": "13417", "title": "Difference of Squares of Sum and Difference", "text": ":$\\forall a, b \\in \\R: \\left({a + b}\\right)^2 - \\left({a - b}\\right)^2 = 4 a b$"}
+{"_id": "13420", "title": "Is there a Limit to the Multiplicative Persistence of a Number?", "text": "It has been conjectured that there may be an upper limit to the multiplicative persistence of a natural number."}
+{"_id": "13421", "title": "Non-Empty Compact Closure is Directed", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a join semilattice. Let $x \\in S$ such that :$x^{\\mathrm{compact} }$ is a non-empty set, where $x^{\\mathrm{compact} }$ denotes the compact closure of $x$. Then :$x^{\\mathrm{compact} }$ is directed."}
+{"_id": "13422", "title": "Set of Cardinality not Greater than Cardinality of Finite Set is Finite", "text": "Let $X, Y$ be sets such that :$\\left\\vert X \\right\\vert \\le \\left\\vert Y \\right\\vert$ and :$Y$ is finite, where $\\left\\vert X \\right\\vert$ denotes the cardinality of $X$. Then $X$ is finite."}
+{"_id": "13423", "title": "Finite iff Cardinality Less than Aleph Zero", "text": "Let $X$ be a set. Then $X$ is finite {{iff}} $\\card X < \\aleph_0$ where: :$\\card X$ denotes the cardinality of $X$ :$\\aleph_0 = \\card \\N$ by Aleph Zero equals Cardinality of Naturals."}
+{"_id": "13424", "title": "P-adic Valuation of Difference of Powers with Coprime Exponent", "text": "Let $x, y \\in \\Z$ be distinct integers. Let $n \\ge 1$ be a natural number. Let $p$ be a prime number. Let: :$p \\divides x - y$ and: :$p \\nmid x y n$. Then  :$\\map {\\nu_p} {x^n - y^n} = \\map {\\nu_p} {x - y}$"}
+{"_id": "13427", "title": "One of 4 Consecutive Numbers Greater than 11 is Divisible by Prime Greater than 11", "text": "Let $n \\in \\Z$ such that $n > 11$. Then at least one of the set: :$\\set {n, n + 1, n + 2, n + 3}$ is divisible by a prime number greater than $11$."}
+{"_id": "13428", "title": "Ratio of Consecutive Lucas Numbers", "text": "For $n \\in \\N$, let $L_n$ be the $n$th Lucas number. Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\frac {L_{n + 1} } {L_n} = \\phi$ where $\\phi = \\dfrac {1 + \\sqrt 5} 2$ is the golden mean."}
+{"_id": "13429", "title": "Ordered Set of All Mappings is Ordered Set", "text": "Let $L = \\struct {S, \\preceq}$ be an ordered set. Let $X$ be a set. Then $L^X$ is also an ordered set."}
+{"_id": "13430", "title": "Closed Form for Lucas Numbers", "text": "The Lucas numbers have a closed-form solution: :$L_n = \\phi^n + \\paren {1 - \\phi}^n = \\paren {\\dfrac {1 + \\sqrt 5} 2}^n + \\paren {\\dfrac {1 - \\sqrt 5} 2}^n$ where $\\phi$ is the golden mean. Putting $\\hat \\phi = 1 - \\phi = -\\dfrac 1 \\phi$ this can be written: :$L_n = \\phi^n + \\hat \\phi^n$"}
+{"_id": "13432", "title": "Fibonacci Number 2n equals Fibonacci Number n by Lucas Number n", "text": "Let $F_n$ denote the $n$th Fibonacci number. Let $L_n$ denote the $n$th Lucas number. Then: :$F_{2 n} = F_n L_n$"}
+{"_id": "13433", "title": "Fibonacci Number 3n in terms of Fibonacci Number n and Lucas Number 2n", "text": "Let $F_n$ denote the $n$th Fibonacci number. Let $L_n$ denote the $n$th Lucas number. Then: :$F_{3 n} = F_n \\paren {L_{2 n} + \\paren {-1}^n}$"}
+{"_id": "13434", "title": "Relation between Square of Fibonacci Number and Square of Lucas Number", "text": "Let $F_n$ denote the $n$th Fibonacci number. Let $L_n$ denote the $n$th Lucas number. Then: :$5 {F_n}^2 + 4 \\paren {-1}^n = {L_n}^2$"}
+{"_id": "13436", "title": "12 Pentominoes", "text": "There exist $12$ distinct free pentominoes: :600px"}
+{"_id": "13437", "title": "18 Fixed Pentominoes", "text": "There exist $18$ distinct fixed pentominoes: :600px"}
+{"_id": "13438", "title": "Differential of Differentiable Functional is Unique", "text": "The differential of a differentiable functional is unique."}
+{"_id": "13439", "title": "Ordered Subset of Ordered Set is Ordered Set", "text": "Let $L = \\struct {S, \\preceq}$ be an ordered set. Let $\\struct {S', \\preceq'}$ be an ordered subset of $L$. Then $\\struct {S', \\preceq'}$ is an ordered set."}
+{"_id": "13440", "title": "Finite Subsets form Ideal", "text": "Let $X$ be a set. Let $\\mathit{Fin}\\left({X}\\right)$ be the set of all finite subsets of $X$. Then $\\mathit{Fin}\\left({X}\\right)$ is ideal in $\\left({\\mathcal P\\left({X}\\right), \\subseteq}\\right)$ where $\\mathcal P\\left({X}\\right)$ denotes the power set of $X$."}
+{"_id": "13441", "title": "Product of Proper Divisors", "text": "Let $n$ be an integer such that $n \\ge 1$. Let $\\map P n$ denote the product of the proper divisors of $n$. Then: :$\\map P n = n^{\\map \\tau n / 2 - 1}$ where $\\map \\tau n$ denotes the $\\tau$ function of $n$."}
+{"_id": "13442", "title": "Product of Divisors", "text": "Let $n$ be an integer such that $n \\ge 1$. Let $\\map D n$ denote the product of the divisors of $n$. Then: :$\\map D n = n^{\\map \\tau n / 2}$ where $\\map \\tau n$ denotes the divisor counting function of $n$."}
+{"_id": "13443", "title": "Square of Reversal of Small-Digit Number", "text": "Let $n$ be an integer whose decimal representation consists of sufficiently small digits. Then the reversal of the square of $n$ is the square of the reversal of $n$."}
+{"_id": "13444", "title": "12 times Sigma of 12 equals 14 times Sigma of 14", "text": "$x = 12$ and $y = 14$ are solutions to the indeterminate equation: :$x \\ \\sigma \\left({x}\\right) = y \\ \\sigma \\left({y}\\right)$ where $\\sigma$ denotes the $\\sigma$ function."}
+{"_id": "13445", "title": "12 Knights to Attack or Occupy All Squares on Chessboard", "text": "On a standard chessboard, it is a maximum of $12$ knights are needed to ensure all squares are either occipied or under attack."}
+{"_id": "13446", "title": "12 Identical Spheres can touch One Other Sphere", "text": "A total of $12$ identical spheres can touch one other such sphere. Each of the outer spheres touch the sphere in the center and $3$ other spheres."}
+{"_id": "13447", "title": "Number of Multidimensional Spheres that can touch One Other Sphere", "text": "Let $S_n$ be a spheres of $n$ dimensions with a given radius $r$. Let $\\map T n$ denote the number of instances of $S_n$ that can touch one other such instance of $S_n$. The sequence of $\\map T n$ begins as follows: :{| border=\"1\" |- ! align=\"right\" style = \"padding: 2px 10px\" | $n$  ! align=\"right\" style = \"padding: 2px 10px\" | $\\map T n$  |- | align=\"right\" style = \"padding: 2px 10px\" | $0$ | align=\"right\" style = \"padding: 2px 10px\" | $0$ |- | align=\"right\" style = \"padding: 2px 10px\" | $1$ | align=\"right\" style = \"padding: 2px 10px\" | $2$ |- | align=\"right\" style = \"padding: 2px 10px\" | $2$ | align=\"right\" style = \"padding: 2px 10px\" | $6$ |- | align=\"right\" style = \"padding: 2px 10px\" | $3$ | align=\"right\" style = \"padding: 2px 10px\" | $12$ |- | align=\"right\" style = \"padding: 2px 10px\" | $4$ | align=\"right\" style = \"padding: 2px 10px\" | $24$ |- | align=\"right\" style = \"padding: 2px 10px\" | $5$ | align=\"right\" style = \"padding: 2px 10px\" | $40$ |- | align=\"right\" style = \"padding: 2px 10px\" | $6$ | align=\"right\" style = \"padding: 2px 10px\" | $72$ |- | align=\"right\" style = \"padding: 2px 10px\" | $7$ | align=\"right\" style = \"padding: 2px 10px\" | $126$ |- | align=\"right\" style = \"padding: 2px 10px\" | $8$ | align=\"right\" style = \"padding: 2px 10px\" | $240$ |- | align=\"right\" style = \"padding: 2px 10px\" | $9$ | align=\"right\" style = \"padding: 2px 10px\" | $272$ |} {{OEIS|A001116}} </onlyinclude>"}
+{"_id": "13451", "title": "Ordering on Closure Operators iff Images are Including", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice. Let $f, g: S \\to S$ be closure operators on $L$. Then: :$f \\preceq g$ {{iff}} $g \\sqbrk S \\subseteq f \\sqbrk S$ where :$\\preceq$ denotes the ordering on mappings :$f \\sqbrk S$ denotes the image of $f$."}
+{"_id": "13452", "title": "Multiple of Perfect Number is Abundant", "text": "Let $n$ be a perfect number. Let $m$ be a positive integer such that $m > 1$. Then $m n$ is abundant."}
+{"_id": "13453", "title": "Multiple of Abundant Number is Abundant", "text": "Let $n$ be an abundant number. Let $m$ be a positive integer such that $m > 1$. Then $m n$ is abundant."}
+{"_id": "13454", "title": "Divisor of Perfect Number is Deficient", "text": "Let $n$ be a perfect number. Let $n = r s$ where $r$ and $s$ are positive integers such that $r > 1$ and $s > 1$. Then $r$ and $s$ are both deficient."}
+{"_id": "13455", "title": "Divisor of Deficient Number is Deficient", "text": "Let $n$ be a perfect number. Let $n = k d$ where $r$ is a positive integer. Then $k$ is deficient."}
+{"_id": "13456", "title": "Abundancy Index of Product is greater than Abundancy Index of Proper Factors", "text": "Let $n \\in \\Z_{>0}$ be a composite number such that $n = r s$, where $r, s \\in \\Z_{>1}$. Then: :$\\dfrac {\\map \\sigma n} n > \\dfrac {\\map \\sigma r} r$ and consequently also: :$\\dfrac {\\map \\sigma n} n > \\dfrac {\\map \\sigma s} s$ where $\\map \\sigma n$ denotes the $\\sigma$ function of $n$. That is, the abundancy index of a composite number is strictly greater than the abundancy index of its divisors."}
+{"_id": "13457", "title": "Condition for Differentiable Functional to have Extremum", "text": "Let $S$ be a set of mappings. Let $y, h \\in S: \\R \\to \\R$ be real functions.  Let $J \\sqbrk y: S \\to \\R$ be a differentiable functional. Then a necessary condition for the differentiable functional $J \\sqbrk {y; h}$ to have an extremum for $y = \\hat y$ is: :$\\bigvalueat {\\delta J \\sqbrk {y; h} } {y \\mathop = \\hat y} = 0$"}
+{"_id": "13459", "title": "Ordered Set of Closure Systems is Ordered Set", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an ordered set. Then $\\operatorname{ClSystems}\\left({L}\\right)$ is an ordered set, where $\\operatorname{ClSystems}\\left({L}\\right)$ denotes the ordered set of closure systems."}
+{"_id": "13460", "title": "Necessary Condition for Integral Functional to have Extremum for given function", "text": "Let $S$ be a set of real mappings such that: :$S = \\set {\\map y x: \\paren {y: S_1 \\subseteq \\R \\to S_2 \\subseteq \\R}, \\paren {\\map y x \\in C^1 \\closedint a b}, \\paren {\\map y a = A, \\map y b = B} }$ Let $J \\sqbrk y: S \\to S_3 \\subseteq \\R$ be a functional of the form: :$\\displaystyle \\int_a^b \\map F {x, y, y'} \\rd x$  Then a necessary condition for $J \\sqbrk y$ to have an extremum (strong or weak) for a given function $\\map y x$ is that $\\map y x$ satisfy Euler's equation: :$F_y - \\dfrac \\d {\\d x} F_{y'} = 0$"}
+{"_id": "13461", "title": "Image of Closure Operator Inherits Infima", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an ordered set. Let $f$ be a closure operator on $L$. Then $R = \\left({f\\left[{S}\\right], \\precsim}\\right)$ inherits infima, where :$\\mathord\\precsim = \\mathord\\preceq \\cap \\left({f\\left[{S}\\right] \\times f\\left[{S}\\right]}\\right)$ :$f\\left[{S}\\right]$ denotes the image of $f$."}
+{"_id": "13462", "title": "Necessary Condition for Integral Functional to have Extremum for given function/Lemma", "text": "Let $\\map \\alpha x, \\map \\beta x$ be real functions. Let $\\map \\alpha x, \\map \\beta x$ be continuous on $\\closedint a b$. Let: :$\\forall \\map h x \\in C^1: \\displaystyle \\int_a^b \\paren {\\map \\alpha x \\map h x + \\map \\beta x \\map {h'} x} \\d x = 0$ subject to the boundary conditions: :$\\map h a = \\map h b = 0$ Then $\\map \\beta x$ is differentiable. Furthermore: :$\\forall x \\in \\closedint a b: \\map {\\beta'} x = \\map \\alpha x$"}
+{"_id": "13463", "title": "If Definite Integral of a(x)h(x) vanishes for any C^0 h(x) then C^0 a(x) vanishes", "text": "Let $\\map \\alpha x$ be a continuous real function on the closed real interval $\\closedint a b$. Let $\\displaystyle \\int_a^b \\map \\alpha x \\map h x \\rd x = 0$ for every real function $\\map h x \\in C^0 \\closedint a b$ such that $\\map h a = 0$ and $\\map h b = 0$. {{explain|the notation $C^0 \\closedint a b$}} Then $\\map \\alpha x = 0$ for all $x \\in \\closedint a b$."}
+{"_id": "13464", "title": "Closure Operator does not Change Infimum of Subset of Image", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an ordered set. Let $c: S \\to S$ be a closure operator on $L$. Let $X$ be a subset of $c\\left[{S}\\right]$ such that :$X$ admits an infimum, where $c\\left[{S}\\right]$ denotes the image of $c$. Then $\\inf X = c\\left({\\inf X}\\right)$"}
+{"_id": "13465", "title": "Infimum in Ordered Subset", "text": "Let $L = \\struct {S, \\preceq}$ be an ordered set. Let $R = \\struct {T, \\preceq'}$ be an ordered subset of $L$. Let $X \\subseteq T$ such that :$X$ admits an infimum in $L$. Then $\\inf_L X \\in T$ {{iff}} :$X$ admits an infimum in $R$ and $\\inf_R X = \\inf_L X$"}
+{"_id": "13466", "title": "Operator Generated by Closure System is Closure Operator", "text": "Let $L = \\left({X, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $S = \\left({T, \\precsim}\\right)$ be a closure system of $L$. Then $\\operatorname{operator}\\left({S}\\right)$ is closure operator, where $\\operatorname{operator}\\left({S}\\right)$ denotes the operator generated by $S$."}
+{"_id": "13467", "title": "Regular Dodecahedron is Dual of Regular Icosahedron", "text": "The regular dodecahedron is the dual of the regular icosahedron."}
+{"_id": "13468", "title": "Image of Operator Generated by Closure System is Set of Closure System", "text": "Let $L = \\left({X, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $S = \\left({T, \\precsim}\\right)$ be a closure system of $L$. Then $\\operatorname{operator}\\left({S}\\right)\\left[{X}\\right] = T$ where $\\operatorname{operator}\\left({S}\\right)$ denotes the operator generated by $S$."}
+{"_id": "13469", "title": "Regular Icosahedron as Pentagonal Antiprism with Pyramidal Endcaps", "text": "The regular icosahedron can be considered as a regular pentagonal antiprism with two regular pentagonal pyramid as end caps. {{stub|Need to construct an icosahedron.}}"}
+{"_id": "13470", "title": "Construction of Rhombic Dodecahedron", "text": "The rhombic dodecahedron can be constructed as follows: Take a cube $K$ embedded in $3$-dimensional space. Place $6$ more cubes, each congruent with $K$, so that one face of each coincides with a different face of $K$. Join the vertices of $K$ to the centers of the adjacent cubes to describe square pyramids whose apices are the centers of the adjacent cubes and whose bases are the faces of $K$. The polyhedron formed by the $6$ square pyramids so formed, together with $K$, is a rhombic dodecahedron."}
+{"_id": "13472", "title": "Recurring Parts of Multiples of One Thirteenth", "text": "The multiples of $\\dfrac 1 {13}$ from $\\dfrac 1 {13}$ to $\\dfrac {12} {13}$ can be divided into two sets of equal size: :one where the digits of the recurring part consists of a cyclic permutation of $076923$ :one where the digits of the recurring part consists of a cyclic permutation of $153846$. :300px"}
+{"_id": "13473", "title": "Twelve Factorial plus One is divisible by 13 Squared", "text": ":$12! + 1$ is divisible by $13^2$."}
+{"_id": "13474", "title": "Torus can be cut into 13 Pieces with 3 Plane Cuts", "text": "A torus can be cut into as many as $13$ separate pieces by $3$ plane cuts."}
+{"_id": "13475", "title": "Operator Generated by Image of Closure Operator is Closure Operator", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice. Let $c: S \\to S$ be a closure operator on $L$. Then $\\map {\\operatorname {operator} } {\\struct {c \\sqbrk S, \\precsim} } = c$ where :$\\mathord \\precsim = \\mathord \\preceq \\cap \\paren {c \\sqbrk S \\times c \\sqbrk S}$ :$\\map {\\operatorname {operator} } {\\struct {c \\sqbrk S, \\precsim} }$ denotes the operator generated by $\\struct {c \\sqbrk S, \\precsim}$"}
+{"_id": "13477", "title": "Pell's Equation/Examples/13", "text": ":$x^2 - 13 y^2 = 1$ has the smallest positive integral solution: :$x = 649$ :$y = 180$"}
+{"_id": "13478", "title": "Pell's Equation/Examples/29", "text": ":$x^2 - 29 y^2 = 1$ has the smallest positive integral solution: :$x = 9801$ :$y = 1820$"}
+{"_id": "13481", "title": "Vanishing First Variational Derivative implies Euler's Equation for Vanishing Variation", "text": "Let $\\map y x$ be a real function such that $\\map y a = A$ and $\\map y b = B$. Let $J \\sqbrk y$ be a functional of the form: :$\\displaystyle J \\sqbrk y = \\int_a^b \\map F {x, y, y'} \\rd x$ Then: :$\\dfrac {\\delta J} {\\delta y} = 0 \\implies F_y - \\dfrac \\d {\\d x} F_{y'} = 0$"}
+{"_id": "13482", "title": "Supremum in Ordered Subset", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an ordered set. Let $R = \\left({T, \\preceq'}\\right)$ be an ordered subset of $L$. Let $X \\subseteq T$ such that :$X$ admits an supremum in $L$. Then $\\sup_L X \\in T$ {{iff}} :$X$ admits an supremum in $R$ and $\\sup_R X = \\sup_L X$"}
+{"_id": "13484", "title": "Partial Quotients of Continued Fraction Expansion of Irrational Square Root", "text": "Let $n \\in \\Z$ such that $n$ is not a square. Let the continued fraction expansion of $\\sqrt n$ be expressed as: :$\\left[{a_0, a_1, a_2, \\ldots}\\right]$ Then the partial quotients of this continued fraction expansion can be calculated as: :$a_r = \\left\\lfloor{\\dfrac{\\left\\lfloor{\\sqrt n}\\right\\rfloor + P_r} {Q_r} }\\right\\rfloor$ where: :$P_r = \\begin{cases} 0 & : r = 0 \\\\ a_{r - 1} Q_{r - 1} - P_{r - 1} & : r > 0 \\\\ \\end{cases}$ :$Q_r = \\begin{cases} 1 & : r = 0 \\\\ \\dfrac {n - {P_r}^2} {Q_{r - 1} } & : r > 0 \\\\ \\end{cases}$"}
+{"_id": "13485", "title": "Operator Generated by Closure System Preserves Directed Suprema iff Closure System Inherits Directed Suprema", "text": "Let $L = \\left({X, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $S = \\left({Y, \\precsim}\\right)$ be a closure system on $L$. Then $\\operatorname{operator}\\left({S}\\right)$ preserves directed suprema {{iff}} $S$ inherits directed suprema. where $\\operatorname{operator}\\left({S}\\right)$ denotes the operator generated by $S$."}
+{"_id": "13487", "title": "Closure Operator Preserves Directed Suprema iff Image of Closure Operator Inherits Directed Suprema", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $h:S \\to S$ be a closure operator on $L$. Then $h$ preserves directed suprema {{iff}} $\\left({h \\left[{S}\\right], \\precsim}\\right)$ inherits directed suprema. where :$h \\left[{S}\\right]$ denotes the image of $h$, :$\\mathord\\precsim = \\mathord\\preceq \\cap \\left({h \\left[{S}\\right] \\times h \\left[{S}\\right]}\\right)$"}
+{"_id": "13488", "title": "Thirteen Archimedean Polyhedra", "text": "There exist exactly $13$ distinct Archimedean polyhedra: :Truncated tetrahedron :Cuboctahedron :Truncated Cube :Truncated octahedron :Rhombicuboctahedron :Truncated cuboctahedron :Snub cube :Icosidodecahedron :Truncated dodecahedron :Truncated icosahedron :Rhombicosidodecahedron :Truncated icosidodecahedron :Snub dodecahedron"}
+{"_id": "13489", "title": "Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions", "text": "Let $\\mathbf y$ be an $n$-dimensional real vector. Let $J \\sqbrk {\\mathbf y}$ be a functional of the form: $\\displaystyle J \\sqbrk {\\mathbf y} = \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ Let: :$\\mathbf y \\in C^1 \\closedint a b$ where $C^1 \\closedint a b$ denotes that $\\mathbf y$ is continuously differentiable in $\\closedint a b$ Let $\\mathbf y$ satisfy boundary conditions: :$\\map {\\mathbf y} a = \\mathbf A$  :$\\map {\\mathbf y} b = \\mathbf B$ where $\\mathbf A$, $\\mathbf B$ are real vectors. Then a necessary condition for $J \\sqbrk {\\mathbf y}$ to have an extremum (strong or weak) for a given $\\mathbf y$ is that they satisfy Euler's equations: $F_{\\mathbf y} - \\dfrac \\d {\\d x} F_{\\mathbf y'} = 0$"}
+{"_id": "13491", "title": "Regular Prism is Semiregular Polyhedron", "text": "A regular prism is a semiregular polyhedron."}
+{"_id": "13492", "title": "Regular Antiprism is Semiregular Polyhedron", "text": "A regular antiprism is a semiregular polyhedron."}
+{"_id": "13493", "title": "Finer Supremum Precedes Supremum", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $X, Y$ be subsets of $S$ such that :$X$ is finer than $Y$. Then $\\sup X \\preceq \\sup Y$ where $\\sup X$ denotes the supremum of $X$."}
+{"_id": "13494", "title": "Thirteen Catalan Polyhedra", "text": "There exist exactly $13$ distinct Catalan polyhedra: :Triakis tetrahedron :Triakis octahedron :Disdyakis dodecahedron :Tetrakis hexahedron :Triakis icosahedron :Disdyakis triacontahedron :Pentakis dodecahedron :Rhombic dodecahedron :Rhombic triacontahedron :Deltoidal icositetrahedron :Deltoidal hexecontahedron :Pentagonal icositetrahedron :Pentagonal hexecontahedron"}
+{"_id": "13495", "title": "Necessary and Sufficient Condition for Integral Parametric Functional to be Independent of Parametric Representation", "text": "Let $x: \\R \\to \\R$ and $y: \\R \\to \\R$ be real functions. Let $J \\sqbrk {x, y}$ be a functional of the form $\\displaystyle J \\sqbrk {x, y} = \\int_{t_0}^{t_1} \\map \\Phi {t, x, y, \\dot x, \\dot y} \\rd t$ where $\\dot y$ denotes the derivative of $y$ {{WRT|Differentiation}} $t$: :$\\dot y = \\dfrac {\\d y} {\\d t}$ Then $J \\sqbrk {x, y}$ depends only on the curve in the $x y$-plane defined by the parametric equations: :$x = \\map x t$, $y = \\map y t$  and not on the choice of the parametric representation of the curve {{iff}} the integrand $\\Phi$ does not involve $t$ explicitly  and is a positive-homogeneous function of degree $1$ in $\\dot x$ and $\\dot y$."}
+{"_id": "13496", "title": "Intersection of Upper Set with Directed Set is Directed Set", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an ordered set. Let $A, B$ be subsets of $S$ such that :$A \\cap B \\ne \\varnothing$ and :$A$ is an upper set, :$B$ is a directed set. Then $A \\cap B$ is a directed set."}
+{"_id": "13497", "title": "Convex Set is Contractible", "text": "Let $V$ be a topological vector space over $\\R$ or $\\C$. Let $A\\subset V$ be a convex subset. Then $A$ is contractible."}
+{"_id": "13498", "title": "Open Subgroup is Closed", "text": "Let $G$ be a topological group. Let $H\\leq G$ be an open subgroup. Then $H$ is closed."}
+{"_id": "13499", "title": "Topological Group is Hausdorff iff Identity is Closed", "text": "Let $G$ be a topological group. Let $e$ be its identity element. Then $G$ is Hausdorff {{iff}} $\\left\\{ {e}\\right\\}$ is closed in $G$."}
+{"_id": "13501", "title": "Jensen's Inequality (Complex Analysis)", "text": "Let $D \\subset \\C$ be an open set with $0 \\in D$. Let $R > 0$ be such that $\\map B {0, R} \\subset D$. Let $f: D \\to \\C$ be analytic with $\\map f 0 \\ne 0$. Let $\\cmod {\\map f z} \\le M$ for $\\cmod z \\le R$. Let $0 < r <R$. Then the number of zeroes of $f$, counted with multiplicity, for $\\cmod z \\le r$, is at most: :$\\dfrac {\\map \\log {M / \\cmod {\\map f 0} } } {\\map \\log {R / r} }$"}
+{"_id": "13502", "title": "Complement of Subset with Property (S) is Closed under Directed Suprema", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an up-complete ordered set. Let $X$ be a subset of $S$ with property (S). Then $\\complement_S\\left({X}\\right)$ is closed under directed suprema."}
+{"_id": "13503", "title": "Lifting The Exponent Lemma for Sums", "text": "Let $x, y \\in \\Z$ be integers with $x + y \\ne 0$. Let $n \\ge 1$ be an odd natural number. Let $p$ be an odd prime. Let: :$p \\divides x + y$ and: :$p \\nmid x y$ where $\\divides$ and $\\nmid$ denote divisibility and non-divisibility respectively. Then: :$\\map {\\nu_p} {x^n + y^n} = \\map {\\nu_p} {x + y} + \\map {\\nu_p} n$ where $\\nu_p$ denotes $p$-adic valuation. </onlyinclude>"}
+{"_id": "13504", "title": "Lifting The Exponent Lemma for p=2", "text": "Let $x, y \\in \\Z$ be distinct odd integers. Let $n \\ge 1$ be a natural number. Let: :$4 \\divides x - y$ where $\\divides$ denotes divisibility. Then :$\\map {\\nu_2} {x^n - y^n} = \\map {\\nu_2} {x - y} + \\map {\\nu_2} n$ where $\\nu_2$ denotes $2$-adic valuation."}
+{"_id": "13505", "title": "Classification of Convex Polyhedra whose Faces are Regular Polygons", "text": "The convex polyhedra whose faces are all regular polygons are as follows: :The $5$ Platonic solids :The regular prisms and regular antiprisms, countably infinite in number :The $13$ Archimedean polyhedra :The $92$ Johnson polyhedra."}
+{"_id": "13506", "title": "Chinese Remainder Theorem/General Result 2", "text": "Let $n_1, n_2, \\ldots, n_k$ be positive integers. Let $b_1, \\ldots, b_k$ be integers such that: : $\\forall i \\ne j: \\gcd \\left({n_i, n_j}\\right) \\mathrel \\backslash b_i - b_j$ where $\\backslash$ denotes divisibility. Then the system of linear congruences: : $x \\equiv b_1 \\pmod {n_1}$ : $x \\equiv b_2 \\pmod {n_2}$ : $\\ldots$ : $x \\equiv b_k \\pmod {n_k}$ has a simultaneous solution which is unique modulo $\\lcm \\left({n_1, \\ldots, n_k}\\right)$."}
+{"_id": "13508", "title": "Necessary Condition for Integral Functional to have Extremum for given Function/Dependent on Nth Derivative of Function", "text": "Let $\\map F {x, y, z_1, \\ldots, z_n}$ be a function in differentiability class $C^2$ {{WRT}} all its variables. Let $y = \\map y x \\in C^n\\openint a b$ such that: :$\\map y a = A_0, \\map {y'} a = A_1, \\ldots, \\map {y^{\\paren {n - 1} } } a = A_{n - 1}$ and: :$\\map y b = B_0, \\map {y'} b = B_1, \\ldots, \\map {y^{\\paren {n - 1} } } b = B_{n - 1}$ Let $J \\sqbrk y$ be a functional of the form: :$\\displaystyle J \\sqbrk y = \\int_a^b \\map F {x, y, y', \\ldots, y^{\\paren n} } \\rd x$ Then a necessary condition for $J \\sqbrk y$ to have an extremum (strong or weak) for a given function $\\map y x$ is that $\\map y x$ satisfy Euler's equation: :$F_y - \\dfrac \\d {\\d x} F_{y'} + \\dfrac {\\d^2} {\\d x^2} F_{y''}  -\\cdots + \\paren {-1}^n \\dfrac {\\d^n} {\\d x^n} F_{y^{\\paren n} } = 0$"}
+{"_id": "13510", "title": "Equivalence of Definitions of Absolute Convergence of Product of Complex Numbers", "text": "Let $\\left\\langle{a_n}\\right\\rangle$ be a sequence of complex numbers. Let $\\log$ denote the complex logarithm. {{TFAE|def = Absolute Convergence of Product}}"}
+{"_id": "13511", "title": "Square of Quadratic Gauss Sum", "text": "Let $p$ be an odd prime. Let $a$ be an integer coprime to $p$. Let $g \\left({a, p}\\right)$ denote the quadratic Gauss sum of $a$ and $p$. Then: :${g \\left({a, p}\\right)}^2 = \\left({\\dfrac {-1} p}\\right) \\cdot p$"}
+{"_id": "13512", "title": "Absolutely Convergent Product is Convergent", "text": "Let $\\struct {\\mathbb K, \\norm {\\,\\cdot\\,} }$ be a valued field. Let $\\mathbb K$ be complete. Let the infinite product $\\ds \\prod_{n \\mathop = 1}^\\infty \\paren {1 + a_n}$ be absolutely convergent. Then it is convergent."}
+{"_id": "13513", "title": "Simplest Variational Problem with Subsidiary Conditions", "text": "Let $J \\sqbrk y$ and $K \\sqbrk y$ be (real) functionals, such that :$\\ds J \\sqbrk y = \\int_a^b \\map F {x, y, y'} \\rd x$ :$\\ds K \\sqbrk y = \\int_a^b \\map G {x, y, y'} \\rd x = l$ where $l$ is a constant. Let $y = \\map y x$ be an extremum of $F \\sqbrk y$, and satisfy boundary conditions: :$\\map y a = A$ :$\\map y b = B$ Then, if $y = \\map y x$ is not an extremal of $K \\sqbrk y$, there exists a constant $\\lambda$ such that $y = \\map y x$ is an extremal of the functional: :$\\ds \\int_a^b \\paren {F + \\lambda G} \\rd x$ or, in other words, $y = \\map y x$ satisfies: :$F_y - \\dfrac {\\d} {\\d x} F_{y'} + \\lambda \\paren {G_y - \\dfrac {\\d} {\\d x} G_{y'} } = 0$"}
+{"_id": "13514", "title": "Complement of Closed under Directed Suprema Subset is Inaccessible by Directed Suprema", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an up-complete ordered set. Let $X$ be a closed under directed suprema subset of $S$. Then $\\complement_S\\left({X}\\right)$ is inaccessible by directed suprema."}
+{"_id": "13515", "title": "Complement of Upper Set is Lower Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $L$ be an upper set. Then $S \\setminus L$ is a lower set."}
+{"_id": "13516", "title": "Closed Set iff Lower and Closed under Directed Suprema in Scott Topological Ordered Set", "text": "Let $T = \\left({S, \\preceq, \\tau}\\right)$ be a relational structure with Scott topology where $\\left({S, \\preceq}\\right)$ is an up-complete ordered set. Let $A$ be a subset of $S$. Then $A$ is closed {{iff}} $A$ is lower and closed under directed suprema."}
+{"_id": "13517", "title": "Complement of Inaccessible by Directed Suprema Subset is Closed under Directed Suprema", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an up-complete ordered set. Let $X$ be an inaccessible by directed suprema subset of $S$. Then $\\complement_S\\left({X}\\right)$ is closed under directed suprema."}
+{"_id": "13518", "title": "Convergence of Complex Sequence in Polar Form", "text": "Let $z \\ne 0$ be a complex number with modulus $r$ and argument $\\theta$. Let $\\sequence {z_n}$ be a sequence of nonzero complex numbers. Let $r_n$ be the modulus of $z_n$ and $\\theta_n$ be an argument of $z_n$. Then $z_n$ converges to $z$ {{iff}} the following hold: :$(1): \\quad r_n$ converges to $r$ :$(2): \\quad$ There exists a sequence $\\sequence {k_n}$ of integers such that $\\theta_n + 2 k_n \\pi$ converges to $\\theta$."}
+{"_id": "13519", "title": "Complex Modulus of Sum of Complex Numbers", "text": "Let $z_1, z_2 \\in \\C$ be complex numbers. Let $\\theta_1$ and $\\theta_2$ be arguments of $z_1$ and $z_2$, respectively. Then: :$\\cmod {z_1 + z_2}^2 = \\cmod {z_1}^2 + \\cmod {z_2}^2 + 2 \\cmod {z_1} \\cmod {z_2} \\, \\map \\cos {\\theta_1 - \\theta_2}$"}
+{"_id": "13520", "title": "Complex Modulus of Difference of Complex Numbers", "text": "Let $z_1, z_2 \\in \\C$ be complex numbers. Let $\\theta_1$ and $\\theta_2$ be arguments of $z_1$ and $z_2$, respectively. Then: :$\\cmod {z_1 - z_2}^2 = \\cmod {z_1}^2 + \\cmod {z_2}^2 - 2 \\cmod {z_1} \\cmod {z_2} \\, \\map \\cos {\\theta_1 - \\theta_2}$"}
+{"_id": "13521", "title": "Uniform Absolute Convergence of Infinite Product of Complex Functions", "text": "Let $X$ be a compact metric space. Let $\\left\\langle{f_n}\\right\\rangle$ be a sequence of continuous functions $X \\to \\C$. Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty f_n$ converge uniformly absolutely on $X$. Then: :$f \\left({x}\\right) = \\displaystyle \\prod_{n \\mathop = 1}^\\infty \\left({1 + f_n \\left({x}\\right)}\\right)$ converges uniformly absolutely on $X$ :$f$ is continuous :there exists $n_0 \\in \\N$ such that $\\displaystyle \\prod_{n \\mathop = n_0}^\\infty \\left({1 + f_n \\left({x}\\right)}\\right)$ has no zeroes."}
+{"_id": "13522", "title": "Simplest Variational Problem with Subsidiary Conditions for Curve on Surface", "text": "Let $J \\sqbrk {y, z}$ be a (real) functional of the form: :$\\ds J \\sqbrk y = \\int_a^b \\map F {x, y, z, y', z'} \\rd x$ Let there exist admissible curves $y, z$ lying on the surface: :$\\map g {x, y, z} = 0$ which satisfy boundary conditions: :$\\map y a = A_1, \\map y b = B_1$ :$\\map z a = A_2, \\map z b = B_2$ Let $J \\sqbrk {y, z}$ have an extremum for the curve $y = \\map y x, z = \\map z x$. Let $g_y$ and $g_z$ not simultaneously vanish at any point of the surface $g = 0$. Then there exists a function $\\map \\lambda x$ such that the curve $y = \\map y x, z = \\map z x$ is an extremal of the functional: :$\\ds \\int_a^b \\paren {F + \\map \\lambda x g} \\rd x$ In other words, $y = \\map y x$ satisfies the differential equations: {{begin-eqn}} {{eqn | l = F_y + \\lambda g_y - \\frac \\d {\\d x} F_{y'}       | r = 0 }} {{eqn | l = F_z + \\lambda g_z - \\frac \\d {\\d x} F_{z'}       | r = 0       | c =  }} {{end-eqn}}"}
+{"_id": "13523", "title": "Empty Intersection iff Subset of Relative Complement", "text": "Let $S$ be a set. Let $A, B$ be subset of $S$. Then $A \\cap B = \\varnothing \\iff A \\subseteq \\complement_S\\left({B}\\right)$"}
+{"_id": "13524", "title": "Subset in Subsets", "text": "Let $S, B$ be sets. Let $A$ be subset of $S$. Then: :$A \\subseteq B \\iff \\forall x \\in S: x \\in A \\implies x \\in B$"}
+{"_id": "13526", "title": "Empty Set as Subset", "text": "Let $S$ be a set. Let $A$ be a subset of $S$. Then: :$A = \\O \\iff \\forall x \\in S: x \\notin A$"}
+{"_id": "13527", "title": "Pythagorean Triangles whose Areas are Repdigit Numbers", "text": "The following Pythagorean triangles have areas consisting of repdigit numbers:"}
+{"_id": "13528", "title": "Exponential of Series Equals Infinite Product", "text": "Let $\\left\\langle{z_n}\\right\\rangle$ be a sequence of complex numbers. Suppose $\\displaystyle \\sum_{n \\mathop = 1}^\\infty z_n$ converges to $z \\in \\C$. Then $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\exp \\left({z_n}\\right)$ converges to $\\exp z$."}
+{"_id": "13530", "title": "Absolute Value of Infinite Product", "text": "Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,}}$ be a valued field. Let $\\sequence{a_n}$ be a sequence in $\\mathbb K$."}
+{"_id": "13531", "title": "Convergence of Series of Complex Numbers by Real and Imaginary Part", "text": "Let $\\sequence {z_n}$ be a sequence of complex numbers. Then: : the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty z_n$ converges to $Z \\in \\C$ {{iff}}: : the series: ::$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\Re \\paren {z_n}$ :and: ::$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\Im \\paren {z_n}$ :converge to $\\Re \\paren Z$ and $\\Im \\paren Z$ respectively."}
+{"_id": "13533", "title": "Lower Closure of Element is Closed under Directed Suprema", "text": "Let $L = \\struct {S, \\preceq}$ be an up-complete ordered set. Let $x \\in S$. Then $x^\\preceq$ is closed under directed suprema, where $x^\\preceq$ denotes the lower closure of $x$."}
+{"_id": "13534", "title": "Pythagorean Triangle whose Hypotenuse and Leg differ by 1", "text": "Let $P$ be a Pythagorean triangle whose sides correspond to the Pythagorean triple $T$. Then: : the hypotenuse of $P$ is $1$ greater than one of its legs {{iff}}: : the generator for $T$ is of the form $G = \\tuple {n, n + 1}$ where $n \\in \\Z_{> 0}$ is a (strictly) positive integer."}
+{"_id": "13538", "title": "General Variation of Integral Functional/Dependent on N Functions", "text": "Let $J$ be a (real) functional of the form $\\ds J \\sqbrk {\\ldots y_i \\ldots} = \\int_{x_0}^{x_1} \\map F {x, \\ldots y_i \\ldots, \\ldots y_i' \\ldots} \\rd x, i = \\openint 1 n$ Then: $\\ds \\delta J = \\int_{x_0}^{x_1} \\sum_{i \\mathop = 1}^n \\paren {F_{y_i} - \\frac \\d {\\d x} F_{y_i'} } \\map {h_i} x + \\intlimits {\\sum_{i \\mathop = 1}^n F_{y_i'} \\delta y_i} {x \\mathop = x_0} {x \\mathop = x_1} + \\intlimits {\\paren {F - \\sum_{i \\mathop = 1}^n y_i'F_{y_i'} } \\delta x} {x \\mathop = x_0} {x \\mathop = x_1}$"}
+{"_id": "13539", "title": "Kusmin-Landau Inequality", "text": "Let $I$ be the half-open interval $\\hointl a b$. Let $f: I \\to R$ be continuously differentiable. Let $f'$ be monotonic. Let $\\norm {f'} \\ge \\lambda$ on $I$ for some $\\lambda \\in \\R_{>0}$, where $\\norm {\\, \\cdot \\,}$ denotes the distance to nearest integer. Then: :$\\displaystyle \\sum_{n \\mathop \\in I} e^{2 \\pi i \\map f n} = \\map O {\\frac 1 \\lambda}$ where the big-O estimate does not depend on $f$."}
+{"_id": "13540", "title": "Closure of Singleton is Lower Closure of Element in Scott Topological Lattice", "text": "Let $T = \\struct {S, \\preceq, \\tau}$ be a up-complete topological lattice with Scott topology. Let $x \\in S$. Then: :$\\set x^- = x^\\preceq$ where :$\\set x^-$ denotes the topological closure of $\\set x$ :$x^\\preceq$ denotes the lower closure of $x$."}
+{"_id": "13541", "title": "Pythagorean Triangle cannot be Isosceles", "text": "Let $P$ be a Pythagorean triangle. Then $P$ is not isosceles."}
+{"_id": "13542", "title": "Brahmagupta-Fibonacci Identity/Corollary", "text": ":$\\left({a^2 + b^2}\\right) \\left({c^2 + d^2}\\right) = \\left({a c - b d}\\right)^2 + \\left({a d + b c}\\right)^2$"}
+{"_id": "13546", "title": "Scott Topological Lattice is T0 Space", "text": "Let $T = \\left({S, \\preceq, \\tau}\\right)$ be a complete topological lattice with Scott topology. Then $T$ is a $T_0$ space."}
+{"_id": "13547", "title": "Lower Closures are Equal implies Elements are Equal", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an ordered set. Let $x, y \\in S$ such that :$x^\\preceq = y^\\preceq$ where $x^\\preceq$ denotes the lower closure of $x$. Then $x = y$"}
+{"_id": "13548", "title": "General Variation of Integral Functional/Dependent on N Functions/Canonical Variables", "text": "Let $\\delta J$ be a general variation of integral functional dependent on n functions. Suppose a following coordinate transformation is done: :$\\set {x, \\ldots y_i, \\ldots, \\ldots, y_i', \\ldots, F} \\to \\set {x, \\ldots, y_i, \\ldots, \\ldots p', \\ldots, H}, i = \\tuple {1, \\ldots, n}$ Then, in canonical variables: :$\\displaystyle \\delta J = \\int_{x_0}^{x_1} \\sum_{i \\mathop = 1}^n \\paren {F_{y_i} - \\dfrac {\\d {p_i} } {\\d x} } \\map {h_i} x \\rd x + \\intlimits {\\sum_{i \\mathop = 1}^n p_i \\delta y_i - H \\delta x} {x \\mathop = x_0} {x \\mathop = x_1}$ where: {{begin-eqn}} {{eqn | l = \\bigvalueat {\\delta x} {x \\mathop = x_j}       | r = \\delta x_j }} {{eqn | l = \\bigvalueat {\\delta y_i} {x \\mathop = x_j}       | r = \\delta_i^j }} {{eqn | l = j       | r = \\tuple {0, 1} }} {{end-eqn}} {{explain|the meaning of $j {{=}} \\tuple {0, 1}$ in this context, and hence the meaning of $x_j$ and $\\delta x_j$, and also $\\delta_i^j$, all of which are obscure}}"}
+{"_id": "13549", "title": "Square of Pythagorean Prime is Hypotenuse of Pythagorean Triangle", "text": "Let $p$ be a Pythagorean prime. Then $p^2$ is the hypotenuse of a Pythagorean triangle."}
+{"_id": "13550", "title": "Square of Hypotenuse of Pythagorean Triangle is Difference of two Cubes/Refutation", "text": "Let $h$ be the hypotenuse of a Pythagorean triangle. Then it is not necessarily the case that: :$h^2 = a^3 - b^3$ for some $a, b \\in \\Z_{>0}$."}
+{"_id": "13551", "title": "Pythagorean Triangle from Sum of Reciprocals of Consecutive Same Parity Integers", "text": "Let $a, b \\in \\Z_{>0}$ be (strictly) positive integers such that they are consecutively of the same parity. Let $\\dfrac p q = \\dfrac 1 a + \\dfrac 1 b$. Then $p$ and $q$ are the legs of a Pythagorean triangle."}
+{"_id": "13553", "title": "Rational Number plus Irrational Number is Irrational", "text": "Rational number plus irrational number is irrational. That is, let $x \\in \\Q$, $y \\in \\R \\setminus \\Q$ and $x + y = z$. Then $z \\in \\R \\setminus \\Q$."}
+{"_id": "13555", "title": "Lower Closure of Element is Topologically Closed in Scott Topological Ordered Set", "text": "Let $T = \\left({S, \\preceq, \\tau}\\right)$ be a relational structure with Scott topology where $\\left({S, \\preceq}\\right)$ is an up-complete ordered set. Let $x \\in S$. Then $x^\\preceq$ is topologically closed, where $x^\\preceq$ denotes the lower closure of $x$."}
+{"_id": "13556", "title": "Complement of Lower Closure of Element is Open in Scott Topological Ordered Set", "text": "Let $T = \\struct {S, \\preceq, \\tau}$ be a relational structure with Scott topology where $\\struct {S, \\preceq}$ is an up-complete ordered set. Let $x \\in S$. Then $\\relcomp S {x^\\preceq}$ is topologically open, where :$x^\\preceq$ denotes the lower closure of $x$, :$\\relcomp S {x^\\preceq}$ denotes the relative complement of $x^\\preceq$."}
+{"_id": "13557", "title": "Open iff Upper and with Property (S) in Scott Topological Lattice", "text": "Let $T = \\left({S, \\preceq, \\tau}\\right)$ be an up-complete topological lattice. Let $A$ be a subset of $S$. Then $A$ is open {{iff}} $A$ is upper and with property (S)."}
+{"_id": "13558", "title": "Necessary Condition for Integral Functional to have Extremum for given Function/Non-differentiable at Intermediate Point", "text": "Let $y, F$ be real functions. Let $y$ be continuously differentiable for $x \\in \\hointr a c \\cap  \\hointl c b$ and satisfy: :$\\map y a = A$ :$\\map y b = B$ Let $J\\sqbrk y$ be a functional of the form :$\\ds J \\sqbrk y = \\int_a^b \\map F {x, y, y'} \\rd x$ Then the functional $J$ has a weak extremum if $y$ satisfies the following system of equations: {{begin-eqn}} {{eqn | l = F_y - \\dfrac \\d {\\d x} F_{y'}       | r = 0 }} {{eqn | l = \\bigvalueat {F_{y'} } {x \\mathop = c \\mathop - 0}       | r = \\bigvalueat {F_{y'} } {x \\mathop = c \\mathop + 0} }} {{eqn | l = \\bigvalueat {\\paren {F - y' F_{y'} } } {x \\mathop = c \\mathop - 0}       | r = \\bigvalueat {\\paren {F - y' F_{y'} } } {x \\mathop = c \\mathop + 0} }} {{end-eqn}} where, by the use of limit from the left and limit from the right, the following abbreviations are denoted as follows: :$\\ds \\bigvalueat {\\map y x} {x \\mathop = c \\mathop + 0} = \\lim_{x \\mathop \\to c^+} \\map y x$ :$\\ds \\bigvalueat {\\map y x} {x \\to x \\mathop = c \\mathop - 0} = \\lim_{x \\mathop \\to c^-} \\map y x$ The last two equations are known as the '''Weierstrass-Erdmann corner conditions'''. {{refactor|move this definition to a separate page once a general principle can be distilled}}"}
+{"_id": "13559", "title": "Euler's Equation for Vanishing Variation in Canonical Variables", "text": "{{refactor|There are a number of pages linking here with the presentation of the link set as \"momenta\". This is going to need a definition of its own, but it is not clear what that is from looking at this page.}} Consider the following system of differential equations: :$(1): \\quad \\begin {cases} F_{y_i} - \\dfrac \\d {\\d x} F_{y_i'} = 0 \\\\ \\dfrac {\\d {y_i} } {\\d x} = y_i'\\end{cases}$ where $i \\in \\set {1, \\ldots, n}$. Let the coordinates $\\tuple {x, \\family {y_i}_{1 \\mathop \\le i \\mathop \\le n}, \\family {y_i'}_{1 \\mathop \\le i \\mathop \\le n}, F}$ be transformed to canonical variables: :$\\tuple {x, \\family {y_i}_{1 \\mathop \\le i \\mathop \\le n}, \\family {p_i}_{1 \\mathop \\le i \\mathop \\le n}, H}$ Then the system $(1)$ is transformed into: :$\\begin {cases} \\dfrac {\\d y_i} {\\d x} = \\dfrac {\\partial H} {\\partial p_i} \\\\ \\dfrac {\\d p_i} {\\d x} = -\\dfrac {\\partial H} {\\partial y_i} \\end {cases}$"}
+{"_id": "13560", "title": "Relational Structure with Topology of Subsets with Property (S) is Topological Space", "text": "Let $T = \\left({S, \\preceq, \\tau}\\right)$ be a relational structure with topology where :$\\left({S, \\preceq}\\right)$ is an up-complete ordered set, :$\\tau$ is the set of all subsets of $S$ with property (S). Then $\\left({S, \\tau}\\right)$ is topological space."}
+{"_id": "13561", "title": "Subgroup of Real Numbers is Discrete or Dense", "text": "Let $G$ be a subgroup of the additive group of real numbers. Then one of the following holds: :$G$ is dense in $\\R$. :$G$ is discrete and there exists $a \\in \\R$ such that $G = a \\Z$, that is, $G$ is cyclic."}
+{"_id": "13562", "title": "Discrete Subgroup of Real Numbers is Closed", "text": "Let $G$ be a subgroup of the additive group of real numbers. Let $G$ be discrete. Then $G$ is closed."}
+{"_id": "13563", "title": "Consecutive Integers with Same Sigma", "text": "Let $\\sigma: \\Z_{>0} \\to \\Z_{>0}$ be the $\\sigma$ function, defined on the strictly positive integers. The equation: :$\\map \\sigma n = \\map \\sigma {n + 1}$ is satisfied by integers in the sequence: :$14, 206, 957, 1334, 1364, 1634, 2685, 2974, 4364, 14841, 18873, \\ldots$ {{OEIS|A002961}}"}
+{"_id": "13565", "title": "Coarser Between Generator Set and Filter is Generator Set of Filter", "text": "Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Let $F$ be a filter on $L$. Let $G$ be a generator set of $F$. Let $A$ be a subset of $S$ such that :$G$ is coarser than $A$ and $A$ is coarser than $F$. Then $A$ is generator set of $F$."}
+{"_id": "13566", "title": "Borsuk-Ulam Theorem", "text": "Let $n$ be a positive integer. Let $f: \\mathbb S^n \\to \\R^n$ be a continuous mapping from an $n$-sphere to $\\R^n$. Then there exists $x \\in \\mathbb S^n$ such that $\\map f x = \\map f {-x}$."}
+{"_id": "13567", "title": "Gershgorin Circle Theorem", "text": "Let $n$ be a positive integer. Let $A = \\sqbrk {a_{i j} }$ be a complex square matrix of order $n$. Let $\\lambda$ be an eigenvalue of $A$. Then there exists $i \\in \\set {1, 2, \\ldots, n}$ such that: :$\\lambda \\in \\map {\\mathbb D} {a_{i i}, R_i}$ where: :$\\displaystyle R_i = \\sum_{j \\mathop \\ne i} \\cmod {a_{ i j} }$ :$\\map {\\mathbb D} {a, R}$ denotes the complex disk of center $a$ and radius $R$."}
+{"_id": "13569", "title": "Finite Infima Set of Coarser Subset is Coarser than Finite Infima Set", "text": "Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Let $A, B$ be subsets of $S$ such that :$A$ is coarser than $B$. Then $\\operatorname{fininfs} \\left({A}\\right)$ is coarser than $\\operatorname{fininfs} \\left({B}\\right)$ where $\\operatorname{fininfs} \\left({B}\\right)$ denotes the finite infima set of $B$."}
+{"_id": "13570", "title": "Approximations to Equilateral Triangles by Heronian Triangles", "text": "The sequence of best approximations to an equilateral triangle by a Heronian triangle begins: :The $\\paren {3, 4, 5}$ triangle, with area $6$ :The $\\paren {13, 14, 15}$ triangle, with area $84$, where $14 = 4^2 - 2$ :The $\\paren {193, 194, 195}$ triangle, where $194 = 14^2 - 2$ :The $\\paren {37 \\, 633, 37 \\, 634, 37 \\, 635}$ triangle, where $37 \\, 634 = 194^2 - 2$ and so on. {{OEIS|A003010}}"}
+{"_id": "13571", "title": "Legendre Transform is Involution", "text": "The Legendre transform is an Involution."}
+{"_id": "13572", "title": "Upper Closure of Coarser Subset is Subset of Upper Closure", "text": "Let $L = \\left({S, \\preceq}\\right)$ be a preordered set. Let $A, B$ be subsets of $S$ such that :$A$ is coarser than $B$. Then $A^\\succeq \\subseteq B^\\succeq$"}
+{"_id": "13573", "title": "Set Coarser than Upper Set is Subset", "text": "Let $\\left({S, \\preceq}\\right)$ be a preordered set. Let $A, B$ be subsets of $S$ such that :$A$ is coarser than $B$ and :$B$ is an upper set. Then $A \\subseteq B$"}
+{"_id": "13574", "title": "Sum of 4 Unit Fractions that equals 1", "text": "There are $14$ ways to represent $1$ as the sum of exactly $4$ unit fractions."}
+{"_id": "13575", "title": "Sum of 5 Unit Fractions that equals 1", "text": "There are $147$ ways to represent $1$ as the sum of exactly $5$ unit fractions."}
+{"_id": "13576", "title": "Sum of 6 Unit Fractions that equals 1", "text": "There are $3462$ ways to represent $1$ as the sum of exactly $6$ unit fractions."}
+{"_id": "13577", "title": "Conditions for Function to be Maximum of its Legendre Transform Two-variable Equivalent", "text": "Let $x, p \\in \\R$. Let $\\map f x$ be a strictly convex real function. Let $f^*$ be a Legendre transformed $f$. Let $\\map g {x, p} = - \\map {f^*} p + x p$ Then: :$\\displaystyle \\map f x = \\max_p \\paren {-\\map {f^*} p + x p}$ where $\\displaystyle \\max_p$ maximises the function with respect to a variable $p$."}
+{"_id": "13578", "title": "Set is Coarser than Image of Mapping of Infima", "text": "Let $\\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Let $f, g:\\N \\to S$ be mappings such that :$\\forall n \\in \\N: g\\left({n}\\right) = \\inf \\left\\{ {f\\left({m}\\right): m \\in \\N \\land m \\le n}\\right\\}$ Then $f\\left[{\\N}\\right]$ is coarser than $g\\left[{\\N}\\right]$ where $f\\left[{\\N}\\right]$ denotes the image of mapping $f$."}
+{"_id": "13579", "title": "Solutions of Ramanujan-Nagell Equation", "text": "Solutions to the Ramanujan-Nagell equation: :$x^2 + 7 = 2^n$ exist for only $5$ values of $n$: :$3, 4, 5, 7, 15$ {{OEIS|A060728}} The corresponding values of $x$ are: :$1, 3, 5, 11, 181$ {{OEIS|A038198}}"}
+{"_id": "13580", "title": "Five Ramanujan-Nagell Numbers", "text": "There exist exactly $5$ Ramanujan-Nagell numbers: positive integers of the form $2^m - 1$ which are triangular: :$0, 1, 3, 15, 4095$ {{OEIS|A076046}}"}
+{"_id": "13581", "title": "Conditions for Functional to be Extremum of Two-variable Functional over Canonical Variable p", "text": "Let $y = \\map y x$ and $\\map F {x, y, y'}$ be real functions. Let $\\dfrac {\\partial^2 F} {\\partial {y'}^2} \\ne 0$. Let $\\ds J \\sqbrk y = \\int_a^b \\map F {x, y, y'} \\rd x$ Let $\\ds J \\sqbrk {y, p} = \\int_a^b \\paren {-\\map H {x, y, p} + p y'} \\rd x$, where $H$ is the Hamiltonian of $J \\sqbrk y$. Then $\\ds J \\sqbrk y = \\bigvalueat {J \\sqbrk {y, p} } {\\frac {\\delta J \\sqbrk{y, p} } {\\delta p} \\mathop = 0}$"}
+{"_id": "13582", "title": "Image of Mapping of Infima is Generator Set of Filter", "text": "Let $\\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Let $f, g:\\N \\to S$ be mappings such that :$\\forall n \\in \\N: g\\left({n}\\right) = \\inf \\left\\{ {f\\left({m}\\right): m \\in \\N \\land m \\le n}\\right\\}$ Let $F$ be a filter such that :$f\\left[{\\N}\\right]$ is generator set of $F$, where $f\\left[{\\N}\\right]$ denotes the image of $f$. Then $g\\left[{\\N}\\right]$ is generator set of $F$."}
+{"_id": "13583", "title": "Subset of Set is Coarser than Set", "text": "Let $\\left({S, \\preceq}\\right)$ be a preordered set. Let $A, B$ be subset of $S$ such that :$A \\subseteq B$ Then $A$ is coarser than $B$."}
+{"_id": "13584", "title": "Definition:Ramanujan-Nagell Number", "text": "A '''Ramanujan-Nagell number''' is a positive integer of the form $2^m - 1$ which is also triangular."}
+{"_id": "13585", "title": "Existence of Product of Three Distinct Primes between n and 2n", "text": "Let $n \\in \\Z$ be an integer such that $n > 15$. Then between $n$ and $2 n$ there exists at least one integer which is the product of $3$ distinct prime numbers."}
+{"_id": "13586", "title": "Second Column and Diagonal of Pascal's Triangle consist of Triangular Numbers", "text": "The $2$nd column and $2$nd diagonal of Pascal's triangle consists of the set of triangular numbers."}
+{"_id": "13587", "title": "Complement of Element is Irreducible implies Element is Meet Irreducible", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $P = \\left({\\tau, \\preceq}\\right)$ be an ordered set where $\\mathord\\preceq = \\mathord\\subseteq \\cap \\left({\\tau \\times \\tau}\\right)$ Let $A \\in \\tau$. Then $\\complement_S\\left({A}\\right)$ is irreducible implies $A$ is meet irreducible in $P$ where $\\complement_S\\left({A}\\right)$ denotes the relative complement of $A$ relative to $S$."}
+{"_id": "13588", "title": "Pentagonal Number as Sum of Triangular Numbers", "text": "Let $P_n$ be the $n$th pentagonal number. Then: :$P_n = T_n + 2 T_{n - 1}$ where $T_n$ is the $n$th triangular number."}
+{"_id": "13590", "title": "Conditions for Integral Functionals to have same Euler's Equations", "text": "Let $\\mathbf y$ be a real $n$-dimensional vector-valued function. Let $\\map F {x, \\mathbf y, \\mathbf y'}$, $\\map \\Phi {x, \\mathbf y}$  be real functions. Let $\\Phi$ be twice differentiable. Let: {{begin-eqn}} {{eqn | l = \\Psi       | r = \\frac {\\d \\Phi} {\\d x} }} {{eqn | r = \\frac {\\partial \\Phi} {\\partial x} + \\sum_{i \\mathop = 1}^n \\frac {\\partial \\Phi} {\\partial y_i} y_i' }} {{end-eqn}} Let $J_1$, $J_2$ be functionals such that: :$\\ds J_1 \\sqbrk {\\mathbf y} = \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ :$\\ds J_2 \\sqbrk {\\mathbf y} = \\int_a^b \\paren {\\map F {x, \\mathbf y, \\mathbf y'} + \\map \\Psi {x, \\mathbf y, \\mathbf y'} } \\rd x$  Then $J_1$ and $J_2$ have same Euler's Equations."}
+{"_id": "13591", "title": "Square of Odd Number as Difference between Triangular Numbers", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Then: :$\\exists a, b \\in \\Z_{\\ge 0}: \\paren {2 n + 1}^2 = T_a - T_b$ where: :$T_a$ and $T_b$ are triangular numbers :$T_a$ and $T_b$ are coprime. That is, the square of every odd number is the difference between two coprime triangular numbers."}
+{"_id": "13592", "title": "Meet in Inclusion Ordered Set", "text": "Let $P = \\left({X, \\subseteq}\\right)$ be an inclusion ordered set. Let $A, B \\in X$ such that :$A \\cap B \\in X$ Then $A \\wedge B = A \\cap B$"}
+{"_id": "13594", "title": "Cube Number as Difference between Squares of Triangular Numbers", "text": "Let $n \\in \\Z_{> 0}$ be a positive integer. Then: :$n^3 = {T_n}^2 - {T_{n - 1} }^2$ where $T_n$ denotes the $n$th triangular number."}
+{"_id": "13595", "title": "Square of Triangular Number equals Sum of Sequence of Cubes", "text": ":$\\displaystyle \\sum_{i \\mathop = 1}^n i^3 = {T_n}^2$ where $T_n$ denotes the $n$th triangular number."}
+{"_id": "13599", "title": "Sum of Sequence of Fifth Powers", "text": ":$\\displaystyle \\sum_{i \\mathop = 1}^n i^5 = \\dfrac { {T_n}^2 \\paren {4 T_n - 1} } 3$ where $T_n$ denotes the $n$th triangular number."}
+{"_id": "13600", "title": "Sum of Adjacent Sequences of Triangular Numbers", "text": "{{begin-eqn}} {{eqn | l = T_1 + T_2 + T_3       | r = T_4       | c =  }} {{eqn | l = T_5 + T_6 + T_7 + T_8       | r = T_9 + T_{10}       | c =  }} {{eqn | l = T_{11} + T_{12} + T_{13} + T_{14} + T_{15}       | r = T_{16} + T_{17} + T_{18}       | c =  }} {{end-eqn}} and so on. The $n$th line of the pattern can be written as: :$\\displaystyle \\sum_{k \\mathop = n^2 + n - 1}^{n^2 + 2 n} T_n = \\sum_{k \\mathop = n^2 + 2 n + 1}^{n^2 + 3 n} T_n$"}
+{"_id": "13602", "title": "Conditions for Transformation to be Canonical", "text": "Let: :$\\ds J_1 \\sqbrk {\\sequence {y_i}_{1 \\mathop \\le i \\mathop \\le n}, \\sequence {p_i}_{1 \\mathop \\le i \\mathop \\le n} } = \\int_a^b \\paren {\\sum_{i \\mathop = 1}^n p_i y_i'-H} \\rd x$ :$\\ds J_2 \\sqbrk {\\sequence {Y_i}_{1 \\mathop \\le i \\mathop \\le n}, \\sequence {P_i}_{1 \\mathop \\le i \\mathop \\le n} } = \\int_a^b \\paren {\\sum_{i \\mathop = 1}^n P_i Y_i'-H^*} \\rd x$ be functionals. Then $\\paren {\\sequence {y_i}_{1 \\mathop \\le i \\mathop \\le n}, \\sequence{p_i}_{1 \\mathop \\le i \\mathop \\le n}, H} \\to \\paren {\\sequence{Y_i}_{1 \\mathop \\le i \\mathop \\le n}, \\sequence {P_i}_{1 \\mathop \\le i \\mathop \\le n}, H^*}$ is a canonical transformation if: :$\\ds \\sum_{i \\mathop = 1}^n p_i y_i' - H = \\sum_{i \\mathop = 1}^n P_i Y_i' - H^* \\pm \\dfrac {\\d \\Phi} {\\d x}$   and: :$\\ds p_i = \\mp \\frac {\\partial \\Phi} {\\d y_i}, \\quad P_i = \\pm \\frac {\\partial \\Phi} {\\d Y_i}, \\quad H = H^* \\mp \\frac {\\partial \\Phi} {\\partial x}$"}
+{"_id": "13603", "title": "Square of Triangular Numbers as Sum of Triangular Numbers", "text": ":${T_n}^2 = T_n + T_{n - 1} T_{n + 1}$ where $T_n$ denotes the $n$th triangular number."}
+{"_id": "13605", "title": "Sum of Sequence of Reciprocals of Triangular Numbers", "text": ":$\\displaystyle \\sum_{k \\mathop \\ge 1} \\dfrac 1 {T_k} = 2$ where $T_k$ denotes the $k$th triangular number."}
+{"_id": "13606", "title": "Triangular Number Pairs with Triangular Sum and Difference", "text": "The sequence of pairs of triangular numbers whose sum and difference are also both triangular begins: :$\\tuple {15, 21}, \\tuple {105, 171}, \\tuple {378, 703}, \\tuple {780, 990}, \\tuple {1485, 4186}, \\tuple {2145, 3741}, \\tuple {5460, 6786}, \\tuple {7875, 8778}$ {{OEIS|A185129|order = first}} {{OEIS|A185128|order = second}}"}
+{"_id": "13607", "title": "Triangular Number whose Square is Triangular", "text": "The only triangular number with less than $660$ digits, whose square is also triangular, is $6$."}
+{"_id": "13608", "title": "Factors of Integer Congruent to 5 modulo 6", "text": "Let $m$ be an positive integer. Let $m \\equiv 5 \\pmod 6$. Then $m$ has two divisors whose sum is divisible by $6$."}
+{"_id": "13609", "title": "Top in Ordered Set of Topology", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $P = \\left({\\tau, \\subseteq}\\right)$ be an inclusion ordered set of $\\tau$. Then $P$ is bounded above and $\\top_P = S$ where $\\top_P$ denotes the greatest element in $P$."}
+{"_id": "13610", "title": "Triangular Numbers which are also Square", "text": "Let $A_n$ be the $n$th non-negative integer whose square is also a triangular number. Then: :$A_n = \\begin{cases} 0 & : n = 0 \\\\ 1 & : n = 1 \\\\ 6 A_{n - 1} - A_{n - 2} & : n > 1 \\end{cases}$"}
+{"_id": "13611", "title": "Index of Square Triangular Number from Preceding", "text": "Let $T_n$ be the $n$th triangular number. Let $T_n$ be square. Then $T_{4 n \\paren {n + 1} }$ is also square."}
+{"_id": "13613", "title": "Triangular Number cannot be Fourth Power", "text": "Let $T_n$ be the $n$th triangular number such that $n > 1$. Then $T_n$ cannot be the $4$th power of an integer."}
+{"_id": "13614", "title": "Triangular Number cannot be Fifth Power", "text": "Let $T_n$ be the $n$th triangular number such that $n > 1$. Then $T_n$ cannot be the $5$th power of an integer."}
+{"_id": "13615", "title": "Hilbert-Waring Theorem/Variant Form", "text": "For each $k \\in \\Z: k \\ge 2$, there exists a positive integer $G \\left({k}\\right)$ such that every sufficiently large positive integer can be expressed as a sum of at most $G \\left({k}\\right)$ $k$th powers."}
+{"_id": "13618", "title": "Equality of Integers to the Power of Each Other", "text": "$2$ and $4$ are the only pair of positive integers $m, n$ such that $m \\ne n$ such that: :$m^n = n^m$"}
+{"_id": "13620", "title": "Square whose Perimeter equals its Area", "text": "The $4 \\times 4$ square is the only square whose area in square units equals its perimeter in units. The area and perimeter of this square are $16$."}
+{"_id": "13622", "title": "Magic Constant of Order 4 Magic Square", "text": "The magic constant of the order $4$ magic square is $34$."}
+{"_id": "13625", "title": "Sum of Cubes on Diagonals of Moessner's Order 4 Magic Square", "text": "The sums of the cubes of the entries on the diagonals of Moessner's order $4$ magic square are equal."}
+{"_id": "13626", "title": "Sum of Squares on Pairs of Rows and Columns of Moessner's Order 4 Magic Square", "text": "The sums of the squares of the entries are equal on the following pairs of rows and columns of Moessner's order $4$ magic square: :Rows $1$ and $4$ :Rows $2$ and $3$ :Columns $1$ and $4$ :Columns $2$ and $3$."}
+{"_id": "13628", "title": "Equivalence of Definitions of Quasiperfect Number", "text": "The following definitions of a quasiperfect number are equivalent:"}
+{"_id": "13629", "title": "Quasiperfect Number is Square of Odd Integer", "text": "Let $n$ be a quasiperfect number. Then: :$n = \\paren {2 k + 1}^2$ for some $k \\in \\Z_{>0}$. That is, a quasiperfect number is the square of an odd integer."}
+{"_id": "13630", "title": "Bottom in Ordered Set of Topology", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $P = \\left({\\tau, \\subseteq}\\right)$ be an inclusion ordered set of $\\tau$. Then $P$ is bounded below and $\\bot_P = \\varnothing$"}
+{"_id": "13631", "title": "Period of Reciprocal of 17 is of Maximal Length", "text": "The decimal expansion of the reciprocal of $17$ has the maximum period, that is: $16$: :$\\dfrac 1 {17} = 0 \\cdotp \\dot 05882 \\, 35294 \\, 11764 \\, \\dot 7$ {{OEIS|A007450}}"}
+{"_id": "13632", "title": "Prime Dudeney Number", "text": "The only prime Dudeney number is $17$:"}
+{"_id": "13633", "title": "Sequence of Dudeney Numbers", "text": "The only Dudeney numbers are: :$0, 1, 8, 17, 18, 26, 27$ two of which are themselves cubes, and one of which is prime."}
+{"_id": "13634", "title": "Derivation of Hamilton-Jacobi Equation", "text": "Let $\\map S {x_0, x_1, \\mathbf y} = \\map S {x, \\mathbf y}$ be the geodetic distance, where $x_0$ is fixed and $x_1=x$. Let $H$ be Hamiltonian. Then the following equation holds: :$\\dfrac {\\partial S} {\\partial x} + \\map H {x, \\mathbf y, \\nabla_{\\mathbf y} S} = 0$ and is known as the Hamilton-Jacobi Equation."}
+{"_id": "13635", "title": "Partial Derivatives of Solution of Hamilton-Jacobi Equation are First Integrals of Euler's Equations", "text": "Let $\\mathbf y = \\sequence {y_i}_{1 \\mathop \\le i \\mathop \\le n}$, $\\boldsymbol \\alpha = \\sequence {\\alpha_i}_{1 \\mathop \\le i \\mathop \\le m}$ be vectors, where $m \\le n$. Let $S = \\map S {x, \\mathbf y, \\boldsymbol \\alpha}$ be a solution of the Hamilton-Jacobi equation, where $\\boldsymbol \\alpha$ are parameters. Then each partial derivative: :$\\dfrac {\\partial S} {\\partial \\alpha_i}$ is a first integral of canonical Euler's equations."}
+{"_id": "13636", "title": "Jacobi's Theorem", "text": "Let $\\mathbf y = \\sequence {y_i}_{1 \\le i \\le n}$, $\\boldsymbol \\alpha = \\sequence {\\alpha_i}_{1 \\le i \\le n}$, $\\boldsymbol \\beta = \\sequence {\\beta_i}_{1 \\le i \\le n}$ be vectors, where $\\alpha_i$ and $ \\beta_i$ are parameters. Let $S = \\map S {x, \\mathbf y, \\boldsymbol \\alpha}$ be a a complete solution of the Hamilton-Jacobi equation. Let: :$\\begin {vmatrix} \\dfrac {\\partial^2 S} {\\partial \\alpha_i \\partial y_k} \\end{vmatrix} \\ne 0$ where $\\begin {vmatrix} \\cdot \\end{vmatrix}$ is a determinant. Let: :$\\dfrac {\\partial S} {\\partial \\alpha_i} = \\beta_i$ Then: :$p_i = \\map {\\dfrac {\\partial S} {\\partial y_i} } {x, \\mathbf y, \\boldsymbol \\alpha}$ :$y_i = \\map {y_i} {x, \\boldsymbol \\alpha, \\boldsymbol \\beta}$ constitute a general solution of the canonical Euler's equations."}
+{"_id": "13637", "title": "Equivalence of Definitions of Principal Ideal", "text": "Let $\\left({S, \\preceq}\\right)$ be a preordered set. Let $I$ be an ideal in $S$. Then :the definitions of principal ideal are equivalent, That means that :$\\exists x \\in I: x$ is upper bound for $I$ {{iff}} :$\\exists x \\in S: I = x^\\preceq$ where $x^\\preceq$ denotes the lower closure of $x$."}
+{"_id": "13638", "title": "Divisions of Numbers in Unit Interval with Numbers in Different Intervals", "text": "Let $I = \\openint 0 1$ be the open unit interval. Let $a_1, a_2, a_3, \\ldots, a_n$ be real numbers chosen in $I$ such that: :$a_1$ and $a_2$ are in different halves of $I$ :$a_1, a_2$ and $a_3$ are in different thirds of $I$ :$a_1, a_2, a_3$ and $a_4$ are in different quarters of $I$ and so on. Then $n \\le 17$. That is, for the conditions to be fulfilled, no more than $17$ numbers can be chosen."}
+{"_id": "13639", "title": "Seventeen Different Wallpaper Patterns", "text": "There are $17$ different symmetry groups for a wallpaper pattern."}
+{"_id": "13641", "title": "Compact Element iff Principal Ideal", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let $P = \\left({\\mathit{Ids}\\left({L}\\right), \\precsim}\\right)$ be an inclusion ordered set where :$\\mathit{Ids}\\left({L}\\right)$ denotes the set of all ideals in $L$, :$\\mathord\\precsim = \\mathord\\subseteq \\cap \\left({\\mathit{Ids}\\left({L}\\right) \\times \\mathit{Ids}\\left({L}\\right)}\\right)$ Let $x \\in \\mathit{Ids}\\left({L}\\right)$ Then $x$ is compact element {{iff}} $x$ is principal ideal in $L$"}
+{"_id": "13642", "title": "Integers as Sum of Three Pairwise Coprime Integers", "text": "Let $n$ be an integer greater than $17$. Then $n$ is the sum of $3$ integers greater than $1$ which are pairwise coprime."}
+{"_id": "13643", "title": "Primes for which Powers to Themselves minus 1 have Common Factors", "text": "Let $p$ and $q$ be prime numbers such that $p^p - 1$ and $q^q - 1$ have a common divisor $d$. The only known $p$ and $q$ such that $d < 400 \\, 000$ are $p = 17, q = 3313$."}
+{"_id": "13644", "title": "Reversal formed by Repdigits of Base minus 1 by Addition and Multiplication", "text": "Let $b \\in \\Z_{>1}$ be an integer greater than $1$. Let $n = b^k - 1$ for some integer $k$ such that $k \\ge 1$. Then: : $n + n$ is the reversal of $\\left({b - 1}\\right) n$ when both are expressed in base $b$ representation."}
+{"_id": "13645", "title": "Power of Base minus 1 is Repdigit Base minus 1", "text": "Let $b \\in \\Z_{>1}$ be an integer greater than $1$. Let $n = b^k - 1$ for some integer $k$ such that $k \\ge 1$. Let $n$ be expressed in base $b$ representation. Then $n$ is a repdigit number consisting of $k$ instances of digit $b - 1$."}
+{"_id": "13646", "title": "Multiple of Repdigit Base minus 1", "text": "Let $b \\in \\Z_{>1}$ be an integer greater than $1$. Let $n$ be a repdigit number of $k$ instances of the digit $b - 1$ for some integer $k$ such that $k \\ge 1$. Let $m \\in \\Z_{>1}$ be an integer such that $1 < m < b$. Then $m \\times n$, when expressed in base $b$, is of the form: :$n = \\left[{r d d \\cdots d s}\\right]_b$ where: :$d = b - 1$ :$r = m - 1$ :$s = b - m$ :there are $k - 1$ occurrences of $d$."}
+{"_id": "13647", "title": "Only Number Twice Sum of Digits is 18", "text": "There exists only one (strictly) positive integer that is exactly twice the sum of its digits."}
+{"_id": "13648", "title": "Period of Reciprocal of 19 is of Maximal Length", "text": "The decimal expansion of the reciprocal of $19$ has the maximum period, that is: $18$: :$\\dfrac 1 {19} = 0 \\cdotp \\dot 05263 \\, 15789 \\, 47368 \\, 42 \\dot 1$ {{OEIS|A021023}}"}
+{"_id": "13649", "title": "Divisibility by 19", "text": "Let $n$ be an integer expressed in the form: :$n = 100 a + b$ Then $n$ is divisible by $19$ {{iff}} $a + 4 b$ is divisible by $19$."}
+{"_id": "13650", "title": "Diameter of Closure of Subset is Diameter of Subset", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $S \\subseteq A$ be bounded in $M$. Then: :$\\map {\\operatorname {diam} } S = \\map {\\operatorname {diam} } {S^-}$ where $\\map {\\operatorname {diam} } S$ denotes the diameter of $S$, and $S^-$ denotes the closure of $S$ in $M$."}
+{"_id": "13652", "title": "Magic Constant of Order 3 Magic Hexagon", "text": "The magic constant of the order 3 magic hexagon is $38$. It is noted that the central cell is $5$, the same as that of the order 3 magic square."}
+{"_id": "13653", "title": "Sum of Sequence of Alternating Positive and Negative Factorials being Prime", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Let: {{begin-eqn}} {{eqn | l = m       | r = \\sum_{k \\mathop = 0}^{n - 1} \\paren {-1}^k \\paren {n - k}!       | c =  }} {{eqn | r = n! - \\paren {n - 1}! + \\paren {n - 2}! - \\paren {n - 3}! + \\cdots \\pm 1       | c =  }} {{end-eqn}} The sequence of $n$ such that $m$ is prime begins: :$3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, \\ldots$ {{OEIS|A001272}} The sequence of those values of $m$ begins: :$5, 19, 101, 619, 4421, 35 \\, 899, 3 \\, 301 \\, 819, 1 \\, 226 \\, 280 \\, 710 \\, 981, \\ldots$ {{OEIS|A071828}}"}
+{"_id": "13656", "title": "Ideals are Continuous Lattice Subframe of Power Set", "text": "Let $L = \\struct {S, \\vee, \\preceq}$ be a bounded below join semilattice. Let $I = \\left({\\mathit{Ids}\\left({L}\\right), \\precsim}\\right)$ be an inclusion ordered set where :$\\mathit{Ids}\\left({L}\\right)$ denotes the set of all ideals in $L$, :$\\mathord\\precsim = \\mathord\\subseteq \\cap \\left({\\mathit{Ids}\\left({L}\\right) \\times \\mathit{Ids}\\left({L}\\right)}\\right)$ Let $P = \\struct {\\mathcal P\\left({S}\\right), \\precsim'}$ be an inclusion ordered set where :$\\mathcal P\\left({S}\\right)$ denotes the power set of $S$, :$\\mathord\\precsim' = \\mathord\\subseteq \\cap \\left({\\mathcal{P}\\left({S}\\right) \\times \\mathcal{P}\\left({S}\\right)}\\right)$ Then $I$ is continuous lattice subframe of $P$."}
+{"_id": "13657", "title": "Smallest Perfect Square Dissection", "text": "The smallest perfect square dissection is of an integer square of side $112$ into $21$ parts."}
+{"_id": "13658", "title": "Smallest Number Expressible as Sum of at most Three Triangular Numbers in 4 ways", "text": "$21$ is the smallest number which can be expressed as the sum of at most $3$ triangular numbers in $4$ ways."}
+{"_id": "13659", "title": "Necessary Condition for Twice Differentiable Functional to have Minimum", "text": "Let $J \\sqbrk y$ be a twice differentiable functional. Let $\\delta J \\sqbrk {\\hat y; h} = 0$. Suppose, for $y = \\hat y$ and all admissible $h$: :$\\delta^2 J \\sqbrk {y; h} \\ge 0$  Then $J$ has a minimum for $y=\\hat y$ if  {{explain|if what?}"}
+{"_id": "13660", "title": "Projective Plane of Order 4 is Unique", "text": "The finite projective plane of order $4$ is unique."}
+{"_id": "13661", "title": "Numbers Equal to Number of Digits in Factorial", "text": "For $n \\in \\set {1, 22, 23, 24}$ the number of digits in the decimal representation of $n!$ is equal to $n$."}
+{"_id": "13662", "title": "Square of Small-Digit Palindromic Number is Palindromic", "text": "Let $n$ be an integer such that the sum of the squares of the digits of $n$ in decimal representation is less than $10$. Let $n$ be palindromic. Then $n^2$ is also palindromic. The sequence of such numbers begins: :$0, 1, 2, 3, 11, 22, 101, 111, 121, 202, 212, 1001, 1111, \\dots$ {{OEIS|A057135}}"}
+{"_id": "13664", "title": "Numbers whose Sigma is Square", "text": "The sequence of positive integers whose $\\sigma$ value is square starts as follows: {{begin-eqn}} {{eqn | l = \\map \\sigma 3       | r = 4       | c =  }} {{eqn | l = \\map \\sigma {22}       | r = 36       | c =  }} {{eqn | l = \\map \\sigma {66}       | r = 144       | c =  }} {{eqn | l = \\map \\sigma {70}       | r = 144       | c =  }} {{eqn | l = \\map \\sigma {81}       | r = 121       | c =  }} {{eqn | l = \\map \\sigma {94}       | r = 144       | c =  }} {{eqn | l = \\map \\sigma {115}       | r = 144       | c =  }} {{eqn | l = \\map \\sigma {119}       | r = 144       | c =  }} {{eqn | l = \\map \\sigma {170}       | r = 324       | c =  }} {{end-eqn}} {{OEIS|A006532}}"}
+{"_id": "13665", "title": "Three times Number whose Sigma is Square", "text": "Let $n \\in \\Z_{>0}$ be a positive integer. Let the $\\sigma$ value of $n$ be square. Let $3$ not be a divisor of $n$. Then the $\\sigma$ value of $3 n$ is square."}
+{"_id": "13666", "title": "Product of Coprime Numbers whose Sigma is Square has Square Sigma", "text": "Let $m, n \\in \\Z_{>0}$ be a positive integer. Let the $\\sigma$ value of $m$ and $n$ both be square. Let $m$ and $n$ be coprime. Then the $\\sigma$ value of $m n$ is square."}
+{"_id": "13668", "title": "Sequence of Differences on Generalized Pentagonal Numbers", "text": "Recall the generalised pentagonal numbers $GP_n$ for $n = 0, 1, 2, \\ldots$ Consider the sequence defined as $\\Delta_n = GP_{n + 1} - GP_n$: :$1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, \\ldots$ {{OEIS|A026741}} Then: :The values of $\\Delta_n$ for odd $n$ consist of the odd numbers :The values of $\\Delta_n$ for even $n$ consist of the natural numbers."}
+{"_id": "13670", "title": "Euler's Pentagonal Numbers Theorem", "text": "Consider the infinite product: :$\\displaystyle P = \\prod_{n \\mathop \\in \\Z_{>0} } \\paren {1 - x^n}$ Then $P$ can be expressed as: :$\\displaystyle P = \\sum_{n \\mathop \\in \\Z_{>0} } \\paren {-1}^{\\ceiling {n / 2} } x^{GP_n}$ where: :$\\ceiling {n / 2}$ denotes the ceiling of $n / 2$ :$GP_n$ denotes the $n$th generalized pentagonal number. That is: :$P = 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + \\cdots$"}
+{"_id": "13671", "title": "Intersection of Semilattice Ideals is Ideal/Set of Sets", "text": "Let $\\struct {S, \\preceq}$ be a bounded below join semilattice. Let $\\II$ be a set of ideals in $\\struct {S, \\preceq}$. Then $\\bigcap \\II$ is an ideal in $\\struct {S, \\preceq}$."}
+{"_id": "13672", "title": "Compact Element iff Existence of Finite Subset that Element equals Intersection and Includes Subset", "text": "Let $X, E$ be sets. Let $P = \\left({\\mathcal P\\left({X}\\right), \\precsim}\\right)$ be an inclusion ordered set where :$\\mathcal P\\left({X}\\right)$ denotes the power set of $X$, :$\\mathord\\precsim = \\mathord\\subseteq \\cap \\left({\\mathcal{P}\\left({X}\\right) \\times \\mathcal{P}\\left({X}\\right)}\\right)$ Let $L = \\left({S, \\preceq}\\right)$ be a continuous lattice subframe of $P$. Then $E$ is compact element in $L$ {{iff}} :$\\exists F \\in \\mathit{Fin}\\left({X}\\right): E = \\bigcap \\left\\{ {I \\in S: F \\subseteq I}\\right\\} \\land F \\subseteq E$ where $\\mathit{Fin}\\left({X}\\right)$ denotes the set of all finite subsets of $X$."}
+{"_id": "13673", "title": "Legendre's Condition", "text": "Let $y =\\map y x$ be a real function, such that: :$\\map y a = A,\\quad \\map y b = B$ Let $J \\sqbrk y$ be a functional, such that: :$\\ds J \\sqbrk y = \\int_a^b \\map F {x, y, y'} \\rd x$ where :$F \\in C^2 \\closedint a b$  {{WRT}} all its variables, and $C$ stands for differentiability class. Then a necessary condition for $J \\sqbrk y$ to have a minimum at $y = \\hat y$ is: :$\\bigintlimits {F_{y'y'} } {y \\mathop = \\hat y} {} \\ge 0 \\quad \\forall x \\in \\closedint a b$"}
+{"_id": "13674", "title": "Euler's Pentagonal Numbers Theorem/Corollary 1", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive integer. Let $\\map \\sigma n$ denote the $\\sigma$ function on $n$. Then: :$\\map \\sigma n = \\displaystyle \\sum_{1 \\mathop \\le n - GP_k \\mathop < n} -\\paren {-1}^{\\ceiling {k / 2} } \\map \\sigma {n - GP_k} + n \\sqbrk {\\exists k \\in \\Z: GP_k = n}$"}
+{"_id": "13675", "title": "Euler's Pentagonal Numbers Theorem/Corollary 2", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive integer. Let $\\map p n$ denote the number of partitions on $n$. Then: :$\\map p n = \\displaystyle \\sum_{1 \\mathop \\le n - GP_k \\mathop < n} -\\paren {-1}^{\\ceiling {k / 2} } \\map p {n - GP_k} + \\sqbrk {\\exists k \\in \\Z: GP_k = n}$"}
+{"_id": "13676", "title": "Integer as Sum of 4 Cubes", "text": "Let $n \\in \\Z$ be an integer. Let $n \\not \\equiv 4 \\pmod 9$ and $n \\not \\equiv 5 \\pmod 9$. Then it is possible to express $n$ as the sum of no more than $4$ cubes which may be either positive or negative."}
+{"_id": "13677", "title": "Period of Reciprocal of 23 is of Maximal Length", "text": "The decimal expansion of the reciprocal of $19$ has the maximum period, that is: $22$: :$\\dfrac 1 {23} = 0 \\cdotp \\dot 04347 \\, 82608 \\, 69565 \\, 21739 \\, 1 \\dot 3$ {{OEIS|A021027}}"}
+{"_id": "13679", "title": "Smallest Integer not Sum of Two Ulam Numbers", "text": "The smallest integer greater than $1$ which is not the sum of two Ulam numbers is $23$."}
+{"_id": "13680", "title": "Numbers with Square-Free Binomial Coefficients", "text": "For every $n$ greater than $23$, there exists a binomial coefficient $\\dbinom n k$ that is not square-free. More specifically, the list of numbers $n$ such that $\\dbinom n k$ are squarefree for all $k = 0, \\dots, n$ is given by: :$1, 2, 3, 5, 7, 11, 23$ {{OEIS|A048278}}"}
+{"_id": "13682", "title": "Infima Inheriting Ordered Subset of Complete Lattice is Complete Lattice", "text": "Let $L = \\struct {X, \\preceq}$ be a complete lattice. Let $S = \\struct {T, \\precsim}$ be an infima inheriting ordered subset of $L$. Then $S$ is a complete lattice."}
+{"_id": "13684", "title": "Gelfond's Constant minus Pi", "text": "Gelfond's constant minus $\\pi$ is very close to $20$: :$e^\\pi - \\pi \\approx 20$"}
+{"_id": "13685", "title": "Sums of Consecutive Sequences of Squares that equal Squares", "text": "The $24$th square pyramidal number is the only one which is square: :$1^2 + 2^2 + 3^2 + \\cdots + 24^2 = 70^2$ while there are several Sum of Sequence of Squares which are square, for example: :$18^2 + 19^2 + \\cdots + 28^2 = 77^2$ and: :$25^2 + 26^2 + \\cdots + 624^2 = 9010^2$"}
+{"_id": "13687", "title": "Smallest Scalene Obtuse Triangle with Integer Sides and Area", "text": "The smallest scalene obtuse triangle with integer sides and area has sides of length $4, 13, 15$."}
+{"_id": "13689", "title": "Image of Compact Subset under Directed Suprema Preserving Closure Operator", "text": "Let $L = \\left({S, \\preceq}\\right)$ be a bounded below algebric lattice. Let $c: S \\to S$ be a closure operator that preserves directed suprema. Then: : $c \\left[{K\\left({L}\\right)}\\right] = K \\left({\\left({c\\left[{S}\\right], \\precsim}\\right)}\\right)$ where :$K \\left({L}\\right)$ denotes the compact subset of $L$, :$c \\left[{S}\\right]$ denotes the image of $S$ under $c$, :$\\mathord \\precsim = \\mathord \\preceq \\cap \\left({c \\left[{S}\\right] \\times c \\left[{S}\\right]}\\right)$"}
+{"_id": "13691", "title": "Smallest Positive Integer with 5 Fibonacci Partitions", "text": "The smallest positive integer which can be partitioned into distinct Fibonacci numbers in $5$ different ways is $24$."}
+{"_id": "13693", "title": "Sigma Function of Non-Square Semiprime", "text": "Let $n \\in \\Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$. Then: :$\\map \\sigma n = \\paren {p + 1} \\paren {q + 1}$ where $\\map \\sigma n$ denotes the sigma function."}
+{"_id": "13696", "title": "Sigma Function of Square-Free Integer", "text": "Let $n$ be an integer such that $n \\ge 2$. Let $n$ be square-free. Let the prime decomposition of $n$ be: :$\\displaystyle n = \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i =  p_1 p_2 \\cdots p_r$ Let $\\sigma \\left({n}\\right)$ be the sigma function of $n$. That is, let $\\sigma \\left({n}\\right)$ be the sum of all positive divisors of $n$. Then: :$\\displaystyle \\sigma \\left({n}\\right) = \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i + 1$"}
+{"_id": "13700", "title": "Spheres in 24 Dimensions in Leech Lattice", "text": "Let a set of identical spheres in a $24$-dimensional space be arranged in a Leech lattice. Then each sphere will touch $196 \\, 560$ other spheres. This is believed to be the densest possible sphere packing in $24$ dimensions."}
+{"_id": "13702", "title": "Cross-Sections of Leech Lattice", "text": "Except for dimensions $10$, $11$ and $13$, the densest possible sphere packing in all dimensions lower than $24$ can be obtained by cross-sections of the $24$-dimensional space Leech lattice arrangement."}
+{"_id": "13703", "title": "Difference Triangle for Sequence of Fifth Powers", "text": "The difference triangle for the sequence of fifth powers ends on the fifth line with instances of $5!$, where $!$ denotes factorial."}
+{"_id": "13704", "title": "Ideals form Arithmetic Lattice", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a bounded below lattice. Let $I = \\left({\\mathit{Ids}\\left({L}\\right), \\precsim}\\right)$ be an inclusion ordered set where :$\\mathit{Ids}\\left({L}\\right)$ denotes the set of all ideals in $L$, :$\\mathord\\precsim = \\mathord\\subseteq \\cap \\left({\\mathit{Ids}\\left({L}\\right) \\times \\mathit{Ids}\\left({L}\\right)}\\right)$ Then $I$ is an arithmetic lattice."}
+{"_id": "13707", "title": "Power of n equalling (n - 1)! + 1", "text": "There is exactly one solution to the equation in the integers: :$\\paren {n - 1}! + 1 = n^k$ for $k > 1$, and that is: :$n = 5$ :$k = 2$"}
+{"_id": "13708", "title": "Square which is 2 Less than Cube", "text": "$25$ is the only square number which is $2$ less than a cube: :$5^2 + 2 = 3^3$"}
+{"_id": "13709", "title": "Ideals form Algebraic Lattice", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let $I = \\left({\\mathit{Ids}\\left({L}\\right), \\precsim}\\right)$ be an inclusion ordered set where :$\\mathit{Ids}\\left({L}\\right)$ denotes the set of all ideals in $L$, :$\\mathord\\precsim = \\mathord\\subseteq \\cap \\left({\\mathit{Ids}\\left({L}\\right) \\times \\mathit{Ids}\\left({L}\\right)}\\right)$ Then $I$ is algebraic lattice."}
+{"_id": "13710", "title": "Definition:Powerful Number", "text": "A '''powerful number''' is a positive integer such that each of its prime factors appears with multiplicity at least $2$. That is, each of its prime factors occurs at least squared."}
+{"_id": "13711", "title": "Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite", "text": "Let   :$\\map P x : \\closedint a b \\to \\R$  :$\\map h x : \\closedint a b \\to \\R$. Let $\\map h x$ be continuously differentiable $\\forall x \\in \\closedint a b$. Suppose: :$\\forall x \\in \\closedint a b: \\map P x > 0$ Then: :$\\ds \\forall \\map h x : \\map h a = \\map h b = 0 : \\int_a^b \\paren {P h'^2 + Q h^2} \\rd x > 0$ {{iff}} the interval $\\closedint a b$ contains no points conjugate to $a$."}
+{"_id": "13712", "title": "Consecutive Odd Powerful Numbers", "text": "$25$ and $27$ are the only known pair of consecutive odd powerful numbers. It is not known whether there are any more."}
+{"_id": "13713", "title": "Square Cullen Numbers", "text": "The numbers: :$1, 9, 25$ are Cullen numbers which are also square."}
+{"_id": "13715", "title": "Numbers Partitioned into Six Hexagonal Numbers", "text": "The integers $11$ and $26$ cannot be represented by the sum of less than $6$ hexagonal numbers."}
+{"_id": "13717", "title": "Continuous Lattice Subframe of Algebraic Lattice is Algebraic Lattice", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a bounded below algebraic lattice. Let $P = \\struct {T, \\precsim}$ be a continuous lattice subframe of $L$. Then $P$ is algebraic lattice."}
+{"_id": "13718", "title": "Period of Reciprocal of 27 is Smallest with Length 3", "text": "$27$ is the smallest positive integer the decimal expansion of whose reciprocal has a period of $3$: :$\\dfrac 1 {27} = 0 \\cdotp \\dot 03 \\dot 7$ {{OEIS|A021027}}"}
+{"_id": "13719", "title": "Period of Reciprocal of 7 is of Maximal Length", "text": "$7$ is the smallest integer $n$ the decimal expansion of whose reciprocal has the maximum period $n - 1$, that is: $6$: :$\\dfrac 1 7 = 0 \\cdotp \\dot 14285 \\dot 7$ {{OEIS|A020806}} :300px"}
+{"_id": "13720", "title": "Sequence of Smallest Numbers whose Reciprocal has Period n", "text": "Let $\\sequence {s_n}$ be the sequence defined as: :$s_n$ is the smallest positive integer the decimal expansion of whose reciprocal has a period of $n$ for $n = 0, 1, 2, \\ldots$ Then $\\sequence {s_n}$ begins: :$1, 3, 11, 27, 101, 41, 7, 239, 73, 81, 451, \\ldots$ {{OEIS|A003060}}"}
+{"_id": "13722", "title": "Sequence of Successive Longest Collatz Sequence Generators", "text": "The sequence of integers which generate a Collatz process which is longer than that of any smaller integers begins: {{begin-eqn}} {{eqn | l = 1       | o = :       | c = $0$ steps }} {{eqn | l = 2       | o = :       | c = $1$ step }} {{eqn | l = 3       | o = :       | c = $7$ steps }} {{eqn | l = 6       | o = :       | c = $8$ steps }} {{eqn | l = 7       | o = :       | c = $16$ steps }} {{eqn | l = 9       | o = :       | c = $19$ steps }} {{eqn | l = 18       | o = :       | c = $20$ steps }} {{eqn | l = 25       | o = :       | c = $23$ steps }} {{eqn | l = 27       | o = :       | c = $111$ steps }} {{end-eqn}} {{OEIS|A006877}}"}
+{"_id": "13723", "title": "Dissection of Nonagon into Triangles with Chords", "text": "There are $27$ different ways to dissect a convex nonagon into triangles using $6$ chords, not counting reflections and rotations as different."}
+{"_id": "13724", "title": "Image of Directed Suprema Preserving Closure Operator is Algebraic Lattice", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below algebraic lattice. Let $c:S \\to S$ be a closure operator that preserves directed suprema. Let $C = \\left({c\\left[{S}\\right], \\precsim}\\right)$ be an ordered subset of $L$. Then $C$ is algebraic lattice."}
+{"_id": "13725", "title": "Dissection of Polygon into Triangles with Chords", "text": "The number of different ways $k$ a convex $n$-sided polygon can be divided into triangles using chords, not counting reflections and rotations as different, is given for the first few $n$ as follows: :{| border=\"1\" |- ! align=\"right\" style = \"padding: 2px 10px\" | $n$  ! align=\"right\" style = \"padding: 2px 10px\" | $k$ |- | align=\"right\" style = \"padding: 2px 10px\" | $3$ | align=\"right\" style = \"padding: 2px 10px\" | $1$ |- | align=\"right\" style = \"padding: 2px 10px\" | $4$ | align=\"right\" style = \"padding: 2px 10px\" | $1$ |- | align=\"right\" style = \"padding: 2px 10px\" | $5$ | align=\"right\" style = \"padding: 2px 10px\" | $1$ |- | align=\"right\" style = \"padding: 2px 10px\" | $6$ | align=\"right\" style = \"padding: 2px 10px\" | $3$ |- | align=\"right\" style = \"padding: 2px 10px\" | $7$ | align=\"right\" style = \"padding: 2px 10px\" | $4$ |- | align=\"right\" style = \"padding: 2px 10px\" | $8$ | align=\"right\" style = \"padding: 2px 10px\" | $12$ |- | align=\"right\" style = \"padding: 2px 10px\" | $9$ | align=\"right\" style = \"padding: 2px 10px\" | $27$ |- | align=\"right\" style = \"padding: 2px 10px\" | $10$ | align=\"right\" style = \"padding: 2px 10px\" | $82$ |- | align=\"right\" style = \"padding: 2px 10px\" | $11$ | align=\"right\" style = \"padding: 2px 10px\" | $228$ |- | align=\"right\" style = \"padding: 2px 10px\" | $12$ | align=\"right\" style = \"padding: 2px 10px\" | $733$ |- | align=\"right\" style = \"padding: 2px 10px\" | $13$ | align=\"right\" style = \"padding: 2px 10px\" | $2282$ |} {{OEIS|A000207}}"}
+{"_id": "13726", "title": "Dissection of Polygon into Triangles with Chords counting Isometries", "text": "The number of different ways $k$ a convex $n$-sided polygon can be divided into triangles using chords, counting reflections and rotations as different, is given for the first few $n$ as follows: :{| border=\"1\" |- ! align=\"right\" style = \"padding: 2px 10px\" | $n$  ! align=\"right\" style = \"padding: 2px 10px\" | $k$ |- | align=\"right\" style = \"padding: 2px 10px\" | $3$ | align=\"right\" style = \"padding: 2px 10px\" | $1$ |- | align=\"right\" style = \"padding: 2px 10px\" | $4$ | align=\"right\" style = \"padding: 2px 10px\" | $2$ |- | align=\"right\" style = \"padding: 2px 10px\" | $5$ | align=\"right\" style = \"padding: 2px 10px\" | $5$ |- | align=\"right\" style = \"padding: 2px 10px\" | $6$ | align=\"right\" style = \"padding: 2px 10px\" | $14$ |- | align=\"right\" style = \"padding: 2px 10px\" | $7$ | align=\"right\" style = \"padding: 2px 10px\" | $42$ |- | align=\"right\" style = \"padding: 2px 10px\" | $8$ | align=\"right\" style = \"padding: 2px 10px\" | $132$ |- | align=\"right\" style = \"padding: 2px 10px\" | $9$ | align=\"right\" style = \"padding: 2px 10px\" | $429$ |- | align=\"right\" style = \"padding: 2px 10px\" | $10$ | align=\"right\" style = \"padding: 2px 10px\" | $1430$ |- | align=\"right\" style = \"padding: 2px 10px\" | $11$ | align=\"right\" style = \"padding: 2px 10px\" | $4862$ |} {{OEIS|A000108}} These are the Catalan numbers."}
+{"_id": "13727", "title": "Boundary of Compact Closed Set is Compact", "text": "Let $X$ be a topological space. Let $K\\subset X$ be a compact subspace of $X$. Let $K$ be closed in $X$. Then its boundary $\\partial K$ is compact."}
+{"_id": "13729", "title": "Discrete Subgroup of Hausdorff Group is Closed", "text": "Let $G$ be a Hausdorff topological group. Let $H$ be a discrete subgroup of $G$. Then $H$ is closed in $G$."}
+{"_id": "13731", "title": "Group is Hausdorff iff Discrete Subgroups are Closed", "text": "A topological group is Hausdorff {{iff}} its discrete subgroups are closed."}
+{"_id": "13733", "title": "Closure of Subgroup is Group", "text": "Let $G$ be a topological group. Let $H\\leq G$ be a subgroup. Let $\\overline H$ denote its closure. Then $\\overline H$ is a subgroup of $G$."}
+{"_id": "13734", "title": "Image of Group Homomorphism is Hausdorff Implies Kernel is Closed", "text": "Let $G$ and $H$ be topological groups. Let $f: G \\to H$ be a morphism. Let its image $\\Img f$ be Hausdorff. Then its kernel $\\map \\ker f$ is closed in $G$."}
+{"_id": "13735", "title": "Group is Connected iff Subgroup and Quotient are Connected", "text": "Let $G$ be a topological group. Let $H \\le G$ be a subgroup. {{TFAE}} :$(1):\\quad$ $G$ is connected :$(2):\\quad$ $H$ is connected and the left quotient space $G / H$ is connected :$(3):\\quad$ $H$ is connected and the right quotient space $G / H$ is connected."}
+{"_id": "13736", "title": "Group Acts by Homeomorphisms Implies Projection on Quotient Space is Open", "text": "Let $G$ be a group acting by homeomorphisms on a topological space $X$. Then the projection map $\\pi: X \\to X / G$ is open."}
+{"_id": "13738", "title": "Neighborhood iff Contains Neighborhood", "text": "Let $X$ be a topological space. Let $x\\in X$. Let $V\\subset X$ be a subset. Then the following are equivalent: :$V$ is a neighborhood of $x$ in $X$ :$V$ contains a neighborhood of $x$ in $X$"}
+{"_id": "13740", "title": "Perfect Number ends in 6 or 28 preceded by Odd Digit", "text": "Let $n$ be an even perfect number. Then $n$ ends either in $6$ or $28$ preceded by an odd digit."}
+{"_id": "13741", "title": "Open Subset of Locally Connected Space is Locally Connected", "text": "Let $T - \\left({S, \\tau}\\right)$ be a locally connected topological space. Let $U \\subset S$ be open in $T$. Then $U$ is locally connected."}
+{"_id": "13742", "title": "Component of Locally Connected Space is Open", "text": "Let $T = \\left({S, \\tau}\\right)$ be a locally connected topological space. Let $G$ be a component of $T$. Then $G$ is open."}
+{"_id": "13743", "title": "Path Component of Locally Path-Connected Space is Open", "text": "Let $T = \\left({S, \\tau}\\right)$ be a locally path-connected topological space. Let $G$ be a path component of $T$. Then $G$ is open in $T$."}
+{"_id": "13744", "title": "Connected and Locally Path-Connected Implies Path Connected", "text": "Let $T$ be a connected and locally path-connected topological space. Then $T$ is path-connected."}
+{"_id": "13745", "title": "Components are Open iff Union of Open Connected Sets", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. {{TFAE}} :$(1): \\quad$ The components of $T$ are open. :$(2): \\quad S$ is a union of open connected sets of $T$."}
+{"_id": "13746", "title": "Path Components are Open iff Union of Open Path-Connected Sets", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. {{TFAE}}: :(1) $\\quad$ The path components of $T$ are open. :(2) $\\quad S$ is a union of open path-connected sets of $T$."}
+{"_id": "13747", "title": "Open Subset of Locally Path-Connected Space is Locally Path-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a locally path-connected topological space. Let $U \\subset S$ be open in $T$. Then $U$ is locally path-connected in $T$."}
+{"_id": "13748", "title": "Even Perfect Number is Hexagonal", "text": "All perfect numbers which are even are hexagonal."}
+{"_id": "13749", "title": "Hexagonal Number is Triangular Number", "text": "Let $H_n$ be the $n$th hexagonal number. Then $H_n$ is the $2 n - 1$th triangular number."}
+{"_id": "13751", "title": "Perfect Number which is Sum of Equal Powers of Two Numbers", "text": "$28$ is the only perfect number which is the sum of equal powers of exactly $2$ positive integers: :$28 = 1^3 + 3^3$"}
+{"_id": "13754", "title": "Congruence Modulo 3 of Power of 2", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Then: :$2^n \\equiv \\paren {-1}^n \\pmod 3$ That is: :$\\exists q \\in \\Z: 2^n = 3 q + \\paren {-1}^n$"}
+{"_id": "13755", "title": "Odd Power of 2 is Congruent to 2 Modulo 3", "text": "Let $n \\in \\Z_{\\ge 0}$ be an odd positive integer. Then: :$2^n \\equiv 2 \\pmod 3$"}
+{"_id": "13758", "title": "Open Set in Open Subspace", "text": "Let $X$ be a topological space. Let $U\\subset X$ be an open subset. Let $V\\subset U$ be a subset. Then $V$ is open in $U$ {{iff}} $V$ is open in $X$."}
+{"_id": "13759", "title": "Equivalence of Definitions of Locally Compact Hausdorff Space", "text": "{{TFAE|def = Locally Compact Hausdorff Space}} Let $T = \\struct{S, \\tau}$ be a Hausdorff topological space."}
+{"_id": "13760", "title": "Neighborhood in Compact Hausdorff Space Contains Compact Neighborhood", "text": "Let $X$ be a compact Hausdorff topological space. Let $x\\in X$. Let $U$ be a neighborhood of $x$. Then $U$ contains a compact neighborhood of $x$."}
+{"_id": "13761", "title": "Are All Perfect Numbers Even?/Progress/Form", "text": "An odd perfect number $n$ is of the form: :$n = p^a q^b r^c \\cdots$ where: :$p, q, r, \\ldots$ are prime numbers of the form $4 k + 1$ for some $k \\in \\Z_{>0}$ :$a$ is also of the form $4 k + 1$ for some $k \\in \\Z_{>0}$ :$b, c, \\ldots$ are all even."}
+{"_id": "13762", "title": "Image of Idempotent and Directed Suprema Preserving Mapping is Complete Lattice", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice. Let $f:S \\to S$ be a mapping that is idempotent and preserves directed suprema. Let $F = \\struct {f \\sqbrk S, \\precsim}$ be an ordered subset of $L$. Then $F$ inherits directed suprema and is complete lattice."}
+{"_id": "13763", "title": "Are All Perfect Numbers Even?/Progress/Minimum Size", "text": "It had been established by $1986$ that an odd perfect number, if one were to exist, would have over $200$ digits. By $1997$ that lower bound had been raised to $300$ digits. By $2012$ that lower bound had been raised again to $1500$ digits."}
+{"_id": "13764", "title": "Are All Perfect Numbers Even?/Progress/Prime Factors", "text": "An odd perfect number has: :at least $8$ distinct prime factors :at least $11$ distinct prime factors if $3$ is not one of them :at least $101$ prime factors (not necessarily distinct) :its greatest prime factor is greater than $1 \\, 000 \\, 000$ :its second largest prime factor is greater than $1000$ :at least one of the prime powers factoring it is greater than $10^{62}$ :if less than $10^{9118}$ then it is divisible by the $6$th power of some prime."}
+{"_id": "13768", "title": "Prime-Generating Quadratics of form 2 a squared plus p/3", "text": "The quadratic form: :$2 a^2 + 3$ yields prime numbers for $a = 0, 1, 2$."}
+{"_id": "13769", "title": "Prime-Generating Quadratics of form 2 a squared plus p/5", "text": "The quadratic form: :$2 a^2 + 5$ yields prime numbers for $a = 0, 1, \\ldots, 4$."}
+{"_id": "13770", "title": "Prime-Generating Quadratics of form 2 a squared plus p/11", "text": "The quadratic form: :$2 a^2 + 11$ yields prime numbers for $a = 0, 1, \\ldots, 10$."}
+{"_id": "13771", "title": "Prime-Generating Quadratics of form 2 a squared plus p/29", "text": "The quadratic form: :$2 a^2 + 29$ yields prime numbers for $a = 0, 1, \\ldots, 28$."}
+{"_id": "13772", "title": "Directed Suprema Preserving Mapping is Increasing", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a join semilattice. Let $f:S \\to S$ be a mapping that preserves directed suprema. Then $f$ is an increasing mapping."}
+{"_id": "13773", "title": "Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Lemma 1", "text": "Let the function $\\map h x$ satisfy the equation: :$-\\map {\\dfrac \\d {\\d x} } {P h'} + Q h = 0$ Let $\\map h x$ have the boundary conditions: :$\\map h a = \\map h b = 0$ Then: :$\\displaystyle \\int_a^b \\paren {P h'^2 + Q h^2} \\rd x = 0$"}
+{"_id": "13775", "title": "Sequence of 4 Consecutive Square-Free Triplets", "text": "The following sets of $4$ consecutive triplets of integers, with one integer between each triplet, are square-free: :$29, 30, 31; 33, 34, 35; 37, 38, 39; 41, 42, 43$ :$101, 102, 103; 105, 106, 107; 109, 110, 111; 113, 114, 115$"}
+{"_id": "13776", "title": "Sequence of Prime Primorial minus 1", "text": "For prime $p$, let $p \\#$ denote the $p$th primorial, defined in the sense that $p \\#$ is the product of all primes less than or equal to $p$. The sequence $\\left\\langle{p}\\right\\rangle$ such that $p \\# - 1$ is prime begins: :$3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, \\ldots$ {{OEIS|A006794}}"}
+{"_id": "13777", "title": "Image of Pair under Mapping", "text": "Let $S, T$ be sets. Let $f: S \\to T$ be a mapping. Then: :$\\forall x, y \\in S: f \\sqbrk {\\set {x, y} } = \\set {\\map f x, \\map f y}$"}
+{"_id": "13778", "title": "Image under Increasing Mapping equal to Special Set is Complete Lattice", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice. Let $f: S \\to S$ be an increasing mapping. Let $P = \\struct {M, \\precsim}$ be an ordered subset of $L$ such that :$M = \\set {x \\in S: x = \\map f x}$ Then $P$ is complete lattice."}
+{"_id": "13779", "title": "Hensel's Lemma/First Form", "text": "Let $p$ be a prime number. Let $k > 0$ be a positive integer. Let $f \\left({X}\\right) \\in \\Z \\left[{X}\\right]$ be a polynomial. Let $x_k \\in \\Z$ such that: :$f \\left({x_k}\\right) \\equiv 0 \\pmod {p^k}$ :$f' \\left({x_k}\\right) \\not \\equiv 0 \\pmod p$ Then for every integer $l \\ge 0$, there exists an integer $x_{k + l}$ such that: :$f \\left({x_{k + l} }\\right) \\equiv 0 \\pmod {p^{k + l} }$ :$x_{k + l}\\equiv x_k \\pmod {p^k}$ and any two integers satisfying these congruences are congruent modulo $p^{k + l}$. Moreover, for all $l\\geq0$ and any solutions $x_{k + l}$ and $x_{k + l + 1}$: :$x_{k + l + 1} \\equiv x_{k + l} - \\dfrac {f \\left({x_{k + l} }\\right)} {f' \\left({x_{k + l} }\\right)} \\pmod {p^{k + l + 1} }$ :$x_{k + l + 1} \\equiv x_{k + l} \\pmod {p^{k + l} }$"}
+{"_id": "13781", "title": "Sigma Function of Power of 2", "text": "Let $n \\in \\Z_{>0}$ be a power of $2$. Then: :$\\map \\sigma n = 2 n - 1$"}
+{"_id": "13782", "title": "Equivalence of Definitions of Saturation Under Equivalence Relation", "text": "Let $\\sim$ be an equivalence relation on a set $S$. Let $T \\subset S$ be a subset. {{TFAE|def = Saturation (Equivalence Relation)|view = saturation}}"}
+{"_id": "13783", "title": "Equivalence of Definitions of Saturated Set Under Equivalence Relation", "text": "Let $\\sim$ be an equivalence relation on a set $S$. Let $T\\subset S$ be a subset. {{TFAE|def = Saturated Set (Equivalence Relation)|view = saturated set|context = Equivalence Relation}}"}
+{"_id": "13784", "title": "Compact Subset is Bounded Below Join Semilattice", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let $P = \\left({K\\left({L}\\right), \\precsim}\\right)$ be an ordered subset of $L$, where $K\\left({L}\\right)$ denotes the compact subset of $L$. Then $P$ is a bounded below join semilattice."}
+{"_id": "13785", "title": "Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Lemma 2", "text": "Let $\\map h x : \\closedint a b \\to \\R$ be continuously differentiable $\\forall x \\in \\closedint a b$. Suppose the function $\\map h x$ satisfies the equation: :$-\\map {\\dfrac \\d {\\d x} } {\\paren {t P + \\paren {1 - t} } h'} + t Q h = 0$ subject to the boundary conditions: :$\\map h {a, t} = \\map h {b, t} = 0$ Then: :$\\displaystyle \\int_a^b \\paren {\\paren {P h'^2 + Q h^2} t + \\paren {1 - t} h'^2} \\rd x = 0$"}
+{"_id": "13786", "title": "Jacobi's Necessary Condition", "text": "Let $J$ be a functional, such that: :$J \\sqbrk y = \\ds \\int_a^b \\map F {x, y, y'} \\rd x$ Let $\\map y x$ correspond to the minimum of $J$. Let: :$F_{y'y'}>0$ along $\\map y x$. Then the open interval $\\openint a b$ contains no points conjugate to $a$."}
+{"_id": "13787", "title": "Stolz-Cesàro Theorem", "text": "Let $\\sequence {a_n}$ be a sequence. {{explain|Domain of $\\sequence {a_n}$ -- $\\R$ presumably but could it be $\\C$?}} Let $\\sequence {b_n}$ be a sequence of (strictly) positive real numbers such that: :$\\displaystyle \\sum_{i \\mathop = 0}^\\infty b_n = \\infty$ If: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\dfrac {a_n} {b_n} = L \\in \\R$ then also: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\dfrac {a_1 + a_2 + \\cdots + a_n} {b_1 + b_2 + \\cdots + b_n} = L$"}
+{"_id": "13788", "title": "Bottom is Compact", "text": "Let $L$ be a bounded below ordered set. Then $\\bot$ is a compact element where $\\bot$ is the smallest element in $L$."}
+{"_id": "13789", "title": "Saturation Under Equivalence Relation in Terms of Graph", "text": "Let $\\RR \\subset S \\times S$ be an equivalence relation on a set $S$. Let $\\pr_1, \\pr_2 : S \\times S \\to S$ denote the projections. Let $T\\subset S$ be a subset. Let $\\overline T$ denote its saturation. Then the following hold: :$\\overline T = \\map {\\pr_1} {\\RR \\cap \\map {\\pr_2^{-1} } T}$ :$\\overline T = \\map {\\pr_2} {\\RR \\cap \\map {\\pr_1^{-1} } T}$"}
+{"_id": "13790", "title": "Nonnegative Quadratic Functional implies no Interior Conjugate Points", "text": "If the quadratic functional  :$\\ds \\int_a^b \\paren {P h'^2 + Q h^2} \\rd x$ where: :$\\forall x \\in \\closedint a b: \\map P x > 0$ is nonnegative for all $\\map h x$: :$\\map h a = \\map h b = 0$ then the closed interval $\\closedint a b$ contains no inside points conjugate to $a$. In other words, the open interval $\\openint a b$ contains no points conjugate to $a$. {{explain|Rewrite the above so it makes better sense. For example, should the \"nonnegative\" comment be above the condition on $\\map P x$?}}"}
+{"_id": "13791", "title": "Projection of Subset is Open iff Saturation is Open", "text": "Let $\\sim$ be an equivalence relation on a topological space $X$. Let $X / \\sim$ be the quotient space. Let $p$ denote the quotient mapping. Let $U \\subset X$. Then the following are equivalent: :$\\map p U$ is open in $X / \\sim$ :The saturation of $U$ is open in $X$"}
+{"_id": "13793", "title": "Multiple of 6 is Semiperfect", "text": "Let $n \\in \\Z_{>0}$ be a multiple of $6$. Then $n$ is semiperfect."}
+{"_id": "13794", "title": "Open Projection and Closed Graph Implies Quotient is Hausdorff", "text": "Let $\\mathcal R\\subset X\\times X$ be an equivalence relation on a topological space $X$. Let $X/\\mathcal R$ be the quotient space. Let $p$ denote the quotient mapping. Let: :$\\mathcal R$ be closed in $X\\times X$ :$p$ be open Then $X/\\mathcal R$ is Hausdorff."}
+{"_id": "13795", "title": "Subgroup is Closed iff Quotient is Hausdorff", "text": "Let $G$ be a topological group. Let $H \\le G$ be a subgroup. Let $G / H$ be their quotient. Then the following are equivalent: :$H$ is closed in $G$ :$G / H$ is Hausdorff"}
+{"_id": "13796", "title": "Higher Homotopy Groups are Abelian", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $x_0 \\in S$. Let $n \\ge 2$ be a integer. Let $\\pi_n \\left({T, x_0}\\right)$ be the $n$th homotopy group with base point $x_0$. Then $\\pi_n \\left({T, x_0}\\right)$ is abelian."}
+{"_id": "13797", "title": "Multiple of Semiperfect Number is Semiperfect", "text": "Let $n \\in \\Z_{>0}$ be a semiperfect number. Let $k \\in \\Z_{>0}$ be a (strictly) positive integer. Then $k n$ is also a semiperfect number."}
+{"_id": "13798", "title": "Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compact Subset", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an algebric lattice. Let $c:S \\to S$ be a closure operator that preserves directed suprema. Then $c\\left[{K\\left({L}\\right)}\\right] \\subseteq K\\left({\\left({c\\left[{S}\\right], \\precsim}\\right)}\\right)$ where :$K\\left({L}\\right)$ denotes the compact subset of $L$, :$c\\left[{S}\\right]$ denotes the image of $S$ under $c$, :$\\mathord\\precsim = \\mathord\\preceq \\cap \\left({c\\left[{S}\\right] \\times c\\left[{S}\\right]}\\right)$"}
+{"_id": "13799", "title": "Integers such that all Coprime and Less are Prime", "text": "The following positive integers have the property that all positive integers less than and coprime to it, excluding $1$, are prime: :$1, 2, 3, 4, 6, 8, 12, 18, 24, 30$ {{OEIS|A048597}} There are no other positive integers with this property."}
+{"_id": "13802", "title": "Separable Extension is Contained in Galois Extension", "text": "Let $E/F$ be a separable finite field extension. Then there exists a finite field extension $L/E$ such that $L/F$ is Galois."}
+{"_id": "13803", "title": "Galois Extension is Galois over Intermediate Field", "text": "Let $L / F$ be a Galois Extension. Let $E$ be an intermediate field. Then $L / E$ is Galois."}
+{"_id": "13804", "title": "Frobenius Endomorphism on Field is Injective", "text": "Let $p$ be a prime number. Let $F$ be a field of characteristic $p$. Then the Frobenius endomorphism $\\Frob: F \\to F$ is injective."}
+{"_id": "13805", "title": "Galois Field is Perfect", "text": "Let $\\GF$ be a Galois field. Then $\\GF$ is perfect."}
+{"_id": "13806", "title": "Algebraically Closed Field is Perfect", "text": "Let $F$ be an algebraically closed field. Then $F$ is perfect."}
+{"_id": "13807", "title": "Area of Smallest Rectangle accommodating Re-Entrant Knight's Tour", "text": "The area of the smallest rectangular chessboard on which a re-entrant knight's tour is possible is $30$ squares. This can be configured either as a $5 \\times 6$ chessboard or a $3 \\times 10$ chessboard."}
+{"_id": "13808", "title": "Area of Smallest Square accommodating Re-Entrant Knight's Tour", "text": "The area of the smallest square chessboard on which a re-entrant knight's tour is possible is $6 \\times 6 = 36$ squares."}
+{"_id": "13809", "title": "Subgroup of Index Least Prime Divisor is Normal", "text": "Let $G$ be a finite group of order $n>1$. Let $p$ be the least prime divisor of $n$. Let $H$ be a subgroup of index $p$. Then $H$ is normal."}
+{"_id": "13810", "title": "Alternating Group is Simple", "text": "Let $n \\ge 5$ be an integer. Then the $n$th alternating group $A_n$ is simple."}
+{"_id": "13811", "title": "Universal Property of Quotient Ring", "text": "Let $R,S$ be commutative rings. Let $I\\trianglelefteq R$ be an ideal. Let $\\pi : R\\to R/I$ be the projection. Let $f:R\\to S$ be a ring homomorphism with $f(I)=0$. Then there exists a unique ring homomorphism $\\overline f:R/I \\to S$ such that $f = \\overline f \\circ \\pi$. :$\\xymatrix{ R \\ar[d]_\\pi \\ar[r]^{\\forall f} & S\\\\ R/I \\ar[ru]_{\\exists ! \\bar f} }$"}
+{"_id": "13812", "title": "Universal Property of Quotient Space", "text": "Let $X$ and $Y$ be topological spaces. Let $\\sim$ be an equivalence relation on $X$. Let $\\pi : X \\to X/\\sim$ be the quotient mapping. Let $f : X \\to Y$ be continuous and $\\sim$-invariant. Then there exists a unique continuous map $\\overline f : X/\\!\\sim \\to Y$ such that $f = \\overline f \\circ \\pi$."}
+{"_id": "13813", "title": "Universal Property of Quotient Group", "text": "Let $G,H$ be a groups. Let $N\\trianglelefteq G$ be an normal subgroup. Let $\\pi : G\\to G/N$ be the projection. Let $f:G\\to H$ be a group homomorphism with $N\\subset\\ker f$. Then there exists a unique group homomorphism $\\overline f:G/N \\to H$ such that $f = \\overline f \\circ \\pi$. $\\xymatrix{ G \\ar[d]^\\pi \\ar[r]^{\\forall f} & H\\\\ G/N \\ar[ru]_{\\exists ! \\bar f} }$"}
+{"_id": "13815", "title": "Sum of Successive Powers in 2 ways", "text": "$31$ and $8191$ can be expressed as the sum of successive powers starting from $1$ in in $2$ different ways."}
+{"_id": "13816", "title": "Period of Reciprocal of 31 is of Odd Length", "text": ":$\\dfrac 1 {31} = 0 \\cdotp \\dot 03225 \\, 80645 \\, 1612 \\dot 9$ {{OEIS|A021035}}"}
+{"_id": "13817", "title": "Properties of Periodic Part of Reciprocal of 31", "text": "We have that the decimal expansion of the reciprocal of $31$ is: {{:Period of Reciprocal of 31 is of Odd Length}} Then: {{begin-eqn}} {{eqn | l = 032258 \\times 2       | r = 64 \\, 516 }} {{eqn | l = 032258 \\times 4       | r = 129 \\, 032 }} {{eqn | l = 032258 \\times 5       | r = 161 \\, 290 }} {{eqn | l = 032258 \\times 7       | r = 225 \\, 806 }} {{eqn | l = 032258 \\times 8       | r = 258 \\, 064 }} {{eqn | l = 032258 \\times 9       | r = 290 \\, 322 }} {{eqn | l = 032258 \\times 14       | r = 451 \\, 612 }} {{eqn | l = 032258 \\times 16       | r = 516 \\, 128 }} {{eqn | l = 032258 \\times 18       | r = 580 \\, 644 }} {{eqn | l = 032258 \\times 19       | r = 612 \\, 902 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 03225 + 80645 + 16129       | r = 99 \\, 999 }} {{eqn | l = 032 + 258 + 065 + 416 + 129       | r = 900 }} {{end-eqn}}"}
+{"_id": "13818", "title": "Sequence of Prime Primorial plus 1", "text": "For prime $p$, let $p \\#$ denote the $p$th primorial, defined in the sense that $p \\#$ is the product of all primes less than or equal to $p$. The sequence $\\left\\langle{p}\\right\\rangle$ such that $p \\# + 1$ is prime begins: :$2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439, 392113, \\ldots$ {{OEIS|A005234}}"}
+{"_id": "13819", "title": "Smallest Adjacent Happy Numbers", "text": "The smallest adjacent happy numbers are $31$ and $32$."}
+{"_id": "13820", "title": "Increasing Mapping Preserves Lower Bounds", "text": "Let $L = \\left({S, \\preceq}\\right)$, $L' = \\left({S', \\preceq'}\\right)$ be ordered sets. Let $f:S \\to S'$ be an increasing mapping. Let $x \\in S$, $X \\subseteq S$ such that :$x$ is lower bound for $X$. Then $f \\left({x}\\right)$ is lower bound for $f \\left[{X}\\right]$."}
+{"_id": "13821", "title": "Order Isomorphism Preserves Lower Bounds", "text": "Let $L = \\left({S, \\preceq}\\right)$, $L' = \\left({S', \\preceq'}\\right)$ be ordered sets. Let $f: S \\to S'$ be an order isomorphism between $L$ and $L'$. Let $x \\in S$, $X \\subseteq S$. Then $x$ is lower bound for $X$ {{iff}} $f \\left({x}\\right)$ is lower bound for $f \\left[{X}\\right]$."}
+{"_id": "13822", "title": "Order Embedding is Increasing Mapping", "text": "Let $\\left({S_1, \\preceq_1}\\right)$, $\\left({S_2, \\preceq_2}\\right)$ be ordered sets. Let $f:S_1 \\to S_2$ be an order embedding. Then $f$ is increasing mapping."}
+{"_id": "13823", "title": "Order Isomorphism Preserves Upper Bounds", "text": "Let $L = \\struct {S, \\preceq}$, $L' = \\struct {S', \\preceq'}$ be ordered sets. Let $f: S \\to S'$ be an order isomorphism between $L$ and $L'$. Let $x \\in S$, $X \\subseteq S$. Then $x$ is an upper bound for $X$ {{iff}} $\\map f x$ is an upper bound for $f \\sqbrk X$."}
+{"_id": "13824", "title": "Increasing Mapping Preserves Upper Bounds", "text": "Let $L = \\left({S, \\preceq}\\right)$, $L' = \\left({S', \\preceq'}\\right)$ be ordered sets. Let $f:S \\to S'$ be an increasing mapping. Let $x \\in S$, $X \\subseteq S$ such that :$x$ is upper bound for $X$. Then $f\\left({x}\\right)$ is upper bound for $f\\left[{X}\\right]$."}
+{"_id": "13825", "title": "Order Isomorphism Preserves Infima and Suprema", "text": "Let $L = \\left({S, \\preceq}\\right)$, $L' = \\left({S', \\preceq'}\\right)$ be ordered sets. Let $f:S \\to S'$ be an order isomorphism between $L$ and $L'$. Then $f$ preserves infima and suprema."}
+{"_id": "13826", "title": "Number of Regions by dividing Circle by Chords", "text": "Let $n$ points be marked on the circumference of a circle $C$. Let chords be drawn between each pair of these points. For each $n$, the maximum number $C \\left({n}\\right)$ of regions into which $C$ can be divided is as follows: :{| border=\"1\" |- ! align=\"right\" style = \"padding: 2px 10px\" | $n$  ! align=\"right\" style = \"padding: 2px 10px\" | $C \\left({n}\\right)$  |- | align=\"right\" style = \"padding: 2px 10px\" | $1$ | align=\"right\" style = \"padding: 2px 10px\" | $1$ |- | align=\"right\" style = \"padding: 2px 10px\" | $2$ | align=\"right\" style = \"padding: 2px 10px\" | $2$ |- | align=\"right\" style = \"padding: 2px 10px\" | $3$ | align=\"right\" style = \"padding: 2px 10px\" | $4$ |- | align=\"right\" style = \"padding: 2px 10px\" | $4$ | align=\"right\" style = \"padding: 2px 10px\" | $8$ |- | align=\"right\" style = \"padding: 2px 10px\" | $5$ | align=\"right\" style = \"padding: 2px 10px\" | $16$ |- | align=\"right\" style = \"padding: 2px 10px\" | $6$ | align=\"right\" style = \"padding: 2px 10px\" | $31$ |- | align=\"right\" style = \"padding: 2px 10px\" | $7$ | align=\"right\" style = \"padding: 2px 10px\" | $57$ |- | align=\"right\" style = \"padding: 2px 10px\" | $8$ | align=\"right\" style = \"padding: 2px 10px\" | $99$ |- | align=\"right\" style = \"padding: 2px 10px\" | $9$ | align=\"right\" style = \"padding: 2px 10px\" | $163$ |- | align=\"right\" style = \"padding: 2px 10px\" | $10$ | align=\"right\" style = \"padding: 2px 10px\" | $256$ |} {{OEIS|A000127}} {{ProofWanted|There is an equation which defines this -- I think it can be found in Graham, Knuth & Patashnik but that's just too far for me to reach at the moment without me getting out of this chair.}}"}
+{"_id": "13827", "title": "Prime Factors of 2^64 - 1", "text": "The prime decomposition of $2^{64} - 1$ is given by: :$2^{64} - 1 = 3 \\times 5 \\times 17 \\times 257 \\times 641 \\times 65 \\, 537 \\times 6 \\, 700 \\, 417$"}
+{"_id": "13828", "title": "Prime Decomposition of 5th Fermat Number", "text": "The prime decomposition of the $5$th Fermat number is given by: {{begin-eqn}} {{eqn | l = 2^{\\paren {2^5} } + 1       | r = 4 \\, 294 \\, 967 \\, 297       | c = Sequence of Fermat Numbers }} {{eqn | r = 641 \\times 6 \\, 700 \\, 417       | c =  }} {{eqn | r = \\paren {5 \\times 2^7 + 1} \\times \\paren {3 \\times 17449 \\times 2^7 + 1}       | c =  }} {{end-eqn}}"}
+{"_id": "13831", "title": "Mapping Assigning to Element Its Compact Closure Preserves Infima and Directed Suprema", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below algebraic lattice. Let $C = \\left({K\\left({L}\\right), \\preceq'}\\right)$ be an ordered subset of $L$ where $K\\left({L}\\right)$ denotes the compact subset of $L$. Let $P = \\left({\\mathcal P\\left({K\\left({L}\\right)}\\right), \\precsim}\\right)$ be an inclusion ordered set of power set of $K\\left({L}\\right)$. Then there exists $f:S \\to \\mathcal P\\left({K\\left({L}\\right)}\\right)$ such that $f$ preserves infima and directed suprema and is an injection and $\\forall x \\in S: f\\left({x}\\right) = x^{\\mathrm{compact} }$ where $x^{\\mathrm{compact} }$ denotes the compact closure of $x$."}
+{"_id": "13832", "title": "Power of 2 is Difference between Two Powers", "text": "Let $n \\in \\Z_{>0}$ be a power of $2$. Then $n$ is the difference between powers of two positive integers greater than or equal to $2$. {{questionable|This is so trivial I wonder whether something got lost in translation.}}"}
+{"_id": "13833", "title": "Smallest Sequence of Three Consecutive Semiprimes", "text": "The smallest triple of consecutive semiprimes is: :$33, 34, 35$"}
+{"_id": "13834", "title": "Integers not Sum of Distinct Triangular Numbers", "text": "The sequence of integers which cannot be expressed as the sum of distinct triangular numbers is: :$2, 5, 8, 12, 23, 33$ {{OEIS|A053614}}"}
+{"_id": "13835", "title": "Powers of 2 and 5 without Zeroes", "text": "The following $n \\in \\Z$ are such that both $2^n$ and $5^n$ have no zeroes in their decimal representation: :$0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 33$ {{OEIS|A007496}}"}
+{"_id": "13836", "title": "Vector Cross Product Distributes over Addition", "text": "The vector cross product is distributive over addition. That is, in general: :$\\mathbf a \\times \\paren {\\mathbf b + \\mathbf c} = \\paren {\\mathbf a \\times \\mathbf b} + \\paren {\\mathbf a \\times \\mathbf c}$ for $\\mathbf a, \\mathbf b, \\mathbf c \\in \\R^3$."}
+{"_id": "13837", "title": "Jacobi's Equation is Variational Equation of Euler's Equation", "text": "The Variational equation of Euler's equation is Jacobi's equation."}
+{"_id": "13838", "title": "Palindromes in Base 10 and Base 2", "text": "The following $n \\in \\Z$ are palindromic in both decimal and binary: :$0, 1, 3, 5, 7, 9, 33, 99, 313, 585, 717, 7447, 9009, 15 \\, 351, 32 \\, 223, 39 \\, 993, \\ldots$ {{OEIS|A007632}}"}
+{"_id": "13840", "title": "Bottom in Compact Closure", "text": "Let $L = \\left({S, \\preceq}\\right)$ be a bounded below ordered set. Let $x \\in S$. Then $\\bot \\in x^{\\mathrm{compact} }$ where $\\bot$ denotes the smallest element in $L$, :$ x^{\\mathrm{compact} }$ denotes the compact closure of $x$."}
+{"_id": "13842", "title": "Product of Two Distinct Primes has 4 Positive Divisors", "text": "Let $n \\in \\Z_{>0}$ be a positive integer which is the product of $2$ distinct primes. Then $n$ has exactly $4$ positive divisors."}
+{"_id": "13843", "title": "Product of Two Distinct Primes is Multiplicatively Perfect", "text": "Let $n \\in \\Z_{>0}$ be a positive integer which is the product of $2$ distinct primes. Then $n$ is multiplicatively perfect."}
+{"_id": "13844", "title": "Cube of Prime has 4 Positive Divisors", "text": "Let $n \\in \\Z_{>0}$ be a positive integer which is the cube of a prime number. Then $n$ has exactly $4$ positive divisors."}
+{"_id": "13845", "title": "Cube of Prime is Multiplicatively Perfect", "text": "Let $n \\in \\Z_{>0}$ be a positive integer which is the cube of a prime number. Then $n$ is multiplicatively perfect."}
+{"_id": "13846", "title": "Integers Differing by 2 with Same Sigma", "text": "Let $\\sigma: \\Z_{>0} \\to \\Z_{>0}$ be the $\\sigma$ function, defined on the strictly positive integers. The equation: :$\\sigma \\left({n}\\right) = \\sigma \\left({n + 2}\\right)$ is satisfied by integers in the sequence: :$33, 54, 284, 366, 834, 848, 918, 1240, 1504, 2910, 2913, 3304, \\ldots$ {{OEIS|A007373}}"}
+{"_id": "13847", "title": "Compact Closure is Directed", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let $x \\in S$. Then $x^{\\mathrm{compact} }$ is directed where $x^{\\mathrm{compact} }$ denotes the compact closure of $x$."}
+{"_id": "13848", "title": "Sum of 2 Lucky Numbers in 4 Ways", "text": "The number $34$ is the smallest positive integer to be the sum of $2$ lucky numbers in $4$ different ways."}
+{"_id": "13849", "title": "35 Hexominoes", "text": "There exist $35$ distinct free hexominoes: :600px"}
+{"_id": "13850", "title": "Fixed Point of Permutation is Fixed Point of Power", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\sigma \\in S_n$. Let $i \\in \\Fix \\sigma$, where $\\Fix \\sigma$ denotes the set of fixed elements of $\\sigma$. Then for all $m \\in \\Z$: :$i \\in \\Fix {\\sigma^m}$"}
+{"_id": "13852", "title": "Hexominoes cannot form Rectangle", "text": "While there are a total of $210$ squares in a complete set of hexominoes, it is impossible to build them into a rectangle of side lengths $a$ and $b$ where $a \\times b = 210$."}
+{"_id": "13853", "title": "Number of Heptominoes", "text": "There exist $108$ distinct free heptominoes, one of which has a hole: :600px"}
+{"_id": "13854", "title": "369 Octominoes", "text": "There exist $369$ distinct free octominoes, $6$ of which have a hole: :600px"}
+{"_id": "13855", "title": "1285 9-ominoes", "text": "There exist $1285$ distinct free $9$-ominoes, $37$ of which have a hole: :600px"}
+{"_id": "13856", "title": "Mapping Assigning to Element Its Compact Closure is Order Isomorphism", "text": "Let $L = \\struct {S, \\vee, \\preceq}$ be a bounded below algebraic join semilattice. Let $C = \\struct {\\map K L, \\preceq'}$ be an ordered subset of $L$ where $\\map K L$ denotes the compact subset of $L$. Let $I = \\struct {\\map {\\mathit {Ids} } C, \\precsim}$ be an inclusion ordered set where $\\map {\\mathit {Ids} } C$ denotes the set of all ideals in $C$. Let $f: S \\to \\map {\\mathit {Ids} } C$ be a mapping such that :$\\forall x \\in S: \\map f x = x^{\\mathrm {compact} }$ where $x^{\\mathrm {compact} }$ denotes the compact closure of $x$. Thus $f$ is order isomorphism between $L$ and $I$."}
+{"_id": "13857", "title": "Maximum Length of Non-Crossing Knight's Move", "text": "The maximum length of a non-crossing knight's tour on a standard chessboard is $35$ moves."}
+{"_id": "13858", "title": "Prime Factors of 35, 36, 4734 and 4735", "text": "The integers: :$35, 4374$ have the same prime factors between them as the integers: :$36, 4375$"}
+{"_id": "13860", "title": "Equivalence Class of Fixed Element", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\sigma \\in S_n$. Let $\\RR_\\sigma$ denote the equivalence defined in Permutation Induces Equivalence Relation. Let $i \\in \\N^*_{\\le n}$. Then: :$i \\in \\Fix \\sigma$ {{iff}} $\\eqclass i {\\RR_\\sigma} = \\set i$ where: :$\\eqclass i {\\RR_\\sigma}$ denotes the equivalence class of $i$ under $\\RR_\\sigma$ :$\\Fix \\sigma$ denotes the set of fixed elements of $\\sigma$."}
+{"_id": "13862", "title": "Compact Closure is Increasing", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $x, y \\in S$ such that :$x \\preceq y$ Then $x^{\\mathrm{compact} } \\subseteq y^{\\mathrm{compact} }$ where $x^{\\mathrm{compact} }$ denotes the compact closure of $x$."}
+{"_id": "13864", "title": "Sum of Entries in Row of Pascal's Triangle", "text": "The sum of all the entries in the $n$th row of Pascal's triangle is equal to $2^n$."}
+{"_id": "13867", "title": "Columns of Pascal's Triangle contain Simplicial Polytopic Numbers", "text": "The columns of Pascal's triangle contain the simplicial polytopic numbers: : Column $0$: repeated instances of number $1$ : Column $1$: the (strictly) positive integers : Column $2$: the triangular numbers : Column $3$: the tetrahedral numbers : Column $4$: the pentatope numbers and so on."}
+{"_id": "13869", "title": "Sum of Entries in Lesser Diagonal of Pascal's Triangle equal Fibonacci Number", "text": "The sum of the entries in the $n$th lesser diagonal of Pascal's triangle equals the $n + 1$th Fibonacci number."}
+{"_id": "13870", "title": "Extension of Infima Preserving Mapping to Complete Lattice Preserves Infima", "text": "Let $L_1 = \\struct {S_1, \\preceq_1}$, $L_2 = \\struct {S_2, \\preceq_2}$ be ordered sets. Let $L_3 = \\struct {S_3, \\preceq_3}$ be a complete lattice. Suppose that. :$L_2$ is an infima inheriting ordered subset of $L_3$. Let $f: S_1 \\to S_2$ be a mapping such that: :$f$ preserves infima. Then $f: S_1 \\to S_3$ preserves infima."}
+{"_id": "13871", "title": "Extension of Directed Suprema Preserving Mapping to Complete Lattice Preserves Directed Suprema", "text": "Let $L_1 = \\left({S_1, \\preceq_1}\\right)$, $L_2 = \\left({S_2, \\preceq_2}\\right)$ be ordered sets. Let $L_3 = \\left({S_3, \\preceq_3}\\right)$ be a complete lattice. Suppose that :$L_2$ is directed suprema inheriting ordered subset of $L_3$. Let $f:S_1 \\to S_2$ be a mapping such that :$f$ preserves directed suprema. Then $f:S_1 \\to S_3$ preserves directed suprema."}
+{"_id": "13875", "title": "Reciprocal as Summation of Binomial Coefficients of Reciprocals", "text": ":$\\forall n \\in \\Z_{>0}: \\dfrac 1 n = \\displaystyle \\sum_{k \\mathop = 0}^{n - 1} \\paren {-1}^k \\dbinom {n - 1} k \\dfrac 1 {k + 1}$ where $\\dbinom {n - 1} k$ denotes a binomial coefficient. That is, for example: {{begin-eqn}} {{eqn | l = \\dfrac 1 1       | r = 1 }} {{eqn | l = \\dfrac 1 2       | r = 1 - \\dfrac 1 2 }} {{eqn | l = \\dfrac 1 3       | r = 1 - 2 \\times \\dfrac 1 2 + \\dfrac 1 3 }} {{eqn | l = \\dfrac 1 4       | r = 1 - 3 \\times \\dfrac 1 2 + 3 \\times \\dfrac 1 3 - \\dfrac 1 4 }} {{eqn | l = \\dfrac 1 5       | r = 1 - 4 \\times \\dfrac 1 2 + 6 \\times \\dfrac 1 3 - 4 \\times \\dfrac 1 4 + \\dfrac 1 5 }} {{end-eqn}}"}
+{"_id": "13877", "title": "Complement of Irreducible Topological Subset is Prime Element", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $X$ be an irreducible subset of $S$ such that :$\\complement_S\\left({X}\\right) \\in \\tau$ Let $L = \\left({\\tau, \\preceq}\\right)$ be an inclusion ordered set of topology $\\tau$. Then $\\complement_S\\left({X}\\right)$ is prime element in $L$."}
+{"_id": "13878", "title": "Element of Leibniz Harmonic Triangle as Sum of Elements on Diagonal from Below", "text": "Consider the Leibniz harmonic triangle: {{:Definition:Leibniz Harmonic Triangle}} Let $\\tuple {n, m}$ be the element in the $n$th row and $m$th column. Then: :$\\tuple {n, m} = \\displaystyle \\sum_{k \\mathop \\ge 0} \\tuple {n + 1 + k, m + k}$"}
+{"_id": "13879", "title": "Element of Pascal's Triangle is Sum of Diagonal or Column starting above it going Upwards", "text": "Consider Pascal's triangle: :{{Definition:Pascal's Triangle}} Let $\\tuple {n, m}$ be the element in the $n$th row and $m$th column. Then: :$\\tuple {n, m} = \\displaystyle \\sum_{k \\mathop \\ge 0} \\tuple {n - k - 1, m - 1}$ and: :$\\tuple {n, m} = \\displaystyle \\sum_{k \\mathop \\ge 0} \\tuple {n - k, m - k - 1}$"}
+{"_id": "13880", "title": "Power of Moved Element is Moved", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\sigma \\in S_n$. Then for all $m \\in \\Z$: :$i \\notin \\Fix \\sigma \\implies \\sigma^m \\paren i \\notin \\Fix \\sigma$ where $\\Fix \\sigma$ denotes the set of fixed elements of $\\sigma$."}
+{"_id": "13881", "title": "Integer both Square and Triangular", "text": "Consider a pair of (strictly) positive integers $a$ and $b$ such that $a < b$. Then: :$\\dfrac a b$ is a best rational approximation to the square root of $2$ {{iff}}: :$\\paren {a b}^2$ is both square and triangular."}
+{"_id": "13882", "title": "Relative Complement is Decreasing", "text": "Let $X, Y, S$ be set such that $X \\subseteq Y \\subseteq S$ Then $\\complement_S\\left({X}\\right) \\supseteq \\complement_S\\left({Y}\\right)$"}
+{"_id": "13884", "title": "Sum of Sequence of Cubes/Sequence", "text": "The sequence of integers which are the sum of the sequence of the first $n$ cubes begins: :$0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, \\ldots$"}
+{"_id": "13885", "title": "Square Numbers which are Sigma values", "text": "The sequence of square numbers which are the $\\sigma$ value of a (strictly) positive integer begins: :$1, 4, 36, 121, 144, 256, 324, 400, 576, 784, 900, 961, \\ldots$ {{OEIS|A038688}}"}
+{"_id": "13886", "title": "Join and Meet in Inclusion Ordered Set of Topology", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $L = \\left({\\tau, \\preceq}\\right)$ be an inclusion ordered set of $\\tau$. Let $X, Y \\in \\tau$. Then $X \\vee Y = X \\cup Y$ and $X \\wedge Y = X \\cap Y$"}
+{"_id": "13887", "title": "Join in Inclusion Ordered Set", "text": "Let $P = \\left({X, \\subseteq}\\right)$ be an inclusion ordered set. Let $A, B \\in X$ such that :$A \\cup B \\in X$ Then $A \\vee B = A \\cup B$"}
+{"_id": "13889", "title": "2-Digit Numbers divisible by both Product and Sum of Digits", "text": "The $2$-digit positive integers which are divisible by both the sum and product of their digits are: :$12, 24, 36$"}
+{"_id": "13890", "title": "Sequence of 7 Consecutive Integers including Multiple of Prime greater than 41", "text": "Every sequence of $7$ consecutive integers greater than $36$ includes a multiple of a prime number greater than $41$."}
+{"_id": "13891", "title": "Cyclic Permutation of 3-Digit Multiple of 37", "text": "Let $n$ be a $3$-digit multiple of $37$. Let $m$ be an integer formed by cyclically permuting the digits of $n$. Then $m$ is also a multiple of $37$."}
+{"_id": "13892", "title": "Definition:Juggler Sequence", "text": "Let $m \\in \\Z_{\\ge 0}$ be a positive integer. The '''juggler sequence on $m$''' is defined recursively as: :$J_m \\left({n}\\right) = \\begin{cases} m & : n = 0 \\\\ \\left\\lfloor{\\sqrt {J_m \\left({n - 1}\\right)} }\\right\\rfloor & : n \\text{ even} \\\\ \\left\\lfloor{\\sqrt {\\left({J_m \\left({n - 1}\\right)}\\right)^3} }\\right\\rfloor & : n \\text{ odd} \\end{cases}$ where: :$\\left\\lfloor{x}\\right\\rfloor$ denotes the floor of $x$ :$\\sqrt x$ denotes the positive square root of $x$."}
+{"_id": "13893", "title": "Period of Reciprocal of 37 has Length 3", "text": "$37$ is the $2$nd positive integer the decimal expansion of whose reciprocal has a period of $3$: :$\\dfrac 1 {37} = 0 \\cdotp \\dot 02 \\dot 7$ {{OEIS|A021041}}"}
+{"_id": "13894", "title": "Centered Hexagonal Number as Sum of Triangular Numbers", "text": "Let $C_n$ be the $n$th centered hexagonal number. Then: :$C_n = 6 T_{n - 1} + 1$ where $T_{n - 1}$ denotes the $n - 1$th triangular number."}
+{"_id": "13896", "title": "Closed Form for Centered Hexagonal Numbers", "text": "Let $C_n$ be the $n$th centered hexagonal number. Then: :$C_n = 3 n \\paren {n - 1} + 1$"}
+{"_id": "13897", "title": "Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers", "text": "The positive even integers which cannot be expressed as the sum of $2$ composite odd numbers are: :$2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 26, 28, 32, 38$ {{OEIS|A118081}}"}
+{"_id": "13898", "title": "Upper Closure is Compact in Topological Lattice", "text": "Let $L = \\left({S, \\preceq, \\tau}\\right)$ be a topological lattice. Suppose that :for every subset $X$ of $S$ if $X$ is open, then $X$ is upper. Let $x \\in S$. Then $x^\\succeq$ is compact where $x^\\succeq$ denotes the upper closure of $x$."}
+{"_id": "13899", "title": "Positive Real Numbers Closed under Multiplication", "text": "The set $\\R_{>0}$ of strictly positive real numbers is closed under multiplication: :$\\forall a, b \\in \\R_{> 0}: a \\times b \\in \\R_{> 0}$"}
+{"_id": "13900", "title": "Positive Power Function on Non-negative Reals is Strictly Increasing", "text": "Let $a \\in \\Q_{> 0}$ be a strictly positive rational number. Let $f_a: \\R_{\\ge 0} \\to \\R$ be the real function defined as: :$\\map {f_a} x = x^a$ Then $f_a$ is strictly increasing."}
+{"_id": "13901", "title": "Smallest Consecutive Even Numbers such that Added to Divisor Count are Equal", "text": "$30$ is the smallest positive even integer $n$ with the property: {{begin-eqn}} {{eqn | l = n + \\map \\tau n       | r = m       | c =  }} {{eqn | l = \\paren {n + 2} + \\map \\tau {n + 2}       | r = m       | c =  }} {{eqn | l = \\paren {n + 4} + \\map \\tau {n + 4}       | r = m       | c =  }} {{end-eqn}} where: :$m \\in \\Z_{>0}$ is some positive integer :$\\map \\tau n$ is the $\\tau$ function: the number of divisors of $n$. In this case, where $n = 30$, we have that $m = 38$."}
+{"_id": "13902", "title": "2-Digit Positive Integer equals Product plus Sum of Digits iff ends in 9", "text": "Let $n$ be a $2$-digit positive integer. Then: :$n$ equals the sum added to the product of its digits {{iff}}: :the last digit of $n$ is $9$."}
+{"_id": "13903", "title": "Number of Convex Polygons from Complete Set of Hexiamonds", "text": "The number of convex polygons that can be assembled from the complete set of $12$ hexiamonds is $39$."}
+{"_id": "13904", "title": "Infimum of Open Set is Way Below Element in Complete Scott Topological Lattice", "text": "Let $T = \\left({S, \\preceq, \\tau}\\right)$ be a complete topological lattice with Scott topology. Let $X$ be an open subset of $S$, Let $x \\in X$. Then $\\inf X \\ll x$ where $\\ll$ denotes the way below relation."}
+{"_id": "13906", "title": "Letters of Names of Numbers in Alphabetical Order/English", "text": "When written in English, the only integer the letters of whose name appear in alphabetical order is: :$40$: '''forty'''"}
+{"_id": "13907", "title": "Letters of Names of Numbers in Alphabetical Order/French", "text": "When written in French, the only integers the letters of whose name appear in alphabetical order are: {{begin-eqn}} {{eqn | o =        | l = 2:       | c = '''deux''' }} {{eqn | o =        | l = 5:       | c = '''cinq''' }} {{eqn | o =        | l = 10:       | c = '''dix''' }} {{eqn | o =        | l = 100:       | c = '''cent''' }} {{end-eqn}}"}
+{"_id": "13908", "title": "Cyclic Permutations of 5-Digit Multiples of 41", "text": "Let $n$ be a multiple of $41$ with $5$ digits. Let $m$ be an integer formed by cyclically permuting the digits of $n$. Then $m$ is a multiple of $41$."}
+{"_id": "13910", "title": "Prime-Generating Quadratic of form 2 x squared minus 1000 x minus 2609", "text": "The quadratic function: :$2 x^2 - 1000 x - 2609$ has $602$ prime values among its first $1000$ values. Some of those prime numbers are negative."}
+{"_id": "13911", "title": "Scott Topology equals to Scott Sigma", "text": "Let $\\left({T, \\preceq, \\tau}\\right)$ be a up-complete topological lattice with Scott topology. Then $\\tau = \\sigma\\left({\\left({T, \\preceq}\\right)}\\right)$ where $\\sigma\\left({L}\\right)$ denotes the Scott sigma of $L$."}
+{"_id": "13913", "title": "Complement of Lower Closure is Prime Element in Inclusion Ordered Set of Scott Sigma", "text": "Let $L = \\struct {S, \\preceq, \\tau}$ be a complete Scott topological lattice. Let $D = \\struct {\\map \\sigma L, \\precsim}$ be an inclusion ordered set of the Scott sigma of $L$. Let $x \\in S$. Then: :$\\relcomp S {x^\\preceq}$ is a prime element in $D$ and: :$\\relcomp S {x^\\preceq} \\ne S$"}
+{"_id": "13914", "title": "Topological Closure of Singleton is Irreducible", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $x$ be a point of $T$. Then: : $\\left\\{ {x}\\right\\}^-$ is irreducible where $\\left\\{ {x}\\right\\}^-$ denotes the topological closure of $\\left\\{ {x}\\right\\}$."}
+{"_id": "13917", "title": "Element equals to Supremum of Infima of Open Sets that Element Belongs implies Topological Lattice is Continuous", "text": "Let $L = \\left({S, \\preceq, \\tau}\\right)$ be a complete Scott topological lattice. Let :$\\forall x \\in S: x = \\sup \\left\\{ {\\inf X: x \\in X \\in \\sigma\\left({L}\\right)}\\right\\}$ Then $L$ is continuous."}
+{"_id": "13918", "title": "Proper Class is not Element of Class", "text": "Let $\\mathrm P$ be a proper class. Then $\\mathrm P$ is not an element of any class, that is: :$\\neg \\exists A : \\mathrm P \\in A$"}
+{"_id": "13922", "title": "Prime-Generating Quadratic of form x squared - 79 x + 1601", "text": "The quadratic function: :$x^2 - 79 x + 1601$ gives prime values for integer $x$ such that $0 \\le x \\le 79$. The primes generated are repeated once each."}
+{"_id": "13923", "title": "Continuous Group Action is by Homeomorphisms", "text": "Let $G$ be a topological group acting continuously on a topological space $X$. Then $G$ acts by homeomorphisms."}
+{"_id": "13924", "title": "Discrete Group Acts Continuously iff Acts by Homeomorphisms", "text": "Let $G$ be a discrete group acting on a topological space $X$. Then the following are equivalent: :$G$ acts continuously :$G$ acts by homeomorphisms"}
+{"_id": "13926", "title": "Euler Phi Function of 3", "text": ":$\\map \\phi 3 = 2$"}
+{"_id": "13927", "title": "Number of Distinct Parenthesizations on Word", "text": "Let $w_n$ denote an arbitrary word of $n$ elements. The number of distinct parenthesizations of $w_n$ is the Catalan number $C_{n - 1}$: :$C_{n - 1} = \\dfrac 1 n \\dbinom {2 \\paren {n - 1} } {n - 1}$"}
+{"_id": "13928", "title": "Number of Paths on Graph along X-axis using Diagonal Steps", "text": "The number of different paths that can be taken on a Cartesian plane from the origin to $\\tuple {2 n + 2, 0}$, using only diagonal steps, and never touching the $x$-axis except at the beginning and the end of the path, is the Catalan number $C_n$: :400px"}
+{"_id": "13929", "title": "Non-Integral Value of Göbel's Sequence", "text": "Consider Göbel's sequence defined recursively as: :$x_n = \\begin{cases} 1 & : n = 0 \\\\ \\ds \\paren {1 + \\sum_{k \\mathop = 0}^{n - 1} {x_k}^2} / n & : n > 0 \\end{cases}$ The smallest $n$ such that $x_n$ is not an integer is $43$."}
+{"_id": "13930", "title": "Non-Integral Value of 3-Göbel Sequence", "text": "Consider the $3$-Göbel sequence defined recursively as: :$x_n = \\begin {cases} 1 & : n = 0 \\\\ \\displaystyle \\paren {1 + \\sum_{k \\mathop = 0}^{n - 1} {x_k}^3} / n & : n > 0 \\end {cases}$ The smallest $n$ such that $x_n$ is not an integer is $88$."}
+{"_id": "13932", "title": "Best Approximation from Below to 1 as Sum of Minimal Number of Unit Fractions", "text": ":$\\dfrac 1 2 + \\dfrac 1 3 + \\dfrac 1 7 + \\dfrac 1 {43} = 1 -  \\dfrac 1 {1806}$ This is the best approximation from below to $1$ as the minimal sum of unit fractions."}
+{"_id": "13933", "title": "Cuboid with Integer Edges and Face Diagonals", "text": "The smallest cuboid whose edges and the diagonals of whose faces are all integers has edge lengths $44$, $117$ and $240$. Its space diagonal, however, is not an integer. {{expand|Add the definition of Definition:Euler Brick, and rewrite and rename as appropriate.}}"}
+{"_id": "13934", "title": "Largest n such that 1 to n can be Partitioned for no Element to be Sum of 2 Elements in Same Set", "text": "$44$ is the largest integer $n$ such that the set of integers from $1$ to $n$ can be partitioned into $4$ subsets such that no integer in any of these subsets is the sum of $2$ other integers in the same subset: :$\\set {1, 3, 5, 15, 17, 19, 26, 28, 40, 42, 44}$ :$\\set {2, 7, 8, 18, 21, 24, 27, 33, 37, 38, 43}$ :$\\set {4, 6, 13, 20, 22, 23, 25, 30, 32, 39, 41}$ :$\\set {9, 10, 11, 12, 14, 16, 29, 31, 34, 35, 36}$"}
+{"_id": "13937", "title": "Conjugate Transpose is Involution", "text": "Let $\\mathbf A$ be a complex-valued matrix. Let $\\mathbf A^*$ denote the Hermitian conjugate of $\\mathbf A$. Then the operation of Hermitian conjugate is an involution: :$\\paren {\\mathbf A^*}^* = \\mathbf A$"}
+{"_id": "13938", "title": "Number as Sum of Distinct Primes greater than 11", "text": "Every number greater than $45$ can be expressed as the sum of distinct primes greater than $11$."}
+{"_id": "13939", "title": "Subclass of Set is Set", "text": "Let $A$ be a set. Let $\\map \\phi x$ be a condition in which $x$ is taken to be a set. Then there exists a set that consists of all of the elements of $A$ that satisfies this condition. In ZF, this result is known as the Axiom of Specification."}
+{"_id": "13940", "title": "Hexagonal Number as 4 times Triangular Number plus n", "text": "Let $H_n$ be the $n$th hexagonal number. Then: :$H_n = 4 T_{n - 1} + n$ where $T_{n - 1}$ is the $n - 1$th triangular number."}
+{"_id": "13941", "title": "Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Dependent on N Functions", "text": "Let $K$ be a (real) functional, such that: :$\\ds K \\sqbrk {\\mathbf h} = \\int_a^b \\paren {\\mathbf h' \\mathbf P \\mathbf h' + \\mathbf h \\mathbf Q \\mathbf h} \\rd x$ where: :$\\mathbf h$ is an $N$-dimensional vector :$\\mathbf Q$ is a $N \\times N$ matrix :$\\mathbf P$ is a $N\\times N$ symmetric positive definite matrix. Let $\\closedint a b$ be such that it does not contain a point conjugate to $a$. Then: :$\\forall \\mathbf h: \\map {\\mathbf h} a = \\map {\\mathbf h} b = 0: K \\sqbrk {\\mathbf h} > 0$ {{iff}} the above holds. {{explain|\"the above\" -- reference the specific statement, there is plenty of \"above\" and it may not be immediately clear exactly what}}"}
+{"_id": "13943", "title": "Class is Proper iff Bijection from Class to Proper Class", "text": "Let $A$ be a class. Let $\\mathrm P$ be a proper class. Then $A$ is proper if and only if there exists a bijection from $A$ to $\\mathrm P$."}
+{"_id": "13947", "title": "Infinite Product of Analytic Functions", "text": "Let $D \\subset \\C$ be an open connected set. Let $\\left\\langle{f_n}\\right\\rangle$ be a sequence of analytic functions $f_n: D \\to \\C$ that are not identically zero. Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\left({f_n - 1}\\right)$ converge locally uniformly absolutely on $D$. Then: :$(1): \\quad f = \\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ converges locally uniformly absolutely on $D$ :$(2): \\quad f$ is analytic :$(3): \\quad$ For each $z \\in D$, $f_n \\left({z}\\right) = 0$ for finitely many $n \\in \\N$ :$(4): \\quad$ For each $z \\in D$, $\\operatorname {mult}_z \\left({f}\\right) = \\displaystyle \\sum_{n \\mathop = 1}^\\infty \\operatorname{mult}_z \\left({f_n}\\right)$ where $\\operatorname{mult}$ denotes multiplicity."}
+{"_id": "13949", "title": "Uniformly Continuous Function Preserves Uniform Convergence", "text": "Let $X$ be a set. Let $M$ and $N$ be metric spaces. Let $(g_n)$ be a sequence of mappings $g_n:X\\to M$. Let $g_n$ converge uniformly to $g:X\\to M$. Let $f:M\\to N$ be uniformly continuous. Then $(f\\circ g_n)$ converges uniformly to $f\\circ g$."}
+{"_id": "13950", "title": "Complex Exponential is Uniformly Continuous on Half-Planes", "text": "Let $a\\in\\R$. Then $\\exp$ is uniformly continuous on the half-plane $\\set {z \\in \\C : \\map \\Re z \\le a}$."}
+{"_id": "13951", "title": "Local Uniform Convergence Implies Compact Convergence", "text": "Let $X$ be a topological space. Let $M$ be a metric space. Let $(f_n)$ be a sequence of mappings $f_n : X\\to M$. Let $f_n$ converge locally uniformly to $f:X\\to M$. Then $f_n$ converges compactly to $f$."}
+{"_id": "13954", "title": "Bounds for Complex Logarithm", "text": "Let $\\ln$ denote the complex logarithm. Let $z \\in \\C$ with $\\cmod z \\le \\dfrac 1 2$. Then: :$\\dfrac 1 2 \\cmod z \\le \\cmod {\\map \\ln {1 + z} } \\le \\dfrac 3 2 \\cmod z$"}
+{"_id": "13955", "title": "Reciprocal of Absolutely Convergent Product is Absolutely Convergent", "text": "Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,}}$ be a valued field. Let $\\sequence {1 + a_n}$ be a sequence of nonzero elements of $\\mathbb K$. Let the infinite product $\\ds \\prod_{n \\mathop = 1}^\\infty \\left({1 + a_n}\\right)$ converge absolutely to $a \\in \\mathbb K \\setminus \\set 0$. Then $\\ds \\prod_{n \\mathop = 1}^\\infty \\frac 1 {1 + a_n}$  converges absolutely to $1 / a$."}
+{"_id": "13957", "title": "Equivalence of Definitions of Absolute Convergence of Product", "text": "Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,}}$ be a valued field. Let $\\sequence{a_n}$ be a sequence in $\\mathbb K$. {{TFAE|def = Absolute Convergence of Product}} === Definition 1 === {{:Definition:Absolute Convergence of Product/General Definition/Definition 1}} === Definition 2 === {{:Definition:Absolute Convergence of Product/General Definition/Definition 2}}"}
+{"_id": "13958", "title": "Factors in Convergent Product Converge to One", "text": "Let $\\struct {\\mathbb K, \\norm {\\,\\cdot\\,} }$ be a valued field. Let the infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ be convergent. Then $a_n \\to 1$."}
+{"_id": "13959", "title": "Logarithm of Convergent Product of Real Numbers", "text": "The following are equivalent: :The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ converges to $a \\in \\R_{\\ne 0}$. :The series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\ln a_n$ converges to $\\ln a$."}
+{"_id": "13960", "title": "Factors in Absolutely Convergent Product Converge to One", "text": "Let $\\struct {\\mathbb K, \\norm {\\, \\cdot \\,} }$ be a valued field. Let the infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\paren {1 + a_n}$ be absolutely convergent. Then: :$a_n \\to 0$"}
+{"_id": "13963", "title": "Logarithm of Divergent Product of Real Numbers", "text": "Let $\\left\\langle{a_n}\\right\\rangle$ be a sequence of strictly positive real numbers."}
+{"_id": "13964", "title": "Logarithm of Infinite Product of Real Numbers", "text": "Let $(a_n)$ be a sequence of strictly positive real numbers."}
+{"_id": "13965", "title": "Absolutely Convergent Product Does not Diverge to Zero", "text": "Let $\\struct {\\mathbb K, \\norm {\\, \\cdot \\,} }$ be a valued field. Let the infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\paren {1 + a_n}$ be absolutely convergent. Then it is not divergent to $0$."}
+{"_id": "13967", "title": "Absolute Value of Absolutely Convergent Product is Absolutely Convergent", "text": "Let the infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ converge absolutely to $a\\in\\mathbb K$. Then $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\norm{a_n}$ converges absolutely to $\\norm{a}$."}
+{"_id": "13971", "title": "Logarithm of Divergent Product of Real Numbers/Infinity", "text": "{{TFAE}}: :$(1): \\quad$ The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ diverges to $+\\infty$. :$(2): \\quad$ The series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\log a_n$ diverges to $+\\infty$."}
+{"_id": "13973", "title": "Product of Convergent Products is Convergent", "text": "Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,} }$ be a valued field. Let $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ converge to $a$. Let $\\displaystyle \\prod_{n \\mathop = 1}^\\infty b_n$ converge to $b$. Then $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_nb_n$ converges to $ab$."}
+{"_id": "13974", "title": "Product of Convergent and Divergent Product is Divergent", "text": "Let $\\struct {\\mathbb K, \\norm {\\, \\cdot \\,} }$ be a valued field. Let $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ be convergent. Let $\\displaystyle \\prod_{n \\mathop = 1}^\\infty b_n$ be divergent. Then $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n b_n$ is divergent."}
+{"_id": "13975", "title": "Trimorphic Number is not necessarily Automorphic", "text": "Let $n \\in \\Z_{>0}$ be a trimorphic number. Then it is not necessarily the case that $n$ is also an automorphic number."}
+{"_id": "13976", "title": "Reciprocal of 49 shows Powers of 2 in Decimal Expansion", "text": "The decimal expansion of the reciprocal of $49$ contains the powers of $2$: :$\\dfrac 1 {49} = 0 \\cdotp \\dot 02040 \\, 81632 \\, 65306 \\, 12244 \\, 89795 \\, 91836 \\, 73469 \\, 38775 \\, 5 \\dot 1$ {{OEIS|A007450}}"}
+{"_id": "13979", "title": "Numbers the Multiple of whose Reciprocal are Cyclic Permutations", "text": "Let $m \\in \\Z_{>0}$. Consider the reciprocal of $m$. Let $n \\in \\Z$ such that $1 \\le n < m$. Then: :The digits in the decimal expansion of the rational number $\\dfrac n m$ form a cyclic permutation of the digits in the decimal expansion of $\\dfrac 1 m$ {{iff}}: :$(1): \\quad m$ is the integer power of a prime number $p$ :$(2): \\quad$ The period of recurrence of the decimal expansion of $1 / p$ is $p - 1$ :$(3): \\quad p$ is not a divisor of $n$."}
+{"_id": "13980", "title": "Uniform Product of Continuous Functions is Continuous", "text": "Let $X$ be a metric space. Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,}}$ be a valued field. Let $\\left\\langle{f_n}\\right\\rangle$ be a sequence of bounded continuous mappings $f_n: X \\to \\mathbb K$. Let the product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ converge uniformly to $f$. Then $f$ is continuous."}
+{"_id": "13981", "title": "Logarithm of Infinite Product of Complex Functions", "text": "Let $X$ be a weakly locally compact topological space. Let $\\sequence {f_n}$ be a sequence of everywhere nonzero continuous mappings $f_n: X \\to \\C$. Then the following are equivalent: :$(1): \\quad$ The product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ converges locally uniformly to $f$. :$(2): \\quad$ The series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\ln f_n$ converges locally uniformly to $\\ln f + 2k\\pi i$ for some mapping $k:K\\to\\Z$."}
+{"_id": "13982", "title": "Logarithmic Derivative of Infinite Product of Analytic Functions", "text": "Let $D\\subset\\C$ be open. Let $(f_n)$ be a sequence of analytic functions $f_n:D\\to\\C$. Let none of the $f_n$ be identically zero on any open subset of $D$. Let the product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ converge locally uniformly to $f$. Then $\\displaystyle \\frac{f'}f = \\sum_{n \\mathop = 1}^\\infty \\frac{f_n'}{f_n}$ and the series converges locally uniformly in $D\\setminus\\{z\\in D : f(z) = 0\\}$."}
+{"_id": "13983", "title": "Galois Connection implies Upper Adjoint is Surjection iff Lower Adjoint is Injection", "text": "Let $L = \\struct {S, \\preceq}$ and $R = \\struct {T, \\precsim}$ be ordered sets. Let $g: S \\to T$ and $d: T \\to S$ be mappings such that $\\struct {g, d}$ is a Galois connection. Then $g$ is a surjection {{iff}} $d$ is an injection."}
+{"_id": "13984", "title": "Derivative of Uniform Limit of Analytic Functions", "text": "Let $U$ be an open subset of $\\C$. Let $\\left\\langle{f_n}\\right\\rangle_{n \\mathop \\in \\N}$ be a sequence of analytic functions $f_n : U \\to \\C$. Let $\\left\\langle{f_n}\\right\\rangle$ converge locally uniformly to $f$ on $U$. Then the sequence $\\left\\langle{f_n'}\\right\\rangle_{n \\mathop \\in \\N}$ converges locally uniformly to $f'$."}
+{"_id": "13985", "title": "Derivative of Infinite Product of Analytic Functions", "text": "Let $D \\subset \\C$ be open. Let $\\left\\langle{f_n}\\right\\rangle$ be a sequence of analytic functions $f_n: D \\to \\C$. Let the product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ converge locally uniformly to $f$. Then: :$\\displaystyle f' = \\sum_{n \\mathop = 1}^\\infty f_n'\\cdot \\prod_{\\substack{k \\mathop = 1 \\\\ k\\mathop \\ne n} }^\\infty f_k$ and the series converges locally uniformly in $D$."}
+{"_id": "13986", "title": "Uniformly Convergent Sequence Multiplied with Function", "text": "Let $X$ be a set. Let $V$ be a normed vector space over $\\mathbb K$. Let $\\sequence {f_n}$ be a sequence of mappings $f_n: X \\to V$. Let $\\sequence {f_n}$ be uniformly convergent. Let $g: X \\to \\mathbb K$ be bounded. Then $\\sequence {f_n g}$ is uniformly convergent."}
+{"_id": "13987", "title": "Logarithmic Derivative of Product of Analytic Functions", "text": "Let $D \\subset \\C$ be open. Let $f, g: D \\to \\C$ be analytic. Let $z \\in D$ with $f \\left({z}\\right) \\ne 0 \\ne g \\left({z}\\right)$. Then: :$\\dfrac{\\left({f g}\\right)' \\left({z}\\right)} {\\left({f g}\\right) \\left({z}\\right)} = \\dfrac{f' \\left({z}\\right)} {f \\left({z}\\right)} + \\dfrac {g' \\left({z}\\right)} {g \\left({z}\\right)}$ {{explain|Link to a clarifying definition of what $\\left({f g}\\right)$ is -- presumably it's the pointwise product, but in the context there's nothing to say it can't be $\\left({f \\circ g}\\right)$, that is, $f$ of $g$ of $z$.}}"}
+{"_id": "13988", "title": "Uniformly Convergent Sequence on Dense Subset", "text": "Let $X$ be a metric space. Let $Y \\subset X$ be dense. Let $V$ be a Banach space. Let $\\sequence {f_n}$ be a sequence of continuous mappings $f_n : X\\to V$. Let $\\sequence {f_n}$ be uniformly convergent on $Y$. Then $\\sequence {f_n}$ is uniformly convergent on $X$."}
+{"_id": "13989", "title": "Tail of Uniformly Convergent Product Converges Uniformly to One", "text": "Let $X$ be a compact topological space. Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,}}$ be a valued field. Let $\\sequence {f_n}$ be a sequence of continuous mappings $f_n: X \\to \\mathbb K$. Let the infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ converge uniformly on $X$. Then for all $N \\in \\N$, $\\displaystyle \\prod_{n \\mathop = N}^\\infty f_n$ converges uniformly and the sequence $\\displaystyle \\prod_{n \\mathop = N}^\\infty f_n$ converges uniformly to $1$."}
+{"_id": "13990", "title": "Factors in Uniformly Convergent Product Converge Uniformly to One", "text": "Let $X$ be a set. Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,}}$ be a valued field. Let $\\left\\langle{f_n}\\right\\rangle$ be a sequence of bounded mappings $f_n: X \\to \\mathbb K$. Let the infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ converge uniformly on $X$. Then $f_n$ converges uniformly to $1$."}
+{"_id": "13991", "title": "Quintuplets of Consecutive Integers which are not Sigma Values", "text": "The elements of the following $5$-tuples of consecutive integers have the property that they are not values of the $\\sigma$ function $\\map \\sigma n$ for any $n$: :$\\tuple {49, 50, 51, 52, 53}$ :$\\tuple {115, 116, 117, 118, 119}$ :$\\tuple {145, 146, 147, 148, 149}$"}
+{"_id": "13992", "title": "Positive Integers which are not Sigma Values", "text": "The following positive integers are not the values of the $\\sigma$ function $\\sigma \\left({n}\\right)$ for any $n$: :$2, 5, 9, 10, 11, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 37, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 58, 59, 61, 64, 65, \\ldots$ {{OEIS|A007369}}"}
+{"_id": "13993", "title": "Uniformly Absolutely Convergent Product is Uniformly Convergent", "text": "Let $X$ be a set. Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,}}$ be a valued field. Let $\\mathbb K$ be complete. Let $\\left\\langle{f_n}\\right\\rangle$ be a sequence of bounded mappings $f_n: X \\to \\mathbb K$. Let the infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ converge uniformly absolutely on $X$. Then it converges uniformly."}
+{"_id": "13994", "title": "Uniformly Convergent Product Satisfies Uniform Cauchy Criterion", "text": "Let $X$ be a compact topological space. Let $\\struct {\\mathbb K, \\norm {\\, \\cdot \\,} }$ be a valued field. Let $\\sequence {f_n}$ be a sequence of continuous mappings $f_n: X \\to \\mathbb K$. Let the infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ converge uniformly on $X$. Then $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ satisfies the uniform Cauchy condition for products."}
+{"_id": "13995", "title": "Absolute Value of Uniformly Convergent Product", "text": "Let $X$ be a compact topological space. Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,}}$ be a valued field. Let $\\left\\langle{f_n}\\right\\rangle$ be a sequence of continuous mappings $f_n: X \\to \\mathbb K$. Let the infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ converge uniformly to $f$. Then $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\norm{f_n}$ converges uniformly to $ \\norm{f}$."}
+{"_id": "13996", "title": "Squares whose Digits can be Separated into 2 other Squares", "text": "The decimal representation of the following square numbers can be split into two parts which are each themselves square: {{begin-eqn}} {{eqn | l = 7^2       | r = 49       | c = $4 = 2^2$,       | cc= $9 = 3^2$ }} {{eqn | l = 13^2       | r = 169       | c = $16 = 4^2$,       | cc= $9 = 3^2$ }} {{eqn | l = 19^2       | r = 361       | c = $36 = 6^2$,       | cc= $1 = 1^2$ }} {{eqn | l = 35^2       | r = 1225       | c = $1 = 1^2$,       | cc= $225 = 15^2$ }} {{eqn | l = 38^2       | r = 1444       | c = $144 = 12^2$,       | cc= $4 = 2^2$ }} {{eqn | l = 41^2       | r = 1681       | c = $16 = 4^2$,       | cc= $81 = 9^2$ }} {{eqn | l = 57^2       | r = 3249       | c = $324 = 18^2$,       | cc= $9 = 3^2$ }} {{eqn | l = 65^2       | r = 4225       | c = $4 = 2^2$,       | cc= $225 = 15^2$ }} {{eqn | l = 70^2       | r = 4900       | c = $4 = 2^2$,       | cc= $900 = 30^2$ }} {{eqn | l = 125^2       | r = 15 \\, 625       | c = $1 = 1^2$,       | cc= $5625 = 75^2$ }} {{eqn | l = 130^2       | r = 16 \\, 900       | c = $16 = 4^2$,       | cc= $900 = 30^2$ }} {{eqn | l = 190^2       | r = 36 \\, 100       | c = $36 = 6^2$,       | cc= $100 = 10^2$ }} {{eqn | l = 205^2       | r = 42 \\, 025       | c = $4 = 2^2$,       | cc= $2025 = 45^2$ }} {{eqn | l = 223^2       | r = 49 \\, 729       | c = $49 = 7^2$,       | cc= $729 = 27^2$ }} {{end-eqn}} {{OEIS|A048375}}"}
+{"_id": "13998", "title": "Zeroes of Infinite Product of Analytic Functions", "text": "Let $D \\subset \\C$ be an open connected set. Let $\\left\\langle{f_n}\\right\\rangle$ be a sequence of analytic functions $f_n: D \\to \\C$. Let $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ converge locally uniformly to $f$. Let $z_0\\in D$. Then: :$(1): \\quad$ $f$ is identically zero {{Iff}} some $f_n$ is identically zero  :$(2): \\quad$ $f_n \\left({z_0}\\right) = 0$ for finitely many $n \\in \\N$ :$(3): \\quad$ If $f$ is not identically zero, $\\operatorname {mult}_{z_0} \\left({f}\\right) = \\displaystyle \\sum_{n \\mathop = 1}^\\infty \\operatorname{mult}_{z_0} \\left({f_n}\\right)$ where $\\operatorname{mult}$ denotes multiplicity."}
+{"_id": "13999", "title": "Infinite Product of Analytic Functions is Analytic", "text": "Let $D \\subset \\C$ be an open set. Let $\\left\\langle{f_n}\\right\\rangle$ be a sequence of analytic functions $f_n: D \\to \\C$. Let $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ converge locally uniformly to $f$. Then $f$ is analytic."}
+{"_id": "14000", "title": "Sum of 2 Squares in 2 Distinct Ways", "text": "Let $m, n \\in \\Z_{>0}$ be distinct positive integers that can be expressed as the sum of two distinct square numbers. Then $m n$ can be expressed as the sum of two square numbers in at least two distinct ways."}
+{"_id": "14001", "title": "Bounds for Finite Product of Real Numbers", "text": "Let $a_1, a_2, \\ldots, a_n$ be positive real numbers. Then: :$\\displaystyle \\sum_{k \\mathop = 1}^n a_k \\le \\prod_{k \\mathop = 1}^n \\paren {1 + a_k} \\le \\map \\exp {\\sum_{k \\mathop = 1}^n a_k}$"}
+{"_id": "14002", "title": "Bounds for Complex Exponential", "text": "Let $\\exp$ denote the complex exponential. Let $z \\in \\C$ with $\\cmod z \\le \\dfrac 1 2$. Then :$\\dfrac 1 2 \\cmod z \\le \\cmod {\\exp z - 1} \\le \\dfrac 3 2 \\cmod z$"}
+{"_id": "14003", "title": "Sum of 2 Squares in 2 Distinct Ways/Sequence", "text": "The sequence of positive integers which can be expressed as the sum of two square numbers in two or more different ways begins: {{begin-eqn}} {{eqn | l = 50       | r = 7^2 + 1^2       | rr= = 5^2 + 5^2 }} {{eqn | l = 65       | r = 8^2 + 1^2       | rr= = 7^2 + 4^2 }} {{eqn | l = 85       | r = 9^2 + 2^2       | rr= = 7^2 + 6^2 }} {{eqn | l = 125       | r = 11^2 + 2^2       | rr= = 10^2 + 5^2 }} {{eqn | l = 130       | r = 11^2 + 3^2       | rr= = 9^2 + 7^2 }} {{eqn | l = 145       | r = 12^2 + 1^2       | rr= = 9^2 + 8^2 }} {{eqn | l = 170       | r = 13^2 + 1^2       | rr= = 11^2 + 7^2 }} {{end-eqn}}"}
+{"_id": "14004", "title": "Sum of 2 Squares in 2 Distinct Ways/Examples/50", "text": "$50$ is the smallest positive integer which can be expressed as the sum of two square numbers in two distinct ways: {{begin-eqn}} {{eqn | l = 50       | r = 5^2 + 5^2 }} {{eqn | r = 7^2 + 1^2 }} {{end-eqn}}"}
+{"_id": "14007", "title": "Lower Adjoint at Element is Minimum of Preimage of Singleton of Element implies Composition is Identity", "text": "Let $L = \\struct {S, \\preceq}, R = \\struct {T, \\precsim}$ be ordered sets. Let $g: S \\to T, d: T \\to S$ be mappings such that :$\\forall t \\in T: \\map d t = \\min \\set {g^{-1} \\sqbrk {\\set t} }$ Then $g \\circ d = I_T$ where $I_T$ denotes the identity mapping of $T$."}
+{"_id": "14008", "title": "Bounds for Weierstrass Elementary Factors", "text": "Let $E_p: \\C \\to \\C$ denote the $p$th Weierstrass elementary factor: :$\\map {E_p} z = \\begin{cases} 1 - z & : p = 0 \\\\ \\paren {1 - z} \\map \\exp {z + \\dfrac {z^2} 2 + \\cdots + \\dfrac {z^p} p} & : \\text{otherwise}\\end{cases}$ Let $z \\in \\C$."}
+{"_id": "14009", "title": "Order of Product of Entire Functions", "text": "Let $f, g: \\C \\to \\C$ be entire functions of order $\\alpha$ and $\\beta$. Then $f g$ has order at most $\\map \\max {\\alpha, \\beta}$."}
+{"_id": "14010", "title": "Numbers Partitioned into up to 4 Squares in 5 Ways", "text": "The following positive integers can be expressed as the sum of no more than $4$ squares in $5$ distinct ways: :$50, 52, 54, 58, \\ldots$"}
+{"_id": "14011", "title": "Order of Product of Entire Function with Polynomial", "text": "Let $f:\\C\\to\\C$ be an entire function of order $\\omega$. Let $P:\\C\\to\\C$ be a nonzero polynomial. Then $f\\cdot P$ has order $\\omega$."}
+{"_id": "14012", "title": "Order of Shifted Entire Function", "text": "Let $f: \\C \\to \\C$ be an entire function of order $\\alpha$. Let $a \\in \\C$. Then $f \\left({z + a}\\right)$ has order $\\alpha$."}
+{"_id": "14013", "title": "Zerofree Entire Function of Finite Order is Exponential of Polynomial", "text": "Let $f:\\C\\to\\C$ be an entire function of finite order. Let $f$ have no zeroes. Then $f=\\exp P$ for some polynomial $P$."}
+{"_id": "14014", "title": "Borel-Carathéodory Lemma", "text": "Let $D \\subset \\C$ be an open set with $0 \\in D$. Let $R > 0$ be such that $\\map B {0, R} \\subset D$. {{explain|Domain of $R$, definition of $B$}} Let $f: D \\to \\C$ be analytic with $\\map f 0 = 0$. Let $\\map \\Re {\\map f z} \\le M$ for $\\cmod z \\le R$. Let $0 < r < R$. Then for $\\cmod z \\le r$: :$(1): \\quad \\cmod {\\map f z} \\le \\dfrac {2 M r} {R - r}$ :$(2): \\quad \\cmod {\\map {f^{\\paren k} } z} \\le \\dfrac {2 M R k!} {\\paren {R - r}^{k + 1} }$ for all $k \\ge 1$"}
+{"_id": "14015", "title": "Order of Sum of Entire Functions", "text": "Let $f, g: \\C \\to \\C$ be entire functions of order $\\alpha$ and $\\beta$. Then $f + g$ has order at most $\\map \\max {\\alpha, \\beta}$, with equality if $\\alpha \\ne \\beta$."}
+{"_id": "14016", "title": "Exponent of Convergence is Less Than Order", "text": "Let $f: \\C \\to \\C$ be an entire function. Let $\\omega$ be its order. Let $\\tau$ be its exponent of convergence. Then $\\tau \\le \\omega$."}
+{"_id": "14017", "title": "Class is Proper iff Bijection from Class to Proper Class/Corollary", "text": "$A$ is proper if and only if there exists a bijection from $\\mathrm P$ to $A$."}
+{"_id": "14019", "title": "Injection from Proper Class to Class", "text": "Let $A$ be a class. Let $\\mathrm P$ be a proper class. Let $f: \\mathrm P \\to A$ be an injection. Then $A$ is proper."}
+{"_id": "14020", "title": "Polynomial has Order Zero", "text": "Let $P:\\C\\to\\C$ be a polynomial function. Then $P$ has order $0$."}
+{"_id": "14022", "title": "Order is Maximum of Exponent of Convergence and Degree", "text": "Let $f: \\C \\to \\C$ be an entire function. Let $\\omega$ be its order. Let $\\tau$ be its exponent of convergence. Let $h$ be the degree of the polynomial in its canonical factorization. Then: :$\\omega = \\map \\max {\\tau, h}$"}
+{"_id": "14023", "title": "Recurring Parts of Multiples of One Fifty-Third", "text": "The multiples of $\\dfrac 1 {53}$ from $\\dfrac 1 {53}$ to $\\dfrac {52} {53}$ can be divided into $4$ sets of equal size: :one where the digits of the recurring part consists of a cyclic permutation of $01886 \\, 79245 \\, 283$ :one where the digits of the recurring part consists of a cyclic permutation of $03773 \\, 58490 \\, 566$ :one where the digits of the recurring part consists of a cyclic permutation of $07547 \\, 16981 \\, 132$ :one where the digits of the recurring part consists of a cyclic permutation of $09433 \\, 96226 \\, 415$."}
+{"_id": "14024", "title": "Order of Reciprocal of Entire Function", "text": "Let $f: \\C \\to \\C$ be an entire function of order $\\rho$. Let $f$ have no zeroes. Then $1/f$ has order $\\rho$."}
+{"_id": "14025", "title": "Jacobi's Necessary Condition/Dependent on N Functions", "text": "Let $J$ be a functional, such that: :$J \\sqbrk {\\mathbf y} = \\displaystyle \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ where $\\mathbf y = \\paren {\\sequence {y_i}_{1 \\le i \\le N} }$ is an N-dimensional real vector. Let $\\map {\\mathbf y} x$ correspond to the minimum of $J$. Let the $N\\times N$ matrix $\\mathbf P = F_{y_i' y_j'}$ be positive definite along $\\map {\\mathbf y} x$. Then the open interval $\\openint a b$ contains no points conjugate to $a$."}
+{"_id": "14027", "title": "Probability of no 2 People out of 53 Sharing the Same Birthday", "text": "Let there be $53$ people in a room. The probability that no $2$ of them have the same birthday is approximately $\\dfrac 1 {53}$."}
+{"_id": "14028", "title": "Primes not Sum of or Difference between Powers of 2 and 3", "text": "The sequence of prime numbers which cannot be expressed as either sum of or the difference between a power of $2$ and a power of $3$ begins: :$53, 71, 103, 107, 109, 149, 151, \\ldots$ {{OEIS|A007644}}"}
+{"_id": "14029", "title": "Triangular Fibonacci Numbers", "text": "The only Fibonacci numbers which are also triangular are: :$0, 1, 3, 21, 55$ {{OEIS|A039595}}"}
+{"_id": "14030", "title": "Repdigit Triangular Numbers", "text": "The only repdigit numbers which are also triangular are: :$55, 66, 666$ {{OEIS|A045914}}"}
+{"_id": "14031", "title": "Increasing and Ordering on Mappings implies Mapping is Composition", "text": "Let $L = \\left({S, \\preceq}\\right), R = \\left({T, \\precsim}\\right)$ be ordered sets. Ley $g:S \\to T, d:T \\to S$ be mappings such that :$g$ and $d$ are increasing mappings and :$d \\circ g \\preceq I_S$ and $I_T \\precsim g \\circ d$ where $\\preceq, \\precsim$ denotes the orderings on mappings. Then $d \\circ \\left({g \\circ d}\\right)$ and $g = \\left({g \\circ d}\\right) \\circ g$"}
+{"_id": "14032", "title": "Closed Form for Square Pyramidal Numbers", "text": "The closed-form expression for the $n$th square pyramidal number is: :$S_n = \\dfrac {n \\paren {n + 1} \\paren {2 n + 1} } 6$"}
+{"_id": "14033", "title": "Closed Form for Pentagonal Pyramidal Numbers", "text": "The closed-form expression for the $n$th pentagonal pyramidal number is: :$Q_n = \\dfrac {n^2 \\paren {n + 1} } 2$"}
+{"_id": "14034", "title": "Surjection from Class to Proper Class", "text": "Let $A$ be a class. Let $\\mathrm P$ be a proper class. Let $f: A \\to \\mathrm P$ be a surjection. Then $A$ is proper."}
+{"_id": "14035", "title": "Image of Set under Mapping is Set", "text": "Let $A$ be a class. Let $\\mathrm U$ denote the universal class. Let $f: A \\to \\mathrm U$ be a class mapping. Let $S$ be a subset of $A$. {{explain|Note that further work is needed on the Subset page to clarify that a \"subset\" of a \"proper class\" is indeed a \"set\". Some words are already on that page, but this needs to be formalised and separated out into its own transcluded \"class theoretic\" page.}} Then the image $f \\sqbrk S$ is also a set. If $A$ is a set, then this result is known as the Axiom of Replacement in Zermelo-Fraenkel set theory."}
+{"_id": "14036", "title": "Square Pyramidal and Triangular Numbers", "text": "The only positive integers which are simultaneously square pyramidal and triangular are: :$1, 55, 91, 208 \\, 335$ {{OEIS|A039596}}"}
+{"_id": "14037", "title": "Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes", "text": "The following positive integers cannot be expressed as the sum of distinct non-pythagorean primes: :$1, 2, 4, 5, 6, 8, 9, 12, 13, 15, 16, 17, 20, 24, 25, 27, 28, 32, 35, 36, 39, 48, 51, 55$ {{OEIS|A048262}} All positive integers greater than $55$ can be so expressed."}
+{"_id": "14038", "title": "Sets of 4 Integers a, b, c, d for which Every Integer is in form ax^2 + by^2 + cz^2 + du^2", "text": "There are exactly $55$ sets of $4$ integers $\\left\\{ {a, b, c, d}\\right\\}$ such that all integers can be written in the form: :$n = a x^2 + b y^2 + c z^2 + d w^2$ for integer $x, y, z, w$."}
+{"_id": "14039", "title": "Palindromes Formed by Multiplying by 55", "text": "$55$ multiplied by any of the odd integers between $91$ and $109$ inclusive produces a palindromic number."}
+{"_id": "14043", "title": "Upper Adjoint of Galois Connection is Surjection implies Lower Adjoint at Element is Minimum of Preimage of Singleton of Element", "text": "Let $L = \\struct {S, \\preceq}, R = \\paren {T, \\precsim}$ be ordered sets. Let $g: S \\to T, d:T \\to S$ be mappings such that: :$\\tuple {g, d}$ is a Galois connection and :$g$ is a surjection. Then :$\\forall t \\in T: \\map d t = \\min \\set {g^{-1} \\sqbrk {\\set t} }$"}
+{"_id": "14046", "title": "Dirichlet Convolution Preserves Multiplicativity/General Result", "text": "Let $S \\subset \\N$ be a set of natural numbers with the property: :$m n \\in S, \\; \\gcd \\left({m, n}\\right) = 1 \\implies m, n \\in S$ Define: :$\\left({f*_S g}\\right) \\left({n}\\right) = \\displaystyle \\sum_{\\substack {d \\mathop \\backslash n \\\\ d \\mathop \\in S} } f \\left({d}\\right) g \\left({n / d}\\right)$ Then $f*_S g$ is multiplicative."}
+{"_id": "14047", "title": "Even Integer with Abundancy Index greater than 9", "text": "Let $n \\in \\Z_{>0}$ have an abundancy index greater than $9$. Then $n$ has at least $35$ distinct prime factors."}
+{"_id": "14049", "title": "Relation Between Rank and Exponent of Convergence", "text": "Let $f: \\C \\to \\C$ be an entire function. Let $k$ be its rank and $\\tau$ be its exponent of convergence. Then: * $k=\\tau=0$ if $f$ has finitely many zeroes. * $k<\\tau\\leq k+1$ otherwise."}
+{"_id": "14052", "title": "Sum of Sequence of Odd Squares/Formulation 1", "text": ":$\\displaystyle \\forall n \\in \\N: \\sum_{i \\mathop = 0}^n \\paren {2 i + 1}^2 = \\frac {\\paren {n + 1} \\paren {2 n + 1} \\paren {2 n + 3} } 3$"}
+{"_id": "14053", "title": "Sum of Sequence of Even Squares", "text": ":$\\displaystyle \\forall n \\in \\N: \\sum_{i \\mathop = 0}^n \\left({2 i}\\right)^2 = \\frac {2 n \\left({n + 1}\\right) \\left({2 n + 1}\\right)} 3$"}
+{"_id": "14054", "title": "Killing Form of Orthogonal Lie Algebra", "text": "Let $\\mathbb K \\in \\left\\{ {\\C, \\R}\\right\\}$. Let $n$ be a positive integer. Let $\\mathfrak{so}_n \\left({\\mathbb K}\\right)$ be the Lie algebra of the special orthogonal group $\\operatorname{SO}_n \\left({\\mathbb K}\\right)$. Then its Killing form is $B: \\left({X, Y}\\right) \\mapsto \\left({n - 2}\\right) \\operatorname {tr} \\left({X Y}\\right)$."}
+{"_id": "14056", "title": "Trace of Alternating Product of Matrices and Almost Zero Matrices", "text": "Let $R$ be a ring with unity. Let $n, m$ be positive integers. Let $E_{ij}$ denote the $n \\times n$ matrix with only zeroes except a $1$ at the $\\tuple {i, j}$th element. Let $A_1, \\ldots, A_m \\in R^{n \\times n}$. Let $i_k, j_k \\in \\set {1, \\ldots, n}$ for $k \\in \\set {1, \\ldots, m}$. Let $i_0 = i_m$ and $j_0 = j_m$. Then: :$\\map \\tr {A_1 E_{i_1, j_1} A_2 E_{i_2, j_2} \\cdots A_m E_{i_m, j_m} } = \\displaystyle \\prod_{k \\mathop = 1}^m \\sqbrk {A_k}_{j_{k - 1} i_k}$"}
+{"_id": "14057", "title": "Trace in Terms of Orthonormal Basis", "text": "Let $\\mathbb K \\subset \\C$ be a field. Let $\\struct {V, \\innerprod {\\, \\cdot \\,} {\\, \\cdot \\,} }$ be an inner product space over $\\mathbb K$ of dimension $n$. Let $\\tuple {e_1, \\ldots, e_n}$ be an orthonormal basis of $V$. Let $f: V \\to V$ be a linear operator. Then its trace equals: :$\\map \\tr f = \\displaystyle \\sum_{i \\mathop = 1}^n \\innerprod {\\map f {e_i} } {e_i}$"}
+{"_id": "14058", "title": "Galois Connection with Upper Adjoint Surjective implies Scond Ordered Set and Image of Lower Adjoint are Isomorphic", "text": "Let $L = \\left({S, \\preceq}\\right), R = \\left({T, \\precsim}\\right)$ be ordered sets. Ley $g:S \\to T, d:T \\to S$ be mappings such that :$\\left({g, d}\\right)$ is Galois connection and :$g$ is a surjection/ Let $N = \\left({d\\left[{T}\\right], \\preceq'}\\right)$ be an ordered subset of $L$. Then $R$ and $N$ are order isomorphic."}
+{"_id": "14059", "title": "Positive Integer Sum of 2 Fourth Powers in 2 Ways", "text": "The smallest positive integer which can be expressed as the sum of $2$ fourth powers in $2$ different ways is $635 \\, 318 \\, 657$: {{begin-eqn}} {{eqn | l = 59^4 + 158^4       | r = 12 \\, 117 \\, 361 + 62 \\, 201 \\, 296       | c =  }} {{eqn | r = 635 \\, 318 \\, 657       | c =  }} {{eqn | r = 312 \\, 900 \\, 721 + 322 \\, 417 \\, 936       | c =  }} {{eqn | r = 133^4 + 134^4       | c =  }} {{end-eqn}}"}
+{"_id": "14060", "title": "Trace in Terms of Dual Basis", "text": "Let $R$ be a ring with unity. Let $M$ be a free $R$-module of dimension $n$. Let $\\tuple {e_1, \\ldots, e_n}$ be a basis of $M$. Let $\\tuple {e_1^*,\\ldots, e_n^*}$ be its dual basis Let $f: M \\to M$ be a linear operator. Then its trace equals: :$\\map \\tr f = \\displaystyle \\sum_{i \\mathop = 1}^n e_i^* \\paren {\\map f {e_i} }$"}
+{"_id": "14061", "title": "Internal Angle of Equilateral Triangle", "text": "The internal angles of an equilateral triangle measure $60^\\circ$ or $\\dfrac \\pi 3$ radians."}
+{"_id": "14062", "title": "Smallest Positive Integer which is Sum of 2 Odd Primes in 6 Ways", "text": "The smallest positive integer which is the sum of $2$ odd primes in $6$ different ways is $60$."}
+{"_id": "14063", "title": "Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways", "text": "The sequence of positive integers $n$ which are the smallest such that they are the sum of $2$ odd primes in $k$ different ways begins as follows: :{| border=\"1\" |- ! align=\"right\" style = \"padding: 2px 10px\" | $k$  ! align=\"right\" style = \"padding: 2px 10px\" | $n$  |- | align=\"right\" style = \"padding: 2px 10px\" | $1$ | align=\"right\" style = \"padding: 2px 10px\" | $6$ |- | align=\"right\" style = \"padding: 2px 10px\" | $2$ | align=\"right\" style = \"padding: 2px 10px\" | $10$ |- | align=\"right\" style = \"padding: 2px 10px\" | $3$ | align=\"right\" style = \"padding: 2px 10px\" | $22$ |- | align=\"right\" style = \"padding: 2px 10px\" | $4$ | align=\"right\" style = \"padding: 2px 10px\" | $34$ |- | align=\"right\" style = \"padding: 2px 10px\" | $5$ | align=\"right\" style = \"padding: 2px 10px\" | $48$ |- | align=\"right\" style = \"padding: 2px 10px\" | $6$ | align=\"right\" style = \"padding: 2px 10px\" | $60$ |} {{OEIS|A001172}}"}
+{"_id": "14064", "title": "Change of Basis Matrix from Basis to Itself is Identity", "text": "Let $R$ be a ring with unity. Let $M$ be a free $R$-module of finite dimension $n>0$. Let $\\mathcal B$ be an ordered basis of $M$. Then the change of basis matrix from $\\mathcal B$ to $\\mathcal B$ is the $n\\times n$ identity matrix: :$\\mathbf M_{\\mathcal B, \\mathcal B} = \\mathbf I$"}
+{"_id": "14065", "title": "Sum of Sequence of Seventh Powers", "text": ":$\\displaystyle \\sum_{j \\mathop = 0}^n j^7 = \\dfrac {n^2 \\paren {n + 1}^2 \\paren {3 n^4 + 6 n^3 - n^2 - 4 n + 2} } {24}$"}
+{"_id": "14066", "title": "Change of Coordinate Vector Under Change of Basis", "text": "Let $R$ be a ring with unity. Let $M$ be a free $R$-module of finite dimension $n>0$. Let $\\BB$ and $\\CC$ be bases of $M$. Let $\\mathbf M_{\\BB, \\CC}$ be the change of basis matrix from $\\BB$ to $\\CC$. Let $m\\in M$. Let $\\sqbrk m_\\BB$ and $\\sqbrk m_\\CC$ be its coordinate vectors relative to $\\BB$ and $\\CC$ respectively. Then $\\sqbrk m_\\BB = \\mathbf M_{\\BB, \\CC} \\cdot \\sqbrk m_\\CC$."}
+{"_id": "14067", "title": "Necessary and Sufficient Condition for Boundary Conditions to be Self-adjoint", "text": "Let $\\mathbf p$ be continuously differentiable. The boundary conditions :$\\bigvalueat {\\map {\\mathbf y} a} {x \\mathop = a} = \\bigvalueat {\\map {\\boldsymbol \\psi} {\\mathbf y} } {x \\mathop = a}$ are self-adjoint {{iff}}: :$\\forall i, k \\in \\N: 1 \\le i, k \\le N: \\valueat {\\dfrac {\\partial p_i \\sqbrk {x, \\mathbf y, \\map {\\boldsymbol \\psi} {\\mathbf y} } } {\\partial y_k} } {x \\mathop = a} = \\valueat {\\dfrac {\\partial p_k \\sqbrk {x, \\mathbf y, \\map {\\boldsymbol \\psi} {\\mathbf y} } } {\\partial y_i} } {x \\mathop = a}$"}
+{"_id": "14068", "title": "Reciprocal of 61", "text": "The decimal expansion of the reciprocal of $61$ has the maximum period, that is: $60$: :$\\dfrac 1 {61} = 0 \\cdotp \\dot 01639 \\, 34426 \\, 22950 \\, 81967 \\, 21311 \\, 47540 \\, 98360 \\, 65573 \\, 77049 \\, 18032 \\, 78688 \\, 5245 \\dot 9$ {{OEIS|A007450}} It also contains an equal number ($6$) of each of the digits from $0$ to $9$."}
+{"_id": "14069", "title": "Composition of Galois Connections is Galois Connection", "text": "Let $L_1 = \\left({S_1, \\preceq_1}\\right), L_2 = \\left({S_2, \\preceq_2}\\right), L_3 = \\left({S_3, \\preceq_3}\\right)$ be ordered sets. Let $g_1:S_1 \\to S_2, g_2:S_2 \\to S_3, d_1:S_2 \\to S_1, d_2:S_3 \\to S_2$ be mappings such that :$\\left({g_1, d_1}\\right)$ and $\\left({g_2, d_2}\\right)$ are Galois connections. Then $\\left({g_2 \\circ g_1, d_1 \\circ d_2}\\right)$ is also Galois connection."}
+{"_id": "14070", "title": "Vector Space has Basis", "text": "Let $K$ be a division ring. Let $V$ be a vector space over $K$. Then $V$ has a basis."}
+{"_id": "14071", "title": "Product of Change of Basis Matrices", "text": "Let $R$ be a ring with unity. Let $M$ be a free $R$-module of finite dimension $n>0$. Let $\\mathcal A$, $\\mathcal B$ and $\\mathcal C$ be ordered bases of $M$. Let $\\mathbf M_{\\mathcal A,\\mathcal B}$, $\\mathbf M_{\\mathcal B,\\mathcal C}$ and $\\mathbf M_{\\mathcal A,\\mathcal C}$ be the change of basis matrices from $\\mathcal A$ to $\\mathcal B$, $\\mathcal B$ to $\\mathcal C$ and $\\mathcal A$ to $\\mathcal C$ respectively. Then $\\mathbf M_{\\mathcal A,\\mathcal C} = \\mathbf M_{\\mathcal A,\\mathcal B} \\cdot \\mathbf M_{\\mathcal B,\\mathcal C}$"}
+{"_id": "14072", "title": "Relative Matrix of Composition of Linear Transformations", "text": "Let $R$ be a ring with unity. Let $M, N, P$ be free $R$-modules of finite dimension $m, n, p > 0$ respectively. Let $\\mathcal A,\\mathcal B,\\mathcal C$ be ordered bases of $M, N, P$. Let $f: M \\to N$ and $g : N \\to P$ be linear transformations, and $g \\circ f$ be their composition. Let $\\mathbf M_{f, \\mathcal B, \\mathcal A}$ and $\\mathbf M_{g, \\mathcal C, \\mathcal B}$ be their matrices relative to $\\mathcal A, \\mathcal B$ and $\\mathcal B, \\mathcal C$ respectively. Then the matrix of $g \\circ f$ relative to $\\mathcal A$ and $\\mathcal C$ is: :$\\mathbf M_{g \\mathop \\circ f, \\mathcal C, \\mathcal A} = \\mathbf M_{g, \\mathcal C, \\mathcal B}\\cdot \\mathbf M_{f, \\mathcal B, \\mathcal A}$"}
+{"_id": "14073", "title": "Change of Coordinate Vectors Under Linear Transformation", "text": "Let $R$ be a ring with unity. Let $M, N$ be free $R$-modules of finite dimension $m, n > 0$ respectively. Let $\\mathcal A, \\mathcal B$ be ordered bases of $M$ and $N$ respectively. Let $f: M \\to N$ be a linear transformation. Let $\\mathbf M_{f, \\mathcal B, \\mathcal A}$ be its matrix relative to $\\mathcal A$ and $\\mathcal B$. Then for all $m \\in M$: :$\\left[{f \\left({m}\\right)}\\right]_{\\mathcal B} = \\mathbf M_{f, \\mathcal B, \\mathcal A} \\cdot \\left[{m}\\right]_{\\mathcal A}$ where $\\left[{\\, \\cdot \\,}\\right]_{-}$ denotes the coordinate vector with respect to a basis."}
+{"_id": "14076", "title": "Matrix of Bilinear Form Under Change of Basis", "text": "Let $R$ be a ring with unity. Let $M$ be a free $R$-module of finite dimension $n>0$. Let $\\mathcal A$ and $\\mathcal B$ be ordered bases of $M$. Let $\\mathbf M_{\\mathcal A, \\mathcal B}$ be the change of basis matrix from $\\mathcal A$ to $\\mathcal B$. Let $f : M\\times M \\to R$ be a bilinear form. Let $\\mathbf M_{f, \\mathcal A}$ be its matrix relative to $\\mathcal A$. Then its matrix relative to $\\mathcal B$ equals: :$\\mathbf M_{f, \\mathcal B} = \\mathbf M_{\\mathcal A, \\mathcal B}^\\intercal \\mathbf M_{f, \\mathcal A} \\mathbf M_{\\mathcal A, \\mathcal B}$"}
+{"_id": "14077", "title": "Symmetric Bilinear Form is Reflexive", "text": "Let $\\mathbb K$ be a field. Let $V$ be a vector space over $\\mathbb K$. Let $b$ be a bilinear form on $V$. Let $b$ be symmetric. Then $b$ is reflexive."}
+{"_id": "14078", "title": "Alternating Bilinear Form is Reflexive", "text": "Let $\\mathbb K$ be a field. Let $V$ be a vector space over $\\mathbb K$. Let $b$ be a bilinear form on $V$. Let $b$ be alternating. Then $b$ is reflexive."}
+{"_id": "14079", "title": "Bilinear Form is Reflexive iff Symmetric or Alternating", "text": "Let $\\mathbb K$ be a field. Let $V$ be a vector space over $\\mathbb K$. Let $b$ be a bilinear form on $V$. Then the following are equivalent: :$(1): \\quad$ $b$ is reflexive :$(2): \\quad$ $b$ is symmetric or alternating"}
+{"_id": "14080", "title": "Reflexive Bilinear Form is Symmetric or Alternating", "text": "Let $\\mathbb K$ be a field. Let $V$ be a vector space over $\\mathbb K$. Let $f$ be a bilinear form on $V$. Let $f$ be reflexive. Then $f$ is symmetric or alternating."}
+{"_id": "14081", "title": "Apéry's Theorem", "text": "Apéry's constant: :$\\map \\zeta 3 = \\displaystyle \\sum_{n \\mathop = 1}^\\infty \\frac 1 {n^3}$ is irrational."}
+{"_id": "14082", "title": "Reciprocals whose Decimal Expansion contain Equal Numbers of Digits from 0 to 9", "text": "The following positive integers $p$ have reciprocals whose decimal expansions: :$(1): \\quad$ have the maximum period, that is: $p - 1$ :$(2): \\quad$ have an equal number, $\\dfrac {p - 1} {10}$, of each of the digits from $0$ to $9$: ::$61, 131,\\ldots$"}
+{"_id": "14083", "title": "Reciprocal of 131", "text": "The decimal expansion of the reciprocal of $131$ has the maximum period, that is: $130$: :$\\dfrac 1 {131} = 0 \\cdotp \\dot 00763 \\, 35877 \\, 86259 \\, 54198 \\, 47328 \\, 24427 \\, 48091 \\, 60305 \\, 34351 \\, 14503 \\, 81679 \\, 38931 \\, 29770 \\, 99236 \\, 64122 \\, 13740 \\, 45801 \\, 52671 \\, 75572 \\, 51908 \\, 39694 \\, 65648 \\, 85496 \\, 18320 \\, 61068 \\, 7022 \\dot 9$ {{OEIS|A007450}} It also contains an equal number ($13$) of each of the digits from $0$ to $9$."}
+{"_id": "14084", "title": "Sequences of 4 Consecutive Integers with Rising Sigma", "text": "The following ordered quadruples of consecutive integers have sigma values which are strictly increasing: :$61, 62, 63, 64$ :$73, 74, 75, 76$"}
+{"_id": "14085", "title": "Symmetric Bilinear Form can be Diagonalized", "text": "Let $\\mathbb K$ be a field. Let $V$ be a vector space over $\\mathbb K$ of finite dimension $n>0$. Let $f$ be a symmetric bilinear form on $V$. Then there exists an ordered basis for which the relative matrix of $f$ is diagonal."}
+{"_id": "14086", "title": "Dimension of Radical of Bilinear Form", "text": "Let $\\mathbb K$ be a field. Let $V$ be a vector space over $\\mathbb K$ of finite dimension $n > 0$. Let $f$ be a bilinear form on $V$. Let $\\operatorname{rad} \\left({V}\\right)$ be the radical of $V$. Let $\\operatorname{rk} \\left({f}\\right)$ be the rank of $f$. Then: : $\\dim \\left({\\operatorname{rad} \\left({V}\\right)}\\right) = n - \\operatorname{rk} \\left({f}\\right)$ where $\\dim$ denotes dimension."}
+{"_id": "14087", "title": "Dimension of Orthogonal Complement With Respect to Bilinear Form", "text": "Let $\\mathbb K$ be a field. Let $V$ be a vector space over $\\mathbb K$ of finite dimension. Let $f$ be a nondegenerate bilinear form on $V$. Let $U\\subset V$ be a subspace. Let $U^\\perp$ be its orthogonal complement. Then: :$\\map \\dim U + \\map \\dim U^\\perp = \\map \\dim V$"}
+{"_id": "14089", "title": "Anisotropic Vector Gives Composition of Bilinear Space", "text": "Let $\\mathbb K$ be a field. Let $\\left({V, f}\\right)$ be a bilinear space over $\\mathbb K$. Let $v \\in V$ be anisotropic. Let $\\left\\langle{v}\\right\\rangle$ be its span. Let $v^\\perp$ be its orthogonal complement. Then $\\left({V, f}\\right)$ is the internal orthogonal sum of $\\left\\langle{v}\\right\\rangle$ and $v^\\perp$: :$V = \\left\\langle{v}\\right\\rangle \\oplus v^\\perp$"}
+{"_id": "14090", "title": "Basis of Free Module is No Greater than Generator", "text": "Let $R$ be a commutative ring with unity. Let $M$ be a free $R$-module with basis $B$. Let $S$ be a generating set for $M$. Then: :$\\size B \\le \\size S$. That is, there exists an injection from $B$ to $S$."}
+{"_id": "14091", "title": "Extended Rolle's Theorem", "text": "Let $f: D \\to \\R$ be differentiable on a closed interval $I \\subseteq \\R$. Let $x_0 < x_1 < \\dots < x_n \\in I$. Let $f \\left({x_i}\\right) = 0$ for $i = 0, \\ldots, n$. Then for all $i = 0, \\ldots, n-1$: : $\\exists \\xi_i \\in \\left({x_i \\,.\\,.\\, x_{i+1}}\\right): f' \\left({\\xi_i}\\right) = 0$"}
+{"_id": "14092", "title": "Pell's Equation/Examples/61", "text": ":$x^2 - 61 y^2 = 1$ has the smallest positive integral solution: :$x = 1 \\, 766 \\, 319 \\, 049$ :$y = 226 \\, 153 \\, 980$"}
+{"_id": "14093", "title": "No Infinitely Descending Membership Chains/Corollary", "text": "There cannot exist a sequence $\\left\\langle{x_n}\\right\\rangle$ whose domain is $\\N_{\\gt 0}$ such that: :$\\forall n \\in \\N_{\\gt 0}: x_{n+1} \\in x_n$"}
+{"_id": "14094", "title": "Sequence of Inconsummate Numbers", "text": "The sequence of inconsummate numbers begins: :$62, 63, 65, 75, 84, 95, 161, 173, 195, 216, 261, 266, 272, 276, \\ldots$ {{OEIS|A003635}}"}
+{"_id": "14095", "title": "Kaprekar's Process for 2-Digit Numbers", "text": "Kaprekar's process, when applied to a non-repdigit $2$-digit positive integer leads to the cycle: :$09 \\to 81 \\to 63 \\to 27 \\to 45 \\to 09$ Note that it is important to retain the leading zero on the $9$, or the process trivially terminates in $0$."}
+{"_id": "14097", "title": "Numbers with 6 or more Prime Factors", "text": "The sequence of positive integers with $6$ or more prime factors (not necessarily distinct) begins: :$64, 96, 128, 144, 160, 192, 216, 224, 240, 256, \\ldots$ {{OEIS|A046305}}"}
+{"_id": "14098", "title": "Cube as Sum of Sequence of Centered Hexagonal Numbers", "text": ":$C_n = \\displaystyle \\sum_{i \\mathop = 1}^n H_i$ where: :$C_n$ denotes the $n$th cube number :$H_i$ denotes the $i$th centered hexagonal number."}
+{"_id": "14099", "title": "Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared", "text": "Let $p$ be a prime number. Then there exists at least one positive integer $n$ greater than $1$ such that: :$n^{p - 1} \\equiv 1 \\pmod {p^2}$"}
+{"_id": "14100", "title": "Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes", "text": "The smallest positive integer which is both the sum of $2$ square numbers in two distinct ways and also the sum of $2$ cube numbers is $65$: {{begin-eqn}} {{eqn | l = 65       | m = 16 + 49       | mo= =       | r = 4^2 + 7^2       | c =  }} {{eqn | m = 1 + 64       | mo= =       | r = 1^2 + 8^2       | c =  }} {{eqn | o =        | mo= =       | r = 1^3 + 4^3       | c =  }} {{end-eqn}}"}
+{"_id": "14101", "title": "Magic Constant of Order 5 Magic Square", "text": "The magic constant of the order $5$ magic square is $65$."}
+{"_id": "14103", "title": "Triple of Triangular Numbers whose Pairwise Sums are Triangular", "text": "The following triplet of triangular numbers has the property that the sum of each pair of them, and their total, are all triangular numbers: :$66, 105, 105$"}
+{"_id": "14106", "title": "Largest Even Integer not expressible as Sum of 2 k Odd Composite Integers", "text": "Let $k \\in \\Z_{>0}$ be a (strictly) positive integer. The largest even integer which cannot be expressed as the sum of $2 k$ odd positive composite integers is $18 k + 20$."}
+{"_id": "14107", "title": "Number whose Square and Cube use all Digits Once", "text": "The only integer whose square and cube use each of the digits from $0$ to $9$ exactly once each is $69$."}
+{"_id": "14108", "title": "Cube of 71 is Odd Integers in Sequence", "text": "The cube of $71$, when expressed in decimal notation, is the odd integers from $3$ to $11$ written in sequence: ::$71^3 = 357 \\, 911$"}
+{"_id": "14109", "title": "Prime Numbers which Divide Sum of All Lesser Primes", "text": "The following sequence of prime numbers has the property that each is a divisor of the sum of all primes smaller than them: :$2, 5, 71, 369 \\, 119, 415 \\, 074 \\, 643$ {{OEIS|A007506}} As of time of writing (April $2020$), no others are known."}
+{"_id": "14110", "title": "4 Positive Integers in Arithmetic Sequence which have Same Euler Phi Value", "text": "The following sets of $4$ positive integers which form an arithmetic sequence are the smallest which all have the same Euler $\\phi$ value: :$72, 78, 84, 90$ :$216, 222, 228, 234$ :$76 \\, 326, 76 \\, 332, 76 \\, 338, 76 \\, 344$"}
+{"_id": "14111", "title": "Positive Integers which are Euler Phi Value for 17 Integers", "text": "There are $17$ positive integers which have an Euler $\\phi$ value of the following: :$72, 96, 120, \\ldots$"}
+{"_id": "14112", "title": "Euler Phi Function of Non-Square Semiprime", "text": "Let $n \\in \\Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$. Let $\\map \\phi n$ denote the Euler $\\phi$ function. Then: :$\\map \\phi n = \\paren {p - 1} \\paren {q - 1}$"}
+{"_id": "14115", "title": "Euler Phi Function of Square-Free Integer", "text": "Let $n$ be an integer such that $n \\ge 2$. Let $n$ be square-free. Let $\\map \\phi n$ be the Euler $\\phi$ function of $n$. That is, let $\\map \\phi n$ be the count of strictly positive integers less than or equal to $n$ which are prime to $n$. Then: :$\\map \\phi n = \\displaystyle \\prod_{\\substack {p \\mathop \\divides n \\\\ p \\mathop > 2} } \\paren {p - 1}$ where $p \\divides n$ denotes the primes which divide $n$."}
+{"_id": "14118", "title": "Euler Phi Function of 2 times Odd Prime", "text": "Let $n \\in \\Z_{>0}$ be a semiprime of the form $2 p$, where $p$ is an odd prime. Let $\\map \\phi n$ denote the Euler $\\phi$ function. Then: :$\\map \\phi n = p - 1$"}
+{"_id": "14119", "title": "Set is Not Element of Itself", "text": "There cannot exist a set which is an element of itself. That is: :$\\neg \\exists a: a \\in a$"}
+{"_id": "14122", "title": "Product of Number of Edges, Edges per Face and Faces of Tetrahedron", "text": "The product of the number of edges, edges per face and faces of a tetrahedron is $72$."}
+{"_id": "14124", "title": "Product of Number of Edges, Edges per Face and Faces of Regular Octahedron", "text": "The product of the number of edges, edges per face and faces of a regular octahedron is $288$."}
+{"_id": "14125", "title": "Ordinals are Well-Ordered/Corollary", "text": "Let $A$ be a set of ordinals. Let $\\Epsilon {\\restriction_A}$ denote the epsilon restriction on $A$. Then $A$ is strictly well-ordered by $\\Epsilon {\\restriction_A}$."}
+{"_id": "14127", "title": "Product of Number of Edges, Edges per Face and Faces of Regular Icosahedron", "text": "The product of the number of edges, edges per face and faces of a regular icosahedron is $1800$."}
+{"_id": "14128", "title": "Smallest 5th Power equal to Sum of 5 other 5th Powers", "text": "The smallest positive integer whose fifth power can be expressed as the sum of $5$ other fifth powers is $72$: :$72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5$"}
+{"_id": "14129", "title": "Smallest Consecutive Even Nontotients", "text": "The smallest pair of consecutive even nontotients is $74$ and $76$."}
+{"_id": "14130", "title": "Positive Integers Expressible by Sum of Integers whose Reciprocals Sum to 1", "text": "Every positive integer over $77$ can be expressed as the sum of positive integers whose reciprocals add up to $1$. The full sequence of numbers that cannot be expressed as such is: :$2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 33, 34, 35, 36, 39, 40, 41, 42, 44, 46, 47, 48, 49, 51, 56, 58, 63, 68, 70, 72, 77$ {{OEIS|A051882}}"}
+{"_id": "14131", "title": "Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers", "text": ":$79 = 15 \\times 1^4 + 4 \\times 2^4$"}
+{"_id": "14133", "title": "Reciprocal of Square of 1 Less than Number Base", "text": "Let $b \\in \\Z$ be an integer such that $b > 2$. Let $n = \\paren {b - 1}^2$. The reciprocal of $n$, expressed in base $b$, recurs with period $b - 1$: :$\\dfrac 1 n = \\sqbrk {0.012 \\ldots cd012 \\ldots}_b$ where: :$c = b - 3$ :$d = b - 1$"}
+{"_id": "14134", "title": "Square of 1 Less than Number Base", "text": "Let $b \\in \\Z$ be an integer such that $b > 2$. Let $n = b - 1$. The square of $n$ is expressed in base $b$ as: :$n^2 = \\left[{c1}\\right]_b$ where $c = b - 2$."}
+{"_id": "14135", "title": "Positive Integers whose Square Root equals Sum of Digits", "text": "The following positive integers have a square root that equals the sum of their digits: :$0, 1, 81$ and there are no more."}
+{"_id": "14137", "title": "Closed Form for Heptagonal Numbers", "text": "The closed-form expression for the $n$th heptagonal number is: :$H_n = \\dfrac {n \\paren {5 n - 3} } 2$"}
+{"_id": "14138", "title": "Square Numbers whose Sigma is Square", "text": "The sequence of square numbers whose $\\sigma$ value is square starts as follows: {{begin-eqn}} {{eqn | l = \\map \\sigma {1^2}       | r = 1^2       | c =  }} {{eqn | l = \\map \\sigma {9^2}       | r = 11^2       | c =  }} {{eqn | l = \\map \\sigma {20^2}       | r = 31^2       | c =  }} {{eqn | l = \\map \\sigma {180^2}       | r = 341^2       | c =  }} {{eqn | l = \\map \\sigma {1306^2}       | r = 1729^2       | c =  }} {{end-eqn}} {{OEIS|A008847}}"}
+{"_id": "14139", "title": "Integers for which Sigma of Phi equals Sigma", "text": "The following positive integers have the property that the $\\sigma$ value of their Euler $\\phi$ value equals their $\\sigma$ value: :$\\map \\sigma {\\map \\phi n} = \\map \\sigma n$ :$1, 87, 362, 1257, 1798, 5002, 9374, \\ldots$ {{OEIS|A033631}}"}
+{"_id": "14140", "title": "Consecutive Triplets not Sum of Pentagonal Numbers", "text": "The following triplets of consecutive positive integers are such that none is the sum of $3$ pentagonal numbers: :$\\tuple {19, 20, 21}$ :$\\tuple {88, 89, 90}$ :$\\tuple {99, 100, 101}$ :$\\tuple {111, 112, 113}$"}
+{"_id": "14141", "title": "Prime Gaps of 8", "text": "The following pairs of consecutive prime numbers are those whose difference is $8$: :$\\left({89, 97}\\right), \\left({359, 367}\\right), \\left({389, 397}\\right), \\left({401, 409}\\right), \\ldots$ {{OEIS|A031926}}"}
+{"_id": "14142", "title": "Smallest Cunningham Chain of the First Kind of Length 6", "text": "The smallest Cunningham chain of the first kind of length $6$ is: :$\\tuple {89, 179, 359, 719, 1439, 2879}$"}
+{"_id": "14143", "title": "Sequence of Sum of Squares of Digits", "text": "For a positive integer $n$, let $\\map f n$ be the integer created by adding the squares of digits of $n$. Let $m \\in \\Z_{>0}$ be expressed in decimal notation. Let $\\sequence {S_m}_{n \\mathop \\in \\Z_{>0} }$ be the sequence defined as follows: :$n_k = \\begin{cases} m & : n = 1 \\\\ \\map f {n_{k - 1} } & : n > 1 \\end{cases}$ Then eventually either $\\sequence {S_m}$ sticks at $1$, or goes into the cycle: :$\\ldots, 4, 16, 37, 58, 89, 145, 42, 20, 4, \\ldots$"}
+{"_id": "14145", "title": "2-Digit Numbers forming Longest Reverse-and-Add Sequence", "text": "Let $m \\in \\Z_{>0}$ be a positive integer expressed in decimal notation. Let $\\map r m$ be the reverse-and-add process on $m$. Let $r$ be applied iteratively to $m$. The $2$-digit integers $m$ which need the largest number of iterations before reaching a palindromic number are $89$ and $98$, both needing $24$ iterations."}
+{"_id": "14147", "title": "Cauchy Sequence is Bounded/Real Numbers", "text": "Every Cauchy sequence in $\\R$ is bounded."}
+{"_id": "14149", "title": "Fermat Pseudoprime/Base 3/Examples/91", "text": "The smallest Fermat pseudoprime to base $3$ is $91$: :$3^{91} \\equiv 3 \\pmod {91}$ despite the fact that $91$ is not prime: :$91 = 7 \\times 13$"}
+{"_id": "14151", "title": "Even Integers not Sum of 2 Twin Primes", "text": "The following even integers cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes: :$2, 4, 94, 96, 98, 400, 402, 404, \\ldots$ {{OEIS|A007534}} {{expand|add a page to the effect that it is conjectured that this list is complete.}}"}
+{"_id": "14153", "title": "Central Field is Field of Functional", "text": "Let $\\mathbf y$ be an $N$-dimensional vector. Let $J$ be a functional, such that: :$\\ds J \\sqbrk {\\mathbf y} = \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ Let the following be a central field: :$\\map {\\mathbf y'} x = \\map {\\boldsymbol \\psi} {x, \\mathbf y}$ Then this central field is a field of functional $J$."}
+{"_id": "14154", "title": "Continued Fraction Expansion of Fourth Power of Pi", "text": "The continued fraction expansion of the $4$th power of $\\pi$ begins: :$\\pi^4 \\approx 97 + \\cfrac 1 {2 + \\cfrac 1 {2 + \\cfrac 1 {3 + \\cfrac 1 {1 + \\cfrac 1 {16 \\, 539} } } } }$ {{OEIS|A058286}}"}
+{"_id": "14155", "title": "Decimal Expansion of Fourth Power of Pi", "text": "The decimal expansion of the $4$th power of $\\pi$ begins: :$97 \\cdotp 40909 \\, 10340 \\, 0 \\ldots$ {{OEIS|A092425}}"}
+{"_id": "14157", "title": "Even Integers not Expressible as Sum of 3, 5 or 7 with Prime", "text": "The even integers that cannot be expressed as the sum of $2$ prime numbers where one of those primes is $3$, $5$ or $7$ begins: :$98, 122, 124, 126, 128, 148, 150, \\ldots$ {{OEIS|A283555}}"}
+{"_id": "14159", "title": "Repdigit Number consisting of Instances of 9 is Kaprekar", "text": "A repdigit number that consists entirely of the digit $9$ is a Kaprekar number."}
+{"_id": "14164", "title": "Numbers equal to Sum of Primes not Greater than its Prime Counting Function Value", "text": "Let $\\pi \\left({n}\\right): \\Z_{\\ge 0} \\to \\Z_{\\ge 0}$ denote the prime-counting function: :$\\pi \\left({n}\\right) =$ the count of the primes less than $n$ Consider the equation: :$n = \\displaystyle \\sum_{p \\mathop \\le \\pi \\left({n}\\right)} p$ where $p \\le \\pi \\left({n}\\right)$ denotes the primes not greater than $\\pi \\left({n}\\right)$. Then $n$ is one of: :$5, 17, 41, 77, 100$ {{OEIS|A091864}}"}
+{"_id": "14165", "title": "Smallest Seventh Power which is Sum of 8 other Seventh Powers", "text": "The smallest seventh power that can be expressed as the sum of $8$ other seventh powers is $102^7$: :$102^7 = 12^7 + 35^7 + 53^7 + 58^7 + 64^7 + 83^7 + 85^7 + 90^7$"}
+{"_id": "14167", "title": "Consecutive Integers with Same Euler Phi Value", "text": "Let $\\phi: \\Z_{>0} \\to \\Z_{>0}$ be the Euler $\\phi$ function, defined on the strictly positive integers. The equation: :$\\map \\phi n = \\map \\phi {n + 1}$ is satisfied by integers in the sequence: :$1, 3, 15, 104, 164, 194, 255, 495, 584, 975, 2204, \\ldots$ {{OEIS|A001274}}"}
+{"_id": "14169", "title": "Sequences of 3 Consecutive Integers with Rising Phi", "text": "The following ordered triples of consecutive integers have $\\phi$ values which are strictly increasing: :$105, 106, 107$ :$165, 166, 167$ :$315, 316, 317$"}
+{"_id": "14170", "title": "Integers whose Sigma equals Half Phi times Tau", "text": "The following positive integers $n$ have the property where: :$\\sigma \\left({n}\\right) = \\dfrac {\\phi \\left({n}\\right) \\times \\tau \\left({n}\\right)} 2$ where: :$\\sigma \\left({n}\\right)$ denotes the $\\sigma$ function: the sum of the divisors of $n$ :$\\phi \\left({n}\\right)$ denotes the Euler $\\phi$ function: the count of positive integers smaller than of $n$ which are coprime to $n$ :$\\tau \\left({n}\\right)$ denotes the $\\tau$ function: the count of the divisors of $n$: These positive integers are: :$35, 105, \\ldots$"}
+{"_id": "14171", "title": "Reciprocals of Odd Numbers adding to 1", "text": "$105$ is the smallest positive integer $n$ such that $1$ can be expressed as the sum of reciprocals of distinct odd integers such that none are less than $\\dfrac 1 n$: :$1 = \\dfrac 1 3 + \\dfrac 1 5 + \\dfrac 1 7 + \\dfrac 1 9 + \\dfrac 1 {11} + \\dfrac 1 {33} + \\dfrac 1 {35} + \\dfrac 1 {45} + \\dfrac 1 {55} + \\dfrac 1 {77} + \\dfrac 1 {105}$ {{OEIS|A238795}} There are $5$ ways of expressing $1$ as the sum of reciprocals of only $9$ distinct odd integers, but then the least term is less than $\\dfrac 1 {105}$. The $9$-term solutions are as follows: :$1 = \\dfrac 1 3 + \\dfrac 1 5 + \\dfrac 1 7 + \\dfrac 1 9 + \\dfrac 1 {11} + \\dfrac 1 {15} + \\dfrac 1 {35} + \\dfrac 1 {45} + \\dfrac 1 {231}$ {{OEIS|A201644}} :$1 = \\dfrac 1 3 + \\dfrac 1 5 + \\dfrac 1 7 + \\dfrac 1 9 + \\dfrac 1 {11} + \\dfrac 1 {15} + \\dfrac 1 {21} + \\dfrac 1 {231} + \\dfrac 1 {315}$ {{OEIS|A201648}} :$1 = \\dfrac 1 3 + \\dfrac 1 5 + \\dfrac 1 7 + \\dfrac 1 9 + \\dfrac 1 {11} + \\dfrac 1 {15} + \\dfrac 1 {33} + \\dfrac 1 {45} + \\dfrac 1 {385}$ {{OEIS|A201649}} :$1 = \\dfrac 1 3 + \\dfrac 1 5 + \\dfrac 1 7 + \\dfrac 1 9 + \\dfrac 1 {11} + \\dfrac 1 {15} + \\dfrac 1 {21} + \\dfrac 1 {165} + \\dfrac 1 {693}$ {{OEIS|A201647}} :$1 = \\dfrac 1 3 + \\dfrac 1 5 + \\dfrac 1 7 + \\dfrac 1 9 + \\dfrac 1 {11} + \\dfrac 1 {15} + \\dfrac 1 {21} + \\dfrac 1 {135} + \\dfrac 1 {10 \\, 395}$ {{OEIS|A201646}}"}
+{"_id": "14173", "title": "Smallest Polyomino with Hole", "text": "The smallest polyomino with a hole is the heptomino in the form of a $3 \\times 3$  square with the center $1 \\times 1$ square and a corner $1 \\times 1$ square missing: :200px"}
+{"_id": "14174", "title": "Integers whose Sigma Value is Cube", "text": "The following positive integers are those whose $\\sigma$ value is a cube: :$1, 7, 102, 110, 142, 159, 187, 381, 690, 714, 770, 994, 1034, \\ldots$ {{OEIS|A020477}}"}
+{"_id": "14175", "title": "Magic Constant of Smallest Prime Magic Square", "text": "The magic constant of the smallest prime magic square is $111$."}
+{"_id": "14177", "title": "Sequence of Square Lucky Numbers", "text": "The sequence of lucky numbers which are also square begins: :$1, 9, 25, 49, 169, 289, 361, 529, \\ldots$ {{OEIS|A031162}}"}
+{"_id": "14178", "title": "Conditions for Extremal Embedding in Field of Functional", "text": "Let $J$ be a functional such that: :$\\ds J \\sqbrk {\\mathbf y} = \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ Let $\\gamma$ be an extremal of $J$, defined by $\\mathbf y = \\map {\\mathbf y} x$ for $x \\in \\closedint a b$. Suppose: :$\\forall x \\in \\closedint a b: \\det \\paren {F_{\\mathbf y' \\mathbf y'} } \\ne 0$ Suppose no points conjugate to $\\paren {a, \\map {\\mathbf y} a}$ lie on $\\gamma$. Then $\\gamma$ can be embedded in a field."}
+{"_id": "14179", "title": "Sequence of Smallest Consecutive Composite Numbers longer than 100", "text": "The $1$st prime gap greater than $100$ is between $370 \\, 261$ and $370 \\, 373$, of length $112$. That is, the sequence of the smallest consecutive composite positive integers longer than $100$ is that of $111$ such, from  $370 \\, 262$ to $370 \\, 372$."}
+{"_id": "14180", "title": "Difference between Two Squares equal to Repunit", "text": "The sequence of differences of two squares that each make a repunit begins: {{begin-eqn}} {{eqn | l = 1^2 - 0^2       | r = 1       | c =  }} {{eqn | l = 6^2 - 5^2       | r = 11       | c =  }} {{eqn | l = 20^2 - 17^2       | r = 111       | c =  }} {{eqn | l = 56^2 - 45^2       | r = 1111       | c =  }} {{eqn | l = 56^2 - 55^2       | r = 111       | c =  }} {{eqn | l = 156^2 - 115^2       | r = 11 \\, 111       | c =  }} {{eqn | l = 340^2 - 67^2       | r = 111 \\, 111       | c =  }} {{eqn | l = 344^2 - 65^2       | r = 111 \\, 111       | c =  }} {{eqn | l = 356^2 - 125^2       | r = 111 \\, 111       | c =  }} {{end-eqn}}"}
+{"_id": "14181", "title": "Smallest Equilateral Triangle with Internal Point at Integer Distances from Vertices", "text": "The smallest equilateral triangle with sides of integer length which contains a point which is an integer distance from each vertex has a side length $112$: :500px There exists a point inside it which is $57$, $65$ and $73$ away from the three vertices."}
+{"_id": "14182", "title": "Smallest 3-Digit Permutable Prime", "text": "The smallest $3$-digit permutable prime is $113$."}
+{"_id": "14183", "title": "Infinite Group has Infinite Number of Subgroups", "text": "Let $\\struct {G, \\circ}$ be an infinite group. Then $\\struct {G, \\circ}$ has an infinite number of distinct subgroups."}
+{"_id": "14184", "title": "Group is Finite iff Finite Number of Subgroups", "text": "Let $\\struct {G, \\circ}$ be a group. Then $G$ is finite {{iff}} $\\struct {G, \\circ}$ has a  finite number of subgroups."}
+{"_id": "14185", "title": "3-Digit Permutable Primes", "text": "The $3$-digit permutable primes are: :$311, 199, 337$ and their anagrams, and no other."}
+{"_id": "14186", "title": "Digits of Permutable Prime", "text": "Let $p$ be a permutable prime with more than $1$ digit. Then $p$ contains only digits from the set: :$\\left\\{ {1, 3, 7, 9}\\right\\}$"}
+{"_id": "14187", "title": "Divisibility by 5", "text": "An integer $N$ expressed in decimal notation is divisible by $5$ {{iff}} the units digit of $N$ is divisible by $5$. That is: :$N = \\sqbrk {a_n \\ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \\cdots + a_n 10^n$ is divisible by $5$  {{iff}}: :$a_0$ is divisible by $5$."}
+{"_id": "14188", "title": "Prime Repdigit Number is Repunit", "text": "Let $b \\in \\Z_{>0}$ be an integer greater than $1$. Let $n \\in \\Z$ expressed in base $b$ be a repdigit number with more than $1$ digit. Let $n$ be prime. Then $n$ is a repunit (in base $b$)."}
+{"_id": "14190", "title": "Prime Gaps of 14", "text": "The following pairs of consecutive prime numbers are those whose difference is $14$: :$\\tuple {113, 127}, \\tuple {293, 307}, \\tuple {317, 331}, \\ldots$ {{OEIS|A031932}}"}
+{"_id": "14191", "title": "Prime Gaps of 18", "text": "After the prime gap of $14$ between the pairs of consecutive prime numbers: :$\\left({113, 127}\\right), \\left({293, 307}\\right), \\left({317, 331}\\right), \\ldots$ the next prime gap which is greater than $14$ is between the pair of consecutive prime numbers: :$\\left({523, 541}\\right)$ for a prime gap of $18$."}
+{"_id": "14192", "title": "Number of Different Ways to Colour the Faces of Cube with 3 Colours", "text": "The number of different ways to colour the faces of a cube with $3$ given colours, one colour per face, is $114$."}
+{"_id": "14193", "title": "Number of Different Ways to Colour the Faces of Cube with 4 Colours", "text": "The number of different ways to colour the faces of a cube with $4$ given colours, one colour per face, is $2652$."}
+{"_id": "14194", "title": "Number of Different Ways to Colour the Faces of Cube with 5 Colours", "text": "The number of different ways to colour the faces of a cube with $5$ given colours, one colour per face, is $29 \\, 660$."}
+{"_id": "14195", "title": "Smallest Number which is Sum of 4 Triples with Equal Products", "text": "The smallest positive integer which is the sum of $4$  distinct ordered triples, each of which has the same product, is $118$: {{begin-eqn}} {{eqn | l = 118       | r = 14 + 50 + 54 }} {{eqn | r = 15 + 40 + 63       | c =  }} {{eqn | r = 18 + 30 + 70       | c =  }} {{eqn | r = 21 + 25 + 72       | c =  }} {{end-eqn}}"}
+{"_id": "14196", "title": "Interval containing Prime Number of forms 4n - 1, 4n + 1, 6n - 1, 6n + 1", "text": "Let $n \\in \\Z$ be an integer such that $n \\ge 118$. Then between $n$ and $\\dfrac {4 n} 3$ there exists at least one prime number of each of the forms: :$4 m - 1, 4 m + 1, 6 m - 1, 6 m + 1$"}
+{"_id": "14197", "title": "Sum of Cubes of 5 Consecutive Integers which is Square", "text": "The following sequences of $5$ consecutive positive integers have cubes that sum to squares: :$1, 2, 3, 4, 5$ :$96, 96, 98, 99, 100$ :$118, 119, 120, 121, 122$ No other such sequence of $5$ consecutive positive integers has the same property."}
+{"_id": "14198", "title": "Smallest Number to appear 6 Times in Pascal's Triangle", "text": "The smallest positive integer greater than $1$ to appear $6$ times in Pascal's Triangle is $120$."}
+{"_id": "14199", "title": "Smallest n such that 6 n + 1 and 6 n - 1 are both Composite", "text": "The smallest positive integer $n$ such that $6 n + 1$ and $6 n - 1$ are both composite is $20$."}
+{"_id": "14200", "title": "Smallest Number with 16 Divisors", "text": "The smallest positive integer with $16$ divisors is $120$."}
+{"_id": "14201", "title": "Fermat Set is Diophantine Quadruple", "text": "The Fermat set $F = \\left\\{{1, 3, 8, 120}\\right\\}$ is a Diophantine quadruple: :$\\forall a, b \\in F: a \\ne b: a b + 1 = n^2$ for some $n \\in \\Z$."}
+{"_id": "14202", "title": "Fermat Set cannot be Extended to Diophantine Quintuple", "text": "The Fermat set $F = \\left\\{{1, 3, 8, 120}\\right\\}$ cannot be extended to a Diophantine quintuple."}
+{"_id": "14203", "title": "Ratio of 360 to Aliquot Sum", "text": "$360$ has the property that its ratio to its aliquot sum is $4 : 9$."}
+{"_id": "14204", "title": "Are All Triperfect Numbers Even?/Progress/Minimum Size", "text": "It has been established that an odd triperfect number, if one were to exist, would be greater than $10^{70}$. If it does not have $3$ as a prime factor, then it is greater than $10^{108}$."}
+{"_id": "14205", "title": "Are All Triperfect Numbers Even?/Progress/Prime Factors", "text": "An odd triperfect number has: :at least $11$ distinct prime factors :at least $32$ distinct prime factors if $3$ is not one of them."}
+{"_id": "14206", "title": "Are All Triperfect Numbers Even?/Progress/Form", "text": "An odd triperfect number is square."}
+{"_id": "14211", "title": "Squares which are 4 Less than Cubes", "text": "The only two square numbers which are $4$ less than a cube are: :$2^2 + 4 = 2^3$ :$11^2 + 4 = 5^3$"}
+{"_id": "14212", "title": "Integer Greater than 121 is Sum of Distinct Primes of form 4 n + 1", "text": "Let $n$ be an integer greater than $121$. Then $n$ can be expressed as the sum of distinct prime numbers of the form $4 n + 1$."}
+{"_id": "14213", "title": "Palindromes in Base 10 and Base 3", "text": "The following $n \\in \\Z$ are palindromic in both decimal and ternary: :$0, 1, 2, 4, 8, 121, 151, 212, 242, 484, 656, 757, \\ldots$ {{OEIS|A007633}}"}
+{"_id": "14214", "title": "Square Number which is Sum of Consecutive Powers", "text": "The only square number which is the sum of consecutive powers of a positive integer is $121$: :$121 = 3^0 + 3^1 + 3^2 + 3^3 + 3^4$"}
+{"_id": "14215", "title": "Numbers whose Difference equals Difference between Cube and Seventh Power", "text": "The following $2$ pairs of integers are the only ones known which exhibit this pattern: :$\\left\\vert {5^3 - 2^7}\\right\\vert = 5 - 2$ :$\\left\\vert {13^3 - 3^7}\\right\\vert = 13 - 3$"}
+{"_id": "14217", "title": "Triangles with Integer Area and Integer Sides in Arithmetical Sequence", "text": "The triangles with the following sides in arithmetic sequence have integer areas: :$3, 4, 5$ :$13, 14, 15$ :$15, 28, 41$ :$15, 26, 37$ Their areas are: :$6, 84, 126, 156$"}
+{"_id": "14218", "title": "Equivalence of Definitions of Weierstrass E-Function", "text": "Let $\\mathbf y, \\mathbf z, \\mathbf w$  be $n$-dimensional vectors. Let $\\mathbf y$ be such that $\\map{\\mathbf y} a=A$ and $\\map{\\mathbf y} b=B$. Let $J$ be a functional such that: :$\\ds J \\sqbrk {\\mathbf y} = \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ {{TFAE|def = Weierstrass E-Function}}"}
+{"_id": "14219", "title": "Conditions for Strong Minimum of Functional", "text": "Let $\\mathbf y$ be an $n$-dimensional vector such that $\\map {\\mathbf y} a = A$ and $\\map {\\mathbf y} b = B$ Let $J$ be a functional such that: :$\\ds J \\sqbrk {\\mathbf y} = \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ Let $\\gamma$ be an extremal curve of $J$. Let the following be the field of the functional $J$: :$\\mathbf y' = \\map {\\boldsymbol \\psi} {x, \\mathbf y}$ Let $R$ be an open region containing $\\gamma$ and have the field $\\boldsymbol \\psi$ defined as $\\forall \\paren {x, \\mathbf y} \\in R$. Let $\\mathbf w$ be a finite vector. Suppose that: :$\\forall \\paren {x, \\mathbf y} \\in R: \\map E {x, \\mathbf y, \\boldsymbol \\psi, \\mathbf w}\\ge 0$ where $E$ is Weierstrass E-Function. Then $J$ has a strong minimum for $\\gamma$. {{explain|What is a strong minimum?}}"}
+{"_id": "14220", "title": "Factor of Mersenne Number Mp is of form 2kp + 1", "text": ":$q = 2 k p + 1$ for some integer $k$."}
+{"_id": "14221", "title": "Factor of Mersenne Number Mp equivalent to 1 mod p", "text": ":$q \\equiv 1 \\pmod p$"}
+{"_id": "14222", "title": "Factor of Mersenne Number equivalent to +-1 mod 8", "text": ":$q \\equiv \\pm 1 \\pmod 8$"}
+{"_id": "14223", "title": "Numbers with 7 or more Prime Factors", "text": "The sequence of positive integers with $7$ or more prime factors (not necessarily distinct) begins: :$128, 192, 256, 288, 320, 384, 432, 448, 480, 512, \\ldots$ {{OEIS|A046307}}"}
+{"_id": "14224", "title": "Numbers not Sum of Distinct Squares", "text": "The positive integers which are not the sum of $1$ or more distinct squares are: :$2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128$ {{OEIS|A001422}}"}
+{"_id": "14225", "title": "132 is Sum of all 2-Digit Numbers formed from its Digits", "text": "$132$ is the smallest sum of all the $2$-digit (positive) integers formed from its own digits."}
+{"_id": "14228", "title": "Harmonic Mean of Divisors in terms of Tau and Sigma", "text": "Let $n \\in \\Z_{>0}$ be a positive integer. The harmonic mean of the divisors of $n$ is given by: :$H \\left({n}\\right) = \\dfrac {n \\, \\tau \\left({n}\\right)} {\\sigma \\left({n}\\right)}$ where: :$\\tau \\left({n}\\right)$ denotes the $\\tau$ (tau) function: the number of divisors of $n$ :$\\sigma \\left({n}\\right)$ denotes the $\\sigma$ (sigma) function: the sum of the divisors of $n$."}
+{"_id": "14229", "title": "Perfect Number is Ore Number", "text": "Let $n \\in \\Z_{>0}$ be a perfect number. Then $n$ is an Ore number."}
+{"_id": "14231", "title": "Square Fibonacci Number", "text": "After $1$, there exists exactly one Fibonacci number which is also square: :$F_{12} = 144 = 12^2$ which is also coincidentally the square of its index."}
+{"_id": "14232", "title": "Carmichael's Theorem", "text": "Let $n \\in \\Z$ such that $n > 12$. Then the $n$th Fibonacci number $F_n$ has at least one prime factor which does not divide any smaller Fibonacci number. The exceptions for $n \\le 12$ are: :$F_1 = 1, F_2 = 1$: neither have any prime factors :$F_6 = 8$ whose only prime factor is $2$ which is $F_3$ :$F_{12} = 144$ whose only prime factors are $2$ (which is $F_3$) and $3$ (which is $F_4$)."}
+{"_id": "14233", "title": "Weierstrass's Necessary Condition", "text": "Let $\\mathbf y: \\R \\to \\R^n$ be an $n$-dimensional vector-valued function such that $\\map {\\mathbf y} a = A$ and $\\map {\\mathbf y} b = B$. Let $J$ be a functional such that: :$\\ds J \\sqbrk {\\mathbf y} = \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ Let $\\mathbf w$ be an $n$-dimensional vector such that $\\mathbf w \\in \\R^n$. Let $\\gamma$ be a strong minimum of $J$. {{explain|exactly what a ''strong'' minimum is, by means of a link to a definition page}} Then along $\\gamma$ and for every $\\mathbf w$: :$\\map E {x, \\mathbf y, \\mathbf y', \\mathbf w} \\ge 0$ {{explain|exactly what ''along'' $\\gamma$ means, by means of a link to a definition page}} where $E$ stands for the Weierstrass E-Function."}
+{"_id": "14234", "title": "Squares Ending in Repeated Digits", "text": "A square number $n^2$ can end in a repeated digit {{iff}} either: :$(1): \\quad n^2$ is a multiple of $100$, in which case $n$ is a multiple of $10$ :$(2): \\quad n^2$ ends in $44$ and $n$ ends in $12, 38, 62$ or $88$."}
+{"_id": "14235", "title": "Magic Constant of Smallest Prime Magic Square with Consecutive Primes", "text": "The magic constant of the smallest prime magic square whose elements are consecutive odd primes is $4 \\, 440 \\, 084 \\, 513$."}
+{"_id": "14236", "title": "Magic Constant of Smallest Prime Magic Square with Consecutive Primes from 3", "text": "The magic constant of the smallest prime magic square whose elements are consecutive odd primes from $3$ upwards is $4514$."}
+{"_id": "14238", "title": "Sum of 2 Squares in 2 Distinct Ways/Examples/145", "text": "$145$ can be expressed as the sum of two square numbers in two distinct ways: {{begin-eqn}} {{eqn | l = 145       | r = 12^2 + 1^2 }} {{eqn | r = 9^2 + 8^2 }} {{end-eqn}}"}
+{"_id": "14239", "title": "Representation of 1 as Sum of n Unit Fractions", "text": "Let $U \\left({n}\\right)$ denote the number of different ways of representing $1$ as the sum of $n$ unit fractions. Then for various $n$, $U \\left({n}\\right)$ is given by the following table: :{| border=\"1\" |- ! align=\"right\" style = \"padding: 2px 10px\" | $n$  ! align=\"right\" style = \"padding: 2px 10px\" | $U \\left({n}\\right)$  |- | align=\"right\" style = \"padding: 2px 10px\" | $1$ | align=\"right\" style = \"padding: 2px 10px\" | $1$ |- | align=\"right\" style = \"padding: 2px 10px\" | $2$ | align=\"right\" style = \"padding: 2px 10px\" | $1$ |- | align=\"right\" style = \"padding: 2px 10px\" | $3$ | align=\"right\" style = \"padding: 2px 10px\" | $3$ |- | align=\"right\" style = \"padding: 2px 10px\" | $4$ | align=\"right\" style = \"padding: 2px 10px\" | $14$ |- | align=\"right\" style = \"padding: 2px 10px\" | $5$ | align=\"right\" style = \"padding: 2px 10px\" | $147$ |- | align=\"right\" style = \"padding: 2px 10px\" | $6$ | align=\"right\" style = \"padding: 2px 10px\" | $3462$ |} {{OEIS|A002966}}"}
+{"_id": "14240", "title": "Sum of 3 Unit Fractions that equals 1", "text": "There are $3$ ways to represent $1$ as the sum of exactly $3$ unit fractions."}
+{"_id": "14241", "title": "Necessary Condition for Integral Functional to have Extremum for given function/Dependent on n Variables", "text": "Let $\\mathbf x$ be an $n$-dimensional vector. Let $\\map u {\\mathbf x}$ be a real function. Let $R$ be a fixed region. Let $J$ be a functional such that :$\\ds J \\sqbrk u = \\idotsint_R \\map F {\\mathbf x, u, u_{\\mathbf x} } \\rd x_1 \\cdots \\rd x_n$ Then a necessary condition for $J \\sqbrk u$ to have an extremum (strong or weak) for a given mapping $\\map u {\\mathbf x}$ is that $\\map u {\\mathbf x}$ satisfies Euler's equation: :$F_u - \\dfrac {\\partial} {\\partial \\mathbf x} F_{u_{\\mathbf x} } = 0$"}
+{"_id": "14242", "title": "Repeated Sum of Cubes of Digits of Multiple of 3", "text": "Let $k \\in \\Z_{>0}$ be a positive integer. Let $f: \\Z_{>0} \\to \\Z_{>0}$ be the mapping defined as: :$\\forall m \\in \\Z_{>0}: \\map f m = $ the sum of the cubes of the digits of $n$. Let $n_0 \\in \\Z_{>0}$ be a (strictly) positive integer which is a multiple of $3$. Consider the sequence: :$s_n = \\begin{cases} n_0 & : n = 0 \\\\ \\map f {s_{n - 1} } & : n > 0 \\end{cases}$ Then: :$\\exists r \\in \\N_{>0}: s_r = 153$ That is, by performing $f$ repeatedly on a multiple of $3$ eventually results in the pluperfect digital invariant $153$."}
+{"_id": "14243", "title": "Numbers for which Euler Phi Function of 2n + 1 is less than that of 2n", "text": "The sequence of positive integers for which: :$\\map \\phi {2 n + 1} < \\map \\phi {2 n}$ begins: :$157, 262, 367, 412, \\ldots$ {{OEIS|A001837}}"}
+{"_id": "14245", "title": "Odd Numbers Not Expressible as Sum of 5 Distinct Non-Zero Coprime Squares", "text": "The largest odd positive integer that cannot be expressed as the sum of exactly $5$ non-zero square numbers all of which are coprime is $245$. {{expand|The full list of those numbers is to be investigated.}}"}
+{"_id": "14246", "title": "159 is not Expressible as Sum of Fewer than 19 Fourth Powers", "text": ":$159 = 14 \\times 1^4 + 4 \\times 2^4 + 3^4$"}
+{"_id": "14247", "title": "Numbers not Expressible as Sum of Distinct Pentagonal Numbers", "text": "The positive integer which cannot be expressed as the sum of distinct pentagonal numbers are: :$2, 3, 4, 7, 8, 9, 10, 11, 14, 15, 16, 19, 20, 21, 24, 25, 26, 29, 30,$ :$31, 32, 33, 37, 38, 42, 43, 44, 45, 46, 49, 50, 54, 55, 59, 60, 61, 65,$ :$66, 67, 72, 77, 80, 81, 84, 89, 94, 95, 96, 100, 101, 102, 107, 112, 116,$ :$124, 136, 137, 141, 142, 147, 159$"}
+{"_id": "14248", "title": "Sum of Distinct Primes of form 6n-1", "text": "$161$ is the largest integer that cannot be expressed as the sum of distinct primes of the form $6 n - 1$."}
+{"_id": "14249", "title": "Equivalence of Definitions of Polynomial Ring in One Variable", "text": "Let $R$ be a commutative ring with unity. The following definitions of polynomial ring are equivalent in the following sense: : For every two constructions, there exists an $R$-isomorphism which sends indeterminates to indeterminates. {{explain|this statement has to be made more precise}} === Definition 1: As a Ring of Sequences === {{:Definition:Polynomial Ring/Sequences}} === Definition 2: As a Monoid Ring on the Natural Numbers === {{:Definition:Polynomial Ring/Monoid Ring on Natural Numbers}}"}
+{"_id": "14250", "title": "Smallest Triplet of Integers whose Product with Tau Value are Equal", "text": "Let $\\map \\tau n$ denote the $\\tau$ function: the number of divisors of $n$. The smallest set of $3$ integers $T$ such that $m \\, \\map \\tau m$ is equal for each $m \\in T$ is: :$\\set {168, 192, 224}$"}
+{"_id": "14251", "title": "General Variation of Integral Functional/Dependent on n Variables", "text": "Let $\\mathbf x$ be an $n$-dimensional vector. Let $u = \\map u {\\mathbf x}$ be a real-valued function. Let $J$ be a functional such that: :$\\ds J \\sqbrk u = \\int_R \\map F {\\mathbf x, u, \\dfrac {\\partial u} {\\partial \\mathbf x} } \\rd x_1 \\dotsm \\rd x_n$ Let $\\mathbf x^*, u^*$ be defined by the following transformations $\\paren \\star$: {{begin-eqn}} {{eqn | l = \\mathbf x^*       | r = \\map {\\boldsymbol \\Phi} {\\mathbf x, u, \\dfrac {\\partial u} {\\partial \\mathbf x}; \\epsilon} = \\mathbf x + \\epsilon \\map {\\boldsymbol \\phi} {\\mathbf x, u, \\dfrac {\\partial u} {\\partial \\mathbf x} } + \\map \\OO {\\epsilon^2} }} {{eqn | l = u^*       | r = \\map \\Psi {\\mathbf x, u, \\dfrac {\\partial u} {\\partial \\mathbf x}; \\epsilon} = \\mathbf x + \\epsilon \\map \\psi {\\mathbf x, u, \\dfrac {\\partial u} {\\partial \\mathbf x } } + \\map \\OO {\\epsilon^2} }} {{end-eqn}} where: :$\\boldsymbol \\Phi$, $\\Psi$ are differentiable {{WRT|Differentiation}} $\\epsilon$ and: {{begin-eqn}} {{eqn | l = \\map {\\boldsymbol \\Phi} {\\mathbf x, u, \\dfrac {\\partial u} {\\partial \\mathbf x}; 0}       | r = \\mathbf x }} {{eqn | l = \\map \\Psi {\\mathbf x, u, \\dfrac {\\partial u} {\\partial \\mathbf x}; 0}       | r = u }} {{eqn | l = \\map {\\boldsymbol \\phi} {\\mathbf x, u, \\dfrac {\\partial u} {\\partial \\mathbf x} }       | r = \\valueat {\\dfrac {\\partial \\boldsymbol \\Phi} {\\partial \\epsilon} } {\\epsilon \\mathop = 0} }} {{eqn | l = \\map \\psi {\\mathbf x, u, \\dfrac {\\partial u} {\\partial \\mathbf x} }       | r = \\valueat {\\dfrac {\\partial \\Psi} {\\partial \\epsilon} } {\\epsilon \\mathop = 0} }} {{end-eqn}} Then the variation of the functional $J$ due to the original mapping being transformed by $\\paren \\star$ reads: {{explain|not clear what \"aforementioned transformation\" refers to. Best to identify it with a label and refer to that label.}} :$\\ds \\delta J = \\epsilon \\int_R \\paren {F_u - \\dfrac {\\partial F_{u_{\\mathbf x} } } {\\partial \\mathbf x} } \\overline \\psi \\rd x_1 \\dotsm \\rd x_n + \\epsilon \\int_R \\map {\\dfrac {\\partial} {\\partial \\mathbf x} } {F_{u_x} \\overline {\\boldsymbol \\psi} + F \\boldsymbol \\phi} \\rd x_1 \\dotsm \\rd x_n$ where: :$\\overline \\psi = \\psi - u_{\\mathbf x} \\boldsymbol \\phi$"}
+{"_id": "14253", "title": "Sequence of Record Peaks in Values of Sigma", "text": "Let $d: \\Z_{>0} \\to \\Z_{>0}$ be the mapping defined as: :$d \\left({n}\\right) = \\sigma \\left({n}\\right) - \\sigma \\left({n'}\\right)$ where: :$n$ denotes a highly abundant number :$n'$ denotes the previous highly abundant number. The following $n \\in \\Z_{>0}$ have the property that they have a higher value of $d \\left({n}\\right)$ than any smaller $n$: :$\\forall m \\in \\Z_{>0}: m < n \\implies \\sigma \\left({m}\\right) < \\sigma \\left({n}\\right)$ That is, they are the peak $\\sigma$ values which exceed the previous peak by a higher number than any previous peak. That is, they are highly abundant number which have $\\sigma$ values  whose difference with that of the $\\sigma$ (sigma) value of the previous highly abundant numbers is greater than that with the previous record difference. :{| border=\"1\" |- ! align=\"right\" style = \"padding: 2px 10px\" | $n$  ! align=\"right\" style = \"padding: 2px 10px\" | $n'$  ! align=\"right\" style = \"padding: 2px 10px\" | $\\sigma \\left({n}\\right)$  ! align=\"right\" style = \"padding: 2px 10px\" | $\\sigma \\left({n'}\\right)$  ! style = \"padding: 2px 10px\" | $d \\left({n}\\right)$  |- | align=\"right\" style = \"padding: 2px 10px\" | $2$ | align=\"right\" style = \"padding: 2px 10px\" | $1$ | align=\"right\" style = \"padding: 2px 10px\" | $3$ | align=\"right\" style = \"padding: 2px 10px\" | $1$ | align=\"right\" style = \"padding: 2px 10px\" | $2$ |- | align=\"right\" style = \"padding: 2px 10px\" | $4$ | align=\"right\" style = \"padding: 2px 10px\" | $3$ | align=\"right\" style = \"padding: 2px 10px\" | $7$ | align=\"right\" style = \"padding: 2px 10px\" | $4$ | align=\"right\" style = \"padding: 2px 10px\" | $3$ |- | align=\"right\" style = \"padding: 2px 10px\" | $6$ | align=\"right\" style = \"padding: 2px 10px\" | $4$ | align=\"right\" style = \"padding: 2px 10px\" | $12$ | align=\"right\" style = \"padding: 2px 10px\" | $7$ | align=\"right\" style = \"padding: 2px 10px\" | $5$ |- | align=\"right\" style = \"padding: 2px 10px\" | $12$ | align=\"right\" style = \"padding: 2px 10px\" | $10$ | align=\"right\" style = \"padding: 2px 10px\" | $28$ | align=\"right\" style = \"padding: 2px 10px\" | $18$ | align=\"right\" style = \"padding: 2px 10px\" | $10$ |- | align=\"right\" style = \"padding: 2px 10px\" | $24$ | align=\"right\" style = \"padding: 2px 10px\" | $20$ | align=\"right\" style = \"padding: 2px 10px\" | $60$ | align=\"right\" style = \"padding: 2px 10px\" | $42$ | align=\"right\" style = \"padding: 2px 10px\" | $18$ |- | align=\"right\" style = \"padding: 2px 10px\" | $36$ | align=\"right\" style = \"padding: 2px 10px\" | $30$ | align=\"right\" style = \"padding: 2px 10px\" | $91$ | align=\"right\" style = \"padding: 2px 10px\" | $72$ | align=\"right\" style = \"padding: 2px 10px\" | $19$ |- | align=\"right\" style = \"padding: 2px 10px\" | $48$ | align=\"right\" style = \"padding: 2px 10px\" | $42$ | align=\"right\" style = \"padding: 2px 10px\" | $124$ | align=\"right\" style = \"padding: 2px 10px\" | $96$ | align=\"right\" style = \"padding: 2px 10px\" | $28$ |- | align=\"right\" style = \"padding: 2px 10px\" | $60$ | align=\"right\" style = \"padding: 2px 10px\" | $48$ | align=\"right\" style = \"padding: 2px 10px\" | $168$ | align=\"right\" style = \"padding: 2px 10px\" | $124$ | align=\"right\" style = \"padding: 2px 10px\" | $44$ |- | align=\"right\" style = \"padding: 2px 10px\" | $120$ | align=\"right\" style = \"padding: 2px 10px\" | $108$ | align=\"right\" style = \"padding: 2px 10px\" | $360$ | align=\"right\" style = \"padding: 2px 10px\" | $280$ | align=\"right\" style = \"padding: 2px 10px\" | $80$ |}"}
+{"_id": "14254", "title": "Order Isomorphism forms Galois Connection", "text": "Let $L_1 = \\left({S_1, \\preceq_1}\\right)$, $L_2 = \\left({S_2, \\preceq_2}\\right)$ be ordered sets. Let $f:S_1 \\to S_2$ be an order isomorphism between $L_1$ and $L_2$. Then $\\left({f, f^{-1} }\\right)$ is a Galois connection."}
+{"_id": "14255", "title": "Numbers of Primes with at most n Digits", "text": "Let $p: \\Z_{>0} \\to \\Z_{>0}$ be the mapping defined as: :$\\forall n \\in \\Z_{>0}: p \\left({n}\\right) = $ the number of prime numbers with no more than $n$ digits Then the value of $p$ for the first few numbers is given below: :{| border=\"1\" |- ! align=\"right\" style = \"padding: 2px 10px\" | $n$  ! align=\"right\" style = \"padding: 2px 10px\" | $p \\left({n}\\right)$  |- | align=\"right\" style = \"padding: 2px 10px\" | $1$ | align=\"right\" style = \"padding: 2px 10px\" | $4$ |- | align=\"right\" style = \"padding: 2px 10px\" | $2$ | align=\"right\" style = \"padding: 2px 10px\" | $25$ |- | align=\"right\" style = \"padding: 2px 10px\" | $3$ | align=\"right\" style = \"padding: 2px 10px\" | $168$ |- | align=\"right\" style = \"padding: 2px 10px\" | $4$ | align=\"right\" style = \"padding: 2px 10px\" | $1229$ |- | align=\"right\" style = \"padding: 2px 10px\" | $5$ | align=\"right\" style = \"padding: 2px 10px\" | $9592$ |- | align=\"right\" style = \"padding: 2px 10px\" | $6$ | align=\"right\" style = \"padding: 2px 10px\" | $78 \\, 498$ |- | align=\"right\" style = \"padding: 2px 10px\" | $7$ | align=\"right\" style = \"padding: 2px 10px\" | $664 \\, 579$ |- | align=\"right\" style = \"padding: 2px 10px\" | $8$ | align=\"right\" style = \"padding: 2px 10px\" | $5 \\, 761 \\, 455$ |- | align=\"right\" style = \"padding: 2px 10px\" | $9$ | align=\"right\" style = \"padding: 2px 10px\" | $50 \\, 847 \\, 534$ |- | align=\"right\" style = \"padding: 2px 10px\" | $10$ | align=\"right\" style = \"padding: 2px 10px\" | $455 \\, 052 \\, 511$ |} {{OEIS|A006880}}"}
+{"_id": "14256", "title": "Sequence of Square Centered Hexagonal Numbers", "text": "The sequence of centered hexagonal numbers which are also square begins: :$1, 169, 32 \\, 761, 6 \\, 355 \\, 441, \\ldots$ {{OEIS|A006051}}"}
+{"_id": "14257", "title": "169 as Sum of up to 155 Squares", "text": "$169$ can be expressed as the sum of $n$ non-zero squares for all $n$ from $1$ to $155$."}
+{"_id": "14258", "title": "Lattice of Power Set is Arithmetic", "text": "Let $X$ be a set. Let $P = \\left({\\mathcal P\\left({X}\\right), \\cup, \\cap, \\subseteq}\\right)$ be a lattice of power set. Then $P$ is arithmetic."}
+{"_id": "14259", "title": "Nth Derivative of Natural Logarithm", "text": "The $n$th derivative of $\\map \\ln x$ for $n \\ge 1$ is: :$\\dfrac {\\d^n} {\\d x^n} \\ln x = \\dfrac {\\paren {n - 1}! \\paren {-1}^{n - 1} } {x^n}$"}
+{"_id": "14260", "title": "Sum of Factorials of Digits of 169", "text": "Let the factorials of the digits of $169$ be added. Let the same process be done on the result. Repeat. After $3$ iterations, the result will be $169$."}
+{"_id": "14261", "title": "Squares which are Difference between Two Cubes", "text": "$169$ is the smallest square number which is the difference between two cubes: :$169 = 8^3 - 7^3$ {{expand|Add the rest of the sequence, having found out what they are.}}"}
+{"_id": "14262", "title": "Cube of 180 is Sum of Sequence of Consecutive Cubes", "text": ":$180^3 = \\displaystyle \\sum_{k \\mathop = 6}^{69} k^3$ That is: :$180^3 = 6^3 + 7^3 + \\cdots + 67^3 + 68^3 + 69^3$"}
+{"_id": "14264", "title": "Numbers Expressible as Sum of Five Distinct Squares", "text": "The largest positive integer which cannot be expressed as the sum of no more than $5$ distinct squares is $188$. Both $188$ and $124$ require as many as $6$ distinct squares to represent them: {{begin-eqn}} {{eqn | l = 124       | r = 1 + 4 + 9 + 25 + 36 + 49       | c =  }} {{eqn | r = 1^2 + 2^2 + 3^2 + 5^2 + 6^2 + 7^2       | c =  }} {{eqn | l = 188       | r = 1 + 4 + 9 + 25 + 49 + 100       | c =  }} {{eqn | r = 1^2 + 2^2 + 3^2 + 5^2 + 7^2 + 10^2       | c =  }} {{end-eqn}}"}
+{"_id": "14266", "title": "Closed Form for Heptagonal Pyramidal Numbers", "text": "The closed-form expression for the $n$th heptagonal pyramidal number is: :$Q_n = \\dfrac {n \\paren {n + 1} \\paren {5 n - 2} } 6$"}
+{"_id": "14267", "title": "Heptagonal Pyramidal Numbers which are Square", "text": "The sequence of heptagonal pyramidal numbers which also have the property of being square begins: :$0, 1, 196, \\ldots$ {{expand|Further research needed}}"}
+{"_id": "14268", "title": "Conditions for Limit Function to be Limit Minimizing Function of Functional", "text": "Let $y$ be a real function. Let $J \\sqbrk y$ be a functional. Let $\\sequence {y_n}$ be a minimizing sequence of $J$. Let: :$\\ds \\lim_{n \\mathop \\to \\infty} y_n = \\hat y$ Suppose $J$ is lower semicontinuous at $y = \\hat y$. Then: :$\\ds J \\sqbrk {\\hat y} = \\lim_{n \\mathop \\to \\infty} J \\sqbrk {y_n}$"}
+{"_id": "14269", "title": "Smallest 10 Primes in Arithmetic Sequence", "text": "The smallest $10$ primes in arithmetic sequence are: :$199 + 210 n$ for $n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9$. These are also the smallest $8$ and $9$ primes in arithmetic sequence."}
+{"_id": "14271", "title": "Sum of Cubes of 3 Consecutive Integers which is Square", "text": "The following sequence of $3$ consecutive positive integers have cubes that sum to a square: :$23, 24, 25$"}
+{"_id": "14272", "title": "Integer Greater than 205 is Sum of Distinct Primes of form 6 n + 1", "text": "Let $n$ be an integer greater than $205$. Then $n$ can be expressed as the sum of distinct prime numbers of the form $6 n + 1$."}
+{"_id": "14274", "title": "Numbers such that Tau divides Phi divides Sigma", "text": "The sequence of integers $n$ with the property that: :$\\map \\tau n \\divides \\map \\phi n \\divides \\map \\sigma n$ where: :$\\divides$ denotes divisibility :$\\tau$ denotes the $\\tau$ (tau) function: the count of divisors of $n$ :$\\phi$ denotes the Euler $\\phi$ (phi) function: the count of smaller integers coprime to $n$ :$\\sigma$ denotes the $\\sigma$ (sigma) function: the sum of divisors of $n$ begins: :$1, 3, 15, 30, 35, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, \\ldots$ {{OEIS|A020493}}"}
+{"_id": "14275", "title": "Number of Representations as Sum of Two Primes", "text": "The number of ways an integer $n$ can be represented as the sum of two primes is no greater than the number of primes in the interval $\\closedint {\\dfrac n 2} {n - 2}$."}
+{"_id": "14276", "title": "Integers whose Number of Representations as Sum of Two Primes is Maximum", "text": "$210$ is the largest integer which can be represented as the sum of two primes in the maximum number of ways. The full list of such numbers is as follows: :$1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 24, 30, 36, 42, 48, 60, 90, 210$ {{OEIS|A141340}} The list contains: :$n \\le 8$ :$n \\le 18$ where $2 \\divides n$ :$n \\le 48$ where $2 \\times 3 \\divides n$ :$n \\le 90$ where $2 \\times 3 \\times 5 \\divides n$ :$210 = 2 \\times 3 \\times 5 \\times 7$ {{WIP|Add this sequence to the number pages}}"}
+{"_id": "14277", "title": "25 as Sum of 4 to 11 Squares", "text": "$25$ can be expressed as the sum of $n$ non-zero squares for all $n$ from $4$ to $11$."}
+{"_id": "14278", "title": "Cubes which are Sum of Three Cubes", "text": "The following cube numbers can be expressed as the sum of $3$ positive cube numbers: :$6^3, 9^3, 12^3, 18^3, 19^3, 20^3, 24^3, 25^3, \\ldots$ {{OEIS|A066890}} with cube roots: :$6, 9, 12, 18, 19, 20, 24, 25, \\ldots$ {{OEIS|A023042}} {{WIP|Add these sequences to the separate pages for the numbers themselves}}"}
+{"_id": "14279", "title": "Ritz Method implies Not Worse Approximation with Increased Number of Functions", "text": "Consider the Ritz method. Let $\\eta_n = \\boldsymbol \\alpha \\boldsymbol \\phi$. Let $J \\sqbrk {\\eta_n} = \\mu_n$. Then: :$\\mu_n \\ge \\mu_{n + 1}$"}
+{"_id": "14280", "title": "Dissection of Cube into 3 Cubes using 8 Pieces", "text": "A cube can be dissected into $3$ smaller cubes by cutting it into $8$ pieces and reassembling them."}
+{"_id": "14282", "title": "Smallest Multiplicative Magic Square is of Order 3", "text": "The order of the smallest multiplicative magic square is $3$, for example: {{:Multiplicative Magic Square/Examples/Order 3/Smallest}}"}
+{"_id": "14286", "title": "230 Fedorov Groups where Chiral Pairs are Distinct", "text": "There are $219$ Fedorov groups, if chiral copies are considered distinct."}
+{"_id": "14287", "title": "Equivalence of Definitions of Amicable Pair", "text": "Let $m \\in \\Z_{>0}$ and $n \\in \\Z_{>0}$ be (strictly) positive integers. {{TFAE|def = Amicable Pair}}"}
+{"_id": "14288", "title": "Thabit's Rule", "text": "Let $n$ be a positive integer such that: {{begin-eqn}} {{eqn | l = a       | r = 3 \\times 2^n - 1       | c =  }} {{eqn | l = b       | r = 3 \\times 2^{n - 1} - 1       | c =  }} {{eqn | l = c       | r = 9 \\times 2^{2 n - 1} - 1       | c =  }} {{end-eqn}} are all prime. Then: :$\\tuple {2^n a b, 2^n c}$ forms an amicable pair."}
+{"_id": "14289", "title": "Smaller of Thabit Pair is Tetrahedral", "text": "Let $\\tuple {m_1, m_2}$ be a Thabit pair such that $m_1 < m_2$. Then $m_1$ is a tetrahedral number."}
+{"_id": "14292", "title": "Amicable Pair with Smallest Common Prime Factor 5", "text": "A amicable pair whose smallest common prime factor is greater than $3$ has the elements: :$m_1 = 5 \\times 7^2 \\times 11^2 \\times 13 \\times 17 \\times 19^3 \\times 23 \\times 37 \\times 181 \\times 101 \\times 8693 \\times 19 \\, 479 \\, 382 \\, 229$ and: :$m_2 = 5 \\times 7^2 \\times 11^2 \\times 13 \\times 17 \\times 19^3 \\times 23 \\times 37 \\times 181 \\times 365 \\, 147 \\times 47 \\, 307 \\, 071 \\, 129$ This was at one point the smallest known counterexample to the observation that: :most amicable pairs consist of even integers :most of the rest, whose elements are odd, have both elements divisible by $3$."}
+{"_id": "14293", "title": "Equivalence of Definitions of Amicable Triplet", "text": "Let $m_1, m_2, m_3 \\in \\Z_{>0}$ be (strictly) positive integers. {{TFAE|def = Amicable Triplet}}"}
+{"_id": "14294", "title": "Largest Number not Expressible as Sum of Less than 37 Positive Fifth Powers", "text": "The largest positive integer which cannot be expressed as the sum of less than $37$ positive fifth powers is $223$: :$223 = 31 \\times 1^5 + 6 \\times 2^5$"}
+{"_id": "14295", "title": "Smallest Triple of Consecutive Sums of Squares", "text": "The smallest triple of consecutive positive integers each of which is the sum of two squares is: :$\\tuple {232, 233, 234}$"}
+{"_id": "14296", "title": "No Quadruple of Consecutive Sums of Squares Exists", "text": "It is not possible for a quadruple of consecutive positive integers each of which is the sum of two squares."}
+{"_id": "14297", "title": "Directed iff Filtered in Dual Ordered Set", "text": "Let $\\struct {S, \\preceq_1}$ be an ordered set. Let $\\struct {S, \\preceq_2}$ be a dual ordered set of $\\struct {S, \\preceq_1}$ Let $X \\subseteq S$. Then: :$X$ is directed in $\\struct {S, \\preceq_1}$ {{iff}}: :$X$ is filtered in $\\struct {S, \\preceq_2}$."}
+{"_id": "14298", "title": "Sum of Two Squares not Congruent to 3 modulo 4", "text": "Let $n \\in \\Z$ such that $n = a^2 + b^2$ where $a, b \\in \\Z$. Then $n$ is not congruent modulo $4$ to $3$."}
+{"_id": "14299", "title": "Numbers not Expressible as Sum of Less than 9 Positive Cubes", "text": "The following are the only positive integers cannot be expressed as the sum of less than $9$ positive cubes: {{begin-eqn}} {{eqn | l = 23       | r = 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3       | c =  }} {{eqn | l = 239       | r = 4^3 + 4^3 + 3^3 + 3^3 + 3^3 + 3^3 + 1^3 + 1^3 + 1^3       | c =  }} {{end-eqn}}"}
+{"_id": "14300", "title": "Largest Prime Factor of n squared plus 1", "text": "Let $n \\in \\Z$ be greater than $239$. Then the largest prime factor of $n^2 + 1$ is at least $17$."}
+{"_id": "14301", "title": "Solution of Ljunggren Equation", "text": "The only solutions of the Ljunggren equation: :$x^2 + 1 = 2 y^4$ are: :$x = 1,  y = 1$ :$x = 239,  y = 13$ {{OEIS|A229384}}"}
+{"_id": "14302", "title": "Solutions of Diophantine Equation x^4 + y^4 = z^2 + 1 for x = 239", "text": "Consider the indeterminate Diophantine equation: :$x^4 + y^4 = z^2 + 1$ When $x = 239$ and $x > y$, there are $3$ solutions: {{begin-eqn}} {{eqn | l = 239^4 + 104^4       | r = 58 \\, 136^2 + 1       | c =  }} {{eqn | l = 239^4 + 143^4       | r = 60 \\, 671^2 + 1       | c =  }} {{eqn | l = 239^4 + 208^4       | r = 71 \\, 656^2 + 1       | c =  }} {{end-eqn}}"}
+{"_id": "14303", "title": "Number with over 240 Divisors is greater than 1,000,000", "text": "Let $n \\in \\Z$ be an integer with over $240$ divisors. Then: :$n > 1 \\, 000 \\, 000$"}
+{"_id": "14305", "title": "Sequence of 4 Consecutive Integers with Equal Number of Divisors", "text": "The following sequence of integers are sets of $4$ consecutive integers which all have the same number of divisors: :$\\map \\tau m = \\map \\tau {m + 1} = \\map \\tau {m + 2} = \\map \\tau {m + 3}$ where $\\map \\tau n$ denotes the $\\tau$ function. :$242, 243, 244, 245, 3655, 3656, 3657, 3658, 4503, 4504, 4505, 4506, \\ldots$ {{OEIS|A039665}}"}
+{"_id": "14306", "title": "Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes/Sequence", "text": "The  sequence of positive integers which are both the sum of $2$ square numbers in two distinct ways and also the sum of $2$ cube numbers begins: :$65, 250, \\ldots$"}
+{"_id": "14307", "title": "Sum of 3 Cubes in 2 Distinct Ways", "text": "The sequence of positive integers which can be expressed as the sum of $3$ cubes numbers in two or more different ways begins: {{begin-eqn}} {{eqn | l = 251       | r = 5^3 + 5^3 + 1^3       | rr= = 6^3 + 3^3 + 2^3 }} {{end-eqn}} {{OEIS|???}}"}
+{"_id": "14308", "title": "78,557 is Sierpiński", "text": "$78 \\, 557$ is a Sierpiński number of the second kind."}
+{"_id": "14309", "title": "Existence of Infinite Number of Numbers that are Riesel, Carmichael and Sierpiński", "text": "There exists an infinite number of integers which are simultaneously Sierpiński, Riesel and Carmichael."}
+{"_id": "14310", "title": "Equivalence of Definitions of Balanced Prime", "text": "The following definitions of a balanced prime are equivalent:"}
+{"_id": "14311", "title": "Divisor of Fermat Number", "text": "Let $F_n$ be a Fermat number. Let $m$ be divisor of $F_n$."}
+{"_id": "14312", "title": "Prime Decomposition of 6th Fermat Number", "text": "The prime decomposition of the $6$th Fermat number is given by: {{begin-eqn}} {{eqn | l = 2^{\\paren {2^6} } + 1       | r = 18 \\, 446 \\, 744 \\, 073 \\, 709 \\, 551 \\, 617       | c = Sequence of Fermat Numbers }} {{eqn | r = 274 \\, 177 \\times 67 \\, 280 \\, 421 \\, 310 \\, 721       | c =  }} {{eqn | r = \\paren {3^2 \\times 7 \\times 17 \\times 2^8 + 1} \\times \\paren {5 \\times 47 \\times 373 \\times 2 \\, 998 \\, 279 \\times 2^8 + 1}       | c =  }} {{end-eqn}}"}
+{"_id": "14313", "title": "Prime Decomposition of 7th Fermat Number", "text": "The prime decomposition of the $7$th Fermat number is given by: {{begin-eqn}} {{eqn | l = 2^{\\paren {2^7} } + 1       | r = 340 \\, 282 \\, 366 \\, 920 \\, 938 \\, 463 \\, 463 \\, 374 \\, 607 \\, 431 \\, 768 \\, 211 \\, 457       | c = Sequence of Fermat Numbers }} {{eqn | r = 59 \\, 649 \\, 589 \\, 127 \\, 497 \\, 217 \\times 5 \\, 704 \\, 689 \\, 200 \\, 685 \\, 129 \\, 054 \\, 721       | c =  }} {{eqn | r = \\paren {116 \\, 503 \\, 103 \\, 764 \\, 643 \\times 2^9 + 1} \\paren {3^5 \\times 5 \\times 12497 \\times 733803 839347 \\times 2^9 + 1}       | c =  }} {{end-eqn}}"}
+{"_id": "14314", "title": "Filtered iff Directed in Dual Ordered Set", "text": "Let $\\left({S, \\preceq_1}\\right)$ be an ordered set. Let $\\left({S, \\preceq_2}\\right)$ be a dual ordered set of $\\left({S, \\preceq_1}\\right)$ Let $X \\subseteq S$. Then $X$ is filtered in $\\left({S, \\preceq_1}\\right)$ {{iff}} $X$ is directed in $\\left({S, \\preceq_2}\\right)$"}
+{"_id": "14315", "title": "Filters equal Ideals in Dual Ordered Set", "text": "Let $L_1 = \\left({S, \\preceq_1}\\right)$ be an ordered set. Let $L_2 = \\left({S, \\preceq_2}\\right)$ be a dual ordered set of $L_1$. Then $\\mathit{Filt}\\left({L_1}\\right) = \\mathit{Ids}\\left({L_2}\\right)$ where :$\\mathit{Filt}\\left({L_1}\\right)$ denotes the set of all filters of $L_1$, :$\\mathit{Ids}\\left({L_2}\\right)$ denotes the set of all ideals of $L_2$."}
+{"_id": "14316", "title": "Pépin's Test", "text": "Let $F_n = 2^{2^n} + 1$ be a Fermat number. Then $F_n$ is prime {{iff}}: :$3^{\\paren {F_n - 1} / 2} \\equiv -1 \\pmod {F_n}$"}
+{"_id": "14317", "title": "Triangular Fermat Number", "text": "The only one Fermat number which is triangular is $3$."}
+{"_id": "14318", "title": "Fermat Number is not Square", "text": "There exist no Fermat numbers which are square."}
+{"_id": "14319", "title": "Fermat Number is not Cube", "text": "There exist no Fermat numbers which are cubes."}
+{"_id": "14321", "title": "Ideals equal Filters in Dual Ordered Set", "text": "Let $L_1 = \\left({S, \\preceq_1}\\right)$ be an ordered set. Let $L_2 = \\left({S, \\preceq_2}\\right)$ be a dual ordered set of $L_1$. Then $\\mathit{Ids}\\left({L_1}\\right) = \\mathit{Filt}\\left({L_2}\\right)$ where :$\\mathit{Filt}\\left({L_2}\\right)$ denotes the set of all filters of $L_2$, :$\\mathit{Ids}\\left({L_1}\\right)$ denotes the set of all ideals of $L_1$."}
+{"_id": "14322", "title": "Power Set is Filter in Lattice of Power Set", "text": "Let $X$ be a set. Let $L = \\left({\\mathcal P\\left({X}\\right), \\cup, \\cap, \\subseteq}\\right)$ be a inclusion lattice of power set of $X$. Then $\\mathcal P\\left({X}\\right)$ is a filter on $L$."}
+{"_id": "14323", "title": "Triplet in Arithmetic Sequence with equal Sigma", "text": "The smallest triple of integers in arithmetic sequence which have the same $\\sigma$ (sigma) value is: :$\\map \\sigma {267} = \\map \\sigma {295} = \\map \\sigma {323} = 360$"}
+{"_id": "14324", "title": "Singleton of Set is Filter in Lattice of Power Set", "text": "Let $X$ be a set. Let $L = \\left({\\mathcal P\\left({X}\\right), \\cup, \\cap, \\subseteq}\\right)$ be an inclusion lattice of power set of $X$. Then $\\left\\{ {X}\\right\\}$ is a filter on $L$."}
+{"_id": "14325", "title": "Existence of Arbitrarily Long Aliquot Sequences", "text": "It is possible to construct an aliquot sequence of arbitrary length which is monotonically increasing."}
+{"_id": "14326", "title": "Consecutive Triple of Repeated Digit-Products", "text": "The triplet of integers $281, 282, 283$ have the property that if their digits are multiplied, and the process repeated on the result until only $1$ digit remains, that final digit is the same for all three, that is, $6$. There does not exist an set of four consecutive integers which also all end up at the same single digit."}
+{"_id": "14327", "title": "Consecutive Powerful Numbers", "text": "The following pairs are of consecutive positive integers both of which are powerful: :$\\left({8, 9}\\right), \\left({288, 289}\\right), \\left({675, 676}\\right), \\left({9800, 9801}\\right), \\left({332 \\, 928, 332 \\, 929}\\right), \\ldots$ {{OEIS|A060355}}"}
+{"_id": "14328", "title": "Smallest Multiple of 9 with all Digits Even", "text": "$288$ is the smallest integer multiple of $9$ all of whose digits are even."}
+{"_id": "14332", "title": "Kaprekar's Symmetry", "text": "Let $n$ be a Kaprekar number with $D$ digits. Then $10^D - n$ is also a Kaprekar number."}
+{"_id": "14333", "title": "Cyclic Permutation of Kaprekar Number", "text": "Let $n$ be a Kaprekar number of $k$ digits. Let $m$ be an integer formed from a cyclic permutation of the digits of $n$. Let $m$ be squared and the result split into $2$ parts, where the $2$nd part is of $k$ digits. Let these two parts be added, in the way of operating on a Kaprekar number. If the result is more than $k$ digits long, split that into $2$ parts, where the $2$nd part is of $k$ digits, and add the parts. The result will be another cyclic permutation of the digits of $n$."}
+{"_id": "14335", "title": "Filters of Lattice of Power Set form Bounded Above Ordered Set", "text": "Let $X$ be a set. Let $L = \\left({\\mathcal P\\left({X}\\right), \\cup, \\cap, \\subseteq}\\right)$ be an inclusion lattice of power set of $X$. Let $F = \\left({\\mathit{Filt}\\left({L}\\right), \\subseteq}\\right)$ be an inclusion ordered set, where $\\mathit{Filt}\\left({L}\\right)$ denotes the set of all filters on $L$. Then $F$ is bounded above and $\\top_F = \\mathcal P\\left({X}\\right)$ where $\\top_F$ denotes the greatest element of $F$."}
+{"_id": "14337", "title": "Palindromic Primes in Base 10 and Base 2", "text": "The following $n \\in \\Z$ are prime numbers which are palindromic in both decimal and binary: :$3, 5, 7, 313, 7 \\, 284 \\, 717 \\, 174 \\, 827, 390 \\, 714 \\, 505 \\, 091 \\, 666 \\, 190 \\, 505 \\, 417 \\, 093, \\ldots$ {{OEIS|A046472}} It is not known whether there are any more."}
+{"_id": "14338", "title": "Ideals form Complete Lattice", "text": "Let $L = \\struct {S, \\vee, \\preceq}$ be a bounded below join semilattice. Let $\\mathcal I = \\struct {\\mathit{Ids}\\left({L}\\right), \\subseteq}$ be an inclusion ordered set, where $\\mathit{Ids}\\left({L}\\right)$ denotes the set of all ideals in $L$. Then $\\mathcal I$ is complete lattice."}
+{"_id": "14339", "title": "Filters form Complete Lattice", "text": "Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be a bounded above meet semilattice. Let $F = \\left({\\mathit{Filt}\\left({L}\\right), \\subseteq}\\right)$ be an inclusion ordered set, where $\\mathit{Filt}\\left({L}\\right)$ denotes set of all filters on $L$. Then $F$ is a complete lattice."}
+{"_id": "14340", "title": "319 is not Expressible as Sum of Fewer than 19 Fourth Powers", "text": ":$319 = 15 \\times 1^4 + 3 \\times 2^4 + 4^4$ or: :$319 = 12 \\times 1^4 + 4 \\times 2^4 + 3 \\times 3^4$"}
+{"_id": "14341", "title": "Positive Integers not Expressible as Sum of Fewer than 19 Fourth Powers", "text": "The following positive integers are the only ones which cannot be expressed as the sum of fewer than $19$ fourth powers: === $79$ as Sum of $19$ Fourth Powers === {{:Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers}} === $159$ as Sum of $19$ Fourth Powers === {{:159 is not Expressible as Sum of Fewer than 19 Fourth Powers}} === $319$ as Sum of $19$ Fourth Powers === {{:319 is not Expressible as Sum of Fewer than 19 Fourth Powers}} === $399$ as Sum of $19$ Fourth Powers === {{:399 is not Expressible as Sum of Fewer than 19 Fourth Powers}}"}
+{"_id": "14343", "title": "Existence of n such that M - 2^n or M + 2^n has no Prime factors less than 331", "text": "Let $M \\in \\Z$ be an integer. Then there exists a positive integer $n \\in \\Z_{\\ge 0}$ such that either $M - 2^n$ or $M + 2^n$ has no prime factors less than $331$."}
+{"_id": "14344", "title": "Completely Irreducible implies Infimum differs from Element", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $p \\in S$ such that :$p$ is completely irreducible. Then $\\map \\inf {p^\\succeq \\setminus \\set p} \\ne p$ where $p^\\succeq$ donotes the upper closure of $p$."}
+{"_id": "14345", "title": "Not Every Number is the Sum or Difference of Two Prime Powers", "text": "Not every positive integer can be expressed in the form $p^m \\pm q^n$ where $p, q$ are prime and $m, n$ are positive integers."}
+{"_id": "14346", "title": "Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element", "text": "Let $L = \\struct {S, \\preceq}$ be an ordered set. Let $p \\in S$. Then $p$ is completely irreducible {{iff}} :$\\exists q \\in S: p \\prec q \\land \\paren {\\forall s \\in S: p \\prec s \\implies q \\preceq s} \\land p^\\succeq = \\set p \\cup q^\\succeq$ where $p^\\succeq$ denotes the upper closure of $p$."}
+{"_id": "14347", "title": "Poulet Number/Examples/341", "text": "The smallest Poulet number is $341$: :$2^{341} \\equiv 2 \\pmod {341}$ despite the fact that $341$ is not prime: :$341 = 11 \\times 31$"}
+{"_id": "14348", "title": "Fourth Powers which are Sum of 4 Fourth Powers", "text": "The following positive integers are such that their fourth powers can be expressed as the sum of the fourth powers of $4$ other positive integers with no common factors: :$353, 651, 2487, 2501, 2829, \\ldots$ {{OEIS|A039664}}"}
+{"_id": "14349", "title": "Sums of both 2 and 3 Consecutive Squares", "text": "The following are the smallest positive integers that are the sum of both $2$ and $3$ consecutive non-zero square numbers: :$365, 35 \\, 645, 3 \\, 492 \\, 725, 342 \\, 251 \\, 285, 33 \\, 537 \\, 133 \\, 085, 3 \\, 286 \\, 296 \\, 790 \\, 925, \\ldots$ {{OEIS|A007667}}"}
+{"_id": "14350", "title": "Sturm-Liouville Problem/Unit Weight Function", "text": "Let $P, Q: \\R \\to \\R$ be real mappings such that $P$ is smooth and positive, while $Q$ is continuous: :$\\map P x \\in C^\\infty$  :$\\map P x > 0$ :$\\map Q x \\in C^0$ Let the Sturm-Liouville equation, with $\\map w x = 1$, be of the form: :$-\\paren {P y'}' + Q y = \\lambda y$ where $\\lambda \\in \\R$. Let it satisfy the following boundary conditions: :$\\map y a = \\map y b = 0$ Then all solutions of the Sturm-Liouville equation, together with their eigenvalues, form infinite sequences $\\sequence {y^{\\paren n} }$ and $\\sequence {\\lambda^{\\paren n} }$. Furthermore, each $\\lambda^{\\paren n}$ corresponds to an eigenfunction $y^{\\paren n}$, unique up to a constant factor."}
+{"_id": "14352", "title": "Discrepancy between Julian Year and Tropical Year", "text": "The Julian year and the tropical year differ such that the Julian calendar becomes $1$ day further out approximately every $128$ years."}
+{"_id": "14354", "title": "Completely Irreducible and Subset Admits Infimum Equals Element implies Element Belongs to Subset", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an ordered set. Let $x \\in S$ such that :$x$ is completely irreducible. Let $X \\subseteq S$ such that :$X$ admits an infimum and $x = \\inf X$ Then $x \\in X$"}
+{"_id": "14355", "title": "Order Generating Includes Completely Irreducible Elements", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $X \\subseteq S$ be an order generating subset of $S$. Let $X \\in S$ be a completely irreducible element of $S$. Then $x \\in X$."}
+{"_id": "14356", "title": "399 is not Expressible as Sum of Fewer than 19 Fourth Powers", "text": ":$399 = 14 \\times 1^4 + 3 \\times 2^4 + 3^4 + 4^4$ or: :$399 = 11 \\times 1^4 + 4 \\times 2^4 + 4 \\times 3^4$"}
+{"_id": "14357", "title": "Maximal implies Difference equals Intersection", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $x, y \\in S$ such that :$x$ is maximal in $S \\setminus y^\\succeq$ Then $x^\\succeq \\setminus \\left\\{ {x}\\right\\} = x^\\succeq \\cap y^\\succeq$"}
+{"_id": "14358", "title": "Square of n such that 2n-1 is Composite is not Sum of Square and Prime", "text": "Let $n^2$ be a square such that $2 n - 1$ is composite. Then $n^2$ cannot be expressed as the sum of a square and a prime."}
+{"_id": "14359", "title": "40 times Heptagonal Numbers plus 9 gives Squares of Numbers ending in 7", "text": "Consider the heptagonal numbers: :$\\displaystyle H_n = \\sum_{k \\mathop = 1}^n \\left({5 k - 4}\\right)$ Let $S_n$ be the sequence defined as: :$\\forall n \\in \\Z_{>1}: S_n = 40 \\times H_n + 9$ Then $S_n$ consists of the squares of all the positive integers which end in a $7$: :$49, 289, 729, 1369, 2209, 3249, 4489, 5929, 7569, \\ldots$ that is: :$7^2, 17^2, 27^2, 37^2, 47^2, 57^2, 67^2, 77^2, 87^2, 97^2, \\ldots$ {{OEIS|A017354}}"}
+{"_id": "14360", "title": "Maximal implies Completely Irreducible", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice. Let $p \\in S$ such that :$\\exists k \\in S: p$ is maximal in $S \\setminus k^\\succeq$ Then $p$ is completely irreducible."}
+{"_id": "14361", "title": "Infimum of Intersection of Upper Closures equals Join Operands", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a join semilattice. Let $x, y \\in S$. Then $\\inf\\left({x^\\succeq \\cap y^\\succeq}\\right) = x \\vee y$"}
+{"_id": "14362", "title": "Sturm-Liouville Problem/Unit Weight Function/Lemma", "text": "Let $\\map \\alpha x: \\R \\to \\R$ such that $\\map \\alpha x \\in C^2 \\closedint a b$. Suppose: :$\\displaystyle \\forall \\map h x \\in C^2 \\closedint a b: \\map h a = \\map h b = \\map {h'} a = \\map {h'} b = 0: \\int_a^b \\map \\alpha x \\, \\map {h''} x \\rd x = 0$ Then: :$\\forall x \\in \\closedint a b: \\map \\alpha x = c_0 + c_1 x$ where $ c_0, c_1 \\in \\R $ are constants."}
+{"_id": "14363", "title": "Intersection of Upper Closures is Upper Closure of Join Operands", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a join semilattice. Let $x, y \\in S$. Then $\\left({x \\vee y}\\right)^\\succeq = x^\\succeq \\cap y^\\succeq$"}
+{"_id": "14364", "title": "Strictly Precede and Step Condition and not Precede implies Joins are equal", "text": "Let $\\struct {S, \\vee, \\preceq}$ be a join semilattice. Let $p, q, u \\in S$ be such that: :$p \\prec q$ and $\\paren {\\forall s \\in S: p \\prec s \\implies q \\preceq s}$ and $u \\npreceq p$ Then: :$p \\vee u = q \\vee u$"}
+{"_id": "14365", "title": "Not Preceding implies Exists Completely Irreducible Element in Algebraic Lattice", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a bounded below algebraic lattice. Let $x, y \\in S$ such that :$y \\npreceq x$ Then :$\\exists p \\in S: p$ is completely irreducible $\\land ~x \\preceq p \\land y \\npreceq p$"}
+{"_id": "14366", "title": "Set of All Completely Irreducible Elements is Smallest Order Generating", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below algebraic lattice. Then $\\mathit{Irr}\\left({L}\\right)$ is order generating and :$\\forall X \\subseteq S: X$ is order generating $\\implies \\mathit{Irr}\\left({L}\\right) \\subseteq X$ where $\\mathit{Irr}\\left({L}\\right)$ denotes the set of all completely irreducible elements of $L$."}
+{"_id": "14368", "title": "Preimage of Lower Set under Increasing Mapping is Lower", "text": "Let $\\struct {S, \\preceq}$, $\\struct {T, \\precsim}$ be preordered sets. Let $f: S \\to T$ be an increasing mapping. Let $X \\subseteq T$ be a lower subset of $T$. Then $f^{-1} \\sqbrk X$ is lower where $f^{-1} \\sqbrk X$ denotes the preimage of $X$ under $f$."}
+{"_id": "14369", "title": "Preimage of Upper Set under Increasing Mapping is Upper", "text": "Let $\\struct {S, \\preceq}$, $\\struct {T, \\precsim}$ be preordered sets. Let $f: S \\to T$ be an increasing mapping. Let $X \\subseteq T$ be a upper subset of $T$. Then $f^{-1} \\sqbrk X$ is upper where $f^{-1} \\sqbrk X$ denotes the preimage of $X$ under $f$."}
+{"_id": "14371", "title": "Directed Suprema Preserving Mapping at Element is Supremum", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ and $\\left({T, \\vee_2, \\wedge_2, \\precsim}\\right)$ be bounded below continuous lattices. Let $f: S \\to T$ be a mapping such that :$f$ preserves directed suprema. Let $x \\in S$. Then $f\\left({x}\\right) = \\sup \\left\\{ {f\\left({w}\\right): w \\in S \\land w \\ll x}\\right\\}$"}
+{"_id": "14372", "title": "Preceding implies Way Below Closure is Subset of Way Below Closure", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $x, y \\in S$ such that :$x \\preceq y$ Then $x^\\ll \\subseteq y^\\ll$ where $x^\\ll$ denotes the way below closure of $x$."}
+{"_id": "14373", "title": "Mapping at Element is Supremum implies Mapping is Increasing", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a lattice. Let $\\left({T, \\vee_2, \\wedge_2, \\precsim}\\right)$ be a complete lattice. Let $f: S \\to T$ be a mapping such that :$\\forall x \\in S: f\\left({x}\\right) = \\sup \\left\\{ {f\\left({w}\\right): w \\in S \\land w \\ll x}\\right\\}$ Then $f$ is an increasing mapping."}
+{"_id": "14374", "title": "Semiperfect Number is not Deficient", "text": "Let $n \\in \\Z_{>0}$ be a semiperfect number. Then $n$ is not deficient."}
+{"_id": "14375", "title": "Perfect Number is Primitive Semiperfect", "text": "Let $n \\in \\Z_{>0}$ be a perfect number. Then $n$ is also a primitive semiperfect number."}
+{"_id": "14376", "title": "Mapping at Element is Supremum implies Way Below iff There Exists Element that Way Below and Way Below", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $\\left({T, \\vee_2, \\wedge_2, \\precsim}\\right)$ be a continuous complete lattice. Let $f: S \\to T$ be a mapping such that :$\\forall x \\in S: f\\left({x}\\right) = \\sup \\left\\{ {f\\left({w}\\right): w \\in S \\land w \\ll x}\\right\\}$ Let $x \\in S, y \\in T$. Then :$y \\ll f\\left({x}\\right) \\iff \\exists w \\in S: w \\ll x \\land y \\ll f\\left({w}\\right)$"}
+{"_id": "14377", "title": "Subset and Image Admit Suprema and Mapping is Increasing implies Supremum of Image Precedes Mapping at Supremum", "text": "Let $\\left({S, \\preceq}\\right)$, $\\left({T, \\precsim}\\right)$ be ordered sets. Let $f: S \\to T$ be a increasing mapping. Let $D \\subseteq S$ such that :$D$ admits a supremum in $S$ and $f\\left[{D}\\right]$ admits a supremum in $T$. Then $\\sup \\left({f\\left[{D}\\right]}\\right) \\precsim f\\left({\\sup D}\\right)$"}
+{"_id": "14378", "title": "Characteristic Subgroup of Normal Subgroup is Normal", "text": "Let $G$ be a group. Let $N\\leq G$ be normal. Let $H\\leq N$ be characteristic. Then $H$ is normal in $G$."}
+{"_id": "14379", "title": "Center is Characteristic Subgroup", "text": "Let $G$ be a group. Then its center $\\map Z G$ is characteristic in $G$."}
+{"_id": "14380", "title": "Semidirect Product of Groups is Group", "text": "Let $H$ and $N$ be groups. Let $\\operatorname{Aut}(N)$ denote the automorphism group of $N$. Let $\\phi : H\\to \\operatorname{Aut}(N)$ be a group homomorphism, that is, let $H$ act on $N$. Let $N\\rtimes_\\phi H$ be the semidirect product of $N$ and $H$ with respect to $\\phi$, that is: :$N\\rtimes_\\phi H = (N\\times H, \\circ)$ where :$(n_1, h_1) \\circ (n_2, h_2) = (n_1\\cdot \\phi(h_1)(n_2), h_1\\cdot h_2)$ Then $N\\rtimes_\\phi H$ is a group."}
+{"_id": "14381", "title": "Inverse of Element in Semidirect Product", "text": "Let $N$ and $H$ be groups. Let $H$ act by automorphisms on $N$ via $\\phi$. Let $N \\rtimes_\\phi H$ be the corresponding (outer) semidirect product. Let $\\tuple {n, h} \\in N \\rtimes_\\phi H$. Then: {{begin-eqn}} {{eqn | l = \\tuple {n, h}^{-1}       | r = \\tuple {\\map {\\phi_{h^{-1} } } {n^{-1} }, h^{-1} }       | c =  }} {{eqn | r = \\tuple {\\paren {\\map {\\phi_{h^{-1} } } n}^{-1}, h^{-1} }       | c =  }} {{eqn | r = \\tuple {\\map { {\\phi_h}^{-1} } {n^{-1} }, h^{-1} }       | c =  }} {{eqn | r = \\tuple {\\paren {\\map { {\\phi_h}^{-1} } n}^{-1}, h^{-1} }       | c =  }} {{end-eqn}}"}
+{"_id": "14382", "title": "Semidirect Product with Trivial Action is Direct Product", "text": "Let $H$ and $N$ be groups. Let $\\Aut N$ denote the automorphism group of $N$. Let $\\phi: H \\to \\Aut N$ be defined as: :$\\forall h \\in H: \\map \\phi h = I_N$ for all $h \\in H$ where $I_N$ denotes the identity mapping on $N$. Let $N \\rtimes_\\phi H$ be the corresponding semidirect product. Then $N \\rtimes_\\phi H$ is the direct product of $N$ and $H$."}
+{"_id": "14383", "title": "Semidirect Product is Abelian iff Components are Abelian and Action is Trivial", "text": "Let $N$ and $H$ be groups. Let $H$ act by automorphisms on $N$ via $\\phi$. Let $N\\rtimes_\\phi H$ be the corresponding semidirect product. Then the following are equivalent: :$(1): \\quad$ $N\\rtimes_\\phi H$ is abelian :$(2): \\quad$ $N$ and $H$ are abelian and $H$ acts trivially"}
+{"_id": "14384", "title": "Integers whose Squares end in 444", "text": "The sequence of positive integers whose square ends in $444$ begins: :$38, 462, 538, 962, 1038, 1462, 1538, 1962, 2038, 2462, 2538, 2962, 3038, 3462, \\ldots$ {{OEIS|A039685}}"}
+{"_id": "14385", "title": "Subset and Image Admit Infima and Mapping is Increasing implies Infimum of Image Succeeds Mapping at Infimum", "text": "Let $\\struct {S, \\preceq}$ and $\\struct {T, \\precsim}$ be ordered sets. Let $f: S \\to T$ be a increasing mapping. Let $D \\subseteq S$ such that :$D$ admits a infimum in $S$ and $f \\sqbrk D$ admits a infimum in $T$. Then $\\map f {\\inf D} \\precsim \\map \\inf {f \\sqbrk D}$"}
+{"_id": "14386", "title": "Equivalence of Definitions of Finite Galois Extension", "text": "Let $L/K$ be a finite field extension. {{TFAE|def = Finite Galois Extension}}"}
+{"_id": "14387", "title": "Finite Field Extension has Finite Galois Group", "text": "Let $E / F$ be a finite field extension. Then its automorphism group is finite."}
+{"_id": "14388", "title": "Primitive Element Theorem", "text": "Let $E/F$ be a separable field extension of finite degree. Then $E/F$ is simple: there exists $\\alpha\\in E$ such that $E=F(\\alpha)$."}
+{"_id": "14389", "title": "Largest Number not Expressible as Sum of Less than 32 Positive Fifth Powers", "text": "The largest positive integer which cannot be expressed as the sum of less than $32$ positive fifth powers is $466$: :$466 = 18 \\times 1^5 + 14 \\times 2^5$"}
+{"_id": "14390", "title": "Smallest Square which is Sum of 3 Fourth Powers", "text": "The smallest positive integer whose square is the sum of $3$ fourth powers is $481$: :$481^2 = 12^4 + 15^4 + 20^4$"}
+{"_id": "14391", "title": "Solutions to p^2 Divides 10^p - 10", "text": "The known prime numbers $p$ which satisfy the equation: :$p^2 \\divides \\paren {10^p - 10}$ where $\\divides$ denotes divisibility, are: :$3, 487, 56 \\, 598 \\, 313$ {{OEIS|A045616}}"}
+{"_id": "14392", "title": "Kaprekar's Process on 3 Digit Number ends in 495", "text": "Let $n$ be a $3$-digit integer whose digits are not all the same. Kaprekar's process, when applied to $n$, results in $495$ after no more than $6$ iterations."}
+{"_id": "14393", "title": "Nilpotent Elements of Commutative Ring form Ideal", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring whose zero is $0_R$ and whose unity is $1_R$. The subset of nilpotent elements of $R$ form an ideal of $R$."}
+{"_id": "14395", "title": "Algebraic Closure of Field is Unique", "text": "Let $F$ be a field. Let $K$ and $L$ be algebraic closures of $F$. Then $K$ and $L$ are $F$-isomorphic."}
+{"_id": "14396", "title": "Multiplicative Group of Galois Field is Cyclic", "text": "Let $\\GF$ be a Galois field of order $q$. Then its multiplicative group is cyclic of order $q-1$: :$\\GF^\\times \\cong C_{q - 1}$"}
+{"_id": "14397", "title": "Automorphism Group of C Over R", "text": "The field extension $\\C / \\R$ of complex numbers $\\C$ over real numbers $\\R$ has automorphism group $\\operatorname{Aut}$: :$\\operatorname{Aut} \\paren {\\C / \\R} = \\set {\\operatorname{id}, \\sigma}$ where: :$\\operatorname{id}$ denotes the identity mapping :$\\sigma$ denotes complex conjugation"}
+{"_id": "14398", "title": "Image under Inclusion Mapping", "text": "Let $X$ be a set. Let $S \\subseteq X$, $Z \\subseteq S$. Then $i_S\\left[{Z}\\right] = Z$ where :$i_S$ denotes the inclusion mapping of $S$, :$i_S\\left[{Z}\\right]$ denotes the image of $Z$ under $i_S$."}
+{"_id": "14399", "title": "Limit Inferior of Inclusion Moore-Smith Sequence is Supremum of Directed Subset", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be an up-complete lattice. Let $D \\subseteq S$ be a directed subset of $S$. Let $\\left({D, \\preceq'}\\right)$ be a directed ordered subset of $L$. Let $i_D: D \\to S$, the inclusion mapping, be a Moore-Smith sequence in $S$. Then $\\liminf i_D = \\sup D$"}
+{"_id": "14400", "title": "Correspondence Between Group Actions and Permutation Representations", "text": "Let $G$ be a group. Let $X$ be a set. There is a one-to-one correspondence between group actions of $G$ on $X$ and permutation representations of $G$ in $X$, as follows: Let $\\phi : G \\times X \\to X$ be a group action. Let $\\rho : G \\to \\struct {\\map \\Gamma X, \\circ}$ be a permutation representation. The following are equivalent: :$(1): \\quad \\rho$ is the permutation representation associated to $\\phi$ :$(2): \\quad \\phi$ is the group action associated to $\\rho$"}
+{"_id": "14402", "title": "Trivial Field Extension is Galois", "text": "Let $F$ be a field. The trivial field extension $F/F$ is Galois."}
+{"_id": "14404", "title": "Automorphism Group Acts Faithfully on Generating Set", "text": "Let $E/F$ be a field extension. Let $\\operatorname{Aut}(E/F)$ be its automorphism group. Let $S\\subset E$ be a generating set of the extension. Let $S$ be stable under the group action of $\\operatorname{Aut}(E/F)$. Then the induced group action on $S$ is faithful."}
+{"_id": "14405", "title": "Upper Closure in Ordered Subset is Intersection of Subset and Upper Closure", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an ordered set. Let $\\left({T, \\precsim}\\right)$ be an ordered subset of $L$. Let $t \\in T$. Then $t^\\succsim = T \\cap t^\\succeq$"}
+{"_id": "14407", "title": "Approximation to Power of 7 by Power of 10", "text": ":$7^{510} \\approx 1 \\cdotp 00000 \\, 09377 \\, 76536 \\ldots \\times 10^{431}$ This is the closest known approximation of a power of $7$ by a power of $10$."}
+{"_id": "14408", "title": "Limit Inferior of Repetition Moore-Smith Sequence", "text": "Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Let $N = \\left({\\N, \\le}\\right)$ be a directed ordered set. Let $a, b \\in S$. Let $f = \\left({c_i}\\right)_{i \\in \\N} = \\left({a, b, a, b, \\dots}\\right):\\N \\to S$ be a Moore-Smith sequence. Then $\\liminf \\left({c_i}\\right)_{i \\in \\N} = a \\wedge b$"}
+{"_id": "14409", "title": "Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping is Increasing", "text": "Let $\\left({S, \\vee_1, \\wedge_1, \\preceq_1}\\right)$ and $\\left({T, \\vee_2, \\wedge_2, \\preceq_2}\\right)$ be lattices. Let $f: S \\to T$ be a mapping such that :for all directed set $\\left({D, \\precsim}\\right)$ and Moore-Smith sequence $N:D \\to S$ in $S$: $f\\left({\\liminf N}\\right) \\preceq_2 \\liminf\\left({f \\circ N}\\right)$ Then $f$ is an increasing mapping."}
+{"_id": "14410", "title": "Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Supremum of Image is Mapping at Supremum of Directed Subset", "text": "Let $\\left({S, \\vee_1, \\wedge_1, \\preceq_1}\\right)$ and $\\left({T, \\vee_2, \\wedge_2, \\preceq_2}\\right)$ be up-complete lattices. Let $f: S \\to T$ be a mapping such that :for all directed set $\\left({D, \\precsim}\\right)$ and Moore-Smith sequence $N:D \\to S$ in $S$: $f\\left({\\liminf N}\\right) \\preceq_2 \\liminf\\left({f \\circ N}\\right)$ Let $D$ be a directed subset of $S$. Then $\\sup \\left({f\\left[{D}\\right]}\\right) = f\\left({\\sup D}\\right)$ where $f\\left[{D}\\right]$ denotes the image of $D$ under $f$."}
+{"_id": "14411", "title": "Prime Values of Double Factorial plus 1", "text": "Let $n!!$ denote the double factorial function. The sequence of positive integers $n$ such that $n!! + 1$ is prime begins: :$0, 1, 2, 518, 33 \\, 416, 37 \\, 310, 52 \\, 608, 123 \\, 998, 220 \\, 502, \\ldots$ {{OEIS|A080778}}"}
+{"_id": "14412", "title": "Limit Inferior of Restriction Moore-Smith Sequence is Supremum of Image of Directed Subset", "text": "Let $L = \\left({S, \\vee_1, \\wedge_1, \\preceq_1}\\right)$ and $\\left({T, \\vee_2, \\wedge_2, \\preceq_2}\\right)$ be up-complete lattices. Let $f:S \\to T$ be an increasing mapping. Let $D \\subseteq S$ be a directed subset of $S$. Let $\\left({D, \\preceq'}\\right)$ be a directed ordered subset of $L$. Let $f \\restriction D: D \\to T$, the restriction of mapping, be a Moore-Smith sequence in $T$. Then $\\liminf \\left({f \\restriction D}\\right) = \\sup \\left({f\\left[{D}\\right]}\\right)$"}
+{"_id": "14413", "title": "Infimum of Image of Upper Closure of Element under Increasing Mapping", "text": "Let $\\left({S, \\preceq}\\right)$ and $\\left({T, \\precsim}\\right)$ be ordered set. Let $f:S \\to T$ be an increasing mapping. Let $x \\in S$. Then $\\inf \\left({f\\left[{x^\\succeq}\\right]}\\right) = f\\left({x}\\right)$"}
+{"_id": "14415", "title": "Composition of Mapping and Inclusion is Restriction of Mapping", "text": "Let $S, T$ be sets. Let $f:S \\to T$ be a mapping. Let $A \\subseteq S$. Then $f \\circ i_A = f \\restriction A$ where :$i_A$ denotes the inclusion mapping of $A$, :$f \\restriction A$ denotes the restriction of $f$ to $A$."}
+{"_id": "14416", "title": "Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping Preserves Directed Suprema", "text": "Let $\\struct {S, \\vee_1, \\wedge_1, \\preceq_1}$ and $\\struct {T, \\vee_2, \\wedge_2, \\preceq_2}$ be complete lattices. Let $f: S \\to T$ be a mapping such that :for all directed sets $\\struct {D, \\precsim}$ and Moore-Smith sequences $N:D \\to S$ in $S$: $\\map f {\\liminf N} \\preceq_2 \\map \\liminf {f \\circ N}$ Then $f$ preserves directed suprema."}
+{"_id": "14417", "title": "Thurston's Geometrization Conjecture", "text": "When a topological manifold of dimension $3$ has been split into its connected sum and the Jaco-Shalen-Johannson torus decomposition, the remaining components each admit exactly one of the following geometries: :$(1): \\quad$ Euclidean geometry :$(2): \\quad$ Hyperbolic geometry :$(3): \\quad$ Spherical geometry :$(4): \\quad$ The geometry of $\\mathbb S^2 \\times \\R$ :$(5): \\quad$ The geometry of $\\mathbb H^2 \\times \\R$ :$(6): \\quad$ The geometry of the universal cover $S L_2 \\R^~$ of the Lie group $S L_2 \\R$ :$(7): \\quad$ Nil geometry :$(8): \\quad$ Sol geometry where: :$\\mathbb S^2$ is the 2-sphere :$\\mathbb H^2$ is the hyperbolic plane."}
+{"_id": "14418", "title": "Poincaré Conjecture/Historical Note", "text": "The Poincaré Conjecture was first posed in $1904$ by {{AuthorRef|Jules Henri Poincaré}}. It was finally resolved by the work of {{AuthorRef|Grigori Perelman}}, who solved Thurston's Geometrization Conjecture in $2003$."}
+{"_id": "14419", "title": "Definition:Stabilizer/Linguistic Note", "text": "The British English spelling for '''stabilizer''' is '''stabiliser'''."}
+{"_id": "14420", "title": "Mapping Preserves Directed Suprema implies Mapping is Continuous", "text": "Let $\\left({S, \\preceq_1, \\tau_1}\\right)$ and $\\left({T, \\preceq_2, \\tau_2}\\right)$ be up-complete ordered sets with Scott topologies. Let $f: S \\to T$ be a directed suprema preserving mapping. Then $f$ is continuous."}
+{"_id": "14421", "title": "Secant of Complex Number", "text": "Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\\sec \\paren {a + b i} = \\dfrac {\\cos a \\cosh b + i \\sin a \\sinh b} {\\cos^2 a \\cosh^2 b + \\sin^2 a \\sinh^2 b}$ where: :$\\sec$ denotes the complex secant function. :$\\sin$ denotes the real sine function :$\\cos$ denotes the real cosine function :$\\sinh$ denotes the hyperbolic sine function :$\\cosh$ denotes the hyperbolic cosine function"}
+{"_id": "14422", "title": "Tangent of Complex Number/Formulation 1", "text": ":$\\tan \\paren {a + b i} = \\dfrac {\\sin a \\cosh b + i \\cos a \\sinh b} {\\cos a \\cosh b - i \\sin a \\sinh b}$"}
+{"_id": "14423", "title": "Tangent of Complex Number/Formulation 2", "text": ":$\\tan \\paren {a + b i} = \\dfrac {\\tan a + i \\tanh b} {1 - i \\tan a \\tanh b}$"}
+{"_id": "14424", "title": "Tangent of Complex Number/Formulation 3", "text": ":$\\tan \\paren {a + b i} = \\dfrac {\\tan a - \\tan a \\tanh ^2 b} {1 + \\tan ^2 a \\tanh ^2 b} + \\dfrac {\\tanh b + \\tan ^2 a \\tanh b} {1 + \\tan ^2 a \\tanh ^2 b} i$"}
+{"_id": "14425", "title": "Cosecant of Complex Number", "text": "Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\\csc \\paren {a + b i} = \\dfrac {\\sin a \\cosh b - i \\cos a \\sinh b} {\\sin^2 a \\cosh^2 b + \\cos^2 a \\sinh^2 b}$ where: :$\\csc$ denotes the complex cosecant function. :$\\sin$ denotes the real sine function :$\\cos$ denotes the real cosine function :$\\sinh$ denotes the hyperbolic sine function :$\\cosh$ denotes the hyperbolic cosine function"}
+{"_id": "14426", "title": "Cotangent of Complex Number", "text": "Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then:"}
+{"_id": "14427", "title": "Cotangent of Complex Number/Formulation 1", "text": ":$\\cot \\paren {a + b i} = \\dfrac {\\cos a \\cosh b - i \\sin a \\sinh b} {\\sin a \\cosh b + i \\cos a \\sinh b}$"}
+{"_id": "14428", "title": "Cotangent of Complex Number/Formulation 2", "text": ":$\\map \\cot {a + b i} = \\dfrac {i \\cot a \\coth b - 1} {\\cot a - i \\coth b}$"}
+{"_id": "14430", "title": "Continuous iff Mapping at Limit Inferior Precedes Limit Inferior of Composition of Mapping and Sequence", "text": "Let $\\left({S, \\preceq_1, \\tau_1}\\right)$ and $\\left({T, \\preceq_2, \\tau_2}\\right)$ be complete topological lattices with Scott topologies. Let $f: S \\to T$ be a mapping. Then $f$ is continuous {{iff}} :for all directed set $\\left({D, \\precsim}\\right)$ and Moore-Smith sequence $N:D \\to S$ in $S$: $f\\left({\\liminf N}\\right) \\preceq_2 \\liminf\\left({f \\circ N}\\right)$"}
+{"_id": "14431", "title": "Mapping is Increasing implies Mapping at Infimum for Sequence Precedes Infimum for Composition of Mapping and Sequence", "text": "Let $\\left({S, \\vee_1, \\wedge_1, \\preceq_1}\\right)$ and $\\left({T, \\vee_2, \\wedge_2, \\preceq_2}\\right)$ be complete lattices. Let $f: S \\to T$ be an increasing mapping. Let $\\left({D, \\precsim}\\right)$ be a directed set. Let $N: D \\to S$ be a Moore-Smith sequence in $S$. Let $j \\in D$. Then $f\\left({\\inf\\left({N\\left[{\\precsim \\left({j}\\right)}\\right]}\\right)}\\right) \\preceq_2 \\inf \\left({\\left({f \\circ N}\\right)\\left[{\\precsim \\left({j}\\right)}\\right]}\\right)$"}
+{"_id": "14432", "title": "Set of Infima for Sequence is Directed", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $\\left({A, \\precsim}\\right)$ be a non-empty directed set. Let $Z: A \\to S$ be a Moore-Smith sequence. Let $D = \\left\\{ {\\inf \\left({Z\\left[{\\precsim \\left({j}\\right)}\\right]}\\right): j \\in A}\\right\\}$ be a subset of $S$. Then $D$ is directed."}
+{"_id": "14433", "title": "Preceding implies Image is Subset of Image", "text": "Let $\\left({S, \\precsim}\\right)$ be a preordered set. Let $x, y \\in S$ such that :$x \\precsim y$ Then $\\precsim\\left({y}\\right) \\subseteq \\mathord\\precsim\\left({x}\\right)$ where $\\precsim\\left({y}\\right)$ denotes the image of $y$ under $\\precsim$."}
+{"_id": "14434", "title": "Universal Property of Group Ring", "text": "Let $R$ be a commutative ring with unity. Let $G$ be a group. Let $R \\sqbrk G$ be the corresponding group ring. Let $S$ be a commutative ring with unity. Let $\\phi: R \\to S$ be a ring homomorphism. Let $\\beta : G \\to R^\\times$ be a group homomorphism, where $R^\\times$ is the multiplicative group of $R$. Then there exists a unique ring homomorphism from $R \\sqbrk G$ to $S$ which extends $\\phi$ and $\\beta$."}
+{"_id": "14435", "title": "Continuous iff Directed Suprema Preserving", "text": "Let $\\left({S, \\preceq_1, \\tau_1}\\right)$ and $\\left({T, \\preceq_2, \\tau_2}\\right)$ be complete topological lattices with Scott topologies. Let $f: S \\to T$ be a mapping. Then $f$ is continuous {{iff}} $f$ preserves directed suprema."}
+{"_id": "14436", "title": "Universal Property of Quotient of Topological Group", "text": "Let $G$ and $H$ be topological groups. Let $N$ be a normal subgroup of $G$. Let $\\pi : G \\to G/N$ be the quotient mapping. Let $f : G \\to H$ be a continuous group homomorphism whose kernel contains $N$. Then there exists a unique continuous group homomorphism $\\overline f : G/N \\to H$ such that $f = \\overline f \\circ \\pi$."}
+{"_id": "14437", "title": "Universal Property of Quotient Set", "text": "Let $X$ and $Y$ be sets. Let $\\sim$ be an equivalence relation on $X$. Let $\\pi : X \\to X/\\sim$ be the quotient mapping to the quotient set. Let $f : X \\to Y$ be $\\sim$-invariant. Then there exists a unique mapping $\\overline f : X/\\!\\sim \\to Y$ such that $f = \\overline f \\circ \\pi$. :$\\xymatrix{ X \\ar[d]_\\pi \\ar[r]^f & Y\\\\ X/\\sim \\ar@{.>}[ru]_{\\overline f} }$"}
+{"_id": "14438", "title": "Continuous iff Way Below iff There Exists Element that Way Below and Way Below", "text": "Let $\\struct {S, \\preceq_1, \\tau_1}$ and $\\struct {T, \\preceq_2, \\tau_2}$ be complete continuous topological lattices with Scott topologies. Let $f: S \\to T$ be a mapping. Then $f$ is continuous {{iff}} :$\\forall x \\in S, y \\in T: y \\ll \\map f x \\iff \\exists w \\in S: w \\ll x \\land y \\ll \\map f w$ {{explain|link to definition of $\\ll$ and $\\gg$}}"}
+{"_id": "14440", "title": "Finite Symmetric Group is Ambivalent", "text": "Let $n$ be a natural number. Let $S_n$ be a symmetric group of order $n$. Then $S_n$ is ambivalent."}
+{"_id": "14441", "title": "Alternating Groups that are Ambivalent", "text": "Let $n$ be a natural number. Then the $n$th alternating group $A_n$ is ambivalent {{iff}} $n\\in \\{1, 2, 5, 6, 10, 14\\}$. {{OEIS|A115200}}"}
+{"_id": "14442", "title": "Transitivity of Big-O Estimates/General", "text": "Let $X$ be a topological space. Let $V$ be a normed vector space over $\\R$ or $\\C$ with norm $\\norm {\\,\\cdot\\,}$. Let $f, g, h: X \\to V$ be functions. Let $x_0 \\in X$. Let $f = \\map \\OO g$ and $g = \\map \\OO h$ as $x \\to x_0$, where $\\OO$ denotes big-O notation. Then $f = \\map \\OO h$ as $x \\to x_0$."}
+{"_id": "14443", "title": "Transitivity of Big-O Estimates/Sequences", "text": "Let $\\sequence {a_n}$, $\\sequence {b_n}$ and $\\sequence {c_n}$ be sequences of real or complex numbers. Let $a_n = \\map \\OO {\\sequence {b_n} }$ and $b_n = \\map \\OO {\\sequence {c_n} }$, where $O$ denotes big-O notation. Then $a_n = \\map \\OO {\\sequence {c_n} }$."}
+{"_id": "14444", "title": "Conjugate of Cycle", "text": "Let $n \\ge 1$ be a natural number. Let $S_n$ be the symmetric group on $n$ letters. Let $\\pi, \\sigma \\in S_n$. Let $\\sigma$ be a cycle of length $k$. Then the conjugate $\\pi \\sigma \\pi^{-1}$ is a cycle of length $k$."}
+{"_id": "14445", "title": "Center of Group is Kernel of Conjugacy Action", "text": "Let $G$ be a group. Let $Z$ be the kernel of the conjugacy action. Then $Z$ is the center of $G$."}
+{"_id": "14446", "title": "Little-O Implies Big-O/General Result", "text": "Let $X$ be a topological space. Let $V$ be a normed vector space over $\\R$ or $\\C$ with norm $\\norm {\\,\\cdot\\,}$ Let $f, g: X \\to V$ be mappings. Let $x_0 \\in X$. Let $f = \\map o g$ as $x \\to x_0$, where $o$ denotes little-O notation. Then $f = \\map \\OO g$ as $x \\to x_0$, where $\\OO$ denotes big-$\\OO$ notation."}
+{"_id": "14447", "title": "Little-O Implies Big-O/Sequences", "text": "Let $\\sequence {a_n}$ and $\\sequence {b_n}$ be sequences of real or complex numbers. Let $a_n = \\map o {b_n}$ where $o$ denotes little-O notation. Then $a_n = \\map \\OO {b_n}$ where $\\OO$ denotes big-$\\OO$ notation."}
+{"_id": "14450", "title": "Substitution in Big-O Estimate/Sequences", "text": "Let $(a_n)$ and $(b_n)$ be sequences of real or complex numbers. Let $a_n = O(b_n)$ where $O$ denotes big-O notation. Let $(n_k)$ be a diverging sequence of natural numbers. Then $a_{n_k} = O(b_{n_k})$."}
+{"_id": "14451", "title": "First Sequence of Three Consecutive Strictly Decreasing Euler Phi Values", "text": "The first sequence of $3$ consecutive positive integers whose Euler $\\phi$ values are strictly decreasing is: :$\\map \\phi {523} > \\map \\phi {524} > \\map \\phi {525}$"}
+{"_id": "14452", "title": "Bases of Finitely Generated Free Module have Equal Cardinality", "text": "Let $R$ be a commutative ring with unity. Let $M$ be a free $R$-module. Let $M$ be finitely generated. Let $B$ and $C$ be bases of $M$. Then $B$ and $C$ are finite and have the same cardinality."}
+{"_id": "14453", "title": "Bases of Free Module have Equal Cardinality", "text": "Let $R$ be a commutative ring with unity. Let $M$ be a free $R$-module. Let $B$ and $C$ be bases of $M$. Then $B$ and $C$ are equinumerous."}
+{"_id": "14454", "title": "Sequences of Three Consecutive Strictly Increasing Euler Phi Values", "text": "The following sequences of $3$ consecutive positive integers have the property that their Euler $\\phi$ values are strictly increasing: :$\\tuple {105, 106, 107}, \\tuple {165, 166, 167}, \\tuple {315, 316, 317}, \\tuple {525, 526, 527}, \\dots$ {{expand|Related sequences: A161962 (superset), A161963}}"}
+{"_id": "14455", "title": "Basis of Vector Space Injects into Generator", "text": "Let $K$ be a division ring. Let $V$ be a vector space over $K$. Let $B$ be a basis of $V$. Let $G$ be a generator of $V$. Then there exists an injection from $B$ to $G$."}
+{"_id": "14456", "title": "Increasing Union of Ideals is Ideal/Sequence", "text": "Let $R$ be a ring. Let $S_0 \\subseteq S_1 \\subseteq S_2 \\subseteq \\dotsb \\subseteq S_i \\subseteq \\dotsb$ be ideals of $R$. Then the increasing union $S$: :$\\displaystyle S = \\bigcup_{i \\mathop \\in \\N} S_i$ is an ideal of $R$."}
+{"_id": "14457", "title": "Increasing Union of Ideals is Ideal/Chain", "text": "Let $R$ be a ring. Let $\\left({P, \\subseteq}\\right)$ be the ordered set consisting of all ideals of $R$, ordered by inclusion. Let $\\left\\{{I_\\alpha}\\right\\}_{\\alpha \\in A}$ be a non-empty chain of ideals in $P$. Let $\\displaystyle I = \\bigcup_{\\alpha \\in A} I_\\alpha$ be their union. Then $I$ is an ideal of $R$."}
+{"_id": "14458", "title": "Union of Chain of Proper Ideals is Proper Ideal", "text": "Let $R$ be a ring with unity. Let $\\struct {P, \\subseteq}$ be the ordered set consisting of all ideals of $R$, ordered by inclusion. Let $\\sequence {I_\\alpha}_{\\alpha \\mathop \\in A}$ be a non-empty chain of proper ideals in $P$. Let $\\displaystyle I = \\bigcup_{\\alpha \\mathop \\in A} I_\\alpha$ be their union. Then $I$ is a proper ideal of $R$."}
+{"_id": "14459", "title": "Dimension of Free Vector Space on Set", "text": "Let $k$ be a division ring. Let $X$ be a set. Let $k^{\\paren X}$ be the free vector space on $X$. The vector space $k^{\\paren X}$ has dimension the cardinality of $X$."}
+{"_id": "14461", "title": "239 is not Expressible as Sum of Fewer than 19 Fourth Powers", "text": ":$239 = 13 \\times 1^4 + 4 \\times 2^4 + 2 \\times 3^4$"}
+{"_id": "14462", "title": "479 is not Expressible as Sum of Fewer than 19 Fourth Powers", "text": ":$479 = 13 \\times 1^4 + 3 \\times 2^4 + 2 \\times 3^4 + 4^4$ or: :$479 = 10 \\times 1^4 + 4 \\times 2^4 + 5 \\times 3^4$"}
+{"_id": "14463", "title": "559 is not Expressible as Sum of Fewer than 19 Fourth Powers", "text": ":$559 = 15 \\times 1^4 + 2 \\times 2^4 + 2 \\times 4^4$ or: :$559 = 9 \\times 1^4 + 4 \\times 2^4 + 6 \\times 3^4$"}
+{"_id": "14465", "title": "Canonical Basis of Free Module on Set is Basis", "text": "Let $R$ be a ring with unity. Let $I$ be a set. Let $R^{(I)}$ be the free $R$-module on $I$. Let $B$ be its canonical basis. Then $B$ is a basis of $R^{(I)}$."}
+{"_id": "14466", "title": "Korselt's Theorem", "text": "Let $n \\ge 2$ be an integer. Then $n$ is a '''Carmichael number''' {{iff}}: : $(1): \\quad n$ is odd and the following conditions hold for every prime factor $p$ of $n$: : $(2): \\quad p^2 \\nmid n$ : $(3): \\quad \\paren {p - 1} \\divides \\paren {n - 1}$ where: :$\\divides$ denotes divisibility :$\\nmid$ denotes non-divisibility."}
+{"_id": "14468", "title": "Intersection of Submodules is Submodule", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $S$ be a set of submodules of $M$. Then the intersection $\\bigcap S$ is a submodule of $M$."}
+{"_id": "14469", "title": "There are Infinitely Many Carmichael Numbers", "text": "There are infinitely many Carmichael numbers."}
+{"_id": "14470", "title": "Product of Big-O Estimates/Sequences", "text": "Let $\\sequence {a_n}, \\sequence {b_n}, \\sequence {c_n}, \\sequence {d_n}$ be sequences of real or complex numbers. Let: :$a_n = \\map \\OO {b_n}$ :$c_n = \\map \\OO {d_n}$ where $\\OO$ denotes big-O notation. Then: :$a_n c_n = \\map \\OO {b_n d_n}$"}
+{"_id": "14471", "title": "Equivalence of Definitions of Order of Entire Function", "text": "Let $f: \\C \\to \\C$ be an entire function. Let $\\ln$ denote the natural logarithm. {{TFAE| def = Order of Entire Function}}"}
+{"_id": "14472", "title": "Little-O Times Big-O is Little-O/Sequences", "text": "Let $\\sequence {a_n}, \\sequence {b_n}, \\sequence {c_n}, \\sequence {d_n}$ be sequences of real or complex numbers. Let: :$a_n = \\map \\OO {b_n}$ :$c_n = \\map o {d_n}$ where: :$\\OO$ denotes big-O notation :$o$ denotes little-o notation. Then: :$a_n c_n = \\map o {b_n d_n}$"}
+{"_id": "14473", "title": "Field Norm of Complex Number Equals Field Norm", "text": "Let $z = a + i b$ be a complex number, where $a, b \\in \\R$. Then the field norm of $z$ is the field norm with respect to the field extension $\\C / \\R$."}
+{"_id": "14474", "title": "Big-O Notation for Sequences Coincides with General Definition", "text": "Let $\\left \\langle {a_n} \\right \\rangle$ and $\\left \\langle {b_n} \\right \\rangle$ be sequences of real or complex numbers. Let $\\N$ be given the discrete topology. {{TFAE}} :$(1): \\quad$ $a_n = O(b_n)$, where $O$ denotes big-O notation for sequences :$(2): \\quad$ $a_n = O(b_n)$, where $O$ stands for the general definition of big-O notation"}
+{"_id": "14476", "title": "Bounded iff Big-O of 1/Sequences", "text": "Let $\\sequence {a_n}$ be a sequence of real or complex numbers. {{TFAE}} :$(1): \\quad a_n$ is bounded :$(2): \\quad a_n = \\map \\OO 1$, where $\\OO$ denotes big-$\\OO$ notation"}
+{"_id": "14478", "title": "Sum of Little-O Estimates/Sequences", "text": "Let $\\sequence {a_n}, \\sequence {b_n}, \\sequence {c_n}, \\sequence {d_n}$ be sequences of real or complex numbers. Let: :$a_n = \\map o {b_n}$ :$c_n = \\map o {d_n}$ where $o$ denotes little-o notation. Then: :$a_n + c_n = \\map o {\\size {b_n} + \\size {d_n} }$"}
+{"_id": "14479", "title": "Equivalence of Definitions of Asymptotically Equal Sequences", "text": "Let $\\sequence {a_n}$ and $\\sequence {b_n}$ be sequences in $\\R$. {{TFAE|def = Asymptotically Equal Sequences}}"}
+{"_id": "14480", "title": "Largest Number not Expressible as Sum of Multiples of 23 and 28", "text": "The largest integer $n$ that cannot be expressed in the form: :$n = 23 x + 28 y$ for $x, y \\in \\Z_{>0}$ is $593$."}
+{"_id": "14481", "title": "Open implies There Exists Way Below Element", "text": "Let $L = \\left({S, \\preceq, \\tau}\\right)$ be a continuous topological lattice with Scott topology. Let $p \\in S, A \\subseteq S$ such that :$A$ is open and $p \\in A$. Then :$\\exists q \\in A: q \\ll p$"}
+{"_id": "14482", "title": "Smallest Fourth Power which is Sum of 5 Fourth Powers", "text": "$625$ is the smallest fourth power which is the sum of $5$ fourth powers: :$625 = 5^4 = 2^4 + 2^4 + 3^4 + 4^4 + 4^4$"}
+{"_id": "14483", "title": "Interior is Union of Way Above Closures", "text": "Let $\\left({S, \\preceq, \\tau}\\right)$ be a complete continuous topological lattice with Scott topology. Let $X \\subseteq S$. Then $X^\\circ = \\bigcup \\left\\{ {x^\\gg: x \\in S \\land x^\\gg \\subseteq X}\\right\\}$ where :$X^\\circ$ denotes the interior of $X$, :$x^\\gg$ denotes the way above closure of $x$."}
+{"_id": "14484", "title": "Way Above Closures Form Basis", "text": "Let $L = \\left({S, \\preceq, \\tau}\\right)$ be a complete continuous topological lattice with Scott topology. Then $\\left\\{ {x^\\gg: x \\in S}\\right\\}$ is basis of $L$."}
+{"_id": "14486", "title": "Record Gaps between Twin Primes", "text": "The gaps between the following pairs of twin primes are larger than those for all smaller pairs: {{begin-eqn}} {{eqn | l = \\tuple {3, 5}       | o = \\to       | r = \\tuple {5, 7}       | c = a gap of $0$ }} {{eqn | l = \\tuple {5, 7}       | o = \\to       | r = \\tuple {11, 13}       | c = a gap of $4$ }} {{eqn | l = \\tuple {17, 19}       | o = \\to       | r = \\tuple {29, 31}       | c = a gap of $10$ }} {{eqn | l = \\tuple {41, 43}       | o = \\to       | r = \\tuple {59, 61}       | c = a gap of $16$ }} {{eqn | l = \\tuple {69, 73}       | o = \\to       | r = \\tuple {101, 103}       | c = a gap of $28$ }} {{eqn | l = \\tuple {311, 313}       | o = \\to       | r = \\tuple {347, 349}       | c = a gap of $34$ }} {{eqn | l = \\tuple {347, 349}       | o = \\to       | r = \\tuple {419, 421}       | c = a gap of $70$ }} {{eqn | l = \\tuple {659, 661}       | o = \\to       | r = \\tuple {809, 811}       | c = a gap of $148$ }} {{end-eqn}} {{OEIS|A036061}} {{finish|Add this list to the Number Sequence}}"}
+{"_id": "14488", "title": "Sophie Germain Prime cannot be 6n+1", "text": "Let $p$ be a Sophie Germain prime. Then $p$ cannot be of the form $6 n + 1$, where $n$ is a positive integer."}
+{"_id": "14489", "title": "Way Above Closure is Open", "text": "Let $L = \\left({S, \\preceq, \\tau}\\right)$ be a complete continuous topological lattice with Scott topology. Let $x \\in S$. Then $x^\\gg$ is open where $x^\\gg$ denotes the way above closure of $x$."}
+{"_id": "14490", "title": "Way Above Closure is Upper", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $x \\in S$. Then $x^\\gg$ is upper where $x^\\gg$ denotes the way above closure of $x$."}
+{"_id": "14491", "title": "Way Above Closures that Way Below Form Local Basis", "text": "Let $L = \\left({S, \\preceq, \\tau}\\right)$ be a complete continuous topological lattice with Scott topology. Let $p \\in S$. Then $\\left\\{ {q^\\gg: q \\in S \\land q \\ll p}\\right\\}$ is local basis at $p$."}
+{"_id": "14492", "title": "Characterization of Analytic Basis by Local Bases", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $P$ be a set of subsets of $S$ such that :$P \\subseteq \\tau$ and :for all $p \\in S$: there exists local basis $B$ at $p: B \\subseteq P$ Then $P$ is basis of $T$."}
+{"_id": "14493", "title": "Approximation to Golden Ratio using 666", "text": "The Golden Ratio is approximated closely by the following formula: :$\\phi \\approx -2 \\sin \\left({666}\\right) = 1.61803 \\, 39887 \\, 5 \\ldots$"}
+{"_id": "14494", "title": "Euler Phi Function of 666 equals Product of Digits", "text": "The number $666$ has the following interesting property: :$\\map \\phi {666} = 6 \\times 6 \\times 6$ where $\\phi$ denotes the Euler $\\phi$ function."}
+{"_id": "14495", "title": "Sum of Sequence of Squares of Primes", "text": "Let $S = \\sequence {s_n}$ be the integer sequence defined as: :$\\displaystyle s_n = \\sum_{i \\mathop = 1}^n {p_i}^2$ where $P_i$ denotes the $i$th prime number. Then $S$ begins: :$4, 13, 38, 87, 208, 377, 666, 1027, 1556, 2397, 3358, 4727, 6408, 8257, 10466, \\ldots$ {{OEIS|A024450}}"}
+{"_id": "14496", "title": "Pair of Consecutive Powerful Numbers whose First is Odd", "text": "The only known pair of consecutive integers which are both powerful numbers such that the first of the pair is odd is: :$\\tuple {675, 676}$"}
+{"_id": "14497", "title": "Interior is Union of Elements of Basis", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $B$ be a basis of $T$. Let $V$ be a subset of $S$. Then $V^\\circ = \\bigcup \\left\\{ {G \\in B: G \\subseteq V}\\right\\}$ where $V^\\circ$ denotes the interior of $V$."}
+{"_id": "14498", "title": "Palindromic Squares with Non-Palindromic Roots", "text": "The sequence of palindromic squares with non-palindromic square roots begins: :$676, 69 \\, 696, 94 \\, 249, 698 \\, 896, 5 \\, 221 \\, 225, 6 \\, 948 \\, 496, 522 \\, 808 \\, 225, \\ldots$ This sequence is not explicitly given in {{OEISLink}}. The sequence of those corresponding non-palindromic square roots begins: :$26, 264, 307, 836, 2285, 2636, 22 \\, 865, 24 \\, 846, 30 \\, 693, \\ldots$ {{OEIS|A251673}}"}
+{"_id": "14499", "title": "Tetrahedral Numbers which are Sum of 2 Tetrahedral Numbers", "text": "The sequence of tetrahedral numbers which are the sum of two other tetrahedral numbers begins: :$680, \\ldots$ {{expand|To be expanded when we have some more.}}"}
+{"_id": "14500", "title": "Open Set is Union of Elements of Basis", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $B$ be a basis of $T$. Let $V$ be an open subset of $S$. Then $V = \\bigcup \\left\\{ {G \\in B: G \\subseteq V}\\right\\}$"}
+{"_id": "14502", "title": "Equivalence of Definitions of Compatible Atlases", "text": "Let $M$ be a topological space. Let $\\mathscr F, \\mathscr G$ be $d$-dimensional atlases of class $C^k$ on $M$. {{TFAE|def = Compatible Atlases}}"}
+{"_id": "14503", "title": "Atlas is Contained in Unique Maximal Atlas", "text": "Let $M$ be a topological space. Let $A$ be a $d$-dimensional atlas of class $C^k$. Then $A$ is contained in a unique maximal atlas of class $C^k$."}
+{"_id": "14504", "title": "Locally Euclidean iff has C0-Atlas", "text": "Let $M$ be a topological space. {{TFAE}} :$(1):\\quad$ $M$ is locally euclidean. :$(2):\\quad$ There exists a $C^0$-atlas on $M$."}
+{"_id": "14505", "title": "Differentiable Structure Contains Unique Maximal Atlas", "text": "Let $M$ be a topological space. Let $k$ and $d$ be natural numbers. Let $S$ be a $d$-dimensional differentiable structure of class $C^k$ on $M$. Then $S$ contains a unique maximal $C^k$-atlas."}
+{"_id": "14506", "title": "Consecutive Integers whose Product is Primorial", "text": "The following primorials can be expressed as the product of consecutive integers: :$2, 6, 30, 210, 510 \\, 510$ {{OEIS|A161620}} No others are known. The corresponding indices of those primorials are: :$2, 3, 5, 7, 17$ {{OEIS|A215658}} The corresponding values of $n$ such that $p\\# = \\paren {n - 1} n$ are: :$2, 3, 6, 15, 715$ {{OEIS|A215659}}"}
+{"_id": "14508", "title": "Integers whose Ratio between Sigma and Phi is Square", "text": "The sequence of integers whose $\\sigma$ value divided by its Euler $\\phi$ value is a square begins: :$1, 14, 30, 105, 248, 264, 418, 714, 1485, 3080, \\ldots$ {{OEIS|A293391}} {{finish|This sequence needs to be added to the individual integer pages.}}"}
+{"_id": "14509", "title": "Implicit Function Theorem/Real Functions", "text": "Let $n$ and $k$ be natural numbers. Let $\\Omega \\subset \\R^{n + k}$ be open. Let $f: \\Omega \\to \\R^k$ be continuous. Let the partial derivatives of $f$ with respect to $\\R^k$ be continuous. Let $\\tuple {a, b} \\in \\Omega$, with $a\\in \\R^n$ and $b\\in \\R^k$. Let $\\map f {a, b} = 0$. For $\\tuple {x_0, y_0} \\in \\Omega$, let $D_2 \\map f {x_0, y_0}$ denote the total derivative of the function $y \\mapsto \\map f {x_0, y}$ at $y_0$. Let the linear map $D_2 \\map f {a, b}$ be invertible. Then there exist neighborhoods $U \\subset \\Omega$ of $a$ and $V \\subset \\R^k$ of $b$ such that there exists a unique function $g: U \\to V$ such that $\\map f {x, \\map g x} = 0$ for all $x \\in U$. Moreover, $g$ is continuous."}
+{"_id": "14510", "title": "Uniform Contraction Mapping Theorem", "text": "Let $M$ and $N$ be metric spaces. Let $M$ be complete. Let $f : M \\times N \\to M$ be a continuous uniform contraction. Then for all $t \\in N$ there exists a unique $g \\left({t}\\right) \\in M$ such that $f(g \\left({t}\\right), t) = g \\left({t}\\right)$, and the mapping $g: N \\to M$ is continuous."}
+{"_id": "14512", "title": "Mapping at Element is Supremum of Compact Elements implies Mapping at Element is Supremum that Way Below", "text": "Let $\\left({S, \\vee_1, \\wedge_1, \\preceq_1}\\right)$ and $\\left({T, \\vee_2, \\wedge_2, \\preceq_2}\\right)$ be complete lattices. Let $f: S \\to T$ be a mapping such that :$\\forall x \\in S: f\\left({x}\\right) = \\sup \\left\\{ {f\\left({w}\\right): w \\in S \\land w \\preceq_1 x \\land w}\\right.$ is compact$\\left.{}\\right\\}$ Then :$\\forall x \\in S: f\\left({x}\\right) = \\sup \\left\\{ {f\\left({w}\\right): w \\in S \\land w \\ll x}\\right\\}$"}
+{"_id": "14513", "title": "Inverse Function Theorem for Real Functions", "text": "Let $n\\geq1$ and $k\\geq1$ be natural numbers. Let $\\Omega\\subset \\R^n$ be open. $f : \\Omega \\to \\R^n$ be a vector-valued function of class $C^k$. Let $a\\in\\Omega$. Let the differential $Df(a)$ of $f$ at $a$ be invertible. Then there exist open sets $U\\subset\\Omega$ and $V\\subset\\R^n$ such that the restriction of $f$ to $U$ is a $C^k$-diffeomorphism $f:U\\to V$."}
+{"_id": "14515", "title": "720 is Product of Consecutive Numbers in Two Ways", "text": ":$720 = 6 \\times 5 \\times 4 \\times 3 \\times 2 = 10 \\times 9 \\times 8$"}
+{"_id": "14516", "title": "Factorial which is Sum of Two Squares", "text": "The only factorial which can be expressed as the sum of two squares is: {{begin-eqn}} {{eqn | l = 6!       | r = 12^2 + 24^2       | c =  }} {{end-eqn}}"}
+{"_id": "14517", "title": "Multiplicity of 720 in 720 Factorial", "text": "The multiplicity of $720$ in $720!$ is $178$. That is: :$720^{178} \\divides 720!$ but: :$720^{179} \\nmid 720!$ where: :$720!$ denotes $720$ factorial :$\\divides$ denotes divisibility :$\\nmid$ denotes non-divisibility."}
+{"_id": "14518", "title": "Continuous iff Mapping at Element is Supremum", "text": "Let $\\left({S, \\preceq_1, \\tau_1}\\right)$ and $\\left({T, \\preceq_2, \\tau_2}\\right)$ be complete continuous topological lattices with Scott topologies. Let $f: S \\to T$ be a mapping. Then $f$ is continuous {{iff}} :$\\forall x \\in S: f\\left({x}\\right) = \\sup \\left\\{ {f\\left({w}\\right): w \\in S \\land w \\ll x}\\right\\}$"}
+{"_id": "14520", "title": "Equivalence of Definitions of Generated Submodule", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $S\\subset M$ be a subset. {{TFAE|def = Generated Submodule}}"}
+{"_id": "14521", "title": "Equivalence of Definitions of Basis of Vector Space", "text": "Let $K$ be a division ring. Let $\\struct {G, +_G, \\circ}_K$ be an vector space over $K$. {{TFAE| def = Basis of Vector Space}}"}
+{"_id": "14522", "title": "Equivalence of Definitions of Differentiable Real Function at Point", "text": "Let $\\openint a b \\subset \\R$ be an open interval. Let $\\xi$ be a point in $\\openint a b$. {{TFAE|def = Differentiable Real Function at Point|view = differentiable real function at a point}}"}
+{"_id": "14524", "title": "Solutions to Approximate Fermat Equation x^3 = y^3 + z^3 Plus or Minus 1", "text": "The approximate Fermat equation: :$x^3 = y^3 + z^3 \\pm 1$ has the solutions: {{begin-eqn}} {{eqn | l = 9^3       | r = 6^3 + 8^3 + 1 }} {{eqn | l = 103^3       | r = 64^3 + 94^3 - 1       | c =  }} {{end-eqn}}"}
+{"_id": "14525", "title": "Cubes which are Sum of Five Cubes", "text": "The following cube numbers can be expressed as the sum of $5$ positive cube numbers: :$9^3, \\ldots$ {{expand|More terms needed. It seems that: </br> $4$ and all numbers $> 8$ can be so expressed </br> only $4, 8, 10, 11, 13$ require repeated cubes}}"}
+{"_id": "14526", "title": "Period of Reciprocal of 729 is 81", "text": "The decimal expansion of the reciprocal of $729$ has $\\dfrac 1 9$ the maximum period, that is, $81$: :$\\dfrac 1 {729} = 0 \\cdotp \\dot 00137 \\, 17421 \\, 12482 \\, 85322 \\, 35939 \\, 64334 \\, 70507 \\, 54458 \\, 16186 \\, 55692 \\, 72976 \\, 68038 \\, 40877 \\, 91495 \\, 19890 \\, 26063 \\, \\dot 1$ The recurring part can be arranged in groups of $9$ digits each, revealing an interesting pattern: {{begin-eqn}} {{eqn | l = 001 \\, 371 \\, 742       | o =  }} {{eqn | l = 112 \\, 482 \\, 853       | o =  }} {{eqn | l = 223 \\, 593 \\, 964       | o =  }} {{eqn | l = 334 \\, 705 \\, 075       | o =  }} {{eqn | l = 445 \\, 816 \\, 186       | o =  }} {{eqn | l = 556 \\, 927 \\, 297       | o =  }} {{eqn | l = 668 \\, 038 \\, 408       | o =  }} {{eqn | l = 779 \\, 149 \\, 519       | o =  }} {{eqn | l = 890 \\, 260 \\, 631       | o =  }} {{end-eqn}} that is, each row (apart from the last) can be obtained from the previous one by adding $111 \\, 111 \\, 111$ to it. {{OEIS|A021733}}"}
+{"_id": "14527", "title": "Implicit Function Theorem for Differentiable Real Functions", "text": "Let $\\Omega \\subset \\R^{n+k}$ be open. Let $f : \\Omega \\to \\R^k$ be differentiable. Let the $i$th partial derivatives of $f$ be continuous in $\\Omega$ for $n+1 \\leq i \\leq n+k$. Let $(a,b) \\in \\Omega$, with $a\\in \\R^n$ and $b\\in \\R^k$. Let $f(a,b) = 0$. For $(x_0,y_0)\\in\\Omega$, let $D_2 f(x_0,y_0)$ denote the differential of the function $y\\mapsto f(x_0, y)$ at $y_0$. Let the linear map $D_2 f(a,b)$ be invertible. Then there exist neighborhoods $U\\subset\\Omega$ of $a$ and $V\\subset\\R^k$ of $b$ such that there exists a unique function $g : U \\to V$ such that $f(x, g(x)) = 0$ for all $x\\in U$. Moreover, $g$ is differentiable, and its differential satisfies: :$dg (x) = - \\left( (D_2f)(x, g(x)) \\right)^{-1} \\circ (D_1 f)(x, g(x))$ for all $x\\in U$."}
+{"_id": "14528", "title": "Implicit Function Theorem for Smooth Real Functions", "text": "Let $\\Omega \\subset \\R^{n+k}$ be open. Let $f : \\Omega \\to \\R^k$ be smooth. Let $(a,b) \\in \\Omega$, with $a\\in \\R^n$ and $b\\in \\R^k$. Let $f(a,b) = 0$. For $(x_0,y_0)\\in\\Omega$, let $D_2 f(x_0,y_0)$ denote the differential of the function $y\\mapsto f(x_0, y)$ at $y_0$. Let the linear map $D_2 f(a,b)$ be invertible. Then there exist neighborhoods $U\\subset\\Omega$ of $a$ and $V\\subset\\R^k$ of $b$ such that there exists a unique function $g : U \\to V$ such that $f(x, g(x)) = 0$ for all $x\\in U$. Moreover, $g$ is smooth, and its differential satisfies: :$dg (x) = - \\left( (D_2f)(x, g(x)) \\right)^{-1} \\circ (D_1 f)(x, g(x))$ for all $x\\in U$."}
+{"_id": "14529", "title": "Sum of 4 Consecutive Binomial Coefficients forming Square", "text": "Consider the Diophantine equation: :$\\dbinom n 0 + \\dbinom n 1 + \\dbinom n 2 + \\dbinom n 3 = m^2$ where: :$\\dbinom a b$ denotes a binomial coefficient :$n$ is an integer :$m$ is a non-negative integer. Then $n$ has one of the following values: :$-1, 0, 2, 7, 15, 74, 767$ {{OEIS|A047694}} The corresponding values of $m$ are: :$0, 1, 2, 8, 24, 260, 8672$ {{OEIS|A047695}}"}
+{"_id": "14530", "title": "Implicit Function Theorem for Lipschitz Contractions", "text": "Let $M$ and $N$ be metric spaces. Let $M$ be complete. Let $f : M \\times N \\to M$ be a Lipschitz continuous uniform contraction. Then for all $t\\in N$ there exists a unique $g(t) \\in M$ such that $f(g(t), t) = g(t)$, and the mapping $g : N \\to M$ is Lipschitz continuous."}
+{"_id": "14532", "title": "Local Normal Form for Immersions", "text": "Let $\\Omega\\subset\\R^k$ be open. Let $f: \\Omega \\to \\R^n$ be an immersion. Let $p \\in \\Omega$. Then: :$k \\le n$ and there exists a local diffeomorphism $\\phi$ around $\\map f p$ such that: :$\\phi \\circ \\map f x = \\tuple {x, 0}$ for all $x$ in a neighborhood of $p$."}
+{"_id": "14533", "title": "Local Normal Form for Submersions", "text": "Let $\\Omega\\subset\\R^n$ be open. Let $f : \\Omega \\to \\R^k$ be an submersion. Let $p\\in\\Omega$. Then $n\\geq k$, and there exists a local diffeomorphism $\\phi$ around $f(p)$ such that :$\\phi\\circ f (x, y) = x$ for all $(x, y)$ in a neighborhood of $p$."}
+{"_id": "14534", "title": "Smallest Square Inscribed in Two Pythagorean Triangles", "text": "The smallest square with integer sides that can be inscribed within two different Pythagorean triangles so that one side of the square lies on the hypotenuse has side length $780$. The two Pythagorean triangles in question have side lengths $\\tuple {1443, 1924, 2405}$ and $\\tuple {1145, 2748, 2977}$."}
+{"_id": "14535", "title": "Sequence of Numbers Divisible by Sequence of Primes", "text": "The integers in this sequence: :$788, 789, 790, 791, 792, 793$ are divisible by: :$2, 3, 5, 7, 11, 13$ respectively."}
+{"_id": "14536", "title": "Numbers whose Square is Palindromic with Even Number of Digits", "text": "The sequence of positive integers whose square is a palindromic number with an even number of digits begins: :$836, 798 \\, 644, 64 \\, 030 \\, 648, 83 \\, 163 \\, 115 \\, 486, 6 \\, 360 \\, 832 \\, 925 \\, 898, \\ldots$ {{OEIS|A016113}}"}
+{"_id": "14537", "title": "Sum of Sequence of Factorials", "text": "The sequence $S = \\sequence {s_n}$ defined as: :$\\displaystyle s_n = \\sum_{k \\mathop = 1}^n k!$ begins: :$1, 3, 9, 33, 153, 873, 5913, 46 \\, 233, 409 \\, 113, 4 \\, 037 \\, 913, \\ldots$ {{OEIS|A007489}}"}
+{"_id": "14538", "title": "Number of Magic Squares of Order 4", "text": "The number of different magic squares of order $4$, up to rotation and reflection, is $880$."}
+{"_id": "14539", "title": "Primitive Semiperfect Numbers which are not Primitive Abundant", "text": "The sequence of primitive semiperfect numbers which are not also primitive abundant starts: :$6, 28, 350, 490, 496, 770, 910, 1190, \\ldots$ These are semiperfect numbers which are either: : perfect or: : whose only abundant aliquot parts are weird."}
+{"_id": "14540", "title": "Sequence of Odd Abundant Numbers", "text": "The sequence of odd abundant numbers begins: :$945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, \\ldots$ {{OEIS|A005231}}"}
+{"_id": "14541", "title": "Set of 5 Triplets whose Sums and Products are Equal", "text": "The following set of $5$ triplets of integers have the property that: :the sum of the integers in each triplet are equal and: :the product of the integers in each triplet are equal: :$\\tuple {6, 480, 495}$, $\\tuple {11, 160, 810}$, $\\tuple {12, 144, 825}$, $\\tuple {20, 81, 880}$, $\\tuple {33, 48, 900}$ The sum is $981$, and the product is $1 \\, 425 \\, 600$. This is the only known such set of $5$ triplets of integers with this property."}
+{"_id": "14543", "title": "Divisibility Test for 7, 11 and 13", "text": "Mark off the integer $N$ being tested into groups of $3$ digits. Because of the standard way of presenting integers, this may already be done, for example: :$N = 22 \\, 846 \\, 293 \\, 462 \\, 733 \\, 356$ Number the groups of $3$ from the right:  :$N = \\underbrace{22}_6 \\, \\underbrace{846}_5 \\, \\underbrace{293}_4 \\, \\underbrace{462}_3 \\, \\underbrace{733}_2 \\, \\underbrace{356}_1$ Considering each group a $3$-digit integer, add the even numbered groups together, and subtract the odd numbered groups: :$22 - 846 + 293 - 462 + 733 - 356 = -616$ where the sign is irrelevant. If the result is divisible by $7$, $11$ or $13$, then so is $N$. In this case: :$616 = 2^3 \\times 7 \\times 11$ and so $N$ is divisible by $7$ and $11$ but not $13$."}
+{"_id": "14544", "title": "Solutions to Diophantine Equation x (x + 1) = y (y + 5) (y + 10) (y + 15)", "text": "The Diophantine equation : $n = x \\left({x + 1}\\right) = y \\left({y + 5}\\right) \\left({y + 10}\\right) \\left({y + 15}\\right)$ has exactly $2$ solutions: {{begin-eqn}} {{eqn | l = 1056       | r = 32 \\times 33 = 1 \\times 6 \\times 11 \\times 16 }} {{eqn | l = 43 \\, 056       | r = 207 \\times 208 = 8 \\times 13 \\times 18 \\times 23 }} {{end-eqn}}"}
+{"_id": "14545", "title": "Numbers Reversed when Multiplying by 9", "text": "Numbers of the form $\\sqbrk {10 (9) 89}_{10}$ are reversed when they are multiplied by $9$: {{begin-eqn}} {{eqn | l = 1089 \\times 9       | r = 9801 }} {{eqn | l = 10 \\, 989 \\times 9       | r = 98 \\, 901 }} {{eqn | l = 109 \\, 989 \\times 9       | r = 989 \\, 901 }} {{end-eqn}} and so on."}
+{"_id": "14547", "title": "Square which is Difference between Square and Square of Reversal", "text": "$33^2 = 65^2 - 56^2$ This is the only square of a $2$-digit number which has this property."}
+{"_id": "14548", "title": "Numbers for which Sixth Power plus 1091 is Composite", "text": "The number $1091$ has the property that: :$x^6 + 1091$ is composite for all integer values of $x$ from $1$ to $3905$."}
+{"_id": "14549", "title": "Wieferich's Criterion", "text": "Suppose Fermat's equation: :$x^p + y^p = z^p$ has a solution in which $p$ is an odd prime that does not divide any of $x$, $y$ or $z$. Then $2^{p - 1} - 1$ is divisible by $p^2$."}
+{"_id": "14550", "title": "1105 as Sum of Two Squares", "text": "$1105$ can be expressed as the sum of two squares in more ways than any smaller integer: {{begin-eqn}} {{eqn | l = 1105       | m = 1089 + 16       | mo= =       | r = 33^2 + 4^2       | c =  }} {{eqn | m = 1024 + 81       | mo= =       | r = 32^2 + 9^2       | c =  }} {{eqn | m = 961 + 144       | mo= =       | r = 31^2 + 12^2       | c =  }} {{eqn | m = 625 + 529       | mo= =       | r = 24^2 + 23^2       | c =  }} {{end-eqn}}"}
+{"_id": "14551", "title": "Integer as Difference between Two Squares", "text": "Let $n$ be a positive integer. Then $n$ can be expressed as: :$n = a^2 - b^2$ {{iff}} $n$ has at least two distinct divisors of the same parity."}
+{"_id": "14552", "title": "Difference between Two Squares equal to Repunit/Corollary 1", "text": "{{begin-eqn}} {{eqn | l = 6^2 - 5^2       | r = 11       | c =  }} {{eqn | l = 56^2 - 45^2       | r = 1111       | c =  }} {{eqn | l = 556^2 - 445^2       | r = 111 \\, 111       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}} and in general for integer $n$: :$R_{2 n} = {\\underbrace {55 \\ldots 56}_{n - 1 \\ 5 \\text{'s} } }^2 - {\\underbrace {44 \\ldots 45}_{n - 1 \\ 4 \\text{'s} } }^2$ that is: :$\\displaystyle \\sum_{k \\mathop = 0}^{2 n - 1} 10^k = \\paren {\\sum_{k \\mathop = 1}^{n - 1} 5 \\times 10^k + 6}^2 - \\paren {\\sum_{k \\mathop = 1}^{n - 1} 4 \\times 10^k + 5}^2$"}
+{"_id": "14553", "title": "Difference between Two Squares equal to Repunit/Corollary 2", "text": "{{begin-eqn}} {{eqn | l = 6^2 - 5^2       | r = 11       | c =  }} {{eqn | l = 56^2 - 45^2       | r = 1111       | c =  }} {{eqn | l = 5056^2 - 5045^2       | r = 111 \\, 111       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}} and in general for integer $n$: :$R_{2 n} = {\\underbrace{5050 \\ldots 56}_{n - 1 \\ 5 \\text{'s} } }^2 - {\\underbrace{5050 \\ldots 45}_{n - 1 \\ 5 \\text{'s} } }^2$ that is: :$\\displaystyle \\sum_{k \\mathop = 0}^{2 n - 1} 10^k = \\left({\\sum_{k \\mathop = 1}^{n - 1} 5 \\times 10^{2 k - 1} + 6}\\right)^2 - \\left({\\sum_{k \\mathop = 1}^{n - 1} 5 \\times 10^{2 k - 1} - 5}\\right)^2$"}
+{"_id": "14554", "title": "Numbers whose Squares are Consecutive Odd or Even Integers Juxtaposed", "text": "Integers whose squares consists of $2$ consecutive odd or even integers juxtaposed include: :$1127^2 = 01 \\, 270 \\, 129$ :$8874^2 = 78 \\, 747 \\, 876$ Such integers come in pairs which add to $1$ more than a power of $10$: :$1127 + 8874 = 10 \\, 001$ {{expand|A great deal more to be done here}}"}
+{"_id": "14555", "title": "Sixth Power as Sum of 7 Sixth Powers", "text": "The smallest known integer whose $6$th power can be expressed as the sum of $7$ smaller $6$th powers is $1141$: :$1141^6 = 74^6 + 234^6 + 402^6 + 474^6 + 702^6 + 894^6 + 1077^6$"}
+{"_id": "14556", "title": "Numbers Not Expressible as Sum of no more than 5 Squares of Composite Numbers", "text": "There are $256$ integers which cannot be expressed as the sum of no more than $5$ squares of composite numbers: :$1, 2, 3, \\ldots, 1167$ {{finish}}"}
+{"_id": "14558", "title": "Product of Injective Spaces is Injective", "text": "Let $I$ be a non-empty set. Let $\\left({\\left({S_i, \\tau_i}\\right)}\\right)_{i \\in I}$ be an indexed family of injective topological spaces. Then $\\displaystyle \\prod_{i \\mathop \\in I} \\left({S_i, \\tau_i}\\right)$ is injective space."}
+{"_id": "14559", "title": "Retract of Injective Space is Injective", "text": "Let $T = \\left({S, \\tau}\\right)$ be an injective topological space. Let $R = \\left({Z, \\tau'}\\right)$ be a retract of $T$. Then $R$ is injective."}
+{"_id": "14560", "title": "Numbers equal to Sum of Squares of two Parts", "text": "Integers that can be split into two parts whose squares add up to it include: :$1233 = 12^2 + 33^2$ :$8833 = 88^2 + 33^2$ {{expand|Need to establish the parameters of this}}"}
+{"_id": "14561", "title": "Triples of Consecutive Sphenic Numbers", "text": "The sequence of triplets of consecutive sphenic numbers starts: :$\\tuple {1309, 1310, 1311}, \\tuple {1885, 1886, 1887}, \\tuple {2013, 2014, 2015}, \\ldots$ {{OEIS|A066509|order = first}} {{OEIS|A248202|order = middle}}"}
+{"_id": "14564", "title": "Smallest Quadruplet of Consecutive Integers Divisible by Cube", "text": "The smallest sequence of quadruplets of consecutive integers each of which is divisible by a cube greater than $1$ is: :$\\tuple {22 \\, 624, 22 \\, 625, 22 \\, 626, 22 \\, 627}$"}
+{"_id": "14565", "title": "Riemann Zeta Function as a Multiple Integral", "text": "For $n \\in \\Z_{> 0}$, the Riemann zeta function is given by: :$\\displaystyle \\map \\zeta n = \\int_{\\closedint 0 1^n} \\frac 1 {1 - \\prod_{i \\mathop = 1}^n x_i} \\prod_{i \\mathop = 1}^n \\rd x_i$ where $\\closedint 0 1^n$ denotes the Cartesian $n$th power of the closed real interval $\\closedint 0 1$."}
+{"_id": "14568", "title": "Closed Form for Hexagonal Pyramidal Numbers", "text": "The closed-form expression for the $n$th hexagonal pyramidal number is: :$S_n = \\dfrac {n \\paren {n + 1} \\paren {4 n - 1} } 6$"}
+{"_id": "14569", "title": "Tetrahedral and Triangular Numbers", "text": "The only positive integers which are simultaneously tetrahedral and triangular are: :$1, 10, 120, 1540, 7140$"}
+{"_id": "14570", "title": "Restriction of Composition is Composition of Restriction", "text": "Let $X, Y, Z$ be sets. Let $f: X \\to Y$ and $g: Y \\to Z$ be mappings. Let $S \\subseteq X$. Then: : $\\left({g \\circ f}\\right) \\restriction S = g \\circ \\left({f \\restriction S}\\right)$"}
+{"_id": "14575", "title": "Smallest Fermat Pseudoprime to Bases 2, 3 and 5", "text": "The smallest Fermat pseudoprime to bases $2$, $3$ and $5$ is $1729$."}
+{"_id": "14578", "title": "Numbers that Factorise into Sum of Digits and Reversal", "text": "The following positive integers can each be expressed as the product of the sum of its digits and the reversal of the sum of its digits: :$1, 81, 1458, 1729$ {{OEIS|A110921}}"}
+{"_id": "14579", "title": "1782 is 3 Times Sum of all 2-Digit Numbers from its Digits", "text": "$1782$ equals $3$ multiplied by the sum of all the $2$-digit integers that can be formed from its digits."}
+{"_id": "14580", "title": "Triple of Consecutive Happy Numbers", "text": "The smallest triple of consecutive integers all of which are happy is: :$\\left({1880, 1881, 1882}\\right)$"}
+{"_id": "14581", "title": "Numbers whose Digits are Unchanged when Subtracting Reversal", "text": "The following sequence consists of the integers which have the property that subtraction of their reversals results in anagrams of them: :$954, 1980, 2961, 3870, 5823, 7641, 9108, 19980, 29880, 29961, 32760, \\ldots$ {{OEIS|A121969}}"}
+{"_id": "14585", "title": "17 Consecutive Integers each with Common Factor with Product of other 16", "text": "The $17$ consecutive integers from $2184$ to $2200$ have the property that each one is not coprime with the product of the other $16$."}
+{"_id": "14589", "title": "Lower and Upper Bounds for Sequences/Warning", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let $x_n \\to l$ as $n \\to \\infty$. Then it is '''not''' the case that: :$(1): \\quad \\forall n \\in \\N: x_n > a \\implies l > a$ :$(2): \\quad \\forall n \\in \\N: x_n < b \\implies l < b$"}
+{"_id": "14591", "title": "Smallest Fourth Power as Sum and Difference of Fourth Powers", "text": "The smallest $4$th power that can be expressed as the sum of $2$ $4$th powers minus a $3$rd is: :$2401 = 7^4 = 227^4 + 157^4 - 239^4$ with all numbers less than $10^4$."}
+{"_id": "14592", "title": "Zsigmondy's Theorem for Sums", "text": "Let $a > b > 0$ be coprime positive integers. Let $n \\ge 1$ be a (strictly) positive integer. Then there is a prime number $p$ such that :$p$ divides $a^n + b^n$ :$p$ does not divide $a^k + b^k$ for all $k < n$ with the following exception: :$n = 3$, $a = 2$, $b = 1$"}
+{"_id": "14593", "title": "Cyclotomic Polynomial of Index times Prime Power", "text": "Let $n, k \\ge 1$ be natural numbers. Let $p$ be a prime number. Let $\\Phi_n$ denote the $n$th cyclotomic polynomial. Then $\\map {\\Phi_{p^k n}} x = \\begin{cases} \\map {\\Phi_n} {x^{p^k}} & \\text{if } p \\divides n\\\\ \\dfrac {\\map {\\Phi_n} {x^{p^k}}} {\\map {\\Phi_n} {x^{p^{k - 1}}}} & \\text{if } p \\nmid n \\end{cases}$"}
+{"_id": "14594", "title": "Multiplicative Order of Roots of Cyclotomic Polynomial Modulo Prime", "text": "Let $n\\geq1$ be a natural number. Let $p$ be a prime number Let $n=p^\\alpha q$ where $\\alpha = \\nu_p(n)$ is the valuation of $p$ in $n$. Let $a\\in\\Z$ with $\\Phi_n(a)\\equiv0\\pmod p$. Then the order of $a$ modulo $p$ is $q$: :$\\operatorname{ord}_p(a) = q$."}
+{"_id": "14595", "title": "Lower Topology is Unique", "text": "Let $T_1 = \\left({S, \\preceq, \\tau_1}\\right)$ and $T_2 = \\left({S, \\preceq, \\tau_2}\\right)$ be relational structures with lower topologies. Then: : $\\tau_1 = \\tau_2$"}
+{"_id": "14596", "title": "Homogeneous Cyclotomic Polynomial is Symmetric", "text": "Let $n>1$ be a natural number. Let $\\Phi_n(x,y)$ be the $n$th homogeneous cyclotomic polynomial. Then $\\Phi_n(x,y) = \\Phi_n(y,x)$, that is, $\\Phi_n(x,y)$ is symmetric."}
+{"_id": "14597", "title": "Cyclotomic Polynomial of Index Power of Two", "text": "Let $n \\ge 1$ be a natural number. Then the $2^n$th cyclotomic polynomial is: :$\\map {\\Phi_{2^n} } x = x^{2^{n - 1} } + 1$"}
+{"_id": "14598", "title": "Trivial Estimate for Cyclotomic Polynomials", "text": "Let $n \\ge 1$ be a natural number. Let $\\Phi_n$ be the $n$th cyclotomic polynomial. Let $\\phi$ be the Euler totient function. Let $z \\in \\C$ be a complex number. Then: :$\\size {\\size z - 1}^{\\map \\phi n} \\le \\size {\\map {\\Phi_n} z} \\le \\paren {\\size z + 1}^{\\map \\phi n}$ where: :the first inequality becomes an equality only if: :::$n = 1$ and $z \\in \\R_{\\ge 0}$ ::or: :::$n = 2$ and $z \\in \\R_{\\le 0}$ :the second inequality becomes an equality only if: :::$n = 1$ and $z \\in \\R_{\\le 0}$ ::or: :::$n = 2$ and $z \\in \\R_{\\ge 0}$"}
+{"_id": "14600", "title": "Complete List of Special Highly Composite Numbers", "text": "There are exactly $6$ special highly composite numbers: :$1, 2, 6, 12, 60, 2520$ {{OEIS|A106037}}"}
+{"_id": "14602", "title": "Prime Decomposition of Highly Composite Number", "text": "Let $n$ be a highly composite number. Let the prime decomposition of $n$ be expressed as: :$n = \\displaystyle \\prod_{k \\mathop \\in \\N} {p_k}^{r_k}$ where $p_k$ denotes the $k$th prime. Then the sequence $\\left\\langle{r_k}\\right\\rangle$ is decreasing. That is: :$\\forall k \\in \\N: r_k \\ge r_{k + 1}$"}
+{"_id": "14606", "title": "Complement of Upper Closure of Element is Open in Lower Topology", "text": "Let $T = \\left({S, \\preceq, \\tau}\\right)$ be a relational structure with lower topology. Let $x \\in S$. Then $\\complement_S\\left({x^\\succeq}\\right)$ is open and $x^\\succeq$ is closed."}
+{"_id": "14607", "title": "Open Subset is Lower in Lower Topology", "text": "Let $T = \\left({S, \\preceq, \\tau}\\right)$ be a transitive relational structure with lower topology. Let $A \\subseteq S$ such that :$A$ is open. Then $A$ is lower."}
+{"_id": "14608", "title": "Ratio between Consecutive Highly Composite Numbers Greater than 2520 is Less than 2", "text": "The ratio between $2$ consecutive highly composite numbers both greater than $2520$ is less than $2$."}
+{"_id": "14609", "title": "Modified Kaprekar Process on 4-Digit Number terminates in 2538", "text": "Let $n$ be a $4$-digit number. Let $n$ be operated on by the modified Kaprekar process. The eventual result is always $2538$."}
+{"_id": "14610", "title": "Fibonacci Number equal to Sum of Sequence of Cubes", "text": "The following Fibonacci number can be expressed as the sum of a sequence of cubes: :$F_{18} = 2584 = 7^3 + 8^3 + 9^3 + 10^3$ {{expand|Any more?}}"}
+{"_id": "14611", "title": "Dudeney's Property of 2592", "text": ":$2592 = 2^5 \\times 9^2$ It is the only number $n$ that has the property that: :$n = \\sqbrk {abcd} = a^b \\times c^d$ where $\\sqbrk {abcd}$ denotes the decimal representation of $n$."}
+{"_id": "14615", "title": "2601 as Sum of 3 Squares in 12 Different Ways", "text": "$2601$ can be expressed as the sum of $3$ squares in $12$ different ways."}
+{"_id": "14616", "title": "Reversal of Number Multiplied by 11", "text": "Let $n \\in \\N$ be a number for which, when written in decimal notation, no two adjacent digits total to more than $9$. Let $n'$ denote the reversal of $n$. Then $n \\times 11$ is the reversal of $n' \\times 11$."}
+{"_id": "14617", "title": "Reduction Formula for Integral of Power of Tangent", "text": "For all $n \\in \\Z_{> 1}$:   :$\\displaystyle \\int \\map {\\tan^n} x \\rd x = \\frac {\\map {\\tan^{n - 1} } x} {n - 1} - \\int \\map {\\tan^{n - 2} } x \\rd x$"}
+{"_id": "14620", "title": "Numbers Appearing 8 Times in Pascal's Triangle", "text": "The number $3003$ is the smallest integer to appear $8$ times in Pascal's triangle. No other number below $2^{23}$ appears as often."}
+{"_id": "14624", "title": "3367 Multiplied by 2-Digit Number", "text": "In order to multiply $3367$ by a $2$-digit integer $\\sqbrk {xy}$: :divide the $6$-digit integer $\\sqbrk {xyxyxy}$ by $3$."}
+{"_id": "14626", "title": "Integer as Sum of 2 Cubes in 3 Ways", "text": "$4104$ is the smallest natural number which can be expressed as the sum of $2$ cubes in $3$ different ways: {{begin-eqn}} {{eqn | l = 4104       | r = 16^3 + 2^3       | c =  }} {{eqn | r = 15^3 + 9^3       | c =  }} {{eqn | r = \\paren {-12}^3 + 18^3       | c =  }} {{end-eqn}}"}
+{"_id": "14627", "title": "Composite Fibonacci Numbers with Prime Index", "text": "The sequence of composite Fibonacci numbers with a prime index begins: :$4181, 1 \\, 346 \\, 269, 24 \\, 157 \\, 817, 165 \\, 580 \\, 141, \\ldots$ {{OEIS|A050937}} The corresponding sequence of prime indices begins: :$19, 31, 37, 41, 53, 59, 61, 67, 71, 73, 79, \\ldots$ {{OEIS|A038672}}"}
+{"_id": "14632", "title": "Closed Subset is Upper in Lower Topology", "text": "Let $T = \\left({S, \\preceq, \\tau}\\right)$ be a transitive relational structure with lower topology. Let $A \\subseteq S$ such that :$A$ is closed. Then $A$ is upper."}
+{"_id": "14642", "title": "Square Pyramidal Number also Square", "text": "$4900$ is the only square pyramidal number which is also square: :$4900 = 70^2 = \\displaystyle \\sum_{k \\mathop = 1}^{24} k^2 = \\dfrac {24 \\paren {24 + 1} \\paren {2 \\times 24 + 1} } 6$"}
+{"_id": "14643", "title": "Mapping Preserves Non-Empty Infima implies Mapping is Continuous in Lower Topological Lattice", "text": "Let $T = \\left({S, \\preceq, \\tau}\\right)$ and $Q = \\left({X, \\preceq', \\tau'}\\right)$ be complete topological lattices with lower topologies. Let $f: S \\to X$ be a mapping such that :for all non-empty subsets $Y$ of $S$: $f$ preserves the infimum of $Y$. Then $f$ is continuous mapping."}
+{"_id": "14650", "title": "5040 is Product of Consecutive Numbers in Two Ways", "text": ":$5040 = 7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 = 10 \\times 9 \\times 8 \\times 7$"}
+{"_id": "14651", "title": "Products of Consecutive Integers in 2 Ways", "text": "The following integers are the product of consecutive integers in $2$ ways: :$-720, 720, 5040$"}
+{"_id": "14653", "title": "8 Mutually Non-Attacking Rooks on Chessboard", "text": "On a standard chessboard, it is possible to arrange a maximum of $8$ rooks so that no rook is attacking any other rook. There are $5282$ such arrangements, up to rotation and reflection."}
+{"_id": "14654", "title": "Triangular Lucas Numbers", "text": "The only Lucas numbers which are also triangular are: :$1, 3, 5778$ {{OEIS|A248506}}"}
+{"_id": "14662", "title": "Subspace of Subspace is Subspace", "text": "Let $T = \\struct{S, \\tau}$ be a topological space. Let $H \\subseteq S$ and $\\tau_H$ be the subspace topology on $H$. Let $K\\subseteq H$. Then the subspace topology on $K$ induced by $\\tau$ equals the subspace topology on $K$ induced by $\\tau_H$."}
+{"_id": "14663", "title": "Positive Integer Sum of 3 Fourth Powers in 2 Ways", "text": "The smallest positive integer which can be expressed as the sum of $3$ fourth powers in $2$ different ways is $6578$: {{begin-eqn}} {{eqn | l = 1^4 + 2^4 + 9^4       | r = 1 + 16 + 6561       | c =  }} {{eqn | r = 6578       | c =  }} {{eqn | r = 81 + 2401 + 4096       | c =  }} {{eqn | r = 3^4 + 7^4 + 8^4       | c =  }} {{end-eqn}}"}
+{"_id": "14666", "title": "Mapping Preserves Infima implies Mapping is Continuous in Lower Topological Lattice", "text": "Let $T = \\struct {S, \\preceq, \\tau}$ and $Q = \\struct {X, \\preceq', \\tau'}$ be complete topological lattices with lower topologies. Let $f: S \\to X$ be a mapping such that :$f$ preserves all infima. Then $f$ is continuous mapping."}
+{"_id": "14669", "title": "Compact in Subspace is Compact in Topological Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $K \\subseteq S$ be a subset. Let $\\tau_K$ be the subspace topology on $K$. Let $T' = \\struct {K, \\tau_K}$ be the topological subspace of $T$ determined by $K$. Let $H \\subseteq K$ be compact in $T'$. Then $H$ is compact in $T$."}
+{"_id": "14670", "title": "If Infimum of Filtered Subset belongs to Element of Sub-Basis then Subset and Element Intersect implies Infimum of Subset belongs to Closure of Subset", "text": "Let $T = \\left({S, \\preceq, \\tau}\\right)$ be a complete topological lattice with lower topology. Let $B$ be an analytic sub-basis of $T$. Let $F$ be a filtered subset of $S$ such that :$\\forall A \\in B: \\inf F \\in A \\implies F \\cap A \\ne \\varnothing$ Then $\\inf F \\in F^-$ where $F^-$ denotes the topological closure of $F$."}
+{"_id": "14674", "title": "Product with Repdigit can be Split into Parts which Add to Repdigit", "text": "Let $n$ be a positive integer with $d_1$ digits. Let $m$ be a repdigit number with $d_2$ digits such that $d_2 > d_1$. Let $r$ consist of the result when the rightmost $d_2$ digits of $m n$ is cut off and added to the remaining left hand portion. Then $r$ is a repdigit number."}
+{"_id": "14675", "title": "Sum of Reciprocals of Squares of Odd Integers as Double Integral", "text": ":$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\frac 1 {\\paren {2 n - 1}^2} = \\int_0^1 \\int_0^1 \\frac 1 {1 - x^2 y^2} \\rd x \\rd y$"}
+{"_id": "14677", "title": "Three Tri-Automorphic Numbers for each Number of Digits", "text": "Let $d \\in \\Z_{>0}$ be a (strictly) positive integer. Then there exist exactly $3$ tri-automorphic numbers with exactly $d$ digits. These tri-automorphic numbers all end in $2$, $5$ or $7$."}
+{"_id": "14680", "title": "Fourier Series/x squared over Minus Pi to Pi", "text": ":$\\displaystyle x^2 = \\frac {\\pi^2} 3 + \\sum_{n \\mathop = 1}^\\infty \\paren {\\paren {-1}^n \\frac 4 {n^2} \\cos n x}$"}
+{"_id": "14682", "title": "Integer and its Double forming Pandigital Pair", "text": "$6729$ and its double contain all the digits from $1$ to $9$ between them."}
+{"_id": "14683", "title": "4-Digit Numbers forming Longest Reverse-and-Add Sequence", "text": "Let $m \\in \\Z_{>0}$ be a positive integer expressed in decimal notation. Let $r \\left({m}\\right)$ be the reverse-and-add process on $m$. Let $r$ be applied iteratively to $m$. The $4$-digit integers $m$ which need the largest number of iterations before reaching a palindromic number are: :$6999, 7998, 8997, 9996$ all of which need $20$ iterations."}
+{"_id": "14684", "title": "Two Consecutive Integers each Product of Four Distinct Primes", "text": "The sequence of pairs of consecutive positive integers which are each the product of exactly $4$ distinct prime numbers begins: :$\\tuple {7314, 7315}, \\tuple {8294, 8295}, \\tuple {8645, 8646}, \\tuple {9009, 9010}, \\ldots$ {{OEIS|A140078|order = first}}"}
+{"_id": "14685", "title": "Fourier Series/Identity Function over Minus Pi to Pi", "text": "For $x \\in \\openint {-\\pi} \\pi$: :$\\displaystyle x = 2 \\sum_{n \\mathop = 1}^\\infty \\frac {\\paren {-1}^{n + 1} } n \\sin n x$"}
+{"_id": "14686", "title": "Fourier Series/Fourth Power of x over Minus Pi to Pi", "text": ":$\\displaystyle x^4 = \\frac {\\pi^4} 5 + \\sum_{n \\mathop = 1}^\\infty \\frac {8 n^2 \\pi^2 - 48} {n^4} \\cos n \\pi \\cos n x$"}
+{"_id": "14689", "title": "Fourier Series/Absolute Value of x over Minus Pi to Pi", "text": "For $x \\in \\openint {-\\pi} \\pi$: :$\\displaystyle \\size x = \\frac \\pi 2 - \\frac 4 \\pi \\sum_{n \\mathop = 1}^\\infty \\frac {\\map \\cos {2 n - 1} x} {\\paren {2 n - 1}^2}$"}
+{"_id": "14690", "title": "Fourier Series/Pi minus x over 0 to 2 Pi", "text": ":$\\displaystyle \\pi - x = 2 \\sum_{n \\mathop = 1}^\\infty \\frac {\\sin n x} n$"}
+{"_id": "14693", "title": "Mapping is Continuous implies Mapping Preserves Filtered Infima in Lower Topological Lattice", "text": "Let $T = \\left({S, \\preceq, \\tau}\\right)$ and $Q = \\left({X, \\preceq', \\tau'}\\right)$ be complete topological lattices with lower topologies. Let $f: S \\to X$ be a mapping such that :$f$ is a continuous mapping. Then $f$ preserves filtered infima."}
+{"_id": "14694", "title": "3 Numbers in A.P. whose 4th Powers are Sum of Four 4th Powers", "text": "The following triplets of integers in arithmetic sequence with common difference of $60$ can all be expressed as the sum of four $4$th powers: :$\\tuple {8373, 8433, 8493}, \\tuple {8517, 8577, 8637}, \\ldots$"}
+{"_id": "14695", "title": "Intersection of Chain of Prime Ideals of Commutative Ring is Prime Ideal", "text": "Let $R$ be a commutative ring. Let $\\Spec R$ be the spectrum of $R$, ordered by inclusion. Let $\\set {P_\\alpha}_{\\alpha \\mathop \\in A}$ be a non-empty chain of prime ideals of $\\Spec R$. Let $\\displaystyle P = \\bigcap_{\\alpha \\mathop \\in A} P_\\alpha$ be their intersection. Then $P$ is a prime ideal of $R$."}
+{"_id": "14696", "title": "Number of Different Ways to play First n Moves in Chess", "text": "The sequence formed from the number of ways to play the first $n$ moves in chess begins: :$20, 400, 8902, 197 \\, 742, \\ldots$ {{OEIS|A007545}} The count for the fourth move is already ambiguous, as it depends on whether only legal moves count, or whether all moves, legal or illegal, are included. The count as given here does include illegal moves in addition to legal ones."}
+{"_id": "14697", "title": "Largest Product of Pandigital Factors", "text": "The largest integer that can be obtained by multiplying $2$ integers which between them use all the digits from $1$ to $9$ is: :$843 \\, 973 \\, 902 = 9642 \\times 87531$"}
+{"_id": "14699", "title": "Smallest Pandigital Square", "text": "The smallest square number which contains all the digits from $1$ to $9$ is: :$11 \\, 826^2 = 139 \\, 854 \\, 276$"}
+{"_id": "14700", "title": "Locally Compact Space is Weakly Locally Compact", "text": "Let $T = \\struct{S, \\tau}$ be a locally compact topological space. Then $T$ is weakly locally compact."}
+{"_id": "14701", "title": "Space is Neighborhood of all its Points", "text": "Let $T = \\struct {X, \\tau}$ be a topological space. Let $x\\in X$. Then $X$ is a neighborhood of $x$."}
+{"_id": "14704", "title": "Local Compactness is Preserved under Open Continuous Surjection", "text": "Let $T_A = \\left({S_A, \\tau_A}\\right)$ and $T_B = \\left({S_B, \\tau_B}\\right)$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a continuous mapping which is also an open mapping and a surjection. If $T_A$ is locally compact, then $T_B$ is also locally compact."}
+{"_id": "14705", "title": "Equivalence of Definitions of Noetherian Module", "text": "Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module. {{TFAE|def = Noetherian Module}}"}
+{"_id": "14710", "title": "Largest Positive Integer not Sum of Distinct Cubes", "text": "$12 \\, 758$ is the largest positive integer that cannot be expressed as the sum of distinct cubes."}
+{"_id": "14711", "title": "Particular Point Space is Locally Compact", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Then $T$ is locally compact."}
+{"_id": "14712", "title": "Product of Summations is Summation Over Cartesian Product of Products", "text": "This is a generalization of the distributive law: :$\\displaystyle \\prod_{a \\mathop \\in A} \\sum_{b \\mathop \\in B_a} t_{a, b} = \\sum_{c \\mathop \\in \\prod_{a \\mathop \\in A} B_a} \\prod_{a \\mathop \\in A} t_{a, c_a}$ where the product of sets $\\prod_{a \\mathop \\in A} B_a$ is taken to be a cartesian product. {{explain|In order to make this comprehensible, the precise nature of $A$ and $B$ need to be defined. Presumably $A$ is a set (of numbers?), and $B$ is a family of sets of numbers indexed by $A$? And the $t$s are elements of $B$ and are numbers? I'm having difficulty.}}"}
+{"_id": "14716", "title": "Largest Number not Sum of Squares of Distinct Primes", "text": "The largest positive integer which cannot be expressed as the sum of the squares of distinct prime numbers is $17 \\, 163$."}
+{"_id": "14717", "title": "Volume of Smallest Rational Tetrahedron", "text": "The only rational tetrahedron whose edge lengths are less than $157$ has: : edges of length $117$, $80$, $53$, $52$, $51$, $84$ : faces of area $1800$, $1890$, $2016$, $1170$ : volume of $18 \\, 144$."}
+{"_id": "14718", "title": "Square and Tetrahedral Numbers", "text": "The only positive integers which are simultaneously tetrahedral and square are: :$1, 4, 19 \\, 600$"}
+{"_id": "14721", "title": "Smallest Pythagorean Quadrilateral with Integer Sides", "text": "The smallest Pythagorean quadrilateral in which the sides of the $4$ right triangles formed by its sides and perpendicular diagonals are all integers has an area of $21 \\, 576$. The sides of the right triangles in question are: :$25, 60, 65$ :$91, 60, 109$ :$91, 312, 325$ :$25, 312, 313$"}
+{"_id": "14722", "title": "Integers whose Tau value equals Cube Root", "text": "There are $3$ positive integers whose $\\tau$ value equals its cube root: {{begin-eqn}} {{eqn | l = 1 = 1^3       | o = :       | r = \\map \\tau 1 = 1       | c = {{TauLink|1}} }} {{eqn | l = 21 \\, 952 = 28^3       | o = :       | r = \\map \\tau {21 \\, 952} = 28       | c = {{TauLink|21,952|21 \\, 952}} }} {{eqn | l = 64 \\, 000 = 40^3       | o = :       | r = \\map \\tau {64 \\, 000} = 40       | c = {{TauLink|64,000|64 \\, 000}} }} {{end-eqn}} {{OEIS|A066693}}"}
+{"_id": "14723", "title": "Smallest Integer which is Product of 4 Triples all with Same Sum", "text": "The smallest integer which can be expressed as the product of $4$ different triplets of integers each of which has the same sum is: {{begin-eqn}} {{eqn | l = 25 \\, 200       | r = 6 \\times 56 \\times 75 }} {{eqn | r = 7 \\times 40 \\times 90 }} {{eqn | r = 9 \\times 28 \\times 100 }} {{eqn | r = 12 \\times 20 \\times 105 }} {{end-eqn}}"}
+{"_id": "14724", "title": "4 Integers whose Euler Phi Value is 10,368", "text": ":$\\map \\phi {25 \\, 930} = \\map \\phi {25 \\, 935} = \\map \\phi {25 \\, 940} = \\map \\phi {25 \\, 942} = 10 \\, 368 = 2^7 \\times 3^4$ where $\\phi$ denotes the Euler $\\phi$ function."}
+{"_id": "14725", "title": "Change of Lead in Prime Number Race 4n+1 vs. 4n-1", "text": "Consider the prime number race between $4 n + 1$ and $4 n - 1$. While the prime numbers of the form $4 n - 1$ appear usually to be in the majority, the lead changes from one to the other an infinite number of times."}
+{"_id": "14726", "title": "Fourth Power expressible as Sum of 6 Fourth Powers", "text": "$28 \\, 561$ can be expressed as the sum of $6$ fourth powers: :$28 \\, 561 = 13^4 = 12^4 + 8^4 + 7^4 + 6^4 + 2^4 + 2^4$"}
+{"_id": "14728", "title": "Smallest Differences between Fractional Parts of Square and Cube Roots", "text": "Apart from $6$th powers, the value of $n$ less than $50 \\, 000$ for which the difference between the fractional parts of $\\sqrt n$ and $\\sqrt [3] n$ is smallest is $30 \\, 739$. The next integer to produce a smaller difference above that is $62 \\, 324$."}
+{"_id": "14729", "title": "Smallest Triplet of Consecutive Integers each Divisible by Fourth Power", "text": "This triplet of consecutive integers has the property that each of them is divisible by a fourth power: :$33 \\, 614, 33 \\, 615, 33 \\, 616$ This is the smallest such triplet."}
+{"_id": "14730", "title": "Abundancy of Integers in form 945 + 630n", "text": "A large number of integers of the form $945 + 630 n$, for $n \\in \\Z_{\\ge 0}$, are abundant. The first counterexample is for $n = 52$."}
+{"_id": "14731", "title": "Sigma Function of Integer/Corollary", "text": ":$\\displaystyle \\map \\sigma n = \\prod_{\\substack {1 \\mathop \\le i \\mathop \\le r \\\\ k_i \\mathop > 1} } \\frac {p_i^{k_i + 1} - 1} {p_i - 1} \\prod_{\\substack {1 \\mathop \\le i \\mathop \\le r \\\\ k_i \\mathop = 1} } \\paren {p_i + 1}$"}
+{"_id": "14732", "title": "Affine Group of One Dimension as Semidirect Product", "text": "Let $\\map {\\operatorname{Af}_1} \\R$ be the $1$-dimensional affine group on $\\R$. Let $\\R^+$ be the additive group of real numbers. Let $\\R^\\times$ be the multiplicative group of real numbers. Let $\\phi: \\R^\\times \\to \\Aut {\\R^+}$ be defined as: :$\\forall b \\in \\R^\\times: \\map \\phi b = \\paren {a \\mapsto a b}$ Let $\\R^+ \\rtimes_\\phi \\R^\\times$ be the corresponding semidirect product. Then: :$\\map {\\operatorname {Af}_1} \\R \\cong \\R^+ \\rtimes_\\phi \\R^\\times$ where $\\cong$ denotes (group) isomorphism."}
+{"_id": "14733", "title": "Sequence of Consecutive Integers with Same Number of Divisors", "text": "The following sequence of consecutive integers all have the same number of divisors, that is, $8$: :$40 \\, 311, 40 \\, 312, 40 \\, 313, 40 \\, 314, 40 \\, 315$ This is the longest such sequence known."}
+{"_id": "14734", "title": "40,585/Historical Note", "text": "The fact that $40 \\, 585$ is a factorion base $10$  was discovered as late as $1964$ by {{AuthorRef|Leigh Janes}}."}
+{"_id": "14737", "title": "Smallest Sequence of 5 Consecutive Numbers which are Happy", "text": "The smallest sequence of $5$ consecutive integers all of which are happy numbers is: :$44 \\, 488, 44 \\, 489, 44 \\, 490, 44 \\, 491, 44 \\, 492$"}
+{"_id": "14738", "title": "Reciprocal of 21", "text": "The decimal expansion of the reciprocal of $21$ has period $6$: :$\\dfrac 1 {21} = 0 \\cdotp \\dot 04761 \\, \\dot 9$"}
+{"_id": "14739", "title": "Properties of 47,619", "text": "If you split $047 \\, 619$ into two halves, they add up to $666$: :$047 + 619 = 666$ which is a multiple of $333$. and: : $047 \\, 619 = 143 \\times 333$ Similarly, you can split $047 \\, 619$ into three thirds, and these add up to $99$: :$04 + 76 + 19 = 99$ and: : $047 \\, 619 = 481 \\times 99$ The square of $47 \\, 619$: :$47 \\, 619^2 = 2 \\, 267 \\, 569 \\, 161$ can itself be split into two $6$-digit halves which together add to the recurring part of $\\dfrac 4 7$: :$2267 + 569 \\, 161 = 571 \\, 428$ This is caused by the fact that $047 \\, 619$ is the recurring part of the Reciprocal of 21: :$\\dfrac 1 {21} = 0 \\cdotp \\dot 04761 \\, \\dot 9$ where $21$ is the product of (the smallest) $2$ distinct primes which do not divide $10$. {{finish|for whatever value of \"finish off\" is determined}}"}
+{"_id": "14740", "title": "Smallest Fourth Power as Sum of 5 Fourth Powers", "text": "The smallest $4$th power which can be expressed as the sum of $5$ $4$th powers is: :$15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4$"}
+{"_id": "14741", "title": "Boolean Interpretation is Well-Defined", "text": "Let $\\LL_0$ be the language of propositional logic. Let $v: \\LL_0 \\to \\set {\\T, \\F}$ be a boolean interpretation. Then $v$ is well-defined."}
+{"_id": "14743", "title": "Pandigital Pairs whose Squares are Pandigital", "text": "The elements of the following pandigital pairs of integers each have squares which are themselves pandigital: :$\\left({35 \\, 172, 60 \\, 984}\\right), \\left({57 \\, 321, 60 \\, 984}\\right), \\left({58 \\, 413, 96 \\, 702}\\right), \\left({59 \\, 403, 76 \\, 182}\\right)$ {{OEIS|A085545|order = first}}"}
+{"_id": "14745", "title": "Language of Propositional Logic has Unique Parsability", "text": "The language of propositional logic $\\mathcal L_0$ has unique parsability."}
+{"_id": "14746", "title": "Powers of 2 not containing Digit Power of 2", "text": "$2^{16} = 65 \\, 536$ is the only known power of $2$, up to $2^{31 \\, 000}$, whose digits do not contain $1$, $2$, $4$ or $8$."}
+{"_id": "14747", "title": "Construction of Regular 65,537-Gon", "text": "It is possible to construct a regular polygon with $65 \\, 537$ sides) using a compass and straightedge construction."}
+{"_id": "14748", "title": "There are no Odd Unitary Perfect Numbers", "text": "No unitary perfect numbers exist which are odd."}
+{"_id": "14749", "title": "Automorphic Numbers with 5 Digits", "text": "The only $5$-digit automorphic number which does not begin with a zero is $90 \\, 625$."}
+{"_id": "14750", "title": "Kaprekar's Process on 5 Digit Number", "text": "Let $n$ be a $5$-digit integer whose digits are not all the same. Kaprekar's process, when applied to $n$, results in one of the following $3$ cycles: :$53 \\, 955 \\to 59 \\, 994 \\to 53 \\, 955$ :$61 \\, 974 \\to 82 \\, 962 \\to 75 \\, 933 \\to 63 \\, 954 \\to 61 \\, 974$ :$62 \\, 964 \\to 71 \\, 973 \\to 83 \\, 952 \\to 74 \\, 943 \\to 62 \\, 964$"}
+{"_id": "14751", "title": "Tableau Extension Lemma/General Statement", "text": "Let $\\mathbf H'$ be another finite set of WFFs. Then there exists a finished finite propositional tableau $T'$ such that: $(1):\\quad$ the root of $T'$ is $\\mathbf H \\cup \\mathbf H'$; $(2):\\quad$ $T$ is a rooted subtree of $T'$."}
+{"_id": "14753", "title": "Tableau Extension Lemma/General Statement/Proof 2", "text": "Let $T$ be a finite propositional tableau. Let its hypothesis set $\\mathbf H$ be finite. {{:Tableau Extension Lemma/General Statement}}"}
+{"_id": "14754", "title": "Factors of Sums of Powers of 100,000", "text": "All integers $n$ of the form: :$n = \\displaystyle \\sum_{k \\mathop = 0}^m 10^{5 k}$ for $m \\in \\Z_{> 0}$ are composite."}
+{"_id": "14756", "title": "Points Defined by Adjacent Pairs of Digits of Reciprocal of 7 lie on Ellipse", "text": "Consider the digits that form the recurring part of the reciprocal of $7$: :$\\dfrac 1 7 = 0 \\cdotp \\dot 14285 \\dot 7$ Take the digits in ordered pairs, and treat them as coordinates of a Cartesian plane. It will be found that they all lie on an ellipse: :400px"}
+{"_id": "14757", "title": "Continuous Function on Compact Subspace of Euclidean Space is Bounded", "text": "Let $\\R^n$ be the $n$-dimensional Euclidean space. Let $S \\subseteq \\R^n$ be a compact subspace of $\\R^n$. Let $f: S \\to \\R$ be a continuous function. Then $f$ is bounded in $\\R$."}
+{"_id": "14758", "title": "Points Defined by Adjacent Pairs of Digits of Reciprocal of 13 lie on Hyperbola", "text": "Consider the digits that form the recurring part of the reciprocal of $13$: :$\\dfrac 1 {13} = 0 \\cdotp \\dot 07692 \\dot 3$ Take the digits in ordered pairs, and treat them as coordinates of a Cartesian plane. It will be found that they all lie on a hyperbola: :600px"}
+{"_id": "14759", "title": "Recurring Part of Fraction times Period gives 9-Repdigit", "text": "Let a (strictly) positive integer $n$ be such that the decimal expansion of its reciprocal has a recurring part of period $d$. Let $m$ be the integer formed from the $d$ digits of the recurring part. Then $m \\times n$ is a $d$-digit repdigit number consisting of $9$s. {{mistake|Counterexample:1/6. Possibly true if n is not a multiple of 2 or 5}}"}
+{"_id": "14760", "title": "Properties of 142,857", "text": "This page gathers together some properties of $142 \\, 857$ which arise through its being the digits of the recurring part of the reciprocal of $7$. Multiplication of $142 \\, 857$ by numbers higher than $7$ produces a similar pattern to when you multiply it by a single digit, but with added complications. For example: {{begin-eqn}} {{eqn | l = 142 \\, 857 \\times 12       | r = 1 \\, 714 \\, 284       | c =  }} {{end-eqn}} This becomes $714 \\, 285$ when you take the $1$ off the front and add it to the back. The exception is when you multiply it by $7$ or a multiple of $7$: {{begin-eqn}} {{eqn | l = 142 \\, 857 \\times 7       | r = 999 \\, 999       | c =  }} {{eqn | l = 142 \\, 857 \\times 14       | r = 1 \\, 999 \\, 998       | c =  }} {{end-eqn}} From Recurring Part of Fraction times Period gives 9-Repdigit, it is seen that this property is shared of all numbers formed from the digits of the recurring part of a recurring fraction. If you divide $142 \\, 857$ into two equal parts and add them, you get $999$: :$142 + 857 = 999$ Thus by Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999, $142 \\, 857$ is divisible by $999$: :$142 \\, 857 = 143 \\times 999$ Also, we have: :$999 \\, 999 = 1001 \\times 999$ and so $999 \\, 999$ is divisible by $999$. But as $999 \\, 999 = 7 \\times 142 \\, 857$ we have that $999 \\, 999$ is divisible by $7$. Thus it follows from Euclid's Lemma that $142 \\, 857$ is divisible by $999$. {{finish|A lot of material from this chapter of Wells has been skipped, because it's just not very interesting. I leave it open for someone else to complete, if they want to.}}"}
+{"_id": "14762", "title": "Quotient of Group by Itself", "text": "Let $G$ be a group. Let $G / G$ be the quotient group of $G$ by itself. Then: :$G / G \\cong \\set e$ That is, the quotient of a group by itself is isomorphic to the trivial group."}
+{"_id": "14763", "title": "Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999", "text": "Let $n$ be an integer which has at least $3$ digits when expressed in decimal notation. Let the digits of $n$ be divided into groups of $3$, counting from the right, and those groups added. Then the result is equal to a multiple of $999$ {{iff}} $n$ is divisible by $999$."}
+{"_id": "14764", "title": "Number which is Sum of Subfactorials of Digits", "text": "The only integer which is the sum of the subfactorials of its digits is $148 \\, 349$: :$148 \\, 349 = \\mathop !1 + \\mathop !4 + \\mathop !8 + \\mathop !3 \\mathop + \\mathop !4 \\mathop + \\mathop !9$"}
+{"_id": "14765", "title": "Integers Representable as Product of both 3 and 4 Consecutive Integers", "text": "There are $3$ integers which can be expressed as both $x \\paren {x + 1} \\paren {x + 2} \\paren {x + 3}$ for some $x$, and $y \\paren {y + 1} \\paren {y + 2}$ for some $y$: :$24, 120, 175 \\, 560$"}
+{"_id": "14766", "title": "Squares whose Digits form Consecutive Increasing Integers", "text": "The sequence of integers whose squares have a decimal representation consisting of the concatenation of $2$ consecutive increasing integers begins: :$428, 573, 727, 846, 7810, 36 \\, 365, 63 \\, 636, 326 \\, 734, \\ldots$ {{OEIS|A030467}}"}
+{"_id": "14767", "title": "Smallest Fifth Power which is Sum of 6 Fifth Powers", "text": "The smallest fifth power which is the sum of $6$ fifth powers is $12^5 = 248 \\, 832$: :$12^5 = 4^5 + 5^5 + 6^5 + 7^5 + 9^5 + 11^5$"}
+{"_id": "14768", "title": "Prime Numbers Embedded in Digits of Pi", "text": "The sequence of prime numbers that can be found starting from the beginning of the decimal expansion of $\\pi$ (pi) begins: :$3, 31, 314 \\, 159, 31 \\, 415 \\, 926 \\, 535 \\, 897 \\, 932 \\, 384 \\, 626 \\, 433 \\, 832 \\, 795 \\, 028 \\, 841, \\ldots$ {{OEIS|A005042}}"}
+{"_id": "14771", "title": "Cube which can be Represented as Sum of 3, 4, 5, 6, 7 or 8 Cubes", "text": ":$351 \\, 120^3$ can be represented as the sum of $3$, $4$, $5$, $6$, $7$ or $8$ cubes."}
+{"_id": "14772", "title": "Prime Gaps of 100", "text": "The following pairs of consecutive prime numbers are those whose difference is $100$: :$\\tuple {396 \\, 733, 396 \\, 833}, \\ldots$ {{expand|Only know the first pair so far. Research needed to find the next one(s).}}"}
+{"_id": "14773", "title": "Property of 490,689", "text": "The number $490 \\, 689$ can be expressed as the sum of $3$ cubes in $2$ different ways: :$490 \\, 689 = 4^3 + 60^3 + 65^3 = 8^3 + 25^3 \\times 78^3$ while at the same time the products of the contributory cube roots of each of those $2$ ways are equal: :$4 \\times 60 \\times 65 = 8 \\times 25 \\times 78$"}
+{"_id": "14774", "title": "510,510 is Product of 4 Consecutive Fibonacci Numbers", "text": "$510 \\, 510$ Can be expressed as the product of $4$ distinct Fibonacci numbers: :$510 \\, 510 = 13 \\times 21 \\times 34 \\times 55$ and is also the $7$th primorial: :$510 \\, 510 = 2 \\times 3 \\times 5 \\times 7 \\times 11 \\times 13 \\times 17$"}
+{"_id": "14775", "title": "Tableau Confutation contains Finite Tableau Confutation", "text": "Let $\\mathbf H$ be a countable set of WFFs of propositional logic. Let $T$ be a tableau confutation of $\\mathbf H$. Then there exists a finite rooted subtree of $T'$ that is also a tableau confutation of $\\mathbf H'$."}
+{"_id": "14778", "title": "Set of 7 Anagrams which are Square", "text": "The following integers are all anagrams, and all square: :$1 \\, 048 \\, 576, 1 \\, 056 \\, 784, 1 \\, 085 \\, 764, 5 \\, 740 \\, 816, 5 \\, 764 \\, 801, 6 \\, 754 \\, 801, 7 \\, 845 \\, 601$"}
+{"_id": "14779", "title": "Burnside's Lemma", "text": "Let $G$ be a finite group acting on a set $X$. Let $X / G$ be the set of orbits under this action. For $x \\in X$, let $\\Stab x$ be the stabilizer of $x$ by $G$. For $g \\in G$, let $X^g$ denotes the set of all elements in $X$ which is fixed by $g$, that is: :$X^g := \\set {x \\in X: g x = x}$ Then: :$\\displaystyle \\size {X / G} = \\frac 1 {\\order G} \\sum_{g \\mathop \\in G} \\size {X^g}$ In words, the number of orbits equals the average number of fixed elements."}
+{"_id": "14780", "title": "Smallest Cunningham Chain of the First Kind of Length 7", "text": "The smallest Cunningham chain of the first kind of length $7$ is: :$\\left({1 \\, 122 \\, 659, 2 \\, 245 \\, 319, 4 \\, 490 \\, 639, 8 \\, 981 \\, 279, 17 \\, 962 \\, 559, 35 \\, 925 \\, 119, 71 \\, 850 \\, 239}\\right)$"}
+{"_id": "14782", "title": "Factorisation of Quintic x^5 - x + n into Irreducible Quadratic and Irreducible Cubic", "text": "The quintic $x^5 - x + n$ can be factorized into the product of an irreducible quadratic and an an irreducible cubic {{iff}} $n$ is in the set: :$\\set {\\pm 15, \\pm 22 \\, 440, \\pm 2 \\, 759 \\, 640}$"}
+{"_id": "14783", "title": "Factorial as Product of Consecutive Factorials", "text": "The only factorials which are the product of consecutive factorials are: {{begin-eqn}} {{eqn | l = 0!       | r = 0! \\times 1!       | c =  }} {{eqn | l = 1!       | r = 0! \\times 1!       | c =  }} {{eqn | l = 2!       | r = 1! \\times 2!       | c =  }} {{eqn | r = 0! \\times 1! \\times 2!       | c =  }} {{eqn | l = 10!       | r = 6! \\times 7!       | c =  }} {{end-eqn}}"}
+{"_id": "14785", "title": "Smallest Solution to Equation p^p times q^q = r^r", "text": "Consider the Diophantine equation: :$p^p \\times q^q = r^r$ Its smallest solution is: {{begin-eqn}} {{eqn | l = p       | r = 12^6       | rr=  = 2 \\, 985 \\, 984 }} {{eqn | l = q       | r = 6^8       | rr=  = 1 \\, 679 \\, 616 }} {{eqn | l = r       | r = 2^{11} \\times 3^7       | rr=  = 4 \\, 478 \\, 976 }} {{end-eqn}}"}
+{"_id": "14786", "title": "Existence of Matrix Logarithm", "text": "Let $T$ be a square matrix of order $n$. Then there exists a real matrix $S$ such that $e^S = T$ {{iff}}: :$(1): \\quad  T$ is not a singular matrix and: :$(2): \\quad $for every negative eigenvalue $\\lambda$ of $T$ and for every positive integer $k$, the Jordan form of $T$ has an even number of $k \\times k$ blocks associated with $\\lambda$. pcu6qfvqi59x43t8w6ubk68j3p2rko5"}
+{"_id": "14787", "title": "Weak Existence of Matrix Logarithm", "text": "Let $T$ be a square matrix of order $n$. Let $\\norm {T - I} < 1$ in the norm on bounded linear operators, where $I$ the identity matrix. Then there is a square matrix $S$ such that: :$e^S = T$ where $e^S$ is the matrix exponential."}
+{"_id": "14788", "title": "Norm on Vector Space is Continuous Function", "text": "Let $V$ be a vector space with norm $\\norm {\\, \\cdot \\,}$. The function $\\norm {\\, \\cdot \\,}: V \\to \\R$ is continuous."}
+{"_id": "14789", "title": "Largest Integer Not Expressible as Sum of Distinct 4th Powers", "text": "The largest integer which cannot be expressed as the sum of distinct $4$th powers is $5 \\, 134 \\, 240$."}
+{"_id": "14791", "title": "Properties of 12,345,679", "text": "$12 \\, 345 \\, 679$ has the following properties: {{begin-eqn}} {{eqn | l = 12 \\, 345 \\, 679 \\times 1       | r = 12 \\, 345 \\, 679       | c = digit $8$ is missing }} {{eqn | l = 12 \\, 345 \\, 679 \\times 2       | r = 24 \\, 691 \\, 358       | c = digit $7$ is missing }} {{eqn | l = 12 \\, 345 \\, 679 \\times 3       | r = 37 \\, 037 \\, 037       | c =  }} {{eqn | l = 12 \\, 345 \\, 679 \\times 4       | r = 49 \\, 382 \\, 716       | c = digit $5$ is missing }} {{eqn | l = 12 \\, 345 \\, 679 \\times 5       | r = 61 \\, 728 \\, 395       | c = digit $4$ is missing }} {{eqn | l = 12 \\, 345 \\, 679 \\times 6       | r = 74 \\, 074 \\, 074       | c =  }} {{eqn | l = 12 \\, 345 \\, 679 \\times 7       | r = 86 \\, 419 \\, 753       | c = digit $2$ is missing }} {{eqn | l = 12 \\, 345 \\, 679 \\times 8       | r = 98 \\, 765 \\, 432       | c = digit $1$ is missing }} {{eqn | l = 12 \\, 345 \\, 679 \\times 9       | r = 111 \\, 111 \\, 111       | c =  }} {{end-eqn}} In each product, the sequence $1$ to $9$, with the one given digit missing, can be read in order by cycling round it, skipping a fixed number of digits (counting an extra one when going from start to end), for example: :$2 \\ (4691) \\ 3 \\ (58?2) \\ 4 \\ (6913) \\ 5 \\ (8?24) \\ 6 \\ (9135) \\ 8 (?246) \\ 9$ {{expand|Add some mathematical analysis explaining this phenomenon}}"}
+{"_id": "14792", "title": "Number of Ways to Tile Standard Chessboard with Dominoes", "text": "The number of ways to tile a standard chessboard with dominoes is $12 \\, 988 \\, 816$"}
+{"_id": "14794", "title": "Sequence of 8 Consecutive Primes with Same Pattern of Differences as from 11", "text": "The $8$ prime numbers starting at $15 \\, 760 \\, 091$ have the same prime gaps as the $8$ primes starting at $11$. Those $8$ primes are: :$15 \\, 760 \\, 091, 15 \\, 760 \\, 093, 15 \\, 760 \\, 097, 15 \\, 760 \\, 099, 15 \\, 760 \\, 103, 15 \\, 760 \\, 109, 15 \\, 760 \\, 111, 15 \\, 760 \\, 117$ Their prime gaps are: :$2, 4, 2, 4, 6, 2, 6$ The $8$ primes starting at $11$: :$11, 13, 17, 19, 23, 29, 31, 37$ Their prime gaps are: :$2, 4, 2, 4, 6, 2, 6$"}
+{"_id": "14795", "title": "Smallest Even Integer whose Euler Phi Value is not the Euler Phi Value of an Odd Integer", "text": "The smallest even integer whose Euler $\\phi$ value is shared by no odd integer is $33 \\, 817 \\, 088$."}
+{"_id": "14796", "title": "Smallest Integer which is Sum of 2 Cubes in 4 Ways", "text": "The smallest positive integer which can be expressed as the sum of $2$ cubes in $4$ different ways is: {{begin-eqn}} {{eqn | l = 42 \\, 549 \\, 416       | r = 348^3 + 74^3       | c =  }} {{eqn | r = 282^3 + 272^3       | c =  }} {{eqn | r = \\left({-2662}\\right)^3 + 2664^3       | c =  }} {{eqn | r = \\left({-475}\\right)^3 + 531^3       | c =  }} {{end-eqn}}"}
+{"_id": "14797", "title": "Squares whose Digits form Consecutive Integers", "text": "The sequence of integers whose squares have a decimal representation consisting of the concatenation of $2$ consecutive integers, either increasing or decreasing begins: :$91, 428, 573, 727, 846, 7810, 9079, 9901, 36 \\, 365, 63 \\, 636, 326 \\, 734, 673 \\, 267, 733 \\, 674, \\ldots$ This sequence can be divided into two subsequences:  Those where the consecutive integers are increasing: :$428, 573, 727, 846, 7810, 36 \\, 365, 63 \\, 636, 326 \\, 734, 673 \\, 267, \\ldots$ {{OEIS|A030467}} Those where the consecutive integers are decreasing: :$91, 9079, 9901, 733 \\, 674, 999 \\, 001, 88 \\, 225 \\, 295, \\ldots$ {{OEIS|A054216}}"}
+{"_id": "14798", "title": "Squares whose Digits form Consecutive Decreasing Integers", "text": "The sequence of integers whose squares have a decimal representation consisting of the concatenation of $2$ consecutive decreasing integers begins: :$91, 9079, 9901, 733 \\, 674, 999 \\, 001, 88 \\, 225 \\, 295, 99 \\, 990 \\, 001, \\ldots$ {{OEIS|A030467}}"}
+{"_id": "14799", "title": "Numbers n whose Euler Phi value Divides n + 1", "text": "The following integers $n$ satisfy the equation: :$\\exists k \\in \\Z: k \\, \\map \\phi n = n + 1$ where $\\phi$ denotes the Euler $\\phi$ function: :$83 \\, 623 \\, 935, 83 \\, 623 \\, 935 \\times 83 \\, 623 \\, 937$"}
+{"_id": "14800", "title": "Hardy-Ramanujan Number/Examples/87,539,319", "text": "The $3$rd Hardy-Ramanujan number $\\operatorname {Ta} \\left({3}\\right)$ is $87 \\, 539 \\, 319$: {{begin-eqn}} {{eqn | l = 87 \\, 539 \\, 319       | r = 167^3 + 436^3       | c =  }} {{eqn | r = 228^3 + 423^3       | c =  }} {{eqn | r = 255^3 + 414^3       | c =  }} {{end-eqn}}"}
+{"_id": "14801", "title": "Polynomial is Linear Combination of Monomials", "text": "Let $R$ be a commutative ring with unity. Let $R \\sqbrk X$ be a polynomial ring over $R$ in the variable $X$. Let $P \\in R \\sqbrk X$. Then $P$ is a linear combination of the monomials of $R \\sqbrk X$, with coefficients in $R$. {{explain|this needs to be made more precise}}"}
+{"_id": "14802", "title": "Sum over Disjoint Union of Finite Sets", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $S$ and $T$ be finite disjoint sets. Let $S \\cup T$ be their union. Let $f: S \\cup T \\to \\mathbb A$ be a mapping. Then we have the equality of summations over finite sets: :$\\displaystyle \\sum_{u \\mathop \\in S \\mathop \\cup T} f \\left({u}\\right) = \\sum_{s \\mathop \\in S} f \\left({s}\\right) + \\sum_{t \\mathop \\in T} f \\left({t}\\right)$"}
+{"_id": "14803", "title": "Finite Summation does not Change under Permutation", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $S$ be a finite set. Let $f : S \\to \\mathbb A$ be a mapping. Let $\\sigma : S\\to S$ be a permutation. Then we have the equality of summations over finite sets: :$\\displaystyle \\sum_{s \\mathop \\in S} \\map f s = \\sum_{s \\mathop \\in S} \\map f {\\map \\sigma s}$"}
+{"_id": "14805", "title": "Change of Variables in Summation over Finite Set", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $S$ and $T$ be finite sets. Let $f: S \\to \\mathbb A$ be a mapping. Let $g: T \\to S$ be a bijection. Then we have an equality of summations over finite sets: :$\\displaystyle \\sum_{s \\mathop \\in S} f \\left({s}\\right) = \\sum_{t \\mathop \\in T} f \\left({g \\left({t}\\right)}\\right)$"}
+{"_id": "14806", "title": "Indexed Summation does not Change under Permutation", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $a$ and $b$ be integers. Let $\\closedint a b$ be the integer interval between $a$ and $b$. Let $f: \\closedint a b \\to \\mathbb A$ be a mapping. Let $\\sigma: \\closedint a b \\to \\closedint a b$ be a permutation. Then we have an equality of indexed summations: :$\\displaystyle \\sum_{i \\mathop = a}^b \\map f i = \\sum_{i \\mathop = a}^b \\map f {\\map \\sigma i}$"}
+{"_id": "14807", "title": "Indexed Summation over Translated Interval", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $a$ and $b$ be integers. Let $\\closedint a b$ be the integer interval between $a$ and $b$. Let $f: \\closedint a b \\to \\mathbb A$ be a mapping. Let $c\\in\\Z$ be an integer. Then we have an equality of indexed summations: :$\\displaystyle \\sum_{i \\mathop = a}^b f(i) = \\sum_{i \\mathop = a + c}^{b + c} \\map f {i - c}$"}
+{"_id": "14808", "title": "Indexed Summation over Adjacent Intervals", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N,\\Z,\\Q,\\R,\\C$. Let $a,b,c$ be integers. Let $\\left[{a \\,.\\,.\\, c}\\right]$ denote the integer interval between $a$ and $c$. Let $b \\in \\left[{a-1 \\,.\\,.\\, c}\\right]$. Let $f : \\left[{a \\,.\\,.\\, c}\\right] \\to \\mathbb A$ be a mapping. Then we have an equality of indexed summations: :$\\displaystyle \\sum_{i \\mathop = a}^c f(i) = \\sum_{i \\mathop = a}^b f(i) + \\sum_{i \\mathop = b+1}^c f(i)$"}
+{"_id": "14809", "title": "Indexed Summation over Interval of Length Two", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N,\\Z,\\Q,\\R,\\C$. Let $a\\in\\Z$ be an integer. Let $f : \\{a, a+1\\} \\to \\mathbb A$ be a real-valued function. Then the indexed summation: :$\\displaystyle \\sum_{i \\mathop = a}^{a+1} f(i) = f(a) + f(a+1)$."}
+{"_id": "14810", "title": "Indexed Summation over Interval of Length One", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N,\\Z,\\Q,\\R,\\C$. Let $a\\in\\Z$ be an integer. Let $f : \\{a\\} \\to \\mathbb A$ be a mapping on the singleton $\\{a \\}$. Then the indexed summation: :$\\displaystyle \\sum_{i \\mathop = a}^{a} f(i) = f(a)$"}
+{"_id": "14811", "title": "Change of Variables in Indexed Summation", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $a, b, c, d$ be integers. Let $\\closedint a b$ denote the integer interval between $a$ and $b$. Let $f: \\closedint a b \\to \\mathbb A$ be a mapping. Let $g: \\closedint c d \\to \\closedint a b$ be a bijection. Then we have an equality of indexed summations: :$\\displaystyle \\sum_{i \\mathop = a}^b \\map f i = \\sum_{i \\mathop = c}^d \\map f {\\map g i}$"}
+{"_id": "14812", "title": "Translation of Integer Interval is Bijection", "text": "Let $a, b, c \\in \\Z$ be integers. Let $\\closedint a b$ denote the integer interval between $a$ and $b$. Then the mapping $T: \\closedint a b \\to \\closedint {a + c} {b + c}$ defined as: :$\\map T k = k + c$ is a bijection."}
+{"_id": "14813", "title": "Indexed Summation without First Term", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N,\\Z,\\Q,\\R,\\C$. Let $a$ and $b$ be integers with $a\\leq b$. Let $\\left[{a \\,.\\,.\\, b}\\right]$ be the integer interval between $a$ and $b$. Let $f : \\left[{a \\,.\\,.\\, b}\\right] \\to \\mathbb A$ be a mapping. Then we have an equality of indexed summations: :$\\displaystyle \\sum_{i \\mathop = a}^b f(i) = f(a) + \\sum_{i \\mathop = a+1}^b f(\\sigma(i))$"}
+{"_id": "14814", "title": "Summation over Interval equals Indexed Summation", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $a, b \\in \\Z$ be integers. Let $\\left[{a \\,.\\,.\\, b}\\right]$ be the integer interval between $a$ and $b$. Let $f: \\left[{a \\,.\\,.\\, b}\\right] \\to \\mathbb A$ be a mapping. Then the summation over the finite set $\\left[{a \\,.\\,.\\, b}\\right]$ equals the indexed summation from $a$ to $b$: :$\\displaystyle\\sum_{k \\mathop \\in \\left[{a \\,.\\,.\\, b}\\right]} f \\left({k}\\right) = \\sum_{k \\mathop = a}^b f \\left({k}\\right)$"}
+{"_id": "14815", "title": "Hardy-Ramanujan Number/Examples/1729", "text": "The $2$nd Hardy-Ramanujan number $\\map {\\operatorname {Ta}} 2$ is $1729$: {{begin-eqn}} {{eqn | l = 1729       | r = 12^3 + 1^3       | c =  }} {{eqn | r = 10^3 + 9^3       | c =  }} {{end-eqn}}"}
+{"_id": "14816", "title": "Cardinality of Integer Interval", "text": "Let $a, b \\in \\Z$ be integers. Let $\\left[{a \\,.\\,.\\, b}\\right]$ denote the integer interval between $a$ and $b$. Then $\\left[{a \\,.\\,.\\, b}\\right]$ is finite and its cardinality equals: :$\\begin{cases}  b - a + 1 & : b \\ge a - 1 \\\\  0 & : b \\le a - 1  \\end{cases}$"}
+{"_id": "14817", "title": "Hardy-Ramanujan Number/Examples/6,963,472,309,248", "text": "The $4$th Hardy-Ramanujan number $\\operatorname {Ta} \\left({4}\\right)$ is $6 \\, 963 \\, 472 \\, 309 \\, 248$: {{begin-eqn}} {{eqn | l = 6 \\, 963 \\, 472 \\, 309 \\, 248       | r = 2421^3 + 19 \\, 083^3       | c =  }} {{eqn | r = 5436^3 + 18 \\, 948^3       | c =  }} {{eqn | r = 10 \\, 200^3 + 18 \\, 072^3       | c =  }} {{eqn | r = 13 \\, 322^3 + 16 \\, 630^3       | c =  }} {{end-eqn}}"}
+{"_id": "14818", "title": "Indexed Summation of Sum of Mappings", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $a, b$ be integers. Let $\\closedint a b$ denote the integer interval between $a$ and $b$. Let $f, g: \\closedint a b \\to \\mathbb A$ be mappings. Let $h = f + g$ be their pointwise sum. Then we have the equality of indexed summations: :$\\displaystyle \\sum_{i \\mathop = a}^b \\map h i = \\sum_{i \\mathop = a}^b \\map f i + \\sum_{i \\mathop = a}^b \\map g i$"}
+{"_id": "14819", "title": "Summation of Sum of Mappings on Finite Set", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $S$ be a finite set. Let $f, g: S \\to \\mathbb A$ be mappings. Let $h = f + g$ be their sum. Then we have the equality of summations on finite sets: :$\\displaystyle \\sum_{s \\mathop \\in S} h \\left({s}\\right) = \\sum_{s \\mathop \\in S} f \\left({s}\\right) + \\sum_{s \\mathop \\in S} g \\left({s}\\right)$"}
+{"_id": "14820", "title": "Indexed Summation of Multiple of Mapping", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N,\\Z,\\Q,\\R,\\C$. Let $a, b$ be integers. Let $\\left[{a \\,.\\,.\\, b}\\right]$ denote the integer interval between $a$ and $b$. Let $f: \\left[{a \\,.\\,.\\, b}\\right] \\to \\mathbb A$ be a mapping. Let $\\lambda \\in \\mathbb A$. Let $g = \\lambda \\cdot f$ be the product of $f$ with $\\lambda$. Then we have the equality of indexed summations: :$\\displaystyle \\sum_{i \\mathop = a}^b g \\left({i}\\right) = \\lambda \\cdot \\sum_{i \\mathop = a}^b f \\left({i}\\right)$"}
+{"_id": "14821", "title": "Summation of Multiple of Mapping on Finite Set", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $S$ be a finite set. Let $f: S \\to \\mathbb A$ be a mapping. Let $\\lambda \\in \\mathbb A$. Let $g = \\lambda \\cdot f$ be the product of $f$ with $\\lambda$. Then we have the equality of summations on finite sets: :$\\displaystyle \\sum_{s \\mathop \\in S} g \\left({s}\\right) = \\lambda \\cdot \\sum_{s \\mathop \\in S} f \\left({s}\\right)$"}
+{"_id": "14822", "title": "Linear Combination of Indexed Summations", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N,\\Z,\\Q,\\R,\\C$. Let $a,b$ be integers. Let $\\left[{a \\,.\\,.\\, b}\\right]$ denote the integer interval between $a$ and $b$. Let $f, g : \\left[{a \\,.\\,.\\, b}\\right] \\to \\mathbb A$ be mappings. Let $\\lambda, \\mu \\in \\mathbb A$. Let $\\lambda \\cdot f + \\mu \\cdot g$ be the sum of the product of $f$ with $\\lambda$ and the product of $g$ with $\\mu$. Then we have the equality of indexed summations: :$\\displaystyle \\sum_{i \\mathop = a}^b \\left( \\lambda \\cdot f(i) + \\mu \\cdot g(i) \\right) = \\lambda \\cdot \\sum_{i \\mathop = a}^b f(i) + \\mu \\cdot \\sum_{i \\mathop = a}^b g(i)$"}
+{"_id": "14823", "title": "Triangle Inequality for Indexed Summations", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N,\\Z,\\Q,\\R,\\C$. Let $a,b$ be integers. Let $\\left[{a \\,.\\,.\\, b}\\right]$ denote the integer interval between $a$ and $b$. Let $f : \\left[{a \\,.\\,.\\, b}\\right] \\to \\mathbb A$ be a mapping. Let $|\\cdot|$ denote the standard absolute value. Let $\\vert f \\vert$ be the absolute value of $f$. Then we have the inequality of indexed summations: :$\\displaystyle \\left\\vert \\sum_{i \\mathop = a}^b f(i) \\right\\vert \\leq \\sum_{i \\mathop = a}^b \\vert f(i) \\vert$"}
+{"_id": "14824", "title": "Triangle Inequality for Summation over Finite Set", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $S$ be a finite set. Let $f : S \\to \\mathbb A$ be a mapping. Let $\\left\\vert{\\, \\cdot\\,}\\right\\vert$ denote the standard absolute value. Let $\\left\\vert{f}\\right\\vert$ be the absoute value of $f$. Then we have the inequality of summations on finite sets: :$\\displaystyle \\left\\vert \\sum_{s \\mathop \\in S} f(s) \\right\\vert \\leq \\sum_{s \\mathop \\in S} \\vert f(s) \\vert$"}
+{"_id": "14826", "title": "Exchange of Order of Indexed Summations", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $a, b, c, d \\in \\Z$ be integers. Let $\\closedint a b$ denote the integer interval between $a$ and $b$."}
+{"_id": "14828", "title": "Exchange of Order of Summations over Finite Sets/Cartesian Product", "text": "Let $f: S \\times T \\to \\mathbb A$ be a mapping. Then we have an equality of summations over finite sets: :$\\displaystyle \\sum_{s \\mathop \\in S} \\sum_{t \\mathop \\in T} \\map f {s, t} = \\sum_{t \\mathop \\in T} \\sum_{s \\mathop \\in S} \\map f {s, t}$"}
+{"_id": "14829", "title": "Exchange of Order of Summations over Finite Sets", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $S, T$ be finite sets. Let $S \\times T$ be their cartesian product."}
+{"_id": "14830", "title": "Sum over Complement of Finite Set", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $S$ be a finite set. Let $f: S \\to \\mathbb A$ be a mapping. Let $T \\subseteq S$ be a subset. Let $S \\setminus T$ be its relative complement. Then we have the equality of summations over finite sets: :$\\displaystyle \\sum_{s \\mathop \\in S \\setminus T} f \\left({s}\\right) = \\sum_{s \\mathop \\in S} f \\left({s}\\right) - \\sum_{t \\mathop \\in T} f \\left({t}\\right)$"}
+{"_id": "14831", "title": "Mapping Defines Additive Function of Subalgebra of Power Set", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $S$ be a finite set. Let $f: S \\to \\mathbb A$ be a mapping. Let $B$ be an algebra of sets over $S$. Define $\\Sigma: B \\to \\mathbb A$ using summation as: :$\\Sigma \\left({T}\\right) = \\displaystyle \\sum_{t \\mathop \\in T} f \\left({t}\\right)$ for $T\\subseteq S$. Then $\\Sigma$ is an additive function on $B$."}
+{"_id": "14833", "title": "Summation over Finite Set Equals Summation over Support", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $S$ be a finite set. Let $f: S \\to \\mathbb A$ be a mapping. Let $\\operatorname{Supp} \\left({f}\\right)$ be its support. Then we have an equality of summations over finite sets: :$\\displaystyle \\sum_{s \\mathop \\in S} f \\left({s}\\right) = \\sum_{s \\mathop \\in \\operatorname{Supp} \\left({f}\\right)} f \\left({s}\\right)$"}
+{"_id": "14834", "title": "Summation of Zero", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N,\\Z,\\Q,\\R,\\C$."}
+{"_id": "14835", "title": "Summation of Zero/Indexed Summation", "text": "Let $a,b$ be integers. Let $\\left[{a \\,.\\,.\\, b}\\right]$ denote the integer interval between $a$ and $b$. Let $0 : \\left[{a \\,.\\,.\\, b}\\right] \\to \\mathbb A$ be the zero mapping. Then the indexed summation of $0$ from $a$ to $b$ equals zero: :$\\displaystyle \\sum_{i \\mathop = a}^b 0(i) = 0$"}
+{"_id": "14836", "title": "Summation of Zero/Finite Set", "text": "Let $S$ be a finite set. Let $0 : S \\to \\mathbb A$ be the zero mapping. {{explain|Presumably the above is a constant mapping on $0$ -- needs to be made explicit.}} Then the summation of $0$ over $S$ equals zero: :$\\displaystyle \\sum_{s \\mathop \\in S} 0 \\left({s}\\right) = 0$"}
+{"_id": "14839", "title": "Universal Property of Polynomial Ring", "text": "Let $R$ be a commutative ring with unity. The different definitions of a polynomial ring $(R(x), \\iota, x)$ on $R$ satisfy the universal property of a polynomial ring:"}
+{"_id": "14840", "title": "Pandigital Properties of 123,456,789", "text": "$123 \\, 456 \\, 789$ has the following properties: It is pandigital, and remains so when multiplied by $2$, $4$, $5$, $7$ and $8$: {{begin-eqn}} {{eqn | l = 123 \\, 456 \\, 789 \\times 1       | r = 123 \\, 456 \\, 789       | c =  }} {{eqn | l = 123 \\, 456 \\, 789 \\times 2       | r = 246 \\, 913 \\, 578       | c =  }} {{eqn | l = 123 \\, 456 \\, 789 \\times 3       | r = 370 \\, 370 \\, 367       | c =  }} {{eqn | l = 123 \\, 456 \\, 789 \\times 4       | r = 493 \\, 827 \\, 156       | c =  }} {{eqn | l = 123 \\, 456 \\, 789 \\times 5       | r = 617 \\, 283 \\, 945       | c =  }} {{eqn | l = 123 \\, 456 \\, 789 \\times 6       | r = 740 \\, 740 \\, 734       | c =  }} {{eqn | l = 123 \\, 456 \\, 789 \\times 7       | r = 864 \\, 197 \\, 523       | c =  }} {{eqn | l = 123 \\, 456 \\, 789 \\times 8       | r = 987 \\, 654 \\, 312       | c =  }} {{eqn | l = 123 \\, 456 \\, 789 \\times 9       | r = 1 \\, 111 \\, 111 \\, 101       | c =  }} {{end-eqn}} {{expand|Add some mathematical analysis explaining this phenomenon}}"}
+{"_id": "14841", "title": "Pandigital Integers remaining Pandigital on Multiplication", "text": "Certain pandigital integers remain pandigital when multiplying them by certain single-digit integers: {{begin-eqn}} {{eqn | l = 1 \\, 098 \\, 765 \\, 432 \\times 1       | r = 1 \\, 098 \\, 765 \\, 432       | c = which is pandigital }} {{eqn | l = 1 \\, 098 \\, 765 \\, 432 \\times 2       | r = 2 \\, 197 \\, 530 \\, 864       | c = which is pandigital }} {{eqn | l = 1 \\, 098 \\, 765 \\, 432 \\times 3       | r = 3 \\, 296 \\, 296 \\, 296       | c =  }} {{eqn | l = 1 \\, 098 \\, 765 \\, 432 \\times 4       | r = 4 \\, 395 \\, 061 \\, 728       | c = which is pandigital }} {{eqn | l = 1 \\, 098 \\, 765 \\, 432 \\times 5       | r = 5 \\, 493 \\, 827 \\, 160       | c = which is pandigital }} {{eqn | l = 1 \\, 098 \\, 765 \\, 432 \\times 6       | r = 6 \\, 592 \\, 592 \\, 592       | c =  }} {{eqn | l = 1 \\, 098 \\, 765 \\, 432 \\times 7       | r = 7 \\, 691 \\, 358 \\, 024       | c = which is pandigital }} {{eqn | l = 1 \\, 098 \\, 765 \\, 432 \\times 8       | r = 8 \\, 790 \\, 123 \\, 456       | c = which is pandigital }} {{eqn | l = 1 \\, 098 \\, 765 \\, 432 \\times 9       | r = 9 \\, 888 \\, 888 \\, 888       | c =  }} {{end-eqn}} The sequence: :$1039675824, 1053826974, 1068253974, 1068379524, 1073968254, 1075396824, 1098765432, 1204756839, 1234567890, 1357802469$ contains all pandigital integers with at least $4$ nontrivial pandigital multiples, of which: :$1098765432, 1234567890$ has $5$. {{OEIS|A167476}} {{expand|Add some mathematical analysis explaining this phenomenon.}}"}
+{"_id": "14843", "title": "Algebra Defined by Ring Homomorphism is Algebra", "text": "Let $R$ be a commutative ring. Let $\\struct {S, +, *}$ be a ring. Let $f : R \\to S$ be a ring homomorphism. Let the image of $f$ be a subset of the center of $S$. Let $\\struct {S_R, *}$ be the algebra defined by the ring homomorphism $f$. Then $\\struct {S_R, *}$ is an algebra over $R$."}
+{"_id": "14844", "title": "Algebra Defined by Ring Homomorphism is Associative", "text": "Let $R$ be a commutative ring. Let $\\struct {S, +, *}$ be a ring with unity. Let $f: R \\to S$ be a ring homomorphism. Let the image of $f$ be a subset of the center of $S$. Let $\\struct {S_R, *}$ be the algebra defined by the ring homomorphism $f$. Then $\\struct {S_R, *}$ is an associative algebra."}
+{"_id": "14845", "title": "Algebra Defined by Ring Homomorphism on Commutative Ring is Commutative", "text": "Let $R$ be a commutative ring. Let $\\left({S, +, *}\\right)$ be a commutative ring. Let $f: R \\to S$ be a ring homomorphism. Let $\\left({S_R, *}\\right)$ be the algebra defined by the ring homomorphism $f$. Then $\\left({S_R, *}\\right)$ is an commutative algebra."}
+{"_id": "14846", "title": "Smallest Integer which is Sum of 3 Sixth Powers in 2 Ways", "text": "The smallest positive integer which can be expressed as the sum of $3$ sixth powers in $2$ different ways is: {{begin-eqn}} {{eqn | l = 160 \\, 426 \\, 514       | r = 3^6 + 19^6 + 22^6       | c =  }} {{eqn | r = 10^6 + 15^6 + 23^6       | c =  }} {{end-eqn}} Also note that: {{begin-eqn}} {{eqn | l = 854       | r = 3^2 + 19^2 + 22^2       | c =  }} {{eqn | r = 10^2 + 15^2 + 23^2       | c =  }} {{end-eqn}}"}
+{"_id": "14847", "title": "Infinite Number of Integers which are Sum of 3 Sixth Powers in 2 Ways", "text": "There exist an infinite number of positive integers which can be expressed as the sum of $3$ sixth powers in $2$ different ways."}
+{"_id": "14849", "title": "Integer which is Sum of 3 Fourth Powers in 2 Ways and Products of Those Roots", "text": "The positive integer $256 \\, 103 \\, 393$ can be expressed as the sum of $3$ fourth powers in $2$ different ways: {{begin-eqn}} {{eqn | l = 256 \\, 103 \\, 393       | r = 22^4 + 93^4 + 116^4       | c =  }} {{eqn | r = 29^4 + 66^4 + 124^4       | c =  }} {{end-eqn}} Also note that: {{begin-eqn}} {{eqn | l = 237 \\, 336       | r = 22 \\times 93 \\times 116       | c =  }} {{eqn | r = 29 \\times 66 \\times 124       | c =  }} {{end-eqn}}"}
+{"_id": "14850", "title": "Groups of Order 8", "text": "Let $G$ be a group of order $8$. Then $G$ is isomorphic to one of the following: :$\\Z_8$ :$\\Z_4 \\oplus \\Z_2$ :$\\Z_2 \\oplus \\Z_2 \\oplus \\Z_2$ :$D_4$ :$\\Dic 2$ where: :$\\Z_n$ is the cyclic group of order $n$ :$D_4$ is the dihedral group of order $8$ :$\\Dic 2$ is the dicyclic group of order $8$."}
+{"_id": "14851", "title": "Triangular Numbers which are Product of 3 Consecutive Integers", "text": "The $6$ triangular numbers which can be expressed as the product of $3$ consecutive integers are: :$6, 120, 210, 990, 185 \\, 836, 258 \\, 474 \\, 216$ {{OEIS|A001219}}"}
+{"_id": "14852", "title": "First Harmonic Number to exceed 20", "text": "The first harmonic number that is greater than $20$ is $H_{272 \\, 400 \\, 600}$. That is, the number of terms of the harmonic series required for its partial sum to exceed $20$ is $272 \\, 400 \\, 600$."}
+{"_id": "14853", "title": "First Harmonic Number to exceed 10", "text": "The first harmonic number that is greater than $10$ is $H_{12 \\, 367}$. That is, the number of terms of the harmonic series required for its partial sum to exceed $10$ is $12 \\, 367$."}
+{"_id": "14854", "title": "First Harmonic Number to exceed 100", "text": "The first harmonic number that is greater than $100$ is $H_n$ where $n \\approx 1.5 \\times 10^{43}$. That is, it takes approximately $1.5 \\times 10^{43}$ terms of the harmonic series required for its partial sum to exceed $100$."}
+{"_id": "14855", "title": "Number of Magic Squares of Order 5", "text": "Up to rotations and reflections, there are $275 \\, 305 \\, 224$ distinct magic squares of order $5$."}
+{"_id": "14857", "title": "Universal Property of Field of Rational Fractions", "text": "Let $R$ be an integral domain. Let $(R(x), \\iota, x)$ be the field of rational fractions over $R$. Let $(K, f, a)$ be an ordered triple, where: :$K$ is a field :$f : R \\to K$ is a unital ring homomorphism :$a$ is an element of $K$. Then there exists a unique unital ring homomorphism $\\bar f : R(x) \\to K$ such that $\\bar f\\circ\\iota = f$ and $\\bar f(x) = a$. :$\\xymatrix{ R \\ar[d]^\\iota \\ar[r]^{\\forall f} & K\\\\ R(x) \\ar[ru]_{\\exists ! \\bar f} }$"}
+{"_id": "14858", "title": "Equivalence of Definitions of Unital Subalgebra", "text": "Let $R$ be a commutative ring. Let $\\struct {A_R, *}$ be an unital algebra over $R$ whose unit is $1_A$. Let $\\struct {B_R, *}$ be a subalgebra of $A_R$. {{TFAE|def = Unital Subalgebra}}"}
+{"_id": "14860", "title": "Polydivisible Number/Examples/381,654,729", "text": "The integer $381 \\, 654 \\, 729$ is the only polydivisible number which is pandigital in the sense of excluding zero."}
+{"_id": "14861", "title": "Pandigital Product of Pandigital Pairs in 3 Ways", "text": "The pandigital integer $0 \\, 429 \\, 315 \\, 678$ can be expressed as the product of a pandigital doubleton in $3$ different ways: {{begin-eqn}} {{eqn | l = 0 \\, 429 \\, 315 \\, 678       | r = 04 \\, 926 \\times 87 \\, 153 }} {{eqn | r = 07 \\, 923 \\times 54 \\, 186}} {{eqn | r = 15 \\, 846 \\times 27 \\, 093}} {{end-eqn}}"}
+{"_id": "14862", "title": "Smallest Integer which is Sum of 2 Fourth Powers in 2 Ways", "text": "The smallest positive integer which can be expressed as the sum of $2$ fourth powers in $2$ different ways is: {{begin-eqn}} {{eqn | l = 635 \\, 318 \\, 657       | r = 59^4 + 158^4       | c =  }} {{eqn | r = 133^4 + 134^4       | c =  }} {{end-eqn}}"}
+{"_id": "14863", "title": "Largest Pandigital Square less Zero", "text": "The largest pandigital square (in the sense where pandigital excludes the zero) is $923 \\, 187 \\, 456$: :$923 \\, 187 \\, 456 = 30 \\, 384^2$"}
+{"_id": "14864", "title": "Pandigital Properties of 987,654,321", "text": "$987 \\, 654 \\, 321$ has the following properties: It is pandigital, and remains so when multiplied by $1$, $2$, $4$, $5$, $7$ and $8$: {{begin-eqn}} {{eqn | l = 987 \\, 654 \\, 321 \\times 1       | r = 987 \\, 654 \\, 321       | c =  }} {{eqn | l = 987 \\, 654 \\, 321 \\times 2       | r = 1 \\, 975 \\, 308 \\, 642       | c =  }} {{eqn | l = 987 \\, 654 \\, 321 \\times 3       | r = 2 \\, 962 \\, 962 \\, 963       | c =  }} {{eqn | l = 987 \\, 654 \\, 321 \\times 4       | r = 3 \\, 950 \\, 617 \\, 284       | c =  }} {{eqn | l = 987 \\, 654 \\, 321 \\times 5       | r = 4 \\, 938 \\, 271 \\, 605       | c =  }} {{eqn | l = 987 \\, 654 \\, 321 \\times 6       | r = 5 \\, 925 \\, 925 \\, 925       | c =  }} {{eqn | l = 987 \\, 654 \\, 321 \\times 7       | r = 6 \\, 975 \\, 308 \\, 642       | c =  }} {{eqn | l = 987 \\, 654 \\, 321 \\times 8       | r = 7 \\, 901 \\, 234 \\, 568       | c =  }} {{eqn | l = 987 \\, 654 \\, 321 \\times 9       | r = 8 \\, 888 \\, 888 \\, 889       | c =  }} {{end-eqn}} {{expand|Add some mathematical analysis explaining this phenomenon}} Also: :$987 \\, 654 \\, 321 - 123 \\, 456 \\, 789 = 864 \\, 197 \\, 532$ which is also pandigital."}
+{"_id": "14865", "title": "Largest 9-Digit Prime Number", "text": "The largest prime number with $9$ digits is $999 \\, 999 \\, 937$."}
+{"_id": "14866", "title": "Smallest Pandigital Square with Zero", "text": "The smallest pandigital square (in the sense where pandigital includes the zero) is $1 \\, 026 \\, 753 \\, 849$: :$1 \\, 026 \\, 753 \\, 849 = 32 \\, 043^2$"}
+{"_id": "14867", "title": "Sound Proof System is Consistent", "text": "Let $\\mathcal L$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\mathcal L$. Let $\\mathscr P$ be a proof system for $\\mathcal L$. Suppose that $\\mathscr P$ is sound for $\\mathscr M$. Then $\\mathscr P$ is consistent."}
+{"_id": "14869", "title": "Rule of Explosion/Variant 3", "text": ":$p, \\neg p \\vdash q$"}
+{"_id": "14870", "title": "Smallest Integer which is Sum of 3 Fifth Powers in 2 Ways", "text": "The smallest positive integer which can be expressed as the sum of $3$ fifth powers in $2$ different ways: The positive integer $1 \\, 375 \\, 298 \\, 099$ can be expressed as the sum of $3$ fifth powers in $2$ different ways: {{begin-eqn}} {{eqn | l = 1 \\, 375 \\, 298 \\, 099       | r = 24^5 + 28^5 + 67^5       | c =  }} {{eqn | r = 3^5 + 54^5 + 62^5       | c =  }} {{end-eqn}}"}
+{"_id": "14871", "title": "Automorphic Numbers with 10 Digits", "text": "The only $10$-digit automorphic numbers are: :$1 \\, 787 \\, 109 \\, 376$ :$8 \\, 212 \\, 890 \\, 625$"}
+{"_id": "14872", "title": "Left-Truncated Automorphic Number is Automorphic", "text": "Let $n$ be an automorphic number, expressed in some conventional number base. Let any number of digits be removed from the left-hand end of $n$. Then what remains is also an automorphic number."}
+{"_id": "14873", "title": "Square whose Sigma is Cubic", "text": "The number $1 \\, 857 \\, 437 \\, 604$ is a square number whose $\\sigma$ value is a cube."}
+{"_id": "14874", "title": "Largest Right-Truncatable Primes allowing 1", "text": "Let $1$ be temporarily considered to be a prime number. Under that consideration, the largest right-truncatable prime numbers are: :$1 \\, 979 \\, 339 \\, 333$ :$1 \\, 979 \\, 339 \\, 339$"}
+{"_id": "14875", "title": "Completely Multiplicative Function is Multiplicative", "text": "Let $f: \\Z \\to \\Z$ be a function on the integers $\\Z$. Let $f$ be completely multiplicative. Then $f$ is multiplicative."}
+{"_id": "14876", "title": "Pandigital Numbers Divisible by All Integers up to 18", "text": "The following pandigital integers are divisible by all the positive integers up to $18$: :$2 \\, 438 \\, 195 \\, 760$ :$3 \\, 785 \\, 942 \\, 160$ :$4 \\, 753 \\, 869 \\, 120$ :$4 \\, 876 \\, 391 \\, 520$"}
+{"_id": "14877", "title": "Smallest n for which 2^n-3 is Divisible by n", "text": "The smallest positive integer $n$ such that $2^n - 3$ is divisible by $n$ is $4 \\, 700 \\, 063 \\, 497$."}
+{"_id": "14878", "title": "Smallest Odd Abundant Number not Divisible by 3", "text": "The smallest odd abundant number not divisible by $3$ is $5 \\, 391 \\, 411 \\, 025$."}
+{"_id": "14881", "title": "Closure of Irreducible Subspace is Irreducible", "text": "Let $X$ be a topological space. Let $Y \\subset X$ be an irreducible subspace. Then its closure $\\overline Y$ is also irreducible."}
+{"_id": "14882", "title": "Point is Contained in Irreducible Component", "text": "Let $X$ be a topological space. Let $x\\in X$ be a point. Then $x$ is contained in some irreducible component of $X$."}
+{"_id": "14883", "title": "Irreducible Subspace is Contained in Irreducible Component", "text": "Let $X$ be a topological space. Let $Y\\subset X$ be an irreducible subspace. Then there exists an irreducible component of $X$ containing $Y$."}
+{"_id": "14885", "title": "Trivial Topological Space is Irreducible", "text": "Let $X$ be a trivial topological space. Then $X$ is irreducible."}
+{"_id": "14886", "title": "Trivial Topological Space is Indiscrete", "text": "Let $X$ be a trivial topological space. Then $X$ is indiscrete."}
+{"_id": "14887", "title": "Irreducible Components of Hausdorff Space are Points", "text": "Let $X$ be a Hausdorff space. Then the irreducible components of $X$ are the singleton sets."}
+{"_id": "14889", "title": "Pandigital Integer Formed by Digits in Alphabetical Order", "text": "The number $8 \\, 549 \\, 176 \\, 320$ is the pandigital integer formed from the digits from $0$ to $9$ arranged in alphabetical order."}
+{"_id": "14891", "title": "Pandigital Properties of 9,876,543,210", "text": "$9 \\, 876 \\, 543 \\, 210$ has the following properties: :$9 \\, 876 \\, 543 \\, 210 - 0 \\, 123 \\, 456 \\, 789 = 9 \\, 753 \\, 086 \\, 421$ all three terms of which are pandigital. {{expand|Add more}}"}
+{"_id": "14894", "title": "Union of Open Irreducible Non-Disjoint Subspaces is Irreducible", "text": "Let $T = \\left({S, \\tau}\\right)$ be an irreducible toplogical space. Let $U$ and $V$ be open irreducible subspaces. Let their intersection $U \\cap V$ be non-empty. Then their union $U \\cup V$ is irreducible."}
+{"_id": "14896", "title": "Closed Set of Ultraconnected Space is Ultraconnected", "text": "Let $T = \\struct {S, \\tau}$ be an ultraconnected topological space. Let $F \\subset S$ be a closed set in $T$. Then $F$ is ultraconnected."}
+{"_id": "14897", "title": "Noetherian Space is Compact", "text": "Let $X$ be a noetherian topological space. Then $X$ is compact."}
+{"_id": "14898", "title": "Subspace of Noetherian Space is Noetherian", "text": "Let $X$ be a noetherian topological space. Let $Y\\subseteq X$ be a subspace. Then $Y$ is noetherian."}
+{"_id": "14899", "title": "Zero Locus of Larger Set is Smaller", "text": "Let $k$ be a field. Let $n \\ge 1$ be a natural number. Let $A = k \\sqbrk {X_1, \\ldots, X_n}$ be the ring of polynomials in $n$ variables over $k$. Let $I, J \\subseteq A$ be subsets, and $\\map V I$ and $\\map V J$ their zero loci. Let $I \\subseteq J$. Then $\\map V I \\supseteq \\map V J$."}
+{"_id": "14900", "title": "Smallest Multiply Perfect Number of Order 5", "text": "The number $14 \\, 182 \\, 439 \\, 040$ is multiply perfect of order $5$: :$\\sigma \\left({14 \\, 182 \\, 439 \\, 040}\\right) = 70 \\, 912 \\, 195 \\, 200 = 5 \\times 14 \\, 182 \\, 439 \\, 040$ It is the smallest positive integer to be so."}
+{"_id": "14901", "title": "Smallest Fourth Power expressible as Sum of 4 Fourth Powers", "text": "$15 \\, 527 \\, 402 \\, 881$ is the smallest fourth power which can be expressed as the sum of $4$ fourth powers: :$15 \\, 527 \\, 402 \\, 881 = 353^4 = 30^4 + 120^4 + 272^4 + 315^4$"}
+{"_id": "14902", "title": "Largest Known Lead by 4n+1 in Prime Number Race", "text": "In the prime number race between prime numbers of the form $4 n - 1$ and $4 n + 1$, the highest known stretch of integers where $4 n + 1$ is not less than $4 n - 1$ is between $18 \\, 465 \\, 126 \\, 293$ and $19 \\, 033 \\, 524 \\, 538$."}
+{"_id": "14903", "title": "Number whose Square is in 2 Identical Halves", "text": "The number $36 \\, 363 \\, 636 \\, 364$ has the property that its square can be split into two identical halves: :$36 \\, 363 \\, 636 \\, 364 = 1 \\, 322 \\, 314 \\, 049 \\, 613 \\, 223 \\, 140 \\, 496$"}
+{"_id": "14904", "title": "Fifth Power expressible as Sum of 4 Fifth Powers", "text": "$61 \\, 917 \\, 364 \\, 224$ can be expressed as the sum of $4$ fifth powers: :$61 \\, 917 \\, 364 \\, 224 = 144^5 = 27^5 + 84^5 + 110^5 + 133^5$"}
+{"_id": "14905", "title": "Common Sum of 3 Distinct Amicable Pairs", "text": "The integer $64 \\, 795 \\, 852 \\, 800$ is the sum of $3$ distinct amicable pairs: :$29 \\, 912 \\, 035 \\, 725$ and $34 \\, 883 \\, 817 \\, 075$ :$31 \\, 695 \\, 652 \\, 275$ and $33 \\, 100 \\, 200 \\, 525$ :$32 \\, 129 \\, 958 \\, 525$ and $32 \\, 665 \\, 894 \\, 275$ all of them odd."}
+{"_id": "14907", "title": "Probability of Receiving Complete Suit as Hand at Bridge", "text": "The probability of being dealt a complete suit in a deal at Bridge is $1$ in $158 \\, 753 \\, 389 \\, 900$."}
+{"_id": "14908", "title": "Polynomial Ring is Generated by Indeterminate over Ground Ring", "text": "Let $R$ be a commutative ring with unity. Let $R \\sqbrk X$ be a polynomial ring over $R$. Let $\\iota: R \\to R \\sqbrk X$ be the embedding. Then $R \\sqbrk X$ is generated by $X$ over $R$."}
+{"_id": "14909", "title": "Smallest Cunningham Chain of the First Kind of Length 12", "text": "The smallest Cunningham chain of the first kind of length $12$ is: :$554 \\, 688 \\, 278 \\, 429$, $1 \\, 109 \\, 376 \\, 556 \\, 859$, $2 \\, 218 \\, 753 \\, 113 \\, 719$, $4 \\, 437 \\, 506 \\, 227 \\, 439$, ::$8 \\, 875 \\, 012 \\, 454 \\, 879$, $17 \\, 750 \\, 024 \\, 909 \\, 759$, $35 \\, 500 \\, 049 \\, 819 \\, 519$, $71 \\, 000 \\, 099 \\, 639 \\, 039$, ::$142 \\, 000 \\, 199 \\, 278 \\, 079$, $284 \\, 000 \\, 398 \\, 556 \\, 159$, $568 \\, 000 \\, 797 \\, 112 \\, 319$, $1 \\, 136 \\, 001 \\, 594 \\, 224 \\, 639$"}
+{"_id": "14911", "title": "Set of Integers Bounded Above by Real Number has Greatest Element", "text": "Let $\\Z$ be the set of integers. Let $\\le$ be the usual ordering on the real numbers $\\R$. Let $\\O \\subset S \\subseteq \\Z$ such that $S$ is bounded above in $\\struct {\\R, \\le}$. Then $S$ has a greatest element."}
+{"_id": "14913", "title": "Set of Integers Bounded Below by Real Number has Smallest Element", "text": "Let $\\Z$ be the set of integers. Let $\\le$ be the usual ordering on the real numbers $\\R$. Let $\\O \\subset S \\subseteq \\Z$ such that $S$ is bounded below in $\\struct {\\R, \\le}$. Then $S$ has a smallest element."}
+{"_id": "14914", "title": "Real Number lies between Unique Pair of Consecutive Integers", "text": "Let $x$ be a real number."}
+{"_id": "14915", "title": "Supremum of Set of Integers is Integer", "text": "Let $S \\subset \\Z$ be a non-empty subset of the set of integers. Let $S$ be bounded above in the set of real numbers. Then its supremum $\\sup S$ is an integer."}
+{"_id": "14916", "title": "Definition:Constructed Semantics/Instance 1/Rule of Idempotence", "text": "The Rule of Idempotence: :$(p \\lor p) \\implies p$ is a tautology in Instance 1 of constructed semantics."}
+{"_id": "14917", "title": "Definition:Constructed Semantics/Instance 1/Rule of Addition", "text": "The Rule of Addition: :$q \\implies (q \\lor p)$ is a tautology in Instance 1 of constructed semantics."}
+{"_id": "14918", "title": "Definition:Constructed Semantics/Instance 1/Rule of Commutation", "text": "The Rule of Commutation: :$\\left({p \\lor q}\\right) \\implies \\left({q \\lor p}\\right)$ is a tautology in Instance 1 of constructed semantics."}
+{"_id": "14919", "title": "Supremum of Set of Integers equals Greatest Element", "text": "Let $S \\subset \\Z$ be a non-empty subset of the set of integers. Let $S$ be bounded above in the set of real numbers $\\R$. Then $S$ has a greatest element, and it is equal to the supremum $\\sup S$."}
+{"_id": "14920", "title": "Definition:Constructed Semantics/Instance 1/Factor Principle", "text": "The Factor Principle: :$\\left({p \\implies q}\\right) \\implies \\left({\\left({r \\lor p}\\right) \\implies \\left ({r \\lor q}\\right)}\\right)$ is a tautology in Instance 1 of constructed semantics."}
+{"_id": "14921", "title": "Weak Inequality of Integers iff Strict Inequality with Integer plus One", "text": "Let $a, b \\in \\Z$ be integers. {{TFAE}} :$(1): \\quad a \\le b$ :$(2): \\quad a < b + 1$ where: :$\\le$ is the ordering on the integers :$<$ is the strict ordering on the integers."}
+{"_id": "14922", "title": "Set of Orbits forms Partition", "text": "Let $G$ be a group. Let $X$ be a set. Let $G$ act on $X$. Then the set of orbits of the group action forms a partition of $X$."}
+{"_id": "14925", "title": "Interior of Closure is Regular Open", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $H \\subseteq S$. Then $H^{- \\circ}$ is regular open."}
+{"_id": "14931", "title": "Jordan's Lemma", "text": "Consider a complex-valued, continuous function $f$ defined on the contour: :$C_r = \\left\\{r e^{i \\theta}: 0 \\le \\theta \\le \\pi \\right\\}, \\ r > 0$ If the function $f$ is of the form: :$f \\left({z}\\right) = e^{i a z} g \\left({z}\\right), \\ a > 0, \\ z \\in C_r$ Then: :$\\displaystyle \\left\\vert{\\int_{C_r} f \\left({z}\\right) \\rd z}\\right\\vert \\le \\frac \\pi a \\max_{0 \\le \\theta \\le \\pi} \\left\\vert{g \\left(re^{i \\theta}\\right)}\\right\\vert$"}
+{"_id": "14932", "title": "Lobachevsky Integral Formula", "text": "Let $f$ be a continuous function, periodic in $\\pi$. Then: :$\\displaystyle \\int_0^\\infty \\frac {\\sin x} x f \\left({x}\\right) \\rd x = \\int_0^{\\frac \\pi 2} f \\left({x}\\right) \\rd x$"}
+{"_id": "14937", "title": "Dirichlet Series of Convolution of Arithmetic Functions", "text": "Let $f, g: \\N \\to \\C$ be arithmetic functions. Let $h = f * g$ be their Dirichlet convolution. Let $F, G, H$ be their Dirichlet series. Let $s$ be a complex number such that $\\map F s$ and $\\map G s$ converge absolutely. Then $\\map H s$ converges absolutely to $\\map F s \\times \\map G s$."}
+{"_id": "14941", "title": "Riemann Zeta Function in terms of Dirichlet Eta Function", "text": "Let $\\zeta$ be the Riemann zeta function. Let $\\eta$ be the Dirichlet eta function. Let $s \\in \\C$ be a complex number with real part $\\sigma > 1$. Then $\\zeta(s) = \\dfrac 1 {1 - 2^{1-s}} \\eta \\left({s}\\right)$."}
+{"_id": "14942", "title": "Functional Equation for Riemann Zeta Function", "text": "Let $\\zeta$ be the Riemann zeta function. Let $\\map \\zeta s$ have an analytic continuation for $\\map \\Re s > 0$. Then: :$\\displaystyle \\map \\Gamma {\\frac s 2} \\pi^{-s/2} \\map \\zeta s = \\map \\Gamma {\\frac {1 - s} 2} \\pi^{\\frac {s - 1} 2} \\map \\zeta {1 - s}$ where $\\Gamma$ is the gamma function"}
+{"_id": "14943", "title": "Integral Representation of Riemann Zeta Function in terms of Fractional Part", "text": "Let $\\zeta$ be the Riemann zeta function. Let $s\\in\\C$ be a complex number with real part $\\sigma>1$. Then  :$\\displaystyle \\zeta \\left({s}\\right) = \\frac s {s - 1} - s \\int_1^\\infty \\left\\{ {x}\\right\\} x^{-s - 1} \\rd x$ where $\\left\\{{x}\\right\\}$ denotes the fractional part of $x$."}
+{"_id": "14944", "title": "Binomial Form of Relation between Riemann Zeta Function and Dirichlet Eta Function", "text": "Let $\\zeta$ be the Riemann zeta function. Let $s\\in\\C$ be a complex number with real part $\\Re(s)>1$. Then: $\\displaystyle \\zeta \\left({s}\\right) = \\frac 1 {1 - 2^{1-s} } \\sum_{n \\mathop = 0}^\\infty \\left({\\frac 1 {2^{n+1} } \\sum_{k \\mathop = 0}^n \\left({-1}\\right)^k \\binom n k \\left({k + 1}\\right)^{-s} }\\right)$"}
+{"_id": "14946", "title": "Analytic Continuations of Riemann Zeta Function to Complex Plane", "text": "The Riemann zeta function $\\zeta$ has a unique analytic continuation to $\\C\\setminus\\{1\\}$."}
+{"_id": "14947", "title": "Analytic Continuations of Riemann Zeta Function to Right Half-Plane", "text": "The Riemann zeta function has a unique analytic continuation to $\\{s \\in \\C : \\Re(s) > 0\\}\\setminus\\{1\\}$, the half-plane $\\Re(s)>0$ minus the point $s=1$."}
+{"_id": "14949", "title": "Dirichlet Series of Inverse of Arithmetic Function", "text": "Let $f : \\N \\to\\C$ be an arithmetic function. Let $g : \\N \\to \\C$ be an Dirichlet inverse of $f$. Let $F, G$ be their Dirichlet series. Let $s \\in \\C$ such that both $F(s)$ and $G(s)$ converge absolutely. Then $F(s) \\cdot G(s) = 1$."}
+{"_id": "14950", "title": "Invertibility of Arithmetic Functions", "text": "Let $f: \\N \\to \\C$ be an arithmetic function. Then $f$ has a Dirichlet inverse {{iff}}: :$\\map f 1 \\ne 0$"}
+{"_id": "14951", "title": "Equivalence of Definitions of Consistent Set of Formulas", "text": "{{TFAE|def = Consistent (Logic)/Proof System/Propositional Logic|view = Consistent Proof System for Propositional Logic}} Let $\\LL_0$ be the language of propositional logic. Let $\\mathscr P$ be a proof system for $\\LL_0$. Let $\\FF$ be a collection of logical formulas."}
+{"_id": "14952", "title": "Analytic Continuation of Riemann Zeta Function using Mellin Transform of Fractional Part", "text": "Let $\\zeta$ denote the Riemann zeta function. The analytic continuation of $\\zeta$ to the half-plane $\\map \\Re s > 0$ is given by: :$\\displaystyle \\frac s {s - 1} - s \\int_1^\\infty \\fractpart x x^{-s - 1} \\rd x$ where $x^{-s - 1}$ takes the principle value $e^{-\\map \\ln x \\paren {s + 1} }$"}
+{"_id": "14954", "title": "Analytic Continuation of Riemann Zeta Function using Dirichlet Eta Function", "text": "Let $\\zeta$ be the Riemann zeta function. Let $\\eta$ be the Dirichlet Eta Function. Then: :$\\dfrac 1 {1 - 2^{1 - s} } \\map \\eta s$ defines an analytic continuation of $\\zeta$ to the half-plane $\\map \\Re s > 0$ minus $s = 1$."}
+{"_id": "14955", "title": "Analytic Continuation of Riemann Zeta Function using Jacobi Theta Function", "text": "Let $\\zeta$ be the Riemann zeta function. Then :$\\displaystyle\\frac{\\pi^{s/2}}{\\Gamma \\left({\\frac s 2}\\right)} \\cdot\\left( - \\frac 1 {s \\left({1 - s}\\right)} + \\int_1^\\infty \\left({x^{s / 2 - 1} + x^{- \\left({s + 1}\\right) / 2} }\\right) \\omega \\left({x}\\right) \\ \\mathrm d x \\right)$ defines an analytic continuation of $\\zeta$ to the half-plane $\\Re(s)>0$ minus $s=1$."}
+{"_id": "14956", "title": "Complex Modulus of Real Number equals Absolute Value", "text": "Let $x \\in \\R$ be a real number. Then the complex modulus of $x$ equals the absolute value of $x$."}
+{"_id": "14957", "title": "Equivalence of Definitions of Ceiling Function", "text": "Let $x$ be a real number. {{TFAE|def = Ceiling Function}}"}
+{"_id": "14958", "title": "Characterizing Property of Infimum of Subset of Real Numbers", "text": "Let $S \\subset \\R$ be a non-empty subset of the real numbers. Let $S$ be bounded below. Let $\\alpha \\in \\R$. {{TFAE}} :$(1): \\quad \\alpha$ is the infimum of $S$ :$(2): \\quad \\alpha$ is a lower bound for $S$ ::and: :::$\\forall \\epsilon \\in \\R_{> 0}$ there exists $x \\in S$ with $x < \\alpha + \\epsilon$"}
+{"_id": "14959", "title": "Characterizing Property of Supremum of Subset of Real Numbers", "text": "Let $S \\subset \\R$ be a non-empty subset of the real numbers. Let $S$ be bounded above. Let $\\omega \\in \\R$. {{TFAE}} :$(1): \\quad \\omega$ is the supremum of $S$ :$(2): \\quad \\omega$ is an upper bound for $S$ ::and: :::$\\forall \\epsilon \\in \\R_{> 0}$ there exists $x \\in S$ with $x > \\omega - \\epsilon$"}
+{"_id": "14961", "title": "Limit of Positive Real Sequence is Positive", "text": "Let $\\sequence {x_n}$ be a sequence of positive real numbers. Let $x_n$ converge to $L$. Then $L \\ge 0$."}
+{"_id": "14962", "title": "Real Sequence with Nonzero Limit is Eventually Nonzero", "text": "Let $\\sequence {x_n}$ be a real sequence. Let $\\sequence {x_n}$ converge to $a \\ne 0$. Then: :$\\exists N \\in \\N: \\forall n \\ge N: x_n \\ne 0$ That is, eventually every term of $\\sequence {x_n}$ becomes non-zero."}
+{"_id": "14963", "title": "Mittag-Leffler Expansion for Cotangent Function", "text": ":$\\displaystyle \\pi \\cot \\pi z = \\frac 1 z + 2 \\sum_{n \\mathop = 1}^\\infty \\frac z {z^2 - n^2}$ where: :$z \\in \\C$ is not an integer :$\\cot$ is the cotangent function."}
+{"_id": "14964", "title": "Laurent Series Expansion for Cotangent Function", "text": ":$\\displaystyle \\pi \\cot \\pi z = \\frac 1 z - 2 \\sum_{n \\mathop = 1}^\\infty \\map \\zeta {2 n} z^{2 n - 1}$ where: :$z \\in \\C$ such that $\\cmod z < 1$ :$\\zeta$ is the Riemann Zeta function."}
+{"_id": "14967", "title": "Equivalence of Definitions of Removable Discontinuity of Real Function", "text": "Let $A \\subseteq \\R$ be a subset of the real numbers. Let $f : A \\to \\R$ be a real function. Let $f$ be discontinuous at $a\\in A$. {{TFAE|def = Removable Discontinuity of Real Function|view = removable discontinuity}}"}
+{"_id": "14968", "title": "Equivalence of Definitions of Polynomial Function on Subset of Ring", "text": "Let $R$ be a commutative ring with unity. Let $S \\subset R$ be a subset. {{TFAE|def = Polynomial Function/Ring |view = polynomial function}}"}
+{"_id": "14969", "title": "Equivalence of Definitions of Polynomial Ring in Multiple Variables", "text": "Let $R$ be a commutative ring with unity. The following definitions of polynomial ring are equivalent in the following sense: : For every two constructions, there exists an $R$-isomorphism which sends indeterminates to indeterminates. {{explain|this statement has to be made more precise}} === Definition 1: As the monoid ring on a free monoid on a set === {{:Definition:Polynomial Ring/Monoid Ring on Free Monoid on Set}} {{ExpandList}}"}
+{"_id": "14970", "title": "Monomials form Basis of Polynomial Ring/One Variable", "text": "Let $R$ be a commutative ring with unity. Let $R \\sqbrk X$ be a polynomial ring over $R$ in the variable $X$. Then the monomials of $R \\sqbrk X$ are a basis of $R \\sqbrk X$ as a module over $R$."}
+{"_id": "14971", "title": "Equivalence of Definitions of Polynomial in Ring Element", "text": "Let $R$ be a commutative ring. Let $S$ be a subring with unity of $R$. Let $x\\in R$. {{TFAE|def = Polynomial in Ring Element}}"}
+{"_id": "14974", "title": "Equality of Monomials of Polynomial Ring in One Variable", "text": "Let $R$ be a commutative ring with unity. Let $R \\sqbrk X$ be a polynomial ring in one variable $X$ over $R$. Let $k, l \\in \\N$ be distinct natural numbers. Then the mononomials $X^k$ and $X^l$ are distinct, where $X^k$ denotes the $k$th power of $X$."}
+{"_id": "14976", "title": "Ring Isomorphic to Polynomial Ring is Polynomial Ring/One Variable", "text": "Let $R$ be a commutative ring with unity. Let $R \\sqbrk X$ be a polynomial ring in one variable $X$ over $R$. Let $\\iota : R \\to R \\sqbrk X$ denote the canonical embedding. Let $S$ be a commutative ring with unity and $f: R \\sqbrk X \\to S$ be a ring isomorphism. Then $\\struct {S, f \\circ \\iota, \\map f X}$ is a polynomial ring in one variable $\\map f X$ over $R$."}
+{"_id": "14977", "title": "Equality of Monomials of Polynomial Ring in Multiple Variables", "text": "Let $R$ be a commutative ring with unity. Let $I$ be a set. Let $R \\sqbrk {\\sequence {x_i}_{i \\mathop \\in I} }$ be a polynomial ring in $I$ variables $\\sequence {x_i}_{i \\mathop \\in I}$ over $R$. Let $a, b : I \\to \\N$ be distinct mappings with finite support. Then the monomials $\\ds \\prod_{i \\mathop \\in I} X_i^{a_i}$ and $\\ds \\prod_{i \\mathop \\in I} X_i^{b_i}$ are distinct, where: :$X_i^k$ denotes the $k$th power of $X_i$ :$\\prod$ denotes product with finite support"}
+{"_id": "14980", "title": "Mittag-Leffler Expansion for Cosecant Function", "text": ":$\\displaystyle \\pi \\cosec \\pi z = \\frac 1 z + 2\\sum_{n \\mathop = 1}^\\infty \\paren {-1}^n \\frac z {z^2 - n^2}$"}
+{"_id": "14981", "title": "Upper Sum Never Smaller than Lower Sum for any Pair of Subdivisions", "text": "Let $\\closedint a b$ be a closed real interval. Let $f$ be a bounded real function defined on $\\closedint a b$. Let $P$ and $Q$ be finite subdivisions of $\\closedint a b$. Let $\\map L P$ be the lower sum of $f$ on $\\closedint a b$ with respect to $P$. Let $\\map U Q$ be the upper sum of $f$ on $\\closedint a b$ with respect to $Q$. Then $\\map L P \\le \\map U Q$."}
+{"_id": "14982", "title": "Set Finite iff Surjection from Initial Segment of Natural Numbers", "text": "Let $S$ be a set. Then $S$ is finite {{iff}} for some $n \\in \\N$ there exists a surjection $f: \\N_{< n} \\to S$. Here, $\\N_{< n}$ denotes an initial segment of $\\N$."}
+{"_id": "14983", "title": "Set Finite iff Injection to Initial Segment of Natural Numbers", "text": "Let $S$ be a set. Then $S$ is finite {{iff}} for some $n \\in \\N$ there exists an injection $f: S \\to \\N_{< n}$. Here, $\\N_{< n}$ denotes an initial segment of $\\N$."}
+{"_id": "14989", "title": "Finite Product of Finite Sets is Finite", "text": "Let $\\sequence {S_n}$ be a sequence of finite sets. Let $\\displaystyle \\prod_{k \\mathop = 1}^n S_k$ be their Cartesian product. Then $\\displaystyle \\prod_{k \\mathop = 1}^n S_k$ is also a finite set."}
+{"_id": "14991", "title": "Existence of Integers with Multiplicative Persistence Greater than 11", "text": "It is not known whether there exists a number $n$ such that: :$P \\left({n}\\right) = 12$ where $P \\left({n}\\right)$ denotes the multiplicative persistence of $n$."}
+{"_id": "14992", "title": "Smallest Strong Fibonacci Pseudoprime of Type I", "text": "The smallest strong Fibonacci pseudoprime of type I is $443 \\, 372 \\, 888 \\, 629 \\, 441$."}
+{"_id": "14993", "title": "Infimum of Upper Sums Never Smaller than Lower Sum", "text": ":$\\inf_P \\map U P \\ge \\map L S$"}
+{"_id": "14994", "title": "Supremum of Lower Sums Never Greater than Upper Sum", "text": ":$\\sup_P \\map L P \\le \\map U S$"}
+{"_id": "14996", "title": "Properties of 5,559,060,566,555,523", "text": "$3^{33} = 5 \\, 559 \\, 060 \\, 566 \\, 555 \\, 523$ has the following properties: : It has a remarkably large number of $5$s (half of its digits). : Multiply it by $2$, $4$ or $6$, and the result has $10$ of a particular digit. : Multiply it by $8$, and $9$ of the digits of the result are $4$."}
+{"_id": "14997", "title": "First Occurrence of Prime Gap of 864", "text": "The first occurrence of a prime gap of $864$ is between $6 \\, 505 \\, 941 \\, 701 \\, 960 \\, 039$ and $6 \\, 505 \\, 941 \\, 701 \\, 960 \\, 903$."}
+{"_id": "14998", "title": "Hardy-Ramanujan Number/Examples/48,988,659,276,962,496", "text": "The $5$th Hardy-Ramanujan number $\\map {\\operatorname {Ta} } 5$ is $48 \\, 988 \\, 659 \\, 276 \\, 962 \\, 496$: {{begin-eqn}} {{eqn | l = 48 \\, 988 \\, 659 \\, 276 \\, 962 \\, 496       | r = 38 \\, 787^3 + 365 \\, 757^3       | c =  }} {{eqn | r = 107 \\, 839^3 + 362 \\, 753^3       | c =  }} {{eqn | r = 205 \\, 292^3 + 342 \\, 952^3       | c =  }} {{eqn | r = 221 \\, 424^3 + 336 \\, 588^3       | c =  }} {{eqn | r = 231 \\, 518^3 + 331 \\, 954^3       | c =  }} {{end-eqn}}"}
+{"_id": "14999", "title": "Reciprocal of 19 from Sum of Powers of 2 Backwards", "text": "The decimal expansion of the reciprocal of $19$ can be constructed by summing the powers of $2$, offset progressively backwards by $1$ digit: <pre>                          1                         2                        4                       8                     16                    32                   64                 128                256               512             1024            2048           4096          8192        16384       32768      65536    131072 + 262144  ...... -------------------------- .......1052631578947368421 </pre> while: :$\\dfrac 1 {19} = 0 \\cdotp \\dot 05263 \\, 15789 \\, 47368 \\, 42 \\dot 1$"}
+{"_id": "15000", "title": "Multiples of Reciprocals with Maximum Period form Magic Square", "text": "Let $p$ be a prime number whose reciprocal has a decimal expansion which has the maximum period, that is, $p - 1$. Let the first $p - 1$ digits of $\\dfrac 1 p$ be expressed as an integer $n$ (with one or more leading zeroes as it is presented). Then the first $p - 1$ multiples of $n$, when listed in order of size, arrange themselves as a magic square such that: :the sum of the elements of each row :the sum of the elements in each column are the same."}
+{"_id": "15001", "title": "General Fibonacci Sequence whose Terms are all Composite", "text": "The general Fibonacci sequence $\\left\\langle{a_n}\\right\\rangle$ defined as: :$a_n = \\begin{cases} r & : n = 0 \\\\ s & : n = 1 \\\\ a_{n - 2} + a_{n - 1} & : n > 1 \\end{cases}$ where: :$r = 62 \\, 638 \\, 280 \\, 004 \\, 239 \\, 857$ :$s = 49 \\, 463 \\, 435 \\, 743 \\, 205 \\, 655$ is such that: : $r$ and $s$ are coprime : all its terms are composite."}
+{"_id": "15002", "title": "Superset of Infinite Set is Infinite", "text": "Let $S$ be an infinite set. Let $T \\supseteq S$ be a superset of $S$. Then $T$ is also infinite."}
+{"_id": "15003", "title": "Prime Decomposition of 2^58+1", "text": "The number $2^{58} + 1$ has the prime decomposition: :$2^{58} + 1 = 5 \\times 107 \\, 367 \\, 629 \\times 536 \\, 903 \\, 681$"}
+{"_id": "15004", "title": "Sum of Squares as Product of Factors with Square Roots", "text": ":$x^2 + y^2 = \\left({x + \\sqrt {2 x y} + y}\\right) \\left({x - \\sqrt {2 x y} + y}\\right)$"}
+{"_id": "15006", "title": "Prime Decomposition of Repunits", "text": "The prime decomposition of the repunits is as follows: {{begin-eqn}} {{eqn | l = R_2       | o =        | r = 11       | c =  }} {{eqn | l = R_3       | r = 3 \\times 37       | c =  }} {{eqn | l = R_4       | r = 11 \\times 101       | c =  }} {{eqn | l = R_5       | r = 41 \\times 271       | c =  }} {{eqn | l = R_6       | r = 11 \\times 111 \\times 91       | c =  }} {{eqn | r = 3 \\times 7 \\times 11 \\times 13 \\times 37       | c =  }} {{eqn | l = R_7       | r = 239 \\times 4649       | c =  }} {{eqn | l = R_8       | r = 1111 \\times 10 \\, 001       | c =  }} {{eqn | r = 11 \\times 73 \\times 101 \\times 137       | c =  }} {{eqn | l = R_9       | r = 111 \\times 1 \\, 001 \\, 001       | c =  }} {{eqn | r = 3^2 \\times 37 \\times 333 \\, 667       | c =  }} {{eqn | l = R_{10}       | r = 11 \\times 11 \\, 111 \\times 9091       | c =  }} {{eqn | r = 11 \\times 41 \\times 271 \\times 9091       | c =  }} {{eqn | l = R_{11}       | r = 21 \\, 649 \\times 513 \\, 239       | c =  }} {{eqn | l = R_{12}       | r = 111 \\times 1111 \\times 900 \\, 991       | c =  }} {{eqn | r = 3 \\times 7 \\times 11 \\times 13 \\times 37 \\times 101 \\times 9901       | c =  }} {{eqn | l = R_{13}       | r = 53 \\times 79 \\times 265 \\, 371 \\, 653       | c =  }} {{eqn | l = R_{14}       | r = 11 \\times 1 \\, 111 \\, 111 \\times 909 \\, 091       | c =  }} {{eqn | r = 11 \\times 239 \\times 4649 \\times 909 \\, 091       | c =  }} {{eqn | l = R_{15}       | r = 111 \\times 11 \\, 111 \\times 90 \\, 090 \\, 991       | c =  }} {{eqn | r = 3 \\times 31 \\times 37 \\times 41 \\times 271 \\times 2 \\, 906 \\, 161       | c =  }} {{eqn | l = R_{16}       | r = 11 \\, 111 \\, 111 \\times 100 \\, 000 \\, 001       | c =  }} {{eqn | r = 11 \\times 17 \\times 73 \\times 101 \\times 137 \\times 5 \\, 882 \\, 353       | c =  }} {{eqn | l = R_{17}       | r = 2 \\, 071 \\, 723 \\times 5 \\, 363 \\, 222 \\, 357       | c =  }} {{eqn | l = R_{18}       | r = 111 \\, 111 \\, 111 \\times 1 \\, 000 \\, 000 \\, 001       | c =  }} {{eqn | r = 3^2 \\times 7 \\times 11 \\times 13 \\times 19 \\times 37 \\times 52 \\, 579 \\times 333 \\, 667       | c =  }} {{eqn | l = R_{19}       | r = 1 \\, 111 \\, 111 \\, 111 \\, 111 \\, 111 \\, 111       | c =  }} {{eqn | l = R_{20}       | r = 1111 \\times 11111 \\times 900 \\, 099 \\, 009 \\, 991       | c =  }} {{eqn | r = 11 \\times 41 \\times 101 \\times 271 \\times 3541 \\times 9091 \\times 27 \\, 961       | c =  }} {{end-eqn}} {{expand|Go as far as we want}}"}
+{"_id": "15007", "title": "Repunit cannot be Square", "text": "A repunit (apart from the trivial $1$) cannot be a square."}
+{"_id": "15008", "title": "Second Principle of Recursive Definition", "text": "Let $\\N$ be the natural numbers. Let $T$ be a set. Let $a \\in T$. For each $n \\in \\N_{>0}$, let $G_n: T^n \\to T$ be a mapping. Then there exists exactly one mapping $f: \\N \\to T$ such that: :$\\forall x \\in \\N: f(x) = \\begin{cases} a & : x = 0\\\\ G_n \\left({f(0), \\ldots, f(n)}\\right) & : x = n + 1 \\end{cases}$"}
+{"_id": "15009", "title": "Repunit cannot be Cube", "text": "A repunit (apart from the trivial $1$) cannot be a cube."}
+{"_id": "15010", "title": "Condition for Repunits to be Coprime", "text": "Let $R_p$ and $R_q$ be repunit numbers with $p$ and $q$ digits respectively. Then $R_p$ and $R_q$ are coprime {{iff}} $p$ and $q$ are coprime."}
+{"_id": "15011", "title": "Divisors of Repunit with Composite Index", "text": "Let $R_n$ be a repunit number with $n$ digits. Let $n$ be composite such that $n = r s$ where $1 < r < n$ and $1 < s < n$. Then $R_r$ and $R_s$ are both divisors of $R_n$."}
+{"_id": "15013", "title": "Square of Small Repunit is Palindromic", "text": "The squares of repunits with up to $9$ digits are palindromic."}
+{"_id": "15014", "title": "Square Number ending in 9 Digits in Reverse Order", "text": "The next integer after $1 \\, 111 \\, 111 \\, 111$ whose square ends in $987 \\, 654 \\, 321$ is: :$2 \\, 380 \\, 642 \\, 361^2 = 5 \\, 667 \\, 458 \\, 050 \\, 987 \\, 654 \\, 321$"}
+{"_id": "15015", "title": "5 Numbers such that Sum of any 3 is Square", "text": "This set of $5$ integers has the property that the sum of any $3$ of them is square: {{begin-eqn}} {{eqn | l = 26 \\, 072 \\, 323 \\, 311 \\, 568 \\, 661 \\, 931       | o =  }} {{eqn | l = 43 \\, 744 \\, 839 \\, 742 \\, 282 \\, 591 \\, 947       | o =  }} {{eqn | l = 118 \\, 132 \\, 654 \\, 413 \\, 675 \\, 138 \\, 222       | o =  }} {{eqn | l = 186 \\, 378 \\, 732 \\, 807 \\, 587 \\, 076 \\, 747       | o =  }} {{eqn | l = 519 \\, 650 \\, 114 \\, 814 \\, 905 \\, 002 \\, 347       | o =  }} {{end-eqn}}"}
+{"_id": "15016", "title": "Total Number of Reachable Positions on Rubik's Cube", "text": "The total number $N$ of reachable positions on Rubik's cube is: {{begin-eqn}} {{eqn | l = N       | r = 43 \\, 252 \\, 003 \\, 274 \\, 489 \\, 856 \\, 000       | c =  }} {{eqn | r = \\dfrac {8! \\times 12! \\times 3^8 \\times 2^{12} } {2 \\times 3 \\times 3}       | c =  }} {{end-eqn}}"}
+{"_id": "15018", "title": "Smallest Number which is Multiplied by 99 by Appending 1 to Each End", "text": "The smallest positive integer which is multiplied by $99$ when $1$ is appended to each end is: :$112 \\, 359 \\, 550 \\, 561 \\, 797  \\, 732 \\, 809$"}
+{"_id": "15019", "title": "Smallest Multiply Perfect Number of Order 6", "text": "The number $154 \\, 345 \\, 556 \\, 085 \\, 770 \\, 649 \\, 600$ is multiply perfect of order $6$: {{begin-eqn}} {{eqn | l = \\sigma \\left({154 \\, 345 \\, 556 \\, 085 \\, 770 \\, 649 \\, 600}\\right)       | r = 926 \\, 073 \\, 336 \\, 514 \\, 623 \\, 897 \\, 600       | c =  }} {{eqn | r = 6 \\times 154 \\, 345 \\, 556 \\, 085 \\, 770 \\, 649 \\, 600       | c =  }} {{end-eqn}} It is the smallest positive integer to be so."}
+{"_id": "15020", "title": "Sequence of 9 Consecutive Integers each with 48 Divisors", "text": "The $9$ integers beginning $17 \\, 796 \\, 126 \\, 877 \\, 482 \\, 329 \\, 126 \\, 044$ each has $48$ divisors."}
+{"_id": "15022", "title": "Polydivisible Number/Examples/3,608,528,850,368,400,786,036,725", "text": "The largest polydivisible number has $25$ digits: :$3 \\, 608 \\, 528 \\, 850 \\, 368 \\, 400 \\, 786 \\, 036 \\, 725$"}
+{"_id": "15023", "title": "No Polydivisible Number with 26 Digits Exists", "text": "There exists no polydivisible number with $26$ digits or more."}
+{"_id": "15024", "title": "Probability of All Players receiving Complete Suit at Bridge", "text": "The probability of all $4$ players in a game of Bridge being dealt a complete suit is $1$ in $2 \\, 235 \\, 197 \\, 406 \\, 895 \\, 366 \\, 368 \\, 301 \\, 560 \\, 000$."}
+{"_id": "15025", "title": "Powers of 10 Expressible as Product of 2 Zero-Free Factors", "text": "The powers of $10$ which can be expressed as the product of $2$ factors neither of which has a zero in its decimal representation are: {{begin-eqn}} {{eqn | l = 10^1       | r = 2 \\times 5       | c =  }} {{eqn | l = 10^2       | r = 4 \\times 25 }} {{eqn | l = 10^3       | r = 8 \\times 125 }} {{eqn | l = 10^4       | r = 16 \\times 625 }} {{eqn | l = 10^5       | r = 32 \\times 3125 }} {{eqn | l = 10^6       | r = 64 \\times 15 \\, 625 }} {{eqn | l = 10^7       | r = 128 \\times 78 \\, 125 }} {{eqn | l = 10^9       | r = 512 \\times 1 \\, 953 \\, 125 }} {{eqn | l = 10^{18}       | r = 262 \\, 144 \\times 3 \\, 814 \\, 697 \\, 265 \\, 625 }} {{eqn | l = 10^{33}       | r = 8 \\, 589 \\, 934 \\, 592 \\times 116 \\, 415 \\, 321 \\, 826 \\, 934 \\, 814 \\, 453 \\, 125 }} {{end-eqn}}"}
+{"_id": "15026", "title": "Sequence of Palindromic Sophie Germain Primes", "text": "The number $N = 191 \\, 918 \\, 080 \\, 818 \\, 091 \\, 909 \\, 090 \\, 909 \\, 190 \\, 818 \\, 080 \\, 819 \\, 191$ has the property that: :$N$ is a palindromic Sophie Germain prime :$2 N + 1$ is also a palindromic Sophie Germain prime :$2 \\left({2 N + 1}\\right) + 1$ is also a palindromic prime, but not a Sophie Germain prime."}
+{"_id": "15029", "title": "Power of 2 containing no Digit 2", "text": "$2^{168}$ contains no $2$ anywhere in its decimal representation."}
+{"_id": "15030", "title": "Order of Fischer-Griess Monster", "text": "The order of the Fischer-Griess Monster $\\mathrm M$ is given as: {{begin-eqn}} {{eqn | l = \\order {\\mathrm M}       | r = 2^{46} \\times 3^{20} \\times 5^9 \\times 7^6 \\times 11^2 \\times 13^3 \\times 17 \\times 19 \\times 23 \\times 29 \\times 31 \\times 41 \\times 47 \\times 59 \\times 71       | c =  }} {{eqn | r = 808 \\, 017 \\, 424 \\, 794 \\, 512 \\, 875 \\, 886 \\, 459 \\, 904 \\, 961 \\, 710 \\, 757 \\, 005 \\, 754 \\, 368 \\, 000 \\, 000 \\, 000       | c =  }} {{end-eqn}}"}
+{"_id": "15031", "title": "Smallest Cube whose Sum of Divisors is Cube", "text": "The smallest cube $N$ such that $\\map \\sigma N$ is also a cube is: :$27 \\, 418 \\, 521 \\, 963 \\, 671 \\, 501 \\, 273 \\, 905 \\, 190 \\, 135 \\, 082 \\, 692 \\, 041 \\, 730 \\, 405 \\, 303 \\, 870 \\, 249 \\, 023 \\, 209$ where $\\map \\sigma N$ denotes the $\\sigma$ function of $N$: the sum of the divisors of $N$"}
+{"_id": "15032", "title": "Uniqueness of Polynomial Ring in One Variable", "text": "Let $R$ be a commutative ring with unity. Let $\\struct {R \\sqbrk X, \\iota, X}$ and $\\struct {R \\sqbrk Y, \\kappa, Y}$ be polynomial rings in one variable over $R$. Then there exists a unique ring homomorphism $f: R \\sqbrk X \\to R \\sqbrk Y$ such that: :$f \\circ \\iota = \\kappa$ :$\\map f X = Y$ and it is an isomorphism."}
+{"_id": "15033", "title": "Monomials of Polynomial Ring are Linearly Independent/One Variable", "text": "Let $R$ be a commutative ring with unity. Let $R \\sqbrk X$ be a polynomial ring in one variable $X$ over $R$. Then the set of monomials $\\set {X^k : k \\in \\N}$ is linearly independent."}
+{"_id": "15034", "title": "Equivalence of Definitions of Constant Polynomial", "text": "Let $R$ be a commutative ring with unity. Let $P\\in R[x]$ be a polynomial in one variable over $R$. {{TFAE|def = Constant Polynomial}}"}
+{"_id": "15036", "title": "Algebraic Element of Field Extension is Root of Unique Monic Polynomial of Minimal Degree", "text": "Let $L / K$ be a field extension. Let $\\alpha \\in L$ be algebraic over $K$. Then there exists a unique monic polynomial $f \\in K \\sqbrk x$ of smallest degree such that $\\map f \\alpha = 0$, called the '''minimal polynomial'''."}
+{"_id": "15037", "title": "Algebraic Element of Field Extension is Root of Unique Monic Irreducible Polynomial", "text": "Let $L/K$ be a field extension. Let $\\alpha \\in L$ be algebraic over $K$. Then there exists a unique irreducible monic polynomial $f\\in K[x]$ such that $f(\\alpha) = 0$, called the '''minimal polynomial'''."}
+{"_id": "15038", "title": "Annihilating Polynomial of Minimal Degree is Irreducible", "text": "Let $L / K$ be a field extension. Let $\\alpha \\in L$ be algebraic over $K$. Let $f \\in K[x]$ be a nonzero polynomial of minimal degree such that $\\map f \\alpha = 0$. Then $f$ is irreducible in $K \\sqbrk x$."}
+{"_id": "15039", "title": "Nonzero Ideal of Polynomial Ring over Field has Unique Monic Generator", "text": "Let $K$ be a field. Let $K \\sqbrk x$ be the polynomial ring in one variable over $K$. Let $I \\subseteq K \\sqbrk x$ be a nonzero ideal. Then $I$ is generated by a unique monic polynomial."}
+{"_id": "15040", "title": "Equivalence of Definitions of Integral Element of Algebra", "text": "{{TFAE|def = Integral Element of Algebra}} Let $A$ be a commutative ring with unity. Let $f : A \\to B$ be a commutative $A$-algebra. Let $b\\in B$."}
+{"_id": "15041", "title": "Product of Closed Sets is Closed", "text": "Let $\\family {\\struct {S_i, \\tau_i} }_{i \\mathop \\in I}$ be a family of topological spaces where $I$ is an arbitrary indexing set. Let $\\displaystyle S = \\prod_{i \\mathop \\in I} S_i$. Let $T = \\struct {S, \\TT}$ be the product space of $\\family {\\struct {S_i, \\tau_i} }_{i \\mathop \\in I}$ with Tychonoff topology $\\TT$. Suppose we have an indexed family of sets $\\family {C_i}_{i \\mathop \\in I}$, where each $C_i$ is closed in $\\struct {S_i, \\tau_i}$. Then $\\displaystyle \\prod_{i \\mathop \\in I} C_i$ is closed in $\\struct {S, \\TT}$."}
+{"_id": "15042", "title": "Prime Decomposition of 9th Fermat Number", "text": "The prime decomposition of the $9$th Fermat number is given by: {{begin-eqn}} {{eqn | l = 2^{\\left({2^9}\\right)} + 1       | r = 13 \\, 407 \\, 807 \\, 929 \\, 942 \\, 597 \\, 099 \\, 574 \\, 024 \\, 998 \\, 205 \\, 846 \\, 127 \\, 479 \\, 365 \\, 820 \\, 592 \\, 393 \\, 377 \\, 723 \\, 561 \\, 443 \\, 721 \\, 764 \\, 030 \\, 073 \\, 546 \\, 976 \\, 801 \\, 874 \\, 298 \\, 166 \\, 903 \\, 427 \\, 690 \\, 031 \\, 858 \\, 186 \\, 486 \\, 050 \\, 853 \\, 753 \\, 882 \\, 811 \\, 946 \\, 569 \\, 946 \\, 433 \\, 649 \\, 006 \\, 084 \\, 097       | c =  }} {{eqn | r = 2 \\, 424 \\, 833       | c =  }} {{eqn | o =       | ro= \\times       | r = 7 \\, 455 \\, 602 \\, 825 \\, 647 \\, 884 \\, 208 \\, 337 \\, 395 \\, 736 \\, 200 \\, 454 \\, 918 \\, 783 \\, 366 \\, 342 \\, 657       | c =  }} {{eqn | o =       | ro= \\times       | r = 741 \\, 640 \\, 062 \\, 627 \\, 530 \\, 801 \\, 524 \\, 787 \\, 141 \\, 901 \\, 937 \\, 474 \\, 059 \\, 940 \\, 781 \\, 097 \\, 519 \\, 023 \\, 905 \\, 821 \\, 316 \\, 144 \\, 415 \\, 759 \\, 504 \\, 705 \\, 008 \\, 092 \\, 818 \\, 711 \\, 693 \\, 940 \\, 737       | c =  }} {{eqn | r = \\paren {2^5 \\times 37 \\times 2^{11} + 1}       | c =  }} {{eqn | o =       | ro= \\times       | r = \\paren {19 \\times 47 \\times 82 \\, 488 \\, 781 \\times 1 \\, 143 \\, 290 \\, 228 \\, 161 \\, 321 \\times 43 \\, 226 \\, 490 \\, 359 \\, 557 \\, 706 \\, 629 \\times 2^{11} + 1}       | c =  }} {{eqn | o =       | ro= \\times       | r = \\paren {1129 \\times 26 \\, 813 \\times 40 \\, 644 \\, 377 \\times 17 \\, 338 \\, 437 \\, 577 \\, 121 \\times 16 \\, 975 \\, 143 \\, 302 \\, 271 \\, 505 \\, 426 \\, 897 \\, 585 \\, 653 \\, 131 \\, 126 \\, 520 \\, 182 \\, 328 \\, 037 \\, 821 \\, 729 \\, 720 \\, 833 \\, 840 \\, 187 \\, 223 \\times 2^{11} + 1}       | c =  }} {{end-eqn}}"}
+{"_id": "15043", "title": "Prime Number Formed by Concatenating Consecutive Integers down to 1", "text": "Let $N$ be an integer whose decimal representation consists of the concatenation of all the integers from a given $n$ in descending order down to $1$. Let the $N$ that is so formed be prime. The only $n$ less than $100$ for which this is true is $82$. That is: :$82 \\, 818 \\, 079 \\, 787 \\, 776 \\ldots 121 \\, 110 \\, 987 \\, 654 \\, 321$ is the only prime number formed this way starting at $100$ or less."}
+{"_id": "15044", "title": "Yoneda Lemma for Covariant Functors", "text": "Let $C$ be a locally small category. Let $\\mathbf{Set}$ be the category of sets. === Bijection === {{:Bijection in Yoneda Lemma for Covariant Functors}} === Naturality === {{:Naturality of Yoneda Lemma for Covariant Functors}}"}
+{"_id": "15045", "title": "Naturality of Yoneda Lemma for Covariant Functors", "text": "Let $[C, \\mathbf{Set}]$ be the covariant functor category. Let $C \\times [C, \\mathbf{Set}] $ be the product category. Let $C \\times [C, \\mathbf{Set}] \\to \\mathbf{Set} : (A, F) \\mapsto \\operatorname{Nat}(h^A, F)$ be the covariant functor defined as the composition of the hom bifunctor and the product of the contravariant Yoneda functor $h^-$ and the identity functor $\\operatorname{id}_{[C, \\mathbf{Set}]}$. Let $\\operatorname{ev} : C \\times [C, \\mathbf{Set}] \\to \\mathbf{Set} : (A, F) \\mapsto F(A)$ be the functor evaluation functor. Then $\\Phi_{(A, F)} : \\operatorname{Nat}(h^A, F) \\to F(A) : \\eta \\mapsto \\eta_A(\\operatorname{id}_A)$ defines a natural isomorphism, where $\\operatorname{id}_A$ is the identity morphism of $A$."}
+{"_id": "15047", "title": "Alternating Even-Odd Digit Palindromic Prime", "text": "Let the notation $\\left({abc}\\right)_n$ be interpreted to mean $n$ consecutive repetitions of a string of digits $abc$ concatenated in the decimal representation of an integer. The integer: :$\\left({10987654321234567890}\\right)_{42} 1$ has the following properties: : it is a palindromic prime with $841$ digits : its digits are alternately odd and even."}
+{"_id": "15050", "title": "Bijection in Yoneda Lemma for Covariant Functors", "text": "Let $F: C \\to \\mathbf {Set}$ be a covariant functor. Let $A \\in C$ be an object. Let $I_A$ be its identity morphism. Let $h^A = \\map {\\operatorname {Hom} } {A, -}$ be its covariant hom-functor. The class of natural transformations $\\map {\\operatorname {Nat} } {h^A, F}$ is a small class, and: :$\\alpha: \\map {\\operatorname {Nat} } {h^A, F} \\to \\map F A: \\eta \\mapsto \\map {\\eta_A} {I_A}$ :$\\beta: \\map F A \\to \\map {\\operatorname {Nat} } {h^A, F}: u \\mapsto \\paren {X \\mapsto \\paren {f \\mapsto \\map {\\paren {\\map F f} } u} }$ are inverses of each other."}
+{"_id": "15051", "title": "Yoneda Lemma for Contravariant Functors", "text": "Let $C$ be a locally small category. Let $\\mathbf{Set}$ be the category of sets. === Bijection === {{:Bijection in Yoneda Lemma for Contravariant Functors}} === Naturality === {{:Naturality of Yoneda Lemma for Contravariant Functors}}"}
+{"_id": "15052", "title": "Naturality of Yoneda Lemma for Contravariant Functors", "text": "Let $[C^{\\operatorname{op}}, \\mathbf{Set}]$ be the contravariant functor category. Let $C^{\\operatorname{op}} \\times [C^{\\operatorname{op}}, \\mathbf{Set}] $ be the product category. Let $C^{\\operatorname{op}} \\times [C^{\\operatorname{op}}, \\mathbf{Set}] \\to \\mathbf{Set} : (A, F) \\mapsto \\operatorname{Nat}(h_A, F)$ be the covariant functor defined as the composition of the hom bifunctor and the product of the opposite of the covariant Yoneda functor $h_-$ and the identity functor $\\operatorname{id}_{[C^{\\operatorname{op}}, \\mathbf{Set}]}$. Let $\\operatorname{ev} : C^{\\operatorname{op}} \\times [C^{\\operatorname{op}}, \\mathbf{Set}] \\to \\mathbf{Set} : (A, F) \\mapsto F(A)$ be the contravariant functor evaluation functor. Then $\\Phi_{(A, F)} : \\operatorname{Nat}(h_A, F) \\to F(A) : \\eta \\mapsto \\eta_A(\\operatorname{id}_A)$ defines a natural isomorphism, where $\\operatorname{id}_A$ is the identity morphism of $A$."}
+{"_id": "15053", "title": "Period of Reciprocal of Repunit 1031 is 1031", "text": "The decimal expansion of the reciprocal of the repunit prime $R_{1031}$ has a period of $1031$. :$\\dfrac 1 {R_{1031}} = 0 \\cdotp \\underbrace{\\dot 000 \\ldots 000}_{1030} \\dot 9$ This is the only prime number to have a period of exactly $1031$."}
+{"_id": "15054", "title": "Titanic Prime whose Digits are all Prime", "text": "The integer defined as: :$7532 \\times \\dfrac {10^{1104} - 1} {10^4 - 1} + 1$ is a titanic prime all of whose digits are themselves prime. That is: :$\\underbrace{7532}_{275} 7533$"}
+{"_id": "15055", "title": "Titanic Prime whose Digits are all 0 or 1", "text": "The integer defined as: :$10^{641} \\times \\dfrac{10^{640} - 1} 9 + 1$ is a titanic prime all of whose digits are either $0$ or $1$. That is: :$\\left({1}\\right)_{640} \\left({0}\\right)_{640} 1$ where $\\left({a}\\right)_b$ means $b$ instances of $a$ in a string."}
+{"_id": "15056", "title": "Titanic Sophie Germain Prime", "text": "The integer defined as: :$39 \\, 051 \\times 2^{6001} - 1$ is a titanic prime which is also a Sophie Germain prime: {{begin-eqn}} {{eqn | o =        | r = 11820 \\, 50794 \\, 19125 \\, 52383 \\, 74423 \\, 53078 \\, 56017 \\, 05024 \\, 84819 \\, 01689 }} {{eqn | o =        | r = 74975 \\, 95139 \\, 68621 \\, 89553 \\, 48654 \\, 81137 \\, 72841 \\, 27658 \\, 52217 \\, 40999 }} {{eqn | o =        | r = 04778 \\, 71896 \\, 78015 \\, 63535 \\, 94741 \\, 82340 \\, 68638 \\, 88011 \\, 18130 \\, 14219 }} {{eqn | o =        | r = 81435 \\, 50235 \\, 73607 \\, 51980 \\, 74200 \\, 04306 \\, 58030 \\, 53360 \\, 79821 \\, 16678 }} {{eqn | o =        | r = 32541 \\, 21729 \\, 72493 \\, 53731 \\, 27605 \\, 59447 \\, 95967 \\, 46064 \\, 11137 \\, 07858 }} {{eqn | o =        | r = 37078 \\, 27755 \\, 33462 \\, 32179 \\, 66482 \\, 80947 \\, 33386 \\, 65681 \\, 87582 \\, 11189 }} {{eqn | o =        | r = 25630 \\, 83169 \\, 50526 \\, 70023 \\, 66301 \\, 83449 \\, 99960 \\, 25913 \\, 90035 \\, 61496 }} {{eqn | o =        | r = 03726 \\, 62661 \\, 50693 \\, 56343 \\, 90085 \\, 30468 \\, 46645 \\, 69888 \\, 03202 \\, 50070 }} {{eqn | o =        | r = 38139 \\, 19172 \\, 69637 \\, 71838 \\, 13812 \\, 48256 \\, 38384 \\, 37787 \\, 83423 \\, 06357 }} {{eqn | o =        | r = 09062 \\, 96393 \\, 13908 \\, 65400 \\, 30048 \\, 07291 \\, 64958 \\, 29772 \\, 97828 \\, 35273 }} {{eqn | o =        | r = 02603 \\, 73947 \\, 05739 \\, 46904 \\, 93564 \\, 50661 \\, 00172 \\, 36892 \\, 20285 \\, 60354 }} {{eqn | o =        | r = 58830 \\, 25332 \\, 20848 \\, 80128 \\, 32451 \\, 94645 \\, 21648 \\, 78503 \\, 66425 \\, 73281 }} {{eqn | o =        | r = 55405 \\, 94426 \\, 29476 \\, 00573 \\, 05011 \\, 86259 \\, 25148 \\, 08537 \\, 31389 \\, 24832 }} {{eqn | o =        | r = 90593 \\, 45279 \\, 70389 \\, 89332 \\, 87614 \\, 90279 \\, 77417 \\, 70009 \\, 37843 \\, 56718 }} {{eqn | o =        | r = 78965 \\, 55090 \\, 40413 \\, 05491 \\, 45610 \\, 39734 \\, 55313 \\, 36378 \\, 82326 \\, 51747 }} {{eqn | o =        | r = 26323 \\, 96872 \\, 58800 \\, 36097 \\, 85595 \\, 50576 \\, 58179 \\, 78961 \\, 56439 \\, 38001 }} {{eqn | o =        | r = 61356 \\, 42993 \\, 82918 \\, 89157 \\, 64818 \\, 24068 \\, 61810 \\, 98754 \\, 13407 \\, 25598 }} {{eqn | o =        | r = 81076 \\, 88939 \\, 65566 \\, 79970 \\, 94454 \\, 12508 \\, 20606 \\, 03037 \\, 82723 \\, 11003 }} {{eqn | o =        | r = 86445 \\, 85147 \\, 95431 \\, 68421 \\, 48123 \\, 63910 \\, 96321 \\, 63833 \\, 76594 \\, 77873 }} {{eqn | o =        | r = 36044 \\, 25100 \\, 46756 \\, 76942 \\, 21197 \\, 98655 \\, 69863 \\, 08993 \\, 13991 \\, 54810 }} {{eqn | o =        | r = 29955 \\, 71299 \\, 30916 \\, 19908 \\, 66968 \\, 53268 \\, 78801 \\, 17165 \\, 95377 \\, 09390 }} {{eqn | o =        | r = 12417 \\, 99779 \\, 38952 \\, 06419 \\, 62790 \\, 94932 \\, 21996 \\, 15477 \\, 09894 \\, 18755 }} {{eqn | o =        | r = 79741 \\, 05192 \\, 62661 \\, 21081 \\, 92384 \\, 45257 \\, 78675 \\, 87928 \\, 74768 \\, 12218 }} {{eqn | o =        | r = 63148 \\, 68786 \\, 76854 \\, 53862 \\, 69957 \\, 63612 \\, 71978 \\, 31119 \\, 74476 \\, 86496 }} {{eqn | o =        | r = 45065 \\, 87748 \\, 91053 \\, 15072 \\, 63384 \\, 65410 \\, 90174 \\, 27502 \\, 19115 \\, 20006 }} {{eqn | o =        | r = 99485 \\, 86281 \\, 23536 \\, 18641 \\, 48374 \\, 90557 \\, 49920 \\, 15285 \\, 92211 \\, 19416 }} {{eqn | o =        | r = 75209 \\, 57766 \\, 75409 \\, 22211 \\, 29543 \\, 79999 \\, 81129 \\, 89523 \\, 59262 \\, 62800 }} {{eqn | o =        | r = 46942 \\, 15484 \\, 08243 \\, 63610 \\, 64351 \\, 53563 \\, 01617 \\, 42451 \\, 12051 \\, 59183 }} {{eqn | o =        | r = 34354 \\, 13049 \\, 42449 \\, 46301 \\, 59875 \\, 51181 \\, 09280 \\, 53716 \\, 57952 \\, 29658 }} {{eqn | o =        | r = 01206 \\, 92006 \\, 20396 \\, 63689 \\, 45859 \\, 75910 \\, 58626 \\, 38955 \\, 88424 \\, 79023 }} {{eqn | o =        | r = 70325 \\, 29477 \\, 90965 \\, 29020 \\, 39505 \\, 24422 \\, 75678 \\, 32327 \\, 27410 \\, 18290 }} {{eqn | o =        | r = 15226 \\, 89958 \\, 01677 \\, 48481 \\, 42430 \\, 49977 \\, 81717 \\, 47239 \\, 67104 \\, 08734 }} {{eqn | o =        | r = 21063 \\, 13953 \\, 69197 \\, 18416 \\, 66197 \\, 78782 \\, 49199 \\, 73757 \\, 81152 \\, 15777 }} {{eqn | o =        | r = 88246 \\, 98396 \\, 88365 \\, 29090 \\, 59197 \\, 96301 \\, 79613 \\, 87838 \\, 71578 \\, 75079 }} {{eqn | o =        | r = 17192 \\, 38121 \\, 06694 \\, 45136 \\, 51899 \\, 17332 \\, 26537 \\, 65466 \\, 92624 \\, 57805 }} {{eqn | o =        | r = 18650 \\, 91862 \\, 60159 \\, 38818 \\, 25424 \\, 40894 \\, 26520 \\, 87364 \\, 29048 \\, 52293 }} {{eqn | o =        | r = 88924 \\, 40043 \\, 51 }} {{end-eqn}}"}
+{"_id": "15057", "title": "Titanic Prime whose Digits are all 9 except for one 1", "text": "The integer defined as: :$2 \\times 10^{3020} - 1$ is a titanic prime all of whose digits are $9$ except one, which is $1$. That is: :$1 \\left({9}\\right)_{3020}$ where $\\left({a}\\right)_b$ means $b$ instances of $a$ in a string."}
+{"_id": "15058", "title": "Titanic Prime consisting of Blocks of 111 of each Digit plus Zeroes", "text": "The integer defined as: :$\\left({1}\\right)_{111} \\left({2}\\right)_{111} \\left({3}\\right)_{111} \\left({4}\\right)_{111} \\left({5}\\right)_{111} \\left({6}\\right)_{111} \\left({7}\\right)_{111} \\left({8}\\right)_{111} \\left({9}\\right)_{111} \\left({0}\\right)_{2284} 1$ where $\\left({a}\\right)_b$ means $b$ instances of $a$ in a string, is a titanic prime."}
+{"_id": "15060", "title": "Pair of Titanic Twin Primes", "text": "The integers defined as: :$190 \\, 116 \\times 3003 \\times 10^{5120} \\pm 1$ are a pair of titanic twin primes. That is: :$570 \\, 918 \\, 347 \\paren 9_{5820}$ and: :$570 \\, 918 \\, 348 \\paren 0_{5819} 1$ where $\\paren a_b$ means $b$ instances of $a$ in a string."}
+{"_id": "15062", "title": "Indexed Iterated Operation does not Change under Permutation", "text": "Let $G$ be a commutative semigroup. Let $a, b \\in\\Z$ be integers. Let $\\closedint a b$ be the integer interval between $a$ and $b$. Let $f: \\closedint a b \\to G$ be a mapping. Let $\\sigma: \\closedint a b \\to \\closedint a b$ be a permutation."}
+{"_id": "15063", "title": "Summation over Cartesian Product as Double Summation", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $S, T$ be finite sets. Let $S \\times T$ be their cartesian product. Then we have an equality of summations over finite sets: :$\\displaystyle \\sum_{s \\mathop \\in S} \\sum_{t \\mathop \\in T} \\map f {s, t} = \\sum_{\\paren {s, t} \\mathop \\in S \\times T} \\map f {s, t}$"}
+{"_id": "15064", "title": "Zeroth Hyperoperation is Successor Function", "text": "The '''zeroth hyperoperation''' is the successor function: :$H_0 \\left({x, y}\\right) = y + 1$"}
+{"_id": "15065", "title": "First Hyperoperation is Addition Operation", "text": "The '''$1$st hyperoperation''' is the addition operation restricted to the positive integers: :$\\forall x, y \\in \\Z_{\\ge 0}: H_1 \\left({x, y}\\right) = x + y$"}
+{"_id": "15066", "title": "Second Hyperoperation is Multiplication Operation", "text": "The '''$2$nd hyperoperation''' is the multiplication operation restricted to the positive integers: :$\\forall x, y \\in \\Z_{\\ge 0}: H_2 \\left({x, y}\\right) = x \\times y$"}
+{"_id": "15067", "title": "Third Hyperoperation is Integer Power Operation", "text": "The '''$3$rd hyperoperation''' is the integer power operation restricted to the positive integers: :$\\forall x, y \\in \\Z_{\\ge 0}: H_3 \\left({x, y}\\right) = x^y$"}
+{"_id": "15068", "title": "Fourth Hyperoperation is Tetration Operation", "text": "The '''$4$th hyperoperation''' is the tetration operation restricted to the positive integers: :$\\forall x, y \\in \\Z_{\\ge 0}: H_4 \\left({x, y}\\right) = x \\uparrow \\uparrow y$ where $\\uparrow \\uparrow$ denotes tetration: :$x \\uparrow \\uparrow n := \\begin{cases} 1 & : n = 0 \\\\ x \\uparrow \\left({x \\uparrow \\uparrow \\left({n - 1}\\right)}\\right) & : n > 0 \\\\ \\end{cases}$ In this context, $x \\uparrow y$ is the Knuth notation for powers: :$x \\uparrow y := x^y$"}
+{"_id": "15069", "title": "Largest Integer Expressible by 3 Digits", "text": "The largest integer that can be represented using no more than $3$ digits, with no additional symbols, is: :$9^{9^9} = 9^{387 \\, 420 \\, 489}$ and (at $369 \\, 693 \\, 100$ digits, is too large to be calculated on a conventional calculator. Note that this does not include the notation for tetration: ${}^9 9$."}
+{"_id": "15070", "title": "Largest Integer Expressible by 3 Digits/Logarithm Base 10", "text": ":$\\map {\\log_{10} } {9^{9^9} } \\approx 369 \\, 693 \\,099 \\cdotp 63157 \\, 03685 \\, 87876 \\, 1$"}
+{"_id": "15071", "title": "Largest Integer Expressible by 3 Digits/Number of Digits", "text": ":$9^{9^9}$ has $369 \\, 693 \\, 100$ digits when expressed in decimal notation."}
+{"_id": "15072", "title": "Number of Primes up to n Approximates to Eulerian Logarithmic Integral", "text": "The prime-counting function approximates to the Eulerian logarithmic integral: :$\\map \\pi n \\approx \\displaystyle \\int_2^n \\frac {\\d x} {\\ln x}$"}
+{"_id": "15076", "title": "Point at which Prime-Counting Function becomes less than Eulerian Logarithmic Integral", "text": "Let $\\map \\pi n$ denote the prime-counting function. Let $a \\uparrow b$ be interpreted as Knuth notation for $a^b$."}
+{"_id": "15077", "title": "Reciprocals of Prime Numbers", "text": "The decimal representations of the reciprocals of the first few prime numbers are as follows: :{| border=\"1\" |- ! align=\"right\" style = \"padding: 2px 10px\" | $n$  ! align=\"left\" style = \"padding: 2px 10px\" | $1 / n$  ! style = \"padding: 2px 10px\" | Also see |- | align=\"right\" style = \"padding: 2px 10px\" | $2$ | align=\"left\" style = \"padding: 2px 10px\" | $0 \\cdotp 5$ | style = \"padding: 2px 10px\" |  |- | align=\"right\" style = \"padding: 2px 10px\" | $3$ | align=\"left\" style = \"padding: 2px 10px\" | $0 \\cdotp \\dot 3$ | style = \"padding: 2px 10px\" |  |- |- | align=\"right\" style = \"padding: 2px 10px\" | $5$ | align=\"left\" style = \"padding: 2px 10px\" | $0 \\cdotp 2$ | style = \"padding: 2px 10px\" |  |- |- | align=\"right\" style = \"padding: 2px 10px\" | $7$ | align=\"left\" style = \"padding: 2px 10px\" | $0 \\cdotp \\dot 14285 \\, \\dot 7$ | style = \"padding: 2px 10px\" | Period of Reciprocal of 7 is of Maximal Length |- | align=\"right\" style = \"padding: 2px 10px\" | $11$ | align=\"left\" style = \"padding: 2px 10px\" | $0 \\cdotp \\dot 0 \\dot 9$ | style = \"padding: 2px 10px\" |  |- | align=\"right\" style = \"padding: 2px 10px\" | $13$ | align=\"left\" style = \"padding: 2px 10px\" | $0 \\cdotp \\dot 07692 \\dot 3$ | style = \"padding: 2px 10px\" |  |- | align=\"right\" style = \"padding: 2px 10px\" | $17$ | align=\"left\" style = \"padding: 2px 10px\" | $0 \\cdotp \\dot 05882 \\, 35294 \\, 11764 \\, \\dot 7$ | style = \"padding: 2px 10px\" | Period of Reciprocal of 17 is of Maximal Length |- | align=\"right\" style = \"padding: 2px 10px\" | $19$ | align=\"left\" style = \"padding: 2px 10px\" | $0 \\cdotp \\dot 05263 \\, 15789 \\, 47368 \\, 42 \\dot 1$ | style = \"padding: 2px 10px\" | Period of Reciprocal of 19 is of Maximal Length |- | align=\"right\" style = \"padding: 2px 10px\" | $23$ | align=\"left\" style = \"padding: 2px 10px\" | $0 \\cdotp \\dot 04347 \\, 82608 \\, 69565 \\, 21739 \\, 1 \\dot 3$ | style = \"padding: 2px 10px\" | Period of Reciprocal of 23 is of Maximal Length |- | align=\"right\" style = \"padding: 2px 10px\" | $29$ | align=\"left\" style = \"padding: 2px 10px\" | $0 \\cdotp \\dot 03448 \\, 27586 \\, 20689 \\, 65517 \\, 24137 \\, 93 \\dot 1$ | style = \"padding: 2px 10px\" |  |- | align=\"right\" style = \"padding: 2px 10px\" | $31$ | align=\"left\" style = \"padding: 2px 10px\" | $0 \\cdotp \\dot 03225 \\, 80645 \\, 1612 \\dot 9$ | style = \"padding: 2px 10px\" | Period of Reciprocal of 31 is of Odd Length |- | align=\"right\" style = \"padding: 2px 10px\" | $37$ | align=\"left\" style = \"padding: 2px 10px\" | $0 \\cdotp \\dot 02 \\dot 7$ | style = \"padding: 2px 10px\" | Period of Reciprocal of 37 has Length 3 |}"}
+{"_id": "15078", "title": "Factors of Repunits", "text": "The prime factors of the first few repunits are as follows: :{| border=\"1\" |- ! align=\"right\" style = \"padding: 2px 10px\" | $R_n$  ! align=\"left\" style = \"padding: 2px 10px\" | Factors of $n$  ! style = \"padding: 2px 10px\" | Also see |- | align=\"right\" style = \"padding: 2px 10px\" | $2$ | align=\"left\" style = \"padding: 2px 10px\" | $11$ | style = \"padding: 2px 10px\" |  |- | align=\"right\" style = \"padding: 2px 10px\" | $3$ | align=\"left\" style = \"padding: 2px 10px\" | $3 \\times 37$ | style = \"padding: 2px 10px\" |  |- |- | align=\"right\" style = \"padding: 2px 10px\" | $4$ | align=\"left\" style = \"padding: 2px 10px\" | $11 \\times 101$ | style = \"padding: 2px 10px\" |  |- |- | align=\"right\" style = \"padding: 2px 10px\" | $5$ | align=\"left\" style = \"padding: 2px 10px\" | $41 \\times 271$ | style = \"padding: 2px 10px\" |  |- | align=\"right\" style = \"padding: 2px 10px\" | $6$ | align=\"left\" style = \"padding: 2px 10px\" | $3 \\times 7 \\times 11 \\times 13 \\times 37$ | style = \"padding: 2px 10px\" |  |- | align=\"right\" style = \"padding: 2px 10px\" | $7$ | align=\"left\" style = \"padding: 2px 10px\" | $239 \\times 4649$ | style = \"padding: 2px 10px\" |  |- | align=\"right\" style = \"padding: 2px 10px\" | $8$ | align=\"left\" style = \"padding: 2px 10px\" | $11 \\times 73 \\times 101 \\times 137$ | style = \"padding: 2px 10px\" |  |- | align=\"right\" style = \"padding: 2px 10px\" | $9$ | align=\"left\" style = \"padding: 2px 10px\" | $3^2 \\times 37 \\times 333 \\, 667$ | style = \"padding: 2px 10px\" |  |- | align=\"right\" style = \"padding: 2px 10px\" | $10$ | align=\"left\" style = \"padding: 2px 10px\" | $11 \\times 41 \\times 271 \\times 9091$ | style = \"padding: 2px 10px\" |  |- | align=\"right\" style = \"padding: 2px 10px\" | $11$ | align=\"left\" style = \"padding: 2px 10px\" | $21 \\, 649 \\times 513 \\, 239$ | style = \"padding: 2px 10px\" |  |- | align=\"right\" style = \"padding: 2px 10px\" | $12$ | align=\"left\" style = \"padding: 2px 10px\" | $3 \\times 7 \\times 11 \\times 13 \\times 37 \\times 101 \\times 9901$ | style = \"padding: 2px 10px\" |  |- | align=\"right\" style = \"padding: 2px 10px\" | $13$ | align=\"left\" style = \"padding: 2px 10px\" | $53 \\times 79 \\times 265 \\, 371 \\, 653$ | style = \"padding: 2px 10px\" |  |}"}
+{"_id": "15080", "title": "Separability is not Weakly Hereditary", "text": "The property of separability is not weakly hereditary."}
+{"_id": "15083", "title": "Filter is Finer iff Sets of Basis are Subsets", "text": "Let $S$ be a set. Let $\\powerset S$ denote the power set of $S$. Let $\\FF, \\FF' \\subset \\powerset S$ be two filters on $S$. Let $\\FF$ have a basis $\\BB$. Let $\\FF'$ have a basis $\\BB'$. $\\FF$ is finer than $\\FF'$ {{iff}} for every set of $\\BB'$, there is a set of $\\BB$ subset to it."}
+{"_id": "15085", "title": "Existence of Topological Space which satisfies no Separation Axioms but T0", "text": "There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_0$ (Kolmogorov) axiom."}
+{"_id": "15086", "title": "Existence of Topological Space which satisfies no Separation Axioms but T0 and T1", "text": "There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_0$ (Kolmogorov) axiom and $T_1$ (Fréchet) axiom."}
+{"_id": "15087", "title": "Equivalence of Definitions of Unital Associative Commutative Algebra", "text": "Let $A$ be a commutative ring with unity."}
+{"_id": "15088", "title": "Existence of Hausdorff Space which is not T3, T4 or T5", "text": "There exists at least one example of a topological space which is a $T_2$ (Hausdorff) space (and hence also $T_0$ (Kolmogorov) and $T_1$ (Fréchet), but is not a $T_3$ space, a $T_4$ space or a $T_5$ space."}
+{"_id": "15089", "title": "Existence of Topological Space which satisfies no Separation Axioms but T3", "text": "There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_3$ axiom."}
+{"_id": "15090", "title": "Existence of Topological Space which satisfies no Separation Axioms but T4", "text": "There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_4$ axiom."}
+{"_id": "15093", "title": "Existence of Tychonoff Space which is not Normal", "text": "There exists at least one example of a topological space which is a Tychonoff space, but is not also a normal space."}
+{"_id": "15094", "title": "Existence of T4 Space which is not T3 1/2", "text": "There exists at least one example of a topological space which is a $T_4$ space, but is not also a $T_{3 \\frac 1 2}$ space."}
+{"_id": "15095", "title": "Existence of Normal Space which is not Completely Normal", "text": "There exists at least one example of a topological space which is a normal space, but is not also a completely normal space."}
+{"_id": "15098", "title": "Separation Properties Not Preserved by Expansion", "text": "These separation properties are not generally preserved under expansion: :$T_3$ Space :Regular Space :$T_4$ Space :Completely Regular Space :$T_5$ Space :Normal Space :Completely Normal Space"}
+{"_id": "15099", "title": "Existence of Completely Normal Space whose Product Space is Not Normal", "text": "There exists at least one example of a completely normal topological space $T$ such that the product space $T \\times T$ is not a normal topological space."}
+{"_id": "15101", "title": "Existence of Completely Normal Space which is not Perfectly Normal", "text": "There exists at least one example of a completely normal topological space which is not perfectly normal."}
+{"_id": "15102", "title": "Existence of Semiregular Topological Space which is not T3", "text": "There exists at least one example of a semiregular topological space which is not a $T_3$ space."}
+{"_id": "15103", "title": "Existence of Semiregular Topological Space which is not Completely Hausdorff", "text": "There exists at least one example of a semiregular topological space which is not a completely Hausdorff space."}
+{"_id": "15105", "title": "Existence of Sigma-Compact Space which is not Compact", "text": "There exists at least one example of a $\\sigma$-compact topological space which is not also a compact space."}
+{"_id": "15106", "title": "Existence of Lindelöf Space which is not Sigma-Compact", "text": "There exists at least one example of a $\\sigma$-compact topological space which is not also a compact space."}
+{"_id": "15108", "title": "Existence of Countably Compact Space which is not Sequentially Compact", "text": "There exists at least one example of a countably compact topological space which is not also a sequentially compact space."}
+{"_id": "15109", "title": "Existence of Weakly Countably Compact Space which is not Countably Compact", "text": "There exists at least one example of a weakly countably compact topological space which is not also a countably compact space."}
+{"_id": "15110", "title": "Existence of Pseudocompact Space which is not Countably Compact", "text": "There exists at least one example of a pseudocompact topological space which is not also a countably compact space."}
+{"_id": "15114", "title": "Vectorization of Product of Three Matrices", "text": "Let $R$ be a ring. Let $A, B, C$ be matrices over $R$ such that the matrix product $ABC$ is defined. Then $\\map {\\operatorname {vec} }{ABC} = \\paren {C^\\intercal \\otimes A} \\cdot \\map {\\operatorname {vec} } B$ where: :$\\operatorname {vec}$ denotes vectorization :$C^\\intercal$ is the transpose of $C$ :$\\otimes$ denotes Kronecker product :$\\cdot$ denotes matrix product"}
+{"_id": "15116", "title": "Existence of Weakly Sigma-Locally Compact Space which is not Strongly Locally Compact", "text": "There exists at least one example of a weakly $\\sigma$-locally compact topological space which is not also a strongly locally compact space."}
+{"_id": "15117", "title": "Existence of First-Countable Space which is not Second-Countable", "text": "There exists at least one example of a first-countable topological space which is not also a second-countable space."}
+{"_id": "15118", "title": "Existence of Lindelöf Space which is not Second-Countable", "text": "There exists at least one example of a Lindelöf space which is not also a second-countable space."}
+{"_id": "15120", "title": "Existence of Topological Space satisfying Countable Chain Condition which is not Separable", "text": "There exists at least one example of a topological space which satisfies the countable chain condition which is not also a separable space."}
+{"_id": "15121", "title": "Equivalence of Definitions of Strongly Locally Compact Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. {{TFAE|def = Strongly Locally Compact Space}}"}
+{"_id": "15122", "title": "Uniqueness of Representing Objects", "text": "Let $C$ be a locally small category. Let $\\mathbf{Set}$ be the category of sets. Let $F : \\mathbf C \\to \\mathbf{Set}$ be a covariant functor. Let $(A, \\eta)$ and $(B, \\xi)$ be representations of $F$. Then there exists a unique isomorphism $f : A \\to B$ such that $\\eta \\circ h^f = \\xi$, where: :$h^f$ is the precomposition natural transformation :$\\circ$ denotes vertical composition of natural transformations"}
+{"_id": "15123", "title": "Existence and Uniqueness of Reduced Form of Group Word", "text": "Let $X$ be a set. Let $w$ be a group word on $X$. Then $w$ has a unique reduced form."}
+{"_id": "15126", "title": "Existence of Metacompact Space which is not Paracompact", "text": "There exists at least one example of a metacompact topological space which is not also a paracompact space."}
+{"_id": "15129", "title": "Existence of Minimal Hausdorff Space which is not Compact", "text": "Let $S$ be a set. Let $\\tau$ be the minimal subset of the power set $\\powerset S$ such that $\\struct {S, \\tau}$ is a Hausdorff space. Then it is not necessarily the case that $\\struct {S, \\tau}$ is compact."}
+{"_id": "15130", "title": "Existence of Maximal Compact Topological Space which is not Hausdorff", "text": "Let $S$ be a set. Let $\\tau$ be the minimal subset of the power set $\\powerset S$ such that $\\struct {S, \\tau}$ is a compact topological space. Then it is not necessarily the case that $\\struct {S, \\tau}$ is a Hausdorff space."}
+{"_id": "15131", "title": "Gauss's Lemma (Polynomial Theory)", "text": "Gauss's lemma on polynomials may refer to any of the following statements."}
+{"_id": "15132", "title": "Infinite Product of Sigma-Compact Spaces is not always Sigma-Compact", "text": "Let $I$ be an indexing set with infinite cardinality. Let $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be a family of topological spaces indexed by $I$. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let each of $\\struct {S_\\alpha, \\tau_\\alpha}$ be $\\sigma$-compact. Then it is not necessarily the case that $\\struct {S, \\tau}$ is also $\\sigma$-compact."}
+{"_id": "15133", "title": "Uncountable Product of Sequentially Compact Spaces is not always Sequentially Compact", "text": "Let $I$ be an indexing set with uncountable cardinality. Let $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be a family of topological spaces indexed by $I$. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let each of $\\struct {S_\\alpha, \\tau_\\alpha}$ be sequentially compact. Then it is not necessarily the case that $\\struct {S, \\tau}$ is also sequentially compact."}
+{"_id": "15134", "title": "Product of Countably Compact Spaces is not always Countably Compact", "text": "Let $I$ be an indexing set. Let $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be a family of topological spaces indexed by $I$. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let each of $\\struct {S_\\alpha, \\tau_\\alpha}$ be countably compact. Then it is not necessarily the case that $\\struct {S, \\tau}$ is also countably compact."}
+{"_id": "15136", "title": "Product of Lindelöf Spaces is not always Lindelöf", "text": "Let $I$ be an indexing set. Let $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be a family of topological spaces indexed by $I$. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let each of $\\struct {S_\\alpha, \\tau_\\alpha}$ be a Lindelöf space. Then it is not necessarily the case that $\\struct {S, \\tau}$ is also Lindelöf space."}
+{"_id": "15138", "title": "Uncountable Product of Second-Countable Spaces is not always Second-Countable", "text": "Let $I$ be an indexing set with uncountable cardinality. Let $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be a family of topological spaces indexed by $I$. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let each of $\\struct {S_\\alpha, \\tau_\\alpha}$ be a second-countable space. Then it is not necessarily the case that $\\struct {S, \\tau}$ is also a second-countable space."}
+{"_id": "15142", "title": "Paracompactness is not always Preserved under Open Continuous Mapping", "text": "Let $T_A = \\struct {X_A, \\tau_A}$ be a topological space which is paracompact. Let $T_B = \\struct {X_B, \\tau_B}$ be another topological space. Let $\\phi: T_A \\to T_B$ be a mapping which is both continuous and open. Then it is not necessarily the case that $T_B$ is also paracompact."}
+{"_id": "15143", "title": "Correspondence Theorem for Ring Epimorphisms", "text": "Let $A$ and $B$ be commutative rings with unity. Let $\\pi : A \\to B$ be a ring epimorphism. Let $I$ be the set of ideals of $A$ containing the kernel $\\operatorname{ker} \\pi$. Let $J$ be the set of ideals of $B$."}
+{"_id": "15144", "title": "Union of Closure with Closure of Complement is Whole Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $H \\subseteq S$ be a subset of $S$. Let $H^-$ denote the closure of $H$ in $T$. Let $S \\setminus H$ denote the complement of $H$ relative to $S$. Then: :$H^- \\cup \\left({S \\setminus H}\\right)^- = S$"}
+{"_id": "15145", "title": "Equivalent Characterisations of Irrational Periodic Continued Fraction", "text": "Let $x \\in \\R \\setminus \\Q$ be an irrational number. Let $(a_n)_{n\\geq 0}$ be its continued fraction. Let $N \\geq 0$ be a natural number. {{TFAE}} :$(1):\\quad$ The sequence of partial quotients $(a_n)_{n\\geq 0}$ is periodic for $n \\geq N$. :$(2):\\quad$ The sequence of complete quotients $(x_n)_{n\\geq 0}$ is periodic for $n \\geq N$. :$(3):\\quad$ There exists $M > N$ such that $x_M = x_N$."}
+{"_id": "15146", "title": "Component of Point is not always Intersection of its Clopen Sets", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $x \\in S$. Let $\\map {\\operatorname {Comp}_x} T$ denote the component of $x$ in $T$. Let $K_x = \\displaystyle \\bigcap_{x \\mathop \\in K} K$ clopen in $T$. Then it is not always the case that $\\map {\\operatorname {Comp}_x} T = K_x$"}
+{"_id": "15147", "title": "Complement of Clopen Set is Clopen", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be a clopen set of $T$. Let $\\relcomp S H$ denote the complement of $H$ relative to $S$. Then $\\relcomp S H$ is also a clopen set of $T$."}
+{"_id": "15148", "title": "Clopen Set and Complement form Separation", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be a clopen set of $T$. Let $\\relcomp S H$ be the complement of $H$ relative to $S$. Then $H$ and $\\relcomp S H$ form a separation of $T$."}
+{"_id": "15149", "title": "Path Component is not necessarily Arc Component", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $P$ be a path component of $T$. Then it is not necessarily the case that $P$ is also an arc component of $T$."}
+{"_id": "15150", "title": "Component is not necessarily Path Component", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $C$ be a component of $T$. Then it is not necessarily the case that $C$ is also an path component of $T$."}
+{"_id": "15151", "title": "Quasicomponent is not necessarily Component", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $Q$ be a quasicomponent of $T$. Then it is not necessarily the case that $C$ is also a component of $T$."}
+{"_id": "15152", "title": "Simple Infinite Continued Fraction is Uniquely Determined by Limit", "text": "Let $(a_n)_{n\\geq 0}$ and $(b_n)_{n\\geq 0}$ be simple infinite continued fractions in $\\R$. Let $(a_n)_{n\\geq 0}$ and $(b_n)_{n\\geq 0}$ have the same limit. Then they are equal."}
+{"_id": "15153", "title": "Continued Fraction Expansion of Irrational Number Converges to Number Itself", "text": "Let $x$ be an irrational number. Then the continued fraction expansion of $x$ converges to $x$."}
+{"_id": "15154", "title": "Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself", "text": "Let Let $(a_n)_{n\\geq 0}$ be a simple infinite continued fractions in $\\R$. Then $(a_n)_{n\\geq 0}$ converges to an irrational number, whose continued fraction expansion is $(a_n)_{n\\geq 0}$."}
+{"_id": "15155", "title": "Correspondence between Irrational Numbers and Simple Infinite Continued Fractions", "text": "Let $\\R \\setminus \\Q$ be the set of irrational numbers. Let $S$ be the set of all simple infinite continued fractions in $\\R$. The mappings: :$\\R \\setminus \\Q \\to S$ that sends an irrational number to its continued fraction expansion :$S \\to \\R \\setminus \\Q$ that sends a simple infinite continued fractions to its value are inverses of each other."}
+{"_id": "15157", "title": "Ultraconnected Space is Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is ultraconnected. Then $T$ is connected."}
+{"_id": "15158", "title": "Irreducible Space is not necessarily Path-Connected", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space which is irreducible. Then $T$ is not necessarily path-connected."}
+{"_id": "15159", "title": "Ultraconnected Space is not necessarily Arc-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is ultraconnected. Then $T$ is not necessarily arc-connected."}
+{"_id": "15160", "title": "Correspondence between Rational Numbers and Simple Finite Continued Fractions", "text": "Let $\\Q$ be the set of rational numbers. Let $S$ be the set of all simple finite continued fractions in $\\Q$, whose last partial quotient is not $1$. The mappings: :$\\Q \\to S$ that sends an rational number to its continued fraction expansion :$S \\to \\Q$ that sends a simple finite continued fractions to its value are inverses of each other."}
+{"_id": "15161", "title": "Existence of Non-Locally Connected Space where Components and Quasicomponents are Equal", "text": "There exists at least one example of a topological space which is not locally connected, but whose components and quasicomponents are equal."}
+{"_id": "15163", "title": "Locally Connected Space is not necessarily Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is locally connected. Then it is not necessarily the case that $T$ is also a connected space."}
+{"_id": "15165", "title": "Irreducible Space with Finitely Many Open Sets is Path-Connected", "text": "Let $T = \\left({S, \\tau}\\right)$ be a irreducible topological space. Let its topology $\\tau$ be finite. Then $T$ is path-connected."}
+{"_id": "15166", "title": "Topological Space with Generic Point is Path-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T$ have a generic point $g \\in S$. Then $T$ is path-connected."}
+{"_id": "15167", "title": "Floor of Simple Finite Continued Fraction", "text": "Let $\\sequence {a_k}_{k \\mathop \\ge 0}$ be a simple finite continued fraction of length $n \\ge 0$. Let $x = [a_0, \\ldots, a_n]$ be its value. Then the floor of $x$ is the partial denominator $a_0$: :$\\floor x = a_0$ unless $n = 1$ and $a_1 = 1$, in which case $x = \\floor x = a_0 + 1$."}
+{"_id": "15168", "title": "Value of Finite Continued Fraction of Strictly Positive Real Numbers is Strictly Positive", "text": "Let $(a_0, \\ldots, a_n)$ be a finite continued fraction in $\\R$ of length $n \\geq 0$. Let all partial quotients $a_k>0$ be strictly positive. Let $x = [a_0, a_1, \\ldots, a_n]$ be its value. Then $x>0$."}
+{"_id": "15169", "title": "Value of Finite Continued Fraction of Real Numbers is at Least First Term", "text": "Let $(a_0, \\ldots, a_n)$ be a finite continued fraction in $\\R$ of length $n \\geq 0$. Let the partial quotients $a_k>0$ be strictly positive for $k>0$. Let $x = [a_0, a_1, \\ldots, a_n]$ be its value. Then $x \\geq a_0$, and $x>a_0$ if the length $n\\geq 1$."}
+{"_id": "15171", "title": "Finite Simple Continued Fraction has Rational Value", "text": "Let $n \\ge 0$ be a natural number. Let $\\paren {a_0, \\ldots, a_n}$ be a simple finite continued fraction of length $n$. Then its value $\\sqbrk {a_0, \\ldots, a_n}$ is a rational number."}
+{"_id": "15172", "title": "Accuracy of Convergents of Convergent Simple Infinite Continued Fraction", "text": "Let $C = (a_0, a_1, \\ldots)$ be an infinite simple continued fraction in $\\R$. Let $C$ converge to $x \\in \\R$. For $n\\geq0$, let $C_n = p_n/q_n$ be the $n$th convergent of $C$, where $p_n$ and $q_n$ are the $n$th numerator and denominator. Then for all $n\\geq 0$: :$\\left\\vert x - \\dfrac {p_n}{q_n}\\right\\vert < \\dfrac 1{q_nq_{n+1}}$."}
+{"_id": "15174", "title": "Locally Path-Connected Space is not necessarily Locally Arc-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is locally path-connected. Then it is not necessarily the case that $T$ is also a locally arc-connected space."}
+{"_id": "15176", "title": "Path-Connected Space is not necessarily Locally Path-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is path-connected. Then it is not necessarily the case that $T$ is also locally path-connected."}
+{"_id": "15177", "title": "Lower Bounds for Denominators of Simple Continued Fraction", "text": "Let $n \\in \\N \\cup \\{\\infty\\}$ be an extended natural number. Let $\\left[{a_0, a_1, a_2, \\ldots}\\right]$ be a simple continued fraction in $\\R$ of length $N$. Let $q_0, q_1, q_2, \\ldots$ be its denominators."}
+{"_id": "15178", "title": "Equality of Rational Numbers", "text": "Let $a, b, c, d$ be integers, with $b$ and $d$ nonzero. {{TFAE}} :$(1): \\quad$ The rational numbers $\\dfrac a b$ and $\\dfrac c d$ are equal. :$(2): \\quad$ The integers $a d$ and $b c$ are equal."}
+{"_id": "15179", "title": "Locally Path-Connected Space is not necessarily Path-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is locally path-connected. Then it is not necessarily the case that $T$ is also path-connected."}
+{"_id": "15180", "title": "Arc-Connected Space is not necessarily Locally Arc-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is arc-connected. Then it is not necessarily the case that $T$ is also locally arc-connected."}
+{"_id": "15181", "title": "Locally Arc-Connected Space is not necessarily Arc-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is locally arc-connected. Then it is not necessarily the case that $T$ is also arc-connected."}
+{"_id": "15182", "title": "Odd Convergents of Simple Continued Fraction are Strictly Decreasing", "text": "The odd convergents satisfy $C_1 > C_3 > C_5 > \\cdots$"}
+{"_id": "15183", "title": "Even Convergents of Simple Continued Fraction are Strictly Increasing", "text": "The even convergents satisfy $C_0 < C_2 < C_4 \\cdots$."}
+{"_id": "15184", "title": "Denominators of Simple Continued Fraction are Strictly Positive", "text": "Let $n \\in \\N \\cup \\{\\infty\\}$ be an extended natural number. Let $(a_0, a_1, \\ldots)$ be a simple continued fraction in $\\R$ of length $n$. Let $q_0, q_1, q_2, \\ldots$ be its denominators. Then for $0 \\leq k \\leq n$ we have $q_k > 0$."}
+{"_id": "15186", "title": "Simple Finite Continued Fraction is Almost Determined by Value", "text": "Let $n,m \\geq 0$ be natural number. Let $(a_k)_{0 \\leq k \\leq m}$ and $(b_k)_{0 \\leq k \\leq n}$ be simple finite continued fractions in $\\R$. Let $(a_k)_{0 \\leq k \\leq m}$ and $(b_k)_{0 \\leq k \\leq n}$ have the same value. Then either: *$n = m$, and the sequences are equal. *$n = m + 1$, $a_k = b_k$ for $k \\leq m$, $a_m = b_m-1$ and $b_{m+1} = 1$ *$m = n + 1$, $a_k = b_k$ for $k \\leq n$, $b_n = a_n-1$ and $a_{n+1} = 1$"}
+{"_id": "15187", "title": "Subgroup Generated by One Element is Set of Powers", "text": "Let $G$ be a group. Let $a \\in G$. Then the subgroup generated by $a$ is the set of powers: :$\\gen a  = \\set {a^n : n \\in \\Z}$"}
+{"_id": "15188", "title": "Finite Product Space is Connected iff Factors are Connected", "text": "Let $T_1 = \\struct {S_1, \\tau_1}, T_2 = \\struct {S_2, \\tau_2}, \\dotsc, T_n = \\struct {S_n, \\tau_n}$ be topological spaces. Let $T = \\displaystyle \\prod_{i \\mathop = 1}^n T_i$ be the product space of $T_1, T_2, \\ldots, T_n$. Then $T$ is connected {{iff}} each of $T_1, T_2, \\ldots, T_n$ are connected."}
+{"_id": "15191", "title": "Binary Coproduct in Preadditive Category is Biproduct", "text": "Let $A$ be a preadditive category. Let $a_1, a_2$ be objects of $A$. Let $(a_1 \\sqcup a_2, i_1, i_2)$ be their binary coproduct, assuming it exists. Let $p_1 : a_1 \\sqcup a_2 \\to a_1$ be the unique morphism with: :$p_1 \\circ i_1 = 1 : a_1 \\to a_1$ :$p_1 \\circ i_2 = 0 : a_1 \\to a_2$ Let $p_2 : a_1 \\sqcup a_2 \\to a_2$ be the unique morphism with: :$p_2 \\circ i_1 = 0 : a_2 \\to a_1$ :$p_2 \\circ i_2 = 1 : a_2 \\to a_2$ where $1$ is the identity morphism and $0$ is the zero morphism. Then $(a_1 \\sqcup a_2, i_1, i_2, p_1, p_2)$ is the binary biproduct of $a_1$ and $a_2$."}
+{"_id": "15195", "title": "Continuous Mapping from Compact Space to Hausdorff Space Preserves Local Connectedness", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ be a compact topological space. Let $T_2 = \\struct {S_2, \\tau_2}$ be a $T_2$ (Hausdorff) space. Let $f: T_1 \\to T_2$ be a continuous mapping. Let $T_1$ be locally connected. Then $T_2$ is also locally connected."}
+{"_id": "15196", "title": "Minimum Degree Bound for Simple Planar Graph", "text": "Let $G$ be a simple connected planar graph. Then:  :$\\map \\delta G \\le 5$ where $\\map \\delta G$ denotes the minimum degree of vertices of $G$."}
+{"_id": "15197", "title": "Linear Bound Lemma", "text": "For a simple connected planar graph $G_n$, where $n \\geq 3$ is a number of vertices: :$m \\leq 3 n − 6$, where $m$ is a number of edges."}
+{"_id": "15198", "title": "Existence of Connected Space which is Totally Pathwise Disconnected", "text": "There exists at least one example of a topological space which is both connected and totally pathwise disconnected."}
+{"_id": "15200", "title": "Zero Dimensional Space is not necessarily T0", "text": "Let $T = \\struct {S, \\tau}$ be a zero dimensional topological space. Then $T$ is not necessarily a $T_0$ (Kolmogorov) space."}
+{"_id": "15201", "title": "Scattered Space is not necessarily T1", "text": "Let $T = \\struct {S, \\tau}$ be a scattered topological space. Then $T$ is not necessarily a $T_1$ (Fréchet) space."}
+{"_id": "15202", "title": "Existence of Connected Non-T1 Scattered Space", "text": "There exists at least one example of a connected topological space which is not a $T_1$ (Fréchet) space, which is also a scattered space."}
+{"_id": "15203", "title": "Sorgenfrey Line is not Second-Countable", "text": "Let $T = \\struct {\\mathbb R, \\tau}$ be the Sorgenfrey line. Then $T$ is not second-countable."}
+{"_id": "15204", "title": "Existence of Biconnected Set without Dispersion Point", "text": "There exists at least one example of a biconnected set which does not have a dispersion point."}
+{"_id": "15205", "title": "Existence of Connected Punctiform Space", "text": "There exists at least one example of a connected topological space which is also punctiform."}
+{"_id": "15207", "title": "Metric Space is Completely Normal", "text": "Let $M = \\struct {A, d}$ be a metric space. Then $M$ is a completely normal space."}
+{"_id": "15209", "title": "Bounded Metric Space is not necessarily Totally Bounded", "text": "Let $M = \\struct {A, d}$ be a bounded metric space. Then it is not necessarily the case that $M$ is totally bounded."}
+{"_id": "15210", "title": "Total Boundedness is not Preserved under Homeomorphism", "text": "Let $M = \\struct {A, d}$ be a totally bounded metric space. Let $M' = \\struct {A', d'}$ be a metric space. Let $M$ be homeomorphic to $M'$. Then it is not necessarily the case that $M'$ is totally bounded."}
+{"_id": "15212", "title": "Topological Completeness is not Hereditary", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is topologically complete. Let $H \\subseteq S$ be a subset of $S$. Let $\\struct {H, \\tau_H}$ be the topological subspace of $T$ induced by $H$. Then it is not necessarily the case that $\\struct {H, \\tau_H}$ is also topologically complete. That is, topological completeness is not hereditary."}
+{"_id": "15213", "title": "Completion Theorem (Metric Space)/Lemma 4", "text": ":$\\tilde M = \\struct {\\tilde A, \\tilde d}$ is unique up to isometry."}
+{"_id": "15214", "title": "Sorgenfrey Line is First-Countable", "text": "Let $\\R$ be the set of real numbers. Let $\\BB = \\set {\\hointr a b: a, b \\in \\R}$. Let $\\tau$ be the topology generated by $\\BB$, that is, the Sorgenfrey line. Then $\\tau$ is first-countable."}
+{"_id": "15215", "title": "Metrizable Space is not necessarily Second-Countable", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is metrizable. Then it is not necessarily the case that $T$ is second-countable."}
+{"_id": "15216", "title": "Regular Paracompact Space is not necessarily Metrizable", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is regular and paracompact. Then it is not necessarily the case that $T$ is metrizable."}
+{"_id": "15217", "title": "Uniform Space whose Topology is Metrizable is not necessarily Metrizable", "text": "Let $\\UU$ be a uniformity on a set $S$. Let $\\struct {\\struct {S, \\UU}, \\tau}$ be the uniform space generated from $\\UU$. Let $T = \\struct {S, \\tau}$ be the uniformizable space yielded by $\\struct {\\struct {S, \\UU}, \\tau}$. Let $T$ be a metrizable space. Then it is not necessarily the case that $\\UU$ is itself a metrizable uniformity."}
+{"_id": "15220", "title": "Open Sets in Indiscrete Topology", "text": "$H$ is an open set of $T$ {{iff}} either $H = S$ or $H = \\O$."}
+{"_id": "15221", "title": "Closed Sets in Indiscrete Topology", "text": "$H$ is a closed set of $T$ {{iff}} either $H = S$ or $H = \\O$."}
+{"_id": "15222", "title": "F-Sigma Sets in Indiscrete Topology", "text": "$H$ is an $F_\\sigma$ ($F$-sigma) set of $T$ {{iff}} either $H = S$ or $H = \\O$."}
+{"_id": "15224", "title": "Subset of Indiscrete Space is Compact", "text": "$H$ is compact in $T$."}
+{"_id": "15227", "title": "Partition Topology is not Completely Hausdorff", "text": "Let $S$ be a set and let $\\PP$ be a partition on $S$ which is not the (trivial) partition of singletons. Let $T = \\struct {S, \\tau}$ be the partition space whose basis is $\\PP$. Then $T$ is not a $T_{2 \\frac 1 2}$ (completely Hausdorff) space."}
+{"_id": "15228", "title": "Partition Topology is not Hausdorff", "text": "Let $S$ be a set and let $\\PP$ be a partition on $S$ which is not the (trivial) partition of singletons. Let $T = \\struct {S, \\tau}$ be the partition space whose basis is $\\PP$. Then $T$ is not a $T_2$ (Hausdorff) space."}
+{"_id": "15229", "title": "Partition Topology is not T1", "text": "Let $S$ be a set and let $\\PP$ be a partition on $S$ which is not the (trivial) partition of singletons. Let $T = \\struct {S, \\tau}$ be the partition space whose basis is $\\PP$. Then $T$ is not a $T_1$ (Fréchet) space."}
+{"_id": "15230", "title": "Odd-Even Topology is Lindelöf", "text": "$T$ is Lindelöf."}
+{"_id": "15231", "title": "Odd-Even Topology is Separable", "text": "$T$ is separable."}
+{"_id": "15232", "title": "Odd-Even Topology is First-Countable", "text": "$T$ is first-countable."}
+{"_id": "15234", "title": "Particular Point Space less Particular Point is Discrete", "text": "Let $T = \\left({S, \\tau_p}\\right)$ be a particular point space, whose particular point is $p$. Let $H = S \\setminus \\left\\{ {p}\\right\\}$ where $\\setminus$ denotes set difference. Then the topological subspace $T_H = \\left({H, \\tau_H}\\right)$ induced on $H$ by $\\tau_p$ is a discrete space."}
+{"_id": "15236", "title": "Zero is Accumulation Point of Sequence in Sierpiński Space", "text": "Let $T = \\struct {\\set {0, 1}, \\tau_0}$ be a Sierpiński space. The sequence in $T$: :$\\sigma = \\sequence {0, 1, 0, 1, \\ldots}$ has $0$ as an accumulation point."}
+{"_id": "15239", "title": "Infinite Particular Point Space is not Countably Paracompact", "text": "Let $T = \\left({S, \\tau_p}\\right)$ be an infinite particular point space. Then $T$ is not countably paracompact."}
+{"_id": "15242", "title": "Closed Extension Topology is not Hausdorff", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_p = \\struct {S^*_p, \\tau^*_p}$ be the closed extension space of $T$. Then $T^*_p$ is not a $T_2$ (Hausdorf) space."}
+{"_id": "15243", "title": "Closed Extension Topology is not T3", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_p = \\struct {S^*_p, \\tau^*_p}$ be the closed extension space of $T$. Then $T^*_p$ is not a $T_3$ space."}
+{"_id": "15246", "title": "Either-Or Topology is T4", "text": "Let $T = \\struct {S, \\tau}$ be the either-or space. Then $T$ is a $T_4$ space."}
+{"_id": "15251", "title": "Limit Points in Fort Space", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space. Let $x \\in S$ such that $x \\ne p$. Then $p$ is the only limit point of $x$."}
+{"_id": "15253", "title": "Rational Numbers in Real Euclidean Plus Space are Open Set", "text": "Let $\\R$ be the set of real numbers. Let $d: \\R \\times \\R \\to \\R$ be the Euclidean plus metric: :$\\map d {x, y} := \\size {x - y} + \\displaystyle \\sum_{i \\mathop = 1}^\\infty 2^{-i} \\map \\inf {1, \\size {\\max_{j \\mathop \\le i} \\frac 1 {\\size {x - r_j} } - \\max_{j \\mathop \\le i} \\frac 1 {\\size {y - r_j} } } }$ Let $\\Q$ be the set of rational numbers. Then $\\Q$ is an open set of $\\struct {\\R, d}$."}
+{"_id": "15254", "title": "Alexandroff Extension is Compact", "text": "Let $T = \\struct {S, \\tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \\cup \\set p$. Let $T^* = \\struct {S^*, \\tau^*}$ be the Alexandroff extension on $S$. Then $T^*$ is a compact topological space."}
+{"_id": "15255", "title": "Condition for Alexandroff Extension to be T1 Space", "text": "Let $T = \\struct {S, \\tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \\cup \\set p$. Let $T^* = \\struct {S^*, \\tau^*}$ be the Alexandroff extension on $S$. Then $T^*$ is a $T_1$ (Fréchet) space {{iff}} $T$ is a $T_1$ (Fréchet) space."}
+{"_id": "15256", "title": "Condition for Alexandroff Extension to be T2 Space", "text": "Let $T = \\struct {S, \\tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \\cup \\set p$. Let $T^* =\\struct {S^*, \\tau^*}$ be the Alexandroff extension on $S$. Then $T^*$ is a $T_2$ (Hausdorff) space {{iff}} $T$ is a locally compact Hausdorff space."}
+{"_id": "15257", "title": "Alexandroff Extension which is T2 Space is also T4 Space", "text": "Let $T = \\struct {S, \\tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \\cup \\set p$. Let $T^* = \\struct {S^*, \\tau^*}$ be the Alexandroff extension on $S$. Let $T^*$ be a $T_2$ (Hausdorff) space. Then $T^*$ is a $T_4$ space."}
+{"_id": "15258", "title": "Alexandroff Extension of Rational Number Space is not Hausdorff", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Let $p$ be a new element not in $\\Q$. Let $\\Q^* := \\Q \\cup \\set p$. Let $T^* = \\struct {\\Q^*, \\tau^*}$ be the Alexandroff extension on $\\struct {\\Q, \\tau_d}$. Then $T^*$ is not a $T_2$ (Hausdorff) space."}
+{"_id": "15259", "title": "Alexandroff Extension of Rational Number Space is T1 Space", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Let $p$ be a new element not in $\\Q$. Let $\\Q^* := \\Q \\cup \\set p$. Let $T^* = \\struct {\\Q^*, \\tau^*}$ be the Alexandroff extension on $\\struct {\\Q, \\tau_d}$. Then $T^*$ is a $T_1$ (Fréchet) space."}
+{"_id": "15261", "title": "Particular Point of Alexandroff Extension of Rational Number Space is Dispersion Point", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Let $p$ be a new element not in $\\Q$. Let $\\Q^* := \\Q \\cup \\set p$. Let $T^* = \\struct {\\Q^*, \\tau^*}$ be the Alexandroff extension on $\\left({\\Q, \\tau_d}\\right)$. Then $p$ is a dispersion point of $T^*$."}
+{"_id": "15262", "title": "Alexandroff Extension of Rational Number Space is Connected", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Let $p$ be a new element not in $\\Q$. Let $\\Q^* := \\Q \\cup \\set p$. Let $T^* = \\struct {\\Q^*, \\tau^*}$ be the Alexandroff extension on $\\struct {\\Q, \\tau_d}$. Then $T^*$ is a connected space."}
+{"_id": "15263", "title": "Alexandroff Extension of Rational Number Space is Biconnected", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Let $p$ be a new element not in $\\Q$. Let $\\Q^* := \\Q \\cup \\set p$. Let $T^* = \\struct {\\Q^*, \\tau^*}$ be the Alexandroff extension on $\\struct {\\Q, \\tau_d}$. Then $T^*$ is a biconnected space."}
+{"_id": "15265", "title": "Hilbert Sequence Space is Complete Metric Space", "text": "Let $A$ be the set of all real sequences $\\left\\langle{x_i}\\right\\rangle$ such that the series $\\displaystyle \\sum_{i \\mathop \\ge 0} x_i^2$ is convergent. Let $\\ell^2 = \\struct {A, d_2}$ be the Hilbert sequence space on $\\R$. Then $\\ell^2$ is a complete metric space."}
+{"_id": "15266", "title": "Hilbert Sequence Space is Separable", "text": "Let $A$ be the set of all real sequences $\\sequence {x_i}$ such that the series $\\displaystyle \\sum_{i \\mathop \\ge 0} x_i^2$ is convergent. Let $\\ell^2 = \\struct {A, d_2}$ be the Hilbert sequence space on $\\R$. Then $\\ell^2$ is a separable space."}
+{"_id": "15267", "title": "Hilbert Sequence Space is Second-Countable", "text": "Let $\\ell^2$ be the Hilbert sequence space on $\\R$. Then $\\ell^2$ is a second-countable space."}
+{"_id": "15268", "title": "Hilbert Sequence Space is Lindelöf", "text": "Let $\\ell^2$ be the Hilbert sequence space on $\\R$. Then $\\ell^2$ is a Lindelöf space."}
+{"_id": "15270", "title": "Compact Subset of Hilbert Sequence Space is Closed", "text": "Let $A$ be the set of all real sequences $\\sequence {x_i}$ such that the series $\\displaystyle \\sum_{i \\mathop \\ge 0} x_i^2$ is convergent. Let $\\ell^2 = \\struct {A, d_2}$ be the Hilbert sequence space on $\\R$. Let $H$ be a compact subset of $\\ell^2$ Then $H$ is closed in $\\ell^2$."}
+{"_id": "15271", "title": "Subset of Hilbert Sequence Space with Non-Empty Interior is not Compact", "text": "Let $A$ be the set of all real sequences $\\left\\langle{x_i}\\right\\rangle$ such that the series $\\displaystyle \\sum_{i \\mathop \\ge 0} x_i^2$ is convergent. Let $\\ell^2 = \\struct {A, d_2}$ be the Hilbert sequence space on $\\R$. Let $H$ be a subset of $\\ell^2$ whose interior  is non-empty. Then $H$ is not compact in $\\ell^2$."}
+{"_id": "15272", "title": "Point in Hilbert Sequence Space has no Compact Neighborhood", "text": "Let $A$ be the set of all real sequences $\\left\\langle{x_i}\\right\\rangle$ such that the series $\\displaystyle \\sum_{i \\mathop \\ge 0} x_i^2$ is convergent. Let $\\ell^2 = \\struct {A, d_2}$ be the Hilbert sequence space on $\\R$. Then no point of $\\ell^2$ has a compact neighborhood."}
+{"_id": "15273", "title": "Compact Subset of Hilbert Sequence Space is Nowhere Dense", "text": "Let $A$ be the set of all real sequences $\\sequence {x_i}$ such that the series $\\displaystyle \\sum_{i \\mathop \\ge 0} x_i^2$ is convergent. Let $\\ell^2 = \\struct {A, d_2}$ be the Hilbert sequence space on $\\R$. Let $H$ be a compact subset of $\\ell^2$. Then $H$ is nowhere dense in $\\ell^2$."}
+{"_id": "15275", "title": "Hilbert Sequence Space is Arc-Connected", "text": "Let $A$ be the set of all real sequences $\\sequence {x_i}$ such that the series $\\displaystyle \\sum_{i \\mathop \\ge 0} x_i^2$ is convergent. Let $\\ell^2 = \\struct {A, d_2}$ be the Hilbert sequence space on $\\R$. Then $\\ell^2$ is arc-connected."}
+{"_id": "15276", "title": "Hilbert Sequence Space is Homeomorphic to Countable Infinite Product of Real Number Spaces", "text": "Let $A$ be the set of all real sequences $\\sequence {x_i}$ such that the series $\\displaystyle \\sum_{i \\mathop \\ge 0} x_i^2$ is convergent. Let $\\ell^2 = \\struct {A, d_2}$ be the Hilbert sequence space on $\\R$. Let $\\struct {\\R, \\tau_d}$ denote the real number line under the Euclidean topology. Let $\\R^\\omega = \\displaystyle \\prod_{i \\mathop \\in \\N} \\struct {\\R, \\tau_d}$ denote the countable-dimensional real Cartesian space under the Tychonoff topology. Then $\\ell^2$ is homeomorphic to $\\R^\\omega$."}
+{"_id": "15277", "title": "Fréchet Space (Functional Analysis) is Metric Space", "text": "Let $\\struct {\\R^\\omega, d}$ be the '''Fréchet space on $\\R^\\omega$'''. Then $\\struct {\\R^\\omega, d}$ is a metric space."}
+{"_id": "15278", "title": "Countable Infinite Product of Real Number Spaces is Homeomorphic to Fréchet Metric Space", "text": "Let $\\struct {\\R, \\tau_d}$ denote the real number line under the Euclidean topology. Let $T = \\struct {\\R^\\omega, \\tau} = \\displaystyle \\prod_{i \\mathop \\in \\N} \\struct {\\R, \\tau_d}$ denote the countable-dimensional real Cartesian space under the product topology $\\tau$. Let $\\struct {\\R^\\omega, d}$ be the '''Fréchet space on $\\R^\\omega$''', where: :$\\map d {x, y} = \\displaystyle \\sum_{i \\mathop \\in \\N} \\dfrac {2^{-i} \\size {x_i - y_i} } {1 + \\size {x_i - y_i} }$ Then the topology induced by $d$ is exactly the Tychonoff topology $\\tau$."}
+{"_id": "15280", "title": "Separable Metric Space is Homeomorphic to Subspace of Fréchet Metric Space", "text": "Let $M = \\struct {A, d}$ be a metric space whose induced topology is separable. Then $M$ is homeomorphic to a subspace of the Fréchet space $\\struct {\\R^\\omega, d}$ on the countable-dimensional real Cartesian space $\\R^\\omega$."}
+{"_id": "15281", "title": "Hilbert Cube is Metric Space", "text": "Let $M = \\struct {I^\\omega, d_2}$ be the Hilbert cube. Then $M$ is a metric space."}
+{"_id": "15282", "title": "Hilbert Cube is Homeomorphic to Countable Infinite Product of Real Number Unit Intervals", "text": "Let $M_1 = \\struct {I^\\omega, d_2}$ be the Hilbert cube: :$M_1 = \\displaystyle \\prod_{k \\mathop \\in \\N} \\closedint 0 {\\dfrac 1 k}$ under the same metric as that of the Hilbert sequence space: :$\\displaystyle \\forall x = \\sequence {x_i}, y = \\sequence {y_i} \\in I^\\omega: \\map {d_2} {x, y} := \\paren {\\sum_{k \\mathop \\ge 0} \\paren {x_k - y_k}^2}^{\\frac 1 2}$ Let $M_2$ be the metric space defined as: :$M_2 = \\displaystyle \\prod_{k \\mathop \\in \\N} \\closedint 0 1$ under the Tychonoff topology. Then $M_1$ and $M_2$ are homeomorphic."}
+{"_id": "15283", "title": "Hilbert Cube is Completely Normal", "text": "Let $M = \\struct {I^\\omega, d_2}$ be the Hilbert cube. Then $M$ is a completely normal space."}
+{"_id": "15284", "title": "Hilbert Cube is Separable", "text": "Let $M = \\struct {I^\\omega, d_2}$ be the Hilbert cube. Then $M$ is a separable space."}
+{"_id": "15285", "title": "Hilbert Cube is Second-Countable", "text": "Let $M = \\struct {I^\\omega, d_2}$ be the Hilbert cube. Then $M$ is a second-countable space."}
+{"_id": "15287", "title": "Hilbert Cube is Arc-Connected", "text": "Let $M = \\struct {I^\\omega, d_2}$ be the Hilbert cube. Then $M$ is an arc-connected space."}
+{"_id": "15290", "title": "Interval of Totally Ordered Set is Convex", "text": "Let $\\struct {S, \\preccurlyeq}$ be a totally ordered set. Let $I \\subseteq S$ be an interval in $S$. Then $I$ is convex."}
+{"_id": "15291", "title": "Convex Set of Ordered Set is not necessarily Interval", "text": "Let $\\struct {S, \\preccurlyeq}$ be an ordered set. Let $C$ be a convex set of $S$. Then it is not necessarily the case that $C$ is an interval of $S$."}
+{"_id": "15292", "title": "Union of Non-Disjoint Convex Sets is Convex Set", "text": "Let $\\struct {S, \\preccurlyeq}$ be an ordered set. Let $\\CC$ be a set of convex sets of $S$ such that their intersection is non-empty: :$\\displaystyle \\bigcap \\CC \\ne \\O$ Then the union $\\displaystyle \\bigcup \\CC$ is also convex."}
+{"_id": "15293", "title": "Subset of Convex Set can be Uniquely Expressed as Partition of Maximal Convex Sets", "text": "Let $\\struct {S, \\preccurlyeq}$ be an ordered set. Then $S$ can be uniquely expressed as the partition whose components are maximally convex sets."}
+{"_id": "15294", "title": "Separated Subsets of Linearly Ordered Space under Order Topology", "text": "Let $T = \\struct {S, \\preceq, \\tau}$ be a linearly ordered space. Let $A$ and $B$ be separated sets of $T$. Let $A^*$ and $B^*$ be defined as: :$A^* := \\ds \\bigcup \\set {\\closedint a b: a, b \\in A, \\closedint a b \\cap B^- = \\O}$ :$B^* := \\ds \\bigcup \\set {\\closedint a b: a, b \\in B, \\closedint a b \\cap A^- = \\O}$ where $A^-$ and $B^-$ denote the closure of $A$ and $B$ in $T$. Then $A^*$ and $B^*$ are themselves separated sets of $T$."}
+{"_id": "15295", "title": "Generators of Special Linear Group of Order 2 over Integers", "text": "Let: :$ S = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix}$ and: :$T = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix}$ Then $S$ and $T$ are generators for the special linear group of order $2$ over $\\Z$."}
+{"_id": "15296", "title": "Partition of Linearly Ordered Space by Convex Components is Linearly Ordered Set", "text": "Let $T = \\struct {S, \\preceq, \\tau}$ be a linearly ordered space. Let $A$ and $B$ be separated sets of $T$. Let $A^*$ and $B^*$ be defined as: :$A^* := \\displaystyle \\bigcup \\set {\\closedint a b: a, b \\in A, \\closedint a b \\cap B^- = \\O}$ :$B^* := \\displaystyle \\bigcup \\set {\\closedint a b: a, b \\in B, \\closedint a b \\cap A^- = \\O}$ where $A^-$ and $B^-$ denote the closure of $A$ and $B$ in $T$. Let $A^*$, $B^*$ and $\\relcomp S {A^* \\cup B^*}$ be expressed as the union of convex components of $S$: :$\\displaystyle A^* = \\bigcup A_\\alpha, \\quad B^* = \\bigcup B_\\beta, \\quad \\relcomp S {A^* \\cup B^*} = \\bigcup C_\\gamma$ where $\\relcomp S X$ denotes the complement of $X$ with respect to $S$. Then the set $M = \\set {A_\\alpha, B_\\beta, C_\\gamma}$ inherits a linear ordering from $S$, and so is a linearly ordered set."}
+{"_id": "15297", "title": "Successor Sets of Linearly Ordered Set Induced by Convex Component Partition", "text": "Let $T = \\struct {S, \\preceq, \\tau}$ be a linearly ordered space. Let $A$ and $B$ be separated sets of $T$. Let $A^*$ and $B^*$ be defined as: :$A^* := \\displaystyle \\bigcup \\set {\\closedint a b: a, b \\in A, \\closedint a b \\cap B^- = \\O}$ :$B^* := \\displaystyle \\bigcup \\set {\\closedint a b: a, b \\in B, \\closedint a b \\cap A^- = \\O}$ where $A^-$ and $B^-$ denote the closure of $A$ and $B$ in $T$. Let $A^*$, $B^*$ and $\\relcomp S {A^* \\cup B^*}$ be expressed as the union of convex components of $S$: :$\\displaystyle A^* = \\bigcup A_\\alpha, \\quad B^* = \\bigcup B_\\beta, \\quad \\relcomp S {A^* \\cup B^*} = \\bigcup C_\\gamma$ where $\\relcomp S X$ denotes the complement of $X$ with respect to $S$. Let $M$ be the linearly ordered set: :$M = \\set {A_\\alpha, B_\\beta, C_\\gamma}$ as defined in Partition of Linearly Ordered Space by Convex Components is Linearly Ordered Set. Then each of the sets $A_\\alpha \\in M$ has an immediate successor in $M$ if $A_\\alpha$ intersects the closure of $S_\\alpha$, the set of strict upper bounds for $A_\\alpha$. Similarly for $B_\\beta$. That immediate successor ${C_\\alpha}^+$ to $A_\\alpha$ is an element in $\\set {C_\\gamma}$."}
+{"_id": "15298", "title": "Linearly Ordered Space is T5", "text": "Let $T = \\struct {S, \\preceq, \\tau}$ be a linearly ordered space. Then $T$ is a $T_5$ space."}
+{"_id": "15299", "title": "Linearly Ordered Space is T1", "text": "Let $T = \\struct {S, \\preceq, \\tau}$ be a linearly ordered space. Then $T$ is a $T_1$ (Fréchet) space."}
+{"_id": "15300", "title": "Linearly Ordered Space is Completely Normal", "text": "Let $T = \\struct {S, \\preceq, \\tau}$ be a linearly ordered space. Then $T$ is a completely normal space."}
+{"_id": "15301", "title": "Linearly Ordered Space is Compact iff Complete", "text": "Let $T = \\struct {S, \\preceq, \\tau}$ be a linearly ordered space. Then $T$ is a compact space {{iff}} it is complete."}
+{"_id": "15303", "title": "Basis for Open Ordinal Topology", "text": "Let $\\Gamma$ be a limit ordinal. Let $\\hointr 0 \\Gamma$ denote the open ordinal space on $\\Gamma$. Consider the set $\\BB$ of subsets of $\\hointr 0 \\Gamma$ of the form: :$\\openint \\alpha {\\beta + 1} = \\hointl \\alpha \\beta = \\set {x \\in \\hointr 0 \\Gamma: \\alpha < x < \\beta + 1}$ for $\\alpha, \\beta \\in \\hointr 0 \\Gamma$. {{explain|It's unclear from the source work used whether this is the basis for the Open Ordinal Topology or Closed Ordinal Topology.}} Then $\\BB$ forms a basis for $\\hointr 0 \\Gamma$."}
+{"_id": "15304", "title": "Omega is Closed in Uncountable Closed Ordinal Space but not G-Delta Set", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\closedint 0 \\Omega$ denote the closed ordinal space on $\\Omega$. Then $\\set \\Omega$ is a closed set of $\\closedint 0 \\Omega$ but not a $G_\\delta$ set."}
+{"_id": "15305", "title": "Uncountable Closed Ordinal Space is not First-Countable", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\closedint 0 \\Omega$ denote the closed ordinal space on $\\Omega$. Then $\\closedint 0 \\Omega$ is not a first-countable space."}
+{"_id": "15306", "title": "Omega as Limit Point of Intervals of Uncountable Closed Ordinal Space", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\closedint 0 \\Omega$ denote the closed ordinal space on $\\Omega$. Then $\\Omega$ is a limit point of the set $\\openint a \\Omega$, but not the limit point of any sequence of points in $\\openint a \\Omega$."}
+{"_id": "15307", "title": "Uncountable Open Ordinal Space is not Separable", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\hointr 0 \\Omega$ denote the open ordinal space on $\\Omega$. Then $\\hointr 0 \\Omega$ is not a separable space."}
+{"_id": "15308", "title": "Uncountable Closed Ordinal Space is not Separable", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\closedint 0 \\Omega$ denote the closed ordinal space on $\\Omega$. Then $\\closedint 0 \\Omega$ is not a separable space."}
+{"_id": "15309", "title": "Uncountable Open Ordinal Space is First-Countable", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\hointr 0 \\Omega$ denote the open ordinal space on $\\Omega$. Then $\\hointr 0 \\Omega$ is a first-countable space."}
+{"_id": "15310", "title": "Ordinal Space is Completely Normal", "text": "Let $\\Gamma$ denote a limit ordinal. Let $\\hointr 0 \\Gamma$ denote the open ordinal space on $\\Gamma$. Let $\\closedint 0 \\Gamma$ denote the closed ordinal space on $\\Gamma$. Then $\\hointr 0 \\Gamma$ and $\\closedint 0 \\Gamma$ are both completely normal."}
+{"_id": "15311", "title": "Uncountable Closed Ordinal Space is not Perfectly Normal", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\closedint 0 \\Omega$ denote the closed ordinal space on $\\Omega$. Then $\\closedint 0 \\Omega$ is not a perfectly normal space."}
+{"_id": "15312", "title": "Uncountable Closed Ordinal Space is not Second-Countable", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\closedint 0 \\Omega$ denote the closed ordinal space on $\\Omega$. Then $\\closedint 0 \\Omega$ is not a second-countable space."}
+{"_id": "15313", "title": "Uncountable Open Ordinal Space is not Second-Countable", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\hointr 0 \\Omega$ denote the open ordinal space on $\\Omega$. Then $\\hointr 0 \\Omega$ is not a second-countable space."}
+{"_id": "15314", "title": "Ring Element is Unit iff Unit in Integral Extension", "text": "let $A$ be a commutative ring with unity. Let $a \\in A$. Let $B$ be an integral ring extension of $A$. {{TFAE}} :$(1): \\quad a$ is a unit of $A$ :$(2): \\quad a$ is a unit of $B$"}
+{"_id": "15315", "title": "Separable Elements Form Field", "text": "Let $E/F$ be an algebraic field extension. Then the subset of separable elements of $E$ form a intermediate field, called the '''relative separable closure'''."}
+{"_id": "15316", "title": "Decomposition of Field Extension as Separable Extension followed by Purely Inseparable", "text": "Let $E/F$ be an algebraic field extension. Then the relative separable closure $K=F^{sep}$ in $E$ is the unique intermediate field with the following properties: * $K/F$ is separable. * $E/K$ is purely inseparable."}
+{"_id": "15317", "title": "Subextensions of Separable Field Extension are Separable", "text": "Let $E/K/F$ be a tower of fields. Let $E/F$ be separable. Then $E/K$ and $K/F$ are separable."}
+{"_id": "15318", "title": "Finite Orbit under Group of Automorphisms of Field implies Separable over Fixed Field", "text": "Let $E$ be a field. Let $G \\le \\Aut E$ be a subgroup of its automorphism group. Let $F = \\map {\\operatorname {Fix}_E} G$ be its fixed field. Let $\\alpha \\in E$ have a finite orbit under $G$. Then $\\alpha$ is separable over $F$."}
+{"_id": "15319", "title": "Countable Closed Ordinal Space is Second-Countable", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\Gamma$ be a limit ordinal which strictly precedes $\\Omega$. Let $\\closedint 0 \\Gamma$ denote the closed ordinal space on $\\Gamma$. Then $\\closedint 0 \\Gamma$ is a second-countable space."}
+{"_id": "15320", "title": "Countable Open Ordinal Space is Second-Countable", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\Gamma$ be a limit ordinal which strictly precedes $\\Omega$. Let $\\hointr 0 \\Gamma$ denote the open ordinal space on $\\Gamma$. Then $\\hointr 0 \\Gamma$ is a second-countable space."}
+{"_id": "15321", "title": "Minimal Polynomial of Element with Finite Orbit under Group of Automorphisms over Fixed Field in terms of Orbit", "text": "Let $E$ be a field. Let $G \\le \\Aut E$ be a subgroup of its automorphism group. Let $F = \\map {\\operatorname {Fix_E} } G$ be its fixed field. Let $\\alpha \\in E$ have a finite orbit under $G$. Then $\\alpha$ is algebraic over $F$ and the product of polynomials :$\\map p x = \\displaystyle \\prod_{\\beta \\mathop \\in \\Lambda} \\paren {x - \\beta}$ is the minimal polynomial of $\\alpha$ over $F$."}
+{"_id": "15322", "title": "Countable Closed Ordinal Space is Metrizable", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\Gamma$ be a limit ordinal which strictly precedes $\\Omega$. Let $\\closedint 0 \\Gamma$ denote the closed ordinal space on $\\Gamma$. Then $\\closedint 0 \\Gamma$ is a metrizable space."}
+{"_id": "15324", "title": "Closed Ordinal Space is Complete Order Space", "text": "Let $\\Gamma$ be a limit ordinal. Let $\\closedint 0 \\Gamma$ denote the closed ordinal space on $\\Gamma$. Then $\\closedint 0 \\Gamma$ is a complete order space."}
+{"_id": "15325", "title": "Closed Ordinal Space is Compact", "text": "Let $\\Gamma$ be a limit ordinal. Let $\\closedint 0 \\Gamma$ denote the closed ordinal space on $\\Gamma$. Then $\\closedint 0 \\Gamma$ is a compact space."}
+{"_id": "15326", "title": "Ordinal Space is Strongly Locally Compact", "text": "Let $T$ denote an ordinal space on a limit ordinal $\\Gamma$. Then $T$ is a strongly locally compact space."}
+{"_id": "15327", "title": "Open Ordinal Space is not Compact in Closed Ordinal Space", "text": "Let $\\Gamma$ be a limit ordinal. Let $\\hointr 0 \\Gamma$ denote the open ordinal space on $\\Gamma$. Consider the compact subspace $\\hointr 0 \\Gamma$. Then $\\hointr 0 \\Gamma$ is not compact in $\\closedint 0 \\Gamma$."}
+{"_id": "15328", "title": "Integral Transform is Mapping", "text": "Let $\\map F p$ be an integral transform: :$\\map F p = \\displaystyle \\int_a^b \\map f x \\map K {p, x} \\rd x$ Let $T$ be the integral operator associated with $\\map F p$. Then $T$ is a mapping from the domain of $T$ to its image. That is, for every $\\map f x$ there exists a unique $\\map F p$."}
+{"_id": "15329", "title": "Integral Operator is Linear", "text": "Let $T$ be an integral operator. Let $f$ and $g$ be integrable real functions on a domain appropriate to $T$. Then $T$ is a linear operator: :$\\forall \\alpha, \\beta \\in \\R: \\map T {\\alpha f + \\beta g} = \\alpha \\map T f + \\beta \\map T g$"}
+{"_id": "15332", "title": "Inverse Integral Operator is Linear if Unique", "text": "Let $T$ be an integral operator. Let $f$ be an integrable real function on a domain appropriate to $T$. Let $F = \\map T f$ and $G = \\map T g$. Let $T$ have a unique inverse $T^{-1}$. Then $T^{-1}$ is a linear operator: :$\\forall p, q \\in \\R: \\map {T^{-1} } {p F + q G} = p \\map {T^{-1} } F + q \\map {T^{-1} } G$"}
+{"_id": "15333", "title": "Trigonometric Series is Convergent if Sum of Absolute Values of Coefficients is Convergent", "text": "Let $\\map S x$ be a trigonometric series: :$\\map S x = \\dfrac {a_0} 2 + \\displaystyle \\sum_{n \\mathop = 1}^\\infty \\paren {a_n \\cos n x + b_n \\sin n x}$ Let the series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\paren {\\size {a_n} + \\size {b_n} }$ be convergent. Then $S$ is a convergent series."}
+{"_id": "15334", "title": "Convergent Trigonometric Series is Periodic", "text": "Let $\\map S x$ be a trigonometric series: :$\\map S x = \\dfrac {a_0} 2 + \\displaystyle \\sum_{n \\mathop = 1}^\\infty \\paren {a_n \\cos n x + b_n \\sin n x}$ Let $S$ be convergent. Then $S$ is periodic: :$\\forall r \\in \\Z: \\map S {x + 2 r \\pi} = \\map S x$"}
+{"_id": "15337", "title": "Integral over 2 pi of Sine of m x by Sine of n x", "text": "Let $m, n \\in \\Z$ be integers. Let $\\alpha \\in \\R$ be a real number. Then: :$\\displaystyle \\int_\\alpha^{\\alpha + 2 \\pi} \\sin m x \\sin n x \\rd x = \\begin{cases} 0 & : m \\ne n \\\\ \\pi & : m = n \\end{cases}$ That is: :$\\displaystyle \\int_\\alpha^{\\alpha + 2 \\pi} \\sin m x \\sin n x \\rd x = \\pi \\delta_{m n}$ where $\\delta_{m n}$ is the Kronecker delta."}
+{"_id": "15338", "title": "Integral over 2 pi of Cosine of m x by Cosine of n x", "text": "Let $m, n \\in \\Z$ be integers. Let $\\alpha \\in \\R$ be a real number. Then: :$\\ds \\int_\\alpha^{\\alpha + 2 \\pi} \\cos m x \\cos n x \\rd x = \\begin {cases} 0 & : m \\ne n \\\\ \\pi & : m = n \\end {cases}$ That is: :$\\ds \\int_\\alpha^{\\alpha + 2 \\pi} \\cos m x \\cos n x \\rd x = \\pi \\delta_{m n}$ where $\\delta_{m n}$ is the Kronecker delta."}
+{"_id": "15339", "title": "Integral over 2 pi of Sine of m x by Cosine of n x", "text": "Let $m, n \\in \\Z$ be integers. Let $\\alpha \\in \\R$ be a real number. Then: :$\\displaystyle \\int_\\alpha^{\\alpha + 2 \\pi} \\sin m x \\cos n x \\rd x = 0$"}
+{"_id": "15340", "title": "Integral over 2 pi of Sine of n x", "text": "Let $m \\in \\Z$ be an integer. Then: :$\\displaystyle \\int_\\alpha^{\\alpha + 2 \\pi} \\sin m x \\rd x = 0$"}
+{"_id": "15341", "title": "Integral over 2 pi of Cosine of n x", "text": "Let $m \\in \\Z$ be an integer. Then: :$\\displaystyle \\int_\\alpha^{\\alpha + 2 \\pi} \\cos m x \\rd x = \\begin {cases} 0 & : m \\ne 0 \\\\ 2 \\pi & : m = 0 \\end {cases}$"}
+{"_id": "15343", "title": "Coefficients of Sine Terms in Convergent Trigonometric Series", "text": "Let $\\map S x$ be a trigonometric series which converges to $\\map f x$ on the interval $\\openint \\alpha {\\alpha + 2 \\pi}$: :$\\map f x = \\dfrac {a_0} 2 + \\displaystyle \\sum_{m \\mathop = 1}^\\infty \\left({a_m \\cos m x + b_m \\sin m x}\\right)$ Then: :$\\forall n \\in \\Z_{\\ge 0}: b_n = \\dfrac 1 \\pi \\displaystyle \\int_\\alpha^{\\alpha + 2 \\pi} \\map f x \\sin n x \\rd x$"}
+{"_id": "15344", "title": "Sine of Angle plus Full Angle/Corollary", "text": "Let $n \\in \\Z$ be an integer. Then: :$\\map \\sin {x + 2 n \\pi} = \\sin x$"}
+{"_id": "15345", "title": "Cosine of Angle plus Full Angle/Corollary", "text": "Let $n \\in \\Z$ be an integer. Then: : $\\cos \\left({x + 2 n \\pi}\\right) = \\cos x$"}
+{"_id": "15346", "title": "Fourier's Theorem", "text": "Let $\\alpha \\in \\R$ be a real number. Let $\\map f x$ be a real function which is defined and bounded on the interval $\\openint \\alpha {\\alpha + 2 \\pi}$. Let $f$ satisfy the Dirichlet conditions on $\\openint \\alpha {\\alpha + 2 \\pi}$: {{:Definition:Dirichlet Conditions}} Outside the interval $\\openint \\alpha {\\alpha + 2 \\pi}$, let $f$ be periodic and defined such that: :$\\map f x = \\map f {x + 2 \\pi}$ Let $f$ be defined by the Fourier series: :$(1): \\quad \\displaystyle \\frac {a_0} 2 + \\sum_{n \\mathop = 1}^\\infty \\paren {a_n \\cos n x + b_n \\sin n x}$ such that: :$\\displaystyle a_n = \\dfrac 1 \\pi \\int_\\alpha^{\\alpha + 2 \\pi} \\map f x \\cos n x \\rd x$ :$\\displaystyle b_n = \\dfrac 1 \\pi \\int_\\alpha^{\\alpha + 2 \\pi} \\map f x \\sin n x \\rd x$ Then for all $a \\in \\R$, $(1)$ converges to the sum: :$\\displaystyle \\frac 1 2 \\paren {\\lim_{x \\mathop \\to a^+} \\map f x + \\lim_{x \\mathop \\to a^-} \\map f x}$ where the $\\lim$ symbols denote the limit from the right and the limit from the left."}
+{"_id": "15347", "title": "Definite Integral of Step Function", "text": "Let $\\alpha, \\beta \\in \\R$ be a real numbers such that $\\alpha < \\beta$. Let $f \\left({x}\\right)$ be a step function defined on the interval $\\left[{\\alpha \\,.\\,.\\, \\beta}\\right]$: :$f \\left({x}\\right) = \\lambda_1 \\chi_{\\mathbb I_1} + \\lambda_2 \\chi_{\\mathbb I_2} + \\cdots + \\lambda_n \\chi_{\\mathbb I_n}$ where: :$\\lambda_1, \\lambda_2, \\ldots, \\lambda_n$ are real constants :$\\mathbb I_1, \\mathbb I_2, \\ldots, \\mathbb I_n$ are intervals, where these intervals partition $\\left[{\\alpha \\,.\\,.\\, \\beta}\\right]$ :$\\chi_{\\mathbb I_1}, \\chi_{\\mathbb I_2}, \\ldots, \\chi_{\\mathbb I_n}$ are characteristic functions of $\\mathbb I_1, \\mathbb I_2, \\ldots, \\mathbb I_n$. Then the definite integral of $f$ {{{WRT|Integration}} $x$ over $\\left[{\\alpha \\,.\\,.\\, \\beta}\\right]$ is given by: :$\\displaystyle \\int_\\alpha^\\beta f \\left({x}\\right) \\, \\mathrm d x = \\sum_{k \\mathop = 1}^n \\lambda_k \\left({\\beta_k - \\alpha_k}\\right)$ where $\\alpha_k, \\beta_k$ are the endpoints of $\\mathbb I_k$ for $1 \\le k \\le n$."}
+{"_id": "15348", "title": "Step Function satisfies Dirichlet Conditions", "text": "Let $\\alpha, \\beta \\in \\R$ be a real numbers such that $\\alpha < \\beta$. Let $\\map f x$ be a step function defined on the interval $\\openint \\alpha \\beta$. Then $f$ satisfies the Dirichlet conditions."}
+{"_id": "15350", "title": "Parseval's Theorem/Formulation 1", "text": "Let $f$ be a real function which is square-integrable over the interval $\\openint {-\\pi} \\pi$. Let $f$ be expressed by the Fourier series:  :$\\map f x \\sim \\dfrac {a_0} 2 + \\displaystyle \\sum_{n \\mathop = 1}^\\infty \\paren {a_n \\cos n x + b_n \\sin n x}$ Then: :$\\displaystyle \\frac 1 \\pi \\int_{-\\pi}^\\pi \\size {\\map {f^2} x} \\rd x = \\frac { {a_0}^2} 2 + \\sum_{n \\mathop = 1}^\\infty \\paren { {a_n}^2 + {b_n}^2}$"}
+{"_id": "15351", "title": "Fourier Series/Square of x minus pi, Square of pi", "text": ":400pxrightthumb$\\map f x$ and $5$th order expansion Let $\\map f x$ be the real function defined on $\\openint 0 {2 \\pi}$ as: :$\\map f x = \\begin{cases} \\paren {x - \\pi}^2 & : 0 < x \\le \\pi \\\\ \\pi^2 & : \\pi < x < 2 \\pi \\end{cases}$ Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \\map f x       | o = \\sim       | r = \\frac {2 \\pi^2} 3 + \\sum_{n \\mathop = 1}^\\infty \\paren {\\frac {2 \\cos n x} {n^2} + \\paren {\\frac {\\paren {-1}^n \\pi} n - \\frac {2 \\paren {1 - \\paren {-1}^n} } {\\pi n^3} } \\sin n x}       | c =  }} {{eqn | r = \\frac {2 \\pi^2} 3 + 2 \\paren {\\cos x + \\frac {\\cos 2 x} {2^2} + \\frac {\\cos 3 x} {3^2} + \\cdots}       | c =  }} {{eqn | o =        | ro= -       | r = \\pi \\sin x + \\frac \\pi 2 \\sin x - \\frac \\pi 3 \\sin x + \\frac \\pi 4 \\sin x - \\cdots       | c =  }} {{eqn | o =        | ro= -       | r = \\dfrac 4 \\pi \\sin x - \\frac 4 {\\pi 3^3} \\sin 3 x - \\frac 4 {\\pi 5^3} \\sin 5 x \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "15354", "title": "Sum of Reciprocals of Squares Alternating in Sign", "text": "{{begin-eqn}} {{eqn | l = \\dfrac {\\pi^2} {12}       | r = \\sum_{n \\mathop = 1}^\\infty \\dfrac {\\paren {-1}^{n + 1} } {n^2}       | c =  }} {{eqn | r = \\frac 1 {1^2} - \\frac 1 {2^2} + \\frac 1 {3^2} - \\frac 1 {4^2} + \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "15358", "title": "Fourier Series/4 minus x squared over Range of 2", "text": "Let $\\map f x$ be the real function defined on $\\openint 0 2$ as: :600pxthumbright$\\map f x$ and its $7$th approximation :$\\map f x = 4 - x^2$ Then its Fourier series can be expressed as: :$\\map f x \\sim \\displaystyle \\frac 8 3 - \\frac 4 {\\pi^2} \\sum_{n \\mathop = 1}^\\infty \\frac {\\cos n \\pi x} {n^2} + \\frac 4 \\pi \\sum_{n \\mathop = 1}^\\infty  \\frac {\\sin n \\pi x} n$"}
+{"_id": "15359", "title": "Fourier Series/1 over -1 to 0, Cosine of Pi x over 0 to 1", "text": "Let $\\map f x$ be the real function defined on $\\openint {-1} 1$ as: :800pxthumbright$\\map f x$ and its $7$th approximation :$\\map f x = \\begin{cases} 1 & : -1 < x < 0 \\\\ \\map \\cos {\\pi x} & : 0 < x < 1 \\end{cases}$ Then its Fourier series can be expressed as: :$\\map f x \\sim \\displaystyle \\dfrac 1 2 + \\frac {\\cos \\pi x} 2 + \\sum_{r \\mathop = 1}^\\infty \\paren {\\dfrac {4 r \\sin 2 r \\pi x} {\\pi \\paren {2 r + 1} \\paren {2 r - 1} } - \\dfrac {2 \\map \\sin {2 r + 1} \\pi x } {\\pi \\paren {2 r + 1} } }$"}
+{"_id": "15360", "title": "Dilogarithm of Square", "text": ":$\\operatorname {Li_2} \\left({z}\\right) + \\operatorname {Li_2} \\left({-z}\\right) = \\frac 1 2 \\operatorname {Li_2} \\left({z^2}\\right)$ where $\\operatorname {Li_2}$ denotes Spence's Function."}
+{"_id": "15361", "title": "Power Series Expansion for Spence's Function", "text": "Spence's function has a power series expansion:  :$\\displaystyle \\operatorname {Li}_2 \\left({z}\\right) = \\sum_{n \\mathop = 1}^\\infty \\frac {z^n} {n^2}$ This converges for $\\left|{z}\\right| \\le 1$. {{explain|The domain needs to be clarified. As it stands, the notation and construction of the proof suggests $\\R$.}}"}
+{"_id": "15362", "title": "Odd Function of Zero is Zero", "text": "Let $f: \\R \\to \\R$ be an odd function. Let $f$ be defined at the point $x = 0$. Then: :$\\map f 0 = 0$"}
+{"_id": "15363", "title": "Fourier Cosine Coefficients for Even Function over Symmetric Range", "text": "Let $\\map f x$ be an even real function defined on the interval $\\openint {-\\lambda} \\lambda$. Let the Fourier series of $\\map f x$ be expressed as: :$\\map f x \\sim \\dfrac {a_0} 2 + \\displaystyle \\sum_{n \\mathop = 1}^\\infty \\paren {a_n \\cos \\frac {n \\pi x} \\lambda + b_n \\sin \\frac {n \\pi x} \\lambda}$ Then for all $n \\in \\Z_{\\ge 0}$: :$a_n = \\dfrac 2 \\lambda \\displaystyle \\int_0^\\lambda \\map f x \\cos \\frac {n \\pi x} \\lambda \\rd x$"}
+{"_id": "15364", "title": "Fourier Sine Coefficients for Even Function over Symmetric Range", "text": "Let $\\map f x$ be an even real function defined on the interval $\\openint {-\\lambda} \\lambda$. Let the Fourier series of $\\map f x$ be expressed as: :$\\displaystyle \\map f x \\sim \\frac {a_0} 2 + \\sum_{n \\mathop = 1}^\\infty \\paren {a_n \\cos \\frac {n \\pi x} \\lambda + b_n \\sin \\frac {n \\pi x} \\lambda}$ Then for all $n \\in \\Z_{> 0}$: :$b_n = 0$"}
+{"_id": "15365", "title": "Fourier Series for Even Function over Symmetric Range", "text": "Let $\\map f x$ be an even real function defined on the interval $\\openint {-\\lambda} \\lambda$. Then the Fourier series of $\\map f x$ can be expressed as: :$\\map f x \\sim \\dfrac {a_0} 2 + \\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n \\cos \\frac {n \\pi x} \\lambda$ where for all $n \\in \\Z_{\\ge 0}$: :$a_n = \\dfrac 2 \\lambda \\displaystyle \\int_0^\\lambda \\map f x \\cos \\frac {n \\pi x} \\lambda \\rd x$"}
+{"_id": "15366", "title": "Fourier Cosine Coefficients for Odd Function over Symmetric Range", "text": "Let $\\map f x$ be an odd real function defined on the interval $\\openint {-\\lambda} \\lambda$. Let the Fourier series of $\\map f x$ be expressed as: :$\\map f x \\sim \\dfrac {a_0} 2 + \\displaystyle \\sum_{n \\mathop = 1}^\\infty \\paren {a_n \\cos \\frac {n \\pi x} \\lambda + b_n \\sin \\frac {n \\pi x} \\lambda}$ Then for all $n \\in \\Z_{\\ge 0}$: :$a_n = 0$"}
+{"_id": "15367", "title": "Fourier Sine Coefficients for Odd Function over Symmetric Range", "text": "Let $\\map f x$ be an odd real function defined on the interval $\\openint {-\\lambda} \\lambda$. Let the Fourier series of $\\map f x$ be expressed as: :$\\displaystyle \\map f x \\sim \\frac {a_0} 2 + \\sum_{n \\mathop = 1}^\\infty \\paren {a_n \\cos \\frac {n \\pi x} \\lambda + b_n \\sin \\frac {n \\pi x} \\lambda}$ Then for all $n \\in \\Z_{> 0}$: :$b_n = \\displaystyle \\frac 2 \\lambda \\int_0^\\lambda \\map f x \\sin \\frac {n \\pi x} \\lambda \\rd x$"}
+{"_id": "15368", "title": "Fourier Series for Odd Function over Symmetric Range", "text": "Let $\\map f x$ be an odd real function defined on the interval $\\openint {-\\lambda} \\lambda$. Then the Fourier series of $\\map f x$ can be expressed as: :$\\map f x \\sim \\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n \\sin \\frac {n \\pi x} \\lambda$ where for all $n \\in \\Z_{> 0}$: :$b_n = \\displaystyle \\frac 2 \\lambda \\int_0^\\lambda \\map f x \\sin \\frac {n \\pi x} \\lambda \\rd x$"}
+{"_id": "15369", "title": "Half-Range Fourier Sine Series over Negative Range", "text": "Let $\\map f x$ be a real function defined on the open real interval $\\openint 0 \\lambda$. Let $f$ be expressed using the half-range Fourier sine series over $\\openint 0 \\lambda$: :$\\displaystyle \\map S x \\sim \\sum_{n \\mathop = 1}^\\infty b_n \\sin \\frac {n \\pi x} \\lambda$ where: :$b_n = \\displaystyle \\frac 2 \\lambda \\int_0^\\lambda \\map f x \\sin \\frac {n \\pi x} \\lambda \\rd x$ for all $n \\in \\Z_{\\ge 0}$. Then over the interval $\\openint {-\\lambda} 0$, $\\map S x$ takes the values: :$\\map S x = -\\map f {-x}$ {{refactor|This following bit depends upon what happens at $x {{=}} 0$ which needs to be carefully considered, so put it here as a corollary}} That is, the real function expressed by the half-range Fourier sine series over $\\openint 0 \\lambda$ is an odd function over $\\openint {-\\lambda} \\lambda$."}
+{"_id": "15371", "title": "Half-Range Fourier Cosine Series over Negative Range", "text": "Let $\\map f x$ be a real function defined on the open real interval $\\openint 0 \\lambda$. Let $f$ be expressed using the half-range Fourier cosine series over $\\openint 0 \\lambda$: :$\\displaystyle \\map C x \\sim \\frac {a_0} 2 + \\sum_{n \\mathop = 1}^\\infty a_n \\cos \\frac {n \\pi x} \\lambda$ where: :$a_n = \\displaystyle \\frac 2 \\lambda \\int_0^\\lambda \\map f x \\cos \\frac {n \\pi x} \\lambda \\rd x$ for all $n \\in \\Z_{\\ge 0}$. Then over the closed real interval $\\openint {-\\lambda} 0$, $\\map C x$ takes the values: :$\\map C x = \\map f {-x}$ {{refactor|This following bit depends upon what happens at $x {{=}} 0$ which needs to be carefully considered, so put it here as a corollary}} That is, the real function expressed by the half-range Fourier cosine series over $\\openint 0 \\lambda$ is an even function over $\\openint {-\\lambda} \\lambda$."}
+{"_id": "15372", "title": "Fourier Series/x over 0 to 2, x-2 over 2 to 4", "text": "Let $\\map f x$ be the real function defined on $\\openint 0 4$ as: :$\\map f x = \\begin{cases} x & : 0 < x \\le 2 \\\\ x - 2 & : 2 < x < 4 \\end{cases}$ Then its Fourier series can be expressed as: :$\\map f x \\sim \\displaystyle 1 + \\frac 4 \\pi \\sum_{n \\mathop = 1}^\\infty \\frac {\\paren {-1}^{r - 1} } {2 r - 1} \\paren {1 + \\frac {4 \\paren {-1}^r} {\\paren {2 r - 1} \\pi} } \\cos \\frac {\\paren {2 r - 1} \\pi x} 4$"}
+{"_id": "15373", "title": "Fourier Series over General Range from Specific", "text": "Let $a, b \\in \\R$ be real numbers. Let $f: \\R \\to \\R$ be a function such that $\\displaystyle \\int_a^b \\map f x \\rd x$ converges absolutely. Then $f$ can be expressed by a Fourier series of the form: :$\\displaystyle \\frac {A_0} 2 + \\sum_{m \\mathop = 1}^\\infty \\paren {A_m \\cos \\frac {2 m \\pi \\paren {x - a} } {b - a} + B_m \\sin \\frac {2 m \\pi \\paren {x - a} } {b - a} }$ where: {{begin-eqn}} {{eqn | l = A_m       | r = \\dfrac 2 {b - a} \\int_a^b \\map f x \\cos \\frac {2 m \\pi \\paren {x - a} } {b - a} \\rd x }} {{eqn | l = B_m       | r = \\dfrac 2 {b - a} \\int_a^b \\map f x \\sin \\frac {2 m \\pi \\paren {x - a} } {b - a} \\rd x }} {{end-eqn}}"}
+{"_id": "15374", "title": "Definite Integral to Infinity of Reciprocal of 1 plus Power of x", "text": ":$\\displaystyle \\int_0^\\infty \\frac 1 {1 + x^n} \\rd x = \\frac \\pi n \\csc \\paren {\\frac \\pi n}$"}
+{"_id": "15375", "title": "Leibniz's Integral Rule", "text": "Let $\\map f {x, t}$, $\\map a t$, $\\map b t$ be continuously differentiable real functions on some region $R$ of the $\\tuple {x, t}$ plane. Then for all $\\tuple {x, t} \\in R$: :$\\displaystyle \\frac \\rd {\\rd t} \\int_{\\map a t}^{\\map b t} \\map f {x, t} \\rd x = \\map f {t, \\map b t} \\frac {\\rd b} {\\rd t} - \\map f {t, \\map a t} \\frac {\\rd a} {\\rd t} + \\int_{\\map a t}^{\\map b t} \\frac {\\partial} {\\partial t} \\map f {x, t} \\rd x$"}
+{"_id": "15376", "title": "Series Expansion of Function over Complete Orthonormal Set", "text": "Let $\\map f x$ be a real function defined over the interval $\\openint a b$. Let $\\map f x$ be able to be expressed in terms of a complete orthonormal set of real functions $S := \\family {\\map {\\phi_i} x}_{i \\mathop \\in I}$ for some indexing set $I$: :$\\map f x = \\displaystyle \\sum_{i \\mathop \\in I} a_i \\map {\\phi_i} x$ Then the coefficients $\\family {a_i}_{i \\mathop \\in I}$ can be determined as: :$\\forall i \\in I: a_i = \\displaystyle \\int_a^b \\map f x \\map {\\phi_i} x \\rd x$"}
+{"_id": "15377", "title": "Vector as Sum of Orthogonal Base Vectors", "text": "Let $\\mathbf v$ be a vector quantity in ordinary $3$-space. Let $\\mathbf i, \\mathbf j, \\mathbf k$ be orthonormal base vectors. Then: :$\\mathbf v = \\paren {\\mathbf v \\cdot \\mathbf i} \\mathbf i + \\paren {\\mathbf v \\cdot \\mathbf j} \\mathbf j + \\paren {\\mathbf v \\cdot \\mathbf k} \\mathbf k$"}
+{"_id": "15378", "title": "Scaled Sine Functions of Integer Multiples form Orthonormal Set", "text": "For all $n \\in \\Z_{>0}$, let $\\map {\\phi_n} x$ be the real function defined on the interval $\\openint 0 \\lambda$ as: :$\\map {\\phi_n} x = \\sqrt {\\dfrac 2 \\lambda} \\sin \\dfrac {n \\pi x} \\lambda$ Let $S$ be the set: :$S = \\set {\\phi_n: n \\in \\Z_{>0} }$ Then $S$ is an orthonormal set."}
+{"_id": "15379", "title": "Correspondence Theorem for Quotient Rings", "text": "Let $A$ be a commutative ring with unity. Let $\\mathfrak a \\subseteq A$ be an ideal. Let $A / \\mathfrak a$ be the quotient ring and $\\pi : A \\to A / \\mathfrak a$ the quotient ring epimorphism."}
+{"_id": "15380", "title": "Correspondence Theorem for Quotient Rings/Bijection", "text": "The direct image mapping $\\pi^\\to$ and the inverse image mapping $\\pi^\\gets$ induce reverse bijections between the ideals of $A$ containing $\\mathfrak a$ and the ideals of $A/\\mathfrak a$, specifically: Let $I$ be the set of ideals of $A$ containing $\\mathfrak a$. Let $J$ be the set of ideals of $A/\\mathfrak a$. Then: #For every ideal $\\mathfrak b \\in I$, its image $\\pi^{\\to}(\\mathfrak b) = \\pi(\\mathfrak b) \\in J$. #For every ideal $\\mathfrak c \\in J$, its preimage $\\pi^{\\gets}(\\mathfrak c) = \\pi^{-1}(\\mathfrak c) \\in I$. #The restrictions $\\pi^\\to : I \\to J$ and $\\pi^\\gets : J \\to I$ are reverse bijections."}
+{"_id": "15381", "title": "Correspondence Theorem for Ring Epimorphisms/Bijection", "text": "The direct image mapping $\\pi^\\to$ and the inverse image mapping $\\pi^\\gets$ induce reverse bijections between $I$ and $J$, specifically: #For every ideal $\\mathfrak a \\in I$, its image $\\pi^{\\to}(\\mathfrak a) = \\pi(\\mathfrak a) \\in J$. #For every ideal $\\mathfrak b \\in J$, its preimage $\\pi^{\\gets}(\\mathfrak b) = \\pi^{-1}(\\mathfrak b) \\in I$. #The restrictions $\\pi^\\to : I \\to J$ and $\\pi^\\gets : J \\to I$ are reverse bijections."}
+{"_id": "15382", "title": "Image of Ideal under Ring Epimorphism is Ideal", "text": "Let $f : A \\to B$ be a ring epimorphism. Let $I \\subseteq A$ be an ideal. Then its image $f(I) \\subseteq B$ is an ideal."}
+{"_id": "15383", "title": "Preimage of Prime Ideal under Ring Homomorphism is Prime Ideal", "text": "Let $A$ and $B$ be commutative rings with unity. Let $f : A \\to B$ be a ring homomorphism. Let $\\mathfrak p \\subseteq B$ be a prime ideal. Then its preimage $\\map {f^{-1} } {\\mathfrak p}$ is a prime ideal of $A$."}
+{"_id": "15384", "title": "Correspondence Theorem for Localizations of Rings", "text": "Let $A$ be a commutative ring with unity. Let $S\\subseteq A$ be a multiplicatively closed subset. Let $A \\overset \\iota \\to A_S$ be the localization at $S$. Let $I$ be the set of saturated ideals of $A$ by $S$. Let $J$ be the set of ideals of $A_S$."}
+{"_id": "15385", "title": "Integral Domain iff Zero Ideal is Prime", "text": "Let $A$ be a commutative ring with unity. {{TFAE}} :$(1): \\quad A$ is an integral domain :$(2): \\quad$ the zero ideal $0 \\subseteq A$ is prime"}
+{"_id": "15388", "title": "Complement of Prime Ideal of Ring is Multiplicatively Closed", "text": "Let $R$ be a commutative ring with unity. Let $P \\subset R$ be a prime ideal of $R$. Then its complement $R \\setminus P$ is multiplicatively closed."}
+{"_id": "15389", "title": "Derivative of Gamma Function", "text": ":$\\displaystyle \\map {\\Gamma'} x = \\int_0^\\infty t^{x - 1} \\ln t \\, e^{-t} \\rd t$ where $\\map {\\Gamma'} x$ denotes the derivative of the Gamma function evaluated at $x$."}
+{"_id": "15390", "title": "Hankel Representation of Riemann Zeta Function", "text": "Let $C$ be the Hankel contour. Then for $s \\in \\C \\setminus \\Z_{>0}$:  :$\\displaystyle \\zeta\\left({s}\\right) = \\frac {i \\Gamma \\left({1 - s}\\right)} {2 \\pi} \\oint_C \\frac {\\left({-z}\\right)^{s - 1} } {e^z - 1} \\, \\mathrm d z$ where: :$\\zeta$ is the Riemann Zeta function :$\\Gamma$ is the Gamma function."}
+{"_id": "15391", "title": "Fourier Series/Minus Pi over 0 to Pi, x minus Pi over Pi to 2 Pi", "text": "Let $\\map f x$ be the real function defined on $\\openint 0 {2 \\pi}$ as: :500pxthumbright$\\map f x$ and its $7$th approximation :$\\map f x = \\begin{cases} -\\pi & : 0 < x \\le \\pi \\\\ x - \\pi & : \\pi < x < 2 \\pi \\end{cases}$ Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \\map f x       | o = \\sim       | r = -\\frac \\pi 4 + \\frac 2 \\pi \\sum_{r \\mathop = 0}^\\infty \\frac {\\cos \\paren {2 r + 1} x} {\\paren {2 r + 1}^2} - \\sum_{n \\mathop = 1}^\\infty \\frac {2 - \\paren {-1}^n \\sin n x} n       | c =  }} {{eqn | r = -\\frac \\pi 4 + \\dfrac 2 \\pi \\paren {\\cos x + \\frac {\\cos 3 x} 9 + \\frac {\\cos 5 x} {25} + \\cdots} - 3 \\paren {\\sin x + \\frac {\\sin 3 x} 3 + \\frac {\\sin 5 x} 5 + \\cdots} - \\paren {\\frac {\\sin 2 x} 2 + \\frac {\\sin 4 x} 4 + \\cdots}       | c =  }} {{end-eqn}}"}
+{"_id": "15393", "title": "Argument Principle", "text": "Let $f$ be a function meromorphic in the interior of some simply connected region $D$. Let $f$ be holomorphic with no zeroes on the boundary of $D$. Let $N$ denote the number of zeroes of $f$ in the interior of $D$, counted up to multiplicity. Let $P$ denote the number of poles of $f$ in the interior of $D$, counted up to order. Then:   :$\\displaystyle N - P = \\frac 1 {2\\pi i} \\oint_D \\frac { f'\\left({z}\\right) } { f\\left({z}\\right) } \\, \\mathrm d z$"}
+{"_id": "15394", "title": "Ring with Unity has Prime Ideal", "text": "Let $A$ be a non-trivial commutative ring with unity. Then $A$ has a prime ideal."}
+{"_id": "15395", "title": "Proper Ideal of Ring is Contained in Maximal Ideal", "text": "Let $A$ be a commutative ring with unity. Let $\\mathfrak a \\subseteq A$ be a proper ideal. Then there exists a maximal ideal $\\mathfrak m$ with $\\mathfrak a \\subseteq \\mathfrak m$."}
+{"_id": "15396", "title": "Proper Ideal iff Quotient Ring is Nontrivial", "text": "Let $A$ be a commutative ring. Let $\\mathfrak a \\subseteq A$ be an ideal. {{TFAE}} :$(1): \\quad \\mathfrak a$ is a proper ideal :$(2): \\quad$ The quotient ring $A / \\mathfrak a$ is nontrivial ring"}
+{"_id": "15398", "title": "Half-Range Fourier Sine Series/Sine of Half x over 0 to Pi, Minus Sine of Half x over Pi to 2 Pi", "text": "Let $\\map f x$ be the real function defined on $\\openint 0 {2 \\pi}$ as: :600pxthumbright$\\map f x$ and its $7$th approximation :$\\map f x = \\begin {cases} \\sin \\dfrac x 2 & : 0 \\le x < \\pi \\\\ -\\sin \\dfrac x 2 & : \\pi < x \\le 2 \\pi \\end {cases}$ Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \\map f x       | o = \\sim       | r = \\frac 8 \\pi \\sum_{n \\mathop = 1}^\\infty \\paren {-1}^{n - 1} \\frac {n \\sin n x} {4 n^2 - 1}       | c =  }} {{eqn | r = \\frac 8 \\pi \\paren {\\frac {\\sin x} {1 \\times 3} - \\frac {2 \\sin 2 x} {3 \\times 5} + \\frac {3 \\sin 3 x} {5 \\times 7} - \\frac {4 \\sin 4 x} {7 \\times 9} + \\frac {5 \\sin 5 x} {9 \\times 11} - \\dotsb}       | c =  }} {{end-eqn}}"}
+{"_id": "15401", "title": "Fourier Series/Cosine of x over Minus Pi to Zero, Minus Cosine of x over Zero to Pi", "text": "Let $\\map f x$ be the real function defined on $\\openint {-\\pi} \\pi$ as: :600pxthumbright$\\map f x$ and its $7$th approximation :$\\map f x = \\begin {cases} \\cos x & : -\\pi < x < 0 \\\\ -\\cos x & : 0 < x < \\pi \\end {cases}$ Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \\map f x       | o = \\sim       | r = -\\frac 8 \\pi \\sum_{r \\mathop = 1}^\\infty \\frac {r \\sin 2 r x} {4 r^2 - 1}       | c =  }} {{eqn | r = -\\frac 8 \\pi \\paren {\\frac {\\sin 2 x} {1 \\times 3} + \\frac {\\sin 4 x} {3 \\times 5} + \\frac {\\sin 6 x} {5 \\times 7}  + \\frac {\\sin 8 x} {7 \\times 9} + \\dotsb}       | c =  }} {{end-eqn}}"}
+{"_id": "15403", "title": "Holomorphic Function is Analytic", "text": "Let $a \\in \\C$ be a complex number. Let $r > 0$ be a real number. Let $f$ be a function holomorphic on some open ball $D = \\map B {a, r}$.  Then $f$ is complex analytic on $D$."}
+{"_id": "15404", "title": "Schwarz's Lemma", "text": "Let $D$ be the unit disk centred at $0$. Let $f : D \\to \\C$ be a holomorphic function. Let $\\map f 0 = 0$ and $\\cmod {\\map f z} \\le 1$ for all $z \\in D$.  Then $\\cmod {\\map {f'} 0} \\le 1$, and $\\cmod {\\map f z} \\le \\cmod z$ for all $z \\in D$."}
+{"_id": "15405", "title": "Ring of Integers of Number Field is Dedekind Domain", "text": "Let $K$ be a number field. Let $\\mathcal O_K$ be its ring of integers. Then $\\mathcal O_K$ is a Dedekind domain."}
+{"_id": "15406", "title": "Fourier Series/Exponential of x over Minus Pi to Pi", "text": "Let $\\map f x$ be the real function defined on $\\R$ as: :200pxthumbright$\\map f x$ and its $7$th approximation :$\\map f x = \\begin{cases} e^x & : -\\pi < x \\le \\pi \\\\ \\map f {x + 2 \\pi} & : \\text{everywhere} \\end{cases}$ Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \\map f x       | o = \\sim       | r = \\frac {\\sinh \\pi} \\pi \\paren {1 + 2 \\sum_{n \\mathop = 1}^\\infty \\frac {\\paren {-1}^n} {1 + n^2} \\paren {\\cos n x - n \\sin n x} }       | c =  }} {{eqn | r = \\frac {\\sinh \\pi} \\pi \\paren {1 + 2 \\paren {-\\dfrac {\\cos x - \\sin x} 2 + \\dfrac {\\cos 2 x - 2 \\sin 2 x} 5 - \\dfrac {\\cos 3 x - 3 \\sin 3 x} {10} + \\dotsb} }       | c =  }} {{end-eqn}}"}
+{"_id": "15407", "title": "Sum of Reciprocals of Squares plus 1", "text": ":$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\frac 1 {n^2 + 1}  = \\frac 1 2 \\paren {\\pi \\coth \\pi - 1}$"}
+{"_id": "15408", "title": "Half-Range Fourier Cosine Series/Cosine of Non-Integer Multiple of x over 0 to Pi", "text": "Let $\\lambda \\in \\R \\setminus \\Z$ be a real number which is not an integer. :500pxthumbright$\\map f x$ for $\\lambda = 4 \\cdotp 6$ and its $5$th approximation Let $\\map f x$ be the real function defined on $\\openint 0 \\pi$ as: :$\\map f x = \\cos \\lambda x$ Then its half-range Fourier cosine series can be expressed as: {{begin-eqn}} {{eqn | l = \\map f x       | o = \\sim       | r = \\frac {2 \\lambda \\sin \\lambda \\pi} \\pi \\paren {\\frac 1 {2 \\lambda^2} + \\sum_{n \\mathop = 1}^\\infty \\paren {-1}^n \\frac {\\cos n x} {\\lambda^2 - n^2} }       | c =  }} {{eqn | r = \\frac {2 \\lambda \\sin \\lambda \\pi} \\pi \\paren {\\frac 1 {2 \\lambda^2} - \\frac {\\cos x} {\\lambda^2 - 1} + \\frac {\\cos 2 x} {\\lambda^2 - 4} - \\frac {\\cos 3 x} {\\lambda^2 - 9} + \\frac {\\cos 4 x} {\\lambda^2 - 16} - \\dotsb}       | c =  }} {{end-eqn}}"}
+{"_id": "15409", "title": "Series Expansion for Pi Cotangent of Pi Lambda", "text": "Let $\\lambda \\in \\R \\setminus \\Z$ be a real number which is not an integer. Then: :$\\displaystyle \\pi \\cot \\pi \\lambda = \\frac 1 \\lambda + \\sum_{n \\mathop = 1}^\\infty \\frac {2 \\lambda} {\\lambda^2 - n^2}$"}
+{"_id": "15410", "title": "Series Expansion for Pi Cosecant of Pi Lambda", "text": "Let $\\lambda \\in \\R \\setminus \\Z$ be a real number which is not an integer. Then: :$\\displaystyle \\pi \\csc \\pi \\lambda = \\sum_{n \\mathop = 1}^\\infty \\paren {-1}^n \\paren {\\frac 1 {n + \\lambda} + \\frac 1 {n - 1 - \\lambda} }$"}
+{"_id": "15413", "title": "Prime Ideal of Principal Ideal Domain is Maximal", "text": "Let $D$ be a principal ideal domain whose zero is $0_D$. Let $J \\subseteq D$ be a nonzero prime ideal. Then $J$ is maximal."}
+{"_id": "15414", "title": "Gauss's Digamma Theorem", "text": "Let $\\dfrac p q$ be a positive rational number with $p < q$. Then:  :$\\displaystyle \\psi \\left({\\frac p q}\\right) = -\\gamma - \\ln 2 q - \\frac \\pi 2 \\cot \\left({\\frac p q \\pi}\\right) + 2 \\sum_{n \\mathop = 1}^{\\left\\lceil{q / 2}\\right\\rceil - 1} \\cos \\left({\\frac {2 \\pi p n} q}\\right) \\ln\\left({\\sin \\left({\\frac {\\pi n} q}\\right)}\\right)$ where: :$\\psi$ is the digamma function :$\\cot$ is the cotangent function :$\\ln$ is the natural logarithm."}
+{"_id": "15416", "title": "Gauss's Integral Form of Digamma Function", "text": "Let $z$ be a complex number with a positive real part, then:  :$\\displaystyle \\psi \\left({z}\\right) = \\int_0^\\infty \\left({\\frac{ e^{-t} } t - \\frac {e^{-zt } } {1 - e^{-t} } }\\right) \\rd t$ where $\\psi$ is the digamma function."}
+{"_id": "15417", "title": "Rodrigues' Formula for Legendre Polynomials", "text": ":$\\map {P_n} x = \\dfrac 1 {2^n n!} \\dfrac {\\d^n} {\\d x^n} \\paren {x^2 - 1}^n$ where $P_n$ is the $n$th Legendre polynomial."}
+{"_id": "15420", "title": "Modulus of Gamma Function of Imaginary Number", "text": "Let $t \\in \\R$ be a real number. Then:  :$\\cmod {\\map \\Gamma {i t} } = \\sqrt {\\dfrac {\\pi \\csch \\pi t} t}$ where: :$\\Gamma$ is the Gamma function :$\\csch$ is the hyperbolic cosecant function."}
+{"_id": "15422", "title": "Existence of Homomorphism between Localizations of Ring", "text": "Let $A$ be a commutative ring with unity. Let $S, T \\subseteq A$ be multiplicatively closed subsets. {{TFAE}} :$(1): \\quad$ There exists an $A$-algebra homomorphism $h : A_S \\to A_T$ between localizations, the '''induced homomorphism'''. :$(2): \\quad S$ is a subset of the saturation of $T$. :$(3): \\quad$  The saturation of $S$ is a subset of the saturation of $T$. :$(4): \\quad$  Every prime ideal meeting $S$ also meets $T$."}
+{"_id": "15423", "title": "Fourier Series/Pi Squared minus x Squared over Minus Pi to Pi", "text": "Let $\\map f x$ be the real function defined on $\\openint {-\\pi} \\pi$ as: :400pxthumbright$\\map f x$ and its $4$th approximation :$\\map f x = \\pi^2 - x^2$ $f$ can be expressed as a half-range Fourier cosine series thus: {{begin-eqn}} {{eqn | l = \\map f x       | o = \\sim       | r = \\frac {2 \\pi^2} 3 + 4 \\sum_{n \\mathop = 1}^\\infty \\paren {-1}^{n - 1} \\frac {\\cos n x} {n^2}       | c =  }} {{eqn | r = \\frac {2 \\pi^2} 3 + 4 \\paren {\\cos x - \\frac 1 4 \\cos 2 x + \\frac 1 9 \\cos 3 x - \\cdots}       | c =  }} {{end-eqn}}"}
+{"_id": "15424", "title": "Half-Range Fourier Cosine Series/Identity Function/0 to Pi", "text": "Let $\\map f x$ be the real function defined on $\\openint 0 \\pi$ as: :400pxthumbright$\\map f x$ and its $4$th approximation :$\\map f x = x$ Then its half-range Fourier cosine series can be expressed as: {{begin-eqn}} {{eqn | l = x       | o = \\sim       | r = \\frac \\pi 2 - \\frac 4 \\pi \\sum_{n \\mathop = 1}^\\infty \\frac {\\cos \\paren {2 n - 1} x} {\\paren {2 n - 1}^2}       | c =  }} {{eqn | r = \\frac \\pi 2 - \\frac 4 \\pi \\paren {\\cos x + \\frac {\\cos 3 x} {3^2} + \\frac {\\cos 5 x} {5^2} + \\cdots}       | c =  }} {{end-eqn}}"}
+{"_id": "15428", "title": "Hilbert's Basis Theorem for Finitely Generated Algebras", "text": "Let $A$ be a Noetherian ring. Let $B$ be a finitely generated algebra over $A$. Then $B$ is Noetherian."}
+{"_id": "15429", "title": "Zariski's Lemma", "text": "Let $L/k$ be a field extension. Let $L$ be finitely generated as an algebra over $k$. Then $L/k$ is a finite field extension."}
+{"_id": "15431", "title": "Cayley-Hamilton Theorem/Finitely Generated Modules", "text": "Let $A$ be a commutative ring with unity. Let $M$ be a finitely generated $A$-module. Let $\\mathfrak a$ be an ideal of $A$. Let $\\phi$ be an endomorphism of $M$ such that $\\phi \\left({M}\\right) \\subseteq \\mathfrak a M$. Then $\\phi$ satisfies an equation of the form: :$\\phi^n + a_{n-1} \\phi^{n-1} + \\cdots + a_1 \\phi + a_0 = 0$ with the $a_i \\in \\mathfrak a$."}
+{"_id": "15432", "title": "Weak Nullstellensatz", "text": "Let $k$ be an algebraically closed field. Let $n \\geq 0$ be an natural number. Let $k \\left[{x_1,\\ldots, x_n}\\right]$ be the polynomial ring in $n$ variables over $k$. Let $I \\subseteq k \\left[{x_1,\\ldots, x_n}\\right]$ be an ideal. {{TFAE}} # $I$ is the unit ideal: $I = (1)$. # Its zero-locus is empty set: $V(I) = \\varnothing$."}
+{"_id": "15433", "title": "Finite-Dimensional Integral Domain over Field is Field", "text": "Let $k$ be a field. Let $R$ be a $k$-algebra of finite dimension which is an integral domain. Then $R$ is a field."}
+{"_id": "15434", "title": "Preimage of Maximal Ideal of Finitely Generated Algebra is Maximal", "text": "Let $k$ be a field. Let $A$ and $B$ be $k$-algebras. Let $f : A \\to B$ be a $k$-algebra homomorphism. Let $B$ be finitely generated over $k$. Let $\\mathfrak m$ be a maximal ideal of $B$. Then its preimage $f^{-1}(\\mathfrak m)$ is a maximal ideal of $A$."}
+{"_id": "15436", "title": "Subalgebra of Finite Field Extension is Field", "text": "Let $E / F$ be an algebraic field extension. Let $A \\subseteq E$ be a unital subalgebra over $F$. Then $A$ is a field."}
+{"_id": "15437", "title": "Weierstrass's Elliptic Function is Doubly Periodic", "text": "Weierstrass's elliptic function $\\map \\wp {z; \\omega_1, \\omega_2}$ is doubly periodic in $2 \\omega_1$ and $2 \\omega_2$."}
+{"_id": "15438", "title": "Differential Equation satisfied by Weierstrass's Elliptic Function", "text": "The differential equation:  :$\\paren {\\dfrac {\\d f} {\\d z} }^2 = 4 f^3 - g_2 f - g_3$ where:  :$\\displaystyle g_2 = 60 \\sum_{\\tuple {n, m} \\mathop \\in \\Z^2 \\setminus \\tuple {0, 0} } \\frac 1 {\\paren {2 m \\omega_1 + 2 n \\omega_2}^4}$ and:  :$\\displaystyle g_3 = 140 \\sum_{\\tuple {n, m} \\mathop \\in \\Z^2 \\setminus \\tuple {0, 0} } \\frac 1 {\\paren {2 m \\omega_1 + 2 n \\omega_2}^6}$ has the general solution:  :$\\map f z = \\map \\wp {z + C; \\omega_1, \\omega_2}$ where: :$\\wp$ is Weierstrass's elliptic function :$C$ is an arbitrary constant :$\\omega_1$, $\\omega_2$ are constants independent of $z$. {{explain|What domain are all the variables and constants in?}}"}
+{"_id": "15439", "title": "Piecewise Continuous Function with Improper Integrals may not be Bounded", "text": "Let $f$ be a real function defined on a closed interval $\\closedint a b$, $a < b$.  Let $f$ be a piecewise continuous function with improper integrals. Then $f$ may not be piecewise continuous and bounded on $\\closedint a b$."}
+{"_id": "15440", "title": "Weierstrass's Elliptic Function is Even in z", "text": "Let $\\omega_1$ and $\\omega_2$ be non-zero complex constants with $\\dfrac {\\omega_1} {\\omega_2}$ having a positive imaginary part.  For $z \\in \\C \\setminus \\left\\{2m \\omega_1 + 2n\\omega_2: \\left({n,m}\\right) \\in \\Z^2\\right\\}$: :$\\displaystyle \\wp\\left({-z; \\omega_1, \\omega_2}\\right) = \\wp\\left({z; \\omega_1, \\omega_2}\\right)$ That is, Weierstrass's elliptic function is even in $z$."}
+{"_id": "15444", "title": "Duplication Formula for Weierstrass's Elliptic Function", "text": "Let $\\omega_1$, $\\omega_2$ be non-zero complex constants with $\\dfrac {\\omega_1} {\\omega_2}$ having a positive imaginary part. Let $z$ be a complex number where $z \\notin \\set {2 m \\omega_1 + 2 n \\omega_2: \\tuple {n, m} \\in \\Z^2}$. Then: :$\\map \\wp {2 z; \\omega_1, \\omega_2} = \\dfrac 1 4 \\paren {\\dfrac {\\map {\\wp''} {z; \\omega_1, \\omega_2} } {\\map {\\wp'} {z; \\omega_1, \\omega_2} } }^2 - 2 \\map \\wp {z; \\omega_1, \\omega_2}$ where: :$\\wp$ is Weierstrass's elliptic function  :$\\wp'$ and $\\wp''$ denote its first and second derivative with respect to $z$."}
+{"_id": "15445", "title": "Integral to Infinity of Sine p x over x", "text": ":$\\displaystyle \\int_0^\\infty \\frac {\\sin p x} x \\rd x = \\begin{cases} \\dfrac \\pi 2 & : p > 0 \\\\ 0 & : p = 0 \\\\ -\\dfrac \\pi 2 & : p < 0 \\end{cases}$"}
+{"_id": "15446", "title": "Radical of Sum of Ideals", "text": "Let $A$ be a commutative ring with unity. Let $\\mathfrak a, \\mathfrak b \\subseteq A$ be ideals. Then for the radical of their sum we have: :$\\operatorname{Rad} \\left({\\mathfrak a + \\mathfrak b}\\right) = \\operatorname{Rad} \\left({\\operatorname{Rad} \\left({\\mathfrak a}\\right) + \\operatorname{Rad} \\left({\\mathfrak b}\\right)}\\right)$"}
+{"_id": "15447", "title": "Ideal of Ring is Contained in Radical", "text": "Let $A$ be a commutative ring with unity. Let $\\mathfrak a \\subseteq A$ be an ideal. Then $\\mathfrak a$ is contained in its radical: :$\\mathfrak a \\subseteq \\operatorname{Rad} \\left({\\mathfrak a}\\right)$"}
+{"_id": "15448", "title": "Radical of Ideal Preserves Inclusion", "text": "Let $A$ be a commutative ring with unity. Let $\\mathfrak a \\subseteq \\mathfrak b \\subseteq A$ be an ideals. Then we have an inclusion of their radicals: :$\\operatorname{Rad} \\left({\\mathfrak a}\\right) \\subseteq \\operatorname{Rad} \\left({\\mathfrak b}\\right)$"}
+{"_id": "15449", "title": "Ideals with Coprime Radicals are Coprime", "text": "Let $A$ be a commutative ring with unity. Let $\\mathfrak a, \\mathfrak b \\subseteq A$ be ideals. Let their radicals be coprime: :$\\operatorname{Rad} \\left({\\mathfrak a}\\right) + \\operatorname{Rad} \\left({\\mathfrak b}\\right) = \\left({1}\\right)$ Then $\\mathfrak a$ and $\\mathfrak b$ are coprime."}
+{"_id": "15450", "title": "Radical of Unit Ideal", "text": "Let $A$ be a commutative ring with unity. Let $\\left({1}\\right)$ be its unit ideal. Then its radical equals $(1)$: :$\\operatorname{Rad} \\left({\\left({1}\\right)}\\right) = \\left({1}\\right)$."}
+{"_id": "15451", "title": "Radical of Power of Prime Ideal", "text": "Let $A$ be a commutative ring with unity. Let $\\mathfrak p \\subseteq A$ be a prime ideal. Let $n > 0$ be a natural number. Then the radical of the $n$th power of $\\mathfrak p$ equals $\\mathfrak p$: :$\\operatorname{Rad} \\left({\\mathfrak p^n}\\right) = \\mathfrak p$."}
+{"_id": "15454", "title": "Coefficients of Product of Two Polynomials", "text": "Let $R$ be a commutative ring with unity. Let $f, g \\in R \\sqbrk x$ be polynomials over $R$. For a natural number $n \\ge 0$, let: :$a_n$ be the coefficient of the monomial $x^n$ in $f$. :$b_n$ be the coefficient of the monomial $x^n$ in $g$."}
+{"_id": "15455", "title": "Equivalence of Definitions of Irreducible Polynomial over Field", "text": "Let $K$ be a field. {{TFAE|def = Irreducible Polynomial}}"}
+{"_id": "15456", "title": "Linear Combination of Integrals/Indefinite", "text": ":$\\displaystyle \\int \\paren {\\lambda \\map f x + \\mu \\map g x} \\rd x = \\lambda \\int \\map f x \\rd x + \\mu \\int \\map g x \\rd x$"}
+{"_id": "15457", "title": "Linear Combination of Integrals/Definite", "text": ":$\\displaystyle \\int_a^b \\paren {\\lambda \\map f t + \\mu \\map g t} \\rd t = \\lambda \\int_a^b \\map f t \\rd t + \\mu \\int_a^b \\map g t \\rd t$"}
+{"_id": "15459", "title": "Correspondence between Abelian Groups and Z-Modules/Isomorphism of Categories", "text": "Let $\\Z$ be the ring of integers. Let $\\mathbf{Ab}$ be the category of abelian groups. Let $\\mathbf{\\mathbb Z-Mod}$ be the category of unitary $\\Z$-modules. Then the: * forgetful functor $\\mathbf{\\mathbb Z-Mod} \\to \\mathbf{Ab}$ * associated Z-module functor $\\mathbf{Ab} \\to \\mathbf{\\mathbb Z-Mod}$ are strict inverse functors. In particular, $\\mathbf{Ab}$ and $\\mathbf{\\mathbb Z-Mod}$ are isomorphic."}
+{"_id": "15460", "title": "Correspondence between Abelian Groups and Z-Modules/Homomorphisms", "text": "Let $G, H$ be abelian groups. Let $f : G \\to H$ be a mapping. {{TFAE}} # $f$ is a group homomorphism. # $f$ is a $\\Z$-module homomorphism between the $\\Z$-modules associated with $G$ and $H$."}
+{"_id": "15466", "title": "Mapping to Singleton is Unique", "text": "Let $S$ be a set. Let $T$ be a singleton. Then there exists a unique mapping $S \\to T$."}
+{"_id": "15470", "title": "Rectangular Formula for Definite Integrals", "text": "Let $f$ be a real function which is integrable on the closed interval $\\closedint a b$. Let $P = \\set {x_0, x_1, x_2, \\ldots, x_{n - 1}, x_n}$ form a normal subdivision of $\\closedint a b$: :$\\forall r \\in \\set {1, 2, \\ldots, n}: x_r - x_{r - 1} = \\dfrac {b - a} n$ Then the definite integral of $f$ {{WRT|Integration}} $x$ from $a$ to $b$ can be approximated as: :$\\displaystyle \\int_a^b \\map f x \\rd x \\approx \\sum_{r \\mathop = 0}^{n - 1} h \\map f {x_r}$ where $h = \\dfrac {b - a} n$."}
+{"_id": "15471", "title": "Trapezoidal Formula for Definite Integrals", "text": "Let $f$ be a real function which is integrable on the closed interval $\\closedint a b$. Let $P = \\set {x_0, x_1, x_2, \\ldots, x_{n - 1}, x_n}$ form a normal subdivision of $\\closedint a b$: :$\\forall r \\in \\set {1, 2, \\ldots, n}: x_r - x_{r - 1} = \\dfrac {b - a} n$ Then the definite integral of $f$ {{WRT|Integration}} $x$ from $a$ to $b$ can be approximated as: :$\\displaystyle \\int_a^b \\map f x \\rd x \\approx \\dfrac h 2 \\paren {\\map f {x_0} + \\map f {x_n} + \\sum_{r \\mathop = 1}^{n - 1} 2 \\map f {x_r} }$ where $h = \\dfrac {b - a} n$."}
+{"_id": "15472", "title": "Simpson's Rule", "text": "Let $f$ be a real function which is integrable on the closed interval $\\closedint a b$. Let $P = \\set {x_0, x_1, x_2, \\ldots, x_{n - 1}, x_n}$ form a normal subdivision of $\\closedint a b$: :$\\forall r \\in \\set {1, 2, \\ldots, n}: x_r - x_{r - 1} = \\dfrac {b - a} n$ where $n$ is even. Then the definite integral of $f$ {{WRT|Integration}} $x$ from $a$ to $b$ can be approximated as: :$\\displaystyle \\int_a^b \\map f x \\rd x \\approx \\dfrac h 3 \\paren {\\map f {x_0} + \\map f {x_n} + \\sum_{r \\mathop = 1}^{m - 1} 2 \\map f {x_{2 m - 1} } + \\sum_{r \\mathop = 1}^{m - 1} 4 \\map f {x_{2 m} } }$ where: :$h = \\dfrac {b - a} n$ :$m = \\dfrac n 2$"}
+{"_id": "15473", "title": "Definite Integral to Infinity of Reciprocal of x Squared plus a Squared", "text": ":$\\ds \\int_0^\\infty \\dfrac {\\d x} {x^2 + a^2} = \\frac \\pi {2 a}$ for $a \\ne 0$."}
+{"_id": "15474", "title": "Definite Integral to Infinity of Reciprocal of 1 plus Power of x/Corollary", "text": ":$\\displaystyle \\int_0^\\infty \\frac 1 {a^n + x^n} \\rd x = \\frac \\pi {n a^{n - 1} } \\csc \\paren {\\frac \\pi n}$"}
+{"_id": "15477", "title": "Definite Integral to Infinity of Power of x over 1 + x", "text": ":$\\displaystyle \\int_0^\\infty \\dfrac {x^{p - 1} \\rd x} {1 + x} = \\frac \\pi {\\sin \\pi p}$ for $0 < p < 1$."}
+{"_id": "15482", "title": "Universal Property of Free Abelian Group on Set", "text": "Let $S$ be a set. Let $(\\Z^{(S)}, \\iota)$ be the free abelian group on $S$. Let $G$ be an abelian group. Let $f : S \\to G$ be a mapping. Then there exists a unique group homomorphism $g : \\Z^{(S)} \\to G$ with $g \\circ \\iota = f$: :$\\xymatrix{ S \\ar[d]_\\iota \\ar[r]^{\\forall f} & G\\\\ \\Z^{(S)} \\ar[ru]_{\\exists ! g} }$"}
+{"_id": "15483", "title": "Definite Integral from 0 to a of Root of a Squared minus x Squared", "text": ":$\\displaystyle \\int_0^a \\sqrt {a^2 - x^2} \\rd x = \\frac {\\pi a^2} 4$ for $a > 0$."}
+{"_id": "15484", "title": "Universal Property of Abelianization of Group", "text": "Let $G$ be a group. Let $G^{\\operatorname {ab} }$ be its abelianization. Let $\\pi : G \\to G^{\\operatorname {ab} }$ be the quotient group epimorphism. Let $H$ be an abelian group. Let $f: G \\to H$ be a group homomorphism. Then there exists a unique group homomorphism $g : G^{\\operatorname {ab}} \\to H$ such that $g \\circ \\pi = f$: :$\\xymatrix { G \\ar[d]_\\pi \\ar[r]^{\\forall f} & H\\\\ G^{\\operatorname {ab} } \\ar[ru]_{\\exists ! g} }$"}
+{"_id": "15486", "title": "Definite Integral from 0 to a of x^m by (a^n - x^n)^p", "text": ":$\\displaystyle \\int_0^a x^m \\paren {a^n - x^n}^p \\rd x = \\frac {a^{m + 1 + n p} \\, \\map \\Gamma {\\frac {m + 1} n} \\map \\Gamma {p + 1} } {n \\map \\Gamma {\\frac {m + 1} n + p + 1} }$"}
+{"_id": "15487", "title": "Equivalence of Definitions of Complement of Subgroup", "text": "Let $G$ be a group with identity $e$. Let $H$ and $K$ be subgroups. {{TFAE|def = Complement of Subgroup}}"}
+{"_id": "15488", "title": "Equivalent Characterizations of Finer Equivalence Relation", "text": "Let $X$ be a set. Let $\\equiv$ and $\\sim$ be equivalence relations on $X$. {{TFAE}} # $\\equiv$ is finer than $\\sim$: #:$\\forall x, y \\in X : x \\equiv y \\implies x \\sim y$ #The graph of $\\equiv$ is contained in the graph of $\\sim$. #Every $\\equiv$-equivalence class is contained in a $\\sim$-equivalence class. #Every $\\sim$-equivalence class is saturated under $\\equiv$."}
+{"_id": "15489", "title": "Equivalence of Definitions of Finer Topology", "text": "Let $S$ be a set. Let $\\tau_1$ and $\\tau_2$ be topologies on $S$. {{TFAE|def = Finer Topology}}"}
+{"_id": "15494", "title": "Definite Integral from 0 to Half Pi of Square of Sine x", "text": ":$\\displaystyle \\int_0^{\\frac \\pi 2} \\sin^2 x \\rd x = \\frac \\pi 4$"}
+{"_id": "15495", "title": "Definite Integral from 0 to Half Pi of Square of Cosine x", "text": ":$\\displaystyle \\int_0^{\\frac \\pi 2} \\cos^2 x \\rd x = \\frac \\pi 4$"}
+{"_id": "15496", "title": "Reduction Formula for Definite Integral of Power of Cosine", "text": "Let $n \\in \\Z_{> 0}$ be a positive integer. Let $I_n$ be defined as: :$\\displaystyle I_n = \\int_0^{\\frac \\pi 2} \\cos^n x \\rd x$ Then $\\left\\langle{I_n}\\right\\rangle$ is a decreasing sequence of real numbers which satisfies: :$n I_n = \\left({n - 1}\\right) I_{n - 2}$ Thus: :$I_n = \\dfrac {n - 1} n  I_{n - 2}$ is a reduction formula for $I_n$."}
+{"_id": "15497", "title": "Definite Integral from 0 to Half Pi of Even Power of Cosine x", "text": ":$\\displaystyle \\int_0^{\\frac \\pi 2} \\cos^{2 n} x \\rd x = \\dfrac {\\paren {2 n}!} {\\paren {2^n n!}^2} \\dfrac \\pi 2$"}
+{"_id": "15498", "title": "Definite Integral from 0 to Half Pi of Odd Power of Cosine x", "text": ":$\\displaystyle \\int_0^{\\frac \\pi 2} \\cos^{2 n + 1} x \\rd x = \\dfrac {\\left({2^n n!}\\right)^2} {\\left({2 n + 1}\\right)!}$"}
+{"_id": "15499", "title": "Arctangent of Zero is Zero", "text": ":$\\arctan 0 = 0$"}
+{"_id": "15500", "title": "Riemann P-symbol in terms of Gaussian Hypergeometric Function", "text": "Let: :$\\displaystyle f\\left({z}\\right) = \\operatorname P \\left\\{ \\begin{matrix} a & b & c \\\\ \\alpha & \\beta & \\gamma & z \\\\ \\alpha' & \\beta' & \\gamma' \\end{matrix} \\right\\}$ where: :$\\operatorname P$ is the Riemann P-symbol :$\\alpha + \\beta + \\gamma + \\alpha' + \\beta' + \\gamma' = 1$  :$\\alpha - \\alpha'$ is not a negative integer.  Then:  :$\\displaystyle f\\left({z}\\right) = \\left({ \\frac {z - a} {z - b} }\\right)^\\alpha \\left({ \\frac {z - c} {z - b} }\\right)^\\gamma {}_2 \\operatorname F_1 \\left({ {\\alpha + \\beta + \\gamma, \\alpha + \\beta' + \\gamma} \\atop {1 + \\alpha - \\alpha'} } \\, \\middle \\vert {\\, \\frac {\\left({z - a}\\right) \\left({c - b}\\right)} {\\left({z - b} \\right) \\left({c - a}\\right)} }\\right)$ where ${}_2 \\operatorname F_1$ is the Gaussian hypergeometric function."}
+{"_id": "15502", "title": "Sum of Powers of Positive Integers", "text": "Let $n, p \\in \\Z_{>0}$ be (strictly) positive integers. Then: {{begin-eqn}} {{eqn | l = \\sum_{k \\mathop = 1}^n k^p       | r = 1^p + 2^p + \\cdots + n^p       | c =  }} {{eqn | r = \\frac {n^{p + 1} } {p + 1} + \\sum_{k \\mathop = 1}^p \\frac {B_k \\, p^{\\underline {k - 1} } \\, n^{p - k + 1} } {k!}       | c =  }} {{eqn | r = \\frac {n^{p + 1} } {p + 1} + \\frac {B_1 \\, n^p} {1!} + \\frac {B_2 \\, p \\, n^{p - 1} } {2!} + \\frac {B_4 \\, p \\paren {p - 1} \\paren {p - 2} n^{p - 3} } {4!} + \\cdots       | c =  }} {{end-eqn}} where: :$B_k$ are the Bernoulli numbers :$p^{\\underline k}$ is the $k$th falling factorial of $p$."}
+{"_id": "15506", "title": "Order of Finite p-Group is Power of p", "text": "Let $G$ be a finite group. Let $p$ be a prime number. Let all elements of $G$ have order a power of $p$. Then $G$ is a $p$-group."}
+{"_id": "15516", "title": "Sum of Reciprocals of Sixth Powers Alternating in Sign", "text": "{{begin-eqn}} {{eqn | l = \\sum_{n \\mathop = 1}^\\infty \\dfrac {\\paren {-1}^{n + 1} } {n^6}       | r = \\frac 1 {1^6} - \\frac 1 {2^6} + \\frac 1 {3^6} - \\frac 1 {4^6} + \\cdots       | c =  }} {{eqn | r = \\frac {31 \\pi^6} {30 \\, 240}       | c =  }} {{end-eqn}}"}
+{"_id": "15517", "title": "Sum of Reciprocals of Fourth Powers of Odd Integers", "text": "{{begin-eqn}} {{eqn | l = \\sum_{n \\mathop = 1}^\\infty \\frac 1 {\\left({2 n - 1}\\right)^4}       | r = 1 + \\dfrac 1 {3^4} + \\dfrac 1 {5^4} + \\dfrac 1 {7^4} + \\dfrac 1 {9^4} + \\cdots       | c =  }} {{eqn | r = \\dfrac {\\pi^4} {96}       | c =  }} {{end-eqn}}"}
+{"_id": "15518", "title": "Sum of Reciprocals of Sixth Powers of Odd Integers", "text": "{{begin-eqn}} {{eqn | l = \\sum_{n \\mathop = 1}^\\infty \\frac 1 {\\left({2 n - 1}\\right)^6}       | r = 1 + \\dfrac 1 {3^6} + \\dfrac 1 {5^6} + \\dfrac 1 {7^6} + \\dfrac 1 {9^6} + \\cdots       | c =  }} {{eqn | r = \\dfrac {\\pi^6} {960}       | c =  }} {{end-eqn}}"}
+{"_id": "15519", "title": "Half-Range Fourier Sine Series/x by Pi minus x over 0 to Pi", "text": "Let $\\map f x$ be the real function defined on $\\openint 0 \\pi$ as: :$\\map f x = x \\paren {\\pi - x}$ Then its half-range Fourier sine series can be expressed as: :$\\displaystyle \\map f x \\sim \\frac 8 \\pi \\sum_{r \\mathop = 0}^\\infty \\frac {\\sin \\paren {2 r + 1} x} {\\paren {2 r + 1}^3}$"}
+{"_id": "15521", "title": "Sum of Reciprocals of Powers of Odd Integers Alternating in Sign", "text": ":$\\displaystyle \\sum_{n \\mathop = 0}^\\infty \\frac {\\paren {-1}^n} {\\paren {2 n + 1}^s} = \\frac 1 {2 \\map \\Gamma s} \\int_0^\\infty x^{s - 1} \\map \\sech x \\rd x$ where: :$\\map \\Re s > 0$ :$\\Gamma$ is the gamma function :$\\sech$ is the hyperbolic secant function."}
+{"_id": "15522", "title": "Sum of Reciprocals of Cubes of Odd Integers Alternating in Sign in Pairs", "text": "{{begin-eqn}} {{eqn | l = \\frac 1 {1^3} + \\frac 1 {3^3} - \\frac 1 {5^3} - \\frac 1 {7^3} + \\cdots       | r = \\frac {3 \\pi^3 \\sqrt 2} {128}       | c =  }} {{end-eqn}}"}
+{"_id": "15523", "title": "Sum of Sequence of Products of Consecutive Odd Reciprocals", "text": "{{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 0}^n \\frac 1 {\\paren {2 j + 1} \\paren {2 j + 3} }       | r = \\frac 1 {1 \\times 3} + \\frac 1 {3 \\times 5} + \\frac 1 {5 \\times 7} + \\frac 1 {7 \\times 9} + \\cdots + \\frac 1 {\\paren {2 n + 1} \\paren {2 n + 3} }       | c =  }} {{eqn | r = \\frac {n + 1} {2 n + 3}       | c =  }} {{end-eqn}}"}
+{"_id": "15524", "title": "Equivalence of Definitions of Local Ring Homomorphism", "text": "Let $(A, \\mathfrak m)$ and $(B, \\mathfrak n)$ be commutative local rings. Let $f : A \\to B$ be a unital ring homomorphism. {{TFAE|def = Local Ring Homomorphism}}"}
+{"_id": "15525", "title": "Sum of Sequence of Products of Consecutive Odd and Consecutive Even Reciprocals", "text": "{{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 1}^n \\frac 1 {j \\left({j + 2}\\right)}       | r = \\frac 1 {1 \\times 3} + \\frac 1 {2 \\times 4} + \\frac 1 {3 \\times 5} + \\frac 1 {4 \\times 6} + \\cdots + \\frac 1 {n \\left({n + 2}\\right)}       | c =  }} {{eqn | r = \\frac 3 4 - \\frac {2 n + 3} {2 \\left({n + 1}\\right) \\left({n + 2}\\right)}       | c =  }} {{end-eqn}}"}
+{"_id": "15527", "title": "Sum of Sequence of Products of Squares of 3 Consecutive Reciprocals", "text": "{{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 1}^\\infty \\frac 1 {j^2 \\paren {j + 1}^2 \\paren {j + 2}^2}       | r = \\frac 1 {1^2 \\times 2^2 \\times 3^2} + \\frac 1 {2^2 \\times 3^2 \\times 4^2} + \\frac 1 {3^2 \\times 4^2 \\times 5^2} + \\frac 1 {4^2 \\times 5^2 \\times 6^2} + \\cdots       | c =  }} {{eqn | r = \\frac {4 \\pi^2 - 39} {16}       | c =  }} {{end-eqn}}"}
+{"_id": "15528", "title": "Definite Integral from 0 to 1 of Power of u over 1 + Power of u", "text": "{{begin-eqn}} {{eqn | l = \\int_0^1 \\dfrac {u^{a - 1} \\rd u} {1 + u^d}       | r = \\sum_{j \\mathop = 0}^\\infty \\frac {\\paren {-1}^j} {a + j d}       | c =  }} {{eqn | r = \\frac 1 a - \\frac 1 {a + d} + \\frac 1 {a + 2 d} - \\frac 1 {a + 3 d} + \\cdots       | c =  }} {{end-eqn}} where $a, d > 0$."}
+{"_id": "15529", "title": "Sum of Reciprocals of Even Powers of Odd Integers", "text": "Let $n \\in \\Z_{> 0}$ be a (strictly) positive integer. {{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 1}^\\infty \\frac 1 {\\paren {2 j - 1}^{2 n} }       | r = \\dfrac 1 {1^{2 n} } + \\dfrac 1 {3^{2 n} } + \\dfrac 1 {5^{2 n} } + \\dfrac 1 {7^{2 n} } + \\cdots       | c =  }} {{eqn | r = \\paren {-1}^{n + 1} \\dfrac {B_{2 n} \\paren {2^{2 n} - 1} \\pi^{2 n} } {2 \\paren {2 n}!}       | c =  }} {{end-eqn}}"}
+{"_id": "15530", "title": "Sum of Reciprocals of Powers of Integers Alternating in Sign", "text": "Let $n \\in \\Z_{> 0}$ be a (strictly) positive integer. {{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 1}^\\infty \\left({-1}\\right)^{j + 1} \\frac 1 {j^{2 n} }       | r = \\dfrac 1 {1^{2 n} } - \\dfrac 1 {2^{2 n} } + \\dfrac 1 {3^{2 n} } - \\dfrac 1 {4^{2 n} } + \\cdots       | c =  }} {{eqn | r = \\left({-1}\\right)^{n + 1} \\dfrac {B_{2 n} \\left({2^{2 n - 1} - 1}\\right) \\pi^{2 n} } {\\left({2 n}\\right)!}       | c =  }} {{end-eqn}}"}
+{"_id": "15531", "title": "Hermite's Formula for Hurwitz Zeta Function", "text": ":$\\displaystyle \\map \\zeta {s, q} = \\frac 1 {2 q^s} + \\frac { q^{1 - s} } {s - 1} + 2 \\int_0^\\infty \\frac {\\map \\sin {s \\arctan \\frac x q} } {\\paren {q^2 + x^2}^{\\frac 1 2 s} \\paren {e^{2 \\pi x} - 1} } \\rd x$ where:  :$\\zeta$ is the Hurwitz zeta function :$\\map \\Re s > 1$ :$\\map \\Re q > 0$."}
+{"_id": "15532", "title": "Binet's Formula for Logarithm of Gamma Function/Formulation 1", "text": "Let $z$ be a complex number with a positive real part. Then:  :$\\displaystyle \\Ln \\map \\Gamma z = \\paren {z - \\frac 1 2} \\Ln z - z + \\frac 1 2 \\ln 2 \\pi + \\int_0^\\infty \\paren {\\frac 1 2 - \\frac 1 t + \\frac 1 {e^t - 1} } \\frac {e^{-t z} } t \\rd t$ where:  :$\\Gamma$ is the Gamma function :$\\Ln$ is the principal branch of the complex logarithm."}
+{"_id": "15533", "title": "Binet's Formula for Logarithm of Gamma Function/Formulation 2", "text": "Let $z$ be a complex number with a positive real part. Then:  :$\\displaystyle \\Ln \\map \\Gamma z = \\paren {z - \\frac 1 2} \\Ln z - z + \\frac 1 2 \\ln 2 \\pi + 2 \\int_0^\\infty \\frac {\\map \\arctan {t / z} } {e^{2 \\pi t} - 1} \\rd t$ where:  :$\\Gamma$ is the Gamma function :$\\Ln$ is the principal branch of the complex logarithm."}
+{"_id": "15534", "title": "Sum of Reciprocals of Odd Powers of Odd Integers Alternating in Sign", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Then: {{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 0}^\\infty \\frac {\\left({-1}\\right)^j} {\\left({2 j + 1}\\right)^{2 n + 1} }       | r = \\left({-1}\\right)^{n + 1} \\frac {\\pi^{2 n + 1} E_{2 n} } {2^{2 n + 2} \\left({2 n}\\right)!}       | c =  }} {{eqn | r = \\frac 1 {1^{2 n + 1} } - \\frac 1 {3^{2 n + 1} } + \\frac 1 {5^{2 n + 1} } - \\frac 1 {7^{2 n + 1} } + \\cdots       | c =  }} {{end-eqn}} where $E_n$ is the $n$th Euler number."}
+{"_id": "15536", "title": "Sum of Infinite Series of Product of Power and Cosine", "text": "Let $r \\in \\R$ such that $\\size r < 1$. Let $z \\in \\R$ such that $z \\ne 2 m \\pi$ for any $m \\in \\Z$. Then: {{begin-eqn}} {{eqn | l = \\sum_{k \\mathop = 0}^\\infty r^k \\cos k x       | r = 1 + r \\cos x + r^2 \\cos 2 x + r^3 \\cos 3 x + \\cdots       | c =  }} {{eqn| r = \\dfrac {1 - r \\cos x} {1 - 2 r \\cos x + r^2}       | c =  }} {{end-eqn}}"}
+{"_id": "15538", "title": "Sum of Series of Product of Power and Sine", "text": "Let $r \\in \\R$. Let $x \\in \\R$ such that $x \\ne 2 m \\pi$ for any $m \\in \\Z$. Then: {{begin-eqn}} {{eqn | l = \\sum_{k \\mathop = 1}^n r^k \\map \\sin {k x}       | r = r \\sin x + r^2 \\sin 2 x + r^3 \\sin 3 x + \\cdots + r^n \\sin n x       | c =  }} {{eqn | r = \\dfrac {r \\sin x - r^{n + 1} \\map \\sin {n + 1} x + r^{n + 2} \\sin n x} {1 - 2 r \\cos x + r^2}       | c =  }} {{end-eqn}}"}
+{"_id": "15539", "title": "Euler-Maclaurin Summation Formula", "text": "Let $f$ be a real function which is appropriately differentiable and integrable. Then: {{begin-eqn}} {{eqn | l = \\sum_{k \\mathop = 1}^{n - 1} \\map f k       | r = \\int_0^n \\map f x \\rd x - \\frac {\\map f 0 + \\map f n} 2 + \\sum_{k \\mathop = 1}^\\infty \\frac {B_{2 k} } {\\paren {2 k}!} \\paren {\\map {f^{\\paren {2 k - 1} } } n - \\map {f^{\\paren {2 k - 1} } } 0}       | c =  }} {{eqn | r = \\int_0^n \\map f x \\rd x - \\frac 1 2 \\paren {\\map f n + \\map f 0}       | c =  }} {{eqn | o =        | ro= +       | r = \\frac 1 {12} \\paren {\\map {f'} n - \\map {f'} 0}       | c =  }} {{eqn | o =        | ro= -       | r = \\frac 1 {720} \\paren {\\map {f'''} n - \\map {f'''} 0}       | c =  }} {{eqn | o =        | ro= +       | r = \\frac 1 {30 \\, 240} \\paren {\\map {f^{\\paren 5} } n - \\map {f^{\\paren 5} } 0}       | c =  }} {{eqn | o =        | ro= -       | r = \\frac 1 {1 \\, 209 \\, 600} \\paren {\\map {f^{\\paren 7} } n - \\map {f^{\\paren 7} } 0}       | c =  }} {{end-eqn}} where: :$f^{\\paren k}$ denotes the $k$th derivative of $f$ :$B_n$ denotes the $n$th Bernoulli number. {{Proofread|Confusion between the various types of Bernoulli numbers means I may have got signs and indices wrong.}}"}
+{"_id": "15544", "title": "Power Series Expansion of Reciprocal of 1 + x", "text": "Let $x \\in \\R$ such that $-1 < x < 1$. Then: {{begin-eqn}} {{eqn | l = \\dfrac 1 {1 + x}       | r = \\sum_{k \\mathop = 0}^\\infty \\left({-1}\\right)^k x^k       | c =  }} {{eqn | r = 1 - x + x^2 - x^3 + x^4 - \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "15549", "title": "Frullani's Integral", "text": ":$\\displaystyle \\int_0^\\infty \\frac {\\map f {a x} - \\map f {b x} } x \\rd x = \\paren {\\map f \\infty - \\map f 0} \\ln \\frac a b$"}
+{"_id": "15551", "title": "Power Series Expansion of Reciprocal of Cube of 1 + x", "text": "Let $x \\in \\R$ such that $-1 < x < 1$. Then: {{begin-eqn}} {{eqn | l = \\dfrac 1 {\\paren {1 + x}^3}       | r = \\sum_{k \\mathop = 0}^\\infty \\paren {-1}^k \\frac {\\paren {k + 2} \\paren {k + 1} } 2 x^k       | c =  }} {{eqn | r = 1 - 3 x + 6 x^2 - 10 x^3 + 15 x^4 - \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "15554", "title": "Power Series Expansion of Square Root of 1 + x", "text": "Let $x \\in \\R$ such that $-1 < x \\le 1$. Then: {{begin-eqn}} {{eqn | l = \\dfrac 1 {\\sqrt {1 + x} }       | r = 1 + \\sum_{k \\mathop = 1}^\\infty \\left({-1}\\right)^{k - 1} \\frac {\\left({2 \\left({k - 1}\\right)}\\right)!} {2^{2 k - 1} k! \\left({k - 1}\\right)!} x^k       | c =  }} {{eqn | r = 1 + \\frac 1 2 x - \\frac 2 {2 \\times 4} x^2 + \\frac {1 \\times 3} {2 \\times 4 \\times 6} x^3 - \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "15558", "title": "Power Series Expansion for Half Logarithm of 1 + x over 1 - x", "text": "{{begin-eqn}} {{eqn | l = \\frac 1 2 \\map \\ln {\\frac {1 + x} {1 - x} }       | r = \\sum_{n \\mathop = 0}^\\infty \\frac {x^{2 n + 1} } {2 n + 1} }} {{eqn | r = x + \\frac {x^3} 3 + \\frac {x^5} 5 + \\frac {x^7} 7 + \\cdots }} {{end-eqn}} valid for all $x \\in \\R$ such that $-1 < x < 1$."}
+{"_id": "15559", "title": "Power Series Expansion for Logarithm of x/Formulation 1", "text": "{{begin-eqn}} {{eqn | l = \\ln x        | r = 2 \\paren {\\sum_{n \\mathop = 0}^\\infty \\frac 1 {2 n + 1} \\paren {\\frac {x - 1} {x + 1} }^{2 n + 1} } }} {{eqn | r = 2 \\paren {\\frac {x - 1} {x + 1} + \\frac 1 3 \\paren {\\frac {x - 1} {x + 1} }^3 + \\frac 1 5 \\paren {\\frac {x - 1} {x + 1} }^5 + \\cdots} }} {{end-eqn}} valid for all $x \\in \\R$ such that $-1 < x < 1$."}
+{"_id": "15562", "title": "Power Series Expansion for Cotangent Function", "text": "The (real) cotangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \\cot x       | r = \\sum_{n \\mathop = 0}^\\infty \\frac {\\left({-1}\\right)^{n} 2^{2 n} B_{2 n} \\, x^{2 n - 1} } {\\left({2 n}\\right)!}       | c =  }} {{eqn | r = \\frac 1 x - \\frac x 3 - \\frac {x^3} {45} - \\frac {2 x^5} {945} + \\cdots       | c =  }} {{end-eqn}} where $B_{2 n}$ denotes the Bernoulli numbers. This converges for $0 < \\left|{x}\\right| < \\pi$."}
+{"_id": "15563", "title": "Power Series Expansion for Secant Function", "text": "The (real) secant function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \\sec x       | r = \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {E_{2 n} x^{2 n} } {\\paren {2 n}!} }} {{eqn | r = 1 + \\frac {x^2} 2 + \\frac {5 x^4} {24} + \\frac {61 x^6} {720} + \\cdots }} {{end-eqn}} where $E_{2 n}$ denotes the Euler numbers. This converges for $\\size x < \\dfrac \\pi 2$."}
+{"_id": "15564", "title": "Power Series Expansion for Real Arccotangent Function", "text": "The arccotangent function has a Taylor series expansion: :$\\operatorname {arccot} x = \\begin{cases} \\displaystyle \\frac \\pi 2 - \\sum_{n \\mathop = 0}^\\infty \\left({-1}\\right)^n \\frac {x^{2 n + 1} } {2 n + 1} & : -1 \\le x \\le 1 \\\\ \\displaystyle \\sum_{n \\mathop = 0}^\\infty \\left({-1}\\right)^n \\frac 1 {\\left({2 n + 1}\\right) x^{2 n + 1} } & : x \\ge 1 \\\\ \\displaystyle \\pi + \\sum_{n \\mathop = 0}^\\infty \\left({-1}\\right)^n \\frac 1 {\\left({2 n + 1}\\right) x^{2 n + 1} } & : x \\le -1 \\end{cases}$ That is: :$\\operatorname {arccot} x = \\begin{cases} \\displaystyle \\frac \\pi 2 - \\left({x - \\frac {x^3} 3 + \\frac {x^5} 5 - \\frac {x^7} 7 + \\cdots}\\right) & : -1 \\le x \\le 1 \\\\ \\displaystyle \\frac 1 x - \\frac 1 {3 x^3} + \\frac 1 {5 x^5} - \\cdots & : x \\ge 1 \\\\ \\displaystyle \\pi + \\frac 1 x - \\frac 1 {3 x^3} + \\frac 1 {5 x^5} - \\cdots & : x \\le -1 \\end{cases}$"}
+{"_id": "15566", "title": "Power Series Expansion for Real Arccosecant Function", "text": "The arccosecant function has a Taylor Series expansion: {{begin-eqn}} {{eqn | l = \\operatorname {arccsc} x       | r = \\sum_{n \\mathop = 0}^\\infty \\frac {\\left({2 n}\\right)!} {2^{2 n} \\left({n!}\\right)^2 \\left({2 n + 1}\\right) x^{2 n + 1} }       | c =  }} {{eqn | r = \\frac 1 x + \\frac 1 2 \\frac 1 {3 x^3} + \\frac {1 \\times 3} {2 \\times 4} \\frac 1 {5 x^5} + \\frac {1 \\times 3 \\times 5} {2 \\times 4 \\times 6} \\frac 1 {7 x^7} + \\cdots       | c =  }} {{end-eqn}} which converges for $\\left\\lvert{x}\\right\\rvert \\ge 1$."}
+{"_id": "15567", "title": "Power Series Expansion for Hyperbolic Sine Function", "text": "The hyperbolic sine function has the power series expansion: {{begin-eqn}} {{eqn | l = \\sinh x       | r = \\sum_{n \\mathop = 0}^\\infty \\frac {x^{2 n + 1} } {\\paren {2 n + 1}!}       | c =  }} {{eqn | r = x + \\frac {x^3} {3!} + \\frac {x^5} {5!} + \\frac {x^7} {7!} + \\cdots       | c =  }} {{end-eqn}} valid for all $x \\in \\R$."}
+{"_id": "15568", "title": "Power Series Expansion for Hyperbolic Cosine Function", "text": "The hyperbolic cosine function has the power series expansion: {{begin-eqn}} {{eqn | l = \\cosh x       | r = \\sum_{n \\mathop = 0}^\\infty \\frac {x^{2 n} } {\\paren {2 n}!}       | c =  }} {{eqn | r = 1 + \\frac {x^2} {2!} + \\frac {x^4} {4!} + \\frac {x^6} {6!} + \\cdots       | c =  }} {{end-eqn}} valid for all $x \\in \\R$."}
+{"_id": "15569", "title": "Power Series Expansion for Hyperbolic Tangent Function", "text": "The hyperbolic tangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \\tanh x       | r = \\sum_{n \\mathop = 1}^\\infty \\frac {2^{2 n} \\left({2^{2 n} - 1}\\right) B_{2 n} \\, x^{2 n - 1} } {\\left({2 n}\\right)!}       | c =  }} {{eqn | r = x - \\frac {x^3} 3 + \\frac {2 x^5} {15} - \\frac {17 x^7} {315} + \\cdots       | c =  }} {{end-eqn}} where $B_{2 n}$ denotes the Bernoulli numbers. This converges for $\\left|{x}\\right| < \\dfrac \\pi 2$."}
+{"_id": "15570", "title": "Power Series Expansion for Hyperbolic Cotangent Function", "text": "The hyperbolic cotangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \\coth x       | r = \\sum_{n \\mathop = 0}^\\infty \\frac {2^{2 n} B_{2 n} \\, x^{2 n - 1} } {\\left({2 n}\\right)!}       | c =  }} {{eqn | r = \\frac 1 x + \\frac x 3 - \\frac {x^3} {45} + \\frac {2 x^5} {45} + \\cdots       | c =  }} {{end-eqn}} where $B_{2 n}$ denotes the Bernoulli numbers. This converges for $0 < \\left|{x}\\right| < \\pi$."}
+{"_id": "15571", "title": "Power Series Expansion for Hyperbolic Secant Function", "text": "The hyperbolic secant function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \\sech x       | r = \\sum_{n \\mathop = 0}^\\infty \\frac {E_{2 n} x^{2 n} } {\\paren {2 n}!}        | c =  }} {{eqn | r = 1 - \\frac {x^2} 2 + \\frac {5 x^4} {24} - \\frac {61 x^6} {720} + \\cdots       | c =  }} {{end-eqn}} where $E_{2 n}$ denotes the Euler numbers. This converges for $\\size x < \\dfrac \\pi 2$."}
+{"_id": "15572", "title": "Power Series Expansion for Hyperbolic Cosecant Function", "text": "<onlyinclude> The hyperbolic cosecant function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \\operatorname {csch} x       | r = \\sum_{n \\mathop = 0}^\\infty \\dfrac {2 \\left({1 - 2^{2 n - 1} }\\right) B_{2 n} \\,  x^{2 n - 1} } {\\left({2 n}\\right)!}       | c =  }} {{eqn | r = \\frac 1 x - \\frac x 6 + \\frac {7 x^3} {360} - \\frac {31 x^5} {15 \\, 120} + \\cdots       | c =  }} {{end-eqn}} where $B_n$ denotes the Bernoulli numbers. This converges for $0 < \\left\\lvert{x}\\right\\rvert < \\pi$."}
+{"_id": "15573", "title": "Sum of Hyperbolic Tangent and Cotangent", "text": ":$\\tanh x + \\coth x = 2 \\coth 2 x$"}
+{"_id": "15574", "title": "Power Series Expansion for Real Inverse Hyperbolic Sine", "text": "The (real) inverse hyperbolic sine function has a Taylor series expansion: :$\\sinh^{-1} x = \\begin{cases} \\displaystyle \\sum_{n \\mathop = 0}^\\infty \\frac {\\left({-1}\\right)^n \\left({2 n}\\right)!} {2^{2 n} \\left({n!}\\right)^2} \\frac {x^{2 n + 1} } {2 n + 1} & : \\left\\lvert{x}\\right\\rvert \\le 1 \\\\  \\displaystyle \\ln 2 x - \\left({\\sum_{n \\mathop = 1}^\\infty \\frac {\\left({-1}\\right)^n \\left({2 n}\\right)!} {2^{2 n + 1} \\left({n!}\\right)^2 n x^{2 n} } }\\right) & : x \\ge 1 \\\\  \\displaystyle -\\ln \\left({-2 x}\\right) + \\left({\\sum_{n \\mathop = 1}^\\infty \\frac {\\left({-1}\\right)^n \\left({2 n}\\right)!} {2^{2 n + 1} \\left({n!}\\right)^2 n x^{2 n} } }\\right) & : x \\le -1 \\\\  \\end{cases}$ That is: :$\\sinh^{-1} x = \\begin{cases} \\displaystyle x - \\dfrac 1 2 \\dfrac {x^3} 3 + \\dfrac {1 \\times 3} {2 \\times 4} \\dfrac {x^5} 5 - \\dfrac {1 \\times 3 \\times 5} {2 \\times 4 \\times 6} \\dfrac {x^7} 7  + \\cdots& : \\left\\lvert{x}\\right\\rvert \\le 1 \\\\  \\displaystyle \\ln 2 x - \\left({\\dfrac 1 2 \\dfrac 1 {2 x^2} + \\dfrac {1 \\times 3} {2 \\times 4} \\dfrac 1 {4 x^4} - \\dfrac {1 \\times 3 \\times 5} {2 \\times 4 \\times 6} \\dfrac 1 {6 x^6} + \\cdots}\\right) & : x \\ge 1 \\\\  \\displaystyle -\\ln \\left({-2 x}\\right) + \\left({\\dfrac 1 2 \\dfrac 1 {2 x^2} + \\dfrac {1 \\times 3} {2 \\times 4} \\dfrac 1 {4 x^4} - \\dfrac {1 \\times 3 \\times 5} {2 \\times 4 \\times 6} \\dfrac 1 {6 x^6} + \\cdots}\\right) & : x \\le -1 \\\\  \\end{cases}$"}
+{"_id": "15575", "title": "Power Series Expansion for Real Inverse Hyperbolic Cosine", "text": "The (real) inverse hyperbolic cosine function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \\cosh^{-1} x       | r = \\ln 2 x - \\left({\\sum_{n \\mathop = 1}^\\infty \\frac {\\left({2 n}\\right)!} {2^{2 n + 1} \\left({n!}\\right)^2 n x^{2 n} } }\\right)       | c =  }} {{eqn | r = \\ln 2 x - \\left({\\dfrac 1 2 \\dfrac 1 {2 x^2} + \\dfrac {1 \\times 3} {2 \\times 4} \\dfrac 1 {4 x^4} + \\dfrac {1 \\times 3 \\times 5} {2 \\times 4 \\times 6} \\dfrac 1 {6 x^6} + \\cdots}\\right)       | c =  }} {{end-eqn}} for $x \\ge 1$."}
+{"_id": "15576", "title": "Power Series Expansion for Real Inverse Hyperbolic Tangent", "text": "The (real) inverse hyperbolic tangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \\tanh^{-1} x       | r = \\sum_{n \\mathop = 0}^\\infty \\frac {x^{2 n + 1} } {2 n + 1}       | c =  }} {{eqn | r = x + \\frac {x^3} 3 + \\frac {x^5} 5 + \\frac {x^7} 7 + \\cdots       | c =  }} {{end-eqn}} for $\\size x < 1$."}
+{"_id": "15577", "title": "Power Series Expansion for Real Inverse Hyperbolic Cotangent", "text": "The (real) inverse hyperbolic cotangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \\coth^{-1} x       | r = \\sum_{n \\mathop = 0}^\\infty \\frac 1 {\\paren {2 n + 1} x^{2 n + 1} }       | c =  }} {{eqn | r = \\frac 1 x + \\frac 1 {3 x^3} + \\frac 1 {5 x^5} + \\frac 1 {7 x^7} + \\cdots       | c =  }} {{end-eqn}} for $\\size x > 1$."}
+{"_id": "15579", "title": "Power Series Expansion for Exponential of Cosine of x", "text": ":$e^{\\cos x} = e \\left({1 - \\dfrac {x^2} 2 + \\dfrac {x^4} 6 - \\dfrac {31 x^6} {720} + \\cdots}\\right)$ for all $x \\in \\R$."}
+{"_id": "15580", "title": "Power Series Expansion for Exponential of Tangent of x", "text": ":$e^{\\tan x} = 1 + x + \\dfrac {x^2} 2 + \\dfrac {x^3} 2 + \\dfrac {3 x^4} 8 + \\cdots$ for all $x \\in \\R$ such that $\\size x < \\frac \\pi 2$."}
+{"_id": "15583", "title": "Power Series Expansion for Exponential of Cosine of x/Proof 2", "text": "For all $x \\in \\R$: :$\\displaystyle e^{\\cos \\left({x}\\right)} = e \\left({e^{cos \\left({x}\\right)- 1 } }\\right) =  e \\left({\\sum_{n \\mathop = 0}^\\infty \\frac{ \\left({-1}\\right)^m P_2 \\left({2 m}\\right)} {2 m!} x^{2 m} }\\right)$ where $P_2 \\left({2 m}\\right)$ is the partition of the set of size 2m into even blocks."}
+{"_id": "15584", "title": "Sine of x plus Cosine of x", "text": "=== Sine Form === {{:Sine of x plus Cosine of x/Sine Form}} === Cosine Form === {{:Sine of x plus Cosine of x/Cosine Form}}"}
+{"_id": "15585", "title": "Sine of x plus Cosine of x/Sine Form", "text": ":$\\sin x + \\cos x = \\sqrt 2 \\sin \\left({x + \\dfrac \\pi 4}\\right)$"}
+{"_id": "15586", "title": "Cosine of x minus Sine of x", "text": "=== Sine Form === {{:Cosine of x minus Sine of x/Sine Form}} === Cosine Form === {{:Cosine of x minus Sine of x/Cosine Form}}"}
+{"_id": "15587", "title": "Cosine of x minus Sine of x/Sine Form", "text": ":$\\cos x - \\sin x = \\sqrt 2 \\, \\map \\sin {x + \\dfrac {3 \\pi} 4}$"}
+{"_id": "15588", "title": "Cosine of x minus Sine of x/Cosine Form", "text": ":$\\cos x - \\sin x = \\sqrt 2 \\, \\map \\cos {x + \\dfrac \\pi 4}$"}
+{"_id": "15589", "title": "Power Series Expansion for Exponential of x by Cosine of x", "text": "{{begin-eqn}} {{eqn | l = e^x \\cos x       | r = \\sum_{n \\mathop = 1}^\\infty \\frac {2^{n / 2} \\, \\cos \\left({n \\pi / 4}\\right) x^n} {n!}       | c =  }} {{eqn | r = 1 + x - \\frac {x^3} 3 - \\frac {x^4} 6 + \\cdots       | c =  }} {{end-eqn}} for all $x \\in \\R$."}
+{"_id": "15591", "title": "Power Series Expansion for Logarithm of Cosine of x", "text": "{{begin-eqn}} {{eqn | l = \\ln \\size {\\cos x}       | r = \\sum_{n \\mathop = 1}^\\infty \\frac {\\paren {-1}^n 2^{2 n - 1} \\{2^{2 n} - 1} B_{2 n} \\, x^{2 n} } {n \\paren {2 n}!}       | c =  }} {{eqn | r = -\\frac {x^2} 2 - \\frac {x^4} {12} - \\frac {x^6} {45} - \\frac {17 x^8} {2520} - \\cdots       | c =  }} {{end-eqn}} for all $x \\in \\R$ such that $\\size x < \\dfrac \\pi 2$."}
+{"_id": "15592", "title": "Expectation of Gaussian Distribution", "text": "Let $X \\sim N \\paren {\\mu, \\sigma^2}$ for some $\\mu \\in \\R, \\sigma \\in \\R_{> 0}$, where $N$ is the Gaussian distribution. Then:  :$\\expect X = \\mu$"}
+{"_id": "15593", "title": "Power Series Expansion for Logarithm of Tangent of x", "text": "{{begin-eqn}} {{eqn | l = \\ln \\left\\lvert{\\tan x}\\right\\rvert       | r = \\ln \\left\\lvert{x}\\right\\rvert + \\sum_{n \\mathop = 1}^\\infty \\frac {\\left({-1}\\right)^{n - 1} 2^{2 n} \\left({2^{2 n - 1} - 1}\\right) B_{2 n} \\, x^{2 n} } {n \\left({2 n}\\right)!}       | c =  }} {{eqn | r = \\ln \\left\\lvert{x}\\right\\rvert + \\frac {x^2} 3 + \\frac {7 x^4} {90} + \\frac {62 x^6} {2835} + \\cdots       | c =  }} {{end-eqn}} for all $x \\in \\R$ such that $0 < \\left|{x}\\right| < \\dfrac \\pi 2$."}
+{"_id": "15595", "title": "Derivative of Logarithm over Power", "text": ":$\\dfrac \\d {\\d x} \\dfrac {\\ln x} {x^n} = \\dfrac {1 - n \\ln x} {x^{n + 1} }$"}
+{"_id": "15596", "title": "Nth Derivative of Natural Logarithm by Reciprocal", "text": ":$\\dfrac {\\d^n} {\\d x^n} \\dfrac {\\ln x} x = \\paren {-1}^{n + 1} n! \\dfrac {H_n - \\ln x} {x^{n + 1} }$ where $H_n$ denotes the $n$th harmonic number: :$H_n = \\displaystyle \\sum_{r \\mathop = 1}^n \\dfrac 1 r = 1 + \\dfrac 1 2 + \\dfrac 1 3 + \\cdots + \\dfrac 1 r$"}
+{"_id": "15597", "title": "Variance of Gaussian Distribution", "text": "Let $X \\sim N \\paren {\\mu, \\sigma^2}$ for some $\\mu \\in \\R, \\sigma \\in \\R_{> 0}$, where $N$ is the Gaussian distribution. Then:  :$\\var X = \\sigma^2$"}
+{"_id": "15598", "title": "Variance as Expectation of Square minus Square of Expectation/Continuous", "text": "Let $X$ be a continuous random variable. Then the variance of $X$ can be expressed as: :$\\var X = \\expect {X^2} - \\paren {\\expect X}^2$ That is, it is the expectation of the square of $X$ minus the square of the expectation of $X$."}
+{"_id": "15599", "title": "Variance as Expectation of Square minus Square of Expectation/Discrete", "text": "Let $X$ be a discrete random variable. Then the variance of $X$ can be expressed as: :$\\var X = \\expect {X^2} - \\paren {\\expect X}^2$ That is, it is the expectation of the square of $X$ minus the square of the expectation of $X$."}
+{"_id": "15600", "title": "Reversion of Power Series", "text": "Let $\\displaystyle y = \\sum_{n \\mathop = 1}^\\infty c_n x^n$ be a power series. Then: :$\\displaystyle x = \\sum_{n \\mathop = 1}^\\infty C_n y^n$ is also a power series, where: {{begin-eqn}} {{eqn | l = c_1 C_1       | r = 1 }} {{eqn | l = {c_1}^3 C_2       | r = -c_2 }} {{eqn | l = {c_1}^5 C_3       | r = 2 {c_2}^2 - c_1 c_3 }} {{eqn | l = {c_1}^7 C_4       | r = 5 c_1 c_2 c_3 - 5 {c_2}^2 - {c_1}^2 c_4 }} {{eqn | l = {c_1}^9 C_5       | r = 6 {c_1}^2 c_2 c_4 + 3 {c_1}^2 {c_3}^2 - {c_1}^3 c_5 + 14 {c_2}^4 - 21 c_1 {c_2}^2 c_3 }} {{eqn | l = {c_1}^{11} C_6       | r = 7 {c_1}^3 c_2 c_5 + 84 c_1 {c_2}^3 c_3 + 7 {c_1}^3 c_3 c_4 - 28 {c_1}^2 c_2 {c_3}^2 - {c_1}^4 c_6 - 28 {c_1}^2 {c_2}^2 c_4 - 42 {c_2}^5 }} {{end-eqn}}"}
+{"_id": "15601", "title": "Expectation of Continuous Uniform Distribution", "text": "Let $X \\sim \\ContinuousUniform a b$ for some $a, b \\in \\R$, $a \\ne b$, where $\\operatorname U$ is the continuous uniform distribution.  Then:  :$\\expect X = \\dfrac {a + b} 2$"}
+{"_id": "15602", "title": "Variance of Continuous Uniform Distribution", "text": "Let $X \\sim \\ContinuousUniform a b$ for some $a, b \\in \\R$, $a \\ne b$, where $\\operatorname U$ is the continuous uniform distribution.  Then:  :$\\var X = \\dfrac {\\paren {b - a}^2} {12}$"}
+{"_id": "15603", "title": "Median of Gaussian Distribution", "text": "Let $X \\sim \\Gaussian \\mu {\\sigma^2}$ for some $\\mu \\in \\R, \\sigma \\in \\R_{> 0}$, where $N$ is the Gaussian distribution. Then the median of $X$ is equal to $\\mu$."}
+{"_id": "15604", "title": "Expectation of Gamma Distribution", "text": "Let $X \\sim \\map \\Gamma {\\alpha, \\beta}$ for some $\\alpha, \\beta > 0$, where $\\Gamma$ is the Gamma distribution. The expectation of $X$ is given by: :$\\expect X = \\dfrac \\alpha \\beta$"}
+{"_id": "15605", "title": "Variance of Gamma Distribution", "text": "Let $X \\sim \\map \\Gamma {\\alpha, \\beta}$ for some $\\alpha, \\beta > 0$, where $\\Gamma$ is the Gamma distribution. The variance of $X$ is given by: :$\\var X = \\dfrac \\alpha {\\beta^2}$"}
+{"_id": "15606", "title": "Sum of Bernoulli Numbers by Power of Two and Binomial Coefficient", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = \\sum_{k \\mathop = 1}^n \\dbinom {2 n + 1} {2 k} 2^{2 k} B_{2 k}       | r = \\binom {2 n + 1} 2 2^2 B_2 + \\binom {2 n + 1} 4 2^4 B_4 + \\binom {2 n + 1} 6 2^6 B_6 + \\cdots       | c =  }} {{eqn | r = 2 n       | c =  }} {{end-eqn}} where $B_n$ denotes the $n$th Bernoulli number."}
+{"_id": "15607", "title": "Moment in terms of Moment Generating Function", "text": "Let $X$ be a random variable.  Let $M_X$ be the moment generating function of $X$. Then:  :$\\expect {X^n} = \\map {M^{\\paren n}_X} 0$ where: :$n$ is a non-negative integer :$M^{\\paren n}_X$ denotes the $n$th derivative of $M_X$ :$\\expect {X^n}$ denotes the expectation of $X^n$."}
+{"_id": "15609", "title": "Sum of Euler Numbers by Binomial Coefficient", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = E_{2 n}       | r = \\sum_{k \\mathop = 0}^{n - 1} \\dbinom {2 n} {2 k} E_{2 n - 2 k}       | c =  }} {{eqn | r = \\binom {2 n} 2 E_{2 n - 2} + \\binom {2 n} 4 E_{2 n - 4} + \\binom {2 n} 6 E_{2 n - 6} + \\cdots + 1       | c =  }} {{end-eqn}} where $E_n$ denotes the $n$th Euler number."}
+{"_id": "15610", "title": "Moment Generating Function of Gaussian Distribution", "text": "Let $X \\sim \\Gaussian \\mu {\\sigma^2}$ for some $\\mu \\in \\R, \\sigma \\in \\R_{> 0}$, where $N$ is the Gaussian distribution. Then the moment generating function $M_X$ of $X$ is given by:  :$\\map {M_X} t = \\map \\exp {\\mu t + \\dfrac 1 2 \\sigma^2 t^2}$"}
+{"_id": "15611", "title": "Bernoulli Number in terms of Euler Numbers", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = B_{2 n}       | r = \\frac {2n} {2^{2 n} \\left({2^{2 n} - 1}\\right)} \\left({\\sum_{k \\mathop = 1}^n \\dbinom {2 n - 1} {2 k - 1} E_{2 n - 2 k} }\\right)       | c =  }} {{eqn | r = \\frac {2n} {2^{2 n} \\left({2^{2 n} - 1}\\right)} \\left({\\binom {2 n - 1} 1 E_{2 n - 2} + \\binom {2 n - 1} 3 E_{2 n - 4} + \\binom {2 n - 1} 5 E_{2 n - 6} + \\cdots + 1}\\right)       | c =  }} {{end-eqn}} where: : $B_n$ denotes the $n$th Bernoulli number : $E_n$ denotes the $n$th Euler number."}
+{"_id": "15617", "title": "Equality of Vector Quantities", "text": "Two vector quantities are equal {{iff}} they have the same magnitude and direction. That is: :$\\mathbf a = \\mathbf b \\iff \\paren {\\size {\\mathbf a} = \\size {\\mathbf b} \\land \\hat {\\mathbf a} = \\hat {\\mathbf b} }$ where: :$\\hat {\\mathbf a}$ denotes the unit vector in the direction of $\\mathbf a$ :$\\size {\\mathbf a}$ denotes the magnitude of $\\mathbf a$."}
+{"_id": "15618", "title": "Vector Addition is Commutative", "text": "Let $\\mathbf a, \\mathbf b$ be vector quantities. Then: :$\\mathbf a + \\mathbf b = \\mathbf b + \\mathbf a$"}
+{"_id": "15619", "title": "Vector Addition is Associative", "text": "Let $\\mathbf a, \\mathbf b, \\mathbf c$ be vectors. Then: :$\\mathbf a + \\paren {\\mathbf b + \\mathbf c} = \\paren {\\mathbf a + \\mathbf b} + \\mathbf c$"}
+{"_id": "15620", "title": "Scalar Multiplication of Vectors is Associative", "text": "Let $\\mathbf a$ be a vector quantity. Let $m, n$ be scalar quantities. Then: :$m \\paren {n \\mathbf a} = \\paren {m n} \\mathbf a = n \\paren {m \\mathbf a}$"}
+{"_id": "15621", "title": "Scalar Multiplication of Vectors is Distributive over Scalar Addition", "text": "Let $\\mathbf a$ be a vector quantity. Let $m, n$ be scalar quantities. Then: :$\\paren {m + n} \\mathbf a = m \\mathbf a + n \\mathbf a$"}
+{"_id": "15622", "title": "Scalar Multiplication of Vectors is Distributive over Vector Addition", "text": "Let $\\mathbf a, \\mathbf b$ be a vector quantities. Let $m$ be a scalar quantity. Then: :$m \\paren {\\mathbf a + \\mathbf b} = m \\mathbf a + m \\mathbf b$"}
+{"_id": "15624", "title": "Scalar Triple Product equals Determinant", "text": "Let $\\mathbf a, \\mathbf b, \\mathbf c$ be vectors in a vector space $\\mathbf V$ of $3$ dimensions: {{begin-eqn}} {{eqn | l = \\mathbf a       | r = a_i \\mathbf i + a_j \\mathbf j + a_k \\mathbf k }} {{eqn | l = \\mathbf b       | r = b_i \\mathbf i + b_j \\mathbf j + b_k \\mathbf k }} {{eqn | l = \\mathbf c       | r = c_i \\mathbf i + c_j \\mathbf j + c_k \\mathbf k }} {{end-eqn}} where $\\left({\\mathbf i, \\mathbf j, \\mathbf k}\\right)$ is the standard ordered basis of $\\mathbf V$. Let $\\mathbf a \\times \\mathbf b$ denote the vector cross product of $\\mathbf a$ with $\\mathbf b$. Let $\\mathbf a \\cdot \\mathbf b$ denote the dot product of $\\mathbf a$ with $\\mathbf b$. Then: :$\\mathbf a \\cdot \\left({\\mathbf b \\times \\mathbf c}\\right) = \\begin{vmatrix} a_i & a_j & a_k \\\\ b_i & b_j & b_k \\\\ c_i & c_j & c_k \\end{vmatrix}$"}
+{"_id": "15625", "title": "Differential Entropy of Gaussian Distribution", "text": "Let $X \\sim \\Gaussian \\mu {\\sigma^2}$ for some $\\mu \\in \\R, \\sigma \\in \\R_{> 0}$, where $N$ is the Gaussian distribution. Then the differential entropy $\\map h X$ of $X$ is given by:  :$\\map h X = \\map \\ln {\\sigma \\sqrt {2 \\pi} } + \\dfrac 1 2$"}
+{"_id": "15627", "title": "Lagrange's Formula/Corollary", "text": ":$\\left({\\mathbf a \\times \\mathbf b}\\right) \\times \\mathbf c = \\left({\\mathbf{a \\cdot c} }\\right) \\mathbf b - \\left({\\mathbf{b \\cdot c} }\\right) \\mathbf a$"}
+{"_id": "15629", "title": "Dot Product of Vector Cross Products", "text": "Let $\\mathbf a, \\mathbf b, \\mathbf c, \\mathbf d$ be vectors in a vector space $\\mathbf V$ of $3$ dimensions: {{begin-eqn}} {{eqn | l = \\mathbf a       | r = a_1 \\mathbf e_1 + a_2 \\mathbf e_2 + a_3 \\mathbf e_3 }} {{eqn | l = \\mathbf b       | r = b_1 \\mathbf e_1 + b_2 \\mathbf e_2 + b_3 \\mathbf e_3 }} {{eqn | l = \\mathbf c       | r = c_1 \\mathbf e_1 + c_2 \\mathbf e_2 + c_3 \\mathbf e_3 }} {{eqn | l = \\mathbf d       | r = d_1 \\mathbf e_1 + d_2 \\mathbf e_2 + d_3 \\mathbf e_3 }} {{end-eqn}} where $\\left({\\mathbf e_1, \\mathbf e_2, \\mathbf e_3}\\right)$ is the standard ordered basis of $\\mathbf V$. Let $\\mathbf a \\times \\mathbf b$ denote the vector cross product of $\\mathbf a$ with $\\mathbf b$. Let $\\mathbf a \\cdot \\mathbf b$ denote the dot product of $\\mathbf a$ with $\\mathbf b$. Then: :$\\left({\\mathbf a \\times \\mathbf b}\\right) \\cdot \\left({\\mathbf c \\times \\mathbf d}\\right) = \\left({\\mathbf a \\cdot \\mathbf c}\\right) \\left({\\mathbf b \\cdot \\mathbf d}\\right) - \\left({\\mathbf a \\cdot \\mathbf d}\\right) \\left({\\mathbf b \\cdot \\mathbf c}\\right)$"}
+{"_id": "15630", "title": "Vector Cross Product of Vector Cross Products", "text": "Let $\\mathbf a, \\mathbf b, \\mathbf c, \\mathbf d$ be vectors in a vector space $\\mathbf V$ of $3$ dimensions: {{begin-eqn}} {{eqn | l = \\mathbf a       | r = a_1 \\mathbf e_1 + a_2 \\mathbf e_2 + a_3 \\mathbf e_3 }} {{eqn | l = \\mathbf b       | r = b_1 \\mathbf e_1 + b_2 \\mathbf e_2 + b_3 \\mathbf e_3 }} {{eqn | l = \\mathbf c       | r = c_1 \\mathbf e_1 + c_2 \\mathbf e_2 + c_3 \\mathbf e_3 }} {{eqn | l = \\mathbf d       | r = d_1 \\mathbf e_1 + d_2 \\mathbf e_2 + d_3 \\mathbf e_3 }} {{end-eqn}} where $\\left({\\mathbf e_1, \\mathbf e_2, \\mathbf e_3}\\right)$ is the standard ordered basis of $\\mathbf V$. Let $\\mathbf a \\times \\mathbf b$ denote the vector cross product of $\\mathbf a$ with $\\mathbf b$. Let $\\mathbf a \\cdot \\mathbf b$ denote the dot product of $\\mathbf a$ with $\\mathbf b$. Then: {{begin-eqn}} {{eqn | l = \\left({\\mathbf a \\times \\mathbf b}\\right) \\times \\left({\\mathbf c \\times \\mathbf d}\\right)       | r = \\mathbf c \\left({\\mathbf a \\cdot \\left({\\mathbf b \\times \\mathbf d}\\right)}\\right) - \\mathbf d \\left({\\mathbf a \\cdot \\left({\\mathbf b \\times \\mathbf c}\\right)}\\right)       | c =  }} {{eqn | r = \\mathbf b \\left({\\mathbf a \\cdot \\left({\\mathbf c \\times \\mathbf d}\\right)}\\right) - \\mathbf a \\left({\\mathbf b \\cdot \\left({\\mathbf c \\times \\mathbf d}\\right)}\\right)       | c =  }} {{end-eqn}}"}
+{"_id": "15631", "title": "Derivative of Scalar Triple Product of Vector-Valued Functions", "text": "Let: : $\\mathbf f: x \\mapsto \\left\\langle{f_1 \\left({x}\\right), f_2 \\left({x}\\right), \\ldots, f_n \\left({x}\\right)}\\right\\rangle$ : $\\mathbf g: x \\mapsto \\left\\langle{g_1 \\left({x}\\right), g_2 \\left({x}\\right), \\ldots, g_n \\left({x}\\right)}\\right\\rangle$ : $\\mathbf h: x \\mapsto \\left\\langle{h_1 \\left({x}\\right), h_2 \\left({x}\\right), \\ldots, h_n \\left({x}\\right)}\\right\\rangle$ be differentiable vector-valued functions. The derivative of their scalar triple product is given by: :$\\dfrac \\d {\\d x} \\left({\\mathbf f \\cdot \\left({\\mathbf g \\times \\mathbf h}\\right)}\\right) = \\dfrac {\\d \\mathbf f} {\\d x} \\cdot \\left({\\mathbf g \\times \\mathbf h}\\right) + \\mathbf f \\cdot \\left({\\dfrac {\\d \\mathbf g} {\\d x} \\times \\mathbf h}\\right) + \\mathbf f \\cdot \\left({\\mathbf g \\times \\dfrac {\\d \\mathbf h} {\\d x} }\\right)$"}
+{"_id": "15632", "title": "Dot Product of Vector-Valued Function with its Derivative", "text": "Let: : $\\mathbf f \\left({x}\\right) = \\displaystyle \\sum_{k \\mathop = 1}^n f_k \\left({x}\\right) \\mathbf e_k$ be a differentiable vector-valued function. The dot product of $\\mathbf f$ with its derivative is given by: :$\\mathbf f \\left({x}\\right) \\cdot \\dfrac {\\d \\mathbf f \\left({x}\\right)} {\\d x} = \\left\\lvert{\\mathbf f \\left({x}\\right)}\\right\\rvert \\dfrac {\\d \\left\\lvert{\\mathbf f \\left({x}\\right)}\\right\\rvert} {\\d x}$ where $\\left\\lvert{\\mathbf f \\left({x}\\right)}\\right\\rvert \\ne 0$."}
+{"_id": "15633", "title": "Dot Product of Constant Magnitude Vector-Valued Function with its Derivative is Zero", "text": "Let: :$\\map {\\mathbf f} x = \\displaystyle \\sum_{k \\mathop = 1}^n \\map {f_k} x \\mathbf e_k$ be a differentiable vector-valued function. Let $\\map {\\mathbf f} x$ be such that its magnitude is constant: :$\\size {\\map {\\mathbf f} x} = c$ for some $c \\in \\R$. Then the dot product of $\\mathbf f$ with its derivative is zero: :$\\map {\\mathbf f} x \\cdot \\dfrac {\\d \\map {\\mathbf f} x} {\\d x} = 0$"}
+{"_id": "15634", "title": "Expectation of Beta Distribution", "text": "Let $X \\sim \\BetaDist \\alpha \\beta$ for some $\\alpha, \\beta > 0$, where $\\operatorname{Beta}$ denotes the beta distribution.  Then: :$\\expect X = \\dfrac \\alpha {\\alpha + \\beta}$"}
+{"_id": "15635", "title": "Variance of Beta Distribution", "text": "Let $X \\sim \\map \\Beta {\\alpha, \\beta}$ for some $\\alpha, \\beta > 0$, where $\\Beta$ is the Beta distribution.  Then: :$\\var X = \\dfrac {\\alpha \\beta} {\\paren {\\alpha + \\beta}^2 \\paren {\\alpha + \\beta + 1} }$"}
+{"_id": "15638", "title": "Moment Generating Function of Gamma Distribution", "text": "Let $X \\sim \\Gamma \\left({\\alpha, \\beta}\\right)$ for some $\\alpha, \\beta > 0$, where $\\Gamma$ is the Gamma distribution. Then the moment generating function of $X$, $M_X$, is given by:  :$\\displaystyle M_X \\left({t}\\right) = \\begin{cases} \\left({1 - \\frac t \\beta}\\right)^{-\\alpha} & t < \\beta \\\\ \\text{does not exist} & t \\ge \\beta \\end{cases}$"}
+{"_id": "15641", "title": "Gradient Operator Distributes over Addition", "text": "Let $\\mathbf V$ be a vector space of $n$ dimensions. Let $\\left({\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}\\right)$ be the standard ordered basis of $\\mathbf V$. Let $f \\left({x_1, x_2, \\ldots, x_n}\\right), g \\left({x_1, x_2, \\ldots, x_n}\\right): \\mathbf V \\to \\R$ be real-valued functions on $\\mathbf V$. Let $\\nabla f$ denote the gradient of $f$. Then: :$\\nabla \\left({f + g}\\right) = \\nabla f + \\nabla g$"}
+{"_id": "15642", "title": "Divergence Operator Distributes over Addition", "text": "Let $\\mathbf V \\left({x_1, x_2, \\ldots, x_n}\\right)$ be a vector space of $n$ dimensions. Let $\\left({\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}\\right)$ be the standard ordered basis of $\\mathbf V$. Let $\\mathbf f$ and $\\mathbf g: \\mathbf V \\to \\mathbf V$ be vector-valued functions on $\\mathbf V$: :$\\mathbf f := \\left({f_1 \\left({\\mathbf x}\\right), f_2 \\left({\\mathbf x}\\right), \\ldots, f_n \\left({\\mathbf x}\\right)}\\right)$ :$\\mathbf g := \\left({g_1 \\left({\\mathbf x}\\right), g_2 \\left({\\mathbf x}\\right), \\ldots, g_n \\left({\\mathbf x}\\right)}\\right)$ Let $\\nabla \\cdot \\mathbf f$ denote the divergence of $f$. Then: :$\\nabla \\cdot \\left({\\mathbf f + \\mathbf g}\\right) = \\nabla \\cdot \\mathbf f + \\nabla \\cdot \\mathbf g$"}
+{"_id": "15643", "title": "Curl Operator Distributes over Addition", "text": "Let $\\map {\\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.. Let $\\tuple {\\mathbf i, \\mathbf j, \\mathbf k}$ be the standard ordered basis on $\\R^3$. Let $\\mathbf f$ and $\\mathbf g: \\R^3 \\to \\R^3$ be vector-valued functions on $\\R^3$: :$\\mathbf f := \\tuple {\\map {f_x} {\\mathbf x}, \\map {f_y} {\\mathbf x}, \\map {f_z} {\\mathbf x} }$ :$\\mathbf g := \\tuple {\\map {g_x} {\\mathbf x}, \\map {g_y} {\\mathbf x}, \\map {g_z} {\\mathbf x} }$ Let $\\nabla \\times \\mathbf f$ denote the curl of $f$. Then: :$\\nabla \\times \\paren {\\mathbf f + \\mathbf g} = \\nabla \\times \\mathbf f + \\nabla \\times \\mathbf g$"}
+{"_id": "15646", "title": "Product Rule for Divergence", "text": "Let $\\mathbf V \\left({x_1, x_2, \\ldots, x_n}\\right)$ be a vector space of $n$ dimensions. Let $\\left({\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}\\right)$ be the standard ordered basis of $\\mathbf V$. Let $\\mathbf f := \\left({f_1 \\left({\\mathbf x}\\right), f_2 \\left({\\mathbf x}\\right), \\ldots, f_n \\left({\\mathbf x}\\right)}\\right): \\mathbf V \\to \\mathbf V$ be a vector-valued function on $\\mathbf V$. Let $g \\left({x_1, x_2, \\ldots, x_n}\\right): \\mathbf V \\to \\R$ be a real-valued function on $\\mathbf V$. Let $\\nabla \\cdot \\mathbf f$ denote the divergence of $f$. Then: :$\\nabla \\cdot \\left({g \\, \\mathbf f}\\right) = g \\left({\\nabla \\cdot \\mathbf f}\\right) + \\left({\\nabla g}\\right) \\cdot \\mathbf f$"}
+{"_id": "15647", "title": "Product Rule for Curl", "text": "Let $\\map {\\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.. Let $\\tuple {\\mathbf i, \\mathbf j, \\mathbf k}$ be the standard ordered basis on $\\R^3$. Let $\\mathbf f := \\tuple {\\map {f_x} {\\mathbf x}, \\map {f_y} {\\mathbf x}, \\map {f_z} {\\mathbf x} }: \\R^3 \\to \\R^3$ be a vector-valued function on $\\R^3$. Let $\\map g {x, y, z}: \\R^3 \\to \\R$ be a real-valued function on $\\R^3$. Let $\\nabla \\times \\mathbf f$ denote the curl of $f$. Then: :$\\nabla \\times \\paren {g \\, \\mathbf f} = \\map g {\\nabla \\times \\mathbf f} + \\paren {\\nabla g} \\times \\mathbf f$"}
+{"_id": "15648", "title": "Moment Generating Function of Beta Distribution", "text": "Let $X \\sim \\operatorname{Beta} \\left({\\alpha, \\beta}\\right)$ for some $\\alpha, \\beta > 0$, where $\\operatorname{Beta}$ is the Beta distribution.  Then the moment generating function $M_X$ of $X$ is given by:  :$\\displaystyle M_X \\left({t}\\right) = 1 + \\sum_{k \\mathop = 1}^\\infty \\left({ \\prod_{r \\mathop = 0}^{k - 1} \\frac {\\alpha + r} {\\alpha + \\beta + r} }\\right) \\frac{t^k} {k!}$"}
+{"_id": "15649", "title": "Raw Moment of Beta Distribution", "text": "Let $X \\sim \\BetaDist \\alpha \\beta$ for some $\\alpha, \\beta > 0$, where $\\operatorname{Beta}$ is the Beta distribution.  Then: :$\\displaystyle \\expect {X^n} = \\prod_{r \\mathop = 0}^{n - 1} \\frac {\\alpha + r} {\\alpha + \\beta + r}$ for positive integer $n$."}
+{"_id": "15650", "title": "Linearity of Expectation Function/Discrete", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $X$ and $Y$ be random variables on $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\expect X$ denote the expectation of $X$. Then: :$\\forall \\alpha, \\beta \\in \\R: \\expect {\\alpha X + \\beta Y} = \\alpha \\, \\expect X + \\beta \\, \\expect Y$"}
+{"_id": "15651", "title": "Linearity of Expectation Function/Continuous", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $X$ and $Y$ be random variables on $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $E$ denote the expectation function. Then: :$\\forall \\alpha, \\beta \\in \\R: \\expect {\\alpha X + \\beta Y} = \\alpha \\expect X + \\beta \\expect Y$"}
+{"_id": "15654", "title": "Derivative of Constant Multiple/Real", "text": "Let $f$ be a real function which is differentiable on $\\R$. Let $c \\in \\R$ be a constant. Then: :$\\map {\\dfrac \\d {\\d x} } {c \\map f x} = c \\map {\\dfrac \\d {\\d x} } {\\map f x}$"}
+{"_id": "15655", "title": "Bound on Complex Values of Gamma Function", "text": "Let $\\map \\Gamma z$ denote the Gamma function. Then for any complex number $z = s + i t$, we have for $\\size b \\le \\size t$: :$\\size {\\map \\Gamma {s + i t} } \\le \\dfrac {\\size {s + i b} } {\\size {s + i t} } \\size {\\map \\Gamma {s + i b} }$"}
+{"_id": "15657", "title": "Set of Vectors defined by Directed Line Segments in Space forms Vector Space", "text": "Let $\\R^3$ be a real cartesian space of $3$ dimensions. Consider the set $S$ of directed line segments in $\\R^3$. Let the equivalence relation $\\sim$ be applied to $\\R^3$ such that: :$\\forall L_1, L_2 \\in \\R^3: L_1 \\sim L_2$ {{iff}} there exists a translation $T$ such that $T \\left({L_1}\\right) = L_2$ Let $\\mathbb V$ denote the set of equivalence classes of $\\sim$ on $S$. The elements of $\\mathbb V$ form a vector space where: :$\\forall \\mathbf v_1, \\mathbf v_2 \\in \\mathbb V: \\mathbf v_1 + \\mathbf v_2$ denotes the element of $\\mathbb V$ exemplified by the result of the operation of joining of a representative element of element of $\\mathbf v_2$ to the end of a similarly representative element of element of $\\mathbf v_1$. :$\\forall \\mathbf v \\in \\mathbb V, \\lambda \\in \\R: \\lambda \\mathbf v$ denotes the element of $\\mathbb V$ exemplified by the result of the operation of mulitplying a representative element of element of $\\mathbf v$ by the scale factor $\\lambda$."}
+{"_id": "15658", "title": "Matrix is Invertible iff Determinant has Multiplicative Inverse/Sufficient Condition", "text": "Let $R$ be a commutative ring with unity. Let $\\mathbf A \\in R^{n \\times n}$ be a square matrix of order $n$. Let  the determinant of $\\mathbf A$ be invertible in $R$. Then $\\mathbf A$ is an invertible matrix."}
+{"_id": "15659", "title": "Chebyshev's Inequality", "text": "Let $X$ be a random variable.  Let $\\expect X = \\mu$ for some $\\mu \\in \\R$.  Let $\\var X = \\sigma^2$ for some $\\sigma^2 \\in \\R_{> 0}$. Then, for all $k > 0$:  :$\\map \\Pr {\\size {X - \\mu} \\ge k \\sigma} \\le \\dfrac 1 {k^2}$"}
+{"_id": "15663", "title": "Sign of Permutation on n Letters is Well-Defined", "text": "Let $n \\in \\N$ be a natural number. Let $S_n$ denote the symmetric group on $n$ letters. Let $\\rho \\in S_n$ be a permutation in $S_n$. Let $\\sgn \\paren \\rho$ denote the sign of $\\rho$. Then $\\sgn \\paren \\rho$ is well-defined, in that it is either $1$ or $-1$."}
+{"_id": "15664", "title": "Odd and Even Permutations of Set are Equivalent", "text": "Let $n \\in \\N_{> 0}$ be a natural number greater than $0$. Let $S$ be a set of cardinality $n$. Let $S_n$ denote the symmetric group on $S$ of order $n$. Let $R_e$ and $R_o$ denote the subsets of $S_n$ consisting of even permutations and odd permutations respectively. Then $R_e$ and $R_o$ are equivalent."}
+{"_id": "15665", "title": "Cardinality of Even and Odd Permutations on Finite Set", "text": "Let $n \\in \\N_{> 0}$ be a natural number greater than $0$. Let $S$ be a set of cardinality $n$. Let $S_n$ denote the symmetric group on $S$ of order $n$. Let $R_e$ and $R_o$ denote the subsets of $S_n$ consisting of even permutations and odd permutations respectively. Then the cardinality of both $R_e$ and $R_o$ is $\\dfrac {n!} 2$."}
+{"_id": "15666", "title": "Congruence Modulo Integer is Equivalence Relation", "text": "For all $z \\in \\Z$, congruence modulo $z$ is an equivalence relation."}
+{"_id": "15672", "title": "Integer Combination of Coprime Integers/General Result", "text": "Let $a_1, a_2, \\ldots, a_n$ be integers. Then $\\gcd \\set {a_1, a_2, \\ldots, a_n} = 1$ {{iff}} there exists an integer combination of them equal to $1$: :$\\exists m_1, m_2, \\ldots, m_n \\in \\Z: \\ds \\sum_{k \\mathop = 1}^n m_k a_k = 1$"}
+{"_id": "15673", "title": "Properties of Beta Function", "text": "Let $\\Beta \\left({x, y}\\right)$ denote the Beta function: {{:Definition:Beta Function/Definition 1}} $\\Beta \\left({x, y}\\right)$ has the following properties:"}
+{"_id": "15674", "title": "Beta Function of x with y+m+1", "text": "Let $\\map \\Beta {x, y}$ denote the Beta function. Then: :$\\map \\Beta {x, y} = \\dfrac {\\map {\\Gamma_m} y m^x} {\\map {\\Gamma_m} {x + y} } \\map \\Beta {x, y + m + 1}$ where $\\Gamma_m$ is the partial Gamma function: :$\\map {\\Gamma_m} y := \\dfrac {m^y m!} {y \\paren {y + 1} \\paren {y + 2} \\cdots \\paren {y + m} }$"}
+{"_id": "15676", "title": "Binomial Coefficient expressed using Beta Function", "text": "Let $\\dbinom r k$ denote a binomial coefficient. Then: :$\\dbinom r k = \\dfrac 1 {\\left({r + 1}\\right) B \\left({k + 1, r - k + 1}\\right)}$"}
+{"_id": "15677", "title": "Symmetry Rule for Binomial Coefficients/Complex Numbers", "text": "For all $z, w \\in \\C$ such that it is not the case that $z$ is a negative integer and $w$ an integer: :$\\dbinom z w = \\dbinom z {z - w}$"}
+{"_id": "15678", "title": "Factors of Binomial Coefficient/Complex Numbers", "text": "For all $z, w \\in \\C$ such that it is not the case that $z$ is a negative integer and $w$ an integer: :$\\dbinom z w = \\dfrac z w \\dbinom {z - 1} {w - 1}$ where $\\dbinom z w$ is a binomial coefficient."}
+{"_id": "15679", "title": "Pascal's Rule/Complex Numbers", "text": "For all $z, w \\in \\C$ such that it is not the case that $z$ is a negative integer and $w$ an integer: :$\\dbinom z {w - 1} + \\dbinom z w = \\dbinom {z + 1} w$ where $\\dbinom z w$ is a binomial coefficient."}
+{"_id": "15682", "title": "Binomial Theorem/Extended", "text": "Let $r, \\alpha \\in \\C$ be complex numbers. Let $z \\in \\C$ be a complex number such that $\\left|{z}\\right| < 1$. Then: :$\\displaystyle \\paren {1 + z}^r = \\sum_{k \\mathop \\in \\Z} \\dbinom r {\\alpha + k} z^{\\alpha + k}$ where $\\dbinom r {\\alpha + k}$ denotes a binomial coefficient."}
+{"_id": "15683", "title": "Chu-Vandermonde Identity/Extended", "text": "Let $r, s, \\alpha, \\beta \\in \\C$ be complex numbers. Then: :$\\displaystyle \\sum_{k \\mathop \\in \\Z} \\dbinom r {\\alpha + k} \\dbinom s {\\beta - k}$ where $\\dbinom r {\\alpha + k}$ denotes a binomial coefficient."}
+{"_id": "15684", "title": "Beta Function of Half with Half", "text": ":$\\Beta \\left({\\dfrac 1 2, \\dfrac 1 2}\\right) = \\pi$ where $\\Beta$ denotes the Beta function."}
+{"_id": "15688", "title": "Binomial Coefficient of Real Number with Half", "text": ":$\\dbinom r {1 / 2} = \\dfrac {2^{2 r + 1} } {\\dbinom {2 r} r \\pi}$ where $\\dbinom r {1 / 2}$ denotes a binomial coefficient."}
+{"_id": "15689", "title": "Limit to Infinity of Binomial Coefficient over Power", "text": ":$\\displaystyle \\lim_{r \\mathop \\to \\infty} \\dfrac {\\dbinom r k} {r^k} = \\frac 1 {\\map \\Gamma {k + 1}}$"}
+{"_id": "15690", "title": "Approximation to x+y Choose y", "text": ":$\\displaystyle \\lim_{x, y \\mathop \\to \\infty} \\dbinom {x + y} y = \\sqrt {\\dfrac 1 {2 \\pi} \\paren {\\frac 1 x + \\frac 1 y} } \\paren {1 + \\dfrac y x}^x \\paren {1 + \\dfrac x y}^y$"}
+{"_id": "15692", "title": "Product of r Choose k with r Minus Half Choose k/Formulation 1", "text": "Let $k \\in \\Z$, $r \\in \\R$. :$\\dbinom r k \\dbinom {r - \\frac 1 2} k = \\dfrac {\\dbinom {2 r} k \\dbinom {2 r - k} k} {4^k}$ where $\\dbinom r k$ denotes a binomial coefficient."}
+{"_id": "15693", "title": "Provable Consequence of Theorems is Theorem", "text": "Let $\\mathcal P$ be a proof system for a formal language $\\mathcal L$. Let $\\mathcal F$ be a collection of theorems of $\\mathcal P$. Denote with $\\mathscr P \\left({\\mathcal F}\\right)$ the proof system obtained from $\\mathscr P$ by adding all the WFFs from $\\mathcal F$ as axioms. Let $\\phi$ be a provable consequence of $\\mathcal F$: :$\\vdash_{\\mathscr P} \\mathcal F$ :$\\mathcal F \\vdash_{\\mathscr P} \\phi$ Then $\\phi$ is also a theorem of $\\mathscr P$: :$\\vdash_{\\mathscr P} \\phi$"}
+{"_id": "15694", "title": "Binomial Coefficient of Minus Half", "text": "Let $k \\in \\Z$. :$\\dbinom {-\\frac 1 2} k = \\dfrac {\\paren {-1}^k} {4^k} \\dbinom {2 k} k$ where $\\dbinom {-\\frac 1 2} k$ denotes a binomial coefficient."}
+{"_id": "15695", "title": "Falling Factorial of Sum of Integers", "text": "Let $r \\in \\R$ be a real number. Let $a, b \\in \\Z$ be (positive) integers. Then: :$r^{\\underline {a + b} } = r^{\\underline a} \\left({r - a}\\right)^{\\underline b}$ where $r^{\\underline a}$ denotes the $a$th falling factorial of $r$."}
+{"_id": "15696", "title": "Approximate Size of Sum of Harmonic Series", "text": "Let $H_n$ denote the sum of the harmonic series: :$H_n = \\displaystyle \\sum_{k \\mathop = 1}^n \\frac 1 k$ Then $H_n$ can be approximated as follows: :$H_n \\approx \\ln n + \\gamma + \\dfrac 1 {2 n} - \\dfrac 1 {12 n^2} + \\dfrac 1 {120 n^4} - \\epsilon$ where: :$\\gamma$ denotes the Euler-Mascheroni constant: $\\gamma \\approx 0 \\cdotp 57721 \\, 56649 \\, \\ldots$ :$0 < \\epsilon < \\dfrac 1 {252 n^6}$"}
+{"_id": "15698", "title": "Riemann Zeta Function of 8", "text": "The Riemann zeta function of $8$ is given by: {{begin-eqn}} {{eqn | l = \\map \\zeta 8       | r = \\dfrac 1 {1^8} +  \\dfrac 1 {2^8} +  \\dfrac 1 {3^8} +  \\dfrac 1 {4^8} + \\cdots       | c =  }} {{eqn | r = \\dfrac {\\pi^8} {9450}       | c =  }} {{eqn | o = \\approx       | r = 1 \\cdotp 00408 \\, 3 \\ldots       | c =  }} {{end-eqn}}"}
+{"_id": "15699", "title": "Sum of Sequence of Harmonic Numbers", "text": ":$\\displaystyle \\sum_{k \\mathop = 1}^n H_k = \\left({n + 1}\\right) H_n - n$ where $H_k$ denotes the $k$th harmonic number."}
+{"_id": "15700", "title": "Sum over k to n of k Choose m by kth Harmonic Number", "text": ":$\\displaystyle \\sum_{k \\mathop = 1}^n \\binom k m H_k = \\binom {n + 1} {m + 1} \\left({H_{n + 1} - \\frac 1 {m + 1} }\\right)$ where: : $\\dbinom k m$ denotes a binomial coefficient : $H_k$ denotes the $k$th harmonic number."}
+{"_id": "15701", "title": "Sum over k of n Choose k by x to the k by kth Harmonic Number", "text": "Let $x \\in \\R_{> 0}$ be a real number. Then: :$\\displaystyle \\sum_{k \\mathop \\in \\Z} \\binom n k x^k H_k = \\left({x + 1}\\right)^n \\left({H_n - \\ln \\left({1 + \\frac 1 x}\\right)}\\right) + \\epsilon$ where: : $\\dbinom n k$ denotes a binomial coefficient : $H_k$ denotes the $k$th harmonic number : $0 < \\epsilon < \\dfrac 1 {x \\left({n + 1}\\right)}$"}
+{"_id": "15706", "title": "Power to Real Number by Decimal Expansion is Uniquely Defined", "text": "Let $r \\in \\R_{> 1}$ be a real number greater than $1$, expressed by its decimal expansion: :$r = n \\cdotp d_1 d_2 d_3 \\ldots$ The power $x^r$ of a (strictly) positive real number $x$ defined as: :$(1): \\quad \\displaystyle \\lim_{k \\mathop \\to \\infty} x^{\\psi_1} \\le \\xi \\le x^{\\psi_2}$ where: {{begin-eqn}} {{eqn | l = \\psi_1       | r = n + \\sum_{j \\mathop = 1}^k \\frac {d_1} {10^k} = n + \\frac {d_1} {10} + \\cdots + \\frac {d_k} {10^k} }} {{eqn | l = \\psi_2       | r = \\psi_1 + \\dfrac 1 {10^k} }} {{end-eqn}} is unique."}
+{"_id": "15707", "title": "Change of Base of Logarithm/Base 2 to Base 10", "text": ":$\\log_{10} x = \\left({\\lg x}\\right) \\left({\\log_{10} 2}\\right) = 0 \\cdotp 30102 \\, 99956 \\, 63981 \\, 19521 \\, 37389 \\ldots \\lg x$"}
+{"_id": "15709", "title": "Ordering on Real Numbers from Decimal Expansion", "text": "Let $x, y \\in \\R$ be real numbers. Let $x$ and $y$ be expressed by their decimal expansions: {{begin-eqn}} {{eqn | l = x       | r = m \\cdotp d_1 d_2 d_3 \\ldots }} {{eqn | l = y       | r = n \\cdotp e_1 e_2 e_3 \\ldots       | c =  }} {{end-eqn}} Let $\\preccurlyeq$ be the lexicographic ordering on $\\R$ defined as: :$x \\preccurlyeq y$ {{iff}}: ::$m \\prec n$ :or: ::$m = n$ and $\\exists k \\in \\Z_{>0}: \\left({\\forall j: 1 \\le j < k: d_j = e_j}\\right) \\land d_k < e_k$ :or: ::$m = n$ and $\\forall j \\in \\Z_{>0}: d_j = e_j$. Then: : $x \\le y$ where $\\le$ denotes the usual ordering on $\\R$."}
+{"_id": "15712", "title": "Necessary Precision for x equal to log base 10 of 2 to determine Decimal expansion of 10 to the x", "text": "Let $b = 10$. Let $x \\approx \\log_{10} 2$. Let it be necessary to calculate the decimal expansion of $x$ to determine the first $3$ decimal places of $b^x$. An infinite number of decimal places of $x$ would in fact be necessary."}
+{"_id": "15713", "title": "Change of Base of Logarithm/Base 10 to Base e/Form 1", "text": ":$\\ln x = \\paren {\\ln 10} \\paren {\\log_{10} x} = 2 \\cdotp 30258 \\, 50929 \\, 94 \\ldots \\log_{10} x$"}
+{"_id": "15714", "title": "Change of Base of Logarithm/Base 10 to Base e/Form 2", "text": ":$\\ln x = \\dfrac {\\log_{10} x} {\\log_{10} e} = \\dfrac {\\log_{10} x} {0 \\cdotp 43429 \\, 44819 \\, 03 \\ldots}$"}
+{"_id": "15715", "title": "Change of Base of Logarithm/Base e to Base 10/Form 1", "text": ":$\\log_{10} x = \\paren {\\log_{10} e} \\paren {\\ln x} = 0 \\cdotp 43429 \\, 44819 \\, 03 \\ldots \\ln x$"}
+{"_id": "15716", "title": "Change of Base of Logarithm/Base e to Base 10/Form 2", "text": ":$\\log_{10} x = \\dfrac {\\ln x} {\\ln 10} = \\dfrac {\\ln x} {2 \\cdotp 30258 \\, 50929 \\, 94 \\ldots}$"}
+{"_id": "15717", "title": "Change of Base of Logarithm/Base 10 to Base 2", "text": ":$\\lg x = \\dfrac {\\log_{10} x} {\\log_{10} 2} = \\dfrac {\\log_{10} x} {0 \\cdotp 30102 \\, 99956 \\, 63981 \\, 19521 \\, 37389 \\ldots}$"}
+{"_id": "15718", "title": "Root of Quotient equals Quotient of Roots", "text": ":$\\sqrt [n] {\\dfrac a b} = \\dfrac {\\sqrt [n] a} {\\sqrt [n] b}$"}
+{"_id": "15719", "title": "Logarithm to Own Base equals 1", "text": "Let $b \\in \\R_{>0}$ be a strictly positive real number such that $b \\ne 1$. Let $\\log_b$ denote the logarithm to base $b$. Then: :$\\log_b b = 1$"}
+{"_id": "15720", "title": "General Logarithm/Examples/Base b of 1", "text": ":$\\log_b 1 = 0$"}
+{"_id": "15721", "title": "General Logarithm/Examples/Base b of -1", "text": ":$\\log_b \\left({-1}\\right)$ is undefined in the real number line."}
+{"_id": "15722", "title": "Change of Base of Logarithm/Base 2 to Base 8", "text": ":$\\log_8 x = \\dfrac {\\lg x} 3$"}
+{"_id": "15723", "title": "Number of Digits to Represent Integer in Given Number Base", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $b \\in \\Z$ be an integer such that $b > 1$. Let $d$ denote the number of digits of $n$ when represented in base $b$. Then: :$d = \\ceiling {\\map {\\log_b} {n + 1} }$ where $\\ceiling {\\, \\cdot \\,}$ denotes the ceiling function."}
+{"_id": "15724", "title": "Number of Bits for Decimal Integer", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $n$ have $m$ digits when expressed in decimal notation. Then $n$ may require as many as $\\ceiling {\\dfrac m {\\log_{10} 2} }$ bits to represent it."}
+{"_id": "15726", "title": "Sum of Summations over Overlapping Domains/Infinite Series", "text": "Let the fiber of truth of both $R$ and $S$ be infinite. Then provided that any $3$ of the $4$ summations converge: :$\\ds \\sum_{\\map R j} a_j + \\sum_{\\map S j} a_j = \\sum_{\\map R j \\mathop \\lor \\map S J} a_j + \\sum_{\\map R j \\mathop \\land \\map S j} a_j$ where $\\lor$ and $\\land$ signify logical disjunction and logical conjunction respectively."}
+{"_id": "15727", "title": "Translation of Index Variable of Summation/Infinite Series", "text": "Let $R: \\Z \\to \\left\\{ {\\mathrm T, \\mathrm F}\\right\\}$ be a propositional function on the set of integers. Let $\\displaystyle \\sum_{R \\left({j}\\right)} a_j$ denote a summation over $R$. Let the fiber of truth of $R$ be infinite. Then: :$\\displaystyle \\sum_{R \\left({j}\\right)} a_j = \\sum_{R \\left({c \\mathop + j}\\right)} a_{c \\mathop + j} = \\sum_{R \\left({c \\mathop - j}\\right)} a_{c \\mathop - j}$ where $c$ is an integer constant which is not dependent upon $j$."}
+{"_id": "15728", "title": "Exchange of Order of Summation/Example", "text": "Let the fiber of truth of both $R$ and $S$ be infinite. Then it is not necessarily the case that: :$\\displaystyle \\sum_{\\map R i} \\sum_{\\map S j} a_{i j} = \\sum_{\\map S j} \\sum_{\\map R i} a_{i j}$"}
+{"_id": "15729", "title": "Sum of Geometric Sequence/Examples/Index to Minus 1", "text": "Let $x$ be an element of one of the standard number fields: $\\Q, \\R, \\C$ such that $x \\ne 1$. Then the formula for Sum of Geometric Sequence: :$\\displaystyle \\sum_{j \\mathop = 0}^n x^j = \\frac {x^{n + 1} - 1} {x - 1}$ still holds when $n = -1$: :$\\displaystyle \\sum_{j \\mathop = 0}^{-1} x^j = \\frac {x^0 - 1} {x - 1}$"}
+{"_id": "15731", "title": "Sum of Geometric Sequence/Examples/Common Ratio 1", "text": "Consider the Sum of Geometric Sequence defined on the standard number fields for all $x \\ne 1$. :$\\displaystyle \\sum_{j \\mathop = 0}^n a x^j = a \\paren {\\frac {1 - x^{n + 1} } {1 - x} }$ When $x = 1$, the formula reduces to: :$\\displaystyle \\sum_{j \\mathop = 0}^n a 1^j = a \\paren {n + 1}$"}
+{"_id": "15732", "title": "Sum of Geometric Sequence/Examples/One Seventh from 1 to n", "text": ":$\\displaystyle \\sum_{j \\mathop = 0}^n \\dfrac 1 {7^j} = \\frac 7 6 \\left({1 - \\frac 1 {7^{n + 1} } }\\right)$"}
+{"_id": "15737", "title": "Summation of Unity over Elements", "text": "Let $S \\subseteq \\Z$ be a set of integers. Let: :$n := \\displaystyle \\sum_{j \\mathop \\in  S} 1$ Then $n$ is equal to the cardinality of $S$."}
+{"_id": "15739", "title": "Hausdorff Maximal Principle Implies Zorn's Lemma", "text": "The Hausdorff Maximal Principle implies Zorn's Lemma."}
+{"_id": "15740", "title": "Zorn's Lemma Implies Well Ordering Theorem", "text": "Zorn's Lemma implies the Well-Ordering Theorem."}
+{"_id": "15741", "title": "Uncountable Sum as Series", "text": "Let $X$ be an uncountable set. Let $f: X \\to \\closedint 0 {+\\infty}$ be an extended real-valued function. The uncountable sum: :$\\displaystyle \\sum_{x \\mathop \\in X} \\map f x = \\sup \\set {\\sum_{x \\mathop \\in F} \\map f x : F \\subseteq X, F \\text{ finite} }$ is $+\\infty$, or can be expressed as a (possibly divergent) series."}
+{"_id": "15742", "title": "Sum with Maximum is Maximum of Sum", "text": "Let $a, b, c \\in \\R$ be real numbers. Then: :$a + \\max \\set {b, c} = \\max \\set {a + b, a + c}$"}
+{"_id": "15745", "title": "Sum of Indexed Suprema", "text": "Let $\\left \\langle {a_i} \\right \\rangle_{i \\mathop \\in I}$ be a family of elements of the real numbers $\\R$ indexed by $I$. Let $\\left \\langle {b_j} \\right \\rangle_{j \\mathop \\in J}$ be a family of elements of the real numbers $\\R$ indexed by $J$. Let $R \\left({i}\\right)$ and $S \\left({j}\\right)$ be propositional functions of $i \\in I$, $j \\in J$. Let $\\displaystyle \\sup_{R \\left({i}\\right)} a_i$ and $\\displaystyle \\sup_{S \\left({j}\\right)} b_j$ be the indexed suprema on $\\left \\langle {a_i} \\right \\rangle$ and $\\left \\langle {b_j} \\right \\rangle$ respectively. Then: :$\\displaystyle \\left({\\sup_{R \\left({i}\\right)} a_i}\\right) + \\left({\\sup_{S \\left({j}\\right)} b_j}\\right) = \\sup_{R \\left({i}\\right)} \\left({\\sup_{S \\left({j}\\right)} \\left({a_i + b_j}\\right)}\\right)$"}
+{"_id": "15746", "title": "Product of Indexed Suprema of Non-Negative Numbers", "text": "Let $\\left \\langle {a_i} \\right \\rangle_{i \\mathop \\in I}$ be a family of elements of the non-negative real numbers $\\R_{\\ge 0}$ indexed by $I$. Let $\\left \\langle {b_j} \\right \\rangle_{j \\mathop \\in J}$ be a family of elements of the non-negative real numbers $\\R_{\\ge 0}$ indexed by $J$. Let $R \\left({i}\\right)$ and $S \\left({j}\\right)$ be propositional functions of $i \\in I$, $j \\in J$. Let $\\displaystyle \\sup_{R \\left({i}\\right)} a_i$ and $\\displaystyle \\sup_{S \\left({j}\\right)} b_j$ be the indexed suprema on $\\left \\langle {a_i} \\right \\rangle$ and $\\left \\langle {b_j} \\right \\rangle$ respectively. Then: :$\\displaystyle \\left({\\sup_{R \\left({i}\\right)} a_i}\\right) \\left({\\sup_{S \\left({j}\\right)} b_j}\\right) = \\sup_{R \\left({i}\\right)} \\left({\\sup_{S \\left({j}\\right)} \\left({a_i b_j}\\right)}\\right)$"}
+{"_id": "15747", "title": "Change of Index Variable of Supremum", "text": "Let $\\family {a_i}{i \\mathop \\in I}$ be a family of elements of the non-negative real numbers $\\R_{\\ge 0}$ indexed by $I$. Let $\\map R i$ be a propositional functions of $i \\in I$. Let $\\displaystyle \\sup_{\\map R i} a_i$ be the indexed supremum on $\\family {a_i}$. Then: :$\\displaystyle \\sup_{\\map R i} a_i = \\sup_{\\map R j} a_j$"}
+{"_id": "15748", "title": "Permutation of Indices of Supremum", "text": "Let $\\left \\langle {a_i} \\right \\rangle_{i \\mathop \\in I}$ be a family of elements of the non-negative real numbers $\\R_{\\ge 0}$ indexed by $I$. Let $R \\left({i}\\right)$ be a propositional functions of $i \\in I$. Let $\\displaystyle \\sup_{R \\left({i}\\right)} a_i$ be the indexed supremum on $\\left \\langle {a_i} \\right \\rangle$. Then: :$\\displaystyle \\sum_{R \\left({i}\\right)} a_i = \\sum_{R \\left({\\pi \\left({i}\\right)}\\right)} a_{\\pi \\left({i}\\right)}$ where $\\pi$ is a permutation on the fiber of truth of $R$."}
+{"_id": "15749", "title": "Exchange of Order of Supremum Operators", "text": "Let $\\left \\langle {a_i} \\right \\rangle_{i \\mathop \\in I}$ be a family of elements of the non-negative real numbers $\\R_{\\ge 0}$ indexed by $I$. Let $\\left \\langle {b_j} \\right \\rangle_{j \\mathop \\in J}$ be a family of elements of the non-negative real numbers $\\R_{\\ge 0}$ indexed by $J$. Let $R \\left({i}\\right)$ and $S \\left({j}\\right)$ be propositional functions of $i \\in I$, $j \\in J$. Let $\\displaystyle \\sup_{R \\left({i}\\right)} a_i$ and $\\displaystyle \\sup_{S \\left({j}\\right)} b_j$ be the indexed suprema on $\\left \\langle {a_i} \\right \\rangle$ and $\\left \\langle {b_j} \\right \\rangle$ respectively. Then: :$\\displaystyle \\sup_{R \\left({i}\\right)} \\left({\\sup_{S \\left({j}\\right)} a_{i j} }\\right) = \\sup_{S \\left({j}\\right)} \\left({\\sup_{R \\left({i}\\right)} a_{i j} }\\right)$"}
+{"_id": "15750", "title": "Supremum of Suprema over Overlapping Domains", "text": "Let $\\family {a_i}_{i \\mathop \\in I}$ be a family of elements of the non-negative real numbers $\\R_{\\ge 0}$ indexed by $I$. Let $\\map R i$ and $\\map S i$ be propositional functions of $i \\in I$. Let $\\displaystyle \\sup_{\\map R i} a_i$ and $\\displaystyle \\sup_{\\map S i} a_i$ be the indexed suprema on $\\family {a_i}$ over $\\map R i$ and $\\map S i$ respectively. Then: :$\\displaystyle \\map \\sup {\\sup_{\\map R i} a_i, \\sup_{\\map S i} a_i} = \\sup_{\\map R i \\mathop \\lor \\map S i} a_i$"}
+{"_id": "15751", "title": "Contour Integration by Substitution", "text": "Let $f$ be a holomorphic function on a simply connected domain $V \\subseteq \\mathbb C$. Let $\\gamma$ be a contour in $V$ starting at $z_1$ and ending at $z_2$. Let $U$ be a connected domain. Let $\\phi: U \\to V$ be a holomorphic function with $\\phi^{-1} \\sqbrk {\\set {z_1, z_2} } \\ne \\O$. Let $\\omega$ be a contour in $U$ starting at $u_1$ and ending at $u_2$, such that $\\map \\phi {u_1} = z_1$ and $\\map \\phi {u_2} = z_2$. Then the contour integral of $f$ over $\\gamma$ satisfies the following substitution: :$\\displaystyle \\int_\\gamma \\map f z \\rd z = \\int_\\omega \\map f {\\map \\phi u} \\, \\map {\\phi'} u \\rd u$"}
+{"_id": "15752", "title": "Infimum of Set of Integers is Integer", "text": "Let $S \\subset \\Z$ be a non-empty subset of the set of integers. Let $S$ be bounded below in the set of real numbers. Then its infimum $\\inf S$ is an integer."}
+{"_id": "15753", "title": "Infimum of Set of Integers equals Smallest Element", "text": "Let $S \\subset \\Z$ be a non-empty subset of the set of integers. Let $S$ be bounded below in the set of real numbers $\\R$. Then $S$ has a smallest element, and it is equal to the infimum $\\sup S$."}
+{"_id": "15754", "title": "Greatest Element is Supremum", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $T \\subseteq S$. Let $T$ have a greatest element $M$. Then $M$ is the supremum of $T$ in $S$."}
+{"_id": "15755", "title": "Smallest Element is Infimum", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T \\subseteq S$. Let $T$ have a smallest element $m$. Then $m$ is the infimum of $T$ in $S$."}
+{"_id": "15757", "title": "Floor Function/Examples/Floor of 1.1", "text": ":$\\floor {1 \\cdotp 1} = 1$"}
+{"_id": "15758", "title": "Floor Function/Examples/Floor of -1.1", "text": ":$\\floor {-1 \\cdotp 1} = -2$"}
+{"_id": "15759", "title": "Ceiling Function/Examples/Ceiling of -1.1", "text": ":$\\ceiling {-1 \\cdotp 1} = -1$"}
+{"_id": "15760", "title": "Floor Function/Examples/Floor of 0.99999", "text": ":$\\floor {0 \\cdotp 99999} = 0$"}
+{"_id": "15761", "title": "Floor Function/Examples/Floor of Binary Logarithm of 35", "text": ":$\\floor {\\lg 35} = 5$ where: : $\\lg x$ denotes the binary logarithm ($\\log_2$) of $x$"}
+{"_id": "15762", "title": "Modulo Operation/Examples/100 mod 3", "text": ":$100 \\bmod 3 = 1$"}
+{"_id": "15764", "title": "Modulo Operation/Examples/-100 mod 7", "text": ":$-100 \\bmod 7 = 5$"}
+{"_id": "15765", "title": "Modulo Operation/Examples/-100 mod 0", "text": ":$-100 \\bmod 0 = -100$"}
+{"_id": "15766", "title": "Modulo Operation/Examples/5 mod -3", "text": ":$5 \\bmod -3 = 5$"}
+{"_id": "15767", "title": "Modulo Operation/Examples/18 mod -3", "text": ":$18 \\bmod -3 = 0$"}
+{"_id": "15768", "title": "Modulo Operation/Examples/-2 mod -3", "text": ":$-2 \\bmod -3 = -2$"}
+{"_id": "15771", "title": "Modulo Operation/Examples/0.11 mod -0.1", "text": ":$0 \\cdotp 11 \\bmod -0 \\cdotp 1 = -0 \\cdotp 09$"}
+{"_id": "15774", "title": "Common Factor Cancelling in Congruence/Corollary 1/Warning", "text": "Let $a$ ''not'' be coprime to $m$. Then it is not necessarily the case that: :$x \\equiv y \\pmod m$"}
+{"_id": "15775", "title": "Chinese Remainder Theorem/Warning", "text": "Let $r$ ''not'' be coprime to $s$. Then it is not necessarily the case that: :$a \\equiv b \\pmod {r s}$ {{iff}} $a \\equiv b \\pmod r$ and $a \\equiv b \\pmod s$ where $a \\equiv b \\pmod r$ denotes that $a$ is congruent modulo $r$ to $b$."}
+{"_id": "15776", "title": "Euler Phi Function of 2", "text": ":$\\map \\phi 2 = 1$"}
+{"_id": "15778", "title": "Existence of Abscissa of Convergence", "text": "Let $\\displaystyle \\map f s = \\sum_{n \\mathop = 1}^\\infty a_n n^{-s}$ be a Dirichlet series. Let the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\size {a_n n^{-s} }$ not converge for all $s \\in \\C$, or diverge for all $s \\in \\C$. Then there exists a real number $\\sigma_c$ such that $\\map f s$ converges for all $s = \\sigma + it$ with $\\sigma > \\sigma_c$, and does not converge for all $s$ with $\\sigma < \\sigma_c$. We call $\\sigma_c$ the '''abscissa of convergence''' of the Dirichlet series."}
+{"_id": "15780", "title": "Factorial as Sum of Series of Subfactorial by Falling Factorial over Factorial", "text": "{{begin-eqn}} {{eqn | l = n!       | r = \\sum_{k \\mathop \\ge 0} \\dfrac { {!k} \\, n^{\\underline k} } {k!}       | c =  }} {{eqn | r = \\dfrac { !0 \\times n^{\\underline 0} } {0!} + \\dfrac { {!1} \\times n^{\\underline 1} } {1!} + \\dfrac { {!2} \\times n^{\\underline 2} } {2!} + \\dfrac { {!3} \\times n^{\\underline 3} } {3!} + \\cdots       | c =  }} {{eqn | r = 1 + \\left({1 - \\dfrac 1 {1 !} }\\right) n + \\left({1 - \\dfrac 1 {1 !} + \\dfrac 1 {2 !} }\\right) n \\left({n - 1}\\right) + \\left({1 - \\dfrac 1 {1 !} + \\dfrac 1 {2 !} - \\dfrac 1 {3 !} }\\right) n \\left({n - 1}\\right) \\left({n - 2}\\right) + \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "15782", "title": "Factorial of Half", "text": ":$\\left({\\dfrac 1 2}\\right)! = \\dfrac {\\sqrt \\pi} 2$"}
+{"_id": "15783", "title": "Laplace Transform of Complex Power", "text": "Let $q$ be a constant complex number with $\\map \\Re q > -1$ Let $t^q: \\R_{>0} \\to \\C$ be a branch of the complex power multifunction chosen such that $f$ is continuous on the half-plane $\\map \\Re s > 0$. Then $f$ has a Laplace transform given by: :$\\laptrans {t^q} = \\dfrac {\\map \\Gamma {q + 1} } {s^{q + 1} }$ where $\\Gamma$ denotes the gamma function."}
+{"_id": "15784", "title": "Local Minimum of Gamma Function on Positive Domain", "text": "The local minimum of the Gamma function on the positive real numbers occurs at the point: :$\\left({1 \\cdotp 46163 21449 68362 34126 26595, 0 \\cdotp 88560 31944 10888 70027 88159}\\right)$ {{OEIS|A030169|order = $x$-coordinate}} {{OEIS|A030171|order = $y$-coordinate}}"}
+{"_id": "15785", "title": "Properties of Falling Factorial", "text": "Let $x^{\\underline n}$ denote the $n$th falling factorial power of $x$. This page gathers together some of the properties of the falling factorial."}
+{"_id": "15786", "title": "Properties of Rising Factorial", "text": "Let $x^{\\underline n}$ denote the $n$th falling factorial power of $x$. This page gathers together some of the properties of the rising factorial."}
+{"_id": "15787", "title": "Integer to Power of Itself Less One Falling is Factorial", "text": ":$n^{\\underline {n - 1} } = n!$"}
+{"_id": "15791", "title": "Recursive Form of Generalized Termial", "text": "The termial function as defined on the real numbers fulfils the identity: :$x? = x + \\left({x - 1}\\right)?$"}
+{"_id": "15792", "title": "Euler Form of Gamma Function at Positive Integers", "text": "The Euler form of the Gamma function: :$\\displaystyle \\Gamma \\left({z}\\right) := \\lim_{m \\mathop \\to \\infty} \\frac {m^z m!} {z \\left({z + 1}\\right) \\left({z + 2}\\right) \\cdots \\left({z + m}\\right)}$ converges to the factorial function at positive integers: :$\\displaystyle \\lim_{m \\mathop \\to \\infty} \\frac {m^n m!} {\\left({n + 1}\\right) \\left({n + 2}\\right) \\cdots \\left({n + m}\\right)} = n!$"}
+{"_id": "15793", "title": "Gamma Function of Minus One Half", "text": ":$\\map \\Gamma {-\\dfrac 1 2} = -2 \\sqrt \\pi$"}
+{"_id": "15795", "title": "Diameter of N-Cube", "text": "Let $Q_n = \\closedint {c - R} {c + R}^n$ be an $n$-cube in Euclidean $n$-Space equipped with the usual metric. Then the diameter of $Q_n$ is given by: :$\\map {\\operatorname {diam} } {Q_n} = 2 R \\sqrt n$"}
+{"_id": "15796", "title": "Legendre's Theorem", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $p$ be a prime number. Let $n$ be expressed in base $p$ representation. Let $r$ be the digit sum of the representation of $n$ in base $p$. Then $n!$ is divisible by $p^\\mu$ but not by $p^{\\mu + 1}$, where: :$\\mu = \\dfrac {n - r} {p - 1}$"}
+{"_id": "15797", "title": "Uniform Convergence of General Dirichlet Series", "text": "Let $\\arg \\left({z}\\right)$ denote the argument of the complex number $z \\in \\C$. Let $\\displaystyle f \\left({s}\\right) = \\sum_{n \\mathop = 1}^\\infty a_n e^{-\\lambda_n \\left({s}\\right)}$ be a general Dirichlet series. Let $f \\left({s}\\right)$ converge at $s_0 = \\sigma_0 + i t_0$. Then $f \\left({s}\\right)$ converges uniformly for all $s$ such that: :$\\left\\vert{\\arg \\left({s - s_0}\\right)}\\right\\vert \\le a < \\dfrac \\pi 2$"}
+{"_id": "15798", "title": "Factorial as Sum of Series of Subfactorial by Falling Factorial over Factorial/Condition for Convergence", "text": "Consider the series: {{:Factorial as Sum of Series of Subfactorial by Falling Factorial over Factorial}} This converges only when $n \\in \\Z_{\\ge 0}$, that is, when $n$ is a non-negative integer."}
+{"_id": "15799", "title": "Bounded Subspace of Euclidean Space is Totally Bounded", "text": "Let $\\struct {\\R^n, \\norm {\\, \\cdot \\,} }$ be a Euclidean space, where $\\norm {\\, \\cdot \\,}$ denotes the usual metric. Let $M$ be a metric subspace of $\\struct {\\R^n, \\norm {\\, \\cdot \\,} }$. Let $M$ be bounded. Then $M$ is totally bounded."}
+{"_id": "15800", "title": "Difference is Rational is Equivalence Relation", "text": "Define $\\sim$ as the relation on real numbers given by: :$x \\sim y \\iff x - y \\in \\Q$ That is, that the difference between $x$ and $y$ is rational. Then $\\sim$ is an equivalence relation."}
+{"_id": "15801", "title": "Infinite Product of Product of Sequence of n plus alpha over Sequence of n plus beta", "text": ":$\\displaystyle \\prod_{n \\mathop \\ge 1} \\dfrac {\\left({n + \\alpha_1}\\right) \\cdots \\left({n + \\alpha_k}\\right)} {\\left({n + \\beta_1}\\right) \\cdots \\left({n + \\beta_k}\\right)} = \\dfrac {\\Gamma \\left({1 + \\beta_1}\\right) \\cdots\\Gamma \\left({1 + \\beta_1}\\right)} {\\Gamma \\left({1 + \\alpha_1}\\right) \\cdots\\Gamma \\left({1 + \\alpha_k}\\right)}$ where: :$\\alpha_1 + \\cdots + \\alpha_k = \\beta_1 + \\cdots + \\beta_k$ :none of the $\\beta$s is a negative integer."}
+{"_id": "15805", "title": "Characterization of N-Cube", "text": "Let $\\struct {\\R^n, d}$ be a Euclidean $n$-Space equipped with the usual metric $d$. Let $x, y \\in \\R^n$, where: :$x = \\tuple {x_1, x_2, \\ldots, x_n}$ :$y = \\tuple {y_1, y_2, \\ldots, y_n}$ Let $R > 0$ be fixed. Let: :$\\displaystyle Q = \\set {x, y \\in \\R^n: \\sup_{x, y} \\max_i \\size {y_i - x_i} \\le R}$ Then $Q$ is an $n$-cube."}
+{"_id": "15806", "title": "Factorial of Integer plus Reciprocal of Integer", "text": "Let $x \\in \\Z$ be a positive integer. Then: :$\\ds \\lim_{n \\mathop \\to \\infty} \\dfrac {\\paren {n + x}!} {n! n^x} = 1$"}
+{"_id": "15807", "title": "Lower and Upper Bound of Factorial", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: :$\\dfrac {n^n} {e^{n - 1} } \\le n! \\le \\dfrac {n^{n + 1} } {e^{n - 1} }$"}
+{"_id": "15808", "title": "Sum of Indices of Falling Factorial", "text": ":$x^{\\underline {m + n} } = x^{\\underline m} \\left({x - m}\\right)^{\\underline n}$"}
+{"_id": "15809", "title": "Sum of Indices of Rising Factorial", "text": ":$x^{\\overline {m + n} } = x^{\\overline m} \\paren {x + m}^{\\overline n}$"}
+{"_id": "15810", "title": "Union of Indexed Family of Sets Equal to Union of Disjoint Sets", "text": "Let $\\family {E_n}_{n \\mathop \\in \\N}$ be a countable indexed family of sets where at least two $E_n$ are distinct. Then there exists a countable indexed family of disjoint sets $\\family {F_n}_{n \\mathop \\in \\N}$ defined by: :$\\displaystyle F_k = E_k \\setminus \\paren {\\bigcup_{j \\mathop = 0}^{k \\mathop - 1} E_j}$ satisfying: :$\\displaystyle \\bigsqcup_{n \\mathop \\in \\N} F_n = \\bigcup_{n \\mathop \\in \\N} E_n$ where $\\bigsqcup$ denotes disjoint union."}
+{"_id": "15811", "title": "Borel Sigma-Algebra Generated by Closed Sets", "text": "Let $\\mathcal B \\left({S, \\tau}\\right)$ be a Borel $\\sigma$-algebra generated by the set of open sets in $S$. Then $\\mathcal B \\left({S, \\tau}\\right)$ is equivalently generated by the set of closed sets in $S$."}
+{"_id": "15813", "title": "Stirling Number of n with n-m is Polynomial in n of Degree 2m/Unsigned First Kind", "text": "Let $m \\in \\Z_{\\ge 0}$. The unsigned Stirling number of the first kind $\\displaystyle \\left[{n \\atop n - m}\\right]$ is a polynomial in $n$ of degree $2 m$."}
+{"_id": "15814", "title": "Unsigned Stirling Number of the First Kind of n with n-2", "text": ":$\\displaystyle \\left[{n \\atop n - 2}\\right] = \\binom n 4 + 2 \\binom {n + 1} 4$"}
+{"_id": "15815", "title": "Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum", "text": "Let $\\MM$ be an infinite $\\sigma$-algebra on a set $X$. Then $\\MM$ is has cardinality at least that of the cardinality of the continuum $\\mathfrak c$: :$\\map \\Card \\MM \\ge \\mathfrak c$"}
+{"_id": "15816", "title": "Stirling Number of the Second Kind of n with n-2", "text": ":$\\displaystyle {n \\brace n - 2} = \\binom {n + 1} 4 + 2 \\binom n 4$"}
+{"_id": "15817", "title": "Unsigned Stirling Number of the First Kind of n with n-3", "text": ":$\\displaystyle \\left[{n \\atop n - 3}\\right] = \\binom n 6 + 8 \\binom {n + 1} 6 + 6 \\binom {n + 2} 6$"}
+{"_id": "15818", "title": "Stirling Number of the Second Kind of n with n-3", "text": ":$\\displaystyle {n \\brace n - 3} = \\binom {n + 2} 6 + 8 \\binom {n + 1} 6 + 6 \\binom n 6$"}
+{"_id": "15819", "title": "Duality Law for Stirling Numbers", "text": "For all integers $n, m \\in \\Z$: :$\\displaystyle \\left\\{ {n \\atop m}\\right\\} = \\left[{-m \\atop -n}\\right]$ where: :$\\displaystyle \\left\\{ {n \\atop m}\\right\\}$ denotes a Stirling number of the second kind :$\\displaystyle \\left[{n \\atop m}\\right]$ denotes an unsigned Stirling number of the first kind."}
+{"_id": "15820", "title": "Power of Complex Number as Summation of Stirling Numbers of Second Kind", "text": "Let $z \\in \\C$ be a complex number whose real part is positive. Then: :$z^r = \\displaystyle \\sum_{k \\mathop \\in \\Z} {r \\brace r - k} z^{\\underline {r - k} }$ where: :$\\displaystyle {r \\brace r - k}$ denotes the extension of the Stirling numbers of the second kind to the complex plane :$z^{\\underline {r - k} }$ denotes $z$ to the $r - k$ falling."}
+{"_id": "15822", "title": "Non-Divisbility of Binomial Coefficients of n by Prime", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Let $p$ be a prime number. Then: :$\\dbinom n k$ is not divisible by $p$ for any $k \\in \\Z_{\\ge 0}$ where $0 \\le k \\le n$ {{iff}}: :$n = a p^m - 1$ where $1 \\le a < p$ for some $m \\in \\Z_{\\ge 0}$."}
+{"_id": "15828", "title": "Summation Formula for Reciprocal of Binomial Coefficient", "text": "{{begin-eqn}} {{eqn | l = \\sum_{k \\mathop \\ge 0} \\binom n k \\dfrac {\\paren {-1}^k} {k + x}       | r = \\dfrac 1 {x \\binom {n + x} n}       | c =  }} {{eqn | r = \\dfrac {n!} {x \\paren {x + 1} \\cdots \\paren {x + n} }       | c =  }} {{end-eqn}} as long as the denominators are not zero."}
+{"_id": "15830", "title": "Sum over k of r Choose k by Minus r Choose m Minus 2k", "text": "Let $r \\in \\R$, $m \\in \\Z$. :$\\displaystyle \\sum_{k \\mathop \\in \\Z} \\binom r k \\binom {-r} {m - 2 k} \\paren {-1}^{m + k} = \\binom r m$"}
+{"_id": "15831", "title": "Sigma-Algebra Contains Generated Sigma-Algebra of Subset", "text": "Let  $\\sigma_\\FF$ be a  be a $\\sigma$-algebra on a set $\\FF$. Let $\\sigma_\\FF$ contain a set of sets $\\EE$. Let $\\map \\sigma \\EE$ be the $\\sigma$-algebra generated by $\\EE$. Then $\\map \\sigma \\EE \\subseteq \\sigma_\\FF$"}
+{"_id": "15832", "title": "Generated Sigma-Algebra Contains Generated Sigma-Algebra of Subset", "text": "Let $\\sigma\\left({\\mathcal F}\\right)$ be the $\\sigma$-algebra generated by $\\mathcal E$. Let $\\sigma\\left({\\mathcal F}\\right)$ contain a set of sets $\\mathcal E$. Let $\\sigma \\left({\\mathcal E}\\right)$ be the $\\sigma$-algebra generated by $\\mathcal E$. Then $\\sigma \\left({\\mathcal E}\\right) \\subseteq \\sigma\\left({\\mathcal F}\\right)$."}
+{"_id": "15833", "title": "Binomial Theorem/Abel's Generalisation/x+y = 0", "text": "Consider Abel's Generalisation of Binomial Theorem: {{:Abel's Generalisation of Binomial Theorem}} This holds in the special case where $x + y = 0$."}
+{"_id": "15837", "title": "Binomial Theorem/Hurwitz's Generalisation", "text": ":$\\displaystyle \\paren {x + y}^n = \\sum x \\paren {x + \\epsilon_1 z_1 + \\cdots + \\epsilon_n z_n}^{\\epsilon_1 + \\cdots + \\epsilon_n - 1} \\paren {y - \\epsilon_1 z_1 - \\cdots - \\epsilon_n z_n}^{n - \\epsilon_1 - \\cdots - \\epsilon_n}$ where the summation ranges over all $2^n$ choices of $\\epsilon_1, \\ldots, \\epsilon_n = 0$ or $1$ independently."}
+{"_id": "15838", "title": "Set of Relations can be Ordered by Inclusion", "text": "Let $S \\times T$ be the product of two sets. Let $\\mathcal R$ be a set of relations on $S \\times T$. Then $\\mathcal R$ can be ordered by inclusion."}
+{"_id": "15839", "title": "Set of Mappings can be Ordered by Inclusion", "text": "Let $S \\times T$ be the product of two sets. Let $\\mathcal F$ be a set of mappings on $S \\times T$. Then $\\mathcal F$ can be ordered by inclusion."}
+{"_id": "15841", "title": "Binomial Theorem/Abel's Generalisation/Negative n", "text": "Abel's Generalisation of Binomial Theorem: {{:Abel's Generalisation of Binomial Theorem}} does not hold for $n \\in \\Z_{< 0}$."}
+{"_id": "15842", "title": "Summation from k to m of r Choose k by s Choose n-k by nr-(r+s)k", "text": ":$\\displaystyle \\sum_{k \\mathop = 0}^m \\dbinom r k \\dbinom s {n - k} \\paren {n r - \\paren {r + s} k} = \\paren {m + 1} \\paren {n - m} \\dbinom r {m + 1} \\dbinom s {n - m}$"}
+{"_id": "15843", "title": "Factors of Binomial Coefficient/Corollary 2", "text": "For all $r \\in \\R, k \\in \\Z$: :$\\dbinom r {k - 1} = k \\paren {r - k} \\dbinom r k$"}
+{"_id": "15844", "title": "Summation from k to m of 2k-1 Choose k by 2n-2k Choose n-k by -1 over 2k-1", "text": ":$\\displaystyle \\sum_{k \\mathop = 0}^m \\binom {2 k - 1} k \\binom {2 n - 2 k} {n - k} \\dfrac {-1} {2 k - 1} = \\dfrac {n - m} {2 n} \\dbinom {2 m} m \\dbinom {2 n - 2 m} {n - m} + \\dfrac 1 2 \\dbinom {2 n} n$"}
+{"_id": "15845", "title": "Inverse of Pascal's Triangle expressed as Matrix", "text": "Consider Pascal's triangle expressed as a (square) matrix $\\mathbf M$, with the top left element holding $\\dbinom 0 0$. :$\\begin{pmatrix} 1 &  0 &  0 &   0 &   0 &   0 &   0 &   0 &   0 & \\cdots \\\\ 1 &  1 &  0 &   0 &   0 &   0 &   0 &   0 &   0 & \\cdots \\\\ 1 &  2 &  1 &   0 &   0 &   0 &   0 &   0 &   0 & \\cdots \\\\ 1 &  3 &  3 &   1 &   0 &   0 &   0 &   0 &   0 & \\cdots \\\\ 1 &  4 &  6 &   4 &   1 &   0 &   0 &   0 &   0 & \\cdots \\\\ 1 &  5 & 10 &  10 &   5 &   1 &   0 &   0 &   0 & \\cdots \\\\ 1 &  6 & 15 &  20 &  15 &   6 &   1 &   0 &   0 & \\cdots \\\\ 1 &  7 & 21 &  35 &  35 &  21 &   7 &   1 &   0 & \\cdots \\\\ 1 &  8 & 28 &  56 &  70 &  56 &  28 &   8 &   1 & \\cdots \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\\\ \\end{pmatrix}$ The inverse $\\mathbf M^{-1}$ of $\\mathbf M$ is the same as $\\mathbf M$ but with alternate elements negated, starting with the elements below the main diagonal: :$\\begin{pmatrix}  1 &  0 &   0 &   0 &   0 &   0 &  0 &  0 & 0 & \\cdots \\\\ -1 &  1 &   0 &   0 &   0 &   0 &  0 &  0 & 0 & \\cdots \\\\  1 & -2 &   1 &   0 &   0 &   0 &  0 &  0 & 0 & \\cdots \\\\ -1 &  3 &  -3 &   1 &   0 &   0 &  0 &  0 & 0 & \\cdots \\\\  1 & -4 &   6 &  -4 &   1 &   0 &  0 &  0 & 0 & \\cdots \\\\ -1 &  5 & -10 &  10 &  -5 &   1 &  0 &  0 & 0 & \\cdots \\\\  1 & -6 &  15 & -20 &  15 &  -6 &  1 &  0 & 0 & \\cdots \\\\ -1 &  7 & -21 &  35 & -35 &  21 & -7 &  1 & 0 & \\cdots \\\\  1 & -8 &  28 & -56 &  70 & -56 & 28 & -8 & 1 & \\cdots \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\\\ \\end{pmatrix}$"}
+{"_id": "15846", "title": "Binomial Coefficient of Half", "text": "Let $k \\in \\Z$. :$\\dbinom {\\frac 1 2} k = \\dfrac {\\left({-1}\\right)^{k - 1} } {4^k \\left({2 k - 1}\\right)} \\dbinom {2 k} k$ where $\\dbinom {\\frac 1 2} k$ denotes a binomial coefficient."}
+{"_id": "15847", "title": "Binomial Coefficient of Half/Corollary", "text": "Let $k \\in \\Z_{\\ge 0}$. :$\\dbinom {\\frac 1 2} k = \\dfrac {\\left({-1}\\right)^{k - 1} } {2^{2 k - 1} \\left({2 k - 1}\\right)} \\dbinom {2 k - 1} k - \\delta_{k 0}$ where $\\dbinom {\\frac 1 2} k$ denotes a binomial coefficient."}
+{"_id": "15849", "title": "Wosets are Isomorphic to Each Other or Initial Segments", "text": "Let $\\struct {S, \\preceq_S}$ and $\\struct {T, \\preceq_T}$ be well-ordered sets. Then precisely one of the following hold: :$\\struct {S, \\preceq_S}$ is order isomorphic to $\\struct {T, \\preceq_T}$ or: :$\\struct {S, \\preceq_S}$ is order isomorphic to an initial segment in $\\struct {T, \\preceq_T}$ or: :$\\struct {T, \\preceq_T}$ is order isomorphic to an initial segment in $\\struct {S, \\preceq_S}$"}
+{"_id": "15852", "title": "Gaussian Binomial Theorem", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. {{begin-eqn}} {{eqn | l = \\sum_{k \\mathop \\in \\Z} \\dbinom n k_q q^{k \\paren {k - 1} / 2} x^k       | r = \\prod_{k \\mathop = 1}^n \\paren {1 + q^{k - 1} x}       | c =  }} {{eqn | r = \\paren {1 + x} \\paren {1 + q x} \\paren {1 + q^2 x} \\cdots \\paren {1 + q^{n - 1} x}       | c =  }} {{end-eqn}} where $\\dbinom n k_q$ denotes a Gaussian binomial coefficient."}
+{"_id": "15853", "title": "Addition Rule for Gaussian Binomial Coefficients/Formulation 1", "text": ":$\\dbinom n m_q = \\dbinom {n - 1} m_q + \\dbinom {n - 1} {m - 1}_q q^{n - m}$"}
+{"_id": "15854", "title": "Addition Rule for Gaussian Binomial Coefficients", "text": "Let $\\dbinom n m_q$ denote a Gaussian binomial coefficient."}
+{"_id": "15856", "title": "Union of Initial Segments is Initial Segment or All of Woset", "text": "Let $\\struct {X, \\preccurlyeq}$ be a well-ordered non-empty set. Let $A \\subseteq X$. Let: :$\\ds J = \\bigcup_{x \\mathop \\in A} S_x$ be a union of initial segments defined by the elements of $A$. Then either: :$J = X$ or: :$J$ is an initial segment of $X$."}
+{"_id": "15857", "title": "Countable Subset of Minimal Uncountable Well-Ordered Set Has Upper Bound", "text": "Let $\\Omega$ denote the minimal uncountable well-ordered set. Let $\\omega$ be a countable subset of $\\Omega$. Then $\\omega$ has an upper bound in $\\Omega$."}
+{"_id": "15858", "title": "Gaussian Binomial Coefficient of 1", "text": ":$\\dbinom 1 m_q = \\delta_{0 m} + \\delta_{1 m}$ That is: :$\\dbinom 1 m_q = \\begin{cases} 1 & : m = 0 \\text { or } m = 1 \\\\ 0 & : \\text{otherwise} \\end{cases}$ where $\\dbinom 1 m_q$ denotes a Gaussian binomial coefficient."}
+{"_id": "15860", "title": "Symmetry Rule for Gaussian Binomial Coefficients", "text": "Let $q \\in \\R_{\\ne 1}, n \\in \\Z_{>0}, k \\in \\Z$. Then: :$\\dbinom n k_q = \\dbinom n {n - k}_q$ where $\\dbinom n k_q$ is a Gaussian binomial coefficient."}
+{"_id": "15862", "title": "Gaussian Binomial Theorem/Real Numbers", "text": "Let $r \\in \\R$ be a real number. :$\\displaystyle \\sum_{k \\mathop \\in \\Z} \\dbinom r k_q q^{k \\left({k - 1}\\right) / 2} x^k = \\prod_{k \\mathop \\ge 0} \\dfrac {1 + q^k x} {1 + q^{r + k} x}$ where: :$\\dbinom r k_q$ denotes a Gaussian binomial coefficient. :$x \\in \\R: \\left\\lvert{x}\\right\\rvert < 1$ :$q \\in \\R: \\left\\lvert{q}\\right\\rvert < 1$."}
+{"_id": "15863", "title": "Gaussian Binomial Theorem/Negation of Upper Index", "text": "Let $r \\in \\R$ be a real number. {{begin-eqn}} {{eqn | l = \\prod_{k \\mathop \\ge 0} \\dfrac {1 + q^{k + r + 1} x} {1 + q^k x}       | r = \\sum_{k \\mathop \\in \\Z} \\dbinom {-r - 1} k_q q^{k \\left({k - 1}\\right) / 2} \\left({-q^{r + 1} x}\\right)^k       | c =  }} {{eqn | r = \\sum_{k \\mathop \\in \\Z} \\dbinom {k + r} x^k       | c =  }} {{end-eqn}} where: :$\\dbinom r k_q$ denotes a Gaussian binomial coefficient. :$x \\in \\R: \\left\\lvert{x}\\right\\rvert < 1$ :$q \\in \\R: \\left\\lvert{q}\\right\\rvert < 1$."}
+{"_id": "15865", "title": "Cardinality of Set of Combinations with Repetition", "text": "Let $S$ be a finite set with $n$ elements The number of $k$-combinations of $S$ with repetition is given by: :$N = \\dbinom {n + k - 1} k$"}
+{"_id": "15866", "title": "Sum over k of Unsigned Stirling Number of the First Kind of n+1 with k+1 by Stirling Number of the Second Kind of k with m by -1^k-m", "text": "Let $m, n \\in \\Z_{\\ge 0}$. :$\\displaystyle \\sum_k {n + 1 \\brack k + 1} {k \\brace m} \\paren {-1}^{k - m} = \\sqbrk {n \\ge m} \\dfrac {n!} {m!}$ where: :$\\sqbrk {n \\ge m}$ is Iverson's convention :$\\displaystyle {n + 1 \\brack k + 1}$ denotes an unsigned Stirling number of the first kind :$\\displaystyle {k \\brace m}$ denotes a Stirling number of the second kind."}
+{"_id": "15867", "title": "Dixon's Identity/General Case", "text": "For $l, m, n \\in \\Z_{\\ge 0}$: :$\\displaystyle \\sum_{k \\mathop \\in \\Z} \\paren {-1}^k \\dbinom {l + m} {l + k} \\dbinom {m + n} {m + k} \\dbinom {n + l} {n + k} = \\dfrac {\\paren {l + m + n}!} {l! \\, m! \\, n!}$"}
+{"_id": "15868", "title": "Dixon's Identity", "text": "For $n \\in \\Z_{\\ge 0}$: :$\\displaystyle \\sum_{k \\mathop \\in \\Z} \\paren {-1}^k \\binom {2 n} {n + k}^3 = \\dfrac {\\paren {3 n}!} {\\paren {n!}^3}$"}
+{"_id": "15869", "title": "Dixon's Identity/Gaussian Binomial Form", "text": "=== Formulation 1 === {{:Dixon's Identity/Gaussian Binomial Form/Formulation 1}} === Formulation 2 === {{:Dixon's Identity/Gaussian Binomial Form/Formulation 2}}"}
+{"_id": "15870", "title": "Dixon's Identity/Gaussian Binomial Form/Formulation 1", "text": "For $l, m, n \\in \\Z_{\\ge 0}$: :$\\displaystyle \\sum_{k \\mathop \\in \\Z} \\paren {-1}^k \\dbinom {m - r - s} k_q \\dbinom {n + r - s} {n - k}_q \\dbinom {r + k} {m + n}_q = \\dbinom r m_q \\dbinom s n_q$ where $\\dbinom r m_q$ denotes a Gaussian binomial coefficient"}
+{"_id": "15871", "title": "Sum over j, k of -1^j+k by j+j Choose k+l by r Choose j by n Choose k by s+n-j-k Choose m-j", "text": "Let $l, m, n \\in \\Z$ be integers such that $n \\ge 0$. Then: :$\\displaystyle \\sum_{j, k \\mathop \\in \\Z} \\left({-1}\\right)^{j + k} \\dbinom {j + k} {k + l} \\dbinom r j \\dbinom n k \\dbinom {s + n - j - k} {m - j} = \\left({-1}\\right)^l \\dbinom {n + r} {n + l} \\dbinom {s - r} {m - n - l}$"}
+{"_id": "15874", "title": "Analytic Continuation of Generating Function of Dirichlet Series", "text": "Let $\\displaystyle \\map \\lambda s = \\sum_{n \\mathop = 1}^\\infty \\frac {a_n} {n^s}$ be a Dirichlet series  Let $c \\in \\R$ be greater than the abscissa of absolute convergence of $\\lambda$ and greater than $0$. Let $\\displaystyle \\map g z = \\sum_{k \\mathop = 1}^\\infty \\map \\lambda {c k} z^k $ be the generating function of $\\map \\lambda {c k}$ Then the analytic continuation of $g$ to $\\C$ is equal to  :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n  \\frac z {n^c - z}$"}
+{"_id": "15875", "title": "Uniqueness of Real z such that x Choose n+1 Equals y Choose n+1 Plus z Choose n", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Let $x, y \\in \\R$ be real numbers which satisfy: :$n \\le y \\le x \\le y + 1$ Then there exists a unique real number $z$ such that: :$\\dbinom x {n + 1} = \\dbinom y {n + 1} + \\dbinom z n$ where $n - 1 \\le z \\le y$."}
+{"_id": "15876", "title": "X Choose n leq y Choose n + z Choose n-1 where n leq y leq x leq y+1 and n-1 leq z leq y", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Let $x, y \\in \\R$ be real numbers which satisfy: :$n \\le y \\le x \\le y + 1$ Let $z$ be the unique real number $z$ such that: :$\\dbinom x {n + 1} = \\dbinom y {n + 1} + \\dbinom z n$ where $n - 1 \\le z \\le y$. Its uniqueness is proved at Uniqueness of Real $z$ such that $\\dbinom x {n + 1} = \\dbinom y {n + 1} + \\dbinom z n$. Then: :$\\dbinom x n \\le \\dbinom y n + \\dbinom z {n - 1}$"}
+{"_id": "15877", "title": "Upper Bound for Binomial Coefficient", "text": "Let $n, k \\in \\Z$ such that $n \\ge k \\ge 0$. Then: :$\\dbinom n k \\le \\left({\\dfrac {n e} k}\\right)^k$ where $\\dbinom n k$ denotes a binomial coefficient."}
+{"_id": "15878", "title": "Lower Bound for Binomial Coefficient", "text": "Let $n, k \\in \\Z$ such that $n \\ge k \\ge 0$. Then: :$\\dbinom n k \\ge \\left({\\dfrac {\\left({n - k}\\right) e} k}\\right)^k \\dfrac 1 {e k}$ where $\\dbinom n k$ denotes a binomial coefficient."}
+{"_id": "15881", "title": "Sum over k of n Choose k by p^k by (1-p)^n-k by Absolute Value of k-np", "text": "Let $n \\in \\Z_{\\ge 0}$ be a non-negative integer. Then: :$\\displaystyle \\sum_{k \\mathop \\in \\Z} \\dbinom n k p^k \\left({1 - p}\\right)^{n - k} \\left\\lvert{k - n p}\\right\\rvert = 2 \\left\\lceil{n p}\\right\\rceil \\dbinom n {\\left\\lceil{n p}\\right\\rceil} p^{\\left\\lceil{n p}\\right\\rceil} \\left({1 - p}\\right)^{n - 1 - \\left\\lceil{n p}\\right\\rceil}$"}
+{"_id": "15882", "title": "Sequence of General Harmonic Numbers Converges for Index Greater than 1", "text": "Let $H_n^{\\paren r}$ denote the general harmonic number: :$\\displaystyle H_n^{\\paren r} = \\sum_{k \\mathop = 1}^n \\frac 1 {k^r}$ for $r \\in \\R_{>0}$. Let $r > 1$. Then as $n \\to \\infty$, $H_n^{\\paren r}$ is convergent with an upper bound of $\\dfrac {2^{r - 1} } {2^{r - 1} - 1}$."}
+{"_id": "15884", "title": "Generating Function of Bernoulli Polynomials", "text": "Let $\\map {B_n} x$ denote the $n$th Bernoulli polynomial. Then the generating function for $B_n$ is: :$\\displaystyle \\frac {t e^{t x} } {e^t - 1} = \\sum_{k \\mathop = 0}^\\infty \\frac {\\map {B_k} x} {k!} t^k$"}
+{"_id": "15885", "title": "Symmetry of Bernoulli Polynomial", "text": "Let $\\map {B_n} x$ denote the nth Bernoulli polynomial. Then: :$\\map {B_n} {1 - x} = \\paren {-1}^n \\map {B_n} x$"}
+{"_id": "15886", "title": "Value of Odd Bernoulli Polynomial at One Half", "text": "Let $B_n \\left({x}\\right)$ denote the $n$th Bernoulli polynomial. Then: : $B_{2 n + 1} \\left({\\dfrac 1 2}\\right) = 0$"}
+{"_id": "15887", "title": "Harmonic Number is Greater than Logarithm plus Gamma", "text": ":$H_n > \\ln n + \\gamma$ where: :$H_n$ denotes the $n$th harmonic number :$\\gamma$ denotes the Euler-Mascheroni constant."}
+{"_id": "15888", "title": "Summation to n of Power of k over k", "text": ":$\\displaystyle \\sum_{k \\mathop = 1}^n \\dfrac {x^k} k = H_n + \\displaystyle \\sum_{k \\mathop = 1}^n \\dbinom n k \\dfrac {\\left({x - 1}\\right)^k} k$ where: :$H_n$ denotes the $n$th harmonic number :$\\dbinom n k$ denotes a binomial coefficient."}
+{"_id": "15889", "title": "Harmonic Number as Unsigned Stirling Number of First Kind over Factorial", "text": ":$H_n = \\dfrac {\\left[{ {n + 1} \\atop 2}\\right]} {n!}$ where: :$H_n$ denotes the $n$th harmonic number :$n!$ denotes the $n$th factorial :$\\displaystyle \\left[{ {n + 1} \\atop 2}\\right]$ denotes an unsigned Stirling number of the first kind."}
+{"_id": "15893", "title": "Summation over k to n of Natural Logarithm of k", "text": ":$\\ds \\sum_{k \\mathop = 1}^n \\ln k = \\map \\ln {n!}$ where $n!$ denotes the $n$th factorial."}
+{"_id": "15894", "title": "Difference between Summation of Natural Logarithms and Summation of Harmonic Numbers", "text": ":$\\displaystyle \\sum_{k \\mathop = 1}^n H_k - \\sum_{k \\mathop = 1}^n \\map \\ln {n!} \\approx \\gamma n + \\dfrac {\\ln n} 2 + 0 \\cdotp 158$ where: :$H_k$ denotes the $k$th harmonic number :$n!$ denotes the $n$th factorial :$\\gamma$ denotes the Euler-Mascheroni constant."}
+{"_id": "15903", "title": "Summation to n of kth Harmonic Number over k", "text": ":$\\displaystyle \\sum_{k \\mathop = 1}^n \\dfrac {H_k} k = \\dfrac { {H_n}^2 + H_n^{\\paren 2} } 2$ where: :$H_n$ denotes the $n$th harmonic number :$H_n^{\\paren 2}$ denotes a general harmonic number."}
+{"_id": "15905", "title": "Principle of Recursive Definition for Well-Ordered Sets", "text": "Let $J$ be a well-ordered set. Let $C$ be any set. Let $\\mathcal F$ be the set of all functions that map initial segments $S_a$ of $J$ into $C$. Then for any function of the form: :$\\rho: \\mathcal F \\to C$ there exists a unique function: :$h: J \\to C$ satisfying: :$\\forall \\alpha \\in J: h\\left({\\alpha}\\right) = \\rho\\left({ h {\\restriction_{S_\\alpha}} }\\right)$ where ${\\restriction}$ denotes the restriction of a mapping."}
+{"_id": "15906", "title": "Angles with Parallel or Perpendicular Arms are Equal or Supplementary", "text": "Let $A$ and $B$ be angles such that their corresponding arms are either parallel or perpendicular. Then $A$ and $B$ are either equal or supplementary."}
+{"_id": "15907", "title": "Sum of External Angles of Polygon equals Four Right Angles", "text": "Let the external angles of a polygon be generated in the same direction going around the polygon. Then the sum of all these external angles equals $4$ right angles."}
+{"_id": "15908", "title": "Bisector of Apex of Isosceles Triangle also Bisects Base", "text": "Let $\\triangle ABC$ be an isosceles triangle whose apex is $A$. Let $AD$ be the bisector of $\\angle BAC$ such that $AD$ intersects $BC$ at $D$. Then $AD$ bisects $BC$."}
+{"_id": "15909", "title": "Bisector of Apex of Isosceles Triangle is Perpendicular to Base", "text": "Let $\\triangle ABC$ be an isosceles triangle whose apex is $A$. Let $AD$ be the bisector of $\\angle BAC$ such that $AD$ intersects $BC$ at $D$. Then $AD$ is perpendicular to $BC$."}
+{"_id": "15910", "title": "Equilateral Triangle is Equiangular", "text": "Let $\\triangle ABC$ be an equilateral triangle. Then $\\triangle ABC$ is also equiangular."}
+{"_id": "15911", "title": "Equiangular Triangle is Equilateral", "text": "Let $\\triangle ABC$ be equiangular. Then $\\triangle ABC$ is an equilateral triangle."}
+{"_id": "15912", "title": "No Order Isomophism Between Distinct Initial Segments of Woset", "text": "Let $E$ be a well-ordered set. Let $S_\\alpha, S_\\beta$ be initial segments of $E$ that are order isomorphic. Then $S_\\alpha = S_\\beta$."}
+{"_id": "15913", "title": "Perpendicular is Shortest Straight Line from Point to Straight Line", "text": "Let $AB$ be a straight line. Let $C$ be a point which is not on $AB$. Let $D$ be a point on $AB$ such that $CD$ is perpendicular to $AB$. Then the length of $CD$ is less than the length of all other line segments that can be drawn from $C$ to $AB$."}
+{"_id": "15914", "title": "Straight Lines which make Equal Angles with Perpendicular to Straight Line are Equal", "text": "Let $AB$ be a straight line. Let $C$ be a point which is not on $AB$. Let $D$ be a point on $AB$ such that $CD$ is perpendicular to $AB$. Let $E, F$ be points on $AB$ such that $\\angle DCE = \\angle DCF$. Then $CE = CF$."}
+{"_id": "15915", "title": "Straight Line making Larger Angle with Perpendicular to Straight Line is Longer", "text": "Let $AB$ be a straight line. Let $C$ be a point which is not on $AB$. Let $D$ be a point on $AB$ such that $CD$ is perpendicular to $AB$. Let $E, F$ be points on $AB$ such that $\\angle DCE > \\angle DCF$. Then $CE > CF$."}
+{"_id": "15916", "title": "Two Equal Straight Lines can be Constructed from Point to Straight Line", "text": "Let $AB$ be a straight line. Let $C$ be a point which is not on $AB$. Then exactly $2$ straight lines $CD$ and $CE$ can be drawn such that $CD = CE$ and $D, E$ on $AB$."}
+{"_id": "15918", "title": "Diagonals of Rhombus Bisect Each Other at Right Angles", "text": "Let $ABCD$ be a rhombus. The diagonals $AC$ and $BD$ of $ABCD$ bisect each other at right angles."}
+{"_id": "15919", "title": "Quadrilateral is Parallelogram iff One Pair of Opposite Sides is Equal and Parallel", "text": "Let $ABCD$ be a quadrilateral. Then: :$ABCD$ is a parallelogram {{iff}}: :$AB = CD$ and $AB \\parallel CD$ where $AB \\parallel CD$ denotes that $AB$ is parallel to $CD$."}
+{"_id": "15920", "title": "Quadrilateral is Parallelogram iff Both Pairs of Opposite Sides are Equal or Parallel", "text": "Let $ABCD$ be a quadrilateral. Then: :$ABCD$ is a parallelogram {{iff}}: :either $AB = CD$ and $AD = BC$ :or $AB \\parallel CD$ and $AD \\parallel BC$ where $AB \\parallel CD$ denotes that $AB$ is parallel to $CD$."}
+{"_id": "15921", "title": "Quadrilateral is Parallelogram iff Both Pairs of Opposite Angles are Equal", "text": "Let $ABCD$ be a quadrilateral. Then: :$ABCD$ is a parallelogram {{iff}}: :$\\angle ABC = \\angle ADC$ and $\\angle BAD = \\angle BCD$."}
+{"_id": "15922", "title": "Quadrilateral is Parallelogram iff Diagonals Bisect each other", "text": "Let $ABCD$ be a quadrilateral. Then: : $ABCD$ is a parallelogram {{iff}}: :both: :: $AD$ is a bisector of $BC$ :and: :: $BC$ is a bisector of $AD$."}
+{"_id": "15923", "title": "Parallelograms are Congruent if Two Adjacent Sides and Included Angle are respectively Equal", "text": "Let $ABCD$ and $EFGH$ be parallelograms. Then $ABCD$ and $EFGH$ are congruent if: : $2$ adjacent sides of $ABCD$ are equal to $2$ corresponding adjacent sides of $EFGH$ : the angle between those $2$ adjacent sides on both $ABCD$ and $EFGH$ are equal."}
+{"_id": "15925", "title": "Parallel Lines which intercept Equal Segments on Transversals", "text": "Let $3$ or more parallel lines intersect equal line segments on one transversal. Then those same $3$ or more parallel lines intersect equal line segments on every transversal."}
+{"_id": "15927", "title": "Line Parallel to Side of Triangle which Bisects One Side also Bisects Other Side", "text": "Let $ABC$ be a triangle. Let $DE$ be a straight line parallel to $BC$. Let $DE$ bisect $AB$. Then $DE$ also bisects $AC$. That is, $DE$ is a midline of $\\triangle ABC$. 400px"}
+{"_id": "15928", "title": "Midline Theorem", "text": "The midline of a triangle is parallel to the third side of that triangle and half its length."}
+{"_id": "15929", "title": "Midline and Median of Triangle Bisect Each Other", "text": "Let $\\triangle ABC$ be a triangle. Let $DE$ be the midline of $\\triangle ABC$ which bisects $AB$ and $AC$. Let $AF$ be the median of $ABC$ which bisects $BC$. Then $AF$ and $DE$ bisect each other."}
+{"_id": "15931", "title": "Diagonals of Rectangle are Equal", "text": "The diagonals of a rectangle are equal."}
+{"_id": "15933", "title": "Summation of Odd Reciprocals in terms of Harmonic Numbers", "text": ":$\\displaystyle \\sum_{k \\mathop = 1}^n \\dfrac 1 {2 k - 1} = H_{2 n} - \\dfrac {H_n} 2$ where $H_n$ denotes the $n$th harmonic number."}
+{"_id": "15934", "title": "Numerator of p-1th Harmonic Number is Divisible by Prime p", "text": "Let $p$ be an odd prime. Consider the harmonic number $H_{p - 1}$ expressed in canonical form. The numerator of $H_{p - 1}$ is divisible by $p$."}
+{"_id": "15937", "title": "Numerator of p-1th Harmonic Number is Divisible by p^2 for Prime Greater than 3", "text": "Let $p$ be a prime number such that $p > 3$. Consider the harmonic number $H_{p - 1}$ expressed in canonical form. The numerator of $H_{p - 1}$ is divisible by $p^2$."}
+{"_id": "15939", "title": "Summation of Power Series by Harmonic Sequence", "text": "Consider the power series: :$\\map f x = \\displaystyle \\sum_{k \\mathop \\ge 0} a_k x^k$ Let $\\map f x$ converge for $x = x_0$. Then: :$\\displaystyle \\sum_{k \\mathop \\ge 0} a_k {x_0}^k H_k = \\int_0^1 \\dfrac {\\map f {x_0} - \\map f {x_0 y} } {1 - y} \\rd y$ where $H_n$ denotes the $n$th harmonic number."}
+{"_id": "15940", "title": "Summation over k to n of Harmonic Numbers over n+1-k", "text": ":$\\displaystyle \\sum_{k \\mathop = 1}^n \\dfrac {H_k} {n + 1 - k} = {H_{n + 1} }^2 - H_{n + 1}^{\\paren 2}$ where $H_k$ denotes the $k$th harmonic number."}
+{"_id": "15941", "title": "Summation over k to n of Harmonic Number k by Harmonic Number n-k", "text": ":$\\displaystyle \\sum_{k \\mathop = 1}^n H_k H_{n - k} = \\paren {n + 1} \\paren { {H_n}^2 - H_n^{\\paren 2} } - 2 n \\paren {n_n - 1}$ where $H_k$ denotes the $k$th harmonic number."}
+{"_id": "15942", "title": "Extension of Harmonic Number to Non-Integer Argument", "text": "Let $\\map H x$ be the real function defined as: :$\\map H x = \\gamma + \\dfrac {\\map {\\Gamma'} {x + 1} } {\\map \\Gamma {x + 1} }$ where: :$\\gamma$ denotes the Euler-Mascheroni constant :$\\Gamma$ denotes the gamma function :$\\Gamma'$ denotes the derivative of the gamma function. Then $H$ is an extension of the mapping $H: \\N \\to \\Q$ defined as: :$\\forall n \\in \\N: \\map H n = H_n$ where $H_n$ denotes the $n$th harmonic number."}
+{"_id": "15943", "title": "Initial Segment Determined by Smallest Element is Empty", "text": "Let $\\left({S, \\preceq}\\right)$ be a well-ordered set, where $S$ is non-empty. Let $s_0 = \\min S$, the smallest element of $S$. Then the initial segment determined by $s_0$, $S_{s_0}$, is empty."}
+{"_id": "15946", "title": "Empty Mapping is Injective", "text": "Let $\\nu: \\varnothing \\to T$ be an empty mapping. Then $\\nu$ is an injection."}
+{"_id": "15947", "title": "Empty Mapping to Empty Set is Bijective", "text": "Let $\\nu: \\varnothing \\to \\varnothing$ be an empty mapping. Then $\\nu$ is a bijection."}
+{"_id": "15948", "title": "Upper and Lower Bound of Fibonacci Number", "text": "For all $n \\in \\N_{> 0}$: :$\\phi^{n - 2} \\le F_n \\le \\phi^{n - 1}$ where: :$F_n$ is the $n$th Fibonacci number :$\\phi$ is the golden section: $\\phi = \\dfrac {1 + \\sqrt 5} 2$"}
+{"_id": "15950", "title": "Generating Function for Fibonacci Numbers", "text": "Let $\\map G z$ be the function defined as: :$\\map G z = \\dfrac z {1 - z - z^2}$ Then $\\map G z$ is a generating function for the Fibonacci numbers."}
+{"_id": "15951", "title": "Summation over k to n of Product of kth with n-kth Fibonacci Numbers", "text": ":$\\displaystyle \\sum_{k \\mathop = 0}^n F_k F_{n - k} = \\dfrac {\\left({n - 1}\\right) F_n + 2n F_{n - 1} } 5$ where $F_n$ denotes the $n$th Fibonacci number."}
+{"_id": "15955", "title": "Fibonacci Number in terms of Smaller Fibonacci Numbers/Negative Indices", "text": "Let $n \\in \\Z_{< 0}$ be a negative integer. Let $F_n$ be the $n$th Fibonacci number (as extended to negative integers). Then Fibonacci Number in terms of Smaller Fibonacci Numbers: :$F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$ continues to hold, whether $m$ or $n$ are positive or negative."}
+{"_id": "15956", "title": "Euler-Binet Formula/Negative Index", "text": "Let $n \\in \\Z_{< 0}$ be a negative integer. Let $F_n$ be the $n$th Fibonacci number (as extended to negative integers). Then the Euler-Binet Formula: :$F_n = \\dfrac {\\phi^n - \\hat \\phi^n} {\\sqrt 5}$ continues to hold."}
+{"_id": "15957", "title": "Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less", "text": "Let $n \\in \\Z$. Then: :$\\phi^n = F_n \\phi + F_{n - 1}$ where: :$F_n$ denotes the $n$th Fibonacci number :$\\phi$ denotes the golden mean."}
+{"_id": "15958", "title": "Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less/Positive Index", "text": "Let $n \\in \\Z_{\\ge 0}$. Then: :$\\phi^n = F_n \\phi + F_{n - 1}$ where: :$F_n$ denotes the $n$th Fibonacci number :$\\phi$ denotes the golden mean."}
+{"_id": "15959", "title": "Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less/Negative Index", "text": "Let $n \\in \\Z_{\\le 0}$. Then: :$\\phi^n = F_n \\phi + F_{n - 1}$ where: :$F_n$ denotes the $n$th Fibonacci number as extended to negative indices :$\\phi$ denotes the golden mean."}
+{"_id": "15960", "title": "Fibonacci Number by One Minus Golden Mean plus Fibonacci Number of Index One Less", "text": "Let $n \\in \\Z$. Then: :$\\hat \\phi^n = F_n \\hat \\phi + F_{n - 1}$ where: :$F_n$ denotes the $n$th Fibonacci number :$\\hat \\phi$ denotes the $1$ minus the golden mean: ::$\\hat \\phi := 1 - \\phi$"}
+{"_id": "15961", "title": "Reciprocal Form of One Minus Golden Mean", "text": ":$\\hat \\phi = - \\dfrac 1 \\phi$ where: :$\\phi$ denotes the golden mean :$\\hat \\phi$ denotes one minus the golden mean: $\\hat \\phi = 1 - \\phi$."}
+{"_id": "15962", "title": "Closed Form of One Minus Golden Mean", "text": ":$\\hat \\phi = \\dfrac {1 - \\sqrt 5} 2$ where: :$\\hat \\phi$ denotes one minus the golden mean: $\\hat \\phi = 1 - \\phi$."}
+{"_id": "15963", "title": "Second Order Fibonacci Number in terms of Fibonacci Numbers", "text": "The second order Fibonacci number $\\mathcal F_n$ can be expressed in terms of Fibonacci numbers as: :$\\dfrac {3 n + 3} 5 F_n - \\dfrac n 5 F_{n + 1}$"}
+{"_id": "15964", "title": "General Fibonacci Number in terms of Fibonacci Numbers", "text": "Let $r$ and $s$ be numbers, usually integers but not necessarily so limited. Let $\\left\\langle{a_n}\\right\\rangle$ be the general Fibonacci sequence : :$a_n = \\begin{cases} r & : n = 0 \\\\ s & : n = 1 \\\\ a_{n - 2} + a_{n - 1} & : n > 1 \\end{cases}$ Then $a_n$ can be expressed in Fibonacci numbers as: :$a_n = F_{n - 1} r + F_n s$"}
+{"_id": "15965", "title": "Fibonacci Number plus Constant in terms of Fibonacci Numbers", "text": "Let $c$ be a number. Let $\\left\\langle{b_n}\\right\\rangle$ be the sequence defined as: :$b_n = \\begin{cases} 0 & : n = 0 \\\\ 1 & : n = 1 \\\\ b_{n - 2} + b_{n - 1} + c & : n > 1 \\end{cases}$ Then $\\left\\langle{b_n}\\right\\rangle$  can be expressed in Fibonacci numbers as: :$b_n = c F_{n - 1} + \\left({c + 1}\\right) F_n - c$"}
+{"_id": "15966", "title": "Fibonacci Number plus Binomial Coefficient in terms of Fibonacci Numbers", "text": "Let $m \\in \\Z_{>0}$ be a positive integer. Let $\\left\\langle{a_n}\\right\\rangle$ be the sequence defined as: :$a_n = \\begin{cases} 0 & : n = 0 \\\\ 1 & : n = 1 \\\\ a_{n - 2} + a_{n - 1} + \\dbinom {n - 2} m & : n > 1 \\end{cases}$ where $\\dbinom {n - 2} m$ denotes a binonial coefficient. Then $\\left\\langle{a_n}\\right\\rangle$  can be expressed in Fibonacci numbers as: :$a_n = F_{m + 1} F_{n - 1} + \\left({F_{m + 2} + 1}\\right) F_n - \\displaystyle \\sum_{k \\mathop = 0}^m \\dbinom {n + m - k} k$"}
+{"_id": "15967", "title": "Fibonacci Number plus Arbitrary Function in terms of Fibonacci Numbers", "text": "Let $f \\left({n}\\right)$ and $g \\left({n}\\right)$ be arbitrary arithmetic functions. Let $\\left\\langle{a_n}\\right\\rangle$ be the sequence defined as: :$a_n = \\begin{cases} 0 & : n = 0 \\\\ 1 & : n = 1 \\\\ a_{n - 1} + a_{n - 2} + f \\left({n - 2}\\right) & : n > 1 \\end{cases}$ Let $\\left\\langle{b_n}\\right\\rangle$ be the sequence defined as: :$b_n = \\begin{cases} 0 & : n = 0 \\\\ 1 & : n = 1 \\\\ b_{n - 1} + b_{n - 2} + g \\left({n - 2}\\right) & : n > 1 \\end{cases}$ Let $\\left\\langle{c_n}\\right\\rangle$ be the sequence defined as: :$c_n = \\begin{cases} 0 & : n = 0 \\\\ 1 & : n = 1 \\\\ c_{n - 1} + c_{n - 2} + x f \\left({n - 2}\\right) + y g \\left({n - 2}\\right) & : n > 1 \\end{cases}$ where $x$ and $y$ are arbitrary. Then $\\left\\langle{c_n}\\right\\rangle$ can be expressed in Fibonacci numbers as: :$c_n = x a_n + y b_n + \\left({1 - x - y}\\right) F_n$"}
+{"_id": "15968", "title": "Vajda's Identity/Formulation 1", "text": ":$F_{n + i} F_{n + j} - F_n F_{n + i + j} = \\paren {-1}^n F_i F_j$"}
+{"_id": "15969", "title": "Vajda's Identity/Formulation 2", "text": ":$F_{n + k} F_{m - k} - F_n F_m = \\left({-1}\\right)^n F_{m - n - k} F_k$"}
+{"_id": "15970", "title": "Sum of Squares of Consecutive Fibonacci Numbers", "text": ":${F_n}^2 + {F_{n + 1} }^2 = F_{2 n + 1}$ where $F_n$ denotes the $n$th Fibonacci number."}
+{"_id": "15971", "title": "Hausdorff Maximal Principle implies Well-Ordering Theorem", "text": "Let the Hausdorff Maximal Principle hold. Then the Well-Ordering Theorem holds."}
+{"_id": "15972", "title": "Cosine of 36 Degrees", "text": ":$\\cos 36 \\degrees = \\cos \\dfrac \\pi 5 = \\dfrac \\phi 2 = \\dfrac {1 + \\sqrt 5} 4$ where $\\phi$ denotes the golden mean."}
+{"_id": "15975", "title": "Sine of 36 Degrees", "text": ":$\\sin 36^\\circ = \\sin \\dfrac \\pi 5 = \\dfrac {\\sqrt {\\sqrt 5 / \\phi} } 2 = \\sqrt {\\dfrac 5 8 - \\dfrac {\\sqrt 5} 8}$ where $\\phi$ denotes the golden mean."}
+{"_id": "15976", "title": "Partial Sums of Power Series with Fibonacci Coefficients", "text": ":$\\displaystyle \\sum_{k \\mathop = 0}^n F_k x^k = \\begin{cases} \\dfrac {x^{n + 1} F_{n + 1} + x^{n + 2} F_n - x} {x^2 + x - 1} & : x^2 + x - 1 \\ne 0 \\\\ \\dfrac {\\left({n + 1}\\right) x^n F_{n + 1} + \\left({n + 2}\\right) x^{n + 1} F_n - 1} {2 x + 1} & : x^2 + x - 1 = 0 \\end{cases}$ where $F_n$ denotes the $n$th Fibonacci number."}
+{"_id": "15978", "title": "Sum over k of n Choose k by Fibonacci t to the k by Fibonacci t-1 to the n-k by Fibonacci m+k", "text": ":$\\displaystyle \\sum_{k \\mathop \\ge 0} \\binom n k {F_t}^k {F_{t - 1} }^{n - k} F_{m + k} = F_{m + t n}$ where: :$\\dbinom n k$ denotes a binomial coefficient :$F_n$ denotes the $n$th Fibonacci number."}
+{"_id": "15979", "title": "Matrix whose Determinant is Fibonacci Number", "text": "The $n \\times n$ determinant: :$D_n = \\begin{vmatrix} 1 & -1 & 0 & 0 & \\cdots & 0 & 0 & 0 \\\\ 1 & 1 & -1 & 0 & \\cdots & 0 & 0 & 0 \\\\ 0 & 1 & 1 & -1 & \\cdots & 0 & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & 0 & 0 & \\cdots & 1 & 1 & -1 \\\\ 0 & 0 & 0 & 0 & \\cdots & 0 & 1 & 1 \\\\ \\end{vmatrix}$ evaluates to $F_{n + 1}$."}
+{"_id": "15980", "title": "Determinant with Unit Element in Otherwise Zero Column", "text": "Let $D$ be the determinant: :$D = \\begin{vmatrix}   1 & b_{12} & \\cdots & b_{1n} \\\\   0 & b_{22} & \\cdots & b_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\   0 & b_{n2} & \\cdots & b_{nn} \\end{vmatrix}$ Then: :$D = \\begin{vmatrix}   b_{22} & \\cdots & b_{2n} \\\\ \\vdots & \\ddots & \\vdots \\\\   b_{n2} & \\cdots & b_{nn} \\end{vmatrix}$"}
+{"_id": "15982", "title": "Well-Ordering Theorem implies Hausdorff Maximal Principle", "text": "Let the Well-Ordering Theorem hold. Then the Hausdorff Maximal Principle holds."}
+{"_id": "15983", "title": "Fibonacci Number with Prime Index 2n+1 is Congruent to 5^n Modulo p", "text": "Let $p = 2 n + 1$ be an odd prime. Then: :$F_p \\equiv 5^n \\pmod p$"}
+{"_id": "15985", "title": "Fibonacci Number n+1 Minus Golden Mean by Fibonacci Number n", "text": ":$F_{n + 1} - \\phi F_n = \\hat \\phi^n$ where: :$F_n$ denotes the $n$th Fibonacci number :$\\phi$ denotes the golden mean."}
+{"_id": "15986", "title": "Recurrence Relation where n+1th Term is A by nth term + B to the n", "text": "Let $\\left\\langle{a_n}\\right\\rangle$ be the sequence defined by the recurrence relation: :$a_n = \\begin{cases} 0 & : n = 0 \\\\ A a_{n - 1} + B^{n - 1} & : n > 0 \\end{cases}$ for numbers $A$ and $B$. Then the closed form for $\\left\\langle{a_n}\\right\\rangle$ is given by: :$a_n = \\begin{cases} \\dfrac {A^n - B^n} {A - B} & : A \\ne B \\\\ n A^{n - 1} & : A = B \\end{cases}$"}
+{"_id": "15987", "title": "Fibonomial Coefficient is Integer", "text": "Let $\\dbinom n k_\\mathcal F$ be a Fibonomial coefficient. Then $\\dbinom n k_\\mathcal F$ is an integer."}
+{"_id": "15988", "title": "Recurrence Relation for Fibonomial Coefficients", "text": ":$\\dbinom n k_\\mathcal F = F_{k - 1} \\dbinom {n - 1} k_\\mathcal F + F_{n - k + 1} \\dbinom {n - 1} {k - 1}_\\mathcal F$ where: :$\\dbinom n k_\\mathcal F$ denotes a Fibonomial coefficient :$F_{k - 1}$ etc. denote Fibonacci numbers."}
+{"_id": "15990", "title": "Golden Mean by One Minus Golden Mean equals Minus 1", "text": ":$\\phi \\hat \\phi = -1$ where: :$\\phi$ denotes the golden mean :$\\hat \\phi := 1 - \\phi$"}
+{"_id": "15993", "title": "Remainder of Fibonacci Number Divided by Fibonacci Number is Plus or Minus Fibonacci Number", "text": "Let $F_n$ and $F_m$ be Fibonacci numbers. By the Division Theorem, let: :$F_n = q F_m + r$ where: :$q \\in \\Z$ :$r \\in \\Z: 0 \\le r < \\size {F_m}$ Then either $r$ or $\\size {F_m} - r$, or both, is a Fibonacci number."}
+{"_id": "15994", "title": "Residue of Fibonacci Number Modulo Fibonacci Number", "text": "Let $F_n$ denote the $n$th Fibonacci number. Let $m, r$ be non-negative integers. Then: :$F_{m n + r} \\equiv \\paren {\\begin{cases} F_r & : m \\bmod 4 = 0 \\\\ \\paren {-1}^{r + 1} F_{n - r} & : m \\bmod 4 = 1 \\\\ \\paren {-1}^n F_r & : m \\bmod 4 = 2 \\\\ \\paren {-1}^{r + 1 + n} F_{n - r} & : m \\bmod 4 = 3 \\end{cases} } \\pmod {F_n}$"}
+{"_id": "15995", "title": "Sine of Multiple of Pi by 2 plus i by Natural Logarithm of Golden Mean", "text": "Let $z = \\dfrac \\pi 2 + i \\ln \\phi$. Then: :$\\dfrac {\\sin n z} {\\sin z} = i^{1 - n} F_n$ where: :$\\phi$ denotes the golden mean :$F_n$ denotes the $n$th Fibonacci number."}
+{"_id": "15999", "title": "Golden Mean as Root of Quadratic", "text": "The golden mean $\\phi$ is one of the roots of the quadratic equation: :$x^2 - x - 1 = 0$ The other root is $\\hat \\phi = 1 - \\phi$."}
+{"_id": "16000", "title": "Proper Subtower is Initial Segment", "text": "Let $\\left({T_1,\\preccurlyeq}\\right)$ be a proper subtower of $\\left({T_2,\\preccurlyeq}\\right)$. Then $\\left({T_1,\\preccurlyeq}\\right)$ is an initial segment of $\\left({T_2,\\preccurlyeq}\\right)$."}
+{"_id": "16001", "title": "Power of Golden Mean as Sum of Smaller Powers", "text": "Let $\\phi$ denote the golden mean. Then: :$\\forall z \\in \\C: \\phi^z = \\phi^{z - 1} + \\phi^{z - 2}$"}
+{"_id": "16002", "title": "100 in Golden Mean Number System is Equivalent to 011", "text": "Consider the golden mean number system. Let $p$ and $q$ be arbitrary strings in $\\left\\{ {0, 1}\\right\\}$. Let $x \\in \\R_{\\ge 0}$ have a representation which includes the string $100$, say: :$x = p100q$ Then $x \\in \\R_{\\ge 0}$ also has the representation: :$x = p011q$ Similarly, let $x \\in \\R_{\\ge 0}$ have a representation which includes the string $011$, say: :$x = p011q$ Then $x \\in \\R_{\\ge 0}$ also has the representation: :$x = p100q$ That is, any instance of $100$ appearing in a representation of a non-negative real number $x$ is equivalent to $011$, and vice versa. Note that the instance of $100$ or $011$ may also include a radix point; the instance of $011$ or $100$ to which it is equivalent will include the radix point in the same location."}
+{"_id": "16003", "title": "Conversion of Number in Golden Mean Number System to Simplest Form", "text": "Let $x \\in \\R_{\\ge 0}$ have a representation $S$ in the golden mean number system. Then $S$ can be converted to its simplest form as follows: :$(1): \\quad$ Replace any infinite string on the right hand end of $S$ of the form $010101 \\ldots$ with $100$ :$(2): \\quad$ Repeatedly replace the leftmost instance of $011$ with $100$."}
+{"_id": "16004", "title": "Simplest Form of Non-Negative Number in Golden Mean Number System is Unique", "text": "Let $x \\in \\R_{\\ge 0}$ be represented in the golden mean number system. Let $S$ be the representation for $x$ in its simplest form. Then $S$ is unique in the sense that there exists no other representation of $x$ in simplest form."}
+{"_id": "16007", "title": "Incidence of Double Letters in Fibonacci String", "text": "Let $S_n$ denote the $n$th Fibonacci string. Then: :$(1):\\quad$ There are no instances of $2$ $\\text a$'s together :$(2):\\quad$ There are no instances of $3$ $\\text b$'s together in $S_n$."}
+{"_id": "16008", "title": "Fibonacci String Begins with ba", "text": "Let $S_n$ be a Fibonacci string of length $n$. Then for $n \\ge 3$, $S_n$ begins with $\\text{ba}$."}
+{"_id": "16009", "title": "Fibonacci String Ends with ab or ba", "text": "Let $S_n$ be a Fibonacci string of length $n$. Then for $n \\ge 3$, $S_n$ ends either with $\\text{ba}$ or with $\\text{ab}$."}
+{"_id": "16010", "title": "Count of a's and b's in Fibonacci String", "text": "Let $S_n$ denote the $n$th Fibonacci string. Then for $n \\ge 3$, $S_n$ has: :$F_{n - 2}$ instances of $\\text a$ :$F_{n - 1}$ instances of $\\text b$."}
+{"_id": "16011", "title": "Initial Part of Fibonacci String", "text": "Let $n \\in \\Z_{>1}$. Let $S_n$ denote the $n$th Fibonacci string. Let $m \\in \\Z$ such that $1 < m \\le n$. Let $F_m$ denote the $m$th Fibonacci number. The initial part of $S_n$ of length $F_m$ is the Fibonacci string $S_m$."}
+{"_id": "16012", "title": "Positions of Instances of a in Fibonacci String", "text": "Let $S_n$ denote the $n$th Fibonacci string. Let $F_n$ denote the $n$th Fibonacci number. Let $m \\in \\Z$ such that $m \\le F_n$. Let $m - 1$ be expressed in Zeckendorf representation as $Z_{m - 1}$. Then the $m$th letter of $S_n$ is $\\text a$ {{iff}}: : $k_r = 2$ where $k_r$ denotes the final digit of $Z_{m - 1}$."}
+{"_id": "16013", "title": "Positions of Instances of b in Fibonacci String", "text": "Let $S_n$ denote the $n$th Fibonacci string. Let $F_n$ denote the $n$th Fibonacci number. Let $k \\in \\Z$ such that $k \\le F_n$. Then the $k$th letter of $S_n$ is $\\text b$ {{iff}}: : $\\left\\lfloor{\\left({k + 1}\\right) \\phi^{-1} }\\right\\rfloor - \\left\\lfloor{k \\phi^{-1} }\\right\\rfloor = 1$ where: : $\\left\\lfloor{\\, \\cdot \\,}\\right\\rfloor$ denotes the floor function : $\\phi$ denotes the golden mean."}
+{"_id": "16014", "title": "Optimal Strategy for Fibonacci Nim", "text": "Consider a game of Fibonacci nim with $n$ counters. Let it be the turn of player $\\text A$. Let the maximum number of counters that can be taken by $\\text A$ be $q$. Let $n$ be expressed in Zeckendorf representation as: :$n = F_{k_1} + F_{k_2} + \\cdots + F_{k_r}$ Then $\\text A$ can force a win {{iff}}: :$F_{k_r} \\le q$ and by taking those $F_{k_r}$ counters."}
+{"_id": "16015", "title": "Recursively Defined Sequence/Examples/Term is Term of Index less 1 plus 6 times Term of Index less 2", "text": "Consider the integer sequence $\\left\\langle{a_n}\\right\\rangle$ defined recursively as: :$a_n = \\begin{cases} 0 & : n = 0 \\\\ 1 & : n = 1 \\\\ a_{n - 1} + 6 a_{n - 2} & : \\text{otherwise} \\end{cases}$ $a_n$ has a closed-form expression: :$a_n = \\dfrac {3^n - \\left({-2}\\right)^n} 5$"}
+{"_id": "16016", "title": "Recursively Defined Sequence/Examples/Minimum over k of Maximum of 1 plus Function of k and 2 plus Function of n-k", "text": "Consider the integer sequence $\\left\\langle{f \\left({n}\\right)}\\right\\rangle$ defined recusrively as: :$f \\left({n}\\right) = \\begin{cases} 0 & : n = 1 \\\\ \\displaystyle \\min_{0 \\mathop < k \\mathop < n} \\max \\left({1 + f \\left({k}\\right), 2 + f \\left({n - k}\\right)}\\right) & : n > 1 \\end{cases}$ $f \\left({n}\\right)$ has a closed-form expression: :$f \\left({n}\\right) = m$ for $F_m < n \\le F_{m + 1}$ where $F_m$ denotes the $m$th Fibonacci number."}
+{"_id": "16017", "title": "Zeckendorf Representation of Integer shifted Left", "text": "Let $\\map f x$ be the real function defined as: :$\\forall x \\in \\R: \\map f x = \\floor {x + \\phi^{-1} }$ where: :$\\floor {\\, \\cdot \\,}$ denotes the floor function :$\\phi$ denotes the golden mean. Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Let $n$ be expressed in Zeckendorf representation: :$n = F_{k_1} + F_{k_2} + \\cdots + F_{k_r}$ with the appropriate restrictions on $k_1, k_2, \\ldots, k_r$. Then: :$F_{k_1 + 1} + F_{k_2 + 1} + \\cdots + F_{k_r + 1} = \\map f {\\phi n}$"}
+{"_id": "16019", "title": "Two Non-Negative Integers have Zeckendorf Representations of which one is Shifted Representation of the Other", "text": "Let $m, n \\in \\Z_{\\ge 0}$ be non-negative integers. Then there exists a unique set of integers: :$\\left\\{ {k_1, k_2, \\ldots, k_r}\\right\\}$ where: : $k_1 \\gg k_2 \\gg \\cdots \\gg k_r$ where $a \\gg b$ denotes that $a - b > 1$ such that: :$m = F_{k_1} + F_{k_2} + \\cdots + F_{k_r}$ and: :$n = F_{k_1 + 1} + F_{k_2 + 1} + \\cdots + F_{k_r + 1}$ Note that: : each of the $k$'s may be negative and: :$r$ may equal $0$."}
+{"_id": "16022", "title": "Generating Function by Power of Parameter", "text": "Let $G \\left({z}\\right)$ be the generating function for the sequence $\\left\\langle{a_n}\\right\\rangle$. Let $m \\in \\Z_{\\ge 0}$ be a non-negative integer. Then $z^m  G \\left({z}\\right)$ is the generating function for the sequence $\\left\\langle{a_{n - m} }\\right\\rangle$."}
+{"_id": "16023", "title": "Generating Function Divided by Power of Parameter", "text": "Let $G \\left({z}\\right)$ be the generating function for the sequence $\\left\\langle{a_n}\\right\\rangle$. Let $m \\in \\Z_{\\ge 0}$ be a non-negative integer. Then $\\dfrac 1 {z^m} \\left({G \\left({z}\\right) - \\displaystyle \\sum_{k \\mathop = 0}^{m - 1} a_k z^k}\\right)$ is the generating function for the sequence $\\left\\langle{a_{n + m} }\\right\\rangle$."}
+{"_id": "16024", "title": "Generating Function for Linearly Recurrent Sequence", "text": "Let $\\sequence {a_n}$ be a linearly recurrent sequence defined as: :$a_n = \\begin{cases} b_n & : 1 \\le n \\le m \\\\ c_1 a_{n - 1} + c_2 a_{n - 2} + \\cdots + c_m a_{n - m} & : n > m \\end{cases}$ where: :$m \\in \\Z_{>0}$ is a (strictly) positive integer :$b_1, \\ldots, b_m$ are constants. Then the generating function for $\\sequence {a_n}$ is of the form: :$\\map G z = \\dfrac {\\map P z} {1 - c_1 z - c_2 z^2 - \\cdots - c_m z^m}$ where $\\map P z$ is a polynomial  in $z$ given by $b_1 z + b_2 z^2 + \\cdots + b_m z^m$."}
+{"_id": "16026", "title": "Product of Generating Functions", "text": "Let $\\map G z$ be the generating function for the sequence $\\sequence {a_n}$. Let $\\map H z$ be the generating function for the sequence $\\sequence {b_n}$. Then $\\map G z \\map H z$ is the generating function for the sequence $\\sequence {c_n}$, where: :$\\forall n \\in \\Z_{\\ge 0}: c_n = \\displaystyle \\sum_{k \\mathop = 0}^n a_k b_{n - k}$"}
+{"_id": "16027", "title": "Generating Function for Sequence of Partial Sums of Series", "text": "Let $s$ be the the series: :$\\displaystyle s = \\sum_{n \\mathop = 1}^\\infty a_n = a_0 + a_1 + a_2 + a_3 + \\cdots$ Let $G \\left({z}\\right)$ be the generating function for the sequence $\\left\\langle{a_n}\\right\\rangle$. Let $\\left\\langle{c_n}\\right\\rangle$ denote the sequence of partial sums of $s$. Then the generating function for $\\left\\langle{c_n}\\right\\rangle$ is given by: :$\\displaystyle \\dfrac 1 {1 - z} G \\left({z}\\right) = \\sum_{n \\mathop \\ge 0} c_n z^n$"}
+{"_id": "16028", "title": "Product of Generating Functions/General Rule", "text": "Let $G_0 \\left({z}\\right), G_1 \\left({z}\\right), G_2 \\left({z}\\right), \\ldots$ be any number of generating functions (up to countably infinite) for the sequences $\\left\\langle{a_0 n}\\right\\rangle, \\left\\langle{a_1 n}\\right\\rangle, \\left\\langle{a_2 n}\\right\\rangle, \\ldots$ Then: {{begin-eqn}} {{eqn | l = \\prod_{j \\mathop \\ge 0} G_j \\left({z}\\right)       | r = \\prod_{j \\mathop \\ge 0} \\sum_{k \\mathop \\ge 0} a_{j k} z^k       | c =  }} {{eqn | r = \\sum_{n \\mathop \\ge 0} z^n \\sum_{\\substack {k_0, k_1, k_2, \\ldots \\mathop \\ge 0 \\\\ k_0 \\mathop + k_1 \\mathop + \\mathop \\cdots \\mathop = n} } \\left({\\prod_{j \\mathop \\ge 0} a_{j k} }\\right)       | c =  }} {{end-eqn}}"}
+{"_id": "16029", "title": "Product of Exponential Generating Functions", "text": "Let $G \\left({z}\\right)$ be the exponential generating function for the sequence $\\left\\langle{\\dfrac {a_n} {n!} }\\right\\rangle$. Let $H \\left({z}\\right)$ be the exponential generating function for the sequence $\\left\\langle{\\dfrac {b_n} {n!} }\\right\\rangle$. Then $G \\left({z}\\right) H \\left({z}\\right)$ is the generating function for the sequence $\\left\\langle{\\dfrac {c_n} {n!} }\\right\\rangle$, where: :$\\forall n \\in \\Z_{\\ge 0}: c_n = \\displaystyle \\sum_{k \\mathop \\in \\Z} \\dbinom n k a_k b_{n - k}$"}
+{"_id": "16030", "title": "Generating Function of Multiple of Parameter", "text": "Let $G \\left({z}\\right)$ be the generating function for the sequence $\\left\\langle{a_n}\\right\\rangle$. Let $c$ be a constant. Then $G \\left({c z}\\right)$ be the generating function for the sequence $\\left\\langle{b_n}\\right\\rangle$ where: :$\\forall n \\in \\Z_{\\ge 0}: b_n = c^n a_n$"}
+{"_id": "16031", "title": "Generating Function for Sequence of Powers of Constant", "text": "Let $c \\in \\R$ be a constant. Let $\\sequence {a_n}$ be the sequence defined as: :$\\forall n \\in \\Z_{\\ge 0}: a_n = c^n$ That is: :$\\sequence {a_n} = 1, c, c^2, c^3, \\ldots$ Then the generating function for $\\sequence {a_n}$ is given as: :$\\map G z = \\dfrac 1 {1 - c z}$"}
+{"_id": "16033", "title": "Generating Function for Odd Terms of Sequence", "text": "Let $\\map G z$ be the generating function for the sequence $\\sequence {a_n}$. Consider the subsequence $\\sequence {b_n} := \\tuple {a_1, a_3, a_5, \\ldots}$ Then the generating function for $\\sequence {b_n}$ is: :$\\dfrac 1 2 \\paren {\\map G z - \\map G {-z} }$"}
+{"_id": "16036", "title": "Derivative of Generating Function", "text": "Let $\\map G z$ be the generating function for the sequence $\\sequence {a_n}$. Then: {{begin-eqn}} {{eqn | l = \\frac \\d {\\d z} \\map G z       | r = \\sum_{k \\mathop \\ge 0} \\left({k + 1}\\right) a_{k + 1} z^k       | c =  }} {{eqn | r = a_1 + 2 a_2 z + 3 a_3 z^3 + \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "16038", "title": "Integral of Generating Function", "text": "Let $G \\left({z}\\right)$ be the generating function for the sequence $\\left\\langle{a_n}\\right\\rangle$. Then: {{begin-eqn}} {{eqn | l = \\int_0^z G \\left({t}\\right) \\rd t       | r = \\sum_{k \\mathop \\ge 1} \\dfrac {a_{k - 1} z^k} k       | c =  }} {{eqn | r = a_0 z + \\dfrac {a_1 z^2} 2 + \\dfrac {a_2 z^3} 3 + \\dfrac {a_3 z^4} 4 + \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "16039", "title": "Generating Function for Sequence of Reciprocals of Natural Numbers", "text": "Let $\\sequence {a_n}$ be the sequence defined as: :$\\forall n \\in \\N_{> 0}: a_n = n$ That is: :$\\sequence {a_n} = 1, \\dfrac 1 2, \\dfrac 1 3, \\dfrac 1 4, \\ldots$ Then the generating function for $\\sequence {a_n}$ is given as: :$\\map G z = \\map \\ln {\\dfrac 1 {1 - z} }$"}
+{"_id": "16040", "title": "Generating Function for Sequence of Harmonic Numbers", "text": "Let $\\sequence {a_n}$ be the sequence defined as: :$\\forall n \\in \\N_{> 0}: a_n = H_n$ where $H_n$ denotes the $n$th harmonic number. That is: :$\\sequence {a_n} = 1, 1 + \\dfrac 1 2, 1 + \\dfrac 1 2 + \\dfrac 1 3, \\ldots$ Then the generating function for $\\sequence {a_n}$ is given as: :$\\map G z = \\dfrac 1 {1 - z} \\map \\ln {\\dfrac 1 {1 - z} }$"}
+{"_id": "16041", "title": "Binomial Theorem for Negative Index and Negative Parameter", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Let $z \\in \\R$ be a real number such that $\\size z < 1$. Then: {{begin-eqn}} {{eqn | l = \\dfrac 1 {\\paren {1 - z}^{n + 1} }       | r = \\sum_{k \\mathop \\ge 0} \\binom {-n - 1} k \\paren {-z}^k       | c =  }} {{eqn | r = \\sum_{k \\mathop \\ge 0} \\binom {n + k} n z^k       | c =  }} {{end-eqn}} where $\\dbinom {n + k} n$ denotes a binomial coefficient."}
+{"_id": "16042", "title": "Power Series Expansion for Integer Power of Exponential Function minus 1", "text": "Let $e^z$ denote the exponential function. Then: {{begin-eqn}} {{eqn | l = \\left({e^z - 1}\\right)^n       | r = z^n + \\dfrac 1 {n + 1} \\left\\{ { {n + 1} \\atop n}\\right\\} z^{n + 1} + \\cdots       | c =  }} {{eqn | r = n! \\sum_{k \\mathop \\in \\Z} \\left\\{ {k \\atop n}\\right\\} \\frac {z^k} {k!}       | c =  }} {{end-eqn}} where $\\displaystyle \\left\\{ {k \\atop n}\\right\\}$ denotes a Stirling number of the second kind."}
+{"_id": "16043", "title": "Power Series Expansion for Reciprocal of 1-z to the m+1 by Logarithm of Reciprocal of 1-z", "text": ":$\\dfrac 1 {\\left({1 - z}\\right)^{m + 1} } \\ln \\left({\\dfrac 1 {1 - z} }\\right) = \\displaystyle \\sum_{k \\mathop \\ge 1} \\left({H_{m + k} - H_m}\\right) \\dbinom {m + k} k z^k$ where: :$\\dbinom {m + k} k$ denotes a binomial coefficient :$H_m$ denotes the $m$th harmonic number."}
+{"_id": "16044", "title": "Power Series Expansion for nth Power of Logarithm of Reciprocal of 1-z", "text": "{{begin-eqn}} {{eqn | l = \\left({\\ln \\dfrac 1 {1 - z} }\\right)^n       | r = z^n + \\dfrac 1 {n + 1} \\left[{ {n + 1} \\atop n}\\right] z^{n + 1} + \\cdots       | c =  }} {{eqn | r = n! \\sum_{k \\mathop \\in \\Z} \\left[{k \\atop n}\\right] \\frac {z^k} {k!}       | c =  }} {{end-eqn}} where $\\displaystyle \\left[{k \\atop n}\\right]$ denotes an unsigned Stirling number of the first kind."}
+{"_id": "16045", "title": "Sum over k of Stirling Numbers of Second Kind by x^k", "text": "{{begin-eqn}} {{eqn | l = \\sum_k \\left\\{ {k \\atop n}\\right\\} z^k       | r = \\dfrac {z^n} {\\prod \\limits_{k \\mathop = 1}^n \\left({1 - k n}\\right)}       | c =  }} {{eqn | r = \\dfrac {z^n} {\\left({1 - z}\\right) \\left({1 - 2 z}\\right) \\cdots \\left({1 - n z}\\right)}       | c =  }} {{end-eqn}} where: :$\\displaystyle \\left\\{ {k \\atop n}\\right\\}$ denotes a Stirling number of the second kind."}
+{"_id": "16046", "title": "Sum over k of r by r+kt to the Power of k-1 over k Factorial by Power of z", "text": "Let $x$ be the continuous real function of $z$ which satisfies: :$\\ln x = z x^t$ where $x = 1$ when $z = 0$. Then: {{begin-eqn}} {{eqn | l = x^r       | r = \\sum_{k \\mathop \\ge 0} \\dfrac {r \\left({r + k t}\\right)^{k - 1} } {k!} z^k       | c =  }} {{eqn | r = 1 + r z + \\dfrac {r \\left({r + 2 t}\\right)} 2 z^2 + \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "16048", "title": "Cauchy's Integral Formula/General Result/Corollary", "text": "Let $\\map G z$ be the generating function for the sequence $\\sequence {a_n}$. Let the coefficient of $z^n$ extracted from $\\map G z$ be denoted: :$\\sqbrk {z^n} \\map G z := a_n$ Let $\\map G z$ be convergent for $z = z_0$ and $0 < r < \\cmod {z_0}$. Then: :$\\sqbrk {z^n} \\map G z = \\displaystyle \\frac 1 {2 \\pi i} \\oint_{\\cmod z \\mathop = r} \\dfrac {\\map G z \\d z} {z^{n + 1} }$"}
+{"_id": "16049", "title": "Summation of Products of n Numbers taken m at a time with Repetitions", "text": "Let $a, b \\in \\Z$ be integers such that $b \\ge a$. Let $U$ be a set of $n = b - a + 1$ numbers $\\left\\{ {x_a, x_{a + 1}, \\ldots, x_b}\\right\\}$. Let $m \\in \\Z_{>0}$ be a (strictly) positive integer. Let: {{begin-eqn}} {{eqn | l = h_m       | r = \\sum_{a \\mathop \\le j_1 \\mathop \\le \\mathop \\cdots \\mathop \\le j_m \\mathop \\le b} \\paren {\\prod_{k \\mathop = 1}^m x_{j_k} }       | c =  }} {{eqn | r = \\sum_{a \\mathop \\le j_1 \\mathop \\le \\mathop \\cdots \\mathop \\le j_m \\mathop \\le b} x_{j_1} \\cdots x_{j_m}       | c =  }} {{end-eqn}} That is, $h_m$ is the product of all $m$-tuples of elements of $U$ taken $m$ at a time, allowing for repetition. For $r \\in \\Z_{> 0}$, let: :$S_r = \\displaystyle \\sum_{k \\mathop = a}^b {x_k}^r$ Then: {{begin-eqn}} {{eqn | l = h_m       | r = \\sum_{\\substack {k_1, k_2, \\ldots, k_m \\mathop \\ge 0 \\\\ k_1 \\mathop + 2 k_2 \\mathop + \\mathop \\cdots \\mathop + m k_m \\mathop = m} } \\paren {\\prod_{j \\mathop = 1}^m \\dfrac { {S_j}^{k_j} } {j^{k_j} k_j !} }       | c =  }} {{eqn | r = \\sum_{\\substack {k_1, k_2, \\ldots, k_m \\mathop \\ge 0 \\\\ k_1 \\mathop + 2 k_2 \\mathop + \\mathop \\cdots \\mathop + m k_m \\mathop = m} } \\dfrac { {S_1}^{k_1} } {1^{k_1} k_1 !} \\dfrac { {S_2}^{k_2} } {2^{k_2} k_2 !} \\cdots \\dfrac { {S_m}^{k_m} } {m^{k_m} k_m !}       | c =  }} {{end-eqn}}"}
+{"_id": "16051", "title": "Derivative of Generating Function for Sequence of Harmonic Numbers", "text": "Let $\\sequence {a_n}$ be the sequence defined as: :$\\forall n \\in \\N_{> 0}: a_n = H_n$ where $H_n$ denotes the $n$th harmonic number. Let $\\map G z$ be the generating function for $\\sequence {a_n}$: :$\\map G z = \\dfrac 1 {1 - z} \\map \\ln {\\dfrac 1 {1 - z} }$ from Generating Function for Sequence of Harmonic Numbers. Then the derivative of $\\map G z$ {{WRT|Differentiation}} $z$ is given by: :$\\map {G'} z = \\dfrac 1 {\\paren {1 - z}^2} \\map \\ln {\\dfrac 1 {1 - z} } + \\dfrac 1 {\\paren {1 - z}^2}$"}
+{"_id": "16061", "title": "Summation of Products of n Numbers taken m at a time with Repetitions/Examples/Order 4", "text": ":$\\displaystyle \\sum_{a \\mathop \\le j_1 \\mathop \\le j_2 \\mathop \\le j_3 \\mathop \\le j_4 \\mathop \\le b} x_{j_1} x_{j_2} x_{j_3} x_{j_4} = \\dfrac { {S_1}^4} {24} + \\dfrac { {S_1}^2 S_2} 4 + \\dfrac { {S_2}^2} 8 + \\dfrac {S_1 S_3} 3 + \\dfrac {S_4} 4$ where: :$\\displaystyle S_r := \\sum_{k \\mathop = a}^b {x_k}^r$."}
+{"_id": "16062", "title": "Summation of Products of n Numbers taken m at a time with Repetitions/Recurrence Formula", "text": "A recurrence relation for $h_n$ can be given as: {{begin-eqn}} {{eqn | l = h_n       | r = \\sum_{k \\mathop = 1}^n \\dfrac {S_k h_{n - k} } n       | c =  }} {{eqn | r = \\dfrac 1 n \\paren {S_1 h_{n - 1} + S_2 h_{n - 2} + \\cdots S_n h_0}       | c =  }} {{end-eqn}} for $n \\ge 1$."}
+{"_id": "16063", "title": "Summation of Products of n Numbers taken m at a time with Repetitions/Lemma 1", "text": "Let $\\map G z$ be the generating function for the sequence $\\sequence {h_m}$. Then: {{begin-eqn}} {{eqn | l = \\map G z       | r = \\prod_{k \\mathop = a}^b \\dfrac 1 {1 - x_k z}       | c =  }} {{eqn | r = \\dfrac 1 {\\paren {1 - x_a z} \\paren {1 - x_{a + 1} z} \\cdots \\paren {1 - x_b z} }       | c =  }} {{end-eqn}}"}
+{"_id": "16064", "title": "Summation of Products of n Numbers taken m at a time with Repetitions/Lemma 2", "text": "{{begin-eqn}} {{eqn | l = \\ln \\left({G \\left({z}\\right)}\\right)       | r = \\sum_{k \\mathop \\ge 1} \\dfrac {S_k z^k} k       | c =  }} {{end-eqn}}"}
+{"_id": "16065", "title": "Generating Function for Elementary Symmetric Function", "text": "Let $U$ be a set of $n$ numbers $\\set {x_1, x_2, \\ldots, x_n}$. Define: {{begin-eqn}} {{eqn | l = \\map {e_m} {U}       | r = \\begin{cases} 1 & m = 0\\\\ \\displaystyle \\sum_{1 \\mathop \\le j_1 \\mathop < \\mathop \\cdots \\mathop < j_m \\mathop \\le n} x_{j_1} x_{j_2} \\cdots x_{j_m} & 1 \\leq m \\leq n \\\\ 0 & m \\gt n \\\\ \\end{cases}      | c = elementary symmetric function }} {{eqn | l = a_m       | r = \\map {e_m} U       | c = for $m=0,1,2,\\ldots$ }} {{eqn | l = \\map G z       | r = \\displaystyle \\sum_{m=0}^\\infty a_m z^m       | c = generating function for $\\set {a_m}_{m=0}^\\infty$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \\map G z       | r = \\displaystyle \\prod_{k \\mathop = 1}^n \\paren {1 + x_k z} }} {{end-eqn}}"}
+{"_id": "16066", "title": "Newton-Girard Formulas/Lemma 1", "text": "Let $G \\left({z}\\right)$ be the generating function for the sequence $\\left\\langle{h_m}\\right\\rangle$. Then: {{begin-eqn}} {{eqn | l = G \\left({z}\\right)       | r = \\prod_{k \\mathop = a}^b \\left({1 + x_k z}\\right)       | c =  }} {{eqn | r = \\left({1 + x_a z}\\right) \\left({1 + x_{a + 1} z}\\right) \\cdots \\left({1 + x_b z}\\right)       | c =  }} {{end-eqn}}"}
+{"_id": "16067", "title": "Newton-Girard Formulas/Lemma 2", "text": "{{begin-eqn}} {{eqn | l = \\ln \\left({G \\left({z}\\right)}\\right)       | r = \\sum_{k \\mathop \\ge 1} \\left({-1}\\right)^{k + 1} \\dfrac {S_k z^k} k       | c =  }} {{end-eqn}}"}
+{"_id": "16069", "title": "Newton-Girard Formulas/Examples/Order 3", "text": ":$\\displaystyle \\sum_{a \\mathop \\le i \\mathop < j \\mathop < k \\mathop \\le b} x_i x_j x_k = \\dfrac { {S_1}^3} 6 - \\dfrac {S_1 S_2} 2 + \\dfrac {S_3} 3$ where: :$\\displaystyle S_r := \\sum_{k \\mathop = a}^b {x_k}^r$"}
+{"_id": "16070", "title": "Newton-Girard Formulas/Examples/Order 4", "text": ":$\\displaystyle \\sum_{a \\mathop \\le j_1 \\mathop < j_2 \\mathop < j_3 \\mathop < j_4 \\mathop \\le b} x_{j_1} x_{j_2} x_{j_3} x_{j_4} = \\dfrac { {S_1}^4} {24} - \\dfrac { {S_1}^2 S_2} 4 + \\dfrac { {S_2}^2} 8 + \\dfrac {S_1 S_3} 3 - \\dfrac {S_4} 4$ where: :$\\displaystyle S_r := \\sum_{k \\mathop = a}^b {x_k}^r$."}
+{"_id": "16071", "title": "Newton-Girard Formulas/Examples/Order 1", "text": ":$\\displaystyle \\sum_{a \\mathop \\le i \\mathop \\le b} x_i = S_1$ where: :$\\displaystyle S_r := \\sum_{k \\mathop = a}^b {x_k}^r$"}
+{"_id": "16072", "title": "Newton-Girard Formulas/Recurrence Formula", "text": "A recurrence relation for $h_n$ can be given as: {{begin-eqn}} {{eqn | l = h_n       | r = \\sum_{k \\mathop = 1}^n \\dfrac {\\left({-1}\\right)^{k + 1} S_k h_{n - k} } n       | c =  }} {{eqn | r = \\dfrac 1 n \\left({S_1 h_{n - 1} - S_2 h_{n - 2} + \\cdots S_n h_0}\\right)       | c =  }} {{end-eqn}} for $n \\ge 1$."}
+{"_id": "16073", "title": "Newton's Identities", "text": "Let $X$ be a set of $n$ numbers $\\set {x_1, x_2, \\ldots, x_n}$. Define: {{begin-eqn}} {{eqn | l = {\\mathbf S}_m        | r = \\set { \\paren {j_1,\\ldots,j_m} : 1 \\le j_1 \\lt \\cdots \\lt j_m \\le n}       | c = $1 \\le m \\le n$ }} {{eqn | l = \\map {e_m} {X}       | r = \\begin{cases} 1 & m = 0\\\\ \\displaystyle \\sum_{ {\\mathbf S}_m } x_{j_1} \\cdots x_{j_m} & 1 \\leq m \\leq n \\\\ 0 & m \\gt n \\\\ \\end{cases}      | c = elementary symmetric function }} {{eqn | l = \\map {p_k} X       | r = \\begin{cases}                \\displaystyle  n   & k = 0 \\\\               \\displaystyle \\sum_{i \\mathop = 1}^n  x_i^k   & k \\ge 1 \\\\             \\end{cases}       | c = power sums }} {{end-eqn}} Then '''Newton's Identities''' are: {{begin-eqn}} {{eqn | n = 1       | l = k \\, \\map {e_k} X       | r = \\displaystyle \\sum_{i \\mathop = 1}^k              \\paren {-1}^{i-1}             \\map {e_{k-i} } X             \\map {p_i} X       | c = for $1 \\leq k \\leq n$ }} {{eqn | n = 2       | l = 0       | r = \\displaystyle \\sum_{i \\mathop = k-n}^k              \\paren {-1}^{i-1}             \\map {e_{k-i} } X             \\map {p_i} X       | c = for $1 \\leq n \\lt k$ }} {{end-eqn}}"}
+{"_id": "16074", "title": "Summation of Products of n Numbers taken m at a time with Repetitions/Inverse Formula", "text": "Let $S_m$ be expressed in the form: :$S_m = \\displaystyle \\sum_{k_1 \\mathop + 2 k_2 \\mathop + \\mathop \\cdots \\mathop + m k_m \\mathop = m} A_m {h_1}^{k_1} {h_2}^{k_2} \\cdots {h_m}^{k_m}$ for $k_1, k_2, \\ldots, k_m \\ge 0$. Then : :$A_m = \\left({-1}\\right)^{k_1 + k_2 + \\cdots + k_m - 1} \\dfrac {m \\left({k_1 + k_2 + \\cdots + k_m - 1}\\right)! } {k_1! \\, k_2! \\, \\cdots k_m!}$"}
+{"_id": "16077", "title": "Law of Subtraction", "text": "On the following number systems: : integers $\\Z$ : rational numbers $\\Q$ : real numbers $\\R$ : complex numbers $\\C$ there exists a unique $x$ such that: :$a + x = b$ for every given $a$ and $b$. $x$ is then defined and denoted: :$x := b - a$"}
+{"_id": "16078", "title": "Law of Division", "text": "Let $\\mathbb F$ denote one of the following number systems: : rational numbers $\\Q$ : real numbers $\\R$ : complex numbers $\\C$ Let $a, b \\in \\mathbb F$ such that $a \\ne 0$. Then there exists a unique $x$ such that: :$a x = b$ $x$ is then defined and denoted: :$x := b / a$"}
+{"_id": "16079", "title": "Limit of Rational Sequence is not necessarily Rational", "text": "Let $S = \\left\\langle{a_n}\\right\\rangle$ be a rational sequence. Let $S$ be convergent to a limit $L$. Then it is not necessarily the case that $L$ is itself a rational number."}
+{"_id": "16083", "title": "Powers of Imaginary Unit", "text": "The (integer) powers of the imaginary unit $i$ are: {{begin-eqn}} {{eqn | l = i^0       | r = 1       | c =  }} {{eqn | l = i^1       | r = i       | c =  }} {{eqn | l = i^2       | r = -1       | c =  }} {{eqn | l = i^3       | r = -i       | c =  }} {{eqn | l = i^4       | r = 1       | c =  }} {{end-eqn}} {{qed}}"}
+{"_id": "16084", "title": "Real Number Multiplied by Complex Number", "text": "Let $a \\in \\R$ be a real number. Let $c + d i \\in \\C$ be a complex number. Then: :$a \\times \\left({c + d i}\\right) = \\left({c + d i}\\right) \\times a = a c + i a d$"}
+{"_id": "16085", "title": "Complex Number equals Negative of Conjugate iff Wholly Imaginary", "text": "Let $z \\in \\C$ be a complex number. Let $\\overline z$ be the complex conjugate of $z$. Then $\\overline z = -z$ {{iff}} $z$ is wholly imaginary."}
+{"_id": "16086", "title": "Square of Complex Conjugate is Complex Conjugate of Square", "text": "Let $z \\in \\C$ be a complex number. Let $\\overline z$ denote the complex conjugate of $z$. Then: : $\\overline {z^2} = \\left({\\overline z}\\right)^2$"}
+{"_id": "16087", "title": "Complex Modulus equals Zero iff Zero", "text": "Let $z = a + i b$ be a complex number. Let $\\cmod z$ be the modulus of $z$. Then: :$\\cmod z = 0 \\iff z = 0$"}
+{"_id": "16088", "title": "Complex Modulus is Non-Negative", "text": "Let $z = a + i b \\in \\C$ be a complex number. Let $\\cmod z$ be the modulus of $z$. Then: :$\\cmod z \\ge 0$"}
+{"_id": "16089", "title": "Complex Modulus equals Complex Modulus of Conjugate", "text": "Let $z \\in \\C$ be a complex number. Let $\\overline z$ denote the complex conjugate of $z$. Let $\\cmod z$ denote the modulus of $z$. Then: :$\\cmod z = \\cmod {\\overline z}$"}
+{"_id": "16091", "title": "Square of Complex Modulus equals Complex Modulus of Square", "text": "Let $z \\in \\C$ be a complex number. Let $\\left\\vert{z}\\right\\vert$ be the modulus of $z$. Then: : $\\left\\vert{z^2}\\right\\vert = \\left\\vert{z}\\right\\vert^2$"}
+{"_id": "16092", "title": "Power of Complex Modulus equals Complex Modulus of Power", "text": "Let $z \\in \\C$ be a complex number. Let $\\left\\vert{z}\\right\\vert$ be the modulus of $z$. Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Then: :$\\left\\vert{z^n}\\right\\vert = \\left\\vert{z}\\right\\vert^n$"}
+{"_id": "16093", "title": "Power of Complex Conjugate is Complex Conjugate of Power", "text": "Let $z \\in \\C$ be a complex number. Let $\\overline z$ denote the complex conjugate of $z$. Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Then: :$\\overline {z^n} = \\left({\\overline z}\\right)^n$"}
+{"_id": "16094", "title": "Conjugate of Polynomial is Polynomial of Conjugate", "text": "Let $f \\left({z}\\right) = a_n z^n + a_{n - 1} z^{n - 1} + \\cdots + a_1 z + a_0$ be a polynomial over complex numbers where $a_0, \\ldots, a_n$ are real numbers. Let $\\alpha \\in \\C$ be a complex number. Then: :$\\overline {f \\left({\\alpha}\\right)} = f \\left({\\overline \\alpha}\\right)$ where $\\overline \\alpha$ denotes the complex conjugate of $\\alpha$."}
+{"_id": "16095", "title": "Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs", "text": "Let $\\map f z = a_n z^n + a_{n - 1} z^{n - 1} + \\cdots + a_1 z + a_0$ be a polynomial over complex numbers where $a_0, \\ldots, a_n$ are real numbers. Let $\\alpha \\in \\C$ be a root of $f$. Then $\\overline \\alpha$ is also a root of $f$, where $\\overline \\alpha$ denotes the complex conjugate of $\\alpha$. That is, all complex roots of $f$ appear as conjugate pairs."}
+{"_id": "16098", "title": "Equation of Circle in Complex Plane/Formulation 1", "text": "Let $\\C$ be the complex plane. Let $C$ be a circle in $\\C$ whose radius is $r \\in \\R_{>0}$ and whose center is $\\alpha \\in \\C$. Then $C$ may be written as: :$\\cmod {z - \\alpha} = r$ where $\\cmod {\\, \\cdot \\,}$ denotes complex modulus."}
+{"_id": "16099", "title": "Equation of Circle in Complex Plane/Formulation 1/Interior", "text": "The points in $\\C$ which correspond to the interior of $C$ can be defined by: :$\\left\\lvert{z - \\alpha}\\right\\rvert < r$"}
+{"_id": "16100", "title": "Equation of Circle in Complex Plane/Formulation 1/Exterior", "text": "The points in $\\C$ which correspond to the exterior of $C$ can be defined by: :$\\left\\lvert{z - \\alpha}\\right\\rvert > r$"}
+{"_id": "16101", "title": "Condition for Complex Number to be in Right Half Plane", "text": "Let $\\C$ be the complex plane. Let $P$ be the half-plane of $\\C$ to the right of the infinite straight line $x = \\lambda$. The points in $P$ can be defined by: :$\\map \\Re z > \\lambda$ where $\\map \\Re z$ denotes the real part of $z$."}
+{"_id": "16103", "title": "Equation of Imaginary Axis in Complex Plane", "text": "Let $\\C$ be the complex plane. Let $z \\in \\C$ be subject to the condition: :$\\cmod {z - 1} = \\cmod {z + 1}$ where $\\cmod {\\, \\cdot \\,}$ denotes complex modulus. Then the locus of $z$ is the imaginary axis."}
+{"_id": "16105", "title": "Conversion between Cartesian and Polar Coordinates in Plane", "text": "Let $S$ be the plane. Let a cartesian plane $C$ be applied to $S$. Let a polar coordinate plane $P$ be superimposed upon $C$ such that: :$(1): \\quad$ The origin of $C$ coincides with the pole of $P$. :$(2): \\quad$ The $x$-axis of $C$ coincides with the polar axis of $P$. Let $p$ be a point in $S$. Let $p$ be specified as $p = \\polar {r, \\theta}$ expressed in the polar coordinates of $P$. Then $p$ is expressed as $\\tuple {r \\cos \\theta, r \\sin \\theta}$ in $C$. Contrariwise, let $p$ be expressed as $\\tuple {x, y}$ in the cartesian coordinates of $C$. Then $p$ is expressed as: :$p = \\polar {\\sqrt {x^2 + y^2}, \\arctan \\dfrac y x + \\pi \\sqbrk {x < 0 \\text{ or } y < 0} + \\pi \\sqbrk {x > 0 \\text{ and } y < 0} }$ where: :$\\sqbrk {\\, \\cdot \\,}$ is Iverson's convention. :$\\arctan$ denotes the arctangent function."}
+{"_id": "16107", "title": "Polar Form of Reciprocal of Complex Number", "text": "Let $z := r \\left({\\cos \\theta + i \\sin \\theta}\\right) \\in \\C$ be a complex number expressed in polar form. Then: :$\\dfrac 1 z = \\dfrac {\\cos \\theta - i \\sin \\theta} r$"}
+{"_id": "16108", "title": "Argument of Complex Conjugate equals Argument of Reciprocal", "text": "Let $z \\in \\C$ be a complex number. Then: :$\\arg {\\overline z} = \\arg \\dfrac 1 z = -\\arg z$ where: :$\\arg$ denotes the argument of a complex number :$\\overline z$ denotes the complex conjugate of $z$."}
+{"_id": "16109", "title": "Multiplication by Imaginary Unit is Equivalent to Rotation through Right Angle", "text": "Let $z \\in \\C$ be a complex number. Let $z$ be interpreted as a vector in the complex plane. Let $w \\in \\C$ be the complex number defined as $z$ multiplied by the imaginary unit $i$: :$w = i z$ Then $w$ can be interpreted as the vector $z$ after being rotated through a right angle in an anticlockwise direction."}
+{"_id": "16110", "title": "Multiplication of Complex Number by -1 is Equivalent to Rotation through Two Right Angles", "text": "Let $z \\in \\C$ be a complex number. Let $z$ be interpreted as a vector in the complex plane. Let $w \\in \\C$ be the complex number defined as $z$ multiplied by $-1$: :$w = \\left({-1}\\right) z$ Then $w$ can be interpreted as the vector $z$ after being rotated through two right angles. The direction of rotation is usually interpreted as being anticlockwise, but a rotated through two right angles is the same whichever direction the rotation is performed."}
+{"_id": "16112", "title": "Geometrical Interpretation of Complex Addition", "text": "Let $a, b \\in \\C$ be complex numbers expressed as vectors $\\mathbf a$ and $\\mathbf b$ respectively. Let $OA$ and $OB$ be two adjacent sides of the parallelogram $OACB$ such that $OA$ corresponds to $\\mathbf a$ and $OB$ corresponds to $\\mathbf b$. Then the diagonal $OC$ of $OACB$ corresponds to $\\mathbf a + \\mathbf b$, the sum of $a$ and $b$ expressed as a vector."}
+{"_id": "16113", "title": "Geometrical Interpretation of Complex Subtraction", "text": "Let $a, b \\in \\C$ be complex numbers expressed as vectors $\\mathbf a$ and $\\mathbf b$ respectively. Let $OA$ and $OB$ be two adjacent sides of the parallelogram $OACB$ such that $OA$ corresponds to $\\mathbf a$ and $OB$ corresponds to $\\mathbf b$. Then the diagonal $BA$ of $OACB$ corresponds to $\\mathbf a - \\mathbf b$, the difference of $a$ and $b$ expressed as a vector."}
+{"_id": "16114", "title": "Condition for Collinearity of Points in Complex Plane/Formulation 1", "text": "Let $z_1$, $z_2$ and $z_3$ be points in the complex plane. Then $z_1$, $z_2$ and $z_3$ are collinear {{iff}}: :$\\dfrac {z_1 - z_3} {z_3 - z_2} = \\lambda$ where $\\lambda \\in \\R$ is a real number. If this is the case, then $z_3$ divides the line segment in the ratio $\\lambda$. If $\\lambda > 0$ then $z_3$ is between $z_1$ and $z_2$, and if $\\lambda < 0$ then $z_3$ is outside the line segment joining $z_1$ to $z_2$."}
+{"_id": "16118", "title": "Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Sum of Series", "text": "The following definition of the concept of the real exponential function:"}
+{"_id": "16119", "title": "Equivalence of Definitions of Real Exponential Function/Inverse of Natural Logarithm implies Limit of Sequence", "text": "The following definition of the concept of the real exponential function:"}
+{"_id": "16120", "title": "Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Extension of Rational Exponential", "text": "The following definition of the concept of the real exponential function:"}
+{"_id": "16121", "title": "Equivalence of Definitions of Real Exponential Function/Extension of Rational Exponential implies Differential Equation", "text": "The following definition of the concept of the real exponential function:"}
+{"_id": "16126", "title": "Exponential of Sum/Complex Numbers/General Result", "text": "Let $m \\in \\N_{>0}$ be a natural number. Let $z_1, z_2, \\ldots, z_m \\in \\C$ be complex numbers. Let $\\exp z$ be the exponential of $z$. Then: :$\\displaystyle \\map \\exp {\\sum_{j \\mathop = 1}^m z_j} = \\prod_{j \\mathop = 1}^m \\paren {\\exp z_j}$"}
+{"_id": "16127", "title": "Exponential of Sum/Complex Numbers/General Result/Corollary", "text": "Let $m \\in \\Z_{>0}$ be a positive integer. Let $z \\in \\C$ be a complex number. Let $\\exp z$ be the exponential of $z$. Then: : $\\displaystyle \\exp \\paren {m z} = \\paren {\\exp z}^m$"}
+{"_id": "16129", "title": "Sum over k from 1 to n of n Choose k by Sine of n Theta", "text": ":$\\displaystyle \\sum_{k \\mathop = 1}^n \\dbinom n k \\sin k \\theta = \\paren {2 \\cos \\dfrac \\theta 2}^n \\sin \\dfrac {n \\theta} 2$"}
+{"_id": "16130", "title": "Point of Perpendicular Intersection on Real Line from Points in Complex Plane", "text": "Let $a, b \\in \\C$ be complex numbers represented by the points $A$ and $B$ respectively in the complex plane. Let $x \\in \\R$ be a real number represented by the point $X$ on the real axis such that $AXB$ is a right triangle with $X$ as the right angle. Then: :$x = \\dfrac {a_x - b_x \\pm \\sqrt {a_x^2 + b_x^2 + 2 a_x b_x - 4 a_y b_y} } 2$ where: : $a = a_x + a_y i, b = b_x + b_y i$ {{mistake|What rubbish. Working on it.}}"}
+{"_id": "16131", "title": "Condition for Points in Complex Plane to form Parallelogram/Examples/2+i, 3+2i, 2+3i, 1+2i", "text": "The points in the complex plane represented by the complex numbers: :$2 + i, 3 + 2 i, 2 + 3 i, 1 + 2 i$ are the vertices of a square."}
+{"_id": "16132", "title": "Equation of Ellipse in Complex Plane", "text": "Let $\\C$ be the complex plane. Let $E$ be an ellipse in $\\C$ whose major axis is $d \\in \\R_{>0}$ and whose foci are at $\\alpha, \\beta \\in \\C$. Then $C$ may be written as: :$\\cmod {z - \\alpha} + \\cmod {z - \\beta} = d$ where $\\cmod {\\, \\cdot \\,}$ denotes complex modulus."}
+{"_id": "16133", "title": "Condition for Collinearity of Points in Complex Plane/Formulation 2", "text": "Let $z_1, z_2, z_3$ be distinct complex numbers. Then: : $z_1, z_2, z_3$ are collinear in the complex plane {{iff}}: ::$\\exists \\alpha, \\beta, \\gamma \\in \\R: \\alpha z_1 + \\beta z_2 + \\gamma z_3 = 0$ :where: ::$\\alpha + \\beta + \\gamma = 0$ ::not all of $\\alpha, \\beta, \\gamma$ are zero."}
+{"_id": "16136", "title": "Quadrilateral in Complex Plane is Cyclic iff Cross Ratio of Vertices is Real", "text": "Let $z_1, z_2, z_3, z_4$ be distinct complex numbers. Then: : $z_1, z_2, z_3, z_4$ define the vertices of a cyclic quadrilateral {{iff}} their cross ratio: : $\\paren {z_1, z_3; z_2, z_4} = \\dfrac {\\paren {z_1 - z_2} \\paren {z_3 - z_4} } {\\paren {z_1 - z_4} \\paren {z_3 - z_2} }$ is wholly real."}
+{"_id": "16137", "title": "Definition:Cross Ratio", "text": "Let $z_1, z_2, z_3, z_4$ be distinct complex numbers. The cross ratio of $z_1, z_2, z_3, z_4$ is defined and denoted: : $\\paren {z_1, z_3; z_2, z_4} = \\dfrac {\\paren {z_1 - z_2} \\paren {z_3 - z_4} } {\\paren {z_1 - z_4} \\paren {z_3 - z_2} }$"}
+{"_id": "16138", "title": "Equation relating Points of Parallelogram in Complex Plane", "text": "Let $ABVU$ be a parallelogram in the complex plane whose vertices correspond to the complex numbers $a, b, v, u$ respectively. Let $\\angle BAU = \\alpha$. Let $\\cmod {UA} = \\lambda \\cmod {AB}$. :510px Then: :$u = \\paren {1 - q} a + q b$ :$v = -q a + \\paren {1 + q} b$ where: :$q = \\lambda e^{i \\alpha}$"}
+{"_id": "16139", "title": "Circle of Apollonius in Complex Plane", "text": "Let $\\C$ be the complex plane. Let $\\lambda \\in \\R$ be a real number such that $\\lambda \\ne 0$ and $\\lambda \\ne 1$. Let $a, b \\in \\C$ such that $a \\ne b$. The equation: :$\\cmod {\\dfrac {z - a} {z - b} } = \\lambda$ decribes a circle of Apollonius $C$ in $\\C$ such that: :if $\\lambda < 0$, then $a$ is inside $C$ and $b$ is outside :if $\\lambda > 0$, then $b$ is inside $C$ and $a$ is outside. If $\\lambda = 1$ then $z$ describes the perpendicular bisector of the line segment joining $a$ to $b$."}
+{"_id": "16140", "title": "P-adic Norm is Non-Archimedean Norm", "text": "The $p$-adic norm forms a non-Archimedean norm on the rational numbers $\\Q$."}
+{"_id": "16141", "title": "Circle of Apollonius is Circle", "text": "Let $A, B$ be distinct points in the plane. Let $\\lambda \\in \\R_{>0}$ be a strictly positive real number. Let $X$ be the locus of points in the plane such that: :$XA = \\lambda \\paren {XB}$ Then $X$ is in the form of a circle, known as a circle of Apollonius. :400px If $\\lambda < 1$, then $A$ is inside the circle, and $B$ is outside. If $\\lambda > 1$, then $B$ is inside the circle, and $A$ is outside."}
+{"_id": "16145", "title": "Cosine of Angle plus Integer Multiple of Pi", "text": ":$\\map \\cos {\\theta + n \\pi} = \\paren {-1}^n \\cos \\theta$"}
+{"_id": "16146", "title": "Sine of Angle plus Integer Multiple of Pi", "text": ":$\\map \\sin {\\theta + n \\pi} = \\paren {-1}^n \\sin \\theta$"}
+{"_id": "16147", "title": "Complex Division/Examples/(1 + sin theta + i cos theta) (1 + sin theta - i cos theta)^-1", "text": ":$\\dfrac {1 + \\sin \\theta + i \\cos \\theta} {1 + \\sin \\theta - i \\cos \\theta} = \\sin \\theta + i \\cos \\theta$"}
+{"_id": "16148", "title": "Sum of 1 + sin pi by 5 plus i cos pi by 5 to Fifth Power plus i times its Conjugate", "text": ":$\\paren {1 + \\sin \\dfrac \\pi 5 + i \\cos \\dfrac \\pi 5}^5 + i \\paren {1 + \\sin \\dfrac \\pi 5 - i \\cos \\dfrac \\pi 5}^5 = 0$"}
+{"_id": "16149", "title": "Complex Roots of Unity include 1", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $U_n = \\set {z \\in \\C: z^n = 1}$ be the set of complex $n$th roots of unity. Then $1 \\in U_n$. That is, $1$ is always one of the complex $n$th roots of unity of any $n$."}
+{"_id": "16150", "title": "Positive Real Complex Root of Unity", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $U_n = \\set {z \\in \\C: z^n = 1}$ be the set of complex $n$th roots of unity. The only $x \\in U_n$ such that $x \\in \\R_{>0}$ is: :$x = 1$ That is, $1$ is the only complex $n$th root of unity which is a positive real number."}
+{"_id": "16152", "title": "Real Complex Roots of Unity for Even Index", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer such that $n$ is even. Let $U_n = \\set {z \\in \\C: z^n = 1}$ be the set of complex $n$th roots of unity. The only $x \\in U_n$ such that $x \\in \\R$ are: :$x = 1$ or $x \\in -1$ That is, $1$ and $-1$ are the only complex $n$th roots of unity which are real number."}
+{"_id": "16153", "title": "Modulus of Complex Root of Unity equals 1", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer such that $n$ is even. Let $U_n = \\set {z \\in \\C: z^n = 1}$ be the set of complex $n$th roots of unity. Let $z \\in U_n$. Then: :$\\cmod z = 1$ where $\\cmod z$ denotes the modulus of $z$."}
+{"_id": "16154", "title": "First Complex Root of Unity is Primitive", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $U_n$ denote the complex $n$th roots of unity: :$U_n = \\set {z \\in \\C: z^n = 1}$ Let $\\alpha_1 = \\exp \\paren {\\dfrac {2 \\pi i} n}$ denote the first complex root of unity. Then $\\alpha_1$ is a primitive complex root of unity."}
+{"_id": "16155", "title": "Powers of Primitive Complex Root of Unity form Complete Set", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $U_n$ denote the complex $n$th roots of unity: :$U_n = \\set {z \\in \\C: z^n = 1}$ Let $\\alpha_k = \\exp \\paren {\\dfrac {2 k \\pi i} n}$ denote the $k$th complex root of unity. Let $\\alpha_k$ be a primitive complex root of unity. Let $V_k = \\set { {\\alpha_k}^r: r \\in \\set {0, 1, \\ldots, n - 1} }$. Then: : $V_k = U_n$ That is, $V_k = \\set { {\\alpha_k}^r: r \\in \\set {0, 1, \\ldots, n - 1} }$ forms the complete set of complex $n$th roots of unity."}
+{"_id": "16156", "title": "Sum of Powers of Primitive Complex Roots of Unity", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $U_n$ denote the complex $n$th roots of unity: :$U_n = \\set {z \\in \\C: z^n = 1}$ Let $\\alpha = \\exp \\paren {\\dfrac {2 k \\pi i} n}$ denote a primitive complex $n$th root of unity. Let $s \\in \\Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 0}^{n - 1} \\alpha^{j s}       | r = 1 + \\alpha^s + \\alpha^{2 s} + \\cdots + \\alpha^{\\paren {n - 1} s}       | c =  }} {{eqn | r = \\begin {cases} n & : n \\divides s \\\\ 0 & : n \\nmid s \\end {cases}       | c =  }} {{end-eqn}} where: :$n \\divides s$ denotes that $n$ is a divisor of $s$ :$n \\nmid s$ denotes that $n$ is not a divisor of $s$."}
+{"_id": "16157", "title": "Difference of Two Powers/Examples/Difference of Two Cubes/Corollary", "text": ":$x^3 - 1 = \\paren {x - 1} \\paren {x^2 + x + 1}$"}
+{"_id": "16158", "title": "Sum of Cube Roots of Unity", "text": "Let $U_3 = \\set {1, \\omega, \\omega^2}$ denote the Cube Roots of Unity. Then: :$1 + \\omega + \\omega^2 = 0$"}
+{"_id": "16160", "title": "Sum of Cubes of Three Indeterminates Minus 3 Times their Product", "text": "For indeterminates $x, y, z$: :$x^3 + y^3 + z^3 - 3 x y z = \\paren {x + y + z} \\paren {x + \\omega y + \\omega^2 z} \\paren {x + \\omega^2 y + \\omega z}$ where $\\omega = -\\dfrac 1 2 + \\dfrac {\\sqrt 3} 2$"}
+{"_id": "16164", "title": "Combination Theorem for Cauchy Sequences/Sum Rule", "text": ":$\\sequence {x_n + y_n}$ is a Cauchy sequence."}
+{"_id": "16165", "title": "Power of Complex Number minus 1", "text": "Let $z \\in \\C$ be a complex number. Then: :$z^n - 1 = \\displaystyle \\prod_{k \\mathop = 0}^{n - 1} \\paren {z - \\alpha^k}$ where $\\alpha$ is a primitive complex $n$th root of unity."}
+{"_id": "16166", "title": "Power of Complex Number minus 1/Corollary", "text": "Let $z \\in \\C$ be a complex number. Then: :$\\displaystyle \\sum_{k \\mathop = 0}^{n - 1} z^k = \\prod_{k \\mathop = 1}^{n - 1} \\paren {z - \\alpha^k}$ where $\\alpha$ is a primitive complex $n$th root of unity."}
+{"_id": "16167", "title": "Product of Differences between 1 and Complex Roots of Unity", "text": "Let $\\alpha$ be a primitive complex $n$th root of unity. Then: :$\\displaystyle \\prod_{k \\mathop = 1}^{n - 1} \\paren {1 - \\alpha^k} = n$"}
+{"_id": "16168", "title": "Complex Roots of Unity occur in Conjugate Pairs", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $U_n$ denote the complex $n$th roots of unity: :$U_n = \\set {z \\in \\C: z^n = 1}$ Let $\\alpha \\in U_n$ be the first complex $n$th root of unity. Then: :$\\forall k \\in \\Z_{>0}, k < \\dfrac n 2: \\overline {\\alpha^k} = \\alpha^{n - k}$ That is, each of the complex $n$th roots of unity occur in conjugate pairs: :$\\tuple {\\alpha, \\alpha^{n - 1} }; \\tuple {\\alpha^2, \\alpha^{n - 2} }; \\ldots; \\tuple {\\alpha^s, \\alpha^{n - s} }$ where: :$s = \\dfrac {n - 1} 2$ for odd $n$ :$s = \\dfrac {n - 2} 2$ for even $n$."}
+{"_id": "16169", "title": "Combination Theorem for Cauchy Sequences/Product Rule", "text": ":$\\sequence {x_n y_n}$ is a Cauchy sequence."}
+{"_id": "16170", "title": "Factorisation of x^(2n+1)-1 in Real Domain", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: :$z^{2 n + 1} - 1 = \\paren {z - 1} \\displaystyle \\prod_{k \\mathop = 1}^n \\paren {z^2 - 2 \\cos \\dfrac {2 \\pi k} {2 n + 1} + 1}$"}
+{"_id": "16172", "title": "Factorisation of z^n-a", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $a \\in \\C$ be a complex number. Then: :$z^n - a = \\displaystyle \\prod_{k \\mathop = 0}^{n - 1} \\paren {z - \\alpha^k b}$ where: :$\\alpha$ is a primitive complex $n$th root of unity :$b$ is any complex number such that $b^n = a$."}
+{"_id": "16174", "title": "Cosine of 144 Degrees", "text": ":$\\cos 144 \\degrees = \\cos \\dfrac {4 \\pi} 5 = -\\dfrac \\phi 2 = -\\dfrac {1 + \\sqrt 5} 4$"}
+{"_id": "16175", "title": "Sine of 144 Degrees", "text": ":$\\sin 144 \\degrees = \\cos \\dfrac {4 \\pi} 5 = \\sqrt {\\dfrac 5 8 - \\dfrac {\\sqrt 5} 8}$"}
+{"_id": "16176", "title": "Cube Root of Unity if Modulus is 1 and Real Part is Minus Half", "text": "Let $z \\in \\C$ be a complex number such that: :$\\cmod z = 1$ :$\\Re \\paren z = -\\dfrac 1 2$ where: :$\\cmod z$ denotes the complex modulus of $z$ :$\\Re \\paren z$ denotes the real part of $z$. Then: :$z^3 = 1$"}
+{"_id": "16177", "title": "Sum of Two Cubes in Complex Domain", "text": ":$a^3 + b^3 = \\paren {a + b} \\paren {a \\omega + b \\omega^2} \\paren {a \\omega^2 + b \\omega}$ where: : $\\omega = -\\dfrac 1 2 + \\dfrac {\\sqrt 3} 2$"}
+{"_id": "16179", "title": "Roots of Complex Number/Examples/z^8 + 1 = 0", "text": "The roots of the polynomial: :$z^8 + 1 = 0$ are: :$\\set {\\cos \\dfrac {\\paren {2 k + 1} \\pi} 8 + i \\sin \\dfrac {\\paren {2 k + 1} \\pi} 8: k \\in \\set {0, 1, \\ldots, 7} }$"}
+{"_id": "16180", "title": "Quadruple Angle Formulas/Cosine/Factor Form", "text": ":$\\cos 4 \\theta = \\paren {\\cos \\theta - \\cos \\dfrac \\pi 8} \\paren {\\cos \\theta - \\cos \\dfrac {3 \\pi} 8} \\paren {\\cos \\theta - \\cos \\dfrac {5 \\pi} 8} \\paren {\\cos \\theta - \\cos \\dfrac {7 \\pi} 8}$"}
+{"_id": "16181", "title": "Factorisation of z^n+1", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: :$z^n + 1 = \\displaystyle \\prod_{k \\mathop = 0}^{n - 1} \\paren {z - \\exp \\dfrac {\\paren {2 k + 1} i \\pi} n}$"}
+{"_id": "16183", "title": "Factorisation of z^(2n)+1 in Real Domain", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: :$z^{2 n} + 1 = \\displaystyle \\prod_{k \\mathop = 1}^n \\paren {z^2 - 2 z \\cos \\dfrac {\\paren {2 k + 1} \\pi} {2 n} + 1}$"}
+{"_id": "16185", "title": "Roots of Complex Number/Examples/z^5 + 1 = 0", "text": "The roots of the polynomial: :$z^5 + 1 = 0$ are: :$\\set {\\cos \\dfrac \\pi 5 \\pm i \\sin \\dfrac \\pi 5, \\cos \\dfrac {3 \\pi} 5 \\pm i \\sin \\dfrac {3 \\pi} 5, -1}$"}
+{"_id": "16186", "title": "Combination Theorem for Cauchy Sequences", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Let $\\sequence {x_n}$,  $\\sequence {y_n}$ be Cauchy sequences in $R$. Let $a, b \\in R$. The following results hold:"}
+{"_id": "16189", "title": "Convergence of Modulus of Convergent Complex Sequence", "text": "Let $\\sequence {z_n}$ be a sequence in $\\C$. Let $\\sequence {z_n}$ converge to a value $c \\in \\C$. Let $\\cmod z$ denote the modulus of a complex number $z$. Then: :$\\sequence {\\cmod {z_n} }$ converges to a value $\\cmod c$."}
+{"_id": "16190", "title": "Combination Theorem for Sequences/Real/Difference Rule", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {x_n - y_n} = l - m$"}
+{"_id": "16198", "title": "Combination Theorem for Sequences/Complex/Sum Rule", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {z_n + w_n} = c + d$"}
+{"_id": "16200", "title": "Combination Theorem for Sequences/Complex", "text": "Let $\\sequence {z_n}$ and $\\sequence {w_n}$ be sequences in $\\C$. Let $\\sequence {z_n}$ and $\\sequence {w_n}$ be convergent to the following limits: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} z_n = c$ :$\\displaystyle \\lim_{n \\mathop \\to \\infty} w_n = d$ Let $\\lambda, \\mu \\in \\C$. Then the following results hold: === Sum Rule === {{:Combination Theorem for Sequences/Complex/Sum Rule}} === Difference Rule === {{:Combination Theorem for Sequences/Complex/Difference Rule}} === Multiple Rule === {{:Combination Theorem for Sequences/Complex/Multiple Rule}} === Combined Sum Rule === {{:Combination Theorem for Sequences/Complex/Combined Sum Rule}} === Product Rule === {{:Combination Theorem for Sequences/Complex/Product Rule}} === Quotient Rule === {{:Combination Theorem for Sequences/Complex/Quotient Rule}}"}
+{"_id": "16201", "title": "Combination Theorem for Sequences/Complex/Difference Rule", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {z_n - w_n} = c - d$"}
+{"_id": "16202", "title": "Combination Theorem for Sequences/Complex/Multiple Rule", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {\\lambda z_n} = \\lambda c$"}
+{"_id": "16204", "title": "Combination Theorem for Sequences/Complex/Combined Sum Rule", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {\\lambda z_n + \\mu w_n} = \\lambda c + \\mu d$"}
+{"_id": "16206", "title": "Combination Theorem for Sequences/Normed Division Ring/Sum Rule", "text": ":$\\sequence {x_n + y_n}$ is convergent and $\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {x_n + y_n} = l + m$"}
+{"_id": "16208", "title": "Absolute Value of Product", "text": "Let $x, y \\in \\R$ be real numbers. Then: :$\\size {x y} = \\size x \\size y$ where $\\size x$ denotes the absolute value of $x$."}
+{"_id": "16209", "title": "Combination Theorem for Sequences/Complex/Product Rule", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {z_n w_n} = c d$"}
+{"_id": "16210", "title": "Combination Theorem for Sequences/Complex/Quotient Rule", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\frac {z_n} {w_n} = \\frac c d$ provided that $d \\ne 0$."}
+{"_id": "16211", "title": "Combination Theorem for Sequences/Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\sequence {x_n}$,  $\\sequence {y_n} $ be sequences in $R$. Let $\\sequence {x_n}$ and $\\sequence {y_n}$ be convergent in the norm $\\norm {\\, \\cdot \\,}$ to the following limits: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ :$\\displaystyle \\lim_{n \\mathop \\to \\infty} y_n = m$ Let $\\lambda, \\mu \\in R$. Then the following results hold: === Sum Rule === {{:Combination Theorem for Sequences/Normed Division Ring/Sum Rule}} === Difference Rule === {{:Combination Theorem for Sequences/Normed Division Ring/Difference Rule}} === Multiple Rule === {{:Combination Theorem for Sequences/Normed Division Ring/Multiple Rule}} === Combined Sum Rule === {{:Combination Theorem for Sequences/Normed Division Ring/Combined Sum Rule}} === Product Rule === {{:Combination Theorem for Sequences/Normed Division Ring/Product Rule}} === Inverse Rule === {{:Combination Theorem for Sequences/Normed Division Ring/Inverse Rule}} === Quotient Rule === {{:Combination Theorem for Sequences/Normed Division Ring/Quotient Rule}}"}
+{"_id": "16212", "title": "Combination Theorem for Sequences/Normed Division Ring/Product Rule", "text": ":$\\sequence {x_n y_n}$ is convergent to the limit $\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {x_n y_n} = l m$"}
+{"_id": "16213", "title": "Combination Theorem for Sequences/Normed Division Ring/Multiple Rule", "text": ":$\\sequence {\\lambda x_n}$ is convergent and $\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {\\lambda x_n} = \\lambda l$"}
+{"_id": "16214", "title": "Combination Theorem for Sequences/Normed Division Ring/Combined Sum Rule", "text": ":$\\sequence {\\lambda x_n + \\mu y_n }$ is convergent and $\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {\\lambda x_n + \\mu y_n} = \\lambda l + \\mu m$"}
+{"_id": "16215", "title": "Combination Theorem for Sequences/Normed Division Ring/Difference Rule", "text": ":$\\sequence {x_n - y_n}$ is convergent and $\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {x_n - y_n} = l - m$"}
+{"_id": "16219", "title": "Cauchy's Convergence Criterion for Series", "text": "A series $\\displaystyle \\sum_{i \\mathop = 0}^\\infty a_i$ is convergent {{iff}} for every $\\epsilon > 0$ there is a number $N \\in \\N$ such that: :$\\size {a_{n + 1} + a_{n + 2} + \\cdots + a_m} < \\epsilon$ holds for all $n \\ge N$ and $m > n$. {{explain|What domain is $\\sequence {a_n}$ in?}}"}
+{"_id": "16222", "title": "Combination Theorem for Sequences/Normed Division Ring/Product Rule/Proof 2", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Let $\\sequence {x_n}$ and $\\sequence {y_n}$ be sequences in $R$. Let $\\sequence {x_n}$ and $\\sequence {y_n}$ be convergent in the norm $\\norm {\\,\\cdot\\,}$ to the following limits: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ :$\\displaystyle \\lim_{n \\mathop \\to \\infty} y_n = m$ Then: {{:Combination Theorem for Sequences/Normed Division Ring/Product Rule}}"}
+{"_id": "16223", "title": "Convergent Sequence in Normed Division Ring is Bounded", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,}}$ be a normed division ring. Let $\\sequence {x_n}$ be a sequence in $R$. Let $\\sequence {x_n}$ be convergent in the norm $\\norm {\\,\\cdot\\,}$ to the following limit: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ Then $\\sequence {x_n}$ is bounded."}
+{"_id": "16224", "title": "Metric Induced by Norm on Normed Division Ring is Metric", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Let $d$ be the metric induced by $\\norm{\\,\\cdot\\,}$. Then $d$ is a metric."}
+{"_id": "16226", "title": "Existence of Radius of Convergence of Complex Power Series/Divergence", "text": "Let ${B_R}^- \\paren \\xi$ denote the closed $R$-ball of $\\xi$. Let $z \\notin {B_R}^- \\paren \\xi$. Then $S \\paren z$ is divergent."}
+{"_id": "16227", "title": "Radius of Convergence of Power Series in Complex Plane", "text": "Consider the complex power series: :$S = \\displaystyle \\sum_{k \\mathop = 0}^\\infty z^n$ The radius of convergence $S$ is $1$."}
+{"_id": "16228", "title": "Radius of Convergence of Power Series Expansion for Cosine Function", "text": "The cosine function has the complex power series expansion: {{begin-eqn}} {{eqn | l = C \\paren z       | r = \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {z^{2 n} } {\\paren {2 n}!}       | c =  }} {{eqn | r = 1 - \\frac {z^2} {2!} + \\frac {z^4} {4!} - \\frac {z^6} {6!} + \\cdots       | c =  }} {{end-eqn}} which is the power series expansion of the cosine function. This is valid for all $z \\in \\C$."}
+{"_id": "16229", "title": "Radius of Convergence of Power Series Expansion for Sine Function", "text": "The sine function has the complex power series expansion: {{begin-eqn}} {{eqn | l = S \\paren z       | r = \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {z^{2 n + 1} } {\\paren {2 n + 1}!}       | c =  }} {{eqn | r = z - \\frac {z^3} {3!} + \\frac {z^5} {5!} - \\frac {z^7} {7!} + \\cdots       | c =  }} {{end-eqn}} which is the power series expansion of the sine function. This is valid for all $z \\in \\C$."}
+{"_id": "16230", "title": "Product of Absolutely Convergent Series", "text": "Let $f \\paren z = \\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ and $g \\paren z = \\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n$ be two real or complex series that are absolutely convergent. Then $f \\paren z g \\paren z$ is an absolutely convergent series, and: :$f \\paren z g \\paren z = \\displaystyle \\sum_{n \\mathop = 1}^\\infty c_n$ where: :$c_n = \\displaystyle \\sum_{k \\mathop = 1}^n a_k b_{n - k}$"}
+{"_id": "16234", "title": "Sum of Infinite Series of Product of nth Power of Cosine by nth Multiple of Cosine", "text": "Let $0 < \\theta < \\dfrac \\pi 2$. Then: {{begin-eqn}} {{eqn | l = \\sum_{n \\mathop = 0}^\\infty \\cos^n \\theta \\, \\map \\cos {n + 1} \\theta       | r = \\cos \\theta + \\cos \\theta \\cos 2 \\theta + \\cos^2 \\theta \\cos 3 \\theta + \\cos^3 \\theta \\cos 4 \\theta + \\cdots       | c =  }} {{eqn | r = 0       | c =  }} {{end-eqn}}"}
+{"_id": "16236", "title": "Euler's Formula/Real Domain", "text": "Let $\\theta \\in \\R$ be a real number. Then: :$e^{i \\theta} = \\cos \\theta + i \\sin \\theta$"}
+{"_id": "16242", "title": "Reverse Triangle Inequality/Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Then: :$\\forall x, y \\in R: \\norm {x - y} \\ge \\bigsize {\\norm x - \\norm y}$"}
+{"_id": "16243", "title": "Sequence Converges to Within Half Limit/Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring with zero $0$. Let $\\sequence {x_n}$ be a sequence in $R$. Let $\\sequence {x_n}$ be convergent in the norm $\\norm {\\, \\cdot \\,}$ to the following limit: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l \\ne 0$ Then: :$\\exists N: \\forall n > N: \\norm {x_n} > \\dfrac {\\norm l} 2$"}
+{"_id": "16246", "title": "Limit of Subsequence equals Limit of Sequence/Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring with zero: $0$. Let $\\sequence {x_n}$ be a sequence in $R$. Let $\\sequence {x_n}$ be convergent in the norm $\\norm {\\, \\cdot \\,}$ to the following limit: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ Let $\\sequence {x_{n_r} }$ be a subsequence of $\\sequence {x_n}$. Then: :$\\sequence {x_{n_r} }$ is convergent and $\\displaystyle \\lim_{r \\mathop \\to \\infty} x_{n_r} = l$"}
+{"_id": "16247", "title": "Combination Theorem for Sequences/Normed Division Ring/Quotient Rule", "text": "Suppose $m \\ne 0$. Then: :$\\exists k \\in \\N : \\forall n \\in \\N: y_{k + n} \\ne 0$ and the sequences: :$\\sequence {x_{k + n} \\ {y_{k + n} }^{-1} }$ and $\\sequence { {y_{k + n} }^{-1} \\ x_{k + n} }$ are well-defined and convergent with: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_{k + n} \\ {y_{k + n} }^{-1} = l m^{-1}$ :$\\displaystyle \\lim_{n \\mathop \\to \\infty} {y_{k + n} }^{-1} \\ x_{k + n} = m^{-1} l$"}
+{"_id": "16250", "title": "Boundedness of Real Sine and Cosine", "text": "Let $x \\in \\R$ be a real number. Then:"}
+{"_id": "16251", "title": "Real Sine Function is Bounded", "text": ":$\\size {\\sin x} \\le 1$"}
+{"_id": "16253", "title": "Complex Cosine Function is Unbounded", "text": "The complex cosine function is unbounded."}
+{"_id": "16256", "title": "Argument of Exponential is Imaginary Part plus Multiple of 2 Pi", "text": "Let $z \\in \\C$ be a complex number. Let $\\exp z$ denote the complex exponential of $z$. Let $\\arg z$ denote the argument of $z$. Then: :$\\map \\arg {\\exp z} = \\set {\\Im z + 2 k \\pi: k \\in \\Z}$ where $\\Im z$ denotes the imaginary part of $z$."}
+{"_id": "16257", "title": "Real Part of Sine of Complex Number", "text": "Let $z = x + i y \\in \\C$ be a complex number, where $x, y \\in \\R$. Let $\\sin z$ denote the complex sine function. Then: :$\\map \\Re {\\sin z} = \\sin x \\cosh y$ where: :$\\Re z$ denotes the real part of a complex number $z$ :$\\sin$ denotes the sine function (real and complex) :$\\cosh$ denotes the hyperbolic cosine function."}
+{"_id": "16261", "title": "Inverse Tangent of i", "text": "The inverse tangent of $i$ is not defined."}
+{"_id": "16262", "title": "Cauchy-Hadamard Theorem/Complex Case", "text": "Let $\\xi \\in \\C$ be a complex number. Let $\\displaystyle S \\paren z = \\sum_{n \\mathop = 0}^\\infty a_n \\paren {z - \\xi}^n$ be a (complex) power series about $\\xi$. Then the radius of convergence $R$ of $S \\paren z$ is given by: :$\\displaystyle \\dfrac 1 R = \\limsup_{n \\mathop \\to \\infty} \\cmod {a_n}^{1/n}$ If: :$\\displaystyle \\limsup_{n \\mathop \\to \\infty} \\cmod {a_n}^{1/n} = 0$ then the radius of convergence is infinite, and $S \\paren z$ is absolutely convergent for all $z \\in \\C$."}
+{"_id": "16265", "title": "Cauchy-Hadamard Theorem/Real Case", "text": "Let $\\xi \\in \\R$ be a real number. Let $\\displaystyle \\map S x = \\sum_{n \\mathop = 0}^\\infty a_n \\paren {x - \\xi}^n$ be a power series about $\\xi$. Then the radius of convergence $R$ of $S \\paren x$ is given by: :$\\displaystyle \\frac 1 R = \\limsup_{n \\mathop \\to \\infty} \\size {a_n}^{1/n}$ If: :$\\displaystyle \\frac 1 R = \\limsup_{n \\mathop \\to \\infty} \\size {a_n}^{1/n} = 0$ then the radius of convergence is infinite and therefore the interval of convergence is $\\R$."}
+{"_id": "16266", "title": "Comparison Test for Convergence of Power Series", "text": "Let $A = \\displaystyle \\sum_{n \\mathop \\ge 0} a_n z^n$ and $B = \\displaystyle \\sum_{n \\mathop \\ge 0} b_n z^n$ be power series in $\\C$. Let $R_A$ and $R_B$ be the radii of convergence of $A$ and $B$ respectively. Let $\\cmod {b_n} \\le \\cmod {a_n}$ for all $n \\in \\N$. Then $R_A \\le R_B$."}
+{"_id": "16268", "title": "Hyperbolic Tangent of Complex Number", "text": "Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then:"}
+{"_id": "16270", "title": "Hyperbolic Cosine of Complex Number", "text": "Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\\cosh \\paren {a + b i} = \\cosh a \\cos b + i \\sinh a \\sin b$ where: :$\\cos$ denotes the real cosine function :$\\sin$ denotes the real sine function :$\\sinh$ denotes the hyperbolic sine function :$\\cosh$ denotes the hyperbolic cosine function"}
+{"_id": "16272", "title": "Hyperbolic Tangent of Complex Number/Formulation 1", "text": ":$\\tanh \\paren {a + b i} = \\dfrac {\\sinh a \\cos b + i \\cosh a \\sin b} {\\cosh a \\cos b + i \\sinh a \\sin b}$"}
+{"_id": "16273", "title": "Hyperbolic Tangent of Complex Number/Formulation 2", "text": ":$\\tanh \\paren {a + b i} = \\dfrac {\\tanh a + i \\tan b} {1 + i \\tanh a \\tan b}$"}
+{"_id": "16274", "title": "Hyperbolic Tangent of Complex Number/Formulation 3", "text": ":$\\tanh \\paren {a + b i} = \\dfrac {\\tanh a - \\tanh a \\tan^2 b} {1 + \\tan^2 a \\tanh^2 b} + \\dfrac {\\tan b + \\tanh^2 a \\tan b} {1 + \\tanh^2 a \\tan^2 b} i$"}
+{"_id": "16275", "title": "Hyperbolic Cosecant of Complex Number", "text": "Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\\map \\csch {a + b i} = \\dfrac {\\sinh a \\cos b - i \\cosh a \\sin b} {\\sinh^2 a \\cos^2 b + \\cosh^2 a \\sin^2 b}$ where: :$\\csch$ denotes the hyperbolic cosecant function. :$\\sin$ denotes the real sine function :$\\cos$ denotes the real cosine function :$\\sinh$ denotes the hyperbolic sine function :$\\cosh$ denotes the hyperbolic cosine function"}
+{"_id": "16276", "title": "Hyperbolic Secant of Complex Number", "text": "Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\\sech \\paren {a + b i} = \\dfrac {\\cosh a \\cos b - i \\sinh a \\sin b} {\\cosh^2 a \\cos^2 b + \\sinh^2 a \\sin^2 b}$ where: :$\\sech$ denotes the hyperbolic secant function. :$\\sin$ denotes the real sine function :$\\cos$ denotes the real cosine function :$\\sinh$ denotes the hyperbolic sine function :$\\cosh$ denotes the hyperbolic cosine function"}
+{"_id": "16277", "title": "Hyperbolic Cotangent of Complex Number", "text": "Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then:"}
+{"_id": "16278", "title": "Hyperbolic Cotangent of Complex Number/Formulation 2", "text": ":$\\map \\coth {a + b i} = \\dfrac {1 + i \\coth a \\cot b} {\\coth a - i \\cot b}$"}
+{"_id": "16279", "title": "Hyperbolic Cotangent of Complex Number/Formulation 3", "text": ":$\\map \\coth {a + b i} = \\dfrac {\\coth a - \\coth a \\cot^2 b} {\\coth^2 a + \\cot^2 b} + \\dfrac {\\coth b + \\coth^2 a \\cot b} {\\coth^2 a + \\cot^2 b} i$"}
+{"_id": "16280", "title": "Hyperbolic Tangent of Complex Number/Formulation 4", "text": ":$\\tanh \\paren {a + b i} = \\dfrac {\\sinh 2 a + i \\sin 2 b} {\\cosh 2 a + \\cos 2 b}$"}
+{"_id": "16281", "title": "Tangent of Complex Number/Formulation 4", "text": ":$\\tan \\paren {a + b i} = \\dfrac {\\sin 2 a + i \\sinh 2 b} {\\cos 2 a + \\cosh 2 b}$"}
+{"_id": "16282", "title": "Characterisation of Cauchy Sequence in Non-Archimedean Norm", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring with non-Archimedean norm $\\norm {\\,\\cdot\\,}$. Let $\\sequence {x_n}$ be a sequence in $R$. Then: :$\\sequence {x_n}$ is a Cauchy sequence {{iff}} $\\displaystyle \\lim_{n \\mathop \\to \\infty} \\norm {x_{n + 1} - x_n} = 0$."}
+{"_id": "16283", "title": "Characterisation of Cauchy Sequence in Non-Archimedean Norm/Necessary Condition", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring with non-Archimedean norm $\\norm {\\,\\cdot\\,}$. Let $\\sequence {x_n}$ be a Cauchy sequence in $R$. Then: :$\\lim_{n \\mathop \\to \\infty} \\norm {x_{n + 1} - x_n} = 0$"}
+{"_id": "16284", "title": "Characterisation of Cauchy Sequence in Non-Archimedean Norm/Sufficient Condition", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring with non-Archimedean norm $\\norm {\\,\\cdot\\,}$. Let $\\sequence {x_n}$ be a sequence in $R$. Let $\\displaystyle \\lim_{n \\mathop \\to \\infty} \\norm {x_{n + 1} - x_n} = 0$. Then: :$\\sequence {x_n}$ is a Cauchy sequence."}
+{"_id": "16285", "title": "Power Series of Sine of Odd Theta", "text": "Let $r \\in \\R$ such that $\\size r < 1$. Let $\\theta \\in \\R$ such that $\\theta \\ne m \\pi$ for any $m \\in \\Z$. Then: {{begin-eqn}} {{eqn | l = \\sum_{k \\mathop \\ge 0} \\map \\sin {2 k + 1} \\theta r^k       | r = \\sin \\theta + r \\sin 3 \\theta + r^2 \\sin 5 \\theta + \\cdots       | c =  }} {{eqn | r = \\dfrac {\\paren {1 + r} \\sin \\theta} {1 - 2 r \\cos 2 \\theta + r^2}       | c =  }} {{end-eqn}}"}
+{"_id": "16287", "title": "Modulus of Limit/Normed Division Ring", "text": "Let $\\struct {R, \\norm { \\, \\cdot \\, } }$ be a normed division ring. Let $\\sequence {x_n}$ be a convergent sequence in $R$ to the limit $l$. That is, let $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$. Then :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\norm {x_n} = \\norm l$"}
+{"_id": "16288", "title": "Three Points in Ultrametric Space have Two Equal Distances", "text": "Let $\\struct {X, d}$ be an ultrametric space. Let $x, y, z \\in X$ with $\\map d {x, z} \\ne \\map d {y, z}$. Then: :$\\map d {x, y} = \\max \\set {\\map d {x, z}, \\map d {y, z} }$"}
+{"_id": "16289", "title": "Three Points in Ultrametric Space have Two Equal Distances/Corollary", "text": "Let $\\struct {X, d}$ be an ultrametric space. Let $x, y, z \\in X$. Then: :at least two of the distances $\\map d {x, y}$, $\\map d {x, z}$ and $\\map d {y, z}$ are equal."}
+{"_id": "16290", "title": "Non-Archimedean Norm iff Non-Archimedean Metric/Necessary Condition", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,}}$ be a non-Archimedean normed division ring. Let $d$ be the metric induced by $\\norm {\\,\\cdot\\,}$. Then $d$ is a non-Archimedean metric."}
+{"_id": "16291", "title": "Non-Archimedean Norm iff Non-Archimedean Metric/Sufficient Condition", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring with zero $0$. Let $d$ be the metric induced by $\\norm {\\, \\cdot \\,}$. Let $d$ be non-Archimedean. Then: :$\\norm {\\, \\cdot \\,}$ is a non-Archimedean norm."}
+{"_id": "16292", "title": "Non-Archimedean Norm iff Non-Archimedean Metric", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring with zero $0$. Let $d$ be the metric induced by $\\norm {\\,\\cdot\\,}$. Then: :$\\norm {\\, \\cdot \\,}$ is a non-Archimedean norm {{iff}} $d$ is a non-Archimedean metric."}
+{"_id": "16293", "title": "Quotients of 3 Unequal Numbers are Unequal", "text": "Let $x, y, z \\in \\R_{\\ne 0}$ be non-zero real numbers which are not all equal. Then $\\dfrac x y, \\dfrac y z, \\dfrac z x$ are also not all equal."}
+{"_id": "16294", "title": "Union of Power Sets not always Equal to Powerset of Union", "text": "The union of the power sets of two sets $S$ and $T$ is not necessarily equal to the power set of their union."}
+{"_id": "16295", "title": "Limit of Intersection of Closed Intervals from Zero to Positive Integer Reciprocal", "text": "For all (strictly) positive integers $n \\in \\Z_{>0}$, let $A_n$ be the closed real interval: :$A_n = \\closedint 0 {\\dfrac 1 n}$ Let $A \\subseteq \\R$ be the subset of the real numbers defined as: :$A = \\displaystyle \\lim_{n \\mathop \\to \\infty} \\bigcap A_n$ Then: :$A = \\set 0$"}
+{"_id": "16296", "title": "Distributive Laws/Examples/A cap B cap (C cup D) subset of (A cap D) cup (B cap C)", "text": "Let: :$P = A \\cap B \\cap \\paren {C \\cup D}$ :$Q = \\paren {A \\cap D} \\cup \\paren {B \\cap C}$ Then: :$P \\subseteq Q$"}
+{"_id": "16297", "title": "Total Number of Set Partitions", "text": "Let $S$ be a finite set of cardinality $n$. Then the number of different partitions of $S$ is $B_n$, where $B_n$ is the $n$th Bell number."}
+{"_id": "16298", "title": "Combination Theorem for Sequences/Normed Division Ring/Inverse Rule", "text": "Suppose $l \\ne 0$. Then: :$\\exists k \\in \\N : \\forall n \\in \\N: x_{k + n} \\ne 0$ and the subsequence $\\sequence { x_{k+n}^{-1} }$ is well-defined and convergent with: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} {x_{k + n} }^{-1} = l^{-1}$."}
+{"_id": "16299", "title": "Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind", "text": "Let $B_n$ be the Bell number for $n \\in \\Z_{\\ge 0}$. Then: :$B_n = \\displaystyle \\sum_{k \\mathop = 0}^n {n \\brace k}$ where $\\displaystyle {n \\brace k}$ denotes a Stirling number of the second kind."}
+{"_id": "16301", "title": "Quotient Group of Solvable Group is Solvable", "text": "Let $G$ be a solvable group. Let $N$ be a normal subgroup of $G$. Then, $G / N$, the quotient group of $G$ by $N$ is solvable."}
+{"_id": "16302", "title": "Equivalence of Well-Ordering Principle and Induction/Proof/PFI implies PCI", "text": "The Principle of Finite Induction implies the Principle of Complete Finite Induction. That is: :Principle of Finite Induction: Given a subset $S \\subseteq \\N$ of the natural numbers which has these properties: ::$0 \\in S$ ::$n \\in S \\implies n + 1 \\in S$ :then $S = \\N$. implies: :Principle of Complete Finite Induction: Given a subset $S \\subseteq \\N$ of the natural numbers which has these properties: ::$0 \\in S$ ::$\\set {0, 1, \\ldots, n} \\subseteq S \\implies n + 1 \\in S$ :then $S = \\N$."}
+{"_id": "16303", "title": "Equivalence of Well-Ordering Principle and Induction/Proof/PCI implies WOP", "text": "The Principle of Complete Induction implies the Well-Ordering Principle. That is: :Principle of Complete Induction: Given a subset $S \\subseteq \\N$ of the natural numbers which has these properties: ::$0 \\in S$ ::$\\set {0, 1, \\ldots, n} \\subseteq S \\implies n + 1 \\in S$ :then $S = \\N$. implies: :Well-Ordering Principle: Every nonempty subset of $\\N$ has a minimal element."}
+{"_id": "16304", "title": "Equivalence of Well-Ordering Principle and Induction/Proof/WOP implies PFI", "text": "The Well-Ordering Principle implies the Principle of Finite Induction. That is: :Well-Ordering Principle: Every non-empty subset of $\\N$ has a minimal element implies: :Principle of Finite Induction: Given a subset $S \\subseteq \\N$ of the natural numbers which has these properties: ::$0 \\in S$ ::$n \\in S \\implies n + 1 \\in S$ :then $S = \\N$."}
+{"_id": "16305", "title": "Floor Function/Examples/Floor of 5 over 2", "text": ":$\\floor {\\dfrac 5 2} = 2$"}
+{"_id": "16306", "title": "Floor Function/Examples/Floor of Minus 5 over 2", "text": ":$\\floor {-\\dfrac 5 2} = -3$"}
+{"_id": "16307", "title": "Floor Function/Examples/Floor of 14", "text": ":$\\floor {14} = 14$"}
+{"_id": "16308", "title": "Floor Function/Examples/Floor of Root 10", "text": ":$\\floor {\\sqrt {10} } = 3$"}
+{"_id": "16309", "title": "Combination Theorem for Cauchy Sequences/Inverse Rule", "text": "Suppose $\\sequence {x_n}$ does not converge to $0$. Then: :$\\exists K \\in \\N: \\forall n > K : x_n \\ne 0$ and the sequence: :$\\sequence {\\paren {x_{K + n} }^{-1} }_{n \\mathop \\in \\N}$ is well-defined and a Cauchy sequence."}
+{"_id": "16312", "title": "Subsequence of Cauchy Sequence in Normed Division Ring is Cauchy Sequence", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring with zero: $0$. Let $\\sequence {x_n}$ be a Cauchy sequence in $R$. Let $\\sequence {x_{n_r} }$ be a subsequence of $\\sequence {x_n}$. Then: :$\\sequence {x_{n_r} }$ is a Cauchy sequence in $R$."}
+{"_id": "16313", "title": "Coprime Divisors of Square Number are Square", "text": "Let $r$ be a square number. Let $r = s t$ where $s$ and $t$ are coprime. Then both $s$ and $t$ are square."}
+{"_id": "16314", "title": "Combination Theorem for Cauchy Sequences/Constant Rule", "text": ":the constant sequence $\\tuple {a, a, a, \\dots}$ is a Cauchy sequence."}
+{"_id": "16315", "title": "Combination Theorem for Cauchy Sequences/Multiple Rule", "text": ":$\\sequence {a x_n}$ is a Cauchy sequence."}
+{"_id": "16316", "title": "Combination Theorem for Cauchy Sequences/Difference Rule", "text": ":$\\sequence {x_n - y_n}$ is a Cauchy sequence."}
+{"_id": "16318", "title": "Congruent Integers in Same Residue Class", "text": "Let $m \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\Z_m$ be the set of residue classes modulo $m$: :$Z_m = \\set {\\eqclass 0 m, \\eqclass 1 m, \\dotsc, \\eqclass {m - 1} m}$ Let $a, b \\in \\set {0, 1, \\ldots, m -1 }$. Then: :$\\eqclass a m = \\eqclass b m \\iff a \\equiv b \\pmod m$"}
+{"_id": "16319", "title": "Residue Classes form Partition of Integers", "text": "Let $m \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\Z_m$ be the set of residue classes modulo $m$: :$\\Z_m = \\set {\\eqclass 0 m, \\eqclass 1 m, \\dotsc, \\eqclass {m - 1} m}$ Then $\\Z_m$ forms a partition of $\\Z$."}
+{"_id": "16320", "title": "Cardinality of Set of Residue Classes", "text": "Let $m \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\Z_m$ be the set of residue classes modulo $m$. Then: :$\\card {Z_m} = m$ where $\\card { \\, \\cdot \\,}$ denotes cardinality."}
+{"_id": "16321", "title": "Structure Induced by Ring with Unity Operations is Ring with Unity", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose unity is $1_R$. Let $S$ be a set. Let $\\struct {R^S, +', \\circ'}$ be the structure on $R^S$ induced by $+'$ and $\\circ'$. Then $\\struct {R^S, +', \\circ'}$ is a ring with unity whose unity is $f_{1_R}: S \\to R$, defined by: :$\\forall s \\in S: \\map {f_{1_R} } s = 1_R$"}
+{"_id": "16322", "title": "Cauchy Sequences form Ring with Unity", "text": "Let $\\struct {R, +, \\circ, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\struct {R^\\N, +, \\circ}$ be the ring of sequences over $R$ with unity $\\tuple {1, 1, 1, \\dotsc}$. Let $\\CC \\subset R^\\N$ be the set of Cauchy sequences on $R$. Then: :$\\struct {\\CC, +, \\circ}$ is a subring of $R^\\N$ with unity $\\tuple {1, 1, 1, \\dotsc}$."}
+{"_id": "16323", "title": "Equivalence Relation on Power Set induced by Intersection with Subset", "text": "Let $A, T$ be sets such that $A \\subseteq T$. Let $S = \\powerset T$ denote the power set of $T$. Let $\\alpha$ denote the relation defined on $S$ by: :$\\forall X, Y \\in S: X \\mathrel \\alpha Y \\iff X \\cap A = Y \\cap A$ Then $\\alpha$ is an equivalence relation."}
+{"_id": "16325", "title": "Equivalence Relation on Power Set induced by Intersection with Subset/Equivalence Class of Empty Set", "text": "The equivalence class of $\\O$ in $S$ with respect to $\\alpha$ is given by: :$\\eqclass \\O \\alpha = \\powerset {T \\setminus A}$"}
+{"_id": "16327", "title": "Reflexive and Symmetric Relation is not necessarily Transitive", "text": "Let $S$ be a set. Let $\\alpha \\subseteq S \\times S$ be a relation on $S$. Let $\\alpha$ be both reflexive and symmetric. Then it is not necessarily the case that $\\alpha$ is also transitive."}
+{"_id": "16332", "title": "Symmetric and Transitive Relation is not necessarily Reflexive", "text": "Let $S$ be a set. Let $\\alpha \\subseteq S \\times S$ be a relation on $S$. Let $\\alpha$ be both symmetric and transitive. Then it is not necessarily the case that $\\alpha$ is also reflexive."}
+{"_id": "16334", "title": "Hilbert Proof System Instance 2 Independence Results", "text": "Let $\\mathscr H_2$ be Instance 2 of the Hilbert proof systems. Then the following independence results hold:"}
+{"_id": "16335", "title": "Equivalence Relation on Natural Numbers such that Quotient is Power of Two", "text": "Let $\\alpha$ denote the relation defined on the natural numbers $\\N$ by: :$\\forall x, y \\in \\N: x \\mathrel \\alpha y \\iff \\exists n \\in \\Z: x = 2^n y$ Then $\\alpha$ is an equivalence relation."}
+{"_id": "16336", "title": "Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Equivalence Class of Prime", "text": "Let $\\eqclass p \\alpha$ be the $\\alpha$-equivalence class of a prime number $p$. Then $\\eqclass p \\alpha$ contains no other prime number other than $p$."}
+{"_id": "16337", "title": "Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Smallest Equivalence Class with no Prime", "text": "Let $\\eqclass x \\alpha$ denote the $\\alpha$-equivalence class of a natural number $x$. Let $r$ be the smallest natural number such that $\\eqclass r \\alpha$ contains no prime number. Then $r = 9$."}
+{"_id": "16338", "title": "Equivalence Relation on Square Matrices induced by Positive Integer Powers", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $S$ be the set of all square matrices of order $n$. Let $\\alpha$ denote the relation defined on $S$ by: :$\\forall \\mathbf A, \\mathbf B \\in S: \\mathbf A \\mathrel \\alpha \\mathbf B \\iff \\exists r, s \\in \\N: \\mathbf A^r = \\mathbf B^s$ Then $\\alpha$ is an equivalence relation."}
+{"_id": "16339", "title": "Equivalence Relation on Integers Modulo 5 induced by Squaring", "text": "Let $\\beta$ denote the relation defined on the integers $\\Z$ by: :$\\forall x, y \\in \\Z: x \\mathrel \\beta y \\iff x^2 \\equiv y^2 \\pmod 5$ Then $\\beta$ is an equivalence relation."}
+{"_id": "16340", "title": "Equivalence Relation on Integers Modulo 5 induced by Squaring/Number of Equivalence Classes", "text": "The number of distinct $\\beta$-equivalence classes is $3$: {{begin-eqn}} {{eqn | l = \\eqclass 0 \\beta       | o =  }} {{eqn | l = \\eqclass 1 \\beta       | r = \\eqclass 4 \\beta       | c =  }} {{eqn | l = \\eqclass 2 \\beta       | r = \\eqclass 3 \\beta       | c =  }} {{end-eqn}}"}
+{"_id": "16341", "title": "Equivalence Relation on Integers Modulo 5 induced by Squaring/Addition Modulo Beta is not Well-Defined", "text": "Let the $+_\\beta$ operator (\"addition\") on the $\\beta$-equivalence classes be defined as: :$\\eqclass a \\beta +_\\beta \\eqclass b \\beta := \\eqclass {a + b} \\beta$ Then such an operation is not well-defined."}
+{"_id": "16343", "title": "Constant Sequence Converges to Constant in Normed Division Ring", "text": ":the constant sequence $\\tuple {\\lambda, \\lambda, \\lambda, \\dots}$ is convergent and $\\displaystyle \\lim_{n \\mathop \\to \\infty} \\lambda = \\lambda$"}
+{"_id": "16344", "title": "Product of Sequence Converges to Zero with Cauchy Sequence Converges to Zero", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring with zero $0$. Let $\\sequence {x_n}$, $\\sequence {y_n} $ be sequences in $R$. Let $\\sequence {x_n}$ converge to $0$. Let $\\sequence {y_n}$ be a Cauchy sequence. Then: :$\\sequence {x_n y_n}$ and $\\sequence {y_n x_n}$ converge to $0$."}
+{"_id": "16345", "title": "Cauchy Sequence is Bounded/Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Every Cauchy sequence in $R$ is bounded."}
+{"_id": "16346", "title": "Cauchy Sequence with Finite Elements Prepended is Cauchy Sequence", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\sequence {x_n}$ be a sequence in $R$. Let $N \\in \\N$ Let $\\sequence {y_n}$ be the sequence defined by: :$\\forall n, y_n = x_{N + n}$ Let $\\sequence {y_n}$ be a Cauchy sequence in $R$. Then: :$\\sequence {x_n}$ is a Cauchy sequence in $R$."}
+{"_id": "16347", "title": "Convergent Sequence with Finite Elements Prepended is Convergent Sequence", "text": "Let $\\struct {R, \\norm { \\, \\cdot \\, } }$ be a normed division ring. Let $\\sequence {x_n}$ be a sequence in $R$. Let $N \\in \\N$ Let $\\sequence {y_n}$ be the sequence defined by: :$\\forall n, y_n = x_{N+n}$ Let $\\sequence {y_n}$ be a convergent sequence in $R$ with limit $l$. Then: :$\\sequence {x_n}$ is a convergent sequence in $R$ with limit $l$."}
+{"_id": "16348", "title": "Maximal Left and Right Ideal iff Quotient Ring is Division Ring", "text": "Let $R$ be a ring with unity. Let $J$ be an ideal of $R$. Then the following are equivalent: :$(1): \\quad J$ is a maximal left ideal :$(2): \\quad J$ is a maximal right ideal :$(3): \\quad$ the quotient ring $R / J$ is a division ring."}
+{"_id": "16353", "title": "Definition:Constructed Semantics/Instance 2/Rule of Addition", "text": "The Rule of Addition: :$q \\implies (q \\lor p)$ is a tautology in Instance 2 of constructed semantics."}
+{"_id": "16354", "title": "Definition:Constructed Semantics/Instance 2/Rule of Commutation", "text": "The Rule of Commutation: :$\\left({p \\lor q}\\right) \\implies \\left({q \\lor p}\\right)$ is a tautology in Instance 2 of constructed semantics."}
+{"_id": "16355", "title": "Definition:Constructed Semantics/Instance 2/Factor Principle", "text": "The Factor Principle: :$\\left({p \\implies q}\\right) \\implies \\left({\\left({r \\lor p}\\right) \\implies \\left ({r \\lor q}\\right)}\\right)$ is a tautology in Instance 2 of constructed semantics."}
+{"_id": "16356", "title": "Test for Left Ideal", "text": "Let $J$ be a subset of a ring $\\struct {R, +, \\circ}$. Then $J$ is an left ideal of $\\struct{R, +, \\circ}$ {{iff}} these all hold: :$(1): \\quad J \\ne \\O$ :$(2): \\quad \\forall x, y \\in J: x + \\paren {-y} \\in J$ :$(3): \\quad \\forall j \\in J, r \\in R: r \\circ j \\in J$"}
+{"_id": "16357", "title": "Test for Right Ideal", "text": "Let $J$ be a subset of a ring $\\struct {R, +, \\circ}$. Then $J$ is a right ideal of $\\struct {R, +, \\circ}$ {{iff}} these all hold: :$(1): \\quad J \\ne \\O$ :$(2): \\quad \\forall x, y \\in J: x + \\paren {-y} \\in J$ :$(3): \\quad \\forall j \\in J, r \\in R: j \\circ r \\in J$"}
+{"_id": "16358", "title": "Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Left Ideal implies Quotient Ring is Division Ring", "text": "Let $R$ be a ring with unity. Let $J$ be an ideal of $R$. If $J$ is a maximal left ideal then the quotient ring $R / J$ is a division ring."}
+{"_id": "16359", "title": "Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Right Ideal implies Quotient Ring is Division Ring", "text": "Let $R$ be a ring with unity. Let $J$ be an ideal of $R$. If $J$ is a maximal right ideal then the quotient ring $R / J$ is a division ring."}
+{"_id": "16363", "title": "Inverse of Injective and Surjective Mapping is Mapping", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping such that: :$(1): \\quad f$ is an injection :$(2): \\quad f$ is a surjection. Then the inverse $f^{-1}$ of $f$ is itself a mapping."}
+{"_id": "16365", "title": "Inverse is Mapping implies Mapping is Injection and Surjection", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let the inverse $f^{-1} \\subseteq T \\times S$ itself be a mapping. Then: :$(1): \\quad f$ is an injection :$(2): \\quad f$ is a surjection."}
+{"_id": "16366", "title": "Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Quotient Ring is Division Ring implies Maximal Right Ideal", "text": "Let $R$ be a ring with unity. Let $J$ be an ideal of $R$. If the quotient ring $R / J$ is a division ring then $J$ is a maximal right ideal."}
+{"_id": "16368", "title": "Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Quotient Ring is Division Ring implies Maximal Left Ideal", "text": "Let $R$ be a ring with unity. Let $J$ be an ideal of $R$. If the quotient ring $R / J$ is a division ring then $J$ is a maximal left ideal."}
+{"_id": "16370", "title": "Quotient Ring of Cauchy Sequences is Division Ring", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\CC$ be the ring of Cauchy sequences over $R$. Let $\\NN$ be the set of null sequences. Then the quotient ring $\\CC / \\NN$ is a division ring."}
+{"_id": "16372", "title": "Hilbert Proof System Instance 2 Independence Results/Independence of A1", "text": "Axiom $(A1)$ is independent from $(A2)$, $(A3)$, $(A4)$."}
+{"_id": "16373", "title": "Embedding Ring into Ring Structure Induced by Ring Operations", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $S$ be a non-empty set. Let $\\struct {R^S, +', \\circ'}$ be the ring of mappings, where $+'$ and $\\circ'$ are the pointwise operations induced on $R^S$ by $+$ and $\\circ$. For each $r \\in R$, let $f_r: S \\to R$ be the mapping defined by: :$\\forall s \\in S, \\map {f_r} s = r$ That is, $f_r$ is the constant mapping from $S$ to $r$. Let $\\phi: R \\to R^S$ be the mapping from the ring $R$ to the ring $R^S$ defined by: :$\\forall r \\in R: \\map \\phi r = f_r$ Then: :$\\phi$ is a ring monomorphism."}
+{"_id": "16374", "title": "Embedding Normed Division Ring into Ring of Cauchy Sequences", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\CC$ be the ring of Cauchy sequences over $R$ Let $\\phi: R \\to \\CC$ be the mapping from $R$ to $\\CC$ defined as: :$\\forall a \\in R: \\map \\phi a = \\tuple {a, a, a, \\dots}$ where $\\tuple {a, a, a, \\dots}$ is the constant sequence. Then $\\phi$ is a ring monomorphism."}
+{"_id": "16375", "title": "Hilbert Proof System Instance 2 Independence Results/Independence of A2", "text": "Axiom $(A2)$ is independent from $(A1)$, $(A3)$, $(A4)$."}
+{"_id": "16376", "title": "Definition:Constructed Semantics/Instance 3/Rule of Idempotence", "text": "The Rule of Idempotence: :$(p \\lor p) \\implies p$ is a tautology in Instance 3 of constructed semantics."}
+{"_id": "16377", "title": "Definition:Constructed Semantics/Instance 3/Rule of Commutation", "text": "The Rule of Commutation: :$\\left({p \\lor q}\\right) \\implies \\left({q \\lor p}\\right)$ is a tautology in Instance 3 of constructed semantics."}
+{"_id": "16378", "title": "Definition:Constructed Semantics/Instance 3/Factor Principle", "text": "The Factor Principle: :$\\left({p \\implies q}\\right) \\implies \\left({\\left({r \\lor p}\\right) \\implies \\left ({r \\lor q}\\right)}\\right)$ is a tautology in Instance 3 of constructed semantics."}
+{"_id": "16379", "title": "Hilbert Proof System Instance 2 Independence Results/Independence of A3", "text": "Axiom $(A3)$ is independent from $(A1)$, $(A2)$, $(A4)$."}
+{"_id": "16380", "title": "Definition:Constructed Semantics/Instance 4/Rule of Idempotence", "text": "The Rule of Idempotence: :$(p \\lor p) \\implies p$ is a tautology in Instance 4 of constructed semantics."}
+{"_id": "16381", "title": "Definition:Constructed Semantics/Instance 4/Rule of Addition", "text": "The Rule of Addition: :$q \\implies (q \\lor p)$ is a tautology in Instance 4 of constructed semantics."}
+{"_id": "16382", "title": "Definition:Constructed Semantics/Instance 4/Factor Principle", "text": "The Factor Principle: :$\\left({p \\implies q}\\right) \\implies \\left({\\left({r \\lor p}\\right) \\implies \\left ({r \\lor q}\\right)}\\right)$ is a tautology in Instance 4 of constructed semantics."}
+{"_id": "16383", "title": "Cardinality of Set of Surjections", "text": "Let $S$ and $T$ be finite sets. Let $\\card S = m, \\card T = n$. Let $C$ be the number of surjections from $S$ to $T$. Then: :$C = n! \\displaystyle {m \\brace n}$ where $\\displaystyle {m \\brace n}$ denotes a Stirling number of the second kind."}
+{"_id": "16384", "title": "Hilbert Proof System Instance 2 Independence Results/Independence of A4", "text": "Axiom $(\\text A 4)$ is independent from $(\\text A 1)$, $(\\text A 2)$, $(\\text A 3)$."}
+{"_id": "16386", "title": "Image of Set Difference under Mapping/Corollary 3", "text": "Let $f: S \\to T$ be a surjection. Let $A \\subseteq S$ be a subset of $S$. Then: :$T \\setminus f \\sqbrk A \\subseteq f \\sqbrk {S \\setminus A}$ where $\\setminus$ denotes set difference."}
+{"_id": "16387", "title": "Definition:Constructed Semantics/Instance 5/Rule of Idempotence", "text": "The Rule of Idempotence: :$(p \\lor p) \\implies p$ is a tautology in Instance 5 of constructed semantics."}
+{"_id": "16388", "title": "Additive Function is Linear for Rational Factors", "text": "Let $f: \\R \\to \\R$ be an additive function. Then: :$\\forall r \\in \\Q, x \\in \\R: \\map f {x r} = r \\map f x$"}
+{"_id": "16389", "title": "Definition:Constructed Semantics/Instance 5/Rule of Addition", "text": "The Rule of Addition: :$q \\implies (q \\lor p)$ is a tautology in Instance 5 of constructed semantics."}
+{"_id": "16390", "title": "Definition:Constructed Semantics/Instance 5/Rule of Commutation", "text": "The Rule of Commutation: :$\\left({p \\lor q}\\right) \\implies \\left({q \\lor p}\\right)$ is a tautology in Instance 5 of constructed semantics."}
+{"_id": "16391", "title": "Additive Function of Zero is Zero", "text": "Let $f: \\R \\to \\R$ be an additive function. Then: :$f \\paren 0 = 0$"}
+{"_id": "16392", "title": "Additive Function is Odd Function", "text": "Let $f: \\R \\to \\R$ be an additive function. Then $f$ is an odd function."}
+{"_id": "16394", "title": "Cross-Relation on Real Numbers is Equivalence Relation", "text": "Let $\\R^2$ denote the cartesian plane. Let $\\alpha$ denote the relation defined on $\\R^2$ by: :$\\tuple {x_1, y_1} \\mathrel \\alpha \\tuple {x_2, y_2} \\iff x_1 + y_2 = x_2 + y_1$ Then $\\alpha$ is an equivalence relation on $\\R^2$."}
+{"_id": "16395", "title": "Cross-Relation on Real Numbers is Equivalence Relation/Geometrical Interpretation", "text": "The equivalence classes of $\\alpha$, when interpreted as points in the plane, are the straight lines of slope $1$."}
+{"_id": "16397", "title": "Subtraction on Numbers is Anticommutative/Natural Numbers", "text": "The operation of subtraction on the natural numbers $\\N$ is anticommutative, and defined only when $a = b$: That is: :$a - b = b - a \\iff a = b$"}
+{"_id": "16398", "title": "Subtraction on Numbers is Anticommutative/Integral Domains", "text": "The operation of subtraction on the numbers is anticommutative. That is: :$a - b = b - a \\iff a = b$"}
+{"_id": "16399", "title": "Natural Number Subtraction is not Closed", "text": "The operation of subtraction on the natural numbers is not closed."}
+{"_id": "16400", "title": "Integers under Addition form Semigroup", "text": "The set of integers under addition $\\struct {\\Z, +}$ forms a semigroup."}
+{"_id": "16401", "title": "Natural Numbers under Multiplication form Subsemigroup of Integers", "text": "Let $\\struct {\\N, \\times}$ denote the set of natural numbers under multiplication. Let $\\struct {\\Z, \\times}$ denote the set of integers under multiplication. Then $\\struct {\\N, \\times}$ is a subsemigroup of $\\struct {\\Z, \\times}$."}
+{"_id": "16402", "title": "Cauchy Sequence Is Eventually Bounded Away From Non-Limit", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\sequence {x_n}$ be a Cauchy sequence in $R$. Suppose $\\sequence {x_n}$ does not converge to $l \\in R$, then: :$\\exists K \\in \\N$ and $C \\in \\R_{>0}: \\forall n > K: C < \\norm {x_n - l}$"}
+{"_id": "16403", "title": "Embedding Division Ring into Quotient Ring of Cauchy Sequences", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\CC$ be the ring of Cauchy sequences over $R$ Let $\\NN = \\set {\\sequence {x_n}: \\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = 0}$ Let $\\norm {\\, \\cdot \\,}: \\CC \\, \\big / \\NN \\to \\R_{\\ge 0}$ be the norm on the quotient ring $\\CC \\, \\big / \\NN$ defined by: :$\\displaystyle \\forall \\sequence {x_n} + \\NN: \\norm {\\sequence {x_n} + \\NN} = \\lim_{n \\mathop \\to \\infty} \\norm {x_n}$ Let $\\phi: R \\to \\CC \\, \\big / \\NN$ be the mapping from $R$ to the quotient ring $\\CC \\, \\big / \\NN$ defined by: :$\\forall a \\in R: \\map \\phi a = \\sequence {a, a, a, \\dotsc} + \\NN$ where $\\sequence {a, a, a, \\dotsc} + \\NN$ is the left coset in $\\CC \\, \\big / \\NN$ that contains the constant sequence $\\sequence {a, a, a, \\dotsc}$. Then: :$\\phi$ is a distance-preserving ring monomorphism."}
+{"_id": "16409", "title": "Non-Zero Real Numbers under Multiplication form Group", "text": "Let $\\R_{\\ne 0}$ be the set of real numbers without zero: :$\\R_{\\ne 0} = \\R \\setminus \\set 0$ The structure $\\struct {\\R_{\\ne 0}, \\times}$ forms a group."}
+{"_id": "16410", "title": "Symmetric Group on n Letters is Isomorphic to Symmetric Group", "text": "The symmetric group on $n$ letters $\\struct {S_n, \\circ}$ is isomorphic to the symmetric group on the $n$ elements of any set $T$ whose cardinality is $n$. That is: :$\\forall T \\subseteq \\mathbb U, \\card T = n: \\struct {S_n, \\circ} \\cong \\struct {\\Gamma \\paren T, \\circ}$"}
+{"_id": "16411", "title": "Symmetric Groups of Same Order are Isomorphic", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $T_1$ and $T_2$ be sets whose cardinality $\\card {T_1}$ and $\\card {T_2}$ are both $n$. Let $\\struct {\\map \\Gamma {T_1}, \\circ}$ and $\\struct {\\map \\Gamma {T_2}, \\circ}$ be the symmetric group on $S$ and $T$ respectively. Then $\\struct {\\map \\Gamma {T_1}, \\circ}$ and $\\struct {\\map \\Gamma {T_2}, \\circ}$ are isomorphic."}
+{"_id": "16412", "title": "Quotient Ring of Cauchy Sequences is Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\CC$ be the ring of Cauchy sequences over $R$ Let $\\NN$ be the set of null sequences. For all $\\sequence {x_n} \\in \\CC$, let $\\eqclass {x_n} {}$ denote the left coset $\\sequence {x_n} + \\NN$ Let $\\norm {\\, \\cdot \\,}_1: \\CC \\,\\big / \\NN \\to \\R_{\\ge 0}$ be defined by: :$\\displaystyle \\forall \\eqclass {x_n} {} \\in \\CC \\,\\big / \\NN: \\norm {\\eqclass {x_n} {} }_1 = \\lim_{n \\mathop \\to \\infty} \\norm {x_n}$ Then: :$\\struct {\\CC \\,\\big / \\NN, \\norm {\\, \\cdot \\,}_1 }$ is a normed division ring."}
+{"_id": "16413", "title": "Inequality Rule for Real Sequences", "text": "Let $\\sequence {x_n}$ and $\\sequence {y_n}$ be sequences in $\\R$. Let $\\sequence {x_n}$ and $\\sequence {y_n}$ be convergent to the following limits: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ :$\\displaystyle \\lim_{n \\mathop \\to \\infty} y_n = m$ Let there exist $N \\in \\N$ such that: :$\\forall n \\ge N: x_n \\le y_n$ Then: :$l \\le m$"}
+{"_id": "16416", "title": "Rule of Association/Disjunction/Formulation 2/Reverse Implication", "text": ":$\\vdash \\left({p \\lor \\left({q \\lor r}\\right)}\\right) \\impliedby \\left({\\left({p \\lor q}\\right) \\lor r}\\right)$"}
+{"_id": "16421", "title": "Left Regular Representation of Subset Product", "text": "Let $\\struct {S, \\circ}$ be a magma. Let $T \\subseteq S$ be a subset of $S$. Let $\\lambda_a: S \\to S$ be the left regular representation of $S$ with respect to $a$. Then: :$\\lambda_a \\sqbrk T = \\set a \\circ T = a \\circ T$ where $a \\circ T$ denotes subset product with a singleton."}
+{"_id": "16422", "title": "Right Regular Representation of Subset Product", "text": "Let $\\struct {S, \\circ}$ be a magma. Let $T \\subseteq S$ be a subset of $S$. Let $\\rho_a: S \\to S$ be the right regular representation of $S$ with respect to $a$. Then: :$\\rho_a \\sqbrk T = T \\circ \\set a = T \\circ a$ where $T \\circ a$ denotes subset product with a singleton."}
+{"_id": "16423", "title": "Order of Cycle is Length of Cycle", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\pi \\in S_n$ be a cyclic permutation of length $k$. Then: :$\\order \\pi = k$ where: :$\\order \\pi$ denotes the order of $\\pi$ in $S_n$."}
+{"_id": "16426", "title": "Norm Sequence of Cauchy Sequence has Limit", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\sequence {x_n}$ be a Cauchy sequence in $R$. Then $\\sequence {\\norm {x_n} }$ has a limit in $\\R$. That is, :$\\exists l \\in \\R: \\displaystyle \\lim_{n \\mathop \\to \\infty} \\norm {x_n} = l$"}
+{"_id": "16427", "title": "Equivalent Cauchy Sequences have Equal Limits of Norm Sequences", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\sequence {x_n}$ and $\\sequence {y_n}$ be Cauchy sequences in $R$. Let $\\displaystyle \\lim_{n \\mathop \\to \\infty} {x_n - y_n} = 0$. Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\norm {x_n} = \\lim_{n \\mathop \\to \\infty} \\norm {y_n}$"}
+{"_id": "16429", "title": "Rule of Conjunction/Sequent Form/Formulation 2", "text": ":$\\vdash p \\implies \\paren{ q \\implies \\paren{ p \\land q } }$"}
+{"_id": "16433", "title": "Subgroup Generated by One Element is Cyclic", "text": "Let $G$ be a group. Let $a \\in G$. Then $\\gen a$, the subgroup generated by $a$, is cyclic:"}
+{"_id": "16438", "title": "Order of Elements in Quaternion Group", "text": "Let $Q = \\Dic 2$ be the quaternion group, whose group presentation is given by: :$\\Dic 2 = \\gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ Then $\\Dic 2$ has: : $1$ element of order $2$ and: : $6$ elements of order $4$."}
+{"_id": "16440", "title": "Reduced Residues Modulo 5 under Multiplication form Cyclic Group", "text": "Let $\\struct {\\Z'_5, \\times_5}$ denote the multiplicative group of reduced residues modulo $5$. Then $\\struct {\\Z'_5, \\times_5}$ is cyclic."}
+{"_id": "16441", "title": "Number of Generators of Cyclic Group whose Order is Power of 2", "text": "Let $G$ be a finite cyclic group. Let the order of $G$ be $2^k$ for some $k \\in \\Z_{>0}$. Then $G$ has $2^{n - 1}$ distinct generators."}
+{"_id": "16446", "title": "Element of Group is in its own Coset/Left", "text": "Let: : $x H$ be the left coset of $x$ modulo $H$. Then: : $x \\in x H$"}
+{"_id": "16447", "title": "Element of Group is in its own Coset/Right", "text": "Let: : $H x$ be the right coset of $x$ modulo $H$. Then: : $x \\in H x$"}
+{"_id": "16449", "title": "Element of Group is in Unique Coset of Subgroup/Right", "text": "There exists a exactly one right coset of $H$ containing $x$, that is: $H x$"}
+{"_id": "16450", "title": "Equivalence of Definitions of Infinite Order Element", "text": "{{TFAE|def = Infinite Order Element}} Let $G$ be a group whose identity is $e_G$. Let $x \\in G$ be an element of $G$."}
+{"_id": "16451", "title": "Equivalence of Definitions of Finite Order Element", "text": "{{TFAE|def = Finite Order Element}} Let $G$ be a group whose identity is $e_G$. Let $x \\in G$ be an element of $G$."}
+{"_id": "16454", "title": "Right Cosets are Equal iff Left Cosets by Inverse are Equal", "text": "Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$. Let $g_1, g_2 \\in G$. Then: :$H g_1 = H g_2 \\iff {g_1}^{-1} H = {g_2}^{-1} H$ where: :${g_1}^{-1}$ and ${g_2}^{-1}$ denote the inverses of $g_1$ and $g_2$ in $G$ :$H g_1$ and $H g_2$ denote the right cosets of $H$ by $g_1$ and $g_2$ respectively :${g_1}^{-1} H$ and ${g_2}^{-1} H$ denote the left cosets of $H$ by ${g_1}^{-1}$ and ${g_2}^{-1}$ respectively."}
+{"_id": "16455", "title": "Subgroup of Subgroup with Prime Index", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H$ be a subgroup of $G$. Let $K$ be a subgroup of $H$. Let: :$\\index G K = p$ where: :$p$ denotes a prime number :$\\index G K$ denotes the index of $K$ in $G$. Then either: :$H = K$ or: :$H = G$"}
+{"_id": "16457", "title": "Completeness Theorem for Hilbert Proof System Instance 2 and Boolean Interpretations", "text": "Instance 2 of the Hilbert proof systems is a complete proof system for boolean interpretations. That is, for every WFF $\\mathbf A$: :$\\models_{\\mathrm{BI}} \\mathbf A$ implies $\\vdash_{\\mathscr H_2} \\mathbf A$"}
+{"_id": "16459", "title": "Klein Four-Group as Subgroup of S4", "text": "Let $G$ be the following subset of the symmetric group on $4$ letters $S_4$, expressed in two-row notation: {{begin-eqn}} {{eqn | l = e       | r = \\begin{bmatrix} 1 & 2 & 3 & 4 \\\\ 1 & 2 & 3 & 4 \\end{bmatrix}       | c =  }} {{eqn | l = a       | r = \\begin{bmatrix} 1 & 2 & 3 & 4 \\\\ 2 & 1 & 4 & 3 \\end{bmatrix}       | c =  }} {{eqn | l = b       | r = \\begin{bmatrix} 1 & 2 & 3 & 4 \\\\ 3 & 4 & 1 & 2 \\end{bmatrix}       | c =  }} {{eqn | l = c       | r = \\begin{bmatrix} 1 & 2 & 3 & 4 \\\\ 4 & 3 & 2 & 1 \\end{bmatrix}       | c =  }} {{end-eqn}} Then $G$ is an example of the Klein $4$-group."}
+{"_id": "16460", "title": "Non-Abelian Order 10 Group has Order 5 Element", "text": "Let $G$ be a non-abelian group of order $10$. Then $G$ has at least one element of order $5$."}
+{"_id": "16461", "title": "Quotient of Cauchy Sequences is Metric Completion", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $d$ be the metric induced by $\\struct {R, \\norm {\\, \\cdot \\,} }$. Let $\\mathcal C$ be the ring of Cauchy sequences over $R$. Let $\\mathcal N$ be the set of null sequences in $R$. Let $\\mathcal C \\,\\big / \\mathcal N$ be the quotient ring of Cauchy sequences of $\\mathcal C$ by the maximal ideal $\\mathcal N$. Let $\\norm {\\, \\cdot \\,}: \\mathcal C \\,\\big / \\mathcal N \\to \\R_{\\ge 0}$ be the norm on the quotient ring $\\mathcal C \\,\\big / \\mathcal N$ defined by: :$\\displaystyle \\forall \\sequence {x_n} + \\mathcal N: \\norm {\\sequence {x_n} + \\mathcal N} = \\lim_{n \\mathop \\to \\infty} \\norm{x_n}$ Let $d'$ be the metric induced by $\\struct {\\mathcal C \\,\\big / \\mathcal N, \\norm {\\, \\cdot \\,} }$ Let $\\phi: R \\to \\mathcal C \\,\\big / \\mathcal N$ be the mapping from $R$ to the quotient ring $\\mathcal C \\,\\big / \\mathcal N$ defined by: :$\\forall a \\in R: \\map \\phi a = \\sequence {a, a, a, \\ldots} + \\mathcal N$ where $\\sequence {a, a, a, \\ldots} + \\mathcal N$ is the left coset in $\\mathcal C \\, \\big / \\mathcal N$ that contains the constant sequence $\\sequence {a, a, a, \\ldots}$. Then: :$\\struct {\\mathcal C \\,\\big / \\mathcal N, d'}$ is the metric completion of $\\struct {R,d}$ and: :$\\map \\phi R$ is a dense subset of $\\mathcal C \\,\\big / \\mathcal N$"}
+{"_id": "16462", "title": "Completion of Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Then: :$\\struct {R, \\norm {\\, \\cdot \\,} }$ has a normed division ring completion $\\struct {R', \\norm {\\, \\cdot \\,}' }$"}
+{"_id": "16463", "title": "Group of Order 27 has Subgroup of Order 3", "text": "Let $G$ be a group whose identity element is $e$. Let $G$ be of order $27$. Then $G$ has at least one subgroup of order $3$."}
+{"_id": "16464", "title": "Group does not Necessarily have Subgroup of Order of Divisor of its Order", "text": "Let $G$ be a finite group whose order is $n$. Let $d$ be a divisor of $n$. Then it is not necessarily the case that $G$ has a subgroup of order $d$."}
+{"_id": "16465", "title": "Characteristic Function of Normal Distribution", "text": "The characteristic function of the normal distribution with mean $\\mu$ and variance $\\sigma^2$ is :$\\map \\phi t = e^{i t \\mu - \\frac 1 2 t^2 \\sigma^2}$"}
+{"_id": "16468", "title": "Product of Subset with Intersection/Equality does not Hold", "text": "While it is the case that: :$X \\circ \\paren {Y \\cap Z} \\subseteq \\paren {X \\circ Y} \\cap \\paren {X \\circ Z}$ it is not necessarily the case that: :$X \\circ \\paren {Y \\cap Z} = \\paren {X \\circ Y} \\cap \\paren {X \\circ Z}$"}
+{"_id": "16471", "title": "Left Cosets are Equal iff Element in Other Left Coset", "text": "Let $x H$ denote the left coset of $H$ by $x$. Then: :$x H = y H \\iff x \\in y H$"}
+{"_id": "16472", "title": "Right Cosets are Equal iff Element in Other Right Coset", "text": "Let $H x$ denote the right coset of $H$ by $x$. Then: :$H x = H y \\iff x \\in H y$"}
+{"_id": "16478", "title": "Order of Element divides Order of Centralizer", "text": "Let $G$ be a finite group. Let $x \\in G$ be an element of $G$. Let $\\map {C_G} x$ denote the centralizer of $x$. Then: :$\\order x \\divides \\order {\\map {C_G} x}$ where: :$\\order x$ denotes the order of $x$ in $G$ :$\\divides$ denotes divisibility :$\\order {\\map {C_G} x}$ denotes the order of $\\map {C_G} x$."}
+{"_id": "16479", "title": "Left Coset of Stabilizer in Group of Transformations", "text": "Let $S$ be a non-empty set. Let $G$ be a group of permutations of $S$. Let $t \\in G$. Let $G_t$ be the set defined as: :$G_t = \\set {g \\in G: \\map g t = t}$ Then each left coset of $G_t$ in $G$ consists of the elements of $G$ that map $t$ to some element of $S$. {{explain|The source work does not discuss group actions, but still defines $G_t$ as the stabilizer of $t$ in $G$. This needs to be reviewed and put into the language of group actions as a result related to transformation group action -- but this area of group theory is not as well covered in {{ProofWiki}} as it ought to be. I need to dig out my college notes on group actions, which were more comprehensive and understandable than any of the other works I have on my shelf, which will also need to be exploited properly.<br>Hence the second part of this question in Whitelaw is not covered yet.}}"}
+{"_id": "16481", "title": "Normed Division Ring is Field iff Completion is Field", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\struct {R', \\norm {\\, \\cdot \\,}' }$ be a normed division ring completion of $\\struct {R, \\norm {\\, \\cdot \\,} }$ Then: :$R$ is a field {{iff}} $R'$ is a field."}
+{"_id": "16482", "title": "Non-Abelian Order 8 Group has Order 4 Element", "text": "Let $G$ be a non-abelian group of order $8$. Then $G$ has at least one element of order $4$."}
+{"_id": "16483", "title": "Group of Prime Order p has p-1 Elements of Order p", "text": "Let $p$ be a prime number. Let $G$ be a group with identity $e$ whose order is $p$. Then $G$ has $p - 1$ elements of order $p$."}
+{"_id": "16484", "title": "Number of Order p Elements in Group with m Order p Subgroups", "text": "Let $G$ be a group whose identity is $e$. Let $G$ have $m$ subgroups of order $p$. The total number of elements of $G$ of order $p$ is $m \\paren {p - 1}$."}
+{"_id": "16486", "title": "Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2", "text": "Let $p$ be a prime number. Let $G$ be a finite abelian group whose identity is $e$. Let $G$ have at least $p$ elements of order $p$. Then: : $p^2 \\divides \\order G$ where: :$\\divides$ denotes divisibility :$\\order G$ denotes the order of $G$."}
+{"_id": "16487", "title": "Abelian Group of Semiprime Order is Cyclic", "text": "Let $p$ and $q$ be distinct prime numbers. Let $G$ be an abelian group such that: :$\\order G = p q$ where $\\order G$ denotes the order of $G$. Then $G$ is cyclic."}
+{"_id": "16489", "title": "General Linear Group to Determinant is Homomorphism/Corollary", "text": "The kernel of the $\\det$ mapping is the special linear group $\\SL {n, \\R}$."}
+{"_id": "16491", "title": "Quotient Group of General Linear Group by Special Linear Group", "text": "Let $\\GL {n, \\R}$ denote the general linear group of degree $n$ over $\\R$. Let $\\SL {n, \\R}$ denote the special linear group of degree $n$ over $\\R$. Then the quotient group $\\GL {n, \\R} / \\SL {n, \\R}$ is the multiplicative group of real numbers $\\struct {\\R_{\\ne 0}, \\times}$."}
+{"_id": "16494", "title": "Normed Division Ring Completions are Isometric and Isomorphic/Lemma 1", "text": "Let $\\psi' = \\phi_2 \\circ \\phi_1^{-1}:\\phi_1 \\paren R \\to \\phi_2 \\paren R$ be the composition of $\\phi_1^{-1}$ with $\\phi_2$.  Then $\\psi': \\struct {\\map {\\phi_1} R, \\norm {\\, \\cdot \\,}_1 } \\to \\struct {\\map {\\phi_2} R, \\norm {\\, \\cdot \\,}_2 }$ is an isometric ring isomorphism."}
+{"_id": "16495", "title": "Normed Division Ring Completions are Isometric and Isomorphic/Lemma 2", "text": "Let $\\psi: S_1 \\to S_2$ be defined by: :$\\forall x \\in S_1: \\map \\psi x = \\displaystyle \\lim_{n \\mathop \\to \\infty} \\map {\\psi'} {x_n}$ where $x = \\displaystyle \\lim_{n \\mathop \\to \\infty} x_n$ for some sequence $\\sequence {x_n} \\subseteq R_1$ Then $\\psi$ is a well-defined mapping."}
+{"_id": "16496", "title": "Normed Division Ring Completions are Isometric and Isomorphic/Lemma 3", "text": ":$\\psi$ is a surjective mapping."}
+{"_id": "16497", "title": "Normed Division Ring Completions are Isometric and Isomorphic/Lemma 4", "text": ":$\\psi$ is an isometry."}
+{"_id": "16498", "title": "Normed Division Ring Completions are Isometric and Isomorphic/Lemma 5", "text": ":$\\psi$ is a ring isomorphism."}
+{"_id": "16499", "title": "Normed Division Ring Completions are Isometric and Isomorphic", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\struct {S_1, \\norm {\\, \\cdot \\,}_1 }$ and $\\struct {S_2, \\norm {\\, \\cdot \\,}_2 }$ be normed division ring completions of $\\struct {R, \\norm {\\, \\cdot \\,} }$ Then there exists an isometric isomorphism $\\psi: \\struct {S_1, \\norm {\\, \\cdot \\,}_1 } \\to \\struct {S_2, \\norm {\\, \\cdot \\,}_2 }$"}
+{"_id": "16500", "title": "Distance-Preserving Image Isometric to Domain for Metric Spaces", "text": "Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces. Let $\\phi: M_1 \\to M_2$ be a distance-preserving mapping. Then: :$\\phi: M_1 \\to \\Img \\phi$ is an isometry."}
+{"_id": "16501", "title": "Distance-Preserving Mapping is Injection of Metric Spaces", "text": "Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces. Let $\\phi: M_1 \\to M_2$ be a distance-preserving mapping. Then $\\phi$ is an injection."}
+{"_id": "16502", "title": "Odd Power Function is Surjective", "text": "Let $n \\in \\Z_{\\ge 0}$ be an odd positive integer. Let $f_n: \\R \\to \\R$ be the real function defined as: :$\\map {f_n} x = x^n$ Then $f_n$ is a surjection."}
+{"_id": "16503", "title": "Residue at Multiple Pole", "text": "Let $f: \\C \\to \\C$ be a function meromorphic on some region, $D$, containing $a$.  Let $f$ have a single pole in $D$, of order $N$, at $a$. Then the residue of $f$ at $a$ is given by: :$\\displaystyle \\Res f a = \\frac 1 {\\paren {N - 1}!} \\lim_{z \\mathop \\to a} \\frac { \\d^{N - 1} } { \\d z^{N - 1} } \\paren {\\paren {z - a}^N \\map f z}$"}
+{"_id": "16504", "title": "Bijection from Cartesian Product of Initial Segments to Initial Segment", "text": "Let $\\N_k$ be used to denote the set of the first $k$ non-zero natural numbers: :$\\N_k := \\set {1, 2, \\ldots, k}$ Then a bijection can be established between $\\N_k \\times \\N_l$ and $\\N_{k l}$, where $\\N_k \\times \\N_l$ denotes the Cartesian product of $\\N_k$ and $\\N_l$."}
+{"_id": "16505", "title": "Bijection between S x T and T x S", "text": "Let $S$ and $T$ be sets. Let $S \\times T$ be the Cartesian product of $S$ and $T$. Then there exists a bijection from $S \\times T$ to $T \\times S$."}
+{"_id": "16509", "title": "Trivial Group is Group", "text": "The trivial group is a group."}
+{"_id": "16511", "title": "Order of Dihedral Group", "text": "The dihedral group $D_n$ is of order $2 n$."}
+{"_id": "16516", "title": "Matrix Entrywise Addition is Commutative", "text": "Let $\\map \\MM {m, n}$ be a $m \\times n$ matrix space over one of the standard number systems. For $\\mathbf A, \\mathbf B \\in \\map \\MM {m, n}$, let $\\mathbf A + \\mathbf B$ be defined as the matrix entrywise sum of $\\mathbf A$ and $\\mathbf B$. The operation $+$ is commutative on $\\map \\MM {m, n}$. That is: :$\\mathbf A + \\mathbf B = \\mathbf B + \\mathbf A$ for all $\\mathbf A$ and $\\mathbf B$ in $\\map \\MM {m, n}$."}
+{"_id": "16517", "title": "Special Linear Group is not Abelian", "text": "Let $K$ be a field whose zero is $0_K$ and unity is $1_K$. Let $\\SL {n, K}$ be the special linear group of order $n$ over $K$. Then $\\SL {n, K}$ is not an abelian group."}
+{"_id": "16519", "title": "Summation Formula (Complex Analysis)/Lemma", "text": "Let $C_N$ be the square with vertices $\\left({N + \\frac 1 2}\\right) \\left({\\pm 1 \\pm i}\\right)$ for $N \\in \\N$.  Then there exists a constant $A$ independent of $N$ such that:  :$\\displaystyle \\left\\vert{\\cot \\left({\\pi z}\\right)}\\right\\vert < A$  for all $z$ on $C_N$."}
+{"_id": "16520", "title": "Product of Generating Elements of Dihedral Group", "text": "Let $D_n$ be the dihedral group of order $2 n$. Let $D_n$ be defined by its group presentation: :$D_n = \\gen {\\alpha, \\beta: \\alpha^n = \\beta^2 = e, \\beta \\alpha \\beta = \\alpha^{−1} }$ Then for all $k \\in \\Z_{\\ge 0}$: :$\\beta \\alpha^k = \\alpha^{n - k} \\beta$"}
+{"_id": "16521", "title": "Center of Dihedral Group", "text": "Let $n \\in \\N$ be a natural number such that $n \\ge 3$. Let $D_n$ be the dihedral group of order $2 n$, given by: :$D_n = \\gen {\\alpha, \\beta: \\alpha^n = \\beta^2 = e, \\beta \\alpha \\beta = \\alpha^{−1} }$ Let $\\map Z {D_n}$ denote the center of $D_n$. Then: :$\\map Z {D_n} = \\begin{cases} e & : n \\text { odd} \\\\ \\set {e, \\alpha^{n / 2} } & : n \\text { even} \\end{cases}$"}
+{"_id": "16522", "title": "Intersection of Additive Groups of Integer Multiples", "text": "Let $m, n \\in \\Z_{> 0}$ be (strictly) positive integers. Let $\\struct {m \\Z, +}$ and $\\struct {n \\Z, +}$ be the corresponding additive groups of integer multiples. Then: :$\\struct {m \\Z, +} \\cap \\struct {n \\Z, +} = \\struct {\\lcm \\set {m, n} \\Z, +}$"}
+{"_id": "16524", "title": "Subgroup of Additive Group of Integers Generated by Two Integers", "text": "Let $m, n \\in \\Z_{> 0}$ be (strictly) positive integers. Let $\\struct {\\Z, +}$ denote the additive group of integers. Let $\\gen {m, n}$ be the subgroup of $\\struct {\\Z, +}$ generated by $m$ and $n$. Then: :$\\gen {m, n} = \\struct {\\gcd \\set {m, n} \\Z, +}$ That is, the additive groups of integer multiples of $\\gcd \\set {m, n}$, where $\\gcd \\set {m, n}$ is the greatest common divisor of $m$ and $n$."}
+{"_id": "16525", "title": "Subgroups of Cartesian Product of Additive Group of Integers", "text": "Let $\\struct {\\Z, +}$ denote the additive group of integers. Let $m, n \\in \\Z_{> 0}$ be (strictly) positive integers. Let $\\struct {\\Z \\times \\Z, +}$ denote the Cartesian product of $\\struct {\\Z, +}$ with itself. The subgroups of $\\struct {\\Z \\times \\Z, +}$ are not all of the form: :$\\struct {m \\Z, +} \\times \\struct {n \\Z, +}$ where $\\struct {m \\Z, +}$ denotes the additive group of integer multiples of $m$."}
+{"_id": "16526", "title": "Equivalence Relation on Symmetric Group by Image of n is Congruence Modulo Subgroup", "text": "Let $S_n$ denote the symmetric group on $n$ letters $\\set {1, \\dots, n}$. Let $\\sim$ be the relation on $S_n$ defined as: :$\\forall \\pi, \\tau \\in S_n: \\pi \\sim \\tau \\iff \\map \\pi n = \\map \\tau n$ Then $\\sim$ is an equivalence relation which is congruence modulo a subgroup. {{explain|Work needed to be done to explain exactly what is happening here.}}"}
+{"_id": "16528", "title": "Order of Multiplicative Group of Reduced Residues", "text": "Let $\\struct {\\Z'_m, \\times_m}$ denote the multiplicative group of reduced residues modulo $m$. The order of $\\struct {\\Z'_m, \\times_m}$ is $\\map \\phi m$, where $\\phi$ denotes the Euler $\\phi$ function."}
+{"_id": "16529", "title": "Circle Group is Uncountably Infinite", "text": "The circle group $\\struct {K, \\times}$ is an uncountably infinite group."}
+{"_id": "16530", "title": "Subgroups of Symmetry Group of Regular Hexagon", "text": "Let $\\mathcal H = ABCDEF$ be a regular hexagon. Let $D_6$ denote the symmetry group of $\\mathcal H$. :520px Let $e$ denote the identity mapping Let $\\alpha$ denote rotation of $\\mathcal H$ anticlockwise through $\\dfrac \\pi 3$ radians ($60 \\degrees$) Let $\\beta$ denote reflection of $\\mathcal H$ in the $AD$ axis. The subsets of $D_6$ which form its subgroups are as follows: ;Order $1$: :$\\set e$ ;Order $2$: :$\\set {e, \\alpha^3}$ :$\\set {e, \\beta}$ :$\\set {e, \\alpha \\beta}$ :$\\set {e, \\alpha^2 \\beta}$ :$\\set {e, \\alpha^3 \\beta}$ :$\\set {e, \\alpha^4 \\beta}$ :$\\set {e, \\alpha^5 \\beta}$ ;Order $3$: :$\\set {e, \\alpha^2, \\alpha^4}$ ;Order $4$: :$\\set {e, \\alpha^3, \\beta, \\alpha^3 \\beta}$ :$\\set {e, \\alpha^3, \\alpha \\beta, \\alpha^4 \\beta}$ :$\\set {e, \\alpha^3, \\alpha^2 \\beta, \\alpha^5 \\beta}$ ;Order $6$: :$\\set {e, \\alpha, \\alpha^2, \\alpha^3, \\alpha^4, \\alpha^5}$ :$\\set {e, \\alpha^2, \\alpha^4, \\beta, \\alpha^2 \\beta, \\alpha^4 \\beta}$ :$\\set {e, \\alpha^2, \\alpha^4, \\alpha \\beta, \\alpha^3 \\beta, \\alpha^5 \\beta}$ ;Order $12$: :$D_6$ itself."}
+{"_id": "16531", "title": "Subgroup of Symmetric Group that Fixes n", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $H$ denote the subgroup of $S_n$ which consists of all $\\pi \\in S_n$ such that: :$\\map \\pi n = n$ Then: :$H = S_{n - 1}$ and the index of $H$ in $S_n$ is given by: :$\\index {S_n} H = n$"}
+{"_id": "16532", "title": "Product of Orders of Abelian Group Elements Divides LCM of Order of Product", "text": "Let $G$ be an abelian group. Let $a, b \\in G$. Then: :$\\order {a b} \\divides \\lcm \\set {\\order a, \\order b}$ where: :$\\order a$ denotes the order of $a$ :$\\divides$ denotes divisibility :$\\lcm$ denotes the lowest common multiple."}
+{"_id": "16534", "title": "Groups of Order 4", "text": "There exist exactly $2$ groups of order $4$, up to isomorphism: :$C_4$, the cyclic group of order $4$ :$K_4$, the Klein $4$-group."}
+{"_id": "16537", "title": "Subgroup of Circle Group Generated by Distinct Roots of Unity", "text": "Let $K$ be the circle group. Let $m, n \\in \\Z_{>0}$ be (strictly) positive integers. Let $d = \\lcm \\set {m, n}$ be the least common multiple of $m$ and $n$. Let $\\alpha$ be a primitive $n$th root of unity. Let $\\beta$ be a primitive $m$th root of unity. Let $\\gamma$ be a primitive $d$th root of unity. Let $H = \\gen {\\alpha, \\beta}$ be the subgroup of $K$ generated by $\\alpha, \\beta$. Then $H = \\gen \\gamma$."}
+{"_id": "16538", "title": "Multiplicative Group of Complex Roots of Unity is Subgroup of Circle Group", "text": "Let $n \\in \\Z$ be an integer such that $n > 0$. Let $\\struct {U_n, \\times}$ denote the multiplicative group of complex $n$th roots of unity. Let $\\struct {K, \\times}$ denote the circle group. Then $\\struct {U_n, \\times}$ is a subgroup of $\\struct {K, \\times}$."}
+{"_id": "16539", "title": "Intersection of Multiplicative Groups of Complex Roots of Unity", "text": "Let $\\struct {K, \\times}$ denote the circle group. Let $m, n \\in \\Z_{>0}$ be (strictly) positive integers. Let $c = \\lcm \\set {m, n}$ be the lowest common multiple of $m$ and $n$. Let $\\struct {U_n, \\times}$ denote the multiplicative group of complex $n$th roots of unity. Let $\\struct {U_m, \\times}$ denote the multiplicative group of complex $m$th roots of unity. Let $H = U_m \\cap U_n$. Then $H = U_c$."}
+{"_id": "16540", "title": "Direct Product of Normal Subgroups is Normal", "text": "Let $G$ and $G'$ be groups. Let: :$H \\lhd G$ :$H' \\lhd G'$ where $\\lhd$ denotes the relation of being a normal subgroup. Then: :$\\paren {H \\times H'} \\lhd \\paren {G \\times G'}$ where $H \\times H'$ denotes the group direct product of $H$ and $H'$"}
+{"_id": "16542", "title": "Normalizer of Reflection in Dihedral Group", "text": "Let $n \\in \\N$ be a natural number such that $n \\ge 3$. Let $D_n$ be the dihedral group of order $2 n$, given by: :$D_n = \\gen {\\alpha, \\beta: \\alpha^n = \\beta^2 = e, \\beta \\alpha \\beta = \\alpha^{−1} }$ Let $\\map {N_{D_n} } {\\set \\beta}$ denote the normalizer of the singleton containing the reflection element $\\beta$. Then: :$\\map {N_{D_n} } {\\set \\beta} = \\begin{cases} \\set {e, \\beta} & : n \\text { odd} \\\\ \\set {e, \\beta, \\alpha^{n / 2}, \\alpha^{n / 2} \\beta} & : n \\text { even} \\end{cases}$"}
+{"_id": "16544", "title": "Center of Quaternion Group", "text": "Let $Q = \\Dic 2 = \\gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ be the quaternion group. Let $\\map Z {\\Dic 2}$ denote the center of $\\Dic 2$. Then: :$\\map Z {\\Dic 2} = \\set {e, a^2}$"}
+{"_id": "16545", "title": "Product of Generating Elements of Quaternion Group", "text": "Let $Q = \\Dic 2$ be the quaternion group: :$\\Dic 2 = \\gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ Then for all $k \\in \\Z_{\\ge 0}$: :$b a^k = a^{-k} b$"}
+{"_id": "16546", "title": "Conjugacy Classes of Quaternion Group", "text": "Let $Q = \\Dic 2 = \\gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ be the quaternion group. The conjugacy classes of $\\Dic 2$ are: :$\\set e, \\set {a^2}, \\set {a, a^3}, \\set {b, a^2 b}, \\set {a b, a^3 b}$"}
+{"_id": "16547", "title": "Conjugacy Action on Subsets is Group Action", "text": "Let $\\powerset G$ be the set of all subgroups of $G$. For any $S \\in \\powerset G$ and for any $g \\in G$, the conjugacy action: :$g * S := g \\circ S \\circ g^{-1}$ is a group action."}
+{"_id": "16549", "title": "Normed Vector Space Requires Multiplicative Norm on Division Ring", "text": "Let $R$ be a normed division ring with a submultiplicative norm $\\norm {\\, \\cdot \\,}_R$. Let $V$ be a vector space that is not a trivial vector space. Let $\\norm {\\, \\cdot \\,}: V \\to \\R_{\\ge 0}$ be a mapping from $V$ to the positive real numbers satisfying the vector space norm axioms. Then $\\norm {\\, \\cdot \\,}_R$ is a multiplicative norm. That is: :$\\forall r, s \\in R: \\norm {r s}_R = \\norm r_R \\norm s_R$"}
+{"_id": "16550", "title": "Ring with Multiplicative Norm has No Proper Zero Divisors", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let its zero be denoted by $0_R$. Let $\\norm {\\,\\cdot\\,}$ be a multiplicative norm on $R$. Then $R$ has no proper zero divisors.  That is: :$\\forall x, y \\in R^*: x \\circ y \\ne 0_R$ where $R^*$ is defined as $R \\setminus \\set {0_R}$."}
+{"_id": "16551", "title": "Finite Ring with Multiplicative Norm is Field", "text": "Let $R$ be a finite ring with a multiplicative norm. Then $R$ is a field."}
+{"_id": "16552", "title": "Composition of Isometries is Isometry", "text": "Let: :$\\struct {X_1, d_1}$ :$\\struct {X_2, d_2}$ :$\\struct {X_3, d_3}$ be metric spaces. Let: :$\\phi: \\struct {X_1, d_1} \\to \\struct {X_2, d_2}$ :$\\psi: \\struct {X_2, d_2} \\to \\struct {X_3, d_3}$ be isometries. Then the composite of $\\phi$ and $\\psi$ is also an isometry."}
+{"_id": "16553", "title": "Subgroup Action is Group Action", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\struct {H, \\circ}$ be a subgroup of $G$. Let $*: H \\times G \\to G$ be the subgroup action defined for all $h \\in H, g \\in G$ as: :$\\forall h \\in H, g \\in G: h * g := h \\circ g$ Then $*$ is a group action."}
+{"_id": "16554", "title": "Orbit of Subgroup Action is Coset", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\struct {H, \\circ}$ be a subgroup of $G$. Let $*: H \\times G \\to G$ be the subgroup action defined for all $h \\in H, g \\in G$ as: :$\\forall h \\in H, g \\in G: h * g := h \\circ g$ The orbit of $x \\in G$ is the right coset by $x$ of $H$: :$\\Orb x = H x$"}
+{"_id": "16555", "title": "Removable Singularity at Infinity implies Constant Function", "text": "Let $f : \\C \\to \\C$ be an entire function. Let $f$ have an removable singularity at $\\infty$.  Then $f$ is constant."}
+{"_id": "16557", "title": "Stabilizers of Elements in Same Orbit are Conjugate Subgroups", "text": "Let $G$ be a group acting on a set $X$. Let: :$y, z \\in \\Orb x$ where $\\Orb x$ denotes the orbit of some $x \\in X$. Then their stabilizers $\\Stab y$ and $\\Stab z$ are conjugate subgroups."}
+{"_id": "16558", "title": "Stabilizer of Subgroup Action on Left Coset Space", "text": "Let $G$ be a group. Let $H$ and $K$ be subgroups of $G$. Let $K$ act on the left coset space $G / H^l$ by: :$\\forall \\tuple {k, g H} \\in K \\times G / H^l: k * g H := \\paren {k g} H$ The stabilizer of $g H$ is $K \\cap H^g$, where $H^g$ denotes the $G$-conjugate of $H$ by $g$."}
+{"_id": "16567", "title": "Cardinality of Set Difference with Subset", "text": "Let $S$ and $T$ be sets such that $T$ is finite. Let $T \\subseteq S$. Then: :$\\card {S \\setminus T} = \\card S - \\card T$ where $\\card S$ denotes the cardinality of $S$."}
+{"_id": "16568", "title": "Riemann Zeta Function at Non-Positive Integers", "text": "Let $n \\ge 0$ be a integer.  Then:  :$\\map \\zeta {-n} = \\paren {-1}^n \\dfrac {B_{n + 1} } {n + 1}$  where: :$B_n$ is the $n$th Bernoulli number :$\\zeta$ is the Riemann Zeta function"}
+{"_id": "16569", "title": "Remainder on Division is Least Positive Residue", "text": "Let $a, b \\in \\Z$ be integers such that $a \\ge 0$ and $b \\ne 0$. Let $r$ be the remainder resulting from the operation of integer division of $a$ by $b$: $a = q b + r, 0 \\le r < \\size b$ Then $r$ is equal to the least positive residue of $a \\pmod b$."}
+{"_id": "16571", "title": "Floor Function/Examples/Floor of 3", "text": ":$\\floor 3 = 3$"}
+{"_id": "16573", "title": "Exponential on Real Numbers is Injection", "text": "Let $\\exp: \\R \\to \\R$ be the exponential function: :$\\map \\exp x = e^x$ Then $\\exp$ is an injection."}
+{"_id": "16574", "title": "Surjective Restriction of Real Exponential Function", "text": "Let $\\exp: \\R \\to \\R$ be the exponential function: :$\\map \\exp x = e^x$ Then the restriction of the codomain of $\\exp$ to the strictly positive real numbers: :$\\exp: \\R \\to \\R_{>0}$ is a surjective restriction. Hence: :$\\exp: \\R \\to \\R_{>0}$ is a bijection."}
+{"_id": "16575", "title": "Maximum Rule for Real Sequences", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\max \\set {x_n, y_n} = \\max \\set {l, m}$"}
+{"_id": "16577", "title": "Division Subring of Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $S$ be a division subring of $R$. Then: :$\\struct {S, \\norm {\\, \\cdot \\,}_S}$ is a normed division subring of $\\struct {R, \\norm {\\, \\cdot \\,} }$ where $\\norm {\\, \\cdot \\,}_S$ is the norm $\\norm{\\,\\cdot\\,}$ restricted to $S$."}
+{"_id": "16578", "title": "Normed Division Ring is Dense Subring of Completion", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\struct {R', \\norm {\\, \\cdot \\,}' }$ be a normed division ring completion of $\\struct {R, \\norm {\\, \\cdot \\,} }$ Then: :$\\struct {R, \\norm {\\, \\cdot \\,} }$ is isometrically isomorphic to a dense normed division subring of $\\struct {R', \\norm {\\, \\cdot \\,}' }$."}
+{"_id": "16579", "title": "Inverse of Isometric Isomorphism", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,}_R}$ and $\\struct {S, \\norm {\\,\\cdot\\,}_S}$ be normed division rings. Let $\\phi:R \\to S$ be a mapping. Then $\\phi:R \\to S$ is an isometric isomorphism {{iff}} $\\phi^{-1}: S \\to R$ is also an isometric isomorphism."}
+{"_id": "16580", "title": "Isometric Isomorphism is Norm-Preserving", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,}_R}$ and $\\struct {S, \\norm {\\,\\cdot\\,}_S}$ be normed division rings. Let $\\phi: R \\to S$ be a ring isomorphism. Then $\\phi: R \\to S$ is an isometric isomorphism {{iff}} $\\phi$ satisfies: :$\\forall x \\in R: \\norm {\\map \\phi x}_S = \\norm x_R $"}
+{"_id": "16581", "title": "Taylor Series of Holomorphic Function", "text": ":$\\displaystyle \\map f z = \\sum_{n \\mathop = 0}^\\infty \\frac {\\map {f^n} a} {n!} \\paren {z - a}^n$"}
+{"_id": "16582", "title": "Equivalence of Definitions of Square Function", "text": "Let $\\F$ denote one of the standard classes of numbers: $\\N$, $\\Z$, $\\Q$, $\\R$, $\\C$. {{TFAE|def = Square Function}}"}
+{"_id": "16583", "title": "Subring of Non-Archimedean Division Ring", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring with non-archimedean norm $\\norm {\\, \\cdot \\,}$. Let $\\struct {S, \\norm {\\, \\cdot \\,}_S }$ be a normed division subring of $R$. Then: :$\\norm {\\, \\cdot \\,}_S$ is a non-archimedean norm."}
+{"_id": "16584", "title": "Isometrically Isomorphic Non-Archimedean Division Rings", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,}_R}$ and $\\struct {S, \\norm {\\,\\cdot\\,}_S}$ be normed division rings. Let $\\phi:R \\to S$ be an isometric isomorphism. Then: :$\\norm {\\,\\cdot\\,}_R$ is a non-archimedean norm {{iff}} $\\norm {\\,\\cdot\\,}_S$ is a non-archimedean norm."}
+{"_id": "16585", "title": "Non-Archimedean Division Ring Iff Non-Archimedean Completion", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\struct {R', \\norm {\\, \\cdot \\,}' }$ be a normed division ring completion of $\\struct {R, \\norm {\\, \\cdot \\,} }$ Then: :$\\norm {\\, \\cdot \\,}$ is non-archimedean {{iff}} $\\norm {\\, \\cdot \\,}'$ is non-archimedean."}
+{"_id": "16586", "title": "Domain of Integer Square Function", "text": "The domain of the integer square function is the entire set of integers $\\Z$."}
+{"_id": "16587", "title": "Image of Integer Square Function", "text": "The image of the integer square function is the set of square numbers."}
+{"_id": "16588", "title": "Restriction of Real Square Mapping to Positive Reals is Bijection", "text": "Let $f: \\R \\to \\R$ be the real square function: :$\\forall x \\in \\R: \\map f x = x^2$ Let $g: \\R_{\\ge 0} \\to R_{\\ge 0} := f {\\restriction_{\\R_{\\ge 0} \\times R_{\\ge 0} } }$ be the restriction of $f$ to the positive real numbers $\\R_{\\ge 0}$. Then $g$ is a bijective restriction of $f$."}
+{"_id": "16589", "title": "Inverse of Real Square Function on Positive Reals", "text": "Let $f: \\R_{\\ge 0} \\to R_{\\ge 0}$ be the restriction of the real square function to the positive real numbers $\\R_{\\ge 0}$. The inverse of $f$ is $f^{-1}: \\R_{\\ge 0} \\times R_{\\ge 0}$ defined as: :$\\forall x \\in \\R_{\\ge 0}: \\map {f^{-1} } x = \\sqrt x$ where $\\sqrt x$ is the positive square root of $x$."}
+{"_id": "16590", "title": "Real Square Function is not Bijective", "text": "Let $f: \\R \\to \\R$ be the real square function: :$\\forall x \\in \\R: \\map f x = x^2$ Then $f$ is not a bijection."}
+{"_id": "16591", "title": "Inverse of Linear Function on Real Numbers", "text": "Let $a, b \\in \\R$ be real numbers such that $a \\ne 0$. Let $f: \\R \\to \\R$ be the real function defined as: :$\\forall x \\in \\R: \\map f x = a x + b$ Then the inverse of $f$ is given by: :$\\forall y \\in \\R: \\map {f^{-1} } y = \\dfrac {y - b} a$"}
+{"_id": "16592", "title": "Linear Function on Real Numbers is Bijection", "text": "Let $a, b \\in \\R$ be real numbers. Let $f: \\R \\to \\R$ be the real function defined as: :$\\forall x \\in \\R: \\map f x = a x + b$ Then $f$ is a bijection {{iff}} $a \\ne 0$."}
+{"_id": "16593", "title": "Composition of Linear Real Functions", "text": "Let $a, b, c, d \\in \\R$ be real numbers. Let $\\theta_{a, b}: \\R \\to \\R$  be the real function defined as: :$\\forall x \\in \\R: \\map {\\theta_{a, b} } x = a x + b$ Let $\\theta_{c, d} \\circ \\theta_{a, b}$ denote the composition of $\\theta_{c, d}$ with $\\theta_{a, b}$. Then: :$\\theta_{c, d} \\circ \\theta_{a, b} = \\theta_{a c, b c + d}$"}
+{"_id": "16594", "title": "Condition for Composition of Linear Real Functions to be Commutative", "text": "Let $a, b, c, d \\in \\R$ be real numbers. Let $\\theta_{a, b}: \\R \\to \\R$  be the real function defined as: :$\\forall x \\in \\R: \\map {\\theta_{a, b} } x = a x + b$ Let $\\theta_{c, d} \\circ \\theta_{a, b}$ denote the composition of $\\theta_{c, d}$ with $\\theta_{a, b}$. Then: :$\\theta_{c, d} \\circ \\theta_{a, b} = \\theta_{a, b} \\circ \\theta_{c, d}$ {{iff}}: :$b c + d = a d + b$"}
+{"_id": "16595", "title": "Composition of Right Inverse with Mapping is Idempotent", "text": "Let $f: S \\to T$ be a mapping. Let $g: T \\to S$ be a right inverse mapping of $f$. Then: :$\\paren {g \\circ f} \\circ \\paren {g \\circ f} = g \\circ f$"}
+{"_id": "16597", "title": "Sets of Permutations of Equivalent Sets are Equivalent", "text": "Let $A$ and $B$ be sets such that: :$A \\sim B$ where $\\sim$ denotes set equivalence. Let $\\map \\Gamma A$ denote the set of permutations on $A$. Then: :$\\map \\Gamma A \\sim \\map \\Gamma B$"}
+{"_id": "16599", "title": "Three Points in Ultrametric Space have Two Equal Distances/Corollary 2", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring with non-Archimedean norm $\\norm{\\,\\cdot\\,}$, Let $x, y \\in R$ and $\\norm x \\ne \\norm y$. Then: :$\\norm {x + y} = \\norm {x - y} = \\norm {y - x} = \\max \\set {\\norm x, \\norm y}$"}
+{"_id": "16600", "title": "Topological Properties of Non-Archimedean Division Rings/Centers of Open Balls", "text": ":$y \\in \\map {B_r} x \\implies \\map {B_r} y = \\map {B_r} x$"}
+{"_id": "16601", "title": "Topological Properties of Non-Archimedean Division Rings/Centers of Closed Balls", "text": ":$y \\in \\map { {B_r}^-} x \\implies \\map { {B_r}^-} y = \\map { {B_r}^-} x$"}
+{"_id": "16602", "title": "Topological Properties of Non-Archimedean Division Rings/Open Balls are Clopen", "text": ":The open $r$-ball of $x$, $\\map {B_r} x$, is both open and closed in the metric induced by $\\norm {\\,\\cdot\\,}$."}
+{"_id": "16603", "title": "Topological Properties of Non-Archimedean Division Rings/Closed Balls are Clopen", "text": ":The closed $r$-ball of $x$, $\\map { {B_r}^-} x$, is both open and closed in the metric induced by $\\norm {\\,\\cdot\\,}$."}
+{"_id": "16606", "title": "Mittag-Leffler Expansion for Hyperbolic Cotangent Function", "text": ":$\\displaystyle \\pi \\, \\map \\coth {\\pi z} = \\frac 1 z + 2 \\sum_{n \\mathop = 1}^\\infty \\frac z {z^2 + n^2}$ where:  :$z \\in \\C$ is not an integer multiple of $i$ :$\\coth$ is the hyperbolic cotangent function."}
+{"_id": "16607", "title": "Mittag-Leffler Expansion for Secant Function", "text": ":$\\displaystyle \\pi \\, \\map \\sec {\\pi z} = 4 \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {2 n + 1} {\\paren {2 n + 1}^2 - 4 z^2}$ where:  :$z \\in \\C$ is not a half-integer :$\\sec$ is the secant function."}
+{"_id": "16608", "title": "Pringsheim's Theorem", "text": "Let $f$ be a holomorphic function defined on a unit disc centered at the origin of the complex plane and is denoted by its Taylor series: :$\\map f z = \\displaystyle \\sum_{n \\mathop = 0}^{\\infty} c_n z^n$ Let: :$(1): \\quad \\forall n \\ge 0: c_n \\ge 0$ :$(2): \\quad$ the radius of convergence of the Taylor series of function $f$ is $1$. Then $z = 1$ is an isolated singularity of $f$."}
+{"_id": "16609", "title": "Mittag-Leffler Expansion for Tangent Function", "text": ":$\\displaystyle \\pi \\map \\tan {\\pi z} = 8 \\sum_{n \\mathop = 0}^\\infty \\frac z {\\paren {2 n + 1}^2 - 4 z^2}$ where:  :$z \\in \\C$ is not a half-integer :$\\tan$ is the tangent function."}
+{"_id": "16610", "title": "Mittag-Leffler Expansion for Hyperbolic Tangent Function", "text": ":$\\displaystyle \\pi \\map \\tanh {\\pi z} = 8 \\sum_{n \\mathop = 0}^\\infty \\frac z {4 z^2 + \\paren {2 n + 1}^2}$ where: :$z \\in \\C$ is not a half-integer multiple of $i$ :$\\tanh$ is the hyperbolic tangent function."}
+{"_id": "16611", "title": "Mittag-Leffler Expansion for Hyperbolic Secant Function", "text": ":$\\displaystyle \\pi \\, \\map \\sech {\\pi z} = 4 \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {2 n + 1} {\\paren {2 n + 1}^2 + 4 z^2}$ where:  :$z \\in \\C$ is not a half-integer multiple of $i$  :$\\sech$ is the hyperbolic secant function."}
+{"_id": "16612", "title": "Mittag-Leffler Expansion for Hyperbolic Cosecant Function", "text": ":$\\ds \\pi \\map \\csch {\\pi z} = \\frac 1 z + 2 \\sum_{n \\mathop = 1}^\\infty \\paren {-1}^n \\frac z {z^2 + n^2}$ where: :$z \\in \\C$ is not an integer multiple of $i$ :$\\csch$ is the hyperbolic cosecant function."}
+{"_id": "16617", "title": "Real Numbers under Subtraction do not form Semigroup", "text": "The set of real numbers under subtraction $\\struct {\\R, -}$ does not form a semigroup."}
+{"_id": "16619", "title": "Count of Binary Operations on Set/Sequence", "text": "The sequence of $N$ for each $n$ begins: $\\begin{array} {c|rr} n & n^2 & N = n^{\\paren {n^2} } \\\\ \\hline 1 & 1 & 1 \\\\ 2 & 4 & 16 \\\\ 3 & 9 & 19 \\ 683 \\\\ 4 & 16 & 4 \\ 294 \\ 967 \\ 296 \\\\ \\end{array}$ There are still only $4$ elements in a set, and already there are over $4$ thousand million different possible algebraic structures."}
+{"_id": "16620", "title": "Count of Commutative Binary Operations on Set/Sequence", "text": "The sequence of $N$ for each $n$ begins: $\\begin{array} {c|cr} n & \\dfrac {n \\paren {n + 1} }2 & n^{\\frac {n \\paren {n + 1} } 2} \\\\ \\hline 1 & 1 & 1 \\\\ 2 & 3 & 8 \\\\ 3 & 6 & 729 \\\\ 4 & 10 & 1 \\ 048 \\ 576 \\\\ \\end{array}$ and so on."}
+{"_id": "16621", "title": "Mittag-Leffler’s Expansion Theorem", "text": "Let $f$ be a meromorphic function with only simple poles continuous, or with a removable singularity, at $0$.  Let $X$ be the set of poles of $f$.  For $N \\in \\N$, let $C_N$ be a circle, centred at the origin, of radius $R_N$, where $R_N \\to \\infty$ as $N \\to \\infty$, such that $\\partial C_N$ contains no poles of $f$ for any $N$.  Let $M > 0$ be a real number independent of $N$ such that for all $z \\in \\partial C_N$, $\\cmod {\\map f z} < M$ , for all $N \\in \\N$. Then: :$\\displaystyle \\map f z = \\map f 0 + \\sum_{n \\mathop \\in X} \\Res f n \\paren {\\frac 1 {z - n} + \\frac 1 n}$  where: :$\\Res f n$ is the residue of $f$ at $n$ :$z$ is not a pole of $f$ :$\\displaystyle \\map f 0 = \\lim_{z \\mathop \\to 0} \\map f z$ if $f$ has a removable singularity at $0$."}
+{"_id": "16622", "title": "Normed Division Ring Operations are Continuous", "text": "Let $\\struct {R, +, *, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Let $d$ be the metric induced by the norm $\\norm {\\,\\cdot\\,}$. Let $p \\in \\R_{\\ge 1} \\cup \\set \\infty$. Let $d_p$ be the $p$-product metric on $R \\times R$. Let $R^* = R \\setminus \\set 0$ Let $d^*$ be the restriction of $d$ to $R^*$. Then the following results hold:"}
+{"_id": "16623", "title": "Normed Division Ring Operations are Continuous/Addition", "text": ":$+ : \\struct {R \\times R, d_p} \\to \\struct{R,d}$  is continuous."}
+{"_id": "16624", "title": "Normed Division Ring Operations are Continuous/Negation", "text": ":$\\eta: \\struct {R, d} \\to \\struct {R, d}: \\map \\eta x = -x$  is continuous."}
+{"_id": "16625", "title": "Normed Division Ring Operations are Continuous/Multiplication", "text": ":$* : \\struct {R \\times R, d_p} \\to \\struct {R, d}$  is continuous."}
+{"_id": "16627", "title": "Normed Division Ring Operations are Continuous/Corollary", "text": "Let $\\tau$ be the topology induced by the metric $d$. Then: :$\\struct {R, \\tau}$ is a topological division ring."}
+{"_id": "16628", "title": "Count of Binary Operations with Identity/Sequence", "text": "The sequence of $N$ for each $n$ begins: $\\begin{array} {c|cr} n & \\paren {n - 1}^2 + 1 & n^{\\paren {n - 1}^2 + 1}\\\\ \\hline 1 & 1 & 1 \\\\ 2 & 2 & 4 \\\\ 3 & 5 & 243 \\\\ 4 & 10 & 1 \\ 048 \\ 576 \\\\ \\end{array}$"}
+{"_id": "16629", "title": "Structure with Element both Identity and Zero has One Element", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $z \\in S$ such that $z$ is both an identity element and a zero element. Then: :$S = \\set z$"}
+{"_id": "16630", "title": "Group/Examples/Linear Functions", "text": "Let $G$ be the set of all real functions $\\theta_{a, b}: \\R \\to \\R$ defined as: :$\\forall x \\in \\R: \\map {\\theta_{a, b} } x = a x + b$ where $a, b \\in \\R$ such that $a \\ne 0$. The algebraic structure $\\struct {G, \\circ}$, where $\\circ$ denotes composition of mappings, is a group. $\\struct {G, \\circ}$ is specifically non-abelian."}
+{"_id": "16631", "title": "Summation Formula for Alternating Series", "text": "Let $C_N$ be the square with vertices $\\paren {N + \\dfrac 1 2} \\paren {\\pm 1 \\pm i}$ for some real $N \\in \\N$.  Let $f$ be a function meromorphic on $C_N$. Let $\\cmod {\\map f z} < \\dfrac M {\\cmod z^k}$, for constants $k > 1$ and $M$ independent of $N$, for all $z \\in \\partial C_N$.  Let $X$ be the set of poles of $f$. Then:  :$\\displaystyle \\sum_{n \\mathop \\in \\Z \\mathop \\setminus X} \\paren {-1}^n \\map f n = -\\sum_{z_0 \\mathop \\in X} \\Res {\\pi \\map \\csc {\\pi z} \\map f z} {z_0}$ If $X \\cap \\Z = \\O$, this becomes:  :$\\displaystyle \\sum_{n \\mathop = -\\infty}^\\infty \\paren {-1}^n \\map f n = -\\sum_{z_0 \\mathop \\in X} \\Res {\\pi \\map \\csc {\\pi z} \\map f z} {z_0}$"}
+{"_id": "16632", "title": "Integer Multiples under Addition form Subgroup of Integers", "text": "Let $\\struct {\\Z, +}$ denote the additive group of integers. Let $n \\Z$ be the set of integer multiples of $n$. Then $\\struct {n \\Z, +}$ is a subgroup of $\\struct {\\Z, +}$. Hence $\\struct {n \\Z, +}$ can be justifiably referred to as the additive group of integer multiples."}
+{"_id": "16634", "title": "Group is Generated by Itself", "text": "Let $G$ be a group. Then: :$G = \\gen G$ where $\\gen G$ denotes the group generated by $G$."}
+{"_id": "16636", "title": "Summation Formula for Alternating Series over Half-Integers", "text": "Let $C_N$ be the square with vertices $N \\paren {\\pm 1 \\pm i}$ for real $N \\in \\N$.  Let $f$ be a function meromorphic on $C_N$. Let $\\cmod {\\map f z} < \\dfrac M {\\cmod z^k}$, for constants $k > 1$ and $M$ independent of $N$, for all $z \\in \\partial C_N$.  Let $X$ be the set of poles of $f$. Let $Y$ be the set of poles of $\\map f {\\dfrac {2 z + 1} 2}$.  Then:  :$\\displaystyle \\sum_{n \\in \\Z \\setminus Y} \\paren {-1}^n \\map f {\\frac {2 n + 1} 2} = \\sum_{z_0 \\in X} \\Res {\\pi \\sec \\paren {\\pi z} \\map f z} {z_0}$ If $Y \\cap \\Z = \\O$, this becomes:  :$\\displaystyle \\sum_{n \\mathop = -\\infty}^\\infty \\paren {-1}^n \\map f {\\frac {2 n + 1} 2} = \\sum_{z_0 \\in X} \\Res {\\pi \\sec \\paren {\\pi z} \\map f z} {z_0}$"}
+{"_id": "16637", "title": "Summation Formula over Half-Integers", "text": "Let $C_N$ be the square with vertices $N \\paren {\\pm 1 \\pm i}$ for real $N \\in \\N$.  Let $f$ be a function meromorphic on $C_N$. Let $\\cmod {\\map f z} < \\dfrac M {\\cmod z^k}$, for constants $k > 1$ and $M$ independent of $N$, for all $z \\in \\partial C_N$.  Let $X$ be the set of poles of $f$. Let $Y$ be the set of poles of $\\map f {\\dfrac {2 z + 1} 2}$. Then:  :$\\displaystyle \\sum_{n \\in \\Z \\setminus Y} \\map f {\\frac {2 n + 1} 2} = \\sum_{z_0 \\in X} \\Res {\\pi \\tan \\paren {\\pi z} \\map f z} {z_0}$ If $Y \\cap \\Z = \\O$, this becomes:  :$\\displaystyle \\sum_{n \\mathop = -\\infty}^\\infty \\map f {\\frac {2 n + 1} 2} = \\sum_{z_0 \\in X} \\Res {\\pi \\tan \\paren {\\pi z} \\map f z} {z_0}$"}
+{"_id": "16638", "title": "Heaviside Expansion Formula", "text": "Let $P, Q$ be polynomials with coefficients in $\\C$.  Let $\\deg Q \\ge \\deg P + 1$. Let $\\map Q z$ have a simple zero for $z \\in X$.  Let $\\map {\\laptrans f} z = \\dfrac {\\map P z} {\\map Q z}$.  Then: :$\\displaystyle \\map f t = \\sum_{z \\mathop \\in X} e^{z t} \\frac {\\map P z} {\\map {Q'} z}$"}
+{"_id": "16639", "title": "Action of Inverse of Group Element", "text": "Let $\\struct {G, \\circ}$ be a group. Let $S$ be a sets. Let $*: G \\times S \\to S$ be a group action. Then: :$g * a = b \\iff g^{-1} * b = a$"}
+{"_id": "16640", "title": "Union Operation on Supersets of Subset is Closed", "text": "Let $S$ be a set. Let $T \\subseteq S$ be a given subset of $S$. Let $\\powerset S$ denote the power set of $S$ Let $\\mathscr S$ be the subset of $\\powerset S$ defined as: :$\\mathscr S = \\set {Y \\in \\powerset S: T \\subseteq Y}$ Then the algebraic structure $\\struct {\\mathscr S, \\cup}$ is closed."}
+{"_id": "16642", "title": "Existence of Magma with no Proper Submagma", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive integer. Let $S$ be a set of cardinality $n$: :$\\card S = n$ Then there exists an operation $\\circ$ on $S$ such that: :$\\struct {S, \\circ}$ is a magma :$\\struct {S, \\circ}$ has no submagma $\\struct {T, \\circ}$ such that $T$ is a non-empty proper subset of $S$."}
+{"_id": "16643", "title": "Subset of Abelian Group Generated by Product of Element with Inverse Element is Subgroup", "text": "Let $\\struct {G, \\circ}$ be an abelian group. Let $S \\subset G$ be a non-empty subset of $G$ such that $\\struct {S, \\circ}$ is closed. Let $H$ be the set defined as: :$H := \\set {x \\circ y^{-1}: x, y \\in S}$ Then $\\struct {H, \\circ}$ is a subgroup of $\\struct {G, \\circ}$."}
+{"_id": "16644", "title": "Local Basis Test", "text": "Let $\\struct {S, \\tau}$ be a topological space. Let $x \\in S$. Let $\\BB$ be a local basis for $x$ in $\\struct {S, \\tau}$. Let $\\CC$ be a set of open neighborhoods of $x$. Then: :$\\CC$ is a local basis {{iff}}: ::$\\forall B \\in \\BB \\implies \\exists C \\in \\CC: C \\subseteq B$"}
+{"_id": "16647", "title": "Union of Subgroups/Corollary 1", "text": "Let $H \\cup K$ be a subgroup of $G$. Then either $H \\subseteq K$ or $K \\subseteq H$."}
+{"_id": "16650", "title": "Stabilizer of Element after Group Action", "text": "Let $\\struct {G, \\circ}$ be a group. Let $S$ be a set. Let $*_S: G \\times S \\to S$ be a group actions. Let $x \\in S, a \\in G$. Then: :$\\Stab {a * x} = a^{-1} \\circ \\Stab x \\circ a$"}
+{"_id": "16651", "title": "Group Action of Symmetric Group/Subset", "text": "Let $r \\in \\N: 0 < r \\le n$. Let $B_r$ denote the set of all subsets of $\\N_n$ of cardinality $r$: :$B_r  := \\set {S \\subseteq \\N_n: \\card S = r}$ Let $*$ be the mapping $*: S_n \\times B_r \\to B_r$ defined as: :$\\forall \\pi \\in S_n, \\forall S \\in B_r: \\pi * B_r = \\pi \\sqbrk S$ where $\\pi \\sqbrk S$ denotes the image of $S$ under $\\pi$. Then $*$ is a group action."}
+{"_id": "16652", "title": "Group Action of Symmetric Group on Subset is Transitive", "text": "Let $r \\in \\N: 0 < r \\le n$. Let $B_r$ denote the set of all subsets of $\\N_n$ of cardinality $r$: :$B_r  := \\set {S \\subseteq \\N_n: \\card S = r}$ Let $*$ be the mapping $*: S_n \\times B_r \\to B_r$ defined as: :$\\forall \\pi \\in S_n, \\forall S \\in B_r: \\pi * B_r = \\pi \\sqbrk S$ where $\\pi \\sqbrk S$ denotes the image of $S$ under $\\pi$. Then $*$ is a transitive group action."}
+{"_id": "16654", "title": "Coset Product on Non-Normal Subgroup is not Well-Defined", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H$ be a subgroup of $G$ which is not normal. Let $a, b \\in G$. Then it is not necessarily the case that the coset product: :$\\paren {a \\circ H} \\circ \\paren {b \\circ H} = \\paren {a \\circ b} \\circ H$ is well-defined."}
+{"_id": "16655", "title": "Klein Four-Group is Normal in A4", "text": "Let $A_4$ denote the alternating group on $4$ letters, whose Cayley table is given as: {{:Alternating Group on 4 Letters/Cayley Table}} <onlyinclude> The subsets of $A_4$ which form subgroups of $A_4$ are as follows: Consider the order $4$ subgroup $V$ of $A_4$, presented by Cayley table: :$\\begin{array}{c|cccc} \\circ & e & t & u & v \\\\ \\hline e & e & t & u & v \\\\ t & t & e & v & u \\\\ u & u & v & e & t \\\\ v & v & u & t & e \\\\ \\end{array}$ Then $V$ is normal in $A_4$. Its index is: :$\\index {A_4} V = \\dfrac {\\order {A_4} } {\\order V} = \\dfrac {12} 4 = 3$ The (left) cosets of $V$ are: :$V$ :$A := a V$ :$P := p V$ and the Cayley table of the quotient group $A_4 / V$ is given by: :$\\begin{array}{c|ccc} \\circ & V & A & P \\\\ \\hline V & V & A & P \\\\ A & A & P & V \\\\ P & P & V & A \\\\ \\end{array}$ Note that while $A_4 / V$ is Abelian, $A_4$ is not."}
+{"_id": "16656", "title": "Coset of Trivial Subgroup is Singleton", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $E := \\struct {\\set e, \\circ}$ denote the trivial subgroup of $\\struct {G, \\circ}$. Let $g \\in G$. Then the left coset and right coset of $E$ by $g$ is $\\set g$."}
+{"_id": "16657", "title": "Inverse Elements of Right Transversal is Left Transversal", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $S \\subseteq G$ be a right transversal for $H$ in $G$. Let $T$ be the set defined as: :$T := \\set {x^{-1}: x \\in S}$ where $x^{-1}$ is the inverse of $x$ in $G$. Then $T$ is a left transversal for $H$ in $G$."}
+{"_id": "16658", "title": "Condition for Subset of Group to be Right Transversal", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$ whose index in $G$ is $n$: :$\\index G H = n$ Let $S \\subseteq G$ be a subset of $G$ of cardinality $n$. Then $S$ is a right transversal for $H$ in $G$ {{iff}}: :$\\forall x, y \\in S: x \\ne y \\implies x y^{-1} \\notin H$"}
+{"_id": "16659", "title": "Group Action on Coset Space is Transitive", "text": "Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$. Let $*: G \\times G / H \\to G / H$ be the action on the (left) coset space: :$\\forall g \\in G, \\forall g' H \\in G / H: g * \\paren {g' H} := \\paren {g g'} H$ Then $G$ is a transitive group action."}
+{"_id": "16660", "title": "Stabilizer of Coset under Group Action on Coset Space", "text": "Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$. Let $*: G \\times G / H \\to G / H$ be the action on the (left) coset space: :$\\forall g \\in G, \\forall g' H \\in G / H: g * \\paren {g' H} := \\paren {g g'} H$ Then the stabilizer of $a H$ under $*$ is given by: :$\\Stab {a H} = a H a^{-1}$"}
+{"_id": "16663", "title": "Index of Subgroup equals Index of Conjugate", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then: :$\\index G H = \\index G {a H a^{-1} }$ where $\\index G H$ denotes the index of $H$ in $G$."}
+{"_id": "16664", "title": "Normality Relation is not Transitive", "text": "Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $K$ be a normal subgroup of $N$. Then it is not necessarily the case that $K$ is a normal subgroup of $G$."}
+{"_id": "16666", "title": "Power of Coset Product is Coset of Power", "text": "Let $\\struct {G, \\circ}$ be a group. Let $N$ be a normal subgroup of $G$. Let $a \\in G$. Then: :$\\forall n \\in \\Z: \\paren {a \\circ N} = \\paren {a^n} \\circ N$"}
+{"_id": "16667", "title": "Condition for Power of Element of Quotient Group to be Identity", "text": "Let $G$ be a group whose identity is $e$. Let $N$ be a normal subgroup of $G$. Let $a \\in G$. Then: :$\\paren {a N}^n$ is the identity of the quotient group $G / N$ {{iff}}: :$a^n \\in N$"}
+{"_id": "16668", "title": "Additive Group of Integers is Normal Subgroup of Reals", "text": "Let $\\struct {\\Z, +}$ be the additive group of integers. Let $\\struct {\\R, +}$ be the additive group of real numbers. Then $\\struct {\\Z, +}$ is a normal subgroup of $\\struct {\\R, +}$."}
+{"_id": "16671", "title": "Mapping from Additive Group of Integers to Powers of Group Element is Homomorphism", "text": "Let $\\struct {G, \\circ}$ be a group. Let $g \\in G$. Let $\\struct {\\Z, +}$ denote the additive group of integers. Let $\\phi_g: \\struct {\\Z, +} \\to \\struct {G, \\circ}$ be the mapping defined as: :$\\forall k \\in \\Z: \\map {\\phi_g} k = g^k$ Then $\\phi_g$ is a (group) homomorphism."}
+{"_id": "16672", "title": "Inner Automorphisms form Subgroup of Symmetric Group", "text": "Let $G$ be a group. Let $\\struct {\\map \\Gamma G, \\circ}$ be the symmetric group on $G$. Let $\\Inn G$ denote the inner automorphism group of $G$. Then: :$\\Inn G \\le \\struct {\\map \\Gamma G, \\circ}$ where $\\le$ denotes the relation of being a subgroup."}
+{"_id": "16673", "title": "Order is Preserved by Group Isomorphism", "text": "Let $G$ and $H$ be groups. Let $\\phi: G \\to H$ be a (group) isomorphism. Then: :$\\order G = \\order H$ where $\\order {\\, \\cdot \\,}$ denotes the order of a group."}
+{"_id": "16674", "title": "Additive Group of Reals is Subgroup of Complex", "text": "Let $\\struct {\\R, +}$ be the additive group of real numbers. Let $\\struct {\\C, +}$ be the additive group of complex numbers. Then $\\struct {\\R, +}$ is a subgroup of $\\struct {\\C, +}$."}
+{"_id": "16677", "title": "Mapping to Power is Endomorphism iff Abelian", "text": "Let $\\struct {G, \\circ}$ be a group. Let $n \\in \\Z$ be an integer. Let $\\phi: G \\to G$ be defined as: :$\\forall g \\in G: \\map \\phi g = g^n$ Then $\\struct {G, \\circ}$ is abelian {{iff}} $\\phi$ is a (group) endomorphism."}
+{"_id": "16679", "title": "Additive Groups of Integers and Integer Multiples are Isomorphic", "text": "Let $n \\in \\Z_{> 0}$ be a strictly positive integer. Let $\\struct {n \\Z, +}$ denote the additive group of integer multiples. Let $\\struct {\\Z, +}$ denote the additive group of integers. Then $\\struct {n \\Z, +}$ is isomorphic to $\\struct {\\Z, +}$."}
+{"_id": "16680", "title": "Additive Group of Real Numbers is Not Isomorphic to Multiplicative Group of Real Numbers", "text": "Let $\\struct {\\R, +}$ denote the additive group of real numbers. Let $\\struct {\\R_{\\ne 0}, \\times}$ denote the multiplicative group of real numbers. Then $\\struct {\\R, +}$ is not isomorphic to $\\struct {\\R_{\\ne 0}, \\times}$."}
+{"_id": "16681", "title": "Normal Subgroup is Kernel of Group Homomorphism", "text": "Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Then there exists a group homomorphism of which $N$ is the kernel."}
+{"_id": "16682", "title": "Homomorphic Image of Cyclic Group is Cyclic Group", "text": "Let $G$ be a cyclic group with generator $g$. Let $H$ be a group. Let $\\phi: G \\to H$ be a (group) homomorphism. Let $\\Img G$ denote the homomorphic image of $G$ under $\\phi$. Then $\\Img G$ is a cyclic group with generator $\\map \\phi g$. That is: :$\\phi \\sqbrk {\\gen g} = \\gen {\\map \\phi g}$"}
+{"_id": "16684", "title": "Kernel of Homomorphism on Cyclic Group", "text": "Let $G = \\gen g$ be a cyclic group with generator $g$. Let $H$ be a group. Let $\\phi: G \\to H$ be a (group) homomorphism. Let $\\map \\ker \\phi$ denote the kernel of $\\phi$. Let $\\Img G$ denote the homomorphic image of $G$ under $\\phi$. Then: :$\\map \\ker \\phi = \\gen {g^m}$ where: :$m = 0$ if $\\Img \\phi$ is an infinite cyclic group :$m = \\order {\\Img \\phi}$ if $\\Img \\phi$ is a finite cyclic group."}
+{"_id": "16686", "title": "Mapping from Group Element to Inner Automorphism is Homomorphism", "text": "Let $G$ be a group. Let $\\kappa: G \\to \\Aut G$ be the mapping from $G$ to the automorphism group of $G$ defined as: :$\\forall x \\in G: \\map \\kappa x := \\kappa_x$ where $\\kappa_x$ is the inner automorphism on $x$: :$\\forall g \\in G: \\map {\\kappa_x} g = x g x^{-1}$ Then $\\kappa$ is a homomorphism."}
+{"_id": "16687", "title": "Image of Mapping from Group Element to Inner Automorphism is Inner Automorphism Group", "text": "Let $G$ be a group. Let $\\kappa: G \\to \\Aut G$ be the mapping from $G$ to the automorphism group of $G$ defined as: :$\\forall x \\in G: \\map \\kappa x := \\kappa_x$ where $\\kappa_x$ is the inner automorphism on $x$: :$\\forall g \\in G: \\map {\\kappa_x} g = x g x^{-1}$ Then $\\Img \\kappa$ is the inner automorphism group of $G$."}
+{"_id": "16688", "title": "Order of Monomorphic Image of Group Element", "text": "Let $G$ and $H$be groups whose identities are $e_G$ and $e_H$ respectively. Let $\\phi: G \\to H$ be a monomorphism. Let $g \\in G$ be of finite order. Then: :$\\forall g \\in G: \\order {\\map \\phi g} = \\order g$"}
+{"_id": "16689", "title": "Image under Epimorphism of Center is Subset of Center", "text": "Let $G$ and $H$ be groups. Let $\\theta: G \\to H$ be an epimorphism. Let $\\map Z G$ denote the center of $G$. Then: :$\\theta \\sqbrk {\\map Z G} \\subseteq \\map Z H$"}
+{"_id": "16690", "title": "Metric Subspace Induces Subspace Topology", "text": "Let $M = \\struct {A,d}$ be a metric space. Let $H \\subseteq A$. Let $\\tau$ be the topology induced by the metric $d$. Let $\\tau_H$ be the subspace topology induced by $\\tau$ on $H$. Let $d_H$ be the subspace metric induced by $d$ on $H$. Let $\\tau_{d_H}$ be the topology induced by the metric $d_H$. Then: :$\\tau_{d_H} = \\tau_H$"}
+{"_id": "16694", "title": "Quotient of Cauchy Sequences is Metric Completion/Lemma 1", "text": ":$\\quad \\mathcal C \\,\\big / \\mathcal N = \\tilde {\\mathcal C}$"}
+{"_id": "16695", "title": "Quotient of Cauchy Sequences is Metric Completion/Lemma 2", "text": ":$\\quad d' = \\tilde d$"}
+{"_id": "16696", "title": "Generator for Quaternion Group", "text": "The Quaternion Group can be generated by the matrices: :$\\mathbf a = \\begin{bmatrix} 0 & 1 \\\\ -1 & 0 \\end{bmatrix} \\qquad \\mathbf b = \\begin{bmatrix} 0 & i \\\\ i & 0 \\end{bmatrix}$ where $i$ is the imaginary unit: :$i^2 = -1$"}
+{"_id": "16697", "title": "Condition for Nu Function to be 1", "text": "Let: :$n = \\displaystyle \\prod_{i \\mathop = 1}^s p_i^{m_i}$ where $p_1, p_2, \\ldots, p_s$ are distinct primes. Then: :$(1): \\quad m_1, m_2, \\ldots, m_s = 1$, that is, $n$ is square-free :$(2): \\quad \\forall i, j \\in \\set {1, 2, \\ldots, s}: p_i \\not \\equiv 1 \\pmod {p_j}$ {{iff}}: :every group $G$ of order $n$ is cyclic and so $\\map \\nu n = 1$."}
+{"_id": "16698", "title": "Group of Order 15 is Cyclic Group", "text": "Let $G$ be a group whose order is $15$. Then $G$ is cyclic."}
+{"_id": "16699", "title": "Number of Abelian Groups", "text": "Let $n \\in \\Z_{\\ge 1}$ be a (strictly) positive integer. Let: :$n = \\displaystyle \\prod_{i \\mathop = 1}^s p_i^{m_i}$ where the $p_i$ are distinct primes. Let $\\map {\\nu_a} n$ denote the number of abelian groups of order $n$. Then: :$\\map {\\nu_a} n = \\displaystyle \\prod_{i \\mathop = 1}^s \\map {\\nu_a} {p_i^{m_i} }$ where: :$\\map {\\nu_a} {p_i^{m_i} }$ is the number of integer partitions of $m_i$."}
+{"_id": "16700", "title": "Order of Quotient Group", "text": "Let $G$ be a finite group. Let $N$ be a normal subgroup of $G$. Let $G / N$ be the quotient group of $G$ by $N$. Then: :$\\dfrac {\\order G} {\\order N} = \\order {G / N}$ where $\\order G$ denotes the order of $G$."}
+{"_id": "16701", "title": "Definition:Composition Series/Composition Length", "text": "Let $\\HH$ be a composition series for $G$. The '''composition length''' of $G$ is the length of $\\HH$."}
+{"_id": "16702", "title": "Definition:Composition Series/Composition Factor", "text": "Let $\\HH = \\set e = G_0 \\lhd G_1 \\lhd \\cdots \\lhd G_{n - 1} \\lhd G_n = G$ be a composition series for $G$. Each of the quotient groups: :$G_1 / G_0, G_2 / G_1, \\ldots, G_n / G_{n - 1}$ are the '''composition factors''' of $G$."}
+{"_id": "16703", "title": "Non-Abelian Simple Finite Groups are Infinitely Many", "text": "There exist infinitely many types of group which are non-abelian and finite."}
+{"_id": "16706", "title": "Upper Bound of Order of Non-Abelian Finite Simple Group", "text": "Let $G$ be a non-abelian finite simple group. Let $t \\in G$ be a self-inverse element of $G$. Let $\\map {C_G} t$ denote the centralizer of $t$ in $G$. Let $m = \\order {\\map {C_G} t}$ be the order of $\\map {C_G} t$. Then: :$\\order G \\le \\paren {\\dfrac {m \\paren {m + 1} } 2}!$"}
+{"_id": "16707", "title": "Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 1", "text": ":$\\norm {\\, \\cdot \\,}_1$ is well-defined. That is,  :$(1): \\quad \\forall \\eqclass {x_n}{}: \\lim_{n \\mathop \\to \\infty} \\norm{x_n}$ exists. :$(2): \\quad \\displaystyle \\forall \\eqclass {x_n}{}, \\eqclass {y_n}{} \\in \\mathcal C \\,\\big / \\mathcal N: \\eqclass {x_n}{} = \\eqclass {y_n}{} \\implies \\lim_{n \\mathop \\to \\infty} \\norm{x_n} = \\lim_{n \\mathop \\to \\infty} \\norm{y_n}$"}
+{"_id": "16708", "title": "Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 2", "text": ":$\\norm {\\, \\cdot \\,}_1$ satisfies {{NormAxiom|1}} That is: :$\\forall \\eqclass {x_n} {} \\in \\CC \\,\\big / \\NN: \\norm {\\eqclass {x_n} {} }_1 = 0 \\iff \\eqclass {x_n} {} = \\eqclass {0_R} {} $"}
+{"_id": "16709", "title": "Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 3", "text": ":$\\norm {\\, \\cdot \\,}_1$ satisfies the {{NormAxiom|2}}. That is: :$\\forall \\eqclass {x_n} {}, \\eqclass {y_n} {} \\in \\CC \\,\\big / \\NN: \\norm {\\eqclass {x_n} {} \\eqclass {y_n} {} }_1 = \\norm {\\eqclass {x_n} {} }_1 \\times \\norm {\\eqclass {y_n} {} }_1$"}
+{"_id": "16710", "title": "Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 4", "text": ":$\\norm {\\, \\cdot \\,}_1$ satisfies the {{NormAxiom|3}}. That is: :$\\forall \\eqclass {x_n} {}, \\eqclass {y_n} {} \\in \\CC \\,\\big / \\NN: \\norm {\\eqclass {x_n} {} +  \\eqclass {y_n} {} }_1 \\le \\norm {\\eqclass {x_n} {} }_1 + \\norm {\\eqclass {y_n} {} }_1$"}
+{"_id": "16711", "title": "Invertible Elements of Monoid form Subgroup", "text": "Let $\\struct {S, \\circ}$ be a monoid whose identity element is $e$. Let $U \\subseteq S$ be the subset of $S$ consisting of the invertible elements of $S$. Then $\\struct {U, \\circ}$ forms a subgroup of $S$."}
+{"_id": "16712", "title": "Combination Theorem for Sequences/Normed Division Ring/Inverse Rule/Lemma", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} y_n^{-1} = l^{-1}$"}
+{"_id": "16714", "title": "Division Ring Norm is Continuous on Induced Metric Space", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,}}$ be a normed division ring. Let $d$ be the metric induced by the norm $\\norm {\\,\\cdot\\,}$. The mapping $\\norm {\\,\\cdot\\,} : \\struct {R, d} \\to \\R$ is continuous."}
+{"_id": "16716", "title": "Properties of Norm on Division Ring", "text": "Let $\\struct {R, +, \\circ}$ be a division ring with zero $0_R$ and unity $1_R$. Let $\\norm {\\,\\cdot\\,}$ be a norm on $R$. Let $x, y \\in R$. Then the following hold:"}
+{"_id": "16717", "title": "Properties of Norm on Division Ring/Norm of Negative", "text": ":$\\norm {-x} = \\norm {x}$"}
+{"_id": "16718", "title": "Properties of Norm on Division Ring/Norm of Unity", "text": ":$\\norm {1_R} = 1$."}
+{"_id": "16719", "title": "Properties of Norm on Division Ring/Norm of Negative of Unity", "text": ":$\\norm{-1_R} = 1$"}
+{"_id": "16720", "title": "Repunit is Zuckerman Number", "text": "Let $n$ be a repunit. Then $n$ is also a Zuckerman number."}
+{"_id": "16721", "title": "Properties of Norm on Division Ring/Norm of Difference", "text": ":$\\norm {x - y} \\le \\norm x + \\norm y$"}
+{"_id": "16722", "title": "Properties of Norm on Division Ring/Norm of Inverse", "text": ":$x \\ne 0_R \\implies \\norm {x^{-1} } = \\dfrac 1 {\\norm x}$"}
+{"_id": "16723", "title": "Properties of Norm on Division Ring/Norm of Quotient", "text": ":$y \\ne 0_R \\implies \\norm{xy^{-1}} = \\norm{y^{-1}x} = \\dfrac {\\norm{x}}{\\norm{y}}$"}
+{"_id": "16724", "title": "Properties of Norm on Division Ring/Norm of Power Equals Unity", "text": ":$\\forall n \\in \\N_{\\gt 0}: \\norm {x^n} = 1 \\implies \\norm x = 1$"}
+{"_id": "16727", "title": "Conversion from Hexadecimal to Binary", "text": "Let $n$ be a (positive) integer expressed in hexadecimal notation as: :$n = \\sqbrk {a_r a_{r - 1} \\dotso a_1 a_0}_H$ Then $n$ can be expressed in binary notation as: :$n = \\sqbrk {b_{r 3} b_{r 2} b_{r 1} b_{r 0} b_{\\paren {r - 1} 3} b_{\\paren {r - 1} 2} b_{\\paren {r - 1} 1} b_{\\paren {r - 1} 0} \\dotso b_{1 3} b_{1 2} b_{1 1} b_{1 0} b_{0 3} b_{0 2} b_{0 1} b_{0 0} }_2$ where $\\sqbrk {b_{j 3} b_{j 2} b_{j 1} b_{j 0} }_2$ is the expression of the hexadecimal digit $a_j$ in binary notation. That is, you take the binary expression of each hexadecimal digit, padding them out with zeroes to make them $4$ bits long, and simply concatenate them."}
+{"_id": "16728", "title": "Birthday Paradox/General/3", "text": "Let $n$ be a set of people. Let the probability that at least $3$ of them have the same birthday be greater than $50 \\%$. Then $n \\ge 88$."}
+{"_id": "16729", "title": "Convergent Sequence is Cauchy Sequence/Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,}} $ be a normed division ring. Every convergent sequence in $R$ is a Cauchy sequence."}
+{"_id": "16730", "title": "Sum of 3 Squares in 2 Distinct Ways", "text": "$27$ is the smallest positive integer which can be expressed as the sum of $3$ square numbers in $2$ distinct ways: {{begin-eqn}} {{eqn | l = 27       | r = 3^2 + 3^2 + 3^2 }} {{eqn | r = 5^2 + 1^2 + 1^2 }} {{end-eqn}}"}
+{"_id": "16731", "title": "Triangular Numbers which are Sum of Two Cubes", "text": "The sequence of triangular numbers which are the sum of $2$ cubes begins: :$28, 91, 351, 2926, 8001, 46971, 58653, 93528, 97461, \\dots$ {{OEIS|A113958}}"}
+{"_id": "16732", "title": "Product of Factors of Perfect Number", "text": "Let $P$ be the perfect number $2^{n - 1} \\paren {2^n - 1}$. Then: :$\\displaystyle \\prod_{d \\mathop \\divides P} d = P^n$"}
+{"_id": "16733", "title": "Sequence is Bounded in Norm iff Bounded in Metric", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} } $ be a normed division ring. Let $d$ be the metric induced on $R$ be the norm $\\norm {\\,\\cdot\\,}$. Let $\\sequence {x_n}$ be a sequence in $R$. Then: :$\\sequence {x_n} $ is a bounded sequence in the normed division ring $\\struct {R, \\norm {\\,\\cdot\\,} }$ {{iff}} $\\sequence {x_n} $ is a bounded sequence in the metric space $\\struct {R, d}$."}
+{"_id": "16734", "title": "Sequence is Bounded in Norm iff Bounded in Metric/Necessary Condition", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,}} $ be a normed division ring. Let $d$ be the metric induced on $R$ be the norm $\\norm {\\,\\cdot\\,}$. Let $\\sequence {x_n}$ be a sequence in $R$. Let $\\sequence {x_n} $ be a bounded sequence in the normed division ring $\\struct {R, \\norm {\\,\\cdot\\,}}$  Then: :$\\sequence {x_n} $ is a bounded sequence in the metric space $\\struct {R, d}$"}
+{"_id": "16735", "title": "Sequence is Bounded in Norm iff Bounded in Metric/Sufficient Condition", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,}} $ be a normed division ring. Let $d$ be the metric induced on $R$ be the norm $\\norm {\\,\\cdot\\,}$. Let $\\sequence {x_n}$ be a sequence in $R$. Let $\\sequence {x_n} $ be a bounded sequence in the metric space $\\struct {R, d}$ Then: :$\\sequence {x_n} $ is a bounded sequence in the normed division ring $\\struct {R, \\norm {\\,\\cdot\\,} }$"}
+{"_id": "16741", "title": "Magic Constant of Magic Cube", "text": "The magic constant of a magic cube of order $n$ is given by: :$C_n = \\dfrac {n \\paren {n^3 + 1} } 2$"}
+{"_id": "16742", "title": "Sum of Terms of Magic Cube", "text": "The total of all the entries in a magic cube of order $n$ is given by: :$T_n = \\dfrac {n^3 \\paren {n^3 + 1} } 2$"}
+{"_id": "16743", "title": "Smallest Magic Cube is of Order 3", "text": "Apart from the trivial order $1$ magic cube: {{:Magic Cube/Examples/Order 1}} the smallest magic cube is the order $3$ magic cube: {{:Magic Cube/Examples/Order 3}}"}
+{"_id": "16748", "title": "Fourth Power as Summation of Groups of Consecutive Integers", "text": "Take the positive integers and group them in sets such that the $m$th set contains the next $m$ positive integers: :$\\set 1, \\set {2, 3}, \\set {4, 5, 6}, \\set {7, 8, 9, 10}, \\set {11, 12, 13, 14, 15}, \\ldots$ Remove all the sets with an even number of elements. Then the sum of all the integers in the first $n$ sets remaining equals $n^4$."}
+{"_id": "16750", "title": "Smallest Number with 2^n Divisors", "text": "The smallest positive integer with $2^n$ divisors is found by multiplying together the first $n$ numbers in this sequence: :$2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, \\ldots$ which consists of all the primes and powers of primes."}
+{"_id": "16751", "title": "Convergent Subsequence of Cauchy Sequence/Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Let $\\sequence{x_n}_{n \\mathop \\in \\N}$ be a Cauchy sequence in $\\struct {R, \\norm {\\,\\cdot\\,} }$.  Let $x \\in R$. Then $\\sequence {x_n}$ converges to $x$ {{iff}} $\\sequence {x_n}$ has a subsequence that converges to $x$."}
+{"_id": "16752", "title": "Null Sequences form Maximal Left and Right Ideal", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\CC$ be the ring of Cauchy sequences over $R$. Let $\\NN$ be the set of null sequences. That is: :$\\NN = \\set {\\sequence {x_n}: \\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = 0 }$ Then $\\NN$ is a ring ideal of $\\CC$ that is a maximal left ideal and a maximal right ideal."}
+{"_id": "16753", "title": "Null Sequences form Maximal Left and Right Ideal/Lemma 1", "text": ":$\\mathcal N$ is an ideal of $\\mathcal C$."}
+{"_id": "16754", "title": "Null Sequences form Maximal Left and Right Ideal/Lemma 2", "text": ":$\\mathcal N$ is a maximal left ideal."}
+{"_id": "16755", "title": "Null Sequences form Maximal Left and Right Ideal/Lemma 3", "text": ":$\\NN$ is a maximal right ideal."}
+{"_id": "16756", "title": "Prime Decomposition of 8th Fermat Number", "text": "The prime decomposition of the $8$th Fermat number is given by: {{begin-eqn}} {{eqn | l = 2^{\\paren {2^8} } + 1       | r = 115 \\, 792 \\, 089 \\, 237 \\, 316 \\, 195 \\, 423 \\, 570 \\, 985 \\, 008 \\, 687 \\, 907 \\, 853 \\, 269 \\, 984 \\, 665 \\, 640 \\, 564 \\, 039 \\, 457 \\, 584 \\, 007 \\, 913 \\, 129 \\, 639 \\, 937       | c =  }} {{eqn | r = 1 \\, 238 \\, 926 \\, 361 \\, 552 \\, 897       | c =  }} {{eqn | o =       | ro= \\times       | r = 93 \\, 461 \\, 639 \\, 715 \\, 357 \\, 977 \\, 769 \\, 163 \\, 558 \\, 199 \\, 606 \\, 896 \\, 584 \\, 051 \\, 237 \\, 541 \\, 638 \\, 188 \\, 580 \\, 280 \\, 321       | c =  }} {{eqn | r = \\paren {2 \\times 157 \\times 3 \\, 853 \\, 149 \\, 761 \\times 2^{10} + 1}       | c =  }} {{eqn | o =       | ro= \\times       | r = \\paren {2 \\times 3 \\times 5 \\times 7 \\times 13 \\times 31 \\, 618 \\, 624 \\, 099 \\, 079 \\times 1 \\, 057 \\, 372 \\, 046 \\, 781 \\, 162 \\, 536 \\, 274 \\, 034 \\, 354 \\, 686 \\, 893 \\, 329 \\, 625 \\, 329 \\times 2^{10} + 1}       | c =  }} {{end-eqn}}"}
+{"_id": "16757", "title": "Product of Sequence of Fermat Numbers plus 2", "text": "Let $F_n$ denote the $n$th Fermat number. Then: {{begin-eqn}} {{eqn | ll= \\forall n \\in \\Z_{>0}:       | l = F_n       | r = \\prod_{j \\mathop = 0}^{n - 1} F_j + 2       | c =  }} {{eqn | r = F_0 F_1 \\dotsm F_{n - 1} + 2       | c =  }} {{end-eqn}}"}
+{"_id": "16758", "title": "492 is Sum of 3 Cubes in 3 Ways", "text": "$492$ can be expressed as the sum of $3$ cubes, either positive or negative in $3$ known ways. {{begin-eqn}} {{eqn | l = 492       | r = 50^3 + \\paren {-19}^3 + \\paren {-49}^3 }} {{eqn | r = 123 \\, 134^3 + 9179^3 + \\paren {-123 \\, 151}^3 }} {{eqn | r = 1 \\, 793 \\, 337 \\, 644^3 + \\paren {-81 \\, 3701 \\, 167}^3 + \\paren {-1 \\, 735 \\, 662 \\, 109}^3 }} {{end-eqn}}"}
+{"_id": "16759", "title": "Smallest n needing 6 Numbers less than n so that Product of Factorials is Square", "text": "Let $n \\in \\Z_{>0}$ be a positive integer. Then it is possible to choose at most $6$ positive integers less than $n$ such that the product of their factorials is square. The smallest $n$ that actually requires $6$ numbers to be chosen is $527$."}
+{"_id": "16761", "title": "Null Sequences form Maximal Left and Right Ideal/Lemma 1/Lemma 1.1", "text": ":$\\mathcal N \\ne \\O$"}
+{"_id": "16762", "title": "Null Sequences form Maximal Left and Right Ideal/Lemma 1/Lemma 1.2", "text": ":$\\forall \\sequence {x_n}, \\sequence {y_n} \\in \\NN: \\sequence {x_n} + \\paren {-\\sequence {y_n} } \\in \\NN$"}
+{"_id": "16763", "title": "Null Sequences form Maximal Left and Right Ideal/Lemma 1/Lemma 1.3", "text": ":$\\quad \\forall \\sequence {x_n} \\in \\NN, \\sequence {y_n} \\in \\CC: \\sequence {x_n} \\sequence {y_n} \\in \\NN, \\sequence {y_n} \\sequence {x_n} \\in \\NN$"}
+{"_id": "16764", "title": "Null Sequences form Maximal Left and Right Ideal/Lemma 2/Lemma 2.1", "text": ":$\\NN \\subsetneq \\CC$."}
+{"_id": "16765", "title": "Null Sequences form Maximal Left and Right Ideal/Lemma 2/Lemma 2.2", "text": ":There is no left ideal $\\JJ$ of $\\CC$ such that $\\NN \\subsetneq \\JJ \\subsetneq \\CC$"}
+{"_id": "16766", "title": "Difference between Two Squares equal to Repdigit", "text": "{{begin-eqn}} {{eqn | l = 6^2 - 5^2       | r = 11       | c =  }} {{eqn | l = 56^2 - 45^2       | r = 1111       | c =  }} {{eqn | l = 556^2 - 445^2       | r = 111 \\, 111       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 7^2 - 4^2       | r = 33       | c =  }} {{eqn | l = 67^2 - 34^2       | r = 3333       | c =  }} {{eqn | l = 667^2 - 334^2       | r = 333 \\, 333       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 8^2 - 3^2       | r = 55       | c =  }} {{eqn | l = 78^2 - 23^2       | r = 5555       | c =  }} {{eqn | l = 778^2 - 223^2       | r = 555 \\, 555       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 9^2 - 2^2       | r = 77       | c =  }} {{eqn | l = 89^2 - 12^2       | r = 7777       | c =  }} {{eqn | l = 889^2 - 112^2       | r = 777 \\, 777       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}}"}
+{"_id": "16767", "title": "Largest Number Not Expressible as Sum of Fewer than 8 Cubes", "text": "$8042$ is (probably) the largest positive integer that cannot be expressed as the sum of fewer than $8$ cubes."}
+{"_id": "16769", "title": "Sequence of 5 Consecutive Non-Primable Numbers by Changing 1 Digit", "text": "The following sequence of $5$ consecutive positive integers cannot be made into prime numbers by changing just one digit: :$872\\,894, 872\\,895, 872\\,896, 872\\,897, 872\\,898$ {{OEIS|A192545}}"}
+{"_id": "16772", "title": "Smallest 17 Primes in Arithmetic Sequence", "text": "The smallest $17$ primes in arithmetic sequence are: :$3\\,430\\,751\\,869 + 87\\,297\\,210 n$ for $n = 0, 1, \\ldots, 16$."}
+{"_id": "16774", "title": "Prime Gap of 654", "text": "There exists a prime gap of $654$ between $11\\,000\\,001\\,446\\,613\\,353$ and $11\\,000\\,001\\,446\\,614\\,007$."}
+{"_id": "16775", "title": "Pair of Large Twin Primes", "text": "The integers defined as: :$1\\,159\\,142\\,985 \\times 2^{2304} \\pm 1$ are a pair of twin primes each with $703$ digits."}
+{"_id": "16777", "title": "Integers under Subtraction do not form Group", "text": "Let $\\struct {\\Z, -}$ denote the algebraic structure formed by the set of integers under the operation of subtraction. Then $\\struct {\\Z, -}$ is not a group."}
+{"_id": "16778", "title": "Sequence of Powers of Number less than One/Necessary Condition", "text": "Let $x \\in \\R$ be such that $\\size{x} < 1$. Let $\\sequence {x_n}$ be the sequence in $\\R$ defined as $x_n = x^n$. Then $\\sequence {x_n}$ is a null sequence."}
+{"_id": "16784", "title": "Sequence of Powers of Number less than One/Sufficient Condition", "text": "Let $x \\in \\R$. Let $\\sequence {x_n}$ be the sequence in $\\R$ defined as $x_n = x^n$. Let $\\sequence {x_n}$ be a null sequence. Then $\\size x < 1$."}
+{"_id": "16786", "title": "Real Sine Function is neither Injective nor Surjective", "text": "The real sine function is neither an injection nor a surjection."}
+{"_id": "16787", "title": "Sequence of Powers of Number less than One/Complex Numbers", "text": "Let $z \\in \\C$. Let $\\sequence {z_n}$ be the sequence in $\\C$ defined as $z_n = z^n$. Then: :$\\size z < 1$ {{iff}} $\\sequence {z_n}$ is a null sequence."}
+{"_id": "16788", "title": "Sequence of Powers of Number less than One/Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,}}$ be a normed division ring Let $x \\in \\R$. Let $\\sequence {x_n}$ be the sequence in $\\R$ defined as $x_n = x^n$. Then: :$\\norm x < 1$ {{iff}} $\\sequence {x_n}$ is a null sequence."}
+{"_id": "16793", "title": "Sequence of Powers of Number less than One", "text": "Let $x \\in \\R$. Let $\\sequence {x_n}$ be the sequence in $\\R$ defined as $x_n = x^n$. Then: :$\\size x < 1$ {{iff}} $\\sequence {x_n}$ is a null sequence."}
+{"_id": "16794", "title": "Set of Rotations is Subgroup of Symmetry Group", "text": "Let $G$ be a symmetry group. Let $H$ be the subset of $G$ consisting of the rotations in $G$ about a given axis. Then $H$ is a subgroup of $G$."}
+{"_id": "16795", "title": "Equivalence of Definitions of Equivalent Division Ring Norms", "text": "Let $R$ be a division ring. Let $\\norm {\\, \\cdot \\,}_1: R \\to \\R_{\\ge 0}$ and $\\norm {\\, \\cdot \\,}_2: R \\to \\R_{\\ge 0}$ be norms on $R$. Let $d_1$ and $d_2$ be the metrics induced by the norms $\\norm {\\, \\cdot \\,}_1$ and $\\norm {\\, \\cdot \\,}_2$ respectively. {{TFAE|def = Equivalent Division Ring Norms}}"}
+{"_id": "16796", "title": "Equivalence of Definitions of Equivalent Division Ring Norms/Topologically Equivalent implies Convergently Equivalent", "text": "Let $d_1$ and $d_2$ be topologically equivalent metrics. Then: :$d_1$ and $d_2$ are convergently equivalent metrics."}
+{"_id": "16797", "title": "Equivalence of Definitions of Equivalent Division Ring Norms/Convergently Equivalent implies Null Sequence Equivalent", "text": "Let $\\norm {\\, \\cdot \\,}_1$ and $\\norm {\\, \\cdot \\,}_2$ satisfy: :for all sequences $\\sequence {x_n}$ in $R:\\sequence {x_n}$ converges to $l$ in $\\norm{\\, \\cdot \\,}_1 \\iff \\sequence {x_n}$ is a converges to $l$ in $\\norm {\\, \\cdot \\,}_2$ Then for all sequences $\\sequence {x_n}$ in $R$: :$\\sequence {x_n}$ is a null sequence in $\\norm {\\, \\cdot \\,}_1 \\iff \\sequence {x_n}$ is a null sequence in $\\norm {\\, \\cdot \\,}_2$"}
+{"_id": "16798", "title": "Equivalence of Definitions of Equivalent Division Ring Norms/Null Sequence Equivalent implies Open Unit Ball Equivalent", "text": "Let $\\norm {\\, \\cdot \\,}_1$ and $\\norm {\\, \\cdot \\,}_2$ satisfy: :for all sequences $\\sequence {x_n}$ in $R:\\sequence {x_n}$ is a null sequence in $\\norm {\\, \\cdot \\,}_1  \\iff \\sequence {x_n}$ is a null sequence in $\\norm {\\, \\cdot \\,}_2$  Then $\\forall x \\in R$: :$\\norm x_1 < 1 \\iff \\norm x_2 < 1$"}
+{"_id": "16799", "title": "Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm", "text": "Let $\\norm {\\, \\cdot \\,}_1$ and $\\norm {\\, \\cdot \\,}_2$ satisfy: :$\\forall x \\in R: \\norm x_1 < 1 \\iff \\norm x_2 < 1$  Then: :$\\exists \\alpha \\in \\R_{> 0}: \\forall x \\in R: \\norm x_1 = \\norm x_2^\\alpha$"}
+{"_id": "16800", "title": "Product of Subgroups of Prime Power Order", "text": "Let $p$ be a prime number. Let $G$ be a group of order $p^a k$, where: :$a \\in \\Z_{>0}$ is a (strictly) positive integer :$p$ is not a divisor of $k$. Let $P \\le G$ be a subgroup of $G$ of order $p^a$. Let $Q \\le G$ be a subgroup of $G$ of order $p^b$, where $0 < b \\le a$. Let it be the case that $Q$ is not a subgroup of $P$. Then $P Q$ is not a subgroup of $G$."}
+{"_id": "16801", "title": "Linearly Independent Set is Contained in some Basis", "text": "There exists a basis $B$ for $E$ such that $H \\subseteq B$."}
+{"_id": "16802", "title": "Linearly Independent Set is Basis iff of Same Cardinality as Dimension", "text": "$H$ is a basis for $E$ {{iff}} it contains exactly $n$ elements."}
+{"_id": "16804", "title": "ISBN-10 is Error-Correcting Code", "text": "ISBN-$10$ is an error-correcting code in the following sense:"}
+{"_id": "16805", "title": "ISBN-10 is Error-Correcting Code/Transposition Error", "text": "If any two of the first $9$ digits are transposed, the check digit will be wrong."}
+{"_id": "16806", "title": "ISBN-10 is Error-Correcting Code/Transmission Error", "text": "If an error has been made in any one of the first $9$ digits, the check digit will be wrong."}
+{"_id": "16807", "title": "Cardinality of Master Code", "text": "Let $\\map V {n, p}$ be a master code of length $n$ modulo $p$. Then there are $p^n$ elements of $\\map V {n, p}$."}
+{"_id": "16808", "title": "Master Code forms Vector Space", "text": "Let $\\map V {n, p}$ be a master code of length $n$ modulo $p$. Then $\\map V {n, p}$ forms a vector space over $\\Z_p$ of $n$ dimensions."}
+{"_id": "16811", "title": "Minimum Distance of Linear Code is Smallest Weight of Non-Zero Codeword", "text": "Let $C$ be a linear $\\tuple {n, k}$-code whose master code is $\\map V {n, p}$. Let $\\map d C$ denote the minimum distance of $C$. Then: :$\\map d C = \\displaystyle \\min_{u \\mathop \\in C} \\map w u$ where $\\map w u$ denotes the weight of $u$."}
+{"_id": "16813", "title": "Error Correction Capability of Linear Code", "text": "Let $C$ be a linear code. Let $C$ have a minimum distance $d$. Then $C$ corrects $e$ transmission errors for all $e$ such that $2 e + 1 \\le d$."}
+{"_id": "16814", "title": "Generation of Linear Code from Standard Generator Matrix", "text": "Let $G$ be a (standard) generator matrix for a linear code. The following methods can be used to generate a linear code from $G$:"}
+{"_id": "16815", "title": "Generation of Linear Code from Standard Generator Matrix/Method 1", "text": "A linear code $C$ can be obtained from $G$ by: :considering the rows of $G$ as codewords :forming all possible linear combinations of those codewords, considering them as vectors of a vector space."}
+{"_id": "16816", "title": "Generation of Linear Code from Standard Generator Matrix/Method 2", "text": "A linear code $C$ can be obtained from $G$ by: :taking the set $U$ of all sequences of length $k$ over $\\Z_p$ and expressing them as $1 \\times k$ matrices :forming all possible matrix products $u G$ for all $u \\in U$."}
+{"_id": "16817", "title": "Golay Ternary Code has Minimum Distance 5", "text": "The Golay ternary code has a minimum distance of $5$."}
+{"_id": "16818", "title": "Golay Ternary Code Corrects 2 Errors", "text": "The Golay ternary code corrects $2$ transmission errors."}
+{"_id": "16819", "title": "Decoding Received Word using Coset Decoding Table", "text": "Let $C$ be a linear code. Let $v$ be a received word, which may have transmission errors. To find out the transmitted codeword $u$ corresponding to $v$: :$(1): \\quad$ Find $v$ in the coset decoding table for $C$. :$(2): \\quad$ The corresponding transmitted codeword $u$ will be found at the top of the column where $v$ can be found."}
+{"_id": "16820", "title": "Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 1", "text": ":$\\norm {\\, \\cdot \\,}_1$ is the trivial norm."}
+{"_id": "16821", "title": "Syndrome is Zero iff Vector is Codeword", "text": "Let $C$ be a linear $\\tuple {n, k}$-code whose master code is $\\map V {n, p}$ Let $G$ be a (standard) generator matrix for $C$. Let $P$ be a standard parity check matrix for $C$. Let $w \\in \\map V {n, p}$. Then the syndrome of $w$ is zero {{iff}} $w$ is a codeword of $C$."}
+{"_id": "16822", "title": "Condition for Vectors to have Same Syndrome", "text": "Let $C$ be a linear $\\tuple {n, k}$-code whose master code is $\\map V {n, p}$ Let $G$ be a (standard) generator matrix for $C$. Let $P$ be a standard parity check matrix for $C$. Let $u, v \\in \\map V {n, p}$. Then $u$ and $v$ have the same syndrome {{iff}} they are in the same coset of $C$."}
+{"_id": "16823", "title": "Syndrome Decoding", "text": "Let $C$ be a linear $\\tuple {n, k}$-code whose master code is $\\map V {n, p}$ To decode a given vector $v$ of $\\map V {n, p}$, the syndrome of $v$ can be used as follows. Create an array $T$ of $2$ column consisting of the following: :The top row contains: ::in column $1$: the zero of $C$  ::in column $2$: its syndrome. :The $r$th row subsequent contains: ::in column $1$: any element of $\\map V {n, p}$ of minimum weight which is not already included in the first $r - 1$ rows ::in column $2$: its syndrome. To decode a given vector $v$ of $\\map V {n, p}$: :Calculate its syndrome :Find it in column $2$ of $T$ :See what is in column $1$ of $T$, and call it $u$, say :Subtract $u$ from $v$."}
+{"_id": "16824", "title": "Euler's Equation/Independent of x", "text": "Let $y$ be a mapping. Let $J$ be a functional such that: :$\\ds J \\sqbrk y = \\int_a^b \\map F {y, y'} \\rd x$ Then the corresponding Euler's Equation can be reduced to: :$F - y' F_{y'} = C$  where $C$ is an arbitrary constant."}
+{"_id": "16825", "title": "Euler's Equation/Independent of y", "text": "Let $y$ be a mapping Let $J$ be a functional such that :$\\ds J \\sqbrk y = \\int_a^b \\map F {x,y'} \\rd x$ Then the corresponding Euler's equation can be reduced to: :$F_{y'} = C$  where $C$ is an arbitrary constant."}
+{"_id": "16828", "title": "Subset of Linear Code with Even Weight Codewords", "text": "Let $C$ be a linear code. Let $C^+$ be the subset of $C$ consisting of all the codewords of $C$ which have even weight. Then $C^+$ is a subgroup of $C$ such that either $C^+ = C$ or such that $\\order {C^+} = \\dfrac {\\order C} 2$."}
+{"_id": "16829", "title": "Minimal Smooth Surface of Revolution", "text": "Let $\\map y x$ be a real mapping in 2-dimensional real Euclidean space. Let $y$ pass through the points $\\tuple {x_0, y_0}$ and $\\tuple {x_1, y_1}$. Consider a surface of revolution constructed by rotating $y$ around the $x$-axis. Suppose this surface is smooth for any $x$ between $x_0$ and $x_1$. Then its surface area is minimized by the following curve, known as a catenoid: :$y = C \\map \\cosh {\\dfrac {x + C_1} C}$ Furthermore, its area is: :$A = \\paren {x_1 - x_0} C \\pi + \\dfrac {\\pi C^2} 2 \\paren {\\map \\sinh {\\dfrac {2 \\paren {x_1 + C_1} } C} - \\map \\sinh {\\dfrac {2 \\paren {x_0 + C_1} } C} }$"}
+{"_id": "16830", "title": "Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2", "text": ":$\\forall x \\in R: \\norm x_1 = \\norm x_2^\\alpha$"}
+{"_id": "16831", "title": "Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.1", "text": "Then: :$\\forall y \\in R: \\norm y_1 > 1 \\iff \\norm y_2 > 1$"}
+{"_id": "16832", "title": "Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.2", "text": "Then: :$\\forall y \\in R:\\norm y_1 = 1 \\iff \\norm y_2 = 1$"}
+{"_id": "16834", "title": "Homomorphism from Group of Cube Roots of Unity to Itself", "text": "Let $\\struct {U_3, \\times}$ denote the multiplicative group of the complex cube roots of unity. Here, $U_3 = \\set {1, \\omega, \\omega^2}$ where $\\omega = e^{2 i \\pi / 3}$. Let $\\phi: U_3 \\to U_3$ be defined as: :$\\forall z \\in U_3: \\map \\phi z = \\begin{cases} 1 & : z = 1 \\\\ \\omega^2 & : z = \\omega \\\\ \\omega & : z = \\omega^2 \\end{cases}$ Then $\\phi$ is a group homomorphism."}
+{"_id": "16835", "title": "Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.3", "text": ":$\\alpha = \\beta$"}
+{"_id": "16838", "title": "Max is Associative", "text": "The Max operation is associative: : $\\max \\left({\\max \\left({x, y}\\right), z}\\right) = \\max \\left({x, \\max \\left({y, z}\\right)}\\right)$ Thus we are justified in writing $\\max \\left({x, y, z}\\right)$."}
+{"_id": "16839", "title": "Min is Associative", "text": "The Min operation is associative: : $\\min \\left({\\min \\left({x, y}\\right), z}\\right) = \\min \\left({x, \\min \\left({y, z}\\right)}\\right)$ Thus we are justified in writing $\\min \\left({x, y, z}\\right)$."}
+{"_id": "16840", "title": "Theoretical Justification for Cycle Notation", "text": "Let $\\N_k$ be used to denote the initial segment of natural numbers: :$\\N_k = \\closedint 1 k = \\set {1, 2, 3, \\ldots, k}$ Let $\\rho: \\N_n \\to \\N_n$ be a permutation of $n$ letters. Let $i \\in \\N_n$. Let $k$ be the smallest (strictly) positive integer for which $\\map {\\rho^k} i$ is in the set: :$\\set {i, \\map \\rho i, \\map {\\rho^2} i, \\ldots, \\map {\\rho^{k - 1} } i}$ Then: :$\\map {\\rho^k} i = i$"}
+{"_id": "16841", "title": "Equivalence of Definitions of Equivalent Division Ring Norms/Norm is Power of Other Norm implies Topologically Equivalent", "text": "Let $\\norm {\\, \\cdot \\,}_1$ and $\\norm {\\, \\cdot \\,}_2$ satisfy: :$\\exists \\alpha \\in \\R_{\\gt 0}: \\forall x \\in R: \\norm x_1 = \\norm x_2^\\alpha$ Then $d_1$ and $d_2$ are topologically equivalent metrics."}
+{"_id": "16842", "title": "Equivalence of Definitions of Equivalent Division Ring Norms/Cauchy Sequence Equivalent implies Open Unit Ball Equivalent", "text": "Let $R$ be a division ring. Let $\\norm {\\, \\cdot \\,}_1: R \\to \\R_{\\ge 0}$ and $\\norm {\\, \\cdot \\,}_2: R \\to \\R_{\\ge 0}$ be norms on $R$. Let $\\norm {\\, \\cdot \\,}_1$ and $\\norm {\\, \\cdot \\,}_2$ satisfy: :for all sequences $\\sequence {x_n}$ in $R$: $\\sequence {x_n}$ is a Cauchy sequence in $\\norm {\\, \\cdot \\,}_1$ {{iff}} $\\sequence {x_n}$ is a Cauchy sequence in $\\norm {\\, \\cdot \\,}_2$ Then $\\forall x \\in R$: :$\\norm x_1 < 1 \\iff \\norm x_2 < 1$"}
+{"_id": "16843", "title": "Equal Order Elements may not be Conjugate", "text": "Let $G$ be a group Let $x, y \\in G$ be elements of $G$ such that: :$\\order x = \\order y$ where $\\order x$ denotes the order of $x$. Then it is not necessarily the case that $x$ and $y$ are conjugates."}
+{"_id": "16844", "title": "Group Action of Symmetric Group on Complex Vector Space", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $S_n$ denote the symmetric group on $n$ letters. Let $V$ denote a vector space over the complex numbers $\\C$. Let $V$ have a basis: :$\\mathcal B := \\set {v_1, v_2, \\ldots, v_n}$ Let $*: S_n \\times V \\to V$ be a group action of $S_n$ on $V$ defined as: :$\\forall \\tuple {\\rho, v} \\in S_n \\times V: \\rho * v := \\lambda_1 v_{\\map \\rho 1} + \\lambda_2 v_{\\map \\rho 2} + \\dotsb + \\lambda_n v_{\\map \\rho n}$ where: : $v = \\lambda_1 v_1 + \\lambda_2 v_2 + \\dotsb + \\lambda_n v_n$ Then $*$ is a group action."}
+{"_id": "16845", "title": "Group Action of Symmetric Group on Complex Vector Space/Orbit", "text": "The orbit of an element $v \\in V$ is: :$\\Orb v = \\displaystyle \\set {w \\in V: \\exists \\rho \\in S_n: w = \\sum_{k \\mathop = 1}^n \\lambda_k v_{\\map \\rho k} }$"}
+{"_id": "16846", "title": "Group Action of Symmetric Group on Complex Vector Space/Stabilizer", "text": "The stabilizer of an element $v \\in V$ is: :$\\Stab v = \\displaystyle \\set {\\rho \\in S_n: \\sum_{k \\mathop = 1}^n \\lambda_k v_k = \\sum_{k \\mathop = 1}^n \\lambda_{\\map \\rho k} v_k}$"}
+{"_id": "16847", "title": "Equivalence of Definitions of Equivalent Division Ring Norms/Norm is Power of Other Norm implies Cauchy Sequence Equivalent", "text": "Let $\\norm {\\, \\cdot \\,}_1$ and $\\norm {\\, \\cdot \\,}_2$ satisfy: :$\\exists \\alpha \\in \\R_{> 0}: \\forall x \\in R: \\norm x_1 = \\norm x_2^\\alpha$ Then for all sequences $\\sequence {x_n}$ in $R$: :$\\sequence {x_n}$ is a Cauchy sequence in $\\norm {\\, \\cdot \\,}_1$ {{iff}} $\\sequence {x_n}$ is a Cauchy sequence in $\\norm {\\, \\cdot \\,}_2$"}
+{"_id": "16848", "title": "Conjugacy Classes of Symmetric Group", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $S_n$ denote the symmetric group on $n$ letters. The conjugacy classes of $S_n$ are determined entirely by the cycle type. That is, the conjugacy class $\\conjclass x$ of an element $x$ of $S_n$ consists of all the elements of $S_n$ whose cycle type is the same as the cycle type of $x$."}
+{"_id": "16849", "title": "Identity of Group is in Center", "text": "Let $G$ be a group. Let $e$ be the identity of $G$. Then $e$ is in the center of $G$: :$e \\in \\map Z G$"}
+{"_id": "16850", "title": "Identity of Group is in Singleton Conjugacy Class", "text": "Let $G$ be a group. Let $e$ be the identity of $G$. Then $e$ is in its own singleton conjugacy class: :$\\conjclass e = \\set e$"}
+{"_id": "16851", "title": "Finite Group with 2 Conjugacy Classes has 2 Elements", "text": "Let $G$ be a finite group. Let $G$ have exactly $2$ conjugacy classes. Then $G$ has exactly $2$ elements."}
+{"_id": "16853", "title": "Group of Order 15 has Cyclic Subgroups of Order 3 and Order 5", "text": "Let $G$ be a group whose order is $15$. Then $G$ has :a cyclic subgroup of order $3$ and: :a cyclic subgroup of order $5$."}
+{"_id": "16854", "title": "Number of Sylow p-Subgroups in Group of Order 15", "text": "Let $G$ be a group whose order is $15$. Then: :the number of Sylow $3$-subgroups is in the set $\\set {1, 4, 7, \\ldots}$ :the number of Sylow $5$-subgroups is in the set $\\set {1, 6, 11, \\ldots}$"}
+{"_id": "16858", "title": "Direct Product of Sylow p-Subgroups is Sylow p-Subgroup", "text": "Let $G_1$ and $G_2$ be groups. Let $H_1$ and $H_2$ be subgroups of $G_1$ and $G_2$ respectively. Let $H_1$ be a Sylow $p$-subgroup of $G_1$. Let $H_2$ be a Sylow $p$-subgroup of $G_2$. Then $H_1 \\times H_2$ is a Sylow $p$-subgroup of $G_1 \\times G_2$."}
+{"_id": "16859", "title": "Direct Product of Unique Sylow p-Subgroups is Unique Sylow p-Subgroup", "text": "Let $G_1$ and $G_2$ be groups. Let $H_1$ and $H_2$ be subgroups of $G_1$ and $G_2$ respectively. Let $G_1$ be such that $H_1$ is the unique Sylow $p$-subgroup of $G_1$. Let $G_2$ be such that $H_2$ is the unique Sylow $p$-subgroup of $G_2$. Then $H_1 \\times H_2$ is the unique Sylow $p$-subgroup of $G_1 \\times G_2$."}
+{"_id": "16860", "title": "Intersection of Sylow p-Subgroup with Subgroup not necessarily Sylow p-Subgroup", "text": "Let $G$ be a group. Let $P$ be a Sylow $p$-subgroup of $G$. Let $H$ be a subgroup of $G$. Then $P \\cap H$ is not necessarily a Sylow $p$-subgroup of $H$."}
+{"_id": "16861", "title": "Sylow p-Subgroups of Group of Order 2p", "text": "Let $p$ be an odd prime. Let $G$ be a group of order $2 p$. Then $G$ has exactly one Sylow $p$-subgroup. This Sylow $p$-subgroup is normal."}
+{"_id": "16864", "title": "Groups of Order 2p", "text": "Let $p$ be a prime number. Let $G$ be a group. Let the order of $G$ be $2 p$. Then $G$ is either: :the cyclic group $C_{2 p}$ or: :the dihedral group $D_p$."}
+{"_id": "16865", "title": "Group of Order p q has Normal Sylow p-Subgroup", "text": "Let $p$ and $q$ be prime numbers such that $p > q$. Let $G$ be a group of order $p q$. Then $G$ has exactly one Sylow $p$-subgroup. This Sylow $p$-subgroup is normal."}
+{"_id": "16867", "title": "Characterisation of Non-Archimedean Division Ring Norms", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring with unity $1_R$. Then $\\norm {\\,\\cdot\\,}$ is non-Archimedean {{iff}}: :$\\forall n \\in \\N_{>0}: \\norm {n \\cdot 1_R} \\le 1$ where: :$n \\cdot 1_R = \\underbrace {1_R + 1_R + \\dotsb + 1_R}_{\\text {$n$ times} }$"}
+{"_id": "16868", "title": "Groups of Order 21", "text": "There exist exactly $2$ groups of order $21$, up to isomorphism: :$(1): \\quad C_{21}$, the cyclic group of order $21$ :$(2): \\quad$ the group whose group presentation is: :::$\\gen {x, y: x^7 = e = y^3, y x y^{-1} = x^2}$"}
+{"_id": "16869", "title": "Groups of Order 21/Matrix Representation of Non-Abelian Instance", "text": "Let $G$ be the group of order $21$ whose group presentation is: :$\\gen {x, y: x^7 = e = y^3, y x y^{-1} = x^2}$ Then $G$ can be instantiated by the following pair of matrices over $\\Z_7$: :$X = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} \\qquad Y = \\begin{pmatrix} 4 & 0 \\\\ 0 & 2 \\end{pmatrix}$"}
+{"_id": "16870", "title": "Normal Sylow p-Subgroups in Group of Order 12", "text": "Let $G$ be of order $12$. Then $G$ has either: :a normal Sylow $2$-subgroup or: :a normal Sylow $3$-subgroup."}
+{"_id": "16871", "title": "Group of Order p^2 q has Normal Sylow p-Subgroup", "text": "Let $p$ and $q$ be prime numbers such that $p \\ne q$. Let $G$ be a group of order $p^2 q$. Then $G$ has a normal Sylow $p$-subgroup."}
+{"_id": "16872", "title": "Group of Order 30 has Normal Cyclic Subgroup of Order 15", "text": "Let $G$ be of order $30$. Then $G$ has a normal subgroup of order $15$ which is cyclic."}
+{"_id": "16873", "title": "Groups of Order 30/Lemma", "text": "Let $G$ be a group of order $30$. Then $G$ is one of the following: :The cyclic group $C_{30}$ :The dihedral group $D_{15}$ :Isomorphic to one of: ::$\\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^4}$ ::$\\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^{11} }$"}
+{"_id": "16874", "title": "Normal Subgroup of Group of Order 24", "text": "Let $G$ be a group of order $24$. Then $G$ has either: :a normal subgroup of order $8$ or: :a normal subgroup of order $4$."}
+{"_id": "16875", "title": "Group of Order 35 is Cyclic Group", "text": "Let $G$ be a group whose order is $35$. Then $G$ is cyclic."}
+{"_id": "16878", "title": "Group of Order 105 has Normal Sylow 5-Subgroup or Normal Sylow 7-Subgroup", "text": "Let $G$ be a group of order $105$. Then $G$ has either: :exactly one normal Sylow $5$-subgroup or: :exactly one normal Sylow $7$-subgroup."}
+{"_id": "16879", "title": "Group of Order 105 has Normal Cyclic Subgroup of Index 3", "text": "Let $G$ be a group of order $105$. Then $G$ has a normal cyclic subgroup $N$ such that: :$\\index G N = 3$ where $\\index G N$ denotes the index of $N$ in $G$."}
+{"_id": "16880", "title": "Diagonal Relation is Reflexive", "text": "The diagonal relation $\\Delta_S$ on a set $S$ is a reflexive relation in $S$."}
+{"_id": "16881", "title": "Diagonal Relation is Symmetric", "text": "The diagonal relation $\\Delta_S$ on a set $S$ is a symmetric relation in $S$."}
+{"_id": "16882", "title": "Diagonal Relation is Transitive", "text": "The diagonal relation $\\Delta_S$ on a set $S$ is a transitive relation in $S$."}
+{"_id": "16883", "title": "Group of Order 56 has Unique Sylow 2-Subgroup or Unique Sylow 7-Subgroup", "text": "Let $G$ be a group of order $56$. Then $G$ has either: :exactly one normal Sylow $2$-subgroup or: :exactly one normal Sylow $7$-subgroup."}
+{"_id": "16884", "title": "Subgroup of Direct Product is not necessarily Direct Product of Subgroups", "text": "Let $G$ and $H$ be groups. Let $G \\times H$ denote the direct product of $G$ and $H$. Let $K$ be a subgroup of $G \\times H$. Then it is not necessarily the case that $K$ is of the form: :$G' \\times H'$ where: :$G'$ is a subgroup of $G$ :$H'$ is a subgroup of $H$."}
+{"_id": "16888", "title": "Groups of Order 30", "text": "Let $G$ be a group of order $30$. Then $G$ is one of the following: :The cyclic group $C_{30}$ :The dihedral group $D_{15}$ :The group direct product $C_5 \\times D_3$ :The group direct product $C_3 \\times D_5$"}
+{"_id": "16889", "title": "Characterisation of Non-Archimedean Division Ring Norms/Necessary Condition", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring with unity $1_R$. Then: :$\\norm {\\,\\cdot\\,}$ is non-Archimedean $\\implies \\forall n \\in \\N_{>0}: \\norm {n \\cdot 1_R} \\le 1$. where: $n \\cdot 1_R = \\underbrace {1_R + 1_R + \\dots + 1_R}_{\\text {$n$ times} }$"}
+{"_id": "16890", "title": "Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring with unity $1_R$. Then: :$\\forall n \\in \\N_{>0}: \\norm {n \\cdot 1_R} \\le 1 \\implies \\norm {\\,\\cdot\\,}$ is Definition:Non-Archimedean Division Ring Norm where: :$n \\cdot 1_R = \\underbrace {1_R + 1_R + \\dots + 1_R}_{\\text {$n$ times} }$"}
+{"_id": "16891", "title": "Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 1", "text": "Let $y \\ne 0_R$ where $0_R$ is the zero of $R$. Then: :$\\norm {x + y} \\le \\max \\set {\\norm x, \\norm y} \\iff \\norm {x y^{-1} + 1_R} \\le \\max \\set {\\norm {x y^{-1} }, 1}$"}
+{"_id": "16892", "title": "Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 2", "text": "Then for all $i$, $0 \\le i \\le n$: :$\\norm x^i \\le \\max \\set {\\norm x^n , 1}$"}
+{"_id": "16893", "title": "Group Epimorphism preserves Central Subgroups", "text": "Let $G$ and $H$ be groups. Let $\\theta: G \\to H$ be an epimorphism. Let $Z \\le G$ be a central subgroup of $G$. Then $\\theta \\sqbrk Z$ is a central subgroup of $H$."}
+{"_id": "16894", "title": "Direct Product of Central Subgroups", "text": "Let $G$ and $H$ be groups. Let $Z$ and $W$ be central subgroups of $G$ and $H$ respectively. Then $Z \\times W$ is a central subgroup of $G \\times H$."}
+{"_id": "16895", "title": "Groups of Order 30/C 5 x D 3", "text": "Let $G$ be a group of order $30$. Let $G$ have the group presentation: :$\\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^{11} }$ Then $G$ is isomorphic to the group direct product of the cyclic group $C_5$ and the dihedral group $D_3$: :$G \\cong C_5 \\times D_3$"}
+{"_id": "16897", "title": "Groups of Order 30/C 3 x D 5", "text": "Let $G$ be a group of order $30$. Let $G$ have the group presentation: :$\\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^4}$ Then $G$ is isomorphic to the group direct product of the cyclic group $C_3$ and the dihedral group $D_5$: :$G \\cong C_3 \\times D_5$"}
+{"_id": "16898", "title": "Dihedral Group D6 is Internal Direct Product of C2 with D3", "text": "The dihedral group $D_6$ is an internal direct product of the cyclic group $C_2$ of order $2$ and the dihedral group $D_3$: :$D_6 = C_2 \\times D_3$"}
+{"_id": "16899", "title": "Sequence of Integers defining Abelian Group", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive integer. Let $C_n$ be a finite abelian group. Then $C_n$ is of the form: :$C_{n_1} \\times C_{n_2} \\times \\cdots \\times C_{n_r}$ such that: :$n = \\displaystyle \\prod_{k \\mathop = 1}^r n_k$ :$\\forall k \\in \\set {2, 3, \\ldots, r}: n_k \\divides n_{k - 1}$ where $\\divides$ denotes divisibility."}
+{"_id": "16902", "title": "Parallelism is Equivalence Relation", "text": "Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \\in S$, let $l_1 \\parallel l_2$ denote that $l_1$ is parallel to $l_2$. Then $\\parallel$ is an equivalence relation on $S$."}
+{"_id": "16903", "title": "Connected Equivalence Relation is Trivial", "text": "Let $S$ be a set. Let $\\mathcal R$ be a relation on $S$ which is both connected and an equivalence relation. Then $\\mathcal R$ is the trivial relation on $S$."}
+{"_id": "16904", "title": "Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 3", "text": "Let $\\sequence {x_n}$ be the real sequence defined as $x_n = \\paren {n + 1}^{1/n}$, using exponentiation. Then $\\sequence {x_n}$ converges with a limit of $1$."}
+{"_id": "16905", "title": "Characterisation of Non-Archimedean Division Ring Norms/Corollary 1", "text": "$\\norm {\\,\\cdot\\,}$ is non-Archimedean {{iff}}: :$\\sup \\set {\\norm {n \\cdot 1_R}: n \\in \\N_{> 0}} = 1$."}
+{"_id": "16906", "title": "Characterisation of Non-Archimedean Division Ring Norms/Corollary 3", "text": "$\\norm {\\,\\cdot\\,}$ is Archimedean {{iff}}: :$\\sup \\set {\\norm {n \\cdot 1_R}: n \\in \\N_{\\gt 0} } = +\\infty$"}
+{"_id": "16907", "title": "Characterisation of Non-Archimedean Division Ring Norms/Corollary 2", "text": "Let $\\sup \\set {\\norm {n \\cdot 1_R}: n \\in \\N_{> 0} } = C < +\\infty$."}
+{"_id": "16909", "title": "Norms Equivalent to Absolute Value on Rational Numbers", "text": "Let $\\alpha \\in \\R_{\\gt 0}$. Let $\\norm{\\,\\cdot\\,}:\\Q \\to \\R$ be the mapping defined by: :$\\forall x \\in \\Q: \\norm{x} = \\size {x}^\\alpha$ where $\\size {x}$ is the absolute value of $x$ in $\\Q$. Then: :$\\norm{\\,\\cdot\\,}$ is a norm on $\\Q$ {{iff}} $\\,\\,\\alpha \\le 1$"}
+{"_id": "16910", "title": "Reflexive Relation on Set of Cardinality 2 is Transitive", "text": "Let $S$ be a set whose cardinality is equal to $2$: :$\\card S = 2$ Let $\\odot \\subseteq S \\times S$ be a reflexive relation on $S$. Then $\\odot$ is also transitive."}
+{"_id": "16911", "title": "Relation on Set of Cardinality 2 cannot be Non-Symmetric and Non-Transitive", "text": "Let $S$ be a set whose cardinality is equal to $2$: :$\\card S = 2$ Let $\\odot \\subseteq S \\times S$ be a relation on $S$. Then it is not possible for $\\odot$ to be not symmetric and also not transitive."}
+{"_id": "16913", "title": "Relation on Set of Cardinality 1 is Symmetric and Transitive", "text": "Let $S$ be a set whose cardinality is equal to $1$: :$\\card S = 1$ Let $\\odot \\subseteq S \\times S$ be a relation on $S$. Then $\\odot$ is both symmetric and transitive."}
+{"_id": "16915", "title": "Congruence of Triangles is Equivalence Relation", "text": "Let $S$ denote the set of all triangles in the plane. Let $\\triangle A \\cong \\triangle B$ denote the relation that $\\triangle A$ is congruent to $\\triangle B$. Then $\\cong$ is an equivalence relation on $S$."}
+{"_id": "16920", "title": "Even Integer Plus 5 is Odd/Indirect Proof", "text": "{{:Even Integer Plus 5 is Odd}}"}
+{"_id": "16921", "title": "Even Integer Plus 5 is Odd/Proof by Contradiction", "text": "{{:Even Integer Plus 5 is Odd}}"}
+{"_id": "16922", "title": "Sum of Sequence of Cubes/Historical Note", "text": "The result '''Sum of Sequence of Cubes''' was documented by {{AuthorRef|Aryabhata the Elder}} in his work ''Āryabhaṭīya'' of $499$ CE."}
+{"_id": "16924", "title": "Sum of Sequence of Odd Squares/Formulation 2", "text": ":$\\displaystyle \\forall n \\in \\Z_{> 0}: \\sum_{i \\mathop = 1}^n \\paren {2 i - 1}^2 = \\frac {4 n^3 - n} 3$"}
+{"_id": "16925", "title": "Sum of Sequence of Products of Consecutive Odd and Consecutive Even Numbers", "text": "{{begin-eqn}} {{eqn | lo= \\forall n \\in \\Z_{>0}:       | l = \\sum_{j \\mathop = 1}^n j \\paren {j + 2}       | r = 1 \\times 3 + 2 \\times 4 + 3 \\times 5 + \\dotsb + n \\paren {n + 2}       | c =  }} {{eqn | r = \\frac {n \\paren {n + 1} \\paren {2 n + 7} } 6       | c =  }} {{end-eqn}}"}
+{"_id": "16926", "title": "Arcsin as an Integral", "text": ":$\\displaystyle \\map \\arcsin x = \\int_0^\\Theta \\frac {\\d x} {\\sqrt {1 - x^2} }$"}
+{"_id": "16927", "title": "Sum of Sequence of Products of Consecutive Fibonacci Numbers", "text": "=== Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers === {{:Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers}} === Sum of Even Sequence of Products of Consecutive Fibonacci Numbers === {{:Sum of Even Sequence of Products of Consecutive Fibonacci Numbers}}"}
+{"_id": "16928", "title": "Lucas Number as Element of Recursive Sequence", "text": "Let $L_k$ be the $k$th Lucas number, defined as the sum of two Fibonacci numbers: :$L_n = F_{n - 1} + F_{n + 1}$ Then $L_n$ can be defined as the $n$th element of the recursive sequence: :$L_n = \\begin{cases} 2 & : n = 0 \\\\ 1 & : n = 1 \\\\ L_{n - 1} + L_{n - 2} & : \\text{otherwise} \\end{cases}$"}
+{"_id": "16929", "title": "Product of nth Lucas and Fibonacci Numbers", "text": "Let $L_k$ be the $k$th Lucas number. Let $F_k$ be the $k$th Fibonacci number. Then: :$\\forall n \\in \\N_{>0}: F_n L_n = F_{2 n}$"}
+{"_id": "16930", "title": "Representation of Integers in Balanced Ternary", "text": "Let $n \\in \\Z$ be an integer. $n$ can be represented uniquely in balanced ternary: :$\\displaystyle n = \\sum_{j \\mathop = 0}^m r_j 3^j$ :$\\sqbrk {r_m r_{m - 1} \\ldots r_2 r_1 r_0}$ such that: where: :$m \\in \\Z_{>0}$ is a strictly positive integer such that $3^m < \\size {2 n} < 3^{m + 1}$ :all the $r_j$ are such that $r_j \\in \\set {\\underline 1, 0, 1}$, where $\\underline 1 := -1$."}
+{"_id": "16931", "title": "Absolute Value is Norm", "text": "The absolute value is a norm on the set of real numbers $\\R$."}
+{"_id": "16932", "title": "Bounds for Integer Expressed in Base k", "text": "Let $n \\in \\Z$ be an integer. Let $k \\in \\Z$ such that $k \\ge 2$. Let $n$ be expressed in base $k$ notation: :$n = \\displaystyle \\sum_{j \\mathop = 1}^s a_j k^j$ where each of the $a_j$ are such that $a_j \\in \\set {0, 1, \\ldots, k - 1}$. Then: :$0 \\le n < k^{s + 1}$"}
+{"_id": "16934", "title": "Existence of q for which j - qk is Positive", "text": "Let $j, k \\in \\Z$ be integers such that $k > 0$. Then there exist $q \\in \\Z$ such that $j - q k > 0$."}
+{"_id": "16935", "title": "Power Function on Base between Zero and One is Strictly Decreasing/Real Number", "text": "Let $a \\in \\R$ be a real number such that $0 \\lt a \\lt 1$. Let $f: \\R \\to \\R$ be the real function defined as: :$\\map f x = a^x$ where $a^x$ denotes $a$ to the power of $x$. Then $f$ is strictly decreasing."}
+{"_id": "16936", "title": "Integral Ideal is Ideal of Ring", "text": "Let $J$ be a non-empty subset of the set of integers $\\Z$. Then: :$J$ is an integral ideal {{iff}}: :$J$ is an ideal of the ring of integers $\\struct {\\Z, +, \\times}$."}
+{"_id": "16937", "title": "Set of Integer Multiples is Integral Ideal", "text": "Let $m \\in \\Z$ be an integer. Let $m \\Z$ denote the set of integer multiples of $m$. Then $m \\Z$ is an integral ideal."}
+{"_id": "16939", "title": "Integral Ideal is Set of Integer Multiples", "text": "Let $J$ be an integral ideal. Then $J$ is in the form of a set of integer multiples $m \\Z$ for some $m \\in \\Z$."}
+{"_id": "16940", "title": "Difference between Odd Squares is Divisible by 8", "text": "Let $a$ and $b$ be odd integers. Then $a^2 - b^2$ is divisible by $8$."}
+{"_id": "16942", "title": "Norms Equivalent to Absolute Value on Rational Numbers/Necessary Condition", "text": "Let $\\alpha \\in \\R_{\\gt 0}$. Let $\\norm{\\,\\cdot\\,}:\\Q \\to \\R$ be the mapping defined by: :$\\forall x \\in \\Q: \\norm{x} = \\size {x}^\\alpha$ where $\\size {x}$ is the absolute value of $x$ in $\\Q$. Then: :$\\norm{\\,\\cdot\\,}$ is a norm on $\\Q \\implies \\,\\,\\alpha \\le 1$"}
+{"_id": "16943", "title": "Norms Equivalent to Absolute Value on Rational Numbers/Sufficient Condition", "text": "Let $\\alpha \\in \\R_{\\gt 0}$. Let $\\norm{\\,\\cdot\\,}:\\Q \\to \\R$ be the mapping defined by: :$\\forall x \\in \\Q: \\norm{x} = \\size {x}^\\alpha$ where $\\size {x}$ is the absolute value of $x$ in $\\Q$. Then: :$\\alpha \\le 1 \\implies \\norm{\\,\\cdot\\,}$ is a norm on $\\Q$"}
+{"_id": "16948", "title": "Intersection of Sets of Integer Multiples", "text": "Let $m, n \\in \\Z$ such that $m n \\ne 0$. Let $m \\Z$ denote the set of integer multiples of $m$. Then: :$m \\Z \\cap n \\Z = \\lcm \\set {m, n} \\Z$ where $\\lcm$ denotes lowest common multiple."}
+{"_id": "16949", "title": "Set of Integer Multiples of GCD", "text": "Let $m, n \\in \\Z$. Let $m \\Z$ denote the set of integer multiples of $m$ Then: :$m \\Z \\cup n \\Z \\subseteq \\gcd \\set {m, n} \\Z$ where $\\gcd$ denotes greatest common divisor."}
+{"_id": "16951", "title": "GCD of Generators of General Fibonacci Sequence is Divisor of All Terms", "text": "Let $\\mathcal F = \\sequence {a_n}$ be a general Fibonacci sequence generated by the parameters $r, s, t, u$: :$a_n = \\begin{cases} r & : n = 0 \\\\ s & : n = 1 \\\\ t a_{n - 2} + u a_{n - 1} & : n > 1 \\end{cases}$ Let: :$d = \\gcd \\set {r, s}$ where $\\gcd$ denotes greatest common divisor. Then: :$\\forall n \\in \\Z_{>0}: d \\divides a_n$"}
+{"_id": "16952", "title": "GCD of Consecutive Integers of General Fibonacci Sequence", "text": "Let $\\mathcal F = \\sequence {a_n}$ be a general Fibonacci sequence generated by the parameters $r, s, t, u$: :$a_n = \\begin{cases} r & : n = 0 \\\\ s & : n = 1 \\\\ t a_{n - 2} + u a_{n - 1} & : n > 1 \\end{cases}$ Let: :$d = \\gcd \\set {r, s}$ where $\\gcd$ denotes greatest common divisor. Let $f = \\gcd \\set {a_m, a_{m - 1} }$ for some $m \\in \\Z$. Let $\\gcd \\set {f, t} = 1$. Then: :$f \\divides d$"}
+{"_id": "16953", "title": "Three Points in Ultrametric Space have Two Equal Distances/Corollary 3", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring with non-Archimedean norm $\\norm{\\,\\cdot\\,}$, Let $x, y \\in R$ and $\\norm x \\lt \\norm y$. Then: :$\\norm {x + y} = \\norm {x - y} = \\norm {y - x} = \\norm y$"}
+{"_id": "16954", "title": "Three Points in Ultrametric Space have Two Equal Distances/Corollary 4", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring with non-Archimedean norm $\\norm{\\,\\cdot\\,}$, Let $x, y \\in R$. Then: :* $\\norm {x + y} \\lt \\norm y \\implies \\norm x = \\norm y$ :* $\\norm {x - y} \\lt \\norm y \\implies \\norm x = \\norm y$ :* $\\norm {y - x} \\lt \\norm y \\implies \\norm x = \\norm y$"}
+{"_id": "16955", "title": "Equivalent Norms are both Non-Archimedean or both Archimedean", "text": "Let $R$ be a division ring with unity $1_R$. Let $\\norm {\\,\\cdot\\,}_1$ and $\\norm {\\,\\cdot\\,}_2$ be equivalent norms on $R$. Then $\\norm {\\,\\cdot\\,}_1$ and $\\norm {\\,\\cdot\\,}_2$ are either both non-Archimedean or both Archimedean."}
+{"_id": "16957", "title": "Perpendicular Distance from Straight Line in Plane to Origin", "text": "Let $L$ be the straight line embedded in the cartesian plane whose equation is given as: :$a x + b y = c$ Then the perpendicular distance $d$ between $L$ and $\\tuple {0, 0}$ is given by: :$d = \\size {\\dfrac c {\\sqrt {a^2 + b^2} } }$"}
+{"_id": "16958", "title": "Necessary Condition for Integral Functional to have Extremum/Two Variables", "text": "Let $D \\subset \\R^2$. Let $\\Gamma$ be the boundary of $D$. Let $S$ be a set of real mappings such that: :$S = \\set {\\map z {x, y}: \\paren {z: S_1 \\subseteq \\R^2 \\to S_2 \\subseteq \\R}, \\paren {\\map z {x, y} \\in \\map {C^2}D}, \\paren {\\map z \\Gamma = 0} }$ Let $J \\sqbrk z: S \\to S_3 \\subseteq \\R$ be a functional of the form: :$\\ds \\iint_D \\map F {x, y, z, z_x, z_y} \\rd x \\rd y$  Then a necessary condition for $J \\sqbrk y$ to have an extremum (strong or weak) for a given function $\\map z {x, y}$ is that $\\map z {x, y}$ satisfy Euler's equation: :$F_z - \\dfrac \\partial {\\partial x} F_{z_x} - \\dfrac \\partial {\\partial y} F_{z_y} = 0$"}
+{"_id": "16961", "title": "Line in Plane is Straight iff Gradient is Constant", "text": "Let $\\mathcal L$ be a curve which can be embedded in the plane. Then $\\mathcal L$ is a straight line {{iff}} it is of constant gradient."}
+{"_id": "16962", "title": "Equation of Straight Line in Plane/General Equation", "text": "A straight line $\\mathcal L$ is the set of all $\\tuple {x, y} \\in \\R^2$, where: :$\\alpha_1 x + \\alpha_2 y = \\beta$ where $\\alpha_1, \\alpha_2, \\beta \\in \\R$ are given, and not both $\\alpha_1, \\alpha_2$ are zero."}
+{"_id": "16963", "title": "Slope of Straight Line joining Points in Cartesian Plane", "text": "Let $p_1 := \\tuple {x_1, y_1}$ and $p_2 := \\tuple {x_2, y_2}$ be points in a cartesian plane. Let $\\mathcal L$ be the straight line passing through $p_1$ and $p_2$. Then the slope of $\\mathcal L$ is given by: :$\\tan \\theta = \\dfrac {y_2 - y_1} {x_2 - x_1}$ where $\\theta$ is the angle made by $\\mathcal L$ with the $x$-axis."}
+{"_id": "16964", "title": "Equation of Straight Line in Plane/Two-Point Form", "text": "Let $p_1 := \\tuple {x_1, y_1}$ and $p_2 := \\tuple {x_2, y_2}$ be points in a cartesian plane. Let $\\LL$ be the straight line passing through $p_1$ and $p_2$. Then $\\LL$ can be described by the equation: :$\\dfrac {y - y_1} {x - x_1} = \\dfrac {y_2 - y_1} {x_2 - x_1}$ or: :$\\dfrac {x - x_1} {x_2 - x_1} = \\dfrac {y - y_1} {y_2 - y_1}$"}
+{"_id": "16965", "title": "Equation of Straight Line in Plane/Slope-Intercept Form", "text": "Let $\\mathcal L$ be the straight line defined by the general equation: :$\\alpha_1 x + \\alpha_2 y = \\beta$ Then $\\mathcal L$ can be described by the equation: :$y = m x + c$ where: {{begin-eqn}} {{eqn | l = m       | r = -\\dfrac {\\alpha_1} {\\alpha_2}       | c =  }} {{eqn | l = c       | r = \\dfrac {\\beta} {\\alpha_2}       | c =  }} {{end-eqn}} such that $m$ is the slope of $\\mathcal L$ and $c$ is the $y$-intercept."}
+{"_id": "16966", "title": "Equation of Straight Line in Plane/Two-Intercept Form", "text": "Let $\\mathcal L$ be a straight line which intercepts the $x$-axis and $y$-axis respectively at $\\tuple {a, 0}$ and $\\tuple {0, b}$, where $a b \\ne 0$. Then $\\mathcal L$ can be described by the equation: :$\\dfrac x a + \\dfrac y a = 1$"}
+{"_id": "16967", "title": "Equation of Straight Line in Plane/Normal Form", "text": "Let $\\mathcal L$ be a straight line such that: :the perpendicular distance from $\\mathcal L$ to the origin is $p$ :the angle made between that perpendicular and the $x$-axis is $\\alpha$. Then $\\mathcal L$ can be defined by the equation: :$x \\cos \\alpha + y \\sin \\alpha = p$"}
+{"_id": "16968", "title": "Perpendicular Distance from Straight Line in Plane to Point", "text": "Let $\\LL$ be a straight line embedded in a cartesian plane, given by the equation: :$a x + b y = c$ Let $P$ be a point in the cartesian plane whose coordinates are given by: :$P = \\tuple {x_0, y_0}$ Then the perpendicular distance $d$ from $P$ to $\\LL$ is given by: :$d = \\dfrac {\\size {a x_0 + b y_0 + c} } {\\sqrt {a^2 + b^2} }$"}
+{"_id": "16969", "title": "Equation of Straight Line in Plane/Point-Slope Form", "text": "Let $\\mathcal L$ be a straight line embedded in a cartesian plane, given in slope-intercept form as: :$y = m x + c$ Let $\\mathcal L$ pass through the point $\\tuple {x_0, y_0}$. Then $\\mathcal L$ can be expressed by the equation: :$y - y_0 = m \\paren {x - x_0}$"}
+{"_id": "16970", "title": "Shortest Possible Distance between Lattice Points on Straight Line in Cartesian Plane", "text": "Let $\\mathcal L$ be the straight line defined by the equation: :$a x - b y = c$ Let $p_1$ and $p_2$ be lattice points on $\\mathcal L$. Then the shortest possible distance $d$ between $p_1$ and $p_2$ is: :$d = \\dfrac {\\sqrt {a^2 + b^2} } {\\gcd \\set {a, b} }$ where $\\gcd \\set {a, b}$ denotes the greatest common divisor of $a$ and $b$."}
+{"_id": "16971", "title": "Decomposition into Even-Odd Integers is not always Unique", "text": "For every even integer $n$ such that $n > 1$, if $n$ can be expressed as the product of one or more even-times odd integers, it is not necessarily the case that this product is unique."}
+{"_id": "16972", "title": "Decomposition into Product of Power of 2 and Odd Integer is Unique", "text": "Let $n \\in \\Z$ be an integer. Then $n$ can be decomposed into the product of a power of $2$ and an odd integer."}
+{"_id": "16974", "title": "Prime Decomposition of Integer is Unique", "text": "Let $n$ be an integer such that $n > 1$. Then the prime decomposition of $n$ is unique."}
+{"_id": "16975", "title": "Expression for Integers as Powers of Same Primes", "text": "Let $a, b \\in \\Z$ be integers. Let their prime decompositions be given by: {{begin-eqn}} {{eqn | l = a       | r = {q_1}^{e_1} {q_2}^{e_2} \\cdots {q_r}^{e_r} }} {{eqn | r = \\prod_{\\substack {q_i \\mathop \\divides a \\\\ \\text {$q_i$ is prime} } } {q_i}^{e_i} }} {{eqn | l = b       | r = {s_1}^{f_1} {s_2}^{f_2} \\cdots {s_u}^{f_u} }} {{eqn | r = \\prod_{\\substack {s_i \\mathop \\divides b \\\\ \\text {$s_i$ is prime} } } {s_i}^{f_i} }} {{end-eqn}} Then there exist prime numbers: :$t_1 < t_2 < \\dotsb < t_v$ such that: {{begin-eqn}} {{eqn | n = 1       | l = a       | r = {t_1}^{g_1} {t_2}^{g_2} \\cdots {t_v}^{g_v} }} {{eqn | n = 2       | l = b       | r = {t_1}^{h_1} {t_2}^{h_2} \\cdots {t_v}^{h_v} }} {{end-eqn}}"}
+{"_id": "16979", "title": "Expression for Integers as Powers of Same Primes/General Result", "text": "Let $a_1, a_2, \\dotsc, a_n \\in \\Z$ be integers. Let their prime decompositions be given by: :$\\displaystyle a_i = \\prod_{\\substack {p_{i j} \\mathop \\divides a_i \\\\ \\text {$p_{i j}$ is prime} } } {p_{i j} }^{e_{i j} }$ Then there exists a set $T$ of prime numbers: :$T = \\set {t_1, t_2, \\dotsc, t_v}$ such that: :$t_1 < t_2 < \\dotsb < t_v$ :$\\displaystyle a_i = \\prod_{j \\mathop = 1}^v {t_j}^{g_{i j} }$"}
+{"_id": "16980", "title": "GCD from Prime Decomposition/General Result", "text": "Let $n \\in \\N$ be a natural number such that $n \\ge 2$. Let $\\N_n$ be defined as: :$\\N_n := \\set {1, 2, \\dotsc, n}$ Let $A_n = \\set {a_1, a_2, \\dotsc, a_n} \\subseteq \\Z$ be a set of $n$ integers. From Expression for Integers as Powers of Same Primes, let: :$\\displaystyle \\forall i \\in \\N_n: a_i = \\prod_{p_j \\mathop \\in T} {p_j}^{e_{i j} }$ where: :$T = \\set {p_j: j \\in \\N_r}$ such that: :$\\forall j \\in \\N_{r - 1}: p_j < p_{j - 1}$ :$\\forall j \\in \\N_r: \\exists i \\in \\N_n: p_j \\divides a_i$ where $\\divides$ denotes divisibility. Then: :$\\displaystyle \\map \\gcd {A_n} = \\prod_{j \\mathop \\in \\N_r} {p_j}^{\\min \\set {e_{i j}: \\, i \\in \\N_n} }$ where $\\map \\gcd {A_n}$ denotes the greatest common divisor of $a_1, a_2, \\dotsc, a_n$."}
+{"_id": "16981", "title": "LCM from Prime Decomposition/General Result", "text": "Let $n \\in \\N$ be a natural number such that $n \\ge 2$. Let $\\N_n$ be defined as: :$\\N_n := \\set {1, 2, \\dotsc, n}$ Let $A_n = \\set {a_1, a_2, \\dotsc, a_n} \\subseteq \\Z$ be a set of $n$ integers. From Expression for Integers as Powers of Same Primes, let: :$\\displaystyle \\forall i \\in \\N_n: a_i = \\prod_{p_j \\mathop \\in T} {p_j}^{e_{i j} }$ where: :$T = \\set {p_j: j \\in \\N_r}$ such that: :$\\forall j \\in \\N_{r - 1}: p_j < p_{j - 1}$ :$\\forall j \\in \\N_r: \\exists i \\in \\N_n: p_j \\divides a_i$ where $\\divides$ denotes divisibility. Then: :$\\displaystyle \\map \\lcm {A_n} = \\prod_{j \\mathop \\in \\N_r} {p_j}^{\\max \\set {e_{i j}: \\, i \\in \\N_n} }$ where $\\map \\lcm {A_n}$ denotes the greatest common divisor of $a_1, a_2, \\dotsc, a_n$."}
+{"_id": "16986", "title": "Alternating Summation of Binomial Coefficient of Summation of Binomial Coefficient of Sequence", "text": "Let $\\sequence a, \\sequence b$ be real sequences which satisfy the condition: :$a_n = \\displaystyle \\sum_{r \\mathop = 0}^n \\binom n r b_r$ Then: :$\\displaystyle \\paren {-1}^n b_n = \\sum_{s \\mathop = 0}^n \\binom n s \\paren {-1}^s a_s$"}
+{"_id": "16988", "title": "Integer and Fifth Power have same Last Digit", "text": "Let $n \\in \\Z$ be an integer. Then $n^5$ has the same last digit as $n$ when both are expressed in conventional decimal notation."}
+{"_id": "16989", "title": "Sufficient Condition for 5 to divide n^2+1", "text": "Let: {{begin-eqn}} {{eqn | l = 5       | o = \\nmid       | r = n - 1 }} {{eqn | l = 5       | o = \\nmid       | r = n }} {{eqn | l = 5       | o = \\nmid       | r = n + 1 }} {{end-eqn}} where $\\nmid$ denotes non-divisibility. Then: :$5 \\divides n^2 + 1$ where $\\divides$ denotes divisibility."}
+{"_id": "16990", "title": "Wilson's Theorem/Necessary Condition", "text": "Let $p$ be a prime number. Then: :$\\paren {p - 1}! \\equiv -1 \\pmod p$"}
+{"_id": "16991", "title": "Wilson's Theorem/Sufficient Condition", "text": "Let $p$ be a (strictly) positive integer such that: :$\\paren {p - 1}! \\equiv -1 \\pmod p$ Then $p$ is a prime number."}
+{"_id": "16994", "title": "Necessary Condition for Integral Functional to have Extremum/Two Variables/Lemma", "text": "Let $D \\subset \\R^2$. Let $\\Gamma$ be the boundary of $D$. Let $\\alpha : D \\to \\R$ be a continuous mapping. Let $h : D \\to \\R$ be a twice differentiable mapping such that $\\map h \\Gamma = 0$. Suppose for every $h$ we have that: :$\\displaystyle \\iint_D \\map \\alpha {x, y} \\map h {x,y} \\rd x \\rd y = 0$. Then: :$\\displaystyle \\forall x, y \\in D : \\map \\alpha {x, y} = 0$"}
+{"_id": "16995", "title": "Number of Different n-gons that can be Inscribed in Circle", "text": "Let $C$ be a circle on whose circumference $n$ points are placed which divide $C$ into $n$ equal arcs. The number of different $n$-gons (either stellated or otherwise) that can be described on $C$ whose vertices are those $n$ points is: :$S_n = \\dfrac {\\paren {n - 1}!} 2$"}
+{"_id": "16996", "title": "Number of Regular Stellated Odd n-gons", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive odd integer. Then there are $\\dfrac {n - 1} 2$ distinct regular stellated $n$-gons."}
+{"_id": "16998", "title": "Partition of Non-Regular Prime Stellated Cyclic Polygons into Rotation Classes", "text": "Let $p$ be an odd prime. Let $C$ be a circle whose center is $O$. Consider the set $P$ of $p$ points on the circumference of $C$ dividing it into $p$ equal arcs. Let $S$ be the set of all non-regular stellated $p$-gons whose vertices are the elements of $P$. Let $\\sim$ denote the equivalence relation on $S$ defined as: :$\\forall \\tuple {a, b} \\in S \\times S: a \\sim b \\iff$ there exists a plane rotation about $O$ transforming $a$ to $b$. Then the $\\sim$-equivalence classes of $S$ into which $S$ can thereby be partitioned all have cardinality $p$."}
+{"_id": "16999", "title": "Square Modulo n Congruent to Square of Inverse Modulo n", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: :$a^2 \\equiv \\paren {n - a}^2 \\pmod n$ where the notation denotes congruence modulo $n$."}
+{"_id": "17004", "title": "Partition of Integer into Powers of 2 for Consecutive Integers", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\map b n$ denote the number of ways $n$ can be partitioned into (integer) powers of $2$. Then: :$\\map b {2 n} = \\map b {2 n + 1}$"}
+{"_id": "17006", "title": "Number of Partitions with no Multiple of 3 equals Number of Partitions where Parts appear No More than Twice", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\map t n$ denote the number of ways $n$ can be partitioned into parts which are specifically not multiples of $3$. Let $\\map v n$ denote the number of ways $n$ can be partitioned such that no part appears twice. Then: :$\\forall n \\in \\Z_{>0}: \\map t n = \\map v n$"}
+{"_id": "17007", "title": "Congruent Numbers are not necessarily Equal", "text": "Let $x, y, z \\in \\R$ be real numbers such that: :$x \\equiv y \\pmod z$ where $x \\equiv y \\pmod z$ denotes congruence modulo $z$. Then it is not necessarily the case that $x = y$."}
+{"_id": "17008", "title": "Congruence Modulo Negative Number", "text": "Let $a, b, c \\in \\R$ be real numbers. Then: :$a \\equiv b \\pmod c \\iff a \\equiv b \\pmod {-c}$"}
+{"_id": "17009", "title": "P-adic Norm and Absolute Value are Not Equivalent", "text": "Let $\\norm {\\,\\cdot\\,}_p$ be the $p$-adic norm on the rationals $\\Q$ for some prime number $p$. Let $\\size{\\,\\cdot\\,}$ be the absolute value on the rationals $\\Q$. Then $\\norm {\\,\\cdot\\,}_p$ and $\\size{\\,\\cdot\\,}$ are not equivalent norms. That is, the topology induced by $\\norm {\\,\\cdot\\,}_p$ does not equal the topology induced by $\\size {\\,\\cdot\\,}$."}
+{"_id": "17010", "title": "P-adic Norm and Absolute Value are Not Equivalent/Proof 1", "text": "Let $\\norm {\\,\\cdot\\,}_p$ be the $p$-adic norm on the rationals $\\Q$ for some prime number $p$. Let $\\size {\\,\\cdot\\,}$ be the absolute value on the rationals $\\Q$. Then $\\norm {\\,\\cdot\\,}_p$ and $\\size {\\,\\cdot\\,}$ are not equivalent norms. That is, the topology induced by $\\norm {\\,\\cdot\\,}_p$ does not equal the topology induced by $\\size {\\,\\cdot\\,}$."}
+{"_id": "17011", "title": "P-adic Norm and Absolute Value are Not Equivalent/Proof 2", "text": "Let $\\norm {\\,\\cdot\\,}_p$ be the $p$-adic norm on the rationals $\\Q$ for some prime number $p$. Let $\\size {\\,\\cdot\\,}$ be the absolute value on the rationals $\\Q$. Then $\\norm {\\,\\cdot\\,}_p$ and $\\size {\\,\\cdot\\,}$ are not equivalent norms. That is, the topology induced by $\\norm {\\,\\cdot\\,}_p$ does not equal the topology induced by $\\size {\\,\\cdot\\,}$."}
+{"_id": "17012", "title": "P-adic Norms are Not Equivalent", "text": "Let $p_1$ and $p_2$ be prime numbers such that $p_1 \\neq p_2$. Let $\\norm {\\,\\cdot\\,}_{p_1}$ and $\\norm {\\,\\cdot\\,}_{p_2}$ be the $p$-adic norms on the rationals $\\Q$. Then $\\norm {\\,\\cdot\\,}_{p_1}$ and $\\norm {\\,\\cdot\\,}_{p_2}$ are not equivalent norms. That is, the topology induced by $\\norm {\\,\\cdot\\,}_{p_1}$ does not equal the topology induced by $\\norm {\\,\\cdot\\,}_{p_2}$."}
+{"_id": "17013", "title": "Polynomials of Congruent Integers are Congruent", "text": "Let $x, y, m \\in \\Z$ be integers where $m \\ne 0$. Let: :$x \\equiv y \\pmod m$ where the notation indicates congruence modlo $m$. Let $a_0, a_1, \\ldots, a_r$ be integers. Then: :$\\displaystyle \\sum_{k \\mathop = 0}^r a_k x^k \\equiv \\sum_{k \\mathop = 0}^r a_k y^k \\pmod m$"}
+{"_id": "17014", "title": "Congruent Integers less than Half Modulus are Equal", "text": "Let $k \\in \\Z_{>0}$ be a strictly positive integer. Let $a, b \\in \\Z$ such that $\\size a < \\dfrac k 2$ and $\\size b < \\dfrac k 2$. Then: :$a \\equiv b \\pmod k \\implies a = b$ where $\\equiv$ denotes congruence modulo $k$."}
+{"_id": "17015", "title": "Complete Residue System Modulo m has m Elements", "text": "Let $m \\in \\Z_{\\ne 0}$ be a non-zero integer. Let $S := \\set {r_1, r_2, \\dotsb, r_s}$ be a complete residue system modulo $m$. Then $s = m$."}
+{"_id": "17016", "title": "Initial Segment of Natural Numbers forms Complete Residue System", "text": "Let $m \\in \\Z_{\\ne 0}$ be a non-zero integer. Let $\\N_m = \\set {0, 1, 2, \\ldots, m - 1}$ denote the initial segment of $\\N$ Then $\\N_m$ is a complete residue system modulo $m$."}
+{"_id": "17018", "title": "Number of Non-Dividing Primes Less than n is Less than Euler Phi Function of n", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive integer. Let $\\map w n$ denote the number of primes strictly less than $n$ which are not divisors of $n$. Let $\\map \\phi n$ denote the Euler $\\phi$ function of $n$. Then: :$\\map w n < \\map \\phi n$"}
+{"_id": "17019", "title": "Schatunowsky's Theorem", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive integer. Let $\\map w n$ denote the number of primes strictly less than $n$ which are not divisors of $n$. Let $\\map \\phi n$ denote the Euler $\\phi$ function of $n$. Then $30$ is the largest integer $n$ such that: :$\\map w n = \\map \\phi n - 1$"}
+{"_id": "17020", "title": "Position of Card after n Modified Perfect Faro Shuffles", "text": "Let $D$ be a deck of cards $D$ of size $2 r$. Let $C$ be a card in position $x$ of $D$. Let $n$ modified perfect faro shuffles be performed on $C$. Then $C$ will be in position $w$, where: :$w \\equiv 2^n x \\pmod {2 r + 1}$"}
+{"_id": "17021", "title": "Deck of 52 Cards returns to Original Order after 52 Modified Perfect Faro Shuffles", "text": "Let $D$ be a deck of $52$ cards. Let $D$ be given a sequence of modified perfect faro shuffles. Then after $52$ such shuffles, the cards of $D$ will be in the same order they started in."}
+{"_id": "17022", "title": "Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order", "text": "Let $D$ be a deck of $2 m$ cards. Let $D$ be given a sequence of modified perfect faro shuffles. Then the cards of $D$ will return to their original order after $n$ such shuffles, where: :$2^n \\equiv 1 \\pmod {2 m + 1}$"}
+{"_id": "17024", "title": "Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 8 Cards", "text": "Let $D$ be a deck of $8$ cards. Let $D$ be given a sequence of modified perfect faro shuffles. Then after $6$ such shuffles, the cards of $D$ will be in the same order they started in."}
+{"_id": "17029", "title": "Sum of Two Odd Powers", "text": "Let $\\F$ be one of the standard number systems, that is $\\Z, \\Q, \\R$ and so on. Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Then: {{begin-eqn}} {{eqn | l = a^{2 n + 1} + b^{2 n + 1}       | r = \\paren {a + b} \\sum_{j \\mathop = 0}^{2 n} \\paren {-1}^j a^{2 n - j} b^j       | c =  }} {{eqn | r = \\paren {a + b} \\paren {a^{2 n} - a^{2 n - 1} b + a^{2 n - 2} b^2 - \\dotsb - a b^{2 n - 1} + b^{2 n} }       | c =  }} {{end-eqn}}"}
+{"_id": "17033", "title": "Difference of Two Fifth Powers", "text": ":$x^5 - y^5 = \\paren {x - y} \\paren {x^4 + x^3 y + x^2 y^2 + x y^3 + y^4}$"}
+{"_id": "17035", "title": "Difference of Two Sixth Powers", "text": ":$x^6 - y^6 = \\paren {x - y} \\paren {x + y} \\paren {x^2 + x y + y^2} \\paren {x^2 - x y + y^2}$"}
+{"_id": "17037", "title": "Difference of Two Odd Powers", "text": "Let $\\mathbb F$ denote one of the standard number systems, that is $\\Z$, $\\Q$, $\\R$ and $\\C$. Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Then for all $a, b \\in \\mathbb F$: {{begin-eqn}} {{eqn | l = a^{2 n + 1} - b^{2 n + 1}       | r = \\paren {a - b} \\sum_{j \\mathop = 0}^{2 n} a^{2 n - j} b^j       | c =  }} {{eqn | r = \\paren {a - b} \\paren {a^{2 n} + a^{2 n - 1} b + a^{2 n - 2} b^2 + \\dotsb + a b^{2 n - 1} + b^{2 n} }       | c =  }} {{end-eqn}}"}
+{"_id": "17039", "title": "Difference of Two Even Powers", "text": "Let $\\mathbb F$ denote one of the standard number systems, that is $\\Z$, $\\Q$, $\\R$ and $\\C$. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then for all $a, b \\in \\mathbb F$: {{begin-eqn}} {{eqn | l = a^{2 n} - b^{2 n}       | r = \\paren {a - b} \\paren {a + b} \\sum_{j \\mathop = 0}^{n - 1} a^{2 \\paren {n - j - 1} } b^{2 j}       | c =  }} {{eqn | r = \\paren {a - b} \\paren {a + b} \\paren {a^{2 n - 2} + a^{2 n - 4} b^2 + a^{2 n - 6} b^4 + \\dotsb + a^2 b^{2 n - 4} + b^{2 n - 2} }       | c =  }} {{end-eqn}}"}
+{"_id": "17040", "title": "Factors of Difference of Two Odd Powers", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = x^{2 n + 1} - y^{2 n + 1}       | r = \\paren {x - y} \\displaystyle \\prod_{k \\mathop = 1}^n \\paren {x^2 - 2 x y \\cos \\dfrac {2 \\pi k} {2 n + 1} + y^2}       | c =  }} {{eqn | r = \\paren {x - y} \\paren {x^2 - 2 x y \\cos \\dfrac {2 \\pi} {2 n + 1} + y^2} \\paren {x^2 - 2 x y \\cos \\dfrac {4 \\pi} {2 n + 1} + y^2} \\dotsm \\paren {x^2 - 2 x y \\cos \\dfrac {2 n \\pi} {2 n + 1} + y^2}       | c =  }} {{end-eqn}}"}
+{"_id": "17041", "title": "Factors of Sum of Two Odd Powers", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = x^{2 n + 1} + y^{2 n + 1}       | r = \\paren {x + y} \\displaystyle \\prod_{k \\mathop = 1}^n \\paren {x^2 + 2 x y \\cos \\dfrac {2 \\pi k} {2 n + 1} + y^2}       | c =  }} {{eqn | r = \\paren {x + y} \\paren {x^2 + 2 x y \\cos \\dfrac {2 \\pi} {2 n + 1} + y^2} \\paren {x^2 + 2 x y \\cos \\dfrac {4 \\pi} {2 n + 1} + y^2} \\dotsm \\paren {x^2 + 2 x y \\cos \\dfrac {2 n \\pi} {2 n + 1} + y^2}       | c =  }} {{end-eqn}}"}
+{"_id": "17042", "title": "Factors of Difference of Two Even Powers", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: :$x^{2 n} - y^{2 n} = \\paren {x - y} \\paren {x + y} \\displaystyle \\prod_{k \\mathop = 1}^{n - 1} \\paren {x^2 - 2 x y \\cos \\dfrac {k \\pi} n + y^2}$"}
+{"_id": "17043", "title": "Factors of Sum of Two Even Powers", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: :$x^{2 n} + y^{2 n} = \\displaystyle \\prod_{k \\mathop = 1}^n \\paren {x^2 + 2 x y \\cos \\dfrac {\\paren {2 k - 1} \\pi} {2 n} + y^2}$"}
+{"_id": "17045", "title": "Topological Properties of Non-Archimedean Division Rings/Spheres are Clopen", "text": ":The $r$-sphere of $x$, $\\map {S_r} x$, is both open and closed in the metric induced by $\\norm {\\,\\cdot\\,}$."}
+{"_id": "17047", "title": "Magnitude of Projection of Complex Number on Another", "text": "Let $z_1$ and $z_2$ denote complex numbers in vector form. Let $\\map {\\pr_1} {z_1, z_2}$ denote the projection of $z_1$ on $z_2$. {{explain|We really need another page to explain the concept of Definition:Projection (Analytic Geometry) in the context of the Definition:Complex Plane}} Then: :$\\cmod {\\map {\\pr_1} {z_1, z_2} } = \\dfrac {\\cmod {z_1 \\circ z_2} } {\\cmod {z_2} }$ where: :$z_1 \\circ z_2$ denotes complex dot product :$\\cmod {z_2}$ denotes complex modulus."}
+{"_id": "17049", "title": "Non-Archimedean Division Ring is Totally Disconnected", "text": "Let $\\struct {R, \\norm{\\,\\cdot\\,} }$ be a non-Archimedean normed division ring. Let $\\tau$ be the topology induced by the norm $\\norm{\\,\\cdot\\,}$. Then the topological space $\\struct {R, \\tau}$ is totally disconnected."}
+{"_id": "17053", "title": "Distance between Points in Complex Plane", "text": "Let $A$ and $B$ be points in the complex plane such that: :$A = \\tuple {x_1, y_1}$ :$B = \\tuple {x_2, y_2}$ Then the distance between $A$ and $B$ is given by: {{begin-eqn}} {{eqn | l = \\size {AB}       | r = \\sqrt {\\paren {x_2 - x_1}^2 + \\paren {y_2 - y_1}^2}       | c =  }} {{eqn | r = \\cmod {z_1 - z_2}       | c =  }} {{end-eqn}} where $z_1$ and $z_2$ are represented by the complex numbers $z_1$ and $z_2$ respectively."}
+{"_id": "17054", "title": "Linear Combination of Non-Parallel Complex Numbers is Zero if Factors are Both Zero", "text": "Let $z_1$ and $z_2$ be complex numbers expressed as vectors such taht $z_1$ is not parallel to $z_2$. Let $a, b \\in \\R$ be real numbers such that: :$a z_1 + b z_2 = 0$ Then $a = 0$ and $b = 0$."}
+{"_id": "17055", "title": "Equation for Line through Two Points in Complex Plane/Formulation 1", "text": "$L$ can be expressed by the equation: :$\\map \\arg {\\dfrac {z - z_1} {z_2 - z_1} } = 0$"}
+{"_id": "17057", "title": "Equation for Line through Two Points in Complex Plane/Parametric Form 1", "text": "$L$ can be expressed by the equation: :$z = z_1 + t \\paren {z_2 - z_1}$ or: :$z = \\paren {1 - t} z_1 + t z_2$ This form of $L$ is known as the '''parametric form''', where $t$ is the '''parameter'''."}
+{"_id": "17058", "title": "Equation for Line through Two Points in Complex Plane/Symmetric Form", "text": "$L$ can be expressed by the equation: :$z = \\dfrac {m z_1 + n z_2} {m + n}$ This form of $L$ is known as the '''symmetric form'''."}
+{"_id": "17060", "title": "Valuation Ring of Non-Archimedean Division Ring is Subring", "text": "Let $\\struct {R, \\norm{\\,\\cdot\\,}}$ be a non-Archimedean normed division ring with zero $0_R$ and unity $1_R$. Let $\\OO$ be the valuation ring induced by the non-Archimedean norm $\\norm {\\,\\cdot\\,}$, that is: :$\\OO = \\set {x \\in R : \\norm{x} \\le 1}$ Then $\\OO$ is a subring of $R$: :with a unity: $1_R$ :in which there are no (proper) zero divisors, that is: :::$\\forall x, y \\in \\OO: x \\circ y = 0_R \\implies x = 0_R \\text{ or } y = 0_R$"}
+{"_id": "17061", "title": "Valuation Ideal is Maximal Ideal of Induced Valuation Ring", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,}}$ be a non-Archimedean normed division ring with zero $0_R$ and unity $1_R$. Let $\\OO$ be the valuation ring induced by the non-Archimedean norm $\\norm {\\,\\cdot\\,}$, that is: :$\\OO = \\set{x \\in R : \\norm x \\le 1}$ Let $\\PP$ be the valuation ideal induced by the non-Archimedean norm $\\norm {\\,\\cdot\\,}$, that is: :$\\PP = \\set{x \\in R : \\norm x < 1}$ Then $\\PP$ is an ideal of $\\OO$: :$(a):\\quad \\PP$ is a maximal left ideal :$(b):\\quad \\PP$ is a maximal right ideal :$(c):\\quad$ the quotient ring $\\OO / \\PP$ is a division ring."}
+{"_id": "17062", "title": "Sequence of Imaginary Reciprocals", "text": "Consider the subset $S$ of the complex plane defined as: :$S := \\set {\\dfrac i n : n \\in \\Z_{>0} }$ That is: :$S := \\set {i, \\dfrac i 2, \\dfrac i 3, \\dfrac i 4, \\ldots}$ where $i$ is the imaginary unit."}
+{"_id": "17063", "title": "Sequence of Imaginary Reciprocals/Boundedness", "text": "The set $S$ is bounded in $\\C$."}
+{"_id": "17064", "title": "Sequence of Imaginary Reciprocals/Limit Points", "text": "The set $S$ has exactly one limit point, and that is $z = 0$."}
+{"_id": "17065", "title": "Sequence of Imaginary Reciprocals/Closedness", "text": "The set $S$ is not closed."}
+{"_id": "17066", "title": "Valuation Ring of P-adic Norm on Rationals", "text": "Let $\\norm {\\,\\cdot\\,}_p$ be the $p$-adic norm on the rationals $\\Q$ for some prime $p$. The induced valuation ring on $\\struct {\\Q,\\norm {\\,\\cdot\\,}_p}$ is the set: :$\\OO = \\Z_{\\paren p} = \\set {\\dfrac a b \\in \\Q : p \\nmid b}$"}
+{"_id": "17067", "title": "Valuation Ideal of P-adic Norm on Rationals", "text": "Let $\\norm {\\,\\cdot\\,}_p$ be the $p$-adic norm on the rationals $\\Q$ for some prime $p$. The induced valuation ideal on $\\struct {\\Q,\\norm {\\,\\cdot\\,}_p}$ is the set: :$\\PP = p \\Z_{\\ideal p} = \\set {\\dfrac a b \\in \\Q : p \\nmid b, p \\divides a}$ where $\\Z_{\\ideal p}$ is the induced valuation ring on $\\struct {\\Q,\\norm {\\,\\cdot\\,}_p}$"}
+{"_id": "17068", "title": "Residue Field of P-adic Norm on Rationals", "text": "Let $\\norm {\\,\\cdot\\,}_p$ be the $p$-adic norm on the rationals $\\Q$ for some prime $p$. The induced residue field on $\\struct {\\Q,\\norm {\\,\\cdot\\,}_p}$ is isomorphic to the field $\\F_p$ of integers modulo $p$."}
+{"_id": "17069", "title": "Sequence of Imaginary Reciprocals/Boundary Points", "text": "Every point of $S$, along with the point $z = 0$, is a boundary point of $S$."}
+{"_id": "17070", "title": "Sequence of Imaginary Reciprocals/Interior", "text": "No point of $S$ is an interior point."}
+{"_id": "17071", "title": "Sequence of Imaginary Reciprocals/Openness", "text": "$S$ is not an open set."}
+{"_id": "17072", "title": "Sequence of Imaginary Reciprocals/Connectedness", "text": "$S$ is not connected."}
+{"_id": "17073", "title": "Sequence of Imaginary Reciprocals/Not an Open Region", "text": "$S$ is not an open region."}
+{"_id": "17075", "title": "Sequence of Imaginary Reciprocals/Not Compact", "text": "The set $S$ is not compact."}
+{"_id": "17076", "title": "Sequence of Imaginary Reciprocals/Closure is Compact", "text": "The closure $S^-$ of the set $S$ is compact."}
+{"_id": "17077", "title": "Product of Complex Conjugates/Examples/3 Arguments", "text": "Let $z_1, z_2, z_3 \\in \\C$ be complex numbers. Let $\\overline z$ denote the complex conjugate of the complex number $z$. Then: :$\\overline {z_1 z_2 z_3} = \\overline {z_1} \\cdot \\overline {z_2} \\cdot \\overline {z_3}$"}
+{"_id": "17079", "title": "Residue Field of P-adic Norm on Rationals/Lemma 1", "text": ":$\\phi$ is a homomorphism."}
+{"_id": "17080", "title": "Residue Field of P-adic Norm on Rationals/Lemma 2", "text": ":$p \\Z = \\map \\ker \\phi$"}
+{"_id": "17081", "title": "Residue Field of P-adic Norm on Rationals/Lemma 3", "text": ":$\\phi : \\Z \\to \\Z_{\\paren p} / p \\Z_{\\paren p}$ is a surjection."}
+{"_id": "17082", "title": "Equation for Line through Two Points in Complex Plane/Parametric Form 2", "text": "$L$ can be expressed by the equations: {{begin-eqn}} {{eqn | l = x - x_1       | r = t \\paren {x_2 - x_1} }} {{eqn | l = y - y_1       | r = t \\paren {y_2 - y_1} }} {{end-eqn}} These are the '''parametric equations of $L$''', where $t$ is the parameter."}
+{"_id": "17083", "title": "Equation for Perpendicular Bisector of Two Points in Complex Plane/Parametric Form 1", "text": "$L$ can be expressed by the equation: :$z = ...$ or: :$z = ...$ This form of $L$ is known as the '''parametric form''', where $t$ is the '''parameter'''."}
+{"_id": "17084", "title": "Geodesic Equation/2d Surface Embedded in 3d Euclidean Space", "text": "Let $\\sigma: U \\subset \\R^2 \\to V \\subset \\R^3$ be a smooth surface specified by a vector-valued function: :$\\mathbf r = \\map {\\mathbf r} {u, v}$ Then a geodesic of $\\sigma$ satisfies the following system of differential equations: :$\\dfrac {E_u u'^2 + 2 F_u u' v' + G_u v'^2} {\\sqrt{E u'^2 + 2 F u' v' + G v'^2} } - \\dfrac \\d {\\d t} \\dfrac {2 \\paren {E u' + F v'} } {\\sqrt{E u'^2 + 2 F u' v' + G v'^2} } = 0$ :$\\dfrac {E_v u'^2 + 2 F_v u' v' + G_v v'^2} {\\sqrt{E u'^2 + 2 F u' v' + G v'^2} } - \\dfrac \\d {\\d t} \\dfrac {2 \\paren {F u' + G v'} } {\\sqrt{E u'^2 + 2 F u' v' + G v'^2} } = 0$ where $E, F, G$ are the functions of the first fundamental form: :$\\displaystyle E = {\\mathbf r}_u \\cdot {\\mathbf r}_u, F = {\\mathbf r}_u \\cdot {\\mathbf r}_v, G = {\\mathbf r}_v \\cdot {\\mathbf r}_v$"}
+{"_id": "17086", "title": "Modulus of Exponential of i z where z is on Circle", "text": "Let $C$ be the circle embedded in the complex plane given by the equation: :$z = R e^{i \\theta}$ Then: :$\\cmod {e^{i z} } = e^{-R \\sin \\theta}$"}
+{"_id": "17087", "title": "Sum of Complex Numbers in Exponential Form/General Result", "text": "Let $n \\in \\Z_{>0}$ be a positive integer. For all $k \\in \\set {1, 2, \\dotsc, n}$, let: :$z_k = r_k e^{i \\theta_k}$ be non-zero complex numbers in exponential form. Let: :$r e^{i \\theta} = \\displaystyle \\sum_{k \\mathop = 1}^n z_k = z_1 + z_2 + \\dotsb + z_k$ Then: {{begin-eqn}} {{eqn | l = r       | r = \\sqrt {\\displaystyle \\sum_{k \\mathop = 1}^n r_k + \\displaystyle \\sum_{1 \\mathop \\le j \\mathop < k \\mathop \\le n} 2 {r_j} {r_k} \\, \\map \\cos {\\theta_j - \\theta_k} } }} {{eqn | l = \\theta       | r = \\map \\arctan {\\dfrac {r_1 \\sin \\theta_1 + r_2 \\sin \\theta_2 + \\dotsb + r_n \\sin \\theta_n} {r_1 \\cos \\theta_1 + r_2 \\cos \\theta_2 + \\dotsb + r_n \\cos \\theta_n} } }} {{end-eqn}}"}
+{"_id": "17088", "title": "Bias of Sample Variance", "text": "Let $X_1, X_2, \\ldots, X_n$ form a random sample from a population with mean $\\mu$ and variance $\\sigma^2$. Let:  :$\\displaystyle \\bar X = \\frac 1 n \\sum_{i \\mathop = 1}^n X_i$ Then: :$\\displaystyle \\hat {\\sigma^2} = \\frac 1 n \\sum_{i \\mathop = 1}^n \\paren {X_i - \\bar X}^2$ is a biased estimator of $\\sigma^2$, with:  :$\\displaystyle \\operatorname{bias} \\paren {\\hat {\\sigma ^2}} = -\\frac {\\sigma^2} n$"}
+{"_id": "17092", "title": "Sample Mean is Unbiased Estimator of Population Mean", "text": "Let $X_1, X_2, \\ldots, X_n$ form a random sample from a population with mean $\\mu$ and variance $\\sigma^2$. Then:  :$\\displaystyle \\bar X = \\frac 1 n \\sum_{i \\mathop = 1}^n X_i$ is an unbiased estimator of $\\mu$."}
+{"_id": "17093", "title": "Ostrowski's Theorem/Archimedean Norm", "text": "Let $\\norm {\\, \\cdot \\,}$ be a non-trivial Archimedean norm on the rational numbers $\\Q$. Then $\\norm {\\, \\cdot \\,}$ is equivalent to the absolute value $\\size {\\, \\cdot \\,}$."}
+{"_id": "17094", "title": "Ostrowski's Theorem/Non-Archimedean Norm", "text": "Let $\\norm {\\, \\cdot \\,}$ be a non-trivial non-Archimedean norm on the rational numbers $\\Q$. Then $\\norm {\\, \\cdot \\,}$ is equivalent to the $p$-adic norm $\\norm {\\, \\cdot \\,}_p$ for some prime $p$."}
+{"_id": "17095", "title": "Ostrowski's Theorem/Archimedean Norm/Lemma 1.1", "text": ":$\\forall n \\in N: \\norm n \\le n^\\alpha$"}
+{"_id": "17096", "title": "Ostrowski's Theorem/Archimedean Norm/Lemma 1.2", "text": ":$\\forall n \\in N: \\norm n \\ge n^\\alpha$"}
+{"_id": "17097", "title": "Equivalent Norms on Rational Numbers", "text": "Let $\\norm {\\, \\cdot \\,}_1$ and $\\norm {\\, \\cdot \\,}_2$ be norms on the rational numbers $\\Q$. Then $\\norm {\\, \\cdot \\,}_1$ and $\\norm {\\, \\cdot \\,}_2$ are equivalent {{iff}}: :$\\exists \\alpha \\in \\R_{\\gt 0}: \\forall n \\in \\N: \\norm n_1 = \\norm n_2^\\alpha$"}
+{"_id": "17098", "title": "Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.1", "text": ":$\\exists n \\in \\N: 0 < \\norm n < 1$."}
+{"_id": "17100", "title": "Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.2", "text": ":$n_0$ is a prime number."}
+{"_id": "17101", "title": "Sum of Cosines of k pi over 5", "text": ":$\\cos 36 \\degrees + \\cos 72 \\degrees + \\cos 108 \\degrees + \\cos 144 \\degrees = 0$"}
+{"_id": "17102", "title": "Products of nth Roots of Unity taken up to n-1 at a Time is Zero", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $U_n = \\set {z \\in \\C: z^n = 1}$ be the complex $n$th roots of unity. Then the sum of the products of the elements of $U_n$ taken $2, 3, 4, \\dotsc n - 1$ at a time is zero."}
+{"_id": "17103", "title": "Absolute Value of Complex Dot Product is Commutative", "text": "Let $z_1$ and $z_2$ be complex numbers. Let $z_1 \\circ z_2$ denote the (complex) dot product of $z_1$ and $z_2$. Then: :$\\size {z_1 \\circ z_2} = \\size {z_2 \\circ z_1}$ where $\\size {\\, \\cdot \\,}$ denotes the absolute value function."}
+{"_id": "17106", "title": "Three Points in Ultrametric Space have Two Equal Distances/Corollary 5", "text": "Let $\\norm {\\, \\cdot \\,}$ be a non-trivial non-Archimedean norm on the rational numbers $\\Q$. Let $a, b \\in \\Z_{\\ne 0}$ be coprime, $a \\perp b$ Then: :$\\norm a = 1$ or $\\norm b = 1$"}
+{"_id": "17107", "title": "Exponential of 2 m i Arccotangent of p", "text": ":$\\map \\exp {2 m i \\arccot p} \\paren {\\dfrac {p i + 1} {p i - 1} }^m = 1$"}
+{"_id": "17108", "title": "Modulus z - 1 Less than Modulus z + 1 iff Real z Greater than Zero", "text": "Let $z \\in \\C$ be a complex number. Then: :$\\cmod {z - 1} < \\cmod {z + 1} \\iff \\map \\Re z > 0$"}
+{"_id": "17109", "title": "Vertices of Equilateral Triangle in Complex Plane/Sufficient Condition", "text": "Let $z_1$, $z_2$ and $z_3$ be complex numbers. Let $z_1$, $z_2$ and $z_3$ represent on the complex plane the vertices of an equilateral triangle. Then: :${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$"}
+{"_id": "17110", "title": "Vertices of Equilateral Triangle in Complex Plane/Necessary Condition", "text": "Let $z_1$, $z_2$ and $z_3$ be complex numbers. Let $z_1$, $z_2$ and $z_3$ fulfil the condition: :${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$ Then $z_1$, $z_2$ and $z_3$ represent on the complex plane the vertices of an equilateral triangle."}
+{"_id": "17112", "title": "Geodesic Equation/2d Surface Embedded in 3d Euclidean Space/Cylinder", "text": "Let $\\sigma$ be the surface of a cylinder. Let $\\sigma$ be embedded in 3-dimensional Euclidean space. Let $\\sigma$ be parameterised by $\\tuple {\\phi, z}$ as :$\\mathbf r = \\tuple {a \\cos \\phi, a \\sin \\phi, z}$ where :$a > 0$ and :$z, \\phi \\in \\R$ Then geodesics on $\\sigma$ are of the following form: :$z = C_1 \\phi + C_2$ where $C_1, C_2$ are real arbitrary constants."}
+{"_id": "17113", "title": "Modulus of Sum equals Modulus of Distance implies Quotient is Imaginary", "text": "Let $z_1$ and $z_2$ be complex numbers such that: :$\\cmod {z_1 + z_2} = \\cmod {z_1 - z_2}$ Then $\\dfrac {z_2} {z_1}$ is wholly imaginary."}
+{"_id": "17114", "title": "Difference of Even Powers of z + a and z - a", "text": "Let $m \\in \\Z$ be an integer such that $m > 1$. Then for all complex number $z$: :$\\paren {z + a}^{2 m} - \\paren {z - a}^{2 m} = 4 m a z \\displaystyle \\prod_{k \\mathop = 1}^{m - 1} \\paren {z^2 + a^2 \\cot^2 \\dfrac {k \\pi} {2 m} }$"}
+{"_id": "17116", "title": "Imaginary Part of Complex Product", "text": "Let $z_1$ and $z_2$ be complex numbers. Then: :$\\map \\Im {z_1 z_2} = \\map \\Re {z_1} \\, \\map \\Im {z_2} + \\map \\Im {z_1} \\, \\map \\Re {z_2}$"}
+{"_id": "17118", "title": "Element of Center in Group whose Order is Power of 2", "text": "Let $n \\in \\Z$ be an integer such that $n \\ge 2$. Let $G$ be a group whose order is $2^n$. Let $x \\in G$ be of order $2^{n - 1}$ in $G$. Then $x^{2^{n - 2} }$ is an element of the center of $G$."}
+{"_id": "17119", "title": "Intersection of Abelian Subgroups is Normal Subgroup of Subgroup Generated by those Subgroups", "text": "Let $G$ be a group. Let $L$ and $M$ be abelian subgroups of $G$. Let $H = \\gen {L, M}$ be the subgroup of $G$ generated by $L$ and $M$. Then $L \\cap M$ is a normal subgroup of $H$."}
+{"_id": "17120", "title": "Characterisation of Non-Archimedean Division Ring Norms/Corollary 5", "text": "If $\\norm {\\, \\cdot \\,}$ is non-Archimedean then: :$\\sup \\set {\\norm {n \\cdot 1_R}: n \\in \\Z} = 1$ where $n \\cdot 1_R =  \\begin{cases} \\underbrace {1_R + 1_R + \\dots + 1_R}_{\\text {$n$ times} } & : n > 0 \\\\ 0 & : n = 0 \\\\ \\\\ -\\underbrace {\\paren {1_R + 1_R + \\dots + 1_R} }_{\\text {$-n$ times} } & : n < 0 \\\\ \\end{cases}$"}
+{"_id": "17121", "title": "Subgroup Containing all Squares of Group Elements is Normal", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$ with the property that: :$\\forall x \\in G: x^2 \\in H$ Then $H$ is normal in $G$."}
+{"_id": "17123", "title": "Commutator is Identity iff Elements Commute", "text": "Let $G$ be a group whose identity is $e$. Let $x, y \\in G$. Let $\\sqbrk {x, y}$ denote the commutator of $x$ and $y$. Then $\\sqbrk {x, y} = e$ {{iff}} $x$ and $y$ commute."}
+{"_id": "17125", "title": "Commutator of Quotient Group Elements", "text": "Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $\\sqbrk {x, y}$ denote the commutator of $x, y \\in G$: :$\\sqbrk {x, y} = x^{-1} y^{-1} x y$ Then: :$\\forall x, y \\in G: \\sqbrk {x N, y N} = \\sqbrk {x, y} N$ where $x N$ and $y N$ are left cosets of $N$, and so elements of the quotient group $G / N$ of $G$ by $N$."}
+{"_id": "17126", "title": "Quotient Group is Abelian iff All Commutators in Divisor", "text": "Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $G / N$ be the quotient group of $G$ by $N$. Then the quotient group $G / N$ is abelian {{iff}}: :$\\forall x, y \\in G: \\sqbrk {x, y} \\in N$ where $\\sqbrk {x, y}$ denotes the commutator of $x$ and $y$."}
+{"_id": "17127", "title": "Sufficient Condition for Quotient Group by Intersection to be Abelian", "text": "Let $G$ be a group. Let $N$ and $K$ be normal subgroups of $G$. Let the quotient groups $G / N$ and $G / K$ be abelian. Then the quotient group $G / \\paren {N \\cap K}$ is also abelian."}
+{"_id": "17128", "title": "Quotient Group by Intersection of Normal Subgroups not necessarily Cyclic if Quotient Groups are", "text": "Let $G$ be a group. Let $N$ and $K$ be normal subgroups of $G$. Let the quotient groups $G / N$ and $G / K$ be cyclic. Then the quotient group $G / \\paren {N \\cap K}$ is not necessarily cyclic."}
+{"_id": "17129", "title": "Order of Boolean Group is Power of 2", "text": "Let $G$ be a Boolean group. Let $\\order G$ denote the order of $G$. Then: :$\\order G = 2^n$ where $n \\in \\Z_{\\ge 0}$ is a positive integer."}
+{"_id": "17130", "title": "Subgroup of Index 2 contains all Squares of Group Elements", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$ whose index is $2$. Then: :$\\forall x \\in G: x^2 \\in H$"}
+{"_id": "17132", "title": "Center of Non-Abelian Group of Order pq is Trivial", "text": "Let $p$ and $q$ be distinct prime numbers. Let $G$ be a non-abelian group of order $p q$ whose identity is $e$. Then the center of $G$ is trivial: :$\\map Z G = \\set e$"}
+{"_id": "17133", "title": "Number of Elements of Order p in Group of Order pq is Multiple of q", "text": "Let $p$ and $q$ be distinct prime numbers. Let $G$ be a non-abelian group of order $p q$. Then the number of elements of $G$ of order $p$ is a multiple of $q$."}
+{"_id": "17135", "title": "Intersection of Normal Subgroup with Center in p-Group", "text": "Let $p$ be a prime number Let $G$ be a $p$-group. Let $N$ be a non-trivial normal subgroup of $G$. Let $\\map Z G$ denote the center of $G$. Then: :$N \\cap \\map Z G$ is a non-trivial normal subgroup of $G$."}
+{"_id": "17136", "title": "Normal Subgroup of p-Group of Order p is Subset of Center", "text": "Let $p$ be a prime number. Let $G$ be a $p$-group. Let $N$ be a normal subgroup of $G$ of order $p$. Then: :$N \\subseteq \\map Z G$ where $\\map Z G$ denotes the center of $G$."}
+{"_id": "17137", "title": "Non-Abelian Group of Order p Cubed has Exactly One Normal Subgroup of Order p", "text": "Let $p$ be a prime number. Let $G$ be a non-abelian group of order $p^3$. Then $G$ contains exactly one normal subgroup of order $p$."}
+{"_id": "17139", "title": "Non-Abelian Order 2p Group has Order p Element", "text": "Let $p$ be an odd prime. Let $G$ be a non-abelian group of order $2 p$. Then $G$ has at least one element of order $p$."}
+{"_id": "17140", "title": "Fourth Power Modulo 5", "text": "Let $n \\in \\Z$ be an integer. Then: :$n^4 \\equiv m \\pmod 5$ where $m \\in \\set {0, 1}$."}
+{"_id": "17141", "title": "Subgroup of Order p in Group of Order 2p is Normal/Corollary", "text": "Let $G$ be non-abelian. Every element of $G \\setminus K$ is of order $2$, and: :$\\forall b \\in G \\setminus K: b a b^{-1} = a^{-1}$"}
+{"_id": "17142", "title": "Subgroup of Order p in Group of Order 2p is Normal", "text": "Let $p$ be an odd prime. Let $G$ be a group of order $2 p$. Let $a \\in G$ be of order $p$. Let $K = \\gen a$ be the subgroup of $G$ generated by $a$. Then $K$ is normal in $G$."}
+{"_id": "17143", "title": "Inner Automorphisms form Subgroup of Automorphism Group", "text": "Let $G$ be a group. Then the set $\\Inn G$ of all inner automorphisms of $G$ forms a normal subgroup of the automorphism group $\\Aut G$ of $G$: :$\\Inn G \\le \\Aut G$"}
+{"_id": "17144", "title": "Property of Group Automorphism which Fixes Identity Only", "text": "Let $G$ be a finite group whose identity is $e$. Let $\\phi: G \\to G$ be a group automorphism. Let $\\phi$ have the property that: :$\\forall g \\in G \\setminus \\set e: \\map \\phi t \\ne t$ That is, the only fixed element of $\\phi$ is $e$. Then: :$\\forall x, y \\in G: x^{-1} \\, \\map \\phi x = y^{-1} \\, \\map \\phi y \\implies x = y$"}
+{"_id": "17145", "title": "Property of Group Automorphism which Fixes Identity Only/Corollary 1", "text": ":$\\forall x \\in G: \\exists g \\in G: x = g^{-1} \\, \\map \\phi g$"}
+{"_id": "17148", "title": "Square Order 2 Matrices over Real Numbers form Ring with Unity", "text": "Let $S$ denote the set of square matrices of order $2$ whose entries are the set of real numbers. Then $S$ forms a ring with unity whose unity is the matrix $\\begin {pmatrix} 1 & 0 \\\\ 0 & 1 \\end {pmatrix}$."}
+{"_id": "17150", "title": "Product Formula for Norms on Non-zero Rationals", "text": "Let $\\Q_{\\ne 0}$ be the set of non-zero rational numbers. Let  $\\Bbb P$ denote the set of prime numbers. Let $a \\in \\Q_{\\ne 0}$. Then the following infinite product converges: :$\\size a \\times \\displaystyle \\prod_{p \\mathop \\in \\Bbb P}^{} \\norm a_p = 1$ where: :$\\size {\\,\\cdot\\,}$ is the absolute value on $\\Q$ :$\\norm {\\,\\cdot\\,}_p$ is the $p$-adic norm on $\\Q$ for prime number $p$"}
+{"_id": "17152", "title": "Null Ring is Subring of Ring", "text": "Let $R$ be a ring. Then the null ring is a subring of $R$."}
+{"_id": "17153", "title": "Integers form Subring of Reals", "text": "The ring of integers $\\struct {\\Z, +, \\times}$ forms a subring of the field of real numbers."}
+{"_id": "17156", "title": "Unity of Subfield is Unity of Field", "text": "The unity of $\\struct {K, +, \\times}$ is also $1$."}
+{"_id": "17157", "title": "Zero of Subring is Zero of Ring", "text": "The zero of $\\struct {S, +, \\times}$ is also $0$."}
+{"_id": "17158", "title": "Zero of Subfield is Zero of Field/Proof 1", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0$. Let $\\struct {K, +, \\times}$ be a subfield of $\\struct {F, +, \\times}$. {{:Zero of Subfield is Zero of Field}}"}
+{"_id": "17159", "title": "Zero of Subfield is Zero of Field/Proof 2", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0$. Let $\\struct {K, +, \\times}$ be a subfield of $\\struct {F, +, \\times}$. {{:Zero of Subfield is Zero of Field}}"}
+{"_id": "17161", "title": "P-adic Norm not Complete on Rational Numbers/Proof 1/Case 1", "text": "Let $\\norm {\\,\\cdot\\,}_p$ be the $p$-adic norm on the rationals $\\Q$ for some prime $p > 3$.   Then: :$\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$ is not a complete normed division ring. That is, there exists a Cauchy sequence in $\\struct {\\Q, \\norm{\\,\\cdot\\,}_p}$ which does not converge to a limit in $\\Q$."}
+{"_id": "17162", "title": "P-adic Norm not Complete on Rational Numbers/Proof 1/Case 2", "text": "Let $\\norm {\\,\\cdot\\,}_p$ be the $p$-adic norm on the rationals $\\Q$ for $p = 2$ or $3$.   Then: :$\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$ is not a complete normed division ring. That is, there exists a Cauchy sequence in $\\struct {\\Q, \\norm{\\,\\cdot\\,}_p}$ which does not converge to a limit in $\\Q$."}
+{"_id": "17163", "title": "Element in Integral Domain is Divisor iff Principal Ideal is Superset", "text": ":$x \\divides y \\iff \\ideal y \\subseteq \\ideal x$"}
+{"_id": "17165", "title": "Residue of Gamma Function", "text": "Let $\\Gamma$ be the Definition:Gamma Function.  Let $n$ be a non-negative integer. Then:  :$\\Res \\Gamma {-n} = \\dfrac {\\paren {-1}^n} {n!}$"}
+{"_id": "17166", "title": "Periodic Element is Multiple of Period", "text": "Let $f: \\R \\to \\R$ be a real periodic function with period $P$. Let $L$ be a periodic element of $f$. Then $P \\divides L$."}
+{"_id": "17167", "title": "Equivalence of Definitions of Associate in Integral Domain", "text": "{{TFAE|def = Associate in Integral Domain|context = Integral Domain|view = Associate}} Let $\\struct {D, +, \\circ}$ be an integral domain. Let $x, y \\in D$."}
+{"_id": "17170", "title": "Finite Set of Elements in Principal Ideal Domain has GCD", "text": "Let $\\struct {D, +, \\circ}$ be a principal ideal domain. Let $a_1, a_2, \\dotsc, a_n$ be non-zero elements of $D$. Then $a_1, a_2, \\dotsc, a_n$ all have a greatest common divisor."}
+{"_id": "17171", "title": "Set of Linear Combinations of Finite Set of Elements of Principal Ideal Domain is Principal Ideal", "text": "Let $\\struct {D, +, \\circ}$ be a principal ideal domain. Let $a_1, a_2, \\dotsc, a_n$ be non-zero elements of $D$. Let $J$ be the set of all linear combinations in $D$ of $\\set {a_1, a_2, \\dotsc, a_n}$ Then for some $x \\in D$: :$J = \\ideal x$ where $\\ideal x$ denotes the principal ideal generated by $x$."}
+{"_id": "17172", "title": "Greatest Common Divisors in Principal Ideal Domain are Associates", "text": "Let $\\struct {D, +, \\circ}$ be a principal ideal domain. Let $S = \\set {a_1, a_2, \\dotsc, a_n}$ be a set of non-zero elements of $D$. Let $y_1$ and $y_2$ be greatest common divisors of $S$. Then $y_1$ and $y_2$ are associates."}
+{"_id": "17173", "title": "Greatest Common Divisor in Principal Ideal Domain is Expressible as Linear Combination", "text": "Let $\\struct {D, +, \\circ}$ be a principal ideal domain. Let $S = \\set {a_1, a_2, \\dotsc, a_n}$ be a set of non-zero elements of $D$. Let $y$ be a greatest common divisor of $S$. Then $y$ is expressible in the form: :$y = d_1 a_1 + d_2 a_2 + \\dotsb + d_n a_n$ where $d_1, d_2, \\dotsc, d_n \\in D$."}
+{"_id": "17174", "title": "Complete Factorizations of Proper Element in Principal Ideal Domain are Equivalent", "text": "Let $\\struct {D, +, \\circ}$ be a principal ideal domain. Let $x \\in D$ be a proper element of $D$. Let there be two complete factorizations of $x$: :$x = u_y \\circ y_1 \\circ y_2 \\circ \\cdots \\circ y_m = F_1$ :$x = u_z \\circ z_1 \\circ z_2 \\circ \\cdots \\circ z_n = F_2$ Then $F_1$ and $F_2$ are equivalent."}
+{"_id": "17175", "title": "Moment Generating Function of Poisson Distribution", "text": "Let $X$ be a discrete random variable with a Poisson distribution with parameter $\\lambda$ for some $\\lambda \\in \\R_{> 0}$. Then the moment generating function $M_X$ of $X$ is given by:  :$\\map {M_X} t = e^{\\lambda \\paren {e^t - 1} }$"}
+{"_id": "17180", "title": "Moment Generating Function of Binomial Distribution", "text": "Let $X$ be a discrete random variable with a binomial distribution with parameters $n$ and $p$ for some $n \\in \\N$ and $0 \\le p \\le 1$: :$X \\sim \\Binomial n p$ Then the moment generating function $M_X$ of $X$ is given by:  :$\\map {M_X} t = \\paren {1 - p + p e^t}^n$"}
+{"_id": "17181", "title": "Moment Generating Function of Exponential Distribution", "text": "Let $X$ be a continuous random variable with an exponential distribution with parameter $\\beta$ for some $\\beta \\in \\R_{> 0}$. Then the moment generating function $M_X$ of $X$ is given by:  :$\\displaystyle \\map {M_X} t = \\frac 1 {1 - \\beta t}$ for $t < \\dfrac 1 \\beta$, and is undefined otherwise."}
+{"_id": "17182", "title": "Moment Generating Function of Geometric Distribution", "text": "Let $X$ be a discrete random variable with a geometric distribution with parameter $p$ for some $0 < p \\le 1$.  Then the moment generating function $M_X$ of $X$ is given by:  :$\\displaystyle \\map {M_X} t = \\frac p {1 - \\paren {1 - p} e^t}$ for $t < -\\map \\ln {1 - p}$, and is undefined otherwise."}
+{"_id": "17184", "title": "Maximal Ideal iff Quotient Ring is Field/Proof 1/Maximal Ideal implies Quotient Ring is Field", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let $J$ be a maximal ideal. Then the quotient ring $R / J$ is a field."}
+{"_id": "17185", "title": "Maximal Ideal iff Quotient Ring is Field/Proof 1/Quotient Ring is Field implies Ideal is Maximal", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let the quotient ring $R / J$ be a field. Then $J$ is a maximal ideal."}
+{"_id": "17193", "title": "Ring of Polynomial Forms is not necessarily Isomorphic to Ring of Polynomial Functions", "text": "Let $D$ be an integral domain. Let $D \\sqbrk X$ be the ring of polynomial forms in $X$ over $D$. Let $\\map P D$ be the ring of polynomial functions over $D$. Then it is not necessarily the case that $D \\sqbrk X$ is isomorphic with $\\map P D$."}
+{"_id": "17194", "title": "Moment Generating Function of Bernoulli Distribution", "text": "Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$ for some $0 \\le p \\le 1$.  Then the moment generating function $M_X$ of $X$ is given by:  :$\\map {M_X} t = q + p e^t$ where $q = 1 - p$."}
+{"_id": "17198", "title": "Double of Antiperiodic Element is Periodic", "text": "Let $f: X \\to X$ be a function, where $X$ is either $\\R$ or $\\C$. Let $L \\in X_{\\ne 0}$ be an anti-periodic element of $f$. Then $2L$ is a periodic element of $f$. In other words, every anti-periodic function is also periodic."}
+{"_id": "17200", "title": "Skewness of Gaussian Distribution", "text": "Let $X$ be a continuous random variable with a Gaussian distribution with parameters $\\mu$ and $\\sigma^2$ for some $\\mu \\in \\R$ and $\\sigma \\in \\R_{> 0}$.  Then the skewness $\\gamma_1$ of $X$ is equal to $0$."}
+{"_id": "17201", "title": "Inverse of Unit in Centralizer of Ring is in Centralizer", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $S$ be a subset of $R$. Let $\\map {C_R} S$ denote the centralizer of $S$ in $R$ Let $u \\in R$ be a unit of $R$. Then: :$u \\in \\map {C_R} S \\implies u^{-1} \\in \\map {C_R} S$"}
+{"_id": "17203", "title": "Skewness in terms of Non-Central Moments", "text": "Let $X$ be a random variable with mean $\\mu$ and standard deviation $\\sigma$. Then the skewness $\\gamma_1$ of $X$ is given by:  :$\\gamma_1 = \\dfrac {\\expect {X^3} - 3 \\mu \\sigma^2 - \\mu^3} {\\sigma^3}$"}
+{"_id": "17204", "title": "Intersection of Ring Ideals is Ideal", "text": "Let $\\struct {R, +, \\circ}$ be a ring Let $\\mathbb L$ be a non-empty set of ideals of $R$. Then the intersection $\\bigcap \\mathbb L$ of the members of $\\mathbb L$ is itself an ideal of $R$."}
+{"_id": "17205", "title": "Set of Ring Elements forming Zero Product with given Element is Ideal", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring whose zero is $0_R$. Let $a \\in R$ be an arbitrary element of $R$. Let $A$ be the subset of $R$ defined as: :$A = \\set {x \\in R: x \\circ a = 0_R}$ Then $A$ is an ideal of $A$."}
+{"_id": "17206", "title": "Skewness of Bernoulli Distribution", "text": "Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$. Then the skewness $\\gamma_1$ of $X$ is given by:  :$\\gamma_1 = \\dfrac {1 - 2 p} {\\sqrt {p q} }$ where $q = 1 - p$."}
+{"_id": "17207", "title": "Intersection of Subrings is Subring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\mathbb L$ be a non-empty set of subrings of $R$. Then the intersection $\\displaystyle \\bigcap \\mathbb L$ of the members of $\\mathbb L$ is itself a subring of $R$."}
+{"_id": "17209", "title": "Intersection of Ring Ideals is Largest Ideal Contained in all Ideals", "text": "Let $\\struct {R, +, \\circ}$ be a ring Let $\\mathbb L$ be a non-empty set of ideals of $R$. Then the intersection $\\bigcap \\mathbb L$ of the members of $\\mathbb L$ is the largest ideal of $R$ contained in each member of $\\mathbb L$."}
+{"_id": "17210", "title": "Intersection of Subrings is Largest Subring Contained in all Subrings", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\mathbb L$ be a non-empty set of subrings of $R$. Then the intersection $\\displaystyle \\bigcap \\mathbb L$ of the members of $\\mathbb L$ is the largest subring of $R$ contained in each member of $\\mathbb L$."}
+{"_id": "17211", "title": "Median of Continuous Uniform Distribution", "text": "Let $X$ be a continuous random variable which is uniformly distributed on a closed real interval $\\closedint a b$.  Then the median $M$ of $X$ is given by:  :$M = \\dfrac {a + b} 2$"}
+{"_id": "17212", "title": "Intersection of Ring Ideals Containing Subset is Smallest", "text": "Let $\\struct {R, +, \\circ}$ be a ring Let $S \\subseteq R$ be a subset of $R$. Let $L$ be the intersection of the set of all ideals of $R$ containing $S$. Then $L$ is the smallest ideal of $R$ containing $S$."}
+{"_id": "17213", "title": "Intersection of Subrings Containing Subset is Smallest", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $S \\subseteq R$ be a subset of $R$. Let $L$ be the intersection of the set of all subrings of $R$ containing $S$. Then $L$ is the smallest subring of $R$ containing $S$."}
+{"_id": "17214", "title": "Intersection of Division Subrings is Division Subring", "text": "Let $\\struct {D, +, \\circ}$ be a division ring. Let $\\mathbb K$ be a non-empty set of division subrings of $D$. Then the intersection $\\bigcap \\mathbb K$ of the members of $\\mathbb K$ is itself a division subring of $D$."}
+{"_id": "17215", "title": "Intersection of Subfields is Subfield", "text": "Let $\\struct {F, +, \\circ}$ be a field. Let $\\mathbb K$ be a non-empty set of subfields of $F$. Then the intersection $\\bigcap \\mathbb K$ of the members of $\\mathbb K$ is itself a subfield of $F$."}
+{"_id": "17216", "title": "Intersection of Division Subrings is Largest Division Subring Contained in all Division Subrings", "text": "Let $\\struct {D, +, \\circ}$ be a division ring. Let $\\mathbb K$ be a non-empty set of division subrings of $D$. Let $\\bigcap \\mathbb K$ be the intersection of the elements of $\\mathbb K$. Then $\\bigcap \\mathbb K$ is the largest division subring of $D$ contained in each element of $\\mathbb K$."}
+{"_id": "17217", "title": "Intersection of Division Subrings Containing Subset is Smallest", "text": "Let $\\struct {D, +, \\circ}$ be a division ring. Let $S \\subseteq D$ be a subset of $D$. Let $L$ be the intersection of the set of all division subrings of $D$ containing $S$. Then $L$ is the smallest division subring of $D$ containing $S$."}
+{"_id": "17218", "title": "Intersection of Subfields Containing Subset is Smallest", "text": "Let $\\struct {F, +, \\circ}$ be a field. Let $S \\subseteq F$ be a subset of $F$. Let $L$ be the intersection of the set of all subfields of $F$ containing $S$. Then $L$ is the smallest subfield of $F$ containing $S$."}
+{"_id": "17219", "title": "Intersection of Subfields is Largest Subfield Contained in all Subfields", "text": "Let $\\struct {F, +, \\circ}$ be a field. Let $\\mathbb K$ be a non-empty set of subfields of $F$. Let $\\bigcap \\mathbb K$ be the intersection of the elements of $\\mathbb K$. Then $\\bigcap \\mathbb K$ is the largest subfield of $F$ contained in each element of $\\mathbb K$."}
+{"_id": "17220", "title": "Equivalent Norms on Rational Numbers/Necessary Condition", "text": "Let $\\norm {\\, \\cdot \\,}_1$ and $\\norm {\\, \\cdot \\,}_2$ be norms on the rational numbers $\\Q$. Let $\\norm {\\, \\cdot \\,}_1$ and $\\norm {\\, \\cdot \\,}_2$ be equivalent norms. Then: :$\\exists \\alpha \\in \\R_{\\gt 0}: \\forall n \\in \\N: \\norm n_1 = \\norm n_2^\\alpha$"}
+{"_id": "17221", "title": "Equivalent Norms on Rational Numbers/Sufficient Condition", "text": "Let $\\norm {\\, \\cdot \\,}_1$ and $\\norm {\\, \\cdot \\,}_2$ be norms on the rational numbers $\\Q$. Let $\\norm {\\, \\cdot \\,}_1$ and $\\norm {\\, \\cdot \\,}_2$ satisfy: :$\\exists \\alpha \\in \\R_{\\gt 0}: \\forall n \\in \\N: \\norm n_1 = \\norm n_2^\\alpha$ Then $\\norm {\\, \\cdot \\,}_1$ and $\\norm {\\, \\cdot \\,}_2$ are equivalent"}
+{"_id": "17222", "title": "Commutative and Unitary Ring with 2 Ideals is Field", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity whose zero is $0_R$. Let $\\struct {R, +, \\circ}$ be such that the only ideals of $\\struct {R, +, \\circ}$ are: :$\\set {0_R}$ and: $\\struct {R, +, \\circ}$ itself. That is, such that $\\struct {R, +, \\circ}$ has no non-null proper ideals. Then $\\struct {R, +, \\circ}$ is a field."}
+{"_id": "17223", "title": "Field has 2 Ideals", "text": "Let $\\struct {F, +, \\circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Then the only ideals of $\\struct {F, +, \\circ}$ are:$\\struct {F, +, \\circ}$ and $\\set {0_F}$. That is, $\\struct {F, +, \\circ}$ has no non-null proper ideals."}
+{"_id": "17225", "title": "General Antiperiodicity Property", "text": "Let $f: X \\to X$ be an antiperiodic function, where $X$ is either $\\R$ or $\\C$. Let $L$ be an antiperiodic element of $f$. Let $n \\in \\Z$ be an integer. :If $n$ is even, then $n L$ is a periodic element of $f$. :If $n$ is odd, then $n L$ is an antiperiodic element of $f$."}
+{"_id": "17226", "title": "Antiperiodic Element is Multiple of Antiperiod", "text": "Let $f: \\R \\to \\R$ be a real anti-periodic function with anti-period $A$. Let $L$ be an anti-periodic element of $f$. Then $A \\divides L$."}
+{"_id": "17227", "title": "Skewness of Poisson Distribution", "text": "Let $X$ be a discrete random variable with a Poisson distribution with parameter $\\lambda$. Then the skewness $\\gamma_1$ of $X$ is given by:  :$\\gamma_1 = \\dfrac 1 {\\sqrt \\lambda}$"}
+{"_id": "17228", "title": "Periodic Element is Multiple of Antiperiod", "text": "Let $f: \\R \\to \\R$ be a real anti-periodic function with anti-period $A$. Let $L$ be a periodic element of $f$. Then $A \\divides L$."}
+{"_id": "17229", "title": "Skewness of Continuous Uniform Distribution", "text": "Let $X$ be a continuous random variable which is uniformly distributed on a closed real interval $\\closedint a b$.  Then the skewness $\\gamma_1$ of $X$ is equal to $0$."}
+{"_id": "17231", "title": "Principal Ideal in Integral Domain generated by Power Plus One is Subset of Principal Ideal generated by Power", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose unity is $1_D$. Let $a \\in D$ be a proper element of $D$. Then: :$\\forall n \\in \\Z_{\\ge 0}: \\ideal {a^{n + 1} } \\subsetneq \\ideal {a_n}$ where $\\ideal x$ denotes the principal ideal of $D$ generated by $x$."}
+{"_id": "17232", "title": "Non-Field Integral Domain has Infinite Number of Ideals", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain which is not a field. Then $\\struct {D, +, \\circ}$ has an infinite number of distinct ideals."}
+{"_id": "17234", "title": "Ring Homomorphism from Ring with Unity to Integral Domain Preserves Unity", "text": "Let $\\struct {R, +_R, \\circ_R}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\\struct {D, +_D, \\circ_D}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $\\phi: R \\to D$ be a ring homomorphism such that: :$\\map \\ker \\phi \\ne R$ where $\\map \\ker \\phi$ denotes the kernel of $\\phi$. Then $\\map \\phi {1_R} = 1_D$."}
+{"_id": "17237", "title": "Unity plus Negative of Nilpotent Ring Element is Unit", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $x \\in R$ be nilpotent. Then $1_R - x$ is a unit of $R$."}
+{"_id": "17238", "title": "Quotient of Commutative Ring by Nilradical is Reduced", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring whose zero is $0_R$ and whose unity is $1_R$. Let $\\struct {N, +, \\circ}$ denote the nilradical of $R$. The quotient ring $R / N$ is a reduced ring."}
+{"_id": "17239", "title": "Self-Inverse Element of Integral Domain is Unity or its Negative", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $x \\in D$ such that $x^2 = 1_D$. Then either $x = 1_D$ or $x = -1_D$."}
+{"_id": "17240", "title": "Product of Units of Integral Domain with Finite Number of Units", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $D$ have a finite number of units. Let $U_D$ be the set of units of $\\struct {D, +, \\circ}$. Then: :$\\displaystyle \\prod_{x \\mathop \\in U_D} x = -1_D$"}
+{"_id": "17243", "title": "Prime Ideals of Ring of Integers", "text": "Let $\\struct {\\Z, +, \\times}$ denote the ring of integers. Let $J$ be a prime ideal of $\\Z$. Then either: :$J = \\set 0$ or: :$J = \\ideal p$ where: :$p$ is a prime number :$\\ideal p$ denotes the principal ideal of $\\Z$ generated by $p$."}
+{"_id": "17245", "title": "Additive Group and Multiplicative Group of Field are not Isomorphic", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $\\struct {F, +}$ denote the additive group of $F$. Let $\\struct {F_{\\ne 0_F}, \\times}$ denote the multiplicative group of $F$. Then $\\struct {F, +}$ and $\\struct {F_{\\ne 0_F}, \\times}$ are not isomorphic to each other."}
+{"_id": "17247", "title": "Field Norm of Complex Number is Positive Definite", "text": "Let $\\C$ denote the set of complex numbers. Let $N: \\C \\to \\R_{\\ge 0}$ denote the field norm on complex numbers: :$\\forall z \\in \\C: \\map N z = \\cmod z^2$ where $\\cmod z$ denotes the complex modulus of $z$. Then $N$ is positive definite on $\\C$."}
+{"_id": "17248", "title": "Field Norm of Complex Number is Multiplicative Function", "text": "Let $\\C$ denote the set of complex numbers. Let $N: \\C \\to \\R_{\\ge 0}$ denote the field norm on complex numbers: :$\\forall z \\in \\C: \\map N z = \\cmod z^2$ where $\\cmod z$ denotes the complex modulus of $z$. Then $N$ is a multiplicative function on $\\C$."}
+{"_id": "17250", "title": "Units of 5th Cyclotomic Ring", "text": "Let $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ denote the $5$th cyclotomic ring. The units of $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ are $1$ and $-1$."}
+{"_id": "17251", "title": "Field Norm on 5th Cyclotomic Ring", "text": "Let $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ denote the $5$th cyclotomic ring. Let $\\alpha = a + i b \\sqrt 5$ be an arbitrary element of $\\Z \\sqbrk {i \\sqrt 5}$. The field norm of $\\alpha$ is given by: :$\\map N \\alpha = a^2 + 5 b^2$"}
+{"_id": "17252", "title": "5th Cyclotomic Ring has no Elements with Field Norm of 2 or 3", "text": "Let $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ denote the $5$th cyclotomic ring. There are no elements of $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ whose field norm is either $2$ or $3$."}
+{"_id": "17253", "title": "Irreducible Elements of 5th Cyclotomic Ring", "text": "Let $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ denote the $5$th cyclotomic ring. The following elements of $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ are irreducible: :$2$ :$3$ :$1 + i \\sqrt 5$ :$1 - i \\sqrt 5$"}
+{"_id": "17254", "title": "Value of Field Norm on 5th Cyclotomic Ring is Integer", "text": "Let $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ denote the $5$th cyclotomic ring. Let $\\alpha = a + i b \\sqrt 5$ be an arbitrary element of $\\Z \\sqbrk {i \\sqrt 5}$. Let $\\map N \\alpha$ denoted the field norm of $\\alpha$. Then $\\map N \\alpha$ is an integer."}
+{"_id": "17255", "title": "Elements of 5th Cyclotomic Ring with Field Norm 1", "text": "Let $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ denote the $5$th cyclotomic ring. The only elements of $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ whose field norm equals $1$ are the units of $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$: $1$ and $-1$."}
+{"_id": "17258", "title": "Polynomials in Integers with Even Constant Term forms Ideal", "text": "Let $\\Z \\sqbrk X$ be the ring of polynomials in $X$ over $\\Z$. Let $S \\subseteq \\Z \\sqbrk X$ be the set of polynomials over $\\Z$ in $X$ which have a constant term which is even. Then $S$ is an ideal of $\\Z \\sqbrk X$."}
+{"_id": "17260", "title": "Polynomials in Integers is Unique Factorization Domain", "text": "Let $\\Z \\sqbrk X$ be the ring of polynomials in $X$ over $\\Z$. Then $\\Z \\sqbrk X$ is a unique factorization domain."}
+{"_id": "17262", "title": "Ring of Gaussian Integers is Principal Ideal Domain", "text": "The ring of Gaussian integers: :$\\struct {\\Z \\sqbrk i, +, \\times}$ forms a principal ideal domain."}
+{"_id": "17264", "title": "Matrix Multiplication is not Commutative", "text": "Let $R$ be a ring with unity. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer such that $n \\ne 1$. Let $\\map {\\MM_R} n$ denote the $n \\times n$ matrix space over $R$. Then (conventional) matrix multiplication over $\\map {\\MM_R} n$ is not commutative: :$\\exists \\mathbf A, \\mathbf B \\in \\map {\\MM_R} n: \\mathbf {A B} \\ne \\mathbf {B A}$ If $R$ is specifically not commutative, then the result holds when $n = 1$ as well."}
+{"_id": "17265", "title": "Matrix Multiplication on Square Matrices over Trivial Ring is Commutative", "text": "Let $\\struct {R, +, \\circ}$ be the trivial ring over an underlying set. Let $\\map {\\MM_R} n$ denote the $n \\times n$ matrix space over $R$. Then (conventional) matrix multiplication is commutative over $\\map {\\MM_R} n$: :$\\forall \\mathbf A, \\mathbf B \\in \\map {\\MM_R} n: \\mathbf {A B} = \\mathbf {B A}$"}
+{"_id": "17266", "title": "Area between Smooth Curve and Line with Fixed Endpoints is Maximized by Arc of Circle", "text": "Let $y$ be a smooth curve, embedded in $2$-dimensional Euclidean space. Let $y$ have a total length of $l$. Let it be contained in the upper halfplane with an exception of endpoints, which are on the x-axis and are given. Suppose, $y$, together with a line segment connecting $y$'s endpoints, maximizes the enclosed area. Then $y$ is an arc of a circle."}
+{"_id": "17270", "title": "Preordering of Products under Operation Compatible with Preordering", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\precsim$ be a preordering on $S$. Then $\\precsim$ is compatible with $\\circ$ {{iff}}: :$\\forall x_1, x_2, y_1, y_2 \\in S: x_1 \\precsim x_2 \\land y_1 \\precsim y_2 \\implies \\paren {x_1 \\circ y_1} \\precsim \\paren {x_2 \\circ y_2}$"}
+{"_id": "17271", "title": "P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 1", "text": ":$\\exists x \\in \\Z_{>0}:  p \\nmid x, x \\ge \\dfrac {p + 1} 2$"}
+{"_id": "17272", "title": "P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 5", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n^k = a$ in $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$"}
+{"_id": "17273", "title": "Characterisation of Cauchy Sequence in Non-Archimedean Norm/Corollary 1", "text": "Let $\\norm {\\,\\cdot\\,}_p$ be the $p$-adic norm on the rationals $\\Q$ for some prime $p$. Let $\\sequence {x_n}$ be a sequence of integers such that: :$\\forall n: x_{n + 1} \\equiv x_n \\pmod {p^n}$ Then: :$\\sequence {x_n}$ is a Cauchy sequence in $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$."}
+{"_id": "17274", "title": "P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 2", "text": ":$a \\in \\Z_{> 0}: \\nexists \\,c \\in \\Z : c^k = a$"}
+{"_id": "17275", "title": "P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 3", "text": ":$\\map f {x_1} \\equiv 0 \\pmod p$"}
+{"_id": "17276", "title": "P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 4", "text": ":$\\map {f'} {x_1} \\not \\equiv 0 \\pmod p$"}
+{"_id": "17277", "title": "Surjection from Natural Numbers iff Countable/Corollary 1", "text": "Let $T$ be a countably infinite set. Let $S$ be a non-empty set. Then $S$ is countable {{iff}} there exists a surjection $f: T \\to S$."}
+{"_id": "17278", "title": "Surjection from Natural Numbers iff Countable/Corollary 2", "text": "Let $T$ be a countably infinite set. Let $S$ be an uncountable set. Let $f:T \\to S$ be a mapping. Then $f$ is not a surjection."}
+{"_id": "17279", "title": "P-adic Numbers are Uncountable", "text": "Let $p$ be any prime number. The set of $p$-adic numbers $\\Q_p$ is an uncountable set."}
+{"_id": "17283", "title": "Ordered Integral Domain is Totally Ordered Ring", "text": "Let $\\struct {D, +, \\times, \\le}$ be an ordered integral domain. Then $\\struct {D, +, \\times, \\le}$ is a totally ordered ring."}
+{"_id": "17284", "title": "Strict Negativity is equivalent to Strictly Preceding Zero", "text": ":$\\map N a \\iff a < 0$"}
+{"_id": "17286", "title": "Sum of Strictly Negative Elements is Strictly Negative", "text": ":$\\map N a, \\map N b \\implies \\map N {a + b}$"}
+{"_id": "17287", "title": "Product of Two Strictly Negative Elements is Strictly Positive", "text": ":$\\map N a, \\map N b \\implies \\map P {a \\times b}$"}
+{"_id": "17288", "title": "Product of Strictly Negative Element with Strictly Positive Element is Strictly Negative", "text": ":$\\map N a, \\map P b \\implies \\map N {a \\times b}$"}
+{"_id": "17290", "title": "Principle of Mathematical Induction/Zero-Based", "text": "Let $\\map P n$ be a propositional function depending on $n \\in \\N$. Suppose that: :$(1): \\quad \\map P 0$ is true :$(2): \\quad \\forall k \\in \\N: k \\ge 0 : \\map P k \\implies \\map P {k + 1}$ Then: :$\\map P n$ is true for all $n \\in \\N$."}
+{"_id": "17291", "title": "Principle of Mathematical Induction/One-Based", "text": "Let $\\map P n$ be a propositional function depending on $n \\in \\N_{>0}$. Suppose that: :$(1): \\quad \\map P 1$ is true :$(2): \\quad \\forall k \\in \\N_{>0}: k \\ge 1 : \\map P k \\implies \\map P {k + 1}$ Then: :$\\map P n$ is true for all $n \\in \\N_{>0}$."}
+{"_id": "17293", "title": "Principle of Finite Induction/One-Based", "text": "Let $S \\subseteq \\N_{>0}$ be a subset of the $1$-based natural numbers. Suppose that: :$(1): \\quad 1 \\in S$ :$(2): \\quad \\forall n \\in \\N_{>0} : n \\in S \\implies n + 1 \\in S$ Then: :$S = \\N_{>0}$"}
+{"_id": "17295", "title": "Second Principle of Finite Induction/One-Based", "text": "Let $S \\subseteq \\N_{>0}$ be a subset of the $1$-based natural numbers. Suppose that: :$(1): \\quad 1 \\in S$ :$(2): \\quad \\forall n \\in \\N_{>0}: \\paren {\\forall k: 1 \\le k \\le n \\implies k \\in S} \\implies n + 1 \\in S$ Then: :$S = \\N_{>0}$"}
+{"_id": "17296", "title": "Second Principle of Mathematical Induction/Zero-Based", "text": "Let $\\map P n$ be a propositional function depending on $n \\in \\N$. Suppose that: :$(1): \\quad \\map P 0$ is true :$(2): \\quad \\forall k \\in \\N: \\map P 0 \\land \\map P 1 \\land \\ldots \\land \\map P {k - 1} \\land \\map P k \\implies \\map P {k + 1}$ Then: :$\\map P n$ is true for all $n \\in \\N$."}
+{"_id": "17297", "title": "Second Principle of Mathematical Induction/One-Based", "text": "Let $\\map P n$ be a propositional function depending on $n \\in \\N_{>0}$. Suppose that: :$(1): \\quad \\map P 1$ is true :$(2): \\quad \\forall k \\in \\N_{>0}: \\map P 1 \\land \\map P 2 \\land \\ldots \\land \\map P {k - 1} \\land \\map P k \\implies \\map P {k + 1}$ Then: :$\\map P n$ is true for all $n \\in \\N_{>0}$."}
+{"_id": "17306", "title": "Null Sequences form Maximal Left and Right Ideal/Corollary 1", "text": "Then $\\NN$ is a maximal ring ideal of $\\CC$."}
+{"_id": "17307", "title": "Non-Zero Integer has Finite Number of Divisors", "text": "Let $n \\in \\Z_{\\ne 0}$ be a non-zero integer. Then $n$ has a finite number of divisors."}
+{"_id": "17308", "title": "Coprimality Relation is Non-Reflexive", "text": ":$\\perp$ is non-reflexive."}
+{"_id": "17312", "title": "Ring of Polynomials over Reals is not Field", "text": "Let $\\R \\sqbrk X$ be the ring of polynomials in an indeterminate $X$ over $\\R$. Then $\\R \\sqbrk X$ is not a field."}
+{"_id": "17319", "title": "Ideals of Ring of Integers Modulo m", "text": "Let $m \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\struct {\\Z_m, +, \\times}$ denote the ring of integers modulo $m$. The ideals of $\\struct {\\Z_m, +, \\times}$ are of the form: :$d \\Z / m \\Z$ where $d$ is a divisor of $m$."}
+{"_id": "17320", "title": "Quotient Ring of Cauchy Sequences is Division Ring/Corollary 1", "text": "Then the quotient ring $\\CC \\,\\big / \\NN$ is a field."}
+{"_id": "17321", "title": "Quotient Ring of Cauchy Sequences is Normed Division Ring/Corollary 1", "text": "Then $\\struct {\\CC \\,\\big / \\NN, \\norm {\\, \\cdot \\,}_1 }$ is a valued field."}
+{"_id": "17322", "title": "Cauchy Sequence Converges Iff Equivalent to Constant Sequence", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,}}$ be a normed division ring. Let $\\CC$ be the ring of Cauchy sequences over $R$ Let $\\NN$ be the set of null sequences. Let $\\\\CC \\,\\big / \\NN$ be the quotient ring of Cauchy sequences of $\\CC$ by the maximal ideal $\\NN$. Let $\\sequence {x_n} \\in \\CC$. Then $\\sequence {x_n}$ converges in $\\struct {R, \\norm {\\,\\cdot\\,} }$ {{iff}} :$\\exists a \\in R: \\sequence {x_n} \\in \\sequence {a, a, a, \\dotsc} + \\NN$ where $\\sequence {a, a, a, \\dotsc} + \\NN$ is the left coset in $\\CC \\, \\big / \\NN$ that contains the constant sequence $\\sequence {a, a, a, \\dotsc}$."}
+{"_id": "17323", "title": "Homomorphism of Ring Subtraction", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring homomorphism. Then: :$\\forall a, b \\in R_1: \\map \\phi {a -_1 b} = \\map \\phi a -_2 \\map \\phi b$ where $a -_1 b$ denotes subtraction of $b$ from $a$."}
+{"_id": "17324", "title": "Limit of Modulo Operation/Limit 1", "text": "Let $x$ and $y$ be real numbers. Let $x \\bmod y$ denote the modulo operation. Then $\\displaystyle \\lim_{y \\mathop \\to 0} x \\bmod y = 0$."}
+{"_id": "17325", "title": "Limit of Modulo Operation/Limit 2", "text": "Let $x$ and $y$ be real numbers. Let $x \\bmod y$ denote the modulo operation. Then $\\displaystyle \\lim_{y \\mathop \\to \\infty} x \\bmod y = x$ if $x \\ge 0$."}
+{"_id": "17326", "title": "Limit of Modulo Operation", "text": "Let $x$ and $y$ be real numbers. Let $x \\bmod y$ denote the modulo operation. Then holding $x$ fixed gives: :$\\displaystyle \\lim_{y \\mathop \\to 0} x \\bmod y = 0$ :$\\displaystyle \\lim_{y \\mathop \\to \\infty} x \\bmod y = x$ if $x \\ge 0$ {{expand|What about the limits with respect to $x$?}}"}
+{"_id": "17327", "title": "Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary", "text": "Let $\\struct {R, \\norm{\\,\\cdot\\,} }$ be a non-Archimedean normed division ring with zero $0_R$ Let $\\sequence {x_n}$ be a Cauchy sequence such that $\\sequence {x_n}$ does not converge to $0_R$. Then: :$\\exists N \\in \\N: \\forall n, m \\ge N: \\norm {x_n} = \\norm {x_m}$"}
+{"_id": "17328", "title": "Equivalence of Definitions of Unit of Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose unity is $1_R$. {{TFAE|def = Unit of Ring}}"}
+{"_id": "17329", "title": "Bézout's Lemma/Euclidean Domain", "text": "Let $\\struct {D, +, \\times}$ be a Euclidean domain whose zero is $0$ and whose unity is $1$. Let $\\nu: D \\setminus \\set 0 \\to \\N$ be the Euclidean valuation on $D$. Let $a, b \\in D$ such that $a$ and $b$ are not both equal to $0$. Let $\\gcd \\set {a, b}$ be the greatest common divisor of $a$ and $b$. Then: :$\\exists x, y \\in D: a \\times x + b \\times y = \\gcd \\set {a, b}$ such that $\\gcd \\set {a, b}$ is the element of $D$ such that: :$\\forall c = a \\times x + b \\times y \\in D: \\map \\nu {\\gcd \\set {a, b} } \\le \\map \\nu c$"}
+{"_id": "17331", "title": "Euclidean Valuation of Non-Unit is less than that of Product", "text": "Let $\\struct {D, +, \\times}$ be a Euclidean domain whose zero is $0$, and unity is $1$. Let the valuation function of $D$ be $\\nu$. Let $b, c \\in D_{\\ne 0}$. Then: :If $c$ is not a unit of $D$ then $\\map \\nu b < \\map \\nu {b c}$"}
+{"_id": "17332", "title": "Element is Unit iff its Euclidean Valuation equals that of 1", "text": "Let $\\struct {D, +, \\times}$ be a Euclidean domain whose zero is $0$, and unity is $1$. Let the valuation function of $D$ be $\\nu$. Let $a \\in D$. Then: :$a$ is a unit of $D$ {{iff}} $\\map \\nu a = \\map \\nu 1$"}
+{"_id": "17333", "title": "Gauss's Lemma on Primitive Polynomials over Ring", "text": "Let $R$ be a commutative ring with unity. Let $f, g \\in R \\sqbrk X$ be primitive polynomials. {{explain|Definition:Primitive Polynomial over Ring}} Then $f g$ is primitive."}
+{"_id": "17334", "title": "Rational Polynomial is Content Times Primitive Polynomial/Existence", "text": "Let $\\Q \\sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\\map f X \\in \\Q \\sqbrk X$. Then: :$\\map f X = \\cont f \\, \\map {f^*} X$ where: :$\\cont f$ is the content of $\\map f X$ :$\\map {f^*} X$ is a primitive polynomial."}
+{"_id": "17335", "title": "Rational Polynomial is Content Times Primitive Polynomial/Uniqueness", "text": "Let $\\Q \\sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\\map f X \\in \\Q \\sqbrk X$ be given. Then there exist unique content $\\cont f$ of $\\map f X$ and primitive polynomial $\\map {f^*} X$ such that: :$\\map f X = \\cont f \\, \\map {f^*} X$"}
+{"_id": "17337", "title": "Content of Polynomial in Dedekind Domain is Multiplicative", "text": "Let $R$ be a Dedekind domain. Let $f, g \\in R \\sqbrk X$ be polynomials. Let $\\cont f$ denote the content of $f$. Then $\\cont {f g} = \\cont f \\cont g$ is the product of $\\cont f$ and $\\cont g$."}
+{"_id": "17339", "title": "Factors of Polynomial with Integer Coefficients have Integer Coefficients", "text": "Let $\\Q \\sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\\map h X \\in \\Q \\sqbrk X$ have coefficients all of which are integers. Let it be possible to express $\\map h X$ as: :$\\map h X = \\map f X \\, \\map g X$ where $\\map f X, \\map g X \\in \\Q \\sqbrk X$. Then it is also possible to express $\\map h X$ as: :$\\map h X = \\map {f'} X \\, \\map {g'} X$ where: :$\\map {f'} X, \\map {g'} X \\in \\Q \\sqbrk X$ :the coefficients of $\\map {f'} X$ and $\\map {g'} X$ are all integers :$\\map {f'} X = a \\map f X$ and $\\map {g'} X = b \\map f X$, for $a, b \\in \\Q$."}
+{"_id": "17340", "title": "Polynomial which is Irreducible over Integers is Irreducible over Rationals", "text": "Let $\\Z \\sqbrk X$ be the ring of polynomial forms over the integers in the indeterminate $X$. Let $\\Q \\sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\\map f X \\in \\Z \\sqbrk X$ be irreducible in $\\Z \\sqbrk X$. Then $\\map f X$ is also irreducible in $\\Q \\sqbrk X$."}
+{"_id": "17343", "title": "Dirichlet Function is Periodic", "text": "Let $D: \\R \\to \\R$ be a Dirichlet function: :$\\forall x \\in \\R: \\map D x = \\begin{cases} c & : x \\in \\Q \\\\ d & : x \\notin \\Q \\end{cases}$ Then $D$ is periodic. Namely, every non-zero rational number is a periodic element of $D$."}
+{"_id": "17344", "title": "Dirichlet Function has no Period", "text": "The Dirichlet functions are periodic by Dirichlet Function is Periodic. However, they do not admit a period. That is, there does not exist a smallest value $L \\in \\R_{> 0}$ such that: :$\\forall x \\in \\R: \\map D x = \\map D {x + L}$"}
+{"_id": "17345", "title": "Existence of Nonconstant Periodic Function with no Period", "text": "There exists a real, non-constant function $f$ such that: :$(1): \\quad f$ is periodic. :$(2): \\quad f$ does '''not''' have a period."}
+{"_id": "17346", "title": "Vector Augend plus Addend equals Augend implies Addend is Zero", "text": "Let $V$ be a vector space over a field $F$. Let $\\mathbf a, \\mathbf b \\in V$. Let $\\mathbf a + \\mathbf b = \\mathbf a$. Then: :$\\mathbf b = \\bszero$ where $\\bszero$ is the zero vector of $V$."}
+{"_id": "17347", "title": "Vector Space of All Mappings is Vector Space", "text": "Let $\\struct {K, +, \\circ}$ be a division ring. Let $\\struct {G, +_G, \\circ}_K$ be a $K$-vector space. Let $S$ be a set. Let $\\struct {G^S, +_G', \\circ}_R$ be the vector space of all mappings from $S$ to $G$. Then $\\struct {G^S, +_G', \\circ}_K$ is a $K$-vector space."}
+{"_id": "17348", "title": "Unitary Module of All Mappings is Unitary Module", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {G, +_G, \\circ}_R$ be a unitary $R$-module. Let $S$ be a set. Let $\\struct {G^S, +_G', \\circ}_R$ be the module of all mappings from $S$ to $G$. Then $\\struct {G^S, +_G', \\circ}_R$ is a unitary module."}
+{"_id": "17349", "title": "Finite Direct Product of Unitary Modules is Unitary Module", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {G_1, +_1, \\circ_1}_R, \\struct {G_2, +_2, \\circ_2}_R, \\ldots, \\struct {G_n, +_n, \\circ_n}_R$ be unitary $R$-modules. Let: :$\\displaystyle G = \\prod_{k \\mathop = 1}^n G_k$ be their direct product. Then $G$ is a unitary module."}
+{"_id": "17354", "title": "Nonconstant Periodic Function with no Period is Discontinuous Everywhere", "text": "Let $f$ be a real periodic function that does not have a period. Then $f$ is either constant or discontinuous everywhere."}
+{"_id": "17355", "title": "Module on Cartesian Product is Module", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $n \\in \\N_{>0}$. Let $\\struct {R^n, +, \\times}_R$ be the '''$R$-module $R^n$'''. Then $\\struct {R^n, +, \\times}_R$ is an $R$-module."}
+{"_id": "17356", "title": "Module on Cartesian Product of Ring with Unity is Unitary Module", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring with unity. Let $n \\in \\N_{>0}$. Let $\\struct {R^n, +, \\times}_R$ be the '''$R$-module $R^n$'''. Then $\\struct {R^n, +, \\times}_R$ is a unitary $R$-module."}
+{"_id": "17361", "title": "Ring of Polynomial Forms over Field is Vector Space", "text": "Let $\\struct {F, +, \\times}$ be a field whose unity is $1_F$. Let $F \\sqbrk X$ be the ring of polynomials over $F$. Then $F \\sqbrk X$ is an vector space over $F$."}
+{"_id": "17363", "title": "Norm is Complete Iff Equivalent Norm is Complete", "text": "Let $R$ be a division ring. Let $\\norm {\\,\\cdot\\,}_1$ and $\\norm {\\,\\cdot\\,}_2$ be equivalent norms on $R$. Then: :$\\struct {R,\\norm {\\,\\cdot\\,}_1}$ is complete {{iff}} $\\struct {R,\\norm {\\,\\cdot\\,}_2}$ is complete."}
+{"_id": "17364", "title": "Ring of Polynomial Forms over Field is Vector Space/Corollary", "text": "Let $S \\subseteq F \\sqbrk X$ denote the subset of $F \\sqbrk X$ defined as: :$S = \\set {\\mathbf x \\in F \\sqbrk X: \\map \\deg {\\mathbf x} < d}$ for some $d \\in \\Z_{>0}$. Then $S$ is an vector space over $F$."}
+{"_id": "17365", "title": "P-adic Norm of p-adic Number is Power of p", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $x \\in \\Q_p: x \\ne 0$. Then: :$\\exists v \\in \\Z: \\norm x_p = p^{-v}$"}
+{"_id": "17367", "title": "Borel-TIS inequality", "text": "Let $T$ be a topological space. Let $\\sequence {f_t}_{t \\mathop \\in T}$ be a centred (i.e. mean zero) Gaussian process on $T$, such that: :$\\norm f_T := \\sup_{t \\mathop \\in T} \\size {f_t}$ is almost surely finite. Let: :$\\sigma_T^2 := \\sup_{t \\mathop \\in T} \\operatorname E \\size {f_t}^2$ Then $\\map {\\operatorname E} {\\norm f_T}$ and $\\sigma_T$ are both finite, and, for each $u > 0$: :$\\map {\\operatorname P} {\\norm f_T > \\map {\\operatorname E} {\\norm f_T} + u} \\le \\map \\exp {\\dfrac {-u^2} {2 \\sigma_T^2} }$"}
+{"_id": "17369", "title": "Negation of Propositional Function in Two Variables", "text": "Let $\\map P {x, y}$ be a propositional function of two Variables. Then: :$\\neg \\forall x: \\exists y: \\map P {x, y} \\iff \\exists x: \\forall y: \\neg \\map P {x, y}$ That is: :''It is not the case that for all $x$ a value of $y$ can be found to satisfy $\\map P {x, y}$'' means the same thing as: :''There exists at least one value of $x$ such that for all $y$ it is not possible to satisfy $\\map P {x, y}$''"}
+{"_id": "17370", "title": "Valuation Ideal of P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. The valuation ideal induced by norm $\\norm {\\,\\cdot\\,}_p$ is the principal ideal: :$p \\Z_p = \\set {x \\in \\Q_p: \\norm x_p < 1}$"}
+{"_id": "17371", "title": "Integers are Arbitrarily Close to P-adic Integers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\Z_p$ be the $p$-adic integers. Let $x \\in \\Z_p$. Then for $n \\in \\N$ there exists unique $\\alpha \\in \\Z$: :$(1): \\quad 0 \\le \\alpha \\le p^n - 1$ :$(2): \\quad \\norm { x -\\alpha}_p \\le p^{-n}$"}
+{"_id": "17372", "title": "Integers are Dense in P-adic Integers", "text": "The integers $\\Z$ are dense in the metric space $\\struct{\\Z_p, d_p}$."}
+{"_id": "17373", "title": "P-adic Integer is Limit of Unique Coherent Sequence of Integers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\Z_p$ be the $p$-adic integers. Let $x \\in \\Z_p$. Then there exists a unique coherent sequence $\\sequence {\\alpha_n}$: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\alpha_n = x$"}
+{"_id": "17379", "title": "Linearly Independent Set is Contained in some Basis/Proof 1", "text": "Let $E$ be a vector space of $n$ dimensions. Let $H$ be a linearly independent subset of $E$. {{:Linearly Independent Set is Contained in some Basis}}"}
+{"_id": "17380", "title": "Linearly Independent Set is Contained in some Basis/Proof 2", "text": "Let $E$ be a vector space of $n$ dimensions. Let $H$ be a linearly independent subset of $E$. {{:Linearly Independent Set is Contained in some Basis}}"}
+{"_id": "17381", "title": "Vector Space over Division Subring is Vector Space", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose unity is $1_R$. Let $S$ be a division subring of $R$, such that $1_R \\in S$. The vector space $\\struct {R, +, \\circ_S}_S$ over $\\circ_S$ is a $S$-vector space."}
+{"_id": "17382", "title": "Vector Space on Field Extension is Vector Space", "text": "Let $\\struct {K, +, \\times}$ be a field. Let $L / K$ be a field extension over $K$. Let $\\struct {L, +, \\times}_K$ be the a vector space of $L$ over $K$. Then $\\struct {L, +, \\times}_K$ is a vector space."}
+{"_id": "17383", "title": "Sheldon Conjecture", "text": "There is only $1$ Sheldon prime, and that is $73$."}
+{"_id": "17384", "title": "Condition for Linear Divisor of Polynomial", "text": "Let $\\map P x$ be a polynomial in $x$. Let $a$ be a constant. Then $x - a$ is a divisor of $\\map P x$ {{iff}} $a$ is a root of $P$."}
+{"_id": "17386", "title": "Polynomial with Algebraic Number as Root is Multiple of Minimal Polynomial", "text": "Let $F$ be a field. Let $\\map P x$ be a polynomial in $F$. Let $z$ be a root of $\\map P x$. Then $\\map P x$ is a multiple of the minimal polynomial $\\map m x$ in $z$ over $F$."}
+{"_id": "17387", "title": "Simple Algebraic Field Extension consists of Polynomials in Algebraic Number", "text": "Let $F$ be a field. Let $\\theta \\in \\C$ be algebraic over $F$. Let $\\map F \\theta$ be the simple field extension of $F$ by $\\theta$. Then $\\map F \\theta$ consists of polynomials that can be written in the form $\\map f \\theta$, where $\\map f x$ is a polynomial over $F$."}
+{"_id": "17388", "title": "Element of Simple Algebraic Field Extension of Degree n is Polynomial in Algebraic Number of Degree Less than n", "text": "Let $F$ be a field. Let $\\theta \\in \\C$ be algebraic over $F$ of degree $n$. Let $\\map F \\theta$ be the simple field extension of $F$ by $\\theta$. Then any element of $\\map F \\theta$ can be written as $\\map f \\theta$, where $\\map f x$ is a polynomial over $F$ of degree at most $n - 1$."}
+{"_id": "17389", "title": "Degree of Simple Algebraic Field Extension equals Degree of Algebraic Number", "text": "Let $F$ be a field. Let $\\theta \\in \\C$ be algebraic over $F$ of degree $n$. Let $\\map F \\theta$ be the simple field extension of $F$ by $\\theta$. Then $\\map F \\theta$ is a finite extension of $F$ whose degree is: :$\\index {\\map F \\theta} F = n$"}
+{"_id": "17390", "title": "Degree of Element of Finite Field Extension divides Degree of Extension", "text": "Let $F$ be a field whose zero is $0$ and whose unity is $1$. Let $K / F$ be a finite field extension of degree $n$. Let $\\alpha \\in K$ be algebraic over $F$. Then the degree of $\\alpha$ is a divisor of $n$."}
+{"_id": "17391", "title": "Algebraic Element of Degree 3 is not Element of Field Extension of Degree Power of 2", "text": "Let $K / F$ be a finite field extension of degree $2^m$. Let $\\alpha \\in K$ be algebraic over $F$ with degree $3$. Then $\\alpha \\notin K$."}
+{"_id": "17392", "title": "Construction of Point in Cartesian Plane with Rational Coordinates", "text": "Let $\\CC$ be a Cartesian plane. Let $P = \\tuple {x, y}$ be a rational point in $\\CC$. Then $P$ is constructible using a compass and straightedge construction."}
+{"_id": "17393", "title": "Construction of Integer Multiple of Line Segment", "text": "Let $AB$ be a line segment in the plane. Let $AC$ be a line segment in the plane through a point $C$ Let $D$ be a point on $AC$ such that $AD = n AB$ for some $n \\in \\Z$. Then $AD$ is constructible using a compass and straightedge construction."}
+{"_id": "17394", "title": "Point in Plane is Constructible iff Coordinates in Extension of Degree Power of 2", "text": "Let $\\CC$ be a Cartesian plane. Let $S$ be a set of points in $\\CC$. Let $F$ be the smallest field containing all the coordinates of the points in $S$. Let $P = \\tuple {a, b}$ be a point in $\\CC$. Then: :$P$ is constructible from $S$ using a compass and straightedge construction {{iff}}: :the coordinates of $P$ are contained in a finite extension $K$ of $F$ whose degree is given by: ::$\\index K F = 2^m$ :for some $m \\in \\Z_{\\ge 0}$."}
+{"_id": "17395", "title": "Construction of Lattice Point in Cartesian Plane", "text": "Let $\\CC$ be a Cartesian plane. Let $P = \\tuple {a, b}$ be a lattice point in $\\CC$. Then $P$ is constructible using a compass and straightedge construction."}
+{"_id": "17396", "title": "Construction of Regular Heptagon by Compass and Straightedge Construction is Impossible", "text": "There is no compass and straightedge construction for a regular heptagon."}
+{"_id": "17397", "title": "Rational Numbers with Denominator Power of Two form Integral Domain", "text": "Let $\\Q$ denote the set of rational numbers. Let $S \\subseteq \\Q$ denote the set of set of rational numbers of the form $\\dfrac p q$ where $q$ is a power of $2$: :$S = \\set {\\dfrac p q: p \\in \\Z, q \\in \\set {2^m: m \\in \\Z_{\\ge 0} } }$ Then $\\struct {S, +, \\times}$ is an integral domain."}
+{"_id": "17402", "title": "Normal Subgroup of Order 25 in Group of Order 100", "text": "Let $G$ be a group of order $100$. Then $G$ has a normal subgroup of order $25$."}
+{"_id": "17403", "title": "Normal p-Subgroup contained in All Sylow p-Subgroups", "text": "Let $G$ be a finite group. Let $p$ be a prime number. Let $H$ be a normal subgroup of $G$ which is a $p$-group. Then $H$ is a subset of every Sylow $p$-subgroup of $G$."}
+{"_id": "17405", "title": "Group of Order p^2 q is not Simple", "text": "Let $p$ and $q$ be prime numbers such that $p \\ne q$. Let $G$ be a group of order $p^2 q$. Then $G$ is not simple."}
+{"_id": "17406", "title": "Simple Group of Order Less than 60 is Prime", "text": "Let $G$ be a simple group. Let $\\order G < 60$, where $\\order G$ denotes the order of $G$. Then $G$ is a prime group."}
+{"_id": "17407", "title": "Group of Order 42 has Normal Subgroup of Order 7", "text": "Let $G$ be of order $42$. Then $G$ has a normal subgroup of order $7$."}
+{"_id": "17408", "title": "Group of Order 54 has Normal Subgroup of Order 27", "text": "Let $G$ be of order $54$. Then $G$ has a normal subgroup of order $27$."}
+{"_id": "17409", "title": "Group of Order 40 has Normal Subgroup of Order 5", "text": "Let $G$ be of order $40$. Then $G$ has a normal subgroup of order $5$."}
+{"_id": "17410", "title": "Finite Group with One Sylow p-Subgroup per Prime Divisor is Isomorphic to Direct Product", "text": "Let $G$ be a finite group whose order is $n$ and whose identity element is $e$. Let $G$ be such that it has exactly $1$ Sylow $p$-subgroup for each prime divisor of $n$. Then $G$ is isomorphic to the internal direct product of all its Sylow $p$-subgroups."}
+{"_id": "17411", "title": "Number of Subgroups of Prime Power Order is Congruent to 1 modulo Prime", "text": "Let $G$ be a finite group whose order is $n$. Let $p$ be a prime number such that $p^k$ is a divisor of $n$. Then the number of subgroups of order $p^k$ is congruent to $1$ modulo $p$."}
+{"_id": "17412", "title": "Count of Distinct Homomorphisms between Additive Groups of Integers Modulo m", "text": "Let $m, n \\in \\Z_{>0}$ be (strictly) positive integers. Let $\\struct {\\Z_m, +}$ denote the additive group of integers modulo $m$. The number of distinct homomorphisms $\\phi: \\struct {\\Z_m, +} \\to \\struct {\\Z_n, +}$ is $\\gcd \\set {m, n}$."}
+{"_id": "17414", "title": "P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 1", "text": ":$\\forall n \\in \\N: \\alpha_{n + 1} \\equiv \\alpha_n \\pmod {p^{n + 1}}$"}
+{"_id": "17415", "title": "P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 3", "text": "$\\sequence {\\alpha_n}$ is a unique sequence satisfying properties $(1)$, $(2)$ and $(3)$ above."}
+{"_id": "17416", "title": "P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 2", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\alpha_n = x$"}
+{"_id": "17417", "title": "Group Types of Order Prime Squared", "text": "Let $p$ be a prime number. Let $G$ be a group of order $p^2$. Then $G$ is isomorphic either to $\\Z_{p^2}$ or to $\\Z_p \\times \\Z_p$, where $\\Z_p$ denotes the additive group of integers modulo $p$."}
+{"_id": "17418", "title": "Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers", "text": "$\\struct {G, \\circ}$ is isomorphic to the additive group of real numbers $\\struct {\\R, +}$."}
+{"_id": "17419", "title": "Open Set Characterization of Denseness/Analytic Basis", "text": "Let $\\mathcal B \\subseteq \\tau$ be an analytic basis for $\\tau$. Then $S$ is (everywhere) dense in $X$ {{iff}} every non-empty open set of $\\mathcal B$ contains an element of $S$."}
+{"_id": "17420", "title": "Open Set Characterization of Denseness/Open Ball", "text": "Let $\\struct{X, d}$ be a metric space. Let $\\tau_d$ be the topology induced by the metric $d$. Let $S \\subseteq X$. Then $S$ is (everywhere) dense in $\\struct{X, \\tau_d}$ {{iff}} every open ball contains an element of $S$."}
+{"_id": "17422", "title": "P-adic Integers is Metric Completion of Integers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $d$ be the subspace metric of the $p$-adic metric on the $p$-adic integers $\\Z_p$. Then $\\struct {\\Z_p, d}$ is the metric completion of the integers $\\Z$."}
+{"_id": "17423", "title": "P-adic Metric on P-adic Numbers is Non-Archimedean Metric", "text": "Let $p \\in \\N$ be a prime. Let $\\norm{\\,\\cdot\\,}_p: \\Q_p \\to \\R_{\\ge 0}$ be the $p$-adic norm on the $p$-adic numbers $\\Q_p$. Let $d_p$ be the $p$-adic metric on $\\Q_p$: :$\\forall x, y \\in \\Q_p: \\map {d_p} {x, y} = \\norm{x - y}_p$ Then $d_p$ is a non-Archimedean metric that extends the $p$-adic metric on the rationals $\\Q$ to $\\Q_p$."}
+{"_id": "17426", "title": "Order of Automorphism Group of Cyclic Group", "text": "Let $C_n$ denote the cyclic group of order $n$. Let $\\Aut {C_n}$ denote the automorphism group of $C_n$. Then: :$\\order {\\Aut {C_n} } = \\map \\phi n$ where: :$\\order {\\, \\cdot \\,}$ denotes the order of a group :$\\map \\phi n$ denotes the Euler $\\phi$ function."}
+{"_id": "17427", "title": "Order of Automorphism Group of Dihedral Group", "text": "Let $D_n$ denote the dihedral group of order $n$. Let $\\Aut {D_n}$ denote the automorphism group of $D_n$. Then: :$\\order {\\Aut {D_n} } = 2 \\map \\phi n$ where: :$\\order {\\, \\cdot \\,}$ denotes the order of a group :$\\map \\phi n$ is the Euler $\\phi$ function."}
+{"_id": "17429", "title": "Valuation Ring of P-adic Norm on Rationals/Corollary 1", "text": "The set of integers $\\Z$ is a subring of $\\OO$."}
+{"_id": "17430", "title": "Valuation Ring of P-adic Norm is Subring of P-adic Integers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\Z_p$ be the $p$-adic integers. Let $\\Z_{\\ideal p}$ be the induced valuation ring on $\\struct {\\Q,\\norm {\\,\\cdot\\,}_p}$. Then: :$(1): \\quad \\Z_{\\ideal p} = \\Q \\cap \\Z_p$. :$(2): \\quad \\Z_{\\ideal p}$ is a subring of $\\Z_p$."}
+{"_id": "17431", "title": "Valuation Ring of P-adic Norm is Subring of P-adic Integers/Corollary 1", "text": "The set of integers $\\Z$ is a subring of $\\Z_p$."}
+{"_id": "17435", "title": "Laplace Transform of 1", "text": "Let $f: \\R \\to \\R$ be the function defined as: :$\\forall t \\in \\R: \\map f t = 1$ Then the Laplace transform of $\\map f t$ is given by: :$\\laptrans {\\map f t} = \\dfrac 1 s$ for $\\map \\Re s > 0$."}
+{"_id": "17444", "title": "Primitive of Exponential of a x by Hyperbolic Sine of b x", "text": ":$\\displaystyle \\int e^{a x} \\sinh b x \\rd x = \\frac {e^{a x} \\paren {a \\sinh b x - b \\cosh b x} } {a^2 - b^2} + C$"}
+{"_id": "17446", "title": "Primitive of Exponential of a x by Hyperbolic Cosine of b x", "text": ":$\\displaystyle \\int e^{a x} \\cosh b x \\rd x = \\frac {e^{a x} \\paren {a \\cosh b x + b \\sinh b x} } {a^2 - b^2} + C$"}
+{"_id": "17449", "title": "Laplace Transform of Derivative/Discontinuity at t = 0", "text": "Let $f$ fail to be continuous at $t = 0$, but let: :$\\displaystyle \\lim_{t \\mathop \\to 0} \\map f t = \\map f {0^+}$ exist. Then $\\laptrans f$ exists for $\\map \\Re s > a$, and: :$\\laptrans {\\map {f'} t} = s \\laptrans {\\map f t} - \\map f {0^+}$"}
+{"_id": "17450", "title": "Laplace Transform of Derivative/Discontinuity at t = a", "text": "Let $f$ have a jump discontinuity at $t = a$. Then: :$\\laptrans {\\map {f'} t} = s \\laptrans {\\map f t} - \\map f 0 - e^{a s} \\paren {\\map f {a^+} - \\map f {a^-} }$"}
+{"_id": "17451", "title": "Laplace Transform of Integral", "text": ":$\\displaystyle \\laptrans {\\int_0^t \\map f u \\rd u} = \\dfrac {\\map F s} s$ wherever $\\laptrans f$ exists."}
+{"_id": "17452", "title": "Integral of Laplace Transform", "text": ":$\\displaystyle \\laptrans {\\dfrac {\\map f t} t} = \\int_s^{\\to \\infty} \\map F u \\rd u$ wherever $\\displaystyle \\lim_{t \\mathop \\to 0} \\dfrac {\\map f t} t$ and $\\laptrans f$ exist."}
+{"_id": "17453", "title": "Inclusion Mapping on Subring is Homomorphism", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\struct {S, +{\\restriction_S}, \\circ {\\restriction_S}}$ be a subring of $R$. Let $i_S: S \\to R$ be the inclusion mapping from $S$ to $R$. Then ${i_S}$ is a ring homomorphism."}
+{"_id": "17454", "title": "Inclusion Mapping on Subring is Monomorphism", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\struct {S, +{\\restriction_S}, \\circ {\\restriction_S} }$ be a subring of $R$. Let $i_S: S \\to R$ be the inclusion mapping from $S$ to $R$. Then $i_S$ is a ring monomorphism."}
+{"_id": "17455", "title": "Negative of Subring is Negative of Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. For each $x \\in R$ let $-x$ denote the ring negative of $x$ in $R$. Let $\\struct {S, + {\\restriction_S}, \\circ {\\restriction_S}}$ be a subring of $R$. For each $x \\in S$ let $\\mathbin \\sim x$ denote the ring negative of $x$ in $S$. Then: :$\\forall x \\in S: \\mathbin \\sim x = -x$"}
+{"_id": "17457", "title": "Limit to Infinity of Laplace Transform", "text": "Let $\\laptrans {\\map f t} = \\map F s$ denote the Laplace transform of the real function $f$. Then: :$\\displaystyle \\lim_{s \\mathop \\to \\infty} \\map F s = 0$"}
+{"_id": "17458", "title": "Initial Value Theorem of Laplace Transform", "text": "Let $\\laptrans {\\map f t} = \\map F s$ denote the Laplace transform of the real function $f$. Then: :$\\displaystyle \\lim_{t \\mathop \\to 0} \\map f t = \\lim_{s \\mathop \\to \\infty} s \\, \\map F s$ if those limits exist."}
+{"_id": "17459", "title": "Final Value Theorem of Laplace Transform", "text": "Let $\\laptrans {\\map f t} = \\map F s$ denote the Laplace transform of the real function $f$. Then: :$\\displaystyle \\lim_{t \\mathop \\to \\infty} \\map f t = \\lim_{s \\mathop \\to 0} s \\, \\map F s$ if those limits exist."}
+{"_id": "17462", "title": "Evaluation of Integral using Laplace Transform", "text": "Let $\\laptrans {\\map f t} = \\map F s$ denote the Laplace transform of the real function $f$. Then: :$\\displaystyle \\int_0^{\\to \\infty} \\map f t \\rd t = \\map F 0$ assuming the integral is convergent."}
+{"_id": "17464", "title": "Valuation Ring of Non-Archimedean Division Ring is Clopen", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a non-Archimedean normed division ring with zero $0_R$. Let $\\mathcal O$ be valuation ring induced by $\\norm{\\,\\cdot\\,}$. Then $\\mathcal O$ is a both open and closed in the metric induced by $\\norm{\\,\\cdot\\,}$."}
+{"_id": "17465", "title": "Valuation Ring of Non-Archimedean Division Ring is Clopen/Corollary 1", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Then the $p$-adic integers $\\Z_p$ is both open and closed in the $p$-adic metric."}
+{"_id": "17467", "title": "Series Expansion of Bessel Function of the First Kind", "text": "{{begin-eqn}} {{eqn | l = \\map {J_n} x       | r = \\dfrac {x^n} {2^n \\, \\map \\Gamma {n + 1} } \\paren {1 - \\dfrac {x^2} {2 \\paren {2 n + 2} } + \\dfrac {x^4} {2 \\times 4 \\paren {2 n + 2} \\paren {2 n + 4} } - \\cdots}       | c =  }} {{eqn | r = \\sum_{k \\mathop = 0}^\\infty \\dfrac {\\paren {-1}^k} {k! \\, \\map \\Gamma {n + k + 1} } \\paren {\\dfrac x 2}^{n + 2 k}       | c =  }} {{end-eqn}}"}
+{"_id": "17468", "title": "Recurrence Formula for Bessel Function of the First Kind", "text": "Let $\\map {J_n} x$ denote the Bessel function of the first kind of order $n$. Then: :$\\map {J_{n + 1} } x = \\dfrac {2 n} x \\map {J_n} x - \\map {J_{n - 1} } x$ And:  :$\\map {J_{n + 1} } x = -2 \\map {J_n'} x + \\map {J_{n - 1} } x$ {{refactor|page per result}}"}
+{"_id": "17469", "title": "Derivative of x^n by Bessel Function of the First Kind of Order n of x", "text": "Let $\\map {J_n} x$ denote the Bessel function of the first kind of order $n$. Then: :$\\map {\\dfrac \\d {\\d x} } {x^n \\map {J_n} x} = x^n \\map {J_{n - 1} } x$"}
+{"_id": "17470", "title": "Generating Function for Bessel Function of the First Kind of Order n of x", "text": "Let $\\map {J_n} x$ denote the Bessel function of the first kind of order $n$. Then: :$\\map \\exp {\\dfrac {x \\paren {t - \\frac 1 t} } 2} = \\displaystyle \\sum_{n \\mathop = -\\infty}^\\infty \\map {J_n} x t^n$"}
+{"_id": "17471", "title": "Series Expansion of Bessel Function of the First Kind/Negative Index", "text": "{{begin-eqn}} {{eqn | l = \\map {J_{-n} } x       | r = \\dfrac {x^{-n} } {2^{-n} \\, \\map \\Gamma {1 - n} } \\paren {1 - \\dfrac {x^2} {2 \\paren {2 - 2 n} } + \\dfrac {x^4} {2 \\times 4 \\paren {2 - 2 n} \\paren {4 - 2 n} } - \\cdots}       | c =  }} {{eqn | r = \\sum_{k \\mathop = 0}^\\infty \\dfrac {\\paren {-1}^k} {k! \\, \\map \\Gamma {k + 1 - n} } \\paren {\\dfrac x 2}^{2 k - n}       | c =  }} {{end-eqn}}"}
+{"_id": "17472", "title": "Bessel Function of the First Kind for Imaginary Argument", "text": "Let $\\map {J_n} x$ denote the Bessel function of the first kind of order $n$. Then: :$\\map {J_n } {i x} = i^{-n} \\, \\map {I_n} x$ where: :$i$ denotes the imaginary unit :$\\map {I_n} x$ denotes the modified Bessel function of the first kind of order $n$."}
+{"_id": "17475", "title": "Integral to Infinity of Shifted Dirac Delta Function by Continuous Function", "text": "Let $\\map \\delta x$ denote the Dirac delta function. Let $g$ be a continuous real function. Let $a \\in \\R_{\\ge 0}$ be a positive real number. Then: :$\\displaystyle \\int_0^{+ \\infty} \\map \\delta {x - a} \\, \\map g x \\rd x = \\map g a$"}
+{"_id": "17476", "title": "Function which is Zero except on Countable Set of Points is Null", "text": "Let $S \\subseteq \\R$ be a subset of $\\R$ such that $S$ is countable, either finite or countably infinite. Let $f: \\R \\to \\R$ be a real function such that: :$\\forall x \\in \\R \\setminus S: \\map f x = 0$ That is, except perhaps for the elements of $S$, the value of $f$ is zero. Then $f$ is a null function."}
+{"_id": "17477", "title": "Laplace Transform of Bessel Function of the First Kind of Order Zero", "text": "Let $J_0$ denote the Bessel function of the first kind of order $0$. Then the Laplace transform of $J_0$ is given as: :$\\laptrans {\\map {J_0} t} = \\dfrac 1 {\\sqrt {s^2 + 1} }$"}
+{"_id": "17478", "title": "Laplace Transform of Bessel Function of the First Kind", "text": "Let $J_n$ denote the Bessel function of the first kind of order $n$. Then the Laplace transform of $J_n$ is given as: :$\\laptrans {\\map {J_n} {a t} } = \\dfrac {\\paren {\\sqrt {s^2 + a^2} - s}^n} {a^n \\sqrt {s^2 + a^2} }$"}
+{"_id": "17479", "title": "Laplace Transform of Sine of Root", "text": ":$\\laptrans {\\sin \\sqrt t} = \\dfrac {\\sqrt \\pi} {2 s^{3/2} } \\map \\exp {-\\dfrac 1 {4 s} }$ where $\\laptrans f$ denotes the Laplace transform of the function $f$."}
+{"_id": "17480", "title": "Laplace Transform of Cosine of Root over Root", "text": ":$\\laptrans {\\dfrac {\\cos \\sqrt t} {\\sqrt t} } = \\sqrt {\\dfrac \\pi s} \\, \\map \\exp {-\\dfrac 1 {4 s} }$ where $\\laptrans f$ denotes the Laplace transform of the function $f$."}
+{"_id": "17482", "title": "Laplace Transform of Error Function of Root", "text": ":$\\laptrans {\\map \\erf {\\sqrt t} } = \\dfrac 1 {s \\sqrt {s + 1} }$ where: :$\\laptrans f$ denotes the Laplace transform of the function $f$ :$\\erf$ denotes the error function"}
+{"_id": "17483", "title": "Laplace Transform of Sine Integral Function", "text": ":$\\laptrans {\\map \\Si t} = \\dfrac 1 s \\arctan \\dfrac 1 s$ where: :$\\laptrans f$ denotes the Laplace transform of the function $f$ :$\\Si$ denotes the sine integral function"}
+{"_id": "17484", "title": "Laplace Transform of Cosine Integral Function", "text": ":$\\laptrans {\\map \\Ci t} = \\dfrac {\\map \\ln {s^2 + 1} } {2 s}$ where: :$\\laptrans f$ denotes the Laplace transform of the function $f$ :$\\Ci$ denotes the cosine integral function."}
+{"_id": "17486", "title": "Laplace Transform of Heaviside Step Function", "text": "Let $\\map {u_c} t$ denote the Heaviside step function: :$\\map {u_c} t = \\begin{cases} 1 & : t > c \\\\ 0 & : t < c \\end{cases}$ The Laplace transform of $\\map {u_c} t$ is given by: :$\\laptrans {\\map {u_c} t} = \\dfrac {e^{-s c} } s$ for $\\map \\Re s > c$."}
+{"_id": "17487", "title": "Laplace Transform of Dirac Delta Function", "text": "Let $\\map \\delta t$ denote the Dirac delta function. The Laplace transform of $\\map \\delta t$ is given by: :$\\laptrans {\\map \\delta t} = 1$"}
+{"_id": "17489", "title": "Laplace Transform of Null Function", "text": "Let $\\mathcal N: \\R \\to \\R$ be a null function. The Laplace transform of $\\map {\\mathcal N} t$ is given by: :$\\laptrans {\\map {\\mathcal N} t} = 0$"}
+{"_id": "17491", "title": "Laplace Transform of Function of t minus a/Proof 1", "text": "Let $f$ be a function such that $\\laptrans f$ exists. Let $\\laptrans {\\map f t} = \\map F s$ denote the Laplace transform of $f$. Let $a \\in \\C$ or $\\R$ be constant. {{:Laplace Transform of Function of t minus a}}"}
+{"_id": "17492", "title": "Laplace Transform of Function of t minus a/Proof 2", "text": "Let $f$ be a function such that $\\laptrans f$ exists. Let $\\laptrans {\\map f t} = \\map F s$ denote the Laplace transform of $f$. Let $a \\in \\C$ or $\\R$ be constant. {{:Laplace Transform of Function of t minus a}}"}
+{"_id": "17493", "title": "Laplace Transform of Sine of t over t", "text": "Let $\\sin$ denote the real sine function. Let $\\laptrans f$ denote the Laplace transform of a real function $f$. Then: :$\\laptrans {\\dfrac {\\sin t} t} = \\arctan \\dfrac 1 s$"}
+{"_id": "17497", "title": "Integral to Infinity of Function over Argument", "text": ":$\\displaystyle \\int_0^\\infty {\\dfrac {\\map f t} t} = \\int_0^{\\to \\infty} \\map F u \\rd u$ provided the integrals converge."}
+{"_id": "17503", "title": "Integral to Infinity of Exponential of -t^2", "text": ":$\\displaystyle \\int_0^\\infty \\map \\exp {-t^2} \\rd t = \\dfrac {\\sqrt \\pi} 2$"}
+{"_id": "17504", "title": "Laplace Transform of Real Power", "text": "Let $n$ be a constant real number such that $n > -1$ Let $f: \\R \\to \\R$ be the real function defined as: :$\\map f t = t^n$ Then $f$ has a Laplace transform given by: {{begin-eqn}} {{eqn\t| l = \\laptrans {\\map f t} \t| r = \\int_0^\\infty e^{-s t} t^n \\rd t }} {{eqn\t| r = \\frac {\\map \\Gamma {n + 1} } {s^{n + 1} } }} {{end-eqn}} where $\\Gamma$ denotes the gamma function."}
+{"_id": "17505", "title": "Laplace Transform of Reciprocal of Square Root", "text": ":$\\laptrans {\\dfrac 1 {\\sqrt t} } = \\sqrt {\\dfrac \\pi s}$ where $\\laptrans f$ denotes the Laplace transform of the real function $f$."}
+{"_id": "17506", "title": "Gamma Function of 3 over 2", "text": ":$\\map \\Gamma {\\dfrac 3 2} = \\dfrac {\\sqrt \\pi} 2$"}
+{"_id": "17507", "title": "Gamma Function of Minus 3 over 2", "text": ":$\\map \\Gamma {-\\dfrac 3 2} = \\dfrac {4 \\sqrt \\pi} 3$"}
+{"_id": "17509", "title": "Gamma Function of Zero", "text": ":$\\map \\Gamma 0$ is not defined."}
+{"_id": "17510", "title": "Gamma Function of Minus 1", "text": ":$\\map \\Gamma {-1}$ is not defined."}
+{"_id": "17511", "title": "Gamma Function of Minus 2", "text": ":$\\map \\Gamma {-2}$ is not defined."}
+{"_id": "17525", "title": "Integral to Infinity of Bessel Function of First Kind order Zero", "text": ":$\\displaystyle \\int_0^\\infty \\map {J_0} t \\rd t = 1$ where $J_0$ denotes theBessel function of the first kind of order $0$."}
+{"_id": "17526", "title": "Integral to Infinity of e^-t by Error Function of Root t", "text": ":$\\displaystyle \\int_0^\\infty e^{-t} \\erf \\sqrt t \\rd t = \\dfrac {\\sqrt 2} 2$"}
+{"_id": "17531", "title": "Second Derivative of Laplace Transform", "text": "Let $f: \\R \\to \\R$ or $\\R \\to \\C$ be a continuous function, twice differentiable on any closed interval $\\closedint 0 a$. Let $\\laptrans f = F$ denote the Laplace transform of $f$. Then, everywhere that $\\dfrac {\\d^2} {\\d s^2} \\laptrans f$ exists: :$\\dfrac {\\d^2} {\\d s^2} \\laptrans {\\map f t} = \\laptrans {t^2 \\, \\map f t}$"}
+{"_id": "17532", "title": "Laplace Transform of Multiple Integral", "text": ":$\\displaystyle \\laptrans {\\underbrace {\\int_0^t \\dotsm \\int_0^t}_{\\text {$n$ times} } \\map f u \\rd u^n} = \\dfrac {\\map F s} {s^n}$ wherever $\\laptrans f$ exists."}
+{"_id": "17537", "title": "Integral of Reciprocal is Divergent/Unbounded Above", "text": ":$\\displaystyle \\int_1^n \\frac {\\d x} x \\to +\\infty$ as $n \\to + \\infty$"}
+{"_id": "17538", "title": "Integral of Reciprocal is Divergent/To Zero", "text": ":$\\displaystyle \\int_\\gamma^1 \\frac {\\d x} x \\to -\\infty$ as $\\gamma \\to 0^+$"}
+{"_id": "17541", "title": "Convergence of P-Series/Lemma", "text": "Let $p = x + i y$ be a complex number where $x, y \\in \\R$ such that: :$x > 0$ :$x \\ne 1$ Then: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\frac 1 {n^x}$ converges {{iff}} $\\displaystyle \\lim_{P \\mathop \\to \\infty} \\dfrac {P^{1 - x} } {1 - x}$ converges."}
+{"_id": "17542", "title": "Integral to Infinity of Reciprocal of Power of x", "text": "The improper integral :$\\displaystyle \\int_1^\\infty \\dfrac {\\d t} {t^x}$ exists {{iff}} $x > 1$."}
+{"_id": "17543", "title": "Convergence of P-Series/Real", "text": "Let $p \\in \\R$ be a real number. Then the $p$-series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty n^{-p}$ is convergent {{iff}} $p > 1$."}
+{"_id": "17546", "title": "Real Number Ordering is Transitive", "text": "Let $a, b, c \\in \\R$ such that $a > b$ and $b > c$. Then: :$a > c$"}
+{"_id": "17548", "title": "Real Number Ordering is Compatible with Multiplication/Positive Factor", "text": ":$\\forall a, b, c \\in \\R: a < b \\land c > 0 \\implies a c < b c$"}
+{"_id": "17549", "title": "Real Number Ordering is Compatible with Multiplication/Negative Factor", "text": ":$\\forall a, b, c \\in \\R: a < b \\land c < 0 \\implies a c > b c$"}
+{"_id": "17550", "title": "Rational Power of Product of Real Numbers", "text": "Let $r, s \\in \\R_{> 0}$ be (strictly) positive real numbers. <onlyinclude> Let $x \\in \\Q$ be a rational number. Let $r^x$ be defined as $r$ to the power of $x$. Then: :$\\paren {r s}^x = r^x s^x$"}
+{"_id": "17551", "title": "Sign of Quadratic Function Between Roots", "text": "Let $a \\in \\R_{>0}$ be a (strictly) positive real number. Let $\\alpha$ and $\\beta$, where $\\alpha < \\beta$, be the roots of the quadratic function: :$\\map Q x = a x^2 + b x + c$ whose discriminant $b^2 - 4 a c$ is (strictly) positive. Then: :$\\begin {cases} \\map Q x < 0 & : \\text {when $\\alpha < x < \\beta$} \\\\ \\map Q x > 0 & : \\text {when $x < \\alpha$ or $x > \\beta$} \\end {cases}$"}
+{"_id": "17552", "title": "Minimum Value of Real Quadratic Function", "text": "Let $a \\in \\R_{>0}$ be a (strictly) positive real number. Consider the quadratic function: :$\\map Q x = a x^2 + b x + c$ $\\map Q x$ achieves a minimum at $x = -\\dfrac b {2 a}$, at which point $\\map Q x = c - \\dfrac {b^2} {4 a}$."}
+{"_id": "17553", "title": "Number of Type Rational r plus s Root 2 is Irrational", "text": "Let $r, s \\in \\Q$ be rational numbers. Then $r + s \\sqrt 2$ is irrational."}
+{"_id": "17554", "title": "Roots of Quadratic with Rational Coefficients of form r plus s Root 2", "text": "Consider the quadratic equation: :$(1): \\quad a^2 x + b x + c = 0$ where $a, b, c$ are rational. Let $\\alpha = r + s \\sqrt 2$ be one of the roots of $(1)$. Then $\\beta = r - s \\sqrt 2$ is the other root of $(1)$."}
+{"_id": "17557", "title": "Supremum is not necessarily Greatest Element", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $T$ admit a supremum in $S$. Then the supremum of $T$ in $S$ is not necessarily the greatest element of $T$."}
+{"_id": "17561", "title": "Closed Interval Defined by Absolute Value", "text": ":$\\set {x \\in \\R: \\size {\\xi - x} \\le \\delta} = \\closedint {\\xi - \\delta} {\\xi + \\delta}$ where $\\closedint {\\xi - \\delta} {\\xi + \\delta}$ is the closed real interval between $\\xi - \\delta$ and $\\xi + \\delta$."}
+{"_id": "17562", "title": "Open Interval Defined by Absolute Value", "text": ":$\\set {x \\in \\R: \\size {\\xi - x} < \\delta} = \\openint {\\xi - \\delta} {\\xi + \\delta}$ where $\\openint {\\xi - \\delta} {\\xi + \\delta}$ is the open real interval between $\\xi - \\delta$ and $\\xi + \\delta$."}
+{"_id": "17565", "title": "Distance from Subset of Real Numbers to Element", "text": ":$x \\in S \\implies \\map d {x, S} = 0$"}
+{"_id": "17566", "title": "Distance from Subset of Real Numbers to Supremum", "text": "Let $S$ be bounded above such that $\\xi = \\sup S$. Then: :$\\map d {\\xi, S} = 0$"}
+{"_id": "17567", "title": "Distance from Subset of Real Numbers to Infimum", "text": "Let $S$ be bounded below such that $\\xi = \\inf S$. Then: :$\\map d {\\xi, S} = 0$"}
+{"_id": "17568", "title": "Real Number at Distance Zero from Closed Real Interval is In Interval", "text": "Let $I \\subseteq \\R$ be a closed real interval. Then: :$\\map d {x, I} = 0 \\implies x \\in I$"}
+{"_id": "17569", "title": "Existence of Real Number at Distance Zero from Open Real Interval not in Interval", "text": "Let $I \\subseteq \\R$ be an open real interval such that $I \\ne \\O$ and $I \\ne \\R$. Then: :$\\exists x \\notin I: \\map d {x, I} = 0$"}
+{"_id": "17570", "title": "Distance from Subset of Real Numbers to Element/Proof 1", "text": "Let $S$ be a subset of the set of real numbers $\\R$. Let $x \\in \\R$ be a real number. Let $\\map d {x, S}$ be the distance between $x$ and $S$. Then: {{:Distance from Subset of Real Numbers to Element}}"}
+{"_id": "17571", "title": "Distance from Subset of Real Numbers to Element/Proof 2", "text": "Let $S$ be a subset of the set of real numbers $\\R$. Let $x \\in \\R$ be a real number. Let $\\map d {x, S}$ be the distance between $x$ and $S$. Then: {{:Distance from Subset of Real Numbers to Element}}"}
+{"_id": "17572", "title": "Distance from Subset of Real Numbers to Supremum/Proof 1", "text": "Let $S$ be a subset of the set of real numbers $\\R$. Let $x \\in \\R$ be a real number. Let $\\map d {x, S}$ be the distance between $x$ and $S$. {{:Distance from Subset of Real Numbers to Supremum}}"}
+{"_id": "17573", "title": "Distance from Subset of Real Numbers to Supremum/Proof 2", "text": "Let $S$ be a subset of the set of real numbers $\\R$. Let $x \\in \\R$ be a real number. Let $\\map d {x, S}$ be the distance between $x$ and $S$. {{:Distance from Subset of Real Numbers to Supremum}}"}
+{"_id": "17574", "title": "Distance from Subset of Real Numbers to Infimum/Proof 1", "text": "Let $S$ be a subset of the set of real numbers $\\R$. Let $x \\in \\R$ be a real number. Let $\\map d {x, S}$ be the distance between $x$ and $S$. {{:Distance from Subset of Real Numbers to Infimum}}"}
+{"_id": "17575", "title": "Distance from Subset of Real Numbers to Infimum/Proof 2", "text": "Let $S$ be a subset of the set of real numbers $\\R$. Let $x \\in \\R$ be a real number. Let $\\map d {x, S}$ be the distance between $x$ and $S$. {{:Distance from Subset of Real Numbers to Infimum}}"}
+{"_id": "17576", "title": "Infimum of Set of Reciprocals of Positive Integers", "text": "Let $S$ be the subset of the set of real numbers defined as: :$S = \\set {\\dfrac 1 n: n \\in \\Z_{>0} }$ Then: :$\\inf S = 0$ where $\\inf S$ denotes the infimum of $S$."}
+{"_id": "17579", "title": "Set of Numbers of form n - 1 over n is Bounded Above", "text": "Let $S$ be the subset of the set of real numbers $\\R$ defined as: :$S = \\set {\\dfrac {n - 1} n: n \\in \\Z_{>0} }$ $S$ is bounded above with supremum $1$. $S$ has no greatest element."}
+{"_id": "17580", "title": "Set of Rational Numbers Strictly between Zero and One has no Greatest or Least Element", "text": "Let $S \\subseteq \\Q$ be the subset of the set of rational numbers defined as: :$S = \\set {r \\in \\Q: 0 < r < 1}$ Then $S$ has no greatest or smallest element. However, $S$ has a supremum $1$ and an infimum $0$."}
+{"_id": "17581", "title": "Between two Real Numbers exists Irrational Number", "text": "Let $a, b \\in \\R$ be real numbers where $a < b$.  Then there exists an irrational number $\\xi \\in \\R \\setminus \\Q$ such that: :$a < \\xi < b$"}
+{"_id": "17583", "title": "Sequence of Powers of Reciprocals is Null Sequence/Real Index", "text": "Let $r \\in \\R_{>0}$ be a strictly positive real number. Let $\\sequence {x_n}$ be the sequence in $\\R$ defined as: : $x_n = \\dfrac 1 {n^r}$ Then $\\sequence {x_n}$ is a null sequence."}
+{"_id": "17584", "title": "Odd Order Derivative of Even Function Vanishes at Zero", "text": "Let $X$ be a symmetric subset of $\\R$ containing $0$.  Let $n$ be a positive integer.  Let $f:X \\to \\R$ be an even function.  Let $f$ be at least $\\paren{2 n + 1}$-times differentiable. Then: :$\\map {f^{\\paren {2 n + 1} } } 0 = 0$"}
+{"_id": "17585", "title": "Reciprocal of Null Sequence/Corollary", "text": ":$x_n \\to \\infty$ as $n \\to \\infty$ {{iff}} $\\size {\\dfrac 1 {x_n} } \\to 0$ as $n \\to \\infty$"}
+{"_id": "17587", "title": "Index of Subsequence not Less than its Index", "text": "Let $\\sequence {x_n}_{n \\mathop \\ge 1}$ be a sequence in a set $S$. Let $\\sequence {x_{n_r} }$ be a subsequence of $\\sequence {x_n}$. Then: :$\\forall n \\in \\N_{>0}: n_r \\ge r$"}
+{"_id": "17591", "title": "Even Order Derivative of Odd Function Vanishes at Zero", "text": "Let $X$ be a symmetric subset of $\\R$ containing $0$.  Let $n$ be a positive integer.  Let $f:X \\to \\R$ be an odd function.  Let $f$ be at least $\\paren {2 n}$-times differentiable. Then:  :$\\map {f^{\\paren {2 n} } } 0 = 0$"}
+{"_id": "17593", "title": "Geometric Mean of Reciprocals is Reciprocal of Geometric Mean", "text": "Let $x_1, x_2, \\ldots, x_n \\in \\R_{> 0}$ be strictly positive real numbers. Let $G_n$ denote the geometric mean of $x_1, x_2, \\ldots, x_n$. Let ${G_n}'$ denote the geometric mean of their reciprocals $\\dfrac 1 {x_1}, \\dfrac 1 {x_2}, \\ldots, \\dfrac 1 {x_n}$. Then: :${G_n}' = \\dfrac 1 {G_n}$"}
+{"_id": "17594", "title": "Geometric Mean is Never Less than Harmonic Mean", "text": "Let $x_1, x_2, \\ldots, x_n \\in \\R_{> 0}$ be strictly positive real numbers. Let $G_n$ be the geometric mean of $x_1, x_2, \\ldots, x_n$. Let $H_n$ be the harmonic mean of $x_1, x_2, \\ldots, x_n$. Then $G_n \\ge H_n$."}
+{"_id": "17595", "title": "Harmonic Mean of two Real Numbers is Between them", "text": "Let $a, b \\in \\R_{\\ne 0}$ be non-zero real numbers such that $a < b$. Let $\\map H {a, b}$ denote the narmonic mean of $a$ and $b$. Then: :$a < \\map H {a, b} < b$"}
+{"_id": "17596", "title": "Arithmetic Mean of two Real Numbers is Between them", "text": "Let $a, b \\in \\R_{\\ne 0}$ be non-zero real numbers such that $a < b$. Let $\\map A {a, b}$ denote the narmonic mean of $a$ and $b$. Then: :$a < \\map A {a, b} < b$"}
+{"_id": "17597", "title": "Raw Moment of Bernoulli Distribution", "text": "Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$. Let $n$ be a strictly positive integer.  Then the $n$th raw moment $\\expect {X^n}$ of $X$ is given by:  :$\\expect {X^n} = p$"}
+{"_id": "17599", "title": "Equivalence of Definitions of Local Basis", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $x$ be an element of $S$. {{TFAE|def = Local Basis}}"}
+{"_id": "17600", "title": "Equivalence of Definitions of Local Basis/Local Basis for Open Sets Implies Neighborhood Basis of Open Sets", "text": "Let $\\BB$ be a set of open neighborhoods of $x$ such that: :$\\forall U \\in \\tau: x \\in U \\implies \\exists H \\in \\BB: H \\subseteq U$"}
+{"_id": "17601", "title": "Equivalence of Definitions of Local Basis/Neighborhood Basis of Open Sets Implies Local Basis for Open Sets", "text": "Let $\\BB$ be a set of open neighborhoods of $x$ such that: :every neighborhood of $x$ contains a set in $\\BB$."}
+{"_id": "17603", "title": "Arcsine of Zero is Zero", "text": ":$\\arcsin 0 = 0$ where $\\arcsin$ is the arcsine function."}
+{"_id": "17604", "title": "Arcsine of One is Half Pi", "text": ":$\\arcsin 1 = \\dfrac \\pi 2$ where $\\arcsin$ is the arcsine function."}
+{"_id": "17605", "title": "Real Sequence (1 + x over n)^n is Convergent", "text": "The sequence $\\sequence {s_n}$ defined as: :$s_n = \\paren {1 + \\dfrac x n}^n$ is convergent."}
+{"_id": "17606", "title": "Raw Moment of Poisson Distribution", "text": "Let $X$ be a discrete random variable with the Poisson distribution with parameter $\\lambda$. Let $n$ be a strictly positive integer.  Then the $n$th raw moment $\\expect {X^n}$ of $X$ is given by:  :$\\displaystyle \\expect {X^n} = \\sum_{k \\mathop = 0}^n \\lambda^k {n \\brace k}$ where $\\displaystyle {n \\brace k}$ is a Stirling number of the second kind."}
+{"_id": "17607", "title": "Raw Moment of Exponential Distribution", "text": "Let $X$ be a continuous random variable of the exponential distribution with parameter $\\beta$ for some $\\beta \\in \\R_{> 0}$ Let $n$ be a strictly positive integer.  Then the $n$th raw moment $\\expect {X^n}$ of $X$ is given by:  :$\\expect {X^n} = n! \\beta^n$"}
+{"_id": "17608", "title": "Central Moment of Exponential Distribution", "text": "Let $X$ be a continuous random variable of the exponential distribution with parameter $\\beta$ for some $\\beta \\in \\R_{> 0}$ Let $n$ be a strictly positive integer.  Then the $n$th central moment $\\mu_n$ of $X$ is given by:  :$\\displaystyle \\mu_n = n! \\beta^n \\sum_{k \\mathop = 0}^n \\frac {\\paren {-1}^k} {k!}$"}
+{"_id": "17610", "title": "Product of Limits of Real Sequences (1 + x over n)^n and (1 - x over n)^n equals 1", "text": "Let $\\sequence {a_n}$ be the sequence defined as: :$a_n = \\paren {1 + \\dfrac x n}^n$ Let $\\sequence {b_n}$ be the sequence defined as: :$b_n = \\paren {1 - \\dfrac x n}^n$ Then the product of the limits of $\\sequence {a_n}$ and $\\sequence {b_n}$ equals $1$"}
+{"_id": "17611", "title": "Farey Sequence is not Convergent", "text": "Consider the Farey sequence: :$F = \\dfrac 1 2, \\dfrac 1 3, \\dfrac 2 3, \\dfrac 1 4, \\dfrac 2 4, \\dfrac 3 4, \\dfrac 1 5, \\dfrac 2 5, \\dfrac 3 5, \\dfrac 4 5, \\dfrac 1 6, \\ldots$ $F$ is not convergent."}
+{"_id": "17613", "title": "Skewness of Exponential Distribution", "text": "Let $X$ be a continuous random variable of the exponential distribution with parameter $\\beta$ for some $\\beta \\in \\R_{> 0}$ Then the skewness $\\gamma_1$ of $X$ is equal to $2$."}
+{"_id": "17614", "title": "Excess Kurtosis of Geometric Distribution", "text": "Let $X$ be a discrete random variable with the geometric distribution with parameter $p$. Then the excess kurtosis $\\gamma_2$ of $X$ is given by:  :$\\gamma_2 = 6 + \\dfrac {p^2} {1 - p}$"}
+{"_id": "17615", "title": "Excess Kurtosis of Poisson Distribution", "text": "Let $X$ be a discrete random variable with a Poisson distribution with parameter $\\lambda$. Then the excess kurtosis $\\gamma_2$ of $X$ is given by:  :$\\gamma_2 = \\dfrac 1 \\lambda$"}
+{"_id": "17618", "title": "Expectation of Chi-Squared Distribution", "text": "Let $n$ be a strictly positive integer.  Let $X \\sim \\chi^2_n$ where $\\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the expectation of $X$ is given by:  :$\\expect X = n$"}
+{"_id": "17619", "title": "Variance of Chi-Squared Distribution", "text": "Let $n$ be a strictly positive integer.  Let $X \\sim \\chi^2_n$ where $\\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the variance of $X$ is given by:  :$\\var X = 2 n$"}
+{"_id": "17620", "title": "Skewness of Chi-Squared Distribution", "text": "Let $n$ be a strictly positive integer.  Let $X \\sim \\chi^2_n$ where $\\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the skewness $\\gamma_1$ of $X$ is given by:  :$\\gamma_1 = \\sqrt{\\dfrac 8 n}$"}
+{"_id": "17622", "title": "Raw Moment of Chi-Squared Distribution", "text": "Let $n$ and $m$ be strictly positive integers.  Let $X \\sim \\chi^2_n$ where $\\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the $m$th raw moment $\\expect {X^m}$ of $X$ is given by:  :$\\displaystyle \\expect {X^m} = \\prod_{k \\mathop = 0}^{m - 1} \\paren {n + 2 k}$"}
+{"_id": "17623", "title": "Sequence of Square Roots of Natural Numbers is not Cauchy", "text": "Let $\\sequence {x_n}_{n \\mathop \\in \\N_{>0} }$ be the sequence in $\\R$ defined as: :$x_n = \\sqrt n$ Then, despite the fact that from Difference Between Adjacent Square Roots Converges: :$\\size {\\sqrt {n + 1} - \\sqrt n} \\to 0$ as $n \\to \\infty$ it is not the case that $\\sequence {x_n}$ is a Cauchy sequence."}
+{"_id": "17624", "title": "Square of Standard Gaussian Random Variable has Chi-Squared Distribution", "text": "Let $X \\sim \\Gaussian 0 1$ where $\\Gaussian 0 1$ is the standard Gaussian distribution.  Then $X^2 \\sim \\chi^2_1$ where $\\chi^2_1$ is the chi-square distribution with $1$ degree of freedom."}
+{"_id": "17625", "title": "Sum of Chi-Squared Random Variables", "text": "Let $n_1, n_2, \\ldots, n_k$ be strictly positive integers which sum to $N$. Let $X_i \\sim \\chi^2_{n_i}$ for $1 \\le i \\le k$, where $\\chi^2_{n_i}$ is the chi-squared distribution with $n_i$ degrees of freedom.  Then:  :$\\displaystyle X = \\sum_{i \\mathop = 1}^k X_i \\sim \\chi^2_N$"}
+{"_id": "17626", "title": "Moment Generating Function of Chi-Squared Distribution", "text": "Let $n$ be a strictly positive integer.  Let $X \\sim \\chi^2_n$ where $\\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the moment generating function of $X$, $M_X$, is given by:  :$\\displaystyle \\map {M_X} t = \\begin{cases} \\paren {1 - 2 t}^{-n / 2} & t < \\frac 1 2 \\\\ \\text{does not exist} & t \\ge \\frac 1 2\\end{cases}$"}
+{"_id": "17627", "title": "Moment Generating Function of Linear Combination of Independent Random Variables", "text": "Let $X_1, X_2, \\ldots, X_n$ be independent random variables.  Let $k_1, k_2, \\ldots, k_n$ be real numbers. Let:  :$\\displaystyle X = \\sum_{i \\mathop = 1}^n k_i X_i$ Let $M_{X_i}$ be the moment generating function of $X_i$ for $1 \\le i \\le n$. Then:  :$\\displaystyle \\map {M_X} t = \\prod_{i \\mathop = 1}^n \\map {M_{X_i}} {k_i t}$ for all $t$ such that $M_{X_i}$ exists for all $1 \\le i \\le n$."}
+{"_id": "17629", "title": "Variance of Random Sample from Gaussian Distribution has Chi-Squared Distribution", "text": "Let $X_1, X_2, \\ldots, X_n$ form a random sample of size $n$ from the Gaussian distribution $\\Gaussian \\mu {\\sigma^2}$ for some $\\mu \\in \\R, \\sigma \\in \\R_{> 0}$.  Let: :$\\displaystyle \\bar X = \\frac 1 n \\sum_{i \\mathop = 1}^n X_i$ and: :$\\displaystyle s^2 = \\frac 1 {n - 1} \\sum_{i \\mathop = 1}^n \\paren {X_i - \\bar X}^2$ Then:  :$\\dfrac {\\paren {n - 1} s^2} {\\sigma^2} \\sim \\chi^2_{n - 1}$ where $\\chi^2_{n - 1}$ is the chi-squared distribution with $n - 1$ degrees of freedom."}
+{"_id": "17630", "title": "Expectation of F-Distribution", "text": "Let $n, m$ be strictly positive integers.  Let $X \\sim F_{n, m}$ where $F_{n, m}$ is the F-distribution with $\\tuple {n, m}$ degrees of freedom. Then the expectation of $X$ is given by: :$\\expect X = \\dfrac m {m - 2}$ for $m > 2$, and does not exist otherwise."}
+{"_id": "17631", "title": "Variance of F-Distribution", "text": "Let $n, m$ be strictly positive integers.  Let $X \\sim F_{n, m}$ where $F_{n, m}$ is the F-distribution with $\\tuple {n, m}$ degrees of freedom. Then the variance of $X$ is given by:  :$\\var X = \\dfrac {2 m^2 \\paren {m + n - 2} } {n \\paren {m - 4} \\paren {m - 2}^2}$ for $m > 4$, and does not exist otherwise."}
+{"_id": "17632", "title": "Differential Equations for Shortest Path on 3d Sphere/Cartesian Coordinates", "text": "Let $M$ be a $3$-dimensional Euclidean space. Let $S$ be a sphere embedded in $M$. Let $\\gamma$ be a curve on $S$. Let the chosen coordinate system be Cartesian. Let $\\gamma$ begin at $\\paren {x_0, y_0, z_0}$ and terminate at $\\paren {x_1, y_1, z_1}$. Let $\\map y x$, $\\map z x$ be real functions. Let $\\gamma$ connecting both endpoints be of minimum length. Then $\\gamma$ satisfies the following equations of motion: :$2 y \\map \\lambda x - \\dfrac \\d {\\d x} \\dfrac {y'} {\\sqrt {1 + y'^2 + z'^2} } = 0$ :$2 z \\map \\lambda x - \\dfrac \\d {\\d x} \\dfrac {z'} {\\sqrt {1 + y'^2 + z'^2} } = 0$"}
+{"_id": "17633", "title": "Skewness of F-Distribution", "text": "Let $n, m$ be strictly positive integers.  Let $X \\sim F_{n, m}$ where $F_{n, m}$ is the F-distribution with $\\tuple {n, m}$ degrees of freedom. Then the skewness $\\gamma_1$ of $X$ is given by:  :$\\gamma_1 = \\dfrac {2 \\paren {m + 2 n - 2} } {m - 6} \\sqrt {\\dfrac {2 \\paren {m - 4} } {n \\paren {m + n - 2} } }$ for $m > 6$, and does not exist otherwise."}
+{"_id": "17636", "title": "Expectation of Chi Distribution", "text": "Let $n$ be a strictly positive integer.  Let $X \\sim \\chi_n$ where $\\chi_n$ is the chi distribution with $n$ degrees of freedom.  Then the expectation of $X$ is given by:  :$\\expect X = \\sqrt 2 \\dfrac {\\map \\Gamma {\\paren {n + 1} / 2} } {\\map \\Gamma {n / 2} }$ where $\\Gamma$ is the gamma function."}
+{"_id": "17637", "title": "Mean of Random Sample from Chi-Squared Distribution has Gamma Distribution", "text": "Let $n$ be a strictly positive integer.  Let $X_1, X_2, \\ldots, X_k$ form a random sample of size $k$ from the chi-squared distribution with $n$ degrees of freedom.  Then:  :$\\displaystyle \\overline X = \\frac 1 k \\sum_{i \\mathop = 1}^k X_i \\sim \\map \\Gamma {\\frac {n k} 2, \\frac k 2}$ where $\\map \\Gamma {\\dfrac {n k} 2, \\dfrac k 2}$ is the gamma distribution with parameters $\\dfrac {n k} 2$ and $\\dfrac k 2$."}
+{"_id": "17638", "title": "Equivalence of Definitions of Separated Sets", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A, B \\subseteq S$. {{TFAE|def = Separated Sets}}"}
+{"_id": "17639", "title": "Equivalence of Definitions of Separated Sets/Definition 1 implies Definition 2", "text": "Let $A, B \\subseteq S$ satisfy: :$A^- \\cap B = A \\cap B^- = \\O$ where $A^-$ denotes the closure of $A$ in $T$, and $\\O$ denotes the empty set."}
+{"_id": "17640", "title": "Equivalence of Definitions of Separated Sets/Definition 2 implies Definition 1", "text": "Let $A, B \\subseteq S$. Let $U, V \\in \\tau$ satisfy: :$A \\subset U$ and $U \\cap B = \\O$ :$B \\subset V$ and $V \\cap A = \\O$"}
+{"_id": "17642", "title": "Excess Kurtosis of Student's t-Distribution", "text": "Let $k$ be a strictly positive integer.  Let $X \\sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom. Then the excess kurtosis $\\gamma_2$ of $X$ is given by:  :$\\gamma_2 = \\dfrac 6 {k - 4}$ for $k > 4$, and does not exist otherwise."}
+{"_id": "17643", "title": "Connected Set in Subspace", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A \\subseteq B \\subseteq S$. Let $T_B = \\struct {B, \\tau_B}$ be the topological space where $\\tau_B$ is the subspace topology on $B$. Then :$A$ is connected in $T_B$ {{iff}} $A$ is connected in $T$."}
+{"_id": "17644", "title": "Expectation of Erlang Distribution", "text": "Let $k$ be a strictly positive integer.  Let $\\lambda$ be a strictly positive real number.  Let $X$ be a continuous random variable with an Erlang distribution with parameters $k$ and $\\lambda$. Then the expectation of $X$ is given by: :$\\expect X = \\dfrac k \\lambda$"}
+{"_id": "17645", "title": "Variance of Erlang Distribution", "text": "Let $k$ be a strictly positive integer.  Let $\\lambda$ be a strictly positive real number.  Let $X$ be a continuous random variable with an Erlang distribution with parameters $k$ and $\\lambda$. Then the variance of $X$ is given by:  :$\\var X = \\dfrac k {\\lambda^2}$"}
+{"_id": "17646", "title": "Covariance as Expectation of Product minus Product of Expectations", "text": "Let $X$ and $Y$ be random variables.  Let the expectations of $X$ and $Y$ exist and be finite.  Then the covariance of $X$ and $Y$ is given by:  :$\\cov {X, Y} = \\expect {X Y} - \\expect X \\expect Y$"}
+{"_id": "17647", "title": "Covariance of Independent Random Variables is Zero", "text": "Let $X$ and $Y$ be independent random variables. Let the expectations of $X$ and $Y$ exist and be finite.  Then the covariance of $X$ and $Y$ is $0$."}
+{"_id": "17648", "title": "Variance of Linear Combination of Random Variables", "text": "Let $X$ and $Y$ be random variables.  Let the variances of $X$ and $Y$ be finite.  Let $a$ and $b$ be real numbers.  Then the variance of $a X + b Y$ is given by:  :$\\var {a X + b Y} = a^2 \\, \\var X + b^2 \\, \\var Y + 2 a b \\, \\cov {X, Y}$ where $\\cov {X, Y}$ is the covariance of $X$ and $Y$."}
+{"_id": "17649", "title": "Square of Expectation of Product is Less Than or Equal to Product of Expectation of Squares", "text": "Let $X$ and $Y$ be random variables.  Let the expectation of $X Y$, $\\expect {X Y}$, exist and be finite.  Then:  :$\\paren {\\expect {X Y} }^2 \\le \\expect {X^2} \\expect {Y^2}$"}
+{"_id": "17650", "title": "Square of Covariance is Less Than or Equal to Product of Variances", "text": "Let $X$ and $Y$ be random variables.  Let the variances of $X$ and $Y$ exist and be finite.  Then:  :$\\paren {\\cov {X, Y} }^2 \\le \\var X \\, \\var Y$ where $\\cov {X, Y}$ denotes the covariance of $X$ and $Y$."}
+{"_id": "17651", "title": "Absolute Value of Pearson Correlation Coefficient is Less Than or Equal to 1", "text": "Let $X$ and $Y$ be random variables.  Let the variances of $X$ and $Y$ exist and be finite.  Then:  :$\\size {\\map \\rho {X, Y} } \\le 1$ where $\\map \\rho {X, Y}$ denotes the Pearson correlation coefficient of $X$ and $Y$."}
+{"_id": "17652", "title": "Equivalence of Definitions of Locally Connected Space/Definition 1 implies Definition 2", "text": "Let each point of $T$ have a local basis consisting entirely of connected sets in $T$."}
+{"_id": "17653", "title": "Equivalence of Definitions of Locally Connected Space/Definition 2 implies Definition 1", "text": "Let $T$ be weakly locally connected at each point of $T$. That is, each point of $T$ has a neighborhood basis consisting of connected sets of $T$."}
+{"_id": "17654", "title": "Equivalence of Definitions of Locally Connected Space/Definition 1 implies Definition 3", "text": "Let each point $x$ of $T$ have a local basis $\\mathcal D_x$ consisting entirely of connected sets in $T$."}
+{"_id": "17655", "title": "Equivalence of Definitions of Locally Connected Space/Definition 3 implies Definition 1", "text": "Let $T$ have a basis $\\mathcal B$ consisting of connected sets in $T$."}
+{"_id": "17657", "title": "Equivalence of Definitions of Locally Connected Space/Definition 4 implies Definition 3", "text": "Let the components of the open sets of $T$ are also open in $T$."}
+{"_id": "17658", "title": "Connected Component is Closed", "text": "Let $T = \\struct{S, \\tau}$ be a topological space. Then every connected component of $T$ is closed."}
+{"_id": "17659", "title": "Constant Mapping is Non-Commutative", "text": "Let $S$ be a set whose cardinality is greater than one. Let $f: S \\to S$ and $g: S \\to S$ be constant mappings on $S$. Then: :$f \\circ g \\ne g \\circ f$ where $\\circ$ denotes composition of mappings."}
+{"_id": "17660", "title": "Quotient Mapping is Injection iff Equality", "text": "Let $\\mathcal R$ be an equivalence relation on $S$. Then the quotient mapping $q_{\\mathcal R}: S \\to S / \\mathcal R$ is an injection {{iff}} $\\mathcal R$ is the equality relation."}
+{"_id": "17661", "title": "Closure of Subset in Subspace", "text": "Let $T = \\struct{S, \\tau}$ be a topological space. Let $H$ be a subset of $S$. Let $T_H = \\struct {H, \\tau_H}$ be the topological subspace on $H$. Let $A$ be a subset of $H$. Then: :$\\map {\\cl_H} A = H \\cap \\map \\cl A$ where  :$\\map {\\cl_H} A$ denotes the closure of $A$ in $T_H$ :$\\map \\cl A$ denotes the closure of $A$ in $T$"}
+{"_id": "17662", "title": "Equivalence of Definitions of Locally Path-Connected Space/Definition 1 implies Definition 2", "text": "Let each point of $T$ have a local basis consisting entirely of path-connected sets in $T$."}
+{"_id": "17663", "title": "Equivalence of Definitions of Locally Path-Connected Space/Definition 2 implies Definition 1", "text": "Let each point of $T$ have a neighborhood basis consisting of path-connected sets in $T$."}
+{"_id": "17664", "title": "Equivalence of Definitions of Locally Path-Connected Space/Definition 1 implies Definition 3", "text": "Let each point $x$ of $T$ have a local basis $\\mathcal D_x$ consisting entirely of path-connected sets in $T$."}
+{"_id": "17665", "title": "Equivalence of Definitions of Locally Path-Connected Space/Definition 3 implies Definition 1", "text": "Let $T$ have a basis $\\mathcal B$ consisting of path-connected sets in $T$."}
+{"_id": "17666", "title": "Equivalence of Definitions of Locally Path-Connected Space/Definition 3 implies Definition 4", "text": "Let $T$ have a basis consisting of path-connected sets in $T$."}
+{"_id": "17667", "title": "Equivalence of Definitions of Locally Path-Connected Space/Definition 4 implies Definition 3", "text": "Let the path components of open sets of $T$ be also open in $T$."}
+{"_id": "17669", "title": "Path-Connected Set in Subspace", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A \\subseteq B \\subseteq S$. Let $T_B = \\struct {B, \\tau_B}$ be the topological space where $\\tau_B$ is the subspace topology on $B$. Then :$A$ is path-connected in $T_B$ {{iff}} $A$ is path-connected in $T$."}
+{"_id": "17670", "title": "Precisely One Function in terms of And, Or and Not", "text": "Let $\\map P {A, B, C}$ denote the precisely one function on the statements $A$, $B$ and $C$. Then: :$\\map P {A, B, C} \\dashv \\vdash \\paren {A \\land \\neg B \\land \\neg C} \\lor \\paren {\\neg A \\land B \\land \\neg C} \\lor \\paren {\\neg A \\land \\neg B \\land C}$ where: :$\\land$ denotes conjunction :$\\lor$ denotes disjunction :$\\neg$ denotes negation"}
+{"_id": "17671", "title": "Lévy's Continuity Theorem", "text": "Let $\\sequence {X_n}_{n \\mathop \\ge 1}$ be a sequence of discrete random variables with characteristic functions $\\map {\\phi_n} t := E \\sqbrk {e^{i t X_n} }$. Let the sequence $\\sequence {\\phi_n}$ converge to some real function $\\phi$: :$\\forall t \\in \\R: \\map {\\phi_n} t \\to \\map \\phi t$. Then the following statements are equivalent: :$(1): \\quad$ The $\\sequence {X_n}$ converges in distribution to some random variable $X$: ::$X_n \\stackrel {dist} {\\to} X$ with characteristic function $\\map {\\phi_X} t := \\map \\phi t$ :$(2): \\quad$ The sequence $\\sequence {X_n}$ is tight, that is: ::$\\displaystyle \\lim_{M \\mathop \\to \\infty} \\sup_{n \\mathop \\ge 1} P \\sqbrk {\\size {X_n} \\ge M} = 0$"}
+{"_id": "17672", "title": "Composition of Three Mappings which form Identity Mapping", "text": "Let $A$, $B$ and $C$ be non-empty sets. Let $f: A \\to B$, $g: B \\to C$ and $h: C \\to A$ be mappings. Let the following hold: {{begin-eqn}} {{eqn | l = h \\circ g \\circ f       | r = I_A }} {{eqn | l = f \\circ h \\circ g       | r = I_B }} {{eqn | l = g \\circ f \\circ h       | r = I_C }} {{end-eqn}} where: :$g \\circ f$ (and so on) denote composition of mappings :$I_A$ (and so on) denote the identity mappings. Then each of $f$, $g$ and $h$ are bijections, and: {{begin-eqn}} {{eqn | l = f^{-1}       | r = h \\circ g }} {{eqn | l = g^{-1}       | r = f \\circ h }} {{eqn | l = h^{-1}       | r = g \\circ f }} {{end-eqn}} where $f^{-1}$ (and so on) denote the inverse mappings."}
+{"_id": "17673", "title": "Composition of Product Mappings on Natural Numbers", "text": "Let $a \\in \\N$ be a natural number. Let $\\mu_a: \\N \\to \\N$ be the mapping defined as: :$\\forall x \\in \\N: \\map {\\mu_a} x = x a$ Then: :$\\mu_{a b} = \\mu_b \\circ \\mu_a$"}
+{"_id": "17675", "title": "Moment Generating Function of Linear Transformation of Random Variable", "text": "Let $X$ be a random variable.  Let $\\alpha$ and $\\beta$ be real numbers. Let $Z = \\alpha X + \\beta$. Let $M_X$ be the moment generating function of $X$.  Then the moment generating function of $Z$, $M_Z$, is given by:  :$\\map {M_Z} t = e^{\\beta t} \\map {M_X} {\\alpha t}$"}
+{"_id": "17676", "title": "Standard Gaussian Random Variable as Transformation of Gaussian Random Variable", "text": "Let $\\mu$ be a real number.  Let $\\sigma$ be a positive real number.  Let $X \\sim \\Gaussian \\mu {\\sigma^2}$ where $\\Gaussian \\mu {\\sigma^2}$ is the Gaussian distribution with parameters $\\mu$ and $\\sigma^2$.  Then:  :$\\dfrac {X - \\mu} \\sigma \\sim \\Gaussian 0 1$ where $\\Gaussian 0 1$ is the standard Gaussian distribution."}
+{"_id": "17677", "title": "Composition of Addition Mappings on Natural Numbers", "text": "Let $a \\in \\N$ be a natural number. Let $\\alpha_a: \\N \\to \\N$ be the mapping defined as: :$\\forall x \\in \\N: \\map {\\alpha_a} x = x + a$ Then: :$\\alpha_{a + b} = \\alpha_b \\circ \\alpha_a$"}
+{"_id": "17678", "title": "Reciprocal of Random Variable with F-Distribution has F-Distribution", "text": "Let $n, m$ be strictly positive integers.  Let $X \\sim F_{n, m}$ where $F_{n, m}$ is the F-distribution with $\\tuple {n, m}$ degrees of freedom. Then:  :$\\dfrac 1 X \\sim F_{m, n}$"}
+{"_id": "17679", "title": "Square of Random Variable with t-Distribution has F-Distribution", "text": "Let $k$ be a strictly positive integer.  Let $X \\sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom. Then:  :$X^2 \\sim F_{1, k}$ where $F_{1, k}$ is the $F$-distribution with $\\tuple {1, k}$ degrees of freedom."}
+{"_id": "17680", "title": "Union of Connected Sets with Common Point is Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\family {B_\\alpha}_{\\alpha \\mathop \\in A}$ be a family of connected sets of $T$. Let $\\exists x \\in \\displaystyle \\bigcap \\family {B_\\alpha}_{\\alpha \\mathop \\in A}$. Then :$\\displaystyle \\bigcup \\family {B_\\alpha}_{\\alpha \\mathop \\in A}$ is a connected set of $T$."}
+{"_id": "17681", "title": "Connected Subset of Union of Disjoint Open Sets", "text": "Let $T = \\struct{S, \\tau}$ be a topological space. Let $A$ be a connected set of $T$. Let $U, V$ be disjoint open sets. Let $A \\subseteq U \\cup V$. Then :either $A \\subseteq U$ or $A \\subseteq V$."}
+{"_id": "17682", "title": "Rearrangement of Variables in Total Differential Equation", "text": "In a total differential equation, any one of the variables can be regarded as an independent variable, while the remainder can be treated as dependent variables."}
+{"_id": "17684", "title": "Archimedean Principle/Variant", "text": "Let $x$ and $y$ be a natural numbers. Then there exists a natural number $n$ such that: :$n x \\ge y$"}
+{"_id": "17687", "title": "Set Intersection Preserves Subsets/Families of Sets/Intersection is Empty Implies Intersection of Subsets is Empty", "text": "Let $I$ be an indexing set. Let $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$ and $\\family {B_\\alpha}_{\\alpha \\mathop \\in I}$ be indexed families of subsets of a set $S$. Let: :$\\forall \\beta \\in I: A_\\beta \\subseteq B_\\beta$ Then: :$\\displaystyle \\bigcap_{\\alpha \\mathop \\in I} B_\\alpha = \\O \\implies \\bigcap_{\\alpha \\mathop \\in I} A_\\alpha = \\O$"}
+{"_id": "17690", "title": "Union of Path-Connected Sets with Common Point is Path-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\family {B_\\alpha}_{\\alpha \\mathop \\in A}$ be a family of path-connected sets of $T$. Let $\\exists x \\in \\displaystyle \\bigcap \\family {B_\\alpha}_{\\alpha \\mathop \\in A}$. Then :$\\displaystyle \\bigcup \\family {B_\\alpha}_{\\alpha \\mathop \\in A}$ is a path-connected set of $T$."}
+{"_id": "17694", "title": "Upper Bound for Lucas Number", "text": "Let $L_n$ denote the $n$th Lucas number. Then: :$L_n < \\paren {\\dfrac 7 4}^n$"}
+{"_id": "17700", "title": "Condition for Increasing Binomial Coefficients", "text": "Let $n \\in \\Z_{> 0}$ be a (strictly) positive integer. Let $\\dbinom n k$ denote a binomial coefficient for $k \\in \\N$. Then: :$\\dbinom n k < \\dbinom n {k + 1} \\iff 0 \\le k < \\dfrac {n - 1} 2$"}
+{"_id": "17702", "title": "Condition for Equality of Adjacent Binomial Coefficients", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\dbinom n k$ denote a binomial coefficient for $k \\in \\Z$. Then: :$\\dbinom n k = \\dbinom n {k + 1}$ {{iff}}: :$n$ is an odd integer :$k = \\dfrac {n - 1} 2$"}
+{"_id": "17705", "title": "Equivalence of Definitions of Component/Equivalence Class equals Union of Connected Sets", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $x \\in T$. Let $\\CC_x = \\set {A \\subseteq S: x \\in A \\land A \\text{ is connected in } T}$ Let $C = \\bigcup \\CC_x$ Let $\\sim$ be the equivalence relation defined by: :$y \\sim z$ {{iff}} $y$ and $z$ are connected in $T$. Let $C’$ be the equivalence class of $\\sim$ containing $x$. Then $C = C'$."}
+{"_id": "17706", "title": "Equivalence of Definitions of Component/Union of Connected Sets is Maximal Connected Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $x \\in T$. Let $\\mathcal C_x = \\set {A \\subseteq S : x \\in A \\land A \\text{ is connected in } T}$ Let $C = \\bigcup \\mathcal C_x$ Then $C$ is a maximal connected set of $T$."}
+{"_id": "17707", "title": "Equivalence of Definitions of Component/Maximal Connected Set is Union of Connected Sets", "text": "Let $\\tilde C$ be a maximal connected set of $T$ that contains $x$."}
+{"_id": "17708", "title": "Binomial Coefficient n Choose j in terms of n-2 Choose r", "text": "Let $n \\in \\Z$ such that $n \\ge 4$. Let $\\dbinom n k$ denote a binomial coefficient for $k \\in \\Z$. Then: :$\\dbinom n k = \\dbinom {n - 2} {k - 2} + 2 \\dbinom {n - 2} {k - 1} + \\dbinom {n - 2} k$ for $2 \\le k \\le n - 2$."}
+{"_id": "17709", "title": "Sum of Sequence of Binomial Coefficients by Powers of 2", "text": "{{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 0}^n 2^j \\binom n j       | r = \\dbinom n 0 + 2 \\dbinom n 1 + 2^2 \\dbinom n 2 + \\dotsb + 2^n \\dbinom n n       | c =  }} {{eqn | r = 3^n       | c =  }} {{end-eqn}}"}
+{"_id": "17712", "title": "Sum of Sequence of n Choose 2", "text": "Let $n \\in \\Z$ be an integer such that $n > 2$. {{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 2}^n \\dbinom j 2       | r = \\dbinom 2 2 + \\dbinom 3 2 + \\dbinom 4 2 + \\dotsb + \\dbinom n 2       | c =  }} {{eqn | r = \\dbinom {n + 1} 3       | c =  }} {{end-eqn}} where $\\dbinom n j$ denotes a binomial coefficient."}
+{"_id": "17714", "title": "If n is Triangular then so is 25n + 3", "text": "Let $n$ be a triangular number. Then $25 n + 3$ is also triangular."}
+{"_id": "17715", "title": "If n is Triangular then so is 49n + 6", "text": "Let $n$ be a triangular number. Then $49 n + 6$ is also triangular."}
+{"_id": "17716", "title": "Sum of Sequence of Triangular Numbers", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $T_n$ denote the $n$th triangular number. Then: {{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 1}^n T_j       | r = T_1 + T_2 + T_3 + \\dotsb + T_n       | c =  }} {{eqn | r = \\dfrac {n \\paren {n + 1} \\paren {n + 2} } 6       | c =  }} {{end-eqn}}"}
+{"_id": "17719", "title": "Square of Odd Multiple of 3 is Difference between Triangular Numbers", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Let $T_n$ denote the $n$th triangular number. Let $m = 2 n + 1$ be an odd integer Then: :$\\paren {3 m}^2 = T_{9 n + 4} - T_{3 n + 1}$"}
+{"_id": "17720", "title": "Square Sum of Three Consecutive Triangular Numbers", "text": "Let $T_n$ denote the $n$th triangular number for $n \\in \\Z_{>0}$ a (strictly) positive integer. Let $T_n + T_{n + 1} + T_{n + 2}$ be a square number. Then at least one value of $n$ fulfils this condition: :$n = 5$"}
+{"_id": "17722", "title": "Sufficient Condition for Square of Product to be Triangular", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $2 n^2 \\pm 1 = m^2$ be a square number. Then $\\paren {m n}^2$ is a triangular number."}
+{"_id": "17723", "title": "Topology on Singleton is Indiscrete Topology", "text": "Let $S$ be a singleton. The only possible topology on $S$ is the indiscrete topology."}
+{"_id": "17724", "title": "Existence of Divisor with Remainder between 2b and 3b", "text": "For every pair of integers $a, b$ where $b > 0$, there exist unique integers $q$ and $r$ where $2 b \\le r < 3 b$ such that: :$a = q b + r$"}
+{"_id": "17725", "title": "Integer of form 6k + 5 is of form 3k + 2 but not Conversely", "text": "Let $n \\in \\Z$ be an integer of the form: :$n = 6 k + 5$ where $k \\in \\Z$. Then $n$ can also be expressed in the form: :$n = 3 k + 2$ for some other $k \\in \\Z$. However it is not necessarily the case that if $n$ can be expressed in the form: :$n = 3 k + 2$ then it can also be expressed in the form: :$n = 6 k + 5$"}
+{"_id": "17726", "title": "Odd Integer Modulo 4", "text": "Let $n$ be an odd integer. Then $n$ can be expressed either as: :$n = 4 k + 1$ or as: :$n = 4 k + 3$"}
+{"_id": "17728", "title": "N (n + 1) (2n + 1) over 6 is Integer", "text": "Let $n \\in \\Z$ be an integer. Then $\\dfrac {n \\paren {n + 1} \\paren {2 n + 1} } 6$ is also an integer."}
+{"_id": "17731", "title": "Number which is Square and Cube Modulo 7", "text": "Let $n \\in \\Z$ be an integer. Let $n$ be both a square and a cube at the same time. Then either: :$n \\equiv 0 \\pmod 7$ or: :$n \\equiv 1 \\pmod 7$"}
+{"_id": "17732", "title": "Weak Law of Large Numbers", "text": "Let $P$ be a population. Let $P$ have mean $\\mu$ and finite variance.  Let $\\sequence {X_n}_{n \\mathop \\ge 1}$ be a sequence of random variables forming a random sample from $P$. Let:  :$\\ds {\\overline X}_n = \\frac 1 n \\sum_{i \\mathop = 1}^n X_i$ Then: :${\\overline X}_n \\xrightarrow p \\mu$ where $\\xrightarrow p$ denotes convergence in probability."}
+{"_id": "17733", "title": "Variance of Linear Combination of Random Variables/Corollary", "text": "Let $X$ and $Y$ be independent random variables."}
+{"_id": "17734", "title": "Singleton is Connected in Topological Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $x \\in S$. Then the singleton $\\set{x}$ is connected."}
+{"_id": "17735", "title": "Equivalence of Definitions of Component/Lemma 1", "text": ":$C$ is connected in $T$ and $C \\in \\mathcal C_x$."}
+{"_id": "17736", "title": "Variance of Sample Mean", "text": "Let $X_1, X_2, \\ldots, X_n$ form a random sample from a population with mean $\\mu$ and variance $\\sigma^2$. Let:  :$\\displaystyle \\overline X = \\frac 1 n \\sum_{i \\mathop = 1}^n X_i$ Then:  :$\\var {\\overline X} = \\dfrac {\\sigma^2} n$"}
+{"_id": "17738", "title": "Common Divisor Divides Integer Combination/General Result", "text": "Let $c$ be a common divisor of a set of integers $A := \\set {a_1, a_2, \\dotsc, a_n}$. That is: :$\\forall x \\in A: c \\divides x$ Then $c$ divides any integer combination of elements of $A$: :$\\forall x_1, x_2, \\dotsc, x_n \\in \\Z: c \\divides \\paren {a_1 x_2 + a_2 x_2 + \\dotsb + a_n x_n}$"}
+{"_id": "17739", "title": "One is Common Divisor of Integers", "text": "Let $a, b \\in \\Z$ be integers. Then $1$ is a common divisor of $a$ and $b$."}
+{"_id": "17741", "title": "Equivalence of Definitions of Path Component", "text": "{{TFAE|def = Path Component|view = Path Component|context = Topology (Mathematical Branch)|contextview = Topology}} Let $T = \\struct {S, \\tau}$ be a topological space. Let $x \\in T$."}
+{"_id": "17742", "title": "GCD of Integer and its Negative", "text": "Let $a \\in \\Z$ be an integer. Then: :$\\gcd \\set {a, -a} = \\size a$ where: :$\\gcd$ denotes greatest common divisor :$\\size a$ denotes the absolute value of $a$."}
+{"_id": "17743", "title": "Equivalence of Definitions of Path Component/Lemma 1", "text": ":$C$ is path-connected in $T$ and  $C \\in \\CC_x$."}
+{"_id": "17744", "title": "Square Divides Product of Multiples", "text": "Let $a, b, c, \\in \\Z$ be integers. Let: :$a \\divides b, a \\divides c$ where $\\divides$ denotes divisibility. Then: :$a^2 \\divides b c$"}
+{"_id": "17745", "title": "Abel's Limit Theorem", "text": "Let $\\sequence {a_i}$ be a convergent series. The limit which is assigned by the Abel summation method exists and equals the sum of the series."}
+{"_id": "17746", "title": "Equivalence of Definitions of Path Component/Equivalence Class equals Union of Path-Connected Sets", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $x \\in T$. Let $\\CC_x = \\left\\{ {A \\subseteq S : x \\in A \\land A }\\right.$ is path-connected in $\\left. {T}\\right\\}$. Let $C = \\bigcup \\CC_x$ Let $\\sim$ be the equivalence relation defined by: :$y \\sim z$ {{iff}} $y$ and $z$ are path-connected in $T$. Let $C'$ be the equivalence class of $\\sim$ containing $x$. Then $C = C'$."}
+{"_id": "17747", "title": "Image of Path is Path-Connected Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $I \\subset \\R$ be the closed real interval $\\closedint a b$. Let $\\gamma: I \\to S$ be a path. Then: :$\\map \\gamma I$ is a path-connected set of $T$."}
+{"_id": "17749", "title": "Closure of Hadamard Product", "text": "Let $\\struct {S, \\cdot}$ be an algebraic structure. Let $\\map {\\MM_S} {m, n}$ be a $m \\times n$ matrix space over $S$. For $\\mathbf A, \\mathbf B \\in \\map {\\MM_S} {m, n}$, let $\\mathbf A \\circ \\mathbf B$ be defined as the Hadamard product of $\\mathbf A$ and $\\mathbf B$. The operation $\\circ$ is closed on $\\map {\\MM_S} {m, n}$ {{iff}} $\\cdot$ is closed on $\\struct {S, \\cdot}$."}
+{"_id": "17750", "title": "Associativity of Hadamard Product", "text": "Let $\\struct {S, \\cdot}$ be an algebraic structure. Let $\\map {\\MM_S} {m, n}$ be a $m \\times n$ matrix space over $S$. For $\\mathbf A, \\mathbf B \\in \\map {\\MM_S} {m, n}$, let $\\mathbf A \\circ \\mathbf B$ be defined as the Hadamard product of $\\mathbf A$ and $\\mathbf B$. The operation $\\circ$ is associative on $\\map {\\MM_S} {m, n}$ {{iff}} $\\cdot$ is associative on $\\struct {S, \\cdot}$."}
+{"_id": "17751", "title": "Commutativity of Hadamard Product", "text": "Let $\\struct {S, \\cdot}$ be an algebraic structure. Let $\\map {\\MM_S} {m, n}$ be a $m \\times n$ matrix space over $S$. For $\\mathbf A, \\mathbf B \\in \\map {\\MM_S} {m, n}$, let $\\mathbf A \\circ \\mathbf B$ be defined as the Hadamard product of $\\mathbf A$ and $\\mathbf B$. The operation $\\circ$ is commutative on $\\map {\\MM_S} {m, n}$ {{iff}} $\\cdot$ is commutative on $\\struct {S, \\cdot}$."}
+{"_id": "17752", "title": "Matrix Entrywise Addition over Ring is Closed", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\map {\\MM_R} {m, n}$ be a $m \\times n$ matrix space over $R$. For $\\mathbf A, \\mathbf B \\in \\map {\\MM_R} {m, n}$, let $\\mathbf A + \\mathbf B$ be defined as the matrix entrywise sum of $\\mathbf A$ and $\\mathbf B$. The operation $+$ is closed on $\\map {\\MM_R} {m, n}$."}
+{"_id": "17753", "title": "Points are Path-Connected iff Contained in Path-Connected Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $x, y \\in S$ Then: :$x, y$ are path-connected points in $T$ {{iff}} there exists a path-connected set of $T$ containing $x$ and $y$."}
+{"_id": "17754", "title": "Compound Angle Formulas", "text": "=== Sine of Sum === {{:Sine of Sum}} === Cosine of Sum === {{:Cosine of Sum}} === Tangent of Sum === {{:Tangent of Sum}}"}
+{"_id": "17755", "title": "Equivalence of Definitions of Path Component/Union of Path-Connected Sets is Maximal Path-Connected Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $x \\in T$. Let $\\CC_x = \\left\\{ {A \\subseteq S : x \\in A \\land A } \\right.$ is path-connected in $\\left. {T} \\right\\}$ Let $C = \\bigcup \\CC_x$ Then $C$ is a maximal path-connected set of $T$."}
+{"_id": "17756", "title": "Solutions to Diophantine Equation 16x^2+32x+20 = y^2+y", "text": "The indeterminate Diophantine equation: :$16x^2 + 32x + 20 = y^2 + y$ has exactly $4$ solutions: :$\\tuple {0, 4}, \\tuple {-2, 4}, \\tuple {0, -5}, \\tuple {-2, -5}$"}
+{"_id": "17757", "title": "Equivalence of Definitions of Path Component/Maximal Path-Connected Set is Union of Path-Connected Sets", "text": "Let $\\tilde C$ be a maximal path-connected set of $T$ that contains $x$."}
+{"_id": "17758", "title": "Power Set is Closed under Set Complement", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Then: :$\\forall A \\in \\powerset S: \\relcomp S A \\in \\powerset S$"}
+{"_id": "17759", "title": "Normal Subgroup of Symmetric Group Order Greater than 4 is Alternating Group", "text": "Let $n \\in \\N$ be a natural number such that $n > 4$. Let $S_n$ denote the symmetric group on $n$ letters. Let $A_n$ denote the alternating group on $n$ letters. $A_n$ is the only proper non-trivial normal subgroup of $S_n$."}
+{"_id": "17760", "title": "Path Components are Open iff Union of Open Path-Connected Sets/Path Components are Open implies Space is Union of Open Path-Connected Sets", "text": "Let the path components of $T$ be open sets."}
+{"_id": "17761", "title": "Path Components are Open iff Union of Open Path-Connected Sets/Space is Union of Open Path-Connected Sets implies Path Components are Open", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $S$ be the union of open path-connected sets of $T$. Then: :The path components of $T$ are open sets."}
+{"_id": "17762", "title": "Path Components are Open iff Union of Open Path-Connected Sets/Lemma 1", "text": "::$U \\cap C \\ne \\O$ {{iff}} $U \\ne \\O$ and $U \\subseteq C$"}
+{"_id": "17763", "title": "Integral Points of Elliptic Curve y^2 = x^3+3x", "text": "The elliptic curve: :$y^2 = x^3 + 3x$ has exactly $7$ lattice points: :$\\tuple {0, 0}, \\tuple {1, \\pm 2}, \\tuple {3, \\pm 6}, \\tuple {12, \\pm 42}$"}
+{"_id": "17769", "title": "Components are Open iff Union of Open Connected Sets/Components are Open implies Space is Union of Open Connected Sets", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let the components of $T$ be open sets. Then: :$S$ is a union of open connected sets of $T$."}
+{"_id": "17770", "title": "Components are Open iff Union of Open Connected Sets/Space is Union of Open Connected Sets implies Components are Open", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $S$ be a union of open connected sets of $T$. Then: :The components of $T$ are open sets."}
+{"_id": "17772", "title": "Components are Open iff Union of Open Connected Sets/Lemma 1", "text": "::$U \\cap C \\neq \\O$ {{iff}} $U \\neq \\O$ and $U \\subseteq C$"}
+{"_id": "17775", "title": "Integral to Infinity of Square of Sine p x over x Squared", "text": ":$\\displaystyle \\int_0^\\infty \\paren {\\frac {\\sin p x} x}^2 \\rd x = \\frac {\\pi \\size p} 2$"}
+{"_id": "17776", "title": "Integral to Infinity of One minus Cosine p x over x Squared", "text": ":$\\displaystyle \\int_0^\\infty \\frac {1 - \\cos p x} {x^2} \\rd x = \\frac {\\pi \\size p} 2$"}
+{"_id": "17777", "title": "1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways", "text": "The number $1$ can be expressed as the sum of $4$ distinct unit fractions in $6$ different ways: {{begin-eqn}} {{eqn | l = 1       | r = \\frac 1 2 + \\frac 1 3 + \\frac 1 7 + \\frac 1 {42} }} {{eqn | r = \\frac 1 2 + \\frac 1 3 + \\frac 1 8 + \\frac 1 {24} }} {{eqn | r = \\frac 1 2 + \\frac 1 3 + \\frac 1 9 + \\frac 1 {18} }} {{eqn | r = \\frac 1 2 + \\frac 1 3 + \\frac 1 {10} + \\frac 1 {15} }} {{eqn | r = \\frac 1 2 + \\frac 1 4 + \\frac 1 5 + \\frac 1 {20} }} {{eqn | r = \\frac 1 2 + \\frac 1 4 + \\frac 1 6 + \\frac 1 {12} }} {{end-eqn}}"}
+{"_id": "17782", "title": "Empty Set is Subset of Power Set", "text": "The empty set is a subset of all power sets: :$\\forall S: \\O \\subseteq \\powerset S$"}
+{"_id": "17783", "title": "Equivalence of Definitions of Weakly Locally Connected at Point/Definition 1 implies Definition 2", "text": "Let $x$ have a neighborhood basis consisting of connected sets."}
+{"_id": "17784", "title": "Equivalence of Definitions of Weakly Locally Connected at Point/Definition 2 implies Definition 1", "text": "Let every open neighborhood $U$ of $x$ contain an open neighborhood $V$ such that every two points of $V$ lie in some connected subset of $U$."}
+{"_id": "17785", "title": "Cycle of Subsets implies Set Equality", "text": "Let $A_1, A_2, \\dotsc, A_n$ be sets. Let: :$\\forall k \\in \\set {2, 3, \\dotsc, n}: A_{k - 1} \\subseteq A_k$ and: :$A_n \\subseteq A_1$ Then: :$\\forall j, k \\in \\set {1, 2, \\dotsc, n}: A_j = A_k$"}
+{"_id": "17792", "title": "Correspondence between Set and Ordinate of Cartesian Product is Mapping", "text": "Let $S$ and $T$ be sets such that $T \\ne \\O$. Let $S \\times T$ denote their cartesian product. Let $t \\in T$ be given. Let $j_t \\subseteq S \\times \\paren {S \\times T}$ be the relation on $S \\times {S \\times T}$ defined as: :$\\forall s \\in \\S: \\map {j_t} s = \\tuple {s, t}$ Then $j_t$ is a mapping."}
+{"_id": "17793", "title": "Mapping from Set to Ordinate of Cartesian Product is Injection", "text": "Let $S$ and $T$ be sets such that $T \\ne \\O$. Let $S \\times T$ denote their cartesian product. Let $t \\in T$ be given. Let $j_t \\subseteq S \\times \\paren {S \\times T}$ be the mapping from $S$ to $S \\times T$ defined as: :$\\forall s \\in \\S: \\map {j_t} s = \\tuple {s, t}$ Then $j_t$ is an injection."}
+{"_id": "17794", "title": "Primitive of Sine Integral Function", "text": ":$\\displaystyle \\int \\map \\Si x \\rd x = x \\map \\Si x + \\cos x + C$"}
+{"_id": "17795", "title": "Preimage of Subset of Cartesian Product under Injection from Factor", "text": "Let $S$ and $T$ be sets such that $T \\ne \\O$. Let $S \\times T$ denote their cartesian product. Let $t \\in T$ be given. Let $j_t \\subseteq S \\times \\paren {S \\times T}$ be the injection from $S$ to $S \\times T$ defined as: :$\\forall s \\in \\S: \\map {j_t} s = \\tuple {s, t}$ Let $W \\subseteq S \\times T$. Let $V = {j_t}^{-1} \\sqbrk W$ denote the preimage of $W$ under $j_t$. Then: :$V = \\set {s: \\tuple {s, t} \\in W}$"}
+{"_id": "17797", "title": "Derivative of Error Function", "text": ":$\\displaystyle \\frac \\d {\\d x} \\paren {\\map \\erf x} = \\frac 2 {\\sqrt \\pi} e^{-x^2}$"}
+{"_id": "17798", "title": "Limit to Infinity of Error Function", "text": ":$\\displaystyle \\lim_{x \\mathop \\to \\infty} \\map \\erf x = 1$"}
+{"_id": "17799", "title": "Dirichlet Beta Function in terms of Hurwitz Zeta Function", "text": ":$\\displaystyle \\map \\beta s = \\frac 1 {4^s} \\paren {\\map \\zeta {s, \\frac 1 4} - \\map \\zeta {s, \\frac 3 4} }$"}
+{"_id": "17800", "title": "Inverse of Right-Total Relation is Left-Total", "text": ":$\\RR$ is right-total {{iff}} $\\RR^{-1}$ is left-total."}
+{"_id": "17801", "title": "Inverse of Left-Total Relation is Right-Total", "text": ":$\\mathcal R$ is left-total {{iff}} $\\mathcal R^{-1}$ is right-total."}
+{"_id": "17802", "title": "Inverse of Mapping is Right-Total Relation", "text": "Let $f$ be a mapping. Then its inverse $f^{-1}$ is a right-total relation."}
+{"_id": "17803", "title": "Inverse of One-to-One Relation is One-to-One", "text": "The inverse of a one-to-one relation is a one-to-one relation."}
+{"_id": "17804", "title": "Inverse of Injection is One-to-One Relation", "text": "Let $f$ be an injective mapping. Then its inverse $f^{-1}$ is a one-to-one relation."}
+{"_id": "17805", "title": "Inverse of Surjection is Relation both Left-Total and Right-Total", "text": "Let $f$ be an surjective mapping. Then its inverse $f^{-1}$ is a relation which is both left-total and right-total."}
+{"_id": "17806", "title": "Inverse is Mapping implies Mapping is Injection", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let the inverse $f^{-1} \\subseteq T \\times S$ itself be a mapping. Then $f$ is an injection."}
+{"_id": "17809", "title": "Inverse is Mapping implies Mapping is Surjection", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let the inverse $f^{-1} \\subseteq T \\times S$ itself be a mapping. Then $f$ is a surjection."}
+{"_id": "17813", "title": "Derivative Function is not Invertible", "text": "Let $\\Bbb I = \\closedint a b$ be a closed interval on the set of real numbers $\\R$ such that $a < b$. Let $A$ denote the set of all continuous real functions $f: \\Bbb I \\to \\R$. Let $B \\subseteq A$ denote the set of all functions differentiable on $\\Bbb I$ whose derivative is continuous on $\\Bbb I$. Let $d: B \\to A$ denote the mapping defined as: :$\\forall \\map f x \\in B: \\map d f = \\map {D_x} f$ where $D_x$ denotes the derivative of $f$ {{WRT|Differentiation}} $x$. Then $d$ is not an invertible mapping."}
+{"_id": "17815", "title": "Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 2 implies Definition 4", "text": "Let $\\sequence {x_k}$ satisfy: :$\\forall \\epsilon > 0: \\exists N \\in \\R_{>0}: \\forall n \\in \\N: n > N \\implies x_n \\in \\map {B_\\epsilon} l$ where $\\map {B_\\epsilon} l$ is the open $\\epsilon$-ball of $l$."}
+{"_id": "17816", "title": "Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 4 implies Definition 2", "text": "Let $\\sequence {x_k}$ satisfy: :for every $\\epsilon \\in \\R{>0}$, the open $\\epsilon$-ball about $l$ contains all but finitely many of the $p_n$."}
+{"_id": "17817", "title": "Definite Integral from 0 to 2 Pi of Reciprocal of a plus b Cosine x", "text": ":$\\displaystyle \\int_0^{2 \\pi} \\frac {\\d x} {a + b \\cos x} = \\frac {2 \\pi} {\\sqrt {a^2 - b^2} }$"}
+{"_id": "17819", "title": "Map from Set of Continuous Functions on Interval to Set of their Integrals", "text": "Let $\\Bbb I = \\closedint a b$ be a closed interval on the set of real numbers $\\R$ such that $a < b$. Let $A$ denote the set of all continuous real functions $f: \\Bbb I \\to \\R$. Let $B \\subseteq A$ denote the set of all functions differentiable on $\\Bbb I$ whose derivative is continuous on $\\Bbb I$. Let $C \\subseteq B$ denote the subset of $B$ which consists of all elements of $B$ such that $\\map f a = 0$. For each $f \\in A$, let $h$ denote the mapping defined as: :$\\map {\\paren {\\map h f} } x = \\displaystyle \\int_a^x \\map f t \\rd t$ Then: :$h: A \\to C$"}
+{"_id": "17821", "title": "Definite Integral from 0 to Half Pi of Reciprocal of One plus Power of Tan x", "text": ":$\\displaystyle \\int_0^{\\pi/2} \\frac {\\d x} {1 + \\tan^m x} = \\frac \\pi 4$"}
+{"_id": "17822", "title": "Möbius Transformation is Bijection", "text": "Let $a, b, c, d \\in \\C$ be complex numbers. Let $f: \\overline \\C \\to \\overline \\C$ be the Möbius transformation: :$\\map f z = \\begin {cases} \\dfrac {a z + b} {c z + d} & : z \\ne -\\dfrac d c \\\\ \\infty & : z = -\\dfrac d c \\\\ \\dfrac a c & : z = \\infty \\\\ \\infty & : z = \\infty \\text { and } c = 0 \\end {cases}$ Then: :$f: \\overline \\C \\to \\overline \\C$ is a bijection {{iff}}: :$a c - b d \\ne 0$"}
+{"_id": "17823", "title": "Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x", "text": ":$\\displaystyle \\int_0^1 x^m \\paren {\\ln x}^n \\rd x = \\frac {\\paren {-1}^n \\map \\Gamma {n + 1} } {\\paren {m + 1}^{n + 1} }$"}
+{"_id": "17824", "title": "Definite Integral from 0 to 1 of x to the x", "text": "{{begin-eqn}} {{eqn\t| l = \\int_0^1 x^x \\rd x \t| r = \\sum_{n \\mathop = 1}^\\infty \\frac {\\paren {-1}^{n + 1} } {n^n} }} {{eqn\t| r = -\\sum_{n \\mathop = 1}^\\infty \\paren {-n}^{-n} }} {{eqn\t| r = 0.78343 \\ 05107 \\ 12\\ldots }} {{end-eqn}}"}
+{"_id": "17825", "title": "Definite Integral from 0 to 1 of x to the minus x", "text": "{{begin-eqn}} {{eqn\t| l = \\int_0^1 x^{-x} \\rd x \t| r = \\sum_{n \\mathop = 1}^\\infty n^{-n} }} {{eqn\t| r = 1.29128 \\ 5997 \\ldots }} {{end-eqn}}"}
+{"_id": "17826", "title": "Definite Integral to Infinity of Exponential of -a x by Sine of b x", "text": ":$\\displaystyle \\int_0^\\infty e^{-a x} \\sin b x \\rd x = \\frac b {a^2 + b^2}$"}
+{"_id": "17827", "title": "Sum of Exponential of i k x", "text": ":$\\displaystyle \\sum_{k \\mathop = 0}^n \\map \\exp {i k x} = \\paren {i \\sin \\frac {n x} 2 + \\cos \\frac {n x} 2} \\frac {\\map \\sin {\\frac {\\paren {n + 1} x} 2} } {\\sin \\frac x 2}$"}
+{"_id": "17829", "title": "Extension of Extension of Mapping is Extension", "text": "Let $A, B, C, S$ be sets such that $A \\subseteq B \\subseteq C$. Let $f: A \\to S$, $g: B \\to S$ and $h: C \\to S$ be mappings such that: :$g$ is an extension of $f$ to $B$ :$h$ is an extension of $g$ to $C$. Then $h$ is an extension of $f$ to $C$."}
+{"_id": "17830", "title": "Indexed Cartesian Space is Set of all Mappings", "text": "Let $I$ be an indexing set. Let $\\displaystyle \\prod_{i \\mathop \\in I} S$ denote the cartesian space of $S$ indexed by $I$. Then $\\displaystyle \\prod_{i \\mathop \\in I} S$ is the set of all mappings from $I$ to $S$, and hence the notation: :$S^I := \\displaystyle \\prod_{i \\mathop \\in I} S$"}
+{"_id": "17832", "title": "Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 2", "text": "Let $M = \\struct {A, d}$ be a metric space or a pseudometric space. Let $\\sequence {x_k}$ be a sequence in $A$. {{TFAE|def = Convergent Sequence|context = Metric Space|contextview = Metric Spaces}}"}
+{"_id": "17833", "title": "Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 3", "text": "Let $M = \\struct {A, d}$ be a metric space or a pseudometric space. Let $\\sequence {x_k}$ be a sequence in $A$. {{TFAE|def = Convergent Sequence|context = Metric Space|contextview = Metric Spaces}}"}
+{"_id": "17834", "title": "Definite Integral from 0 to 1 of Logarithm of x over One minus x", "text": ":$\\displaystyle \\int_0^1 \\frac {\\ln x} {1 - x} \\rd x = -\\frac {\\pi^2} 6$"}
+{"_id": "17835", "title": "Definite Integral from 0 to 1 of Logarithm of One plus x over x", "text": ":$\\displaystyle \\int_0^1 \\frac {\\map \\ln {1 + x} } x \\rd x = \\frac {\\pi^2} {12}$"}
+{"_id": "17836", "title": "Definite Integral from 0 to 1 of Logarithm of One minus x over x", "text": ":$\\displaystyle \\int_0^1 \\frac {\\map \\ln {1 - x} } x \\rd x = -\\frac {\\pi^2} 6$"}
+{"_id": "17837", "title": "Definite Integral to Infinity of Exponential of -a x by Cosine of b x", "text": ":$\\displaystyle \\int_0^\\infty e^{-a x} \\cos b x \\rd x = \\frac a {a^2 + b^2}$"}
+{"_id": "17839", "title": "Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x", "text": ":$\\displaystyle \\int_0^\\infty \\frac {e^{-a x} - e^{-b x} } x \\rd x = \\ln \\frac b a$"}
+{"_id": "17840", "title": "Definite Integral to Infinity of Exponential of -a x^2", "text": ":$\\displaystyle \\int_0^\\infty e^{-a x^2} \\rd x = \\frac 1 2 \\sqrt {\\frac \\pi a}$"}
+{"_id": "17841", "title": "Definite Integral from 0 to 1 of Difference of Powers of x over Logarithm of x", "text": ":$\\displaystyle \\int_0^1 \\frac {x^m - x^n} {\\ln x} \\rd x = \\map \\ln {\\frac {m + 1} {n + 1} }$"}
+{"_id": "17842", "title": "Union is Commutative/Family of Sets", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be an indexed family of sets. Let $\\displaystyle I = \\bigcup_{i \\mathop \\in I} S_i$ denote the union of  $\\family {S_i}_{i \\mathop \\in I}$. Let $J \\subseteq I$ be a subset of $I$. Then: :$\\displaystyle \\bigcup_{i \\mathop \\in I} S_i = \\bigcup_{j \\mathop \\in J} S_j \\cup \\bigcup_{k \\mathop \\in \\relcomp I J} S_k = \\bigcup_{k \\mathop \\in \\relcomp I J} S_k \\cup \\bigcup_{j \\mathop \\in J} S_j$ where $\\relcomp I J$ denotes the complement of $J$ relative to $I$."}
+{"_id": "17843", "title": "Intersection is Commutative/Family of Sets", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be an indexed family of sets. Let $\\displaystyle I = \\bigcap_{i \\mathop \\in I} S_i$ denote the intersection of $\\family {S_i}_{i \\mathop \\in I}$. Let $J \\subseteq I$ be a subset of $I$. Then: :$\\displaystyle \\bigcap_{i \\mathop \\in I} S_i = \\bigcap_{j \\mathop \\in J} S_j \\cap \\bigcap_{k \\mathop \\in \\relcomp I J} S_k = \\bigcap_{k \\mathop \\in \\relcomp I J} S_k \\cap \\bigcap_{j \\mathop \\in J} S_j$ where $\\relcomp I J$ denotes the complement of $J$ relative to $I$."}
+{"_id": "17846", "title": "Definite Integral to Infinity of Power of x by Logarithm of x over One plus x", "text": ":$\\displaystyle \\int_0^\\infty \\frac {x^{p - 1} \\ln x} {1 + x} \\rd x = -\\pi^2 \\csc p \\pi \\cot p \\pi$"}
+{"_id": "17847", "title": "Definite Integral from 0 to Half Pi of Logarithm of Sine x", "text": ":$\\displaystyle \\int_0^{\\pi/2} \\map \\ln {\\sin x} \\rd x = -\\frac \\pi 2 \\ln 2$"}
+{"_id": "17848", "title": "Definite Integral from 0 to Half Pi of Logarithm of Cosine x", "text": ":$\\displaystyle \\int_0^{\\pi/2} \\map \\ln {\\cos x} \\rd x = -\\frac \\pi 2 \\ln 2$"}
+{"_id": "17849", "title": "Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary/P-adic Norm", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\sequence {x_n}$ be a Cauchy sequence such that $\\sequence {x_n}$ does not converge to $0$. Then: :$\\exists N \\in \\N: \\forall n, m \\ge N: \\norm {x_n}_p = \\norm {x_m}_p$"}
+{"_id": "17850", "title": "Definite Integral to Infinity of Logarithm of Exponential of x plus One over Exponential of x minus One", "text": ":$\\displaystyle \\int_0^\\infty \\map \\ln {\\frac {e^x + 1} {e^x - 1} } \\rd x = \\frac {\\pi^2} 4$"}
+{"_id": "17851", "title": "Definite Integral from 0 to Quarter Pi of Logarithm of One plus Tan x", "text": ":$\\displaystyle \\int_0^{\\pi/4} \\map \\ln {1 + \\tan x} \\rd x = \\frac \\pi 8 \\ln 2$"}
+{"_id": "17852", "title": "Primitive of Sine x by Logarithm of Sine x", "text": ":$\\ds \\int \\sin x \\map \\ln {\\sin x} \\rd x = \\cos x \\paren {1 - \\map \\ln {\\sin x} } + \\ln \\size {\\tan \\frac x 2} + C$"}
+{"_id": "17853", "title": "Euler-Mascheroni Constant as Difference of Integrals involving Cosine", "text": ":$\\displaystyle \\int_0^1 \\frac {1 - \\cos x} x \\rd x - \\int_1^\\infty \\frac {\\cos x} x \\rd x = \\gamma$"}
+{"_id": "17854", "title": "Definite Integral from 0 to 1 of Arcsine of x over x", "text": ":$\\displaystyle \\int_0^1 \\frac {\\arcsin x} x = \\frac \\pi 2 \\ln 2$"}
+{"_id": "17855", "title": "Definite Integral from 0 to 1 of Arctangent of x over x", "text": ":$\\displaystyle \\int_0^1 \\frac {\\arctan x} x \\rd x = G$"}
+{"_id": "17857", "title": "Definite Integral from 0 to Half Pi of x over Sine x", "text": ":$\\displaystyle \\int_0^{\\pi/2} \\frac x {\\sin x} \\rd x = 2 G$"}
+{"_id": "17858", "title": "Definite Integral to Infinity of Arctangent of p x minus Arctangent of q x over x", "text": ":$\\displaystyle \\int_0^\\infty \\frac {\\arctan p x - \\arctan q x} x \\rd x = \\frac \\pi 2 \\ln \\frac p q$"}
+{"_id": "17859", "title": "Definite Integral from 0 to Half Pi of Sine x by Logarithm of Sine x", "text": ":$\\displaystyle \\int_0^{\\pi/2} \\sin x \\map \\ln {\\sin x} \\rd x = \\ln 2 - 1$"}
+{"_id": "17862", "title": "Definite Integral to Infinity of Exponential of -(a x^2 plus b x plus c)", "text": ":$\\displaystyle \\int_0^\\infty \\map \\exp {-\\paren {a x^2 + b x + c} } \\rd x = \\frac 1 2 \\sqrt {\\frac \\pi a} \\map \\exp {\\frac {b^2 - 4 a c} {4 a} } \\map \\erfc {\\frac b {2 \\sqrt a} }$"}
+{"_id": "17863", "title": "Set is Subset of Intersection of Supersets", "text": "Let $S$, $T_1$ and $T_2$ be sets. Let $S$ be a subset of both $T_1$ and $T_2$. Then: :$S \\subseteq T_1 \\cap T_2$ That is: :$\\paren {S \\subseteq T_1} \\land \\paren {S \\subseteq T_2} \\implies S \\subseteq \\paren {T_1 \\cap T_2}$"}
+{"_id": "17867", "title": "Definite Integral to Infinity of Power of x by Exponential of -a x^2", "text": ":$\\displaystyle \\int_0^\\infty x^m e^{-a x^2} \\rd x = \\frac {\\map \\Gamma {\\paren {m + 1}/2} } {2 a^{\\paren {m + 1}/2} }$"}
+{"_id": "17870", "title": "Integral Representation of Dirichlet Eta Function in terms of Gamma Function", "text": ":$\\displaystyle \\map \\eta s = \\frac 1 {\\map \\Gamma s} \\int_0^\\infty \\frac {x^{s - 1} } {e^x + 1} \\rd x$"}
+{"_id": "17871", "title": "Definite Integral to Infinity of x over Exponential of x plus One", "text": ":$\\displaystyle \\int_0^\\infty \\frac x {e^x + 1} \\rd x = \\frac {\\pi^2} {12}$"}
+{"_id": "17872", "title": "Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x", "text": ":$\\displaystyle \\int_0^1 \\ln x \\map \\ln {1 - x} \\rd x = 2 - \\frac {\\pi^2} 6$"}
+{"_id": "17873", "title": "Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One plus x", "text": ":$\\displaystyle \\int_0^1 \\ln x \\map \\ln {1 + x} \\rd x = 2 - 2 \\ln 2 - \\frac {\\pi^2} {12}$"}
+{"_id": "17875", "title": "Power Series Expansion for Error Function", "text": ":$\\displaystyle \\map \\erf x = \\frac 2 {\\sqrt \\pi} \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {x^{2 n + 1} } {n! \\paren {2 n + 1} }$"}
+{"_id": "17876", "title": "Set is Subset of Intersection of Supersets/General Result", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of sets indexed by $I$. Let $X$ be a set such that: :$\\forall i \\in I: X \\subseteq S_i$ Then: :$X \\subseteq \\bigcup_{i \\mathop \\in I} S_i$ where $\\displaystyle \\bigcup_{i \\mathop \\in I} S_i$ is the intersection of $\\family {S_i}$."}
+{"_id": "17877", "title": "Error Function is Odd", "text": ":$\\map \\erf {-x} = -\\map \\erf x$"}
+{"_id": "17878", "title": "Error Function of Zero", "text": ":$\\map \\erf 0 = 0$"}
+{"_id": "17881", "title": "P-adic Expansion is a Cauchy Sequence in P-adic Norm", "text": "Let $p$ be a prime number. Let $\\norm {\\,\\cdot\\,}_p$ be the $p$-adic norm on the rationals numbers $\\Q$. Let $\\displaystyle \\sum_{n \\mathop = m}^\\infty d_n p^n$ be a $p$-adic expansion. Then the sequence of partial sums of the series: :$\\displaystyle \\sum_{n \\mathop = m}^\\infty d_n p^n$ is a Cauchy sequence in the valued field  $\\struct{\\Q, \\norm{\\,\\cdot\\,}_p}$."}
+{"_id": "17882", "title": "P-adic Expansion is a Cauchy Sequence in P-adic Norm/Converges to P-adic Number", "text": "Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers."}
+{"_id": "17883", "title": "Characterization of Exponential Integral Function", "text": ":$\\displaystyle \\map \\Ei x = -\\gamma - \\ln x + \\int_0^x \\frac {1 - e^{-u} } u \\rd u$"}
+{"_id": "17886", "title": "Membership Relation is Not Reflexive", "text": "Let $\\Bbb S$ be a set of sets in the context of pure set theory Let $\\RR$ denote the membership relation on $\\Bbb S$: :$\\forall \\tuple {a, b} \\in \\Bbb S \\times \\Bbb S: \\tuple {a, b} \\in \\RR \\iff a \\in b$ $\\RR$ is not in general a reflexive relation."}
+{"_id": "17888", "title": "Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\sequence{x_n}$ be a sequence in the rational numbers $\\Q$. Then $\\sequence {x_n}$ is a Cauchy sequence in $\\struct{\\Q, \\norm{\\,\\cdot\\,}_p}$ {{iff}} $\\sequence {x_n}$ is a Cauchy sequence in $\\struct{\\Q_p, \\norm{\\,\\cdot\\,}_p}$"}
+{"_id": "17889", "title": "Definite Integral to Infinity of Sine x over Root x", "text": ":$\\displaystyle \\int_0^\\infty \\frac {\\sin x} {\\sqrt x} \\rd x = \\sqrt {\\frac \\pi 2}$"}
+{"_id": "17890", "title": "Digamma Function of One Half", "text": ":$\\displaystyle \\map \\psi {\\frac 1 2} = -\\gamma - 2 \\ln 2$"}
+{"_id": "17891", "title": "Definite Integral to Infinity of Exponential of -x^2 by Logarithm of x", "text": ":$\\displaystyle \\int_0^\\infty e^{-x^2} \\ln x \\rd x = -\\frac {\\sqrt \\pi} 4 \\paren {\\gamma + 2 \\ln 2}$"}
+{"_id": "17892", "title": "Definite Integral to Infinity of Cosine x over Root x", "text": ":$\\displaystyle \\int_0^\\infty \\frac {\\cos x} {\\sqrt x} \\rd x = \\sqrt {\\frac \\pi 2}$"}
+{"_id": "17893", "title": "Definite Integral to Infinity of Cube of Sine x over x Cubed", "text": ":$\\displaystyle \\int_0^\\infty \\frac {\\sin^3 x} {x^3} \\rd x = \\frac {3 \\pi} 8$"}
+{"_id": "17894", "title": "Equivalence of Definitions of Composition of Mappings", "text": "{{TFAE|def = Composition of Mappings}} Let $f_1: S_1 \\to S_2$ and $f_2: S_2 \\to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$."}
+{"_id": "17896", "title": "P-adic Numbers are Generated Ring Extension of P-adic Integers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\Z_p$ be the $p$-adic integers. Then: :$Q_p = \\Z_p \\sqbrk {1 / p}$ where $\\Z_p \\sqbrk {1 / p}$ denotes the ring extension generated by $1 / p$."}
+{"_id": "17898", "title": "Limit to Infinity of Exponential Integral Function", "text": ":$\\displaystyle \\lim_{x \\mathop \\to \\infty} \\map \\Ei x = 0$"}
+{"_id": "17899", "title": "Definite Integral to Infinity of Power of x over Hyperbolic Sine of a x", "text": ":$\\displaystyle \\int_0^\\infty \\frac {x^n} {\\sinh a x} \\rd x = \\frac {2^{n + 1} - 1} {2^n a^{n + 1} } \\map \\Gamma {n + 1} \\map \\zeta {n + 1}$"}
+{"_id": "17901", "title": "Fourier Series/Identity Function over Symmetric Range", "text": "Let $\\lambda \\in \\R_{>0}$ be a strictly positive real number. Let $\\map f x: \\openint {-\\lambda} \\lambda \\to \\R$ be the identity function on the open real interval $\\openint {-\\lambda} \\lambda$: :$\\forall x \\in \\openint {-\\lambda} \\lambda: \\map f x = x$ The Fourier series of $f$ over $\\openint {-\\lambda} \\lambda$ can be given as: :$\\map f x \\sim \\dfrac {2 \\lambda} \\pi \\displaystyle \\sum_{n \\mathop = 1}^\\infty \\frac {\\paren {-1}^{n + 1} } n \\sin \\frac {n \\pi x} \\lambda$"}
+{"_id": "17902", "title": "Restriction of Mapping is its Intersection with Cartesian Product of Subset with Image", "text": "Let $f: S \\to T$ be a mapping. Let $X \\subseteq S$. Let $f {\\restriction_X}$ be the restriction of $f$ to $X$. Then: :$f {\\restriction_X} = f \\cap \\paren {X \\times \\Img f}$ where: :$\\Img f$ denotes the image of $f$, defined as: ::$\\Img f = \\set {t \\in T: \\exists s \\in S: t = \\map f s}$ :$X \\times \\Img f$ denotes the cartesian product of $X$ with $\\Img f$."}
+{"_id": "17903", "title": "Primitive of Reciprocal of One plus Fourth Power of x", "text": ":$\\displaystyle \\int \\frac 1 {1 + x^4} \\rd x = \\frac 1 {2 \\sqrt 2} \\paren {\\map \\arctan {\\frac 1 {\\sqrt 2} \\paren {x - \\frac 1 x} } + \\frac 1 2 \\ln \\size {\\frac {x^2 + \\sqrt 2 x + 1} {x^2 - \\sqrt 2 x + 1} } } + C$"}
+{"_id": "17905", "title": "Sine Integral Function is Odd", "text": ":$\\displaystyle \\map \\Si {-x} = -\\map \\Si x$"}
+{"_id": "17906", "title": "Sine Integral Function of Zero", "text": ":$\\map \\Si 0 = 0$"}
+{"_id": "17907", "title": "Arctangent of Root 3 over 3", "text": ":$\\map \\arctan {\\dfrac {\\sqrt 3} 3} = \\dfrac \\pi 6$"}
+{"_id": "17908", "title": "Cardinality of Set Union/2 Sets", "text": "Let $S_1$ and $S_2$ be finite sets. Then: :$\\card {S_1 \\cup S_2} = \\card {S_1} + \\card {S_2} - \\card {S_1 \\cap S_2}$"}
+{"_id": "17909", "title": "Cardinality of Set Union/3 Sets", "text": "Let $S_1$, $S_2$ and $S_3$ be finite sets. Then: {{begin-eqn}}      {{eqn | l = \\card {S_1 \\cup S_2 \\cup S_3}       | r = \\card {S_1} + \\card {S_2} + \\card {S_3}       | c = }} {{eqn | o =        | ro= -       | r = \\card {S_1 \\cap S_2} - \\card {S_1 \\cap S_3} - \\card {S_2 \\cap S_3}       | c = }} {{eqn | o =        | ro= +       | r = \\card {S_1 \\cap S_2 \\cap S_3}       | c = }} {{end-eqn}}"}
+{"_id": "17910", "title": "Definite Integral to Infinity of Exponential of -i x^2", "text": ":$\\displaystyle \\int_0^\\infty \\map \\exp {-i x^2} \\rd x = \\frac 1 2 \\sqrt {\\frac \\pi 2} \\paren {1 - i}$"}
+{"_id": "17911", "title": "P-adic Number times Integer Power of p is P-adic Integer", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\Z_p$ be the $p$-adic integers. Then: :$\\forall a \\in \\Q_p: \\exists n \\in \\N: p^n a \\in \\Z_p$"}
+{"_id": "17912", "title": "Definite Integral to Infinity of Sine of a x^2", "text": ":$\\ds \\int_0^\\infty \\map \\sin {a x^2} \\rd x = \\frac 1 2 \\sqrt {\\frac \\pi {2 a} }$"}
+{"_id": "17913", "title": "Definite Integral to Infinity of Cosine of a x^2", "text": ":$\\displaystyle \\int_0^\\infty \\map \\cos {a x^2} \\rd x = \\frac 1 2 \\sqrt {\\frac \\pi {2 a} }$"}
+{"_id": "17915", "title": "Definite Integral to Infinity of Hyperbolic Sine of a x over Exponential of b x minus One", "text": ":$\\displaystyle \\int_0^\\infty \\frac {\\sinh a x} {e^{b x} - 1} \\rd x = \\frac 1 {2 a} - \\frac \\pi {2 b} \\cot \\frac {a \\pi} b$"}
+{"_id": "17918", "title": "Union with Intersection equals Intersection with Union iff Subset", "text": "Let $A$, $B$ and $C$ be sets. Then: :$\\paren {A \\cap B} \\cup C = A \\cap \\paren {B \\cup C} \\iff C \\subseteq A$"}
+{"_id": "17919", "title": "(A cap C) cup (B cap Complement C) = Empty iff B subset C subset Complement A", "text": "Let $A$, $B$ and $C$ be subsets of a universe $\\Bbb U$. Then: :$\\paren {A \\cap C} \\cup \\paren {B \\cap \\map \\complement C} = \\O \\iff B \\subseteq C \\subseteq \\map \\complement A$ where $\\map \\complement C$ denotes the complement of $C$ in $\\Bbb U$."}
+{"_id": "17920", "title": "Union of Intersections of 2 from 3 equals Intersection of Unions of 2 from 3", "text": "Let $A$, $B$ and $C$ be sets. Then: :$\\paren {A \\cap B} \\cup \\paren {B \\cap C} \\cup \\paren {C \\cap A} = \\paren {A \\cup B} \\cap \\paren {B \\cup C} \\cap \\paren {C \\cup A}$"}
+{"_id": "17922", "title": "Definite Integral to Infinity of Cosine of a x over Hyperbolic Cosine of b x", "text": ":$\\displaystyle \\int_0^\\infty \\frac {\\cos a x} {\\cosh b x} \\rd x = \\frac \\pi {2 b} \\sech \\frac {a \\pi} {2 b}$"}
+{"_id": "17923", "title": "Definite Integral to Infinity of Hyperbolic Sine of a x over Exponential of b x plus One", "text": ":$\\displaystyle \\int_0^\\infty \\frac {\\sinh a x} {e^{b x} + 1} \\rd x = \\frac \\pi {2 b} \\csc \\frac {a \\pi} b - \\frac 1 {2 a}$"}
+{"_id": "17926", "title": "Intersection Complement of Set with Itself is Complement", "text": "Let $A$ and $B$ be subsets of a universal set $\\Bbb U$. Let $\\uparrow$ denote the operation on $A$ and $B$ defined as: :$\\paren {A \\uparrow B} \\iff \\paren {\\relcomp {\\Bbb U} {A \\cap B} }$ where $\\relcomp {\\Bbb U} A$ denotes the complement of $A$ in $\\Bbb U$. Then: :$A \\uparrow A = \\relcomp {\\Bbb U} A$"}
+{"_id": "17928", "title": "Set Union expressed as Intersection Complement", "text": "Let $A$ and $B$ be subsets of a universal set $\\Bbb U$. Let $\\uparrow$ denote the operation on $A$ and $B$ defined as: :$\\paren {A \\uparrow B} \\iff \\paren {\\relcomp {\\Bbb U} {A \\cap B} }$ where $\\relcomp {\\Bbb U} A$ denotes the complement of $A$ in $\\Bbb U$. Then: :$A \\cup B = \\paren {A \\uparrow A} \\uparrow \\paren {B \\uparrow B}$"}
+{"_id": "17929", "title": "Element in Set iff Singleton in Powerset", "text": "Let $S$ be a set. Then: :$x \\in S \\iff \\set x \\in \\powerset S$ where $\\powerset S$ denotes the power set of $S$."}
+{"_id": "17931", "title": "Set Consisting of Empty Set is not Empty", "text": "Let $S$ be the set defined as: :$S = \\set \\O$ Then $S$ is not the empty set. That is: :$\\O \\ne \\set \\O$"}
+{"_id": "17932", "title": "Elements of Ordered Pair do not Commute", "text": "Let $\\set {a, b}$ be a doubleton, so that $a$ and $b$ are distinct objects. Let $\\tuple {a, b}$ denote the ordered pair such that the first coordinate is $a$ and the second coordinate is $b$. Then: :$\\tuple {a, b} \\ne \\tuple {b, a}$"}
+{"_id": "17933", "title": "Definite Integral to Infinity of Cosine m x over x Squared plus a Squared", "text": ":$\\displaystyle \\int_0^\\infty \\frac {\\cos m x} {x^2 + a^2} \\rd x = \\frac \\pi {2 a} e^{-m a}$"}
+{"_id": "17935", "title": "Combination Theorem for Continuous Mappings/Normed Division Ring", "text": "Let $\\struct{S, \\tau_{_S}}$ be a topological space. Let $\\struct{R, +, *, \\norm{\\,\\cdot\\,}}$ be a normed division ring. Let $\\tau_{_R}$ be the topology induced by the norm $\\norm{\\,\\cdot\\,}$. Let $\\lambda \\in R$. Let $f,g : \\struct{S, \\tau_{_S}} \\to \\struct{R, \\tau_{_R}}$ be continuous mappings. Let $U = S \\setminus \\set{x : \\map g x = 0}$ Let $g^{-1} : U \\to R$ denote the mapping defined by: :$\\forall x \\in U : \\map {g^{-1}} x = \\map g x^{-1}$ Let $\\tau_{_U}$ be the subspace topology on $U$. Then the following results hold: === Sum Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Sum Rule}} === Translation Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Translation Rule}} === Negation Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Negation Rule}} === Product Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Product Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Multiple Rule}} === Inverse Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Inverse Rule}}"}
+{"_id": "17936", "title": "Combination Theorem for Continuous Mappings/Normed Division Ring/Sum Rule", "text": ":$f + g: \\struct {S, \\tau_{_S} } \\to \\struct{R, \\tau_{_R} }$ is continuous."}
+{"_id": "17937", "title": "Combination Theorem for Continuous Mappings/Normed Division Ring/Multiple Rule", "text": ":$\\lambda * f: \\struct {S, \\tau_{_S} } \\to \\struct {R, \\tau_{_R} }$ is continuous :$f * \\lambda: \\struct {S, \\tau_{_S} } \\to \\struct {R, \\tau_{_R} }$ is continuous."}
+{"_id": "17938", "title": "Combination Theorem for Continuous Mappings/Normed Division Ring/Product Rule", "text": ":$f * g: \\struct {S, \\tau_{_S} } \\to \\struct {R, \\tau_{_R} }$ is continuous."}
+{"_id": "17939", "title": "Combination Theorem for Continuous Mappings/Normed Division Ring/Inverse Rule", "text": ":$g^{-1}: \\struct {U, \\tau_{_U} } \\to \\struct {R, \\tau_{_R} }$ is continuous."}
+{"_id": "17945", "title": "Derivative of Cosine Integral Function", "text": ":$\\displaystyle \\frac \\d {\\d x} \\paren {\\map \\Ci x} = -\\frac {\\cos x} x$"}
+{"_id": "17946", "title": "Derivative of Exponential Integral Function", "text": ":$\\displaystyle \\frac \\d {\\d x} \\paren {\\map \\Ei x} = -\\frac {e^{-x} } x$"}
+{"_id": "17947", "title": "Derivative of Sine Integral Function", "text": ":$\\displaystyle \\frac \\d {\\d x} \\paren {\\map \\Si x} = \\frac {\\sin x} x$"}
+{"_id": "17949", "title": "Primitive of Exponential Integral Function", "text": ":$\\displaystyle \\int \\map \\Ei x \\rd x = x \\map \\Ei x - e^{-x} + C$"}
+{"_id": "17950", "title": "Direct Image Mapping of Domain is Image Set of Mapping", "text": "Let $S$ and $T$ be sets. Let $\\powerset S$ and $\\powerset T$ be their power sets. Let $f \\subseteq S \\times T$ be a mapping from $S$ to $T$. Let $f^\\to: \\powerset S \\to \\powerset T$ be the direct image mapping of $f$: :$\\forall X \\in \\powerset S: \\map {f^\\to} X = \\begin {cases} \\set {t \\in T: \\exists s \\in X: \\map f s = t} & : X \\ne \\O \\\\ \\O & : X = \\O \\end {cases}$ Then: :$\\map {f^\\to} S = \\Img f$ where $\\Img f$ is the image set of $f$."}
+{"_id": "17951", "title": "Direct Image Mapping of Domain is Image Set of Relation", "text": "Let $S$ and $T$ be sets. Let $\\powerset S$ and $\\powerset T$ be their power sets. Let $\\mathcal R \\subseteq S \\times T$ be a relation on $S \\times T$. Let $\\mathcal R^\\to: \\powerset S \\to \\powerset T$ be the direct image mapping of $\\mathcal R$: :$\\forall X \\in \\powerset S: \\map {\\mathcal R^\\to} X = \\begin {cases} \\set {t \\in T: \\exists s \\in X: \\tuple {x, t} \\in \\mathcal R} & : X \\ne \\O \\\\ \\O & : X = \\O \\end {cases}$ Then: :$\\map {\\mathcal R^\\to} {\\Dom {\\mathcal R} } = \\Img {\\mathcal R}$ where: :$\\Dom {\\mathcal R}$ is the domain of $\\mathcal R$ :$\\Img {\\mathcal R}$ is the image set of $\\mathcal R$."}
+{"_id": "17952", "title": "Inverse Image Mapping of Codomain is Preimage Set of Mapping", "text": "Let $S$ and $T$ be sets. Let $\\powerset S$ and $\\powerset T$ be their power sets. Let $f \\subseteq S \\times T$ be a mapping from $S$ to $T$. Let $f^\\gets: \\powerset T \\to \\powerset S$ be the inverse image mapping of $f$: :$\\forall Y \\in \\powerset T: \\map {f^\\gets} Y = \\begin {cases} \\set {s \\in S: \\exists t \\in Y: \\map f s = t} & : \\Img f \\cap Y \\ne \\O \\\\ \\O & : \\Img f \\cap Y = \\O \\end {cases}$ Then: :$\\map {f^\\gets} T = \\Preimg f$ where $\\Preimg f$ is the preimage set of $f$."}
+{"_id": "17953", "title": "Inverse Image Mapping of Codomain is Preimage Set of Relation", "text": "Let $S$ and $T$ be sets. Let $\\powerset S$ and $\\powerset T$ be their power sets. Let $\\mathcal R \\subseteq S \\times T$ be a relation on $S \\times T$. Let $\\mathcal R^\\gets: \\powerset T \\to \\powerset S$ be the inverse image mapping of $\\mathcal R$: :$\\forall X \\in \\powerset S: \\map {\\mathcal R^\\to} X = \\begin {cases} \\set {t \\in T: \\exists s \\in X: \\tuple {x, t} \\in \\mathcal R} & : X \\ne \\O \\\\ \\O & : X = \\O \\end {cases}$ Then: :$\\map {\\mathcal R^\\gets} T = \\Preimg {\\mathcal R}$ where $\\Preimg {\\mathcal R}$ is the preimage of $\\mathcal R$."}
+{"_id": "17956", "title": "Direct Image Mapping of Mapping is Mapping", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping on $S \\times T$. Let $f^\\to: \\powerset S \\to \\powerset T$ be the direct image mapping of $f$: :$\\forall X \\in \\powerset S: \\map {f^\\to} X = \\set {t \\in T: \\exists s \\in X: \\map f s = t}$ Then $f^\\to$ is indeed a mapping."}
+{"_id": "17957", "title": "Inverse Image Mapping of Mapping is Mapping", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping from $S$ to $T$. Let $f^\\gets$ be the inverse image mapping of $f$: :$f^\\gets: \\powerset T \\to \\powerset S: \\map {f^\\gets} Y = f^{-1} \\sqbrk Y$ Then $f^\\gets$ is indeed a mapping."}
+{"_id": "17960", "title": "Direct Image Mapping of Mapping is Empty iff Argument is Empty", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping from $S$ to $T$. Let $f^\\to$ be the direct image mapping of $f$: :$f^\\to: \\powerset S \\to \\powerset T: \\map {f^\\to} X = \\set {t \\in T: \\exists s \\in X: \\map f s = t}$ Then: :$\\map {f^\\to} X = \\O \\iff X = \\O$"}
+{"_id": "17961", "title": "Derivative of Fresnel Sine Integral Function", "text": ":$\\displaystyle \\frac {\\d \\operatorname S} {\\d x} = \\sqrt {\\frac 2 \\pi} \\sin x^2$"}
+{"_id": "17962", "title": "Derivative of Fresnel Cosine Integral Function", "text": ":$\\displaystyle \\frac {\\d \\operatorname C} {\\d x} = \\sqrt {\\frac 2 \\pi} \\cos x^2$"}
+{"_id": "17964", "title": "Power Series Expansion for Fresnel Cosine Integral Function", "text": ":$\\displaystyle \\map {\\operatorname C} x = \\sqrt {\\frac 2 \\pi} \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {x^{4 n + 1} } {\\paren {4 n + 1} \\paren {2 n}!}$"}
+{"_id": "17965", "title": "Fresnel Sine Integral Function is Odd", "text": ":$\\map {\\operatorname S} {-x} = -\\map {\\operatorname S} x$"}
+{"_id": "17966", "title": "Fresnel Sine Integral Function of Zero", "text": ":$\\map {\\operatorname S} 0 = 0$"}
+{"_id": "17969", "title": "Fresnel Cosine Integral Function is Odd", "text": ":$\\map {\\operatorname C} {-x} = -\\map {\\operatorname C} x$"}
+{"_id": "17970", "title": "Fresnel Cosine Integral Function of Zero", "text": ":$\\map {\\operatorname C} 0 = 0$"}
+{"_id": "17971", "title": "Direct Image Mapping of Left-Total Relation is Empty iff Argument is Empty", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R: S \\to T$ be a left-total relation on $S \\times T$. Let $\\mathcal R^\\to$ be the direct image mapping of $\\mathcal R$: :$\\mathcal R^\\to: \\powerset S \\to \\powerset T: \\map {\\mathcal R^\\to} X = \\set {t \\in T: \\exists s \\in X: \\tuple {s, t} \\in \\mathcal R}$ Then: :$\\map {\\mathcal R^\\to} X = \\O \\iff X = \\O$"}
+{"_id": "17972", "title": "Limit to Infinity of Fresnel Cosine Integral Function", "text": ":$\\displaystyle \\lim_{x \\mathop \\to \\infty} \\map {\\operatorname C} x = \\frac 1 2$"}
+{"_id": "17974", "title": "Composition of Direct Image Mappings of Mappings", "text": "Let $A, B, C$ be non-empty sets. Let $f: A \\to B$ and $g: B \\to C$ be mappings. Let: :$f^\\to: \\powerset A \\to \\powerset B$ and :$g^\\to: \\powerset B \\to \\powerset C$ be the direct image mappings of $f$ and $g$. Then: :$\\paren {g \\circ f}^\\to = g^\\to \\circ f^\\to$"}
+{"_id": "17975", "title": "Composition of Inverse Image Mappings of Mappings", "text": "Let $A, B, C$ be non-empty sets. Let $f: A \\to B, g: B \\to C$ be mappings. Let: :$f^\\gets: \\powerset B \\to \\powerset A$ and :$g^\\gets: \\powerset C \\to \\powerset B$ be the inverse image mappings of $f$ and $g$. Then: :$\\paren {g \\circ f}^\\gets = f^\\gets \\circ g^\\gets$"}
+{"_id": "17977", "title": "Unit of Ring of Mappings iff Image is Subset of Ring Units", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity $1_R$. Let $U_R$ be the set of units in $R$. Let $S$ be a set. Let $\\struct {R^S, +', \\circ'}$ be the ring of mappings on the set of mappings $R^S$. Then: :$f \\in R^S$ is a unit of $R^S$ {{iff}} $\\Img f \\subseteq U_R$ where $\\Img f$ is the image of $f$. In this case, the inverse of $f$ is the mapping $f^{-1} : S \\to U_R$ defined by: :$\\forall x \\in S : \\map {f^{-1} } x = \\map f x^{-1}$"}
+{"_id": "17979", "title": "Relative Complement Mapping on Powerset is Bijection", "text": "Let $S$ be a set. Let $\\complement_S: \\powerset S \\to \\powerset S$ denote the relative complement mapping on the power set of $S$. Then $\\complement_S$ is a bijection. Thus each $T \\subseteq S$ is in one-to-one correspondence with its relative complement."}
+{"_id": "17981", "title": "Power of One plus x in terms of Gaussian Hypergeometric Function", "text": ":$\\displaystyle {}_2 \\map {F_1} {-p, 1; 1; -x} = \\paren {1 + x}^p$"}
+{"_id": "17983", "title": "Laplace Transform of Exponential times Sine", "text": ":$\\map {\\laptrans {e^{b t} \\sin a t} } s = \\dfrac a {\\paren {s - b}^2 + a^2}$"}
+{"_id": "17987", "title": "Inverse Image of Direct Image of Inverse Image equals Inverse Image Mapping", "text": "Let $f: S \\to T$ be a mapping. Let: :$f^\\to: \\powerset S \\to \\powerset T$ denote the direct image mapping of $f$ :$f^\\gets: \\powerset T \\to \\powerset S$ denote the inverse image mapping of $f$ where $\\powerset S$ denotes the power set of $S$. Then: :$f^\\gets \\circ f^\\to \\circ f^\\gets = f^\\gets$ where $\\circ$ denotes composition of mappings."}
+{"_id": "17989", "title": "Graph of Real Function in Cartesian Plane intersects Vertical at One Point", "text": "Let $f: \\R \\to \\R$ be a real function. Let its graph be embedded in the Cartesian plane $\\mathcal C$: :520px Every vertical line through a point $a$ in the domain of $f$ intersects the graph of $f$ at exactly one point $P = \\tuple {a, \\map f a}$."}
+{"_id": "17990", "title": "Equation of Vertical Line", "text": "Let $\\mathcal L$ be a vertical line embedded in the Cartesian plane $\\mathcal C$. Then the equation of $\\mathcal L$ can be given by: :$x = a$ where $\\tuple {a, 0}$ is the point at which $\\mathcal L$ intersects the $x$-axis. :520px"}
+{"_id": "17991", "title": "Equation of Horizontal Line", "text": "Let $\\mathcal L$ be a horizontal line embedded in the Cartesian plane $\\mathcal C$. Then the equation of $\\mathcal L$ can be given by: :$y = b$ where $\\tuple {0, b}$ is the point at which $\\mathcal L$ intersects the $y$-axis. :520px"}
+{"_id": "17992", "title": "Unit of Ring of Mappings iff Image is Subset of Ring Units/Image is Subset of Ring Units implies Unit of Ring of Mappings", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity $1_R$. Let $U_R$ be the set of units in $R$. Let $S$ be a set. Let $\\struct {R^S, +', \\circ'}$ be the ring of mappings on the set of mappings $R^S$. Let $\\Img f \\subseteq U_R$ where $\\Img f$ is the image of $f$. Then: :$f \\in R^S$ is a unit of $R^S$ and the inverse of $f$ is the mapping $f^{-1} : S \\to U_R$ defined by: :$\\forall x \\in S : \\map {f^{-1}} {x} = \\map f x^{-1}$"}
+{"_id": "17993", "title": "Unit of Ring of Mappings iff Image is Subset of Ring Units/Unit of Ring of Mappings implies Image is Subset of Ring Units", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity $1_R$. Let $U_R$ be the set of units in $R$. Let $S$ be a set. Let $\\struct {R^S, +', \\circ'}$ be the ring of mappings on the set of mappings $R^S$. Let $f \\in R^S$ be a unit of $R^S$. Then: :$\\Img f \\subseteq U_R$ where $\\Img f$ is the image of $f$. In which case, the inverse of $f$ is the mapping $f^{-1} : S \\to U_R$ defined by: :$\\forall x \\in S : \\map {f^{-1} } x = \\map f x^{-1}$"}
+{"_id": "17994", "title": "Graph of Real Surjection in Coordinate Plane intersects Every Horizontal Line", "text": "Let $f: \\R \\to \\R$ be a real function which is surjective. Let its graph be embedded in the Cartesian plane $\\mathcal C$: :520px Every horizontal line through a point $b$ in the codomain of $f$ intersects the graph of $f$ on at least one point $P = \\tuple {a, b}$ where $b = \\map f a$."}
+{"_id": "17995", "title": "Graph of Real Injection in Coordinate Plane intersects Horizontal Line at most Once", "text": "Let $f: \\R \\to \\R$ be a real function which is injective. Let its graph be embedded in the Cartesian plane $\\mathcal C$: :520px Let $\\mathcal L$ be a horizontal line through a point $b$ in the codomain of $f$. Then $\\mathcal L$ intersects the graph of $f$ on at most one point $P = \\tuple {a, b}$ where $b = \\map f a$."}
+{"_id": "17996", "title": "Graph of Real Bijection in Coordinate Plane intersects Horizontal Line at One Point", "text": "Let $f: \\R \\to \\R$ be a real function which is bijective. Let its graph be embedded in the Cartesian plane $\\mathcal C$: :520px Every horizontal line through a point $b$ in the codomain of $f$ intersects the graph of $f$ on exactly one point $P = \\tuple {a, b}$ where $b = \\map f a$."}
+{"_id": "17998", "title": "Composition of 3 Mappings where Pairs of Mappings are Bijections", "text": "Let $A$, $B$, $C$ and $D$ be sets. Let: :$f: A \\to B$ :$g: B \\to C$ :$h: C \\to D$ be mappings. Let $g \\circ f$ and $h \\circ g$ be bijections. Then $f$, $g$ and $h$ are all bijections."}
+{"_id": "17999", "title": "Mapping Composed with Bijection forming Bijection is Bijection", "text": "Let $A$, $B$ and $C$ be sets. Let $f: A \\to B$ and $g: B \\to C$ be mappings. Let the composite mapping $g \\circ f$ be a bijection. Let either $f$ or $g$ be a bijection. Then both $f$ and $g$ are bijections."}
+{"_id": "18000", "title": "Composite of Three Mappings in Cycle forming Injections and Surjection", "text": "Let $A$, $B$ and $C$ be non-empty sets. Let $f: A \\to B$, $g: B \\to C$ and $h: C \\to A$ be mappings. Let the following hold: :$h \\circ g \\circ f$ is an injection :$f \\circ h \\circ g$ is an injection :$g \\circ f \\circ h$ is a surjection. where: :$g \\circ f$ (and so on) denote composition of mappings. Then each of $f$, $g$ and $h$ are bijections."}
+{"_id": "18001", "title": "Component Mappings of Set Coproduct are Injective", "text": "Let $S_1$ and $S_2$ be sets. Let $\\struct {C, i_1, i_2}$ be a coproduct of $S_1$ and $S_2$. Then $i_1$ and $i_2$ are injections."}
+{"_id": "18003", "title": "Existence of Bijection between Coproducts of two Sets", "text": "Let $S_1$ and $S_2$ be sets. Let $\\struct {C, i_1, i_2}$ and $\\struct {D, j_1, j_2}$ be two coproducts on $S_1$ and $S_2$. Then there exists a unique bijection $\\theta: D \\to C$ such that: :$\\theta \\circ j_i = i_1$ :$\\theta \\circ j_2 = i_2$"}
+{"_id": "18004", "title": "Number of Friday 13ths in a Year", "text": "In any given year, there are between $1$ and $3$ (inclusive) months in which the $13$th falls on a Friday."}
+{"_id": "18005", "title": "Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Equivalence Class Contains 1 Odd Number", "text": "Let $\\eqclass n \\alpha$ be the $\\alpha$-equivalence class of a natural number $n$. Then $\\eqclass n \\alpha$ contains exactly $1$ odd number."}
+{"_id": "18006", "title": "Equivalence Relation on Natural Numbers such that Quotient is Power of Two/One of Pair of Equivalent Elements is Divisor of the Other", "text": "Let $c, d \\in \\N$ such that $c \\mathrel \\alpha d$. Then either: :$c \\divides d$ or: :$d \\divides c$ where $\\divides$ denotes divisibility."}
+{"_id": "18007", "title": "Exists Divisor in Set of n+1 Natural Numbers no greater than 2n", "text": "Let $S$ be a set of $n + 1$ non-non-zero natural numbers all less than or equal to $2 n$. Then there exists $a, b \\in S$ such that :$a \\divides b$ where $\\divides$ denotes divisibility."}
+{"_id": "18008", "title": "Equality of Squares Modulo Integer is Equivalence Relation", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\RR_n$ be the relation on the set of integers $\\Z$ defined as: :$\\forall x, y \\in \\Z: x \\mathrel {\\RR_n} y \\iff x^2 \\equiv y^2 \\pmod n$ Then $\\RR_n$ is an equivalence relation."}
+{"_id": "18009", "title": "Left Ideal is Left Module over Ring", "text": "Let $\\struct {R, +, \\times}$ be a ring. Let $J \\subseteq R$ be a left ideal of $R$. Let $\\circ : R \\times J \\to J$ be the restriction of $\\times$ to $R \\times J$. Then $\\struct {J, +, \\circ}$ is a left module over $\\struct {R, +, \\times}$."}
+{"_id": "18010", "title": "Right Ideal is Right Module over Ring", "text": "Let $\\struct {R, +, \\times}$ be a ring. Let $J \\subseteq R$ be a right ideal of $R$. Let $\\circ : J \\times R \\to J$ be the restriction of $\\times$ to $J \\times R$. Then $\\struct {J, +, \\circ}$ is a right module over $\\struct {R, +, \\times}$."}
+{"_id": "18011", "title": "Opposite Ring is Ring", "text": "Let $\\struct {R, +, \\times}$ be a ring. Let $\\struct {R, +, *}$ be the opposite ring of $\\struct {R, +, \\times}$. Then $\\struct {R, +, *}$ is a ring."}
+{"_id": "18012", "title": "Opposite Ring of Opposite Ring", "text": "Let $\\struct {R, +, \\times}$ be a ring. Let $\\struct {R, +, *}$ be the opposite ring of $\\struct {R, +, \\times}$. Let $\\struct {R, +, \\circ}$ be the opposite ring of $\\struct {R, +, *}$. Then $\\struct {R, +, \\circ} = \\struct {R, +, \\times}$."}
+{"_id": "18013", "title": "Left Module over Ring Induces Right Module over Opposite Ring", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {R, +_R, *_R}$ be the opposite ring of $\\struct {R, +_R, \\times_R}$. Let $\\struct{G, +_G, \\circ}$ be a left module over $\\struct {R, +_R, \\times_R}$. Let $\\circ’ : G \\times R \\to G$ be the binary operation defined by: :$\\forall \\lambda \\in R: \\forall x \\in G: x \\circ’ \\lambda = \\lambda \\circ x$ Then $\\struct{G, +_G, \\circ’}$ is a right module over $\\struct {R, +_R, *_R}$."}
+{"_id": "18014", "title": "Right Module over Ring Induces Left Module over Opposite Ring", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {R, +_R, *_R}$ be the opposite ring of $\\struct {R, +_R, \\times_R}$. Let $\\struct{G, +_G, \\circ}$ be a right module over $\\struct {R, +_R, \\times_R}$. Let $\\circ’ : R \\times G \\to G$ be the binary operation defined by: :$\\forall \\lambda \\in R: \\forall x \\in G: \\lambda \\circ’ x = x \\circ \\lambda $ Then $\\struct{G, +_G, \\circ’}$ is a left module over $\\struct {R, +_R, *_R}$."}
+{"_id": "18015", "title": "Ring is Commutative iff Opposite Ring is Itself", "text": "Let $\\struct {R, +, \\times}$ be a ring. Let $\\struct {R, +, *}$ be the opposite ring of $\\struct {R, +, \\times}$. Then $\\struct {R, +, \\times}$ is a commutative ring {{iff}}: :$\\struct {R, +, \\times} = \\struct {R, +, *}$"}
+{"_id": "18017", "title": "Left Module induces Right Module over same Ring iff Actions are Commutative", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {G, +_G, \\circ}$ be a left module over $\\struct {R, +_R, \\times_R}$. Let $\\circ': G \\times R \\to G$ be the binary operation defined by: :$\\forall \\lambda \\in R: \\forall x \\in G: x \\circ' \\lambda = \\lambda \\circ x$ Then $\\struct {G, +_G, \\circ'}$ is a right module over $\\struct {R, +_R, \\times_R}$ {{iff}}: :$\\forall \\lambda, \\mu \\in R: \\forall x \\in G: \\paren {\\lambda \\times_R \\mu} \\circ x = \\paren {\\mu \\times_R \\lambda} \\circ x$"}
+{"_id": "18018", "title": "Right Module induces Left Module over same Ring iff Actions are Commutative", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {G, +_G, \\circ}$ be a right module over $\\struct {R, +_R, \\times_R}$. Let $\\circ' : R \\times G \\to G$ be the binary operation defined by: :$\\forall \\lambda \\in R: \\forall x \\in G: \\lambda \\circ' x = x \\circ \\lambda $ Then $\\struct {G, +_G, \\circ'}$ is a left module over $\\struct {R, +_R, \\times_R}$ {{iff}}: :$\\forall \\lambda, \\mu \\in R: \\forall x \\in G: x \\circ \\paren{ \\lambda \\times_R \\mu} = x \\circ \\paren {\\mu \\times_R \\lambda}$"}
+{"_id": "18019", "title": "Euler's Theorem for Planar Graphs", "text": "Let $G = \\struct {V, E}$ be a connected planar graph with $V$ vertices and $E$ edges. Let $F$ be the number of faces of $G$. Then: :$V - E + F = 2$"}
+{"_id": "18020", "title": "Fermat Problem", "text": "Let $\\triangle ABC$ be a triangle Let the vertices of $\\triangle ABC$ all have angles less than $120 \\degrees$. Let $\\triangle ABG$, $\\triangle BCE$ and $\\triangle ACF$ be equilateral triangles constructed on the sides of $ABC$. Let $AE$, $BF$ and $CG$ be constructed. Let $P$ be the point at which $AE$, $BF$ and $CG$ meet. :500px Then $P$ is the Fermat-Torricelli point of $\\triangle ABC$. If one of vertices of $\\triangle ABC$ be of $120 \\degrees$ or more, then that vertex is itself the Fermat-Torricelli point of $\\triangle ABC$."}
+{"_id": "18022", "title": "Left Module over Commutative Ring induces Right Module", "text": "Let $\\struct {R, +_R, \\times_R}$ be a commutative ring. Let $\\struct{G, +_G, \\circ}$ be a left module over $\\struct {R, +_R, \\times_R}$. Let $\\circ’ : G \\times R \\to G$ be the binary operation defined by: :$\\forall \\lambda \\in R: \\forall x \\in G: x \\circ’ \\lambda = \\lambda \\circ x$ Then $\\struct{G, +_G, \\circ’}$ is a right module over $\\struct {R, +_R, \\times_R}$."}
+{"_id": "18023", "title": "Right Module over Commutative Ring induces Left Module", "text": "Let $\\struct {R, +_R, \\times_R}$ be a commutative ring. Let $\\struct{G, +_G, \\circ}$ be a right module over $\\struct {R, +_R, \\times_R}$. Let $\\circ' : R \\times G \\to G$ be the binary operation defined by: :$\\forall \\lambda \\in R: \\forall x \\in G: \\lambda \\circ’ x = x \\circ \\lambda$ Then $\\struct{G, +_G, \\circ'}$ is a left module over $\\struct {R, +_R, \\times_R}$."}
+{"_id": "18024", "title": "Right Ideal is Right Module over Ring/Ring is Right Module over Ring", "text": "Let $\\struct {R, +, \\times}$ be a ring. Then $\\struct {R, +, \\times}$ is a right module over $\\struct {R, +, \\times}$."}
+{"_id": "18028", "title": "Right Module Does Not Necessarily Induce Left Module over Ring", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct{G, +_G, \\circ}$ be a right module over $\\struct {R, +_R, \\times_R}$. Let $\\circ' : R \\times G \\to G$ be the binary operation defined by: :$\\forall \\lambda \\in R: \\forall x \\in G: \\lambda \\circ' x = x \\circ \\lambda$ Then $\\struct{G, +_G, \\circ'}$ is not necessarily a left module over $\\struct {R, +_R, \\times_R}$"}
+{"_id": "18029", "title": "Left Module Does Not Necessarily Induce Right Module over Ring/Lemma", "text": ":$G$ is a left ideal of $\\struct {\\map {\\MM_S} 2, +, \\times}$."}
+{"_id": "18030", "title": "Right Module Does Not Necessarily Induce Left Module over Ring/Lemma", "text": ":$G$ is a right ideal of $\\struct {\\map {\\MM_S} 2, +, \\times}$."}
+{"_id": "18031", "title": "Ideal is Bimodule over Ring", "text": "Let $\\struct {R, +, \\times}$ be a ring. Let $J \\subseteq R$ be an ideal of $R$. Let $\\circ_l : R \\times J \\to J$ be the restriction of $\\times$ to $R \\times J$. Let $\\circ_r : J \\times R \\to J$ be the restriction of $\\times$ to $J \\times R$. Then $\\struct {J, +, \\circ_l, \\circ_r}$ is a bimodule over $\\struct {R, +, \\times}$."}
+{"_id": "18032", "title": "Ideal is Bimodule over Ring/Ring is Bimodule over Ring", "text": "Let $\\struct {R, +, \\times}$ be a ring. Then $\\struct {R, +, \\times, \\times}$ is a bimodule over $\\struct {R, +, \\times}$."}
+{"_id": "18033", "title": "Left Module over Commutative Ring induces Bimodule", "text": "Let $\\struct {R, +_R, \\times_R}$ be a commutative ring. Let $\\struct{G, +_G, \\circ}$ be a left module over $\\struct {R, +_R, \\times_R}$. Let $\\circ’ : G \\times R \\to G$ be the binary operation defined by: :$\\forall \\lambda \\in R: \\forall x \\in G: x \\circ’ \\lambda = \\lambda \\circ x$ Then $\\struct{G, +_G, \\circ, \\circ’}$ is a bimodule over $\\struct {R, +_R, \\times_R}$."}
+{"_id": "18034", "title": "Right Module over Commutative Ring induces Bimodule", "text": "Let $\\struct {R, +_R, \\times_R}$ be a commutative ring. Let $\\struct{G, +_G, \\circ}$ be a right module over $\\struct {R, +_R, \\times_R}$. Let $\\circ’ : R \\times G \\to G$ be the binary operation defined by: :$\\forall \\lambda \\in R: \\forall x \\in G: \\lambda \\circ’ x = x \\circ \\lambda $ Then $\\struct{G, +_G, \\circ’, \\circ}$ is a bimodule over $\\struct {R, +_R, \\times_R}$."}
+{"_id": "18035", "title": "Diagonals of Kite are Perpendicular", "text": "Let $ABCD$ be a kite such that $AC$ and $BD$ are its diagonals. Then $AC$ and $BD$ are perpendicular."}
+{"_id": "18036", "title": "One Diagonal of Kite Bisects the Other", "text": "Let $ABCD$ be a kite such that: :$AC$ and $BD$ are its diagonals :$AB = BC$ :$AD = DC$ Then $BD$ is the perpendicular bisector of $AC$."}
+{"_id": "18037", "title": "Theorem of Even Perfect Numbers/Historical Note", "text": "The first part of this proof of the '''Theorem of Even Perfect Numbers''' was documented by {{AuthorRef|Euclid}} in {{BookLink|The Elements|Euclid}}: {{EuclidPropLink|book = IX|prop = 36|title = Theorem of Even Perfect Numbers/Sufficient Condition}}. The second part was achieved by {{AuthorRef|Leonhard Paul Euler|Euler}}."}
+{"_id": "18039", "title": "Combination Theorem for Continuous Mappings/Topological Group", "text": "Let $\\struct{S, \\tau_{_S}}$ be a topological space. Let $\\struct{G, *, \\tau_{_G}}$ be a topological group. Let $\\lambda \\in G$. Let $f,g : \\struct{S, \\tau_{_S}} \\to \\struct{G, \\tau_{_G}}$ be continuous mappings. Then the following results hold: === Product Rule === {{:Combination Theorem for Continuous Mappings/Topological Group/Product Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Mappings/Topological Group/Multiple Rule}} === Inverse Rule === {{:Combination Theorem for Continuous Mappings/Topological Group/Inverse Rule}}"}
+{"_id": "18040", "title": "Combination Theorem for Continuous Mappings/Topological Group/Product Rule", "text": ":$f * g : \\struct {S, \\tau_{_S} } \\to \\struct {G, \\tau_{_G} }$ is a continuous mapping."}
+{"_id": "18041", "title": "Combination Theorem for Continuous Mappings/Topological Group/Multiple Rule", "text": ":$\\lambda * f: \\struct {S, \\tau_{_S} } \\to \\struct {G, \\tau_{_G} }$ is a continuous mapping :$f * \\lambda: \\struct {S, \\tau_{_S} } \\to \\struct {G, \\tau_{_G} }$ is a continuous mapping."}
+{"_id": "18042", "title": "Combination Theorem for Continuous Mappings/Topological Group/Inverse Rule", "text": ":$g^{-1}: \\struct {S, \\tau_{_S} } \\to \\struct {G, \\tau_{_G} }$ is a continuous mapping."}
+{"_id": "18043", "title": "Continuous Mapping to Topological Product/Corollary", "text": "Let $T = T_1 \\times T_2$ be a product space of two topological spaces $T_1$ and $T_2$. Let $T'$ be a topological space. Let $f: T' \\to T_1$ be a mapping. Let $g: T' \\to T_2$ be a mapping. Let $f \\times g : T’ \\to T$ be the mapping defined by: :$\\forall x \\in T’ : \\map {\\paren {f \\times g}} x = \\tuple{ \\map f x, \\map g x}$ Then $f \\times g$ is continuous {{iff}} $f$ and $g$ are continuous."}
+{"_id": "18044", "title": "Combination Theorem for Continuous Mappings/Topological Semigroup", "text": "Let $\\struct{S, \\tau_{_S}}$ be a topological space. Let $\\struct{G, *, \\tau_{_G}}$ be a topological semigroup. Let $\\lambda \\in G$. Let $f,g : \\struct{S, \\tau_{_S}} \\to \\struct{G, \\tau_{_G}}$ be continuous mappings. Then the following results hold: === Product Rule === {{:Combination Theorem for Continuous Mappings/Topological Semigroup/Product Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Mappings/Topological Semigroup/Multiple Rule}}"}
+{"_id": "18045", "title": "Combination Theorem for Continuous Mappings/Topological Semigroup/Product Rule", "text": ":$f * g: \\struct{S, \\tau_{_S} } \\to \\struct {G, \\tau_{_G} }$ is a continuous mapping."}
+{"_id": "18046", "title": "Combination Theorem for Continuous Mappings/Topological Semigroup/Multiple Rule", "text": ":$\\lambda * f: \\struct {S, \\tau_{_S} } \\to \\struct {G, \\tau_{_G} }$ is a continuous mapping :$f * \\lambda: \\struct {S, \\tau_{_S} } \\to \\struct {G, \\tau_{_G} }$ is a continuous mapping."}
+{"_id": "18047", "title": "Volume of Solid of Revolution/Parametric Form", "text": "Let $x: \\R \\to \\R$ and $y: \\R \\to \\R$ be real functions defined on the interval $\\closedint a b$. Let $y$ be integrable on the (closed) interval $\\closedint a b$. Let $x$ be differentiable on the (open) interval $\\openint a b$. Let the points be defined: :$A = \\tuple {\\map x a, \\map y a}$ :$B = \\tuple {\\map x b, \\map y b}$ :$C = \\tuple {\\map x b, 0}$ :$D = \\tuple {\\map x a, 0}$ Let the figure $ABCD$ be defined as being bounded by the straight lines $y = 0$, $x = a$, $x = b$ and the curve defined by: :$\\set {\\tuple {\\map x t, \\map y t}: a \\le t \\le b}$ Let the solid of revolution $S$ be generated by rotating $ABCD$ around the $x$-axis (that is, $y = 0$). Then the volume $V$ of $S$ is given by: :$\\displaystyle V = \\pi \\int_a^b \\paren {\\map y t}^2 \\map {x'} t \\rd t$"}
+{"_id": "18048", "title": "Combination Theorem for Continuous Mappings/Topological Ring", "text": "Let $\\struct{S, \\tau_{_S}}$ be a topological space. Let $\\struct{R, +, *, \\tau_{_R}}$ be a topological ring. Let $\\lambda \\in R$. Let $f,g : \\struct{S, \\tau_{_S}} \\to \\struct{R, \\tau_{_R}}$ be continuous mappings. Then the following results hold: === Sum Rule === {{:Combination Theorem for Continuous Mappings/Topological Ring/Sum Rule}} === Translation Rule === {{:Combination Theorem for Continuous Mappings/Topological Ring/Translation Rule}} === Negation Rule === {{:Combination Theorem for Continuous Mappings/Topological Ring/Negation Rule}} === Product Rule === {{:Combination Theorem for Continuous Mappings/Topological Ring/Product Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Mappings/Topological Ring/Multiple Rule}}"}
+{"_id": "18049", "title": "Combination Theorem for Continuous Mappings/Topological Division Ring", "text": "Let $\\struct{S, \\tau_{_S}}$ be a topological space. Let $\\struct{R, +, *, \\tau_{_R}}$ be a topological division ring. Let $\\lambda \\in R$. Let $f,g : \\struct{S, \\tau_{_S}} \\to \\struct{R, \\tau_{_R}}$ be continuous mappings. Let $U = S \\setminus \\set{x : \\map g x = 0}$ Let $g^{-1} : U \\to R$ denote the mapping defined by: :$\\forall x \\in U : \\map {g^{-1}} x = \\map g x^{-1}$ Let $\\tau_{_U}$ be the subspace topology on $U$. Then the following results hold: === Sum Rule === {{:Combination Theorem for Continuous Mappings/Topological Division Ring/Sum Rule}} === Translation Rule === {{:Combination Theorem for Continuous Mappings/Topological Division Ring/Translation Rule}} === Negation Rule === {{:Combination Theorem for Continuous Mappings/Topological  Division Ring/Negation Rule}} === Product Rule === {{:Combination Theorem for Continuous Mappings/Topological Division Ring/Product Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Mappings/Topological  Division Ring/Multiple Rule}} === Inverse Rule === {{:Combination Theorem for Continuous Mappings/Topological Division Ring/Inverse Rule}}"}
+{"_id": "18050", "title": "Weight of Body at Earth's Surface", "text": "Let $B$ be a body of mass $m$ situated at (or near) the surface of Earth. Then the weight of $B$ is given by: :$W = m g$ where $g$ is the value of the acceleration due to gravity at the surface of Earth."}
+{"_id": "18051", "title": "Combination Theorem for Continuous Mappings/Topological Ring/Sum Rule", "text": ":$f + g: \\struct {S, \\tau_{_S} } \\to \\struct {R, \\tau_{_R} }$ is continuous."}
+{"_id": "18052", "title": "Combination Theorem for Continuous Mappings/Topological Ring/Negation Rule", "text": ":$-g : \\struct {S, \\tau_{_S} } \\to \\struct {R, \\tau_{_R} }$ is continuous."}
+{"_id": "18053", "title": "Combination Theorem for Continuous Mappings/Topological Ring/Multiple Rule", "text": ":$\\lambda * f: \\struct {S, \\tau_{_S} } \\to \\struct {R, \\tau_{_R} }$ is continuous :$f * \\lambda: \\struct {S, \\tau_{_S} } \\to \\struct {R, \\tau_{_R} }$ is continuous."}
+{"_id": "18054", "title": "Combination Theorem for Continuous Mappings/Topological Ring/Product Rule", "text": ":$f * g: \\struct {S, \\tau_{_S} } \\to \\struct {R, \\tau_{_R} }$ is continuous."}
+{"_id": "18055", "title": "Combination Theorem for Continuous Mappings/Topological Ring/Combined Rule", "text": ":$\\lambda * f + \\mu * g: \\struct{S, \\tau_{_S}} \\to \\struct{R, \\tau_{_R}}$ is a continuous mapping :$f * \\lambda + g * \\mu: \\struct{S, \\tau_{_S}} \\to \\struct{R, \\tau_{_R}}$ is a continuous mapping."}
+{"_id": "18056", "title": "Combination Theorem for Continuous Mappings/Topological Division Ring/Sum Rule", "text": ":$f + g: \\struct {S, \\tau_{_S} } \\to \\struct {R, \\tau_{_R} }$ is continuous."}
+{"_id": "18057", "title": "Combination Theorem for Continuous Mappings/Topological Division Ring/Multiple Rule", "text": ":$\\lambda * f: \\struct {S, \\tau_{_S} } \\to \\struct {R, \\tau_{_R} }$ is continuous :$f * \\lambda: \\struct {S, \\tau_{_S} } \\to \\struct {R, \\tau_{_R} }$ is continuous."}
+{"_id": "18058", "title": "Combination Theorem for Continuous Mappings/Topological Division Ring/Product Rule", "text": ":$f * g: \\struct {S, \\tau_{_S} } \\to \\struct {R, \\tau_{_R} }$ is continuous."}
+{"_id": "18059", "title": "Combination Theorem for Continuous Mappings/Topological Division Ring/Inverse Rule", "text": ":$g^{-1}: \\struct {U, \\tau_{_U} } \\to \\struct {R, \\tau_{_R} }$ is continuous."}
+{"_id": "18060", "title": "Combination Theorem for Continuous Mappings/Topological Division Ring/Negation Rule", "text": ":$-g: \\struct {S, \\tau_{_S} } \\to \\struct {R, \\tau_{_R} }$ is continuous."}
+{"_id": "18061", "title": "Combination Theorem for Continuous Mappings/Topological Ring/Translation Rule", "text": ":$\\lambda + f: \\struct {S, \\tau_{_S} } \\to \\struct {R, \\tau_{_R} }$ is continuous."}
+{"_id": "18062", "title": "Combination Theorem for Continuous Mappings/Topological Division Ring/Translation Rule", "text": ":$\\lambda + f: \\struct {S, \\tau_{_S} } \\to \\struct {R, \\tau_{_R} }$ is continuous."}
+{"_id": "18063", "title": "Pointwise Operation is Composite of Operation with Mapping to Cartesian Product", "text": "Let $S$ be a set. Let $\\struct {T, *}$ be an algebraic structure. Let $T^S$ be the set of all mappings from $S$ to $T$. Let the algebraic structure $\\struct {T^S, \\oplus}$ be the algebraic structure on $T^S$ induced by $*$. Let $f, g \\in T^S$, that is, let $f: S \\to T$ and $g: S \\to T$ be mappings. Let $f \\times g : S \\to T \\times T$ be the mapping from $S$ to the cartesian product $T \\times T$ Defined by: :$\\forall x \\in S : \\map {\\paren {f \\times g}} x = \\tuple {\\map f x, \\map g x}$ Then: :$f \\oplus g = * \\circ \\paren {f \\times g}$ That is, $f \\oplus g$ is the composition of the binary operation $*$ with the mapping $f \\times g : S \\to T \\times T$."}
+{"_id": "18064", "title": "Integral Representation of Bessel Function of the First Kind", "text": "Let $\\map {J_n} x$ denote the Bessel function of the first kind of order $n$."}
+{"_id": "18065", "title": "Integral Representation of Bessel Function of the First Kind/Integer Order", "text": "Let $n \\in \\Z$ be an integer. Then: :$\\displaystyle \\map {J_n} x = \\dfrac 1 \\pi \\int_0^\\pi \\map \\cos {n \\theta - x \\sin \\theta} \\rd \\theta$"}
+{"_id": "18066", "title": "Integral Representation of Bessel Function of the First Kind/Non-Integer Order", "text": "Let $n \\in \\Z$ be an integer. Then: :$\\displaystyle \\map {J_n} x = \\dfrac {x^n} {2^n \\sqrt \\pi \\map \\Gamma {n + \\frac 1 2} } \\int_0^\\pi \\map \\cos {x \\sin \\theta} \\cos^{2 n} \\theta \\rd \\theta$"}
+{"_id": "18067", "title": "Combination Theorem for Continuous Mappings/Normed Division Ring/Translation Rule", "text": ":$\\lambda + f: \\struct {S, \\tau_{_S} } \\to \\struct {R, \\tau_{_R} }$ is continuous."}
+{"_id": "18068", "title": "Combination Theorem for Continuous Mappings/Normed Division Ring/Negation Rule", "text": ":$- g : \\struct{S, \\tau_{_S}} \\to \\struct{R, \\tau_{_R}}$ is continuous."}
+{"_id": "18069", "title": "Legendre Transform of Strictly Convex Real Function is Strictly Convex", "text": "Let $\\map f x$ be a strictly convex real function. Then the function $\\map {f^*} p$ acquired through the Legendre Transform is also strictly convex."}
+{"_id": "18072", "title": "Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms/Lemma 1", "text": ":$\\forall y \\in R: \\lambda_y = y * I_{_R} \\text { and } \\rho_y = I_{_R} * y$"}
+{"_id": "18073", "title": "Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms/Lemma 2", "text": ":$x * I_{_R}$ is a bijection and $x^{-1} * I_{_R}$ is the inverse of $x * I_{_R}$ :$I_{_R} * x$ is a bijection and $I_{_R} * x^{-1}$ is the inverse of $I_{_R} * x$"}
+{"_id": "18075", "title": "Open Balls of P-adic Number", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\Z_p$ be the $p$-adic integers. Let $a \\in \\Q_p$. For all $\\epsilon \\in \\R_{>0}$, let $\\map {B_\\epsilon} a$ denote the open ball of $a$ of radius $\\epsilon$.  Then: :$\\forall n \\in Z : \\map {B_{p^{-n} } } a = a + p^{n + 1} \\Z_p$"}
+{"_id": "18076", "title": "Local Basis of P-adic Number", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. Then the set of open balls $\\set {\\map {B_{p^{-n} } } a : n \\in Z}$ is a local basis of $a$ consisting of clopen sets."}
+{"_id": "18077", "title": "Closed Ball of P-adic Number", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\Z_p$ denote the $p$-adic integers. Let $a \\in \\Q_p$. For all $\\epsilon \\in \\R_{>0}$, let $\\map { {B_\\epsilon}^-} a$ denote the closed ball of $a$ of radius $\\epsilon$.  Then: :$\\forall n \\in Z : \\map {B^-_{p^{-n} } } a = a + p^n \\Z_p$ where $a + p^n \\Z_p$ denotes the left coset of the principal ideal $p^n \\Z_p$ containing $a$ in the subring $\\Z_p$. That is, the closed ball $\\map { {B_\\epsilon}^-} a$ is the set: :$a + p^n \\Z_p = \\set{a + p^n z : z \\in \\Z_p}$"}
+{"_id": "18078", "title": "Closed Ball is Disjoint Union of Open Balls in P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. For all $\\epsilon \\in \\R_{>0}$: :let $\\map { {B_\\epsilon}^-} a$ denote the closed $\\epsilon$-ball of $a$.  :let $\\map {B_\\epsilon} a$ denote the open $\\epsilon$-ball of $a$.  Then: :$(1): \\quad \\forall n \\in Z : \\map {B^-_{p^{-n} } } a = \\displaystyle \\bigcup_{i \\mathop = 0}^{p - 1} \\map {B_{p^{-n} } } {a + i p^n}$ :$(2): \\quad \\forall n \\in Z : \\set {\\map {B_{p^{-n} } } {a + i p^n} : i = 0, \\dotsc, p - 1}$ is a set of pairwise disjoint open balls."}
+{"_id": "18079", "title": "Sphere is Disjoint Union of Open Balls in P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\Z_p$ be the $p$-adic integers. Let $a \\in \\Q_p$. For all $\\epsilon \\in \\R_{>0}$: :let $\\map {S_\\epsilon} a$ denote the sphere of $a$ of radius $\\epsilon$.  :let $\\map {B_\\epsilon} a$ denote the open ball of $a$ of radius $\\epsilon$.  Then: :$\\forall n \\in Z : \\map {S_{p^{-n} } } a = \\displaystyle \\bigcup_{i \\mathop = 1}^{p - 1} \\map {B_{p^{-n} } } {a + i p^n}$"}
+{"_id": "18080", "title": "Null Sequence induces Local Basis in Metric Space", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $a \\in A$. Let $\\sequence{x_n}$ be a real null sequence such that: :$\\forall n \\in N: x_n > 0$ Let $\\map {B_\\epsilon} a$ denote the open $\\epsilon$-ball of $a$ in $M$. Then: :$\\mathcal B_{\\sequence{x_n}} = \\set{\\map {B_{x_n}} a : n \\in \\N}$ is a local basis at $a$."}
+{"_id": "18081", "title": "Null Sequence induces Local Basis in Metric Space/Sequence of Reciprocals", "text": ":$\\mathcal B = \\set {\\map {B_{1/n}} a : n \\in \\N}$ is a local basis at $a$."}
+{"_id": "18083", "title": "Consecutive Integers which are Powers of 2 or 3", "text": "The only pairs of consecutive positive integers which are powers of $2$ or $3$ are: :$\\tuple {1, 2}$, $\\tuple {2, 3}$, $\\tuple {3, 4}$, $\\tuple {8, 9}$"}
+{"_id": "18085", "title": "Altitudes of Triangle Bisect Angles of Orthic Triangle", "text": "Let $\\triangle ABC$ be a triangle. Let $\\triangle DEF$ be its orthic triangle. The altitudes of $\\triangle ABC$ are the angle bisectors of $\\triangle DEF$."}
+{"_id": "18086", "title": "Open Ball in Normed Division Ring is Open Ball in Induced Metric", "text": "Let $\\struct{R, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Let $d$ be the metric induced by the norm $\\norm {\\,\\cdot\\,}$. Let $a \\in R$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. Let $\\map {B_\\epsilon} {a; \\norm {\\,\\cdot\\,} }$ denote the open ball in the normed division ring $\\struct {R, \\norm {\\,\\cdot\\,} }$. Let $\\map {B_\\epsilon} {a; d }$ denote the open ball in the metric space $\\struct {R, d}$. Then: :$\\map {B_\\epsilon} {a; \\norm {\\,\\cdot\\,} }$ = $\\map {B_\\epsilon} {a; d }$"}
+{"_id": "18087", "title": "Closed Ball in Normed Division Ring is Closed Ball in Induced Metric", "text": "Let $\\struct{R, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Let $d$ be the metric induced by the norm $\\norm {\\,\\cdot\\,}$. Let $a \\in R$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. Let $\\map {{B_\\epsilon}^-} {a; \\norm {\\,\\cdot\\,} }$ denote the closed ball in the normed division ring $\\struct {R, \\norm {\\,\\cdot\\,} }$. Let $\\map {{B_\\epsilon}^-} {a; d }$ denote the closed ball in the metric space $\\struct {R, d}$. Then: :$\\map {{B_\\epsilon}^-} {a; \\norm {\\,\\cdot\\,} }$ = $\\map {{B_\\epsilon}^-} {a; d }$"}
+{"_id": "18088", "title": "Sphere in Normed Division Ring is Sphere in Induced Metric", "text": "Let $\\struct{R, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Let $d$ be the metric induced by the norm $\\norm {\\,\\cdot\\,}$. Let $a \\in R$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. Let $\\map {S_\\epsilon} {a; \\norm {\\,\\cdot\\,} }$ denote the sphere in the normed division ring $\\struct {R, \\norm {\\,\\cdot\\,} }$. Let $\\map {S_\\epsilon} {a; d }$ denote the sphere in the metric space $\\struct {R, d}$. Then: :$\\map {S_\\epsilon} {a; \\norm {\\,\\cdot\\,} }$ = $\\map {S_\\epsilon} {a; d }$"}
+{"_id": "18089", "title": "Sphere is Set Difference of Closed Ball with Open Ball", "text": "Let $M = \\struct{A, d}$ be a metric space or pseudometric space. Let $a \\in A$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. Let $\\map {{B_\\epsilon}^-} {a; d}$ denote the $\\epsilon$-closed ball of $a$ in $M$. Let $\\map {B_\\epsilon} {a; d}$ denote the $\\epsilon$-open ball of $a$ in $M$. Let $\\map {S_\\epsilon} {a; d}$ denote the $\\epsilon$-sphere of $a$ in $M$. Then: :$\\map {S_\\epsilon} {a; d} = \\map {{B_\\epsilon}^-} {a; d} \\setminus \\map {B_\\epsilon} {a; d}$"}
+{"_id": "18090", "title": "Sphere is Set Difference of Closed Ball with Open Ball/Normed Division Ring", "text": "Let $\\struct{R, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Let $a \\in R$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. Let $\\map {{B_\\epsilon}^-} {a; \\norm {\\,\\cdot\\,} }$ denote the $\\epsilon$-closed ball of $a$ in $\\struct {R, \\norm {\\,\\cdot\\,} }$. Let $\\map {B_\\epsilon} {a; \\norm {\\,\\cdot\\,} }$ denote the $\\epsilon$-open ball of $a$ in $\\struct {R, \\norm {\\,\\cdot\\,} }$. Let $\\map {S_\\epsilon} {a; \\norm {\\,\\cdot\\,} }$ denote the $\\epsilon$-sphere of $a$ in $\\struct {R, \\norm {\\,\\cdot\\,} }$. Then: :$\\map {S_\\epsilon} {a; \\norm {\\,\\cdot\\,} } = \\map { {B_\\epsilon}^-} {a; \\norm {\\,\\cdot\\,} } \\setminus \\map {B_\\epsilon} {a; \\norm {\\,\\cdot\\,} }$"}
+{"_id": "18092", "title": "Conservation of Momentum", "text": "Let $P$ be a physical system. Let it have the action $S$: :$\\displaystyle S = \\int_{t_0}^{t_1} L \\rd t$ where $L$ is the standard Lagrangian, and $t$ is time. Suppose $L$ does not depend on one of the coordinates explicitly: :$\\dfrac {\\partial L} {\\partial x_j} = 0$ Then the total momentum of $P$ along the axis $x_j$ is conserved."}
+{"_id": "18093", "title": "Conservation of Angular Momentum (Lagrangian Mechanics)", "text": "Let $P$ be a physical system composed of finite number of particles. Let it have the action $S$: :$\\displaystyle S = \\int_{t_0}^{t_1} L \\rd t$ where $L$ is the standard Lagrangian, and $t$ is time. Suppose $L$ is invariant {{WRT}} rotation around $z$-axis. Then the total angular momentum of $P$ along $z$-axis is conserved."}
+{"_id": "18094", "title": "Leibniz's Law for Sets", "text": "Let $S$ be an arbitrary set. Then: :$x = y \\dashv \\vdash x \\in S \\iff y \\in S$ for all $S$ in the universe of discourse. This is therefore the justification behind the notion of the definition of set equality."}
+{"_id": "18095", "title": "Area of Annulus", "text": "Let $A$ be an annulus whose inner radius is $r$ and whose outer radius is $R$. The area of $A$ is given by: :$\\map \\Area A = \\pi \\paren {R^2 - r^2}$"}
+{"_id": "18096", "title": "Area of Annulus as Area of Rectangle", "text": "Let $A$ be an annulus whose inner radius is $r$ and whose outer radius is $R$. The area of $A$ is given by: :$\\map \\Area A = 2 \\pi \\paren {r + \\dfrac w 2} \\times w$ where $w$ denotes the width of $A$. That is, it is the area of the rectangle contained by: :the width of $A$ :the circle midway in radius between the inner radius and outer radius of $A$."}
+{"_id": "18098", "title": "Characterization of Closed Ball in P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\Z_p$ be the $p$-adic integers. For any $\\epsilon \\in \\R_{>0}$ and $a \\in \\Q_p$ let $\\map {{B_\\epsilon}^-} a$ denote the closed ball of center $a$ of radius $\\epsilon$.  Let $x, y \\in \\Q_p$. Let $n \\in Z$. {{TFAE}} ::$(1)\\quad x \\in \\map {B^{\\,-}_{p^{-n}}} y$ ::$(2)\\quad \\norm{x -y}_p \\le p^{-n}$ ::$(3)\\quad \\map {B^{\\,-}_{p^{-n}}} x = \\map {B^{\\,-}_{p^{-n}}} y$ ::$(4)\\quad x - y \\in p^n \\Z_p$ ::$(5)\\quad x + p^n \\Z_p = y + p^n \\Z_p$"}
+{"_id": "18099", "title": "Characterization of Open Ball in P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\Z_p$ be the $p$-adic integers. For any $\\epsilon \\in \\R_{>0}$ and $a \\in \\Q_p$ let $\\map {B_\\epsilon} a$ denote the closed ball of center $a$ of radius $\\epsilon$.  Let $n \\in Z$. Let $x, y \\in \\Q_p$. {{TFAE}}: :$(1): \\quad x \\in \\map {B_{p^{-n} } } y$ :$(2): \\quad \\norm{x - y}_p < p^{-n}$ :$(3): \\quad \\map {B_{p^{-n} } } x = \\map {B_{p^{-n} } } y$ :$(4): \\quad x - y \\in p^{n + 1} \\Z_p$ :$(5): \\quad x + p^{n + 1} \\Z_p = y + p^{n + 1} \\Z_p$"}
+{"_id": "18100", "title": "Complete Archimedean Valued Field is Real or Complex Numbers", "text": "Let $\\struct{k, \\norm{\\,\\cdot\\,}}$ be a complete valued field with Archimedean norm $\\norm{\\,\\cdot\\,}$. Then either: :$k$ is isomorphic to the real numbers $\\R$ and $\\norm{\\,\\cdot\\,}$ is equivalent to the absolute value $\\size{\\,\\cdot\\,}$ on $\\R$ or :$k$ is isomorphic to the complex numbers $\\C$ and $\\norm{\\,\\cdot\\,}$ is equivalent to the complex modulus $\\size{\\,\\cdot\\,}$ on $\\C$"}
+{"_id": "18102", "title": "Equivalence of Definitions of Convergent P-adic Sequence", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\sequence {x_n} $ be a sequence in $\\Q_p$. {{TFAE}} === Definition 1 === {{:Definition:Convergent Sequence/P-adic Numbers/Definition 1|Definition 1}} === Definition 2 === {{:Definition:Convergent Sequence/P-adic Numbers/Definition 2}} === Definition 3 === {{:Definition:Convergent Sequence/P-adic Numbers/Definition 3}} === Definition 4 === {{:Definition:Convergent Sequence/P-adic Numbers/Definition 4}}"}
+{"_id": "18103", "title": "Equivalence of Definitions of Convergence in Normed Division Rings", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\sequence {x_n}$ be a sequence in $R$. {{TFAE|def = Convergent Sequence in Normed Division Ring}} === Definition 1 === {{:Definition:Convergent Sequence/Normed Division Ring/Definition 1}} === Definition 2 === {{:Definition:Convergent Sequence/Normed Division Ring/Definition 2}} === Definition 3 === {{:Definition:Convergent Sequence/Normed Division Ring/Definition 3}}"}
+{"_id": "18104", "title": "Finite Complement Topology is not Metrizable", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is not a metrizable space."}
+{"_id": "18105", "title": "Poisson Brackets of Classical Particle in Radial Potential on Plane", "text": "Let $P$ be a classical particle embedded in a 2-dimensional Euclidean manifold. Let the real-valued functions $\\map r t$, $\\map \\theta t$ denote the position of $P$ in polar coordinates, where $t$ is time. Suppose, the potential energy of $P$ depends only on $r$. Then $P$ has the following Poisson brackets: {{begin-eqn}} {{eqn| l = \\sqbrk {r, p_r}       | r = 1 }} {{eqn| l = \\sqbrk {\\theta, p_\\theta}      | r = 1 }} {{eqn| l = \\sqbrk {r, H}      | r = \\dfrac {p_r} m }} {{eqn| l = \\sqbrk {\\theta, H}       | r = \\dfrac {p_\\theta} {m r^2} }} {{eqn| l = \\sqbrk {p_r, H}       | r = -\\dfrac {\\partial U} {\\partial r} }} {{eqn| l = \\sqbrk {p_\\theta, H}       | r = 0 }} {{end-eqn}}"}
+{"_id": "18106", "title": "Multiplication by Power of 10 by Moving Decimal Point", "text": "Let $n \\in \\R$ be a real number. Let $n$ be expressed in decimal notation. Let $10^d$ denote a power of $10$ for some integer $d$ Then $n \\times 10^d$ can be expressed in decimal notation by shifting the decimal point $d$ places to the right. Thus, if $d$ is negative, and so $10^d = 10^{-e}$ for some $e \\in \\Z_{>0}$, $n \\times 10^d$ can be expressed in decimal notation by shifting the decimal point $e$ places to the left."}
+{"_id": "18107", "title": "Number of Significant Figures in Result of Multiplication", "text": "Let $m$ and $n$ be numbers which are presented to $d_m$ and $d_n$ significant figures respectively. Then the most significant figures that $m \\times n$ can have is $\\min \\set {d_m, d_n}$."}
+{"_id": "18108", "title": "Number of Significant Figures in Result of Division", "text": "Let $m$ and $n$ be numbers which are presented to $d_m$ and $d_n$ significant figures respectively. Then the most significant figures that $\\dfrac m n$ can have is $\\min \\set {d_m, d_n}$."}
+{"_id": "18109", "title": "Number of Significant Figures in Result of Square Root", "text": "Let $m$ be a numbers which is presented to $d$ significant figures. Then the most significant figures that $\\sqrt m$ can have is also $d$."}
+{"_id": "18110", "title": "Number of Significant Figures in Result of Addition or Subtraction", "text": "Let $m$ and $n$ be numbers. Let $d_m$ and $d_n$ be the position of the least significant digit of $m$ and $n$ respectively. Then the least significant digit in either $m + n$ or $m - n$ is in the position corresponding to the greater significant digit of $d_m$ and $d_n$."}
+{"_id": "18111", "title": "Range of Common Logarithm of Number between 1 and 10", "text": "Let $x \\in \\R$ be a real number such that: :$1 \\le x < 10$ Then: :$0 \\le \\log_{10} x \\le 1$ where $\\log_{10}$ denotes the common logarithm function."}
+{"_id": "18112", "title": "Common Logarithm of Number in Scientific Notation", "text": "Let $n$ be a positive real number which is presented (possibly approximated) in scientific notation as: :$n = a \\times 10^d$ where: :$1 \\le a < 10$ :$d \\in \\Z$ is an integer. Then: :$\\log_{10} n = \\log_{10} a + d$ where: :$0 \\le \\log_{10} a < 1$"}
+{"_id": "18113", "title": "Characteristic of Common Logarithm of Number Greater than 1", "text": "Let $x \\in \\R_{>1}$ be a (strictly) positive real number greater than $1$. The characteristic of its common logarithm $\\log_{10} x$ is equal to one less than the number of digits to the left of the decimal point of $x$."}
+{"_id": "18114", "title": "Characteristic of Common Logarithm of Number Less than 1", "text": "Let $x \\in \\R_{>0}$ be a (strictly) positive real number such that $x < 1$. The characteristic of its common logarithm $\\log_{10} x$ is equal to one less than the number of zero digits to the immediate right of the decimal point of $x$."}
+{"_id": "18116", "title": "P-adic Norm satisfies Non-Archimedean Norm Axioms", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers with $p$-adic norm $\\norm {\\,\\cdot\\,}_p : \\Q_p \\times \\Q_p \\to \\R_{\\ge 0}$. Then $\\norm {\\,\\cdot\\,}_p$ satisfies the non-Archimedean norm axioms: {{begin-axiom}} {{axiom | n = \\text N 1         | lc= Positive Definiteness:         | q = \\forall x \\in \\Q_p         | ml= \\norm x_p = 0         | mo= \\iff         | mr= x = 0 }} {{axiom | n = \\text N 2         | lc= Multiplicativity:         | q = \\forall x, y \\in \\Q_p         | ml= \\norm {x \\cdot y}_p         | mo= =         | mr= \\norm x_p \\times \\norm y_p }} {{axiom | n = \\text N 4         | lc= Ultrametric Inequality:         | q = \\forall x, y \\in Q_p         | ml= \\norm {x + y}_p         | mo= \\le         | mr= \\max \\set {\\norm x_p, \\norm y_p} }} {{end-axiom}}"}
+{"_id": "18117", "title": "Circles with Same Poles are Parallel", "text": "Let $S$ be a sphere. Let $C$ and $D$ be circles on $S$ (either great circles or small circles). Let $C$ and $D$ both have the same pair of poles. Then $C$ and $D$ are parallel."}
+{"_id": "18118", "title": "Three Points on Sphere in Same Hemisphere", "text": "Let $S$ be a sphere. Let $A$, $B$ and $C$ be points on $S$ which do not all lie on the same great circle. Then it is possible to divide $S$ into two hemispheres such that $A$, $B$ and $C$ all lie on the same hemisphere."}
+{"_id": "18119", "title": "Side of Spherical Triangle is Less than 2 Right Angles", "text": "Let $ABC$ be a spherical triangle on a sphere $S$. Let $AB$ be a side of $ABC$. The '''length''' of $AB$ is less than $2$ right angles."}
+{"_id": "18120", "title": "Center is Element of Closed Ball", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $a \\in A$. Let $\\epsilon \\in \\R_{>0}$ be a positive real number. Let $\\map { {B_\\epsilon}^-} a$ be the closed $\\epsilon$-ball of $a$ in $M$. Then: :$a \\in \\map {{B_\\epsilon}^-} a$"}
+{"_id": "18121", "title": "Center is Element of Closed Ball/Normed Division Ring", "text": "Let $\\struct{R, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Let $a \\in R$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. Let $\\map { {B_\\epsilon}^-} a$ be the closed $\\epsilon$-ball of $a$ in $\\struct{R, \\norm {\\,\\cdot\\,} }$. Then: :$a \\in \\map { {B_\\epsilon}^-} a$"}
+{"_id": "18122", "title": "Center is Element of Closed Ball/P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. Let $\\map { {B_\\epsilon}^-} a$ be the closed $\\epsilon$-ball of $a$ in $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$. Then: :$a \\in \\map { {B_\\epsilon}^-} a$"}
+{"_id": "18123", "title": "Length of Arc of Small Circle", "text": "Let $S$ be a sphere. Let $\\bigcirc FCD$ be a small circle on $S$. Let $C$ and $D$ be the points on $\\bigcirc FCD$ such that $CD$ is the arc of $\\bigcirc FCD$ whose length is to be determined."}
+{"_id": "18124", "title": "Definition:Geographical Coordinates", "text": "Let $J$ be a point on Earth's surface. The '''geographical coordinates''' of $J$ are the definition of the position of $J$ with respect to the equator and the principal meridian."}
+{"_id": "18126", "title": "Two Parallel Lines lie in Same Plane", "text": "Let $L_1$ and $L_2$ be two lines which are parallel. Then $L_1$ and $L_2$ both lie in the same plane."}
+{"_id": "18127", "title": "Center is Element of Open Ball", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $a \\in A$. Let $\\epsilon \\in \\R_{>0}$ be a positive real number. Let $\\map {B_\\epsilon} a$ be the open $\\epsilon$-ball of $a$ in $M$. Then: :$a \\in \\map {B_\\epsilon} a$"}
+{"_id": "18128", "title": "Center is Element of Open Ball/Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Let $a \\in R$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. Let $\\map {B_\\epsilon} a$ be the open $\\epsilon$-ball of $a$ in $\\struct{R, \\norm {\\,\\cdot\\,} }$. Then: :$a \\in \\map {B_\\epsilon} a$"}
+{"_id": "18129", "title": "Center is Element of Open Ball/P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. Let $\\map {B_\\epsilon} a$ be the open $\\epsilon$-ball of $a$ in $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$. Then: :$a \\in \\map {B_\\epsilon} a$"}
+{"_id": "18130", "title": "Equivalence of Definitions of Non-Archimedean Division Ring Norm", "text": "Let $\\struct {R, +, \\circ}$ be a division ring whose zero is denoted $0_R$. {{TFAE}} === Definition 1 === {{:Definition:Non-Archimedean/Norm (Division Ring)/Definition 1}} === Definition 2 === {{:Definition:Non-Archimedean/Norm (Division Ring)/Definition 2}}"}
+{"_id": "18132", "title": "Sine of Half Angle for Spherical Triangles", "text": ":$\\sin \\dfrac A 2 = \\sqrt {\\dfrac {\\map \\sin {s - b} \\, \\map \\sin {s - c} } {\\sin b \\sin c} }$ where $s = \\dfrac {a + b + c} 2$."}
+{"_id": "18133", "title": "Open Ball in P-adic Numbers is Closed Ball", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. For all $\\epsilon \\in \\R_{>0}$:  :Let $\\map {B_\\epsilon} a$ denote the open $\\epsilon$-ball of $a$ :Let $\\map {B^-_\\epsilon} a$ denote the closed $\\epsilon$-ball of $a$.  Then: :$\\forall n \\in Z : \\map {B_{p^{-n} } } a = \\map {B^-_{p^{-\\paren {n + 1} } } } a$"}
+{"_id": "18134", "title": "Tangent of Half Angle for Spherical Triangles", "text": ":$\\tan \\dfrac A 2 = \\sqrt {\\dfrac {\\map \\sin {s - b} \\, \\map \\sin {s - c} } {\\sin s \\, \\map \\sin {s - a} } }$ where $s = \\dfrac {a + b + c} 2$."}
+{"_id": "18135", "title": "Countable Basis for P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\tau_p$ be the topology induced by the non-Archimedean norm $\\norm {\\,\\cdot\\,}_p$. For any $\\epsilon \\in \\R_{>0}$ and $a \\in \\Q_p$ let $\\map {B_\\epsilon} a$ denote the open $\\epsilon$-ball of $a$.   Then: :$\\BB_p = \\set {\\map {B_{p^{-n} } } q : q \\in \\Q, n \\in \\Z}$ is a countable basis for $\\struct{\\Q_p, \\tau_p}$."}
+{"_id": "18136", "title": "P-adic Numbers is Second Countable Topological Space", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\tau_p$ be the topology induced by the non-Archimedean norm $\\norm {\\,\\cdot\\,}_p$. Then the topological space $\\struct {\\Q_p, \\tau_p}$ is second-countable."}
+{"_id": "18137", "title": "P-adic Numbers is Totally Disconnected Topological Space", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\tau_p$ be the topology induced by the non-Archimedean norm $\\norm {\\,\\cdot\\,}_p$. Then the topological space $\\struct {\\Q_p, \\tau_p}$ is totally disconnected."}
+{"_id": "18138", "title": "P-adic Numbers is Hausdorff Topological Space", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\tau_p$ be the topology induced by the non-Archimedean norm $\\norm {\\,\\cdot\\,}_p$. Then the topological space $\\struct{\\Q_p, \\tau_p}$ is Hausdorff."}
+{"_id": "18139", "title": "P-adic Numbers is Locally Compact Topological Space", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\tau_p$ be the topology induced by the non-Archimedean norm $\\norm {\\,\\cdot\\,}_p$. Then the topological space $\\struct {\\Q_p, \\tau_p}$ is locally compact."}
+{"_id": "18140", "title": "Open and Closed Balls in P-adic Numbers are Compact Subspaces", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. Let $n \\in \\Z$. Then the open ball $\\map {B_{p^{-n} } } a$ and closed ball $\\map {B^-_{p^{-n}}} a$ are compact. {{disambiguate|Definition:Compact}}"}
+{"_id": "18142", "title": "Ambiguous Case", "text": "Let $\\triangle ABC$ be a triangle. Let the sides $a, b, c$ of $\\triangle ABC$ be opposite $A, B, C$ respectively. Let the sides $a$ and $b$ be known. Let the angle $\\angle B$ also be known. Then it may not be possible to know the value of $\\angle A$. This is known as the '''ambiguous case'''."}
+{"_id": "18143", "title": "Analogue Formula for Spherical Law of Cosines", "text": "Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Then: {{begin-eqn}} {{eqn | l = \\sin a \\cos B       | r = \\cos b \\sin c - \\sin b \\cos c \\cos A }} {{eqn | l = \\sin a \\cos C       | r = \\cos c \\sin b - \\sin c \\cos b \\cos A }} {{end-eqn}}"}
+{"_id": "18146", "title": "Four-Parts Formula", "text": "Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. We have: :$\\cos a \\cos C = \\sin a \\cot b - \\sin C \\cot B$ That is: :$\\map \\cos {\\text {inner side} } \\cdot \\map \\cos {\\text {inner angle} } = \\map \\sin {\\text {inner side} } \\cdot \\map \\cot {\\text {other side} } - \\map \\sin {\\text {inner angle} } \\cdot \\map \\cot {\\text {other angle} }$ This is known as the '''four-parts formula''', as it defines the relationship between each of four consecutive parts of $\\triangle ABC$."}
+{"_id": "18151", "title": "Countable Basis for P-adic Numbers/Closed Balls", "text": "For any $\\epsilon \\in \\R_{>0}$ and $a \\in \\Q_p$ let $\\map {B_\\epsilon^-} a$ denote the closed $\\epsilon$-ball of $a$.   Then: :$\\BB_p = \\set {\\map {B^-_{p^{-n} } } q : q \\in \\Q, n \\in \\Z}$ is a countable basis for $\\struct{\\Q_p, \\tau_p}$."}
+{"_id": "18152", "title": "Countable Basis for P-adic Numbers/Cosets", "text": "Let $\\Z_p$ be the $p$-adic integers. Then: :$\\BB_p = \\set {q + p^n \\Z_p : q \\in \\Q, n \\in \\Z}$ is a countable basis for $\\struct{\\Q_p, \\tau_p}$."}
+{"_id": "18155", "title": "Napier's Tangent Rule for Right Spherical Triangles", "text": "Let $\\triangle ABC$ be a right spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let the angle $\\sphericalangle C$ be a right angle. Let the remaining parts of $\\triangle ABC$ be arranged according to the '''interior''' of this circle, where the symbol $\\Box$ denotes a right angle. :410px Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Then the sine of the '''middle part''' equals the product of the tangents of the '''adjacent parts'''."}
+{"_id": "18156", "title": "Napier's Cosine Rule for Quadrantal Triangles", "text": "Let $\\triangle ABC$ be a quadrantal triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let the side $c$ be a right angle. Let the remaining parts of $\\triangle ABC$ be arranged according to the '''exterior''' of this circle, where the symbol $\\Box$ denotes a right angle. :410px Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Then the sine of the '''middle part''' equals the product of the cosine of the '''opposite parts'''."}
+{"_id": "18159", "title": "Napier's Rules for Right Spherical Triangles", "text": "'''Napier's Rules for Right Spherical Triangles''' are the special cases of the Spherical Law of Cosines for a spherical triangle one of whose angles is a right angle. :410px Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let angle $\\angle C$ be a right angle. Let the remaining parts of $\\triangle ABC$ be arranged in a circle as the '''interior''' of the above, where the symbol $\\Box$ denotes a right angle. Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Let the remaining two parts be called '''opposite parts'''. Then: :The sine of the '''middle part''' equals the product of the tangents of the '''adjacent parts'''. :The sine of the '''middle part''' equals the product of the cosines of the '''opposite parts'''."}
+{"_id": "18160", "title": "Napier's Rules for Quadrantal Triangles", "text": "'''Napier's Rules for Quadrantal Triangles''' are the special cases of the Spherical Law of Cosines for a spherical triangle one of whose sides is a right angle. :410px Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let side $c$ be a right angle. Let the remaining parts of $\\triangle ABC$ be arranged in a circle as the '''exterior''' of the above, where the symbol $\\Box$ denotes a right angle. Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Let the remaining two parts be called '''opposite parts'''. Then: :The sine of the '''middle part''' equals the product of the tangents of the '''adjacent parts'''. :The sine of the '''middle part''' equals the product of the cosines of the '''opposite parts'''."}
+{"_id": "18161", "title": "Spherical Triangle is Polar Triangle of its Polar Triangle", "text": "Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let $\\triangle A'B'C'$ be the polar triangle of $\\triangle ABC$. Then $\\triangle ABC$ is the polar triangle of $\\triangle A'B'C'$."}
+{"_id": "18162", "title": "Side of Spherical Triangle is Supplement of Angle of Polar Triangle", "text": "Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let $\\triangle A'B'C'$ be the polar triangle of $\\triangle ABC$. Then $A'$ is the supplement of $a$. That is: :$A' = \\pi - a$ and it follows by symmetry that: :$B' = \\pi - b$ :$C' = \\pi - c$"}
+{"_id": "18163", "title": "Integers are Dense in P-adic Integers/Unit Ball", "text": "The integers $\\Z$ are dense in the closed ball $\\map {B^-_1} 0$."}
+{"_id": "18164", "title": "Analogue Formula for Spherical Law of Cosines/Corollary", "text": "{{begin-eqn}} {{eqn | l = \\sin A \\cos b       | r = \\cos B \\sin C + \\sin B \\cos C \\cos a }} {{eqn | l = \\sin A \\cos c       | r = \\cos C \\sin B + \\sin C \\cos B \\cos a }} {{end-eqn}}"}
+{"_id": "18165", "title": "Four-Parts Formula/Corollary", "text": ":$\\cos A \\cos c = \\sin A \\cot B - \\sin c \\cot b$"}
+{"_id": "18169", "title": "Cosine in Terms of Haversine", "text": ":$\\cos \\theta = 1 - 2 \\hav \\theta$ where $\\cos$ denotes cosine and $\\hav$ denotes haversine."}
+{"_id": "18170", "title": "Spherical Law of Haversines", "text": "Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Then: :$\\hav a = \\map \\hav {b - c} + \\sin b \\sin c \\hav A$ where $\\hav$ denotes haversine."}
+{"_id": "18173", "title": "Sign of Haversine", "text": "The haversine is non-negative for all $\\theta \\in \\R$."}
+{"_id": "18174", "title": "Haversine Function is Even", "text": "The haversine is an even function: :$\\forall \\theta \\in \\R: \\map \\hav {-\\theta} = \\hav \\theta$"}
+{"_id": "18175", "title": "Delambre's Analogies", "text": "Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively."}
+{"_id": "18177", "title": "Napier's Analogies", "text": "Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively."}
+{"_id": "18178", "title": "Half Angle Formulas for Spherical Triangles", "text": "Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively."}
+{"_id": "18179", "title": "Half Side Formulas for Spherical Triangles", "text": "Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively."}
+{"_id": "18181", "title": "Tangent of Half Side for Spherical Triangles", "text": ":$\\tan \\dfrac a 2 = \\sqrt {\\dfrac {-\\cos S \\, \\map \\cos {S - A} } {\\map \\cos {S - B} \\, \\map \\cos {S - C} } }$ where $S = \\dfrac {a + b + c} 2$."}
+{"_id": "18182", "title": "Zenith Distance is Complement of Celestial Altitude", "text": "Let $X$ be the position of a star (or other celestial body) on the celestial sphere. The zenith distance $z$ of $X$ is the complement of the altitude $a$ of $X$: :$z = 90 \\degrees - a$"}
+{"_id": "18183", "title": "P-adic Metric on P-adic Numbers is Non-Archimedean Metric/Corollary 1", "text": "Then: :$\\forall x, y, z \\in R: \\norm {x - y}_p \\le \\max \\set {\\norm {x - z}_p, \\norm {y - z}_p}$"}
+{"_id": "18184", "title": "Zenith Distance of North Celestial Pole equals Colatitude of Observer", "text": "Let $O$ be an observer of the celestial sphere. Let $P$ be the position of the north celestial pole with respect to $O$. Let $z$ denote the zenith distance of $P$. Let $\\psi$ denote the (terrestrial) colatitude of $O$. Then: :$z = \\psi$"}
+{"_id": "18185", "title": "Altitude of North Celestial Pole equals Latitude of Observer", "text": "Let $O$ be an observer of the celestial sphere. Let $P$ be the position of the north celestial pole with respect to $O$. Let $a$ denote the altitude of $P$. Let $\\phi$ denote the (terrestrial) latitude of $O$. Then: :$a = \\phi$"}
+{"_id": "18187", "title": "Celestial Body moves along Parallel of Declination", "text": "Consider the celestial sphere $C$ whose observer is $O$. Let $B$ be a celestial body on $C$. Let $\\delta$ be the declination of $B$. Let $P_\\delta$ be the parallel of declination whose declination is $\\delta$. Then $B$ appears to move along the path of $P_\\delta$, in the direction from the north to east to south to west. It takes $1$ sidereal day for $B$ to travel around $P_\\delta$."}
+{"_id": "18188", "title": "Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. For all $\\epsilon \\in \\R_{>0}$, let $\\map { {B_\\epsilon}^-} a$ denote the closed $\\epsilon$-ball of $a$.  Let $n, m \\in Z$, such that $n < m$. Then: :$(1) \\quad \\map {B^-_{p^{-n}}} a = \\displaystyle \\bigcup_{i \\mathop = 0}^{p^\\paren {m - n} - 1} \\map {B^-_{p^{-m} } } {a + i p^n}$ :$(2) \\quad \\set {\\map {B^-_{p^{-m} } } {a + i p^n} : i = 0, \\dots, p^\\paren {m - n} - 1}$ is a set of pairwise disjoint closed balls"}
+{"_id": "18189", "title": "Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Union of Closed Balls", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. For all $\\epsilon \\in \\R_{>0}$, let $\\map {{B_\\epsilon}^-} a$ denote the closed $\\epsilon$-ball of $a$.  Let $n, m \\in Z$, such that $n < m$. Then: :$\\map {B^-_{p^{-n} } } a = \\displaystyle \\bigcup_{i \\mathop = 0}^{p^\\paren {m - n} - 1} \\map {B^-_{p^{-m} } } {a + i p^\\paren {m - n} }$"}
+{"_id": "18190", "title": "Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Disjoint Closed Balls", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. For all $\\epsilon \\in \\R_{>0}$, let $\\map {B_\\epsilon} a$ denote the open $\\epsilon$-ball of $a$.  Then: :$\\forall n \\in Z : \\set{\\map {B^-_{p^{-m} } } {a + i p^n} : i = 0, \\dotsc, p^\\paren {m - n} - 1}$ is a set of pairwise disjoint open balls."}
+{"_id": "18191", "title": "Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1", "text": ":$\\forall y \\in \\Q_p: \\norm y_p \\le p^{-n}$ {{iff}} there exists $i \\in \\Z$ such that: ::$(1)\\quad 0 \\le i \\le p^\\paren {m - n} - 1$ ::$(2)\\quad \\norm {y - i p^n}_p \\le p^{-m}$"}
+{"_id": "18192", "title": "Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Necessary Condition", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $n, m \\in Z$, such that $n < m$. Let $y \\in \\Q_p$. Let $\\norm{y}_p \\le p^{-n}$. Then there exists $i \\in \\Z$ such that: ::$(1) \\quad 0 \\le i \\le p^\\paren {m - n} - 1$ ::$(2) \\quad \\norm {y - i p^n}_p \\le p^{-m}$"}
+{"_id": "18193", "title": "Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Sufficient Condition", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $y \\in \\Q_p$ Let $n, m \\in Z$, such that $n < m$. Let there exist $i \\in \\Z$: :$(1): \\quad 0 \\le i \\le p^\\paren {m - n} - 1$ :$(2): \\quad \\norm {y - i p^n}_p \\le p^{-m}$ Then: :$\\norm y_p \\le p^{-n}$"}
+{"_id": "18194", "title": "Open and Closed Balls in P-adic Numbers are Totally Bounded", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. Let $n \\in \\Z$. Then the open ball $\\map {B_{p^{-n} } } a$ and closed ball $\\map {B^-_{p^{-n} } } a$ are totally bounded subspaces."}
+{"_id": "18195", "title": "Cofactor Sum Identity", "text": "Let $J_n$ be the $n \\times n$ matrix of all ones. Let $A$ be an $n \\times n$ matrix. Let $A_{ij}$ denote the cofactor of element $\\paren {i,j}$ in $\\det\\paren A$, $1\\le i,j \\le n$.                  Then: :$\\displaystyle \\det \\paren {A -J_n} = \\det \\paren A - \\sum_{i \\mathop  = 1}^n \\sum_{j \\mathop = 1}^n A_{ij} $"}
+{"_id": "18196", "title": "Sum of Elements of Invertible Matrix", "text": "Let $\\mathbf J_n$ be the $n \\times n$ square ones matrix. Let $\\mathbf B$ be an $n\\times n$ invertible matrix with entries $b_{i j}$, $1 \\le i, j \\le n$. Then: :$\\displaystyle \\sum_{i \\mathop = 1}^n \\sum_{j \\mathop = 1}^n b_{i j} = 1 -  \\map \\det {\\mathbf B} \\map \\det {\\mathbf B^{-1} - \\mathbf J_n}$"}
+{"_id": "18197", "title": "Open and Closed Balls in P-adic Numbers are Clopen in P-adic Metric", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. Let $n \\in \\Z$. Then the open ball $\\map {B_{p^{-n}}} a$ and closed ball $\\map {B^-_{p^{-n}}} a$ are clopen in the $p$-adic metric."}
+{"_id": "18198", "title": "Closed Balls Centered on P-adic Number is Countable", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. Then the set of all closed balls centered on $a$ is the countable set: :$\\BB^- = \\set {\\map {B^-_{p^{-n} } } a: n \\in \\Z}$"}
+{"_id": "18199", "title": "Closed Balls Centered on P-adic Number is Countable/Open Balls", "text": "Then the set of all open balls centered on $a$ is the countable set: :$\\BB = \\set {\\map {B_{p^{-n} } } a : n \\in \\Z}$"}
+{"_id": "18200", "title": "Closed Balls Centered on P-adic Number is Countable/Lemma", "text": ":$\\exists n \\in \\Z : p^{-n} \\le \\epsilon < p^{-\\paren {n - 1} }$"}
+{"_id": "18201", "title": "Closed Balls Centered on P-adic Number is Countable/Open Balls/Lemma", "text": ":$\\exists n \\in \\Z : p^{-\\paren {n + 1} } < \\epsilon \\le p^{-n}$"}
+{"_id": "18202", "title": "Open Ball contains Smaller Open Ball", "text": "Let $M = \\struct{A, d}$ be a metric space. Let $a \\in A$. Let $\\epsilon, \\delta \\in \\R_{>0}$ such that $\\epsilon \\le \\delta$. Let $\\map {B_\\epsilon} a$ be the open $\\epsilon$-ball on $a$. Let $\\map {B_\\delta} a$ be the open $\\delta$-ball on $a$. Then: :$\\map {B_\\epsilon} a \\subseteq \\map {B_\\delta} a$"}
+{"_id": "18203", "title": "Closed Ball contains Smaller Closed Ball", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $a \\in A$. Let $\\epsilon, \\delta \\in \\R_{> 0}$ such that $\\epsilon \\le \\delta$. Let $\\map {B^-_\\epsilon} a$ be the closed $\\epsilon$-ball on $a$. Let $\\map {B^-_\\delta} a$ be the closed $\\delta$-ball on $a$. Then: :$\\map {B^-_\\epsilon} a \\subseteq \\map {B^-_\\delta} a$"}
+{"_id": "18204", "title": "Open Ball contains Strictly Smaller Closed Ball", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $a \\in A$. Let $\\epsilon, \\delta \\in \\R_{>0}$ such that $\\epsilon < \\delta$. Let $\\map {B^-_\\epsilon} a$ be the closed $\\epsilon$-ball on $a$. Let $\\map {B_\\delta} a$ be the open $\\delta$-ball on $a$. Then: :$\\map {B^-_\\epsilon} a \\subseteq \\map {B_\\delta} a$"}
+{"_id": "18205", "title": "Closed Ball contains Smaller Open Ball", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $a \\in A$. Let $\\epsilon, \\delta \\in \\R_{> 0}$ such that $\\epsilon \\le \\delta$. Let $\\map {B_\\epsilon} a$ be the open $\\epsilon$-ball on $a$. Let $\\map {B^-_\\delta} a$ be the closed $\\delta$-ball on $a$. Then: :$\\map {B_\\epsilon} a \\subseteq \\map {B^-_\\delta} a$"}
+{"_id": "18207", "title": "Open and Closed Balls in P-adic Numbers are Compact Subspaces/P-adic Integers", "text": "Then the set of $p$-adic integers $\\Z_p$ is compact."}
+{"_id": "18208", "title": "Open Balls form Local Basis for Point of Metric Space", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $x \\in A$. Let $\\BB_x$ be the set of all open balls of $M$ centered on $x$. That is: :$\\BB_x = \\set {\\map {B_\\epsilon} x : \\epsilon \\in \\R_{>0}}$ Then $\\BB$ is a local basis of $x$."}
+{"_id": "18209", "title": "Local Basis of P-adic Number/Cosets", "text": "Let $\\Z_p$ be the $p$-adic integers. Then the set $\\set {a + p^n \\Z_p: n \\in Z}$ is a local basis of $a$ consisting of clopen sets."}
+{"_id": "18210", "title": "Local Basis of P-adic Number/Closed Balls", "text": "Then the set of closed balls $\\set {\\map {B^-_{p^{-n} } } a: n \\in Z}$ is a local basis of $a$ consisting of clopen sets."}
+{"_id": "18212", "title": "Measurement of Terrestrial Longitude", "text": "To measure the (terrestrial) longitude of a position: :$(1): \\quad$ Identify the exact moment of noon at the location in question :$(2): \\quad$ Work out the precise time of day $t$ that this happens. Let $N$ be the time of noon at the principal meridian on the day in question. Let $x$ be the number of hours before $N$ at the principal meridian. If $t$ is after $N$, then $x$ is treated as negative. Then the (terrestrial) longitude is $15 x \\degrees$. If $x$ is positive, the (terrestrial) longitude is east. If $x$ is negative, the (terrestrial) longitude is west."}
+{"_id": "18213", "title": "Coherent Sequence is Partial Sum of P-adic Expansion", "text": "Let $p$ be a prime number. Let $\\sequence{\\alpha_n}$ be a coherent sequence. Then there exists a unique $p$-adic expansion of the form: :$\\displaystyle \\sum_{n \\mathop = 0}^\\infty d_n p^n$ such that: :$\\forall n \\in \\N: \\alpha_n = \\displaystyle \\sum_{i \\mathop = 0}^n d_i p^i$"}
+{"_id": "18215", "title": "Difference of Consecutive terms of Coherent Sequence", "text": "Let $p$ be a prime number. Let $\\sequence {\\alpha_n}$ be a coherent sequence. Then: :for all $n \\in \\N_{>0}$ there exists $c_n \\in \\N$ such that: ::$0 \\le c_n < p$ ::$\\alpha_n - \\alpha_{n - 1} = c_n p^n$"}
+{"_id": "18216", "title": "Vandermonde Matrix Identity for Cauchy Matrix", "text": "Assume values $\\set { x_1,\\ldots,x_n,y_1,\\ldots,y_n }$ are distinct in matrix {{begin-eqn}} {{eqn   | l = C    | r = \\paren {\\begin{smallmatrix}         \\dfrac {1} {x_1 - y_1} & \\dfrac {1} {x_1 - y_2} & \\cdots & \\dfrac {1} {x_1 - y_n} \\\\         \\dfrac {1} {x_2 - y_1} & \\dfrac 1 {x_2 - y_2}   & \\cdots & \\dfrac {1} {x_2 - y_n} \\\\         \\vdots                 & \\vdots                 & \\cdots & \\vdots \\\\         \\dfrac {1} {x_n - y_1} & \\dfrac {1} {x_n - y_2} & \\cdots & \\dfrac {1} {x_n - y_n} \\\\ \\end{smallmatrix} }   | c = Cauchy matrix of order $n$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn   | l = C   | r = -P V_x^{-1} V_y Q^{-1}   | c = Vandermonde matrix identity for a Cauchy matrix  }} {{end-eqn}}"}
+{"_id": "18217", "title": "Function that Satisfies Axioms of Uncertainty", "text": "Let $n \\in \\N$ be a natural number. Let $p_1, p_2, \\dotsc, p_n$ be real numbers such that: :$\\forall i \\in \\set {1, 2, \\dotsc, n}: p_i \\ge 0$ :$\\displaystyle \\sum_{i \\mathop = 1}^n p_i = 1$ Let $\\map H {p_1, p_2, \\ldots, p_n}$ be a real-valued function which satisfies the axioms of uncertainty. Then: :$\\map H {p_1, p_2, \\ldots, p_n} = \\displaystyle -\\lambda \\sum_{i \\mathop = 1}^n p_i \\log_b p_i$ where: :$\\lambda \\in \\R_{>0}$ :$b \\in \\R_{>1}$"}
+{"_id": "18218", "title": "Uncertainty Function satisfies Axioms of Uncertainty", "text": "Let $X$ be a random variable. Let $X$ take a finite number of values with probabilities $p_1, p_2, \\dotsc, p_n$. Let $\\map H X$ be the '''uncertainty function''' of $X$: :$\\map H X = \\displaystyle -\\sum_k p_k \\lg p_k$ where: :$\\lg$ denotes logarithm base $2$ :the summation is over those $k$ where $p_k > 0$. Then the uncertainty function satisfies the Axioms of Uncertainty."}
+{"_id": "18222", "title": "Number of k-Cycles in Symmetric Group", "text": "Let $n \\in \\N$ be a natural number. Let $S_n$ denote the symmetric group on $n$ letters. Let $k \\in N$ such that $k \\le n$. The number of elements $m$ of $S_n$ which are $k$-cycles is given by: :$m = \\paren {k - 1}! \\dbinom n k = \\dfrac {n!} {k \\paren {n - k}!}$"}
+{"_id": "18223", "title": "Number of k-Cycles on Set of k Elements", "text": "Let $k \\in \\N$ be a natural number. Let $S_k$ denote the symmetric group on $k$ letters. The number of elements of $S_k$ which are $k$-cycles is $\\paren {k - 1}!$."}
+{"_id": "18225", "title": "Zero Padded Basis Representation", "text": "Let $b \\in \\Z: b > 1$. Let $m \\in \\Z_{> 0}$. For every $n \\in \\Z_{\\ge 0}$ such that $n < b^m$, there exists one and only one sequence $\\sequence {r_j}_{0 \\mathop \\le j \\mathop \\le m - 1}$ such that: : $(1): \\quad \\displaystyle n = \\sum_{j \\mathop = 0}^{m - 1} r_j b^j$ : $(2): \\quad \\displaystyle \\forall j \\in \\closedint 0 {m - 1}: r_j \\in \\N_b$"}
+{"_id": "18226", "title": "Sum of Elements of Inverse of Matrix with Column of Ones", "text": "Let $\\mathbf B = \\sqbrk b_n$ denote the inverse of a square matrix $\\mathbf A$ of order $n$. Let $\\mathbf A$ be such that it has a row or column of all ones. Then the sum of elements in $\\mathbf B$ is one: :$\\displaystyle \\sum_{i \\mathop  = 1}^n \\sum_{j \\mathop = 1}^n b_{ij}  = 1$"}
+{"_id": "18227", "title": "Coherent Sequence is Partial Sum of P-adic Expansion/Lemma", "text": ":$\\forall n \\in \\N: \\alpha_n = \\displaystyle \\sum_{i \\mathop = 0}^n b_{i, i} p^i$"}
+{"_id": "18229", "title": "Limit to Infinity of Number of p-Groups of Order p^m", "text": "Let $p$ be a prime number. Let $m \\in \\N$ be a natural number Let $\\map \\nu {p^n}$ denote the $\\nu$ function of $p^n$: the number of group types of order $p^m$. Then: :$\\map \\nu {p^m} = p^{A m^3}$ where: :$\\displaystyle \\lim_{m \\mathop \\to \\infty} A = \\dfrac 2 {27}$"}
+{"_id": "18230", "title": "Copeland-Erdős Constant is Normal", "text": "The Copeland-Erdős constant whose decimal expansion is formed by concatenating the prime numbers in ascending order: :$0 \\cdotp 23571 \\, 11317 \\, 1923 \\ldots$ is normal with respect to base $10$."}
+{"_id": "18231", "title": "Koebe Quarter Theorem", "text": "Let $f: \\C \\to \\C$ be a schlicht function, that is, a univalent complex function such that $\\map f 0 = 0$ and $\\map {f'} 0 = 1$. Then the image of the unit disk contains the closed disk of radius $\\dfrac 1 4$. Hence for any $w \\in f \\sqbrk {\\Bbb D}$ we have that $\\cmod w \\le \\dfrac 1 4$. The constant $\\dfrac 1 4$ is sharp and so cannot be improved."}
+{"_id": "18233", "title": "Limit to Infinity of Summation of Euler Phi Function over Square", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\dfrac {\\map \\Phi n} {n^2} = \\dfrac 3 {\\pi^2}$ where: :$\\map \\Phi n = \\displaystyle \\sum_{k \\mathop = 1}^n \\map \\phi k$ :$\\map \\phi k$ is the Euler $\\phi$ function of $k$. Numerically, this evaluates to: :$\\dfrac 3 {\\pi^2} \\approx 0 \\cdotp 30396 35509 \\ldots$ {{OEIS|A104141}}"}
+{"_id": "18234", "title": "Blaschke's Theorem", "text": "Let $C$ be a closed convex curve. Let the minimum width $w$ of $C$ be such that $w \\ge 1$. Then $C$ can contain a circle whose radius $\\dfrac 1 3$."}
+{"_id": "18235", "title": "At Least One Third of Zeros of Riemann Zeta Function on Critical Line", "text": "At least $\\dfrac 1 3$ of the nontrivial zeroes of the Riemann $\\zeta$ function lie on the critical line."}
+{"_id": "18237", "title": "Complex Sine Function is Entire", "text": "Let $\\sin: \\C \\to \\C$ be the complex sine function.  Then $\\sin$ is entire."}
+{"_id": "18238", "title": "Summation of Reciprocal of Zero of Order 1 Bessel Function by Order 0 Bessel Function of it", "text": ":$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\dfrac 1 {x_n \\map {J_0 } {x_n} } = 0 \\cdotp 38479 \\ldots$ where: :$x_n$ is the $n$th zero of the order $1$ Bessel function of the first kind :$\\map {J_0 } {x_n}$ is the order $0$ Bessel function of the first kind of $x_n$."}
+{"_id": "18239", "title": "Bloch's Theorem", "text": "Let $f: \\C \\to \\C$ be a holomorphic function in the unit disk $\\cmod z \\le 1$. Let $\\cmod {\\map {f'} 0} = 1$. Then there exists: :a disk $D$ of radius $B$ :an analytic function $\\phi$ in $D$ such that $\\map f {\\map \\phi z} = z$ for all $z$ in $D$ where $B > \\dfrac 1 {72}$ is an absolute constant."}
+{"_id": "18240", "title": "Complex Cosine Function is Entire", "text": "Let $\\cos: \\C \\to \\C$ be the complex cosine function.  Then $\\cos$ is entire."}
+{"_id": "18241", "title": "Arccosecant Logarithmic Formulation", "text": "Let $x$ be a real number.  Let $x \\in \\hointl {-\\infty} {-1} \\cup \\hointr 1 {\\infty}$. Then:  :$\\displaystyle \\arccsc x = -i \\map \\Ln {\\sqrt {1 - \\frac 1 {x^2} } + \\frac i x}$ where: :$\\arccsc$ is the arccosecant function :$\\Ln$ is the principal branch of the complex logarithm whose imaginary part lies in $\\hointl {-\\pi} \\pi$."}
+{"_id": "18243", "title": "Landau's Theorem", "text": "Let $f: \\C \\to \\C$ be a holomorphic function in the unit disk $\\cmod z \\le 1$. Let $\\cmod {\\map {f'} 0} = 1$. Then there exists: :a disk $D$ of radius $B$ :an analytic function $\\phi$ in $D$ such that $\\map f {\\map \\phi z} = z$ for all $z$ in $D$ such that $L$ is an absolute constant where: :$L > B$ where $B$ is Bloch's constant."}
+{"_id": "18244", "title": "Lower Bound of Bloch's Constant", "text": "Bloch's constant has a lower bound as follows: :$\\dfrac {\\sqrt 3} 4 + \\dfrac 2 {10 \\, 000} \\le B$"}
+{"_id": "18245", "title": "Upper Bound of Bloch's Constant", "text": "Bloch's constant has an upper bound as follows: :$B \\le \\sqrt {\\dfrac {\\sqrt 3 -1} 2} \\times \\dfrac {\\map \\Gamma {\\frac 1 3} \\map \\Gamma {\\frac {11} {12} } } {\\map \\Gamma {\\frac 1 4} }$"}
+{"_id": "18246", "title": "Limit of Difference between Consecutive Prime Numbers", "text": "The Prime Number Theorem indicates that the average value of the difference between two consecutive prime numbers is of the order of $\\log p_n$. Let $E = \\displaystyle \\liminf_{n \\mathop \\to \\infty} \\dfrac {p_{n + 1} - p_n} {\\log p_n}$. If there are infinitely many twin primes, then $E = 0$. If not, then it is not known what the value of $E$ is."}
+{"_id": "18248", "title": "Distribution of Numbers with More than 2 Prime Factors", "text": "For sufficiently large $x$, there always exists an integer with more than $2$ prime factors between $\\paren {x - x^\\alpha}$ and $x$, where: :$\\alpha \\ge 0 \\cdotp 477 \\ldots$"}
+{"_id": "18249", "title": "All Nontrivial Zeroes of Riemann Zeta Function are on Critical Strip", "text": "All of the nontrivial zeroes of the Riemann $\\zeta$ function lie on the critical strip."}
+{"_id": "18250", "title": "Nontrivial Zeroes of Riemann Zeta Function are Symmetrical with respect to Critical Line", "text": "The nontrivial zeroes of the Riemann $\\zeta$ function are distributed symmetrically with respect to the  critical line. That is, suppose $s_1 = \\sigma_1 + i t$ is a nontrivial zero of $\\zeta$. Then there exists another nontrivial zero $s_2$ of $\\zeta$ such that: :$s_2 = 1 - \\sigma_1 + i t$"}
+{"_id": "18251", "title": "Critical Line Theorem", "text": "There exist an infinite number of nontrivial zeroes of the Riemann $\\zeta$ function on the critical line."}
+{"_id": "18252", "title": "Prime-Counting Function in terms of Eulerian Logarithmic Integral", "text": "Let $\\map \\pi x$ denote the prime-counting function of a number $x$. Let $\\map \\Li x$ denote the Eulerian logarithmic integral of $x$: :$\\map \\Li x := \\displaystyle \\int_2^x \\dfrac {\\d t} {\\ln t}$ Then: :$\\map \\pi x = \\map \\Li x + \\map {\\mathcal O} {x \\, \\map \\exp {-c \\sqrt {\\ln x} } }$ where: :$\\mathcal O$ is the big-O notation :$c$ is some constant."}
+{"_id": "18254", "title": "Mertens' Third Theorem", "text": ":$\\displaystyle \\lim_{x \\mathop \\to \\infty} \\ln x \\prod_{\\substack {p \\mathop \\le x \\\\ \\text {$p$ prime} } } \\paren {1 - \\dfrac 1 p} = e^{-\\gamma}$ where $\\gamma$ denotes the Euler-Mascheroni constant."}
+{"_id": "18255", "title": "Abi-Khuzam Inequality", "text": "Let $\\triangle ABC$ be a triangle. Then: :$\\sin A \\cdot \\sin B \\cdot \\sin C \\le k A \\cdot B \\cdot C$ where: :$A, B, C$ are measured in radians :$k = \\paren {\\dfrac {3 \\sqrt 3} {2 \\pi} }^3 \\approx 0 \\cdotp 56559 \\, 56245 \\ldots$"}
+{"_id": "18257", "title": "Limit to 1 of Zeta of s minus Reciprocal of s-1", "text": ":$\\displaystyle \\lim_{s \\mathop \\to 1} \\paren {\\map \\zeta s - \\dfrac 1 {s - 1} } = \\gamma$ where: :$\\zeta$ denotes the Riemann $\\zeta$ (zeta) function :$\\gamma$ denotes the Euler-Mascheroni constant."}
+{"_id": "18258", "title": "Limit to Infinity of x minus Gamma of Reciprocal of x", "text": ":$\\displaystyle \\lim_{x \\mathop \\to \\infty} \\paren {x - \\map \\Gamma {\\dfrac 1 x} } = \\gamma$ where: :$\\Gamma$ denotes the $\\Gamma$ (Gamma) function :$\\gamma$ denotes the Euler-Mascheroni constant."}
+{"_id": "18259", "title": "Jung's Theorem", "text": "Let $S \\subseteq \\R^n$ be a compact subspace of an $n$-dimensional Euclidean space. Let $d = \\displaystyle \\max_{x, y \\mathop \\in S} \\map d {x, y}$ be the diameter of $S$. Then there exists a closed ball ${B_r}^-$ with radius $r$ such that: :$r = d \\sqrt {\\dfrac n {2 \\paren {n + 1} } }$ such that $S \\subseteq {B_r}^-$."}
+{"_id": "18260", "title": "Jung's Theorem in the Plane", "text": "Let $S \\subseteq \\R^2$ be a compact region in a Euclidean plane. Let $d$ be the diameter of $S$. Then there exists a circle $C$ with radius $r$ such that: :$r = d \\dfrac {\\sqrt 3} 3$ such that $S \\subseteq C$."}
+{"_id": "18263", "title": "Alternating Sum and Difference of Factorials to Infinity", "text": "According to {{AuthorRef|Leonhard Paul Euler}}: {{begin-eqn}} {{eqn | l = \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n n!       | r = \\int_0^\\infty \\dfrac {e^{-u} } {1 + u} \\rd u       | c =  }} {{eqn | r = G       | c = the Euler-Gompertz constant }} {{eqn | o = \\approx       | r = 0 \\cdotp 59634 \\, 73623 \\, 23194 \\, 07434 \\, 10784 \\, 99369 \\, 27937 \\, 6074 \\ldots       | c =  }} {{end-eqn}} {{explain|Clarify meaning of this equality. Using naive manipulations (carelessly swapping integration/sum and using geometric series results) it's quite straightforward to see why someone might think this is true but we should establish the sense in which it is, since the sum on the LHS diverges.<br/>What we need to do is go back to Euler's original statement of this and see what he meant. He did lots of stuff like this, plugging values into formulae that they weren't applicable to, like e.g. $1 + 2 + 3 + \\ldots {{=}} \\dfrac 1 {12}$, I presume it's like one of those.}}"}
+{"_id": "18264", "title": "Arcsine Function in terms of Gaussian Hypergeometric Function", "text": ":$\\displaystyle \\arcsin x = x \\, {}_2 \\map {F_1} {\\frac 1 2, \\frac 1 2; \\frac 3 2; x^2}$"}
+{"_id": "18269", "title": "Mean Distance between Two Random Points in Cuboid", "text": "Let $B$ be a cuboid in the Cartesian $3$-space $\\R^3$ as: :$\\size x \\le a$, $\\size y \\le b$, $\\size z \\le c$ Let $E$ denote the mean distance $D$ between $2$ points chosen at random from the interior of $B$. Then: {{begin-eqn}} {{eqn | l = E       | r = \\dfrac {2 r} {15} - \\dfrac 7 {45} \\paren {\\paren {r - r_1} \\paren {\\dfrac {r_1} a}^2 + \\paren {r - r_2} \\paren {\\dfrac {r_2} b}^2 + \\paren {r - r_3} \\paren {\\dfrac {r_3} c}^2}       | c =  }} {{eqn | o =       | ro= +       | r = \\dfrac 8 {315 a^2 b^2 c^2} \\paren {a^7 + b^7 + c^7 - {r_1}^7 - {r_2}^7 - {r_3}^7 + r^7}       | c =  }} {{eqn | o =       | ro= +       | r = \\dfrac 1 {15 a b^2 c^2} \\paren {b^6 \\sinh^{-1} \\dfrac a b + c^6 \\sinh^{-1} \\dfrac a c - {r_1}^2 \\paren { {r_1}^4 - 8 b^2 c^2} \\sinh^{-1} \\dfrac a {r_1} }       | c =  }} {{eqn | o =       | ro= +       | r = \\dfrac 1 {15 a^2 b c^2} \\paren {c^6 \\sinh^{-1} \\dfrac b c + a^6 \\sinh^{-1} \\dfrac b a - {r_2}^2 \\paren { {r_2}^4 - 8 c^2 a^2} \\sinh^{-1} \\dfrac b {r_2} }       | c =  }} {{eqn | o =       | ro= +       | r = \\dfrac 1 {15 a^2 b^2 c} \\paren {a^6 \\sinh^{-1} \\dfrac c a + b^6 \\sinh^{-1} \\dfrac c b - {r_3}^2 \\paren { {r_3}^4 - 8 a^2 b^2} \\sinh^{-1} \\dfrac c {r_3} }       | c =  }} {{eqn | o =       | ro= -       | r = \\dfrac 4 {15 a b c} \\paren {a^4 \\arcsin \\dfrac {b c} {r_2 r_3} + b^4 \\arcsin \\dfrac {a c} {r_3 r_1} + c^4 \\arcsin \\dfrac {a b} {r_1 r_2} }       | c =  }} {{end-eqn}} where: {{begin-eqn}} {{eqn | l = r       | r = \\sqrt {a^2 + b^2 + c^2} }} {{eqn | l = r_1       | r = \\sqrt {b^2 + c^2} }} {{eqn | l = r_2       | r = \\sqrt {a^2 + c^2} }} {{eqn | l = r_3       | r = \\sqrt {a^2 + b^2} }} {{end-eqn}}"}
+{"_id": "18270", "title": "Mean Distance between Two Random Points in Unit Cube", "text": "The mean distance $R$ between $2$ points chosen at random from the interior of a unit cube is given by: {{begin-eqn}} {{eqn | l = R       | r = \\frac {4 + 17 \\sqrt 2 - 6 \\sqrt3 - 7 \\pi} {105} + \\frac {\\map \\ln {1 + \\sqrt 2 } } 5 + \\frac {2 \\, \\map \\ln {2 + \\sqrt 3} } 5 }} {{eqn | o = \\approx       | r = 0 \\cdotp 66170 \\, 71822 \\, 67176 \\, 23515 \\, 582 \\ldots       | c =  }} {{end-eqn}} The value $R$ is known as the Robbins constant."}
+{"_id": "18271", "title": "Uniform Matroid is Matroid", "text": "Let $S$ be a finite set of cardinality $n$. Let $0 \\le k \\le n$. Let $U_{k,n} = \\struct{S, \\mathscr I}$ be the uniform matroid of rank $k$. Then $U_{k,n}$ is a matroid."}
+{"_id": "18273", "title": "Matroid Induced by Linear Independence in Vector Space is Matroid", "text": "Let $V$ be a vector space. Let $S$ be a finite subset of $V$. Let $\\struct{S, \\mathscr I}$ be the matroid induced on $S$ by linear independence in $V$. That is, $\\mathscr I$ is the set of linearly independent subsets of $S$. Then $\\struct{S, \\mathscr I}$ is a matroid."}
+{"_id": "18275", "title": "Matroid Induced by Algebraic Independence is Matroid", "text": "Let $L / K$ be a field extension. Let $S \\subseteq L$ be a finite subset of $L$. Let $\\struct{S, \\mathscr I}$ be the matroid induced by algebraic independence over $K$ on $S$. That is, $\\mathscr I$ is the set of algebraically independent subsets of $S$. Then $\\struct{S, \\mathscr I}$ is a matroid."}
+{"_id": "18276", "title": "Matroid Induced by Affine Independence is Matroid", "text": "Let $\\R^n$ be the $n$-dimensional real Euclidean space. Let $S = \\set{x_1, \\dots, x_r}$ be a finite subset of $\\R^n$. Let $\\struct{S, \\mathscr I}$ be the matroid induced by affine independence on $S$. That is, $\\mathscr I$ is the set of affinely independent subsets of $S$. Then $\\struct{S, \\mathscr I}$ is a matroid."}
+{"_id": "18277", "title": "Matroid Induced by Linear Independence in Abelian Group is Matroid", "text": "Let $\\struct{G, +}$ be a torsion-free Abelian group. Let $\\struct{G, +, \\times}$ be the $\\Z$-module associated with $G$. Let $S$ be a finite subset of $G$. Let $\\struct{S, \\mathscr I}$ be the matroid induced by linear independence in $G$ on $S$. That is, $\\mathscr I$ is the set of linearly independent subsets of $S$. Then $\\struct{S, \\mathscr I}$ is a matroid."}
+{"_id": "18278", "title": "Largest Mutually Coprime Subset of Initial Segment of Natural Numbers", "text": "Let $n \\in \\N$ be a natural number. Consider the set $\\N_n$ defined as: :$\\N_n = \\closedint 1 n = \\set {1, 2, \\ldots n}$ Let $Q_n$ be the largest subset of $\\N_n$ such that no element of $Q_n$ is the divisor of another element of $Q_n$. Let $\\map f n$ be the cardinality of $Q_n$. Then for sufficiently large $n$: :$0 \\cdotp 6725 \\ldots \\le \\dfrac {\\map f n} n \\le 0 \\cdotp 6736 \\ldots$"}
+{"_id": "18279", "title": "Independent Set can be Augmented by Larger Independent Set", "text": "Let $M = \\struct{S, \\mathscr I}$ be a matroid. Let $X, Y \\in \\mathscr I$ such that: :$\\size X < \\size Y$ Then there exists non-empty $Z \\subseteq Y \\setminus X$ such that: :$X \\cup Z \\in \\mathscr I$ :$\\size {X \\cup Z} = \\size Y$"}
+{"_id": "18280", "title": "All Bases of Matroid have same Cardinality", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $\\rho: \\powerset S \\to \\Z$ be the rank function of $M$. Let $B$ be a base of $M$. Then: :$\\size B = \\map \\rho S$ That is, all bases of $M$ have the same cardinality, which is the rank of $M$."}
+{"_id": "18281", "title": "Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom", "text": "Let $S$ be a finite set. Let $\\mathscr B$ be a non-empty set of subsets of $S$. Then $\\mathscr B$ is the set of bases of a matroid on $S$ {{iff}} $\\mathscr B$ satisfies the base axiom: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1}}"}
+{"_id": "18282", "title": "Number of Integer Partitions into Sum of Consecutive Primes", "text": "Let $n$ be a natural number. Let $\\map f n$ denote the number of integer partitions of $n$ where the parts are consecutive prime numbers. For example: :$\\map f {41} = 3$ because: :$41 = 11 + 13 + 17 = 2 + 3 + 5 + 7 + 11 + 13$ Then: :$\\displaystyle \\lim_{x \\mathop \\to \\infty} \\dfrac 1 x \\sum_{n \\mathop = 1}^x \\map f n = \\ln 2$"}
+{"_id": "18287", "title": "Dixon's Theorem (Group Theory)", "text": "Let $P_1$ and $P_2$ be distinct elements of the symmetric group on $n$ letters. The probability that $\\set {P_1, P_2}$ forms a generator of $S_n$ approaches $\\dfrac 3 4$ as $n$ tends to infinity."}
+{"_id": "18288", "title": "P-adic Integer is Limit of Unique P-adic Expansion", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\Z_p$ be the $p$-adic integers. Let $x \\in \\Z_p$. Then $x$ is the limit of a unique $p$-adic expansion of the form: :$\\displaystyle \\sum_{n \\mathop = 0}^\\infty d_n p^n$"}
+{"_id": "18289", "title": "P-adic Number is Limit of Unique P-adic Expansion", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $x \\in \\Q_p$. Then $x$ is the limit of a unique $p$-adic expansion."}
+{"_id": "18291", "title": "Number of Natural Numbers Less than x which are Squares or Sums of Two Squares", "text": "Let $x$ be a real number. The number of natural numbers smaller than $x$ which are either square or the sum of $2$ squares is given by the expression: :$\\map N x \\approx \\dfrac {k x} {\\sqrt {\\ln x} }$ where $k$ is given by: :$k = \\sqrt {\\dfrac 1 2 \\displaystyle \\prod_{\\substack {r \\mathop = 4 n \\mathop + 3 \\\\ \\text {$r$ prime} } } \\paren {1 - \\dfrac 1 {r^2} }^{-1} }$ The number $k$ is known as the Landau-Ramanujan constant: {{:Definition:Landau-Ramanujan Constant}}"}
+{"_id": "18292", "title": "Finite Set Contains Subset of Smaller Cardinality", "text": "Let $S$ be a finite sets. Let :$\\size S = n$ where $\\size {\\, \\cdot \\,}$ denotes cardinality. Let $0 \\le m \\le n$. Then there exists a subset $X \\subseteq S$ such that: :$\\size X = m$"}
+{"_id": "18293", "title": "Bounds on Number of Odd Terms in Pascal's Triangle", "text": "Let $P_n$ be the number of odd elements in the first $n$ rows of Pascal's triangle. Then: :$0 \\cdotp 812 \\ldots < \\dfrac {P_n} {n^{\\lg 3} } < 1$ where $\\lg 3$ denotes logarithm base $2$ of $3$. The lower bound $0 \\cdotp 812 \\ldots$ is known as the Stolarsky-Harborth constant."}
+{"_id": "18295", "title": "Integral from 0 to 1 of Complete Elliptic Integral of First Kind", "text": "Let $G$ denote Catalan's constant. Then: :$2 G = \\displaystyle \\int_0^1 \\map K k \\rd k$ where $\\map K k$ denotes the complete elliptic integral of the first kind: :$\\map K k = \\displaystyle \\int \\limits_0^{\\pi / 2} \\dfrac {\\d \\phi} {\\sqrt {1 - k^2 \\sin^2 \\phi} }$"}
+{"_id": "18296", "title": "Trivial Group is Smallest Group", "text": "Let $G = \\struct {\\set e, \\circ}$ be a trivial group. Then $G$ is the smallest group possible, in that there exists no set with lower cardinality which is the underlying set of a group."}
+{"_id": "18298", "title": "Definition:Salem Constant", "text": "The '''Salem constant''' is the greatest real root of Lehmer's polynomial: :$x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1 = 0$ Its value is approximately: :$1 \\cdotp 17628 \\, 08182 \\, 599 \\ldots$ {{OEIS|A073011}}"}
+{"_id": "18300", "title": "Partial Sums of P-adic Expansion forms Coherent Sequence", "text": "Let $p$ be a prime number. Let $\\displaystyle \\sum_{n \\mathop = 0}^\\infty d_n p^n$ be a $p$-adic expansion. Let $\\sequence{\\alpha_n}$ be the sequence of partial sums;  that is: :$\\forall n \\in \\N :\\alpha_n = \\displaystyle \\sum_{i \\mathop = 0}^n d_i p^i$. Then $\\sequence{\\alpha_n}$ is a coherent sequence."}
+{"_id": "18302", "title": "Gibbs Phenomenon", "text": "The Fourier series overshoots at a jump discontinuity, and adding more terms to the sum does not cause this overshoot to die out. :500px :500px {{finish|Anyone care to flesh this out? I've got bored with it.}}"}
+{"_id": "18305", "title": "Mergelyan-Wesler Theorem", "text": "Let $P = \\sequence {D_1, D_2, \\dotsc}$ be an infinite sequence of disjoint open disks whose union is the unit disk $D$ except for a set of measure zero. Let $r_n$ be the radius of $D_n$. Then: :$\\displaystyle \\sum_{k \\mathop = 1}^\\infty r_k = +\\infty$"}
+{"_id": "18306", "title": "Definition:Solid-Packing Constant for Circles", "text": "Let $P = \\sequence {D_1, D_2, \\dotsc}$ be an infinite sequence of disjoint open disks whose union is the unit disk $D$ except for a set of measure zero. Let $r_n$ be the radius of $D_n$. Let $x \\in \\R_{>0}$ be a (strictly) positive real number. Let $\\map {M_x} P$ be defined as: :$\\map {M_x} P = \\displaystyle \\sum_{k \\mathop = 1}^\\infty {r_k}^x$ For each $P$, there exists a (real) number $\\map e P$ such that: :$\\map {M_x} P$ is divergent for $x < \\map e P$ :$\\map {M_x} P$ is convergent for $x > \\map e P$ From the Mergelyan-Wesler Theorem: :$1 < \\map e P < 2$ for all $P$. The constant $S$ such that: :$S < \\map e P$ is known as the '''solid-packing constant for circles'''. It can be interpreted as the fractal dimension of the set of points of $P$ which are not covered by the $D_n$ open disks."}
+{"_id": "18307", "title": "Value of Solid-Packing Constant for Circles", "text": "The value of the solid-packing constant for circles is estimated as being: :$S \\approx 1 \\cdotp 306951 \\ldots$"}
+{"_id": "18308", "title": "Integer Arbitrarily Close to Rational in Valuation Ring of P-adic Norm", "text": "Let $\\norm {\\,\\cdot\\,}_p$ be the $p$-adic norm on the rationals $\\Q$ for some prime number $p$. Let $x \\in \\Q$ such that $\\norm x_p \\le 1$. Then for all $i \\in \\N$ there exists $\\alpha \\in \\Z$ such that: :$\\norm{x - \\alpha}_p \\le p^{-i}$"}
+{"_id": "18309", "title": "Equivalence Class in P-adic Integers Contains Unique Coherent Sequence", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. Let $\\mathbf a$ be an equivalence class in $\\Q_p$ such that $\\norm{\\mathbf a}_p \\le 1$. Then $\\mathbf a$ has exactly one representative that is a coherent sequence."}
+{"_id": "18310", "title": "Equivalence Class in P-adic Integers Contains Unique Coherent Sequence/P-adic Expansion", "text": "Then $\\mathbf a$ has exactly one representative that is a $p$-adic expansion of the form: :$\\displaystyle \\sum_{n \\mathop = 0}^\\infty d_n p^n$"}
+{"_id": "18311", "title": "Equivalence Class in P-adic Numbers Contains Unique P-adic Expansion", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. Let $\\mathbf a$ be an equivalence class in $\\Q_p$. Then $\\mathbf a$ has exactly one representative that is a $p$-adic expansion."}
+{"_id": "18312", "title": "Value of Plastic Constant", "text": "The plastic constant $P$ is evaluated as: {{begin-eqn}} {{eqn | l = P       | r = \\sqrt [3] {\\frac {9 + \\sqrt {69} } {18} } + \\sqrt [3] {\\frac {9 - \\sqrt {69} } {18} }       | c =  }} {{eqn | r = 1 \\cdotp 32471 \\, 79572 \\, 44746 \\, 02596 \\, 09088 \\, 54 \\ldots       | c =  }} {{end-eqn}}"}
+{"_id": "18313", "title": "Plastic Constant is Smallest Pisot-Vijayaraghavan Number", "text": "The plastic constant is the smallest Pisot-Vijayaraghavan number."}
+{"_id": "18314", "title": "Unique Integer Close to Rational in Valuation Ring of P-adic Norm", "text": "Let $\\norm {\\,\\cdot\\,}_p$ be the $p$-adic norm on the rationals $\\Q$ for some prime number $p$. Let $x \\in \\Q$ such that $\\norm{x}_p \\le 1$. Then for all $i \\in \\N$ there exists a unique $\\alpha \\in \\Z$ such that: :$(1): \\quad \\norm {x - \\alpha}_p \\le p^{-i}$ :$(2): \\quad  0 \\le \\alpha \\le p^i - 1$"}
+{"_id": "18315", "title": "Hermite Constant for Dimension 2", "text": "The Hermite constant for dimension $2$ is: :$\\gamma_2 = \\dfrac 2 {\\sqrt 3}$ or, as it is often presented: :$\\paren {\\gamma_2}^2 = \\dfrac 4 3$"}
+{"_id": "18316", "title": "Upper Bound of Hermite Constant", "text": "Let $\\gamma_n$ be the Hermite constant of dimension $n$. Then: :$\\gamma_n \\le \\dfrac {\\paren {1 + \\epsilon_n} n} {\\pi e}$ where $e_n \\to 0$"}
+{"_id": "18317", "title": "Angle between Straight Lines in Plane", "text": "Let $L_1$ and $L_2$ be straight lines embedded in a cartesian plane, given by the equations: {{begin-eqn}} {{eqn | ll= L_1:       | l = y       | r = m_1 x + c_1 }} {{eqn | ll= L_2:       | l = y       | r = m_2 x + c_2 }} {{end-eqn}} Then the angle $\\psi$ between $L_1$ and $L_2$ is given by: :$\\psi = \\arctan \\dfrac {m_1 - m_2} {1 + m_1 m_2}$"}
+{"_id": "18318", "title": "Parallel Straight Lines have Same Slope", "text": "Let $L_1$ and $L_2$ be straight lines in the plane. Let $L_1$ and $L_2$ have slopes of $m_1$ and $m_2$ respectively. Then $L_1$ and $L_2$ are parallel {{iff}} $m_1 = m_2$."}
+{"_id": "18319", "title": "Product of Slopes of Perpendicular Lines is Minus 1", "text": "Let $L_1$ and $L_2$ be straight lines in the plane. Let $L_1$ and $L_2$ have slopes of $m_1$ and $m_2$ respectively. Then $L_1$ and $L_2$ are perpendicular {{iff}} $m_1 m_2 = -1$."}
+{"_id": "18321", "title": "Slope of Tangent to Lemniscate at Origin", "text": "Consider the lemniscate of Bernoulli $M$ embedded in a Cartesian plane such that its foci are at $\\tuple {a, 0}$ and $\\tuple {-a, 0}$ respectively. Let $O$ denote the origin. The tangents to $M$ at $O$ are at an angle of $45 \\degrees = \\dfrac \\pi 4$ to the $x$-axis. :630px"}
+{"_id": "18322", "title": "Area of Lobe of Lemniscate of Bernoulli", "text": "Consider the lemniscate of Bernoulli $M$ embedded in a Cartesian plane such that its foci are at $\\tuple {a, 0}$ and $\\tuple {-a, 0}$ respectively. Let $O$ denote the origin. The area of one lobe of $M$ is $a^2$."}
+{"_id": "18324", "title": "Equation of Cardioid/Polar", "text": "Let $C$ be a cardioid embedded in a polar coordinate plane such that: :its stator of radius $a$ is positioned with its center at $\\polar {a, 0}$ :there is a cusp at the origin. The polar equation of $C$ is: :$r = 2 a \\paren {1 + \\cos \\theta}$"}
+{"_id": "18326", "title": "Length of Perimeter of Cardioid", "text": "Consider the cardioid $C$ embedded in a polar plane given by its polar equation: :$r = 2 a \\paren {1 + \\cos \\theta}$ where $a > 0$. The length of the perimeter of $C$ is $16 a$."}
+{"_id": "18327", "title": "Arc Length for Polar Curve", "text": "Let $a$ and $b$ be real numbers. Let $\\mathcal C$ be a simple curve continuous on $\\closedint a b$ and continuously differentiable on $\\openint a b$. Let $\\mathcal C$ be described by the parametric equations: :$\\begin {cases} x & = r \\cos \\theta \\\\ y & = r \\sin \\theta \\end {cases}$ where: :$r$ is a function of $\\theta$ :$\\theta \\in \\closedint a b$. Then the length $s$ of $\\mathcal C$ is given by: :$\\displaystyle s = \\int_a^b \\sqrt {r^2 + \\paren {\\frac {\\d r} {\\d \\theta} }^2} \\rd \\theta$"}
+{"_id": "18328", "title": "Number of Petals of Odd Index Rhodonea Curve", "text": "Let $n$ be an odd positive integer. Let $R$ be a '''rhodonea curve''' defined by one of the polar equations: {{begin-eqn}} {{eqn | l = r       | r = a \\cos n \\theta }} {{eqn | l = r       | r = a \\sin n \\theta }} {{end-eqn}} Then $R$ has $n$ petals."}
+{"_id": "18329", "title": "Number of Petals of Even Index Rhodonea Curve", "text": "Let $n$ be an even (strictly) positive integer. Let $R$ be a '''rhodonea curve''' defined by one of the polar equations: {{begin-eqn}} {{eqn | l = r       | r = a \\cos n \\theta }} {{eqn | l = r       | r = a \\sin n \\theta }} {{end-eqn}} Then $R$ has $2 n$ petals."}
+{"_id": "18331", "title": "Equation of Tractrix", "text": "Let $S$ be a cord of length $a$ situated as a (straight) line segment whose endpoints are $P$ and $T$. Let $S$ be aligned along the $x$-axis of a cartesian plane with $T$ at the origin and $P$ therefore at the point $\\tuple {a, 0}$. Let $T$ be dragged along the $y$-axis. :File:Tractrix.png"}
+{"_id": "18332", "title": "Equation of Witch of Agnesi/Parametric Form", "text": "The equation of the Witch of Agnesi can be presented in paremetric form as: :$\\begin {cases} x = 2 a \\cot \\theta \\\\ y = a \\paren {1 - \\cos 2 \\theta} \\end {cases}$"}
+{"_id": "18334", "title": "Area of Loop of Folium of Descartes", "text": "Consider the folium of Descartes $F$, given in parametric form as: :$\\begin {cases} x = \\dfrac {3 a t} {1 + t^3} \\\\ y = \\dfrac {3 a t^2} {1 + t^3} \\end {cases}$ The area $\\AA$ of the loop of $F$ is given as: :$\\AA = \\dfrac {3 a^2} 2$"}
+{"_id": "18335", "title": "Asymptote to Folium of Descartes", "text": "Consider the folium of Descartes $F$, given in parametric form as: :$\\begin {cases} x = \\dfrac {3 a t} {1 + t^3} \\\\ y = \\dfrac {3 a t^2} {1 + t^3} \\end {cases}$ The straight line whose equation is given by: :$x + y + a = 0$ is an asymptote to $F$."}
+{"_id": "18336", "title": "Behaviour of Parametric Equations for Folium of Descartes according to Parameter", "text": "Consider the folium of Descartes $F$, given in parametric form as: :$\\begin {cases} x = \\dfrac {3 a t} {1 + t^3} \\\\ y = \\dfrac {3 a t^2} {1 + t^3} \\end {cases}$ Then: :$F$ has a discontinuity at $t = -1$. :For $t < -1$, the section in the $4$th quadrant is generated :For $-1 < t \\le 0$, the section in the $2$nd quadrant is generated  :For $0 \\le t$, the section in the $1$st quadrant is generated."}
+{"_id": "18338", "title": "Evolute of Ellipse/Cartesian Form", "text": "The evolute of $E$ is given by the Cartesian equation: :$\\paren {a x}^{2 / 3} + \\paren {b y}^{2 / 3} = \\paren {a^2 - b^2}^{2 / 3}$"}
+{"_id": "18339", "title": "Evolute of Ellipse/Parametric Form", "text": "The evolute of $E$ can be expressed using the parametric equation: :$\\begin {cases} a x = \\paren {a^2 - b^2} \\cos^3 \\theta \\\\ b y = \\paren {a^2 - b^2} \\sin^3 \\theta \\end {cases}$"}
+{"_id": "18340", "title": "Evolute of Ellipse", "text": "Let $E$ be an ellipse embedded in a Cartesian plane with the equation: :$\\dfrac {x^2} {a^2} + \\dfrac {y^2} {b^2} = 1$"}
+{"_id": "18341", "title": "Equation of Ovals of Cassini", "text": "Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$. Let $b$ be a real constant."}
+{"_id": "18342", "title": "Equation of Ovals of Cassini/Cartesian Form", "text": "The Cartesian equation: :$\\paren {x^2 + y^2 + a^2}^2 - 4 a^2 x^2 = b^4$ describes the '''ovals of Cassini'''."}
+{"_id": "18343", "title": "Equation of Ovals of Cassini/Polar Form", "text": "The polar equation: :$r^4 + a^4 - 2 a^2 r^2 \\cos 2 \\theta = b^4$ describes the '''ovals of Cassini'''."}
+{"_id": "18344", "title": "Lemniscate of Bernoulli is Special Case of Ovals of Cassini", "text": "The lemniscate of Bernoulli is a special case of the ovals of Cassini."}
+{"_id": "18347", "title": "Cramer's Rule", "text": "Let $n \\in \\N$. Let $b_1, b_2, \\dots, b_n$ be real numbers. Let $\\mathbf b = \\tuple {b_1, b_2, \\dots, b_n}^T$. Let $x_1, x_2, \\dots, x_n$ be real numbers. Let $\\mathbf x = \\tuple {x_1, x_2, \\dots, x_n}^T$. Let $A$ be an invertible $n \\times n$ matrix with coefficients in $\\R$. For each $i \\in \\set {1, \\dots, n}$, let $A_i$ be the matrix obtained by replacing the $i$th column with $\\mathbf b$. Let: :$A \\mathbf x = \\mathbf b$ Then: :$\\mathbf x_i = \\dfrac {\\map \\det {A_i} } {\\map \\det A}$ for each $i \\in \\set {1, \\dots, n}$."}
+{"_id": "18348", "title": "Equation of Limaçon of Pascal", "text": "Let $C$ be a circle of diameter $a$ whose circumference passes through the origin $O$ and whose diameter through $O$ lies on the horizontal. Let $b$ be a real constant."}
+{"_id": "18349", "title": "Equation of Limaçon of Pascal/Polar Form", "text": "The limaçon of Pascal can be defined by the polar equation: :$r = b + a \\cos \\theta$"}
+{"_id": "18350", "title": "Equation of Cissoid of Diocles", "text": "Let $C$ be a circle of radius $a$ whose circumference passes through the origin $O$ and whose diameter through $O$ lies on the horizontal."}
+{"_id": "18351", "title": "Equation of Cissoid of Diocles/Polar Form", "text": "The cissoid of Diocles can be defined by the polar equation: :$r = 2 a \\sin \\theta \\tan \\theta$"}
+{"_id": "18353", "title": "Equation of Cissoid of Diocles/Parametric Form", "text": "The cissoid of Diocles can be defined by the parametric equation: :$\\begin {cases} x = 2 a \\sin^2 \\theta \\\\ y = \\dfrac {2 a \\sin^3 \\theta} {\\cos \\theta} \\end {cases}$"}
+{"_id": "18354", "title": "Primitive of One plus x Squared over One plus Fourth Power of x", "text": ":$\\displaystyle \\int \\frac {x^2 + 1} {x^4 + 1} \\rd x = \\frac 1 {\\sqrt 2} \\map \\arctan {\\frac 1 {\\sqrt 2} \\paren {x - \\frac 1 x} } + C$"}
+{"_id": "18355", "title": "Primitive of Minus One plus x Squared over One plus Fourth Power of x", "text": ":$\\displaystyle \\int \\frac {x^2 - 1} {x^4 + 1} \\rd x = \\frac 1 {2 \\sqrt 2} \\ln \\size {\\frac {x^2 - \\sqrt 2 x + 1} {x^2 + \\sqrt 2 x + 1} } + C$"}
+{"_id": "18362", "title": "Dissection of Square into 8 Acute Triangles", "text": "A square can be dissected into $8$ acute triangles."}
+{"_id": "18363", "title": "Dissection of Square into 9 Acute Triangles", "text": "A square can be dissected into $9$ acute triangles."}
+{"_id": "18364", "title": "Length of Chord Projected from Point on Intersecting Circle", "text": "Let $C_1$ and $C_2$ be two circles which intersect at $A$ and $B$. Let $T$ be a point on $C_1$. Let $P$ and $Q$ be the points $TA$ and $TB$ intersect $C_2$. :400px Then $PQ$ is constant, wherever $T$ is positioned on $C_1$."}
+{"_id": "18366", "title": "First Projection on Ordered Pair of Sets", "text": "Let $a$ and $b$ be sets. Let $w = \\tuple {a, b}$ denote the ordered pair of $a$ and $b$. Let $\\map {\\pr_1} w$ denote the first projection on $w$. Then: :$\\displaystyle \\map {\\pr_1} w = \\bigcup \\bigcap w$ where $\\displaystyle \\bigcup$ and $\\displaystyle \\bigcap$ denote union and intersection respectively."}
+{"_id": "18367", "title": "Intersection of Doubleton", "text": "Let $\\set {x, y}$ be a doubleton. Then: :$\\displaystyle \\bigcap \\set {x, y} = x \\cup y$"}
+{"_id": "18368", "title": "Second Projection on Ordered Pair of Sets", "text": "Let $a$ and $b$ be sets. Let $w = \\tuple {a, b}$ denote the ordered pair of $a$ and $b$. Let $\\map {\\pr_2} w$ denote the second projection on $w$. Then: :$\\displaystyle \\map {\\pr_2} w = \\begin {cases} \\displaystyle \\map \\bigcup {\\bigcup w \\setminus \\bigcap w} & : \\displaystyle \\bigcup w \\ne \\bigcap w \\\\ \\displaystyle \\bigcup \\bigcup w & : \\displaystyle \\bigcup w = \\bigcap w \\end {cases}$ where: :$\\displaystyle \\bigcup$ and $\\displaystyle \\bigcap$ denote union and intersection respectively. :$\\setminus$ denotes the set difference operator."}
+{"_id": "18369", "title": "Order Isomorphism between Tosets is not necessarily Unique", "text": "Let $\\struct {S_1, \\preccurlyeq_1}$ and $\\struct {S_2, \\preccurlyeq_2}$ be tosets. Let $\\struct {S_1, \\preccurlyeq_1} \\cong \\struct {S_2, \\preccurlyeq_2}$, that is, let $\\struct {S_1, \\preccurlyeq_1}$ and $\\struct {S_2, \\preccurlyeq_2}$ be order isomorphic. Then it is not necessarily the case that there is exactly one mapping $f: S_1 \\to S_2$ such that $f$ is an order isomorphism."}
+{"_id": "18370", "title": "Pi as Sum of Sequence of Reciprocal of Product of Three Consecutive Integers/Lemma", "text": ":$\\displaystyle \\iiint \\dfrac x {x^2 + 1} \\rd x \\rd x \\rd x = x \\map \\arctan x + \\dfrac {\\paren {x^2 - 1} \\map \\ln {x^2 + 1} - 3 x^2} 4$ with all integration constants at $0$."}
+{"_id": "18371", "title": "Supremum of Set Equals Maximum of Suprema of Subsets", "text": "Let $S$ be a non-empty real set. Let $\\set {S_i: i \\in \\set {1, 2, \\ldots, n} }$, $n \\in \\N_{>0}$, be a set of non-empty subsets of $S$. Let $S = \\bigcup S_i$. Then: :$S_i$ has a supremum for every $i$ in $\\set {1, 2, \\ldots, n}$ {{iff}}: :$S$ has a supremum and, in either case: :$\\sup S = \\max \\set {\\sup S_1, \\sup S_2, \\ldots, \\sup S_n}$"}
+{"_id": "18372", "title": "Equivalence Class in P-adic Integers Contains Unique Coherent Sequence/Lemma 1", "text": ":$\\forall j \\in \\N: \\norm {\\beta_{\\map N {j + 1} } - \\beta_{\\map N j} }_p \\le p^{-\\paren {j + 1} }$"}
+{"_id": "18373", "title": "Equivalence Class in P-adic Integers Contains Unique Coherent Sequence/Lemma 2", "text": ":$\\forall j \\in \\N: \\norm {\\alpha_{j + 1} - \\alpha_j }_p \\le p^{-\\paren {j + 1} }$"}
+{"_id": "18377", "title": "Differential Equation defining Confocal Conics", "text": "Consider the equation: :$(1): \\quad \\dfrac {x^2} {a^2 + \\lambda} + \\dfrac {y^2} {b^2 + \\lambda} = 1$ where $a^2 > b^2$ and $-\\lambda < a^2$. defining the set of confocal conics whose foci are at $\\tuple {\\pm \\sqrt {a^2 - b^2}, 0}$. The differential equation defining these confocal conics is: :$x y \\paren {\\paren {y'}^2 - 1} + \\paren {x^2 - y^2 - a^2 + b^2} y' = 0$"}
+{"_id": "18378", "title": "Equivalence Class in P-adic Integers Contains Unique Coherent Sequence/Lemma 3", "text": ":$\\sequence {\\alpha_n}$ and $\\sequence {\\beta_n}$ are representatives of the same equivalence class in $\\Q_p$."}
+{"_id": "18379", "title": "Equivalence Class in P-adic Integers Contains Unique Coherent Sequence/Lemma 4", "text": ":$\\sequence {\\alpha_j}$ is the only coherent sequence that represents $\\mathbf a$."}
+{"_id": "18380", "title": "Elimination of Constants by Partial Differentiation", "text": "Let $x_1, x_2, \\dotsc, x_m$ be independent variables. Let $c_1, c_2, \\dotsc, c_n$ be arbitrary constants. Let this equation: :$(1): \\quad \\map f {x_1, x_2, \\dotsc, x_m, z, c_1, c_2, \\dotsc, c_n} = 0$ define a dependent variable $z$ via the implicit function $f$. Then it may be possible to eliminate the constants by successive partial differentiation of $(1)$."}
+{"_id": "18381", "title": "Representatives of same P-adic Number iff Difference is Null Sequence", "text": "Let $p$ be a prime number. Let $\\norm {\\,\\cdot\\,}_p$ be the $p$-adic norm on the rational numbers $\\Q$. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. Let $\\sequence{\\alpha_n}$ and $\\sequence{\\beta_n}$ be Cauchy sequences in $\\struct {Q, \\norm {\\,\\cdot\\,}_p}$. Then:  :$\\sequence {\\alpha_n}$ and $\\sequence {\\beta_n}$ are representatives of the same equivalence class in $\\Q_p$ {{iff}}: :the sequence $\\sequence {\\alpha_n - \\beta_n}$ is a null sequence."}
+{"_id": "18385", "title": "Inverse of Vandermonde Matrix/Eisinberg Formula", "text": "Let: {{begin-eqn}} {{eqn | l = \\prod_{k \\mathop = 1}^n \\paren {x - x_k}       | r = a_nx^n + \\sum_{m \\mathop = 0}^{n-1} a_m x^m       | c = Polynomial expansion in powers of $x$ }} {{eqn | r = x^n + \\sum_{m \\mathop = 0}^{n - 1} \\paren {-1}^{n - m} \\map {e_{n-m} } {x_1, \\ldots, x_n} \\, x^m       | c =  Viete's Formulas and {{Defof|Elementary Symmetric Function}} }} {{eqn | l = W_n        | r =  \\begin{bmatrix}   1         &  x_1      & \\cdots & x_1^{n-1} \\\\   1         & x_2       & \\cdots & x_2^{n-1} \\\\ \\vdots      & \\vdots    & \\ddots & \\vdots    \\\\   1         & x_1^{n-1} & \\cdots & x_n^{n-1} \\\\ \\end{bmatrix}       | c = {{Defof|Vandermonde Matrix}} of Order $n$ }} {{end-eqn}} Let $W_n$ have a matrix inverse $W_n^{-1} = \\begin {bmatrix} d_{ij} \\end {bmatrix}$.  Let $a_n = \\map {e_0} {x_1, \\ldots, x_n} = 1$. Then: {{begin-eqn}} {{eqn | n = 1       | l = d_{ij}       | r = \\dfrac {\\displaystyle \\sum_{k \\mathop = 0}^{n - i} a_{i + k} \\, x_j^k} {\\displaystyle \\prod_{m \\mathop = 1, m \\mathop \\ne j }^n \\paren {x_j - x_m} }       | c = for $i, j = 1, \\ldots, n$ }} {{eqn | r = \\dfrac {\\displaystyle \\sum_{k \\mathop = 0}^{n - i} \\paren {-1}^{n - i - k} \\map {e_{n - i - k} } {x_1, \\ldots, x_n} \\, x_j^k} {\\displaystyle \\prod_{m \\mathop = 1, m \\mathop \\ne j }^n \\paren {x_j - x_m} }       | c = for $i, j = 1, \\ldots, n$ }} {{end-eqn}}"}
+{"_id": "18386", "title": "Linear First Order ODE/dy = f(x) dx", "text": "Let $f: \\R \\to \\R$ be an integrable real function. The linear first order ODE: :$(1): \\quad \\dfrac {\\d y} {\\d x} = \\map f x$ has the general solution: :$y = \\displaystyle \\int \\map f x \\rd x + C$ where $\\displaystyle \\int \\map f x \\rd x$ denotes the primitive of $f$."}
+{"_id": "18387", "title": "Linear First Order ODE/dy = f(x) dx/Initial Condition", "text": "Consider the linear first order ODE: :$(1): \\quad \\dfrac {\\d y} {\\d x} = \\map f x$ subject to the initial condition: :$y = y_0$ when $x = x_0$ $(1)$ has the particular solution: :$y = y_0 + \\displaystyle \\int_{x_0}^x \\map f \\xi \\rd \\xi$ where $\\displaystyle \\int \\map f x \\rd x$ denotes the primitive of $f$."}
+{"_id": "18389", "title": "Solution by Integrating Factor/Examples/y' - 3y = sin x", "text": "The linear first order ODE: :$\\dfrac {\\d y} {\\d x} - 3 y = \\sin x$ has the general solution: :$y = \\dfrac 1 {10} \\paren {3 \\sin x - \\cos x} + C e^{3 x}$"}
+{"_id": "18390", "title": "Solution to Linear First Order ODE with Constant Coefficients", "text": "A linear first order ODE with constant coefficients in the form: :$(1): \\quad \\dfrac {\\d y} {\\d x} + a y = \\map Q x$ has the general solution: :$\\ds y = e^{-a x} \\paren {\\int e^{a x} \\map Q x \\rd x + C}$"}
+{"_id": "18393", "title": "Solution to Linear First Order ODE with Constant Coefficients/With Initial Condition", "text": "Consider the linear first order ODE with constant coefficients in the form: :$(1): \\quad \\dfrac {\\d y} {\\d x} + a y = \\map Q x$ with initial condition $\\tuple {x_0, y_0}$ Then $(1)$ has the particular solution: :$\\displaystyle y = e^{-a x} \\int_{x_0}^x e^{a \\xi} \\map Q \\xi \\rd \\xi + y_0 e^{a \\paren {x - x_0} }$"}
+{"_id": "18396", "title": "Solution by Integrating Factor/Examples/y' + y = x^-1", "text": "Consider the linear first order ODE: :$(1): \\quad \\dfrac {\\d y} {\\d x} + y = \\dfrac 1 x$ with the initial condition $\\tuple {1, 0}$. This has the particular solution: :$y = \\displaystyle e^{-x} \\int_1^x \\dfrac {e^\\xi \\rd \\xi} \\xi$"}
+{"_id": "18397", "title": "General Solution equals Particular Solution plus Complementary Function", "text": "Consider the linear first order ODE with constant coefficients: :$(1): \\quad \\dfrac {\\d y} {\\d x} + a y = \\map Q x$ The general solution to $(1)$ consists of: :the particular solution to $(1)$ for which the arbitrary constant is $0$ plus: :the complementary function to $(1)$."}
+{"_id": "18398", "title": "Derivation of Auxiliary Equation to Constant Coefficient LSOODE", "text": "Consider the linear Second Order ODE with Constant Coefficients: :$(1): \\quad \\dfrac {\\d^2 y} {\\d x^2} + p \\dfrac {\\d y} {\\d x} + q y = \\map R x$ and its auxiliary equation: :$(2): \\quad m^2 + p m + q = 0$ The fact that the solutions of $(2)$ dictate the general solution of $(1)$ can be derived."}
+{"_id": "18399", "title": "Linear Second Order ODE/y'' + 2 y' + 2 y = 0", "text": "The second order ODE: :$(1): \\quad y'' + 2 y' + 2 y = 0$ has the general solution: :$y = e^{-x} \\paren {A \\cos x + B \\sin x}$"}
+{"_id": "18400", "title": "Linear Second Order ODE/y'' + 2 y' + 2 y = 0/Verification", "text": "The equation: :$(1): \\quad y = e^{-x} \\paren {A \\cos x + B \\sin x}$ is a set of solutions to the second order ODE: :$y'' + 2 y' + 2 y = 0$"}
+{"_id": "18401", "title": "Linear Second Order ODE/y'' - 7 y' - 5 y = x^3 - 1", "text": "The second order ODE: :$(1): \\quad y'' - 7 y' - 5 y = x^3 - 1$ has the general solution: :$y = C_1 \\, \\map \\exp {\\paren {\\dfrac 7 2 + \\dfrac {\\sqrt {69} } 2} x} + C_2 \\, \\map \\exp {\\paren {\\dfrac 7 2 - \\dfrac {\\sqrt {69} } 2} x} + \\dfrac 1 {625} \\paren {-125 x^3 + 525 x^2 - 1620 x + 2603}$"}
+{"_id": "18402", "title": "Linear Second Order ODE/y'' - 7 y' - 5 y = 0", "text": "The second order ODE: :$(1): \\quad y'' - 7 y' - 5 y = 0$ has the general solution: :$y = C_1 \\, \\map \\exp {\\paren {\\dfrac 7 2 + \\dfrac {\\sqrt {69} } 2} x} + C_2 \\, \\map \\exp {\\paren {\\dfrac 7 2 - \\dfrac {\\sqrt {69} } 2} x}$"}
+{"_id": "18406", "title": "Convergent Sequence is Cauchy Sequence/Normed Vector Space", "text": "Let $\\struct{X, \\norm{\\,\\cdot\\,} }$ be a normed vector space. Every convergent sequence in $X$ is a Cauchy sequence."}
+{"_id": "18410", "title": "Linear Second Order ODE/y'' - 2 y' - 5 y = 0", "text": "The second order ODE: :$(1): \\quad y'' - 2 y' - 5 y = 0$ has the general solution: :$y = C_1 \\, \\map \\exp {\\paren {1 + \\sqrt 6} x} + C_2 \\, \\map \\exp {\\paren {1 - \\sqrt 6} x}$"}
+{"_id": "18411", "title": "Linear Second Order ODE/y'' - 2 y' - 5 y = 2 cos 3 x - sin 3 x", "text": "The second order ODE: :$(1): \\quad y'' - 2 y' - 5 y = 2 \\cos 3 x - \\sin 3 x$ has the general solution: :$y = C_1 \\, \\map \\exp {\\paren {1 + \\sqrt 6} x} + C_2 \\, \\map \\exp {\\paren {1 - \\sqrt 6} x} + \\dfrac 1 {116} \\paren {\\sin 3 x - 17 \\cos 3 x}$"}
+{"_id": "18412", "title": "Linear Second Order ODE/y'' - 2 y' - 5 y = 2 cos 3 x - sin 3 x/Particular Solution", "text": "The second order ODE: :$(1): \\quad y'' - 2 y' - 5 y = 2 \\cos 3 x - \\sin 3 x$ has a particular solution: :$y_p = \\dfrac 1 {116} \\paren {\\sin 3 x - 17 \\cos 3 x}$"}
+{"_id": "18415", "title": "Linear Second Order ODE/y'' + 4 y = 3 sin 2 x", "text": "The second order ODE: :$(1): \\quad y'' + 4 y = 3 \\sin 2 x$ has the general solution: :$y = C_1 \\sin k x + C_2 \\cos k x - \\dfrac 3 4 x \\cos 2 x$"}
+{"_id": "18416", "title": "Cauchy Sequence is Bounded/Normed Vector Space", "text": "Let $V = \\struct {X, \\norm {\\,\\cdot\\,} }$ be a normed vector space. Every Cauchy sequence in $X$ is bounded."}
+{"_id": "18417", "title": "Linear Second Order ODE/y'' - 4 y' - 5 y = 0", "text": "The second order ODE: :$(1): \\quad y'' - 4 y' - 5 y = 0$ has the general solution: :$y = C_1 e^{5 x} + C_2 e^{-x}$"}
+{"_id": "18418", "title": "Linear Second Order ODE/y'' - 4 y' - 5 y = x^2/y(0) = 1, y'(0) = -1", "text": "Consider the second order ODE: :$(1): \\quad y'' - 4 y' - 5 y = x^2$ whose initial conditions are: :$y = 1$ when $x = 0$ :$y' = -1$ when $x = 0$ $(1)$ has the particular solution: :$y = \\dfrac {e^{5 x} } {375} + \\dfrac {4 e^{-x} } 3 - \\dfrac {x^2} 5 + \\dfrac {8 x} {25} - \\dfrac {42} {125}$"}
+{"_id": "18421", "title": "P-adic Unit has Norm Equal to One", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\Z_p$ denote the $p$-adic integers. Let $x \\in \\Q_p$. Then x is a $p$-adic unit {{iff}} $\\norm x_p = 1$"}
+{"_id": "18422", "title": "P-adic Number times P-adic Norm is P-adic Unit", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\Z_p^\\times$ be the $p$-adic units. Let $a \\in \\Q_p$. Then there exists $n \\in \\Z$ such that: :$p^n a \\in \\Z_p^\\times$ where :$p^n = \\norm a_p$"}
+{"_id": "18423", "title": "Solution to Simultaneous Homogeneous Linear First Order ODEs with Constant Coefficients", "text": "Consider the system of linear first order ordinary differential equations with constant coefficients: {{begin-eqn}} {{eqn | n = 1       | l = \\dfrac {\\d y} {\\d x} + a y + b z       | r = 0 }} {{eqn | n = 2       | l = \\dfrac {\\d x} {\\d z} + c y + d z       | r = 0 }} {{end-eqn}} The general solution to $(1)$ and $(2)$ consists of the linear combinations of the following: {{begin-eqn}} {{eqn | l = y       | r = A_1 e^{k_1 x} }} {{eqn | l = z       | r = B_1 e^{k_1 x} }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = y       | r = A_2 e^{k_2 x} }} {{eqn | l = z       | r = B_2 e^{k_2 x} }} {{end-eqn}} where $A_1 : B_1 = A_2 : B_2 = r$ where $r$ is either of the roots of the quadratic equation: :$\\paren {k + a} \\paren {k + d} - b c = 0$"}
+{"_id": "18424", "title": "Trivial Solution of Homogeneous Linear 1st Order ODE", "text": "The homogeneous linear first order ODE: :$\\dfrac {\\d y} {\\d x} + \\map Q x y = 0$ has the particular solution: :$\\map y x = 0$ that is, the zero constant function. This particular solution is referred to as the '''trivial solution'''."}
+{"_id": "18425", "title": "Simultaneous Homogeneous Linear First Order ODEs/Examples/y' - 3y + 2z = 0, y' + 4y - z = 0", "text": "Consider the system of linear first order ordinary differential equations with constant coefficients: {{begin-eqn}} {{eqn | n = 1       | l = \\dfrac {\\d y} {\\d x} - 3 y + 2 z       | r = 0 }} {{eqn | n = 2       | l = \\dfrac {\\d x} {\\d z} + 4 y - z       | r = 0 }} {{end-eqn}} The general solution to $(1)$ and $(2)$ consists of the linear combinations of the following: {{begin-eqn}} {{eqn | l = y       | r = C_1 e^{5 x} + C_2 e^{-x} }} {{eqn | l = z       | r = -C_1 e^{5 x} + 2 C_2 e^{-x} }} {{end-eqn}}"}
+{"_id": "18426", "title": "Particular Solution of Constant Coefficient Linear nth Order ODE", "text": "Consider the linear $n$th order ODE with constant coefficients: :$(1): \\quad \\displaystyle \\sum_{k \\mathop = 0}^n a_k \\dfrac {\\d^k y} {d x^k} = \\map R x$ Let $(1)$ have the following $n$ initial conditions: :$(2): \\quad y = y_0, \\dfrac {\\d y} {\\d x} = y_1, \\dotsc, \\dfrac {\\d^{n - 1} y} {\\d x^{n - 1} } = y_{n - 1}$ when $x = x_0$. Then there exists exactly one particular solution of $(1)$ which satisfies $(2)$."}
+{"_id": "18427", "title": "Particular Solution of System of Constant Coefficient Linear 1st Order ODEs", "text": "Consider the system of linear first order ordinary differential equations with constant coefficients: {{begin-eqn}} {{eqn | n = 1       | l = \\dfrac {\\d y} {\\d x} + a y + b z       | r = 0 }} {{eqn | n = 2       | l = \\dfrac {\\d x} {\\d z} + c y + d z       | r = 0 }} {{end-eqn}} Let $(1)$ and $(2)$ have the following $n$ initial conditions: :$(3): \\quad y = y_0, z = z_0$ when $x = x_0$. Then there exists exactly one particular solution of $(1)$ and $(2)$ which satisfies $(3)$."}
+{"_id": "18429", "title": "Linear First Order ODE/y' - y = e^x/y(0) = 0", "text": "Consider the linear first order ODE: :$(1): \\quad \\dfrac {\\d y} {\\d x} - y = e^x$ subject to the initial condition: :$\\map y 0 = 0$ $(1)$ has the particular solution: :$y = x e^x$"}
+{"_id": "18430", "title": "Linear Second Order ODE/y'' + 4 y' + 5 y = 0", "text": "The second order ODE: :$(1): \\quad y'' + 4 y' + 5 y = 0$ has the general solution: :$y = e^{-2 x} \\paren {C_1 \\cos x + C_2 \\sin x}$"}
+{"_id": "18433", "title": "Linear Second Order ODE/y'' - 5 y' + 6 y = 0", "text": "The linear second order ODE: :$(1): \\quad y'' - 5 y' + 6 y = 0$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{3 x}$"}
+{"_id": "18434", "title": "Linear Second Order ODE/y'' - 5 y' + 6 y = cos x + sin x", "text": "The second order ODE: :$(1): \\quad y'' - 5 y' + 6 y = \\cos x + \\sin x$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{3 x} + \\dfrac {\\cos x} 5$"}
+{"_id": "18435", "title": "Lindelöf's Lemma/Lemma", "text": "Let $C$ be a set of open real sets. Then there is a countable subset $D$ of $C$ such that: :$\\displaystyle \\bigcup_{O \\mathop \\in D} O = \\bigcup_{O \\mathop \\in C} O$"}
+{"_id": "18436", "title": "Linear Second Order ODE/y'' - 4 y = x^2 - 3 x - 4", "text": "The second order ODE: :$(1): \\quad y'' - 4 y = x^2 - 3 x - 4$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{-2 x} - \\dfrac {x^2} 4 + \\dfrac {3 x} 4 + \\dfrac 7 8$"}
+{"_id": "18437", "title": "Linear Second Order ODE/y'' + 2 y' + y = x exp -x", "text": "The second order ODE: :$(1): \\quad y'' + 2 y' + y = x e^{-x}$ has the general solution: :$y = e^{-x} \\paren {C_1 + C_2 x + \\dfrac {x^3} 6}$"}
+{"_id": "18439", "title": "Angular Momentum Commutation Rules", "text": "Let $J_x$, $J_y$ and $J_z$ denote the angular momentum operators. Then: {{begin-eqn}} {{eqn | l = \\sqbrk {J_x, J_y}       | r = i J_z }} {{eqn | l = \\sqbrk {J_y, J_z}       | r = i J_x }} {{eqn | l = \\sqbrk {J_z, J_x}       | r = i J_y }} {{end-eqn}} where $\\sqbrk {\\, \\cdot, \\cdot \\,}$ denotes the commutator operator."}
+{"_id": "18440", "title": "P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. Let $\\displaystyle \\sum_{i \\mathop = m}^\\infty d_i p^i$ be a $p$-adic expansion. Let $\\mathbf a$ be the equivalence class in $\\Q_p$ containing $\\displaystyle \\sum_{i \\mathop = m}^\\infty d_i p^i$. Let $l$ be the index of the first non-zero coefficient in the $p$-adic expansion: :$l = \\min \\set {i: i \\ge m \\land d_i \\ne 0}$ Then: :$\\norm {\\mathbf a}_p = p^{-l}$"}
+{"_id": "18441", "title": "Eventually Constant Sequence Converges to Constant", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = \\lambda$"}
+{"_id": "18443", "title": "Non-Zero Complex Numbers under Multiplication form Group", "text": "Let $\\C_{\\ne 0}$ be the set of complex numbers without zero, that is: :$\\C_{\\ne 0} = \\C \\setminus \\set 0$ The structure $\\struct {\\C_{\\ne 0}, \\times}$ is a group."}
+{"_id": "18444", "title": "Real Numbers under Addition form Group", "text": "Let $\\R$ be the set of real numbers. The structure $\\struct {\\R, +}$ is an infinite abelian group."}
+{"_id": "18448", "title": "General Linear Group is not Abelian", "text": "Let $K$ be a field whose zero is $0_K$ and unity is $1_K$. Let $\\GL {n, K}$ be the general linear group of order $n$ over $K$. Then $\\GL {n, K}$ is not an abelian group."}
+{"_id": "18449", "title": "Group of Unitary Matrices under Multiplication is not Abelian", "text": "Let $n > 1$ be a natural number. Then the group of unitary matrices $\\map U n$ is not abelian."}
+{"_id": "18452", "title": "Closure in Subspace/Corollary 1", "text": "Let $W \\subseteq S$ and let $\\map {\\cl_T} W$ denote the closure of $W$ in $T$. Let $\\map {\\cl_H} {W \\cap H}$ denote the closure of $W \\cap H$ in $T_H$. Then: :$\\map {\\cl_H} {W \\cap H} \\subseteq \\map {\\cl_T} W \\cap H$"}
+{"_id": "18453", "title": "Complex Numbers form Vector Space over Themselves", "text": "The set of complex numbers $\\C$, with the operations of addition and multiplication, forms a vector space."}
+{"_id": "18454", "title": "Quaternions form Vector Space over Themselves", "text": "The set of quaternions $\\H$, with the operations of addition and multiplication, forms a vector space."}
+{"_id": "18455", "title": "Definition:Differential Operator", "text": "Let $A$ be a mapping from a function space $\\FF_1$ to another function space $\\FF_2$. Let $f \\in \\FF_2$ be a real function such that $f$ is the image of $u \\in \\FF_1$ that is:　$f = A \\sqbrk u$ A '''differential operator''' is represented as a linear combination, finitely generated by $u$ and its derivatives containing higher degree such as :$\\displaystyle \\map P {x, D} = \\sum _{\\size \\alpha \\mathop \\le m} \\map {a_\\alpha} x D^\\alpha$ where: :$\\alpha = \\set {\\alpha_1, \\alpha_2, \\dotsc \\alpha_n}$ is a set of non-negative integers forming a multi-index :$\\size \\alpha = \\alpha_1 + \\alpha_2 + \\dotsb + \\alpha_n$ is the length of $\\alpha$ :the $\\map {a_\\alpha} x$ are real functions on a open domain in a real cartesian space of $n$ dimensions :$D^\\alpha = D_1^{\\alpha_1} D_2^{\\alpha_2} \\dotsm D_n^{\\alpha_n}$. {{Proofread|This definition copied from Wikipedia and made more or less coherent. Please correct as necessary.}} Category:Definitions/Operator Theory Category:Definitions/Differential Calculus dkbpexztmfvuwqouudtsa0vqyna6v7q"}
+{"_id": "18456", "title": "Simple Events are Mutually Exclusive", "text": "Let $\\EE$ be an experiment. Let $e_1$ and $e_2$ be distinct simple events in $\\EE$. Then $e_1$ and $e_2$ are mutually exclusive."}
+{"_id": "18457", "title": "Non-Trivial Event is Union of Simple Events", "text": "Let $\\EE$ be an experiment. Let $e$ be an event in $\\EE$ such that $e \\ne \\O$. That is, such that $e$ is non-trivial. Then $e$ can be expressed as the union of a set of simple events in $\\EE$."}
+{"_id": "18458", "title": "Sample Space is Union of All Distinct Simple Events", "text": "Let $\\EE$ be an experiment. Let $\\Omega$ denote the sample space of $\\EE$. Then $\\Omega$ is the union of the set of simple events in $\\EE$."}
+{"_id": "18460", "title": "Equivalence of Definitions of Probability Measure", "text": "Let $\\EE$ be an experiment. {{TFAE|def = Probability Measure}}"}
+{"_id": "18463", "title": "Product Space is Product in Category of Topological Spaces", "text": "Let $\\mathbf{Top}$ be the category of topological spaces. Let $\\family {\\struct {X_i, \\tau_i} }_{i \\mathop \\in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\\struct{X, \\tau}$ be the product space of $\\family {\\struct {X_i, \\tau_i} }_{i \\mathop \\in I}$. Then $\\struct{X, \\tau}$ is the product of $\\family {\\struct {X_i, \\tau_i} }_{i \\mathop \\in I}$ in $\\mathbf{Top}$."}
+{"_id": "18464", "title": "Strictly Positive Rational Numbers are Closed under Addition", "text": ":$\\forall a, b \\in \\Q_{>0}: a + b \\in \\Q_{>0}$"}
+{"_id": "18465", "title": "Strictly Positive Rational Numbers are Closed under Multiplication", "text": ":$\\forall a, b \\in \\Q_{>0}: a b \\in \\Q_{>0}$"}
+{"_id": "18467", "title": "Set of Rationals Greater than Root 2 has no Smallest Element", "text": "Let $B$ be the set of all positive rational numbers $p$ such that $p^2 > 2$. Then $B$ has no smallest element."}
+{"_id": "18468", "title": "Rational Number Not in Cut is Greater than Element of Cut", "text": "Let $\\alpha$ be a cut. Let $p \\in \\alpha$. Let $q \\in \\Q$ such that $q \\notin \\alpha$. Then $q > p$."}
+{"_id": "18470", "title": "Natural Basis of Tychonoff Topology", "text": "Let $\\family {\\struct {X_i, \\tau_i} }_{i \\mathop \\in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $X$ be the cartesian product of $\\family {X_i}_{i \\mathop \\in I}$: :$\\displaystyle X := \\prod_{i \\mathop \\in I} X_i$ Then the natural basis on $X$ is the set $\\BB$ of cartesian products of the form $\\displaystyle \\prod_{i \\mathop \\in I} U_i$ where: :for all $i \\in I : U_i \\in \\tau_i$ :for all but finitely many indices $i : U_i = X_i$"}
+{"_id": "18471", "title": "Natural Basis of Tychonoff Topology/Finite Product", "text": "Let $n \\in \\N$. For all $k \\in \\set {1, \\ldots, n}$, let $\\struct {X_k, \\tau_k}$ be topological spaces. Let $\\displaystyle X = \\prod_{k \\mathop = 1}^n X_k$ be the cartesian product of $X_1, \\ldots, X_n$. Then the natural basis on $X$ is: :$\\BB = \\set {\\displaystyle \\prod_{k \\mathop = 1}^n U_k : \\forall k : U_k \\in \\tau_k}$"}
+{"_id": "18472", "title": "Box Topology may not be Coarsest Topology such that Projections are Continuous", "text": "Let $\\family {\\struct{X_i, \\tau_i}}_{i \\mathop \\in I}$ be an $I$-indexed family of topological spaces. Let $X$ be the cartesian product of $\\family {X_i}_{i \\mathop \\in I}$, that is: :$\\displaystyle X := \\prod_{i \\mathop \\in I} X_i$ Let $\\tau$ be the box topology on $X$. For each $i \\in I$, let $\\pr_i: X \\to X_i$ denote the $i$th projection on $X$: :$\\forall \\family {x_j}_{j \\mathop \\in I} \\in X: \\map {\\pr_i} {\\family {x_j}_{j \\mathop \\in I} } = x_i$ Then $\\tau$ may not be the coarsest topology on $X$ for which the projections $\\family{\\pr_i} _{i \\mathop \\in I}$ are continuous."}
+{"_id": "18473", "title": "Box Topology may not form Categorical Product in the Category of Topological Spaces", "text": "Let $\\family {\\struct{X_i, \\tau_i}}_{i \\mathop \\in I}$ be an $I$-indexed family of topological spaces. Let $X$ be the cartesian product of $\\family {X_i}_{i \\mathop \\in I}$, that is: :$\\displaystyle X := \\prod_{i \\mathop \\in I} X_i$ Let $\\tau$ be the box topology on $X$. Then $\\tau$ may not be the categorical product in the category of topological spaces."}
+{"_id": "18474", "title": "Ordering on Cuts satisfies Trichotomy Law", "text": "Let $\\alpha$ and $\\beta$ be cuts. Then exactly one of the following applies: {{begin-eqn}} {{eqn | n = 1       | l = \\alpha       | o = <       | r = \\beta }} {{eqn | n = 2       | l = \\alpha       | r = \\beta }} {{eqn | n = 3       | l = \\alpha       | o = >       | r = \\beta }} {{end-eqn}} where $<$ and so $>$ denote the strict ordering of cuts: :$\\alpha < \\beta \\iff \\exists p \\in \\Q: p \\in \\alpha, p \\notin \\beta$ Hence the ordering of cuts $\\le$ is a total ordering."}
+{"_id": "18475", "title": "Ordering on Cuts is Transitive", "text": "Let $\\alpha$, $\\beta$ and $\\gamma$ be cuts. Let: {{begin-eqn}} {{eqn | n = 1       | l = \\alpha       | o = <       | r = \\beta }} {{eqn | n = 2       | l = \\beta       | o = <       | r = \\gamma }} {{end-eqn}} where $<$ denotes the strict ordering of cuts: :$\\alpha < \\beta \\iff \\exists p \\in \\Q: p \\in \\alpha, p \\notin \\beta$ Then: :$\\alpha < \\gamma$ Hence the ordering of cuts $\\le$ is a transitive relation."}
+{"_id": "18476", "title": "Ordering on Cuts is Total", "text": "Let $\\CC$ denote the set of cuts. Let $<$ denote the strict ordering on cuts defined as: :$\\forall \\alpha, \\beta \\in \\CC: \\alpha < \\beta \\iff \\exists p \\in \\Q: p \\in \\alpha, p \\notin \\beta$ Then $<$ is a (strict) total ordering on $\\CC$."}
+{"_id": "18477", "title": "Sum of Cuts is Cut", "text": "Let $\\alpha$ and $\\beta$ be cuts. Let $\\gamma$ be the set of all rational numbers $r$ such that: :$\\exists p \\in \\alpha, q \\in \\beta: r = p + q$ Then $\\gamma$ is also a cut. Thus the operation of addition on the set of cuts is closed."}
+{"_id": "18478", "title": "Addition of Cuts is Commutative", "text": "Let $\\alpha$ and $\\beta$ be cuts. Let the operation of $\\alpha + \\beta$ be the sum of $\\alpha$ and $\\beta$. Then: :$\\alpha + \\beta = \\beta + \\alpha$"}
+{"_id": "18479", "title": "Addition of Cuts is Associative", "text": "Let $\\alpha$, $\\beta$ and $\\gamma$ be cuts. Let the operation of $\\alpha + \\beta$ be the sum of $\\alpha$ and $\\beta$. Then: :$\\paren {\\alpha + \\beta} + \\gamma = \\alpha + \\paren {\\beta + \\gamma}$"}
+{"_id": "18480", "title": "Identity Element for Addition of Cuts", "text": "Let $\\alpha$ be a cut. Let $0^*$ be the rational cut associated with the (rational) number $0$: :$0^* = \\set {r \\in \\Q: r < 0}$ Then: :$\\alpha + 0^* = \\alpha$ where $+$ denotes the operation of addition of cuts."}
+{"_id": "18481", "title": "Existence of Upper and Lower Numbers of Cut whose Difference equal Given Rational", "text": "Let $\\alpha$ be a cut. Let $r \\in \\Q_{>0}$ be a (strictly) positive rational number. Then there exist rational numbers $p$ and $q$ such that: :$p \\in \\alpha, q \\notin \\alpha$ :$q - p = r$ such that $q$ is not the smallest upper number of $\\alpha$."}
+{"_id": "18482", "title": "Existence of Unique Inverse Element for Addition of Cuts", "text": "Let $\\alpha$ be a cut. Let $0^*$ be the rational cut associated with the (rational) number $0$: :$0^* = \\set {r \\in \\Q: r < 0}$ Then there exists a unique cut $\\beta$ such that: :$\\alpha + \\beta = 0^*$ where $+$ denotes the operation of addition of cuts."}
+{"_id": "18483", "title": "Ordering on Cuts is Compatible with Addition of Cuts", "text": "Let $\\alpha$, $\\beta$ and $\\gamma$ be cuts. Let the operation of $\\alpha + \\beta$ be the sum of $\\alpha$ and $\\beta$. Let $\\beta < \\gamma$ denotes the strict ordering on cuts defined as: :$\\beta < \\gamma \\iff \\exists p \\in \\Q: p \\in \\beta, p \\notin \\gamma$ Then: :$\\beta < \\gamma \\implies \\alpha + \\beta < \\alpha + \\gamma$"}
+{"_id": "18484", "title": "Ordering on Cuts is Compatible with Addition of Cuts/Corollary", "text": "Let $0^*$ denote the rational cut associated with the (rational) number $0$. If: :$\\alpha > 0^*$ and $\\gamma > 0^*$ then: :$\\alpha + \\gamma > 0^*$"}
+{"_id": "18485", "title": "Existence of Unique Difference between Cuts", "text": "Let $\\alpha$ and $\\beta$ be cuts. Then there exists exactly one cut $\\gamma$ such that: :$\\alpha + \\gamma = \\beta$"}
+{"_id": "18486", "title": "Set of Cuts under Addition forms Abelian Group", "text": "Let $\\CC$ denote the set of cuts. Let $\\struct {\\CC, +}$ denote the algebraic structure formed from $\\CC$ and the operation $+$ of addition of cuts. Then $\\struct {\\CC, +}$ forms an abelian group."}
+{"_id": "18487", "title": "Product of Positive Cuts is Positive Cut", "text": "Let $0^*$ denote the rational cut associated with the (rational) number $0$. Let $\\alpha$ and $\\beta$ be cuts such that $\\alpha \\ge 0^*$ and $\\beta \\ge 0^*$, where $\\ge$ denotes the ordering on cuts. Let $\\gamma$ be the set of all rational numbers $r$ such that either: :$r < 0$ or: :$\\exists p \\in \\alpha, q \\in \\beta: r = p q$ where $p \\ge 0$ and $q \\ge 0$. Then $\\gamma$ is also a cut. Thus the operation of multiplication on the set of positive cuts is closed."}
+{"_id": "18488", "title": "Product of Cuts is Cut", "text": "Let $\\alpha$ and $\\beta$ be cuts. Let $\\alpha \\beta$ denote the product of cuts. Then $\\alpha \\beta$ is also a cut. Thus the operation of multiplication on the set of cuts is closed."}
+{"_id": "18489", "title": "Factorisation of z^n+a", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $a \\in \\C$ be a complex number. Then: :$z^n + a = \\displaystyle \\prod_{k \\mathop = 0}^{n - 1} \\paren {z - \\alpha_k b}$ where: :$\\alpha_k$ are the complex $n$th roots of negative unity :$b$ is any complex number such that $b^n = a$."}
+{"_id": "18490", "title": "Multiplication of Cuts is Commutative", "text": "Let $\\alpha$ and $\\beta$ be cuts. Let the $\\alpha \\beta$ be the product of $\\alpha$ and $\\beta$. Then: :$\\alpha \\beta = \\beta \\alpha$"}
+{"_id": "18491", "title": "Multiplication of Cuts is Associative", "text": "Let $\\alpha$, $\\beta$ and $\\gamma$ be cuts. Let $\\alpha \\beta$ denote the product of $\\alpha$ and $\\beta$. Then: :$\\paren {\\alpha \\beta} \\gamma = \\alpha \\paren {\\beta \\gamma}$"}
+{"_id": "18492", "title": "Natural Basis of Tychonoff Topology/Lemma 1", "text": ":$\\SS \\subseteq \\BB$"}
+{"_id": "18493", "title": "Natural Basis of Tychonoff Topology/Lemma 2", "text": ":$\\forall B_1, B_2 \\in \\BB : B_1 \\cap B_2 \\in \\BB$"}
+{"_id": "18494", "title": "Natural Basis of Tychonoff Topology/Lemma 3", "text": ":$\\displaystyle \\forall B \\in \\BB : B = \\bigcap_{j \\mathop \\in J} \\pr_j^{-1} \\sqbrk {U_j}$ where: :$\\displaystyle B = \\prod_{i \\mathop \\in I} U_i$ :$J = \\set{j \\in I : U_i \\ne X_i}$ is finite."}
+{"_id": "18495", "title": "Multiplication of Cuts Distributes over Addition", "text": "Let $\\alpha$, $\\beta$ and $\\gamma$ be cuts. Let: :$\\alpha + \\beta$ denote the sum of $\\alpha$ and $\\beta$. :$\\alpha \\beta$ denote the product of $\\alpha$ and $\\beta$. Then: :$\\alpha \\paren {\\beta + \\gamma} = \\alpha \\beta + \\alpha \\gamma$"}
+{"_id": "18496", "title": "Product of Cut with Zero Cut equals Zero Cut", "text": "Let $\\alpha$ be a cut. Let $0^*$ denote the rational cut associated with the (rational) number $0$. Then: :$\\alpha 0^* = 0^*$ where $\\alpha 0^*$ denote the product of $\\alpha$ and $0^*$."}
+{"_id": "18497", "title": "Product of Cuts is Zero Cut iff Either Factor equals Zero Cut", "text": "Let $\\alpha$ and $\\beta$ be cuts. Let $0^*$ denote the rational cut associated with the (rational) number $0$. Then: :$\\alpha \\beta = 0^*$ {{iff}}: :$\\alpha = 0^*$ or $\\beta = 0^*$ where $\\alpha \\beta$ denotes the product of $\\alpha$ and $\\beta$."}
+{"_id": "18498", "title": "Cut Associated with 1 is Identity for Multiplication of Cuts", "text": "Let $\\alpha$ be a cut. Let $1^*$ denote the rational cut (rational) number $1$. Then: :$\\alpha 1^* = \\alpha$ where $\\alpha 1^*$ denote the product of $\\alpha$ and $1^*$."}
+{"_id": "18499", "title": "Multiplication of Positive Cuts preserves Ordering", "text": "Let $0^*$ denote the rational cut associated with the (rational) number $0$. Let $\\alpha$, $\\beta$ and $\\gamma$ be cuts such that: :$0^* < \\alpha < \\beta$ :$0^* < \\gamma$ where $<$ denotes the strict ordering on cuts. Then :$\\alpha \\gamma < \\beta \\gamma$ where $\\alpha \\gamma$ denotes the product of $\\alpha$ and $\\gamma$."}
+{"_id": "18500", "title": "Existence of Unique Inverse Element for Multiplication of Cuts", "text": "Let $0^*$ be the rational cut associated with the (rational) number $0$: :$0^* = \\set {r \\in \\Q: r < 0}$ Let $\\alpha$ be a cut such that $\\alpha \\ne 0^*$. Then for every cut $\\beta$, there exists a unique cut $\\gamma$ such that: :$\\alpha \\gamma = \\beta$ where $\\alpha \\gamma$ denotes the operation of product of $\\alpha$ and $\\gamma$. In this context, $\\gamma$ can be expressed as $\\beta / \\alpha$."}
+{"_id": "18501", "title": "Sum of Rational Cuts is Rational Cut", "text": "Let $p \\in\\ Q$ and $q \\in \\Q$ be rational numbers. Let $p^*$ and $q^*$ denote the rational cuts associated with $p$ and $q$. Then: :$p^* + q^* = \\paren {p + q}^*$ Thus the operation of addition on the set of rational cuts is closed."}
+{"_id": "18503", "title": "Ordering of Rational Cuts preserves Ordering of Associated Rational Numbers", "text": "Let $p \\in\\ Q$ and $q \\in \\Q$ be rational numbers. Let $p^*$ and $q^*$ denote the rational cuts associated with $p$ and $q$. Then: :$p^* < q^* \\iff p < q$ where $p^* < q^*$ denotes the strict ordering on cuts defined as: :$\\beta < \\gamma \\iff \\exists p \\in \\Q: p \\in \\beta, p \\notin \\gamma$"}
+{"_id": "18504", "title": "Exists Rational Cut Between two Cuts", "text": "Let $\\alpha$ and $\\beta$ be cuts. Let $\\alpha < \\beta$, where $<$ denotes the strict ordering on cuts. Then there exists a rational cut $r^*$ associated with the rational number $r$ such that: :$\\alpha < r^* < \\beta$"}
+{"_id": "18505", "title": "Condition for Rational Cut to be Less than Given Cut", "text": "Let $\\alpha$ be a cut. Let $p^*$ be the rational cut associated with a rational number $p$. Then: :$p \\in \\alpha$ {{iff}}: :$p^* < \\alpha$ where $<$ denotes the strict ordering on cuts."}
+{"_id": "18506", "title": "Continuous Mapping to Topological Product/General Result", "text": "Let $X$ be a topological space. Let $\\family {Y_i}_{i \\mathop \\in I}$ be an indexed family of topological spaces for some indexing set $I$. Let $\\displaystyle Y = \\prod_{i \\mathop \\in I} Y_i$ be the product space of $\\family {Y_i}_{i \\mathop \\in I}$.  For each $i \\in I$, let $\\pr_i: X \\to X_i$ denote the $i$th projection on $X$: :$\\forall \\family {x_j}_{j \\mathop \\in I} \\in X: \\map {\\pr_i} {\\family {x_j}_{j \\mathop \\in I} } = x_i$ Let $f$ be a mapping from $X$ to $Y$.  Then $f$ is continuous {{iff}} $\\pr_i \\circ f$ is continuous for all $i \\in I$."}
+{"_id": "18507", "title": "Set of Cuts forms Ordered Field", "text": "Let $\\CC$ denote the set of cuts. Let $\\struct {\\CC, +, \\times, \\le}$ denote the ordered structure formed from $\\CC$ and: :the operation $+$ of addition of cuts :the operation $\\times$ of multiplication of cuts :the ordering $\\le$ of cuts. Then $\\struct {\\CC, +, \\times, \\le}$ is an ordered field."}
+{"_id": "18509", "title": "Box Topology contains Tychonoff Topology", "text": "Let $\\family {\\struct {X_i, \\tau_i} }_{i \\mathop \\in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $X$ be the cartesian product of $\\family {X_i}_{i \\mathop \\in I}$: :$\\displaystyle X := \\prod_{i \\mathop \\in I} X_i$ Let $\\tau$ be the Tychonoff topology on $X$. Let $\\tau'$ be the box topology on $X$. Then: :$\\tau \\subseteq \\tau'$"}
+{"_id": "18510", "title": "Ordered Field of Rational Cuts is Isomorphic to Rational Numbers", "text": "Let $\\struct {\\RR, +, \\times, \\le}$ denote the ordered field of rational cuts. Let $\\struct {\\Q, +, \\times, \\le}$ denote the field of rational numbers. Then $\\struct {\\RR, +, \\times, \\le}$ and $\\struct {\\Q, +, \\times, \\le}$ are isomorphic."}
+{"_id": "18512", "title": "Supremum of Subset of Real Numbers May or May Not be in Subset", "text": "Let $S \\subset \\R$ be a proper subset of the set $\\R$ of real numbers. Let $S$ admit a supremum $M$. Then $M$ may or may not be an element of $S$."}
+{"_id": "18516", "title": "Uniqueness of Positive Root of Positive Real Number/Positive Exponent", "text": "Let $x \\in \\R$ be a real number such that $x > 0$. Let $n \\in \\Z$ be an integer such that $n > 0$. Then there is at most one $y \\in \\R: y \\ge 0$ such that $y^n = x$."}
+{"_id": "18518", "title": "Basis Condition for Coarser Topology", "text": "Let $S$ be a set. Let $\\BB_1$ and $\\BB_2$ be two bases on $S$. Let $\\tau_1$ and $\\tau_2$ be the topologies generated by $\\BB_1$ and $\\BB_2$ respectively. If $\\BB_1$ and $\\BB_2$ satisfy: :$\\forall U \\in \\BB_1 : \\exists \\AA \\subseteq \\BB_2: U = \\bigcup \\AA$ then $\\tau_1$ is coarser than $\\tau_2$."}
+{"_id": "18520", "title": "Basis Condition for Coarser Topology/Corollary 2", "text": "If $\\BB_1 \\subseteq \\BB_2$ then $\\tau_1$ is coarser than $\\tau_2$."}
+{"_id": "18521", "title": "Continuous Mapping on Finer Domain and Coarser Codomain Topologies is Continuous", "text": "Let $\\struct {X, \\tau_1}$ and $\\struct {Y, \\tau_2}$ be topological spaces. Let $f: \\struct {X, \\tau_1} \\to \\struct {Y, \\tau_2}$ be a continuous mapping. Let $\\tau'_1$ be a finer topology on $X$ than $\\tau_1$, that is, $\\tau_1 \\subseteq \\tau'_1$. Let $\\tau'_2$ be a coarser topology on $Y$ than $\\tau_2$, that is, $\\tau'_2 \\subseteq \\tau_2$. Then: :$f: \\struct {X, \\tau'_1} \\to \\struct {Y, \\tau'_2}$ is a continuous mapping."}
+{"_id": "18522", "title": "Projection from Box Topology is Continuous", "text": "Let $\\family {\\struct{X_i, \\tau_i}}_{i \\mathop \\in I}$ be an $I$-indexed family of topological spaces. Let $X$ be the cartesian product of $\\family {X_i}_{i \\mathop \\in I}$, that is: :$\\displaystyle X := \\prod_{i \\mathop \\in I} X_i$ Let $\\tau$ be the box topology on $X$. For each $i \\in I$, let $\\pr_i: X \\to X_i$ denote the $i$th projection on $X$: :$\\forall \\family {x_j}_{j \\mathop \\in I} \\in X: \\map {\\pr_i} {\\family {x_j}_{j \\mathop \\in I} } = x_i$ Then $\\pr_i: \\struct{X,\\tau} \\to \\struct{X_i,\\tau_i}$ is continuous for all $i \\in I$."}
+{"_id": "18523", "title": "Domain Topology Contains Initial Topology iff Mappings are Continuous", "text": "Let $\\struct{Y, \\tau}$ be a topological space. Let $\\family {\\struct{X_i, \\tau_i}}_{i \\mathop \\in I}$ be a family of topological spaces. Let $\\family {f_i}_{i \\mathop \\in I}$ be a family of mappings $f_i : Y \\to X_i$. Let $\\tau'$ be the initial topology on $Y$ with respect to $\\family {f_i}_{i \\mathop \\in I}$. Then: :$\\tau' \\subseteq \\tau$ {{iff}} $\\forall i \\in I : f_i: \\struct{Y, \\tau} \\to \\struct{X_i, \\tau_i}$ is $\\tuple{\\tau, \\tau_i}$-continuous."}
+{"_id": "18524", "title": "Final Topology Contains Codomain Topology iff Mappings are Continuous", "text": "Let $\\struct{Y, \\tau}$ be a topological space. Let $\\family {\\struct{X_i, \\tau_i}}_{i \\mathop \\in I}$ be a family of topological spaces. Let $\\family {f_i}_{i \\mathop \\in I}$ be a family of mappings $f_i : X_i \\to Y$. Let $\\tau'$ be the final topology on $Y$ with respect to $\\family {f_i}_{i \\mathop \\in I}$. Then: :$\\tau \\subseteq \\tau'$ {{iff}} $\\forall i \\in I : f_i: \\struct{X_i, \\tau_i} \\to \\struct{Y, \\tau}$ is continuous."}
+{"_id": "18525", "title": "Equivalence of Definitions of Final Topology", "text": "Let $X$ be a set. Let $I$ be an indexing set. Let $\\family {\\struct {Y_i, \\tau_i} }_{i \\mathop \\in I}$ be an indexed family of topological spaces indexed by $I$. Let $\\family {f_i: Y_i \\to X}_{i \\mathop \\in I}$ be an indexed family of mappings indexed by $I$. {{TFAE|def = Final Topology}}"}
+{"_id": "18526", "title": "Condition for Trivial Relation to be Mapping", "text": "Let $S$ and $T$ be sets. Let $\\RR = S \\times T$ be the trivial relation in $S$ to $T$. Then $\\RR$ is a mapping {{iff}} either: :$(2): \\card S = 0$ or: :$(1): \\card T = 1$ where $\\card {\\, \\cdot \\,}$ denotes cardinality."}
+{"_id": "18527", "title": "Relation to Empty Set is Mapping iff Domain is Empty", "text": "Let $S$ be a set Let $S \\times \\O$ denote the cartesian product of $S$ with the empty set $\\O$. Let $\\RR \\subseteq S \\times \\O$ be a relation in $S$ to $\\O$. Then $\\RR$ is a mapping {{iff}} $S = \\O$."}
+{"_id": "18528", "title": "Equivalence of Definitions of Initial Topology/Definition 1 Implies Definition 2", "text": "Let: :$\\SS = \\set{\\map {f_i^{-1}} U: i \\in I, U \\in \\tau_i} \\subseteq \\map \\PP X$ where $\\map {f_i^{-1}} U$ denotes the preimage of $U$ under $f_i$. Let $\\tau$ be the topology on $X$ generated by the subbase $\\SS$."}
+{"_id": "18529", "title": "Equivalence of Definitions of Initial Topology/Definition 2 Implies Definition 1", "text": "Let $\\tau$ be the coarsest topology on $X$ such that each $f_i: X \\to Y_i$ is $\\tuple{\\tau, \\tau_i}$-continuous. Let: :$\\SS = \\set {\\map {f_i^{-1} } U: i \\in I, U \\in \\tau_i} \\subseteq \\map \\PP X$ where $\\map {f_i^{-1} } U$ denotes the preimage of $U$ under $f_i$."}
+{"_id": "18530", "title": "Even Natural Numbers are Infinite", "text": "The set of even natural numbers is infinite."}
+{"_id": "18531", "title": "Set of Points on Line Segment is Infinite", "text": "The set of points on a line segment is infinite."}
+{"_id": "18533", "title": "Finite Sets are Comparable", "text": "Let $S$ and $T$ be finite sets. Then $S$ and $T$ are comparable by size."}
+{"_id": "18534", "title": "Equivalence of Definitions of Final Topology/Definition 1 Implies Definition 2", "text": "Let: :$\\tau = \\set{U \\subseteq X: \\forall i \\in I: \\map {f_i^{-1}} U \\in \\tau_i} \\subseteq \\powerset X$"}
+{"_id": "18535", "title": "Equivalence of Definitions of Final Topology/Definition 2 Implies Definition 1", "text": "Let $\\tau$ be the finest topology on $X$ such that each $f_i: Y_i \\to X$ is $\\tuple{\\tau_i, \\tau}$-continuous."}
+{"_id": "18536", "title": "Strictly Positive Integers have same Cardinality as Natural Numbers", "text": "Let $\\Z_{>0} := \\set {1, 2, 3, \\ldots}$ denote the set of strictly positive integers. Let $\\N := \\set {0, 1, 2, \\ldots}$ denote the set of natural numbers. Then $\\Z_{>0}$ has the same cardinality as $\\N$."}
+{"_id": "18537", "title": "Power Set of Natural Numbers is Cardinality of Continuum", "text": "Let $\\N$ denote the set of natural numbers. Let $\\powerset \\N$ denote the power set of $\\N$. Let $\\card {\\powerset \\N}$ denote the cardinality of $\\powerset \\N$. Then $\\card {\\powerset \\N}$ is the cardinality of the continuum."}
+{"_id": "18538", "title": "Continuum Hypothesis is Independent of ZFC", "text": "The Continuum Hypothesis can be neither proved nor disproved from the axioms of either Zermelo-Fraenkel set theory (ZF) or ZFC."}
+{"_id": "18539", "title": "Axiom of Choice is Independent of ZF", "text": "The Axiom of Choice can be neither proved nor disproved from the axioms of Zermelo-Fraenkel set theory."}
+{"_id": "18540", "title": "Empty Set can be Derived from Comprehension Principle", "text": "The '''empty set''' can be formed by application of the comprehension principle. Hence the '''empty set''' can be derived as a valid object in Frege set theory."}
+{"_id": "18542", "title": "Power Set can be Derived using Comprehension Principle", "text": "Let $a$ be a set. By application of the comprehension principle, the power set $\\powerset a$ can be formed. Hence the '''power set''' $\\powerset a$ can be derived as a valid object in Frege set theory."}
+{"_id": "18543", "title": "Set Union can be Derived using Comprehension Principle", "text": "Let $a$ be a set of sets. By application of the comprehension principle, the union $\\bigcup a$ can be formed. Hence the '''union''' $\\bigcup a$ can be derived as a valid object in Frege set theory."}
+{"_id": "18544", "title": "Set of Natural Numbers can be Derived using Comprehension Principle", "text": "Let $\\N$ denote the set of natural numbers. By application of the comprehension principle, $\\N$ can be derived as a valid object in Frege set theory."}
+{"_id": "18546", "title": "Exists Subset which is not Element", "text": "Let $S$ be a set. Then there exists at least one subset of $S$ which is not an element of $S$."}
+{"_id": "18547", "title": "Box Topology may not be Coarsest Topology such that Projections are Continuous/Lemma", "text": "Let $\\struct{X, \\tau}$ be a topological space. Let $U \\in \\tau$ such that $U \\neq \\O$ and $U \\neq X$. Let: :$Y = \\displaystyle \\prod_{n \\mathop \\in \\N } X = X \\times X \\times X \\times \\ldots$ be the countable Cartesian product of $\\family {X}_{n \\in \\N}$. Let $\\tau_T$ be the Tychonoff topology on $Y$. Let $\\tau_b$ be the box topology on $Y$. Let: :$V = \\displaystyle \\prod_{n \\mathop \\in \\N } U = U \\times U \\times U \\times \\ldots$ be the countable Cartesian product of $\\family {U}_{n \\in \\N}$. Then: :$V$ is an element of the box topology $\\tau_b$ :$V$ is not an element of the Tychonoff topology $\\tau_T$"}
+{"_id": "18550", "title": "P-adic Expansion Representative of P-adic Number is Unique", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. Let $\\mathbf a$ be an equivalence class in $\\Q_p$. Let $\\displaystyle \\sum_{i \\mathop = m}^\\infty d_i p^i$ and $\\displaystyle \\sum_{i \\mathop = k}^\\infty e_i p^i$ be $p$-adic expansions that represent $\\mathbf a$. Then: :$(1) \\quad m = k$ :$(2) \\quad \\forall i \\ge m : d_i = e_i$ That is, the $p$-adic expansions $\\displaystyle \\sum_{i \\mathop = m}^\\infty d_i p^i$ and $\\displaystyle \\sum_{i \\mathop = k}^\\infty e_i p^i$ are identical."}
+{"_id": "18551", "title": "Class is Subclass of Universal Class", "text": "Let $V$ denote the universal class. Let $A$ be a class. Then $A$ is a subclass of $V$."}
+{"_id": "18552", "title": "Not Every Class is a Set", "text": "Let $A$ be a class. Then it is not necessarily the case that $A$ is also a set."}
+{"_id": "18554", "title": "Class has Subclass which is not Element", "text": "Let $A$ be a class. Then $A$ has at least one subclass $B$ which is not an element of $A$."}
+{"_id": "18555", "title": "Basic Universe is Supercomplete", "text": "Let $V$ be a basic universe. Then $V$ is supercomplete."}
+{"_id": "18556", "title": "Basic Universe is not Set", "text": "Let $V$ be a basic universe. Then $V$ is not a set."}
+{"_id": "18558", "title": "Empty Class is Subclass of All Classes", "text": "The empty class is a subclass of all classes."}
+{"_id": "18559", "title": "Empty Class is Supercomplete", "text": "The empty class is supercomplete."}
+{"_id": "18560", "title": "Basic Universe is not Empty", "text": "Let $V$ be a basic universe Then $V$ is not the empty class."}
+{"_id": "18562", "title": "Singleton Class can be Formed from Set", "text": "Let $V$ be a basic universe. Let $a \\in V$ be a set. Then the singleton class $\\set a$ can be formed, which is a subclass of $V$."}
+{"_id": "18563", "title": "Singleton Class of Empty Set is Supercomplete", "text": "Let $\\O$ denote the empty set. Then the singleton $\\set \\O$ is supercomplete."}
+{"_id": "18564", "title": "Singleton Classes are Equal iff Sets are Equal", "text": "Let $a$ and $b$ be sets. Let $\\set a$ and $\\set b$ denote the singleton classes of $a$ and $b$. Then: :$\\set a = \\set b \\iff a = b$"}
+{"_id": "18565", "title": "Doubleton Class can be Formed from Two Sets", "text": "Let $V$ be a basic universe. Let $a, b \\in V$ be sets. Then the doubleton class $\\set {a, b}$ can be formed, which is a subclass of $V$."}
+{"_id": "18566", "title": "Doubleton Class of Equal Sets is Singleton Class", "text": "Let $V$ be a basic universe. Let $a, b \\in V$ be sets. Consider the doubleton class $\\set {a, b}$. Let $a = b$. Then: :$\\set {a, b} = \\set a$ where $\\set a$ denotes the singleton class of $a$."}
+{"_id": "18567", "title": "Equivalence of Definitions of Axiom of Pairing", "text": "The following formulations of the '''axiom of pairing''' in the context of '''axiomatic set theory''' are equivalent:"}
+{"_id": "18568", "title": "Singleton Class of Set is Set", "text": "Let $x$ be a set. Then the singleton class $\\set x$ is likewise a set."}
+{"_id": "18569", "title": "Basic Universe has Infinite Number of Elements", "text": "Let $V$ be a basic universe. Then $V$ has an infinite number of elements."}
+{"_id": "18570", "title": "Equivalence of Definitions of Axiom of Pairing for Classes", "text": "The following formulations of the '''axiom of pairing''' in the context of '''class theory''' are equivalent:"}
+{"_id": "18571", "title": "Equality of Ordered Pairs/Lemma", "text": "Let $\\set {a, b}$ and $\\set {a, d}$ be doubletons such that $\\set {a, b} = \\set {a, d}$. Then: :$b = d$"}
+{"_id": "18572", "title": "Equality of Ordered Pairs/Necessary Condition", "text": "Let $\\tuple {a, b}$ and $\\tuple {c, d}$ be ordered pairs such that $\\tuple {a, b} = \\tuple {c, d}$. Then $a = c$ and $b = d$."}
+{"_id": "18573", "title": "Equality of Ordered Pairs/Sufficient Condition", "text": "Let $\\tuple {a, b}$ and $\\tuple {c, d}$ be ordered pairs. Let $a = c$ and $b = d$. Then: :$\\tuple {a, b} = \\tuple {c, d}$"}
+{"_id": "18577", "title": "Intersection of Class Exists and is Unique", "text": "Let $V$ be a basic universe. Let $A \\subseteq V$ be a class. Let $\\displaystyle \\bigcap A$ denote the intersection of $A$. Then $\\displaystyle \\bigcap A$ is guaranteed to exist and is unique."}
+{"_id": "18578", "title": "Intersection of Non-Empty Class is Set", "text": "Let $V$ be a basic universe. Let $A \\subseteq V$ be a non-empty class. Let $\\displaystyle \\bigcap A$ denote the intersection of $A$. Then $\\displaystyle \\bigcap A$ is a set."}
+{"_id": "18579", "title": "Intersection of Empty Set/Class Theory", "text": "Let $V$ be a basic universe. Let $\\O$ denote the empty class. Then the intersection of $\\O$ is $V$: :$\\displaystyle \\bigcap \\O = V$"}
+{"_id": "18580", "title": "Union of Subclass is Subset of Union of Class", "text": "Let $A$ and $B$ be classes. Let $\\displaystyle \\bigcup A$ and $\\displaystyle \\bigcup B$ denote the union of $A$ and union of $B$ respectively. Let $A$ be a subclass of $B$: :$A \\subseteq B$ Then $\\displaystyle \\bigcup A$ is a subset of $\\displaystyle \\bigcup B$: :$\\displaystyle \\bigcup A \\subseteq \\displaystyle \\bigcup B$"}
+{"_id": "18582", "title": "Union of Transitive Class is Subset", "text": "Let $A$ be a transitive class. Let $\\displaystyle \\bigcup A$ denote the union of $A$. Then: :$\\displaystyle \\bigcup A \\subseteq A$"}
+{"_id": "18583", "title": "Union of Class is Subset implies Class is Transitive", "text": "Let $A$ be a class. Let $\\displaystyle \\bigcup A$ denote the union of $A$. Let: :$\\displaystyle \\bigcup A \\subseteq A$ Then $A$ is transitive."}
+{"_id": "18584", "title": "Union of Transitive Class is Transitive", "text": "Let $A$ be a class. Let $\\displaystyle \\bigcup A$ denote the union of $A$. Let $A$ be transitive. Then $\\displaystyle \\bigcup A$ is also transitive."}
+{"_id": "18585", "title": "Union of Class is Transitive if Every Element is Transitive", "text": "Let $A$ be a class. Let $\\bigcup A$ denote the union of $A$. Let $A$ be such that every element of $A$ is transitive. Then $\\bigcup A$ is also transitive."}
+{"_id": "18586", "title": "Set Difference with Set Difference is Union of Set Difference with Intersection/Corollary", "text": ":$T \\setminus \\paren {S \\setminus T} = T$"}
+{"_id": "18587", "title": "Power Set Exists and is Unique", "text": "Let $V$ be a basic universe. Let $x \\in V$ be a set. Let $\\powerset x$ denote the power set of $x$. Then $\\powerset x$ is guaranteed to exist and is unique."}
+{"_id": "18588", "title": "Element of Class is Subset of Union of Class", "text": "Let $A$ be a class. Let $\\displaystyle \\bigcup A$ denote the union of $A$. Let $x \\in A$. Then: :$x \\subseteq \\displaystyle \\bigcup A$"}
+{"_id": "18590", "title": "Set equals Union of Power Set", "text": "Let $x$ be a set of sets. Let $\\powerset x$ denote the power set of $x$. Let $\\displaystyle \\map \\bigcup {\\powerset x}$ denote the union of $\\powerset x$. Then: :$x = \\displaystyle \\map \\bigcup {\\powerset x}$"}
+{"_id": "18591", "title": "Cartesian Product Exists and is Unique", "text": "Let $A$ and $B$ be classes. Let $A \\times B$ be the '''cartesian product''' of $A$ and $B$. Then $A \\times B$ exists and is unique."}
+{"_id": "18592", "title": "Cartesian Product of Sets is Set", "text": "Let $V$ be a basic universe. Let $A$ and $B$ be sets in $V$. Then $A \\times B$ is also a set."}
+{"_id": "18595", "title": "Real Function with Positive Derivative is Increasing", "text": "If $\\forall x \\in \\openint a b: \\map {f'} x \\ge 0$, then $f$ is increasing on $\\closedint a b$."}
+{"_id": "18596", "title": "Real Function with Strictly Positive Derivative is Strictly Increasing", "text": "If $\\forall x \\in \\openint a b: \\map {f'} x > 0$, then $f$ is strictly increasing on $\\closedint a b$."}
+{"_id": "18599", "title": "Continuous Real Function Differentiable on Borel Set", "text": "Let $\\map \\BB {\\R, \\size {\\, \\cdot \\,} }$ be the Borel Sigma-Algebra on $\\R$ with the usual topology. Let $f: \\R \\to \\R$ be a continuous real function. Let $\\map D f$ be the set of all points at which $f$ is differentiable. Then $\\map D f$ is a Borel Set with respect to $\\map \\BB {\\R, \\size {\\, \\cdot \\,} }$."}
+{"_id": "18601", "title": "Limit with Rational Epsilon and Delta", "text": "Let $\\openint a b$ be an open real interval. Let $c \\in \\openint a b$. Let $f: \\openint a b \\setminus \\set c \\to \\R$ be a real function. Let $L \\in \\R$. Suppose that: :$\\forall \\epsilon > 0 \\in \\Q_{>0}: \\exists \\delta \\in \\Q_{>0}: \\forall x \\in \\R: 0 < \\size {x - c} < \\delta \\implies \\size {\\map f x - L} < \\epsilon$ Then the limit of $f$ exists as $x$ tends to $c$, and is equal to $L$."}
+{"_id": "18602", "title": "Chain Rule for Probability", "text": "Let $\\EE$ be an experiment with probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $A, B \\in \\Sigma$ be events of $\\EE$. The '''conditional probability of $A$ given $B$''' is: :$\\map \\Pr {A \\mid B} = \\dfrac {\\map \\Pr {A \\cap B} } {\\map \\Pr B}$"}
+{"_id": "18603", "title": "Limit with Epsilon Powers of 2", "text": "Let $\\openint a b$ be an open real interval. Let $c \\in \\openint a b$. Let $f: \\openint a b \\setminus \\set c \\to \\R$ be a real function. Let $L \\in \\R$. Suppose that: :$\\forall n > 0 \\in \\N: \\exists \\delta \\in \\R_{>0}: \\forall x \\in \\R: 0 < \\size {x - c} < \\delta \\implies \\size {\\map f x - L} < 2^{-n} $ Then the limit of $f$ exists as $x$ tends to $c$, and is equal to $L$."}
+{"_id": "18606", "title": "Characterization of Probability Density Function", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $X: \\Omega \\to \\R$ be a continuous random variable on $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\Omega_X = \\Img X$, the image of $X$. Let the '''probability density function''' of $X$ is the mapping $f_X: \\R \\to \\closedint 0 1$ be defined as: :$\\forall x \\in \\R: \\map {f_X} x = \\begin {cases} \\displaystyle \\lim_{\\epsilon \\mathop \\to 0^+} \\frac {\\map \\Pr {x - \\frac \\epsilon 2 \\le X \\le x + \\frac \\epsilon 2} } \\epsilon & : x \\in \\Omega_X \\\\ 0 & : x \\notin \\Omega_X \\end {cases}$ Suppose that the cumulative distribution function of $X$ defines a continuously differentiable real function $F_X: x \\mapsto \\map \\Pr {X \\le x}$. Then the '''probability density function''' of $X$ is the mapping $f_X: \\R \\to \\closedint 0 1$ satisfies: :$\\dfrac{\\d}{\\d x} \\map{F_X}{x} = \\map {f_X} x$."}
+{"_id": "18607", "title": "Abel's Test", "text": "Let $\\displaystyle \\sum a_n$ be a convergent real series. Let $\\sequence {b_n}$ be a decreasing sequence of positive real numbers. Then the series $\\displaystyle \\sum a_n b_n$ is also convergent."}
+{"_id": "18608", "title": "Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 2", "text": "Let $S$ be a set. Let $\\BB$ be a synthetic basis on $S$. Let $\\tau$ be the topology on $S$ generated by the synthetic basis $\\mathcal B$: :$\\tau = \\set{\\bigcup \\AA: \\AA \\subseteq \\BB}$ Then: :$\\forall U \\subseteq S: U \\in \\tau \\iff U = \\bigcup \\set {B \\in \\BB: B \\subseteq U}$"}
+{"_id": "18609", "title": "Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 3", "text": "Let $S$ be a set. Let $\\mathcal B$ be a synthetic basis on $S$. Let $\\tau$ be the topology on $S$ generated by the synthetic basis $\\mathcal B$: :$\\tau = \\left\\{{\\bigcup \\mathcal A: \\mathcal A \\subseteq \\mathcal B}\\right\\}$ Then: :$\\forall U \\subseteq S: U \\in \\tau \\iff \\forall x \\in U: \\exists B \\in \\mathcal B: x \\in B \\subseteq U$"}
+{"_id": "18610", "title": "Projection from Product Topology is Open and Continuous", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ and $T_2 = \\struct {S_2, \\tau_2}$ be topological spaces. Let $T = \\struct {T_1 \\times T_2, \\tau}$ be the product space of $T_1$ and $T_2$, where $\\tau$ is the Tychonoff topology on $S$. Let $\\pr_1: T \\to T_1$ and $\\pr_2: T \\to T_2$ be the first and second projections from $T$ onto its factors. Then both $\\pr_1$ and $\\pr_2$ are open and continuous."}
+{"_id": "18611", "title": "Projection from Product Topology is Open and Continuous/General Result", "text": "Let $\\family {T_i}_{i \\mathop \\in I} = \\family {\\struct{S_i, \\tau_i}}_{i \\mathop \\in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\\displaystyle S = \\prod_{i \\mathop \\in I} S_i$ be the corresponding product space. Let $\\tau$ denote the Tychonoff topology on $S$. Let $\\pr_i: S \\to S_i$ be the corresponding projection from $S$ onto $S_i$. Then $\\pr_i$ is open and continuous for all $i \\in I$."}
+{"_id": "18612", "title": "Projection from Product Topology is Continuous/General Result", "text": "Let $\\family {T_i}_{i \\mathop \\in I} = \\family {\\struct{S_i, \\tau_i}}_{i \\mathop \\in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\\displaystyle S = \\prod_{i \\mathop \\in I} S_i$ be the corresponding product space. Let $\\tau$ denote the Tychonoff topology on $S$. Let $\\pr_i: S \\to S_i$ be the corresponding projection from $S$ onto $S_i$. Then $\\pr_i$ is continuous for all $i \\in I$."}
+{"_id": "18613", "title": "Projection from Product Topology is Open/General Result", "text": "Let $\\family {T_i}_{i \\mathop \\in I} = \\family {\\struct {S_i, \\tau_i} }_{i \\mathop \\in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\\displaystyle S = \\prod_{i \\mathop \\in I} S_i$ be the corresponding product space. Let $\\tau$ denote the Tychonoff topology on $S$. Let $\\pr_i: S \\to S_i$ be the corresponding projection from $S$ onto $S_i$. Then $\\pr_i$ is open for all $i \\in I$."}
+{"_id": "18614", "title": "Product Space Basis Induced from Factor Space Bases", "text": "Let $\\family {\\struct{S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be an indexed family of topological spaces for $\\alpha$ in some indexing set $I$. Let $\\BB_\\alpha$ be a basis for the topology $\\tau_\\alpha$ for each $\\alpha \\in I$. Let $\\struct {S, \\tau} = \\displaystyle \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let $\\BB$ be the set of all cartesian products of the form $\\displaystyle \\prod_{\\alpha \\mathop \\in I} U_\\alpha$ where: :for all but finitely many indices $\\alpha : U_\\alpha = S_\\alpha$ :for all $\\alpha \\in I : U_\\alpha \\ne S_\\alpha \\implies U_\\alpha \\in \\BB_\\alpha$ Then $\\BB$ is a basis for the topology on the product space $\\struct{S, \\tau}$."}
+{"_id": "18615", "title": "Existence and Uniqueness of Domain of Relation", "text": "Let $V$ be a basic universe. Let $\\RR \\subseteq V \\times V$ be a relation. Then the domain $\\Dom \\RR$ of $\\RR$ exists and is unique."}
+{"_id": "18616", "title": "Existence and Uniqueness of Image of Relation", "text": "Let $V$ be a basic universe. Let $\\RR \\subseteq V \\times V$ be a relation. Then the image $\\Img \\RR$ of $\\RR$ exists and is unique."}
+{"_id": "18617", "title": "Union of Union of Relation is Union of Domain with Image", "text": "Let $V$ be a basic universe. Let $\\RR \\subseteq V \\times V$ be a relation. Let $\\Dom \\RR$ denote the domain of $\\RR$. Then: :$\\map \\bigcup {\\bigcup \\RR} = \\Dom \\RR \\cup \\Img \\RR$ where: :$\\bigcup \\RR$ denotes the union of $\\RR$ :$\\Dom \\RR$ denotes the domain of $\\RR$ :$\\Img \\RR$ denotes the image of $\\RR$."}
+{"_id": "18618", "title": "Domain of Relation is Subset of Union of Union of Relation", "text": "Let $V$ be a basic universe. Let $\\RR \\subseteq V \\times V$ be a relation. Let $\\Dom \\RR$ denote the domain of $\\RR$. Then: :$\\Dom \\RR \\subseteq \\map \\bigcup {\\bigcup \\RR}$ where $\\bigcup \\RR$ denotes the union of $\\RR$."}
+{"_id": "18619", "title": "Image of Relation is Subset of Union of Union of Relation", "text": "Let $V$ be a basic universe. Let $\\RR \\subseteq V \\times V$ be a relation. Let $\\Img \\RR$ denote the image of $\\RR$. Then: :$\\Img \\RR \\subseteq \\map \\bigcup {\\bigcup \\RR}$ where $\\bigcup \\RR$ denotes the union of $\\RR$."}
+{"_id": "18620", "title": "Relation is Set implies Domain and Image are Sets", "text": "Let $V$ be a basic universe. Let $\\RR \\subseteq V \\times V$ be a relation. Let $\\RR$ be a set. Then $\\Dom \\RR$ and $\\Img \\RR$ are also sets."}
+{"_id": "18622", "title": "Definition:Mapping/Class Theory", "text": "Let $V$ be a basic universe. A '''mapping''' $f$ in the context of Class Theory is a relation such that: :$f \\subseteq V \\times V$: :$\\forall x \\in \\Dom f: \\exists! y \\in \\Img f: \\tuple {x, y} \\in f$ That is, for every $x$ in the domain of $f$, there exists exactly one $y$ in the image of $f$ such that $\\tuple {x, y} \\in f$."}
+{"_id": "18625", "title": "Set is Transitive iff Subset of Power Set", "text": "A set $S$ is transitive {{iff}}: :$S \\subseteq \\powerset S$ where $\\powerset S$ denotes the power set of $S$."}
+{"_id": "18627", "title": "Universal Class less Set is not Transitive", "text": "Let $V$ be a basic universe. Let $a \\in V$ be a set. Then: :$V \\setminus \\set a$ is not a transitive class where $\\setminus$ denotes class difference."}
+{"_id": "18629", "title": "Equivalence of Formulations of Axiom of Infinity for Zermelo Universe", "text": "The following formulations of the '''Axiom of Infinity''' in the context of '''class theory''' are equivalent:"}
+{"_id": "18630", "title": "Inductive Construction of Natural Numbers fulfils Peano's Axioms", "text": "Let $P$ denote the set of natural numbers by definition as an inductive set. Then $P$ fulfils Peano's axioms."}
+{"_id": "18631", "title": "Natural Number is Transitive Set", "text": "Let $n$ be a natural number. Then $n$ is a transitive set."}
+{"_id": "18632", "title": "Natural Number is Ordinary Set", "text": "Let $n$ be a natural number. Then $n$ is an ordinary set."}
+{"_id": "18634", "title": "Product Space Local Basis Induced from Factor Spaces Local Bases", "text": "Let $\\family {\\struct{S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be an indexed family of topological spaces for $\\alpha$ in some indexing set $I$. Let $\\struct {S, \\tau} = \\displaystyle \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let $x = \\family{x_\\alpha} \\in S$. Let $\\BB_\\alpha$ be a local basis for $x_\\alpha$ in the topological space $\\struct {S_\\alpha, \\tau_\\alpha}$ for each $\\alpha \\in I$. Let $\\BB_x$ be the set of all cartesian products of the form $\\displaystyle \\prod_{\\alpha \\mathop \\in I} U_\\alpha$ where: :for all but finitely many indices $\\alpha : U_\\alpha = S_\\alpha$ :for all $\\alpha \\in I : U_\\alpha \\ne S_\\alpha \\implies U_\\alpha \\in \\BB_\\alpha$ Then $\\BB_x$ is a local basis for $x$ in the product space $\\struct{S, \\tau}$."}
+{"_id": "18635", "title": "Natural Numbers cannot be Elements of Each Other", "text": "Let $m$ and $n$ be natural numbers. Then it cannot be the case that both $m \\in n$ and $n \\in m$."}
+{"_id": "18636", "title": "Inductive Construction of Natural Numbers fulfils Peano's Axiom of Injectivity", "text": "Let $P$ denote the set of natural numbers by definition as an inductive set. Then $P$ fulfils: :{{PeanoAxiom|3}} where $s$ denotes the successor mapping."}
+{"_id": "18637", "title": "Element of Natural Number is Natural Number", "text": "Let $n$ be a natural number. Let $m \\in n$. Then $m$ is also a natural number."}
+{"_id": "18639", "title": "Natural Number is Superset of its Union", "text": "Let $n \\in \\N$ be a natural number as defined by the von Neumann construction. Then: :$\\bigcup n \\subseteq n$"}
+{"_id": "18640", "title": "Subspace of Product Space Homeomorphic to Factor Space/Product with Singleton", "text": "Let $T_1$ and $T_2$ be non-empty topological spaces. Let $b \\in T_2$. Let $T_1 \\times T_2$ be the product space of $T_1$ and $T_2$. Let $T_2 \\times T_1$ be the product space of $T_2$ and $T_1$. Then: :$T_1$ is homeomorphic to the subspace $T_1 \\times \\set b$ of $T_1 \\times T_2$ :$T_1$ is homeomorphic to the subspace $\\set b \\times T_1$ of $T_2 \\times T_1$"}
+{"_id": "18642", "title": "Set of Natural Numbers Equals its Union", "text": "Let $\\omega$ denote the set of natural numbers as defined by the von Neumann construction on a Zermelo universe $V$. Then: :$\\bigcup \\omega = \\omega$"}
+{"_id": "18643", "title": "Set of Natural Numbers Equals Union of its Successor", "text": "Let $\\omega$ denote the set of natural numbers as defined by the von Neumann construction on a Zermelo universe $V$. Then: :$\\bigcup \\omega^+ = \\omega$"}
+{"_id": "18645", "title": "Equivalence of Definitions of Minimally Inductive Class", "text": "Let $A$ be a class. Let $g$ be a mapping on $A$. {{TFAE|def = Minimally Inductive Class under General Mapping|view = minimally inductive class under $g$}}"}
+{"_id": "18646", "title": "Principle of General Induction", "text": "Let $M$ be a class. Let $g: M \\to M$ be a mapping on $M$. Let $M$ be minimally inductive under $g$. Let $P: M \\to \\set {\\T, \\F}$ be a propositional function on $M$. Suppose that: :$(1): \\quad \\map P \\O = \\T$ :$(2): \\quad \\forall x \\in M: \\map P x = \\T \\implies \\map P {\\map g x} = \\T$ Then: :$\\forall x \\in M: \\map P x = \\T$"}
+{"_id": "18647", "title": "Von Neumann Construction of Natural Numbers is Minimally Inductive", "text": "Let $\\omega$ denote the set of natural numbers as defined by the von Neumann construction. $\\omega$ is a minimally inductive class under the successor mapping."}
+{"_id": "18648", "title": "Cartesian Product of Subsets/Family of Subsets", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of sets where $I$ is an arbitrary index set. Let $S = \\displaystyle \\prod_{i \\mathop \\in I} S_i$ be the Cartesian product of $\\family {S_i}_{i \\mathop \\in I}$. Let $\\family {T_i}_{i \\mathop \\in I}$ be a family of sets. Let $T = \\displaystyle \\prod_{i \\mathop \\in I} T_i$ be the Cartesian product of $\\family {T_i}_{i \\mathop \\in I}$. Then: :$\\paren{\\forall i \\in I: T_i \\subseteq S_i} \\implies T \\subseteq S$."}
+{"_id": "18649", "title": "Double Induction Principle", "text": "Let $M$ be a class. Let $g: M \\to M$ be a mapping on $M$. Let $M$ be a minimally inductive class under $g$. Let $\\RR$ be a relation on $M$ which satisfies: {{begin-axiom}} {{axiom | n = \\text D_1         | q = \\forall x \\in M         | m = \\map \\RR {x, \\O} }} {{axiom | n = \\text D_2         | q = \\forall x, y \\in M         | m = \\map \\RR {x, y} \\land \\map \\RR {y, x} \\implies \\map \\RR {x, \\map g y} }} {{end-axiom}} Then $\\map \\RR {x, y}$ holds for all $x, y \\in M$."}
+{"_id": "18650", "title": "Double Induction Principle/Lemma", "text": "Let $x$ be a right normal element of $M$ with respect to $\\RR$. Then $x$ is also a left normal element of $M$ with respect to $\\RR$."}
+{"_id": "18652", "title": "Cartesian Product of Subsets/Family of Nonempty Subsets", "text": "Let $T_i \\ne \\O$ for all $i \\in I$. Then: :$T \\subseteq S \\iff \\forall i \\in I: T_i \\subseteq S_i$."}
+{"_id": "18653", "title": "Projection is Injection iff Factor is Singleton/Family of Sets", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be a non-empty family of non-empty sets where $I$ is an arbitrary index set. Let $S = \\displaystyle \\prod_{i \\mathop \\in I} S_i$ be the Cartesian product of $\\family {S_i}_{i \\mathop \\in I}$. Let $\\pr_j: S \\to S_j$ be the $j$th projection on $S$. Then $\\pr_j$ is an injection {{iff}} $S_i$ is a singleton for all $i \\in I \\setminus \\set j$."}
+{"_id": "18654", "title": "Progressing Function Lemma", "text": "Let $A$ be a class. Let $g$ be a progressing mapping on $A$. Let $\\RR$ be the relation defined as: :$\\map \\RR {x, y} \\iff \\map g x \\subseteq y \\lor y \\subseteq x$ where $\\lor$ denotes disjunction (inclusive \"or\"). Then: {{begin-axiom}} {{axiom | n = 1         | q = \\forall y \\in \\Dom g         | ml= \\map \\RR {y, \\O} }} {{axiom | n = 2         | q = \\forall x, y \\in \\Dom g         | ml= \\map \\RR {x, y} \\land \\map \\RR {y, x}         | mo= \\implies         | mr= \\map \\RR {x, \\map g y} }} {{end-axiom}} where $\\land$ denotes conjunction (\"and\")."}
+{"_id": "18655", "title": "Minimally Inductive Class under Progressing Mapping induces Nest", "text": "Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Then $M$ is a nest in which: :$\\forall x, y \\in M: \\map g x \\subseteq y \\lor y \\subseteq x$"}
+{"_id": "18656", "title": "Sandwich Principle", "text": "Let $A$ be a class. Let $g: A \\to A$ be a mapping on $A$ such that: :for all $x, y \\in A$, either $\\map g x \\subseteq y$ or $y \\subseteq x$. Then: :$\\forall x, y \\in A: x \\subseteq y \\subseteq \\map g x \\implies x = y \\lor y = \\map g x$ That is, there is no element $y$ of $A$ such that: :$x \\subset y \\subset \\map g x$ where $\\subset$ denotes a proper subset."}
+{"_id": "18657", "title": "Sandwich Principle/Corollary 1", "text": "Let: :$x \\subset y$ where $\\subset$ denotes a proper subset. Then: :$\\map g x \\subseteq y$"}
+{"_id": "18658", "title": "Sandwich Principle/Corollary 2", "text": "Let $g$ be a progressing mapping. Let $x \\subseteq y$. Then: :$\\map g x \\subseteq \\map g y$"}
+{"_id": "18659", "title": "Class under Progressing Mapping such that Elements are Sandwiched is Nest", "text": "Let $A$ be a class. Let $g: A \\to A$ be a progressing mapping on $A$ such that: :$\\forall x, y \\in A: \\map g x \\subseteq y \\lor y \\subseteq x$ Then $A$ is a nest: :$\\forall x, y \\in A: x \\subseteq y \\lor y \\subseteq x$"}
+{"_id": "18662", "title": "Characteristics of Minimally Inductive Class under Progressing Mapping", "text": "Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Then for all $x, y \\in M$:"}
+{"_id": "18663", "title": "Characteristics of Minimally Inductive Class under Progressing Mapping/Sandwich Principle", "text": ":$x \\subseteq y \\subseteq \\map g x \\implies x = y \\lor y = \\map g x$"}
+{"_id": "18664", "title": "Characteristics of Minimally Inductive Class under Progressing Mapping/Image of Proper Subset is Subset", "text": ":$x \\subset y \\implies \\map g x \\subseteq y$"}
+{"_id": "18665", "title": "Characteristics of Minimally Inductive Class under Progressing Mapping/Mapping Preserves Subsets", "text": ":$x \\subseteq y \\implies \\map g x \\subseteq \\map g y$"}
+{"_id": "18667", "title": "Non-Empty Bounded Subset of Minimally Inductive Class under Progressing Mapping has Greatest Element", "text": "Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Then every non-empty bounded subset of $M$ has a greatest element."}
+{"_id": "18668", "title": "Projection is Injection iff Factor is Singleton/Family of Sets/Necessary Condition", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be a non-empty family of non-empty sets where $I$ is an arbitrary index set. Let $S = \\displaystyle \\prod_{i \\mathop \\in I} S_i$ be the Cartesian product of $\\family {S_i}_{i \\mathop \\in I}$. Let $\\pr_j: S \\to S_j$ be the $j$th projection on $S$. Let $\\pr_j$ be an injection. Then $S_i$ is a singleton for all $i \\in I \\setminus \\set j$."}
+{"_id": "18669", "title": "Projection is Injection iff Factor is Singleton/Family of Sets/Sufficient Condition", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be a non-empty family of non-empty sets where $I$ is an arbitrary index set. Let $S = \\displaystyle \\prod_{i \\mathop \\in I} S_i$ be the Cartesian product of $\\family {S_i}_{i \\mathop \\in I}$. Let $\\pr_j: S \\to S_j$ be the $j$th projection on $S$. Let $S_i$ be a singleton for all $i \\in I \\setminus \\set {j}$. Then $\\pr_j$ is an injection. </onlyinclude>"}
+{"_id": "18670", "title": "Product Space of Subspaces is Subspace of Product Space", "text": "Let $\\family {\\struct {X_i, \\tau_i} }_{i \\mathop \\in I}$ be a family of topological spaces where $I$ is an arbitrary index set. Let $\\displaystyle T = \\struct {X, \\tau} = \\prod_{i \\mathop \\in I} \\struct {X_i, \\tau_i}$ be the product space of $\\family {\\struct {X_i, \\tau_i} }_{i \\mathop \\in I}$. Let $\\family {\\struct {Y_i, \\upsilon_i} }_{i \\mathop \\in I}$ be a family of topological spaces such that: :$\\forall i \\in I : \\struct {Y_i, \\upsilon_i}$ is a topological subspace of $\\struct {X_i, \\tau_i}$ Let $\\displaystyle S = \\struct {Y, \\upsilon} = \\prod_{i \\mathop \\in I} \\struct {Y_i, \\upsilon_i}$ be the product space of $\\family {\\struct {Y_i, \\upsilon_i} }_{i \\mathop \\in I}$. Let $T_Y = \\struct {Y, \\tau_Y}$ be the topological subspace of $T$. Then $S = T_Y$."}
+{"_id": "18671", "title": "Difference of Complex Conjugates", "text": "Let $z_1, z_2 \\in \\C$ be complex numbers. Let $\\overline z$ denote the complex conjugate of the complex number $z$. Then: :$\\overline {z_1 - z_2} = \\overline {z_1} - \\overline {z_2}$"}
+{"_id": "18672", "title": "Nonzero natural number is another natural number successor", "text": "Let $\\N$ be the 0-based natural numbers: :$\\N = \\left\\{{0, 1, 2, \\ldots}\\right\\}$ Let $s: \\N \\to \\N: \\map s n = n + 1$ be the successor function. Then: :$\\forall n \\in \\N \\setminus \\set 0 \\paren {\\exists m \\in \\N: \\map s m = n }$"}
+{"_id": "18674", "title": "Double Angle Formulas/Hyperbolic Functions", "text": "=== Double Angle Formula for Hyperbolic Sine === {{:Double Angle Formulas/Hyperbolic Sine}} === Double Angle Formula for Hyperbolic Cosine === {{:Double Angle Formulas/Hyperbolic Cosine}} === Double Angle Formula for Hyperbolic Tangent === {{:Double Angle Formulas/Hyperbolic Tangent}} where $\\sinh, \\cosh, \\tanh$ denote hyperbolic sine, hyperbolic cosine and hyperbolic tangent respectively."}
+{"_id": "18675", "title": "Subspace of Product Space Homeomorphic to Factor Space/Proof 1/Lemma 2", "text": ":$\\pr_i {\\restriction_{Y_i} } = p_i$"}
+{"_id": "18676", "title": "Subspace of Product Space Homeomorphic to Factor Space/Proof 1/Lemma 1", "text": ":$Y_i = \\prod_{j \\mathop \\in I} Z_j$"}
+{"_id": "18677", "title": "Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element", "text": "Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Let $x$ be a fixed point of $g$. Then $x$ is the greatest element of $M$."}
+{"_id": "18678", "title": "Closed Class under Progressing Mapping Lemma", "text": "Let $N$ be a class which is closed under a progressing mapping $g$. Let $g$ be such that: :$\\forall x, y \\in N: \\map g x \\subseteq y \\lor y \\subseteq x$ :if $\\map g x = x$, then $x$ is the greatest element of $N$. Let the following hold: :$A \\subseteq N$ is a subclass of $N$ :$x \\in N$ is an element of $N$ Let $x$ be: :a proper subset of all elements of $A$ and: :the greatest element of $A$ with that property. Then $\\map g x \\in A$ and is the smallest element of $A$."}
+{"_id": "18679", "title": "Minimally Inductive Class under Progressing Mapping is Well-Ordered under Inclusion", "text": "Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Let $x$ be a fixed point of $g$. Then $M$ is well-ordered under the inclusion relation."}
+{"_id": "18680", "title": "Minimally Closed Class under Progressing Mapping induces Nest", "text": "For all $x, y \\in N$: :either $\\map g x \\subseteq y$ or $y \\subseteq x$ and $N$ forms a nest: :$\\forall x, y \\in N: x \\subseteq y$ or $y \\subseteq x$"}
+{"_id": "18681", "title": "Minimally Closed Class under Progressing Mapping", "text": "Statement of Conditions: {{:Minimally Closed Class under Progressing Mapping/Statement}} Then the following results hold:"}
+{"_id": "18683", "title": "Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element", "text": "$g$ has no fixed point, unless possibly the greatest element, if there is one."}
+{"_id": "18685", "title": "Smallest Element of Minimally Closed Class under Progressing Mapping", "text": "$b$ is the smallest element of $N$."}
+{"_id": "18691", "title": "Equivalence of Definitions of Minimally Closed Class", "text": "Let $A$ be a class. Let $g$ be a mapping on $A$. {{TFAE|def = Minimally Closed Class|view = minimally closed class under $g$}}"}
+{"_id": "18692", "title": "Principle of General Induction/Minimally Closed Class", "text": "Let $M$ be a class. Let $g: M \\to M$ be a mapping on $M$. Let $b \\in M$ such that $M$ is minimally closed under $g$ with respect to $b$. Let $P: M \\to \\set {\\T, \\F}$ be a propositional function on $M$. Suppose that: :$(1): \\quad \\map P b = \\T$ :$(2): \\quad \\forall x \\in M: \\map P x = \\T \\implies \\map P {\\map g x} = \\T$ Then: :$\\forall x \\in M: \\map P x = \\T$"}
+{"_id": "18693", "title": "Double Induction Principle/Minimally Closed Class", "text": "Let $M$ be a class which is closed under a progressing mapping $g$. Let $b$ be an element of $M$ such that $M$ is minimally closed under $g$ with respect to $b$. Let $\\RR$ be a relation on $M$ which satisfies: {{begin-axiom}} {{axiom | n = \\text D_1         | q = \\forall x \\in M         | m = \\map \\RR {x, b} }} {{axiom | n = \\text D_2         | q = \\forall x, y \\in M         | m = \\map \\RR {x, y} \\land \\map \\RR {y, x} \\implies \\map \\RR {x, \\map g y} }} {{end-axiom}} Then $\\map \\RR {x, y}$ holds for all $x, y \\in M$."}
+{"_id": "18694", "title": "Double Induction Principle/Minimally Closed Class/Lemma", "text": "Let $x$ be a right normal element of $M$ with respect to $\\RR$. Then $x$ is also a left normal element of $M$ with respect to $\\RR$."}
+{"_id": "18697", "title": "Sandwich Principle for Minimally Closed Class", "text": "Let $N$ be a class which is closed under a progressing mapping $g$. Let $b$ be an element of $N$ such that $N$ is minimally closed under $g$ with respect to $b$. Then for all $x, y \\in N$: :$x \\subseteq y \\subseteq \\map g x \\implies x = y \\lor y = \\map g x$"}
+{"_id": "18700", "title": "Image of Proper Subset under Progressing Mapping on Minimally Closed Class", "text": "Let $N$ be a class which is closed under a progressing mapping $g$. Let $b$ be an element of $N$ such that $N$ is minimally closed under $g$ with respect to $b$. Then: :$x \\subset y \\implies \\map g x \\subseteq y$"}
+{"_id": "18705", "title": "Equation of Cornu Spiral/Parametric", "text": "Let $K$ be a Cornu spiral embedded in a Cartesian coordinate plane such that the origin coincides with the point at which $s = 0$. Then $K$ can be expressed by the parametric equations: :$\\begin {cases} x = a \\sqrt 2 \\map {\\operatorname C} {\\dfrac s {a \\sqrt 2} } \\\\ y = a \\sqrt 2 \\map {\\operatorname S} {\\dfrac s {a \\sqrt 2} } \\end {cases}$ where: :$\\operatorname C$ denotes the Fresnel cosine integral function :$\\operatorname S$ denotes the Fresnel sine integral function."}
+{"_id": "18707", "title": "Sampling Function is its own Fourier Transform", "text": "Consider the sampling function $\\operatorname {III}: \\R \\to \\R$. Then: :$\\map \\FF {\\operatorname {III} } = \\operatorname {III}$ where $\\FF$ denotes the Fourier transform."}
+{"_id": "18708", "title": "Even Impulse Pair is Fourier Transform of Cosine Function", "text": "Consider the (real) cosine function $\\cos: \\R \\to \\R$. Then: :$\\map \\FF {\\cos} = \\operatorname {II}$ where: :$\\FF$ denotes the Fourier transform :$\\operatorname {II}$ denotes the even impulse pair function."}
+{"_id": "18709", "title": "Convolution of Real Function with Rectangle Function", "text": "Let $f: \\R \\to \\R$ be a real function. Consider the rectangle function $\\Pi: \\R \\to \\R$. Then: :$\\forall x \\in \\R: \\map \\Pi x * \\map f x = \\displaystyle \\int_{x \\mathop - \\frac 1 2}^{x \\mathop + \\frac 1 2} \\map f u \\rd u$ where $*$ denotes the convolution integral."}
+{"_id": "18710", "title": "Repeated Fourier Transform of Even Function", "text": "Let $f: \\R \\to \\R$ be an even real function which is Lebesgue integrable. Let $\\displaystyle \\map \\FF {\\map f t} = \\map F s = \\int_{-\\infty}^\\infty e^{-2 \\pi i s t} \\map f t \\rd t$ be the Fourier transform of $f$. Let $\\displaystyle \\map \\FF {\\map F s} = \\map g t = \\int_{-\\infty}^\\infty e^{-2 \\pi i t s} \\map F s \\rd s$ be the Fourier transform of $F$. Then: :$\\map g t = \\map f t$"}
+{"_id": "18711", "title": "Repeated Fourier Transform of Odd Function", "text": "Let $f: \\R \\to \\R$ be an odd real function which is Lebesgue integrable. Let $\\displaystyle \\map \\FF {\\map f t} = \\map F s = \\int_{-\\infty}^\\infty e^{-2 \\pi i s t} \\map f t \\rd t$ be the Fourier transform of $f$. Let $\\displaystyle \\map \\FF {\\map F s} = \\map g t = \\int_{-\\infty}^\\infty e^{-2 \\pi i t s} \\map F s \\rd s$ be the Fourier transform of $F$. Then: :$\\map g t = -\\map f t$"}
+{"_id": "18712", "title": "Repeated Fourier Transform of Real Function", "text": "Let $f: \\R \\to \\R$ be a real function which is Lebesgue integrable. Let $\\displaystyle \\map \\FF {\\map f t} = \\map F s = \\int_{-\\infty}^\\infty e^{-2 \\pi i s t} \\map f t \\rd t$ be the Fourier transform of $f$. Let $\\displaystyle \\map \\FF {\\map F s} = \\map g t = \\int_{-\\infty}^\\infty e^{-2 \\pi i t s} \\map F s \\rd s$ be the Fourier transform of $F$. Then: :$\\map g t = \\map f {-t}$"}
+{"_id": "18713", "title": "Fourier's Theorem/Integral Form", "text": "Let $f: \\R \\to \\R$ be a real function which satisfies the Dirichlet conditions on $\\R$. Then: :$\\dfrac {\\map f {t^+} + \\map f {t^-} } 2 = \\displaystyle \\int_{-\\infty}^\\infty e^{2 \\pi i t s} \\paren {\\int_{-\\infty}^\\infty e^{-2 \\pi i d t} \\map f t \\rd t} \\rd s$ where: :$\\map f {t^+}$ and $\\map f {t^-}$ denote the limit from above and the limit from below of $f$ at $t$."}
+{"_id": "18715", "title": "Subspace of Product Space Homeomorphic to Factor Space/Proof 2/Lemma 1", "text": ":$\\map {p_i^\\to} {\\map{\\pr_k^\\gets } {V_k} \\cap Y_i}$ is open in $\\struct{X_i, \\tau_i}$"}
+{"_id": "18716", "title": "Subspace of Product Space Homeomorphic to Factor Space/Proof 2/Injection", "text": "Let $\\family {X_i}_{i \\mathop \\in I}$ be a family of sets where $I$ is an arbitrary index set. Let $\\displaystyle X = \\prod_{i \\mathop \\in I} X_i$ be the Cartesian product of $\\family {X_i}_{i \\mathop \\in I}$. Let $z \\in X$. Let $i \\in I$.  Let $Y_i = \\set {x \\in X: \\forall j \\in I \\setminus \\set i: x_j = z_j}$. Let $p_i = \\pr_i {\\restriction_{Y_i}}$, where $\\pr_i$ is the projection from $X$ to $X_i$. Then: :$p_i$ is an injection."}
+{"_id": "18717", "title": "Subspace of Product Space Homeomorphic to Factor Space/Proof 2/Surjection", "text": "Let $\\family {X_i}_{i \\mathop \\in I}$ be a family of sets where $I$ is an arbitrary index set. Let $\\displaystyle X = \\prod_{i \\mathop \\in I} X_i$ be the Cartesian product of $\\family {X_i}_{i \\mathop \\in I}$. Let $z \\in X$. Let $i \\in I$.  Let $Y_i = \\set {x \\in X: \\forall j \\in I \\setminus \\set i: x_j = z_j}$. Let $p_i = \\pr_i {\\restriction_{Y_i} }$, where $\\pr_i$ is the projection from $X$ to $X_i$. Then: :$p_i$ is a surjection."}
+{"_id": "18718", "title": "Subspace of Product Space Homeomorphic to Factor Space/Proof 2/Continuous Mapping", "text": "Let $\\family {\\struct {X_i, \\tau_i} }_{i \\mathop \\in I}$ be a family of topological spaces where $I$ is an arbitrary index set. Let $\\displaystyle \\struct {X, \\tau} = \\prod_{i \\mathop \\in I} \\struct {X_i, \\tau_i}$ be the product space of $\\family {\\struct {X_i, \\tau_i} }_{i \\mathop \\in I}$. Let $z \\in X$. Let $i \\in I$.  Let $Y_i = \\set {x \\in X: \\forall j \\in I \\setminus \\set i: x_j = z_j}$. Let $\\upsilon_i$ be the subspace topology of $Y_i$ relative to $\\tau$.  Let $p_i = \\pr_i {\\restriction_{Y_i}}$, where $\\pr_i$ is the projection from $X$ to $X_i$. Then: :$p_i$ is continuous."}
+{"_id": "18719", "title": "Subspace of Product Space Homeomorphic to Factor Space/Proof 2/Open Mapping", "text": "Let $\\family {\\struct {X_i, \\tau_i} }_{i \\mathop \\in I}$ be a family of topological spaces where $I$ is an arbitrary index set. Let $\\displaystyle \\struct {X, \\tau} = \\prod_{i \\mathop \\in I} \\struct {X_i, \\tau_i}$ be the product space of $\\family {\\struct {X_i, \\tau_i} }_{i \\mathop \\in I}$. Let $z \\in X$. Let $i \\in I$.  Let $Y_i = \\set {x \\in X: \\forall j \\in I \\setminus \\set i: x_j = z_j}$. Let $\\upsilon_i$ be the subspace topology of $Y_i$ relative to $\\tau$.  Let $p_i = \\pr_i {\\restriction_{Y_i}}$, where $\\pr_i$ is the projection from $X$ to $X_i$. Then: :$p_i$ is an open mapping."}
+{"_id": "18720", "title": "Triple Angle Formulas/Sine/Historical Note", "text": "The Triple Angle Formula for Sine is often attributed to {{AuthorRef|François Viète}}, although it was in fact discovered by {{AuthorRef|Jamshīd al-Kāshī}} a century or more earlier."}
+{"_id": "18721", "title": "Exponential Distribution in terms of Continuous Uniform Distribution", "text": "Let $X \\sim \\mathrm U \\hointl 0 1$ where $\\mathrm U \\hointl 0 1$ is the continuous uniform distribution on $\\hointl 0 1$. Let $\\beta$ be a positive real number. Then:  :$-\\beta \\ln X \\sim \\Exponential \\lambda$ where $\\operatorname {Exp}$ is the exponential distribution."}
+{"_id": "18723", "title": "Expectation of Non-Negative Random Variable is Non-Negative", "text": "Let $X$ be a random variable.  Let $\\map \\Pr {X \\ge 0} = 1$.  Then $\\expect X \\ge 0$, where $\\expect X$ denotes the expectation of $X$."}
+{"_id": "18724", "title": "Expectation of Non-Negative Random Variable is Non-Negative/Discrete", "text": "Let $X$ be a discrete random variable.  Let $\\map \\Pr {X \\ge 0} = 1$.  Then $\\expect X \\ge 0$, where $\\expect X$ denotes the expectation of $X$."}
+{"_id": "18725", "title": "Expectation of Non-Negative Random Variable is Non-Negative/Continuous", "text": "Let $X$ be a continuous random variable.  Let $\\map \\Pr {X \\ge 0} = 1$.  Then $\\expect X \\ge 0$, where $\\expect X$ denotes the expectation of $X$."}
+{"_id": "18726", "title": "Expectation of Almost Surely Constant Random Variable", "text": "Let $X$ be an almost surely constant random variable. That is, there exists some $c \\in \\R$ such that:  :$\\map \\Pr {X = c} = 1$ Then:  :$\\expect X = c$"}
+{"_id": "18727", "title": "Expectation Preserves Inequality", "text": "Let $X$, $Y$ be random variables. Let $\\map \\Pr {X \\ge Y} = 1$. Then:  :$\\expect X \\ge \\expect Y$"}
+{"_id": "18729", "title": "Covariance of Random Variable with Itself", "text": "Let $X$ be a random variable. Then $\\cov {X, X} = \\var X$."}
+{"_id": "18730", "title": "Covariance is Symmetric", "text": "Let $X$ and $Y$ be random variables. Then $\\cov {X, Y} = \\cov {Y, X}$."}
+{"_id": "18731", "title": "Covariance of Sums of Random Variables", "text": "Let $n$ be a strictly positive integer. Let $\\sequence {X_i}_{1 \\le i \\le n}$, $\\sequence {Y_j}_{1 \\le j \\le n}$ be sequences of random variables. Then:  :$\\displaystyle \\cov {\\sum_{i \\mathop = 1}^n X_i, \\sum_{j \\mathop = 1}^n Y_j} = \\sum_{i \\mathop = 1}^n \\sum_{j \\mathop = 1}^n \\cov {X_i, Y_j}$"}
+{"_id": "18732", "title": "Expectation of Linear Transformation of Random Variable/Discrete", "text": "Let $X$ be a discrete random variable.  Let $a, b$ be real numbers.  Then we have:  :$\\expect {a X + b} = a \\expect X + b$ where $\\expect X$ denotes the expectation of $X$."}
+{"_id": "18733", "title": "Expectation of Linear Transformation of Random Variable/Continuous", "text": "Let $X$ be a continuous random variable.  Let $a, b$ be real numbers.  Then we have:  :$\\expect {a X + b} = a \\expect X + b$ where $\\expect X$ denotes the expectation of $X$."}
+{"_id": "18734", "title": "Cauchy's Mean Theorem/Proof of Equality Condition", "text": "Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers which are all positive. Let $A_n$ be the arithmetic mean of $x_1, x_2, \\ldots, x_n$. Let $G_n$ be the geometric mean of $x_1, x_2, \\ldots, x_n$. Then: :$A_n = G_n$ {{iff}}: :$\\forall i, j \\in \\set {1, 2, \\ldots, n}: x_i = x_j$ That is, {{iff}} all terms are equal. Then:"}
+{"_id": "18736", "title": "Fourier Series/Square Wave", "text": "600pxthumbrightSquare Wave and $9$th Approximation Let $\\map S x$ be the square wave defined on the real numbers $\\R$ as: :$\\forall x \\in \\R: \\map S x = \\begin {cases} 1 & : x \\in \\openint 0 l \\\\ -1 & : x \\in \\openint {-l} 0 \\\\ \\map S {x + 2 l} & : x < -l \\\\ \\map S {x - 2 l} & : x > +l \\end {cases}$ Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \\map S x       | o = \\sim       | r = \\frac 4 \\pi \\sum_{r \\mathop = 0}^\\infty \\frac 1 {2 r + 1} {\\sin \\frac {\\pi x} l}       | c =  }} {{eqn | r = \\frac 4 \\pi \\paren {\\sin \\frac {\\pi x} l + \\dfrac 1 3 \\sin \\frac {3 \\pi x} l + \\dfrac 1 5 \\sin \\frac {5 \\pi x} l + \\dotsb}       | c =  }} {{end-eqn}}"}
+{"_id": "18737", "title": "Fourier Series/Triangle Wave", "text": "600pxthumbrightTriangle Wave and $9$th Approximation <onlyinclude> Let $\\map T x$ be the triangle wave defined on the real numbers $\\R$ as: :$\\forall x \\in \\R: \\map T x = \\begin {cases} \\size x & : x \\in \\closedint {-l} l \\\\ \\map T {x + 2 l} & : x < -l \\\\ \\map T {x - 2 l} & : x > +l \\end {cases}$ where: :$l$ is a given real constant :$\\size x$ denotes the absolute value of $x$. Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \\map T x       | o = \\sim       | r = \\frac l 2 - \\frac {4 l} {\\pi^2} \\sum_{n \\mathop = 0}^\\infty \\frac 1 {\\paren {2 n + 1}^2} \\cos \\dfrac {\\paren {2 n + 1} \\pi x} l       | c =  }} {{eqn | r = \\frac l 2 - \\frac {4 l} {\\pi^2} \\paren {\\cos \\dfrac {\\pi x} l + \\frac 1 {3^2} \\cos \\dfrac {3 \\pi x} l + \\frac 1 {5^2} \\cos \\dfrac {5 \\pi x} l + \\dotsb}       | c =  }} {{end-eqn}}"}
+{"_id": "18741", "title": "Subspace of Product Space Homeomorphic to Factor Space/Product with Singleton/Lemma", "text": ":$f$ is a bijection."}
+{"_id": "18742", "title": "Standard Continuous Uniform Distribution in terms of Exponential Distribution", "text": "Let $X$ and $Y$ be independent random variables. Let $\\beta$ be a strictly positive real number. Let $X$ and $Y$ be random samples from the exponential distribution with parameter $\\beta$.  Then: :$\\dfrac X {X + Y} \\sim \\operatorname U \\openint 0 1$ where $\\operatorname U \\openint 0 1$ is the uniform distribution on $\\openint 0 1$."}
+{"_id": "18745", "title": "Identity Function is Odd Function", "text": "Let $I_\\R: \\R \\to \\R$ denote the identity function on $\\R$. Then $I_\\R$ is an odd function."}
+{"_id": "18747", "title": "Half-Range Fourier Series/Identity Function", "text": "Let $\\lambda \\in \\R_{>0}$ be a strictly positive real number. Let $\\map f x: \\openint 0 \\lambda \\to \\R$ be the identity function on the open real interval $\\openint 0 \\lambda$: :$\\forall x \\in \\openint 0 \\lambda: \\map f x = x$ The half-range Fourier series of $f$ over $\\openint 0 \\lambda$ can be given in the following forms:"}
+{"_id": "18748", "title": "Half-Range Fourier Series/Identity Function/Cosine", "text": "The half-range Fourier cosine series for $\\map f x$ can be expressed as: {{begin-eqn}} {{eqn | l = \\map f x       | o = \\sim       | r = \\frac \\lambda 2 - \\frac {4 \\lambda} {\\pi^2} \\sum_{n \\mathop = 0}^\\infty \\frac 1 {\\paren {2 n + 1}^2} \\cos \\dfrac {\\paren {2 n + 1} \\pi x} \\lambda       | c =  }} {{eqn | r = \\frac \\lambda 2 - \\frac {4 \\lambda} {\\pi^2} \\paren {\\cos \\dfrac {\\pi x} \\lambda + \\frac 1 {3^2} \\cos \\dfrac {3 \\pi x} \\lambda + \\frac 1 {5^2} \\cos \\dfrac {5 \\pi x} \\lambda + \\dotsb}       | c =  }} {{end-eqn}}"}
+{"_id": "18749", "title": "Half-Range Fourier Series/Identity Function/Sine", "text": "The half-range Fourier sine series for $\\map f x$ can be expressed as: {{begin-eqn}} {{eqn | l = \\map f x       | o = \\sim       | r = \\dfrac {2 \\lambda} \\pi \\displaystyle \\sum_{n \\mathop = 1}^\\infty \\frac {\\paren {-1}^{n + 1} } n \\sin \\frac {n \\pi x} \\lambda       | c =  }} {{eqn | r = \\dfrac {2 \\lambda} \\pi \\paren {\\sin \\dfrac {\\pi x} \\lambda - \\frac 1 2 \\sin \\dfrac {2 \\pi x} \\lambda + \\frac 1 3 \\sin \\dfrac {3 \\pi x} \\lambda - \\dotsb}       | c =  }} {{end-eqn}}"}
+{"_id": "18750", "title": "Absolute Value Function is Even Function", "text": "Let $\\size {\\, \\cdot \\,} : \\R \\to \\R$ denote the absolute value function on $\\R$: Then $\\size {\\, \\cdot \\,}$ is an even function."}
+{"_id": "18751", "title": "Fourier Series/Absolute Value Function over Symmetric Range", "text": "Let $\\lambda \\in \\R_{>0}$ be a strictly positive real number. Let $\\map f x: \\openint {-\\lambda} \\lambda \\to \\R$ be the absolute value function on the open real interval $\\openint {-\\lambda} \\lambda$: :$\\forall x \\in \\openint {-\\lambda} \\lambda: \\map f x = \\size x$ The Fourier series of $f$ over $\\openint {-\\lambda} \\lambda$ can be given as: {{begin-eqn}} {{eqn | l = \\map f x       | o = \\sim       | r = \\frac \\lambda 2 - \\frac {4 \\lambda} {\\pi^2} \\sum_{n \\mathop = 0}^\\infty \\frac 1 {\\paren {2 n + 1}^2} \\cos \\dfrac {\\paren {2 n + 1} \\pi x} \\lambda       | c =  }} {{eqn | r = \\frac \\lambda 2 - \\frac {4 \\lambda} {\\pi^2} \\paren {\\cos \\dfrac {\\pi x} \\lambda + \\frac 1 {3^2} \\cos \\dfrac {3 \\pi x} \\lambda + \\frac 1 {5^2} \\dfrac {5 \\pi x} \\lambda + \\dotsb}       | c =  }} {{end-eqn}}"}
+{"_id": "18755", "title": "Factor Spaces of Hausdorff Product Space are Hausdorff", "text": "Let $\\SS = \\family {\\struct {S_\\alpha, \\tau_\\alpha} }$ be an indexed family of topological spaces for $\\alpha$ in some indexing set $I$. Let $\\displaystyle T = \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\SS$. Let $T$ be a $T_2$ (Hausdorff) space. Then: :for each $\\alpha \\in I$, $\\struct {S_\\alpha, \\tau_\\alpha}$ is a $T_2$ (Hausdorff) space."}
+{"_id": "18756", "title": "Cartesian Product is Empty iff Factor is Empty/Family of Sets", "text": "Let $I$ be an indexing set. Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of sets indexed by $I$. Let $\\displaystyle S = \\prod_{i \\mathop \\in I} S_i$ be the Cartesian product of $\\family {S_i}_{i \\mathop \\in I}$. Then: :$S = \\O$  {{iff}} $S_i = \\O$ for some $i \\in I$"}
+{"_id": "18757", "title": "Derivative of P-Norm wrt P", "text": "Let $p \\ge 1$ be a real number. Let $\\ell^p$ denote the $p$-sequence space. Let $\\mathbf x = \\sequence {x_n} \\in \\ell^p$. Let $\\norm {\\mathbf x}_p$ be a p-norm. Suppose, $\\norm {\\mathbf x}_p \\ne 0$. Then: :$\\displaystyle \\dfrac \\d {\\d p} \\norm {\\mathbf x}_p = \\frac {\\norm {\\mathbf x}_p} p \\paren { \\frac {\\sum_{n \\mathop = 0}^\\infty \\size {x_n}^p \\map \\ln {\\size {x_n} } } {\\norm {\\bf x}_p^p} - \\map \\ln {\\norm {\\bf x}_p} }$"}
+{"_id": "18759", "title": "Extension of Half-Range Fourier Sine Function to Symmetric Range", "text": "Let $\\map f x$ be a real function defined on the interval $\\openint 0 \\lambda$. Let $\\map f x$ be represented by the half-range Fourier sine series $\\map S x$: :$\\map f x \\sim \\map S x = \\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n \\sin \\frac {n \\pi x} \\lambda$ where for all $n \\in \\Z_{> 0}$: :$b_n = \\displaystyle \\frac 2 \\lambda \\int_0^\\lambda \\map f x \\sin \\frac {n \\pi x} \\lambda \\rd x$ Then $\\map S x$ also represents the extension $g: \\openint {-\\lambda} \\lambda \\to \\R$ of $f$, defined as: :$\\forall x \\in \\openint {-\\lambda} \\lambda: \\map g x = \\begin {cases} \\map f x & : x > 0 \\\\ -\\map f {-x} & : x < 0 \\\\ 0 & : x = 0 \\end {cases}$"}
+{"_id": "18762", "title": "Primitive of x by Logarithm of x squared plus a squared", "text": ":$\\displaystyle \\int x \\map \\ln {x^2 + a^2} \\rd x = \\frac {\\paren {x^2 + a^2} \\map \\ln {x^2 + a^2} - x^2} 2 + C$"}
+{"_id": "18763", "title": "Equivalence of Definitions of Sets Separated by Neighborhoods", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. {{TFAE|def = Separated by Neighborhoods/Sets|view = Sets Separated by Neighborhoods}}"}
+{"_id": "18764", "title": "Equivalence of Definitions of Points Separated by Neighborhoods", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. {{TFAE|def = Points Separated by Neighborhoods}}"}
+{"_id": "18765", "title": "Product Space is T3 iff Factor Spaces are T3/Product Space is T3 implies Factor Spaces are T3", "text": "Let $\\mathbb S = \\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be an indexed family of non-empty topological spaces for $\\alpha$ in some indexing set $I$. Let $\\displaystyle T = \\struct{S, \\tau} = \\displaystyle \\prod_{\\alpha \\mathop \\in I} \\struct{S_\\alpha, \\tau_\\alpha}$ be the product space of $\\mathbb S$. Let $T$ be a $T_3$ space. Then for each $\\alpha \\in I$, $\\struct {S_\\alpha, \\tau_\\alpha}$ is a $T_3$ space."}
+{"_id": "18766", "title": "Product Space is T3 iff Factor Spaces are T3/Factor Spaces are T3 implies Product Space is T3", "text": "Let $\\mathbb S = \\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be an indexed family of non-empty topological spaces for $\\alpha$ in some indexing set $I$. Let $\\displaystyle T = \\struct{S, \\tau} = \\displaystyle \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\mathbb S$. For each $\\alpha \\in I$, let $\\struct{S_\\alpha, \\tau_\\alpha}$ be a $T_3$ space. Then $T$ is a $T_3$ space."}
+{"_id": "18770", "title": "Mittag-Leffler Expansion for Cosecant Function/Real Domain", "text": ":$\\pi \\cosec \\pi \\alpha = \\dfrac 1 \\alpha + \\displaystyle 2 \\sum_{n \\mathop \\ge 1} \\paren {-1}^n \\dfrac {\\alpha} {\\alpha^2 - n^2}$"}
+{"_id": "18771", "title": "Sum from -m to m of 1 minus Cosine of n + alpha of theta over n + alpha", "text": "For $0 < \\theta < 2 \\pi$: :$\\displaystyle \\sum_{n \\mathop = -m}^m \\dfrac {1 - \\cos \\paren {n + \\alpha} \\theta} {n + \\alpha} = \\int_0^\\theta \\map \\sin {\\alpha u} \\dfrac {\\sin \\paren {m + \\frac 1 2} u \\rd u} {\\sin \\frac 1 2 u}$"}
+{"_id": "18772", "title": "Sum over Integers of Cosine of n + alpha of theta over n + alpha", "text": "Let $\\alpha \\in \\R$ be a real number which is specifically not an integer. For $0 < \\theta < 2 \\pi$: :$\\displaystyle \\dfrac 1 \\alpha + \\sum_{n \\mathop \\ge 1} \\dfrac {2 \\alpha} {\\alpha^2 - n^2} = \\sum_{n \\mathop \\in \\Z} \\dfrac {\\cos \\paren {n + \\alpha} \\theta} {n + \\alpha}$"}
+{"_id": "18774", "title": "Sum of Complex Exponentials of i times Arithmetic Sequence of Angles", "text": "Let $\\alpha \\in \\R$ be a real number such that $\\alpha \\ne 2 \\pi k$ for $k \\in \\Z$. Then:"}
+{"_id": "18775", "title": "Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 1", "text": ":$\\displaystyle \\sum_{k \\mathop = 0}^n e^{i \\paren {\\theta + k \\alpha} } = \\paren {\\map \\cos {\\theta + \\frac {n \\alpha} 2} + i \\map \\sin {\\theta + \\frac {n \\alpha} 2} } \\frac {\\map \\sin {\\alpha \\paren {n + 1} / 2} } {\\map \\sin {\\alpha / 2} }$"}
+{"_id": "18776", "title": "Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 2", "text": ":$\\displaystyle \\sum_{k \\mathop = 1}^n e^{i \\paren {\\theta + k \\alpha} } = \\paren {\\map \\cos {\\theta + \\frac {n + 1} 2 \\alpha} + i \\map \\sin {\\theta + \\frac {n + 1} 2 \\alpha} } \\frac {\\map \\sin {n \\alpha / 2} } {\\map \\sin {\\alpha / 2} }$"}
+{"_id": "18777", "title": "Sum of Sines of Arithmetic Sequence of Angles/Formulation 1", "text": "{{begin-eqn}} {{eqn | l = \\sum_{k \\mathop = 0}^n \\map \\sin {\\theta + k \\alpha}       | r = \\sin \\theta + \\map \\sin {\\theta + \\alpha} + \\map \\sin {\\theta + 2 \\alpha} + \\map \\sin {\\theta + 3 \\alpha} + \\dotsb }} {{eqn | r = \\frac {\\map \\sin {\\alpha \\paren {n + 1} / 2} } {\\map \\sin {\\alpha / 2} } \\map \\sin {\\theta + \\frac {n \\alpha} 2} }} {{end-eqn}}"}
+{"_id": "18778", "title": "Sum of Sines of Arithmetic Sequence of Angles/Formulation 2", "text": "{{begin-eqn}} {{eqn | l = \\sum_{k \\mathop = 1}^n \\map \\sin {\\theta + k \\alpha}       | r = \\map \\sin {\\theta + \\alpha} + \\map \\sin {\\theta + 2 \\alpha} + \\map \\sin {\\theta + 3 \\alpha} + \\dotsb }} {{eqn | r = \\map \\sin {\\theta + \\frac {n + 1} 2 \\alpha}\\frac {\\map \\sin {n \\alpha / 2} } {\\map \\sin {\\alpha / 2} } }} {{end-eqn}}"}
+{"_id": "18780", "title": "Sum of Cosines of Arithmetic Sequence of Angles/Formulation 2", "text": "{{begin-eqn}} {{eqn | l = \\sum_{k \\mathop = 1}^n \\map \\cos {\\theta + k \\alpha}       | r = \\map \\cos {\\theta + \\alpha} + \\map \\cos {\\theta + 2 \\alpha} + \\map \\cos {\\theta + 3 \\alpha} + \\dotsb }} {{eqn | r = \\map \\cos {\\theta + \\frac {n + 1} 2 \\alpha} \\frac {\\map \\sin {n \\alpha / 2} } {\\map \\sin {\\alpha / 2} } }} {{end-eqn}}"}
+{"_id": "18781", "title": "Weierstrass Approximation Theorem/Lemma 1", "text": ":$\\displaystyle \\sum_{k \\mathop = 0}^n k \\map {p_{n,k} } t = n t$"}
+{"_id": "18782", "title": "Weierstrass Approximation Theorem/Lemma 2", "text": ":$\\displaystyle \\sum_{k \\mathop = 0}^n \\paren {k - nt}^2 \\map {p_{n,k} } t = n t \\paren {1 - t}$"}
+{"_id": "18783", "title": "Convergent Sequence in Normed Vector Space has Unique Limit", "text": "Let $\\struct {X, \\norm {\\,\\cdot\\,}}$ be a normed vector space. Let $\\sequence {x_n}$ be a sequence in $\\struct {X, \\norm {\\,\\cdot\\,} }$. Then $\\sequence {x_n}$ can have at most one limit."}
+{"_id": "18784", "title": "Product Space is T3 1/2 iff Factor Spaces are T3 1/2/Product Space is T3 1/2 implies Factor Spaces are T3 1/2", "text": "Let $\\mathbb S = \\family{\\struct{S_\\alpha, \\tau_\\alpha}}_{\\alpha \\mathop \\in I}$ be an indexed family of non-empty topological spaces for $\\alpha$ in some indexing set $I$. Let $\\displaystyle T = \\struct{S, \\tau} = \\displaystyle \\prod_{\\alpha \\mathop \\in I} \\struct{S_\\alpha, \\tau_\\alpha}$ be the product space of $\\mathbb S$. Let $T$ be a $T_{3 \\frac 1 2}$ space. Then for each $\\alpha \\in I$, $\\struct{S_\\alpha, \\tau_\\alpha}$ is a $T_{3 \\frac 1 2}$ space."}
+{"_id": "18785", "title": "Product Space is T3 1/2 iff Factor Spaces are T3 1/2/Factor Spaces are T3 1/2 implies Product Space is T3 1/2", "text": "Let $\\mathbb S = \\family {\\struct{S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be an indexed family of topological spaces for $\\alpha$ in some indexing set $I$ with $S_\\alpha \\neq \\O$ for every $\\alpha \\in I$. Let $\\displaystyle T = \\struct{S, \\tau} = \\displaystyle \\prod_{\\alpha \\mathop \\in I} \\struct{S_\\alpha, \\tau_\\alpha}$ be the product space of $\\mathbb S$. For each $\\alpha \\in I$, let $\\struct{S_\\alpha, \\tau_\\alpha}$ be a $T_{3 \\frac 1 2}$ space. Then $T$ is a $T_{3 \\frac 1 2}$ space."}
+{"_id": "18786", "title": "Half-Range Fourier Sine Series/Sine of Non-Integer Multiple of x over 0 to Pi", "text": "Let $\\lambda \\in \\R \\setminus \\Z$ be a real number which is not an integer. Let $\\map f x$ be the real function defined on $\\openint 0 \\pi$ as: :$\\map f x = \\sin \\lambda x$ Then its half-range Fourier sine series can be expressed as: {{begin-eqn}} {{eqn | l = \\map f x       | o = \\sim       | r = \\frac {2 \\sin \\lambda \\pi} \\pi \\paren {\\sum_{n \\mathop = 1}^\\infty \\paren {-1}^n \\frac {n \\sin n x} {\\lambda^2 - n^2} }       | c =  }} {{eqn | r = \\frac {2 \\sin \\lambda \\pi} \\pi \\paren {-\\frac {\\sin x} {\\lambda^2 - 1} + \\frac {2 \\sin 2 x} {\\lambda^2 - 4} - \\frac {3 \\sin 3 x} {\\lambda^2 - 9}  + \\frac {4 \\sin 4 x} {\\lambda^2 - 16} - \\dotsb}       | c =  }} {{end-eqn}}"}
+{"_id": "18787", "title": "Minimum Rule for Continuous Functions", "text": "Let $\\struct {S, \\tau}$ be a topological space. Let $f, g: S \\to \\R$ be continuous real-valued functions. Let $\\min \\set {f, g}: S \\to \\R$ denote the pointwise minimum of $f$ and $g$. Then: :$\\min \\set {f, g}$ is continuous."}
+{"_id": "18788", "title": "Min is Half of Sum Less Absolute Difference", "text": "For all numbers $a, b$ where $a, b$ in $\\N, \\Z, \\Q$ or $\\R$: :$\\min \\set {a, b} = \\dfrac 1 2 \\paren {a + b - \\size {a - b} }$"}
+{"_id": "18789", "title": "Continuity Test for Real-Valued Functions", "text": "Let $\\struct{S, \\tau}$ be a topological space. Let $f: S \\to \\R$ be a real-valued function. Let $x \\in S$. Then $f$ is continuous at $x$ {{iff}}: :$\\forall \\epsilon \\in \\R_{>0} : \\exists U \\in \\tau : x \\in U : \\map {f^\\to} U \\subseteq \\openint {\\map f x - \\epsilon} {\\map f x + \\epsilon}$"}
+{"_id": "18790", "title": "Continuity Test for Real-Valued Functions/Everywhere Continuous", "text": "Let $\\struct{S, \\tau}$ be a topological space. Let $f: S \\to \\R$ be a real-valued function. Then $f$ is everywhere continuous {{iff}}: :$\\forall x \\in S : \\forall \\epsilon \\in \\R_{>0} : \\exists U \\in \\tau : x \\in U : \\map {f^\\to} U \\subseteq \\openint {\\map f x - \\epsilon} {\\map f x + \\epsilon}$"}
+{"_id": "18791", "title": "Oesterlé-Masser Conjecture", "text": "Let $\\epsilon \\in \\R$ be a strictly positive real number."}
+{"_id": "18792", "title": "Oesterlé-Masser Conjecture/Formulation 1", "text": "There exists only a finite number of triples of (strictly) positive integers $\\tuple {a, b, c}$ with the conditions: :$a + b = c$ :$a$, $b$ and $c$ are pairwise coprime such that: :$c > \\map {\\operatorname {rad} } {a b c}^{1 + \\epsilon}$ where $\\operatorname {rad}$ denotes the radical of an integer."}
+{"_id": "18793", "title": "Definite Integral of Function satisfying Dirichlet Conditions is Continuous", "text": "Let $f: \\R \\to \\R$ be a real function defined in the open interval $\\openint {-\\pi} \\pi$. Let $f$ fulfil the Dirichlet conditions in $\\openint {-\\pi} \\pi$. Let $a_0, a_1, \\dotsc; b_1, \\dotsc$ be the Fourier coefficients of $f$ in $\\openint {-\\pi} \\pi$. Then the real function: :$\\map F x = \\displaystyle \\int_{-\\pi}^x \\map f t \\rd t - \\dfrac {a_0} 2 x$ is continuous on $\\openint {-\\pi} \\pi$."}
+{"_id": "18795", "title": "Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers/Proof 1", "text": "Let $G := \\set {x \\in \\R: -1 < x < 1}$ be the set of all real numbers whose absolute value is less than $1$. Let $\\circ: G \\times G \\to G$ be the binary operation defined as: :$\\forall x, y \\in G: x \\circ y = \\dfrac {x + y} {1 + x y}$ {{:Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers}}"}
+{"_id": "18796", "title": "Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers/Proof 2", "text": "Let $G := \\set {x \\in \\R: -1 < x < 1}$ be the set of all real numbers whose absolute value is less than $1$. Let $\\circ: G \\times G \\to G$ be the binary operation defined as: :$\\forall x, y \\in G: x \\circ y = \\dfrac {x + y} {1 + x y}$ {{:Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers}}"}
+{"_id": "18798", "title": "Triangle Inequality/Complex Numbers/Corollary 2", "text": "Let $z_1, z_2 \\in \\C$ be complex numbers. Let $\\cmod z$ be the modulus of $z$. Then: :$\\cmod {z_1 + z_2} \\ge \\cmod {\\cmod {z_1} - \\cmod {z_2} }$"}
+{"_id": "18799", "title": "Continuity Test for Real-Valued Functions/Necessary Condition", "text": "Let $\\struct {S, \\tau}$ be a topological space. Let $f: S \\to \\R$ be a real-valued function. Let $x \\in S$. Let $f$ be continuous at $x$ Then: :$\\forall \\epsilon \\in \\R_{>0} : \\exists U \\in \\tau : x \\in U : \\map {f^\\to} U \\subseteq \\openint {\\map f x - \\epsilon} {\\map f x + \\epsilon}$"}
+{"_id": "18800", "title": "Continuity Test for Real-Valued Functions/Sufficient Condition", "text": "Let $\\struct {S, \\tau}$ be a topological space. Let $f: S \\to \\R$ be a real-valued function. Let $x \\in S$. Let $f$ satisfy: :$\\forall \\epsilon \\in \\R_{>0} : \\exists U \\in \\tau : x \\in U : \\map {f^\\to} U \\subseteq \\openint {\\map f x - \\epsilon} {\\map f x + \\epsilon}$ Then $f$ is continuous at $x$"}
+{"_id": "18803", "title": "Equivalence of Definitions of Completely Hausdorff Space", "text": "{{TFAE|def = Completely Hausdorff Space|view = a completely Hausdorff space}} Let $T = \\struct {S, \\tau}$ be a topological space."}
+{"_id": "18804", "title": "Product Space is Completely Hausdorff iff Factor Spaces are Completely Hausdorff/Necessary Condition", "text": "Let $\\mathbb S = \\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be an indexed family of non-empty topological spaces for $\\alpha$ in some indexing set $I$. Let $\\displaystyle T = \\struct {S, \\tau} = \\displaystyle \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\mathbb S$. Let $T$ be a completely Hausdorff space. Then for each $\\alpha \\in I$, $\\struct {S_\\alpha, \\tau_\\alpha}$ is a completely Hausdorff space."}
+{"_id": "18805", "title": "Metric Space is T4", "text": "Let $M = \\struct {A, d}$ be a metric space. Then $M$ is $T_4$."}
+{"_id": "18806", "title": "Product Space is Completely Hausdorff iff Factor Spaces are Completely Hausdorff/Sufficient Condition", "text": "Let $\\SS = \\family {\\struct {S_\\alpha, \\tau_\\alpha} }$ be an indexed family of topological spaces for $\\alpha$ in some indexing set $I$. Let $\\displaystyle T = \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\SS$. Let each of $\\struct {S_\\alpha, \\tau_\\alpha}$ for $\\alpha \\in I$ be completely Hausdorff spaces. Then $T$ is a completely Hausdorff spaces."}
+{"_id": "18807", "title": "P-adic Expansion Less Intial Zero Terms Represents Same P-adic Number", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. Let $\\mathbf a$ be an equivalence class in $\\Q_p$. Let $\\displaystyle \\sum_{i \\mathop = m}^\\infty d_i p^i$ be a $p$-adic expansion that represents $\\mathbf a$. Let $l$ be the first index $i \\ge m$ such that $d_i \\ne 0$ Then the series: :$\\displaystyle \\sum_{i \\mathop = l}^\\infty d_i p^i$ also represents $\\mathbf a$."}
+{"_id": "18808", "title": "Subsequence is Equivalent to Cauchy Sequence", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\sequence {x_n}$ be a Cauchy sequence in $R$. Let $\\sequence {x_{m_n} }$ be a subsequence of $\\sequence {x_n}$. Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} {x_n - x_{m_n} } = 0$"}
+{"_id": "18809", "title": "Numbers with Square-Free Binomial Coefficients/Lemma", "text": "Let $n$ be a (strictly) positive integer.  Let $p$ be a prime number. By Basis Representation Theorem, there is a unique sequence $\\sequence {a_j}_{0 \\mathop \\le j \\mathop \\le r}$ such that: :$(1): \\quad \\displaystyle n = \\sum_{k \\mathop = 0}^r a_k p^k$ :$(2): \\quad \\displaystyle \\forall k \\in \\closedint 0 r: a_k \\in \\N_b$ :$(3): \\quad r_t \\ne 0$ Suppose $r \\ge 2$ and $p^2 \\nmid \\dbinom n m$ for all $0 \\le m \\le n$.  Then: :$p^{r - 1} \\divides \\paren {n + 1}$ that is: :$p^{r - 1}$ divides $\\paren {n + 1}$."}
+{"_id": "18810", "title": "Derivative of Generating Function/General Result/Corollary", "text": "Let the coefficient of $z^n$ extracted from $\\map G z$ be denoted: :$\\sqbrk {z^n} \\map G z := a_n$ Then: :$\\sqbrk {z^m} \\map G z = \\dfrac 1 {m!} \\map {G^{\\paren m} } 0$ where $G^{\\paren m}$ denotes the $m$th derivative of $G$."}
+{"_id": "18811", "title": "Derivative of Generating Function/General Result", "text": "Let $m$ be a positive integer. Then: :$\\dfrac {\\d^m} {\\d z^m} \\map G z = \\displaystyle \\sum_{k \\mathop \\ge 0} \\dfrac {\\paren {k + m}!} {k!} a_{k + m} z^k$ === Corollary === {{:Derivative of Generating Function/General Result/Corollary}}"}
+{"_id": "18813", "title": "Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm form Banach Space", "text": "Let $I = \\closedint a b$ be a closed real interval. Let $\\map \\CC I$ be the space of real-valued functions, continuous on $I$. Let $\\norm {\\,\\cdot\\,}_\\infty$ be the supremum norm on real-valued functions, continuous on $I$. Then $\\struct {\\map \\CC I, \\norm {\\,\\cdot\\,}_\\infty}$ is a Banach space."}
+{"_id": "18814", "title": "Inductive Set under Progressing Mapping has Minimally Inductive Subset", "text": "Let $A$ be an inductive class under a mapping $g$. Let $A$ be a set. Then there exists some subset $S$ of $A$ such that $S$ is minimally inductive under $g$."}
+{"_id": "18815", "title": "Recurrence Relation for Bell Numbers", "text": "Let $B_n$ be the Bell number for $n \\in \\Z_{\\ge 0}$. Then: :$B_{n + 1} = \\displaystyle \\sum_{k \\mathop = 0}^n \\dbinom n k B_k$ where $\\dbinom n k$ are binomial coefficients."}
+{"_id": "18820", "title": "Natural Number m is Less than n implies n is not Greater than Successor of n", "text": "Let $\\N$ be the natural numbers. Let $m, n \\in \\N$. Then: :$m < n \\implies m + 1 \\le n$"}
+{"_id": "18823", "title": "Natural Number Ordering is Preserved by Successor Mapping", "text": "Let $\\N$ be the natural numbers. Let $m, n \\in \\N$. Then: :$n \\le m \\implies n^+ \\le m^+$"}
+{"_id": "18824", "title": "Non-Empty Bounded Subset of Natural Numbers has Greatest Element", "text": "Let $\\omega$ be the set of natural numbers defined as the von Neumann construction. Then every non-empty bounded subset of $\\omega$ has a greatest element."}
+{"_id": "18826", "title": "Natural Number Less than or Equal to Successor of Another", "text": "Let $\\N$ be the natural numbers. Let $m, n \\in \\N$ such that $m \\le n^+$. Then either: :$(1): \\quad m \\le n$ or: :$(2): \\quad m = n^+$"}
+{"_id": "18827", "title": "Mapping whose Image of Natural Number n is Subset of Image of Successor", "text": "Let $f: \\N \\to A$ be a mapping from the set of natural numbers $\\N$ to a class $A$. Let $f$ have the property that: :$\\forall n \\in \\N: \\map f n \\subseteq \\map f {n^+}$ where $n^+$ is the successor of $n$. Then: :$\\forall n, m \\in N: n \\le m \\implies \\map f n \\subseteq \\map f m$"}
+{"_id": "18828", "title": "Mapping whose Image of Natural Number n is Subset of Image of Successor/Corollary", "text": "Let $f$ have the property that: :$\\forall n \\in \\N: \\map f n \\subsetneq \\map f {n^+}$ where $n^+$ is the successor of $n$. Then: :$\\forall n, m \\in N: n < m \\implies \\map f n \\subsetneq \\map f m$"}
+{"_id": "18831", "title": "Non-Empty Finite Set of Natural Numbers has Greatest Element", "text": "Let $A$ be a non-empty finite set of natural numbers. Then $A$ has a greatest element."}
+{"_id": "18832", "title": "Set of Subsets of Element of Minimally Inductive Class is Finite", "text": "Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Let $x \\in M$. Let $S$ be the set of all $y \\in M$ such that $y \\subseteq x$. Then $S$ is finite."}
+{"_id": "18833", "title": "Minimally Inductive Class with Fixed Element is Finite", "text": "Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Let there exist an element $x \\in M$ such that $x = \\map g x$. Then $M$ is a finite class."}
+{"_id": "18834", "title": "Set of Natural Numbers is either Finite or Denumerable", "text": "Let $S$ be a subset of the natural numbers $\\N$. Then $S$ is either finite or denumerable."}
+{"_id": "18837", "title": "Unlike Electric Charges Attract", "text": "Let $a$ and $b$ be stationary particles, each carrying an electric charge of $q_a$ and $q_b$ respectively. Let $q_a$ and $q_b$ be of the opposite sign. That is, let $q_a$ and $q_b$ be unlike charges. Then the forces exerted by $a$ on $b$, and by $b$ on $a$, are such as to cause $a$ and $b$ to attract each other."}
+{"_id": "18838", "title": "Value of Vacuum Permittivity", "text": "The value of the '''vacuum permittivity''' is calculated as: :$\\varepsilon_0 = 8 \\cdotp 85418 \\, 78128 (13) \\times 10^{-12} \\, \\mathrm F \\, \\mathrm m^{-1}$ (farads per metre) with a relative uncertainty of $1 \\cdotp 5 \\times 10^{-10}$."}
+{"_id": "18841", "title": "Successor Mapping on Natural Numbers has no Fixed Element", "text": "Let $\\N$ denote the set of natural numbers. Then: :$\\forall n \\in \\N: n + 1 \\ne n$"}
+{"_id": "18843", "title": "Residue of Fibonacci Number Modulo Fibonacci Number/Lemma", "text": ":$F_{m n + 1} \\equiv \\paren {\\begin{cases} F_1 & : m \\bmod 4 = 0 \\\\ F_{n - 1} & : m \\bmod 4 = 1 \\\\ \\paren {-1}^n F_1 & : m \\bmod 4 = 2 \\\\ \\paren {-1}^n F_{n - 1} & : m \\bmod 4 = 3 \\end{cases} } \\pmod {F_n}$"}
+{"_id": "18844", "title": "Complement of Interval Defined by Absolute Value", "text": "Let $\\xi, \\delta \\in \\R$ be real numbers. Let $\\delta > 0$. Then:"}
+{"_id": "18845", "title": "Complement of Open Interval Defined by Absolute Value", "text": ":$\\set {x \\in \\R: \\size {\\xi - x} \\ge \\delta} = \\R \\setminus \\openint {\\xi - \\delta} {\\xi + \\delta}$ where: :$\\openint {\\xi - \\delta} {\\xi + \\delta}$ is the open real interval between $\\xi - \\delta$ and $\\xi + \\delta$ :$\\setminus$ denotes the set difference operator."}
+{"_id": "18846", "title": "Complement of Closed Interval Defined by Absolute Value", "text": ":$\\set {x \\in \\R: \\size {\\xi - x} > \\delta} = \\R \\setminus \\closedint {\\xi - \\delta} {\\xi + \\delta}$ where: :$\\closedint {\\xi - \\delta} {\\xi + \\delta}$ is the closed real interval between $\\xi - \\delta$ and $\\xi + \\delta$ :$\\setminus$ denotes the set difference operator."}
+{"_id": "18848", "title": "Size of y-1 lt n and Size of y+1 gt 1 over n", "text": "Let $T_n \\subseteq \\R$ be the subset of the set of real numbers $\\R$ defined as: :$T_n = \\set {y: \\size {y - 1} < n \\land \\size {y + 1} > \\dfrac 1 n}$ Then: :$T_n = \\openint {1 - n} {-1 - \\dfrac 1 n} \\cup \\openint {-1 + \\dfrac 1 n} {1 + n}$"}
+{"_id": "18852", "title": "Real Part as Mapping is Surjection", "text": "Let $f: \\C \\to \\R$ be the projection from the complex numbers to the real numbers defined as: :$\\forall z \\in \\C: \\map f z = \\map \\Re z$ where $\\map \\Re z$ denotes the real part of $z$. Then $f$ is a surjection."}
+{"_id": "18853", "title": "Imaginary Part as Mapping is Surjection", "text": "Let $f: \\C \\to \\R$ be the projection from the complex numbers to the real numbers defined as: :$\\forall z \\in \\C: \\map f z = \\map \\Im z$ where $\\map \\Im z$ denotes the imaginary part of $z$. Then $f$ is a surjection."}
+{"_id": "18854", "title": "Condition for Mapping from Quotient Set to be Well-Defined", "text": "Let $S$ and $T$ be sets. Let $\\RR$ be an equivalence relation on $S$. Let $f: S \\to T$ be a mapping from $S$ to $T$. Let $S / \\RR$ be the quotient set of $S$ induced by $\\RR$. Let $q_\\RR: S \\to S / \\RR$ be the quotient mapping induced by $\\RR$. Then: :there exists a mapping $\\phi: S / \\RR \\to T$ such that $\\phi \\circ q_\\RR = f$ {{iff}}: :$\\forall x, y \\in S: \\tuple {x, y} \\in \\RR \\implies \\map f x = \\map f y$ ::$\\begin {xy} \\xymatrix@L + 2mu@ + 1em {  S \\ar[r]^*{f}    \\ar[d]_*{q_\\RR} & T \\\\ S / \\RR \\ar@{-->}[ur]_*{\\phi} } \\end {xy}$"}
+{"_id": "18856", "title": "Condition for Mapping from Quotient Set to be Surjection", "text": "Let the mapping $\\phi: S / \\RR \\to T$ defined as: :$\\phi \\circ q_\\RR = f$ be well-defined. Then: :$\\phi$ is a surjection {{iff}}: :$f$ is a surjection."}
+{"_id": "18857", "title": "Condition for Mapping from Quotient Set to be Injection", "text": "Let the mapping $\\phi: S / \\RR \\to T$ defined as: :$\\phi \\circ q_\\RR = f$ be well-defined. Then: :$\\phi$ is an injection {{iff}}: :$\\forall x, y \\in S: \\tuple {x, y} \\in \\RR \\iff \\map f x = \\map f y$"}
+{"_id": "18858", "title": "Mapping from Quotient Set when Defined is Unique", "text": "Let the mapping $\\phi: S / \\RR \\to T$ defined as: :$\\phi \\circ q_\\RR = f$ be well-defined. Then $\\phi$ is unique."}
+{"_id": "18862", "title": "Conditions for Commutative Diagram on Quotient Mappings between Mappings", "text": "Let $A$ and $B$ be sets. Let $\\RR_S$ and $\\RR_T$ be equivalence relations on $S$ and $T$ respectively. Let $f: S \\to T$ be a mapping from $S$ to $T$. Let $S / \\RR_S$ and $T / \\RR_T$ be the quotient sets of $S$ and $T$ induced by $\\RR_S$ and $\\RR_T$ respectively. Let $q_S: S \\to S / \\RR_S$ and $q_T: T \\to T / \\RR_T$ be the quotient mappings induced by $\\RR_S$ and $\\RR_T$ respectively. Then a mapping $g: S / \\RR_S \\to T / \\RR_T$ exists such that: :$q_T \\circ f = g \\circ q_S$ {{iff}}: :$\\forall x, y \\in S: x \\mathrel {\\RR_S} y \\implies \\map f x \\mathrel {\\RR_T} \\map f y$ ::$\\begin {xy} \\xymatrix@L + 2mu@ + 1em {  S \\ar[r]^*{f}    \\ar[d]_*{q_S} & T \\ar[d]^*{q_T} \\\\ S / \\RR_S \\ar@{-->}[r]_*{g} & T / \\RR_T } \\end {xy}$"}
+{"_id": "18864", "title": "Subset of Finite Dimensional Normed Vector Space is Compact iff Closed and Bounded/Sufficient Condition", "text": "Let $\\struct {X, \\norm {\\,\\cdot\\,}}$ be a finite-dimensional normed vector space. Let $K \\subset X$ be a compact subset. Then $K$ is closed and bounded."}
+{"_id": "18867", "title": "Factors of Group Direct Product are not Subgroups", "text": "Let $\\struct {G, \\circ_1}$ and $\\struct {H, \\circ_2}$ be groups. Let $\\struct {G \\times H, \\circ}$ be the group direct product of $\\struct {G, \\circ_1}$ and $\\struct {H, \\circ_2}$. Then neither $\\struct {G, \\circ_1}$ nor $\\struct {H, \\circ_2}$ is a subgroup of $\\struct {G \\times H, \\circ}$."}
+{"_id": "18869", "title": "Additive Group of Complex Numbers is Direct Product of Reals with Reals", "text": "Let $\\struct {\\C, +}$ be the additive group of complex numbers. Let $\\struct {\\R, +}$ be the additive group of real numbers. Then the direct product $\\struct {\\R, +} \\times \\struct {\\R, +}$ is isomorphic with $\\struct {\\C, +}$."}
+{"_id": "18871", "title": "Imaginary Numbers under Addition form Group", "text": "Let $\\II$ denote the set of complex numbers of the form $0 + i y$ That is, let $\\II$ be the set of all wholly imaginary numbers. Then the algebraic structure $\\struct {\\II, +}$ is a group."}
+{"_id": "18872", "title": "Imaginary Numbers under Multiplication do not form Group", "text": "Let $\\II$ denote the set of complex numbers of the form $0 + i y$ for $y \\in \\R_{\\ne 0}$. That is, let $\\II$ be the set of all wholly imaginary non-zero numbers. Then the algebraic structure $\\struct {\\II, \\times}$ is not a group."}
+{"_id": "18873", "title": "Set of Isometries in Complex Plane under Composition forms Group", "text": "Let $S$ be the set of all complex functions $f: \\C \\to \\C$ which preserve distance when embedded in the complex plane. That is: :$\\size {\\map f a - \\map f b} = \\size {a - b}$ Let $\\struct {S, \\circ}$ be the algebraic structure formed from $S$ and the composition operation $\\circ$. Then $\\struct {S, \\circ}$ is a group."}
+{"_id": "18874", "title": "Set of Affine Mappings on Real Line under Composition forms Group", "text": "Let $S$ be the set of all real functions $f: \\R \\to \\R$ of the form: :$\\forall x \\in \\R: \\map f x = r x + s$ where $r \\in \\R_{\\ne 0}$ and $s \\in \\R$ Let $\\struct {S, \\circ}$ be the algebraic structure formed from $S$ and the composition operation $\\circ$. Then $\\struct {S, \\circ}$ is a group."}
+{"_id": "18875", "title": "Arbitrary Cyclic Group of Order 4", "text": "Let $S = \\set {1, 2, 3, 4}$. Consider the algebraic structure $\\struct {S, \\circ}$ given by the Cayley table: :$\\begin{array}{r|rrrr} \\circ & 2 & 3 & 4 & 1 \\\\ \\hline 2 & 2 & 3 & 4 & 1  \\\\ 3 & 3 & 4 & 1 & 2 \\\\ 4 & 4 & 1 & 2 & 3 \\\\ 1 & 1 & 2 & 3 & 4 \\\\ \\end{array}$ Then $\\struct {S, \\circ}$ is a group. Specifically, $\\struct {S, \\circ}$ is the cyclic group of order $4$."}
+{"_id": "18882", "title": "Factors of Sums of Powers of 100,000/General Result", "text": "All integers $n$ of the form: :$n = \\displaystyle \\sum_{k \\mathop = 0}^m 10^{r k}$ for $m \\in \\Z_{> 0}$ are composite for $r \\ge 2$. The only exceptions are $r = 2^k, m = 1$ for some $k \\in \\N$, and $r = m + 1 =$ some odd prime, where $n$ could be prime."}
+{"_id": "18883", "title": "Odd Integers under Addition do not form Subgroup of Integers", "text": "Let $S$ denote the set of odd integers. Then $\\struct {S, +}$ is not a subgroup of the additive group of integers $\\struct {\\Z, +}$."}
+{"_id": "18887", "title": "Equivalence of Definitions of Closed Set in Normed Vector Space", "text": "{{TFAE|def = Closed Set in Normed Vector Space|view = Closed Set|context = Normed Vector Space|contextview = Normed Vector Spaces}} Let $V = \\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $F \\subseteq X$."}
+{"_id": "18888", "title": "Cosets of Positive Reals in Multiplicative Group of Complex Numbers", "text": "Let $S$ be the positive real axis of the complex plane: :$S = \\set {z \\in \\C: z = x + 0 i, x \\in \\R_{>0} }$ Consider the algebraic structure $\\struct {S, \\times}$ as a subgroup of the multiplicative group of complex numbers $\\struct {\\C_{\\ne 0}, \\times}$. The cosets of $\\struct {S, \\times}$ are the sets of the form: :$\\set {z \\in \\C: \\exists r \\in \\R_{>0}: z = r e^{i \\theta}}$ for some $\\theta \\in \\hointr 0 {2 \\pi}$ That is, the sets of all complex numbers with a constant argument."}
+{"_id": "18889", "title": "Morphism from Multiplicative Group of Complex Numbers to Unit Circle", "text": "Let $\\struct {\\C_{\\ne 0}, \\times}$ denote the multiplicative group of complex numbers. Let $f: \\C_{\\ne 0} \\to \\C_{\\ne 0}$ be the mapping defined as: :$\\forall z \\in \\C_{\\ne 0}: \\map f z = \\dfrac z {\\cmod z}$ where $\\cmod z$ denotes the modulus of $z$. Then $f$ is an endomorphism on $\\struct {\\C_{\\ne 0}, \\times}$ whose kernel is the positive real axis: :$\\set {z \\in \\C: z = x + 0 i, x \\in \\R_{>0} }$ and whose image is the unit circle: :$\\set {z \\in \\C: \\cmod z = 1}$"}
+{"_id": "18890", "title": "Leigh.Samphier/Sandbox/P-adic Expansion Representative of P-adic Number is Unique", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. That is, $\\Q_p$ is the quotient ring $\\CC \\, \\big / \\NN$ where: :$\\CC$ denotes the commutative ring of Cauchy sequences over $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$ :$\\NN$ denotes the set of null sequences in $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$. Let $\\mathbf a$ be an equivalence class in $\\Q_p$. Let $\\displaystyle \\sum_{i \\mathop = m}^\\infty d_i p^i$ and $\\displaystyle \\sum_{i \\mathop = k}^\\infty e_i p^i$ be $p$-adic expansions that represent $\\mathbf a$. Then: :$(1) \\quad m = k$ :$(2) \\quad \\forall i \\ge m : d_i = e_i$ That is, the $p$-adic expansions $\\displaystyle \\sum_{i \\mathop = m}^\\infty d_i p^i$ and $\\displaystyle \\sum_{i \\mathop = k}^\\infty e_i p^i$ are identical."}
+{"_id": "18891", "title": "Rational Numbers are Dense Subfield of P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\norm {\\,\\cdot\\,}_p$ be the p-adic norm on the rationals $\\Q$. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. That is, $\\Q_p$ is the quotient ring $\\CC \\, \\big / \\NN$ where: :$\\CC$ denotes the commutative ring of Cauchy sequences over $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$ :$\\NN$ denotes the set of null sequences in $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$. Let $\\phi: \\Q \\to \\Q_p$ be the mapping defined by: :$\\map \\phi r = \\sequence {r, r, r, \\dotsc} + \\NN$ where $\\sequence {r, r, r, \\dotsc} + \\NN$ is the left coset in $\\CC \\, \\big / \\NN$ that contains the constant sequence $\\sequence {r, r, r, \\dotsc}$. Then: :$\\Q$ is isometrically isomorphic to $\\map \\phi \\Q$ which is a dense subfield of $\\Q_p$."}
+{"_id": "18892", "title": "Graph is 0-Regular iff Edgeless", "text": "Let $G$ be a graph. Then $G$ is $0$-regular graph {{iff}} $G$ is edgeless."}
+{"_id": "18894", "title": "Edgeless Graph of Order n has n Components", "text": "Let $N_n$ denote the edgeless graph with $n$ vertices. Then $N_n$ has $n$ components."}
+{"_id": "18895", "title": "Complete Graph of Order 1 is Edgeless", "text": "The complete graph $K_1$ of order $1$ is the edgeless graph $N_1$."}
+{"_id": "18899", "title": "Edgeless Graph of Order 1 is Tree", "text": "Let $N_1$ denote the edgeless graph with $1$ vertex. Then $N_1$ is a tree."}
+{"_id": "18900", "title": "Edgeless Graph of Order Greater than 1 is Forest", "text": "Let $N_n$ denote the edgeless graph with $n$ vertices such that $n > 1$. Then $N_n$ is a forest."}
+{"_id": "18903", "title": "Size of Complete Graph", "text": "Let $K_n$ denote the complete graph of order $n$ where $n \\ge 0$. The size of $K_n$ is given by: :$\\size {K_n} = \\dfrac {n \\paren {n - 1} } 2$"}
+{"_id": "18904", "title": "Field Operations of P-adic Numbers as Quotient of Cauchy Sequences", "text": "Let $p$ be a prime number. Let $\\norm {\\,\\cdot\\,}_p$ be the p-adic norm on the rationals $\\Q$. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. That is, $\\Q_p$ is the quotient ring $\\CC \\, \\big / \\NN$ where: :$\\CC$ denotes the commutative ring of Cauchy sequences over $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$ :$\\NN$ denotes the set of null sequences in $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$. Then the field operations on $\\Q_p$ are defined by: :$+ : \\quad \\forall \\sequence {x_n} + \\NN, \\sequence {y_n} + \\NN \\in \\CC \\, \\big / \\NN$: :::$\\quad \\paren {\\sequence {x_n} + \\NN} + \\paren {\\sequence {y_n} + \\NN} = \\sequence {x_n + y_n} + \\NN$ :$\\circ : \\quad \\forall \\sequence {x_n} + \\NN, \\sequence {y_n} + \\NN \\in \\CC \\, \\big / \\NN$:  :::$\\quad \\paren {\\sequence {x_n} + \\NN} \\paren {\\sequence {y_n} + \\NN} = \\sequence {x_n y_n} + \\NN$"}
+{"_id": "18906", "title": "Size of Graph/Examples/Order 3 Graphs", "text": "Let $G$ be a simple graph of order $3$. Then it is possible for $G$ to have a size of $0$, $1$, $2$ or $3$. Examples of each are presented below: :600px"}
+{"_id": "18907", "title": "Simple Graph where All Vertices and All Edges are Adjacent", "text": "Let $G$ be a simple graph in which: :every vertex is adjacent to every other vertex and: :every edge is adjacent to every other edge. Then $G$ is of order no greater than $3$."}
+{"_id": "18908", "title": "Simple Graph whose Vertices all Incident but Edges not Adjacent", "text": "Let $G = \\struct {V, E}$ be a simple graph such that: :every vertex is incident with at least one edge :no two edges are adjacent to each other. Then $G$ has an even number of vertices."}
+{"_id": "18909", "title": "Size of Star Graph", "text": "Let $S_n$ denote the star graph of order $n$ where $n > 0$. The size of $S_n$ is given by: :$\\size {S_n} = n - 1$"}
+{"_id": "18910", "title": "Smallest Simple Graph with One Vertex Adjacent to All Others", "text": "Let $G = \\struct {V, E}$ be a simple graph of order $n$. Let $G$ be the simple graph with the smallest size such that one vertex is adjacent to all other vertices of $G$. Then $G$ is the star graph of order $n$ and is of size $n - 1$."}
+{"_id": "18912", "title": "Heine-Borel Theorem/Normed Vector Space", "text": "Let $\\struct {X, \\norm {\\,\\cdot\\,}}$ be a finite-dimensional normed vector space. A subset $K \\subset X$ is compact {{iff}} $K$ is closed and bounded."}
+{"_id": "18913", "title": "Heine-Borel Theorem/Necessary Condition", "text": "Let $\\struct {X, \\norm {\\,\\cdot\\,}}$ be a finite-dimensional normed vector space. Let $K \\subseteq X$ be closed and bounded . Then $K$ is a compact subset."}
+{"_id": "18914", "title": "Heine-Borel Theorem/Sufficient Condition", "text": "Let $\\struct {X, \\norm {\\,\\cdot\\,}}$ be a finite-dimensional normed vector space. Let $K \\subset X$ be a compact subset. Then $K$ is closed and bounded."}
+{"_id": "18916", "title": "Maximum Number of Arcs in Digraph", "text": "Let $D_n$ be a digraph of order $n$ such that $n \\ge 1$. Let $D_n$ have the greatest number of arcs of all digraphs of order $n$. The number of arcs in $D$ is given by: :$\\size {D_n} = n \\paren {n - 1}$"}
+{"_id": "18919", "title": "Leigh.Samphier/Sandbox/P-adic Norm of p-adic Number is Power of p", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $x \\in \\Q_p: x \\ne 0$. Then: :$\\exists v \\in \\Z: \\norm x_p = p^{-v}$ === Lemma === {{:Leigh.Samphier/Sandbox/P-adic Norm of p-adic Number is Power of p/Lemma}}"}
+{"_id": "18920", "title": "Leigh.Samphier/Sandbox/P-adic Norm of p-adic Number is Power of p/Lemma", "text": "Let $\\norm {\\,\\cdot\\,}_p$ be the $p$-adic norm on the rationals $\\Q$ for some prime $p$. Let $\\sequence {x_n}$ be a Cauchy sequence in $\\struct{\\Q, \\norm {\\,\\cdot\\,}_p}$ such that $\\sequence {x_n}$ does not converge to $0$. Then: :$\\exists v \\in \\Z: \\displaystyle \\lim_{n \\mathop \\to \\infty} \\norm{x_n}_p = p^{-v}$"}
+{"_id": "18925", "title": "Cycle Graph is Hamiltonian", "text": "Let $n \\in \\Z$ be an integer such that $n \\ge 3$. Let $C_n$ denote the cycle graph of order $n$. Then $C_n$ is Hamiltonian."}
+{"_id": "18926", "title": "Maximum Degree of Vertex in Simple Graph", "text": "Let $G = \\struct {V, E}$ be a simple graph. Let $\\card V$ denote the order of $G$. Then no vertex of $G$ has a degree higher than $\\card V - 1$."}
+{"_id": "18927", "title": "Degrees of Vertices determine Order and Size of Graph", "text": "Let $G = \\struct {V, E}$ be a simple graph. Let the degrees of each of the vertices of $G$ be given. Then it is possible to determine both the order $\\card V$ and size $\\card E$ of $G$."}
+{"_id": "18928", "title": "Order and Size of Graph do not determine Degrees of Vertices", "text": "Let $G = \\struct {V, E}$ be a simple graph. Let both the order $\\card V$ and size $\\card E$ of $G$ be given. Then it is not always possible to determine the degrees of each of the vertices of $G$."}
+{"_id": "18929", "title": "Size of Regular Graph in terms of Degree and Order", "text": "Let $G = \\struct {V, E}$ be an $r$-regular graph of order $p$. Let $q$ denote the size of $G$. Then: :$q = \\dfrac {p r} 2$ when such an $r$-regular graph exists. If an $r$-regular graph of order $p$ does exist, then $p r$ is an even integer."}
+{"_id": "18930", "title": "Same Degrees of Vertices does not imply Graph Isomorphism", "text": "Let $G = \\struct {\\map V G, \\map E G}$ and $H = \\struct {\\map V H, \\map E H}$ be graphs such that: :$\\card {\\map V G} = \\card {\\map V H}$ where $\\card {\\map V G}$ denotes the order of $G$. Let $\\phi: G \\to H$ be a mapping which preserves the degrees of the vertices: :$\\forall v \\in \\map V G: \\map {\\deg_H} {\\map \\phi v} = \\map {\\deg_G} v$ Then it is not necessarily the case that $\\phi$ is an isomorphism."}
+{"_id": "18935", "title": "Isomorphism Classes for Order 2 Simple Graphs", "text": "There are $2$ equivalence classes for simple graphs of order $2$ under graph isomorphism: :the edgeless graph of order $2$ and :the complete graph of order $2$."}
+{"_id": "18936", "title": "Isomorphism Classes for Order 4 Size 3 Simple Graphs", "text": "There are $3$ equivalence classes for simple graphs of order $4$ and size $3$ under isomorphism: :400px"}
+{"_id": "18937", "title": "Differential Equation governing First-Order Reaction", "text": "Let a substance decompose spontaneously in a '''first-order reaction'''. The differential equation which governs this reaction is given by: :$-\\dfrac {\\d x} {\\d t} = k x$ where: :$x$ determines the quantity of substance at time $t$. :$k \\in \\R_{>0}$."}
+{"_id": "18938", "title": "Formula for Radiocarbon Dating", "text": "Let $Q$ be a quantity of a sample of dead organic material (usually wood) whose time of death is to be determined. Let $t$ years be the age of $Q$ which is to be determined. Let $r$ denote the ratio of the quantity of carbon-14 remaining in $Q$ after time $t$ to the quantity of carbon-14 in $Q$ at the time of its death. Then the number of years that have elapsed since the death of $Q$ is given by: :$t = -8060 \\ln r$"}
+{"_id": "18940", "title": "Leigh.Samphier/Sandbox/Representative of P-adic Sum", "text": "Let $p$ be any prime number. Let $\\Q_p$ be the $p$-adic numbers as a quotient of Cauchy sequences. That is, $\\Q_p$ is the quotient ring $\\CC \\, \\big / \\NN$ where: :$\\CC$ denotes the commutative ring of Cauchy sequences over $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$ :$\\NN$ denotes the set of null sequences in $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$. and $\\norm {\\,\\cdot\\,}_p$ denotes the p-adic norm on the rationals $\\Q$. Let $x, y$ be any left cosets of $\\Q_p$. Let $\\sequence{x_n}$ and $\\sequence{y_n}$ be any repesentatives of $x$ and $y$ respectively. Then: :the sequence $\\sequence{x_n + y_n}$ is a repesentative of $x + y$"}
+{"_id": "18941", "title": "Leigh.Samphier/Sandbox/Representative of P-adic Product", "text": "Let $p$ be any prime number. Let $\\Q_p$ be the $p$-adic numbers as a quotient of Cauchy sequences. That is, $\\Q_p$ is the quotient ring $\\CC \\, \\big / \\NN$ where: :$\\CC$ denotes the commutative ring of Cauchy sequences over $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$ :$\\NN$ denotes the set of null sequences in $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$. and $\\norm {\\,\\cdot\\,}_p$ denotes the p-adic norm on the rationals $\\Q$. Let $x, y$ be any left cosets of $\\Q_p$. Let $\\sequence{x_n}$ and $\\sequence{y_n}$ be any repesentatives of $x$ and $y$ respectively. Then: :the sequence $\\sequence{x_n y_n}$ is a repesentative of $x y$"}
+{"_id": "18942", "title": "Leigh.Samphier/Sandbox/Complete Normed Division Ring is Completion of Dense Subring", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a complete normed division ring. Let $\\struct {S, \\norm {\\, \\cdot \\,}}$ be a dense normed division subring of $\\struct {R, \\norm {\\, \\cdot \\,}}$. Then: :$\\struct {R, \\norm {\\, \\cdot \\,} }$ is a completion of $\\struct {S, \\norm {\\, \\cdot \\,}}$ where the inclusion mapping $i : S \\to R$ is the required distance-preserving ring monomorphism."}
+{"_id": "18943", "title": "Leigh.Samphier/Sandbox/Distance-Preserving Homomorphism Preserves Norm", "text": "Let $\\struct {R_1, \\norm {\\, \\cdot \\,}_1 }, \\struct {R_2, \\norm {\\, \\cdot \\,}_2 }$ be normed division rings. Let $\\phi: R_1 \\to R_2$ be a distance-preserving ring homomorphism. Then: :$\\forall x \\in R_1 : \\norm{\\map \\phi x}_2 = \\norm x_1$"}
+{"_id": "18944", "title": "Leigh.Samphier/Sandbox/Element of Completion is Limit of Sequence in Normed Division Ring", "text": "Let $\\struct {R_1, \\norm {\\, \\cdot \\,}_1 }, \\struct {R_2, \\norm {\\, \\cdot \\,}_2 }$ be normed division rings. Let $\\struct {R_2, \\norm {\\, \\cdot \\,}_2 }$ be a completion of $\\struct {R_1, \\norm {\\, \\cdot \\,}_1 }$ with distance-preserving ring monomorphism $\\phi: R_1 \\to R_2$. Then for all $x \\in R_2$, there exists a sequence $\\sequence{x_n}$ in $R_1$: :$x = \\displaystyle \\lim_{n \\mathop \\to \\infty} \\map \\phi {x_n}$"}
+{"_id": "18945", "title": "Leigh.Samphier/Sandbox/Normed Division Ring Determines Norm on Completion", "text": "Let $\\struct {R_1, \\norm {\\, \\cdot \\,}_1 }$ be a normed division ring. Let $\\struct {R_2, \\norm {\\, \\cdot \\,}_2 }$ be a normed division ring completion of $\\struct {R_1, \\norm {\\, \\cdot \\,}_1 }$ with distance-preserving ring monomorphism $\\phi: R_1 \\to R_2$. Then for all $x \\in R_2$, there exists a sequence $\\sequence{x_n}$ in $R_1$: :$x = \\displaystyle \\lim_{n \\mathop \\to \\infty} \\map \\phi {x_n}$ and :$\\norm x_2 = \\displaystyle \\lim_{n \\mathop \\to \\infty} \\norm {x_n}_1$"}
+{"_id": "18947", "title": "Leigh.Samphier/Sandbox/Normed Division Ring Determines Norm on Completion/Corollary", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a complete normed division ring. Let $\\struct {S, \\norm {\\, \\cdot \\,}}$ be a dense normed division subring of $\\struct {R, \\norm {\\, \\cdot \\,}}$. Then for all $x \\in R$, there exists a sequence $\\sequence{x_n}$ in $S$: :$x = \\displaystyle \\lim_{n \\mathop \\to \\infty} x_n$ and :$\\norm x = \\displaystyle \\lim_{n \\mathop \\to \\infty} \\norm {x_n}$"}
+{"_id": "18948", "title": "Leigh.Samphier/Sandbox/Inclusion Mapping on Normed Division Subring is Distance Preserving Monomorphism", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\struct {S, \\norm {\\, \\cdot \\,}_S}$ be a normed division subring of $\\struct {R, \\norm {\\, \\cdot \\,}}$. Then the inclusion mapping $i : S \\to R$ is a distance-preserving ring monomorphism."}
+{"_id": "18950", "title": "Period of Reciprocal of Prime", "text": "Consider the decimal expansion of the reciprocal $\\dfrac 1 p$ of a prime $p$. If $p \\nmid a$, the decimal expansion of $\\dfrac 1 p$ is periodic in base $a$ and its period of recurrence is the order of $a$ modulo $p$. If $p \\divides a$, the decimal expansion of $\\dfrac 1 p$ in base $a$ terminates."}
+{"_id": "18951", "title": "Maximum Period of Reciprocal of Prime", "text": "Let $p$ be a prime number such that $p$ is not a divisor of $10$. The period of recurrence of the reciprocal of $p$ when expressed in decimal notation is less than or equal to $p - 1$."}
+{"_id": "18953", "title": "Maximum Abscissa for Loop of Folium of Descartes", "text": "Consider the folium of Descartes defined in parametric form as: :$\\begin {cases} x = \\dfrac {3 a t} {1 + t^3} \\\\ y = \\dfrac {3 a t^2} {1 + t^3} \\end {cases}$ :500px The point on the loop at which the $x$ value is at a maximum occurs when $t = \\sqrt [3] {\\dfrac 1 2}$, corresponding to the point $P$ defined as: :$P = \\tuple {2^{2/3} a, 2^{1/3} a}$"}
+{"_id": "18957", "title": "Condition for Denesting of Square Root/Lemma", "text": "Let $a, b, c, d \\in \\Q_{\\ge 0}$. Suppose $\\sqrt b \\notin \\Q$. Then: :$\\sqrt {a + \\sqrt b} = \\sqrt {c + \\sqrt d} \\implies a = c, b = d$"}
+{"_id": "18958", "title": "First Order ODE/y' + y = 0", "text": "The first order ODE: :$\\dfrac {\\d y} {\\d x} + y = 0$ has the general solution: :$y = C e^{-x}$ where $C$ is an arbitrary constant."}
+{"_id": "18961", "title": "Order-Preserving Mapping Not Always Semilattice Homomorphism", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be semilattices. Let $\\preceq_1$ be the ordering on $S$ defined by: :$a \\preceq_1 b \\iff \\paren {a \\circ b} = b$ Let $\\preceq_2$ be the ordering on $T$ defined by: :$x \\preceq_2 y \\iff \\paren {x * y} = y$ Let $\\phi: \\struct {S, \\preceq_1} \\to \\struct {T, \\preceq_2}$ be an order-preserving mapping. Then: : $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ may not be a semilattice homomorphism"}
+{"_id": "18962", "title": "Factorial as Product of Consecutive Factorials/Lemma 2", "text": "Let $n \\in \\N$. Then $\\paren {2 n - 2}! \\, \\paren {2 n - 1}! > \\paren {3 n - 1}!$ for all $n \\ge 7$."}
+{"_id": "18963", "title": "Factorial as Product of Consecutive Factorials/Lemma 1", "text": "Let $n \\in \\N$. Then $\\paren {2 n - 1}! \\, \\paren {2 n}! > \\paren {3 n - 1}!$ for all $n > 1$."}
+{"_id": "18964", "title": "Entire Function with Bounded Real Part is Constant", "text": "Let $f : \\C \\to \\C$ be an entire function.  Let the real part of $f$ be bounded. That is, there exists a positive real number $M$ such that:  :$\\cmod {\\map \\Re {\\map f z} } < M$ for all $z \\in \\C$, where $\\map \\Re {\\map f z}$ denotes the real part of $\\map f z$. Then $f$ is constant."}
+{"_id": "18965", "title": "Mass of Mole of Isotope of Element", "text": "Let $S$ be a substance made up entirely of a particular isotope $Q$ of a particular element. Let one atom of $Q$ contain $n$ neutrons and $p$ protons. Then one mole of $S$ has a mass of approximately $n + p$ grams."}
+{"_id": "18966", "title": "Mass of Mole of Substance", "text": "Let $S$ be a substance with molecular weight $W_S$. Then one mole of $S$ has a mass of $W_S$ grams."}
+{"_id": "18967", "title": "Linear Second Order ODE/y'' = y'", "text": "The second order ODE: :$(1): \\quad y'' = y'$ has the general solution: :$y = A_1 e^x + A_2$"}
+{"_id": "18968", "title": "First Order ODE/x y' = 2 y", "text": "The first order ODE: :$x y' = 2 y$ has the general solution: :$y = C x^2$ where $C$ is an arbitrary constant."}
+{"_id": "18970", "title": "Singleton in Normed Vector Space is Closed", "text": "Let $X$ be a vector space over $\\R$ or $\\C$. Let $\\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $x \\in X$ be a singleton. Then $x$ is closed."}
+{"_id": "18971", "title": "Singleton Set is Nowhere Dense in Rational Space", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Then every singleton subset of $\\Q$ is nowhere dense in $\\struct {\\Q, \\tau_d}$."}
+{"_id": "18972", "title": "Real Function of Two Variables represents Surface in Cartesian 3-Space", "text": "Let $S$ and $T$ be subsets of the set of real numbers $\\R$. Let $f: S \\times T \\to \\R$ be a real function of two variables. Then the locus of $f$ describes a surface embedded in the Cartesian space $\\R^3$."}
+{"_id": "18973", "title": "Cauchy's Lemma (Number Theory)", "text": "Let $a$ and $b$ be odd positive integers.  Suppose $a$ and $b$ satisfy:  :$b^2 < 4 a$ :$3 a < b^2 + 2 b + 4$ Then there exist non-negative integers $s, t, u, v$ such that: {{begin-eqn}} {{eqn | l = a       | r = s^2 + t^2 + u^2 + v^2 }} {{eqn | l = b       | r = s + t + u + v }} {{end-eqn}}"}
+{"_id": "18974", "title": "Finite Subset of Normed Vector Space is Closed", "text": "Let $M = \\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $F \\subseteq X$ be finite. Then $F$ is closed in $M$."}
+{"_id": "18976", "title": "Primitive of x fourth by Cosine of a x", "text": ":$\\ds \\int x^4 \\cos a x \\rd x = \\frac {\\sin a x} a x^4 + \\frac {4 \\cos a x} {a^2} x^3 - \\frac {12 \\sin a x} {a^3} x^2 - \\frac {24 \\cos a x} {a^4} x + \\frac {24 \\sin a x} {a^5} + C$"}
+{"_id": "18977", "title": "Primitive of x sixth by Cosine of a x", "text": ":$\\displaystyle \\int x^6 \\cos a x \\rd x = \\frac {\\sin a x} a x^6 + \\frac {6 \\cos a x} {a^2} x^5 - \\frac {30 \\sin a x} {a^3} x^4 - \\frac {120 \\cos a x} {a^4} x^3 + \\frac {360 \\sin a x} {a^5} x^2 + \\frac {720 \\cos a x} {a^6} x - \\frac {720 \\sin a x} {a^7} + C$"}
+{"_id": "18978", "title": "Fourier Series/Sixth Power of x over Minus Pi to Pi", "text": ":$\\displaystyle x^6 = \\frac {\\pi^6} 7 + \\sum_{n \\mathop = 1}^\\infty \\frac {12 n^4 \\pi^4 - 240 n^2 \\pi^2 + 1440} {n^6} \\cos n \\pi \\cos n x$"}
+{"_id": "18979", "title": "Derivative of x to the a x", "text": ":$\\dfrac \\d {\\d x} x^{a x} = a x^{a x} \\paren {\\ln x + 1}$"}
+{"_id": "18981", "title": "Integer as Sum of Polygonal Numbers/Lemma 1", "text": "Let $n, m \\in \\N_{>0}$ such that $m \\ge 3$. Let $n < 116 m$. Then $n$ can be expressed as a sum of at most $m + 2$ polygonal numbers of order $m + 2$."}
+{"_id": "18982", "title": "Integer as Sum of Polygonal Numbers/Lemma 3", "text": "Let $n, m, r \\in \\R_{>0}$. Suppose $\\dfrac n m > 1$. Let $b \\in \\openint {\\dfrac 2 3 + \\sqrt {8 \\paren {\\dfrac n m} - 8} } {\\dfrac 1 2 + \\sqrt {6 \\paren {\\dfrac n m} - 3} }$. Define: :$a = 2 \\paren {\\dfrac {n - b - r} m} + b = \\paren {1 - \\dfrac 2 m} b + 2 \\paren {\\dfrac {n - r} m}$ Then $a, b$ satisfy: :$b^2 < 4 a$ :$3 a < b^2 + 2 b + 4$"}
+{"_id": "18983", "title": "Integer as Sum of Polygonal Numbers/Lemma 2", "text": "Let $n, m \\in \\R_{>0}$ such that $\\dfrac n m \\ge 1$. Define $I$ to be the open real interval: :$I = \\openint {\\dfrac 2 3 + \\sqrt {8 \\paren {\\dfrac n m} - 8} } {\\dfrac 1 2 + \\sqrt {6 \\paren {\\dfrac n m} - 3} }$ Then: :For $\\dfrac n m \\ge 116$, the length of $I$ is greater than $4$."}
+{"_id": "18984", "title": "Integer as Sum of Three Odd Squares", "text": "Let $r$ be a positive integer. Then: :$r \\equiv 3 \\pmod 8$ {{iff}}: :$r$ is the sum of $3$ odd squares. {{explain|Could this be an {{iff}} proof? Could also be named more precisely.</br>It is now. I referred to Integer as Sum of Three Squares when naming this. Could be classified as a corollary}}"}
+{"_id": "18986", "title": "Join Semilattice Ordered Subset Not Always Subsemilattice", "text": "Let $\\struct {S, \\circ}$ be a semilattices. Let $\\preceq$ be the ordering on $S$ defined by: :$a \\preceq b \\iff \\paren {a \\circ b} = b$ Let $T$ be a subset of $S$. Let the ordered subset $\\struct{T, \\preceq \\restriction_T}$ be a join semilattice. Let $\\vee$ be the binary operation on $S$ defined by: :for all $a, b \\in S$, $a \\vee b$ is the join of $a$ and $b$ with respect to $\\preceq$. Then: :$\\struct{T, \\vee}$ may not be a subsemilattice of $\\struct {S, \\circ}$."}
+{"_id": "18987", "title": "Closure of Subset of Closed Set of Topological Space is Subset", "text": "Let $T$ = $\\struct {S, \\tau}$ be a topological space. Let $F$ be a closed set of $T$. Let $H \\subseteq F$ be a subset of $F$. Let $H^-$ denote the closure of $H$. Then $H^- \\subseteq F$."}
+{"_id": "18990", "title": "Number of Partial Derivatives of Order n", "text": "Let $u = \\map f {x_1, x_2, \\ldots, x_m}$ be a function of the $m$ independent variables $x_1, x_2, \\ldots, x_m$. There are $m^n$ partial derivatives of $u$ of order $n$."}
+{"_id": "18991", "title": "Normed Vector Space is Open in Itself", "text": "Let $M = \\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Then the set $X$ is an open set of $M$."}
+{"_id": "18992", "title": "Normed Vector Space is Open in Itself/Proof 1", "text": "Let $M = \\struct{X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Then the set $X$ is an open set of $M$."}
+{"_id": "18993", "title": "Normed Vector Space is Open in Itself/Proof 2", "text": "Let $M = \\struct{X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Then the set $X$ is an open set of $M$."}
+{"_id": "18996", "title": "Empty Set is Closed in Normed Vector Space", "text": "Let $M = \\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Then the empty set $\\O$ is closed in $M$."}
+{"_id": "18997", "title": "Empty Set is Open in Normed Vector Space", "text": "Let $M = \\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Then the empty set $\\O$ is an open set of $M$."}
+{"_id": "18998", "title": "Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent", "text": "Let $S \\subseteq \\R$. Let $\\sequence {f_n}$ be a sequence of real functions $S \\to \\R$.  Then $\\sequence {f_n}$ is uniformly Cauchy on $S$ {{iff}} $\\sequence {f_n}$ converges uniformly on $S$."}
+{"_id": "18999", "title": "Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent/Sufficient Condition", "text": "Let $S \\subseteq \\R$. Let $\\sequence {f_n}$ be a sequence of real functions $S \\to \\R$. Let $\\sequence {f_n}$ be uniformly Cauchy on $S$.  Then $\\sequence {f_n}$ is uniformly convergent on $S$."}
+{"_id": "19000", "title": "Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent/Necessary Condition", "text": "Let $S \\subseteq \\R$. Let $\\sequence {f_n}$ be a sequence of real functions $S \\to \\R$. Let $\\sequence {f_n}$ be uniformly convergent on $S$. Then $\\sequence {f_n}$ is uniformly Cauchy on $S$."}
+{"_id": "19001", "title": "Open Ball of Point Inside Open Ball/Normed Vector Space", "text": "Let $M = \\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $\\map {B_\\epsilon} x$ be an open $\\epsilon$-ball in $M = \\struct{X, \\norm {\\, \\cdot \\,}}$. Let $y \\in \\map {B_\\epsilon} x$. Then: : $\\exists \\delta \\in \\R: \\map {B_\\delta} y \\subseteq \\map {B_\\epsilon} x$ That is, for every point in an open $\\epsilon$-ball in a normed vector space, there exists an open $\\delta$-ball of that point entirely contained within that open $\\epsilon$-ball."}
+{"_id": "19002", "title": "Uniformly Convergent Sequence of Continuous Functions Converges to Continuous Function", "text": "Let $S \\subseteq \\R$. Let $x \\in S$. Let $\\sequence {f_n}$ be a sequence of real functions $S \\to \\R$ converging uniformly to $f : S \\to \\R$. Let $f_n$ be continuous at $x$ for all $n \\in \\N$. Then $f$ is continuous at $x$."}
+{"_id": "19003", "title": "Open Ball is Open Set/Normed Vector Space", "text": "Let $M = \\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $x \\in X$. Let $\\epsilon \\in \\R_{>0}$. Let $\\map {B_\\epsilon} x$ be an open $\\epsilon$-ball of $x$ in $M$. Then $\\map {B_\\epsilon} x$ is an open set of $M$."}
+{"_id": "19004", "title": "Finite Intersection of Open Sets of Normed Vector Space is Open", "text": "Let $M = \\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $U_1, U_2, \\ldots, U_n$ be open in $M$. Then $\\displaystyle \\bigcap_{i \\mathop = 1}^n U_i$ is open in $M$. That is, a finite intersection of open subsets is open."}
+{"_id": "19005", "title": "Union of Open Sets of Normed Vector Space is Open", "text": "Let $M = \\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. The union of a set of open sets of $M$ is open in $M$."}
+{"_id": "19006", "title": "Definite Integral of Limit of Uniformly Convergent Sequence of Integrable Functions", "text": "Let $a, b \\in \\R$ with $a < b$.  Let $\\sequence {f_n}$ be a sequence of Riemann integrable real functions $\\closedint a b \\to \\R$ converging uniformly to $f : \\closedint a b \\to \\R$. Then $f$ is integrable and:  :$\\displaystyle \\int_a^b \\map f x \\rd x = \\lim_{n \\to \\infty} \\int_a^b \\map {f_n} x \\rd x$"}
+{"_id": "19009", "title": "Intersection of Closed Sets is Closed/Normed Vector Space", "text": "Let $M = \\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Then the intersection of an arbitrary number of closed sets of $M$ (either finitely or infinitely many) is itself closed."}
+{"_id": "19010", "title": "Normed Vector Space is Closed in Itself", "text": "Let $M = \\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Then $X$ is closed in $M$."}
+{"_id": "19011", "title": "Infinite Series of Functions is Uniformly Convergent iff Sequence of Partial Sums is Uniformly Cauchy", "text": "Let $S \\subseteq \\R$. Let $\\sequence {f_n}$ be a sequence of real functions $S \\to \\R$. Then the infinite series:  :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty f_n$ converges uniformly on $S$ {{iff}} for all $\\varepsilon \\in \\R_{> 0}$ there exists $N \\in \\N$ such that: :$\\displaystyle \\size {\\sum_{k \\mathop = m + 1}^n \\map {f_k} x} < \\varepsilon$ for all $x \\in S$ and $n > m > N$."}
+{"_id": "19012", "title": "De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection/Corollary", "text": ":$T_1 \\cap T_2 = \\overline {\\overline T_1 \\cup \\overline T_2}$"}
+{"_id": "19013", "title": "LCM of 3 Integers in terms of GCDs of Pairs of those Integers/Lemma", "text": "Let $a, b, c \\in \\Z_{>0}$ be strictly positive integers. Then: :$\\gcd \\set {\\gcd \\set {a, b}, \\gcd \\set {a, c} } = \\gcd \\set {a, b, c}$"}
+{"_id": "19014", "title": "Events One of Which equals Intersection", "text": "Let the probability space of an experiment $\\EE$ be $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $A, B \\in \\Sigma$ be events of $\\EE$, so that $A \\subseteq \\Omega$ and $B \\subseteq \\Omega$. Let $A$ and $B$ be such that: :$A \\cap B = A$ Then whenever $A$ occurs, it is always the case that $B$ occurs as well."}
+{"_id": "19015", "title": "Union of Event with Complement is Certainty", "text": "Let the probability space of an experiment $\\EE$ be $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $A \\in \\Sigma$ be an events of $\\EE$, so that $A \\subseteq \\Omega$. Then: :$A \\cup \\overline A = \\Omega$ where $\\overline A$ is the complementary event to $A$. That is, $A \\cup \\overline A$ is a certainty."}
+{"_id": "19016", "title": "Equivalence of Definitions of Adherent Point/Definition 1 iff Definition 2", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A \\subseteq S$. {{TFAE|def = Adherent Point|view = adherent point of $A$}}"}
+{"_id": "19017", "title": "Equivalence of Definitions of Adherent Point/Definition 1 iff Definition 3", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A \\subseteq S$. {{TFAE|def = Adherent Point|view = adherent point of $A$}}"}
+{"_id": "19018", "title": "Compact Sets in Fortissimo Space", "text": "A set in Fortissimo space is compact {{iff}} it is finite."}
+{"_id": "19020", "title": "Uniformly Convergent Series of Continuous Functions Converges to Continuous Function", "text": "Let $S \\subseteq \\R$. Let $x \\in S$. Let $\\sequence {f_n}$ be a sequence of real functions. Let $f_n$ be continuous at $x$ for all $n \\in \\N$. Let the infinite series:  :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty f_n$ be uniformly convergent to a real function $f : S \\to \\R$. Then $f$ is continuous at $x$."}
+{"_id": "19021", "title": "Uniformly Convergent Series of Continuous Functions Converges to Continuous Function/Corollary", "text": "Let $S \\subseteq \\R$. Let $\\sequence {f_n}$ be a sequence of real functions. Let $f_n$ be continuous for all $n \\in \\N$. Let the infinite series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty f_n$ be uniformly convergent to a real function $f : S \\to \\R$. Then $f$ is continuous."}
+{"_id": "19022", "title": "Euclidean Space is Banach Space/Proof 1", "text": "Let $m$ be a positive integer. Then the Euclidean space $\\R^m$, along with the Euclidean norm, forms a Banach space over $\\R$."}
+{"_id": "19023", "title": "Euclidean Space is Banach Space/Proof 2", "text": "Let $m$ be a positive integer. Then the Euclidean space $\\R^m$, along with the Euclidean norm, forms a Banach space over $\\R$."}
+{"_id": "19024", "title": "Monotone Function is of Bounded Variation", "text": "Let $a, b$ be real numbers with $a < b$. Let $f : \\closedint a b \\to \\R$ be a monotone function. Then $f$ is of bounded variation."}
+{"_id": "19028", "title": "Norm Equivalence is Equivalence", "text": "Let $X$ be a vector space. Let $\\norm {\\, \\cdot \\,}_a$ and $\\norm {\\, \\cdot \\,}_b$ be equivalent norms on $X$. Denote this relation by $\\sim$:  :$\\norm {\\, \\cdot \\,}_a \\sim \\norm {\\, \\cdot \\,}_b$. Then $\\sim$ is an equivalence relation."}
+{"_id": "19031", "title": "Continuous Non-Negative Real Function with Zero Integral is Zero Function", "text": "Let $a, b$ be real numbers with $a < b$.  Let $f : \\closedint a b \\to \\R$ be a continuous function.  Let:  :$\\map f x \\ge 0$ for all $x \\in \\closedint a b$. Let:  :$\\displaystyle \\int_a^b \\map f x \\rd x = 0$ Then $\\map f x = 0$ for all $x \\in \\closedint a b$."}
+{"_id": "19036", "title": "Sequential Continuity is Equivalent to Continuity in the Reals/Sufficient Condition", "text": "Let $A \\subseteq \\R$ be a subset of the real numbers. Let $c \\in A$. Let $f : A \\to \\R$ be a real function. Then if $f$ is continuous at $c$: :for each sequence $\\sequence {x_n}$ in $A$ that converges to $c$, the sequence $\\sequence {\\map f {x_n} }$ converges to $\\map f c$."}
+{"_id": "19037", "title": "Sequential Continuity is Equivalent to Continuity in the Reals/Necessary Condition", "text": "Let $A \\subseteq \\R$ be a subset of the real numbers. Let $c \\in A$. Let $f : A \\to \\R$ be a real function. Then if $f$ is continuous at $c$: :for each sequence $\\sequence {x_n}$ in $A$ that converges to $c$, the sequence $\\sequence {\\map f {x_n} }$ converges to $\\map f c$."}
+{"_id": "19039", "title": "Metric Closure and Topological Closure of Subset are Equivalent", "text": "Let $M = \\struct{A, d}$ be a metric space. Let $T = \\struct{A, \\tau}$ be the topological space with the topology induced by $d$. Let $H \\subseteq A$. Then: :the metric closure of $H$ in $M$ equals the topological closure of $H$ in $T$"}
+{"_id": "19041", "title": "Set together with Condensation Points is not necessarily Closed", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$. Let $\\CC$ denote the set of condensation points of $H$. Then it is not necessarily the case that $H \\cup \\CC$ is a closed set of $T$."}
+{"_id": "19045", "title": "Basis Test for Limit Point", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\BB$ be a synthetic basis of $T$. Let $H \\subseteq S$. Then $x \\in S$ is a limit point of $H$ {{iff}}: :$\\forall U \\in \\BB : x \\in U$ satisfies $H \\cap U \\setminus \\set x \\ne \\O$"}
+{"_id": "19046", "title": "Basis Test for Adherent Point", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\BB$ be a synthetic basis of $T$. Let $H \\subseteq S$. Then $x \\in S$ is an adherent point of $H$ {{iff}}: :$\\forall U \\in \\BB : x \\in U$ satisfies $H \\cap U \\ne \\O$"}
+{"_id": "19047", "title": "Set of Liouville Numbers is Uncountable", "text": "The set of Liouville numbers is uncountable."}
+{"_id": "19048", "title": "Isolated Point in Metric Space iff Isolated Point in Topological Space", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $T = \\struct {A, \\tau}$ be the topological space with the topology induced by $d$. Let $H \\subseteq A$. Let $x \\in H$ Then: :$x$ is an isolated point of $H$ in $M$ {{iff}} $x$ is an isolated point of $H$ in $T$"}
+{"_id": "19049", "title": "Limit Point in Metric Space iff Limit Point in Topological Space", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $T = \\struct {A, \\tau}$ be the topological space with the topology induced by $d$. Let $H \\subseteq A$. Then: :$x \\in H$ is a limit point in $M$ {{iff}} $x$ is a limit point in $T$"}
+{"_id": "19051", "title": "Sum of Functions of Bounded Variation is of Bounded Variation", "text": "Let $a, b$ be real numbers with $a < b$. Let $f, g : \\closedint a b \\to \\R$ be functions of bounded variation. Let $V_f$ and $V_g$ be the total variations of $f$ and $g$ respectively.  Then $f + g$ is of bounded variation with: :$V_{f + g} \\le V_f + V_g$ where $V_{f + g}$ denotes the total variation of $f + g$."}
+{"_id": "19054", "title": "Open Sets in Vector Spaces with Equivalent Norms Coincide", "text": "Let $M_a = \\struct {X, \\norm {\\, \\cdot \\, }_a}$ and $M_b = \\struct {X, \\norm {\\, \\cdot \\,}_b}$ be normed vector spaces. Let $U \\subseteq X$ be an open set in $M_a$. Suppose, $\\norm {\\, \\cdot \\, }_a$ and $\\norm {\\, \\cdot \\,}_b$ are equivalent norms, i.e. $\\norm {\\, \\cdot \\, }_a \\sim \\norm {\\, \\cdot \\,}_b$. Then $U$ is also open in $M_b$."}
+{"_id": "19055", "title": "Local Basis Test for Isolated Point", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$. Let $x \\in H$. Let $\\BB_x$ be a local basis of $x$. Then $x$ is an isolated point of $H$ {{iff}}: :$\\exists U \\in \\BB_x : U \\cap H = \\set x$"}
+{"_id": "19056", "title": "Local Basis Test for Limit Point", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$. Let $x \\in S$. Let $\\BB_x$ be a local basis of $x$. Then $x \\in S$ is a limit point of $H$ {{iff}}: :$\\forall U \\in \\BB_x : H \\cap U \\setminus \\set x \\ne \\O$"}
+{"_id": "19057", "title": "Local Basis Test for Adherent Point", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$. Let $x \\in S$. Let $\\BB_x$ be a local basis of $x$. Then $x \\in S$ is an adherent point of $H$ {{iff}}: :$\\forall U \\in \\BB_x : H \\cap U \\ne \\O$"}
+{"_id": "19060", "title": "Closure of Subset of Metric Space is Closed", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $H \\subseteq A$ be a subset of $A$. Let $H^-$ denote the closure of $H$. Then $H^-$ is a closed set of $M$."}
+{"_id": "19061", "title": "Convergent Sequences in Vector Spaces with Equivalent Norms Coincide", "text": "Let $M_a = \\struct {X, \\norm {\\, \\cdot \\, }_a}$ and $M_b = \\struct {X, \\norm {\\, \\cdot \\,}_b}$ be normed vector spaces. Let $\\sequence {x_n}_{n \\mathop \\in \\N}$ be an convergent sequence in $M_a$. Suppose, $\\norm {\\, \\cdot \\, }_a$ and $\\norm {\\, \\cdot \\,}_b$ are equivalent norms, i.e. $\\norm {\\, \\cdot \\, }_a \\sim \\norm {\\, \\cdot \\,}_b$. Then $\\sequence {x_n}_{n \\mathop \\in \\N}$ is also convergent in $M_b$."}
+{"_id": "19062", "title": "Open Mapping is not necessarily Closed Mapping", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ and $T_2 = \\struct {S_2, \\tau_2}$ be topological spaces. Let $f: T_1 \\to T_2$ be a mapping which is not a bijection. Let $f$ be an open mapping. Then it is not necessarily the case that $f$ is also a closed mapping."}
+{"_id": "19065", "title": "Equivalence of Definitions of Limit Point of Filter Basis", "text": "{{TFAE|def = Limit Point of Filter Basis}} Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\FF$ be a filter on the underlying set $S$ of $T$. Let $\\BB$ be a filter basis of $\\FF$."}
+{"_id": "19066", "title": "Richert's Theorem", "text": "Let $S = \\set {s_1, s_2, \\dots}$ be an infinite set of (strictly) positive integers, with the property: :$s_n < s_{n + 1}$ for every $n \\in \\N$ Suppose there exists some integers $N, k$ such that every integer $n$ with $N < n \\le N + s_{k + 1}$: :$n$ can be expressed as a sum of distinct elements in $\\set {s_1, s_2, \\dots, s_k}$ :$s_{i + 1} \\le 2 s_i$ for every $i \\ge k$ Then for any $n > N$, $n$ can be expressed as a sum of distinct elements in $S$."}
+{"_id": "19068", "title": "Number as Sum of Distinct Primes", "text": "For $n \\ne 1, 4, 6$, $n$ can be expressed as the sum of distinct primes."}
+{"_id": "19072", "title": "Differentiable Function of Bounded Variation may not have Bounded Derivative", "text": "Let $a, b$ be real numbers with $a < b$. Let $f : \\closedint a b \\to \\R$ be a continuous function of bounded variation. Let $f$ be differentiable on $\\openint a b$. Then $f'$ is not necessarily bounded."}
+{"_id": "19073", "title": "Existence of Urysohn Function does not guarantee Normal Space", "text": "Let $T = \\struct {S, \\tau}$ be a regular space. Let $T$ have the property that: :For all closed sets $A, B \\subseteq S$ of $T$ such that $A \\cap B = \\O$, there exists an Urysohn function for $A$ and $B$. Then it is not necessarily the case that $T$ is a normal space."}
+{"_id": "19075", "title": "Existence of Compact Space which Satisfies No Separation Axioms", "text": "There exists at least one example of a compact space for which none of the Tychonoff separation axioms are satisfied."}
+{"_id": "19076", "title": "Everywhere Dense iff Interior of Complement is Empty", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A \\subset S$. Then $A$ is everywhere dense {{iff}}: :$\\paren {\\relcomp S A}^\\circ = \\O$ where $A^\\circ$ is the interior of $A$."}
+{"_id": "19077", "title": "Greatest Set is Unique", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Let $\\TT \\subseteq \\powerset S$ be a subset of $\\powerset S$. Then the greatest set of $\\TT$, if it exists, must be unique."}
+{"_id": "19078", "title": "Greatest Set may not Exist", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Let $\\TT \\subseteq \\powerset S$ be a subset of $\\powerset S$. The greatest set of $\\TT$ may not exist."}
+{"_id": "19079", "title": "Smallest Set is Unique", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Let $\\TT \\subseteq \\powerset S$ be a subset of $\\powerset S$. Then the smallest set of $\\TT$, if it exists, must be unique."}
+{"_id": "19080", "title": "Smallest Set may not Exist", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Let $\\TT \\subseteq \\powerset S$ be a subset of $\\powerset S$. The smallest set of $\\TT$ may not exist."}
+{"_id": "19081", "title": "Mapping is Surjection if its Direct Image Mapping is Surjection", "text": "Let $f: S \\to T$ be a mapping. Let $f^\\to: \\powerset S \\to \\powerset T$ be the direct image mapping of $f$. Let $f^\\to$ be a surjection. Then $f: S \\to T$ is also a surjection."}
+{"_id": "19083", "title": "Nested Sequences in Complete Metric Space not Tending to Zero may be Disjoint", "text": "Let $M = \\struct {A, d}$ be a complete metric space. Let $\\family {S_k}_{k \\mathop \\in \\N}$ be a nested sequence of closed balls in $M$. Let the radii of $\\family {S_k}_{k \\mathop \\in \\N}$ be convergent in $M$, but not to zero. Then it is not necessarily the case that their intersection $\\displaystyle \\bigcap S_k$ is non-empty."}
+{"_id": "19086", "title": "Union of Interiors is Subset of Interior of Union", "text": "Let $T$ be a topological space. Let $\\H$ be a set of subsets of $T$. That is, let $\\H \\subseteq \\powerset T$ where $\\powerset T$ is the power set of $T$. Then the union of the interiors of the elements of $\\H$ is a subset of the interior of the union of $\\H$. :$\\displaystyle \\bigcup_{H \\mathop \\in \\H} H^\\circ \\subseteq \\paren {\\bigcup_{H \\mathop \\in \\H} H}^\\circ $"}
+{"_id": "19087", "title": "Closed Ball is Closed/Normed Vector Space", "text": "Let $M = \\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $x \\in X$. Let $\\epsilon \\in \\R_{> 0}$. Let $\\map {B_\\epsilon^-} x$ be the closed $\\epsilon$-ball of $x$ in $M$. Then $\\map {B_\\epsilon^-} x$ is a closed set of $M$."}
+{"_id": "19088", "title": "Unit Sphere is Closed/Normed Vector Space", "text": "Let $M = \\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $\\Bbb S := \\set {x \\in X : \\norm {x} = 1}$ be a unit sphere in $M$. Then $\\Bbb S$ is closed in $M$."}
+{"_id": "19090", "title": "Continued Fraction Expansion of Euler's Number/Proof 1/Lemma", "text": ":For $n \\in \\Z , n \\ge 0$: {{begin-eqn}} {{eqn | l = A_n       | r = q_{3 n} e - p_{3 n} }} {{eqn | l = B_n       | r = p_{3 n + 1} - q_{3 n + 1} e }} {{eqn | l = C_n       | r = p_{3 n + 2} - q_{3 n + 2} e }} {{end-eqn}}"}
+{"_id": "19091", "title": "Limit Points in Particular Point Space/Subset", "text": "Let $U \\subseteq S$ such that $p \\in U$. Let $x \\in S$ such that $x \\ne p$. Then $x$ is a limit point of $U$."}
+{"_id": "19092", "title": "Limit Points in Closed Extension Space/Subset", "text": "Let $U \\subseteq S^*_p$ such that $p \\in U$. Let $x \\in S$. Then $x$ is a limit point of $U$."}
+{"_id": "19093", "title": "Convergent Sequence in Particular Point Space", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Let $\\sequence {a_i}$ be a convergent sequence in $T$. Except for a finite number of indices, the terms of $\\sequence {a_i}$ for which $a_i \\ne p$ are all equal."}
+{"_id": "19094", "title": "Accumulation Points for Sequence in Particular Point Space", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Let $\\sequence {a_i}$ be an infinite sequence in $T$. Let $\\beta$ be an accumulation point of $\\sequence {a_i}$. Then $\\beta$ is such that an infinite number of terms of $\\sequence {a_i}$ are equal either to $\\beta$ or to $p$."}
+{"_id": "19095", "title": "Absolute Value of Absolutely Continuous Function is Absolutely Continuous", "text": "Let $I \\subseteq \\R$ be a real interval. Let $f : I \\to \\R$ be an absolutely continuous function. Then $\\size f$ is absolutely continuous."}
+{"_id": "19096", "title": "Limit Points in Particular Point Space/Subset/Proof 2", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. {{:Limit Points in Particular Point Space/Subset}}"}
+{"_id": "19097", "title": "Limit Points in Particular Point Space/Subset/Proof 1", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. {{:Limit Points in Particular Point Space/Subset}}"}
+{"_id": "19098", "title": "Sum of Absolutely Continuous Functions is Absolutely Continuous", "text": "Let $I \\subseteq \\R$ be a real interval. Let $f, g : I \\to \\R$ be absolutely continuous functions. Then $f + g$ is absolutely continuous."}
+{"_id": "19099", "title": "Norms on Finite-Dimensional Real Vector Space are Equivalent", "text": "Norms on finite-dimensional real vector space are equivalent."}
+{"_id": "19100", "title": "Integral of Distribution Function", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space and $f$ be a $\\mu$-measurable function. Let $p > 0, r \\geq 0$. For $\\lambda > 0$, let $E_\\lambda = \\set {x \\in X: \\size {\\map f x} > \\lambda}$, so that $\\map m \\lambda = \\map \\mu {E_\\lambda}$ is the distribution function of $f$. Then: :$\\displaystyle \\int_0^\\infty p \\lambda^{p - 1} \\int_{E_\\lambda} \\size f^r \\rd \\mu \\rd \\lambda = \\int_X \\size f^{p + r} \\rd \\mu$ and in particular: :$\\displaystyle \\int_0^\\infty p \\lambda^{p - 1} \\map m \\lambda \\rd \\lambda = \\int_X \\size f^p \\rd \\mu$"}
+{"_id": "19103", "title": "Either-Or Topology is Compact", "text": "Let $T = \\struct {S, \\tau}$ be the either-or space. Then $T$ is a compact space."}
+{"_id": "19105", "title": "Compact Space is Lindelöf", "text": "Every compact space is Lindelöf."}
+{"_id": "19112", "title": "Limit Point of Underlying Set of Sequence of Reciprocals and Reciprocals + 1", "text": "Let $\\sequence {a_n}$ denote the sequence defined as: {{begin-eqn}} {{eqn | l = a_n       | r = \\begin {cases} \\dfrac 2 {n + 1} & : \\text {$n$ odd} \\\\ 1 + \\dfrac 2 n & : \\text {$n$ even} \\end {cases}       | c =  }} {{eqn | r = \\sequence {\\dfrac 1 1, 1 + \\dfrac 1 1, \\dfrac 1 2, 1 + \\dfrac 1 2, \\dfrac 1 3, 1 + \\dfrac 1 3, \\dotsb}       | c =  }} {{end-eqn}} Let $\\struct {\\R, \\tau}$ denote the real number line under the usual (Euclidean) topology. Let $S$ denote the set of terms of $\\sequence {a_n}$ considered as a subset of $\\struct {\\R, \\tau_d}$. Then $0$ is a limit point of $S$."}
+{"_id": "19113", "title": "Omega-Accumulation Point of Underlying Set of Sequence of Reciprocals and Reciprocals + 1", "text": "Let $\\sequence {a_n}$ denote the sequence defined as: {{begin-eqn}} {{eqn | l = a_n       | r = \\begin {cases} \\dfrac 2 {n + 1} & : \\text {$n$ odd} \\\\ 1 + \\dfrac 2 n & : \\text {$n$ even} \\end {cases}       | c =  }} {{eqn | r = \\sequence {\\dfrac 1 1, 1 + \\dfrac 1 1, \\dfrac 1 2, 1 + \\dfrac 1 2, \\dfrac 1 3, 1 + \\dfrac 1 3, \\dotsb}       | c =  }} {{end-eqn}} Let $\\struct {\\R, \\tau}$ denote the real number line under the usual (Euclidean) topology. Let $S$ denote the set of terms of $\\sequence {a_n}$ considered as a subset of $\\struct {\\R, \\tau_d}$. Then $0$ is an $\\omega$-accumulation point of $S$."}
+{"_id": "19114", "title": "Accumulation Point of Sequence of Reciprocals and Reciprocals + 1", "text": "Let $\\struct {\\R, \\tau}$ denote the real number line under the usual (Euclidean) topology. Let $\\sequence {a_n}$ denote the sequence in $\\struct {\\R, \\tau}$ defined as: {{begin-eqn}} {{eqn | l = a_n       | r = \\begin {cases} \\dfrac 2 {n + 1} & : \\text {$n$ odd} \\\\ 1 + \\dfrac 2 n & : \\text {$n$ even} \\end {cases}       | c =  }} {{eqn | r = \\sequence {\\dfrac 1 1, 1 + \\dfrac 1 1, \\dfrac 1 2, 1 + \\dfrac 1 2, \\dfrac 1 3, 1 + \\dfrac 1 3, \\dotsb}       | c =  }} {{end-eqn}} Then $0$ is an accumulation point of $\\sequence {a_n}$."}
+{"_id": "19115", "title": "Zero is not a Limit Point of Sequence of Reciprocals and Reciprocals + 1", "text": "Let $\\struct {\\R, \\tau}$ denote the real number line under the usual (Euclidean) topology. Let $\\sequence {a_n}$ denote the sequence in $\\struct {\\R, \\tau}$ defined as: {{begin-eqn}} {{eqn | l = a_n       | r = \\begin {cases} \\dfrac 2 {n + 1} & : \\text {$n$ odd} \\\\ 1 + \\dfrac 2 n & : \\text {$n$ even} \\end {cases}       | c =  }} {{eqn | r = \\sequence {\\dfrac 1 1, 1 + \\dfrac 1 1, \\dfrac 1 2, 1 + \\dfrac 1 2, \\dfrac 1 3, 1 + \\dfrac 1 3, \\dotsb}       | c =  }} {{end-eqn}} Then $0$ is not a limit point of $\\sequence {a_n}$."}
+{"_id": "19117", "title": "1 plus Power of 2 is not Perfect Power except 9", "text": "The only solution to: :$1 + 2^n = a^b$ is: :$\\tuple {n, a, b} = \\tuple {3, 3, 2}$ for positive integers $n, a, b$ with $b > 1$."}
+{"_id": "19119", "title": "Fermat Number is not Perfect Power", "text": "There exist no Fermat numbers which are perfect powers."}
+{"_id": "19121", "title": "Uncountable Closed Ordinal Space is Countably Compact", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\closedint 0 \\Omega$ denote the closed ordinal space on $\\Omega$. Then $\\closedint 0 \\Omega$ is a countably compact space."}
+{"_id": "19122", "title": "Sum of Euler Numbers by Binomial Coefficients Vanishes", "text": "$\\forall n \\in \\Z_{>0}: \\displaystyle \\sum_{k \\mathop = 0}^n \\binom {2 n} {2 k} E_{2 k} = 0$ where $E_k$ denotes the $k$th Euler number. == Corollary == Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = E_{2 n}       | r = -\\sum_{k \\mathop = 0}^{n - 1} \\dbinom {2 n} {2 k} E_{2 k}       | c =  }} {{eqn | r = -\\paren {\\binom {2 n} 0 E_0 + \\binom {2 n} 2 E_2 + \\binom {2 n} 4 E_4 + \\cdots + \\binom {2 n} {2 n - 2} E_{2 n - 2} }       | c =  }} {{end-eqn}} where $E_n$ denotes the $n$th Euler number."}
+{"_id": "19123", "title": "Uncountable Open Ordinal Space is Countably Compact", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\hointr 0 \\Omega$ denote the open ordinal space on $\\Omega$. Then $\\hointr 0 \\Omega$ is a countably compact space."}
+{"_id": "19124", "title": "Uncountable Open Ordinal Space is not Metacompact", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\hointr 0 \\Omega$ denote the open ordinal space on $\\Omega$. Then $\\hointr 0 \\Omega$ is not a metacompact space."}
+{"_id": "19125", "title": "Uncountable Open Ordinal Space is not Paracompact", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\hointr 0 \\Omega$ denote the open ordinal space on $\\Omega$. Then $\\hointr 0 \\Omega$ is not a paracompact space."}
+{"_id": "19126", "title": "Uncountable Open Ordinal Space is not Lindelöf", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\hointr 0 \\Omega$ denote the open ordinal space on $\\Omega$. Then $\\hointr 0 \\Omega$ is not a Lindelöf space."}
+{"_id": "19127", "title": "Uncountable Open Ordinal Space is not Sigma-Compact", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\hointr 0 \\Omega$ denote the open ordinal space on $\\Omega$. Then $\\hointr 0 \\Omega$ is not a $\\sigma$-compact space."}
+{"_id": "19128", "title": "Uncountable Closed Ordinal Space is Lindelöf", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\closedint 0 \\Omega$ denote the closed ordinal space on $\\Omega$. Then $\\closedint 0 \\Omega$ is a Lindelöf space."}
+{"_id": "19130", "title": "Uncountable Open Ordinal Space is Sequentially Compact", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\hointr 0 \\Omega$ denote the open ordinal space on $\\Omega$. Then $\\hointr 0 \\Omega$ is a sequentially compact space."}
+{"_id": "19131", "title": "Sum of Euler Numbers by Binomial Coefficients Vanishes/Corollary", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = E_{2 n}       | r = -\\sum_{k \\mathop = 0}^{n - 1} \\dbinom {2 n} {2 k} E_{2 k}       | c =  }} {{eqn | r = -\\paren {\\binom {2 n} 0 E_0 + \\binom {2 n} 2 E_2 + \\binom {2 n} 4 E_4 + \\cdots + \\binom {2 n} {2 n - 2} E_{2 n - 2} }       | c =  }} {{end-eqn}} where $E_n$ denotes the $n$th Euler number."}
+{"_id": "19132", "title": "Integer to Power of Multiple of Order/Corollary", "text": "Then $\\map \\phi n$ is a multiple of $c$, where $\\map \\phi n$ is the Euler phi function of $n$."}
+{"_id": "19133", "title": "Divisor of Fermat Number/Euler's Result", "text": "Then $m$ is in the form: :$k \\, 2^{n + 1} + 1$ where $k \\in \\Z_{>0}$ is an integer."}
+{"_id": "19134", "title": "Divisor of Fermat Number/Refinement by Lucas", "text": "Let $n \\ge 2$. Then $m$ is in the form: :$k \\, 2^{n + 2} + 1$"}
+{"_id": "19135", "title": "Definition:Double Pointed Real Number Line", "text": "Let $T_\\R = \\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $T_D = \\struct {D, \\tau_D}$ be the indiscrete topology on the doubleton $D = \\set {a, b}$. Let $T = T_\\R \\times T_D$ be theproduct space of $T_\\R$ and $T_D$. $T$ is known as the '''double pointed real number line'''."}
+{"_id": "19136", "title": "Set Closure is Smallest Closed Set/Normed Vector Space", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,} }$ be a normed vector space. Let $S$ be a subset of $X$: :$S \\subseteq X$ Let $S^-$ be the closure of $S$. Then $S^-$ is the smallest closed set which contains $S$."}
+{"_id": "19137", "title": "Bounded Real Function may not be of Bounded Variation", "text": "Let $a, b$ be real numbers with $a < b$. Let $f : \\closedint a b \\to \\R$ be a bounded function. Then $f$ is not necessarily of bounded variation."}
+{"_id": "19138", "title": "Element is Loop iff Member of Closure of Empty Set", "text": "Let $M = \\struct{S, \\mathscr I}$ be a matroid. Let $x \\in S$. Then: :$x$ is a loop {{iff}} $x \\in \\map \\sigma \\O$ where $\\map \\sigma \\O$ denotes the closure of the empty set."}
+{"_id": "19139", "title": "Singleton is Dependent implies Rank is Zero/Corollary", "text": ":$x$ is a loop {{iff}} $\\map \\rho {\\set x} = 0$"}
+{"_id": "19140", "title": "Superset of Dependent Set is Dependent/Corollary", "text": "Let $A \\subseteq S$. Let $x \\in A$. If $x$ is a loop then $A$ is dependent."}
+{"_id": "19141", "title": "Closure of Subset contains Loop", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $x$ be a loop of $M$. Let $A \\subseteq S$. Then: :$x \\in \\map \\sigma A$ where $\\map \\sigma A$ denotes the closure of $A$."}
+{"_id": "19142", "title": "Element is Loop iff Singleton is Circuit", "text": "Let $M = \\struct{S, \\mathscr I}$ be a matroid. Let $x \\in S$. Then: :$x$ is a loop {{iff}} $\\set x$ is a circuit"}
+{"_id": "19143", "title": "Element is Member of Base iff Not Loop", "text": "Let $M = \\struct{S, \\mathscr I}$ be a matroid. Let $\\mathscr B$ denote the set of all bases of $M$. Let $x \\in S$. Then: :$\\exists B \\in \\mathscr B: x \\in B$ {{iff}} $x$ is not a loop"}
+{"_id": "19145", "title": "Parallel Relationship is Transitive", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $x, y, z \\in S : x \\ne y, x \\ne z, y \\ne z$. If $x$ is parallel to $y$ and $y$ is parallel to $z$ then $x$ is parallel to $z$."}
+{"_id": "19146", "title": "Distinct Matroid Elements are Parallel iff Each is in Closure of Other", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $\\sigma: \\powerset S \\to \\powerset S$ denote the closure operator of $M$. Let $x, y \\in S : x \\ne y$. Then $x$ is parallel to $y$ {{iff}}: :$(1)\\quad x$ and $y$ are not loops :$(2)\\quad x \\in \\map \\sigma {\\set y}$ :$(3)\\quad y \\in \\map \\sigma {\\set x}$"}
+{"_id": "19147", "title": "Closure of Subset Contains Parallel Elements", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $\\sigma: \\powerset S \\to \\powerset S$ denote the closure operator of $M$. Let $A \\subseteq S$. Let $x, y \\in S$. If $x \\in \\map \\sigma A$ and $y$ is parallel to $x$ then: :$y \\in \\map \\sigma A$"}
+{"_id": "19149", "title": "Loop Belongs to Every Flat", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $A \\subseteq S$. Let $x \\in S$. If $x$ is a loop and $A$ is a flat subset then $x \\in A$."}
+{"_id": "19150", "title": "Parallel Elements Depend on Same Subsets", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $A \\subseteq S$. Let $x, y \\in S$. Let $x$ be parallel to $y$. Then: :$x$ depends on $A$ {{iff}} $y$ depends on $A$"}
+{"_id": "19152", "title": "Absolutely Continuous Real Function is Uniformly Continuous", "text": "Let $I \\subseteq \\R$ be a real interval. Let $f : I \\to \\R$ be an absolutely continuous function. Then $f$ is uniformly continuous."}
+{"_id": "19154", "title": "Closure of Convex Subset in Normed Vector Space is Convex", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $C \\subseteq X$ be a convex subset of $X$. Let $C^-$ be the closure of $C$. Then $C^- \\subseteq X$ is also a convex subset of $X$."}
+{"_id": "19155", "title": "Minimal Number of Distinct Prime Factors for Integer to have Abundancy Index Exceed Given Number", "text": "Let $r \\in \\R$. Let $\\mathbb P^-$ be the set of prime numbers with possibly finitely many numbers removed. Define: :$M = \\min \\set {m \\in \\N: \\displaystyle \\prod_{i \\mathop = 1}^m \\frac {p_i} {p_i - 1} > r}$ where $p_i$ is the $i$th element of $\\mathbb P^-$, ordered by size. Then $M$ satisfies: :$(1): \\quad$ Every number formed with fewer than $M$ distinct prime factors in $\\mathbb P^-$ has abundancy index less than $r$ :$(2): \\quad$ There exists some number formed with $M$ distinct prime factors in $\\mathbb P^-$ with abundancy index at least $r$ So $M$ is the minimal number of distinct prime factors in $\\mathbb P^-$ a number must have for it to have abundancy index at least $r$. For $r$ an integer greater than $1$: If $\\mathbb P^-$ is taken to be the set of all prime numbers, the values of $M$ are: :$2, 3, 4, 6, 9, 14, 22, 35, 55, 89, 142, \\cdots$ {{OEIS|A005579}} This theorem shows that this sequence is a subsequence of the sequence A256969 in the OEIS, only differing by an offset. If we require the numbers to be odd, we remove $2$ from $\\mathbb P^-$. The sequence of values of $M$ are: :$3, 8, 21, 54, 141, 372, 995, 2697, 7397, 20502, \\cdots$ {{OEIS|A005580}}"}
+{"_id": "19156", "title": "1 plus Square is not Perfect Power", "text": "The equation: :$x^p = y^2 + 1$ has no solution in the integers for $x, y, p > 1$."}
+{"_id": "19157", "title": "1 plus Perfect Power is not Prime Power except for 9", "text": "The only solution to: :$x^m = y^n + 1$ is: :$\\tuple {x, m, y, n} = \\tuple {3, 2, 2, 3}$ for positive integers $x, y, m, n > 1$, and $x$ is a prime number. This is a special case of Catalan's Conjecture."}
+{"_id": "19158", "title": "Field is Principal Ideal Domain", "text": "Let $F$ be a field. Then $F$ is a principal ideal domain."}
+{"_id": "19159", "title": "Complex Vector Space is Vector Space", "text": "Let $\\C$ denote the set of complex numbers. Then the complex vector space $\\C^n$ is a vector space."}
+{"_id": "19161", "title": "Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Necessary Condition", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,}}$ is a normed vector space. Let $D \\subseteq X$ be a subset of $X$. Let $D^-$ be the closure of $D$. Then $D$ is dense iff $D^- = X$."}
+{"_id": "19162", "title": "Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Sufficient Condition", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,}}$ is a normed vector space. Let $D \\subseteq X$ be a subset of $X$. Let $D^-$ be the closure of $D$. Then $D$ is dense iff $D^- = X$."}
+{"_id": "19163", "title": "Modulus of Limit/Normed Vector Space", "text": "Let $\\struct {X, \\norm { \\, \\cdot \\, } }$ be a normed vector space. Let $\\sequence {x_n}$ be a convergent sequence in $R$ to the limit $x$. That is, let $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = x$. Then :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\norm {x_n} = \\norm x$"}
+{"_id": "19164", "title": "Matrix Entrywise Addition forms Abelian Group", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Let $\\map {\\MM_R} {m, n}$ be a $m \\times n$ matrix space over $\\struct {R, +, \\circ}$. Then $\\struct {\\map {\\MM_R} {m, n}, +}$, where $+$ is matrix entrywise addition, is a group."}
+{"_id": "19165", "title": "Definition:Negative Matrix/General Group", "text": "Let $\\struct {G, \\cdot}$ be a group. Let $\\map {\\MM_G} {m, n}$ denote the $m \\times n$ matrix space over $\\struct {G, \\cdot}$. Let $\\mathbf A = \\sqbrk a_{m n}$ be an element of $\\struct {\\map {\\MM_G} {m, n}, \\circ}$, where $\\circ$ is the Hadamard product. Then the '''negative (matrix) of $\\mathbf A = \\sqbrk a_{m n}$''' is denoted and defined as: :$-\\mathbf A := \\sqbrk {a^{-1} }_{m n}$ where $a^{-1}$ is the inverse element of $a \\in G$."}
+{"_id": "19166", "title": "Negative Matrix is Inverse for Matrix Entrywise Addition over Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Let $\\map {\\MM_R} {m, n}$ be a $m \\times n$ matrix space over $\\struct {R, +, \\circ}$. Let $\\mathbf A$ be an element of $\\map {\\MM_R} {m, n}$. Let $-\\mathbf A$ be the negative of $\\mathbf A$. Then $-\\mathbf A$ is the inverse for the operation $+$, where $+$ is matrix entrywise addition."}
+{"_id": "19167", "title": "Zero Matrix is Identity for Matrix Entrywise Addition over Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\map {\\MM_R} {m, n}$ be a $m \\times n$ matrix space over $R$. Let $\\mathbf 0_R = \\sqbrk {0_R}_{m n}$ be the zero matrix of $\\map {\\MM_R} {m, n}$. Then $\\mathbf 0_R$ is the identity element for matrix entrywise addition."}
+{"_id": "19168", "title": "Properties of Matrix Entrywise Addition over Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Let $\\map {\\MM_R} {m, n}$ be a $m \\times n$ matrix space over $S$ over an algebraic structure $\\struct {R, +, \\circ}$. Let $\\mathbf A, \\mathbf B \\in \\map {\\MM_R} {m, n}$. Let $\\mathbf A + \\mathbf B$ be defined as the matrix entrywise sum of $\\mathbf A$ and $\\mathbf B$. The operation of matrix entrywise addition satisfies the following properties: :$+$ is closed on $\\map {\\MM_R} {m, n}$ :$+$ is associative on $\\map {\\MM_R} {m, n}$ :$+$ is commutative on $\\map {\\MM_R} {m, n}$."}
+{"_id": "19171", "title": "Matrix Entrywise Addition is Associative", "text": "Let $\\map \\MM {m, n}$ be a $m \\times n$ matrix space over one of the standard number systems. For $\\mathbf A, \\mathbf B \\in \\map \\MM {m, n}$, let $\\mathbf A + \\mathbf B$ be defined as the matrix entrywise sum of $\\mathbf A$ and $\\mathbf B$. The operation $+$ is associative on $\\map \\MM {m, n}$. That is: :$\\paren {\\mathbf A + \\mathbf B} + \\mathbf C = \\mathbf A + \\paren {\\mathbf B + \\mathbf C}$ for all $\\mathbf A$, $\\mathbf B$ and $\\mathbf C$ in $\\map \\MM {m, n}$."}
+{"_id": "19172", "title": "Matrix Entrywise Addition over Ring is Commutative", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\map {\\MM_R} {m, n}$ be a $m \\times n$ matrix space over $R$. For $\\mathbf A, \\mathbf B \\in \\map {\\MM_R} {m, n}$, let $\\mathbf A + \\mathbf B$ be defined as the matrix entrywise sum of $\\mathbf A$ and $\\mathbf B$. The operation $+$ is commutative on $\\map {\\MM_R} {m, n}$. That is: :$\\mathbf A + \\mathbf B = \\mathbf B + \\mathbf A$ for all $\\mathbf A$ and $\\mathbf B$ in $\\map {\\MM_R} {m, n}$."}
+{"_id": "19178", "title": "Zero Matrix is Identity for Matrix Entrywise Addition", "text": "Let $\\map \\MM {m, n}$ be a $m \\times n$ matrix space over one of the standard number systems. Let $\\mathbf 0 = \\sqbrk 0_{m n}$ be the zero matrix of $\\map \\MM {m, n}$. Then $\\mathbf 0$ is the identity element for matrix entrywise addition."}
+{"_id": "19185", "title": "Matrix Scalar Product is Associative", "text": "Let $\\Bbb F$ denote one of the standard number systems. Let $\\map \\MM {m, n}$ be a $m \\times n$ matrix space over $\\Bbb F$. For $\\mathbf A \\in \\map \\MM {m, n}$ and $\\lambda$ \\in $\\Bbb F$, let $\\lambda \\mathbf A$ be defined as the matrix scalar product of $\\lambda$ and $\\mathbf A$. The matrix scalar product is associative on $\\map \\MM {m, n}$, in the following sense: For all $\\mathbf A$ in $\\map \\MM {m, n}$ and $\\lambda, \\mu \\in \\Bbb F$: :$\\lambda \\paren {\\mu \\mathbf A} = \\paren {\\lambda \\mu} \\mathbf A$"}
+{"_id": "19186", "title": "Matrix Scalar Product Distributes over Number Addition", "text": "Let $\\Bbb F$ denote one of the standard number systems. Let $\\map \\MM {m, n}$ be a $m \\times n$ matrix space over $\\Bbb F$. For $\\mathbf A \\in \\map \\MM {m, n}$ and $\\lambda$ \\in $\\Bbb F$, let $\\lambda \\mathbf A$ be defined as the matrix scalar product of $\\lambda$ and $\\mathbf A$. The matrix scalar product is associative on $\\map \\MM {m, n}$, in the following sense: For all $\\mathbf A$ in $\\map \\MM {m, n}$ and $\\lambda, \\mu \\in \\Bbb F$: :$\\paren {\\lambda + \\mu} \\mathbf A = \\lambda \\mathbf A + \\mu \\mathbf A$"}
+{"_id": "19187", "title": "Negative Matrix is Inverse for Matrix Entrywise Addition", "text": "Let $\\Bbb F$ denote one of the standard number systems. Let $\\map \\MM {m, n}$ be a $m \\times n$ matrix space over $\\Bbb F$. Let $\\mathbf A$ be an element of $\\map \\MM {m, n}$. Let $-\\mathbf A$ be the negative of $\\mathbf A$. Then $-\\mathbf A$ is the inverse for the operation $+$, where $+$ is matrix entrywise addition."}
+{"_id": "19188", "title": "Zero Matrix is Zero for Matrix Multiplication", "text": "Let $\\struct {R, +, \\times}$ be a ring. Let $\\mathbf A$ be a matrix over $R$ of order $m \\times n$ Let $\\mathbf 0$ be a zero matrix whose order is such that either: :$\\mathbf 0 \\mathbf A$ is defined or: :$\\mathbf A \\mathbf 0$ is defined or both. Then: :$\\mathbf 0 \\mathbf A = \\mathbf 0$ or: :$\\mathbf A \\mathbf 0 = \\mathbf 0$ whenever they are defined. The order of $\\mathbf 0$ will be according to the orders of the factor matrices."}
+{"_id": "19189", "title": "Unit Matrix is Identity for Matrix Multiplication", "text": "Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\map {\\MM_R} n$ denote the metric space of square matrices of order $n$ over $R$. Let $\\mathbf I_n$ denote the unit matrix of order $n$: Then: :$\\forall \\mathbf A \\in \\map {\\MM_R} n: \\mathbf A \\mathbf I_n = \\mathbf A = \\mathbf I_n \\mathbf A$ That is, the unit matrix $\\mathbf I_n$ is the identity element for (conventional) matrix multiplication over $\\map {\\MM_R} n$."}
+{"_id": "19190", "title": "Unit Matrix is Identity for Matrix Multiplication/Left", "text": "Let $\\map {\\MM_R} {m, n}$ denote the $m \\times n$ metric space over $R$. Let $I_m$ denote the unit matrix of order $m$. Then: :$\\forall \\mathbf A \\in \\map {\\MM_R} {m, n}: \\mathbf I_m \\mathbf A  = \\mathbf A$"}
+{"_id": "19191", "title": "Unit Matrix is Identity for Matrix Multiplication/Right", "text": "Let $\\map {\\MM_R} {m, n}$ denote the $m \\times n$ metric space over $R$. Let $I_n$ denote the unit matrix of order $n$. Then: :$\\forall \\mathbf A \\in \\map {\\MM_R} {m, n}: \\mathbf A \\mathbf I_n  = \\mathbf A$"}
+{"_id": "19193", "title": "Inverse of Square Matrix over Field is Unique", "text": "Let $\\Bbb F$ be a field, usually one of the standard number fields $\\Q$, $\\R$ or $\\C$. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\map \\MM n$ denote the matrix space of order $n$ square matrices over $\\Bbb F$. Let $\\mathbf B$ be an inverse matrix of $\\mathbf A$. Then $\\mathbf B$ is the only inverse matrix of $\\mathbf A$."}
+{"_id": "19194", "title": "Rank of Empty Set is Zero", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $\\rho : \\powerset S \\to \\Z$ be the rank function of $M$. Then: :$\\map \\rho \\O = 0$"}
+{"_id": "19195", "title": "Rank Function is Increasing", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $\\rho: \\powerset S \\to \\Z$ be the rank function of $M$. Let $A, B \\subseteq S$ be subsets of $S$ such that $A \\subseteq B$. Then: :$\\map \\rho A \\le \\map \\rho B$"}
+{"_id": "19198", "title": "Inverse of Transpose of Matrix is Transpose of Inverse", "text": "Let $\\mathbf A$ be a matrix over a field. Let $\\mathbf A^\\intercal$ denote the transpose of $\\mathbf A$. Let $\\mathbf A$ be an invertible matrix. Then $\\mathbf A^\\intercal$ is also invertible and: :$\\paren {\\mathbf A^\\intercal}^{-1} = \\paren {\\mathbf A^{-1} }^\\intercal$ where $\\mathbf A^{-1}$ denotes the inverse of $\\mathbf A$."}
+{"_id": "19199", "title": "Max yields Supremum of Parameters/General Case", "text": "Let $x_1, x_2, \\dots ,x_n \\in S$ for some $n \\in \\N_{>0}$. Then: :$\\max \\set {x_1, x_2, \\dotsc, x_n} = \\sup \\set {x_1, x_2, \\dotsc, x_n}$"}
+{"_id": "19200", "title": "Max of Subfamily of Operands Less or Equal to Max", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. Let $x_1, x_2, \\dotsc, x_n \\in S$ for some $n \\in \\N_{>0}$. Let $\\set{k_1, k_2, \\dotsc, k_m} \\subseteq \\set{1, 2, \\dotsc, n}$ Then: :$\\max \\set {x_{k_1}, x_{k_2}, \\dotsc, x_{k_m}} \\preceq \\max \\set {x_1, x_2, \\dotsc, x_n}$ where: :$\\max$ denotes the max operation"}
+{"_id": "19201", "title": "Elementary Matrix corresponding to Elementary Row Operation", "text": "Let $\\mathbf I$ denote the unit matrix of order $m$ over a field $K$. Let $e$ be an elementary row operation on $\\mathbf I$. Let $\\mathbf E$ be the elementary row matrix of order $m$ uniquely defined as: :$\\mathbf E = e \\paren {\\mathbf I}$ where $\\mathbf I$ is the unit matrix. Let $r_k$ denote the $k$th row of $\\mathbf I$ for $1 \\le k \\le m$."}
+{"_id": "19202", "title": "Elementary Matrix corresponding to Elementary Row Operation/Scale Row", "text": "Let $e$ be the elementary row operation acting on $\\mathbf I$ as: {{begin-axiom}} {{axiom | n = \\text {ERO} 1         | t = For some $\\lambda \\in K_{\\ne 0}$, multiply row $k$ of $\\mathbf I$ by $\\lambda$         | m = r_k \\to \\lambda r_k }} {{end-axiom}}"}
+{"_id": "19203", "title": "Elementary Matrix corresponding to Elementary Row Operation/Scale Row and Add", "text": "Let $e$ be the elementary row operation acting on $\\mathbf I$ as: {{begin-axiom}} {{axiom | n = \\text {ERO} 2         | t = For some $\\lambda \\in K$, add $\\lambda$ times row $j$ to row $i$         | m = r_i \\to r_i + \\lambda r_j }} {{end-axiom}}"}
+{"_id": "19204", "title": "Elementary Matrix corresponding to Elementary Row Operation/Exchange Rows", "text": "Let $e$ be the elementary row operation acting on $\\mathbf I$ as: {{begin-axiom}} {{axiom | n = \\text {ERO} 3         | t = Interchange rows $i$ and $j$         | m = r_i \\leftrightarrow r_j }} {{end-axiom}}"}
+{"_id": "19205", "title": "Power Set of Singleton", "text": "Let $x$ be an object. Then the power set of the singleton $\\set x$ is: :$\\powerset {\\set x} = \\set {\\O, \\set x}$"}
+{"_id": "19206", "title": "Row Operation to Clear First Column of Matrix", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be an $m \\times n$ matrix over a field $K$. Then there exists a row operation to convert $\\mathbf A$ into another $m \\times n$ matrix $\\mathbf B = \\sqbrk b_{m n}$ with the following properties: :$(1): \\quad$ Except possibly for element $b_{1 1}$, all the elements of column $1$ are $0$ :$(2): \\quad$ If $b_{1 1} \\ne 0$, then $b_{1 1} = 1$. This process is referred to as '''clearing the first column'''."}
+{"_id": "19208", "title": "Matrix is Row Equivalent to Echelon Matrix", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be a matrix of order $m \\times n$ over a field $F$. Then $A$ is row equivalent to an echelon matrix of order $m \\times n$."}
+{"_id": "19210", "title": "Singleton is Independent implies Rank is One", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $x \\in S$. Let $\\set x$ be independent. Then: :$\\map \\rho {\\set x} = 1$ where $\\rho$ denotes the rank function of $M$."}
+{"_id": "19211", "title": "Singleton is Dependent implies Rank is Zero", "text": ":$\\map \\rho {\\set x} = 0$"}
+{"_id": "19212", "title": "System of Simultaneous Equations may have No Solution", "text": "Let $S$ be a system of simultaneous equations. Then it is possible that $S$ may have a solution set which is empty."}
+{"_id": "19213", "title": "System of Simultaneous Equations may have Unique Solution", "text": "Let $S$ be a system of simultaneous equations. Then it is possible that $S$ may have a solution set which is a singleton."}
+{"_id": "19215", "title": "Absolutely Convergent Series is Convergent iff Normed Vector Space is Banach", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be an absolutely convergent series in $X$. Then $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ is convergent {{iff}} $X$ is a Banach space."}
+{"_id": "19216", "title": "Absolutely Convergent Series is Convergent iff Normed Vector Space is Banach/Necessary Condition", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be an absolutely convergent series in $X$. Suppose $X$ is a Banach space. Then $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ is convergent."}
+{"_id": "19217", "title": "Absolutely Convergent Series is Convergent iff Normed Vector Space is Banach/Sufficient Condition", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $\\ds \\sum_{n \\mathop = 1}^\\infty a_n$ be an absolutely convergent series in $X$. Suppose $\\ds \\sum_{n \\mathop = 1}^\\infty a_n$ is convergent. Then $X$ is a Banach space."}
+{"_id": "19220", "title": "Trivial Solution to System of Homogeneous Simultaneous Linear Equations is Solution", "text": "Let $S$ be a '''system of homogeneous simultaneous linear equations''': :$\\displaystyle \\forall i \\in \\set {1, 2, \\ldots, m}: \\sum_{j \\mathop = 1}^n \\alpha_{i j} x_j = 0$ Consider the trivial solution to $A$: :$\\tuple {x_1, x_2, \\ldots, x_n}$ such that: :$\\forall j \\in \\set {1, 2, \\ldots, n}: x_j = 0$ Then the trivial solution is indeed a solution to $S$."}
+{"_id": "19221", "title": "Sine of Integer Multiple of Argument/Formulation 3", "text": "{{begin-eqn}} {{eqn | l = \\sin n \\theta       | r = \\sin \\theta \\cos^{n - 1} \\theta \\paren {1 + 1 + \\frac {\\cos 2 \\theta} {\\cos^2 \\theta} + \\frac {\\cos 3 \\theta} {\\cos^3 \\theta} + \\cdots + \\frac {\\cos \\paren {n - 1} \\theta} {\\cos^{n - 1} \\theta} }       | c =  }} {{eqn | r = \\sin \\theta \\cos^{n - 1} \\theta \\sum_{k \\mathop = 0}^{n - 1} \\frac {\\cos k \\theta} {\\cos^k \\theta}       | c =  }} {{end-eqn}}"}
+{"_id": "19222", "title": "Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations", "text": "Let $S$ be a system of simultaneous linear equations: :$\\displaystyle \\forall i \\in \\set {1, 2, \\ldots, m}: \\sum_{j \\mathop = 1}^n \\alpha_{i j} x_j = \\beta_i$ Let $\\begin {pmatrix} \\mathbf A & \\mathbf b \\end {pmatrix}$ denote the augmented matrix of $S$. Let $\\begin {pmatrix} \\mathbf A' & \\mathbf b' \\end {pmatrix}$ be obtained from $\\begin {pmatrix} \\mathbf A & \\mathbf b \\end {pmatrix}$ by means of an elementary row operation. Let $S'$ be the system of simultaneous linear equations of which $\\begin {pmatrix} \\mathbf A' & \\mathbf b' \\end {pmatrix}$ is the augmented matrix. Then $S$ and $S'$ are equivalent."}
+{"_id": "19223", "title": "Existence of Inverse Elementary Row Operation", "text": "Let $\\map \\MM {m, n}$ be a metric space of order $m \\times n$ over a field $K$. Let $\\mathbf A \\in \\map \\MM {m, n}$ be a matrix.  Let $\\map e {\\mathbf A}$ be an elementary row operation which transforms $\\mathbf A$ to a new matrix $\\mathbf A' \\in \\map \\MM {m, n}$. Let $\\map {e'} {\\mathbf A'}$ be the inverse of $e$. Then $e'$ is an elementary row operation which always exists and is unique."}
+{"_id": "19224", "title": "Superset of Dependent Set is Dependent", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $A, B \\subseteq S$ such that $A \\subseteq B$ If $A$ is a dependent subset then $B$ is a dependent subset."}
+{"_id": "19225", "title": "Powers of 16 Modulo 20", "text": "Let $n \\in \\Z_{> 0}$ be a strictly positive integer. Then: :$16^n \\equiv 16 \\pmod {20}$"}
+{"_id": "19227", "title": "Sine of Integer Multiple of Argument/Formulation 1/Lemma", "text": ":For $n \\in \\Z$: {{begin-eqn}} {{eqn | l = \\map \\cos {n \\theta} \\map \\sin {\\theta}        | r = \\map \\sin {n \\theta} \\map \\cos {\\theta} - \\map \\sin {\\paren {n - 1 } \\theta}  }} {{end-eqn}}"}
+{"_id": "19231", "title": "Conjugacy Class of Identity is only Conjugacy Class which is Subgroup", "text": "Let $G$ be a group. Let $e$ denote the identity of $G$. Let $\\conjclass g$ denote the conjugacy class of the element $g$. Then conjugacy class of identity is the only conjugacy class which is a subgroup of $G$: :$\\conjclass g < G \\iff g = e$"}
+{"_id": "19232", "title": "Empty Group Word is Reduced", "text": "Let $S$ be a set Let $\\epsilon$ be the empty group word on $S$. Then $\\epsilon$ is reduced."}
+{"_id": "19234", "title": "Closed Unit Ball is Convex Set", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,} }$ be a normed vector space. Let $\\map {B_1^-} 0$ be a closed unit ball in $X$. Then $\\map {B_1^-} 0$ is convex."}
+{"_id": "19236", "title": "Uncountable Sum as Series/Corollary", "text": "Let $f: X \\to \\closedint 0 {+\\infty}$ have uncountably infinite support. Then: :$\\displaystyle \\sum_{x \\mathop \\in X} \\map f x = +\\infty$"}
+{"_id": "19237", "title": "Identity Matrix from Upper Triangular Matrix", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be an upper triangular matrix of order $m \\times n$ with no zero diagonal elements. Let $k = \\min \\set {m, n}$. Then $\\mathbf A$ can be transformed into a matrix such that the first $k$ rows and columns form the unit matrix of order $k$."}
+{"_id": "19238", "title": "Simultaneous Linear Equations have Solution iff Ranks of Matrix of Coefficients and Augmented Matrix are Equal", "text": "Let $S$ be a system of simultaneous linear equations: :$\\displaystyle \\forall i \\in \\set {1, 2, \\ldots, m} : \\sum_{j \\mathop = 1}^n \\alpha_{i j} x_j = \\beta_i$ Let $S$ be expressed in matrix form as: :$\\mathbf A \\mathbf x = \\mathbf b$ where: :$\\mathbf A = \\begin {pmatrix} \\alpha_{1 1} & \\alpha_{1 2} & \\cdots & \\alpha_{1 n} \\\\ \\alpha_{2 1} & \\alpha_{2 2} & \\cdots & \\alpha_{2 n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\alpha_{m 1} & \\alpha_{m 2} & \\cdots & \\alpha_{m n} \\\\ \\end {pmatrix}$,  $\\mathbf x = \\begin {pmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{pmatrix}$, $\\mathbf b = \\begin {pmatrix} \\beta_1 \\\\ \\beta_2 \\\\ \\vdots \\\\ \\beta_m \\end {pmatrix}$ Then $S$ has at least one solution {{iff}}: :$\\map \\rho {\\mathbf A} = \\map \\rho {\\begin {array} {c|c} \\mathbf A & \\mathbf b \\end {array} }$ where: :$\\map \\rho {\\mathbf A}$ denotes the rank of $\\mathbf A$ :$\\paren {\\begin {array} {c|c} \\mathbf A & \\mathbf b \\end {array} }$ denotes the augmented matrix of $S$."}
+{"_id": "19239", "title": "Simultaneous Linear Equations has Unique Solution iff Rank of Matrix of Coefficients equals Number of Columns", "text": "Let $S$ be a system of $m$ simultaneous linear equations in $n$ variables: :$\\displaystyle \\forall i \\in \\set {1, 2, \\ldots, m} : \\sum_{j \\mathop = 1}^n \\alpha_{i j} x_j = \\beta_i$ Let $S$ be expressed in matrix form as: :$\\mathbf A \\mathbf x = \\mathbf b$ where: :$\\mathbf A = \\begin {pmatrix} \\alpha_{1 1} & \\alpha_{1 2} & \\cdots & \\alpha_{1 n} \\\\ \\alpha_{2 1} & \\alpha_{2 2} & \\cdots & \\alpha_{2 n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\alpha_{m 1} & \\alpha_{m 2} & \\cdots & \\alpha_{m n} \\\\ \\end {pmatrix}$,  $\\mathbf x = \\begin {pmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{pmatrix}$, $\\mathbf b = \\begin {pmatrix} \\beta_1 \\\\ \\beta_2 \\\\ \\vdots \\\\ \\beta_m \\end {pmatrix}$ Then $S$ has exactly one solution {{iff}}: :$\\map \\rho {\\mathbf A} = n$ where $\\map \\rho {\\mathbf A}$ denotes the rank of $\\mathbf A$."}
+{"_id": "19240", "title": "Max Equals an Operand", "text": "Let $x_1, x_2, \\dotsc, x_n \\in S$ for some $n \\in \\N_{>0}$. Then: :$\\exists i \\in \\closedint 1 n : x_i = \\max \\set {x_1, x_2, \\dotsc, x_n}$"}
+{"_id": "19243", "title": "Similar Matrices have same Traces", "text": "Let $\\mathbf A = \\sqbrk a_n$ and $\\mathbf B = \\sqbrk b_n$ be square matrices of order $n$. Let $\\mathbf A$ and $\\mathbf B$ be similar. Then: :$\\map \\tr {\\mathbf A} = \\map \\tr {\\mathbf B}$ where $\\map \\tr {\\mathbf A}$ denotes the trace of $\\mathbf A$."}
+{"_id": "19245", "title": "Cosine of Integer Multiple of Argument/Formulation 2", "text": "{{begin-eqn}} {{eqn | l = \\cos n \\theta       | r = \\cos^n \\theta \\paren {1 - \\dbinom n 2 \\paren {\\tan \\theta}^2 + \\dbinom n 4 \\paren {\\tan \\theta}^4 - \\cdots}       | c =  }} {{eqn | r = \\cos^n \\theta \\sum_{k \\mathop \\ge 0} \\paren {-1}^k \\dbinom n {2 k } \\paren {\\tan^{2 k } \\theta}       | c =  }} {{end-eqn}}"}
+{"_id": "19246", "title": "Polygamma Reflection Formula", "text": ":$\\map {\\psi_n} z - \\paren {-1}^n \\map {\\psi_n} {1 - z} = -\\pi \\dfrac {\\d^n} {\\d z^n} \\cot \\pi z$"}
+{"_id": "19247", "title": "Polygamma Function in terms of Hurwitz Zeta Function", "text": ":$\\displaystyle \\map {\\psi_n} z = \\paren {-1}^{n + 1} \\map \\Gamma {n + 1} \\map \\zeta {n + 1, z}$"}
+{"_id": "19248", "title": "Area of Parallelogram from Determinant", "text": "Let $OABC$ be a parallelogram in the Cartesian plane whose vertices are located at: {{begin-eqn}} {{eqn | l = O       | r = \\tuple {0, 0} }} {{eqn | l = A       | r = \\tuple {a, c} }} {{eqn | l = B       | r = \\tuple {a + b, c + d} }} {{eqn | l = C       | r = \\tuple {b, d} }} {{end-eqn}} The area of $OABC$ is given by: :$\\map \\Area {OABC} = \\begin {vmatrix} a & b \\\\ c & d \\end {vmatrix}$ where $\\begin {vmatrix} a & b \\\\ c & d \\end {vmatrix}$ denotes the determinant of order $2$."}
+{"_id": "19249", "title": "Matrix is Invertible iff Rank equals Order", "text": "Let $R$ be a commutative ring with unity. Let $\\mathbf A \\in R^{n \\times n}$ be a square matrix of order $n$. Then $\\mathbf A$ is invertible {{iff}} its rank also equals $n$."}
+{"_id": "19250", "title": "Determinant of Upper Triangular Matrix", "text": "Let $\\mathbf T_n$ be an upper triangular matrix of order $n$. Let $\\map \\det {\\mathbf T_n}$ be the determinant of $\\mathbf T_n$. Then $\\map \\det {\\mathbf T_n}$ is equal to the product of all the diagonal elements of $\\mathbf T_n$. That is: :$\\displaystyle \\map \\det {\\mathbf T_n} = \\prod_{k \\mathop = 1}^n a_{k k}$"}
+{"_id": "19251", "title": "Cosine of Integer Multiple of Argument/Formulation 3", "text": "{{begin-eqn}} {{eqn | l = \\cos n \\theta       | r = \\cos \\paren {n - 1} \\theta \\cos \\theta + \\paren {1 - \\sec^2 \\theta } \\cos^n \\theta \\paren {1 + 1 + \\frac {\\cos 2 \\theta} {\\cos^2 \\theta} + \\frac {\\cos 3 \\theta} {\\cos^3 \\theta} + \\cdots + \\frac {\\cos \\paren {n - 2} \\theta} {\\cos^{n - 2} \\theta} }       | c =  }} {{eqn | r = \\cos \\paren {n - 1} \\theta \\cos \\theta + \\paren {1 - \\sec^2 \\theta } \\cos^n \\theta \\sum_{k \\mathop = 0}^{n - 2} \\frac {\\cos k \\theta} {\\cos^k \\theta}       | c =  }} {{end-eqn}}"}
+{"_id": "19252", "title": "Determinant of Elementary Row Matrix/Scale Row", "text": "Let $e_1$ be the elementary row operation $\\text {ERO} 1$: {{begin-axiom}} {{axiom | n = \\text {ERO} 1         | t = For some $\\lambda \\ne 0$, multiply row $k$ by $\\lambda$         | m = r_k \\to \\lambda r_k }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\\mathbf E_1$ be the elementary row matrix corresponding to $e_1$. The determinant of $\\mathbf E_1$ is: :$\\map \\det {\\mathbf E_1} = \\lambda$"}
+{"_id": "19253", "title": "Determinant of Elementary Row Matrix/Scale Row and Add", "text": "Let $e_2$ be the elementary row operation $\\text {ERO} 2$: {{begin-axiom}} {{axiom | n = \\text {ERO} 2         | t = For some $\\lambda$, add $\\lambda$ times row $j$ to row $i$         | m = r_i \\to r_i + \\lambda r_j }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\\mathbf E_2$ be the elementary row matrix corresponding to $e_2$. The determinant of $\\mathbf E_2$ is: :$\\map \\det {\\mathbf E_2} = 1$"}
+{"_id": "19254", "title": "Determinant of Elementary Row Matrix/Exchange Rows", "text": "Let $e_3$ be the elementary row operation $\\text {ERO} 3$: {{begin-axiom}} {{axiom | n = \\text {ERO} 3         | t = Exchange rows $i$ and $j$         | m = r_i \\leftrightarrow r_j }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\\mathbf E_3$ be the elementary row matrix corresponding to $e_3$. The determinant of $\\mathbf E_3$ is: :$\\map \\det {\\mathbf E_3} = -1$"}
+{"_id": "19255", "title": "Elementary Matrix corresponding to Elementary Column Operation", "text": "Let $\\mathbf I$ denote the unit matrix of order $n$ over a field $K$. Let $e$ be an elementary column operation on $\\mathbf I$. Let $\\mathbf E$ be the elementary column matrix of order $n$ uniquely defined as: :$\\mathbf E = e \\paren {\\mathbf I}$ where $\\mathbf I$ is the unit matrix. Let $\\kappa_k$ denote the $k$th column of $\\mathbf I$ for $1 \\le k \\le n$."}
+{"_id": "19256", "title": "Elementary Matrix corresponding to Elementary Column Operation/Scale Column", "text": "Let $e$ be the elementary column operation acting on $\\mathbf I$ as: {{begin-axiom}} {{axiom | n = \\text {ECO} 1         | t = For some $\\lambda \\in K_{\\ne 0}$, multiply column $k$ of $\\mathbf I$ by $\\lambda$         | m = \\kappa_k \\to \\lambda \\kappa_k }} {{end-axiom}}"}
+{"_id": "19257", "title": "Elementary Matrix corresponding to Elementary Column Operation/Scale Column and Add", "text": "Let $e$ be the elementary column operation acting on $\\mathbf I$ as: {{begin-axiom}} {{axiom | n = \\text {ECO} 2         | t = For some $\\lambda \\in K$, add $\\lambda$ times column $j$ to row $i$         | m = \\kappa_i \\to \\kappa_i + \\lambda r_j }} {{end-axiom}}"}
+{"_id": "19258", "title": "Elementary Matrix corresponding to Elementary Column Operation/Exchange Columns", "text": "Let $e$ be the elementary column operation acting on $\\mathbf I$ as: {{begin-axiom}} {{axiom | n = \\text {ECO} 3         | t = Interchange columns $i$ and $j$         | m = \\kappa_i \\leftrightarrow \\kappa_j }} {{end-axiom}}"}
+{"_id": "19260", "title": "Row Operation has Inverse", "text": "Let $\\map \\MM {m, n}$ be a metric space of order $m \\times n$ over a field $K$. Let $\\mathbf A \\in \\map \\MM {m, n}$ be a matrix.  Let $\\Gamma$ be a row operation which transforms $\\mathbf A$ to a new matrix $\\mathbf B \\in \\map \\MM {m, n}$. Then there exists another row operation $\\Gamma'$ which transforms $\\mathbf B$ back to $\\mathbf A$."}
+{"_id": "19261", "title": "Real Numbers with Absolute Value form Normed Vector Space", "text": "Let $\\R$ be the set of real numbers. Let $\\size {\\, \\cdot \\,}$ be the absolute value. Then $\\struct {\\R, \\size {\\, \\cdot \\,}}$ is a normed vector space."}
+{"_id": "19263", "title": "Row Operation is Equivalent to Pre-Multiplication by Product of Elementary Matrices", "text": "Let $\\map \\MM {m, n}$ be a metric space of order $m \\times n$ over a field $K$. Let $\\mathbf A \\in \\map \\MM {m, n}$ be a matrix.  Let $\\Gamma$ be a row operation which transforms $\\mathbf A$ to a new matrix $\\mathbf B \\in \\map \\MM {m, n}$. Then there exists a unique invertible square matrix $\\mathbf R$ of order $m$ such that: :$\\mathbf R \\mathbf A = \\mathbf B$ where $\\mathbf R$ is the product of a finite sequence of elementary row matrices."}
+{"_id": "19264", "title": "Product of Matrices is Invertible iff Matrices are Invertible", "text": "Let $\\mathbf A$ and $\\mathbf B$ be square matrices of order $n$. Let $\\mathbf A \\mathbf B$ denote the matrix product of $\\mathbf A$ and $\\mathbf B$. Let $\\mathbf I$ be the $n \\times n$ unit matrix. Let $\\mathbf A$ and $\\mathbf B$ be invertible. Then: :$\\mathbf A \\mathbf B$ is invertible {{iff}} :both $\\mathbf A$ and $\\mathbf B$ are invertible."}
+{"_id": "19265", "title": "Square Root of Number Plus or Minus Square Root", "text": "Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$. Then:"}
+{"_id": "19266", "title": "Elementary Row Matrix is Invertible", "text": "Let $\\mathbf E$ be an elementary row matrix. Then $\\mathbf E$ is invertible."}
+{"_id": "19267", "title": "Existence of Inverse Elementary Column Operation", "text": "Let $\\map \\MM {m, n}$ be a metric space of order $m \\times n$ over a field $K$. Let $\\mathbf A \\in \\map \\MM {m, n}$ be a matrix.  Let $\\map e {\\mathbf A}$ be an elementary column operation which transforms $\\mathbf A$ to a new matrix $\\mathbf A' \\in \\map \\MM {m, n}$. Let $\\map {e'} {\\mathbf A'}$ be the inverse of $e$. Then $e'$ is an elementary column operation which always exists and is unique."}
+{"_id": "19268", "title": "Column Operation has Inverse", "text": "Let $\\map \\MM {m, n}$ be a metric space of order $m \\times n$ over a field $K$. Let $\\mathbf A \\in \\map \\MM {m, n}$ be a matrix.  Let $\\Gamma$ be a column operation which transforms $\\mathbf A$ to a new matrix $\\mathbf B \\in \\map \\MM {m, n}$. Then there exists another column operation $\\Gamma'$ which transforms $\\mathbf B$ back to $\\mathbf A$."}
+{"_id": "19269", "title": "Elementary Column Matrix is Invertible", "text": "Let $\\mathbf E$ be an elementary column matrix. Then $\\mathbf E$ is invertible."}
+{"_id": "19270", "title": "Column Operation is Equivalent to Post-Multiplication by Product of Elementary Matrices", "text": "Let $\\map \\MM {m, n}$ be a metric space of order $m \\times n$ over a field $K$. Let $\\mathbf A \\in \\map \\MM {m, n}$ be a matrix.  Let $\\Gamma$ be a column operation which transforms $\\mathbf A$ to a new matrix $\\mathbf B \\in \\map \\MM {m, n}$. Then there exists a unique invertible square matrix $\\mathbf K$ of order $n$ such that: :$\\mathbf A \\mathbf K = \\mathbf B$ where $\\mathbf K$ is the product of a finite sequence of elementary column matrices."}
+{"_id": "19271", "title": "Existence of Inverse Elementary Row Operation/Scalar Product of Row", "text": "Let $\\map \\MM {m, n}$ be a metric space of order $m \\times n$ over a field $K$. Let $\\mathbf A \\in \\map \\MM {m, n}$ be a matrix.  Let $\\map e {\\mathbf A}$ be the elementary row operation which transforms $\\mathbf A$ to a new matrix $\\mathbf A' \\in \\map \\MM {m, n}$. {{begin-axiom}} {{axiom | n = \\text {ERO} 1         | t = For some $\\lambda \\in K_{\\ne 0}$, multiply row $i$ by $\\lambda$         | m = r_i \\to \\lambda r_i }} {{end-axiom}} Let $\\map {e'} {\\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary row operation: :$e' := r_i \\to \\dfrac 1 \\lambda r_i$"}
+{"_id": "19272", "title": "Existence of Inverse Elementary Row Operation/Add Scalar Product of Row to Another", "text": "Let $\\map \\MM {m, n}$ be a metric space of order $m \\times n$ over a field $K$. Let $\\mathbf A \\in \\map \\MM {m, n}$ be a matrix.  Let $\\map e {\\mathbf A}$ be the elementary row operation which transforms $\\mathbf A$ to a new matrix $\\mathbf A' \\in \\map \\MM {m, n}$. {{begin-axiom}} {{axiom | n = \\text {ERO} 2         | t = For some $\\lambda \\in K$, add $\\lambda$ times row $k$ to row $l$         | m = r_k \\to r_k + \\lambda r_l }} {{end-axiom}} Let $\\map {e'} {\\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary row operation: :$e' := r'_k \\to r'_k - \\lambda r'_l$"}
+{"_id": "19273", "title": "Existence of Inverse Elementary Row Operation/Exchange Rows", "text": "Let $\\map \\MM {m, n}$ be a metric space of order $m \\times n$ over a field $K$. Let $\\mathbf A \\in \\map \\MM {m, n}$ be a matrix.  Let $\\map e {\\mathbf A}$ be the elementary row operation which transforms $\\mathbf A$ to a new matrix $\\mathbf A' \\in \\map \\MM {m, n}$. {{begin-axiom}} {{axiom | n = \\text {ERO} 3         | t = Exchange rows $k$ and $l$         | m = r_k \\leftrightarrow r_l }} {{end-axiom}} Let $\\map {e'} {\\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary row operation: :$e' := r_k \\leftrightarrow r_l$ That is: :$e' = e$"}
+{"_id": "19276", "title": "Existence of Inverse Elementary Column Operation/Scalar Product of Column", "text": "Let $\\map \\MM {m, n}$ be a metric space of order $m \\times n$ over a field $K$. Let $\\mathbf A \\in \\map \\MM {m, n}$ be a matrix.  Let $\\map e {\\mathbf A}$ be the elementary column operation which transforms $\\mathbf A$ to a new matrix $\\mathbf A' \\in \\map \\MM {m, n}$. {{begin-axiom}} {{axiom | n = \\text {ECO} 1         | t = For some $\\lambda \\in K_{\\ne 0}$, multiply column $k$ by $\\lambda$         | m = \\kappa_k \\to \\lambda \\kappa_k }} {{end-axiom}} Let $\\map {e'} {\\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary column operation: :$e' := \\kappa_k \\to \\dfrac 1 \\lambda \\kappa_k$"}
+{"_id": "19277", "title": "Existence of Inverse Elementary Column Operation/Add Scalar Product of Column to Another", "text": "Let $\\map \\MM {m, n}$ be a metric space of order $m \\times n$ over a field $K$. Let $\\mathbf A \\in \\map \\MM {m, n}$ be a matrix.  Let $\\map e {\\mathbf A}$ be the elementary column operation which transforms $\\mathbf A$ to a new matrix $\\mathbf A' \\in \\map \\MM {m, n}$. {{begin-axiom}} {{axiom | n = \\text {ECO} 2         | t = For some $\\lambda \\in K$, add $\\lambda$ times column $l$ to column $k$         | m = \\kappa_k \\to \\kappa_k + \\lambda \\kappa_l }} {{end-axiom}} Let $\\map {e'} {\\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary column operation: :$e' := \\kappa_k \\to \\kappa_k - \\lambda \\kappa_l$"}
+{"_id": "19278", "title": "Existence of Inverse Elementary Column Operation/Exchange Columns", "text": "Let $\\map \\MM {m, n}$ be a metric space of order $m \\times n$ over a field $K$. Let $\\mathbf A \\in \\map \\MM {m, n}$ be a matrix.  Let $\\map e {\\mathbf A}$ be the elementary column operation which transforms $\\mathbf A$ to a new matrix $\\mathbf A' \\in \\map \\MM {m, n}$. {{begin-axiom}} {{axiom | n = \\text {ECO} 3         | t = Interchange columns $k$ and $l$         | m = \\kappa_k \\leftrightarrow \\kappa_l }} {{end-axiom}} Let $\\map {e'} {\\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary column operation: :$e' := \\kappa_k \\leftrightarrow \\kappa_l$ That is: :$e' = e$"}
+{"_id": "19279", "title": "Elementary Column Operations as Matrix Multiplications", "text": "Let $e$ be an elementary column operation. Let $\\mathbf E$ be the elementary column matrix of order $n$ defined as: :$\\mathbf E = e \\paren {\\mathbf I}$ where $\\mathbf I$ is the unit matrix. Then for every $m \\times n$ matrix $\\mathbf A$: :$e \\paren {\\mathbf A} = \\mathbf A \\mathbf E$ where $\\mathbf A \\mathbf E$ denotes the conventional matrix product."}
+{"_id": "19280", "title": "Square Root of Number Plus Square Root/Proof 1", "text": "Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$. Then: {{:Square Root of Number Plus Square Root}}"}
+{"_id": "19281", "title": "Square Root of Number Plus Square Root/Proof 2", "text": "Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$. Then: {{:Square Root of Number Plus Square Root}}"}
+{"_id": "19283", "title": "Square Root of Number Minus Square Root/Proof 1", "text": "Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$. Then: {{:Square Root of Number Plus Square Root}}"}
+{"_id": "19284", "title": "Exchange of Columns as Sequence of Other Elementary Column Operations", "text": "Let $\\mathbf A$ be an $m \\times n$ matrix. Let $i, j \\in \\closedint 1 m: i \\ne j$ Let $\\kappa_k$ denote the $k$th column of $\\mathbf A$ for $1 \\le k \\le n$: :$\\kappa_k = \\begin {pmatrix} a_{1 k} \\\\ a_{2 k} \\\\ \\vdots \\\\ a_{m k} \\end {pmatrix}$ Let $e$ be the elementary column operation acting on $\\mathbf A$ as: {{begin-axiom}} {{axiom | n = \\text {ERO} 3         | t = Interchange columns $i$ and $j$         | m = \\kappa_i \\leftrightarrow \\kappa_j }} {{end-axiom}} Then $e$ can be expressed as a finite sequence of exactly $4$ instances of the other two elementary column operations. {{begin-axiom}} {{axiom | n = \\text {ERO} 1         | t = For some $\\lambda \\in K_{\\ne 0}$, multiply column $i$ by $\\lambda$         | m = \\kappa_i \\to \\lambda \\kappa_i }} {{axiom | n = \\text {ERO} 2         | t = For some $\\lambda \\in K$, add $\\lambda$ times column $j$ to column $i$         | m = \\kappa_i \\to \\kappa_i + \\lambda \\kappa_j }} {{end-axiom}}"}
+{"_id": "19286", "title": "Determinant of Elementary Column Matrix", "text": "Let $\\mathbf E$ be an elementary column matrix. The determinant of $\\mathbf E$ is as follows:"}
+{"_id": "19287", "title": "Determinant of Elementary Column Matrix/Scale Column", "text": "Let $e_1$ be the elementary column operation $\\text {ECO} 1$: {{begin-axiom}} {{axiom | n = \\text {ECO} 1         | t = For some $\\lambda \\ne 0$, multiply column $k$ by $\\lambda$         | m = \\kappa_k \\to \\lambda \\kappa_k }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\\mathbf E_1$ be the elementary column matrix corresponding to $e_1$. The determinant of $\\mathbf E_1$ is: :$\\map \\det {\\mathbf E_1} = \\lambda$"}
+{"_id": "19288", "title": "Determinant of Elementary Column Matrix/Scale Column and Add", "text": "Let $e_2$ be the elementary column operation $\\text {ECO} 2$: {{begin-axiom}} {{axiom | n = \\text {ECO} 2         | t = For some $\\lambda$, add $\\lambda$ times column $j$ to column $i$         | m = \\kappa_i \\to \\kappa_i + \\lambda \\kappa_j }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\\mathbf E_2$ be the elementary column matrix corresponding to $e_2$. The determinant of $\\mathbf E_2$ is: :$\\map \\det {\\mathbf E_2} = 1$"}
+{"_id": "19290", "title": "Determinant with Column Multiplied by Constant", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $\\map \\det {\\mathbf A}$ be the determinant of $\\mathbf A$. Let $\\mathbf B$ be the matrix resulting from one column of $\\mathbf A$ having been multiplied by a constant $c$. Then: :$\\map \\det {\\mathbf B} = c \\map \\det {\\mathbf A}$ That is, multiplying one column of a square matrix by a constant multiplies its determinant by that constant."}
+{"_id": "19291", "title": "Determinant of Elementary Column Matrix/Exchange Columns", "text": "Let $e_3$ be the elementary column operation $\\text {ECO} 3$: {{begin-axiom}} {{axiom | n = \\text {ECO} 3         | t = Exchange columns $i$ and $j$         | m = \\kappa_i \\leftrightarrow \\kappa_j }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\\mathbf E_3$ be the elementary column matrix corresponding to $e_3$. The determinant of $\\mathbf E_3$ is: :$\\map \\det {\\mathbf E_3} = -1$"}
+{"_id": "19292", "title": "Determinant of Rescaling Matrix/Corollary", "text": "Let $\\mathbf A$ be a square matrix of order $n$. Let $\\lambda$ be a scalar. Let $\\lambda \\mathbf A$ denote the scalar product of $\\mathbf A$ by $\\lambda$. Then: :$\\map \\det {\\lambda \\mathbf A} = \\lambda^n \\map \\det {\\mathbf A}$ where $\\det$ denotes determinant."}
+{"_id": "19293", "title": "Sequence of Row Operations is Row Operation", "text": "Let $\\map \\MM {m, n}$ be a metric space of order $m \\times n$ over a field $K$. Let $\\mathbf A \\in \\map \\MM {m, n}$ be a matrix.  Let $\\Gamma_1$ be a row operation which transforms $\\mathbf A$ to a new matrix $\\mathbf B \\in \\map \\MM {m, n}$. Let $\\Gamma_2$ be a row operation which transforms $\\mathbf B$ to another new matrix $\\mathbf C \\in \\map \\MM {m, n}$. Then there exists another row operation $\\Gamma$ which transforms $\\mathbf A$ back to $\\mathbf C$ such that $\\Gamma$ consists of $\\Gamma_1$ followed by $\\Gamma_2$."}
+{"_id": "19294", "title": "Sequence of Column Operations is Column Operation", "text": "Let $\\map \\MM {m, n}$ be a metric space of order $m \\times n$ over a field $K$. Let $\\mathbf A \\in \\map \\MM {m, n}$ be a matrix.  Let $\\Gamma_1$ be a column operation which transforms $\\mathbf A$ to a new matrix $\\mathbf B \\in \\map \\MM {m, n}$. Let $\\Gamma_2$ be a column operation which transforms $\\mathbf B$ to another new matrix $\\mathbf C \\in \\map \\MM {m, n}$. Then there exists another column operation $\\Gamma$ which transforms $\\mathbf A$ back to $\\mathbf C$ such that $\\Gamma$ consists of $\\Gamma_1$ followed by $\\Gamma_2$."}
+{"_id": "19295", "title": "Equivalence of Definitions of Determinant", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. {{TFAE|def = Determinant of Matrix|view = the determinant of $\\mathbf A$}}"}
+{"_id": "19296", "title": "Intersection With Singleton is Disjoint if Not Element", "text": "Let $S$ be a set. Let $\\set x$ be the singleton of $x$. Then: :$x \\notin S$ {{iff}} $\\set x \\cap S = \\O$"}
+{"_id": "19300", "title": "Product of Matrix with Adjugate equals Determinant by Unit Matrix", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $\\map \\det {\\mathbf A}$ be the determinant of $\\mathbf A$. Let $\\adj {\\mathbf A}$ be the determinant of $\\mathbf A$. Then: :$\\paren {\\adj {\\mathbf A} } \\mathbf A = \\map \\det {\\mathbf A} \\mathbf I = \\mathbf A \\paren {\\adj {\\mathbf A} }$ where $\\mathbf I$ denotes the unit matrix of order $n$."}
+{"_id": "19301", "title": "Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant", "text": "Let $\\mathbf A = \\sqbrk a_n$ be an invertible square matrix of order $n$. Let $\\map \\det {\\mathbf A}$ be the determinant of $\\mathbf A$. Let $\\adj {\\mathbf A}$ be the determinant of $\\mathbf A$. Then: :$\\mathbf A^{-1} = \\dfrac 1 {\\map \\det {\\mathbf A} } \\cdot \\adj {\\mathbf A}$ where $\\mathbf A^{-1}$ denotes the inverse of $\\mathbf A$"}
+{"_id": "19302", "title": "Greedy Algorithm yields Maximal Set", "text": "Let $\\struct{S,\\mathscr F}$ be an independence system. Let $w : S \\to \\R_{\\ge 0}$ be a weight function. Then the Greedy Algorithm selects a maximal set $A_0$ in $\\mathscr F$."}
+{"_id": "19303", "title": "Greedy Algorithm may not yield Maximum Weight", "text": "Let $\\struct{S,\\mathscr F}$ be an independence system. Let $w : S \\to \\R_{\\ge 0}$ be a weight function. Then the maximal set $A_0 \\in \\mathscr F$ selected by the Greedy Algorithm may not have maximum weight."}
+{"_id": "19304", "title": "Greedy Algorithm guarantees Maximum Weight iff Matroid", "text": "Let $S$ be a finite set. Let $\\mathscr I$ be a non-empty set of subsets of $S$. Then $\\mathscr I$ is the set of independent subsets of a matroid on $S$ {{iff}}: :$(1) \\quad \\struct{S, \\mathscr I}$ is an independence system :$(2) \\quad$ for all non-negative weight functions $w : S \\to \\R_{\\ge 0}$, the Greedy Algorithm selects $A_w \\in \\mathscr I$: :::::$\\forall B \\in \\mathscr I: \\map {w^+} {A_w} \\ge \\map {w^+} B$ :where $w^+$ denotes the extended weight function of $w$."}
+{"_id": "19305", "title": "Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication form Vector Space", "text": "Let $I := \\closedint a b$ be a closed real interval. Let $\\map \\CC I$ be a space of continuous on closed interval real-valued functions. Let $\\struct {\\R, +_\\R, \\times_\\R}$ be the field of real numbers. Let $\\paren +$ be the pointwise addition of real-valued functions. Let $\\paren {\\, \\cdot \\,}$ be the pointwise scalar multiplication of real-valued functions. Then $\\struct {\\map \\CC I, +, \\, \\cdot \\,}_\\R$ is a vector space."}
+{"_id": "19306", "title": "Independent Subset is Contained in Base", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $\\mathscr B$ denote the set of all bases of $M$. Let $A \\in \\mathscr I$. Then: :$\\exists B \\in \\mathscr B : A \\subseteq B$"}
+{"_id": "19307", "title": "Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements/Corollary", "text": "Let $\\struct{S, \\preceq}$ be a finite ordered set. Let $x \\in S$. Then there exists a maximal element $M \\in S$ and a minimal element $m \\in S$ such that: :$m \\preceq x \\preceq M$"}
+{"_id": "19308", "title": "Equivalent Conditions for Element is Loop", "text": "Let $M = \\struct{S, \\mathscr I}$ be a matroid. Let $\\sigma$ denote the closure operator on $M$. Let $\\rho$ denote the rank function of $M$. Let $\\mathscr B$ denote the set of all bases of $M$. Let $x \\in S$. {{TFAE}} :$(1)\\quad x$ is a loop :$(2)\\quad x \\in \\map \\sigma \\O$  :$(3)\\quad \\map \\rho {\\set x} = 0$ :$(4)\\quad \\set x$ is a circuit :$(5)\\quad x$ is not an element of any $B \\in \\mathscr B$"}
+{"_id": "19309", "title": "Power Set of Doubleton", "text": "Let $x, y$ be distinct objects. Then the power set of the doubleton $\\set {x, y}$ is: :$\\powerset {\\set {x, y}} = \\big \\{ \\O, \\set x, \\set y, \\set {x,y} \\big \\}$"}
+{"_id": "19310", "title": "Doubleton of Elements is Subset", "text": "Let $S$ be a set. Let $\\set {x,y}$ be the doubleton of distinct $x$ and $y$. Then: :$x, y \\in S \\iff \\set {x,y} \\subseteq S$"}
+{"_id": "19311", "title": "Sum of Unitary Divisors of Power of Prime", "text": "Let $n = p^k$ be the power of a prime number $p$. Then the sum of all positive unitary divisors of $n$ is $1 + n$."}
+{"_id": "19312", "title": "Sum of Unitary Divisors is Multiplicative", "text": "Let $\\map {\\sigma^*} n$ denote the sum of unitary divisors of $n$. Then the function: :$\\displaystyle \\sigma^*: \\Z_{>0} \\to \\Z_{>0}: \\map {\\sigma^*} n = \\sum_{\\substack d \\mathop \\divides n \\\\ d \\mathop \\perp \\frac n d} d$ is multiplicative."}
+{"_id": "19313", "title": "Sum of Unitary Divisors of Integer", "text": "Let $n$ be an integer such that $n \\ge 2$. Let $\\map {\\sigma^*} n$ be the sum of all positive unitary divisors of $n$. Let the prime decomposition of $n$ be: :$\\displaystyle n = \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i} =  p_1^{k_1} p_2^{k_2} \\cdots p_r^{k_r}$ Then: :$\\displaystyle \\map {\\sigma^*} n = \\prod_{1 \\mathop \\le i \\mathop \\le r} \\paren {1 + p_i^{k_i}}$"}
+{"_id": "19314", "title": "Range of Infinite Sequence may be Finite", "text": "Let $\\sequence {x_n}_{n \\mathop \\in \\N}$ be an infinite sequence. Then it is possible for the range of $\\sequence {x_n}$ to be finite."}
+{"_id": "19318", "title": "Largest Number not Expressible as Sum of Multiples of Coprime Integers", "text": "Let $a, b$ be coprime integers, each greater than $1$. Then the largest number not expressible as a sum of multiples of $a$ and $b$ is the number: :$a b - a - b = \\paren {a - 1} \\paren {b - 1} - 1$"}
+{"_id": "19319", "title": "Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication form Vector Space", "text": "Let $I := \\closedint a b$ be a closed real interval. Let $\\map \\CC I$ be a space of continuous on closed interval real-valued functions. Let $\\map {\\CC^1} I$ be a space of continuously differentiable functions on closed interval $I$. Let $\\struct {\\R, +_\\R, \\times_\\R}$ be the field of real numbers. Let $\\paren +$ be the pointwise addition of real-valued functions. Let $\\paren {\\, \\cdot \\,}$ be the pointwise scalar multiplication of real-valued functions. Then $\\struct {\\map {\\CC^1} I, +, \\, \\cdot \\,}_\\R$ is a vector space."}
+{"_id": "19320", "title": "Fermat Quotient of 2 wrt p is Square iff p is 3 or 7/Generalization", "text": "The Fermat quotient of $2$ with respect to $p$: :$\\map {q_p} 2 = \\dfrac {2^{p - 1} - 1} p$ is a perfect power {{iff}} $p = 3$ or $p = 7$."}
+{"_id": "19321", "title": "Taxicab Norm is Norm", "text": "The taxicab norm is a norm on the real and complex numbers."}
+{"_id": "19322", "title": "Multiplication by 2 over 3 in Egyptian Fractions", "text": "Let $\\dfrac 1 n$ be an Egyptian fraction not equal to $\\dfrac 2 3$. In order to multiply $\\dfrac 1 n$ by $\\dfrac 2 3$ and have it that $\\dfrac 1 n \\times \\dfrac 2 3$ is also expressed in Egyptian form, we have: :$\\dfrac 1 n \\times \\dfrac 2 3 = \\dfrac 1 {2 n} + \\dfrac 1 {6 n}$"}
+{"_id": "19323", "title": "Proper Fraction can be Expressed as Finite Sum of Unit Fractions", "text": "Let $\\dfrac p q$ denote a proper fraction expressed in canonical form. Then it is always possible to express $\\dfrac p q$ as the sum of a finite number of distinct unit fractions: {{begin-eqn}} {{eqn | l = \\dfrac p q       | r = \\sum_{\\substack {1 \\mathop \\le k \\mathop \\le m \\\\ n_j \\mathop \\le n_{j + 1} } } \\dfrac 1 {n_k}       | c =  }} {{eqn | r = \\dfrac 1 {n_1} + \\dfrac 1 {n_2} + \\dotsb + \\dfrac 1 {n_m}       | c =  }} {{end-eqn}}"}
+{"_id": "19325", "title": "Upper Limit of Number of Unit Fractions to express Proper Fraction from Greedy Algorithm", "text": "Let $\\dfrac p q$ denote a proper fraction expressed in canonical form. Let $\\dfrac p q$ be expressed as the sum of a finite number of distinct unit fractions using Fibonacci's Greedy Algorithm. Then $\\dfrac p q$ is expressed using no more than $p$ unit fractions."}
+{"_id": "19326", "title": "Smallest n for which 3 over n produces 3 Egyptian Fractions using Greedy Algorithm when 2 Sufficient", "text": "Consider proper fractions of the form $\\dfrac 3 n$ expressed in canonical form. Let Fibonacci's Greedy Algorithm be used to generate a sequence $S$ of Egyptian fractions for $\\dfrac 3 n$. The smallest $n$ for which $S$ consists of $3$ terms, where $2$ would be sufficient, is $25$."}
+{"_id": "19327", "title": "Supremum Norm is Norm/Continuous on Closed Interval Real-Valued Function", "text": "Let $I = \\closedint a b$ be a closed interval. Let $\\struct {\\map \\CC I, +, \\, \\cdot \\,}_\\R$ be the vector space of real-valued functions, continuous on $I$. Let $\\map x t \\in \\map \\CC I$ be a continuous real function. Let $\\size {\\, \\cdot \\,}$ be the absolute value. Let $\\norm {\\, \\cdot \\,}_\\infty$ be the supremum norm on real-valued functions, continuous on $I$. Then $\\norm {\\, \\cdot \\,}_\\infty$ is a norm over $\\struct {\\map \\CC I, +, \\, \\cdot \\,}_\\R$."}
+{"_id": "19328", "title": "Union with Disjoint Singleton is Dependent if Element Depends on Subset", "text": "Let $M = \\struct{S, \\mathscr I}$ be a matroid. Let $A \\subseteq S$. Let $x \\in S : x \\notin A$. If $x$ depends on $A$ then $A \\cup \\set x$ is dependent"}
+{"_id": "19329", "title": "Element Depends on Independent Set iff Union with Singleton is Dependent", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $X \\in \\mathscr I$. Let $x \\in S : x \\notin X$. Then: :$x \\in \\map \\sigma X$ {{iff}} $X \\cup \\set x$ is dependent."}
+{"_id": "19330", "title": "Element Depends on Independent Set iff Union with Singleton is Dependent/Lemma", "text": "Let $A \\in \\mathscr I$ such that $A \\subseteq X \\cup \\set x$. Then: :$\\size A \\le \\size X$"}
+{"_id": "19331", "title": "Rank of Independent Subset Equals Cardinality", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $\\rho : \\powerset S \\to \\Z$ be the rank function of $M$. Let $X \\in \\mathscr I$ Then: :$\\map \\rho X = \\size X$"}
+{"_id": "19332", "title": "Generating Function for Lucas Numbers", "text": "Let $\\map G z$ be the function defined as: :$\\map G z = \\dfrac {2 - z} {1 - z - z^2}$ Then $\\map G z$ is a generating function for the Lucas numbers."}
+{"_id": "19333", "title": "492 Cubed is Sum of 3 Positive Cubes in 13 Ways", "text": "The cube of $492$ can be expressed as the sum of $3$ positive cubes in $13$ different ways: {{begin-eqn}} {{eqn | l = 492^3       | r = 24^3 + 204^3 + 480^3 }} {{eqn | r = 48^3 + 85^3 + 491^3 }} {{eqn | r = 72^3 + 384^3 + 396^3 }} {{eqn | r = 113^3 + 264^3 + 463^3 }} {{eqn | r = 114^3 + 360^3 + 414^3 }} {{eqn | r = 149^3 + 336^3 + 427^3 }} {{eqn | r = 176^3 + 204^3 + 472^3 }} {{eqn | r = 190^3 + 279^3 + 449^3 }} {{eqn | r = 207^3 + 297^3 + 438^3 }} {{eqn | r = 226^3 + 332^3 + 414^3 }} {{eqn | r = 243^3 + 358^3 + 389^3 }} {{eqn | r = 246^3 + 328^3 + 410^3 }} {{eqn | r = 281^3 + 322^3 + 399^3 }} {{end-eqn}}"}
+{"_id": "19334", "title": "Maximum Area of Isosceles Triangle", "text": "Consider two line segments $A$ and $B$ of equal length $a$ which are required to be the legs of an isosceles triangle $T$. Then the area of $T$ is greatest when the apex of $T$ is a right angle. The area of $T$ in this situation is equal to $\\dfrac {a^2} 2$."}
+{"_id": "19335", "title": "Inscribing Equilateral Triangle inside Square with a Coincident Vertex", "text": "Let $\\Box ABCD$ be a square. It is required that $\\triangle DGH$ be an equilateral triangle inscribed within $\\Box ABCD$ such that vertex $D$ of $\\triangle DGH$ coincides with vertex $D$ of $\\Box ABCD$."}
+{"_id": "19336", "title": "Construction of Perpendicular using Rusty Compass", "text": "Let $AB$ be a line segment. Using a straightedge and rusty compass, it is possible to construct a straight line at right angles to $AB$ from the endpoint $A$, without extending $AB$ past $A$."}
+{"_id": "19337", "title": "Division of Straight Line into Equal Parts using Rusty Compass", "text": "Let $AB$ be a line segment. Using a straightedge and rusty compass, it is possible to divide $AB$ into as many equal parts as required."}
+{"_id": "19338", "title": "Construction of Regular Pentagon using Rusty Compass", "text": "Using a straightedge and rusty compass, it is possible to inscribe a regular pentagon inside a circle."}
+{"_id": "19342", "title": "Sum to Infinity of 2x^2n over n by 2n Choose n", "text": "For $\\cmod x < 1$: :$\\displaystyle \\frac {2 x \\arcsin x} {\\sqrt {1 - x^2} } = \\sum_{n \\mathop = 1}^\\infty \\frac {\\paren {2 x}^{2 n} } {n \\dbinom {2 n} n}$"}
+{"_id": "19344", "title": "Definite Integral from 0 to Pi of Logarithm of a plus b Cosine x", "text": ":$\\displaystyle \\int_0^\\pi \\map \\ln {a + b \\cos x} \\rd x = \\pi \\map \\ln {\\frac {a + \\sqrt {a^2 - b^2} } 2}$"}
+{"_id": "19345", "title": "Arccosine in terms of Arctangent", "text": ":$\\displaystyle \\arccos x = 2 \\map \\arctan {\\sqrt {\\frac {1 - x} {1 + x} } }$"}
+{"_id": "19346", "title": "Definite Integral from 0 to Half Pi of Reciprocal of a plus b Cosine x", "text": ":$\\displaystyle \\int_0^{\\pi/2} \\frac 1 {a + b \\cos x} \\rd x = \\frac 1 {\\sqrt {a^2 - b^2} } \\map \\arccos {\\frac b a}$"}
+{"_id": "19347", "title": "Definite Integral to Infinity of Exponential of -a x by Sine of b x over x", "text": ":$\\displaystyle \\int_0^\\infty \\frac {e^{-a x} \\sin b x} x \\rd x = \\map \\arctan {\\frac b a}$"}
+{"_id": "19348", "title": "Definite Integral to Infinity of Sine of m x over Exponential of 2 Pi x minus One", "text": ":$\\displaystyle \\int_0^\\infty \\frac {\\sin m x} {e^{2 \\pi x} - 1} \\rd x = \\frac 1 4 \\coth \\frac m 2 - \\frac 1 {2 m}$"}
+{"_id": "19349", "title": "Definite Integral to Infinity of Exponential of -a x^2 by Cosine of b x", "text": ":$\\displaystyle \\int_0^\\infty e^{-a x^2} \\cos b x \\rd x = \\frac 1 2 \\sqrt {\\frac \\pi a} \\map \\exp {-\\frac {b^2} {4 a} }$"}
+{"_id": "19350", "title": "Fourier Series/Logarithm of Sine of x over 0 to Pi", "text": ":$\\displaystyle \\map \\ln {\\sin x} = -\\ln 2 - \\sum_{n \\mathop = 1}^\\infty \\frac {\\cos 2 n x} n$"}
+{"_id": "19351", "title": "Subset of Set Difference iff Disjoint Set", "text": "Let $S, T$ be sets. Let $A \\subseteq S$ Then: :$A \\cap T = \\varnothing \\iff A \\subseteq S \\setminus T$ where: :$A \\cap T$ denotes set intersection :$\\varnothing$ denotes the empty set :$S \\setminus T$ denotes set difference."}
+{"_id": "19352", "title": "Set Difference of Doubleton and Singleton is Singleton", "text": "Let $x, y$ be distinct objects. Then: :$\\set{x, y} \\setminus \\set x = \\set y$"}
+{"_id": "19353", "title": "Egyptian Formula for Area of Quadrilateral", "text": "Let $\\Box ABCD$ be a quadrilateral. Let the sides of $\\Box ABCD$ be $a$, $b$, $c$ and $d$ such that $a$ is opposite $c$ and $b$ is opposite $d$. Then the area of $\\Box ABCD$ can be approximated by: :$\\map \\Area {\\Box ABCD} \\approx \\dfrac {a + c} 2 \\times \\dfrac {b + d} 2$ The closer $\\Box ABCD$ is to a rectangle, the better the approximation."}
+{"_id": "19354", "title": "1 plus Perfect Power is not Power of 2", "text": "The equation: :$1 + a^n = 2^m$ has no solutions in the integers for $n, m > 1$. This is an elementary special case of Catalan's Conjecture."}
+{"_id": "19355", "title": "Definite Integral from 0 to Half Pi of Logarithm of Sine x by Cosine of 2nx", "text": "For $n \\in \\N_{>0}$: :$\\displaystyle \\int_0^{\\pi/2} \\map \\ln {\\sin x} \\cos 2 n x \\ \\d x = -\\frac \\pi {4 n}$"}
+{"_id": "19356", "title": "Distinct Matroid Elements are Parallel iff Each is in Closure of Other/Lemma", "text": "Let $a, b \\in S$. Let $\\set a$ and $\\set b$ be independent. Then $\\set {a, b}$ is dependent {{iff}}: :$a \\in \\map \\sigma {\\set b}$ and :$b \\in \\map \\sigma {\\set a}$"}
+{"_id": "19357", "title": "Definite Integral to Infinity of Sine m x over x by x Squared plus a Squared", "text": ":$\\displaystyle \\int_0^\\infty \\frac {\\sin m x} {x \\paren {x^2 + a^2} } \\rd x = \\frac \\pi {2 a^2} \\paren {1 - e^{-m a} }$"}
+{"_id": "19358", "title": "Definite Integral from 0 to Half Pi of Square of Logarithm of Sine x", "text": ":$\\displaystyle \\int_0^{\\pi/2} \\paren {\\map \\ln {\\sin x} }^2 \\rd x = \\frac \\pi 2 \\paren {\\ln 2}^2 + \\frac {\\pi^3} {24}$"}
+{"_id": "19361", "title": "Definite Integral from 0 to Pi of Sec x by Logarithm of One plus b Cosine x over One plus a Cosine x", "text": ":$\\displaystyle \\int_0^{\\pi/2} \\sec x \\map \\ln {\\frac {1 + b \\cos x} {1 + a \\cos x} } \\rd x = \\frac 1 2 \\paren {\\paren {\\arccos a}^2 - \\paren {\\arccos b}^2}$"}
+{"_id": "19363", "title": "Definite Integral to Infinity of Cosine p x minus Cosine q x over x Squared", "text": ":$\\displaystyle \\int_0^\\infty \\frac {\\cos p x - \\cos q x} {x^2} \\rd x = \\frac {\\pi \\paren {\\size q - \\size p} } 2$"}
+{"_id": "19364", "title": "Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x by Cosecant of p x", "text": ":$\\displaystyle \\int_0^\\infty \\frac {e^{-a x} - e^{-b x} } {x \\csc p x} \\rd x = \\arctan \\frac b p - \\arctan \\frac a p$"}
+{"_id": "19365", "title": "Definite Integral of Periodic Function", "text": "Let $f$ be a Darboux integrable periodic function with period $L$. Let $\\alpha \\in \\R$ and $n \\in \\Z$. Then: :$\\displaystyle \\int_\\alpha^{\\alpha + n L} \\map f x \\d x = n \\int_0^L \\map f x \\d x$"}
+{"_id": "19366", "title": "Independent Subset is Contained in Maximal Independent Subset", "text": "Let $M = \\struct{S, \\mathscr I}$ be a matroid. Let $A \\subseteq S$. Let $X \\in \\mathscr I$ such that $X \\subseteq A$. Then: :$\\exists Y \\in \\mathscr I : X \\subseteq Y \\subseteq A : \\size Y = \\map \\rho A$ where $\\rho$ is the rank function on $M$."}
+{"_id": "19367", "title": "Automorphic Numbers in Base 10", "text": "If leading zeroes are allowed, there are exactly $4$ $n$-digit automorphic numbers in base $10$: :$00 \\dots 00$ :$00 \\dots 01$ :$5^{2^{n - 1} } \\pmod {10^n}$ :$6^{5^{n - 1} } \\pmod {10^n}$"}
+{"_id": "19368", "title": "Seventeen Horses/General Problem 1", "text": "A man dies, leaving $n$ indivisible and indistinguishable objects to be divided among $3$ heirs. They are to be distributed in the ratio $\\dfrac 1 a : \\dfrac 1 b : \\dfrac 1 c$. Let $\\dfrac 1 a + \\dfrac 1 b + \\dfrac 1 c < 1$. Then there are $7$ possible values of $\\tuple {n, a, b, c}$ such that the required shares are: :$\\dfrac {n + 1} a, \\dfrac {n + 1} b, \\dfrac {n + 1} c$ These values are: :$\\tuple {7, 2, 4, 8}, \\tuple {11, 2, 4, 6}, \\tuple {11, 2, 3, 12}, \\tuple {17, 2, 3, 9}, \\tuple {19, 2, 4, 5}, \\tuple {23, 2, 3, 8}, \\tuple {41, 2, 3, 7}$ leading to shares, respectively, of: :$\\tuple {4, 2, 1}, \\tuple {6, 3, 2}, \\tuple {6, 4, 1}, \\tuple {9, 6, 2}, \\tuple {10, 5, 4}, \\tuple {12, 8, 3}, \\tuple {21, 14, 6}$"}
+{"_id": "19369", "title": "Seventeen Horses/General Problem 2", "text": "A man dies, leaving $n$ indivisible and indistinguishable objects to be divided among $m$ heirs. They are to be distributed in the ratio $\\dfrac 1 {a_1} : \\dfrac 1 {a_2} : \\cdots : \\dfrac 1 {a_m}$. Let $t = \\dfrac q r = \\displaystyle \\sum_{k \\mathop = 1}^m \\dfrac 1 {a_k}$ expressed in canonical form. Let $t \\ne 1$. Then it is possible to achieve the required share by adding $s$ objects to the existing $n$ such that: :$s + q = r$ when $q = n$. This still works whether $q$ is positive or negative."}
+{"_id": "19370", "title": "Sum of Sequence of Products of 3 Consecutive Reciprocals", "text": ":$\\displaystyle \\sum_{j \\mathop = 1}^n \\frac 1 {j \\paren {j + 1} \\paren {j + 2} } = \\frac {n \\paren {n + 3} } {4 \\paren {n + 1} \\paren {n + 2} }$"}
+{"_id": "19371", "title": "Sum of Sequence of Products of 3 Consecutive Reciprocals/Corollary", "text": ":$\\displaystyle \\sum_{j \\mathop = 1}^\\infty \\frac 1 {j \\paren {j + 1} \\paren {j + 2} } = \\frac 1 4$"}
+{"_id": "19376", "title": "Heronian Triangle whose Altitude and Sides are Consecutive Integers", "text": "There exists exactly one Heronian triangle one of whose altitudes and its sides are all consecutive integers. This is the Heronian triangle whose sides are $\\tuple {13, 14, 15}$ and which has an altitude $12$."}
+{"_id": "19377", "title": "Integer Heronian Triangle can be Scaled so Area equals Perimeter", "text": "Let $T_1$ be an integer Heronian triangle whose sides are $a$, $b$ and $c$. Then there exists a rational number $k$ such that the Heronian triangle $T_2$ whose sides are $k a$, $k b$ and $k c$ such that the perimeter of $T$ is equal to the area of $T$."}
+{"_id": "19379", "title": "Heronian Triangle is Similar to Integer Heronian Triangle", "text": "Let $\\triangle {ABC}$ be a Heronian triangle. Then there exists an integer Heronian triangle $\\triangle {A'B'C'}$ such that $\\triangle {ABC}$ and $\\triangle {A'B'C'}$ are similar."}
+{"_id": "19380", "title": "Triple with Sum and Product Equal", "text": "For $a, b, c \\in \\Z$, $a \\le b \\le c$, the solutions to the equation: :$a + b + c = a b c$ are: :$\\tuple {1, 2, 3}$ :$\\tuple {-3, -2, -1}$ and the trivial solution set: :$\\set {\\tuple {-z, 0, z}: z \\in \\N}$"}
+{"_id": "19381", "title": "Triple with Product Quadruple the Sum", "text": "Let $a, b, c \\in \\N$ such that $a \\le b \\le c$. Then the solutions to: :$a b c = 4 \\paren {a + b + c}$ are: :$\\tuple {0, 0, 0}, \\tuple {1, 5, 24}, \\tuple {1, 6, 14}, \\tuple {1, 8, 9}, \\tuple {2, 3, 10}, \\tuple {2, 4, 6}$"}
+{"_id": "19383", "title": "Proper Integer Heronian Triangle whose Area is 24", "text": "There exists exactly one proper integer Heronian triangle whose area equals $24$. That is, the obtuse triangle whose sides are of length $4$, $13$ and $15$."}
+{"_id": "19384", "title": "Semiperimeter of Integer Heronian Triangle is Composite", "text": "The semiperimeter of an integer Heronian triangle is always a composite number."}
+{"_id": "19385", "title": "Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Necessary Condition", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $\\mathscr B$ be the set of bases of the matroid on $M$. Then $\\mathscr B$ satisfies the base axiom: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1}}"}
+{"_id": "19386", "title": "Independent Subset is Base if Cardinality Equals Rank of Matroid", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $\\rho: \\powerset S \\to \\Z$ be the rank function of $M$. Let $B \\in \\mathscr I$ such that: :$\\size B = \\map \\rho S$ Then: :$B$ is a base of $M$."}
+{"_id": "19387", "title": "Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition", "text": "Let $S$ be a finite set. Let $\\mathscr B$ be a non-empty set of subsets of $S$ satisfying the base axiom: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1}} Then $\\mathscr B$ is the set of bases of a matroid on $S$."}
+{"_id": "19388", "title": "Morley's Trisector Theorem", "text": "Let $\\triangle ABC$ be a triangle. Let the internal angles of $\\triangle ABC$ be trisected. Let the points where these angle trisectors first intersect be $D$, $E$ and $F$. :500px Then $\\triangle EDF$ is equilateral."}
+{"_id": "19392", "title": "Mean Number of Elements Fixed by Self-Map", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive integer. Let $S$ be a set of cardinality $n$. Let $S^S$ be the set of all mappings from $S$ to itself. Let $\\map \\mu n$ denote the arithmetic mean of the number of fixed points of all the mappings in $S^S$. Then: :$\\map \\mu n = 1$"}
+{"_id": "19393", "title": "Condition for 3 over n producing 3 Egyptian Fractions using Greedy Algorithm when 2 Sufficient", "text": "Consider proper fractions of the form $\\dfrac 3 n$ expressed in canonical form. Let Fibonacci's Greedy Algorithm be used to generate a sequence $S$ of Egyptian fractions for $\\dfrac 3 n$. Then $S$ consists of $3$ terms, where $2$ would be sufficient {{iff}} the following conditions hold: :$n \\equiv 1 \\pmod 6$ :$\\exists d: d \\divides n$ and $d \\equiv 2 \\pmod 3$"}
+{"_id": "19394", "title": "P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences form Vector Space", "text": "Let $\\ell^p$ be the p-sequence space. Let $\\struct {\\R, +_\\R, \\times_\\R}$ be the field of real numbers. Let $\\paren +$ be the pointwise addition on the ring of sequences. Let $\\paren {\\, \\cdot \\,}$ be the pointwise multiplication on the ring of sequences. Then $\\struct {\\ell^p, +, \\, \\cdot \\,}_\\R$ is a vector space."}
+{"_id": "19395", "title": "Orthogonal Latin Squares of Order 6 do not Exist", "text": "Two orthogonal Latin squares of order $6$ do not exist."}
+{"_id": "19396", "title": "Divisor of Product", "text": "Let $a, b, c \\in \\Z$ be integers. Let the symbol $\\divides$ denote the divisibility relation. Let $a \\divides b c$. Then there exist integers $r, s$ such that: :$a = r s$, where $r \\divides b$ and $s \\divides c$."}
+{"_id": "19397", "title": "Equivalence of Definitions of Matroid", "text": "Let $M = \\struct {S, \\mathscr I}$ be an independence system. {{TFAE|def=Matroid}}"}
+{"_id": "19398", "title": "P-adic Norm is Well Defined", "text": "P-adic norm $\\norm {\\, \\cdot \\,}_p$ is well defined."}
+{"_id": "19399", "title": "Equivalence of Definitions of Matroid/Definition 1 implies Definition 2", "text": "Let $M = \\struct {S, \\mathscr I}$ be an independence system. Let $M$ also satisfy: {{begin-axiom}} {{axiom | n = \\text I 3         | q = \\forall U, V \\in \\mathscr I         | mr= \\size V < \\size U \\implies \\exists x \\in U \\setminus V : V \\cup \\set x \\in \\mathscr I }} {{end-axiom}} Then $M$ satisfies: {{begin-axiom}} {{axiom | n = \\text I 3'         | q = \\forall U, V \\in \\mathscr I         | mr= \\size U = \\size V + 1 \\implies \\exists x \\in U \\setminus V : V \\cup \\set x \\in \\mathscr I }} {{end-axiom}}"}
+{"_id": "19400", "title": "Equivalence of Definitions of Matroid/Definition 2 implies Definition 3", "text": "Let $M = \\struct {S, \\mathscr I}$ be an independence system. Let $M$ also satisfy: {{begin-axiom}} {{axiom | n = \\text I 3'         | q = \\forall U, V \\in \\mathscr I         | mr= \\size U = \\size V + 1 \\implies \\exists x \\in U \\setminus V : V \\cup \\set x \\in \\mathscr I }} {{end-axiom}} Then $M$ satisfies: {{begin-axiom}} {{axiom | n = \\text I 3''         | q = \\forall U, V \\in \\mathscr I         | mr= \\size V < \\size U \\implies \\exists Z \\subseteq U \\setminus V : \\paren{V \\cup Z \\in \\mathscr I} \\land \\paren{ \\size {V \\cup Z} = \\size U} }} {{end-axiom}}"}
+{"_id": "19401", "title": "Equivalence of Definitions of Matroid/Definition 3 implies Definition 1", "text": "Let $M = \\struct {S, \\mathscr I}$ be an independence system. Let $M$ also satisfy: {{begin-axiom}} {{axiom | n = \\text I 3''         | q = \\forall U, V \\in \\mathscr I         | mr= \\size V < \\size U \\implies \\exists Z \\subseteq U \\setminus V : \\paren {V \\cup Z \\in \\mathscr I} \\land \\paren {\\size {V \\cup Z} = \\size U} }} {{end-axiom}} Then $M$ satisfies: {{begin-axiom}} {{axiom | n = \\text I 3         | q = \\forall U, V \\in \\mathscr I         | mr= \\size V < \\size U \\implies \\exists x \\in U \\setminus V : V \\cup \\set x \\in \\mathscr I }} {{end-axiom}}"}
+{"_id": "19402", "title": "Equivalence of Definitions of Matroid/Definition 1 implies Definition 4", "text": "Let $M = \\struct {S, \\mathscr I}$ be an independence system. Let $M$ also satisfy: {{begin-axiom}} {{axiom | n = \\text I 3         | q = \\forall U, V \\in \\mathscr I         | mr= \\size V < \\size U \\implies \\exists x \\in U \\setminus V : V \\cup \\set x \\in \\mathscr I }} {{end-axiom}} Then $M$ satisfies: {{begin-axiom}} {{axiom | n = \\text I 3'''         | q = \\forall A \\subseteq S         | mr= \\text{ all maximal subsets } Y \\subseteq A \\text{ with } Y \\in \\mathscr I \\text{ have the same cardinality} }} {{end-axiom}}"}
+{"_id": "19403", "title": "Equivalence of Definitions of Matroid/Definition 4 implies Definition 1", "text": "Let $M = \\struct {S, \\mathscr I}$ be an independence system. Let $M$ also satisfy: {{begin-axiom}} {{axiom | n = \\text I 3'''         | q = \\forall A \\subseteq S         | mr= \\text{ all maximal subsets } Y \\subseteq A \\text{ with } Y \\in \\mathscr I \\text{ have the same cardinality} }} {{end-axiom}} Then $M$ satisfies: {{begin-axiom}} {{axiom | n = \\text I 3         | q = \\forall U, V \\in \\mathscr I         | mr= \\size V < \\size U \\implies \\exists x \\in U \\setminus V : V \\cup \\set x \\in \\mathscr I }} {{end-axiom}}"}
+{"_id": "19404", "title": "Cardinality of Set Difference", "text": "Let $S$ and $T$ be sets such that $T$ is finite. Then: :$\\card {S \\setminus T} = \\card S - \\card {S \\cap T}$ where $\\card S$ denotes the cardinality of $S$."}
+{"_id": "19405", "title": "Set Difference and Intersection are Disjoint", "text": "Let $S$ and $T$ be sets. Then:  :$S \\setminus T$ and $S \\cap T$ are disjoint  where $S \\setminus T$ denotes set difference and $S \\cap T$ denotes set intersection."}
+{"_id": "19406", "title": "Straight Line has Zero Curvature", "text": "A straight lines has zero curvature."}
+{"_id": "19407", "title": "Partial Differential Equation of Spheres in 3-Space", "text": "The set of spheres in real Cartesian $3$-dimensional space can be described by the system of partial differential equations: :$\\dfrac {1 + z_x^2} {z_{xx} } = \\dfrac {z_x z_x} {z_{xy} } = \\dfrac {1 + z_y^2} {z_{yy} }$ and if the spheres are expected to be real: :$z_{xx} z_{yy} > z_{xy}$"}
+{"_id": "19408", "title": "Fermat's Right Triangle Theorem", "text": "$x^4 + y^4 = z^2$ has no solutions in the (strictly) positive integers."}
+{"_id": "19411", "title": "Taylor's Theorem/One Variable with Two Functions", "text": "Let $f$ and $g$ be real functions satisfying following conditions: :$(1): \\quad f$ is $n + 1$ times differentiable on the open interval $\\openint a x$ :$(2): \\quad f$ is of differentiability class $C^n$ on the closed interval $\\closedint a x$ :$(3): \\quad g$ is $k + 1$ times differentiable on the open interval $\\openint a x$ :$(4): \\quad g$ is of differentiability class $C^k$ on the closed interval $\\closedint a x$ :$(5): \\quad \\map {g^{\\paren {k + 1}}} t \\ne 0$ for any $t \\in \\openint a x$ Then the following equation holds for some real number $\\xi \\in \\openint a x$: :$\\dfrac {\\map {f^{\\paren {n + 1} } } \\xi /n!} {\\map {g^{\\paren {k + 1} } } \\xi /k!} \\paren {x - \\xi}^{n - k} = \\dfrac {\\map f x - \\map f a - \\map {f'} a \\paren {x - a} - \\dfrac {\\map {f''} a} {2!} \\paren {x - a}^2 - \\dotsb - \\dfrac {\\map {f^{\\paren n} } a} {n!} \\paren {x - a}^n} {\\map g x - \\map g a - \\map {g'} a \\paren {x - a} - \\dfrac {\\map {g''} a} {2!} \\paren {x - a}^2 - \\dotsb - \\dfrac {\\map {g^{\\paren k} } a} {k!} \\paren {x - a}^k}$ or equivalently: {{begin-eqn}} {{eqn | l = \\map f x       | r = \\map f a + \\map {f'} a \\paren {x - a} + \\dfrac {\\map {f''} a} {2!} \\paren {x - a}^2 + \\dotsb + \\dfrac {\\map {f^{\\paren n} } a} {n!} \\paren {x - a}^n + R_n }} {{eqn | l = R_n       | r = \\dfrac {\\map {f^{\\paren {n + 1} } } \\xi / n!} {\\map {g^{\\paren {k + 1} } } \\xi / k!} \\paren {x - \\xi}^{n - k} \\paren {\\map g x - \\map g a - \\map {g'} a \\paren {x - a} - \\dfrac {\\map {g''} a} {2!} \\paren {x - a}^2 - \\dotsb - \\dfrac {\\map {g^{\\paren k} } a} {k!} \\paren {x - a}^k} }} {{end-eqn}}"}
+{"_id": "19412", "title": "Set of Even Integers is Countably Infinite", "text": "Let $\\Bbb E$ be the set of even integers. Then $\\Bbb E$ is countably infinite."}
+{"_id": "19413", "title": "Basis Expansion of Rational Number", "text": "Let $x$ be a rational number. {{WIP|Time has caught up with me, I'll continue this later. I'm late for work.}} q2llusuyprk7567f7o3l3jnktxipxcq"}
+{"_id": "19414", "title": "Integration by Parts/Definite Integral", "text": ":$\\displaystyle \\int_a^b \\map f t \\map G t \\rd t = \\bigintlimits {\\map F t \\map G t} a b - \\int_a^b \\map F t \\map g t \\rd t$"}
+{"_id": "19415", "title": "Integration by Parts/Primitive", "text": ":$\\ds \\int \\map f t \\map G t \\rd t = \\map F t \\map G t - \\int \\map F t \\map g t \\rd t$ on $\\closedint a b$."}
+{"_id": "19416", "title": "Integration by Substitution/Primitive", "text": "The primitive of $f$ can be evaluated by: :$\\ds \\int \\map f x \\rd x = \\int \\map f {\\map \\phi u} \\dfrac \\d {\\d u} \\map \\phi u \\rd u$ where $x = \\map \\phi u$."}
+{"_id": "19418", "title": "Product Formula for Norms on Non-zero Rationals/Lemma", "text": "Let $z \\in \\Z_{\\ne 0}$. Then the following infinite product converges: :$\\size z \\times \\displaystyle\\prod_{p \\mathop \\in \\Bbb P}^{} \\norm z_p = 1$"}
+{"_id": "19419", "title": "P-adic Open Ball is Instance of Open Ball of a Norm", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. Let $B \\subseteq \\Q_p$. Then:  :$B$ is an open ball in $p$-adic numbers with radius $\\epsilon$ and centre $a$ {{iff}}: :$B$ is an open ball of the normed division ring $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ with radius $\\epsilon$ and centre $a$ . That is, the definition of an open ball in $p$-adic numbers is a specific instance of the general definition of an open ball in a normed division ring."}
+{"_id": "19420", "title": "P-adic Closed Ball is Instance of Closed Ball of a Norm", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. Let $B \\subseteq \\Q_p$. Then:  :$B$ is a closed ball in $p$-adic numbers with radius $\\epsilon$ and centre $a$ {{iff}}: :$B$ is a closed ball of the normed division ring $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ with radius $\\epsilon$ and centre $a$  That is, the definition of a closed ball in $p$-adic numbers is a specific instance of the general definition of a closed ball in a normed division ring."}
+{"_id": "19421", "title": "P-adic Sphere is Instance of Sphere of a Norm", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. Let $S \\subseteq \\Q_p$. Then:  :$S$ is a sphere in $p$-adic numbers with radius $\\epsilon$ and centre $a$ {{iff}}: :$S$ is a sphere of the normed division ring $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ with radius $\\epsilon$ and centre $a$  That is, the definition of a closed ball in $p$-adic numbers is a specific instance of the general definition of a sphere in a normed division ring."}
+{"_id": "19422", "title": "Sphere is Set Difference of Closed Ball with Open Ball/P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\Q_p$ be the $p$-adic numbers. Let $a \\in \\Q_p$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. Let $\\map {{B_\\epsilon}^-} a$ denote the $\\epsilon$-closed ball of $a$ in $\\Q_p$. Let $\\map {B_\\epsilon} a$ denote the $\\epsilon$-open ball of $a$ in $\\Q_p$. Let $\\map {S_\\epsilon} a$ denote the $\\epsilon$-sphere of $a$ in $\\Q_p$. Then: :$\\map {S_\\epsilon} a = \\map { {B_\\epsilon}^-} a \\setminus \\map {B_\\epsilon} a$"}
+{"_id": "19423", "title": "Set is Closed in Metric Space iff Closed in Induced Topological Space", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $\\tau$ be the topology induced by the metric $d$. Let $F$ be a subset of $M$. Then: :$F$ is closed in $M$ {{iff}} $F$ is closed in $\\struct {A, \\tau}$"}
+{"_id": "19428", "title": "Primitive of Reciprocal of Root of a squared minus x squared/Arccosine Form", "text": ":$\\displaystyle \\int \\frac 1 {\\sqrt {a^2 - x^2} } \\rd x = -\\arccos \\frac x a + C$"}
+{"_id": "19429", "title": "Negative of Logarithm of x plus Root x squared minus a squared", "text": "Let $x \\in \\R: \\size x > 1$. Let $x > 1$. Then: :$-\\map \\ln {x + \\sqrt {x^2 - a^2} } = \\map \\ln {x - \\sqrt {x^2 - a^2} } - \\map \\ln {a^2}$"}
+{"_id": "19430", "title": "Supremum Norm on Vector Space of Real Matrices is Norm", "text": "Supremum Norm forms a norm on the vector space of real matrices."}
+{"_id": "19432", "title": "Arccotangent Logarithmic Formulation", "text": "For any real number $x$: :$\\arccot x = \\dfrac 1 2 i \\, \\map \\ln {\\dfrac {1 + i x} {1 - i x} }$ where $\\arccot x$ is the arccotangent and $i^2 = -1$."}
+{"_id": "19433", "title": "Primitive of Reciprocal of x squared plus a squared/Arccotangent Form", "text": ":$\\ds \\int \\frac {\\d x} {x^2 + a^2} = -\\frac 1 a \\arccot \\frac x a + C$"}
+{"_id": "19434", "title": "Primitive of Reciprocal of x by Root of x squared plus a squared/Reciprocal Logarithm Form", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\sqrt {x^2 + a^2} } = \\frac 1 a \\map \\ln {\\frac x {a + \\sqrt {x^2 + a^2} } } + C$"}
+{"_id": "19435", "title": "Derivative of Hyperbolic Sine", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\sinh x} = \\cosh x$"}
+{"_id": "19436", "title": "Open Ball is Open Set/Metric Space", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $x \\in A$. Let $\\epsilon \\in \\R_{>0}$. Let $B_\\epsilon \\left({x}\\right)$ be an open $\\epsilon$-ball of $x$ in $M$. Then $B_\\epsilon \\left({x}\\right)$ is an open set of $M$."}
+{"_id": "19437", "title": "Derivative of Hyperbolic Cosine", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\cosh x} = \\sinh x$"}
+{"_id": "19438", "title": "Derivative of Hyperbolic Tangent", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\tanh x} = \\sech^2 x$"}
+{"_id": "19439", "title": "Derivative of Hyperbolic Cotangent", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\coth x} = -\\csch^2 x$"}
+{"_id": "19440", "title": "Derivative of Hyperbolic Secant", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\sech x} = -\\sech x \\tanh x$"}
+{"_id": "19441", "title": "Derivative of Hyperbolic Cosecant", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\csch x} = -\\csch x \\coth x$"}
+{"_id": "19442", "title": "Derivative of Inverse Hyperbolic Sine Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\sinh^{-1} u} = \\dfrac 1 {\\sqrt {1 + u^2} } \\dfrac {\\d u} {\\d x}$"}
+{"_id": "19443", "title": "Derivative of Inverse Hyperbolic Cosine Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\cosh^{-1} u} = \\dfrac 1 {\\sqrt {u^2 - 1} } \\dfrac {\\d u} {\\d x}$  where $u > 1$"}
+{"_id": "19444", "title": "Derivative of Inverse Hyperbolic Tangent Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\tanh^{-1} u} = \\dfrac 1 {1 - u^2} \\dfrac {\\d u} {\\d x}$  where $\\size u < 1$"}
+{"_id": "19447", "title": "Derivative of Inverse Hyperbolic Cosecant Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\csch^{-1} u} = \\dfrac {-1} {\\size u \\sqrt {1 + u^2} } \\dfrac {\\d u} {\\d x}$"}
+{"_id": "19448", "title": "Intersection of Closed Sets is Closed/Topology", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then the intersection of an arbitrary number of closed sets of $T$ (either finitely or infinitely many) is itself closed."}
+{"_id": "19452", "title": "Finite Union of Closed Sets is Closed/Topology", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then the union of finitely many closed sets of $T$ is itself closed."}
+{"_id": "19453", "title": "Closed Ball is Closed/Metric Space", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $x \\in A$. Let $\\epsilon \\in \\R_{>0}$. Let $\\map {B_\\epsilon^-} x$ be the closed $\\epsilon$-ball of $x$ in $M$. Then $\\map {B_\\epsilon^-} x$ is a closed set of $M$."}
+{"_id": "19455", "title": "Derivative of Sine of Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\sin u} = \\cos u \\dfrac {\\d u} {\\d x}$"}
+{"_id": "19459", "title": "Derivative of Secant of Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\sec u} = \\sec u \\tan u \\dfrac {\\d u} {\\d x}$"}
+{"_id": "19460", "title": "Derivative of Cosecant of Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\csc u} = \\csc u \\cot u \\dfrac {\\d u} {\\d x}$"}
+{"_id": "19462", "title": "Derivative of Natural Logarithm of Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\ln u} = \\dfrac 1 u \\dfrac {\\d u} {\\d x}$"}
+{"_id": "19463", "title": "Derivative of Constant to Power of Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {a^u} = a^u \\ln a \\dfrac {\\d u} {\\d x}$"}
+{"_id": "19465", "title": "Derivative of Arcsine of Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\arcsin u} = \\dfrac 1 {\\sqrt {1 - u^2} } \\dfrac {\\d u} {\\d x}$"}
+{"_id": "19466", "title": "Derivative of Arccosine of Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\arccos u} = -\\dfrac 1 {\\sqrt {1 - u^2} } \\dfrac {\\d u} {\\d x}$"}
+{"_id": "19467", "title": "Derivative of Arctangent of Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\arctan u} = \\dfrac 1 {1 + u^2} \\dfrac {\\d u} {\\d x}$"}
+{"_id": "19468", "title": "Derivative of Arccotangent of Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\arccot u} = -\\dfrac 1 {1 + u^2} \\dfrac {\\d u} {\\d x}$"}
+{"_id": "19469", "title": "Derivative of Arcsecant of Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\arcsec u} = \\dfrac 1 {\\size u \\sqrt {u^2 - 1} } \\dfrac {\\d u} {\\d x}$"}
+{"_id": "19470", "title": "Derivative of Arccosecant of Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\arccsc u} = -\\dfrac 1 {\\size u \\sqrt {u^2 - 1} } \\dfrac {\\d u} {\\d x}$"}
+{"_id": "19472", "title": "Derivative of Odd Function is Even", "text": "Let $f$ be a differentiable real function such that $f$ is odd. Then its derivative $f'$ is an even function."}
+{"_id": "19473", "title": "Form of Prime Sierpiński Number of the First Kind", "text": "Suppose $S_n = n^n + 1$ is a prime Sierpiński number of the first kind. Then: :$n = 2^{2^k}$ for some integer $k$."}
+{"_id": "19474", "title": "Motion of Body Falling through Air", "text": "The motion of a body $B$ falling through air can be described using the following differential equation: :$m \\dfrac {\\d^2 y} {\\d t^2} = m g - k \\dfrac {d y} {d t}$ where: :$m$ denotes mass of $B$ :$y$ denotes the height of $B$ from an arbitrary reference :$t$ denotes time elapsed from an arbitrary reference :$g$ denotes the Acceleration Due to Gravity of $B$ :$k$ denotes the coefficient of resistive force exerted on $B$ by the air (assumed to be proportional to the speed of $B$)"}
+{"_id": "19476", "title": "Rationals are Everywhere Dense in Reals/Normed Vector Space", "text": "Let $\\struct {\\R, \\size {\\, \\cdot \\,}}$ be the normed vector space of real numbers. Let $\\Q$ be the set of rational numbers. Then $\\Q$ are everywhere dense in $\\struct {\\R, \\size {\\, \\cdot \\,}}$"}
+{"_id": "19477", "title": "Number of Parameters of Autoregressive Model", "text": "Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \\dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \\dotsc$ Let $\\tilde z_t, \\tilde z_{t - 1}, \\tilde z_{t - 2}, \\dotsc$ be deviations from a constant mean level $\\mu$: :$\\tilde z_t = z_t - \\mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \\dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \\dotsc$ Let $M$ be an '''autoregessive model''' on $S$ of order $p$: :$\\tilde z_t = \\phi_1 \\tilde z_{t - 1} + \\phi_2 \\tilde z_{t - 2} + \\dotsb + \\phi_p \\tilde z_{t - p} + a_t$ Then $M$ has $p + 2$ parameters. </onlyinclude>"}
+{"_id": "19478", "title": "Autoregressive Model is Special Case of Linear Filter Model", "text": "Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \\dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \\dotsc$ Let $\\tilde z_t, \\tilde z_{t - 1}, \\tilde z_{t - 2}, \\dotsc$ be deviations from a constant mean level $\\mu$: :$\\tilde z_t = z_t - \\mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \\dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \\dotsc$ Let $M$ be an '''autoregessive model''' on $S$ of order $p$: :$(1): \\quad \\tilde z_t = \\phi_1 \\tilde z_{t - 1} + \\phi_2 \\tilde z_{t - 2} + \\dotsb + \\phi_p \\tilde z_{t - p} + a_t$ Then $M$ is a special case of a linear filter model. </onlyinclude>"}
+{"_id": "19479", "title": "Irrationals are Everywhere Dense in Reals/Topology", "text": "Let $T = \\struct {\\R, \\tau}$ denote the real number line with the usual (Euclidean) topology. Let $\\R \\setminus \\Q$ be the set of irrational numbers. Then $\\R \\setminus \\Q$ is everywhere dense in $T$."}
+{"_id": "19480", "title": "Irrationals are Everywhere Dense in Reals/Normed Vector Space", "text": "Let $\\struct {\\R, \\size {\\, \\cdot \\,}}$ be the normed vector space of real numbers. Let $\\R \\setminus \\Q$ be the set of irrational numbers. Then $\\R \\setminus \\Q$ are everywhere dense in $\\struct {\\R, \\size {\\, \\cdot \\,}}$"}
+{"_id": "19481", "title": "Real Number Subtracted from Itself leaves Zero", "text": "Let $x \\in \\R$ be a real number. Then: :$x - x = 0$ where $x - x$ denotes the operation of real subtraction."}
+{"_id": "19482", "title": "Real Number Ordering is Compatible with Multiplication/Positive Factor/Corollary", "text": ":$\\forall a, b, c, d \\in \\R: 0 < a < b \\land 0 < c < d \\implies a c < b d$"}
+{"_id": "19483", "title": "Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom", "text": "Let $S$ be a finite set. Let $\\mathscr B$ be a non-empty set of subsets of $S$. {{TFAE|def=Base Axiom (Matroid)|view = Matroid Base Axiom}}"}
+{"_id": "19484", "title": "Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Lemma", "text": "Let $B_1, B_2 \\subseteq S$. Let $x \\in B_1 \\setminus B_2$. Let $y \\in B_2 \\setminus B_1$. Then: :$\\paren{B_1 \\setminus \\set x} \\cup \\set y = \\paren{B_1 \\cup \\set y} \\setminus \\set x$"}
+{"_id": "19485", "title": "Number of Parameters of Moving Average Model", "text": "Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \\dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \\dotsc$ Let $\\tilde z_t$ be the deviation from a constant mean level $\\mu$: :$\\tilde z_t = z_t - \\mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \\dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \\dotsc$ Let $M$ be an '''moving average model''' on $S$ of order $q$: :$\\tilde z_t = a_t - \\theta_1 a_{t - 1} - \\theta_2 a_{t - 2} - \\dotsb - \\theta_q a_{t - q}$ Then $M$ has $q + 2$ parameters. </onlyinclude>"}
+{"_id": "19487", "title": "Characteristic of Field by Annihilator/Characteristic Zero", "text": "Suppose that: :$\\map {\\mathrm {Ann} } F = \\set 0$ That is, the annihilator of $F$ consists of the zero only. Then: :$\\Char F = 0$ That is, the characteristic of $F$ is zero."}
+{"_id": "19488", "title": "Characteristic of Field by Annihilator/Prime Characteristic", "text": "Suppose that: :$\\exists n \\in \\map {\\mathrm {Ann} } F: n \\ne 0$ That is, there exists (at least one) non-zero integer in the annihilator of $F$. If this is the case, then the characteristic of $F$ is non-zero: :$\\Char F = p \\ne 0$ and the annihilator of $F$ consists of the set of integer multiples of $p$: :$\\map {\\mathrm {Ann} } F = p \\Z$ where $p$ is a prime number."}
+{"_id": "19489", "title": "Necessary Condition for Autoregressive Process to be Stationary", "text": "Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \\dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \\dotsc$ Let $\\tilde z_t, \\tilde z_{t - 1}, \\tilde z_{t - 2}, \\dotsc$ be deviations from a constant mean level $\\mu$: :$\\tilde z_t = z_t - \\mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \\dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \\dotsc$ Let $M$ be an '''autoregessive model''' on $S$ of order $p$: :$\\map \\phi B \\tilde z_t = a_t$ where $\\map \\phi B := 1 - \\phi_1 B - \\phi_2 B^2 - \\dotsb - \\phi_p B^p$ is the autoregressive operator of order $p$. Consider the polynomial equation in $B$ of degree $p$: :$(1): \\quad \\map \\phi B = 0$ Let $\\map R \\phi \\subseteq \\C$ denote the set of roots of $(1)$, considered as a polynomial of degree $p$. It is noted that the elements of $\\map R \\phi$ may be real or complex. For $S$ modelled by $M$ to be a stationary process, it is necessary that the elements of $\\map R \\phi$ have a complex modulus greater than $1$: :$\\forall z \\in \\map R \\phi: \\size z > 1$"}
+{"_id": "19491", "title": "ARIMA Model subsumes ARMA Model", "text": "Let $S$ be a stochastic process based on an equispaced time series. Let $M$ be an ARMA model for $S$. Then $M$ is also an implementation of an ARIMA model."}
+{"_id": "19492", "title": "ARIMA Model subsumes Autoregressive Model", "text": "Let $S$ be a stochastic process based on an equispaced time series. Let $M$ be an autoregressive model for $S$. Then $M$ is also an implementation of an ARIMA model."}
+{"_id": "19501", "title": "Leigh.Samphier/Sandbox/Matroid Base Axiom Implies Sets Have Same Cardinality", "text": "Let $S$ be a finite set. Let $\\mathscr B$ be a non-empty set of subsets of $S$. Let $\\mathscr B$ satisfy the base axiom: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1}} Then: :$\\forall B_1, B_2 \\in \\mathscr B : \\card{B_1} = \\card{B_2}$ where $\\card{B_1}$ and $\\card{B_2}$ denote the cardinality of the sets $B_1$ and $B_2$ respectively."}
+{"_id": "19502", "title": "Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 4 Iff Definition 5/Lemma", "text": ":$\\forall B_1, B_2 \\in \\mathscr B : \\card{B_1} = \\card{B_2}$"}
+{"_id": "19503", "title": "Negative of Field Negative", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0_F$. Let $a \\in F$ and let $-a$ be the field negative of $a$. Then: :$-\\paren {-a} = a$"}
+{"_id": "19504", "title": "Field Product with Zero", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0_F$. Let $a \\in F$. Then: :$a \\times 0_F = 0_F$"}
+{"_id": "19505", "title": "Product with Field Negative", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $a, b \\in F$. Then: :$-\\paren {a \\times b} = a \\times \\paren {-b} = \\paren {-a} \\times b$"}
+{"_id": "19506", "title": "Product with Field Negative/Corollary", "text": ":$\\paren {-1_F} \\times a = \\paren {-a}$"}
+{"_id": "19507", "title": "Condition for Division by Field Elements to be Unity", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $a, b \\in F$. Then: :$\\dfrac a b = 1_F$ {{iff}} :$a = b$ where $\\dfrac a b$ denotes division."}
+{"_id": "19508", "title": "Field Product with Non-Zero Element yields Unique Solution", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $a, b, x \\in F$ such that $b \\ne 0_F$. Let: :$b \\times x = a$ Then: :$x = a b^{-1}$ That is: :$x = \\dfrac a b$ where $\\dfrac a b$ denotes division."}
+{"_id": "19509", "title": "Field Unity Divided by Element equals Multiplicative Inverse", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $a \\in F$. Then: :$\\dfrac {1_F} a = a^{-1}$ where $\\dfrac {1_F} a$ denotes division."}
+{"_id": "19512", "title": "Set Difference of Larger Set with Smaller is Not Empty", "text": "Let $S$ and $T$ be finite sets. Let $\\card S > \\card T$. Then: :$S \\setminus T \\ne \\O$"}
+{"_id": "19513", "title": "Autocovariance Matrix for Stationary Process is Variance by Autocorrelation Matrix", "text": "Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\\sequence {s_n} = \\tuple {z_1, z_2, \\dotsb, z_n}$ Let $\\boldsymbol \\Gamma_n$ denote the '''autocovariance matrix''' associated with $S$ for $\\sequence {s_n}$. Let $\\mathbf P_n$ denote the '''autocorrelation matrix''' associated with $S$ for $\\sequence {s_n}$. Then: :$\\boldsymbol \\Gamma_n = \\sigma_z^2 \\mathbf P_n$ where $\\sigma_z^2$ denotes the variance of $S$."}
+{"_id": "19514", "title": "Variance of Linear Function of Observations of Stationary Process", "text": "Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\\sequence {s_n} = \\tuple {z_1, z_2, \\dotsb, z_n}$ Let $L_t$ be a linear function of $\\sequence {s_n}$: :$L_t = l_1 z_t + l_2 z_{t - 1} + \\dotsb + l_n z_{t - n + 1}$ Then the variance of $L_t$ is given by: :$\\var {L_t} = \\displaystyle \\sum_{i \\mathop = 1}^n \\sum_{j \\mathop = 1}^n l_i l_j \\gamma {\\size {j - i} }$ where $\\gamma_k$ is the autocovariance of $S$ at lag $k$."}
+{"_id": "19515", "title": "Autocorrelation Matrix is Positive Definite", "text": "Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\\sequence {s_n} = \\tuple {z_1, z_2, \\dotsb, z_n}$ Let $\\mathbf P_n$ denote the '''autocorrelation matrix''' associated with $S$ for $\\sequence {s_n}$. Then $\\mathbf P_n$ is a positive definite matrix."}
+{"_id": "19516", "title": "Determinant of Autocorrelation Matrix is Strictly Positive", "text": "Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\\sequence {s_n} = \\tuple {z_1, z_2, \\dotsb, z_n}$ Let $\\mathbf P_n$ denote the '''autocorrelation matrix''' associated with $S$ for $\\sequence {s_n}$. The determinant of $\\mathbf P_n$ is strictly positive."}
+{"_id": "19517", "title": "Sum of Wholly Real Numbers is Wholly Real", "text": "Let $x = \\tuple {a, 0}$ and $y = \\tuple {b, 0}$ be wholly real complex numbers. Then $x + y$ is also wholly real."}
+{"_id": "19518", "title": "Product of Wholly Real Numbers is Wholly Real", "text": "Let $x = \\tuple {a, 0}$ and $y = \\tuple {b, 0}$ be wholly real complex numbers. Then $x y$ is also wholly real."}
+{"_id": "19519", "title": "Product of Imaginary Unit with Itself", "text": "Let $\\tuple {0, 1}$ denote the imaginary unit. Then: :$\\tuple {0, 1} \\times \\tuple {0, 1} = \\tuple {-1, 0}$ where $\\times$ denotes complex multiplication."}
+{"_id": "19521", "title": "Element of Matroid Base and Circuit has a Substitute", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $B \\subseteq S$ be a base of $M$. Let $C \\subseteq S$ be a circuit of $M$. Let $x \\in B \\cap C$. Then: :$\\exists y \\in C \\setminus B : \\paren{B \\setminus \\set x} \\cup \\set y$ is a base of $M$"}
+{"_id": "19522", "title": "Set Difference with Proper Subset is Proper Subset", "text": "Let $S$ be a set. Let $T \\subsetneq S$ be a proper subset of $S$.   Let $S \\setminus T$ denote the set difference between $S$ and $T$. Then: :$S \\setminus T$ is a proper subset of $S$"}
+{"_id": "19523", "title": "Element of Matroid Base and Circuit has a Substitute/Lemma 1", "text": ":$C \\setminus \\set x$ is an independent proper subset of $C$"}
+{"_id": "19525", "title": "All Bases of Matroid have same Cardinality/Corollary", "text": "Let $X \\subseteq S$ be any independent subset of $M$. Then: :$\\card X \\le \\card B$"}
+{"_id": "19526", "title": "Independent Set can be Augmented by Larger Independent Set/Corollary", "text": "Let $B \\subseteq S$ be a base of $M$. Then: :$\\exists Z \\subseteq B \\setminus X : \\card{X \\cup Z} = \\card B  : X \\cup Z$ is a base of $M$"}
+{"_id": "19527", "title": "Independent Subset is Base if Cardinality Equals Rank of Matroid/Corollary", "text": "Let $B \\subseteq S$ be a base of $M$. Let $X \\subseteq S$ be any independent subset of $M$. Let $\\card X = \\card B$. Then: :$X$ is a base of $M$."}
+{"_id": "19528", "title": "Element of Matroid Base and Circuit has a Substitute/Lemma 2", "text": ":$x \\notin \\paren{ C \\setminus \\set x} \\cup X$"}
+{"_id": "19529", "title": "Element of Matroid Base and Circuit has a Substitute/Lemma 3", "text": ":$\\exists y \\in \\paren{\\paren{C \\setminus \\set x} \\cup X} \\setminus \\paren{B \\setminus \\set x} : \\paren{B \\setminus \\set x} \\cup \\set y \\in \\mathscr I : \\card{\\paren{B \\setminus \\set x} \\cup \\set y} = \\card {\\paren{ C \\setminus \\set x} \\cup X}$"}
+{"_id": "19530", "title": "Matroid Unique Circuit Property", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $X \\subseteq S$ be an independent subset of $M$. Let $x \\in S$ such that: :$X \\cup \\set x$ is a dependent subset of $M$. Then there exists a unique circuit $C$ such that: :$x \\in C \\subseteq X \\cup \\set x$"}
+{"_id": "19531", "title": "Absolute Value of Negative", "text": "Let $x \\in \\R$ be a real number. Then: :$\\size x = \\size {-x}$ where $\\size x$ denotes the absolute value of $x$."}
+{"_id": "19532", "title": "Dependent Subset Contains a Circuit", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $\\mathscr C$ denote the set of all circuits of $M$. Let $A$ be a dependent subset. Then: :$\\exists C \\in \\mathscr C : C \\subseteq A$"}
+{"_id": "19533", "title": "Condition for Linear Operation on Complex Numbers to be of Finite Order", "text": "Let $A$ be the operation on the complex numbers $\\C$ defined as: :$\\map A x = \\alpha x + \\beta$ Then $A$ is of finite order greater than $1$ {{iff}} $\\alpha$ is a root of unity other than $1$."}
+{"_id": "19534", "title": "Positive Rational Numbers under Division do not form Group", "text": "Let $\\struct {\\Q, /}$ denote the algebraic structure consisting of the set of rational numbers $\\Q$ under the operation $/$ of division. We have that $\\struct {\\Q, /}$ is not a group."}
+{"_id": "19535", "title": "Derivative of Inverse Hyperbolic Sine of x over a/Corollary 1", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\ln \\size {x + \\sqrt {x^2 + a^2} } } = \\dfrac 1 {\\sqrt {x^2 + a^2} }$"}
+{"_id": "19536", "title": "Negative of Logarithm of x plus Root x squared plus a squared", "text": "Let $x \\in \\R$ be a real number. Then: :$-\\map \\ln {x + \\sqrt {x^2 + a^2} } = \\map \\ln {-x + \\sqrt {x^2 + a^2} } - \\map \\ln {a^2}$"}
+{"_id": "19538", "title": "Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form/Corollary", "text": ":$\\displaystyle \\int \\frac {\\d x} {-\\sqrt {x^2 + a^2} } = \\ln \\size {x - \\sqrt {x^2 + a^2} } + C$"}
+{"_id": "19539", "title": "Derivative of Inverse Hyperbolic Cosine of x over a/Corollary 1", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\map \\ln {x + \\sqrt {x^2 - a^2} } } = \\dfrac 1 {\\sqrt {x^2 - a^2} }$ for $x > a$."}
+{"_id": "19540", "title": "Derivative of Inverse Hyperbolic Cosine of x over a/Corollary 2", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\map \\ln {x - \\sqrt {x^2 - a^2} } } = -\\dfrac 1 {\\sqrt {x^2 - a^2} }$ for $x > a$."}
+{"_id": "19541", "title": "Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form/Corollary", "text": ":$\\displaystyle \\int \\frac {\\d x} {-\\sqrt {x^2 - a^2} } = \\ln \\size {x - \\sqrt {x^2 - a^2} } + C$"}
+{"_id": "19542", "title": "Derivative of Inverse Hyperbolic Tangent of x over a/Corollary", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\dfrac 1 {2 a} \\map \\ln {\\dfrac {a + x} {a - x} } } = \\dfrac 1 {a^2 - x^2}$ where $\\size x < a$."}
+{"_id": "19543", "title": "Derivative of Inverse Hyperbolic Cotangent of x over a/Corollary", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\dfrac 1 {2 a} \\map \\ln {\\dfrac {x + a} {x - a} } } = \\dfrac 1 {a^2 - x^2}$ where $\\size x > a$."}
+{"_id": "19544", "title": "Leigh.Samphier/Sandbox/Set Difference of Matroid Circuit with Element is Independent", "text": ":$C \\setminus \\set x$ is an independent subset of $C$"}
+{"_id": "19545", "title": "Union of Matroid Base with Element of Complement is Dependent", "text": ":$B \\cup \\set x$ is a dependent superset of $B$"}
+{"_id": "19547", "title": "Primitive of Power of a x + b/Proof 2", "text": "{{:Primitive of Power of a x + b}} where $n \\ne 1$."}
+{"_id": "19549", "title": "Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form", "text": ":$\\displaystyle \\int \\dfrac {\\d x} {a^2 - x^2} = \\begin {cases} \\dfrac 1 a \\tanh^{-1} \\dfrac x a + C & : \\size x < a \\\\ & \\\\ \\dfrac 1 a \\coth^{-1} \\dfrac x a + C & : \\size x > a \\\\ & \\\\ \\text {undefined} & : x = a \\end {cases}$"}
+{"_id": "19550", "title": "Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form", "text": ":$\\displaystyle \\int \\frac {\\d x} {a^2 - x^2} = \\frac 1 a \\coth^{-1} \\frac x a + C$ where $\\size x > a$."}
+{"_id": "19552", "title": "Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a", "text": "Let $\\size x < a$. Then: :$\\displaystyle \\int \\frac {\\d x} {a^2 - x^2} = \\dfrac 1 {2 a} \\map \\ln {\\dfrac {a + x} {a - x} } + C$"}
+{"_id": "19556", "title": "Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x greater than a", "text": "Let $\\size x > a$. Then: :$\\displaystyle \\int \\frac {\\d x} {a^2 - x^2} = \\dfrac 1 {2 a} \\map \\ln {\\dfrac {x + a} {x - a} } + C$"}
+{"_id": "19558", "title": "Primitive of Reciprocal of a squared minus x squared/Logarithm Form", "text": "Let $a \\in \\R_{>0}$ be a strictly positive real constant. Let $x \\in \\R$ such that $\\size x \\ne a$."}
+{"_id": "19560", "title": "Primitive of Reciprocal of x squared minus a squared/Logarithm Form", "text": "Let $a \\in \\R_{>0}$ be a strictly positive real constant. Let $x \\in \\R$ such that $\\size x \\ne a$."}
+{"_id": "19563", "title": "Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^2 - a^2} = \\begin {cases} \\dfrac 1 {2 a} \\map \\ln {\\dfrac {a - x} {a + x} } + C & : \\size x < a\\\\ & \\\\ \\dfrac 1 {2 a} \\map \\ln {\\dfrac {x - a} {x + a} } + C & : \\size x > a \\\\ & \\\\ \\text {undefined} & : \\size x = a \\end {cases}$"}
+{"_id": "19564", "title": "Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x less than a", "text": "Let $\\size x < a$. Then: :$\\displaystyle \\int \\frac {\\d x} {x^2 - a^2} = \\dfrac 1 {2 a} \\map \\ln {\\dfrac {a - x} {a + x} } + C$"}
+{"_id": "19565", "title": "Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x greater than a", "text": "Let $\\size x > a$. Then: :$\\ds \\int \\frac {\\d x} {x^2 - a^2} = \\dfrac 1 {2 a} \\map \\ln {\\dfrac {x - a} {x + a} } + C$"}
+{"_id": "19566", "title": "Power Set is Closed under Complement", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Then: :$\\forall A \\in \\powerset S: \\relcomp S A \\in \\powerset S$ where $\\relcomp S A$ denotes the complement of $A$ relative to $S$."}
+{"_id": "19568", "title": "Parallelism is Reflexive Relation", "text": "Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \\in S$, let $l_1 \\parallel l_2$ denote that $l_1$ is parallel to $l_2$. Then $\\parallel$ is a reflexive relation on $S$."}
+{"_id": "19569", "title": "Parallelism is Symmetric Relation", "text": "Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \\in S$, let $l_1 \\parallel l_2$ denote that $l_1$ is parallel to $l_2$. Then $\\parallel$ is a symmetric relation on $S$."}
+{"_id": "19570", "title": "Parallelism is Equivalence Relation/Transitivity", "text": "Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \\in S$, let $l_1 \\parallel l_2$ denote that $l_1$ is parallel to $l_2$. Then $\\parallel$ is a transitive relation on $S$."}
+{"_id": "19572", "title": "Approximation/Examples/22 over 7", "text": "$\\dfrac {22} 7$ is a convenient approximation to $\\pi$: :$\\dfrac {22} 7 = 3 \\cdotp \\dot 14285 \\dot 7$"}
+{"_id": "19574", "title": "Perpendicularity is Antireflexive Relation", "text": "Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \\in S$, let $l_1 \\perp l_2$ denote that $l_1$ is perpendicular to $l_2$. Then $\\perp$ is an antireflexive relation on $S$."}
+{"_id": "19575", "title": "Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma", "text": ":$\\exists B_3 \\in \\mathscr B$: :::$V \\subseteq B_3$ :::$\\card{B_1 \\cap B_3} > \\card{B_1 \\cap B_2}$"}
+{"_id": "19577", "title": "Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 2", "text": "{{begin-eqn}} {{eqn | l = \\card {B_1}       | r = \\card{B_1 \\cap B_2} + \\card{B_1 \\setminus B_2} }} {{eqn | l = \\card {B_2}       | r = \\card{B_2 \\cap B_1} + \\card{V \\setminus B_1} + \\card{\\paren{B_2 \\setminus B_1} \\setminus V} }} {{end-eqn}}"}
+{"_id": "19578", "title": "Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 3", "text": ":$\\card{B_1 \\cap B_3} = \\card{B_1 \\cap B_2} + 1$"}
+{"_id": "19580", "title": "Equivalence of Versions of Axiom of Choice/Formulation 2 implies Formulation 1", "text": "The following formulation of the Axiom of Choice:"}
+{"_id": "19581", "title": "Equivalence of Versions of Axiom of Choice/Formulation 1 implies Formulation 3", "text": "The following formulation of the Axiom of Choice:"}
+{"_id": "19582", "title": "Equivalence of Versions of Axiom of Choice/Formulation 3 implies Formulation 1", "text": "The following formulation of the Axiom of Choice:"}
+{"_id": "19583", "title": "C^k Norm is Norm", "text": "Let $I = \\closedint a b$ be a closed real interval. Let $\\struct {\\map {\\CC^k} I, +, \\, \\cdot \\,}_\\R$ be the vector space of real-valued functions, k-times differentiable on $I$. Let $x \\in \\map {\\CC^k} I$ be a real-valued function of differentiability class $k$. Let $\\norm {\\, \\cdot \\,}_{\\map {C^k} I}$ be the $C^k$ norm on $I$. Then $\\norm {\\, \\cdot \\,}_{\\map {C^k} I}$ is a norm on $\\struct {\\map {\\CC^k} I, +, \\, \\cdot \\,}_\\R$."}
+{"_id": "19584", "title": "Existence of Minimal Polynomial for Square Matrix over Field", "text": "Let $K$ be a field. Let $n$ be a natural number. Let $K^{n \\times n}$ be the set of $n \\times n$ matrices over $K$.  Let $A \\in K^{n \\times n}$. Then the minimal polynomial of $A$ exists and has degree at most $n^2$."}
+{"_id": "19585", "title": "Homeomorphic Topologies on Same Set may not be Identical", "text": "Let $S$ be a set. Let $\\tau_1$ and $\\tau_2$ both be topologies on $S$ such that the topological spaces $\\struct {S, \\tau_1}$ and $\\struct {S, \\tau_2}$ are homeomorphic. Then it is not necessarily the case that $\\struct {S, \\tau_1} = \\struct {S, \\tau_1}$."}
+{"_id": "19587", "title": "Clopen Sets in Indiscrete Topology", "text": "The only subsets of $S$ which are both closed and open in $T$ are $S$ and $\\O$."}
+{"_id": "19588", "title": "Equivalence of Definitions of Connected Topological Space/No Separation iff No Clopen Sets", "text": "{{TFAE|def = Connected Topological Space}} Let $T = \\struct {S, \\tau}$ be a topological space."}
+{"_id": "19589", "title": "Dependent Subset of Independent Set Union Singleton Contains Singleton", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $X$ be an independent subset of $M$. Let $x \\in S$. Let $C$ be a dependent subset of $M$ such that: :$C \\subseteq X \\cup \\set x$. Then: :$x \\in C$"}
+{"_id": "19591", "title": "Matroid Unique Circuit Property/Corollary", "text": "Let $B$ be a base of $M$. Let $x \\in S \\setminus B$. Then there exists a unique circuit $C$ such that: :$x \\in C \\subseteq B \\cup \\set x$ That is, $C$ is the fundamental circuit of $x$ in $B$."}
+{"_id": "19592", "title": "Leigh.Samphier/Sandbox/Matroid Base Substitution From Fundamental Circuit", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $B$ be a base of $M$. Let $y \\in S \\setminus B$. Let $\\map C {y,B}$ denote the fundamental circuit of $y$ in $B$. Let $x \\in B$. Then: :$\\paren{B \\setminus \\set x} \\cup \\set y$ is a base of $M$ {{iff}} $x \\in \\map C {y,B}$"}
+{"_id": "19593", "title": "Socrates is Mortal/Variant", "text": ":$(1): \\quad$ ''If {{AuthorRef|Socrates}} is a man then {{AuthorRef|Socrates}} is mortal.'' :$(2): \\quad$ ''{{AuthorRef|Socrates}} is a man.'' :$(3): \\quad$ ''Therefore {{AuthorRef|Socrates}} is mortal.''"}
+{"_id": "19594", "title": "Self-Distributive Law for Conditional/Forward Implication/Formulation 2", "text": ":$\\paren {p \\implies \\paren {q \\implies r} } \\implies \\paren {\\paren {p \\implies q} \\implies \\paren {p \\implies r} }$"}
+{"_id": "19595", "title": "Self-Distributive Law for Conditional/Reverse Implication/Formulation 2", "text": ":$\\vdash \\paren {\\paren {p \\implies q} \\implies \\paren {p \\implies r} } \\implies \\paren {p \\implies \\paren {q \\implies r} }$"}
+{"_id": "19599", "title": "Odd Integers under Addition do not form Group", "text": "Let $S$ be the set of odd integers: :$S = \\set {x \\in \\Z: \\exists n \\in \\Z: x = 2 n + 1}$ Let $\\struct {S, +}$ denote the algebraic structure formed by $S$ under the operation of addition. Then $\\struct {S, +}$ is not a group."}
+{"_id": "19600", "title": "Number of Digits in Number", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive integer. Let $b \\in \\Z_{>1}$ be an integer greater than $1$. Let $n$ be expressed in base $b$. Then the number of digits $d$ in this expression for $n$ is: :$d = 1 + \\floor {\\log_b n}$ where: :$\\floor {\\, \\cdot \\,}$ denotes the floor function :$\\log_b$ denotes the logarithm to base $b$."}
+{"_id": "19601", "title": "Every Tenth Power of Two Minus Every Third Power of Ten is Divisible By Three", "text": "Let $x \\in \\Z_{\\ge 0}$ be a non-negative integer. Then $2^{10 x} - 10^{3 x}$ is divisible by $3$. That is: :$2^{10 x} - 10^{3 x} \\equiv 0 \\pmod 3$"}
+{"_id": "19603", "title": "Gödel's Incompleteness Theorems/Second", "text": "Let $T$ be the set of theorems of some recursive set of sentences in the language of arithmetic such that $T$ contains minimal arithmetic. Let $\\map {\\mathrm {Cons} } T$ be the propositional function which states that $T$ is consistent. Then it is not possible to prove $\\map {\\mathrm {Cons} } T$ by means of formal statements within $T$ itself."}
+{"_id": "19604", "title": "Linear Function on Stationary Stochastic Model is Stationary", "text": "Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\\sequence {s_n} = \\tuple {z_1, z_2, \\dotsb, z_n}$ Let $L_t$ be a linear function of $\\sequence {s_n}$: :$L_t = l_1 z_t + l_2 z_{t - 1} + \\dotsb + l_n z_{t - n + 1}$ Then $L_t$ is itself stationary."}
+{"_id": "19608", "title": "Strict Ordering on Integers is Well-Defined", "text": "Let $\\eqclass {a, b} {}$ denote an integer, as defined by the formal definition of integers. Let: {{begin-eqn}} {{eqn | l = \\eqclass {a, b} {}       | r = \\eqclass {a', b'} {}       | c =  }} {{eqn | l = \\eqclass {c, d} {}       | r = \\eqclass {c', d'} {}       | c =  }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \\eqclass {a, b} {}       | o = <       | r = \\eqclass {c, d} {}       | c =  }} {{eqn | ll= \\iff       | l = \\eqclass {a', b'} {}       | o = <       | r = \\eqclass {c', d'} {}       | c =  }} {{end-eqn}}"}
+{"_id": "19609", "title": "Definition:Gaussian Process", "text": "Let $S$ be a stochastic process giving rise to a time series $T$. Let the probability distribution of $T$ be a multivariate normal distribution. Then $S$ is called a '''Gaussian process'''."}
+{"_id": "19612", "title": "Second Order Weakly Stationary Gaussian Stochastic Process is Strictly Stationary", "text": "Let $S$ be a Gaussian stochastic process giving rise to a time series $T$. Let $S$ be weakly stationary of order $2$. Then $S$ is strictly stationary of order $2$."}
+{"_id": "19613", "title": "Ordering on Integers is Transitive", "text": "Let $\\eqclass {a, b} {}$ denote an integer, as defined by the formal definition of integers. Then: {{begin-eqn}} {{eqn | l = \\eqclass {a, b} {}       | o = \\le       | r = \\eqclass {c, d} {}       | c =  }} {{eqn | lo= \\land       | l = \\eqclass {c, d} {}       | o = \\le       | r = \\eqclass {e, f} {}       | c =  }} {{eqn | ll= \\implies       | l = \\eqclass {a, b} {}       | o = \\le       | r = \\eqclass {e, f} {}       | c =  }} {{end-eqn}} That is, ordering on the integers is transitive."}
+{"_id": "19614", "title": "Strict Ordering on Integers is Asymmetric", "text": "Let $\\eqclass {a, b} {}$ denote an integer, as defined by the formal definition of integers. Then: {{begin-eqn}} {{eqn | l = \\eqclass {a, b} {}       | o = <       | r = \\eqclass {c, d} {}       | c =  }} {{eqn | lo= \\implies       | l = \\eqclass {c, d} {}       | o = \\not <       | r = \\eqclass {a, b} {}       | c =  }} {{end-eqn}} That is, strict ordering on the integers is asymmetric."}
+{"_id": "19617", "title": "Negative of Integer", "text": "Let $x \\in \\Z$ be an integer. Let $x = \\eqclass {a, b} {}$ be defined from the formal definition of integers, where $\\eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers. Then: :$-x = \\eqclass {b, a} {}$"}
+{"_id": "19618", "title": "Symmetric Function Theorem", "text": "Let $f$ be a polynomial in $n$ variables. Let $f$ be of degree $r$ in each of its $n$ variables. Then $f$ is equal to a polynomial of total degree $r$ with integer coefficients in the elementary symmetric functions: :$ds \\sum x_i \\sim x_i x_j, \\dotsc, \\prod x_j$ and the coefficients of $f$."}
+{"_id": "19619", "title": "Euler's Integral Theorem", "text": ":$\\ds H_n = \\ln n + \\gamma + \\map \\OO {\\dfrac 1 n}$ where: :$H_n$ denotes the $n$th harmonic number :$\\gamma$ denotes the Euler-Mascheroni constant."}
+{"_id": "19625", "title": "Ordering on Positive Integers is Equivalent to Ordering on Natural Numbers", "text": "Let $u, v \\in \\Z_{>0}$ be natural numbers. Consider the mapping $\\phi: \\N_{>0} \\to \\Z_{>0}$ defined as: :$\\forall u \\in \\N_{>0}: \\map \\phi u = u'$ where $u' \\in \\Z$ denotes the (strictly) positive integer $\\eqclass {b + u, b} {}$. Let $u', v' \\in \\Z_{>0}$ be strictly positive integers. Then: :$u > v \\iff u' > v'$"}
+{"_id": "19626", "title": "Product of Absolute Values of Integers", "text": "Let $a, b \\in \\Z$ be integers. Let $\\size a$ denote the absolute value of $a$: :$\\size a = \\begin {cases} a & : a \\ge 0 \\\\ -a : a < 0 \\end {cases}$ Then: :$\\size a \\times \\size b = \\size {a \\times b}$"}
+{"_id": "19628", "title": "Displacement of Particle under Force", "text": "Let $P$ be a particle of constant mass $m$. Let the position of $P$ at time $t$ be specified by the position vector $\\mathbf r$. Let a force applied to $P$ be represented by the vector $\\mathbf F$. Then the motion of $P$ can be given by the differential equation: :$\\mathbf F = m \\dfrac {\\d^2 \\mathbf r} {\\d t^2}$ or using Newtonian notation: :$\\mathbf F = m \\ddot {\\mathbf r}$"}
+{"_id": "19629", "title": "1-Sequence Space is Separable", "text": "$\\ell^1$ space is a separable space."}
+{"_id": "19630", "title": "Primitive of Root of Function under Half its Derivative", "text": "Let $f$ be a real function which is integrable. Then: :$\\ds \\int \\frac {\\map {f'} x} {2 \\sqrt {\\map f x} } \\rd x = \\sqrt {\\map f x} + C$ where $C$ is an arbitrary constant."}
+{"_id": "19635", "title": "Primitive of Hyperbolic Secant Function/Arctangent of Half Hyperbolic Tangent Form", "text": ":$\\ds \\int \\sech x \\rd x =  2 \\map \\arctan {\\tanh \\dfrac x 2} + C$"}
+{"_id": "19636", "title": "First Order ODE/y' + 2 x y = 1", "text": "The first order ODE: :$y' + 2 x y = 1$ has the general solution: :$y = e^{-{x^2} } \\ds \\int_a^x e^{t^2} \\rd t$ where $a$ is an arbitrary constant."}
+{"_id": "19637", "title": "Electric Potential over Conducting Surface is Constant", "text": "Let $S$ be a conducting surface. The electric potential $V$ over $S$ is constant. This can be expressed using the Laplacian: :$\\nabla^2 V = 0$ and is thus seen to satisfy Laplace's equation."}
+{"_id": "19638", "title": "Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 1 Iff Definition 4", "text": "Let $S$ be a finite set. Let $\\mathscr B$ be a non-empty set of subsets of $S$. {{TFAE|def=Base Axiom (Matroid)|view = Matroid Base Axiom}}"}
+{"_id": "19639", "title": "Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 4 Iff Definition 5", "text": "Let $S$ be a finite set. Let $\\mathscr B$ be a non-empty set of subsets of $S$. {{TFAE|def=Base Axiom (Matroid)|view = Matroid Base Axiom}}"}
+{"_id": "19640", "title": "Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 3 Iff Definition 7", "text": "Let $S$ be a finite set. Let $\\mathscr B$ be a non-empty set of subsets of $S$. {{TFAE|def=Base Axiom (Matroid)|view = Matroid Base Axiom}}"}
+{"_id": "19641", "title": "Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 1 Iff Definition 3", "text": "Let $S$ be a finite set. Let $\\mathscr B$ be a non-empty set of subsets of $S$. {{TFAE|def=Base Axiom (Matroid)|view = Matroid Base Axiom}}"}
+{"_id": "19644", "title": "P-Norm is Norm/P-Sequence Space", "text": "The $p$-norm on the $p$-sequence space is a norm."}
+{"_id": "19645", "title": "P-Norm is Norm/Complex Numbers", "text": "The $p$-norm on the complex numbers is a norm."}
+{"_id": "19646", "title": "P-Norm is Norm/Real Numbers", "text": "The $p$-norm on the real numbers is a norm."}
+{"_id": "19650", "title": "Space of Bounded Sequences with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space", "text": "Let $\\ell^\\infty$ be the space of bounded sequences. Let $\\struct {\\C, +_\\C, \\times_\\C}$ be the field of complex numbers. Let $\\paren +$ be the pointwise addition on the ring of sequences. Let $\\paren {\\, \\cdot \\,}$ be the pointwise multiplication on the ring of sequences. Then $\\struct {\\ell^\\infty, +, \\, \\cdot \\,}_\\C$ is a vector space."}
+{"_id": "19651", "title": "Zero Vector has no Direction", "text": "A zero vector has no direction."}
+{"_id": "19652", "title": "Power Set is Closed under Countable Unions", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Then: :$\\forall A_n \\in \\powerset S: n = 1, 2, \\ldots: \\ds \\bigcup_{n \\mathop = 1}^\\infty A_n \\in \\powerset S$"}
+{"_id": "19655", "title": "Probability of Union of Disjoint Events is Sum of Individual Probabilities", "text": "Let $\\EE$ be an experiment. Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability measure on $\\EE$. Then: :$\\forall A, B \\in \\Sigma: A \\cap B = \\O \\implies \\map \\Pr {A \\cup B} = \\map \\Pr A + \\map \\Pr B$"}
+{"_id": "19657", "title": "Probability Measure on Finite Sample Space", "text": "Let $\\Omega = \\set {\\omega_1, \\omega_2, \\ldots, \\omega_n}$ be a finite set. Let $\\Sigma$ be a $\\sigma$-algebra on $\\Omega$. Let $p_1, p_2, \\ldots, p_n$ be non-negative real numbers such that: :$p_1 + p_2 + \\cdots + p_n = 1$ Let $Q: \\Sigma \\to \\R$ be the mapping defined as: :$\\forall A \\in \\Sigma: \\map Q A = \\ds \\sum_{i: \\omega_i \\in A} p_i$ Then $\\struct {\\Omega, \\Sigma, Q}$ constitutes a probability space. That is, $Q$ is a probability measure on $\\struct {\\Omega, \\Sigma}$."}
+{"_id": "19660", "title": "Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms", "text": "Let $S$ be a finite set. Let $\\rho : \\powerset S \\to \\Z$ be a mapping from the power set of $S$ to the integers. {{TFAE|def=Rank Axioms (Matroid)|view=Matroid Rank Axioms}}"}
+{"_id": "19661", "title": "Length of Element of Arc in Orthogonal Curvilinear Coordinates", "text": "Let $\\tuple {q_1, q_2, q_3}$ denote a set of orthogonal curvilinear coordinates. Let the relation between those orthogonal curvilinear coordinates and Cartesian coordinates be expressed as: {{begin-eqn}} {{eqn | l = x       | r = \\map x {q_1, q_2, q_3} }} {{eqn | l = y       | r = \\map y {q_1, q_2, q_3} }} {{eqn | l = z       | r = \\map z {q_1, q_2, q_3} }} {{end-eqn}} where $\\tuple {x, y, z}$ denotes the Cartesian coordinates. The length $\\d l$ of a small arc is given by: :$\\d l = {h_1}^2 {\\d q_1}^2 + {h_2}^2 {\\d q_2}^2 + {h_3}^2 {\\d q_3}^2$ where: :${h_i}^2 = \\paren {\\dfrac {\\partial x} {\\partial q_i} }^2 + \\paren {\\dfrac {\\partial y} {\\partial q_i} }^2 + \\paren {\\dfrac {\\partial z} {\\partial q_i} }^2$"}
+{"_id": "19662", "title": "Laplacian of Function in Orthogonal Curvilinear Coordinates", "text": "Let $\\map \\psi {q_1, q_2, q_3}$ denote a real-valued function embedded in an orthogonal curvilinear coordinate system. Then the Laplacian of $\\psi$ can be expressed as: :$\\nabla^2 \\psi = \\dfrac 1 {h_1 h_2 h_3} \\paren {\\map {\\dfrac \\partial {\\partial q_1} } {\\dfrac {h_2 h_3} {h_1} \\dfrac {\\partial \\psi} {\\partial q_1} } + \\map {\\dfrac \\partial {\\partial q_2} } {\\dfrac {h_3 h_1} {h_2} \\dfrac {\\partial \\psi} {\\partial q_2} } + \\map {\\dfrac \\partial {\\partial q_3} } {\\dfrac {h_1 h_2} {h_3} \\dfrac {\\partial \\psi} {\\partial q_3} } }$ where: :${h_i}^2 = \\paren {\\dfrac {\\partial x} {\\partial q_i} }^2 + \\paren {\\dfrac {\\partial y} {\\partial q_i} }^2 + \\paren {\\dfrac {\\partial z} {\\partial q_i} }^2$"}
+{"_id": "19663", "title": "Pressure of Gas for Density and Temperature", "text": "The pressure, density and temperature of a body of gas are linked by the equation: :$p = k \\rho \\paren {1 + \\alpha t}$ where: :$p$ denotes the pressure of the gas :$\\rho$ denotes the density of the gas :$t$ denotes thetemperature of the gas :$k$ and $\\alpha$ are constants."}
+{"_id": "19666", "title": "Leigh.Samphier/Sandbox/Matroid Satisfies Circuit Axioms", "text": "Let $S$ be a finite set. Let $\\mathscr C$ be a non-empty set of subsets of $S$. Then $\\mathscr C$ is the set of circuits of a matroid on $S$ {{iff}} $\\mathscr C$ satisfies the circuit axioms: {{: Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 1}}"}
+{"_id": "19667", "title": "Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms", "text": "Let $S$ be a finite set. Let $\\mathscr C$ be a non-empty set of subsets of $S$. {{TFAE|def=Circuit Axioms (Matroid)|view=Matroid Circuit Axioms}}"}
+{"_id": "19668", "title": "Leigh.Samphier/Sandbox/Rank of Matroid Circuit is One Less Than Cardinality", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $C \\subseteq S$ be a circuit of $M$. Let $\\rho: \\powerset S \\to \\Z$ denote the rank function of $M$. Then: :$\\map \\rho C = \\card C -1$"}
+{"_id": "19669", "title": "Leigh.Samphier/Sandbox/Bound for Cardinality of Matroid Circuit", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $C \\subseteq S$ be a circuit of $M$. Let $\\rho: \\powerset S \\to \\Z$ denote the rank function of $M$. Then: :$\\card C \\le \\map \\rho S + 1$"}
+{"_id": "19670", "title": "Leigh.Samphier/Sandbox/Matroid with No Circuits Has Single Base", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid with no circuits. Then: :$S$ is the only base on $M$."}
+{"_id": "19671", "title": "Leigh.Samphier/Sandbox/Proper Subset of Matroid Circuit is Independent", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $C \\subseteq S$ be a circuit of $M$. Then: :every proper subset $A$ of $C$ is independent"}
+{"_id": "19672", "title": "Supremum Norm is Norm/Space of Bounded Sequences", "text": "The supremum norm on the space of bounded sequences is a norm."}
+{"_id": "19673", "title": "Parallelogram Law for Vector Subtraction", "text": "Let $\\mathbf u$ and $\\mathbf v$ be vectors. Consider a parallelogram, two of whose adjacent sides represent $\\mathbf y$ and $\\mathbf v$ (in magnitude and direction). :400px Then the diagonal of the parallelogram connecting the terminal points of $\\mathbf u$ and $\\mathbf v$ represents the magnitude and direction of $\\mathbf u - \\mathbf v$, the difference of $\\mathbf u$ and $\\mathbf v$."}
+{"_id": "19674", "title": "Space of Bounded Sequences with Supremum Norm forms Normed Vector Space", "text": "Space of bounded sequences with supremum norm forms a normed vector space."}
+{"_id": "19675", "title": "Riesz's Lemma", "text": "Let $X$ be a normed vector space. Let $Y$ be a proper closed linear subspace of $X$.  Let $\\alpha \\in \\openint 0 1$. Then there exists $x_\\alpha \\in X$ such that:  :$\\norm {x_\\alpha} = 1$ with: :$\\norm {x_\\alpha - y} > \\alpha$ for all $y \\in Y$."}
+{"_id": "19676", "title": "Normed Vector Space is Finite Dimensional iff Unit Sphere is Compact", "text": "Let $X$ be a normed vector space. Let $S = \\map {S_1} 0$ be the unit sphere centred at $0$ in $X$. Then $X$ is finite dimensional {{iff}} $S$ is compact."}
+{"_id": "19677", "title": "Finite Dimensional Subspace of Normed Vector Space is Closed", "text": "Let $V$ be a normed vector space. Let $W$ be a finite dimensional subspace of $V$.  Then $W$ is closed."}
+{"_id": "19678", "title": "Catalan's Conjecture", "text": "The only solution to the Diophantine equation: :$x^a - y^b = 1$ for $a, b > 1$ and $x, y > 0$, is: :$x = 3, a = 2, y = 2, b = 3$"}
+{"_id": "19679", "title": "Total Force on Charged Particle from 2 Charged Particles", "text": "Let $p_1$, $p_2$ and $p_3$ be charged particles. Let $q_1$, $q_2$ and $q_3$ be the electric charges on $p_1$, $p_2$ and $p_3$ respectively. Let $\\mathbf F_{ij}$ denote the force exerted on $q_j$ by $q_i$. Let $\\mathbf F_i$ denote the force exerted on $q_i$ by the combined action of the other two charged particles. Then the force $\\mathbf F_1$ exerted on $q_1$ by the combined action of $q_2$ and $q_3$ is given by: {{begin-eqn}} {{eqn | l = \\mathbf F_1       | r = \\mathbf F_{21} + \\mathbf F_{31}       | c =  }} {{eqn | r = \\dfrac {q_2 q_1} {4 \\pi \\varepsilon_0 r_{2 1}^3} \\mathbf r_{2 1} + \\dfrac {q_3 q_1} {4 \\pi \\varepsilon_0 r_{3 1}^3} \\mathbf r_{3 1}       | c =  }} {{end-eqn}} where: :$\\mathbf F_{21} + \\mathbf F_{31}$ denotes the vector sum of $\\mathbf F_{21}$ and $\\mathbf F_{31}$ :$\\mathbf r_{ij}$ denotes the displacement from $p_i$ to $p_j$ :$r_{ij}$ denotes the distance between $p_i$ and $p_j$ :$\\varepsilon_0$ denotes the vacuum permittivity."}
+{"_id": "19680", "title": "Total Force on Charged Particle from Multiple Charged Particles", "text": "Let $p_1, p_2, \\ldots, p_n$ be charged particles. Let $q_1, q_2, \\ldots, q_n$ be the electric charges on $p_1, p_2, \\ldots, p_n$ respectively. For all $i$ in $\\set {1, 2, \\ldots, n}$ where $i \\ne j$, let $\\mathbf F_{i j}$ denote the force exerted on $q_j$ by $q_i$. For all $i$ in $\\set {1, 2, \\ldots, n}$, let $\\mathbf F_i$ denote the force exerted on $q_i$ by the combined action of all the other charged particles. Then the force $\\mathbf F_i$ exerted on $q_i$ by the combined action of all the other charged particles is given by: {{begin-eqn}} {{eqn | l = \\mathbf F_i       | r = \\sum_{\\substack {1 \\mathop \\le j \\mathop \\le n \\\\ i \\mathop \\ne j} } \\mathbf F_{j i}       | c =  }} {{eqn | r = \\dfrac 1 {4 \\pi \\varepsilon_0} \\sum_{\\substack {1 \\mathop \\le j \\mathop \\le n \\\\ i \\mathop \\ne j} } \\dfrac {q_i q_j} {r_{j i}^3} \\mathbf r_{j i}       | c =  }} {{end-eqn}} where: :the summation denotes the vector sum of $\\mathbf F_{21}$ and $\\mathbf F_{31}$ :$\\mathbf r_{ij}$ denotes the displacement from $p_i$ to $p_j$ :$r_{ij}$ denotes the distance between $p_i$ and $p_j$ :$\\varepsilon_0$ denotes the vacuum permittivity."}
+{"_id": "19682", "title": "Space of Continuously Differentiable on Closed Interval Real-Valued Functions with C^1 Norm forms Normed Vector Space", "text": "Space of Continuously Differentiable on Closed Interval Real-Valued Functions with $C^1$ norm forms a normed vector space."}
+{"_id": "19683", "title": "Ring of Endomorphisms is Ring with Unity", "text": "Let $\\struct {G, \\oplus}$ be an abelian group. Let $\\mathbb G$ be the set of all group endomorphisms of $\\struct {G, \\oplus}$. Let $\\struct {\\mathbb G, \\oplus, *}$ denote the '''ring of endomorphisms''' on $\\struct {G, \\oplus}$. Then $\\struct {\\mathbb G, \\oplus, *}$ is a ring with unity."}
+{"_id": "19687", "title": "Space of Continuously Differentiable on Closed Interval Real-Valued Functions with C^1 Norm is Banach Space", "text": "Let $I := \\closedint a b$ be a closed real interval. Let $\\map \\CC I$ be the space of real-valued functions, continuous on $I$. Let $\\map {\\CC^1} I$ be the space of real-valued functions, continuously differentiable on $I$. Let $\\norm {\\, \\cdot \\,}_{1, \\infty}$ be the $\\CC^1$ norm. $\\struct {\\map {\\CC^1} I, \\norm {\\, \\cdot \\,}_{1, \\infty} }$ be the normed space of real-valued functions, continuously differentiable on $I$. Then $\\struct {\\map {\\CC^1} I, \\norm {\\, \\cdot \\,}_{1, \\infty} }$ is a Banach space."}
+{"_id": "19690", "title": "Set of Order m times n Matrices does not form Ring", "text": "Let $m, n \\in \\N_{>0}$ be non-zero natural numbers such that $m > n$. Let $S$ be the set of all matrices of order $m \\times n$. Then the algebraic structure $\\struct {S, +, \\times}$ is not a ring. Note that $\\times$ denotes conventional matrix multiplication."}
+{"_id": "19691", "title": "Set of Order 3 Vectors under Cross Product does not form Ring", "text": "Let $S$ be the set of all vectors in a vector space of dimension $3$. Let $\\times$ denote the cross product operation. Then the algebraic structure $\\struct {S, +, \\times}$ is not a ring."}
+{"_id": "19692", "title": "Integers under Subtraction do not form Semigroup", "text": "Let $\\struct {\\Z, -}$ denote the algebraic structure formed by the set of integers under the operation of subtraction. Then $\\struct {\\Z, -}$ is not a semigroup."}
+{"_id": "19693", "title": "Leigh.Samphier/Sandbox/Matroid satisfies Rank Axioms/Necessary Condition", "text": "Let $S$ be a finite set. Let $\\rho : \\powerset S \\to \\Z$ be the rank function of a matroid on $S$. Then $\\rho$ satisfies the rank axioms: {{:Definition:Rank Axioms (Matroid)/Definition 1}}"}
+{"_id": "19694", "title": "Leigh.Samphier/Sandbox/Matroid satisfies Rank Axioms/Sufficient Condition", "text": "Let $S$ be a finite set. Let $\\rho : \\powerset S \\to \\Z$ be a mapping from the power set of $S$ to the integers. Let $\\rho$ satisfy the rank axioms: {{:Definition:Rank Axioms (Matroid)/Definition 1}} Then $\\rho$ is the rank function of a matroid on $S$."}
+{"_id": "19695", "title": "Leigh.Samphier/Sandbox/Independent Subset is Contained in Maximal Independent Subset/Corollary", "text": "Let $M = \\struct{S, \\mathscr I}$ be a matroid. Let $A \\subseteq S$. Let $X$ be a maximal independent subset of $A$. Then: :$\\card X = \\map \\rho A$ where $\\rho$ is the rank function on $M$."}
+{"_id": "19696", "title": "Motion of Body with Constant Mass", "text": "Let $B$ be a body with constant mass $m$ undergoing a force $\\mathbf F$. Then the equation of motion of $B$ is given by: :$\\mathbf F = m \\mathbf a$ where $\\mathbf a$ is the acceleration of $B$."}
+{"_id": "19697", "title": "Like Vector Quantities are Multiples of Each Other", "text": "Let $\\mathbf a$ and $\\mathbf b$ be like vector quantities. Then: :$\\mathbf a = \\dfrac {\\size {\\mathbf a} } {\\size {\\mathbf b} } \\mathbf b$ where: :$\\size {\\mathbf a}$ denotes the magnitude of $\\mathbf a$ :$\\dfrac {\\size {\\mathbf a} } {\\size {\\mathbf b} } \\mathbf b$ denotes the scalar product of $\\mathbf b$ by $\\dfrac {\\size {\\mathbf a} } {\\size {\\mathbf b} }$."}
+{"_id": "19698", "title": "Direct Product Norm is Norm", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,}}$ and $\\struct {Y, \\norm {\\, \\cdot \\,}}$ be normed vector spaces. Let $V = X \\times Y$ be a direct product of vector spaces $X$ and $Y$ together with induced component-wise operations. Let $\\norm {\\tuple {x, y} }$ be the direct product norm. Then $\\norm {\\tuple {x, y} }$ is a norm on $V$."}
+{"_id": "19699", "title": "Vector Quantity as Scalar Product of Unit Vector Quantity", "text": "Let $\\mathbf a$ be a vector quantity. Then: :$\\mathbf a = \\size {\\mathbf a} \\mathbf {\\hat a}$ where: :$\\size {\\mathbf a}$ denotes the magnitude of $\\mathbf a$ :$\\mathbf {\\hat a}$ denotes the unit vector in the direction $\\mathbf a$."}
+{"_id": "19700", "title": "Scalar Product of Magnitude by Unit Vector Quantity", "text": "Let $\\mathbf a$ be a vector quantity. Let $m$ be a scalar quantity. Then: :$m \\mathbf a = m \\paren {\\size {\\mathbf a} \\hat {\\mathbf a} } = \\paren {m \\size {\\mathbf a} } \\hat {\\mathbf a}$ where: :$\\size {\\mathbf a}$ denotes the magnitude of $\\mathbf a$ :$\\hat {\\mathbf a}$ denotes the unit vector in the direction $\\mathbf a$."}
+{"_id": "19702", "title": "Vector Quantity can be Expressed as Sum of 3 Non-Coplanar Vectors", "text": "Let $\\mathbf r$ be a vector quantity embedded in space. Let $\\mathbf a$, $\\mathbf b$ and $\\mathbf c$ be non-coplanar. Then $\\mathbf r$ can be expressed uniquely as the resultant of $3$ vector quantities which are each parallel to one of $\\mathbf a$, $\\mathbf b$ and $\\mathbf c$."}
+{"_id": "19703", "title": "Vectors are Equal iff Components are Equal", "text": "Two vector quantities are equal {{iff}} they have the same components."}
+{"_id": "19705", "title": "Components of Vector in terms of Direction Cosines", "text": "Let $\\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space. Let $\\mathbf i$, $\\mathbf j$ and $\\mathbf k$ be the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively. Let $\\cos \\alpha$, $\\cos \\beta$ and $\\cos \\gamma$ be the direction cosines of $\\mathbf r$ with respect to the $x$-axis, $y$-axis and $z$-axis respectively. Let $x$, $y$ and $z$ be the components of $\\mathbf r$ in the $\\mathbf i$, $\\mathbf j$ and $\\mathbf k$ directions respectively. Let $r$ denote the magnitude of $\\mathbf r$, that is: :$r := \\size {\\mathbf r}$ Then: {{begin-eqn}} {{eqn | l = x       | r = r \\cos \\alpha }} {{eqn | l = y       | r = r \\cos \\beta }} {{eqn | l = z       | r = r \\cos \\gamma }} {{end-eqn}}"}
+{"_id": "19706", "title": "Magnitude of Vector Quantity in terms of Components", "text": "Let $\\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space. Let $\\mathbf r$ be expressed in terms of its components: :$\\mathbf r = x \\mathbf i + y \\mathbf j + z \\mathbf k$ where $\\mathbf i$, $\\mathbf j$ and $\\mathbf k$ denote the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively. Then: :$\\size {\\mathbf r} = \\sqrt {x^2 + y^2 + z^2}$ where $\\size {\\mathbf r}$ denotes the magnitude of $\\mathbf r$."}
+{"_id": "19707", "title": "Components of Zero Vector Quantity are Zero", "text": "Let $\\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space. Let $\\mathbf r$ be expressed in terms of its components: :$\\mathbf r = x \\mathbf i + y \\mathbf j + z \\mathbf k$ Let $\\mathbf r$ be the zero vector. Then: :$x = y = z = 0$"}
+{"_id": "19708", "title": "Unit Vector in terms of Direction Cosines", "text": "Let $\\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space. Let $\\mathbf i$, $\\mathbf j$ and $\\mathbf k$ be the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively. Let $\\cos \\alpha$, $\\cos \\beta$ and $\\cos \\gamma$ be the direction cosines of $\\mathbf r$ with respect to the $x$-axis, $y$-axis and $z$-axis respectively. Let $\\mathbf {\\hat r}$ denote the unit vector in the direction of $\\mathbf r$. Then: :$\\mathbf {\\hat r} = \\paren {\\cos \\alpha} \\mathbf i + \\paren {\\cos \\beta} \\mathbf j + \\paren {\\cos \\gamma} \\mathbf k$"}
+{"_id": "19709", "title": "Unit Vectors of Cartesian 3-Space form Basis", "text": "Consider the Cartesian $3$-space $C$. Let $\\mathbf i$, $\\mathbf j$ and $\\mathbf k$ denote the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively. Then $\\set {\\mathbf i, \\mathbf j, \\mathbf k}$ forms a basis for $C$."}
+{"_id": "19710", "title": "Vectors from Sum and Difference", "text": "Let $\\mathbf a$ and $\\mathbf b$ be vector quantities. Let $\\mathbf c = \\mathbf a + \\mathbf b$ and $\\mathbf d = \\mathbf a - \\mathbf b$ be given. Then: {{begin-eqn}} {{eqn | l = \\mathbf a       | r = \\dfrac 1 2 \\paren {\\mathbf c + \\mathbf d} }} {{eqn | l = \\mathbf b       | r = \\dfrac 1 2 \\paren {\\mathbf c - \\mathbf d} }} {{end-eqn}}"}
+{"_id": "19713", "title": "Leigh.Samphier/Sandbox/Matroid satisfies Rank Axioms/Sufficient Condition/Lemma", "text": ":$\\forall Y \\subseteq S: \\map \\rho Y \\le \\card Y$"}
+{"_id": "19714", "title": "Dot Product of Perpendicular Vectors", "text": "Let $\\mathbf a$ and $\\mathbf b$ be vector quantities. Let $\\mathbf a$ and $\\mathbf b$ be perpendicular. Then: :$\\mathbf a \\cdot \\mathbf b = 0$ where $\\cdot$ denotes dot product."}
+{"_id": "19715", "title": "Dot Product of Orthogonal Basis Vectors", "text": "Let $\\tuple {\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}$ be an orthonormal basis of a vector space $V$. Then: :$\\forall i, j \\in \\set {1, 2, \\ldots, n}: \\mathbf e_i \\cdot \\mathbf e_j = \\delta_{i j}$ where: :$\\mathbf e_i \\cdot \\mathbf e_j$ denotes the dot product of $\\mathbf e_i$ and $\\mathbf e_j$ :$\\delta_{i j}$ denotes the Kronecker delta."}
+{"_id": "19719", "title": "Second Partial Derivative wrt r of ln (r^2 + s)", "text": ":$\\dfrac {\\partial^2} {\\partial r^2} \\map \\ln {r^2 + s} = \\dfrac {2 \\paren {s - r^2} } {\\paren {r^2 + s}^2}$"}
+{"_id": "19721", "title": "Derivative of Square of Tangent", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\tan^2 x} = 2 \\tan x \\sec^2 x$ when $\\cos x \\ne 0$."}
+{"_id": "19728", "title": "Partial Derivatives of x^y^z", "text": "Let: :$u = x^{\\paren {y^z} }$ Then: {{begin-eqn}} {{eqn | l = \\dfrac {\\partial u} {\\partial x}       | r = y^z x^{\\paren {y^z - 1} } }} {{eqn | l = \\dfrac {\\partial u} {\\partial y}       | r = x^{y^z} z y^{z - 1} \\ln x }} {{eqn | l = \\dfrac {\\partial u} {\\partial z}       | r = x^{\\paren {y^z} } y^z \\ln x \\ln y }} {{end-eqn}}"}
+{"_id": "19729", "title": "Partial Derivatives of x^u + u^y", "text": "Let: :$u = x^u + u^y$ Then: {{begin-eqn}} {{eqn | l = \\dfrac {\\partial u} {\\partial x}       | r = \\frac {u^2} {x \\paren {1 - u \\ln x - y} } }} {{eqn | l = \\dfrac {\\partial u} {\\partial y}       | r = \\frac {u \\ln u} {1 - u \\ln x - y} }} {{end-eqn}}"}
+{"_id": "19731", "title": "Third Partial Derivatives of x^y", "text": "Let: :$u = x^y$ Then: :$\\dfrac {\\partial^3 u} {\\partial x^2 \\partial y} = \\dfrac {\\partial^3 u} {\\partial x \\partial y \\partial x}$"}
+{"_id": "19732", "title": "Number of Terms in Homogeneous Polynomial", "text": "The number of terms in a homogeneous polynomial of degree $n$ in $m$ indeterminates is given by: :$N = \\dbinom {n + m - 1} n = \\dfrac {\\paren {n + m - 1}!}{n! \\, \\paren {m - 1}!}$"}
+{"_id": "19734", "title": "Definition:Set", "text": "A '''set''' is intuitively defined as any aggregation of objects, called elements, which can be precisely defined in some way or other. We can think of each set as a single entity in itself, and we can denote it (and usually do) by means of a single symbol. That is, ''anything you care to think of'' can be a set. This concept is known as the comprehension principle. However, there are problems with the comprehension principle. If we allow it to be used without any restrictions at all, paradoxes arise, the most famous example probably being Russell's Paradox. Hence some sources define a '''set''' as a ''' 'well-defined' collection of objects''', leaving the concept of what constitutes well-definition to later in the exposition."}
+{"_id": "19735", "title": "Definition:Element", "text": "Let $S$ be a set. An '''element of $S$''' is a member of $S$."}
+{"_id": "19736", "title": "Definition:Zermelo-Fraenkel Axioms", "text": "=== The Axiom of Extension === {{:Axiom:Axiom of Extension/Set Theory}} === The Axiom of the Empty Set === {{:Axiom:Axiom of Empty Set/Set Theory}} === The Axiom of Pairing === {{:Axiom:Axiom of Pairing/Set Theory}} === The Axiom of Specification === {{:Axiom:Axiom of Specification/Set Theory}} === The Axiom of Unions === {{:Axiom:Axiom of Unions/Set Theory}} === The Axiom of Powers === {{:Axiom:Axiom of Powers/Set Theory}} === The Axiom of Infinity === {{:Axiom:Axiom of Infinity/Set Theory}} === The Axiom of Replacement === {{:Axiom:Axiom of Replacement}} === The Axiom of Foundation === {{:Axiom:Axiom of Foundation}}"}
+{"_id": "19737", "title": "Definition:Empty Set", "text": "The '''empty set''' is a set which has no elements. That is, $x \\in \\O$ is false, whatever $x$ is."}
+{"_id": "19738", "title": "Definition:Singleton", "text": "A '''singleton''' is a set that contains exactly one element."}
+{"_id": "19739", "title": "Definition:Subset", "text": "Let $S$ and $T$ be sets. $S$ is a '''subset''' of a set $T$ {{iff}} all of the elements of $S$ are also elements of $T$. This is denoted: :$S \\subseteq T$ That is: :$S \\subseteq T \\iff \\forall x: \\paren {x \\in S \\implies x \\in T}$ If the elements of $S$ are not all also elements of $T$, then $S$ is not a '''subset''' of $T$: :$S \\nsubseteq T$ means $\\neg \\paren {S \\subseteq T}$"}
+{"_id": "19740", "title": "Definition:Set Union", "text": "Let $S$ and $T$ be sets. The '''(set) union''' of $S$ and $T$ is the set $S \\cup T$, which consists of all the elements which are contained in either (or both) of $S$ and $T$: :$x \\in S \\cup T \\iff x \\in S \\lor x \\in T$"}
+{"_id": "19741", "title": "Definition:Set Intersection", "text": "Let $S$ and $T$ be sets. The '''(set) intersection''' of $S$ and $T$ is written $S \\cap T$. It means the set which consists of all the elements which are contained in both of $S$ and $T$: :$x \\in S \\cap T \\iff x \\in S \\land x \\in T$"}
+{"_id": "19742", "title": "Definition:Set Difference", "text": "The '''(set) difference''' between two sets $S$ and $T$ is written $S \\setminus T$, and means the set that consists of the elements of $S$ which are not elements of $T$: :$x \\in S \\setminus T \\iff x \\in S \\land x \\notin T$"}
+{"_id": "19743", "title": "Definition:Set Equality", "text": "Let $S$ and $T$ be sets. === Definition 1 === {{:Definition:Set Equality/Definition 1}} === Definition 2 === {{:Definition:Set Equality/Definition 2}}"}
+{"_id": "19744", "title": "Definition:Statement", "text": "A '''statement''' is a sentence which has objective and logical meaning."}
+{"_id": "19745", "title": "Definition:Proposition", "text": "A '''proposition''' is a statement which is offered up for investigation as to its truth or falsehood. Loosely, a '''proposition''' is a statement which is about to be proved (or disproved)."}
+{"_id": "19746", "title": "Definition:Truth Value", "text": "=== Aristotelian Logic === {{:Definition:Truth Value/Aristotelian Logic}} {{expand|Other logic types}}"}
+{"_id": "19747", "title": "Definition:Logic", "text": "'''Logic''' is the study of the structure of statements and their truth values, divorced from their conceptual content. It has frequently been defined as ''' the science of reasoning'''."}
+{"_id": "19748", "title": "Definition:Atom (Logic)", "text": "In a particular branch of logic, certain concepts are at such a basic level of simplicity they can not be broken down into anything simpler. Those concepts are called '''atoms''' or described as '''atomic'''. Different branches of logic admit different '''atoms'''. === Propositional Logic === In propositional logic, the '''atoms''' are statements."}
+{"_id": "19749", "title": "Definition:Subject", "text": "The '''subject''' of a simple statement in logic is the part of the statement specifying the object being talked about. The '''subject''' of a simple statement is atomic in predicate logic. The '''subject''' and predicate of a simple statement are referred to as its terms."}
+{"_id": "19750", "title": "Definition:Predicate", "text": "The '''predicate''' of a simple statement in logic is the part of the statement which defines ''what is being said'' about the subject. It is a word or phrase which, when combined with one or more names of objects, turns into a meaningful sentence. The predicates of simple statements are atomic in predicate logic. The subject and predicate of a simple statement are referred to as its terms."}
+{"_id": "19751", "title": "Definition:Simple Statement", "text": "A '''simple statement''' is a statement which has one subject and one predicate. For example, the statement: :'''London is the capital of England''' is a '''simple statement'''. '''London''' is the subject and '''is the capital of England''' is the predicate."}
+{"_id": "19752", "title": "Definition:Logical Term", "text": "Both the subject and the predicate of a simple statement are referred to as its '''(logical) terms'''. Note that this use of the word '''term''' is a more specialized use of the word term as used in algebra."}
+{"_id": "19753", "title": "Definition:Object", "text": "An '''object''' is a ''thing''. Everything that can be talked about in mathematics and logic can be referred to as an '''object'''."}
+{"_id": "19754", "title": "Definition:Symbol", "text": "In its broadest possible sense: :A '''symbol''' is an object used to represent another object. In a narrower and more \"mathematical\" sense, a '''symbol''' is a sign of a particular shape to which is assigned a meaning, and is used to represent a concept or identify a particular object. It is generally much more convenient to use a symbol than the plain speaking that it replaces, because it is invariably more compact. One character can replace a large number of words. As definitions become more complex, the symbols tend to convey more information -- but by the same coin, understanding exactly what a symbol means becomes more difficult. Symbols may mean different things in different contexts. A symbol that means something in one context may mean something completely different in another. This is because the number of different concepts is greater than human ingenuity can create symbols for, and some of them naturally have to be used more than once. This does not matter as long as, before we use any symbol, we define exactly what we mean by it. Some symbols are standard and rarely need defining, but in some contexts there are subtle differences to the ''exact'' meaning of a \"standard\" symbol. Therefore all fields of mathematics generally introduce themselves with a rash of definitions, many of which are symbols."}
+{"_id": "19755", "title": "Definition:Variable/Predicate Logic", "text": "In the context of predicate logic, a variable is often called an '''object variable''' or '''arbitrary name'''. As such, it is a symbol which is assigned to an ''arbitrarily selected'' object from a given universe of discourse. The understanding is that (during the scope of the argument to which it is relevant) the arbitrary name could apply equally well to ''any'' of the objects in that universe."}
+{"_id": "19756", "title": "Definition:Operation/Arity", "text": "The '''arity''' of an operation is the number of operands it uses. The '''arity''' of an operation may be, in general, any number. It may even be infinite."}
+{"_id": "19757", "title": "Definition:Function", "text": "The process which is symbolised by an operation is called a '''function'''. The operand(s) of the operation can be considered to be the input(s). The output of the '''function''' is whatever the operation is defined as doing with the operand(s). A '''function''' is in fact another name for a mapping, but while the latter term is used in the general context of set theory and abstract algebra, the term '''function''' is generally reserved for mappings between sets of numbers."}
+{"_id": "19758", "title": "Definition:Equals", "text": ":$x = y$ means '''$x$ is the same object as $y$''', and is read '''$x$ equals $y$''', or '''$x$ is equal to $y$'''. :$x \\ne y$ means '''$x$ is not the same object as $y$''', and is read '''$x$ is not equal to $y$'''. The expression: :$a = b$ means: :$a$ and $b$ are names for the same object."}
+{"_id": "19759", "title": "Definition:Symbolic Logic", "text": "'''Symbolic logic''' is the study of logic in which the logical form of statements is analyzed by using symbols as tools. Instead of explicit statements, logical formulas are investigated, which are symbolic representations of statements, and compound statements in particular. In '''symbolic logic''', the rules of reasoning and logic are investigated by means of formal systems, which form a good foundation for the symbolic manipulations performed in this field."}
+{"_id": "19760", "title": "Definition:Propositional Logic", "text": "'''Propositional logic''' is a sub-branch of symbolic logic in which the truth values of propositional formulas are investigated and analysed. The atoms of propositional logic are simple statements. There are various systems of propositional logic for determining the truth values of propositional formulas, for example: * Natural deduction is a technique for deducing valid sequents from other valid sequents by applying precisely defined proof rules, each of which themselves are either \"self-evident\" axioms or themselves derived from other valid sequents. * The Method of Truth Tables, which consists of the construction of one or more truth tables which exhaustively list all the possible truth values of all the statement variables with a view to determining the required answer by inspection."}
+{"_id": "19761", "title": "Definition:Predicate Logic", "text": "'''Predicate logic''' is a sub-branch of symbolic logic. It is an extension of propositional logic in which the internal structure of simple statements is analyzed. Thus in '''predicate logic''', simple statements are no longer atomic. The atoms of '''predicate logic''' are subjects and predicates of simple statements. There are various formal systems allowing for rigid determination of the theorems of '''predicate logic''':"}
+{"_id": "19762", "title": "Definition:Statement Label", "text": "A '''statement label''' is a symbol which is assigned to a ''particular'' statement, so that it can be identified without the need to write it out in full. For such conventions to make sense, different statements must always be given different '''statement labels'''. The citing of a statement label can be interpreted as an assertion that the statement represented by that symbol is true. That is: :'''$P$''' means :'''$P \\text { is true}$'''"}
+{"_id": "19763", "title": "Definition:Logical Connective", "text": "A '''logical connective''' is an object which either modifies a statement, or combines existing statements into a new statement, called a compound statement. It is almost universal to identify a '''logical connective''' with the symbol representing it. Thus, '''logical connective''' may also, particularly in symbolic logic, be used to refer to that symbol, rather than speaking of a '''connective symbol''' separately. In mathematics, '''logical connectives''' are considered to be '''truth-functional'''.  That is, the truth value of a compound statement formed using the '''connective''' is assumed to depend ''only'' on the truth value of the comprising statements. Thus, as far as the '''connective''' is concerned, it does not matter what the comprising statements precisely ''are''. As a consequence of this truth-functionality, a '''connective''' has a corresponding truth function, which goes by the same name as the '''connective''' itself. The arity of this truth function is the number of statements the '''logical connective''' combines into a single compound statement. === Unary Logical Connective === {{:Definition:Logical Connective/Unary}} === Binary Logical Connective === {{:Definition:Logical Connective/Binary}}"}
+{"_id": "19764", "title": "Definition:Natural Language", "text": "A '''natural language''' is one of the conventional, everyday languages in which people usually communicate. Although there are many natural languages in the world, we are not generally going to distinguish between them, merely lumping them all into the one concept.  When '''natural language''' is referred to on {{ProofWiki}}, it will usually mean English."}
+{"_id": "19765", "title": "Definition:Compound Statement", "text": "A '''compound statement''' is a statement which results from the application of one or more logical connectives to a collection of simple statements. === Substatement === {{:Definition:Compound Statement/Substatement}} === Ill-Formed Compound Statement === {{:Definition:Compound Statement/Ill-Formed}}"}
+{"_id": "19766", "title": "Definition:Compound Statement/Substatement", "text": "A '''substatement''' of a compound statement is one of the statements that comprise it."}
+{"_id": "19767", "title": "Definition:Statement Form", "text": "A '''statement form''' is a symbolic representation of a compound statement. It consists of statement variables along with logical connectives joining them. It is traditional, particularly in the field of mathematical logic, to use lowercase Greek letters to stand for general formulas (the usual ones being $\\phi, \\psi$ and $\\chi$), but more modern treatments are starting to use ordinary lowercase letters of the English alphabet, usually $p, q, r$ etc. === Specific Form === {{:Definition:Statement Form/Specific Form}}"}
+{"_id": "19768", "title": "Definition:Logical Formula", "text": "Let $\\mathcal L$ be a formal language used in the field of symbolic logic. Then the well-formed formulas of $\\mathcal L$ are often referred to as '''logical formulas'''. They are symbolic representations of statements, and often of compound statements in particular."}
+{"_id": "19769", "title": "Definition:Logical Connective/Unary", "text": "A '''unary logical connective''' (or '''one-place connective''') is a connective whose effect on its compound statement is determined by the truth value of ''one'' substatement. In standard Aristotelian logic, there are four of these. The only non-trivial one is logical not, as shown on Unary Truth Functions."}
+{"_id": "19770", "title": "Definition:Logical Connective/Binary", "text": "A '''binary logical connective''' (or '''two-place connective''') is a connective whose effect on its compound statement is determined by the truth value of ''two'' substatements. In standard Aristotelian logic, there are 16 '''binary logical connectives''', cf. Binary Truth Functions. In the field of symbolic logic, the following four (symbols for) '''binary logical connectives''' are commonly used: * Conjunction: the '''And''' connective $p \\land q$: '''$p$ is true ''and'' $q$ is true'''. * Disjunction: the '''Or''' connective $p \\lor q$: '''$p$ is true ''or'' $q$ is true, ''or possibly both'''''. * The conditional connective $p \\implies q$: '''''If'' $p$ is true, ''then'' $q$ is true'''. * The biconditional connective $p \\iff q$: '''$p$ is true ''if and only if'' $q$ is true''', or '''$p$ ''is equivalent to'' $q$'''."}
+{"_id": "19771", "title": "Definition:Axiom", "text": "In all contexts, the definition of the term '''axiom''' is by and large the same. That is, an '''axiom''' is a statement which is ''accepted'' as being true. A statement that is considered an '''axiom''' can be described as being '''axiomatic'''."}
+{"_id": "19772", "title": "Definition:Logical Argument", "text": "A '''logical argument''' (or just '''argument''') is a process of creating a new statement from one or more existing statements. An '''argument''' proceeds from a set of premises to a conclusion, by means of logical implication, via a procedure called logical inference. An '''argument''' may have more than one premise, but only one conclusion. While statements may be classified as either '''true''' or '''false''', an '''argument''' may be classified as either valid or invalid."}
+{"_id": "19773", "title": "Definition:Assumption", "text": "An '''assumption''' is a statement which is introduced into an argument, whose truth value is (temporarily) accepted as True. In mathematics, the keyword '''let''' is often the indicator here that an assumption is going to be introduced. For example: :'''Let $p$''' (be true) ... can be interpreted, in natural language, as: :'''Let us assume, for the sake of argument, that $p$ is true''' ..."}
+{"_id": "19774", "title": "Definition:Premise", "text": "A '''premise''' is an assumption that is used as a basis from which to start to construct an argument. When the validity or otherwise of a proof is called into question, one may request the arguer to \"check your premises\"."}
+{"_id": "19775", "title": "Definition:Conclusion", "text": "A '''conclusion''' is a statement that is obtained as the result of the process of an argument."}
+{"_id": "19776", "title": "Definition:Valid Argument", "text": "A '''valid argument''' is a logical argument in which the premises provide conclusive reasons for the conclusion. When a proof is valid, we may say one of the following: * The conclusion '''follows from''' the premises; * The premises '''entail''' the conclusion; * The conclusion is true '''on the strength of''' the premises; * The conclusion is '''drawn from''' the premises; * The conclusion is '''deduced from''' the premises; * The conclusion is '''derived from''' the premises."}
+{"_id": "19777", "title": "Definition:Proof", "text": "A '''proof''' is another name for a valid argument, but in this context the assumption is made that the premises are all true. That is, a valid argument that has one or more false premises is not a proof."}
+{"_id": "19778", "title": "Definition:Invalid Argument", "text": "An '''invalid argument''' is a argument in which the premises do ''not'' provide conclusive reasons for the conclusion."}
+{"_id": "19779", "title": "Definition:Depend", "text": "A statement $q$ '''depends upon''' another statement $p$ if the truth value of $q$ is ''influenced'' (in some way) by that of $p$, but is ''not necessarily'' implied by that of $p$. Some authors prefer '''rests on''' for '''depends upon'''."}
+{"_id": "19780", "title": "Definition:Therefore", "text": "If statement $p$ logically implies statement $q$, then we may say: :'''$p$, therefore $q$'''. The symbology: :$p, q \\vdash r$ means: :'''Given as premises $p$ and $q$, we may validly conclude $r$''' So the symbol $\\vdash$ is interpreted to mean '''therefore'''. Thus, $p, q \\vdash r$ reads as: :'''$p$ and $q$, therefore $r$.''' A fallacy may be indicated by $p, q \\not \\vdash r$, which can be interpreted as: :'''Given as premises $p$ and $q$, we may ''not'' validly conclude $r$.'''"}
+{"_id": "19781", "title": "Definition:Because", "text": "If statement $p$ logically implies statement $q$, then we may say: : '''$q$, because $p$.''' The symbol $\\dashv$ is interpreted to mean '''because'''. Thus: : $r \\dashv p, q$ means: :'''Given as premises $p$ and $q$, we may validly conclude $r$''' or :'''$r$, because $p$ and $q$.'''"}
+{"_id": "19782", "title": "Definition:Fallacy", "text": "A '''(logical) fallacy''' is a mistake caused by an application of an invalid argument. '''Logical fallacies''' abound. Most of the documented '''fallacies''' that can be found in the literature arise from linguistic imprecision or deliberately misleading statements, or both. Frequently the commission of '''fallacies''' is as deliberate as the telling of lies."}
+{"_id": "19783", "title": "Definition:Sequent", "text": "A '''sequent''' is an expression in the form: :$\\phi_1, \\phi_2, \\ldots, \\phi_n \\vdash \\psi$ where $\\phi_1, \\phi_2, \\ldots, \\phi_n$ are premises (any number of them), and $\\psi$ the conclusion (only one), of an argument."}
+{"_id": "19785", "title": "Definition:Zermelo-Fraenkel Set Theory", "text": "'''Zermelo-Fraenkel Set Theory''' is a system of axiomatic set theory upon which the whole of (at least conventional) mathematics can be based. Its basis consists of a system of Aristotelian logic, appropriately axiomatised, together with the Zermelo-Fraenkel axioms of set theory. These are as follows: {{:Definition:Zermelo-Fraenkel Axioms}}"}
+{"_id": "19786", "title": "Definition:Axiomatic Set Theory", "text": "'''Axiomatic set theory''' is a system of set theory which differs from so-called naive set theory in that the sets which are allowed to be generated are strictly constrained by the axioms."}
+{"_id": "19787", "title": "Definition:ZFC", "text": "'''Zermelo-Fraenkel Set Theory with the Axiom of Choice''' is a system of axiomatic set theory upon which the whole of (at least conventional) mathematics can be based. Its basis consists of a system of Aristotelian logic, appropriately axiomatised, together with the Zermelo-Fraenkel axioms of Set Theory and the (controversial) Axiom of Choice. These are as follows: {{:Definition:Zermelo-Fraenkel Axioms}} === The Axiom of Choice === {{:Axiom:Axiom of Choice}}"}
+{"_id": "19788", "title": "Definition:Set Theory", "text": "'''Set Theory''' is the branch of mathematics which studies sets."}
+{"_id": "19789", "title": "Definition:Logical Not", "text": "The '''logical not''' or '''negation''' operator is a unary connective whose action is to reverse the truth value of the statement on which it operates. :$\\neg p$ is defined as: '''$p$ is not true''', or '''It is not the case that $p$ is true'''. Thus the statement $\\neg p$ is called the '''negation''' of $p$. $\\neg p$ is voiced '''not $p$'''."}
+{"_id": "19790", "title": "Definition:Conjunction", "text": "'''Conjunction''' is a binary connective written symbolically as $p \\land q$ whose behaviour is as follows: : $p \\land q$ is defined as: :'''$p$ is true ''and'' $q$ is true.''' This is called the '''conjunction''' of $p$ and $q$. The statements $p$ and $q$ are known as: : the '''conjuncts''' : the '''members of the conjunction'''. $p \\land q$ is voiced: :'''$p$ and $q$'''."}
+{"_id": "19791", "title": "Definition:Disjunction", "text": "'''Disjunction''' is a binary connective written symbolically as $p \\lor q$ whose behaviour is as follows: : $p \\lor q$ is defined as: :'''Either $p$ is true ''or'' $q$ is true ''or possibly both''.''' This is called the '''disjunction''' of $p$ and $q$. $p \\lor q$ is voiced: :'''$p$ or $q$'''"}
+{"_id": "19792", "title": "Definition:Conditional", "text": "The '''conditional''' or '''implication''' is a binary connective: :$p \\implies q$ defined as: :'''''If'' $p$ is true, ''then'' $q$ is true.''' This is known as a '''conditional statement'''. A '''conditional statement''' is also known as a '''conditional proposition''' or just a '''conditional'''. $p \\implies q$ is voiced: :'''if $p$ then $q$''' or: :'''$p$ implies $q$'''"}
+{"_id": "19793", "title": "Definition:Biconditional", "text": "The '''biconditional''' is a binary connective: :$p \\iff q$ defined as: :$\\paren {p \\implies q} \\land \\paren {q \\implies p}$ That is: :'''''If'' $p$ is true, ''then'' $q$ is true, ''and if'' $q$ is true, ''then'' $p$ is true'''."}
+{"_id": "19794", "title": "Definition:Parenthesis", "text": "'''Parenthesis''' is a syntactical technique to disambiguate the meaning of a logical formula. It allows one to specify that a logical formula should (temporarily) be regarded as being a single entity, being on the same level as a statement variable. Such a formula is referred to as being '''in parenthesis'''. Typically, a formal language, in defining its formal grammar, ensures by means of '''parenthesis''' that all of its well-formed words are uniquely readable. Generally, '''brackets''' are used to indicate that certain formulas are '''in parenthesis'''. The brackets that are mostly used are round ones, the '''left (round) bracket''' $($ and the '''right (round) bracket''' $)$."}
+{"_id": "19795", "title": "Definition:Binding Priority", "text": "The '''binding priority''' is the convention defining the order of '''binding strength''' of the individual connectives in a logical formula. '''Binding priorities''' can be overridden by using parenthesis in appropriate places. Parenthesis always takes priority over conventional '''binding priorities'''."}
+{"_id": "19796", "title": "Definition:True", "text": "A statement has a truth value of '''true''' {{iff}} what it says matches the way that things are."}
+{"_id": "19797", "title": "Definition:False", "text": "A statement has a truth value of '''false''' {{iff}} what it says does not match the way that things are."}
+{"_id": "19798", "title": "Definition:Contingent Statement", "text": "A '''contingent statement''' is a statement form which is neither a tautology, nor unsatisfiable, but whose truth value depends upon the truth value of its component substatements. <!-- == Formal Definition == Let $\\mathfrak M$ be a collection of models for a particular formal language $\\LL$. A well-formed word $\\mathbf A$ of $\\LL$ is said to be '''contingent (for $\\mathfrak M$)''' {{iff}}: :$\\exists \\MM_1, \\MM_2 \\in \\mathfrak M: \\MM_1 \\models \\mathbf A \\land \\MM_2 \\not \\models \\mathbf A$ that is, {{iff}} some models in $\\mathfrak M$ approve, and others disapprove of $\\mathbf A$. == Also known as == {{refactor|Seems to be replaceable by Also see entries. But, will need source review}} In the context of propositional formulas the term '''satisfiable''' is usually used: A propositional formula is '''satisfiable''' if its value is True in at least ''one'' boolean interpretation. A propositional formula is '''not-valid''' or '''falsifiable''' if its value is False in at least ''one'' boolean interpretation. === Set of Logical Formulas === Let $U = \\set {P_1, P_2, \\ldots, P_n}$ be a set of propositional formulas. Let $U' = \\set {p_1, p_2, \\ldots, p_m}$ be the set of all the atoms of all the propositional formulas in $U$. (Some of these atoms, and indeed this will most likely be the case, may be in more than one logical formula.) Then $U$ is '''(mutually) satisfiable''' if there exists a boolean interpretation $v$ for all the atoms in $U'$ such that: :$\\map v {P_1} = \\map v {P_2} = \\cdots = \\map v {P_n} = \\T$. -->"}
+{"_id": "19799", "title": "Definition:Natural Deduction", "text": "'''Natural deduction''' is a technique for deducing valid sequents from other valid sequents by applying precisely defined proof rules, each of which themselves are either \"self-evident\" axioms or themselves derived from other valid sequents, by a technique called logical inference."}
+{"_id": "19800", "title": "Definition:Natural Deduction/Proof Rule", "text": "A '''proof rule''' is a rule in natural deduction which allows one to infer the validity of propositional formulas from other propositional formulas. === Rule of Substitution === {{:Rule of Substitution}} === Rule of Sequent Introduction === {{:Rule of Sequent Introduction}} === Rule of Theorem Introduction === {{:Rule of Theorem Introduction}}"}
+{"_id": "19801", "title": "Definition:Logical Inference", "text": "'''Logical inference''' is the process used in natural deduction to deduce the validity of statement forms from other statement forms by use of proof rules. Given a set of logical formulae and the proof rules, we '''(logically) infer''' other formulas."}
+{"_id": "19802", "title": "Definition:Pool of Assumptions", "text": "The '''pool of assumptions''', for a formula deduced in a particular proof by natural deduction, is the collection of all the assumptions upon which the formula depends. These consist of premises and possibly some intermediate assumptions. The assumptions in this pool are called, naturally enough, the '''pooled assumptions'''. The '''pooled assumptions''' at the end of a proof (its conclusion) are thus seen to be the premises."}
+{"_id": "19803", "title": "Definition:Aristotelian Logic", "text": "'''Aristotelian logic''' is a system of logic which is based upon the philosophy of {{AuthorRef|Aristotle}}. It forms the cornerstone of the entirety of classical logic. The school of '''Aristotelian logic''' consists almost entirely of the relationships between the various categorical syllogisms. This school of philosophy forms the basis of mainstream mathematics, although, for example, mathematicians of the intuitionistic school do not accept the Law of the Excluded middle value."}
+{"_id": "19804", "title": "Definition:Hypothesis", "text": "A '''hypothesis''' is a statement that is made whose truth value has not been established, but which is ''believed'' to be true. The underlying idea is that its truth or falsehood is '''about to be investigated'''. A hypothesis is considered by some branches of philosophy as being synonymous with the words '''wild guess'''[http://www.experiencefestival.com/a/Hypothesis/id/183571]. Possibly the most famous still-unproved hypothesis is the Riemann Hypothesis."}
+{"_id": "19806", "title": "Definition:Discharged Assumption", "text": "The context of this definition is in the process of deducing the validity of sequents using logical inference in the framework of natural deduction. Assumptions that are made during the course of using certain proof rules remain in force only so long as the use of those rules requires them. Once the appropriate rules have been completed, these are known as '''discharged assumptions''', and are not included in the pool of assumptions on which the conclusion of the rule depends."}
+{"_id": "19807", "title": "Definition:Corollary", "text": "A '''corollary''' is a proof which is a direct result, or a direct application, of another proof. It can be considered as being a '''proof for free''' on the back of a proof which has been paid for with blood, sweat and tears."}
+{"_id": "19809", "title": "Definition:Corresponding Conditional", "text": "Let $p_1, p_2, p_3, \\ldots, p_n \\vdash q$ be a sequent of propositional logic. It can be expressed as a theorem as follows: : $\\vdash p_1 \\implies \\left({p_2 \\implies \\left({p_3 \\implies \\left({\\ldots \\implies \\left({p_n \\implies q}\\right) \\ldots }\\right)}\\right)}\\right)$ This is known as the sequent's '''corresponding conditional'''. This is proved in the Extended Rule of Implication. {{wtd|This pertains probably to the classical interpretation of propositional calculus, and needs to be reformulated and put in the correct category in due time}} {{SUBPAGENAME}} fpshpqhmzdo2bsytg3qk4e0pvlsyucd"}
+{"_id": "19810", "title": "Definition:By Hypothesis", "text": "When demonstrating a proof, it is frequently necessary to refer to the specification of the result which is to be proved. This result often contains wording along the lines '''Suppose that ...''' or '''Let ...''' It is convenient to refer back to these specifications during the course of the proof. To do that, the term '''by hypothesis''' is often used."}
+{"_id": "19811", "title": "Definition:Proper Name", "text": "A '''proper name''' (or just '''name''') is a symbol or collection of symbols used to identify a ''particular'' object uniquely. In contrast with natural language, a proper name has a wider range than being the particular identifying label attached to a particular single entity (be it a person, or a place, or whatever else). For example: : '''Sloth''' is a '''proper name''' for ''the concept'' of ''being lazy''. : '''Rain''' is a '''proper name''' for ''the meteorological phenomenon of water falling from the sky''."}
+{"_id": "19812", "title": "Definition:Property", "text": "A '''property''' is a concept that specifies an aspect of an object. In the phrase ''living thing'', ''living'' is a property that ''thing'' has. It creates a distinction between ''things'' that are ''living'' and ''things'' that are ''not living''."}
+{"_id": "19813", "title": "Definition:Abelian Group", "text": "=== Definition 1 === {{:Definition:Abelian Group/Definition 1}} === Definition 2 === {{:Definition:Abelian Group/Definition 2}}"}
+{"_id": "19815", "title": "Definition:Commutative", "text": "=== Commuting Elements === {{:Definition:Commutative/Elements}} === Commutative Operation === {{:Definition:Commutative/Operation}} === Commuting Set of Elements === {{:Definition:Commutative/Set}} === Commutative Algebraic Structure === {{:Definition:Commutative/Algebraic Structure}}"}
+{"_id": "19816", "title": "Definition:Anticommutative", "text": "=== Structure with One Operation === {{:Definition:Anticommutative/Structure with One Operation}} === Structure with Two Operations === {{:Definition:Anticommutative/Structure with Two Operations}}"}
+{"_id": "19817", "title": "Definition:Normal Subgroup", "text": "Let $G$ be a group. Let $N$ be a subgroup of $G$. $N$ is a '''normal subgroup of $G$''' {{iff}}: === Definition 1 === {{:Definition:Normal Subgroup/Definition 1}} === Definition 2 === {{:Definition:Normal Subgroup/Definition 2}} === Definition 3 === {{:Definition:Normal Subgroup/Definition 3}} === Definition 4 === {{:Definition:Normal Subgroup/Definition 4}} === Definition 5 === {{:Definition:Normal Subgroup/Definition 5}} === Definition 6 === {{:Definition:Normal Subgroup/Definition 6}} === Definition 7 === {{:Definition:Normal Subgroup/Definition 7}}"}
+{"_id": "19818", "title": "Definition:Idempotence", "text": "'''Idempotence''' is a property of an algebraic system, or an element of an algebraic system such that: :$E \\circledcirc E = E$ for an object $E$ and an operation $\\circledcirc$."}
+{"_id": "19819", "title": "Definition:Subgroup", "text": "Let $\\struct {G, \\circ}$ be an algebraic structure. $\\struct {H, \\circ}$ is a '''subgroup''' of $\\struct {G, \\circ}$ {{iff}}: :$(1): \\quad \\struct {H, \\circ}$ is a group :$(2): \\quad H$ is a subset of $G$."}
+{"_id": "19820", "title": "Definition:Group", "text": "A '''group''' is a semigroup with an identity (that is, a monoid) in which every element has an inverse."}
+{"_id": "19821", "title": "Definition:Center (Abstract Algebra)/Group", "text": "The '''center of a group''' $G$, denoted $\\map Z G$, is the subset of elements in $G$ that commute with every element in $G$. Symbolically: :$\\map Z G = \\map {C_G} G = \\set {g \\in G: g x = x g, \\forall x \\in G}$ That is, the '''center''' of $G$ is the centralizer of $G$ in $G$ itself."}
+{"_id": "19822", "title": "Definition:Distributive Operation", "text": "Let $S$ be a set on which is defined ''two'' binary operations, defined on all the elements of $S \\times S$, which we will denote as $\\circ$ and $*$. The operation $\\circ$ '''is distributive over''' $*$, or '''distributes over''' $*$, {{iff}}: :$\\circ$ is right distributive over $*$ and: :$\\circ$ is left distributive over $*$."}
+{"_id": "19823", "title": "Definition:Self Distributive", "text": "Let $\\circ$ be a binary operation on the set $S$. Then $\\circ$ is '''self-distributive''' iff: * $\\forall a, b, c \\in S: \\left({a \\circ b}\\right) \\circ c = \\left ({a \\circ c}\\right) \\circ \\left({b \\circ c}\\right)$ * $\\forall a, b, c \\in S: a \\circ \\left({b \\circ c}\\right) = \\left({a \\circ b}\\right) \\circ \\left({a \\circ c}\\right)$"}
+{"_id": "19824", "title": "Definition:Ordered Pair", "text": "The definition of a set does not take any account of the order in which the elements are listed. That is, $\\set {a, b} = \\set {b, a}$, and the elements $a$ and $b$ have the same status - neither is distinguished above the other as being more \"important\". === Informal Definition === {{:Definition:Ordered Pair/Informal Definition}} === Kuratowski Formalization === {{:Definition:Ordered Pair/Kuratowski Formalization}} === Empty Set Formalization === {{:Definition:Ordered Pair/Empty Set Formalization}} === Wiener Formalization === {{:Definition:Ordered Pair/Wiener Formalization}}"}
+{"_id": "19825", "title": "Definition:Cartesian Product", "text": "Let $S$ and $T$ be sets or classes. The '''cartesian product''' $S \\times T$ of $S$ and $T$ is the set (or class) of ordered pairs $\\tuple {x, y}$ with $x \\in S$ and $y \\in T$: :$S \\times T = \\set {\\tuple {x, y}: x \\in S \\land y \\in T}$"}
+{"_id": "19826", "title": "Definition:Relation", "text": "Let $S \\times T$ be the cartesian product of two sets $S$ and $T$. A '''relation''' on $S \\times T$ is an ordered triple: :$\\RR = \\tuple {S, T, R}$ where $R \\subseteq S \\times T$ is a subset of the Cartesian product of $S$ and $T$."}
+{"_id": "19827", "title": "Definition:Mapping", "text": "Let $S$ and $T$ be sets. Let $S\\times T$ be their cartesian product. === Definition 1 === {{:Definition:Mapping/Definition 1}} === Definition 2 === {{:Definition:Mapping/Definition 2}} === Definition 3 === {{:Definition:Mapping/Definition 3}} === Definition 4 === {{:Definition:Mapping/Definition 4}}"}
+{"_id": "19828", "title": "Definition:Composition of Relations", "text": "Let $\\RR_1 \\subseteq S_1 \\times T_1$ and $\\RR_2 \\subseteq S_2 \\times T_2$ be relations. Then the '''composite of $\\RR_1$ and $\\RR_2$''' is defined and denoted as: :$\\RR_2 \\circ \\RR_1 := \\set {\\tuple {x, z} \\in S_1 \\times T_2: \\exists y \\in S_2 \\cap T_1: \\tuple {x, y} \\in \\RR_1 \\land \\tuple {y, z} \\in \\RR_2}$"}
+{"_id": "19829", "title": "Definition:Algebraic Structure", "text": "An '''algebraic structure''' is an ordered tuple: :$\\struct {S, \\circ_1, \\circ_2, \\ldots, \\circ_n}$ where $S$ is a set which has one or more binary operations $\\circ_1, \\circ_2, \\ldots, \\circ_n$ defined on all the elements of $S \\times S$. An '''algebraic structure''' with one (binary) operation is thus an ordered pair which can be denoted $\\struct {S, \\circ}$ or $\\struct {T, *}$, and so on."}
+{"_id": "19830", "title": "Definition:Restriction", "text": "=== Restriction of a Relation === {{:Definition:Restriction/Relation}} === Restriction of a Mapping === {{:Definition:Restriction/Mapping}} === Restriction of an Operation === In the same way that a restriction is defined on a relation, it can be defined on a binary operation. {{:Definition:Restriction/Operation}}"}
+{"_id": "19831", "title": "Definition:Closure (Abstract Algebra)", "text": "=== Algebraic Structures === {{:Definition:Closure (Abstract Algebra)/Algebraic Structure}} === Scalar Product === {{:Definition:Closure (Abstract Algebra)/Scalar Product}} Category:Definitions/Abstract Algebra Category:Definitions/Linear Algebra cx9tvgqnuptsin06zh38wl1671k2lmh"}
+{"_id": "19832", "title": "Definition:Magma", "text": "A '''magma''' is an algebraic structure $\\struct {S, \\circ}$ such that $S$ is closed under $\\circ$."}
+{"_id": "19833", "title": "Definition:Submagma", "text": "Let $\\struct {S, \\circ}$ be a magma. Let $T \\subseteq S$ such that $\\struct {T, \\circ}$ is a magma. Then $\\struct {T, \\circ}$ is a '''submagma''' of $\\struct {S, \\circ}$. This relation can be denoted: : $\\struct {T, \\circ} \\subseteq \\struct {S, \\circ}$ === Induced Operation === {{:Definition:Operation Induced by Restriction}}"}
+{"_id": "19834", "title": "Definition:Cancellable Element", "text": "Let $\\left ({S, \\circ}\\right)$ be an algebraic structure."}
+{"_id": "19835", "title": "Definition:Semigroup", "text": ": A '''semigroup''' is an algebraic structure which is closed and whose operation is associative."}
+{"_id": "19836", "title": "Definition:Subsemigroup", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $T \\subseteq S$ such that $\\struct {T, \\circ {\\restriction_T} }$, where $\\circ {\\restriction_T}$ is the restriction of $\\circ$ to $T$, is a semigroup. Then $\\struct {T, \\circ {\\restriction_T} }$ is a '''subsemigroup''' of $S$."}
+{"_id": "19837", "title": "Definition:Commutative Semigroup", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroup such that the operation $\\circ$ is a commutative operation. Then $\\left({S, \\circ}\\right)$ is a '''commutative semigroup'''."}
+{"_id": "19838", "title": "Definition:Regular Representations", "text": "Let $\\struct {S, \\circ}$ be a magma. === Left Regular Representation === {{:Definition:Regular Representations/Left Regular Representation}} === Right Regular Representation === {{:Definition:Regular Representations/Right Regular Representation}}"}
+{"_id": "19839", "title": "Definition:Subset Product", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. We can define an operation on the power set $\\powerset S$ as follows: :$\\forall A, B \\in \\powerset S: A \\circ_\\PP B = \\set {a \\circ b: a \\in A, b \\in B}$ This is called the '''operation induced on $\\powerset S$ by $\\circ$''', and $A \\circ_\\PP B$ is called the '''subset product''' of $A$ and $B$."}
+{"_id": "19840", "title": "Definition:Cyclic Group", "text": "=== Definition 1 === {{:Definition:Cyclic Group/Definition 1}} === Definition 2 === {{:Definition:Cyclic Group/Definition 2}}"}
+{"_id": "19841", "title": "Definition:Inverse Relation", "text": "Let $\\RR \\subseteq S \\times T$ be a relation. The '''inverse relation to''' (or '''of''') $\\RR$ is defined as: :$\\RR^{-1} := \\set {\\tuple {t, s}: \\tuple {s, t} \\in \\RR}$"}
+{"_id": "19842", "title": "Definition:Reflexivity", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation in $S$. === Reflexive === {{:Definition:Reflexive Relation}} === Coreflexive === {{:Definition:Coreflexive Relation}} === Antireflexive === {{:Definition:Antireflexive Relation}} === Non-reflexive === {{Definition:Non-reflexive Relation}}"}
+{"_id": "19843", "title": "Definition:Transitivity (Relation Theory)", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation in $S$."}
+{"_id": "19844", "title": "Definition:Symmetry (Relation)", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation in $S$. === Symmetric === {{:Definition:Symmetric Relation}} === Asymmetric === {{:Definition:Asymmetric Relation}} === Antisymmetric === {{:Definition:Antisymmetric Relation}} === Non-symmetric === {{:Definition:Non-symmetric Relation}}"}
+{"_id": "19845", "title": "Definition:Equivalence Relation", "text": "Let $\\RR$ be a relation on a set $S$."}
+{"_id": "19846", "title": "Definition:Equivalence Class", "text": "Let $S$ be a set. Let $\\RR \\subseteq S \\times S$ be an equivalence relation on $S$. Let $x \\in S$. Then the '''equivalence class of $x$ under $\\RR$''' is the set: :$\\eqclass x \\RR = \\set {y \\in S: \\tuple {x, y} \\in \\RR}$"}
+{"_id": "19847", "title": "Definition:Power Set", "text": "The '''power set''' of a set $S$ is the set defined and denoted as: :$\\powerset S := \\set {T: T \\subseteq S}$ That is, the set whose elements are all of the subsets of $S$."}
+{"_id": "19848", "title": "Definition:Universal Quantifier", "text": "The symbol $\\forall$ is called the '''universal quantifier'''. It expresses the fact that, in a particular universe of discourse, all objects have a particular property. That is: :$\\forall x:$ means: :'''For all objects $x$, it is true that ...''' In the language of set theory, this can be formally defined: :$\\forall x \\in S: \\map P x := \\set {x \\in S: \\map P x} = S$ where $S$ is some set and $\\map P x$ is a propositional function on $S$."}
+{"_id": "19849", "title": "Definition:Existential Quantifier", "text": "The symbol $\\exists$ is called the '''existential quantifier'''. It expresses the fact that, in a particular universe of discourse, there exists (at least one) object having a particular property. That is: :$\\exists x:$ means: :'''There exists at least one object $x$ such that ...''' In the language of set theory, this can be formally defined: :$\\exists x \\in S: \\map P x := \\set {x \\in S: \\map P x} \\ne \\O$ where $S$ is some set and $\\map P x$ is a propositional function on $S$."}
+{"_id": "19850", "title": "Definition:Propositional Expansion", "text": "Suppose our universe of discourse consists of the objects $\\mathbf X_1, \\mathbf X_2, \\mathbf X_3, \\ldots$ and so on. (There may be an infinite number of objects in this universe.) === Universal Quantifier === {{:Definition:Propositional Expansion/Universal Quantifier}} === Existential Quantifier === {{:Definition:Propositional Expansion/Existential Quantifier}} Category:Definitions/Quantifiers fymh785in8ghd2ze9r80zld20a5k451"}
+{"_id": "19851", "title": "Definition:Disjoint Sets", "text": "Two sets $S$ and $T$ are '''disjoint''' {{iff}}: :$S \\cap T = \\O$ That is, '''disjoint sets''' are such that their intersection is the empty set -- they have no elements in common."}
+{"_id": "19852", "title": "Definition:Pairwise Disjoint", "text": "=== Set of Sets === {{:Definition:Pairwise Disjoint/Set of Sets}} === Family of Sets === {{:Definition:Pairwise Disjoint/Family}}"}
+{"_id": "19853", "title": "Definition:Set Partition", "text": "Let $S$ be a set. === Definition 1 === {{:Definition:Set Partition/Definition 1}} === Definition 2 === {{:Definition:Set Partition/Definition 2}}"}
+{"_id": "19854", "title": "Definition:Relative Complement", "text": "Let $S$ be a set, and let $T \\subseteq S$, that is: let $T$ be a subset of $S$. Then the set difference $S \\setminus T$ can be written $\\relcomp S T$, and is called the '''relative complement of $T$ in $S$''', or the '''complement of $T$ relative to $S$'''. Thus: :$\\relcomp S T = \\set {x \\in S : x \\notin T}$"}
+{"_id": "19855", "title": "Definition:Identity (Abstract Algebra)", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure. === Left Identity === {{:Definition:Identity (Abstract Algebra)/Left Identity}} === Right Identity === {{:Definition:Identity (Abstract Algebra)/Right Identity}} === Two-Sided Identity === {{:Definition:Identity (Abstract Algebra)/Two-Sided Identity}}"}
+{"_id": "19856", "title": "Definition:Symmetric Group", "text": "Let $S$ be a set. Let $\\map \\Gamma S$ denote the set of permutations on $S$. Let $\\struct {\\map \\Gamma S, \\circ}$ be the algebraic structure such that $\\circ$ denotes the composition of mappings. Then $\\struct {\\map \\Gamma S, \\circ}$ is called the '''symmetric group on $S$'''. If $S$ has $n$ elements, then $\\struct {\\map \\Gamma S, \\circ}$ is often denoted $S_n$."}
+{"_id": "19857", "title": "Definition:Inverse (Abstract Algebra)", "text": "Let $\\struct {S, \\circ}$ be a monoid whose identity is $e_S$. === Left Inverse === {{:Definition:Inverse (Abstract Algebra)/Left Inverse}} === Right Inverse === {{:Definition:Inverse (Abstract Algebra)/Right Inverse}} === Inverse === {{:Definition:Inverse (Abstract Algebra)/Inverse}}"}
+{"_id": "19858", "title": "Definition:Symmetric Difference", "text": "The '''symmetric difference''' between two sets $S$ and $T$ is written $S * T$ and is defined as: === Definition 1 === {{:Definition:Symmetric Difference/Definition 1}} === Definition 2 === {{:Definition:Symmetric Difference/Definition 2}} === Definition 3 === {{:Definition:Symmetric Difference/Definition 3}} === Definition 4 === {{:Definition:Symmetric Difference/Definition 4}} === Definition 5 === {{:Definition:Symmetric Difference/Definition 5}}"}
+{"_id": "19859", "title": "Definition:Set Complement", "text": "The '''set complement''' (or, when the context is established, just '''complement''') of a set $S$ in a universe $\\mathbb U$ is defined as: :$\\map \\complement S = \\relcomp {\\mathbb U} S = \\mathbb U \\setminus S$ See the definition of Relative Complement for the definition of $\\relcomp {\\mathbb U} S$."}
+{"_id": "19860", "title": "Definition:Left Hand Side", "text": "In an equation: :$\\text {Expression $1$} = \\text {Expression $2$}$ the term $\\text {Expression $1$}$ is the '''left hand side'''."}
+{"_id": "19861", "title": "Definition:Right Hand Side", "text": "In an equation: :$\\text {Expression $1$} = \\text {Expression $2$}$ the term $\\text {Expression $2$}$ is the '''right hand side'''."}
+{"_id": "19862", "title": "Definition:Domain (Set Theory)", "text": "=== Relation === {{:Definition:Domain (Set Theory)/Relation}} === Mapping === The term '''domain''' is usually seen when the relation in question is actually a mapping. {{:Definition:Domain (Set Theory)/Mapping}} === Binary Operation === {{:Definition:Domain (Set Theory)/Binary Operation}}"}
+{"_id": "19863", "title": "Definition:Range of Relation", "text": "Let $\\RR \\subseteq S \\times T$ be a relation, or (usually) a mapping (which is, of course, itself a relation). The '''range''' of $\\RR$, denoted  is defined as one of two things, depending on the source. On {{ProofWiki}} it is denoted $\\Rng \\RR$, but this may non-standard. === Range as Codomain === The '''range''' of a relation $\\RR \\subseteq S \\times T$ can be defined as the set $T$. As such, it is the same thing as the term '''codomain''' of $\\RR$. === Range as Image === The '''range''' of a relation $\\RR \\subseteq S \\times T$ can also be defined as: :$\\Rng \\RR = \\set {t \\in T: \\exists s \\in S: \\tuple {s, t} \\in \\RR}$ Defined like this, it is the same as what is defined as the '''image set''' of $\\RR$."}
+{"_id": "19864", "title": "Definition:Image (Set Theory)", "text": "== Relation == {{:Definition:Image (Set Theory)/Relation}} == Mapping == {{:Definition:Image (Set Theory)/Mapping}}"}
+{"_id": "19866", "title": "Definition:Trivial Relation", "text": "The '''trivial relation''' is the relation $\\RR \\subseteq S \\times T$ in $S$ to $T$ such that ''every'' element of $S$ relates to ''every'' element in $T$: :$\\RR: S \\times T: \\forall \\tuple {s, t} \\in S \\times T: \\tuple {s, t} \\in \\RR$"}
+{"_id": "19867", "title": "Definition:Null Relation", "text": "The '''null relation ''' is a relation $\\RR$ in $S$ to $T$ such that $\\RR$ is the empty set: :$\\RR \\subseteq S \\times T: \\RR = \\O$ That is, ''no'' element of $S$ relates to ''any'' element in $T$: :$\\RR: S \\times T: \\forall \\tuple {s, t} \\in S \\times T: \\neg s \\mathrel \\RR t$"}
+{"_id": "19868", "title": "Definition:Diagonal Relation", "text": "Let $S$ be a set. The '''diagonal relation on $S$''' is a relation $\\Delta_S$ on $S$ such that: :$\\Delta_S = \\set {\\tuple {x, x}: x \\in S} \\subseteq S \\times S$ Alternatively: :$\\Delta_S = \\set {\\tuple {x, y}: x, y \\in S: x = y}$"}
+{"_id": "19869", "title": "Definition:Proper Subset", "text": "If a set $S$ is a subset of another set $T$, that is, $S \\subseteq T$, and also: :$S \\ne T$ :$S \\ne \\O$ then $S$ is referred to as a '''proper subset''' of $T$. The set $T$ '''properly contains''', or '''strictly contains''', the set $S$. If $S \\subseteq T$ and $S \\ne T$, then the notation $S \\subsetneqq T$ is used. If we wish to refer to a set which we specifically require not to be empty, we can denote it like this: :$\\O \\subsetneqq S$ and one which we want to specify as possibly being empty, we write: :$\\O \\subseteq S$ Thus for $S$ to be a '''proper subset''' of $T$, we can write it as $\\O \\subsetneqq S \\subsetneqq T$. === Proper Superset === {{:Definition:Proper Subset/Proper Superset}}"}
+{"_id": "19870", "title": "Definition:Complement of Relation", "text": "Let $\\RR \\subseteq S \\times T$ be a relation. The '''complement of $\\RR$''' is the relative complement of $\\RR$ with respect to $S \\times T$: :$\\relcomp {S \\times T} \\RR := \\set {\\tuple {s, t} \\in S \\times T: \\tuple {s, t} \\notin \\RR}$ If the sets $S$ and $T$ are implicit, then $\\map \\complement \\RR$ can be used."}
+{"_id": "19871", "title": "Definition:Extension of Relation", "text": "Let: : $\\mathcal R_1 \\subseteq X \\times Y$ be a relation on $X \\times Y$ : $\\mathcal R_2 \\subseteq S \\times T$ be a relation on $S \\times T$ : $X \\subseteq S$ : $Y \\subseteq T$ : $\\mathcal R_2 \\restriction_{X \\times Y}$ be the restriction of $\\mathcal R_2$ to $X \\times Y$. Let $\\mathcal R_2 \\restriction_{X \\times Y} = \\mathcal R_1$. Then $\\mathcal R_2$ '''extends''' or '''is an extension of''' $\\mathcal R_1$."}
+{"_id": "19872", "title": "Definition:Agreement", "text": "=== Relations === {{:Definition:Agreement/Relations}} === Mappings === The concept is usually seen in the context of mappings: {{:Definition:Agreement/Mappings}} Category:Definitions/Relation Theory Category:Definitions/Mapping Theory qv7evisfim8za8vixzcutf0u8sgo5sj"}
+{"_id": "19873", "title": "Definition:Combinable", "text": "=== Relations === {{:Definition:Combinable/Relations}} === Mappings === The concept is usually seen in the context of mappings: {{:Definition:Combinable/Mappings}} Category:Definitions/Relation Theory Category:Definitions/Mapping Theory 6xigoakjky7ou6jg6gg0lnsowcl5esl"}
+{"_id": "19874", "title": "Definition:Union Relation", "text": "Let: :$(1): \\quad \\mathcal R_1 \\subseteq S_1 \\times T_1$ be a relation on $S_1 \\times T_1$ :$(2): \\quad \\mathcal R_2 \\subseteq S_2 \\times T_2$ be a relation on $S_2 \\times T_2$ Let $\\mathcal R_1$ and $\\mathcal R_2$ be combinable, that is, that they agree on $S_1 \\cap S_2$. Then the '''union relation''' (or '''combined relation''') $\\mathcal R$ of $\\mathcal R_1$ and $\\mathcal R_2$ is: :$\\mathcal R \\subseteq \\left({S_1 \\cup S_2}\\right) \\times \\left({T_1 \\cup T_2}\\right): \\mathcal R \\left({s}\\right) = \\begin{cases}  \\mathcal R_1 \\left({s}\\right) : & s \\in S_1 \\\\  \\mathcal R_2 \\left({s}\\right) : & s \\in S_2 \\end{cases}$"}
+{"_id": "19875", "title": "Definition:Relational Structure", "text": "A '''relational structure''' is an ordered pair $\\struct {S, \\RR}$, where: :$S$ is a set :$\\RR$ is an endorelation on $S$."}
+{"_id": "19876", "title": "Definition:Many-to-One Relation", "text": "A relation $\\RR \\subseteq S \\times T$ is '''many-to-one''' {{iff}}: :$\\forall x \\in \\Dom \\RR: \\forall y_1, y_2 \\in \\Cdm \\RR: \\tuple {x, y_1} \\in \\RR \\land \\tuple {x, y_2} \\in \\RR \\implies y_1 = y_2$ That is, every element of the domain of $\\RR$ relates to no more than one element of its codomain."}
+{"_id": "19877", "title": "Definition:One-to-Many Relation", "text": "A relation $\\RR \\subseteq S \\times T$ is '''one-to-many''' {{iff}}: :$\\RR \\subseteq S \\times T: \\forall y \\in \\Img \\RR: \\tuple {x_1, y} \\in \\RR \\land \\tuple {x_2, y} \\in \\RR \\implies x_1 = x_2$ That is, every element of the image of $\\RR$ is related to by exactly one element of its domain. Note that the condition on $t$ concerns the elements in the ''image'', not the codomain. Thus a '''one-to-many relation''' may leave some element(s) of the codomain unrelated."}
+{"_id": "19878", "title": "Definition:One-to-One Relation", "text": "A relation $\\RR \\subseteq S \\times T$ is '''one-to-one''' if it is both many-to-one and one-to-many. That is, every element of the domain of $\\RR$ relates to no more than one element of its codomain, and every element of the image is related to by exactly one element of its domain."}
+{"_id": "19879", "title": "Definition:Many-to-Many Relation", "text": "A relation $\\mathcal R \\subseteq S \\times T$ is '''many-to-many''' if it is neither many-to-one nor one-to-many. That is, there is no restriction to the number of elements relating to or being related to by any individual element. {{SUBPAGENAME}} 9pp6l117o3zdptpdhqwbjgxldvc5l89"}
+{"_id": "19880", "title": "Definition:Graph (Graph Theory)", "text": "A '''graph''' is intuitively defined as a pair consisting of a set of vertices and a set of edges. 400px"}
+{"_id": "19882", "title": "Definition:Quotient Set", "text": "Let $\\RR$ be an equivalence relation on a set $S$. For any $x \\in S$, let $\\eqclass x \\RR$ be the $\\RR$-equivalence class of $x$. The '''quotient set of $S$ induced by $\\RR$''' is the set $S / \\RR$ of $\\RR$-classes of $\\RR$: :$S / \\RR := \\set {\\eqclass x \\RR: x \\in S}$"}
+{"_id": "19883", "title": "Definition:Proper Coloring", "text": "=== Proper Vertex Coloring === A '''proper (vertex) $k$-coloring''' of a simple graph $G = \\left({V, E}\\right)$ is defined as a vertex coloring from a set of $k$ colors such that no two adjacent vertices share a common color. That is, a $k$-coloring of the graph $G = \\left({V, E}\\right)$ is a mapping $c: V \\to \\left\\{{1, 2, \\ldots k}\\right\\}$ such that: :$\\forall e = \\left\\{{u, v}\\right\\} \\in E: c \\left({u}\\right) \\ne \\left({v}\\right)$ === Proper Edge Coloring === A '''proper (edge) $k$-coloring''' of a simple graph $G = \\left({V, E}\\right)$ is defined as an edge coloring from a set of $k$ colors such that no two adjacent edges share a common color. That is, a $k$-coloring of the graph $G = \\left({V, E}\\right)$ is a mapping $c: E \\to \\left\\{{1, 2, \\ldots k}\\right\\}$ such that: :$\\forall v \\in V: \\forall e = \\left\\{{u_k, v}\\right\\} \\in E: c \\left\\{{u_i, v}\\right\\} \\ne c \\left\\{{u_j, v}\\right\\}$ {{SUBPAGENAME}} gso9y1cj4k3eofv3qpouxubl0xume6b"}
+{"_id": "19884", "title": "Definition:Chromatic Number", "text": "The '''chromatic number''' $\\map \\chi G$ of a graph $G$ is the smallest positive integer $k$ such that there exists a proper vertex $k$-coloring of $G$."}
+{"_id": "19885", "title": "Definition:Graph of Mapping", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. The '''graph''' of $f$ is the relation  $\\RR \\subseteq S \\times T$ defined as $\\RR = \\set {\\tuple {x, \\map f x}: x \\in S}$ Alternatively, this can be expressed: :$G_f = \\set {\\tuple {s, t} \\in S \\times T: \\map f s = t}$ where $G_f$ is the '''graph of $f$'''."}
+{"_id": "19886", "title": "Definition:Composition of Mappings", "text": "Let $S_1$, $S_2$ and $S_3$ be sets. Let $f_1: S_1 \\to S_2$ and $f_2: S_2 \\to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$. === Definition 1 === {{:Definition:Composition of Mappings/Definition 1}} === Definition 2 === {{:Definition:Composition of Mappings/Definition 2}} === Definition 3 === {{:Definition:Composition of Mappings/Definition 3}}"}
+{"_id": "19887", "title": "Definition:Inverse of Mapping", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping The '''inverse''' of $f$ is its inverse relation, defined as: :$f^{-1} := \\set {\\tuple {t, s}: \\map f s = t}$ That is: :$f^{-1} := \\set {\\tuple {t, s}: \\tuple {s, t} \\in f}$ That is, $f^{-1} \\subseteq T \\times S$ is the relation which satisfies: :$\\forall s \\in S: \\forall t \\in T: \\tuple {t, s} \\in f^{-1} \\iff \\tuple {s, t} \\in f$"}
+{"_id": "19888", "title": "Definition:Empty Mapping", "text": "Let $T$ be a set. Then the mapping whose domain is the empty set and whose codomain is $T$ is called the '''empty mapping''': :$\\O \\subseteq \\O \\times T = \\O$"}
+{"_id": "19890", "title": "Definition:Identity Mapping", "text": "The '''identity mapping''' of a set $S$ is the mapping $I_S: S \\to S$ defined as: :$I_S = \\set {\\tuple {x, y} \\in S \\times S: x = y}$ or alternatively: :$I_S = \\set {\\tuple {x, x}: x \\in S}$ That is: :$I_S: S \\to S: \\forall x \\in S: \\map {I_S} x = x$"}
+{"_id": "19891", "title": "Definition:Inverse Mapping", "text": "Let $S$ and $T$ be sets."}
+{"_id": "19892", "title": "Definition:Injection", "text": "=== Definition 1 === {{:Definition:Injection/Definition 1}} === Definition 2 === {{:Definition:Injection/Definition 2}} === Definition 3 === {{:Definition:Injection/Definition 3}} === Definition 4 === {{:Definition:Injection/Definition 4}} === Definition 5 === {{:Definition:Injection/Definition 5}} === Definition 6 === {{:Definition:Injection/Definition 6}}"}
+{"_id": "19893", "title": "Definition:Left Cancellable Mapping", "text": "A mapping $f: Y \\to Z$ is '''left cancellable''' (or '''left-cancellable''') {{iff}}: :$\\forall X: \\forall \\struct {g_1, g_2: X \\to Y}: f \\circ g_1 = f \\circ g_2 \\implies g_1 = g_2$ That is, for any set $X$, if $g_1$ and $g_2$ are mappings from $X$ to $Y$: :If $f \\circ g_1 = f \\circ g_2$ :then $g_1 = g_2$."}
+{"_id": "19894", "title": "Definition:Inclusion Mapping", "text": "Let $T$ be a set. Let $S\\subseteq T$ be a subset. The '''inclusion mapping''' $i_S: S \\to T$ is the mapping defined as: :$i_S: S \\to T: \\forall x \\in S: \\map {i_S} x = x$"}
+{"_id": "19895", "title": "Definition:Surjection", "text": "Let $S$ and $T$ be sets or classes. Let $f: S \\to T$ be a mapping from $S$ to $T$."}
+{"_id": "19896", "title": "Definition:Right Cancellable Mapping", "text": "A mapping $f: X \\to Y$ is '''right cancellable''' (or '''right-cancellable''') {{iff}}: :$\\forall Z: \\forall \\paren {h_1, h_2: Y \\to Z}: h_1 \\circ f = h_2 \\circ f \\implies h_1 = h_2$ That is, {{iff}} for any set $Z$: :If $h_1$ and $h_2$ are mappings from $Y$ to $Z$ :then $h_1 \\circ f = h_2 \\circ f$ implies $h_1 = h_2$."}
+{"_id": "19897", "title": "Definition:Bijection", "text": "=== Definition 1 === {{:Definition:Bijection/Definition 1}} === Definition 2 === {{:Definition:Bijection/Definition 2}} === Definition 3 === {{:Definition:Bijection/Definition 3}} === Definition 4 === {{:Definition:Bijection/Definition 4}} === Definition 5 === {{:Definition:Bijection/Definition 5}}"}
+{"_id": "19898", "title": "Definition:Permutation", "text": "A bijection $f: S \\to S$ from a set $S$ to itself is called a '''permutation on''' (or '''of''') '''$S$'''."}
+{"_id": "19899", "title": "Definition:Inverse Mapping/Definition 2", "text": "Let $f: S \\to T$ and $g: T \\to S$ be mappings. Let: :$g \\circ f = I_S$ :$f \\circ g = I_T$ where: :$g \\circ f$ and $f \\circ g$ denotes the composition of $f$ with $g$ in either order :$I_S$ and $I_T$ denote the identity mappings on $S$ and $T$ respectively. That is, $f$ and $g$ are both left inverse mappings and right inverse mappings of each other. Then: :$g$ is '''the inverse (mapping) of $f$''' :$f$ is '''the inverse (mapping) of $g$'''."}
+{"_id": "19900", "title": "Definition:Set Equivalence", "text": "Let $S$ and $T$ be sets. Then $S$ and $T$ are '''equivalent''' {{iff}}: :there exists a bijection $f: S \\to T$ between the elements of $S$ and those of $T$. That is, if they have the '''same cardinality'''. This can be written $S \\sim T$. If $S$ and $T$ are not '''equivalent''' we write $S \\nsim T$."}
+{"_id": "19901", "title": "Definition:Injective Restriction", "text": "Let $f: S \\to T$ be a mapping which is not injective. Then an '''injective restriction''' of $f$ is a restriction $g {\\restriction_{S'}}: S' \\to T$ of $f$ which is an injection. Category:Definitions/Injections Category:Definitions/Restrictions bm55fhzivy70xxq051q4xt7egqr3kda"}
+{"_id": "19902", "title": "Definition:Surjective Restriction", "text": "Let $f: S \\to T$ be a mapping which is not surjective. Then a '''surjective restriction''' of $f$ is a restriction $g: S' \\to T'$ of $f$ such that $g \\restriction_{S'}: S' \\to T'$ is a surjection."}
+{"_id": "19903", "title": "Definition:Bijective Restriction", "text": "Let $f: S \\to T$ be a mapping which is not bijective. A '''bijective restriction''' of $f$ is a restriction $f {\\restriction_{S' \\times T'} }: S' \\to T'$ of $f$ such that $f {\\restriction_{S' \\times T'} }$ is a bijection."}
+{"_id": "19904", "title": "Definition:Set of All Mappings", "text": "Let $S$ and $T$ be sets. The '''set of (all) mappings from $S$ to $T$''' is: :$T^S := \\set {f \\subseteq S \\times T: f: S \\to T \\text { is a mapping} }$"}
+{"_id": "19905", "title": "Definition:Projection (Mapping Theory)", "text": "Let $S_1, S_2, \\ldots, S_j, \\ldots, S_n$ be sets. Let $\\displaystyle \\prod_{i \\mathop = 1}^n S_i$ be the Cartesian product of $S_1, S_2, \\ldots, S_n$. For each $j \\in \\set {1, 2, \\ldots, n}$, the '''$j$th projection on $\\displaystyle S = \\prod_{i \\mathop = 1}^n S_i$''' is the mapping $\\pr_j: S \\to S_j$ defined by: :$\\map {\\pr_j} {s_1, s_2, \\ldots, s_j, \\ldots, s_n} = s_j$ for all $\\tuple {s_1, s_2, \\ldots, s_n} \\in S$."}
+{"_id": "19906", "title": "Definition:Quotient Mapping", "text": "Let $\\RR \\subseteq S \\times S$ be an equivalence on a set $S$. Let $\\eqclass s \\RR$ be the $\\RR$-equivalence class of $s$. Let $S / \\RR$ be the quotient set of $S$ determined by $\\RR$. Then $q_\\RR: S \\to S / \\RR$ is the '''quotient mapping induced by $\\RR$''', and is defined as: :$q_\\RR: S \\to S / \\RR: \\map {q_\\RR} s = \\eqclass s \\RR$"}
+{"_id": "19907", "title": "Definition:Equivalence Relation Induced by Mapping", "text": "Let $f: S \\to T$ be a mapping. Let $\\RR_f \\subseteq S \\times S$ be the relation defined as: :$\\tuple {s_1, s_2} \\in \\RR_f \\iff \\map f {s_1} = \\map f {s_2}$ Then $\\RR_f$ is known as the '''equivalence (relation) induced by $f$'''."}
+{"_id": "19908", "title": "Definition:Well-Defined", "text": "=== Well-Defined Mapping === {{:Definition:Well-Defined/Mapping}} === Well-Defined Relation === The concept can be generalized to include the general relation $\\RR: S \\to T$. {{:Definition:Well-Defined/Relation}} === Well-Defined Operation === {{:Definition:Well-Defined/Operation}}"}
+{"_id": "19909", "title": "Definition:Dominate (Set Theory)", "text": "Let $S$ and $T$ be sets. === Definition 1 === {{:Definition:Dominate (Set Theory)/Definition 1}} === Definition 2 === {{:Definition:Dominate (Set Theory)/Definition 2}} === Strictly Dominated === {{:Definition:Dominate (Set Theory)/Strictly Dominate}}"}
+{"_id": "19910", "title": "Definition:Ordering", "text": "Let $S$ be a set. === Definition 1 === {{:Definition:Ordering/Definition 1}} === Definition 2 === {{:Definition:Ordering/Definition 2}}"}
+{"_id": "19911", "title": "Definition:Partially Ordered Set", "text": "A '''partially ordered set''' is a relational structure $\\left({S, \\preceq}\\right)$ such that $\\preceq$ is a partial ordering. The '''partially ordered set''' $\\left({S, \\preceq}\\right)$ is said to be '''partially ordered by $\\preceq$'''."}
+{"_id": "19912", "title": "Definition:Maximal", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $T \\subseteq S$ be a subset of $S$. === Ordered Set === {{:Definition:Maximal/Ordered Set}} === Maximal Set === {{:Definition:Maximal/Set}}"}
+{"_id": "19913", "title": "Definition:Minimal", "text": "=== Relation === {{:Definition:Minimal/Relation}} === Ordered Set === Let $\\struct {S, \\preceq}$ be an ordered set. Let $T \\subseteq S$ be a subset of $S$. {{:Definition:Minimal/Ordered Set}} === Minimal Set === {{:Definition:Minimal/Set}}"}
+{"_id": "19914", "title": "Definition:Non-Comparable", "text": "Let $\\struct {S, \\RR}$ be a relational structure. Two elements $x, y \\in S, x \\ne y$ are '''non-comparable''' if neither $x \\mathrel \\RR y$ nor $y \\mathrel \\RR x$. If $x$ and $y$ are not non-comparable then they are '''comparable''', but the latter term is not so frequently encountered."}
+{"_id": "19915", "title": "Definition:Order Isomorphism", "text": "=== Definition 1 === {{:Definition:Order Isomorphism/Definition 1}} === Definition 2 === {{:Definition:Order Isomorphism/Definition 2}} Two ordered sets $\\struct {S, \\preceq_1}$ and $\\struct {T, \\preceq_2}$ are '''(order) isomorphic''' if there exists such an '''order isomorphism''' between them. $\\struct {S, \\preceq_1}$ is described as '''(order) isomorphic to''' (or '''with''') $\\struct {T, \\preceq_2}$, and vice versa. This may be written $\\struct {S, \\preceq_1} \\cong \\struct {T, \\preceq_2}$. Where no confusion is possible, it may be abbreviated to $S \\cong T$. === Well-Ordered Sets === When $\\struct {S, \\preceq_1}$ and $\\struct {T, \\preceq_2}$ are well-ordered sets, the condition on the order preservation can be relaxed: {{:Definition:Order Isomorphism/Wosets}}"}
+{"_id": "19923", "title": "Definition:Immediate Predecessor Element", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $a, b \\in S$. Then $a$ is an '''immediate predecessor (element)''' to $b$ {{iff}}: :$(1): \\quad a \\prec b$ :$(2): \\quad \\neg \\exists c \\in S: a \\prec c \\prec b$ That is, there exists no element strictly between $a$ and $b$ in the ordering $\\preceq$. That is: :$a \\prec b$ and $\\openint a b = \\O$ where $\\openint a b$ denotes the open interval from $a$ to $b$. We say that '''$a$ ''immediately'' precedes $b$'''."}
+{"_id": "19924", "title": "Definition:Immediate Successor Element", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $a, b \\in S$. Then $a$ is an '''immediate successor (element)''' to $b$ {{iff}} $b$ is an immediate predecessor (element) to $a$. That is, {{iff}}: :$(1): \\quad b \\prec a$ :$(2): \\quad \\nexists c \\in S: b \\prec c \\prec a$ That is, there exists no element strictly between $b$ and $a$ in the ordering $\\preceq$. That is: :$a \\prec b$ and $\\openint a b = \\O$ where $\\openint a b$ denotes the open interval from $a$ to $b$. We say that '''$a$ ''immediately'' succeeds $b$'''."}
+{"_id": "19925", "title": "Definition:Lattice Ordering", "text": "Let $\\left({S, \\preceq}\\right)$ be a lattice. Then the ordering $\\preceq$ is referred to as a '''lattice ordering'''."}
+{"_id": "19926", "title": "Definition:Complete Lattice", "text": "=== Definition 1 === {{:Definition:Complete Lattice/Definition 1}} === Definition 2 === {{:Definition:Complete Lattice/Definition 2}}"}
+{"_id": "19927", "title": "Definition:Total Ordering", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation on a set $S$."}
+{"_id": "19928", "title": "Definition:Totally Ordered Set", "text": "Let $\\struct {S, \\preceq}$ be a relational structure. Then $\\struct {S, \\preceq}$ is a '''totally ordered set''' {{iff}} $\\preceq$ is a total ordering."}
+{"_id": "19929", "title": "Definition:Chain (Set Theory)", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. A '''chain in $S$''' is a totally ordered subset of $S$. Thus a totally ordered set is itself a '''chain''' in its own right. === Length === {{Definition:Length of Chain}}"}
+{"_id": "19930", "title": "Definition:Maximal Chain", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $\\left({T, \\preceq}\\right) \\subseteq \\left({S, \\preceq}\\right)$ be a chain in $\\left({S, \\preceq}\\right)$ such that there is no other chain in $\\left({S, \\preceq}\\right)$ which has $\\left({T, \\preceq}\\right)$ as a proper subset. Then $\\left({T, \\preceq}\\right)$ is a '''maximal chain in $S$'''. Category:Definitions/Order Theory 0e8z7g0a1z5089hyq1817vtbb4royah"}
+{"_id": "19931", "title": "Definition:Order Embedding", "text": "Let $\\struct {S, \\preceq_1}$ and $\\struct {T, \\preceq_2}$ be ordered sets. Let $\\phi: S \\to T$ be a mapping. === Definition 1 === {{:Definition:Order Embedding/Definition 1}} === Definition 2 === {{:Definition:Order Embedding/Definition 2}} === Definition 3 === {{:Definition:Order Embedding/Definition 3}} === Definition 4 === {{:Definition:Order Embedding/Definition 4}}"}
+{"_id": "19932", "title": "Definition:Strictly Increasing", "text": "=== Ordered Sets === {{:Definition:Strictly Increasing/Mapping}} === Real Functions === This definition continues to hold when $S = T = \\R$. {{:Definition:Strictly Increasing/Real Function}} === Sequences === {{:Definition:Strictly Increasing Sequence}}"}
+{"_id": "19933", "title": "Definition:Strictly Decreasing", "text": "=== Ordered Sets === {{:Definition:Strictly Decreasing/Mapping}} === Real Functions === This definition continues to hold when $S = T = \\R$. {{:Definition:Strictly Decreasing/Real Function}} === Sequences === {{:Definition:Strictly Decreasing Sequence}}"}
+{"_id": "19934", "title": "Definition:Strictly Monotone", "text": "=== Ordered Sets === {{:Definition:Strictly Monotone/Mapping}} === Real Functions === This definition continues to hold when $S = T = \\R$: {{:Definition:Strictly Monotone/Real Function}} === Sequences === {{:Definition:Strictly Monotone/Sequence}}"}
+{"_id": "19935", "title": "Definition:Increasing", "text": "=== Ordered Sets === {{:Definition:Increasing/Mapping}} === Real Functions === This definition continues to hold when $S = T = \\R$. {{:Definition:Increasing/Real Function}} === Sequences === {{:Definition:Increasing/Sequence}}"}
+{"_id": "19936", "title": "Definition:Decreasing", "text": "=== Ordered Sets === {{:Definition:Decreasing/Mapping}} === Real Functions === This definition continues to hold when $S = T = \\R$; thus: {{:Definition:Decreasing/Real Function}} === Sequences === {{:Definition:Decreasing Sequence}}"}
+{"_id": "19937", "title": "Definition:Monotone (Order Theory)", "text": "=== Ordered Sets === {{:Definition:Monotone (Order Theory)/Mapping}} === Real Functions === {{:Definition:Monotone (Order Theory)/Real Function}} === Sequences === {{:Definition:Monotone (Order Theory)/Sequence}}"}
+{"_id": "19938", "title": "Definition:Well-Ordered Set", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then $\\struct {S, \\preceq}$ is a '''well-ordered set''' if the ordering $\\preceq$ is a well-ordering. That is, if ''every'' non-empty subset of $S$ has a smallest element: :$\\forall T \\in \\powerset S: \\exists a \\in T: \\forall x \\in T: a \\preceq x$ where $\\powerset S$ denotes the power set of $S$. Or, such that $\\preceq$ is a well-founded total ordering."}
+{"_id": "19939", "title": "Definition:Entropic", "text": "=== Entropic Operation === {{:Definition:Entropic Operation}} === Entropic Structure === {{:Definition:Entropic Structure}}"}
+{"_id": "19940", "title": "Definition:Constant Operation", "text": "Let $S$ be a set. For a given $c \\in S$, a '''constant operation''' on $S$ is defined as: :$\\forall x, y \\in S: x \\left[{c}\\right] y = c$"}
+{"_id": "19941", "title": "Definition:Left Operation", "text": "Let $S$ be a set. For any $x, y \\in S$, the '''left operation''' on $S$ is the binary operation defined as: :$\\forall x, y \\in S: x \\leftarrow y = x$"}
+{"_id": "19942", "title": "Definition:Right Operation", "text": "Let $S$ be a set. For any $x, y \\in S$, the '''right operation''' on $S$ is the binary operation defined as: :$\\forall x, y \\in S: x \\rightarrow y = y$"}
+{"_id": "19943", "title": "Definition:Max Operation", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. The '''max operation''' is the binary operation on $\\left({S, \\preceq}\\right)$ defined as: :$\\forall x, y \\in S: \\map \\max {x, y} = \\begin{cases} y & : x \\preceq y \\\\ x & : y \\preceq x \\end{cases}$ === General Definition === {{:Definition:Max Operation/General Definition}}"}
+{"_id": "19944", "title": "Definition:Min Operation", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. The '''min operation''' is the binary operation on $\\struct {S, \\preceq}$ defined as: :$\\forall x, y \\in S: \\map \\min {x, y} = \\begin{cases} x & : x \\preceq y \\\\ y & : y \\preceq x \\end{cases}$"}
+{"_id": "19945", "title": "Definition:Underlying Set", "text": "=== Abstract Algebra === {{:Definition:Underlying Set/Abstract Algebra}} === Relational Structure === {{:Definition:Underlying Set/Relational Structure}} === Metric Space === {{:Definition:Underlying Set/Metric Space}} === Topological Space === {{:Definition:Underlying Set/Topological Space}} Category:Definitions/Category Theory a7mb18de9mttrahnxjpc0qw88z2jsxf"}
+{"_id": "19947", "title": "Definition:Free Semigroup", "text": "A semigroup which has a non-commutative product in which no product can ever be expressed more simply in terms of the other elements is called a '''free semigroup'''. {{wtd|add formal definition}} Category:Definitions/Semigroups 5x3mhh7kk70r8suh77znqcgbbq4ia1w"}
+{"_id": "19948", "title": "Definition:Monoid", "text": "A '''monoid''' is a semigroup with an identity element."}
+{"_id": "19949", "title": "Definition:Commutative Monoid", "text": "A monoid whose operation is commutative is a '''commutative monoid'''."}
+{"_id": "19950", "title": "Definition:Free Idempotent Monoid", "text": "Let $\\left({S, \\circ}\\right)$ be a monoid which is generated by a set of elements, and such that: :$\\forall x \\in S: x = x \\circ x$ Then $\\left({S, \\circ}\\right)$ is a '''free idempotent monoid'''. Category:Definitions/Monoids e9oprbgcas2ctxk70wfg41gmz4it0i7"}
+{"_id": "19951", "title": "Definition:Submonoid", "text": "A '''submonoid''' of a monoid $\\struct {S, \\circ}$ is a monoid $\\struct {T, \\circ}$ such that $T \\subseteq S$."}
+{"_id": "19952", "title": "Definition:Inverse Semigroup", "text": "An '''inverse semigroup''' is a semigroup $\\struct {S, \\circ}$ such that: :$\\forall a \\in S: \\exists! b \\in S: a = a \\circ b \\circ a, b = b \\circ a \\circ b$"}
+{"_id": "19953", "title": "Definition:Invertible Element", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure which has an identity $e_S$. If $x \\in S$ has an inverse, then $x$ is said to be '''invertible for $\\circ$'''. That is, $x$ is '''invertible''' {{iff}}: :$\\exists y \\in S: x \\circ y = e_S = y \\circ x$ === Invertible Operation === {{:Definition:Invertible Operation}}"}
+{"_id": "19954", "title": "Definition:Trivial Group", "text": "A '''trivial group''' is a group with only one element $e$."}
+{"_id": "19955", "title": "Definition:Quasigroup", "text": "A '''quasigroup''' is a magma $\\struct {S, \\circ}$ which has the Latin square property. That is, such that $\\forall a \\in S$, the left and right regular representations $\\lambda_a$ and $\\rho_a$ are permutations on $S$."}
+{"_id": "19956", "title": "Definition:Algebra Loop", "text": "An '''algebra loop''' $\\struct {S, \\circ}$ is a quasigroup with an identity element."}
+{"_id": "19957", "title": "Definition:Relation Compatible with Operation", "text": "Let $\\struct {S, \\circ}$ be a closed algebraic structure. Let $\\RR$ be a relation on $S$. Then $\\RR$ is '''compatible with $\\circ$''' {{iff}}: :$\\forall x, y, z \\in S: x \\mathrel \\RR y \\implies \\paren {x \\circ z} \\mathrel \\RR \\paren {y \\circ z}$ :$\\forall x, y, z \\in S: x \\mathrel \\RR y \\implies \\paren {z \\circ x} \\mathrel \\RR \\paren {z \\circ y}$"}
+{"_id": "19958", "title": "Definition:Congruence Relation", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\mathcal R$ be an equivalence relation on $S$. Then $\\mathcal R$ is a '''congruence relation for $\\circ$''' {{iff}}: :$\\forall x_1, x_2, y_1, y_2 \\in S: \\paren {x_1 \\mathrel {\\mathcal R} x_2} \\land \\paren {y_1 \\mathrel {\\mathcal R} y_2} \\implies \\paren {x_1 \\circ y_1} \\mathrel {\\mathcal R} \\paren {x_2 \\circ y_2}$"}
+{"_id": "19959", "title": "Definition:Universally Compatible Relation", "text": "A relation $\\RR$ is '''universally compatible''' on a set $S$ {{iff}} it is compatible with every closed operation that can be defined on $S$. Category:Definitions/Abstract Algebra twhewcuarnswdtdw7zeva9pgo97zit0"}
+{"_id": "19960", "title": "Definition:Universally Congruent", "text": "A equivalence $\\RR$ is '''universally congruent''' on a set $S$ {{iff}} it is a congruence for every closed operation that can be defined on $S$. Category:Definitions/Abstract Algebra 9z1lpx8mkuih5wpyapcmr1bebin32g8"}
+{"_id": "19961", "title": "Definition:Operation Induced on Quotient Set", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\mathcal R$ be a congruence relation on $\\struct {S, \\circ}$. Let $S / \\mathcal R$ be the quotient set of $S$ by $\\mathcal R$. The '''operation $\\circ_\\mathcal R$ induced on $S / \\mathcal R$ by $\\circ$''' is defined as: :$\\eqclass x {\\mathcal R} \\circ_\\mathcal R \\eqclass y {\\mathcal R} = \\eqclass {x \\circ y} {\\mathcal R}$"}
+{"_id": "19962", "title": "Definition:Quotient Structure", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure. Let $\\mathcal R$ be a congruence relation on $\\left({S, \\circ}\\right)$. Let $S / \\mathcal R$ be the quotient set of $S$ by $\\mathcal R$. Let $\\circ_\\mathcal R$ be the operation induced on $S / \\mathcal R$ by $\\circ$. The '''quotient structure defined by $\\mathcal R$''' is the algebraic structure: : $\\left({S / \\mathcal R, \\circ_\\mathcal R}\\right)$"}
+{"_id": "19963", "title": "Definition:External Direct Product", "text": "Let $\\left({S, \\circ_1}\\right)$ and $\\left({T, \\circ_2}\\right)$ be algebraic structures. The '''(external) direct product''' $\\left({S \\times T, \\circ}\\right)$ of two algebraic structures $\\left({S, \\circ_1}\\right)$ and $\\left({T, \\circ_2}\\right)$ is the set of ordered pairs: :$\\left({S \\times T, \\circ}\\right) = \\left\\{{\\left({s, t}\\right): s \\in S, t \\in T}\\right\\}$ where the operation $\\circ$ is defined as: :$\\left({s_1, t_1}\\right) \\circ \\left({s_2, t_2}\\right) = \\left({s_1 \\circ_1 s_2, t_1 \\circ_2 t_2}\\right)$"}
+{"_id": "19964", "title": "Definition:Algebraic Substructure", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure. Let $T \\subseteq S$. Then $\\left({T, \\circ}\\right)$ is an '''algebraic substructure''' of $\\left({S, \\circ}\\right)$."}
+{"_id": "19965", "title": "Definition:Internal Direct Product", "text": "Let $\\left({S_1, \\circ {\\restriction_{S_1}} }\\right), \\left({S_2, \\circ {\\restriction_{S_2}} }\\right)$ be closed algebraic substructures of an algebraic structure $\\left({S, \\circ}\\right)$ where $\\circ {\\restriction_{S_1}}, \\circ {\\restriction_{S_2}}$ are the operations induced by the restrictions of $\\circ$ to $S_1, S_2$ respectively. The structure $\\left({S, \\circ}\\right)$ is the '''internal direct product of $S_1$ and $S_2$''' if the mapping: :$C: S_1 \\times S_2 \\to S: C \\left({\\left({s_1, s_2}\\right)}\\right) = s_1 \\circ s_2$ is an isomorphism from the cartesian product $\\left({S_1, \\circ {\\restriction_{S_1}}}\\right) \\times \\left({S_2, \\circ {\\restriction_{S_2}}}\\right)$ onto $\\left({S, \\circ}\\right)$. The operation $\\circ$ on $S$ is the operation induced on $S$ by $\\circ {\\restriction_{S_1}}$ and $\\circ {\\restriction_{S_2}}$."}
+{"_id": "19966", "title": "Definition:Homomorphism (Abstract Algebra)", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be magmas. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be a mapping from $\\struct {S, \\circ}$ to $\\struct {T, *}$. Let $\\circ$ have the morphism property under $\\phi$, that is: :$\\forall x, y \\in S: \\map \\phi {x \\circ y} = \\map \\phi x * \\map \\phi y$ Then $\\phi$ is a '''homomorphism'''."}
+{"_id": "19967", "title": "Definition:Morphism Property", "text": "Let $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ be a mapping from one algebraic structure $\\left({S, \\circ}\\right)$ to another $\\left({T, *}\\right)$. Then $\\circ$ has the '''morphism property''' under $\\phi$ iff: :$\\forall x, y \\in S: \\phi \\left({x \\circ y}\\right) = \\phi \\left({x}\\right) * \\phi \\left({y}\\right)$"}
+{"_id": "19968", "title": "Definition:Epimorphism (Abstract Algebra)", "text": "A homomorphism which is a surjection is described as '''epic''', or called an '''epimorphism'''."}
+{"_id": "19969", "title": "Definition:Endomorphism", "text": "An '''endomorphism''' is a homomorphism from an algebraic structure into itself."}
+{"_id": "19970", "title": "Definition:Monomorphism (Abstract Algebra)", "text": "A homomorphism which is an injection is descibed as '''monic''', or called a '''monomorphism'''."}
+{"_id": "19971", "title": "Definition:Isomorphism (Abstract Algebra)", "text": "An '''isomorphism''' is a homomorphism which is a bijection. That is, it is a mapping which is both a monomorphism and an epimorphism."}
+{"_id": "19972", "title": "Definition:Automorphism (Abstract Algebra)", "text": "An '''automorphism''' is an isomorphism from an algebraic structure to itself."}
+{"_id": "19973", "title": "Definition:Canonical Injection (Abstract Algebra)", "text": "Let $\\struct {S_1, \\circ_1}$ and $\\struct {S_2, \\circ_2}$ be algebraic structures with identities $e_1, e_2$ respectively. The following mappings: :$\\inj_1: \\struct {S_1, \\circ_1} \\to \\struct {S_1, \\circ_1} \\times \\struct {S_2, \\circ_2}: \\forall x \\in S_1: \\map {\\inj_1} x = \\tuple {x, e_2}$ :$\\inj_2: \\struct {S_2, \\circ_2} \\to \\struct {S_1, \\circ_1} \\times \\struct {S_2, \\circ_2}: \\forall x \\in S_2: \\map {\\inj_2} x = \\tuple {e_1, x}$ are called the '''canonical injections'''. === General Definition === {{:Definition:Canonical Injection (Abstract Algebra)/General Definition}}"}
+{"_id": "19974", "title": "Definition:Inverse Completion", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Let $\\struct {C, \\circ} \\subseteq \\struct {S, \\circ}$ be the subsemigroup of cancellable elements of $\\struct {S, \\circ}$. Let $\\struct {T, \\circ'}$ be a semigroup defined such that: :$(1): \\quad \\struct {S, \\circ}$ is a subsemigroup of $\\struct {T, \\circ'}$ :$(2): \\quad$ Every element of $C$ has an inverse in $T$ for $\\circ'$ :$(3): \\quad \\gen {S \\cup C^{-1} } = \\struct {T, \\circ'}$ where: :$\\gen {S \\cup C^{-1} }$ denotes the semigroup generated by $S \\cup C^{-1}$ :$C^{-1}$ denotes the inverse of $C$. Then $\\struct {T, \\circ'}$ is called an '''inverse completion''' of $\\struct {S, \\circ}$."}
+{"_id": "19975", "title": "Definition:Inverse of Subset", "text": "=== Monoid === {{Definition:Inverse of Subset/Monoid}} === Group === {{Definition:Inverse of Subset/Group}}"}
+{"_id": "19976", "title": "Definition:Automorphism Group/Group", "text": "Let $\\struct {S, *}$ be an algebraic structure. Let $\\mathbb S$ be the set of automorphisms of $S$. Then the algebraic structure $\\struct {\\mathbb S, \\circ}$, where $\\circ$ denotes composition of mappings, is called the '''automorphism group''' of $S$. The structure $\\struct {S, *}$ is usually a group. However, this is not necessary for this definition to be valid. The '''automorphism group of $S$''' is denoted on {{ProofWiki}} as $\\Aut S$."}
+{"_id": "19977", "title": "Definition:Inner Automorphism", "text": "Let $G$ be a group. Let $x \\in G$. Let the mapping $\\kappa_x: G \\to G$ be defined such that: :$\\forall g \\in G: \\map {\\kappa_x} g = x g x^{-1}$ $\\kappa_x$ is called the '''inner automorphism of $G$ (given) by $x$'''."}
+{"_id": "19978", "title": "Definition:Proper Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group. Then $\\struct {H, \\circ}$ is a '''proper subgroup of $\\struct {G, \\circ}$''' {{iff}}: : $(1): \\quad \\struct {H, \\circ}$ is a subgroup of $\\struct {G, \\circ}$ : $(2): \\quad H \\ne G$, i.e. $H \\subset G$. The notation $H < G$, or $G > H$, means: : '''$H$ is a proper subgroup of $G$'''. If $H$ is a subgroup of $G$, but it is not specified whether $H = G$ or not, then we write $H \\le G$, or $G \\ge H$. === Non-Trivial Proper Subgroup === {{:Definition:Proper Subgroup/Non-Trivial}}"}
+{"_id": "19979", "title": "Definition:Trivial Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Then the algebraic structure $\\struct {\\set e, \\circ}$ is called '''the trivial subgroup''' of $\\struct {G, \\circ}$."}
+{"_id": "19980", "title": "Definition:Conjugate (Group Theory)", "text": "Let $\\left({G, \\circ}\\right)$ be a group. === Conjugate of an Element === {{:Definition:Conjugate (Group Theory)/Element}} === Conjugate of a Set === {{:Definition:Conjugate (Group Theory)/Subset}} Category:Definitions/Conjugacy Category:Definitions/Group Theory 6rvsq6cjwk0uolx4vxip5h1ihz56stg"}
+{"_id": "19981", "title": "Definition:Conjugacy Class", "text": "The equivalence classes into which the conjugacy relation divides its group into are called '''conjugacy classes'''. The '''conjugacy class''' of an element $x \\in G$ can be denoted $\\conjclass x$."}
+{"_id": "19982", "title": "Definition:Kernel (Abstract Algebra)", "text": "=== Kernel of Magma Homomorphism === {{:Definition:Kernel of Magma Homomorphism}} === Kernel of Group Homomorphism === {{:Definition:Kernel of Group Homomorphism}} === Kernel of Ring Homomorphism === {{:Definition:Kernel of Ring Homomorphism}} === Kernel of Linear Transformation === {{:Definition:Kernel of Linear Transformation}} === Kernel of Homomorphism of Differential Complexes === {{:Definition:Kernel of Homomorphism of Differential Complexes}} Category:Definitions/Abstract Algebra 0imy2su25o141xy7qlspg0qh3nw1sf7"}
+{"_id": "19983", "title": "Definition:Entropic Structure", "text": "An '''entropic structure''' is an algebraic structure $\\struct {S, \\circ}$ such that $\\circ$ is an entropic operation."}
+{"_id": "19984", "title": "Definition:Ringoid (Abstract Algebra)", "text": "A '''ringoid''' is a triple $\\struct {S, *, \\circ}$ where: :$S$ is a set :$*$ and $\\circ$ are binary operations on $S$ :the operation $\\circ$ distributes over $*$. That is: :$\\forall a, b, c \\in S: a \\circ \\paren {b * c} = \\paren {a \\circ b} * \\paren {a \\circ c}$ :$\\forall a, b, c \\in S: \\paren {a * b} \\circ c = \\paren {a \\circ c} * \\paren {b \\circ c}$"}
+{"_id": "19985", "title": "Definition:Semiring (Abstract Algebra)", "text": "A '''semiring''' is a ringoid $\\struct {S, *, \\circ}$ in which: :$(1): \\quad \\struct {S, *}$ forms a semigroup :$(2): \\quad \\struct {S, \\circ}$ forms a semigroup. That is, such that $\\struct {S, *, \\circ}$ has the following properties: {{begin-axiom}} {{axiom | n = \\text A 0         | q = \\forall a, b \\in S         | m = a * b \\in S         | c = Closure under $*$ }} {{axiom | n = \\text A 1         | q = \\forall a, b, c \\in S         | m = \\paren {a * b} * c = a * \\paren {b * c}         | c = Associativity of $*$ }} {{axiom | n = \\text M 0         | q = \\forall a, b \\in S         | m = a \\circ b \\in S         | c = Closure under $\\circ$ }} {{axiom | n = \\text M 1         | q = \\forall a, b, c \\in S         | m = \\paren {a \\circ b} \\circ c = a \\circ \\paren {b \\circ c}         | c = Associativity of $\\circ$ }} {{axiom | n = \\text D         | q = \\forall a, b, c \\in S         | m = a \\circ \\paren {b * c} = \\paren {a \\circ b} * \\paren {a \\circ c} }} {{axiom | m = \\paren {a * b} \\circ c = \\paren {a \\circ c} * \\paren {b \\circ c}         | c = $\\circ$ is distributive over $*$ }} {{end-axiom}} These are called the '''semiring axioms'''."}
+{"_id": "19986", "title": "Definition:Cancellable Semiring", "text": "A '''cancellable semiring''' is a semiring $\\left({S, *, \\circ}\\right)$ in which all the elements of $S$ are cancellable for $*$."}
+{"_id": "19987", "title": "Definition:Ring (Abstract Algebra)", "text": "A '''ring''' $\\struct {R, *, \\circ}$ is a semiring in which $\\struct {R, *}$ forms an abelian group. That is, in addition to $\\struct {R, *}$ being closed, associative and commutative under $*$, it also has an identity, and each element has an inverse."}
+{"_id": "19988", "title": "Definition:Ring Zero", "text": "Let $\\struct {R, +, \\circ}$ be a ring. The identity for ring addition is called the '''ring zero''' (of $\\struct {R, +, \\circ}$). It is denoted $0_R$ (or just $0$ if there is no danger of ambiguity)."}
+{"_id": "19989", "title": "Definition:Ring Negative", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Let $x \\in R$. The inverse of $x$ with respect to the addition operation $+$ in the additive group $\\struct {R, +}$ of $R$ is referred to as the '''(ring) negative''' of $x$ and is denoted $-x$. That is, the '''(ring) negative''' of $x$ is the element $-x$ of $R$ such that: :$x + \\paren {-x} = 0_R$"}
+{"_id": "19990", "title": "Definition:Commutative Ring", "text": "A '''commutative ring''' is a ring $\\struct {R, +, \\circ}$ in which the ring product $\\circ$ is commutative."}
+{"_id": "19991", "title": "Definition:Unity (Abstract Algebra)/Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. If the semigroup $\\struct {R, \\circ}$ has an identity, this identity is referred to as '''the unity of the ring $\\struct {R, +, \\circ}$'''. It is (usually) denoted $1_R$, where the subscript denotes the particular ring to which $1_R$ belongs (or often $1$ if there is no danger of ambiguity)."}
+{"_id": "19992", "title": "Definition:Trivial Ring", "text": "A ring $\\struct {R, +, \\circ}$ is a '''trivial ring''' {{iff}}: :$\\forall x, y \\in R: x \\circ y = 0_R$"}
+{"_id": "19993", "title": "Definition:Null Ring", "text": "A ring with one element is called '''the null ring'''. That is, the '''null ring''' is $\\struct {\\set {0_R}, +, \\circ}$, where ring addition and the ring product are defined as: :$0_R + 0_R = 0_R$ :$0_R \\circ 0_R = 0_R$"}
+{"_id": "19994", "title": "Definition:Ring with Unity", "text": "Let $\\struct {R, +, \\circ}$ be a non-null ring. Then $\\struct {R, +, \\circ}$ is a '''ring with unity''' {{iff}} the multiplicative semigroup $\\struct {R, \\circ}$ has an identity element. Such an identity element is known as a unity."}
+{"_id": "19995", "title": "Definition:Unit of Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose unity is $1_R$. === Definition 1 === {{:Definition:Unit of Ring/Definition 1}} === Definition 2 === {{:Definition:Unit of Ring/Definition 2}}"}
+{"_id": "19996", "title": "Definition:Group of Units", "text": "=== Group of Units of Monoid === {{:Definition:Group of Units/Monoid}} === Group of Units of Ring === {{:Definition:Group of Units/Ring}} Category:Definitions/Ring Theory Category:Definitions/Monoids n24ojfgrw9lt9htgvmq9928d9jdx6sg"}
+{"_id": "19997", "title": "Definition:Division Product", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $\\struct {U_R, \\circ}$ be the group of units of $\\struct {R, +, \\circ}$. Then we define the following notation: :$\\forall x \\in U_R, y \\in R$, we have: ::$\\dfrac y x := y \\circ \\paren {x^{-1} } = \\paren {x^{-1} } \\circ y$ $\\dfrac y x$ is a '''division product''', and $\\dfrac y x$ is voiced '''$y$ divided by $x$'''. We also write (out of space considerations) $y / x$ for $\\dfrac y x$. This notation is usually used when $\\struct {R, +, \\circ}$ is a field."}
+{"_id": "19998", "title": "Definition:Proper Element of Ring", "text": "A non-zero element of a ring which does not have a product inverse is called a '''proper element'''."}
+{"_id": "19999", "title": "Definition:Zero Divisor", "text": "=== Rings === {{:Definition:Zero Divisor/Ring}} === Commutative Rings === The definition is usually made when the ring in question is commutative: {{:Definition:Zero Divisor/Commutative Ring}} === Algebras === {{:Definition:Zero Divisor/Algebra}}"}
+{"_id": "20000", "title": "Definition:Division Ring", "text": "A '''division ring''' is a ring with unity $\\struct {R, +, \\circ}$ with the following properties: === Definition 1 === {{:Definition:Division Ring/Definition 1}} === Definition 2 === {{:Definition:Division Ring/Definition 2}} === Definition 3 === {{:Definition:Division Ring/Definition 3}}"}
+{"_id": "20001", "title": "Definition:Field (Abstract Algebra)", "text": "A '''field''' is a non-trivial division ring whose ring product is commutative. Thus, let $\\struct {F, +, \\times}$ be an algebraic structure. Then $\\struct {F, +, \\times}$ is a '''field''' {{iff}}: :$(1): \\quad$ the algebraic structure $\\struct {F, +}$ is an abelian group :$(2): \\quad$ the algebraic structure $\\struct {F^*, \\times}$ is an abelian group where $F^* = F \\setminus \\set 0$ :$(3): \\quad$ the operation $\\times$ distributes over $+$."}
+{"_id": "20002", "title": "Definition:Skew Field", "text": "A '''skew field''' is a division ring whose ring product is specifically '''not''' commutative."}
+{"_id": "20003", "title": "Definition:Integral Domain", "text": "=== Definition 1 === {{:Definition:Integral Domain/Definition 1}} === Definition 2 === {{:Definition:Integral Domain/Definition 2}}"}
+{"_id": "20004", "title": "Definition:Subring", "text": "Let $\\struct {R, +, \\circ}$ be an algebraic structure with two operations. A '''subring of $\\struct {R, +, \\circ}$''' is a subset $S$ of $R$ such that $\\struct {S, +_S, \\circ_S}$ is a ring."}
+{"_id": "20005", "title": "Definition:Subdomain", "text": "Let $\\left({R, +, \\circ}\\right)$ be an algebraic structure with two operations. A '''subdomain of $\\left({R, +, \\circ}\\right)$''' is a subset $S$ of $R$ such that $\\left({S, +_S, \\circ_S}\\right)$ is an integral domain."}
+{"_id": "20006", "title": "Definition:Division Subring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. A '''division subring''' of $\\struct {R, +, \\circ}$ is a subset $S$ of $R$ such that $\\struct {S, +_S, \\circ_S}$ is a division ring."}
+{"_id": "20007", "title": "Definition:Centralizer", "text": "=== Centralizer of a Group Element === {{:Definition:Centralizer/Group Element}} === Centralizer of a Subset of a Group === {{:Definition:Centralizer/Group Subset}} === Centralizer of a Subgroup === {{:Definition:Centralizer/Subgroup}} === Centralizer of a Ring Subset === {{:Definition:Centralizer/Ring Subset}}"}
+{"_id": "20008", "title": "Definition:Ideal of Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\struct {J, +}$ be a subgroup of $\\struct {R, +}$. Then $J$ is an '''ideal of $R$''' {{iff}}: :$\\forall j \\in J: \\forall r \\in R: j \\circ r \\in J \\land r \\circ j \\in J$ that is, {{iff}}: :$\\forall r \\in R: J \\circ r \\subseteq J \\land r \\circ J \\subseteq J$"}
+{"_id": "20009", "title": "Definition:Congruence Modulo an Ideal", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring, and let $J$ be an ideal of $R$. The notation: :$a \\equiv b \\pmod J$ is used to mean: :$a + \\left({-b}\\right) \\in J$"}
+{"_id": "20010", "title": "Definition:Quotient Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $J$ be an ideal of $R$. Let $R / J$ be the (left) coset space of $R$ modulo $J$ with respect to $+$. Define an operation $+$ on $R / J$ by: :$\\forall x, y: \\paren {x + J} + \\paren {y + J} := \\paren {x + y} + J$ Also, define the operation $\\circ$ on $R / J$ by: :$\\forall x, y: \\paren {x + J} \\circ \\paren {y + J} := \\paren {x \\circ y} + J$ The algebraic structure $\\struct {R / J, +, \\circ}$ is called the '''quotient ring of $R$ by $J$'''."}
+{"_id": "20011", "title": "Definition:Divisor (Algebra)", "text": "=== Ring with Unity === {{:Definition:Divisor (Algebra)/Ring with Unity}} === Natural Numbers === {{:Definition:Divisor (Algebra)/Natural Numbers}} === Integers === As the set of integers form an integral domain, the concept '''divides''' is fully applicable to the integers. {{:Definition:Divisor (Algebra)/Integer}} === Gaussian Integers === As the set of Gaussian integers form an integral domain, the concept '''divides''' is also fully applicable to the Gaussian integers. {{:Definition:Divisor (Algebra)/Gaussian Integer}} === Real Numbers === The concept of '''divisibility''' can also be applied to the real numbers $\\R$. {{:Definition:Divisor (Algebra)/Real Number}}"}
+{"_id": "20013", "title": "Definition:Multiple", "text": "=== Integral Domain === {{:Definition:Multiple/Integral Domain}} === Integers === As the set of integers form an integral domain, the concept of being a '''multiple''' is fully applicable to the integers. {{:Definition:Multiple/Integer}}"}
+{"_id": "20014", "title": "Definition:Associate", "text": "{{transclude:Definition:Associate/Integral Domain   |title = Integral Domain   |section = def   |link = true   |increase = 1   |header = 3 }} === Integers === As the integers form an integral domain, the definition can be applied directly to the set of integers $\\Z$: {{:Definition:Associate/Integers}} === Commutative and Unitary Ring === The concept of associatehood can also be applied to the general commutative and unitary ring, even though there may be (proper) zero divisors in the latter: {{:Definition:Associate/Commutative and Unitary Ring}}"}
+{"_id": "20015", "title": "Definition:Trivial Factorization", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain. Let $\\struct {U_D, \\circ}$ be the group of units of $\\struct {D, +, \\circ}$. A factorization in $\\struct {D, +, \\circ}$ of the form $x = u \\circ y$, where $u \\in U_D$ (that is, where $x$ is an associate of $y$) is called a '''trivial factorization'''. === Non-Trivial Factorization === {{:Definition:Trivial Factorization/Non-Trivial Factorization}}"}
+{"_id": "20016", "title": "Definition:Irreducible Element of Ring", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose zero is $0_D$. Let $\\struct {U_D, \\circ}$ be the group of units of $\\struct {D, +, \\circ}$. Let $x \\in D: x \\notin U_D, x \\ne 0_D$, that is, $x$ is non-zero and not a unit. === Definition 1 === {{:Definition:Irreducible Element of Ring/Definition 1}} === Definition 2 === {{:Definition:Irreducible Element of Ring/Definition 2}}"}
+{"_id": "20017", "title": "Definition:Tidy Factorization", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose unity is $1_D$. Let $\\struct {U_D, \\circ}$ be the group of units of $\\struct {D, +, \\circ}$. Any factorization of $x \\in D$ can always be '''tidied''' into the form: :$x = u \\circ y_1 \\circ y_2 \\circ \\cdots \\circ y_n$ where $u \\in \\struct {U_D, \\circ}$, and may be $1_D$, and $y_1, y_2, \\ldots, y_n$ are all non-zero and non-units. This is done by forming the ring product of all units of a factorization into one unit, and rearranging all the remaining factors as necessary. Such a factorization is called '''tidy'''."}
+{"_id": "20018", "title": "Definition:Equivalent Factorizations", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain. Let $x$ be a non-zero non-unit element of $D$. Let there be two tidy factorizations of $x$: :$x = u_y \\circ y_1 \\circ y_2 \\circ \\cdots \\circ y_m$ :$x = u_z \\circ z_1 \\circ z_2 \\circ \\cdots \\circ z_n$ These two factorizations are '''equivalent''' if one of the following equivalent statements holds:   :$(1): \\quad$ There exists a bijection $\\pi: \\set {1, \\ldots, m} \\to \\set {1, \\ldots, n}$ such that $y_i$ and $z_{\\map \\pi i}$ are associates of each other for each $i \\in \\set {1, \\ldots, m}$. :$(2): \\quad$ The multisets of principal ideals $\\multiset {\\ideal {y_i}: i = 1, \\ldots, m}$ and $\\multiset {\\ideal {z_i}: i = 1, \\ldots, n}$ are equal. The equivalence of the definitions is shown by part $(3)$ of Principal Ideals in Integral Domain."}
+{"_id": "20019", "title": "Definition:Complete Factorization", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain. Let $x$ be a non-zero non-unit element of $D$. A '''complete factorization''' of $x$ in $D$ is a tidy factorization: :$x = u \\circ y_1 \\circ y_2 \\circ \\cdots \\circ y_n$ such that: :$u$ is a unit of $D$ :all of $y_1, y_2, \\ldots, y_n$ are irreducible in $D$."}
+{"_id": "20020", "title": "Definition:Unique Factorization Domain", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain. If, for all $x \\in D$ such that $x$ is non-zero and not a unit of $D$: :$(1): \\quad x$ possesses a complete factorization in $D$ :$(2): \\quad$ Any two complete factorizations of $x$ in $D$ are equivalent then $D$ is a '''unique factorization domain'''."}
+{"_id": "20021", "title": "Definition:Subfield", "text": "Let $\\struct {S, *, \\circ}$ be an algebraic structure with $2$ operations. Let $T$ be a subset of $S$ such that $\\struct {T, *, \\circ}$ is a field. Then $\\struct {T, *, \\circ}$ is a '''subfield''' of $\\struct {S, *, \\circ}$."}
+{"_id": "20022", "title": "Definition:Congruence Modulo Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. We can use $H$ to define relations on $G$ as follows: === Left Congruence Modulo Subgroup === {{:Definition:Congruence Modulo Subgroup/Left Congruence}} === Right Congruence Modulo Subgroup === {{:Definition:Congruence Modulo Subgroup/Right Congruence}}"}
+{"_id": "20023", "title": "Definition:Coset", "text": "=== Left Coset === {{:Definition:Coset/Left Coset}} === Right Coset === {{:Definition:Coset/Right Coset}}"}
+{"_id": "20024", "title": "Definition:Coset Space", "text": "Let $G$ be a group, and let $H$ be a subgroup of $G$. === Left Coset Space === {{:Definition:Coset Space/Left Coset Space}} === Right Coset Space === {{:Definition:Coset Space/Right Coset Space}} === Note === If we are (as is usual) concerned at a particular time with only the left or the right coset space, then the superscript is usually dropped and the notation $G / H$ is used for both the left and right coset space. If, in addition, $H$ is a normal subgroup of $G$, then $G / H^l = G / H^r$ and the notation $G / H$ is then unambiguous anyway."}
+{"_id": "20025", "title": "Definition:Quotient Group", "text": "Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Then the left coset space $G / N$ is a group, where the group operation is defined as: :$\\paren {a N} \\paren {b N} = \\paren {a b} N$ $G / N$ is called the '''quotient group of $G$ by $N$'''."}
+{"_id": "20026", "title": "Definition:Quotient Epimorphism", "text": "=== Group === {{:Definition:Quotient Epimorphism/Group}} === Ring === {{:Definition:Quotient Epimorphism/Ring}}"}
+{"_id": "20027", "title": "Definition:Central Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$ which is a subset of the center of $G$. Then $H$ is a '''central subgroup of $G$'''."}
+{"_id": "20028", "title": "Definition:Normalizer", "text": "Let $G$ be a group. Let $S$ be a subset of $G$. Then the '''normalizer of $S$ in $G$''' is the set $\\map {N_G} S$ defined as: :$\\map {N_G} S := \\set {a \\in G: S^a = S}$ where $S^a$ is the $G$-conjugate of $S$ by $a$."}
+{"_id": "20029", "title": "Definition:Ordered Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\preceq$ be an ordering compatible with the ring structure of $\\struct {R, +, \\circ}$. Then $\\struct {R, +, \\circ, \\preceq}$ is an '''ordered ring'''."}
+{"_id": "20030", "title": "Definition:Positive", "text": "Let $\\struct {R, +, \\circ, \\le}$ be an ordered ring whose zero is $0_R$. Then $x \\in R$ is '''positive''' {{iff}} $0_R \\le x$. The set of all '''positive elements of $R$''' is denoted: :$R_{\\ge 0_R} := \\set {x \\in R: 0_R \\le x}$"}
+{"_id": "20031", "title": "Definition:Strictly Positive", "text": "Let $\\struct {R, +, \\circ, \\le}$ be an ordered ring whose zero is $0_R$. Then $x \\in R$ is '''strictly positive''' {{iff}} $0_R \\le x$ and $x \\ne 0_R$. One may write (more conveniently) that $0_R < x$ or $x > 0_R$ to express that $x$ is '''strictly positive'''. Thus, the set of all '''strictly positive''' elements of $R$ can be denoted: :$R_{>0_R} := \\set {x \\in R: x > 0_R}$"}
+{"_id": "20032", "title": "Definition:Ordering Compatible with Ring Structure", "text": "{{begin-axiom}} {{axiom | n = \\text {OR} 1         | lc= $\\preccurlyeq$ is compatible with $+$:         | q = \\forall a, b, c \\in  R         | ml= a \\preccurlyeq b         | mo= \\implies         | mr= \\paren {a + c} \\preccurlyeq \\paren {b + c} }} {{axiom | n = \\text {OR} 2         | lc= Product of Positive Elements is Positive         | q = \\forall a, b \\in R         | ml= 0_R \\preccurlyeq x, 0_R \\preccurlyeq y         | mo= \\implies         | mr= 0_R \\preccurlyeq x \\circ y }} {{end-axiom}}"}
+{"_id": "20033", "title": "Definition:Ring of Endomorphisms", "text": "Let $\\struct {G, \\oplus}$ be an abelian group. Let $\\mathbb G$ be the set of all group endomorphisms of $\\struct {G, \\oplus}$. Let $*: \\mathbb G \\times \\mathbb G \\to \\mathbb G$ be the operation defined as: :$\\forall u, v \\in \\mathbb G: u * v = u \\circ v$ where $u \\circ v$ is defined as composition of mappings. Then $\\struct {\\mathbb G, \\oplus, *}$ is called the '''ring of endomorphisms''' of the abelian group $\\struct {G, \\oplus}$."}
+{"_id": "20034", "title": "Definition:Field of Quotients", "text": "Let $D$ be an integral domain. Let $F$ be a field. === Definition 1 === {{:Definition:Field of Quotients/Definition 1}} === Definition 2 === {{:Definition:Field of Quotients/Definition 2}} === Definition 3 === {{:Definition:Field of Quotients/Definition 3}} === Definition 4 === {{:Definition:Field of Quotients/Definition 4}}"}
+{"_id": "20035", "title": "Definition:Ordered Field", "text": "Let $\\left({R, +, \\circ, \\preceq}\\right)$ be an ordered ring. Let $\\left({R, +, \\circ}\\right)$ be a field. Then $\\left({R, +, \\circ, \\preceq}\\right)$ is an '''ordered field'''."}
+{"_id": "20036", "title": "Definition:Ordered Structure", "text": "An '''ordered structure''' $\\left({S, \\circ, \\preceq}\\right)$ is a set $S$ such that: : $(1): \\quad \\left({S, \\circ}\\right)$ is an algebraic structure : $(2): \\quad \\left({S, \\preceq}\\right)$ is an ordered set : $(3): \\quad \\preceq$ is compatible with $\\circ$."}
+{"_id": "20037", "title": "Definition:Zero (Number)", "text": "The number '''zero''' is defined as being the cardinal of the empty set."}
+{"_id": "20038", "title": "Definition:Multiplication", "text": "'''Multiplication''' is the basic operation $\\times$ everyone is familiar with. For example: :$3 \\times 4 = 12$ :$13 \\cdotp 2 \\times 7 \\cdotp 7 = 101 \\cdotp 64$"}
+{"_id": "20039", "title": "Definition:Subtraction", "text": "=== Natural Numbers === {{:Definition:Subtraction/Natural Numbers}} === Integers === {{:Definition:Subtraction/Integers}} === Rational Numbers === {{:Definition:Subtraction/Rational Numbers}} === Real Numbers === {{:Definition:Subtraction/Real Numbers}} === Complex Numbers === {{:Definition:Subtraction/Complex Numbers}} === Extended Real Subtraction === {{:Definition:Extended Real Subtraction}}"}
+{"_id": "20040", "title": "Definition:Finite", "text": "=== Finite Cardinal === {{:Definition:Finite Cardinal}} === Finite Set === {{:Definition:Finite Set}} === Finite Extended Real Number === {{:Definition:Finite Extended Real Number}}"}
+{"_id": "20041", "title": "Definition:Infinite Set", "text": "A set which is not finite is called '''infinite'''. That is, it is a set for which there is no bijection between it and any $\\N_n$, where $\\N_n$ is the the set of all elements of $n$ less than $n$, no matter how big we make $n$."}
+{"_id": "20042", "title": "Definition:Cardinality", "text": "Two sets (either '''finite''' or '''infinite''') which are '''equivalent''' are said to have the same '''cardinality'''. The '''cardinality''' of a set $S$ is written $\\card S$."}
+{"_id": "20043", "title": "Definition:Countable Set", "text": "Let $S$ be a set. === Definition 1 === {{:Definition:Countable Set/Definition 1}} === Definition 2 === {{:Definition:Countable Set/Definition 2}} === Definition 3 === {{:Definition:Countable Set/Definition 3}}"}
+{"_id": "20044", "title": "Definition:Uncountable Set", "text": "A set which is infinite but not countable is described as '''uncountable'''."}
+{"_id": "20046", "title": "Definition:Ordered Tuple", "text": "Let $n \\in \\N$ be a natural number. Let $\\N^*_n$ be the first $n$ non-zero natural numbers: :$\\N^*_n := \\set {1, 2, \\ldots, n}$"}
+{"_id": "20047", "title": "Definition:Operation", "text": "An '''operation''' is an object, identified by a symbol, which can be interpreted as a process which, from a number of objects, creates a new object."}
+{"_id": "20048", "title": "Definition:Iterated Binary Operation", "text": "=== Indexed Iteration === {{Definition:Indexed Iterated Binary Operation}} === Iteration over Finite Set === {{:Definition:Iterated Binary Operation over Finite Set}} === Iteration over Set with Finite Support === {{Definition:Iterated Binary Operation over Set with Finite Support}}"}
+{"_id": "20049", "title": "Definition:Summation", "text": "Let $\\struct {S, +}$ be an algebraic structure where the operation $+$ is an operation derived from, or arising from, the addition operation on the natural numbers. Let $\\tuple {a_1, a_2, \\ldots, a_n} \\in S^n$ be an ordered $n$-tuple in $S$. === Definition by Index === {{:Definition:Summation/Indexed}} === Definition by Inequality === {{:Definition:Summation/Inequality}} === Definition by Propositional Function === {{:Definition:Summation/Propositional Function}}"}
+{"_id": "20050", "title": "Definition:Product Notation (Algebra)", "text": "Let $\\struct {S, \\times}$ be an algebraic structure where the operation $\\times$ is an operation derived from, or arising from, the multiplication operation on the natural numbers. Let $\\tuple {a_1, a_2, \\ldots, a_n} \\in S^n$ be an ordered $n$-tuple in $S$. === Definition by Index === {{:Definition:Product Notation (Algebra)/Index}}"}
+{"_id": "20051", "title": "Definition:Addition", "text": "'''Addition''' is the basic operation $+$ everyone is familiar with. For example: :$2 + 3 = 5$ :$47 \\cdotp 3 + 191\\cdotp 4 = 238 \\cdotp 7$"}
+{"_id": "20052", "title": "Definition:Power of Element", "text": "=== Magma === {{:Definition:Power of Element/Magma}} === Magma with Identity === {{:Definition:Power of Element/Magma with Identity}} === Semigroup === {{:Definition:Power of Element/Semigroup}} === Monoid === {{:Definition:Power of Element/Monoid}} === Group === {{:Definition:Power of Element/Group}} === Ring === {{:Definition:Power of Element/Ring}}"}
+{"_id": "20053", "title": "Definition:Indexing Set", "text": "Let $I$ and $S$ be sets. Let $x: I \\to S$ be a mapping. Let $x_i$ denote the image of an element $i \\in I$ of the domain $I$ of $x$. Let $\\family {x_i}_{i \\mathop \\in I}$ denote the set of the images of all the element $i \\in I$ under $x$. When a mapping is used in this context, the domain $I$ of $x$ is called the '''indexing set''' of the terms $\\family {x_i}_{i \\mathop \\in I}$. === Index === {{:Definition:Indexing Set/Index}} === Indexed Set === {{:Definition:Indexing Set/Indexed Set}} === Indexing Function === {{:Definition:Indexing Set/Function}} === Family === {{:Definition:Indexing Set/Family}} === Term === {{:Definition:Indexing Set/Term}} === Family of Distinct Elements === {{:Definition:Indexing Set/Family of Distinct Elements}} === Family of Sets === {{:Definition:Indexing Set/Family of Sets}} === Family of Subsets === {{:Definition:Indexing Set/Family of Subsets}}"}
+{"_id": "20054", "title": "Definition:Integer", "text": "The numbers $\\set {\\ldots, -3, -2, -1, 0, 1, 2, 3, \\ldots}$ are called the '''integers'''. This set is usually denoted $\\Z$. An individual element of $\\Z$ is called '''an integer'''."}
+{"_id": "20055", "title": "Definition:Additive Group of Integers", "text": "The '''additive group of integers''' $\\struct {\\Z, +}$ is the set of integers under the operation of addition."}
+{"_id": "20056", "title": "Definition:Rational Number", "text": "A number in the form $\\dfrac p q$, where both $p$ and $q$ are integers ($q$ non-zero), is called a '''rational number'''. The set of all '''rational numbers''' is usually denoted $\\Q$. Thus: :$\\Q = \\set {\\dfrac p q: p \\in \\Z, q \\in \\Z_{\\ne 0} }$"}
+{"_id": "20057", "title": "Definition:Irrational Number", "text": "An '''irrational number''' is a real number which is not rational. That is, an '''irrational number''' is one that can not be expressed in the form $\\dfrac p q$ such that $p$ and $q$ are both integers. The set of '''irrational numbers''' can therefore be expressed as $\\R \\setminus \\Q$, where: :$\\R$ is the set of real numbers :$\\Q$ is the set of rational numbers :$\\setminus$ denotes set difference."}
+{"_id": "20058", "title": "Definition:Factorial", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. === Definition 1 === {{:Definition:Factorial/Definition 1}} === Definition 2 === {{:Definition:Factorial/Definition 2}}"}
+{"_id": "20059", "title": "Definition:Binomial Coefficient", "text": "=== Definition 1 === {{:Definition:Binomial Coefficient/Integers/Definition 1}} === Definition 2 === {{:Definition:Binomial Coefficient/Integers/Definition 2}} === Definition 3 === {{:Definition:Binomial Coefficient/Integers/Definition 3}}"}
+{"_id": "20060", "title": "Definition:Number", "text": "There are five main classes of number: :$(1): \\quad$ The natural numbers: $\\N = \\set {0, 1, 2, 3, \\ldots}$ :$(2): \\quad$ The integers: $\\Z = \\set {\\ldots, -3, -2, -1, 0, 1, 2, 3, \\ldots}$ :$(3): \\quad$ The rational numbers: $\\Q = \\set {p / q: p, q \\in \\Z, q \\ne 0}$ :$(4): \\quad$ The real numbers: $\\R = \\set {x: x = \\sequence {s_n} }$ where $\\sequence {s_n}$ is a Cauchy sequence in $\\Q$ :$(5): \\quad$ The complex numbers: $\\C = \\set {a + i b: a, b \\in \\R, i^2 = -1}$"}
+{"_id": "20061", "title": "Definition:Absolute Value", "text": "=== Definition 1 === {{:Definition:Absolute Value/Definition 1}} === Definition 2 === {{:Definition:Absolute Value/Definition 2}}"}
+{"_id": "20062", "title": "Definition:Remainder", "text": "Let $a, b \\in \\Z$ be integers such that $b \\ne 0$. From the Division Theorem, we have that: :$\\forall a, b \\in \\Z, b \\ne 0: \\exists! q, r \\in \\Z: a = q b + r, 0 \\le r < \\left|{b}\\right|$ The value $r$ is defined as the '''remainder of $a$ on division by $b$''', or the '''remainder of $\\dfrac a b$'''. === Real Arguments === When $x, y \\in \\R$ the '''remainder''' is still defined: {{:Definition:Remainder/Real}}"}
+{"_id": "20063", "title": "Definition:Even Integer", "text": "=== Definition 1 === {{:Definition:Even Integer/Definition 1}} === Definition 2 === {{:Definition:Even Integer/Definition 2}} === Definition 3 === {{:Definition:Even Integer/Definition 3}}"}
+{"_id": "20064", "title": "Definition:Odd Integer", "text": "=== Definition 1 === {{:Definition:Odd Integer/Definition 1}} === Definition 2 === {{:Definition:Odd Integer/Definition 2}} === Definition 3 === {{:Definition:Odd Integer/Definition 3}} === Euclid's Definition === {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book VII/7 - Odd Number}}'' {{EuclidDefRefNocat|VII|7|Odd Number}}"}
+{"_id": "20065", "title": "Definition:Integer Combination", "text": "Let $a, b \\in \\Z$. An integer $n$ of the form: :$n = p a + q b: p, q \\in \\Z$ is an '''integer combination''' of $a$ and $b$."}
+{"_id": "20066", "title": "Definition:Common Divisor", "text": "=== Integral Domain === {{:Definition:Common Divisor/Integral Domain}} === Integers === The definition is usually applied when the integral domain in question is the set of integers $\\Z$, thus: {{:Definition:Common Divisor/Integers}} === Real Numbers === The definition can also be applied when the integral domain in question is the real numbers $\\R$, thus: {{:Definition:Common Divisor/Real Numbers}}"}
+{"_id": "20067", "title": "Definition:Coprime", "text": "=== GCD Domain === {{:Definition:Coprime/GCD Domain}} === Euclidean Domain === {{:Definition:Coprime/Euclidean Domain}} === Integers === {{:Definition:Coprime/Integers}}"}
+{"_id": "20068", "title": "Definition:Integers Modulo m", "text": "Let $m \\in \\Z$ be an integer. The '''integers modulo $m$''' are the set of least positive residues of the set of residue classes modulo $m$: :$\\Z_m = \\set {0, 1, \\ldots, m - 1}$"}
+{"_id": "20069", "title": "Definition:Prime Number", "text": "=== Definition 1 === {{:Definition:Prime Number/Definition 1}} === Definition 2 === {{:Definition:Prime Number/Definition 2}} === Definition 3 === {{:Definition:Prime Number/Definition 3}} === Definition 4 === {{:Definition:Prime Number/Definition 4}} === Definition 5 === {{:Definition:Prime Number/Definition 5}} === Definition 6 === {{:Definition:Prime Number/Definition 6}} === Definition 7 === {{:Definition:Prime Number/Definition 7}} === Euclid's Definition === {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book VII/11 - Prime Number}}'' {{EuclidDefRefNocat|VII|11|Prime Number}}"}
+{"_id": "20070", "title": "Definition:Composite Number", "text": "A '''composite number''' $c$ is a positive integer that has more than two positive divisors. That is, an integer greater than $1$ which is not prime is defined as composite."}
+{"_id": "20071", "title": "Definition:Sylow p-Subgroup", "text": "Let $p$ be prime. Let $G$ be a finite group whose order is denoted by $\\left\\vert{G}\\right\\vert$. === Definition 1 === {{:Definition:Sylow p-Subgroup/Definition 1}} === Definition 2 === {{:Definition:Sylow p-Subgroup/Definition 2}} === Definition 3 === {{:Definition:Sylow p-Subgroup/Definition 3}}"}
+{"_id": "20072", "title": "Definition:Prime Decomposition", "text": "Let $n > 1 \\in \\Z$. From the Fundamental Theorem of Arithmetic, $n$ has a unique factorization of the form: {{begin-eqn}} {{eqn | l = n       | r = \\prod_{p_i \\mathop \\divides n} {p_i}^{k_i}       | c =  }} {{eqn | r = {p_1}^{k_1} {p_2}^{k_2} \\cdots {p_r}^{k_r}       | c =  }} {{end-eqn}} where: :$p_1 < p_2 < \\cdots < p_r$ are distinct primes :$k_1, k_2, \\ldots, k_r$ are (strictly) positive integers. This unique expression is known as the '''prime decomposition of $n$'''."}
+{"_id": "20073", "title": "Definition:Möbius Function", "text": "Let $n \\in \\Z_{>0}$, that is, a strictly positive integer. The '''Möbius function''' is the function $\\mu: \\Z_{>0} \\to \\Z_{>0}$ defined as: :$\\map \\mu n = \\begin{cases} 1 & : n = 1 \\\\ 0 & : \\exists p \\in \\mathbb P: p^2 \\divides n\\\\ \\left({-1}\\right)^k & : n = p_1 p_2 \\ldots p_k: p_i \\in \\mathbb P \\end{cases}$"}
+{"_id": "20074", "title": "Definition:Square", "text": "=== Geometry === {{:Definition:Quadrilateral/Square}} === Abstract Algebra === {{:Definition:Square/Mapping}}"}
+{"_id": "20075", "title": "Definition:Fibonacci Number", "text": "The '''Fibonacci numbers''' are a sequence $\\sequence {F_n}$ of integers which is formally defined recursively as: :$F_n = \\begin {cases} 0 & : n = 0 \\\\ 1 & : n = 1 \\\\ F_{n - 1} + F_{n - 2} & : \\text {otherwise} \\end {cases}$ for all $n \\in \\Z_{\\ge 0}$."}
+{"_id": "20076", "title": "Definition:Cauchy Sequence", "text": "=== Metric Space === {{:Definition:Cauchy Sequence/Metric Space}} === Normed Vector Space === {{:Definition:Cauchy Sequence/Normed Vector Space}} === Normed Division Ring === {{:Definition:Cauchy Sequence/Normed Division Ring}}"}
+{"_id": "20077", "title": "Definition:Real Number", "text": "{{:Definition:Real Number/Number Line Definition}}"}
+{"_id": "20078", "title": "Definition:Real Interval", "text": "Informally, the set of all real numbers between any two given real numbers $a$ and $b$ is called a '''(real) interval'''."}
+{"_id": "20079", "title": "Definition:Mediant", "text": "Let $r, s \\in \\Q$, i.e. let $r, s$ be rational numbers. Let $r$ and $s$ be expressed as $r = \\dfrac a b, s = \\dfrac c d$ where $a, b, c, d$ are integers such that $b > 0, d > 0$ (this is always possible by Divided by Positive Element of Field of Quotients). Then the '''mediant''' of $r$ and $s$ is $\\dfrac {a + c} {b + d}$."}
+{"_id": "20080", "title": "Definition:Close Packed", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Then $\\left({S, \\preceq}\\right)$ is defined as '''close packed (in itself)''' {{iff}} between every two elements of $S$ there exists another element of $S$: : $\\forall a, b \\in S: a \\prec b \\implies \\exists c \\in S: a \\prec c \\prec b$ === Close Packed Subset === {{:Definition:Close Packed/Subset}}"}
+{"_id": "20081", "title": "Definition:Floor Function", "text": "Let $x$ be a real number. Informally, the '''floor function of $x$''' is the greatest integer less than or equal to $x$. === Definition 1 === {{Definition:Floor Function/Definition 1}} === Definition 2 === {{Definition:Floor Function/Definition 2}} === Definition 3 === {{Definition:Floor Function/Definition 3}}"}
+{"_id": "20082", "title": "Definition:Ceiling Function", "text": "Let $x$ be a real number. Informally, the '''ceiling function of $x$''' is the smallest integer greater than or equal to $x$. === Definition 1 === {{Definition:Ceiling Function/Definition 1}} === Definition 2 === {{Definition:Ceiling Function/Definition 2}} === Definition 3 === {{Definition:Ceiling Function/Definition 3}}"}
+{"_id": "20083", "title": "Definition:Real Function", "text": "A '''real function''' is a mapping or function whose domain and codomain are subsets of the set of real numbers $\\R$."}
+{"_id": "20084", "title": "Definition:Implicit Function", "text": "Consider a (real) function of two independent variables $z = \\map f {x, y}$. Let a relation between $x$ and $y$ be expressed in the form $\\map f {x, y} = 0$ defined on some interval $\\mathbb I$. If there exists a function: :$y = \\map g x$ defined on $\\mathbb I$ such that: :$\\forall x \\in \\mathbb I: \\map f {x, \\map g x} = 0$ then the relation $\\map f {x, y} = 0$ defines $y$ as an '''implicit function''' of $x$."}
+{"_id": "20085", "title": "Definition:Order of Structure", "text": "The '''order''' of an algebraic structure $\\struct {S, \\circ}$ is the cardinality of its underlying set, and is denoted $\\order S$. Thus, for a finite set $S$, the '''order of $\\struct {S, \\circ}$''' is the number of elements in $S$."}
+{"_id": "20086", "title": "Definition:Word (Abstract Algebra)", "text": "Let $\\struct {G, \\circ}$ be a magma. Let $S \\subseteq G$ be a subset. A '''word in $S$''' is the product of a finite number of elements of $S$. The '''set of words in $S$''' is denoted $\\map W S$: :$\\map W S := \\set {s_1 \\circ s_2 \\circ \\cdots \\circ s_n: n \\in \\N_{>0}: s_i \\in S, 1 \\le i \\le n}$"}
+{"_id": "20087", "title": "Definition:Truth Table", "text": "A '''truth table''' is a tabular array that represents the computation of a truth function, that is, a function of the form: :$f : \\mathbb B^k \\to \\mathbb B$ where: :$k$ is a non-negative integer :$\\mathbb B$ is a set of truth values, usually $\\set {0, 1}$ or $\\set {T, F}$."}
+{"_id": "20088", "title": "Definition:Index of Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. The '''index of $H$ (in $G$)''', denoted $\\index G H$, is the cardinality of the left (or right) coset space $G / H$. === Finite Index === {{:Definition:Index of Subgroup/Finite}} === Infinite Index === {{:Definition:Index of Subgroup/Infinite}}"}
+{"_id": "20089", "title": "Definition:Parametric Operator", "text": "Let $A$ be an index set. Let $\\Omega$ be a indexed family of operators $\\left({\\Omega_{\\alpha} }\\right)_A = \\left\\{ {\\Omega_\\alpha: \\alpha \\in A}\\right\\}$ indexed by the index $\\alpha$. {{explain|Definition of \"operator\" in this context.}} Then $\\Omega$ is a '''parametric operator''' with '''parameter $\\alpha$''' in the '''parameter set $A$'''. Category:Definitions/Set Theory 2qv55f8elqvw9z1t799m5q40r3h0865"}
+{"_id": "20090", "title": "Definition:Order of Group Element", "text": "Let $G$ be a group whose identity is $e_G$. Let $x \\in G$ be an element of $G$. === Definition 1 === {{:Definition:Order of Group Element/Definition 1}} === Definition 2 === {{:Definition:Order of Group Element/Definition 2}} === Definition 3 === {{:Definition:Order of Group Element/Definition 3}}"}
+{"_id": "20091", "title": "Definition:Derangement", "text": "A '''derangement''' is a permutation $f: S \\to S$ from a set $S$ to itself where $\\map f s \\ne s$ for any $s \\in S$.  If $S$ is finite, the number of '''derangements''' is denoted by $D_n$ where $n = \\card S$ (the cardinality of $S$.)"}
+{"_id": "20092", "title": "Definition:Hamiltonian Group", "text": "A group $G$ is described as '''Hamiltonian''' {{iff}} $G$ is a non-abelian group such that every subgroup is normal."}
+{"_id": "20093", "title": "Definition:Simple Group", "text": "A group $G$ is '''simple''' {{iff}} it has only $G$ and the trivial group as normal subgroups. That is, if the composition length of $G$ is $1$."}
+{"_id": "20094", "title": "Definition:Group Direct Product", "text": "Let $\\struct {G, \\circ_1}$ and $\\struct {H, \\circ_2}$ be groups. Let $G \\times H: \\set {\\tuple {g, h}: g \\in G, h \\in H}$ be their cartesian product. The '''(external) direct product''' of $\\struct {G, \\circ_1}$ and $\\struct {H, \\circ_2}$ is the group $\\struct {G \\times H, \\circ}$ where the operation $\\circ$ is defined as: :$\\tuple {g_1, h_1} \\circ \\tuple {g_2, h_2} = \\tuple {g_1 \\circ_1 g_2, h_1 \\circ_2 h_2}$ This is usually referred to as the '''group direct product''' of $G$ and $H$."}
+{"_id": "20095", "title": "Definition:Independent Subgroups", "text": "Let $G$ be a group whose identity is $e$. Let $\\left \\langle {H_n} \\right \\rangle$ be a sequence of subgroups of $G$. === Definition 1 === {{:Definition:Independent Subgroups/Definition 1}} === Definition 2 === {{:Definition:Independent Subgroups/Definition 2}}"}
+{"_id": "20096", "title": "Definition:Group Presentation", "text": "Let $G$ be a group. A '''presentation for $G$''' is a triple $\\tuple {S, R, f}$ where: :$S$ is a set :$R$ is a set of relations on $S$ :$f: \\gen {S \\mid R} \\to G$ is a group isomorphism from the group defined by $\\struct {S, R}$."}
+{"_id": "20097", "title": "Definition:Infinite Cyclic Group", "text": "=== Definition 1 === {{:Definition:Infinite Cyclic Group/Definition 1}} === Definition 2 === {{:Definition:Infinite Cyclic Group/Definition 2}}"}
+{"_id": "20098", "title": "Definition:Free Group", "text": "=== Definition 1 === {{:Definition:Free Group/Definition 1}} === Definition 2 === {{:Definition:Free Group/Definition 2}}"}
+{"_id": "20099", "title": "Definition:Ring of Integers Modulo m", "text": "Let $m \\in \\Z: m \\ge 2$. Let $\\Z_m$ be the set of integers modulo $m$. Let $+_m$ and $\\times_m$ denote addition modulo $m$ and multiplication modulo $m$ respectively. The algebraic structure $\\struct {\\Z_m, +_m, \\times_m}$ is '''the ring of integers modulo $m$'''."}
+{"_id": "20100", "title": "Definition:Symmetry Mapping", "text": "A '''symmetry mapping''' (or just '''symmetry''') of a geometric figure is a bijection from the figure to itself which preserves the distance between points. In other words, it is a self-congruence. Intuitively and informally, a '''symmetry''' is a movement of the figure so that it looks exactly the same after it has been moved."}
+{"_id": "20101", "title": "Definition:Geometric Figure", "text": "A '''geometric figure''' is intuitively defined as a set of points and lines in space. {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book I/14 - Geometric Figure}}'' {{EuclidDefRefNocat|I|14|Geometric Figure}} The boundary may or may not be included in a particular figure. If this is important (and in the study of topology it usually is), then whether it is included or not needs to be specified. === Plane Figure === {{:Definition:Geometric Figure/Plane Figure}} === Three-Dimensional Figure === {{:Definition:Geometric Figure/Three-Dimensional Figure}} === Rectilineal Figure === {{:Definition:Geometric Figure/Rectilineal Figure}}"}
+{"_id": "20102", "title": "Definition:Permutation on n Letters", "text": "Let $\\N_k$ be used to denote the initial segment of natural numbers: : $\\N_k = \\closedint 1 k = \\set {1, 2, 3, \\ldots, k}$ A '''permutation on $n$ letters''' is a '''permutation''': : $\\pi: \\N_n \\to \\N_n$ The usual symbols for denoting a general '''permutation''' are $\\pi$ (not to be confused with the famous circumference over diameter), $\\rho$ and $\\sigma$."}
+{"_id": "20103", "title": "Definition:Fixed Element of Permutation", "text": "Let $S$ be a set. Let $\\pi: S \\to S$ be a permutation on $S$. Let $x \\in S$. $x$ is '''fixed by $\\pi$''' {{iff}}: :$\\map \\pi x = x$"}
+{"_id": "20104", "title": "Definition:Cyclic Permutation", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\rho \\in S_n$ be a permutation on $S$. Then $\\rho$ is a '''cyclic permutation of length $k$''' {{iff}} there exists $k \\in \\Z: k > 0$ and $i \\in \\Z$ such that: :$(1): \\quad k$ is the smallest such that $\\map {\\rho^k} i = i$ :$(2): \\quad \\rho$ fixes each $j$ not in $\\set {i, \\map \\rho i, \\ldots, \\map {\\rho^{k - 1} } i}$."}
+{"_id": "20105", "title": "Definition:Disjoint Permutations", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\pi, \\rho \\in S_n$ both be permutations on $S_n$. Then $\\pi$ and $\\rho$ are '''disjoint''' {{iff}}: :$(1): \\quad i \\notin \\Fix \\pi \\implies i \\in \\Fix \\rho$ :$(2): \\quad i \\notin \\Fix \\rho \\implies i \\in \\Fix \\pi$ That is, each element moved by $\\pi$ is fixed by $\\rho$ and (equivalently) each element moved by $\\rho$ is fixed by $\\pi$. That is, {{iff}} their supports are disjoint sets. We may say that: : $\\pi$ is '''disjoint from $\\rho$''' : $\\rho$ is '''disjoint from $\\pi$''' : $\\pi$ and $\\rho$ are '''(mutually) disjoint'''. Note of course that it is perfectly possible for $i \\in \\Fix \\pi$ and also $i \\in \\Fix \\rho$, that is, there may well be elements fixed by more than one of a pair of '''disjoint permutations'''."}
+{"_id": "20106", "title": "Definition:Group Action", "text": "Let $X$ be a set. Let $\\struct {G, \\circ}$ be a group whose identity is $e$. === Left Group Action === {{:Definition:Group Action/Left Group Action}} === Right Group Action === {{:Definition:Group Action/Right Group Action}} The group $G$ thus '''acts on''' the set $X$. The group $G$ can be referred to as the '''group of transformations''', or a '''transformation group'''."}
+{"_id": "20107", "title": "Definition:Orbit (Group Theory)", "text": "Let $G$ be a group acting on a set $X$. === Definition 1 === {{:Definition:Orbit (Group Theory)/Definition 1}} === Definition 2 === {{:Definition:Orbit (Group Theory)/Definition 2}}"}
+{"_id": "20108", "title": "Definition:Effective Transformation Group", "text": "Let $G$ be a group whose identity is $e$. Let $X$ be a set. Let $\\phi: G \\times X \\to X$ be a group action. Then $G$ is an '''effective transformation group for $\\phi$''' {{iff}} $\\phi$ is faithful."}
+{"_id": "20109", "title": "Definition:Transposition", "text": "Let $S$ be a set. A '''transposition''' on $S$ is a $2$-cycle. That is, a '''transposition''' is a permutation $\\rho$ on $S$ which exchanges, or '''transposes''', exactly two elements of $S$. Thus if $\\rho$ is a '''transposition''' which '''transposes''' two elements $r, s \\in S$, it follows from the definition of fixed elements that: :$\\Fix \\rho = S \\setminus \\set {r, s}$"}
+{"_id": "20111", "title": "Definition:R-Algebraic Structure", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {S, \\ast_1, \\ast_2, \\ldots, \\ast_n}$ be an algebraic structure with $n$ operations. Let $\\circ: R \\times S \\to S$ be a binary operation. Then $\\struct {S, \\ast_1, \\ast_2, \\ldots, \\ast_n, \\circ}_R$ is an '''$R$-algebraic structure with $n$ operations'''. If the number of operations in $S$ is either understood or general, it is just called an '''$R$-algebraic structure''', and the structure can be denoted $\\struct {S, \\circ}_R$."}
+{"_id": "20112", "title": "Definition:Module", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {G, +_G}$ be an abelian group. A module over $R$ is an $R$-algebraic structure with one operation $\\struct {G, +_G, \\circ}_R$ which is both a left module and a right module: === Left Module === {{:Definition:Left Module}} === Right Module === {{:Definition:Right Module}}"}
+{"_id": "20113", "title": "Definition:Linear Transformation", "text": "A '''linear transformation''' is a homomorphism from one module to another."}
+{"_id": "20114", "title": "Definition:Module on Cartesian Product", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $n \\in \\N_{>0}$. Let $+: R^n \\times R^n \\to R^n$ be defined as: :$\\tuple {\\alpha_1, \\ldots, \\alpha_n} + \\tuple {\\beta_1, \\ldots, \\beta_n} = \\tuple {\\alpha_1 +_R \\beta_1, \\ldots, \\alpha_n +_R \\beta_n}$ Let $\\times: R \\times R^n \\to R^n$ be defined as: :$\\lambda \\times \\tuple {\\alpha_1, \\ldots, \\alpha_n} = \\tuple {\\lambda \\times_R \\alpha_1, \\ldots, \\lambda \\times_R \\alpha_n}$ Then $\\struct {R^n, +, \\times}_R$ is '''the $R$-module $R^n$'''."}
+{"_id": "20115", "title": "Definition:Trivial Module", "text": "Let $\\struct {G, +_G}$ be an abelian group whose identity is $e_G$. Let $\\struct {R, +_R, \\circ_R}$ be a ring. Let $\\circ$ be defined as: :$\\forall \\lambda \\in R: \\forall x \\in G: \\lambda \\circ x = e_G$ Then $\\struct {G, +_G, \\circ}_R$ is an $R$-module. Such a module is called a '''trivial module'''."}
+{"_id": "20116", "title": "Definition:Submodule", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\struct {S, +, \\circ}_R$ be an $R$-algebraic structure with one operation. Let $T$ be a closed subset of $S$. Let $\\struct {T, +_T, \\circ_T}_R$ be an $R$-module where: :$+_T$ is the restriction of $+$ to $T \\times T$ :$\\circ_T$ is the restriction of $\\circ$ to $R \\times T$. Then $\\struct {T, +_T, \\circ_T}_R$ is a '''submodule''' of $\\struct {S, +, \\circ}_R$."}
+{"_id": "20117", "title": "Definition:Null Module", "text": "Let $\\left({R, +_R, \\circ_R}\\right)$ be a ring. Let $G$ be the trivial group. Then the $R$-module $\\left({G, +_G, \\circ}\\right)_R$ is known as the '''null module'''."}
+{"_id": "20118", "title": "Definition:P-Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. If $H$ is a $p$-group, then $H$ is a '''$p$-subgroup''' of $G$. {{SUBPAGENAME}} e6bq56il8qbgbcyygojvbcz1jag6lue"}
+{"_id": "20119", "title": "Definition:Finitely Generated Module", "text": "Let $R$ be a ring. Let $G$ be a module over $R$. Then $G$ is '''finitely generated''' {{iff}} there is a generator for $G$ which is finite."}
+{"_id": "20120", "title": "Definition:Linear Combination", "text": "Let $R$ be a ring. === Linear Combination of Sequence === {{:Definition:Linear Combination/Sequence}} === Linear Combination of Subset === {{:Definition:Linear Combination/Subset}} === Linear Combination of Empty Set === {{:Definition:Linear Combination/Empty Set}}"}
+{"_id": "20121", "title": "Definition:Linearly Independent", "text": "Let $G$ be an abelian group whose identity is $e$. Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\\left({G, +_G, \\circ}\\right)_R$ be a unitary $R$-module. === Sequence === {{:Definition:Linearly Independent/Sequence}} === Set === {{:Definition:Linearly Independent/Set}}"}
+{"_id": "20122", "title": "Definition:Basis (Topology)", "text": "=== Analytic Basis === {{:Definition:Basis (Topology)/Analytic Basis}} === Synthetic Basis === {{:Definition:Basis (Topology)/Synthetic Basis}}"}
+{"_id": "20123", "title": "Definition:Free Module", "text": "Let $R$ be a ring with unity. Let $G$ be a unitary $R$-module. Then $G$ is described as '''free''' if there exists a basis of $G$."}
+{"_id": "20124", "title": "Definition:Ordered Basis", "text": "Let $R$ be a ring with unity. Let $G$ be a free $R$-module. An '''ordered basis''' of $G$ is a sequence $\\left \\langle {a_k} \\right \\rangle_{1 \\mathop \\le k \\mathop \\le n}$ of elements of $G$ such that $\\left\\{{a_1, \\ldots, a_n}\\right\\}$ is a basis of $G$."}
+{"_id": "20125", "title": "Definition:Topological Manifold/Differentiable Manifold", "text": "Let $M$ be a second-countable locally Euclidean space of dimension $d$.  Let $\\mathscr F$ be a $d$-dimensional differentiable structure on $M$ of class $\\mathcal C^k$, where $k \\ge 1$. Then $\\left({M, \\mathscr F}\\right)$ is a '''differentiable manifold of class $C^k$ and dimension $d$'''."}
+{"_id": "20126", "title": "Definition:Dimension (Linear Algebra)", "text": "=== Module === {{:Definition:Dimension of Module}} === Vector Space === {{:Definition:Dimension of Vector Space}} {{wtd|Infinite dimensional case}}"}
+{"_id": "20127", "title": "Definition:Metric Space", "text": "A '''metric space''' $M = \\struct {A, d}$ is an ordered pair consisting of: :$(1): \\quad$ a non-empty set $A$ together with: :$(2): \\quad$ a real-valued function $d: A \\times A \\to \\R$ which acts on $A$, satisfying the metric space axioms: {{:Definition:Metric Space Axioms}}"}
+{"_id": "20128", "title": "Definition:Topological Space", "text": "Let $S$ be a set. Let $\\tau$ be a topology on $S$. That is, let $\\tau \\subseteq \\powerset S$ satisfy the open set axioms: {{:Definition:Open Set Axioms}} Then the ordered pair $\\struct {S, \\tau}$ is called a '''topological space'''. The elements of $\\tau$ are called open sets of $\\struct {S, \\tau}$."}
+{"_id": "20129", "title": "Definition:Topology", "text": "{{:Definition:Topology/Definition 1}}"}
+{"_id": "20130", "title": "Definition:Smooth Real Function", "text": "A real function is '''smooth''' if it is of differentiability class $C^\\infty$. That is, if it admits of continuous derivatives of all orders."}
+{"_id": "20131", "title": "Definition:Almost Everywhere", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. A property $\\map P x$ of elements of $X$ is said to hold '''($\\mu$-)almost everywhere''' if the set: :$\\set {x \\in X: \\neg \\map P x}$ of elements of $X$ such that $P$ does not hold is contained in a $\\mu$-null set."}
+{"_id": "20132", "title": "Definition:Regular Value", "text": "Let $X$ and $Y$ be smooth manifolds. Let $f: X \\to Y$ be a smooth mapping. Then a point $y \\in Y$ is called a '''regular value''' of $f$ {{iff}} the pushforward of $f$ at $x$: : $f_* \\vert_x: T_x X \\to T_y Y$ {{explain|What do all the symbols mean in this context? Presume $\\vert$ might mean restriction, but this is not obvious (if so then use $\\restriction$); if \"pushforward\" actually means $f_* \\vert_x: T_x X \\to T_y Y$ then set up the page to define it, thus doing all the hard work of defining that concept all in one place.}} is surjective for every $x \\in f^{-1} \\left({y}\\right) \\subseteq X$."}
+{"_id": "20133", "title": "Definition:Kronecker Delta", "text": "Let $\\Gamma$ be a set. Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Then $\\delta_{\\alpha \\beta}: \\Gamma \\times \\Gamma \\to R$ is the mapping on the cartesian square of $\\Gamma$ defined as: :$\\forall \\tuple {\\alpha, \\beta} \\in \\Gamma \\times \\Gamma: \\delta_{\\alpha \\beta} := \\begin{cases} 1_R & : \\alpha = \\beta \\\\ 0_R & : \\alpha \\ne \\beta \\end{cases}$"}
+{"_id": "20134", "title": "Definition:Linear Form", "text": "Let $R$ be a commutative ring. Let $\\struct {G, +_G, \\circ}_R$ be a module over $R$. Let $\\phi: \\struct {G, +_G, \\circ}_R \\to \\struct {R, +_R, \\circ}_R$ be a linear transformation from $G$ to the $R$-module $R$. Then $\\phi$ is called a '''linear form on $G$'''."}
+{"_id": "20135", "title": "Definition:Algebraic Dual", "text": "Let $R$ be a commutative ring. Let $G$ be a module over $R$. The $R$-module $\\map {\\LL_R} {G, R}$ of all linear forms on $G$ is usually denoted $G^*$ and is called the '''algebraic dual''' of $G$. === Double Dual === {{:Definition:Algebraic Dual/Double Dual}}"}
+{"_id": "20136", "title": "Definition:Ordered Dual Basis", "text": "Let $R$ be a commutative ring. Let $\\struct {G, +_G, \\circ}_R$ be an $n$-dimensional module over $R$. Let $\\sequence {a_n}$ be an ordered basis of $G$. Let $G^*$ be the algebraic dual of $G$. Then there is an ordered basis $\\sequence {a'_n}$ of $G^*$ satisfying $\\forall i, j \\in \\closedint 1 n: \\map {a'_i} {a_j} = \\delta_{i j}$. This ordered basis $\\sequence {a'_n}$ of $G^*$ is called the '''ordered basis of $G^*$ dual to $\\sequence {a_n}$''', or the '''ordered dual basis of $G^*$'''."}
+{"_id": "20137", "title": "Definition:Evaluation Linear Transformation", "text": "Let $R$ be a commutative ring. Let $G$ be an $R$-module. Let $G^*$ be the algebraic dual of $G$. Let $G^{**}$ be the algebraic dual of $G^*$. For each $x \\in G$, we define the mapping $x^\\wedge: G^* \\to R$ as: :$\\forall t' \\in G^*: x^\\wedge \\left({t'}\\right) = t' \\left({x}\\right)$ Then $x^\\wedge \\in G^{**}$. The mapping $J: G \\to G^{**}$ defined as: :$\\forall x \\in G: J \\left({x}\\right) = x^\\wedge$ is called the '''evaluation linear transformation from $G$ into $G^{**}$'''. It is usual to denote the mapping $t': G^* \\to G$ as follows: :$\\forall x \\in G, t' \\in G^*: \\left \\langle {x, t'} \\right \\rangle := t' \\left({x}\\right)$"}
+{"_id": "20138", "title": "Definition:Evaluation Isomorphism", "text": "Let $R$ be a commutative ring. Let $G$ be a unitary $R$-module whose dimension is finite. Then the evaluation linear transformation $J: G \\to G^{**}$ is called the '''evaluation isomorphism from $G$ to $G^{**}$."}
+{"_id": "20139", "title": "Definition:Annihilator", "text": "Let $R$ be a commutative ring. Let $M$ and $N$ be modules over $R$. Let $B : M \\times N \\to R$ be a bilinear mapping. The '''annihilator of $D \\subseteq M$''', denoted $\\operatorname{Ann}_N \\left({D}\\right)$ is the set: :$\\left\\{{n \\in N : \\forall d \\in D: B \\left({d, n}\\right) = 0}\\right\\}$"}
+{"_id": "20140", "title": "Definition:Transpose of Linear Transformation", "text": "Let $R$ be a commutative ring. Let $G$ and $H$ be $R$-modules. Let $G^*$ and $H^*$ be the algebraic duals of $G$ and $H$ respectively. Let $\\mathcal L_R \\left({G, H}\\right)$ be the set of all linear transformations from $G$ to $H$. Let $u \\in \\mathcal L_R \\left({G, H}\\right)$. The '''transpose''' of $u$ is the mapping $u^t: H^* \\to G^*$ defined as: :$\\forall y' \\in H^*: u^t \\left({y'}\\right) = y' \\circ u$ where $y' \\circ u$ is the composition of $y'$ and $u$."}
+{"_id": "20141", "title": "Definition:Vector Space", "text": "{{:Definition:Vector Space/Definition 1}}"}
+{"_id": "20142", "title": "Definition:Vector Quantity", "text": "A '''vector quantity''' is a is a real-world concept that needs for its model a mathematical object with more than one component to specify it. Formally, a '''vector quantity''' is an element of a vector space, often the real vector space $\\R^n$. The usual intellectual frame of reference is to interpret a '''vector quantity''' as having: :A magnitude :A direction."}
+{"_id": "20143", "title": "Definition:Vector Space of All Mappings", "text": "Let $\\struct {K, +, \\circ}$ be a division ring. Let $\\struct {G, +_G, \\circ}_K$ be a $K$-vector space. Let $S$ be a set. Let $G^S$ be the set of all mappings from $S$ to $G$. Then $\\struct {G^S, +_G', \\circ}_K$ is a $K$-vector space, where: :$+_G'$ is the operation induced on $G^S$ by $+_G$ :$\\forall \\lambda \\in K: \\forall f \\in G^S: \\forall x \\in S: \\map {\\paren {\\lambda \\circ f} } x = \\lambda \\circ \\paren {\\map f x}$ This is the $K$-vector space $G^S$ of all mappings from $S$ to $G$."}
+{"_id": "20144", "title": "Definition:Real Vector Space", "text": "Let $\\R$ be the set of real numbers. Then the $\\R$-module $\\R^n$ is called the '''real ($n$-dimensional) vector space'''."}
+{"_id": "20145", "title": "Definition:Vector Subspace", "text": "Let $K$ be a division ring. Let $\\struct {S, +, \\circ}_K$ be a $K$-algebraic structure with one operation. Let $T$ be a closed subset of $S$. Let $\\struct {T, +_T, \\circ_T}_K$ be a $K$-vector space where: :$+_T$ is the restriction of $+$ to $T \\times T$ and :$\\circ_T$ is the restriction of $\\circ$ to $K \\times T$. Then $\\struct {T, +_T, \\circ_T}_K$ is a '''(vector) subspace''' of $\\struct {S, +, \\circ}_K$."}
+{"_id": "20146", "title": "Definition:Rank (Linear Algebra)", "text": "=== Linear Transformation === {{:Definition:Rank/Linear Transformation}} === Matrix === {{:Definition:Rank/Matrix}} {{expand|These definitions can be proved compatible (by viewing a matrix as a lin. transform., and maybe vice versa); such is a typical PW entry. Also, cf. Definition:Finite Rank Operator}} {{SUBPAGENAME}} e3nadl00pqmmtky0opygemoaw128urn"}
+{"_id": "20147", "title": "Definition:Nullity", "text": "=== Linear Transformation === {{:Definition:Nullity/Linear Transformation}} === Matrix === {{:Definition:Nullity/Matrix}} Category:Definitions/Linear Algebra lc760v5s457b9y1705qimrz55m6xrua"}
+{"_id": "20148", "title": "Definition:Coordinate System/Coordinate", "text": "Let $\\sequence {a_n}$ be a coordinate system of a unitary $R$-module $G$. Let $\\displaystyle x \\in G: x = \\sum_{k \\mathop = 1}^n \\lambda_k a_k$. The scalars $\\lambda_1, \\lambda_2, \\ldots, \\lambda_n$ can be referred to as the '''coordinates of $x$ relative to $\\sequence {a_n}$'''."}
+{"_id": "20149", "title": "Definition:Real Number Plane", "text": "The points on the plane are in one-to-one correspondence with the (real) Cartesian plane $\\R^2$. So from the definition of an ordered $n$-tuple, the general element of $\\R^2$ can be defined as an ordered couple $\\tuple {x_1, x_2}$ where $x_1, x_2 \\in \\R$, or, conventionally, $\\tuple {x, y}$. Thus, we can identify the elements of $\\R^2$ with points in the plane and refer to the point ''as'' its coordinates. Thus we can refer to $\\R^2$ ''as'' '''the plane'''."}
+{"_id": "20151", "title": "Definition:Point", "text": "{{EuclidDefinition|book = I|def = 1|name = Point}} This is interpreted to mean that a '''point''' is something that can not be divided into anything smaller. That is, it is an object in space which has zero size and is therefore the smallest thing that can be imagined. The only property possessed by a '''point''' is position."}
+{"_id": "20152", "title": "Definition:Angle", "text": "Given two intersecting lines or line segments, the amount of rotation about the intersection required to bring one into correspondence with the other is called the '''angle''' between them."}
+{"_id": "20153", "title": "Definition:Right Angle", "text": "A '''right angle''' is an angle that is equal to half of a straight angle."}
+{"_id": "20154", "title": "Definition:Straight Angle", "text": "A '''straight angle''' is defined to be the angle formed by the two parts of a straight line from a point on that line."}
+{"_id": "20155", "title": "Definition:Zero Angle", "text": "The '''zero angle''' is an angle the measure of which is $0$ regardless of the unit of measurement."}
+{"_id": "20156", "title": "Definition:Acute Angle", "text": "An '''acute angle''' is an angle which has a measure between that of a right angle and that of a zero angle."}
+{"_id": "20157", "title": "Definition:Obtuse Angle", "text": "An '''obtuse angle''' is an angle which has a measurement between those of a right angle and a straight angle."}
+{"_id": "20158", "title": "Definition:Full Angle", "text": "A '''full angle''' is an angle equivalent to one full rotation."}
+{"_id": "20159", "title": "Definition:Angle/Unit/Degree", "text": "The '''degree (of arc)''' is a measurement of plane angles, symbolized by $\\degrees$. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''degree''' }} {{eqn | r = 60       | c = minutes }} {{eqn | r = 60 \\times 60 = 3600       | c = seconds }} {{eqn | r = \\dfrac 1 {360}       | c = full angle (by definition) }} {{end-eqn}}"}
+{"_id": "20160", "title": "Definition:Reflex Angle", "text": "A '''reflex angle''' is an angle which has a measure between that of a straight angle and that of a full angle."}
+{"_id": "20161", "title": "Definition:Angle/Unit/Radian", "text": "The '''radian''' is a measure of plane angles symbolized either by the word $\\radians$ or without any unit. '''Radians''' are pure numbers, as they are ratios of lengths. The addition of $\\radians$ is merely for clarification. $1 \\radians$ is the angle subtended at the center of a circle by an arc whose length is equal to the radius: :360px"}
+{"_id": "20162", "title": "Definition:Line", "text": "{{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book I/2 - Line}}'' {{EuclidDefRefNocat|I|2|Line}} This can be interpreted to mean that a line is a construct that has no thickness. This mathematical abstraction can not of course be actualised in reality because however thin you make your line, it will have some finite width. It can be considered as a continuous succession of points."}
+{"_id": "20163", "title": "Definition:Circle", "text": "{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book I/15 - Circle}}'' {{EuclidDefRefNocat|I|15|Circle}} :300px"}
+{"_id": "20164", "title": "Definition:Boundary (Geometry)", "text": "{{EuclidDefinition|book = I|def = 13|name = Boundary}} For example, the endpoints of a line segment are its boundaries. === Containment === {{:Definition:Boundary (Geometry)/Containment}}"}
+{"_id": "20165", "title": "Definition:Surface (Geometry)", "text": "{{EuclidDefinition|book = I|def = 5|name = Surface}} and: :''{{:Definition:Euclid's Definitions - Book XI/2 - Extremity of Solid}}'' {{EuclidDefRefNocat|XI|2|Extremity of Solid}}"}
+{"_id": "20166", "title": "Definition:Linear Measure", "text": "'''Linear measure''' is the means of measurement of physical displacement. === Dimension === {{:Definition:Linear Measure/Dimension}} === Units === {{:Definition:Linear Measure/Units}} === Length === {{:Definition:Linear Measure/Length}} === Breadth === {{:Definition:Linear Measure/Breadth}} === Depth === {{:Definition:Linear Measure/Depth}} === Height === {{:Definition:Linear Measure/Height}} === Thickness === {{:Definition:Linear Measure/Thickness}} === Distance === {{:Definition:Linear Measure/Distance}}"}
+{"_id": "20167", "title": "Definition:Segment of Circle", "text": ":300px {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book III/6 - Segment of Circle}}'' {{EuclidDefRefNocat|III|6|Segment of Circle}} === Base === {{:Definition:Segment of Circle/Base}} === Angle of a Segment === {{:Definition:Segment of Circle/Angle of Segment}} === Angle in a Segment === {{:Definition:Segment of Circle/Angle in Segment}} === Similar Segments === {{:Definition:Segment of Circle/Similar}}"}
+{"_id": "20168", "title": "Definition:Diameter of Bounded Metric Subspace", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $S \\subseteq A$ be bounded in $M$. Then the '''diameter''' of $S$ is defined as: :$\\map {\\operatorname {diam} } S := \\sup \\set {\\map d {x, y}: x, y \\in S}$ That is, by the definition of the supremum, $\\map {\\operatorname {diam} } S$ is the smallest real number $D$ such that any two points of $S$ are at most a distance $D$ apart."}
+{"_id": "20169", "title": "Definition:Region", "text": "=== Metric Space === {{:Definition:Region/Metric Space}} === Complex === {{:Definition:Region/Complex}} === Region in the Plane === The usual usage of '''region''' is in the real number plane or complex plane. {{:Definition:Region/Plane}} ==== Interior ==== The boundary of a region separates its '''interior''' from the '''exterior'''. The '''interior''' consists of the points of the plane which are the elements of the region. Such points are called '''interior points''' of the region. It is \"usual\" that the interior is the \"smaller bit\" which is visually apparently on the ''inside'' as it appears on the page or screen, but this is of course not necessarily the case. Also see the definition of interior and boundary from a topological perspective. ==== Bounded ==== A region in the the plane is '''bounded''' if there is a circle in the plane which encloses it. Also see the definition of bounded in the context of a metric space."}
+{"_id": "20170", "title": "Definition:Cartesian Coordinate System", "text": "A '''Cartesian coordinate system''' is a coordinate system in which the position of a point is determined by its relation to a set of perpendicular straight lines. These straight lines are referred to as coordinate axes."}
+{"_id": "20171", "title": "Definition:Axis", "text": "An '''axis''' is the name used for a general infinite straight line which is particularly significant in some particular way in the study of linear transformations of a real vector space. === Coordinate Axes === {{:Definition:Axis/Coordinate Axes}}"}
+{"_id": "20172", "title": "Definition:Analytic Geometry", "text": "'''Analytic geometry''' is the study of geometry by algebraic manipulation of systems of ordered pairs of variables representing points in Cartesian space."}
+{"_id": "20174", "title": "Definition:Parallel (Geometry)", "text": "=== Lines === {{:Definition:Parallel (Geometry)/Lines}} === Planes === {{:Definition:Parallel (Geometry)/Planes}} === Line Parallel to Plane === {{:Definition:Parallel (Geometry)/Line to Plane}}"}
+{"_id": "20175", "title": "Definition:Geometry", "text": "'''Geometry''' is a branch of mathematics which studies such matters as form, position, dimension and various other properties of ordinary space. It has been suggested that '''geometry''' can be divided into $3$ main branches: :'''Metrical geometry''', that is to say, what is understood as '''geometry''' proper :'''Projective geometry''' :'''Analytic geometry'''"}
+{"_id": "20177", "title": "Definition:Homogeneous (Analytic Geometry)", "text": "A straight line or plane is '''homogeneous''' if it contains the origin."}
+{"_id": "20178", "title": "Definition:Matrix", "text": "Let $S$ be a set. Let $m, n \\in \\Z_{>0}$ be strictly positive integers. An '''$m \\times n$ matrix over $S$''' (said '''$m$ times $n$''' or '''$m$ by $n$''') is a mapping from the cartesian product of two integer intervals $\\closedint 1 m \\times \\closedint 1 n$ into $S$."}
+{"_id": "20179", "title": "Definition:Matrix Space", "text": "Let $m, n \\in \\Z_{>0}$ be (strictly) positive integers. Let $S$ be a set. The '''$m \\times n$ matrix space over $S$''' is defined as the set of all $m \\times n$ matrices over $S$, and is denoted $\\map {\\MM_S} {m, n}$."}
+{"_id": "20180", "title": "Definition:Matrix Entrywise Addition", "text": "Let $\\mathbf A$ and $\\mathbf B$ be matrices of numbers. Let the orders of $\\mathbf A$ and $\\mathbf B$ both be $m \\times n$. Then the '''matrix entrywise sum of $\\mathbf A$ and $\\mathbf B$''' is written $\\mathbf A + \\mathbf B$, and is defined as follows: Let $\\mathbf A + \\mathbf B = \\mathbf C = \\sqbrk c_{m n}$. Then: :$\\forall i \\in \\closedint 1 m, j \\in \\closedint 1 n: c_{i j} = a_{i j} + b_{i j}$ Thus $\\mathbf C = \\sqbrk c_{m n}$ is the $m \\times n$ matrix whose entries are made by performing the adding corresponding entries of $\\mathbf A$ and $\\mathbf B$. That is, the '''matrix entrywise sum of $\\mathbf A$ and $\\mathbf B$''' is the '''Hadamard product''' of $\\mathbf A$ and $\\mathbf B$ with respect to addition of numbers. This operation is called '''matrix entrywise addition'''."}
+{"_id": "20181", "title": "Definition:Zero Matrix/General Monoid", "text": "Let $\\struct {S, \\circ}$ be a monoid whose identity is $e$. Let $\\map {\\MM_S} {m, n}$ be an $m \\times n$ matrix space over $S$. The '''zero matrix of $\\map {\\MM_S} {m, n}$''', denoted $\\mathbf e$, is the $m \\times n$ matrix whose elements are all $e$, and can be written $\\sqbrk e_{m n}$."}
+{"_id": "20182", "title": "Definition:Negative Matrix", "text": "Let $\\GF$ denote one of the standard number systems. Let $\\map \\MM {m, n}$ be a $m \\times n$ matrix space over $\\GF$. Let $\\mathbf A = \\sqbrk a_{m n}$ be an element of $\\map \\MM {m, n}$. Then the '''negative (matrix) of $\\mathbf A$''' is denoted and defined as: :$-\\mathbf A := \\sqbrk {-a}_{m n}$ {{begin-eqn}} {{eqn | l = -\\mathbf A       | r = -1 \\mathbf A       | c =  }} {{eqn | r = \\sqbrk {-a}_{m n}       | c =  }} {{end-eqn}} where: :$-1 \\mathbf A$ denotes the matrix scalar product of $-1$ with $\\mathbf A$ :$-a$ is the negative of $a$."}
+{"_id": "20183", "title": "Definition:Matrix Scalar Product", "text": "Let $\\GF$ denote one of the standard number systems. Let $\\map \\MM {m, n}$ be the $m \\times n$ matrix space over $\\GF$. Let $\\mathbf A = \\sqbrk a_{m n} \\in \\map \\MM {m, n}$. Let $\\lambda \\in \\GF$ be any element of $\\Bbb F$. The operation of '''scalar multiplication of $\\mathbf A$ by $\\lambda$''' is defined as follows. Let $\\lambda \\mathbf A = \\mathbf C$. Then: :$\\forall i \\in \\closedint 1 m, j \\in \\closedint 1 n: c_{i j} = \\lambda a_{i j}$ $\\lambda \\mathbf A$ is the '''scalar product of $\\lambda$ and $\\mathbf A$'''. Thus $\\mathbf C = \\sqbrk c_{m n}$ is the $m \\times n$ matrix composed of the product of $\\lambda$ with the corresponding elements of $\\mathbf A$."}
+{"_id": "20184", "title": "Definition:Matrix Product (Conventional)", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\mathbf A = \\sqbrk a_{m n}$ be an $m \\times n$ matrix over $R$. Let $\\mathbf B = \\sqbrk b_{n p}$ be an $n \\times p$ matrix over $R$. Then the '''matrix product of $\\mathbf A$ and $\\mathbf B$''' is written $\\mathbf A \\mathbf B$ and is defined as follows. Let $\\mathbf A \\mathbf B = \\mathbf C = \\sqbrk c_{m p}$. Then: :$\\displaystyle \\forall i \\in \\closedint 1 m, j \\in \\closedint 1 p: c_{i j} = \\sum_{k \\mathop = 1}^n a_{i k} \\circ b_{k j}$ Thus $\\sqbrk c_{m p}$ is the $m \\times p$ matrix where each entry $c_{i j}$ is built by forming the (ring) product of each entry in the $i$'th row of $\\mathbf A$ with the corresponding entry in the $j$'th column of $\\mathbf B$ and adding up all those products. This operation is called '''matrix multiplication''', and $\\mathbf C$ is the '''matrix product''' of $\\mathbf A$ with $\\mathbf B$."}
+{"_id": "20185", "title": "Definition:Relative Matrix", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring with unity. Let $G$ be a free $R$-module of finite dimension $n>0$ Let $H$ be a free $R$-module of finite dimension $m>0$ Let $\\left \\langle {a_n} \\right \\rangle$ be an ordered basis of $G$. Let $\\left \\langle {b_m} \\right \\rangle$ be an ordered basis of $H$. Let $u : G \\to H$ be a linear transformation. The '''matrix of $u$ relative to $\\left \\langle {a_n} \\right \\rangle$ and $\\left \\langle {b_m} \\right \\rangle$''' is the $m \\times n$ matrix $\\left[{\\alpha}\\right]_{m n}$ where: :$\\displaystyle \\forall \\left({i, j}\\right) \\in \\left[{1 \\,.\\,.\\, m}\\right] \\times \\left[{1 \\,.\\,.\\, n}\\right]:u \\left({a_j}\\right) = \\sum_{i \\mathop = 1}^m \\alpha_{i j} \\circ b_i$ That is, the matrix whose columns are the coordinate vectors of the image of the basis elements of $\\mathcal A$ relative to the basis $\\mathcal B$. The matrix of such a linear transformation $u$ relative to the ordered bases $\\left \\langle {a_n} \\right \\rangle$ and $\\left \\langle {b_m} \\right \\rangle$ is denoted: : $\\left[{u; \\left \\langle {b_m} \\right \\rangle, \\left \\langle {a_n} \\right \\rangle}\\right]$ If $u$ is an automorphism on an $n$-dimensional module $G$, we can write $\\left[{u; \\left \\langle {a_n} \\right \\rangle, \\left \\langle {a_n} \\right \\rangle}\\right]$ as $\\left[{u; \\left \\langle {a_n} \\right \\rangle}\\right]$."}
+{"_id": "20186", "title": "Definition:Invertible Matrix", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\mathbf A$ be an element of the ring of square matrices $\\struct {\\map {\\MM_R} n, +, \\times}$. Then $\\mathbf A$ is '''invertible''' {{iff}}: :$\\exists \\mathbf B \\in \\struct {\\map {\\MM_R} n, +, \\times}: \\mathbf A \\mathbf B = \\mathbf I_n = \\mathbf B \\mathbf A$ where $\\mathbf I_n$ denotes the unit matrix of order $n$. Such a $\\mathbf B$ is the inverse of $\\mathbf A$. It is usually denoted $\\mathbf A^{-1}$."}
+{"_id": "20187", "title": "Definition:Change of Basis Matrix", "text": "Let $R$ be a ring with unity. Let $G$ be a finite-dimensional free $R$-module. Let $A = \\left \\langle {a_n} \\right \\rangle$ and $B = \\left \\langle {b_n} \\right \\rangle$ be ordered bases of $G$. === Definition 1 === {{:Definition:Change of Basis Matrix/Definition 1}} === Definition 2 === {{:Definition:Change of Basis Matrix/Definition 2}}"}
+{"_id": "20188", "title": "Definition:Matrix Equivalence", "text": "Let $R$ be a ring with unity. Let $\\mathbf A, \\mathbf B$ be $m \\times n$ matrices over $R$. Let there exist: :an invertible square matrix $\\mathbf P$ of order $n$ over $R$ :an invertible square matrix $\\mathbf Q$ of order $m$ over $R$ such that $\\mathbf B = \\mathbf Q^{-1} \\mathbf A \\mathbf P$. Then $\\mathbf A$ and $\\mathbf B$ are '''equivalent''', and we can write $\\mathbf A \\equiv \\mathbf B$. Thus, from Matrix Corresponding to Change of Basis under Linear Transformation, two matrices are '''equivalent''' {{iff}} they are the matrices of the same linear transformation, relative to (possibly) different ordered bases."}
+{"_id": "20189", "title": "Definition:Matrix Similarity", "text": "Let $R$ be a ring with unity. Let $n\\geq1$ be a natural number. Let $\\mathbf A, \\mathbf B$ be square matrices of order $n$ over $R$. Let there exist an invertible square matrix $\\mathbf P$ of order $n$ over $R$ such that $\\mathbf B = \\mathbf P^{-1} \\mathbf A \\mathbf P$. Then $\\mathbf A$ and $\\mathbf B$ are '''similar''', and we can write $\\mathbf A \\sim \\mathbf B$. Thus, from the corollary to Matrix Corresponding to Change of Basis under Linear Transformation, two matrices are similar {{iff}} they are the matrices of the same linear operator, relative to (possibly) different ordered bases."}
+{"_id": "20190", "title": "Definition:Transpose of Matrix", "text": "Let $\\mathbf A = \\sqbrk \\alpha_{m n}$ be an $m \\times n$ matrix over a set. Then the '''transpose''' of $\\mathbf A$ is denoted $\\mathbf A^\\intercal$ and is defined as: :$\\mathbf A^\\intercal = \\sqbrk \\beta_{n m}: \\forall i \\in \\closedint 1 n, j \\in \\closedint 1 m: \\beta_{i j} = \\alpha_{j i}$"}
+{"_id": "20192", "title": "Definition:Linear Equation", "text": "A '''linear equation''' is an equation in the form: :$b = a_1 x_1 + a_2 x_2 + \\cdots + a_n x_n$ where all of $a_1, \\ldots, a_n, x_1, \\ldots x_n, b$ are elements of a given field."}
+{"_id": "20193", "title": "Definition:Simultaneous Equations", "text": "A '''system of simultaneous equations''' is a set of equations: :$\\forall i \\in \\set {1, 2, \\ldots, m} : \\map {f_i} {x_1, x_2, \\ldots x_n} = \\beta_i$ That is: {{begin-eqn}} {{eqn | l = \\beta_1       | r = \\map {f_1} {x_1, x_2, \\ldots x_n} }} {{eqn | l = \\beta_2       | r = \\map {f_2} {x_1, x_2, \\ldots x_n} }} {{eqn | o = \\cdots}} {{eqn | l = \\beta_m       | r = \\map {f_m} {x_1, x_2, \\ldots x_n} }} {{end-eqn}}"}
+{"_id": "20194", "title": "Definition:Principal Ideal of Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity. Let $a \\in R$. We define: :$\\ideal a = \\displaystyle \\set {\\sum_{i \\mathop = 1}^n r_i \\circ a \\circ s_i: n \\in \\N, r_i, s_i \\in R}$ The ideal $\\ideal a$ is called the '''principal ideal of $R$ generated by $a$'''."}
+{"_id": "20195", "title": "Definition:Principal Ideal Domain", "text": "A '''principal ideal domain''' is an integral domain in which every ideal is a principal ideal."}
+{"_id": "20196", "title": "Definition:Maximal Ideal of Ring", "text": "Let $R$ be a ring. An ideal $J$ of $R$ is '''maximal''' {{iff}}: :$(1): \\quad J \\subsetneq R$ :$(2): \\quad$ There is no ideal $K$ of $R$ such that $J \\subsetneq K \\subsetneq R$. That is, {{iff}} $J$ is a maximal element of the set of all proper ideals of $R$ ordered by inclusion."}
+{"_id": "20197", "title": "Definition:Iff", "text": "The logical connective '''iff''' is a convenient shorthand for '''if and only if'''."}
+{"_id": "20198", "title": "Definition:Characteristic of Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. === Definition 1 === {{:Definition:Characteristic of Ring/Definition 1}} === Definition 2 === {{:Definition:Characteristic of Ring/Definition 2}} === Definition 3 === {{:Definition:Characteristic of Ring/Definition 3}}"}
+{"_id": "20199", "title": "Definition:Prime Field", "text": "A '''prime field''' is a field with no proper subfields."}
+{"_id": "20200", "title": "Definition:Prime Subfield", "text": "Let $F$ be a field. From Intersection of Subfields is Subfield, the intersection of all the subfields of $F$ is itself a subfield which is unique and minimal. This unique minimal subfield in a field $F$ is called the '''prime subfield of $F$'''."}
+{"_id": "20201", "title": "Definition:Polynomial over Ring", "text": "Let $R$ be a commutative ring with unity. === One Variable === {{Definition:Polynomial over Ring/One Variable}} === Multiple Variables === {{Definition:Polynomial over Ring/Multiple Variables}}"}
+{"_id": "20202", "title": "Definition:Transcendental (Abstract Algebra)", "text": "=== Field Extension === {{:Definition:Transcendental (Abstract Algebra)/Field Extension}} === Transcendental over Integral Domain === {{:Definition:Transcendental (Abstract Algebra)/Ring}}"}
+{"_id": "20203", "title": "Definition:Elementary Function", "text": "An '''elementary function''' is one of the following: * The constant function: $\\map {f_c} x = c$ where $c \\in \\R$ * Powers of $x$: $\\map f x = x^y$, where $y \\in \\R$ * Exponentials: $\\map f x = e^x$ * Natural logarithms: $\\map f x = \\ln x$ * Trigonometric functions: $\\map f x = \\sin x$, $\\map f x = \\cos x$ * Inverse trigonometric functions: $\\map f x = \\arcsin x$, $\\map f x = \\arccos x$ * All functions that are compositions of the above, for example $\\map f x = \\ln \\sin x$, $\\map f x = e^{\\cos x}$ * All functions obtained by adding, subtracting, multiplying and dividing any of the above types any finite number of times."}
+{"_id": "20204", "title": "Definition:Derivative", "text": "Informally, a '''derivative''' is the rate of change of one variable with respect to another. === Real Function === {{:Definition:Derivative/Real Function/Derivative at Point}} === Complex Function === {{:Definition:Derivative/Complex Function/Point}} === Vector-Valued Function === {{:Definition:Derivative/Vector-Valued Function/Point}}"}
+{"_id": "20205", "title": "Definition:Differentiation", "text": "The process of obtaining the derivative of a differentiable function $f$ with respect to $x$ is known as: : '''differentiation (of $f$) with respect to $x$''' or : '''differentiation WRT $x$'''"}
+{"_id": "20206", "title": "Definition:Continuous Function", "text": "=== Continuous Complex Function === {{:Definition:Continuous Complex Function}} === Continuous Real Function === {{:Definition:Continuous Real Function}}"}
+{"_id": "20207", "title": "Definition:Subdivision (Real Analysis)", "text": "Let $\\closedint a b$ be a closed interval of the set $\\R$ of real numbers. === Finite === {{:Definition:Subdivision (Real Analysis)/Finite}} === Infinite === {{:Definition:Subdivision (Real Analysis)/Infinite}}"}
+{"_id": "20208", "title": "Definition:Lower Sum", "text": "Let $\\closedint a b$ be a closed real interval. Let $f: \\closedint a b \\to \\R$ be a bounded real function. Let $P = \\set {x_0, x_1, x_2, \\ldots, x_n}$ be a finite subdivision of $\\closedint a b$. For all $\\nu \\in \\set {1, 2, \\ldots, n}$, let $m_\\nu^{\\paren f}$ be the infimum of $f$ on the interval $\\closedint {x_{\\nu - 1} } {x_\\nu}$. Then: :$\\displaystyle \\map {L^{\\paren f} } P = \\sum_{\\nu \\mathop = 1}^n m_\\nu^{\\paren f} \\paren {x_\\nu - x_{\\nu - 1} }$ is called the '''lower sum of $f$ on $\\closedint a b$ belonging''' (or '''with respect''') '''to (the subdivision) $P$'''. If there is no ambiguity as to what function is under discussion, $m_\\nu$ and $\\map L P$ are usually used."}
+{"_id": "20209", "title": "Definition:Upper Sum", "text": "Let $\\closedint a b$ be a closed real interval. Let $f: \\closedint a b \\to \\R$ be a bounded real function. Let $P = \\set {x_0, x_1, x_2, \\ldots, x_n}$ be a finite subdivision of $\\closedint a b$. For all $\\nu \\in \\set {1, 2, \\ldots, n}$, let $M_\\nu^{\\paren f}$ be the supremum of $f$ on the interval $\\closedint {x_{\\nu - 1} } {x_\\nu}$. Then: :$\\displaystyle \\map {U^{\\paren f} } P = \\sum_{\\nu \\mathop = 1}^n M_\\nu^{\\paren f} \\paren {x_\\nu - x_{\\nu - 1} }$ is called the '''upper sum of $f$ on $\\closedint a b$ belonging''' (or '''with respect''') '''to (the subdivision) $P$'''. If there is no ambiguity as to what function is under discussion, $M_\\nu$ and $\\map U P$ are often seen."}
+{"_id": "20210", "title": "Definition:Definite Integral", "text": "Let $\\closedint a b$ be a closed real interval. Let $f: \\closedint a b \\to \\R$ be a real function. === Riemann Integral === {{:Definition:Definite Integral/Riemann|Riemann Integral}} === Darboux Integral === {{:Definition:Definite Integral/Darboux|Darboux Integral}}"}
+{"_id": "20211", "title": "Definition:Dedekind Cut", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. === Definition 1 === {{:Definition:Dedekind Cut/Definition 1}} === Definition 2 === {{:Definition:Dedekind Cut/Definition 2}}"}
+{"_id": "20212", "title": "Definition:Complex Number", "text": "{{:Definition:Complex Number/Definition 1}}"}
+{"_id": "20214", "title": "Definition:Determinant/Matrix", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. That is, let: :$\\mathbf A = \\begin {bmatrix} a_{1 1} & a_{1 2} & \\cdots & a_{1 n} \\\\ a_{2 1} & a_{2 2} & \\cdots & a_{2 n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ a_{n 1} & a_{n 2} & \\cdots & a_{n n} \\\\ \\end {bmatrix}$ === Definition 1 === {{:Definition:Determinant/Matrix/Definition 1}} === Definition 2 === {{:Definition:Determinant/Matrix/Definition 2}}"}
+{"_id": "20215", "title": "Definition:Cofactor", "text": "Let $R$ be a commutative ring with unity. Let $\\mathbf A \\in R^{n \\times n}$ be a square matrix of order $n$.  Let: : $D = \\begin{vmatrix}  a_{11} & a_{12} & \\cdots & a_{1n} \\\\   a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\   a_{n1} & a_{n2} & \\cdots & a_{nn}\\end{vmatrix}$ be a determinant of order $n$. === Cofactor of an Element === {{:Definition:Cofactor/Element}} === Cofactor of a Minor === {{:Definition:Cofactor/Minor}}"}
+{"_id": "20216", "title": "Definition:Discriminant of Polynomial", "text": "Let $k$ be a field. Let $\\map f X \\in k \\sqbrk X$ be a polynomial of degree $n$. Let $\\overline k$ be an algebraic closure of $k$. Let the roots of $f$ in $\\overline k$ be $\\alpha_1, \\alpha_2, \\ldots, \\alpha_n$. Then the '''discriminant''' $\\map \\Delta f$ of $f$ is defined as: :$\\displaystyle \\map \\Delta f := \\prod_{1 \\mathop \\le i \\mathop < j \\mathop \\le n} \\paren {\\alpha_i - \\alpha_j}^2$ === Quadratic Equation === The concept is usually encountered in the context of a quadratic equation $a x^2 + b x + c$: {{:Definition:Discriminant of Polynomial/Quadratic Equation}} === Cubic Equation === In the context of a cubic equation $a x^3 + b x^2 + c x + d$: {{Definition:Discriminant of Polynomial/Cubic Equation}}"}
+{"_id": "20217", "title": "Definition:Minor of Determinant", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Consider the order $k$ square submatrix $\\mathbf B$ obtained by deleting $n - k$ rows and $n - k$ columns from $\\mathbf A$. Let $\\map \\det {\\mathbf B}$ denote the determinant of $\\mathbf B$. Then $\\map \\det {\\mathbf B}$ is an '''order-$k$ minor''' of $\\map \\det {\\mathbf A}$. Thus a '''minor''' is a determinant formed from the elements (in the same relative order) of $k$ specified rows and columns."}
+{"_id": "20219", "title": "Definition:Diagonal Matrix", "text": "Let $\\mathbf A = \\begin{bmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{n1} & a_{n2} & \\cdots & a_{nn} \\\\ \\end{bmatrix}$ be a square matrix of order $n$. Then $\\mathbf A$ is a '''diagonal matrix''' {{iff}} all elements of $\\mathbf A$ are zero except for possibly its diagonal elements."}
+{"_id": "20220", "title": "Definition:Triangular Matrix", "text": "Let $\\mathbf T = \\begin {bmatrix} a_{1 1} & a_{1 2} & \\cdots & a_{1 n} \\\\ a_{2 1} & a_{2 2} & \\cdots & a_{2 n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ a_{m 1} & a_{m 2} & \\cdots & a_{m n} \\\\ \\end {bmatrix}$ be a matrix of order $m \\times n$. Then $\\mathbf T$ is a '''triangular matrix''' {{iff}} all the elements either above or below the diagonal are zero. === Upper Triangular Matrix === {{:Definition:Triangular Matrix/Upper Triangular Matrix}} === Lower Triangular Matrix === {{:Definition:Triangular Matrix/Lower Triangular Matrix}}"}
+{"_id": "20221", "title": "Definition:Leading Coefficient of Matrix", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be an $m \\times n$ matrix. The '''leading coefficient''' of each row of $\\mathbf A$ is the leftmost non-zero element of that row. A zero row has no '''leading coefficient'''."}
+{"_id": "20222", "title": "Definition:Echelon Matrix", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be an $m \\times n$ matrix. === Echelon Form === {{:Definition:Echelon Matrix/Echelon Form}} === Reduced Echelon Form === {{:Definition:Echelon Matrix/Reduced Echelon Form}}"}
+{"_id": "20223", "title": "Definition:Complex Conjugate", "text": "Let $z = a + i b$ be a complex number. Then the '''(complex) conjugate''' of $z$ is denoted $\\overline z$ and is defined as: :$\\overline z := a - i b$"}
+{"_id": "20225", "title": "Definition:Root (Analysis)", "text": "Let $x, y \\in \\R_{\\ge 0}$ be positive real numbers. Let $n \\in \\Z$ be an integer such that $n \\ne 0$. Then $y$ is the '''positive $n$th root of $x$''' {{iff}}: :$y^n = x$ and we write: :$y = \\sqrt[n] x$ Using the power notation, this can also be written: :$y = x^{1/n}$"}
+{"_id": "20226", "title": "Definition:Power (Algebra)", "text": "=== Natural Numbers === {{:Definition:Power (Algebra)/Natural Number}} === Integers === {{:Definition:Power (Algebra)/Integer}} === Rational Numbers === {{:Definition:Power (Algebra)/Rational Number}} === Real Numbers === {{:Definition:Power (Algebra)/Real Number/Definition 1}} === Complex Numbers === {{Definition:Power (Algebra)/Complex Number}}"}
+{"_id": "20227", "title": "Definition:Exponential Function", "text": "The '''exponential function''' is denoted $\\exp$ and can be defined in several ways, as described below."}
+{"_id": "20228", "title": "Definition:Square Root", "text": "A '''square root''' of a number $n$ is a number $z$ such that $z$ squared equals $n$."}
+{"_id": "20229", "title": "Definition:Arithmetic Mean", "text": "Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\displaystyle A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a set of numbers, add them all up and divide by how many there are."}
+{"_id": "20230", "title": "Definition:Geometric Mean", "text": "Let $x_1, x_2, \\ldots, x_n \\in \\R_{>0}$ be (strictly) positive real numbers. The '''geometric mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\displaystyle G_n := \\paren {\\prod_{k \\mathop = 1}^n x_k}^{1/n}$"}
+{"_id": "20231", "title": "Definition:Harmonic Mean", "text": "Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers which are all positive. The '''harmonic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\displaystyle H_n := \\paren {\\frac 1 n \\paren {\\sum_{k \\mathop = 1}^n \\frac 1 {x_k} } }^{-1}$ That is, to find the '''harmonic mean''' of a set of $n$ numbers, take the reciprocal of the arithmetic mean of their reciprocals."}
+{"_id": "20232", "title": "Definition:Reciprocal", "text": "Let $x \\in \\R$ be a real number such that $x \\ne 0$. Then $\\dfrac 1 x$ is called the '''reciprocal of $x$'''."}
+{"_id": "20235", "title": "Definition:Euler's Number", "text": "=== As the Limit of a Sequence === {{:Definition:Euler's Number/Limit of Sequence}} === As the Limit of a Series === {{:Definition:Euler's Number/Limit of Series}} === As the Base of the Natural Logarithm === As the base of the Natural Logarithm: {{:Definition:Euler's Number/Base of Logarithm}} === In Terms of the Exponential Function === In terms of the exponential function: {{:Definition:Euler's Number/Exponential Function}} === As the Base of the Exponential with Derivative One at Zero === {{:Definition:Euler's Number/Base of Exponential}}"}
+{"_id": "20236", "title": "Definition:Field Extension", "text": "Let $F$ be a field. A '''field extension over $F$''' is a field $E$ where $F \\subseteq E$. That is, such that $F$ is a subfield of $E$."}
+{"_id": "20237", "title": "Definition:Unbounded Divergent Sequence", "text": "=== Real Sequence === {{:Definition:Unbounded Divergent Sequence/Real Sequence}} === Complex Sequence === As the Complex Numbers cannot be Ordered Compatibly with Ring Structure, there is no concept of $-\\infty$ in discussions relating to $\\C$. So we can use only the following definition: {{:Definition:Unbounded Divergent Sequence/Complex Sequence}}"}
+{"_id": "20238", "title": "Definition:Oscillating Sequence", "text": "Let $S$ be one of the standard number fields $\\Q$, $\\R$ or $\\C$. Let $\\sequence {x_n}$ be a sequence in $S$. Let $\\sequence {x_n}$ be divergent. Suppose $\\sequence {x_n}$ is ''not'' unbounded. That is, let: :$\\neg x_n \\to \\infty$ as $n \\to \\infty$ Then $\\sequence {x_n}$ is said to '''oscillate'''."}
+{"_id": "20240", "title": "Definition:Field Adjoined Element", "text": "{{delete|think it fits more naturally here}} Let $E/F$ be a field extension, $\\alpha \\in E$. Then: :$F[\\alpha] $ denotes the smallest subring of $E$ containing $F \\cup \\alpha$. :$F(\\alpha) $ denotes the smallest subfield of $E$ containing $F \\cup \\alpha$.  We say this as '''$F$ adjoined with $\\alpha$'''. Category:Definitions/Field Theory 53di6f01mi92rlorjbozg4cb00lpybf"}
+{"_id": "20241", "title": "Definition:Simple Field Extension", "text": "Let $E / F$ be a field extension. Then $E$ is a '''simple extension over $F$''' {{iff}}: : $\\exists \\alpha \\in E: E = F \\sqbrk \\alpha$ where $F \\sqbrk \\alpha$ is the field extension generated by $\\alpha$."}
+{"_id": "20242", "title": "Definition:Subsequence", "text": "Let $\\sequence {x_n}$ be a sequence in a set $S$. Let $\\sequence {n_r}$ be a strictly increasing sequence in $\\N$. Then the composition $\\sequence {x_{n_r} }$ is called a '''subsequence of $\\sequence {x_n}$'''."}
+{"_id": "20243", "title": "Definition:Limit Superior", "text": "Let $\\sequence {x_n}$ be a bounded sequence in $\\R$. === Definition 1 === {{:Definition:Limit Superior/Definition 1}} === Definition 2 === {{:Definition:Limit Superior/Definition 2}}"}
+{"_id": "20244", "title": "Definition:Limit Inferior", "text": "Let $\\sequence {x_n}$ be a bounded sequence in $\\R$. === Definition 1 === {{:Definition:Limit Inferior/Definition 1}} === Definition 2 === {{:Definition:Limit Inferior/Definition 2}}"}
+{"_id": "20245", "title": "Definition:Series", "text": "=== General Definition === {{:Definition:Series/General}} === Series in a Standard Number Field === The usual context for the definition of a series occurs when $S$ is one of the standard number fields $\\Q, \\R, \\C$. {{:Definition:Series/Number Field}} === Finite === {{:Definition:Series/Finite}}"}
+{"_id": "20246", "title": "Definition:Cauchy Criterion", "text": "The '''Cauchy criterion''' is the necessary and sufficient condition for an infinite sequence in a complete metric space to converge that the absolute difference between successive terms with sufficiently large indices tends to zero."}
+{"_id": "20247", "title": "Definition:Absolutely Convergent Series", "text": "=== General Definition === {{:Definition:Absolutely Convergent Series/General}} === Real Numbers === {{:Definition:Absolutely Convergent Series/Real Numbers}} === Complex Numbers === {{:Definition:Absolutely Convergent Series/Complex Numbers}}"}
+{"_id": "20248", "title": "Definition:Rational Function", "text": "Let $F$ be a field. Let $P: F \\to F$ and $Q: F \\to F$ be polynomial functions on $F$. Let $S$ be the set $F$ from which all the roots of $Q$ have been removed. That is: :$S = F \\setminus \\set {x \\in F: \\map Q x = 0}$ Then the equation $y = \\dfrac {\\map P x} {\\map Q x}$ defines a mapping from $S$ to $F$. Such a mapping is called a '''rational function'''."}
+{"_id": "20249", "title": "Definition:Semiperimeter", "text": "The '''semiperimeter''' of a plane figure is defined to be one half of the perimeter."}
+{"_id": "20250", "title": "Definition:Circumradius", "text": "=== Polygon === {{:Definition:Circumradius/Polygon}} === Polyhedron === {{:Definition:Circumradius/Polyhedron}} Category:Definitions/Geometry mx2rosmsqqf5hj1aog5ceifz94uowrp"}
+{"_id": "20251", "title": "Definition:Incircle of Triangle", "text": "The '''incircle''' of a triangle is the circle tangent to each side of the triangle. This circle is said to be inscribed in the triangle. :410px === Incenter === {{:Definition:Incircle of Triangle/Incenter}} === Inradius === {{:Definition:Incircle of Triangle/Inradius}}"}
+{"_id": "20252", "title": "Definition:Excircle of Triangle", "text": "Given a triangle, extend two sides in the direction opposite their common vertex. The circle tangent to both of these lines and to the third side of the triangle is called an '''excircle'''. :500px There are three '''excircles''' for every triangle. === Excenter === {{:Definition:Excircle of Triangle/Excenter}} === Exradius === {{:Definition:Excircle of Triangle/Exradius}}"}
+{"_id": "20253", "title": "Definition:Circumcircle of Triangle", "text": "The '''circumcircle of a triangle''' is the circle that passes through all three of the triangle's vertices. This circle is said to circumscribe the triangle. :320px === Circumcenter === {{:Definition:Circumcircle of Triangle/Circumcenter}} === Circumradius === {{:Definition:Circumcircle of Triangle/Circumradius}}"}
+{"_id": "20254", "title": "Definition:Triangle (Geometry)", "text": ":300px A '''triangle''' is a polygon with exactly three sides."}
+{"_id": "20255", "title": "Definition:Altitude of Triangle", "text": "Let $\\triangle ABC$ be a triangle. Let a perpendicular be dropped from $\\angle A$ to its opposite side $a$: :400px The line $h_a$ so constructed is called the '''altitude''' of $\\angle A$. === Foot of Altitude === {{:Definition:Altitude of Triangle/Foot}}"}
+{"_id": "20256", "title": "Definition:Polygon", "text": "A '''polygon''' is a closed plane figure made up of an unspecified number of non-crossing straight line segments that join in pairs at their endpoints. For example: :400px"}
+{"_id": "20257", "title": "Definition:Quadrilateral", "text": "A '''quadrilateral''' is a polygon with exactly four sides."}
+{"_id": "20258", "title": "Definition:Perimeter", "text": "The '''perimeter''' of a plane figure is the length of its boundary."}
+{"_id": "20259", "title": "Definition:Bisection", "text": "To '''bisect''' a finite geometrical object is to cut it into two equal parts."}
+{"_id": "20262", "title": "Definition:Differentiable Mapping", "text": "=== Real Function === {{:Definition:Differentiable Mapping/Real Function}} === Complex Function === {{:Definition:Differentiable Mapping/Complex Function}} === Real-Valued Function === {{:Definition:Differentiable Mapping/Real-Valued Function}} === Vector-Valued Function === {{:Definition:Differentiable Mapping/Vector-Valued Function}} === Between Differentiable Manifolds === {{Definition:Differentiable Mapping between Manifolds}}"}
+{"_id": "20263", "title": "Definition:Compact Space", "text": "=== Euclidean Space === {{:Definition:Compact Space/Euclidean Space}} === Topology === {{:Definition:Compact Space/Topology}} === Metric Space === {{:Definition:Compact Space/Metric Space}} === Normed Vector Space === {{:Definition:Compact Space/Normed Vector Space}}"}
+{"_id": "20264", "title": "Definition:Open Cover", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\CC$ be a cover for $S$. Then $\\CC$ is an '''open cover (of $T$)''' {{iff}}: :$\\CC \\subseteq \\tau$ That is, {{iff}} all the elements of $\\CC$ are open sets. === Open Cover of Subset === {{:Definition:Open Cover/Subset}}"}
+{"_id": "20265", "title": "Definition:Closed Set", "text": "=== Topology === Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$. {{:Definition:Closed Set/Topology}} === Metric Space === In the context of metric spaces, the same definition applies: {{:Definition:Closed Set/Metric Space}} === Normed Vector Space === {{:Definition:Closed Set/Normed Vector Space}} === Complex Analysis === {{:Definition:Closed Set/Complex Analysis}} === Real Analysis === {{:Definition:Closed Set/Real Analysis/Real Numbers}}"}
+{"_id": "20266", "title": "Definition:Submanifold", "text": "A '''submanifold''' is a manifold which is completely contained in some other manifold."}
+{"_id": "20267", "title": "Definition:Codimension", "text": "Let $X$ be a manifold. Let $Z$ be a submanifold of $X$. The '''codimension of $Z$ in $X$''' is the dimension of $X$ minus the dimension of $Z$. Category:Definitions/Topology 28d2r6r1yrbtw22lu96vpui26vgff5r"}
+{"_id": "20268", "title": "Definition:Critical Point (Topology)", "text": "Let $f: X \\to Y$ be a smooth map of manifolds. A point $x \\in X$ is called a '''critical point''' of $f$ {{iff}} $\\d f_x: T_x \\sqbrk X \\to T_y \\sqbrk Y$ is not surjective at $x$. {{explain|What are $\\d f_x$, $T_x$ and $T_y$ in this context?}} Category:Definitions/Topology e5niy2x3798s0kyysfe4v2x207hvzt6"}
+{"_id": "20270", "title": "Definition:Homeomorphism", "text": "=== Topological Spaces === Let $T_\\alpha = \\struct {S_\\alpha, \\tau_\\alpha}$ and $T_\\beta = \\struct {S_\\beta, \\tau_\\beta}$ be topological spaces. Let $f: T_\\alpha \\to T_\\beta$ be a bijection. {{:Definition:Homeomorphism/Topological Spaces/Definition 1}} === Metric Spaces === The same definition applies to metric spaces: {{:Definition:Homeomorphism/Metric Spaces/Definition 1}} === Manifolds === The same definition applies to manifolds: {{:Definition:Homeomorphism/Manifolds}}"}
+{"_id": "20271", "title": "Definition:Hausdorff Space", "text": "{{:Definition:Hausdorff Space/Definition 1}}"}
+{"_id": "20272", "title": "Definition:Tychonoff Separation Axioms", "text": "The '''Tychonoff separation axioms''' are a classification system for topological spaces. They are not axiomatic as such, but conditions that may or may not apply to general or specific topological spaces."}
+{"_id": "20274", "title": "Definition:Differential Form", "text": "Let $X$ be a smooth manifold. Let $\\map {T_x} X$ denotes the tangent space of $X$. A '''$p$-form''' on $X$ is a function $\\omega: \\map {T_x} X^p \\to \\R$ defined at each point of $X$ which takes $p$ vectors as inputs, and outputs a real number."}
+{"_id": "20275", "title": "Definition:Euclidean Space", "text": "Let $S$ be one of the standard number fields $\\Q$, $\\R$, $\\C$. Let $S^n$ be a cartesian space for $n \\in \\N_{\\ge 1}$. Let $d: S^n \\times S^n \\to \\R$ be the usual (Euclidean) metric on $S^n$. Then $\\tuple {S^n, d}$ is a '''Euclidean space'''."}
+{"_id": "20278", "title": "Definition:Tangent Vector", "text": "Let $M$ be a smooth manifold. Let $m \\in M$ be a point.  Let $V$ be an open neighborhood of $m$.  Let $C^\\infty \\left({V, \\R}\\right)$ be defined as the set of all smooth mappings $f: V \\to \\R$. === Definition 1 === {{:Definition:Tangent Vector/Definition 1}} === Definition 2 === {{:Definition:Tangent Vector/Definition 2}}"}
+{"_id": "20279", "title": "Definition:Tangent Bundle", "text": "Let $X$ be a differentiable manifold. The '''tangent bundle''' of $X$ is the disjoint union of all the tangent spaces of $X$ at $x$ as the base point $x$ ranges over the entire manifold: :$\\displaystyle \\map T X = \\coprod_{x \\mathop \\in X} \\map {T_x} X$ {{explain|probably also need to mention the natural differential structure on the tangent bundle.}} Category:Definitions/Topology nvuraom6bujfdc60gq0lg9ek5cpxewn"}
+{"_id": "20280", "title": "Definition:P-adic Norm", "text": "=== Definition 1 === {{:Definition:P-adic Norm/Definition 1}} === Definition 2 === {{:Definition:P-adic Norm/Definition 2}} === $p$-adic Metric === {{:Definition:P-adic Norm/P-adic Metric}} === $p$-adic Numbers === Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. {{:Definition:P-adic Norm/P-adic Numbers}}"}
+{"_id": "20281", "title": "Definition:Homotopy", "text": "=== Free Homotopy === {{:Definition:Homotopy/Free}} === Relative Homotopy === {{:Definition:Homotopy/Relative}} === Path-Homotopy === {{:Definition:Homotopy/Path}}"}
+{"_id": "20282", "title": "Definition:Homotopy Group", "text": "Let $X$ be a topological space, and $x_0 \\in X$. Let $\\mathbb S^n \\subseteq \\R^{n+1}$ be the $n$-sphere, and $a \\in \\mathbb S^n$. Let $\\pi_n \\left({X, x_0}\\right)$ be the set of homotopy classes relative to $a$ of continuous mappings $c: \\mathbb S^n \\to X$ such that $c(a) = x_0$. Let $* : \\pi_n \\left({X, x_0}\\right) \\times \\pi_n \\left({X, x_0}\\right) \\to \\pi_n \\left({X, x_0}\\right)$ denote the concatenation of homotopy classes of paths. That is, if $\\overline{c_1}, \\overline{c_2}$ are two elements of $\\pi_n \\left({X, x_0}\\right)$, then: :$\\overline{c_1} * \\overline{c_2} = \\overline{c_1 \\cdot c_2}$ where $\\cdot$ denotes the usual concatenation of paths. Then $\\left({\\pi_n \\left({X, x_0}\\right), *}\\right)$ is the '''$n$th fundamental group''' of $X$. The first homotopy group is usually called the '''fundamental group''' when higher homotopy groups are not in sight. For a path-connected manifold, by Fundamental Group is Independent of Base Point for Path-Connected Space, the isomorphism class of $\\pi_1 \\left({X, x_0}\\right)$ does not depend on $x_0$ and we just write $\\pi_1 \\left({X}\\right)$."}
+{"_id": "20283", "title": "Definition:Concatenation (Topology)", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $c_1, c_2: \\left[{0  \\,.\\,.\\, 1}\\right]^n \\to S$ be maps. Let $c_1$ and $c_2$ both satisfy the concatenation criterion $c \\left({\\partial \\left[{0 \\,.\\,.\\, 1}\\right]^n}\\right) = x_0$. <!-- For two maps $c_1, c_2: \\left[{0 \\,.\\,.\\, 1}\\right]^n \\to X$ which satisfy the concatenation criterion $c \\left({\\partial \\left[{0 \\,.\\,.\\, 1}\\right]^n}\\right) = x_0$--> Then the '''concatenation''' $c_1 * c_2$ is defined as: :$\\left({c_1 * c_2}\\right) \\left({t_1, t_2, \\ldots, t_n}\\right) = \\begin{cases} c_1 \\left({2t_1, t_2, \\ldots, t_n}\\right) & : t_1 \\in \\left[{0 \\,.\\,.\\, 1/2}\\right] \\\\  c_2 \\left({2t_1-1, t_2, \\ldots, t_n}\\right) & : t_1 \\in \\left[{1/2 \\,.\\,.\\, 1}\\right] \\end{cases} $  where $\\left({t_1, \\ldots, t_n}\\right)$ are coordinates in the $n$-cube. By Continuous Mapping on Finite Union of Closed Sets, $c_1*c_2$ is continuous. This resulting map is continuous, since: * $2 \\left({\\dfrac 1 2}\\right) = 1$ and $2 \\left({\\dfrac 1 2}\\right) - 1 = 0$; * anywhere any coordinate of $\\hat t$ is either $1$ or $0$, $\\left({c_1*c_2}\\right) \\left({\\hat t}\\right) = x_0$. The resulting map also clearly satisfies the concatenation criteria itself."}
+{"_id": "20284", "title": "Definition:Fundamental Group", "text": "Let $(X, x_0)$ be a pointed topological space with base point $x_0$. The '''fundamental group''' $\\pi_1 \\left({X, x_0}\\right)$ of $X$ at the base point $x_0$ is the set of homotopy classes of loops with base point $x_0$ with multiplication of homotopy classes of paths."}
+{"_id": "20285", "title": "Definition:Maximum Value of Real Function/Local", "text": "Let $f$ be a real function defined on an open interval $\\openint a b$. Let $\\xi \\in \\openint a b$. Then $f$ has a '''local maximum at $\\xi$''' {{iff}}: :$\\exists \\openint c d \\subseteq \\openint a b: \\forall x \\in \\openint c d: \\map f x \\le \\map f \\xi$ That is, {{iff}} there is some subinterval on which $f$ attains a maximum within that interval."}
+{"_id": "20286", "title": "Definition:Minimum Value of Real Function/Local", "text": "Let $f$ be a real function defined on an open interval $\\openint a b$. Let $\\xi \\in \\openint a b$. Then $f$ has a '''local minimum at $\\xi$''' {{iff}}: :$\\exists \\openint c d \\subseteq \\openint a b: \\forall x \\in \\openint c d: \\map f x \\ge \\map f \\xi$ That is, {{iff}} there is some subinterval on which $f$ attains a minimum within that interval."}
+{"_id": "20287", "title": "Definition:Stationary Point", "text": "Let $f$ be a real function which is differentiable on the open interval $\\openint a b$. Let $\\exists \\xi \\in \\openint a b: \\map {f'} \\xi = 0$, where $\\map {f'} \\xi$ is the derivative of $f$ at $\\xi$. Then $\\xi$ is known as a '''stationary point of $f$'''."}
+{"_id": "20288", "title": "Definition:Cauchy Equivalent Metrics", "text": "Let $X$ be a set upon which there are two metrics $d_1$ and $d_2$. That is, $\\struct {X, d_1}$ and $\\struct {X, d_2}$ are two different metric spaces on the same underlying set $X$. Then $d_1$ and $d_2$ are said to be '''Cauchy equivalent''' {{iff}} for every sequence $\\sequence {x_n}$ of points in $X$: :$\\sequence {x_n}$ is a Cauchy sequence in $\\struct {X, d_1} \\iff \\sequence {x_n}$ is a  is a Cauchy sequence in $\\struct {X, d_2}$ {{NamedforDef|Augustin Louis Cauchy|cat = Cauchy}}"}
+{"_id": "20290", "title": "Definition:Homology Group", "text": "Let $X$ be a topological space. Let the standard n-simplex be denoted: :$\\displaystyle \\Delta^n := \\left\\{ {\\left({x_0, \\ldots, x_n}\\right) \\in {\\R_{\\ge 0} }^{n + 1}: \\sum_{i \\mathop = 1}^{n + 1} x_i = 1}\\right\\}$ Let $\\mathcal C \\left({\\Delta^n, X}\\right)$ be the set of continuous mappings from $\\Delta^n$ to $X$.  For $n \\ge 0$, let $C_n \\left({X}\\right)$ be the free abelian group generated by $\\mathcal C \\left({\\Delta^n, X}\\right)$.  Then there is a boundary map $\\partial_n: C_n \\left({X}\\right) \\to C_{n - 1} \\left({X}\\right)$ defined as follows. First, there are maps $s^i_n: \\Delta^{n - 1} \\to \\Delta^n$, where $n > 0$ and $0 \\le i \\le n$, defined by: :$s^i_n \\left({x_0, \\ldots, x_{n - 1} }\\right) = \\left({x_0, \\ldots, x_{i - 1}, 0, x_i, \\ldots, x_{n-1} }\\right)$ These can be considered as the inclusion of $\\Delta^{n-1}$ as a 'face' of $\\Delta^n$. For a continuous function $\\phi: \\Delta^n \\to X$ we define: :$\\displaystyle \\partial_n \\left({\\phi}\\right) = \\sum_{i \\mathop = 0}^n \\left({-1}\\right)^i \\phi \\circ s^i_n$ This definition, along with the requirement that $\\partial_n$ be a group homomorphism, uniquely specifies $\\partial_n$. In addition, one has $\\partial_{n-1} \\partial_n = 0$, meaning that the sequence of groups and morphisms: :$0 \\gets C_0 \\left({X}\\right) \\stackrel {\\partial_1} {\\longleftarrow} C_1 \\left({X}\\right) \\stackrel {\\partial_2} {\\longleftarrow} C_2 \\left({X}\\right) \\stackrel {\\partial_3} {\\longleftarrow} \\cdots$ are a chain complex. This is demonstrated in Singular Chains form Chain Complex. Let $\\partial_0$ denote the map $C_0 \\left({X}\\right) \\to 0$. Thus the $n$th singular homology group of $X$ is defined as the $n$th homology group of this chain complex. Explicitly: Let $B_n \\left({X}\\right) \\subset C_n \\left({X}\\right)$ denote the image of $\\partial_{n+1}$. Let $Z_n \\left({X}\\right)$ denote the kernel of $\\partial_n$. Since $\\partial_n \\partial_{n + 1} = 0$: :$B_n \\left({X}\\right) \\subseteq Z_n \\left({X}\\right)$ Then define: :$H_n \\left({X}\\right) = \\dfrac {Z_n \\left({X}\\right)} {B_n \\left({X}\\right)}$"}
+{"_id": "20291", "title": "Definition:Convex Real Function", "text": "Let $f$ be a real function which is defined on a real interval $I$. === Definition 1 === {{:Definition:Convex Real Function/Definition 1}} === Definition 2 === {{:Definition:Convex Real Function/Definition 2}} === Definition 3 === {{:Definition:Convex Real Function/Definition 3}}"}
+{"_id": "20292", "title": "Definition:Concave Real Function", "text": "Let $f$ be a real function which is defined on a real interval $I$. === Definition 1 === {{:Definition:Concave Real Function/Definition 1}} === Definition 2 === {{:Definition:Concave Real Function/Definition 2}} === Definition 3 === {{:Definition:Concave Real Function/Definition 3}}"}
+{"_id": "20294", "title": "Definition:Simply Connected", "text": "A path-connected topological space $T = \\left({S, \\tau}\\right)$ is said to be '''simply connected''' if the fundamental group $\\pi_1 \\left({T}\\right)$ is trivial. {{stub|Expand to include the context of regions in the plane.}}"}
+{"_id": "20295", "title": "Definition:Primitive (Calculus)", "text": "=== Primitive of Real Function === {{:Definition:Primitive (Calculus)/Real}}"}
+{"_id": "20296", "title": "Definition:Transcendental Number", "text": "A number (either real or complex) is '''transcendental''' {{iff}} it is not algebraic."}
+{"_id": "20297", "title": "Definition:Improper Integral", "text": "An '''improper integral''' is a definite integral over an interval which is not closed, that is, open or half open, and whose limits of integration are the end points of that interval. When the end point is not actually in the interval, the conventional definition of the definite integral is not valid."}
+{"_id": "20299", "title": "Definition:Lebesgue Measure", "text": "Let $\\JJ_{ho}^n$ be the set of half-open $n$-rectangles. Let $\\map \\BB {\\R^n}$ be the Borel $\\sigma$-algebra on $\\R^n$. Let $\\lambda^n$ be the $n$-dimensional Lebesgue pre-measure on $\\JJ_{ho}^n$. Any measure $\\mu$ extending $\\lambda^n$ to $\\map \\BB {\\R^n}$ is called '''$n$-dimensional Lebesgue measure'''."}
+{"_id": "20300", "title": "Definition:Measurable Set", "text": "Let $\\struct {X, \\Sigma}$ be a measurable space. A subset $S \\subseteq X$ is said to be '''($\\Sigma$-)measurable''' {{iff}} $S \\in \\Sigma$. === Measurable Set of an Arbitrary Outer Measure === {{:Definition:Measurable Set/Arbitrary Outer Measure}} === Measurable Subset of the Reals === {{:Definition:Measurable Set/Subset of Reals}} === Measurable Subset of $\\R^n$ === {{:Definition:Measurable Set/Subsets of Real Space}}"}
+{"_id": "20301", "title": "Definition:Strictly Negative", "text": "Let $\\struct {R, +, \\circ, \\le}$ be an ordered ring whose zero is $0_R$. Then $x \\in R$ is '''strictly negative''' {{iff}}: :$x \\le 0_R$ and $x \\ne 0_R$ That $x$ is strictly negative may be (more conveniently) denoted $0_R < x$ or $x > 0_R$. Thus, the set of all strictly negative elements of $R$ is denoted: :$R_{< 0_R} := \\set {x \\in R: x < 0_R}$"}
+{"_id": "20302", "title": "Definition:Algebra of Sets", "text": "{{:Definition:Algebra of Sets/Definition 1}}"}
+{"_id": "20303", "title": "Definition:Quotient Topology/Quotient Space", "text": "The '''quotient space of $S$ by $\\RR$''' is the topological space whose points are elements of the quotient set of $\\RR$ and whose topology is $\\tau_\\RR$: :$T_\\RR := \\struct {S / \\RR, \\tau_\\RR}$"}
+{"_id": "20304", "title": "Definition:Torus (Topology)", "text": "The '''$n$-dimensional torus''' (or '''$n$-torus)''' $\\Bbb T^n$ is defined as the space whose points are those of the cross product of $n$ circles: :$\\Bbb T^n = \\underbrace{\\Bbb S^1 \\times \\Bbb S^1 \\times \\ldots \\times \\Bbb S^1}_{n \\text{ times}}$ and whose topology $\\tau_{\\Bbb T^n}$ is defined as: :$U \\in \\tau_{\\Bbb T^n} \\iff \\exists U_1, U_2, \\ldots, U_n \\in \\tau_{\\Bbb S^1} : U = U_1 \\times U_2 \\times \\ldots \\times U_n$ where $\\tau_{\\Bbb S^1}$ is the topology of the circle. {{explain|In this context, what is the \"topology of the circle\"?}} Category:Definitions/Algebraic Topology 2hjek1yjp7hakl3p0qgvhxiw3md3yve"}
+{"_id": "20305", "title": "Definition:Extended Real Number Line", "text": "=== Definition 1 === {{:Definition:Extended Real Number Line/Definition 1}} === Definition 2 === {{:Definition:Extended Real Number Line/Definition 2}}"}
+{"_id": "20307", "title": "Definition:Characteristic Function (Set Theory)", "text": "=== Set === {{:Definition:Characteristic Function (Set Theory)/Set}} === Relation === {{:Definition:Characteristic Function (Set Theory)/Relation}}"}
+{"_id": "20308", "title": "Definition:Simple Function", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. A real-valued function $f: X \\to \\R$ is said to be a '''simple function''' {{iff}} it is a finite linear combination of characteristic functions: :$\\displaystyle f = \\sum_{k \\mathop = 1}^n a_k \\chi_{S_k}$ where $a_1, a_2, \\ldots, a_n$ are real numbers and each of the sets $S_k$ is $\\Sigma$-measurable."}
+{"_id": "20309", "title": "Definition:Lebesgue Integral", "text": "Let $\\lambda^n$ be a Lebesgue measure on $\\R^n$. Let $f: \\R^n \\to \\overline \\R$ be a Lebesgue integrable function. Then the $\\lambda^n$-integral of $f$: :$\\displaystyle \\int f \\rd \\lambda^n$ is called the '''Lebesgue integral of $f$'''."}
+{"_id": "20310", "title": "Definition:Skolem Sequence", "text": "A '''Skolem sequence of order $n$''' is a (finite) sequence $S = \\left\\{{s_1, s_2, \\ldots, s_{2 n}}\\right\\}$ of $2 n$ integers for which: :$(1): \\quad$ For every $k \\in \\left\\{{1, 2, \\ldots, n}\\right\\}$ there exist exactly two elements $s_i, s_j \\in S$ so that $s_i = s_j = k$ and: :$(2): \\quad$ If $s_i = s_j = k$ and $i < j$ then $j - i = k$. {{NamedforDef|Thoralf Albert Skolem|cat = Skolem}} Category:Definitions/Number Theory f7hnlsqeuoxpb7zqpn3nkg6eyl7o8i8"}
+{"_id": "20311", "title": "Definition:Counting Measure", "text": "Let $\\struct {X, \\Sigma}$ be a measurable space. The '''counting measure (on $X$)''', denoted $\\size {\\, \\cdot \\,}$, is the measure defined by: :$\\size {\\, \\cdot \\,}: \\Sigma \\to \\overline \\R, \\ \\size E := \\begin {cases} \\map \\# E & : \\text {$E$ is finite} \\\\ +\\infty & : \\text {$E$ is infinite} \\end{cases}$ where $\\overline \\R$ denotes the extended real numbers, and $\\#$ denotes cardinality. That $\\size {\\, \\cdot \\,}$ is actually a measure is shown on Counting Measure is Measure."}
+{"_id": "20312", "title": "Definition:Standard Discrete Metric", "text": "The '''standard discrete metric''' on a set $S$ is the metric satisfying: :$\\map d {x, y} = \\begin{cases} 0 & : x = y  \\\\  1 & : x \\ne y \\end{cases}$ This can be expressed using the Kronecker delta notation as: :$\\map d {x, y} = 1 - \\delta_{x y}$ The resulting metric space $M = \\struct {S, d}$ is '''the standard discrete metric space on $S$'''."}
+{"_id": "20313", "title": "Definition:Euclidean Metric/Real Vector Space", "text": "The '''Euclidean metric''' on $\\R^n$ is defined as: :$\\displaystyle d_2 \\left({x, y}\\right) := \\left({\\sum_{i \\mathop = 1}^n \\left({x_i - y_i}\\right)^2}\\right)^{1 / 2}$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({y_1, y_2, \\ldots, y_n}\\right) \\in \\R^n$."}
+{"_id": "20314", "title": "Definition:Connected Sum", "text": "The '''connected sum''' of two manifolds $A^n, B^n$ of dimension $n$ is defined as follows: Let $\\Bbb D^n$ be a closed n-disk. Let $\\alpha: \\Bbb D^n \\to A^n$ be a continuous (or, in the case of smooth manifolds, a smooth) injection. Let $\\beta: \\Bbb D^n \\to B^n$ be a similar function.   Define the set: :$S = \\paren {A^n \\setminus \\map \\alpha {\\paren {\\Bbb D^n}^\\circ} } \\cup \\paren {B^n \\setminus \\map \\beta {\\paren {\\Bbb D^n}^\\circ} }$ where: :$\\setminus$ denotes set difference :$\\paren {\\Bbb D^n}^\\circ$ denotes the interior of $B^n$. Define an equivalence relation $\\sim$ on $S$ as: :$x \\sim y \\iff \\paren {\\paren {x = y} \\lor \\paren {\\map {\\alpha^{-1} } x = \\map {\\beta^{-1} } y} }$ Since the interiors of the disks were removed from the manifolds, it necessarily follows that: :$\\map {\\alpha^{-1} } x, \\map {\\beta^{-1} } y \\in \\partial \\Bbb D^n$ The '''connected sum''' $A^n \\# B^n$ is defined as the quotient space of $S$ under $\\sim$. Category:Definitions/Topology mqyj5hjf7j2aem5z5zo7otzc6peyd2b"}
+{"_id": "20315", "title": "Definition:Sphere", "text": "=== Geometry === {{:Definition:Sphere/Geometry}} === Topology === {{:Definition:Sphere/Topology}} === Metric Space === {{:Definition:Sphere/Metric Space}} === Normed Division Ring === {{:Definition:Sphere/Normed Division Ring}} === Normed Vector Space === {{:Definition:Sphere/Normed Vector Space}} === P-adic Numbers === The definition of an '''sphere''' in the context of the $p$-adic numbers is a direct application of the definition of an sphere in a normed division ring: {{:Definition:Sphere/P-adic Numbers}} Category:Definitions/Topology Category:Definitions/Metric Spaces Category:Definitions/Normed Division Rings tntunx5oet71ckpg8veon8tihn7moi3"}
+{"_id": "20316", "title": "Definition:Argument of Complex Number", "text": "Let $z = x + i y$ be a complex number. An '''argument of $z$''', or $\\arg z$, is formally defined as a solution to the pair of equations: :$(1): \\quad \\dfrac x {\\cmod z} = \\map \\cos {\\arg z}$ :$(2): \\quad \\dfrac y {\\cmod z} = \\map \\sin {\\arg z}$ where $\\cmod z$ is the modulus of $z$. From Sine and Cosine are Periodic on Reals, it follows that if $\\theta$ is an '''argument''' of $z$, then so is $\\theta + 2 k \\pi$ where $k \\in \\Z$ is ''any'' integer."}
+{"_id": "20318", "title": "Definition:Cantor Set", "text": "=== As a Limit of Intersections === {{:Definition:Cantor Set/Limit of Intersections}} === From Ternary Representation === {{:Definition:Cantor Set/Ternary Representation}} === As a Limit of a Decreasing Sequence === {{:Definition:Cantor Set/Limit of Decreasing Sequence}} These definitions are all equivalent, as shown on Equivalence of Definitions of Cantor Set."}
+{"_id": "20319", "title": "Definition:Dihedral Group", "text": "The '''dihedral group''' $D_n$ of order $2 n$ is the group of symmetries of the regular $n$-gon."}
+{"_id": "20320", "title": "Definition:Euler-Mascheroni Constant", "text": "The '''Euler-Mascheroni constant''' $\\gamma$ is the real number that is defined as: {{begin-eqn}} {{eqn | l = \\gamma       | o = :=       | r = \\lim_{n \\mathop \\to +\\infty} \\paren {\\sum_{k \\mathop = 1}^n \\frac 1 k - \\int_1^n \\frac 1 x \\rd x} }} {{eqn | r = \\lim_{n \\mathop \\to +\\infty} \\paren {H_n - \\ln n} }} {{end-eqn}} where $H_n$ is the harmonic series and $\\ln$ is the natural logarithm."}
+{"_id": "20321", "title": "Definition:Alternating Group", "text": "Let $S_n$ denote the symmetric group on $n$ letters. For any $\\pi \\in S_n$, let $\\map \\sgn \\pi$ be the sign of $\\pi$. The kernel of the mapping $\\sgn: S_n \\to C_2$ is called the '''alternating group on $n$ letters''' and denoted $A_n$."}
+{"_id": "20322", "title": "Definition:Dicyclic Group", "text": "For even $n$, the '''dicyclic group''' $\\Dic n$ of order $4 n$ is the group having the presentation: :$\\Dic n = \\gen {x, y: x^{2 n} = e, y^2 = x^n, y^{-1} x y = x^{-1} }$"}
+{"_id": "20323", "title": "Definition:Quaternion Group", "text": "The dicyclic group $\\Dic 2$ is known as the '''quaternion group'''. The elements of $\\Dic 2$ are: :$\\Dic 2 = \\set {e, a, a^2, a^3, b, a b, a^2 b, a^3 b}$"}
+{"_id": "20324", "title": "Definition:Power Series", "text": "=== Real Domain === {{:Definition:Power Series/Real Domain}} === Complex Domain === {{:Definition:Power Series/Complex Domain}}"}
+{"_id": "20325", "title": "Definition:Interval of Convergence", "text": "Let $\\xi \\in \\R$ be a real number. Let $\\displaystyle \\map S x = \\sum_{n \\mathop = 0}^\\infty a_n \\paren {x - \\xi}^n$ be a power series in $x$ about $\\xi$. Then the set of values of $x$ for which $\\map S x$ converges is called the '''interval of convergence''' of $S$."}
+{"_id": "20326", "title": "Definition:Radius of Convergence", "text": "=== Real Domain === {{:Definition:Radius of Convergence/Real Domain}} === Complex Domain === {{:Definition:Radius of Convergence/Complex Domain}}"}
+{"_id": "20327", "title": "Definition:Conditionally Convergent Series", "text": "A series which is convergent, but not absolutely convergent, is said to be '''conditionally convergent'''."}
+{"_id": "20328", "title": "Definition:Divergent (Analysis)", "text": "=== Divergent Sequence === {{:Definition:Divergent Sequence}} === Divergent Series === {{:Definition:Divergent Series}} === Divergent Function === {{:Definition:Divergent Function}} === Divergent Improper Integral === {{:Definition:Divergent Improper Integral}}"}
+{"_id": "20329", "title": "Definition:Taylor Series", "text": "Let $f$ be a real function which is smooth on the open interval $\\openint a b$. Let $\\xi \\in \\openint a b$. Then the '''Taylor series expansion of $f$''' about the point $\\xi$ is: :$\\displaystyle \\sum_{n \\mathop = 0}^\\infty \\frac {\\paren {x - \\xi}^n} {n!} \\map {f^{\\paren n} } \\xi$ It is ''not'' necessarily the case that this power series is convergent with sum $\\map f x$."}
+{"_id": "20330", "title": "Definition:Analytic Function", "text": "=== Real Analytic Function === {{:Definition:Analytic Function/Real Numbers}} === Complex Analytic Function === {{:Definition:Analytic Function/Complex Plane}}"}
+{"_id": "20331", "title": "Definition:Sine", "text": "=== Definition from Triangle === {{:Definition:Sine/Definition from Triangle}} === Definition from Circle === The sine of an angle in a right triangle can be extended to the full circle as follows: {{:Definition:Sine/Definition from Circle}} === Real Numbers === {{:Definition:Sine/Real Function}} === Complex Numbers === {{:Definition:Sine/Complex Function}}"}
+{"_id": "20332", "title": "Definition:Cosine", "text": "=== Definition from Triangle === {{:Definition:Cosine/Definition from Triangle}} === Definition from Circle === {{:Definition:Cosine/Definition from Circle}} === Real Numbers === {{:Definition:Cosine/Real Function}} === Complex Numbers === {{:Definition:Cosine/Complex Function}}"}
+{"_id": "20333", "title": "Definition:Trigonometric Function", "text": "There are six basic '''trigonometric functions''': sine, cosine, tangent, cotangent, secant, and cosecant."}
+{"_id": "20334", "title": "Definition:Secant Function", "text": "=== Definition from Triangle === {{:Definition:Secant Function/Definition from Triangle}} === Definition from Circle === {{:Definition:Secant Function/Definition from Circle}} === Real Function === {{:Definition:Secant Function/Real}} === Complex Function === {{:Definition:Secant Function/Complex}}"}
+{"_id": "20335", "title": "Definition:Cosecant", "text": "=== Definition from Triangle === {{:Definition:Cosecant/Definition from Triangle}} === Definition from Circle === {{:Definition:Cosecant/Definition from Circle}} === Real Function === {{:Definition:Cosecant/Real Function}} === Complex Function === {{:Definition:Cosecant/Complex Function}}"}
+{"_id": "20336", "title": "Definition:Cotangent", "text": "=== Definition from Triangle === {{:Definition:Cotangent/Definition from Triangle}} === Definition from Circle === {{:Definition:Cotangent/Definition from Circle}} === Real Function === {{:Definition:Cotangent/Real Function}} === Complex Function === {{:Definition:Cotangent/Complex Function}}"}
+{"_id": "20337", "title": "Definition:Holomorphic Function", "text": "=== Complex Function === {{:Definition:Holomorphic Function/Complex Plane}} === Vector-Valued Function === {{definition wanted}}"}
+{"_id": "20338", "title": "Definition:Complex Function", "text": "A '''complex function''' is a function whose domain and codomain are subsets of the set of complex numbers $\\C$."}
+{"_id": "20339", "title": "Definition:Periodic Function", "text": "=== Periodic Real Function === {{:Definition:Periodic Function/Real}} === Periodic Complex Function === {{:Definition:Periodic Function/Complex}}"}
+{"_id": "20340", "title": "Definition:Pi", "text": "The real number $\\pi$ (pronounced '''pie''') is an irrational number (see proof here) whose value is approximately $3.14159\\ 26535\\ 89793\\ 23846\\ 2643 \\ldots$"}
+{"_id": "20341", "title": "Definition:Decomposable Set", "text": "A set $S \\subset \\R^n$ is '''decomposable in $m$ sets $A_1, \\ldots, A_m \\subset \\R^n$''' if there exist isometries $\\phi_1, \\ldots, \\phi_m: \\R^n \\to \\R^n$ such that: :$(1):\\quad \\displaystyle S = \\bigcup_{k \\mathop = 1}^m \\phi_k \\left({A_k}\\right)$  :$(2):\\quad \\forall i \\ne j: \\phi_i \\left({A_i}\\right) \\cap \\phi_j \\left({A_j}\\right) = \\varnothing$ Such a union is known as a '''decomposition'''. {{explain|The definition of Definition:Irreducible Space suggests that the definition may apply to a general topological space, not only $\\R^n$. This needs to be clarified.}}"}
+{"_id": "20342", "title": "Definition:Isometry (Metric Spaces)", "text": "=== Definition 1 === {{:Definition:Isometry (Metric Spaces)/Definition 1}} === Definition 2 === {{:Definition:Isometry (Metric Spaces)/Definition 2}} Such metric spaces $M_1$ and $M_2$ are defined as being '''isometric'''. === Isometry Into === {{:Definition:Isometry (Metric Spaces)/Into}}"}
+{"_id": "20343", "title": "Definition:Equidecomposable", "text": "Two sets $S, T \\subset \\R^n$ are said to be '''equidecomposable''' if there exists a set: :$X = \\set {A_1, \\ldots, A_m} \\subset \\powerset {\\R^n}$ where $\\powerset {\\R^n}$ is the power set of $\\R^n$, such that both $S$ and $T$ are decomposable into the elements of $X$."}
+{"_id": "20345", "title": "Definition:Asymptotically Equal", "text": "=== Sequences === {{:Definition:Asymptotically Equal/Sequences/Definition 1}} === Functions === {{:Definition:Asymptotically Equal/Functions}} === General Definition === {{:Definition:Asymptotically Equal/General Definition/Point}}"}
+{"_id": "20346", "title": "Definition:Inverse Sine/Arcsine", "text": "{{:Definition:Real Arcsine}}"}
+{"_id": "20347", "title": "Definition:Inverse Cosine/Real/Arccosine", "text": "{{:Graph of Arccosine Function|Graph}} From Shape of Cosine Function, we have that $\\cos x$ is continuous and strictly decreasing on the interval $\\closedint 0 \\pi$. From Cosine of Multiple of Pi, $\\cos \\pi = -1$ and $\\cos 0 = 1$. Therefore, let $g: \\closedint 0 \\pi \\to \\closedint {-1} 1$ be the restriction of $\\cos x$ to $\\closedint 0 \\pi$. Thus from Inverse of Strictly Monotone Function, $\\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\\closedint {-1} 1$. This function is called '''arccosine of $x$''' and is written $\\arccos x$. Thus: :The domain of $\\arccos x$ is $\\closedint {-1} 1$ :The image of $\\arccos x$ is $\\closedint 0 \\pi$."}
+{"_id": "20348", "title": "Definition:Inverse Tangent/Real/Arctangent", "text": "{{:Graph of Arctangent Function}} From Shape of Tangent Function, we have that $\\tan x$ is continuous and strictly increasing on the interval $\\openint {-\\dfrac \\pi 2} {\\dfrac \\pi 2}$. From the same source, we also have that: :$\\tan x \\to + \\infty$ as $x \\to \\dfrac \\pi 2 ^-$ :$\\tan x \\to - \\infty$ as $x \\to -\\dfrac \\pi 2 ^+$ Let $g: \\openint {-\\dfrac \\pi 2} {\\dfrac \\pi 2} \\to \\R$ be the restriction of $\\tan x$ to $\\openint {-\\dfrac \\pi 2} {\\dfrac \\pi 2}$. Thus from Inverse of Strictly Monotone Function, $\\map g x$ admits an inverse function, which will be continuous and strictly increasing on $\\R$. This function is called '''arctangent''' of $x$ and is written $\\arctan x$. Thus: :The domain of $\\arctan x$ is $\\R$ :The image of $\\arctan x$ is $\\openint {-\\dfrac \\pi 2} {\\dfrac \\pi 2}$."}
+{"_id": "20350", "title": "Definition:Limit Point/Topology", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. === Limit Point of Set === {{:Definition:Limit Point/Topology/Set}} === Limit Point of Point === The concept of a '''limit point''' can be sharpened to apply to individual points, as follows: {{:Definition:Limit Point/Topology/Point}} === Limit Point of Sequence === {{:Definition:Limit Point/Topology/Sequence}}"}
+{"_id": "20351", "title": "Definition:Vacuous Truth", "text": "Let $P \\implies Q$ be a conditional statement. Suppose that $P$ is false. Then the statement $P \\implies Q$ is a '''vacuous truth''', or '''is vacuously true'''. It is frequently encountered in the form: :$\\forall x: \\map P x \\implies \\map Q x$ when the propositional function $\\map P x$ is false for all $x$. Such a statement is also a '''vacuous truth'''. For example, the statement: :'''All cats who are expert chess-players are also fluent in ancient Sanskrit''' is '''(vacuously) true''', because (as far as the author knows) there ''are'' no cats who are expert chess-players."}
+{"_id": "20352", "title": "Definition:Metric Subspace", "text": "Let $\\struct {A, d}$ be a metric space. Let $H \\subseteq A$. Let $d_H: H \\times H \\to \\R$ be the restriction $d \\restriction_{H \\times H}$ of $d$ to $H$. That is, let $\\forall x, y \\in H: \\map {d_H} {x, y} = \\map d {x, y}$. Then $d_H$ is the '''metric induced on $H$ by $d$''' or the '''subspace metric of $d$ (with respect to $H$)'''. The metric space $\\struct {H, d_H}$ is called a '''metric subspace of $\\struct {A, d}$'''."}
+{"_id": "20353", "title": "Definition:Equivalent Metrics", "text": "Let $X$ be a set upon which there are two metrics $d_1$ and $d_2$. That is, $\\struct {X, d_1}$ and $\\struct {X, d_2}$ are two different metric spaces on the same set $X$. Let $\\sequence {x_n}$ be a sequence in $X$. Let $n \\to \\infty$. Suppose that $x_n \\to x$ in $\\struct {X, d_1}$ {{iff}} $x_n \\to x$ in $\\struct {X, d_2}$. Then $d_1$ and $d_2$ are '''equivalent metrics'''."}
+{"_id": "20355", "title": "Definition:Taxicab Metric", "text": "The '''taxicab metric''' on $A_{1'} \\times A_{2'}$ is defined as: : $d_1 \\left({x, y}\\right) := d_{1'} \\left({x_1, y_1}\\right) + d_{2'} \\left({x_2, y_2}\\right)$ where $x = \\left({x_1, x_2}\\right), y = \\left({y_1, y_2}\\right) \\in A_{1'} \\times A_{2'}$."}
+{"_id": "20357", "title": "Definition:Topologically Equivalent Metrics", "text": "Let $M_1 = \\struct {A, d_1}$ and $M_2 = \\struct {A, d_2}$ be metric spaces on the same underlying set $A$. === Definition 1 === {{:Definition:Topologically Equivalent Metrics/Definition 1}} === Definition 2 === {{:Definition:Topologically Equivalent Metrics/Definition 2}}"}
+{"_id": "20358", "title": "Definition:Lipschitz Equivalence", "text": "=== Metric Spaces === {{:Definition:Lipschitz Equivalence/Metric Spaces}} === Metrics === {{:Definition:Lipschitz Equivalence/Metrics/Definition 1}}"}
+{"_id": "20359", "title": "Definition:Inverse Cotangent/Real/Arccotangent", "text": "{{:Graph of Arccotangent Function}} From Shape of Cotangent Function, we have that $\\cot x$ is continuous and strictly decreasing on the interval $\\openint 0 \\pi$. From the same source, we also have that: :$\\cot x \\to + \\infty$ as $x \\to 0^+$ :$\\cot x \\to - \\infty$ as $x \\to \\pi^-$ Let $g: \\openint 0 \\pi \\to \\R$ be the restriction of $\\cot x$ to $\\openint 0 \\pi$. Thus from Inverse of Strictly Monotone Function, $\\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\\R$. This function is called '''arccotangent''' of $x$ and is written $\\arccot x$. Thus: :The domain of $\\arccot x$ is $\\R$ :The image of $\\arccot x$ is $\\openint 0 \\pi$."}
+{"_id": "20360", "title": "Definition:Natural Logarithm", "text": "=== Positive Real Numbers === {{:Definition:Natural Logarithm/Positive Real}} === Complex Numbers === {{:Definition:Natural Logarithm/Complex}}"}
+{"_id": "20362", "title": "Definition:Discrete Topology", "text": "Let $S \\ne \\O$ be a set. Let $\\tau = \\powerset S$ be the power set of $S$. That is, let $\\tau$ be the set of all subsets of $S$: :$\\tau := \\set {H: H \\subseteq S}$ Then $\\tau$ is called '''the discrete topology on $S$''' and $\\struct {S, \\tau} = \\struct {S, \\powerset S}$ '''the discrete space on $S$''', or just '''a discrete space'''. === Finite Discrete Topology === {{:Definition:Discrete Topology/Finite}} === Infinite Discrete Topology === {{:Definition:Discrete Topology/Infinite}}"}
+{"_id": "20363", "title": "Definition:Indiscrete Topology", "text": "Let $S \\ne \\O$ be a set. Let $\\tau = \\set {S, \\O}$. Then $\\tau$ is called the '''indiscrete topology''' on $S$."}
+{"_id": "20364", "title": "Definition:Fourier Transform", "text": "The '''Fourier transform''' of a Lebesgue integrable function $f: \\R^N \\to \\C$ is the function $\\map \\FF f: \\R^N \\to \\C$ given by: :$\\displaystyle \\map \\FF {\\map f \\xi} := \\int_{\\R^N} \\map f {\\mathbf x} e^{-2 \\pi i \\mathbf x \\cdot \\xi} \\rd \\mathbf x$ for $\\xi \\in \\R^N$. Here, the product $\\mathbf x \\cdot \\xi$ in the exponential is the dot product of the vectors $\\mathbf x$ and $\\mathbf \\xi$."}
+{"_id": "20365", "title": "Definition:Metrizable Topology", "text": "Let $\\struct {S, d}$ be a metric space. Let $\\struct {S, \\tau}$ be the topological space induced by $d$. Then for any topological space which is homeomorphic to such a $\\struct {S, \\tau}$, it and its topology are defined as '''metrizable'''."}
+{"_id": "20366", "title": "Definition:Finer Topology", "text": "Let $S$ be a set. Let $\\tau_1$ and $\\tau_2$ be topologies on $S$. === Definition 1 === {{Definition:Finer Topology/Definition 1}} === Definition 2 === {{Definition:Finer Topology/Definition 2}} This can be expressed as: :$\\tau_1 \\ge \\tau_2 := \\tau_1 \\supseteq \\tau_2$ === Strictly Finer === {{:Definition:Finer Topology/Strictly Finer}}"}
+{"_id": "20367", "title": "Definition:Coarser Topology", "text": "Let $S$ be a set. Let $\\tau_1$ and $\\tau_2$ be topologies on $S$. Let $\\tau_1 \\subseteq \\tau_2$. Then $\\tau_1$ is said to be '''coarser''' than $\\tau_2$. This can be expressed as: :$\\tau_1 \\le \\tau_2 := \\tau_1 \\subseteq \\tau_2$ === Strictly Coarser === {{:Definition:Coarser Topology/Strictly Coarser}}"}
+{"_id": "20369", "title": "Definition:Convolution (Measure Theory)", "text": "Let $\\BB^n$ be the Borel $\\sigma$-algebra on $\\R^n$, and let $\\lambda^n$ be Lebesgue measure on $\\R^n$. === Convolution of Measurable Functions === {{:Definition:Convolution of Measurable Functions}} === Convolution of Measurable Function and Measure === {{:Definition:Convolution of Measurable Function and Measure}} === Convolution of Measures === {{:Definition:Convolution of Measures}}"}
+{"_id": "20370", "title": "Definition:Meromorphic Function", "text": "Let $U \\subset \\C$ be an open set. A '''meromorphic function on $U$''' is a holomorphic function on the complement of a discrete subset $P \\subset U$ such that: :for all $p\\in P$, $f$ has either a pole or removable singularity at $p$."}
+{"_id": "20372", "title": "Definition:Uniform Convergence", "text": "=== Metric Space === {{:Definition:Uniform Convergence/Metric Space}} === Real Numbers === The above definition can be applied directly to the real numbers treated as a metric space: {{:Definition:Uniform Convergence/Real Numbers}} === Infinite Series === {{:Definition:Uniform Convergence/Infinite Series}}"}
+{"_id": "20373", "title": "Definition:Pointwise Convergence", "text": "Let $\\sequence {f_n}$ be a sequence of real functions defined on $D \\subseteq \\R$. Suppose that: :$\\displaystyle \\forall x \\in D: \\lim_{n \\mathop \\to \\infty} \\map {f_n} x = \\map f x$ That is: :$\\forall x \\in D: \\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\R: \\forall n > N: \\size {\\map {f_n} x - \\map f x} < \\epsilon$ Then '''$\\sequence {f_n}$ converges to $f$ pointwise on $D$ as $n \\to \\infty$'''. (See the definition of convergence of a sequence)."}
+{"_id": "20374", "title": "Definition:Sub-Basis", "text": "=== Analytic Sub-Basis === {{:Definition:Sub-Basis/Analytic Sub-Basis}} === Synthetic Sub-Basis === {{:Definition:Sub-Basis/Synthetic Sub-Basis}}"}
+{"_id": "20375", "title": "Definition:Entire Function", "text": "A holomorphic self-map $f$ of the complex plane is called an '''entire function'''."}
+{"_id": "20376", "title": "Definition:Topological Subspace", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be a non-empty subset of $S$. Define: :$\\tau_H := \\set {U \\cap H: U \\in \\tau} \\subseteq \\powerset H$ where $\\powerset H$ denotes the power set of $H$. Then the topological space $T_H = \\struct {H, \\tau_H}$ is called a '''(topological) subspace''' of $T$."}
+{"_id": "20377", "title": "Definition:Closure (Topology)", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$. === Definition 1 === {{:Definition:Closure (Topology)/Definition 1}} === Definition 2 === {{:Definition:Closure (Topology)/Definition 2}} === Definition 3 === {{:Definition:Closure (Topology)/Definition 3}} === Definition 4 === {{:Definition:Closure (Topology)/Definition 4}} === Definition 5 === {{:Definition:Closure (Topology)/Definition 5}} === Definition 6 === {{:Definition:Closure (Topology)/Definition 6}}"}
+{"_id": "20378", "title": "Definition:Sigma-Algebra", "text": "{{:Definition:Sigma-Algebra/Definition 1}}"}
+{"_id": "20379", "title": "Definition:Borel Sigma-Algebra", "text": "=== Topological Spaces === Let $\\struct {S, \\tau}$ be a topological space The '''Borel sigma-algebra''' $\\map {\\mathcal B} {S, \\tau}$ of $\\struct {S, \\tau}$ is the $\\sigma$-algebra generated by $\\tau$. That is, it is the $\\sigma$-algebra generated by the set of open sets in $S$. === Metric Spaces === Let $\\struct {X, \\norm {\\,\\cdot\\,} }$ be a metric space. The '''Borel sigma-algebra''' (or '''$\\sigma$-algebra''') on $\\struct {X, \\norm {\\,\\cdot\\,} }$ is the $\\sigma$-algebra generated by the open sets in $\\powerset X$. By the definition of a topology induced by a metric, this definition is a particular instance of the definition of a Borel $\\sigma$-algebra on topological spaces. === Borel Set === {{:Definition:Borel Sigma-Algebra/Borel Set}}"}
+{"_id": "20380", "title": "Definition:Lebesgue Sigma-Algebra", "text": "Let $\\tau$ be the Euclidean topology on $\\R^n$. Let $\\mathcal N$ be the collection of null sets in $\\R^n$. The '''Lebesgue sigma-algebra''' on $\\R^n$ is the sigma-algebra generated by the set $\\tau \\cup \\mathcal N$. {{namedfor|Henri Léon Lebesgue}} {{SUBPAGENAME}} {{SUBPAGENAME}} 74sbdzjrem7wc3tly1t0hqfqiuwfr5y"}
+{"_id": "20381", "title": "Definition:Null Set", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. A set $N \\in \\Sigma$ is called a '''($\\mu$-)null set''' {{iff}} $\\mu \\left({N}\\right) = 0$. === Family of Null Sets === The family of '''$\\mu$-null sets''', $\\left\\{{N \\in \\Sigma: \\mu \\left({N}\\right) = 0}\\right\\}$, is denoted $\\mathcal N_{\\mu}$."}
+{"_id": "20382", "title": "Definition:Complex Modulus", "text": "Let $z = a + i b$ be a complex number, where $a, b \\in \\R$. Then the '''(complex) modulus of $z$''' is written $\\cmod z$ and is defined as the square root of the sum of the squares of the real and imaginary parts: :$\\cmod z := \\sqrt {a^2 + b^2}$"}
+{"_id": "20383", "title": "Definition:Real-Valued Function", "text": "Let $f: S \\to T$ be a function. Let $S_1 \\subseteq S$ such that $\\map f {S_1} \\subseteq \\R$. Then $f$ is said to be '''real-valued on $S_1$'''. That is, $f$ is defined as '''real-valued on $S_1$''' {{iff}} the image of $S_1$ under $f$ lies entirely within the set of real numbers $\\R$. A '''real-valued function''' is a function $f: S \\to \\R$ whose codomain is the set of real numbers $\\R$. That is, $f$ is '''real-valued''' {{iff}} it is '''real-valued''' over its entire domain."}
+{"_id": "20384", "title": "Definition:Complex-Valued Function", "text": "Let $f: S \\to T$ be a function. Let $S_1 \\subseteq S$ such that $f \\left({S_1}\\right) \\subseteq \\C$. Then $f$ is defined as '''complex-valued on $S_1$'''. That is, $f$ is defined as '''complex-valued on $S_1$''' if the image of $S_1$ under $f$ lies entirely within the set of complex numbers $\\C$. A '''complex-valued function''' is a function $f: S \\to \\C$ whose codomain is the set of complex numbers $\\C$. That is $f$ is complex-valued iff it is complex-valued over its entire domain."}
+{"_id": "20386", "title": "Definition:Usual Metric", "text": "Let $X$ be one of the standard number fields $\\Q$, $\\R$, $\\C$. Let $X^n$ be a cartesian space on $X$. The '''usual metric''' on $X^n$ is the Euclidean metric on $X^n$: === Real Numbers === {{:Definition:Euclidean Metric/Real Vector Space}} === Rational Numbers === {{:Definition:Euclidean Metric/Rational Space}} === Complex Plane === {{:Definition:Euclidean Metric/Complex Plane}} Category:Definitions/Examples of Metric Spaces Category:Definitions/Euclidean Metric qy3nckivsilneywbe9yipzdhkmtyufq"}
+{"_id": "20387", "title": "Definition:Absolute Continuity", "text": "Let $I \\subseteq \\R$ be a real interval. A real function $f: I \\to \\R$ is said to be '''absolutely continuous''' if it satisfies the following property: :For every $\\epsilon > 0$ there exists $\\delta > 0$ such that the following property holds: ::For every finite set of disjoint closed real intervals $\\closedint {a_1} {b_1}, \\dotsc, \\closedint {a_n} {b_n} \\subseteq I$ such that: :::$\\displaystyle \\sum_{i \\mathop = 1}^n \\size {b_i - a_i} < \\delta$ ::it holds that: :::$\\displaystyle \\sum_{i \\mathop = 1}^n \\size {\\map f {b_i} - \\map f {a_i} } < \\epsilon$"}
+{"_id": "20388", "title": "Definition:Uniform Continuity", "text": "=== Metric Spaces === {{:Definition:Uniform Continuity/Metric Space}} === Real Numbers === {{:Definition:Uniform Continuity/Real Numbers}}"}
+{"_id": "20389", "title": "Definition:Inner Product", "text": "Let $\\C$ be the field of complex numbers. Let $\\F$ be a subfield of $\\C$. Let $V$ be a vector space over $\\F$. An '''inner product''' is a mapping $\\innerprod \\cdot \\cdot: V \\times V \\to \\mathbb F$ that satisfies the following properties: {{begin-axiom}} {{axiom | n = 1         | lc= Conjugate Symmetry         | q = \\forall x, y \\in V         | m = \\quad \\innerprod x y = \\overline {\\innerprod y x} }} {{axiom | n = 2         | lc= Bilinearity         | q = \\forall x, y \\in V, \\forall a \\in \\mathbb F         | m = \\quad \\innerprod {a x + y} z = a \\innerprod x z + \\innerprod y z }} {{axiom | n = 3         | lc= Non-Negative Definiteness         | q = \\forall x \\in V         | m = \\quad \\innerprod x x \\in \\R_{\\ge 0} }} {{axiom | n = 4         | lc= Positiveness         | q = \\forall x \\in V         | m = \\quad \\innerprod x x = 0 \\implies x = \\mathbf 0_V }} {{end-axiom}} That is, an inner product is a semi-inner product with the additional condition $(4)$. If $\\mathbb F$ is a subfield of the field of real numbers $\\R$, it follows from Complex Number equals Conjugate iff Wholly Real that $\\overline {\\innerprod y x} = \\innerprod y x$ for all $x, y \\in V$. Then $(1)$ above may be replaced by: {{begin-axiom}} {{axiom | n = 1'         | lc= Symmetry         | q = \\forall x, y \\in V         | m = \\innerprod x y = \\innerprod y x }} {{end-axiom}} === Inner Product Space === {{:Definition:Inner Product Space}}"}
+{"_id": "20390", "title": "Definition:Inner Product Space", "text": "An '''inner product space''' is a vector space together with an associated inner product."}
+{"_id": "20392", "title": "Definition:Isolated Point (Topology)", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. === Isolated Point of Subset === {{:Definition:Isolated Point (Topology)/Subset}} === Isolated Point of Space === When $H = S$ the definition applies to the entire topological space $T = \\struct {S, \\tau}$: {{:Definition:Isolated Point (Topology)/Space}}"}
+{"_id": "20393", "title": "Definition:Everywhere Dense", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be a subset. === Definition 1 === {{:Definition:Everywhere Dense/Definition 1}} === Definition 2 === {{:Definition:Everywhere Dense/Definition 2}}"}
+{"_id": "20394", "title": "Definition:Sequentially Compact Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $H \\subseteq S$. Then $H$ is '''sequentially compact in $T$''' {{iff}} every infinite sequence in $H$ has a subsequence which converges to a point in $S$."}
+{"_id": "20395", "title": "Definition:Totally Bounded Metric Space", "text": "=== Definition 1 === {{:Definition:Totally Bounded Metric Space/Definition 1}} === Definition 2 === {{:Definition:Totally Bounded Metric Space/Definition 2}}"}
+{"_id": "20397", "title": "Definition:Interior (Topology)", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$. === Definition 1 === {{:Definition:Interior (Topology)/Definition 1}} === Definition 2 === {{:Definition:Interior (Topology)/Definition 2}} === Definition 3 === {{:Definition:Interior (Topology)/Definition 3}}"}
+{"_id": "20398", "title": "Definition:Nowhere Dense", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$. === Definition 1 === {{:Definition:Nowhere Dense/Definition 1}} === Definition 2 === {{:Definition:Nowhere Dense/Definition 2}}"}
+{"_id": "20399", "title": "Definition:Quotient Topology", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $\\mathcal R \\subseteq S \\times S$ be an equivalence relation on $S$. Let $q_\\mathcal R: S \\to S / \\mathcal R$ be the quotient mapping induced by $\\mathcal R$. === Definition 1 === {{:Definition:Quotient Topology/Definition 1}} === Definition 2 === {{:Definition:Quotient Topology/Definition 2}}"}
+{"_id": "20400", "title": "Definition:Topological Property", "text": "Let $P$ be a property whose domain is the set of all topological spaces. Suppose that whenever $\\map P T$ holds, then so does $\\map P {T'}$, where $T$ and $T'$ are topological spaces which are homeomorphic. Then $P$ is known as a '''topological property'''. Loosely, a topological property is one which is preserved under homeomorphism."}
+{"_id": "20401", "title": "Definition:Cover of Set", "text": "Let $S$ be a set. A '''cover for $S$''' is a set of sets $\\CC$ such that: :$\\displaystyle S \\subseteq \\bigcup \\CC$ where $\\bigcup \\CC$ denotes the union of $\\CC$."}
+{"_id": "20402", "title": "Definition:Subcover", "text": "Let $S$ be a set. Let $\\mathcal U$ be a cover for $S$. A '''subcover of $\\mathcal U$ for $S$''' is a set $\\mathcal V \\subseteq \\mathcal U$ such that $\\mathcal V$ is also a cover for $S$. === Finite Subcover === {{:Definition:Subcover/Finite}} === Countable Subcover === {{:Definition:Subcover/Countable}}"}
+{"_id": "20403", "title": "Definition:Relatively Compact Subspace", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $T_H = \\left({H, \\tau_H}\\right)$ be a subspace of $T$. Let $\\operatorname{cl} \\left({H}\\right)$ be the closure of $H$ in $T$. Then $T_H$ is '''relatively compact in $T$''' {{iff}} $\\operatorname{cl} \\left({H}\\right)$ is compact."}
+{"_id": "20404", "title": "Definition:Inscribe", "text": "Let a geometric figure $A$ be constructed in the interior of another geometric figure $B$ such that: :$(1): \\quad$ $A$ and $B$ have points in common :$(2): \\quad$ No part of $A$ is outside $B$. Then $A$ is '''inscribed''' inside $B$. === Circle in Polygon === {{:Definition:Inscribe/Circle in Polygon}} === Polygon in Circle === {{:Definition:Inscribe/Polygon in Circle}} === Polygon in Polygon === {{:Definition:Inscribe/Polygon in Polygon}}"}
+{"_id": "20405", "title": "Definition:Circumscribe", "text": "=== Circle around Polygon === {{:Definition:Circumscribe/Circle around Polygon}} === Polygon around Circle === {{:Definition:Circumscribe/Polygon around Circle}} === Polygon around Polygon === {{:Definition:Circumscribe/Polygon around Polygon}}"}
+{"_id": "20406", "title": "Definition:Fixed Point", "text": "Let $f: S \\to T$ be a mapping. Then a '''fixed point''' (or '''fixed element''') '''of $S$ under $f$''' is an $x \\in S$ such that $\\map f x = x$."}
+{"_id": "20408", "title": "Definition:Coefficient", "text": "A '''coefficient''' is a constant which is used in a particular context to be multiplied by a variable that is under consideration. === Binomial Coefficient === {{:Definition:Binomial Coefficient/Integers/Definition 3}} === Polynomial Coefficient === For the usage of this term in the context of polynomial theory: {{:Definition:Polynomial Coefficient}}"}
+{"_id": "20409", "title": "Definition:Lemma", "text": "A '''lemma''' is a statement which is proven during the course of reaching the proof of a theorem. Logically there is no qualitative difference between a '''lemma''' and a theorem. They are both statements whose value is either true or false. However, a '''lemma''' is seen more as a stepping-stone than a theorem in itself (and frequently takes a lot more work to prove than the theorem to which it leads). Some lemmas are famous enough to be named after the mathematician who proved them (for example: Abel's Lemma and Urysohn's Lemma), but they are still categorised as second-class citizens in the aristocracy of mathematics. :''Always the lemma, never the theorem.''"}
+{"_id": "20410", "title": "Definition:Euclidean Geometry", "text": "Euclidean geometry is the branch of geometry in which the parallel postulate applies. An assumption which is currently under question is whether or not ordinary space is itself Euclidean. '''Euclidean geometry''' adheres to Euclid's postulates."}
+{"_id": "20411", "title": "Definition:Uniform Equivalence", "text": "=== Metric Spaces === {{:Definition:Uniform Equivalence/Metric Spaces}} === Metrics === {{:Definition:Uniform Equivalence/Metrics}} Category:Definitions/Metric Spaces rbfy2keax03l4o8uavh8o5ccqlogn3b"}
+{"_id": "20412", "title": "Definition:A Priori", "text": "'''A priori''' knowledge is the sort of knowledge which comes from reason alone. That is, it does not require the exercise of experience to know it. For example: :''If Fred Bloggs has committed a crime, then he is guilty'' as opposed to: :''Fred Bloggs has committed the crime of usury.''"}
+{"_id": "20413", "title": "Definition:Clopen Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $X \\subseteq S$ such that $X$ is both open in $T$ and closed in $T$. Then $X$ is described as '''clopen'''."}
+{"_id": "20414", "title": "Definition:Connected (Topology)", "text": "=== Topological Space === {{Definition:Connected (Topology)/Topological Space}} === Set of Topological Space === {{Definition:Connected (Topology)/Set}} === Points in Topological Space === {{:Definition:Connected (Topology)/Points}}"}
+{"_id": "20415", "title": "Definition:Position", "text": "'''Position''' is an abstract geometrical concept that defines '''where something is'''. It is only possible to define the '''position''' of an object in relation to the '''position''' of another object. Even when one tries to consider an '''''absolute'' position''', one has to describe it in relation to a frame of reference, which itself needs to have a '''position''' (which ultimately is itself arbitrarily defined)."}
+{"_id": "20416", "title": "Definition:Angle Bisector", "text": ":300px Let $\\angle ABC$ be an angle. The '''angle bisector''' of $\\angle ABC$ is the straight line which bisects $\\angle ABC$. In the above diagram, $BD$ is the '''angle bisector''' of $\\angle ABC$. Thus $\\angle ABD \\cong \\angle DBC$ and $\\angle ABD + \\angle DBC = \\angle ABC$."}
+{"_id": "20417", "title": "Definition:Darboux Function", "text": "Let $S \\subseteq \\R$. Let $f: S \\to \\R$ be a real function. Then $f$ is '''Darboux''' {{iff}}, given any $a, b \\in S$ and $y \\in \\R$ such that $a < b$ and $y$ is between $\\map f a$ and $\\map f b$, there exists a $c \\in S$ such that $a \\le c \\le b$ and $\\map f c = y$. That is, for every '''intermediate value''' between $\\map f a$ and $\\map f b$, that value is the image of some '''intermediate value''' between $a$ and $b$."}
+{"_id": "20418", "title": "Definition:Path (Topology)", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $I \\subset \\R$ be the closed real interval $\\closedint a b$. A '''path in $T$''' is a continuous mapping $\\gamma: I \\to S$. The mapping $\\gamma$ can be referred as: :a '''path (in $T$) joining $\\map \\gamma a$ and $\\map \\gamma b$''' or: :a '''path (in $T$) from $\\map \\gamma a$ to $\\map \\gamma b$'''. It is common to refer to a point $z = \\map \\gamma t$ as a '''point on the path $\\gamma$''', even though $z$ is in fact on the image of $\\gamma$. === Initial Point === {{:Definition:Path (Topology)/Initial Point}} === Final Point === {{:Definition:Path (Topology)/Final Point}} === Endpoint === {{:Definition:Path (Topology)/Endpoint}}"}
+{"_id": "20419", "title": "Definition:Congruence (Geometry)", "text": "In the field of Euclidean geometry, two geometric figures are '''congruent''' if they are, informally speaking, both \"the same size and shape\". That is, one figure can be overlaid on the other figure with a series of rotations, translations, and reflections. Specifically: : all corresponding angles of the '''congruent''' figures must have the same measurement : all corresponding sides of the '''congruent''' figures must be be the same length."}
+{"_id": "20420", "title": "Definition:WLOG", "text": "Suppose there are several cases which need to be investigated. If the same argument can be used to dispose of two or more of these cases, then it is acceptable in a proof to pick just one of these cases, and announce this fact with the words: '''Without loss of generality, ...''', or just '''WLOG'''."}
+{"_id": "20421", "title": "Definition:Sector", "text": "A '''sector''' of a circle is the area bounded by two radii and an arc. {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book III/10 - Sector of Circle}}'' {{EuclidDefRefNocat|III|10|Sector of Circle}} In the diagram below, $BAC$ is a '''sector'''.  :300px In fact there are two '''sectors''', together making up the whole of the circle. When describing a sector by means of its three defining points (as in $BAC$ above), the convention is to report them in the following order: :$(1):\\quad$ The point on the circumference at the end of the clockwise radius :$(2):\\quad$ The point at the center of the circle :$(3):\\quad$ The point on the circumference at the end of the anticlockwise radius Thus in the '''sector''' above, $BAC$ describes the '''sector''' indicated by $\\theta$, while the '''sector''' comprising the rest of the circle is described by $CAB$. === Angle of Sector === {{:Definition:Sector/Angle}}"}
+{"_id": "20422", "title": "Definition:Component (Topology)", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let the relation $\\sim $ be defined on $T$ as follows: :$x \\sim y$ {{iff}} $x$ and $y$ are connected in $T$. That is, {{iff}} there exists a connected set of $T$ that contains both $x$ and $y$. === Definition 1 === {{:Definition:Component (Topology)/Definition 1}} === Definition 2 === {{:Definition:Component (Topology)/Definition 2}} === Definition 3 === {{:Definition:Component (Topology)/Definition 3}}"}
+{"_id": "20424", "title": "Definition:Totally Disconnected Space", "text": "A topological space $T = \\struct {S, \\tau}$ is a '''totally disconnected space''' {{iff}} all components of $T$ are singletons."}
+{"_id": "20425", "title": "Definition:Extremal Length", "text": "Let $\\Gamma$ be a set of rectifiable curves in the complex plane $\\C$. We consider conformal metrics of the form $\\map \\rho z \\cmod {\\d z}$, where: :$\\rho: \\C \\to \\hointr 0 \\to$ is Borel measurable and: :the area $\\displaystyle \\map A \\rho := \\iint \\rho^2 \\paren {x + i y} \\rd x \\rd y$ is finite and positive. Every $\\gamma \\in \\Gamma$ has a distance function with respect to such a metric, defined by: :$\\displaystyle \\map L {\\gamma, \\rho} := \\int_\\gamma \\map \\rho z \\size {\\d z}$ We define: :$\\displaystyle \\map L {\\Gamma, \\rho} := \\inf_{\\gamma \\mathop \\in \\Gamma} \\map L {\\gamma, \\rho}$ and: :$\\displaystyle \\map \\lambda \\Gamma := \\sup_\\rho \\frac {\\map L {\\Gamma, \\rho}^2} {\\map A \\rho}$ The quantity $\\map \\lambda \\Gamma$ is called the '''extremal length''' of the curve family $\\Gamma$. Its reciprocal: :$\\mod \\Gamma := \\dfrac 1 {\\map \\lambda \\Gamma}$ is called the '''modulus''' of $\\Gamma$."}
+{"_id": "20426", "title": "Definition:Converse Statement", "text": "The '''converse''' of the conditional: :$p \\implies q$ is the statement: :$q \\implies p$"}
+{"_id": "20427", "title": "Definition:Complete Metric Space", "text": "=== Definition 1 === {{:Definition:Complete Metric Space/Definition 1}} === Definition 2 === {{:Definition:Complete Metric Space/Definition 2}}"}
+{"_id": "20428", "title": "Definition:Golden Mean", "text": "=== Definition 1 === {{:Definition:Golden Mean/Definition 1}} === Definition 2 === {{:Definition:Golden Mean/Definition 2}}"}
+{"_id": "20430", "title": "Definition:Teichmüller Annulus", "text": "Let $R \\in \\R_{>0}$. The set: :$A := \\C \\setminus \\paren {\\closedint{-1} 0 \\cup \\hointr R {+\\infty} }$ is a '''Teichmüller annulus'''. The modulus of $A$ is denoted $\\map \\Lambda R$."}
+{"_id": "20431", "title": "Definition:Degenerate Connected Set", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $H \\subseteq S$ be a subset of $T$. $H$ is a '''degenerate connected set''' of $T$ {{iff}} it is a connected set of $T$ containing exactly one element. === Non-Degenerate Connected Set === {{:Definition:Degenerate Connected Set/Non-Degenerate}} === Degenerate Connected Space === When $H = S$ itself, the entire space can be referred to in this way: {{:Definition:Degenerate Connected Set/Topological Space}}"}
+{"_id": "20432", "title": "Definition:Riemann Zeta Function", "text": "The '''Riemann Zeta Function''' $\\zeta$ is the complex function defined on the half-plane $\\map \\Re s > 1$ as the series: :$\\displaystyle \\map \\zeta s = \\sum_{n \\mathop = 1}^\\infty \\frac 1 {n^s}$"}
+{"_id": "20434", "title": "Definition:Cevian", "text": "Let $\\triangle ABC$ be a triangle. :400px A '''cevian''' is a line segment from a vertex of the triangle to the opposite side. In the above diagram, $CD$ is a '''cevian'''."}
+{"_id": "20435", "title": "Definition:Median of Triangle", "text": "Let $\\triangle ABC$ be a triangle. :400px A '''median''' is a cevian which bisects the opposite."}
+{"_id": "20436", "title": "Definition:Planar Graph", "text": "A '''planar graph''' is a graph which can be drawn in the plane (for example, on a piece of paper) without any of the edges crossing over, that is, meeting at points other than the vertices. This is a '''planar graph''': :300px === Face === {{:Definition:Planar Graph/Face}}"}
+{"_id": "20437", "title": "Definition:Graph (Graph Theory)/Vertex", "text": "Let $G = \\struct {V, E}$ be a graph. The '''vertices''' (singular: '''vertex''') are the elements of $V$. Informally, the '''vertices''' are the points that are connected by the edges."}
+{"_id": "20438", "title": "Definition:Area", "text": "'''Area''' is the measure of the extent of a surface. It has $2$ dimensions and is specified in units of length squared."}
+{"_id": "20439", "title": "Definition:Euler Characteristic of Finite Graph", "text": "Let $X = \\struct {V, E}$ be a graph. Let $X$ be embedded in a surface. The '''Euler characteristic''' of $X$ is written $\\map \\chi X$ and is defined as: :$\\map \\chi x = v - e + f$ where: :$v = \\size V$ is the number of vertices :$e = \\size E$ is the number of edges :$f$ is the number of faces. === Generalized Formula === {{DefinitionWanted}}"}
+{"_id": "20440", "title": "Definition:Deleted Neighborhood", "text": "=== Real Analysis === {{:Definition:Deleted Neighborhood/Real Analysis}} === Complex Analysis === {{:Definition:Deleted Neighborhood/Complex Analysis}} === Metric Space === {{:Definition:Deleted Neighborhood/Metric Space}} === Normed Vector Space === {{:Definition:Deleted Neighborhood/Normed Vector Space}} === Topology === {{:Definition:Deleted Neighborhood/Topology}}"}
+{"_id": "20442", "title": "Definition:Depressed Polynomial", "text": "Let $f \\left({x}\\right)$ be a polynomial over a field $k$: :$f \\left({x}\\right) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \\cdots + a_1 x + a_0$ If $a_{n-1} = 0_k$, then we call $f$ a '''depressed polynomial'''. It has been suggested that a polynomial with further zero terms might be referred to as \"downright despondent\", though this convention has yet to gain widespread usage by the community. {{refactor|Put the below into its own page}}"}
+{"_id": "20443", "title": "Definition:Cubic Equation", "text": "A '''cubic equation''' is a polynomial equation of the form: : $a x^3 + b x^2 + c x + d = 0$"}
+{"_id": "20444", "title": "Definition:Distinct", "text": "=== Of Two or More Objects === {{:Definition:Distinct/Plural}} === Of a Single Object === {{:Definition:Distinct/Singular}}"}
+{"_id": "20445", "title": "Definition:Quartic Equation", "text": "A '''quartic equation''' is a polynomial equation of the form: :$a x^4 + b x^3 + c x^2 + d x + e$"}
+{"_id": "20446", "title": "Definition:Quintic Equation", "text": "Let $\\map f x = a x^5 + b x^4 + c x^3 + d x^2 + e x + f$ be a polynomial function over a field $\\mathbb k$ of degree $5$. Then the equation $\\map f x = 0$ is the general '''quintic equation''' over $\\mathbb k$."}
+{"_id": "20447", "title": "Definition:Mersenne Prime", "text": "A '''Mersenne prime''' is a Mersenne number which happens to be prime. That is, it is a prime number of the form $2^p - 1$. The number $2^p - 1$ is, in this context, often denoted $M_p$."}
+{"_id": "20448", "title": "Definition:Net (Metric Space)", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. An '''$\\epsilon$-net for $M$''' is a subset $S \\subseteq A$ such that: :$\\displaystyle A \\subseteq \\bigcup_{x \\mathop \\in S} \\map {B_\\epsilon} x$ where $\\map {B_\\epsilon} x$ denotes the open $\\epsilon$-ball of $x$ in $M$."}
+{"_id": "20449", "title": "Definition:Lebesgue Number", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $\\mathcal U$ be an open cover of $M$. A fixed strictly positive real number $\\epsilon \\in \\R_{>0}$ is called a '''Lebesgue number for $\\mathcal U$''' {{iff}}: :$\\forall x \\in A: \\exists U \\left({x}\\right) \\in \\mathcal U: B_\\epsilon \\left({x}\\right) \\subseteq U \\left({x}\\right)$ where $B_\\epsilon \\left({x}\\right)$ is the open $\\epsilon$-ball of $x$ in $M$."}
+{"_id": "20450", "title": "Definition:Circular Definition", "text": "A '''circular definition''' is a definition of a concept $A$ in terms of another concept $B$ such that the definition of $B$ itself refers (either directly or indirectly) to the definition of $A$. It is a truism that '''circular definitions''' are always to be avoided."}
+{"_id": "20451", "title": "Definition:Basic Null Sequence", "text": "The sequences defined as: :$\\sequence {\\dfrac 1 {n^r} }_{n \\mathop \\in \\N}$ for $r \\in \\R: r > 0$ :$\\sequence {\\alpha^n}_{n \\mathop \\in \\N}$ for $\\alpha \\in \\C: \\cmod {\\alpha} < 1$ are known as the '''basic null sequences'''."}
+{"_id": "20452", "title": "Definition:Dot Product", "text": "Let $\\mathbf a$ and $\\mathbf b$ be vectors in a vector space $\\mathbf V$ of $n$ dimensions: :$\\mathbf a = \\displaystyle \\sum_{k \\mathop = 1}^n a_k \\mathbf e_k$ :$\\mathbf b = \\displaystyle \\sum_{k \\mathop = 1}^n b_k \\mathbf e_k$ where $\\tuple {\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}$ is the standard ordered basis of $\\mathbf V$."}
+{"_id": "20453", "title": "Definition:Legendre Symbol", "text": "Let $p$ be an odd prime. Let $a \\in \\Z$ be an integer."}
+{"_id": "20454", "title": "Definition:Diophantine Equation", "text": "A '''Diophantine equation''' is an indeterminate polynomial equation that allows the variables to take integer values only."}
+{"_id": "20455", "title": "Definition:Perfect Number", "text": "=== Definition 1 === {{:Definition:Perfect Number/Definition 1}} === Definition 2 === {{:Definition:Perfect Number/Definition 2}} === Definition 3 === {{:Definition:Perfect Number/Definition 3}} === Definition 4 === {{:Definition:Perfect Number/Definition 4}}"}
+{"_id": "20456", "title": "Definition:Fermat Prime", "text": "A '''Fermat prime''' is a Fermat number, that is a number of the form $2^{\\paren {2^n} } + 1$, which happens to be prime."}
+{"_id": "20457", "title": "Definition:Mersenne Number", "text": "A '''Mersenne number''' is a natural number of the form $2^p - 1$, where $p$ is prime. The number $2^p - 1$, in this context, can be denoted $M_p$."}
+{"_id": "20458", "title": "Definition:Carmichael Number", "text": "An integer $n > 0$ is a '''Carmichael number''' {{iff}}: : $(1): \\quad n$ is composite : $(2): \\quad \\forall a \\in \\Z: a \\perp n: a^n \\equiv a \\pmod n$, or, equivalently, that $a^{n - 1} \\equiv 1 \\pmod n$. That is, a '''Carmichael number''' is a composite number $n$ which satisfies $a^n \\equiv a \\pmod n$ for all integers $a$ which are coprime to it."}
+{"_id": "20459", "title": "Definition:Divisor Counting Function", "text": "Let $n$ be an integer such that $n \\ge 1$. The '''divisor counting function''' is defined on $n$ as being the total number of positive integer divisors of $n$. It is denoted on {{ProofWiki}} as $\\tau$ (the Greek letter '''tau'''). That is: :$\\displaystyle \\map \\tau n = \\sum_{d \\mathop \\divides n} 1$ where $\\displaystyle \\sum_{d \\mathop \\divides n}$ is the sum over all divisors of $n$."}
+{"_id": "20460", "title": "Definition:Sum Over Divisors", "text": "Let $n$ be a positive integer. Let $f: \\Z_{>0} \\to \\Z_{>0}$ be a mapping on the positive integers. Let $d \\divides n$ denote that $d$ is a divisor of $n$. Then the '''sum of $\\map f d$ over all the divisors of $n$''' is denoted: :$\\displaystyle \\sum_{d \\mathop \\divides n} \\map f d$ Thus, for example: :$\\displaystyle \\sum_{d \\mathop \\divides 10} \\map f d = \\map f 1 + \\map f 2 + \\map f 5 + \\map f {10}$ Category:Definitions/Number Theory iatkl4ppzsm04lgginpa6k7bligu0hl"}
+{"_id": "20461", "title": "Definition:Sigma Function", "text": "Let $n$ be an integer such that $n \\ge 1$. The '''sigma function''' $\\map \\sigma n$ is defined on $n$ as being the sum of all the positive integer divisors of $n$. That is: :$\\displaystyle \\map \\sigma n = \\sum_{d \\mathop \\divides n} d$ where $\\displaystyle \\sum_{d \\mathop \\divides n}$ is the sum over all divisors of $n$."}
+{"_id": "20462", "title": "Definition:Abundancy Index", "text": "Let $n$ be a positive integer. Let $\\map \\sigma n$ be the sigma function of $n$. That is, let $\\map \\sigma n$ be the sum of all positive divisors of $n$. Then the '''abundancy index''' of $n$ is defined as $\\dfrac {\\map \\sigma n} n$."}
+{"_id": "20463", "title": "Definition:Abundance", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Let $\\sigma \\left({n}\\right)$ be the sigma function of $n$. That is, let $\\sigma \\left({n}\\right)$ be the sum of all positive divisors of $n$. Then the '''abundance''' of $n$ is defined as $A \\left({n}\\right) = \\sigma \\left({n}\\right) - 2 n$."}
+{"_id": "20464", "title": "Definition:Multiplicative Order of Integer", "text": "Let $a$ and $n$ be integers. Let there exist a positive integer $c$ such that: :$a^c \\equiv 1 \\pmod n$ Then the least such integer is called '''order of $a$ modulo $n$'''."}
+{"_id": "20465", "title": "Definition:Euler Phi Function", "text": "Let $n \\in \\Z_{>0}$, that is, a strictly positive integer. The '''Euler $\\phi$ (phi) function''' is the arithmetic function $\\phi: \\Z_{>0} \\to \\Z_{>0}$ defined as: :$\\map \\phi n = $ the number of strictly positive integers less than or equal to $n$ which are prime to $n$ That is: :$\\map \\phi n = \\card {S_n}: S_n = \\set {k: 1 \\le k \\le n, k \\perp n}$"}
+{"_id": "20466", "title": "Definition:Reduced Residue System/Least Positive", "text": "The '''least positive reduced residue system modulo $m$''' is the set of integers: :$\\set {a_1, a_2, \\ldots, a_{\\map \\phi m} }$ with the following properties: :$\\map \\phi m$ is the Euler $\\phi$ function :$\\forall i: 0 < a_i < m$ :each of which is prime to $m$ :no two of which are congruent modulo $m$."}
+{"_id": "20467", "title": "Definition:Divisor Function", "text": "The '''divisor function''': :$\\displaystyle \\map {\\sigma_\\alpha} n = \\sum_{m \\mathop \\divides n} m^\\alpha$ (meaning the summation is taken over all $m \\le n$ such that $m$ divides $n$)."}
+{"_id": "20468", "title": "Definition:Von Mangoldt Function", "text": "The '''von Mangoldt function''' $\\Lambda: \\N \\to \\R$ is defined as: :$\\Lambda \\left({n}\\right) = \\begin{cases} \\ln \\left({p}\\right) & : \\exists m \\in \\N, p \\in \\mathbb P: n = p^m \\\\ 0 & : \\text{otherwise} \\end{cases}$ where $\\mathbb P$ is the set of all prime numbers."}
+{"_id": "20469", "title": "Definition:Knot (Knot Theory)", "text": "Let $Y$ be a manifold and $X \\subset Y$ a submanifold of $Y$. Let $i: X \\to Y$ be an inclusion, that is a mapping such that $i \\sqbrk X = X$. Then a '''knotted embedding''' is an embedding $\\phi: X \\to Y$ (or the image of such an embedding) such that $\\phi \\sqbrk X$ is not freely homotopic to $i \\sqbrk X$."}
+{"_id": "20470", "title": "Definition:Reidemeister Move", "text": "The '''Reidemeister moves''' are one of three operations on knots which encapsulate the intuitive notion of '''equivalent knots'''.   The three moves are shown below. :File:Reidemeister move 1.png  :File:Reidemeister move 2.png  :File:Reidemeister move 3.png {{namedfor|Kurt Reidemeister}} {{SUBPAGENAME}} 768qxtheox2rkueocc9cn88uv7mu0ff"}
+{"_id": "20471", "title": "Definition:Alexander Polynomial", "text": "For a knot $K$ with Seifert matrix $V$, the '''Alexander polynomial''' of $K$ is defined as: :$\\Delta_K \\left({t}\\right) = \\det \\left({V - t V^T}\\right)$ {{NamedforDef|James Waddell Alexander II|cat=Alexander}} Category:Definitions/Knot Theory 46i64iuexv28nredfqs7sslyuvx0ksf"}
+{"_id": "20472", "title": "Definition:Seifert Matrix", "text": "For a knot $K$ with Seifert surface $S$, the '''Seifert matrix''' $V$ of $K$ is defined by its entries as: :$v_{ij} = \\operatorname{lk} \\left({ x_i, x_k^* }\\right)$ where: : the $x_a$ are the generators of the fundamental group $\\pi_1(S)$ : $x_a^*$ is the positive push-off of $x_a$ : $\\operatorname{lk}$ is the linking number of the two loops. {{NamedforDef|Karl Johannes Herbert Seifert}} Category:Definitions/Knot Theory d1dkw982gw80dm2m2egtdme1ebvceii"}
+{"_id": "20473", "title": "Definition:Completely Multiplicative Function", "text": "Let $K$ be a field. Let $f: K \\to K$ be a function on $K$. Then $f$ is described as '''completely multiplicative''' {{iff}}: :$\\forall m, n \\in K: \\map f {m n} = \\map f m \\map f n$"}
+{"_id": "20474", "title": "Definition:Embedding (Topology)", "text": "Let $A, B$ be topological spaces. Let $f: A \\to B$ be a mapping. Let the image of $f$ be given the subspace topology. Let the restriction $f {\\restriction_{A \\times f \\left({A}\\right)}}$ of $f$ to its image be a homeomorphism. Then $f$ is an '''embedding''' (of $A$ into $B$)."}
+{"_id": "20475", "title": "Definition:Primitive Root", "text": "Let $a, n \\in \\Z_{>0}$, that is, let $a$ and $n$ be strictly positive integers. Let the multiplicative order of $a$ modulo $n$ be $\\phi \\left({n}\\right)$, where $\\phi \\left({n}\\right)$ is the Euler phi function of $n$. Then $a$ is a '''primitive root of $n$''' or a '''primitive root modulo $n$'''."}
+{"_id": "20476", "title": "Definition:Square-Free Integer", "text": "Let $n \\in \\Z$. Then $n$ is '''square-free''' {{iff}} $n$ has no divisor which is the square of a prime. That is, {{iff}} the prime decomposition $n = p_1^{k_1} p_2^{k_2} \\ldots p_r^{k_r}$ is such that: :$\\forall i: 1 \\le i \\le r: k_i = 1$"}
+{"_id": "20477", "title": "Definition:Quadratic Residue", "text": "Let $p$ be an odd prime. Let $a \\in \\Z$ be an integer such that $a \\not \\equiv 0 \\pmod p$. Then $a$ is a '''quadratic residue of $p$''' {{iff}} $x^2 \\equiv a \\pmod p$ has a solution. That is, {{iff}}: :$\\exists x \\in \\Z: x^2 \\equiv a \\pmod p$"}
+{"_id": "20478", "title": "Definition:Set of Residue Classes/Least Positive", "text": "Let $\\eqclass a m$ be the residue class of $a$ (modulo $m$). Let $r$ be the smallest non-negative integer in $\\eqclass a m$. Then from Integer is Congruent to Integer less than Modulus: :$0 \\le r < m$ and: :$a \\equiv r \\pmod m$ Then $r$ is called the '''least positive residue''' of $a \\pmod m$."}
+{"_id": "20479", "title": "Definition:Lowest Common Multiple", "text": "=== Integral Domain === {{:Definition:Lowest Common Multiple/Integral Domain}} === Integers === {{:Definition:Lowest Common Multiple/Integers}}"}
+{"_id": "20480", "title": "Definition:Greatest Common Divisor", "text": "=== Integral Domain === {{:Definition:Greatest Common Divisor/Integral Domain}} === Integers === When the integral domain in question is the integers $\\Z$, the GCD is often defined differently, as follows: {{:Definition:Greatest Common Divisor/Integers/Definition 1}} === Polynomial Ring over Field === {{:Definition:Greatest Common Divisor/Polynomial Ring over Field}}"}
+{"_id": "20481", "title": "Definition:Prime-Counting Function", "text": "The '''prime-counting function''' is the function $\\pi: \\R \\to \\Z$ which counts the number of primes less than or equal to some real number. That is: :$\\displaystyle \\forall x \\in \\R: \\map \\pi x = \\sum_{\\substack {p \\mathop \\in \\mathbb P \\\\ p \\mathop \\le x} } 1$ where $\\mathbb P$ denotes the set of prime numbers."}
+{"_id": "20482", "title": "Definition:Primorial", "text": "There are two definitions for '''primorials''': one for primes and one for positive integers. === Definition for Primes === {{:Definition:Primorial/Prime}} === Definition for Positive Integers === {{:Definition:Primorial/Positive Integer}}"}
+{"_id": "20483", "title": "Definition:Euclid Number", "text": "A '''Euclid number''' is a natural number of the form: :$E_n := p_n\\# + 1$ where $p_n\\#$ is the primorial of the $n$th prime number."}
+{"_id": "20484", "title": "Definition:O Notation", "text": "'''$\\OO$ notation''' is a type of order notation, typically used in computer science for comparing 'run-times' of algorithms, or in analysis for comparing growth rates between two growth functions."}
+{"_id": "20485", "title": "Definition:Residue (Complex Analysis)", "text": "Let $f: \\C \\to \\C$ be a complex function. Let $z_0 \\in U \\subset \\C$ such that $f$ is analytic in $U \\setminus \\left\\{{z_0}\\right\\}$. Then by Existence of Laurent Series, there is a Laurent series: :$\\displaystyle \\sum_{j \\mathop = -\\infty}^\\infty a_j \\left({z - z_0}\\right)^j$ such that the sum converges to $f$ in $U - \\left\\{{z_0}\\right\\}$.   The '''residue''' at a point $z = z_0$ of a function $f: \\C \\to \\C$ is defined as $a_{-1}$ in that Laurent series. It is denoted $\\operatorname{Res} \\left({f, z_0}\\right)$ or just $\\operatorname{Res}\\left({z_0}\\right)$ when $f$ is understood. Category:Definitions/Complex Analysis 9wpg3ebiqofw56oz9g2cyuu4velr859"}
+{"_id": "20486", "title": "Definition:Laurent Series", "text": "Let $f: \\C \\to \\C$ be a complex function. Let $z_0 \\in U \\subset \\C$ such that $f$ is analytic in $U \\setminus \\left\\{{z_0}\\right\\}$. A '''Laurent series''' is a sum: :$\\displaystyle \\sum_{j \\mathop = -\\infty}^\\infty a_j \\left({z - z_0}\\right)^j$ such that the sum converges to $f$ in $U \\setminus \\left\\{{z_0}\\right\\}$. {{NamedforDef|Pierre Alphonse Laurent|cat = Laurent}}"}
+{"_id": "20487", "title": "Definition:Residue Class", "text": "Let $m \\in \\Z$. Let $\\mathcal R_m$ be the congruence relation modulo $m$ on the set of all $a, b \\in \\Z$: :$\\mathcal R_m = \\set {\\tuple {a, b} \\in \\Z \\times \\Z: \\exists k \\in \\Z: a = b + k m}$ We have that congruence modulo $m$ is an equivalence relation. So for any $m \\in \\Z$, we denote the equivalence class of any $a \\in \\Z$ by $\\eqclass a m$, such that: {{begin-eqn}} {{eqn | l = \\eqclass a m       | r = \\set {x \\in \\Z: a \\equiv x \\pmod m}       | c =  }} {{eqn | r = \\set {x \\in \\Z: \\exists k \\in \\Z: x = a + k m}       | c =  }} {{eqn | r = \\set {\\ldots, a - 2 m, a - m, a, a + m, a + 2 m, \\ldots}       | c =  }} {{end-eqn}} The equivalence class $\\eqclass a m$ is called the '''residue class of $a$ (modulo $m$)'''."}
+{"_id": "20488", "title": "Definition:Integral Polynomial", "text": "An '''integral polynomial''' is a polynomial over the ring of integers $\\Z$. Category:Definitions/Polynomial Theory 9r6gum4pxtxzllljm37cc6hw64ps770"}
+{"_id": "20489", "title": "Definition:Polynomial Congruence", "text": "Let $P \\left({x}\\right)$ be an integral polynomial. Then the expression: :$P \\left({x}\\right) \\equiv 0 \\pmod n$ is known as a '''polynomial congruence'''. === Linear Congruence === {{:Definition:Linear Congruence}}"}
+{"_id": "20490", "title": "Definition:Linear Congruence", "text": "A '''linear congruence''' is a polynomial congruence of the form: :$a_0 + a_1 x \\equiv 0 \\pmod n$ That is, one where the degree of the integral polynomial is $1$. Such a congruence is frequently encountered in the equivalent form: :$a x \\equiv b \\pmod n$"}
+{"_id": "20491", "title": "Definition:Simultaneous Congruences", "text": "A '''system of simultaneous congruences''' is a set of polynomial congruences: :$\\forall i \\in \\left[{1 \\,.\\,.\\, r}\\right]: P_i \\left({x}\\right) \\equiv 0 \\pmod {n_i}$ That is: {{begin-eqn}} {{eqn | l = P_1 \\left({x}\\right)       | r = 0       | rr= \\pmod {n_1}       | c =  }} {{eqn | l = P_2 \\left({x}\\right)       | r = 0       | rr= \\pmod {n_2}       | c =  }} {{eqn | o = \\cdots       | c =  }} {{eqn | l = P_r \\left({x}\\right)       | r = 0       | rr= \\pmod {n_r}       | c =  }} {{end-eqn}}"}
+{"_id": "20492", "title": "Definition:Jacobi Symbol", "text": "Let $m \\in \\Z$ be any integer and $n \\in \\Z$ be any odd integer such that $n \\ge 3$. Let the prime decomposition of $n$ be: : $\\displaystyle n = \\prod_{i \\mathop = 1}^r p_i^{k_i}$. Then the '''Jacobi symbol''' $\\left({\\dfrac m n}\\right)$ is defined as: :$\\displaystyle \\left({\\frac m n}\\right) = \\prod_{i \\mathop = 1}^r \\left({\\frac m {p_i}}\\right)^{k_i}$ where $\\left({\\dfrac m {p_i}}\\right)$ is defined as the Legendre symbol."}
+{"_id": "20493", "title": "Definition:Continued Fraction", "text": "{{:Definition:Continued Fraction/Finite}}"}
+{"_id": "20494", "title": "Definition:Exterior Derivative", "text": "Let an exact $n$-form $\\omega$ be given on an $m$-manifold, with local coordinates $x_1, x_2, \\dots, x_m$. Let a local coordinate expression for $\\omega$ be given: :$\\omega = \\map f {x_1, \\ldots, x_m} \\rd x_{\\map \\phi 1} \\wedge \\d x_{\\map \\phi 2} \\wedge \\cdots \\wedge \\d x_{\\map \\phi n}$ where: :$\\phi: \\set {1, \\ldots, n} \\to \\set {1, \\ldots, m}$ is an injection which determines which coordinate vectors $\\omega$ acts on. :$\\wedge$ denotes the wedge product. The '''exterior derivative''' $\\d \\omega$ is the $\\paren {n + 1}$-form defined as: :$\\displaystyle \\d \\omega = \\paren {\\sum_{k \\mathop = 1}^m \\frac {\\partial f} {\\partial x_k} \\rd x_k} \\wedge \\d x_{\\map \\phi 1} \\wedge \\d x_{\\map \\phi 2} \\wedge \\dots \\wedge \\d x_{\\map \\phi n}$ For inexact forms: :$\\map \\d {a + b} = \\d a + \\d b$"}
+{"_id": "20495", "title": "Definition:Wedge Product", "text": "Let $\\alpha$ and $\\beta$ be two differential forms. Let $\\alpha$ be an $x$-form. Let $\\beta$ be a $y$-form. The '''wedge product''' $\\alpha \\wedge \\beta$ is defined as the linear antisymmetric map from $F^x \\times F^y \\to F^{x + y}$, where $F^a$ is the set of $a$-forms in some manifold. Let $x_0$ be a specific point in a manifold $X$. Let $\\alpha$ be an $x$-form. Let $\\phi$ be a $1$-form. Let there be a set of vectors $\\mathbf u_1, \\mathbf u_2, \\dotsc, \\mathbf u_x, \\mathbf v \\in \\map {T_{x_0} } X$. The '''wedge product''' is defined as: :$\\alpha \\wedge \\map \\phi {\\mathbf u_1, \\mathbf u_2, \\dotsc, \\mathbf u_x, \\mathbf v} := \\sum_P \\map \\varepsilon P \\, \\map \\alpha {P_1} \\, \\map \\phi {P_2}$ where: :$P$ is some permutation of $\\mathbf u_1, \\mathbf u_2, \\dots, \\mathbf u_x, \\mathbf v$ :$P_1$ is the first $x$ terms of the permutation $P$ :$P_2$ the final term of permutation $P$ :$\\varepsilon$ is the permutation symbol of $P$. The sum is taken over all possible permutations. This definition extends to wedge products of arbitrary forms through the linearity and antisymmetric conditions. Category:Definitions/Topology 0iso3pdnb5hi2zud1lsrw02j5zjl4la"}
+{"_id": "20496", "title": "Definition:Permutation Symbol", "text": "The '''permutation symbol''' $\\varepsilon$ of a permutation $P$ of a set of elements is defined as: :$+1$ for even permutations (permutations that are an even number of pair swaps) :$-1$ for odd permutations :$0$ if the list of elements is not a permutation (that is, contains a repeated value). Frequently, the permutation will be explicit, for example: :$\\varepsilon_{i j k \\ldots}$ :$\\varepsilon^{i j \\ldots}_{k l \\ldots}$ :$\\varepsilon^{i j k \\ldots}$ This notation is especially useful when raising and lowering indices (that is, converting between forms and vectors)."}
+{"_id": "20497", "title": "Definition:Local Coordinates", "text": "Let $X$ be an $n$-dimensional manifold. Let $p \\in X$, and let $U \\subset X$ be a neighbourhood of $p$. Then a set of mappings $x_i: U \\to \\R$, $1 \\le i \\le n$, satisfying: :$a = b \\iff \\forall i: \\map {x_i} a = \\map {x_i} b$ is called a set of '''local coordinates'''. When the neighbourhood $U$ is to be stressed, one may also say '''local coordinates for $U$'''. Similarly, when the element $p$ is to be stressed, one may also say '''local coordinates around $p$'''."}
+{"_id": "20498", "title": "Definition:Partial Derivative", "text": "=== Real Analysis === {{:Definition:Partial Derivative/Real Analysis}} === Complex Analysis === {{:Definition:Partial Derivative/Complex Analysis}}"}
+{"_id": "20499", "title": "Definition:Riemann Surface", "text": "A '''Riemann surface''' is a connected complex manifold of dimension $1$."}
+{"_id": "20500", "title": "Definition:Riemann Sphere", "text": "Let $f_1: \\C \\to \\R^2$ be defined as: :$\\forall z \\in \\C: \\map {f_1} z = \\tuple {\\map \\Re z, \\map \\Im z}$ Let $f_2: \\R^2 \\to \\R^3$ be the inclusion map: :$\\forall \\tuple {a, b} \\in \\C^2: \\map {f_2} {a, b} = \\tuple {a, b, 0}$ Let $f = f_2 \\circ f_1$. Let $F: \\C \\to \\map {\\PP} {\\R^3}$ be defined as the mapping which takes $z$ to the closed line interval from $\\tuple {0, 0, 1}$ to $\\map f z$ for all $z \\in \\C$. Let $G = \\set {x, y, z: x^2 + y^2 + z^2 = 1}$. Then the Riemann map $R: \\C \\to \\mathbb S^2$ is defined as: :$\\map R x = \\map F z \\cap G$ The set $R \\sqbrk \\C \\cup \\set {\\tuple {0, 0, 1} } $ is called the '''Riemann sphere''', with the understanding that $\\map f \\infty = \\tuple {0, 0, 1}$. {{expand|According to the definition in Clapham & Nicholson, include the definition as the extended complex plane under a stereographic projection}}"}
+{"_id": "20501", "title": "Definition:Measure (Measure Theory)", "text": "Let $\\struct {X, \\Sigma}$ be a measurable space. Let $\\mu: \\Sigma \\to \\overline \\R$ be a mapping, where $\\overline \\R$ denotes the set of extended real numbers. Then $\\mu$ is called a '''measure''' on $\\Sigma$ {{iff}} $\\mu$ has the following properties: {{begin-axiom}} {{axiom | n = 1         | q = \\forall E \\in \\Sigma         | ml= \\map \\mu E         | mo= \\ge         | mr= 0 }} {{axiom | n = 2         | q = \\forall \\sequence {S_n}_{n \\mathop \\in \\N} \\subseteq \\Sigma: \\forall i, j \\in \\N: S_i \\cap S_j = \\O         | ml= \\map \\mu {\\bigcup_{n \\mathop = 1}^\\infty S_n}         | mo= =         | mr= \\sum_{n \\mathop = 1}^\\infty \\map \\mu {S_n}         | rc= that is, $\\mu$ is a countably additive function }} {{axiom | n = 3         | q = \\exists E \\in \\Sigma         | ml= \\map \\mu E         | mo= \\in         | mr= \\R         | rc= that is, there exists at least one $E \\in \\Sigma$ such that $\\map \\mu E$ is finite }} {{end-axiom}}"}
+{"_id": "20502", "title": "Definition:Periodic Continued Fraction", "text": "Let $\\left[{a_1, a_2, a_3, \\ldots}\\right]$ be a simple infinite continued fraction. Let the partial quotients be of the form: :$\\left[{r_1, r_2, \\ldots, r_m, s_1, s_2, \\ldots, s_n, s_1, s_2, \\ldots, s_n, s_1, s_2, \\ldots, s_n, \\ldots}\\right]$ that is, ending in a block of partial quotients which repeats itself indefinitely. Such a SICF is known as a '''periodic continued fraction'''. The notation used for this is $\\left[{r_1, r_2, \\ldots, r_m, \\left \\langle{s_1, s_2, \\ldots, s_n}\\right \\rangle}\\right]$, where the repeating block is placed in angle brackets. === Purely Periodic Continued Fraction === {{:Definition:Periodic Continued Fraction/Purely Periodic}} === Cycle === {{:Definition:Periodic Continued Fraction/Cycle}} Category:Definitions/Continued Fractions d8e3wu8mzkg2wgroy2qfbtovdmhlsus"}
+{"_id": "20503", "title": "Definition:Fractional Part", "text": "Let $x \\in \\R$ be a real number. Let $\\floor x$ be the floor function of $x$. The '''fractional part''' of $x$ is the difference: :$\\fractpart x := x - \\floor x$"}
+{"_id": "20504", "title": "Definition:Analytic Continuation", "text": "Let $U \\subset \\C$ be an open set. Let $f: U \\to \\C$ be an analytic function. Let $V$ be an open subset of $\\C$ such that $U \\subset V$. An '''analytic continuation of $f$ to $V$''' is an analytic function $F: V \\to \\C$ such that $\\map F z = \\map f z$ for $z \\in U$."}
+{"_id": "20505", "title": "Definition:Gamma Function", "text": "=== Integral Form === {{:Definition:Gamma Function/Integral Form}} === Weierstrass Form === {{:Definition:Gamma Function/Weierstrass Form}} === Hankel Form === {{:Definition:Gamma Function/Hankel Form}} === Euler Form === {{:Definition:Gamma Function/Euler Form}}"}
+{"_id": "20506", "title": "Definition:Absolute Convergence of Product", "text": "=== Complex Numbers === Let $\\sequence {a_n}$ be a sequence in $\\C$. {{:Definition:Absolute Convergence of Product/Complex Numbers/Definition 1}} === General Definition === Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,} }$ be a valued field. Let $\\sequence {a_n}$ be a sequence in $\\mathbb K$. {{:Definition:Absolute Convergence of Product/General Definition/Definition 1}}"}
+{"_id": "20507", "title": "Definition:Fourier Series", "text": "=== Formulation 1 === {{:Definition:Fourier Series/Formulation 1}} === Formulation 2 === {{:Definition:Fourier Series/Formulation 2}}"}
+{"_id": "20508", "title": "Definition:Quadratic Irrational", "text": "A '''quadratic irrational''' is an irrational number of the form: :$r + s \\sqrt n$ where $r, s$ are rational and $n$ is a positive integer which is not a square."}
+{"_id": "20509", "title": "Definition:Pythagorean Triple", "text": "A '''Pythagorean triple''' is an ordered triple of positive integers $\\tuple {x, y, z}$ such that $x^2 + y^2 = z^2$. That is, a '''Pythagorean triple''' is a solution to the Pythagorean equation."}
+{"_id": "20510", "title": "Definition:Projective Space", "text": "=== Real Projective Space === {{:Definition:Projective Space/Real Projective Space}} === Over a Field === {{:Definition:Projective Space/Over a Field}} Category:Definitions/Topology Category:Definitions/Projective Geometry t7mbpaqdvrj4t5lg8ftdxiyrain9yiz"}
+{"_id": "20511", "title": "Definition:Dirichlet Series", "text": "Let $a_n: \\N \\to \\C$ be an arithmetic function. Its '''Dirichlet series''' is a complex function $f: \\C \\to \\C$ defined by a series: :$\\displaystyle \\map f s = \\sum_{n \\mathop = 1}^\\infty a_n n^{-s}$"}
+{"_id": "20512", "title": "Definition:Pythagorean Equation", "text": "The '''Pythagorean equation''' is the Diophantine equation: :$x^2 + y^2 = z^2$ where $x, y, z$ are integers such that $x, y, z > 0$. Solutions of this equation are known as Pythagorean triples. If, in addition, $\\left({x, y, z}\\right)$ is a primitive Pythagorean triple, then $\\left({x, y, z}\\right)$ is known as a '''primitive solution''' of $x^2 + y^2 = z^2$."}
+{"_id": "20514", "title": "Definition:Triangular Number", "text": "'''Triangular numbers''' are those denumerating a collection of objects which can be arranged in the form of an equilateral triangle."}
+{"_id": "20515", "title": "Definition:Geometric Sequence", "text": "A '''geometric sequence''' is a sequence $\\sequence {x_n}$ in $\\R$ defined as: :$x_n = a r^n$ for $n = 0, 1, 2, 3, \\ldots$ Thus its general form is: :$a, ar, ar^2, ar^3, \\ldots$ and the general term can be defined recursively as: :$x_n = \\begin{cases} a & : n = 0 \\\\ r x_n & : n > 0 \\\\ \\end{cases}$"}
+{"_id": "20516", "title": "Definition:Square Number", "text": "'''Square numbers''' are those denumerating a collection of objects which can be arranged in the form of a square. They can be denoted: :$S_1, S_2, S_3, \\ldots$ === Definition 1 === {{:Definition:Square Number/Definition 1}} === Definition 2 === {{:Definition:Square Number/Definition 2}} === Definition 3 === {{:Definition:Square Number/Definition 3}} === Definition 4 === {{:Definition:Square Number/Definition 4}}"}
+{"_id": "20517", "title": "Definition:Arithmetic Sequence", "text": "An '''arithmetic sequence''' is a finite sequence $\\sequence {a_k}$ in $\\R$ or $\\C$ defined as: :$a_k = a_0 + k d$ for $k = 0, 1, 2, \\ldots, n - 1$ Thus its general form is: :$a_0, a_0 + d, a_0 + 2 d, a_0 + 3 d, \\ldots, a_0 + \\paren {n - 1} d$"}
+{"_id": "20518", "title": "Definition:Polygonal Number", "text": "'''Polygonal numbers''' are those denumerating a collection of objects which can be arranged in the form of an regular polygon."}
+{"_id": "20519", "title": "Definition:Pentagonal Number", "text": "'''Pentagonal numbers''' are those denumerating a collection of objects which can be arranged in the form of a regular pentagon."}
+{"_id": "20520", "title": "Definition:Hexagonal Number", "text": "'''Hexagonal numbers''' are those denumerating a collection of objects which can be arranged in the form of a regular hexagon."}
+{"_id": "20521", "title": "Definition:Pyramidal Number", "text": "The '''pyramidal numbers''' are integers defined as sums of sequences of polygonal numbers as follows. Let $P \\left({k, n}\\right)$ be the $n$th $k$-gonal number. Then the '''$n$th $k$gonal-based pyramidal number''' $Y \\left({k, n}\\right)$ is defined as: :$\\displaystyle Y \\left({k, n}\\right) = \\sum_{i \\mathop = 1}^n P \\left({k, i}\\right)$"}
+{"_id": "20522", "title": "Definition:Set of Residue Classes/Least Absolute", "text": "Let $\\eqclass a m$ be the residue class of $a$ (modulo $m$). Except when $r = \\dfrac m 2$, we can choose $r$ to be the integer in $\\eqclass a m$ which has the smallest absolute value. In that exceptional case we have: : $-\\dfrac m 2 + m = \\dfrac m 2$ and so: : $-\\dfrac m 2 \\equiv \\dfrac m 2 \\pmod m$ Thus $r$ is defined as the '''least absolute residue''' of $a$ (modulo $m$) {{iff}}: : $-\\dfrac m 2 < r \\le \\dfrac m 2$"}
+{"_id": "20523", "title": "Definition:Balanced Incomplete Block Design", "text": "A '''Balanced Incomplete Block Design''' or '''BIBD''' with parameters $v, b, r, k, \\lambda$ is a block design where: * $v$ is the number of points in the design * $b$ is the number of blocks * $k$ is the size of each block * $r$ is the number of blocks any point can be in * $\\lambda$ is the number of times any two points can occur in the same block and has the following properties: * Each block is of size $k$ * All of the $\\displaystyle \\binom v 2$ pairs occur together in exactly $\\lambda$ blocks. A BIBD with parameters $v, b, r, k, \\lambda$ is commonly written several ways, for example: * $\\operatorname{BIBD} \\left({v, k, \\lambda}\\right)$ * $\\left ({v, k, \\lambda}\\right)$-$\\operatorname{BIBD}$"}
+{"_id": "20524", "title": "Definition:Odd Prime", "text": "Every even integer is divisible by $2$ (because this is the definition of '''even'''). Therefore, apart from $2$ itself, all primes are odd. So, referring to an '''odd prime''' is a convenient way of specifying that a number is a prime number, but not equal to $2$."}
+{"_id": "20525", "title": "Definition:Combination", "text": "Let $S$ be a set containing $n$ elements. An '''$r$-combination of $S$''' is a subset of $S$ which has $r$ elements."}
+{"_id": "20526", "title": "Definition:Generating Function", "text": "Let $A = \\left \\langle {a_n}\\right \\rangle$ be a sequence in $\\R$. Then $\\displaystyle G_A \\left({z}\\right) = \\sum_{n \\mathop \\ge 0} a_n z^n$ is called the '''generating function''' for the sequence $A$."}
+{"_id": "20527", "title": "Definition:Fermat Pseudoprime", "text": "Let $q$ be a composite number such that $\\exists n \\in N: n^q \\equiv n \\pmod q$. Then $q$ is a '''Fermat pseudoprime to base $n$'''."}
+{"_id": "20529", "title": "Definition:Algorithm", "text": "An '''algorithm''' is a finite set of '''instructions''' (or '''rules''') that defines a sequence of '''operations''' for solving a particular '''computational problem''' for all '''problem instances''' for some '''problem set'''."}
+{"_id": "20530", "title": "Definition:Algorism", "text": "'''Algorism''' is an archaic term which means '''the process of doing arithmetic using Arabic numerals'''. === Algorist === {{:Definition:Algorism/Algorist}}"}
+{"_id": "20531", "title": "Definition:Abacus", "text": "An '''abacus''' (plural: '''abacuses''' or '''abaci''') is a tool for performing arithmetical calculations. It consists of: : a series of lines (for example: grooves in sand, or wires on a frame), upon which are: : a number of items (for example: pebbles in the grooves, or beads on the wires), which are manipulated by hand so as to represent numbers. As such, it is the earliest known machine for mathematics, and can be regarded as the earliest ancestor of the electronic computer."}
+{"_id": "20532", "title": "Definition:Abacism", "text": "'''Abacism''' means '''the process of doing arithmetic using an abacus'''. === Abacist === {{:Definition:Abacism/Abacist}}"}
+{"_id": "20533", "title": "Definition:Arabic Numerals", "text": "The '''Arabic numerals''' are: {{begin-eqn}} {{eqn | l = 1:       | o = \\)'''one'''\\(:       | r = \\circ       | c =  }} {{eqn | l = 2:       | o = \\)'''two'''\\(:       | r = \\circ \\, \\circ       | c =  }} {{eqn | l = 3:       | o = \\)'''three'''\\(:       | r = \\circ \\circ \\circ       | c =  }} {{eqn | l = 4:       | o = \\)'''four'''\\(:       | r = \\circ \\circ \\circ \\, \\circ       | c =  }} {{eqn | l = 5:       | o = \\)'''five'''\\(:       | r = \\circ \\circ \\circ \\circ \\circ       | c =  }} {{eqn | l = 6:       | o = \\)'''six'''\\(:       | r = \\circ \\circ \\circ \\circ \\circ \\, \\circ       | c =  }} {{eqn | l = 7:       | o = \\)'''seven'''\\(:       | r = \\circ \\circ \\circ \\circ \\circ \\circ \\circ       | c =  }} {{eqn | l = 8:       | o = \\)'''eight'''\\(:       | r = \\circ \\circ \\circ \\circ \\circ \\circ \\circ \\, \\circ       | c =  }} {{eqn | l = 9:       | o = \\)'''nine'''\\(:       | r = \\circ \\circ \\circ \\circ \\circ \\circ \\circ \\circ \\circ       | c =  }} {{eqn | l = 0:       | o = \\)'''zero'''\\(:       | r = \\)or '''nought''', '''naught''', '''nil''', '''nothing''', '''cypher''', '''zilch''', '''kabutnik''', '''sweet Fanny Adams''', etc.\\(       | c =  }} {{end-eqn}} They are used in conjunction with the positional decimal system of numeric representation: see Basis Representation Theorem."}
+{"_id": "20534", "title": "Definition:Roman Numerals", "text": "The '''Roman numerals''' are: {{begin-eqn}} {{eqn | l = \\mathrm I:       | o = \\)'''unus'''\\(:       | r = 1 }} {{eqn | l = \\mathrm V:       | o = \\)'''quinque'''\\(:       | r = 5 }} {{eqn | l = \\mathrm X:       | o = \\)'''decem'''\\(:       | r = 10 }} {{eqn | l = \\mathrm L:       | o = \\)'''quinquaginta'''\\(:       | r = 50 }} {{eqn | l = \\mathrm C:       | o = \\)'''centum'''\\(:       | r = 100 }} {{eqn | l = \\mathrm D:       | o = \\)'''quingenti'''\\(:       | r = 500 }} {{eqn | l = \\mathrm M:       | o = \\)'''mille'''\\(:       | r = 1000 }} {{end-eqn}}"}
+{"_id": "20535", "title": "Definition:Unlimited Register Machine", "text": "An '''unlimited register machine''', abbreviated URM, is an abstraction (or idealization) of a computing device with the following characteristics: === Registers === A '''URM''' has a number of locations called '''registers''' which can store natural numbers: $\\set {0, 1, 2, \\ldots}$. Any given URM program may make use of only a finite number of registers. Registers are usually referred to by the subscripted uppercase letters $R_1, R_2, R_3, \\ldots$. The subscript (which is a natural number) is called the '''index''' of the register. The number held at any one time by a register is usually referred to by the corresponding lowercase letter $r_1, r_2, r_3, \\ldots$. The registers are '''unlimited''' in the following two senses: :$(1): \\quad$ Although a URM program may make use of only a finite number of registers, there is no actual upper bound on how many a particular program ''can'' actually use. :$(2): \\quad$ There is no upper bound on the size of the natural numbers that may be stored in any register. === Program === The numbers held in the registers of a URM are manipulated according to a '''program'''. A '''URM program''' is a finite list of basic instructions. The instructions are written in a fixed order and numbered $1, 2, 3, \\ldots$. For historical reasons, the number of the instruction is called its '''line''' number. We can refer either to the '''line of the program''' or the '''line in the URM'''. It is convenient to use $\\Bbb U$ to stand for the set of all URM programs. ==== Length of Program ==== Let $P \\in \\Bbb U$ be a URM program. By definition, $P$ is a finite list of basic instructions. We define the function $\\lambda: \\Bbb U \\to \\N$ as follows: :$\\forall P \\in \\Bbb U: \\map \\lambda P = \\text { the number of basic instructions that comprise } P$. ==== Number of Registers Used ==== Let $P \\in \\Bbb U$ be a URM program. By definition, $P$ uses a finite number of registers. We define the function $\\rho: \\Bbb U \\to \\N$ as follows: :$\\forall P \\in \\Bbb U: \\map \\rho P = \\text{ the highest register number used by } P$. So in any URM program $P$, no instruction refers to any register with index greater than $R_u$, where $u = \\map \\rho P$. === Basic Instructions === {| border=\"1\" |- ! align=\"left\" | Name  ! align=\"left\" | Notation ! align=\"left\" | Effect ! align=\"left\" | Description |- | <tt>Zero</tt>  | $\\map Z n$ | $0 \\to R_n$ | Replace the number in $R_n$ by $0$. |- | <tt>Successor</tt>  | $\\map S n$ | $r_n + 1 \\to R_n$ | Add $1$ to the number in $R_n$. |- | <tt>Copy</tt>  | $\\map C {m, n}$ | $r_m \\to R_n$ | Replace the number in $R_n$ by the number in $R_m$ (leaving the one in $R_m$ as it was). |- | <tt>Jump</tt>  | $\\map J {m, n, q}$ | $r_m = r_n ? \\Rightarrow q$ | If the numbers in $R_m$ and $R_n$ are equal, go to instruction number $q$, otherwise go to the next instruction. |} Basic instructions are also (and more commonly) known as '''commands''', because the word's shorter and quicker to say. === Operation === When a URM runs a program, it always starts by executing the first instruction of the program. When it has carried out an instruction, it moves to the next instruction and executes that one, unless required otherwise by a <tt>Jump</tt> instruction. ==== Instruction Pointer ==== The line number which is currently about to be executed is known as the '''instruction pointer'''. It can be imagined as a special-purpose register in the URM that holds the line number. ==== Stage of Computation ==== The '''stage of computation''' (or just '''stage''') of a URM program is the count of how many basic instructions have been carried out. Thus each stage corresponds to the processing of one instruction. ==== State ==== The '''state''' (or '''situation''') of a URM program at a particular point in time is defined as: :the value of the instruction pointer :the value, at that point, of each of the registers that are used by the program. === Termination === A URM program '''stops''', or '''terminates''', or '''halts''', when there are no more instructions to carry out. This can happen in either of two ways: :$(1): \\quad$ If the program carries out the last instruction, and this does not involve a <tt>Jump</tt> to an earlier instruction, the program will stop. :$(2): \\quad$ If the program carries out a <tt>Jump</tt> instruction to a non-existent instruction, the program will stop. Such a <tt>Jump</tt> instruction is known as an '''exit jump''' . The line on which a particular run of a URM program stops is called the '''exit line'''. If the program, when running, never reaches such a state, then it is said to be in an '''endless loop''' and will '''never terminate'''. Note that whether a program terminates or not may depend on its input. It may terminate perfectly well for one input, but go into an endless loop on another. === Input === The '''input''' to a URM program is: :either an ordered $k$-tuple $\\tuple {n_1, n_2, \\ldots, n_k} \\in \\N^k$ :or a natural number $n \\in \\N$. In the latter case, it is convenient to consider a single natural number as an ordered $1$-tuple $\\tuple {n_1} \\in \\N^1 = \\N$. Hence we can discuss inputs to URM programs solely as instances of tuples, and not be concerned with cumbersome repetition for the cases where $k = 1$ and otherwise. The convention usually used is for a URM program $P$ to start computation with: :the input $\\left({n_1, n_2, \\ldots, n_k}\\right)$ in registers $R_1, R_2, \\ldots, R_k$ :$0$ in all other registers used by $P$. That is, the initial state of the URM is: :$\\forall i \\in \\closedint 1 k: r_i = n_i$ :$\\forall i > k: r_i = 0$. It is usual for the input (either all or part) to be overwritten during the course of the operation of a program. That is, at the end of a program, $R_1, R_2, \\ldots, R_k$ are not guaranteed still to contain $n_1, n_2, \\ldots, n_k$ unless the program has been explicitly written so as to ensure that this is the case. === Output === At the end of the running of a URM program, the '''output''' will be found in register $R_1$. === Null Program === A '''null program''' or '''empty program''' is a URM program which contains no instructions."}
+{"_id": "20536", "title": "Definition:URM Computability", "text": "Let $P$ be a URM program, and let $k$ be any positive integer. === Program === $P$ is said to '''compute the function $f: \\N^k \\to \\N$''' {{iff}}: :for all ordered $k$-tuples $\\tuple {n_1, n_2, \\ldots, n_k} \\in \\N^k$, the computation of a URM using the program $P$ with input $\\tuple {n_1, n_2, \\ldots, n_k}$ produces the output $\\map f {n_1, n_2, \\ldots, n_k}$. If there are any inputs such that either of the following happens: :the output fails to equal $\\map f {n_1, n_2, \\ldots, n_k}$ :the program will never terminate, then the program does ''not'' compute the function $f: \\N^k \\to \\N$. === Function === The function $f: \\N^k \\to \\N$ is said to be '''URM computable''' if there exists a URM program which computes it. ==== Partial Function ==== $P$ is said to '''compute the partial function $f: \\N^k \\to \\N$''' {{iff}}: :For all ordered $k$-tuples $\\tuple {n_1, n_2, \\ldots, n_k} \\in \\N^k$: ::If the computation of $P$ with input $\\tuple {n_1, n_2, \\ldots, n_k}$ '''halts''', it produces the output $\\map f {n_1, n_2, \\ldots, n_k}$. ::If the computation of $P$ with input $\\tuple {n_1, n_2, \\ldots, n_k}$ '''does not halt''', $\\map f {n_1, n_2, \\ldots, n_k}$ is undefined. The partial function $f: \\N^k \\to \\N$ is said to be '''URM computable''' if there exists a URM program which computes it. Note that a URM program can be used with any number of input variables. For any positive integer $k$, the input consists of the state of the registers $R_1, R_2, \\ldots, R_k$. Thus a given URM program $P$ computes a partial function $f: \\N^k \\to \\N$ for ''each'' positive integer $k$. In this context, it is convenient to use the notation $f^k_P$ to denote the partial function of $k$ variables computed by $P$. === Set === Let $A \\subseteq \\N$. Then $A$ is a '''URM computable set''' {{iff}} its characteristic function $\\chi_A$ is a URM computable function. === Relation === Let $\\mathcal R \\subseteq \\N^k$ be an $n$-ary relation on $\\N^k$. Then $\\mathcal R$ is a '''URM computable relation''' {{iff}} its characteristic function $\\chi_\\mathcal R$ is a URM computable function. Category:Definitions/Mathematical Logic 76rt3tca4y9ub6otxrsed9y89sdhumh"}
+{"_id": "20537", "title": "Definition:Indeterminate Variable", "text": "An '''indeterminate variable''' is a variable whose  value has not been specified. Informally, it \"could be anything\". Category:Definitions/Algebra 8l2sa3neea2rgq1qtpuzgbspyqe9oea"}
+{"_id": "20538", "title": "Definition:Turing Machine", "text": "A '''Turing machine''' is an idealization of a computing machine. The idea goes as follows. A '''Turing machine''' works by manipulating symbols on an imaginary piece of paper by means of a specific set of algorithmic rules."}
+{"_id": "20539", "title": "Definition:Substitution (Mathematical Logic)", "text": "=== Mapping === Let $S$ be a set. Let $f: S^t \\to S$ be a mapping. Let $\\left\\{{g_1: S^k \\to S, g_2: S^k \\to S, \\ldots, g_t: S^k \\to S}\\right\\}$ be a set of mappings. Let the mapping $h: S^k \\to S$ be defined as: :$h \\left({s_1, s_2, \\ldots, s_k}\\right) = f \\left({g_1 \\left({s_1, s_2, \\ldots, s_k}\\right), g_2 \\left({s_1, s_2, \\ldots, s_k}\\right), \\ldots, g_t \\left({s_1, s_2, \\ldots, s_k}\\right)}\\right)$ Then $h$ is said to be '''obtained from $f, g_1, g_2, \\ldots, g_k$ by substitution'''. The definition can be generalized in the following ways: * It can apply to mappings which operate on variously different sets. * Each of $g_1, g_2, \\ldots, g_t$ may have different arities. If $g$ is a mapping of $m$ variables where $m > k$, we can always consider it a mapping of $k$ variables in which the additional variables play no part. So if $g_i$ is a mapping of $k_i$ variables, we can take $k = \\max \\left\\{{k_i: i = 1, 2, \\ldots, t}\\right\\}$ and then each $g_i$ is then a mapping of $k$ variables. === Partial Function === Let $f: \\N^t \\to \\N$ be a partial function. Let $\\left\\{{g_1: \\N^k \\to \\N, g_2: \\N^k \\to \\N, \\ldots, g_t: \\N^k \\to \\N}\\right\\}$ be a set of partial functions. Let the partial function $h: \\N^k \\to \\N$ be defined as: :$h \\left({n_1, n_2, \\ldots, n_k}\\right) \\approx f \\left({g_1 \\left({n_1, n_2, \\ldots, n_k}\\right), g_2 \\left({n_1, n_2, \\ldots, n_k}\\right), \\ldots, g_t \\left({n_1, n_2, \\ldots, n_k}\\right)}\\right)$ where $\\approx$ is as defined in Partial Function Equality. Then $h$ is said to be '''obtained from $f, g_1, g_2, \\ldots, g_k$ by substitution'''. Note that $h \\left({n_1, n_2, \\ldots, n_k}\\right)$ is defined only when: * All of $g_1 \\left({n_1, n_2, \\ldots, n_k}\\right), g_2 \\left({n_1, n_2, \\ldots, n_k}\\right), \\ldots, g_t \\left({n_1, n_2, \\ldots, n_k}\\right)$ are defined * $f \\left({g_1 \\left({n_1, n_2, \\ldots, n_k}\\right), g_2 \\left({n_1, n_2, \\ldots, n_k}\\right), \\ldots, g_t \\left({n_1, n_2, \\ldots, n_k}\\right)}\\right)$ is defined. {{SUBPAGENAME}} 902scgeuilpski6x8f3xrrjypbjez3o"}
+{"_id": "20540", "title": "Definition:Legendre's Constant", "text": "'''Legendre's constant''' (or the '''Legendre constant''') is a mathematical constant conjectured by {{AuthorRef|Adrien-Marie Legendre}} to specify the prime-counting function $\\map \\pi n$. Legendre conjectured in $1796$ that $\\map \\pi n$ satisfies: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map \\pi n - \\frac n {\\map \\ln n} = B$ where $B$ is '''Legendre's constant'''. If such a number $B$ exists, then this implies the Prime Number Theorem. {{AuthorRef|Adrien-Marie Legendre|Legendre}}'s guess for $B$ was about $1 \\cdotp 08366$. {{OEIS|A228211}} Later, {{AuthorRef|Carl Friedrich Gauss}} looked at this problem and thought that $B$ might actually be lower. In $1896$, {{AuthorRef|Jacques Salomon Hadamard}} and {{AuthorRef|Charles de la Vallée Poussin}} independently proved the Prime Number Theorem and showed that $B$ is in fact equal to $1$. {{AuthorRef|Adrien-Marie Legendre|Legendre}}'s first guess of $1 \\cdotp 08366 \\ldots$ is still (incorrectly) referred to as '''Legendre's constant''', even though its \"correct\" value is in fact exactly $1$. Hence it is only now of historical importance. {{NamedforDef|Adrien-Marie Legendre|cat = Legendre}}"}
+{"_id": "20541", "title": "Definition:Concatenation of URM Programs", "text": "Let $P$ and $Q$ be URM programs. The '''concatenation of $P$ and $Q$''' is denoted $P * Q$. It consists of program $P$ followed immediately by program $Q$. If $P$ and $Q$ compute particular functions, then it is usually necessary to make modifications to either $P$ or $Q$, or both, to make sure that the concatenated program continues to compute a function. The simplest concatenation is illustrated in Composition of One-Variable URM Computable Functions."}
+{"_id": "20542", "title": "Definition:Primitive Recursion", "text": "=== Primitive Recursion on Several Variables === {{:Definition:Primitive Recursion/Several Variables}} === Primitive Recursion on One Variable === {{:Definition:Primitive Recursion/One Variable}} === Primitive Recursion on Partial Functions === {{:Definition:Primitive Recursion/Partial Function}} Category:Definitions/Mathematical Logic nuimqic7c09jqixl4gfoq47l6ub7fur"}
+{"_id": "20543", "title": "Definition:Basic Primitive Recursive Function", "text": "The '''basic primitive recursive functions''' are: === Zero Function === {{:Definition:Basic Primitive Recursive Function/Zero Function}} === Successor Function === {{:Definition:Basic Primitive Recursive Function/Successor Function}} === Projection Function === {{:Definition:Basic Primitive Recursive Function/Projection Function}} === Identity Function === {{:Definition:Basic Primitive Recursive Function/Identity Function}}"}
+{"_id": "20544", "title": "Definition:Primitive Recursive", "text": "=== Function === {{:Definition:Primitive Recursive/Function}} === Set === {{:Definition:Primitive Recursive/Set}} === Relation === {{:Definition:Primitive Recursive/Relation}} Category:Definitions/Mathematical Logic 4ff3875born2il0kz9vdw2xhm4au51x"}
+{"_id": "20545", "title": "Definition:Logarithmic Integral", "text": "The '''logarithmic integral''' is defined as: :$\\displaystyle \\map {\\operatorname {li} } x = \\PV_0^x \\frac {\\d t} {\\map \\ln t}$ where: :$\\ln$ denotes the natural logarithm function :$\\operatorname {PV}$ denotes the Cauchy principal value of the proceeding integral. {{mistake|\"Proceeding\" is wrong here. Suspected malapropism. Recommended a rewrite to make it more accurate.}} That is, as $\\dfrac 1 {\\ln t}$ has discontinuities at $t = 0$ and $t = 1$: :$\\map {\\operatorname {li} } x = \\begin{cases}\\displaystyle \\lim_{\\varepsilon \\to 0^+} \\paren {\\int_{\\varepsilon}^x \\frac {\\rd t} {\\ln t} } & 0 < x < 1 \\\\ \\displaystyle \\lim_{\\varepsilon \\mathop \\to 0^+} \\paren {\\int_\\varepsilon^{1 - \\varepsilon} \\frac {\\rd t} {\\ln t} + \\int_{1 + \\varepsilon}^x \\frac {\\rd t} {\\ln t} } & x > 1\\end{cases}$ === Eulerian Logarithmic Interval === {{:Definition:Logarithmic Integral/Eulerian}}"}
+{"_id": "20546", "title": "Definition:Big-Omega", "text": "'''Big-Omega notation''' is a type of order notation for typically comparing 'run-times' or growth rates between two growth functions. Let $f, g$ be two functions. {{explain|Domain and range of $f$ and $g$ to be specified: reals?}} Then: : $f \\left({n}\\right) \\in \\Omega \\left({g \\left({n}\\right)}\\right)$ iff: : $\\exists c > 0, k \\ge 0: \\forall n > k: f \\left({n}\\right) \\ge c g \\left({n}\\right)$ This is read as: :'''$f \\left({n}\\right)$ is big omega of $g \\left({n}\\right)$'''. {{refactor|Put the below into separate pages, either as a 2nd definition or just an equivalence proof}} Another method of determining the condition is the following limit: :$\\displaystyle \\lim_{n \\to \\infty} {\\frac{f \\left({n}\\right)} {g \\left({n}\\right)}} = c > 0$ where $0 < c \\le \\infty$. If such a $c$ does exist, then: : $f \\left({n}\\right) \\in \\Omega \\left({g \\left({n}\\right)}\\right)$ {{refactor|The below into its own page}} To say that $f \\left({n}\\right) \\in \\Omega \\left({g \\left({n}\\right)}\\right)$ is equivalent to: : $g \\left({n}\\right) \\in \\mathcal O \\left({f \\left({n}\\right)}\\right)$ where $\\mathcal O$ is the big-O notation."}
+{"_id": "20547", "title": "Definition:Big-Theta", "text": "'''Big-Theta notation''' is a type of order notation for typically comparing 'run-times' or growth rates between two growth functions. Big-Theta is a stronger statement than big-O and big-omega. Suppose $f: \\N \\to \\R, g: \\N \\to \\R$ are two functions. Then: :$f \\left({n}\\right) \\in \\Theta \\left({g \\left({n}\\right)}\\right)$  iff: :$(f \\left({n}\\right) \\in O \\left({g \\left({n}\\right)}\\right) \\land (f \\left({n}\\right) \\in \\Omega \\left({g \\left({n}\\right)}\\right)$ where $O \\left({g \\left({n}\\right)}\\right)$ is big-O and $\\Omega \\left({g \\left({n}\\right)}\\right)$ is Big-Omega. This is read as: : $f \\left({n}\\right)$ is '''big-theta''' of $g \\left({n}\\right)$. Another method of determining the condition is the following limit: :$\\displaystyle \\lim_{n \\to \\infty} \\frac{f \\left({n}\\right)}{g \\left({n}\\right)} = c$, where $0 < c < \\infty$ If such a $c$ does exist, then $f \\left({n}\\right) \\in \\Theta (g \\left({n}\\right))$. Category:Definitions/Order Notation gq89r79059rqjllgejs4j80517jrrpx"}
+{"_id": "20548", "title": "Definition:Signum Function", "text": "Let $X \\subseteq \\R$ be a subset of the real numbers. The '''signum function''' $\\sgn: X \\to \\set {-1, 0, 1}$ is defined as: :$\\forall x \\in X: \\map \\sgn x := \\sqbrk {x > 0} - \\sqbrk {x < 0}$ where $\\sqbrk {x > 0}$ etc. denotes Iverson's convention. That is: :$\\forall x \\in X: \\map \\sgn x := \\begin{cases} -1 & : x < 0 \\\\ 0 & : x = 0 \\\\ 1 & : x > 0 \\end{cases}$"}
+{"_id": "20549", "title": "Definition:Hall Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a Hall subgroup of $G$ {{iff}} the index and order of $H$ in $G$ are coprime: :$\\index G H \\perp \\order H$"}
+{"_id": "20550", "title": "Definition:Hall Divisor", "text": "Let $n \\in \\Z$ be an integer. A '''Hall divisor''' of $n$ is a divisor $d$ of $n$ such that $n$ and $\\dfrac n d$ are coprime. {{NamedforDef|Philip Hall|cat=Hall}} Category:Definitions/Number Theory nz9dq89jb25nbrz7jpbr6i1lz2yh7y2"}
+{"_id": "20551", "title": "Definition:Fermat Number", "text": "A '''Fermat number''' is a natural number of the form $2^{\\paren {2^n} } + 1$, where $n = 0, 1, 2, \\ldots$. The number $2^{\\paren {2^n} } + 1$ is, in this context, often denoted $F_n$."}
+{"_id": "20552", "title": "Definition:Bounded Minimization", "text": "=== Function === Let $f: \\N^{k+1} \\to \\N$ be a function. Let $\\left({n_1, n_2, \\ldots, n_k}\\right) \\in \\N^k$ and let $z \\in \\N$ be fixed. Then the '''bounded minimization operation on $f$''' is written as: :$\\mu y \\le z \\left({f \\left({n_1, n_2, \\ldots, n_k, y}\\right) = 0}\\right)$ and is specified as follows: :$\\mu y \\le z \\left({f \\left({n_1, n_2, \\ldots, n_k, y}\\right) = 0}\\right) = \\begin{cases} \\text{the smallest } y \\in \\N \\text{ such that } f \\left({n_1, n_2, \\ldots, n_k, y}\\right) = 0 & : \\exists y \\in \\N: y \\le z \\\\ z + 1 & : \\text{otherwise} \\end{cases}$ === Relation === Let $\\mathcal R \\left({n_1, n_2, \\ldots, n_k, y}\\right) $ be a $k+1$-ary relation on $\\N^{k+1}$. Let $\\left({n_1, n_2, \\ldots, n_k}\\right) \\in \\N^k$ and let $z \\in \\N$ be fixed. Then the '''bounded minimization operation on $\\mathcal R$''' is written as: :$\\mu y \\le z \\mathcal R\\left({n_1, n_2, \\ldots, n_k, y}\\right)$ and is specified as follows: :$\\mu y \\le z \\mathcal R\\left({n_1, n_2, \\ldots, n_k, y}\\right) = \\begin{cases} \\text{the smallest } y \\in \\N \\text{ for which } \\mathcal R \\left({n_1, n_2, \\ldots, n_k, y}\\right) \\text{ holds} & : \\exists y \\in \\N: y \\le z \\\\ z + 1 & : \\text{otherwise} \\end{cases}$ We can consider the definition for a function to be a special case of this. === The no-solution case === The choice of $z + 1$ for the value when there is no solution $y$ less than or equal to $z$ is arbitrary, but convenient. It ensures a well-defined solution for every $z$. {{SUBPAGENAME}} rc6242xnjliugkuy1a959k2vrh36zdn"}
+{"_id": "20553", "title": "Definition:Sequence Coding", "text": "Let $\\left \\langle {a_1, a_2, \\ldots, a_k}\\right \\rangle$ be a finite sequence in $\\N^*$ (that is, $\\forall i \\in \\left\\{{1, 2, \\ldots, k}\\right\\}: a_i > 0$). Let $p_i$ be the $i$th prime number, so that: * $p_1 = 2$ * $p_2 = 3$ * $p_3 = 5$ * $p_4 = 7$ etc. Let $n = p_1^{a_1} p_2^{a_2} \\cdots p_k^{a_k}$ where $p_i$ is the $i$th prime number. Then $n \\in \\N$ '''codes the sequence $\\left \\langle {a_1, a_2, \\ldots, a_k}\\right \\rangle$''', or $n$ is the '''code number for the sequence $\\left \\langle {a_1, a_2, \\ldots, a_k}\\right \\rangle$'''. The set of all code numbers of finite sequences in $\\N$ is denoted $\\operatorname{Seq}$. Note that $n \\in \\operatorname{Seq}$ {{iff}} $n$ is divisible by all the primes $p_1, p_2, \\ldots, p_k$ , where $p_k$ is the ''largest'' prime dividing $n$. Category:Definitions/Mathematical Logic 8nm3rv1b0qojnbcrur1aqlhaj5ey4pr"}
+{"_id": "20554", "title": "Definition:Length of Integer", "text": "Let $n \\in \\Z$ such that $n \\ge 2$. The '''length of $n$''' is defined as the number of distinct prime factors of $n$ and is denoted $\\operatorname{len} \\left({n}\\right)$. Category:Definitions/Number Theory bypkuv4z8a6bx5f0ieg97dy06q1s1ll"}
+{"_id": "20555", "title": "Definition:Prime Enumeration Function", "text": "Let the function $p: \\N \\to \\N$ be defined as: {| | align=\"left\" | $p(0) = 1$, |  |- | align=\"left\" | $p(n) =$ the $n^\\text{th}$ prime number, | align=\"left\" | if $n > 0$ |} Thus for example: * $p \\left({0}\\right) = 1$ * $p \\left({1}\\right) = 2$ * $p \\left({2}\\right) = 3$ * $p \\left({3}\\right) = 5 \\ldots$ This function is called the '''prime enumeration function'''. Note, of course, that although $p \\left({0}\\right) = 1$, there is no suggestion of treating $1$ as prime (it definitely isn't). {{SUBPAGENAME}} 5znwmmg9j7fhvw5u2zzwk6wi00uvbsc"}
+{"_id": "20556", "title": "Definition:Prime Exponent Function", "text": "Let $n \\in \\N$ be a natural number. Let the prime decomposition of $n$ be given as: :$\\displaystyle n = \\prod_{j \\mathop = 1}^k \\left({p \\left({j}\\right)}\\right)^{a_j}$ where $p \\left({j}\\right)$ is the prime enumeration function. Then the exponent $a_j$ of $p \\left({j}\\right)$ in $n$ is denoted $\\left({n}\\right)_j$. If $p \\left({j}\\right)$ does not divide $n$, then $\\left({n}\\right)_j = 0$. We also define: : $\\forall n \\in \\N: \\left({n}\\right)_0 = 0$ : $\\forall j \\in \\N: \\left({0}\\right)_j = 0$ : $\\forall j \\in \\N: \\left({1}\\right)_j = 0$ Category:Definitions/Mathematical Logic omf6e7p52tjekowf8q9ct16nkt68wv4"}
+{"_id": "20557", "title": "Definition:Partial Function", "text": "Let $S \\subset \\N^k$. Let $f: S \\to \\N$ be a function. Suppose that $\\forall x \\in \\N^k \\setminus S$, $f$ is undefined at $x$. Then $f$ is known as a '''partial function from $\\N^k$ to $\\N$'''. Thus we can specify a function that has values for some, but not all, elements of $\\N$. It can be seen that the definition of a '''partial function''' as given here is compatible with that of a partial mapping."}
+{"_id": "20558", "title": "Definition:Fibonacci Prime", "text": "A '''Fibonacci prime''' is a Fibonacci number which happens to be prime."}
+{"_id": "20559", "title": "Definition:Minimization", "text": "=== Function === {{:Definition:Minimization/Function}} === Partial Function === {{:Definition:Minimization/Partial Function}} === Relation === {{:Definition:Minimization/Relation}} Category:Definitions/Mathematical Logic k03ox0qvog152icfb0q4bgpixvoifwz"}
+{"_id": "20560", "title": "Definition:Total Function", "text": "Let $f: \\N^k \\to \\N$ be a partial function such that $f$ is defined on $S \\subseteq \\N^k$. Now let $S = \\N^k$. That is, the domain of $f$ is then the ''whole'' of $\\N^k$. Then $f$ is a '''total function'''. {{SUBPAGENAME}} 9cdhchwm3nvl55y2i8wd6ja4wi7ryyk"}
+{"_id": "20564", "title": "Definition:Partial Function Equality", "text": "Let $g: \\N^k \\to \\N$ and $h: \\N^k \\to \\N$ be partial functions. We write: :$\\map g {n_1, n_2, \\ldots, n_k} \\approx \\map h {n_1, n_2, \\ldots, n_k}$ {{iff}} either: :both $\\map g {n_1, n_2, \\ldots, n_k}$ and $\\map h {n_1, n_2, \\ldots, n_k}$ are defined and equal or: :neither $\\map g {n_1, n_2, \\ldots, n_k}$ nor $\\map h {n_1, n_2, \\ldots, n_k}$ are defined. That is, {{iff}} $\\map g {n_1, n_2, \\ldots, n_k} = \\map h {n_1, n_2, \\ldots, n_k}$ wherever either are defined. Thus, '''$g$ is equal to $h$''', and we can write $g = h$, {{iff}}: :$\\forall x \\in \\N^k: \\map g x \\approx \\map h x$ Category:Definitions/Mathematical Logic 61x59x4xn76f8r3x33886hhbxaeoql8"}
+{"_id": "20566", "title": "Definition:Degree (Vertex)", "text": "Let $G = \\struct {V, E}$ be an undirected graph. Let $v \\in V$ be a vertex of $G$. The '''degree of $v$ in $G$''' is the number of edges to which it is incident. It is denoted $\\map {\\deg_G} v$, or just $\\map \\deg v$ if it is clear from the context which graph is being referred to. That is: :$\\map {\\deg_G} v = \\card {\\set {u \\in V : \\set {u, v} \\in E} }$"}
+{"_id": "20567", "title": "Definition:Recursive", "text": "=== Function === {{:Definition:Recursive/Function}} === Set === {{:Definition:Recursive/Set}} === Relation === {{:Definition:Recursive/Relation}}"}
+{"_id": "20568", "title": "Definition:Total Recursive Function", "text": "In general, a recursive function is a partial function. A '''total recursive function''' is a recursive function which is total. {{SUBPAGENAME}} d62xoedqu8ncmk22ggtbm02u9t3xw08"}
+{"_id": "20569", "title": "Definition:Trace Table", "text": "Let $P$ be a URM program. The '''trace table''' of $P$ consists of: * The stage of computation; * The number of the instruction of $P$ that is about to be performed; * A list of the contents of all the registers used by $P$ at this point. Thus the trace table is a list of the states of the URM program at each stage."}
+{"_id": "20570", "title": "Definition:Algorithmic Computability", "text": "The concept of '''algorithmic computability''' is an intuitive one. An '''algorithmically computable function''' is a function which can be carried by means of an algorithm, theoretically by a person using pencil and paper. The concept arose in the decades before the invention of digital computers. Much of the theoretical groundwork was done in the 1930s by such as {{AuthorRef|Alan Mathison Turing}}, {{AuthorRef|Alonzo Church}} and {{AuthorRef|Stephen Cole Kleene}}. The term used by {{AuthorRef|Alonzo Church|Church}} when he discussed the issue in his how famous Church's Thesis was '''effectively calculable'''. Category:Definitions/Mathematical Logic 46qfqqk5ue9dtu54ziryz5971matx0s"}
+{"_id": "20571", "title": "Definition:Mistake", "text": "A '''mistake''' (as opposed to an error) is an inaccuracy or (deduced) falsehood caused by (ultimately) human fault. Synonyms include such terms as '''boob''', '''bloomer''', '''blooper''', '''blunder''', '''cock-up''', '''bug''' (computer science), '''mess-up''', '''oversight''', '''howler''' and other less polite words frequently encountered at the backs of schoolrooms worldwide. Please feel free to add your own."}
+{"_id": "20572", "title": "Definition:Error", "text": "Let $x$ be an approximation to a (true) value $X$. The '''error''' $\\varepsilon$ is a measure of how much difference there is between $x$ and $X$."}
+{"_id": "20573", "title": "Definition:Transversal (Geometry)", "text": "A '''transversal''' of two lines lying in the same plane is a line which intersects them in two different points. The transversal is said to '''cut''' the two lines that it crosses. :400px In the above diagram, $EF$ is a '''transversal''' of the lines $AB$ and $CD$."}
+{"_id": "20574", "title": "Definition:Supplementary Angles", "text": ":500px Let $\\angle ACB$ be a straight angle. Let $\\angle BCD + \\angle DCA = \\angle ACB$. That is, $\\angle DCA = \\angle ACB - \\angle BCD$. Then $\\angle DCA$ is the '''supplement''' of $\\angle BCD$. Hence, for any angle $\\alpha$ (whether less than a straight angle or not), the '''supplement''' of $\\alpha$ is $\\pi - \\alpha$. Measured in degrees, the '''supplement''' of $\\alpha$ is $180^\\circ - \\alpha$. If $\\alpha$ is the '''supplement''' of $\\beta$, then it follows that $\\beta$ is the '''supplement''' of $\\alpha$. Hence we can say that $\\alpha$ and $\\beta$ are '''supplementary'''. It can be seen from this that the '''supplement''' of a reflex angle is negative. Thus, '''supplementary angles''' are two angles whose measures add up to the measure of $2$ right angles. That is, their measurements add up to $180$ degrees or $\\pi$ radians. Another (equivalent) definition is to say that two angles are '''supplementary''' which, when set next to each other, form a straight angle."}
+{"_id": "20575", "title": "Definition:Complementary Angles", "text": ":300px Let $\\angle BAC$ be a right angle. Let $\\angle BAD + \\angle DAC = \\angle BAC$. That is, $\\angle DAC = \\angle BAC - \\angle BAD$. Then $\\angle DAC$ is the '''complement''' of $\\angle BAD$. Hence, for any angle $\\alpha$ (whether less than a right angle or not), the complement of $\\alpha$ is $\\dfrac \\pi 2 - \\alpha$. Measured in degrees, the complement of $\\alpha$ is $90^\\circ - \\alpha$. If $\\alpha$ is the complement of $\\beta$, then it follows that $\\beta$ is the complement of $\\alpha$. Hence we can say that $\\alpha$ and $\\beta$ are '''complementary'''. It can be seen from this that the '''complement''' of an angle greater than a right angle is negative. Thus '''complementary angles''' are two angles whose measures add up to the measure of a right angle. That is, their measurements add up to $90$ degrees or $\\dfrac \\pi 2$ radians."}
+{"_id": "20577", "title": "Definition:Lucas Number", "text": "=== Definition 1 === {{:Definition:Lucas Number/Definition 1}} === Definition 2 === {{:Definition:Lucas Number/Definition 2}} === Sequence === {{:Definition:Lucas Number/Sequence}}"}
+{"_id": "20580", "title": "Definition:Exclusive Or", "text": "'''Exclusive Or''' is a binary connective which can be written symbolically as $p \\oplus q$ whose behaviour is as follows: :$p \\oplus q$ means: : '''Either $p$ is true ''or'' $q$ is true ''but not both''.''' or symbolically: : $p \\oplus q := \\paren {p \\lor q} \\land \\neg \\paren {p \\land q}$ where $\\land$ denotes the ''and'' operator and $\\lor$ denotes the ''or'' operator."}
+{"_id": "20581", "title": "Definition:Doubleton", "text": "A '''doubleton''' is a set that contains exactly two elements."}
+{"_id": "20584", "title": "Definition:Connected Relation", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation on a set $S$. Then $\\mathcal R$ is '''connected''' {{iff}}: :$\\forall a, b \\in S: a \\ne b \\implies \\tuple {a, b} \\in \\mathcal R \\lor \\tuple {b, a} \\in \\mathcal R$ That is, {{iff}} every pair of distinct elements is related (either or both ways round)."}
+{"_id": "20585", "title": "Definition:Well-Founded Ordered Set", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then $\\struct {S, \\preceq}$ is '''well-founded''' {{iff}} it satisfies the '''minimal condition''': :Every non-empty subset of $S$ has a minimal element. The term '''well-founded''' can equivalently be said to apply to the ordering $\\preceq$ itself rather than to the ordered set $\\struct {S, \\preceq}$ as a whole."}
+{"_id": "20586", "title": "Definition:Partial Ordering", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then the ordering $\\preceq$ is a '''partial ordering''' on $S$ {{iff}} $\\preceq$ is not connected. That is, {{iff}} $\\struct {S, \\preceq}$ has at least one pair which is non-comparable: :$\\exists x, y \\in S: x \\npreceq y \\land y \\npreceq x$"}
+{"_id": "20587", "title": "Definition:Well-Ordering", "text": "{{:Definition:Well-Ordering/Definition 1}}"}
+{"_id": "20589", "title": "Definition:Ordinal", "text": "{{:Definition:Ordinal/Definition 2}}"}
+{"_id": "20590", "title": "Definition:Mathematical Theory", "text": "A '''mathematical theory''', or just '''theory''', is a concept in mathematical logic. Let $U$ be a set of logical formulas. Let $\\map \\TT U$ be the set of all logical formulas $P$ such that $P$ is a semantic consequence of $U$. That is, let $\\map \\TT U = \\set {P: U \\models P}$. Then $\\TT$ is called '''the (mathematical) theory of $U$'''. The elements of $\\map \\TT U$ are called theorems of $U$. The elements of $U$ are called the axioms of $\\map \\TT U$."}
+{"_id": "20591", "title": "Definition:Sign (Mathematical Theory)", "text": "A '''sign''' of a mathematical theory is a symbol which has a specific meaning in the theory, as follows: # A logical sign; # A letter; # A specific sign. A logical sign is a sign which has a specific meaning and definition in a particular class of mathematical theory. A letter is a more or less arbitrary symbol whose definition depends on the specific context. For a given mathematical theory, there are signs which are specific to the use to which the theory is being put. These are the specific signs. {{wtd|This may need to go, as it covers the same ground as Sign (Formal Systems).}}"}
+{"_id": "20592", "title": "Definition:Logical Sign (Mathematical Theory)", "text": "A '''logical sign''' of a mathematical theory is a symbol which is specifically pre-defined for the class of the theory."}
+{"_id": "20593", "title": "Definition:Letter (Mathematical Theory)", "text": "A '''letter''' of a mathematical theory is a more or less arbitrary symbol whose definition depends on the specific context. Depending on the nature of the mathematical theory, and the particular location in whatever assembly in which it occurs, the interpretation of the letter will vary."}
+{"_id": "20594", "title": "Definition:Specific Sign (Mathematical Theory)", "text": "A '''specific sign''' of a mathematical theory is a symbol which is defined according to the use to which the theory is being put."}
+{"_id": "20595", "title": "Definition:Assembly (Mathematical Theory)", "text": "An '''assembly''' in a mathematical theory is a succession of signs written one after another, along with other delineating marks according to the specific nature of the theory under consideration."}
+{"_id": "20596", "title": "Definition:Link (Bourbaki Theory)", "text": "An assembly in {{AuthorRef|Nicolas Bourbaki|Bourbaki}}'s exposition of a mathematical theory is specified as being: * A succession of signs written one after another; * Certain signs which are not letters can be joined in pairs by '''links''', as follows: :$\\overbrace {\\tau A \\Box}^{} A'$<ref>The specific symbology used by {{AuthorRef|Nicolas Bourbaki|Bourbaki}} in {{BookLink|Theory of Sets|Nicolas Bourbaki}} has not been rendered accurately here, as the author of this page has not been able to establish a method by which to do it. However, the intent has been expressed as accurately as possible.</ref> {{stub}}"}
+{"_id": "20598", "title": "Definition:Word (Formal Systems)", "text": "Let $\\mathcal A$ be an alphabet. Then a '''word in $\\mathcal A$''' is a juxtaposition of finitely many (primitive) symbols of $\\mathcal A$. '''Words''' are the most ubiquitous of collations used for formal languages."}
+{"_id": "20600", "title": "Definition:Concatenation (Formal Systems)", "text": "Let $\\AA$ be an alphabet of symbols. '''Concatenation''' is the process of placing elements of $\\AA$ and words in $\\AA$ next to each other to form a longer word."}
+{"_id": "20601", "title": "Definition:String", "text": "Let $\\mathcal A$ be an alphabet of symbols. A '''string (in $\\mathcal A$)''' is a sequence of symbols from $\\mathcal A$. There is no limit to the number of times a particular symbol may appear in a given '''string'''. === Finite String === {{:Definition:String/Finite}} === Infinite String === {{:Definition:String/Infinite}}"}
+{"_id": "20602", "title": "Definition:Formal Grammar", "text": "Let $\\LL$ be a formal language whose alphabet is $\\AA$. The '''formal grammar''' of $\\LL$ comprises of rules of formation, which determine whether collations in $\\AA$ belong to $\\LL$ or not. Roughly speaking, there are two types of '''formal grammar''': top-down grammar and bottom-up grammar."}
+{"_id": "20603", "title": "Definition:Well-Formed Formula", "text": "Let $\\FF$ be a formal language whose alphabet is $\\AA$. A '''well-formed formula''' is a collation in $\\AA$ which can be built by using the rules of formation of the formal grammar of $\\FF$. That is, a collation in $\\AA$ is a '''well-formed formula''' in $\\FF$ {{iff}} it has a parsing sequence in $\\FF$."}
+{"_id": "20604", "title": "Definition:Formal System", "text": "A '''formal system''' is a formal language $\\mathcal L$ together with a deductive apparatus for $\\mathcal L$."}
+{"_id": "20605", "title": "Definition:Deductive Apparatus", "text": "Let $\\mathcal L$ be a formal language. A '''deductive apparatus for $\\mathcal L$''' is a formally specified system for deriving conclusions about the well-formed formulas of $\\mathcal L$. In mathematics and logic, '''deductive apparatuses''' can by and large be divided into proof systems and formal semantics. === Proof System === {{:Definition:Proof System}} === Formal Semantics === {{:Definition:Formal Semantics}}"}
+{"_id": "20606", "title": "Definition:Axiom/Formal Systems/Axiom Schema", "text": "An '''axiom schema''' is a well-formed formula $\\phi$ of $\\LL$, except for it containing one or more variables which are ''outside'' $\\LL$ itself. This formula can then be used to represent an infinite number of individual axioms in one statement. Namely, each of these variables is allowed to take a specified range of values, most commonly WFFs. Each WFF $\\psi$ that results from $\\phi$ by a valid choice of values for all the variables is then an axiom of $\\mathscr P$."}
+{"_id": "20607", "title": "Definition:Proof System/Rule of Inference", "text": "A '''rule of inference''' is a specification of a valid means to conclude new theorems in $\\mathscr P$ from given theorems and axioms of $\\mathscr P$. Often, the formulation of '''rules of inference''' also appeals to the notion of provable consequence."}
+{"_id": "20608", "title": "Definition:LAST", "text": "'''LAST''' stands for '''LAnguage of Set Theory'''. It is a formal system designed for the description of sets."}
+{"_id": "20609", "title": "Definition:Zenzizenzizenzic", "text": "The '''zenzizenzizenzic''' of a number is its eighth power."}
+{"_id": "20611", "title": "Definition:Predicate Symbol", "text": "Let $\\mathcal L$ be a formal language (for example, the language of predicate logic $\\mathcal L_1$). A '''predicate symbol''' is a letter of $\\mathcal L$ used to describe a predicate or a relation. The name '''predicate symbol''' is a gesture to the reader to make clear what such a symbol should (intuitively) represent in the formal language $\\mathcal L$."}
+{"_id": "20613", "title": "Definition:Singular Statement", "text": "A '''singular statement''' is a statement whose subject is identified by means of a proper name. More generally, it is a statement which contains no variables, either bound or free. === Individuating Description === {{:Definition:Singular Statement/Individuating Description}} === Designatory Function === {{:Definition:Singular Statement/Designatory Function}}"}
+{"_id": "20614", "title": "Definition:Universal Statement", "text": "A '''universal statement''' is one which expresses the fact that all objects (in a particular universe of discourse) have a particular property. That is, a statement of the form: :$\\forall x: P \\paren x$ where: : $\\forall$ is the universal quantifier : $P$ is a predicate symbol. It means: :All $x$ (in some given universe of discourse) have the property $P$. Note that if there exist no $x$ in this particular universe, $\\forall x: P \\paren x$ is always true: see vacuous truth. === Bound Variable === {{:Definition:Bound Variable/Examples/Universal Statement}}"}
+{"_id": "20615", "title": "Definition:Existential Statement", "text": "An '''existential statement''' is one which expresses the existence of at least one object (in a particular universe of discourse) which has a particular property. That is, a statement of the form: :$\\exists x: P \\paren x$ where: : $\\exists$ is the existential quantifier : $P$ is a predicate symbol. It means: :There exists at least one $x$ (in some given universe of discourse) which has the property $P$. === Bound Variable === {{:Definition:Bound Variable/Examples/Existential Statement}}"}
+{"_id": "20616", "title": "Definition:Universe of Discourse", "text": "The '''universe of discourse''', or just '''universe''', is the term used to mean '''everything we are talking about'''. When introducing the symbols $\\forall$ (the universal quantifier) or $\\exists$ (the existential quantifier), it is understood that the objects referred to are those in the specified '''universe'''. It is usual to define that '''universe'''."}
+{"_id": "20620", "title": "Definition:Logical Complement", "text": "The '''(logical) complement''' of a propositional formula $\\mathbf A$ is the negation of $\\mathbf A$, that is, $\\neg \\mathbf A$. Conversely, the '''complement''' of $\\neg \\mathbf A$ is defined to be $\\mathbf A$. === Complementary Pair === {{:Definition:Logical Complement/Complementary Pair}}"}
+{"_id": "20621", "title": "Definition:Logical NAND", "text": "'''NAND''' (that is, '''not and'''), is a binary connective, written symbolically as $p \\uparrow q$, whose behaviour is as follows: :$p \\uparrow q$ is defined as: :'''it is not the case that $p$ and $q$ are both true.''' $p \\uparrow q$ is voiced: :'''$p$ nand $q$'''"}
+{"_id": "20622", "title": "Definition:Logical NOR", "text": "'''NOR''' (that is, '''not or'''), is a binary connective, written symbolically as $p \\downarrow q$, whose behaviour is as follows: :$p \\downarrow q$ is defined as:  :'''neither $p$ nor $q$ is true.''' $p \\downarrow q$ is voiced: : '''$p$ nor $q$'''"}
+{"_id": "20623", "title": "Definition:Backus-Naur Form", "text": "'''Backus-Naur Form''' (abbrevated '''BNF''') is a (formal) metalanguage for defining the syntax of a formal language $\\LL$. As such, it is a formal grammar for $\\LL$. '''BNF''' is only applicable to formal languages that use the collation system of words and concatenation. {{transclude:Definition:Backus-Naur Form/Alphabet   | section  = definition   | increase = 1   | title    = Alphabet   | header   = 3   | link     = true }} {{transclude:Definition:Backus-Naur Form/Rules of Formation   | section  = definition   | increase = 1   | title    = Rules of Formation   | header   = 3   | link     = true }} {{transclude:Definition:Backus-Naur Form/Specification   | section  = definition   | increase = 1   | title    = Specification   | header   = 3   | link     = true }} === Terminals and Non-Terminals === ==== Non-Terminal ==== {{:Definition:Backus-Naur Form/Non-Terminal}} ==== Terminal ==== {{:Definition:Backus-Naur Form/Terminal}} That is, non-terminals are analogous to grammatical clauses of a natural language, while terminals are analogous to its words."}
+{"_id": "20624", "title": "Definition:Metalanguage", "text": "A '''metalanguage''' is a language (either formal or natural) which is used to make statements about another language (again, either formal or natural). === Formal Systems === {{:Definition:Metalanguage/Formal Systems}} === Object Language === {{:Definition:Metalanguage/Object Language}} === Metasyntax === {{:Definition:Metalanguage/Metasyntax}} === Metasymbol === {{:Definition:Metalanguage/Metasymbol}}"}
+{"_id": "20625", "title": "Definition:Syntax", "text": "The '''syntax''' of a language (either natural or formal) is its ''structure''."}
+{"_id": "20626", "title": "Definition:Parsing Sequence", "text": "Let $\\FF$ be a formal language with alphabet $\\AA$. Let $S$ be a collation in $\\AA$. A '''parsing sequence''' for $S$ in $\\FF$ is a sequence of collations in $\\FF$ formed by application of rules of formation of $\\FF$ from previous collations in this sequence, and ending in the collation $S$. If $S$ has no '''parsing sequence''' in $\\FF$, then it is not a well-formed formula in $\\FF$. A parsing sequence for a given well-formed formula in any formal language is usually not unique. Thus, we can determine whether $S$ is a well-formed formula in any formal language by using a sequence of rules of formation of that language. To '''parse''' a collation in a formal language is to find a parsing sequence for that collation, and thereby to determine whether or not it is a well-formed formula."}
+{"_id": "20627", "title": "Definition:Metatheorem", "text": "A '''metatheorem''' of a formal language $\\mathcal F$ is a theorem about $\\mathcal F$ which is not part of $\\mathcal F$. {{mistake|It would seem that a formal system is intended instead of formal language.}} {{NoSources}} Category:Definitions/Formal Systems ltkl28rbseo5rcqxg0q5hjai2kkrg1c"}
+{"_id": "20628", "title": "Definition:Multigraph", "text": "A '''multigraph''' is a graph that can have more than one edge between a pair of vertices. That is, $G = \\left({V, E}\\right)$ is a '''multigraph''' if $V$ is a set and $E$ is a multiset of 2-element subsets of $V$."}
+{"_id": "20629", "title": "Definition:Block Incidence Matrix of a Design", "text": "The '''block incidence matrix''' of a $(v,b,r,k,\\lambda)$-BIBD, is a $v\\times b$ incidence matrix $A=(a_{ij})$ such that: $a_{ij}=\\begin{cases} 1 & : \\text{ if the } i^{th} \\text { point is in the } j^{th} \\text { block} \\\\ 0 & : \\text{ otherwise} \\end{cases}$ {{SUBPAGENAME}} gdc8tmjisbj7dr1loz33eonpe8ox3kz"}
+{"_id": "20630", "title": "Definition:Contrapositive Statement", "text": "The '''contrapositive''' of the conditional: :$p \\implies q$ is the statement: :$\\neg q \\implies \\neg p$"}
+{"_id": "20631", "title": "Definition:Walk (Graph Theory)", "text": "A '''walk''' on a graph is: : an alternating series of vertices and edges : beginning and ending with a vertex : in which each edge is incident with the vertex immediately preceding it and the vertex immediately following it. A '''walk''' between two vertices $u$ and $v$ is called a '''$u$-$v$ walk'''."}
+{"_id": "20632", "title": "Definition:Connected (Graph Theory)", "text": "=== Vertices === {{:Definition:Connected (Graph Theory)/Vertices}} === Graph === {{:Definition:Connected (Graph Theory)/Graph}}"}
+{"_id": "20633", "title": "Definition:Isomorphism (Graph Theory)", "text": "Let $G = \\struct {\\map V G, \\map E G}$ and $H = \\struct {\\map V H, \\map E H}$ be graphs. Let there exist a bijection $F: \\map V G \\to \\map V H$ such that for each edge $\\set {u, v} \\in \\map E G$, there is an edge $\\set {\\map F u, \\map F v} \\in \\map E H$. That is, that: :$F: \\map V G \\to \\map V H$ is a homomorphism, and :$F^{-1}: \\map V H \\to \\map V G$ is a homomorphism. Then $G$ and $H$ are '''isomorphic''', and this is denoted $G \\cong H$. The function $F$ is called an '''isomorphism''' from $G$ to $H$."}
+{"_id": "20634", "title": "Definition:Eulerian Graph", "text": "A loop-multigraph or loop-multidigraph is called '''Eulerian''' {{iff}} it contains an Eulerian circuit."}
+{"_id": "20636", "title": "Definition:Symmetric Design", "text": "A block design with parameters $v, b, r, k, \\lambda$ is said to be '''symmetric''' if $b = v$, and (consequently) $r = k$. === Note === The '''symmetry''' does not mean the incidence matrix is symmetric (it often is not). {{explain|Why $b = v$ has as a consequence $r = k$.}}"}
+{"_id": "20637", "title": "Definition:Isomorphism (Abstract Algebra)/Group Isomorphism", "text": "Let $\\struct {G, \\circ}$ and $\\struct {H, *}$ be groups. Let $\\phi: G \\to H$ be a (group) homomorphism. Then $\\phi$ is a group isomorphism {{iff}} $\\phi$ is a bijection."}
+{"_id": "20638", "title": "Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism", "text": "Let $\\struct {R, +, \\circ}$ and $\\struct {S, \\oplus, *}$ be rings. Let $\\phi: R \\to S$ be a (ring) homomorphism. Then $\\phi$ is a ring isomorphism {{iff}} $\\phi$ is a bijection."}
+{"_id": "20639", "title": "Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism", "text": "Let $\\struct {S, \\ast_1, \\ast_2, \\ldots, \\ast_n, \\circ}_R$ and $\\struct {T, \\odot_1, \\odot_2, \\ldots, \\odot_n, \\otimes}_R$ be $R$-algebraic structures. Let $\\phi: S \\to T$ be an $R$-algebraic structure homomorphism. Then $\\phi$ is an $R$-algebraic structure isomorphism {{iff}} $\\phi$ is a bijection."}
+{"_id": "20640", "title": "Definition:Isomorphism (Abstract Algebra)/Isomorphic Copy", "text": "Let $\\phi: S \\to T$ be an isomorphism. Let $x \\in S$. Then $\\map \\phi x \\in T$ is known as '''the isomorphic copy of $x$ (under $\\phi$)'''."}
+{"_id": "20641", "title": "Definition:Hamiltonian Graph", "text": "A '''Hamiltonian graph''' is an undirected graph that contains a Hamilton cycle."}
+{"_id": "20642", "title": "Definition:Subgraph", "text": "A graph $H = \\struct {\\map V H, \\map E H}$ is called a '''subgraph''' of a graph $G = \\struct {\\map V G, \\map E G}$ {{iff}} $\\map V H \\subseteq \\map V G$ and $\\map E H \\subseteq \\map E G$.  That is to say, it contains no vertices or edges that are not in the original graph. If $H$ is a '''subgraph''' of $G$, then: :$G$ '''contains''' $H$; :$H$ '''is contained in''' $G$"}
+{"_id": "20643", "title": "Definition:Hamilton Cycle", "text": "A '''Hamilton cycle''' in a graph is a cycle that contains every vertex of the graph (but not necessarily every edge)."}
+{"_id": "20644", "title": "Definition:Eulerian Circuit", "text": "An '''Eulerian circuit''' (or '''eulerian circuit''') is a circuit that passes through every vertex of a graph and uses every edge exactly once. It follows that every Eulerian circuit is also an Eulerian trail."}
+{"_id": "20646", "title": "Definition:Decision Procedure", "text": "Let $S$ be a set of propositional formulas. A '''decision procedure for $S$''' is an algorithm which, given a propositional formula $\\mathbf A$, always terminates, returning the answer: * '''Yes''' if $\\mathbf A \\in S$; * '''No''' if $\\mathbf A \\notin S$. === Decision Procedure for Satisfiability === {{:Definition:Decision Procedure/Satisfiability}} === Decision Procedure for Tautologies === {{:Definition:Decision Procedure/Tautologies}} === Refutation Procedure === {{:Definition:Decision Procedure/Refutation Procedure}}"}
+{"_id": "20647", "title": "Definition:Semantic Consequence", "text": "Let $\\mathscr M$ be a formal semantics for a formal language $\\mathcal L$. Let $\\mathcal F$ be a collection of WFFs of $\\mathcal L$. Let $\\map {\\mathscr M} {\\mathcal F}$ be the formal semantics obtained from $\\mathscr M$ by retaining only the structures of $\\mathscr M$ that are models of $\\mathcal F$. Let $\\phi$ be a tautology for $\\map {\\mathscr M} {\\mathcal F}$. Then $\\phi$ is called a '''semantic consequence of $\\mathcal F$''', and this is denoted as: :$\\mathcal F \\models_{\\mathscr M} \\phi$ That is to say, $\\phi$ is a '''semantic consequence of $\\mathcal F$''' {{iff}}, for each $\\mathscr M$-structure $\\mathcal M$: :$\\mathcal M \\models_{\\mathscr M} \\mathcal F$ implies $\\mathcal M \\models_{\\mathscr M} \\phi$ where $\\models_{\\mathscr M}$ is the models relation. Note in particular that for $\\mathcal F = \\O$, the notation agrees with the notation for a $\\mathscr M$-tautology: :$\\models_{\\mathscr M} \\phi$ The concept naturally generalises to sets of formulas $\\mathcal G$ on the {{RHS}}: :$\\mathcal F \\models_{\\mathscr M} \\mathcal G$ {{iff}} $\\mathcal F \\models_{\\mathscr M} \\phi$ for every $\\phi \\in \\mathcal G$."}
+{"_id": "20648", "title": "Definition:Basis Element", "text": "Let $S^n$ be a cartesian space. An element $s \\in S$ is known as a '''basis element of $S^n$'''. Alternatively, let the ordered tuple $X = \\left({x_1, x_2, \\ldots, x_n}\\right)$ be any element of $S^n$. Let $x_j \\in \\left({x_1, x_2, \\ldots, x_n}\\right)$. Then $x_j$ is a '''basis element of $S^n$'''. Category:Definitions/Set Theory hrgjfky62qrvwcm3mrgw1dcs7onr003"}
+{"_id": "20650", "title": "Definition:Logical Complement/Complementary Pair", "text": "For any propositional formula $\\mathbf A$, the set $\\left\\{{\\mathbf A, \\neg \\mathbf A}\\right\\}$ is called a '''complementary pair of formulas'''."}
+{"_id": "20651", "title": "Definition:Disjunctive Normal Form", "text": "A propositional formula $P$ is in '''disjunctive normal form''' {{iff}} it consists of a disjunction of: :$(1):\\quad$ conjunctions of literals and/or: :$(2):\\quad$ literals."}
+{"_id": "20652", "title": "Definition:Multiset", "text": "A '''multiset''' is a pair $\\struct {S, \\mu}$ where: :$S$ is a set :$\\mu: S \\to \\N_{>0}$ is a mapping to the strictly positive natural numbers For $s \\in S$ the natural number $\\map \\mu s$ is called the '''multiplicity''' of $s$. Note that the '''multiplicities''' of elements is finite: we do not allow infinitely many occurrences of the same element, though the set $S$ itself may be finite, countably infinite or uncountably infinite."}
+{"_id": "20654", "title": "Definition:Semantics", "text": "The '''semantics''' of a language (either natural or formal) is its '''meaning''' in a linguistic sense."}
+{"_id": "20655", "title": "Definition:Trail", "text": "A '''trail''' is a walk in which all edges are distinct. A '''trail''' between two vertices $u$ and $v$ is called a '''$u$-$v$ trail'''. The set of vertices and edges which go to make up a '''trail''' form a subgraph. This subgraph itself is also referred to as a '''trail'''."}
+{"_id": "20656", "title": "Definition:Circuit", "text": "A '''circuit''' is a closed trail with at least one edge. The set of vertices and edges which go to make up a circuit form a subgraph. This subgraph itself is also referred to as a '''circuit'''."}
+{"_id": "20657", "title": "Definition:Walk (Graph Theory)/Closed", "text": "A '''closed walk''' is a walk whose first vertex is the same as the last. That is, it is a walk which ends where it starts."}
+{"_id": "20658", "title": "Definition:Tree (Graph Theory)", "text": "=== Definition 1=== {{:Definition:Tree (Graph Theory)/Definition 1}} === Definition 2=== {{:Definition:Tree (Graph Theory)/Definition 2}}"}
+{"_id": "20660", "title": "Definition:Path (Graph Theory)", "text": "A '''path''' is a trail in which all vertices (except perhaps the first and last ones) are distinct. A path between two vertices $u$ and $v$ is called a '''$u$-$v$ path'''. The set of vertices and edges which go to make up a '''path''' form a subgraph. This subgraph itself is also referred to as a '''path'''. === Open Path === {{:Definition:Path (Graph Theory)/Open}}"}
+{"_id": "20661", "title": "Definition:Rooted Tree", "text": "A '''rooted tree''' is a tree with a countable number of nodes, in which a particular node is distinguished from the others and called the '''root node''': :300px"}
+{"_id": "20663", "title": "Definition:Cycle (Graph Theory)", "text": "A '''cycle''' is a circuit in which no vertex except the first (which is also the last) appears more than once. An '''$n$-cycle''' is a cycle with $n$ vertices. The set of vertices and edges which go to make up a cycle form a subgraph. This subgraph itself is also referred to as a '''cycle'''. === Odd Cycle === {{:Definition:Cycle (Graph Theory)/Odd}} === Even Cycle === {{:Definition:Cycle (Graph Theory)/Even}}"}
+{"_id": "20666", "title": "Definition:Walk (Graph Theory)/Length", "text": "The '''length''' of a walk is the number of edges it has, counting repeated edges as many times as they appear. A walk is said to be of '''infinite length''' {{iff}} it has infinitely many edges."}
+{"_id": "20667", "title": "Definition:Finite Graph", "text": "A '''finite graph''' is a graph with a finite number of edges and a finite number of vertices."}
+{"_id": "20668", "title": "Definition:Simple Graph", "text": "A '''simple graph''' is a graph which is: :An undirected graph, that is, the edges are defined as doubleton sets of vertices and not ordered pairs :Not a multigraph, that is, there is no more than one edge between each pair of vertices :Not a loop-graph, that is, there are no loops, that is, edges which start and end at the same vertex :Not a weighted graph, that is, the edges are not mapped to a number."}
+{"_id": "20670", "title": "Definition:Root of Polynomial", "text": "Let $R$ be a commutative ring with unity. Let $f \\in R \\sqbrk x$ be a polynomial over $R$. A '''root''' in $R$ of $f$ is an element $x \\in R$ for which $\\map f x = 0$, where $\\map f x$ denotes the image of $f$ under the evaluation homomorphism at $x$."}
+{"_id": "20671", "title": "Definition:Root of Mapping", "text": "Let $f: R \\to R$ be a mapping on a ring $R$. Let $x \\in R$. Then the values of $x$ for which $f \\left({x}\\right) = 0_R$ are known as the '''roots of the mapping $f$'''."}
+{"_id": "20672", "title": "Definition:Tree (Graph Theory)/Leaf Node", "text": "Let $v$ be a node of a tree $T$. Then $v$ is a '''leaf node''' of a $T$ {{iff}} $v$ is of degree $1$. If $T$ is a rooted tree, this is equivalent to saying that $v$ has no child nodes."}
+{"_id": "20673", "title": "Definition:Rooted Tree/Ancestor Node", "text": "Let $T$ be a rooted tree with root $r_T$. Let $t$ be a node of $T$. An '''ancestor node''' of $t$ is a node in the path from $t$ to $r_T$. This path is indeed unique, by Path in Tree is Unique. === Proper Ancestor === {{:Definition:Rooted Tree/Ancestor Node/Proper}}"}
+{"_id": "20674", "title": "Definition:Propositional Tableau", "text": "'''Propositional tableaus''' are a specific subset of the labeled trees for propositional logic. === Identification === {{:Definition:Propositional Tableau/Identification}} {{transclude:Definition:Propositional Tableau/Construction  |section = def  |increase = 1  |link = true  |title = Construction  |header = 3 }}"}
+{"_id": "20675", "title": "Definition:Rooted Tree/Parent Node", "text": "Let $T$ be a rooted tree whose root is $r_T$. Let $t$ be a node of $T$. From Path in Tree is Unique, there is only one path from $t$ to $r_T$. Let $\\pi: T \\setminus \\left\\{{r_T}\\right\\} \\to T$ be the mapping defined by: :$\\pi \\left({t}\\right) := \\text{the node adjacent to $t$ on the path to $r_T$}$ Then $\\pi \\left({t}\\right)$ is known as the '''parent node''' of $t$. The mapping $\\pi$ is called the '''parent mapping'''."}
+{"_id": "20676", "title": "Definition:Labeled Tree for Propositional Logic/Hypothesis Set", "text": "The countable set $\\mathbf H$ of WFFs of propositional logic is called the '''hypothesis set'''. The elements of $\\mathbf H$ are known as '''hypothesis WFFs'''. The '''hypothesis set''' $\\mathbf H$ is considered to be attached to the root node of $T$."}
+{"_id": "20677", "title": "Definition:Finished Set of WFFs of Propositional Logic", "text": "Let $\\Delta$ be a set of WFFs of propositional logic. Then $\\Delta$ is '''finished''' iff: * $\\Delta$ is not contradictory * For each WFF $\\mathbf C \\in \\Delta$, either $\\mathbf C$ is basic or one of the following is true: ** $\\mathbf C$ has the form $\\neg \\neg \\mathbf A$ where $\\mathbf A \\in \\Delta$ ** $\\mathbf C$ has the form $\\left({\\mathbf A \\land \\mathbf B}\\right)$ where both $\\mathbf A \\in \\Delta$ and $\\mathbf B \\in \\Delta$ ** $\\mathbf C$ has the form $\\neg \\left({\\mathbf A \\land \\mathbf B}\\right)$ where either $\\neg \\mathbf A \\in \\Delta$ or $\\neg \\mathbf B \\in \\Delta$ ** $\\mathbf C$ has the form $\\left({\\mathbf A \\lor \\mathbf B}\\right)$ where either $\\mathbf A \\in \\Delta$ or $\\mathbf B \\in \\Delta$ ** $\\mathbf C$ has the form $\\neg \\left({\\mathbf A \\lor \\mathbf B}\\right)$ where both $\\neg \\mathbf A \\in \\Delta$ and $\\neg \\mathbf B \\in \\Delta$ ** $\\mathbf C$ has the form $\\left({\\mathbf A \\implies \\mathbf B}\\right)$ where either $\\neg \\mathbf A \\in \\Delta$ or $\\mathbf B \\in \\Delta$ ** $\\mathbf C$ has the form $\\neg \\left({\\mathbf A \\implies \\mathbf B}\\right)$ where both $\\mathbf A \\in \\Delta$ and $\\neg \\mathbf B \\in \\Delta$ ** $\\mathbf C$ has the form $\\left({\\mathbf A \\iff \\mathbf B}\\right)$ where either: *** both $\\mathbf A \\in \\Delta$ and $\\mathbf B \\in \\Delta$, or: *** both $\\neg \\mathbf A \\in \\Delta$ and $\\neg \\mathbf B \\in \\Delta$; ** $\\mathbf C$ has the form $\\neg \\left({\\mathbf A \\iff \\mathbf B}\\right)$ where either: *** both $\\mathbf A \\in \\Delta$ and $\\neg \\mathbf B \\in \\Delta$ ** or: *** both $\\neg \\mathbf A \\in \\Delta$ and $\\mathbf B \\in \\Delta$."}
+{"_id": "20678", "title": "Definition:Finished Branch of Propositional Tableau", "text": "Let $T$ be a propositional tableau. Let $\\Gamma$ be a branch of $T$. Then $\\Gamma$ is '''finished''' iff: :$(1): \\quad \\Gamma$ is not contradictory :$(2): \\quad$ Every non-basic WFF on $\\Gamma$ is used at some node of $\\Gamma$. That is, $\\Gamma$ is '''finished''' iff the set $\\Delta$ of WFFs of propositional logic which occur along $\\Gamma$ is a finished set."}
+{"_id": "20679", "title": "Definition:Finished Propositional Tableau", "text": "Let $T$ be a propositional tableau. Then $T$ is '''finished''' iff every branch of $T$ is either finished or contradictory."}
+{"_id": "20682", "title": "Definition:Infix Notation", "text": "=== Binary Relations === {{:Definition:Infix Notation/Binary Relation}} === Binary Operations === {{:Definition:Infix Notation/Binary Operation}} Category:Definitions/Language Definitions tk1e76yudsuwl9jb67v7rmy4rymavvw"}
+{"_id": "20683", "title": "Definition:Steiner Triple System", "text": "A '''Steiner triple system of order $v$''' is a BIBD with block size $3$, and each pair of points occurring together in exactly $1$ block (called a '''triple''')."}
+{"_id": "20684", "title": "Definition:Quantifier", "text": "The universal quantifier $\\forall$ and the existential quantifier $\\exists$ are referred to collectively as '''quantifiers'''."}
+{"_id": "20685", "title": "Definition:Atomic WFF of Predicate Logic", "text": "An '''atomic WFF''' of predicate logic is a WFF not containing any connectives or quantifiers. In the formal grammar of predicate logic, the '''atomic WFFs''' are precisely those  WFFs which can be formed by only applying the rule $\\mathbf W ~ \\mathcal P_n$."}
+{"_id": "20686", "title": "Definition:Modulus of Complex-Valued Function", "text": "Let $f: S \\to \\C$ be a complex-valued function. Then the '''(complex) modulus of $f$''' is written $\\left|{f}\\right|: S \\to \\R$ and is the real-valued function defined as: :$\\forall z \\in S: \\left|{f}\\right| \\left({z}\\right) = \\left|{f \\left({z}\\right)}\\right|$."}
+{"_id": "20687", "title": "Definition:Modulus (Geometric Function Theory)", "text": "In geometric function theory, the term '''modulus''' is used to denote certain conformal invariants of configurations or curve families. More precisely, the modulus of a curve family $\\Gamma$ is the reciprocal of its extremal length: :$\\mod \\Gamma := \\dfrac 1 {\\map \\lambda \\Gamma}$ === Modulus of a Quadrilateral === Consider a quadrilateral; that is, a Jordan domain $Q$ in the complex plane (or some other Riemann surface), together with two disjoint closed boundary arcs $\\alpha$ and $\\alpha'$. Then the '''modulus''' of the quadrilateral $\\map Q {\\alpha, \\alpha'}$ is the extremal length of the family of curves in $Q$ that connect $\\alpha$ and $\\alpha'$. Equivalently, there exists a rectangle $R = \\set {x + i y: \\cmod x < a, \\cmod y < b}$ and a conformal isomorphism between $Q$ and $R$ under which $\\alpha$ and $\\alpha'$ correspond to the vertical sides of $R$. Then the modulus of $\\map Q {\\alpha, \\alpha'}$ is equal to the ratio $a/b$. See Modulus of a Quadrilateral. === Modulus of an Annulus === Consider an annulus $A$; that is, a domain whose boundary consists of two Jordan curves. Then the '''modulus''' $\\mod A$ is the extremal length of the family of curves in $A$ that connect the two boundary components of $A$. Equivalently, there is a round annulus $\\tilde A = \\set {z \\in \\C: r < \\cmod z < R}$ that is conformally equivalent to $A$. Then: :$\\mod A := \\dfrac 1 {2 \\pi} \\map \\ln {\\dfrac R r}$ The modulus of $A$ can also be denoted $\\map M R$."}
+{"_id": "20688", "title": "Definition:Occurrence (Predicate Logic)", "text": "Let $\\mathbf A$ be a WFF of predicate logic. Let $S$ be a string in the alphabet of predicate logic. Each place where $S$ appears in $\\mathbf A$ is called an '''occurrence of $S$ in $\\mathbf A$'''. Note that $S$ may consist of a single symbol, but may not be null. === Scope === {{:Definition:Scope (Logic)/Quantifier}} === Bound Occurrence === {{:Definition:Bound Occurrence}} === Free Occurrence === {{:Definition:Free Occurrence}} === Alphabetic Substitution === {{:Definition:Alphabetic Substitution}}"}
+{"_id": "20689", "title": "Definition:Well-Formed Part", "text": "Let $\\FF$ be a formal language with alphabet $\\AA$. Let $\\mathbf A$ be a well-formed formula of $\\FF$. Let $\\mathbf B$ be a subcollation of $\\mathbf A$. Then $\\mathbf B$ is a '''well-formed part of $\\mathbf A$''' {{iff}} $\\mathbf B$ is a well-formed formula of $\\FF$. === Proper Well-Formed Part === {{:Definition:Well-Formed Part/Proper Well-Formed Part}}"}
+{"_id": "20690", "title": "Definition:Scope (Logic)/Quantifier", "text": "Let $\\mathbf A$ be a WFF of the language of predicate logic. Let $Q$ be an occurrence of a quantifier in $\\mathbf A$. Let $\\mathbf B$ be a well-formed part of $\\mathbf A$ such that $\\mathbf B$ begins (omitting outer parentheses) with $Q x$. That is, such that $\\mathbf B = \\paren {Q x: \\mathbf C}$ for some WFF $\\mathbf C$. $\\mathbf B$ is called the '''scope of the quantifier $Q$'''."}
+{"_id": "20691", "title": "Definition:Bound Occurrence", "text": "Let $Q x$ be an occurrence of a quantifier in $\\mathbf A$. Any occurrence of the variable $x$ in the scope of $Q$ is called a '''bound occurrence'''."}
+{"_id": "20692", "title": "Definition:Freely Substitutable", "text": "Let $\\mathbf C$ be a WFF of predicate logic. Let $x$ be a variable in $\\mathbf C$. Let $\\phi \\left({y_1, \\ldots, y_n}\\right)$ be a term in which the variables $y_1, \\ldots y_n$ occur. Then $\\phi \\left({y_1, \\ldots, y_n}\\right)$ is '''freely substitutable for $x$ in $\\mathbf C$''' {{iff}} no free occurrence of $x$ occurs in a well-formed part of $\\mathbf C$ which is of the form: :$( Q y_i: \\mathbf B )$ where $Q$ is a quantifier and $\\mathbf B$ is a WFF."}
+{"_id": "20693", "title": "Definition:Free Occurrence", "text": "An occurrence of a variable $x$ in $\\mathbf A$ is said to be a '''free occurrence''' {{iff}} it is not bound."}
+{"_id": "20694", "title": "Definition:Classes of WFFs/Plain WFF", "text": "A '''plain WFF''' of predicate logic is a WFF with no parameters. Thus $WFF \\left({\\mathcal P, \\mathcal F, \\varnothing}\\right)$ is the set of all '''plain WFFs''' with relation symbols from $\\mathcal P$ and function symbols from $\\mathcal F$. {{refactor|However immediate, this goes on a proof page}} It is immediate that a '''plain WFF''' is a WFF with parameters from $\\mathcal K$ for ''all'' choices of $\\mathcal K$."}
+{"_id": "20695", "title": "Definition:Free Variable", "text": "Let $x$ be a variable in an expression $E$. $x$ is a '''free variable in $E$''' {{iff}} it is not a bound variable. In the context of predicate logic, $x$ is a '''free variable in $E$''' {{iff}} it has not been introduced by a quantifier, either: :the universal quantifier $\\forall$ or :the existential quantifier $\\exists$."}
+{"_id": "20696", "title": "Definition:Wilson Prime", "text": "A '''Wilson prime''' is a prime number $p$ such that: :$p^2 \\divides \\paren {p - 1}! + 1$ where: :$\\divides$ signifies divisibility :$!$ is the factorial operator."}
+{"_id": "20697", "title": "Definition:Instance", "text": "Let $\\mathbf C$ be a plain WFF in the language of predicate logic. Let $x_1, x_2, \\ldots, x_n$ be the free variables of $\\mathbf C$. Let $\\mathcal M$ be a structure for predicate logic of type $\\mathcal P$ whose universe set is $M$. Then an '''instance of $\\mathbf C$ in $M$''' is the sentence with parameters from $M$ formed by choosing $a_1, a_2, \\ldots, a_n \\in M$ and replacing all free occurrences of $x_k$ in $\\mathbf C$ by $a_k$ for $k = 1, \\ldots, n$. The resulting sentence is denoted: :$\\mathbf C \\left({x_1, \\ldots, x_n \\,//\\, a_1, \\ldots, a_n}\\right)$ Thus $\\mathbf C \\left({x_1, \\ldots, x_n \\,//\\, a_1, \\ldots, a_n}\\right) \\in SENT \\left({\\mathcal P, M}\\right)$. If $\\mathbf C$ is a plain sentence, then no parameters are needed, and $\\mathbf C$ is already an instance of itself."}
+{"_id": "20698", "title": "Definition:Heronian Triangle", "text": "=== Definition 1 === {{:Definition:Heronian Triangle/Definition 1}} === Definition 2 === {{:Definition:Heronian Triangle/Definition 2}}"}
+{"_id": "20699", "title": "Definition:Pseudometric", "text": "A '''pseudometric''' on a set $A$ is a real-valued function $d: A \\times A \\to \\R$ which satisfies the following conditions: {{begin-axiom}} {{axiom | n = \\text M 1         | q = \\forall x \\in A         | m = \\map d {x, x} = 0 }} {{axiom | n = \\text M 2         | q = \\forall x, y, z \\in A         | m = \\map d {x, y} + \\map d {y, z} \\ge \\map d {x, z} }} {{axiom | n = \\text M 3         | q = \\forall x, y \\in A         | m = \\map d {x, y} = \\map d {y, x} }} {{end-axiom}} The difference between a '''pseudometric''' and a metric is that a '''pseudometric''' does not insist that the distance function between distinct elements is ''strictly'' positive. === Pseudometric Space === {{:Definition:Pseudometric/Pseudometric Space}}"}
+{"_id": "20700", "title": "Definition:Derived Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $X \\subseteq S$ be a subset of $S$. The '''derived set''' of $X$ is the set of all limit points of $X$. It is often denoted $X'$."}
+{"_id": "20701", "title": "Definition:Perfect Set", "text": "=== Definition 1 === {{:Definition:Perfect Set/Definition 1}} === Definition 2 === {{:Definition:Perfect Set/Definition 2}} === Definition 3 === {{:Definition:Perfect Set/Definition 3}}"}
+{"_id": "20703", "title": "Definition:Independent Variable", "text": "=== Real Function === {{:Definition:Independent Variable/Real Function}} === Complex Function === {{:Definition:Independent Variable/Complex Function}}"}
+{"_id": "20704", "title": "Definition:Differential Equation", "text": "A '''differential equation''' is a mathematical equation for an unknown function of one or several variables relating: :$(1): \\quad$ The values of the function itself :$(2): \\quad$ Its derivatives of various orders."}
+{"_id": "20705", "title": "Definition:First Order Ordinary Differential Equation", "text": "A '''first order ordinary differential equation''' is an ordinary differential equation in which any derivatives with respect to the independent variable have order no greater than $1$."}
+{"_id": "20706", "title": "Definition:Dependent Variable", "text": "=== Real Function === {{:Definition:Dependent Variable/Real Function}} === Complex Function === {{:Definition:Dependent Variable/Complex Function}}"}
+{"_id": "20707", "title": "Definition:Initial Condition", "text": "Let $\\Phi = \\map F {x, y, y', y'', \\ldots, y^{\\paren n} }$ be an ordinary differential equation. An '''initial condition''' is an ordered pair $\\tuple {x_0, y_0}$ which any solution of $\\Phi$ must satisfy. That is, an '''initial condition''' is the additional imposition that a solution $y = \\map y x$ of $\\Phi$ satisfy: :$\\map y {x_0} = y_0$"}
+{"_id": "20708", "title": "Definition:Initial Value Problem", "text": "Let $\\map y x$ be a solution to the first order ordinary differential equation: :$\\dfrac {\\d y} {\\d x} = \\map f {x, y}$ which is subject to an initial condition: $\\tuple {a, b}$. That is, it is required that $y = b$ when $x = a$, that is $\\map y a = b$. The finding of the resulting particular solution is known as an '''initial value problem'''."}
+{"_id": "20710", "title": "Definition:Homogeneous Function", "text": "Let $V$ and $W$ be two vector spaces over a field $F$. Let $f: V \\to W$ be a function from $V$ to $W$. Then $f$ is '''homogeneous of degree $n$''' {{iff}}: :$f \\left({\\alpha \\mathbf v}\\right) = \\alpha^n f \\left({\\mathbf v}\\right)$ for all nonzero $\\mathbf v \\in V$ and $\\alpha \\in F$."}
+{"_id": "20711", "title": "Definition:Homogeneous Differential Equation", "text": "A '''homogeneous differential equation''' is a first order ordinary differential equation of the form: :$\\map M {x, y} + \\map N {x, y} \\dfrac {\\d y} {\\d x} = 0$ where both $M$ and $N$ are homogeneous functions of the same degree."}
+{"_id": "20712", "title": "Definition:Exact Differential Equation", "text": "Let a first order ordinary differential equation be expressible in this form: :$\\map M {x, y} + \\map N {x, y} \\dfrac {\\d y} {\\d x} = 0$ such that $M$ and $N$ are ''not'' homogeneous functions of the same degree. However, suppose there happens to exist a function $\\map f {x, y}$ such that: :$\\dfrac {\\partial f} {\\partial x} = M, \\dfrac {\\partial f} {\\partial y} = N$ such that the second partial derivatives of $f$ exist and are continuous. Then the expression $M \\rd x + N \\rd y$ is called an '''exact differential''', and the differential equation is called an '''exact differential equation'''."}
+{"_id": "20713", "title": "Definition:Integrating Factor", "text": "Consider the first order ordinary differential equation: :$(1): \\quad \\map M {x, y} + \\map N {x, y} \\dfrac {\\d y} {\\d x} = 0$ such that $M$ and $N$ are real functions of two variables which are ''not'' homogeneous functions of the same degree. Suppose also that: :$\\dfrac {\\partial M} {\\partial y} \\ne \\dfrac {\\partial N} {\\partial x}$ Then from Solution to Exact Differential Equation, $(1)$ is not exact, and that method can not be used to solve it. However, suppose we can find a real function of two variables $\\map \\mu {x, y}$ such that: :$\\map \\mu {x, y} \\paren {\\map M {x, y} + \\map N {x, y} \\dfrac {\\d y} {\\d x} } = 0$ is exact. Then the solution of $(1)$ ''can'' be found by the technique defined in Solution to Exact Differential Equation. The function $\\map \\mu {x, y}$ is called an '''integrating factor'''."}
+{"_id": "20714", "title": "Definition:Pervushin's Number", "text": "'''Pervushin's number''' is the Mersenne number $M_{61} = 2^{61} - 1$. {{NamedforDef|Ivan Mikheevich Pervushin|cat = Pervushin}}"}
+{"_id": "20715", "title": "Definition:Linear First Order Ordinary Differential Equation", "text": "A '''linear first order ordinary differential equation''' is a differential equation which is in (or can be manipulated into) the form: :$\\dfrac {\\d y} {\\d x} + \\map P x y = \\map Q x$ where $\\map P x$ and $\\map Q x$ are functions of $x$."}
+{"_id": "20716", "title": "Definition:Bernoulli's Equation", "text": "'''Bernoulli's equation''' is a first order ordinary differential equation which can be put into the form: :$\\dfrac {\\d y} {\\d x} + \\map P x y = \\map Q x y^n$ where $n \\ne 0$ and $n \\ne 1$."}
+{"_id": "20718", "title": "Definition:Boundary (Topology)", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$. === Definition from Closure and Interior === {{:Definition:Boundary (Topology)/Definition 1}}"}
+{"_id": "20719", "title": "Definition:Probability Space", "text": "A '''probability space''' is a measure space $\\struct {\\Omega, \\Sigma, \\Pr}$ in which $\\map \\Pr \\Omega = 1$. A '''probability space''' is used to define the parameters determining the outcome of an experiment $\\EE$. In this context, the elements of a '''probability space''' are generally referred to as follows: :$\\Omega$ is called the sample space of $\\EE$ :$\\Sigma$ is called the event space of $\\EE$ :$\\Pr$ is called the probability measure on $\\EE$.  Thus it is a measurable space $\\struct {\\Omega, \\Sigma}$ with a probability measure $\\Pr$. === Discrete Probability Space === {{:Definition:Probability Space/Discrete}} === Continuous Probability Space === {{:Definition:Probability Space/Continuous}}"}
+{"_id": "20720", "title": "Definition:Measure Space", "text": "A '''measure space''' is a triple $\\struct {X, \\Sigma, \\mu}$ where: :$X$ is a set :$\\Sigma$ is a $\\sigma$-algebra on $X$ :$\\mu$ is a measure on $\\Sigma$. Thus it is a measurable space $\\struct {X, \\Sigma}$ with a measure."}
+{"_id": "20721", "title": "Definition:Outer Measure", "text": "Let $X$ be a set, and let $\\mathcal P \\left({X}\\right)$ be its power set. An '''outer measure (on $X$)''' is a mapping: : $\\mu^*: \\mathcal P \\left({X}\\right) \\to \\overline \\R_{\\ge 0}$ that satisfies the following conditions: :$(1): \\quad \\mu^* \\left({\\varnothing}\\right) = 0$ :$(2): \\quad \\mu^* \\left({A}\\right) \\le \\mu^* \\left({B}\\right)$ for all $A, B \\in \\mathcal P \\left({X}\\right)$ with $A \\subseteq B$ (that is, $\\mu^*$ is monotone) :$(3): \\quad \\displaystyle \\mu^* \\left({\\bigcup_{i \\mathop \\in \\N} A_i}\\right) \\le \\sum_{i \\mathop = 1}^\\infty \\mu^* \\left({A_i}\\right)$ for all sequences $\\left({A_i}\\right)_{i \\mathop \\in \\N} \\in \\mathcal P \\left({X}\\right)$ (that is, $\\mu^*$ is countably subadditive) where $\\overline{\\R}_{\\ge 0}$ denotes the set of positive extended real numbers."}
+{"_id": "20722", "title": "Definition:Limit Superior of Sequence of Sets", "text": "=== Definition 1 === {{:Definition:Limit Superior of Sequence of Sets/Definition 1}} === Definition 2 === {{:Definition:Limit Superior of Sequence of Sets/Definition 2}}"}
+{"_id": "20723", "title": "Definition:Limit Inferior of Sequence of Sets", "text": "=== Definition 1 === {{:Definition:Limit Inferior of Sequence of Sets/Definition 1}} === Definition 2 === {{:Definition:Limit Inferior of Sequence of Sets/Definition 2}}"}
+{"_id": "20724", "title": "Definition:Limit of Sets", "text": "Let $\\Bbb S = \\set {E_n : n \\in \\N}$ be a sequence of sets. Let the limit superior of $\\Bbb S$ be equal to the limit inferior of $\\Bbb S$. Then the '''limit of $\\Bbb S$''', denoted $\\ds \\lim_{n \\mathop \\to \\infty} E_n$, is defined as: :$\\ds \\lim_{n \\mathop \\to \\infty} E_n := \\limsup_{n \\mathop \\to \\infty} E_n$ and so also: :$\\ds \\lim_{n \\mathop \\to \\infty} E_n := \\liminf_{n \\mathop \\to \\infty} E_n$ and '''$\\Bbb S$ converges to the limit'''."}
+{"_id": "20725", "title": "Definition:Convergence Almost Everywhere", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space.  Let $D \\in \\Sigma$. Let $f: D \\to \\R$ be a $\\Sigma$-measurable function. Let $\\left({f_n}\\right)_{n \\in \\N}$ be a sequence of $\\Sigma$-measurable functions $f_n: D \\to \\R$. Then $\\left({f_n}\\right)_{n \\in \\N}$ is said to '''converge almost everywhere''' (or '''converge a.e.''') on $D$ to $f$ {{iff}}: :$\\mu \\left({ \\left\\{{x \\in D : f_n \\left({x}\\right) \\text{ does not converge to } f \\left({x}\\right) }\\right\\} }\\right) = 0$ and we write $f_n \\stackrel{a.e.}{\\to} f$. In other words, the sequence of functions converges pointwise outside of a $\\mu$-null set."}
+{"_id": "20726", "title": "Definition:Almost Uniform Convergence", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $D \\in \\Sigma$. Let $\\sequence {f_n}_{n \\mathop \\in \\N}, f_n: D \\to \\R$ be a sequence of $\\Sigma$-measurable functions. Then $\\sequence {f_n}_{n \\mathop \\in \\N}$ is said to '''converge almost uniformly''' (or '''converge a.u.''') on $D$ {{iff}}: :For all $\\epsilon > 0$, there is a measurable subset $E_\\epsilon \\subseteq D$ of $D$ such that: ::$(1): \\quad \\map \\mu {E_\\epsilon} < \\epsilon$; ::$(2): \\quad \\sequence {f_n}_{n \\mathop \\in \\N}$ converges uniformly to $f$ on $D \\setminus E_\\epsilon$."}
+{"_id": "20727", "title": "Definition:Convergence in Measure", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $\\left({f_n}\\right)_{n \\in \\N}, f_n: X \\to \\R$ be a sequence of $\\Sigma$-measurable functions. Then $f_n$ is said to '''converge in measure''' to a measurable function $f: X \\to \\R$ iff: :$\\displaystyle \\forall \\epsilon > 0: \\lim_{n \\to \\infty} \\mu \\left({ \\left\\{ {x \\in D : \\left|{ f_n \\left({ x }\\right) - f \\left({ x }\\right) }\\right| \\ge \\epsilon }\\right\\} }\\right) = 0$ for all $D \\in \\Sigma$ with $\\mu \\left({D}\\right) < + \\infty$. To express that $f_n$ '''converges to $f$ in measure''' one writes $f_n \\stackrel{\\mu}{\\longrightarrow} f$ or $\\displaystyle \\operatorname{\\mu-\\!\\lim\\,} \\limits_{n \\to \\infty} f_n = f$."}
+{"_id": "20728", "title": "Definition:Vector Length", "text": "The '''length''' of a vector $\\mathbf v$ in a vector space $\\struct {G, +_G, \\circ}_K$ is defined as $\\norm V$, the norm of $V$."}
+{"_id": "20730", "title": "Definition:Dimension (Topology)/Locally Euclidean Space", "text": "Let $M$ be a locally Euclidean space.  Let $\\left({U, \\kappa}\\right)$ be a coordinate chart such that:  :$\\kappa: U \\to \\kappa \\left({U}\\right) \\subseteq \\R^n$ for some $n \\in \\N$. Then the natural number $n$ is called the '''dimension of $M$'''."}
+{"_id": "20735", "title": "Definition:Congruence (Metric Spaces)", "text": "Two subsets $A, B \\subset X$ of a space $X$ are said to be '''congruent''' if there exists an isometry $f: X \\to X$ such that $f \\left({A}\\right) = B$. Such an isometry is called a '''congruence'''. {{SUBPAGENAME}} byxjucn6es49hx80reyrkzddur0unqq"}
+{"_id": "20737", "title": "Definition:Outer Product", "text": "Given two vectors $\\mathbf u = \\tuple {u_1, u_2, \\ldots, u_m}$ and $\\mathbf v = \\tuple {v_1, v_2, \\ldots, v_n}$, their '''outer product''' $\\mathbf u \\otimes \\mathbf v$ is defined as: :$\\mathbf u \\otimes \\mathbf v = A = \\begin{bmatrix} u_1 v_1 & u_1 v_2 & \\dots & u_1 v_n \\\\  u_2 v_1 & u_2 v_2 & \\dots & u_2 v_n \\\\  \\vdots & \\vdots & \\ddots & \\vdots \\\\ u_m v_1 & u_m v_2 & \\dots & u_m v_n \\end{bmatrix}$"}
+{"_id": "20742", "title": "Definition:Harshad Number", "text": "A '''harshad number''' is a positive integer which is divisible by the sum of its digits base $10$. Hence a '''harshad number''' is a Niven number base $10$."}
+{"_id": "20743", "title": "Definition:Number Base", "text": "=== Integers === {{:Definition:Number Base/Integers}} === Real Numbers === {{:Definition:Number Base/Real Numbers}} === Integer Part === {{:Definition:Number Base/Integer Part}} === Fractional Part === {{:Definition:Number Base/Fractional Part}} === Radix Point === {{:Definition:Number Base/Radix Point}}"}
+{"_id": "20744", "title": "Definition:Digit", "text": "Let $n$ be a number expressed in a particular number base, $b$ for example. Then $n$ can be expressed as: :$\\sqbrk {r_m r_{m - 1} \\ldots r_2 r_1 r_0 . r_{-1} r_{-2} \\ldots}_b$ where: :$m$ is such that $b^m \\le n < b^{m+1}$; :all the $r_i$ are such that $0 \\le r_i < b$. Each of the $r_i$ are known as the '''digits of $n$ (base $b$)'''."}
+{"_id": "20745", "title": "Definition:Niven Number", "text": "A '''Niven number''' (in a given number base $b$) is a positive integer which is divisible by the sum of its digits in that given base $b$. That is, $N$ is a '''Niven number base $b$''' {{iff}}: :$\\displaystyle \\exists A \\in \\Z: N = \\sum_{k \\mathop = 0}^m r_k b^k = A \\sum_{k \\mathop = 0}^m r_k$ where $\\displaystyle \\sum_{k \\mathop = 0}^m r_k b^k$ is the representation of $N$ in base $b$ as defined according to the Basis Representation Theorem."}
+{"_id": "20746", "title": "Definition:All-Harshad Number", "text": "An '''all-Harshad number''' (or '''all-Niven number''') is a positive integer which is a Niven number in '''all''' number bases."}
+{"_id": "20748", "title": "Definition:Total Derivative", "text": "Let $f \\left({x_1, x_2, \\ldots, x_n}\\right)$ be a continuous real function of multiple variables. Let each of $x_1, x_2, \\ldots, x_n$ be continuous real functions of a single independent variable $t$. Then the '''total derivative of $f$ with respect to $t$''' is defined as: :$\\displaystyle \\frac {\\mathrm d f}{\\mathrm d t} = \\sum_{k \\mathop = 1}^n \\frac {\\partial f} {\\partial x_k} \\frac {\\mathrm d x_k}{\\mathrm d t} = \\frac {\\partial f} {\\partial x_1} \\frac {\\mathrm d x_1}{\\mathrm d t} + \\frac {\\partial f} {\\partial x_2} \\frac {\\mathrm d x_2}{\\mathrm d t} + \\cdots + \\frac {\\partial f} {\\partial x_n} \\frac {\\mathrm d x_n}{\\mathrm d t}$ where $\\dfrac {\\partial f} {\\partial x_k}$ is the partial derivative of $f$ with respect to $x_k$. Note that in the above definition, nothing precludes $t$ from being one of the instances of $x_k$ itself. So we have that the '''total derivative of $f$ with respect to $x_k$''' is defined as: :$\\dfrac {\\mathrm d f} {\\mathrm d x_k} = \\dfrac {\\partial f} {\\partial x_1} \\dfrac {\\mathrm d x_1}{\\mathrm d x_k} + \\dfrac {\\partial f} {\\partial x_2} \\dfrac {\\mathrm d x_2} {\\mathrm d x_k} + \\cdots + \\dfrac {\\partial f} {\\partial x_k} + \\cdots + \\dfrac {\\partial f} {\\partial x_n} \\dfrac {\\mathrm d x_n}{\\mathrm d x_k}$"}
+{"_id": "20750", "title": "Definition:Second Chebyshev Function", "text": "The '''Second Chebyshev Function''' $\\psi \\left({x}\\right)$ is defined as follows: :$\\displaystyle \\psi \\left({x}\\right) = \\sum_{k \\mathop \\ge 1} \\sum_{p^k \\mathop \\le x} \\ln p$ where, for each $k$, the sum extends over all powers of prime numbers $p$ such that $p^k \\le x$."}
+{"_id": "20751", "title": "Definition:First Chebyshev Function", "text": "The '''First Chebyshev Function''' $\\vartheta \\left({x}\\right)$, or $\\theta \\left({x}\\right)$, is defined as:  :$\\displaystyle \\vartheta \\left({x}\\right) = \\sum_{p \\mathop \\le x} \\ln p$  where the sum extends over all prime numbers $p$ such that $p \\le x$. {{NamedforDef|Pafnuty Lvovich Chebyshev|cat=Chebyshev}} Category:Definitions/Number Theory 55hzinvcscyby3ei76iikbxdag52fiq"}
+{"_id": "20752", "title": "Definition:Kronecker Product", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ and $\\mathbf B = \\sqbrk b_{p q}$ be matrices. The '''Kronecker product''' of $\\mathbf A$ and $\\mathbf B$ is denoted $\\mathbf A \\otimes \\mathbf B$ and is defined as the block matrix: :$\\mathbf A \\otimes \\mathbf B = \\begin{bmatrix} a_{11} \\mathbf B & a_{12} \\mathbf B & \\cdots & a_{1n} \\mathbf B \\\\ a_{21} \\mathbf B & a_{22} \\mathbf B & \\cdots & a_{2n} \\mathbf B \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{m1} \\mathbf B & a_{m2} \\mathbf B & \\cdots & a_{mn} \\mathbf B \\end{bmatrix}$ Writing this out in full: :$\\mathbf A \\otimes \\mathbf B = \\begin{bmatrix} a_{11} b_{11} & a_{11} b_{12} & \\cdots & a_{11} b_{1q} & \\cdots & \\cdots & a_{1n} b_{11} & a_{1n} b_{12} & \\cdots & a_{1n} b_{1q} \\\\ a_{11} b_{21} & a_{11} b_{22} & \\cdots & a_{11} b_{2q} & \\cdots & \\cdots & a_{1n} b_{21} & a_{1n} b_{22} & \\cdots & a_{1n} b_{2q} \\\\ \\vdots & \\vdots & \\ddots & \\vdots & & & \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{11} b_{p1} & a_{11} b_{p2} & \\cdots & a_{11} b_{pq} & \\cdots & \\cdots & a_{1n} b_{p1} & a_{1n} b_{p2} & \\cdots & a_{1n} b_{pq} \\\\ \\vdots & \\vdots & & \\vdots & \\ddots & & \\vdots & \\vdots & & \\vdots \\\\ \\vdots & \\vdots & & \\vdots & & \\ddots & \\vdots & \\vdots & & \\vdots \\\\ a_{m1} b_{11} & a_{m1} b_{12} & \\cdots & a_{m1} b_{1q} & \\cdots & \\cdots & a_{mn} b_{11} & a_{mn} b_{12} & \\cdots & a_{mn} b_{1q} \\\\ a_{m1} b_{21} & a_{m1} b_{22} & \\cdots & a_{m1} b_{2q} & \\cdots & \\cdots & a_{mn} b_{21} & a_{mn} b_{22} & \\cdots & a_{mn} b_{2q} \\\\ \\vdots & \\vdots & \\ddots & \\vdots & & & \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{m1} b_{p1} & a_{m1} b_{p2} & \\cdots & a_{m1} b_{pq} & \\cdots & \\cdots & a_{mn} b_{p1} & a_{mn} b_{p2} & \\cdots & a_{mn} b_{pq}  \\end{bmatrix}$ Thus, if: :$\\mathbf A$ is a matrix with order $m \\times n$ :$\\mathbf B$ is a matrix with order $p \\times q$ then $\\mathbf A \\otimes \\mathbf B$ is a matrix with order $m p \\times n q$."}
+{"_id": "20753", "title": "Definition:Kronecker Sum", "text": "Let $\\mathbf A = \\sqbrk a_n$ and $\\mathbf B = \\sqbrk b_m$ be square matrices with orders $n$ and $m$ respectively. The '''Kronecker sum''' of $\\mathbf A$ and $\\mathbf B$ is denoted $\\mathbf A \\oplus \\mathbf B$ and is defined as: :$\\mathbf A \\oplus \\mathbf B = \\paren {\\mathbf A \\otimes \\mathbf I_m} + \\paren {\\mathbf I_n \\otimes \\mathbf B}$ where: :$\\otimes$ denotes the Kronecker product :$+$ denotes conventional matrix entrywise addition :$\\mathbf I_m$ and $\\mathbf I_n$ are the unit matrices of order $m$ and $n$ respectively."}
+{"_id": "20754", "title": "Definition:Column Matrix", "text": "A '''column matrix''' is an $m \\times 1$ matrix: :$\\mathbf C = \\begin {bmatrix} c_{1 1} \\\\ c_{2 1} \\\\ \\vdots \\\\ c_{m 1} \\end {bmatrix}$ That is, it is a matrix with only one column."}
+{"_id": "20755", "title": "Definition:Matrix Direct Sum", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ and $\\mathbf B = \\sqbrk b_{p q}$ be matrices. The '''matrix direct sum''' of $\\mathbf A$ and $\\mathbf B$ is denoted $\\mathbf A \\oplus \\mathbf B$ and is defined as: :$\\mathbf A \\oplus \\mathbf B := \\begin {bmatrix} \\mathbf A & \\mathbf 0 \\\\ \\mathbf 0 & \\mathbf B \\end {bmatrix}$ where $\\mathbf 0$ is a zero matrix, the upper-right $\\mathbf 0$ being $m \\times q$ and the lower left $\\mathbf 0$ being $n \\times p$. Thus, if: :$\\mathbf A$ is a matrix with order $m \\times n$ :$\\mathbf B$ is a matrix with order $p \\times q$ then $\\mathbf A \\oplus \\mathbf B$ is a matrix with order $\\paren {m + p} \\times \\paren {n + q}$."}
+{"_id": "20756", "title": "Definition:Matrix Addition", "text": "=== Matrix Entrywise Addition === This is the usual operation when '''matrix addition''' is specified without qualification): {{:Definition:Matrix Entrywise Addition}} === Matrix Direct Sum === {{:Definition:Matrix Direct Sum}} === Kronecker Sum === {{:Definition:Kronecker Sum}}"}
+{"_id": "20757", "title": "Definition:Matrix Product", "text": "=== Matrix Product (Conventional) === {{:Definition:Matrix Product (Conventional)}} === Matrix Scalar Product === {{:Definition:Matrix Scalar Product}} === Commutative Matrix Product === {{:Definition:Commutative Matrix Product}} === Kronecker Product === Also known as matrix direct product: {{:Definition:Kronecker Product}} === Hadamard Product === Also known as Matrix Entrywise Product or Schur Product: {{:Definition:Hadamard Product}} === Frobenius Inner Product === {{:Definition:Frobenius Inner Product}} === Cracovian === {{:Definition:Cracovian}}"}
+{"_id": "20759", "title": "Definition:Graph (Graph Theory)/Order", "text": "Let $G = \\struct {V, E}$ be a graph. The '''order''' of $G$ is the cardinality of its vertex set."}
+{"_id": "20760", "title": "Definition:Graph (Graph Theory)/Size", "text": "Let $G = \\struct {V, E}$ be a graph. The '''size''' of $G$ is the count of its edges."}
+{"_id": "20761", "title": "Definition:Directed Graph", "text": "{{:Definition:Directed Graph/Formal Definition}}"}
+{"_id": "20762", "title": "Definition:Directed Graph/Arc", "text": "Let $G = \\struct {V, E}$ be a digraph. The '''arcs''' are the elements of $E$. Informally, an '''arc''' is a line that joins one vertex to another."}
+{"_id": "20763", "title": "Definition:Network", "text": "A '''network''' $N = \\struct {G, w}$ is: :a graph or digraph $G = \\struct {V, E}$ together with: :a mapping $w: E \\to \\R$ from the edge set $E$ of $G$ into the set $\\R$ of real numbers. === Weight Function === {{:Definition:Network/Weight Function}} A general '''network''' can be denoted $N = \\struct {V, E, w}$ where the elements are understood to be expressed in the order: vertex set, edge set, weight function. === Weight === {{:Definition:Network/Weight}}"}
+{"_id": "20764", "title": "Definition:Edge Set", "text": "Let $G = \\struct {V, E}$ be a graph. The set $E$ of edges in $G$ is called the '''edge set'''. It consists of (unordered) pairs of elements of the vertex set $V$. It is often convenient to refer to the '''edge set''' for a given graph $G$ as $\\map E G$, especially if there is at any one time more than one graph under consideration."}
+{"_id": "20765", "title": "Definition:Loop (Graph Theory)", "text": "Let $G = \\struct {V, E}$ be a loop-graph. A '''loop''' is an edge $e$ of $G$ whose endvertices are the same vertex. Thus a '''loop''' $e$ on the vertex $v$ would be written: :$e = vv$"}
+{"_id": "20766", "title": "Definition:Loop-Graph/Loop-Digraph", "text": "A '''loop-digraph''' is a directed graph which allows an arc to start and end at the same vertex: :380px"}
+{"_id": "20767", "title": "Definition:Signed Graph", "text": "A '''signed graph''' is an undirected network whose functional values are $\\pm 1$. The edges of such a graph are known as '''positive edges''' and '''negative edges'''. It is usual to indicate the functional values by means of just the $+$ or $-$ sign. === Example === :320px Another way to represent a '''signed graph''' is to use two different line styles: one to represent '''positive edges''' and one for the '''negative edges'''. Thus the above example can be rendered as: :320px"}
+{"_id": "20768", "title": "Definition:Vertex Set", "text": "Let $G = \\struct {V, E}$ be a graph. The set $V$ of vertices in $G$ is called the '''vertex set'''. It is often convenient to refer to the '''vertex set''' for a given graph $G$ as $\\map V G$, especially if there is at any one time more than one graph under consideration."}
+{"_id": "20769", "title": "Definition:Regular Graph", "text": "Let $G = \\struct {V, E}$ be an simple graph whose vertices all have the same degree $r$. Then $G$ is called '''regular of degree $r$''', or '''$r$-regular'''."}
+{"_id": "20770", "title": "Definition:Complete Graph", "text": "Let $G = \\struct {V, E}$ be a simple graph such that every vertex is adjacent to every other vertex. Then $G$ is called '''complete'''. The '''complete graph''' of order $p$ is denoted $K_p$."}
+{"_id": "20771", "title": "Definition:Relation Isomorphism", "text": "Let $\\struct {S_1, \\RR_1}$ and $\\struct {S_2, \\RR_2}$ be relational structures. Let there exist a bijection $\\phi: S_1 \\to S_2$ such that: :$(1): \\quad \\forall \\tuple {s_1, t_1} \\in \\RR_1: \\tuple {\\map \\phi {s_1}, \\map \\phi {t_1} } \\in \\RR_2$ :$(2): \\quad \\forall \\tuple {s_2, t_2} \\in \\RR_2: \\tuple {\\map {\\phi^{-1} } {s_2}, \\map {\\phi^{-1} } {t_2} } \\in \\RR_1$ Then $\\struct {S_1, \\RR_1}$ and $\\struct {S_2, \\RR_2}$ are '''isomorphic''', and this is denoted $S_1 \\cong S_2$. The function $\\phi$ is called a '''relation isomorphism''', or just an '''isomorphism''', from $\\struct {S_1, \\RR_1}$ to $\\struct {S_2, \\RR_2}$."}
+{"_id": "20773", "title": "Definition:Separation (Topology)", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A$ and $B$ be open sets of $T$. $A$ and $B$ form a '''separation of $T$''' {{iff}}: :$(1): \\quad A$ and $B$ are non-empty :$(2): \\quad A \\cup B = S$ :$(3): \\quad A \\cap B = \\O$"}
+{"_id": "20775", "title": "Definition:Component of Graph", "text": "Let $G$ be a graph. Let $H$ be a subgraph of $G$ such that: :$H$ is connected :$H$ is not contained in any connected subgraph of $G$ which has more vertices or edges than $H$ has. Then $H$ is a '''component''' of $G$."}
+{"_id": "20776", "title": "Definition:Cut-Vertex", "text": "Let $G = \\struct {V, E}$ be a connected graph. Let $v$ be a vertex of $G$. Then $v$ is a '''cut-vertex''' of $G$ {{iff}} the vertex deletion $G - v$ is a vertex cut of $G$. That is, such that $G - v$ is disconnected. Thus, a '''cut-vertex''' of $G$ is a singleton vertex cut of $G$."}
+{"_id": "20777", "title": "Definition:Bridge", "text": "Let $G = \\struct {V, E}$ be a connected graph. Let $e \\in E$ be an edge of $G$ such that the edge deletion $G - e$ is disconnected. Then $e$ is known as a '''bridge''' of $G$."}
+{"_id": "20778", "title": "Definition:Semi-Eulerian Graph", "text": "A graph is called '''semi-Eulerian''' if it contains an Eulerian trail. Note that the definition of graph here includes: * Simple graph * Loop-graph * Multigraph * Loop-multigraph. Note that an Eulerian graph is also '''semi-Eulerian''', as an Eulerian circuit is still a path, and therefore an Eulerian trail."}
+{"_id": "20779", "title": "Definition:Eulerian Trail", "text": "An '''Eulerian trail''' is a trail $T$ that passes through every vertex of a graph $G$ and uses every edge of $G$ exactly once."}
+{"_id": "20780", "title": "Definition:Loop-Graph", "text": "A '''loop-graph''' is a graph which allows an edge to start and end at the same vertex:"}
+{"_id": "20781", "title": "Definition:Multigraph/Multiple Edge", "text": "Let $G = \\struct {V, E}$ be a multigraph. A '''multiple edge''' is an edge of $G$ which has another edge with the same endvertices. That is, where there is more than one edge that joins any pair of vertices, each of those edges is called a multiple edge."}
+{"_id": "20782", "title": "Definition:Edge-Disjoint Trails", "text": "Let $G = \\struct {V, E}$ be an undirected graph. Let $T_1$ and $T_2$ be trails in $G$. Then $T_1$ and $T_2$ are '''edge-disjoint''' {{iff}} they have no edges in common. Category:Definitions/Graph Theory 8469m5sgbg3g6u72kp32xrsk59hnf37"}
+{"_id": "20783", "title": "Definition:Degree Sequence", "text": "Let $G = \\struct {V, E}$ be a graph. The '''degree sequence''' of $G$ is the sequence obtained by listing all the degrees of all the vertices of $G$ in ascending order, with repeats as needed."}
+{"_id": "20784", "title": "Definition:Edgeless Graph", "text": "An '''edgeless graph''' is a graph with no edges. That is, an '''edgeless graph''' is a graph of size zero. Equivalently, an '''edgeless graph''' is a graph whose vertices are all isolated. The '''edgeless graph''' of order $n$ is denoted $N_n$ and can be referred to as the '''$n$-edgeless graph'''."}
+{"_id": "20785", "title": "Definition:Cycle Graph", "text": "A '''cycle graph''' is a graph which consists of a single cycle. The cycle graph of order $n$ is denoted $C_n$. A cycle graph is $2$-regular."}
+{"_id": "20786", "title": "Definition:Cubic Graph", "text": "A '''cubic graph''' is a $3$-regular graph, that is, a graph whose vertices all have degree $3$. Category:Definitions/Graph Theory pq0840pba83c94wu707geqzrc2qyyue"}
+{"_id": "20787", "title": "Definition:Platonic Graph", "text": "The '''Platonic graphs''' are a set of graphs representing the edges and vertices of the five Platonic solids. === Tetrahedron === {{:Definition:Platonic Graph/Tetrahedron}} === Octahedron === {{:Definition:Platonic Graph/Octahedron}} === Cube === {{:Definition:Platonic Graph/Cube}} === Icosahedron === {{:Definition:Platonic Graph/Icosahedron}} === Dodecahedron === {{:Definition:Platonic Graph/Dodecahedron}} Category:Definitions/Graph Theory 1jcdrumlmzh2kvngkaxwcxvdkx0qo84"}
+{"_id": "20788", "title": "Definition:Graph (Graph Theory)/Edge", "text": "Let $G = \\struct {V, E}$ be a graph. The '''edges''' are the elements of $E$."}
+{"_id": "20789", "title": "Definition:Petersen Graph", "text": "The '''Petersen graph''' is a $3$-regular graph of order $10$ and size $15$, as follows: :300px {{NamedforDef|Julius Petersen|cat=Petersen}} Category:Definitions/Graph Theory bicspd0k48dgehu31cc38obfr3smtdv"}
+{"_id": "20790", "title": "Definition:Bipartite Graph", "text": "A '''bipartite graph''' is a graph $G = \\left({V, E}\\right)$ where: :$V$ is partitioned into two sets $A$ and $B$ such that: :each edge is incident to a vertex in $A$ and a vertex in $B$."}
+{"_id": "20791", "title": "Definition:Complete Bipartite Graph", "text": "A '''complete bipartite graph''' is a bipartite graph $G = \\left({A \\mid B, E}\\right)$ in which every vertex in $A$ is adjacent to every vertex in $B$. The complete bipartite graph where $A$ has $m$ vertices and $B$ has $n$ vertices is denoted $K_{m, n}$. Note that $K_{m, n}$ is the same as $K_{n, m}$."}
+{"_id": "20792", "title": "Definition:Path Graph", "text": "A '''path graph''' is a tree which has a path which passes through all its vertices. The path graph with $n$ vertices is denoted $P_n$."}
+{"_id": "20793", "title": "Definition:Hamiltonian Path", "text": "A '''Hamiltonian path''' in a graph is a path (not a cycle) that contains every vertex of the graph (but not necessarily every edge)."}
+{"_id": "20794", "title": "Definition:Semi-Hamiltonian Graph", "text": "A '''semi-Hamiltonian graph''' is a graph that contains a Hamiltonian path, but ''not'' a Hamilton cycle."}
+{"_id": "20795", "title": "Definition:Kolmogorov Space", "text": "{{:Definition:Kolmogorov Space/Definition 1}}"}
+{"_id": "20796", "title": "Definition:Balanced Signed Graph", "text": "Let $G = \\left({V, E}\\right)$ be a signed graph. Then $G$ is '''balanced''' if it is possible to partition $V$ into two subsets $A, B$ such that: * the positive edges have both ends in $A$ or both ends in $B$; * the negative edges have one end in $A$ and the other end $B$. It is a frequent practice to apply a different colour to the vertices in each set to enhance clarity."}
+{"_id": "20797", "title": "Definition:Forest", "text": "A '''forest''' is a simple graph whose components are all trees. A '''connected forest''' is just a single tree."}
+{"_id": "20798", "title": "Definition:Labeled Graph", "text": "A '''labeled graph''' is a graph whose vertices are each assigned an element from a set of symbols (letters, usually, but this is unimportant). The important thing to note is that the vertices can be distinguished one from another. Note that it is not necessary for all the vertices to be assigned different labels. A vertex-colored graph can be considered as a labeled graph, in which the labels assigned are elements from a set of colors. In this context it is then rare that all vertices are required to be assigned different colors. A graph which has no such labeling is called an '''unlabeled graph'''. Category:Definitions/Graph Theory 4b2l3t5iposhx9ll8xnw8edw2drmyzw"}
+{"_id": "20799", "title": "Definition:Binary Tree", "text": "A '''binary tree''' is a rooted tree which has at most two branches at any node. That is, every node in a '''binary tree''' has (apart from the root) degree of either $1$ (for the leaf nodes) or $3$ (one for the parent, two for the children). The branches at any particular node are frequently called the '''left-hand branch''' and the '''right-hand branch''', and a distinction is made between them."}
+{"_id": "20801", "title": "Definition:Prüfer Sequence", "text": "A '''Prüfer sequence''' of order $n$ is a (finite) sequence of integers: :$\\left({\\mathbf a_1, \\mathbf a_2, \\ldots, \\mathbf a_{n-2}}\\right)$ such that $\\forall i: 1 \\le i \\le n-2: 1 \\le \\mathbf a_i \\le n$. That is, it is a (finite) sequence of $n - 2$ integers between $1$ and $n$."}
+{"_id": "20802", "title": "Definition:Sierpiński Number of the Second Kind", "text": "A '''Sierpiński number of the second kind''' is an odd positive integer $k$ such that integers of the form $k2^n + 1$ are composite for all positive integers $n$. That is, when $k$ is a '''Sierpiński number of the second kind''', all members of the set: :$\\left\\{{k 2^n + 1}\\right\\}$ are composite."}
+{"_id": "20804", "title": "Definition:Center (Abstract Algebra)/Ring", "text": "The '''center of a ring''' $\\struct {R, +, \\circ}$, denoted $\\map Z R$, is the subset of elements in $R$ that commute with every element in $R$. Symbolically: :$\\map Z R = \\map {C_R} R = \\set {x \\in R: \\forall s \\in R: s \\circ x = x \\circ s}$ That is, the '''center of $R$''' is the centralizer of $R$ in $R$ itself. It is clear that the '''center of a ring''' $\\struct {R, +, \\circ}$ can be defined as the center of the group $\\struct {R, \\circ}$. {{explain|I think the group is not $R$, but $R^*$, to avoid finding an inverse for the zero under multiplication.<br>Taking this question to Discussion.}}"}
+{"_id": "20805", "title": "Definition:Center of Tree", "text": "Take a tree, and remove all the nodes whose degree is $1$, along with all their incident edges. Repeat the process until either: :One node is left or: :Two nodes are left, joined by a single edge. If one node is left, it is called the '''center''' of the tree. If two nodes are left, joined by a single edge, this is called the '''bicenter''' of the tree. === Central Tree === {{:Definition:Central Tree}} === Bicentral Tree === {{:Definition:Bicentral Tree}}"}
+{"_id": "20806", "title": "Definition:Sophie Germain Prime", "text": "A '''Sophie Germain prime''' is a prime number $p$ such that $2 p + 1$ is also prime."}
+{"_id": "20807", "title": "Definition:Spanning Tree", "text": "Let $G$ be a connected graph. A '''spanning tree for $G$''' is a spanning subgraph of $G$ which is also a tree. {{refactor|Move the below into its own page}} Clearly a tree is its own spanning tree: As a tree $T$ of order $n$ has $n - 1$ edges, its spanning tree must also contain $n-1$ edges, and those must be the same ones as in $T$."}
+{"_id": "20808", "title": "Definition:Minimum Spanning Tree", "text": "Let $G$ be a weighted graph. The '''minimum spanning tree''' for $G$ is a spanning tree for $G$ which has a minimum total weight. For a given $G$, the '''minimum spanning tree''' may not be unique. It can also be called the '''minimum connector''' for $G$."}
+{"_id": "20809", "title": "Definition:Maximum Spanning Tree", "text": "Let $G$ be a weighted graph. The '''maximum spanning tree''' for $G$ is a spanning tree for $G$ which has a maximum total weight. For a given $G$, the '''maximum spanning tree''' may not be unique. It can also be called the '''maximum connector''' for $G$. Category:Definitions/Graph Theory 3lrmza7lzie58zcyrmb6czhcbeu7ssf"}
+{"_id": "20810", "title": "Definition:Greedy Algorithm", "text": "A '''greedy algorithm''' is an algorithm whose decision strategy is such that any choice made at any stage takes no consideration of any future state. The strategy is to make the choice which has the greatest short-term effect towards the long-term goal. In some problems this may not produce the optimum solution."}
+{"_id": "20811", "title": "Definition:Adjacency Matrix", "text": "An '''adjacency matrix''' is a matrix which describes a graph by representing which vertices are adjacent to which other vertices. If $G$ is a graph of order $n$, then its '''adjacency matrix''' is a square matrix of order $n$, where each row and column corresponds to a vertex of $G$. The element $a_{i j}$ of such a matrix specifies the number of edges from vertex $i$ to vertex $j$. An '''adjacency matrix''' for a simple graph and a loop-digraph is a logical matrix, that is, one whose elements are all either $0$ or $1$. An '''adjacency matrix''' for an undirected graph is symmetrical about the main diagonal. This is because if vertex $i$ is adjacent to vertex $j$, then $j$ is adjacent to $i$. An '''adjacency matrix''' for a weighted graph or network contains the weights of the edges."}
+{"_id": "20812", "title": "Definition:Witch of Agnesi", "text": ":620px Let $OAM$ be a circle of radius $a$ whose center is at $\\tuple {0, a}$. Let $M$ be the point such that $OM$ is a diameter of $OAM$. Let $OA$ be extended to cut the tangent to the circle through $M$ at $N$. Generate $NP$ perpendicular to $MN$ and $AP$ parallel to $MN$. As $A$ moves around the circle $OAM$, the point $P$ traces the curve known as the '''Witch of Agnesi'''."}
+{"_id": "20813", "title": "Definition:Preordering", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation on a set $S$. === Definition 1 === {{:Definition:Preordering/Definition 1}} === Definition 2 === {{:Definition:Preordering/Definition 2}}"}
+{"_id": "20814", "title": "Definition:Partial Preordering", "text": "Let $S$ be a set. Let $\\precsim$ be a preordering on $S$. Then $\\precsim$ is a '''partial preordering''' on $S$ iff $\\precsim$ is not connected. That is, iff there is at least one pair of elements of $S$ which is non-comparable: :$\\exists x, y \\in S: x \\not \\precsim y \\land y \\not \\precsim x$ Category:Definitions/Preorder Theory 5bfsqd25kqx0ntuksh9o4xk4bvw0d32"}
+{"_id": "20815", "title": "Definition:Total Preordering", "text": "Let $S$ be a set. Let $\\precsim$ be a preordering on $S$. Then $\\precsim$ is a '''total preordering''' on $S$ {{iff}} $\\precsim$ is connected. That is, {{iff}} there is no pair of elements of $S$ which is non-comparable: :$\\forall x, y \\in S: x \\precsim y \\lor y \\precsim x$"}
+{"_id": "20816", "title": "Definition:Trichotomy", "text": "Let $S$ be a set. A '''trichotomy''' on $S$ is a relation $\\RR$ on $S$ such that for every pair of elements $a, b \\in S$, '''exactly''' one of the following three conditions applies: :$a \\mathrel \\RR b$ :$a = b$ :$b \\mathrel \\RR a$"}
+{"_id": "20817", "title": "Definition:Right-Total Relation", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R \\subseteq S \\times T$ be a relation in $S$ to $T$. Then $\\mathcal R$ is '''right-total''' {{iff}}: :$\\forall t \\in T: \\exists s \\in S: \\tuple {s, t} \\in \\mathcal R$ That is, {{iff}} every element of $T$ is related to by some element of $S$. That is, {{iff}}: :$\\Img {\\mathcal R} = T$ where $\\Img {\\mathcal R}$ denotes the image of $\\mathcal R$."}
+{"_id": "20818", "title": "Definition:Left-Total Relation", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R \\subseteq S \\times T$ be a relation in $S$ to $T$. Then $\\mathcal R$ is '''left-total''' {{iff}}: :$\\forall s \\in S: \\exists t \\in T: \\tuple {s, t} \\in \\mathcal R$ That is, {{iff}} every element of $S$ relates to some element of $T$."}
+{"_id": "20819", "title": "Definition:Endorelation", "text": "Let $S \\times S$ be the cartesian product of a set or class $S$ with itself. Let $\\RR$ be a relation on $S \\times S$. Then $\\RR$ is referred to as an '''endorelation on $S$'''."}
+{"_id": "20820", "title": "Definition:Correspondence", "text": "A relation $\\RR \\subseteq S \\times T$ is a '''correspondence''' {{iff}} $\\RR$ is both left-total and right-total."}
+{"_id": "20821", "title": "Definition:Coreflexive Relation", "text": "{{:Definition:Coreflexive Relation/Definition 1}}"}
+{"_id": "20822", "title": "Definition:Reflexive Relation", "text": "{{:Definition:Reflexive Relation/Definition 1}}"}
+{"_id": "20823", "title": "Definition:Antireflexive Relation", "text": "$\\RR$ is '''antireflexive''' {{iff}}: :$\\forall x \\in S: \\tuple {x, x} \\notin \\RR$"}
+{"_id": "20824", "title": "Definition:Symmetric Relation", "text": "{{:Definition:Symmetric Relation/Definition 1}}"}
+{"_id": "20825", "title": "Definition:Asymmetric Relation", "text": "{{:Definition:Asymmetric Relation/Definition 1}}"}
+{"_id": "20826", "title": "Definition:Antisymmetric Relation", "text": "{{:Definition:Antisymmetric Relation/Definition 1}}"}
+{"_id": "20827", "title": "Definition:Transitive Relation", "text": "{{:Definition:Transitive Relation/Definition 1}}"}
+{"_id": "20828", "title": "Definition:Antitransitive Relation", "text": "$\\mathcal R$ is '''antitransitive''' {{iff}}: :$\\left({x, y}\\right) \\in \\mathcal R \\land \\left({y, z}\\right) \\in \\mathcal R \\implies \\left({x, z}\\right) \\notin \\mathcal R$ that is: : $\\left\\{ {\\left({x, y}\\right), \\left({y, z}\\right)}\\right\\} \\subseteq \\mathcal R \\implies \\left({x, z}\\right) \\notin \\mathcal R$"}
+{"_id": "20829", "title": "Definition:Euclidean Relation", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation in $S$."}
+{"_id": "20830", "title": "Definition:Serial Relation", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation in $S$. $\\mathcal R$ is '''serial''' {{iff}}: :$\\forall x \\in S: \\exists y \\in S: \\tuple {x, y} \\in \\mathcal R$"}
+{"_id": "20831", "title": "Definition:Totally Ordered Ring", "text": "Let $\\struct {R, +, \\circ, \\preceq}$ be an ordered ring. If the ordering $\\preceq$ is a total ordering, then $\\struct {R, +, \\circ, \\preceq}$ is a '''totally ordered ring'''."}
+{"_id": "20832", "title": "Definition:Isolated Vertex", "text": "If the degree of $v$ is zero, then $v$ is an '''isolated vertex'''."}
+{"_id": "20833", "title": "Definition:Locally Finite Graph", "text": "A '''locally finite graph''' $G$ is an infinite graph where every vertex of $G$ has finite degree."}
+{"_id": "20834", "title": "Definition:Infinite Graph", "text": "A graph which has either an infinite number of edges or vertices is called an '''infinite graph'''."}
+{"_id": "20835", "title": "Definition:Graph (Graph Theory)/Edge/Endvertex", "text": "Let $G = \\struct {V, E}$ be a graph or digraph. Let $e = u v$ be an edge of $G$, that is, $e \\in E$. The '''endvertices''' of $e$ are the vertices $u$ and $v$."}
+{"_id": "20837", "title": "Definition:Link (Graph Theory)", "text": "Let $G = \\left({V, E}\\right)$ be a graph. A '''link''' is a edge of $G$ which is ''not'' a loop. That is, a '''link''' is an edge whose endvertices are  distinct. When $G$ is not a loop-graph, then ''all'' edges are '''links'''. Category:Definitions/Graph Theory Category:Definitions/Loop-Graphs f9wb2ctf7wc7s87eboen67avjnjs8w5"}
+{"_id": "20838", "title": "Definition:Complement (Graph Theory)", "text": "=== Simple Graph === {{:Definition:Complement (Graph Theory)/Simple Graph}} === Loop-Graph === {{:Definition:Complement (Graph Theory)/Loop-Graph}}"}
+{"_id": "20839", "title": "Definition:Null Graph", "text": "The '''null graph''' is the graph which has no vertices. That is, the '''null graph''' is the graph of order zero. It is called '''''the'' null graph''' because, from Empty Set is Unique, there is only one such entity."}
+{"_id": "20841", "title": "Definition:Group Homomorphism", "text": "Let $\\struct {G, \\circ}$ and $\\struct {H, *}$ be groups. Let $\\phi: G \\to H$ be a mapping such that $\\circ$ has the morphism property under $\\phi$. That is, $\\forall a, b \\in G$: :$\\map \\phi {a \\circ b} = \\map \\phi a * \\map \\phi b$ Then $\\phi: \\struct {G, \\circ} \\to \\struct {H, *}$ is a group homomorphism."}
+{"_id": "20842", "title": "Definition:Ring Homomorphism", "text": "Let $\\struct {R, +, \\circ}$ and $\\struct{S, \\oplus, *}$ be rings. Let $\\phi: R \\to S$ be a mapping such that both $+$ and $\\circ$ have the morphism property under $\\phi$. That is, $\\forall a, b \\in R$: {{begin-eqn}} {{eqn | n = 1       | l = \\map \\phi {a + b}       | r = \\map \\phi a \\oplus \\map \\phi b }} {{eqn | n = 2       | l = \\map \\phi {a \\circ b}       | r = \\map \\phi a * \\map \\phi b }} {{end-eqn}} Then $\\phi: \\struct {R, +, \\circ} \\to \\struct {S, \\oplus, *}$ is a ring homomorphism."}
+{"_id": "20843", "title": "Definition:R-Algebraic Structure Homomorphism", "text": "Let $R$ be a ring. Let $\\struct {S, \\ast_1, \\ast_2, \\ldots, \\ast_n, \\circ}_R$ and $\\struct {T, \\odot_1, \\odot_2, \\ldots, \\odot_n, \\otimes}_R$ be $R$-algebraic structures. Let $\\phi: S \\to T$ be a mapping. Then $\\phi$ is an '''$R$-algebraic structure homomorphism''' {{iff}}: :$(1): \\quad \\forall k \\in \\closedint 1 n: \\forall x, y \\in S: \\map \\phi {x \\ast_k y} = \\map \\phi x \\odot_k \\map \\phi y$ :$(2): \\quad \\forall x \\in S: \\forall \\lambda \\in R: \\map \\phi {\\lambda \\circ x} = \\lambda \\otimes \\map \\phi x$ where $\\closedint 1 n = \\set {1, 2, \\ldots, n}$ denotes an integer interval. Note that this definition also applies to modules and vector spaces."}
+{"_id": "20844", "title": "Definition:Ring Monomorphism", "text": "Let $\\struct {R, +, \\circ}$ and $\\struct {S, \\oplus, *}$ be rings. Let $\\phi: R \\to S$ be a (ring) homomorphism. Then $\\phi$ is a ring monomorphism {{iff}} $\\phi$ is an injection."}
+{"_id": "20845", "title": "Definition:Group Monomorphism", "text": "Let $\\struct {G, \\circ}$ and $\\struct {H, *}$ be groups. Let $\\phi: G \\to H$ be a (group) homomorphism. Then $\\phi$ is a group monomorphism {{iff}} $\\phi$ is an injection."}
+{"_id": "20846", "title": "Definition:Ring Epimorphism", "text": "Let $\\struct {R, +, \\circ}$ and $\\struct {S, \\oplus, *}$ be rings. Let $\\phi: R \\to S$ be a (ring) homomorphism. Then $\\phi$ is a ring epimorphism {{iff}} $\\phi$ is a surjection."}
+{"_id": "20847", "title": "Definition:Totally Ordered Field", "text": "Let $\\struct {F, +, \\circ, \\preceq}$ be an ordered ring. Let $\\struct {F, +, \\circ}$ be a field. Let the ordering $\\preceq$ be a total ordering. Then $\\struct {F, +, \\circ, \\preceq}$ is a '''totally ordered field'''."}
+{"_id": "20848", "title": "Definition:Group Epimorphism", "text": "Let $\\struct {G, \\circ}$ and $\\struct {H, *}$ be groups. Let $\\phi: G \\to H$ be a (group) homomorphism. Then $\\phi$ is a group epimorphism {{iff}} $\\phi$ is a surjection."}
+{"_id": "20849", "title": "Definition:F-Homomorphism", "text": "Let $R, S$ be rings with unity. Let $F$ be a subfield of both $R$ and $S$. Then a ring homomorphism $\\varphi: R \\to S$ is called an '''$F$-homomorphism''' if: :$\\forall a \\in F: \\map \\phi a = a$ That is, $\\phi \\restriction_F = I_F$ where: :$\\phi \\restriction_F$ is the restriction of $\\phi$ to $F$ :$I_F$ is the identity mapping on $F$."}
+{"_id": "20850", "title": "Definition:Homomorphism (Graph Theory)", "text": "Let $G = \\struct {\\map V G, \\map E G}$ and $H = \\struct {\\map V H, \\map E H}$ be graphs. Let there exist a mapping $F: \\map V G \\to \\map V H$ such that: :for each edge $\\set {u, v} \\in \\map E G$ :there exists an edge $\\set {\\map F u, \\map F v} \\in \\map E H$. Then $G$ and $H$ are '''homomorphic'''. The mapping $F$ is called a '''homomorphism''' from $G$ to $H$."}
+{"_id": "20851", "title": "Definition:Spanning Subgraph", "text": "Let $G$ be a graph. A '''spanning subgraph for $G$''' is a subgraph of $G$ which contains every vertex of $G$. This is also called a '''factor of $G$'''. {{questionable|Should it not also be specified that $G$ should be connected, and so should such a subgraph?}} Category:Definitions/Graph Theory 5q0yh81fvqx7br0i540r006k03z4scq"}
+{"_id": "20852", "title": "Definition:Induced Subgraph", "text": "Let $G = \\struct {\\map V G, \\map E G}$ be a graph. Let $V' \\subseteq V$ be a subset of vertices of $G$. The '''subgraph of $G$ induced by $V'$''' is the subgraph $G' = \\struct {\\map {V'} {G'}, \\map {E'} {G'} }$ of $G$ such that: :$G'$ has the vertex set $V'$ :For all $u, v \\in V'$, $e = u v \\in \\map E G$ {{iff}} $e \\in \\map {E'} {G'}$. That is, it contains all the edges of $G$ that connect elements of the given subset of the vertex set of $G$, and only those edges. Category:Definitions/Graph Theory s3rv5i6adxz714n5952txh488944nv0"}
+{"_id": "20853", "title": "Definition:Universal Graph", "text": "Let $\\Bbb G$ be a set of graphs. A '''universal graph in $\\Bbb G$''' is a graph $K$ in which every element of $\\Bbb G$ can be embedded in $K$. Note that $K$ need not itself be an element of $\\Bbb G$. {{SUBPAGENAME}} psjyq6jsu0vkx9s97sgljejl6sxgyxr"}
+{"_id": "20854", "title": "Definition:Undirected Graph", "text": "An '''undirected graph''' is a graph whose edge set considered as a relation is symmetric. That is, if $G$ is an '''undirected graph''' and $e = u v$ is an edge in $G$, then $v u$ is also an edge in $G$. In other words, if you can \"go from $u$ to $v$\", you can also \"go from $v$ to $u$\"."}
+{"_id": "20855", "title": "Definition:Acyclic Graph", "text": "An '''acyclic graph''' is a graph or digraph with no cycles. An acyclic connected undirected graph is a tree. An acyclic disconnected undirected graph is a forest. Category:Definitions/Graph Theory msa04pz6xvmbm8pnm72qmm2q3eqybl5"}
+{"_id": "20856", "title": "Definition:Girth", "text": "Let $G$ be a graph. The '''girth''' of $G$ is the smallest length of any cycle in $G$. An acyclic graph is defined as having a girth of infinity. Category:Definitions/Graph Theory lx5d97tx9bvd2oei5nq2w1ezn00wsae"}
+{"_id": "20858", "title": "Definition:Circumference (Graph Theory)", "text": "Let $G$ be a graph. The '''circumference''' of $G$ is the longest length of any cycle in $G$. An acyclic graph is defined as having a circumference of infinity. Category:Definitions/Graph Theory if2yu8uzvefxxd38tq9v3qkbii32gnx"}
+{"_id": "20859", "title": "Definition:Strict Ordering", "text": "=== Definition 1 === {{:Definition:Strict Ordering/Asymmetric and Transitive}} === Definition 2 === {{:Definition:Strict Ordering/Antireflexive and Transitive}}"}
+{"_id": "20860", "title": "Definition:Strict Partial Ordering", "text": "Let $\\left({S, \\prec}\\right)$ be a relational structure. Let $\\prec$ be a strict ordering. Then $\\prec$ is a '''strict partial ordering''' on $S$ iff $\\prec$ is not connected. That is, iff $\\left({S, \\prec}\\right)$ has at least one pair which is non-comparable: :$\\exists x, y \\in S: x \\not \\prec y \\land y \\not \\prec x$"}
+{"_id": "20861", "title": "Definition:Strict Total Ordering", "text": "Let $\\struct {S, \\prec}$ be a relational structure. Let $\\prec$ be a strict ordering. Then $\\prec$ is a '''strict total ordering''' on $S$ {{iff}} $\\struct {S, \\prec}$ has no non-comparable pairs: :$\\forall x, y \\in S: x \\ne y \\implies x \\prec y \\lor y \\prec x$ That is, {{iff}} $\\prec$ is connected."}
+{"_id": "20862", "title": "Definition:Strict Weak Ordering", "text": "A '''strict weak ordering''' on a set $S$ is a relation $\\mathcal R$ such that: :$(1): \\quad \\mathcal R$ is a strict partial ordering :$(2): \\quad$ The incomparability relation $\\mathcal R'$ defined as: ::::$a \\mathrel {\\mathcal R'} b  := \\neg \\left({a \\mathrel {\\mathcal R} b}\\right) \\land \\neg \\left({b \\mathrel {\\mathcal R} a}\\right)$ :::is transitive."}
+{"_id": "20864", "title": "Definition:Heronian Mean", "text": "The '''Heronian mean''' of two numbers $x$ and $y$ is defined as: :$\\displaystyle H = \\frac {x + \\sqrt{xy} + y} 3$ It can also be defined as: :$\\displaystyle H = \\frac 2 3 \\left({\\frac {x + y} 2}\\right) + \\frac 1 3 \\sqrt {x y}$ Thus it is seen to be a weighted mean of their arithmetic mean and geometric mean. {{NamedforDef|Heron of Alexandria|cat=Heron}} Category:Definitions/Algebra 31ngnwoz9cx69aihedjxw56an5tqgp2"}
+{"_id": "20865", "title": "Definition:Weighted Mean", "text": "Let $S = \\sequence {x_1, x_2, \\ldots, x_n}$ be a sequence of real numbers. Let $\\map W x$ be a weight function to be applied to the terms of $S$. The '''weighted mean''' of $S$ is defined as: :$\\bar x := \\dfrac {\\displaystyle \\sum_{i \\mathop = 1}^n \\map W {x_i} x_i} {\\displaystyle \\sum_{i \\mathop = 1}^n \\map W {x_i} }$ This means that elements of $S$ with a larger weight contribute more to the '''weighted mean''' than those with a smaller weight. If we write: :$\\forall i: 1 \\le i \\le n: w_i = \\map W {x_i}$ we can write this '''weighted mean''' as: :$\\bar x := \\dfrac {w_1 x_1 + w_2 x_2 + \\cdots + w_n x_n} {w_1 + w_2 + \\cdots + w_n}$ From the definition of the weight function, none of the weights can be negative. While some of the weights may be zero, not ''all'' of them can, otherwise we would be dividing by zero. === Normalized Weighted Mean === {{:Definition:Weighted Mean/Normalized}}"}
+{"_id": "20866", "title": "Definition:Reflexive Closure", "text": "=== Definition 1 === {{:Definition:Reflexive Closure/Union with Diagonal}} === Definition 2 === {{:Definition:Reflexive Closure/Smallest Reflexive Superset}} === Definition 3 === {{:Definition:Reflexive Closure/Intersection of Reflexive Supersets}}"}
+{"_id": "20867", "title": "Definition:Reflexive Reduction", "text": "Let $\\RR$ be a relation on a set $S$. The '''reflexive reduction''' of $\\RR$ is denoted $\\RR^\\ne$, and is defined as: :$\\RR^\\ne := \\RR \\setminus \\set {\\tuple {x, x}: x \\in S}$"}
+{"_id": "20868", "title": "Definition:Transitive Closure (Relation Theory)", "text": "=== Definition 1 === {{:Definition:Transitive Closure (Relation Theory)/Smallest Transitive Superset}} === Definition 2 === {{:Definition:Transitive Closure (Relation Theory)/Intersection of Transitive Supersets}} === Definition 3 === {{:Definition:Transitive Closure (Relation Theory)/Finite Chain}} === Definition 4 === {{:Definition:Transitive Closure (Relation Theory)/Union of Compositions}}"}
+{"_id": "20870", "title": "Definition:Symmetric Closure", "text": "Let $\\mathcal R$ be a relation on a set $S$. === Definition 1 === {{:Definition:Symmetric Closure/Definition 1}} === Definition 2 === {{:Definition:Symmetric Closure/Definition 2}}"}
+{"_id": "20871", "title": "Definition:Reflexive Transitive Closure", "text": "Let $\\mathcal R$ be a relation on a set $S$. === Smallest Reflexive Transitive Superset === {{:Definition:Reflexive Transitive Closure/Smallest Reflexive Transitive Superset}} === Reflexive Closure of Transitive Closure === {{:Definition:Reflexive Transitive Closure/Reflexive Closure of Transitive Closure}} === Transitive Closure of Reflexive Closure === {{:Definition:Reflexive Transitive Closure/Transitive Closure of Reflexive Closure}}"}
+{"_id": "20872", "title": "Definition:Continuous Real Function", "text": "=== Continuity at a Point === {{Definition:Continuous Real Function/Point}} === Continuous Everywhere === {{Definition:Continuous Real Function/Everywhere}} === Continuity on a Subset of Domain === {{Definition:Continuous Real Function/Subset}}"}
+{"_id": "20873", "title": "Definition:Continuous Mapping (Topology)", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ and $T_2 = \\struct {S_2, \\tau_2}$ be topological spaces. Let $f: S_1 \\to S_2$ be a mapping from $S_1$ to $S_2$. == Continuous at a Point == {{Definition:Continuous Mapping (Topology)/Point}} == Continuous on a Set == {{Definition:Continuous Mapping (Topology)/Set}} == Continuous Everywhere == {{Definition:Continuous Mapping (Topology)/Everywhere}}"}
+{"_id": "20874", "title": "Definition:Discontinuous Mapping", "text": "=== Discontinuous Real Function === {{:Definition:Discontinuous Mapping/Real Function}} === Discontinuous Topological Space === {{:Definition:Discontinuous Mapping/Topological Space}}"}
+{"_id": "20875", "title": "Definition:Continuous Mapping (Metric Space)", "text": "Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \\to A_2$ be a mapping from $A_1$ to $A_2$. Let $a \\in A_1$ be a point in $A_1$. === Continuous at a Point === {{Definition:Continuous Mapping (Metric Space)/Point}} === Continuous on a Space === {{Definition:Continuous Mapping (Metric Space)/Space}}"}
+{"_id": "20877", "title": "Definition:Universe (Set Theory)", "text": "Sets are considered to be subsets of some large '''universal set''', also called the '''universe'''. Exactly what this '''universe''' is will vary depending on the subject and context. When discussing particular sets, it should be made clear just what that '''universe''' is. However, note that from There Exists No Universal Set, this '''universe''' cannot be ''everything that there is''. The traditional symbol used to signify the '''universe''' is $\\mathfrak A$. However, this is old-fashioned and inconvenient, so some newer texts have taken to using $\\mathbb U$ or just $U$ instead. With this notation, this definition can be put into symbols as: :$\\forall S: S \\subseteq \\mathbb U$ The use of $\\mathbb U$ or a variant is not universal: some sources use $X$."}
+{"_id": "20878", "title": "Definition:Ring of Sets", "text": "{{:Definition:Ring of Sets/Definition 1}}"}
+{"_id": "20880", "title": "Definition:Ring (Abstract Algebra)/Ring Axioms", "text": "A ring is an algebraic structure $\\struct {R, *, \\circ}$, on which are defined two binary operations $\\circ$ and $*$, which satisfy the following conditions: {{begin-axiom}} {{axiom | n = \\text A 0         | q = \\forall a, b \\in R         | m = a * b \\in R         | lc= Closure under addition }} {{axiom | n = \\text A 1         | q = \\forall a, b, c \\in R         | m = \\paren {a * b} * c = a * \\paren {b * c}         | lc= Associativity of addition }} {{axiom | n = \\text A 2         | q = \\forall a, b \\in R         | m = a * b = b * a         | lc= Commutativity of addition }} {{axiom | n = \\text A 3         | q = \\exists 0_R \\in R: \\forall a \\in R         | m = a * 0_R = a = 0_R * a         | lc= Identity element for addition: the zero }} {{axiom | n = \\text A 4         | q = \\forall a \\in R: \\exists a' \\in R         | m = a * a' = 0_R = a' * a         | lc= Inverse elements for addition: negative elements }} {{axiom | n = \\text M 0         | q = \\forall a, b \\in R         | m = a \\circ b \\in R         | lc= Closure under product }} {{axiom | n = \\text M 1         | q = \\forall a, b, c \\in R         | m = \\paren {a \\circ b} \\circ c = a \\circ \\paren {b \\circ c}         | lc= Associativity of product }} {{axiom | n = \\text D         | q = \\forall a, b, c \\in R         | m = a \\circ \\paren {b * c} = \\paren {a \\circ b} * \\paren {a \\circ c}         | lc= Product is distributive over addition }} {{axiom | m = \\paren {a * b} \\circ c = \\paren {a \\circ c} * \\paren {b \\circ c} }} {{end-axiom}} These criteria are called the '''ring axioms'''."}
+{"_id": "20881", "title": "Definition:Group Axioms", "text": "A group is an algebraic structure $\\struct {G, \\circ}$ which satisfies the following four conditions: {{begin-axiom}} {{axiom | n = \\text G 0         | lc= Closure         | q = \\forall a, b \\in G         | m = a \\circ b \\in G }} {{axiom | n = \\text G 1         | lc= Associativity         | q = \\forall a, b, c \\in G         | m = a \\circ \\paren {b \\circ c} = \\paren {a \\circ b} \\circ c }} {{axiom | n = \\text G 2         | lc= Identity         | q = \\exists e \\in G: \\forall a \\in G         | m = e \\circ a = a = a \\circ e }} {{axiom | n = \\text G 3         | lc= Inverse         | q = \\forall a \\in G: \\exists b \\in G         | m = a \\circ b = e = b \\circ a }} {{end-axiom}} These four stipulations are called the '''group axioms'''."}
+{"_id": "20882", "title": "Definition:Field Axioms", "text": "The properties of a field are as follows. For a given field $\\struct {F, +, \\circ}$, these statements hold true: {{begin-axiom}} {{axiom | n = \\text A 0         | lc= Closure under addition         | q = \\forall x, y \\in F         | m = x + y \\in F }} {{axiom | n = \\text A 1         | lc= Associativity of addition         | q = \\forall x, y, z \\in F         | m = \\paren {x + y} + z = x + \\paren {y + z} }} {{axiom | n = \\text A 2         | lc= Commutativity of addition         | q = \\forall x, y \\in F         | m = x + y = y + x }} {{axiom | n = \\text A 3         | lc= Identity element for addition         | q = \\exists 0_F \\in F: \\forall x \\in F         | m = x + 0_F = x = 0_F + x         | rc= $0_F$ is called the zero }} {{axiom | n = \\text A 4         | lc= Inverse elements for addition         | q = \\forall x: \\exists x' \\in F         | m = x + x' = 0_F = x' + x         | rc= $x'$ is called a negative element }} {{axiom | n = \\text M 0         | lc= Closure under product         | q = \\forall x, y \\in F         | m = x \\circ y \\in F }} {{axiom | n = \\text M 1         | lc= Associativity of product         | q = \\forall x, y, z \\in F         | m = \\paren {x \\circ y} \\circ z = x \\circ \\paren {y \\circ z} }} {{axiom | n = \\text M 2         | lc= Commutativity of product         | q = \\forall x, y \\in F         | m = x \\circ y = y \\circ x }} {{axiom | n = \\text M 3         | lc= Identity element for product         | q = \\exists 1_F \\in F, 1_F \\ne 0_F: \\forall x \\in F         | m = x \\circ 1_F = x = 1_F \\circ x         | rc= $1_F$ is called the unity }} {{axiom | n = \\text M 4         | lc= Inverse elements for product         | q = \\forall x \\in F^*: \\exists x^{-1} \\in F^*         | m = x \\circ x^{-1} = 1_F = x^{-1} \\circ x }} {{axiom | n = \\text D         | lc= Product is distributive over addition         | q = \\forall x, y, z \\in F         | m = x \\circ \\paren {y + z} = \\paren {x \\circ y} + \\paren {x \\circ z} }} {{end-axiom}} These are called the '''field axioms'''."}
+{"_id": "20883", "title": "Definition:Unit of System of Sets", "text": "Let $\\mathcal S$ be a system of sets. Let $U \\in \\mathcal S$ such that: :$\\forall A \\in \\mathcal S: A \\cap U = A$ Then $U$ is '''the unit of $\\mathcal S$'''. Note that, for a given system of sets, if $U$ exists then it is unique. {{SUBPAGENAME}} 72s55wkujcn21d1czz8k1faomsc1q94"}
+{"_id": "20886", "title": "Definition:Semiring of Sets", "text": "A '''semiring of sets''' or '''semi-ring of sets''' is a system of sets $\\SS$, subject to: :$(1):\\quad \\O \\in \\SS$ :$(2):\\quad A, B \\in \\SS \\implies A \\cap B \\in \\SS$; that is, $\\SS$ is $\\cap$-stable :$(3):\\quad$ If $A, A_1 \\in \\SS$ such that $A_1 \\subseteq A$, then there exists a finite sequence $A_2, A_3, \\ldots, A_n \\in \\SS$ such that: ::$(3a):\\quad \\displaystyle A = \\bigcup_{k \\mathop = 1}^n A_k$ ::$(3b):\\quad$ The $A_k$ are pairwise disjoint Alternatively, criterion $(3)$ can be replaced by: :$(3'):\\quad$ If $A, B \\in \\SS$, then there exists a finite sequence of pairwise disjoint sets $A_1, A_2, \\ldots, A_n \\in \\SS$ such that $\\displaystyle A \\setminus B = \\bigcup_{k \\mathop = 1}^n A_k$."}
+{"_id": "20887", "title": "Definition:Continuous Complex Function", "text": "As the complex plane is a metric space, the same definition of continuity applies to complex functions as to metric spaces."}
+{"_id": "20889", "title": "Definition:Triangle (Geometry)/Isosceles", "text": "An '''isosceles triangle''' is a triangle in which two sides are the same length. :300px"}
+{"_id": "20890", "title": "Definition:Collinear", "text": "Three (or more) points are '''collinear''' {{iff}} there exists a straight line that passes through all the points."}
+{"_id": "20891", "title": "Definition:Triangle (Geometry)/Right-Angled", "text": "A '''right-angled triangle''' is a triangle in which one of the vertices is a right angle. :450px Note that in order to emphasise the nature of the right angle in such a triangle, a small square is usually drawn inside it."}
+{"_id": "20892", "title": "Definition:Chord", "text": "=== Chord of a Circle === :300px {{:Definition:Chord of Circle}} === Chord of a Polygon === {{:Definition:Chord of Polygon}} === Chord of a Parabola === {{:Definition:Chord of Parabola}} === Chord of a general Curve === {{:Definition:Chord of Curve}} There are other sorts of chords, still to be documented."}
+{"_id": "20894", "title": "Definition:Row Matrix", "text": "A '''row matrix''' is a $1 \\times n$ matrix: :$\\mathbf R = \\begin{bmatrix} r_{1 1} & r_{1 2} & \\cdots & r_{1 n} \\end{bmatrix}$ That is, it is a matrix with only one row."}
+{"_id": "20895", "title": "Definition:Complements of Parallelograms", "text": "Let $ABDC$ and $EFHG$ be two parallelograms with the same angles, which share a diagonal, such that $ABDC \\cap EFHG \\ne \\O$. :400px Then the two parallelograms $CIGK$ and $BJHL$ are known as the '''complements''' of the parallelograms $ABDC$ and $EFHG$."}
+{"_id": "20896", "title": "Definition:Diameter of Parallelogram", "text": "Let $ABCD$ be a parallelogram: :400px The '''diameters''' of $ABCD$ are the lines $AC$ and $BD$ joining their opposite vertices."}
+{"_id": "20900", "title": "Definition:Conjugate Angles", "text": "The '''conjugate''' of an angle $\\theta$ is the angle $\\phi$ such that: :$\\theta + \\phi = 2 \\pi$ where $\\theta$ and $\\pi$ are expressed in radians. That is, it is the angle that makes the given angle equal to a full angle. Equivalently, the '''conjugate''' of an angle $\\theta$ is the angle $\\phi$ such that: :$\\theta + \\phi = 360 \\degrees$ where $\\theta$ and $\\pi$ are expressed in degrees. Thus, '''conjugate angles''' are two angles whose measures add up to the measure of $4$ right angles. That is, their measurements add up to $360$ degrees or $2 \\pi$ radians."}
+{"_id": "20901", "title": "Definition:Porism", "text": "The '''porism''' is a term whose precise definition is difficult to pin down. It arises from the works of {{AuthorRef|Euclid}}, and appears to have two different meanings: # A result which follows directly, or in passing, during the course of proving something else; that is, a corollary. # A proposition that asserts that there may be conditions under which a particular problem has either no solutions or an indeterminate number of them."}
+{"_id": "20902", "title": "Definition:Perpendicular Bisector", "text": "Let $AB$ be a line segment. The '''perpendicular bisector''' of $AB$ is the straight line which: :is perpendicular to $AB$ :passes through the point which bisects $AB$. :300px"}
+{"_id": "20903", "title": "Definition:Concentric Circles", "text": "Circles are said to be '''concentric''' if they both have the same point as the center. :300px"}
+{"_id": "20904", "title": "Definition:Baire Space (Topology)", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. === Definition 1 === {{:Definition:Baire Space (Topology)/Definition 1}} === Definition 2 === {{:Definition:Baire Space (Topology)/Definition 2}} === Definition 3 === {{:Definition:Baire Space (Topology)/Definition 3}} === Definition 4 === {{:Definition:Baire Space (Topology)/Definition 4}}"}
+{"_id": "20906", "title": "Definition:Sequence/Infinite Sequence", "text": "An '''infinite sequence''' is a sequence whose domain is infinite."}
+{"_id": "20907", "title": "Definition:Baire Space (Set Theory)", "text": "The '''Baire space''' $\\mathbf B$ is defined as the set of all infinite sequences of natural numbers. It can also be defined as the Cartesian product of a countably infinite number of copies of the set of natural numbers."}
+{"_id": "20908", "title": "Definition:Lindelöf Space", "text": "A topological space $T = \\struct {S, \\tau}$ is a '''Lindelöf space''' {{iff}} every open cover of $S$ has a countable subcover."}
+{"_id": "20909", "title": "Definition:Concyclic Points", "text": "A set of $4$ or more points $S = \\set {P_1, P_2, \\ldots, P_n}$ is '''concyclic''' if they all lie on the circumference of a circle. :320px In the above diagram, points $A, B, C, D$ are '''concyclic''', as they all lie on the circumference of the circle centered at $O$."}
+{"_id": "20910", "title": "Definition:Ramsey Number", "text": "Ramsey's Theorem states that in any coloring of the edges of a sufficiently large complete graph, one will find monochromatic complete subgraphs. More precisely, for any given number of colors $c$, and any given integers $n_1, \\ldots, n_c$, there is a number $\\map R {n_1, \\ldots, n_c}$ such that: :if the edges of a complete graph of order $\\map R {n_1, \\ldots, n_c}$ are colored with $c$ different colours, then for some $i$ between $1$ and $c$, it must contain a complete subgraph of order $n_i$ whose edges are all color $i$. This number $\\map R {n_1, \\ldots, n_c}$ is called the '''Ramsey number''' for $n_1, \\ldots, n_c$. {{NamedforDef|Frank Plumpton Ramsey|cat = Ramsey}} Category:Definitions/Ramsey Theory ss1epz5od4r49erb7jxpvryp77w1lej"}
+{"_id": "20911", "title": "Definition:Triangle (Graph Theory)", "text": "The complete graph $K_3$ of order $3$ is called a '''triangle'''. :250px Category:Definitions/Graph Theory tarxgbvw4ub9izysid9scvqkk2vm13d"}
+{"_id": "20915", "title": "Definition:Claw", "text": "The complete bipartite graph $K_{1, 3}$ is known as a '''claw'''. :250px Category:Definitions/Examples of Graphs Category:Definitions/Examples of Trees Category:Definitions/Star Graphs if50yz7bco8hmj3w5ybm8q8ue795blv"}
+{"_id": "20916", "title": "Definition:Group Product", "text": "=== Group Law === {{:Definition:Group Product/Group Law}} === Product Element === {{:Definition:Group Product/Product Element}}"}
+{"_id": "20917", "title": "Definition:Coloring", "text": "=== Vertex Coloring === {{:Definition:Coloring/Vertex Coloring}} === Edge Coloring === {{:Definition:Coloring/Edge Coloring}}"}
+{"_id": "20919", "title": "Definition:Incident (Graph Theory)/Planar Graph", "text": "Let $G = \\left({V, E}\\right)$ be a planar graph: Then a face of $G$ is '''incident to''' an edge $e$ of $G$ if $e$ is one of those which surrounds the face. Similarly, a face of $G$ is '''incident to''' a vertex $v$ of $G$ if $v$ is at the end of one of those incident edges. In the above graph, for example, the face $ABHC$ is incident to: : the edges $AB, BH, HC, CA$ : the vertices $A, B, H, C$."}
+{"_id": "20920", "title": "Definition:Monochromatic Graph", "text": "A '''monochromatic graph''' is a colored graph (either vertex-colored or edge-colored, depending on the context) in which each of the vertices or edges is assigned the same color. Category:Definitions/Graph Theory Category:Definitions/Ramsey Theory 3ooml52187mqsm3jya2goo6e8dlmogt"}
+{"_id": "20921", "title": "Definition:Indeterminate Equation", "text": "An '''indeterminate equation''' is an equation for which there is more than one unknown. Hence, unless other restrictions are imposed, there may be an infinite number of solutions."}
+{"_id": "20922", "title": "Definition:Equation", "text": "An '''equation''' is a mathematical statement that states that two expressions are equal. For expressions $A$ and $B$, this would usually be portrayed: :$A = B$ where $A$ is known as the {{LHS}} and $B$ the {{RHS}}."}
+{"_id": "20923", "title": "Definition:Expression", "text": "An '''expression''' is a well-formed word of mathematical symbols which together can be considered as a single object. {{expand|Match to natural language use of the word}} Category:Definitions/Language Definitions rneabjlvkc4874mwc2gbp8s0rlip1tf"}
+{"_id": "20924", "title": "Definition:Algebraic Number", "text": "An algebraic number is an algebraic element of the field extension $\\C / \\Q$. That is, it is a complex number that is a root of a polynomial with rational coefficients."}
+{"_id": "20926", "title": "Definition:Cyclic Quadrilateral", "text": "A '''cyclic quadrilateral''' is a quadrilateral which can be circumscribed: :300px"}
+{"_id": "20931", "title": "Definition:Recursive Sequence", "text": "A '''recursive sequence''' is a sequence where each term is defined from earlier terms in the sequence. A famous example of a recursive sequence is the Fibonacci sequence: :$F_n = F_{n-1} + F_{n-2}$ The equation which defines this sequence is called a '''recurrence relation''' or '''difference equation'''. === Initial Terms === {{:Definition:Recursive Sequence/Initial Terms}}"}
+{"_id": "20932", "title": "Definition:Lebesgue Space", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space, and let $p \\in \\R$, $p \\ge 1$. The '''(real) Lebesgue $p$-space of $\\mu$''' is defined as: :$\\map {\\mathcal L^p} \\mu := \\set {f: X \\to \\R: f \\in \\map {\\mathcal M} \\Sigma, \\displaystyle \\int \\size f^p \\rd \\mu < \\infty}$ where $\\map {\\mathcal M} \\Sigma$ denotes the space of $\\Sigma$-measurable functions. On $\\map {\\mathcal L^p} \\mu$, we can introduce the $p$-seminorm $\\norm {\\, \\cdot \\,}_p$ by: :$\\forall f \\in \\mathcal L^p: \\norm f_p := \\paren {\\displaystyle \\int \\size f^p \\rd \\mu}^{1 / p}$ Next, define the equivalence $\\sim$ by: :$f \\sim g \\iff \\norm {f - g}_p = 0$ The resulting quotient space: :$\\map {L^p} \\mu := \\map {\\mathcal L^p} \\mu / \\sim$ is also called '''(real) Lebesgue $p$-space'''."}
+{"_id": "20933", "title": "Definition:Closed-Form Expression", "text": "An expression is a '''closed-form expression''' {{iff}} it can be expressed in terms of a bounded number of elementary functions. Informally is it in apposition to a recurrence relation, which defines a sequence of terms from earlier terms in that sequence."}
+{"_id": "20934", "title": "Definition:Closed-Form Solution", "text": "A system of equations has a '''closed-form solution''' {{iff}} at least one solution can be expressed as a closed-form expression. Category:Definitions/Analysis Category:Definitions/Number Theory 5lp48p7qalkeo0oz6hglvonfh7wk32e"}
+{"_id": "20935", "title": "Definition:Lexicographic Order", "text": "Let $\\struct {S_1, \\preceq_1}$ and $\\struct {S_2, \\preceq_2}$ be ordered sets. The '''lexicographic order on $S_1 \\times S_2$''' is the relation $\\preccurlyeq$ defined on $S_1 \\times S_2$ as: :$\\tuple {x_1, x_2} \\preccurlyeq \\tuple {y_1, y_2} \\iff \\tuple {x_1 \\prec_1 y_1} \\lor \\paren {x_1 = y_1 \\land x_2 \\preceq_2 y_2}$"}
+{"_id": "20936", "title": "Definition:Basis Expansion", "text": "=== Positive Real Numbers === {{:Definition:Basis Expansion/Positive Real Numbers}} === Negative Real Numbers === {{:Definition:Basis Expansion/Negative Real Numbers}}"}
+{"_id": "20937", "title": "Definition:Decimal Notation", "text": "'''Decimal notation''' is the quotidian technique of expressing numbers in base $10$. Every number $x \\in \\R$ is expressed in the form: :$\\displaystyle x = \\sum_{j \\mathop \\in \\Z} r_j 10^j$ where: :$\\forall j \\in \\Z: r_j \\in \\set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$"}
+{"_id": "20938", "title": "Definition:Decimal Expansion", "text": "Let $x \\in \\R$ be a real number. The '''decimal expansion''' of $x$ is the expansion of $x$ in base $10$. $x = \\floor x + \\displaystyle \\sum_{j \\mathop \\ge 1} \\frac {d_j} {10^j}$: :$\\sqbrk {s \\cdotp d_1 d_2 d_3 \\ldots}_{10}$ where: :$s = \\floor x$, the floor of $x$ :it is not the case that there exists $m \\in \\N$ such that $d_M = 9$ for all $M \\ge m$. (That is, the sequence of digits does not end with an infinite sequence of $9$s.)"}
+{"_id": "20940", "title": "Definition:Neo-Pythagoreanism", "text": "'''Neo-Pythagoreanism''' was a school of philosophy which revived the teachings of {{AuthorRef|Pythagoras of Samos|Pythagoras}}. It flourished mainly in the 1st and 2nd centuries C.E. Its influence on mathematics was to associate the numbers with mystical properties, effectively adding a religious aspect to the subject. Its influence upon the Western mathematical tradition was considerably greater than it might have been, in part because of the popularity of {{AuthorRef|Nicomachus of Gerasa|Nicomachus's}} {{BookLink|Introduction to Arithmetic|Nicomachus of Gerasa}}. Category:Definitions/Schools 5sx9x7pn4a8u0bmron9i6upxur2md08"}
+{"_id": "20941", "title": "Definition:Iverson's Convention", "text": "'''Iverson's Convention''' is a notation which allows a compact means of assigning a value of $1$ or $0$ to a proposition $P$, depending on whether $P$ is true or false: :$\\sqbrk P = \\begin{cases} 1 & : P \\ \\text { is true} \\\\ 0 & : P \\ \\text { is false} \\end{cases}$ It is sometimes seen specified as: :$\\sqbrk P = \\begin{cases} 1 & : P \\ \\text { is true} \\\\ 0 & : P \\ \\text { otherwise} \\end{cases}$ which can be useful in fields of mathematics where the Law of the Excluded Middle does not apply."}
+{"_id": "20942", "title": "Definition:Combinatorial Matrix", "text": "A '''combinatorial matrix''' is a square matrix specified as: :$a_{ij} = y + \\delta_{ij} x$ where $\\delta_{ij}$ is the Kronecker delta."}
+{"_id": "20943", "title": "Definition:Vandermonde Matrix", "text": "The '''Vandermonde matrix''' of order $n$ is a square matrix specified variously as: :$a_{ij} = x_i^{j - 1}$ :$a_{ij} = x_j^i$ :$a_{ij} = x_i^{n - j}$ etc."}
+{"_id": "20944", "title": "Definition:Cauchy Matrix", "text": "The '''Cauchy matrix''', commonly denoted $C_n$, can be found defined in two forms. The '''Cauchy matrix''' is an $m \\times n$ matrix whose elements are in the form: : either $a_{ij} = \\dfrac 1 {x_i + y_j}$ : or $a_{ij} = \\dfrac 1 {x_i - y_j}$. where $x_1, x_2, \\ldots, x_m$ and $y_1, y_2, \\ldots, y_n$ are elements of a field $F$."}
+{"_id": "20945", "title": "Definition:Cauchy Determinant", "text": "A '''Cauchy determinant of order $n$''' is the determinant of a square Cauchy matrix of order $n$: :$\\det \\left({C_n}\\right) = \\begin{vmatrix} \\dfrac 1 {x_1 + y_1} & \\dfrac 1 {x_1 + y_2} & \\cdots & \\dfrac 1 {x_1 + y_n} \\\\ \\dfrac 1 {x_2 + y_1} & \\dfrac 1 {x_2 + y_2} & \\cdots & \\dfrac 1 {x_2 + y_n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\dfrac 1 {x_n + y_1} & \\dfrac 1 {x_n + y_2} & \\cdots & \\dfrac 1 {x_n + y_n} \\\\ \\end{vmatrix}$ Its value is given by: :$\\det \\left({C_n}\\right) = \\dfrac {\\displaystyle \\prod_{1 \\mathop \\le i \\mathop < j \\mathop \\le n} \\left({x_j - x_i}\\right) \\left({y_j - y_i}\\right)} {\\displaystyle \\prod_{1 \\mathop \\le i, \\, j \\mathop \\le n} \\left({x_i + y_j}\\right)}$ If $C_n$ is given by: :$\\det \\left({C_n}\\right) = \\begin{bmatrix} \\dfrac 1 {x_1 - y_1} & \\dfrac 1 {x_1 - y_2} & \\cdots & \\dfrac 1 {x_1 - y_n} \\\\ \\dfrac 1 {x_2 - y_1} & \\dfrac 1 {x_2 - y_2} & \\cdots & \\dfrac 1 {x_2 - y_n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\dfrac 1 {x_n - y_1} & \\dfrac 1 {x_n - y_2} & \\cdots & \\dfrac 1 {x_n - y_n} \\\\ \\end{bmatrix}$ then its value is given by: :$\\det \\left({C_n}\\right) = \\dfrac {\\displaystyle \\prod_{1 \\mathop \\le i \\mathop < j \\mathop \\le n} \\left({x_j - x_i}\\right) \\left({y_j - y_i}\\right)} {\\displaystyle \\prod_{1 \\mathop \\le i, \\, j \\mathop \\le n} \\left({x_i - y_j}\\right)}$"}
+{"_id": "20946", "title": "Definition:Ordered Semigroup", "text": "An '''ordered semigroup''' is an ordered structure $\\left({S, \\circ, \\preceq}\\right)$ such that $\\left({S, \\circ}\\right)$ is a semigroup."}
+{"_id": "20947", "title": "Definition:Ordered Monoid", "text": "An '''ordered monoid''' is an ordered structure $\\left({S, \\circ, \\preceq}\\right)$ such that $\\left({S, \\circ}\\right)$ is a monoid."}
+{"_id": "20948", "title": "Definition:Ordered Group", "text": "An '''ordered group''' is an ordered structure $\\left({G, \\circ, \\preceq}\\right)$ such that $\\left({G, \\circ}\\right)$ is a group."}
+{"_id": "20949", "title": "Definition:Unitary Module", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring with unity whose unity is $1_R$. Let $\\struct {G, +_G}$ be an abelian group. A unitary module over $R$ is an $R$-algebraic structure with one operation $\\struct {G, +_G, \\circ}_R$  which satisfies the unitary module axioms: {{:Definition:Unitary Module Axioms}}"}
+{"_id": "20950", "title": "Definition:Scalar Ring", "text": "Let $\\struct {S, *_1, *_2, \\ldots, *_n, \\circ}_R$ be an $R$-algebraic structure with $n$ operations, where: :$\\struct {R, +_R, \\times_R}$ is a ring :$\\struct {S, *_1, *_2, \\ldots, *_n}$ is an algebraic structure with $n$ operations :$\\circ: R \\times S \\to S$ is a binary operation. Then the ring $\\struct {R, +_R, \\times_R}$ is called the scalar ring of $\\struct {S, *_1, *_2, \\ldots, *_n, \\circ}_R$. If the scalar ring is understood, then $\\struct {S, *_1, *_2, \\ldots, *_n, \\circ}_R$ can be rendered $\\struct {S, *_1, *_2, \\ldots, *_n, \\circ}$. === Scalar === {{:Definition:Scalar/R-Algebraic Structure}} === Zero Scalar === {{:Definition:Scalar Ring/Zero Scalar}}"}
+{"_id": "20951", "title": "Definition:Zero Vector", "text": "The identity of $\\struct {G, +_G}$ is usually denoted $\\bszero$, or some variant of this, and called the '''zero vector''': :$\\forall \\mathbf a \\in \\struct {G, +_G, \\circ}_R: \\bszero +_G \\mathbf a = \\mathbf a = \\mathbf a +_G \\bszero$ Note that on occasion it is advantageous to denote the '''zero vector''' differently, for example by $e$, or $\\bszero_V$ or $\\bszero_G$, in order to highlight the fact that the '''zero vector''' is not the same object as the zero scalar."}
+{"_id": "20952", "title": "Definition:Ones Matrix", "text": "A '''ones matrix''' is a matrix all of whose elements are $1$. === Square Ones Matrix === {{:Definition:Ones Matrix/Square}} Category:Definitions/Examples of Matrices 5cjlislse9yofpn5hab0ex9o4j10tl3"}
+{"_id": "20953", "title": "Definition:Infinite Hilbert Matrix", "text": "The '''infinite Hilbert matrix''' is the infinite matrix whose elements are defined as: :$a_{ij} = \\dfrac 1 {i + j - 1}$ In full, it appears as: :$\\begin{bmatrix} 1 & \\tfrac 1 2 & \\tfrac 1 3 & \\tfrac 1 4 & \\tfrac 1 5 & \\tfrac 1 6 & \\ldots \\\\ \\tfrac 1 2 & \\tfrac 1 3 & \\tfrac 1 4 & \\tfrac 1 5 & \\tfrac 1 6 & \\tfrac 1 7 & \\ldots \\\\ \\tfrac 1 3 & \\tfrac 1 4 & \\tfrac 1 5 & \\tfrac 1 6 & \\tfrac 1 7 & \\tfrac 1 8 & \\ldots \\\\ \\tfrac 1 4 & \\tfrac 1 5 & \\tfrac 1 6 & \\tfrac 1 7 & \\tfrac 1 8 & \\tfrac 1 9 & \\ldots \\\\ \\tfrac 1 5 & \\tfrac 1 6 & \\tfrac 1 7 & \\tfrac 1 8 & \\tfrac 1 9 & \\tfrac 1 {10} & \\ldots \\\\ \\tfrac 1 6 & \\tfrac 1 7 & \\tfrac 1 8 & \\tfrac 1 9 & \\tfrac 1 {10} & \\tfrac 1 {11} & \\ldots \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots  \\end{bmatrix}$ {{NamedforDef|David Hilbert|cat = Hilbert}}"}
+{"_id": "20954", "title": "Definition:Submatrix", "text": "Let $\\mathbf A$ be a matrix with $m$ rows and $n$ columns. A '''submatrix''' of $\\mathbf A$ is a matrix formed by selecting from $\\mathbf A$: :a subset of the rows and: :a subset of the columns and forming a new matrix by using those entries, in the same relative positions, that appear in both the rows and columns of those selected."}
+{"_id": "20955", "title": "Definition:Infinite Matrix", "text": "An '''infinite matrix''' is a matrix whose order is infinite. Category:Definitions/Matrices Category:Definitions/Infinite Sets ewbagrpivov0hm03vovayq0jx2rchdd"}
+{"_id": "20956", "title": "Definition:Hilbert Matrix", "text": "A '''Hilbert matrix''' is an order $n$ square submatrix of the infinite Hilbert matrix, consisting of the elements in the first $n$ rows and columns of that matrix. Thus it is an $n \\times n$ matrix whose elements are defined as: :$a_{i j} = \\dfrac 1 {i + j - 1}$"}
+{"_id": "20957", "title": "Definition:Closure Operator/Power Set", "text": "Let $S$ be a set. Let $\\powerset S$ denote the power set of $S$. A '''closure operator''' on $S$ is a mapping: :$\\cl: \\powerset S \\to \\powerset S$ which satisfies the following conditions for all sets $X, Y \\subseteq S$: {{begin-axiom}} {{axiom | n = 1         | lc= $\\cl$ is inflationary         | q = \\forall X \\subseteq S         | ml= X         | mo= \\subseteq         | mr= \\map \\cl X }} {{axiom | n = 2         | lc= $\\cl$ is increasing         | q = \\forall X, Y \\subseteq S         | ml= X \\subseteq Y         | mo= \\implies         | mr= \\map \\cl X \\subseteq \\map \\cl Y }} {{axiom | n = 3         | lc= $\\cl$ is idempotent         | q = \\forall X \\subseteq S         | ml= \\map \\cl {\\map \\cl X}         | mo= =         | mr= \\map \\cl X }} {{end-axiom}}"}
+{"_id": "20958", "title": "Definition:Prime Factor", "text": "Let $n \\in \\Z$ be an integer. Then $p$ is a '''prime factor''' of $n$ {{iff}}: : $(1): \\quad p$ is a prime number : $(2): \\quad p$ is a divisor (that is, factor) of $n$."}
+{"_id": "20959", "title": "Definition:Modulo Operation", "text": "Let $x, y \\in \\R$ be real numbers. Then the '''modulo operation''' is defined and denoted as: :$x \\bmod y := \\begin{cases} x - y \\floor {\\dfrac x y} & : y \\ne 0 \\\\ x & : y = 0 \\end{cases}$ where $\\floor {\\, \\cdot \\,}$ denotes the floor function."}
+{"_id": "20960", "title": "Definition:Integral Multiple", "text": "=== Rings and Fields === {{:Definition:Integral Multiple/Rings and Fields}} === Real Numbers === This concept is often seen when $F$ is the set of real numbers $\\R$. {{:Definition:Integral Multiple/Real Numbers}}"}
+{"_id": "20961", "title": "Definition:Quotient (Algebra)", "text": "Let $a, b \\in \\Z$ be integers such that $b \\ne 0$. From the Division Theorem: :$\\forall a, b \\in \\Z, b \\ne 0: \\exists_1 q, r \\in \\Z: a = q b + r, 0 \\le r < \\size b$ The value $q$ is defined as the '''quotient of $a$ on division by $b$''', or the '''quotient of $\\dfrac a b$'''. === Real Arguments === When $x, y \\in \\R$ the quotient is still defined: {{:Definition:Quotient (Algebra)/Real}}"}
+{"_id": "20962", "title": "Definition:Pre-Socratic", "text": "The term '''pre-Socratic''' refers to various schools of Greek philosophy before {{AuthorRef|Socrates}}."}
+{"_id": "20963", "title": "Definition:Eleatic School", "text": "The '''Eleatic School''' was a pre-Socratic school of Greek philosophy. Its guiding tenet was the concept of Oneness. God was the eternal Unity, who permeated the universe and guided it with his thought."}
+{"_id": "20964", "title": "Definition:Steiner Inellipse", "text": "Let $\\triangle ABC$ be a triangle whose sides have midpoints $D, E, F$. Then the unique ellipse which can be drawn wholly inside $\\triangle ABC$ tangent to $D, E, F$ is called the '''Steiner inellipse''': :400px {{namedfor|Jakob Steiner}} Category:Definitions/Geometry gnl34qb94fbxkknnk6j0c2nbp18fsw3"}
+{"_id": "20965", "title": "Definition:Boundary (Graph Theory)", "text": "=== Simple Graph === Let $G = \\tuple {V, E}$ be a simple graph. Let $v \\in V$ be a vertex of $G$. Then the '''boundary''' of $v$ is the set of all vertices of $G$ which are adjacent to $v$: :$\\map B v = \\set {u \\in V: \\set {u, v} \\in E}$ {{NoSources}} Category:Definitions/Graph Theory 0gncwb2r43l2nyhgh8hawis4u4igswy"}
+{"_id": "20966", "title": "Definition:Rectifiable Curve", "text": "A '''rectifiable curve''' is a curve of finite length."}
+{"_id": "20968", "title": "Definition:Ray (Topology)", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. A '''ray''' in $T$ is an embedding $R_{>0} \\to S$. Category:Definitions/Topology h89tok1dvfpmc6d0xrmzp40eszkad2b"}
+{"_id": "20969", "title": "Definition:Weight Function", "text": "A '''weight function''' on a set $S$ is a mapping from $S$ to the real numbers: :$w: S \\to \\R$ It is common for the requirements of a specific application under discussion for the codomain of $w$ to be restricted to the positive reals: :$w: S \\to \\R_{\\ge 0}$ The thing that determines whether a given mapping is a '''weight function''' depends more on how it is used."}
+{"_id": "20970", "title": "Definition:Rising Factorial", "text": "Let $x$ be a real number (but usually an integer). Let $n$ be a positive integer. Then '''$x$ to the (power of) $n$ rising''' is defined as: :$\\displaystyle x^{\\overline n} := \\prod_{j \\mathop = 0}^{n - 1} \\paren {x + j} = x \\paren {x + 1} \\cdots \\paren {x + n - 1}$"}
+{"_id": "20971", "title": "Definition:Falling Factorial", "text": "Let $x$ be a real number (but usually an integer). Let $n$ be a positive integer. Then '''$x$ to the (power of) $n$ falling''' is: :$\\displaystyle x^{\\underline n} := \\prod_{j \\mathop = 0}^{n - 1} \\left({x - j}\\right) = x \\left({x - 1}\\right) \\cdots \\left({x - n + 1}\\right)$"}
+{"_id": "20972", "title": "Definition:Stirling Numbers of the First Kind", "text": "=== Unsigned Stirling Numbers of the First Kind === {{:Definition:Stirling Numbers of the First Kind/Unsigned}} === Signed Stirling Numbers of the First Kind === {{:Definition:Stirling Numbers of the First Kind/Signed}}"}
+{"_id": "20973", "title": "Definition:Stirling Numbers", "text": "'''Stirling numbers''' come in various forms. In the below: : $\\delta_{n k}$ is the Kronecker delta : $n$ and $k$ are non-negative integers. === Unsigned Stirling Numbers of the First Kind === {{:Definition:Stirling Numbers of the First Kind/Unsigned}} === Signed Stirling Numbers of the First Kind === {{:Definition:Stirling Numbers of the First Kind/Signed}} === Stirling Numbers of the Second Kind === {{:Definition:Stirling Numbers of the Second Kind|Stirling Numbers of the Second Kind}}"}
+{"_id": "20974", "title": "Definition:Stirling Numbers of the Second Kind", "text": "{{:Definition:Stirling Numbers of the Second Kind/Definition 1}}"}
+{"_id": "20975", "title": "Definition:Pascal's Triangle", "text": ":$\\begin{array}{r|rrrrrrrrrr} n & \\binom n 0 & \\binom n 1 & \\binom n 2 & \\binom n 3 & \\binom n 4 & \\binom n 5 & \\binom n 6 & \\binom n 7 & \\binom n 8 & \\binom n 9 & \\binom n {10} & \\binom n {11} & \\binom n {12} \\\\ \\hline 0  & 1 &  0 &  0 &   0 &   0 &   0 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\\\ 1  & 1 &  1 &  0 &   0 &   0 &   0 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\\\ 2  & 1 &  2 &  1 &   0 &   0 &   0 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\\\ 3  & 1 &  3 &  3 &   1 &   0 &   0 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\\\ 4  & 1 &  4 &  6 &   4 &   1 &   0 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\\\ 5  & 1 &  5 & 10 &  10 &   5 &   1 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\\\ 6  & 1 &  6 & 15 &  20 &  15 &   6 &   1 &   0 &   0 &  0  &  0 &  0 & 0 \\\\ 7  & 1 &  7 & 21 &  35 &  35 &  21 &   7 &   1 &   0 &  0  &  0 &  0 & 0 \\\\ 8  & 1 &  8 & 28 &  56 &  70 &  56 &  28 &   8 &   1 &  0  &  0 &  0 & 0 \\\\ 9  & 1 &  9 & 36 &  84 & 126 & 126 &  84 &  36 &   9 &  1  &  0 &  0 & 0 \\\\ 10 & 1 & 10 & 45 & 120 & 210 & 252 & 210 & 120 &  45 &  10 &  1 &  0 & 0 \\\\ 11 & 1 & 11 & 55 & 165 & 330 & 462 & 462 & 330 & 165 &  55 & 11 &  1 & 0 \\\\ 12 & 1 & 12 & 66 & 220 & 495 & 792 & 924 & 792 & 495 & 220 & 66 & 12 & 1 \\\\ \\end{array}$"}
+{"_id": "20976", "title": "Definition:Stirling's Triangles", "text": "'''Stirling's Triangles''' are the arrays formed by arranging Stirling's Numbers of the first and second kind, as follows:"}
+{"_id": "20977", "title": "Definition:Array", "text": "An '''array''' is an arrangement (usually rectangular) of objects (usually numbers) such that the relations between the entries borne by their relative positions in that arrangement have a particular significance."}
+{"_id": "20979", "title": "Definition:Congruence (Number Theory)", "text": "Let $z \\in \\R$. === Definition by Remainder after Division === {{:Definition:Congruence (Number Theory)/Remainder after Division}} === Definition by Modulo Operation === {{:Definition:Congruence (Number Theory)/Modulo Operation}} === Definition by Integer Multiple === {{:Definition:Congruence (Number Theory)/Integer Multiple}}"}
+{"_id": "20981", "title": "Definition:Filter on Set", "text": "Let $S$ be a set. Let $\\powerset S$ denote the power set of $S$. === Definition 1 === {{:Definition:Filter on Set/Definition 1}} === Definition 2 === {{:Definition:Filter on Set/Definition 2}} === Filtered Set === {{:Definition:Filter on Set/Filtered Set}} === Trivial Filter === {{:Definition:Filter on Set/Trivial Filter}}"}
+{"_id": "20982", "title": "Definition:Ultrafilter on Set", "text": "=== Definition 1 === {{:Definition:Ultrafilter on Set/Definition 1}} === Definition 2 === {{:Definition:Ultrafilter on Set/Definition 2}} === Definition 3 === {{:Definition:Ultrafilter on Set/Definition 3}} === Definition 4 === {{:Definition:Ultrafilter on Set/Definition 4}}"}
+{"_id": "20983", "title": "Definition:Filter Basis", "text": "Let $S$ be a set. Let $\\powerset S$ denote the power set of $S$. === Definition 1 === {{:Definition:Filter Basis/Definition 1}} === Definition 2 === {{:Definition:Filter Basis/Definition 2}}"}
+{"_id": "20984", "title": "Definition:Addition/Modulo Addition", "text": "{{:Definition:Addition/Modulo Addition/Definition 1}}"}
+{"_id": "20985", "title": "Definition:Multiplication/Modulo Multiplication", "text": "{{:Definition:Multiplication/Modulo Multiplication/Definition 1}}"}
+{"_id": "20987", "title": "Definition:Topology Generated by Synthetic Sub-Basis", "text": "Let $X$ be a set. Let $\\mathcal S \\subseteq \\mathcal P \\left({X}\\right)$ be a synthetic sub-basis on $X$. === Definition 1 === {{:Definition:Topology Generated by Synthetic Sub-Basis/Definition 1}} === Definition 2 === {{:Definition:Topology Generated by Synthetic Sub-Basis/Definition 2}}"}
+{"_id": "20988", "title": "Definition:Initial Topology", "text": "Let $X$ be a set. Let $I$ be an indexing set. Let $\\left \\langle {\\left({Y_i, \\tau_i}\\right)} \\right \\rangle_{i \\mathop \\in I}$ be an indexed family of topological spaces indexed by $I$. Let $\\left \\langle {f_i: X \\to Y_i} \\right \\rangle_{i \\mathop \\in I}$ be an indexed family of mappings indexed by $I$."}
+{"_id": "20989", "title": "Definition:Image Filter", "text": "Let $X, Y$ be sets. Let $\\mathcal P \\left({X}\\right)$ and $\\mathcal P \\left({Y}\\right)$ be the power sets of $X$ and $Y$ respectively. Let $f: X \\to Y$ a mapping. Let $\\mathcal F \\subset \\mathcal P \\left({X}\\right)$ be a filter on $X$. Then :$f \\left({\\mathcal F}\\right) := \\left\\{{U \\subseteq Y: f^{-1} \\left({U}\\right) \\in \\mathcal F}\\right\\}$ is a filter on $Y$, called the '''image filter''' of $\\mathcal F$ with respect to $f$."}
+{"_id": "20990", "title": "Definition:Open Mapping", "text": "Let $\\struct {S_1, \\tau_1}$ and $\\struct {S_2, \\tau_2}$ be topological spaces. Let $f: S_1 \\to S_2$ be a mapping.  Then $f$ is said to be an '''open mapping''' {{iff}}: :$\\forall U \\in \\tau_1: f \\sqbrk U \\in \\tau_2$ where $f \\sqbrk U$ denotes the image of $U$ under $f$."}
+{"_id": "20991", "title": "Definition:Replicative Function", "text": "A '''replicative function''' is a real function $f$ such that: :$\\displaystyle \\forall n \\in \\Z_{\\ge 0}: \\sum_{k \\mathop = 0}^{n - 1} f \\left({x + \\frac k n}\\right) = f \\left({n x}\\right)$ where $\\sum$ denotes indexed summation."}
+{"_id": "20992", "title": "Definition:Dimension (Geometry)", "text": "The '''dimension''' of a (geometrical) space is the minimum number of coordinates needed to specify a point in it."}
+{"_id": "20994", "title": "Definition:Infinity", "text": "Informally, the term '''infinity''' is used to mean '''some infinite number''', but this concept falls very far short of a usable definition. The symbol $\\infty$ (supposedly invented by {{AuthorRef|John Wallis}}) is often used in this context to mean '''an infinite number'''. However, outside of its formal use in the definition of limits its use is strongly discouraged until you know what you're talking about. It is defined as having the following properties: :$\\forall n \\in \\Z: n < \\infty$ :$\\forall n \\in \\Z: n + \\infty = \\infty$ :$\\forall n \\in \\Z: n \\times \\infty = \\infty$ :$\\infty^2 = \\infty$ Similarly, the quantity written as $-\\infty$ is defined as having the following properties: :$\\forall n \\in \\Z: -\\infty< n$ :$\\forall n \\in \\Z: -\\infty + n = -\\infty$ :$\\forall n \\in \\Z: -\\infty \\times n = -\\infty$ :$\\paren {-\\infty}^2 = -\\infty$ The latter result seems wrong when you think of the rule that a negative number square equals a positive one, but remember that infinity is not exactly a number as such."}
+{"_id": "20996", "title": "Definition:Veridical Paradox", "text": "A '''veridical paradox''' is a counter-intuitive result which can be demonstrated to be true."}
+{"_id": "20997", "title": "Definition:Falsidical Paradox", "text": "A '''falsidical paradox''' is a result which, as well as being absurd on the surface, is the result of faulty reasoning and so is genuinely false. Such apparent paradoxes can be classified as fallacies."}
+{"_id": "20998", "title": "Definition:Antinomy", "text": "An '''antinomy''' is a self-contradictory statement which arises as the result of a valid argument from a set of premises which contains an inherent ambiguity or contradiction. Such apparent paradoxes can also often be classified as fallacies."}
+{"_id": "20999", "title": "Definition:Dialetheia", "text": "A '''dialetheia''' is a statement which is paradoxical through the fact that it is both true and false at the same time. As such, it is incompatible with {{AuthorRef|Aristotle}}'s Law of the Excluded Middle and Principle of Non-Contradiction, and therefore does not in general appear in the context of conventional mathematics."}
+{"_id": "21000", "title": "Definition:Finite Difference Operator", "text": "Let $f: \\R \\to \\R$ be a real function. The '''(finite) difference operator''' on $f$ comes in a number of forms, as follows."}
+{"_id": "21001", "title": "Definition:Chen Prime", "text": "A '''Chen Prime''' is a prime number $p$ such that $p + 2$ is either a prime number or a semiprime. The even integer $2 p + 2$ then satisfies Chen's Theorem. The first few '''Chen primes''' are :$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, \\ldots$ {{OEIS|A109611}} The first few '''non-Chen primes''' are :$43, 61, 73, 79, 97, 103, 151, 163, 173, 193, 223, 229, 241, \\ldots$ {{OEIS|A102540}}"}
+{"_id": "21002", "title": "Definition:Semiprime Number", "text": "A '''semiprime (number)''' is an integer which is the product of two (not necessarily distinct) primes."}
+{"_id": "21003", "title": "Definition:Sufficiently Large", "text": "Let $P$ be a property of real numbers. Then: : '''$P \\left({x}\\right)$ holds for all sufficiently large $x$''' {{iff}}: :$\\exists a \\in \\R: \\forall x \\in \\R: x \\ge a: P \\left({x}\\right)$ That is: :''There exists a real number $a$ such that for every (real) number not less than $a$, the property $P$ holds.'' It is not necessarily the case, for a given property $P$ about which such a statement is made, that the value of $a$ actually needs to be known, merely that such a value can be demonstrated to exist."}
+{"_id": "21004", "title": "Definition:Inverse Trigonometric Function", "text": "As there are six basic trigonometric functions, so each of these has its inverse functions."}
+{"_id": "21005", "title": "Definition:Harmonic Numbers", "text": "The '''harmonic numbers''' are denoted $H_n$ and are defined for positive integers $n$: :$\\displaystyle \\forall n \\in \\Z, n \\ge 0: H_n = \\sum_{k \\mathop = 1}^n \\frac 1 k$"}
+{"_id": "21006", "title": "Definition:Vector Cross Product", "text": "Let $\\mathbf a$ and $\\mathbf b$ be vectors in a vector space $\\mathbf V$ of $3$ dimensions: :$\\mathbf a = a_i \\mathbf i + a_j \\mathbf j + a_k \\mathbf k$ :$\\mathbf b = b_i \\mathbf i + b_j \\mathbf j + b_k \\mathbf k$ where $\\left({\\mathbf i, \\mathbf j, \\mathbf k}\\right)$ is the standard ordered basis of $\\mathbf V$."}
+{"_id": "21007", "title": "Definition:Digit Sum", "text": "Let $n \\in \\Z: n \\ge 0$. The '''digit sum of $n$ to base $b$''' is the sum of all the digits of $n$ when expressed in base $b$. That is, if: :$\\displaystyle n = \\sum_{k \\mathop \\ge 0} r_k b^k$ where $0 \\le r_k < b$, then: :$\\displaystyle s_b \\left({n}\\right) = \\sum_{k \\mathop \\ge 0} r_k$"}
+{"_id": "21008", "title": "Definition:Digital Root", "text": "Let $n \\in \\Z: n \\ge 0$. Let $n$ be expressed in base $b$ notation. Let $n_1 = \\map {s_b} n$ be the digit sum of $n$ to base $b$. Then let $n_2 = \\map {s_b} {n_1}$ be the digit sum of $n_1$ to base $b$. Repeat the process, until $n_m$ has only one digit, that is, that $1 \\le n_m < b$. Then $n_m$ is the '''digital root of $n$ to the base $b$'''."}
+{"_id": "21009", "title": "Definition:Gaussian Integer", "text": "A '''Gaussian integer''' is a complex number whose real and imaginary parts are both integers. That is, a '''Gaussian integer''' is a number in the form: :$a + b i: a, b \\in \\Z$ The set of all '''Gaussian integers''' can be denoted $\\Z \\sqbrk i$, and hence can be defined as: :$\\Z \\sqbrk i = \\set {a + b i: a, b \\in \\Z}$"}
+{"_id": "21010", "title": "Definition:Gaussian Rational", "text": "A '''Gaussian rational''' is a complex number whose real and imaginary parts are both rational numbers. That is, a '''Gaussian rational''' is a number in the form: :$a + b i: a, b \\in \\Q$ The set of all '''Gaussian rationals''' can be denoted $\\Q \\sqbrk i$, and hence can be defined as: :$\\Q \\sqbrk i = \\set {a + b i: a, b \\in \\Q}$"}
+{"_id": "21011", "title": "Definition:Field Automorphism", "text": "Let $\\struct {F, +, \\circ}$ be a field. Let $\\phi: F \\to F$ be a (field) isomorphism from $F$ to itself. Then $\\phi$ is a field automorphism."}
+{"_id": "21012", "title": "Definition:Field Homomorphism", "text": "Let $\\struct {F, +, \\times}$ and $\\struct {K, \\oplus, \\otimes}$ be fields. Let $\\phi: F \\to K$ be a mapping such that both $+$ and $\\times$ have the morphism property under $\\phi$. That is, $\\forall a, b \\in F$: {{begin-eqn}} {{eqn | n = 1       | l = \\map \\phi {a + b}       | r = \\map \\phi a \\oplus \\map \\phi b }} {{eqn | n = 2       | l = \\map \\phi {a \\times b}       | r = \\map \\phi a \\otimes \\map \\phi b }} {{end-eqn}} Then $\\phi: \\struct {F, +, \\times} \\to \\struct {K, \\oplus, \\otimes}$ is a field homomorphism."}
+{"_id": "21013", "title": "Definition:Monic Polynomial", "text": "Let $R$ be a commutative ring with unity. Let $f \\in R \\sqbrk x$ be a polynomial in one variable over $R$. Then $f$ is '''monic''' {{iff}} $f$ is nonzero and its leading coefficient is $1$."}
+{"_id": "21015", "title": "Definition:Leading Coefficient of Polynomial", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\struct {S, +, \\circ}$ be a subring of $R$. Let $\\displaystyle f = \\sum_{k \\mathop = 0}^n a_k \\circ x^k$ be a polynomial in $x$ over $S$. The coefficient $a_n \\ne 0_R$ is called the '''leading coefficient''' of $f$. === Polynomial Form === {{:Definition:Leading Coefficient of Polynomial/Polynomial Form}}"}
+{"_id": "21016", "title": "Definition:Experiment", "text": "An '''experiment''' can be defined in natural language as a '''test to see what happens'''. === Informal Definition === {{:Definition:Experiment/Informal Definition}} === Formal Definition === {{:Definition:Experiment/Formal Definition}}"}
+{"_id": "21017", "title": "Definition:Sample Space", "text": "Let $\\EE$ be an experiment. The '''sample space''' of $\\EE$ is usually denoted $\\Omega$ (Greek capital '''omega'''), and is defined as '''the set of all possible outcomes of $\\EE$'''."}
+{"_id": "21018", "title": "Definition:Elementary Event", "text": "Let $\\EE$ be an experiment. An '''elementary event''' of $\\EE$, often denoted $\\omega$ (Greek lowercase '''omega''') is one of the elements of the sample space $\\Omega$ (Greek capital '''omega''') of $\\EE$."}
+{"_id": "21019", "title": "Definition:Event Space", "text": "Let $\\EE$ be an experiment whose probability space is $\\struct {\\Omega, \\Sigma, \\Pr}$. The '''event space''' of $\\EE$ is usually denoted $\\Sigma$ (Greek capital '''sigma'''), and is '''the set of all outcomes of $\\EE$ which are interesting'''. By definition, $\\struct {\\Omega, \\Sigma}$ is a measurable space. Hence the '''event space''' $\\Sigma$ is a sigma-algebra on $\\Omega$."}
+{"_id": "21020", "title": "Definition:Event", "text": "Let $\\EE$ be an experiment. An '''event in $\\EE$''' is an element of the event space $\\Sigma$ of $\\EE$."}
+{"_id": "21021", "title": "Definition:Probability Measure", "text": "Let $\\EE$ be an experiment. === Definition 1 === {{:Definition:Probability Measure/Definition 1}} === Definition 2 === {{:Definition:Probability Measure/Definition 2}} === Definition 3 === {{:Definition:Probability Measure/Definition 3}} === Definition 4 === {{:Definition:Probability Measure/Definition 4}}"}
+{"_id": "21023", "title": "Definition:Percentage", "text": "A '''percentage''' is a convenient way of expressing (usually) a fraction as an easily-grasped number, by multiplying it by $100$ and appending the symbol $\\%$ which means '''per cent'''. As conventionally understood, the number being rendered as a '''percentage''' is between $0$ and $1$, thus determining the range of a '''percentage''' to be $0 \\%$ to $100 \\%$. However in many contexts that range is frequently extended (in particular, in the field of rhetoric): :''I want every member of this team to give one hundred and ten percent.''"}
+{"_id": "21024", "title": "Definition:Sigma-Ring", "text": "{{:Definition:Sigma-Ring/Definition 1}}"}
+{"_id": "21025", "title": "Definition:Delta-Ring", "text": "A '''delta-ring''' (which can conveniently be written '''$\\delta$-ring''') is a ring of sets which is closed under countable intersections. That is, a ring of sets $\\RR$ is a '''delta-ring''' {{iff}}: :$\\displaystyle A_1, A_2, \\ldots \\in \\RR \\implies \\bigcap_{n \\mathop = 1}^\\infty A_n \\in \\RR$"}
+{"_id": "21026", "title": "Definition:Delta-Algebra", "text": "A '''delta-algebra''' is a delta-ring with a unit. Thus, a '''delta-algebra''' is an algebra of sets which is closed under countable intersections."}
+{"_id": "21027", "title": "Definition:Borel Algebra", "text": "A '''Borel algebra''' is sometimes used as another name for a $\\sigma$-algebra or $\\delta$-algebra. Can be confused with Borel $\\sigma$-Algebra, which is a specific kind of $\\sigma$-algebra."}
+{"_id": "21028", "title": "Definition:Involution (Mapping)", "text": "An '''involution''' is a mapping which is its own inverse. === Definition 1 === {{:Definition:Involution (Mapping)/Definition 1}} === Definition 2 === {{:Definition:Involution (Mapping)/Definition 2}} === Definition 3 === {{:Definition:Involution (Mapping)/Definition 3}}"}
+{"_id": "21030", "title": "Definition:Additive Function (Conventional)", "text": "Let $f: S \\to S$ be a mapping on an algebraic structure $\\struct {S, +}$. Then $f$ is an '''additive function''' {{iff}} it preserves the addition operation: :$\\forall x, y \\in S: \\map f {x + y} = \\map f x + \\map f y$"}
+{"_id": "21031", "title": "Definition:Completely Additive Function", "text": "Let $\\left({R, +, \\times}\\right)$ be a ring. Let $f: R \\to R$ be a mapping on $R$. Then $f$ is described as '''completely additive''' {{iff}}: :$\\forall m, n \\in R: f \\left({m \\times n}\\right) = f \\left({m}\\right) + f \\left({n}\\right)$ That is, a '''completely additive function''' is one where the value of a product of two numbers equals the sum of the value of each one individually."}
+{"_id": "21032", "title": "Definition:Additive Function (Measure Theory)", "text": "Let $\\SS$ be an algebra of sets. Let $f: \\SS \\to \\overline \\R$ be a function, where $\\overline \\R$ denotes the set of extended real numbers. Then $f$ is defined to be '''additive''' {{iff}}: :$\\forall S, T \\in \\SS: S \\cap T = \\O \\implies \\map f {S \\cup T} = \\map f S + \\map f T$ That is, for any two disjoint elements of $\\SS$, $f$ of their union equals the sum of $f$ of the individual elements. Note from Finite Union of Sets in Additive Function that: :$\\displaystyle \\map f {\\bigcup_{i \\mathop = 1}^n S_i} = \\sum_{i \\mathop = 1}^n \\map f {S_i}$ where $S_1, S_2, \\ldots, S_n$ is any finite collection of pairwise disjoint elements of $\\SS$."}
+{"_id": "21033", "title": "Definition:Countably Additive Function", "text": "Let $\\Sigma$ be a $\\sigma$-algebra. Let $f: \\Sigma \\to \\overline \\R$ be a function, where $\\overline \\R$ denotes the set of extended real numbers. Then $f$ is defined as '''countably additive''' {{iff}}: :$\\displaystyle \\map f {\\bigcup_{n \\mathop \\in \\N} E_n} = \\sum_{n \\mathop \\in \\N} \\map f {E_n}$ where $\\left \\langle {E_n} \\right \\rangle$ is any sequence of pairwise disjoint elements of $\\Sigma$. That is, for any countably infinite set of pairwise disjoint elements of $\\Sigma$, $f$ of their union equals the sum of $f$ of the individual elements."}
+{"_id": "21034", "title": "Definition:Subadditive Function (Conventional)", "text": "Let $\\left({S, +_S}\\right)$ and $\\left({T, +_T, \\preceq}\\right)$ be semigroups such that $\\left({T, +_T, \\preceq}\\right)$ is ordered. Let $f: S \\to T$ be a mapping from $S$ to $T$ which satisfies the relation: :$\\forall a, b \\in S: f \\left({a +_S b}\\right) \\preceq f \\left({a}\\right) +_T f \\left({b}\\right)$ Then $f$ is defined as being '''subadditive'''. The usual context in which this is encountered is where $S$ and $T$ are both the set of real numbers $\\R$ (or a subset of them)."}
+{"_id": "21036", "title": "Definition:Subadditive Function (Measure Theory)", "text": "Let $\\mathcal S$ be an algebra of sets. Let $f: \\mathcal S \\to \\overline \\R$ be a function, where $\\overline \\R$ denotes the extended set of real numbers. Then $f$ is defined to be '''subadditive''' (or '''sub-additive''') iff: :$\\forall S, T \\in \\mathcal S: f \\left({S \\cup T}\\right) \\le f \\left({S}\\right) + f \\left({T}\\right)$ That is, for any two elements of $\\mathcal S$, $f$ applied to their union is not greater than the sum of $f$ of the individual elements."}
+{"_id": "21037", "title": "Definition:Countably Subadditive Function", "text": "Let $\\Sigma$ be a $\\sigma$-algebra over a set $X$. Let $f: \\Sigma \\to \\overline \\R$ be a function, where $\\overline \\R$ denotes the set of extended real numbers. Then $f$ is defined as '''countably subadditive''' iff, for any sequence $\\left \\langle {E_n} \\right \\rangle_{n \\in \\N}$ of elements of $\\Sigma$: :$\\displaystyle f \\left({\\bigcup_{n \\mathop = 0}^\\infty E_n}\\right) \\le \\sum_{n \\mathop = 0}^\\infty f \\left({E_n}\\right)$"}
+{"_id": "21038", "title": "Definition:Measurable Space", "text": "Let $\\Sigma$ be a $\\sigma$-algebra on a set $X$. Then the pair $\\struct {X, \\Sigma}$ is called a '''measurable space'''."}
+{"_id": "21040", "title": "Definition:Monotone (Measure Theory)", "text": "Let $\\mathcal S$ be an algebra of sets. Let $f: \\mathcal S \\to \\overline \\R$ be an extended real-valued function, where $\\overline \\R$ denotes the set of extended real numbers. Then $f$ is defined as '''monotone''' or '''monotonic''' iff: :$\\forall A, B \\in \\mathcal S: A \\subseteq B \\iff f \\left({A}\\right) \\le f \\left({B}\\right)$ Category:Definitions/Set Systems o9ucnp6vtw43gssg2efgwqknu392hel"}
+{"_id": "21042", "title": "Definition:Disjoint Union (Set Theory)", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be an $I$-indexed family of sets. The '''disjoint union''' of $\\family {S_i}_{i \\mathop \\in I}$ is defined as the set: :$\\displaystyle \\bigsqcup_{i \\mathop \\in I} S_i = \\bigcup_{i \\mathop \\in I} \\set {\\tuple {x, i}: x \\in S_i}$ where $\\bigcup$ denotes union. Each of the sets $S_i$ is canonically embedded in the '''disjoint union''' as the set: :$S_i^* = \\set {\\tuple {x, i}: x \\in S_i}$"}
+{"_id": "21043", "title": "Definition:Disjoint Union (Probability Theory)", "text": "Let $\\mathcal C$ be a collection of pairwise disjoint sets. That is, for all sets $A, B \\in \\mathcal C: A \\ne B \\implies A \\cap B = \\varnothing$. Then the union of all sets in $\\mathcal C$ is called their '''disjoint union''': :$\\displaystyle \\bigsqcup_{A \\mathop \\in \\mathcal C} A \\equiv \\bigcup_{A \\mathop \\in \\mathcal C} A$ That is, in this context the term '''disjoint union''' means '''union of sets which are pairwise disjoint'''."}
+{"_id": "21044", "title": "Definition:Equiprobable Outcomes", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a finite probability space. Let $\\Omega = \\set {\\omega_1, \\omega_1, \\ldots, \\omega_n}$. Suppose that $\\map \\Pr {\\omega_i} = \\map \\Pr {\\omega_j}$ for all the $\\omega_i, \\omega_j \\in \\Omega$. Then from Probability Measure on Equiprobable Outcomes: :$\\forall \\omega \\in \\Omega: \\map \\Pr \\omega = \\dfrac 1 n$ :$\\forall A \\subseteq \\Omega: \\map \\Pr A = \\dfrac {\\card A} n$ Such a probability space is said to have '''equiprobable outcomes''', and is sometimes referred to as an equiprobability space."}
+{"_id": "21045", "title": "Definition:Logical Matrix", "text": "A '''logical matrix''' (or '''boolean matrix''') is a matrix whose entries are all either $0$ or $1$."}
+{"_id": "21046", "title": "Definition:Finite Probability Space", "text": "A '''finite probability space''' is a discrete probability space $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ such that the sample space $\\Omega$ is finite. {{SUBPAGENAME}} 3fpbykva2knibxeo21kkyel1xvzyh8i"}
+{"_id": "21047", "title": "Definition:Die", "text": "thumb300pxrightTwo six-sided dice A '''die''' (plural: '''dice''') is a device for generating random numbers, traditionally between $1$ and $6$. A die usually consists of a cube on which each face denotes a different number, usually (again) from $1$ to $6$. Traditionally, opposite faces add to $7$, so: :$1$ is opposite $6$ :$2$ is opposite $5$ :$3$ is opposite $4$. The idea is that when you roll it, or throw it, or otherwise cause it to land upon a flattish surface, one of the sides will be face-up, and (with a good design of die) which side that is can not be precisely predicted. Hence it serves as a generator of a discrete uniform distribution. By using dice with different shapes (usually either platonic solids or prisms), a different probability space can be generated. {{DefinitionWanted|Describe general polyhedral dice, or link to a Wikipedia article on FRP or something.}}"}
+{"_id": "21048", "title": "Definition:Equiprobability Space", "text": "An '''equiprobability space''' is a finite probability space $\\struct {\\Omega, \\Sigma, \\Pr}$ with equiprobable outcomes. That is, for all $\\omega_i, \\omega_j \\in \\Omega$: :$\\map \\Pr {\\omega_i} = \\map \\Pr {\\omega_j}$ From Probability Measure on Equiprobable Outcomes, we have that: :$\\forall \\omega \\in \\Omega: \\map \\Pr \\omega = \\dfrac 1 n$ :$\\forall A \\subseteq \\Omega: \\map \\Pr A = \\dfrac {\\card A} n$ Category:Definitions/Probability Theory 4q5j6b2l1h3stji0utitop5kg1lzl5l"}
+{"_id": "21049", "title": "Definition:Binary Notation", "text": "'''Binary notation''' is the technique of expressing numbers in base $2$. That is, every number $x \\in \\R$ is expressed in the form: :$\\displaystyle x = \\sum_{j \\mathop \\in \\Z} r_j 2^j$ where $\\forall j \\in \\Z: r_j \\in \\set {0, 1}$."}
+{"_id": "21050", "title": "Definition:Hexadecimal Notation", "text": "'''Hexadecimal notation''' is the technique of expressing numbers in base $16$. Every number $x \\in \\R$ is expressed in the form: :$\\displaystyle x = \\sum_{j \\mathop \\in \\Z} r_j 16^j$ where: :$\\forall j \\in \\Z: r_j \\in \\left\\{ {0, 1, \\ldots, 15}\\right\\}$"}
+{"_id": "21051", "title": "Definition:General Logarithm", "text": "=== Positive Real Numbers === {{:Definition:General Logarithm/Positive Real}} === Complex Numbers === {{:Definition:General Logarithm/Complex}}"}
+{"_id": "21052", "title": "Definition:Conditional Probability", "text": "Let $\\EE$ be an experiment with probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $A, B \\in \\Sigma$ be events of $\\EE$. We write the '''conditional probability of $A$ given $B$''' as $\\condprob A B$, and define it as: :''the probability that $A$ has occurred, given that $B$ has occurred.''"}
+{"_id": "21053", "title": "Definition:Independent Events", "text": "Let $\\EE$ be an experiment with probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $A, B \\in \\Sigma$ be events of $\\EE$ such that $\\map \\Pr A > 0$ and $\\map \\Pr B > 0$. === Definition 1 === {{:Definition:Independent Events/Definition 1}} === Definition 2 === {{:Definition:Independent Events/Definition 2}}"}
+{"_id": "21054", "title": "Definition:Disjoint Events", "text": "Let $A$ and $B$ be events in a probability space. Then $A$ and $B$ are '''disjoint''' {{iff}}: :$A \\cap B = \\O$ It follows by definition of probability measure that $A$ and $B$ are '''disjoint''' {{iff}}: :$\\map \\Pr {A \\cap B} = 0$"}
+{"_id": "21055", "title": "Definition:Can't Happen", "text": "=== Probability Theory === {{:Definition:Can't Happen/Probability Theory}} === Computer Science === {{:Definition:Can't Happen/Computer Science}} Category:Definitions/Language Definitions 3rytkxheuo3ss7q7wnp18lgbfwcv0im"}
+{"_id": "21057", "title": "Definition:Partition (Probability Theory)", "text": "Let $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ be a probability space. A '''partition of $\\Omega$''' is a family $\\left\\{{B_i: i \\in I}\\right\\}$ of disjoint events such that $\\displaystyle \\bigcup_i B_i = \\Omega$."}
+{"_id": "21058", "title": "Definition:Increasing Sequence of Events", "text": "Let $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ be a probability space. Let $\\left \\langle {A_n}\\right \\rangle$ be a sequence of events in $\\Sigma$. Then $\\left \\langle {A_n}\\right \\rangle$ is described as '''increasing''' iff: :$\\forall i \\in \\N: A_i \\subseteq A_{i+1}$"}
+{"_id": "21059", "title": "Definition:Limit of Sequence of Events", "text": "Let $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ be a probability space. === Increasing Sequence of Events === {{:Definition:Limit of Sequence of Events/Increasing}} === Decreasing Sequence of Events === {{:Definition:Limit of Sequence of Events/Decreasing}} Category:Definitions/Probability Theory 7j4gf0u73j8c1bj99gn7ek0m2kh3jsr"}
+{"_id": "21060", "title": "Definition:Decreasing Sequence of Events", "text": "Let $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ be a probability space. Let $\\left \\langle {A_n}\\right \\rangle$ be a sequence of events in $\\Sigma$. Then $\\left \\langle {A_n}\\right \\rangle$ is described as '''decreasing''' iff: :$\\forall i \\in \\N: A_{i+1} \\subseteq A_i$"}
+{"_id": "21061", "title": "Definition:Random Variable/Discrete", "text": "Let $\\EE$ be an experiment with a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. <!-- A '''discrete random variable''' on $\\struct {\\Omega, \\Sigma, \\Pr}$ is a random variable on $\\struct {\\Omega, \\Sigma, \\Pr}$ such that $\\Omega_X$, the image of $X$, is a countable subset of $\\R$. --> A '''discrete random variable''' on $\\struct {\\Omega, \\Sigma, \\Pr}$ is a mapping $X: \\Omega \\to \\R$ such that: :$(1): \\quad$ The image of $X$ is a countable subset of $\\R$ :$(2): \\quad$ $\\forall x \\in \\R: \\set {\\omega \\in \\Omega: \\map X \\omega = x} \\in \\Sigma$ Alternatively (and meaning exactly the same thing), the second condition can be written as: :$(2)': \\quad$ $\\forall x \\in \\R: \\map {X^{-1} } x \\in \\Sigma$ where $\\map {X^{-1} } x$ denotes the preimage of $x$. Note that if $x \\in \\R$ is not the image of any elementary event $\\omega$, then $\\map {X^{-1} } x = \\O$ and of course by definition of event space as a sigma-algebra, $\\O \\in \\Sigma$. Note that a discrete random variable also fulfils the conditions for it to be a random variable."}
+{"_id": "21062", "title": "Definition:Probability Mass Function", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $X: \\Omega \\to \\R$ be a discrete random variable on $\\struct {\\Omega, \\Sigma, \\Pr}$. Then the '''(probability) mass function''' of $X$ is the (real-valued) function $p_X: \\R \\to \\closedint 0 1$ defined as: :$\\forall x \\in \\R: \\map {p_X} x = \\begin{cases} \\map \\Pr {\\set {\\omega \\in \\Omega: \\map X \\omega = x} } & : x \\in \\Omega_X \\\\ 0 & : x \\notin \\Omega_X \\end{cases}$ where $\\Omega_X$ is defined as $\\Img X$, the image of $X$. That is, $\\map {p_X} x$ is the probability that the discrete random variable $X$ takes the value $x$."}
+{"_id": "21063", "title": "Definition:Bernoulli Distribution", "text": "Let $X$ be a discrete random variable on a probability space. Then $X$ has the '''Bernoulli distribution with parameter $p$''' {{iff}}: : $(1): \\quad X$ has exactly two possible values, for example $\\Img X = \\set {a, b}$ : $(2): \\quad \\map \\Pr {X = a} = p$ : $(3): \\quad \\map \\Pr {X = b} = 1 - p$ where $0 \\le p \\le 1$."}
+{"_id": "21064", "title": "Definition:Bernoulli Trial", "text": "A '''Bernoulli trial''' is an experiment whose sample space has two elements, which can be variously described, for example, as: :'''Success''' and '''failure''' :'''True''' and '''False''' :$1$ and $0$ :the classic '''heads''' and '''tails'''. Formally, a '''Bernoulli trial''' is modelled by a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$ such that: :$\\Omega = \\set {a, b}$ :$\\Sigma = \\powerset \\Omega$ :$\\map \\Pr a = p, \\map \\Pr b = 1 - p$ where: :$\\powerset \\Omega$ denotes the power set of $\\Omega$ :$0 \\le p \\le 1$ That is, $\\Pr$ obeys a Bernoulli distribution. === Bernoulli Variable === {{:Definition:Bernoulli Trial/Bernoulli Variable}}"}
+{"_id": "21065", "title": "Definition:Binomial Distribution", "text": "Let $X$ be a discrete random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Then $X$ has the '''binomial distribution with parameters $n$ and $p$''' {{iff}}: :$\\Img X = \\set {0, 1, \\ldots, n}$ :$\\map \\Pr {X = k} = \\dbinom n k p^k \\paren {1 - p}^{n - k}$ where $0 \\le p \\le 1$."}
+{"_id": "21066", "title": "Definition:Poisson Distribution", "text": "Let $X$ be a discrete random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Then $X$ has the '''Poisson distribution with parameter $\\lambda$''' (where $\\lambda > 0$) {{iff}}: :$\\Img X = \\set {0, 1, 2, \\ldots} = \\N$ :$\\map \\Pr {X = k} = \\dfrac 1 {k!} \\lambda^k e^{-\\lambda}$ It is written: :$X \\sim \\Poisson \\lambda$"}
+{"_id": "21067", "title": "Definition:Geometric Distribution", "text": "Let $X$ be a discrete random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. $X$ has the '''geometric distribution with parameter $p$''' {{iff}}: :$\\map X \\Omega = \\set {0, 1, 2, \\ldots} = \\N$ :$\\map \\Pr {X = k} = \\paren {1 - p} p^k$ where $0 < p < 1$."}
+{"_id": "21068", "title": "Definition:Negative Binomial Distribution", "text": "Let $X$ be a discrete random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. There are two forms of the '''negative binomial distribution''', as follows: === First Form === {{:Definition:Negative Binomial Distribution/First Form}} === Second Form === {{:Definition:Negative Binomial Distribution/Second Form}}"}
+{"_id": "21069", "title": "Definition:Bernoulli Process", "text": "A '''Bernoulli process''' is a sequence (either finite or infinite) of Bernoulli trials, each of which has the same parameter $p$. That is, '''Bernoulli process (with parameter $p$)''' is a sequence $\\left \\langle {X_i}\\right \\rangle$ (either finite or infinite) such that: * The value of each $X_i$ is one of two values (for example: $a$ or $b$). * The probability that $X_i = a$ is the same for all $i$ (for example: $p$). That is, it is a sequence of experiments, all of which can be modelled by the same Bernoulli distribution. Note: The assumption is that the outcomes of all the Bernoulli trials are independent. {{NamedforDef|Jacob Bernoulli|cat=Bernoulli, Jacob}} Category:Definitions/Probability Theory 6576ce743b258dwovtfzmckcrt9i5kw"}
+{"_id": "21070", "title": "Definition:Expectation", "text": "The '''expectation''' of a random variable is the arithmetic mean of its values."}
+{"_id": "21071", "title": "Definition:Normalized Weight Function", "text": "Let $S = \\left \\langle {x_1, x_2, \\ldots, x_n}\\right \\rangle$ be a sequence of real numbers. Let $W \\left({x}\\right)$ be a weight function to be applied to the elements of $S$. Then $W$ is defined as being '''normalized''' if: :$\\displaystyle \\sum_x W \\left({x}\\right) = 1$ {{SUBPAGENAME}} {{SUBPAGENAME}} 2r5oz2hi2xbnc4etn6rnpkl1y4pooah"}
+{"_id": "21072", "title": "Definition:Variance/Discrete", "text": "Let $X$ be a discrete random variable. Then the '''variance of $X$''', written $\\var X$, is a measure of how much the values of $X$ varies from the expectation $\\expect X$, and is defined as:"}
+{"_id": "21073", "title": "Definition:Conditional Expectation", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $X$ be a discrete random variable on $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $B$ be an event in $\\struct {\\Omega, \\Sigma, \\Pr}$ such that $\\map \\Pr B > 0$. The '''conditional expectation of $X$ given $B$''' is written $\\expect {X \\mid B}$ and defined as: :$\\expect {X \\mid B} = \\displaystyle \\sum_{x \\mathop \\in \\image X} x \\, \\map \\Pr {X = x \\mid B}$ where: :$\\map \\Pr {X = x \\mid B}$ denotes the conditional probability that $X = x$ given $B$ whenever this sum converges absolutely."}
+{"_id": "21075", "title": "Definition:Independent Random Variables", "text": "Let $\\EE$ be an experiment with probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $X$ and $Y$ be random variables on $\\struct {\\Omega, \\Sigma, \\Pr}$. Then $X$ and $Y$ are defined as '''independent (of each other)''' {{iff}}: :$\\map \\Pr {X = x, Y = y} = \\map \\Pr {X = x} \\map \\Pr {Y = y}$ where $\\map \\Pr {X = x, Y = y}$ is the joint probability mass function of $X$ and $Y$."}
+{"_id": "21076", "title": "Definition:Uniform Distribution/Discrete", "text": "Let $X$ be a discrete random variable on a probability space. Then $X$ has a '''discrete uniform distribution with parameter $n$''' {{iff}}: :$\\Img X = \\set {1, 2, \\ldots, n}$ :$\\map \\Pr {X = k} = \\dfrac 1 n$ That is, there is a number of outcomes with an equal probability of occurrence.  This is written: :$X \\sim \\DiscreteUniform n$"}
+{"_id": "21077", "title": "Definition:Cumulative Distribution Function", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $X$ be a random variable on $\\struct {\\Omega, \\Sigma, \\Pr}$. The '''cumulative distribution function''' (or '''c.d.f.''') of $X$ is denoted $\\map F X$, and defined as: :$\\forall x \\in \\R: \\map {\\map F X} x := \\map \\Pr {X \\le x}$"}
+{"_id": "21078", "title": "Definition:Random Variable/Continuous", "text": "Let $\\mathcal E$ be an experiment with a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. A '''continuous random variable''' on $\\struct {\\Omega, \\Sigma, \\Pr}$ is a random variable $X: \\Omega \\to \\R$ whose cumulative distribution function is continuous for all $x \\in \\R$."}
+{"_id": "21080", "title": "Definition:Degenerate Distribution", "text": "Let $X$ be a discrete random variable on a probability space. Then $X$ has a '''degenerate distribution with parameter $r$''' if: :$\\Omega_X = \\set r$ :$\\map \\Pr {X = k} = \\begin{cases} 1 & : k = r \\\\ 0 & : k \\ne r \\end{cases}$ That is, there is only value that $X$ can take, namely $r$, which it takes with certainty."}
+{"_id": "21081", "title": "Definition:Probability Generating Function", "text": "Let $X$ be a discrete random variable whose codomain, $\\Omega_X$, is a subset of the natural numbers $\\N$. Let $p_X$ be the probability mass function for $X$. The '''probability generating function for $X$''', denoted $\\map {\\Pi_X} s$, is the formal power series defined by: :$\\displaystyle \\map {\\Pi_X} s := \\sum_{n \\mathop = 0}^\\infty \\map {p_X} n s^n \\in \\R \\left[\\left[{s}\\right]\\right]$"}
+{"_id": "21082", "title": "Definition:Convergent Series", "text": "Let $\\left({S, \\circ, \\tau}\\right)$ be a topological semigroup. Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be a series in $S$. This series is said to be '''convergent''' {{iff}} its sequence of partial sums $\\left \\langle {s_N} \\right \\rangle$ converges in the topological space $\\left({S, \\tau}\\right)$. If $s_N \\to s$ as $N \\to \\infty$, the series '''converges to the sum $s$''', and one writes $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n = s$. === Convergent Series in a Normed Vector Space (Definition 1) === {{:Definition:Convergent Series/Normed Vector Space/Definition 1}} === Convergent Series in a Normed Vector Space (Definition 2) === {{:Definition:Convergent Series/Normed Vector Space/Definition 2}} === Convergent Series in a Number Field === {{:Definition:Convergent Series/Number Field}}"}
+{"_id": "21084", "title": "Definition:Probability Distribution", "text": "Let $\\tuple {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $X: \\Omega \\to \\R$ be a random variable on $\\tuple {\\Omega, \\Sigma, \\Pr}$. Then the '''probability distribution of $X$''' is the pushforward $X_* \\Pr$ of $\\Pr$ on $\\tuple {\\R, \\map \\BB \\R}$, where $\\map \\BB \\R$ denotes the Borel $\\sigma$-algebra on $\\R$."}
+{"_id": "21086", "title": "Definition:Marginal Probability Mass Function", "text": "Let $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ be a probability space. Let $X: \\Pr \\to \\R$ and $Y: \\Pr \\to \\R$ both be discrete random variables on $\\left({\\Omega, \\Sigma, \\Pr}\\right)$. Let $p_{X, Y}$ be the joint probability mass function of $X$ and $Y$. Then the probability mass functions $p_X$ and $p_Y$ are called the '''marginal (probability) mass functions''' of $X$ and $Y$ respectively. The marginal mass function can be obtained from the joint mass function: {{begin-eqn}} {{eqn | l=p_X \\left({x}\\right)       | r=\\Pr \\left({X = x}\\right)       | c= }} {{eqn | r=\\sum_{y \\mathop \\in \\operatorname{Im} \\left({Y}\\right)} \\Pr \\left({X = x, Y = y}\\right)       | c= }} {{eqn | r=\\sum_y p_{X, Y} \\left({x, y}\\right)       | c= }} {{end-eqn}}"}
+{"_id": "21088", "title": "Definition:Moment of Discrete Random Variable", "text": "Let $X$ be a discrete random variable. Then the '''$n$th moment of $X$''' is denoted $\\mu'_n$ and defined as: :$\\mu'_n = \\expect {X^n}$ where $\\expect {\\, \\cdot \\,}$ denotes the expectation function."}
+{"_id": "21089", "title": "Definition:Diameter of Quadrilateral", "text": "Let $ABCD$ be a quadrilateral: :400px The '''diameters''' of $ABCD$ are the lines $AC$ and $BD$ joining their opposite vertices."}
+{"_id": "21091", "title": "Definition:Matrix Exponential", "text": "The '''matrix exponential''', denoted $e^{t \\mathbf A}$, of the order $n$ constant square matrix $\\mathbf A$, is defined to be the unique solution to the initial value problem: :$X' = \\mathbf A X, \\ \\map X {\\mathbf 0} = \\mathbf I$ where: :$\\mathbf I$ is the unit matrix :$X$ is an order $n$ square matrix which is a function of $t$."}
+{"_id": "21092", "title": "Definition:Stability (Differential Equations)", "text": "For first-order autonomous systems, define $\\phi \\left({t, x_0}\\right)$ to be the unique solution with initial condition $x \\left({0}\\right) = x_0$. {{explain|Make this more precise: \"for first-order autonomous systems\" should be replaced by a statement specifying something like, \"Let $S$ be a first-order autonomous system\" (or whatever letter is appropriate for such an object in this context), and then \"Let $\\phi \\left({t, x_0}\\right)$ be ... of $S$\" or whatever.}} Then a solution with initial condition $x_0$ is '''stable''' on $\\left[{0 \\,.\\,.\\, \\to}\\right)$ if: {{explain|Makes it easier to follow if it is introduced as \"Let $\\mathbf s$ be a solution ... with initial condition $x_0$.}} :given any $\\epsilon > 0$, there exists a $\\delta > 0$ such that $\\left\\Vert{x - x_0}\\right\\Vert < \\delta \\implies \\left\\Vert {\\phi \\left({t, x}\\right) - \\phi \\left({t, x_0}\\right)}\\right\\Vert < \\epsilon$ {{explain|Establish links to define the above notation}} {{refactor|Extract these out into separate pages.}} An equilibrium $x_0$ is '''unstable''' {{iff}} it is not stable. An equilibrium $x_0$ is '''asymptotically stable''' {{iff}}: :For any $x$ in a sufficiently small neighborhood of $x_0$: ::$\\displaystyle \\lim_{t \\to \\infty} \\phi \\left({t, x}\\right) = x_0$ {{MissingLinks}} {{definition wanted|This page might be better approached by using the machinery of Convergence.}} Category:Definitions/Differential Equations b042k7x1lii3wuk3yj7ngz836nagvbt"}
+{"_id": "21093", "title": "Definition:Nonvanishing", "text": "A function $f$ is said to be '''nonvanishing''' if it has no zeroes in its domain. That is, $f$ is '''nonvanishing''' {{iff}}: :$\\forall x \\in \\Dom f: \\map f x \\ne 0$ In this context, $f$ is (usually) either real-valued or complex-valued. In any case, its codomain needs to contain a zero, so at the very least its codomain needs to be a ring. Category:Definitions/Analysis Category:Definitions/Complex Analysis dyl4eurqrshy64dqvb6662uqmu9sc3q"}
+{"_id": "21094", "title": "Definition:Fundamental Matrix", "text": "Let $\\mathbf x' = A \\left({t}\\right) \\mathbf x$ be a system of $n$ linear first order ODEs. Let $\\Phi \\left({t}\\right)$ be an $n \\times n$ matrix function. Then $\\Phi \\left({t}\\right)$ is a '''fundamental matrix''' of the system $\\mathbf x' = A \\left({t}\\right) \\mathbf x$ {{iff}}: :it solves the matrix system $\\mathbf X'=A(t) \\mathbf X$ :$\\det \\Phi \\left({t}\\right)$ is nonvanishing"}
+{"_id": "21095", "title": "Definition:Lyapunov Function", "text": "Let $x_0$ be an equilibrium point of the system $x' = f \\left({x}\\right)$. Then a function $V$ is a '''Lyapunov function''' of the system on an open set $U$ containing the equilibrium iff: * $V \\left({x_0}\\right) = 0$ * $V \\left({x}\\right) > 0$ if $x \\in U \\setminus \\left\\{{x_0}\\right\\}$ * $\\nabla V \\cdot f \\le 0$ for $x \\in U$. {{explain|What $\\nabla$ means in this context.}} If the inequality is strict except at $x_0$, then $V$ is '''strict'''."}
+{"_id": "21096", "title": "Definition:Continuously Differentiable", "text": "A differentiable function $f$ is '''continuously differentiable''' {{iff}} $f$ is of differentiability class $C^1$. That is, if the first order derivative of $f$ (and possibly higher) is continuous."}
+{"_id": "21097", "title": "Definition:Homotopy Class", "text": "Let $X$ and $Y$ be topological spaces. Let $K \\subseteq X$ be any subset. Let $f : X \\to Y$ be a continuous mapping. The '''homotopy class''' or '''$K$-homotopy class''' of $f$ is the equivalence class of $f$ under the equivalence relation defined by homotopy relative to $K$. === Homotopy class of path === {{:Definition:Homotopy Class/Path}}"}
+{"_id": "21098", "title": "Definition:Cycle (Periodic Solution)", "text": "A '''cycle''', or '''periodic solution''', is a solution of a differential equation which is a periodic function. Category:Definitions/Differential Equations 5k43z9413cja1n5zxw3a74fp5wy6jte"}
+{"_id": "21099", "title": "Definition:Differentiability Class", "text": "Let $f: \\R \\to \\R$ be a real function. Then $\\map f x$ is of '''differentiability class''' $C^k$ {{iff}} $\\dfrac {\\d^k} {\\d x^k} \\map f x$ is continuous. That, $f$ is in '''differentiability class $k$''' {{iff}} there exists a $k$th derivative of $f$ which is continuous. If $\\dfrac {\\d^k} {\\d x^k} \\map f x$ is continuous for all $k \\in \\N$, then $\\map f x$ is of '''differentiability class''' $C^\\infty$."}
+{"_id": "21101", "title": "Definition:Direct Image Mapping", "text": "Let $S$ and $T$ be sets. Let $\\powerset S$ and $\\powerset T$ be their power sets. === Relation === {{:Definition:Direct Image Mapping/Relation}} === Mapping === {{:Definition:Direct Image Mapping/Mapping}}"}
+{"_id": "21103", "title": "Definition:Intersection (Geometry)", "text": "The '''intersection''' of two lines $AB$ and $CD$ is denoted by $AB \\cap CD$. The '''intersection''' of two geometric figures is the set of points shared by both figures. Note that this use of $\\cap$ is consistent with that of its more usual context of set intersection. When two lines '''intersect''', they are said to '''cut''' each other."}
+{"_id": "21104", "title": "Definition:Generalized Inverse Gaussian Distribution", "text": "The '''generalized inverse Gaussian distribution''' ('''GIG''')  is a three-parameter family of continuous probability distributions with probability density function: :$\\displaystyle \\forall x > 0: f \\left({x}\\right) = \\frac { \\left({a/b}\\right)^{p/2} } {2 K_p \\left({\\sqrt{a b} }\\right)} x^{\\left({p - 1}\\right)} e^{-\\left({a x + b / x}\\right) / 2}$ where: : $K_p$ is a modified Bessel function of the second kind : $a > 0, b > 0, p$ are real."}
+{"_id": "21105", "title": "Definition:Entropy (Probability Theory)", "text": "Let $X$ be a discrete random variable that takes on the values of $\\set {x_1, x_2, \\ldots, x_n}$ and has a probability mass function of $\\map p {x_i}$.  Then the '''entropy''' of $X$ is:  :$\\ds \\map H X := -\\sum_{i \\mathop = 1}^n \\map p {x_i} \\log_2 \\map p {x_i}$ and is measured in units of '''bits'''."}
+{"_id": "21106", "title": "Definition:Limit of Real Function/Left", "text": "Let $\\openint a b$ be an open real interval. Let $f: \\openint a b \\to \\R$ be a real function. Let $L \\in \\R$. Suppose that: :$\\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: \\forall x \\in \\R: b - \\delta < x < b \\implies \\size {\\map f x - L} < \\epsilon$ where $\\R_{>0}$ denotes the set of strictly positive real numbers. That is, for every real strictly positive $\\epsilon$ there exists a real strictly positive $\\delta$ such that ''every'' real number in the domain of $f$, less than $b$ but within $\\delta$ of $b$, has an image within $\\epsilon$ of $L$. :400px Then $\\map f x$ is said to '''tend to the limit $L$ as $x$ tends to $b$ from the left''', and we write: :$\\map f x \\to L$ as $x \\to b^-$ or :$\\displaystyle \\lim_{x \\mathop \\to b^-} \\map f x = L$ This is voiced: :'''the limit of $\\map f x$ as $x$ tends to $b$ from the left''' and such an $L$ is called: :'''a limit from the left'''."}
+{"_id": "21107", "title": "Definition:Limit of Real Function/Right", "text": "Let $\\Bbb I = \\openint a b$ be an open real interval. Let $f: \\Bbb I \\to \\R$ be a real function. Let $L \\in \\R$. Suppose that: :$\\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: \\forall x \\in \\Bbb I: a < x < a + \\delta \\implies \\size {\\map f x - L} < \\epsilon$ where $\\R_{>0}$ denotes the set of strictly positive real numbers. That is, for every real strictly positive $\\epsilon$ there exists a real strictly positive $\\delta$ such that ''every'' real number in the domain of $f$, greater than $a$ but within $\\delta$ of $a$, has an image within $\\epsilon$ of $L$. :400px Then $\\map f x$ is said to '''tend to the limit $L$ as $x$ tends to $a$ from the right''', and we write: :$\\map f x \\to L$ as $x \\to a^+$ or :$\\displaystyle \\lim_{x \\mathop \\to a^+} \\map f x = L$ This is voiced : '''the limit of $\\map f x$ as $x$ tends to $a$ from the right''' and such an $L$ is called: : '''a limit from the right'''."}
+{"_id": "21109", "title": "Definition:Limit of Sequence", "text": "=== Topological Space === {{:Definition:Limit Point/Topology/Sequence}} === Metric Space === {{:Definition:Limit of Sequence/Metric Space}} === Normed Division Ring === {{:Definition:Limit of Sequence/Normed Division Ring}} === Normed Vector Space === {{:Definition:Limit of Sequence/Normed Vector Space}}"}
+{"_id": "21111", "title": "Definition:Bounded Sequence", "text": "A special case of a bounded mapping is a '''bounded sequence''', where the domain of the mapping is $\\N$. Let $\\struct {T, \\preceq}$ be an ordered set. Let $\\sequence {x_n}$ be a sequence in $T$. Then $\\sequence {x_n}$ is '''bounded''' {{iff}} $\\exists m, M \\in T$ such that $\\forall i \\in \\N$: :$(1): \\quad m \\preceq x_i$ :$(2): \\quad x_i \\preceq M$ That is, {{iff}} it is bounded above and bounded below."}
+{"_id": "21112", "title": "Definition:Bézout Numbers", "text": "Let $a, b \\in \\Z$ such that $a \\ne 0$ or $b \\ne 0$. Let $d$ be the greatest common divisor of $a$ and $b$. By Bézout's Lemma: :$\\exists x, y \\in \\Z: a x + b y = d$ The numbers $x$ and $y$ are known as '''Bézout numbers''' of $a$ and $b$."}
+{"_id": "21113", "title": "Definition:Parity of Integer", "text": "Let $z \\in \\Z$ be an integer. The '''parity''' of $z$ is whether it is even or odd. That is: :an integer of the form $z = 2 n$, where $n$ is an integer, is of '''even parity'''; :an integer of the form $z = 2 n + 1$, where $n$ is an integer, is of '''odd parity'''. :If $z_1$ and $z_2$ are either both even or both odd, $z_1$ and $z_2$ have '''the same parity'''.  :If $z_1$ is even and $z_2$ is odd, then $z_1$ and $z_2$ have '''opposite parity'''."}
+{"_id": "21114", "title": "Definition:Parity (Permutation)", "text": "Let $n \\in \\N$ be a natural number. Let $S_n$ denote the symmetric group on $n$ letters. Let $\\rho \\in S_n$, that is, let $\\rho$ be a permutation of $S_n$. The '''parity''' of $\\rho$ is defined as follows: === Even Permutation === {{:Definition:Even Permutation}} === Odd Permutation === {{:Definition:Odd Permutation}} where $\\map \\sgn \\rho$ denotes the sign of $\\rho$."}
+{"_id": "21115", "title": "Definition:Linear Operator", "text": "A '''linear operator''' is a linear transformation from a module into itself."}
+{"_id": "21116", "title": "Definition:Codomain (Set Theory)", "text": "=== Relation === {{:Definition:Codomain (Set Theory)/Relation}} === Mapping === The term '''codomain''' is usually seen when the relation in question is actually a mapping: {{:Definition:Codomain (Set Theory)/Mapping}}"}
+{"_id": "21119", "title": "Definition:Kernel of Group Homomorphism", "text": "Let $\\struct {G, \\circ}$ and $\\struct {H, *}$ be groups. Let $\\phi: \\struct {G, \\circ} \\to \\struct {H, *}$ be a group homomorphism. The '''kernel''' of $\\phi$ is the subset of the domain of $\\phi$ defined as: :$\\map \\ker \\phi := \\phi \\sqbrk {e_G} = \\set {x \\in G: \\map \\phi x = e_H}$ where $e_H$ is the identity of $H$. That is, $\\map \\ker \\phi$ is the subset of $G$ that maps to the identity of $H$."}
+{"_id": "21120", "title": "Definition:Kernel of Ring Homomorphism", "text": "Let $\\struct {R_1, +_1, \\circ_1}$ and $\\struct {R_2, +_2, \\circ_2}$ be rings. Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring homomorphism. The '''kernel''' of $\\phi$ is the subset of the domain of $\\phi$ defined as: :$\\map \\ker \\phi = \\set {x \\in R_1: \\map \\phi x = 0_{R_2} }$ where $0_{R_2}$ is the zero of $R_2$. That is, $\\map \\ker \\phi$ is the subset of $R_1$ that maps to the zero of $R_2$. From Ring Homomorphism Preserves Zero it follows that $0_{R_1} \\in \\map \\ker \\phi$ where $0_{R_1}$ is the zero of $R_1$."}
+{"_id": "21122", "title": "Definition:Range of Sequence", "text": "Let $\\sequence {x_n}_{n \\mathop \\in A}$ be a sequence. The '''range of $\\sequence {x_n}$''' is the set: :$\\set {x_n: n \\mathop \\in A}$"}
+{"_id": "21123", "title": "Definition:Real Sequence", "text": "A '''real sequence''' is a sequence (usually infinite) whose codomain is the set of real numbers $\\R$."}
+{"_id": "21124", "title": "Definition:Increasing/Sequence", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. Let $A$ be a subset of the natural numbers $\\N$. Then a sequence $\\sequence {a_k}_{k \\mathop \\in A}$ of terms of $S$ is '''increasing''' {{iff}}: :$\\forall j, k \\in A: j < k \\implies a_j \\preceq a_k$"}
+{"_id": "21125", "title": "Definition:Decreasing/Sequence", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. Then a sequence $\\sequence {a_k}_{k \\mathop \\in A}$ of terms of $S$ is '''decreasing''' {{iff}}: :$\\forall j, k \\in A: j < k \\implies a_k \\preceq a_j$"}
+{"_id": "21126", "title": "Definition:Strictly Increasing/Sequence", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. Then a sequence $\\sequence {a_k}_{k \\mathop \\in A}$ of terms of $S$ is '''strictly increasing''' {{iff}}: :$\\forall j, k \\in A: j < k \\implies a_j \\prec a_k$"}
+{"_id": "21127", "title": "Definition:Strictly Decreasing/Sequence", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. Then a sequence $\\sequence {a_k}_{k \\mathop \\in A}$ of terms of $S$ is '''strictly decreasing''' {{iff}}: :$\\forall j, k \\in A: j < k \\implies a_k \\prec a_j$"}
+{"_id": "21128", "title": "Definition:Monotone (Order Theory)/Sequence", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. A sequence $\\sequence {a_k}_{k \\mathop \\in A}$ of elements of $S$ is '''monotone''' {{iff}} it is either increasing or decreasing."}
+{"_id": "21129", "title": "Definition:Rational Sequence", "text": "A '''rational sequence''' is a sequence (usually infinite) whose codomain is the set of rational numbers $\\Q$."}
+{"_id": "21131", "title": "Definition:Self-Map", "text": "Let $S$ be a set. A '''self-map on $S$''' is a mapping from $S$ to itself: :$f: S \\to S$"}
+{"_id": "21132", "title": "Definition:Renaming Mapping", "text": "Let $f: S \\to T$ be a mapping. The '''renaming mapping''' $r: S / \\RR_f \\to \\Img f$ is defined as: :$r: S / \\RR_f \\to \\Img f: \\map r {\\eqclass x {\\RR_f} } = \\map f x$ where: :$\\RR_f$ is the equivalence induced by the mapping $f$ :$S / \\RR_f$ is the quotient set of $S$ determined by $\\RR_f$ :$\\eqclass x {\\RR_f}$ is the equivalence class of $x$ under $\\RR_f$."}
+{"_id": "21134", "title": "Definition:Fixed Element", "text": "=== Fixed Point of General Mapping === {{:Definition:Fixed Point}} === Fixed Element of Permutation === The concept is particularly important when studying permutations in the context of group theory: {{:Definition:Fixed Element of Permutation}} Category:Definitions/Mapping Theory Category:Definitions/Group Theory c94dtj5srdlqtg13xmb7pc8mw3q3a5q"}
+{"_id": "21135", "title": "Definition:Peano Structure", "text": "A '''Peano structure''' $\\struct {P, 0, s}$ comprises a set $P$ with a successor mapping $s: P \\to P$ and a non-successor element $0$. These three together are required to satisfy Peano's axioms."}
+{"_id": "21137", "title": "Definition:Ordered Set", "text": "An '''ordered set''' is a relational structure $\\struct {S, \\preceq}$ such that the relation $\\preceq$ is an ordering. Such a structure may be: :A partially ordered set (poset) :A totally ordered set (toset) :A well-ordered set (woset) depending on whether the ordering $\\preceq$ is: :A partial ordering :A total ordering :A well-ordering."}
+{"_id": "21138", "title": "Definition:Hasse Diagram", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. A '''Hasse diagram''' is a method of representing $\\struct {S, \\preceq}$ as a graph $G$, in which: :$(1):\\quad$ The vertices of $G$ represent the elements of $S$ :$(2):\\quad$ The edges of $G$ represent the elements of $\\preceq$ :$(3):\\quad$ If $x, y \\in S: x \\preceq y$ then the edge representing $x \\preceq y$ is drawn so that $x$ is lower down the page than $y$. ::::That is, the edge ascends (usually obliquely) from $x$ to $y$ :$(4):\\quad$ If $x \\preceq y$ and $y \\preceq z$, then as an ordering is transitive it follows that $x \\preceq z$. ::::But in a '''Hasse diagram''', the relation $x \\preceq z$ is not shown. ::::Transitivity is implicitly expressed by the fact that $z$ is higher up than $x$, and can be reached by tracing a path from $x$ to $z$ completely through ascending edges."}
+{"_id": "21139", "title": "Definition:Conditional/Antecedent", "text": "In a conditional $p \\implies q$, the statement $p$ is the '''antecedent'''."}
+{"_id": "21140", "title": "Definition:Conditional/Consequent", "text": "In a conditional $p \\implies q$, the statement $q$ is the '''consequent'''."}
+{"_id": "21141", "title": "Definition:Cardinal", "text": "Let $S$ be a set. Associated with $S$ there exists a set $\\map \\Card S$ called the '''cardinal of $S$'''. It has the properties: : $(1): \\quad \\map \\Card S \\sim S$ that is, $\\map \\Card S$ is (set) equivalent to $S$ : $(2): \\quad S \\sim T \\iff \\map \\Card S = \\map \\Card T$"}
+{"_id": "21142", "title": "Definition:Philosophical Position", "text": "A '''philosophical position''' is, broadly speaking, a belief that a particular statement is either true or false. In the context of mathematics the term is usually reserved for particular fundamental questions whose truth values have not been explicitly determined in any chosen frame of reference or axiom schema. Hence, in order to progress down a particular logical path, one needs to make a decision as to which it is. Usually for such questions, its answer is of far-reaching importance. It having been necessary to take such a position, one then finds that diverging schools of thought arise: one which '''accepts''' and one which '''rejects''' the truth of the question. The answer to the question itself is added to the particular axiom schema for that school to form an expanded schema."}
+{"_id": "21143", "title": "Definition:Set of Sets", "text": "A '''set of sets''' is a set, whose elements are themselves all sets. Those elements can themselves be assumed to be subsets of some particular fixed set which is frequently referred to as the universe."}
+{"_id": "21144", "title": "Definition:Well-Orderable Set", "text": "Let $S$ be a set. If it is possible to construct an ordering $\\preceq$ on $S$ such that $\\preceq$ is a well-ordering, then $S$ is defined as being '''well-orderable'''."}
+{"_id": "21145", "title": "Definition:Choice Function", "text": "Let $\\mathbb S$ be a set of sets such that: :$\\forall S \\in \\mathbb S: S \\ne \\O$ that is, none of the sets in $\\mathbb S$ may be empty. A '''choice function on $\\mathbb S$''' is a mapping $f: \\mathbb S \\to \\bigcup \\mathbb S$ satisfying: :$\\forall S \\in \\mathbb S: \\map f S \\in S$. That is, for any set in $\\mathbb S$, a '''choice function''' selects an element from that set.   The domain of $f$ is $\\mathbb S$."}
+{"_id": "21147", "title": "Definition:Distributive Operation/Right", "text": "The operation $\\circ$ is '''right distributive''' over the operation $*$ {{iff}}: :$\\forall a, b, c \\in S: \\paren {a * b} \\circ c = \\paren {a \\circ c} * \\paren {b \\circ c}$"}
+{"_id": "21148", "title": "Definition:Distributive Operation/Left", "text": "The operation $\\circ$ is '''left distributive''' over the operation $*$ {{iff}}: :$\\forall a, b, c \\in S: a \\circ \\paren {b * c} = \\paren {a \\circ b} * \\paren {a \\circ c}$"}
+{"_id": "21149", "title": "Definition:Relation Induced by Partition", "text": "Let $S$ be a set. Let $\\Bbb S$ be a partition of a set $S$. Let $\\mathcal R \\subseteq S \\times S$ be the relation defined as: :$\\forall \\tuple {x, y} \\in S \\times S: \\tuple {x, y} \\in \\mathcal R \\iff \\exists T \\in \\Bbb S: \\set {x, y} \\subseteq T$ Then $\\mathcal R$ is the '''(equivalence) relation induced by (the partition) $\\Bbb S$'''."}
+{"_id": "21150", "title": "Definition:Definition", "text": "A '''definition''' lays down the meaning of a concept. It is a statement which tells the reader '''what something is'''. It can be understood as an equation in (usually) natural language."}
+{"_id": "21151", "title": "Definition:Stipulative Definition", "text": "A '''stipulative definition''' is a definition which defines how to interpret the meaning of a symbol. It '''stipulates''', or lays down, the meaning of a symbol in terms of previously defined symbols or concepts. The symbol used for a stipulative definition is: :$\\text {(the symbol being defined)} := \\text {(the meaning of that symbol)}$ This can be written the other way round: :$\\text {(a concept being assigned a symbol)} =: \\text {(the symbol for it)}$ when it is necessary to emphasise that the symbol has been crafted to abbreviate the notation for the concept."}
+{"_id": "21152", "title": "Definition:Ostensive Definition", "text": "An '''ostensive definition''' is a definition which ''shows'' what a symbol is, rather than use words to ''explain'' what it is or what it does. As an example of an '''ostensive definition''', we offer up: :The symbol used for a stipulative definition is $:=$, as in: ::$\\text {(the symbol being defined)} := \\text {(the meaning of that symbol)}$"}
+{"_id": "21153", "title": "Definition:Formal Grammar/Bottom-Up/Extremal Clause", "text": "Let $\\mathcal F$ be a formal language. Let the rules of formation of $\\mathcal F$ be defined in a bottom-up manner. <section notitle=\"true\" name=\"tc\"> The '''extremal clause''' of a bottom-up grammar is the final rule which excludes all collations other than those specified in the formation rules from being well-formed formulas. </section>"}
+{"_id": "21154", "title": "Definition:Occurrence (Formal Systems)", "text": "Let $\\mathcal F$ be a formal language. Let $S, T$ be collations in the alphabet of $\\mathcal F$. Each place where $S$ appears in $T$ is called an '''occurrence of $S$ in $T$'''."}
+{"_id": "21156", "title": "Definition:Null String", "text": "A '''null string''' (or '''empty string''') is a string with no symbols in it. In particular, the '''null string''' is a word. A null string has a length of $0$."}
+{"_id": "21157", "title": "Definition:Substring", "text": "Let $\\mathcal L$ be a formal language with alphabet $\\mathcal A$. Let $S$ be a string in $\\mathcal A$. Let $T$ be a string in $\\mathcal A$ such that: :$S = S_1 T S_2$ where: * $S_1$ and $S_2$ are strings in $\\mathcal A$ (possibly null); * $S_1 T S_2$ is the concatenation of $S_1$, $T$ and $S_2$. Then $T$ is called a '''substring of $S$'''. It follows from this definition that $S$ is a substring of itself (by considering $S_1$ and $S_2$ as both null)."}
+{"_id": "21158", "title": "Definition:Subordinate", "text": "Let $\\mathbf A$ be a WFF of propositional logic. Let $\\circ$ and $\\ast$ be connectives. Then $\\circ$ is '''subordinate to $\\ast$ (in $\\mathbf A$)''' iff the scope of $\\circ$ is a well-formed part of the scope of $\\ast$."}
+{"_id": "21159", "title": "Definition:Venn Diagram", "text": "A '''Venn diagram''' is a technique for the graphic depiction of the interrelationship between a small number (usually $3$ or fewer) of sets. The following diagram illustrates the various operations between three sets. :500px The circles represent the sets $S_1$, $S_2$ and $S_3$. The '''white''' surrounding box represents the universal set $\\mathbb U$. Each of the areas inside the various circle represents an intersection between the various sets and their complements, as follows: :The '''gray''' area represents $S_1 \\cap S_2 \\cap S_3$. :The '''purple''' area represents $S_1 \\cap S_2 \\cap \\overline {S_3}$. :The '''orange''' area represents $S_1 \\cap \\overline {S_2} \\cap S_3$. :The '''green''' area represents $\\overline {S_1} \\cap S_2 \\cap S_3$. :The '''red''' area represents $S_1 \\cap \\overline {S_2} \\cap \\overline {S_3}$. :The '''blue''' area represents $\\overline {S_1} \\cap S_2 \\cap \\overline {S_3}$. :The '''yellow''' area represents $\\overline {S_1} \\cap \\overline {S_2} \\cap S_3$. :The surrounding '''white''' area represents $\\overline {S_1} \\cap \\overline {S_2} \\cap \\overline {S_3}$. The notation $\\overline {S_1}$ denotes set complement. If it is required to show on a diagram that a particular intersection is empty, then it is generally shaded '''black'''."}
+{"_id": "21161", "title": "Definition:Order Type", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be totally ordered sets. Then $S$ and $T$ '''have the same (order) type''' {{iff}} they are order isomorphic."}
+{"_id": "21162", "title": "Definition:Order Type of Natural Numbers", "text": "The order type of $\\struct {\\N, \\le}$ is denoted $\\omega$ ('''omega''')."}
+{"_id": "21163", "title": "Definition:Ordered Sum", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be tosets. Let: * the order type of $\\left({S, \\preceq_1}\\right)$ be $\\theta_1$ * the order type of $\\left({T, \\preceq_2}\\right)$ be $\\theta_2$. Let $S \\cup T$ be the union of $S$ and $T$. We define the ordering $\\preceq$ on $S$ and $T$ as: :$\\forall s \\in S, t \\in T: a \\preceq b \\iff \\begin{cases} a \\preceq_1 b & : a \\in S \\land b \\in S \\\\ a \\preceq_2 b & : \\neg \\left({a \\in S \\land b \\in S}\\right) \\land \\left({a \\in T \\land b \\in T}\\right) \\\\ & : a \\in S, b \\in T \\end{cases}$ That is: : If $a$ and $b$ are both in $S$, they are ordered as they are in $S$. : If $a$ and $b$ are not both in $S$, but they ''are'' both in $T$, they are ordered as they are in $T$. : Otherwise, that is if $a$ and $b$ are in ''both'' sets, their ordering in $S$ takes priority over that in $T$. The ordered set $\\left({S \\cup T, \\preceq}\\right)$ is called the '''ordered sum''' of $S$ and $T$, and is denoted $S + T$."}
+{"_id": "21164", "title": "Definition:Ordered Product", "text": "Let $\\left({S_1, \\preceq_1}\\right)$ and $\\left({S_2, \\preceq_2}\\right)$ be tosets. Let: * the order type of $\\left({S_1, \\preceq_1}\\right)$ be $\\theta_1$; * the order type of $\\left({S_2, \\preceq_2}\\right)$ be $\\theta_2$. Let $T = S_1 \\times S_2$ be the cartesian product of $S_1$ and $S_2$. Consider the relation $\\preceq$ defined on $T$ as follows. Let $a_1$ and $a_2$ be arbitrary elements of $S_1$, and $b_1$ and $b_2$ be arbitrary elements of $S_2$. Then: * $b_1 \\prec b_2 \\implies \\left({a_1, b_1}\\right) \\prec \\left({a_2, b_2}\\right)$ * $b_1 = b_2, a_1 \\prec a_2 \\implies \\left({a_1, b_1}\\right) \\prec \\left({a_2, b_2}\\right)$ * $b_1 = b_2, a_1 = a_2 \\implies \\left({a_1, b_1}\\right) = \\left({a_2, b_2}\\right)$ The ordered set $\\left({S_1 \\times S_2, \\preceq}\\right)$ is called the '''ordered product''' of $S_1$ and $S_2$, and is denoted $S_1 \\cdot S_2$."}
+{"_id": "21165", "title": "Definition:Euler Diagram", "text": "An '''Euler diagram''' is a graphical technique for illustrating the relationships between sets. It differs from a Venn diagram in that whereas the latter illustrates all possible intersections between a number of general sets, an '''Euler diagram''' depicts only those which are relevant for the situation being depicted."}
+{"_id": "21166", "title": "Definition:Scope (Logic)", "text": "The '''scope''' of a logical connective is defined as the statements that it connects, whether this be simple or compound. In the case of a unary connective, there will be only one such statement. === Connective of Propositional Logic === {{:Definition:Scope (Logic)/Connective}} === Quantifier of Predicate Logic === {{:Definition:Scope (Logic)/Quantifier}}"}
+{"_id": "21167", "title": "Definition:Main Connective", "text": "In a compound statement, exactly '''one''' of its logical connectives has the largest scope. That connective is called the '''main connective'''. The scope of the '''main connective''' comprises the entire compound statement. {{transclude:Definition:Main Connective/Propositional Logic   |section = def   |title = Propositional Logic   |link = true   |header = 3   |increase = 1 }}"}
+{"_id": "21168", "title": "Definition:Codomain of Sequence", "text": "The codomain of a sequence can be elements of a set of any objects. If the codomain of a sequence $f$ is $S$, then the sequence is said to be a '''sequence of elements of $S$''', or a '''sequence in $S$'''."}
+{"_id": "21169", "title": "Definition:Length of String", "text": "The '''length''' of a finite string in a formal language is the number of symbols it contains. The '''length''' of a string $S$ can be denoted $\\map \\len S$ or $\\size S$."}
+{"_id": "21170", "title": "Definition:Initial Part", "text": "A string $T$ is an '''initial part''' of a string $S$ {{iff}} $S$ can be formed by concatenating $T$ with another string $T'$: :$S = TT'$"}
+{"_id": "21171", "title": "Definition:Trivial Quotient", "text": "Let $\\Delta_S$ be the diagonal relation on a set $S$. As $\\Delta_S$ is an equivalence, we can form the quotient mapping: :$q_{\\Delta_S}: S \\to S / \\Delta_S$. This quotient mapping is called the '''trivial quotient of $S$'''."}
+{"_id": "21172", "title": "Definition:Conflation", "text": "A '''conflation''' is a mistake in which two or more separate but similar ideas become confused with one another."}
+{"_id": "21173", "title": "Definition:Left Inverse Mapping", "text": "Let $S, T$ be sets where $S \\ne \\O$, i.e. $S$ is not empty. Let $f: S \\to T$ be a mapping. Let $g: T \\to S$ be a mapping such that: :$g \\circ f = I_S$ where: :$g \\circ f$ denotes the composite mapping $f$ followed by $g$; :$I_S$ is the identity mapping on $S$. Then $g: T \\to S$ is called '''a left inverse (mapping)'''."}
+{"_id": "21174", "title": "Definition:Right Inverse Mapping", "text": "Let $S, T$ be sets where $S \\ne \\O$, that is, $S$ is not empty. Let $f: S \\to T$ be a mapping. Let $g: T \\to S$ be a mapping such that: :$f \\circ g = I_T$ where: :$f \\circ g$ denotes the composite mapping $g$ followed by $f$ :$I_T$ is the identity mapping on $T$. Then $g: T \\to S$ is called '''a right inverse (mapping) of $f$'''."}
+{"_id": "21176", "title": "Definition:Commutative Diagram", "text": "A '''commutative diagram''' is a graphical technique designed to illustrate the construction of composite mappings. It is also a widespread tool in category theory, where it deals with morphisms instead of mappings. ::$\\begin{xy} \\xymatrix@L+2mu@+1em{  S_1 \\ar[r]^*{f_1}      \\ar@{-->}[rd]_*{f_2 \\circ f_1} &  S_2 \\ar[d]^*{f_2} \\\\ &  S_3 }\\end{xy}$ It consists of: :$(1): \\quad$ A collection of points representing the various domains and codomains of the mappings in question :$(2): \\quad$ Arrows representing the mappings themselves. The diagram is properly referred to as '''commutative''' {{iff}} all the various paths from the base of one arrow to the head of another represent equal mappings. A mapping which is uniquely determined by the rest of the diagram may be indicated by a dotted arrow. It is however generally advisable not to use more than one dotted arrow per diagram, so as to avoid confusion."}
+{"_id": "21177", "title": "Definition:Extension of Mapping", "text": "As a mapping is, by definition, also a relation, the definition of an '''extension of a mapping''' is the same as that for an extension of a relation: Let: :$f_1 \\subseteq X \\times Y$ be a mapping on $X \\times Y$ :$f_2 \\subseteq S \\times T$ be a mapping on $S \\times T$ :$X \\subseteq S$ :$Y \\subseteq T$ :$f_2 \\restriction_{X \\times Y}$ be the restriction of $f_2$ to $X \\times Y$. Let $f_2 \\restriction_{X \\times Y} = f_1$. Then $f_2$ '''extends''' or '''is an extension of''' $f_1$."}
+{"_id": "21179", "title": "Definition:Extension of Operation", "text": "Let $\\left({S, \\circ}\\right)$ be a magma. Let $\\left({T, \\circ \\restriction_T}\\right)$ be a submagma of $\\left({S, \\circ}\\right)$, where $\\circ \\restriction_T$ denotes the restriction of $\\circ$ to $T$. Then: : '''$\\left({S, \\circ}\\right)$ is an extension of $\\left({T, \\circ \\restriction_T}\\right)$''' or : '''$\\left({S, \\circ}\\right)$ extends $\\left({T, \\circ \\restriction_T}\\right)$''' We can use the term directly to the operation itself and say: : '''$\\circ$ is an extension of $\\circ \\restriction_T$''' or: : '''$\\circ$ extends $\\circ \\restriction_T$'''"}
+{"_id": "21180", "title": "Definition:Union Mapping", "text": "Let: :$(1): \\quad f_1: S_1 \\to T_1$ be a mapping from $S_1$ to $T_1$ :$(2): \\quad f_2: S_2 \\to T_2$ be a mapping from $S_2$ to $T_2$ Let $f_1$ and $f_2$ be combinable, that is, that they agree on $S_1 \\cap S_2$. Then the '''union mapping''' $f = f_1 \\cup f_2$ of $f_1$ and $f_2$ is: :$f: S_1 \\cup S_2 \\to T_1 \\cup T_2: \\map f s = \\begin{cases} \\map {f_1} s : & s \\in S_1 \\\\ \\map {f_2} s : & s \\in S_2 \\end{cases}$"}
+{"_id": "21181", "title": "Definition:Increasing Union", "text": "Let $S_0, S_1, S_2, \\ldots, S_i, \\ldots$ be a nested sequence of sets, that is: :$S_0 \\subseteq S_1 \\subseteq S_2 \\subseteq \\ldots \\subseteq S_i \\subseteq \\ldots$ Let $S$ be the set: :$\\displaystyle S = \\bigcup_{i \\mathop \\in \\N} S_i$ where $\\bigcup$ denotes set union. Then $S$ is called the '''increasing union''' of $S_0, S_1, S_2, \\ldots, S_i, \\ldots$"}
+{"_id": "21183", "title": "Definition:Bound Variable", "text": "A '''bound variable''' is a variable which, when it occurs in an expression, can be replaced with another variable without changing the meaning of the statement."}
+{"_id": "21184", "title": "Definition:F-Sigma Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. An '''$F_\\sigma$ set ($F$-sigma set)''' is a set which can be written as a countable union of closed sets of $T$."}
+{"_id": "21185", "title": "Definition:G-Delta Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. A '''$G_\\delta$ set ($G$-delta set)''' is a set which can be written as a countable intersection of open sets of $T$."}
+{"_id": "21186", "title": "Definition:Equivalent Topological Bases", "text": "Let $X$ be a set. Let $\\mathbb S_1$ and $\\mathbb S_2$ be subsets of $\\powerset X$, the power set of $X$. Let $\\mathbb S_1$ and $\\mathbb S_2$ be used as a synthetic basis or synthetic sub-basis to generate topologies for $X$. Let $\\tau_1$ and $\\tau_2$ be the topologies arising from $\\mathbb S_1$ and $\\mathbb S_2$ respectively. Then $\\mathbb S_1$ and $\\mathbb S_2$ are '''equivalent''' {{iff}} $\\tau_1 = \\tau_2$."}
+{"_id": "21187", "title": "Definition:Local Basis", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $x$ be an element of $S$. === Local Basis for Open Sets === {{:Definition:Local Basis/Local Basis for Open Sets}} === Neighborhood Basis of Open Sets === {{:Definition:Local Basis/Neighborhood Basis of Open Sets}}"}
+{"_id": "21188", "title": "Definition:Weakly Hereditary Property", "text": "Let $\\xi$ be a property whose domain is the set of all topological spaces. Then $\\xi$ is a '''weakly hereditary property''' {{iff}}: :$\\map \\xi X \\implies \\map \\xi Y$ where $Y$ is any closed set of $X$ when considered as a subspace."}
+{"_id": "21189", "title": "Definition:Hereditary Property (Topology)", "text": "Let $\\xi$ be a property whose domain is the set of all topological spaces. Then $\\xi$ is a '''hereditary property''' {{iff}}: :$\\map \\xi X \\implies \\map \\xi Y$ where $Y$ is a subspace of $X$. That is, whenever a topological space has $\\xi$, then so does any subspace."}
+{"_id": "21190", "title": "Definition:Adherent Point", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A \\subseteq S$. === Definition from Neighborhood === {{:Definition:Adherent Point/Definition 3}} === Definition from Open Neighborhood === {{:Definition:Adherent Point/Definition 1}} === Definition from Closure === {{:Definition:Adherent Point/Definition 2}}"}
+{"_id": "21191", "title": "Definition:Omega-Accumulation Point", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A \\subseteq S$. An '''$\\omega$-accumulation point''' of $A$ is a limit point $x$ of $A$ such that every open set containing $x$ also contains an infinite number of points of $A$."}
+{"_id": "21192", "title": "Definition:Condensation Point", "text": "Let $T = \\struct {X, \\tau}$ be a topological space. Let $A \\subseteq X$. A '''condensation point''' of $A$ is a limit point $x$ of $A$ such that every open set containing $x$ also contains an uncountable number of points of $A$."}
+{"_id": "21194", "title": "Definition:Limit Point/Topology/Sequence", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A \\subseteq S$. Let $\\sequence {x_n}$ be a sequence in $A$. Let $\\sequence {x_n}$ converge to a value $\\alpha \\in A$. Then $\\alpha$ is known as a '''limit (point) of $\\sequence {x_n}$ (as $n$ tends to infinity)'''."}
+{"_id": "21195", "title": "Definition:Accumulation Point/Sequence", "text": "Let $\\sequence {x_n}_{n \\mathop \\in \\N}$ be an infinite sequence in $A$. Let $x \\in S$. Suppose that: :$\\forall U \\in \\tau: x \\in U \\implies \\set {n \\in \\N: x_n \\in U}$ is infinite Then $x$ is an '''accumulation point''' of $\\sequence {x_n}$."}
+{"_id": "21197", "title": "Definition:Regular Open Set", "text": "Let $T$ be a topological space. Let $A \\subseteq T$. Then $A$ is '''regular open in $T$''' {{iff}}: :$A = A^{- \\circ}$"}
+{"_id": "21198", "title": "Definition:Regular Closed Set", "text": "Let $T$ be a topological space. Let $A \\subseteq T$. Then $A$ is '''regular closed in $T$''' {{iff}}: :$A = A^{\\circ -}$"}
+{"_id": "21199", "title": "Definition:Exterior (Topology)", "text": "Let $T$ be a topological space. Let $H \\subseteq T$. === Definition 1 === {{:Definition:Exterior (Topology)/Definition 1}} === Definition 2 === {{:Definition:Exterior (Topology)/Definition 2}}"}
+{"_id": "21201", "title": "Definition:Dense-in-itself", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$. Then $H$ is dense-in-itself {{iff}} it contains no isolated points."}
+{"_id": "21203", "title": "Definition:Meager Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A \\subseteq S$. $A$ is '''meager in $T$''' {{iff}} it is a countable union of subsets of $S$ which are nowhere dense in $T$."}
+{"_id": "21204", "title": "Definition:Separable Space", "text": "A topological space $T = \\struct {S, \\tau}$ is '''separable''' {{iff}} there exists a countable subset of $S$ which is everywhere dense in $T$."}
+{"_id": "21206", "title": "Definition:Second-Countable Space", "text": "A topological space $T = \\struct {S, \\tau}$ is '''second-countable''' or '''satisfies the Second Axiom of Countability''' {{iff}} its topology has a countable basis."}
+{"_id": "21207", "title": "Definition:First-Countable Space", "text": "A topological space $T = \\struct {S, \\tau}$ is '''first-countable''' or '''satisfies the First Axiom of Countability''' {{iff}} every point in $S$ has a countable local basis."}
+{"_id": "21208", "title": "Definition:Neighborhood Basis", "text": "Let $\\struct{X, \\tau}$ be a topological space. Let $x \\in X$. Let $\\BB$ be a set of neighborhoods of $x$. Then $\\BB$ is a '''neighborhood basis''' at $x$ {{iff}}: :For each neighborhood $N$ of $x$, there is an $M \\in \\BB$ such that $M \\subseteq N$."}
+{"_id": "21209", "title": "Definition:Closed Mapping", "text": "Let $X, Y$ be topological spaces. Let $f: X \\to Y$ be a mapping.  If, for any closed set $V \\subseteq X$, the image $\\map f V$ is closed in $Y$, then $f$ is referred to as a '''closed mapping'''."}
+{"_id": "21210", "title": "Definition:Continuity/Metric Subspace", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f: A_1 \\to A_2$ be a mapping from $A_1$ to $A_2$. Let $Y \\subseteq A_1$. By definition, $\\left({Y, d_Y}\\right)$ is a metric subspace of $A_1$. Let $a \\in Y$ be a point in $Y$. Then $f$ is '''$\\left({d_Y, d_2}\\right)$-continuous at $a$''' {{iff}}: : $\\forall \\epsilon > 0: \\exists \\delta > 0: d_Y \\left({x, a}\\right) < \\delta \\implies d_2 \\left({f \\left({x}\\right), f \\left({a}\\right)}\\right) < \\epsilon$ Similarly, $f$ is '''$\\left({d_Y, d_2}\\right)$-continuous''' {{iff}}: : $\\forall a \\in Y: f$ is $\\left({d_Y, d_2}\\right)$-continuous at $a$"}
+{"_id": "21211", "title": "Definition:Open Set/Topology", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Then the elements of $\\tau$ are called the '''open sets of $T$'''. Thus: : '''$U \\in \\tau$''' and: : '''$U$ is open in $T$''' are equivalent statements."}
+{"_id": "21212", "title": "Definition:Open Set/Metric Space", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $U \\subseteq A$. Then $U$ is an '''open set in $M$''' {{iff}} it is a neighborhood of each of its points. That is: :$\\forall y \\in U: \\exists \\epsilon \\in \\R_{>0}: \\map {B_\\epsilon} y \\subseteq U$ where $\\map {B_\\epsilon} y$ is the open $\\epsilon$-ball of $y$."}
+{"_id": "21213", "title": "Definition:Open Set/Complex Analysis", "text": "{{:Definition:Open Set/Complex Analysis/Definition 1}}"}
+{"_id": "21214", "title": "Definition:Open Set/Real Analysis", "text": "=== Real Numbers === {{:Definition:Open Set/Real Analysis/Real Numbers}} === Real Euclidean Space === {{:Definition:Open Set/Real Analysis/Real Euclidean Space}}"}
+{"_id": "21215", "title": "Definition:Open Region", "text": "=== Complex Analysis === {{:Definition:Open Region/Complex}} === Open Region in the Plane === {{:Definition:Open Region/Plane}}"}
+{"_id": "21216", "title": "Definition:Neighborhood (Topology)", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. === Neighborhood of a Set === {{:Definition:Neighborhood (Topology)/Set}} === Neighborhood of a Point === The set $A$ can be a singleton, in which case the definition is of the '''neighborhood of a point'''. {{:Definition:Neighborhood (Topology)/Point}}"}
+{"_id": "21217", "title": "Definition:Open Ball", "text": "Let $M = \\struct {A, d}$ be a metric space or pseudometric space. Let $a \\in A$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. The '''open $\\epsilon$-ball of $a$ in $M$''' is defined as: :$\\map {B_\\epsilon} a := \\set {x \\in A: \\map d {x, a} < \\epsilon}$ If it is necessary to show the metric or pseudometric itself, then the notation $\\map {B_\\epsilon} {a; d}$ can be used."}
+{"_id": "21218", "title": "Definition:Neighborhood (Complex Analysis)", "text": "Let $z_0 \\in \\C$ be a complex number. Let $\\epsilon \\in \\R_{>0}$ be a (strictly) positive real number. The '''$\\epsilon$-neighborhood''' of $z_0$ is defined as: :$\\map {N_\\epsilon} {z_0} := \\set {z \\in \\C: \\cmod {z - z_0} < \\epsilon}$"}
+{"_id": "21219", "title": "Definition:Neighborhood (Real Analysis)/Epsilon", "text": "On the real number line with the usual metric, the '''$\\epsilon$-neighborhood''' of $\\alpha$ is defined as the open interval: :$\\map {N_\\epsilon} \\alpha := \\openint {\\alpha - \\epsilon} {\\alpha + \\epsilon}$ where $\\epsilon \\in \\R_{>0}$ is a (strictly) positive real number."}
+{"_id": "21220", "title": "Definition:Limit of Real Function", "text": "Let $\\openint a b$ be an open real interval. Let $c \\in \\openint a b$. Let $f: \\openint a b \\setminus \\set c \\to \\R$ be a real function. Let $L \\in \\R$. Suppose that: :$\\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: \\forall x \\in \\R: 0 < \\size {x - c} < \\delta \\implies \\size {\\map f x - L} < \\epsilon$ where $\\R_{>0}$ denotes the set of strictly positive real numbers. That is: :For every (strictly) positive real number $\\epsilon$, there exists a (strictly) positive real number $\\delta$ such that ''every'' real number $x \\ne c$ in the domain of $f$ within $\\delta$ of $c$ has an image within $\\epsilon$ of $L$. $\\epsilon$ is usually considered as having the connotation of being \"small\" in magnitude, but this is a misunderstanding of its intent: the point is that (in this context) $\\epsilon$ ''can be made'' arbitrarily small. :400px Then $\\map f x$ is said to '''tend to the limit $L$ as $x$ tends to $c$''', and we write: :$\\map f x \\to L$ as $x \\to c$ or :$\\displaystyle \\lim_{x \\mathop \\to c} \\map f x = L$ This is voiced: :'''the limit of $\\map f x$ as $x$ tends to $c$'''. It can directly be seen that this definition is the same as that for a general metric space."}
+{"_id": "21221", "title": "Definition:Limit of Function (Metric Space)", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $c$ be a limit point of $M_1$. Let $f: A_1 \\to A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ ''except possibly'' at $c$. Let $L \\in M_2$. $f \\left({x}\\right)$ is said to '''tend to the limit $L$ as $x$ tends to $c$''' and is written: :$f \\left({x}\\right) \\to L$ as $x \\to c$ or :$\\displaystyle \\lim_{x \\mathop \\to c} f \\left({x}\\right) = L$ {{iff}} the following equivalent conditions hold: === $\\epsilon$-$\\delta$ Condition === {{:Definition:Limit of Function (Metric Space)/Epsilon-Delta Condition}} === $\\epsilon$-Ball Condition === {{:Definition:Limit of Function (Metric Space)/Epsilon-Ball Condition}}"}
+{"_id": "21222", "title": "Definition:Limit of Complex Function", "text": "The definition for the limit of a complex function is exactly the same as that for the general metric space."}
+{"_id": "21223", "title": "Definition:Limit of Sequence/Metric Space", "text": "Let $M = \\left({A, d}\\right)$ be a metric space or pseudometric space. Let $\\left \\langle {x_n} \\right \\rangle$ be a sequence in $M$. Let $\\left \\langle {x_n} \\right \\rangle$ converge to a value $l \\in A$. Then $l$ is a '''limit of $\\left \\langle {x_n} \\right \\rangle$ as $n$ tends to infinity'''. If $M$ is a metric space, this is usually written: :$\\displaystyle l = \\lim_{n \\mathop \\to \\infty} x_n$"}
+{"_id": "21224", "title": "Definition:Limit of Sequence (Number Field)", "text": "=== Real Numbers === {{:Definition:Limit of Sequence/Real Numbers}} === Rational Numbers === {{:Definition:Limit of Sequence/Rational Numbers}} === Complex Numbers === {{:Definition:Limit of Sequence/Complex Numbers}}"}
+{"_id": "21226", "title": "Definition:Divergent Sequence", "text": "A sequence which is not convergent is '''divergent'''."}
+{"_id": "21227", "title": "Definition:Divergent Series", "text": "A series which is not convergent is '''divergent'''."}
+{"_id": "21228", "title": "Definition:Divergent Function", "text": "A function which is not convergent is '''divergent'''."}
+{"_id": "21229", "title": "Definition:Convergent Mapping", "text": "=== Metric Space === {{:Definition:Convergent Mapping/Metric Space}} === Real Function === As the real number line $\\R$ under the usual (Euclidean) metric forms a metric space, the definition also holds for real functions: {{:Definition:Convergent Mapping/Real Function}} === Complex Function === As the complex plane $\\C$ under the usual (Euclidean) metric forms a metric space, the definition also holds for complex functions: {{:Definition:Convergent Mapping/Complex Function}}"}
+{"_id": "21230", "title": "Definition:Filter", "text": "Let $\\left({S, \\preccurlyeq}\\right)$ be an ordered set. A '''filter of $\\left({S, \\preccurlyeq}\\right)$''' (or a '''filter on $\\left({S, \\preccurlyeq}\\right)$''') is a subset $\\mathcal F \\subseteq S$ which satisfies the following conditions: {{begin-axiom}} {{axiom | n = 1         | m = \\mathcal F \\ne \\varnothing }} {{axiom | n = 2         | m = x, y \\in \\mathcal F \\implies \\exists z \\in \\mathcal F: z \\preccurlyeq x, z \\preccurlyeq y }} {{axiom | n = 3         | m = \\forall x \\in \\mathcal F: \\forall y \\in S: x \\preccurlyeq y \\implies y \\in \\mathcal F }} {{end-axiom}} === Proper Filter === {{:Definition:Filter/Proper Filter}}"}
+{"_id": "21231", "title": "Definition:Convergent Filter", "text": "Let $\\struct {S, \\tau}$ be a topological space. Let $\\FF$ be a filter on $S$. Then $\\FF$ '''converges''' to a point $x \\in S$ {{iff}}: :$\\forall N_x \\subseteq S: N_x \\in \\FF$ where $N_x$ is a neighborhood of $x$. That is, a filter '''converges''' to a point $x$ {{iff}} every neighborhood of $x$ is an element of that filter. If there is a point $x \\in S$ such that $\\FF$ '''converges''' to $x$, then $\\FF$ is '''convergent'''."}
+{"_id": "21232", "title": "Definition:Standard Number Field", "text": "The '''standard number fields''' are the following sets of numbers: :The rational numbers: $\\Q = \\set {p / q: p, q \\in \\Z, q \\ne 0}$ :The real numbers: $\\R = \\set {x: x = \\sequence {s_n} }$ where $\\sequence {s_n}$ is a Cauchy sequence in $\\Q$ :The complex numbers: $\\C = \\set {a + i b: a, b \\in \\R, i^2 = -1}$. These sets are indeed fields: :$\\struct {\\Q, +, \\times, \\le}$ is an ordered field, and also a metric space. :$\\struct {\\R, +, \\times, \\le}$ is an ordered field, and also a complete metric space. :$\\struct {\\C, +, \\times}$ is a field, but cannot be ordered compatibly with $+$ and $\\times$. However, it can be treated as a metric space."}
+{"_id": "21233", "title": "Definition:Continuous Invariant", "text": "Let $P$ be a property whose domain is the set of all topological spaces. Suppose that whenever $\\map P T$ holds, then so does $\\map P {T'}$, where: :$T$ and $T'$ are topological spaces :$\\phi: T \\to T'$ is a continuous mapping from $T$ to $T'$ :$\\phi \\sqbrk T = T'$, where $\\phi \\sqbrk T$ denotes the image of $\\phi$. Then $P$ is a '''continuous invariant'''."}
+{"_id": "21234", "title": "Definition:Image (Set Theory)/Relation/Subset", "text": "Let $X \\subseteq S$ be a subset of $S$. Then the '''image set (of $X$ by $\\RR$)''' is defined as: :$\\RR \\sqbrk X := \\set {t \\in T: \\exists s \\in X: \\tuple {s, t} \\in \\RR}$"}
+{"_id": "21235", "title": "Definition:Open Invariant", "text": "Let $P$ be a property whose domain is the set of all topological spaces. Suppose that whenever $\\map P T$ holds, then so does $\\map P {T'}$, where: :$T$ and $T'$ are topological spaces :$\\phi: T \\to T'$ is a mapping from $T$ to $T'$ :$\\phi \\sqbrk T = T'$, where $\\phi \\sqbrk T$ denotes the image of $\\phi$ :$T'$ is an open set. Then $P$ is an '''open invariant'''."}
+{"_id": "21236", "title": "Definition:Closed Invariant", "text": "Let $P$ be a property whose domain is the set of all topological spaces. Suppose that whenever $\\map P T$ holds, then so does $\\map P {T'}$, where: :$T$ and $T'$ are topological spaces :$\\phi: T \\to T'$ is a mapping from $T$ to $T'$ :$\\phi \\sqbrk T = T'$, where $\\phi \\sqbrk T$ denotes the image of $\\phi$ :$T'$ is a closed set. Then $P$ is a '''closed invariant'''."}
+{"_id": "21237", "title": "Definition:Product Space (Topology)", "text": "Let $\\struct {S_1, \\tau_1}$ and $\\struct {S_2, \\tau_2}$ be topological spaces. Let $S_1 \\times S_2$ be the cartesian product of $S_1$ and $S_2$. Let $\\tau$ be the Tychonoff topology on $S_1 \\times S_2$. From Natural Basis of Tychonoff Topology of Finite Product, $\\tau$ is the topology generated by the natural basis: :$\\BB = \\set {U_1 \\times U_2: U_1 \\in \\tau_1, U_2 \\in \\tau_2}$ The topological space $\\struct {S_1 \\times S_2, \\tau}$ is called  the '''product space''' of $\\struct {S_1, \\tau_1}$ and $\\struct {S_2, \\tau_2}$."}
+{"_id": "21238", "title": "Definition:P-Product Metric", "text": "Let $M_{1'} = \\struct {A_{1'}, d_{1'} }$ and $M_{2'} = \\struct {A_{2'}, d_{2'} }$ be metric spaces. Let $A_{1'} \\times A_{2'}$ be the cartesian product of $A_{1'}$ and $A_{2'}$. Let $p \\in \\R_{\\ge 1}$. The '''$p$-product metric''' on $A_{1'} \\times A_{2'}$ is defined as: :$\\map {d_p} {x, y} := \\paren {\\paren {\\map {d_{1'} } {x_1, y_1} }^p + \\paren {\\map {d_{2'} } {x_2, y_2} }^p}^{1/p}$ where $x = \\tuple {x_1, x_2}, y = \\tuple {y_1, y_2} \\in A_{1'} \\times A_{2'}$. The metric space $\\mathcal M_p := \\struct {A_{1'} \\times A_{2'}, d_p}$ is the '''$p$-product (space)''' of $M_{1'}$ and $M_{2'}$."}
+{"_id": "21239", "title": "Definition:Identification Topology", "text": "Let $\\struct {S_1, \\tau_1}$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \\to S_2$ be a mapping. The '''identification topology on $S_2$ with respect to $f$ and $\\struct {S_1, \\tau_1}$''' is defined as: :$\\tau_2 = \\set {V \\in \\powerset {S_2}: f^{-1} \\sqbrk V \\in \\tau_1}$"}
+{"_id": "21240", "title": "Definition:Topological Sum", "text": "Let $\\struct {X, \\tau_1}$ and $\\struct {Y, \\tau_2}$ be topological spaces. The '''topological sum''' $\\struct {Z, \\tau_3}$ of $X$ and $Y$ is defined as: :$Z = X \\sqcup Y$ where: :$X \\sqcup Y$ denotes the disjoint union of $X$ and $Y$ :$\\tau_3$ is the topology generated by $\\tau_1$ and $\\tau_2$."}
+{"_id": "21241", "title": "Definition:Finite Intersection Property", "text": "Let $\\Bbb S$ be a set of sets. Let $\\Bbb S$ have the property that: :the intersection of any finite number of sets in $\\Bbb S$ is not empty. Then $\\Bbb S$ satisfies the '''finite intersection property'''."}
+{"_id": "21242", "title": "Definition:Filter Sub-Basis", "text": "Let $S$ be a set. Let $\\powerset S$ denote the power set of $S$. Let $\\BB \\subset \\powerset S$ be a set of subsets of $\\powerset S$ which satisfies the finite intersection property. That is, the intersection of any finite number of sets in $\\BB$ is not empty. Then $\\BB$, together with the finite intersections of all its elements, is a basis for a filter $\\FF$ on $S$. Thus $\\BB$ is a '''sub-basis''' for $\\FF$."}
+{"_id": "21243", "title": "Definition:Cluster Point of Filter", "text": "Let $S$ be a set. Let $\\powerset S$ denote the power set of $S$. Let $\\FF \\subset \\powerset X$ be a filter on $S$. Let $x \\in S$ be an element of every set in $\\FF$: :$x \\in X: \\forall U \\in \\FF: x \\in U$ Then $x$ is a '''cluster point of $\\FF$'''."}
+{"_id": "21244", "title": "Definition:Principal Ultrafilter", "text": "Let $S$ be a set. Let $\\powerset S$ denote the power set of $S$. Let $\\FF \\subset \\powerset S$ be an ultrafilter on $S$ with a cluster point. Then $\\FF$ is a '''principal ultrafilter on $S$'''."}
+{"_id": "21246", "title": "Definition:Neighborhood Filter", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. === Set === {{:Definition:Neighborhood Filter/Set}} === Point === {{:Definition:Neighborhood Filter/Point}} Category:Definitions/Neighborhoods Category:Definitions/Filter Theory i2b1xtpqf0z3iaiel1kxg70lobnic8w"}
+{"_id": "21247", "title": "Definition:Group Automorphism", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\phi: G \\to G$ be a (group) isomorphism from $G$ to itself. Then $\\phi$ is a group automorphism."}
+{"_id": "21248", "title": "Definition:Scheme of Abbreviation", "text": "When translating from a series of sentences in natural language into a collection of propositional formulas in symbolic logic, it is necessary to replace each simple statement with a statement label. As stated in the definition of statement label, it is imperative that each sentence be replaced with a different label. A '''scheme of abbreviation''' is the list of simple statements, together with their associated statement labels, for reference and unambiguous key in order to translate any conclusion reached back into natural language."}
+{"_id": "21249", "title": "Definition:Negation Normal Form", "text": "A propositional formula $P$ is in '''negation normal form''' ('''NNF''') {{iff}}: * The only logical connectives connecting substatements of $P$ are Not, And and Or, that is, elements of the set $\\left\\{{\\neg, \\land, \\lor}\\right\\}$; * The Not sign $\\neg$ appears only in front of atomic statements. That is $P$ is in '''negation normal form''' iff it consists of literals, conjunctions and disjunctions."}
+{"_id": "21250", "title": "Definition:Finite Group", "text": "A '''finite group''' is a group of finite order. That is, a group $\\struct {G, \\circ}$ is a '''finite group''' {{iff}} its underlying set $G$ is finite."}
+{"_id": "21251", "title": "Definition:Cayley Table", "text": "A '''Cayley table''' is a technique for describing an algebraic structure (usually a finite group) by putting all the products in a square array: :$\\begin {array} {c|cccc} \\circ & a & b & c & d \\\\ \\hline a & a & a & b & a \\\\ b & b & c & a & d \\\\ c & d & e & f & a \\\\ d & c & d & a & b \\\\ \\end {array}$"}
+{"_id": "21252", "title": "Definition:Latin Square", "text": "Let $n \\in \\Z_{>0}$ be some given (strictly) positive integer $n$. A '''Latin square''' of order $n$ is a square array of size $n \\times n$ containing $n$ different symbols, such that every row and column contains '''exactly one''' of each symbol. That is, each row and column is a permutation of the same $n$ symbols."}
+{"_id": "21253", "title": "Definition:Infinite Group", "text": "A group which is not finite is an '''infinite group'''."}
+{"_id": "21254", "title": "Definition:General Linear Group", "text": "Let $K$ be a field. The set of all invertible order-$n$ square matrices over $K$ is a group under (conventional) matrix multiplication. This group is called the '''general linear group (of degree $n$)''' and is denoted $\\GL {n, K}$, or $\\GL n$ if the field is implicit. The field itself is usually $\\R$, $\\Q$ or $\\C$, but can be ''any'' field."}
+{"_id": "21255", "title": "Definition:Noetherian Ring", "text": "=== Definition 1 === {{Definition:Noetherian Ring/Definition 1}} === Definition 2 === {{Definition:Noetherian Ring/Definition 2}} === Definition 3 === {{Definition:Noetherian Ring/Definition 3}} === Definition 4 === {{Definition:Noetherian Ring/Definition 4}}"}
+{"_id": "21256", "title": "Definition:Product (Abstract Algebra)", "text": "=== General Operation === {{:Definition:Operation/Binary Operation/Product}} === Group Product === {{:Definition:Product (Abstract Algebra)/Group}} === Ring Product === {{:Definition:Product (Abstract Algebra)/Ring}} === Field Product === {{:Definition:Product (Abstract Algebra)/Field}} Category:Definitions/Operations Category:Definitions/Abstract Algebra Category:Definitions/Multiplication bnffxibu0958x3dqn2zydtradw911qk"}
+{"_id": "21258", "title": "Definition:Product Inverse", "text": "The inverse of $x \\in U_R$ by $\\circ$ is called the '''(ring) product inverse of $x$'''. The usual means of denoting the product inverse of an element $x$ is by $x^{-1}$. Thus it is distinguished from the additive inverse of $x$, that is, the (ring) negative of $x$, which is usually denoted $-x$."}
+{"_id": "21259", "title": "Definition:Affine Algebraic Variety", "text": "Let $k$ be a field. Then a subset $X \\subseteq k^n$ is an '''affine algebraic variety''' if the following hold: * $X$ is an affine algebraic set  * $X$ is irreducible"}
+{"_id": "21260", "title": "Definition:Zariski Topology", "text": "=== On an Affine Space === {{:Definition:Zariski Topology/Affine Space}} === On the spectrum of a ring === {{:Definition:Zariski Topology/Spectrum of Ring}}"}
+{"_id": "21263", "title": "Definition:Radical of Integer", "text": "The '''radical of an integer''' $n \\in \\Z$ is the product of the individual prime factors of $n$. The radicals of the first few integers are given here: {| |- ! align=\"right\" | $n$ !! ! align=\"right\" | Decomposition !! ! align=\"right\" | $\\map {\\operatorname {rad} } n$ |- | align=\"right\" | $1$ || | align=\"right\" | $1$ || | align=\"right\" | $1$ |- | align=\"right\" | $2$ || | align=\"right\" | $2$ || | align=\"right\" | $2$ |- | align=\"right\" | $3$ || | align=\"right\" | $3$ || | align=\"right\" | $3$ |- | align=\"right\" | $4$ || | align=\"right\" | $2^2$ || | align=\"right\" | $2$ |- | align=\"right\" | $5$ || | align=\"right\" | $5$ || | align=\"right\" | $5$ |- | align=\"right\" | $6$ || | align=\"right\" | $2 \\times 3$ || | align=\"right\" | $6$ |- | align=\"right\" | $7$ || | align=\"right\" | $7$ || | align=\"right\" | $7$ |- | align=\"right\" | $8$ || | align=\"right\" | $2^3$ || | align=\"right\" | $2$ |- | align=\"right\" | $9$ || | align=\"right\" | $3^2$ || | align=\"right\" | $3$ |- | align=\"right\" | $10$ || | align=\"right\" | $2 \\times 5$ || | align=\"right\" | $10$ |- | align=\"right\" | $11$ || | align=\"right\" | $11$ || | align=\"right\" | $11$ |- | align=\"right\" | $12$ || | align=\"right\" | $2^2 \\times 3$ || | align=\"right\" | $6$ |- | align=\"right\" | $13$ || | align=\"right\" | $13$ || | align=\"right\" | $13$ |- | align=\"right\" | $14$ || | align=\"right\" | $2 \\times 7$ || | align=\"right\" | $14$ |- | align=\"right\" | $15$ || | align=\"right\" | $3 \\times 5$ || | align=\"right\" | $15$ |- | align=\"right\" | $16$ || | align=\"right\" | $2^4$ || | align=\"right\" | $2$ |} {{OEIS|A007947}} The radical of $n$ can alternatively be described as the largest square-free integer which divides $n$."}
+{"_id": "21265", "title": "Definition:Generator of Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group. Let $S \\subseteq G$. Let $H$ be the subgroup generated by $S$. Then '''$S$ is a generator of $H$''', denoted $H = \\gen S$, {{iff}} $H$ is the subgroup generated by $S$."}
+{"_id": "21266", "title": "Definition:Join of Subgroups", "text": "Let $\\struct {G, \\circ}$ be a group. Let $A$ and $B$ be subgroups of $G$. The '''join''' of $A$ and $B$ is written and defined as: :$A \\vee B := \\gen {A \\cup B}$ where $\\gen {A \\cup B}$ is the subgroup generated by $A \\cup B$. By the definition of subgroup generator, this can alternatively be written: :$\\displaystyle A \\vee B := \\bigcap \\set {T: T \\text { is a subgroup of } G: A \\cup B \\subseteq T}$ === General Definition === {{:Definition:Join of Subgroups/General Definition}}"}
+{"_id": "21268", "title": "Definition:Argand Diagram", "text": "An '''Argand diagram''' is a graphical representation of a set of complex numbers on the complex plane: :400px"}
+{"_id": "21269", "title": "Definition:Topological Manifold/Smooth Manifold", "text": "Let $M$ be a second-countable locally Euclidean space of dimension $d$.  Let $\\mathscr F$ be a smooth differentiable structure on $M$. Then $\\left({M, \\mathscr F}\\right)$ is called a '''smooth manifold of dimension $d$'''."}
+{"_id": "21270", "title": "Definition:Kampyle of Eudoxus", "text": ":300px {{refactor|Separate pages for Cartesian and Polar equations}} The '''Kampyle of Eudoxus''' has the Cartesian equation: :$x^4 = x^2 + y^2$ where the point $x = y = 0$ is specifically excluded. In polar coordinates, it is described by the equation: :$r = \\sec^2 \\theta$ {{stub}} {{NamedforDef|Eudoxus of Cnidus|cat = Eudoxus}}"}
+{"_id": "21271", "title": "Definition:Cube Number", "text": "A '''cube number''' (or just '''cube''') is a number which can be expressed as the third power of an integer."}
+{"_id": "21272", "title": "Definition:One", "text": "The immediate successor element of zero in the set of natural numbers $\\N$ is called '''one''' and has the symbol $1$. === Naturally Ordered Semigroup === {{:Definition:Unit (One)/Naturally Ordered Semigroup}}"}
+{"_id": "21274", "title": "Definition:Ordered Tuple as Ordered Set", "text": "The rigorous definition of an ordered tuple is as a finite sequence whose domain is $\\N^*_n$. However, it is possible to treat an ordered tuple as an extension of the concept of an ordered pair. === Ordered Triple === {{:Definition:Ordered Tuple as Ordered Set/Ordered Triple}} === Ordered Quadruple === {{:Definition:Ordered Tuple as Ordered Set/Ordered Quadruple}} === Ordered Tuple === {{:Definition:Ordered Tuple as Ordered Set/Ordered Tuple}} === Ordered Singleton === {{:Definition:Ordered Tuple as Ordered Set/Ordered Singleton}} Category:Definitions/Ordered Tuples 8watrb6opfg28e0ug7ndutke9mj0gcg"}
+{"_id": "21275", "title": "Definition:Zero Element", "text": "An element $z \\in S$ is called a '''two-sided zero element''' (or simply '''zero element''' or '''zero''') {{iff}} it is both a '''left zero''' and a '''right zero''': :$\\forall x \\in S: x \\circ z = z = z \\circ x$"}
+{"_id": "21276", "title": "Definition:Invertible Operation", "text": "The operation $\\circ$ is '''invertible''' {{iff}}: :$\\forall a, b \\in S: \\exists r, s \\in S: a \\circ r = b = s \\circ a$"}
+{"_id": "21277", "title": "Definition:Mathematical System", "text": "A '''mathematical system''' is a set $\\SS = \\struct {E, O, A}$ where: :$E$ is a non-empty set of elements :$O$ is a set of relations and operations on the elements of $E$ :$A$ is a set of axioms concerning the elements of $E$ and $O$. === Abstract System === A '''mathematical system''' $\\SS = \\struct {E, O, A}$ is classed as '''abstract''' if the elements of $E$ and $O$ are defined only by their properties as specified in $A$. === Concrete System === A '''mathematical system''' $\\SS = \\struct {E, O, A}$ is classed as '''concrete''' if the elements of $E$ and $O$ are understood as objects independently of their existence in $\\SS$ itself. The distinction between '''abstract''' and '''concrete''' is of questionable value from a modern standpoint, as it is a moot point, for example, as to whether the natural numbers exist independently of Peano's axioms or are specifically '''defined''' by them. === Algebraic System === A '''mathematical system''' $\\SS = \\struct {E, O, A}$ is classed as '''algebraic''' if it has many of the properties of the set of integers. This is usually because such a system is itself an abstraction of certain properties of the integers. The axioms are usually not considered as separate entities from the operations, as their nature is implicit in the operations themselves. Specifically, an algebraic system can be defined as follows: {{:Definition:Algebraic System}}"}
+{"_id": "21278", "title": "Definition:Scalar Field", "text": "Let $\\struct {G, +_G, \\circ}_K$ be a vector space, where: :$\\struct {K, +_K, \\times_K}$ is a field :$\\struct {G, +_G}$ is an abelian group $\\struct {G, +_G}$ :$\\circ: K \\times G \\to G$ is a binary operation. Then the field $\\struct {K, +_K, \\times_K}$ is called the '''scalar field''' of $\\struct {G, +_G, \\circ}_K$."}
+{"_id": "21279", "title": "Definition:Bilinear Mapping", "text": "Let $\\left({R, +_R, \\times_R}\\right)$ be a commutative ring. Let $\\left({A_1, +_1, \\circ_1}\\right)_R, \\left({A_2, +_2, \\circ_2}\\right)_R, \\left({A_3, +_3, \\circ_3}\\right)_R$ be $R$-modules. Let $\\oplus: A_1 \\times A_2 \\to A_3$ be a binary operator with the property that: $\\forall \\left({a_1, a_2}\\right) \\in A_1 \\times A_2$: : $a_1 \\mapsto a_1 \\oplus a_2$ is a linear transformation from $A_1$ to $A_3$ : $a_2 \\mapsto a_1 \\oplus a_2$ is a linear transformation from $A_2$ to $A_3$ Then $\\oplus$ is a '''bilinear mapping'''. That is, $\\forall a, b \\in R, \\forall x, y \\in A_2, z \\in A_3$: : $\\left({\\left({a \\circ_1 x}\\right) +_1 \\left({y \\circ_1 b}\\right)}\\right) \\oplus z = \\left({a \\circ_3 \\left({x \\oplus z}\\right)}\\right) +_3 \\left({\\left({y \\oplus z}\\right) \\circ_3 b}\\right)$ and for all $z \\in A_1, x,y \\in A_2$: : $z \\oplus \\left({\\left({a \\circ_2 x}\\right) +_2 \\left({y \\circ_2 b}\\right)}\\right) = \\left({a \\circ_3 \\left({z \\oplus x}\\right)}\\right) +_3 \\left({\\left({z \\oplus y}\\right) \\circ_3 b}\\right)$ Equivalently, this can be expressed: : $\\left({x +_1 y}\\right) \\oplus z = \\left({x \\oplus z}\\right) +_3 \\left({y \\oplus z}\\right)$ : $z \\oplus \\left({x +_2 y}\\right) = \\left({z \\oplus x}\\right) +_3 \\left({z \\oplus y}\\right)$ : $\\left({a \\circ_1 x}\\right) \\oplus z = a \\circ_3 \\left({x \\oplus z}\\right)$ : $z \\oplus \\left({y \\circ_2 b}\\right) = \\left({z \\oplus y}\\right) \\circ_3 b$ If $\\left({A, +, \\circ}\\right)_R = A_1 = A_2 = A_3$, the notation simplifies considerably: : $\\left({\\left({a \\circ x}\\right) + \\left({b \\circ y}\\right)}\\right) \\oplus z = \\left({a \\circ \\left({x \\oplus z}\\right)}\\right) + \\left({b \\circ \\left({y \\oplus z}\\right)}\\right)$ : $z \\oplus \\left({\\left({a \\circ x}\\right) + \\left({y \\circ b}\\right)}\\right) = \\left({a \\circ \\left({z \\oplus x}\\right)}\\right) + \\left({\\left({z \\oplus y}\\right) \\circ b}\\right)$ or equivalently, more easily digested: : $\\left({x + y}\\right) \\oplus z = \\left({x \\oplus z}\\right) + \\left({y \\oplus z}\\right)$ : $z \\oplus \\left({x + y}\\right) = \\left({z \\oplus x}\\right) + \\left({z \\oplus y}\\right)$ : $\\left({a \\circ x}\\right) \\oplus z = a \\circ \\left({x \\oplus z}\\right)$ : $z \\oplus \\left({y \\circ b}\\right) = \\left({z \\oplus y}\\right) \\circ b$ === Non-Commutative Ring === {{:Definition:Bilinear Mapping/Non-Commutative Ring}}"}
+{"_id": "21280", "title": "Definition:Algebra over Field", "text": "Let $F$ be a field. An '''algebra over $F$''' is an ordered pair $\\struct {A, *}$ where: :$A$ is a vector space over $F$ :$* : A^2 \\to A$ is a bilinear mapping That is, it is an algebra $\\struct {A, *}$ over the ring $F$ where: :$F$ is a field :the $F$-module $A$ is a vector space."}
+{"_id": "21282", "title": "Definition:Algebra over Ring", "text": "Let $R$ be a commutative ring. An '''algebra over $R$''' is an ordered pair $\\left({A, *}\\right)$ where: :$A$ is an $R$-module :$*: A^2 \\to A$ is an $R$-bilinear mapping"}
+{"_id": "21283", "title": "Definition:Second Order Ordinary Differential Equation", "text": "A '''second order ordinary differential equation''' is an ordinary differential equation in which any derivatives with respect to the independent variable have order no greater than $2$."}
+{"_id": "21284", "title": "Definition:Physical Universe", "text": "The '''physical universe''', or usually just '''universe''', is commonly defined as, and understood to be, '''{{WP|Universe|the totality of everything that exists}}'''. === Real-World === Used to describe a phenomenon or object that has a genuine existence in the physical universe."}
+{"_id": "21285", "title": "Definition:Body", "text": "In the context of physics and applied mathematics, a '''body''' is an object (possibly idealized) which is imagined to have a material existence in the physical universe. A '''body''' can be considered as an aggregate of particles."}
+{"_id": "21286", "title": "Definition:Physical Law", "text": "A '''physical law''' is a statement about the physical universe whose truth value stems from observations."}
+{"_id": "21287", "title": "Definition:Force", "text": "A '''force''' is an influence which causes a body to undergo a change in velocity. '''Force''' is a vector quantity."}
+{"_id": "21288", "title": "Definition:Velocity", "text": "The '''velocity''' $\\mathbf v$ of a body $M$ is defined as the first derivative of the displacement $\\mathbf s$ of $M$ from a given point of reference {{WRT|Differentiation}} time $t$: :$\\mathbf v = \\dfrac {\\d \\mathbf s} {\\d t}$"}
+{"_id": "21289", "title": "Definition:Frame of Reference", "text": "A '''frame of reference''' is a coordinate system which is more or less arbitrarily or conveniently disposed in a model of the physical universe. === Point of Reference === {{:Definition:Frame of Reference/Point of Reference}}"}
+{"_id": "21290", "title": "Definition:Mathematical Model", "text": "A '''mathematical model''' is an equation, or a system of equations, whose purpose is to provide an approximation to the behavior of a real-world phenomenon."}
+{"_id": "21291", "title": "Definition:Displacement", "text": "The '''(physical) displacement''' of a body is a measure of its position relative to a given point of reference in a particular frame of reference. '''Displacement''' is a vector quantity, so it specifies a magnitude and direction from the point of reference."}
+{"_id": "21292", "title": "Definition:Scalar Quantity", "text": "A '''scalar quantity''' is a real-world concept that needs for its model a mathematical object which contains only one (usually numeric) component."}
+{"_id": "21293", "title": "Definition:Magnitude", "text": "The '''magnitude''' (or '''size''') of a quantity (either vector or scalar) is a measure of how big it is. It is usually encountered explicitly in the context of vectors: If $\\mathbf v$ is the vector quantity in question, then its '''magnitude''' is denoted: :$\\size {\\mathbf v}$ or :$v$"}
+{"_id": "21294", "title": "Definition:Direction", "text": "The '''direction''' of a vector quantity is a measure of which way it is pointing, relative to a particular frame of reference."}
+{"_id": "21295", "title": "Definition:Speed", "text": "The '''speed''' of a body is a measure of the magnitude of its velocity, taking no account of its direction. It is, therefore, a scalar quantity."}
+{"_id": "21296", "title": "Definition:Acceleration", "text": "The '''acceleration''' $\\mathbf a$ of a body $M$ is defined as the first derivative of the velocity $\\mathbf v$ of $M$ relative to a given point of reference {{WRT|Differentiation}} time $t$: :$\\mathbf a = \\dfrac {\\d \\mathbf v} {\\d t}$"}
+{"_id": "21298", "title": "Definition:Linear Momentum", "text": "The '''linear momentum''' of a body is its mass multiplied by its velocity. :$\\mathbf p = m \\mathbf v$"}
+{"_id": "21299", "title": "Definition:Mass", "text": "The '''mass''' of a body is a measure of how much matter it contains. '''Mass''' is equivalent to inertia. '''Mass''' also determines the degree to which a body creates or is affected by a gravitational field. It is a scalar quantity."}
+{"_id": "21301", "title": "Definition:Second Order Partial Differential Equation", "text": "A '''second order partial differential equation''' is a partial differential equation in which any derivatives with respect to the independent variable have order no greater than $2$. Category:Definitions/Differential Equations dj1g50oyvq5f2s9tfxf8qo3ebx5n9fl"}
+{"_id": "21304", "title": "Definition:Proportion", "text": "Two real variables $x$ and $y$ are '''proportional''' {{iff}} one is a constant multiple of the other: :$\\forall x, y \\in \\R: x \\propto y \\iff \\exists k \\in \\R, k \\ne 0: x = k y$"}
+{"_id": "21306", "title": "Definition:Aggregate", "text": "An '''aggregate''' is an archaic word for an '''infinite set''', as used by {{AuthorRef|Georg Cantor}}. It is used in current parlance to mean a general accumulation of objects, but the concept is vague."}
+{"_id": "21308", "title": "Definition:Ideal (Physics)", "text": "An '''ideal''' (or '''idealized''') object is one in which certain attributes are approximated to zero or infinity."}
+{"_id": "21309", "title": "Definition:Particle", "text": "A '''particle''' is a representation of an object in the physical university which is idealized as having no magnitude. That is, it is modelled as being a mass concentrated at a single point."}
+{"_id": "21311", "title": "Definition:One-Parameter Family of Curves", "text": "Consider the implicit function $\\map f {x, y, c} = 0$ in the $\\tuple {x, y}$-plane where $c$ is a constant. For each value of $c$, we have that $\\map f {x, y, c} = 0$ defines a relation between $x$ and $y$ which can be graphed in the cartesian plane. Thus, each value of $c$ defines a particular curve. The complete set of all these curve for each value of $c$ is called a '''one-parameter family of curves'''. === Parameter === {{:Definition:One-Parameter Family of Curves/Parameter}}"}
+{"_id": "21312", "title": "Definition:Orthogonal Trajectories", "text": "Let $\\map f {x, y, c}$ define a one-parameter family of curves $F$. Let $\\map g {x, y, c}$ also define a one-parameter family of curves $G$, with the property that: :Every curve in $F$ is orthogonal to every curve in $G$. Then $F$ is a '''family of (reciprocal) orthogonal trajectories''' of $G$, and contrariwise."}
+{"_id": "21313", "title": "Definition:Orthogonal (Analytic Geometry)", "text": "Two curves are '''orthogonal''' if they intersect at right angles. The term perpendicular can also be used, but the latter term is usual when the intersecting lines are straight."}
+{"_id": "21314", "title": "Definition:Quadrature", "text": "'''Quadrature''' is an archaic term meaning '''the process of finding area'''."}
+{"_id": "21315", "title": "Definition:Lune", "text": "A '''lune''' is a geometric figure formed by the boundary of two intersecting circles: :300px The area coloured red is a '''lune'''. Category:Definitions/Plane Geometry 3jqvkjclp95b23vs1zi0n1flb3p6fsh"}
+{"_id": "21317", "title": "Definition:Physical Quantity", "text": "A '''physical quantity''' is a physical property which can be quantified."}
+{"_id": "21318", "title": "Definition:Rest Mass", "text": "The '''rest mass''' of a body is its '''mass''' when it is '''at rest'''."}
+{"_id": "21319", "title": "Definition:Physical Property", "text": "A '''physical property''' is any measurable property whose describes the state of a physical system."}
+{"_id": "21320", "title": "Definition:Measurable Property", "text": "A '''measurable property''' is a property of a physical system which can be compared against a standard unit of measurement to provide a numerical value defining the quantity of that property."}
+{"_id": "21321", "title": "Definition:Physical System", "text": "A '''physical system''' is a portion of the physical universe which has been chosen for investigation for a particular purpose."}
+{"_id": "21322", "title": "Definition:Quantitative Property", "text": "A '''quantitative property''' is a property of a physical system which exists in a range of values corresponding with a number, such that the system can be described as having a particular number of a unit of measurement of that property."}
+{"_id": "21323", "title": "Definition:Unit of Measurement", "text": "A '''unit of measurement''' is a specified magnitude of a given physical quantity, defined by convention. It is used as a standard for measurement of that physical quantity. Any other value of the physical quantity can be expressed as a multiple of that '''unit of measurement'''."}
+{"_id": "21324", "title": "Definition:Inertia", "text": "'''Inertia''' is the tendency of a body to maintain the same velocity in the absence of an external force, in accordance with Newton's First Law of Motion. Equivalently put, '''inertia''' is the resistance of a body to a change in its motion. '''Inertia''' is equivalent to mass."}
+{"_id": "21325", "title": "Definition:Friction", "text": "'''Friction''' is a force whose tendency is to reduce the velocity of a body relative to its surroundings. It is caused by interaction of the matter in the body with that of the environment in which it is traveling. '''Friction''' is (at a fundamental level) an electromagnetic force. That is, bits of the medium in or on which the body is traveling get in the way (on a molecular level) of bits of the body. The less friction, the less the reduction in velocity. If there were no '''friction''', then motion would continue for ever."}
+{"_id": "21326", "title": "Definition:Force of Gravity", "text": "The '''force of gravity''' or '''gravitational force''' is the force on a body as a result of Newton's Law of Universal Gravitation. When used in an unqualified sense, it is usual for this to mean the force on a body at the surface of the Earth. From Gravity at Earth's Surface, this is approximately $9.8 \\ \\mathrm N \\ \\mathrm{kg}^{-1}$ The force of gravity varies across the earth's surface, and therefore it makes little sense to use it as a standard. However, the {{WP|General_Conference_on_Weights_and_Measures|CGPM}} adopted a standard acceleration of gravity of $9.806 \\, 65  \\ \\mathrm N \\ \\mathrm{kg}^{-1}$ in 1901. Category:Definitions/Gravity 0qw8377izdwvjztb4f6nhkuaquyo923"}
+{"_id": "21328", "title": "Definition:Dimension (Measurement)", "text": "Every physical quantity has a '''dimension''' associated with it. No attempt is made here to provide an abstract definition of this term. Instead, it will be defined by example."}
+{"_id": "21329", "title": "Definition:Rate", "text": "The '''rate''' of a physical process is its (first) derivative with respect to time. Loosely speaking, it means '''how fast something progresses''', with a wider scope than change in physical displacement. === Dimension === {{:Definition:Rate/Dimension}} Category:Definitions/Dimensions of Measurement tt43pngp41nnkp4tbffuxe13olzp1e3"}
+{"_id": "21330", "title": "Definition:Physical Process", "text": "A '''physical process''' is a physical system whose importance is in the way it changes over time."}
+{"_id": "21331", "title": "Definition:Periodic Process", "text": "A '''periodic process''' is a physical process whose state can be modeled by a periodic function. === Example === The canonical example of such a process is the motion of a pendulum. Category:Definitions/Physics Category:Definitions/Applied Mathematics 0s0oo3vz3ydmblst61a9dup16j9frke"}
+{"_id": "21332", "title": "Definition:Time", "text": "A true definition of what '''time''' actually ''is'' has baffled philosophers from the dawn of, er, time. Therefore it will be left as an exercise for the reader. It is usually treated as a scalar quantity."}
+{"_id": "21333", "title": "Definition:Time Period", "text": "The '''time period''' of a periodic process is the period of the periodic function which models that process. Category:Definitions/Physics Category:Definitions/Applied Mathematics Category:Definitions/Time m4o5wsjh9xnfpa78bufkhus4rnf4jh7"}
+{"_id": "21334", "title": "Definition:Fundamental Unit (Physics)", "text": "A '''fundamental unit''' a unit of measurement for a measurable physical property from which every other unit for that quantity can be derived. The '''fundamental unit''' for a particular measurable quantity is chosen by convention. They are the '''units''' of the fundamental dimensions of physics."}
+{"_id": "21335", "title": "Definition:SI Units", "text": "The '''SI Units''' are the elements of the [http://en.wikipedia.org/wiki/International_System_of_Units International System of Units]."}
+{"_id": "21338", "title": "Definition:Time/Unit/Second", "text": "The '''second''' is the SI base unit of time, and also therefore of the MKS system. It is also the base unit of time for the FPS and CGS systems. The '''second''' is defined as: :the duration of $9 \\ 192 \\ 631 \\ 770$ periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium $133$ atom at rest at $0 \\ \\mathrm K$."}
+{"_id": "21339", "title": "Definition:Kilogram", "text": "The '''kilogram''' is the SI base unit of mass. It is defined as being equal to: :The fixed numerical value of the Planck constant $h$ to be $6 \\cdotp 62607015 \\times 10^{-34}$ when expressed in the unit Joule seconds. The Joule second is equal to $1 \\ \\mathrm {kg} \\mathrm m^2 \\mathrm s^{−1}$, where the metre and the second are defined in terms of: :the speed of light $c$ :the time of transition between the two hyperfine levels of the ground state of the caesium $133$ atom at rest at $0 \\ \\mathrm K$."}
+{"_id": "21340", "title": "Definition:Gram", "text": "The '''gram''' is the CGS base unit of mass. It is defined as the mass of a cubic centimetre water at the temperature of melting ice."}
+{"_id": "21342", "title": "Definition:Temperature", "text": "'''Temperature''' is a physical property of matter that quantifies how hot or cold a body is. It is a scalar quantity which can be mapped directly to the real number line."}
+{"_id": "21343", "title": "Definition:Kelvin", "text": "The '''kelvin''' is the SI base unit of temperature. Its two reference points are: :$0 \\ \\mathrm K$, which is set at absolute zero, the temperature at which all thermal motion stops :$273 \\cdotp 16 \\ \\mathrm K$, which is set at the {{WP|Triple_point|triple point}} of water."}
+{"_id": "21344", "title": "Definition:Absolute Zero", "text": "'''Absolute zero''' is the lowest temperature which can theoretically be achieved. It is the temperature where all motion due to thermal effects stops. Before that temperature can be reached, quantum effects come into play."}
+{"_id": "21345", "title": "Definition:Electric Charge", "text": "'''Electric charge''' is a physical quantity of matter which causes it to experience a force when near other electrically charged matter. It is a scalar quantity."}
+{"_id": "21346", "title": "Definition:Electric Current", "text": "'''Electric current''' is the physical process caused by the flow of electrically charged particles, usually electrons. Thus it is defined as the rate of flow of electric charge: :$I = \\dfrac {\\d Q} {\\d t}$"}
+{"_id": "21347", "title": "Definition:Ampere (Unit)", "text": "The '''ampere''' is the SI base unit of electric current. It is defined as being: :The constant current which will produce a force of attraction whose value is $2 \\times 10^{–7}$ newtons per metre of length between two straight, parallel conductors of infinite length and negligible circular cross section placed one metre apart in a vacuum. This arises from Ampère's Force Law, which states that such an attractive force exists. === Symbol === {{:Symbols:A/Ampere}} {{NamedforDef|André-Marie Ampère|cat = Ampère}}"}
+{"_id": "21349", "title": "Definition:Newton (Unit)", "text": "The '''newton''' is the SI unit of force. It is defined as being: :The amount of force required to accelerate a mass of one kilogram at a rate of one metre per second squared."}
+{"_id": "21350", "title": "Definition:Coulomb (Unit)", "text": "The '''coulomb''' is the SI unit of electric charge. It is defined as being: :The amount of electric charge carried in one second by a constant current of one ampere."}
+{"_id": "21351", "title": "Definition:Cube/Geometry", "text": "A '''cube''' is a hexahedron whose $6$ faces are all congruent squares."}
+{"_id": "21352", "title": "Definition:CGS", "text": "'''CGS''' is the centimetre-gram-second standard system of units of measurement. This system is rarely used nowadays, the SI units having largely taken over."}
+{"_id": "21353", "title": "Definition:MKS", "text": "'''MKS''' is the metre-kilogram-second standard system of units of measurement."}
+{"_id": "21354", "title": "Definition:FPS", "text": "'''FPS''' is the foot-pound-second standard system of units of measurement. It is derived from the imperial system, and was used throughout the British Empire until the mid-20th century."}
+{"_id": "21355", "title": "Definition:Imperial", "text": "The '''imperial system''' is a system of measurement based on traditional established folk measures, adopted by the British empire and used in Britain until the mid-20th century. Its base units can be understood as being the FPS Base Units"}
+{"_id": "21356", "title": "Definition:Free Fall", "text": "A body $B$ influenced by a gravitational field $M$ is in '''free fall''' {{iff}} the force on it caused by $M$ is the only force on $B$."}
+{"_id": "21357", "title": "Definition:Gravitational Field", "text": "Every body which has mass influences every other body which has mass, according to Newton's Law of Universal Gravitation. Thus any body can be considered as being surrounded by a field given rise to by its mass, called a '''gravitational field'''. Its value $\\mathbf g$ at any point is given by: :$\\mathbf g = \\dfrac {G M} {d^3} \\mathbf d$ where: :$G$ is the gravitational constant; :$M$ is the mass of the body; :$\\mathbf d$ is the displacement vector from the point to the center of gravity of the body, whose magnitude is $d$. Category:Definitions/Gravity a6klacxrfb4shh9tbts35rik9o2b3gs"}
+{"_id": "21359", "title": "Definition:Field (Physics)", "text": "A '''field''', in the context of physics, is a physical quantity associated with every point of spacetime. The physical quantity in question may be either in vector or scalar form. Category:Definitions/Physics tcg1synkvpwd2bx0r2lh1zhpm5qoiku"}
+{"_id": "21360", "title": "Definition:Spacetime", "text": "'''Spacetime''' (or''' space–time''', or '''space/time''') is a mathematical model that combines space and time into a single object. The form this model takes is often a four-dimensional vector space: three dimensions for space and one for time. More complicated models are used for more specialized applications. {{SUBPAGENAME}} ibycmme5pk6pp2aeslpiylu7v6owaax"}
+{"_id": "21361", "title": "Definition:Cube Root", "text": "{{:Definition:Cube Root/Real}}"}
+{"_id": "21362", "title": "Definition:Unit Vector", "text": "Let $\\mathbf v$ be a vector quantity. The '''unit vector''' in the direction of $\\mathbf v$ is defined and denoted as: :$\\mathbf {\\hat v} = \\dfrac {\\mathbf v} {\\size {\\mathbf v} }$ where $\\size {\\mathbf v}$ is the magnitude of $\\mathbf v$."}
+{"_id": "21363", "title": "Definition:Spring", "text": "A '''spring''' is a mechanical device for storing elastic potential energy. It works by resisting a force which acts so as to deform it. The greater the deformation, the greater the force. === Ideal Spring === {{:Definition:Spring/Ideal}} === Equilibrium Position === {{:Definition:Spring/Equilibrium Position}}"}
+{"_id": "21365", "title": "Definition:Density (Physics)", "text": "'''Density''' is a physical quantity. The '''density''' of a body is its mass per unit volume."}
+{"_id": "21366", "title": "Definition:Volume", "text": "'''Volume''' is the measure of the extent of a body. It has three dimensions and is specified in units of length cubed."}
+{"_id": "21367", "title": "Definition:Homogeneous (Physics)", "text": "A body is said to be '''homogeneous''' {{iff}} the substance of any part of it is indistinguishable from any other part."}
+{"_id": "21368", "title": "Definition:Weight (Physics)", "text": "The '''weight''' of a body is the magnitude of the force exerted on it by the influence of a gravitational field. The context is that the gravitational field in question is usually that of the Earth."}
+{"_id": "21372", "title": "Definition:Chain (Physics)", "text": "A '''chain''' is an inelastic thread whose stiffness and width are approximated to zero. The mass of a '''chain''' is usually defined in terms of linear density."}
+{"_id": "21373", "title": "Definition:Cord (Physics)", "text": "A '''cord''' is an inelastic thread whose mass, stiffness and width are approximated to zero. That is, it is a chain whose mass is zero."}
+{"_id": "21374", "title": "Definition:Cone", "text": "A '''cone''' is a three-dimensional geometric figure which consists of the set of all straight lines joining the boundary of a plane figure $PQR$ to a point $A$ not in the same plane of $PQR$: :300px"}
+{"_id": "21375", "title": "Definition:Right Circular Cone", "text": "A '''right circular cone''' is a cone: : whose base is a circle : in which there is a line perpendicular to the base through its center which passes through the apex of the cone: : which is made by having a right-angled triangle turning along one of the sides that form the right angle. :300px {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book XI/18 - Cone}}'' {{EuclidDefRefNocat|XI|18|Cone}}"}
+{"_id": "21376", "title": "Definition:Cross-Section", "text": "A '''cross-section''' of a 3-dimensional figure $F$ is the intersection of $F$ with a plane. {{SUBPAGENAME}} ogr1fkh0dxg0lfa72ibeihamehx0jxp"}
+{"_id": "21377", "title": "Definition:Power of Mapping", "text": "Let $f: S \\to S$ be a mapping from $S$ to itself. Because the domain of $f$ is equal to the codomain of $f$ (both are $S$), the composite mapping $f \\circ f$ is defined. We define the '''$n$th power of $f$''' as: :$\\forall n \\in \\N: f^n = \\begin{cases} I_S & : n = 0 \\\\ f \\circ f^{n - 1} & : n > 0 \\end{cases}$ where $I_S$ is the identity mapping."}
+{"_id": "21378", "title": "Definition:Initial Segment of Natural Numbers", "text": "=== Zero-Based === {{:Definition:Initial Segment of Natural Numbers/Zero-Based}} === One-Based === {{:Definition:Initial Segment of Natural Numbers/One-Based}}"}
+{"_id": "21379", "title": "Definition:Count", "text": "To '''count''' a set $S$ is to establish a bijection between $S$ and an initial segment $\\N_n$ of the natural numbers $\\N$. If $S \\sim \\N_n$ (where $\\sim$ denotes set equivalence) then we have '''counted''' $S$ and found it has $n$ elements. If $S \\sim \\N$ then $S$ is infinite but countable."}
+{"_id": "21380", "title": "Definition:Multiplicative Notation", "text": ":$x y$ is used to indicate the result of the operation on $x$ and $y$. There is no symbol used to define the operation itself. :$e$ or $1$ is used for the identity element. :$x^{-1}$ is used for the inverse element. :$x^n$ is used to indicate the $n$th power of $x$."}
+{"_id": "21382", "title": "Definition:Additive Notation", "text": ":$x + y$ is used to indicate the result of the operation $+$ on $x$ and $y$. :$e$ or $0$ is used for the identity element. Note that in this context, $0$ is '''not''' a zero element. :$-x$ is used for the inverse element. :$n x$ is used to indicate the $n$th power of $x$."}
+{"_id": "21383", "title": "Definition:Set of Integer Multiples", "text": "The set $n \\Z$ is defined as: :$\\set {x \\in \\Z: n \\divides x}$ for some $n \\in \\Z_{>0}$. That is, it is the set of all integers which are divisible by $n$, that is, the '''set of integer multiples of $n$'''. Thus we have: :$n \\Z = \\set {\\ldots, -3 n, -2 n, -n, 0, n, 2 n, 3 n, \\ldots}$"}
+{"_id": "21385", "title": "Definition:Transversal (Group Theory)", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $S \\subseteq G$ be a subset of $G$. === Left Transversal === {{:Definition:Transversal (Group Theory)/Left Transversal}} === Right Transversal === {{:Definition:Transversal (Group Theory)/Right Transversal}} === Transversal === A '''transversal for $H$ in $G$''' is either a left transversal or a right transversal."}
+{"_id": "21386", "title": "Definition:Product of Differences", "text": "Let $n \\in \\Z_{> 0}$ be a strictly positive integer. Let $\\tuple {x_1, x_2, \\ldots, x_n}$ be an ordered $n$-tuple of real numbers. The '''product of differences''' of $\\tuple {x_1, x_2, \\ldots, x_n}$ is defined and denoted as: :$\\map {\\Delta_n} {x_1, x_2, \\ldots, x_n} = \\displaystyle \\prod_{1 \\mathop \\le i \\mathop < j \\mathop \\le n} \\paren {x_i - x_j}$ When the underlying ordered $n$-tuple is understood, the notation is often abbreviated to $\\Delta_n$. Thus $\\Delta_n$ is the product of the difference of all ordered pairs of $\\tuple {x_1, x_2, \\ldots, x_n}$ where the index of the first is less than the index of the second."}
+{"_id": "21387", "title": "Definition:Ordinary Space", "text": "'''Ordinary space''' (or just '''space''') is a word used to mean '''the universe we live in'''. The intuitive belief is that space is $3$-dimensional and therefore isomorphic to the real vector space $\\R^3$. Hence '''ordinary space''' is usually taken as an alternative term for Euclidean $3$-dimensional space."}
+{"_id": "21390", "title": "Definition:Computational Method", "text": "A '''computational method''' is an ordered quadruple $\\left({Q, I, \\Omega, f}\\right)$ in which: : $Q$ is a set representing the '''states of the computation''' : $I$ is a set representing the '''input to the computation''' : $\\Omega$ is a set representing the '''output from the computation''' : $f: Q \\to Q$ is a mapping representing the '''computational rule''' subject to the following constraints: : $I \\subseteq Q$ and $\\Omega \\subseteq Q$ : $\\forall x \\in \\Omega: f \\left({x}\\right) = x$ === Computational Sequence === {{:Definition:Computational Method/Computational Sequence}}"}
+{"_id": "21391", "title": "Definition:Set of All Linear Transformations", "text": "Let $G$ and $H$ be $R$-modules. Then $\\map {\\mathrm {Hom}_R} {G, H}$ is the '''set of all linear transformations''' from $G$ to $H$: :$\\map {\\mathrm {Hom}_R} {G, H} := \\set {\\phi: G \\to H: \\phi \\mbox{ is a linear transformation} }$ If it is clear (and therefore does not need to be stated) that the scalar ring is $R$, then this can be written $\\map {\\mathrm {Hom} } {G, H}$. Similarly, $\\map {\\mathrm {Hom}_R} G$ is the set of all linear operators on $G$: :$\\map {\\mathrm {Hom}_R} G := \\set {\\phi: G \\to G: \\phi \\text{ is a linear operator} }$ Again, this can also be written $\\map {\\mathrm {Hom} } G$."}
+{"_id": "21392", "title": "Definition:Inverse Matrix", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\mathbf A$ be a square matrix of order $n$. Let there exist a square matrix $\\mathbf B$ of order $n$ such that: :$\\mathbf A \\mathbf B = \\mathbf I_n = \\mathbf B \\mathbf A$ where $\\mathbf I_n$ denotes the unit matrix of order $n$. Then $\\mathbf B$ is called the '''inverse of $\\mathbf A$''' and is usually denoted $\\mathbf A^{-1}$."}
+{"_id": "21394", "title": "Definition:Stirling's Triangle of the Second Kind", "text": "$\\begin{array}{r|rrrrrrrrrr} n & {n \\brace 0} & {n \\brace 1} & {n \\brace 2} & {n \\brace 3} & {n \\brace 4} & {n \\brace 5} & {n \\brace 6} & {n \\brace 7} & {n \\brace 8} & {n \\brace 9} \\\\ \\hline 0 & 1 & 0 &   0 &    0 &    0 &    0 &    0 &   0 &  0  & 0 \\\\ 1 & 0 & 1 &   0 &    0 &    0 &    0 &    0 &   0 &  0  & 0 \\\\ 2 & 0 & 1 &   1 &    0 &    0 &    0 &    0 &   0 &  0  & 0 \\\\ 3 & 0 & 1 &   3 &    1 &    0 &    0 &    0 &   0 &  0  & 0 \\\\ 4 & 0 & 1 &   7 &    6 &    1 &    0 &    0 &   0 &  0  & 0 \\\\ 5 & 0 & 1 &  15 &   25 &   10 &    1 &    0 &   0 &  0  & 0 \\\\ 6 & 0 & 1 &  31 &   90 &   65 &   15 &    1 &   0 &  0  & 0 \\\\ 7 & 0 & 1 &  63 &  301 &  350 &  140 &   21 &   1 &  0  & 0 \\\\ 8 & 0 & 1 & 127 &  966 & 1701 & 1050 &  266 &  28 &  1  & 0 \\\\ 9 & 0 & 1 & 255 & 3025 & 7770 & 6951 & 2646 & 462 & 36  & 1 \\\\ \\end{array}$"}
+{"_id": "21395", "title": "Definition:Stirling's Triangle of the First Kind (Signed)", "text": "$\\begin{array}{r|rrrrrrrrrr} n & \\map s {n, 0} & \\map s {n, 1} & \\map s {n, 2} & \\map s {n, 3} & \\map s {n, 4} & \\map s {n, 5} & \\map s {n, 6} & \\map s {n, 7} & \\map s {n, 8} & \\map s {n, 9} \\\\ \\hline 0 & 1 &     0 &       0 &      0 &      0 &     0 &     0 &   0 &   0  & 0 \\\\ 1 & 0 &     1 &       0 &      0 &      0 &     0 &     0 &   0 &   0  & 0 \\\\ 2 & 0 &    -1 &       1 &      0 &      0 &     0 &     0 &   0 &   0  & 0 \\\\ 3 & 0 &     2 &      -3 &      1 &      0 &     0 &     0 &   0 &   0  & 0 \\\\ 4 & 0 &    -6 &      11 &     -6 &      1 &     0 &     0 &   0 &   0  & 0 \\\\ 5 & 0 &    24 &     -50 &     35 &    -10 &     1 &     0 &   0 &   0  & 0 \\\\ 6 & 0 &  -120 &     274 &   -225 &     85 &   -15 &     1 &   0 &   0  & 0 \\\\ 7 & 0 &   720 &   -1764 &   1624 &   -735 &   175 &   -21 &   1 &   0  & 0 \\\\ 8 & 0 & -5040 &   13068 & -13132 &   6769 & -1960 &   322 & -28 &   1  & 0 \\\\ 9 & 0 & 40320 & −109584 & 118124 & −67284 & 22449 & −4536 & 546 & −36  & 1 \\\\ \\end{array}$"}
+{"_id": "21396", "title": "Definition:Root of Unity", "text": "Let $n \\in \\Z_{> 0}$ be a strictly positive integer. Let $F$ be a field. The '''$n$th roots of unity of $F$''' are defined as: :$U_n = \\left\\{{z \\in F: z^n = 1}\\right\\}$"}
+{"_id": "21397", "title": "Definition:Internal Group Direct Product", "text": "Let $\\struct {H_1, \\circ {\\restriction_{H_1} } }, \\struct {H_2, \\circ {\\restriction_{H_2} } }$ be subgroups of a group $\\struct {G, \\circ}$ where $\\circ {\\restriction_{H_1} }, \\circ {\\restriction_{H_2} }$ are the restrictions of $\\circ$ to $H_1, H_2$ respectively. === Definition 1 === {{:Definition:Internal Group Direct Product/Definition 1}} === Definition 2 === {{:Definition:Internal Group Direct Product/Definition 2}} === Definition 3 === {{:Definition:Internal Group Direct Product/Definition 3}}"}
+{"_id": "21398", "title": "Definition:Permutable Subgroups", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H$ and $K$ be subgroups of $G$. Let $H \\circ K$ denote the subset product of $H$ and $K$. Then $H$ and $K$ are '''permutable''' {{iff}}: :$H \\circ K = K \\circ H$"}
+{"_id": "21399", "title": "Definition:Cycle Type", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\rho \\in S_n$. From Existence and Uniqueness of Cycle Decomposition, every $\\rho$ may be uniquely expressed as a product of disjoint cycles: :$\\rho = \\tau_1, \\tau_2, \\ldots, \\tau_r$ up to the order of factors. Let $\\tau_1, \\tau_2, \\ldots, \\tau_r$ be arranged in increasing order of cycle length. Let the length of the cycle $\\tau_i$ be $k_i$. The resulting ordered tuple of cycle lengths $\\tuple {k_1, k_2, \\ldots, k_r}$ is called the '''cycle type''' of $\\rho$. Thus $\\sigma$ and $\\rho$ have the same '''cycle type''' if they have the same number of cycles of equal length."}
+{"_id": "21400", "title": "Definition:Permutation on Polynomial", "text": "Let $\\map f {x_1, x_2, \\ldots, x_n}$ denote a polynomial in $n$ variables $x_1, x_2, \\ldots, x_n$. Let $S_n$ denote the symmetric group on $n$ letters. Let $\\pi, \\rho \\in S_n$. Then $\\pi * f$ is the polynomial obtained by applying the permutation $\\pi$ to the subscripts on the variables of $f$. This is called the '''permutation on the polynomial $f$ by $\\pi$''', or the $f$-permutation by $\\pi$."}
+{"_id": "21401", "title": "Definition:Pullback of Quotient Group Isomorphism", "text": "Let $G, H$ be groups. Let $N \\lhd G, K \\lhd H$ be normal subgroups of $G$ and $H$ respectively. Let: :$G / N \\cong H / K$ where: :$G / N$ denotes the quotient of $G$ by $N$ :$\\cong$ denotes group isomorphism. Let $\\theta: G / N \\to H / K$ be such a group isomorphism. The '''pullback $G \\times^\\theta H$ of $G$ and $H$ via $\\theta$''' is the subset of $G \\times H$ of elements of the form $\\tuple {g, h}$ where $\\map \\theta {g N} = h K$."}
+{"_id": "21402", "title": "Definition:Permutation Group", "text": "A '''permutation group''' on a set $S$ is a subgroup of the symmetric group $\\struct {\\map \\Gamma S, \\circ}$ on $S$."}
+{"_id": "21403", "title": "Definition:Commutative and Unitary Ring", "text": "A '''commutative and unitary ring''' $\\struct {R, +, \\circ}$ is a ring with unity which is also commutative. That is, it is a ring such that the ring product $\\struct {R, \\circ}$ is commutative and has an identity element. That is, such that the multiplicative semigroup $\\struct {R, \\circ}$ is a commutative monoid."}
+{"_id": "21404", "title": "Definition:K-Algebra", "text": "Let $k$ be a field. A '''$k$-algebra''' is a unital associative commutative algebra over $k$. Category:Definitions/Commutative Algebra Category:Definitions/Algebras jn4ahs1333akdt43rqk74peimcev5md"}
+{"_id": "21405", "title": "Definition:Entropic Operation", "text": "Let $\\circ$ be an operation on a set $S$ such that: :$\\forall a, b, c, d \\in S: \\left({a \\circ b}\\right) \\circ \\left({c \\circ d}\\right) = \\left({a \\circ c}\\right) \\circ \\left({b \\circ d}\\right)$ Then $\\circ$ is '''entropic'''."}
+{"_id": "21406", "title": "Definition:Finite Ring Homomorphism", "text": "Let $A$ and $B$ be commutative rings with unity. Let $\\phi: A \\to B$ be a ring homomorphism. === Definition 1 === $\\phi$ is '''finite''' {{iff}} $B$ is finite as an algebra over $A$ via $\\phi$. === Definition 2 === $\\phi$ is '''finite''' {{iff}} there exists a finite number of $b_1, \\ldots, b_n$ such that every $b \\in B$ can be written as: :$\\displaystyle b = \\sum_{i \\mathop = 1}^n \\phi \\left({a_i}\\right) b_i$ where $a_i \\in A$."}
+{"_id": "21407", "title": "Definition:Composition (Combinatorics)", "text": "A $k$-composition of a (strictly) positive integer $n \\in \\Z_{> 0}$ is an ordered $k$-tuple: :$c = \\left({c_1, c_2, \\ldots, c_k}\\right)$ such that: : $(1): \\quad c_1 + c_2 + \\cdots + c_k = n$ : $(2): \\quad \\forall i \\in \\left[{1 \\,.\\,.\\, k}\\right]: c_i \\in \\Z_{> 0}$, that is, all the $c_i$ are strictly positive integers. Category:Definitions/Combinatorics 2iv1b2c8grfnqiu47q6bt64v06zxjsv"}
+{"_id": "21408", "title": "Definition:Ideal of Ring/Proper Ideal", "text": "A '''proper ideal''' $J$ of $\\left({R, +, \\circ}\\right)$ is an ideal of $R$ such that $J$ is a proper subset of $R$. That is, such that $J \\subseteq R$ and $J \\ne R$."}
+{"_id": "21409", "title": "Definition:Generated Ideal of Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $S \\subseteq R$ be a subset. === Definition 1 === {{Definition:Generated Ideal of Ring/Definition 1}} === Definition 2: for commutative rings with unity === {{Definition:Generated Ideal of Ring/Definition 2}} === Definition 3: for rings with unity === {{Definition:Generated Ideal of Ring/Definition 3}}"}
+{"_id": "21410", "title": "Definition:Internal Direct Sum of Rings", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring. Let $S_1, S_2, \\ldots, S_n$ be a finite sequence of subrings of $R$. Let $\\displaystyle S = \\prod_{j \\mathop = 1}^n S_j$ be the cartesian product of $S_1$ to $S_n$. Then $S$ is the '''(ring) direct sum''' of $S_1, S_2, \\ldots, S_n$ {{iff}} the mapping $\\phi: S \\to R$ defined as: :$\\phi\\left({\\left({x_1, x_2, \\ldots, x_n}\\right)}\\right) = x_1 + x_2 + \\cdots x_n$ is an isomorphism from $S$ to $R$."}
+{"_id": "21411", "title": "Definition:Prime Ideal of Ring", "text": "Let $R$ be a ring. A '''prime ideal''' of $R$ is a proper ideal $P$ such that: : $I \\circ J \\subseteq P \\implies I \\subseteq P \\text { or } J \\subseteq P$ for any ideals $I$ and $J$ of $R$."}
+{"_id": "21412", "title": "Definition:Generated Division Subring", "text": "Let $\\left({D, +, \\circ}\\right)$ be a division ring. Let $S \\subseteq D$. The '''division subring generated by $S$''' is the smallest division subring of $D$ containing $S$."}
+{"_id": "21413", "title": "Definition:Isomorphism (Abstract Algebra)/Field Isomorphism", "text": "Let $\\struct {F, +, \\circ}$ and $\\struct {K, \\oplus, *}$ be fields. Let $\\phi: F \\to K$ be a (field) homomorphism. Then $\\phi$ is a field isomorphism {{iff}} $\\phi$ is a bijection."}
+{"_id": "21414", "title": "Definition:Cossist", "text": "The '''cossist''' tradition was a school of abacist mathematics developed in Germany in the 15th century. Notable cossists include {{AuthorRef|Johann Faulhaber}}."}
+{"_id": "21415", "title": "Definition:Scalar Multiplication", "text": "=== $R$-Algebraic Structure === {{:Definition:Scalar Multiplication/R-Algebraic Structure}} === Module === {{:Definition:Scalar Multiplication/Module}} === Vector Space === {{:Definition:Scalar Multiplication/Vector Space}}"}
+{"_id": "21416", "title": "Definition:R-Algebraic Structure Automorphism", "text": "Let $\\left({S, \\ast_1, \\ast_2, \\ldots, \\ast_n, \\circ}\\right)_R$ be an $R$-algebraic structure. Let $\\phi: S \\to S$ be an $R$-algebraic structure isomorphism from $S$ to itself. Then $\\phi$ is an $R$-algebraic structure automorphism. This definition continues to apply when $S$ is a module, and also when it is a vector space."}
+{"_id": "21417", "title": "Definition:Generator of Module", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $S \\subseteq M$ be a subset. === Definition 1 === {{:Definition:Generator of Module/Definition 1}} === Definition 2 === {{:Definition:Generator of Module/Definition 2}} === Definition 3 === {{:Definition:Generator of Module/Definition 3}}"}
+{"_id": "21418", "title": "Definition:Kernel of Linear Transformation", "text": "Let $\\phi: G \\to H$ be a linear transformation where $G$ and $H$ are $R$-modules. Let $e_H$ be the identity of $H$. The '''kernel''' of $\\phi$ is defined as: :$\\map \\ker \\phi := \\phi^{-1} \\sqbrk {\\set {e_H} }$. where $\\phi^{-1} \\sqbrk S$ denotes the preimage of $S$ under $\\phi$. === In Vector Space === {{:Definition:Kernel of Linear Transformation/Vector Space}}"}
+{"_id": "21419", "title": "Definition:R-Algebraic Structure Monomorphism", "text": "Let $\\left({S, \\ast_1, \\ast_2, \\ldots, \\ast_n, \\circ}\\right)_R$ and $\\left({T, \\odot_1, \\odot_2, \\ldots, \\odot_n, \\otimes}\\right)_R$ be $R$-algebraic structures. Then $\\phi: S \\to T$ is an '''$R$-algebraic structure monomorphism''' {{iff}}: : $(1): \\quad \\phi$ is an injection : $(2): \\quad \\forall k: k \\in \\left[{1 \\,.\\,.\\, n}\\right]: \\forall x, y \\in S: \\phi \\left({x \\ast_k y}\\right) = \\phi \\left({x}\\right) \\odot_k \\phi \\left({y}\\right)$ : $(3): \\quad \\forall x \\in S: \\forall \\lambda \\in R: \\phi \\left({\\lambda \\circ x}\\right) = \\lambda \\otimes \\phi \\left({x}\\right)$. That is, {{iff}}: : $(1): \\quad \\phi$ is an injection : $(2): \\quad \\phi$ is an $R$-algebraic structure homomorphism. This definition continues to apply when $S$ and $T$ are modules, and also when they are vector spaces. === Vector Space Monomorphism === {{:Definition:Vector Space Monomorphism}}"}
+{"_id": "21420", "title": "Definition:Vector Space Monomorphism", "text": "Let $V$ and $W$ be $K$-vector spaces. Then $\\phi: V \\to W$ is a '''vector space monomorphism''' {{iff}}: : $(1): \\quad \\phi$ is an injection : $(2): \\quad \\forall \\mathbf x, \\mathbf y \\in V: \\phi \\left({\\mathbf x + \\mathbf y}\\right) = \\phi \\left({\\mathbf x}\\right) + \\phi \\left({\\mathbf y}\\right)$ : $(3): \\quad \\forall \\mathbf x \\in V: \\forall \\lambda \\in K: \\phi \\left({\\lambda \\mathbf x}\\right) = \\lambda \\phi \\left({\\mathbf x}\\right)$"}
+{"_id": "21421", "title": "Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Vector Space Isomorphism", "text": "Let $\\struct {V, +, \\circ }$ and $\\struct {W, +', \\circ'}$ be $K$-vector spaces. Then $\\phi: V \\to W$ is a '''vector space isomorphism''' {{iff}}: : $(1): \\quad \\phi$ is a bijection : $(2): \\quad \\forall \\mathbf x, \\mathbf y \\in V: \\map \\phi {\\mathbf x + \\mathbf y} = \\map \\phi {\\mathbf x} +' \\map \\phi {\\mathbf y}$ : $(3): \\quad \\forall \\mathbf x \\in V: \\forall \\lambda \\in K: \\map \\phi {\\lambda \\mathbf x} = \\lambda \\map \\phi {\\mathbf x}$"}
+{"_id": "21422", "title": "Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Module Isomorphism", "text": "Let $\\struct {G, +_G, \\circ}_R$ and $\\struct {H, +_H, \\circ}_R$ be $R$-modules. Let $\\phi: G \\to H$ be a module homomorphism. Then $\\phi$ is a '''module isomorphism''' {{iff}} $\\phi$ is a bijection."}
+{"_id": "21424", "title": "Definition:Quaternion", "text": "A '''quaternion''' is a number in the form: : $a \\mathbf 1 + b \\mathbf i + c \\mathbf j + d \\mathbf k$ where: : $a, b, c, d$ are real numbers : $\\mathbf 1, \\mathbf i, \\mathbf j, \\mathbf k$ are entities related to each other in the following way: {{begin-eqn}}      {{eqn | l = \\mathbf i \\mathbf j = -\\mathbf j \\mathbf i       | r = \\mathbf k       | c =   }} {{eqn | l = \\mathbf j \\mathbf k = -\\mathbf k \\mathbf j       | r = \\mathbf i       | c =  }} {{eqn | l = \\mathbf k \\mathbf i = -\\mathbf i \\mathbf k       | r = \\mathbf j       | c =  }} {{eqn | l = \\mathbf i^2 = \\mathbf j^2 = \\mathbf k^2 = \\mathbf i \\mathbf j \\mathbf k       | r = -\\mathbf 1       | c =  }} {{end-eqn}}"}
+{"_id": "21425", "title": "Definition:Conjugate Quaternion", "text": "Let $\\mathbf x = a \\mathbf 1 + b \\mathbf i + c \\mathbf j + d \\mathbf k$ be a quaternion. The conjugate quaternion of $\\mathbf x$ is defined as: :$\\overline {\\mathbf x} = a \\mathbf 1 - b \\mathbf i - c \\mathbf j - d \\mathbf k$. === Matrix Form === {{:Definition:Conjugate Quaternion/Matrix Form}} === Ordered Pair of Complex Numbers === {{:Definition:Conjugate Quaternion/Ordered Pair}}"}
+{"_id": "21426", "title": "Definition:Subsemiring", "text": "Let $\\left({S, *, \\circ}\\right)$ be an algebraic structure with two operations. A '''subsemiring of $\\left({S, *, \\circ}\\right)$''' is a subset $T$ of $S$ such that $\\left({T, *_T, \\circ_T}\\right)$ is a semiring. {{SUBPAGENAME}} 6corsnq5jtxaxkicgktu8idhrahdz68"}
+{"_id": "21428", "title": "Definition:Additive Group of Ring", "text": "The group $\\struct {R, +}$ is known as the '''additive group of $R$'''."}
+{"_id": "21429", "title": "Definition:Additive Subgroup", "text": "Let $\\struct {R, +, \\times}$ be a ring. Let $\\struct {R, +}$ be the additive group of $\\struct {R, +, \\times}$. Let $\\struct {S, +}$ be a subgroup of $\\struct {R, +}$ Then $\\struct {S, +}$ is an '''additive subgroup of ''' $\\struct {R, +, \\times}$."}
+{"_id": "21430", "title": "Definition:Generated Subring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $S \\subseteq R$ be a subset. The '''subring generated by $S$''' is the smallest subring of $R$ containing $S$; that is, it is the intersection of all subrings of $R$ containing $S$."}
+{"_id": "21431", "title": "Definition:Proper Zero Divisor", "text": "Let $\\struct {R, +, \\circ}$ be a ring. A proper zero divisor of $R$ is an element $x \\in R^*$ such that: :$\\exists y \\in R^*: x \\circ y = 0_R$ where $R^*$ is defined as $R \\setminus \\set {0_R}$. That is, it is a zero divisor of $R$ which is '''specifically not $0_R$'''."}
+{"_id": "21433", "title": "Definition:Perfect Graph", "text": "A graph is '''perfect''' if no two vertices have the same degree."}
+{"_id": "21434", "title": "Definition:Term of Sequence", "text": "The elements of a sequence are known as its '''terms'''. Let $\\sequence {x_n}$ be a sequence. Then the '''$k$th term''' of $\\sequence {x_n}$ is the ordered pair $\\tuple {k, x_k}$."}
+{"_id": "21435", "title": "Definition:Totally Ordered Group", "text": "A '''totally ordered group''' is a totally ordered structure $\\left({G, \\circ, \\preceq}\\right)$ such that $\\left({G, \\circ}\\right)$ is a group."}
+{"_id": "21436", "title": "Definition:Jacobson Radical", "text": "Let $R$ be a commutative ring with unity. Let $\\operatorname{maxspec} \\left({R}\\right)$ be the set of maximal ideals of $R$. Then the '''Jacobson radical''' of $R$ is: :$\\displaystyle \\operatorname{Jac} \\left({R}\\right) = \\bigcap_{m \\mathop \\in \\operatorname{maxspec} \\left({R}\\right)} m$ That is, it is the intersection of all maximal ideals of $R$."}
+{"_id": "21437", "title": "Definition:Jacobson Ring", "text": "Let $\\left({R, +, \\circ}\\right)$ be a commutative ring with unity. Then $\\left({R, +, \\circ}\\right)$ is a '''Jacobson ring''' {{iff}}: : every prime ideal of $\\left({R, +, \\circ}\\right)$ is an intersection of  maximal ideals."}
+{"_id": "21438", "title": "Definition:Adjugate Matrix", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $\\mathbf C$ be its cofactor matrix. The '''adjugate matrix''' of $\\mathbf A$ is the transpose of $\\mathbf C$: :$\\adj {\\mathbf A} = \\mathbf C^\\intercal$"}
+{"_id": "21439", "title": "Definition:Radical of Ideal of Ring", "text": "Let $A$ be a commutative ring with unity. Let $I$ be an ideal of $A$. === Definition 1 === {{Definition:Radical of Ideal of Ring/Definition 1}} === Definition 2 === {{Definition:Radical of Ideal of Ring/Definition 2}}"}
+{"_id": "21440", "title": "Definition:Radical Ideal of Ring", "text": "Let $A$ be a commutative ring with unity. Let $I$ be an ideal of $A$. Then $I$ is a '''radical ideal''' {{iff}} it is equal to its radical."}
+{"_id": "21441", "title": "Definition:Nilpotent Ring Element", "text": "Let $R$ be a ring with zero $0_R$. An element $x \\in R$ is '''nilpotent''' {{iff}}: :$\\exists n \\in \\Z_{>0}: x^n = 0_R$"}
+{"_id": "21442", "title": "Definition:Reduced Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Then $R$ is a '''reduced ring''' {{iff}} it contains no nilpotent elements except the zero."}
+{"_id": "21443", "title": "Definition:Nilradical of Ring", "text": "Let $A$ be a commutative ring with unity. === Definition 1 === {{:Definition:Nilradical of Ring/Definition 1}} === Definition 2 === {{:Definition:Nilradical of Ring/Definition 2}}"}
+{"_id": "21444", "title": "Definition:Polynomial Ring/Monoid Ring on Free Monoid on Set", "text": "Let $R \\sqbrk {\\family {X_i: i \\in I} }$ be the ring of polynomial forms in $\\family {X_i: i \\in I}$. The '''polynomial ring in $I$ indeterminates over $R$''' is the ordered triple $\\struct {\\struct {A, +, \\circ}, \\iota, \\family {X_i: i \\in I} }$ {{DefinitionWanted|define the inclusion and indeterminates in this case}}"}
+{"_id": "21445", "title": "Definition:Monomial of Free Commutative Monoid", "text": "A '''monomial''' in the indexed set $\\family {X_j: j \\in J}$ is a possibly infinite product: :$\\ds \\prod_{j \\mathop \\in J} X_j^{k_j}$ with integer exponents $k_j \\ge 0$ such that $k_j = 0$ for all but finitely many $j$. Let $\\mathbf X = \\family {X_j}_{j \\mathop \\in J}$ and for a multiindex $k = \\paren {k_j}_{j \\mathop \\in J}$ over $J$ define: :$\\ds \\mathbf X^k = \\prod_{j \\mathop \\in J} X_j^{k_j}$ Then a '''monomial''' is an object of the form $\\mathbf X^k$, where $k$ is a multiindex."}
+{"_id": "21446", "title": "Definition:Free Commutative Monoid", "text": "The '''free commutative monoid''' on an indexed set $X = \\family {X_j: j \\in J}$ is the set $M$ of all monomials under the standard multiplication. That is, it is the set $M$ of all finite sequences of $X$."}
+{"_id": "21447", "title": "Definition:Multiindex", "text": "=== Definition 1 === Let $\\ds m = \\prod_{j \\mathop \\in J} X_j^{k_j}$ be a monomial in the indexed set $\\set {X_j: j \\mathop \\in J}$. Such a monomial can be expressed implicitly and more compactly by referring only to the sequence of indices: :$k = \\sequence {k_j}_{j \\mathop \\in J}$ and write $m = \\mathbf X^k$ without explicit reference to the indexing set. Such an expression is called a '''multiindex''' (or '''multi-index'''). === Definition 2 === Let $J$ be a set. A '''$J$-multiindex''' is a sequence of natural numbers indexed by $J$: :$\\displaystyle k = \\sequence {k_j}_{j \\mathop \\in J}$ with only finitely many of the $k_j$ non-zero. === Definition 3 === A '''multiindex''' is an element of $\\Z^J$, the free $\\Z$-module on $J$, an abelian group of rank over $\\Z$ equal to the cardinality of $J$. === Modulus === {{:Definition:Multiindex/Modulus}}"}
+{"_id": "21448", "title": "Definition:Pointwise Addition of Mappings", "text": "Let $S$ be a non-empty set. Let $\\struct {G, \\circ}$ be a commutative semigroup. Let $G^S$ be the set of all mappings from $S$ to $G$. Then '''pointwise addition''' on $G^S$ is the binary operation $\\circ: G^S \\times G^S \\to G^S$ (the $\\circ$ is the same as for $G$) defined by: :$\\forall f, g \\in G^S: \\forall s \\in S: \\map {\\paren {f \\circ g} } s := \\map f s \\circ \\map g s$ The double use of $\\circ$ is justified as $\\struct {G^S, \\circ}$ inherits all abstract-algebraic properties $\\struct {G, \\circ}$ might have. This is rigorously formulated and proved on Mappings to Algebraic Structure form Similar Algebraic Structure. === Pointwise Multiplication === Let $\\circ$ be used with multiplicative notation. Then the operation defined above is called '''pointwise multiplication''' instead."}
+{"_id": "21449", "title": "Definition:Ring of Polynomial Functions", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $R \\sqbrk {\\set {X_j: j \\in J} }$ be the ring of polynomial forms over $R$ in the indeterminates $\\set {X_j: j \\in J}$. Let $R^J$ be the free module on $J$. Let $A$ be the set of all polynomial functions $R^J \\to R$. Then the operations $+$ and $\\circ$ on $R$ induce pointwise operations on $A$. We denote these operations by the same symbols: :$\\forall x \\in R^J: \\map {\\paren {f + g} } x = \\map f x + \\map g x$ :$\\forall x \\in R^J: \\map {\\paren {f \\circ g} } x = \\map f x \\circ \\map g x$ The '''ring of polynomial functions''' is the resulting algebraic structure."}
+{"_id": "21450", "title": "Definition:Arithmetic Function", "text": "An '''arithmetic function''' or '''arithmetical function''' is a complex valued function defined on the strictly positive integers. That is, a mapping: :$f: \\Z_{>0} \\to \\C$"}
+{"_id": "21451", "title": "Definition:Category", "text": "A '''category''' is an interpretation of the metacategory axioms within set theory. Because a metacategory is a metagraph, this means that a '''category''' is a graph. Let $\\mathfrak U$ be a class of sets. A metacategory $\\mathbf C$ is a '''category''' {{iff}}: :$(1): \\quad$ The objects form a subset $\\mathbf C_0$ or $\\operatorname {ob} \\ \\CC \\subseteq \\mathfrak U$ :$(2): \\quad$ The morphisms form a subset $\\mathbf C_1$ or $\\operatorname{mor} \\ \\mathbf C$ or $\\operatorname{Hom} \\ \\mathbf C \\subseteq \\mathfrak U$"}
+{"_id": "21452", "title": "Definition:Abuse of Notation", "text": "Mathematical notation can be considered in two ways: * As an aid to mathematical understanding, no more and no less than a useful convention to encapsulate more or less complicated ideas in a completely unambiguous format; * As the reason for mathematical effort, so as to encapsulate a truth as a documented piece of aesthetic beauty in its own right. '''Abuse of notation''' is a technique of using a system of symbology in a way different from that for which it was originally defined. Such abuse may make a train of thought more streamlined, as it is often possible to save considerable redefinition of one's terms. However, such abuse is frequently considered to be incorrect, improper and (in the eyes of many mathematicians) illegal. Philosophers of the various schools practising pragmatism tend to consider that if a notation has been adequately explained, then one should be allowed to use it in whatever way is most useful to communicate one's ideas. On the other hand, such an attitude causes indignation, rage and fury among philosophers whose attraction to mathematics is purely aesthetic."}
+{"_id": "21453", "title": "Definition:Completion (Metric Space)", "text": "Let $M_1 = \\struct {A, d}$ and $M_2 = \\struct {\\tilde A, \\tilde d}$ be metric spaces. Then $M_2$ is a '''completion''' of $M_1$, or $M_2$ '''completes''' $M_1$, {{iff}}: :$(1): \\quad M_2$ is a complete metric space :$(2): \\quad A \\subseteq \\tilde A$ :$(3): \\quad A$ is dense in $M_2$ :$(4): \\quad \\forall x, y \\in A : \\map {\\tilde d} {x, y} = \\map d {x, y}$. In terms of restriction of functions, this says that $\\map {\\tilde d {\\restriction_A} } = d$. It is immediate from this definition that a '''completion''' of a metric space $M_1$ consists of: :A complete metric space $M_2$ :An isometry $\\phi : A \\to \\tilde A$ such that $\\map \\phi A = \\set {\\map \\phi x: x \\in A}$ is dense in $M_2$. An isometry is often required to be bijective, so here one should consider $\\phi$ as a mapping from $A$ to the image of $\\phi$. Therefore to insist that $\\phi$ be an isometry, ''in this context'', is to say that $\\phi$ must be an injection that preserves the metric of $M_1$."}
+{"_id": "21454", "title": "Definition:Metacategory", "text": "A '''metacategory''' is a metagraph subject to extra restrictions. As such, a '''metacategory''' $\\mathbf C$ consists of: * A collection $\\mathbf C_0$ of objects $X, Y, Z, \\ldots$ * A collection $\\mathbf C_1$ of morphisms $f, g, h, \\ldots$ between its objects The morphisms of $\\mathbf C$ are subjected to: {{begin-axiom}} {{axiom | n = C1         | lc= ''Composition''         | t = For objects $X, Y, Z$ and morphisms $X \\stackrel {f} {\\longrightarrow} Y \\stackrel {g} {\\longrightarrow} Z$ with $\\operatorname{cod} f = \\operatorname{dom} \\, g$, there exists a morphism: :$g \\circ f : X \\to Z$ called the composition of $f$ and $g$. }} {{axiom | n = C2         | lc= ''Identity''          | t = For every object $X$, there is a morphism ${\\operatorname{id}_X}: X \\to X$, called the identity morphism, such that: :$f \\circ {\\operatorname{id}_X} = f$, and ${\\operatorname{id}_X} \\circ g = g$ for any object $Y$ and morphisms $f : X \\to Y$ and $g : Y \\to X$ }} {{axiom | n = C3         | lc= ''Associativity''         | t = For any three morphisms $f, g, h$: :$f \\circ \\left({g \\circ h}\\right) = \\left({f \\circ g}\\right) \\circ h$ whenever these compositions are defined (according to $(C1)$). }} {{end-axiom}} To describe a '''metacategory''' it is necessary to specify: :* The collection $\\mathbf C_0$ of objects :* The collection $\\mathbf C_1$ of morphisms :* For each object $X$, an identity morphism ${\\operatorname{id}_X}: X \\to X$ :* For every appropriate pair of morphisms $f, g$, the composite morphism $g \\circ f$ However, the last two are often taken to be implicit when the objects and morphisms are familiar. Of course, after defining these, it is still to be shown that $(C1)$ up to $(C3)$ are satisfied. A '''metacategory''' is purely axiomatic, and does not use set theory. For example, the objects are not \"elements of the set of objects\", because these axioms are (without further interpretation) unfounded in set theory. For some purposes, it is convenient to have a different description of a '''metacategory'''. Two such descriptions are found on: * Characterization of Metacategory via Equations * Morphisms-Only Metacategory"}
+{"_id": "21455", "title": "Definition:Non-Archimedean", "text": "=== Non-Archimedean Norm (Vector Space) === {{:Definition:Non-Archimedean/Norm (Vector Space)}} === Non-Archimedean Norm (Division Ring) === {{:Definition:Non-Archimedean/Norm (Division Ring)}} === Non-Archimedean Metric === {{:Definition:Non-Archimedean/Metric}} Category:Definitions/Norm Theory dicdyvqyec5b4uu6ym7vxwb2mloc6j5"}
+{"_id": "21456", "title": "Definition:Orthogonal (Linear Algebra)", "text": "Let $\\struct {V, \\innerprod \\cdot \\cdot}$ be an inner product space. Let $u, v \\in V$. Then $u$ and $v$ are '''orthogonal''' {{iff}}: :$\\innerprod u v = 0$ === Orthogonal Set === {{:Definition:Orthogonal (Linear Algebra)/Set}} === Vectors in $\\R^n$ === {{:Definition:Orthogonal (Linear Algebra)/Real Vector Space}}"}
+{"_id": "21457", "title": "Definition:Valued Field", "text": "Let $\\struct {K, +, \\circ}$ be a field. Let $\\norm {\\,\\cdot\\,}$ be a norm on $K$. Then $\\struct {K, \\norm{\\,\\cdot\\,} }$ is a '''valued field'''."}
+{"_id": "21458", "title": "Definition:Minkowski Functional", "text": "Let $E$ be a vector space over $\\R$. A functional $p: E \\to \\R$ is called a '''Minkowski functional''' if it satisfies: {{begin-axiom}} {{axiom | n = 1         | q = \\forall x \\in E, \\forall \\lambda \\in \\R_{>0}         | ml= \\map p {\\lambda x}         | mo= =         | mr= \\lambda \\map p x         | rc= that is, $p$ is positive homogeneous }} {{axiom | n = 2         | q = \\forall x, y \\in E         | ml= \\map p {x + y}         | mo= \\le         | mr= \\map p x + \\map p y         | rc= that is, $p$ is sub-additive }} {{end-axiom}} {{NamedforDef|Hermann Minkowski|cat = Minkowski}} Category:Definitions/Functional Analysis Category:Definitions/Topology 4kxgn0joo47z28y6odqekq891lzuff1"}
+{"_id": "21459", "title": "Definition:Linear Functional", "text": "Let $E$ be a vector space over a field $\\Bbb F$. Let $D$ be a linear subspace of $E$. A mapping $f : D \\to \\Bbb F$ is called a '''linear functional''' iff: :$ f \\left({\\alpha x + \\beta y}\\right) = \\alpha f(x) + \\beta f(y)$ holds for all $x, y$ in $L$ and for all $\\alpha, \\beta$ in $\\Bbb F$."}
+{"_id": "21461", "title": "Definition:Euclidean Domain", "text": "Let $\\struct {R, +, \\circ}$ be an integral domain with zero $0_R$. Let there exist a mapping $\\nu: R \\setminus \\set {0_R} \\to \\N$ with the properties: :$(1): \\quad$ For all $a, b \\in R, b \\ne 0_R$, there exist $q, r \\in R$ with $\\map \\nu r < \\map \\nu b$, or $r = 0_R$ such that: ::::$a = q \\circ b + r$ :$(2): \\quad$ For all $a, b \\in R, b \\ne 0_R$: ::::$\\map \\nu a \\le \\map \\nu {a \\circ b}$ Then $\\nu$ is called a '''Euclidean valuation''' and $R$ is called a '''Euclidean domain'''."}
+{"_id": "21463", "title": "Definition:Farey Sequence", "text": "The '''Farey sequence''' is a chain of subsets of the reduced rational numbers lying in $\\Q \\cap \\closedint 0 1$. For $Q \\in \\Z_{>0}$, the '''Farey set''' $F_Q$ is the set of all reduced rational numbers with denominators not larger than $Q$: :$F_Q = \\set {\\dfrac p q: p = 0, \\ldots, Q,\\ q = 1, \\ldots, Q,\\ p \\perp q}$ where $p \\perp q$ denotes that $p$ and $q$ are coprime."}
+{"_id": "21464", "title": "Definition:Linear Representation", "text": "=== Groups === {{:Definition:Linear Representation/Group}} === Algebras === {{:Definition:Linear Representation/Algebra}}"}
+{"_id": "21465", "title": "Definition:Dimension (Representation Theory)", "text": "Let $\\struct {k, +, \\circ}$ be a field. Let $V$ be a vector space over $k$ of finite dimension. Let $\\GL V$ be the general linear group of $V$. Let $\\struct {G, \\cdot}$ be a finite group. Let $\\rho: G \\to \\GL V$ be a linear representation of $G$ on $V$. The '''dimension''' or '''degree''' of $\\rho$, written $\\map \\deg \\rho$ is the dimension of the vector space $V$. Category:Definitions/Representation Theory c66h1ktiokjtucgu3oi4bqj4i1jxurb"}
+{"_id": "21466", "title": "Definition:Faithful Linear Representation of Group", "text": "Let $ \\left({k, +, \\circ}\\right)$ be a field. Let $V$ be a vector space over $k$ of finite dimension. Let $\\operatorname{GL} \\left({V}\\right)$ be the general linear group of $V$. Let $\\left({G, \\cdot}\\right)$ be a finite group. Let $\\rho : G \\to \\operatorname{GL} \\left({V}\\right)$ be a linear representation of $G$ on $V$. Then $\\rho$ is '''faithful''' if the kernel of $\\rho$ is trivial. {{SUBPAGENAME}} 2gwgnr1htwebwxkr89qk9ty3psbwpvz"}
+{"_id": "21467", "title": "Definition:Linear Group Action", "text": "Let $\\left({V, +, \\cdot}\\right)$ be a vector space over a field $\\left({k, \\oplus, \\circ}\\right)$. Let $G$ be a group. Let $\\phi : G \\times V \\to V$ be an action of $G$ on $V$. Then $\\phi$ is a '''(left) linear group action''' {{iff}} it is compatible with the linear structure of $V$ in the following sense: : $(1): \\quad \\forall v_1, v_2 \\in V: g \\in G: \\phi \\left({g, v_1 + v_2}\\right) = \\phi \\left({g, v_1}\\right) + \\phi \\left({g, v_2}\\right)$ : $(2): \\quad \\forall \\lambda \\in k, g \\in G, v \\in V: \\phi \\left({g, \\lambda \\cdot v}\\right) = \\lambda \\cdot \\phi \\left({g, v}\\right)$ === Right Linear Group Action === {{:Definition:Linear Group Action/Right Linear Group Action}}"}
+{"_id": "21468", "title": "Definition:Group Algebra", "text": "Let $\\left({k, + ,\\circ}\\right)$ be a field. Let $\\left({G, *}\\right)$ be a finite group. Then the '''group algebra''' $k G$ or $k \\left[{G}\\right]$ is the set of all formal sums: :$\\displaystyle \\sum_{g \\in G} \\alpha_g g\\ :\\ \\alpha_g \\in k$ That is, $k \\left[{G}\\right]$ is the free vector space over $k$ with basis $G$. {{SUBPAGENAME}} nne3ovn9w8raya0d71gehml4l2xsazi"}
+{"_id": "21470", "title": "Definition:Krull Dimension of Ring", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. The '''Krull dimension''' of $R$ is the supremum of lengths of chains of prime ideals, ordered by inclusion: :$\\map {\\operatorname {dim_{Krull} } } R = \\sup \\set {n \\in \\N: \\exists p_0, \\ldots, p_n \\in \\Spec R: \\mathfrak p_0 \\subsetneqq \\mathfrak p_1 \\subsetneqq \\cdots \\subsetneqq \\mathfrak p_n}$ where $\\Spec R$ is the prime spectrum of $R$. That is, the '''Krull dimension''' is $\\infty$ if there exist arbitrarily long chains."}
+{"_id": "21471", "title": "Definition:Exact Sequence of Modules", "text": "Let $\\left({R, +, \\cdot}\\right)$ be a ring. Let: :$(1): \\quad \\cdots \\longrightarrow M_i \\stackrel{d_i} {\\longrightarrow} M_{i + 1} \\stackrel {d_{i + 1}} {\\longrightarrow} M_{i + 2} \\stackrel {d_{i + 2}} {\\longrightarrow} \\cdots$ be a sequence of $R$-modules $M_i$ and $R$-module homomorphisms $d_i$. Then the sequence $(1)$ is '''exact''' {{iff}}: : $\\forall i: \\operatorname{Im} d_i = \\ker d_{i + 1}$ where $\\operatorname{Im}$ and $\\ker$ denote the image and kernel of mappings respectively."}
+{"_id": "21472", "title": "Definition:Differential Module", "text": "Let $R$ be a commutative ring with unity. Let $M$ be an $R$-module. A '''differential''' on $M$ is a homomorphism $d:M\\to M$ of $R$-modules such that $d^2 = d\\circ d = 0$. The pair $(M,d)$ is called a '''differential module'''. {{SUBPAGENAME}} 3bgo6tem59m2851njehp4d3uddkt2p1"}
+{"_id": "21473", "title": "Definition:Graded Ring", "text": "Let $M$ be a semigroup. An '''$M$-graded ring''' is a pair $\\struct {R, f}$ where: :$R$ is a ring :$f$ is a gradation of type $M$ on the additive group of $R$ which is compatible with the ring structure of $R$."}
+{"_id": "21474", "title": "Definition:Graded Module", "text": "Let $R$ be a graded commutative ring with unity. Let $M$ be an $R$-module. We say $M$ is $\\N$-'''graded''' if it has a decomposition as a direct sum of abelian groups :$\\displaystyle R = \\bigoplus_{n \\in \\N} R_n $ such that for all $m, n \\in \\N$ we have $R_n M_m \\subseteq M_{n+m}$. Similarly, we say $M$ is $\\Z$-'''graded''' if it has a decomposition as a direct sum of abelian groups :$\\displaystyle R = \\bigoplus_{n \\in \\Z} R_n $ such that for all $m,n \\in \\Z$ we have $R_nM_m \\subseteq M_{n+m}$. {{SUBPAGENAME}} q984qkm4593t5sqg0wbyu6xtemvda2q"}
+{"_id": "21475", "title": "Definition:Differential Complex", "text": "Let $R$ be a commutative ring with unity. Let $\\displaystyle M = \\bigoplus_{n \\in \\Z} M^n$ be a $\\Z$-graded $R$-module that is also a differential module with differential $d$. Then $M$ is a '''differential complex''' if the differential $d$ satisfies: :$ d \\left({M^n}\\right) \\subseteq M^{n+1}$ for all $n \\in \\Z$. The notation $d_n := d \\restriction_{M_n}$ is often seen."}
+{"_id": "21476", "title": "Definition:Cohomology Groups", "text": "{{refactor|various definitions to separate}} Let $\\left({M, d}\\right)$ be a differential complex with grading: :$\\displaystyle M = \\bigoplus_{n \\in \\Z} M^n$ Let $d_n := d \\restriction_{M_n}$. Elements of the module $M$ are called '''cochains'''. Elements of the submodule $Z^n \\left({M}\\right) = \\ker \\left({d_n}\\right)$ are called '''cocycles'''. Elements of the submodule $B^n \\left({M}\\right) = \\operatorname{Im} \\left({d_{n-1}}\\right)$ are called '''coboundaries'''. The modules (and hence groups) $H^n \\left({M}\\right) = Z^n \\left({M}\\right)/B^n \\left({M}\\right)$ are called the '''cohomology groups''' of the differential complex $\\left({M, d}\\right)$. Category:Definitions/Homological Algebra 8zswo67pms9v3ccmsdcahy2jkut4mak"}
+{"_id": "21477", "title": "Definition:Algebraic Number Field", "text": "An '''algebraic number field''' is a finite extension of the field of rational numbers $\\Q$."}
+{"_id": "21478", "title": "Definition:Algebraic Integer", "text": "Let $K / \\Q$ be a number field. Then $\\alpha \\in K$ is an '''algebraic integer''' if it satisfies a monic polynomial $f \\in \\Z \\left[{X}\\right]$. The set of all '''algebraic integers''' in $K$ is denoted $\\mathfrak o_K$ or $\\mathcal O_K$. By Ring of Algebraic Integers it is a ring, hence usually referred to as the '''ring of algebraic integers''' of $K$. === Quadratic Integer === {{:Definition:Algebraic Integer/Quadratic}}"}
+{"_id": "21479", "title": "Definition:Field Norm", "text": "Let $K$ be a field and $L / K$ a finite field extension of $K$. Let $\\alpha\\in L$. === Definition 1 === {{Definition:Field Norm/Definition 1}} === Definition 2: for Galois extensions === {{Definition:Field Norm/Definition 2}}"}
+{"_id": "21480", "title": "Definition:Trace (Field Theory)", "text": "Let $K$ be a field and $L / K$ a finite field extension of $K$. Then by Vector Space on Field Extension is Vector Space, $L$ is naturally a vector space over $K$. Let $\\alpha \\in L$, and $\\theta_\\alpha$ be the linear operator: :$\\theta_\\alpha: L \\to L: \\beta \\mapsto \\alpha\\beta$ The '''trace''' $\\map {\\operatorname {Tr}_{L / K} } \\alpha$ of $\\alpha$ is the trace of this linear operator."}
+{"_id": "21481", "title": "Definition:Content of Polynomial", "text": "=== Integer Polynomial === {{:Definition:Content of Polynomial/Integer}} === Rational Polynomial === {{:Definition:Content of Polynomial/Rational}} === Polynomial in GCD Domain === {{:Definition:Content of Polynomial/GCD Domain}} == Commutative Ring with Unity == {{:Definition:Content of Polynomial/Commutative and Unitary Ring}}"}
+{"_id": "21482", "title": "Definition:Ordered Integral Domain", "text": "=== Definition 1 === {{:Definition:Ordered Integral Domain/Definition 1}} === Definition 2 === {{:Definition:Ordered Integral Domain/Definition 2}} An ordered integral domain can be denoted: :$\\struct {D, +, \\times \\le}$ where $\\le$ is the total ordering induced by the strict positivity property."}
+{"_id": "21483", "title": "Definition:Strict Positivity Property", "text": "{{begin-axiom}} {{axiom | n = \\text P 1         | lc= Closure under Ring Addition:         | q = \\forall a, b \\in D         | m = \\map P a \\land \\map P b \\implies \\map P {a + b} }} {{axiom | n = \\text P 2         | lc= Closure under Ring Product:         | q = \\forall a, b \\in D         | m = \\map P a \\land \\map P b \\implies \\map P {a \\times b} }} {{axiom | n = \\text P 3         | lc= Trichotomy Law:         | q = \\forall a \\in D         | m = \\paren {\\map P a} \\lor \\paren {\\map P {-a} } \\lor \\paren {a = 0_D} }} {{axiom | lc= For $\\text P 3$, '''exactly one''' condition applies for all $a \\in D$. }} {{end-axiom}}"}
+{"_id": "21484", "title": "Definition:Strict Negativity Property", "text": "Let $\\struct {D, +, \\times}$ be an ordered integral domain, whose (strict) positivity property is denoted $P$. The '''strict negativity property''' $N$ is defined as: :$\\forall a \\in D: \\map N a \\iff \\map P {-a}$ This is compatible with the trichotomy law: :$\\forall a \\in D: \\map P a \\lor \\map P {-a} \\lor a = 0_D$ which can therefore be rewritten: :$\\forall a \\in D: \\map P a \\lor \\map N a \\lor a = 0_D$ or even: :$\\forall a \\in D: \\map N a \\lor \\map N {-a} \\lor a = 0_D$"}
+{"_id": "21485", "title": "Definition:Repdigit Number", "text": "Let $b \\in \\Z_{>1}$ be an integer greater than $1$. Let a (positive) integer $n$, greater than $b$, be expressed in base $b$. $n$ is a '''repdigit base $b$''' {{iff}} all of the digits of $n$ are the same digit. For example: :$55555$ or: :$3333_4$ that second one being in base 4."}
+{"_id": "21486", "title": "Definition:Octal Notation", "text": "'''Octal''' is another word for base $8$. That is, every number $x \\in \\R$ is expressed in the form: :$\\displaystyle x = \\sum_{j \\mathop \\in \\Z} r_j 8^j$ where: :$\\forall j \\in \\Z: r_j \\in \\set {0, 1, 2, 3, 4, 5, 6, 7}$"}
+{"_id": "21487", "title": "Definition:Well-Ordered Integral Domain", "text": "Let $\\struct {D, +, \\times \\le}$ be an ordered integral domain whose zero is $0_D$. === Definition 1 === {{:Definition:Well-Ordered Integral Domain/Definition 1}} === Definition 2 === {{:Definition:Well-Ordered Integral Domain/Definition 2}}"}
+{"_id": "21488", "title": "Definition:Gaussian Integral", "text": "=== Gaussian Integral of Two Variables === {{:Definition:Gaussian Integral/Two Variables}} === Gaussian Integral of One Variable === {{:Definition:Gaussian Integral/One Variable}}"}
+{"_id": "21489", "title": "Definition:Completed Riemann Zeta Function", "text": "The '''completed Riemann zeta function''' $\\xi: \\C \\to \\C$ is defined on the complex plane $\\C$ as: :$\\displaystyle \\forall s \\in \\C: \\map \\xi x := \\begin{cases} \\dfrac 1 2 s \\paren {s - 1} \\pi^{-s/2} \\map \\Gamma {\\dfrac s 2} \\map \\zeta s & : \\map \\Re s > 0 \\\\ \\map \\xi {1 - s} & : \\map \\Re s \\le 0 \\end{cases}$ where $\\map \\zeta s$ is the Riemann zeta function."}
+{"_id": "21490", "title": "Definition:Order of Entire Function", "text": "Let $f: \\C \\to \\C$ be an entire function. Let $\\ln$ denote the natural logarithm. === Definition 1 === {{:Definition:Order of Entire Function/Definition 1}} === Definition 2 === {{:Definition:Order of Entire Function/Definition 2}} === Definition 3 === {{:Definition:Order of Entire Function/Definition 3}}"}
+{"_id": "21491", "title": "Definition:Proper Subring", "text": "A subring $S$ of $R$ is a '''proper subring of $R$''' {{iff}} $S$ is neither the null ring nor $R$ itself."}
+{"_id": "21492", "title": "Definition:Schwarz Function", "text": "Let $f: \\R \\to \\C$ be a function. $f$ is a '''Schwarz function''' {{iff}}: :$\\forall c \\in \\R, n \\in \\N_0: \\left\\vert{f^{\\left({n}\\right)} \\left({x}\\right)}\\right\\vert = o \\left({\\left\\vert{x}\\right\\vert^c}\\right)$ where: : $f^{\\left({n}\\right)}$ denotes the $n$th derivative : $o$ is the little-o notation. {{NamedforDef|Karl Hermann Amandus Schwarz|cat = Schwarz}} Category:Definitions/Harmonic Analysis 88ei8d6x22yif1z50vprz1dhfiunrq5"}
+{"_id": "21493", "title": "Definition:Common Multiple", "text": "Let $S$ be a finite set of non-zero integers, that is: :$S = \\set {x_1, x_2, \\ldots, x_n: \\forall k \\in \\N^*_n: x_k \\in \\Z, x_k \\ne 0}$ Let $m \\in \\Z$ such that all the elements of $S$ divide $m$, that is: :$\\forall x \\in S: x \\divides m$ Then $m$ is a '''common multiple''' of all the elements in $S$. {{expand|Expand the concept to general algebraic expressions, for example that $\\paren {2 x + 1} \\paren {x - 2}$ is a '''common multiple''' of $\\paren {2 x + 1}$ and $\\paren {x - 2}$}}"}
+{"_id": "21494", "title": "Definition:Ring Automorphism", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\phi: R \\to R$ be a (ring) isomorphism. Then $\\phi$ is a ring automorphism."}
+{"_id": "21495", "title": "Definition:Even Function", "text": "Let $X \\subset \\R$ be a symmetric set of real numbers: :$\\forall x \\in X: -x \\in X$ A real function $f: X \\to \\R$ is an '''even function''' {{iff}}: :$\\forall x \\in X: \\map f {-x} = \\map f x$"}
+{"_id": "21496", "title": "Definition:Odd Function", "text": "Let $X \\subset \\R$ be a symmetric set of real numbers: :$\\forall x \\in X: -x \\in X$ A real function $f: X \\to \\R$ is an '''odd function''' {{iff}}: :$\\forall x \\in X: \\map f {-x} = -\\map f x$"}
+{"_id": "21497", "title": "Definition:Symmetric Set", "text": "Let $G$ be a group. Let $S \\subseteq G$ such that: :$\\forall x \\in S: x^{-1} \\in S$ That is, for every element in $S$, its inverse is also in $S$. Then $S$ is a '''symmetric subset of $G$''', or (if $G$ is implicit) $S$ is a '''symmetric set'''. Equivalently, $S \\subseteq G$ is a '''symmetric set''' {{iff}}: :$S = S^{-1}$ where $S^{-1}$ is the inverse of $S$. === Symmetric Set of Real Numbers === {{:Definition:Symmetric Set/Real Numbers}} Category:Definitions/Group Theory l1ii5pww7bkn0tufbxo1uud41x0sdy1"}
+{"_id": "21498", "title": "Definition:Proper Divisor", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $U$ be the group of units of $D$. Let $x, y \\in D$. Then $x$ is a '''proper divisor''' of $y$ {{iff}}: :$(1): \\quad x \\divides y$ :$(2): \\quad y \\nmid x$ :$(3): \\quad x \\notin U$ That is: :$(1): \\quad x$ is a divisor of $y$ :$(2): \\quad x$ is not an associate of $y$ :$(3): \\quad x$ is not a unit of $D$ === Integers === As the set of integers form an integral domain, the concept of a '''proper divisor''' is fully applicable to the integers. {{:Definition:Proper Divisor/Integer}}"}
+{"_id": "21499", "title": "Definition:Order of Zero", "text": "Let $f: \\C \\to \\C$ be a complex function. Let $U \\subset \\C$ be such that $f$ is analytic in $U$. Let $x \\in U$ be a zero of $f$. That is, let $x$ be such that $f \\left({x}\\right) = 0$. Let $n \\in \\Z_{\\ge 0}$ be the least positive integer such that: : $f^{(n)} \\left({x}\\right) \\ne 0$ where $f^{(n)}$ denotes the $n$th derivative of $f$. Then $n$ is the '''order of the zero''' at $x$. === Simple Zero === {{:Definition:Order of Zero/Simple Zero}} Category:Definitions/Complex Analysis 0p3aj8fbmmcjakfupmr6rqrd6f1we0x"}
+{"_id": "21501", "title": "Definition:Order of Pole", "text": "Let $f: \\C \\to \\C$ be a complex function. Let $x \\in U \\subset \\C$ be such that $f$ is analytic in $U \\setminus \\set x$, with a pole at $x$. By Existence of Laurent Series there is a series: :$\\displaystyle \\map f z = \\sum_{n \\mathop \\ge n_0}^\\infty a_j \\paren {z - x}^n$ The '''order''' of the pole at $x$ is defined to be $\\size {n_0} > 0$. === Simple Pole === {{:Definition:Order of Pole/Simple Pole}} Category:Definitions/Complex Analysis eget6rdxjtf764d0in5c9ejer94a5m8"}
+{"_id": "21502", "title": "Definition:Locally Uniform Convergence/Complex Functions", "text": "Let $U \\subseteq \\C$ be an open set. Let $\\sequence {f_n}$ be a be a sequence of functions $f_n : U \\to \\C$. For $z \\in U$, let $\\map {D_r} z$ be the disk of radius $r$ about $z$. {{explain|Given that \"disk\" is just another term for \"ball\", it remains to be specified whether that ball is open or closed.}} Then $f_n$ '''converges to $f$ locally uniformly''' {{iff}}: :for each $z \\in U$, there is an $r > 0$ such that $f_n$ converges uniformly to $f$ on $\\map {D_r} z$ and: :$\\map {D_r} z \\subseteq U$"}
+{"_id": "21503", "title": "Definition:Primitive Polynomial (Ring Theory)", "text": "Let $\\Q \\sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $f \\in \\Q \\sqbrk X$ be such that: :$\\cont f = 1$ where $\\cont f$ is the content of $f$. That is: :The greatest common divisor of the coefficients of $f$ is equal to $1$. Then $f$ is described as '''primitive'''."}
+{"_id": "21505", "title": "Definition:Vector Sum", "text": "Let $\\mathbf u$ and $\\mathbf v$ be vector quantities of the same physical property. === Component Definition === {{:Definition:Vector Sum/Component Definition}} === Triangle Law === {{:Definition:Vector Sum/Triangle Law}}"}
+{"_id": "21506", "title": "Definition:Linearly Dependent", "text": "Let $G$ be an abelian group whose identity is $e$. Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\\left({G, +_G, \\circ}\\right)_R$ be a unitary $R$-module. === Sequence === {{:Definition:Linearly Dependent/Sequence}} === Set === {{:Definition:Linearly Dependent/Set}}"}
+{"_id": "21507", "title": "Definition:Valuation Ring", "text": "Let $\\struct {D, +, \\circ}$ be a integral domain. Let $K$ be the field of quotients of $D$. Let $K$ be such that: :for all $x \\in K$, either $x \\in D$ or $x^{-1} \\in D$. Then $D$ is a '''valuation ring''' of $K$."}
+{"_id": "21508", "title": "Definition:Tychonoff Topology", "text": "Let $\\family {\\struct {X_i, \\tau_i} }_{i \\mathop \\in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $X$ be the cartesian product of $\\family {X_i}_{i \\mathop \\in I}$: :$\\displaystyle X := \\prod_{i \\mathop \\in I} X_i$ For each $i \\in I$, let $\\pr_i: X \\to X_i$ denote the $i$th projection on $X$: :$\\forall \\family {x_j}_{j \\mathop \\in I} \\in X: \\map {\\pr_i} {\\family {x_j}_{j \\mathop \\in I} } = x_i$ The '''Tychonoff topology''' on $X$ is defined as the initial topology $\\tau$ on $X$ with respect to $\\family {\\pr_i}_{i \\mathop \\in I}$."}
+{"_id": "21509", "title": "Definition:Fréchet Space (Topology)", "text": "{{:Definition:Fréchet Space (Topology)/Definition 1}}"}
+{"_id": "21510", "title": "Definition:Regular Space", "text": "$\\struct {S, \\tau}$ is a '''regular space''' {{iff}}: :$\\struct {S, \\tau}$ is a $T_3$ space :$\\struct {S, \\tau}$ is a $T_0$ (Kolmogorov) space."}
+{"_id": "21511", "title": "Definition:Normal Space", "text": "$\\struct {S, \\tau}$ is a '''normal space''' {{iff}}: :$\\struct {S, \\tau}$ is a $T_4$ space :$\\struct {S, \\tau}$ is a $T_1$ (Fréchet) space."}
+{"_id": "21512", "title": "Definition:T3 Space", "text": "{{:Definition:T3 Space/Definition 1}}"}
+{"_id": "21513", "title": "Definition:T4 Space", "text": "{{:Definition:T4 Space/Definition 1}}"}
+{"_id": "21514", "title": "Definition:Topologically Distinguishable", "text": "Let $T = \\left({X, \\tau}\\right)$ be a topological space. Let $x, y \\in X$. Then $x$ and $y$ are '''topologically distinguishable''' {{iff}} they do not have exactly the same neighborhoods. That is, either: :$\\exists U \\in \\tau: x \\in U \\subseteq N_x \\subseteq X: y \\notin N_x$ or: :$\\exists V \\in \\tau: y \\in V \\subseteq N_y \\subseteq X: x \\notin N_y$ or both. That is, at least one of the points $x$ and $y$ has a neighborhood that is not a neighborhood of the other. === Topologically Indistinguishable === {{:Definition:Topologically Distinguishable/Indistinguishable}} Category:Definitions/Separation Axioms Category:Definitions/Distinct r06vk8aij0zkgl3mnwiy71x8lu4q3ez"}
+{"_id": "21515", "title": "Definition:T5 Space", "text": "{{:Definition:T5 Space/Definition 1}}"}
+{"_id": "21516", "title": "Definition:Completely Normal Space", "text": "$\\struct {S, \\tau}$ is a '''completely normal space''' {{iff}}: :$\\struct {S, \\tau}$ is a $T_5$ space :$\\struct {S, \\tau}$ is a $T_1$ (Fréchet) space."}
+{"_id": "21517", "title": "Definition:Separated by Neighborhoods", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. === Sets === {{:Definition:Separated by Neighborhoods/Sets}} === Points === {{:Definition:Separated by Neighborhoods/Points}} Category:Definitions/Separation Axioms Category:Definitions/Separated by Neighborhoods kd1ha0mqgbk3v94ybukvzye15f6cscn"}
+{"_id": "21518", "title": "Definition:Urysohn Space", "text": "$\\struct {S, \\tau}$ is an '''Urysohn space''' {{iff}}: :For any distinct elements $x, y \\in S$ (that is, $x \\ne y$), there exists an Urysohn function for $\\set x$ and $\\set y$."}
+{"_id": "21519", "title": "Definition:Separated by Closed Neighborhoods", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. === Sets === {{:Definition:Separated by Closed Neighborhoods/Sets}} === Points === {{:Definition:Separated by Closed Neighborhoods/Points}}"}
+{"_id": "21520", "title": "Definition:Completely Hausdorff Space", "text": "{{:Definition:Completely Hausdorff Space/Definition 1}}"}
+{"_id": "21521", "title": "Definition:Urysohn Function", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A, B \\subseteq S$ such that $A \\cap B = \\O$. Let $f: S \\to \\closedint 0 1$ be a continuous mapping where $\\closedint 0 1$ is the closed unit interval. Then $f$ is an '''Urysohn function for $A$ and $B$''' {{iff}}: :$f {\\restriction_A} = 0, f {\\restriction_B} = 1$ that is: :$\\forall a \\in A: \\map f a = 0$ :$\\forall b \\in B: \\map f b = 1$"}
+{"_id": "21522", "title": "Definition:Dyadic Rational", "text": "A '''dyadic rational''' is a rational number whose denominator, when expressed in canonical form, is an integral power of $2$. That is, $a \\in \\Q$ is '''dyadic''' iff: :$\\exists p \\in \\Z, q \\in \\Z_{>0}: a = \\dfrac p {2^q}$ Category:Definitions/Rational Numbers gq5146yf091y149gjw1gjapawaz37on"}
+{"_id": "21523", "title": "Definition:Tychonoff Space", "text": "$\\struct {S, \\tau}$ is a '''Tychonoff Space''' or '''completely regular space''' {{iff}}: :$\\struct {S, \\tau}$ is a $T_{3 \\frac 1 2}$ space :$\\struct {S, \\tau}$ is a $T_0$ (Kolmogorov) space."}
+{"_id": "21524", "title": "Definition:Limit Point/Topology/Point", "text": "Let $a \\in S$. A point $x \\in S, x \\ne a$ is a '''limit point of $a$''' {{iff}} every open neighborhood of $x$ contains $a$. That is, it is a limit point of the singleton $\\set a$."}
+{"_id": "21525", "title": "Definition:Expansion of Topology", "text": "Let $S$ be a set. Let $\\tau_1$ and $\\tau_2$ be topologies on $S$ such that $\\tau_1 \\subseteq \\tau_2$. Then $\\tau_2$ is an '''expansion''' of $\\tau_1$."}
+{"_id": "21527", "title": "Definition:T3 1/2 Space", "text": "$\\struct {S, \\tau}$ is a '''$T_{3 \\frac 1 2}$ space''' {{iff}}: :For any closed set $F \\subseteq S$ and any point $y \\in S$ such that $y \\notin F$, there exists an Urysohn function for $F$ and $\\set y$."}
+{"_id": "21528", "title": "Definition:Separated by Function", "text": "Let $\\left({X, \\vartheta}\\right)$ be a topological space. Let $A, B \\subseteq X$. Then $A$ and $B$ are '''separated by function''' iff there exists an Urysohn function for $A$ and $B$. $A$ and $B$ may well be singleton sets $A = \\left\\{{a}\\right\\}, B = \\left\\{{b}\\right\\}$. In this case $a$ and $b$ are '''separated by function''' iff $A$ and $B$ are '''separated by function'''. {{SUBPAGENAME}} o11tt4onkwpoxop44h4ziff3vtb20rt"}
+{"_id": "21529", "title": "Definition:Perfectly T4 Space", "text": "$T$ is a '''perfectly $T_4$ space''' {{iff}}: :$(1): \\quad T$ is a $T_4$ space :$(2): \\quad$ Every closed set in $T$ is a $G_\\delta$ set. That is: :Every closed set in $T$ can be written as a countable intersection of open sets of $T$."}
+{"_id": "21530", "title": "Definition:Perfectly Normal Space", "text": "$\\struct {S, \\tau}$ is a '''perfectly normal space''' {{iff}}: :$\\struct {S, \\tau}$ is a perfectly $T_4$ space :$\\struct {S, \\tau}$ is a $T_1$ (Fréchet) space."}
+{"_id": "21531", "title": "Definition:Semiregular Space", "text": "$\\struct {S, \\tau}$ is a '''semiregular space''' {{iff}}: :$\\struct {S, \\tau}$ is a Hausdorff ($T_2$) space :The regular open sets of $T$ form a basis for $T$."}
+{"_id": "21532", "title": "Definition:Almost All", "text": "=== Measure Space === {{:Definition:Almost Everywhere}} === Set Theory === {{:Definition:Almost All/Set Theory}}"}
+{"_id": "21533", "title": "Definition:Finite Intersection Axiom", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T$ be such that: :Every set $V_\\alpha$ of closed sets of $T$ such that $\\displaystyle \\bigcap V_\\alpha = \\O$ contains a finite subset $V_\\beta \\subseteq V_\\alpha$ such that $\\displaystyle \\bigcap V_\\beta = \\O$. Then $T$ satisfies the '''finite intersection axiom'''."}
+{"_id": "21534", "title": "Definition:Sigma-Compact Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. $T$ is '''$\\sigma$-compact''' {{iff}} $S$ is the union of the underlying sets of countably many compact subspaces of $T$."}
+{"_id": "21535", "title": "Definition:Countably Compact Space", "text": "=== Definition 1 === {{:Definition:Countably Compact Space/Definition 1}} === Definition 2 === {{:Definition:Countably Compact Space/Definition 2}} === Definition 3 === {{:Definition:Countably Compact Space/Definition 3}} === Definition 4 === {{:Definition:Countably Compact Space/Definition 4}} === Definition 5 === {{:Definition:Countably Compact Space/Definition 5}}"}
+{"_id": "21536", "title": "Definition:Countable Finite Intersection Axiom", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T$ be such that: :Every countable set $V_\\alpha$ of closed sets of $T$ such that $\\displaystyle \\bigcap V_\\alpha = \\O$ contains a finite subset $V_\\beta \\subseteq V_\\alpha$ such that $\\displaystyle \\bigcap V_\\beta = \\O$. Then $T$ satisfies the '''countable finite intersection axiom'''."}
+{"_id": "21537", "title": "Definition:Weakly Countably Compact Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. $T$ is '''weakly countably compact''' {{iff}} every infinite subset of $S$ has a limit point in $S$."}
+{"_id": "21538", "title": "Definition:Pseudocompact Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. $T$ is '''pseudocompact''' {{iff}} every continuous real-valued function on $S$ is bounded."}
+{"_id": "21539", "title": "Definition:Locally Compact Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Then $T$ is '''locally compact''' {{iff}}: : every point of $S$ has a neighborhood basis $\\mathcal B$ such that all elements of $\\mathcal B$ are compact."}
+{"_id": "21540", "title": "Definition:Strongly Locally Compact Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. === Definition 1 === {{Definition:Strongly Locally Compact Space/Definition 1}} === Definition 2 === {{Definition:Strongly Locally Compact Space/Definition 2}}"}
+{"_id": "21541", "title": "Definition:Convergent Sequence/Topology", "text": "{{:Definition:Convergent Sequence/Topology/Definition 1}}"}
+{"_id": "21542", "title": "Definition:Convergent Sequence/Analysis", "text": "=== Real Numbers === {{:Definition:Convergent Sequence/Real Numbers}} === Rational Numbers === {{:Definition:Convergent Sequence/Rational Numbers}} === Complex Numbers === {{:Definition:Convergent Sequence/Complex Numbers/Definition 1}}"}
+{"_id": "21543", "title": "Definition:Weakly Sigma-Locally Compact Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then $T$ is '''weakly $\\sigma$-locally compact''' {{iff}}: :$T$ is $\\sigma$-compact :$T$ is weakly locally compact."}
+{"_id": "21544", "title": "Definition:Countable Chain Condition", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. $T$ satisfies the '''countable chain condition''' {{iff}} every disjoint set of open sets of $T$ is countable."}
+{"_id": "21545", "title": "Definition:Refinement of Cover", "text": "Let $S$ be a set. Let $\\UU = \\set {U_\\alpha}$ and $\\VV = \\set {V_\\beta}$ be covers of $S$. Then $\\VV$ is a '''refinement''' of $\\UU$ {{iff}}: :$\\forall V_\\beta \\in \\VV: \\exists U_\\alpha \\in \\UU: V_\\beta \\subseteq U_\\alpha$ That is, {{iff}} every element of $\\VV$ is the subset of some element of $\\UU$. === Finer Cover === {{:Definition:Refinement of Cover/Finer Cover}} === Coarser Cover === {{:Definition:Refinement of Cover/Coarser Cover}}"}
+{"_id": "21546", "title": "Definition:Point Finite", "text": "Let $S$ be a set. Let $\\CC$ be a set of subsets of $S$. Then $\\CC$ is '''point finite''' {{iff}} each element of $S$ is an element of finitely many sets in $\\CC$: :$\\forall s \\in S: \\card {\\set {C \\in \\CC: s \\in C} } < \\infty$"}
+{"_id": "21547", "title": "Definition:Locally Finite Cover", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\CC$ be a cover of $S$. Then $\\CC$ is '''locally finite''' {{iff}} each element of $S$ has a neighborhood which intersects a finite number of sets in $\\CC$."}
+{"_id": "21549", "title": "Definition:Star (Topology)", "text": "Let $S$ be a set. Let $\\CC$ be a cover for $S$. Let $x \\in S$. The '''star of $x$ (with respect to $\\CC$)''' is defined as: :$\\displaystyle x^* := \\bigcup \\set {U \\in \\CC: x \\in U}$ That is, the union of all sets in $\\CC$ which contain $x$."}
+{"_id": "21550", "title": "Definition:Star Refinement", "text": "Let $S$ be a set. Let $\\CC$ be a cover for $S$. Let $\\VV$ be a cover for $S$ such that: :$ \\forall x \\in S: \\exists U \\in \\CC: x^* \\subseteq U$ where $x^*$ is the star of $x$ with respect to $\\VV$. That is: :$\\displaystyle x^* := \\bigcup \\set {V \\in \\VV: x \\in V}$ the union of all sets in $\\VV$ which contain $x$. Then $\\VV$ is a '''star refinement''' of $\\CC$."}
+{"_id": "21551", "title": "Definition:Multiplicity (Complex Analysis)", "text": "Let $f: \\C \\to \\C$ be a function. Suppose there is $a \\in \\C$ such that $\\map f a = 0$. Then $a$ is said to be a '''zero of multiplicity $k$''' if there exists non-zero $L \\in \\R$ such that: :$\\displaystyle \\lim_{z \\mathop \\to a} \\dfrac {\\cmod {\\map f z} } {\\cmod {z - a}^k} = L$ {{definition wanted|Needs to be expanded so as to be relevant to other fields, not just $\\C$.}} Category:Definitions/Complex Analysis muauvhfddwgkk8yblkjl4b26455wrai"}
+{"_id": "21552", "title": "Definition:Dirichlet Convolution", "text": "Let $f, g$ be arithmetic functions. === Definition 1 === {{:Definition:Dirichlet Convolution/Definition 1}} === Definition 2 === {{:Definition:Dirichlet Convolution/Definition 2}}"}
+{"_id": "21553", "title": "Definition:Unit Arithmetic Function", "text": "The '''unit arithmetic function''' $u: S \\to \\Z$ is defined by $u \\left({n}\\right) = 1$ for all $n$: :$\\forall n \\in S: u \\left({n}\\right) = 1$ where $S$ is (in theory) any set, but in this context is usually one of the standard number sets $\\Z, \\Q, \\R, \\C$."}
+{"_id": "21554", "title": "Definition:Identity Arithmetic Function", "text": "The '''identity arithmetic function''' $\\iota: S \\to \\Z$ is defined for $n \\geq 1$ by: :$\\forall n \\in S: \\map \\iota n = \\delta_{n 1}$ where: :$S$ is (in theory) any set, but in this context is usually one of the standard number sets $\\Z, \\Q, \\R, \\C$. :$\\delta$ is the Kronecker delta. That is: :$\\forall n \\in S: \\map \\iota n = \\begin {cases} 1 & : n = 1\\\\ 0 & : n \\ne 1 \\end {cases}$"}
+{"_id": "21555", "title": "Definition:Metacompact Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. $T$ is '''metacompact''' {{iff}} every open cover of $S$ has an open refinement which is point finite."}
+{"_id": "21556", "title": "Definition:Paracompact Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. $T$ is '''paracompact''' {{iff}} every open cover of $S$ has an open refinement which is locally finite."}
+{"_id": "21557", "title": "Definition:Fully T4 Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. $T$ is '''fully $T_4$''' {{iff}} every open cover of $S$ has a star refinement."}
+{"_id": "21558", "title": "Definition:Fully Normal Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. $T$ is '''fully normal''' {{iff}}: :Every open cover of $S$ has a star refinement :All points of $T$ are closed."}
+{"_id": "21559", "title": "Definition:Countably Metacompact Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. $T$ is '''countably metacompact''' {{iff}} every countable open cover of $S$ has an open refinement which is point finite."}
+{"_id": "21560", "title": "Definition:Countably Paracompact Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. $T$ is '''countably paracompact''' {{iff}} every countable open cover of $S$ has an open refinement which is locally finite."}
+{"_id": "21561", "title": "Definition:Open Refinement", "text": "Let $T = \\struct {X, \\tau}$ be a topological space. Let $\\UU$ and $\\VV$ be covers of $X$. Then $\\VV$ is an '''open refinement''' of $\\UU$ {{iff}}: :$(1): \\quad \\forall V \\in \\VV: \\exists U \\in \\UU: V \\subseteq U$ :$(2): \\quad \\VV \\subseteq \\tau$ That is: :$(1): \\quad \\VV$ is a refinement of $\\UU$ :$(2): \\quad$ All elements of $\\VV$ are open in $T$ Category:Definitions/Covers c777nta7lbjehhhn3u0t0lntbpkebx7"}
+{"_id": "21562", "title": "Definition:Little-Omega", "text": "Let $f$ and $g$ be real functions. Then: :$f \\left({n}\\right) \\in \\omega \\left({g \\left({n}\\right)}\\right)$ is equivalent to: :$\\displaystyle \\lim_{n \\to \\infty} {\\frac{f \\left({n}\\right)} {g \\left({n}\\right)}} = \\infty$ {{refactor|Extract this into a separate page with proof}} A function $f$ is $\\omega \\left({g}\\right)$ iff $f$ is not $\\mathcal O \\left({g}\\right)$ where $\\mathcal O$ is the big-O notation."}
+{"_id": "21563", "title": "Definition:Dirichlet Character", "text": "Let $q \\in \\Z_{>1}$. Let $\\paren {\\Z / q \\Z}$ denote the ring of integers modulo $q$. Let $G = \\paren {\\Z / q \\Z}^\\times$ be the group of units of $\\paren {\\Z / q \\Z}$. Let $\\C^\\times$ be the group of units of $\\C$. A '''Dirichlet character modulo q''' is a group homomorphism: :$\\chi: G \\to \\C^\\times$ {{refactor|Presumably the following statement is part of the justification for the generalised definition, which needs to be extracted and put into its own page.}} By Reduced Residue System under Multiplication forms Abelian Group, $a + q \\Z \\in G$ {{iff}} $\\map \\gcd {a, q} = 1$. It is standard practice to extend $\\chi$ to a function on $\\Z$ by setting: :$\\map \\chi A = \\begin{cases} \\map \\chi {a + q \\Z} & : \\map \\gcd {a, q} = 1 \\\\ 0 & : \\text{otherwise} \\end{cases}$ {{explain|The nature of $\\chi$ when invoked is not apparent from its notation.  How does one understand, when encountering an instance of $\\chi$ whether it means the mapping from $G$ (for some $q$) of just $\\Z$?}} === Primitive Character === Let $\\chi_0$ be the trivial character modulo $q$. Let $q^*$ be the least divisor of $q$ such that: : $\\chi = \\chi_0 \\chi^*$ where $\\chi^*$ is some character modulo $q^*$. If $q = q^*$ then $\\chi$ is called '''primitive''', otherwise $\\chi$ is '''imprimitive'''."}
+{"_id": "21564", "title": "Definition:Dirichlet L-function", "text": "Let $\\chi : \\left({\\Z / q \\Z}\\right)^\\times \\to \\C^\\times$ be a Dirichlet character. {{explain|the notation $\\left({\\Z / q \\Z}\\right)^\\times$}} A '''Dirichlet $L$-function (associated to $\\chi$)''' is a Dirichlet series: :$\\displaystyle L \\left({s, \\chi}\\right) = \\sum_{n \\mathop \\ge 1} \\chi \\left({n}\\right) n^{-s}$ for all $s \\in \\C$ such that the sum converges."}
+{"_id": "21565", "title": "Definition:Dirichlet Density", "text": "Let $\\mathcal P$ be a set of prime numbers. For $s \\in \\C$, let $\\displaystyle f \\left({s}\\right) = \\sum_{p \\mathop \\in \\mathcal P}\\: p^{-s}$. $S$ has '''Dirichlet density''' $\\alpha$ {{iff}}: :$\\displaystyle \\lim_{s \\mathop \\to 1^+} \\left\\{{\\frac {f \\left({s}\\right)} {\\ln \\left({s - 1}\\right)}}\\right\\} = - \\alpha$ where $1^+$ indicates a limit from above along the real line. {{NamedforDef|Johann Peter Gustav Lejeune Dirichlet|cat = Dirichlet}} Category:Definitions/Analytic Number Theory otagch3ceq5s8l44g2zapaawo3cer6m"}
+{"_id": "21566", "title": "Definition:Character (Number Theory)", "text": "Let $\\left({G, +}\\right)$ be a finite abelian group. Let $\\left({\\C_{\\ne 0}, \\times}\\right)$ be the multiplicative group of complex numbers. A '''character''' of $G$ is a group homomorphism: :$\\chi: G \\to \\C_{\\ne 0}$"}
+{"_id": "21568", "title": "Definition:Completed Dirichlet L-Function", "text": "Let $\\chi$ be a primitive Dirichlet character to the modulus $q \\geq 1$. Let $\\displaystyle \\kappa = \\frac12\\left(1 - \\chi(-1)\\right)$. Let $\\delta = 1$ if $\\chi$ is the principal character, and $0$ otherwise. The '''completed Dirichlet $L$-function''' for $\\chi$ is defined to be :$\\displaystyle \\Lambda(s,\\chi) = \\frac{1 + \\kappa}{2}\\left( s(1-s) \\right)^\\delta \\Gamma\\left( \\frac{s+\\kappa}2 \\right)L(s,\\chi)$ where $L(s,\\chi)$ is the Dirichlet $L$-function for $\\chi$. {{SUBPAGENAME}} sg0f5771dwwj52bt574bbvohaja0j5n"}
+{"_id": "21569", "title": "Definition:Limit Point/Filter", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\FF$ be a filter on $S$. A point $x \\in S$ is called a '''limit point of $\\FF$''' {{iff}} $\\FF$ is finer than the neighborhood filter of $x$."}
+{"_id": "21570", "title": "Definition:Limit Point/Filter Basis", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\FF$ be a filter on the underlying set $S$ of $T$. Let $\\BB$ be a filter basis of $\\FF$. === Definition 1 === {{:Definition:Limit Point/Filter Basis/Definition 1}} === Definition 2 === {{:Definition:Limit Point/Filter Basis/Definition 2}}"}
+{"_id": "21571", "title": "Definition:Convergent Filter Basis", "text": "Let $\\struct {S, \\tau}$ be a topological space. Let $\\BB$ be a filter basis of a filter $\\FF$ on $S$. Then $\\BB$ '''converges''' to a point $x \\in S$ {{iff}}: :$\\forall N_x \\subseteq S: \\exists B \\in \\BB: B \\subseteq N_x$ where $N_x$ is a neighborhood of $x$. That is, a filter basis is '''convergent''' to a point $x$ if every neighborhood of $x$ contains some set of that filter basis."}
+{"_id": "21572", "title": "Definition:Integral Element of Ring Extension", "text": "Let $A$ be a commutative ring with unity. Let $R \\subseteq A$ be a subring. Then $a \\in A$ is said to be '''integral''' over $R$ {{iff}} is is a root of a monic nonzero polynomial over $R$."}
+{"_id": "21573", "title": "Definition:Ring Extension", "text": "Let $R$ and $S$ be commutative rings with unity. Let $\\phi : R \\to S$ be a ring monomorphism. Then $\\phi : R \\to S$ is a '''ring extension''' of $R$."}
+{"_id": "21574", "title": "Definition:Faithful Module", "text": "Let $R$ be a commutative ring with unity. Let $M$ be an $R$-module. Then $M$ is '''faithful''' {{iff}} its annihilator is zero."}
+{"_id": "21575", "title": "Definition:Integral Closure", "text": "Let $A$ be an extension of a commutative ring with unity $R$. Let $C$ be the set of all elements of $A$ that are integral over $R$. Then $C$ is called the '''integral closure''' of $R$ in $A$."}
+{"_id": "21576", "title": "Definition:Integrally Closed", "text": "=== Ring Extension === {{Definition:Integrally Closed in Ring Extension}} === Integral Domain === {{Definition:Integrally Closed Integral Domain}}"}
+{"_id": "21577", "title": "Definition:Multiplicatively Closed Subset of Ring", "text": "Let $\\left({A, +, \\circ}\\right)$ be a ring with unity $1_A$ and zero $0_A$. Let $S \\subseteq A$ be a subset. Then $S$ is '''multiplicatively closed''' {{iff}}: :$(1): \\quad 1_A \\in S$ :$(2): \\quad x, y \\in S \\implies x \\circ y \\in S$"}
+{"_id": "21578", "title": "Definition:Localization of Ring", "text": "Let $A$ be a commutative ring with unity. Let $S \\subseteq A$ be a multiplicatively closed subset of $A$. A '''localization of $A$ at $S$''' is a pair $\\struct {A_S, \\iota}$ where: :$A_S$ is a commutative ring with unity, the actual '''localization''' :$\\iota: A \\to A_S$ is a ring homomorphism, the '''localization homomorphism''' such that: :$(1): \\quad \\map \\iota S \\subseteq A_S^\\times$, where $A_S^\\times$ is the group of units of $A_S$ :$(2): \\quad$ For every pair $\\tuple {B, g}$ where: ::::$B$ is any ring with unity ::::$g: A \\to B$ is a ring homomorphism such that $\\map g S \\subseteq B^\\times$ :::there exists a unique ring homomorphism $h: A_S \\to B$ such that: ::::$g = h \\circ \\iota$ That is, the following diagram commutes:  :325px"}
+{"_id": "21579", "title": "Definition:Prime Spectrum of Ring", "text": "Let $A$ be a commutative ring with unity. The '''prime spectrum''' or '''spectrum''' of $A$ is the set of prime ideals $\\mathfrak p$ of $A$: :$\\Spec A = \\set {\\mathfrak p \\lhd A: \\mathfrak p \\text{ is prime} }$"}
+{"_id": "21580", "title": "Definition:Maximal Spectrum of Ring", "text": "Let $A$ be a commutative ring with unity. The '''maximal spectrum''' of $A$ is the set of maximal ideals of $A$: :$\\operatorname{Max} \\: \\Spec A = \\set {\\mathfrak m \\lhd A : \\mathfrak m \\text { is maximal} }$ where $I \\lhd A$ indicates that $I$ is an ideal of $A$. The notation $\\operatorname {Max} \\: \\Spec A$ is also a shorthand for the locally ringed space :$\\struct {\\operatorname {Max} \\: \\Spec A, \\tau, \\mathcal O_{\\map {\\operatorname {Max Spec} } A} }$ where: :$\\tau$ is the Zariski topology on $\\map {\\operatorname {Max Spec} } A$ :$\\mathcal O_{\\map {\\operatorname {Max Spec} } A}$ is the structure sheaf of $\\map {\\operatorname {Max Spec} } A$"}
+{"_id": "21581", "title": "Definition:Affine Algebraic Set", "text": "Let $k$ be a field. Let $A = k[X_1,\\ldots,X_n]$ be the ring of polynomial functions in $n$ variables over $k$. Then a subset $X \\subseteq k^n$ is an '''affine algebraic set''' if it is the zero locus of some set $T \\subseteq A$."}
+{"_id": "21582", "title": "Definition:Irreducible Space", "text": "=== Definition 1 === {{Definition:Irreducible Space/Definition 1}} === Definition 2 === {{Definition:Irreducible Space/Definition 2}} === Definition 3 === {{Definition:Irreducible Space/Definition 3}} === Definition 4 === {{Definition:Irreducible Space/Definition 4}} === Definition 5 === {{Definition:Irreducible Space/Definition 5}} === Definition 6 === {{Definition:Irreducible Space/Definition 6}} === Definition 7 === {{Definition:Irreducible Space/Definition 7}}"}
+{"_id": "21584", "title": "Definition:Zero Locus of Set of Polynomials", "text": "Let $I \\subseteq A$ be a set. Then the '''zero locus''' of $I$ is the set: :$\\map V I = \\set {x \\in k^n : \\forall f \\in I: \\map f x = 0}$"}
+{"_id": "21585", "title": "Definition:Locally Euclidean Space", "text": "Let $M$ be a topological space. Let $d \\in \\N$ be a natural number. Then $M$ is a '''locally Euclidean space''' of dimension $d$ if each point in $M$ has an open neighbourhood homeomorphic to an open subset of Euclidean space $\\R^d$. === Complex Locally Euclidean Space === {{:Definition:Locally Euclidean Space/Complex}}"}
+{"_id": "21586", "title": "Definition:Chart", "text": "Let $M$ be a locally Euclidean space, or complex locally Euclidean space of dimension $d$. A '''$d$-dimensional chart''' of $M$ is an ordered pair $\\struct {U, \\phi}$, where: :$U$ is an open subset of $M$ :$\\phi: U \\to D$ is a homeomorphism of $U$ onto an open subset $D$ of Euclidean space $\\R^d$."}
+{"_id": "21587", "title": "Definition:Differentiable Structure", "text": "Let $M$ be a topological space. Let $d$ be a natural number. Let $k \\ge 1$ be a natural number. A '''$d$-dimensional differentiable structure of class $\\mathcal C^k$''' on $M$ is a non-empty equivalence class of the set of $d$-dimensional $\\mathcal C^k$-atlases on $M$ under the equivalence relation of compatibility."}
+{"_id": "21588", "title": "Definition:Atlas", "text": "Let $M$ be a topological space. An '''atlas of class $C^k$ and dimension $d$''' on $M$ is a set of $d$-dimensional charts $\\mathscr F = \\family {\\struct {U_\\alpha, \\phi_\\alpha}: \\alpha \\in A}$ indexed by some set $A$ such that: :$(1): \\quad \\displaystyle \\bigcup_{\\alpha \\mathop \\in A} U_\\alpha = M$ :$(2): \\quad$ Every two charts $\\struct {U, \\phi}$ and $\\struct {V, \\psi}$ are $C^k$-compatible. === Smooth Atlas === {{:Definition:Atlas/Smooth Atlas}}"}
+{"_id": "21589", "title": "Definition:Connected Between Two Points", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $a, b \\in S$. $T$ is '''connected between (the) two points $a$ and $b$''' {{iff}} each separation of $T$ includes a single open set $U \\in \\tau$ which contains both $a$ and $b$."}
+{"_id": "21590", "title": "Definition:Quasicomponent", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let the relation $\\sim$ be defined on $T$ as follows: :$x \\sim y \\iff T$ is connected between the two points $x$ and $y$ That is, {{iff}} each separation of $T$ includes a single open set $U \\in \\tau$ which contains both $x$ and $y$. We have that $\\sim$ is an equivalence relation, so from the Fundamental Theorem on Equivalence Relations, the points in $T$ can be partitioned into equivalence classes. These equivalence classes are called the '''quasicomponents''' of $T$."}
+{"_id": "21591", "title": "Definition:Arc (Topology)", "text": "Let $T$ be a topological space. Let $\\mathbb I \\subset \\R$ be the closed unit interval $\\closedint 0 1$. Let $a, b \\in T$. An '''arc from $a$ to $b$''' is a path $f: \\mathbb I \\to T$ such that $f$ is injective. That is, an '''arc from $a$ to $b$''' is a continuous injection $f: \\mathbb I \\to T$ such that $\\map f 0 = a$ and $\\map f 1 = b$. The mapping $f$ can be described as an '''arc (in $T$) joining $a$ and $b$'''."}
+{"_id": "21592", "title": "Definition:Path Component", "text": "Let $T$ be a topological space."}
+{"_id": "21593", "title": "Definition:Arc-Connected", "text": "=== Points in Topological Space === {{:Definition:Arc-Connected/Points}} === Subset of Topological Space === {{:Definition:Arc-Connected/Subset}} === Topological Space === {{:Definition:Arc-Connected/Topological Space}}"}
+{"_id": "21594", "title": "Definition:Arc Component", "text": "Let $T$ be a topological space. Let us define the relation $\\sim$ on $T$ as follows: :$x \\sim y \\iff x$ and $y$ are arc-connected. We have that $\\sim $ is an equivalence relation, so from the Fundamental Theorem on Equivalence Relations, the points in $T$ can be partitioned into equivalence classes. These equivalence classes are called the '''arc components of $T$'''."}
+{"_id": "21595", "title": "Definition:Ring of Integers of Number Field", "text": "Let $K$ be a number field. The '''ring of integers''' of $K$, usually denoted $\\mathcal O_K$ or $\\mathfrak o_K$, is the integral closure of $\\Z$ in $K$."}
+{"_id": "21596", "title": "Definition:GCD Domain", "text": "A '''GCD domain''' is an integral domain in which any two non-zero elements have a greatest common divisor."}
+{"_id": "21597", "title": "Definition:Ascending Chain Condition/Module", "text": "Let $R$ be a commutative ring with unity. Let $M$ be an $R$-module. Let $\\left({D, \\subseteq}\\right)$ be a set of submodules of $M$ ordered by inclusion. Then $M$ is said to have the '''ascending chain condition on submodules''' {{iff}}: :''Every increasing sequence $N_1 \\subseteq N_2 \\subseteq N_3 \\subseteq \\cdots$ with $N_i \\in D$ eventually stabilizes: $\\exists k \\in \\N: \\forall n \\in \\N, n \\ge k: N_n = N_{n+1}$''"}
+{"_id": "21598", "title": "Definition:Descending Chain Condition", "text": "Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module. Let $(D,\\supseteq)$ be a set of submodules of $M$ ordered by inclusion. Then the hypothesis :''Every increasing sequence $N_1 \\supseteq N_2 \\supseteq N_3 \\supseteq \\cdots$ with $N_i \\in D$ eventually terminates: there is $k \\in \\N$ such that $N_k = N_{k+1} = \\cdots$'' is called the '''descending chain condition''' on the submodules in $D$."}
+{"_id": "21599", "title": "Definition:Maximal Condition", "text": "Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module. Let $(D,\\subseteq)$ be the set of submodules of $M$ ordered by inclusion. Then the hypothesis :''Every non-empty subset of $D$ has a maximal element'' is called the '''maximal condition''' on submodules."}
+{"_id": "21600", "title": "Definition:Minimal Condition", "text": "=== Ordered  set === {{Definition:Minimal Condition/Ordered Set}} === Minimal condition on subsets === Let $S$ be a set. Let $F$ be a set of subsets of $S$. Then $S$ '''satisfies the minimal condition on $F$''' {{iff}} $F$, ordered by inclusion satisfies the minimal condition. ==== Minimal condition on submodules ==== Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module. Let $(D,\\supseteq)$ be the set of submodules of $M$ ordered by inclusion. Then the hypothesis :''Every non-empty subset of $D$ has a minimal element'' is called the '''minimal condition''' on submodules."}
+{"_id": "21601", "title": "Definition:Noetherian Module", "text": "Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module. === Definition 1 === {{:Definition:Noetherian Module/Definition 1}} === Definition 2 === {{:Definition:Noetherian Module/Definition 2}} === Definition 3 === {{:Definition:Noetherian Module/Definition 3}}"}
+{"_id": "21602", "title": "Definition:Artinian Module", "text": "Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module Then $M$ is a '''Artinian module''' if either of the following conditions hold: :$(1): \\quad$ $M$ satisfies the descending chain condition on submodules :$(2): \\quad$ $M$ satisfies the minimal condition on submodules."}
+{"_id": "21603", "title": "Definition:Artinian Ring", "text": "Let $A$ be a commutative ring with unity. Then $A$ is '''Artinian''' if it is Artinian as an $A$-module {{NamedforDef|Emil Artin|cat=Artin}} Category:Definitions/Ring Theory 1816gmd3wdbxexj3pmb5o0i5lc7svrd"}
+{"_id": "21604", "title": "Definition:Prime Element of Ring", "text": "Let $R$ be a commutative ring. Let $p \\in R \\setminus \\set 0$ be any non-zero element of $R$. Then $p$ is a '''prime element of $R$''' {{iff}}: :$(1): \\quad p$ is not a unit of $R$  :$(2): \\quad$ whenever $a, b \\in R$ such that $p$ divides $a b$, then either $p$ divides $a$ or $p$ divides $b$."}
+{"_id": "21605", "title": "Definition:Factorization Domain", "text": "A '''factorization domain''' is an integral domain in which every element can be written as a finite product of irreducible elements."}
+{"_id": "21606", "title": "Definition:Primal", "text": "Let $A$ be an integral domain. An element $p \\in A$ is called '''primal''' if whenever $a, b \\in A$ are such that $p$ divides $a b$, then we can write $p = q_1 q_2$, with $q_1$ divides $a$ and $q_2$ divides $B$. Category:Definitions/Ring Theory Category:Definitions/Factorization pgmsxqkkoncitqb13emce2k8nz7syok"}
+{"_id": "21607", "title": "Definition:Schreier Domain", "text": "A ring $A$ is a '''Schreier domain''' if it is an integrally closed integral domain in which every non-zero element is primal. {{NamedforDef|Otto Schreier|cat = Schreier}} Category:Definitions/Ring Theory Category:Definitions/Factorization br9ia0erqzair0nwjvtxcqj2o30eg8h"}
+{"_id": "21609", "title": "Definition:Nagata Criterion", "text": "Let $A$ be a ring. The '''Nagata criterion''' reads as follows: : ''Let $S \\subseteq A$ be a multiplicatively closed subset of $A$ generated by prime elements. If the localization $A_S$ is a UFD, then so is $A$''. By Localization of UFD is UFD, this is equivalent to: : ''Let $S \\subseteq A$ be a multiplicatively closed subset of $A$ generated by prime elements. Then the localization $A_S$ is a UFD {{iff}} $A$ is a UFD''. {{NamedforDef|Masayoshi Nagata|cat = Nagata M}} Category:Definitions/Factorization efl4gt8fnhjlcqc6mzvqnlt3upkzmrm"}
+{"_id": "21610", "title": "Definition:Product of Ideals of Ring", "text": "Let $\\left({R, +, \\circ}\\right)$ be a commutative ring. Let $I,J$ be ideals of $R$. === Definition 1 === The '''product of $I$ and $J$''' is the set of all finite sums: :$IJ = \\{a_1 b_1 + \\cdots + a_r b_r : a_i \\in I, b_i \\in J, r \\in \\N \\}$ === Definition 2 === The '''product of $I$ and $J$''' is the ideal generated by their product as subsets."}
+{"_id": "21611", "title": "Definition:Bézout Domain", "text": "=== Definition 1 === {{:Definition:Bézout Domain/Definition 1}} === Definition 2 === {{:Definition:Bézout Domain/Definition 2}}"}
+{"_id": "21612", "title": "Definition:Direct Sum of Modules", "text": "Let $A$ be a commutative ring with unity. {{explain|For this concept to be more easily understood, it is suggested that the ring be defined using its full specification, i.e. complete with operators, $\\left({A, +, \\circ}\\right)$ and so on, and similarly that a symbol be used to make the module scalar product equally explicit.}} Let $\\left\\{ {M_i}\\right\\}_{i \\in I}$ be a family of $A$-modules indexed by $I$. {{explain|Worth specifically stating exactly what $I$ is.}} Let $M = \\displaystyle \\prod_{i \\mathop \\in I} M_i$ be their direct product. The '''direct sum''' $\\displaystyle \\bigoplus_{i \\mathop \\in I} M_i$ is the submodule of $M$ the consisting of the elements of finite support."}
+{"_id": "21613", "title": "Definition:Projective Module", "text": "Let $A$ be a ring with unity. Let $M$ be an $A$-module. $M$ is '''projective''' {{iff}} there exists an $A$-module $N$ such that the direct sum $M \\oplus N$ is a free module."}
+{"_id": "21614", "title": "Definition:Hereditary Ring", "text": "Let $A$ be a commutative ring. We say that $A$ is '''hereditary''' if for every module $M$ over $A$, if $M$ is projective, then so is each submodule of $M$."}
+{"_id": "21615", "title": "Definition:Semihereditary Ring", "text": "Let $A$ be a commutative ring. We say that $A$ is '''semihereditary''' if for every module $M$ over $A$, if $M$ is projective, then so is each finitely generated submodule of $M$."}
+{"_id": "21616", "title": "Definition:Prüfer Domain", "text": "Let $A$ be an integral domain. Then $A$ is a '''Prüfer domain''' if every finitely generated ideal of $A$ is projective. {{NamedforDef|Heinz Prüfer|cat = Prüfer}} Category:Definitions/Integral Domains c7x5em8c9sysqq9sni7q5de9tjhs3d9"}
+{"_id": "21617", "title": "Definition:Height (Ring Theory)", "text": "Let $A$ be a commutative ring. Let $\\mathfrak p$ be a prime ideal in $A$. The '''height''' of $\\mathfrak p$ is the supremum over all $n$ such that there exists a chain of prime ideals: :$\\mathfrak p_0 \\subsetneqq \\mathfrak p_1 \\subsetneqq \\cdots \\subsetneqq \\mathfrak p$ {{explain|make this precise, as in Definition:Krull Dimension of Ring}} Category:Definitions/Ring Theory jmeuk9fqjcj18r00abrfbfsh5hs9p1t"}
+{"_id": "21618", "title": "Definition:Ultraconnected Space", "text": "=== Definition 1 === {{:Definition:Ultraconnected Space/Definition 1}} === Definition 2 === {{:Definition:Ultraconnected Space/Definition 2}} === Definition 3 === {{:Definition:Ultraconnected Space/Definition 3}}"}
+{"_id": "21619", "title": "Definition:Addition of Polynomials/Polynomial Forms", "text": "Let: : $\\displaystyle f = \\sum_{k \\mathop \\in Z} a_k \\mathbf X^k$ : $\\displaystyle g = \\sum_{k \\mathop \\in Z} b_k \\mathbf X^k$ be polynomials in the indeterminates $\\left\\{{X_j: j \\in J}\\right\\}$ over $R$. {{explain|What is $Z$ in the above? Presumably the integers, in which case they need to be denoted $\\Z$ and limited in domain to non-negative? However, because $Z$ is used elsewhere in the exposition of polynomials to mean something else (I will need to hunt around to find out exactly what), I can not take this assumption for granted.}} The operation '''polynomial addition''' is defined as: :$\\displaystyle f + g := \\sum_{k \\mathop \\in Z} \\left({a_k + b_k}\\right) \\mathbf X^k$ The expression $f + g$ is known as the '''sum''' of $f$ and $g$."}
+{"_id": "21620", "title": "Definition:Multiplication of Polynomials/Polynomial Forms", "text": "Let $\\displaystyle f = \\sum_{k \\mathop \\in Z} a_k \\mathbf X^k$, $\\displaystyle g = \\sum_{k \\mathop \\in Z} b_k \\mathbf X^k$ be polynomial forms in the indeterminates $\\left\\{{X_j: j \\in J}\\right\\}$ over $R$. The '''product of $f$ and $g$''' is defined as: :$\\displaystyle f \\circ g := \\sum_{k \\mathop \\in Z} c_k \\mathbf X^k$  where: :$\\displaystyle c_k = \\sum_{\\substack{p + q \\mathop = k \\\\ p, q \\mathop \\in Z}} a_p b_q$"}
+{"_id": "21622", "title": "Definition:Trivial Topological Space", "text": "A '''trivial topological space''' is a topological space with only one element. The open sets of a '''trivial topological space''' $T = \\left({\\left\\{{s}\\right\\}, \\tau}\\right)$ are $\\varnothing$ and $\\left\\{{s}\\right\\}$."}
+{"_id": "21623", "title": "Definition:Locally Connected Space", "text": "=== Definition 1: Using Local Bases === {{:Definition:Locally Connected Space/Definition 1}} === Definition 2: Using Neighborhood Bases === {{:Definition:Locally Connected Space/Definition 2}} === Definition 3: Using (Global) Basis === {{:Definition:Locally Connected Space/Definition 3}} === Definition 4: Using Open Components === {{:Definition:Locally Connected Space/Definition 4}}"}
+{"_id": "21625", "title": "Definition:Locally Path-Connected Space", "text": "=== Definition 1: using Local Bases === {{:Definition:Locally Path-Connected Space/Definition 1}} === Definition 2: using Neighborhood Bases === {{:Definition:Locally Path-Connected Space/Definition 2}} === Definition 3: using (Global) Basis === {{:Definition:Locally Path-Connected Space/Definition 3}} === Definition 4: using Open Path Components === {{:Definition:Locally Path-Connected Space/Definition 4}}"}
+{"_id": "21626", "title": "Definition:Locally Arc-Connected Space", "text": "A topological space $T = \\struct {S, \\tau}$ is '''locally arc-connected''' {{iff}} it has a basis consisting entirely of arc-connected sets."}
+{"_id": "21627", "title": "Definition:Partition of Unity (Topology)", "text": "Let $X$ be a topological space. Let $\\AA = \\set {\\phi_\\alpha : \\alpha \\in A}$ be a collection of continuous mappings $X \\to \\R$ such that: :$(1): \\quad$ The set $\\set {\\map {\\operatorname {supp} } {\\phi_\\alpha}^\\circ: \\alpha \\in A}$ of interiors of the supports is a locally finite cover of $X$ :$(2): \\quad \\forall x \\in X: \\forall \\alpha \\in A: \\map {\\phi_\\alpha} x \\ge 0$ :$(3): \\quad \\displaystyle \\forall x \\in X: \\sum_{\\alpha \\mathop \\in A} \\map {\\phi_\\alpha} x = 1$ Then $\\set {\\phi_\\alpha: \\alpha \\in A}$ is a '''partition of unity''' on $X$. === Subordinate to Cover === {{:Definition:Partition of Unity (Topology)/Subordinate}} Category:Definitions/Topology j29zrl0omgdcxalwqgdxp3rcvl5o32m"}
+{"_id": "21628", "title": "Definition:Vector Space on Field Extension", "text": "Let $\\struct {K, +, \\times}$ be a field. Let $L / K$ be a field extension over $K$. Then $\\struct {L, +, \\times}_K$ is the '''vector space on $L$ over $K$'''."}
+{"_id": "21629", "title": "Definition:Totally Pathwise Disconnected Space", "text": "=== Definition 1 === {{:Definition:Totally Pathwise Disconnected Space/Definition 1}} === Definition 2 === {{:Definition:Totally Pathwise Disconnected Space/Definition 2}}"}
+{"_id": "21630", "title": "Definition:Characteristic Polynomial", "text": "Let $K$ be a field. Let $L / K$ be a finite field extension of $K$. Then by Vector Space on Field Extension is Vector Space, $L$ is naturally a vector space over $K$. Let $\\alpha \\in L$, and $\\theta_\\alpha$ be the linear operator: :$\\theta_\\alpha: L \\to L : \\beta \\mapsto \\alpha \\beta$ The '''characteristic polynomial''' of $\\alpha$ with respect to the extension $L / K$ is: :$\\det \\sqbrk {X I_L - \\theta_\\alpha}$ where: :$\\det$ denotes the determinant of a linear operator :$X$ is an indeterminate  :$I_L$ is the identity mapping on $L$."}
+{"_id": "21631", "title": "Definition:Determinant/Linear Operator", "text": "Let $V$ be a finite-dimensional vector space over a field $K$. Let $A: V \\to V$ be a linear operator of $V$. The '''determinant''' $\\det \\left({A}\\right)$ of $A$ is defined to be the determinant of any matrix of $A$ relative to some basis."}
+{"_id": "21632", "title": "Definition:Trace (Linear Algebra)", "text": "=== Matrix === {{:Definition:Trace (Linear Algebra)/Matrix}} === Linear Operator === {{:Definition:Trace (Linear Algebra)/Linear Operator}} Category:Definitions/Linear Algebra Category:Definitions/Matrix Algebra iwd100re67e8wjvyj3pv3ss0y9xwzxv"}
+{"_id": "21633", "title": "Definition:Totally Separated Space", "text": "=== Definition 1 === {{:Definition:Totally Separated Space/Definition 1}} === Definition 2 === {{:Definition:Totally Separated Space/Definition 2}}"}
+{"_id": "21634", "title": "Definition:Grothendieck Universe", "text": "A '''Grothendieck universe''' is a set (not a class) which has the properties expected of the universe $\\mathbb U$ of sets in the sense of the Zermelo-Fraenkel axioms with the following properties: :$(1): \\quad \\mathbb U$ is a transitive set: if $u \\in \\mathbb U$ and $x \\in u$ then $x \\in \\mathbb U$ :$(2): \\quad$ If $ u, v \\in \\mathbb U$ then $\\set {u, v} \\in \\mathbb U$ :$(3): \\quad$ If $u \\in \\mathbb U$ then the power set $\\powerset u \\in \\mathbb U$ :$(4): \\quad$ If $A \\in \\mathbb U$, and $\\set {u_\\alpha: \\alpha \\in A}$ is a collection of elements of $\\mathbb U$, then $\\displaystyle \\bigcup_{\\alpha \\mathop \\in A} u_\\alpha \\in \\mathbb U$ {{explain|Tidy up language of that last axiom -- when \"collection\" is used, does this mean set? If so, say so -- if not, then link to exactly what concept is meant. This area of mathematics is such that differences between sets and e.g. classes is important to distinguish, and undefined terms such as \"collection\" are not to be tolerated.}}"}
+{"_id": "21635", "title": "Definition:Morphism", "text": "Let $\\mathbf C$ be a metacategory. A '''morphism''' of $\\mathbf C$ is an object $f$, together with: * A domain $\\operatorname{dom} f$, which is an object of $\\mathbf C$ * A codomain $\\operatorname{cod} f$, also an object of $\\mathbf C$"}
+{"_id": "21636", "title": "Definition:Hom Class", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ and $D$ be objects of $\\mathbf C$. The collection of morphisms $f: C \\to D$ is called a '''hom class''' and is denoted $\\operatorname{Hom}_{\\mathbf C} \\left({C, D}\\right)$."}
+{"_id": "21637", "title": "Definition:Small Category", "text": "Let $\\mathbf C$ be a metacategory. Then $\\mathbf C$ is said to be '''small''' {{iff}} both of the following hold: :The collection of objects $\\mathbf C_0$ is a set; :The collection of morphisms $\\mathbf C_1$ is a set."}
+{"_id": "21638", "title": "Definition:Functor", "text": "Informally, a '''functor''' is a morphism of categories. It may be described as what one must define in order to define a '''natural transformation'''. This is formalized by defining the category of categories. === Covariant Functor === {{:Definition:Functor/Covariant}} === Contravariant Functor === {{:Definition:Functor/Contravariant}}"}
+{"_id": "21639", "title": "Definition:Composition of Morphisms", "text": "Let $\\mathbf C$ be a metacategory. Let $\\left({g, f}\\right)$ be a pair of composable morphisms. Then the '''composition of $f$ and $g$''' is a morphism $g \\circ f$ of $\\mathbf C$ subject to: :$\\operatorname{dom} \\left({g \\circ f}\\right) = \\operatorname{dom} f$ :$\\operatorname{cod} \\left({g \\circ f}\\right) = \\operatorname{cod} g$ This '''composition of morphisms''' can be thought of as an abstraction of both composition of mappings and transitive relations."}
+{"_id": "21640", "title": "Definition:Identity Morphism", "text": "Let $\\mathbf C$ be a metacategory. Let $X$ be an object of $\\mathbf C$. The '''identity morphism''' of $X$, denoted $\\operatorname{id}_X$, is a morphism of $\\mathbf C$ subject to: :$\\operatorname{dom} \\operatorname{id}_X = \\operatorname{cod} \\operatorname{id}_X = X$ :$f \\circ \\operatorname{id}_X = f$ :$\\operatorname{id}_X \\circ g = g$ whenever $X$ is the domain of $f$ or the codomain of $g$, respectively. In most metacategories, the '''identity morphisms''' can be viewed as a representation of \"doing nothing\", in a sense suitable to the metacategory under consideration."}
+{"_id": "21641", "title": "Definition:Product Category", "text": "Let $\\mathbf C$ and $\\mathbf D$ be metacategories. The '''product category''' $\\mathbf C \\times \\mathbf D$ is the category with: {{DefineCategory | ob = $\\tuple {X, Y}$, for all $X \\in \\operatorname {ob} \\mathbf C$, $Y \\in \\operatorname {ob} \\mathbf D$ | mor = $\\tuple {f, g}: \\tuple {X, Y} \\to \\tuple {X', Y'}$ for all $f: X \\to X'$ in $\\mathbf C_1$ and $g: Y \\to Y'$ in  $\\mathbf D_1$ | comp = $\\tuple {f, g} \\circ \\tuple {h, k} := \\tuple {f \\circ h, g \\circ k}$, whenever this is defined | id = $\\operatorname {id}_{\\tuple {X, Y} } := \\tuple {\\operatorname {id}_X, \\operatorname {id}_Y}$ }}"}
+{"_id": "21642", "title": "Definition:Extremally Disconnected Space", "text": "=== Definition 1 === {{:Definition:Extremally Disconnected Space/Definition 1}} === Definition 2 === {{:Definition:Extremally Disconnected Space/Definition 2}} === Definition 3 === {{:Definition:Extremally Disconnected Space/Definition 3}}"}
+{"_id": "21643", "title": "Definition:Zero Dimensional Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then $T$ is zero dimensional {{iff}} it has a basis whose sets are all both closed and open."}
+{"_id": "21644", "title": "Definition:Scattered Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. === Definition 1 === {{:Definition:Scattered Space/Definition 1}} === Definition 2 === {{:Definition:Scattered Space/Definition 2}}"}
+{"_id": "21645", "title": "Definition:Operator Domain", "text": "An operator domain is a notational tool for indexing a collection of operations on a set. Formally, an '''operator domain''' is a set $\\Omega$  together with a mapping $a : \\Omega \\to \\N_0$. We also use the notation, for $n \\in \\N_0$, $\\Omega(n) = \\{\\omega \\in \\Omega : a(\\omega) = n \\}$. The elements of $\\Omega$ are '''operators'''. If $\\omega \\in \\Omega$, then $a(\\omega)$ is the '''arity''' of the operator $\\omega$. The elements of $\\Omega(n)$ are the '''$n$-ary''' operators of $\\Omega$. Note that an operator domain is an abstract labelling set, so the definitions here are purely formal; we have not identified any concrete operations. The latter is done using an Omega algebra. {{SUBPAGENAME}} 16yueicrvs4xcp7n0b9p6dmg2elbm7e"}
+{"_id": "21646", "title": "Definition:Biconnected Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be a connected set in $T$. Then $H$ is '''biconnected''' {{iff}} it is not the union of two disjoint non-degenerate connected sets."}
+{"_id": "21647", "title": "Definition:Cut Point", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be a connected set in $T$ and let $p \\in H$. Let $p \\in H$ such that $H \\setminus \\set p$ is disconnected, where $\\setminus$ denotes set difference. Then $p$ is a cut point of $H$."}
+{"_id": "21648", "title": "Definition:Dispersion Point", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be a connected set in $T$ and let $p \\in H$. Let $p \\in H$ such that $H \\setminus \\set p$ is totally disconnected, where $\\setminus$ denotes set difference. Then $p$ is a dispersion point of $H$."}
+{"_id": "21649", "title": "Definition:Continuum (Topology)", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be both compact and connected in $T$. Then $H$ is a '''continuum'''."}
+{"_id": "21650", "title": "Definition:Degenerate Continuum", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. A '''degenerate continuum''' of $T$ is a continuum in $T$ containing exactly one element. === Non-Degenerate Continuum === {{:Definition:Degenerate Continuum/Non-Degenerate}} Category:Definitions/Continua nt93zsnk4vlipk91bmmucao1s968w4j"}
+{"_id": "21651", "title": "Definition:Subcontinuum", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be a continuum in $T$. Let $K \\subseteq H$ also be a continuum in $T$. Then $K$ is a '''subcontinuum''' of $H$. === Proper Subcontinuum === {{:Definition:Subcontinuum/Proper Subcontinuum}}"}
+{"_id": "21652", "title": "Definition:Indecomposable Continuum", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be a continuum in $T$. Then $H$ is indecomposable {{iff}} it is not the union of two different non-degenerate proper subcontinua."}
+{"_id": "21653", "title": "Definition:Composant", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $H \\subseteq S$ be a continuum in $T$. Let $C \\subseteq H$ be a subset of $H$. === Composant of a Continuum === {{:Definition:Composant/Continuum}} === Composant of a Point === {{:Definition:Composant/Point}} Category:Definitions/Continua aho2kjjba1m5v0etttg9qp3d19e5alq"}
+{"_id": "21654", "title": "Definition:Punctiform Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be a subset of $T$. Then $H$ is punctiform if it contains no non-degenerate continua."}
+{"_id": "21655", "title": "Definition:Metric Space Axioms", "text": "{{begin-axiom}} {{axiom | n = \\text M 1         | q = \\forall x \\in A         | m = \\map d {x, x} = 0 }} {{axiom | n = \\text M 2         | q = \\forall x, y, z \\in A         | m = \\map d {x, y} + \\map d {y, z} \\ge \\map d {x, z} }} {{axiom | n = \\text M 3         | q = \\forall x, y \\in A         | m = \\map d {x, y} = \\map d {y, x} }} {{axiom | n = \\text M 4         | q = \\forall x, y \\in A         | m = x \\ne y \\implies \\map d {x, y} > 0 }} {{end-axiom}}"}
+{"_id": "21656", "title": "Definition:Quasimetric", "text": "A '''quasimetric''' on a set $X$ is a real-valued function $d: A \\times A \\to \\R$ which satisfies the following conditions: {{begin-axiom}} {{axiom | n = \\text M 1         | q = \\forall x \\in A         | m = \\map d {x, x} = 0 }} {{axiom | n = \\text M 2         | q = \\forall x, y, z \\in A         | m = \\map d {x, y} + \\map d {y, z} \\ge \\map d {x, z} }} {{axiom | n = \\text M 4         | q = \\forall x, y \\in A         | m = x \\ne y \\implies \\map d {x, y} > 0 }} {{end-axiom}} Note the numbering system of these conditions. They are numbered this way so as to retain consistency with the metric space axioms, of which these are a subset. The difference between a '''quasimetric''' and a metric is that a '''quasimetric''' does not insist that the distance function between distinct elements is commutative, that is, that $\\map d {x, y} = \\map d {y, x}$. === Quasimetric Space === {{:Definition:Quasimetric/Quasimetric Space}}"}
+{"_id": "21657", "title": "Definition:Trivial Character", "text": "Let $G$ be a finite abelian group. The character $\\chi_0: G \\to \\C_{\\ne 0}$ defined as: :$\\forall g \\in G: \\chi_0 \\left({g}\\right) = 1$ is the '''trivial character on $G$'''."}
+{"_id": "21658", "title": "Definition:Bounded Metric Space", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $M' = \\struct {B, d_B}$ be a subspace of $M$. === Definition 1 === {{:Definition:Bounded Metric Space/Definition 1}} === Definition 2 === {{:Definition:Bounded Metric Space/Definition 2}}"}
+{"_id": "21659", "title": "Definition:Indicator of Group Element", "text": "Let $G$ be a finite group. Let $a \\in G$. Let $H$ be a subgroup of $G$. The '''indicator of $a$ in $H$''' is the least strictly positive integer $n$ such that $a^n \\in H$."}
+{"_id": "21660", "title": "Definition:Nested Sequence", "text": "Let $S$ be a set. Let $\\SS = \\powerset S$ be the power set of $S$. Let $\\family {S_k}_{k \\mathop \\in \\N}$ indexed family of subsets of $S$ such that either: :$\\forall k \\in \\N: S_k \\subseteq S_{k + 1}$ or: :$\\forall k \\in \\N: S_k \\supseteq S_{k + 1}$ Then $\\family {S_k}_{k \\mathop \\in \\N}$ is a '''nested sequence''' of sets."}
+{"_id": "21661", "title": "Definition:Topologically Complete Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $M = \\struct {S, d}$ be a complete metric space such that $\\struct {S, \\tau}$ is the topological space induced by $d$. If there exists such a complete metric space, then $T$ is described as '''topologically complete'''."}
+{"_id": "21662", "title": "Definition:Sigma-Locally Finite Basis", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. A '''$\\sigma$-locally finite basis''' for $T$ is a basis which is the countable union of locally finite covers. {{explain|Is this the same thing as \"countably locally finite\"?}}"}
+{"_id": "21663", "title": "Definition:Quasiuniformity", "text": "Let $S$ be a set. A '''quasiuniformity''' on $S$ is a set of subsets $\\UU$ of the cartesian product $S \\times S$ satisfying the following '''quasiuniformity axioms''': {{:Definition:Quasiuniformity Axioms}} That is, a '''quasiuniformity''' on $S$ is a filter on the cartesian product $S \\times S$ (from $(\\text U 1)$ to $(\\text U 3)$) which also fulfils the condition: :$\\forall u \\in \\UU: \\exists v \\in \\UU$ such that whenever $\\tuple {x, y} \\in v$ and $\\tuple {y, z} \\in v$, then $\\tuple {x, z} \\in u$ which can be seen to be an equivalent statement to $(\\text U 4)$. $u \\circ v$ in this context can be seen to be equivalent to composition of relations. Thus a '''quasiuniformity''' on $S$ is a filter on $S \\times S$ which also fulfils the condition that every element is the composition of another element with itself."}
+{"_id": "21664", "title": "Definition:Uniformity", "text": "Let $S$ be a set. A '''uniformity''' on $S$ is a set of subsets $\\UU$ of the cartesian product $S \\times S$ satisfying the '''quasiuniformity axioms''': {{:Definition:Quasiuniformity Axioms}} and also: :$(\\text U 5): \\forall u \\in \\UU: u^{-1} \\in \\UU$ where $u^{-1}$ is defined as: ::$u^{-1} := \\set {\\tuple {y, x}: \\tuple {x, y} \\in u}$ :That is, all elements of $\\UU$ are symmetric."}
+{"_id": "21665", "title": "Definition:Entourage", "text": "Let $S$ be a set. Let $\\UU \\subseteq \\powerset {S \\times S}$ be a quasiuniformity on $S$. An element $u \\in \\UU$ of such a quasiuniformity is known as an '''entourage''' of $\\UU$'''. Note that an entourage of $\\UU$ actually ''is'' a relation on $S$."}
+{"_id": "21666", "title": "Definition:Separated Quasiuniformity", "text": "Let $\\UU$ be a quasiuniformity on a set $S$. Then $\\UU$ is '''separated''' {{iff}}: :$\\forall u, v \\in \\UU: u \\cap v = \\Delta_S$ That is, {{iff}} the intersection of all its entourages is the diagonal relation."}
+{"_id": "21667", "title": "Definition:Symmetric Filter Basis", "text": "Let $S$ be a set. Let $\\UU$ be a quasiuniformity on $S$. From the definition, a quasiuniformity on $S$ is also a filter on the cartesian product $S \\times S$. Let $\\BB \\subset \\powerset {S \\times S}$ be a filter basis of $\\UU$. Then $\\BB$ is a '''symmetric filter basis''' of $\\UU$ {{iff}} every element of $\\BB$ is symmetric."}
+{"_id": "21668", "title": "Definition:Inverse Entourage", "text": "Let $S$ be a set. Let $\\UU$ be a quasiuniformity on $S$. Let $u \\in \\UU$ be an entourage of $\\UU$. Then the '''inverse entourage''' $u^{-1}$ is defined as: :$u^{-1} := \\set {\\tuple {y, x}: \\tuple {x, y} \\in u}$"}
+{"_id": "21669", "title": "Definition:Symmetric Entourage", "text": "Let $S$ be a set. Let $\\UU$ be a quasiuniformity on $S$. Let $u \\in \\UU$ be an entourage of $\\UU$ such that: :$u = u^{-1}$ where $u^{-1}$ is the inverse of $u$. Then $u$ is '''symmetric'''."}
+{"_id": "21670", "title": "Definition:Quasiuniform Space", "text": "Let $\\UU$ be a quasiuniformity on a set $S$. Then a topology $\\tau$ can be created from $\\UU$ by: :$\\tau := \\set {\\map u x: u \\in \\UU, x \\in S}$ where: :$\\forall x \\in S: \\map u x := \\set {y: \\tuple {x, y} \\in u}$ The resulting topological space $T = \\struct {S, \\tau}$ is called a quasiuniform space. It can be denoted $\\struct {\\struct {S, \\UU}, \\tau}$, or just $\\struct {S, \\UU}$ if it is understood that $\\tau$ is the topology created from $\\UU$."}
+{"_id": "21671", "title": "Definition:Compatible Quasiuniformities", "text": "Let $\\UU_1$ and $\\UU_2$ be quasiuniformities on a set $S$. Let $\\struct {\\struct {S, \\UU_1}, \\tau_1}$ and $\\struct {\\struct {S, \\UU_2}, \\tau_2}$ be the quasiuniform spaces generated by $\\UU_1$ and $\\UU_2$. Then $\\UU_1$ and $\\UU_2$ are '''compatible (with each other)''' {{iff}} their topologies are equal. That is, {{iff}} $\\tau_1 = \\tau_2$."}
+{"_id": "21672", "title": "Definition:Quasiuniformizable Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then $T$ is '''quasiuniformizable''' if there exists a quasiuniformity $\\UU$ on $S$ such that $\\struct {\\struct {S, \\UU}, \\tau}$ is a quasiuniform space."}
+{"_id": "21673", "title": "Definition:Uniform Space", "text": "Let $\\UU$ be a uniformity on a set $S$. The quasiuniform space $\\struct {\\struct {S, \\UU}, \\tau}$ generated from $\\UU$ is a '''uniform space'''."}
+{"_id": "21674", "title": "Definition:Uniformizable Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then $T$ is '''uniformizable''' if there exists a uniformity $\\UU$ on $S$ such that $\\struct {\\struct {S, \\UU}, \\tau}$ is a uniform space."}
+{"_id": "21675", "title": "Definition:Metrizable Uniformity", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $\\UU$ be the uniformity on $X$ defined as: :$\\UU := \\set {u_\\epsilon: \\epsilon \\in \\R_{>0} }$ where: :$\\R_{>0}$ is the set of strictly positive real numbers :$u_\\epsilon$ is defined as: ::$u_\\epsilon := \\set {\\paren {x, y}: \\map d {x, y} < \\epsilon}$ Then $\\UU$ is defined as '''metrizable'''."}
+{"_id": "21676", "title": "Definition:Superfunction", "text": "Let $C, D \\subseteq \\C$ with $z \\in C \\implies z + 1 \\in C$. Let $F: C \\to D$ and $H: D \\to D$ be holomorphic functions. Let $\\map H {\\map F z} = \\map F {z + 1}$ for all $z \\in C$. Then $F$ is said to be a '''superfunction''' of $H$, and $H$ is called a '''transfer function''' of $F$."}
+{"_id": "21678", "title": "Definition:Tetration", "text": "500pxrightthumb$y = \\operatorname{tet}_b \\left({x}\\right)$ versus $x$ for various $b$ 500pxrightthumb$f = \\operatorname{tet}_b \\left({x}\\right)$ in the $x, b$ plane with levels $f = \\text{const}$. === Definition for Integers === For all $x \\in \\R$, $n \\in \\Z_{\\ge 0}$: :${}^n x := \\begin{cases} 1 & : n = 0 \\\\ x^{ \\left({{}^{n - 1} x}\\right)} & : n > 0 \\\\ \\end{cases}$ Using Knuth uparrow notation: :$x \\uparrow \\uparrow n := \\begin{cases} 1 & : n = 0 \\\\ x \\uparrow \\left({x \\uparrow \\uparrow \\left({n - 1}\\right)}\\right) & : n > 0 \\\\ \\end{cases}$ === Definition for base $b \\ge \\exp \\left({1 / e}\\right)$ === Let $b \\in \\R$ such that $b \\ge \\exp \\left({\\dfrac 1 e}\\right)$. Let $L \\in \\C$ be a fixed point of $\\log_b$ such that $\\Im \\left({L}\\right) \\ge 0$. Let $C = \\C \\setminus \\left\\{ {x \\in \\R: x \\le -2}\\right\\}$. Let $\\operatorname{tet}_b: C \\mapsto \\C$ be the superfunction of $z \\mapsto b^z$ such that: : $\\operatorname{tet}_b \\left({0}\\right) = 1$ : $\\forall z \\in C: \\operatorname{tet}_b \\left({z^*}\\right) = \\operatorname{tet}_b \\left({z}\\right)^*$ : $\\displaystyle \\forall x \\in \\R: \\lim_{y \\to +\\infty} \\operatorname{tet}_b \\left({x + \\mathrm i y}\\right) = L$ Then the function $\\operatorname{tet}_b$ is called '''tetration to base $b$'''. === Definition for $0 < b < \\exp \\left({1 / e}\\right)$ === Let $b \\in \\R$ such that $1 < b < \\exp \\left({\\dfrac 1 e}\\right)$. Let $L_1, L_2 \\in \\R: L_1 < L_2$ be the fixed points of $\\log_b$. Let $T = \\dfrac{2 \\pi i} {\\ln \\left({L_1 \\ln \\left({b}\\right)}\\right)}$ Let $C = \\C \\setminus \\left\\{ {x + T m, x \\in \\R: x \\le -2, m \\in \\Z}\\right \\}$ Let $\\operatorname{tet}_b: C \\mapsto \\C$ be the superfunction of $z \\mapsto b^z$ such that: : $\\operatorname{tet}_b (0) = 1$ : $\\forall z \\in C: \\operatorname{tet}_b(z^*) = \\operatorname{tet}_b(z)^*$ : $\\forall z \\in C: \\operatorname{tet}_b(z) = \\operatorname{tet}_b(z+T)$ : $\\displaystyle \\forall y \\in \\R: \\lim_{x \\to -\\infty} \\operatorname{tet}_b (x + \\mathrm i y) = L_2 ~~ $ : $\\displaystyle \\forall \\varepsilon \\in \\R_{>0}: \\exists X \\in \\R$ such that: :: $\\forall x \\in \\R: x > X: \\left|{\\operatorname {tet}_b \\left({x + i y}\\right) - L_1}\\right| < \\varepsilon$   Then the function $\\operatorname{tet}_b$ is called '''tetration to base $b$'''."}
+{"_id": "21679", "title": "Definition:Abundant Number", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Let $\\map A n$ denote the abundance of $n$. === Definition 1 === {{:Definition:Abundant Number/Definition 1}} === Definition 2 === {{:Definition:Abundant Number/Definition 2}} === Definition 3 === {{:Definition:Abundant Number/Definition 3}}"}
+{"_id": "21680", "title": "Definition:Deficient Number", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. === Definition 1 === {{:Definition:Deficient Number/Definition 1}} === Definition 2 === {{:Definition:Deficient Number/Definition 2}} === Definition 3 === {{:Definition:Deficient Number/Definition 3}}"}
+{"_id": "21681", "title": "Definition:Discrete Uniformity", "text": "Let $S$ be a set. The '''discrete uniformity''' on $S$ is the uniformity $\\UU$ defined as: :$\\UU := \\set {u \\subseteq S \\times S: \\Delta_S \\subseteq u}$ that is, all subsets of the cartesian product on $S$ which contain the diagonal relation on $S$."}
+{"_id": "21682", "title": "Definition:Quasiuniformity Axioms", "text": "{{begin-axiom}} {{axiom | n = \\text U 1         | q = \\forall u \\in \\UU         | m = \\Delta_S \\subseteq u }} {{axiom | n = \\text U 2         | q = \\forall u, v \\in \\UU         | m = u \\cap v \\in \\UU }} {{axiom | n = \\text U 3         | q = \\forall u \\in \\UU         | m = u \\subseteq v \\subseteq S \\times S \\implies v \\in \\UU }} {{axiom | n = \\text U 4         | q = \\forall u \\in \\UU         | m = \\exists v \\in \\UU: v \\circ v \\subseteq u }} {{end-axiom}} where: :$\\Delta_S$ is the diagonal relation on $S$, that is: $\\Delta_S = \\set {\\tuple {x, x}: x \\in S}$ :$\\circ$ is defined as: ::$u \\circ v := \\set {\\tuple {x, z}: \\exists y \\in S: \\tuple {x, y} \\in v, \\tuple {y, z} \\in u}$"}
+{"_id": "21683", "title": "Definition:Uniformity Axioms", "text": "{{begin-axiom}} {{axiom | n = \\text U 1         | q = \\forall u \\in \\UU         | m = \\Delta_S \\subseteq u }} {{axiom | n = \\text U 2         | q = \\forall u, v \\in \\UU         | m = u \\cap v \\in \\UU }} {{axiom | n = \\text U 3         | q = \\forall u \\in \\UU         | m = u \\subseteq v \\subseteq S \\times S \\implies v \\in \\UU }} {{axiom | n = \\text U 4         | q = \\forall u \\in \\UU         | m = \\exists v \\in \\UU: v \\circ v \\subseteq u }} {{axiom | n = \\text U 5         | q = \\forall u \\in \\UU         | m = u^{-1} \\in \\UU }} {{end-axiom}} where: :$\\Delta_S$ is the diagonal relation on $S$, that is: $\\Delta_S = \\set {\\tuple {x, x}: x \\in S}$ :$\\circ$ is defined as: ::$u \\circ v := \\set {\\tuple {x, z}: \\exists y \\in S: \\tuple {x, y} \\in v, \\tuple {y, z} \\in u}$ :$u^{-1}$ is defined as: ::$u^{-1} := \\set {\\tuple {y, x}: \\tuple {x, y} \\in u}$ :That is, all elements of $\\UU$ are symmetric."}
+{"_id": "21684", "title": "Definition:Comparable Topologies", "text": "Let $S$ be a set. Let $\\tau_1$ and $\\tau_2$ be topologies on $S$. Then $\\tau_1$ and $\\tau_2$ are '''comparable''' {{iff}} either: :$\\tau_1$ is coarser than $\\tau_2$ or :$\\tau_1$ is finer than $\\tau_2$ That is, by definition of coarser and finer, either: :$\\tau_1 \\subseteq \\tau_2$ or :$\\tau_1 \\supseteq \\tau_2$"}
+{"_id": "21685", "title": "Definition:Partition Topology", "text": "Let $S$ be a set. Let $\\PP$ be a partition of $S$. Let $\\tau$ be the set of subsets of $S$ defined as: :$a \\in \\tau \\iff a$ is the union of sets of $\\PP$ Then $\\tau$ is a '''partition topology''' on $S$, and $\\struct {S, \\tau}$ is a '''partition (topological) space'''."}
+{"_id": "21686", "title": "Definition:Trivial Partition", "text": "Let $S$ be a set such that $S \\ne \\varnothing$. There are two partitions on $S$ which are referred to as '''the trivial partitions on $S$''': === Singleton Partition === {{:Definition:Trivial Partition/Singleton}} === Partition of Singletons === {{:Definition:Trivial Partition/Partition of Singletons}} Category:Definitions/Set Partitions brf8rp4cz616jnoxspgpv6wf5g1lhi0"}
+{"_id": "21687", "title": "Definition:Odd-Even Topology", "text": "Let $\\Z_{>0}$ denote the set of strictly positive integers: :$\\Z_{>0} = \\set {x \\in \\Z: x > 0}$ Let $\\PP$ be the partition on $\\Z_{>0}$ defined as: :$\\PP = \\set {\\set {2 k - 1, 2 k}: k \\in \\Z_{>0} }$ That is: :$\\PP = \\set {\\set {1, 2}, \\set {3, 4}, \\set {5, 6}, \\ldots}$ Then the topology whose basis is $\\PP$ is called the '''odd-even topology'''."}
+{"_id": "21688", "title": "Definition:Deleted Integer Topology", "text": "Let $\\PP$ be the set: :$\\PP = \\set {\\openint {n - 1} n: n \\in \\Z_{> 0} }$ that is, the set of all open real intervals of the form: :$\\openint 0 1, \\openint 1 2, \\openint 2 3, \\ldots$ Let $S$ be the set defined as: :$S = \\displaystyle \\bigcup \\PP = \\R_{\\ge 0} \\setminus \\Z$ that is, the positive real numbers minus the integers. Let $T = \\struct {S, \\tau}$ be the partition topology whose basis is $\\PP$. Then $T$ is called '''the deleted integer topology'''."}
+{"_id": "21691", "title": "Definition:Pseudometrizable Topology", "text": "Let $\\left({S, d}\\right)$ be a pseudometric space. Let $\\left({S, \\tau_d}\\right)$ be the topological space induced by $d$. Then for any topological space which is homeomorphic to such a $\\left({S, \\tau_d}\\right)$, it and its topology are defined as '''pseudometrizable'''."}
+{"_id": "21692", "title": "Definition:Nondeterministic Turing Machine", "text": "A '''Nondeterministic''' (or '''Non-deterministic''') '''Turing Machine''' (NTM) is a variation of the classical Turing machine that relaxes the restriction that all the steps in the machine must be definite. That is, given a single internal state and a single character being read on the tape the machine may have more then one possible response. If any sequence of choices puts the machine into a halting state, then the machine stops after $n$ steps, where $n$ is the minimum number of steps needed to put the machine into a halting state."}
+{"_id": "21693", "title": "Definition:Double Pointed Topology", "text": "Let $T = \\struct {S, \\tau_S}$ be a topological space. Let $A = \\set {a, b}$ be a doubleton. Let $D = \\struct {A, \\set {\\O, A} }$ be the indiscrete space on $A$. Let $\\struct {T \\times D, \\tau}$ be the product space of $T$ and $D$. Then $T \\times D$ is known as the '''double pointed topology''' on $T$."}
+{"_id": "21694", "title": "Definition:NP Complexity Class", "text": "In computation theory, $NP$ is a complexity class in which there exists some nondeterministic Turing machine that will either: : halt within $\\map p {\\size x}$ steps or: :run forever where $p$ is a polynomial and $\\size x$ is the length of the input to the Machine."}
+{"_id": "21695", "title": "Definition:NP-Complete", "text": "A problem $L$ is '''NP-complete''' if any problem in the complexity class NP can be reduced in polynomial time and space to $L$. {{NoSources}} Category:Definitions/Mathematical Logic Category:Definitions/Computer Science dom5wx5bpdsj6xzl16vb1capojqru0r"}
+{"_id": "21696", "title": "Definition:Particular Point Topology", "text": "Let $S$ be a set which is non-empty. Let $p \\in S$ be some '''particular point''' of $S$. We define a subset $\\tau_p$ of the power set $\\powerset S$ as: :$\\tau_p = \\set {A \\subseteq S: p \\in A} \\cup \\set \\O$ that is, all the subsets of $S$ which include $p$, along with the empty set. Then $\\tau_p$ is a topology called the '''particular point topology on $S$ by $p$''', or just '''a particular point topology'''."}
+{"_id": "21697", "title": "Definition:Finitely Generated Field Extension", "text": "Let $E / F$ be a field extension. Then $E$ is said to be '''finitely generated over $F$''' {{iff}}, for some $\\alpha_1, \\ldots, \\alpha_n \\in E$: :$E = F \\left({\\alpha_1, \\ldots, \\alpha_n}\\right)$  where $F \\left({\\alpha_1, \\ldots, \\alpha_n}\\right)$ is the field in $E$ generated by $F \\cup \\left\\{{\\alpha_1, \\ldots, \\alpha_n}\\right\\}$."}
+{"_id": "21698", "title": "Definition:Primitive Element of Field Extension", "text": "Let $F / K$ be a simple field extension such that $F = K \\left({\\alpha}\\right)$. Then $\\alpha$ is a '''primitive element''' of $F$."}
+{"_id": "21700", "title": "Definition:Relative Algebraic Closure", "text": "Let $L / K$ be a field extension. The '''relative algebraic closure''' of $K$ contained in $L$ is the set of all elements of $L$ that are algebraic over $K$."}
+{"_id": "21701", "title": "Definition:Algebraically Closed Field", "text": "Let $K$ be a field. Then $K$ is '''algebraically closed''' {{iff}}: === Definition 1 === {{:Definition:Algebraically Closed/Definition 1}} === Definition 2 === {{:Definition:Algebraically Closed/Definition 2}} === Definition 3 === {{:Definition:Algebraically Closed/Definition 3}}"}
+{"_id": "21702", "title": "Definition:Tower of Fields", "text": "A '''tower of fields''': :$F_k / F_{k-1} / \\dotsb / F_2 / F_1$ is a finite sequence of field extensions: :$F_{j+1} / F_j$ where $j = 1, \\dotsc, k-1$. Category:Definitions/Field Extensions l9bqsmhlpxtxsdmtioa1d46jzyx7lc6"}
+{"_id": "21703", "title": "Definition:Formal Derivative of Polynomial", "text": "Let $R$ be a ring. Let $R \\left[{X}\\right]$ be the polynomial ring over $R$. Let $f = a_0 + a_1 X + \\cdots + a_n X^n \\in R \\left[{X}\\right]$. The '''formal derivative''' $f'$ of $f$ is the polynomial: :$f' \\left({X}\\right) = a_1 + 2 a_2 X + \\cdots + n a_n X^{n-1}$ Category:Definitions/Polynomial Theory c5nbdn36wiio1alfwfu9ngpdral8jip"}
+{"_id": "21704", "title": "Definition:Symmetric Polynomial", "text": "Let $K$ be a field. Let $K \\sqbrk {X_1, \\ldots, X_n}$ be the ring of polynomial forms over $K$. A polynomial $f \\in K \\sqbrk {X_1, \\ldots, X_n}$ is '''symmetric''' {{iff}} for every permutation $\\pi$ of $\\set {1, 2, \\ldots, n}$: :$\\map f {X_1, \\dotsc, X_n} = \\map f {X_{\\map \\pi 1}, \\dotsc, X_{\\map \\pi n} }$"}
+{"_id": "21705", "title": "Definition:Elementary Symmetric Polynomial", "text": "Let $K$ be a field. Let $K \\left[{X_1, \\ldots, X_n}\\right]$ be the ring of polynomial forms over $K$. The '''elementary symmetric polynomials in $n$ variables''' are: :$\\displaystyle f_r \\left({X_1, \\ldots, X_n}\\right) = \\sum_{1 \\mathop \\le i_1 \\mathop < \\cdots \\mathop < i_r \\mathop \\le n} x_{i_1} \\cdots x_{i_r}: \\quad r = 1, \\ldots, n$ Category:Definitions/Polynomial Theory 4zvnd1r1knn8xbx8y0ossuy4fdpxqfj"}
+{"_id": "21706", "title": "Definition:Splitting Field", "text": "=== Of a polynomial === Let $K$ be a field. Let $f$ be a polynomial over $K$. A '''splitting field''' of $f$ over $K$ is a field extension $L / K$ such that: :$f = k \\paren {X - \\alpha_1} \\cdots \\paren {X - \\alpha_n}$ for some $k \\in K$, $\\alpha_1, \\ldots, \\alpha_n \\in L$. We say that $f$ '''splits''' over $L$. === Of a set of polynomials === Let $K$ be a field. Let $\\FF$ be a set of polynomials over $K$. A '''splitting field''' of $\\FF$ over $K$ is a field extension $L / K$ such that for any $f \\in \\FF$: :$f = k \\paren {X - \\alpha_1} \\cdots \\paren {X - \\alpha_n}$ for some $k \\in K$, $\\alpha_1, \\ldots, \\alpha_n \\in L$. We say that $\\FF$ '''splits''' over $L$."}
+{"_id": "21707", "title": "Definition:Sierpiński Space", "text": "The '''Sierpiński space''' is a particular point space with exactly two elements. Its usual presentation is: :$T = \\struct {\\set {0, 1}, \\set {\\O, \\set 0, \\set {0, 1} } }$ that is, as a particular point topology on the set $\\set {0, 1}$ where the particular point is $0$. It can also immediately be seen to be an excluded point topology on the set $\\set {0, 1}$ where the excluded point is $1$. The '''Sierpiński space''' is considered to be a '''trivial instance''' of both the particular point topology and the excluded point topology."}
+{"_id": "21708", "title": "Definition:Closed Extension Topology", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $p$ be a new element for $S$ such that $S^*_p := S \\cup \\set p$. Let $\\tau^*_p$ be the set defined as: :$\\tau^*_p := \\set {U \\cup \\set p: U \\in \\tau} \\cup \\set \\O$ That is, $\\tau^*_p$ is the set of all sets formed by adding $p$ to all the open sets of $\\tau$ and including the empty set. Then: :$\\tau^*_p$ is the closed extension topology of $\\tau$ and: :$T^*_p := \\struct {S^*_p, \\tau^*_p}$ is the closed extension space of $T = \\struct {S, \\tau}$."}
+{"_id": "21711", "title": "Definition:Ultraproduct", "text": "Let $\\mathcal L$ be a first-order language and let $I$ be an infinite set. Let $\\mathcal U$ be an ultrafilter on $I$. Let $\\mathcal M_i$ be an $\\mathcal L$-structure for each $i\\in I$ The '''ultraproduct''': :$\\mathcal M := \\displaystyle \\left({\\prod_{i \\mathop \\in I} \\mathcal M_i }\\right) / \\mathcal U$ is an $\\mathcal L$-structure defined as follows: :$(1): \\quad$ The universe of $\\mathcal M$: Let $X$ be the cartesian product: :$\\prod_{i \\mathop \\in I} \\mathcal M_i$ Define an equivalence relation $\\sim$ on $X$ by: :$\\left({a_i}\\right)_{i \\mathop \\in I} \\sim \\left({b_i}\\right)_{i \\mathop \\in I}$ {{iff}} $\\left\\{ {i \\in I: a_i = b_i}\\right \\} \\in \\mathcal U$ The universe of $\\mathcal M$ is the set of equivalence classes of $X$ modulo $\\sim$. These are essentially sequences taken modulo the equivalence relation above, and are sometimes denoted $\\left({m_i}\\right)_\\mathcal U$. :$(2): \\quad$ Interpretation of non-logical symbols of $\\mathcal L$ in $\\mathcal M$: For each constant symbol $c$, we define $c^\\mathcal M$ to be $\\left({c^{\\mathcal M_i} }\\right)_\\mathcal U$. For each $n$-ary function symbol $f$, we define $f^\\mathcal M$ by setting: :$f^\\mathcal M \\left({\\left({m_{1, i} }\\right)_\\mathcal U, \\dotsc, \\left({m_{n, i} }\\right)_\\mathcal U}\\right)$ to be: :$\\left({f^{\\mathcal M_i} \\left({m_{1, i}, \\dotsc, m_{n, i} }\\right)}\\right)_\\mathcal U$ For each $n$-ary relation symbol $R$, we define $R^\\mathcal M$ to be the set of $n$-tuples: :$\\left({\\left({m_{1, i} }\\right)_\\mathcal U, \\dots, \\left({m_{n, i} }\\right)_\\mathcal U}\\right)$ from $\\mathcal M$ such that: :$\\left\\{ {i \\in I: \\left({m_{1, i}, \\dotsc, m_{n, i} }\\right) \\in R^\\mathcal M_i}\\right\\} \\in \\mathcal U$"}
+{"_id": "21712", "title": "Definition:Pure Transcendental Extension", "text": "Let $K$ be a field. A '''pure transcendental extension''' is a field extension $L / K$ such that: :for some set $\\left\\{ {X_\\alpha: \\alpha \\in A}\\right\\}$, $L$ is the field of rational functions in the indeterminates $\\left\\{ {X_\\alpha}\\right\\}$. Category:Definitions/Field Extensions oolabd8eoks5ushyuqg7tckjyhiloz3"}
+{"_id": "21713", "title": "Definition:Algebraically Independent", "text": "Let $L / K$ be a field extension. Let $A \\subseteq L$ be a subset of $L$. Let $\\map K {\\set {X_\\alpha}_{\\alpha \\mathop \\in A} }$ be the Field of Rational Functions in the indeterminates $\\set {X_\\alpha: \\alpha \\mathop \\in A}$. Then $A$ is '''algebraically independent''' over $K$ if there exists a homomorphism: :$\\phi: \\map K {set {X_\\alpha}_{\\alpha \\mathop \\in A} } \\to L$ such that: :$\\map \\phi {X_\\alpha} = \\alpha$"}
+{"_id": "21714", "title": "Definition:Transcendence Degree", "text": "Let $K$ be a field, and let $L/K$ be a field extension of $K$. The '''transcendence degree''' of $L/K$ is the largest cardinality of an algebraically independent subset $A \\subseteq L$. Category:Definitions/Field Extensions l3ehwhvnm2jmnfvlr5laombdga98ywb"}
+{"_id": "21715", "title": "Definition:Successor Ordinal", "text": "A '''successor ordinal''' is the successor set of an ordinal."}
+{"_id": "21716", "title": "Definition:Limit Ordinal", "text": "An ordinal $\\lambda$ is a '''limit ordinal''' {{iff}} it is neither the zero ordinal nor a successor ordinal."}
+{"_id": "21717", "title": "Definition:Excluded Point Topology", "text": "Let $S$ be a set which is non-empty. Let $p \\in S$ be some '''particular point''' of $S$. We define a subset $\\tau_{\\bar p}$ of the power set $\\powerset S$ as: :$\\tau_{\\bar p} = \\set {A \\subseteq S: p \\notin A} \\cup \\set S$ That is, all the subsets of $S$ which do not include $p$, along with the set $S$. Then $\\tau_{\\bar p}$ is a topology called the '''excluded point topology on $S$ by $p$''', or just '''an excluded point topology'''."}
+{"_id": "21718", "title": "Definition:Open Extension Topology", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $p$ be a new element which is not in $S$. Let $S^*_p = S \\cup \\set p$. Let $\\tau^*_p$ be the set defined as: :$\\tau^*_{\\bar p} = \\set {U: U \\in \\tau} \\cup \\set {S^*_p}$ That is, $\\tau^*_{\\bar p}$ is the set of all sets formed by taking all the open sets of $\\tau$ and adding to them the set $S^*_p$. Then: :$\\tau^*_{\\bar p}$ is the open extension topology of $\\tau$ and: :$T^*_{\\bar p} = \\struct {S^*_p, \\tau^*_{\\bar p} }$ is the open extension space of $T = \\struct {S, \\tau}$."}
+{"_id": "21719", "title": "Definition:Boubaker Polynomials", "text": "The Boubaker polynomials are the components of the following sequence of polynomials: {{begin-eqn}} {{eqn | l = B_0 \\left({x}\\right)       | r = 1      }} {{eqn | l = B_1 \\left({x}\\right)       | r = x      }} {{eqn | l = B_2 \\left({x}\\right)       | r = x^2 + 2      }} {{eqn | l = B_3 \\left({x}\\right)       | r = x^3 + x      }} {{eqn | l = B_4 \\left({x}\\right)       | r = x^4 - 2      }} {{eqn | l = B_5 \\left({x}\\right)       | r = x^5 - x^3 - 3x      }} {{eqn | l = B_6 \\left({x}\\right)       | r = x^6 - 2x^4 - 3x^2 + 2      }} {{eqn | l = B_7 \\left({x}\\right)       | r = x^7 - 3x^5 - 2x^3 + 5x      }} {{eqn | l = B_8 \\left({x}\\right)       | r = x^8 - 4x^6 + 8x^2 - 2      }} {{eqn | l = B_9 \\left({x}\\right)       | r = x^9 - 5x^7 + 3x^5 + 10x^3 - 7x      }} {{eqn | o = \\vdots}} {{end-eqn}}"}
+{"_id": "21720", "title": "Definition:Supremum Norm", "text": "Let $S$ be a set. Let $\\struct {X, \\norm {\\,\\cdot\\,} }$ be a normed vector space. Let $\\mathcal B$ be the set of bounded mappings $S \\to X$. For $f \\in \\mathcal B$ the '''supremum norm''' of $f$ on $S$ is defined as: :$\\norm f_\\infty = \\sup \\set {\\norm {\\map f x}: x \\in S}$ === Supremum Norm on Continuous on Closed Interval Real-Valued Functions === {{:Definition:Supremum Norm/Continuous on Closed Interval Real-Valued Function}}"}
+{"_id": "21721", "title": "Definition:Normed Vector Space", "text": "Let $\\struct {K, +, \\circ}$ be a normed division ring. Let $V$ be a vector space over $K$. Let $\\norm {\\,\\cdot\\,}$ be a norm on $V$. Then $\\struct {V, \\norm {\\,\\cdot\\,} }$ is a '''normed vector space'''."}
+{"_id": "21722", "title": "Definition:Banach Space", "text": "A '''Banach space''' is a normed vector space where every Cauchy sequence is convergent."}
+{"_id": "21723", "title": "Definition:Contraction Mapping", "text": "Let $\\struct {X, d_1}$ and $\\struct {Y, d_2}$ be metric spaces. Let $f: X \\to Y$ be a mapping. Then $f$ is a '''contraction (mapping)''' {{iff}} there exists $\\kappa \\in \\R: 0 \\le \\kappa < 1$ such that: :$\\forall x, y \\in X: \\map {d_2} {\\map f x, \\map f y} \\le \\kappa \\, \\map {d_1} {x, y}$ That is, $f$ is Lipschitz continuous for a Lipschitz constant less than $1$."}
+{"_id": "21724", "title": "Definition:Type", "text": "Let $\\MM$ be an $\\LL$-structure. Let $A$ be a subset of the universe of $\\MM$. Let $\\LL_A$ be the language consisting of $\\LL$ along with constant symbols for each element of $A$. Viewing $\\MM$ as an $\\LL_A$-structure by interpreting each new constant as the element for which it is named, let $\\map {\\operatorname {Th}_A} \\MM$ be the set of $\\LL_A$-sentences satisfied by $\\MM$. An '''$n$-type over $A$''' is a set $p$ of $\\LL_A$-formulas in $n$ free variables such that $p \\cup \\map {\\operatorname {Th}_A} \\MM$ is satisfiable by some $\\LL_A$-structure. {{Disambiguate|Definition:Logical Formula}} === Complete Type === We say that an $n$-type $p$ is '''complete (over $A$)''' {{iff}}: :for every $\\LL_A$-formula $\\phi$ in $n$ free variables, either $\\phi \\in p$ or $\\phi \\notin p$. The set of '''complete $n$-types over $A$''' is often denoted by $\\map {S_n^\\MM} A$. Given an $n$-tuple $\\bar b$ of elements from $\\MM$, the '''type of $\\bar b$ over $A$''' is the complete $n$-type consisting of those $\\LL_A$-formulas $\\map \\phi {x_1, \\dotsc, x_n}$ such that $\\MM \\models \\map \\phi {\\bar b}$. It is often denoted by $\\map {\\operatorname {tp}^\\MM} {\\bar b / A}$. === Realization === Given an $\\LL_A$-structure $\\NN$, a type $p$ is '''realized''' by an element $\\bar b$ of $\\NN$ {{iff}}: :$\\forall \\phi \\in p: \\NN \\models \\map \\phi {\\bar b}$. Such an element $\\bar b$ of $\\NN$ is a '''realization''' of $p$. === Omission === We say that $\\NN$ '''omits''' $p$ if $p$ is not realized in $\\NN$. Then $p$ is an '''omission''' from $\\NN$."}
+{"_id": "21725", "title": "Definition:Linear Recurrence Relation", "text": "A '''linear recurrence relation''' is a recurrence relation which has the form: :$a_n y_{n+k} + a_{n-1} y_{n+k-1} + \\cdots + a_0 y_k = b \\left({k}\\right)$ where $a_0, \\ldots, a_n$ are constants. === Homogeneous Linear Recurrence Relation === If in a linear recurrence relation $b \\left({k}\\right) = 0$, then the equation is '''homogeneous'''. Otherwise it is '''inhomogeneous'''. {{SUBPAGENAME}} {{SUBPAGENAME}} 3nfpbstuhbph1pt4r5euye1w8jnhmok"}
+{"_id": "21726", "title": "Definition:Elementary Equivalence", "text": "Let $\\mathcal{M},\\mathcal{N}$ be $\\mathcal{L}$-structures. We say that $\\mathcal{M}$ and $\\mathcal{N}$ are '''elementarily equivalent''' if for all $\\mathcal{L}$-sentences $\\phi$, we have $\\mathcal{M}\\models \\phi$ if and only if $\\mathcal{N}\\models \\phi$. {{expand}} {{MissingLinks}} {{NoSources}} Category:Definitions/Model Theory s315nv2o7ka885ufob4r45ebb4ooj2p"}
+{"_id": "21727", "title": "Definition:Embedded", "text": "Let $V$ and $W$ be normed vector spaces. Then $V$ is '''embedded''' in $W$ {{iff}}: : $V \\subseteq W$ The inclusion mapping $i_V : V \\to W$ is an '''embedding''' of $V$ into $W$. Category:Definitions/Functional Analysis dtnujhyil86qa3mx9up04gs50vebour"}
+{"_id": "21728", "title": "Definition:Continuously Embedded", "text": "Let $\\operatorname{id} : V \\to W$ be an embedding of normed vector spaces. Then $V$ is '''continuously embedded''' in $W$ and $\\operatorname{id}$ is a '''continuous embedding''' if the identity $\\operatorname{id}$ is a continuous mapping. {{explain|Link to the specific context in which continuous mapping is defined}} Category:Definitions/Functional Analysis b9o0pp32jce8l9y8csui48ciz2f9aay"}
+{"_id": "21729", "title": "Definition:Excluded Set Topology", "text": "Let $S$ be a set which is non-empty. Let $H \\subseteq S$ be some subset of $S$. We define a subset $\\tau_{\\bar H}$ of the power set $\\powerset S$ as: :$\\tau_{\\bar H} = \\set {A \\subseteq S: A \\cap H = \\O} \\cup \\set S$ that is, all the subsets of $S$ which are disjoint from $H$, along with the set $S$. Then $\\tau_{\\bar H}$ is a topology called the '''excluded set topology on $S$ by $H$''', or just '''an excluded set topology'''."}
+{"_id": "21730", "title": "Definition:Either-Or Topology", "text": "Let $S = \\closedint {-1} 1$ be the closed interval on the real number line from $-1$ to $1$. Let $\\tau \\subseteq \\powerset S$ be a subset of the power set of $S$ such that, for any $H \\subseteq S$: :$H \\in \\tau \\iff \\paren {\\set 0 \\nsubseteq H \\lor \\openint {-1} 1 \\subseteq H}$ where $\\lor$ is the inclusive-or logical connective. Then $\\tau$ is the '''either-or topology''', and $T = \\struct {S, \\tau}$ is the '''either-or space'''"}
+{"_id": "21731", "title": "Definition:Type Space", "text": "Let $\\MM$ be an $\\LL$-structure, and let $A$ be a subset of the universe of $\\MM$. Let $\\map {S_n^\\MM} A$ be the set of complete $n$-types over $A$. The '''space of $n$-types over $A$''' is the topological space formed by the set $\\map {S_n^\\MM} A$ together with the topology arising from the basis which consists of the sets: :$\\sqbrk \\phi := \\set {p \\in \\map {S_n^\\MM} A:\\phi \\in p}$ for each $\\LL_A$-formula $\\phi$ with $n$ free variables. Note that each $\\sqbrk \\phi$ is also closed in this topology, since $\\sqbrk \\phi$ is the complement of $\\sqbrk {\\neg \\phi}$ in $\\map {S_n^\\MM} A$. {{LinkWanted|Definition of the notation $\\sqbrk \\phi$}}"}
+{"_id": "21732", "title": "Definition:Isolated Type", "text": "Let $T$ be an $\\mathcal L$-theory. Let $\\phi \\left({\\bar v}\\right)$ be an $\\mathcal L$-formula in $n$ free variables $\\bar v$ such that $T \\cup \\phi \\left({\\bar v}\\right)$ is satisfiable. {{Disambiguate|Definition:Logical Formula}} Let $p$ be an $n$-type of $T$. We say that $\\phi$ '''isolates''' $p$ if for all $\\psi \\in p$, we have: :$T \\models \\forall \\bar{v} \\left({ \\phi \\left({\\bar v}\\right) \\rightarrow \\psi \\left({\\bar v}\\right) }\\right)$ that is, all $\\psi$ are semantic consequences of $\\phi$. Category:Definitions/Model Theory aufuya5baqc2yul5d7w8w9n3memfcg9"}
+{"_id": "21733", "title": "Definition:Witness Property", "text": "{{MissingLinks}} An $\\LL$-theory $T$ is said to have the '''witness property''' if for every $\\LL$-formula $\\map \\phi v$ with one free variable, there is a constant symbol $c$ in $\\LL$ such that: {{Disambiguate|Definition:Logical Formula}} :$T \\models \\paren {\\exists v \\map \\phi v} \\to \\map \\phi c$ that is, $\\paren {\\exists v \\map \\phi v} \\to \\map \\phi c$ is a semantic consequence of $T$. That is, every existential statement satisfied by $T$ is witnessed by a constant. {{NoSources}} Category:Definitions/Model Theory b2x1fd0kttej9lgwie7mff26hbiu8p0"}
+{"_id": "21734", "title": "Definition:Theory (Logic)", "text": "Let $\\mathcal L$ be a logical language. An '''$\\mathcal L$-theory''' $T$ is a set of $\\mathcal L$-sentences. === Theory of Structure === {{:Definition:Theory (Logic)/Structure}} === Complete Theory === {{:Definition:Theory (Logic)/Complete}} === Maximal Theory === {{:Definition:Theory (Logic)/Maximal}} {{NoSources}} Category:Definitions/Mathematical Logic 6jhezd78oww4rjb0ki3dkmgmgibz9xi"}
+{"_id": "21735", "title": "Definition:Distance to Nearest Integer Function", "text": "The '''distance to nearest integer function''' $\\norm \\cdot: \\R \\to \\closedint 0 {\\dfrac 1 2}$ is defined in the following ways: === Definition 1 === {{:Definition:Distance to Nearest Integer Function/Definition 1}} === Definition 2 === {{:Definition:Distance to Nearest Integer Function/Definition 2}}"}
+{"_id": "21736", "title": "Definition:Stability (Model Theory)", "text": "Let $T$ be a complete $\\mathcal L$-theory where $\\mathcal L$ is countable. Let $\\kappa$ be an infinite cardinal. === $\\kappa$-Stable === {{:Definition:Stability (Model Theory)/Kappa-Stable Theory}} === Stable === {{:Definition:Stability (Model Theory)/Stable Theory}} === Unstable === {{:Definition:Stability (Model Theory)/Unstable Theory}}"}
+{"_id": "21737", "title": "Definition:Finite Complement Topology", "text": "Let $S$ be a set whose cardinality is usually specified as being infinite. Let $\\tau$ be the set of subsets of $S$ defined as: :$H \\in \\tau \\iff \\relcomp S H \\text { is finite, or } H = \\O$ where $\\relcomp S H$ denotes the complement of $H$ relative to $S$. Then $\\tau$ is the '''finite complement topology on $S$''', and the topological space $T = \\struct {S, \\tau}$ is a '''finite complement space'''."}
+{"_id": "21738", "title": "Definition:Ramanujan Sum", "text": "Let $e: \\R \\to \\C$ be the mapping defined as: :$\\forall \\alpha \\in \\R: \\map e \\alpha := \\map \\exp {2 \\pi i \\alpha}$ For $q \\in \\N_{>0}$, $n \\in \\N$, the '''Ramanujan sum''' is defined as: :$\\displaystyle \\map {c_q} n = \\sum_{\\substack {1 \\mathop \\le a \\mathop \\le q \\\\ \\gcd \\set {a, q} \\mathop = 1} } \\map e {\\frac {a n} q}$"}
+{"_id": "21739", "title": "Definition:Countable Complement Topology", "text": "Let $S$ be an infinite set whose cardinality is usually taken to be uncountable. Let $\\tau$ be the set of subsets of $S$ defined as: :$H \\in \\tau \\iff \\relcomp S H$ is countable, or $H = \\O$ where $\\relcomp S H$ denotes the complement of $H$ relative to $S$. In this definition, countable is used in its meaning that includes finite. Then $\\tau$ is the '''countable complement topology on $S$''', and the topological space $T = \\struct {S, \\tau}$ is a '''countable complement space'''."}
+{"_id": "21740", "title": "Definition:Higher-Aleph Complement Topology", "text": "Let $S$ be a set whose cardinality is $\\aleph_n$ where $n > 0$. Let $\\tau \\subseteq \\powerset S$ be the set of subsets of $S$ defined as: :$\\tau = \\set {U \\subseteq S: \\size {\\relcomp S U} = \\aleph_m: m < n} \\cup \\set {U \\subseteq S: \\relcomp S U \\text { is finite} } \\cup \\O$ That is, $\\tau$ is the set of subsets of $S$ whose complements relative to $S$ are of a cardinality strictly less than $S$. Then $\\tau$ is an '''$\\aleph_m$ complement topology on $S$''', and the topological space $T = \\struct {S, \\tau}$ is an '''$\\aleph_m$ complement space'''. This construction is an extension of the concept of the finite complement topology and the countable complement topology."}
+{"_id": "21741", "title": "Definition:Saturated Model", "text": "Let $T$ be an $\\mathcal{L}$-theory. Let $\\kappa$ be an infinite cardinal. A model $\\mathcal{M}$ of $T$ is $\\kappa$-'''saturated''' if for every subset $A$ of the  universe of $\\mathcal{M}$ of cardinality strictly less than $\\kappa$, and for every  $n\\in\\mathbb N$, every complete $n$-type $p$ over $A$ is realized in $\\mathcal{M}$. That is, $\\mathcal{M}$ is $\\kappa$-saturated if for all $A  \\subseteq \\mathcal{M}$ with $A<\\kappa$, and for all $n\\in\\mathbb N$, each $p \\in S_{n}^{\\mathcal{M}} (A)$ is realized in $\\mathcal{M}$. We say $\\mathcal{M}$ is '''saturated''' if it is $\\kappa$-saturated where $\\kappa$ is the cardinality of the universe of $\\mathcal{M}$. {{SUBPAGENAME}} 1lsodr3oexq9bc7rhx6nzqgdth5h3hq"}
+{"_id": "21742", "title": "Definition:Universal Model", "text": "Let $T$ be an $\\mathcal L$-theory. Let $\\kappa$ be an infinite cardinal. A model $\\mathcal M$ of $T$ is $\\kappa$-'''universal''' {{iff}}: :for every model $\\mathcal N$ of $T$ whose universe has cardinality strictly less than $\\kappa$, there is an elementary embedding of $\\mathcal N$ into $\\mathcal M$. That is, $\\mathcal M$ is '''$\\kappa$-universal''' {{iff}}: :for all models $\\mathcal N \\models T$ with  cardinality $\\left\\vert{\\mathcal N}\\right\\vert < \\kappa$, there is an elementary embedding $j: \\mathcal N \\to \\mathcal M$. We say $\\mathcal M$ is '''universal''' if it is '''$\\kappa^+$-universal''' where $\\kappa$ is the cardinality of the universe of $\\mathcal M$ and $\\kappa^+$ is the successor cardinal of $\\kappa$. Definitions/Model Theory bklciljojrl7we8ux5qi6os3dtoh5jr"}
+{"_id": "21743", "title": "Definition:Categorical (Model Theory)", "text": "Let $T$ be an $\\mathcal{L}$-theory. Let $\\kappa$ be a cardinal. $T$ is $\\kappa$-'''categorical''' if whenever $\\mathcal{M}$ and $\\mathcal{N}$ are models of $T$ of cardinality $\\kappa$, then $\\mathcal{M}$ and $\\mathcal{N}$ are isomorphic."}
+{"_id": "21744", "title": "Definition:Compact Complement Topology", "text": "Let $T = \\struct {\\R, \\tau}$ be the real number line with the usual (Euclidean) topology. Let $\\tau^*$ be the set defined as: :$\\tau^* = \\leftset {S \\subseteq \\R: S = \\O \\text { or } \\relcomp \\R S}$ is compact in $\\rightset {\\struct {\\R, \\tau} }$ where $\\relcomp \\R S$ denotes the complement of $S$ in $\\R$. Then $\\tau^*$ is the '''compact complement topology''' on $\\R$, and $T^* = \\struct {\\R, \\tau^*}$ is the '''compact complement space''' on $\\R$."}
+{"_id": "21745", "title": "Definition:Graph (Category Theory)", "text": "A '''graph''' is an interpretation of a metagraph within set theory. Let $\\mathfrak U$ be a class of sets. A metagraph $\\mathcal G$ is a '''graph''' if: : $(1): \\quad$ The objects form a subset $\\operatorname{vert}\\mathcal G \\subseteq \\mathfrak U$ : $(2): \\quad$ The morphisms form a subset $\\operatorname{edge}\\mathcal G \\subseteq \\mathfrak U$ If the class $\\mathfrak U$ is a set, then morphisms are functions, and the domain and codomain in the definition of a morphism are those familiar from set theory. If $\\mathfrak U$ is a proper class this is not the case, for example the morphisms of $\\mathcal C$ need not be functions. Category:Definitions/Category Theory 145b347zy1dfv7fjzlb3t9u0j2nel1e"}
+{"_id": "21746", "title": "Definition:Indiscernible", "text": "Let $\\mathcal M$ be an $\\mathcal L$-structure. Let $I$ be an infinite set. Let $X = \\left\\{ {x_i \\in \\mathcal M: i \\in I}\\right\\}$ be an infinite subset of the universe of $\\mathcal M$ indexed by $I$. For: : every $n \\in \\N$ and: : every pair of subsets $\\left\\{ {i_1, \\ldots, i_n}\\right\\}$ and $\\left\\{ {j_1, \\ldots, j_n}\\right\\}$ of $I$ each with $n$ distinct elements, let: : $\\mathcal M \\models \\phi \\left({x_{i_1}, \\ldots, x_{i_n} }\\right) \\leftrightarrow \\phi \\left({x_{j_1}, \\ldots, x_{j_n} }\\right)$ for all $\\mathcal L$-formulas $\\phi$ with $n$ free variables. Then $X$ is '''(an) indiscernible (set) in $\\mathcal M$'''. {{Disambiguate|Definition:Logical Formula}} Informally, $X$ is '''indiscernible (set)''' if $\\mathcal M$ cannot distinguish between same-sized ordered tuples over $X$ using $\\mathcal L$-formulas."}
+{"_id": "21747", "title": "Definition:Order Indiscernible", "text": "Let $\\mathcal M$ be an $\\mathcal L$-structure. Let $\\left({I, \\le}\\right)$ be an infinite ordered set. Let $X = \\left\\{ {x_i \\in \\mathcal M: i \\in I}\\right\\}$ be an infinite subset of the universe of $\\mathcal M$ indexed by $I$. Let $A$ be a subset of the universe of $\\mathcal M$. $X$ is '''(an) order indiscernible (set) over $A$ in $\\mathcal M$''' {{iff}}: : For every $n \\in \\N$ and every pair of chains $i_1 < \\cdots < i_n$ and $j_1 < \\cdots < j_n$ in $I$ each with $n$ distinct elements, we have: :: $\\mathcal M \\models \\phi \\left({x_{i_1}, \\ldots, x_{i_n} }\\right) \\iff \\phi \\left({x_{j_1}, \\ldots, x_{j_n} }\\right)$ : for all $\\mathcal L$-formulas $\\phi$ with $n$ free variables and parameters from $A$. {{Disambiguate|Definition:Logical Formula}} Informally, $X$ is '''order indiscernible''' {{iff}} $\\mathcal M$ cannot distinguish between same-sized increasing ordered tuples over $X$ using $\\mathcal L$-formulas."}
+{"_id": "21748", "title": "Definition:Fort Space", "text": "Let $S$ be an infinite set. Let $p \\in S$ be a particular point of $S$. Let $\\tau_p \\subseteq \\powerset S$ be a subset of the power set of $S$ defined as: :$\\tau_p = \\leftset {U \\subseteq S: p \\in \\relcomp S U} \\text { or } \\set {U \\subseteq S: \\relcomp S U}$ is finite$\\rightset{}$ That is, $\\tau_p$ is the set of all subsets of $S$ whose complement in $S$ either contains $p$ or is finite. Then $\\tau_p$ is a '''Fort topology''' on $S$, and the topological space $T = \\struct {S, \\tau_p}$ is a '''Fort space'''."}
+{"_id": "21749", "title": "Definition:Gödel-Bernays Axioms", "text": "The '''Gödel-Bernays axioms''' are a conservative extension of the Zermelo-Fraenkel axioms with the axiom of choice (ZFC) that allow comprehension of classes. Although not the standard axioms of set theory, particularly in category theory they spare us any set of all sets-type paradoxes. == Axioms for Sets == The first five axioms are identical to the axioms of the same names from ZFC. The quantified variables range over the universe of sets. === The Axiom of Extension === {{:Axiom:Axiom of Extension/Set Theory}} === The Axiom of Pairing === {{:Axiom:Axiom of Pairing/Set Theory}} === The Axiom of Unions === {{:Axiom:Axiom of Unions/Set Theory}} === The Axiom of Powers === {{:Axiom:Axiom of Powers/Set Theory}} === The Axiom of Infinity === {{:Axiom:Axiom of Infinity/Set Theory}} == Axioms for Classes == In the remaining axioms, the quantified variables range over classes. The first two differ from the ZFC axioms with the same names in this way only. The last two have no analogue among the ZFC axioms. === The Axiom of Extension === {{:Axiom:Axiom of Extension (Classes)}} === The Axiom of Foundation === {{:Axiom:Axiom of Foundation (Classes)}} === Class Comprehension === {{:Axiom:Class Comprehension Schema}} === The Axiom of Limitation of Size === {{:Axiom:Axiom of Limitation of Size}}"}
+{"_id": "21750", "title": "Definition:Class (Class Theory)", "text": "A '''class''' is a collection of all sets such that a particular condition holds. In class builder notation, this is written as: :$\\set {x: \\map p x}$ where $\\map p x$ is a statement containing $x$ as a free variable.   This is read: :'''All $x$ such that $\\map p x$ holds.'''"}
+{"_id": "21751", "title": "Definition:Metagraph", "text": "A '''metagraph''' $\\mathcal G$ consists of: * objects $X, Y, Z, \\ldots$ * morphisms $f, g, h, \\ldots$ between its objects These are subjected to the following two axioms: {{begin-axiom}} {{axiom | n = 1         | lc= ''Domains''         | t = Every morphism $f$ has associated an object $\\operatorname{dom} f$, called the '''domain''' of $f$ }} {{axiom | n = 2         | lc= ''Codomains''         | t = Every morphism $f$ has associated an object $\\operatorname{cod} f$, called the '''codomain''' of $f$ }} {{end-axiom}} A '''metagraph''' is purely axiomatic, and does not use set theory. For example, the objects are not \"elements of the set of objects\", because these axioms are (without further interpretation) unfounded in set theory."}
+{"_id": "21752", "title": "Definition:Path (Category Theory)", "text": "Let $\\mathcal G$ be a graph. A (non-empty) '''path''' in $\\mathcal G$ is a sequence $(i,a_1,\\ldots,a_n,j)$ such that: *$n > 1$ *$a_1,\\ldots, a_n$ are edges of $\\mathcal G$ *$i$ is the source of $a_1$ and $j$ is the destination of $a_n$ *For $1 \\leq k < n$ the destination of $a_k$ is the source of $a_{k+1}$ This is usually written in the form: :$\\displaystyle i \\stackrel{a_1}{\\longrightarrow} * \\stackrel{a_2}{\\longrightarrow} \\cdots \\stackrel{a_{n-1}}{\\longrightarrow} * \\stackrel{a_n}{\\longrightarrow} j$ For any vertex $i$ the '''empty path''' from $i$ to $i$ is the pair $(i,i)$. Paths are '''composed''' by concatenation: :$\\displaystyle (j,b_1,\\ldots,b_n,k)(i,a_1,\\ldots,a_m,j) = (i,a_1,\\ldots,a_m,b_1,\\ldots,b_n,k)$ Here we have used the somewhat awkward but more common right-to-left notation for composition. {{SUBPAGENAME}} mtbopkxz4y7tctz5weuq28ngbss9blw"}
+{"_id": "21753", "title": "Definition:Minimal Topology", "text": "Let $S$ be a set and let $\\mathcal P \\left({S}\\right)$ be the power set of $S$. Let $\\Theta_S$ be the set of all topologies on $S$: :$\\Theta_S = \\left\\{{\\tau \\in \\mathcal P \\left({S}\\right): \\tau}\\right.$ is a topology on $\\left.{S}\\right\\}$ Let $\\Phi: \\Theta_S \\to \\left\\{{T, F}\\right\\}$ be a propositional function on $\\Theta_S$. Let $\\vartheta \\in \\Theta_S$ have the property that $\\Phi \\left({\\vartheta}\\right)$ and: :$\\forall \\tau \\in \\Theta_S: \\Phi \\left({\\tau}\\right) \\implies \\vartheta \\subseteq \\tau$ That is, $\\vartheta$ is the coarsest topology on $S$ which satisfies the propositional function $\\Phi$. Then $\\vartheta$ is the '''minimal topology satisfying $\\Phi$'''. {{SUBPAGENAME}} htipkwppnixkuhckesep3patmjqucql"}
+{"_id": "21754", "title": "Definition:Minimal (Model Theory)", "text": "Let $\\mathcal M$ be an $\\mathcal L$-structure. Let $M$ be the universe of $\\mathcal M$. Let $A$ be a subset of $M$. Let $D \\subseteq M^n$ be an infinite $A$-definable set. Let $\\phi \\left({\\bar x, \\bar a}\\right)$ be an $\\mathcal L$-formula with parameters $\\bar a$ from $A\\subseteq M$ and  free variables $\\bar x$ which defines $D$. {{Disambiguate|Definition:Logical Formula}} $D$ is '''minimal''' in $\\mathcal M$ {{iff}} every definable subset of $D$ is either finite or cofinite. $\\phi$ is '''minimal''' in $\\mathcal M$ if $D$ is minimal in $\\mathcal M$. $D$ and $\\phi$ are '''strongly minimal''' in $\\mathcal M$ {{iff}} $\\phi$ is minimal in any elementary extension $\\mathcal N$ of $\\mathcal M$. An $\\mathcal L$-theory $T$ is '''strongly minimal''' if for every model $\\mathcal N$ of $T$ with universe $N$, the set $N$ is strongly minimal. {{NoSources}} Category:Definitions/Model Theory q7voaf1drllxp3qsiprs092mvfdaebc"}
+{"_id": "21755", "title": "Definition:Cofinite Set", "text": "Let $S$ be a set. Let $A$ be a subset of $S$. $A$ is '''cofinite in $S$''' {{iff}} the complement of $A$ in $S$ is finite. Category:Definitions/Set Theory m3q8cemkegb05gdyzr23zyx6e9whs5q"}
+{"_id": "21756", "title": "Definition:Fortissimo Space", "text": "Let $S$ be an uncountably infinite set. Let $p \\in S$ be a particular point of $S$. Let $\\tau_p \\subseteq \\powerset S$ be a subset of the power set of $S$ defined as: :$\\tau_p = \\leftset {U \\subseteq S: p \\in \\relcomp S U}$ or $\\set {U \\subseteq S: \\relcomp S U}$ is countable (either finitely or infinitely)$\\rightset {}$ That is, $\\tau_p$ is the set of all subsets of $S$ whose complement in $S$ either contains $p$ or is countable. Then $\\tau_p$ is a '''Fortissimo topology''' on $S$, and the topological space $T = \\struct {S, \\tau_p}$ is a '''Fortissimo space'''."}
+{"_id": "21757", "title": "Definition:Divide (Model Theory)", "text": "Let $T$ be a complete $\\mathcal L$-theory. Let $\\mathfrak C$ be a monster model for $T$. Let $A$ be a subset of the universe of $\\mathfrak C$. Let $\\phi \\left({\\bar x, \\bar b}\\right)$ be an $\\mathcal L$-formula with free variables $\\bar x$ and parameters $\\bar b$ from the universe of $\\mathfrak C$. {{Disambiguate|Definition:Logical Formula}} Let $\\operatorname{tp} \\left({\\bar b / A}\\right)$ denote the type of $\\bar b$ over $A$. Then $\\phi \\left({\\bar x, \\bar b}\\right)$ '''$k$-divides in $\\mathfrak C$ over $A$''' {{iff}}: :there exists a sequence $\\left\\langle{\\bar b_i}\\right\\rangle_{i \\mathop \\in \\N}$ such that $\\operatorname{tp} \\left({\\bar b_i / A}\\right) = \\operatorname{tp} \\left({\\bar b / A}\\right)$ for each $i \\in \\N$ and: :for any distinct $k$-many terms $\\bar b_{i_1}, \\ldots, \\bar b_{i_k}$ of the sequence, the set $\\left\\{{\\phi \\left({\\bar x, \\bar b_{i_1} }\\right), \\ldots, \\phi \\left({\\bar x, \\bar b_{i_k} }\\right)}\\right\\}$ is not satisfiable in $\\mathfrak C$. Let $\\pi \\left({\\bar x, \\bar b}\\right)$ be a set of formulas with parameters $\\bar b$. $\\pi \\left({\\bar x, \\bar b}\\right)$ '''$k$-divides''' over $A$ if it implies a formula $\\phi \\left({x, \\bar c}\\right)$ which $k$-divides over $A$. Formulas and sets of formulas are said to '''divide''' if they $k$-divide for some $k$."}
+{"_id": "21758", "title": "Definition:Fork", "text": "Let $T$ be a complete $\\mathcal{L}$-theory. Let $\\mathfrak{C}$ be a monster model for $T$. Let $A$ be a subset of the universe of $\\mathfrak{C}$. Let  $\\pi (\\bar{x}, \\bar{b})$ be a set of $\\mathcal{L}$-formulas with free variables $\\bar{x}$ and parameters $\\bar{b}$  from the universe of $\\mathfrak{C}$. {{Disambiguate|Definition:Logical Formula}} $\\pi(\\bar{x}, \\bar{b})$ '''forks''' over $A$ if it implies some disjunction $\\phi_1 (\\bar x, \\bar c_1) \\vee \\cdots \\vee \\phi_n (\\bar x, \\bar c_n)$ where each $\\phi_i (\\bar x, \\bar c_i)$ divides over $A$. An individual formula $\\phi (\\bar x, \\bar b)$ is said to '''fork''' if the singleton $\\{\\phi (\\bar x, \\bar b)\\}$ forks."}
+{"_id": "21759", "title": "Definition:Forking Extension", "text": "Let $T$ be a complete $\\mathcal{L}$-theory. Let $\\mathfrak{C}$ be a monster model for $T$. Let $A\\subseteq B$ be subsets of the universe of $\\mathfrak{C}$. Let $p(\\bar x)$ be a complete $n$-type over $B$. Denote by $p\\restriction A$ the subset of $p$ consisting of those formulas which involve only parameters from $A$. $p$ is a '''non-forking extension''' of $p\\restriction A$ if $p(\\bar x)$ does not fork over $A$. {{SUBPAGENAME}} gdkrozd2nkd7dv0v00jxtvtxhtzt815"}
+{"_id": "21760", "title": "Definition:Monster Model", "text": "Let $T$ be a complete $\\mathcal L$-theory. A '''monster model''' of $T$ whose cardinality $\\kappa$ is infinite is a model of $T$ which is saturated and homogeneous."}
+{"_id": "21762", "title": "Definition:Finite Set", "text": "A set $S$ is defined as '''finite''' {{iff}}: :$\\exists n \\in \\N: S \\sim \\N_{<n}$ where $\\sim$ denotes set equivalence. That is, if there exists an element $n$ of the set of natural numbers $\\N$ such that the set of all elements of $\\N$ less than $n$ is equivalent to $S$. Equivalently, a '''finite set''' is a set with a count."}
+{"_id": "21763", "title": "Definition:Embedding (Model Theory)", "text": "Let $\\MM$ and $\\NN$ be $\\LL$-structures with universes $M$ and $N$ respectively. $j: \\MM \\to \\NN$ is an '''$\\LL$-embedding''' {{iff}} it is an injective map $M \\to N$ which preserves interpretations of all symbols in $\\LL$; that is, such that: :$\\map j {\\map {f^\\MM} {a_1, \\dots, a_{n_f} } } = \\map {f^\\NN} {\\map j {a_1}, \\ldots, \\map j {a_{n_f} } }$ for all function symbols $f$ in $\\LL$ and $a_1, \\dots, a_{n_f}$ in $M$ :$\\tuple {a_1, \\ldots, a_{n_R} } \\in R^\\MM \\iff \\tuple {\\map j {a_1}, \\dots, \\map j {a_{n_R} } } \\in R^\\NN$ for all relation symbols $R$ in $\\LL$ and $a_1, \\dots, a_{n_R}$ in $M$ :$\\map j {c^\\MM} = c^\\NN$ for all constant symbols $c$ in $\\LL$. === Partial Embedding === A common method of constructing isomorphisms and elementary embeddings in proofs is to recursively define them a finite number of elements at a time.  For this  purpose, it is useful to have a definition of embeddings using functions which are only defined on a subset of $M$: Let $A \\subseteq M$ be a subset of $M$. $j: A \\to \\NN$ is a '''partial $\\LL$-embedding''' {{iff}} it is an injective map $A \\to N$ which preserves interpretations of all symbols in $\\LL$ applied to elements of $A$; that is, such that: :$\\map j {\\map {f^\\MM} {a_1, \\dots, a_{n_f} } } = \\map {f^\\NN} {\\map j {a_1}, \\ldots, \\map j {a_{n_f} } }$ for  all function symbols $f$ in $\\LL$ and $a_1, \\dots, a_{n_f}$ in $A$ :$\\tuple {a_1, \\ldots, a_{n_R} } \\in R^\\MM \\iff \\tuple {\\map j {a_1}, \\dots, \\map j {a_{n_R} } } \\in R^\\NN$ for all relation symbols $R$ in $\\LL$ and $a_1, \\dots, a_{n_R}$ in $A$ :$\\map j {c^\\MM} = c^\\NN$ for all constant symbols $c$ in $\\LL$. === Isomorphism === $j: \\MM \\to \\NN$ is an '''$\\LL$-isomorphism''' {{iff}} it is a bijective $\\LL$-embedding. === Automorphism === $j: \\MM \\to \\NN$ is an '''$\\LL$-automorphism''' {{iff}} it is an $\\LL$-isomorphism and $\\MM = \\NN$. It is often useful to talk about automorphisms which are constant on subsets of $M$.  So, there is a definition and a notation for doing so: Let $A \\subseteq M$ be a subset of $M$, and let $b \\in M$. An $\\LL$-automorphism $j$ is an '''$A$-automorphism''' {{iff}} $\\map j a = a$ for all $a\\in A$. An $\\LL$-automorphism $j$ is an '''$A, b$-automorphism''' {{iff}} it is an $\\paren {A \\cup \\set b}$-automorphism; that is: $\\map j a = a$ for all $a \\in A$ and also $\\map j b = b$. Category:Definitions/Model Theory ph5f7arezx633z7f7ypn6jyva7grsru"}
+{"_id": "21764", "title": "Definition:Homogeneous (Model Theory)", "text": "Let $T$ be an $\\mathcal{L}$-theory. Let $\\kappa$ be an infinite cardinal. A model $\\mathcal{M}$ of $T$ is $\\kappa$-'''homogeneous''' if for every subset $A$ and element $b$ in the universe of $\\mathcal{M}$ with the cardinality of $A$ strictly less than $\\kappa$, if $f:A\\to \\mathcal{M}$ is partial elementary, then $f$ extends to an elementary map $f^*: A\\cup\\{b\\} \\to \\mathcal{M}$. That  is, $\\mathcal{M}$ is $\\kappa$-homogeneous if for all $A  \\subseteq \\mathcal{M}$ with  $A<\\kappa$ and all $b\\in \\mathcal{M}$, every elementary $f:A\\to \\mathcal{M}$ extends to an elementary $f^*:A\\cup\\{b\\}\\to\\mathcal{M}$. We say $\\mathcal{M}$ is  '''homogeneous''' if it is $\\kappa$-homogeneous where $\\kappa$ is the cardinality of the universe of $\\mathcal{M}$."}
+{"_id": "21766", "title": "Definition:Arens-Fort Space", "text": "Let $S$ be the set $\\Z_{\\ge 0} \\times \\Z_{\\ge 0}$ be the Cartesian product of the set of positive integers $\\Z_{\\ge 0}$: :$S = \\set {0, 1, 2, \\ldots} \\times \\set {0, 1, 2, \\ldots}$ Let $\\tau \\subseteq \\powerset S$ be a subset of the power set of $S$ such that: :$(1): \\quad \\forall H \\subseteq S: \\tuple {0, 0} \\notin H \\implies H \\in \\tau$ :$(2): \\quad H \\subseteq S: \\tuple {0, 0} \\in H$ and, for all but a finite number of $m \\in \\Z_{\\ge 0}$, the sets $S_m$ defined as: ::::$S_m = \\set {n: \\tuple {m, n} \\notin H}$ :::are finite. That is, $H$ is allowed to be in $\\tau$ if, considering $S = \\Z_{\\ge 0} \\times \\Z_{\\ge 0}$ as the lattice points of the first quadrant of a Cartesian plane: Either: :$H$ does not contain $\\tuple {0, 0}$ :$H$ contains $\\tuple {0, 0}$, and only a finite number of the columns of $S$ are allowed to omit an infinite number of points in $H$. Then $\\tau$ is the '''Arens-Fort topology''' on $S = \\Z_{\\ge 0} \\times \\Z_{\\ge 0}$, and the topological space $T = \\struct {S, \\tau}$ is the '''Arens-Fort space'''."}
+{"_id": "21767", "title": "Definition:Elementary Embedding", "text": "Let $\\mathcal{M}$ and $\\mathcal{N}$ be $\\mathcal{L}$-structures with universes $M$ and $N$ respectively. An $\\mathcal{L}$-embedding $j:\\mathcal{M}\\to \\mathcal{N}$ is an '''elementary embedding''' {{iff}} it preserves truth; that is: :$\\mathcal{M} \\models \\phi(a_1,\\dots, a_n) \\iff \\mathcal{N} \\models \\phi(j(a_1),\\dots, j(a_n))$ holds for all $n\\in\\mathbb N$, all $\\mathcal{L}$-formulas $\\phi$ with $n$ free variables, and for all $a_1,\\dots,a_n \\in M$. === Partial Elementary Embedding === {{:Definition:Elementary Embedding/Partial Elementary Embedding}} {{NoSources}} Category:Definitions/Model Theory 8i39p8d5esovaf6leawjz7ytpscjl32"}
+{"_id": "21768", "title": "Definition:Modified Fort Space", "text": "Let $N$ be an infinite set. Let $\\set a$ and $\\set b$ be singleton sets such that $a \\ne b$ and $a, b \\notin N$. Let $S = N \\cup \\set a \\cup \\set b$. Let $\\tau_{a, b}$ be the set of subsets of $S$ defined as: :$\\tau_{a, b} = \\set {H \\subseteq N} \\cup \\set {H \\subseteq S: \\paren {a \\in H \\lor b \\in H} \\land N \\setminus H \\text { is finite} }$ That is, a subset $H$ of $S$ is in $\\tau_{a, b}$ {{iff}} either: :$(1): \\quad H$ is any subset of $N$ or: :$(2): \\quad$ if $a$ or $b$ or both are in $H$, then $H$ is in $S$ only if it is cofinite in $S$, that is, that it contains all but a finite number of points of $S$ (or $N$, equivalently). Then $\\tau_{a, b}$ is a '''modified Fort topology''' on $a$ and $b$, and the topological space $T = \\struct {S, \\tau_{a, b} }$ is a '''modified Fort space'''."}
+{"_id": "21769", "title": "Definition:Algebraic (Model Theory)", "text": "Let $\\mathcal{M}$ be an $\\mathcal{L}$-structure with universe $M$. Let $A$ be a subset of $M$. and let $\\bar b$ be an ordered $n$-tuple of elements from $M$. Let $\\mathcal{L}_A$ be the language formed by adding constant symbols to $\\mathcal{L}$ for each element of $A$. $\\bar b$ is '''algebraic''' over $A$ if there is an $\\mathcal{L}_A$-formula $\\phi(\\bar x)$ with $n$ free variables such that $\\mathcal{M}\\models \\phi(\\bar b)$ and the set $\\{\\bar m \\in M^n : \\mathcal{M}\\models \\phi(\\bar m)\\}$ has only finitely many elements. {{Disambiguate|Definition:Logical Formula}}"}
+{"_id": "21771", "title": "Definition:Open Point", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $p \\in S$ such that $\\left\\{{p}\\right\\} \\in \\tau$. That is, let $\\left\\{{p}\\right\\}$ be an open set of $T$. Then $p$ is an open point of $T$. {{SUBPAGENAME}} rs9ypa7mfatx7czrg2prm3peck65rh5"}
+{"_id": "21772", "title": "Definition:Age (Model Theory)", "text": "Let $\\MM$ be an $\\LL$-structure. An '''age''' of $\\MM$ is a class $K$ of $\\LL$-structures such that: * if $\\AA$ is a finitely generated $\\LL$-structure such that there is an $\\LL$-embedding $\\AA \\to \\MM$, then $\\AA$ is isomorphic to some structure in $K$, * no two structures in $K$ are isomorphic, and * $K$ does not contain any structures which are not finitely generated or do not embed into $\\MM$. That is, $K$ is an age of $\\MM$ if it contains exactly one representative from each isomorphism type of the finitely-generated structures that embed into $\\MM$. {{MissingLinks|finitely generated structure}} Category:Definitions/Model Theory 5hf61yvyjc9bw34ihvj4d6x21weg2x6"}
+{"_id": "21773", "title": "Definition:Isolated Point (Metric Space)", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. === Isolated Point in Subset === {{:Definition:Isolated Point (Metric Space)/Subset}} === Isolated Point in Space === When $S = A$ this reduces to: {{:Definition:Isolated Point (Metric Space)/Space}}"}
+{"_id": "21774", "title": "Definition:Isolated Point", "text": "=== Topology === {{:Definition:Isolated Point (Topology)/Space}} === Metric Space === {{:Definition:Isolated Point (Metric Space)/Space}} === Complex Analysis === {{:Definition:Isolated Point (Complex Analysis)}} === Real Analysis === {{:Definition:Isolated Point (Real Analysis)}} Category:Definitions/Topology Category:Definitions/Isolated Points ij9snn0nz9qu0psws311i158x6bnib9"}
+{"_id": "21775", "title": "Definition:Isolated Point (Real Analysis)", "text": "Let $S \\subseteq \\R$ be a subset of the set of real numbers. Let $\\alpha \\in S$. Then $\\alpha$ is an '''isolated point of $S$''' {{iff}} there exists an open interval of $\\R$ whose midpoint is $\\alpha$ which contains no points of $S$ except $\\alpha$: :$\\exists \\epsilon \\in \\R_{>0}: \\openint {\\alpha - \\epsilon} {\\alpha + \\epsilon} \\cap S = \\set \\alpha$"}
+{"_id": "21776", "title": "Definition:Limit Point/Topology/Set", "text": "Let $A \\subseteq S$. ==== Definition from Open Neighborhood ==== {{:Definition:Limit Point/Topology/Set/Definition 1}} ==== Definition from Closure ==== {{:Definition:Limit Point/Topology/Set/Definition 2}} ==== Definition from Adherent Point ==== {{:Definition:Limit Point/Topology/Set/Definition 3}} ==== Definition from Relative Complement ==== {{:Definition:Limit Point/Topology/Set/Definition 4}}"}
+{"_id": "21777", "title": "Definition:Limit Point/Complex Analysis", "text": "Let $S \\subseteq \\C$ be a subset of the set of complex numbers. Let $z_0 \\in \\C$.  Let $\\map {N_\\epsilon} {z_0}$ be the $\\epsilon$-neighborhood of $z_0$ for a given $\\epsilon \\in \\R$ such that $\\epsilon > 0$. Then $z_0$ is a '''limit point of $S$''' {{iff}} ''every'' deleted $\\epsilon$-neighborhood $\\map {N_\\epsilon} {z_0} \\setminus \\set {z_0}$ of $z_0$ contains a point in $S$: :$\\forall \\epsilon \\in \\R_{>0}: \\paren {\\map {N_\\epsilon} {z_0} \\setminus \\set {z_0} } \\cap S \\ne \\O$ that is: :$\\forall \\epsilon \\in \\R_{>0}: \\set {z \\in S: 0 < \\cmod {z - z_0} < \\epsilon} \\ne \\O$"}
+{"_id": "21778", "title": "Definition:Limit Point/Metric Space", "text": "Let $M = \\struct {S, d}$ be a metric space. Let $A \\subseteq S$ be a subset of $S$. Let $\\alpha \\in S$.  Then $\\alpha$ is a '''limit point of $A$''' {{iff}} ''every'' deleted $\\epsilon$-neighborhood $\\map {B_\\epsilon} \\alpha \\setminus \\set \\alpha$ of $\\alpha$ contains a point in $A$: :$\\forall \\epsilon \\in \\R_{>0}: \\map {B_\\epsilon} \\alpha \\setminus \\set \\alpha \\cap A \\ne \\O$ that is: :$\\forall \\epsilon \\in \\R_{>0}: \\set {x \\in A: 0 < \\map d {x, \\alpha} < \\epsilon} \\ne \\O$ Note that $\\alpha$ does not have to be an element of $A$ to be a '''limit point'''."}
+{"_id": "21779", "title": "Definition:Deleted Neighborhood/Topology", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $x \\in S$. Let $V \\subseteq S$ be a neighborhood of $x$. Then $V \\setminus \\left\\{{x}\\right\\}$ is called a '''deleted neighborhood of $x$'''. That is, it is a neighborhood of $x$ with $x$ itself ''removed''."}
+{"_id": "21780", "title": "Definition:Deleted Neighborhood/Metric Space", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $x \\in A$. Let $B_\\epsilon \\left({x}\\right)$ be the open $\\epsilon$-ball neighborhood of $x$. Then the '''deleted $\\epsilon$-neighborhood''' of $x$ is defined as $B_\\epsilon \\left({x}\\right) \\setminus \\left\\{{x}\\right\\}$. That is, it is the open $\\epsilon$-ball neighborhood of $x$ with $x$ itself ''removed''. It can also be defined as: : $\\left\\{{y \\in A: 0 < d \\left({x, y}\\right) < \\epsilon}\\right\\}$"}
+{"_id": "21781", "title": "Definition:Deleted Neighborhood/Complex Analysis", "text": "Let $z_0 \\in \\C$ be a point in the complex plane. Let $\\map {N_\\epsilon} {z_0}$ be the $\\epsilon$-neighborhood of $z_0$. Then the '''deleted $\\epsilon$-neighborhood''' of $z_0$ is defined as $\\map {N_\\epsilon} {z_0} \\setminus \\set {z_0}$. That is, it is the $\\epsilon$-neighborhood of $z_0$ with $z_0$ itself ''removed''. It can also be defined as: :$\\map {N_\\epsilon} {z_0} \\setminus \\set {z_0} : = \\set {z \\in A: 0 < \\cmod {z_0 - z} < \\epsilon}$ from the definition of $\\epsilon$-neighborhood."}
+{"_id": "21782", "title": "Definition:Deleted Neighborhood/Real Analysis", "text": "Let $\\alpha \\in \\R$ be a real number. Let $N_\\epsilon \\left({\\alpha}\\right)$ be the $\\epsilon$-neighborhood of $\\alpha$: :$N_\\epsilon \\left({\\alpha}\\right) := \\left({\\alpha - \\epsilon \\,.\\,.\\, \\alpha + \\epsilon}\\right)$ Then the '''deleted $\\epsilon$-neighborhood''' of $\\alpha$ is defined as $N_\\epsilon \\left({\\alpha}\\right) \\setminus \\left\\{{\\alpha}\\right\\}$. That is, it is the $\\epsilon$-neighborhood of $\\alpha$ with $\\alpha$ itself ''removed''. It can also be defined as: : $N_\\epsilon \\left({\\alpha}\\right) \\setminus \\left\\{{\\alpha}\\right\\} : = \\left\\{{x \\in \\R: 0 < \\left \\vert{\\alpha - x}\\right \\vert < \\epsilon}\\right\\}$ or : $N_\\epsilon \\left({\\alpha}\\right) \\setminus \\left\\{{\\alpha}\\right\\} : = \\left({\\alpha - \\epsilon \\,.\\,.\\, \\alpha}\\right) \\cup \\left({\\alpha \\,.\\,.\\, \\alpha + \\epsilon}\\right)$ from the definition of $\\epsilon$-neighborhood."}
+{"_id": "21783", "title": "Definition:Isolated Point (Complex Analysis)", "text": "Let $S \\subseteq \\C$ be a subset of the set of real numbers. Let $z \\in S$. Then $z$ is an '''isolated point of $S$''' {{iff}} there exists a neighborhood of $z$ in $\\C$ which contains no points of $S$ except $z$: :$\\exists \\epsilon \\in \\R_{>0}: \\map {N_\\epsilon} z \\cap S = \\set z$"}
+{"_id": "21784", "title": "Definition:Lipschitz Quaternion", "text": "A '''Lipschitz quaternion''' is a quaternion whose components are all integers. The set $\\Bbb L$ of all '''Lipschitz quaternions''' can therefore be defined as: :$\\Bbb L = \\left\\{{a + b \\mathbf i + c \\mathbf j + d \\mathbf k \\in \\Bbb H: a, b, c, d \\in \\Z}\\right\\}$ {{NamedforDef|Rudolf Lipschitz|cat=Lipschitz}}"}
+{"_id": "21785", "title": "Definition:Hurwitz Quaternion", "text": "A '''Hurwitz quaternion''' is a quaternion whose components are all either: :integers or: :integers plus a half, that is, halves of odd integers. The set $H$ of all '''Hurwitz quaternions''' can therefore be defined as: :$H = \\left\\{{a + b \\mathbf i + c \\mathbf j + d \\mathbf k \\in \\Bbb H: \\left({a, b, c, d \\in \\Z}\\right) \\text { or } \\left({a, b, c, d \\in \\Z + \\dfrac 1 2}\\right)}\\right\\}$"}
+{"_id": "21786", "title": "Definition:Multiple Pointed Topology", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A$ be a finite set whose cardinality is greater than $1$. Let $D = \\struct {A, \\set {\\O, A} }$ be the indiscrete space on $A$. Let $T \\times D$ be the product space of $T$ and $D$. Then $T \\times D$ is known as the '''multiple pointed topology''' on $T$. It is seen that $T \\times D$ is conceptually equivalent to taking the space $T$ and replacing each point with a finite set of topologically indistinguishable points."}
+{"_id": "21787", "title": "Definition:Box Topology", "text": "Let $\\family {\\struct{X_i, \\tau_i}}_{i \\mathop \\in I}$ be an $I$-indexed family of topological spaces. Let $X$ be the cartesian product of $\\family {X_i}_{i \\mathop \\in I}$, that is: :$\\displaystyle X := \\prod_{i \\mathop \\in I} X_i$ Define: :$\\displaystyle \\BB := \\set {\\prod_{i \\mathop \\in I} U_i: \\forall i \\in I: U_i \\in \\tau_i}$ Then $\\BB$ is a synthetic basis on $X$, as shown on Basis for Box Topology. The '''box topology''' on $X$ is defined as the topology $\\tau$ generated by the synthetic basis $\\BB$."}
+{"_id": "21788", "title": "Definition:Big Model", "text": "Let $\\mathcal{M}$ be an $\\mathcal{L}$-structure with universe $M$. Let $\\kappa$ be a cardinal. $\\mathcal{M}$ is '''$\\kappa$-big''' if for every subset $A\\subset M$ with cardinality $|A| < \\kappa$, the following holds: :if $\\mathcal{L}_A$ is the language obtained from $\\mathcal{L}$ by adding new constant symbols for each $a\\in A$, then :for every language $\\mathcal{L}_A^*$ obtained by adding a new relation symbol $R$ to $\\mathcal{L}_A$, and :for every $\\mathcal{L}_A^*$-structure $\\mathcal{N}$ such that $\\mathcal{M}$ and $\\mathcal{N}$ are elementary equivalent as $\\mathcal{L}_A$-structures, :there is a relation $R^\\mathcal{M}$ on $M$ such that $(\\mathcal{M},R^\\mathcal{M})$ is elementary equivalent to $\\mathcal{N}$ as an $\\mathcal{L}_A^*$-structure."}
+{"_id": "21789", "title": "Definition:Word Metric", "text": "Let $\\struct {G, \\circ}$ be a group. Let $S$ be a generating set for $G$ which is closed under inverses (that is, $x^{-1} \\in S \\iff x \\in S$). The '''word metric''' on $G$ with respect to $S$ is the metric $d_S$ defined as follows: :For any $g, h \\in G$, let $\\map {d_S} {g, h}$ be the minimum length among the finite sequences $\\tuple {x_1, \\dots, x_n}$ with each $x_i \\in S$ such that $g \\circ x_1 \\circ \\cdots \\circ x_n = h$. Informally, $\\map {d_S} {g, h}$ is the smallest number of elements from $S$ that one needs to multiply by to get from $g$ to $h$."}
+{"_id": "21790", "title": "Definition:Continuous Extension", "text": "Let $T_1 = \\left({S_1, \\tau_1}\\right)$ and $T_2 = \\left({S_2, \\tau_2}\\right)$ be topological spaces. Let $A, B \\subseteq S_1$ be subsets of $S_1$ such that $A \\subseteq B$. Let $f: A \\to S_2$ and $g: B \\to S_2$ be continuous mappings. Then $g$ is a '''continuous extension''' of $f$ {{iff}}: :$\\forall s \\in A: f \\left({s}\\right) = g \\left({s}\\right)$ That is, a '''continuous extension''' of $f$ is a continuous mapping on a superset which agrees with $f$ on the domain of $f$. Simply, it is a continuous mapping which is an extension. === Real Function === {{:Definition:Continuous Extension/Real Function}} Category:Definitions/Continuity s7yn9e1314qrkrgqebxi7io2vrl0bb7"}
+{"_id": "21791", "title": "Definition:Diagonal Mapping", "text": "Let $S$ be a set. Let $S \\times S$ be the Cartesian product of $S$ with itself. Then the '''diagonal mapping on $S$''' is defined as $\\Delta: S \\to S \\times S$: :$\\forall x \\in S: \\Delta \\left({x}\\right) = \\left({x, x}\\right)$ Clearly $\\Delta$ is an injection, and is not a surjection unless $S$ is a singleton. {{SUBPAGENAME}} tgmqf7v43mpneequ5iqkxn1v1zkho6g"}
+{"_id": "21792", "title": "Definition:Minimal Arithmetic", "text": "'''Minimal arithmetic''' is the set $Q$ of theorems of the recursive set of sentences in the language of arithmetic containing exactly: {{begin-axiom}} {{axiom | n = \\text M 1         | q = \\forall x         | m = \\map s x \\ne 0 }} {{axiom | n = \\text M 2         | q = \\forall x, y         | m = \\map s x = \\map s y \\implies x = y }} {{axiom | n = \\text M 3         | q = \\forall x         | m = x + 0 = x }} {{axiom | n = \\text M 4         | q = \\forall x, y         | m = x + \\map s y = \\map s {x + y} }} {{axiom | n = \\text M 5         | q = \\forall x         | m = x \\cdot 0 = 0 }} {{axiom | n = \\text M 6         | q = \\forall x, y         | m = x \\cdot \\map s y = \\paren {x \\cdot y} + x }} {{axiom | n = \\text M 7         | q = \\forall x         | m = \\neg x < 0 }} {{axiom | n = \\text M 8         | q = \\forall x, y         | m = x < \\map s y \\iff \\paren {x < y \\lor x = y} }} {{axiom | n = \\text M 9         | q = \\forall x         | m = 0 < x \\iff x \\ne 0 }} {{axiom | n = \\text M 10         | q = \\forall x, y         | m = \\map s x < y \\iff \\paren {x < y \\land y \\ne \\map s x} }} {{end-axiom}}"}
+{"_id": "21793", "title": "Definition:Language of Arithmetic", "text": "A '''language of arithmetic''' is a signature for predicate logic consisting of: * the binary operation symbols: $+$ and $\\cdot$ * the unary function symbol: $s$ * the binary relation symbol: $<$ * the constant symbol: $0$"}
+{"_id": "21799", "title": "Definition:Euclid's Definitions - Book I/2 - Line", "text": "A '''line''' is breadthless length."}
+{"_id": "21800", "title": "Definition:Euclid's Definitions - Book I/4 - Straight Line", "text": "A '''straight line''' is a line which lies evenly with the points on itself."}
+{"_id": "21801", "title": "Definition:Euclid's Definitions - Book I/3 - Line Extremities", "text": "The extremities of a line are points."}
+{"_id": "21802", "title": "Definition:Euclid's Definitions - Book XI/2 - Extremity of Solid", "text": "An extremity of a solid is a surface."}
+{"_id": "21803", "title": "Definition:Euclid's Definitions - Book I/7 - Plane Surface", "text": "A '''plane surface''' is a surface which lies evenly with the straight lines on itself."}
+{"_id": "21804", "title": "Definition:Euclid's Definitions - Book I/6 - Surface Extremities", "text": "The extremities of a surface are lines."}
+{"_id": "21805", "title": "Definition:Limit Point/Real Analysis", "text": "Let $S \\subseteq \\R$ be a subset of the real numbers. Let $\\xi \\in \\R$ and let $S_\\xi$ be the set defined as: :$S_\\xi := \\left\\{{x: x \\in S, x \\ne \\xi}\\right\\}$ Then $\\xi$ is a '''limit point''' of $S$ {{iff}} $\\xi$ is at zero distance from $S_\\xi$."}
+{"_id": "21806", "title": "Definition:Euclid's Definitions - Book I/1 - Point", "text": "A '''point''' is that which has no part."}
+{"_id": "21807", "title": "Definition:Euclid's Definitions - Book I/5 - Surface", "text": "A '''surface''' is that which has length and breadth only."}
+{"_id": "21808", "title": "Definition:Euclid's Definitions - Book I/8 - Plane Angle", "text": "A '''plane angle''' is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line."}
+{"_id": "21809", "title": "Definition:Euclid's Definitions - Book I/9 - Rectilineal Angle", "text": "And when the lines containing the angle are straight, the angle is called '''rectilineal'''."}
+{"_id": "21810", "title": "Definition:Euclid's Definitions - Book I/10 - Right Angle", "text": "When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is '''right''', and the straight line standing on the other is called a '''perpendicular''' to that on which it stands."}
+{"_id": "21811", "title": "Definition:Euclid's Definitions - Book I/11 - Obtuse Angle", "text": "An '''obtuse angle''' is an angle greater than a right angle."}
+{"_id": "21812", "title": "Definition:Euclid's Definitions - Book I/12 - Acute Angle", "text": "An '''acute angle''' is an angle less than a right angle."}
+{"_id": "21813", "title": "Definition:Euclid's Definitions - Book I/13 - Boundary", "text": "A '''boundary''' is that which is an extremity of anything."}
+{"_id": "21814", "title": "Definition:Euclid's Definitions - Book I/14 - Geometric Figure", "text": "A '''figure''' is that which is contained by any boundary or boundaries."}
+{"_id": "21816", "title": "Definition:Euclid's Definitions - Book I/15 - Circle", "text": "A '''circle''' is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;"}
+{"_id": "21817", "title": "Definition:Euclid's Definitions - Book I/16 - Center of Circle", "text": "And the point is called the '''center of the circle'''."}
+{"_id": "21818", "title": "Definition:Euclid's Definitions - Book I/17 - Diameter of Circle", "text": "A '''diameter of the circle''' is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the center."}
+{"_id": "21819", "title": "Definition:Euclid's Definitions - Book I/18 - Semicircle", "text": "A '''semicircle''' is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle."}
+{"_id": "21820", "title": "Definition:Euclid's Definitions - Book I/19 - Rectilineal Figure", "text": "'''Rectilineal figures''' are those which are contained by straight lines, '''trilateral''' figures being those contained by three, '''quadrilateral''' those contained by four, and '''multi-lateral''' those contained by more than four straight lines."}
+{"_id": "21821", "title": "Definition:Euclid's Definitions - Book I/22 - Quadrilaterals", "text": "Of quadrilateral figures, a '''square''' is that which is both equilateral and right-angled; an '''oblong''' that which is right-angled but not equilateral; a '''rhombus''' that which is equilateral but not right-angled; and a '''rhomboid''' that which has its opposite sides equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called '''trapezia'''."}
+{"_id": "21822", "title": "Definition:Euclid's Definitions - Book I/20 - Triangles by Sides", "text": "Of trilateral figures, an '''equilateral triangle''' is that which has its three sides equal, an '''isosceles triangle''' that which has two of its sides alone equal, and a '''scalene triangle''' that which has its three sides unequal."}
+{"_id": "21823", "title": "Definition:Euclid's Definitions - Book I/21 - Triangles by Angles", "text": "Further, of trilateral figures, a '''right-angled triangle''' is that which has a right angle, an '''obtuse-angled triangle''' that which has an obtuse angle, and an '''acute-angled triangle''' that which has its three angles acute."}
+{"_id": "21824", "title": "Definition:Euclid's Definitions - Book I/23 - Parallel Lines", "text": "'''Parallel''' straight lines are straight lines which, being in the same plane and being produced indefinitely in either direction, do not meet one another in either direction."}
+{"_id": "21825", "title": "Definition:Euclid's Definitions - Book II/1 - Containment of Rectangle", "text": "Any rectangular parallelogram is said to be '''contained''' by the two straight lines containing the right angle."}
+{"_id": "21826", "title": "Definition:Euclid's Definitions - Book II/2 - Gnomon", "text": "And in any parallelogrammic area let any one whatever of the parallelograms about its diameter with the two complements be called a '''gnomon'''."}
+{"_id": "21827", "title": "Definition:Euclid's Definitions - Book III/1 - Equal Circles", "text": "'''Equal circles''' are those the diameters of which are equal, or the radii of which are equal."}
+{"_id": "21828", "title": "Definition:Euclid's Definitions - Book III/2 - Tangent to Circle", "text": "A straight line is said to '''touch a circle''' which, meeting the circle and being produced, does not cut the circle."}
+{"_id": "21829", "title": "Definition:Euclid's Definitions - Book III/3 - Tangent Circles", "text": "Circles are said to '''touch one another''' which, meeting one another, do not cut one another."}
+{"_id": "21830", "title": "Definition:Euclid's Definitions - Book III/4 - Equally Distant in Circle", "text": "In a circle straight lines are said to be '''equally distant from the center''' when the perpendiculars drawn to them from the center are equal."}
+{"_id": "21831", "title": "Definition:Euclid's Definitions - Book III/5 - Greater Distance in Circle", "text": "And that straight line is said to be '''at a greater distance''' on which the greater perpendicular falls."}
+{"_id": "21832", "title": "Definition:Euclid's Definitions - Book III/6 - Segment of Circle", "text": "A '''segment of a circle''' is the figure contained by a straight line and a circumference of a circle."}
+{"_id": "21833", "title": "Definition:Euclid's Definitions - Book III/7 - Angle of Segment", "text": "An '''angle of a segment''' is that contained by a straight line and a circumference of a circle."}
+{"_id": "21834", "title": "Definition:Euclid's Definitions - Book III/8 - Angle in Segment", "text": "An '''angle in a segment''' is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line which is the base of the segment, is contained by the straight lines so joined."}
+{"_id": "21835", "title": "Definition:Euclid's Definitions - Book III/9 - Stand on Circumference", "text": "And, when the straight lines containing the angle cut off a circumference, the angle is said to '''stand upon''' that circumference."}
+{"_id": "21836", "title": "Definition:Euclid's Definitions - Book III/10 - Sector of Circle", "text": "A '''sector of a circle''' is the figure which, when an angle is constructed at the center of the circle, is contained by the straight lines containing the angle and the circumference cut off by them."}
+{"_id": "21837", "title": "Definition:Euclid's Definitions - Book III/11 - Similar Segments", "text": "'''Similar segments of circles''' are those which admit equal angles, or in which the angles are equal to one another."}
+{"_id": "21838", "title": "Definition:Euclid's Definitions - Book IV/1 - Inscribe", "text": "A rectilineal figure is said to be '''inscribed in a rectilineal figure''' when the respective angles of the inscribed figure lie on the respective sides of that in which it is inscribed."}
+{"_id": "21839", "title": "Definition:Euclid's Definitions - Book IV/2 - Circumscribe", "text": "Similarly a figure is said to be '''circumscribed about a figure''' when the respective sides of the circumscribed figure pass through the respective angles of that about which it is circumscribed."}
+{"_id": "21840", "title": "Definition:Euclid's Definitions - Book IV/3 - Inscribed in Circle", "text": "A rectilineal figure is said to be '''inscribed in a circle''' when each angle of the inscribed figure lies on the circumference of the circle."}
+{"_id": "21841", "title": "Definition:Euclid's Definitions - Book IV/4 - Circumscribed about Circle", "text": "A rectilineal figure is said to be '''circumscribed about a circle''', when each side of the circumscribed figure touches the circumference of the circle."}
+{"_id": "21842", "title": "Definition:Euclid's Definitions - Book IV/5 - Circle Inscribed", "text": "Similarly a circle is said to be '''inscribed in a figure''' when the circumference of the circle touches each side of the figure in which it is inscribed."}
+{"_id": "21843", "title": "Definition:Euclid's Definitions - Book IV/6 - Circle Circumscribed", "text": "A circle is said to be '''circumscribed about a figure''' when the circumference of the circle passes through each angle of the figure about which it is circumscribed."}
+{"_id": "21844", "title": "Definition:Euclid's Definitions - Book IV/7 - Fitted Into Circle", "text": "A straight line is said to be '''fitted into a circle''' when its extremities are on the circumference of the circle."}
+{"_id": "21845", "title": "Definition:Measure (Geometry)", "text": "A geometric quantity $A$ is said to '''measure''' another quantity $B$ when the size of $A$ is a divisor of the size of $B$. Category:Definitions/Geometry 7pv73327m91o2y8shdhh3f65fao156f"}
+{"_id": "21846", "title": "Definition:Part", "text": "=== Of Integer === {{:Definition:Divisor (Algebra)/Integer/Aliquot Part}} === Of Real Number === {{:Definition:Divisor (Algebra)/Real Number/Part}}"}
+{"_id": "21847", "title": "Definition:Euclid's Definitions - Book V/1 - Part", "text": "A magnitude is a '''part''' of a magnitude, the less of the greater, when it measures the greater."}
+{"_id": "21848", "title": "Definition:Euclid's Definitions - Book V/2 - Multiple", "text": "The greater is a '''multiple''' of the less when it is measured by the less."}
+{"_id": "21849", "title": "Definition:Euclid's Definitions - Book V/3 - Ratio", "text": "A '''ratio''' is a sort of relation in respect of size between two magnitudes of the same kind."}
+{"_id": "21850", "title": "Definition:Euclid's Definitions - Book V/4 - Existence of Ratio", "text": "Magnitudes are said to '''have a ratio''' to one another which are capable, when multiplied, of exceeding one another."}
+{"_id": "21851", "title": "Definition:Euclid's Definitions - Book V/5 - Equality of Ratios", "text": "Magnitudes are said to '''be in the same ratio''', the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order."}
+{"_id": "21852", "title": "Definition:Euclid's Definitions - Book V/6 - Proportion", "text": "Let magnitudes which have the same ratio be called '''proportional'''."}
+{"_id": "21853", "title": "Definition:Euclid's Definitions - Book V/7 - Greater Ratio", "text": "When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to '''have a greater ratio''' to the second than the third has to the fourth."}
+{"_id": "21854", "title": "Definition:Euclid's Definitions - Book V/8 - Proportion in Three Terms", "text": "A proportion in three terms is the least possible."}
+{"_id": "21855", "title": "Definition:Euclid's Definitions - Book V/9 - Duplicate Ratio", "text": "When three magnitudes are proportional, the first is said to have to the third the '''duplicate ratio''' of that which it has to the second."}
+{"_id": "21856", "title": "Definition:Euclid's Definitions - Book V/10 - Triplicate Ratio", "text": "When four magnitudes are $<$ continuously $>$ proportional, the first is said to have to the fourth the '''triplicate ratio''' of that which it has to the second, and so on continually, whatever be the proportion."}
+{"_id": "21857", "title": "Definition:Term (Algebra)", "text": "A '''term''' is either a variable or a constant. Let $a \\circ b$ be an expression. Then each of $a$ and $b$ are known as the '''terms''' of the expression. The word '''term''' is usually used when the operation $\\circ$ is addition, that is $+$."}
+{"_id": "21859", "title": "Definition:Term (Natural Language)", "text": "A '''term''' is a noun which is assigned a specified definition within mathematics. In a more specialized context, the word '''term''' is used for an element of an expression. Category:Definitions/Language Definitions 13knjhv833n7iyw3wt2kfec2jupvprb"}
+{"_id": "21860", "title": "Definition:Euclid's Definitions - Book V/11 - Corresponding Magnitudes", "text": "The term '''corresponding magnitudes''' is used of antecedents in relation to antecedents, and of consequents in relation to consequents."}
+{"_id": "21861", "title": "Definition:Euclid's Definitions - Book V/12 - Alternate Ratio", "text": "'''Alternate ratio''' means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent."}
+{"_id": "21862", "title": "Definition:Euclid's Definitions - Book V/13 - Inverse Ratio", "text": "'''Inverse ratio''' means taking the consequent as antecedent in relation to the antecedent as consequent."}
+{"_id": "21863", "title": "Definition:Euclid's Definitions - Book V/14 - Composition of Ratio", "text": "'''Composition of a ratio''' means taking the antecedent together with the consequent as one in relation to the consequent by itself."}
+{"_id": "21864", "title": "Definition:Euclid's Definitions - Book V/15 - Separation of Ratio", "text": "'''Separation of a ratio''' means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself."}
+{"_id": "21865", "title": "Definition:Euclid's Definitions - Book V/17 - Ratio Ex Aequali", "text": "A '''ratio ex aequali''' arises when, there being several magnitudes and another set equal to them in multitude which makes two and two are in the same proportion, as the first is to the last among the first magnitudes, so is the first to the last among the second magnitudes;<br/>Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms."}
+{"_id": "21866", "title": "Definition:Euclid's Definitions - Book V/16 - Conversion of Ratio", "text": "'''Conversion of a ratio''' means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent."}
+{"_id": "21867", "title": "Definition:Euclid's Definitions - Book V/18 - Perturbed Proportion", "text": "A '''perturbed proportion''' arises when, there being three magnitudes and another set equal to them in multitude, as antecedent is to consequent among the first magnitudes, so is antecedent to consequent among the second magnitudes, while, as the consequent is to a third among the first magnitudes, so is a third to the antecedent among the second magnitudes."}
+{"_id": "21868", "title": "Definition:Euclid's Definitions - Book VI/1 - Similar Rectilineal Figures", "text": "'''Similar rectilineal figures''' are such as have their angles severally equal and the sides about the equal angles proportional."}
+{"_id": "21869", "title": "Definition:Euclid's Definitions - Book VI/2 - Reciprocally Related Figures", "text": "Two figures are '''reciprocally related''' when there are in each of the two figures antecedent and consequent ratios."}
+{"_id": "21870", "title": "Definition:Euclid's Definitions - Book VI/3 - Extreme and Mean Ratio", "text": "A straight line is said to have been '''cut in extreme and mean ratio''' when, as the whole line is to the greater segment, so is the greater to the less."}
+{"_id": "21871", "title": "Definition:Euclid's Definitions - Book VI/4 - Height", "text": "The '''height''' of any figure is the perpendicular drawn from the vertex to the base."}
+{"_id": "21872", "title": "Definition:Euclid's Definitions - Book VII/1 - Unit", "text": "An '''unit''' is that of which each of the things that exist is called one."}
+{"_id": "21873", "title": "Definition:Euclid's Definitions - Book VII/2 - Number", "text": "A '''number''' is a multitude composed of units."}
+{"_id": "21874", "title": "Definition:Euclid's Definitions - Book VII/3 - Part", "text": "A number is '''a part''' of a number, the less of the greater, when it measures the greater;"}
+{"_id": "21875", "title": "Definition:Euclid's Definitions - Book VII/4 - Parts", "text": "but '''parts''' when it does not measure it."}
+{"_id": "21876", "title": "Definition:Euclid's Definitions - Book VII/5 - Multiple", "text": "The greater number is a '''multiple''' of the less when it is measured by the less."}
+{"_id": "21877", "title": "Definition:Euclid's Definitions - Book VII/6 - Even Number", "text": "An '''even number''' is that which is divisible into two equal parts."}
+{"_id": "21878", "title": "Definition:Euclid's Definitions - Book VII/7 - Odd Number", "text": "An '''odd number''' is that which is not divisible into two equal parts, or that which differs by an unit from an even number."}
+{"_id": "21879", "title": "Definition:Euclid's Definitions - Book VII/8 - Even Times Even", "text": "An '''even-times even number''' is that which is measured by an even number according to an even number."}
+{"_id": "21880", "title": "Definition:Euclid's Definitions - Book VII/9 - Even Times Odd", "text": "An '''even-times odd number''' is that which is measured by an even number according to an odd number."}
+{"_id": "21881", "title": "Definition:Euclid's Definitions - Book VII/10 - Odd Times Odd", "text": "An '''odd-times odd number''' is that which is measured by an odd number according to an odd number."}
+{"_id": "21882", "title": "Definition:Euclid's Definitions - Book VII/11 - Prime Number", "text": "A '''prime number''' is that which is measured by an unit alone."}
+{"_id": "21883", "title": "Definition:Euclid's Definitions - Book VII/12 - Relatively Prime", "text": "Numbers '''prime to one another''' are those which are measured by an unit alone as a common measure."}
+{"_id": "21884", "title": "Definition:Euclid's Definitions - Book VII/13 - Composite Number", "text": "A '''composite number''' is that which is measured by some number."}
+{"_id": "21885", "title": "Definition:Euclid's Definitions - Book VII/14 - Relatively Composite", "text": "Numbers '''composite to one another''' are those which are measured by some number as a common measure."}
+{"_id": "21886", "title": "Definition:Euclid's Definitions - Book VII/15 - Multiply", "text": "A number is said to '''multiply''' a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced."}
+{"_id": "21887", "title": "Definition:Euclid's Definitions - Book VII/16 - Plane Number", "text": "And, when two numbers having multiplied one another make some number, the number so produced is called '''plane''', and its '''sides''' are the numbers which have multiplied one another."}
+{"_id": "21888", "title": "Definition:Euclid's Definitions - Book VII/17 - Solid Number", "text": "And, when three numbers having multiplied one another make some number, the number so produced is '''solid''', and its '''sides''' are the numbers which have multiplied one another."}
+{"_id": "21889", "title": "Definition:Euclid's Definitions - Book VII/18 - Square Number", "text": "A '''square number''' is equal multiplied by equal, or a number which is contained by two equal numbers."}
+{"_id": "21890", "title": "Definition:Euclid's Definitions - Book VII/19 - Cube Number", "text": "And a '''cube''' is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers."}
+{"_id": "21891", "title": "Definition:Euclid's Definitions - Book VII/20 - Proportional", "text": "Numbers are '''proportional''' when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth."}
+{"_id": "21892", "title": "Definition:Euclid's Definitions - Book VII/21 - Similar Numbers", "text": "'''Similar plane''' and '''solid''' numbers are those which have their sides proportional."}
+{"_id": "21893", "title": "Definition:Euclid's Definitions - Book VII/22 - Perfect Number", "text": "A '''perfect number''' is that which is equal to its own parts."}
+{"_id": "21894", "title": "Definition:Euclid's Definitions - Book X/1 - Commensurable", "text": "Those magnitudes are said to be '''commensurable''' which are measured by the same same measure, and those '''incommensurable''' which cannot have any common measure."}
+{"_id": "21895", "title": "Definition:Euclid's Definitions - Book X/2 - Commensurable in Square", "text": "Straight lines are '''commensurable in square''' when the squares on them are measured by the same area, and '''incommensurable in square''' when the squares on them cannot possibly have any area as a common measure."}
+{"_id": "21896", "title": "Definition:Euclid's Definitions - Book X/3 - Rational Line Segment", "text": "With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called '''rational''', and those straight lines which are commensurable with it, whether in length and in square or square only, '''rational''', but those which are incommensurable with it '''irrational'''."}
+{"_id": "21897", "title": "Definition:Euclid's Definitions - Book X/4 - Rational Area", "text": "And let the square on the assigned straight line be called '''rational''' and those areas which are commensurable with it '''rational''', but those which are incommensurable with it '''irrational''', and the straight lines which produce them '''irrational''', that is, in case the areas are squares, the sides themselves, but in case they are any other rectilineal figures, the straight lines on which are described squares equal to them."}
+{"_id": "21898", "title": "Definition:Euclid's Definitions - Book X (II)/1 - First Binomial", "text": "Given a rational straight line and a binomial, divided into its terms, such that the square on the greater term is greater than the square on the lesser by the square on a straight line commensurable in length with the greater, then, if the greater term be commensurable in length with the rational straight line set out, let the whole be called a '''first binomial''' straight line;"}
+{"_id": "21901", "title": "Definition:Euclid's Definitions - Book X (II)/2 - Second Binomial", "text": "but if the lesser term be commensurable in length with the rational straight line set out, let the whole be called a '''second binomial''';"}
+{"_id": "21902", "title": "Definition:Euclid's Definitions - Book X (II)/3 - Third Binomial", "text": "and if neither of the terms be commensurable in length with the rational straight line set out, let the whole be called a '''third binomial'''."}
+{"_id": "21903", "title": "Definition:Euclid's Definitions - Book X (II)/4 - Fourth Binomial", "text": "Again, if the square on the greater term be greater than the square on the lesser by the square on a straight line incommensurable in length with the greater, then, if the greater term be commensurable in length with the rational straight line set out, let the whole be called a '''fourth binomial''';"}
+{"_id": "21904", "title": "Definition:Euclid's Definitions - Book X (II)/5 - Fifth Binomial", "text": "if the lesser, a '''fifth binomial''';"}
+{"_id": "21905", "title": "Definition:Euclid's Definitions - Book X (II)/6 - Sixth Binomial", "text": "and if neither, a '''sixth binomial'''."}
+{"_id": "21906", "title": "Definition:Euclid's Definitions - Book X (III)/1 - First Apotome", "text": "Given a rational straight line and an apotome, if the square on the whole be greater than the square on the annex by the square on a straight line commensurable in length with the whole, and the whole be commensurable in length with the rational straight line set out, let the apotome be called a '''first apotome'''."}
+{"_id": "21907", "title": "Definition:Euclid's Definitions - Book X (III)/2 - Second Apotome", "text": "But if the annex be commensurable in length with the rational straight line set out, and the square on the whole be greater than that on the annex by the square on a straight line commensurable in length with the whole, let the apotome be called a '''second apotome'''."}
+{"_id": "21908", "title": "Definition:Euclid's Definitions - Book X (III)/3 - Third Apotome", "text": "But if neither be commensurable in length with the rational straight line set out, and the square on the whole be greater than the square on the annex by the square on a straight line commensurable with the whole, let the apotome be called a '''third apotome'''."}
+{"_id": "21909", "title": "Definition:Euclid's Definitions - Book X (III)/4 - Fourth Apotome", "text": "Again, if the square on the whole be greater than the square on the annex by the square on a straight line incommensurable with the whole, then, if the whole be commensurable in length with the rational straight line set out, let the apotome be called a '''fourth apotome''';"}
+{"_id": "21910", "title": "Definition:Euclid's Definitions - Book X (III)/5 - Fifth Apotome", "text": "if the annex be so commensurable, a '''fifth''';"}
+{"_id": "21911", "title": "Definition:Euclid's Definitions - Book X (III)/6 - Sixth Apotome", "text": "and if neither, a '''sixth'''."}
+{"_id": "21912", "title": "Definition:Euclid's Definitions - Book XI/1 - Solid", "text": "A '''solid''' is that which has length, breadth, and depth."}
+{"_id": "21913", "title": "Definition:Euclid's Definitions - Book XI/3 - Line at Right Angles to Plane", "text": "A '''straight line''' is '''at right angles to a plane''' when it makes right angles with all the straight lines which meet it and are in the plane."}
+{"_id": "21914", "title": "Definition:Euclid's Definitions - Book XI/4 - Plane at Right Angles to Plane", "text": "A '''plane''' is '''at right angles to a plane''' when the straight lines drawn, in one of the planes, at right angles to the common section of the planes are at right angles to the remaining plane."}
+{"_id": "21915", "title": "Definition:Euclid's Definitions - Book XI/5 - Inclination of Straight Line", "text": "The '''inclination of a straight line to a plane''' is, assuming a perpendicular drawn from the extremity of the straight line which is elevated above the plane to the plane, and a straight line joined from the point thus arising to the extremity of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up."}
+{"_id": "21916", "title": "Definition:Euclid's Definitions - Book XI/6 - Inclination of Plane", "text": "The '''inclination of a plane to a plane''' is the acute angle contained by the straight lines drawn at right angles to the common section at the same point, one in each of the planes."}
+{"_id": "21917", "title": "Definition:Euclid's Definitions - Book XI/7 - Similarly Inclined", "text": "A plane is said to be '''similarly inclined''' to a plane as another is to another when the said angles of the inclinations are equal to one another."}
+{"_id": "21918", "title": "Definition:Euclid's Definitions - Book XI/8 - Parallel Planes", "text": "'''Parallel planes''' are those which do not meet."}
+{"_id": "21919", "title": "Definition:Euclid's Definitions - Book XI/9 - Similar Solid Figures", "text": "'''Similar solid figures''' are those contained by similar planes equal in multitude."}
+{"_id": "21920", "title": "Definition:Euclid's Definitions - Book XI/10 - Similar Equal Solid Figures", "text": "'''Equal and similar solid figures''' are those contained by similar planes equal in multitude and in magnitude."}
+{"_id": "21921", "title": "Definition:Euclid's Definitions - Book XI/11 - Solid Angle", "text": "A '''solid angle''' is the inclination constituted by more than two lines which meet one another and are not in the same surface, towards all the lines.<br/>Otherwise: A '''solid angle''' is that which is contained by more than two plane angles which are not in the same plane and are constructed to one point."}
+{"_id": "21922", "title": "Definition:Euclid's Definitions - Book XI/12 - Pyramid", "text": "A '''pyramid''' is a solid figure, contained by planes, which is constructed from one plane to one point."}
+{"_id": "21923", "title": "Definition:Euclid's Definitions - Book XI/13 - Prism", "text": "A '''prism''' is a solid figure contained by planes of which, namely those which are opposite, are equal, similar and parallel, while the rest are parallelograms."}
+{"_id": "21924", "title": "Definition:Euclid's Definitions - Book XI/14 - Sphere", "text": "When, the diameter of a semicircle remaining fixed, the semicircle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a '''sphere'''."}
+{"_id": "21925", "title": "Definition:Euclid's Definitions - Book XI/15 - Axis of Sphere", "text": "The '''axis of the sphere''' is the straight line which remains fixed about which the semicircle is turned."}
+{"_id": "21926", "title": "Definition:Euclid's Definitions - Book XI/16 - Center of Sphere", "text": "The '''center of the sphere''' is the same as that of the semicircle."}
+{"_id": "21927", "title": "Definition:Euclid's Definitions - Book XI/17 - Diameter of Sphere", "text": "A '''diameter of the sphere''' is any straight line drawn through the centre and terminated in both directions by the surface of the sphere."}
+{"_id": "21928", "title": "Definition:Euclid's Definitions - Book XI/18 - Cone", "text": "When, one side of those about the right angle in a right-angled triangle remaining fixed, the triangle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a '''cone'''.<br>And, if the straight line which remains fixed be equal to the remaining side about the right angle which is carried round, the cone will be '''right-angled'''; if less, '''obtuse-angled'''; and if greater, '''acute-angled'''."}
+{"_id": "21929", "title": "Definition:Euclid's Definitions - Book XI/19 - Axis of Cone", "text": "The '''axis of the cone''' is the straight line which remains fixed and about which the triangle is turned."}
+{"_id": "21930", "title": "Definition:Euclid's Definitions - Book XI/20 - Base of Cone", "text": "And the '''base''' is the circle described by the straight line which is carried round."}
+{"_id": "21931", "title": "Definition:Euclid's Definitions - Book XI/21 - Cylinder", "text": "When, one side of those about the right angle in a rectangular parallelogram remaining fixed, the parallelogram is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a '''cylinder'''."}
+{"_id": "21932", "title": "Definition:Euclid's Definitions - Book XI/22 - Axis of Cylinder", "text": "The '''axis of the cylinder''' is the straight line which remains fixed and about which the parallelogram is turned."}
+{"_id": "21933", "title": "Definition:Euclid's Definitions - Book XI/23 - Base of Cylinder", "text": "And the '''bases''' are the circles described by the two sides opposite to one another which are carried round."}
+{"_id": "21934", "title": "Definition:Euclid's Definitions - Book XI/24 - Similar Cones and Cylinders", "text": "'''Similar cones and cylinders''' are those in which the axes and the diameters of the bases are proportional."}
+{"_id": "21935", "title": "Definition:Euclid's Definitions - Book XI/25 - Cube", "text": "A '''cube''' is a solid figure contained by six equal squares."}
+{"_id": "21936", "title": "Definition:Euclid's Definitions - Book XI/26 - Octahedron", "text": "An '''octahedron''' is a solid figure contained by eight equal and equilateral triangles."}
+{"_id": "21937", "title": "Definition:Euclid's Definitions - Book XI/27 - Icosahedron", "text": "An '''icosahedron''' is a solid figure contained by twenty equal and equilateral triangles."}
+{"_id": "21938", "title": "Definition:Euclid's Definitions - Book XI/28 - Dodecahedron", "text": "A '''dodecahedron''' is a solid figure contained by twelve equal, equilateral, and equiangular pentagons."}
+{"_id": "21939", "title": "Definition:Conjugate of Quadratic Irrational", "text": "Let $\\alpha = r + s \\sqrt n$ be a quadratic irrational. Then its '''conjugate''' is defined as: :$\\tilde \\alpha = r - s \\sqrt n$ Thus $\\alpha$ and $\\tilde \\alpha$ are known as '''conjugate quadratic irrationals'''. Notation may vary."}
+{"_id": "21940", "title": "Definition:Pentagon", "text": "A '''pentagon''' is a polygon with exactly $5$ sides: :300px"}
+{"_id": "21941", "title": "Definition:Hexagon", "text": "A '''hexagon''' is a polygon with exactly $6$ sides. :300px"}
+{"_id": "21942", "title": "Definition:Equimultiples", "text": "Let $a$ and $b$ be numbers. Then '''equimultiples''' of $a$ and $b$ are numbers of the form $c a$ and $c b$ where $c$ is also a number. That is, they are products arising from the multiplications of two numbers by the same common number."}
+{"_id": "21943", "title": "Definition:Ratio", "text": "Let $x$ and $y$ be quantities which have the same dimensions. Let $\\dfrac x y = \\dfrac a b$ for two numbers $a$ and $b$. Then the '''ratio of $x$ to $y$''' is defined as: :$x : y = a : b$ It explicitly specifies how many times the first number contains the second."}
+{"_id": "21945", "title": "Definition:Archimedean Property", "text": "Let $\\struct  {S, \\circ}$ be a closed algebraic structure on which there exists either an ordering or a norm. Let $\\cdot: \\Z_{>0} \\times S \\to S$ be the operation defined as: :$m \\cdot a = \\begin{cases} a & : m = 1 \\\\ a \\circ \\paren {\\paren {m - 1} \\cdot a} & : m > 1 \\end {cases}$ === Archimedean Property on Norm === {{:Definition:Archimedean Property/Norm}} === Archimedean Property on Ordering === {{:Definition:Archimedean Property/Ordering}}"}
+{"_id": "21946", "title": "Definition:Alternate Ratio", "text": "Let $a : b$ and $c : d$ where $a : b$ denotes the ratio of $a$ to $b$. Then the '''alternate ratio''' is defined as $a : c$ and $b : d$. {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book V/12 - Alternate Ratio}}'' {{EuclidDefRefNocat|V|12|Alternate Ratio}} Category:Definitions/Euclidean Algebra piabn5rg6ig22rgmo41i3zxjb9hcgt9"}
+{"_id": "21947", "title": "Definition:Ratio Ex Aequali", "text": "{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book V/17 - Ratio Ex Aequali}}'' {{EuclidDefRefNocat|V|17|Ratio Ex Aequali}} Category:Definitions/Euclidean Algebra 94lau22sv5d0itmfa7ufokv79qlyzlv"}
+{"_id": "21948", "title": "Definition:Proportion/Perturbed", "text": "Let $a, b, c$ and $A, B, C$ be magnitudes. $a, b, c$ are '''in perturbed proportion''' to $A, B, C$ {{iff}}: :$a : b = B : C$ :$b : c = A : B$ {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book V/18 - Perturbed Proportion}}'' {{EuclidDefRefNocat|V|18|Perturbed Proportion}}"}
+{"_id": "21949", "title": "Definition:Duplicate Ratio", "text": "Let $a, b, c$ be magnitudes such that: :$a : b = b : c$ Then $a$ has the '''duplicate ratio''' to $c$ of the ratio it has to $b$. That is: :$a : c$ is the '''duplicate ratio''' of $a : b$. From the definition of ratio: {{begin-eqn}} {{eqn | l=a : b       | r=b : c       | c= }} {{eqn | ll=\\implies       | l=\\dfrac a b       | r=\\dfrac b c       | c= }} {{eqn | ll=\\implies       | l=\\dfrac a c       | r=\\dfrac a b \\dfrac b c       | c= }} {{eqn | ll=\\implies       | l=\\dfrac a c       | r=\\left({\\dfrac a b}\\right)^2       | c= }} {{end-eqn}} That is: : $a : c = \\left({a : b}\\right)^2$ {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book V/9 - Duplicate Ratio}} {{EuclidDefRefNocat|V|9|Duplicate Ratio}} Category:Definitions/Euclidean Algebra 746epqd5j2n81g9dzz4z7nsjb2hwbr5"}
+{"_id": "21950", "title": "Definition:Triplicate Ratio", "text": "Let $a, b, c, d$ be magnitudes such that: :$a : b = b : c = c : d$ Then $a$ has the '''triplicate ratio''' to $d$ of the ratio it has to $b$. That is: :$a : d$ is the '''triplicate ratio''' of $a : b$. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book V/10 - Triplicate Ratio}} {{EuclidDefRefNocat|V|10|Triplicate Ratio}} Category:Definitions/Euclidean Algebra dhtr67om9nqt1ghro0y9v2eg2faj15o"}
+{"_id": "21951", "title": "Definition:Corresponding Magnitudes", "text": "Let $R_1 = a : b$ and $R_2 = c : d$ be ratios. Then the '''corresponding magnitudes''' of $a$ and $b$ in $R_2$ are $c$ and $d$ respectively. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book V/11 - Corresponding Magnitudes}} {{EuclidDefRefNocat|V|11|Corresponding Magnitudes}} Category:Definitions/Euclidean Algebra o82jjujsnjdiaptw7gincdz3rsfbct3"}
+{"_id": "21952", "title": "Definition:Inverse Ratio", "text": "Let $R = a : b$ be a ratio. Then the '''inverse ratio''' to $R$ is $b : a$. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book V/13 - Inverse Ratio}} {{EuclidDefRefNocat|V|13|Inverse Ratio}}"}
+{"_id": "21953", "title": "Definition:Composition of Ratio", "text": "Let $R = a : b$ be a ratio. Then the '''composition of $R$''' is the ratio $a + b : b$. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book V/14 - Composition of Ratio}} {{EuclidDefRefNocat|V|14|Composition of Ratio}} Category:Definitions/Euclidean Algebra 2cu4arsolrxnoen6gc1bp5i2iut11o9"}
+{"_id": "21954", "title": "Definition:Separation of Ratio", "text": "Let $R = a : b$ be a ratio. Then the '''separation of $R$''' is the ratio $a - b : b$. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book V/15 - Separation of Ratio}} {{EuclidDefRefNocat|V|15|Separation of Ratio}} Category:Definitions/Euclidean Algebra bugz6tpgo3owyiqguubjszvwcneh1qm"}
+{"_id": "21955", "title": "Definition:Conversion of Ratio", "text": "Let $R = a : b$ be a ratio. Then the '''conversion of $R$''' is the ratio $a : a - b$. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book V/16 - Conversion of Ratio}} {{EuclidDefRefNocat|V|16|Conversion of Ratio}} Category:Definitions/Euclidean Algebra p48tjawh10chh0rk1xgurfmzezaevut"}
+{"_id": "21957", "title": "Definition:Section of Line by Line", "text": "Let a geometrical line $A$ cross over (or intersect) another line $B$. The point where they cross is called the '''section''' of the $B$ by $A$ (or equivalently, of $A$ by $B$)."}
+{"_id": "21958", "title": "Definition:Similar Figures", "text": "Two rectilineal figures are '''similar''' {{iff}}: :They have corresponding angles, all of which are equal :They have corresponding sides, all of which are proportional."}
+{"_id": "21959", "title": "Definition:Reciprocal Relation", "text": "{{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book VI/2 - Reciprocally Related Figures}} {{EuclidDefRefNocat|VI|2|Reciprocally Related Figures}}"}
+{"_id": "21961", "title": "Definition:Similar Triangles", "text": "Similar triangles are triangles whose corresponding angles are the same, but whose corresponding sides may be of different lengths. :300px Thus $\\triangle ABC$ is similar to $\\triangle DEF$: :$\\angle ABC = \\angle EFD$ :$\\angle BCA = \\angle EDF$ :$\\angle CAB = \\angle DEF$"}
+{"_id": "21962", "title": "Definition:Rectification", "text": "'''Rectification''' is an archaic term meaning '''the process of finding length''', specifically in the context of arcs."}
+{"_id": "21963", "title": "Definition:Reciprocal Proportion", "text": "Let $P$ and $Q$ be geometric figures of the same type (that is, having the same number and configuration of sides). Let $A$ and $B$ be sides of $P$, and let $C$ and $D$ be sides of $Q$, such that $A$ and $C$ are corresponding sides, and $B$ and $D$ also be corresponding sides. Then $P$ and $Q$ have sides which are '''in reciprocal proportion''', or are '''reciprocally proportional''', if: :$A : D = B : C$ where $A : D$ is the ratio of the lengths of $A$ and $D$."}
+{"_id": "21964", "title": "Definition:Compound Ratio", "text": "Let $R_1 = a : b$ and $R_2 = c : d$ be ratios. The '''compound ratio''' of $R_1$ and $R_2$ is defined as: :$R_1 R_2 = a c : b d$ That is, it is the product of the two ratios."}
+{"_id": "21965", "title": "Definition:Minimal Polynomial", "text": "Let $L / K$ be a field extension. Let $\\alpha \\in L$ be algebraic over $K$. === Definition 1 === {{Definition:Minimal Polynomial/Definition 1}} === Definition 2 === {{Definition:Minimal Polynomial/Definition 2}} === Definition 3 === {{Definition:Minimal Polynomial/Definition 3}}"}
+{"_id": "21966", "title": "Definition:Algebraic Closure", "text": "Let $K$ be a field. An '''algebraic closure''' of $K$ is an algebraically closed algebraic field extension of $K$. An '''algebraic closure''' of $K$ can be denoted $\\overline K$."}
+{"_id": "21967", "title": "Definition:Normal Extension", "text": "=== Definition 1 === {{:Definition:Normal Extension/Definition 1}} === Definition 2 === {{:Definition:Normal Extension/Definition 2}}"}
+{"_id": "21968", "title": "Definition:Separable Extension", "text": "Let $K$ be a field. Let $L/K$ be an algebraic field extension. Then $L/K$ is a '''separable extension''' {{iff}} every $\\alpha\\in L$ is separable over $K$."}
+{"_id": "21969", "title": "Definition:Galois Extension/Finite", "text": "Let $L/K$ be a finite field extension. === Definition 1 === {{:Definition:Galois Extension/Finite/Definition 1}} === Definition 2 === {{:Definition:Galois Extension/Finite/Definition 2}} === Definition 3 === {{:Definition:Galois Extension/Finite/Definition 3}}"}
+{"_id": "21970", "title": "Definition:Intermediate Field", "text": "Let $L / K$ be a field extension. Let $F$ be a field such that: : $K \\subset F \\subset L$ Then $F$ is an '''intermediate field'''. Category:Definitions/Field Extensions dgtlfaiy3osbe2h6rjje26ygd3hsfy4"}
+{"_id": "21971", "title": "Definition:Embedding (Galois Theory)", "text": "Let $K$ and $L$ be fields. A (field) monomorphism $\\phi: K \\to L$ is called an '''embedding''' of $K$ in $L$. Category:Definitions/Field Theory 0qbhx2myjtxif74p84yor3alybnkzrx"}
+{"_id": "21972", "title": "Definition:Inclusion-Reversing Mapping", "text": "Let $A, B$ be sets of sets and $\\phi: A \\to B$ be a mapping. Then $\\phi$ is '''inclusion-reversing''' {{iff}}: : for every pair of sets $a_1, a_2 \\in A$ such that $a_1 \\subseteq a_2$: :: $\\phi \\left({a_2}\\right) \\subseteq \\phi \\left({a_1}\\right)$"}
+{"_id": "21973", "title": "Definition:Galois Group of Field Extension", "text": "Let $L / K$ be a field extension. The '''Galois group''' of $L / K$ is the subgroup of the automorphism group of $L$ consisting of field automorphisms that fix $K$ point-wise: :$\\Gal {L / K} = \\set{\\sigma \\in \\Aut L: \\forall k \\in K: \\map \\sigma k = k}$ === As a topological group === The notation $\\Gal {L / K}$ is also a shorthand for the topological group: :$\\struct {\\Gal {L / K}, \\tau}$ where $\\tau$ is the Krull topology. === Alternative Definition === More generally, we can abandon the condition that $L / K$ be Galois if we choose an algebraic closure $\\overline K$ such that $L \\subseteq \\overline K$ and define: :$\\Gal {L / K} = \\left\\{ {\\sigma: L \\to \\overline K: \\sigma}\\right.$ is an embedding of $L$ such that $\\sigma$ fixes $K$ point-wise$\\left.\\right\\}$ This set will form a group {{iff}} $L / K$ is normal."}
+{"_id": "21974", "title": "Definition:Gradation Compatible with Ring Structure", "text": "Let $\\left({M, \\cdot}\\right)$ be a semigroup. Let $\\left({R, +, \\circ}\\right)$ be a ring. Let $(R_n)_{n\\in M}$ be an gradation of type $M$ on the additive group of $R$. The gradation is '''compatible with the ring structure''' {{iff}} :$\\forall m, n \\in M : \\forall x \\in S_m, y \\in S_n: x \\circ y \\in S_{m \\cdot n}$ and so: :$S_m S_n \\subseteq S_{m\\cdot n}$"}
+{"_id": "21975", "title": "Definition:Filtered Algebra", "text": "A '''filtered algebra''' is a generalization of the notion of a graded algebra. A '''filtered algebra''' over the field $k$ is an algebra $\\left({A_k, \\oplus}\\right)$ over $k$ which has an increasing sequence $\\left\\{{0}\\right\\} \\subset F_0 \\subset F_1 \\subset \\cdots \\subset F_i \\subset \\cdots \\subset A$ of substructures of $A$ such that: :$\\displaystyle A = \\bigcup_{i \\mathop \\in \\N} F_i$ and that is compatible with the multiplication in the following sense:   :$\\forall m, n \\in \\N: F_m \\cdot F_n \\subset F_{n+m}$"}
+{"_id": "21976", "title": "Definition:Graded Algebra", "text": "Let $R$ be a graded commutative ring with unity. An algebra $A$ over $R$ is a '''graded algebra'''. That is, a '''graded algebra''' is an algebra over a ring where the ring is a graded ring."}
+{"_id": "21977", "title": "Definition:Quadratic Algebra", "text": "A '''quadratic algebra''' $A$ is a filtered algebra whose generator consists of degree one elements, with defining relations of degree 2. A '''quadratic algebra''' $A$ is determined by a vector space of generators $V = A_1$ and a subspace of homogeneous quadratic relations $S \\subseteq V \\times V$. Thus : :$A = T \\left({V}\\right) / \\left \\langle {S}\\right \\rangle$ and inherits its grading from the tensor algebra $T \\left({V}\\right)$. If the subspace of relations may also contain inhomogeneous degree 2 elements, $S \\subseteq k \\times V \\times \\left({V \\times V}\\right)$, this construction results in a '''filtered quadratic algebra'''. A graded quadratic algebra $A$ as above admits a '''quadratic dual''': the quadratic algebra generated by $V^*$ and with quadratic relations forming the orthogonal complement of $S$ in $V^* \\times V^*$. {{Help}} {{MissingLinks}} Category:Definitions/Algebras 9mntgrc4e50kba8fm3dwi3zh9n282ca"}
+{"_id": "21979", "title": "Definition:Division Algebra", "text": "Let $\\left({A_F, \\oplus}\\right)$ be an algebra over field $F$ such that $A_F$ does not consist solely of the zero vector $\\mathbf 0_A$ of $A_F$. === Definition 1 === {{:Definition:Division Algebra/Definition 1}} === Definition 2 === {{:Definition:Division Algebra/Definition 2}}"}
+{"_id": "21980", "title": "Definition:Real Algebra", "text": "A '''real algebra''' is an algebra over a field where the field in question is the field of real numbers $\\R$. Category:Definitions/Algebras ct1zb8hp39391t6npjz6qln9cli5sgw"}
+{"_id": "21981", "title": "Definition:Associative Algebra", "text": "Let $R$ be a commutative ring. Let $\\left({A_R, *}\\right)$ be an algebra over $R$. Then $\\left({A_R, *}\\right)$ is an '''associative algebra''' {{iff}} $*$ is an associative operation. That is: :$\\forall a, b, c \\in A_R: \\left({a * b}\\right) * c = a * \\left({b * c}\\right)$"}
+{"_id": "21982", "title": "Definition:Unital Algebra", "text": "Let $R$ be a commutative ring. Let $\\left({A, *}\\right)$ be an algebra over $R$.  Then $\\left({A, *}\\right)$ is a '''unital algebra''' {{iff}} the algebraic structure $\\left({A, \\oplus}\\right)$ has an identity element. That is: :$\\exists 1_A \\in A: \\forall a \\in A: a * 1_A = 1_A * a = a$"}
+{"_id": "21983", "title": "Definition:Quadratic Real Algebra", "text": "A '''quadratic real algebra''' is a quadratic algebra over the field of real numbers. Category:Definitions/Algebras 65ruwah4xeb28o70pjrtklydsbtu0qr"}
+{"_id": "21986", "title": "Definition:Normed Division Algebra", "text": "Let $\\left({A_F, \\oplus}\\right)$ be a unitary algebra where $A_F$ is a vector space over a field $F$. Then $\\left({A_F, \\oplus}\\right)$ is a normed divison algebra iff $A_F$ is a normed vector space such that: :$\\forall a, b \\in A_F: \\left \\Vert{a \\oplus b}\\right \\Vert = \\left \\Vert{a}\\right \\Vert \\left \\Vert{b}\\right \\Vert$ where $\\left \\Vert{a}\\right \\Vert$ denotes the norm of $a$."}
+{"_id": "21987", "title": "Definition:Unitary Division Algebra", "text": "Let $\\left({A_F, \\oplus}\\right)$ be a division algebra. Then $\\left({A_F, \\oplus}\\right)$ is a '''unitary division algebra''' {{iff}} it has an identity element $1_A$ called a '''unit''' for $\\oplus$: :$\\exists 1_A \\in A_R: \\forall a \\in A_R: a \\oplus 1_A = 1_A \\oplus a = a$ The unit is usually denoted $1$ when there is no source of confusion with the identity elements of the underlying structures of the algebra."}
+{"_id": "21988", "title": "Definition:Unit of Algebra", "text": "Let $R$ be a commutative ring. Let $\\left({A_R, \\oplus}\\right)$ be a unitary algebra over $R$. The '''unit''' of $\\left({A_R, \\oplus}\\right)$, denoted $1_A$, is the identity element of the operation $\\oplus$: :$\\forall a \\in A_R: a \\oplus 1_A = 1_A \\oplus a = a$ It is sometimes referred to as the '''multiplicative identity''' of $\\left({A_R, \\oplus}\\right)$. It is usually denoted $1$ when there is no source of confusion with the identity elements of the underlying structures of the algebra."}
+{"_id": "21989", "title": "Definition:Multiplicative Inverse", "text": "Let $\\strut {A_F, \\oplus}$ be a unitary algebra whose unit is $1$ and whose zero is $0$. Let $a \\in A_F$ such that $a \\ne 0$. A '''multiplicative inverse''' of $a$ is an element $b \\in A_F$ such that: :$a \\oplus b = 1 = b \\oplus a$ === Field === {{:Definition:Multiplicative Inverse/Field}} === Multiplicative Inverse of Number === {{:Definition:Multiplicative Inverse/Number}}"}
+{"_id": "21990", "title": "Definition:Power-Associative Operation", "text": "Let $\\circ$ be a binary operation. Then $\\circ$ is defined as being '''power-associative on $S$''' {{iff}}: :$\\forall x \\in S: \\paren {x \\circ x} \\circ x = x \\circ \\paren {x \\circ x}$"}
+{"_id": "21991", "title": "Definition:Alternative Operation", "text": "Let $\\circ$ be a binary operation. Then $\\circ$ is defined as being '''alternative on $S$''' {{iff}}: :$\\forall T := \\set {x, y} \\subseteq S: \\forall x, y, z \\in T: \\paren {x \\circ y} \\circ z = x \\circ \\paren {y \\circ z}$ That is, $\\circ$ is associative over any two elements of $S$. For example, for any $x, y \\in S$: :$\\paren {x \\circ y} \\circ x = x \\circ \\paren {y \\circ x}$ :$\\paren {x \\circ x} \\circ y = x \\circ \\paren {x \\circ y}$ and so on."}
+{"_id": "21992", "title": "Definition:Multilinear Mapping", "text": "Let $\\left({R, +_R, \\times_R}\\right)$ be a commutative ring. Let $\\left({A_1, +_1, \\circ_1}\\right)_R, \\left({A_2, +_2, \\circ_2}\\right)_R, \\ldots, \\left({A_n, +_n, \\circ_n}\\right)_R, \\left({A_{n+1}, +_{n+1}, \\circ_{n+1}}\\right)_R$ be $R$-modules. Let $\\oplus: A_1 \\times A_2 \\times \\cdots \\times A_n \\to A_{n+1}$ be a multiary operator with the property that: $\\forall \\left({a_1, a_2, \\ldots, a_n}\\right) \\in A_1 \\times A_2 \\times \\cdots \\times A_n$: * $a_1 \\mapsto a_1 \\oplus a_2 \\oplus \\cdots \\oplus a_n$ is a linear transformation from $A_1$ to $A_{n+1}$ * $a_2 \\mapsto a_1 \\oplus a_2 \\oplus \\cdots \\oplus a_n$ is a linear transformation from $A_2$ to $A_{n+1}$ * $\\vdots$ * $a_n \\mapsto a_1 \\oplus a_2 \\oplus \\cdots \\oplus a_n$ is a linear transformation from $A_n$ to $A_{n+1}$ Then $\\oplus$ is a '''multilinear mapping'''."}
+{"_id": "21993", "title": "Definition:Associator", "text": "Let $\\left({A_R, \\oplus}\\right)$ be an algebra over a ring. Consider the trilinear mapping $\\left[{\\cdot, \\cdot, \\cdot}\\right]: A_R^3 \\to A_R$ defined as: :$\\forall a, b, c \\in A_R: \\left[{a, b, c}\\right] := \\left({a \\oplus b}\\right) \\oplus c - a \\oplus \\left({b \\oplus c}\\right)$ Then $\\left[{\\cdot, \\cdot, \\cdot}\\right]$ is known as the '''associator''' of $\\left({A_R, \\oplus}\\right)$. It can be considered a measure of how much associativity of $\\oplus$ fails in $\\left({A_R, \\oplus}\\right)$. Note that trivially if $\\left({A_R, \\oplus}\\right)$ is an associative algebra, then: :$\\forall a, b, c \\in A_R: \\left[{a, b, c}\\right] = \\mathbf 0_R$"}
+{"_id": "21994", "title": "Definition:Commutator", "text": "The '''commutator''' of an algebraic structure can be considered a measure of how commutative the structure is. === Groups === {{:Definition:Commutator/Group}} === Rings === {{:Definition:Commutator/Ring}} === Algebras === {{:Definition:Commutator/Algebra}}"}
+{"_id": "21995", "title": "Definition:Commutative Algebra", "text": "Let $R$ be a commutative ring. Let $\\struct {A_R, \\oplus}$ be an algebra over $R$. Then $\\struct {A_R, \\oplus}$ is a commutative algebra {{iff}} $\\oplus$ is a commutative operation. That is: :$\\forall a, b \\in A_R: a \\oplus b = b \\oplus a$"}
+{"_id": "21996", "title": "Definition:Alternating Bilinear Mapping", "text": "Let $\\left({A_R, \\oplus}\\right)$ be an algebra over a ring. By definition, $\\oplus$ is a bilinear mapping. Then $\\oplus$ is an '''alternating bilinear mapping''' {{iff}}: :$\\forall a \\in A_R: a \\oplus a = 0$ For rings with characteristic other than two, the following condition is equivalent: :$\\forall a, b \\in A_R: a \\oplus b = - b \\oplus a$"}
+{"_id": "21997", "title": "Definition:Commutator/Group", "text": "Let $\\struct {G, \\circ}$ be a group. Let $g, h \\in G$. The '''commutator''' of $g$ and $h$ is the operation: :$\\sqbrk {g, h} := g^{-1} \\circ h^{-1} \\circ g \\circ h$"}
+{"_id": "21998", "title": "Definition:Commutator/Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $a, b \\in R$. The '''commutator''' of $a$ and $b$ is the operation: :$\\sqbrk {a, b} := a \\circ b + \\paren {-b \\circ a}$ or more compactly: :$\\sqbrk {a, b} := a \\circ b - b \\circ a$"}
+{"_id": "21999", "title": "Definition:Commutator/Algebra", "text": "Let $\\struct {A_R, \\oplus}$ be an algebra over a ring. Consider the bilinear mapping $\\sqbrk {\\, \\cdot, \\cdot \\,}: A_R^2 \\to A_R$ defined as: :$\\forall a, b \\in A_R: \\sqbrk {a, b} := a \\oplus b - b \\oplus a$ Then $\\sqbrk {\\, \\cdot, \\cdot \\,}$ is known as the '''commutator''' of $\\struct {A_R, \\oplus}$. Note that trivially if $\\struct {A_R, \\oplus}$ is a commutative algebra, then: :$\\forall a, b \\in A_R: \\sqbrk {a, b} = \\mathbf 0_R$"}
+{"_id": "22000", "title": "Definition:Star-Algebra", "text": "Let $A = \\left({A_F, \\oplus}\\right)$ be an unitary division algebra. Let $A$ have a mapping $*: A \\to A$ such that: :$\\forall a \\in A: \\left({a^*}\\right)^* = a$ :$\\forall a, b \\in A: \\left({a \\oplus b}\\right)^* = b^* \\oplus a^*$ Then $A$ is a '''$*$-algebra''' (usually voiced '''star-algebra'''). The mapping $*: A \\to A$ is a conjugation on $A$."}
+{"_id": "22001", "title": "Definition:Antihomomorphism", "text": "Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be a mapping from one algebraic structure $\\struct {S, \\circ}$ to another $\\struct {T, *}$. Then $\\phi$ is an '''antihomomorphism''' {{iff}}: :$\\forall x, y \\in S: \\map \\phi {x \\circ y} = \\map \\phi y * \\map \\phi x$ For structures with more than one operation, $\\phi$ may be '''antihomomorphic''' for a subset of those operations."}
+{"_id": "22002", "title": "Definition:Antihomomorphism/Ring Antihomomorphism", "text": "Let $\\struct {R, +, \\circ}$ and $\\struct {S, \\oplus, *}$ be rings. Then $\\phi: R \\to S$ is a '''ring antihomomorphism''' {{iff}}: {{begin-eqn}} {{eqn | lo= \\forall a, b \\in R:       | l = \\map \\phi {a + b}       | r = \\map \\phi a \\oplus \\map \\phi b }} {{eqn | lo= \\forall a, b \\in R:       | l = \\map \\phi {a \\circ b}       | r = \\map \\phi b * \\map \\phi a }} {{end-eqn}}"}
+{"_id": "22003", "title": "Definition:Conjugation (Abstract Algebra)", "text": "Let $A = \\left({A_F, \\oplus}\\right)$ be an algebra over a field $F$. Let $C: A_F \\to A_F$ be a mapping such that: :$\\forall a \\in A: C \\left({C \\left({a}\\right)}\\right) = a$ :$\\forall a, b \\in A: C \\left({a \\oplus b}\\right) = C \\left({b}\\right) \\oplus C \\left({a}\\right)$ Then $C$ is called a '''conjugation''' on $A$. === Conjugate === {{:Definition:Conjugation (Abstract Algebra)/Conjugate}}"}
+{"_id": "22004", "title": "Definition:Nicely Normed Star-Algebra", "text": "Let $A = \\left({A_F, \\oplus}\\right)$ be a star-algebra whose conjugation is denoted $*$. Then $A$ is a '''nicely normed $*$-algebra''' {{iff}}: :$\\forall a \\in A: a + a^* \\in \\R$ :$\\forall a \\in A, a \\ne 0: 0 < a \\oplus a^* = a^* \\oplus a \\in \\R$ === Real Part === Let $a \\in A$ be an element of a '''nicely normed $*$-algebra'''. Then the '''real part''' of $a$ is given by: :$\\Re \\left({a}\\right) = \\dfrac {a + a^*} 2$   === Imaginary Part === Let $a \\in A$ be an element of a '''nicely normed $*$-algebra'''. Then the '''imaginary part''' of $a$ is given by: :$\\Im \\left({a}\\right) = \\dfrac {a - a^*} 2$ === Norm === Let $a \\in A$ be an element of a '''nicely normed $*$-algebra'''. Then we can define a norm on $a$ by: :$\\left\\Vert{a}\\right\\Vert^2 = a \\oplus a^*$"}
+{"_id": "22005", "title": "Definition:Real Star-Algebra", "text": "Let $A = \\left({A_F, \\oplus}\\right)$ be an star-algebra whose conjugation is denoted $*$. Then $A$ is a '''real $*$-algebra''' {{iff}}: :$\\forall a \\in A: a^*= a$"}
+{"_id": "22006", "title": "Definition:Subalgebra", "text": "Let $R$ be a commutative ring. Let $\\left({A, *}\\right)$ be an algebra over $R$. A '''subalgebra''' of $A$ is a submodule $B \\subseteq A$ such that: :$\\forall x, y \\in B: x * y \\in B$ That is, such that $B$ is closed under $*$."}
+{"_id": "22007", "title": "Definition:Unital Subalgebra", "text": "Let $R$ be a commutative ring. Let $\\left({A_R, *}\\right)$ be an unital algebra over $R$ whose unit is $1_A$. Let $\\left({B_R, *}\\right)$ be a subalgebra of $A_R$. === Definition 1 === {{:Definition:Unital Subalgebra/Definition 1}} === Definition 2 === {{:Definition:Unital Subalgebra/Definition 2}}"}
+{"_id": "22008", "title": "Definition:Generator of Field", "text": "Let $F$ be a field. Let $S \\subseteq F$ be a subset and $K \\le F$ a subfield. The '''field generated by $S$''' is the smallest subfield of $F$ containing $S$. The '''subring of $F$ generated by $K \\cup S$''', written $K \\left[{S}\\right]$, is the smallest subring of $F$ containing $K \\cup S$. The '''subfield of $F$ generated by $K \\cup S$''', written $K \\left({S}\\right)$, is the smallest subfield of $F$ containing $K \\cup S$."}
+{"_id": "22009", "title": "Definition:Generator of Semigroup", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroup. Let $\\varnothing \\subset X \\subseteq S$. Let $\\left({T, \\circ}\\right)$ be the smallest subsemigroup of $\\left({S, \\circ}\\right)$ such that $X \\subseteq T$. Then: : $X$ is a '''generator'''  of $\\left({T, \\circ}\\right)$ : $X$ '''generates''' $\\left({T, \\circ}\\right)$ : $\\left({T, \\circ}\\right)$ is the '''subsemigroup of $\\left({S, \\circ}\\right)$ generated by $X$'''. This is written $T = \\left \\langle {X} \\right \\rangle$."}
+{"_id": "22010", "title": "Definition:Generated Algebraic Substructure", "text": "Let $\\left({A, \\circ}\\right)$ be an algebraic structure. Let $G \\subseteq A$ be any subset of $A$. The '''algebraic substructure generated''' by $G$ is the smallest substructure of $\\left({A, \\circ}\\right)$ which contains $G$. It is written $\\left \\langle {G} \\right \\rangle$."}
+{"_id": "22011", "title": "Definition:Generator of Algebra", "text": "Let $\\left({A_R, \\oplus}\\right)$ be an algebra over a ring $R$. Let $S \\subseteq A_R$ be a subset of $A_R$. The '''subalgebra generated by $S$''' is the smallest subalgebra $B_R$ of $A_R$ which contains $S$."}
+{"_id": "22012", "title": "Definition:Alternative Algebra", "text": "Let $\\struct {A_R, \\oplus}$ be an algebra over a ring $R$. Then $\\struct {A_R, \\oplus}$ is an alternative algebra {{iff}} $\\oplus$ is an alternative operation. That is: :For all $a, b \\in A_R$, the subalgebra generated by $\\set {a, b}$ is an associative algebra."}
+{"_id": "22013", "title": "Definition:Power-Associative Algebra", "text": "Let $\\left({A_R, \\oplus}\\right)$ be an algebra over a ring $R$. Then $\\left({A_R, \\oplus}\\right)$ is a power-associative algebra {{iff}} $\\oplus$ is power-associative. That is: :For all $a \\in A_R$, the subalgebra generated by $\\left\\{{a}\\right\\}$ is an associative algebra."}
+{"_id": "22014", "title": "Definition:Cayley-Dickson Construction", "text": "Let $A = \\left({A_F, \\oplus}\\right)$ be a $*$-algebra. The '''Cayley-Dickson Construction''' on $A$ is the procedure which generates a new algebra $A'$ from $A$ as follows. Let: : $A' = \\left({A'_F, \\oplus'}\\right) = \\left({A, \\oplus}\\right)^2$ where $\\left({A, \\oplus}\\right)^2$ denotes the Cartesian product of $\\left({A, \\oplus}\\right)$ with itself. Then $\\oplus'$ and $*'$ are defined on $A'$ as follows: :$\\left({a, b}\\right) \\oplus' \\left({c, d}\\right) = \\left({a \\oplus c - d \\oplus b^*, a^* \\oplus d + c \\oplus b}\\right)$ :${\\left({a, b}\\right)^*}' = \\left({a^*, -b}\\right)$ where: : $\\left({a, b}\\right), \\left({c, d}\\right) \\in A'$ : $a^*$ is the conjugation of $a \\in A$."}
+{"_id": "22015", "title": "Definition:Class Membership", "text": "To define membership not only for sets, but also for proper classes, we will extend the membership relation to include specific behaviors with proper classes and sets alike: :$\\forall A, B: \\paren {A \\in B \\iff \\exists x: \\paren {A = x \\land x \\in B } }$ With this definition, no proper classes is a member of any other class, proper or not."}
+{"_id": "22016", "title": "Definition:Universal Class", "text": "The '''universal class''' is the class of which all sets are members. The '''universal class''' is defined most commonly in literature as: :$V = \\set {x: x = x}$ where $x$ ranges over all sets. It can be briefly defined as the '''class of all sets'''."}
+{"_id": "22017", "title": "Definition:Real Element in Star-Algebra", "text": "Let $A = \\left({A_F, \\oplus}\\right)$ be a $*$-algebra. Let $A' = \\left({A_F, \\oplus'}\\right)$ be constructed from $A$ using the Cayley-Dickson construction. Let $a \\in A$ be real. Then $\\left({a, 0}\\right)$ is defined as real in $A'$. {{SUBPAGENAME}} hiap5z48d0c7np6b1ydn6qgk1gnjqd8"}
+{"_id": "22018", "title": "Definition:Octonion", "text": "The set of '''octonions''', usually denoted $\\Bbb O$, can be defined by using the Cayley-Dickson construction from the quaternions $\\Bbb H$ as follows: From Quaternions form Algebra, $\\Bbb H$ forms a nicely normed $*$-algebra. Let $a, b \\in \\Bbb H$. Then $\\tuple {a, b} \\in \\Bbb O$, where: :$\\tuple {a, b} \\tuple {c, d} = \\tuple {a c - d \\overline b, \\overline a d + c b}$ :$\\overline {\\tuple {a, b} } = \\tuple {\\overline a, -b}$ where: :$\\overline a$ is the conjugate on $a$ and  :$\\overline {\\tuple {a, b} }$ is the conjugation operation on $\\Bbb O$."}
+{"_id": "22019", "title": "Definition:Quaternion/Construction from Cayley-Dickson Construction", "text": "The set of quaternions $\\Bbb H$ can be defined by the Cayley-Dickson construction from the set of complex numbers $\\C$. From Complex Numbers form Algebra, $\\C$ forms a nicely normed $*$-algebra. Let $a, b \\in \\C$. Then $\\left({a, b}\\right) \\in \\Bbb H$, where: :$\\left({a, b}\\right) \\left({c, d}\\right) = \\left({a c - d \\overline b, \\overline a d + c b}\\right)$ :$\\overline {\\left({a, b}\\right)} = \\left({\\overline a, -b}\\right)$ where: :$\\overline a$ is the complex conjugate of $a$ and  :$\\overline {\\left({a, b}\\right)}$ is the conjugation operation on $\\Bbb H$. It is clear by direct comparison with the Construction from Complex Pairs that this construction genuinely does generate the Quaternions."}
+{"_id": "22020", "title": "Definition:Complex Number/Construction from Cayley-Dickson Construction", "text": "The complex numbers can be defined by the Cayley-Dickson construction from the set of real numbers $\\R$. From Real Numbers form Algebra, $\\R$ forms a nicely normed $*$-algebra. Let $a, b \\in \\R$. Then $\\left({a, b}\\right) \\in \\C$, where: :$\\left({a, b}\\right) \\left({c, d}\\right) = \\left({a c - d \\overline b, \\overline a d + c b}\\right)$ :$\\overline {\\left({a, b}\\right)} = \\left({\\overline a, -b}\\right)$ where: :$\\overline a$ is the conjugate of $a$ and  :$\\overline {\\left({a, b}\\right)}$ is the conjugation operation on $\\C$. From Real Numbers form Algebra, $\\overline a = a$ and so the above translate into: :$\\left({a, b}\\right) \\left({c, d}\\right) = \\left({a c - d b, a d + c b}\\right)$ :$\\overline {\\left({a, b}\\right)} = \\left({a, -b}\\right)$ It is clear by direct comparison with the formal definition that this construction genuinely does generate the complex numbers."}
+{"_id": "22021", "title": "Definition:Monomial", "text": "=== Monomial of Polynomial Ring === {{:Definition:Monomial of Polynomial Ring}} === Monomial of Free Commutative Monoid === {{:Definition:Monomial of Free Commutative Monoid}}"}
+{"_id": "22022", "title": "Definition:Homogeneous Polynomial", "text": "A '''homogeneous polynomial''' is a polynomial whose monomials with nonzero coefficients all have the same total degree."}
+{"_id": "22023", "title": "Definition:Quadratic Form", "text": "Let $\\mathbb K$ be a field of characteristic $\\Char {\\mathbb K} \\ne 2$. Let $V$ be a vector space over $\\mathbb K$. A '''quadratic form''' on $V$ is a mapping $q : V \\mapsto \\mathbb K$ such that: :$\\forall v \\in V : \\forall \\kappa \\in \\mathbb K : \\map q {\\kappa v} = \\kappa^2 \\map q v$ :$b: V \\times V \\to \\mathbb K: \\tuple {v, w} \\mapsto \\map q {v + w} - \\map q v - \\map q w$ is a bilinear form"}
+{"_id": "22024", "title": "Definition:Galois Field", "text": "A '''Galois field''' $\\struct {\\GF, +, \\circ}$ is a field such that $\\GF$ is a finite set."}
+{"_id": "22025", "title": "Definition:Infinite Field", "text": "An infinite field $\\struct {F, +, \\cdot}$ is a field such that $F$ is a infinite set."}
+{"_id": "22026", "title": "Definition:Subtraction/Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. The operation of '''subtraction''' $a - b$ on $R$ is defined as: :$\\forall a, b \\in R: a - b := a + \\paren {-b}$ where $-b$ is the (ring) negative of $b$."}
+{"_id": "22027", "title": "Definition:Zero Divisor/Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. A '''zero divisor (in $R$)''' is an element $x \\in R$ such that either: :$\\exists y \\in R^*: x \\circ y = 0_R$ or: :$\\exists y \\in R^*: y \\circ x = 0_R$ where $R^*$ is defined as $R \\setminus \\set {0_R}$. That is, such that $x$ is either a left zero divisor or a right zero divisor. The expression: : '''$x$ is a zero divisor''' can be written: : $x \\divides 0_R$"}
+{"_id": "22028", "title": "Definition:Zero Divisor/Algebra", "text": "Let $\\struct {A_R, \\oplus}$ be an algebra over a ring $\\struct {R, +, \\cdot}$. Let the zero vector of $A_R$ be $\\mathbf 0_R$. Let $a, b \\in A_R$ such that $a \\ne \\mathbf 0_R$ and $b \\ne \\mathbf 0_R$. Then $a$ and $b$ are '''zero divisors of $A_R$''' {{iff}}: :$a \\oplus b = \\mathbf 0_R$"}
+{"_id": "22029", "title": "Definition:Multiplicative Group", "text": "Let $\\struct {F, +, \\times}$ be a field. Let $F^* := F \\setminus \\set 0$ be the set $F$ less its zero. The group $\\struct {F^*, \\times}$ is known as the '''multiplicative group of $F$'''."}
+{"_id": "22030", "title": "Definition:Multiplicative Identity", "text": "Let $\\struct {F, +, \\times}$ be a field. Then the identity element of the multiplicative group $\\struct {F^*, \\times}$ of $F$ is called the multiplicative identity of $F$. It is often denoted $e_F$ or $1_F$, or, if there is no danger of ambiguity, $e$ or $1$. {{refactor|Either merge this concept with that of Definition:Unity of Field or enter this statement as a page in its own right.}} Note that the multiplicative identity of $F$ is the unity of the ring that $\\struct {F, +, \\times}$ is by definition of a field."}
+{"_id": "22031", "title": "Definition:Multiplicative Inverse/Field", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0_F$. Let $a \\in F$ such that $a \\ne 0_F$. Then the inverse element of $a$ with respect to the $\\times$ operator is called the '''multiplicative inverse''' of $F$. It is usually denoted $a^{-1}$ or $\\dfrac 1 a$."}
+{"_id": "22032", "title": "Definition:Affine Transformation", "text": "Let $\\mathcal E$ and $\\mathcal F$ be affine spaces with difference spaces $E$ and $F$ respectively. Let $\\mathcal L: \\mathcal E \\to \\mathcal F$ be a mapping. Then $\\mathcal L$ is an '''affine transformation''' or '''affine mapping''' if there exists a linear transformation $L: E \\to F$ such that for every pair of points $p, q \\in \\mathcal E$: :$\\map {\\mathcal L} q = \\map {\\mathcal L} p + \\map L {\\vec {p q} }$"}
+{"_id": "22033", "title": "Definition:Annihilator of Ring", "text": "Let $B: R \\times \\Z$ be a bilinear mapping defined as: :$B: R \\times \\Z: \\tuple {r, n} \\mapsto n \\cdot r$ where $n \\cdot r$ defined as an integral multiple of $r$: :$n \\cdot r = r + r + \\cdots \\paren n \\cdots r$ Note the change of order of $r$ and $n$: :$\\map B {r, n} = n \\cdot r$ Let $D \\subseteq R$ be a subring of $R$. Then the '''annihilator''' of $D$ is defined as: :$\\map {\\mathrm {Ann} } D = \\set {n \\in \\Z: \\forall d \\in D: n \\cdot d = 0_R}$ or, when $D = R$: :$\\map {\\mathrm {Ann} } R = \\set {n \\in \\Z: \\forall r \\in R: n \\cdot r = 0_R}$ It is seen to be, therefore, the set of all integers whose integral multiples, with respect to the elements of a ring or a field, are all equal to the zero of that ring or field. === Trivial Annihilator === {{:Definition:Trivial Annihilator}}"}
+{"_id": "22034", "title": "Definition:Ring Endomorphism", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\phi: R \\to R$ be a (ring) homomorphism from $R$ to itself. Then $\\phi$ is a ring endomorphism."}
+{"_id": "22035", "title": "Definition:Zero Homomorphism", "text": "Let $\\struct {R_1, +_1, \\circ_1}$ and $\\struct {R_2, +_2, \\circ_2}$ be rings with zeroes $0_1$ and $0_2$ respectively. Consider the mapping $\\zeta: R_1 \\to R_2$ defined as: :$\\forall r \\in R_1: \\map \\zeta r = 0_2$ Then $\\zeta$ is '''the zero homomorphism from $R_1$ to $R_2$'''."}
+{"_id": "22036", "title": "Definition:Identity Automorphism", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Consider the identity mapping $I_R: R \\to R$: :$\\forall r \\in R: \\map {I_R} r = r$ Then $I_R$ is '''the identity automorphism on $R$'''."}
+{"_id": "22037", "title": "Definition:Field Monomorphism", "text": "Let $\\struct {F, +, \\circ}$ and $\\struct {K, \\oplus, *}$ be fields. Let $\\phi: F \\to K$ be a (field) homomorphism. Then $\\phi$ is a field monomorphism {{iff}} $\\phi$ is an injection."}
+{"_id": "22038", "title": "Definition:Isomorphism (Abstract Algebra)/F-Isomorphism", "text": "Let $R, S$ be rings with unity. Let $F$ be a subfield of both $R$ and $S$. Let $\\phi: R \\to S$ be an $F$-homomorphism such that $\\phi$ is bijective. Then $\\phi$ is an '''$F$-isomorphism'''. The relationship between $R$ and $S$ is denoted $R \\cong_F S$."}
+{"_id": "22039", "title": "Definition:R-Algebraic Structure Epimorphism", "text": "Let $\\left({S, \\ast_1, \\ast_2, \\ldots, \\ast_n, \\circ}\\right)_R$ and $\\left({T, \\odot_1, \\odot_2, \\ldots, \\odot_n, \\otimes}\\right)_R$ be $R$-algebraic structures. Then $\\phi: S \\to T$ is an '''$R$-algebraic structure epimorphism''' {{iff}}: : $(1): \\quad \\phi$ is a surjection : $(2): \\quad \\forall k: k \\in \\left[{1 \\,.\\,.\\, n}\\right]: \\forall x, y \\in S: \\phi \\left({x \\ast_k y}\\right) = \\phi \\left({x}\\right) \\odot_k \\phi \\left({y}\\right)$ : $(3): \\quad \\forall x \\in S: \\forall \\lambda \\in R: \\phi \\left({\\lambda \\circ x}\\right) = \\lambda \\otimes \\phi \\left({x}\\right)$ This definition also applies to modules, and also to vector spaces."}
+{"_id": "22040", "title": "Definition:Field Epimorphism", "text": "Let $\\left({F, +, \\circ}\\right)$ and $\\left({K, \\oplus, *}\\right)$ be fields. Let $\\phi: R \\to S$ be a (field) homomorphism. Then $\\phi$ is a field epimorphism {{iff}} $\\phi$ is a surjection."}
+{"_id": "22041", "title": "Definition:Group Endomorphism", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\phi: G \\to G$ be a (group) homomorphism from $G$ to itself. Then $\\phi$ is a group endomorphism."}
+{"_id": "22042", "title": "Definition:R-Algebraic Structure Endomorphism", "text": "Let $\\struct {S, \\ast_1, \\ast_2, \\ldots, \\ast_n, \\circ}_R$ be an $R$-algebraic structure. Let $\\phi: S \\to S$ be an $R$-algebraic structure homomorphism from $S$ to itself. Then $\\phi$ is an $R$-algebraic structure endomorphism. This definition continues to apply when $S$ is a module, and also when it is a vector space."}
+{"_id": "22043", "title": "Definition:Field Endomorphism", "text": "Let $\\struct {F, +, \\circ}$ be a field. Let $\\phi: F \\to F$ be a (field) homomorphism from $F$ to itself. Then $\\phi$ is a field endomorphism."}
+{"_id": "22044", "title": "Definition:Homogeneous Linear Equations", "text": "A '''system of homogeneous linear equations''' is a set of simultaneous linear equations: :$\\displaystyle \\forall i \\in \\closedint 1 m: \\sum_{j \\mathop = 1}^n \\alpha_{i j} x_j = \\beta_i$ such that all the $\\beta_i$ are equal to zero: :$\\displaystyle \\forall i \\in \\closedint 1 m : \\sum_{j \\mathop = 1}^n \\alpha_{i j} x_j = 0$ That is: {{begin-eqn}} {{eqn | l = 0       | r = \\alpha_{11} x_1 + \\alpha_{12} x_2 + \\cdots + \\alpha_{1n} x_n }} {{eqn | l = 0       | r = \\alpha_{21} x_1 + \\alpha_{22} x_2 + \\cdots + \\alpha_{2n} x_n }} {{eqn | o = \\cdots}} {{eqn | l = 0       | r = \\alpha_{m1} x_1 + \\alpha_{m2} x_2 + \\cdots + \\alpha_{mn} x_n }} {{end-eqn}} === Matrix Representation === {{:Definition:Homogeneous Linear Equations/Matrix Representation}}"}
+{"_id": "22045", "title": "Definition:Set/Explicit Set Definition", "text": "A (finite) set can be defined by '''explicitly''' specifying ''all'' of its elements between the famous curly brackets, known as '''set braces''': $\\set {}$. When a set is defined like this, note that ''all'' and ''only'' the elements in it are listed. This is called '''explicit (set) definition'''. It is possible for a set to contain other sets. For example: :$S = \\set {a, \\set a}$"}
+{"_id": "22046", "title": "Definition:Set/Implicit Set Definition", "text": "If the elements in a set have an ''obvious'' pattern to them, we can define the set '''implicitly''' by using an ellipsis ($\\ldots$). For example, suppose $S = \\set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$. A more compact way of defining this set is: :$S = \\set {1, 2, \\ldots, 10}$ With this notation we are asked to suppose that the numbers count up uniformly, and we can read this definition as: :'''$S$ is the set containing $1$, $2$, and so on, up to $10$.'''"}
+{"_id": "22047", "title": "Definition:Set/Definition by Predicate", "text": "An object can be specified by means of a '''predicate''', that is, in terms of a property (or properties) that it possesses. Whether an object $x$ possesses a particular (characteristic) property $P$ is either true or false (in Aristotelian logic) and so can be the subject of a propositional function $\\map P x$. Hence a set can be specified by means of such a propositional function: :$S = \\set {x: \\map P x}$ which means: :'''$S$ is the set of all objects which have the property $P$''' or, more formally: :'''$S$ is the set of all $x$ such that $\\map P x$ is true.''' In this context, we see that the symbol $:$ is interpreted as '''such that'''."}
+{"_id": "22048", "title": "Definition:Real Number/Operations on Real Numbers", "text": "We interpret the following symbols: {{begin-axiom}} {{axiom | n = \\text R 1         | lc= Negative         | q = \\forall a \\in \\R         | m = \\exists ! \\paren {-a} \\in \\R: a + \\paren {-a} = 0 }} {{axiom | n = \\text R 2         | lc= Minus         | q = \\forall a, b \\in \\R         | m = a - b = a + \\paren {-b} }} {{axiom | n = \\text R 3         | lc= Reciprocal         | q = \\forall a \\in \\R \\setminus \\set 0         | m = \\exists ! a^{-1} \\in \\R: a \\times \\paren {a^{-1} } = 1 = \\paren {a^{-1} } \\times a         | rc= it is usual to write $1/a$ or $\\dfrac 1 a$ for $a^{-1}$ }} {{axiom | n = \\text R 4         | lc= Divided by         | q = \\forall a \\in \\R \\setminus \\set 0         | m = a \\div b = \\dfrac a b = a / b = a \\times \\paren {b^{-1} }         | rc= it is usual to write $1/a$ or $\\dfrac 1 a$ for $a^{-1}$ }} {{end-axiom}} The validity of all these operations is justified by Real Numbers form Field."}
+{"_id": "22049", "title": "Definition:Real Number/Real Number Line", "text": "From the Cantor-Dedekind Hypothesis, the set of real numbers is isomorphic to any infinite straight line. The '''real number line''' is an arbitrary infinite straight line each of whose points is identified with a real number such that the distance between any two real numbers is consistent with the length of the line between those two points. :800px"}
+{"_id": "22050", "title": "Definition:Bounded Above Set", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. A subset $T \\subseteq S$ is '''bounded above (in $S$)''' {{iff}} $T$ admits an upper bound (in $S$)."}
+{"_id": "22051", "title": "Definition:Bounded Above Mapping", "text": "Let $f: S \\to T$ be a mapping whose codomain is an ordered set $\\struct {T, \\preceq}$. Then $f$ is '''bounded above on $S$''' by the upper bound $H$ {{iff}}: :$\\forall x \\in S: \\map f x \\preceq H$ That is, {{iff}} $f \\sqbrk S = \\set {\\map f x: x \\in S}$ is bounded above by $H$."}
+{"_id": "22052", "title": "Definition:Supremum of Set", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $T \\subseteq S$. An element $c \\in S$ is the '''supremum of $T$ in $S$''' {{iff}}: :$(1): \\quad c$ is an upper bound of $T$ in $S$ :$(2): \\quad c \\preceq d$ for all upper bounds $d$ of $T$ in $S$. If there exists a '''supremum''' of $T$ (in $S$), we say that: :'''$T$ admits a supremum (in $S$)''' or :'''$T$ has a supremum (in $S$)'''."}
+{"_id": "22053", "title": "Definition:Supremum of Mapping/Real-Valued Function", "text": "{{:Definition:Supremum of Mapping/Real-Valued Function/Definition 1}}"}
+{"_id": "22054", "title": "Definition:Maximal/Ordered Set", "text": "{{:Definition:Maximal/Ordered Set/Definition 1}}"}
+{"_id": "22055", "title": "Definition:Maximum Value of Real Function/Absolute", "text": "Let $f: \\R \\to \\R$ be a real function. Let $f$ be bounded above by a supremum $B$. It may or may not be the case that $\\exists x \\in \\R: \\map f x = B$. If such a value exists, it is called the '''maximum''' of $f$ on $S$, and that this '''maximum''' is '''attained at $x$'''."}
+{"_id": "22056", "title": "Definition:Bounded Below Set", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. A subset $T \\subseteq S$ is '''bounded below (in $S$)''' {{iff}} $T$ admits a lower bound (in $S$)."}
+{"_id": "22057", "title": "Definition:Bounded Below Mapping", "text": "Let $f: S \\to T$ be a mapping whose codomain is an ordered set $\\struct {T, \\preceq}$. Then $f$ is said to be '''bounded below (in $T$)''' by the lower bound $L$ {{iff}}: :$\\forall x \\in S: L \\preceq \\map f x$ That is, iff $f \\sqbrk S = \\set {\\map f x: x \\in S}$ is bounded below by $L$."}
+{"_id": "22058", "title": "Definition:Upper Bound of Set", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $T$ be a subset of $S$. An '''upper bound for $T$ (in $S$)''' is an element $M \\in S$ such that: :$\\forall t \\in T: t \\preceq M$ That is, $M$ succeeds every element of $T$."}
+{"_id": "22059", "title": "Definition:Upper Bound of Mapping", "text": "Let $f: S \\to T$ be a mapping whose codomain is an ordered set $\\struct {T, \\preceq}$. Let $f$ be bounded above in $T$ by $H \\in T$. Then $H$ is an '''upper bound of $f$'''."}
+{"_id": "22060", "title": "Definition:Lower Bound of Set", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $T$ be a subset of $S$. A '''lower bound for $T$ (in $S$)''' is an element $m \\in S$ such that: :$\\forall t \\in T: m \\preceq t$ That is, $m$ precedes every element of $T$."}
+{"_id": "22061", "title": "Definition:Lower Bound of Mapping", "text": "Let $f: S \\to T$ be a mapping whose codomain is an ordered set $\\struct {T, \\preceq}$. Let $f$ be bounded below in $T$ by $H \\in T$. Then $H$ is a '''lower bound of $f$'''."}
+{"_id": "22062", "title": "Definition:Infimum of Set", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $T \\subseteq S$. An element $c \\in S$ is the '''infimum of $T$ in $S$''' {{iff}}: :$(1): \\quad c$ is a lower bound of $T$ in $S$ :$(2): \\quad d \\preceq c$ for all lower bounds $d$ of $T$ in $S$. If there exists an '''infimum''' of $T$ (in $S$), we say that '''$T$ admits an infimum (in $S$)'''."}
+{"_id": "22063", "title": "Definition:Infimum of Mapping", "text": "Let $S$ be a set. Let $\\struct {T, \\preceq}$ be an ordered set. Let $f: S \\to T$ be a mapping from $S$ to $T$. Let $f \\sqbrk S$, the image of $f$, admit an infimum. Then the '''infimum''' of $f$ (on $S$) is defined by: :$\\displaystyle \\inf_{x \\mathop \\in S} \\map f x = \\inf f \\sqbrk S$ === Real-Valued Function === {{:Definition:Infimum of Mapping/Real-Valued Function}}"}
+{"_id": "22064", "title": "Definition:Bounded Ordered Set", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $T \\subseteq S$ be both bounded below and bounded above in $S$. Then $T$ is '''bounded in $S$'''."}
+{"_id": "22065", "title": "Definition:Bounded Mapping", "text": "Let $\\struct {T, \\preceq}$ be an ordered set. Let $f: S \\to T$ be a mapping. Let the image of $f$ be bounded. Then $f$ is '''bounded'''. That is, $f$ is '''bounded''' {{iff}} it is both bounded above and bounded below."}
+{"_id": "22066", "title": "Definition:Minimal/Ordered Set", "text": "{{:Definition:Minimal/Ordered Set/Definition 1}}"}
+{"_id": "22067", "title": "Definition:Minimum Value of Real Function/Absolute", "text": "Let $f: \\R \\to \\R$ be a real function. Let $f$ be bounded below by an infimum $B$. It may or may not be the case that $\\exists x \\in \\R: \\map f x = B$. If such a value exists, it is called the '''minimum value''' of $f$ on $S$, and this '''minimum''' is '''attained at $x$'''."}
+{"_id": "22068", "title": "Definition:Minimal/Set", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Let $\\mathcal T \\subseteq \\powerset S$ be a subset of $\\powerset S$. Let $\\struct {\\mathcal T, \\subseteq}$ be the ordered set formed on $\\mathcal T$ by $\\subseteq$ considered as an ordering. Then $T \\in \\mathcal T$ is a '''minimal set''' of $\\mathcal T$ {{iff}} $T$ is a minimal element of $\\struct {\\mathcal T, \\subseteq}$. That is: :$\\forall X \\in \\mathcal T: X \\subseteq T \\implies X = T$"}
+{"_id": "22069", "title": "Definition:Continuous Real Function/Interval", "text": "=== Open Interval === This is a straightforward application of continuity on a set. {{Definition:Continuous Real Function/Open Interval}} === Closed Interval === {{Definition:Continuous Real Function/Closed Interval}} === Half Open Intervals === Similar definitions apply to half open intervals: {{Definition:Continuous Real Function/Half Open Interval}}"}
+{"_id": "22070", "title": "Definition:Derivative/Complex Function", "text": "The definition for a complex function is similar to that for real functions. === At a Point === {{:Definition:Derivative/Complex Function/Point}} === On an Open Set === {{:Definition:Derivative/Complex Function/Open Set}}"}
+{"_id": "22071", "title": "Definition:Derivative/Real Function/Derivative at Point", "text": "Let $I$ be an open real interval. Let $f: I \\to \\R$ be a real function defined on $I$. Let $\\xi \\in I$ be a point in $I$. Let $f$ be differentiable at the point $\\xi$. ==== Definition 1 ==== {{:Definition:Derivative/Real Function/Derivative at Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Derivative/Real Function/Derivative at Point/Definition 2}}"}
+{"_id": "22072", "title": "Definition:Derivative/Real Function/Derivative on Interval", "text": "Let $I\\subset\\R$ be an open interval. Let $f : I \\to \\R$ be a real function. Let $f$ be differentiable on the interval $I$. Then the '''derivative of $f$''' is the real function  $f': I \\to \\R$ whose value at each point $x \\in I$ is the derivative $f' \\left({x}\\right)$: :$\\displaystyle \\forall x \\in I: f' \\left({x}\\right) := \\lim_{h \\mathop \\to 0} \\frac {f \\left({x + h}\\right) - f \\left({x}\\right)} h$"}
+{"_id": "22073", "title": "Definition:Derivative/Real Function", "text": "=== At a Point === {{:Definition:Derivative/Real Function/Derivative at Point}} === On an Open Interval === {{:Definition:Derivative/Real Function/Derivative on Interval}} === With Respect To === {{:Definition:Derivative/Real Function/With Respect To}}"}
+{"_id": "22074", "title": "Definition:Derivative/Higher Derivatives/Order of Derivative", "text": "The '''order''' of a derivative is the '''number of times it has been differentiated'''. For example: :a first derivative is of '''first order''', or '''order $1$''' :a second derivative is of '''second order''', or '''order $2$''' and so on."}
+{"_id": "22075", "title": "Definition:Derivative/Higher Derivatives", "text": "=== Second Derivative === {{:Definition:Derivative/Higher Derivatives/Second Derivative}} === Third Derivative === {{:Definition:Derivative/Higher Derivatives/Third Derivative}} === Higher Order Derivatives === Higher order derivatives are defined in similar ways: {{:Definition:Derivative/Higher Derivatives/Higher Order}} === First Derivative === If derivatives of various orders are being discussed, then what has been described here as the derivative is frequently referred to as the '''first derivative''': {{:Definition:Derivative/Real Function/Derivative on Interval}} == Order of Derivative == {{:Definition:Derivative/Higher Derivatives/Order of Derivative}}"}
+{"_id": "22076", "title": "Definition:Derivative/Real Function/With Respect To", "text": "Let $f$ be a real function which is differentiable on an open interval $I$. Let $f$ be defined as an equation: $y = \\map f x$. Then the '''derivative of $y$ with respect to $x$''' is defined as: :$\\displaystyle y^\\prime = \\lim_{h \\mathop \\to 0} \\frac {\\map f {x + h} - \\map f x} h = D_x \\, \\map f x$ This is frequently abbreviated as '''derivative of $y$''' '''WRT''' or '''w.r.t.''' '''$x$''', and often pronounced something like '''wurt'''. We introduce the quantity $\\delta y = \\map f {x + \\delta x} - \\map f x$. This is often referred to as '''the small change in $y$ consequent on the small change in $x$'''. Hence the motivation behind the popular and commonly-seen notation: :$\\displaystyle \\dfrac {\\d y} {\\d x} := \\lim_{\\delta x \\mathop \\to 0} \\dfrac {\\map f {x + \\delta x} - \\map f x} {\\delta x} = \\lim_{\\delta x \\mathop \\to 0} \\dfrac {\\delta y} {\\delta x}$ Hence the notation $\\map {f^\\prime} x = \\dfrac {\\d y} {\\d x}$. This notation is useful and powerful, and emphasizes the concept of a derivative as being the limit of a ratio of very small changes. However, it has the disadvantage that the variable $x$ is used ambiguously: both as the point at which the derivative is calculated and as the variable with respect to which the derivation is done. For practical applications, however, this is not usually a problem."}
+{"_id": "22077", "title": "Definition:O Notation/Big-O Notation", "text": "'''Big-O notation''' occurs in a variety of contexts. === Sequences === {{:Definition:O Notation/Big-O Notation/Sequence}} === Real Analysis === {{:Definition:O Notation/Big-O Notation/Real/Infinity}} === Complex Analysis === {{:Definition:O Notation/Big-O Notation/Complex/Infinity}} === General Definition === {{:Definition:O Notation/Big-O Notation/General Definition/Infinity}}"}
+{"_id": "22078", "title": "Definition:O Notation/Little-O Notation", "text": "'''Little-O notation''' occurs in a variety of contexts. === Sequences === Let $\\sequence {a_n}$ and $\\sequence {b_n}$ be sequences of real or complex numbers. {{:Definition:O Notation/Little-O Notation/Sequence/Definition 1}} === Real Functions === Let $f$ and $g$ be real-valued or complex-valued functions on a subset of $\\R$ containing all sufficiently large real numbers. {{:Definition:O Notation/Little-O Notation/Real Functions/Definition 1}} === Point Estimate === {{:Definition:O Notation/Little-O Notation/Real Point}} === Complex Functions === {{:Definition:O Notation/Little-O Notation/Complex Functions}} === Complex Point Estimate === {{:Definition:O Notation/Little-O Notation/Complex Point}} === General Definition for point estimates === {{:Definition:O Notation/Little-O Notation/General Definition/Point}}"}
+{"_id": "22079", "title": "Definition:Differential", "text": "=== Real Function === {{:Definition:Differential/Real Function/Point}} === Real-Valued Function === {{:Definition:Differential/Real-Valued Function/Point}} === Vector-Valued Function === {{:Definition:Differential/Vector-Valued Function/Point}} === Manifolds === {{:Definition:Differential/Manifolds}} === Functional === {{:Definition:Differential/Functional}}"}
+{"_id": "22080", "title": "Definition:Strictly Monotone/Sequence", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. Then a sequence $\\sequence {a_k}_{k \\mathop \\in A}$ of elements of $S$ is '''strictly monotone''' {{iff}} it is either strictly increasing or strictly decreasing."}
+{"_id": "22081", "title": "Definition:Increasing/Mapping", "text": "Let $\\struct {S, \\preceq_1}$ and $\\struct {T, \\preceq_2}$ be ordered sets. Let $\\phi: S \\to T$ be a mapping. Then $\\phi$ is '''increasing''' {{iff}}: :$\\forall x, y \\in S: x \\preceq_1 y \\implies \\map \\phi x \\preceq_2 \\map \\phi y$ Note that this definition also holds if $S = T$."}
+{"_id": "22082", "title": "Definition:Increasing/Real Function", "text": "Let $f$ be a real function. Then $f$ is '''increasing''' {{iff}}: :$x \\le y \\implies \\map f x \\le \\map f y$"}
+{"_id": "22083", "title": "Definition:Strictly Increasing/Mapping", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be ordered sets. Let $\\phi: \\left({S, \\preceq_1}\\right) \\to \\left({T, \\preceq_2}\\right)$ be a mapping. Then $\\phi$ is '''strictly increasing''' if: :$\\forall x, y \\in S: x \\prec_1 y \\implies \\phi \\left({x}\\right) \\prec_2 \\phi \\left({y}\\right)$ Note that this definition also holds if $S = T$."}
+{"_id": "22084", "title": "Definition:Strictly Increasing/Real Function", "text": "Let $f$ be a real function. Then $f$ is '''strictly increasing''' {{iff}}: : $x < y \\implies f \\left({x}\\right) < f \\left({y}\\right)$"}
+{"_id": "22085", "title": "Definition:Strictly Decreasing/Mapping", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be ordered sets. Let $\\phi: \\left({S, \\preceq_1}\\right) \\to \\left({T, \\preceq_2}\\right)$ be a mapping. Then $\\phi$ is '''strictly decreasing''' {{iff}}: :$\\forall x, y \\in S: x \\prec_1 y \\implies \\phi \\left({y}\\right) \\prec_2 \\phi \\left({x}\\right)$ Note that this definition also holds if $S = T$."}
+{"_id": "22086", "title": "Definition:Strictly Decreasing/Real Function", "text": "Let $f$ be a real function. Then $f$ is '''strictly decreasing''' {{iff}}: :$x < y \\implies \\map f y < \\map f x$"}
+{"_id": "22087", "title": "Definition:Decreasing/Mapping", "text": "Let $\\struct {S, \\preceq_1}$ and $\\struct {T, \\preceq_2}$ be ordered sets. Let $\\phi: \\struct {S, \\preceq_1} \\to \\struct {T, \\preceq_2}$ be a mapping. Then $\\phi$ is '''decreasing''' {{iff}}: :$\\forall x, y \\in S: x \\preceq_1 y \\implies \\map \\phi y \\preceq_2 \\map \\phi x$ Note that this definition also holds if $S = T$."}
+{"_id": "22088", "title": "Definition:Decreasing/Real Function", "text": "Let $f$ be a real function. Then $f$ is '''decreasing''' {{iff}}: :$x \\le y \\implies \\map f y \\le \\map f x$."}
+{"_id": "22089", "title": "Definition:Monotone (Order Theory)/Mapping", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be ordered sets. Let $\\phi: \\left({S, \\preceq_1}\\right) \\to \\left({T, \\preceq_2}\\right)$ be a mapping. Then $\\phi$ is '''monotone''' {{iff}} it is either increasing or decreasing. Note that this definition also holds if $S = T$."}
+{"_id": "22090", "title": "Definition:Monotone (Order Theory)/Real Function", "text": "This definition continues to hold when $S = T = \\R$. Thus, let $f$ be a real function. Then $f$ is '''monotone''' {{iff}} it is either increasing or decreasing."}
+{"_id": "22091", "title": "Definition:Strictly Monotone/Mapping", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be ordered sets. Let $\\phi: \\left({S, \\preceq_1}\\right) \\to \\left({T, \\preceq_2}\\right)$ be a mapping. Then $\\phi$ is '''strictly monotone''' {{iff}} it is either strictly increasing or strictly decreasing. Note that this definition also holds if $S = T$."}
+{"_id": "22092", "title": "Definition:Strictly Monotone/Real Function", "text": "Let $f: S \\to \\R$ be a real function, where $S \\subseteq \\R$. Then $f$ is '''strictly monotone''' {{iff}} it is either strictly increasing or strictly decreasing."}
+{"_id": "22093", "title": "Definition:Analytic Function/Real Numbers", "text": "Let $f$ be a real function which is smooth on the open interval $\\openint a b$. Let $\\xi \\in \\openint a b$. Let $\\openint c d \\subseteq \\openint a b$ be an open interval such that: :$(1): \\quad \\xi \\in \\openint c d$ :$(2): \\quad \\displaystyle \\forall x \\in \\openint c d: \\map f x = \\sum_{n \\mathop = 0}^\\infty \\frac {\\paren {x - \\xi}^n} {n!} \\map {f^{\\paren n} } x$ Then $f$ is described as being '''analytic''' at the point $\\xi$. That is, a function is analytic at a point if it equals its Taylor series expansion in some interval containing that point."}
+{"_id": "22094", "title": "Definition:Analytic Function/Complex Plane", "text": "Let $U \\subset \\C$ be an open set. Let $f : U \\to \\C$ be a complex function. Then $f$ is '''analytic''' in $U$ {{iff}} for every $z_0 \\in U$ there exists a sequence $(a_n) : \\N \\to \\C$ such that the series: :$\\displaystyle \\sum_{n\\mathop = 0}^\\infty a_n(z-z_0)^n$ converges to $f(z)$ in a neighborhood of $z_0$ in $U$."}
+{"_id": "22095", "title": "Definition:Tangent/Geometry", "text": "=== Tangent Line === {{:Definition:Tangent/Geometry/Tangent Line}} === Tangent Circles === {{:Definition:Tangent/Geometry/Tangent Circles}}"}
+{"_id": "22096", "title": "Definition:Cosecant/Analysis", "text": "==== Real Function ==== {{:Definition:Cosecant/Real Function}} ==== Complex Function ==== {{:Definition:Cosecant/Complex Function}}"}
+{"_id": "22097", "title": "Definition:Propositional Expansion/Universal Quantifier", "text": "Let $\\forall$ be the universal quantifier. What $\\forall x: \\map P x$ means is: :$\\mathbf X_1$ has property $P$, and $\\mathbf X_2$ has property $P$, and $\\mathbf X_3$ has property $P$, and ... This translates into propositional logic as: :$\\map P {\\mathbf X_1} \\land \\map P {\\mathbf X_2} \\land \\map P {\\mathbf X_3} \\land \\ldots$ This expression of $\\forall x$ as a conjunction is known as the '''propositional expansion''' of $\\forall x$.  The '''propositional expansion''' for the universal quantifier can exist in actuality only when the number of objects in the universe is finite. If the universe is infinite, then the '''propositional expansion''' can exist only conceptually, and the universal quantifier cannot be eliminated."}
+{"_id": "22098", "title": "Definition:Propositional Expansion/Existential Quantifier", "text": "Let $\\exists$ be the existential quantifier. What $\\exists x: \\map P x$ means is: :At least one of $\\mathbf X_1, \\mathbf X_2, \\mathbf X_3, \\ldots$ has property $P$. This means: :Either $\\mathbf X_1$ has property $P$, or $\\mathbf X_2$ has property $P$, or $\\mathbf X_3$ has property $P$, or ... This translates into propositional logic as: :$\\map P {\\mathbf X_1} \\lor \\map P {\\mathbf X_2} \\lor \\map P {\\mathbf X_3} \\lor \\ldots$ This expression of $\\exists x$ as a disjunction is known as the '''propositional expansion''' of $\\exists x$. The propositional expansion for the existential quantifier can exist in actuality only when the number of objects in the universe is finite. If the universe is infinite, then the propositional expansion can exist only conceptually, and the existential quantifier cannot be eliminated."}
+{"_id": "22099", "title": "Definition:Inverse Statement", "text": "The '''inverse''' of the conditional: : $p \\implies q$ is the statement: :$\\neg p \\implies \\neg q$"}
+{"_id": "22100", "title": "Definition:Non-reflexive Relation", "text": "$\\mathcal R$ is '''non-reflexive''' {{iff}} it is neither reflexive nor antireflexive."}
+{"_id": "22101", "title": "Definition:Non-symmetric Relation", "text": "$\\RR$ is '''non-symmetric''' {{iff}} it is neither symmetric nor asymmetric."}
+{"_id": "22102", "title": "Definition:Non-transitive Relation", "text": "$\\RR$ is '''non-transitive''' {{iff}} it is neither transitive nor antitransitive."}
+{"_id": "22104", "title": "Definition:Restriction/Operation", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure, and let $T \\subseteq S$. The '''restriction of $\\circ$ to $T \\times T$''' is denoted $\\circ {\\restriction_T}$, and is defined as: :$\\forall t_1, t_2 \\in T: t_1 \\mathbin{\\circ {\\restriction_T}} t_2 = t_1 \\circ t_2$ The notation $\\circ {\\restriction_T}$ is generally used only if it is necessary to emphasise that $\\circ {\\restriction_T}$ is strictly different from $\\circ$ (through having a different domain). When no confusion is likely to result, $\\circ$ is generally used for both. Thus in this context, $\\left({T, \\circ {\\restriction_T}}\\right)$ and $\\left({T, \\circ}\\right)$ mean the same thing."}
+{"_id": "22105", "title": "Definition:Restriction/Relation", "text": "Let $\\RR$ be a relation on $S \\times T$. Let $X \\subseteq S$, $Y \\subseteq T$. The '''restriction of $\\RR$ to $X \\times Y$''' is the relation on $X \\times Y$ defined as: :$\\RR {\\restriction_{X \\times Y} }: = \\RR \\cap \\paren {X \\times Y}$ If $Y = T$, then we simply call this the '''restriction of $\\RR$ to $X$''', and denote it as $\\RR {\\restriction_X}$."}
+{"_id": "22106", "title": "Definition:Restriction/Mapping", "text": "Let $f: S \\to T$ be a mapping. Let $X \\subseteq S$. Let $f \\sqbrk X \\subseteq Y \\subseteq T$. The '''restriction of $f$ to $X \\times Y$''' is the mapping $f {\\restriction_{X \\times Y} }: X \\to Y$ defined as: :$f {\\restriction_{X \\times Y} } = f \\cap \\paren {X \\times Y}$ If $Y = T$, then we simply call this the '''restriction of $f$ to $X$''', and denote it as $f {\\restriction_X}$."}
+{"_id": "22107", "title": "Definition:Identity (Abstract Algebra)/Left Identity", "text": "An element $e_L \\in S$ is called a '''left identity''' {{iff}}: :$\\forall x \\in S: e_L \\circ x = x$"}
+{"_id": "22108", "title": "Definition:Identity (Abstract Algebra)/Right Identity", "text": "An element $e_R \\in S$ is called a '''right identity''' {{iff}}: :$\\forall x \\in S: x \\circ e_R = x$"}
+{"_id": "22109", "title": "Definition:Identity (Abstract Algebra)/Two-Sided Identity", "text": "An element $e \\in S$ is called an '''identity (element)''' {{iff}} it is both a left identity and a right identity: :$\\forall x \\in S: x \\circ e = x = e \\circ x$ In Identity is Unique it is established that an identity element, if it exists, is unique within $\\struct {S, \\circ}$. Thus it is justified to refer to it as '''the''' identity (of a given algebraic structure). This identity is often denoted $e_S$, or $e$ if it is clearly understood what structure is being discussed."}
+{"_id": "22110", "title": "Definition:Inverse (Abstract Algebra)/Left Inverse", "text": "An element $x_L \\in S$ is called a '''left inverse''' of $x$ {{iff}}: :$x_L \\circ x = e_S$"}
+{"_id": "22111", "title": "Definition:Inverse (Abstract Algebra)/Right Inverse", "text": "An element $x_R \\in S$ is called a '''right inverse''' of $x$ {{iff}}: :$x \\circ x_R = e_S$"}
+{"_id": "22112", "title": "Definition:Inverse (Abstract Algebra)/Inverse", "text": "The element $y$ is an '''inverse of $x$''' {{iff}}: :$y \\circ x = e_S = x \\circ y$ that is, {{iff}} $y$ is both: :a left inverse of $x$ and: :a right inverse of $x$."}
+{"_id": "22113", "title": "Definition:Idempotence/Mapping", "text": "Let $f: S \\to S$ be a mapping. Then $f$ is '''idempotent''' {{iff}}: :$\\forall x \\in S: f \\left({f \\left({x}\\right)}\\right) = f \\left({x}\\right)$ That is, {{iff}} applying the same mapping a second time to an argument gives the same result as applying it once. And of course, that means the same as applying it as many times as you want. The condition for '''idempotence''' can also be written: :$f \\circ f = f$ where $\\circ$ denotes composition of mappings."}
+{"_id": "22114", "title": "Definition:Idempotence/Operation", "text": "Let $\\circ: S \\times S \\to S$ be a binary operation on $S$. If ''all'' the elements of $S$ are idempotent under $\\circ$, then the term can be applied to the operation itself: The binary operation $\\circ$ is '''idempotent''' {{iff}}: : $\\forall x \\in S: x \\circ x = x$"}
+{"_id": "22115", "title": "Definition:Idempotence/Element", "text": "Let $\\circ: S \\times S \\to S$ be a binary operation on $S$. Let $x \\in S$ have the property that $x \\circ x = x$. Then $x \\in S$ is described as '''idempotent under the operation $\\circ$'''."}
+{"_id": "22116", "title": "Definition:Unique", "text": "Suppose $A$ and $B$ are two objects whose definition is in terms of a given set of properties. If it can be demonstrated that, in order for both $A$ and $B$ to fulfil those properties, it is necessary for $A$ to be equal to $B$, then $A$ (and indeed $B$) is '''unique'''. Equivalently, there is '''one and only one''', or '''exactly one''', such object. Thus, intuitively, an object is '''unique''' if there is precisely one such object. === Unique Existential Quantifier === In the language of predicate logic, '''uniqueness''' can be defined as follows: Let $\\map P x$ be a propositional function and let $x$ and $y$ be objects.  {{:Definition:Existential Quantifier/Unique}}"}
+{"_id": "22117", "title": "Definition:Paradox", "text": "A '''paradox''' is a statement or group of statements that leads to one of the following: * a contradiction * a situation which defies intuition * a result that is merely \"puzzling\"."}
+{"_id": "22118", "title": "Definition:Regular Representations/Left Regular Representation", "text": "The mapping $\\lambda_a: S \\to S$ is defined as: :$\\forall x \\in S: \\map {\\lambda_a} x = a \\circ x$ This is known as the '''left regular representation of $\\struct {S, \\circ}$ with respect to $a$'''."}
+{"_id": "22119", "title": "Definition:Regular Representations/Right Regular Representation", "text": "The mapping $\\rho_a: S \\to S$ is defined as: :$\\forall x \\in S: \\map {\\rho_a} x = x \\circ a$ This is known as the '''right regular representation of $\\struct {S, \\circ}$ with respect to $a$'''."}
+{"_id": "22120", "title": "Definition:Centralizer/Group Element", "text": "Let $\\struct {G, \\circ}$ be a group. Let $a \\in \\struct {G, \\circ}$. The '''centralizer of $a$ (in $G$)''' is defined as: :$\\map {C_G} a = \\set {x \\in G: x \\circ a = a \\circ x}$ That is, the centralizer of $a$ is the set of elements of $G$ which commute with $a$."}
+{"_id": "22121", "title": "Definition:Centralizer/Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H \\le \\struct {G, \\circ}$. The '''centralizer of $H$ (in $G$)''' is the set of elements of $G$ which commute with all $h \\in H$: :$\\map {C_G} H = \\set {g \\in G: \\forall h \\in H: g \\circ h = h \\circ g}$"}
+{"_id": "22122", "title": "Definition:Centralizer/Ring Subset", "text": "Let $S$ be a subset of a ring $\\struct {R, +, \\circ}$. The '''centralizer of $S$ in $R$''' is defined as: :$\\map {C_R} S = \\set {x \\in R: \\forall s \\in S: s \\circ x = x \\circ s}$ That is, the '''centralizer of $S$''' is the set of elements of $R$ which commute with all elements of $S$."}
+{"_id": "22123", "title": "Definition:Center (Abstract Algebra)", "text": "=== Group === {{:Definition:Center (Abstract Algebra)/Group}} === Ring === {{:Definition:Center (Abstract Algebra)/Ring}}"}
+{"_id": "22125", "title": "Definition:Conjugate (Group Theory)/Element", "text": "Let $\\struct {G, \\circ}$ be a group. === Definition 1 === {{:Definition:Conjugate (Group Theory)/Element/Definition 1}} === Definition 2 === {{:Definition:Conjugate (Group Theory)/Element/Definition 2}}"}
+{"_id": "22126", "title": "Definition:Conjugate (Group Theory)/Subset", "text": "Let $S \\subseteq G, a \\in G$. Then the '''$G$-conjugate of $S$ by $a$''' is: :$S^a := \\set {y \\in G: \\exists x \\in S: y = a \\circ x \\circ a^{-1} } = a \\circ S \\circ a^{-1}$ That is, $S^a$ is the set of all elements of $G$ that are the conjugates of elements of $S$ by $a$. When $G$ is the only group under consideration, we usually just refer to the '''conjugate of $S$ by $a$'''."}
+{"_id": "22127", "title": "Definition:Algebra (Abstract Algebra)", "text": "In the context of abstract algebra, in particular ring theory and linear algebra, the following varieties of '''algebra''' exist: * Definition:Boolean Algebra * Definition:Algebra over Ring: an $R$-module $G_R$ over a commutative ring $R$ with a bilinear mapping $\\oplus: G^2 \\to G$. * Definition:Algebra over Field: a vector space $G_F$ over a field $F$ with a bilinear mapping $\\oplus: G^2 \\to G$. * Definition:Real Algebra: an algebra over a field where the field in question is the field of real numbers $\\R$. * Definition:Division Algebra: an algebra over a field $\\left({A_F, \\oplus}\\right)$ such that $\\forall a, b \\in A_F, b \\ne \\mathbf 0_A: \\exists_1 x \\in A_F, y \\in A_F: a = b \\oplus x, a = y \\oplus b$. * Definition:Associative Algebra: an algebra over a ring in which the bilinear mapping $\\oplus$ is associative.  * Definition:Unitary Algebra, also known as a Unital Algebra: an algebra over a ring $\\left({A_R, \\oplus}\\right)$ in which there exists an identity element, that is, a '''unit''', usually denoted $1$, for $\\oplus$. * Definition:Unitary Division Algebra: a division algebra $\\left({A_F, \\oplus}\\right)$ in which there exists an identity element, that is, a '''unit''', usually denoted $1$, for $\\oplus$. * Definition:Graded Algebra: an algebra over a ring where the ring has a gradation, that is, is a graded ring. * Definition:Filtered Algebra: an algebra over a field which has a sequence of subalgebras which constitute a gradation. * Definition:Quadratic Algebra: a filtered algebra whose generator consists of degree one elements, with defining relations of degree 2. * Definition:Lie Algebra"}
+{"_id": "22128", "title": "Definition:Ideal of Ring/Left Ideal", "text": "$J$ is a '''left ideal of $R$''' {{iff}}: :$\\forall j \\in J: \\forall r \\in R: r \\circ j \\in J$ that is, {{iff}}: :$\\forall r \\in R: r \\circ J \\subseteq J$"}
+{"_id": "22129", "title": "Definition:Ideal of Ring/Right Ideal", "text": "$J$ is a '''right ideal of $R$''' {{iff}}: :$\\forall j \\in J: \\forall r \\in R: j \\circ r \\in J$ that is, {{iff}}: :$\\forall r \\in R: J \\circ r \\subseteq J$"}
+{"_id": "22130", "title": "Definition:Divisor (Algebra)/Ring with Unity", "text": "Let $\\struct {R, +, \\circ}$ be an ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $x, y \\in D$. We define the term '''$x$ divides $y$ in $R$''' as follows: :$x \\mathrel {\\divides_R} y \\iff \\exists t \\in R: y = t \\circ x$ When no ambiguity results, the subscript is usually dropped, and '''$x$ divides $y$ in $R$''' is just written $x \\divides y$."}
+{"_id": "22131", "title": "Definition:Divisor (Algebra)/Integer", "text": "Let $\\struct {\\Z, +, \\times}$ be the ring of integers. Let $x, y \\in \\Z$. Then '''$x$ divides $y$''' is defined as: :$x \\divides y \\iff \\exists t \\in \\Z: y = t \\times x$"}
+{"_id": "22132", "title": "Definition:Divisor (Algebra)/Factorization", "text": "Let $x, y \\in D$ where $\\struct {D, +, \\times}$ is an integral domain. Let $x$ be a divisor of $y$. Then by definition it is possible to find some $t \\in D$ such that $y = t \\times x$. The act of breaking down such a $y$ into the product $t \\circ x$ is called '''factorization'''."}
+{"_id": "22133", "title": "Definition:Divisor (Algebra)/Integer/Aliquot Part", "text": "An '''aliquot part''' of an integer $n$ is a divisor of $n$ which is strictly less than $n$."}
+{"_id": "22134", "title": "Definition:Divisor (Algebra)/Integer/Aliquant Part", "text": "An '''aliquant part''' of an integer $n$ is a positive integer which is less than $n$ but is not a divisor of $n$."}
+{"_id": "22135", "title": "Definition:Common Divisor/Integral Domain", "text": "Let $\\struct {D, +, \\times}$ be an integral domain. Let $S \\subseteq D$ be a finite subset of $D$. Let $c \\in D$ such that $c$ divides all the elements of $S$, that is: :$\\forall x \\in S: c \\divides x$ Then $c$ is a '''common divisor''' (or '''common factor''') of all the elements in $S$."}
+{"_id": "22136", "title": "Definition:Common Divisor/Integers", "text": "Let $S$ be a finite set of integers, that is: :$S = \\set {x_1, x_2, \\ldots, x_n: \\forall k \\in \\N^*_n: x_k \\in \\Z}$ Let $c \\in \\Z$ such that $c$ divides all the elements of $S$, that is: :$\\forall x \\in S: c \\divides x$ Then $c$ is a '''common divisor''' (or '''common factor''') of all the elements in $S$."}
+{"_id": "22137", "title": "Definition:Greatest Common Divisor/Integral Domain", "text": "Let $\\struct {D, +, \\times}$ be an integral domain whose zero is $0$. Let $a, b \\in D: a \\ne 0 \\lor b \\ne 0$. Let $d \\divides a$ denote that $d$ is a divisor of $a$. Let $d \\in D$ have the following properties: : $(1): \\quad d \\divides a \\land d \\divides b$ : $(2): \\quad c \\divides a \\land c \\divides b \\implies c \\divides d$ Then $d$ is called '''a greatest common divisor of $a$ and $b$''' (abbreviated '''GCD''' or '''gcd''') and denoted $\\gcd \\set {a, b}$. That is, in the integral domain $D$, $d$ is the GCD of $a$ and $b$ {{iff}}: : $d$ is a common divisor of $a$ and $b$ : Any other common divisor of $a$ and $b$ also divides $d$. We see that, trivially: :$\\gcd \\set {a, b} = \\gcd \\set {b, a}$ so the set notation is justified."}
+{"_id": "22138", "title": "Definition:Greatest Common Divisor/Integers", "text": "Let $a, b \\in \\Z: a \\ne 0 \\lor b \\ne 0$. === Definition 1 === {{:Definition:Greatest Common Divisor/Integers/Definition 1}} === Definition 2 === {{:Definition:Greatest Common Divisor/Integers/Definition 2}} This is denoted $\\gcd \\set {a, b}$."}
+{"_id": "22139", "title": "Definition:Compact Space/Topology", "text": "=== Definition 1 === {{:Definition:Compact Space/Topology/Definition 1}} === Definition 2 === {{:Definition:Compact Space/Topology/Definition 2}} === Definition 3 === {{:Definition:Compact Space/Topology/Definition 3}} === Definition 4 === {{:Definition:Compact Space/Topology/Definition 4}} === Definition 5 === {{:Definition:Compact Space/Topology/Definition 5}}"}
+{"_id": "22140", "title": "Definition:Compact Space/Real Analysis", "text": "Let $\\R$ be the real number space considered as a topological space under the Euclidean topology. Let $H \\subseteq \\R$. Then $H$ is '''compact in $\\R$''' {{iff}} $H$ is closed and bounded."}
+{"_id": "22141", "title": "Definition:Ordered Tree", "text": "Let $T$ be a rooted tree whose root is $v$. Let $O$ be an orchard whose trees are subtrees of $v$. {{explain|Cannot find a reference to an \"orchard\" in this context -- probably means \"ordered forest\".}} Then $T$ is an '''ordered tree'''. Category:Definitions/Rooted Trees ogux298rvf516io4ev39g5353mzk3xb"}
+{"_id": "22142", "title": "Definition:Closed Ball", "text": "{{:Definition:Closed Ball/Metric Space}}"}
+{"_id": "22143", "title": "Definition:Average Value of Function", "text": "Let $f$ be an integrable function on some closed interval $\\closedint a b$. The '''average value of $f$''' (or '''mean value of $f$''') on $\\closedint a b$ is defined as: :$\\displaystyle \\frac 1 {b - a} \\int_a^b \\map f x \\rd x$"}
+{"_id": "22144", "title": "Definition:Unit Circle", "text": "The '''unit circle''' is a circle with radius equal to $1$."}
+{"_id": "22145", "title": "Definition:Cantor Set/Limit of Intersections", "text": "Define, for $n \\in \\N$, subsequently: :$\\map k n := \\dfrac {3^n - 1} 2$ :$\\displaystyle A_n := \\bigcup_{i \\mathop = 1}^{\\map k n} \\openint {\\frac {2 i - 1} {3^n} } {\\frac {2 i} {3^n} }$ Since $3^n$ is always odd, $\\map k n$ is always an integer, and hence the union will always be perfectly defined. Consider the closed interval $\\closedint 0 1 \\subset \\R$. Define: :$\\CC_n := \\closedint 0 1 \\setminus A_n$ The '''Cantor set''' $\\CC$ is defined as: :$\\displaystyle \\CC = \\bigcap_{n \\mathop = 1}^\\infty \\CC_n$"}
+{"_id": "22146", "title": "Definition:Ternary Notation", "text": "'''Ternary notation''' is the technique of expressing numbers in base $3$. That is, every number $x \\in \\R$ is expressed in the form: :$\\displaystyle x = \\sum_{j \\mathop \\in \\Z} r_j 3^j$ where: :$\\forall j \\in \\Z: r_j \\in \\set {0, 1, 2}$"}
+{"_id": "22147", "title": "Definition:Cantor Set/Ternary Representation", "text": "Consider the closed interval $\\closedint 0 1 \\subset \\R$. The '''Cantor set''' $\\CC$ consists of all the points in $\\closedint 0 1$ which can be expressed in base $3$ without using the digit $1$. From Representation of Ternary Expansions, if any number has two different ternary representations, for example: :$\\dfrac 1 3 = 0.10000 \\ldots = 0.02222$ then at most one of these can be written without any $1$'s in it. Therefore this representation of points of $\\CC$ is unique."}
+{"_id": "22148", "title": "Definition:Finite Path", "text": "Let $G$ be a graph. A '''finite path''' in $G$ is a path consisting of a finite number of edges."}
+{"_id": "22149", "title": "Definition:Cantor Space", "text": "Let $\\CC$ be the Cantor set. Let $\\tau_d$ be the Euclidean topology on $\\R$. Then since $\\CC \\subseteq \\R$, we can endow $\\CC$ with the subspace topology $\\tau_\\CC$. The topological space $\\struct {\\CC, \\tau_\\CC}$ is referred to as the '''Cantor space'''."}
+{"_id": "22150", "title": "Definition:Hyperbolic Sine", "text": "The '''hyperbolic sine''' function is defined on the complex numbers as: :$\\sinh: \\C \\to \\C$: :$\\forall z \\in \\C: \\sinh z := \\dfrac {e^z - e^{-z} } 2$"}
+{"_id": "22151", "title": "Definition:Hyperbolic Cosine", "text": "The '''hyperbolic cosine''' function is defined on the complex numbers as: :$\\cosh: \\C \\to \\C$: :$\\forall z \\in \\C: \\cosh z := \\dfrac {e^z + e^{-z} } 2$"}
+{"_id": "22152", "title": "Definition:Norm", "text": "A '''norm''' is a measure which describes a sense of the '''size''' or '''length''' of a mathematical object. In its various contexts: === Ring === {{:Definition:Norm/Ring}} === Division Ring === {{:Definition:Norm/Division Ring}} === Vector Space === {{:Definition:Norm/Vector Space}} === Algebra === {{:Definition:Norm/Algebra}} === Unital Algebra === Let $R$ be a division ring with norm $\\norm {\\,\\cdot\\,}_R$. {{:Definition:Norm/Unital Algebra}}"}
+{"_id": "22153", "title": "Definition:Hyperbolic Tangent", "text": "{{:Definition:Hyperbolic Tangent/Definition 1}}"}
+{"_id": "22155", "title": "Definition:Domain (Category Theory)", "text": "Let $f: X \\to Y$ be a morphism. Then the '''domain''' of $f$ is defined to be the object $X$. This is usually denoted $X = \\Dom f$ or $X = \\map D f$. Category:Definitions/Morphisms mv70ukx8v5i1q4bf5wdjcx8hdzuqnll"}
+{"_id": "22157", "title": "Definition:Codomain (Category Theory)", "text": "Let $f: X \\to Y$ be a morphism. Then the '''codomain''' of $f$ is defined to be the object $Y$. This is usually denoted $Y = \\Cdm f$. Category:Definitions/Morphisms 6ezwuussj083nlrkxcn5m7hvdirbrji"}
+{"_id": "22158", "title": "Definition:Codomain (Set Theory)/Relation", "text": "The '''codomain''' of a relation $\\RR \\subseteq S \\times T$ is the set $T$. It can be denoted $\\Cdm \\RR$."}
+{"_id": "22159", "title": "Definition:Codomain (Set Theory)/Mapping", "text": "Let $f: S \\to T$ be a mapping. The '''codomain''' of $f$ is the set $T$. It is denoted on {{ProofWiki}} by $\\Cdm f$."}
+{"_id": "22160", "title": "Definition:Domain (Set Theory)/Relation", "text": "Let $\\RR \\subseteq S \\times T$ be a relation. The '''domain''' of $\\RR$ is defined and denoted as: :$\\Dom \\RR := \\set {s \\in S: \\exists t \\in T: \\tuple {s, t} \\in \\RR}$ That is, it is the same as what is defined here as the preimage of $\\RR$."}
+{"_id": "22161", "title": "Definition:Domain (Set Theory)/Mapping", "text": "Let $f: S \\to T$ be a mapping. The '''domain''' of $f$ is $S$, and can be denoted $\\Dom f$. In the context of mappings, the '''domain''' and the preimage of a mapping are the same set."}
+{"_id": "22162", "title": "Definition:Domain (Set Theory)/Binary Operation", "text": "Let $\\circ: S \\times S \\to T$ be a binary operation. The '''domain''' of $\\circ$ is the set $S$ and can be denoted $\\Dom \\circ$."}
+{"_id": "22163", "title": "Definition:Finite Sequence", "text": "A '''finite sequence''' is a sequence whose domain is finite. === Length of Sequence === {{transclude:Definition:Length of Sequence|increase=1|section=tc}}"}
+{"_id": "22164", "title": "Definition:Operation/Binary Operation", "text": "A binary operation is the special case of an operation where the operation has exactly two operands. A '''binary operation''' is a mapping $\\circ$ from the Cartesian product of two sets $S \\times T$ to a universe $\\mathbb U$: :$\\circ: S \\times T \\to \\mathbb U: \\map \\circ {s, t} = y \\in \\mathbb U$ If $S = T$, then $\\circ$ can be referred to as a '''binary operation on''' $S$."}
+{"_id": "22165", "title": "Definition:Locus", "text": "A '''locus''' is a set of points which satisfy a particular condition. Such points usually form a curve or a surface, depending on the context."}
+{"_id": "22166", "title": "Definition:Vertical Tangent Line", "text": "Let $P = \\left({c, f \\left({c}\\right)}\\right)$ be a point on the graph of a real function $f$. The vertical line $x = c$ is a vertical tangent line to the graph of $f$ at $P$ {{iff}} any of the following hold: : $(1): \\quad f$ is right continuous at $c$ and $\\displaystyle \\lim_{x \\mathop \\to c ^+} f' \\left({x}\\right) = +\\infty$ : $(2): \\quad f$ is right continuous at $c$ and $\\displaystyle \\lim_{x \\mathop \\to c ^+} f' \\left({x}\\right) = -\\infty$ : $(3): \\quad f$ is left continuous at $c$ and $\\displaystyle \\lim_{x \\mathop \\to c ^-} f' \\left({x}\\right) = +\\infty$ : $(4): \\quad f$ is left continuous at $c$ and $\\displaystyle \\lim_{x \\mathop \\to c ^-} f' \\left({x}\\right) = -\\infty$"}
+{"_id": "22167", "title": "Definition:Left Zero", "text": "An element $z_L \\in S$ is called a '''left zero element''' (or just '''left zero''') {{iff}}: :$\\forall x \\in S: z_L \\circ x = z_L$"}
+{"_id": "22168", "title": "Definition:Right Zero", "text": "An element $z_R \\in S$ is called a '''right zero element''' (or just '''right zero''') {{iff}}: :$\\forall x \\in S: x \\circ z_R = z_R$"}
+{"_id": "22169", "title": "Definition:Divergent Improper Integral", "text": "An improper integral of a real function $f$ is said to '''diverge''' if any of the following hold: : $(1): \\quad f$ is continuous on $\\left[{a \\,.\\,.\\, +\\infty}\\right)$ and the limit $\\displaystyle \\lim_{b \\mathop \\to +\\infty} \\int_a^b f \\left({x}\\right) \\ \\mathrm d x$ does not exist : $(2): \\quad f$ is continuous on $\\left({-\\infty  \\,.\\,.\\, b}\\right]$ and the limit $\\displaystyle \\lim_{a \\mathop \\to -\\infty} \\int_a^b f \\left({x}\\right) \\ \\mathrm d x$ does not exist : $(3): \\quad f$ is continuous on $\\left[{a \\,.\\,.\\, b}\\right)$, has an infinite discontinuity at $b$, and the limit $\\displaystyle \\lim_{c \\mathop \\to b^-} \\int_a^c f \\left({x}\\right) \\ \\mathrm dx$ does not exist : $(4): \\quad f$ is continuous on $\\left({a \\,.\\,.\\, b}\\right]$, has an infinite discontinuity at $a$, and the limit $\\displaystyle \\lim_{c \\mathop \\to a^+} \\int_c^b f \\left({x}\\right) \\ \\mathrm dx$ does not exist."}
+{"_id": "22170", "title": "Definition:Definite Integral/Limits of Integration", "text": "In the expression $\\displaystyle \\int_a^b \\map f x \\rd x$, the values $a$ and $b$ are called the '''limits of integration'''. If there is no danger of confusing the concept with limit of a function or of a sequence, just '''limits'''. Thus $\\displaystyle \\int_a^b \\map f x \\rd x$ can be voiced: : '''The integral of (the function) $f$ of $x$ {{WRT|Integration}} $x$ (evaluated) between the limits (of integration) $a$ and $b$.''' More compactly (and usually), it is voiced: : '''The integral of $f$ of $x$ {{WRT|Integration}} $x$ between $a$ and $b$''' or: : '''The integral of $f$ of $x$ dee $x$ from $a$ to $b$''' === Lower Limit === {{:Definition:Definite Integral/Limits of Integration/Lower Limit}} === Upper Limit === {{:Definition:Definite Integral/Limits of Integration/Upper Limit}} == Also known as == The interval defined by the '''limits of integration''' can be referred to as the '''range of integration'''. Some sources refer to it as the '''interval of integration'''. == Also see == From the Fundamental Theorem of Calculus, we have that: :$\\displaystyle \\int_a^b \\map f x \\rd x = \\map F b - \\map F a$ where $F$ is a primitive of $f$, that is: :$\\map f x = \\dfrac \\d {\\d x} \\map F x$ Then $\\map F b - \\map F a$ is usually written: :$\\bigintlimits {\\map F x} {x \\mathop = a} {x \\mathop = b} := \\map F b - \\map F a$ or, when there is no chance of ambiguity as to the independent variable: :$\\bigintlimits {\\map F x} a b := \\map F b - \\map F a$ Some sources use: :$\\Big.{\\map F x}\\Big|_a^b := \\map F b - \\map F a$ but this is not recommended, as it is not so clear exactly where the expression being evaluated actually starts."}
+{"_id": "22171", "title": "Definition:Cofinal Subset", "text": "Let $\\left({S, \\mathcal R}\\right)$ be a relational structure, that is, a set $S$ endowed with a binary relation $\\mathcal R$. Let $T \\subseteq S$ be a subset of $S$. Then $T$ is a '''cofinal subset of $S$ with respect to $\\mathcal R$''' {{iff}}: :$\\forall x \\in S: \\exists t \\in T: x \\mathop {\\mathcal R} t$"}
+{"_id": "22172", "title": "Definition:Net (Preordered Set)", "text": "Let $X$ be a nonempty set. Let $\\left({\\Lambda, \\precsim}\\right)$ be a preordered set. Let $F: \\Lambda \\to X$ be a mapping. Then $f$ is referred to as a '''net'''."}
+{"_id": "22174", "title": "Definition:Inner Limit", "text": "Let $\\left({\\mathcal X, \\tau}\\right)$ be a Hausdorff topological space. Let $\\left \\langle {C_n}\\right \\rangle_{n \\in \\N}$ be a sequence of sets in $\\mathcal X$. The '''inner limit''' of $\\left \\langle {C_n}\\right \\rangle_{n \\in \\N}$ is defined as: : $\\displaystyle \\liminf_{n \\to\\infty} \\ C_n := \\left\\{{x : \\exists N \\text{ cofinite set of }\\N, \\exists x_v \\in C_v \\left({v \\in N}\\right) \\text{ such that } x_v \\to x}\\right\\}$ where $x_v \\to x$ denotes convergence in the topology $\\tau$."}
+{"_id": "22175", "title": "Definition:Horizontal Asymptote", "text": "The horizontal line $y = L$ is a '''horizontal asymptote''' of the graph of a real function $f$ if either of the following limits exist: :$\\displaystyle \\lim_{x \\mathop \\to +\\infty} \\map f x = L_1$ :$\\displaystyle \\lim_{x \\mathop \\to -\\infty} \\map f x = L_2$"}
+{"_id": "22176", "title": "Definition:Strict Well-Ordering", "text": "=== Definition 1 === {{:Definition:Strict Well-Ordering/Definition 1}} === Definition 2 === {{:Definition:Strict Well-Ordering/Definition 2}}"}
+{"_id": "22177", "title": "Definition:Foundational Relation", "text": "Let $\\left({A, \\mathcal R}\\right)$ be a relational structure where $A$ is either a proper class or a set. Then $\\mathcal R$ is a '''foundational relation on $A$''' {{iff}} every non-empty subset of $A$ has an $\\mathcal R$-minimal element."}
+{"_id": "22178", "title": "Definition:Coset/Right Coset", "text": "The '''right coset of $y$ modulo $H$''', or '''right coset of $H$ by $y$''', is: :$H y = \\set {x \\in G: \\exists h \\in H: x = h y}$ This is the equivalence class defined by right congruence modulo $H$. That is, it is the subset product with singleton: :$H y = H \\set y$"}
+{"_id": "22179", "title": "Definition:Coset Space/Right Coset Space", "text": "The '''right coset space (of $G$ modulo $H$)''' is the quotient set of $G$ by right congruence modulo $H$, denoted $G / H^r$. It is the set of all the right cosets of $H$ in $G$."}
+{"_id": "22180", "title": "Definition:Coset/Left Coset", "text": "The '''left coset of $x$ modulo $H$''', or '''left coset of $H$ by $x$''', is: :$x H = \\set {y \\in G: \\exists h \\in H: y = x h}$ This is the equivalence class defined by left congruence modulo $H$. That is, it is the subset product with singleton: :$x H = \\set x H$"}
+{"_id": "22181", "title": "Definition:Coset Space/Left Coset Space", "text": "The '''left coset space (of $G$ modulo $H$)''' is the quotient set of $G$ by left congruence modulo $H$, denoted $G / H^l$. It is the set of all the left cosets of $H$ in $G$."}
+{"_id": "22182", "title": "Definition:Congruence Modulo Subgroup/Right Congruence", "text": ":$\\mathcal R^r_H = \\set {\\tuple {x, y} \\in G \\times G: x y^{-1} \\in H}$ This is called '''right congruence modulo $H$'''."}
+{"_id": "22183", "title": "Definition:Congruence Modulo Subgroup/Left Congruence", "text": ":$\\mathcal R^l_H := \\set {\\tuple {x, y} \\in G \\times G: x^{-1} y \\in H}$ This is called '''left congruence modulo $H$'''."}
+{"_id": "22184", "title": "Definition:Transversal (Group Theory)/Left Transversal", "text": "$S$ is a '''left transversal for $H$ in $G$''' {{iff}} every left coset of $H$ contains '''exactly one''' element of $S$."}
+{"_id": "22185", "title": "Definition:Transversal (Group Theory)/Right Transversal", "text": "$S$ is a '''right transversal for $H$ in $G$''' {{iff}} every right coset of $H$ contains '''exactly one''' element of $S$."}
+{"_id": "22186", "title": "Definition:Transitive Class", "text": "Let $A$ denote a class, which can be either a set or a proper class. Then $A$ is '''transitive''' {{iff}} every element of $A$ is also a subclass of $A$. That is, $A$ is '''transitive''' {{iff}}: :$x \\in A \\implies x \\subseteq A$ or: :$\\forall x: \\forall y: \\paren {x \\in y \\land y \\in A \\implies x \\in A}$"}
+{"_id": "22187", "title": "Definition:Epsilon Relation", "text": "In the language of set theory $\\in$, the membership primitive, is neither a class nor a set, but a primitive predicate. To simplify formulations, it is useful to introduce a class which behaves identically to the standard membership relation $\\in$ for sets. This class, denoted $\\Epsilon$, will be referred to as the '''epsilon relation'''. In class-builder notation: :$\\Epsilon := \\left\\{{ \\left({ x , y }\\right) : x \\in y }\\right\\}$ Thus, explicitly, $\\Epsilon$ is a relation, taking arguments from ordered pairs of sets $x$ and $y$. It consists of precisely those ordered pairs $\\left({ x , y }\\right)$ satisfying $x \\in y$. The behavior is thus seen to be identical to regular membership with sets. It is '''not''' the same as class membership, because $x$ and $y$ must be set variables. === Restriction of Epsilon Relation === {{:Definition:Epsilon Relation/Restriction}}"}
+{"_id": "22188", "title": "Definition:Point-to-Set Distance", "text": "Let $\\mathcal X$ be a normed space. The '''point-to-set distance''' on $\\mathcal X$ is a mapping: :$d: \\mathcal X \\times \\mathcal P \\left({\\mathcal X}\\right) \\to \\left[{0 \\,.\\,.\\, \\infty}\\right)$ where $P \\left({\\mathcal X}\\right)$ is the power set of $\\mathcal X$. This mapping is defined as: :$d \\left({x, C}\\right) := \\inf \\left\\{{\\left\\|{x - y}\\right\\|: y \\in C}\\right\\}$ with the convention that for any $x \\in \\mathcal X$: :$d \\left({x, \\varnothing}\\right) = +\\infty$ or, to be more mathematically rigorous: :$\\forall C \\in \\mathcal P \\left({\\mathcal X}\\right) \\setminus \\left\\{{\\varnothing}\\right\\}: d \\left({x, \\varnothing}\\right) > d \\left({x, C}\\right)$ Category:Definitions/Normed Spaces 9okxhvq12y3hx8b7toz44iwi4jn1cj6"}
+{"_id": "22189", "title": "Definition:Minkowski Sum", "text": "Let $V$ be a vector space. Let $A, B$ be two subsets of $V$. Then the '''Minkowski sum''' of $A$ and $B$, denoted as $A + B$, is defined as: :$A + B := \\set {a + b: a \\in A, b \\in B}$ where the operation $+$ is vector addition. The '''Minkowski sum''' is therefore a relation in the power set of $V$ in the sense that it is a mapping: :$+: \\powerset V^2 \\to \\powerset V$ {{NamedforDef|Hermann Minkowski|cat = Minkowski}} Category:Definitions/Linear Algebra 7vwxhez7dklqrfg3es0op1k4cg19psl"}
+{"_id": "22190", "title": "Definition:Well-Defined/Mapping", "text": "Let $f: S \\to T$ be a mapping. Let $\\RR$ be an equivalence relation on $S$. Let $S / \\RR$ be the quotient set determined by $\\RR$. Let $\\phi: S / \\RR \\to T$ be a mapping such that: :$\\map \\phi {\\eqclass x \\RR} = \\map f x$ Then $\\phi: S / \\RR \\to T$ is '''well-defined''' {{iff}}: :$\\forall \\tuple {x, y} \\in \\RR: \\map f x = \\map f y$"}
+{"_id": "22191", "title": "Definition:Well-Defined/Relation", "text": "{{stub}}"}
+{"_id": "22192", "title": "Definition:Well-Defined/Operation", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\RR$ be a congruence for $\\circ$. Let $\\circ_\\RR$ be the operation induced on $S / \\RR$ by $\\circ$. Let $\\struct {S / \\RR, \\circ_\\RR}$ be the quotient structure defined by $\\RR$, where $\\circ_\\RR$ is defined as: :$\\eqclass x \\RR \\circ_\\RR \\eqclass y \\RR = \\eqclass {x \\circ y} \\RR$ Then $\\circ_\\RR$ is '''well-defined (on $S / \\RR$)''' {{iff}}: :$x, x' \\in \\eqclass x \\RR, y, y' \\in \\eqclass y \\RR \\implies x \\circ y = x' \\circ y'$"}
+{"_id": "22193", "title": "Definition:Topological Vector Space", "text": "Let $V$ be a vector space over a topological field $K$. Let $\\tau$ be a topology on $V$. Then $\\left({V, \\tau}\\right)$ is called a '''topological vector space''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | t = $\\tau$ is a Hausdorff topology }} {{axiom | n = 2         | t = $+: V \\times V \\to V$ is continuous with respect to $\\tau$ }} {{axiom | n = 3         | t = $\\cdot: K \\times V \\to V$ is continuous with respect to $\\tau$ }} {{end-axiom}} Category:Definitions/Vector Spaces Category:Definitions/Topology p891asntxlcepkrmxk66n6z83g66t2u"}
+{"_id": "22194", "title": "Definition:Star Shaped Set", "text": "Let $V$ be a vector space over a field $K$. Let $W \\subseteq V$ be a subset of $V$. Then $W$ is called a '''star shaped set''' {{iff}}: :$\\forall x \\in W: -x \\in W$ where $-x$ is the negative of $x$."}
+{"_id": "22195", "title": "Definition:Absorbent Set", "text": "Let $V$ be a vector space over a field $K$. Let $W \\subseteq V$ be a subset of $V$. Let $a \\in K$. Let the set $a \\cdot W$ be defined as: : $a \\cdot W := \\left\\{{a \\cdot y: y \\in W} \\right\\}$ Then $W$ is an '''absorbent set in $V$''' {{iff}}: : $\\displaystyle \\bigcup_{a \\mathop \\in K} a \\cdot W = V$ which symbolically can be represented as: : $K \\cdot W = V$"}
+{"_id": "22197", "title": "Definition:Closure (Abstract Algebra)/Algebraic Structure", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Then $S$ has the property of '''closure under $\\circ$''' {{iff}}: :$\\forall \\tuple {x, y} \\in S \\times S: x \\circ y \\in S$ $S$ is said to be '''closed under $\\circ$''', or just that '''$\\struct {S, \\circ}$ is closed'''."}
+{"_id": "22198", "title": "Definition:Closure (Abstract Algebra)/Scalar Product", "text": "Let $\\struct {S, \\circ}_R$ be an $R$-algebraic structure over a ring $R$. Let $T \\subseteq S$ such that $\\forall \\lambda \\in R: \\forall x \\in T: \\lambda \\circ x \\in T$. Then $T$ is '''closed for scalar product'''. If $T$ is also closed for operations on $S$, then it is called a '''closed subset''' of $S$."}
+{"_id": "22199", "title": "Definition:Directed Set", "text": "Let $\\struct {S, \\precsim}$ be a preordered set. Then $\\struct {S, \\precsim}$ is a '''directed set''' {{iff}} every pair of elements of $S$ has an upper bound in $S$: :$\\forall x, y \\in S: \\exists z \\in S: x \\precsim z$ and $y \\precsim z$"}
+{"_id": "22200", "title": "Definition:Epigraph", "text": "Let $f: S \\to \\overline \\R$ be an extended real-valued function. The '''epigraph''' of $f$ is the set: :$\\map {\\operatorname {epi} } f := \\set {\\tuple {x, \\alpha} \\in S \\times \\R: \\map f x \\le \\alpha}$"}
+{"_id": "22201", "title": "Definition:Lower Level Set", "text": "Let $f: S \\to \\overline \\R$ be an extended real-valued function. Let $\\alpha \\in \\R$. The '''$\\alpha$-lower level set of $f$''' is the set: :$\\displaystyle \\operatorname{lev} \\limits_{\\mathop \\le \\alpha} f := \\left\\{ {x \\in S: f \\left({x}\\right) \\le \\alpha}\\right\\}$"}
+{"_id": "22202", "title": "Definition:Lower Semicontinuous", "text": "Let $f: S \\to \\R \\cup \\left\\{{-\\infty, \\infty}\\right\\}$ be an extended real valued function. Let $S$ be endowed with a topology $\\tau$.  Then $f$ is '''lower semicontinuous''' at $\\bar x \\in S$ {{iff}}: : $\\displaystyle \\liminf_{x \\mathop \\to \\bar x} f \\left({x}\\right) = f \\left({\\bar x}\\right)$ where $\\displaystyle \\liminf_{x \\mathop \\to \\bar x} f \\left({x}\\right)$ stands for the lower limit of $f$ at $\\bar x$. === Lower Semicontinuous on Subset === {{:Definition:Lower Semicontinuous/Subset}}"}
+{"_id": "22203", "title": "Definition:Ordinal Class", "text": "The '''Ordinal Class''' is defined as the class of all ordinals: :$\\operatorname{On} = \\{ x : x$ is an ordinal $\\}$"}
+{"_id": "22204", "title": "Definition:Trivial Norm", "text": "{{#switch: 3|1 = =|2 = ==|3 = ===|4 = ====|5 = =====|6 = ======|#default = }} Division Ring {{#switch: 3|1 = =|2 = ==|3 = ===|4 = ====|5 = =====|6 = ======|#default = }} {{:Definition:Trivial Norm/Division Ring}} {{#switch: 3|1 = =|2 = ==|3 = ===|4 = ====|5 = =====|6 = ======|#default = }} Vector Space {{#switch: 3|1 = =|2 = ==|3 = ===|4 = ====|5 = =====|6 = ======|#default = }} {{:Definition:Trivial Norm/Vector Space}}"}
+{"_id": "22205", "title": "Definition:Lower Limit (Topological Space)", "text": "Let $\\struct {S, \\tau}$ be a topological space. Let $f: S \\to \\R \\cup \\set {-\\infty, \\infty}$ be an extended real-valued function. The '''lower limit''' of $f$ at some $x_0 \\in S$ is defined as: :$\\displaystyle \\liminf_{x \\mathop \\to x_0} \\map f x := \\sup_{V \\mathop \\in \\map \\mho {x_0} } \\set {\\inf_{x \\mathop \\in V} \\map f x}$ where $\\map \\mho {x_0}$ stands for the set of open neighborhoods of $x_0$."}
+{"_id": "22206", "title": "Definition:Outer Limit", "text": "Let $\\left({\\mathcal X, \\tau}\\right)$ be a Hausdorff topological space. Let $\\left \\langle {C_n}\\right \\rangle_{n \\in \\N}$ be a sequence of sets in $\\mathcal X$. The '''outer limit''' of $\\left \\langle {C_n}\\right \\rangle_{n \\in \\N}$ is defined as: : $\\displaystyle \\limsup_{n \\to\\infty} \\ C_n := \\left\\{{x : \\exists N \\text{ cofinal set of }\\N, \\exists x_v \\in C_v \\left({v \\in N}\\right) \\text{ such that } x_v \\to x}\\right\\}$ where $x_v \\to x$ denotes convergence in the topology $\\tau$."}
+{"_id": "22207", "title": "Definition:Painlevé-Kuratowski Convergence", "text": "Let $T = \\struct {S, \\tau}$ be a Hausdorff topological space. Let $\\sequence {C_n}_{n \\mathop \\in \\N}$ be a sequence of sets in $T$. Let $\\sequence {C_n}_{n \\mathop \\in \\N}$ be such that: :$\\displaystyle \\liminf_n C_n = \\limsup_n C_n = C$ where: :$\\displaystyle \\liminf_n C_n$ denotes the inner limit of $\\sequence {C_n}_{n \\mathop \\in \\N}$ :$\\displaystyle \\limsup_n C_n$ denotes the outer limit of $\\sequence {C_n}_{n \\mathop \\in \\N}$ Then $\\sequence {C_n}_{n \\mathop \\in \\N}$ is said to be '''convergent in the sense of Painlevé-Kuratowski'''. It can be denoted as: :$C_n \\overset K \\to C$ or: :$\\operatorname {K-lim} \\limits_{n \\mathop \\to \\infty} C_n = C$ or simply: :$\\displaystyle \\lim_n C_n = C$ {{NamedforDef|Paul Painlevé|name2 = Kazimierz Kuratowski|cat = Painlevé|cat2 = Kuratowski}} Category:Definitions/Hausdorff Spaces Category:Definitions/Convergence a3gln6xeea3jcurnylyvfie89koyxxa"}
+{"_id": "22209", "title": "Definition:Ambiguity", "text": "An '''ambiguity''' is a statement which has more than one distinct meaning. Thus a statement is '''ambiguous''' if, without extraneous clarification, it can be interpreted in more than one way."}
+{"_id": "22210", "title": "Definition:Classical Probability Model", "text": "Let $\\EE$ be an experiment The '''classical probability model''' on $\\EE$ is a mathematical model that defines the probability space $\\struct {\\Omega, \\Sigma, \\Pr}$ of $\\EE$ as follows: :$(1) \\quad$ All outcomes of $\\EE$ are equally likely :$(2) \\quad$ There are a finite number of outcomes. Then: :$\\map \\Pr {\\text {event occurring} } := \\dfrac {\\paren {\\text {number of outcomes favorable to event} } } {\\paren {\\text {total number of outcomes possible} } }$ or formally: :$\\map \\Pr \\omega := \\dfrac {\\card \\Sigma} {\\card \\Omega}$ where: :$\\card {\\, \\cdot \\,}$ denotes the cardinality of a set :$\\omega$ denotes an event: $\\omega \\in \\Omega$ :$\\Sigma$ denotes the event space: $\\Sigma \\subseteq \\Omega$ :$\\Omega$ denotes the sample space."}
+{"_id": "22211", "title": "Definition:Equivocation", "text": "An '''equivocation''' is a shift in meaning of an ambiguous term. If an argument's persuasive force depends on utilizing an '''equivocation''' it is fallacious. To make such an argument is to commit the '''fallacy of equivocation'''."}
+{"_id": "22212", "title": "Definition:Relative Frequency Model", "text": "The '''relative frequency model''' is a mathematical model that defines the probability of an event occurring as follows: :$\\map \\Pr {\\text {event occurring} } := \\dfrac {\\paren {\\text {observed number of times event has occurred in the past} } } {\\paren {\\text {observed number of times event has occurred or not occurred} } }$ That is, the probability of an event happening is defined as the '''relative frequency''' of a finite number of events of a particular type in some finite reference class of events.  Symbolically: :$\\map \\Pr \\omega := \\dfrac {f_{\\omega} } n$ where: :$\\omega$ is an elementary event :$f_{\\omega}$ is how many times $\\omega$ occurred :$n$ is the number of trials observed."}
+{"_id": "22213", "title": "Definition:Relative Frequency", "text": "Let $S$ be a sample or a finite population. Let $\\omega$ be a qualitative variable, or a class of a quantitative variable. The '''relative frequency''' of $\\omega$ is defined as: :$\\map {\\operatorname {RF} } \\omega := \\dfrac {f_\\omega} n$ where: :$f_\\omega$ is the (absolute) frequency of $\\omega$ :$n$ is the number of individuals in $S$."}
+{"_id": "22214", "title": "Definition:Sorgenfrey Line", "text": "Let $\\R$ be the set of real numbers Let $\\BB$ be the set: :$\\BB = \\set {\\hointr a b: a, b \\in \\R}$ where $\\hointr a b$ is the half-open interval $\\set {x \\in \\R: a \\le x < b}$. Then $\\BB$ is the basis for a topology $\\tau$ on $\\R$. The topological space $T = \\struct {\\R, \\tau}$ is referred to as the '''Sorgenfrey line'''."}
+{"_id": "22215", "title": "Definition:Sequence of Distinct Terms", "text": "A '''sequence of distinct terms of $S$''' is an injection from a subset of $\\N$ into $S$. Thus a sequence $\\sequence {a_k}_{k \\mathop \\in A}$ is a '''sequence of distinct terms''' {{iff}}: :$\\forall j, k \\in A: j \\ne k \\implies a_j \\ne a_k$"}
+{"_id": "22216", "title": "Definition:Categorical Statement", "text": "Let $S$ and $P$ be predicates. A '''categorical statement''' is a statement that can be expressed in one of the following ways in natural language: {{begin-axiom}} {{axiom | n  = A         | lc = Universal Affirmative:         | t  = Every $S$ is $P$ }} {{axiom | n  = E         | lc = Universal Negative:         | t  = No $S$ is $P$ }} {{axiom | n  = I         | lc = Particular Affirmative:         | t  = Some $S$ is $P$ }} {{axiom | n  = O         | lc = Particular Negative:         | t  = Some $S$ is not $P$ }} {{end-axiom}}"}
+{"_id": "22217", "title": "Definition:Topological Group", "text": "Let $\\left({G, \\odot}\\right)$ be a group. On its underlying set $G$, let $\\left({G, \\tau}\\right)$ be a topological space. === Definition 1 === {{:Definition:Topological Group/Definition 1}} === Definition 2 === {{:Definition:Topological Group/Definition 2}}"}
+{"_id": "22218", "title": "Definition:Inversion Mapping/Topology", "text": "Let $T = \\left({G, \\circ, \\tau}\\right)$ be a topological group. Let $\\phi: G \\to G$ be the mapping defined as: : $\\forall x \\in G: \\phi \\left({x}\\right) = x^{-1}$ Then $\\phi$ is the '''inversion mapping''' of $T$."}
+{"_id": "22220", "title": "Definition:Semi-Inner Product", "text": "Let $\\C$ be the field of complex numbers. Let $\\F$ be a subfield of $\\C$. Let $V$ be a vector space over $\\F$ A '''semi-inner product''' is a mapping $\\innerprod \\cdot \\cdot: V \\times V \\to \\mathbb F$ that satisfies the following properties: {{begin-axiom}} {{axiom | n = 1         | lc= Conjugate Symmetry         | q = \\forall x, y \\in V         | m = \\quad \\innerprod x y = \\overline {\\innerprod y x} }} {{axiom | n = 2         | lc= Sesquilinearity         | q = \\forall x, y, z \\in V, \\forall a \\in \\mathbb F         | m = \\quad \\innerprod {a x + y} z = a \\innerprod x z + \\innerprod y z }} {{axiom | n = 3         | lc= Non-Negative Definiteness         | q = \\forall x \\in V         | m = \\quad \\innerprod x x \\in \\R_{\\ge 0} }} {{end-axiom}} {{refactor|This section merits a separate page: \"Semi-Inner Product in Real Number Field\", perhaps.}} If $\\mathbb F$ is a subfield of the field of real numbers $\\R$, it follows from Complex Number equals Conjugate iff Wholly Real that $\\overline {\\innerprod y x} = \\innerprod y x$ for all $x, y \\in V$. Then $(1)$ above may be replaced by: {{begin-axiom}} {{axiom | n = 1^\\prime         | lc= Symmetry         | q = \\forall x, y \\in V         | m = \\innerprod x y = \\innerprod y x }} {{end-axiom}} === Semi-Inner Product Space === {{:Definition:Semi-Inner Product Space}}"}
+{"_id": "22221", "title": "Definition:Semi-Inner Product Space", "text": "A '''semi-inner product space''' is a vector space together with an associated semi-inner product."}
+{"_id": "22222", "title": "Definition:Inner Product Norm", "text": "Let $V$ be an inner product space over a subfield $\\Bbb F$ of $\\C$. Let $\\left \\langle{\\cdot, \\cdot}\\right \\rangle$ be the inner product of $V$. Then the '''inner product norm''' on $V$ is the mapping $\\left\\Vert{\\cdot}\\right\\Vert: V \\to \\R_{\\ge 0}$ given by :$\\left\\Vert{x}\\right\\Vert := \\left\\langle{x,x}\\right\\rangle^{1/2}$."}
+{"_id": "22223", "title": "Definition:Metric Induced by Norm", "text": "Let $V$ be a normed vector space. Let $\\norm{\\,\\cdot\\,}$ be the norm of $V$. Then the '''induced metric''' or the '''metric induced by $\\norm{\\,\\cdot\\,}$''' is the map $d: V \\times V \\to \\R_{\\ge 0}$ defined as: :$d \\left({x, y}\\right) = \\left\\Vert{x - y}\\right\\Vert$"}
+{"_id": "22224", "title": "Definition:Hilbert Space", "text": "Let $V$ be an inner product space over $\\Bbb F \\in \\set {\\R, \\C}$. Let $d: V \\times V \\to \\R_{\\ge 0}$ be the metric induced by the inner product norm $\\norm {\\,\\cdot\\,}_V$. If $\\struct {V, d}$ is a complete metric space, $V$ is said to be a '''Hilbert space'''."}
+{"_id": "22226", "title": "Definition:Orthogonal (Hilbert Space)", "text": "Let $H$ be a Hilbert space. Let $f, g \\in H$. Let $\\innerprod f g = 0$, where $\\innerprod \\cdot \\cdot$ denotes the inner product. Then $f$ and $g$ are defined as being '''orthogonal''': :$f \\perp g$ === Sets === Let $A, B \\subseteq H$. Then $A$ and $B$ are defined as '''orthogonal''' {{iff}}: :$\\forall a \\in A, b \\in B: a \\perp b$ This is denoted by $A \\perp B$.  When $A, B$ are closed linear subspaces of $H$, they are called '''orthogonal subspaces'''. If $A$ consists of only one element $a$, also $a \\perp B$ is encountered."}
+{"_id": "22227", "title": "Definition:Convex Set (Vector Space)", "text": "Let $V$ be a vector space over $\\R$ or $\\C$. A subset $A \\subseteq V$ is said to be '''convex''' {{iff}}: :$\\forall x, y \\in A: \\forall t \\in \\closedint 0 1: t x + \\paren {1 - t} y \\in A$ === Line Segment === {{:Definition:Convex Set (Vector Space)/Line Segment}}"}
+{"_id": "22228", "title": "Definition:Distance/Points", "text": "=== Metric Space === Let $\\struct {X, d}$ be a metric space. {{:Definition:Metric Space/Distance Function}} === Normed Vector Space === {{:Definition:Distance/Points/Normed Vector Space}} === Real Numbers === {{:Definition:Distance/Points/Real Numbers}} === Complex Numbers === {{:Definition:Distance/Points/Complex Numbers}}"}
+{"_id": "22229", "title": "Definition:Distance/Sets", "text": "=== Real Numbers === {{:Definition:Distance/Sets/Real Numbers}} === Metric Spaces === {{:Definition:Distance/Sets/Metric Spaces}}"}
+{"_id": "22230", "title": "Definition:Outer Automorphism", "text": "Let $G$ be a group. Let $\\phi: G \\to G$ be an automorphism which is not an inner automorphism. Then $\\phi$ is an '''outer automorphism'''."}
+{"_id": "22231", "title": "Definition:Group of Outer Automorphisms", "text": "Let $G$ be a group. Let $\\Aut G$ be the automorphism group of $G$. Let $\\Inn G$ be the inner automorphism group of $G$. Let $\\dfrac {\\Aut G} {\\Inn G}$ be the quotient group of $\\Aut G$ by $\\Inn G$. Then $\\dfrac {\\Aut G} {\\Inn G}$ is called the '''group of outer automorphisms''' of $G$. This group $\\dfrac {\\Aut G} {\\Inn G}$ is often denoted $\\Out G$."}
+{"_id": "22232", "title": "Definition:Characteristic Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup such that: :$\\forall \\phi \\in \\Aut G: \\phi \\sqbrk H = H$ where $\\Aut G$ is the automorphism group of $G$. Then $H$ is  '''characteristic (in $G$)''', or '''a characteristic subgroup of $G$'''."}
+{"_id": "22234", "title": "Definition:Projection (Analytic Geometry)", "text": "Let $M$ and $N$ be distinct lines through the origin in the plane. The '''projection on $M$ along $N$''' is the mapping $\\pr_{M, N}$ such that: :$\\forall x \\in \\R^2: \\map {\\pr_{M, N} } x =$ the intersection of $M$ with the line through $x$ parallel to $N$."}
+{"_id": "22235", "title": "Definition:Orthogonal Projection", "text": "Let $H$ be a Hilbert space. Let $K$ be a closed linear subspace of $H$. Then the '''orthogonal projection''' on $K$ is the map $P_K: H \\to H$ defined by :$k = \\map {P_K} h \\iff k \\in K$ and $\\map d {h, k} = \\map d {h, K}$ where the latter $d$ signifies distance to a set. {{refactor|Extract the 3 statements below and put them into their own pages}} That $P_K$ is well-defined follows from Unique Point of Minimal Distance. The name orthogonal projection stems from the fact that $\\paren {h - \\map {P_K} h} \\perp K$. This and other properties of $P_K$ are collected in Properties of Orthogonal Projection."}
+{"_id": "22237", "title": "Definition:Normal Series", "text": "Let $G$ be a group whose identity is $e$. A '''normal series''' for $G$ is a sequence of (normal) subgroups of $G$: :$\\set e = G_0 \\lhd G_1 \\lhd \\cdots \\lhd G_n = G$ where $G_{i - 1} \\lhd G_i$ denotes that $G_{i - 1}$ is a proper normal subgroup of $G_i$."}
+{"_id": "22239", "title": "Definition:Refinement of Normal Series", "text": "Let $G$ be a group whose identity is $e$. Let $\\sequence {G_i}_{i \\mathop \\in \\closedint 0 n}$ be a normal series for $G$: :$\\sequence {G_i}_{i \\mathop \\in \\closedint 0 n} = \\tuple {\\set e = G_0 \\lhd G_1 \\lhd \\cdots \\lhd G_{n - 1} \\lhd G_n = G}$ Let $\\sequence {H_j}_{j \\mathop \\in \\closedint 0 m}$ be another normal series for $G$: :$\\sequence {H_j}_{j \\mathop \\in \\closedint 0 m} = \\tuple {\\set e = H_0 \\lhd H_1 \\lhd \\cdots \\lhd H_{m - 1} \\lhd H_m = G}$ such that $\\sequence {G_i}_{i \\mathop \\in \\closedint 0 n} \\subseteq \\sequence {H_j}_{j \\mathop \\in \\closedint 0 m}$ Then $\\sequence {H_j}_{j \\mathop \\in \\closedint 0 m}$ is a '''refinement''' of $\\sequence {G_i}_{i \\mathop \\in \\closedint 0 n}$. That is, a '''refinement''' of a normal series is a normal series which contains all the (normal) subgroups of the original normal series, and may contain more. === Proper Refinement === {{:Definition:Refinement of Normal Series/Proper Refinement}}"}
+{"_id": "22240", "title": "Definition:Composition Series", "text": "Let $G$ be a finite group. A '''composition series for $G$''' is a normal series for $G$ which has no proper refinement. === Composition Length === {{:Definition:Composition Series/Composition Length}} === Composition Factor === {{:Definition:Composition Series/Composition Factor}}"}
+{"_id": "22241", "title": "Definition:Contradictory Statements", "text": "Two statements $p$ and $q$ are said to be '''contradictory''' iff: : whenever $p$ is true, $q$ is false. and: : whenever $q$ is true, $p$ is false."}
+{"_id": "22242", "title": "Definition:Contrary Statements", "text": "Two statements are said to be '''contrary''' if they can both be false, but they cannot both be true."}
+{"_id": "22243", "title": "Definition:Subcontrary Statements", "text": "Two statements are said to be '''subcontrary''' if they can both be true, but they cannot both be false."}
+{"_id": "22244", "title": "Definition:Closed Linear Span", "text": "Let $H$ be a Hilbert space, and let $A \\subseteq H$ be a subset. Then the '''closed linear span''' of $A$, denoted $\\vee A$, is defined in the following ways: :$(1): \\qquad \\displaystyle \\vee A = \\bigcap \\Bbb M$, where $\\Bbb M$ consists of all closed linear subspace $M$ of $H$ with $A \\subseteq M$ :$(2): \\qquad \\vee A$ is the smallest closed linear subspace $M$ of $H$ with $A \\subseteq M$ :$(3): \\qquad \\vee A = \\map \\cl {\\map {\\operatorname {span} } A}$, where $\\cl$ denotes closure, and $\\operatorname {span}$ denotes linear span."}
+{"_id": "22245", "title": "Definition:Zero Subspace", "text": "Let $V$ be a vector space with zero vector $\\mathbf 0$. Then the set $(\\mathbf 0) := \\left\\{{\\mathbf 0}\\right\\}$ is called the '''zero subspace''' of $V$. This name is appropriate as $(\\mathbf 0)$ is in fact a subspace of $V$, as proved in Zero Subspace is Subspace. {{Explain|I'm not sure about the nomenclature, but this deserves a name. Feel free to adapt if you have sources to back it up.}} Category:Definitions/Vector Spaces o6xb6n4gd224dpk0cmgqrc2o7idp9ro"}
+{"_id": "22246", "title": "Definition:Length of Group", "text": "Let $G$ be a finite group. The '''length''' of $G$ is the length of a composition series for $G$. That is, the '''length''' of $G$ is the number of factors in a composition series for $G$ (not including $G$ itself). The '''length''' of $G$ can be denoted $\\map l G$. By the Jordan-Hölder Theorem, all composition series for $G$ have the same length. Therefore, the '''length''' of a finite group $G$ is well-defined."}
+{"_id": "22247", "title": "Definition:Bounded Linear Functional", "text": "Let $H$ be a Hilbert space, and let $L$ be a linear functional on $H$. Then $L$ is said to be a '''bounded linear functional''' iff :$\\exists c > 0: \\forall h \\in H: \\left|{Lh}\\right| \\le c \\left\\|{h}\\right\\|$ In view of Continuity of Linear Functionals, a linear functional on a Hilbert space is bounded if and only if it is continuous."}
+{"_id": "22248", "title": "Definition:Orthonormal Subset", "text": "Let $\\struct {V, \\innerprod \\cdot \\cdot}$ be an inner product space. Let $S \\subseteq V$ be a subset of $V$. Then $S$ is an '''orthonormal subset''' {{iff}}: :$(1): \\quad \\forall u \\in S: \\norm u = 1$ where $\\norm {\\, \\cdot \\,}$ is the inner product norm. :$(2): \\quad S$ is an orthogonal set: ::$\\forall u, v \\in S: u \\ne v \\implies \\innerprod u v = 0$"}
+{"_id": "22249", "title": "Definition:Basis (Hilbert Space)", "text": "Let $H$ be a Hilbert space. A '''basis for $H$''' is a maximal orthonormal subset of $H$. Thus, $B$ is a '''basis''' for $H$ {{iff}} for all orthonormal subsets $B'$ of $H$: :$B \\subseteq B' \\implies B = B'$"}
+{"_id": "22250", "title": "Definition:Linear Span", "text": "Let $V$ be a vector space over $K$. Let $A \\subseteq V$ be a subset of $V$. Then the '''linear span''' of $A$, denoted $\\operatorname{span} A$ or $\\map {\\operatorname{span} } A$, is the set: :$\\displaystyle \\set {\\sum_{k \\mathop = 1}^n \\alpha_k f_k: n \\in \\N_{\\ge 1}, \\alpha_i \\in K, f_i \\in A}$ The '''linear span''' can be interpreted as the set of all linear combinations (of finite length) of these vectors. === Definition for $\\R^n$ === In $\\R^n$ (where $n \\in \\N_{>0}$), above definition translates to: :$\\displaystyle \\map {\\operatorname{span} } {\\mathbf v_1, \\mathbf v_2, \\dotsc, \\mathbf v_k} = \\set {\\sum_{i \\mathop = 1}^k \\ c_i \\ \\mathbf v_i: c_i \\in \\R, \\mathbf v_i \\in \\R^n, 1 \\le i \\le k}$"}
+{"_id": "22251", "title": "Definition:Prime Group", "text": "Let $G$ be a group. Then $G$ is a '''prime group''' {{iff}} the order of $G$ is a prime number."}
+{"_id": "22252", "title": "Definition:Solvable Group", "text": "Let $G$ be a finite group. Then $G$ is a '''solvable group''' {{iff}} it has a composition series in which each factor is a cyclic group."}
+{"_id": "22253", "title": "Definition:Bayesian Probability Model", "text": "The '''Bayesian probability model''' is a mathematical model that defines the probability of an event occurring as a degree of belief. That is, that probability is defined as the degree of which it is rational to believe certain statements based on intuition, experience, judgment, or opinion. {{DefinitionWanted}} {{NamedforDef|Thomas Bayes|cat = Bayes}}"}
+{"_id": "22254", "title": "Definition:Amphiboly", "text": "An '''amphiboly''' is an ambiguity resulting from poor syntax."}
+{"_id": "22255", "title": "Definition:Permutation on n Letters/Two-Row Notation", "text": "Let $\\pi$ be a permutation on $n$ letters. The '''two-row notation''' for $\\pi$ is written as two rows of elements of $\\N_n$, as follows: :$\\pi = \\begin{pmatrix} 1 & 2 & 3 & \\ldots & n \\\\ \\map \\pi 1 & \\map \\pi 2 & \\map \\pi 3 & \\ldots & \\map \\pi n \\end{pmatrix}$ The bottom row contains the effect of $\\pi$ on the corresponding entries in the top row."}
+{"_id": "22256", "title": "Definition:Inductive Argument", "text": "An '''inductive argument''' is a form of argument in which, if all the premises are true, the conclusion is ''probably true'', but might not be.  Such lines of reasoning are ubiquitous in everyday life and in most human endeavors. However, '''inductive arguments''' are only conjectures in the field of mathematics. Such arguments are not truth preserving and therefore they are not proofs."}
+{"_id": "22257", "title": "Definition:Syllogism", "text": "A '''syllogism''' is an argument with exactly two premises and one conclusion."}
+{"_id": "22258", "title": "Definition:Membership Relation", "text": "Let $S$ be a set. Let $\\powerset S$ denote the power set of $S$. Let $\\mathcal R \\subseteq S \\times \\powerset S$ be the relation defined as: :$\\tuple {x, A} \\in \\mathcal R \\iff x \\in A$ Thus $\\mathcal R$ is the relation between elements of $S$ and subsets of $S$ expressing '''membership'''."}
+{"_id": "22259", "title": "Definition:Image (Set Theory)/Relation/Element", "text": "Let $s \\in S$. The '''image of $s$ by''' (or '''under''') '''$\\RR$''' is defined as: :$\\map \\RR s := \\set {t \\in T: \\tuple {s, t} \\in \\RR}$ That is, $\\map \\RR s$ is the set of all elements of the codomain of $\\RR$ related to $s$ by $\\RR$."}
+{"_id": "22260", "title": "Definition:Image (Set Theory)/Relation", "text": "=== Image of a Relation === {{:Definition:Image (Set Theory)/Relation/Relation}} === Image of an Element === {{:Definition:Image (Set Theory)/Relation/Element}} === Image of a Subset === {{:Definition:Image (Set Theory)/Relation/Subset}}"}
+{"_id": "22261", "title": "Definition:Image (Set Theory)/Mapping", "text": "{{transclude:Definition:Image (Set Theory)/Mapping/Mapping   |section = tc   |title = Image of a Mapping   |link = true   |header = 3   |increase = 1 }} === Image of an Element === {{:Definition:Image (Set Theory)/Mapping/Element}} === Image of a Subset === {{:Definition:Image (Set Theory)/Mapping/Subset}}"}
+{"_id": "22262", "title": "Definition:Image (Set Theory)/Mapping/Element", "text": "Let $s \\in S$. The '''image of $s$ (under $f$)''' is defined as: :$\\Img s = \\map f s = \\displaystyle \\bigcup \\set {t \\in T: \\tuple {s, t} \\in f}$ That is, $\\map f s$ is the element of the codomain of $f$ related to $s$ by $f$."}
+{"_id": "22264", "title": "Definition:Image (Set Theory)/Mapping/Subset", "text": "Let $f: S \\to T$ be a mapping. Let $X \\subseteq S$ be a subset of $S$."}
+{"_id": "22266", "title": "Definition:Tree (Set Theory)", "text": "Let $\\struct {T, \\preceq}$ be an ordered set. Let $\\struct {T, \\preceq}$ be such that for every $t \\in T$, the lower closure of $t$: :$t^\\preceq := \\set {s \\in T: s \\preceq t}$  is well-ordered by $\\preceq$. Then $\\struct {T, \\preceq}$ is a '''tree'''. === Branch === A '''branch''' of a tree $T$ is a maximal chain in $T$. === Subtree === A '''subtree''' of a tree $\\struct {T, \\preceq}$ is an ordered subset $\\struct {S, \\preceq}$ with the property that: :for every $\\forall s \\in S: \\forall t \\in T: t \\preceq s \\implies t \\in S$ Category:Definitions/Set Theory djo29gkqm3npxpyyu1pnk44i2estlzc"}
+{"_id": "22267", "title": "Definition:Basic Open Set", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $\\mathcal B \\subseteq \\tau$ be a basis for $T$. Let $U \\in \\mathcal B$. Then $U$ is a '''basic open set''' of $T$. That is, a '''basic open set''' of a topology is an open set of that topology which is an element of a basis for that topology. The basis itself needs to be specified for this definition to make sense. Category:Definitions/Topology Category:Definitions/Topological Bases p16ojsxmpoxw5t9h7f3eos4sdcjv7po"}
+{"_id": "22269", "title": "Definition:Sine Integral Function", "text": "The '''sine integral function''' is the real function $\\Si: \\R \\to \\R$ defined as: :$\\map \\Si x = \\begin{cases} \\displaystyle \\int_{t \\mathop \\to 0}^{t \\mathop = x} \\frac {\\sin t} t \\rd t & : x \\ne 0 \\\\ \\vphantom x \\\\ 0 & : x = 0 \\\\ \\end{cases}$"}
+{"_id": "22270", "title": "Definition:Greatest", "text": "=== Ordered Set === {{:Definition:Greatest/Ordered Set}} === Greatest Set === {{:Definition:Greatest Set by Set Inclusion}} === Mapping === {{:Definition:Greatest/Mapping}}"}
+{"_id": "22271", "title": "Definition:Smallest", "text": "=== Ordered Set === {{:Definition:Smallest/Ordered Set}} === Smallest Set === {{:Definition:Smallest Set by Set Inclusion}} === Mapping === {{:Definition:Smallest/Mapping}}"}
+{"_id": "22272", "title": "Definition:Arc Length", "text": "Let $y = \\map f x$ be a real function which is: : continuous on the closed interval $\\closedint a b$ and: : continuously differentiable on the open interval $\\openint a b$. The '''arc length''' $s$ of $f$ between $a$ and $b$ is defined as:  :$s := \\displaystyle \\int_a^b \\sqrt {1 + \\paren {\\frac {\\d y} {\\d x} }^2} \\rd x$"}
+{"_id": "22273", "title": "Definition:Greatest/Ordered Set", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. An element $x \\in S$ is '''the greatest element (of $S$)''' {{iff}}: :$\\forall y \\in S: y \\preceq x$ That is, every element of $S$ '''precedes, or is equal to,''' $x$. The Greatest Element is Unique, so calling it '''''the'' greatest element''' is justified. Thus for an element $x$ to be the '''greatest element''', all $y \\in S$ must be comparable to $x$."}
+{"_id": "22274", "title": "Definition:Smallest/Ordered Set", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. An element $x \\in S$ is '''the smallest element''' {{iff}}: :$\\forall y \\in S: x \\preceq y$ That is, $x$ '''strictly precedes, or is equal to,''' every element of $S$. The Smallest Element is Unique, so calling it '''''the'' smallest element''' is justified. The '''smallest element''' of $S$ is denoted $\\min S$. For an element to be the '''smallest element''', all $y \\in S$ must be comparable to $x$."}
+{"_id": "22275", "title": "Definition:Radicand", "text": "Let $\\sqrt[n] P$ denote the $n$th root of an expression. Then $P$ is the '''radicand''' of $\\sqrt[n] P$. That is, it is '''the object whose $n$th root is to be taken'''."}
+{"_id": "22276", "title": "Definition:Finite Ring", "text": "A finite ring is a ring $\\left({R, +, \\circ}\\right)$ such that $R$ is a finite set."}
+{"_id": "22277", "title": "Definition:Removable Discontinuity", "text": "=== Real Function === {{Definition:Removable Discontinuity of Real Function}} {{definition wanted|discuss functions of $n$ variables, complex functions}}"}
+{"_id": "22279", "title": "Definition:Generalized Sum", "text": "Let $\\left({G, +}\\right)$ be a commutative topological semigroup. Let $\\left({g_i}\\right)_{i \\in I}$ be an indexed subset of $G$. Consider the set $\\mathcal F$ of finite subsets of $I$. Let $\\subseteq$ denote the subset relation on $\\mathcal F$. By virtue of Finite Subsets form Directed Set, $\\left({\\mathcal F, \\subseteq}\\right)$ is a directed set. Define the net: :$\\phi: \\mathcal F \\to G$ by: : $\\displaystyle \\phi \\left({F}\\right) = \\sum_{i \\mathop \\in F} g_i$ Then $\\phi$ is denoted: : $\\displaystyle \\sum \\left\\{{g_i: i \\in I}\\right\\}$ and referred to as a '''generalized sum'''."}
+{"_id": "22281", "title": "Definition:Dimension (Hilbert Space)", "text": "Let $H$ be a Hilbert space, and let $E$ be a basis of $H$. Then the '''dimension''' $\\dim H$ of $H$ is defined as $\\card E$, the cardinality of $E$."}
+{"_id": "22282", "title": "Definition:Successor Set", "text": "Let $S$ be a set. The '''successor (set) of $S$''' is defined and denoted: :$S^+ := S \\cup \\set S$"}
+{"_id": "22283", "title": "Definition:Infinite Successor Set", "text": "Let $S$ be a set. Then $S$ is an '''infinite successor set''' {{iff}}: :$\\O \\in S$ :$x \\in S \\implies x^+ \\in S$ where $\\O$ is the empty set, and $x^+$ denotes the successor set of $x$. === Axiomatic Set Theory === The concept of '''infinite successor set''' is axiomatised in the Axiom of Infinity in Zermelo-Fraenkel set theory: :$\\exists x: \\paren {\\paren {\\exists y: y \\in x \\land \\forall z: \\neg \\paren {z \\in y} } \\land \\forall u: u \\in x \\implies \\paren {u \\cup \\set u \\in x} }$"}
+{"_id": "22284", "title": "Definition:Undefined Term", "text": "For a definition to not be circular, the definer must use already defined terms.  However, this process cannot go on indefinitely. If we were to insist on ''everything'' being defined only using previously defined terms, we would enter an infinite regress. Concepts that are not defined in terms of previously defined concepts are called '''undefined terms'''. An undefined term is frequently explained by using an ostensive definition: that is, a statement that ''shows'' what something is, rather than ''explains''."}
+{"_id": "22285", "title": "Definition:Power (Algebra)/Natural Number", "text": "Let $\\N$ denote the natural numbers. For each $m \\in \\N$, recursively define $e_m: \\N \\to \\N$ to be the mapping: :$e_m \\left({n}\\right) = \\begin{cases} 1 & : n = 0 \\\\ m \\times e_m \\left({x}\\right) & : n = x + 1 \\end{cases}$ where: : $+$ denotes natural number addition. : $\\times$ denotes natural number multiplication. $e_m \\left({n}\\right)$ is then expressed as a binary operation in the form: :$m^n := e_m \\left({n}\\right)$ and is called '''$m$ to the power of $n$'''."}
+{"_id": "22286", "title": "Definition:Topological Semigroup", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroup. On that same underlying set $S$, let $\\left({S, \\tau}\\right)$ be a topological space. Then $\\left({S, \\circ, \\tau}\\right)$ is said to be a '''topological semigroup''' if: :$\\circ: \\left({S, \\tau}\\right) \\times \\left({S, \\tau}\\right) \\to \\left({S, \\tau}\\right)$ is a continuous mapping where $\\left({S, \\tau}\\right) \\times \\left({S, \\tau}\\right)$ is considered as $S \\times S$ with the product topology."}
+{"_id": "22287", "title": "Definition:Isomorphism (Hilbert Spaces)", "text": "Let $H, K$ be Hilbert spaces. Denote by $\\innerprod \\cdot \\cdot_H$ and $\\innerprod \\cdot \\cdot_K$ their respective inner products. An '''isomorphism''' between $H$ and $K$ is a map $U: H \\to K$, such that: :$(1): \\quad U$ is a linear map :$(2): \\quad U$ is surjective :$(3): \\quad \\forall g, h \\in H: \\innerprod g h_H = \\innerprod {U g} {U h}_K$ These three requirements may be summarized by stating that $U$ be a surjective isometry. Furthermore, Surjection that Preserves Inner Product is Linear shows that requirement $(1)$ is superfluous. If such an '''isomorphism''' $U$ exists, $H$ and $K$ are said to be '''isomorphic'''. As the name '''isomorphism''' suggests, Hilbert Space Isomorphism is Equivalence Relation."}
+{"_id": "22288", "title": "Definition:Isometry (Hilbert Spaces)", "text": "Let $H, K$ be Hilbert spaces, and denote by $\\innerprod \\cdot \\cdot_H$ and $\\innerprod \\cdot \\cdot_K$ their respective inner products. A linear map $U: H \\to K$ is called an '''isometry''' {{iff}}: :$\\forall g,h \\in H: \\innerprod g h_H = \\innerprod {U g} {U h}_K$ {{explain|Notation $U g, U h$ may need to be amended so as to make it clear that it means the application of the mapping $U$ on both of $g$ and $h$. If this is indeed what it means (as is probable) then suggest $\\map U g$ and $\\map U h$.}}"}
+{"_id": "22289", "title": "Definition:Unitary Operator", "text": "A '''unitary operator''' is an isomorphism from a Hilbert space $H$ to itself."}
+{"_id": "22290", "title": "Definition:Riemann Sum", "text": "Let $f$ be a real function defined on the closed interval $\\mathbb I = \\closedint a b$. Let $\\Delta$ be a subdivision of $\\mathbb I$. For $1 \\le i \\le n$: :let $\\Delta x_i = x_i - x_{i - 1}$ :let $c_i \\in \\closedint {x_{i - 1} } {x_i}$. The summation: :$\\displaystyle \\sum_{i \\mathop = 1}^n \\map f {c_i} \\Delta x_i$ is called a '''Riemann sum''' of $f$ for the subdivision $\\Delta$. === Geometric Interpretation === {{stub}}"}
+{"_id": "22292", "title": "Definition:Total Relation", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation on a set $S$. Then $\\mathcal R$ is defined as '''total''' {{iff}}: :$\\forall a, b \\in S: \\tuple {a, b} \\in \\mathcal R \\lor \\tuple {b, a} \\in \\mathcal R$ That is, {{iff}} every pair of elements is related (either or both ways round)."}
+{"_id": "22293", "title": "Definition:Signed Area", "text": "Consider the plane region described by a graph in the $xy$-plane. The '''signed area''' of the region is the area such that: : The area of the graph on or above the $x$-axis is defined as positive : The area of the graph below the $x$-axis is defined as negative. {{explain|As this is a definition which requires the use of spatial relationships, might be good to have a diagram of this.}}"}
+{"_id": "22294", "title": "Definition:Initial Segment", "text": "Let $\\struct {S, \\preceq}$ be a well-ordered set. Let $a \\in S$. The '''initial segment (of $S$) determined by $a$''' is defined as: :$S_a := \\set {b \\in S: b \\preceq a \\land b \\ne a}$ which can also be rendered as: :$S_a := \\set {b \\in S: b \\prec a}$ That is, $S_a$ is the set of all elements of $S$ that strictly precede $a$. That is, $S_a$ is the strict lower closure of $a$ (in $S$). By extension, $S_a$ is described as '''an initial segment (of $S$)'''."}
+{"_id": "22295", "title": "Definition:Preordering/Preordered Set", "text": "Let $S$ be a set. Let $\\precsim$ be a preordering on $S$. Then the relational structure $\\left({S, \\precsim}\\right)$ is called a '''preordered set'''."}
+{"_id": "22296", "title": "Definition:Convergent Net", "text": "Let $\\left({X, \\tau}\\right)$ be a topological space, and let $\\left({I, \\leq}\\right)$ be a directed set. Let $\\left({x_i}\\right)_{i \\in I}$ be a net. The net $\\left({x_i}\\right)$ is said to '''converge to''' $x \\in X$, denoted $\\displaystyle x_i \\to x$ or $\\lim x_i = x$, iff: :$\\forall U \\in \\tau: x \\in U \\implies \\exists i_0 \\in I: \\forall i \\ge i_0: x_i \\in U$ That is, for every open $U$ with $x \\in U$, there exists an $i_0 \\in I$ such that forall $i \\ge i_0$, $x_i \\in U$. If $x_i \\to x$, then $x$ is called a '''limit (point) of $\\left({x_i}\\right)$'''. A net $\\left({x_i}\\right)_{i \\in I}$ is called '''convergent''' if there is an $x \\in X$ such that $x_i \\to x$. If such an $x$ does not exist, the net is said to be '''divergent'''. === Cluster Point === The net $\\left({x_i}\\right)$ is said to '''cluster at''' $x \\in X$, denoted $\\displaystyle x_i \\mathop{\\longrightarrow}_{\\text{cl}} x$, iff: :$\\forall U \\in \\tau, i_0 \\in I: x \\in U \\implies \\exists i \\ge i_0: x_i \\in U$ That is, for every open $U$ with $x \\in U$, and for every $i_0 \\in I$, there is an $i \\ge i_0$ such that $x_i \\in U$. If $\\displaystyle x_i \\mathop{\\longrightarrow}_{\\text{cl}} x$, then $x$ is called a '''cluster point of $\\left({x_i}\\right)$'''."}
+{"_id": "22297", "title": "Definition:Precede", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a, b \\in S$ such that $a \\preceq b$. Then '''$a$ precedes $b$'''."}
+{"_id": "22298", "title": "Definition:Strictly Precede", "text": "=== Definition 1 === {{:Definition:Strictly Precede/Definition 1}} === Definition 2 === {{:Definition:Strictly Precede/Definition 2}}"}
+{"_id": "22299", "title": "Definition:Strictly Succeed", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a \\prec b$. That is, let $a$ strictly precede $b$. Then '''$b$ strictly succeeds $a$'''. This can be expressed symbolically as: :$b \\succ a$"}
+{"_id": "22300", "title": "Definition:Succeed", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a, b \\in S$ such that $a \\preceq b$. Then '''$b$ succeeds $a$'''."}
+{"_id": "22301", "title": "Definition:Between", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a, b, c \\in S$ such that $a \\preceq b$ and $b \\preceq c$. Then '''$b$ is between $a$ and $c$'''."}
+{"_id": "22302", "title": "Definition:Strictly Between", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a, b, c \\in S$ such that $a \\prec b$ and $b \\prec c$. That is, such that: :$a \\preceq b$ and $a \\ne b$ :$b \\preceq c$ and $b \\ne c$ Then '''$b$ is strictly between $a$ and $c$'''."}
+{"_id": "22303", "title": "Definition:Circular Proof", "text": "A '''circular proof''' is a proof $A$ in terms of another proof $B$ such that the truth value of $B$ depends (either directly or indirectly) on the truth value of $A$. A '''circular proof''' is valid, as every statement implies itself.  The problem of a '''circular proof''' is that if the truth of a conclusion is [http://en.wiktionary.org/wiki/dubious dubious], premises that are equally dubious provide no reason to support the conclusion."}
+{"_id": "22304", "title": "Definition:Idempotent Semigroup", "text": "An '''idempotent semigroup''' is a semigroup whose operation is idempotent. That is, a semigroup $\\struct {S, \\circ}$ is '''idempotent''' {{iff}}: :$\\forall x \\in S: x \\circ x = x$"}
+{"_id": "22305", "title": "Definition:Commutative Semigroup with respect to Equivalence Relation", "text": "Let $C$ be a class. Let $\\thickapprox$ be an equivalence relation on $C$. Let $\\left({C, \\cdot}\\right)$ be a large magma. Then $\\left({C, \\cdot}\\right)$ is a '''commutative semigroup with respect to $\\thickapprox$''' iff: :it is a semigroup with respect to $\\thickapprox$ :$\\forall x, y \\in C: x \\cdot y \\thickapprox y \\cdot x$"}
+{"_id": "22306", "title": "Definition:Bounded Linear Transformation", "text": "Let $H, K$ be Hilbert spaces. Let $A: H \\to K$ be a linear transformation. Then $A$ is a '''bounded linear transformation''' {{iff}} :$\\exists c > 0: \\forall h \\in H: \\left\\Vert{A h}\\right\\Vert_K \\le c \\left\\Vert{h}\\right\\Vert_H$"}
+{"_id": "22307", "title": "Definition:Space of Bounded Linear Transformations", "text": "Let $H, K$ be Hilbert spaces. Let $\\Bbb F \\in \\set {\\R, \\C}$ be the ground field of $K$. Then the '''space of bounded linear transformations from $H$ to $K$''', $\\map B {H, K}$, is the set of all bounded linear transformations: :$\\map B {H, K} := \\set {A: H \\to K: A \\text{ linear}, \\norm A < \\infty}$ endowed with pointwise addition and ($\\F$)-scalar multiplication. Then $\\map B {H, K}$ is a vector space over $\\Bbb F$. Furthermore, let $\\norm {\\,\\cdot\\,}$ denote the norm on bounded linear transformations. Then $\\norm{\\,\\cdot\\,}$ is a norm on $\\map B {H, K}$, and it even turns $\\map B {H, K}$ into a Banach space. These results are proved in Space of Bounded Linear Transformations is Banach Space. === Space of Bounded Linear Operators === When $H = K$, one denotes $\\map B H$ for $\\map B {H, K}$. In line with the definition of linear operator, $\\map B H$ is called the '''space of bounded linear operators on $H$'''."}
+{"_id": "22308", "title": "Definition:Semigroup with respect to Equivalence Relation", "text": "Let $C$ be a class. Let $\\thickapprox$ be an equivalence relation on $C$. Let $\\left({C, \\cdot}\\right)$ be a large magma. Then $\\left({C, \\cdot}\\right)$ is a '''semigroup with respect to $\\thickapprox$'''  iff: :$\\forall x, y,z \\in C: \\left( x \\cdot y \\right)\\cdot z \\thickapprox x \\cdot \\left(y \\cdot z\\right)$"}
+{"_id": "22309", "title": "Definition:Ground Field", "text": "Let $K$ be a field. Let $V$ be a vector space over $K$. Then $K$ is referred to as the '''ground field''' of $V$. Category:Definitions/Abstract Algebra 3uzqk57qlqusrsghl3rh9u4fi0j70wj"}
+{"_id": "22310", "title": "Definition:Pointwise Scalar Multiplication of Mappings", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring, and let $\\struct {S, \\circ}_R$ be an $R$-algebraic structure. Let $X$ be a non-empty set, and let $S^X$ be the set of all mappings from $X$ to $S$. Then '''pointwise ($R$)-scalar multiplication''' on $S^X$ is the binary operation $\\circ: R \\times S^X \\to S^X$ (the $\\circ$ is the same as for $S$) defined by: :$\\forall \\lambda \\in R: \\forall f \\in S^X: \\forall x \\in X: \\map {\\paren {\\lambda \\circ f} } x := \\lambda \\circ \\map f x$"}
+{"_id": "22311", "title": "Definition:Degenerate Case", "text": "A '''degenerate case''' is a specific manifestation of a particular type of object being included in another, usually simpler, type of object. {{WIP|Include links to the various types of degenerate objects that we have already defined.}}"}
+{"_id": "22312", "title": "Definition:Sesquilinear Form", "text": "Let $H, K$ be Hilbert spaces over $\\Bbb F \\in \\left\\{{\\R, \\C}\\right\\}$. A '''sesquilinear form''' is a function $u: H \\times K \\to \\Bbb F$ subject to: :$(1): \\qquad \\forall \\alpha \\in \\Bbb F, h_1, h_2 \\in H, k \\in K: u \\left({\\alpha h_1 + h_2, k}\\right) = \\alpha u \\left({h_1, k}\\right) + u \\left({h_2, k}\\right)$ :$(2): \\qquad \\forall \\alpha \\in \\Bbb F, h \\in H, k_1, k_2 \\in K: u \\left({h, \\alpha k_1 + k_2}\\right) = \\bar{\\alpha} u \\left({h, k_1}\\right) + u \\left({h, k_2}\\right)$ That is, $u$ is linear in the first argument, and conjugate linear in the second. If $\\Bbb F = \\R$, then a sesquilinear form is the same as a bilinear map."}
+{"_id": "22313", "title": "Definition:Equidistance", "text": "Define the following coordinates in the $xy$-plane: :$a = \\tuple {x_1, x_2}$ :$b = \\tuple {y_1, y_2}$ :$c = \\tuple {z_1, z_2}$ :$d = \\tuple {u_1, u_2}$ where $a, b, c, d \\in \\R^2$ 400px :$a b \\equiv c d \\dashv \\vdash \\paren {\\paren {x_1 - y_1}^2 + \\paren {x_2 - y_2}^2 = \\paren {z_1 - u_1}^2 + \\paren {z_2 - u_2}^2}$"}
+{"_id": "22314", "title": "Definition:Bounded Sesquilinear Form", "text": "Let $H, K$ be Hilbert spaces over $\\Bbb F \\in \\left\\{{\\R, \\C}\\right\\}$. Let $u: H \\times K \\to \\Bbb F$ be a sesquilinear form. Then $u$ is said to be a '''bounded sesquilinear form''', or to be '''bounded''', iff: :$\\exists M \\in \\R: \\forall h \\in H, k \\in K: \\left\\vert{u \\left({h, k}\\right)}\\right\\vert \\le M \\left\\Vert{h}\\right\\Vert_H \\left\\Vert{k}\\right\\Vert_K$ A constant $M$ satisfying the above is called a '''bound''' for $u$."}
+{"_id": "22315", "title": "Definition:Adjoint Linear Transformation", "text": "Let $H, K$ be Hilbert spaces. Let $A \\in \\map B {H, K}$ be a bounded linear transformation. Let $B \\in \\map B {K, H}$ be the unique bounded linear transformation provided by Existence and Uniqueness of Adjoint. Then $B$ is called the '''adjoint''' of $A$, and denoted $A^*$."}
+{"_id": "22317", "title": "Definition:Upper Semilattice on Classical Set", "text": "Let $\\struct {S, \\le}$ be an ordered set with the property that: :$\\forall x, y \\in S: \\sup \\set {x, y} \\in S$ where $\\sup$ denotes supremum. Then $\\struct {S, \\vee}$ is called an '''upper semilattice''', where $\\vee: S \\times S \\to S$ is defined by: :$x \\vee y := \\sup \\set {x, y}$ An '''upper semilattice''' hence is a particular kind of algebraic structure."}
+{"_id": "22319", "title": "Definition:Tarski's Axioms", "text": "'''Tarski's Axioms''' are a series of axioms whose purpose is to provide a rigorous basis for the definition of Euclidean geometry entirely within the framework of first order logic. In the following: : $\\equiv$ denotes the relation of equidistance. : $\\mathsf{B}$ denotes the relation of betweenness. : $=$ denotes the relation of equality. The axioms are as follows:"}
+{"_id": "22320", "title": "Definition:Directed Line Segment", "text": "A '''directed line segment''' is a line segment endowed with the additional property of direction. It is often used in the context of applied mathematics to represent a vector quantity. {{expand|Perhaps the above statement should also be expanded to allow a D.L.S. to be defined as a '''vector quantity''' applied at a particular point. There is a danger (as pointed out on the Definition:Vector Quantity page) of implying  / believing that a vector, in general, is applied at a particular point, for example usually the origin. Thus, this page allows the opportunity to consider a definition of an object which consists of a vector \"rooted\" at a particular point, as a convenient fiction for what is actually happening in the context of physics.}} {{stub|needs a picture It may be worthwhile to point out that this can be formalized with an ordered pair. Establish connection with Definition:Affine Space}}"}
+{"_id": "22321", "title": "Definition:Vector (Euclidean Space)", "text": "A vector is defined as an element of a vector space. We have that $\\R^n$, with the operations of vector addition and scalar multiplication, form a real vector space. Hence a '''vector in $\\R^n$''' is defined as any element of $\\R^n$. {{expand|While it is possible to identify a vector by a tuple in $\\R^n$ we need to explain that the vector is not the point.}} === $\\R^2$: Plane Vector === {{:Definition:Vector (Euclidean Space)/Plane Vector}} === $\\R^3$: Space Vector === {{:Definition:Vector (Euclidean Space)/Space Vector}}"}
+{"_id": "22322", "title": "Definition:Engineering Notation", "text": "=== Euclidean 2-space === Define the ordered 2-tuples: {{begin-eqn}} {{eqn | l = \\mathbf i       | r = \\left\\langle{1, 0}\\right\\rangle }} {{eqn | l = \\mathbf j       | r = \\left\\langle{0, 1}\\right\\rangle }} {{end-eqn}} From Standard Ordered Basis is Basis, we have that any vector in $\\R^2$ can be represented by: :$c_1 \\mathbf i + c_2 \\mathbf j$ where $c_1, c_2 \\in \\R$. This way of presenting vectors is called '''engineering notation'''. === Euclidean 3-space === Define the ordered 3-tuples: {{begin-eqn}} {{eqn | l = \\mathbf i       | r = \\left\\langle{1, 0, 0}\\right\\rangle }} {{eqn | l = \\mathbf j       | r = \\left\\langle{0, 1, 0}\\right\\rangle }} {{eqn | l = \\mathbf k       | r = \\left\\langle{0, 0, 1}\\right\\rangle }} {{end-eqn}} By the same logic as the above definition, we can write any vector in $\\R^3$ as: :$c_1 \\mathbf i + c_2 \\mathbf j + c_3 \\mathbf k$ where $c_1, c_2, c_3 \\in \\R$. Note that $\\mathbf i$ and $\\mathbf j$ take on a different meaning in $3$-space than in $2$-space. === Euclidean $n$-space === In higher dimensions, rather than writing $\\mathbf l, \\mathbf m, \\mathbf n$, and so on, the convention is to use: {{begin-eqn}} {{eqn | l = \\mathbf e_1       | r = \\left\\langle{1, 0, 0, \\ldots, 0, 0}\\right\\rangle }} {{eqn | l = \\mathbf e_2       | r = \\left\\langle{0, 1, 0, \\ldots, 0, 0}\\right\\rangle }} {{eqn | o = \\vdots }} {{eqn | l = \\mathbf e_n       | r = \\left\\langle{0, 0, 0, \\ldots, 0, 1}\\right\\rangle }} {{end-eqn}} Then any vector in $\\R^n$ can be expressed as: :$c_1 \\mathbf e_1 + c_2 \\mathbf e_2 + \\cdots + c_n \\mathbf e_n$ where $c_1, c_2, \\cdots, c_n \\in \\R$. This convention is also frequently seen for $2$-space and $3$-space."}
+{"_id": "22324", "title": "Definition:Vector (Linear Algebra)", "text": "Let $V$ be a vector space. Any element $v$ of $V$ is called a '''vector'''."}
+{"_id": "22325", "title": "Definition:Self-Adjoint Operator", "text": "Let $H$ be a Hilbert space. Let $A \\in B \\left({H}\\right)$ be a bounded linear operator. Then $A$ is said to be '''self-adjoint''' or '''hermitian''' iff: :$A = A^*$ That is, if it equals its adjoint $A^*$. {{namedfor|Charles Hermite}}"}
+{"_id": "22326", "title": "Definition:Normal Operator", "text": "Let $H$ be a Hilbert space. Let $A \\in B \\left({H}\\right)$ be a bounded linear operator. Then $A$ is said to be '''normal''' {{iff}}: :$A^* A = A A^*$ where $A^*$ denotes the adjoint of $A$."}
+{"_id": "22327", "title": "Definition:Real Part (Linear Operator)", "text": "Let $H$ be a Hilbert space over $\\C$. Let $A \\in B \\left({H}\\right)$ be a bounded linear operator. Then the '''real part of $A$''' is the self-adjoint operator: :$\\operatorname{Re} A := \\dfrac 1 2 \\left({A + A^*}\\right)$ The real part of $A$ may be denoted by $\\operatorname{Re} \\left({A}\\right)$, $\\operatorname{re} \\left({A}\\right)$ or $\\Re \\left({A}\\right)$. This resembles the notation for the real part of a complex number."}
+{"_id": "22328", "title": "Definition:Imaginary Part (Linear Operator)", "text": "Let $H$ be a Hilbert space over $\\C$. Let $A \\in B \\left({H}\\right)$ be a bounded linear operator. Then the '''imaginary part of $A$''' is the self-adjoint operator: :$\\operatorname{Im} A := \\dfrac 1 {2i} \\left({A - A^*}\\right)$ The imaginary part of $A$ may be denoted by $\\operatorname{Im} \\left({A}\\right)$, $\\operatorname{im} \\left({A}\\right)$ or $\\Im \\left({A}\\right)$. This resembles the notation for the imaginary part of a complex number."}
+{"_id": "22329", "title": "Definition:Absurd", "text": "'''Absurd''' is a word used mainly in logic meaning either '''meaningless''', '''contradictory''' or '''internally inconsistent'''."}
+{"_id": "22330", "title": "Definition:Idempotent Operator", "text": "Let $H$ be a Hilbert space. Let $A \\in B \\left({H}\\right)$ be a bounded linear operator. Then $A$ is said to be '''idempotent''', or an '''idempotent operator''', iff $A^2 = A$."}
+{"_id": "22331", "title": "Definition:Projection (Hilbert Spaces)", "text": "Let $H$ be a Hilbert space. Let $P \\in \\map B H$ be an idempotent operator. Then $P$ is said to be a '''projection''' {{iff}}: :$\\ker P = \\paren {\\Img P}^\\perp$ where: :$\\ker P$ denotes the kernel of $P$ :$\\Img P$ denotes the image of $P$ :$\\perp$ denotes orthocomplementation."}
+{"_id": "22332", "title": "Definition:Complementary Idempotent", "text": "Let $H$ be a Hilbert space. Let $A \\in B \\left({H}\\right)$ be an idempotent operator. Then the '''complementary idempotent (operator) of $A$''' is the bounded linear operator $I - A$, where $I$ is the identity operator on $H$. The name is appropriate, by Complementary Idempotent is Idempotent and the fact that $I - \\left({I - A}\\right) = A$. === Complementary Projection === If $A$ is a projection, $I - A$ is called the '''complementary projection'''. This name is justified by Complementary Projection is Projection."}
+{"_id": "22333", "title": "Definition:Closed Linear Subspace", "text": "Let $V$ be a topological vector space. A linear subspace of $V$ that is closed is called a '''closed linear subspace'''."}
+{"_id": "22334", "title": "Definition:Internal Hilbert Space Direct Sum", "text": "Let $H$ be a Hilbert space. Let $\\left\\{{M_i : i \\in I}\\right\\}$ be an $I$-indexed collection of pairwise orthogonal subspaces of $H$. Then the '''internal (Hilbert space) direct sum''' of the $M_i$ is their closed linear span $\\vee_i M_i$. It is denoted by $\\bigoplus_i M_i$, or $\\displaystyle \\bigoplus_{i \\in I} M_i$ if the set $I$ is to be stressed. When $I$ is finite, by Closed Linear Subspaces Closed under Setwise Addition, have that: :$\\displaystyle \\bigoplus_{i \\in I} M_i = \\sum_{i \\in I} M_i$, where $\\displaystyle \\sum$ signifies setwise addition."}
+{"_id": "22335", "title": "Definition:Orthogonal Difference", "text": "Let $H$ be a Hilbert space. Let $M, N$ be closed linear subspaces of $H$. Then the '''orthogonal difference of $M$ and $N$''', denoted $M \\ominus N$, is the set $M \\cap N^\\perp$. It is in fact a closed linear subspace of $H$, as proven on Orthogonal Difference is Closed Linear Subspace."}
+{"_id": "22336", "title": "Definition:Maximal/Set", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Let $\\mathcal T \\subseteq \\powerset S$ be a subset of $\\powerset S$. Let $\\struct {\\mathcal T, \\subseteq}$ be the ordered set formed on $\\mathcal T$ by $\\subseteq$ considered as an ordering. Then $T \\in \\mathcal T$ is a '''maximal set''' of $\\mathcal T$ {{iff}} $T$ is a maximal element of $\\struct {\\mathcal T, \\subseteq}$. That is: :$\\forall X \\in \\mathcal T: T \\subseteq X \\implies T = X$"}
+{"_id": "22337", "title": "Definition:Invariant Subspace", "text": "Let $H$ be a Hilbert space. Let $A \\in B \\left({H}\\right)$ be a bounded linear operator. Let $M$ be a closed linear subspace of $H$. Then $M$ is said to be an '''invariant subspace for $A$''' iff $h \\in M \\implies Ah \\in M$. That is, if $AM \\subseteq M$."}
+{"_id": "22338", "title": "Definition:Reducing Subspace", "text": "Let $H$ be a Hilbert space. Let $A \\in B \\left({H}\\right)$ be a bounded linear operator. Let $M$ be a closed linear subspace of $H$; denote by $M^\\perp$ its orthocomplement. Then $M$ is said to be a '''reducing subspace for $A$''' iff both $M$ and $M^\\perp$ are invariant subspaces for $A$. That is, if $AM \\subseteq M$ and $A M^\\perp \\subseteq M^\\perp$."}
+{"_id": "22339", "title": "Definition:Matrix Notation (Bounded Linear Operator)", "text": "Let $H$ be a Hilbert space. Let $A \\in B \\left({H}\\right)$ be a bounded linear operator on $H$. Let $M$ be a closed linear subspace of $H$; denote by $M^\\perp$ its orthocomplement. Observe that $M \\oplus M^\\perp \\cong H$ by Direct Sum of Subspace and Orthocomplement. The associated isomorphism $U$ induces a bounded linear operator $A'$ on $M \\oplus M^\\perp$. $A'$ can be written as the matrix:  :$\\begin{pmatrix}  W & X \\\\  Y & Z \\end{pmatrix}$  where $W \\in B \\left({M}\\right)$, $X \\in B \\left({M^\\perp, M}\\right)$, $Y \\in B \\left({M, M^\\perp}\\right)$, $Z \\in B \\left({M^\\perp}\\right)$. In the above matrix, linear transformations $W, X, Y, Z$ are uniquely determined by the requirement that: :$\\begin{pmatrix} Wm + Xm^\\perp \\\\ Ym + Zm^\\perp\\end{pmatrix} = \\begin{pmatrix}  W & X \\\\  Y & Z \\end{pmatrix} \\begin{pmatrix}m \\\\m^\\perp \\end{pmatrix} = U^{-1} \\left({Ah}\\right)$ whenever $h = U \\left({m, m^\\perp}\\right)$. By abuse of notation, one writes $A$ instead of $A'$ and calls it the '''matrix notation for $A$ (with respect to $M$)'''."}
+{"_id": "22340", "title": "Definition:Inverse (Bounded Linear Transformation)", "text": "Let $H, K$ be Hilbert spaces. Let $A \\in \\map B {H, K}$ be a bounded linear transformation. An '''inverse''' for $A$ is a bounded linear transformation $A^{-1} \\in \\map B {K, H}$ satisfying: :$AA^{-1} = I_K$ :$A^{-1}A = I_H$ where $I_K, I_H$ denote the identity operators on $K, H$, respectively. If such a $A^{-1}$ exists, $A$ is said to be an '''invertible (bounded) linear transformation'''. The operation assigning $A^{-1}$ to $A$ is referred to as '''inverting'''. === Invertible Bounded Linear Operator === When $H = K$, the notation simplifies considerably, and $A$ is said to be a '''invertible (bounded) linear operator'''."}
+{"_id": "22342", "title": "Definition:Coprime/GCD Domain", "text": "Let $\\struct {D, +, \\times}$ be a GCD domain. Let $U \\subseteq D$ be the group of units of $D$. Let $a, b \\in D$ such that $a \\ne 0_D$ and $b \\ne 0_D$ Let $d = \\gcd \\set {a, b}$ be the greatest common divisor of $a$ and $b$. Then $a$ and $b$ are '''coprime''' {{iff}} $d \\in U$. That is, two elements of a GCD domain are '''coprime''' {{iff}} their greatest common divisor is a unit of $D$."}
+{"_id": "22343", "title": "Definition:Coprime/Euclidean Domain", "text": "Let $\\struct {D, +, \\times}$ be a Euclidean domain. Let $U \\subseteq D$ be the group of units of $D$. Let $a, b \\in D$ such that $a \\ne 0_D$ and $b \\ne 0_D$ Let $d = \\gcd \\set {a, b}$ be the greatest common divisor of $a$ and $b$. Then $a$ and $b$ are '''coprime''' {{iff}} $d \\in U$. That is, two elements of a Euclidean domain are '''coprime''' {{iff}} their greatest common divisor is a unit of $D$."}
+{"_id": "22344", "title": "Definition:Coprime/Integers", "text": "Let $a$ and $b$ be integers such that $b \\ne 0$ and $a \\ne 0$ (that is, they are both non-zero). Let $\\gcd \\set {a, b}$ denote the greatest common divisor of $a$ and $b$. Then $a$ and $b$ are '''coprime''' {{iff}} $\\gcd \\set {a, b} = 1$."}
+{"_id": "22345", "title": "Definition:Monotonicity", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be ordered sets. Let $\\phi: \\left({S, \\preceq_1}\\right) \\to \\left({T, \\preceq_2}\\right)$ be a monotone mapping. Then whether $\\phi$ is increasing or decreasing is known as the '''monotonicity of $\\phi$'''.  That is: :if $\\phi$ is such that $x \\mathop{\\preceq_1} y \\implies \\phi \\left({x}\\right) \\mathop{\\preceq_2} \\phi \\left({y}\\right)$, then the '''monotonicity''' of $\\phi$ is increasing :if $\\phi$ is such that $x \\mathop{\\prec_1} y \\implies \\phi \\left({x}\\right) \\mathop{\\prec_2} \\phi \\left({y}\\right)$, then the '''monotonicity''' of $\\phi$ is strictly increasing :if $\\phi$ is such that $x \\mathop{\\preceq_1} y \\implies \\phi \\left({y}\\right) \\mathop{\\preceq_2} \\phi \\left({x}\\right)$, then the '''monotonicity''' of $\\phi$ is decreasing :if $\\phi$ is such that $x \\mathop{\\prec_1} y \\implies \\phi \\left({y}\\right) \\mathop{\\prec_2} \\phi \\left({x}\\right)$, then the '''monotonicity''' of $\\phi$ is strictly decreasing."}
+{"_id": "22347", "title": "Definition:Continuation of Woset", "text": "Let $\\left({S, \\preccurlyeq}\\right)$ be a woset. Let $\\left({T, \\preccurlyeq}\\right)$ be a set with an ordering such that: :$(1): \\quad T$ is an initial segment of $S$ :$(2): \\quad$ The ordering of the elements of $T$ is the same as their ordering in $S$. Then $S$ is a '''continuation''' of $T$."}
+{"_id": "22348", "title": "Definition:Addition/Summand", "text": "Let $a + b$ denote the operation of addition on two objects. The objects $a$ and $b$ are known as the '''summands''' of $a + b$."}
+{"_id": "22349", "title": "Definition:Between (Geometry)", "text": "Define the following coordinates in the $xy$-plane: {{begin-eqn}} {{eqn | l = a       | r = \\left({x_1, x_2}\\right) }} {{eqn | l = b       | r = \\left({y_1, y_2}\\right) }} {{eqn | l = c       | r = \\left({z_1, z_2}\\right) }} {{end-eqn}} where $a, b, c \\in \\R^2$. Let: {{begin-eqn}} {{eqn | l = \\Delta x_1       | r = x_3 - x_2 }} {{eqn | l = \\Delta x_2       | r = x_2 - x_1 }} {{eqn | l = \\Delta y_1       | r = y_2 - y_1 }} {{eqn | l = \\Delta y_2       | r = y_3 - y_2 }} {{end-eqn}} Then: :File:Betweenness(Analytic Def'n).png :$\\mathsf{B}abc \\dashv \\vdash \\left({\\Delta x_1 \\Delta y_1 = \\Delta x_2 \\Delta y_2}\\right) \\land$ :$\\left({0 \\le \\Delta x_1 \\Delta y_1 \\land 0 \\le \\Delta x_2 \\Delta y_2}\\right)$"}
+{"_id": "22350", "title": "Definition:Ordering Induced by Injection", "text": "Let $\\left({T, \\le}\\right)$ be an ordered set, and let $S$ be a set. Let $f: S \\to T$ be an injection. Define $\\le_f$ as the '''ordering induced by $f$ on $S$''' by: :$\\forall s_1, s_2 \\in S: s_1 \\le_f s_2 \\iff f \\left({s_1}\\right) \\le f \\left({s_2}\\right)$ That $\\le_f$ is in fact an ordering is shown on Ordering Induced by Injection is Ordering."}
+{"_id": "22352", "title": "Definition:Dominate (Analysis)", "text": "Let $\\left \\langle {a_n} \\right \\rangle$ be a sequence in $\\R$. Let $\\left \\langle {z_n} \\right \\rangle$ be a sequence in $\\C$. Then $\\left \\langle {a_n} \\right \\rangle$ '''dominates''' $\\left \\langle {z_n} \\right \\rangle$ iff: :$\\forall n \\in \\N: \\left|{z_n}\\right| \\le a_n$ {{SUBPAGENAME}} {{SUBPAGENAME}} dpeqz6xo6qbc2m40xp0hhgcrjj7wbro"}
+{"_id": "22354", "title": "Definition:Vertex Cut", "text": "Let $G$ be a graph. A '''vertex cut''' of $G$ is a set of vertices $W \\subseteq V \\left({G}\\right)$ such that the vertex deletion $G \\setminus W$ is disconnected."}
+{"_id": "22355", "title": "Definition:K-Connected", "text": "Let $G$ be a graph. Let $k \\in \\Z_{>0}$. Then $G$ is '''$k$-connected''' {{iff}}: : $\\left\\lvert{V \\left({G}\\right) }\\right\\rvert > k$ : $G$ is connected : $G$ has no vertex cut $W$ with $\\left\\lvert{W}\\right\\rvert < k$ Informally: : $G$ is connected and has more than $k$ vertices : There is no vertex cut of $G$ containing less than $k$ vertices."}
+{"_id": "22356", "title": "Definition:Connectivity", "text": "Let $G$ be a graph. Then $\\kappa \\left({G}\\right)$, the '''connectivity''' of $G$, is the smallest $k \\in \\Z_{>0}$ such that $G$ is $k$-connected.  If $G$ is disconnected, or if $G$ has fewer than 2 vertices, then $\\kappa \\left({G}\\right) = 0$. {{SUBPAGENAME}} 8n9tlxeii5l0iw6ad6hzbwkzdkislw1"}
+{"_id": "22357", "title": "Definition:Vertex Deletion", "text": "Let $G = \\left({V, E}\\right)$ be an (undirected) graph. Let $W \\subseteq V$ be a set of vertices of $G$. Then the '''graph obtained by deleting $W$ from $G$''', denoted by $G - W$, is the subgraph induced by $V \\setminus W$. Alternatively: : $G - W = \\left({V \\setminus W, \\left\\{{e \\in E : e \\cap W = \\varnothing}\\right\\}}\\right)$ Informally, $G - W$ is the graph obtained from $G$ by removing all vertices in $W$ and all edges incident to those vertices. If $W$ is a singleton such that $W = \\left\\{{v}\\right\\}$, then $G - W$ may be expressed $G - v$."}
+{"_id": "22358", "title": "Definition:Minimum Degree", "text": "Let $G = \\left({V, E}\\right)$ be a graph. Then the '''minimum degree''' of $G$ is: : $\\delta\\left({G}\\right) = \\min\\left\\{{\\deg_G\\left({v}\\right) : v \\in V}\\right\\}$ That is, it is the minimum degree of all the vertices of $G$. {{SUBPAGENAME}} pdfmm4z4qjqku87z11orh4durbra7v7"}
+{"_id": "22359", "title": "Definition:Neighborhood (Graph Theory)", "text": "Let $G = \\struct {V, E}$ be a graph. Let $v \\in V$ be a vertex of $G$. The '''neighborhood of $v$ in $G$''' is: :$\\map {\\Gamma_G} v = \\set {u \\in V : u v \\in E}$ That is, it is the set of all vertices which are adjacent to $v$. {{Languages|Neighborhood}} {{Language|Dutch|nabuurschap}} {{Language|French|voisinage}} {{End-languages}}"}
+{"_id": "22361", "title": "Definition:Halmos Symbol", "text": "The '''Halmos symbol''' is the character: $\\blacksquare$ used to indicate the end of a proof. It replaces the old-fashioned and embarrassingly uncool Q.E.D. which muggles sometimes use when pretending to be clever."}
+{"_id": "22362", "title": "Definition:Included Set Topology", "text": "Let $S$ be a set which is non-null. Let $H \\subseteq S$ be some subset of $S$. We define a subset $\\tau_H$ of the power set $\\powerset S$ as: :$\\tau_H = \\set {A \\subseteq S: H \\subseteq A} \\cup \\set \\O$ that is, all the subsets of $S$ which are supersets of $H$, along with the empty set $\\O$. Then $\\tau_H$ is a topology called the '''included set topology on $S$ by $H$''', or just '''an included set topology'''. The topological space $T = \\struct {S, \\tau_H}$ is called the '''included set space on $S$ by $H$''', or just '''an included set space'''."}
+{"_id": "22363", "title": "Definition:Compact Linear Transformation", "text": "Let $H, K$ be Hilbert spaces. Let $T: H \\to K$ be a linear transformation. Let $\\operatorname{ball} H$ be the closed unit ball of $H$. Then $T$ is said to be a '''compact linear transformation''', or simply '''compact''' {{iff}} $\\operatorname{cl} \\left({T \\left({\\operatorname{ball} H}\\right) }\\right)$ is compact in $K$, where $\\operatorname{cl}$ denotes closure. === Compact Operator === When $H$ and $K$ are equal, one speaks about '''compact (linear) operators''' instead. This is in line with the definition of a linear operator."}
+{"_id": "22364", "title": "Definition:Closed Unit Ball", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $a \\in X$. The '''closed unit ball''' of $X$, denoted $\\operatorname{ball} X$, is the set: :$\\map {B_1^-} a := \\set {x \\in X: \\norm {x - a} \\le 1}$"}
+{"_id": "22365", "title": "Definition:Space of Compact Linear Transformations", "text": "Let $H, K$ be Hilbert spaces. Let $\\Bbb F \\in \\left\\{{\\R, \\C}\\right\\}$ be the ground field of $K$. The '''space of compact linear transformations from $H$ to $K$''',  $B_0 \\left({H, K}\\right)$, is the set of all compact linear transformations: :$B_0 \\left({H, K}\\right):= \\left\\{{T: H \\to K: T \\text{ compact}}\\right\\}$ endowed with pointwise addition and ($\\F$)-scalar multiplication. It is a Banach space, as proven on Space of Compact Linear Transformations is Banach Space. The notation resembles that for the space of bounded linear transformations $B \\left({H, K}\\right)$. This is appropriate as a Compact Linear Transformation is Bounded; i.e., $B_0 \\left({H, K}\\right) \\subseteq B \\left({H, K}\\right)$. === Space of Compact Linear Operators === When $H$ is equal to $K$, one speaks about the '''space of compact (linear) operators''' instead. One writes $B_0 \\left({H}\\right)$ for $B_0 \\left({H, H}\\right)$."}
+{"_id": "22366", "title": "Definition:Finite Rank Operator", "text": "Let $H, K$ be Hilbert spaces. Let $T: H \\to K$ be a linear transformation. Then $T$ is said to be a '''finite rank operator''', or of '''finite rank''', {{iff}} its range, $\\Rng T$, is finite dimensional. Note that a finite rank operator is not necessarily bounded."}
+{"_id": "22367", "title": "Definition:Condensed Series", "text": "Let $\\left \\langle {a_n} \\right \\rangle: n \\mapsto a\\left({n}\\right)$ be a decreasing sequence of strictly positive terms in $\\R$ which converges with a limit of zero. That is, for every $n$ in the domain of $\\left \\langle {a_n} \\right \\rangle$: $a_n > 0, a_{n+1} \\le a_n$, and $a_n \\to 0$ as $n \\to +\\infty$. The series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty 2^n a\\left({2^n}\\right)$ is called the '''condensed''' form of the series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$"}
+{"_id": "22368", "title": "Definition:Space of Continuous Finite Rank Operators", "text": "Let $H, K$ be Hilbert spaces. Then the '''space of continuous finite rank operators from $H$ to $K$''', denoted $B_{00} \\left({H, K}\\right)$, is the set: :$B_{00} \\left({H, K}\\right) := \\left\\{{A \\in B \\left({H, K}\\right): A \\text{ is of finite rank} }\\right\\}$ of all bounded linear transformations of finite rank. By definition, it is a subset of the space of bounded linear transformations $B \\left({H, K}\\right)$. In fact, by Finite Rank Operator is Compact, it is contained in $B_0 \\left({H, K}\\right)$, the space of compact linear transformations."}
+{"_id": "22369", "title": "Definition:Elementary Matrix", "text": "=== Elementary Row Matrix === {{:Definition:Elementary Matrix/Row Operation}} === Elementary Column Matrix === {{:Definition:Elementary Matrix/Column Operation}}"}
+{"_id": "22370", "title": "Definition:Diagonalizable Operator", "text": "Let $H$ be a Hilbert space over $\\Bbb F \\in \\set {\\R, \\C}$. Let $A:H \\to H$ be a linear operator on $H$. The following two definitions of '''diagonalizable operator''' are equivalent: === By a Basis === $A$ is said to be '''diagonalizable''' {{iff}} there exist: :a basis $E$ for $H$ :an indexed set $\\family {\\alpha_e}_{e \\mathop \\in E} \\subseteq \\Bbb F$ of scalars (with $E$ as indexing set) such that: :$\\forall e \\in E: Ae = \\alpha_e e$ ==== Value Set ==== The indexed set $\\family {\\alpha_e}_{e \\mathop \\in E}$ may be called the '''value set of $A$ (with respect to the basis $E$)'''. === By a Partition of Unity === $A$ is said to be '''diagonalizable''' iff there exist: :a partition of unity $\\family {P_i}_{i \\mathop \\in I}$ on $H$ :an indexed set $\\family {\\alpha_i}_{i \\mathop \\in I} \\subseteq \\Bbb F$ of scalars (with the same $I$ as indexing set) such that: :$\\forall i \\in I: \\forall h \\in \\Rng {P_i}: A h = \\alpha_i h$ To express that $A$ is diagonalizable, one writes $A = \\displaystyle \\sum_{i \\in I} \\alpha_i P_i$ or $A = \\displaystyle \\bigoplus_{i \\in I} \\alpha_i P_i$. ==== Value Set ==== The indexed set $\\family {\\alpha_i}_{i \\mathop \\in I}$ may be called the '''value set of $A$ (with respect to the partition of unity $\\family {P_i}_{i \\mathop \\in I}$)'''."}
+{"_id": "22371", "title": "Definition:Row Equivalence", "text": "Two matrices $\\mathbf A = \\sqbrk a_{m n}, \\mathbf B = \\sqbrk b_{m n}$ are '''row equivalent''' if one can be obtained from the other by a finite sequence of elementary row operations. This relationship can be denoted $\\mathbf A \\sim \\mathbf B$."}
+{"_id": "22372", "title": "Definition:Parity Group", "text": ": The group $\\struct {\\Z_2, +_2}$ : $C_2$, the cyclic group of order 2 : The group $\\struct {\\set {1, -1}, \\times}$ : The quotient group $\\dfrac {S_n} {A_n}$ of the symmetric group of order $n$ with the alternating group of order $n$ etc."}
+{"_id": "22373", "title": "Definition:Eigenvector", "text": "Let $H$ be a Hilbert space over $\\Bbb F \\in \\set {\\R, \\C}$. Let $A \\in \\map B H$ be a bounded linear operator. Let $\\alpha \\in \\Bbb F$ be an eigenvalue of $A$. A nonzero vector $h \\in H$ is said to be an '''eigenvector for $\\alpha$''' {{iff}}: :$h \\in \\map \\ker {A - \\alpha I}$ That is, {{iff}} $Ah = \\alpha h$. === Eigenspace === {{refactor|Separate page for this}} The '''eigenspace''' for an eigenvalue $\\alpha$ is the set $\\map \\ker {A - \\alpha I}$. By Kernel of Linear Transformation is Closed Linear Subspace, it is a closed linear subspace of $H$."}
+{"_id": "22375", "title": "Definition:Partition of Unity (Hilbert Space)", "text": "Let $H$ be a Hilbert space. A '''partition of unity''' or '''partition of identity''' on $H$ is a family $\\family {P_i}_{i \\mathop \\in I}$ of projections, subject to: :If $i \\ne j$, then $P_i P_j = P_j P_i = 0$ :$\\vee \\set {\\Img {P_i}: i \\in I} = H$, where $\\vee$ signifies closed linear span One may encounter the notations $1 = \\sum_i P_i$ and $1 = \\bigoplus_i P_i$. Here, $1$ signifies the identity operator on $H$."}
+{"_id": "22377", "title": "Definition:Inflationary Mapping", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $\\phi: S \\to S$ be a mapping. Then $\\phi$ is '''inflationary''' {{iff}}: :$\\forall s \\in S: s \\preceq \\map \\phi s$"}
+{"_id": "22378", "title": "Definition:Hartogs Number", "text": "Let $S$ be a set. The '''Hartogs number''' of $S$ is the smallest ordinal $\\alpha$ such that there exists no injection from $\\alpha$ to $S$."}
+{"_id": "22379", "title": "Definition:Lyndon Word", "text": "A '''Lyndon word''' is a string that is strictly smaller in lexicographic order than all of its rotations. {{NamedforDef|Roger Conant Lyndon|cat = Lyndon}} Category:Definitions/Combinatorics og9zhmasfrj91zns892whlznip07t25"}
+{"_id": "22380", "title": "Definition:Rotation (Permutation Theory)", "text": "Let $\\tuple {a_1, \\ldots, a_n}$ be a string over an alphabet $A$. A '''rotation''' is a mapping $r: A^n \\to A^n$ given by: :$\\tuple {a_1, \\ldots, a_n} \\mapsto \\tuple {a_{\\map \\phi 1}, \\cdots, a_{\\map \\phi n} }$ where $\\phi$ is a permutation on n letters. Category:Definitions/Permutation Theory 4tkxx9i8aqrxcshizqgmrkh3c3jbwir"}
+{"_id": "22381", "title": "Definition:Isomorphism (Category Theory)", "text": "Let $\\mathbf C$ be a category, and let $X, Y$ be objects of $\\mathbf C$. A morphism $f: X \\to Y$ is an '''isomorphism''' if there exists a morphism $g: Y \\to X$ such that: :$g \\circ f = I_X$ :$f \\circ g = I_Y$ where $I_X$ denotes the identity morphism on $X$. It can be seen that this is equivalent to $g$ being both a retraction and a section of $f$."}
+{"_id": "22382", "title": "Definition:Isomorphism (Abstract Algebra)/Ordered Structure Isomorphism", "text": "An '''ordered structure isomorphism''' from an ordered structure $\\struct {S, \\circ, \\preceq}$ to another $\\struct {T, *, \\preccurlyeq}$ is a mapping $\\phi: S \\to T$ that is both: :$(1): \\quad$ An isomorphism, that is a bijective homomorphism, from the structure $\\struct {S, \\circ}$ to the structure $\\struct {T, *}$ :$(2): \\quad$ An order isomorphism from the ordered set $\\struct {S, \\preceq}$ to the ordered set $\\struct {T, \\preccurlyeq}$."}
+{"_id": "22383", "title": "Definition:Symmetry Group", "text": "Let $P$ be a geometric figure. Let $S_P$ be the set of all symmetries of $P$. Let $\\struct {S_P, \\circ}$ be the algebraic structure such that $\\circ$ denotes the composition of mappings. Then $\\struct {S_P, \\circ}$ is called the '''symmetry group of $P$'''."}
+{"_id": "22384", "title": "Definition:Cancellable Element/Left Cancellable", "text": "An element $x \\in \\struct {S, \\circ}$ is '''left cancellable''' {{iff}}: :$\\forall a, b \\in S: x \\circ a = x \\circ b \\implies a = b$"}
+{"_id": "22385", "title": "Definition:Cancellable Element/Right Cancellable", "text": "An element $x \\in \\left ({S, \\circ}\\right)$ is '''right cancellable''' iff: :$\\forall a, b \\in S: a \\circ x = b \\circ x \\implies a = b$"}
+{"_id": "22387", "title": "Definition:Circle Group", "text": "Let $K$ be the set of all complex numbers of unit modulus: :$K = \\set {z \\in \\C: \\cmod z = 1}$ Consider the algebraic structure $\\struct {K, \\times}$ where $\\times$ denotes the operation of complex multiplication. Then $\\struct {K, \\times}$ is called the '''circle group'''."}
+{"_id": "22388", "title": "Definition:Power Series/Real Domain", "text": "Let $\\xi \\in \\R$ be a real number. Let $\\sequence {a_n}$ be a sequence in $\\R$. The series $\\displaystyle \\sum_{n \\mathop = 0}^\\infty a_n \\paren {x - \\xi}^n$, where $x \\in \\R$ is a variable, is called a '''power series in $x$ about the point $\\xi$'''."}
+{"_id": "22389", "title": "Definition:Power Series/Complex Domain", "text": "Let $\\xi \\in \\C$ be a complex number. Let $\\sequence {a_n}$ be a sequence in $\\C$. The series $\\displaystyle \\sum_{n \\mathop = 0}^\\infty a_n \\paren {z - \\xi}^n$, where $z \\in \\C$ is a variable, is called a '''(complex) power series in $z$ about the point $\\xi$'''."}
+{"_id": "22390", "title": "Definition:Radius of Convergence/Real Domain", "text": "Let $\\xi \\in \\R$ be a real number. Let $\\displaystyle \\map S x = \\sum_{n \\mathop = 0}^\\infty a_n \\paren {x - \\xi}^n$ be a power series about $\\xi$. Let $I$ be the interval of convergence of $\\map S x$. Let the endpoints of $I$ be $\\xi - R$ and $\\xi + R$. (This follows from the fact that $\\xi$ is the midpoint of $I$.) Then $R$ is called the '''radius of convergence''' of $\\map S x$. If $\\map S x$ is convergent over the whole of $\\R$, then $I = \\R$ and thus the radius of convergence is infinite."}
+{"_id": "22391", "title": "Definition:Radius of Convergence/Complex Domain", "text": "Let $\\xi \\in \\C$ be a complex number. For $z \\in \\C$, let: :$\\displaystyle \\map f z = \\sum_{n \\mathop = 0}^\\infty a_n \\paren {z - \\xi}^n$ be a power series about $\\xi$. The '''radius of convergence''' is the extended real number $R \\in \\overline \\R$ defined by: :$R = \\displaystyle \\inf \\set {\\cmod {z - \\xi}: z \\in \\C, \\sum_{n \\mathop = 0}^\\infty a_n \\paren {z - \\xi}^n \\text{ is divergent} }$ where a divergent series is a series that is not convergent. As usual, $\\inf \\O = +\\infty$."}
+{"_id": "22394", "title": "Definition:Operation Induced by Restriction", "text": "Let $\\struct {S, \\circ}$ be a magma. Let $\\struct {T, \\circ} \\subseteq \\struct {S, \\circ}$. That is, let $T$ be a subset of $S$ such that $\\circ$ is closed in $T$. Then the restriction of $\\circ$ to $T$, namely $\\circ {\\restriction_T}$, is called the '''(binary) operation induced on $T$ by $\\circ$'''."}
+{"_id": "22395", "title": "Definition:Euclidean Norm", "text": "Let $\\mathbf v = \\tuple {v_1, v_2, \\ldots, v_n}$ be a vector in the Euclidean $n$-space $\\R^n$. The '''Euclidean norm''' of $\\mathbf v$ is defined as: :$\\displaystyle \\norm {\\mathbf v} = \\paren {\\sum_{k \\mathop = 1}^n v_k^2}^{1/2}$"}
+{"_id": "22396", "title": "Definition:Complex Sequence", "text": "A '''complex sequence''' is a sequence (usually infinite) whose codomain is the set of complex numbers $\\C$."}
+{"_id": "22397", "title": "Definition:Length of Sequence", "text": "The '''length''' of a finite sequence is the number of terms it contains, or equivalently, the cardinality of its domain. === Sequence of $n$ Terms === {{:Definition:Sequence of n Terms}}"}
+{"_id": "22398", "title": "Definition:Sequence of n Terms", "text": "A '''sequence of $n$ terms''' is a (finite) sequence whose length is $n$."}
+{"_id": "22399", "title": "Definition:Sequence/Empty Sequence", "text": "An '''empty sequence''' is a (finite) sequence containing no terms. Thus an '''empty sequence''' is a mapping from $\\varnothing$ to $S$, that is, the empty mapping."}
+{"_id": "22400", "title": "Definition:Bounded Below Sequence", "text": "Let $\\struct {T, \\preceq}$ be an ordered set. Let $\\sequence {x_n}$ be a sequence in $T$. Then $\\sequence {x_n}$ is '''bounded below''' {{iff}}: :$\\exists m \\in T: \\forall i \\in \\N: m \\preceq x_i$"}
+{"_id": "22401", "title": "Definition:Bounded Above Sequence", "text": "Let $\\struct {T, \\preceq}$ be an ordered set. Let $\\sequence {x_n}$ be a sequence in $T$. Then $\\sequence {x_n}$ is '''bounded above''' {{iff}}: :$\\exists M \\in T: \\forall i \\in \\N: x_i \\preceq M$"}
+{"_id": "22402", "title": "Definition:Continuous Mapping", "text": "The concept of '''continuity''' makes precise the intuitive notion that a function has no \"jumps\" at a given point. Loosely speaking, in the case of a real function, continuity at a point is defined as the property that the graph of the function does not have a \"break\" at the point. This concept appears throughout mathematics and correspondingly has many variations and generalizations."}
+{"_id": "22403", "title": "Definition:Unbounded Divergent Sequence/Real Sequence", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. === Divergent to Positive Infinity === {{:Definition:Unbounded Divergent Sequence/Real Sequence/Positive Infinity}} === Divergent to Negative Infinity === {{:Definition:Unbounded Divergent Sequence/Real Sequence/Negative Infinity}} === Divergent to Infinity === Consider the case where $\\sequence {x_n}$ is both unbounded above and unbounded below. {{:Definition:Unbounded Divergent Sequence/Real Sequence/Infinity}}"}
+{"_id": "22404", "title": "Definition:Unbounded Divergent Sequence/Complex Sequence", "text": "Let $\\sequence {z_n}$ be a sequence in $\\C$. Then $\\sequence {z_n}$ '''tends to $\\infty$''' or '''diverges to $\\infty$''' {{iff}}: :$\\forall H > 0: \\exists N: \\forall n > N: \\cmod {z_n} > H$ where $\\cmod {z_n}$ denotes the modulus of $z_n$. We write: :$x_n \\to \\infty$ as $n \\to \\infty$."}
+{"_id": "22405", "title": "Definition:Strictly Convex Real Function", "text": "Let $f$ be a real function which is defined on a real interval $I$. === Definition 1 === {{:Definition:Convex Real Function/Definition 1/Strictly}} === Definition 2 === {{:Definition:Convex Real Function/Definition 2/Strictly}} === Definition 3 === {{:Definition:Convex Real Function/Definition 3/Strictly}}"}
+{"_id": "22406", "title": "Definition:Hölder Mean", "text": "Let $x_1, x_2, \\ldots, x_n \\in \\R_{\\ge 0}$ be positive real numbers. Let $p$ be an extended real number. The '''Hölder mean with exponent $p$ of $x_1, x_2, \\ldots, x_n$''' is denoted $\\map {M_p} {x_1, x_2, \\ldots, x_n}$. For real $p \\ne 0$, it is defined as: :$\\displaystyle \\map {M_p} {x_1, x_2, \\ldots, x_n} = \\paren {\\frac 1 n \\sum_{k \\mathop = 1}^n x_k^p}^{1/p}$ whenever the above expression is defined. For $p = 0$, it is defined as: :$\\map {M_0} {x_1, x_2, \\ldots, x_n} = \\paren {x_1 x_2 \\cdots x_n}^{1/n}$ the geometric mean of $x_1, x_2, \\ldots, x_n$. For $p = \\infty$, it is defined as: :$\\map {M_\\infty} {x_1, x_2, \\ldots, x_n} = \\max {\\set {x_1, x_2, \\ldots, x_n} }$ For $p = -\\infty$, it is defined as: :$\\map {M_{-\\infty} } {x_1, x_2, \\ldots, x_n} = \\min {\\set {x_1, x_2, \\ldots, x_n} }$"}
+{"_id": "22407", "title": "Definition:Vector Addition", "text": "=== Vector Addition on Module === {{:Definition:Vector Addition/Module}} === Vector Addition on Vector Space === {{:Definition:Vector Addition/Vector Space}} === Vector Sum === {{:Definition:Vector Sum}}"}
+{"_id": "22408", "title": "Definition:Inductive Ordered Set", "text": "An '''inductive ordered set''' is an ordered set in which every chain has an upper bound."}
+{"_id": "22409", "title": "Definition:P-Norm", "text": "Let $p \\ge 1$ be a real number. Let $\\ell^p$ denote the $p$-sequence space. Let $\\mathbf x = \\sequence {x_n} \\in \\ell^p$. Then the '''$p$-norm''' of $\\mathbf x$ is defined as: :$\\displaystyle \\norm {\\mathbf x}_p = \\paren {\\sum_{n \\mathop = 0}^\\infty \\size {x_n}^p}^{1/p}$"}
+{"_id": "22411", "title": "Definition:Left-Hand Derivative", "text": "Let $B$ be a Banach space over the set of real numbers $\\R$. Let $f: \\R \\to B$ be a mapping from $\\R$ to $B$. The '''left-hand derivative''' of $f$ is defined as the left-hand limit: :$\\displaystyle \\map {f'_-} x = \\lim_{h \\mathop \\to 0^-} \\frac {\\map f {x + h} - \\map f x} h$ If the '''left-hand derivative exists''', then $f$ is said to be '''left-hand differentiable''' at $x$."}
+{"_id": "22412", "title": "Definition:Right-Hand Derivative", "text": "Let $B$ be a Banach space over the set of real numbers $\\R$. Let $f: \\R \\to B$ be a mapping from $\\R$ to $B$. The '''right-hand derivative''' of $f$ is defined as the right-hand limit: :$\\displaystyle \\map {f'_+} x = \\lim_{h \\mathop \\to 0^+} \\frac {\\map f {x + h} - \\map f x} h$ If the '''right-hand derivative''' exists, then $f$ is said to be '''right-hand differentiable''' at $x$. === Real Functions === {{:Definition:Right-Hand Derivative/Real Function}}"}
+{"_id": "22413", "title": "Definition:One-Sided Derivative", "text": "A '''one-sided derivative''' is a right-hand derivative or a left-hand derivative. Category:Definitions/Differential Calculus f96ff9uj9rfx4inafni66xo7pjwpr7z"}
+{"_id": "22414", "title": "Definition:Difference Quotient", "text": "Let $V$ be a vector space over the real numbers $\\R$. Let $f: \\R \\to V$ be a function. A '''difference quotient''' is an expression of the form: :$\\dfrac {\\map f {x + h} - \\map f x} h$ where $h \\ne 0$ is a real number."}
+{"_id": "22415", "title": "Definition:Left Difference Quotient", "text": "Let $V$ be a vector space over the real numbers $\\R$. Let $f: \\R \\to V$ be a function. A '''left difference quotient''' is an expression of the form: :$\\dfrac {\\map f {x + h} - \\map f x} h$ where $h < 0$ is a strictly negative real number."}
+{"_id": "22416", "title": "Definition:Right Difference Quotient", "text": "Let $V$ be a vector space over the real numbers $\\R$. Let $f: \\R \\to V$ be a function. A '''right difference quotient''' is an expression of the form: :$\\dfrac {\\map f {x + h} - \\map f x} h$ where $h > 0$ is a strictly positive real number."}
+{"_id": "22417", "title": "Definition:Strictly Concave Real Function", "text": "Let $f$ be a real function which is defined on a real interval $I$. === Definition 1 === {{:Definition:Concave Real Function/Definition 1/Strictly}} === Definition 2 === {{:Definition:Concave Real Function/Definition 2/Strictly}} === Definition 3 === {{:Definition:Concave Real Function/Definition 3/Strictly}}"}
+{"_id": "22418", "title": "Definition:Co-Countable Set", "text": "Let $S$ be a set, and let $A \\subseteq S$. Then $A$ is said to be '''co-countable (in $S$)''' {{iff}} its relative complement $\\complement_S \\left({A}\\right)$ is countable."}
+{"_id": "22419", "title": "Definition:Trace Sigma-Algebra", "text": "Let $X$ be a set, and let $\\Sigma$ be a $\\sigma$-algebra on $X$. Let $E \\subseteq X$ be a subset of $X$. Then the '''trace $\\sigma$-algebra (of $E$ in $\\Sigma$)''', $\\Sigma_E$, is defined as: :$\\Sigma_E := \\set {E \\cap S: S \\in \\Sigma}$ It is a $\\sigma$-algebra on $E$, as proved on Trace Sigma-Algebra is Sigma-Algebra."}
+{"_id": "22420", "title": "Definition:Pre-Image Sigma-Algebra/Domain", "text": "Let $X, X'$ be sets, and let $f: X \\to X'$ be a mapping. Let $\\Sigma'$ be a $\\sigma$-algebra on $X'$. Then the '''pre-image $\\sigma$-algebra (of $\\Sigma'$) on the domain of $f$''' is defined as: :$f^{-1} \\left({\\Sigma'}\\right) := \\left\\{{f^{-1} \\left({E'}\\right): E' \\in \\Sigma'}\\right\\}$"}
+{"_id": "22421", "title": "Definition:Sigma-Algebra Generated by Collection of Subsets", "text": "Let $X$ be a set. Let $\\GG \\subseteq \\powerset X$ be a collection of subsets of $X$. === Definition 1 === {{:Definition:Sigma-Algebra Generated by Collection of Subsets/Definition 1}} === Definition 2 === {{:Definition:Sigma-Algebra Generated by Collection of Subsets/Definition 2}} === Generator === {{:Definition:Sigma-Algebra Generated by Collection of Subsets/Generator}}"}
+{"_id": "22422", "title": "Definition:Open Rectangle", "text": "Let $n \\ge 1$ be a natural number. Let $a_1, \\ldots, a_n, b_1, \\ldots, b_n$ be real numbers. The Cartesian product: :$\\displaystyle \\prod_{i \\mathop = 1}^n \\openint {a_i} {b_i} = \\openint {a_1} {b_1} \\times \\cdots \\times \\openint {a_n} {b_n} \\subseteq \\R^n$ is called an '''open rectangle in $\\R^n$''' or '''open $n$-rectangle'''. The collection of all '''open $n$-rectangles''' is denoted $\\JJ_o$, or $\\JJ_o^n$ if the dimension $n$ is to be emphasized. === Degenerate Case === In case $a_i \\ge b_i$ for some $i$, the rectangle is taken to be the empty set $\\O$. This is in accordance with the result Cartesian Product is Empty iff Factor is Empty for general Cartesian products."}
+{"_id": "22424", "title": "Definition:Atom of Sigma-Algebra", "text": "Let $\\struct {X, \\Sigma}$ be a measurable space. Let $E \\in \\Sigma$ be non-empty. $E$ is said to be an '''atom (of $\\Sigma$)''' {{iff}} it satisfies: :$\\forall F \\in \\Sigma: F \\subsetneq E \\implies F = \\O$ Thus, '''atoms''' are the minimal non-empty sets in $\\Sigma$ with respect to the subset ordering."}
+{"_id": "22426", "title": "Definition:Character (Representation Theory)", "text": "Let $\\left({G, \\cdot}\\right)$ be a finite group. Let $V$ be a finite dimensional $k$-vector space. Consider a linear representation $\\rho: G \\to \\operatorname{GL}\\left({V}\\right)$ of $G$. The '''character associated with $\\rho$''' is defined as: : $\\chi:G \\to k$ where $\\chi \\left({g}\\right) = \\operatorname{Tr} \\left({\\rho\\left({g}\\right)}\\right)$, the trace of $\\rho\\left({g}\\right)$; which is a linear automorphism of $V$."}
+{"_id": "22428", "title": "Definition:Reducible Linear Representation", "text": "Let $\\rho: G \\to \\operatorname{GL} \\left({V}\\right)$ be a linear representation. $\\rho$ is '''reducible''' {{iff}} there exists a non-trivial proper vector subspace $W$ of $V$ such that: :$\\forall g \\in G: \\rho \\left({g}\\right) \\left({W}\\right) \\subseteq W$ That is, such that $W$ is invariant for every linear operator in the set $\\left\\{{\\rho \\left({g}\\right): g \\in G}\\right\\}$."}
+{"_id": "22429", "title": "Definition:Vector-Valued Function", "text": "Let $f_1, f_2, \\ldots, f_n$ be real functions of $t$. Let $\\mathbb T \\subseteq \\R, \\mathbb Y \\subseteq \\R^n$ (where usually $n \\ge 2$). Let $\\mathbf r$ be a mapping from $\\mathbb T \\to \\mathbb Y$ that maps each $t \\in \\mathbb T$ to a vector $\\left \\langle{f_1 \\left({t}\\right), f_2 \\left({t}\\right), \\ldots, f_n \\left({t}\\right)}\\right \\rangle \\in \\mathbb Y$. Then $\\mathbf r$ is said to be a '''vector-valued function''' (of the '''parameter''' $t$). If $\\mathbb T$ is not explicitly defined, it is taken to be the intersection of all the domains of $f_1 ,f_2, \\cdots, f_n$. === Component Function === {{:Definition:Vector-Valued Function/Component Function}}"}
+{"_id": "22430", "title": "Definition:Limit of Vector-Valued Function", "text": "=== Definition 1 === {{:Definition:Limit of Vector-Valued Function/Definition 1}} === Definition 2 === {{:Definition:Limit of Vector-Valued Function/Definition 2}}"}
+{"_id": "22431", "title": "Definition:Derivative/Vector-Valued Function", "text": "=== Derivative at a Point === {{:Definition:Derivative/Vector-Valued Function/Point}} === Derivative on an Open Set === {{:Definition:Derivative/Vector-Valued Function/Open Set}}"}
+{"_id": "22432", "title": "Definition:Irreducible (Representation Theory)/Linear Representation", "text": "Let $\\rho: G \\to \\operatorname{GL} \\left({V}\\right)$ be a linear representation. Then $\\rho$ is '''irreducible''' {{iff}} it is not reducible. That is, {{iff}} there exists '''no''' non-trivial proper vector subspace $W$ of $V$ such that: : $\\forall g \\in G: \\rho \\left({g}\\right) \\left({W}\\right) \\subseteq W$"}
+{"_id": "22433", "title": "Definition:Monotone Class", "text": "Let $X$ be a set, and let $\\powerset X$ be its power set. Let $\\MM \\subseteq \\powerset X$ be a collection of subsets of $X$. Then $\\MM$ is said to be a '''monotone class (on $X$)''' {{iff}} for every countable, nonempty, index set $I$, it holds that: :$\\displaystyle \\family {A_i}_{i \\mathop \\in I} \\in \\MM \\implies \\bigcup_{i \\mathop \\in I} A_i \\in \\MM$ :$\\displaystyle \\family {A_i}_{i \\mathop \\in I} \\in \\MM \\implies \\bigcap_{i \\mathop \\in I} A_i \\in \\MM$ that is, {{iff}} $\\MM$ is closed under countable unions and countable intersections."}
+{"_id": "22434", "title": "Definition:Monotone Class Generated by Collection of Subsets", "text": "Let $X$ be a set. Let $\\mathcal G \\subseteq \\mathcal P \\left({X}\\right)$ be a collection of subsets of $X$. Then the '''monotone class generated by $\\mathcal G$''', $\\mathfrak m \\left({\\mathcal G}\\right)$, is the smallest monotone class on $X$ that contains $\\mathcal G$. That is, $\\mathfrak m \\left({\\mathcal G}\\right)$ is subject to: :$(1):\\quad \\mathcal G \\subseteq \\mathfrak m \\left({\\mathcal G}\\right)$ :$(2):\\quad \\mathcal G \\subseteq \\mathcal M \\implies \\mathfrak m \\left({\\mathcal G}\\right) \\subseteq \\mathcal M$ for any monotone class $\\mathcal M$ on $X$ In fact, $\\mathfrak m \\left({\\mathcal G}\\right)$ always exists, and is unique, as proved on Existence and Uniqueness of Monotone Class Generated by Collection of Subsets. === Generator === One says that $\\mathcal G$ is a '''generator''' for $\\mathfrak m \\left({\\mathcal G}\\right)$."}
+{"_id": "22436", "title": "Definition:Topology Induced by Metric", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. === Definition 1 === {{:Definition:Topology Induced by Metric/Definition 1}} === Definition 2 === {{:Definition:Topology Induced by Metric/Definition 2}}"}
+{"_id": "22437", "title": "Definition:Pre-Measure", "text": "Let $X$ be a set. Let $\\mathcal S \\subseteq \\mathcal P \\left({X}\\right)$ be a collection of subsets of $X$. Let $\\varnothing \\in \\mathcal S$. Let $\\mu: \\mathcal S \\to \\overline{\\R}_{\\ge 0}$ be a mapping, where $\\overline{\\R}_{\\ge 0}$ denotes the set of positive extended real numbers. Then $\\mu$ is said to be a '''pre-measure''' iff it satisfies the following conditions: :$(1):\\quad \\mu \\left({\\varnothing}\\right) = 0$ :$(2):\\quad$ For every sequence $\\left({A_n}\\right)_{n \\in \\N}$ of pairwise disjoint sets in $\\mathcal S$ with $\\displaystyle \\bigcup_{n \\mathop \\in \\N} A_n \\in \\mathcal S$: ::$\\displaystyle \\mu \\left({\\bigcup_{n \\mathop \\in \\N} A_n}\\right) = \\sum_{n \\mathop \\in \\N} \\mu \\left({A_n}\\right)$ :that is, that $\\mu$ is countably additive."}
+{"_id": "22438", "title": "Definition:Finite Measure", "text": "Let $\\mu$ be a measure on a measurable space $\\struct {X, \\Sigma}$. Then $\\mu$ is said to be a '''finite measure''' {{iff}}: :$\\map \\mu X < \\infty$"}
+{"_id": "22439", "title": "Definition:Finite Measure Space", "text": "A measure space $\\struct {X, \\Sigma, \\mu}$ is said to be '''finite''' {{iff}} $\\mu$ is a finite measure."}
+{"_id": "22440", "title": "Definition:Exhausting Sequence of Sets", "text": "Let $\\left\\langle{S_k}\\right \\rangle_{k \\in \\N}$ be a nested sequence of subsets of $S$ such that: :$(1):\\quad \\forall k \\in \\N: S_k \\subseteq S_{k + 1}$ :$(2):\\quad \\displaystyle \\bigcup_{k \\mathop \\in \\N} S_k = S$ Then $\\left\\langle{S_k}\\right \\rangle_{k \\in \\N}$ is an '''exhausting sequence of sets (in $\\mathcal S$)'''. That is, it is an increasing sequence of subsets of $S$, whose union is $S$. It is common to write $\\left\\langle{S_k}\\right\\rangle_{k \\in \\N} \\uparrow S$ to indicate an '''exhausting sequence of sets'''. Here, the $\\uparrow$ denotes a limit of an increasing sequence."}
+{"_id": "22441", "title": "Definition:Increasing Sequence of Sets", "text": "Let $\\sequence {S_k}_{k \\mathop \\in \\N}$ be a nested sequence of subsets of $S$ such that: :$\\forall k \\in \\N: S_k \\subseteq S_{k + 1}$ Then $\\sequence {S_k}_{k \\mathop \\in \\N}$ is an '''increasing sequence of sets (in $\\SS$)'''."}
+{"_id": "22442", "title": "Definition:Sigma-Finite Measure", "text": "Let $\\mu$ be a measure on a measurable space $\\left({X, \\Sigma}\\right)$. Then $\\mu$ is said to be a '''$\\sigma$-finite''' (or '''sigma-finite''') '''measure''' iff there exists an exhausting sequence $\\left({E_n}\\right)_{n \\in \\N}$ in $\\Sigma$, subject to: :$\\forall n \\in \\N: \\mu \\left({E_n}\\right) < \\infty$"}
+{"_id": "22443", "title": "Definition:Sigma-Finite Measure Space", "text": "A measure space $\\left({X, \\Sigma, \\mu}\\right)$ is said to be '''$\\sigma$-finite''' (or '''sigma-finite''') iff $\\mu$ is a $\\sigma$-finite measure."}
+{"_id": "22444", "title": "Definition:G-Submodule", "text": "Let $\\left({G, \\cdot}\\right)$ be a finite group. Let $\\left({V, \\phi}\\right)$ be a $G$-module. Let $W$ be a vector subspace of $V$. If $\\phi$ is a linear group action when restricted to $G \\times W \\subseteq G \\times V$. Then $(W,\\phi_W)$; where $\\phi_W$ is the restriction of $\\phi$ on $G\\times W$, is called a $G$-submodule of $(V,\\phi)$."}
+{"_id": "22445", "title": "Definition:G-Module Homomorphism", "text": "Let $\\left({G, \\cdot}\\right)$ be a group. Let $\\left({V, \\phi}\\right)$ and $\\left({W, \\mu}\\right)$ be $G$-modules. Then a linear mapping $f: V \\to W$ is called a '''$G$-module homomorphism''' {{iff}}: :$\\forall g \\in G: \\forall v \\in V: f \\left({\\phi \\left({g, v}\\right)}\\right) = \\mu \\left({g, f \\left({v}\\right)}\\right)$"}
+{"_id": "22446", "title": "Definition:Null Space", "text": "Let: :$ \\mathbf A_{m \\times n} = \\begin{bmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} \\\\ \\end{bmatrix}$,  $\\mathbf x_{n \\times 1} = \\begin{bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{bmatrix}$, $\\mathbf 0_{m \\times 1} = \\begin{bmatrix} 0 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{bmatrix}$ be matrices where each column is a member of a real vector space. The set of all solutions to $\\mathbf A \\mathbf x = \\mathbf 0$: :$\\map {\\operatorname N} {\\mathbf A} = \\set {\\mathbf x \\in \\R^n : \\mathbf {A x} = \\mathbf 0}$ is called the '''null space''' of $\\mathbf A$."}
+{"_id": "22447", "title": "Definition:G-Module", "text": "Let $\\left({V, +, \\cdot}\\right)$ be a vector space over a field $\\left({k, \\oplus, \\circ}\\right)$. Let $G$ be a group. Let $\\phi : G \\times V \\to V$ be an linear group action of $G$ on $V$. Then $(V,\\phi)$ is called a $G$-module. Category:Definitions/Group Actions {{SUBPAGENAME}} bof7sirmz3j9f62dk4a07bdtg1h1qf7"}
+{"_id": "22448", "title": "Definition:Limit of Increasing Sequence of Sets", "text": "Let $\\left({S_n}\\right)_{n \\in \\N}$ be an increasing sequence of sets. Let $S = \\displaystyle \\bigcup_{n \\mathop \\in \\N} S_n$. Then $S$ is said to be the '''limit''' of $\\left({S_n}\\right)_{n \\in \\N}$, and one writes $S_n \\uparrow S$."}
+{"_id": "22449", "title": "Definition:Decreasing Sequence of Sets", "text": "Let $\\sequence {S_k}_{k \\mathop \\in \\N}$ be a nested sequence of subsets of $S$ such that: :$\\forall k \\in \\N: S_k \\supseteq S_{k + 1}$ Then $\\sequence {S_k}_{k \\mathop \\in \\N}$ is a '''decreasing sequence of sets (in $\\SS$)'''."}
+{"_id": "22450", "title": "Definition:Limit of Decreasing Sequence of Sets", "text": "Let $\\left({S_n}\\right)_{n \\in \\N}$ be a decreasing sequence of sets. Let $S = \\displaystyle \\bigcap_{n \\mathop \\in \\N} S_n$. Then $S$ is said to be the '''limit''' of $\\left({S_n}\\right)_{n \\in \\N}$, and one writes $S_n \\downarrow S$."}
+{"_id": "22451", "title": "Definition:Reducible G-Module", "text": "Let $M$ be a $G$-module. Then $M$ is '''reducible''' {{iff}} the corresponding linear representation is reducible."}
+{"_id": "22452", "title": "Definition:Edge Contraction", "text": "Let $G$ be an undirected graph. Let $e \\in \\map E G$ be an edge of $G$. Then the '''graph obtained by contracting $e$ in $G$''', denoted by $G / e$, is the graph $H$ defined by: :$\\map V H = \\paren {\\map V G \\setminus e} \\cup \\set v$ :$\\map E H = \\set {f \\in \\map E G: f \\cap e = \\O} \\cup \\set {u v: \\exists f \\in \\map E G: u \\in f \\setminus e, f \\cap e \\ne \\O}$ where $v \\notin \\map V G$ is a fresh object. {{explain|A lot of the above - see talk page (in prep)}} Informally, it is the graph obtained from $G$ by replacing the vertices incident to $e$ with a single vertex adjacent to all their neighbors."}
+{"_id": "22453", "title": "Definition:Dirac Measure", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $x \\in X$ be any point in $X$. Then the '''Dirac measure at $x$''', denoted $\\delta_x$ is the measure defined by: :$\\delta_x: \\Sigma \\to \\overline \\R, \\ \\delta_x \\left({E}\\right) := \\begin{cases}0 & \\text{if } x \\notin E \\\\ 1 & \\text{if } x \\in E\\end{cases}$ where $\\overline \\R$ denotes the extended set of real numbers. That $\\delta_x$ actually is a measure is shown on Dirac Measure is Measure. In fact, Dirac measure is a probability measure."}
+{"_id": "22454", "title": "Definition:Enumeration", "text": "=== Finite Sets === {{:Definition:Enumeration/Finite}} === Countably Infinite Sets === {{:Definition:Enumeration/Countably Infinite}}"}
+{"_id": "22455", "title": "Definition:Discrete Probability Measure", "text": "Let $\\Omega$ be a countable set. Let $\\mathcal P \\left({\\Omega}\\right)$ be its power set, regarded as a $\\sigma$-algebra. Let $\\left({p_\\omega}\\right)_{\\omega \\in \\Omega} \\subseteq \\left[{0 \\,.\\,.\\, 1}\\right]$ be a subset of the closed unit interval in $\\R$, indexed by $\\Omega$. Suppose that $\\displaystyle \\sum_{\\omega \\mathop \\in \\Omega} p_\\omega = 1$. {{explain|link to definition of sum over countable set}} The '''discrete probability measure on $\\Omega$''', denoted $P$, is the mapping defined by: :$\\displaystyle P: \\mathcal P \\left({\\Omega}\\right) \\to \\overline \\R, \\ P \\left({S}\\right) = \\sum_{\\omega \\mathop \\in \\Omega} p_\\omega \\delta_\\omega \\left({S}\\right)$ where $\\overline \\R$ denotes the extended real numbers, and $\\delta_\\omega$ is the Dirac measure at $\\omega$. From this definition, it is seen that the name '''discrete probability measure''' is compatible with the notion of discrete measure, as $\\Omega$ is countable. {{MissingLinks|to eg. Definition:Discrete Sample Space and connected stuff}} === Discrete Probability Space === The measure space $\\left({\\Omega, \\mathcal P \\left({\\Omega}\\right), P}\\right)$ is called '''discrete probability space'''."}
+{"_id": "22456", "title": "Definition:Proper G-Submodule", "text": "Let $\\left({V, \\phi}\\right)$ be a $G$-module. A $G$-submodule of $V$ is called '''proper''' iff it is a proper vector subspace of $V$. {{SUBPAGENAME}} hu0luq718d6r29qzya71z71pi3o2iro"}
+{"_id": "22457", "title": "Definition:Trivial G-Module", "text": "Let $\\left({V, \\phi}\\right)$ be a $G$-module. Then $\\left({V, \\phi}\\right)$ is said to be '''trivial''' iff $V = \\left\\{{\\mathbf 0}\\right\\}$. {{SUBPAGENAME}} bhlwwt3n0a6zqpyzdyhlznjnsw4gdk3"}
+{"_id": "22458", "title": "Definition:Irreducible (Representation Theory)/G-Module", "text": "A $G$-module is '''irreducible''' {{iff}} the corresponding linear representation is irreducible."}
+{"_id": "22459", "title": "Definition:Equivalent Linear Representations", "text": "Let $ \\struct {G, \\cdot}$ be a group. Consider two linear representations $\\rho: G \\to \\GL V$ and $\\rho': G \\to \\GL W$ of $G$. Then $\\rho$ and $\\rho'$ are called '''equivalent (linear representations)''' {{iff}} their correspondent $G$-modules using Correspondence between Linear Group Actions and Linear Representations are isomorphic. {{rewrite|above line needs rewriting, but I can't come up with a suitable replacement}} {{explain|The \"isomorphic\" link goes to a generic \"abstract algebra\" page, but I believe no actual definition has been made for an isomorphism between two G-modules.}} Category:Definitions/Representation Theory i9p93ofh5e70d1hkt0tg49v26zq2kgm"}
+{"_id": "22461", "title": "Definition:Infinite Measure", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Then the '''infinite measure''' is the measure defined by: :$\\mu: \\Sigma \\to \\overline \\R, \\ \\mu \\left({E}\\right) := \\begin{cases} 0 & : \\text{if $E = \\varnothing$} \\\\ +\\infty & : \\text{otherwise}\\end{cases}$ where $\\overline \\R$ denotes the extended real numbers."}
+{"_id": "22462", "title": "Definition:Null Measure", "text": "Let $\\struct {X, \\Sigma}$ be a measurable space. Then the '''null measure''' is the measure defined by: :$\\mu: \\Sigma \\to \\overline \\R: \\map \\mu E := 0$ where $\\overline \\R$ denotes the extended real numbers."}
+{"_id": "22463", "title": "Definition:Preimage/Relation", "text": "Let $\\RR \\subseteq S \\times T$ be a relation. Let $\\RR^{-1} \\subseteq T \\times S$ be the inverse relation to $\\RR$, defined as: :$\\RR^{-1} = \\set {\\tuple {t, s}: \\tuple {s, t} \\in \\RR}$ === Preimage of Element === {{:Definition:Preimage/Relation/Element}} === Preimage of Subset === {{:Definition:Preimage/Relation/Subset}} === Preimage of Relation === {{:Definition:Preimage/Relation/Relation}}"}
+{"_id": "22464", "title": "Definition:Preimage/Mapping", "text": "Let $f: S \\to T$ be a mapping. Let $f^{-1} \\subseteq T \\times S$ be the inverse of $f$, considered as a relation: :$f^{-1} = \\set {\\tuple {t, s}: \\map f s = t}$ === Preimage of Element === {{:Definition:Preimage/Mapping/Element}} === Preimage of Subset === {{:Definition:Preimage/Mapping/Subset}} === Preimage of Mapping === {{:Definition:Preimage/Mapping/Mapping}}"}
+{"_id": "22465", "title": "Definition:Preimage/Relation/Element", "text": "Every $s \\in S$ such that $\\tuple {s, t} \\in \\RR$ is called '''a preimage of $t$'''. In some contexts, it is not individual elements that are important, but ''all'' elements of $S$ which are of interest. Thus '''the preimage of $t \\in T$''' is defined as: :$\\map {\\RR^{-1} } t := \\set {s \\in S: \\tuple {s, t} \\in \\RR}$ This can also be written: :$\\map {\\RR^{-1} } t := \\set {s \\in \\Img {\\RR^{-1} }: \\tuple {t, s} \\in \\RR^{-1} }$ That is, '''the preimage of $t$ under $\\RR$''' is the image of $t$ under $\\RR^{-1}$."}
+{"_id": "22466", "title": "Definition:Preimage/Relation/Subset", "text": "Let $Y \\subseteq T$. The '''preimage of $Y$ under $\\RR$''' is defined as: :$\\RR^{-1} \\sqbrk Y := \\set {s \\in S: \\exists t \\in Y: \\tuple {s, t} \\in \\RR}$ That is, the '''preimage of $Y$ under $\\RR$''' is the image of $Y$ under $\\RR^{-1}$: :$\\RR^{-1} \\sqbrk Y := \\set {s \\in S: \\exists t \\in Y: \\tuple {t, s} \\in \\RR^{-1} }$ If no element of $Y$ has a '''preimage''', then $\\RR^{-1} \\sqbrk Y = \\O$."}
+{"_id": "22467", "title": "Definition:Preimage/Relation/Relation", "text": "The '''preimage''' of $\\RR \\subseteq S \\times T$ is: :$\\Preimg \\RR := \\RR^{-1} \\sqbrk T = \\set {s \\in S: \\exists t \\in T: \\tuple {s, t} \\in \\RR}$"}
+{"_id": "22470", "title": "Definition:Preimage/Mapping/Element", "text": "Every $s \\in S$ such that $\\map f s = t$ is called '''a preimage''' of $t$. '''The preimage''' of an element $t \\in T$ is defined as: :$\\map {f^{-1} } t := \\set {s \\in S: \\map f s = t}$ This can also be expressed as: :$\\map {f^{-1} } t := \\set {s \\in \\Img {f^{-1} }: \\tuple {t, s} \\in f^{-1} }$ That is, the '''preimage of $t$ under $f$''' is the image of $t$ under $f^{-1}$."}
+{"_id": "22471", "title": "Definition:Preimage/Mapping/Subset", "text": "Let $Y \\subseteq T$."}
+{"_id": "22472", "title": "Definition:Preimage/Mapping/Mapping", "text": "The '''preimage of $f$''' is defined as: :$\\Preimg f := \\set {s \\in S: \\exists t \\in T: f \\paren s = t}$ That is: :$\\Preimg f := f^{-1} \\sqbrk T$ where $f^{-1} \\sqbrk T$ is the image of $T$ under $f^{-1}$. In this context, $f^{-1} \\subseteq T \\times S$ is the the inverse of $f$. It is a relation but not necessarily itself a mapping."}
+{"_id": "22473", "title": "Definition:Image (Set Theory)/Relation/Relation", "text": "The '''image''' of $\\RR$ is the set: :$\\Img \\RR := \\RR \\sqbrk S = \\set {t \\in T: \\exists s \\in S: \\tuple {s, t} \\in \\RR}$"}
+{"_id": "22474", "title": "Definition:Image (Set Theory)/Mapping/Mapping", "text": "Let $f: S \\to T$ be a mapping. <section notitle=\"true\" name=\"tc\"> === Definition 1 === {{:Definition:Image (Set Theory)/Mapping/Mapping/Definition 1}} === Definition 2 === {{:Definition:Image (Set Theory)/Mapping/Mapping/Definition 2}} </section>"}
+{"_id": "22475", "title": "Definition:Arbitrarily Large", "text": "Let $P$ be a property of real numbers. We say that '''$P \\left({x}\\right)$ holds for arbitrarily large $x$''' (or '''there exist arbitrarily large $x$ such that $P \\left({x}\\right)$ holds''') {{iff}}: :$\\forall a \\in \\R: \\exists x \\in \\R: x \\ge a: P \\left({x}\\right)$ That is: :''For any real number $a$, there exists a (real) number not less than $a$ such that the property $P$ holds.'' or, more informally and intuitively: :''However large a number you can think of, there will be an even larger one for which $P$ still holds.''"}
+{"_id": "22476", "title": "Definition:Valuation", "text": "Let $\\left({R, +, \\cdot}\\right)$ be a ring. A '''valuation on $R$''' is a mapping: : $\\nu: R \\to \\Z \\cup \\left\\{{+\\infty}\\right\\}$ which fulfils the '''valuation axioms''': {{:Definition:Valuation Axioms}}"}
+{"_id": "22477", "title": "Definition:P-adic Valuation", "text": "Let $p \\in \\N$ be a prime number. === Integers === {{:Definition:P-adic Valuation/Integers}} === Rational Numbers === {{:Definition:P-adic Valuation/Rational Numbers}}"}
+{"_id": "22478", "title": "Definition:Angle Between Vectors", "text": "Let $\\mathbf v, \\mathbf w$ be two non-zero vectors in $\\R^n$. === Case 1 === Suppose that $\\mathbf v$ and $\\mathbf w$ are not scalar multiples of each other: :$\\neg \\exists \\lambda \\in \\R: \\mathbf v = \\lambda \\mathbf w$ Then the '''angle between $\\mathbf v$ and $\\mathbf w$''' is defined as follows: Describe a triangle with lengths corresponding to: :$\\norm {\\mathbf v}, \\norm {\\mathbf w}, \\norm {\\mathbf v - \\mathbf w}$ where $\\norm {\\, \\cdot \\,}$ denotes vector length: :600px The angle formed between the two sides with lengths $\\norm {\\mathbf v}$ and $\\norm {\\mathbf w}$ is called the '''angle between vectors $\\mathbf v$ and $\\mathbf w$'''. By convention, the angle is taken between $0$ and $\\pi$. === Case 2 === Suppose that $\\mathbf v$ and $\\mathbf w$ ''are'' scalar multiples of each other: :$\\exists \\lambda \\in \\R: \\mathbf v = \\lambda \\mathbf w$ As $\\mathbf v$ and $\\mathbf w$ as non-zero, $\\lambda \\ne 0$. If $\\lambda > 0$, then the angle between $\\mathbf v$ and $\\mathbf w$ is defined as a zero angle, that is: :$\\theta = 0$ If $\\lambda < 0$, then the angle between $\\mathbf v$ and $\\mathbf w$ is defined as a straight angle, that is: :$\\theta = \\pi$"}
+{"_id": "22479", "title": "Definition:P-adic Number", "text": "Let $p$ be any prime number. Let $\\norm {\\,\\cdot\\,}_p$ be the p-adic norm on the rationals $\\Q$. By P-adic Norm is Non-Archimedean Norm then $\\norm {\\,\\cdot\\,}_p$ is a non-archimedean norm on $\\Q$ and the pair $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$ is a valued field. === $p$-adic Norm Completion of Rational Numbers === {{:Definition:P-adic Number/P-adic Norm Completion of Rational Numbers}} === Quotient of Cauchy Sequences in $p$-adic Norm === {{:Definition:P-adic Number/Quotient of Cauchy Sequences in P-adic Norm}} === $p$-adic Norm on $\\Q_p$ === {{:Definition:P-adic Norm on P-adic Numbers}}"}
+{"_id": "22480", "title": "Definition:Darboux Integral/Geometric Interpretation", "text": "The expression $\\displaystyle \\int_a^b \\map f x \\rd x$ can be (and frequently is) interpreted as '''the area under the graph'''.  This follows from the definition of the definite integral as a sum of the product of the lengths of intervals and the \"height\" of the function being integrated in that interval and the formula for the area of a rectangle. A depiction of the lower and upper sums illustrates this: :350px  350px It can intuitively be seen that as the number of points in the subdivision increases, the more \"accurate\" the lower and upper sums become. Also note that if the graph is below the $x$-axis, the signed area under the graph becomes negative."}
+{"_id": "22481", "title": "Definition:Intersection Measure", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $F \\in \\Sigma$. Then the '''intersection measure (of $\\mu$ by $F$)''' is the mapping $\\mu_F: \\Sigma \\to \\overline{\\R}$, defined by: :$\\mu_F \\left({E}\\right) = \\mu \\left({E \\cap F}\\right)$ It is in fact a measure on $\\left({X, \\Sigma}\\right)$, as shown on Intersection Measure is Measure. {{expand|verify name, Schilling gives none}}"}
+{"_id": "22482", "title": "Definition:Perpendicular (Linear Algebra)", "text": "Let $\\mathbf u$, $\\mathbf v$ be non-zero vectors in the Euclidean space $\\R^n$. :$\\mathbf u$ and $\\mathbf v$ are '''perpendicular''' {{iff}} the angle between them is a right angle."}
+{"_id": "22483", "title": "Definition:Complete Measure Space", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Suppose that the family of $\\mu$-null sets $\\mathcal{N}_{\\mu}$ satisfies the following condition: :$\\forall N \\in \\mathcal{N}_{\\mu}: \\forall M \\subseteq N: M \\in \\mathcal{N}_{\\mu}$ That is, any subset of a $\\mu$-null set is again a $\\mu$-null set. Then $\\left({X, \\Sigma, \\mu}\\right)$ is said to be a '''complete measure space'''."}
+{"_id": "22484", "title": "Definition:Completion (Measure Space)", "text": "Let $\\left({X, \\Sigma, \\mu}\\right), \\left({\\tilde X, \\Sigma^*, \\bar \\mu}\\right)$ be measure spaces. Then '''$\\left({\\tilde X, \\Sigma^*, \\bar \\mu}\\right)$ is a completion of $\\left({X, \\Sigma, \\mu}\\right)$''' or '''$\\left({\\tilde X, \\Sigma^*, \\bar \\mu}\\right)$ completes $\\left({X, \\Sigma, \\mu}\\right)$''' iff the following conditions hold: :$(1):\\quad \\left({\\tilde X, \\Sigma^*, \\bar \\mu}\\right)$ is a complete measure space :$(2):\\quad \\tilde X = X$ :$(3):\\quad \\Sigma$ is a sub-$\\sigma$-algebra of $\\Sigma^*$ :$(4):\\quad \\forall E \\in \\Sigma: \\bar \\mu \\left({E}\\right) = \\mu \\left({E}\\right)$, i.e. $\\bar \\mu \\restriction_{\\Sigma} = \\mu$"}
+{"_id": "22485", "title": "Definition:Restricted Measure", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $\\Sigma'$ be a sub-$\\sigma$-algebra of $\\Sigma$. Then the '''restricted measure on $\\Sigma'$''' or the '''restriction of $\\mu$ to $\\Sigma'$''' is the mapping $\\nu: \\Sigma \\to \\overline{\\R}$ defined by: :$\\forall E' \\in \\Sigma': \\nu \\left({E'}\\right) = \\mu \\left({E'}\\right)$ That is, $\\nu$ is the restriction $\\mu \\restriction_{\\Sigma'}$."}
+{"_id": "22486", "title": "Definition:Dynkin System", "text": "Let $X$ be a set, and let $\\mathcal D \\subseteq \\mathcal P \\left({X}\\right)$ be a collection of subsets of $X$. Then $\\mathcal D$ is called a '''Dynkin system (on $X$)''' {{iff}} it satisfies the following conditions: :$(1):\\quad X \\in \\mathcal D$ :$(2):\\quad \\forall D \\in \\mathcal D: X \\setminus D \\in \\mathcal D$ :$(3):\\quad$ For all pairwise disjoint sequences $\\left({D_n}\\right)_{n \\in \\N}$ in $\\mathcal D$, $\\displaystyle \\bigcup_{n \\mathop \\in \\N} D_n \\in \\mathcal D$"}
+{"_id": "22487", "title": "Definition:Dynkin System Generated by Collection of Subsets", "text": "Let $X$ be a set. Let $\\mathcal G \\subseteq \\mathcal P \\left({X}\\right)$ be a collection of subsets of $X$. Then the '''Dynkin system generated by $\\mathcal G$''', denoted $\\delta \\left({\\mathcal G}\\right)$, is the smallest Dynkin system on $X$ that contains $\\mathcal G$. That is, $\\delta \\left({\\mathcal G}\\right)$ is subject to: :$(1):\\quad \\mathcal G \\subseteq \\delta \\left({\\mathcal G}\\right)$ :$(2):\\quad \\mathcal G \\subseteq \\mathcal D \\implies \\delta \\left({\\mathcal G}\\right) \\subseteq \\mathcal D$ for any Dynkin system $\\mathcal D$ on $X$ In fact, $\\delta \\left({\\mathcal G}\\right)$ always exists, and is unique, as proved on Existence and Uniqueness of Dynkin System Generated by Collection of Subsets. === Generator === One says that $\\mathcal G$ is a '''generator''' for $\\delta \\left({\\mathcal G}\\right)$."}
+{"_id": "22488", "title": "Definition:Central Product", "text": "Let $G$ and $H$ be groups. Let $Z$ and $W$ be central subgroups of $G$ and $H$ respectively. Let: :$Z \\cong W$ where $\\cong$ denotes isomorphism. Let such a group isomorphism be $\\theta: Z \\to W$. Let $X$ be the set defined as: :$X = \\set {\\tuple {x, \\map \\theta x^{-1} }: x \\in Z}$ Then the quotient group $\\struct {G \\times H} / X$ is denoted $\\struct {G \\times_\\theta H}$ and is called the '''central product of $G$ and $H$ via $\\theta$'''."}
+{"_id": "22489", "title": "Definition:Coset Product", "text": "Let $\\struct {G, \\circ}$ be a group. Let $N$ be a normal subgroup of $G$. Let $a, b \\in G$. The '''coset product''' of $a \\circ N$ and $b \\circ N$ is defined as: :$\\paren {a \\circ N} \\circ \\paren {b \\circ  N} = \\paren {a \\circ b} \\circ N$ where $a \\circ N$ and $b \\circ N$ are the left cosets of $a$ and $b$ by $N$."}
+{"_id": "22490", "title": "Definition:Translation-Invariant Measure", "text": "Let $\\mu$ be a measure on $\\R^n$ equipped with the Borel $\\sigma$-algebra $\\mathcal B \\left({\\R^n}\\right)$. Then $\\mu$ is said to be '''translation-invariant''' or '''invariant under translations''' iff: :$\\forall x \\in \\R^n, \\forall B \\in \\mathcal B: \\mu \\left({x + B}\\right) = \\mu \\left({B}\\right)$ where $x + B$ is the set $\\left\\{{x + b: b \\in B}\\right\\}$. {{SUBPAGENAME}} nid2a2b6pzp4gwqrs3jjup0l3wgnkzz"}
+{"_id": "22491", "title": "Definition:Invariant Measure", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $\\theta: X \\to X$ be an $\\Sigma / \\Sigma$-measurable mapping. Then $\\mu$ is said to be a '''$\\theta$-invariant measure''' or to be '''invariant under $\\theta$''' iff: :$\\forall E \\in \\Sigma: \\mu \\left({\\theta^{-1} \\left({E}\\right) }\\right) = \\mu \\left({E}\\right)$ In terms of a pushforward measure, this can be concisely formulated as: :$\\theta_* \\mu = \\mu$"}
+{"_id": "22492", "title": "Definition:Independent Sigma-Algebras", "text": "Let $\\struct {\\Omega, \\EE, \\Pr}$ be a probability space. Let $\\Sigma$ and $\\Sigma'$ be sub-$\\sigma$-algebras of $\\EE$. Then $\\Sigma$ and $\\Sigma'$ are said to be '''($\\Pr$-)independent''' {{iff}}: :$\\forall E \\in \\Sigma, E' \\in \\Sigma': \\map \\Pr {E \\cap E'} = \\map \\Pr E \\map \\Pr {E'}$"}
+{"_id": "22493", "title": "Definition:Stable under Intersection", "text": "Let $X$ be a set, and let $\\SS \\subseteq \\powerset X$ be a collection of subsets of $X$. Then $\\SS$ is said to be '''stable under intersection(s)''', or simply '''$\\cap$-stable''', {{iff}}: :$\\forall S, T \\in \\SS: S \\cap T \\in \\SS$ Category:Definitions/Set Systems pnx3xh2mygzcfi5yx75kaq9xnpjf63e"}
+{"_id": "22494", "title": "Definition:Sub-Sigma-Algebra", "text": "Let $X$ be a set, and let $\\mathcal A, \\mathcal B$ be $\\sigma$-algebras on $X$. Then $\\mathcal B$ is said to be a '''sub-sigma-algebra''' or '''sub-$\\sigma$-algebra''' of $\\mathcal A$ iff $\\mathcal B \\subseteq \\mathcal A$. {{SUBPAGENAME}} 501i2ym6ibfjwwspmzl7ljcqmez1jpm"}
+{"_id": "22495", "title": "Definition:Lebesgue Pre-Measure", "text": "Let $\\mathcal J_{ho}$ be the collection of half-open $n$-rectangles. '''$n$-dimensional Lebesgue pre-measure''' is the mapping $\\lambda^n: \\mathcal J_{ho} \\to \\overline \\R_{\\ge 0}$ given by: :$\\displaystyle \\lambda^n \\left({ \\left[[{\\mathbf a \\,.\\,.\\, \\mathbf b}\\right)) }\\right) = \\prod_{i \\mathop = 1}^n \\left({b_i - a_i}\\right)$ where $\\overline \\R_{\\ge 0}$ denotes the set of positive extended real numbers."}
+{"_id": "22496", "title": "Definition:Diffuse Measure", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. The measure $\\mu$ is said to be '''diffuse''' or '''non-atomic''' {{iff}} there are no $\\mu$-atoms."}
+{"_id": "22497", "title": "Definition:Half-Open Rectangle", "text": "Let $a_1, \\ldots, a_n, b_1, \\ldots, b_n$ be real numbers. The set: :$\\ds \\prod_{i \\mathop = 1}^n \\hointr {a_i} {b_i} = \\hointr {a_1} {b_1} \\times \\cdots \\times \\hointr {a_n} {b_n} \\subseteq \\R^n$ is called an '''half-open rectangle in $\\R^n$''' or '''half-open $n$-rectangle'''. Here, $\\times$ denotes Cartesian product. The collection of all '''half-open $n$-rectangles''' is denoted $\\JJ_{ho}$, or $\\JJ_{ho}^n$ if the dimension $n$ is to be emphasized. In case $a_i \\ge b_i$ for some $i$, the rectangle is taken to be the empty set $\\O$. This is in accordance with the result Cartesian Product is Empty iff Factor is Empty for general Cartesian products."}
+{"_id": "22498", "title": "Definition:Discrete Measure", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Then $\\mu$ is said to be a '''discrete measure''' {{iff}} it is a series of Dirac measures. That is, {{iff}} there exist: :A sequence $\\sequence {x_n}_{n \\mathop \\in \\N}$ in $X$ :A sequence $\\sequence {\\lambda_n}_{n \\mathop \\in \\N}$ in $\\R$ such that: :$(1):\\quad \\forall E \\in \\Sigma: \\map \\mu E = \\displaystyle \\sum_{n \\mathop \\in \\N} \\lambda_n \\, \\map {\\delta_{x_n} } E$ where $\\delta_{x_n}$ denotes the Dirac measure at $x_n$. By Series of Measures is Measure, defining $\\mu$ by $(1)$ yields a measure."}
+{"_id": "22499", "title": "Definition:Series of Measures", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $\\left({\\mu_n}\\right)_{n \\in \\N}$ be a sequence of measures on $\\left({X, \\Sigma}\\right)$. Let $\\left({\\lambda_n}\\right)_{n \\in \\N}$ be a sequence of positive real numbers. Then the mapping $\\mu: \\Sigma \\to \\overline \\R$, defined by: :$\\displaystyle \\mu \\left({E}\\right) := \\sum_{n \\mathop \\in \\N} \\lambda_n \\mu_n \\left({E}\\right)$ is called a '''series of measures'''."}
+{"_id": "22500", "title": "Definition:Induced Outer Measure", "text": "Let $\\mathcal S$ be a collection of subsets of a set $X$, and suppose that $\\varnothing \\in \\mathcal S$. Let $\\mu$ be a pre-measure on $\\mathcal S$. The '''outer measure induced by the pre-measure $\\mu$''' is the mapping $\\mu^* : \\mathcal P \\left({X}\\right) \\to \\overline \\R_{\\ge 0}$ defined as: :$\\displaystyle \\mu^* \\left({S}\\right) = \\inf \\ \\left\\{ {\\sum_{n \\mathop = 1}^\\infty \\mu \\left({A_n}\\right) : \\forall n \\in \\N: A_n \\in \\mathcal S, \\ S \\subseteq \\bigcup_{n \\mathop = 1}^\\infty A_n} \\right\\}$ Here, $\\mathcal P \\left({X}\\right)$ denotes the power set of $X$, and $\\overline \\R_{\\ge 0}$ denotes the set of positive extended real numbers. The infimum of the empty set is taken to be $+\\infty$. It follows immediately by Construction of Outer Measure that the induced outer measure is an outer measure."}
+{"_id": "22501", "title": "Definition:Homomorphism (Abstract Algebra)/Cartesian Product", "text": "Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be a mapping from one algebraic structure $\\struct {S, \\circ}$ to another $\\struct {T, *}$. We define the cartesian product $\\phi \\times \\phi: S \\times S \\to T \\times T$ as: :$\\forall \\tuple {x, y} \\in S \\times S: \\map {\\paren {\\phi \\times \\phi} } {x, y} = \\tuple {\\map \\phi x, \\map \\phi y}$ Hence we can state that $\\phi$ is a homomorphism {{iff}}: :$\\map \\ast {\\map {\\paren {\\phi \\times \\phi} } {x, y} } = \\map \\phi {\\map \\circ {x, y} }$ using the notation $\\map \\circ {x, y}$ to denote the operation $x \\circ y$. The point of doing this is so we can illustrate what is going on in a commutative diagram: ::$\\begin{xy} \\xymatrix@L+2mu@+1em{  S \\times S \\ar[r]^*{\\circ}      \\ar[d]_*{\\phi \\times \\phi} &  S \\ar[d]^*{\\phi} \\\\  T \\times T  \\ar[r]_*{\\ast} &  T }\\end{xy}$ Thus we see that $\\phi$ is a homomorphism {{iff}} both of the composite mappings from $S \\times S$ to $T$ have the same effect on all elements of $S \\times S$."}
+{"_id": "22503", "title": "Definition:Epimorphism (Category Theory)", "text": "Let $\\mathbf C$ be a metacategory. An '''epimorphism''' is a morphism $f \\in \\mathbf C_1$ such that: : $g \\circ f = h \\circ f \\implies g = h$ for all morphisms $g, h \\in \\mathbf C_1$ for which these compositions are defined. That is, an '''epimorphism''' is a morphism which is right cancellable.  One writes $f: C \\twoheadrightarrow D$ to denote that $f$ is an '''epimorphism'''."}
+{"_id": "22505", "title": "Definition:Ordered Structure Monomorphism", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ and $\\left({T, *, \\preccurlyeq}\\right)$ be ordered structures. An '''ordered structure monomorphism''' from $\\left({S, \\circ, \\preceq}\\right)$ to $\\left({T, *, \\preccurlyeq}\\right)$ is a mapping $\\phi: S \\to T$ that is both: :$(1): \\quad$ A monomorphism, i.e. an injective homomorphism, from the structure $\\left({S, \\circ}\\right)$ to the structure $\\left({T, *}\\right)$ :$(2): \\quad$ An order embedding from the ordered set $\\left({S, \\preceq}\\right)$ to the ordered set $\\left({T, \\preccurlyeq}\\right)$."}
+{"_id": "22506", "title": "Definition:Monomorphism (Category Theory)", "text": "Let $\\mathbf C$ be a metacategory. A '''monomorphism''' is a morphism $f \\in \\mathbf C_1$ such that: :$f \\circ g = f \\circ h \\implies g = h$ for all morphisms $g, h \\in \\mathbf C_1$ for which these compositions are defined. That is, a '''monomorphism''' is a morphism which is left cancellable.  One writes $f: C \\rightarrowtail D$ to denote that $f$ is a '''monomorphism'''."}
+{"_id": "22507", "title": "Definition:Complex Number/Real Part", "text": "Let $z = a + i b$ be a complex number. The '''real part''' of $z$ is the coefficient $a$. The '''real part''' of a complex number $z$ is usually denoted on {{ProofWiki}} by $\\map \\Re z$ or $\\Re z$."}
+{"_id": "22508", "title": "Definition:Complex Number/Imaginary Part", "text": "Let $z = a + i b$ be a complex number. The '''imaginary part''' of $z$ is the coefficient $b$ ('''note:''' not $i b$). The '''imaginary part''' of a complex number $z$ is usually denoted on {{ProofWiki}} by $\\map \\Im z$ or $\\Im z$."}
+{"_id": "22509", "title": "Definition:Strict Lower Closure", "text": "=== Strict Lower Closure of Element === {{:Definition:Strict Lower Closure/Element}} === Strict Lower Closure of Subset === {{:Definition:Strict Lower Closure/Set}}"}
+{"_id": "22510", "title": "Definition:Strict Upper Closure", "text": "=== Strict Upper Closure of Element === {{:Definition:Strict Upper Closure/Element}} === Strict Upper Closure of Subset === {{:Definition:Strict Upper Closure/Set}}"}
+{"_id": "22511", "title": "Definition:Upper Closure/Element", "text": "Let $\\struct {S, \\preccurlyeq}$ be an ordered set. Let $a \\in S$. The '''upper closure of $a$ (in $S$)''' is defined as: :$a^\\succcurlyeq := \\set {b \\in S: a \\preccurlyeq b}$ That is, $a^\\succcurlyeq$ is the set of all elements of $S$ that succeed $a$."}
+{"_id": "22512", "title": "Definition:Order Topology", "text": "=== Definition 1 === {{:Definition:Order Topology/Definition 1}} === Definition 2 === {{:Definition:Order Topology/Definition 2}}"}
+{"_id": "22513", "title": "Definition:Upper Closure", "text": "{{:Definition:Upper Closure/Element}}"}
+{"_id": "22514", "title": "Definition:Lower Closure", "text": "{{:Definition:Lower Closure/Element}}"}
+{"_id": "22515", "title": "Definition:Projection (Mapping Theory)/First Projection", "text": "The '''first projection on $S \\times T$''' is the mapping $\\pr_1: S \\times T \\to S$ defined by: :$\\forall \\tuple {x, y} \\in S \\times T: \\map {\\pr_1} {x, y} = x$"}
+{"_id": "22516", "title": "Definition:Projection (Mapping Theory)/Second Projection", "text": "The '''second projection on $S \\times T$''' is the mapping $\\pr_2: S \\times T \\to T$ defined by: :$\\forall \\tuple {x, y} \\in S \\times T: \\map {\\pr_2} {x, y} = y$"}
+{"_id": "22517", "title": "Definition:Ordering on Extended Real Numbers", "text": "Let $\\overline \\R$ denote the extended real numbers. Extend the natural ordering $\\le_\\R$ on $\\R$ to $\\overline \\R = \\R \\cup \\set {+\\infty, -\\infty}$ by imposing: :$\\forall x \\in \\overline \\R: -\\infty \\le x$ :$\\forall x \\in \\overline \\R: x \\le +\\infty$ That is, considering the relations $\\le$ and $\\le_\\R$ as subsets of $\\overline \\R \\times \\overline \\R$: :${\\le} := {\\le_\\R} \\cup \\set {\\tuple {x, +\\infty}: x \\in \\overline \\R} \\cup \\set {\\tuple {-\\infty, x}: x \\in \\overline \\R}$ where $\\tuple {x, +\\infty}$ and $\\tuple {-\\infty, x}$ denote ordered pairs in $\\overline \\R \\times \\overline \\R$. The ordering $\\le$ is called the '''(usual) ordering on $\\overline \\R$'''."}
+{"_id": "22518", "title": "Definition:Topology on Extended Real Numbers", "text": "Let $\\overline \\R$ denote the extended real numbers. The '''(standard) topology on $\\overline \\R$''' is the order topology $\\tau$ associated to the ordering on $\\overline \\R$."}
+{"_id": "22519", "title": "Definition:External Direct Product/General Definition", "text": "Let $\\left({S_1, \\circ_1}\\right), \\left({S_2, \\circ_2}\\right), \\ldots, \\left({S_n, \\circ_n}\\right)$ be algebraic structures. Let $\\displaystyle \\mathcal S_n = \\prod_{k \\mathop = 1}^n S_k$ be the cartesian product of $S_1, S_2, \\ldots, S_n$. Let $\\circledcirc_n$ be the operation induced on $\\mathcal S_n$ by $\\circ_1, \\ldots, \\circ_n$ defined as: :$\\left({s_1, s_2, \\ldots, s_n}\\right) \\circledcirc_n \\left({t_1, t_2, \\ldots, t_n}\\right) := \\begin{cases} s_1 \\circ_1 t_1 & : n = 1 \\\\ \\left({s_1 \\circ_1 t_1, s_2 \\circ_2 t_2}\\right) & : n = 2 \\\\ \\left({\\left({s_1, s_2, \\ldots, s_{n-1} }\\right) \\circledcirc_{n-1} \\left({t_1, t_2, \\ldots, t_{n-1}}\\right), s_n \\circ_n t_n}\\right) & : n > 2 \\end{cases}$ for all ordered $n$-tuples in $\\mathcal S_n$. That is: :$\\left({s_1, s_2, \\ldots, s_n}\\right) \\circledcirc_n \\left({t_1, t_2, \\ldots, t_n}\\right) := \\left({s_1 \\circ_1 t_1, s_2 \\circ_2 t_2, \\ldots, s_n \\circ_n t_n}\\right)$ The algebraic structure $\\left({\\mathcal S_n, \\circledcirc_n}\\right)$  is called the '''(external) direct product''' of $\\left({S_1, \\circ_1}\\right), \\left({S_2, \\circ_2}\\right), \\ldots, \\left({S_n, \\circ_n}\\right)$."}
+{"_id": "22520", "title": "Definition:External Direct Product/Structures with Two Operations", "text": "Let $\\left({S_1, +_1 ,\\circ_1}\\right), \\left({S_2, +_2 ,\\circ_2}\\right), \\ldots, \\left({S_n, +_n ,\\circ_n}\\right)$ be algebraic structures with two operations. Let $\\displaystyle S = \\prod_{k \\mathop = 1}^n S_k$ be as defined in cartesian product. The operation $+$ induced on $S$ by $+_1, \\ldots, +_n$ is defined as: :$\\left({s_1, s_2, \\ldots, s_n}\\right) + \\left({t_1, t_2, \\ldots, t_n}\\right) = \\left({s_1 +_1 t_1, s_2 +_2 t_2, \\ldots, s_n +_n t_n}\\right)$ The operation $\\circ $ induced on $S$ by $\\circ_1, \\ldots, \\circ_n$ is defined as: :$\\left({s_1, s_2, \\ldots, s_n}\\right) \\circ \\left({t_1, t_2, \\ldots, t_n}\\right) = \\left({s_1 \\circ_1 t_1, s_2 \\circ_2 t_2, \\ldots, s_n \\circ_n t_n}\\right)$ for all ordered $n$-tuples in $S$. The algebraic structure $\\left({S, +, \\circ}\\right)$  is called the '''(external) direct product''' of $\\left({S_1, +_1 ,\\circ_1}\\right), \\left({S_2, +_2 ,\\circ_2}\\right), \\ldots, \\left({S_n, +_n ,\\circ_n}\\right)$."}
+{"_id": "22521", "title": "Definition:Cartesian Product/Finite", "text": "Let $\\sequence {S_n}$ be a sequence of sets.  The '''cartesian product''' of $\\sequence {S_n}$ is defined as: :$\\displaystyle \\prod_{k \\mathop = 1}^n S_k = \\set {\\tuple {x_1, x_2, \\ldots, x_n}: \\forall k \\in \\N^*_n: x_k \\in S_k}$ It is also denoted $S_1 \\times S_2 \\times \\cdots \\times S_n$. Thus $S_1 \\times S_2 \\times \\cdots \\times S_n$ is the set of all ordered $n$-tuples $\\tuple {x_1, x_2, \\ldots, x_n}$ with $x_k \\in S_k$."}
+{"_id": "22522", "title": "Definition:Canonical Injection (Abstract Algebra)/General Definition", "text": "Let $\\struct {S_1, \\circ_1}, \\struct {S_2, \\circ_2}, \\dotsc, \\struct {S_j, \\circ_j}, \\dotsc, \\struct {S_n, \\circ_n}$ be algebraic structures with identities $e_1, e_2, \\dotsc, e_j, \\dotsc, e_n$ respectively. Then the '''canonical injection''' $\\displaystyle \\inj_j: \\struct {S_j, \\circ_j} \\to \\prod_{i \\mathop = 1}^n \\struct {S_i, \\circ_i}$ is defined as: :$\\map {\\inj_j} x = \\tuple {e_1, e_2, \\dotsc, e_{j - 1}, x, e_{j + 1}, \\dotsc, e_n}$"}
+{"_id": "22523", "title": "Definition:Sturm-Liouville Theory", "text": "'''Sturm-Liouville theory''' is the branch of mathematical physics concerned with the eigenvalues and eigenfunctions arising from the Sturm-Liouville equation: :$\\map {\\dfrac \\d {\\d x} } {\\map p x \\dfrac {\\d y} {\\d x} } + \\map q x y = -\\lambda \\map w x y$"}
+{"_id": "22524", "title": "Definition:Sturm-Liouville Equation", "text": "A classical '''Sturm-Liouville equation''' is a real second order ordinary linear differential equation of the form: :$ (1): \\quad \\displaystyle - \\frac {\\mathrm d} {\\mathrm d x} \\left({p \\left({x}\\right) \\frac {\\mathrm d y} {\\mathrm d x}}\\right) + q \\left({x}\\right) y = \\lambda w \\left({x}\\right) y$ where $y$ is a function of the free variable $x$. The functions $p \\left({x}\\right)$, $q \\left({x}\\right)$ and $w \\left({x}\\right)$ are specified. In the simplest cases they are continuous on the closed interval $\\left[{a \\,.\\,.\\, b}\\right]$. In addition: :$(1a): \\quad p \\left({x}\\right) > 0$ has a continuous derivative :$(1b): \\quad w \\left({x}\\right) > 0$ :$(1c): \\quad y$ is typically required to satisfy some boundary conditions at $a$ and $b$. === Weight Function === The function $w \\left({x}\\right)$, which is sometimes called $r \\left({x}\\right)$, is called the '''weight function''' or '''density function'''.  === Eigenvalues === The value of $\\lambda$ is not specified in the equation. Finding the values of $\\lambda$ for which there exists a non-trivial solution of $(1)$ satisfying the boundary conditions is part of the problem called the Sturm-Liouville problem (S-L).  Such values of $\\lambda$ when they exist are called the '''eigenvalues''' of the boundary value problem defined by $(1)$ and the prescribed set of boundary conditions. The corresponding solutions (for such a $\\lambda$) are the '''eigenfunctions''' of this problem."}
+{"_id": "22525", "title": "Definition:Column Space", "text": "Let $R$ be a ring. Let: :$\\mathbf A_{m \\times n} = \\begin{bmatrix} a_{1 1} & a_{1 2} & \\cdots & a_{1 n} \\\\ a_{2 1} & a_{2 2} & \\cdots & a_{2 n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{m 1} & a_{m 2} & \\cdots & a_{m n} \\\\ \\end{bmatrix}$ be a matrix over $R$ such that every column is defined as a vector: :$\\forall i: 1 \\le i \\le m: \\begin {bmatrix} a_{1 i} \\\\ a_{2 i} \\\\ \\vdots \\\\ a_{m i} \\end {bmatrix} \\in \\mathbf V$ where $\\mathbf V$ is some vector space. Then the '''column space of $\\mathbf A$''' is the linear span of all such column vectors: :$\\map {\\mathrm C} {\\mathbf A} = \\map {\\mathrm {span} } {\\begin {bmatrix} a_{1 1} \\\\ a_{2 1} \\\\ \\vdots \\\\ a_{m 1} \\end {bmatrix}, \\begin {bmatrix} a_{1 2} \\\\ a_{2 2} \\\\ \\vdots \\\\ a_{m 2} \\end {bmatrix}, \\cdots, \\begin {bmatrix} a_{1 n} \\\\ a_{2 n} \\\\ \\vdots \\\\ a_{ mn} \\end {bmatrix} }$"}
+{"_id": "22526", "title": "Definition:Row Space", "text": "Let $R$ be a ring. Let $\\mathbf A$ be a matrix over $R$. Let $\\mathbf A^\\intercal$ be the transpose of $\\mathbf A$.   Let the columns of $\\mathbf A^\\intercal$ be members of a vector space. The '''row space of $\\mathbf A$''' is defined as the column space of $\\mathbf A^\\intercal$."}
+{"_id": "22527", "title": "Definition:Simultaneous Linear Equations/Matrix Representation", "text": "A '''system of simultaneous linear equations''' can be expressed as: :$\\mathbf A \\mathbf x = \\mathbf b$ where: :$\\mathbf A = \\begin {bmatrix} \\alpha_{1 1} & \\alpha_{1 2} & \\cdots & \\alpha_{1 n} \\\\ \\alpha_{2 1} & \\alpha_{2 2} & \\cdots & \\alpha_{2 n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\alpha_{m 1} & \\alpha_{m 2} & \\cdots & \\alpha_{m n} \\\\ \\end {bmatrix}$,  $\\mathbf x = \\begin {bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{bmatrix}$, $\\mathbf b = \\begin {bmatrix} \\beta_1 \\\\ \\beta_2 \\\\ \\vdots \\\\ \\beta_m \\end {bmatrix}$ are matrices."}
+{"_id": "22528", "title": "Definition:Simultaneous Equations/Solution Set", "text": "Consider the system of $m$ simultaneous equations in $n$ variables: :$\\mathbb S := \\forall i \\in \\set {1, 2, \\ldots, m} : \\map {f_i} {x_1, x_2, \\ldots x_n} = \\beta_i$ Let $\\mathbb X$ be the set of ordered $n$-tuples: :$\\set {\\sequence {x_j}_{j \\mathop \\in \\set {1, 2, \\ldots, n} }: \\forall i \\in \\set {1, 2, \\ldots, m}: \\map {f_i} {\\sequence {x_j} } = \\beta_i}$ which satisfies each of the equations in $\\mathbb S$. Then $\\mathbb X$ is called the '''solution set''' of $\\mathbb S$."}
+{"_id": "22530", "title": "Definition:Order Completion", "text": "Let $\\struct {S, \\preceq_S}$ be an ordered set. An ordered set $\\struct {T, \\preceq_T}$ is an '''order completion of $S$''' {{iff}}: :$(1):\\quad S \\subseteq T$ :$(2):\\quad {\\preceq_T \\restriction_S} = {\\preceq_S}$, where $\\restriction$ denotes restriction :$(3):\\quad \\struct {T, \\preceq_T}$ is a complete ordered set :$(4):\\quad$ For all ordered sets $\\struct {T', \\preceq_{T'} }$ satisfying $(1), (2)$ and $(3)$, there is a unique order-preserving injection $\\phi: T' \\to T$"}
+{"_id": "22531", "title": "Definition:Complete Lattice/Definition 2", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then $\\struct {S, \\preceq}$ is a '''complete lattice''' {{iff}}: :$\\forall S' \\subseteq S: \\inf S', \\sup S' \\in S$ That is, {{iff}} all subsets of $S$ have both a supremum and an infimum."}
+{"_id": "22532", "title": "Definition:Left Null Space", "text": "Let $R$ be a ring. Let $\\mathbf A$ be a matrix in the matrix space $\\mathcal M_{m, n}\\left({R}\\right)$. Let $\\mathbf A^\\intercal$ be the transpose of $\\mathbf A$. The '''left null space''' '''$\\mathbf A$''' is defined as the null space of $\\mathbf A^\\intercal$."}
+{"_id": "22533", "title": "Definition:Extended Real Addition", "text": "Let $\\overline \\R$ denote the extended real numbers. Define '''extended real addition''' or '''addition on $\\overline \\R$''', denoted $+_{\\overline \\R}: \\overline \\R \\times \\overline \\R \\to \\overline \\R$, by: :$\\forall x,y \\in \\R: x +_{\\overline \\R} y := x +_\\R y$ where $+_\\R$ denotes real addition :$\\forall x \\in \\R: x +_{\\overline \\R} \\left({+\\infty}\\right) = \\left({+\\infty}\\right) +_{\\overline \\R} x := +\\infty$ :$\\forall x \\in \\R: x +_{\\overline \\R} \\left({-\\infty}\\right) = \\left({-\\infty}\\right) +_{\\overline \\R} x := -\\infty$ :$\\left({+\\infty}\\right) +_{\\overline \\R} \\left({+\\infty}\\right) := +\\infty$ :$\\left({-\\infty}\\right) +_{\\overline \\R} \\left({-\\infty}\\right) := -\\infty$ In particular, the expressions: :$\\left({+\\infty}\\right) +_{\\overline \\R} \\left({-\\infty}\\right)$ :$\\left({-\\infty}\\right) +_{\\overline \\R} \\left({+\\infty}\\right)$ are considered '''void''' and should be avoided. When no danger of confusion arises, $+_{\\overline \\R}$ is usually replaced with the more familiar $+$. From the definition of $+_{\\overline \\R}$ on bona fide real numbers, the name '''extended real addition''' is appropriate: the real addition is indeed extended."}
+{"_id": "22534", "title": "Definition:Extended Real Subtraction", "text": "Let $\\overline \\R$ denote the extended real numbers. Define '''extended real subtraction''' or '''subtraction on $\\overline \\R$''', denoted $-_{\\overline \\R}: \\overline \\R \\times \\overline \\R \\to \\overline \\R$, by: :$\\forall x, y \\in \\R: x -_{\\overline \\R} y := x -_{\\R} y$ where $-_\\R$ denotes real subtraction :$\\forall x \\in \\R: x -_{\\overline \\R} \\paren {+\\infty} = \\paren {-\\infty} -_{\\overline \\R} x := -\\infty$ :$\\forall x \\in \\R: x -_{\\overline \\R} \\paren {-\\infty} = \\paren {+\\infty} -_{\\overline \\R} x := +\\infty$ :$\\paren {-\\infty} -_{\\overline \\R} \\paren {+\\infty} := -\\infty$ :$\\paren {+\\infty} -_{\\overline \\R} \\paren {-\\infty} := +\\infty$ In particular, the expressions: :$\\paren {+\\infty} -_{\\overline \\R} \\paren {+\\infty}$ :$\\paren {-\\infty} -_{\\overline \\R} \\paren {-\\infty}$ are considered '''void''' and should be avoided."}
+{"_id": "22535", "title": "Definition:Extended Real Multiplication", "text": "Let $\\overline \\R$ denote the extended real numbers. Define '''extended real multiplication''' or '''multiplication on $\\overline \\R$''', denoted $\\times_{\\overline \\R}: \\overline \\R \\times \\overline \\R \\to \\overline \\R$, by: :$\\forall x \\in \\overline \\R: x \\times_{\\overline \\R} 0 = 0 \\times_{\\overline \\R} x = 0$ :$\\forall x, y \\in \\R: x \\times_{\\overline \\R} y := x \\times_\\R y$ where $\\times_\\R$ denotes real multiplication :$\\forall x \\in \\R_{>0}: x \\times_{\\overline \\R} \\paren {+\\infty} = \\paren {+\\infty} \\times_{\\overline \\R} x := +\\infty$ :$\\forall x \\in \\R_{<0}: x \\times_{\\overline \\R} \\paren {+\\infty} = \\paren {+\\infty} \\times_{\\overline \\R} x := -\\infty$ :$\\forall x \\in \\R_{>0}: x \\times_{\\overline \\R} \\paren {-\\infty} = \\paren {-\\infty} \\times_{\\overline \\R} x := -\\infty$ :$\\forall x \\in \\R_{<0}: x \\times_{\\overline \\R} \\paren {-\\infty} = \\paren {-\\infty} \\times_{\\overline \\R} x := +\\infty$ :$\\paren {+\\infty} \\times_{\\overline \\R} \\paren {+\\infty} := +\\infty$ :$\\paren {-\\infty} \\times_{\\overline \\R} \\paren {-\\infty} := +\\infty$ :$\\paren {+\\infty} \\times_{\\overline \\R} \\paren {-\\infty} := -\\infty$ :$\\paren {-\\infty} \\times_{\\overline \\R} \\paren {+\\infty} := -\\infty$ When no danger of confusion arises, $\\times_{\\overline \\R}$ is usually replaced with the more familiar $\\times$, or even suppressed. From the definition of $\\times_{\\overline \\R}$ on bona fide real numbers, the name '''extended real multiplication''' is appropriate: the real multiplication is indeed extended."}
+{"_id": "22536", "title": "Definition:Internal Direct Product/General Definition", "text": "Let $\\left({S_1, \\circ {\\restriction_{S_1}}}\\right), \\ldots, \\left({S_n, \\circ {\\restriction_{S_n}}}\\right)$ be closed algebraic substructures of an algebraic structure $\\left({S, \\circ}\\right)$ where $\\circ {\\restriction_{S_1}}, \\ldots, \\circ {\\restriction_{S_n}}$ are the operations induced by the restrictions of $\\circ$ to $S_1, \\ldots, S_n$ respectively. The structure $\\left({S, \\circ}\\right)$ is the '''internal direct product of $\\left \\langle {S_n} \\right \\rangle$''' if the mapping: :$\\displaystyle C: \\prod_{k \\mathop = 1}^n S_k \\to S: C \\left({s_1, \\ldots, s_n}\\right) = \\prod_{k \\mathop = 1}^n s_k$ is an isomorphism from the cartesian product $\\left({S_1, \\circ {\\restriction_{S_1}}}\\right) \\times \\cdots \\times \\left({S_n, \\circ {\\restriction_{S_n}}}\\right)$ onto $\\left({S, \\circ}\\right)$. The operation $\\circ$ on $S$ is the operation induced on $S$ by $\\circ {\\restriction_{S_1}}, \\circ {\\restriction_{S_2}}, \\ldots, \\circ {\\restriction_{S_n}}$."}
+{"_id": "22537", "title": "Definition:Measurable Mapping", "text": "Let $\\left({X, \\Sigma}\\right)$ and $\\left({X', \\Sigma'}\\right)$ be measurable spaces. A mapping $f: X \\to X'$ is said to be '''$\\Sigma \\, / \\, \\Sigma'$-measurable''' iff: :$\\forall E' \\in \\Sigma': f^{-1} \\left({E'}\\right) \\in \\Sigma$ That is, iff the preimage of every measurable set under $f$ is again measurable."}
+{"_id": "22539", "title": "Definition:Operation Induced by Direct Product", "text": "Let $\\left({S_1, \\circ_1}\\right)$ and $\\left({S_2, \\circ_2}\\right)$ be algebraic structures. Let $S = S_1 \\times S_2$ be the cartesian product of $S_1$ and $S_2$. Then the '''operation induced on $S$ by $\\circ_1$ and $\\circ_2$''' is the operation $\\circ$ defined as: :$\\left({s_1, s_2}\\right) \\circ \\left({t_1, t_2}\\right) := \\left({s_1 \\circ_1 t_1, s_2 \\circ_2 t_2}\\right)$ for all ordered pairs in $S$. === General Definition === {{:Definition:Operation Induced by Direct Product/General Definition}}"}
+{"_id": "22540", "title": "Definition:Translation Mapping", "text": "=== In an Abelian Group === {{:Definition:Translation Mapping/Abelian Group}} === In a Euclidean Space === {{:Definition:Translation Mapping/Euclidean Space}} === In an Affine Space === {{:Definition:Translation Mapping/Affine Space}}"}
+{"_id": "22541", "title": "Definition:Operation Induced by Direct Product/General Definition", "text": "Let $\\left({S_1, \\circ_1}\\right), \\left({S_2, \\circ_2}\\right), \\ldots, \\left({S_n, \\circ_n}\\right)$ be algebraic structures. Let $\\displaystyle \\mathcal S_n = \\prod_{k \\mathop = 1}^n S_k$ be the cartesian product of $S_1, S_2, \\ldots, S_n$. Then the '''operation induced on $\\mathcal S_n$ by $\\circ_1, \\ldots, \\circ_n$''' is the operation $\\circledcirc_n$ defined as: :$\\left({s_1, s_2, \\ldots, s_n}\\right) \\circledcirc_n \\left({t_1, t_2, \\ldots, t_n}\\right) := \\begin{cases} s_1 \\circ_1 t_1 & : n = 1 \\\\ \\left({s_1 \\circ_1 t_1, s_2 \\circ_2 t_2}\\right) & : n = 2 \\\\ \\left({\\left({s_1, s_2, \\ldots, s_{n-1} }\\right) \\circledcirc_{n-1} \\left({t_1, t_2, \\ldots, t_{n-1}}\\right), s_n \\circ_n t_n}\\right) & : n > 2 \\end{cases}$ for all ordered $n$-tuples in $\\mathcal S_n$. That is: :$\\left({s_1, s_2, \\ldots, s_n}\\right) \\circledcirc_n \\left({t_1, t_2, \\ldots, t_n}\\right) := \\left({s_1 \\circ_1 t_1, s_2 \\circ_2 t_2, \\ldots, s_n \\circ_n t_n}\\right)$"}
+{"_id": "22542", "title": "Definition:Pre-Image Sigma-Algebra", "text": "=== On Domain === {{:Definition:Pre-Image Sigma-Algebra/Domain}} === On Codomain === {{:Definition:Pre-Image Sigma-Algebra/Codomain}}"}
+{"_id": "22543", "title": "Definition:Pre-Image Sigma-Algebra/Codomain", "text": "Let $X, X'$ be sets, and let $f: X \\to X'$ be a mapping. Let $\\Sigma$ be a $\\sigma$-algebra on $X$. Then the '''pre-image $\\sigma$-algebra (of $\\Sigma$) on the codomain of $f$''' is defined as: :$\\Sigma' := \\left\\{{E' \\subseteq X': f^{-1} \\left({E'}\\right) \\in \\Sigma}\\right\\}$"}
+{"_id": "22544", "title": "Definition:Pointwise Operation", "text": "Let $S$ be a set. Let $\\struct {T, \\circ}$ be an algebraic structure. Let $T^S$ be the set of all mappings from $S$ to $T$. Let $f, g \\in T^S$, that is, let $f: S \\to T$ and $g: S \\to T$ be mappings. Then the operation $f \\oplus g$ is defined on $T^S$ as follows: :$f \\oplus g: S \\to T: \\forall x \\in S: \\map {\\paren {f \\oplus g} } x = \\map f x \\circ \\map g x$ The operation $\\oplus$ is called the '''pointwise operation on $T^S$ induced by $\\circ$'''. === Induced Structure === {{:Definition:Pointwise Operation/Induced Structure}}"}
+{"_id": "22545", "title": "Definition:Sigma-Algebra Generated by Collection of Mappings", "text": "Let $\\left({X_i, \\Sigma_i}\\right)$ be measurable spaces, with $i \\in I$ for some index set $I$. Let $X$ be a set, and let, for $i \\in I$, $f_i: X \\to X_i$ be a mapping. Then the '''$\\sigma$-algebra generated by $\\left({f_i}\\right)_{i \\in I}$''', $\\sigma \\left({f_i: i \\in I}\\right)$, is the smallest $\\sigma$-algebra on $X$ such that every $f_i$ is $\\sigma \\left({f_i: i \\in I}\\right) \\, / \\, \\Sigma_i$-measurable. That is, $\\sigma \\left({f_i: i \\in I}\\right)$ is subject to: :$(1):\\quad \\forall i \\in I: \\forall E_i \\in \\Sigma_i: f_i^{-1} \\left({E_i}\\right) \\in \\sigma \\left({f_i: i \\in I}\\right)$ :$(2):\\quad \\sigma \\left({f_i: i \\in I}\\right) \\subseteq \\Sigma$ for all $\\sigma$-algebras $\\Sigma$ on $X$ satisfying $(1)$ In fact, $\\sigma \\left({f_i: i \\in I}\\right)$ always exists, and is unique, as proved on Existence and Uniqueness of Sigma-Algebra Generated by Collection of Mappings."}
+{"_id": "22546", "title": "Definition:Gradient Operator/Real Cartesian Space", "text": "Let $\\R^n$ denote the real Cartesian space of $n$ dimensions. Let $\\map f {x_1, x_2, \\ldots, x_n}$ denote a real-valued function on $\\R^n$. Let $\\tuple {\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}$ be the standard ordered basis on $\\R^n$. Let $\\mathbf u = u_1 \\mathbf e_1 + u_2 \\mathbf e_2 + \\cdots + u_n \\mathbf e_n = \\displaystyle \\sum_{k \\mathop = 1}^n u_k \\mathbf e_k$ be a vector in $\\R^n$. Let the partial derivative of $f$ with respect to $u_k$ exist for all $u_k$. The '''gradient of $f$''' (at $\\mathbf u$) is defined as: {{begin-eqn}} {{eqn | l = \\grad f       | o = :=       | r = \\nabla f }} {{eqn | r = \\paren {\\sum_{k \\mathop = 1}^n \\mathbf e_k \\dfrac \\partial {\\partial x_k} } \\map f {\\mathbf u}       | c = {{Defof|Del Operator}} }} {{eqn | r = \\sum_{k \\mathop = 1}^n \\dfrac {\\map {\\partial f} {\\mathbf u} } {\\partial x_k} \\mathbf e_k       | c =  }} {{end-eqn}}"}
+{"_id": "22547", "title": "Definition:Directional Derivative", "text": "Let: :$f: \\R^n \\to \\R, \\mathbf x \\mapsto \\map f {\\mathbf x}$  be a real-valued function such that the gradient: :$\\nabla \\map f {\\mathbf x}$ exists. Let: :$\\mathbf u$ be a unit vector in $\\R^n$. The '''directional derivative of $f$ in the direction of $\\mathbf u$''' is defined as: {{begin-eqn}} {{eqn | l = D_{\\mathbf u} \\map f {\\mathbf x}       | r = \\nabla \\map f {\\mathbf x} \\cdot \\mathbf u }} {{end-eqn}} where $\\cdot$ denotes the dot product. {{expand|Geometric Interpretation}}"}
+{"_id": "22548", "title": "Definition:Pushforward Measure", "text": "Let $\\struct {X, \\Sigma}$ and $\\struct {X', \\Sigma'}$ be measurable spaces. Let $\\mu$ be a measure on $\\struct {X, \\Sigma}$. Let $f: X \\to X'$ be a $\\Sigma \\, / \\, \\Sigma'$-measurable mapping. Then the '''pushforward of $\\mu$ under $f$''' is the mapping $f_* \\mu: \\Sigma' \\to \\overline \\R$ defined by: :$\\forall E' \\in \\Sigma': \\map {f_* \\mu} {E'} := \\map \\mu {\\map {f^{-1} } {E'} }$ where $\\overline \\R$ denotes the extended real numbers."}
+{"_id": "22549", "title": "Definition:Orthogonal Group", "text": "Let $k$ be a field. The '''($n$th) orthogonal group (on $k$)''', denoted $\\map {\\mathrm O} {n, k}$, is the following subset of the general linear group $\\GL {n, k}$: :$\\map {\\mathrm O} {n, k} := \\set {M \\in \\GL {n, k}: M^\\intercal = M^{-1} }$ where $M^\\intercal$ denotes the transpose of $M$."}
+{"_id": "22550", "title": "Definition:Stieltjes Function", "text": "Let $f: \\R \\to \\overline \\R$ be a real function, where $\\overline \\R$ denotes the extended real numbers. Then $f$ is said to be a '''Stieltjes function''' {{iff}}: :$(1): \\quad f$ is increasing :$(2): \\quad f$ is left-continuous."}
+{"_id": "22551", "title": "Definition:Stieltjes Function of Measure on Real Numbers", "text": "Let $\\mu$ be a measure on $\\R$ with the Borel $\\sigma$-algebra $\\mathcal B \\left({\\R}\\right)$. The '''Stieltjes function of $\\mu$''' is the mapping $F_\\mu: \\R \\to \\overline \\R$ defined by: :$F_\\mu \\left({x}\\right) := \\begin{cases} \\mu \\left({ \\left[{0 \\,.\\,.\\, x}\\right) \\, }\\right) & \\text{if } x > 0\\\\ 0 & \\text{if } x = 0\\\\ - \\mu \\left({ \\left[{x \\,.\\,.\\, 0}\\right) \\, }\\right) & \\text{if } x < 0 \\end{cases}$ where $\\overline \\R$ denotes the extended real numbers."}
+{"_id": "22552", "title": "Definition:Pre-Measure of Finite Stieltjes Function", "text": "Let $\\mathcal J_{ho}$ denote the collection of half-open intervals in $\\R$. Let $f: \\R \\to \\R$ be a finite Stieltjes function. The '''pre-measure of $f$''' is the mapping $\\mu_f: \\mathcal J_{ho} \\to \\overline \\R_{\\ge 0}$ defined by: :$\\mu_f \\left({ \\left[{a \\,.\\,.\\, b}\\right) \\, }\\right) := \\begin{cases} f \\left({b}\\right) - f\\left({a}\\right) & \\text{if } b \\ge a \\\\ 0 & \\text{otherwise} \\end{cases}$ where $\\overline \\R_{\\ge 0}$ denotes the set of positive extended real numbers."}
+{"_id": "22553", "title": "Definition:Measure of Finite Stieltjes Function", "text": "Let $f: \\R \\to \\R$ be a finite Stieltjes function. Let $\\mu_f$ be the pre-measure of $f$. Let $\\mu$ be the unique measure extending $\\mu_f$ provided on Pre-Measure of Finite Stieltjes Function Extends to Unique Measure. Then $\\mu$ is called the '''measure of $f$'''. This definition makes $\\mu$ a measure on $\\mathcal B \\left({\\R}\\right)$, the Borel $\\sigma$-algebra of $\\R$."}
+{"_id": "22554", "title": "Definition:Extended Real-Valued Function", "text": "Let $S$ be a set, and let $\\overline \\R$ denote the extended real numbers. A mapping $f: S \\to \\overline \\R$ is said to be an '''extended real-valued function'''."}
+{"_id": "22555", "title": "Definition:Extended Real Sigma-Algebra", "text": "Let $\\left({\\overline{\\R}, \\tau}\\right)$ be the extended real number space. The '''extended real $\\sigma$-algebra''' $\\overline{\\mathcal B}$ is the Borel $\\sigma$-algebra $\\mathcal B \\left({\\overline{\\R}, \\tau}\\right)$."}
+{"_id": "22556", "title": "Definition:Space of Measurable Functions", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Then the '''space of $\\Sigma$-measurable, real-valued functions''' $\\mathcal M \\left({\\Sigma}\\right)$ is the collection of all $\\Sigma$-measurable, real-valued functions: :$\\mathcal M \\left({\\Sigma}\\right) := \\left\\{{f: X \\to \\R: f \\text{ is $\\Sigma$-measurable}}\\right\\}$ Similarly, the '''space of $\\Sigma$-measurable, extended real-valued functions''' $\\mathcal{M}_{\\overline{\\R}} \\left({\\Sigma}\\right)$ is the collection of all $\\Sigma$-measurable, extended real-valued functions: :$\\mathcal{M}_{\\overline{\\R}} \\left({\\Sigma}\\right) := \\left\\{{f: X \\to \\overline{\\R}: f \\text{ is $\\Sigma$-measurable}}\\right\\}$ === Space of Positive Measurable Functions === The '''space of $\\Sigma$-measurable, positive real-valued functions''' $\\mathcal{M}^+ \\left({\\Sigma}\\right)$ is the subset of positive $\\Sigma$-measurable functions in $\\mathcal M \\left({\\Sigma}\\right)$: :$\\mathcal{M}^+ \\left({\\Sigma}\\right) := \\left\\{{f: X \\to \\R: f \\text{ is positive $\\Sigma$-measurable}}\\right\\}$ Analogously, the '''space of $\\Sigma$-measurable, positive extended real-valued functions''' $\\mathcal{M}_{\\overline{\\R}}^+ \\left({\\Sigma}\\right)$ is defined as: :$\\mathcal{M}_{\\overline{\\R}}^+ \\left({\\Sigma}\\right) := \\left\\{{f: X \\to \\overline{\\R}: f \\text{ is positive $\\Sigma$-measurable}}\\right\\}$"}
+{"_id": "22557", "title": "Definition:Standard Representation of Simple Function", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $f: X \\to \\R$ be a simple function. A '''standard representation of $f$''' consists of: :a finite sequence $a_1, \\ldots, a_n$ of real numbers :a partition $E_0, E_1, \\ldots, E_n$ of $\\Sigma$-measurable sets subject to: :$f = \\displaystyle \\sum_{j \\mathop = 0}^n a_j \\chi_{E_j}$ where $a_0 := 0$, and $\\chi$ denotes characteristic function."}
+{"_id": "22558", "title": "Definition:Space of Simple Functions", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Then the '''space of simple functions on $\\left({X, \\Sigma}\\right)$''', denoted $\\mathcal E \\left({\\Sigma}\\right)$, is the collection of all simple functions $f: X \\to \\R$: :$\\mathcal E \\left({\\Sigma}\\right) := \\left\\{{f: X \\to \\R: f \\text{ is a simple function}}\\right\\}$ === Space of Positive Simple Functions === The '''space of positive simple functions on $\\left({X, \\Sigma}\\right)$''', denoted $\\mathcal{E}^+ \\left({\\Sigma}\\right)$ is the subset of positive simple functions in $\\mathcal E \\left({\\Sigma}\\right)$: :$\\mathcal{E}^+ \\left({\\Sigma}\\right) := \\left\\{{f: X \\to \\R: f \\text{ is a positive simple function}}\\right\\}$"}
+{"_id": "22559", "title": "Definition:Positive Part", "text": "Let $X$ be a set. Let $f: X \\to \\overline \\R$ be an extended real-valued function. Then the '''positive part of $f$''', $f^+: X \\to \\overline \\R$, is the extended real-valued function defined by: :$\\forall x \\in X: \\map {f^+} x := \\max \\set {0, \\map f x}$ where the maximum is taken with respect to the extended real ordering. That is: :$\\forall x \\in X: \\map {f^+} x := \\begin {cases} \\map f x & : \\map f x \\ge 0 \\\\ 0 & : \\map f x < 0 \\end {cases}$"}
+{"_id": "22560", "title": "Definition:Negative Part", "text": "Let $X$ be a set, and let $f: X \\to \\overline \\R$ be an extended real-valued function. Then the '''negative part of $f$''', $f^-: X \\to \\overline \\R$, is the extended real-valued function defined by: :$\\forall x \\in X: \\map {f^-} x := -\\min \\set {0, \\map f x}$ where the minimum is taken with respect to the extended real ordering. That is: :$\\forall x \\in X: \\map {f^-} x := \\begin {cases} \\map f x & : \\map f x \\le 0 \\\\ 0 & : \\map f x > 0 \\end {cases}$"}
+{"_id": "22561", "title": "Definition:Dedekind-Infinite", "text": "A set is said to be '''Dedekind-infinite''' {{iff}} it is equivalent to (at least) one of its proper subsets."}
+{"_id": "22562", "title": "Definition:Continuous Real Function/Left-Continuous/Point", "text": "Let $x_0 \\in A$.  Then $f$ is said to be '''left-continuous at $x_0$''' {{iff}} the limit from the left of $\\map f x$ as $x \\to x_0$ exists and: :$\\displaystyle \\lim_{\\substack {x \\mathop \\to x_0^- \\\\ x_0 \\mathop \\in A} } \\map f x = \\map f {x_0}$ where $\\displaystyle \\lim_{x \\mathop \\to x_0^-}$ is a limit from the left."}
+{"_id": "22563", "title": "Definition:Continuous Real Function/Right-Continuous/Point", "text": "Let $x_0 \\in S$.  Then $f$ is said to be '''right-continuous at $x_0$''' {{iff}} the limit from the right of $f \\left({x}\\right)$ as $x \\to x_0$ exists and: :$\\displaystyle \\lim_{\\substack{x \\mathop \\to x_0^+ \\\\ x_0 \\mathop \\in A}} f \\left({x}\\right) = f \\left({x_0}\\right)$ where $\\displaystyle \\lim_{x \\mathop \\to x_0^+}$ is a limit from the right."}
+{"_id": "22565", "title": "Definition:Pointwise Operation/Real-Valued Functions", "text": "Let $\\R^S$ be the set of all mappings $f: S \\to \\R$, where $\\R$ is the set of real numbers. Let $\\oplus$ be a binary operation on $\\R$. Define $\\oplus: \\R^S \\times \\R^S \\to \\R^S$, called '''pointwise $\\oplus$''', by: :$\\forall f, g \\in \\R^S: \\forall s \\in S: \\map {\\paren {f \\oplus g} } s := \\map f s \\oplus \\map g s$ In the above expression, the operator on the {{RHS}} is the given $\\oplus$ on the real numbers."}
+{"_id": "22566", "title": "Definition:Pointwise Scalar Multiplication of Mappings/Real-Valued Functions", "text": "Let $S$ be a non-empty set, and let $\\R^S$ be the set of all mappings $f: S \\to \\R$. Then '''pointwise ($\\R$-)scalar multiplication''' on $\\R^S$ is the binary operation $\\cdot: \\R \\times \\R^S \\to \\R^S$ defined by: :$\\forall \\lambda \\in \\R: \\forall f \\in \\R^S: \\forall s \\in S: \\map {\\paren {\\lambda \\cdot f} } s := \\lambda \\cdot \\map f s$ where the $\\cdot$ on the right is real multiplication."}
+{"_id": "22567", "title": "Definition:Pointwise Addition of Real-Valued Functions", "text": "Let $f, g: S \\to \\R$ be real-valued functions. Then the '''pointwise sum of $f$ and $g$''' is defined as: :$f + g: S \\to \\R:$ ::$\\forall s \\in S: \\map {\\paren {f + g} } s := \\map f s + \\map g s$ where the $+$ on the {{RHS}} is real-number addition."}
+{"_id": "22568", "title": "Definition:Pointwise Multiplication", "text": "The (binary) operation of '''pointwise multiplication''' is defined on $\\mathbb F^S$ as: :$\\times: \\mathbb F^S \\times \\mathbb F^S \\to \\mathbb F^S: \\forall f, g \\in \\mathbb F^S:$ ::$\\forall s \\in S: \\map {\\paren {f \\times g} } s := \\map f s \\times \\map g s$ where the $\\times$ on the {{RHS}} is conventional arithmetic multiplication."}
+{"_id": "22569", "title": "Definition:Pointwise Limit", "text": "Let $S$ be a set. Let $\\sequence {f_n}_{n \\mathop \\in \\N}$, $f_n: S \\to \\R$ be a sequence of real-valued functions. Suppose that for all $s \\in S$, the limit: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map {f_n} s$ exists. {{definition wanted|Incorporate infinite limits}} Then the '''pointwise limit of $\\sequence {f_n}_{n \\mathop \\in \\N}$''', denoted $\\displaystyle \\lim_{n \\mathop \\to \\infty} f_n: S \\to \\R$, is defined as: :$\\forall s \\in S: \\displaystyle \\map {\\paren {\\lim_{n \\mathop \\to \\infty} f_n} } s := \\lim_{n \\mathop \\to \\infty} \\map {f_n} s$ '''Pointwise limit''' thence is an instance of a pointwise operation on real-valued functions. {{definition wanted|of course, makes sense in arbitrary metric/topological space. cover this}}"}
+{"_id": "22570", "title": "Definition:Absolute Value of Mapping/Real-Valued Function", "text": "Let $S$ be a set. Let $f: S \\to \\R$ be a real-valued function. Then the '''absolute value of $f$''', denoted $\\size f: S \\to \\R$, is defined as: :$\\forall s \\in S: \\map {\\size f} s := \\size {\\map f s}$ where $\\size {\\map f s}$ denotes the absolute value function on $\\R$. '''Absolute value''' thence is an instance of a pointwise operation on real-valued functions."}
+{"_id": "22571", "title": "Definition:Extended Absolute Value", "text": "Let $\\overline \\R$ denote the extended real numbers. Extend the absolute value $\\left\\vert{\\cdot}\\right\\vert$ on $\\R$ to $\\overline \\R = \\R \\cup \\left\\{{+\\infty, -\\infty}\\right\\}$ by defining: :$\\left\\vert{-\\infty}\\right\\vert = \\left\\vert{+\\infty}\\right\\vert = +\\infty$ Thus, the '''extended absolute value''' is a mapping $\\left\\vert{\\cdot}\\right\\vert: \\overline \\R \\to \\overline \\R$."}
+{"_id": "22572", "title": "Definition:Absolute Value of Mapping/Extended Real-Valued Function", "text": "Let $S$ be a set, and let $f: S \\to \\overline \\R$ be an extended real-valued function. Then the '''absolute value of $f$''', denoted $\\size f: S \\to \\overline \\R$, is defined as: :$\\forall s \\in S: \\map {\\size f} s := \\size {\\map f s}$ where $\\size {\\map f s}$ denotes the extended absolute value function on $\\overline \\R$. '''Absolute value''' thence is an instance of a pointwise operation on extended real-valued functions. Since extended absolute value coincides on $\\R$ with the standard ordering, this definition incorporates the definition for real-valued functions."}
+{"_id": "22573", "title": "Definition:Absolute Value of Mapping", "text": "Let $D$ be an ordered integral domain. Let $\\size {\\, \\cdot \\,}_D$ denote the absolute value function on $D$. Let $S$ be a set. Let $f: S \\to D$ be a mapping. Then the '''absolute value of $f$''', denoted $\\size f_D: S \\to D$, is defined as: :$\\forall s \\in S: \\map {\\size f_D} s := \\size {\\map f s}_D$ '''Absolute value''' thence is an instance of a pointwise operation on a mapping."}
+{"_id": "22574", "title": "Definition:Pointwise Operation on Extended Real-Valued Functions", "text": "Let $S$ be a set, and let $f, g : S \\to \\overline \\R$ be extended real-valued functions. Let $\\lambda \\in \\R$. Then extended real-valued functions can be formed by defining (for all $s \\in S$): :$\\lambda \\cdot f: S \\to \\R, \\map {\\paren {\\lambda \\cdot f} } s := \\lambda \\cdot \\map f s$ :$f + g: S \\to \\R, \\map {\\paren {f + g} } s := \\map f s + \\map g s$ :$f \\cdot g: S \\to \\R, \\map {\\paren {f \\cdot g} } s := \\map f s \\cdot \\map g s$ as is done on Pointwise Scalar Multiplication, Pointwise Addition and Pointwise Multiplication, respectively. More generally, let $\\oplus$ be a binary operation on $\\overline \\R$. Define $f \\oplus g: S \\to \\overline \\R$, called '''pointwise $\\oplus$''', by: :$\\map {\\paren {f \\oplus g} } s := \\map f s \\oplus \\map g s$ In above expressions, the subscript ${\\overline \\R}$ of an operator expresses that its operands are extended real numbers. Next, let $\\family {f_i}_{i \\mathop \\in I}, f_i: S \\to \\overline \\R$, be any $I$-indexed collection of extended real-valued functions, where $I$ is some index set. Suppose that $\\oplus^I$ is an $I$-ary operation on $\\overline \\R$. Then define $\\oplus^I \\family {f_i}_{i \\mathop \\in I}: S \\to \\overline \\R$, called '''pointwise $\\oplus^I$''', by: :$\\map {\\oplus^I \\family {f_i}_{i \\mathop \\in I} } s := \\oplus^I \\family {\\map {f_i} s}_{i \\mathop \\in I}$"}
+{"_id": "22576", "title": "Definition:Ordered Subsemigroup", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be an ordered structure. Let $T \\subseteq S$ be a subset of $S$ such that: :$\\left({T, \\circ_T, \\preceq_T}\\right)$ is an ordered semigroup where: :$\\circ_T$ is the operation induced on $H$ by $\\circ$ :$\\preceq_t$ is the restriction of $\\preceq$ to $T \\times T$. Then $\\left({T, \\circ_T, \\preceq_T}\\right)$ is an '''ordered subsemigroup''' of $\\left({S, \\circ, \\preceq}\\right)$."}
+{"_id": "22577", "title": "Definition:Ordered Subgroup", "text": "An '''ordered subgroup''' $\\struct {T, \\circ, \\preceq}$ of an ordered structure $\\struct {S, \\circ, \\preceq}$ is an ordered group such that the group $\\struct {T, \\circ}$ is a subgroup of $\\struct {S, \\circ}$."}
+{"_id": "22578", "title": "Definition:Totally Ordered Structure", "text": "When the ordering $\\preceq$ is a total ordering, the structure $\\left({S, \\circ, \\preceq}\\right)$ is then a '''totally ordered structure'''."}
+{"_id": "22579", "title": "Definition:Integral of Positive Simple Function", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $\\mathcal E^+$ denote the space of positive simple functions Let $f: X \\to \\R, f \\in \\mathcal E^+$ be a positive simple function. Suppose that $f$ admits the following standard representation: :$\\displaystyle f = \\sum_{i \\mathop = 0}^n a_i \\chi_{E_i}$ where $a_0 := 0$, and $\\chi$ denotes characteristic function. Then the '''$\\mu$-integral of $f$''', denoted $I_\\mu \\left({f}\\right)$, is defined by: :$I_\\mu \\left({f}\\right) := \\displaystyle \\sum_{i \\mathop = 0}^n a_i \\mu \\left({E_i}\\right)$"}
+{"_id": "22580", "title": "Definition:Ordered Semigroup Isomorphism", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ and $\\left({T, *, \\preccurlyeq}\\right)$ be ordered semigroups. An '''ordered semigroup isomorphism''' from $\\left({S, \\circ, \\preceq}\\right)$ to $\\left({T, *, \\preccurlyeq}\\right)$ is a mapping $\\phi: S \\to T$ that is both: :$(1): \\quad$ A semigroup isomorphism from the semigroup $\\left({S, \\circ}\\right)$ to the semigroup $\\left({T, *}\\right)$ :$(2): \\quad$ An order isomorphism from the ordered set $\\left({S, \\preceq}\\right)$ to the ordered set $\\left({T, \\preccurlyeq}\\right)$."}
+{"_id": "22581", "title": "Definition:Isomorphism (Abstract Algebra)/Semigroup Isomorphism", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be semigroups. Let $\\phi: S \\to T$ be a (semigroup) homomorphism. Then $\\phi$ is a semigroup isomorphism {{iff}} $\\phi$ is a bijection."}
+{"_id": "22582", "title": "Definition:Semigroup Homomorphism", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be semigroups. Let $\\phi: S \\to T$ be a mapping such that $\\circ$ has the morphism property under $\\phi$. That is, $\\forall a, b \\in S$: :$\\phi \\left({a \\circ b}\\right) = \\phi \\left({a}\\right) * \\phi \\left({b}\\right)$ Then $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ is a semigroup homomorphism."}
+{"_id": "22583", "title": "Definition:Semigroup Epimorphism", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be semigroups. Let $\\phi: S \\to T$ be a (semigroup) homomorphism. Then $\\phi$ is a semigroup epimorphism {{iff}} $\\phi$ is a surjection."}
+{"_id": "22584", "title": "Definition:Semigroup Monomorphism", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be semigroups. Let $\\phi: S \\to T$ be a (semigroup) homomorphism. Then $\\phi$ is a semigroup monomorphism {{iff}} $\\phi$ is an injection."}
+{"_id": "22585", "title": "Definition:Integral of Positive Measurable Function", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Define the '''$\\mu$-integral of positive measurable functions''', denoted $\\displaystyle \\int \\cdot \\rd \\mu: \\MM_{\\overline \\R}^+ \\to \\overline \\R_{\\ge 0}$, as: :$\\forall f \\in \\MM_{\\overline \\R}^+: \\displaystyle \\int f \\rd \\mu := \\sup \\set {\\map {I_\\mu} g: g \\le f, g \\in \\EE^+}$ where: :$\\MM_{\\overline \\R}^+$ denotes the space of positive $\\Sigma$-measurable functions :$\\overline \\R_{\\ge 0}$ denotes the positive extended real numbers :$\\sup$ is a supremum in the extended real ordering :$\\map {I_\\mu} g$ denotes the $\\mu$-integral of the positive simple function $g$ :$g \\le f$ denotes pointwise inequality :$\\EE^+$ denotes the space of positive simple functions === Integral Sign === {{:Definition:Integral Sign}}"}
+{"_id": "22586", "title": "Definition:Semigroup Endomorphism", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroups. Let $\\phi: S \\to S$ be a (semigroup) homomorphism from $S$ to itself. Then $\\phi$ is a semigroup endomorphism."}
+{"_id": "22587", "title": "Definition:Semigroup Automorphism", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroup. Let $\\phi: S \\to S$ be a (semigroup) isomorphism from $S$ to itself. Then $\\phi$ is a semigroup automorphism."}
+{"_id": "22588", "title": "Definition:Ordered Semigroup Automorphism", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be an ordered semigroup. An '''ordered semigroup automorphism''' from $\\left({S, \\circ, \\preceq}\\right)$ to itself is a mapping $\\phi: S \\to S$ that is both: :$(1): \\quad$ A semigroup automorphism, that is, a semigroup isomorphism from the semigroup $\\left({S, \\circ}\\right)$ to itself :$(2): \\quad$ An order isomorphism from the ordered set $\\left({S, \\preceq}\\right)$ to itself."}
+{"_id": "22589", "title": "Definition:Totally Ordered Semigroup", "text": "A '''totally ordered semigroup''' is a totally ordered structure $\\left({S, \\circ, \\preceq}\\right)$ such that $\\left({S, \\circ}\\right)$ is a semigroup."}
+{"_id": "22590", "title": "Definition:Ordered Semigroup Monomorphism", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ and $\\left({T, *, \\preccurlyeq}\\right)$ be ordered semigroups. An '''ordered semigroup monomorphism''' from $\\left({S, \\circ, \\preceq}\\right)$ to $\\left({T, *, \\preccurlyeq}\\right)$ is a mapping $\\phi: S \\to T$ that is both: :$(1): \\quad$ A (semigroup) monomorphism from the semigroup $\\left({S, \\circ}\\right)$ to the semigroup $\\left({T, *}\\right)$ :$(2): \\quad$ An order embedding from the ordered set $\\left({S, \\preceq}\\right)$ to the ordered set $\\left({T, \\preccurlyeq}\\right)$."}
+{"_id": "22591", "title": "Definition:Zero (Number)/Naturally Ordered Semigroup", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a naturally ordered semigroup. Then from axiom $(NO 1)$, $\\left({S, \\circ, \\preceq}\\right)$ has a smallest element. This smallest element of $\\left({S, \\circ, \\preceq}\\right)$ is called '''zero''' and has the symbol $0$. That is: :$\\forall n \\in S: 0 \\preceq n$"}
+{"_id": "22592", "title": "Definition:Closed Interval/Integer Interval", "text": "The '''integer interval between $m$ and $n$''' is denoted and defined as: : $\\closedint m n = \\begin{cases} \\set {x \\in S: m \\le x \\le n} & : m \\le n \\\\ \\O & : n < m \\end{cases}$ where $\\O$ is the empty set."}
+{"_id": "22593", "title": "Definition:Restriction of Ordering", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T \\subseteq S$ be a subset of $S$. Then the '''restriction of $\\preceq$ to $T$''', denoted $\\preceq \\restriction_T$, is defined as: :${\\preceq \\restriction_T} := {\\preceq} \\cap \\left({T \\times T}\\right)$ viewing ${\\preceq} \\subseteq S \\times S$ as a relation on $S$. Here, $\\times$ denotes Cartesian product. Thence the '''restriction of $\\preceq$ to $T$''' is an instance of a restriction of a relation. {{MissingLinks|Undoubtedly there are similar things to this, which need to be put in 'Also see'}}"}
+{"_id": "22594", "title": "Definition:Multiplication/Natural Numbers", "text": "Let $\\N$ be the natural numbers. '''Multiplication''' on $\\N$ is the basic operation $\\times$ everyone is familiar with. For example: :$3 \\times 4 = 12$ :$13 \\times 7 = 91$"}
+{"_id": "22595", "title": "Definition:Distributive Operation/Distributand", "text": "Let $\\circ$ be distributive over $*$. Then $*$ is a '''distributand''' of $\\circ$."}
+{"_id": "22596", "title": "Definition:Distributive Operation/Distributor", "text": "Let $\\circ$ be distributive over $*$. Then $\\circ$ is a '''distributor''' of $*$."}
+{"_id": "22597", "title": "Definition:Increasing Sequence of Mappings", "text": "Let $S$ be a set, and let $\\left({T, \\preceq}\\right)$ be an ordered set. Let $\\left({f_n}\\right)_{n \\mathop \\in \\N}, f_n: S \\to T$ be a sequence of mappings. Then $\\left({f_n}\\right)_{n \\mathop \\in \\N}$ is said to be an '''increasing sequence (of mappings)''' {{iff}}: :$\\forall s \\in S: \\forall m, n \\in \\N: m \\le n \\implies f_m \\left({s}\\right) \\preceq f_n \\left({s}\\right)$ That is, {{iff}} $m \\le n \\implies f_m \\preceq f_n$, where $\\preceq$ denotes pointwise inequality."}
+{"_id": "22598", "title": "Definition:Pointwise Inequality", "text": "Let $S$ be a set, and let $\\struct {T, \\preceq}$ be an ordered set. Let $f, g: S \\to T$ be mappings. Then '''$f$ pointwise precedes $g$''', denoted $f \\preceq g$, {{iff}}: :$\\forall s \\in S: \\map f s \\preceq \\map g s$ Thence it can be seen that '''pointwise precedence''' is an instance of an induced relation on mappings. {{wtd|Not entirely happy with the naming}}"}
+{"_id": "22599", "title": "Definition:Pointwise Inequality of Real-Valued Functions", "text": "Let $S$ be a set. Let $f, g: S \\to \\R$ be real-valued functions. Then '''pointwise inequality of $f$ and $g$''', denoted $f \\le g$, is defined to hold {{iff}}: :$\\forall s \\in S: f \\left({s}\\right) \\le g \\left({s}\\right)$ where $\\le$ denotes the usual ordering on the real numbers $\\R$. Thence '''pointwise inequality''' of real-valued functions is an instance of an induced relation on mappings."}
+{"_id": "22600", "title": "Definition:Pointwise Inequality of Extended Real-Valued Functions", "text": "Let $S$ be a set, and let $f,g: S \\to \\overline{\\R}$ be extended real-valued functions. Then '''pointwise inequality of $f$ and $g$''', denoted $f \\le g$, is defined to hold iff: :$\\forall s \\in S: f \\left({s}\\right) \\le g \\left({s}\\right)$ where $\\le$ denotes the usual ordering on the extended real numbers $\\overline{\\R}$. Thence '''pointwise inequality''' of extended real-valued functions is an instance of an induced relation on mappings."}
+{"_id": "22601", "title": "Definition:Increasing Sequence of Extended Real-Valued Functions", "text": "Let $S$ be a set, and let $\\left({f_n}\\right)_{n \\in \\N}, f_n: S \\to \\overline{\\R}$ be a sequence of extended real-valued functions. Then $\\left({f_n}\\right)_{n \\in \\N}$ is said to be an '''increasing sequence (of extended real-valued functions)''' iff: :$\\forall s \\in S: \\forall m, n \\in \\N: m \\le n \\implies f_m \\left({s}\\right) \\le f_n \\left({s}\\right)$ That is, iff $m \\le n \\implies f_m \\le f_n$, where the second $\\le$ denotes pointwise inequality."}
+{"_id": "22602", "title": "Definition:Increasing Sequence of Real-Valued Functions", "text": "Let $S$ be a set, and let $\\left({f_n}\\right)_{n \\in \\N}, f_n: S \\to \\R$ be a sequence of real-valued functions. Then $\\left({f_n}\\right)_{n \\in \\N}$ is said to be an '''increasing sequence (of real-valued functions)''' {{iff}}: :$\\forall s \\in S: \\forall m, n \\in \\N: m \\le n \\implies f_m \\left({s}\\right) \\le f_n \\left({s}\\right)$ That is, {{iff}} $m \\le n \\implies f_m \\le f_n$, where the second $\\le$ denotes pointwise inequality."}
+{"_id": "22603", "title": "Definition:Pointwise Supremum of Extended Real-Valued Functions", "text": "Let $S$ be a set. Let $\\family {f_i}_{i \\mathop \\in I}, f_i: S \\to \\overline \\R$ be an $I$-indexed collection of extended real-valued functions. Then the '''pointwise supremum of $\\family {f_i}_{i \\mathop \\in I}$''', denoted $\\displaystyle \\sup_{i \\mathop \\in I} f_i: S \\to \\overline \\R$, is defined by: :$\\displaystyle \\map {\\paren {\\sup_{i \\mathop \\in I} f_i} } s := \\sup_{i \\mathop \\in I} \\map {f_i} s$ where the latter supremum is taken in the extended real numbers $\\overline \\R$. By Extended Real Numbers form Complete Poset, this supremum is guaranteed to exist. Thence it can be seen that '''pointwise supremum''' is an instance of a pointwise operation on extended real-valued functions."}
+{"_id": "22604", "title": "Definition:Pointwise Supremum of Real-Valued Functions", "text": "Let $S$ be a set. Let $\\family {f_i}_{i \\mathop \\in I}, f_i: S \\to \\R$ be an $I$-indexed collection of real-valued functions. Then the '''pointwise supremum of $\\family {f_i}_{i \\mathop \\in I}$''', denoted $\\displaystyle \\sup_{i \\mathop \\in I} f_i: S \\to \\overline \\R$, is defined by: :$\\displaystyle \\map {\\paren {\\sup_{i \\mathop \\in I} f_i} } s := \\sup_{i \\mathop \\in I} \\map {f_i} s$ where the latter supremum is taken in the extended real numbers $\\overline \\R$. By Extended Real Numbers form Complete Poset, this supremum is guaranteed to exist. Thence it can be seen that '''pointwise supremum''' is an instance of a pointwise operation on real-valued functions. However, mind that this '''pointwise supremum''' need not be a real-valued function."}
+{"_id": "22605", "title": "Definition:Pointwise Supremum", "text": "Let $S$ be a set. Let $\\struct {T, \\preceq}$ be an ordered set. Let $\\family {f_i}_{i \\mathop \\in I}, f_i: S \\to T$ be an $I$-indexed collection of mappings. Suppose that for all $s \\in S$, it holds that: :$\\displaystyle \\sup_{i \\mathop \\in I} \\map {f_i} s \\in T$ where the supremum is taken in $T$. Then the '''pointwise supremum of $\\family {f_i}_{i \\mathop \\in I}$''', denoted $\\displaystyle \\sup_{i \\mathop \\in I} f_i: S \\to T$, is defined by: :$\\displaystyle \\map {\\paren {\\sup_{i \\mathop \\in I} f_i} } s := \\sup_{i \\mathop \\in I} \\map {f_i} s$ where the latter supremum is again taken in $T$. By assumption, this supremum is guaranteed to exist. Thence it can be seen that '''pointwise supremum''' is an instance of a pointwise operation."}
+{"_id": "22606", "title": "Definition:Ordered Commutative Semigroup", "text": "An '''ordered commutative semigroup''' is an ordered semigroup $\\left({S, \\circ, \\preceq}\\right)$ such that $\\left({S, \\circ}\\right)$ is a commutative semigroup."}
+{"_id": "22607", "title": "Definition:Clearly", "text": "A statement is '''clearly''' true if its proof is apparent. Because of its subjective nature and the high likelihood of mistakes, this website does not endorse using such handwavery in any but the most trivial and boring of cases. Category:Definitions/Language Definitions 1a9f9v01qmxwuwray0m49i2v8ezycv9"}
+{"_id": "22608", "title": "Definition:Divisor (Algebra)/Natural Numbers", "text": "Let $\\N$ be the natural numbers. Let $n \\in \\N$ and $m \\in \\N_{>0}$. Then '''$m$ divides $n$''' is defined as: :$m \\divides n \\iff \\exists p \\in \\N: m \\times p = n$"}
+{"_id": "22609", "title": "Definition:Ordered Tuple/Defined by Sequence", "text": "Let $\\sequence {a_k}_{k \\mathop \\in A}$ be a finite sequence of $n$ terms. Let $\\sigma$ be a permutation of $A$. Then the '''ordered $n$-tuple defined by the sequence $\\sequence {a_{\\map \\sigma k} }_{k \\mathop \\in A}$''' is the ordered $n$-tuple: :$\\sequence {a_{\\map \\sigma {\\map \\tau j} } }_{1 \\mathop \\le j \\mathop \\le n}$ where $\\tau$ is the unique isomorphism from the totally ordered set $\\closedint 1 n$ onto the totally ordered set $A$."}
+{"_id": "22610", "title": "Definition:Operation/N-Ary Operation", "text": "Let $S_1, S_2, \\dots, S_n$ be sets. Let $\\circ: S_1 \\times S_2 \\times \\ldots \\times S_n \\to \\mathbb U$ be a mapping from the cartesian product $S_1 \\times S_2 \\times \\ldots \\times S_n$ to a universal set $\\mathbb U$: That is, suppose that: :$\\circ: S_1 \\times S_2 \\times \\ldots \\times S_n \\to \\mathbb U: \\forall \\tuple {s_1, s_2, \\ldots, s_n} \\in S_1 \\times S_2 \\times \\ldots \\times S_n: \\map \\circ {s_1, s_2, \\ldots, s_n} \\in \\mathbb U$ Then $\\circ$ is an '''$n$-ary operation'''."}
+{"_id": "22611", "title": "Definition:Operation/Operation on Set", "text": "An '''$n$-ary operation on a set $S$''' is an $n$-ary operation where: :the domain is the cartesian space $S^n$ :the codomain is $S$: :$\\odot: S^n \\to S: \\forall \\tuple {s_1, s_2, \\ldots, s_n} \\in S^n: \\map \\odot {s_1, s_2, \\ldots, s_n} \\in S$ That is: :an '''$n$-ary operation on $S$''' needs to be defined for ''all'' tuples in $S^n$ :the image of $\\odot$ is itself in $S$."}
+{"_id": "22612", "title": "Definition:Indexing Set/Index", "text": "An element of the domain $I$ of $x$ is called an '''index'''."}
+{"_id": "22613", "title": "Definition:Indexing Set/Indexed Set", "text": "The image of $x$, that is, $x \\sqbrk I$ or $\\Img x$, is called an '''indexed set'''. That is, it is the '''set indexed by $I$'''"}
+{"_id": "22614", "title": "Definition:Indexing Set/Function", "text": "When used in this context, the mapping $x$ is referred to as an '''indexing function for $S$'''."}
+{"_id": "22615", "title": "Definition:Indexing Set/Term", "text": "The image of $x$ at an index $i$ is referred to as a '''term''' of the (indexed) family, and is denoted $x_i$."}
+{"_id": "22616", "title": "Definition:Indexing Set/Family of Distinct Elements", "text": "Let $x$ be an injection, that is: :$\\forall \\alpha, \\beta \\in I: \\alpha \\ne \\beta \\implies x_\\alpha \\ne x_\\beta$ Then $\\family {x_i} _{i \\mathop \\in I}$ is called a '''family of distinct elements of $S$'''."}
+{"_id": "22617", "title": "Definition:Kernel (Measure Theory)", "text": "Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $\\overline{\\R}_{\\ge0}$ be the set of positive extended real numbers. A '''kernel''' is a mapping $N: X \\times \\Sigma \\to \\overline{\\R}_{\\ge0}$ such that: :$(1):\\quad \\forall x \\in X: N_x: \\Sigma \\to \\overline{\\R}_{\\ge0}, E \\mapsto N(x, E)$ is a measure :$(2):\\quad \\forall E \\in \\Sigma: N_E: X \\to \\overline{\\R}_{\\ge0}, x \\mapsto N(x, E)$ is a positive $\\Sigma$-measurable function {{expand|I found a more general approach on [http://www.physicsforums.com/showthread.php?t{{=}}352389 this website]}}"}
+{"_id": "22618", "title": "Definition:Kernel Transformation of Measure", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $N: X \\times \\Sigma \\to \\overline{\\R}_{\\ge0}$ be a kernel. The '''transformation of $\\mu$ by $N$''' is the mapping $\\mu N: \\Sigma \\to \\overline{\\R}$ defined by: :$\\displaystyle \\forall E \\in \\Sigma: \\mu N \\left({E}\\right) := \\int N_E \\left({x}\\right) \\, d\\mu \\left({x}\\right)$ where $N_E \\left({x}\\right) = N \\left({x, E}\\right)$."}
+{"_id": "22619", "title": "Definition:Kernel Transformation of Positive Measurable Function", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $N: X \\times \\Sigma \\to \\overline{\\R}_{\\ge0}$ be a kernel. Let $f: X \\to \\overline{\\R}$ be a positive measurable function. The '''transformation of $f$ by $N$''' is the mapping $N f: X \\to \\overline{\\R}$ defined by: :$\\forall x \\in X: N f \\left({x}\\right) := \\displaystyle \\int f \\, \\mathrm dN_x$ where $N_x$ is the measure $E \\mapsto N(x, E)$."}
+{"_id": "22621", "title": "Definition:Integrable Function", "text": "=== Darboux === {{:Definition:Definite Integral/Darboux}} === Riemann === {{:Definition:Definite Integral/Riemann}} === Unbounded Above Positive Real Function === {{:Definition:Integrable Function/Unbounded Above}} === Unbounded Real Function === {{:Definition:Integrable Function/Unbounded}} === Measure Space === {{:Definition:Integrable Function/Measure Space}} === $p$-Integrable Function === {{:Definition:Integrable Function/p-Integrable}} === Lebesgue === {{:Definition:Integrable Function/Lebesgue}}"}
+{"_id": "22622", "title": "Definition:Internal Group Direct Product/General Definition", "text": "Let $\\sequence {H_n} = \\struct {H_1, \\circ {\\restriction_{H_1} } }, \\ldots, \\struct {H_n, \\circ {\\restriction_{H_n} } }$ be a (finite) sequence of subgroups of a group $\\struct {G, \\circ}$ where $\\circ {\\restriction_{H_1} }, \\ldots, \\circ {\\restriction_{H_n} }$ are the restrictions of $\\circ$ to $H_1, \\ldots, H_n$ respectively. === Definition 1 === {{:Definition:Internal Group Direct Product/General Definition/Definition 1}} === Definition 2 === {{:Definition:Internal Group Direct Product/General Definition/Definition 2}}"}
+{"_id": "22623", "title": "Definition:Binomial Coefficient/Real Numbers", "text": "Let $r \\in \\R, k \\in \\Z$. Then $\\dbinom r k$ is defined as: :$\\dbinom r k = \\begin{cases} \\dfrac {r^{\\underline k} } {k!} & : k \\ge 0 \\\\ & \\\\ 0 & : k < 0 \\end{cases}$ where $r^{\\underline k}$ denotes the falling factorial. That is, when $k \\ge 0$: :$\\displaystyle \\binom r k = \\frac {r \\paren {r - 1} \\cdots \\paren {r - k + 1} } {k \\paren {k - 1} \\cdots 1} = \\prod_{j \\mathop = 1}^k \\frac {r + 1 - j} j$ It can be seen that this agrees with the definition for integers when $r$ is an integer. For most applications the integer form is sufficient."}
+{"_id": "22624", "title": "Definition:Integer/Formal Definition", "text": "Let $\\struct {\\N, +}$ be the commutative semigroup of natural numbers under addition. From Inverse Completion of Natural Numbers, we can create $\\struct {\\N', +'}$, an inverse completion of $\\struct {\\N, +}$. From Construction of Inverse Completion, this is done as follows: Let $\\boxtimes$ be the cross-relation defined on $\\N \\times \\N$ by: :$\\tuple {x_1, y_1} \\boxtimes \\tuple {x_2, y_2} \\iff x_1 + y_2 = x_2 + y_1$ From Cross-Relation is Congruence Relation, $\\boxtimes$ is a congruence relation. Let $\\struct {\\N \\times \\N, \\oplus}$ be the external direct product of $\\struct {\\N, +}$ with itself, where $\\oplus$ is the operation on $\\N \\times \\N$ induced by $+$ on $\\N$: :$\\tuple {x_1, y_1} \\oplus \\tuple {x_2, y_2} = \\tuple {x_1 + x_2, y_1 + y_2}$ Let the quotient structure defined by $\\boxtimes$ be $\\struct {\\dfrac {\\N \\times \\N} \\boxtimes, \\oplus_\\boxtimes}$ where $\\oplus_\\boxtimes$ is the operation induced on $\\dfrac {\\N \\times \\N} \\boxtimes$ by $\\oplus$. Let us use $\\N'$ to denote the quotient set $\\dfrac {\\N \\times \\N} \\boxtimes$. Let us use $+'$ to denote the operation $\\oplus_\\boxtimes$. Thus $\\struct {\\N', +'}$ is the Inverse Completion of Natural Numbers. As the Inverse Completion is Unique up to isomorphism, it follows that we can ''define'' the structure $\\struct {\\Z, +}$ which is isomorphic to $\\struct {\\N', +'}$. An element of $\\N'$ is therefore an equivalence class of the congruence relation $\\boxtimes$. So an element of $\\Z$ is the isomorphic image of an element $\\eqclass {\\tuple {a, b} } \\boxtimes$ of $\\dfrac {\\N \\times \\N} \\boxtimes$. The set of elements $\\Z$ is called '''the integers'''. === Natural Number Difference === {{:Construction of Inverse Completion/Natural Number Difference}}"}
+{"_id": "22625", "title": "Definition:Totally Ordered Commutative Semigroup", "text": "A '''totally ordered commutative semigroup''' is a totally ordered structure $\\left({S, \\circ, \\preceq}\\right)$ such that $\\left({S, \\circ}\\right)$ is a commutative semigroup."}
+{"_id": "22626", "title": "Definition:Multiplication/Integers", "text": "The multiplication operation in the domain of integers $\\Z$ is written $\\times$. Let us define $\\eqclass {\\tuple {a, b} } \\boxtimes$ as in the formal definition of integers. That is, $\\eqclass {\\tuple {a, b} } \\boxtimes$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\\boxtimes$. $\\boxtimes$ is the congruence relation defined on $\\N \\times \\N$ by $\\tuple {x_1, y_1} \\boxtimes \\tuple {x_2, y_2} \\iff x_1 + y_2 = x_2 + y_1$. In order to streamline the notation, we will use $\\eqclass {a, b} {}$ to mean $\\eqclass {\\tuple {a, b} } \\boxtimes$, as suggested. As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of $\\Z$ are the isomorphic images of the elements of equivalence classes of $\\N \\times \\N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same. Thus multiplication can be formally defined on $\\Z$ as the operation induced on those equivalence classes as specified in the definition of integers. That is, the integers being defined as all the difference congruence classes, integer multiplication can be defined directly as the operation induced by natural number multiplication on these congruence classes. It follows that: :$\\forall a, b, c, d \\in \\N: \\eqclass {a, b} {} \\times \\eqclass {c, d} {} = \\eqclass {a \\times c + b \\times d, a \\times d + b \\times c} {}$ or, more compactly, as $\\eqclass {a c + b d, a d + b c} {}$. This can also be defined as: :$n \\times m = +^n m = \\underbrace {m + m + \\cdots + m}_{\\text{$n$ copies of $m$} }$ and the validity of this is proved in Index Laws for Monoids."}
+{"_id": "22627", "title": "Definition:Ring (Abstract Algebra)/Product", "text": "The distributive operation $\\circ$ in $\\struct {R, *, \\circ}$ is known as the '''(ring) product'''."}
+{"_id": "22628", "title": "Definition:Ring (Abstract Algebra)/Ring Less Zero", "text": "It is convenient to have a symbol for $R \\setminus \\set 0$, that is, the set of all elements of the ring without the zero. Thus we usually use: :$R_{\\ne 0} = R \\setminus \\set 0$"}
+{"_id": "22629", "title": "Definition:Non-Null Ring", "text": "A '''non-null ring''' is a ring with more than one element."}
+{"_id": "22630", "title": "Definition:Integral of Integrable Function", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f: X \\to \\overline \\R$, $f \\in \\map {\\LL^1} \\mu$ be a $\\mu$-integrable function. Then the '''$\\mu$-integral of $f$''' is defined by: :$\\displaystyle \\int f \\rd \\mu := \\int f^+ \\rd \\mu - \\int f^- \\rd \\mu$ where $f^+$ and $f^-$ are the positive and negative parts of $f$, respectively."}
+{"_id": "22631", "title": "Definition:Space of Integrable Functions", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Then the '''space of $\\mu$-integrable, real-valued functions''' $\\map {\\LL^1} \\mu$ is the collection of all $\\mu$-integrable real-valued functions: :$\\map {\\LL^1} \\mu := \\set {f: X \\to \\R: \\text {$f$ is $\\mu$-integrable} }$ Similarly, the '''space of $\\mu$-integrable, extended real-valued functions''' $\\map {\\LL^1_{\\overline \\R} } \\mu$ is the collection of all $\\mu$-integrable extended real-valued functions: :$\\map {\\LL^1_{\\overline \\R} } \\mu := \\set {f: X \\to \\overline \\R: \\text {$f$ is $\\mu$-integrable} }$"}
+{"_id": "22632", "title": "Definition:Integrable Function/Lebesgue", "text": "Let $\\lambda^n$ be a Lebesgue measure on $\\R^n$ for some $n > 0$. Let $f: \\R^n \\to \\overline \\R$ be an extended real-valued function. Then $f$ is said to be '''Lebesgue integrable''' {{iff}} it is $\\lambda^n$-integrable. Similarly, for all real numbers $p \\ge 1$, $f$ is said to be '''Lebesgue $p$-integrable''' {{iff}} it is $p$-integrable under $\\lambda^n$."}
+{"_id": "22633", "title": "Definition:Symmetric Matrix", "text": "Let $\\mathbf A$ be a square matrix over a set $S$. $\\mathbf A$ is '''symmetric''' {{iff}}: :$\\mathbf A = \\mathbf A^\\intercal$ where $\\mathbf A^\\intercal$ is the transpose of $\\mathbf A$."}
+{"_id": "22634", "title": "Definition:Vector Space Axioms", "text": "The '''vector space axioms''' consist of the abelian group axioms: {{begin-axiom}} {{axiom | n = \\text V 0         | lc= Closure Axiom         | q = \\forall \\mathbf x, \\mathbf y \\in G         | m = \\mathbf x +_G \\mathbf y \\in G }} {{axiom | n = \\text V 1         | lc= Commutativity Axiom         | q = \\forall \\mathbf x, \\mathbf y \\in G         | m = \\mathbf x +_G \\mathbf y = \\mathbf y +_G \\mathbf x }} {{axiom | n = \\text V 2         | lc= Associativity Axiom         | q = \\forall \\mathbf x, \\mathbf y, \\mathbf z \\in G         | m = \\paren {\\mathbf x +_G \\mathbf y} +_G \\mathbf z = \\mathbf x +_G \\paren {\\mathbf y +_G \\mathbf z} }} {{axiom | n = \\text V 3         | lc= Identity Axiom         | q = \\exists \\mathbf 0 \\in G: \\forall \\mathbf x \\in G         | m = \\mathbf 0 +_G \\mathbf x = \\mathbf x = \\mathbf x +_G \\mathbf 0 }} {{axiom | n = \\text V 4         | lc= Inverse Axiom         | q = \\forall \\mathbf x \\in G: \\exists \\paren {-\\mathbf x} \\in G         | m = \\mathbf x +_G \\paren {-\\mathbf x} = \\mathbf 0 }} {{end-axiom}} together with the properties of a unitary module: {{begin-axiom}} {{axiom | n = \\text V 5         | lc= Distributivity over Scalar Addition         | q = \\forall \\lambda, \\mu \\in K: \\forall \\mathbf x \\in G         | m = \\paren {\\lambda + \\mu} \\circ \\mathbf x = \\lambda \\circ \\mathbf x +_G \\mu \\circ \\mathbf x }} {{axiom | n = \\text V 6         | lc= Distributivity over Vector Addition         | q = \\forall \\lambda \\in K: \\forall \\mathbf x, \\mathbf y \\in G         | m = \\lambda \\circ \\paren {\\mathbf x +_G \\mathbf y} = \\lambda \\circ \\mathbf x +_G \\lambda \\circ \\mathbf y }} {{axiom | n = \\text V 7         | lc= Associativity with Scalar Multiplication         | q = \\forall \\lambda, \\mu \\in K: \\forall \\mathbf x \\in G         | m = \\lambda \\circ \\paren {\\mu \\circ \\mathbf x} = \\paren {\\lambda \\cdot \\mu} \\circ \\mathbf x }} {{axiom | n = \\text V 8         | lc= Identity for Scalar Multiplication         | q = \\forall \\mathbf x \\in G         | m = 1_K \\circ \\mathbf x = \\mathbf x }} {{end-axiom}}"}
+{"_id": "22635", "title": "Definition:Pairwise Disjoint/Set of Sets", "text": "A set of sets $\\Bbb S$ is said to be '''pairwise disjoint''' {{iff}}: :$\\forall X, Y \\in \\Bbb S: X \\ne Y \\implies X \\cap Y = \\O$ Here, $\\cap$ denotes intersection, and $\\O$ denotes the empty set. Hence we can say that the elements of $\\Bbb S$ are '''pairwise disjoint'''."}
+{"_id": "22636", "title": "Definition:Pairwise Disjoint/Family", "text": "An indexed family of sets $\\family {S_i}_{i \\mathop \\in I}$ is said to be '''pairwise disjoint''' {{iff}}: :$\\forall i, j \\in I: i \\ne j \\implies S_i \\cap S_j = \\O$ Hence the indexed sets $S_i$ themselves, where $i \\in I$, are referred to as being '''pairwise disjoint'''."}
+{"_id": "22638", "title": "Definition:Oscillation", "text": "Let $X$ be a set. Let $\\left({Y, d}\\right)$ be a metric space. Let $f: X \\to Y$ be a mapping. === Oscillation on a Set === {{:Definition:Oscillation/Oscillation on Set}} === Oscillation at a Point === {{:Definition:Oscillation at Point}}"}
+{"_id": "22639", "title": "Definition:Null Ideal", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. The null ring $\\struct {\\set {0_R}, +, \\circ}$ is called the '''null ideal''' of $R$."}
+{"_id": "22640", "title": "Definition:Non-Null Ideal", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Let $J$ be an ideal of $R$ such that $J \\ne \\set {0_R}$. Then $J$ is a '''non-null ideal''' of $R$."}
+{"_id": "22641", "title": "Definition:Only If", "text": "To say that: :''a statement $p$ is true '''only if''' $q$ is true'' is to say that: :''$p$ is a sufficient condition for $q$'': Thus: :$p \\ \\text{only if} \\ q \\dashv \\vdash p \\implies q$"}
+{"_id": "22642", "title": "Definition:If", "text": "To say that: :a statement $p$ is true '''if''' $q$ is true ...is to say that $p$ is a necessary condition for $q$: :$p \\ \\text{if} \\ q \\dashv \\vdash q \\implies p$"}
+{"_id": "22643", "title": "Definition:Internal Direct Sum of Rings/Direct Summand", "text": "In Conditions for Internal Ring Direct Sum it is proved that for this to be the case, then $S_1, S_2, \\ldots, S_n$ must be ideals of $R$. Such ideals are known as '''direct summands''' of $R$."}
+{"_id": "22644", "title": "Definition:Finite Integral Domain", "text": "A finite integral domain is an integral domain $\\left({D, +, \\circ}\\right)$ such that $D$ is a finite set. {{SUBPAGENAME}} k6e73sa570prcefg8p2zj9yaswiy4ir"}
+{"_id": "22645", "title": "Definition:Standard Ordered Basis", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n$ be a positive integer. For each $j \\in \\closedint 1 n$, let $e_j = \\tuple {0_R, 0_R, \\ldots, 1_R, \\cdots, 0_R}$ be the ordered $n$-tuple of elements of $R$ whose $j$th term is $1_R$ and all of whose other entries is $0_R$. Then the ordered $n$-tuple $\\sequence {e_k}_{1 \\mathop \\le k \\mathop \\le n} = \\tuple {e_1, e_2, \\ldots, e_n}$ is called the '''standard ordered basis (of the $R$-module $R^n$)'''. === Vector Space === The concept of a '''standard ordered basis''' is often found in the context of vector spaces. {{:Definition:Standard Ordered Basis/Vector Space}}"}
+{"_id": "22646", "title": "Definition:Standard Basis", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\\sequence {e_k}_{1 \\mathop \\le k \\mathop \\le n}$ be the standard ordered basis of the $R$-module $R^n$. The corresponding (unordered) set $\\set {e_1, e_2, \\ldots, e_n}$ is called the '''standard basis of $R^n$'''. === Vector Space === The concept of a '''standard basis''' is often found in the context of vector spaces. {{:Definition:Standard Basis/Vector Space}}"}
+{"_id": "22647", "title": "Definition:Unit Matrix", "text": "Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\struct {\\map {\\MM_R} n, +, \\times}$ be the ring of order $n$ square matrices over $R$. Then the '''unit matrix (of order $n$)''' of $\\struct {\\map {\\MM_R} n, +, \\times}$ is defined as: :$\\mathbf I_n := \\sqbrk a_n: a_{i j} = \\delta_{i j}$ where $\\delta_{i j}$ is the Kronecker delta for $\\map {\\MM_R} n$."}
+{"_id": "22648", "title": "Definition:Positive/Integer", "text": "Informally, the '''positive integers''' are the set: :$\\Z_{\\ge 0} = \\set {0, 1, 2, 3, \\ldots}$ As the set of integers $\\Z$ is the Inverse Completion of Natural Numbers, it follows that elements of $\\Z$ are the isomorphic images of the elements of equivalence classes of $\\N \\times \\N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same. Thus '''positive''' can be formally defined on $\\Z$ as a relation induced on those equivalence classes as specified in the definition of integers. That is, the integers being defined as all the difference congruence classes, '''positive''' can be defined directly as the relation specified as follows: :The integer $z \\in \\Z: z = \\eqclass {\\tuple {a, b} } \\boxminus$ is '''positive''' {{iff}} $b \\le a$. The set of '''positive integers''' is denoted $\\Z_{\\ge 0}$. An element of $\\Z$ can be specifically indicated as being '''positive''' by prepending a $+$ sign: :$+x := x \\in \\Z_{\\ge 0}$. === Ordering on Integers === {{:Definition:Ordering on Integers}}"}
+{"_id": "22649", "title": "Definition:Positive/Number", "text": "The concept of positive can be applied to the following sets of numbers: :$(1): \\quad$ The integers $\\Z$ :$(2): \\quad$ The rational numbers $\\Q$ :$(3): \\quad$ The real numbers $\\R$ The Complex Numbers cannot be Ordered Compatibly with Ring Structure, so there is no such concept as a positive complex number. As for the natural numbers, they are all positive by dint of their being the non-negative integers. === Integers === {{:Definition:Positive/Integer}} === Rational Numbers === {{:Definition:Positive/Rational Number}} === Real Numbers === {{:Definition:Positive/Real Number}}"}
+{"_id": "22650", "title": "Definition:Ordering on Integers", "text": "=== Definition 1 === {{:Definition:Ordering on Integers/Definition 1}} === Definition 2 === {{:Definition:Ordering on Integers/Definition 2}}"}
+{"_id": "22651", "title": "Definition:Ordered Integral Domain/Trichotomy Law", "text": "The property: :$\\forall a \\in D: \\map P a \\lor \\map P {-a} \\lor a = 0_D$ is known as the '''trichotomy law'''."}
+{"_id": "22652", "title": "Definition:Positive/Rational Number", "text": "The '''positive rational numbers''' are the set defined as: :$\\Q_{\\ge 0} := \\set {x \\in \\Q: x \\ge 0}$ That is, all the rational numbers that are greater than or equal to zero."}
+{"_id": "22653", "title": "Definition:Positive/Real Number", "text": "The '''positive real numbers''' are the set: :$\\R_{\\ge 0} = \\set {x \\in \\R: x \\ge 0}$ That is, all the real numbers that are greater than or equal to zero."}
+{"_id": "22654", "title": "Definition:Strictly Positive/Number", "text": "The concept of strictly positive can be applied to the following sets of numbers: :$(1): \\quad$ The integers $\\Z$ :$(2): \\quad$ The rational numbers $\\Q$ :$(3): \\quad$ The real numbers $\\R$ === Integers === {{:Definition:Strictly Positive/Integer}} === Rational Numbers === {{:Definition:Strictly Positive/Rational Number}} === Real Numbers === {{:Definition:Strictly Positive/Real Number/Definition 1}}"}
+{"_id": "22655", "title": "Definition:Strictly Positive/Integer", "text": "The '''strictly positive integers''' are the set defined as: :$\\Z_{> 0} := \\set {x \\in \\Z: x > 0}$ That is, all the integers that are strictly greater than zero: :$\\Z_{> 0} := \\set {1, 2, 3, \\ldots}$"}
+{"_id": "22656", "title": "Definition:Strictly Positive/Rational Number", "text": "The '''strictly positive rational numbers''' are the set defined as: :$\\Q_{>0} := \\set {x \\in \\Q: x > 0}$ That is, all the rational numbers that are strictly greater than zero."}
+{"_id": "22657", "title": "Definition:Strictly Positive/Real Number", "text": "=== Definition 1 === {{:Definition:Strictly Positive/Real Number/Definition 1}} === Definition 2 === {{:Definition:Strictly Positive/Real Number/Definition 2}}"}
+{"_id": "22658", "title": "Definition:Negative/Number", "text": "The concept of negative can be applied to the following sets of numbers: :$(1): \\quad$ The integers $\\Z$ :$(2): \\quad$ The rational numbers $\\Q$ :$(3): \\quad$ The real numbers $\\R$ The Complex Numbers cannot be Ordered Compatibly with Ring Structure, so there is no such concept as a negative complex number. === Integers === {{:Definition:Negative/Integer}} === Rational Numbers === {{:Definition:Negative/Rational Number}} === Real Numbers === {{:Definition:Negative/Real Number}} === Complex Numbers === {{:Definition:Negative/Complex Number}}"}
+{"_id": "22659", "title": "Definition:Negative/Rational Number", "text": "The '''negative rational numbers''' are the set defined as: :$\\Q_{\\le 0} := \\set {x \\in \\Q: x \\le 0}$ That is, all the rational numbers that are less than or equal to zero."}
+{"_id": "22660", "title": "Definition:Negative/Real Number", "text": "The '''negative real numbers''' are the set defined as: :$\\R_{\\le 0} := \\set {x \\in \\R: x \\le 0}$ That is, all the real numbers that are less than or equal to zero."}
+{"_id": "22661", "title": "Definition:Negative/Integer", "text": "The '''negative integers''' comprise the set: :$\\set {0, -1, -2, -3, \\ldots}$ As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of $\\Z$ are the isomorphic images of the elements of equivalence classes of $\\N \\times \\N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same. Thus '''negative''' can be formally defined on $\\Z$ as a relation induced on those equivalence classes as specified in the definition of integers. That is, the integers being defined as all the difference congruence classes, '''negative''' can be defined directly as the relation specified as follows: The integer $z \\in \\Z: z = \\eqclass {\\tuple {a, b} } \\boxminus$ is '''negative''' {{iff}} $b > a$. The set of '''negative integers''' is denoted $\\Z_{\\le 0}$. An element of $\\Z$ can be specifically indicated as being '''negative''' by prepending a $-$ sign: :$-x \\in \\Z_{\\le 0} \\iff x \\in \\Z_{\\ge 0}$"}
+{"_id": "22662", "title": "Definition:Strictly Negative/Number", "text": "The concept of strictly negative can be applied to the following sets of numbers: :$(1): \\quad$ The integers $\\Z$ :$(2): \\quad$ The rational numbers $\\Q$ :$(3): \\quad$ The real numbers $\\R$ === Integers === {{:Definition:Strictly Negative/Integer}} === Rational Numbers === {{:Definition:Strictly Negative/Rational Number}} === Real Numbers === {{:Definition:Strictly Negative/Real Number}}"}
+{"_id": "22663", "title": "Definition:Strictly Negative/Integer", "text": "The '''strictly negative integers''' are the set defined as: {{begin-eqn}} {{eqn | l = \\Z_{< 0}       | o = :=       | r = \\set {x \\in \\Z: x < 0} }} {{eqn | r = \\set {-1, -2, -3, \\ldots} }} {{end-eqn}} That is, all the integers that are strictly less than zero."}
+{"_id": "22664", "title": "Definition:Strictly Negative/Rational Number", "text": "The '''strictly negative rational numbers''' are the set defined as: :$\\Q_{< 0} := \\set {x \\in \\Q: x < 0}$ That is, all the rational numbers that are strictly less than zero."}
+{"_id": "22665", "title": "Definition:Strictly Negative/Real Number", "text": "The '''strictly negative real numbers''' are the set defined as: :$\\R_{<0} := \\set {x \\in \\R: x < 0}$ That is, all the real numbers that are strictly less than zero."}
+{"_id": "22666", "title": "Definition:Ordering Induced by Positivity Property", "text": "Let $\\struct {R, +, \\circ, \\le}$ be an ordered ring whose zero is $0_R$ and whose unity is $1_R$. Let $P \\subseteq R$ such that: :$(1): \\quad P + P \\subseteq P$ :$(2): \\quad P \\cap \\paren {-P} = \\set {0_R}$ :$(3): \\quad P \\circ P \\subseteq P$ Then the ordering $\\le$ compatible with the ring structure of $R$ is called the '''ordering induced by (the positivity property) $P$'''."}
+{"_id": "22667", "title": "Definition:Rational Number/Formal Definition", "text": "The field $\\struct {\\Q, +, \\times}$ of rational numbers is the field of quotients of the integral domain $\\struct {\\Z, +, \\times}$ of integers. This is shown to exist in Existence of Field of Quotients. In view of Field of Quotients is Unique, we construct the field of quotients of $\\Z$, give it a label $\\Q$ and call its elements '''rational numbers'''."}
+{"_id": "22668", "title": "Definition:Ordered Ring Isomorphism", "text": "Let $\\struct {S, +, \\circ, \\preceq}$ and $\\struct {T, \\oplus, *, \\preccurlyeq}$ be ordered rings. An '''ordered ring isomorphism''' from $\\struct {S, +, \\circ, \\preceq}$ to $\\struct {T, \\oplus, *, \\preccurlyeq}$ is a mapping $\\phi: S \\to T$ that is both: :$(1): \\quad$ An ordered group isomorphism from the ordered group $\\struct {S, +, \\preceq}$ to the ordered group $\\struct {T, \\oplus, \\preccurlyeq}$ :$(2): \\quad$ A semigroup isomorphism from the semigroup $\\struct {S, \\circ}$ to the semigroup $\\struct {T, *}$."}
+{"_id": "22669", "title": "Definition:Ordered Group Isomorphism", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ and $\\left({T, *, \\preccurlyeq}\\right)$ be ordered groups. An '''ordered group isomorphism''' from $\\left({S, \\circ, \\preceq}\\right)$ to $\\left({T, *, \\preccurlyeq}\\right)$ is a mapping $\\phi: S \\to T$ that is both: :$(1): \\quad$ A group isomorphism from the group $\\left({S, \\circ}\\right)$ to the group $\\left({T, *}\\right)$ :$(2): \\quad$ An order isomorphism from the ordered set $\\left({S, \\preceq}\\right)$ to the ordered set $\\left({T, \\preccurlyeq}\\right)$."}
+{"_id": "22670", "title": "Definition:Ordered Group Monomorphism", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ and $\\left({T, *, \\preccurlyeq}\\right)$ be ordered groups. An '''ordered group monomorphism''' from $\\left({S, \\circ, \\preceq}\\right)$ to $\\left({T, *, \\preccurlyeq}\\right)$ is a mapping $\\phi: S \\to T$ that is both: :$(1): \\quad$ A group monomorphism from the group $\\left({S, \\circ}\\right)$ to the group $\\left({T, *}\\right)$ :$(2): \\quad$ An order embedding from the ordered set $\\left({S, \\preceq}\\right)$ to the ordered set $\\left({T, \\preccurlyeq}\\right)$."}
+{"_id": "22671", "title": "Definition:Ordered Ring Monomorphism", "text": "Let $\\left({S, +, \\circ, \\preceq}\\right)$ and $\\left({T, \\oplus, *, \\preccurlyeq}\\right)$ be ordered rings. An '''ordered ring monomorphism''' from $\\left({S, +, \\circ, \\preceq}\\right)$ to $\\left({T, \\oplus, *, \\preccurlyeq}\\right)$ is a mapping $\\phi: S \\to T$ that is both: :$(1): \\quad$ An ordered group monomorphism from the ordered group $\\left({S,+, \\preceq}\\right)$ to the ordered group $\\left({T, \\oplus, \\preccurlyeq}\\right)$ :$(2): \\quad$ A semigroup monomorphism from the semigroup $\\left({S, \\circ}\\right)$ to the semigroup $\\left({T, *}\\right)$."}
+{"_id": "22673", "title": "Definition:Ordered Structure Automorphism", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be an ordered structures. Let $\\phi: S \\to S$ be an ordered structure isomorphism from $S$ to itself. Then $\\phi$ is an ordered structure automorphism."}
+{"_id": "22674", "title": "Definition:Ordered Group Automorphism", "text": "Let $\\left({G, \\circ, \\preceq}\\right)$ be an ordered group. An '''ordered group automorphism''' from $\\left({G, \\circ, \\preceq}\\right)$ to itself is a mapping $\\phi: G \\to G$ that is both: :$(1): \\quad$ A group automorphism, that is, a group isomorphism from the group $\\left({G, \\circ}\\right)$ to itself :$(2): \\quad$ An order isomorphism from the ordered set $\\left({G, \\preceq}\\right)$ to itself."}
+{"_id": "22675", "title": "Definition:Ordered Ring Automorphism", "text": "Let $\\struct {R, +, \\circ, \\preceq}$ be an ordered ring. An '''ordered ring automorphism''' from $\\struct {R, +, \\circ, \\preceq}$ to itself is a mapping $\\phi: R \\to R$ that is both: :$(1): \\quad$ An ordered group automorphism from the ordered group $\\struct {R, +, \\preceq}$ to itself :$(2): \\quad$ A semigroup automorphism from the semigroup $\\struct {R, \\circ}$ to itself."}
+{"_id": "22676", "title": "Definition:Abelian Group Axioms", "text": "{{begin-axiom}} {{axiom | n = \\text G 0         | lc= Closure         | q = \\forall x, y \\in G         | m = x + y \\in G }} {{axiom | n = \\text G 1         | lc= Associativity         | q = \\forall x, y, z \\in G         | m = x + \\paren {y + z} = \\paren {x + y} + z }} {{axiom | n = \\text G 2         | lc= Identity         | q = \\exists 0 \\in G: \\forall x \\in G         | m = 0 + x = x = x + 0 }} {{axiom | n = \\text G 3         | lc= Inverse         | q = \\forall x \\in G: \\exists \\paren {-x}\\in G         | m = x + \\paren {-x} = 0 = \\paren {-x} + x }} {{axiom | n = \\text C         | lc= Commutativity         | q = \\forall x, y \\in G         | m = x + y = y + x }} {{end-axiom}}"}
+{"_id": "22677", "title": "Definition:Scalar/R-Algebraic Structure", "text": "The elements of the scalar ring $\\struct {R, +_R, \\times_R}$ are called '''scalars'''."}
+{"_id": "22678", "title": "Definition:Scalar Multiplication/R-Algebraic Structure", "text": "Let $\\struct {S, *_1, *_2, \\ldots, *_n, \\circ}_R$ be an $R$-algebraic structure with $n$ operations, where: :$\\struct {R, +_R, \\times_R}$ is a ring :$\\struct {S, *_1, *_2, \\ldots, *_n}$ is an algebraic structure with $n$ operations The operation $\\circ: R \\times S \\to S$ is called '''scalar multiplication'''."}
+{"_id": "22679", "title": "Definition:Scalar Ring/Zero Scalar", "text": "The zero of the scalar ring is called the '''zero scalar''' and usually denoted $0$, or, if it is necessary to distinguish it from the identity of $\\struct {G, +_G}$, by $0_R$."}
+{"_id": "22680", "title": "Definition:Vector Subspace/Proper Subspace", "text": "If $T$ is a proper subset of $S$, then $\\struct {T, +_T, \\circ_T}_K$ is a '''proper (vector) subspace''' of $\\struct {S, +, \\circ}_K$."}
+{"_id": "22681", "title": "Definition:Vector Subspace/Hilbert Spaces", "text": "When considering Hilbert spaces, one wants to deal with projections onto subspaces. These projections however require the linear subspace to be closed in topological sense in order to be well-defined. Therefore, in treatises of Hilbert spaces, one encounters the terminology '''linear manifold''' for the concept of '''vector subspace''' defined above. The adapted definition of '''linear subspace''' is then that it is a topologically closed '''linear manifold'''."}
+{"_id": "22682", "title": "Definition:Polynomial over Ring as Function on Free Monoid on Set", "text": "Let $M$ be the free commutative monoid on the indexed set $\\left\\{{X_j: j \\in J}\\right\\}$. A '''polynomial''' in $\\left\\{{X_j: j \\in J}\\right\\}$ can be defined as a mapping $f: M \\to R$ of finite support. That is, it is an element of the ring of polynomial forms."}
+{"_id": "22684", "title": "Definition:Polynomial Equation", "text": "A '''polynomial equation''' is an equation in the form: :$\\map f x = 0$ where $f$ is a polynomial function."}
+{"_id": "22685", "title": "Definition:Strongly Additive Function", "text": "Let $\\mathcal S$ be an algebra of sets. Let $f: \\mathcal S \\to \\overline{\\R}$ be a function, where $\\overline{\\R}$ denotes the extended set of real numbers. Then $f$ is defined to be '''strongly additive''' {{iff}}: :$\\forall S, T \\in \\mathcal S: f \\left({S \\cup T}\\right) + f \\left({S \\cap T}\\right) = f \\left({S}\\right) + f \\left({T}\\right)$"}
+{"_id": "22686", "title": "Definition:Polynomial Function/General Definition", "text": "Let $f = a_1 \\mathbf X^{k_1} + \\cdots + a_r \\mathbf X^{k_r}$ be a polynomial form over $R$ in the indeterminates $\\left\\{{X_j: j \\in J}\\right\\}$. For each $x = \\left({x_j}\\right)_{j \\in J} \\in R^J$, let $\\phi_x: R \\left[{\\left\\{{X_j: j \\in J}\\right\\}}\\right] \\to R$ be the evaluation homomorphism from the ring of polynomial forms at $x$. Then the set: :$\\left\\{{\\left({x, \\phi_x \\left({f}\\right)}\\right): x \\in R^J}\\right\\} \\subseteq R^J \\times R$ defines a '''polynomial function''' $R^J \\to R$."}
+{"_id": "22688", "title": "Definition:Linear Combination/Sequence", "text": "Let $G$ be an $R$-module. Let $\\left \\langle {a_n} \\right \\rangle := \\left \\langle {a_j} \\right \\rangle_{1 \\mathop \\le j \\mathop \\le n}$ be a sequence of elements of $G$ of length $n$. An element $b \\in G$ is a '''linear combination''' of $\\left \\langle {a_n} \\right \\rangle$ {{iff}}: : $\\displaystyle \\exists \\left \\langle {\\lambda_n} \\right \\rangle \\subseteq R: b = \\sum_{k \\mathop = 1}^n \\lambda_k a_k$"}
+{"_id": "22689", "title": "Definition:Linear Combination/Subset", "text": "Let $G$ be an $R$-module. Let $\\O \\subset S \\subseteq G$. Let $b \\in G$ be a linear combination of some sequence $\\sequence {a_n}$ of elements of $S$. Then $b$ is a '''linear combination of $S$'''."}
+{"_id": "22690", "title": "Definition:Linear Combination/Empty Set", "text": "Let $G$ be an $R$-module. $b$ is a linear combination of $\\varnothing$ {{iff}}: : $b = e_G$"}
+{"_id": "22691", "title": "Definition:Linearly Independent/Sequence", "text": "Let $\\left \\langle {a_n} \\right \\rangle$ be a sequence of elements of $G$ such that: : $\\displaystyle \\forall \\left \\langle {\\lambda_n} \\right \\rangle \\subseteq R: \\sum_{k \\mathop = 1}^n \\lambda_k \\circ a_k = e \\implies \\lambda_1 = \\lambda_2 = \\cdots = \\lambda_n = 0_R$ That is, the only way to make $e$ with a linear combination of $\\left \\langle {a_n} \\right \\rangle$ is by making all the elements of $\\left \\langle {\\lambda_n} \\right \\rangle$ equal to $0_R$. Such a sequence is '''linearly independent'''. === Linearly Independent Sequence on a Real Vector Space === {{:Definition:Linearly Independent/Sequence/Real Vector Space}}"}
+{"_id": "22692", "title": "Definition:Linearly Independent/Set", "text": "Let $S \\subseteq G$. Then $S$ is a '''linearly independent set (over $R$)''' {{iff}} every finite sequence of distinct terms in $S$ is a linearly independent sequence. That is, such that: : $\\displaystyle \\forall \\sequence {\\lambda_n} \\subseteq R: \\sum_{k \\mathop = 1}^n \\lambda_k \\circ a_k = e \\implies \\lambda_1 = \\lambda_2 = \\cdots = \\lambda_n = 0_R$ where $a_1, a_2, \\ldots, a_k$ are distinct elements of $S$. === Linearly Independent Set on a Real Vector Space === {{:Definition:Linearly Independent/Set/Real Vector Space}} === Linearly Independent Set on a Complex Vector Space === {{:Definition:Linearly Independent/Set/Complex Vector Space}}"}
+{"_id": "22693", "title": "Definition:Linearly Independent/Sequence/Real Vector Space", "text": "Let $\\struct {\\R^n, +, \\cdot}_\\R$ be a real vector space. Let $\\sequence {\\mathbf v_n}$ be a sequence of vectors in $\\R^n$. Then $\\sequence {\\mathbf v_n}$ is '''linearly independent''' {{iff}}: :$\\ds \\forall \\sequence {\\lambda_n} \\subseteq \\R: \\sum_{k \\mathop = 1}^n \\lambda_k \\mathbf v_k = \\mathbf 0 \\implies \\lambda_1 = \\lambda_2 = \\cdots = \\lambda_n = 0$ where $\\mathbf 0 \\in \\R^n$ is the zero vector and $0 \\in \\R$ is the zero scalar."}
+{"_id": "22694", "title": "Definition:Linearly Independent/Set/Real Vector Space", "text": "Let $\\struct {\\R^n, +, \\cdot}_\\R$ be a real vector space. Let $S \\subseteq \\R^n$. Then $S$ is a '''linearly independent set of real vectors''' if every finite sequence of distinct terms in $S$ is a linearly independent sequence. That is, such that: :$\\displaystyle \\forall \\set {\\lambda_k: 1 \\le k \\le n} \\subseteq \\R: \\sum_{k \\mathop = 1}^n \\lambda_k \\mathbf v_k = \\mathbf 0 \\implies \\lambda_1 = \\lambda_2 = \\cdots = \\lambda_n = 0$ where $\\mathbf v_1, \\mathbf v_2, \\ldots, \\mathbf v_n$ are distinct elements of $S$."}
+{"_id": "22695", "title": "Definition:Linearly Dependent/Sequence", "text": "Let $\\left \\langle {a_k} \\right \\rangle_{1 \\mathop \\le k \\mathop \\le n}$ be a sequence of elements of $G$ such that: : $\\displaystyle \\exists \\left \\langle {\\lambda_k} \\right \\rangle_{1 \\mathop \\le k \\mathop \\le n} \\subseteq R: \\sum_{k \\mathop = 1}^n \\lambda_k \\circ a_k = e$ where not all of $\\lambda_k$ are equal to $0_R$. That is, it is possible to find a linear combination of $\\left \\langle {a_k} \\right \\rangle_{1 \\mathop \\le k \\mathop \\le n}$ which equals $e$. Such a sequence is '''linearly dependent'''. === Linearly Dependent Sequence on a Real Vector Space === {{:Definition:Linearly Dependent/Sequence/Real Vector Space}}"}
+{"_id": "22696", "title": "Definition:Linearly Dependent/Sequence/Real Vector Space", "text": "Let $\\left({\\R^n,+,\\cdot}\\right)_{\\R}$ be a real vector space. Let $\\mathbf 0 \\in \\R^n$ be the zero vector. Let $\\left \\langle {\\mathbf v_k} \\right \\rangle_{1 \\mathop \\le k \\mathop \\le n}$ be a sequence of vectors in $\\R^n$. Then $\\left \\langle {\\mathbf v_k} \\right \\rangle_{1 \\mathop \\le k \\mathop \\le n}$ is '''linearly dependent''' iff: : $\\displaystyle \\exists \\left \\langle {\\lambda_k} \\right \\rangle_{1 \\mathop \\le k \\mathop \\le n} \\subseteq \\R: \\sum_{k \\mathop = 1}^n \\lambda_k \\mathbf v_k = \\mathbf 0$ where not all $\\lambda_k$ are equal to $0$. That is, it is possible to find a linear combination of $\\left \\langle {\\mathbf v_k} \\right \\rangle_{1 \\mathop \\le k \\mathop \\le n}$ which equals $\\mathbf 0$."}
+{"_id": "22697", "title": "Definition:Linearly Dependent/Set", "text": "Let $S \\subseteq G$. Then $S$ is a '''linearly dependent set''' if there exists a sequence of distinct terms in $S$ which is a linearly dependent sequence. That is, such that: :$\\displaystyle \\exists \\set {\\lambda_k: 1 \\le k \\le n} \\subseteq R: \\sum_{k \\mathop = 1}^n \\lambda_k \\circ a_k = e$ where $a_1, a_2, \\ldots, a_n$ are distinct elements of $S$, and where at least one of $\\lambda_k$ is not equal to $0_R$."}
+{"_id": "22698", "title": "Definition:Linearly Dependent/Set/Real Vector Space", "text": "Let $\\struct {\\R^n, +, \\cdot}_\\R$ be a real vector space. Let $S \\subseteq \\R^n$. Then $S$ is a '''linearly dependent set''' if there exists a sequence of distinct terms in $S$ which is a linearly dependent sequence. That is, such that: :$\\displaystyle \\exists \\set {\\lambda_k: 1 \\le k \\le n} \\subseteq \\R: \\sum_{k \\mathop = 1}^n \\lambda_k \\mathbf v_k = \\mathbf 0$ where $\\set {\\mathbf v_1, \\mathbf v_2, \\ldots, \\mathbf v_n} \\subseteq S$, and such that at least one of $\\lambda_k$ is not equal to $0$."}
+{"_id": "22699", "title": "Definition:Atom of Measure", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. An element $x \\in X$ is said to be an '''atom (of $\\mu$)''' {{iff}}: :$(1): \\quad \\set x \\in \\Sigma$ :$(2): \\quad \\map \\mu {\\set x} > 0$"}
+{"_id": "22700", "title": "Definition:Standard Ordered Basis/Vector Space", "text": "Let $\\struct {\\mathbf V, +, \\circ}_{\\mathbb F}$ be a vector space over a field $\\mathbb F$, as defined by the vector space axioms. Let the unity of $\\mathbb F$ be denoted $1_{\\mathbb F}$, and its zero $0_{\\mathbb F}$. Let $\\mathbf e_i$ be a vector whose $i$th term is $1_{\\mathbb F}$ and with entries $0_{\\mathbb F}$ elsewhere. Then the ordered $n$-tuple $\\tuple {\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}$ is the '''standard ordered basis of $\\mathbf V$'''."}
+{"_id": "22701", "title": "Definition:Standard Basis/Vector Space", "text": "Let $\\left({\\mathbf V, +, \\circ}\\right)_{\\mathbb F}$ be a vector space over $\\mathbb F$. Let $\\left({\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}\\right)$ be the standard ordered basis on $\\mathbf V$. The corresponding (unordered) set $\\left\\{{\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}\\right\\}$ is called the '''standard basis of $\\mathbf V$'''"}
+{"_id": "22702", "title": "Definition:Coordinate System", "text": "Let $R$ be a ring with unity. Let $\\sequence {a_k}_{1 \\mathop \\le k \\mathop \\le n}$ be an ordered basis of a free $R$-module $G$. Then $\\sequence {a_k}_{1 \\mathop \\le k \\mathop \\le n}$ can be referred to as a '''coordinate system'''."}
+{"_id": "22703", "title": "Definition:Dimension of Module", "text": "Let $R$ be a ring with unity. Let $G$ be a free $R$-module which has a basis of $n$ elements. Then $G$ is said to have a '''dimension of $n$''' or to be '''$n$-dimensional'''. The dimension of a free $R$-module $G$ is denoted $\\map {\\dim_R} G$, or just $\\map \\dim G$."}
+{"_id": "22704", "title": "Definition:Dimension of Vector Space", "text": "Let $K$ be a division ring. Let $V$ be a vector space over $K$. === Definition 1 === {{:Definition:Dimension of Vector Space/Definition 1}} === Definition 2 === {{:Definition:Dimension of Vector Space/Definition 2}}"}
+{"_id": "22705", "title": "Definition:Dimension of Module/Finite", "text": "Let $G$ be a (unitary) module which is $n$-dimensional for some $n \\in \\N_{>0}$. Then $G$ is '''finite dimensional'''."}
+{"_id": "22706", "title": "Definition:Dimension of Vector Space/Finite", "text": "Let $V$ be a vector space which is $n$-dimensional for some $n \\in \\N_{>0}$. Then $V$ is '''finite dimensional'''. The '''dimension''' of a '''finite-dimensional $K$-vector space''' $V$ is denoted $\\map {\\dim_K} V$, or just $\\map \\dim V$."}
+{"_id": "22707", "title": "Definition:Homomorphism (Abstract Algebra)/Image", "text": "As a homomorphism is a mapping, the '''homomorphic image''' of $\\phi$ is defined in the same way as the image of a mapping: :$\\Img \\phi = \\set {t \\in T: \\exists s \\in S: t = \\map \\phi s}$"}
+{"_id": "22708", "title": "Definition:Linear Operator/Vector Space", "text": "A '''linear operator''' on a vector space is a linear transformation from a vector space into itself."}
+{"_id": "22709", "title": "Definition:Linear Transformation/Vector Space", "text": "Let $V, W$ be vector spaces over a field (or, more generally, division ring) $K$. A mapping $A: V \\to W$ is a '''linear transformation''' {{iff}}: :$\\forall v_1, v_2 \\in V, \\lambda \\in K: \\map A {\\lambda v_1 + v_2} = \\lambda \\map A {v_1} + \\map A {v_2}$ That is, a homomorphism from one vector space to another."}
+{"_id": "22710", "title": "Definition:Kernel of Linear Transformation/Vector Space", "text": "Let $\\struct {\\mathbf V, +, \\times}$ be a vector space. Let $\\struct {\\mathbf V', +, \\times}$ be a vector space whose zero vector is $\\mathbf 0'$. Let $T: \\mathbf V \\to \\mathbf V'$ be a linear transformation. Then the '''kernel''' of $T$ is defined as: :$\\map \\ker T := T^{-1} \\sqbrk {\\set {\\mathbf 0'} } = \\set {\\mathbf x \\in \\mathbf V: \\map T {\\mathbf x} = \\mathbf 0'}$"}
+{"_id": "22711", "title": "Definition:Rank/Linear Transformation", "text": "Let $\\phi$ be a linear transformation from one vector space to another. Let the image of $\\phi$ be finite-dimensional. Then its dimension is called the '''rank of $\\phi$''' and is denoted $\\map \\rho \\phi$."}
+{"_id": "22712", "title": "Definition:Rank/Matrix", "text": "=== Definition 1 === {{:Definition:Rank/Matrix/Definition 1}} === Definition 2 === {{:Definition:Rank/Matrix/Definition 2}} === Definition 3 === {{:Definition:Rank/Matrix/Definition 3}}"}
+{"_id": "22713", "title": "Definition:Nullity/Linear Transformation", "text": "Let $K$ be a division ring. Let $V$ and $W$ be $K$-vector spaces. Let $\\phi: V \\to W$ be a linear transformation. Let the kernel $\\ker \\phi$ be finite dimensional. Then the '''nullity of $\\phi$''' is the dimension of $\\ker \\phi$ and is denoted $\\map \\nu \\phi$."}
+{"_id": "22714", "title": "Definition:Nullity/Matrix", "text": "Let $\\mathbf A$ be a matrix. Then the '''nullity of $\\mathbf A$''' is defined to be the dimension of the null space of $\\mathbf A$."}
+{"_id": "22715", "title": "Definition:Vector Notation", "text": "Several conventions are found in the literature for annotating a general vector quantity in a style that distinguishes it from a scalar quantity, as follows. Let $\\set {x_1, x_2, \\ldots, x_n}$ be a collection of scalars which form the components of an $n$-dimensional vector. The vector $\\tuple {x_1, x_2, \\ldots, x_n}$ can be annotated as: {{begin-eqn}} {{eqn | l = \\bsx       | r = \\tuple {x_1, x_2, \\ldots, x_n} }} {{eqn | l = \\vec x       | r = \\tuple {x_1, x_2, \\ldots, x_n} }} {{eqn | l = \\hat x       | r = \\tuple {x_1, x_2, \\ldots, x_n} }} {{eqn | l = \\underline x       | r = \\tuple {x_1, x_2, \\ldots, x_n} }} {{eqn | l = \\tilde x       | r = \\tuple {x_1, x_2, \\ldots, x_n} }} {{end-eqn}} To emphasize the arrow interpretation of a vector, we can write: :$\\bsv = \\sqbrk {x_1, x_2, \\ldots, x_n}$ or: :$\\bsv = \\sequence {x_1, x_2, \\ldots, x_n}$ In printed material the '''boldface''' $\\bsx$ or $\\mathbf x$ is common. This is the style encouraged and endorsed by {{ProofWiki}}. However, for handwritten material (where boldface is difficult to render) it is usual to use the '''underline''' version $\\underline x$.  Also found in handwritten work are the '''tilde''' version $\\tilde x$ and '''arrow''' version $\\vec x$, but as these are more intricate than the simple underline (and therefore more time-consuming and tedious to write), they will only usually be found in fair copy. It is also noted that the '''tilde''' over $\\tilde x$ does not render well in {{MathJax}} under all browsers, and differs little visually from an '''overline''': $\\overline x$. The '''hat''' version $\\hat x$ usually has a more specialized meaning, namely to symbolize a unit vector. In computer-rendered materials, the '''arrow''' version $\\vec x$ is popular, as it is descriptive and relatively unambiguous, and in $\\LaTeX$ it is straightforward. However, it does not render well in all browsers, and is therefore (reluctantly) not recommended for use on this website."}
+{"_id": "22716", "title": "Definition:Annihilator on Algebraic Dual", "text": "Let $R$ be a commutative ring. Let $G$ be a module over $R$. Let $G^*$ be the algebraic dual of $G$. Let $M$ be a submodule of $G$. The '''annihilator of $M$''', denoted $M^\\circ$, is defined as: :$\\ M^\\circ := \\set {t' \\in G^*: \\forall x \\in M: \\map {t'} x = 0}$"}
+{"_id": "22717", "title": "Definition:Matrix/Order", "text": "Let $\\sqbrk a_{m n}$ be an $m \\times n$ matrix. Then the parameters $m$ and $n$ are known as the '''order''' of the matrix."}
+{"_id": "22718", "title": "Definition:Matrix/Row", "text": "Let $\\mathbf A$ be an $m \\times n$ matrix. For each $i \\in \\closedint 1 m$, the '''rows''' of $\\mathbf A$ are the ordered $n$-tuples: :$r_i = \\tuple {a_{i 1}, a_{i 2}, \\ldots, a_{i n} }$ where $r_i$ is called the '''$i$th row of $\\mathbf A$'''. A '''row''' of an $m \\times n$ matrix can also be treated as a $1 \\times n$ row matrix in its own right: :$r_i = \\begin {bmatrix} a_{i 1} & a_{i 2} & \\cdots & a_{i n} \\end {bmatrix}$ for $i = 1, 2, \\ldots, m$."}
+{"_id": "22719", "title": "Definition:Matrix/Column", "text": "Let $\\mathbf A$ be an $m \\times n$ matrix. For each $j \\in \\closedint 1 n$, the '''columns''' of $\\mathbf A$ are the ordered $m$-tuples: : $c_j = \\tuple {a_{1 j}, a_{2 j}, \\ldots, a_{m j} }$ where $c_j$ is called the '''$j$th column of $\\mathbf A$'''. A '''column''' of an $m \\times n$ matrix can also be treated as a $m \\times 1$ column matrix in its own right: :$c_j = \\begin {bmatrix} a_{1 j} \\\\ a_{2 j} \\\\ \\vdots \\\\ a_{m j} \\end {bmatrix}$ for $j = 1, 2, \\ldots, n$."}
+{"_id": "22720", "title": "Definition:Matrix/Element", "text": "Let $\\mathbf A$ be an $m \\times n$ matrix over a set $S$. The individual $m \\times n$ elements of $S$ that go to form $\\mathbf A = \\sqbrk a_{m n}$ are known as the '''elements of the matrix'''. The '''element''' at row $i$ and column $j$ is called '''element $\\tuple {i, j}$ of $\\mathbf A$''', and can be written $a_{i j}$, or $a_{i, j}$ if $i$ and $j$ are of more than one character. If the indices are still more complicated coefficients and further clarity is required, then the form $a \\tuple {i, j}$ can be used. Note that the first subscript determines the row, and the second the column, of the matrix where the '''element''' is positioned."}
+{"_id": "22721", "title": "Definition:Matrix/Square Matrix", "text": "An $n \\times n$ matrix is called a '''square matrix'''. That is, a '''square matrix''' is a matrix which has the same number of rows as it has columns. A '''square matrix''' is usually denoted $\\sqbrk a_n$ in preference to $\\sqbrk a_{n n}$. In contrast, a non-'''square matrix''' can be referred to as a '''rectangular matrix'''."}
+{"_id": "22722", "title": "Definition:Matrix/Diagonal Elements", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be a matrix. The elements $a_{j j}: j \\in \\closedint 1 {\\min \\set {m, n} }$ constitute the '''main diagonal''' of the matrix. The elements themselves are called the '''diagonal elements'''."}
+{"_id": "22723", "title": "Definition:Matrix/Underlying Structure", "text": "Let $\\mathbf A$ be a matrix over a set $S$. The set $S$ can be referred to as the '''underlying set of $\\mathbf A$'''. In the context of matrices, however, it is usual for $S$ itself to be the underlying set of an algebraic structure in its own right. If this is the case, then the structure $\\struct {S, \\circ_1, \\circ_2, \\ldots, \\circ_n}$ (which may also be an ordered structure) can be referred to as the '''underlying structure of $\\mathbf A$'''. When the '''underlying structure''' is not specified, it is taken for granted that it is one of the standard number systems, usually the real numbers $\\R$."}
+{"_id": "22724", "title": "Definition:Matrix/Zero Row or Column", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be an $m \\times n$ matrix whose underlying structure is a ring or field (usually numbers). If a row or column of $\\mathbf A$ contains only zeroes, then it is a '''zero row''' or a '''zero column'''."}
+{"_id": "22725", "title": "Definition:Matrix/Indices", "text": "Let $\\mathbf A$ be an $m \\times n$ matrix. Let $a_{i j}$ be the element in row $i$ and column $j$ of $\\mathbf A$. Then the subscripts $i$ and $j$ are referred to as the '''indices''' (singular: '''index''') of $a_{i j}$."}
+{"_id": "22726", "title": "Definition:Module Automorphism", "text": "Let $\\left({G, +_G, \\circ}\\right)_R$ be an $R$-module. Let $\\phi: G \\to G$ be a module isomorphism to itself. Then $\\phi$ is a '''module automorphism'''."}
+{"_id": "22727", "title": "Definition:Vector Space Automorphism", "text": "Let $V$ be a $K$-vector space. Let $\\phi: V \\to V$ be a vector space isomorphism to itself. Then $\\phi$ is a '''vector space automorphism'''."}
+{"_id": "22728", "title": "Definition:Pointwise Maximum of Mappings/Extended Real-Valued Functions", "text": "Let $X$ be a set, and let $f, g: X \\to \\overline{\\R}$ be extended real-valued functions. Let $\\max$ be the max operation on $\\overline{\\R}$ (Ordering on Extended Real Numbers is Total Ordering ensures it is in fact defined). Then the '''pointwise maximum of $f$ and $g$''', denoted $\\max \\left({f, g}\\right)$, is defined by: :$\\max \\left({f, g}\\right): X \\to \\overline{\\R}: \\max \\left({f, g}\\right) \\, \\left({x}\\right) := \\max \\left({f \\left({x}\\right), g \\left({x}\\right)}\\right)$ '''Pointwise maximum''' thence is an instance of a pointwise operation on extended real-valued functions. Since the ordering on $\\overline{\\R}$ coincides on $\\R$ with the standard ordering, this definition incorporates the definition for real-valued functions."}
+{"_id": "22729", "title": "Definition:Pointwise Maximum of Mappings/Real-Valued Functions", "text": "Let $S$ be a set. Let $f, g: S \\to \\R$ be real-valued functions. Let $\\max$ be the max operation on $\\R$ (Ordering on Real Numbers is Total Ordering ensures it is in fact defined). Then the '''pointwise maximum of $f$ and $g$''', denoted $\\map \\max {f, g}$, is defined by: :$\\map \\max {f, g}: S \\to \\R: \\map {\\map \\max {f, g} } x := \\map \\max {\\map f x, \\map g x}$ '''Pointwise maximum''' thence is an instance of a pointwise operation on real-valued functions."}
+{"_id": "22730", "title": "Definition:Non-Invertible Matrix", "text": "{{:Definition:Non-Invertible Matrix/Definition 1}}"}
+{"_id": "22731", "title": "Definition:Pointwise Maximum of Mappings", "text": "Let $X$ be a set. Let $\\struct {S, \\preceq}$ be a toset. Let $f, g: X \\to S$ be mappings. Let $\\max$ be the max operation on $\\struct {S, \\preceq}$. Then the '''pointwise maximum of $f$ and $g$''', denoted $\\map \\max {f, g}$, is defined by: :$\\map \\max {f, g}: X \\to S: \\map {\\map \\max {f, g} } x := \\map \\max {\\map f x, \\map g x}$ Hence '''pointwise maximum''' is an instance of a pointwise operation on mappings."}
+{"_id": "22732", "title": "Definition:Pointwise Minimum of Mappings", "text": "Let $X$ be a set. Let $\\struct {S, \\preceq}$ be a toset. Let $f, g: X \\to S$ be mappings. Let $\\min$ be the min operation on $\\struct {S, \\preceq}$. Then the '''pointwise minimum of $f$ and $g$''', denoted $\\map \\min {f, g}$, is defined by: :$\\map \\min {f, g}: X \\to S: \\map {\\map \\min {f, g} } x := \\map \\min {\\map f x, \\map g x}$ Hence '''pointwise minimum''' is an instance of a pointwise operation on mappings."}
+{"_id": "22733", "title": "Definition:Pointwise Minimum of Mappings/Real-Valued Functions", "text": "Let $X$ be a set, and let $f, g: X \\to \\R$ be real-valued functions. Let $\\min$ be the min operation on $\\R$ (Ordering on Real Numbers is Total Ordering ensures it is in fact defined). Then the '''pointwise minimum of $f$ and $g$''', denoted $\\min \\left({f, g}\\right)$, is defined by: :$\\min \\left({f, g}\\right): X \\to \\R: \\min \\left({f, g}\\right) \\, \\left({x}\\right) := \\min \\left({f \\left({x}\\right), g \\left({x}\\right)}\\right)$ '''Pointwise minimum''' thence is an instance of a pointwise operation on real-valued functions."}
+{"_id": "22734", "title": "Definition:Pointwise Minimum of Mappings/Extended Real-Valued Functions", "text": "Let $X$ be a set, and let $f, g: X \\to \\overline{\\R}$ be extended real-valued functions. Let $\\min$ be the min operation on $\\overline{\\R}$ (Ordering on Extended Real Numbers is Total Ordering ensures it is in fact defined). Then the '''pointwise minimum of $f$ and $g$''', denoted $\\min \\left({f, g}\\right)$, is defined by: :$\\min \\left({f, g}\\right): X \\to \\overline{\\R}: \\min \\left({f, g}\\right) \\, \\left({x}\\right) := \\min \\left({f \\left({x}\\right), g \\left({x}\\right)}\\right)$ '''Pointwise minimum''' thence is an instance of a pointwise operation on extended real-valued functions. Since the ordering on $\\overline{\\R}$ coincides on $\\R$ with the standard ordering, this definition incorporates the definition for real-valued functions."}
+{"_id": "22735", "title": "Definition:Simultaneous Equations/Linear Equations", "text": "A '''system of simultaneous linear equations''' is a set of equations: :$\\displaystyle \\forall i \\in \\set {1, 2, \\ldots, m} : \\sum_{j \\mathop = 1}^n \\alpha_{i j} x_j = \\beta_i$ That is: {{begin-eqn}} {{eqn | l = \\beta_1       | r = \\alpha_{1 1} x_1 + \\alpha_{1 2} x_2 + \\cdots + \\alpha_{1 n} x_n }} {{eqn | l = \\beta_2       | r = \\alpha_{2 1} x_1 + \\alpha_{2 2} x_2 + \\cdots + \\alpha_{2 n} x_n }} {{eqn | o = \\cdots}} {{eqn | l = \\beta_m       | r = \\alpha_{m 1} x_1 + \\alpha_{m 2} x_2 + \\cdots + \\alpha_{m n} x_n }} {{end-eqn}}"}
+{"_id": "22736", "title": "Definition:Set of Residue Classes", "text": "The quotient set of congruence modulo $m$ denoted $\\Z_m$ is: :$\\Z_m = \\dfrac \\Z {\\RR_m}$"}
+{"_id": "22737", "title": "Definition:Pointwise Scalar Multiplication of Extended Real-Valued Functions", "text": "Let $S$ be a non-empty set, and let ${\\overline \\R}^S$ be the set of all mappings $f: S \\to \\overline \\R$. Here $\\overline \\R$ denotes the extended set of real numbers. Then '''pointwise ($\\overline \\R$-)scalar multiplication''' on ${\\overline \\R}^S$ is the binary operation $\\cdot: \\overline \\R \\times {\\overline \\R}^S \\to {\\overline \\R}^S$ defined by: :$\\forall \\lambda \\in \\overline \\R: \\forall f \\in {\\overline \\R}^S: \\forall s \\in S: \\map {\\paren {\\lambda \\cdot f} s := \\lambda \\cdot \\map f s$ where the $\\cdot$ on the right is extended real multiplication. '''Pointwise scalar multiplication''' thence is an instance of a pointwise operation on extended real-valued functions."}
+{"_id": "22738", "title": "Definition:Addition/Real Numbers", "text": "The addition operation in the domain of real numbers $\\R$ is written $+$. From the definition, the real numbers are the set of all equivalence classes $\\eqclass {\\sequence {x_n} } {}$ of Cauchy sequences of rational numbers. Let $x = \\eqclass {\\sequence {x_n} } {}, y = \\eqclass {\\sequence {y_n} } {}$, where $\\eqclass {\\sequence {x_n} } {}$ and $\\eqclass {\\sequence {y_n} } {}$ are such equivalence classes. Then $x + y$ is defined as: :$\\eqclass {\\sequence {x_n} } {} + \\eqclass {\\sequence {y_n} } {} = \\eqclass {\\sequence {x_n + y_n} } {}$"}
+{"_id": "22739", "title": "Definition:Multiplication/Real Numbers", "text": "The multiplication operation in the domain of real numbers $\\R$ is written $\\times$. From the definition, the real numbers are the set of all equivalence classes $\\eqclass {\\sequence {x_n} } {}$ of Cauchy sequences of rational numbers. Let $x = \\eqclass {\\sequence {x_n} } {}, y = \\eqclass {\\sequence {y_n} } {}$, where $\\eqclass {\\sequence {x_n} } {}$ and $\\eqclass {\\sequence {y_n} } {}$ are such equivalence classes. Then $x \\times y$ is defined as: : $\\eqclass {\\sequence {x_n} } {} \\times \\eqclass {\\sequence {y_n} } {} = \\eqclass {\\sequence {x_n \\times y_n} } {}$"}
+{"_id": "22740", "title": "Definition:Ring (Abstract Algebra)/Addition", "text": "The distributand $*$ of a ring $\\struct {R, *, \\circ}$ is referred to as '''ring addition''', or just '''addition'''. The conventional symbol for this operation is $+$, and thus a general ring is usually denoted $\\struct {R, +, \\circ}$."}
+{"_id": "22741", "title": "Definition:Rig", "text": "A '''rig''' is an additive semiring $\\struct {S, *, \\circ}$ in which $\\struct {S, *}$ is a monoid. Alternatively, this is a semiring in which $\\struct {S, *}$ is a commutative monoid. That is, $\\struct {S, *, \\circ}$ has the following properties: {{begin-axiom}} {{axiom | n = \\text A 0         | q = \\forall a, b \\in S         | m = a * b \\in S         | lc= Closure under $*$ }} {{axiom | n = \\text A 1         | q = \\forall a, b, c \\in S         | m = \\paren {a * b} * c = a * \\paren {b * c}         | lc= Associativity of $*$ }} {{axiom | n = \\text A 2         | q = \\forall a, b \\in S         | m = a * b = b * a         | lc= Commutativity of $*$ }} {{axiom | n = \\text A 3         | q = \\exists 0_S \\in S: \\forall a \\in F         | m = a * 0_S = a = 0_S * a         | lc= Identity element for $*$: the zero }} {{axiom | n = \\text M 0         | q = \\forall a, b \\in S         | m = a \\circ b \\in S         | lc= Closure under $\\circ$ }} {{axiom | n = \\text M 1         | q = \\forall a, b, c \\in S         | m = \\paren {a \\circ b} \\circ c = a \\circ \\paren {b \\circ c}         | lc= Associativity of $\\circ$ }} {{axiom | n = \\text M 2         | q = \\forall a \\in S         | m = a \\circ 0_S = 0_S = 0_S \\circ a         | lc= The zero is a zero element for $\\circ$ }} {{axiom | n = \\text D         | q = \\forall a, b, c \\in S         | m = a \\circ \\paren {b * c} = \\paren {a \\circ b} * \\paren {a \\circ c}, \\paren {a * b} \\circ c = \\paren {a \\circ c} * \\paren {a \\circ c}         | lc= $\\circ$ is distributive over $*$ }} {{end-axiom}} Note that the zero element needs to be specified here as an axiom: $\\text M 2$. By Ring Product with Zero, in a ring, the property $\\text M 2$ of the zero element follows as a consequence of the ring axioms."}
+{"_id": "22742", "title": "Definition:Congruence (Number Theory)/Modulo Operation", "text": "Let $\\bmod$ be defined as the modulo operation: :$x \\bmod y := \\begin{cases} x - y \\left \\lfloor {\\dfrac x y}\\right \\rfloor & : y \\ne 0 \\\\ x & : y = 0 \\end{cases}$ Then '''congruence modulo $z$''' is the relation on $\\R$ defined as: :$\\forall x, y \\in \\R: x \\equiv y \\pmod z \\iff x \\bmod z = y \\bmod z$"}
+{"_id": "22743", "title": "Definition:Congruence (Number Theory)/Integer Multiple", "text": "Let $x, y \\in \\R$. Then '''$x$ is congruent to $y$ modulo $z$''' {{iff}} their difference is an integer multiple of $z$: :$x \\equiv y \\pmod z \\iff \\exists k \\in \\Z: x - y = k z$"}
+{"_id": "22745", "title": "Definition:Congruence (Number Theory)/Integers", "text": "Let $m \\in \\Z_{> 0}$. === Definition by Remainder after Division === {{:Definition:Congruence (Number Theory)/Integers/Remainder after Division}} === Definition by Modulo Operation === {{:Definition:Congruence (Number Theory)/Integers/Modulo Operation}} === Definition by Integer Multiple === We also see that $a$ is congruent to $b$ modulo $m$ if their difference is a multiple of $m$: {{:Definition:Congruence (Number Theory)/Integers/Integer Multiple}}"}
+{"_id": "22746", "title": "Definition:Congruence (Number Theory)/Residue", "text": "Let $a, b \\in \\Z$. Let $a \\equiv b \\pmod m$. Then $b$ is a '''residue of $a$ modulo $m$'''. '''Residue''' is another word meaning remainder, and is ''any'' integer congruent to $a$ modulo $m$."}
+{"_id": "22747", "title": "Definition:Congruence (Number Theory)/Integers/Modulo Operation", "text": "Let $\\bmod$ be defined as the modulo operation: :$x \\bmod m := \\begin{cases} x - m \\left \\lfloor {\\dfrac x m}\\right \\rfloor & : m \\ne 0 \\\\ x & : m = 0 \\end{cases}$ Then '''congruence modulo $m$''' is the relation on $\\Z$ defined as: :$\\forall x, y \\in \\Z: x \\equiv y \\pmod m \\iff x \\bmod m = y \\bmod m$"}
+{"_id": "22748", "title": "Definition:Congruence (Number Theory)/Integers/Integer Multiple", "text": "Let $x, y \\in \\Z$. '''$x$ is congruent to $y$ modulo $m$''' {{iff}} their difference is an integer multiple of $m$: :$x \\equiv y \\pmod m \\iff \\exists k \\in \\Z: x - y = k m$"}
+{"_id": "22749", "title": "Definition:Integral Multiple/Rings and Fields", "text": "Let $\\struct {F, +, \\times}$ be a ring or a field. Let $a \\in F$. Let $n \\in \\Z$ be an integer. Then $n \\cdot a$ is an integral multiple of $a$ where $n \\cdot a$ is defined as in Powers of Ring Elements: :$n \\cdot a := \\begin {cases} 0_F & : n = 0 \\\\ \\paren {\\paren {n - 1} \\cdot a} + a & : n > 1 \\\\ \\size n \\cdot \\paren {-a} & : n < 0 \\\\ \\end {cases}$ where $\\size n$ is the absolute value of $n$."}
+{"_id": "22750", "title": "Definition:Integral Multiple/Real Numbers", "text": "Let $x, y \\in \\R$ be real numbers. Then $x$ is an '''integral multiple''' of $y$ {{iff}} $x$ is congruent to $0$ modulo $y$: :$x \\equiv 0 \\pmod y$ That is: :$\\exists k \\in \\Z: x = 0 + k y$"}
+{"_id": "22751", "title": "Definition:Trivial Annihilator", "text": "From Annihilator of Ring Always Contains Zero, we have that $0 \\in \\map {\\mathrm {Ann} } R$ whatever the ring $R$ is. $R$ is said to have a '''trivial annihilator''' {{iff}} its annihilator $\\map {\\mathrm {Ann} } R$ consists '''only''' of the integer $0$."}
+{"_id": "22752", "title": "Definition:Pendant Vertex", "text": "Let $G$ be a graph. A vertex $v$ of $G$ is said to be a '''pendant vertex''' {{iff}} it has degree $1$."}
+{"_id": "22753", "title": "Definition:Pendant Edge", "text": "Let $G$ be a graph. An edge $e$ of $G$ is said to be a '''pendant edge''' {{iff}} it is incident on a pendant vertex."}
+{"_id": "22754", "title": "Definition:Parallel Edges", "text": "Let $G$ be a graph or a digraph. A pair of edges $e$ and $e'$ of $G$ are said to be a '''parallel''' {{iff}} they are incident on precisely the same vertices."}
+{"_id": "22755", "title": "Definition:Even Vertex (Graph Theory)", "text": "If the degree of $v$ is even, then $v$ is called an '''even vertex'''."}
+{"_id": "22756", "title": "Definition:Odd Vertex (Graph Theory)", "text": "If the degree of $v$ is odd, then $v$ is an '''odd vertex'''."}
+{"_id": "22757", "title": "Definition:Out-Degree", "text": "The '''out-degree of $v$ in $G$''' is the number of arcs which are incident from $v$. It is denoted $\\map {\\operatorname {outdeg}_G} v$, or just $\\map {\\operatorname {outdeg} } v$ if it is clear from the context which digraph is being referred to."}
+{"_id": "22758", "title": "Definition:In-Degree", "text": "The '''in-degree of $v$ in $G$''' is the number of arcs which are incident to $v$.  It is denoted $\\map {\\operatorname {indeg}_G} v$, or just $\\map {\\operatorname {indeg} } v$ if it is clear from the context which digraph is being referred to."}
+{"_id": "22759", "title": "Definition:Non-Empty Set", "text": "Let $S$ be a set. Then $S$ is said to be '''non-empty''' {{iff}} $S$ has at least one element. By the Axiom of Extension, this may also be phrased as: :$S \\ne \\O$ where $\\O$ denotes the empty set. Many mathematical theorems and definitions require sets to be '''non-empty''' in order to avoid erratic results and inconsistencies."}
+{"_id": "22760", "title": "Definition:Left Cancellable Operation", "text": "The operation $\\circ$ in $\\struct {S, \\circ}$ is '''left cancellable''' {{iff}}: :$\\forall a, b, c \\in S: a \\circ b = a \\circ c \\implies b = c$ That is, {{iff}} all elements of $\\struct {S, \\circ}$ are left cancellable."}
+{"_id": "22761", "title": "Definition:Right Cancellable Operation", "text": "The operation $\\circ$ in $\\struct {S, \\circ}$ is '''right cancellable''' {{iff}}: :$\\forall a, b, c \\in S: a \\circ c = b \\circ c \\implies a = b$ That is, {{iff}} all elements of $\\struct {S, \\circ}$ are right cancellable."}
+{"_id": "22762", "title": "Definition:Cancellable Operation", "text": "Let $\\left ({S, \\circ}\\right)$ be an algebraic structure. === Left Cancellable Operation === {{:Definition:Left Cancellable Operation}} === Right Cancellable Operation === {{:Definition:Right Cancellable Operation}} === Cancellable Operation === The operation $\\circ$ in $\\struct {S, \\circ}$ is '''cancellable''' {{iff}}: * $\\forall a, b, c \\in S: a \\circ b = a \\circ c \\implies b = c$ * $\\forall a, b, c \\in S: a \\circ c = b \\circ c \\implies a = b$ That is, {{iff}} it is both a left cancellable operation and a right cancellable operation."}
+{"_id": "22763", "title": "Definition:Algebraic System", "text": "An '''algebraic system''' is a mathematical system $\\SS = \\struct {E, O}$ where: :$E$ is a non-empty set of elements :$O$ is a set of finitary operations on $E$."}
+{"_id": "22764", "title": "Definition:Successor Mapping on Natural Numbers", "text": "Let $\\N$ be the set of natural numbers. Let $s: \\N \\to \\N$ be the mapping defined as: :$s = \\set {\\tuple {x, y}: x \\in \\N, y = x + 1}$ Considering $\\N$ defined as a Peano structure, this is seen to be an instance of a successor mapping."}
+{"_id": "22765", "title": "Definition:Formal Grammar/Top-Down", "text": "A '''top-down grammar''' for $\\mathcal L$ is a formal grammar which allows well-formed formulas to be built from a single metasymbol. Such a grammar can be made explicit by declaring that: * A metasymbol may be replaced by a letter of $\\mathcal A$. * A metasymbol may be replaced by certain collations labeled with metasymbols and signs of $\\mathcal A$. From the words thus generated, those not containing any metasymbols are the well-formed formulas."}
+{"_id": "22766", "title": "Definition:Formal Grammar/Bottom-Up", "text": "Let $\\mathcal L$ be a formal language whose alphabet is $\\mathcal A$. <section notitle=\"true\" name=\"tc\"> A '''bottom-up grammar''' for $\\mathcal L$ is a formal grammar whose rules of formation allow the user to build well-formed formulas from primitive symbols, in the following way: * Letters are well-formed formulas. * A collection of specified collations of well-formed formulas, possibly labeled with additional signs, are also well-formed formulas. In certain use cases, the first clause is adjusted to allow for more complex situations, for example in the bottom-up specification of predicate logic {{transclude:Definition:Formal Grammar/Bottom-Up/Extremal Clause   |section = tc   |title = Extremal Clause   |header = 3   |link = true }} </section>"}
+{"_id": "22767", "title": "Definition:Finite Extended Real Number", "text": "An extended real number is defined as '''finite''' {{iff}} it is a real number."}
+{"_id": "22768", "title": "Definition:Infinite", "text": "=== Infinite Cardinal === {{:Definition:Infinite Cardinal}} === Infinite Set === {{:Definition:Infinite Set}} === Infinity === {{:Definition:Infinity}}"}
+{"_id": "22769", "title": "Definition:Cross-Relation", "text": "Let $\\struct {S, \\circ}$ be a commutative semigroup. Let $\\struct {S_1, \\circ_{\\restriction_1} }, \\struct {S_2, \\circ_{\\restriction_2} }$ be subsemigroups of $S$, where $\\circ_{\\restriction_1}$ and $\\circ_{\\restriction_2}$ are the restrictions of $\\circ$ to $S_1$ and $S_2$ respectively. Let $\\struct {S_1 \\times S_2, \\oplus}$ be the (external) direct product of $\\struct {S_1, \\circ_{\\restriction_1} }$ and $\\struct {S_2, \\circ_{\\restriction_2} }$, where $\\oplus$ is the operation on $S_1 \\times S_2$ induced by $\\circ_{\\restriction_1}$ on $S_1$ and $\\circ_{\\restriction_2}$ on $S_2$. Let $\\boxtimes$ be the relation on $S_1 \\times S_2$ defined as: :$\\tuple {x_1, y_1} \\boxtimes \\tuple {x_2, y_2} \\iff x_1 \\circ y_2 = x_2 \\circ y_1$ This relation $\\boxtimes$ is referred to as the '''cross-relation on $\\struct {S_1 \\times S_2, \\oplus}$'''."}
+{"_id": "22770", "title": "Definition:Additive Semiring", "text": "An '''additive semiring''' is a semiring with a commutative distributand. That is, an '''additive semiring''' is a ringoid $\\left({S, *, \\circ}\\right)$ in which: : $(1): \\quad \\left({S, *}\\right)$ forms a commutative semigroup : $(2): \\quad \\left({S, \\circ}\\right)$ forms a semigroup. === Additive Semiring Axioms === {{:Definition:Additive Semiring/Axioms}}"}
+{"_id": "22771", "title": "Definition:A Fortiori", "text": "'''A fortiori''' knowledge arises from stronger facts already established. For example, in: :''I am too old for this game; therefore, my father is also too old.'' the knowledge: :''My father is too old for this game.'' arises '''a fortiori''' from the stronger knowledge that: :''I am too old for this game.'' An '''a fortiori''' argument is most commonly used by applying a general fact in a particular case."}
+{"_id": "22772", "title": "Definition:Rule of Formation", "text": "Let $\\mathcal F$ be a formal language whose alphabet is $\\mathcal A$. The '''rules of formation''' of $\\mathcal F$ are the rules which define how to construct collations in $\\mathcal A$ which are well-formed. That is, the '''rules of formation''' tell you how to build collations featuring symbols from the alphabet $\\mathcal A$ which are part of the formal language $\\mathcal F$. The '''rules of formation''' of a formal language together constitute its formal grammar. There are no strict guidelines on how a '''rule of formation''' should look like, since they are employed to ''produce'' such strict guidelines. Thus, these '''rules of formation''' are often phrased in natural language, but their exact form is to some extent arbitrary."}
+{"_id": "22773", "title": "Definition:Cross-Relation on Natural Numbers", "text": "Consider the commutative semigroup $\\left({\\N, +}\\right)$ composed of the natural numbers $\\N$ and addition $+$. Let $\\left({\\N \\times \\N, \\oplus}\\right)$ be the (external) direct product of $\\left({\\N, +}\\right)$ with itself, where $\\oplus$ is the operation on $\\N \\times \\N$ induced by $+$ on $\\N$. Let $\\boxtimes$ be the relation on $\\N \\times \\N$ defined as: :$\\left({x_1, y_1}\\right) \\boxtimes \\left({x_2, y_2}\\right) \\iff x_1 + y_2 = x_2 + y_1$ This relation $\\boxtimes$ is referred to as the '''cross-relation on $\\left({\\N \\times \\N, \\oplus}\\right)$'''."}
+{"_id": "22774", "title": "Definition:Definitional Abbreviation", "text": "When discussing a formal language, some particular WFFs may occur very often. If such WFFs are very unwieldy to write and obscure what the author tries to express, it is convenient to introduce a shorthand for them. Such a shorthand is called a '''definitional abbreviation'''. It does ''not'' in any way alter the meaning or formal structure of a sentence, but is purely a method to keep expressions readable to human eyes."}
+{"_id": "22775", "title": "Definition:Function Symbol", "text": "Let $\\mathcal L$ be a formal language (for example, the language of predicate logic $\\mathcal L_1$). A '''function symbol''' is a letter of $\\mathcal L$ used to describe a function. The name '''function symbol''' is a gesture to the reader to make clear what such a symbol should (intuitively) represent in the formal language $\\mathcal L$."}
+{"_id": "22776", "title": "Definition:Alphabetic Substitution", "text": "Consider the (abbreviated) WFF $Q x: \\mathbf C$. Let $y$ be another variable such that $y$ does not occur in $\\mathbf C$. Let $\\mathbf C'$ be the WFF resulting from replacing all free occurrences of $x$ in $\\mathbf C$ with $y$. Then to all intents and purposes, the WFFs: :$Q x: \\mathbf C$ :$Q y: \\mathbf C'$ will have the same interpretation. Thus we may change the free occurrences of any variable for another variable symbol. This change is called '''alphabetic substitution'''."}
+{"_id": "22777", "title": "Definition:Classes of WFFs", "text": "The set of all WFFs of $\\mathcal L_1$ formed with relation symbols from $\\mathcal P$ and function symbols from $\\mathcal F$ can be denoted $WFF \\left({\\mathcal P, \\mathcal F}\\right)$. If so desired, the parameters can also be emphasized by writing $WFF \\left({\\mathcal P, \\mathcal F, \\mathcal K}\\right)$ instead. To specify $\\mathcal P$, one speaks of '''WFFs with relation symbols from $\\mathcal P$'''. To specify $\\mathcal F$, one speaks of '''WFFs with function symbols from $\\mathcal F$'''. To specify $\\mathcal K$, one speaks of '''WFFs with parameters from $\\mathcal K$'''. Of course, combinations of these are possible. Several classes of WFFs are often considered and have special names. === Plain WFF === {{:Definition:Classes of WFFs/Plain WFF}} === Sentence === {{:Definition:Classes of WFFs/Sentence}} === Plain Sentence === {{:Definition:Classes of WFFs/Plain Sentence}}"}
+{"_id": "22778", "title": "Definition:Classes of WFFs/Sentence", "text": "A WFF is said to be a '''sentence''' {{iff}} it contains no free variables. To denote particular classes of '''sentences''', $SENT \\left({\\mathcal P, \\mathcal F, \\mathcal K}\\right)$ and analogues may be used, similar to the notation for classes of WFFs."}
+{"_id": "22779", "title": "Definition:Classes of WFFs/Plain Sentence", "text": "A WFF is said to be a '''plain sentence''' iff it is both plain and a sentence. That is, if it contains free variables nor parameters. Thus, '''plain sentences''' are those WFFs which are in $SENT \\left({\\mathcal P, \\mathcal F, \\varnothing}\\right)$."}
+{"_id": "22781", "title": "Definition:Compound Statement/Ill-Formed", "text": "The substatements in a compound statement, which are joined by a connective, may be compound statements themselves.  It is clearly necessary that the interpretation of such a compound statement is unambiguous.  A compound statement is said to be '''ill-formed''' if it is ambiguous as to how its substatements are grouped by the action of the connectives. For example, in natural language: :''I would like some juice or water with ice.'' can mean either: :''I would like some juice, or water with ice.'' or: :''I would like some juice with ice, or water with ice.''"}
+{"_id": "22782", "title": "Definition:Set/Implicit Set Definition/Multipart Infinite Set", "text": "Let $S$ be a set. Suppose $S$ is to contain: :$(1): \\quad$ a never-ending list of elements and :$(2): \\quad$ other elements which are unrelated to that list (perhaps another never-ending list). Then a '''semicolon''' is used to separate the various conceptual parts: :$S = \\set {1, 3, 5, \\ldots; 2, 4, 6, \\ldots; \\text{red}, \\text{orange}, \\text{green} }$ Note that ''without'' the semicolon it would appear as though the first list (of odd numbers) '''continued''' as the second list (of even numbers) which in turn '''continued''' as a list of colours, which is absurd."}
+{"_id": "22783", "title": "Definition:Mathematical Logic", "text": "'''Mathematical logic''' is a sub-branch of symbolic logic in which the foundations of the assumptions upon which rest mathematics itself are investigated and made rigorous."}
+{"_id": "22784", "title": "Definition:Multi-Value Logic", "text": "'''Multi-value logic''' is a branch of logic in which it is admissible for a statements to have a truth value other than just true or false."}
+{"_id": "22785", "title": "Definition:Modal Logic", "text": "'''Modal logic''' is a branch of logic in which truth values are more complex than being merely true or false, and which distinguishes between different \"modes\" of truth."}
+{"_id": "22786", "title": "Definition:Naive Set Theory", "text": "'''Naïve set theory''', in contrast with axiomatic set theory, is an approach to set theory which assumes the existence of a universal set, despite the fact that such an assumption leads to paradoxes. A popular alternative (and inaccurate) definition describes this as a : ''non-formalized definition of set theory which describes sets and the relations between them using natural language.'' However, the discipline is founded upon quite as rigid a set of axioms, namely, those of propositional and predicate logic."}
+{"_id": "22787", "title": "Definition:Pure Set Theory", "text": "'''Pure set theory''' is a system of set theory in which all elements of sets are themselves sets."}
+{"_id": "22788", "title": "Definition:Set Intersection/Set of Sets", "text": "Let $\\Bbb S$ be a set of sets The '''intersection of $\\Bbb S$''' is: :$\\ds \\bigcap \\Bbb S := \\set {x: \\forall S \\in \\Bbb S: x \\in S}$ That is, the set of all objects that are elements of all the elements of $\\Bbb S$. Thus: :$\\ds \\bigcap \\set {S, T} := S \\cap T$"}
+{"_id": "22789", "title": "Definition:Set Intersection/Countable Intersection", "text": "Let $\\mathbb S$ be a set of sets. Let $\\left\\langle{S_n}\\right\\rangle_{n \\mathop \\in \\N}$ be a sequence in $\\mathbb S$. Let $S$ be the intersection of $\\left\\langle{S_n}\\right\\rangle_{n \\mathop \\in \\N}$: :$\\displaystyle S = \\bigcap_{n \\mathop \\in \\N} S_n$ Then $S$ is a '''countable intersection''' of sets in $\\mathbb S$."}
+{"_id": "22790", "title": "Definition:Set Union/Set of Sets", "text": "Let $\\mathbb S$ be a set of sets. The '''union of $\\mathbb S$''' is: :$\\ds \\bigcup \\mathbb S := \\set {x: \\exists X \\in \\mathbb S: x \\in X}$ That is, the set of all elements of all elements of $\\mathbb S$. Thus the general union of two sets can be defined as: :$\\ds \\bigcup \\set {S, T} = S \\cup T$"}
+{"_id": "22791", "title": "Definition:Set Union/Countable Union", "text": "Let $\\mathbb S$ be a set of sets. Let $\\left\\langle{S_n}\\right\\rangle_{n \\mathop \\in \\N}$ be a sequence in $\\mathbb S$. Let $S$ be the union of $\\left\\langle{S_n}\\right\\rangle_{n \\mathop \\in \\N}$: :$\\displaystyle S = \\bigcup_{n \\mathop \\in \\N} S_n$ Then $S$ is a '''countable union''' of sets in $\\mathbb S$."}
+{"_id": "22792", "title": "Definition:Set Union/Finite Union", "text": "Let $S = S_1 \\cup S_2 \\cup \\ldots \\cup S_n$. Then: :$\\displaystyle S = \\bigcup_{i \\mathop \\in \\N^*_n} S_i = \\set {x: \\exists i \\in \\N^*_n: x \\in S_i}$ where $\\N^*_n = \\set {1, 2, 3, \\ldots, n}$. If it is clear from the context that $i \\in \\N^*_n$, we can also write $\\displaystyle \\bigcup_{\\N^*_n} S_i$."}
+{"_id": "22793", "title": "Definition:Proof System", "text": "A '''proof system''' $\\mathscr P$ for $\\mathcal L$ comprises: * '''Axioms''' and/or '''axiom schemata'''; * '''Rules of inference''' for deriving theorems. It is usual that a '''proof system''' does this by declaring certain arguments concerning $\\mathcal L$ to be valid. Informally, a '''proof system''' amounts to a precise account of what constitutes a '''(formal) proof'''."}
+{"_id": "22794", "title": "Definition:Set Intersection/Finite Intersection", "text": "Let $S = S_1 \\cap S_2 \\cap \\ldots \\cap S_n$. Then: :$\\displaystyle S = \\bigcap_{i \\mathop \\in \\N^*_n} S_i := \\set {x: \\forall i \\in \\N^*_n: x \\in S_i}$ where $\\N^*_n = \\set {1, 2, 3, \\ldots, n}$. If it is clear from the context that $i \\in \\N^*_n$, we can also write $\\displaystyle \\bigcap_{\\N^*_n} S_i$."}
+{"_id": "22795", "title": "Definition:Set Union/Family of Sets", "text": "Let $I$ be an indexing set. Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of sets indexed by $I$. Then the '''union''' of $\\family {S_i}$ is defined as: :$\\ds \\bigcup_{i \\mathop \\in I} S_i := \\set {x: \\exists i \\in I: x \\in S_i}$"}
+{"_id": "22796", "title": "Definition:Set Intersection/Family of Sets", "text": "Let $I$ be an indexing set. Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of sets indexed by $I$. Then the '''intersection''' of $\\family {S_i}$ is defined as: :$\\displaystyle \\bigcap_{i \\mathop \\in I} S_i := \\set {x: \\forall i \\in I: x \\in S_i}$"}
+{"_id": "22797", "title": "Definition:Zero (Ordinal)", "text": "The '''zero ordinal''', denoted $0$, is the empty set $\\O$."}
+{"_id": "22799", "title": "Definition:Generalized Momentum", "text": "The '''generalized momentum of analytical (Lagrangian, Hamiltonian) formulations of classical mechanics''' is defined as the partial derivative of the Lagrangian with regards to the time derivative of generalized coordinates: :$p_i = \\dfrac {\\partial\\mathcal L} {\\partial \\dot q_i}$ where: : $p_i$ is the $i$th coordinate of the generalized momenta : $\\mathcal L$ is the Lagrangian : $\\dot q_i$ is the time derivative of the generalized coordinates $q_i$. Category:Definitions/Dimensions of Measurement 5trw1vacbts59y03ekptzhiw2a21xus"}
+{"_id": "22800", "title": "Definition:Action Applied by System", "text": "The '''action applied by a system''' from state $1$ to state $2$ is defined as the definite integral of the Lagrangian over time from state $1$ to state $2$: :$\\displaystyle S_{12} = \\int_{t_1}^{t_2} \\mathcal L \\ \\mathrm d t$ where: : $S_{12}$ is the '''action''' from $1$ to $2$ : $t$ is time : $\\mathcal L$ is the Lagrangian. Category:Definitions/Dimensions of Measurement l2q6txwc7jphfgt8q8sil12fyacu1iy"}
+{"_id": "22801", "title": "Definition:Set Product", "text": "Let $S$ and $T$ be sets. Let $P$ be a set and let $\\phi_1: P \\to S$ and $\\phi_2: P \\to T$ be mappings such that: :For all sets $X$ and all mappings $f_1: X \\to S$ and $f_2: X \\to T$ there exists a unique mapping $h: X \\to P$ such that: ::$\\phi_1 \\circ h = f_1$ ::$\\phi_2 \\circ h = f_2$ :that is, such that: ::$\\begin{xy}\\xymatrix@+1em@L+3px{ &  X   \\ar[ld]_*+{f_1}   \\ar@{-->}[d]^*+{h}   \\ar[rd]^*+{f_2} \\\\  S &  P   \\ar[l]^*+{\\phi_1}   \\ar[r]_*+{\\phi_2} &  T }\\end{xy}$ :is a commutative diagram. Then $P$, together with the mappings $\\phi_1$ and $\\phi_2$, is called '''a product of $S$ and $T$'''."}
+{"_id": "22802", "title": "Definition:Integral of Integrable Function over Measurable Set", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space, and let $E \\in \\Sigma$. Let $f: X \\to \\overline \\R$ be a $\\mu$-integrable function. Then the '''$\\mu$-integral of $f$ over $E$''' is defined by: :$\\displaystyle \\int_E f \\rd \\mu := \\int \\chi_E \\cdot f \\rd \\mu$ where: :$\\chi_E$ is the characteristic function of $E$ :$\\chi_E \\cdot f$ is the pointwise product of $\\chi_E$ and $f$ :the integral sign on the {{RHS}} denotes $\\mu$-integration of the function $\\chi_E \\cdot f$."}
+{"_id": "22803", "title": "Definition:Measure with Density", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $f: X \\to \\overline{\\R}_{\\ge 0}$ be a positive $\\mu$-measurable function. Then the '''measure with density $f$ with respect to $\\mu$''', denoted $f \\mu$, is defined by: :$f \\mu \\left({E}\\right) := \\displaystyle \\int_E f \\, \\mathrm d \\mu$ where $\\displaystyle \\int_E f \\, \\mathrm d \\mu$ is the $\\mu$-integral of $f$ over $E$."}
+{"_id": "22804", "title": "Definition:Indexing Set/Family of Sets", "text": "Let $\\mathcal S$ be a set of sets. Let $I$ be an indexing set. Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of elements of $\\mathcal S$ indexed by $I$. Then $\\family {S_i}_{i \\mathop \\in I}$ is referred to as an '''indexed family of sets'''."}
+{"_id": "22805", "title": "Definition:Set Partition/Finite Expansion", "text": "Let $S$ be a set. Let $\\Bbb S = \\set {S_1, S_2, \\ldots, S_n}$ form a partition of $S$. Then the representation by such a partition $\\displaystyle \\bigcup_{k \\mathop = 1}^n S_k = S$ is also called a '''finite expansion''' of $S$. The notations: :$S = S_1 \\mid S_2 \\mid \\cdots \\mid S_n$ or: :$\\Bbb S = \\set {S_1 \\mid S_2 \\mid \\cdots \\mid S_n}$ are sometimes seen."}
+{"_id": "22806", "title": "Definition:Partitioning", "text": "Let $S$ be a set. Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of subsets of $S$ such that: :$(1): \\quad \\forall i \\in I: S_i \\ne \\O$, that is, none of $S_i$ is empty :$(2): \\quad \\ds S = \\bigcup_{i \\mathop \\in I} S_i$, that is, $S_i$ is the union of $\\family {S_i}_{i \\mathop \\in I}$ :$(3): \\quad \\forall i, j \\in I: i \\ne j \\implies S_i \\cap S_j = \\O$, that is, the elements of $\\family {S_i}_{i \\mathop \\in I}$ are pairwise disjoint. Then $\\family {S_i}_{i \\in I}$ is a '''partitioning''' of $S$. The image of this '''partitioning''' is the set $\\set {S_i: i \\in I}$ and is called a partition of $S$. Note the difference between: :the '''partitioning''', which is an indexing function (that is a mapping) and :the partition, which is the effect (that is, the image) of that mapping."}
+{"_id": "22807", "title": "Definition:Set Partition/Component", "text": "The elements $S_1, S_2, \\ldots \\in \\mathbb S$ are known as the '''components''' of the partition."}
+{"_id": "22808", "title": "Definition:P-Sequence Space", "text": "Let $p \\in \\R$, $p \\ge 1$. The '''$p$-sequence space''', denoted $\\ell^p$ or $\\ell^p \\left({\\N}\\right)$, is defined as: :$\\displaystyle \\ell^p := \\left\\{{\\left\\langle{z_n}\\right\\rangle_{n \\in \\N} \\in \\C^\\N: \\sum_{n \\mathop = 0}^\\infty \\left\\vert{z_n}\\right\\vert^p < \\infty}\\right\\}$ As such, $\\ell^p$ is a subspace of $\\C^\\N$, the space of all complex sequences. {{stub}}"}
+{"_id": "22809", "title": "Definition:Set Product/Family of Sets", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be an indexed family of sets. Let $P$ be a set. Let $\\family {\\phi_i}_{i \\mathop \\in I}$ be an indexed family of mappings $\\phi_i: P \\to S_i$ for all $i \\in I$ such that: :For all sets $X$ and all indexed families $\\family {f_i}_{i \\mathop \\in I}$ of mappings $f_i: X \\to S_i$ there exists a unique mapping $h: X \\to P$ such that: ::$\\forall i \\in I: \\phi_i \\circ h = f_i$ :that is, such that for all $i \\in I$: ::$\\begin{xy}\\xymatrix@+1em@L+3px{  X   \\ar@{-->}[d]_*+{h}   \\ar[dr]^*+{f_i} \\\\  P   \\ar[r]_*{\\phi_i} &  S_i }\\end{xy}$ :is a commutative diagram. Then $P$, together with the family of mappings $\\family {\\phi_i}_{i \\mathop \\in I}$, is called '''a product of (the family) $\\family {S_i}_{i \\mathop \\in I}$'''. This '''product of $\\family {S_i}_{i \\mathop \\in I}$''' can be denoted $\\struct {P, \\family {\\phi_i}_{i \\mathop \\in I} }$."}
+{"_id": "22810", "title": "Definition:Cartesian Product/Family of Sets", "text": "=== Definition 1 === {{:Definition:Cartesian Product/Family of Sets/Definition 1}} === Definition 2 === {{:Definition:Cartesian Product/Family of Sets/Definition 2}} === Uncountable Cartesian Product === Using this notation, it is then possible to define the Cartesian product of an uncountable family: {{:Definition:Cartesian Product/Uncountable}}"}
+{"_id": "22811", "title": "Definition:Cartesian Product/Cartesian Space", "text": "Let $S$ be a set. The '''cartesian $n$th power of $S$''', or '''$S$ to the power of $n$''', is defined as: :$\\displaystyle S^n = \\prod_{k \\mathop = 1}^n S = \\set {\\tuple {x_1, x_2, \\ldots, x_n}: \\forall k \\in \\N^*_n: x_k \\in S}$ Thus $S^n = \\underbrace {S \\times S \\times \\cdots \\times S}_{\\text{$n$ times} }$ Alternatively it can be defined recursively: :$S^n = \\begin{cases} S: & n = 1 \\\\ S \\times S^{n - 1} & n > 1 \\end{cases}$ The set $S^n$ called a '''cartesian space'''. An element $x_j$ of an ordered tuple $\\tuple {x_1, x_2, \\ldots, x_n}$ of a '''cartesian space''' $S^n$ is known as a basis element of $S^n$."}
+{"_id": "22812", "title": "Definition:Projection (Mapping Theory)/Family of Sets", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of sets. Let $\\displaystyle \\prod_{i \\mathop \\in I} S_i$ be the Cartesian product of $\\family {S_i}_{i \\mathop \\in I}$. For each $j \\in I$, the '''$j$th projection on $\\displaystyle S = \\prod_{i \\mathop \\in I} S_i$''' is the mapping $\\pr_j: S \\to S_j$ defined by: :$\\map {\\pr_j} {\\family {s_i}_{i \\mathop \\in I} } = s_j$ where $\\family {s_i}_{i \\mathop \\in I}$ is an arbitrary element of $\\displaystyle \\prod_{i \\mathop \\in I} S_i$."}
+{"_id": "22813", "title": "Definition:Seminorm", "text": "Let $\\struct{K, +, \\circ}$ be a division ring with norm $\\norm{\\,\\cdot\\,}_K$. Let $V$ be a vector space over $\\struct{K, \\norm{\\,\\cdot\\,}_K}$, with zero $0_V$. A '''seminorm''' on $V$ is a map from $V$ to the positive reals $\\norm{\\cdot}: V \\to \\R_{\\ge 0}$ satisfying the following properties (for all $x,y \\in V$ and $\\lambda \\in K$): {| | align=\"left\" | '''N2:''' | align=\"left\" | Positive homogeneity: | align=\"left\" | $\\norm{\\lambda x} = \\norm{\\lambda}_K \\times \\norm{x}$ |- | align=\"left\" | '''N3:''' | align=\"left\" | Triangle inequality: | align=\"left\" | $\\norm {x + y} \\leq \\norm{x} + \\norm{y}$ |} {{refactor|Use Axiom template}} These may be referred to as the '''seminorm axioms'''. The '''N2''' and '''N3''' markings originate from the fact that these axioms are also used in defining norms."}
+{"_id": "22814", "title": "Definition:P-Seminorm", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $p \\in \\R$, $p \\ge 1$. Let $\\mathcal{L}^p \\left({\\mu}\\right)$ be Lebesgue $p$-space for $\\mu$. The '''$p$-seminorm''' on $\\mathcal{L}^p \\left({\\mu}\\right)$ is the mapping $\\left\\Vert{\\cdot}\\right\\Vert_p : \\mathcal{L}^p \\left({\\mu}\\right) \\to \\R_{\\ge 0}$ defined by: :$\\displaystyle \\forall f \\in \\mathcal{L}^p \\left({\\mu}\\right): \\left\\Vert{f}\\right\\Vert_p := \\left({\\int \\left\\vert{f}\\right\\vert^p \\, \\mathrm d \\mu}\\right)^{1/p}$ That the '''$p$-seminorm''' is in fact a seminorm is proved on $p$-Seminorm is Seminorm."}
+{"_id": "22815", "title": "Definition:Friedman Number", "text": "A '''Friedman number (base $n$)''' is a (positive) integer which is the result of an expression in base $n$ arithmetic which contains exactly its digits. The expression is subject to the following constraints: :$(1): \\quad$ The arithmetic operators $+$, $-$, $\\times$, $\\div$ and exponentiation are the only operators which are allowed. :$(2): \\quad$ Parentheses are allowed, but only in order to override the default operator precedence, otherwise every number would trivially be Friedman by $n = (n)$. :$(3): \\quad$ Leading zeroes are not allowed, otherwise other numbers would trivially be Friedman by, for example, $011 = 10 + 1$."}
+{"_id": "22816", "title": "Definition:Lebesgue Space/L-Infinity", "text": "The '''Lebesgue $\\infty$-space for $\\mu$''', denoted $\\mathcal{L}^\\infty \\left({\\mu}\\right)$, is defined as: :$\\displaystyle \\mathcal{L}^\\infty \\left({\\mu}\\right) := \\left\\{{f \\in \\mathcal M \\left({\\Sigma}\\right): \\text{$f$ is a.e. bounded}}\\right\\}$ and so consists of all $\\Sigma$-measurable $f: X \\to \\R$ that are almost everywhere bounded, that is, subject to: :$\\exists c \\in \\R: \\mu \\left({\\left\\{{\\left\\vert{f}\\right\\vert > c}\\right\\}}\\right) = 0$ {{refactor|The following can be split off into separate pages, as they need proving in their own right.}} $\\mathcal{L}^\\infty \\left({\\mu}\\right)$ can be endowed with the supremum seminorm $\\left\\Vert{\\cdot}\\right\\Vert_\\infty$ by: :$\\displaystyle \\forall f \\in \\mathcal{L}^\\infty \\left({\\mu}\\right): \\left\\Vert{f}\\right\\Vert_\\infty := \\inf \\, \\left\\{{c \\ge 0: \\mu \\left({\\left\\{{\\left\\vert{f}\\right\\vert > c}\\right\\}}\\right) = 0}\\right\\}$ If, subsequently, we introduce the equivalence $\\sim$ by: :$f \\sim g \\iff \\left\\Vert{f - g}\\right\\Vert_\\infty = 0$ we obtain the quotient space $L^\\infty \\left({\\mu}\\right) := \\mathcal{L}^\\infty \\left({\\mu}\\right) / \\sim$, which is also called '''Lebesgue $\\infty$-space for $\\mu$'''."}
+{"_id": "22817", "title": "Definition:Product Sigma-Algebra", "text": "Let $\\struct {X, \\Sigma_1}$ and $\\struct {Y, \\Sigma_2}$ be measurable spaces. The '''product $\\sigma$-algebra''' of $\\Sigma_1$ and $\\Sigma_2$ is denoted $\\Sigma_1 \\otimes \\Sigma_2$, and defined as: :$\\Sigma_1 \\otimes \\Sigma_2 := \\map \\sigma {\\set {S_1 \\times S_2: S_1 \\in \\Sigma_1 \\land S_2 \\in \\Sigma_2} }$ where: :$\\sigma$ denotes generated $\\sigma$-algebra :$\\times$ denotes Cartesian product. This '''product $\\sigma$-algebra''' $\\Sigma_1 \\otimes \\Sigma_2$ is a $\\sigma$-algebra on $X \\times Y$. === Product of Measurable Spaces === {{:Definition:Product of Measurable Spaces}}"}
+{"_id": "22818", "title": "Definition:Product of Measurable Spaces", "text": "The '''product of $\\left({X, \\Sigma_1}\\right)$ and $\\left({Y, \\Sigma_2}\\right)$''' is the measurable space: :$\\left({X \\times Y, \\Sigma_1 \\otimes \\Sigma_2}\\right)$"}
+{"_id": "22819", "title": "Definition:Product Measure", "text": "Let $\\left({X, \\Sigma_1, \\mu}\\right)$ and $\\left({Y, \\Sigma_2, \\nu}\\right)$ be $\\sigma$-finite measure spaces. Let $\\left({X \\times Y, \\Sigma_1 \\otimes \\Sigma_2}\\right)$ be the product measurable space of $\\left({X, \\Sigma_1}\\right)$ and $\\left({Y, \\Sigma_2}\\right)$. The '''product measure of $\\mu$ and $\\nu$''', denoted $\\mu \\times \\nu$, is the measure defined by: :$\\forall E_1 \\in \\Sigma_1, E_2 \\in \\Sigma_2: \\mu \\times \\nu \\left({E_1 \\times E_2}\\right) = \\mu \\left({E_1}\\right) \\nu \\left({E_2}\\right)$ That this uniquely defines a measure on $\\Sigma_1 \\otimes \\Sigma_2$ is shown on Uniqueness of Product Measures and Existence of Product Measures. === Product Measure Space === {{:Definition:Product Measure Space}}"}
+{"_id": "22820", "title": "Definition:Product Measure Space", "text": "The '''product of $\\left({X, \\Sigma_1, \\mu}\\right)$ and $\\left({Y, \\Sigma_2, \\nu}\\right)$''' is the measure space: :$\\left({X \\times Y, \\Sigma_1 \\otimes \\Sigma_2, \\mu \\times \\nu}\\right)$"}
+{"_id": "22821", "title": "Definition:Survival Function", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $f: X \\to \\overline{\\R}$ be a positive $\\Sigma$-measurable function. The '''survival function of $f$''' is the mapping $F: \\R \\to \\R$ defined by: :$\\forall t \\in \\R: F \\left({t}\\right) := \\mu \\left({\\left\\{{f \\ge t}\\right\\}}\\right)$ where $\\left\\{{f > t}\\right\\}$ is the set $\\left\\{{x \\in X: f \\left({x}\\right) \\ge t}\\right\\}$."}
+{"_id": "22822", "title": "Definition:Characteristic Function (Set Theory)/Set", "text": "Let $E \\subseteq S$. The '''characteristic function of $E$''' is the function $\\chi_E: S \\to \\set {0, 1}$ defined as: :$\\map {\\chi_E} x = \\begin {cases} 1 & : x \\in E  \\\\  0 & : x \\notin E \\end {cases}$ That is: :$\\map {\\chi_E} x = \\begin {cases} 1 & : x \\in E  \\\\  0 & : x \\in \\relcomp S E \\end {cases}$ where $\\relcomp S E$ denotes the complement of $E$ relative to $S$."}
+{"_id": "22823", "title": "Definition:Characteristic Function (Set Theory)/Relation", "text": "The concept of a characteristic function of a subset carries over directly to relations. Let $\\RR \\subseteq S \\times T$ be a relation. The '''characteristic function of $\\RR$''' is the function $\\chi_\\RR: S \\times T \\to \\set {0, 1}$ defined as: :$\\map {\\chi_\\RR} {x, y} = \\begin {cases} 1 & : \\tuple {x, y} \\in \\RR  \\\\  0 & : \\tuple {x, y} \\notin \\RR \\end{cases}$ It can be expressed in Iverson bracket notation as: :$\\map {\\chi_\\RR} {x, y} = \\sqbrk {\\tuple {x, y} \\in \\RR}$ More generally, let $\\displaystyle \\mathbb S = \\prod_{i \\mathop = 1}^n S_i = S_1 \\times S_2 \\times \\ldots \\times S_n$ be the cartesian product of $n$ sets $S_1, S_2, \\ldots, S_n$. Let $\\RR \\subseteq \\mathbb S$ be an $n$-ary relation on $\\mathbb S$. The '''characteristic function of $\\RR$''' is the function $\\chi_\\RR: \\mathbb S \\to \\set {0, 1}$ defined as: :$\\map {\\chi_\\RR} {s_1, s_2, \\ldots, s_n} = \\begin {cases} 1 & : \\tuple {s_1, s_2, \\ldots, s_n} \\in \\RR  \\\\  0 & : \\tuple {s_1, s_2, \\ldots, s_n} \\notin \\RR \\end {cases}$ It can be expressed in Iverson bracket notation as: :$\\map {\\chi_\\RR} {s_1, s_2, \\ldots, s_n} = \\sqbrk {\\tuple {s_1, s_2, \\ldots, s_n} \\in \\RR}$"}
+{"_id": "22824", "title": "Definition:Zero (Cardinal)", "text": "The cardinal associated with the empty set $\\O$ is called '''zero''', and is denoted $0$. More informally, this means that '''zero''' is defined as being '''the number of elements in the empty set'''."}
+{"_id": "22825", "title": "Definition:One (Cardinal)", "text": "The cardinal associated with the singleton $\\left\\{{\\varnothing}\\right\\}$ is called '''one''', and is denoted $1$. More informally, this means that '''one''' is defined as being '''the number of elements in a singleton set'''."}
+{"_id": "22826", "title": "Definition:Product of Cardinals", "text": "Let $A$ and $B$ be sets. Let $\\mathbf a$ and $\\mathbf b$ be the cardinals associated respectively with $A$ and $B$. Then the '''product''' of $\\mathbf a$ and $\\mathbf b$ is defined as: :$\\mathbf a \\mathbf b := \\operatorname{Card} \\left({A \\times B}\\right)$ where: :$A \\times B$ denotes the Cartesian product of $A$ and $B$ :$\\operatorname{Card} \\left({A \\times B}\\right)$ denotes the cardinal associated with $A \\times B$."}
+{"_id": "22828", "title": "Definition:Quasigroup/Right Quasigroup", "text": "$\\left({S, \\circ}\\right)$ is a '''right quasigroup''' {{iff}}: :for all $a \\in S$, the right regular representation $\\rho_a$ is a permutations on $S$. That is: :$\\forall a, b \\in S: \\exists ! x: x \\circ a = b$"}
+{"_id": "22829", "title": "Definition:(1-3) Parastrophe", "text": "$\\struct {S, *}$ is a '''$(1-3)$ parastrophe of $\\struct {S, \\circ}$''' {{iff}}: :$\\forall x_1, x_2, x_3 \\in S: x_1 \\circ x_2 = x_3 \\iff x_3 * x_2 = x_1$"}
+{"_id": "22830", "title": "Definition:Quasigroup/Left Quasigroup", "text": "$\\left({S, \\circ}\\right)$ is a '''left quasigroup''' {{iff}}: :for all $a \\in S$, the left regular representation $\\lambda_a$ is a permutations on $S$. That is: :$\\forall a, b \\in S: \\exists ! x: a \\circ x = b$"}
+{"_id": "22831", "title": "Definition:(2-3) Parastrophe", "text": "$\\struct {S, *}$ is a '''$(2-3)$ parastrophe of $\\struct {S, \\circ}$''' {{iff}}: :$\\forall x_1, x_2, x_3 \\in S: x_1 \\circ x_2 = x_3 \\iff x_1 * x_3 = x_2$"}
+{"_id": "22832", "title": "Definition:B-Algebra", "text": "Let $\\struct {X, \\circ}$ be an algebraic structure. Then $\\struct {X, \\circ}$ is a '''$B$-algebra''' {{iff}}: {{begin-axiom}} {{axiom | n = \\text {AC}         | q = \\forall x, y \\in X         | m = x \\circ y \\in X }} {{axiom | n = \\text A 0         | m = \\exists 0 \\in X }} {{axiom | n = \\text A 1         | q = \\forall x \\in X         | m = x \\circ x = 0 }} {{axiom | n = \\text A 2         | q = \\forall x \\in X         | m = x \\circ 0 = x }} {{axiom | n = \\text A 3         | q = \\forall x, y, z \\in X         | m = \\paren {x \\circ y} \\circ z = x \\circ \\paren {z \\circ \\paren {0 \\circ y} } }} {{end-axiom}}"}
+{"_id": "22833", "title": "Definition:Power (B-Algebra)", "text": "Let $\\struct {X, \\circ}$ be a $B$-algebra. For any $x \\in X$ and $n \\in \\N$, define the '''$n$th power of $x$''', denoted $x^n$, inductively: :$x^n = \\begin{cases} 0 & \\text {if $n = 0$} \\\\ x^{n - 1} \\circ \\paren {0 \\circ x} & \\text {if $n \\ge 1$} \\end{cases}$"}
+{"_id": "22834", "title": "Definition:Commutative B-Algebra", "text": "Let $\\struct {X, \\circ}$ be a $B$-algebra. Then $\\struct {X, \\circ}$ is said to be '''$0$-commutative''' (or just '''commutative''') {{iff}}: :$\\forall x, y \\in X: x \\circ (0 \\circ y) = y \\circ (0 \\circ x)$"}
+{"_id": "22835", "title": "Definition:Sum of Cardinals", "text": "Let $A$ and $B$ be sets. Let $\\mathbf a$ and $\\mathbf b$ be the cardinals associated with $A$ and $B$ respectively. Then the '''sum of $\\mathbf a$ and $\\mathbf b$''' is defined as: :$\\mathbf a + \\mathbf b = \\operatorname{Card} \\left({A \\sqcup B}\\right)$ where: :$A \\sqcup B$ denotes the disjoint union of $A$ and $B$ :$\\operatorname{Card} \\left({A \\sqcup B}\\right)$ denotes the cardinal associated with $A \\cup B$."}
+{"_id": "22836", "title": "Definition:Self-Inverse Element", "text": "Let $\\struct {S, \\circ}$ be a monoid whose identity element is $e$. Let $x \\in S$ be an element of $S$. === Definition 1 === {{:Definition:Self-Inverse Element/Definition 1}} === Definition 2 === {{:Definition:Self-Inverse Element/Definition 2}}"}
+{"_id": "22837", "title": "Definition:Finite Cardinal", "text": "Let $\\mathbf a$ be a cardinal. Then $\\mathbf a$ is described as '''finite''' {{iff}}: :$\\mathbf a < \\mathbf a + \\mathbf 1$ where $\\mathbf 1$ is (cardinal) one. That is, such that $\\mathbf a \\ne \\mathbf a + \\mathbf 1$."}
+{"_id": "22838", "title": "Definition:Infinite Cardinal", "text": "Let $\\mathbf a$ be a cardinal. Then $\\mathbf a$ is described as '''infinite''' {{iff}}: :$\\mathbf a = \\mathbf a + \\mathbf 1$ where $\\mathbf 1$ is (cardinal) one."}
+{"_id": "22839", "title": "Definition:Convolution of Measurable Functions", "text": "Let $f, g: \\R^n \\to \\R$ be $\\mathcal B^n$-measurable functions such that for all $x \\in \\R^n$: :$\\displaystyle \\int_{\\R^n} f \\left({x - y}\\right) g \\left({y}\\right) \\, \\mathrm d \\lambda^n \\left({y}\\right)$ is finite. The '''convolution of $f$ and $g$''', denoted $f * g$, is the mapping defined by: :$\\displaystyle f * g: \\R^n \\to \\R, f * g \\left({x}\\right) := \\int_{\\R^n} f \\left({x - y}\\right) g \\left({y}\\right) \\, \\mathrm d \\lambda^n \\left({y}\\right)$"}
+{"_id": "22840", "title": "Definition:Convolution of Measurable Function and Measure", "text": "Let $\\mu$ be a measure on the Borel $\\sigma$-algebra $\\mathcal B^n$ on $\\R^n$. Let $f: \\R^n \\to \\R$ be a $\\mathcal B^n$-measurable function such that for all $x \\in \\R^n$: :$\\displaystyle \\int_{\\R^n} f \\left({x - y}\\right) \\, \\mathrm d \\mu \\left({y}\\right)$ is finite. The '''convolution of $f$ and $\\mu$''', denoted $f * \\mu$, is the mapping defined by: :$\\displaystyle f * \\mu: \\R^n \\to \\R, f * \\mu \\left({x}\\right) := \\int_{\\R^n} f \\left({x - y}\\right) \\, \\mathrm d \\mu \\left({y}\\right)$"}
+{"_id": "22841", "title": "Definition:Convolution of Measures", "text": "Let $\\mu$ and $\\nu$ be measures on the Borel $\\sigma$-algebra $\\mathcal B^n$ on $\\R^n$. The '''convolution of $\\mu$ and $\\nu$''', denoted $\\mu * \\nu$, is the measure defined by: :$\\displaystyle \\mu * \\nu: \\mathcal B^n \\to \\overline \\R, \\mu * \\nu \\left({B}\\right) := \\int_{\\R^n} \\chi_B \\left({x + y}\\right) \\, \\mathrm d \\mu \\left({x}\\right) \\, \\mathrm d \\nu \\left({y}\\right)$ where $\\chi_B$ is the characteristic function of $B$."}
+{"_id": "22842", "title": "Definition:Group Direct Product/General Definition", "text": "Let $\\family {\\struct {G_i, \\circ_i} }_{i \\mathop \\in I}$ be a family of groups. Let $\\displaystyle G = \\prod_{i \\mathop \\in I} G_i$ be their cartesian product. Let $\\circ$ be the operation defined on $G$ as: :$\\circ := \\family {g_i}_{i \\mathop \\in I} \\circ \\family {h_i}_{i \\mathop \\in I} = \\family {g_i \\circ_i h_i}_{i \\mathop \\in I}$ for all sequences in $G$. The group $\\struct {G, \\circ}$  is called the '''(external) direct product''' of $\\family {\\struct {G_i, \\circ_i} }_{i \\mathop \\in I}$."}
+{"_id": "22843", "title": "Definition:Hölder Continuous", "text": "Let $\\left({M_1, d_1}\\right)$ and $\\left({M_2, d_2}\\right)$ be metric spaces. Let $\\alpha \\in \\R_{\\ge 0}$ be a positive real number. A mapping $f: M_1 \\to M_2$ is said to be '''$\\alpha$-Hölder continuous''' {{iff}}: :$\\exists L \\in \\R_{\\ge 0}: \\forall x, y \\in M_1: d_2 \\left({f \\left({x}\\right), f \\left({y}\\right)}\\right) \\le L \\left({d_1 \\left({x, y}\\right)}\\right)^\\alpha$ Further, $f$ is said to be '''Hölder continuous''' {{iff}} it is '''$\\alpha$-Hölder continuous''' for some $\\alpha \\in \\R_{\\ge 0}$."}
+{"_id": "22844", "title": "Definition:Permutation/Ordered Selection", "text": "Let $S$ be a set of $n$ elements. Let $r \\in \\N: r \\le n$. An '''$r$-permutation of $S$''' is an ordered selection of $r$ elements of $S$."}
+{"_id": "22845", "title": "Definition:Subsequential Limit", "text": "Let $\\sequence {x_n}$ be a sequence. Let $\\sequence {x_{n_r} }$ be a subsequence of $\\sequence {x_n}$. Suppose that $\\sequence {x_{n_r} }$ converges to a limit $x$. Then $x$ is called a '''subsequential limit''' of $\\sequence {x_n}$."}
+{"_id": "22846", "title": "Definition:Ordinal Function", "text": "Let $A$ be an ordinal (we shall allow $A$ to be a proper class). Then the mapping $G : A \\to \\operatorname{On}$ is called an ''ordinal function''."}
+{"_id": "22847", "title": "Definition:Power of Element/Monoid/Invertible Element", "text": "Let $b \\in S$ be invertible for $\\circ$. Let $n \\in \\Z$. The definition $b^n = \\map {\\circ^n} b$ as the $n$th power of $b$ in $\\left({S, \\circ}\\right)$ can be extended to include the inverse of $b$: :$b^{-n} = \\paren {b^{-1} }^n$"}
+{"_id": "22848", "title": "Definition:Power of Element/Monoid", "text": "Let $\\struct {S, \\circ}$ be a monoid whose identity element is $e$. Let $a \\in S$. Let $n \\in \\N$. The definition $a^n = \\map {\\circ^n} a$ as the $n$th power of $a$ in a semigroup can be extended to allow an exponent of $0$: :$a^n = \\begin {cases} e & : n = 0 \\\\ a^{n - 1} \\circ a & : n > 0 \\end{cases}$ or :$n \\cdot a = \\begin {cases} e & : n = 0 \\\\ \\paren {n - 1} a \\circ a & : n > 0 \\end{cases}$ The validity of this definition follows from the fact that a monoid has an identity element. === Invertible Element === {{:Definition:Power of Element/Monoid/Invertible Element}}"}
+{"_id": "22849", "title": "Definition:Power of Element/Group", "text": "Let $\\struct {G, \\circ}$ be a group whose identity element is $e$. Let $g \\in G$. Let $n \\in \\Z$. The definition $g^n = \\map {\\circ^n} g$ as the $n$th power of $g$ in a monoid can be extended to allow negative values of $n$: :$g^n = \\begin{cases} e & : n = 0 \\\\ g^{n - 1} \\circ g & : n > 0 \\\\ \\paren {g^{-n} }^{-1} & : n < 0 \\end{cases}$ or :$n \\cdot g = \\begin{cases} e & : n = 0 \\\\ \\paren {\\paren {n - 1} \\cdot g} \\circ g & : n > 0 \\\\ -\\paren {-n \\cdot g} & : n < 0 \\end{cases}$ The validity of this definition follows from the group axioms: $g$ has an inverse element."}
+{"_id": "22850", "title": "Definition:Canonical Order", "text": "Let $\\operatorname{On}$ be the ordinal class. Let $<$ be the ordinal ordering. Let $\\operatorname{Le}$ be the lexicographic ordering on $\\operatorname{On} \\times \\operatorname{On}$. The '''canonical order''' on $\\operatorname{On} \\times \\operatorname{On}$, denoted $R_0$, is defined as follows, for ordinals $\\alpha, \\beta, \\gamma, \\delta$: ::$\\left({\\alpha, \\beta}\\right) \\mathrel{R_0} \\left({\\gamma, \\delta}\\right)$ :{{iff}}: ::$\\max \\left({\\alpha, \\beta}\\right) < \\max \\left({\\gamma, \\delta}\\right)$ or $\\left({\\max \\left({ \\alpha, \\beta}\\right) = \\max \\left({\\gamma, \\delta}\\right) \\land \\left({\\alpha, \\beta}\\right) \\mathrel{\\operatorname{Le}} \\left({\\gamma, \\delta}\\right)}\\right)$"}
+{"_id": "22851", "title": "Definition:Ordinal Addition", "text": "Let $x$ and $y$ be ordinals. The operation of '''ordinal addition''' $x + y$ is defined using transfinite recursion on $y$, as follows. === Base Case === When $y = \\varnothing$, define: :$x + \\varnothing := x$ === Inductive Case === For a successor ordinal $y^+$, define: :$x + y^+ := \\left({x + y}\\right)^+$ === Limit Case === Let $y$ be a limit ordinal. Then: :$\\displaystyle x + y := \\bigcup_{z \\mathop \\in y} \\left({x + z}\\right)$"}
+{"_id": "22852", "title": "Definition:Relational Loop", "text": "Let $\\RR$ be a relation on a set $S$. Let $a_1, a_2, \\ldots a_n$ be elements of $S$. A '''relational loop on $S$''' takes the form: :$\\tuple {a_1 \\mathrel \\RR a_2 \\land a_2 \\mathrel \\RR a_3 \\dots \\land a_{n - 1} \\mathrel \\RR a_n \\land a_n \\mathrel \\RR a_1}$ That is, it is a subset of $\\RR$ of the form: :$\\set {\\tuple {a_1, a_2}, \\tuple {a_2, a_3}, \\ldots, \\tuple {a_{n - 1}, a_n}, \\tuple {a_n, a_1} }$"}
+{"_id": "22853", "title": "Definition:Real Matrix", "text": "A '''real matrix''' is a matrix whose elements are all real numbers."}
+{"_id": "22854", "title": "Definition:Smooth Homotopy", "text": "Let $X$ and $Y$ be topological spaces. Let $f: X \\to Y$, $g: X \\to Y$ be smooth mappings. Then $f$ and $g$ are '''smoothly homotopic''' {{iff}} there exists a smooth mapping: : $H: X \\times \\left[{0 \\,.\\,.\\, 1}\\right] \\to Y$ such that: : $H \\left({x, 0}\\right) = f \\left({x}\\right)$ and: : $H \\left({x, 1}\\right) = g \\left({x}\\right)$ $H$ is called a '''smooth homotopy between $f$ and $g$'''."}
+{"_id": "22855", "title": "Definition:Basic Primitive Recursive Function/Zero Function", "text": "The '''zero function''' $\\Zero: \\N \\to \\N$ is a basic primitive recursive function, defined as: :$\\forall n \\in \\N: \\map \\Zero n = 0$"}
+{"_id": "22856", "title": "Definition:Basic Primitive Recursive Function/Successor Function", "text": "The '''successor function''' $\\Succ: \\N \\to \\N$ is a basic primitive recursive function, defined as: :$\\forall n \\in \\N: \\map \\Succ n = n + 1$"}
+{"_id": "22857", "title": "Definition:Basic Primitive Recursive Function/Projection Function", "text": "The '''projection functions''' $\\pr_j^k: \\N^k \\to \\N$ are basic primitive recursive functions, defined as: :$\\forall \\tuple {n_1, n_2, \\ldots, n_k} \\in \\N^k: \\map {\\pr_j^k} {n_1, n_2, \\ldots, n_k} = n_j$<ref>The usual notation for the projection function omits the superscript that defines the arity of the particular instance of the projection in question at the time, for example: $\\pr_j$. However, in the context of computability theory, it is a ''very good idea'' to be ''completely certain'' of ''exactly'' which projection function is under discussion.</ref> where $j \\in \\closedint 1 k$."}
+{"_id": "22858", "title": "Definition:Basic Primitive Recursive Function/Identity Function", "text": "The identity function $I_\\N: \\N \\to \\N$ is a basic primitive recursive function, defined as: :$\\forall n \\in \\N: \\map {I_\\N} n = n$ Note that this is an implementation of the projection function: :$\\pr_1^1: \\N \\to \\N: \\map {\\pr_1^1} {n_1} = n_1$"}
+{"_id": "22859", "title": "Definition:Basic Primitive Recursive Function/URM Computability", "text": "The basic primitive recursive functions: * Zero Function * Successor Function * Projection Function * Identity Function are each URM computable by a single-instruction URM program."}
+{"_id": "22860", "title": "Definition:Primitive Recursive/Set", "text": "Let $A \\subseteq \\N$. Then $A$ is a '''primitive recursive set''' iff its characteristic function $\\chi_A$ is a primitive recursive function."}
+{"_id": "22861", "title": "Definition:Primitive Recursive/Relation", "text": "Let $\\mathcal R \\subseteq \\N^k$ be an $n$-ary relation on $\\N^k$. Then $\\mathcal R$ is a '''primitive recursive relation''' {{iff}} its characteristic function $\\chi_\\mathcal R$ is a primitive recursive function."}
+{"_id": "22862", "title": "Definition:Primitive Recursive/Function", "text": "A function is '''primitive recursive''' if it can be obtained from basic primitive recursive functions using the operations of substitution and primitive recursion a finite number of times."}
+{"_id": "22863", "title": "Definition:Primitive Recursion/One Variable", "text": "Let $a \\in \\N$ be a natural number. Let $g: \\N^2 \\to \\N$ be a function. Then the function $h: \\N \\to \\N$ is '''obtained from the constant $a$ and $g$ by primitive recursion''' {{iff}}: :$\\forall n \\in \\N: \\map h n = \\begin {cases} a & : n = 0 \\\\ \\map g {n - 1, \\map h {n - 1} } & : n > 0  \\end{cases}$"}
+{"_id": "22864", "title": "Definition:Primitive Recursion/Partial Function", "text": "Let $f: \\N^k \\to \\N$ and $g: \\N^{k+2} \\to \\N$ be partial functions. Let $\\tuple {n_1, n_2, \\ldots, n_k} \\in \\N^k$. Then the partial function $h: \\N^{k + 1} \\to \\N$ is '''obtained from $f$ and $g$ by primitive recursion''' {{iff}}: :$\\forall n \\in \\N: \\map h {n_1, n_2, \\ldots, n_k, n} \\approx \\begin {cases} \\map f {n_1, n_2, \\ldots, n_k} & : n = 0 \\\\ \\map g {n_1, n_2, \\ldots, n_k, n - 1, \\map h {n_1, n_2, \\ldots, n_k, n - 1} } & : n > 0  \\end{cases}$ where $\\approx$ is as defined in Partial Function Equality. Note that $\\map h {n_1, n_2, \\ldots, n_k, n}$ is defined only when: :$\\map h {n_1, n_2, \\ldots, n_k, n - 1}$ is defined :$\\map g {n_1, n_2, \\ldots, n_k, n - 1, \\map h {n_1, n_2, \\ldots, n_k, n - 1} }$ is defined."}
+{"_id": "22865", "title": "Definition:Primitive Recursion/Several Variables", "text": "Let $f: \\N^k \\to \\N$ and $g: \\N^{k + 2} \\to \\N$ be functions. Let $\\tuple {n_1, n_2, \\ldots, n_k} \\in \\N^k$. Then the function $h: \\N^{k + 1} \\to \\N$ is '''obtained from $f$ and $g$ by primitive recursion''' {{iff}}: :$\\forall n \\in \\N: \\map h {n_1, n_2, \\ldots, n_k, n} = \\begin {cases} \\map f {n_1, n_2, \\ldots, n_k} & : n = 0 \\\\ \\map g {n_1, n_2, \\ldots, n_k, n - 1, \\map h {n_1, n_2, \\ldots, n_k, n - 1} } & : n > 0  \\end {cases}$"}
+{"_id": "22867", "title": "Definition:Minimization/Partial Function", "text": "Let $f: \\N^{k+1} \\to \\N$ be a partial function. Let $n = \\left({n_1, n_2, \\ldots, n_k}\\right) \\in \\N^k$ be fixed. Then the '''minimization operation on $f$''' is written as: :$\\mu y \\left({f \\left({n, y}\\right) = 0}\\right)$ and is specified as follows: :$\\mu y \\left({f \\left({n, y}\\right) = 0}\\right) = \\begin{cases} z & : f \\left({n, z}\\right) = 0 \\text { and } f \\left({n, y}\\right) \\text{ defined and } \\forall y: 0 \\le y < z: f \\left({n, y}\\right) \\ne 0 \\\\ \\text{undefined} & : \\text{otherwise} \\end{cases}$ The partial function: :$g \\left({n}\\right) \\approx \\mu y \\left({f \\left({n, y}\\right) = 0}\\right)$ obtained in this way (see Partial Function Equality) is said to be obtained from $f$ '''by minimization'''. ==== Note ==== It is not enough for there to exist $z$ such that $f \\left({n, z}\\right) = 0$. We need to insist that $f \\left({n, y}\\right)$ is actually defined for all $y$ less than $z$. Otherwise, if we were to try and find $z$ by the recursive technique of trying all $z$ from $0$ up, we would never actually get up as far as $z$ because the undefined value of $f$ for some $y$ is getting in the way. In the context of URM programs, this is significant, as an undefined output from a function is determined by a non-terminating program."}
+{"_id": "22869", "title": "Definition:Recursive/Function", "text": "A function is '''recursive''' if it can be obtained from basic primitive recursive functions using the operations of: : substitution : primitive recursion, and  : minimization on a function a finite number of times."}
+{"_id": "22870", "title": "Definition:Recursive/Set", "text": "Let $A \\subseteq \\N$. Then $A$ is a '''recursive set''' iff its characteristic function $\\chi_A$ is a recursive function."}
+{"_id": "22871", "title": "Definition:Recursive/Relation", "text": "Let $\\mathcal R \\subseteq \\N^k$ be an $n$-ary relation on $\\N^k$. Then $\\mathcal R$ is a '''recursive relation''' {{iff}} its characteristic function $\\chi_\\mathcal R$ is a recursive function."}
+{"_id": "22872", "title": "Definition:Special Linear Group", "text": "Let $R$ be a commutative ring with unity whose zero is $0$ and unity is $1$. The '''special linear group of order $n$ on $R$''' is the set of square matrices of order $n$ whose determinant is $1$. It is a group under (conventional) matrix multiplication."}
+{"_id": "22873", "title": "Definition:Antiassociative Operation", "text": "Let $\\circ$ be a binary operation on the set $S$. $\\circ$ is an '''antiassociative operation''' on $S$ {{iff}}: :$\\forall x, y, z \\in S: \\left({x \\circ y}\\right)\\circ z \\ne x \\circ \\left({y \\circ z}\\right)$"}
+{"_id": "22874", "title": "Definition:Antiassociative Structure", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic strcuture. Then $\\left({S, \\circ}\\right)$ is an '''antiassociative structure''' {{iff}} $\\circ$ is an antiassociative operation. That is, {{iff}}: :$\\forall x, y, z \\in S: \\left({x \\circ y}\\right)\\circ z \\ne x \\circ \\left({y \\circ z}\\right)$"}
+{"_id": "22875", "title": "Definition:Hadamard Product", "text": "Let $\\struct {S, \\cdot}$ be an algebraic structure. Let $\\mathbf A = \\sqbrk a_{m n}$ be an $m \\times n$ matrix over $S$. Let $\\mathbf B = \\sqbrk b_{m n}$ be an $m \\times n$ matrix over $S$. The '''Hadamard product of $\\mathbf A$ and $\\mathbf B$''' is written $\\mathbf A \\circ \\mathbf B$ and is defined as follows: :$\\mathbf A \\circ \\mathbf B := \\mathbf C = \\sqbrk c_{m n}$ where: :$\\forall i \\in \\closedint 1 m, j \\in \\closedint 1 n: c_{i j} = a_{i j} \\cdot_R b_{i j}$"}
+{"_id": "22876", "title": "Definition:Magma of Sets", "text": "Let $X$ be a set. Let $\\SS \\subseteq \\powerset X$ be a set of subsets of $X$. Let $I$ be an index set. For every $i \\in I$, let $J_i$ be an index set, and let: :$\\phi_i: \\powerset X^{J_i} \\to \\powerset X$ be a partial mapping. Then $\\SS$ is a '''magma of sets for $\\set {\\phi_i: i \\in I}$ on $X$''' {{iff}}: :$\\forall i \\in I: \\map {\\phi_i} {\\family {S_j}_{j \\mathop \\in J_i} } \\in \\SS$ for every indexed family $\\family {S_j}_{j \\mathop \\in J_i} \\in \\SS^{J_i}$ in the domain of $\\phi$."}
+{"_id": "22877", "title": "Definition:Magma of Sets Generated by Collection of Subsets", "text": "Let $X$ be a set, and let $\\Phi := \\left\\{{\\phi_i: i \\in I}\\right\\}$ be a collection of partial mappings with codomain $\\mathcal P \\left({X}\\right)$, the power set of $X$. Let $\\mathcal G \\subseteq \\mathcal P \\left({X}\\right)$ be a collection of subsets of $X$. Then the '''magma of sets for $\\Phi$ generated by $\\mathcal G$''' is the unique magma of sets $\\mathcal S \\subseteq \\mathcal P \\left({X}\\right)$ satisfying: :$(1): \\quad \\mathcal G \\subseteq \\mathcal S$ :$(2): \\quad \\mathcal G \\subseteq \\mathcal T$ implies that $\\mathcal S \\subseteq \\mathcal T$ for every magma of sets $\\mathcal T$ To speak of ''the unique'' '''magma of sets generated by $\\mathcal G$''' is justified by Existence and Uniqueness of Magma of Sets Generated by Collection of Subsets."}
+{"_id": "22878", "title": "Definition:Cantor Set/Limit of Decreasing Sequence", "text": "Let $\\map {I_c} \\R$ denote the set of all closed real intervals. Define the mapping $t_1: \\map {I_c} \\R \\to \\map {I_c} \\R$ by: :$\\map {t_1} {\\closedint a b} := \\closedint a {\\dfrac 1 3 \\paren {a + b} }$ and similarly $t_3: \\map {I_c} \\R \\to \\map {I_c} \\R$ by: :$\\map {t_3} {\\closedint a b} := \\closedint {\\dfrac 2 3 \\paren {a + b} } b$ Note in particular how: :$\\map {t_1} {\\closedint a b} \\subseteq \\closedint a b$ :$\\map {t_3} {\\closedint a b} \\subseteq \\closedint a b$ Subsequently, define inductively: :$S_0 := \\set {\\closedint 0 1}$ :$S_{n + 1} := \\map {t_1} {C_n} \\cup \\map {t_3} {C_n}$ and put, for all $n \\in \\N$: :$C_n := \\displaystyle \\bigcup S_n$ Note that $C_{n + 1} \\subseteq C_n$ for all $n \\in \\N$, so that this forms a decreasing sequence of sets. Then the '''Cantor set''' $\\mathcal C$ is defined as its limit, that is: :$\\mathcal C := \\displaystyle \\bigcap_{n \\mathop \\in \\N} C_n$"}
+{"_id": "22879", "title": "Definition:Ordinal Subtraction", "text": "Let $x$ and $y$ be ordinals such that $x \\le y$. Then the operation of '''ordinal subtraction''' is defined as: :$\\displaystyle y - x = \\bigcup \\left\\{{z: x + z = y}\\right\\}$"}
+{"_id": "22880", "title": "Definition:Ordinal Multiplication", "text": "Let $x$ and $y$ be ordinals. The operation of '''ordinal multiplication''' $x \\times y$ is defined using transfinite recursion as follows: === Base Case === {{begin-eqn}} {{eqn | l = \\left({x \\times \\varnothing}\\right)       | r = \\varnothing       | c = if $y$ is equal to $\\varnothing$ }} {{end-eqn}} === Inductive Case === {{begin-eqn}} {{eqn | l = \\left({x \\times z^+}\\right)       | r = \\left({x \\times z}\\right) + x       | c = if $y$ is the successor of some ordinal $z$ }} {{end-eqn}} === Limit Case === {{begin-eqn}} {{eqn | l = \\left({x \\times y}\\right)       | r = \\bigcup_{z \\mathop \\in y} \\left({x \\times z}\\right)       | c = if $y$ is a limit ordinal }} {{end-eqn}}"}
+{"_id": "22881", "title": "Definition:Ordinal Exponentiation", "text": "Let $x$ and $y$ be ordinals. '''Ordinal exponentiation''' $x^y$ is defined using Transfinite Recursion: :$\\displaystyle x^y = \\begin{cases}  0 & : x = 0, \\ y \\ne 0 \\\\  & \\\\ 1 & : x = 0, \\ y = 0 \\\\  & \\\\ 1 & : x \\ne 0, \\ y = 0 \\\\  & \\\\ \\left({x^z \\cdot x}\\right) & : x \\ne 0, \\ y = z^+ \\\\  & \\\\ \\bigcup_{z \\mathop \\in y} x^z & : x \\ne 0, \\ y \\in K_{II} \\\\ \\end{cases}$ where: * $K_{II}$ is the class of all limit ordinals * $0$ denotes the zero ordinal * $1$ denotes the ordinal $1$, that is: $0^+$, the successor of $0$."}
+{"_id": "22882", "title": "Definition:Ordinal Sum", "text": "Let $y$ be an ordinal. Let $\\left\\langle{A_x}\\right\\rangle$ be a sequence of ordinals. The '''ordinal sum of $A_x$''' is denoted $\\displaystyle \\sum_{x \\mathop = 1}^y A_x$ and defined using Transfinite Recursion on $y$ as follows: :$\\displaystyle \\sum_{x \\mathop = 1}^\\varnothing A_x = \\varnothing$ :$\\displaystyle \\sum_{x \\mathop = 1}^{z^+} A_x = \\sum_{x \\mathop = 1}^z \\left({A_x}\\right) + A_{z^+}$ :$\\displaystyle \\sum_{x \\mathop = 1}^y A_x = \\bigcup_{z \\mathop \\in y} \\left({\\sum_{x \\mathop = 1}^z A_x}\\right)$ for limit ordinals $y$. {{NoSources}} Category:Definitions/Ordinal Arithmetic 8t1l44d9ybtngfihv3ohh31hb978bjt"}
+{"_id": "22883", "title": "Definition:Anticommutative/Structure with One Operation", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure Then $\\circ$ is '''anticommutative on $S$''' {{iff}}: :$\\forall x, y \\in S: x \\circ y = y \\circ x \\iff x = y$ Equivalently, it can be defined as: :$\\forall x, y \\in S: x \\ne y \\iff x \\circ y \\ne y \\circ x$"}
+{"_id": "22884", "title": "Definition:Anticommutative/Structure with Two Operations", "text": "Let $\\left({S, +, \\circ}\\right)$ be an algebraic structure. Suppose every element $x$ in $\\left({S, +}\\right)$ has an inverse element $-x$. Then $\\circ$ is '''anticommutative on $S$ with respect to $+$''' {{iff}}: :$\\forall x, y \\in S: x \\circ y = -\\left({y \\circ x}\\right)$"}
+{"_id": "22885", "title": "Definition:Middle Cancellation Property", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroup. Then $\\circ$ is said to have the '''middle cancellation property''' iff: :$a \\circ x \\circ b = c \\circ x \\circ d \\implies a \\circ b = c \\circ d$ holds for all $a,b,c,d,x \\in S$. Category:Definitions/Semigroups rx7p8mvlzf6qo6km7e4d8vpdy9ntb9h"}
+{"_id": "22886", "title": "Definition:Cross Cancellation Property", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure. Then $\\circ$ is said to have the '''cross cancellation property''' iff: :$a \\circ b = c \\circ a \\implies b = c$ holds for all $a, b, c \\in S$. Category:Definitions/Abstract Algebra t0eoxvczi15juggfax1qedcpu11zi0p"}
+{"_id": "22887", "title": "Definition:Mutatis Mutandis", "text": "'''Mutatis mutandis''' is Latin for '''what needs changing having been changed'''. It can be used in place of words like ''analogously'' and ''similarly'', and expresses that an argument can be applied to a new situation in an intuitive fashion. For example the proof of Union is Associative can be made into that for Intersection is Associative by '''mutatis mutandis'''. In this case, this comes down to replacing $\\cup$ by $\\cap$, and $\\lor$ by $\\land$. In particular, the essential flow of the argument was not changed in adapting the proof."}
+{"_id": "22888", "title": "Definition:Uniformly Integrable", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $\\mathcal F \\subseteq \\mathcal M \\left({\\Sigma}\\right)$ be a collection of measurable functions. Then $\\mathcal F$ is said to be '''uniformly integrable (with respect to $\\mu$)''' iff: :$\\forall \\epsilon > 0: \\exists g_\\epsilon \\in \\mathcal{L}^1_+ \\left({\\mu}\\right): \\displaystyle \\sup_{f \\mathop \\in \\mathcal F} \\int_{\\left\\{{\\left\\vert{f}\\right\\vert > g_\\epsilon}\\right\\}} \\left\\vert{f}\\right\\vert \\, \\mathrm d \\mu < \\epsilon$ where: :$\\mathcal{L}^1_+ \\left({\\mu}\\right)$ is the space of positive $\\mu$-integrable functions :$\\left\\{{\\left\\vert{f}\\right\\vert > g_\\epsilon}\\right\\}$ is short for $\\left\\{{x \\in X: \\left\\vert{f \\left({x}\\right)}\\right\\vert > g_\\epsilon \\left({x}\\right)}\\right\\}$"}
+{"_id": "22889", "title": "Definition:Structure (Set Theory)", "text": "Let $A$ be a class. Let $\\mathcal R$ be a relation. The '''relational structure $\\left[{A, \\mathcal R}\\right]$ satisfies well-formed formula $p$''', denoted $\\left[{A, \\mathcal R}\\right] \\models p$, shall be defined on the well-formed parts of $p$: {{begin-eqn}} {{eqn | l = \\left[{A, \\mathcal R}\\right] \\models x \\in y       | o = \\iff       | r = \\left({x \\in A \\land y \\in A \\land x \\mathrel {\\mathcal R} y}\\right) }} {{eqn | l = \\left[{A, \\mathcal R}\\right] \\models \\neg p       | o = \\iff       | r = \\neg \\left[{A, \\mathcal R}\\right] \\models p }} {{eqn | l = \\left[{A, \\mathcal R}\\right] \\models \\left({p \\land q}\\right)       | o = \\iff       | r = \\left({\\left[{A, \\mathcal R}\\right] \\models p \\land \\left[{A, \\mathcal R}\\right] \\models q}\\right) }} {{eqn | l = \\left[{A, \\mathcal R}\\right] \\models \\forall x: P \\left({x}\\right)       | o = \\iff       | r = \\forall x \\in A: \\left[{A, \\mathcal R}\\right] \\models P \\left({x}\\right) }} {{end-eqn}}"}
+{"_id": "22890", "title": "Definition:Standard Structure", "text": "The structure $\\left[{A,R}\\right]$ is called a ''standard structure'' iff: :$\\displaystyle R = \\Epsilon \\cap \\left({ A \\times A }\\right)$ where $\\Epsilon$ denotes the epsilon relation and $\\times$ denotes the Cartesian product. With a standard structure, $\\left[{A,R}\\right] \\models p$ shall be abbreviated: :$\\displaystyle A \\models p \\iff \\left[{A , E \\cap \\left({ A \\times A }\\right)}\\right] \\models p$"}
+{"_id": "22891", "title": "Definition:Relativisation", "text": "Let $p$ be a well-formed formula of the language of set theory. Let $A$ be a class. The '''relativisation of $p$ to $A$''' shall be denoted $p^A$ and shall be defined recursively on the symbols in $p$: :$x \\in y ^A \\iff x \\in y$ :$\\left({\\neg p}\\right)^A \\iff \\neg p^A$ :$\\left({p \\land q}\\right)^A \\iff \\left({p^A \\land q^A}\\right)$ :$\\left({\\forall x: P \\left({x}\\right)}\\right)^A \\iff \\forall x: \\left({ x \\in A \\implies P \\left({x}\\right)^A}\\right)$ Thus, the relativisation of $p$ is simply the well-formed formula achieved when replacing all instances of $\\forall x$ with $\\forall x \\in A$."}
+{"_id": "22892", "title": "Definition:Restricted Universal Quantifier", "text": "Let $A$ be a class in ZF. The '''restricted universal quantifier''' is denoted $\\forall x \\in A$ and is defined as the following definitional abbreviation: :$\\forall x \\in A: P \\left({x}\\right) \\quad \\text{for} \\quad \\forall x: \\left({x \\in A \\implies P \\left({x}\\right)}\\right)$ where $P \\left({x}\\right)$ is any well-formed formula of the language of set theory. Category:Definitions/Predicate Logic Category:Definitions/Zermelo-Fraenkel Class Theory flsam9ljgwf1h557uj8ohqeqyhkdcua"}
+{"_id": "22893", "title": "Definition:Restricted Existential Quantifier", "text": "Let $A$ be a class in ZF. The '''restricted existential quantifier''' is denoted $\\exists x \\in A$ and is defined as the following definitional abbreviation: :$\\exists x \\in A: P \\left({x}\\right) \\quad \\text{for} \\quad \\exists x: \\left({x \\in A \\land P \\left({x}\\right)}\\right)$ where $P \\left({x}\\right)$ is any well-formed formula of the language of set theory. Category:Definitions/Predicate Logic Category:Definitions/Zermelo-Fraenkel Class Theory re80hztvvvqlai1nkih1dwbwjkeauzu"}
+{"_id": "22894", "title": "Definition:Language of Set Theory", "text": "{{refactor}} The language of set theory consists of the language of predicate logic with the binary predicate symbol $\\in$, denoting membership. === Predicate Symbols === The language of set theory uses only one predicate symbol, $\\in$, the membership sign. It is a binary predicate symbol. Using $\\in$, other symbols such as $=$ can be defined (see Definition:Set Equality). === Connectives === The language of set theory borrows the connectives from the language of predicate logic.  However, some of the connectives can be considered definitional abbreviations. $\\land$ and $\\neg$ can be taken as primitive connectives with the following definitional abbreviations: {{begin-eqn}} {{eqn | l = P \\implies Q       | o = \\operatorname{for}       | r = \\neg ( P \\land \\neg Q )       | c =  }} {{eqn | l = P \\lor Q       | o = \\operatorname{for}       | r = \\neg ( \\neg P \\land \\neg Q )       | c =  }} {{eqn | l = P \\iff Q       | o = \\operatorname{for}       | r = ( ( P \\implies Q ) \\land ( Q \\implies P ) )       | c = justified because $\\implies$ is already a definitional abbreviation }} {{end-eqn}} $\\implies$ and $\\neg$ can be taken as primitive connectives with the following definitional abbreviations: {{begin-eqn}} {{eqn | l = ( P \\land Q )       | o = \\operatorname{for}       | r = \\neg ( P \\implies \\neg Q )       | c =  }} {{eqn | l = ( P \\lor Q )       | o = \\operatorname{for}       | r = ( \\neg P \\implies Q )       | c =  }} {{eqn | l = ( P \\iff Q )       | o = \\operatorname{for}       | r = \\neg ( ( P \\implies Q ) \\implies \\neg ( Q \\implies P ) )       | c =  }} {{end-eqn}} The Sheffer stroke $\\mid$ can be taken as a sole primitive connective with the following definitional abbreviations: {{begin-eqn}} {{eqn | l = \\neg P       | o = \\operatorname{for}       | r = ( P \\mid P )       | c =  }} {{eqn | l = ( P \\implies Q )       | o = \\operatorname{for}       | r = ( P \\mid ( Q \\mid Q ) )       | c =  }} {{end-eqn}} The other connectives can be defined using $\\neg P$ and $( P \\implies Q )$ as the \"new\" primitive connectives. === Quantifiers === The language of set theory adopts the same quantifiers as those in the language of predicate logic. However, only $\\forall$ is necessary to adopt as a primitive symbol, and $\\exists$ can be defined: :$\\exists x: P(x) \\operatorname{for} \\neg \\forall x: \\neg P(x)$"}
+{"_id": "22895", "title": "Definition:Russell Class", "text": "The '''Russell class''', denoted $\\operatorname{Ru}$, equals the class of all sets $x$ that are not elements of themselves: :$\\operatorname{Ru} = \\set {x: x \\notin x}$"}
+{"_id": "22896", "title": "Definition:Class Equality", "text": "Let $A$ and $B$ be classes. === Definition 1 === {{:Definition:Class Equality/Definition 1}} === Definition 2 === {{:Definition:Class Equality/Definition 2}}"}
+{"_id": "22897", "title": "Definition:Class/Zermelo-Fraenkel", "text": "A '''class''' in $\\textrm{ZF}$ is a formal vehicle capturing the intuitive notion of a class, namely a collection of all sets such that a particular condition $P$ holds. In $\\textrm{ZF}$, '''classes''' are written using class builder notation: :$\\set {x : \\map P x}$ {{Questionable|Wouldn't you need the Class Comprehension Axiom for this?}} where $\\map P x$ is a statement containing $x$ as a free variable.   More formally, a '''class''' $\\set {x: \\map P x}$ serves to define the following definitional abbreviations involving the membership symbol: {{begin-eqn}} {{eqn | l = y \\in \\set {x: \\map P x}       | o = \\quad \\text{for} \\quad       | r = \\map P y }} {{eqn | l = \\set {x: \\map P x} \\in y       | o = \\quad \\text{for} \\quad       | r = \\exists z \\in y: \\forall x: \\paren {x \\in z \\iff \\map P x} }} {{eqn | l = \\set {x: \\map P x} \\in \\set {y: \\map Q y}       | o = \\quad \\text{for} \\quad       | r = \\exists z: \\paren {\\map Q z \\land \\forall x: \\paren {x \\in z \\iff \\map P x} } }} {{end-eqn}} where: :$x, y ,z$ are variables of $\\textrm{ZF}$ :$P, Q$ are propositional functions. Through these \"rules\", every statement involving $\\set {x: \\map P x}$ can be reduced to a simpler statement involving only the basic language of set theory."}
+{"_id": "22898", "title": "Definition:Small Class", "text": "Let $A$ denote an arbitrary class. Then $A$ is said to be '''small''' {{iff}}: :$\\exists x: x = A$ where $=$ denotes class equality and $x$ is a set variable. That is, a class is '''small''' {{iff}} it is equal to some set variable. To denote that a class $A$ is '''small''', the notation $\\mathcal M \\left({A}\\right)$ may be used. Thus: : $\\mathcal M \\left({A}\\right) \\iff \\exists x: x = A$"}
+{"_id": "22900", "title": "Definition:Set Variable", "text": "In the language of set theory, '''set variables''' are the variables. '''Set variables''' are sometimes referred to as sets. {{NoSources}} Category:Definitions/Set Theory ppbnk9rv8ysy1yvinsaiakf1mtaqdgs"}
+{"_id": "22901", "title": "Definition:Unordered Tuple", "text": "An '''unordered tuple''' is a small class in Zermelo-Fraenkel set theory that contains a finite number of elements. It is written using roster notation and is denoted: :$\\set {a_1, a_2, \\ldots, a_n}$ that is: :$\\set {a_1, a_2, \\ldots, a_n} = \\set {x: x = a_1 \\lor x = a_2 \\lor \\dots \\lor x = a_n}$"}
+{"_id": "22902", "title": "Definition:Category of Sets", "text": "The '''category of sets''', denoted $\\mathbf{Set}$ or $\\mathbf{Sets}$ is the metacategory with: {{DefineCategory |ob   = sets |mor  = mappings |comp = composition of mappings |id   = identity mappings }}"}
+{"_id": "22903", "title": "Definition:Category of Finite Sets", "text": "The '''category of finite sets''', denoted $\\mathbf{Finset}$ or $\\mathbf{Sets}_{\\text{fin}}$ is the metacategory with: {{DefineCategory |ob  = All finite sets; |mor = All mappings between finite sets. }}"}
+{"_id": "22904", "title": "Definition:Category of Ordered Sets", "text": "The '''category of ordered sets''', denoted $\\mathbf{OrdSet}$, is the metacategory with: {{DefineCategory | ob = ordered sets | mor = increasing mappings | comp = composition of mappings | id = identity mappings }}"}
+{"_id": "22905", "title": "Definition:Category of Relations", "text": "The '''category of (binary) relations''', denoted $\\mathbf{Rel}$, is the metacategory with: {{DefineCategory |ob   = sets |mor  = binary relations $\\mathcal R \\subseteq A \\times B$. |comp = composition of relations |id   = identity mappings }}"}
+{"_id": "22906", "title": "Definition:One (Category)", "text": "The '''category one''' $\\mathbf 1$, is the category with: {{DefineCategory |ob = One object: $*$ |mor = One morphism: the identity morphism $\\operatorname{id}_*$. }}"}
+{"_id": "22907", "title": "Definition:Two (Category)", "text": "The category $\\mathbf 2$, '''two''', is the category: :$\\quad * \\longrightarrow \\star$ with: :Two objects, $*$ and $\\star$  :Three morphisms: ::The identity morphism $I_*$ ::The identity morphism $I_\\star$ ::One non-identity morphism $* \\to \\star$"}
+{"_id": "22908", "title": "Definition:Three (Category)", "text": "The category $\\mathbf 3$, '''three''', is the category: ::$\\begin{xy} <0em,0em>*+{\\circledast} = \"x\", <5em,0em>*+{\\star} = \"y\", <5em,-5em>*+{\\bullet} = \"z\", \"x\";\"y\" **@{-} ?>*@{>}, \"y\";\"z\" **@{-} ?>*@{>}, \"x\";\"z\" **@{-} ?>*@{>} \\end{xy}$ with: :Three objects, $\\circledast$, $\\star$ and $\\bullet$; and  :Six morphisms: :* The three identity morphisms $\\operatorname{id}_\\circledast$, $\\operatorname{id}_\\star$, $\\operatorname{id}_\\bullet$; :* One non-identity morphism $\\circledast \\to \\star$; :* One non-identity morphism $\\star \\to \\bullet$; :* One non-identity morphism $\\circledast \\to \\bullet$."}
+{"_id": "22909", "title": "Definition:Zero (Category)", "text": "The category $\\mathbf 0$, '''zero''', is the empty category: :$\\qquad$ with: :No objects and consequently: :No morphisms."}
+{"_id": "22910", "title": "Definition:Functor/Covariant", "text": "Let $\\mathbf C$ and $\\mathbf D$ be metacategories. A '''covariant functor''' $F: \\mathbf C \\to \\mathbf D$ consists of: * An '''object functor''' $F_0$ that assigns to each object $X$ of $\\mathbf C$ an object $FX$ of $\\mathbf D$. * An '''arrow functor''' $F_1$ that assigns to each arrow $f: X \\to Y$ of $\\mathbf C$ an arrow $Ff : FX \\to FY$ of $\\mathbf D$. These '''functors''' must satisfy, for any morphisms $X \\stackrel f \\longrightarrow Y \\stackrel g \\longrightarrow Z$ in $\\mathbf C$: :$\\map F {g \\circ f} = F g \\circ F f$ and: :$\\map F {\\operatorname {id}_X} = \\operatorname{id}_{F X}$ where $\\operatorname {id}_W$ denotes the identity arrow on an object $W$, and $\\circ$ is the composition of morphisms. The behaviour of a '''covariant functor''' can be pictured as follows: ::$\\begin{xy} <4em,4em>*{\\mathbf C} = \"C\", <0em,0em>*+{X} = \"a\", <4em,0em>*+{Y} = \"b\", <4em,-4em>*+{Z}= \"c\", \"a\";\"b\" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{f}, \"b\";\"c\" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{g}, \"a\";\"c\" **@{-} ?>*@{>} ?<>(.5)*!/^1em/{g \\circ f}, \"C\"+/r9em/*{\\mathbf D}, \"C\"+/r2em/;\"C\"+/r6em/ **@{-} ?>*@{>} ?*!/_1em/{F}, \"b\"+/r2em/+/_2em/;\"b\"+/r6em/+/_2em/ **@{~} ?>*@2{>} ?<>(.5)*!/_.6em/{F}, \"a\"+/r13em/*+{FX}=\"Fa\",  \"b\"+/r13em/*+{FY}=\"Fb\",  \"c\"+/r13em/*+{FZ}=\"Fc\", \"Fa\";\"Fb\" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{Ff}, \"Fb\";\"Fc\" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{Fg}, \"Fa\";\"Fc\" **@{-} ?>*@{>} ?<>(.7)*!/r3em/{F \\left({g \\circ f}\\right) = \\\\ Fg \\circ Ff}, \\end{xy}$"}
+{"_id": "22911", "title": "Definition:Functor/Contravariant", "text": "{{:Definition:Functor/Contravariant/Definition 1}}"}
+{"_id": "22912", "title": "Definition:Composition of Functors", "text": "Let $\\mathbf C, \\mathbf D$ and $\\mathbf E$ be metacategories. Let $F: \\mathbf C \\to \\mathbf D$ and $G: \\mathbf D \\to \\mathbf E$ be (covariant) functors. The '''composition of $G$ with $F$''' is the functor $GF: \\mathbf C \\to \\mathbf E$ defined by: :For all objects $C$ of $\\mathbf C$: $\\hskip{2.9cm} GF \\left({C}\\right) := G \\left({FC}\\right)$ :For all morphisms $f: C_1 \\to C_2$ of $\\mathbf C$: $\\quad GF \\left({f}\\right) := G \\left({Ff}\\right)$ $GF$ is said to be a '''composite functor'''."}
+{"_id": "22913", "title": "Definition:Identity Functor", "text": "Let $\\mathbf C$ be a metacategory. The '''identity functor on $\\mathbf C$''' is the functor $\\operatorname{id}_{\\mathbf C}: \\mathbf C \\to \\mathbf C$ defined by: :For all objects $C$ of $\\mathbf C$: $\\hskip{2.9cm} \\operatorname{id}_{\\mathbf C} C := C$ :For all morphisms $f: C_1 \\to C_2$ of $\\mathbf C$: $\\quad \\operatorname{id}_{\\mathbf C} f := f$ That $\\operatorname{id}_{\\mathbf C}$ constitutes a functor is shown on Identity Functor is Functor."}
+{"_id": "22914", "title": "Definition:Category of Categories", "text": "The '''category of categories''', denoted $\\mathbf{Cat}$, is the metacategory with: {{DefineCategory |ob = small categories |mor = functors |comp = composition of functors |id = identity functors }}"}
+{"_id": "22915", "title": "Definition:Preorder Category", "text": "=== Definition 1 === {{:Definition:Preorder Category/Definition 1}} === Definition 2 === {{:Definition:Preorder Category/Definition 2}}"}
+{"_id": "22916", "title": "Definition:Order Category", "text": "=== Definition 1 === {{:Definition:Order Category/Definition 1}} === Definition 2 === {{:Definition:Order Category/Definition 2}}"}
+{"_id": "22917", "title": "Definition:Discrete Category", "text": "Let $\\CC$ be a metacategory. Then $\\CC$ is said to be '''discrete''' {{iff}} it comprises only identity morphisms. If the collection $\\CC$ constitutes the objects of $\\mathbf C$, then $\\mathbf C$ may also be denoted $\\map {\\mathbf {Dis} } \\CC$."}
+{"_id": "22918", "title": "Definition:Monoid Category", "text": "Let $\\left({S, \\circ}\\right)$ be a monoid with identity $e_S$. One can interpret $\\left({S, \\circ}\\right)$ as being a category, with: {{DefineCategory | ob = Only one, say $*$ | mor = $a: * \\to *$, for all $a \\in S$ | comp =  $a \\circ b: * \\to *$ is defined using the operation $\\circ$ of the monoid $S$ | id = $\\operatorname{id}_* := e_S: * \\to *$ }} The category that so arises is called a '''monoid category'''."}
+{"_id": "22919", "title": "Definition:Category of Monoids", "text": "The '''category of monoids''', denoted $\\mathbf{Mon}$, is the metacategory with: {{DefineCategory | ob = monoids | mor = monoid homomorphisms }}"}
+{"_id": "22920", "title": "Definition:Monoid Homomorphism", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be monoids. Let $\\phi: S \\to T$ be a mapping such that $\\circ$ has the morphism property under $\\phi$. That is, $\\forall a, b \\in S$: :$\\phi \\left({a \\circ b}\\right) = \\phi \\left({a}\\right) * \\phi \\left({b}\\right)$ Suppose further that $\\phi$ preserves identities, i.e.: :$\\phi \\left({e_S}\\right) = e_T$ Then $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ is a monoid homomorphism."}
+{"_id": "22921", "title": "Definition:Prime Power", "text": "A '''prime power''' $p^n$ is a positive integer which is a prime number $p$ raised to the power $n$. The list of prime powers begins: :$1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, \\ldots$ {{OEIS|A000961}} Category:Definitions/Number Theory Category:Definitions/Prime Numbers 1zfofa8rlmrr3xzwh7e6k0lmr49jy8e"}
+{"_id": "22922", "title": "Definition:Locally Small Category", "text": "Let $\\mathbf C$ be a metacategory. Then $\\mathbf C$ is said to be '''locally small''' iff all of its hom classes are sets. That is, iff for all objects $X, Y \\in \\mathbf C_0$ of $\\mathbf C$: :$\\operatorname{Hom}_{\\mathbf C} \\left({X, Y}\\right) = \\left\\{{f \\in \\mathbf C_1 }\\ \\middle\\vert \\ {f: X \\to Y}\\right\\}$ is a set."}
+{"_id": "22923", "title": "Definition:Projection Functor", "text": "Let $\\mathbf C$ and $\\mathbf D$ be metacategories, and let $\\mathbf C \\times \\mathbf D$ be their product. The '''first projection functor''' $\\operatorname{pr}_1: \\mathbf C \\times \\mathbf D \\to \\mathbf C$ is defined by: :$\\operatorname{pr}_1 \\left({C, D}\\right) := C$ for all objects $\\left({C, D}\\right) \\in \\operatorname{ob} \\mathbf C \\times \\mathbf D$ :$\\operatorname{pr}_1 \\left({f, g}\\right) := f$ for all morphisms $\\left({f, g}\\right) \\in \\operatorname{mor} \\mathbf C \\times \\mathbf D$ The '''second projection functor''' $\\operatorname{pr}_2: \\mathbf C \\times \\mathbf D \\to \\mathbf D$ is defined by: :$\\operatorname{pr}_2 \\left({C, D}\\right) := D$ for all objects $\\left({C, D}\\right) \\in \\operatorname{ob} \\mathbf C \\times \\mathbf D$ :$\\operatorname{pr}_2 \\left({f, g}\\right) := g$ for all morphisms $\\left({f, g}\\right) \\in \\operatorname{mor} \\mathbf C \\times \\mathbf D$ That these constitute functors is shown on Projection Functor is Functor."}
+{"_id": "22924", "title": "Definition:Dual Category", "text": "Let $\\mathbf C$ be a metacategory. Its '''dual category''', denoted $\\mathbf C^{\\text{op} }$, is defined as follows: {{DefineCategory |ob = $X^{\\text{op} }$, for all $X \\in \\operatorname{ob}\\mathbf C$ |mor = $f^{\\text{op} }: D^{\\text{op} } \\to C^{\\text{op} }$ for all $f: C \\to D$ in $\\mathbf C_1$ |comp = $\\left({f^{\\text{op} } \\circ g^{\\text{op} } }\\right) := \\left({g \\circ f}\\right)^{\\text{op} }$, whenever this is defined |id = $\\operatorname{id}_{X^{\\text{op} } } := \\operatorname{id}_X^{\\text{op} }$ }}"}
+{"_id": "22925", "title": "Definition:Morphism Category", "text": "Let $\\mathbf C$ be a metacategory. Its '''morphism category''', denoted $\\mathbf C^\\to$, is defined as follows: {{DefineCategory |ob = The morphisms $\\mathbf C_1$ of $\\mathbf C$ |mor = $g: f \\to f'$ is a pair $\\left({g_1, g_2}\\right)$ of morphisms of $\\mathbf C$ such that $g_2 \\circ f = f' \\circ g_1$ |comp = $\\left({h_1, h_2}\\right) \\circ \\left({g_1, g_2}\\right) := \\left({h_1 \\circ g_1, h_2 \\circ g_2}\\right)$, whenever this is defined |id = $\\operatorname{id}_{f} := \\left({\\operatorname{id}_C, \\operatorname{id}_D}\\right)$ for $f: C \\to D$ }}"}
+{"_id": "22926", "title": "Definition:Domain Functor", "text": "Let $\\mathbf C$ be a metacategory. Let $\\mathbf C^\\to$ be its morphism category. The '''domain functor''' is the functor $\\operatorname{\\mathbf{dom}}: \\mathbf C^\\to \\to \\mathbf C$ defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \\operatorname{\\mathbf{dom} } f := \\operatorname{dom} f }} {{axiom|lc= Morphism functor:        |m = \\operatorname{\\mathbf{dom} } \\left({g_1, g_2}\\right) := g_1 }} {{end-axiom}} That it is in fact a functor is shown on Domain Functor is Functor. The functor $\\mathbf{dom}$ can be represented as follows: ::$\\begin{xy} <0em,2em>*+{A} = \"A\", <0em,-2em>*+{B} = \"B\", <4em,2em>*+{A'} = \"A2\", <4em,-2em>*+{B'} = \"B2\", \"A\";\"B\" **@{-} ?>*@{>} ?*!/^1em/{f}, \"A\";\"A2\" **@{-} ?>*@{>} ?*!/_1em/{g_1}, \"A2\";\"B2\" **@{-} ?>*@{>} ?*!/_1em/{f'}, \"B\";\"B2\" **@{-} ?>*@{>} ?*!/^1em/{g_2}, <6em,0em>;<10em,0em> **@{~} ?>*@2{>} ?*!/_1em/{\\mathbf{dom}}, <12em,2em>*+{A} = \"AA\", <16em,2em>*+{A'} = \"AA2\", \"AA\";\"AA2\" **@{-} ?>*@{>} ?*!/_1em/{g_1}, \\end{xy}$ It is thus seen to be an example of a forgetful functor."}
+{"_id": "22927", "title": "Definition:Codomain Functor", "text": "Let $\\mathbf C$ be a metacategory. Let $\\mathbf C^\\to$ be its morphism category. The '''codomain functor''' is the functor $\\operatorname{\\mathbf{cod}}: \\mathbf C^\\to \\to \\mathbf C$ defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \\operatorname{\\mathbf{cod} } f := \\operatorname{cod} f }} {{axiom|lc= Morphism functor:        |m = \\operatorname{\\mathbf{cod} } \\left({g_1, g_2}\\right) := g_2 }} {{end-axiom}} That it is in fact a functor is shown on Codomain Functor is Functor. The functor $\\mathbf{cod}$ can be represented as follows: ::$\\begin{xy} <0em,2em>*+{A} = \"A\", <0em,-2em>*+{B} = \"B\", <4em,2em>*+{A'} = \"A2\", <4em,-2em>*+{B'} = \"B2\", \"A\";\"B\" **@{-} ?>*@{>} ?*!/^1em/{f}, \"A\";\"A2\" **@{-} ?>*@{>} ?*!/_1em/{g_1}, \"A2\";\"B2\" **@{-} ?>*@{>} ?*!/_1em/{f'}, \"B\";\"B2\" **@{-} ?>*@{>} ?*!/^1em/{g_2}, <6em,0em>;<10em,0em> **@{~} ?>*@2{>} ?*!/_1em/{\\mathbf{cod}}, <12em,-2em>*+{B} = \"BB\", <16em,-2em>*+{B'} = \"BB2\", \"BB\";\"BB2\" **@{-} ?>*@{>} ?*!/^1em/{g_2}, \\end{xy}$ It is thus seen to be an example of a forgetful functor."}
+{"_id": "22928", "title": "Definition:Slice Category", "text": "Let $\\mathbf C$ be a metacategory. Let $C \\in \\mathbf C_0$ be an object of $\\mathbf C$. The '''slice category''' of $\\mathbf C$ over $C$, denoted $\\mathbf C / C$, is defined as follows: {{DefineCategory |ob = $f: X \\to C$, i.e. the morphisms of $\\mathbf C$ with codomain $C$ |mor = $a: f \\to f'$, for all morphisms $a \\in \\mathbf C_1$ with $f' \\circ a = f$ |comp = $a \\circ b$ is defined precisely as in $\\mathbf C$ |id = $\\operatorname{id}_f := \\operatorname{id}_X$, for $f: X \\to C$ }} The morphisms can be displayed using a commutative diagram as follows: ::$\\begin{xy} <-3em,0em>*+{X} = \"X\", <3em,0em>*+{X'} = \"X2\", <0em,-4em>*+{C} = \"C\", \"X\";\"X2\" **@{-} ?>*@{>} ?*!/_1em/{a}, \"X\";\"C\" **@{-} ?>*@{>} ?<>(.3)*!/^1em/{f}, \"X2\";\"C\" **@{-} ?>*@{>} ?<>(.3)*!/_1em/{f'}, \\end{xy}$"}
+{"_id": "22929", "title": "Definition:Permutation on n Letters/Cycle Notation", "text": "The $k$-cycle $\\rho$ is denoted: :$\\begin {pmatrix} i & \\map \\rho i & \\ldots & \\map {\\rho^{k - 1} } i \\end{pmatrix}$ From Existence and Uniqueness of Cycle Decomposition, all permutations can be defined as the product of disjoint cycles. As Disjoint Permutations Commute, the order in which they are performed does not matter. So, for a given permutation $\\rho$, the '''cycle notation''' for $\\rho$ consists of all the disjoint cycles into which $\\rho$ can be decomposed, concatenated as a product. It is conventional to omit $1$-cycles from the expression, and to write those cycles with lowest starting number first."}
+{"_id": "22930", "title": "Definition:Signum Function/Natural Numbers", "text": "The '''signum function''' $\\sgn: \\N \\to \\set {0, 1}$ is the restriction of the signum function to the natural numbers, defined as: :$\\forall n \\in \\N: \\map \\sgn n := \\begin{cases} 0 & : n = 0 \\\\ 1 & : n > 0 \\end{cases}$"}
+{"_id": "22931", "title": "Definition:Signum Function/Signum Complement", "text": "Let $\\sgn: \\N \\to \\set {0, 1}$ be the signum function on the natural numbers. The '''signum complement function''' $\\overline \\sgn: \\N \\to \\set {0, 1}$ is defined as: :$\\forall n \\in \\N: \\map {\\overline \\sgn} n := \\begin{cases} 1 & : n = 0 \\\\ 0 & : n > 0 \\end{cases}$"}
+{"_id": "22933", "title": "Definition:Sign of Permutation", "text": "Let $n \\in \\N$ be a natural number. Let $\\N_n$ denote the set of natural numbers $\\set {1, 2, \\ldots, n}$. Let $\\tuple {x_1, x_2, \\ldots, x_n}$ be an ordered $n$-tuple of real numbers. Let $\\pi$ be a permutation of $\\N_n$. Let $\\map {\\Delta_n} {x_1, x_2, \\ldots, x_n}$ be the product of differences of $\\tuple {x_1, x_2, \\ldots, x_n}$. Let $\\pi \\cdot \\map {\\Delta_n} {x_1, x_2, \\ldots, x_n}$ be defined as: :$\\pi \\cdot \\map {\\Delta_n} {x_1, x_2, \\ldots, x_n} := \\map {\\Delta_n} {x_{\\map \\pi 1}, x_{\\map \\pi 2}, \\ldots, x_{\\map \\pi n} }$ The '''sign of $\\pi \\in S_n$''' is defined as: :$\\map \\sgn \\pi = \\begin{cases} \\dfrac {\\Delta_n} {\\pi \\cdot \\Delta_n} & : \\Delta_n \\ne 0 \\\\ 0 & : \\Delta_n = 0 \\end{cases}$"}
+{"_id": "22934", "title": "Definition:Composition Functor on Slice Categories", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ and $D$ be objects of $\\mathbf C$. Let $\\mathbf C / C$ and $\\mathbf C / D$ be the associated slice categories. Let $g: C \\to D$ be a morphism of $\\mathbf C$. Then $g$ defines a '''composition functor''' $g_* : \\mathbf C / C \\to \\mathbf C / D$: {{begin-axiom}} {{axiom|lc= Object functor:        |m = g_* f := g \\circ f        |rc= The composition $\\circ$ is taken in $\\mathbf C$ }} {{axiom|lc= Morphism functor:        |m = g_* a := a }} {{end-axiom}} That it is in fact a functor is shown on Composition Functor on Slice Categories is Functor. The effect of $g_*$ is captured in the following commutative diagram: ::$\\begin{xy} <-3em,0em>*+{X} = \"X\", <3em,0em>*+{X'} = \"X2\", <0em,-4em>*+{C} = \"C\", <0em,-8em>*+{D} = \"D\", \"X\";\"X2\" **@{-} ?>*@{>} ?*!/_1em/{a}, \"X\";\"C\" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{f}, \"X2\";\"C\" **@{-} ?>*@{>} ?<>(.4)*!/_.6em/{f'}, \"C\";\"D\" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{g}, \"X\";\"D\" **\\crv{<-5em,-4em>} ?>*@{>} ?*!/^1.6em/{g_* f = \\\\ g \\circ f}, \"X2\";\"D\" **\\crv{<5em,-4em>} ?>*@{>} ?*!/_1.6em/{g_* f' = \\\\ g \\circ f'}, \\end{xy}$"}
+{"_id": "22935", "title": "Definition:Slice Functor", "text": "Let $\\mathbf C$ be a metacategory. Let $\\mathbf{Cat}$ be the category of categories. The '''slice functor''' is the functor $\\mathbf C / \\cdot: \\mathbf C \\to \\mathbf{Cat}$ defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \\mathbf C / C := \\mathbf C / C }} {{axiom|lc= Morphism functor:        |m = \\mathbf C / f := f_* }} {{end-axiom}} where $\\mathbf C / C$ is a slice category, and $f_*$ is the composition functor defined by $f$. The effect of $\\mathbf C / \\cdot$ is captured in the following diagram: ::$\\begin{xy} <0em,0em>*+{A} = \"a\", <4em,0em>*+{B} = \"b\", <4em,-4em>*+{C}= \"c\", \"a\";\"b\" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{f}, \"b\";\"c\" **@{-} ?>*@{>} ?<>(.5)*!/_.6em/{g}, \"a\";\"c\" **@{-} ?>*@{>} ?<>(.4)*!/^1em/{g \\circ f}, \"b\"+/r4em/+/_3em/;\"b\"+/r8em/+/_3em/ **@{~} ?>*@2{>} ?*!/_1em/{\\mathbf C / \\cdot}, \"a\"+/r13em/*+{\\mathbf C / A}=\"Fa\",  \"b\"+/r14em/*+{\\mathbf C / B}=\"Fb\",  \"c\"+/r14em/+/_1em/*+{\\mathbf C / C}=\"Fc\", \"Fa\";\"Fb\" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{f_*}, \"Fb\";\"Fc\" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{g_*}, \"Fa\";\"Fc\" **@{-} ?>*@{>} ?<>(.7)*!/r3em/{\\left({g \\circ f}\\right)_* \\\\ = g_* f_*}, \\end{xy}$ where $g_* f_*$ denotes a composite functor."}
+{"_id": "22936", "title": "Definition:Coslice Category", "text": "Let $\\mathbf C$ be a metacategory. Let $C \\in \\mathbf C_0$ be an object of $\\mathbf C$. The '''coslice category''' of $\\mathbf C$ under $C$, denoted $C / \\mathbf C$, is the category with: {{DefineCategory |ob = $f: C \\to X$, i.e. the morphisms of $\\mathbf C$ with domain $C$ |mor  = $a: f \\to f'$, for all morphisms $a \\in \\mathbf C_1$ with $a \\circ f' = f$ |comp = $a \\circ b$ is defined precisely as in $\\mathbf C$ |id = $\\operatorname{id}_f := \\operatorname{id}_X$, for $f: C \\to X$ }} The morphisms can be displayed using a commutative diagram as follows: ::$\\begin{xy} <-3em,0em>*+{X} = \"X\", <3em,0em>*+{X'} = \"X2\", <0em,4em>*+{C} = \"C\", \"X\";\"X2\" **@{-} ?>*@{>} ?*!/^1em/{a}, \"C\";\"X\" **@{-} ?>*@{>} ?*!/^.6em/{f}, \"C\";\"X2\" **@{-} ?>*@{>} ?<>(.6)*!/_1em/{f'}, \\end{xy}$"}
+{"_id": "22937", "title": "Definition:Category of Pointed Sets", "text": "The '''category of pointed sets''', denoted $\\mathbf {Set}_*$, is defined as follows: {{DefineCategory | ob = pointed sets $\\struct {A, a}$ | mor = pointed mappings | comp = Standard composition of mappings | id = $\\map {\\mathrm {id}_{\\struct {A, a} } } := \\mathrm {id}_A$, the identity mapping on $A$ }}"}
+{"_id": "22938", "title": "Definition:Kleene Closure", "text": "Let $S$ be a set. The '''Kleene closure''' of $S$, denoted $S^*$, is the set of all ordered tuples in $S$."}
+{"_id": "22939", "title": "Definition:Concatenation of Ordered Tuples", "text": "Let $S$ be a set. Let $w, w'$ be finite sequences in $S$ of lengths $n$ and $n'$, respectively. Then the '''concatenation of $w$ and $w'$''', denoted $w * w'$ or simply $w w'$, is the sequence of $n + n'$ terms defined by: :$\\map {w * w'} i := \\begin{cases} \\map w i & : \\text {if $1 \\le i \\le n$} \\\\ \\map {w'} {i - n} & : \\text {if $n < i \\le n + n'$} \\end{cases}$ === Algebraic Structure === {{refactor}} Let $S^*$ be the Kleene closure of $S$. Then $\\struct {S^*, *}$ is an algebraic structure, and by definition a magma."}
+{"_id": "22940", "title": "Definition:Insertion of Generators", "text": "Let $S$ be a set, and let $S^*$ be its Kleene closure. The '''insertion of generators (into $S^*$)''' is the mapping $i: S \\to S^*$ defined by: :$i \\left({s}\\right) := \\left\\langle{s}\\right\\rangle$ that is, it sends any element $s$ of $S$ to the one-term sequence containing only $s$."}
+{"_id": "22941", "title": "Definition:Underlying Set Functor", "text": "Let $\\mathbf{Set}$ be the category of sets. Let $\\mathbf C$ be a metacategory with: :A collection of sets as objects; :A collection of mappings as morphisms. The '''underlying set functor''' $\\left\\vert{\\cdot}\\right\\vert: \\mathbf C \\to \\mathbf{Set}$ is defined in the following cases. {{expand|more to come}} === Category of Monoids === {{:Definition:Underlying Set Functor/Category of Monoids}} Category:Definitions/Category Theory 3ri175rmtt6vzp0uekdndgk3fhoahuu"}
+{"_id": "22942", "title": "Definition:Underlying Set Functor/Category of Monoids", "text": "Let $\\mathbf{Mon}$ be the category of monoids. The '''underlying set functor''' $\\left\\vert{\\cdot}\\right\\vert : \\mathbf{Mon} \\to \\mathbf{Set}$ is the functor defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \\left\\vert{\\left({M, \\circ}\\right)}\\right\\vert := M }} {{axiom|lc= Morphism functor:        |m = \\left\\vert{f}\\right\\vert := f }} {{end-axiom}}"}
+{"_id": "22943", "title": "Definition:Free Monoid", "text": "Let $S$ be a set. A '''free monoid over $S$''' is a monoid $M$ together with a mapping $i: S \\to M$, subject to: :For all monoids $N$, for all mappings $f: S \\to N$, there is a unique monoid homomorphism $\\bar f: M \\to N$, such that: ::$\\bar f \\circ i = f$ This condition is called the '''universal (mapping) property''' or '''UMP''' of the '''free monoid over $S$'''."}
+{"_id": "22944", "title": "Definition:Cantor Normal Form", "text": "Let $x$ be an ordinal. The '''Cantor normal form''' of $x$ is an ordinal summation: :$x = \\omega^{a_1} n_1 + \\dots + \\omega^{a_k} n_k$ where: :$k \\in \\N$ is a natural number :$\\omega$ is the minimal infinite successor set :$\\langle a_i \\rangle$ is a strictly decreasing finite sequence of ordinals. :$\\langle n_i \\rangle$ is a finite sequence of finite ordinals In summation notation: :$x = \\displaystyle \\sum_{i \\mathop = 1}^k \\omega^{a_i} n_i$ {{explain|It still needs to be explained why, when used in pages that link to this, that the summation does not include the object $\\omega$ in it, just some ordinal $x$ instead. It is unclear exactly what this definition means, because $\\omega$, as currently defined on this website, is the Definition:Minimal Infinite Successor Set. Thus this definition appears to be saying: \"Every ordinal (which of course has to include finite ones) can be expressed as finite sums of infinite ordinals.\" How can a finite number (an ordinal is a number, right?) be expressed as the sum of infinite numbers?}}"}
+{"_id": "22945", "title": "Definition:Free Category", "text": "Let $G$ be a digraph. The '''free category''' on $G$, denoted $\\map {\\mathbf C} G$, is the category with: {{DefineCategory | ob = The vertices of $G$ | mor = The walks in $G$ | comp = concatenation of walks | id = $\\operatorname{id}_v$ is the empty walk at $v$ }}"}
+{"_id": "22946", "title": "Definition:Graph Functor", "text": "Let $\\mathbf{Set}$ and $\\mathbf{Rel}$ be the category of sets and the category of relations, respectively. The '''graph functor''' $G: \\mathbf{Set} \\to \\mathbf{Rel}$ is defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |t = $GX := X$ }} {{axiom|lc= Morphism functor:        |t = $Gf := G_f$, the graph of $f$ }} {{end-axiom}}"}
+{"_id": "22947", "title": "Definition:Standard Transitive Model", "text": "Let $p$ be a WFF in the language of set theory. Let $A$ be a class. Then $A$ is a '''standard transitive model''' of $p$ {{iff}}: :$A \\ne \\varnothing$ :$A$ is a transitive class and: :$A \\models p$"}
+{"_id": "22948", "title": "Definition:Inverse Relation Functor", "text": "Let $\\mathbf{Rel}$ be the category of relations. The '''inverse relation (contravariant) functor''' $C: \\mathbf{Rel} \\to \\mathbf{Rel}$ is the contravariant functor defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |t = $CX := X$ }} {{axiom|lc= Morphism functor:        |t = $C \\mathcal R := \\mathcal R^{-1}$, the inverse relation to $\\mathcal R$ }} {{end-axiom}} That it is in fact a contravariant functor is shown on Inverse Relation Functor is Contravariant Functor."}
+{"_id": "22949", "title": "Definition:Transitive Closure (Set Theory)", "text": "=== Definition 1 === {{:Definition:Transitive Closure (Set Theory)/Definition 1}} The following is not equivalent to the above, but they are almost the same. === Definition 2 === {{:Definition:Transitive Closure (Set Theory)/Definition 2}}"}
+{"_id": "22951", "title": "Definition:Split Monomorphism", "text": "Let $\\mathbf C$ be a metacategory. Let $f: C \\to D$ be a morphism of $\\mathbf C$. Then $f$ is said to be a '''split monomorphism''' iff for some $g: D \\to C$, one has: :$g \\circ f = \\operatorname{id}_C$ where $\\operatorname{id}_C$ is the identity morphism of $C$."}
+{"_id": "22952", "title": "Definition:Split Epimorphism", "text": "Let $\\mathbf C$ be a metacategory. Let $f: C \\to D$ be a morphism of $\\mathbf C$. Then $f$ is said to be a '''split epimorphism''' {{iff}} for some $g: D \\to C$, one has: :$f \\circ g = \\operatorname{id}_D$ where $\\operatorname{id}_D$ is the identity morphism of $D$. That is, {{iff}} $f$ has a section."}
+{"_id": "22953", "title": "Definition:Section (Category Theory)", "text": "Let $\\mathbf C$ be a metacategory. Let $f: C \\to D$ be a morphism of $\\mathbf C$. A '''section of $f$''' is a morphism $g: D \\to C$ such that: :$f \\circ g = \\operatorname{id}_D$"}
+{"_id": "22954", "title": "Definition:Retraction", "text": "Let $\\mathbf C$ be a metacategory. Let $f: C \\to D$ be a morphism of $\\mathbf C$. A '''retraction of $f$''' is a morphism $g: D \\to C$ such that: :$g \\circ f = \\operatorname{id}_C$ === Retract === {{:Definition:Retract}}"}
+{"_id": "22955", "title": "Definition:Retract", "text": "Let $g: D \\to C$ be a retraction of $f$. Then $D$ is said to be a '''retract''' of $C$."}
+{"_id": "22956", "title": "Definition:Projective Object", "text": "Let $\\mathbf C$ be a metacategory. Let $P \\in \\mathbf C_0$ be an object of $\\mathbf C$. Then $P$ is said to be '''projective''' iff: :For all epimorphisms $e: E \\twoheadrightarrow X$ and morphisms $f: P \\to X$, there exists $\\bar f: P \\to E$ such that: ::$\\begin{xy} <0em,0em>*+{P} = \"P\", <4em,0em>*+{X} = \"X\", <4em,4em>*+{E} = \"E\", \"P\";\"E\" **@{-} ?>*@{>} ?*!/_.6em/{\\bar f}, \"P\";\"X\" **@{-} ?>*@{>} ?*!/^.6em/{f}, \"E\";\"X\" **@{-} ?>*@2{>} ?<>(.7)*{\\vee} ?*!/_.6em/{e}, \\end{xy}$ :is a commutative diagram, i.e. such that $f = e \\circ \\bar f$. In this situation, $f$ is said to '''lift across $e$'''."}
+{"_id": "22957", "title": "Definition:Transitive with Respect to a Relation", "text": "Let $A$ be a class. Let $\\RR$ be a relation on $A$. Let $S$ be a set. Then $S$ is '''transitive with respect to $\\RR$''' {{iff}}: :$\\forall x \\in A: \\forall y \\in S: \\paren {x \\mathrel \\RR y \\implies x \\in S}$"}
+{"_id": "22958", "title": "Definition:Initial Object", "text": "Let $\\mathbf C$ be a metacategory. An '''initial object''' of $\\mathbf C$ is an object $0 \\in \\mathbf C_0$ of $\\mathbf C$ such that: :For all $C \\in \\mathbf C_0$, there is a unique morphism $0 \\to C$."}
+{"_id": "22959", "title": "Definition:Terminal Object", "text": "Let $\\mathbf C$ be a metacategory. A '''terminal object''' of $\\mathbf C$ is an object $1 \\in \\mathbf C_0$ of $\\mathbf C$ such that: :For all $C \\in \\mathbf C_0$, there is a unique morphism $C \\to 1$."}
+{"_id": "22960", "title": "Definition:Orbit (Group Theory)/Length", "text": "The '''length''' of the orbit $\\Orb x$ of $x$ is the number of elements of $X$ it contains: :$\\size {\\Orb x}$"}
+{"_id": "22961", "title": "Definition:Orbit (Group Theory)/Set of Orbits", "text": "The quotient set $X / \\mathcal R_G$ is called the '''set of orbits of $X$ under the action of $G$'''."}
+{"_id": "22962", "title": "Definition:Relational Closure", "text": "Let $A$ be a class. Let $\\RR \\subseteq A \\times A$ be a relation. Let $x$ be a small class that is a subset of $A$. The '''relational closure''' of $x$ is the smallest small class containing $x$ that is also $\\RR$-transitive."}
+{"_id": "22963", "title": "Definition:Closed Relation", "text": "Let $A$ be a class. Let $\\mathcal R$ be an arbitrary relation. Then $\\mathcal R$ is '''closed''' with respect to $A$ iff $A$ is transitive with respect to $\\mathcal R^{-1}$."}
+{"_id": "22964", "title": "Definition:Supertransitive Class", "text": "Let $A$ be a transitive class. Then $A$ is said to be a '''supertransitive class''' {{iff}}: :$\\forall x \\in A: \\powerset x \\subseteq A$ That is, if $A$ contains the power set of all of its elements. {{explain|This phrasing is a bit awkward; element and subset are bound to be confused}}"}
+{"_id": "22965", "title": "Definition:Von Neumann Hierarchy", "text": "Let $U$ denote the universal class. The '''von Neumann hierarchy''' is a mapping $V: \\On \\to U$ on the ordinals, defined via transfinite recursion: :$\\map V x = \\begin{cases} \\O & : x = 0 \\\\ & \\\\ \\powerset {\\map V n} & : x = n^+ \\\\ & \\\\ \\displaystyle \\bigcup_{y \\mathop \\in x} \\map V y & : x \\in \\operatorname {Lim} \\\\ \\end{cases}$ where: :$\\powerset x$ denotes the power set of $x$ :$\\operatorname {Lim}$ denotes the set of limit ordinals."}
+{"_id": "22966", "title": "Definition:Boolean Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Then $R$ is called a '''Boolean ring''' {{iff}} $R$ is an idempotent ring with unity."}
+{"_id": "22967", "title": "Definition:Well-Founded Set", "text": "Let $S$ be a small class. Let $V \\left({ x }\\right)$ denote the von Neumann hierarchy. Then $S$ is a '''well-founded set''' {{iff}} there is some ordinal $x$ such that $S \\in V\\left({x}\\right)$."}
+{"_id": "22968", "title": "Definition:Rank (Set Theory)", "text": "Let $A$ be a set. Let $V$ denote the von Neumann hierarchy. Then the '''rank''' of $A$ is the smallest ordinal $x$ such that $A \\in V \\left({x+1}\\right)$, given that $x$ exists."}
+{"_id": "22969", "title": "Definition:Prime Number/Negative Prime", "text": "A '''negative prime''' is an integer of the form $-p$ where $p$ is a (positive) prime number."}
+{"_id": "22970", "title": "Definition:Composite Number/Negative Composite", "text": "A negative integer $n$ is composite {{iff}} $\\left|{n}\\right|$ is composite."}
+{"_id": "22971", "title": "Definition:Prime Triplet", "text": "A '''prime triplet''' is a set of three (positive) prime numbers $\\left\\{{n, n+2, n+4}\\right\\}$."}
+{"_id": "22972", "title": "Definition:Power (Algebra)/Integer", "text": "Let $x \\in \\R$ be a real number. Let $n \\in \\Z$ be an integer. The expression $x^n$ is called '''$x$ to the power of $n$'''. $x^n$ is defined recursively as: :$x^n = \\begin{cases} 1 & : n = 0 \\\\ & \\\\ x \\times x^{n - 1} & : n > 0 \\\\ & \\\\ \\dfrac {x^{n + 1} } x & : n < 0 \\end{cases}$ where $\\dfrac{x^{n + 1} } x$ denotes quotient."}
+{"_id": "22973", "title": "Definition:Power (Algebra)/Even Power", "text": "Let $x \\in \\R$ be a real number. Let $n \\in \\Z$ be an even integer. Then $x^n$ is called an '''even power of $x$'''."}
+{"_id": "22974", "title": "Definition:Power (Algebra)/Odd Power", "text": "Let $x \\in \\R$ be a real number. Let $n \\in \\Z$ be an odd integer. Then $x^n$ is called an '''odd power of $x$'''"}
+{"_id": "22975", "title": "Definition:Power (Algebra)/Rational Number", "text": "Let $x \\in \\R$ be a real number such that $x > 0$. Let $m \\in \\Z$ be an integer. Let $y = \\sqrt [m] x$ be the $m$th root of $x$. Then we can write $y = x^{1/m}$ which means the same thing as $y = \\sqrt [m] x$. Thus we can define the power to a positive rational number: Let $r = \\dfrac p q \\in \\Q$ be a positive rational number where $p \\in \\Z_{\\ge 0}, q \\in \\Z_{> 0}$. Then $x^r$ is defined as: :$x^r = x^{p/q} = \\paren {\\sqrt [q] x}^p = \\sqrt [q] {\\paren {x^p} }$. When $r = \\dfrac {-p} q \\in \\Q: r < 0$ we define: :$x^r = x^{-p/q} = \\dfrac 1 {x^{p/q}}$ analogously for the negative integer definition."}
+{"_id": "22976", "title": "Definition:Power (Algebra)/Real Number", "text": "Let $x \\in \\R_{>0}$ be a (strictly) positive real number. Let $r \\in \\R$ be a real number. === Definition 1 === {{:Definition:Power (Algebra)/Real Number/Definition 1}} === Definition 2 === {{:Definition:Power (Algebra)/Real Number/Definition 2}} === Definition 3 === {{:Definition:Power (Algebra)/Real Number/Definition 3}}"}
+{"_id": "22977", "title": "Definition:Power (Algebra)/Complex Number", "text": "Let $z, k \\in \\C$ be any complex numbers. '''$z$ to the power of $k$''' is defined as the multifunction: :$z^k := e^{k \\ln \\paren z}$ where $e^z$ is the exponential function and $\\ln$ is the  natural logarithm multifunction."}
+{"_id": "22978", "title": "Definition:Power (Algebra)/Power of Zero", "text": "Let $r \\in \\R$ be a real number. (This includes the situation where $r \\in \\Z$ or $r \\in \\Q$.) When $x=0$, $x^r$ is defined as follows: :$0^r = \\begin{cases} 1 & : r = 0 \\\\ 0 & : r > 0 \\\\ \\text{Undefined} & : r < 0 \\\\ \\end{cases}$ This takes account of the awkward case $0^0$: it is \"generally accepted\" that $0^0 = 1$ as this convention agrees with certain general results which would otherwise need a special case."}
+{"_id": "22979", "title": "Definition:Power (Algebra)/Exponent", "text": "In the expression $x^r$, the number $r$ is known as the '''exponent''' of $x$, particularly for $r \\in \\R$."}
+{"_id": "22980", "title": "Definition:Two (Boolean Algebra)", "text": "Denote with $\\top$ the canonical tautology. Denote with $\\bot$ the canonical contradiction. Define $\\mathbf 2 := \\set {\\bot, \\top}$, read '''two'''. When endowed with the logical operations $\\lor$, $\\land$ and $\\neg$, $\\mathbf 2$ becomes a Boolean algebra. These operations have the following Cayley tables: :$\\begin{array}{c|cc}  \\lor & \\bot & \\top \\\\ \\hline  \\bot & \\bot & \\top \\\\  \\top & \\top & \\top \\end{array} \\qquad \\begin{array}{c|cc}  \\land & \\bot & \\top \\\\ \\hline  \\bot  & \\bot & \\bot \\\\  \\top  & \\bot & \\top \\end{array} \\qquad \\begin{array}{c|cc}       & \\bot & \\top \\\\ \\hline  \\neg & \\top & \\bot \\end{array}$ That $\\mathbf 2$ thus becomes a Boolean algebra is shown on Two is Boolean Algebra."}
+{"_id": "22982", "title": "Definition:Category of Boolean Algebras", "text": "The '''category of Boolean algebras''', denoted $\\mathbf{BA}$, is the metacategory with: {{DefineCategory |ob  = Boolean algebras |mor =  Boolean homomorphisms }}"}
+{"_id": "22983", "title": "Definition:Operation/Unary Operation", "text": "A '''unary operation''' is the special case of an operation where the operation has exactly one operand. Thus, a '''unary operation''' on a set $S$ is a mapping whose domain and codomain are both $S$."}
+{"_id": "22985", "title": "Definition:Constant (Category Theory)", "text": "Let $\\mathbf C$ be a metacategory, and let $1$ be a terminal object of $\\mathbf C$. A '''constant of $\\mathbf C$''' is a morphism $f: 1 \\to C$ of $\\mathbf C$ which has $1$ as its domain."}
+{"_id": "22986", "title": "Definition:Variable Element", "text": "Let $\\mathbf C$ be a metacategory, and let $C$ be an object of $\\mathbf C$. A '''variable element''' of $C$ is a morphism $f: B \\to C$ with codomain $C$."}
+{"_id": "22987", "title": "Definition:Enough Constants", "text": "Let $\\mathbf C$ be a metacategory. Then $\\mathbf C$ is said to have '''enough constants''' iff: :For all morphisms $f,g : C \\to D$ with $f \\ne g$, there is a constant $c: 1 \\to C$ such that $f \\circ c \\ne g \\circ c$"}
+{"_id": "22989", "title": "Definition:Product (Category Theory)", "text": "=== Binary Product === {{:Definition:Product (Category Theory)/Binary Product}} === General Definition === {{:Definition:Product (Category Theory)/General Definition}}"}
+{"_id": "22990", "title": "Definition:Cardinal Number", "text": "Let $S$ be a set. The '''cardinal number''' of $S$ is defined as follows: :$\\card S = \\displaystyle \\bigcap \\set {x \\in \\On : x \\sim S}$ where $\\On$ is the class of all ordinals. Compare cardinality."}
+{"_id": "22991", "title": "Definition:Aleph Mapping", "text": "Let $\\mathcal N'$ denote the class of all infinite cardinal numbers. Then $\\aleph$ (that is: '''aleph''') is defined as the unique order isomorphism between the two ordered structures $\\left({\\operatorname{On}, \\in}\\right)$ and $\\left({\\mathcal N', \\in}\\right)$ where $\\operatorname{On}$ denotes the class of ordinal numbers. {{explain|A link is needed to a proof that such a unique order isomorphism exists.}}"}
+{"_id": "22992", "title": "Definition:Class of Cardinals", "text": "The '''class of cardinals''' is the class consisting of all cardinal numbers: :$\\NN = \\set {x \\in \\On: \\exists y: x = \\size y}$ where $\\size y$ denotes the cardinal number corresponding to $y$."}
+{"_id": "22993", "title": "Definition:Infinite Cardinal Class", "text": "The '''infinite cardinal class''' is the class of all infinite cardinal numbers. That is: :$\\mathcal N’ = \\mathcal N \\setminus \\omega$"}
+{"_id": "22994", "title": "Definition:Definiendum", "text": "An object being defined in a definition is called a '''definiendum'''."}
+{"_id": "22995", "title": "Definition:Definiens", "text": "The part of a definition that explains the meaning of the definiendum is the '''definiens'''."}
+{"_id": "22996", "title": "Definition:Group Axioms/Right", "text": "A '''group''' is an algebraic structure $\\struct {G, \\circ}$ which satisfies the following four conditions: {{begin-axiom}} {{axiom | n = \\text G 0         | lc= Closure Axiom         | q = \\forall a, b \\in G         | m = a \\circ b \\in G }} {{axiom | n = \\text G 1         | lc= Associativity Axiom         | q = \\forall a, b, c \\in G         | m = a \\circ \\paren {b \\circ c} = \\paren {a \\circ b} \\circ c }} {{axiom | n = \\text G_{\\text R} 2         | lc= Right Identity Axiom         | q = \\exists e \\in G: \\forall a \\in G         | m = a \\circ e = a }} {{axiom | n = \\text G_{\\text R} 3         | lc= Right Inverse Axiom         | q = \\forall a \\in G: \\exists b \\in G         | m = a \\circ b = e }} {{end-axiom}}"}
+{"_id": "22997", "title": "Definition:Group Axioms/Left", "text": "A '''group''' is an algebraic structure $\\struct {G, \\circ}$ which satisfies the following four conditions: {{begin-axiom}} {{axiom | n = \\text G 0         | lc= Closure Axiom         | q = \\forall a, b \\in G         | m = a \\circ b \\in G }} {{axiom | n = \\text G 1         | lc= Associativity Axiom         | q = \\forall a, b, c \\in G         | m = a \\circ \\paren {b \\circ c} = \\paren {a \\circ b} \\circ c }} {{axiom | n = \\text G_{\\text L} 2         | lc= Left Identity Axiom         | q = \\exists e \\in G: \\forall a \\in G         | m = e \\circ a = a }} {{axiom | n = \\text G_{\\text L} 3         | lc= Left Inverse Axiom         | q = \\forall a \\in G: \\exists b \\in G         | m = b \\circ a = e }} {{end-axiom}}"}
+{"_id": "22998", "title": "Definition:Product of Morphisms", "text": "Let $\\mathbf C$ be a metacategory. Let $A, A'$ and $B, B'$ be pairs of objects admitting binary products: ::$\\begin{xy}\\xymatrix@R-1em@C+1em@L+3px{  A &  A \\times A'   \\ar[l]_*+{p_1}   \\ar[r]^*+{p_2} &  A' \\\\  B &  B \\times B'   \\ar[l]_*+{q_1}   \\ar[r]^*+{q_2} &  B' }\\end{xy}$ Let $f: A \\to B$ and $f': A' \\to B'$ be morphisms. The '''product morphism of $f$ and $f'$''', denoted $f \\times f'$, is the unique morphism making the following diagram commute: ::$\\begin{xy}\\xymatrix@+1em@L+3px{  A   \\ar[d]_*+{f} &  A \\times A'   \\ar[l]_*+{p_1}   \\ar[r]^*+{p_2}   \\ar@{-->}[d]^*+{\\hskip{1.3em} f \\times f'} &  A'   \\ar[d]^*+{f'} \\\\  B &  B \\times B'   \\ar[l]^*+{q_1}   \\ar[r]_*+{q_2} &  B' }\\end{xy}$ Thus we see that $f \\times f'$ is the morphism $\\left\\langle{fp_1, f'p_2}\\right\\rangle$."}
+{"_id": "22999", "title": "Definition:Fixed Element of Permutation/Moved", "text": "$x$ '''moved by $\\pi$''' {{iff}}: :$\\map \\pi x \\ne x$"}
+{"_id": "23000", "title": "Definition:Fixed Element of Permutation/Set of Fixed Elements", "text": "The set of elements of $S$ which are fixed by $\\pi$ can be denoted $\\Fix \\pi$."}
+{"_id": "23001", "title": "Definition:Permutation on n Letters/Set of Permutations", "text": "The '''set of permutations of $\\N_n$''' is denoted $S_n$."}
+{"_id": "23002", "title": "Definition:Product Functor", "text": "Let $\\mathbf C$ be a metacategory with binary products. Let $\\mathbf C \\times \\mathbf C$ be the product category of $\\mathbf C$ with itself. The '''product functor''' on $\\mathbf C$ is the functor $\\times: \\mathbf C \\times \\mathbf C \\to \\mathbf C$ defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \\times \\left({C, D}\\right) := C \\times D }} {{axiom|lc= Morphism functor:        |m = \\times \\left({f, f'}\\right) := f \\times f' }} {{end-axiom}} where $C \\times D$ is a binary product of $C$ and $D$, and $f \\times f'$ is the product of $f$ and $f'$. That it is in fact a functor is shown on Product Functor is Functor."}
+{"_id": "23003", "title": "Definition:One (Ordinal)", "text": "The ordinal $1$ is defined as: :$1 = 0^+$ where $0^+$ denotes the successor of the ordinal $0$. Category:Definitions/Ordinals lm8rlyljlz9gn8x9nhximmti7zfq0if"}
+{"_id": "23004", "title": "Definition:Two (Ordinal)", "text": "The ordinal $2$ is defined as: :$2 = 1^+$ where $1^+$ denotes the successor of the ordinal $1$. Category:Definitions/Ordinals hbior4iy1b4mjhw9j2d374orzwnd996"}
+{"_id": "23005", "title": "Definition:Composable Morphisms", "text": "Let $\\mathbf C$ be a metacategory. Let $f, g \\in \\mathbf C_1$ be morphisms of $\\mathbf C$. Then $f$ is said to be '''composable with $g$''' iff: :$\\operatorname{cod} f = \\operatorname{dom} g$ that is, iff the codomain of $f$ is the domain of $g$. When the order of composition is to be made more explicit, one says that '''$\\left({g, f}\\right)$ is a composable pair'''. The collection of all such '''composable pairs''' in $\\mathbf C$ is denoted $\\mathbf C_2$."}
+{"_id": "23006", "title": "Definition:Category with Products/Binary", "text": "Let $\\mathbf C$ be a metacategory. Then $\\mathbf C$ is said to '''have binary products''' or to be a '''(meta)category with binary products''' iff: :For all objects $C, D \\in \\mathbf C_0$, there is a binary product $C \\times D$ for $C$ and $D$."}
+{"_id": "23007", "title": "Definition:Cofinal Relation on Ordinals", "text": "Let $x$ and $y$ be ordinals. Then $y$ is said to be '''cofinal''' with respect to $x$ {{iff}} there exists a mapping $f: y \\to x$ such that: :$(1): \\quad y \\le x$ :$(2): \\quad f$ is strictly increasing. :$(3): \\quad$ For all $a \\in x$, there is some $b \\in y$ such that $f \\left({b}\\right) \\ge a$."}
+{"_id": "23009", "title": "Definition:Regular Cardinal", "text": "Let $\\kappa$ be an infinite cardinal. Then $\\kappa$ is a '''regular cardinal''' {{iff}}: :$\\operatorname{cf} \\left({\\kappa}\\right) = \\kappa$ That is, {{iff}} the cofinality of $\\kappa$ is equal to itself."}
+{"_id": "23010", "title": "Definition:Singular Cardinal", "text": "Let $\\kappa$ be an infinite cardinal. Then $\\kappa$ is a '''singular cardinal''' {{iff}} $\\operatorname{cf} \\left({\\kappa}\\right) < \\kappa$. That is, the cofinality of $\\kappa$ is less than itself."}
+{"_id": "23011", "title": "Definition:Weakly Inaccessible Cardinal", "text": "An infinite cardinal $\\aleph_\\kappa$ is '''weakly inaccessible''' {{iff}}: :$(1): \\quad \\kappa$ is a limit ordinal. :$(2): \\quad \\aleph_\\kappa$ is a regular cardinal."}
+{"_id": "23012", "title": "Definition:Strongly Inaccessible Cardinal", "text": "An infinite cardinal $\\aleph_\\kappa$ is called a '''strongly inaccessible cardinal''' {{iff}}: :$(1): \\quad \\aleph_\\kappa$ is a weakly inaccessible cardinal :$(2): \\quad \\forall x \\in \\kappa: \\left|{\\mathcal P \\left({x}\\right)}\\right| \\in \\kappa$"}
+{"_id": "23013", "title": "Definition:Greatest Common Divisor/Integers/General Definition", "text": "Let $S = \\set {a_1, a_2, \\ldots, a_n} \\subseteq \\Z$ such that $\\exists x \\in S: x \\ne 0$ (that is, at least one element of $S$ is non-zero). Then the '''greatest common divisor''' of $S$: :$\\gcd \\paren S = \\gcd \\set {a_1, a_2, \\ldots, a_n}$ is defined as the largest $d \\in \\Z_{>0}$ such that: :$\\forall x \\in S: d \\divides x$"}
+{"_id": "23014", "title": "Definition:Inverse Morphism", "text": "A morphism $g: Y \\to X$ is said to be an '''inverse (morphism)''' for $f$ {{iff}}: :$g \\circ f = I_X$ :$f \\circ g = I_Y$ where $I_X$ denotes the identity morphism on $X$."}
+{"_id": "23015", "title": "Definition:Population", "text": "The '''population''' of a statistical study is everything in the universe of discourse that is relevant to the study. It includes all objects, measures, and observations under discussion. === Finite Population === {{:Definition:Population/Finite}} === Infinite Population === {{:Definition:Population/Infinite}}"}
+{"_id": "23016", "title": "Definition:Sample", "text": "A '''sample''' is a non-empty, finite, proper subset of a population."}
+{"_id": "23017", "title": "Definition:Individual", "text": "An '''individual''' is an object in a population."}
+{"_id": "23018", "title": "Definition:Object (Category Theory)", "text": "Let $\\mathbf C$ be a metacategory. An '''object''' of $\\mathbf C$ is an object which is considered to be atomic from a category theoretic perspective. It is a conceptual device introduced mainly to make the discussion of morphisms more convenient. '''Objects''' in a general metacategory are usually denoted with capital letters like $A,B,C,X,Y,Z$. The collection of '''objects''' of $\\mathbf C$ is denoted $\\mathbf C_0$. That '''objects''' don't play an important role in category theory is apparent from the fact that the notion of a metacategory can be described while avoiding to mention '''objects''' altogether. {{wtd|That is a big claim which needs a page to back it up; this is demonstrated eg. in 'Categories for the Working Mathematician' by MacLane}} Nonetheless the notion of '''object''' is one of the two basic concepts of metacategories and as such of category theory."}
+{"_id": "23019", "title": "Definition:Maximal Subgroup", "text": "Let $G$ be a group. Let $M \\le G$ be a proper subgroup of $G$. Then $M$ is a '''maximal subgroup of $G$''' {{iff}}: :For every subgroup $H$ of $G$, $M \\subseteq H \\subseteq G$ means $M = H$ or $H = G$. That is, if there is no subgroup of $G$, except $M$ and $G$ itself, which contains $M$."}
+{"_id": "23020", "title": "Definition:Join of Subgroups/General Definition", "text": "Let $H_1, H_2, \\ldots, H_n$ be subgroups of $G$. Then the '''join''' of $H_1, H_2, \\ldots, H_n$ is defined as: :$\\displaystyle \\bigvee_{k \\mathop = 1}^n H_k := \\left \\langle {\\bigcup_{k \\mathop = 1}^n H_k}\\right \\rangle$ or: :$\\displaystyle \\bigvee_{k \\mathop = 1}^n H_k := \\bigcap \\left\\{{T: T \\text { is a subgroup of } G: \\bigcup_{k \\mathop = 1}^n H_k \\subseteq T}\\right\\}$"}
+{"_id": "23021", "title": "Definition:Simple Random Sample", "text": "Let $P$ be a population. Let $S \\subsetneq P$ be a sample. Then $S$ is a '''simple random sample''' (of size $n$) iff it fulfils the following criteria: * The process used to select the individuals of $S$ from $P$ was random; * Every individual in $P$ had an equal chance of being selected to be in $S$; * Every $n$-combination of $P$ had an equal chance of being constructed as a potential $S$."}
+{"_id": "23022", "title": "Definition:Stratified Random Sample", "text": "Let $P$ be a population. Let $S \\subsetneq P$ be a sample. Then $S$ is a '''stratified random sample''' {{iff}}: :there exists a finite expansion $P = P_1 \\mid P_2 \\mid \\cdots \\mid P_k$ of $P$ such that $S$ is the union of $k$ simple random samples, one taken from each $P_i$."}
+{"_id": "23023", "title": "Definition:Convenience Sample", "text": "Let $P$ be a population in the physical universe. Let $S \\subsetneq P$ be a sample. Suppose that data about some individuals in $P$ are more easily accessible than data about others. Then $S$ is a '''convenience sample''' {{iff}} the primary criterion for the construction of $S$ is the higher accessibility of data of certain individuals in $P$ over others. That is, the objects in $S$ are the ones conveniently at hand."}
+{"_id": "23024", "title": "Definition:Unambiguous", "text": "A statement is '''unambiguous''' if it is not ambiguous."}
+{"_id": "23025", "title": "Definition:Stratified Random Sample/Strata", "text": "The components of $P$ are called '''strata''', singular '''stratum'''."}
+{"_id": "23026", "title": "Definition:Dual Statement (Category Theory)", "text": "=== Morphisms-Only Category Theory === Let $\\Sigma$ be a statement in the language of category theory. The '''dual statement''' $\\Sigma^*$ of $\\Sigma$ is the statement obtained from substituting: {{begin-eqn}} {{eqn|l = R_\\circ \\left({y, x, z}\\right)      |o = \\) for \\(      |r = R_\\circ \\left({x, y, z}\\right) }} {{eqn|l = \\operatorname{dom}      |o = \\) for \\(      |r = \\operatorname{cod} }} {{eqn|l = \\operatorname{cod}      |o = \\) for \\(      |r = \\operatorname{dom} }} {{end-eqn}} === Object Category Theory === In the more convenient description of metacategories by using objects, the '''dual statement''' $\\Sigma^*$ of $\\Sigma$ then becomes the statement obtained from substituting: {{begin-eqn}} {{eqn|l = f \\circ g      |o = \\) for \\(      |r = g \\circ f }} {{eqn|l = \\operatorname{cod}      |o = \\) for \\(      |r = \\operatorname{dom} }} {{eqn|l = \\operatorname{dom}      |o = \\) for \\(      |r = \\operatorname{cod} }} {{end-eqn}} {{finish|a page proving that these indeed are the resulting substitutions}} === Example === For example, if $\\Sigma$ is the statement: :$\\exists g: g \\circ f = \\operatorname{id}_{\\operatorname{dom} f}$ describing that $f$ is a split mono, then $\\Sigma^*$ becomes: :$\\exists g: f \\circ g = \\operatorname{id}_{\\operatorname{cod} f}$ which precisely expresses $f$ to be a split epi. For a set $\\mathcal E$ of statements, write: :$\\mathcal E^* := \\left\\{{\\Sigma^*: \\Sigma \\in \\mathcal E}\\right\\}$  for the set comprising of the dual statement of those in $\\mathcal E$."}
+{"_id": "23027", "title": "Definition:Cartesian Product/Factors", "text": "Let $S \\times T$ be the cartesian product of $S$ and $T$. Then the sets $S$ and $T$ are called the '''factors''' of $S \\times T$."}
+{"_id": "23028", "title": "Definition:Metamodel", "text": "Let $\\mathcal L$ be a formal system with deductive apparatus $\\mathcal D$. A '''metamodel''' $\\mathcal M$ for $\\mathcal L$ is an interpretation for $\\mathcal L$. Thus, it assigns meanings to all of the undefined terms of $\\mathcal L$. Further, it specifies a means for deciding if a sentence of $\\mathcal L$ is to be considered true in $\\mathcal M$. It is then required that all the sentences derivable by the deductive apparatus $\\mathcal D$ are true in $\\mathcal M$. From the above specification, it follows that a '''metamodel''' $\\mathcal M$ is a conceptual device, in that it does not require to be founded on rigorous mathematical foundations. Thus, $\\mathcal M$ need not be a set or some other rigid notion, just something that one can contemplate, denote and convey to others. For example, even without rigid mathematical foundations it is clear to all sane people that there is a concept \"set\" that one can reason about. One could then try to determine if the collection of these \"sets\" is a '''metamodel''' for the language of set theory. As another example there is the notion of metacategory, which can be described as being any '''metamodel''' for the language of category theory."}
+{"_id": "23029", "title": "Definition:QED", "text": "The initials of '''Quod Erat Demonstrandum''', which is {{WP|Latin|Latin}} for '''which was to be demonstrated'''. These initials were traditionally added to the end of a proof, after the last line which is supposed to contain the statement that was to be proved."}
+{"_id": "23030", "title": "Definition:Real Interval/Unit Interval", "text": "A '''unit interval''' is a real interval whose endpoints are $0$ and $1$: {{begin-eqn}} {{eqn | l = \\openint 0 1       | o = :=       | r = \\set {x \\in \\R: 0 < x < 1} }} {{eqn | l = \\hointr 0 1       | o = :=       | r = \\set {x \\in \\R: 0 \\le x < 1} }} {{eqn | l = \\hointl 0 1       | o = :=       | r = \\set {x \\in \\R: 0 < x \\le 1} }} {{eqn | l = \\closedint 0 1       | o = :=       | r = \\set {x \\in \\R: 0 \\le x \\le 1} }} {{end-eqn}}"}
+{"_id": "23031", "title": "Definition:Inductive Set/Subset of Real Numbers", "text": "Let $I$ be a subset of the real numbers $\\R$. Then $I$ is an '''inductive set''' {{iff}}: :$1 \\in I$  and :$x \\in I \\implies \\paren {x + 1} \\in I$"}
+{"_id": "23032", "title": "Definition:Language of Category Theory", "text": "The language of '''(morphisms-only) category theory''' is a specific instance of the language of predicate logic. As such, it is considered to have: * unary function symbols $\\operatorname{dom}$ and $\\operatorname{cod}$, called the '''domain''' and '''codomain''' symbols * A ternary relation symbol $R_\\circ$, called the '''composition relation''' symbol All further descriptions required to determine a formal language are inherited from the definition of the language of predicate logic."}
+{"_id": "23033", "title": "Definition:Morphisms-Only Metacategory", "text": "A '''morphisms-only metacategory''' is a metamodel for the language of category theory subject to the following axioms: {{:Axiom:Axioms for Morphisms-Only Category Theory}}"}
+{"_id": "23034", "title": "Definition:Mapping/Defined", "text": "A mapping $f \\subseteq S \\times T$ is '''defined''' at $x \\in S$ {{iff}}: :$\\exists y \\in T: \\tuple {x, y} \\in f$ If for some $x \\in S$, one has: :$\\forall y \\in T: \\tuple {x, y} \\notin f$ then $f$ is '''not defined''' or ('''undefined''') at $x$, and indeed, $f$ is not technically a mapping at all."}
+{"_id": "23035", "title": "Definition:Relation/Truth Set", "text": "Let $\\RR$ be a relation on $S \\times T$. The '''truth set''' of $\\RR$ is the set of all ordered pairs $\\tuple {s, t}$ of $S \\times T$ such that $s \\mathrel \\RR t$: : $\\map \\TT \\RR = \\set {\\tuple {s, t}: s \\mathrel \\RR t}$"}
+{"_id": "23036", "title": "Definition:Relation/General Definition", "text": "Let $\\displaystyle \\Bbb S = \\prod_{i \\mathop = 1}^n S_i = S_1 \\times S_2 \\times \\ldots \\times S_n$ be the cartesian product of $n$ sets $S_1, S_2, \\ldots, S_n$. An '''$n$-ary relation on $\\Bbb S$''' is an ordered $n + 1$-tuple $\\RR$ defined as: :$\\RR := \\struct {S_1, S_2, \\ldots, S_n, R}$ where $R$ is an arbitrary subset $R \\subseteq \\Bbb S$. To indicate that $\\tuple {s_1, s_2, \\ldots, s_n} \\in R$, we write: :$\\map \\RR {s_1, s_2, \\ldots, s_n}$"}
+{"_id": "23037", "title": "Definition:Relation/Unary Relation", "text": "As a special case of an $n$-ary relation on $S$, note that when $n = 1$ we define a '''unary relation''' on $S$ as: :$\\RR \\subseteq S$ That is, a '''unary relation''' is a subset of $S$."}
+{"_id": "23038", "title": "Definition:Random Selection", "text": "A manner of selecting objects from some larger collection of objects is said to be '''random''' if the selection is made according to chance. That is, there is no strict rule or procedure that predictably determines the outcome of the selection.  See experiment and random variable for a  precise mathematical treatment of randomness."}
+{"_id": "23039", "title": "Definition:Agreement/Relations", "text": "Let: :$(1): \\quad \\mathcal R_1 \\subseteq S_1 \\times T_1$ be a relation on $S_1 \\times T_1$ :$(2): \\quad \\mathcal R_2 \\subseteq S_2 \\times T_2$ be a relation on $S_2 \\times T_2$ :$(3): \\quad X \\subseteq S_1 \\cap S_2$ Let: :$\\forall s \\in X: \\mathcal R_1 \\left ({s}\\right) = \\mathcal R_2 \\left ({s}\\right)$ where $\\mathcal R_1 \\left ({s}\\right)$ denotes the image of $s$ under $\\mathcal R$: :$\\mathcal R_1 \\left ({s}\\right) := \\left\\{ {t \\in T: s \\mathrel{\\mathcal R_1} t}\\right\\}$ Then the relations $\\mathcal R_1$ and $\\mathcal R_2$ are said to '''agree on''' or '''be in agreement on''' $X$."}
+{"_id": "23040", "title": "Definition:Agreement/Mappings", "text": "Let: :$(1): \\quad f_1: S_1 \\to T_1$ be a mapping from $S_1$ to $T_1$ :$(2): \\quad f_2: S_2 \\to T_2$ be a mapping from $S_2$ to $T_2$ :$(3): \\quad X \\subseteq S_1 \\cap S_2$ Let: :$\\forall s \\in X: \\map {f_1} s = \\map {f_2} s$ Then the mappings $f_1$ and $f_2$ are said to '''agree on''' or '''be in agreement on''' $X$."}
+{"_id": "23042", "title": "Definition:Combinable/Mappings", "text": "Let: :$(1): \\quad f_1: S_1 \\to T_1$ be a mapping from $S_1$ to $T_1$ :$(2): \\quad f_2: S_2 \\to T_2$ be a mapping from $S_2$ to $T_2$ If $f_1$ and $f_2$ agree on $S_1 \\cap S_2$, they are said to be '''combinable'''."}
+{"_id": "23043", "title": "Definition:Dedekind Complete Set", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then $\\struct {S, \\preceq}$ is '''Dedekind complete''' {{iff}} every non-empty subset of $S$ that is bounded above admits a supremum (in $S$)."}
+{"_id": "23044", "title": "Definition:Dedekind-MacNeille Completion", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. For a subset $A \\subseteq S$, let $A_+$ and $A_-$ be the sets of all upper and lower bounds for $A$ in $S$, respectively. {{explain|Review notation for upper and lower bounds}} The '''Dedekind-MacNeille completion''' of $\\struct {S, \\preceq}$ is defined as the set: :$\\widehat S := \\set {A \\subseteq S: A = \\paren {A_+}_-}$ ordered by inclusion ($\\subseteq$). {{explain|ordered by inclusion in this context - exactly how does this work?}}"}
+{"_id": "23045", "title": "Definition:Dedekind Completion", "text": "Let $S$ be an ordered set. A '''Dedekind completion of $S$''' is a Dedekind complete ordered set $\\tilde S$ together with an order embedding $\\phi: S \\to \\tilde S$, subject to: :For all Dedekind complete ordered sets $X$, and for all order embeddings $f: S \\to X$, there exists a unique order embedding $\\tilde f: \\tilde S \\to X$ such that: ::$\\tilde f \\circ \\phi = f$"}
+{"_id": "23046", "title": "Definition:Abnormal Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is an '''abnormal subgroup''' {{iff}}: :$\\forall g \\in G: g \\in \\gen {H, H^g}$ where $\\gen {H, H^g}$ is the subgroup of $G$ generated by $H$ and the conjugate of $H$ by $g$."}
+{"_id": "23047", "title": "Definition:Self-Normalizing Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''self-normalizing subgroup''' {{iff}}: :$N_G \\left({H}\\right) = H$ where $N_G \\left({H}\\right)$ is the normalizer of $H$ in $G$."}
+{"_id": "23050", "title": "Definition:Contranormal Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''contranormal subgroup''' {{iff}} the normal closure of $H$ is the whole of $G$."}
+{"_id": "23051", "title": "Definition:Subnormal Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''$k$-subnormal subgroup of $G$''' {{iff}} there is a finite sequence of subgroups of $G$: :$H = H_0, H_1, H_2, \\ldots, H_k = G$ such that $H_i$ is a normal subgroup of $H_{i+1}$ for all $i \\in \\left\\{{0, 1, \\ldots, k-1}\\right\\}$. That is, {{iff}} there exists a normal series: :$H = H_0 \\lhd H_1 \\lhd H_2 \\lhd \\cdots \\lhd H_k = G$ where $\\lhd$ denotes the relation of normality. Thus $H$ is a '''subnormal subgroup of $G$''' {{iff}} there exists $k \\in \\Z_{>0}$ such that $H$ is a '''$k$-subnormal subgroup of $G$'''."}
+{"_id": "23052", "title": "Definition:Pronormal Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''pronormal subgroup in $G$''' iff each of its conjugates in $G$ is conjugate to it already in the subgroup generated by $H$ and its conjugate. That is, $H$ is '''pronormal in $G$''' iff: : $\\forall g \\in G: \\exists k \\in \\left\\langle{H, H^g}\\right\\rangle: H^k = H^g$ where: : $\\left\\langle{H, H^g}\\right\\rangle$ is the subgroup generated by $H$ and $H^g$ : $H^g$ is the conjugate of $H$ by $g$."}
+{"_id": "23053", "title": "Definition:Paranormal Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''paranormal subgroup in $G$''' {{iff}} the subgroup generated by $H$ and any conjugate of $H$ is also generated by $H$ and a conjugate of $H$ within that generated subgroup. That is, $H$ is '''paranormal in $G$''' {{iff}}: : $\\forall g \\in G: \\exists k \\in \\left\\langle{H, H^g}\\right\\rangle: \\left\\langle{H, H^k}\\right\\rangle = \\left\\langle{H, H^g}\\right\\rangle$ where: : $\\left\\langle{H, H^g}\\right\\rangle$ is the subgroup generated by $H$ and $H^g$ : $H^g$ is the conjugate of $H$ by $g$."}
+{"_id": "23054", "title": "Definition:Polynormal Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''polynormal subgroup''' of $G$ {{iff}} for all $g \\in G$, the conjugate closure of $H$ in $\\langle H, g \\rangle$ is equal to the conjugate closure of $H$ in $H^{\\langle g \\rangle}$. That is, $H$ is '''polynormal in $G$''' {{iff}}: : $\\forall g \\in G: H^{\\left\\langle g \\right\\rangle} = H^{H^{\\left\\langle g \\right\\rangle}}$ where: : $H^{\\left\\langle g \\right\\rangle}$ is the subgroup generated by the set of all elements of the form $g^nhg^{-n}$ where $h \\in H, n \\in \\mathbb{Z}$, : $H^{H^{\\left\\langle g \\right\\rangle}}$ is the subgroup generated by the set of all elements of the form $khk^{-1}$ where $k \\in H^{\\left\\langle g \\right\\rangle}$."}
+{"_id": "23055", "title": "Definition:Underlying Set/Abstract Algebra", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Then the '''underlying set''' of $\\struct {S, \\circ}$ is the set $S$."}
+{"_id": "23056", "title": "Definition:Underlying Set/Metric Space", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Then the '''underlying set''' of $M$ is the set $A$."}
+{"_id": "23057", "title": "Definition:Underlying Set/Topological Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Then the '''underlying set''' of $T$ is the set $S$."}
+{"_id": "23058", "title": "Definition:Multiplication/Complex Numbers", "text": "The '''multiplication operation''' in the domain of complex numbers $\\C$ is written $\\times$. Let $z = a + i b, w = c + i d$ where $a, b, c, d \\in \\R$. Then $z \\times w$ is defined as: :$\\paren {a + i b} \\times \\paren {c + i d} = \\paren {a c - b d} + i \\paren {a d + b c}$"}
+{"_id": "23059", "title": "Definition:Coproduct", "text": "Let $\\mathbf C$ be a metacategory. Let $A$ and $B$ be objects of $\\mathbf C$. A '''coproduct diagram''' for $A$ and $B$ comprises an object $P$ and morphisms $i_1: A \\to P$, $i_2: B \\to P$: ::$\\begin{xy} <-4em,0em>*+{A} = \"A\", <0em,0em>*+{P}  = \"P\", <4em,0em>*+{B}  = \"B\", \"A\";\"P\" **@{-} ?>*@{>} ?*!/^.8em/{i_1}, \"B\";\"P\" **@{-} ?>*@{>} ?*!/_.8em/{i_2}, \\end{xy}$ subjected to the following universal mapping property: :For any object $X$ and morphisms $x_1, x_2$ like so: ::$\\begin{xy} <-4em,0em>*+{A} = \"A\", <0em,0em>*+{X}  = \"X\", <4em,0em>*+{B}  = \"B\", \"A\";\"X\" **@{-} ?>*@{>} ?*!/^.8em/{x_1}, \"B\";\"X\" **@{-} ?>*@{>} ?*!/_.8em/{x_2}, \\end{xy}$ :there is a unique morphism $u: P \\to X$ such that: ::$\\begin{xy} <0em,5em>*+{X}  = \"X\", <-5em,0em>*+{A} = \"A\", <0em,0em>*+{P}  = \"P\", <5em,0em>*+{B}  = \"B\", \"A\";\"X\" **@{-}  ?>*@{>} ?*!/^.8em/{x_1}, \"B\";\"X\" **@{-}  ?>*@{>} ?*!/_.8em/{x_2}, \"P\";\"X\" **@{--} ?>*@{>} ?*!/_.6em/{u}, \"A\";\"P\" **@{-}  ?>*@{>} ?*!/_.8em/{i_1}, \"B\";\"P\" **@{-}  ?>*@{>} ?*!/^.8em/{i_2}, \\end{xy}$ :is a commutative diagram. That is: :$x_1 = u \\circ i_i$ and $x_2 = u \\circ i_2$ In this situation, $P$ is called a '''coproduct of $A$ and $B$''' and may be denoted $A + B$. We generally write $\\sqbrk {x_1, x_2}$ for the unique morphism $u$ determined by above diagram. The morphisms $i_1$ and $i_2$ are often taken to be implicit. They are called '''injections'''; if necessary, $i_1$ can be called the '''first injection''' and $i_2$ the '''second injection'''. {{expand|the injection definition may merit its own, separate page}} {{definition wanted|general definition}} === Coproduct of Sets === When the objects $A$ and $B$ are sets, the definition of '''coproduct''' takes on the following form. {{:Definition:Coproduct/Sets}}"}
+{"_id": "23060", "title": "Definition:Variable", "text": "A '''variable''' is a label which is used to refer to an unspecified object. A '''variable''' can be identified by means of a symbol, for example: :$x, y, z, A, B, C, \\phi, \\psi, \\aleph$ It is often convenient to append a subscript letter or number to distinguish between different objects of a similar type: :$a_0, a_1, a_2, \\ldots, a_n; S_\\phi, S_{\\phi_x}, \\ldots$ The type of symbol used to define a '''variable''' is purely conventional. Particular types of object, as they are introduced, frequently have a particular range of symbols specified to define them, but there are no strict rules on the subject."}
+{"_id": "23061", "title": "Definition:Variable/Descriptive Statistics", "text": "A '''variable''' is a characteristic property of all individuals in a population or sample. It is a categorization of the population such that each individual can be unambiguously described with respect to said variable. === Quantitative Variable === {{:Definition:Variable/Descriptive Statistics/Quantitative Variable}} === Qualitative Variable === {{:Definition:Variable/Descriptive Statistics/Qualitative Variable|Qualitative Variable}}"}
+{"_id": "23062", "title": "Definition:Variable/Descriptive Statistics/Quantitative Variable", "text": "A '''quantitative variable''' is a variable such that: : The variable can be described by numbers : The performing of arithmetical operations on the data is meaningful."}
+{"_id": "23063", "title": "Definition:Variable/Descriptive Statistics/Qualitative Variable", "text": "A '''qualitative variable''' is a variable such that: : The variable is not a quantitative variable : The variable describes each individual as either having, or not having, some specific property."}
+{"_id": "23064", "title": "Definition:Basis (Topology)/Analytic Basis", "text": "{{:Definition:Basis (Topology)/Analytic Basis/Definition 1}}"}
+{"_id": "23065", "title": "Definition:Basis (Topology)/Synthetic Basis", "text": "Let $S$ be a set. ==== Definition 1 ==== {{:Definition:Basis (Topology)/Synthetic Basis/Definition 1}} ==== Definition 2 ==== {{:Definition:Basis (Topology)/Synthetic Basis/Definition 2}}"}
+{"_id": "23066", "title": "Definition:Countable Basis", "text": "A '''countable (analytic) basis''' for a topology $\\tau$ is an analytic basis for $\\tau$ which is a countable set."}
+{"_id": "23067", "title": "Definition:Sub-Basis/Analytic Sub-Basis", "text": "Let $\\struct {S, \\tau}$ be a topological space. Let $\\SS \\subseteq \\tau$. Define: :$\\displaystyle \\BB = \\set {\\bigcap \\FF: \\FF \\subseteq \\SS, \\FF \\text{ is finite} }$ That is, $\\BB$ is the set of all finite intersections of sets in $\\SS$. Note that $\\FF$ is allowed to be empty in the above definition. Define: :$\\displaystyle \\tau' = \\set {\\bigcup \\AA: \\AA \\subseteq \\BB}$ Suppose that $\\tau \\subseteq \\tau'$. That is, suppose that every $U \\in \\tau$ is a union of finite intersections of sets in $\\SS$, together with $\\O$ and $S$ itself. Then $\\SS$ is called an '''analytic sub-basis''' for $\\tau$."}
+{"_id": "23068", "title": "Definition:Sub-Basis/Synthetic Sub-Basis", "text": "Let $X$ be a set. A '''synthetic sub-basis on $X$''' is ''any'' subset $\\mathcal S \\subseteq \\mathcal P \\left({X}\\right)$ of the power set of $X$."}
+{"_id": "23069", "title": "Definition:Differential/Real Function", "text": "=== At a Point === {{:Definition:Differential/Real Function/Point}} === On an Open Set === {{:Definition:Differential/Real Function/Open Set}} Also: :$f \\left({x + h}\\right) - f \\left({x}\\right) - \\mathrm d f \\left({x; h}\\right) = o \\left({h}\\right)$ as $h \\to 0$. In the above, $o \\left({h}\\right)$ is interpreted as little-O of $h$. {{refactor|put the above somewhere else}}"}
+{"_id": "23070", "title": "Definition:Differential/Real-Valued Function", "text": "{{:Definition:Differential/Real-Valued Function/Point}}"}
+{"_id": "23071", "title": "Definition:Differential/Vector-Valued Function", "text": "=== At a Point === {{:Definition:Differential/Vector-Valued Function/Point}} === On an Open Set === Let $O \\subseteq \\R^n$ be an open set. Let $f = \\tuple {f_1, \\ldots, f_m}^\\intercal: O \\to \\R^m$ be a vector valued function, differentiable at $x \\in O$. The '''differential''' $\\d f$ is a function of two variables, defined as: :$\\map {\\d f} {x; h} = \\map {J_f} x \\cdot h$ where $\\map {J_f} x$ be the Jacobian matrix of $f$ at $x$. That is, if $h = \\tuple {h_1, \\ldots, h_n}$: :$\\map {\\d f} {x; h} = \\begin {pmatrix} \\map {\\dfrac {\\partial f_1} {\\partial x_1} } x & \\cdots & \\map {\\dfrac {\\partial f_1} {\\partial x_n} } x \\\\ \\vdots & \\ddots & \\vdots \\\\ \\map {\\dfrac {\\partial f_m} {\\partial x_1} } x & \\cdots & \\map {\\dfrac {\\partial f_m} {\\partial x_n} } x \\end {pmatrix} \\begin {pmatrix} h_1 \\\\ \\vdots \\\\ h_n \\end {pmatrix}$ </onlyinclude>"}
+{"_id": "23072", "title": "Definition:Dependent Variable/Real Function", "text": "Let $f: \\R \\to \\R$ be a real function. Let $\\map f x = y$. Then $y$ is referred to as a '''dependent variable'''."}
+{"_id": "23073", "title": "Definition:Independent Variable/Real Function", "text": "Let $f: \\R \\to \\R$ be a real function. Let $\\map f x = y$. Then $x$ is referred to as an '''independent variable'''."}
+{"_id": "23074", "title": "Definition:Jacobian", "text": "Let $U$ be an open subset of $\\R^n$. Let $\\mathbf f = \\paren {f_1, f_2, \\ldots, f_m}^\\intercal: U \\to \\R^m$ be a vector valued function, differentiable at $\\mathbf x = \\paren {x_1, x_2, \\ldots, x_n}^\\intercal \\in U$."}
+{"_id": "23075", "title": "Definition:Jacobian/Matrix", "text": "The '''Jacobian matrix''' of $\\mathbf f$ at $\\mathbf x$ is defined to be the matrix of partial derivatives: :$\\displaystyle \\mathbf J_{\\mathbf f} := \\begin{pmatrix}  \\map {\\dfrac {\\partial f_1} {\\partial x_1} } {\\mathbf x} & \\cdots & \\map {\\dfrac {\\partial f_1} {\\partial x_n} } {\\mathbf x} \\\\  \\vdots & \\ddots & \\vdots \\\\  \\map {\\dfrac {\\partial f_m} {\\partial x_1} } {\\mathbf x} & \\cdots & \\map {\\dfrac {\\partial f_m} {\\partial x_n} } {\\mathbf x} \\end{pmatrix}$"}
+{"_id": "23076", "title": "Definition:Jacobian/Determinant", "text": "The '''Jacobian determinant''' of $\\mathbf f$ at $\\mathbf x$ is defined to be the determinant of the Jacobian matrix: :$\\displaystyle \\map \\det {\\mathbf J_{\\mathbf f} } := \\begin{vmatrix}  \\map {\\dfrac {\\partial f_1} {\\partial x_1} } {\\mathbf x} & \\cdots & \\map {\\dfrac {\\partial f_1} {\\partial x_n} } {\\mathbf x} \\\\  \\vdots & \\ddots & \\vdots \\\\  \\map {\\dfrac {\\partial f_m} {\\partial x_1} } {\\mathbf x} & \\cdots & \\map {\\dfrac {\\partial f_m} {\\partial x_n} } {\\mathbf x} \\end{vmatrix}$"}
+{"_id": "23077", "title": "Definition:Differentiable Mapping/Real Function", "text": "{{:Definition:Differentiable Mapping/Real Function/Point}}"}
+{"_id": "23078", "title": "Definition:Differentiable Mapping/Complex Function", "text": "{{:Definition:Differentiable Mapping/Complex Function/Point}}"}
+{"_id": "23079", "title": "Definition:Differentiable Mapping/Real-Valued Function", "text": "{{:Definition:Differentiable Mapping/Real-Valued Function/Point}}"}
+{"_id": "23080", "title": "Definition:Differentiable Mapping/Vector-Valued Function", "text": "{{:Definition:Differentiable Mapping/Vector-Valued Function/Point}}"}
+{"_id": "23081", "title": "Definition:Differentiable Mapping/Real Function/Point", "text": "Let $f$ be a real function defined on an open interval $\\openint a b$. Let $\\xi$ be a point in $\\openint a b$. ==== Definition 1 ==== {{:Definition:Differentiable Mapping/Real Function/Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Differentiable Mapping/Real Function/Point/Definition 2}} These limits, if they exist, are called the derivative of $f$ at $\\xi$."}
+{"_id": "23082", "title": "Definition:Differentiable Mapping/Real Function/Interval/Closed Interval", "text": "Let $f$ be differentiable on the open interval $\\openint a b$. If the following limit from the right exists: :$\\displaystyle \\lim_{x \\mathop \\to a^+} \\frac {\\map f x - \\map f a} {x - a}$ as well as this limit from the left: :$\\displaystyle \\lim_{x \\mathop \\to b^-} \\frac {\\map f x - \\map f b} {x - b}$ then $f$ is '''differentiable on the closed interval $\\closedint a b$'''."}
+{"_id": "23083", "title": "Definition:Differentiable Mapping/Real Function/Interval", "text": "Let $f$ be a real function defined on an open interval $\\openint a b$. Then $f$ is '''differentiable on $\\openint a b$''' {{iff}} $f$ is differentiable at each point of $\\openint a b$."}
+{"_id": "23084", "title": "Definition:Differentiable Mapping/Real Function/Real Number Line", "text": "Let $f$ be a real function defined on $\\R$. By definition, $\\R$ is an (unbounded) open interval. Let $f$ be differentiable on the open interval $\\R$. That is, let $f$ be differentiable at every point of $\\R$. Then $f$ is '''differentiable everywhere (on $\\R$)'''."}
+{"_id": "23085", "title": "Definition:Differentiable Mapping/Real-Valued Function/Point", "text": "Let $U$ be an open subset of $\\R^n$.  Let $f: U \\to \\R$ be a real-valued function. Let $x \\in U$. ==== Definition 1 ==== {{:Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 2}}"}
+{"_id": "23086", "title": "Definition:Differentiable Mapping/Real-Valued Function/Open Set", "text": "Let $\\mathbb X$ be an open subset of $\\R^n$.  Let $f: \\mathbb X \\to \\R$ be a real-valued function. Then $f$ is '''differentiable in the open set $\\mathbb X$''' {{iff}} $f$ is differentiable at each point of $\\mathbb X$."}
+{"_id": "23087", "title": "Definition:Differentiable Mapping/Vector-Valued Function/Point", "text": "Let $\\mathbb X$ be an open subset of $\\R^n$.  Let $f = \\tuple {f_1, f_2, \\ldots, f_m}^\\intercal: \\mathbb X \\to \\R^m$ be a vector valued function. ==== Definition 1 ==== {{:Definition:Differentiable Mapping/Vector-Valued Function/Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Differentiable Mapping/Vector-Valued Function/Point/Definition 2}}"}
+{"_id": "23088", "title": "Definition:Differentiable Mapping/Vector-Valued Function/Region", "text": "Let $S \\subseteq \\mathbb X$. Then $f$ is '''differentiable in the open set $S$''' {{iff}} $f$ is differentiable at each $x$ in $S$. This can be denoted $f \\in \\map {\\CC^1} {S, \\R^m}$."}
+{"_id": "23089", "title": "Definition:Vector-Valued Function/Component Function", "text": "Each $f_1, f_2, \\ldots, f_n$ is a '''component function''' of $\\mathbf r$."}
+{"_id": "23090", "title": "Definition:Subdivision (Real Analysis)/Normal Subdivision", "text": "$P$ is a '''normal subdivision of $\\closedint a b$''' {{iff}}: :the length of every interval of the form $\\closedint {x_i} {x_{i + 1} }$ is the same as every other. That is, {{iff}}: :$\\exists c \\in \\R_{> 0}: \\forall i \\in \\N_{< n}: x_{i + 1} - x_i = c$"}
+{"_id": "23091", "title": "Definition:Topology Generated by Synthetic Basis", "text": "Let $S$ be a set. Let $\\BB$ be a synthetic basis of $S$. === Definition 1 === {{:Definition:Topology Generated by Synthetic Basis/Definition 1}} === Definition 2 === {{:Definition:Topology Generated by Synthetic Basis/Definition 2}} === Definition 3 === {{:Definition:Topology Generated by Synthetic Basis/Definition 3}}"}
+{"_id": "23092", "title": "Definition:Complex Conjugate/Complex Conjugation", "text": "The operation of '''complex conjugation''' is the mapping: : $\\overline \\cdot: \\C \\to \\C: z \\mapsto \\overline z$. where $\\overline z$ is the complex conjugate of $z$. That is, it maps a complex number to its complex conjugate."}
+{"_id": "23093", "title": "Definition:Addition/Complex Numbers", "text": "The '''addition operation''' in the domain of complex numbers $\\C$ is written $+$. Let $z = a + i b, w = c + i d$ where $a, b, c, d \\in \\R, i^2 = -1$. Then $z + w$ is defined as: :$\\paren {a + i b} + \\paren {c + i d} = \\paren {a + c} + i \\paren {b + d}$"}
+{"_id": "23094", "title": "Definition:Category of Vector Spaces", "text": "The '''category of vector spaces''', denoted $\\mathbf{Vect}$, is the metacategory with: {{DefineCategory |ob = vector spaces |mor = linear transformations }} {{DefinitionWanted|Explicate dependence on base field}}"}
+{"_id": "23095", "title": "Definition:Class (Descriptive Statistics)", "text": "Let $D$ be a finite collection of $n$ data regarding some quantitative variable."}
+{"_id": "23096", "title": "Definition:Class (Descriptive Statistics)/Integer Data", "text": "Let the data in $D$ be described by natural numbers or by integers. Let $d_{\\min}$ be the value of the smallest datum in $D$. Let $d_{\\max}$ be the value of the largest datum in $D$.   Let $P = \\left\\{{x_0, x_1, x_2, \\ldots, x_{n-1}, x_n}\\right\\} \\subseteq \\Z$ be a subdivision of $\\left[{a \\,.\\,.\\, b}\\right]$, where $a \\le x_0 \\le x_n \\le b$. The integer interval $\\left[{a \\,.\\,.\\, b}\\right]$, where $a \\le d_{\\min} \\le d_\\max \\le b$, is said to be divided into '''classes''' of integer intervals of the forms $\\left[{x_i \\,.\\,.\\, x_{i+1}}\\right]$ or $\\left[{x_i \\,.\\,.\\, x_i}\\right]$ {{iff}}: : Every datum is assigned into exactly one class : Every class is disjoint from every other : The union of all classes contains the entire integer interval $\\left[{x_0 \\,.\\,.\\, x_n}\\right]$ By convention, the first and last classes are not empty classes."}
+{"_id": "23097", "title": "Definition:Class (Descriptive Statistics)/Real Data", "text": "Let the data in $D$ be described by rational numbers or by real numbers. Let $d_{\\min}$ be the value of the smallest datum in $D$. Let $d_{\\max}$ be the value of the largest datum in $D$.   Let $P = \\left\\{{x_0, x_1, x_2, \\ldots, x_{n-1}, x_n}\\right\\} \\subseteq \\R$ be a subdivision of $\\left[{a \\,.\\,.\\, b}\\right]$, where $a \\le x_0 \\le x_n \\le b$. The closed real interval $\\left[{a \\,.\\,.\\, b}\\right]$, where $a \\le d_{\\text{min}} \\le d_{\\text{max}} \\le b$, is said to be divided into '''classes''' of real intervals with endpoints $x_i$ and $x_{i+1}$ {{iff}}: : Every datum is assigned into exactly one class : Every class is disjoint from every other : The union of all classes contains the entire real interval $\\left[{x_0 \\,.\\,.\\, x_n}\\right]$ The classes may be any combination of open, closed, or half-open intervals that fulfill the above criteria, but usually: : Every class except the last is of the form $\\left[{x_i \\,.\\,.\\, x_{i+1}}\\right)$ : The last class is of the form  $\\left[{x_{n-1} \\,.\\,.\\, x_n}\\right]$  By convention, the first and last classes are not empty classes."}
+{"_id": "23098", "title": "Definition:Pointwise Addition", "text": "The (binary) operation of '''pointwise addition''' is defined on $\\mathbb F^S$ as: :$+: \\mathbb F^S \\times \\mathbb F^S \\to \\mathbb F^S: \\forall f, g \\in \\mathbb F^S:$ ::$\\forall s \\in S: \\map {\\paren {f + g} } s := \\map f s + \\map g s$ where the $+$ on the {{RHS}} is conventional arithmetic addition."}
+{"_id": "23099", "title": "Definition:Pointwise Addition of Complex-Valued Functions", "text": "Let $f, g: S \\to \\C$ be complex-valued functions. Then the '''pointwise sum of $f$ and $g$''' is defined as: :$f + g: S \\to \\C:$ ::$\\forall s \\in S: \\map {\\paren {f + g} } s := \\map f s + \\map g s$ where $+$ on the {{RHS}} is complex addition."}
+{"_id": "23100", "title": "Definition:Pointwise Operation on Complex-Valued Functions", "text": "Let $\\C^S$ be the set of all mappings $f: S \\to \\C$, where $\\C$ is the set of complex numbers. Let $\\oplus$ be a binary operation on $\\C$. Define $\\oplus: \\C^S \\times \\C^S \\to \\C^S$, called '''pointwise $\\oplus$''', by: :$\\forall f, g \\in \\C^S: \\forall s \\in S: \\map {\\paren {f \\oplus g} } s := \\map f s \\oplus \\map g s$ In the above expression, the operator on the {{RHS}} is the given $\\oplus$ on the complex numbers."}
+{"_id": "23101", "title": "Definition:Closed under Mapping", "text": "Let $f: S \\to T$ be a mapping. Let $S' \\subseteq S$. Then $S'$ is '''closed under $f$''' {{iff}}: :$f \\sqbrk {S'} \\subseteq S'$ where $f \\sqbrk {S'}$ is the image of $S'$ under $f$. That is: :$x \\in S' \\implies \\map f x \\in S'$ === Arbitrary Product === {{:Definition:Closed under Mapping/Arbitrary Product}}"}
+{"_id": "23102", "title": "Definition:Pointwise Operation on Number-Valued Functions", "text": "Let $S$ be a non-empty set. Let $\\mathbb F$ be one of the standard number sets: $\\Z, \\Q, \\R$ or $\\C$. Let $\\mathbb F^S$ be the set of all mappings $f: S \\to \\mathbb F$. Let $\\oplus$ be a binary operation on $\\mathbb F$. The (binary) operation '''pointwise $\\oplus$''' is defined on $\\mathbb F^S$ as: :$\\oplus: \\mathbb F^S \\times \\mathbb F^S \\to \\mathbb F^S: \\forall f, g \\in \\mathbb F^S:$ ::$\\forall s \\in S: \\map {\\paren {f \\oplus g} } s := \\map f s \\oplus \\map g s$"}
+{"_id": "23103", "title": "Definition:Pointwise Operation on Integer-Valued Functions", "text": "Let $\\Z^S$ be the set of all mappings $f: S \\to \\Z$, where $\\Z$ is the set of integers. Let $\\oplus$ be a binary operation on $\\Z$. Define $\\oplus: \\Z^S \\times \\Z^S \\to \\Z^S$, called '''pointwise $\\oplus$''', by: :$\\forall f, g \\in \\Z^S: \\forall s \\in S: \\map {\\paren {f \\oplus g} } s := \\map f s \\oplus \\map g s$ In the above expression, the operator on the {{RHS}} is the given $\\oplus$ on the integers."}
+{"_id": "23104", "title": "Definition:Pointwise Operation on Rational-Valued Functions", "text": "Let $\\Q^S$ be the set of all mappings $f: S \\to \\Q$, where $\\Q$ is the set of rational numbers. Let $\\oplus$ be a binary operation on $\\Q$. Define $\\oplus: \\Q^S \\times \\Q^S \\to \\Q^S$, called '''pointwise $\\oplus$''', by: :$\\forall f, g \\in \\Q^S: \\forall s \\in S: \\map {\\paren {f \\oplus g} } s := \\map f s \\oplus \\map g s$ In the above expression, the operator on the {{RHS}} is the given $\\oplus$ on the rational numbers."}
+{"_id": "23105", "title": "Definition:Pointwise Addition of Integer-Valued Functions", "text": "Let $f, g: S \\to \\Z$ be integer-valued functions. Then the '''pointwise sum of $f$ and $g$''' is defined as: :$f + g: S \\to \\Z:$ ::$\\forall s \\in S: \\map {\\paren {f + g} } s := \\map f s + \\map g s$ where the $+$ on the {{RHS}} is integer addition."}
+{"_id": "23106", "title": "Definition:Integer-Valued Function", "text": "Let $f: S \\to T$ be a function. Let $S_1 \\subseteq S$ such that $f \\left({S_1}\\right) \\subseteq \\Z$. Then $f$ is said to be '''integer-valued on $S_1$'''. That is, $f$ is defined as integer-valued on $S_1$ {{iff}} the image of $S_1$ under $f$ lies entirely within the set of integers $\\Z$. An '''integer-valued function''' is a function $f: S \\to \\Z$ whose codomain is the set of integers $\\Z$. That is, $f$ is '''integer-valued''' {{iff}} it is '''integer-valued''' over its entire domain. Category:Definitions/Integers rp9b74fhi6296cvp5f347ess83n70oq"}
+{"_id": "23107", "title": "Definition:Pointwise Addition of Rational-Valued Functions", "text": "Let $f, g: S \\to \\Q$ be rational-valued functions. Then the '''pointwise sum of $f$ and $g$''' is defined as: :$f + g: S \\to \\Q:$ ::$\\forall s \\in S: \\map {\\paren {f + g} } s := \\map f s + \\map g s$ where the $+$ on the {{RHS}} is integer addition."}
+{"_id": "23108", "title": "Definition:Rational-Valued Function", "text": "Let $f: S \\to T$ be a function. Let $S_1 \\subseteq S$ such that $f \\left({S_1}\\right) \\subseteq \\Q$. Then $f$ is said to be '''rational-valued on $S_1$'''. That is, $f$ is defined as rational-valued on $S_1$ iff the image of $S_1$ under $f$ lies entirely within the set of rational numbers $\\Q$. A '''rational-valued function''' is a function $f: S \\to \\Q$ whose codomain is the set of rational numbers $\\Q$. That is, $f$ is '''rational-valued''' {{iff}} it is rational-valued over its entire domain. Category:Definitions/Rational Numbers 5lll7qnf2lmi2z6ki66v4o4j4uu7v4s"}
+{"_id": "23109", "title": "Definition:Pointwise Multiplication of Integer-Valued Functions", "text": "Let $f, g: S \\to \\Z$ be integer-valued functions. Then the '''pointwise product of $f$ and $g$''' is defined as: :$f \\times g: S \\to \\Z:$ ::$\\forall s \\in S: \\map {\\paren {f \\times g} } s := \\map f s \\times \\map g s$ where the $\\times$ on the {{RHS}} is integer multiplication."}
+{"_id": "23110", "title": "Definition:Pointwise Multiplication of Rational-Valued Functions", "text": "Let $f, g: S \\to \\Q$ be rational-valued functions. Then the '''pointwise product of $f$ and $g$''' is defined as: :$f \\times g: S \\to \\Q:$ ::$\\forall s \\in S: \\map {\\paren {f \\times g} } s := \\map f s \\times \\map g s$ where the $\\times$ on the {{RHS}} is rational multiplication."}
+{"_id": "23111", "title": "Definition:Pointwise Multiplication of Real-Valued Functions", "text": "Let $f, g: S \\to \\R$ be real-valued functions. Then the '''pointwise product of $f$ and $g$''' is defined as: :$f \\times g: S \\to \\R:$ ::$\\forall s \\in S: \\map {\\paren {f \\times g} } s := \\map f s \\times \\map g s$ where $\\times$ on the {{RHS}} denotes real multiplication."}
+{"_id": "23112", "title": "Definition:Pointwise Multiplication of Complex-Valued Functions", "text": "Let $f, g: S \\to \\Z$ be complex-valued functions. Then the '''pointwise product of $f$ and $g$''' is defined as: :$f \\times g: S \\to \\Z:$ ::$\\forall s \\in S: \\map {\\paren {f \\times g} } s := \\map f s \\times \\map g s$ where the $\\times$ on the {{RHS}} is complex multiplication."}
+{"_id": "23113", "title": "Definition:Pointwise Scalar Multiplication of Number-Valued Function", "text": "When one of the functions is the constant mapping $f_\\lambda: S \\to \\mathbb F: \\map {f_\\lambda} s = \\lambda$, the following definition arises: The (binary) operation of '''pointwise scalar multiplication''' is defined on $\\mathbb F \\times \\mathbb F^S$ as: :$\\times: \\mathbb F \\times \\mathbb F^S \\to \\mathbb F^S: \\forall \\lambda \\in \\mathbb F, f \\in \\mathbb F^S:$ ::$\\forall s \\in S: \\map {\\paren {\\lambda \\times f} } s := \\lambda \\times \\map f s$ where the $\\times$ on the {{RHS}} is conventional arithmetic multiplication."}
+{"_id": "23114", "title": "Definition:Pointwise Scalar Multiplication of Integer-Valued Function", "text": "Let $f: S \\to \\Z$ be an integer-valued function. Let $\\lambda \\in \\Z$ be an integer. Then the '''pointwise scalar product of $f$ by $\\lambda$''' is defined as: :$\\lambda \\times f: S \\to \\Z:$ ::$\\forall s \\in S: \\map {\\paren {\\lambda \\times f} } s := \\lambda \\times \\map f s$ where $\\times$ on the {{RHS}} is integer multiplication."}
+{"_id": "23115", "title": "Definition:Pointwise Scalar Multiplication of Rational-Valued Function", "text": "Let $f: S \\to \\Q$ be an rational-valued function. Let $\\lambda \\in \\Q$ be an rational number. Then the '''pointwise scalar product of $f$ by $\\lambda$''' is defined as: :$\\lambda \\times f: S \\to \\Q:$ ::$\\forall s \\in S: \\map {\\paren {\\lambda \\times f} } s := \\lambda \\times \\map f s$ where the $\\times $ on the {{RHS}} is rational multiplication."}
+{"_id": "23116", "title": "Definition:Pointwise Scalar Multiplication of Real-Valued Function", "text": "Let $f: S \\to \\R$ be an real-valued function. Let $\\lambda \\in \\R$ be an real number. Then the '''pointwise scalar product of $f$ by $\\lambda$''' is defined as: :$\\lambda \\times f: S \\to \\R:$ ::$\\forall s \\in S: \\map {\\paren {\\lambda \\times f} } s := \\lambda \\times \\map f s$ where the $\\times $ on the {{RHS}} is real multiplication."}
+{"_id": "23117", "title": "Definition:Pointwise Scalar Multiplication of Complex-Valued Function", "text": "Let $f: S \\to \\C$ be an complex-valued function. Let $\\lambda \\in \\C$ be an complex number. Then the '''pointwise scalar product of $f$ by $\\lambda$''' is defined as: :$\\lambda \\times f: S \\to \\C:$ ::$\\forall s \\in S: \\map {\\paren {\\lambda \\times f} } s := \\lambda \\times \\map f s$ where $\\times$ on the {{RHS}} denotes complex multiplication."}
+{"_id": "23118", "title": "Definition:Addition/Rational Numbers", "text": "The addition operation in the domain of rational numbers $\\Q$ is written $+$. Let: :$a = \\dfrac p q, b = \\dfrac r s$ where: :$p, q \\in \\Z$ :$r, s \\in \\Z_{\\ne 0}$ Then $a + b$ is defined as: :$\\dfrac p q + \\dfrac r s = \\dfrac {p s + r q} {q s}$ This definition follows from the definition of and proof of existence of the field of quotients of any integral domain, of which the set of integers is an example."}
+{"_id": "23119", "title": "Definition:Quaternion/Addition", "text": "The '''sum''' of two quaternions $\\mathbf x_1 = a_1 \\mathbf 1 + b_1 \\mathbf i + c_1 \\mathbf j + d_1 \\mathbf k$ and $\\mathbf x_2 = a_2 \\mathbf 1 + b_2 \\mathbf i + c_2 \\mathbf j + d_2 \\mathbf k$ is defined as: :$\\mathbf x_1 + \\mathbf x_2 := \\paren {a_1 + a_2} \\mathbf 1 + \\paren {b_1 + b_2} \\mathbf i + \\paren {c_1 + c_2} \\mathbf j + \\paren {d_1 + d_2} \\mathbf k$"}
+{"_id": "23120", "title": "Definition:Quaternion/Multiplication", "text": "The '''product''' of two quaternions $\\mathbf x_1 = a_1 \\mathbf 1 + b_1 \\mathbf i + c_1 \\mathbf j + d_1 \\mathbf k$ and $\\mathbf x_2 = a_2 \\mathbf 1 + b_2 \\mathbf i + c_2 \\mathbf j + d_2 \\mathbf k$ is defined as: {{begin-eqn}}      {{eqn | l = \\mathbf x_1 \\mathbf x_2       | o = :=       | r = \\paren {a_1 a_2 - b_1 b_2 - c_1 c_2 - d_1 d_2} \\mathbf 1       | c =  }} {{eqn | o =        | ro= +       | r = \\paren {a_1 b_2 + b_1 a_2 + c_1 d_2 - d_1 c_2} \\mathbf i       | c =  }} {{eqn | o =        | ro= +       | r = \\paren {a_1 c_2 - b_1 d_2 + c_1 a_2 + d_1 b_2} \\mathbf j       | c =  }} {{eqn | o =        | ro= +       | r = \\paren {a_1 d_2 + b_1 c_2 - c_1 b_2 + d_1 a_2} \\mathbf k       | c =  }} {{end-eqn}}"}
+{"_id": "23122", "title": "Definition:Quaternion/Algebra over Field", "text": "An algebra of quaternions can be defined over any field as follows: Let $\\mathbb K$ be a field, and $a$, $b \\in \\mathbb K$. Define the '''quaternion algebra''' $\\left\\langle{ a,b }\\right\\rangle_\\mathbb K$ to be the $\\mathbb K$-vector space with basis $\\{1, i, j, k\\}$ subject to: : $i^2 = a$ : $j^2 = b$ : $ij = k = -ji$ Formally this could be achieved as a multiplicative presentation of a suitable group, or as a linear subspace of a finite extension of $\\mathbb K$. Taking $\\mathbb K = \\R$ and $a = b = -1$ we see that this generalises  Hamilton's quaternions."}
+{"_id": "23123", "title": "Definition:Field Extension/Degree", "text": "Let $E / F$ be a field extension. The '''degree of $E / F$''', denoted $\\index E F$, is the dimension of $E / F$ when $E$ is viewed as a vector space over $F$. === Finite === {{:Definition:Field Extension/Degree/Finite}} === Infinite === {{:Definition:Field Extension/Degree/Infinite}}"}
+{"_id": "23124", "title": "Definition:Class/Proper Class", "text": "A '''proper class''' is a class which is not a set. That is, $A$ is a '''proper class''' {{iff}}: : $\\neg \\exists x: x = A$ where $x$ is a set. A class which is not a proper class is a small class."}
+{"_id": "23125", "title": "Definition:Normal Series/Factor Group", "text": "Let $\\sequence {G_i}_{i \\mathop \\in \\closedint 0 n}$ be a normal series for $G$: :$\\sequence {G_i}_{i \\mathop \\in \\closedint 0 n} = \\tuple {\\set e = G_0 \\lhd G_1 \\lhd \\cdots \\lhd G_{n - 1} \\lhd G_n = G}$ The '''factor groups''' of $\\sequence {G_i}_{i \\mathop \\in \\closedint 0 n}$: :$\\set e = G_0 \\lhd G_1 \\lhd \\cdots \\lhd G_n = G$ are the quotient groups: :$G_1 / G_0, G_2 / G_1, \\ldots, G_i / G_{i - 1}, \\ldots, G_n / G_{n-1}$"}
+{"_id": "23126", "title": "Definition:Normal Series/Sequence of Homomorphisms", "text": "Let $\\sequence {G_i}_{i \\mathop \\in \\closedint 0 n}$ be a normal series for $G$: :$\\sequence {G_i}_{i \\mathop \\in \\closedint 0 n} = \\tuple {\\set e = G_0 \\lhd G_1 \\lhd \\cdots \\lhd G_{n - 1} \\lhd G_n = G}$ whose factor groups are: :$H_1 = G_1 / G_0, H_2 = G_2 / G_1, \\ldots, H_i = G_i / G_{i - 1}, \\ldots, H_n = G_n / G_{n - 1}$ By Kernel of Group Homomorphism Corresponds with Normal Subgroup of Domain, such a series can also be expressed as a sequence $\\phi_1, \\ldots, \\phi_n$ of group homomorphisms: :$\\set e \\stackrel {\\phi_1} {\\to} H_1 \\stackrel {\\phi_2} {\\to} H_2 \\stackrel {\\phi_3} {\\to} \\cdots \\stackrel {\\phi_n} {\\to} H_n$"}
+{"_id": "23127", "title": "Definition:Normal Series/Infinite", "text": "A normal series may or may not terminate at either end: :$\\cdots \\stackrel {\\phi_{i - 1} } {\\longrightarrow} H_{i - 1} \\stackrel {\\phi_i} {\\longrightarrow} H_i \\stackrel {\\phi_{i + 1} } {\\longrightarrow} H_{i + 1} \\stackrel {\\phi_{i + 2} } {\\longrightarrow} \\cdots$ Such a series is referred to as an '''infinite normal series'''. The context will determine which end, if either, it terminates."}
+{"_id": "23128", "title": "Definition:Normal Series/Length", "text": "Let $\\sequence {G_i}_{i \\mathop \\in \\closedint 0 n}$ be a normal series for $G$: :$\\sequence {G_i}_{i \\mathop \\in \\closedint 0 n} = \\tuple {\\set e = G_0 \\lhd G_1 \\lhd \\cdots \\lhd G_{n-1} \\lhd G_n = G}$ The '''length''' of $\\sequence {G_i}_{i \\mathop \\in \\closedint 0 n}$ is the number of (normal) subgroups which make it. In this context, the '''length''' of $\\sequence {G_i}_{i \\mathop \\in \\closedint 0 n}$ is $n$. If such a normal series is infinite, then its '''length''' is not defined."}
+{"_id": "23129", "title": "Definition:Non-Trivial Group", "text": "A '''non-trivial group''' is a group which is not the trivial group. It is often used in arguments to exclude the sometimes erratic behaviour of the trivial group."}
+{"_id": "23130", "title": "Definition:Exact Normal Series", "text": "Let $\\sequence {H_i}_{i \\mathop \\in I \\mathop \\subseteq \\Z}$ be a normal series: :$\\cdots \\stackrel {\\phi_{i - 1} } {\\longrightarrow} H_{i - 1} \\stackrel {\\phi_i} {\\longrightarrow} H_i \\stackrel {\\phi_{i + 1} } {\\longrightarrow} H_{i + 1} \\stackrel {\\phi_{i + 2} } {\\longrightarrow} \\cdots$ Suppose that, for some $i \\in I$: :$\\Img {\\phi_i} = \\map \\ker {\\phi_{i + 1} }$ That is, the image of one homomorphism is the kernel of the next. Then $\\sequence {H_i}$ is referred to as '''exact at $H_i$'''. If $\\sequence {H_i}$ is '''exact''' for all $i \\in I$, then $\\sequence {H_i}$ itself is an '''exact normal series'''. </onlyinclude>"}
+{"_id": "23131", "title": "Definition:Equalizer", "text": "Let $\\mathbf C$ be a metacategory. Let $f, g: C \\to D$ be morphisms with common domain and codomain. An '''equalizer''' for $f$ and $g$ is the limit of the diagram $C \\underset f{\\overset g\\rightrightarrows} D$, that is, a morphism $e: E \\to C$ such that: :$f \\circ e = g \\circ e$ and subject to the following UMP: :For any $a: A \\to C$ such that $f \\circ a = g \\circ a$, there is a unique $u: A \\to E$ such that: :::$\\begin{xy}\\xymatrix{  E    \\ar[r]^*{e} &  C   \\ar[r]<2pt>^*{f}   \\ar[r]<-2pt>_*{g}  &  D \\\\  A   \\ar@{.>}[u]^*{u}   \\ar[ur]_*{a} }\\end{xy}$ :is a commutative diagram. I.e., $a = e \\circ u$."}
+{"_id": "23132", "title": "Definition:Metric Space/Distance Function", "text": "The mapping $d: A \\times A \\to \\R$ is referred to as a '''distance function on $A$''' or simply '''distance'''."}
+{"_id": "23133", "title": "Definition:Metric Space/Triangle Inequality", "text": "Axiom $\\text M 2$ is referred to as the '''triangle inequality''', as it is a generalization of the Triangle Inequality which holds on the real number line and complex plane."}
+{"_id": "23134", "title": "Definition:Metric Space/Point", "text": "The elements of $A$ are called the '''points''' of the space."}
+{"_id": "23135", "title": "Definition:Quasimetric/Quasimetric Space", "text": "A '''quasimetric space''' $M = \\left({A, d}\\right)$ is an ordered pair consisting of a set $A \\ne \\varnothing$ followed by a quasimetric $d: A \\times A \\to \\R$ which acts on that set."}
+{"_id": "23136", "title": "Definition:Distance/Points/Real Numbers", "text": "Let $x, y \\in \\R$ be real numbers. Let $\\size {x - y}$ be the absolute value of $x - y$. Then the function $d: \\R^2 \\to \\R$: :$\\map d {x, y} = \\size {x - y}$ is called the '''distance between $x$ and $y$'''."}
+{"_id": "23137", "title": "Definition:Pseudometric/Pseudometric Space", "text": "A '''pseudometric space''' $M = \\left({A, d}\\right)$ is an ordered pair consisting of a set $A \\ne \\varnothing$ followed by a pseudometric $d: A \\times A \\to \\R$ which acts on that set."}
+{"_id": "23138", "title": "Definition:Operation Induced by Injection", "text": "Let $\\struct {T, \\circ}$ be an algebraic structure. Let $f: S \\to T$ be an injection. Then the '''operation induced on $S$ by $f$ and $\\circ$''' is the binary operation $\\circ_f$ on $S$ defined by: :$\\circ_f: S \\times S \\to S: x \\circ_f y := \\map {f^{-1} } {\\map f x \\circ \\map f y}$"}
+{"_id": "23139", "title": "Definition:Continuous Real Function/Point", "text": "Let $A \\subseteq \\R$ be any subset of the real numbers. Let $f: A \\to \\R$ be a real function. Let $x \\in A$ be a point of $A$. === Definition by Epsilon-Delta === {{:Definition:Continuous Real Function/Point/Definition by Epsilon-Delta}} === Definition by Neighborhood === {{:Definition:Continuous Real Function/Point/Definition by Neighborhood}}"}
+{"_id": "23140", "title": "Definition:Continuous Real Function/Subset", "text": "Let $A \\subseteq \\R$ be any subset of the real numbers. Let $f: A \\to \\R$ be a real function. Then '''$f$ is continuous on $A$''' {{iff}} $f$ is continuous at every point of $A$."}
+{"_id": "23141", "title": "Definition:Continuous Real Function/One Side", "text": "=== Continuity from the Left at a Point === {{Definition:Continuous Real Function/Left-Continuous/Point}} === Continuity from the Right at a Point === {{Definition:Continuous Real Function/Right-Continuous/Point}}"}
+{"_id": "23142", "title": "Definition:Continuous Mapping (Metric Space)/Point", "text": "{{Definition:Continuous Mapping (Metric Space)/Point/Definition 1}}"}
+{"_id": "23143", "title": "Definition:Continuous Mapping (Metric Space)/Space", "text": "{{Definition:Continuous Mapping (Metric Space)/Space/Definition 1}}"}
+{"_id": "23144", "title": "Definition:Frequency (Descriptive Statistics)", "text": "Let $S$ be a sample or a population. Let $\\omega$ be a qualitative variable, or a class of a quantitative variable. The '''frequency''' of $\\omega$ is the number of individuals in $S$ satisfying $\\omega$."}
+{"_id": "23145", "title": "Definition:Class (Descriptive Statistics)/Class Mark", "text": "The midpoint of a class is called the '''class mark'''."}
+{"_id": "23146", "title": "Definition:Mean of Grouped Data", "text": "Let $\\mathsf C_1, \\mathsf C_2, \\cdots, \\mathsf C_n$ be classes of data. Let $m_i$ be the class mark of the $i$th class. Let $f_i$ be the frequency of the $i$th class. The '''mean of the grouped data''' for $\\mathsf C_1, \\mathsf C_2, \\cdots, \\mathsf C_n$ is defined as: :$\\displaystyle \\bar x_{\\mathsf C} := \\frac 1 n \\sum_{i \\mathop = 1}^n m_i f_i$"}
+{"_id": "23147", "title": "Definition:Class (Descriptive Statistics)/Empty Class", "text": "A class is '''empty''' if it is of frequency zero."}
+{"_id": "23149", "title": "Definition:Euclidean Metric/Real Number Line", "text": "On the real number line, the Euclidean metric can be seen to degenerate to: :$\\map d {x, y} := \\sqrt {\\paren {x - y}^2} = \\size {x - y}$ where $\\size {x - y}$ denotes the absolute value of $x - y$."}
+{"_id": "23150", "title": "Definition:Open Ball/Radius", "text": "In $\\map {B_\\epsilon} a$, the value $\\epsilon$ is referred to as the '''radius''' of the open $\\epsilon$-ball."}
+{"_id": "23151", "title": "Definition:Affine Space", "text": "=== Associativity Axioms === {{:Definition:Affine Space/Associativity Axioms}} === Group Action === {{:Definition:Affine Space/Group Action}} === Weyl's Axioms === {{:Definition:Affine Space/Weyl's Axioms}} === Addition === {{:Definition:Addition (Affine Geometry)}} === Subtraction === {{:Definition:Subtraction (Affine Geometry)}} === Tangent Space === {{:Definition:Tangent Space (Affine Geometry)}} === Vector === {{:Definition:Vector (Affine Geometry)}} === Point === {{:Definition:Point (Affine Geometry)}}"}
+{"_id": "23152", "title": "Definition:Sequentially Compact Space/In Itself", "text": "A subspace $H \\subseteq S$ is '''sequentially compact in itself''' {{iff}} every infinite sequence in $H$ has a subsequence which converges to a point in $H$. This is understood to mean that $H$ is sequentially compact when we consider it as a topological space with the induced topology of $T$."}
+{"_id": "23153", "title": "Definition:P-Product Metric/General Definition", "text": "Let $M_{1'} = \\left({A_{1'}, d_{1'}}\\right), M_{2'} = \\left({A_{2'}, d_{2'}}\\right), \\ldots, M_{n'} = \\left({A_{n'}, d_{n'}}\\right)$ be metric spaces. Let $\\displaystyle \\mathcal A = \\prod_{i \\mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \\ldots, A_{n'}$. Let $p \\in \\R_{\\ge 1}$. The '''$p$-product metric''' on $\\mathcal A$ is defined as: : $\\displaystyle d_p \\left({x, y}\\right) := \\left({\\sum_{i \\mathop = 1}^n \\left({d_{i'} \\left({x_i, y_i}\\right)}\\right)^p}\\right)^{\\frac 1 p}$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({y_1, y_2, \\ldots, y_n}\\right) \\in \\mathcal A$. The metric space $\\mathcal M_p := \\left({\\mathcal A, d_p}\\right)$ is the '''$p$-product (space)''' of $M_{1'}, M_{2'}, \\ldots, M_{n'}$."}
+{"_id": "23154", "title": "Definition:Affine Geometry", "text": "'''Affine geometry''' is the study of the geometry of affine spaces. It provides a modern axiomatic approach to the study of configurations of lines, planes and hypersurfaces. In particular an affine space can be thought of as a finite dimensional vector space with no distinguished origin, and its affine transformations are those that preserve collinearity."}
+{"_id": "23155", "title": "Definition:Path-Connected/Topology/Points", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $a, b \\in S$ be such that there exists a path from $a$ to $b$. That is, there exists a continuous mapping $f: \\left[{0 \\,.\\,.\\, 1}\\right] \\to S$ such that $f \\left({0}\\right) = a$ and $f \\left({1}\\right) = b$. Then $a$ and $b$ are '''path-connected in $T$'''."}
+{"_id": "23156", "title": "Definition:Path-Connected/Topology/Set", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $U \\subseteq S$ be a subset of $S$. Let $T' = \\left({U, \\tau_U}\\right)$ be the subspace of $T$ induced by $U$. Then $U$ is a '''path-connected set in $T$''' {{iff}} every two points in $U$ are path-connected in $T\\,'$. That is, $U$ is a '''path-connected set in $T$''' {{iff}}: :for every $x, y \\in U$, there exists a continuous mapping $f: \\left[{0 \\,.\\,.\\, 1}\\right] \\to U$ such that $f \\left({0}\\right) = x$ and $f \\left({1}\\right) = y$."}
+{"_id": "23157", "title": "Definition:Path-Connected/Topology/Topological Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then $T$ is a '''path-connected space''' {{iff}} $S$ is a path-connected set of $T$. That is, $T$ is a '''path-connected space''' {{iff}}: :for every $x, y \\in S$, there exists a continuous mapping $f: \\closedint 0 1 \\to S$ such that $\\map f 0 = x$ and $\\map f 1 = y$."}
+{"_id": "23158", "title": "Definition:Path-Connected/Topology", "text": "=== Points in Topological Space === {{:Definition:Path-Connected/Topology/Points}} === Set of Topological Space === {{:Definition:Path-Connected/Topology/Set}} === Topological Space === {{:Definition:Path-Connected/Topology/Topological Space}}"}
+{"_id": "23159", "title": "Definition:Population Parameter", "text": "A '''population parameter''' is a numerical description of a population."}
+{"_id": "23160", "title": "Definition:Sample Statistic", "text": "A '''sample statistic''' is a numerical description of a sample."}
+{"_id": "23161", "title": "Definition:Lagrange Basis Polynomial", "text": "Let $x_0, \\ldots, x_n \\in \\R$ be real numbers. The '''Lagrange basis polynomials''' associated to the $x_i$ are the polynomials: :$\\displaystyle \\map {L_j} X := \\prod_{\\substack {0 \\mathop \\le i \\mathop \\le n \\\\ i \\mathop \\ne j} } \\frac {X - x_i} {x_j - x_i} \\in \\R \\sqbrk X$ {{explain|$\\R \\sqbrk X$}} {{mistake|Not sure if it's a mistake or a different way of defining it, but {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|entry = Lagrange interpolation formula}} has this wrapped up in another product symbol}} {{NamedforDef|Joseph Louis Lagrange|cat = Lagrange}}"}
+{"_id": "23162", "title": "Definition:Final Topology", "text": "Let $X$ be a set. Let $I$ be an indexing set. Let $\\family {\\struct{Y_i, \\tau_i}}_{i \\mathop \\in I}$ be an $I$-indexed family of topological spaces. Let $\\family {f_i: Y_i \\to X}_{i \\mathop \\in I}$ be an $I$-indexed family of mappings. === Definition 1 === {{:Definition:Final Topology/Definition 1}}"}
+{"_id": "23163", "title": "Definition:Convex Set (Order Theory)", "text": "=== Definition 1 === {{:Definition:Convex Set (Order Theory)/Definition 1}} === Definition 2 === {{:Definition:Convex Set (Order Theory)/Definition 2}}"}
+{"_id": "23165", "title": "Definition:Path-Connected/Metric Space", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. $M$ is defined as '''path-connected''' iff: :$\\forall m, n \\in A: \\exists f: \\left[{0 \\,.\\,.\\, 1}\\right] \\to A: f \\left({0}\\right) = m, f \\left({1}\\right) = n$ where $f$ is a continuous mapping. === Subset of Metric Space === {{:Definition:Path-Connected/Metric Space/Subset}}"}
+{"_id": "23166", "title": "Definition:Path-Connected/Metric Space/Subset", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $S \\subseteq A$ be a subset of $M$. Then $S$ is '''path-connected (in $M$)''' {{iff}}: :$\\forall m, n \\in S: \\exists f: \\closedint 0 1 \\to S: \\map f 0 = m, \\map f 1 = n$ where $f$ is a continuous mapping."}
+{"_id": "23168", "title": "Definition:Percentile", "text": "Let $D$ be a finite collection of data regarding some quantitative variable. Let $D$ be divided into precisely $100$ classes. A $P$th '''percentile''' is a value in the interval defined by the $P$th class such that: :$\\dfrac P {100}$ of the data in $D$ fall at or below the value chosen :$1 - \\dfrac P {100}$ of the data in $D$ fall at or above the value chosen. Arbitrarily more precise definitions may be contrived as necessary to define a unique percentile for a given study."}
+{"_id": "23169", "title": "Definition:Quartile", "text": "Let $D$ be a finite collection of data regarding some quantitative variable. Let $D$ be divided into precisely $4$ classes. A $Q$th '''quartile''' is a value in the interval defined by the $Q$th class such that: :$\\dfrac Q 4$ of the data in $D$ fall at or below the value chosen; :$1 - \\dfrac Q 4$ of the data in $D$ fall at or above the value chosen. Arbitrarily more precise definitions may be contrived as necessary to define a unique quartile for a given study. A common convention is: * The '''second quartile''', $Q_2$, is defined as the median of $D$ * The '''first quartile''', $Q_1$, is defined as the median of the data values below and not including $Q_2$  * The '''third quartile''', $Q_3$, is defined as the median of the data values above and not including $Q_2$"}
+{"_id": "23170", "title": "Definition:Interquartile Range", "text": "Let $Q_1$ and $Q_3$ be first and third quartiles. The '''interquartile range''' is defined and denoted as: :$\\operatorname {IQR} := Q_3 - Q_1$"}
+{"_id": "23171", "title": "Definition:Coequalizer", "text": "Let $\\mathbf C$ be a metacategory. Let $f, g: C \\to D$ be morphisms with common domain and codomain. An '''equalizer''' for $f$ and $g$ is a morphism $q: D \\to Q$ such that: :$q \\circ f = q \\circ g$ and subject to the following UMP: :For any $z: D \\to Z$ such that $z \\circ f = z \\circ f$, there is a unique $u: Q \\to Z$ such that: :::::$\\begin{xy}\\xymatrix{  C   \\ar[r]<2pt>^*{f}   \\ar[r]<-2pt>_*{g} &   D   \\ar[r]^*{q}   \\ar[rd]_*{z} &  Q   \\ar@{.>}[d]^*{u} \\\\ & &  Z }\\end{xy}$ :is a commutative diagram. I.e., $z = u \\circ q$."}
+{"_id": "23172", "title": "Definition:Ordering on Multiindices", "text": "Let $Z$ be the set of multiindices. We define an ordering on $Z$ as follows. If $k = \\left \\langle {k_j}\\right \\rangle_{j \\in J}$ and $\\ell = \\left \\langle {\\ell_j}\\right \\rangle_{j \\in J}$ are multiindices, then we write $k \\preceq \\ell$ if: :$\\left( \\forall j \\in J \\right) \\left( k_j \\le \\ell_j \\right)$ By abuse of notation, it is standard to use $\\leq$ to denote the ordering on the set of multiindices as well."}
+{"_id": "23173", "title": "Definition:Addition/Multiindices", "text": "The addition operation in the domain of multiindices $Z$ is written $+$. Let $k = \\sequence {k_j}_{j \\mathop \\in J}$ and $l = \\sequence {l_j}_{j \\mathop \\in J}$ be multiindices. Then $k + l$ is defined as: :$\\paren {\\forall j \\in J} \\paren {\\paren {k + l}_j = k_j + l_j}$"}
+{"_id": "23174", "title": "Definition:Multiindex/Modulus", "text": "Let $k = \\sequence {k_j}_{j \\mathop \\in J}$ be a multiindex. The '''modulus''' of such a multiindex $k$  is defined by: :$\\displaystyle \\size k = \\sum_{j \\mathop \\in J} k_j$"}
+{"_id": "23175", "title": "Definition:Binomial Coefficient/Multiindices", "text": "Let $k = \\left \\langle {k_j}\\right \\rangle_{j \\mathop \\in J}$ and $\\ell = \\left \\langle {\\ell_j}\\right \\rangle_{j \\mathop \\in J}$ be multiindices. Let $\\ell \\le k$. Then $\\dbinom k \\ell$ is defined as: :$\\displaystyle \\binom k \\ell = \\prod_{j \\mathop \\in J} \\binom {k_j} {\\ell_j}$ Note that since by definition only finitely many of the $k_j$ are non-zero, the product in the definition of $\\dbinom k \\ell$ is convergent."}
+{"_id": "23176", "title": "Definition:Power (Algebra)/Multiindices", "text": "Let $k = \\left \\langle {k_j}\\right \\rangle_{j = 1, \\ldots, n}$ be a multiindex indexed by $\\left\\{{1, \\ldots, n}\\right\\}$. Let $x = \\left({x_1, \\ldots, x_n}\\right) \\in \\R^n$ be an ordered tuple of real numbers. Then $x^k$ is defined as: :$\\displaystyle x^k := \\prod_{j \\mathop = 1}^n x_j^{k_j}$ where the powers on the right hand side are integer powers."}
+{"_id": "23177", "title": "Definition:Coordinate System/Origin", "text": "The '''origin''' of a coordinate system is the zero vector. In the $x y$-plane, it is the point: :$O = \\tuple {0, 0}$ and in general, in the Euclidean space $\\R^n$: :$O = \\underbrace {\\tuple {0, 0, \\ldots, 0} }_{\\text{$n$ coordinates} }$ Thus it is the point where the axes cross over each other."}
+{"_id": "23178", "title": "Definition:Neighborhood (Metric Space)", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $S \\subseteq A$ be a subset of $A$. Let $x \\in S$. Let there exist $\\epsilon \\in \\R_{>0}$ such that the open $\\epsilon$-ball at $x$ lies completely in $S$, that is: :$\\map {B_\\epsilon} x \\subseteq S$ Then $S$ is a '''neighborhood of $x$ in $M$'''."}
+{"_id": "23179", "title": "Definition:Opposite Magma", "text": "$\\struct {S, *}$ is the '''opposite magma of $\\struct {S, \\circ}$''' {{iff}}: :$\\forall x_1, x_2, x_3 \\in S: x_1 \\circ x_2 = x_3 \\iff x_2 * x_1 = x_3$"}
+{"_id": "23180", "title": "Definition:Partial Differential Operator", "text": "Let $U \\subseteq \\R^n$ be a open set. Let $\\CC \\subseteq \\map {\\CC^k} {U, \\R}$ be a set of $k$-times continuously differentiable functions. Let $\\displaystyle \\partial_i = \\frac {\\partial} {\\partial x_i}$ denote the partial derivative, $i = 1, \\ldots, n$. For a multiindex $\\alpha = \\tuple {\\alpha_1, \\ldots, \\alpha_n}$ indexed by $\\set {1, \\ldots, n}$ let $\\partial^\\alpha = \\partial_1^{\\alpha_1} \\cdots \\partial_n^{\\alpha_n}$. A mapping $T : \\CC \\to \\map {\\CC^k} {U, \\R}$ is a '''partial differential operator''' if there exist $r \\in \\N$ and functions $f_\\alpha : \\R^n \\to \\C$ for each multiindex $\\alpha$ with $\\cmod \\alpha \\le r$ such that for all $g \\in \\CC$: :$\\displaystyle \\map T g = \\sum_{\\cmod \\alpha \\mathop \\le r} f_\\alpha \\partial^\\alpha g$ Category:Definitions/Partial Differentiation ssux2u3lv8vz8e0zdzfrq79l0aq2xei"}
+{"_id": "23181", "title": "Definition:Region/Metric Space", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. A '''region''' of $M$ is a subset $U$ of $M$ such that $U$ is: :$(1): \\quad$ non-empty :$(2): \\quad$ path-connected."}
+{"_id": "23182", "title": "Definition:Open Set/Pseudometric Space", "text": "Let $P = \\struct {A, d}$ be a pseudometric space. An '''open set''' in $P$ is defined in exactly the same way as for a metric space: $U$ is an '''open set in $P$''' {{iff}}: :$\\forall y \\in U: \\exists \\map \\epsilon y > 0: \\map {B_\\epsilon} y \\subseteq U$ where $\\map {B_\\epsilon} y$ is the open $\\epsilon$-ball of $y$."}
+{"_id": "23183", "title": "Definition:Relatively Closed Set", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A \\subseteq B \\subseteq S$. === Definition 1 === {{:Definition:Relatively Closed Set/Definition 1}} === Definition 2 === {{:Definition:Relatively Closed Set/Definition 2}}"}
+{"_id": "23184", "title": "Definition:Closed Point", "text": "Let $a \\in S$. Then '''$a$ is closed (in $T\\,$)''' {{iff}} $\\left\\{{a}\\right\\}$ is a closed set (in $T\\,$)."}
+{"_id": "23185", "title": "Definition:Closed Set/Metric Space", "text": "{{:Definition:Closed Set/Metric Space/Definition 1}}"}
+{"_id": "23186", "title": "Definition:Bounded Mapping/Metric Space", "text": "Let $M$ be a metric space. Let $f: X \\to M$ be a mapping from any set $X$ into $M$. Then $f$ is a '''bounded mapping''' {{iff}} $\\map f X$ is bounded in $M$."}
+{"_id": "23187", "title": "Definition:Bounded Metric Space/Unbounded", "text": "Let $M = \\left({X, d}\\right)$ be a metric space. Let $M' = \\left({Y, d_Y}\\right)$ be a subspace of $M$. Then '''$M'$ is unbounded (in $M$)''' {{iff}} $M'$ is not bounded in $M$."}
+{"_id": "23188", "title": "Definition:Number Field", "text": "A '''number field''' is a subfield of the field of complex numbers."}
+{"_id": "23189", "title": "Definition:Real Interval/Endpoints", "text": "The numbers $a, b \\in \\R$ are known as the '''endpoints''' of the interval. $a$ is sometimes called the '''left hand endpoint''' and $b$ the '''right hand endpoint''' of the interval."}
+{"_id": "23190", "title": "Definition:Real Interval/Length", "text": "The difference $b - a$ between the endpoints is called the '''length''' of the interval."}
+{"_id": "23191", "title": "Definition:Real Interval/Midpoint", "text": "The '''midpoint''' of a real interval is the number: : $\\dfrac {a + b} 2$"}
+{"_id": "23192", "title": "Definition:Real Interval/Open", "text": "The '''open (real) interval''' from $a$ to $b$ is defined as: :$\\openint a b := \\set {x \\in \\R: a < x < b}$"}
+{"_id": "23193", "title": "Definition:Real Interval/Half-Open", "text": "There are two '''half-open (real) intervals''' from $a$ to $b$. ==== Right half-open ==== {{Definition:Real Interval/Half-Open/Right}} ==== Left half-open ==== {{Definition:Real Interval/Half-Open/Left}}"}
+{"_id": "23194", "title": "Definition:Real Interval/Closed", "text": "The '''closed (real) interval''' from $a$ to $b$ is defined as: :$\\closedint a b = \\set {x \\in \\R: a \\le x \\le b}$"}
+{"_id": "23195", "title": "Definition:Real Interval/Unbounded Closed", "text": "There are two '''unbounded closed intervals''' involving a real number $a \\in \\R$, defined as: {{begin-eqn}} {{eqn | l = \\hointr a \\to       | o = :=       | r = \\set {x \\in \\R: a \\le x} }} {{eqn | l = \\hointl \\gets a       | o = :=       | r = \\set {x \\in \\R: x \\le a} }} {{end-eqn}}"}
+{"_id": "23196", "title": "Definition:Real Interval/Unbounded Open", "text": "There are two '''unbounded open intervals''' involving a real number $a \\in \\R$, defined as: {{begin-eqn}} {{eqn | l = \\openint a \\to       | o = :=       | r = \\set {x \\in \\R: a < x} }} {{eqn | l = \\openint \\gets a       | o = :=       | r = \\set {x \\in \\R: x < a} }} {{end-eqn}} Using the same symbology, the set $\\R$ can be represented as an '''unbounded open interval''' with no endpoints: :$\\openint \\gets \\to = \\R$"}
+{"_id": "23197", "title": "Definition:Real Interval/Empty", "text": "When $a > b$: {{begin-eqn}} {{eqn | l = \\left [{a \\,.\\,.\\, b} \\right]       | m = \\left\\{ {x \\in \\R: a \\le x \\le b}\\right\\}       | mo = =       | r = \\varnothing }} {{eqn | l = \\left [{a \\,.\\,.\\, b} \\right)       | m = \\left\\{ {x \\in \\R: a \\le x < b}\\right\\}       | mo = =       | r = \\varnothing }} {{eqn | l = \\left ({a \\,.\\,.\\, b} \\right]       | m = \\left\\{ {x \\in \\R: a < x \\le b}\\right\\}       | mo = =       | r = \\varnothing }} {{eqn | l = \\left ({a \\,.\\,.\\, b} \\right)       | m = \\left\\{ {x \\in \\R: a < x < b}\\right\\}       | mo = =       | r = \\varnothing }} {{end-eqn}} When $a = b$: : $\\left [{a \\,.\\,.\\, b} \\right) = \\left [{a \\,.\\,.\\, a} \\right) = \\left\\{{x \\in \\R: a \\le x < a}\\right\\} = \\varnothing$ : $\\left ({a \\,.\\,.\\, b} \\right] = \\left ({a \\,.\\,.\\, a} \\right] = \\left\\{{x \\in \\R: a < x \\le a}\\right\\} = \\varnothing$ : $\\left ({a \\,.\\,.\\, b} \\right) = \\left ({a \\,.\\,.\\, a} \\right) = \\left\\{{x \\in \\R: a < x < a}\\right\\} = \\varnothing$ Such empty sets are referred to as '''empty intervals'''."}
+{"_id": "23198", "title": "Definition:Real Interval/Singleton", "text": "When $a = b$: :$\\closedint a b = \\closedint a a = \\set {x \\in \\R: a \\le x \\le a} = \\set a$"}
+{"_id": "23199", "title": "Definition:Interval/Notation/Wirth", "text": "{{begin-eqn}} {{eqn | l = \\openint a b       | o = :=       | r = \\set {s \\in S: \\paren {a \\prec s} \\land \\paren {s \\prec b} }       | c = Open Interval }} {{eqn | l = \\hointr a b       | o = :=       | r = \\set {s \\in S: \\paren {a \\preccurlyeq s} \\land \\paren {s \\prec b} }       | c = Right Half-Open Interval }} {{eqn | l = \\hointl a b       | o = :=       | r = \\set {s \\in S: \\paren {a \\prec s} \\land \\paren {s \\preccurlyeq b} }       | c = Left Half-Open Interval }} {{eqn | l = \\closedint a b       | o = :=       | r = \\set {s \\in S: \\paren {a \\preccurlyeq s} \\land \\paren {s \\preccurlyeq b} }       | c = Closed Interval }} {{end-eqn}}"}
+{"_id": "23200", "title": "Definition:Real Interval/Notation/Conventional", "text": "These are the notations usually seen for real intervals: {{begin-eqn}} {{eqn | l = \\left ({a, b}\\right)       | o = :=       | r = \\set {x \\in \\R: a < x < b}       | c = Open real interval }} {{eqn | l = \\left [{a, b}\\right)       | o = :=       | r = \\set {x \\in \\R: a \\le x < b}       | c = Half-open real interval }} {{eqn | l = \\left ({a, b}\\right]       | o = :=       | r = \\set {x \\in \\R: a < x \\le b}       | c = Half-open real interval }} {{eqn | l = \\left [{a, b}\\right]       | o = :=       | r = \\set {x \\in \\R: a \\le x \\le b}       | c = Closed real interval }} {{end-eqn}} but they can be confused with other usages for this notation. In particular, there exists the danger of taking $\\paren {a, b}$ to mean an ordered pair."}
+{"_id": "23201", "title": "Definition:Real Interval/Notation/Reverse-Bracket", "text": "In order to avoid the ambiguity problem arising from the conventional notation for intervals where an open real interval can be confused with an ordered pair, some authors use the '''reverse-bracket notation''' for open and half-open intervals: {{begin-eqn}} {{eqn | l = \\left ] {\\, a, b} \\right [       | o = :=       | r = \\set {x \\in \\R: a < x < b}       | c = Open real interval }} {{eqn | l = \\left [ {a, b} \\right [       | o = :=       | r = \\set {x \\in \\R: a \\le x < b}       | c = Half-open on the right }} {{eqn | l = \\left ] {\\, a, b} \\right ]       | o = :=       | r = \\set {x \\in \\R: a < x \\le b}       | c = Half-open on the left }} {{end-eqn}} These are often considered to be both ugly ''and'' confusing, and hence are limited in popularity."}
+{"_id": "23202", "title": "Definition:Interval/Notation/Unbounded Intervals", "text": "In Wirth interval notation, unbounded intervals of an ordered set $\\struct {S, \\preccurlyeq}$ are written as follows: {{begin-eqn}} {{eqn | l = \\hointr a \\to       | o = :=       | r = \\set {x \\in S: a \\preccurlyeq x} }} {{eqn | l = \\hointl \\gets a       | o = :=       | r = \\set {x \\in S: x \\preccurlyeq a} }} {{eqn | l = \\openint a \\to       | o = :=       | r = \\set {x \\in S: a \\prec x} }} {{eqn | l = \\openint \\gets a       | o = :=       | r = \\set {x \\in S: x \\prec a} }} {{eqn | l = \\openint \\gets \\to       | o = :=       | r = \\set {x \\in S} = S }} {{end-eqn}}"}
+{"_id": "23203", "title": "Definition:Real Interval/Notation", "text": "An arbitrary (real) interval is frequently denoted $\\mathbb I$. Sources which use the $\\textbf {boldface}$ font for the number sets $\\N, \\Z, \\Q, \\R, \\C$ tend also to use $\\mathbf I$ for this entity. Some sources merely use the ordinary $\\textit {italic}$ font $I$. === Wirth Interval Notation === {{:Definition:Real Interval/Notation/Wirth}}"}
+{"_id": "23204", "title": "Definition:Hereditarily Compact Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. === Definition 1 === {{:Definition:Hereditarily Compact Space/Definition 1}} === Definition 2 === {{:Definition:Hereditarily Compact Space/Definition 2}}"}
+{"_id": "23205", "title": "Definition:Compact Space/Topology/Subspace", "text": "{{:Definition:Compact Space/Topology/Subspace/Definition 2}}"}
+{"_id": "23206", "title": "Definition:Compact Space/Euclidean Space", "text": "Let $\\R^n$ denote Euclidean $n$-space. Let $H \\subseteq \\R^n$. Then $H$ is '''compact in $\\R^n$''' {{iff}} $H$ is closed and bounded. === Real Analysis === The same definition applies when $n = 1$, that is, for the real number line: {{:Definition:Compact Space/Real Analysis}} === Complex Analysis === {{:Definition:Compact Space/Complex Analysis}}"}
+{"_id": "23207", "title": "Definition:Subobject", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ be an object of $\\mathbf C$. A '''subobject''' of $C$ is a monomorphism $m: B \\rightarrowtail C$ with codomain $C$."}
+{"_id": "23208", "title": "Definition:Category of Subobjects", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ be an object of $\\mathbf C$. The '''category of subobjects of $C$''', denoted $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$, is defined as follows: {{DefineCategory |ob = Subobjects $m: B \\to C$ of $C$ |mor = Morphisms $f: \\operatorname{dom} m \\to \\operatorname{dom} m'$ of $\\mathbf C$ such that $m' \\circ f = m$ in $\\mathbf C$ |comp = Inherited from $\\mathbf C$ |id = $\\operatorname{id}_m := \\operatorname{id}_{\\operatorname{dom} m}$, the identity morphism in $\\mathbf C$ of the domain of $m$ }} The behaviour of the morphisms is shown in the following commutative diagram in $\\mathbf C$: ::$\\begin{xy}\\xymatrix@+1em{  B   \\ar[r]^*+{f}   \\ar[rd]_*+{m} &  B'   \\ar[d]^*+{m'} \\\\ &  C }\\end{xy}$"}
+{"_id": "23209", "title": "Definition:Sphere/Geometry", "text": "A '''sphere''' is a surface in solid geometry such that all straight lines falling upon it from one particular point inside it are equal. {{EuclidDefinition|book = XI|def = 14|name = Sphere}}"}
+{"_id": "23210", "title": "Definition:Sphere/Geometry/Center", "text": "That point is called the '''center''' of the sphere."}
+{"_id": "23211", "title": "Definition:Sphere/Geometry/Radius", "text": "A '''radius''' of a sphere is a straight line segment whose endpoints are the center and the surface of the sphere. '''The radius''' of a sphere is the length of one such radius."}
+{"_id": "23212", "title": "Definition:Sphere/Topology", "text": "The $n$-dimensional '''sphere''', or '''$n$-sphere''', is the set: :$\\Bbb S^n = \\set {x \\in \\R^{n + 1} : \\size {x - y} = r}$ where $\\size {\\, \\cdot \\, }$ denotes the Euclidean distance."}
+{"_id": "23215", "title": "Definition:Sphere/Topology/Unit Sphere", "text": "The $n$-dimensional '''unit sphere''', or '''unit $n$-sphere''', is the $n$-sphere of radius $1$ and center the origin: :$\\Bbb S^n = \\left\\{{x \\in \\R^{n+1} : \\left\\lvert{x}\\right\\rvert = 1}\\right\\}$"}
+{"_id": "23216", "title": "Definition:Inclusion Relation on Subobjects", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ be an object of $\\mathbf C$. Let $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$ be the category of subobjects of $C$. The '''inclusion relation $\\subseteq$''' on subobjects of $C$ is defined as follows: :$m \\subseteq m'$ iff there exists a morphism $f: m \\to m'$"}
+{"_id": "23217", "title": "Definition:Homotopy/Relative", "text": "Let $K \\subseteq X$ be a subset of $X$. We say that $f$ and $g$ are '''homotopic relative to $K$''' if there exists a free homotopy $H$ between $f$ and $g$, and: :$(1): \\quad \\forall x \\in K: f \\left({x}\\right) = g \\left({x}\\right)$ :$(2): \\quad \\forall x \\in K, t \\in \\left[{0 \\,.\\,.\\, 1}\\right]: H \\left({x, t}\\right) = f \\left({x}\\right)$"}
+{"_id": "23218", "title": "Definition:Equivalent Subobjects", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ be an object of $\\mathbf C$. Let $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$ be the category of subobjects of $C$. Two subobjects $m, m'$ of $C$ are said to be '''equivalent''' iff: :$m \\subseteq m'$ and $m' \\subseteq m$ where $\\subseteq$ denotes the inclusion relation on subobjects."}
+{"_id": "23219", "title": "Definition:Subobject Class", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ be an object of $\\mathbf C$. Let $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$ be the category of subobjects of $C$. A '''subobject class of $C$''' is an equivalence class of subobjects of $C$ under the equivalence of subobjects. If $m$ is a subobject, its associated '''subobject class''' may be denoted by $\\overline m$ or $\\left[\\!\\left[{m}\\right]\\!\\right]$. === Morphism Class === {{:Definition:Subobject Class/Morphism Class}}"}
+{"_id": "23220", "title": "Definition:Homotopy/Free", "text": "Let $X$ and $Y$ be topological spaces. Let $f: X \\to Y$, $g: X \\to Y$ be continuous mappings. Then $f$ and $g$ are '''(freely) homotopic''' {{iff}} there exists a continuous mapping: : $H: X \\times \\left[{0 \\,.\\,.\\, 1}\\right] \\to Y$ such that, for all $x \\in X$: : $H \\left({x, 0}\\right) = f \\left({x}\\right)$ and: : $H \\left({x, 1}\\right) = g \\left({x}\\right)$ $H$ is called a '''(free) homotopy between $f$ and $g$'''."}
+{"_id": "23221", "title": "Definition:Category of Subobject Classes", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ be an object of $\\mathbf C$. Let $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$ be the category of subobjects of $C$. The '''category of subobject classes of $C$''', denoted $\\overline{\\mathbf{Sub}}_{\\mathbf C} \\left({C}\\right)$, is defined as follows: {{DefineCategory |ob = Subobject classes $\\left[\\!\\left[{m}\\right]\\!\\right]$ of $C$ |mor = Morphism classes $\\left[\\!\\left[{f}\\right]\\!\\right]: \\left[\\!\\left[{m}\\right]\\!\\right] \\to \\left[\\!\\left[{m'}\\right]\\!\\right]$ |comp = Inherited from $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$, i.e. $\\left[\\!\\left[{g}\\right]\\!\\right] \\circ \\left[\\!\\left[{f}\\right]\\!\\right] := \\left[\\!\\left[{g \\circ f}\\right]\\!\\right]$ |id = $\\operatorname{id}_{\\left[\\!\\left[{m}\\right]\\!\\right]} := \\left[\\!\\left[{\\operatorname{id}_m}\\right]\\!\\right]$, the morphism class of the identity morphism of $m$ in $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$ }}"}
+{"_id": "23222", "title": "Definition:Subobject Class/Morphism Class", "text": "Define the equivalence $\\sim$ on the morphisms of $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$ as follows. For morphisms $f: m \\to n$ and $g: m' \\to n'$ of $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$: :$f \\sim g$ iff $m \\sim m'$ and $n \\sim n'$ where $m \\sim m'$ signifies equivalence of subobjects. That $\\sim$ in fact is an equivalence is shown on Morphism Class Equivalence is Equivalence. A '''morphism class''' is an equivalence class $\\left[\\!\\left[{f}\\right]\\!\\right]$ under $\\sim$ of a morphism $f: m \\to m'$. The domain and codomain of $\\left[\\!\\left[{f}\\right]\\!\\right]$ are taken to be $\\left[\\!\\left[{m}\\right]\\!\\right]$ and $\\left[\\!\\left[{m'}\\right]\\!\\right]$, respectively."}
+{"_id": "23223", "title": "Definition:Inclusion Relation on Subobject Classes", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ be an object of $\\mathbf C$. The '''inclusion relation $\\subseteq$''' on subobject classes of $C$ is defined as follows: :$\\left[\\!\\left[{m}\\right]\\!\\right] \\subseteq \\left[\\!\\left[{m'}\\right]\\!\\right]$ iff there exists a morphism $\\left[\\!\\left[{f}\\right]\\!\\right]: \\left[\\!\\left[{m}\\right]\\!\\right] \\to \\left[\\!\\left[{m'}\\right]\\!\\right]$"}
+{"_id": "23224", "title": "Definition:Linear Continuum", "text": "A totally ordered set $\\struct {S, \\preceq}$ is a '''linear continuum''' {{iff}}: :$(1): \\quad \\struct {S, \\preceq}$ is densely ordered :$(2): \\quad \\struct {S, \\preceq}$ is Dedekind complete"}
+{"_id": "23225", "title": "Definition:Close Packed/Subset", "text": "A subset $T \\subseteq S$ is said to be '''close packed in $\\left({S, \\preceq}\\right)$''' {{iff}}: :$\\forall a, b \\in S: a \\prec b \\implies \\exists c \\in T: a \\prec c \\prec b$"}
+{"_id": "23226", "title": "Definition:Increasing/Sequence/Real Sequence", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Then $\\sequence {x_n}$ is '''increasing''' {{iff}}: :$\\forall n \\in \\N: x_n \\le x_{n + 1}$"}
+{"_id": "23227", "title": "Definition:Decreasing/Sequence/Real Sequence", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Then $\\sequence {x_n}$ is '''decreasing''' {{iff}}: :$\\forall n \\in \\N: x_{n + 1} \\le x_n$"}
+{"_id": "23228", "title": "Definition:Strictly Increasing/Sequence/Real Sequence", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Then $\\sequence {x_n}$ is '''strictly increasing''' {{iff}}: :$\\forall n \\in \\N: x_n < x_{n + 1}$"}
+{"_id": "23229", "title": "Definition:Strictly Decreasing/Sequence/Real Sequence", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Then $\\sequence {x_n}$ is '''strictly decreasing''' {{iff}}: :$\\forall n \\in \\N: x_{n + 1} < x_n$"}
+{"_id": "23230", "title": "Definition:Monotone (Order Theory)/Sequence/Real Sequence", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Then $\\sequence {x_n}$ is '''monotone''' {{iff}} it is either increasing or decreasing."}
+{"_id": "23231", "title": "Definition:Strictly Monotone/Sequence/Real Sequence", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Then $\\sequence {x_n}$ is '''strictly monotone''' {{iff}} it is either strictly increasing or strictly decreasing."}
+{"_id": "23232", "title": "Definition:Archimedean Property/Norm", "text": "Let $n: S \\to \\R$ be a norm on $S$. {{explain|What is a norm on a general algebraic structure?}} {{Disambiguate|Definition:Norm}} Then $n$ satisfies the '''Archimedean property on $S$''' {{iff}}: :$\\forall a, b \\in S: n \\paren a < n \\paren b \\implies \\exists m \\in \\N: n \\paren {m \\cdot a} > n \\paren b$ Using the more common symbology for a norm: :$\\forall a, b \\in S: \\norm a < \\norm b \\implies \\exists m \\in \\Z_{>0}: \\norm {m \\cdot a} > \\norm b$"}
+{"_id": "23233", "title": "Definition:Archimedean Property/Ordering", "text": "Let $\\preceq$ be an ordering on $S$. Then $\\preceq$ satisfies the '''Archimedean property on $S$''' {{iff}}: :$\\forall a, b \\in S: a \\prec b \\implies \\exists m \\in \\Z_{>0}: b \\prec m \\cdot a$"}
+{"_id": "23234", "title": "Definition:Cauchy Sequence/Metric Space", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $\\sequence {x_n}$ be a sequence in $M$. Then $\\sequence {x_n}$ is a '''Cauchy sequence''' {{iff}}: :$\\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\N: \\forall m, n \\in \\N: m, n \\ge N: \\map d {x_n, x_m} < \\epsilon$"}
+{"_id": "23235", "title": "Definition:Net (Metric Space)/Finite Net", "text": "A '''finite $\\epsilon$-net for $M$''' is an $\\epsilon$-net for $M$ which is a finite set."}
+{"_id": "23236", "title": "Definition:Cauchy Sequence/Real Numbers", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Then $\\sequence {x_n}$ is a '''Cauchy sequence''' {{iff}}: : $\\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\N: \\forall m, n \\in \\N: m, n \\ge N: \\size {x_n - x_m} < \\epsilon$"}
+{"_id": "23237", "title": "Definition:Cauchy Sequence/Rational Numbers", "text": "Let $\\left \\langle {x_n} \\right \\rangle$ be a rational sequence. Then $\\left \\langle {x_n} \\right \\rangle$ is a '''Cauchy sequence''' {{iff}}: : $\\forall \\epsilon \\in \\Q_{>0}: \\exists N \\in \\N: \\forall m, n \\in \\N: m, n \\ge N: \\left|{x_n - x_m}\\right| < \\epsilon$ where $\\Q_{>0}$ denotes the set of all strictly positive rational numbers."}
+{"_id": "23238", "title": "Definition:Euclidean Space/Real", "text": "Let $\\R^n$ be an $n$-dimensional real vector space. Let the Euclidean metric $d$ be applied to $\\R^n$. Then $\\struct {\\R^n, d}$ is a '''Euclidean $n$-space'''."}
+{"_id": "23239", "title": "Definition:Euclidean Space/Rational", "text": "Let $\\Q^n$ be an $n$-dimensional vector space of rational numbers. Let the Euclidean Metric $d$ be applied to $\\Q^n$. Then $\\left({\\Q^n, d}\\right)$ is a Euclidean $n$-space."}
+{"_id": "23240", "title": "Definition:Euclidean Space/Complex", "text": "Let $\\C$ be the complex plane. Let $d$ be the Euclidean metric on $\\C$. Then $\\left({\\C, d}\\right)$ is a Euclidean space."}
+{"_id": "23241", "title": "Definition:Euclidean Space/Euclidean Topology", "text": "Let $S$ be one of the standard number fields $\\Q$, $\\R$, $\\C$. Let $S^n$ be a cartesian space for $n \\in \\N_{\\ge 1}$. Let $M = \\left({S^n, d}\\right)$ be a Euclidean space. The topology $\\tau_d$ induced by the Euclidean metric $d$ is called the '''Euclidean topology'''."}
+{"_id": "23242", "title": "Definition:Discriminant of Polynomial/Quadratic Equation", "text": "The expression $b^2 - 4 a c$ is called the '''discriminant''' of the equation."}
+{"_id": "23243", "title": "Definition:Linearly Ordered Space", "text": "Let $\\struct {S, \\preceq}$ be a linearly ordered set. Let $\\tau$ be the order topology on $S$. Then $\\struct {S, \\preceq, \\tau}$ is a '''linearly ordered space'''."}
+{"_id": "23244", "title": "Definition:Bounded Mapping/Real-Valued/Attaining its Bounds", "text": "Let $f: S \\to \\R$ be a bounded real-valued function. Let $T$ be a subset of $S$. Suppose that: :$\\exists a, b \\in T: \\forall x \\in S: f \\left({a}\\right) \\le f \\left({x}\\right) \\le f \\left({b}\\right)$ Then $f$ '''attains its bounds on $T$'''."}
+{"_id": "23245", "title": "Definition:Bounded Mapping/Real-Valued", "text": "{{:Definition:Bounded Mapping/Real-Valued/Definition 1}}"}
+{"_id": "23246", "title": "Definition:Bounded Mapping/Complex-Valued", "text": "Let $f: S \\to \\C$ be a complex-valued function. Then $f$ is '''bounded''' {{iff}} the real-valued function $\\cmod f: S \\to \\R$ is bounded, where $\\cmod f$ is the modulus of $f$. That is, $f$ is '''bounded''' if there is a constant $K \\ge 0$ such that $\\cmod {f \\paren z} \\le K$ for all $z \\in S$."}
+{"_id": "23247", "title": "Definition:Continuous Mapping (Metric Space)/Space/Definition 2", "text": "$f$ is '''continuous from $\\left({A_1, d_1}\\right)$ to $\\left({A_2, d_2}\\right)$''' {{iff}}: :for every $U \\subseteq A_2$ which is open in $M_2$, $f^{-1} \\left[{U}\\right]$ is open in $M_1$."}
+{"_id": "23249", "title": "Definition:Composition Functor on Categories of Subobjects", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ and $D$ be objects of $\\mathbf C$. Let $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$ and $\\mathbf{Sub}_{\\mathbf C} \\left({D}\\right)$ be the associated categories of subobjects. Let $g: C \\to D$ be a monomorphism of $\\mathbf C$. Then $g$ defines a '''composition functor''' $g_* : \\mathbf{Sub}_{\\mathbf C} \\left({C}\\right) \\to \\mathbf{Sub}_{\\mathbf C} \\left({D}\\right)$: {{begin-axiom}} {{axiom|lc= Object functor:        |m = g_* f := g \\circ f        |rc= The composition $\\circ$ is taken in $\\mathbf C$ }} {{axiom|lc= Morphism functor:        |m = g_* a := a }} {{end-axiom}} That it is in fact a functor is shown on Composition Functor on Categories of Subobjects is Functor. The effect of $g_*$ is captured in the following commutative diagram: ::$\\begin{xy} <-3em,0em>*+{X} = \"X\", <3em,0em>*+{X'} = \"X2\", <0em,-4em>*+{C} = \"C\", <0em,-8em>*+{D} = \"D\", \"X\";\"X2\" **@{-} ?>*@{>} ?*!/_1em/{a}, \"X\";\"C\" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{f}, \"X2\";\"C\" **@{-} ?>*@{>} ?<>(.4)*!/_.6em/{f'}, \"C\";\"D\" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{g}, \"X\";\"D\" **\\crv{<-5em,-4em>} ?>*@{>} ?*!/^1.6em/{g_* f = \\\\ g \\circ f}, \"X2\";\"D\" **\\crv{<5em,-4em>} ?>*@{>} ?*!/_1.6em/{g_* f' = \\\\ g \\circ f'}, \\end{xy}$"}
+{"_id": "23250", "title": "Definition:Local Membership Relation", "text": "Let $\\mathbf C$ be a metacategory. Let $C$ be an object of $\\mathbf C$. The '''local membership relation''' $\\in_C$ between variable elements $z: Z \\to C$ and subobjects $m: M \\to C$ of $C$ is defined by: :$z \\in_C m$ iff there exists an $f: Z \\to M$: $z = m \\circ f$. If $z \\in_C m$, one says that $z$ is a '''local member''' of $m$."}
+{"_id": "23251", "title": "Definition:Lipschitz Continuity", "text": "Let $M = \\struct {A, d}$ and $M' = \\struct {A', d'}$ be metric spaces. Let Let $f: A \\to A'$ be a mapping. Then $f$ is a '''Lipschitz continuous mapping''' {{iff}} there exists a positive real number $K \\in \\R_{\\ge 0}$ such that: :$\\forall x, y \\in A: \\map {d'} {\\map f x, \\map f y} \\le K \\map d {x, y}$ That is, the distance between the images of two points lies within a fixed multiple of the distance between the points. === At a Point === {{:Definition:Lipschitz Continuity/Point}} === Lipschitz Constant === {{:Definition:Lipschitz Continuity/Lipschitz Constant}} === Real Function === The concept can be directly applied to the real numbers considered as a metric space under the usual topology: {{:Definition:Lipschitz Continuity/Real Function}}"}
+{"_id": "23252", "title": "Definition:Lipschitz Continuity/Lipschitz Constant", "text": "Let $f: A \\to A'$ be a (Lipschitz continuous) mapping such that: : $\\forall x, y \\in A: d\\,' \\left({f \\left({x}\\right), f \\left({y}\\right)}\\right) \\le K d \\left({x, y}\\right)$ where $K \\in \\R_{\\ge 0}$ is a positive real number. Then $K$ is '''a Lipschitz constant for $f$'''."}
+{"_id": "23253", "title": "Definition:Lipschitz Continuity/Real Function", "text": "Let $A \\subseteq \\R$. Let $f: A \\to \\R$ be a real function. Let $I \\subseteq A$ be a real interval on which: :$\\exists K \\in \\R_{\\ge 0}: \\forall x, y \\in I: \\size {\\map f x - \\map f y} \\le K \\size {x - y}$ Then $f$ is '''Lipschitz continuous on $I$'''."}
+{"_id": "23254", "title": "Definition:Lipschitz Equivalence/Metrics", "text": "=== Definition 1 === {{:Definition:Lipschitz Equivalence/Metrics/Definition 1}} === Definition 2 === {{:Definition:Lipschitz Equivalence/Metrics/Definition 2}}"}
+{"_id": "23255", "title": "Definition:Pullback (Category Theory)", "text": "Let $\\mathbf C$ be a metacategory. Let $f: A \\to C$ and $g: B \\to C$ be morphisms with common codomain. A '''pullback''' of $f$ and $g$ is a commutative diagram: ::$\\begin{xy}\\xymatrix{  P   \\ar[r]^*+{p_1}   \\ar[d]_*+{p_2} &  A   \\ar[d]^*+{f} \\\\  B   \\ar[r]_*+{g} &  C }\\end{xy}$ such that $f \\circ p_1 = g \\circ p_2$, subject to the following UMP: :For any commutative diagram: :::$\\begin{xy}\\xymatrix{  Q   \\ar[r]^*+{q_1}   \\ar[d]_*+{q_2} &  A   \\ar[d]^*+{f} \\\\  B   \\ar[r]_*+{g} &  C }\\end{xy}$ :there is a unique morphism $u: Q \\to P$ making the following diagram commute: :::$\\begin{xy}\\xymatrix@+1em{  Q   \\ar@/^/[rrd]^*+{q_1}   \\ar@/_/[ddr]_*+{q_2}   \\ar@{-->}[rd]^*+{u} \\\\ &  P   \\ar[r]_*+{p_1}   \\ar[d]^*+{p_2} &  A   \\ar[d]^*+{f} \\\\ &  B   \\ar[r]_*+{g} &  C }\\end{xy}$ In this situation, $p_1$ is called the '''pullback of $f$ along $g$''' and may be denoted as $g^* f$. Similarly, $p_2$ is called the '''pullback of $g$ along $f$''' and may be denoted $f^* g$."}
+{"_id": "23257", "title": "Definition:Homeomorphism/Topological Spaces", "text": "Let $T_\\alpha = \\left({S_\\alpha, \\tau_\\alpha}\\right)$ and $T_\\beta = \\left({S_\\beta, \\tau_\\beta}\\right)$ be topological spaces. Let $f: T_\\alpha \\to T_\\beta$ be a bijection. === Definition 1 === {{:Definition:Homeomorphism/Topological Spaces/Definition 1}} === Definition 2 === {{:Definition:Homeomorphism/Topological Spaces/Definition 2}} === Definition 3 === {{:Definition:Homeomorphism/Topological Spaces/Definition 3}} === Definition 4 === {{:Definition:Homeomorphism/Topological Spaces/Definition 4}}"}
+{"_id": "23258", "title": "Definition:Homeomorphism/Manifolds", "text": "Let $X$ and $Y$ be manifolds. A '''homeomorphism''' of $X$ to $Y$ is a continuous bijection such that the inverse is also continuous."}
+{"_id": "23259", "title": "Definition:Lipschitz Equivalence/Metric Spaces", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f: M_1 \\to M_2$ be a mapping such that $\\exists h, k \\in \\R_{>0}$ such that: :$\\forall x, y \\in A: h d_2 \\left({f \\left({x}\\right), f \\left({y}\\right)}\\right) \\le d_1 \\left({x, y}\\right) \\le k d_2 \\left({f \\left({x}\\right), f \\left({y}\\right)}\\right)$ Then $f$ is a '''Lipschitz equivalence''', and $M_1$ and $M_2$ are  described as '''Lipschitz equivalent'''."}
+{"_id": "23260", "title": "Definition:Open Set Axioms", "text": "{{begin-axiom}} {{axiom | n = \\text O 1         | t = The union of an arbitrary subset of $\\tau$ is an element of $\\tau$. }} {{axiom | n = \\text O 2         | t = The intersection of any two elements of $\\tau$ is an element of $\\tau$. }} {{axiom | n = \\text O 3         | t = $S$ is an element of $\\tau$. }} {{end-axiom}}"}
+{"_id": "23261", "title": "Definition:Continuous Mapping (Topology)/Point", "text": "Let $x \\in S_1$. === Definition using Open Sets === {{Definition:Continuous Mapping (Topology)/Point/Open Sets}} === Definition using Filters === {{Definition:Continuous Mapping (Topology)/Point/Filters}} If necessary, we can say that '''$f$ is $\\left({\\tau_1, \\tau_2}\\right)$-continuous at $x$'''."}
+{"_id": "23262", "title": "Definition:Continuous Mapping (Topology)/Set", "text": "Let $S$ be a subset of $S_1$. The mapping $f$ is '''continuous on $S$''' {{iff}} $f$ is continuous at every point $x \\in S$."}
+{"_id": "23263", "title": "Definition:Continuous Mapping (Topology)/Everywhere", "text": "=== Definition by Pointwise Continuity === {{Definition:Continuous Mapping (Topology)/Everywhere/Pointwise}} === Definition by Open Sets === {{Definition:Continuous Mapping (Topology)/Everywhere/Open Sets}}"}
+{"_id": "23264", "title": "Definition:Continuous Mapping (Topology)/Everywhere/Open Sets", "text": "The mapping $f$ is '''continuous on $S_1$''' {{iff}}: :$U \\in \\tau_2 \\implies f^{-1} \\sqbrk U \\in \\tau_1$ where $f^{-1} \\sqbrk U$ denotes the preimage of $U$ under $f$."}
+{"_id": "23265", "title": "Definition:Discrete Topology/Finite", "text": "Let $S$ be a finite set. Then $\\tau = \\powerset S$ is a '''finite discrete topology''', and $\\struct {S, \\tau} = \\struct {S, \\powerset S}$ is a '''finite discrete space'''."}
+{"_id": "23266", "title": "Definition:Discrete Topology/Countable", "text": "Let $S$ be a countably infinite set. Then $\\tau = \\powerset S$ is a '''countable discrete topology''', and $\\struct {S, \\tau} = \\struct {S, \\powerset S}$ is a '''countable discrete space'''."}
+{"_id": "23267", "title": "Definition:Discrete Topology/Uncountable", "text": "Let $S$ be an uncountably infinite set. Then $\\tau = \\powerset S$ is an '''uncountable discrete topology''', and $\\struct {S, \\tau} = \\struct {S, \\powerset S}$ is an '''uncountable discrete space'''."}
+{"_id": "23268", "title": "Definition:Discrete Topology/Infinite", "text": "Let $S$ be an infinite set. Then $\\tau = \\powerset S$ is an '''infinite discrete topology''', and $\\struct {S, \\tau} = \\struct {S, \\powerset S}$ is an '''infinite discrete space'''."}
+{"_id": "23269", "title": "Definition:Finer Topology/Strictly Finer", "text": "Let $\\tau_1 \\supsetneq \\tau_2$. $\\tau_1$ is said to be '''strictly finer''' than $\\tau_2$. This can be expressed as: :$\\tau_1 > \\tau_2 := \\tau_1 \\supsetneq \\tau_2$"}
+{"_id": "23270", "title": "Definition:Coarser Topology/Strictly Coarser", "text": "Let $\\tau_1 \\subsetneq \\tau_2$. Then $\\tau_1$ is said to be '''strictly coarser''' than $\\tau_2$. This can be expressed as: :$\\tau_1 < \\tau_2 := \\tau_1 \\subsetneq \\tau_2$"}
+{"_id": "23271", "title": "Definition:Uniform Continuity/Metric Space", "text": "Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces. Then a mapping $f: A_1 \\to A_2$ is '''uniformly continuous on $A_1$''' {{iff}}: :$\\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: \\forall x, y \\in A_1: \\map {d_1} {x, y} < \\delta \\implies \\map {d_2} {\\map f x, \\map f y} < \\epsilon$ where $\\R_{>0}$ denotes the set of all strictly positive real numbers."}
+{"_id": "23272", "title": "Definition:Uniform Continuity/Real Numbers", "text": "Let $I \\subseteq \\R$ be a real interval. A real function $f: I \\to \\R$ is said to be '''uniformly continuous''' on $I$ {{iff}}: :for every $\\epsilon > 0$ there exists $\\delta > 0$ such that the following property holds: ::for every $x, y \\in I$ such that $\\size {x - y} < \\delta$ it happens that $\\size {\\map f x - \\map f y} < \\epsilon$. Formally: $f: I \\to \\R$ is '''uniformly continuous''' {{iff}} the following property holds: :$\\forall \\epsilon > 0: \\exists \\delta > 0: \\paren {x, y \\in I, \\size {x - y} < \\delta \\implies \\size {\\map f x - \\map f y} < \\epsilon}$"}
+{"_id": "23273", "title": "Definition:Compact Space/Metric Space", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $\\tau$ denote the topology on $A$ induced by $d$. Then $M$ is '''compact''' {{iff}} $\\left({A, \\tau}\\right)$ is a compact topological space."}
+{"_id": "23274", "title": "Definition:Upper Integral", "text": "Let $\\closedint a b$ be a closed real interval. Let $f: \\closedint a b \\to \\R$ be a bounded real function. The '''upper integral of $f$ over $\\closedint a b$''' is defined as: :$\\displaystyle \\overline {\\int_a^b} \\map f x \\rd x = \\inf_P \\map U P$ where: :the infimum is taken over all subdivisions $P$ of $\\closedint a b$ :$\\map U P$ denotes the upper sum of $f$ on $\\closedint a b$ belonging to $P$."}
+{"_id": "23275", "title": "Definition:Lower Integral", "text": "Let $\\closedint a b$ be a closed real interval. Let $f: \\closedint a b \\to \\R$ be a bounded real function. The '''lower integral of $f$ over $\\closedint a b$''' is defined as: :$\\displaystyle \\underline {\\int_a^b} \\map f x \\rd x = \\sup_P \\map L P$ where: :the supremum is taken over all subdivisions $P$ of $\\closedint a b$ :$\\map L P$ denotes the lower sum of $f$ on $\\closedint a b$ belonging to $P$."}
+{"_id": "23276", "title": "Definition:Abstract Geometry", "text": "Let $P$ be a set and $L$ be a set of subsets of $P$. Then $\\left({P, L}\\right)$ is an '''abstract geometry''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | q = \\forall A, B \\in P         | m = \\exists l \\in L: A, B \\in l }} {{axiom | n = 2         | q = \\forall l \\in L         | m = \\exists A, B \\in P: A, B \\in l \\land A \\ne B }} {{end-axiom}}"}
+{"_id": "23277", "title": "Definition:Euclidean Plane", "text": "For any real number $a$ let: :$L_a = \\set {\\tuple {x, y} \\in \\R^2: x = a}$ Furthermore, define: :$L_A = \\set {L_a: a \\in \\R}$ For any two real numbers $m$ and $b$ let: :$L_{m, b} = \\set {\\tuple {x, y} \\in \\R^2: y = m x + b}$ Furthermore, define: :$L_{M, B} = \\set {L_{m, b}: m, b \\in \\R}$ Finally let: :$L_E = L_A \\cup L_{M, B}$ The abstract geometry $\\struct {\\R^2, L_E}$ is called the '''Euclidean plane'''."}
+{"_id": "23278", "title": "Definition:Multiple Edge/Multiplicity", "text": "The '''multiplicity''' of an edge is the number of edges having the same pair of endvertices."}
+{"_id": "23279", "title": "Definition:Poincaré Plane", "text": "Let: :$\\H = \\set {\\tuple {x, y} \\in \\R^2: y > 0}$ Let $a \\in \\R$ be a real number. Let: :${}_a L := \\set {\\tuple {x, y} \\in \\H: x = a}$ Define: :${}_A L := \\set{ {}_a L: a \\in \\R}$ Let $c \\in \\R$ be a real number and $r \\in \\R_{>0}$ be a strictly positive real number. Let: :${}_c L_r := \\set {\\tuple {x, y} \\in \\H: \\paren {x - c}^2 + y^2 = r^2}$ Define: :${}_C L_R := \\set { {}_c L_r: c \\in \\R \\land r \\in \\R_{>0} }$ Finally let: :$L_H = {}_A L \\cup {}_C L_R$ The abstract geometry $\\struct {\\H, L_H}$ is called the '''Poincaré plane'''. This is shown to be an abstract geometry in Poincaré Plane is Abstract Geometry."}
+{"_id": "23280", "title": "Definition:Topology Generated by Synthetic Sub-Basis/Definition 1", "text": "Define: :$\\displaystyle \\BB = \\set {\\bigcap \\FF: \\FF \\subseteq \\SS, \\FF \\text{ is finite} }$ That is, $\\BB$ is the set of all finite intersections of sets in $\\SS$. Note that $\\FF$ is allowed to be empty in the above definition. The '''topology generated by $\\SS$''', denoted $\\map \\tau \\SS$, is defined as: :$\\displaystyle \\map \\tau \\SS = \\set {\\bigcup \\AA: \\AA \\subseteq \\BB}$"}
+{"_id": "23281", "title": "Definition:Topology Generated by Synthetic Sub-Basis/Definition 2", "text": "The '''topology generated by $\\mathcal S$''', denoted $\\tau \\left({\\mathcal S}\\right)$, is defined as the unique topology on $X$ that satisfies the following axioms: :$\\left({1}\\right): \\quad \\mathcal S \\subseteq \\tau \\left({\\mathcal S}\\right)$ :$\\left({2}\\right): \\quad$ For any topology $\\mathcal T$ on $X$, the implication $\\mathcal S \\subseteq \\mathcal T \\implies \\tau \\left({\\mathcal S}\\right) \\subseteq \\mathcal T$ holds. That is, $\\tau \\left({\\mathcal S}\\right)$ is the coarsest topology on $X$ for which every element of $\\mathcal S$ is open."}
+{"_id": "23282", "title": "Definition:Balanced String", "text": "Let $S$ be a string in an alphabet containing the left and right brackets $($ and $)$. Then $S$ is said to be '''balanced''' iff it contains equally many left and right brackets."}
+{"_id": "23283", "title": "Definition:Sequential Continuity", "text": "Let $X$ and $Y$ be topological spaces. Let $f: X \\to Y$ be a mapping. === At a Point === {{:Definition:Sequential Continuity/Point}} === On a Domain === {{:Definition:Sequential Continuity/Domain}} Category:Definitions/Continuity Category:Definitions/Topology fc8tazwc9qttrg0qldxdvgcg2yxtqr9"}
+{"_id": "23284", "title": "Definition:Initial Topology/Definition 1", "text": "Let: :$\\mathcal S = \\left\\{{f_i^{-1} \\left[{U}\\right]: i \\in I, U \\in \\tau_i}\\right\\} \\subseteq \\mathcal P \\left({X}\\right)$ where $f_i^{-1} \\left[{U}\\right]$ denotes the preimage of $U$ under $f_i$. The topology $\\tau$ on $X$ generated by $\\mathcal S$ is called the '''initial topology on $X$ with respect to $\\left \\langle {f_i}\\right \\rangle_{i \\mathop \\in I}$'''."}
+{"_id": "23285", "title": "Definition:Initial Topology/Definition 2", "text": "Let $\\tau$ be the coarsest topology on $X$ such that each $f_i: X \\to Y_i$ is $\\left({\\tau, \\tau_i}\\right)$-continuous. Then $\\tau$ is known as the '''initial topology on $X$ with respect to $\\left \\langle {f_i} \\right \\rangle_{i \\mathop \\in I}$'''."}
+{"_id": "23286", "title": "Definition:Product Space (Topology)/General Definition", "text": "Let $\\family {\\struct {S_i, \\tau_i} }_{i \\mathop \\in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $S$ be the cartesian product of $\\family {S_i}_{i \\mathop \\in I}$: :$\\displaystyle S := \\prod_{i \\mathop \\in I} S_i$ Let $\\tau$ be the Tychonoff topology on $S$. From Natural Basis of Tychonoff Topology, $\\tau$ is generated from: :the basis $\\BB$ of cartesian products of the form $\\displaystyle \\prod_{i \\mathop \\in I} U_i$ where: ::for all $i \\in I : U_i \\in \\tau_i$ ::for all but finitely many indices $i : U_i = S_i$ The topological space $\\struct{X, \\tau}$ is called the '''product space''' of $\\family {\\struct {S_i, \\tau_i} }_{i \\mathop \\in I}$."}
+{"_id": "23287", "title": "Definition:Neighborhood (Topology)/Set", "text": "Let $A \\subseteq S$ be a subset of $S$. A '''neighborhood''' of $A$, which can be denoted $N_A$, is any subset of $S$ containing an open set of $T$ which itself contains $A$. That is: :$\\exists U \\in \\tau: A \\subseteq U \\subseteq N_A \\subseteq S$"}
+{"_id": "23288", "title": "Definition:Neighborhood (Topology)/Point", "text": "Let $z \\in S$ be a point in a $S$. Let $N_z$ be a subset of $S$ which contains (as a subset) an open set of $T$ which itself contains (as an element) $z$. Then $N_z$ is a '''neighborhood''' of $z$. That is: :$\\exists U \\in \\tau: z \\in U \\subseteq N_z \\subseteq S$"}
+{"_id": "23290", "title": "Definition:Closed Neighborhood", "text": "If $\\relcomp S {N_A} \\in \\tau$, that is if $N_A$ is closed in $S$, then $N_A$ is called a '''closed neighborhood'''."}
+{"_id": "23291", "title": "Definition:Limit Point/Topology/Set/Definition 1", "text": "A point $x \\in S$ is a '''limit point of $A$''' {{iff}} every open neighborhood $U$ of $x$ satisfies: :$A \\cap \\paren {U \\setminus \\set x} \\ne \\O$ That is, {{iff}} every open set $U \\in \\tau$ such that $x \\in U$ contains some point of $A$ distinct from $x$."}
+{"_id": "23292", "title": "Definition:Limit Point/Topology/Set/Definition 2", "text": "A point $x \\in S$ is a '''limit point of $A$''' {{iff}} : $x$ belongs to the closure of $A$ but is not an isolated point of $A$."}
+{"_id": "23293", "title": "Definition:Limit Point/Topology/Set/Definition 3", "text": "A point $x \\in S$ is a '''limit point of $A$''' {{iff}} $x$ is an adherent point of $A$ but is not an isolated point of $A$."}
+{"_id": "23294", "title": "Definition:Limit Point/Topology/Set/Definition 4", "text": "A point $x \\in S$ is a '''limit point of $A$''' {{iff}} $\\left({S \\setminus A}\\right) \\cup \\left\\{{x}\\right\\}$ is ''not'' a neighborhood of $x$."}
+{"_id": "23295", "title": "Definition:Adherent Point/Definition 1", "text": "A point $x \\in S$ is an '''adherent point of $A$''' {{iff}} every open neighborhood $U$ of $x$ satisfies: :$A \\cap U \\ne \\O$"}
+{"_id": "23296", "title": "Definition:Adherent Point/Definition 2", "text": "A point $x \\in S$ is an '''adherent point of $A$''' {{iff}} $x$ is an element of the closure of $A$."}
+{"_id": "23297", "title": "Definition:Limit Point/Topology/Set/Definition 5", "text": "{{questionable|this definition is not equivalent to the others}} A point $x \\in S$ is a '''limit point of $A$''' if there is a sequence $\\left\\langle{x_n}\\right\\rangle$ in $A$ such that $x$ is a limit point of $\\left\\langle{x_n}\\right\\rangle$, considered as sequence in $S$."}
+{"_id": "23298", "title": "Definition:Convergent Sequence/Topology/Definition 1", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\sequence {x_n}_{n \\mathop \\in \\N}$ be an infinite sequence in $S$. Then $\\sequence {x_n}$ '''converges to the limit $\\alpha \\in S$''' {{iff}}: :$\\forall U \\in \\tau: \\alpha \\in U \\implies \\paren {\\exists N \\in \\R_{>0}: \\forall n \\in \\N: n > N \\implies x_n \\in U}$"}
+{"_id": "23299", "title": "Definition:Convergent Sequence/Topology/Definition 2", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\sequence {x_n}_{n \\mathop \\in \\N}$ be an infinite sequence in $S$. Then $\\sequence {x_n}$ '''converges to the limit $\\alpha \\in S$''' {{iff}}: :$\\forall U \\in \\tau: \\alpha \\in U \\implies \\set {n \\in \\N: x_n \\notin U}$ is finite."}
+{"_id": "23300", "title": "Definition:Closure (Topology)/Definition 2", "text": "The '''closure of $H$ (in $T$)''' is defined as: :$\\displaystyle H^- := \\bigcap \\left\\{{K \\supseteq H: K}\\right.$ is closed in $\\left.{T}\\right\\}$"}
+{"_id": "23301", "title": "Definition:Closure (Topology)/Definition 1", "text": "The '''closure of $H$ (in $T$)''' is defined as: :$H^- := H \\cup H'$ where $H'$ is the derived set of $H$."}
+{"_id": "23302", "title": "Definition:Pullback Functor", "text": "Let $\\mathbf C$ be a metacategory having all pullbacks. Let $f: C \\to D$ be a morphism of $\\mathbf C$. Let $\\mathbf C \\mathop / C$ and $\\mathbf C \\mathop / D$ be the slice categories over $C$ and $D$, respectively. The '''pullback functor''' $f^* : \\mathbf C \\mathop / D \\to \\mathbf C \\mathop / C$ associated to $f$ is defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = f^* \\alpha := f^* \\alpha        |rc= $f^* \\alpha$ is the pullback of $\\alpha$ along $f$ }} {{axiom|lc= Morphism functor:        |m = f^* \\gamma := \\gamma'        |rc= $\\gamma': f^* \\alpha \\to f^* \\beta$ is as on Pullback of Commutative Triangle }} {{end-axiom}} Explicitly, $f^* \\gamma$ is defined as the unique morphism fitting: ::$\\begin{xy}\\xymatrix@+1em@L+4px{  A'   \\ar[rr]^*{f_\\alpha}   \\ar[dd]_*{f^* \\alpha}   \\ar@{-->}[rd]_*{f^* \\gamma} & &  A   \\ar[rd]^*{\\gamma}   \\ar[dd]^(.4)*{\\alpha} \\\\ &  B'   \\ar[ld]^*{f^* \\beta}   \\ar[rr] |{\\hole} ^(.3)*{f_\\beta} & &  B   \\ar[ld]^*{\\beta} \\\\  C   \\ar[rr]_*{f} & &  D }\\end{xy}$"}
+{"_id": "23303", "title": "Definition:Closure (Topology)/Definition 3", "text": "The '''closure of $H$ (in $T$)''', denoted $H^-$, is defined as the smallest closed set of $T$ that contains $H$."}
+{"_id": "23304", "title": "Definition:Closure (Topology)/Definition 4", "text": "The '''closure of $H$ (in $T$)''' is defined as the union of $H$ and its boundary in $T$: :$H^- := H \\cup \\partial H$"}
+{"_id": "23305", "title": "Definition:Closure (Topology)/Definition 5", "text": "The '''closure of $H$ (in $T$)''' is the union of the set of all isolated points of $H$ and the set of all limit points of $H$: :$H^- := H^i \\cup H'$"}
+{"_id": "23306", "title": "Definition:Diagram (Category Theory)", "text": "Let $\\mathbf J$ and $\\mathbf C$ be metacategories. A '''diagram of type $\\mathbf J$''' in $\\mathbf C$ is a functor $D: \\mathbf J \\to \\mathbf C$. === Index Category === In this context, $\\mathbf J$ is referred to as the '''index category'''. Its objects are typically denoted by lowercase letters, $i, j$ etc. Furthermore, one writes $D_i$ in place of the formally more correct $D \\left({i}\\right)$. Similarly, for $\\alpha: i \\to j$ a morphism one writes $D_\\alpha$ in place of $D \\left({\\alpha}\\right)$."}
+{"_id": "23307", "title": "Definition:Cone (Category Theory)", "text": "Let $\\mathbf C$ be a metacategory. Let $D: \\mathbf J \\to \\mathbf C$ be a $\\mathbf J$-diagram in $\\mathbf C$. A '''cone to $D$''' comprises an object $C$ of $\\mathbf C$, and a morphism: :$c_j: C \\to D_j$ for each object of $\\mathbf J$, such that for each morphism $\\alpha: i \\to j$ of $\\mathbf J$: ::$\\begin{xy}\\xymatrix@+0.5em@L+2px{  C   \\ar[d]_*+{c_i}   \\ar[dr]^*+{c_j} \\\\  D_i   \\ar[r]_*+{D_\\alpha} &  D_j }\\end{xy}$ is a commutative diagram. {{expand|when time comes, add def \"cone is natural trafo $\\kappa_C \\to D$ with $\\kappa_C$ constant functor\"}}"}
+{"_id": "23308", "title": "Definition:Morphism of Cones", "text": "Let $\\mathbf C$ be a metacategory. Let $D: \\mathbf J \\to \\mathbf C$ be a $\\mathbf J$-diagram in $\\mathbf C$. Let $\\left({C, c_j}\\right)$ and $\\left({C', c'_j}\\right)$ be cones to $D$. Let $f: C \\to C'$ be a morphism of $\\mathbf C$. Then $f$ is a '''morphism of cones''' iff, for all objects $j$ of $\\mathbf J$: ::$\\begin{xy}\\xymatrix@+0.5em@L+2px{  C   \\ar[r]^*+{f}   \\ar[dr]_*+{c_j} &  C'   \\ar[d]^*+{c'_j} \\\\ &  D_j }\\end{xy}$ is a commutative diagram."}
+{"_id": "23309", "title": "Definition:Interior (Topology)/Definition 1", "text": "The '''interior''' of $H$ is the union of all subsets of $H$ which are open in $T$. That is, the '''interior''' of $H$ is defined as: :$\\displaystyle H^\\circ := \\bigcup_{K \\mathop \\in \\mathbb K} K$ where $\\mathbb K = \\set {K \\in \\tau: K \\subseteq H}$."}
+{"_id": "23310", "title": "Definition:Interior (Topology)/Definition 2", "text": "The '''interior''' of $H$ is defined as the largest open set of $T$ which is contained in $H$."}
+{"_id": "23311", "title": "Definition:Interior Point (Topology)", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$. === Definition 1 === {{:Definition:Interior Point (Topology)/Definition 1}} === Definition 2 === {{:Definition:Interior Point (Topology)/Definition 2}} === Definition 3 === {{:Definition:Interior Point (Topology)/Definition 3}}"}
+{"_id": "23312", "title": "Definition:Nowhere Dense/Definition 1", "text": "$H$ is '''nowhere dense''' in $T$ {{iff}}: :$\\paren {H^-}^\\circ = \\O$ where $H^-$ denotes the closure of $H$ and $H^\\circ$ denotes its interior."}
+{"_id": "23313", "title": "Definition:Nowhere Dense/Definition 2", "text": "$H$ is '''nowhere dense''' in $T$ {{iff}}: :$H^-$ contains no open set of $T$ which is non-empty where $H^-$ denotes the closure of $H$."}
+{"_id": "23314", "title": "Definition:Category of Cones", "text": "Let $\\mathbf C$ be a metacategory. Let $D: \\mathbf J \\to \\mathbf C$ be a $\\mathbf J$-diagram in $\\mathbf C$. The '''category of cones to $D$''', denoted $\\mathbf{Cone} \\left({D}\\right)$, is defined as follows: {{DefineCategory | ob = The cones to $D$ | mor = The morphisms of cones | comp = Composition in $\\mathbf C$ |id = $\\operatorname{id}_{\\left({C, c_j}\\right)} := \\operatorname{id}_C$, for a cone $\\left({C, c_j}\\right)$ }}"}
+{"_id": "23315", "title": "Definition:Limit (Category Theory)", "text": "Let $\\mathbf C$ be a metacategory. Let $D: \\mathbf J \\to \\mathbf C$ be a $\\mathbf J$-diagram in $\\mathbf C$. Let $\\mathbf{Cone} \\left({D}\\right)$ be the category of cones to $D$. A '''limit for $D$''' is a terminal object in $\\mathbf{Cone} \\left({D}\\right)$. It is denoted by $\\varprojlim_j D_j$; the associated morphisms $p_i: \\varprojlim_j D_j \\to D_i$ are usually left implicit."}
+{"_id": "23316", "title": "Definition:Identification Topology/Identification Mapping", "text": "The mapping $f: S_1 \\to S_2$ in this context is called the '''identification mapping'''."}
+{"_id": "23318", "title": "Definition:Product (Category Theory)/General Definition", "text": "Let $\\mathbf C$ be a metacategory. Let $\\mathcal C$ be any collection of objects of $\\mathbf C$. Let $\\mathbf{Dis} \\left({\\mathcal C}\\right)$ be the discrete category on $\\mathcal C$, considered as a subcategory of $\\mathbf C$. A '''product''' for $\\mathcal C$, denoted $\\displaystyle \\prod \\mathcal C$, is a limit for the inclusion functor $D: \\mathbf{Dis} \\left({\\mathcal C}\\right) \\to \\mathbf C$, considered as a diagram. For an object $C$ in $\\mathcal C$, the associated morphism $\\displaystyle \\prod \\mathcal C \\to C$ is denoted $\\operatorname{pr}_C$ and called the '''projection on $C$'''. The whole construction is pictured in the following commutative diagram: ::$\\begin{xy}\\xymatrix@R-.5em@L+3px{ & &  A   \\ar@{-->}[dd]   \\ar[dddl]_*+{a_C}   \\ar[dddr]^*+{a_C'} \\\\ \\\\ & &  \\displaystyle \\prod \\mathcal C   \\ar[dl]^*{\\operatorname{pr}_C}   \\ar[dr]_*{\\operatorname{pr}_{C'}} \\\\  \\mathbf{Dis} \\left({\\mathcal C}\\right) &  C &  \\dots \\quad \\dots &  C' }\\end{xy}$ === Finite Product === {{:Definition:Product (Category Theory)/General Definition/Finite Product}}"}
+{"_id": "23319", "title": "Definition:Product (Category Theory)/Binary Product", "text": "Let $\\mathbf C$ be a metacategory. Let $A$ and $B$ be objects of $\\mathbf C$. A '''(binary) product diagram''' for $A$ and $B$ comprises an object $P$ and morphisms $p_1: P \\to A$, $p_2: P \\to B$: ::$\\begin{xy}\\xymatrix@+1em@L+3px{  A &  P   \\ar[l]_*+{p_1}   \\ar[r]^*+{p_2} &  B }\\end{xy}$ subjected to the following universal mapping property: :For any object $X$ and morphisms $x_1, x_2$ like so: ::$\\begin{xy}\\xymatrix@+1em@L+3px{  A &  X   \\ar[l]_*+{x_1}   \\ar[r]^*+{x_2} &  B }\\end{xy}$ :there is a unique morphism $u: X \\to P$ such that: ::$\\begin{xy}\\xymatrix@+1em@L+3px{ &  X   \\ar[ld]_*+{x_1}   \\ar@{-->}[d]^*+{u}   \\ar[rd]^*+{x_2} \\\\  A &  P   \\ar[l]^*+{p_1}   \\ar[r]_*+{p_2} &  B }\\end{xy}$ :is a commutative diagram, i.e., $x_1 = p_1 \\circ u$ and $x_2 = p_2 \\circ u$. In this situation, $P$ is called a '''(binary) product of $A$ and $B$''' and may be denoted $A \\times B$. Generally, one writes $\\left\\langle{x_1, x_2}\\right\\rangle$ for the unique morphism $u$ determined by above diagram. The morphisms $p_1$ and $p_2$ are often taken to be implicit. They are called '''projections'''; if necessary, $p_1$ can be called the '''first projection''' and $p_2$ the '''second projection'''. {{expand|the projection definition may merit its own, separate page}}"}
+{"_id": "23320", "title": "Definition:Product (Category Theory)/General Definition/Finite Product", "text": "Let $\\displaystyle \\prod \\mathcal C$ be a product for a finite set $\\mathcal C$ of objects of $\\mathbf C$. Then $\\displaystyle \\prod \\mathcal C$ is called a '''finite product'''."}
+{"_id": "23321", "title": "Definition:Finite Category", "text": "Let $\\mathbf C$ be a metacategory. Then $\\mathbf C$ is said to be a '''finite category''' iff both the collection of its objects $\\mathbf C_0$ and the collection of its morphisms $\\mathbf C_1$ are finite sets."}
+{"_id": "23322", "title": "Definition:Limit (Category Theory)/Finite Limit", "text": "Let $\\varprojlim_j D_j$ be a limit for $D$. Then $\\varprojlim_j D_j$ is called a '''finite limit''' {{iff}} $\\mathbf J$ is a finite category."}
+{"_id": "23323", "title": "Definition:Functor Preserving Limits", "text": "Let $\\mathbf C$, $\\mathbf D$ and $\\mathbf J$ be metacategories. Let $F: \\mathbf C \\to \\mathbf D$ be a functor. Then $F$ '''preserves limits of type $\\mathbf J$''' iff for all diagrams $D: \\mathbf J \\to \\mathbf C$ with limit ${\\varprojlim \\,}_j \\, D_j$: :$F \\left({{\\varprojlim \\,}_j \\, D_j}\\right) \\cong {\\varprojlim \\,}_j \\, F D_j$ where $F D: \\mathbf J \\to \\mathbf D$ is the composition of $F$ with $D$."}
+{"_id": "23324", "title": "Definition:Continuous Functor", "text": "Let $\\mathbf C$, $\\mathbf D$ be metacategories. Let $F: \\mathbf C \\to \\mathbf D$ be a functor. Then $F$ is '''continuous''' iff for all diagrams $D: \\mathbf J \\to \\mathbf C$ with limit ${\\varprojlim \\,}_j \\, D_j$: :$F \\left({{\\varprojlim \\,}_j \\, D_j}\\right) \\cong {\\varprojlim \\,}_j \\, F D_j$ where $F D: \\mathbf J \\to \\mathbf D$ is the diagram obtained by composition of $F$ with $D$, and $\\mathbf J$ is an arbitrary metacategory."}
+{"_id": "23325", "title": "Definition:Hausdorff Space/Definition 1", "text": "$\\struct {S, \\tau}$ is a '''Hausdorff space''' or '''$T_2$ space''' {{iff}}: :$\\forall x, y \\in S, x \\ne y: \\exists U, V \\in \\tau: x \\in U, y \\in V: U \\cap V = \\O$  That is: :for any two distinct elements $x, y \\in S$ there exist disjoint open sets $U, V \\in \\tau$ containing $x$ and $y$ respectively."}
+{"_id": "23326", "title": "Definition:Hausdorff Space/Definition 2", "text": "$\\struct {S, \\tau}$ is a '''Hausdorff space''' or '''$T_2$ space''' {{iff}} each point is the intersection of all its closed neighborhoods."}
+{"_id": "23327", "title": "Definition:Cocone", "text": "Let $\\mathbf C$ be a metacategory. Let $D: \\mathbf J \\to \\mathbf C$ be a $\\mathbf J$-diagram in $\\mathbf C$. A '''cocone from $D$''' comprises an object $C$ of $\\mathbf C$, and a morphism: :$c_j: D_j \\to C$ for each object of $\\mathbf J$, such that for each morphism $\\alpha: i \\to j$ of $\\mathbf J$: ::$\\begin{xy}\\xymatrix@+0.5em@L+2px{  D_i   \\ar[r]^*+{D_\\alpha}   \\ar[dr]_*+{c_i} &  D_j   \\ar[d]^*+{c_j} \\\\ &  C }\\end{xy}$ is a commutative diagram. {{expand|when time comes, add def \"cocone is natural trafo $D \\to \\kappa_C$ with $\\kappa_C$ constant functor\"}}"}
+{"_id": "23328", "title": "Definition:Morphism of Cocones", "text": "Let $\\mathbf C$ be a metacategory. Let $D: \\mathbf J \\to \\mathbf C$ be a $\\mathbf J$-diagram in $\\mathbf C$. Let $\\left({C, c_j}\\right)$ and $\\left({C', c'_j}\\right)$ be cocones from $D$. Let $f: C \\to C'$ be a morphism of $\\mathbf C$. Then $f$ is a '''morphism of cones''' iff, for all objects $j$ of $\\mathbf J$: ::$\\begin{xy}\\xymatrix@+0.5em@L+3px{  D_j   \\ar[d]_*+{c_j}   \\ar[dr]^*+{c'_j} \\\\  C   \\ar[r]_*+{f} &  C' }\\end{xy}$ is a commutative diagram."}
+{"_id": "23329", "title": "Definition:Category of Cocones", "text": "Let $\\mathbf C$ be a metacategory. Let $D: \\mathbf J \\to \\mathbf C$ be a $\\mathbf J$-diagram in $\\mathbf C$. The '''category of cocones from $D$''', denoted $\\mathbf{Cocone} \\left({D}\\right)$, is the category with: {{DefineCategory |ob = cocones to $D$ |mor = morphisms of cocones |comp = Composition in $\\mathbf C$ |id = $\\operatorname{id}_{\\left({C, c_j}\\right)} := \\operatorname{id}_C$, for a cocone $\\left({C, c_j}\\right)$ }}"}
+{"_id": "23330", "title": "Definition:Colimit", "text": "Let $\\mathbf C$ be a metacategory. Let $D: \\mathbf J \\to \\mathbf C$ be a $\\mathbf J$-diagram in $\\mathbf C$. Let $\\mathbf{Cocone} \\left({D}\\right)$ be the category of cocones from $D$. A '''colimit for $D$''' is an initial object in $\\mathbf{Cocone} \\left({D}\\right)$. It is denoted by $\\varinjlim_j D_j$; the associated morphisms $\\iota_i: D_i \\to \\varinjlim_j D_j$ are usually left implicit. === Finite Colimit === {{:Definition:Colimit/Finite Colimit}}"}
+{"_id": "23331", "title": "Definition:Colimit/Finite Colimit", "text": "Let $\\varinjlim_j D_j$ be a colimit for $D$. Then $\\varinjlim_j D_j$ is called a '''finite colimit''' {{iff}} $\\mathbf J$ is a finite category."}
+{"_id": "23332", "title": "Definition:T3 Space/Definition 1", "text": "$T = \\struct {S, \\tau}$ is a '''$T_3$ space''' {{iff}}: :$\\forall F \\subseteq S: \\relcomp S F \\in \\tau, y \\in \\relcomp S F: \\exists U, V \\in \\tau: F \\subseteq U, y \\in V: U \\cap V = \\O$  That is, for any closed set $F \\subseteq S$ and any point $y \\in S$ such that $y \\notin F$ there exist disjoint open sets $U, V \\in \\tau$ such that $F \\subseteq U$, $y \\in V$."}
+{"_id": "23333", "title": "Definition:T3 Space/Definition 2", "text": "$T = \\struct {S, \\tau}$ is '''$T_3$''' {{iff}} each open set contains a closed neighborhood around each of its points: :$\\forall U \\in \\tau: \\forall x \\in U: \\exists N_x: \\relcomp S {N_x} \\in \\tau: \\exists V \\in \\tau: x \\in V \\subseteq N_x \\subseteq U$ where $N_x$ denotes a neighborhood of $x$."}
+{"_id": "23334", "title": "Definition:T3 Space/Definition 3", "text": "$T = \\struct {S, \\tau}$ is '''$T_3$''' {{iff}} each of its closed sets is the intersection of its closed neighborhoods: :$\\forall H \\subseteq S: \\relcomp S H \\in \\tau: H = \\bigcap \\set {N_H: \\relcomp S H \\in \\tau, \\exists V \\in \\tau: H \\subseteq V \\subseteq N_H}$"}
+{"_id": "23335", "title": "Definition:Fréchet Space (Topology)/Definition 2", "text": "$\\struct {S, \\tau}$ is a '''Fréchet space''' or '''$T_1$ space''' {{iff}} all points of $S$ are closed in $T$."}
+{"_id": "23337", "title": "Definition:Separated Sets", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A, B \\subseteq S$. === Definition 1 === {{:Definition:Separated Sets/Definition 1}} === Definition 2 === {{:Definition:Separated Sets/Definition 2}} $A$ and $B$ are said to be '''separated sets (of $T$)'''."}
+{"_id": "23338", "title": "Definition:Separated Points", "text": "Let $\\struct {S, \\tau}$ be a topological space. Let $x, y \\in S$ such that both of the following hold: :$\\exists U \\in \\tau: x \\in U, y \\notin U$ :$\\exists V \\in \\tau: y \\in V, x \\notin V$ Then $x$ and $y$ are '''separated points'''."}
+{"_id": "23340", "title": "Definition:Kolmogorov Space/Definition 2", "text": "$\\struct {S, \\tau}$ is a '''Kolmogorov space''' or '''$T_0$ space''' {{iff}} no two points can be limit points of each other."}
+{"_id": "23341", "title": "Definition:Direct Limit of Sequence of Groups", "text": "=== Definition 1 === {{:Definition:Direct Limit of Sequence of Groups/Definition 1}} === Definition 2 === {{:Definition:Direct Limit of Sequence of Groups/Definition 2}}"}
+{"_id": "23342", "title": "Definition:T4 Space/Definition 1", "text": "$T = \\struct {S, \\tau}$ is a '''$T_4$ space''' {{iff}}: :$\\forall A, B \\in \\map \\complement \\tau, A \\cap B = \\O: \\exists U, V \\in \\tau: A \\subseteq U, B \\subseteq V, U \\cap V = \\O$  That is, for any two disjoint closed sets $A, B \\subseteq S$ there exist disjoint open sets $U, V \\in \\tau$ containing $A$ and $B$ respectively."}
+{"_id": "23343", "title": "Definition:T4 Space/Definition 2", "text": "$T = \\struct {S, \\tau}$ is '''$T_4$''' {{iff}} each open set $U$ contains a closed neighborhood of each closed set contained in $U$."}
+{"_id": "23344", "title": "Definition:T5 Space/Definition 1", "text": "$\\struct {S, \\tau}$ is a '''$T_5$ space''' {{iff}}: :$\\forall A, B \\subseteq S, A^- \\cap B = A \\cap B^- = \\O: \\exists U, V \\in \\tau: A \\subseteq U, B \\subseteq V, U \\cap V = \\O$    That is: :$\\struct {S, \\tau}$ is a '''$T_5$ space''' when for any two separated sets $A, B \\subseteq S$ there exist disjoint open sets $U, V \\in \\tau$ containing $A$ and $B$ respectively."}
+{"_id": "23345", "title": "Definition:T5 Space/Definition 2", "text": "$\\struct {S, \\tau}$ is a '''$T_5$ space''' {{iff}}: :$\\forall Y,A \\subseteq S: (A \\subseteq Y^\\circ \\wedge A^- \\subseteq Y) \\implies \\exists N \\subseteq Y: \\relcomp S N \\in \\tau: \\exists U \\in \\tau: A \\subseteq U \\subseteq N$ That is: :$\\struct {S, \\tau}$ is a '''$T_5$ space''' {{iff}} every subset $Y \\subseteq S$ contains a closed neighborhood of each $A \\subseteq Y^\\circ$ for which $A^- \\subseteq Y$. In the above, $Y^\\circ$ denotes the interior of $Y$ and $A^-$ denotes the closure of $A$."}
+{"_id": "23346", "title": "Definition:Homeomorphism/Topological Spaces/Definition 1", "text": "$f$ is a '''homeomorphism''' {{iff}} both $f$ and $f^{-1}$ are continuous."}
+{"_id": "23347", "title": "Definition:Homeomorphism/Topological Spaces/Definition 2", "text": "$f$ is a '''homeomorphism''' {{iff}}: :$\\forall U \\subseteq S_\\alpha: U \\in \\tau_\\alpha \\iff f \\sqbrk U \\in \\tau_\\beta$"}
+{"_id": "23348", "title": "Definition:Homeomorphism/Topological Spaces/Definition 3", "text": "$f$ is a '''homeomorphism''' {{iff}} $f$ is both an open mapping and a continuous mapping."}
+{"_id": "23349", "title": "Definition:Homeomorphism/Topological Spaces/Definition 4", "text": "$f$ is a '''homeomorphism''' {{iff}} $f$ is both a closed mapping and a continuous mapping."}
+{"_id": "23350", "title": "Definition:Category of Groups", "text": "The '''category of groups''', denoted $\\mathbf{Grp}$, is the metacategory with: {{DefineCategory |ob   = groups |mor  = group homomorphisms |id   = identity mappings |comp = composition of mappings }}"}
+{"_id": "23351", "title": "Definition:Cover of Set/Finite", "text": "A cover $\\mathcal C$ for $S$ is a '''finite cover''' {{iff}} $\\mathcal C$ is a finite set."}
+{"_id": "23352", "title": "Definition:Cover of Set/Countable", "text": "A cover $\\mathcal C$ for $S$ is a '''countable cover''' {{iff}} $\\mathcal C$ is a countable set."}
+{"_id": "23353", "title": "Definition:Subcover/Finite", "text": "A '''finite subcover of $\\mathcal U$ for $S$''' is a subcover $\\mathcal V \\subseteq \\mathcal U$ which is finite."}
+{"_id": "23354", "title": "Definition:Subcover/Countable", "text": "A '''countable subcover of $\\mathcal U$ for $S$''' is a subcover $\\mathcal V \\subseteq \\mathcal U$ which is countable."}
+{"_id": "23355", "title": "Definition:Compact Space/Topology/Subspace/Definition 1", "text": "The topological subspace $T_H = \\left({H, \\tau_H}\\right)$ is '''compact in $T$''' {{iff}} $T_H$ is itself a compact topological space."}
+{"_id": "23356", "title": "Definition:Compact Space/Topology/Subspace/Definition 2", "text": "$H$ is '''compact in $T$''' {{iff}} every open cover $\\mathcal C \\subseteq \\tau$ for $H$ has a finite subcover."}
+{"_id": "23357", "title": "Definition:Direct Limit of Sequence of Groups/Definition 1", "text": "Let $\\sequence {G_n}_{n \\mathop \\in \\N}$ be a sequence of groups. For each $n \\in \\N$, let $g_n: G_n \\to G_{n + 1}$ be a group homomorphism. A '''direct limit''' for the sequences $\\sequence {G_n}_{n \\mathop \\in \\N}$ and $\\sequence {g_n}_{n \\mathop \\in \\N}$ comprises: :$(1): \\quad$ a group $G_\\infty$ :$(2): \\quad$ for each $n \\in \\N$, a group homomorphism $u_n: G_n \\to G_\\infty$ such that, for all $n \\in \\N$: :$u_{n + 1} \\circ g_n = u_n$ and, for all groups $H$ together with group homomorphisms $h_n: G_n \\to H$ satisfying $h_{n + 1} \\circ g_n = h_n$, there exists a unique group homomorphism: :$h_\\infty: G_\\infty \\to H$ such that for all $n \\in \\N$: :$h_n = h_\\infty \\circ u_n$"}
+{"_id": "23358", "title": "Definition:Direct Limit of Sequence of Groups/Definition 2", "text": "Let $\\N$ be the order category on the natural numbers. Let $\\mathbf{Grp}$ be the category of groups. Let $G: \\N \\to \\mathbf{Grp}$ be an $\\N$-diagram in $\\mathbf{Grp}$. A '''direct limit''' for $G$ is a colimit ${\\varinjlim \\,}_n \\, G_n$, and is denoted $G_\\infty$."}
+{"_id": "23359", "title": "Definition:Functor Creating Limits", "text": "Let $\\mathbf C, \\mathbf D$ and $\\mathbf J$ be metacategories. Let $F: \\mathbf C \\to \\mathbf D$ be a functor. Then $F$ is said to '''create limits of type $\\mathbf J$''' iff: :For all $\\mathbf J$-diagrams $C: \\mathbf J \\to \\mathbf C$ in $\\mathbf C$, given a limit $\\left({{\\varprojlim \\,}_j \\, FC_j, q_j}\\right)$ for $FC: \\mathbf J \\to \\mathbf D$ in $\\mathbf D$, the limit: ::$\\left({{\\varprojlim \\,}_j \\, C_j, p_j}\\right)$ :exists, and furthermore: ::$F \\left({{\\varprojlim \\,}_j \\, C_j}\\right) = {\\varprojlim \\,}_j \\, FC_j$ ::$F p_j = q_j$ :for all objects $j$ of $\\mathbf J$."}
+{"_id": "23360", "title": "Definition:Functor Creating Colimits", "text": "Let $\\mathbf C, \\mathbf D$ and $\\mathbf J$ be metacategories. Let $F: \\mathbf C \\to \\mathbf D$ be a functor. Then $F$ is said to '''create colimits of type $\\mathbf J$''' iff: :For all $\\mathbf J$-diagrams $C: \\mathbf J \\to \\mathbf C$ in $\\mathbf C$, given a colimit $\\left({{\\varinjlim \\,}_j \\, FC_j, q_j}\\right)$ for $FC: \\mathbf J \\to \\mathbf D$ in $\\mathbf D$, the colimit: ::$\\left({{\\varinjlim \\,}_j \\, C_j, p_j}\\right)$ :exists, and furthermore: ::$F \\left({{\\varinjlim \\,}_j \\, C_j}\\right) = {\\varinjlim \\,}_j \\, FC_j$ ::$F p_j = q_j$ :for all objects $j$ of $\\mathbf J$."}
+{"_id": "23361", "title": "Definition:Uniform Equivalence/Metrics", "text": "Let $A$ be a set on which there are two metrics imposed: $d_1$ and $d_2$. Then $d_1$ and $d_2$ are '''uniformly equivalent''' iff the identity mapping of $A$ is uniformly $\\left({d_1, d_2}\\right)$-continuous and also uniformly $\\left({d_2, d_1}\\right)$-continuous."}
+{"_id": "23362", "title": "Definition:Uniform Equivalence/Metric Spaces", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Then the mapping $f: A_1 \\to A_2$ is a '''uniform equivalence''' of $M_1$ with $M_2$ iff $f$ is a bijection such that $f$ and $f^{-1}$ are both uniformly continuous."}
+{"_id": "23363", "title": "Definition:Evaluation Mapping", "text": "Let $S, T$ be sets, and let $S^T$ be the set of all mappings from $T$ to $S$. The '''evaluation mapping for $S^T$''' is the mapping $\\operatorname{ev}: S^T \\times T \\to S$ defined by: :$\\operatorname{ev} \\left({f, t}\\right) := f \\left({t}\\right)$"}
+{"_id": "23364", "title": "Definition:Exponential (Category Theory)", "text": "Let $\\mathbf C$ be a metacategory with binary products. Let $B$ and $C$ be objects of $\\mathbf C$. An '''exponential of $C$ by $B$''' consists of an object $C^B$ of $\\mathbf C$ and a morphism: :$\\epsilon: C^B \\times B \\to C$ subject to the following UMP: :For all objects $A$ and morphisms $f: A \\times B \\to C$, there exists a unique morphism: ::$\\tilde f: A \\to C^B$ :such that: :::$\\begin{xy}\\xymatrix@+1em@L+3px{  C^B \\times B   \\ar[r]^*+{\\epsilon} &  C \\\\  A \\times B   \\ar[u]_*+{\\tilde f \\times \\operatorname{id}_B \\hskip{2cm}}   \\ar[ur]_*+{f} }\\end{xy}$ :is a commutative diagram, i.e. $\\epsilon \\circ \\left({\\tilde f \\times \\operatorname{id}_B}\\right) = f$. === Evaluation Morphism === {{:Definition:Exponential (Category Theory)/Evaluation}} === Exponential Transpose === {{:Definition:Exponential (Category Theory)/Transpose}} === Category with Exponentials === {{:Definition:Category with Exponentials}}"}
+{"_id": "23365", "title": "Definition:Topological Manifold/Complex Manifold", "text": "Let $M$ be a second-countable, complex locally Euclidean space of dimension $d$.  Let $\\mathscr F$ be a complex analytic differentiable structure on $M$. Then $\\left({M, \\mathscr F}\\right)$ is called a '''complex manifold of dimension $d$'''."}
+{"_id": "23366", "title": "Definition:Smooth Differentiable Structure", "text": "Let $M$ be a topological space. Let $d$ be a natural number. A '''$d$-dimensional smooth differentiable structure''' $\\mathscr F$ on $M$ is a $d$-dimensional differentiable structure on $M$ which is of class $\\mathcal C^k$ for every $k \\in \\N$. Category:Definitions/Manifolds nq6z6g1sej00ysl0zvoltq30dtnffsn"}
+{"_id": "23367", "title": "Definition:Complex Analytic Differentiable Structure", "text": "Let $M$ be a locally Euclidean space of dimension $d$. Then a '''complex analytic structure''' $\\mathscr F$ on $M$ is a collection of charts $\\{(U_\\alpha,\\phi_\\alpha) : \\alpha \\in A\\}$ such that: :$(1): \\quad \\displaystyle \\bigcup_{\\alpha \\in A} U_\\alpha = M$ :$(2): \\quad \\phi_\\alpha \\circ \\phi_\\beta^{-1}$ is of biholomorphic as a map $\\phi_\\beta\\left(U_\\alpha \\cap U_\\beta\\right) \\to \\phi_\\alpha\\left(U_\\alpha \\cap U_\\beta\\right)$ for all $\\alpha,\\beta \\in A$ :$(3): \\quad$ If $(U,\\phi)$ is a chart such that $\\phi \\circ \\phi_\\alpha^{-1}$ and $\\phi_\\alpha \\circ \\phi^{-1}$ are biholomorphic for all $\\alpha \\in A$, then $(U,\\phi) \\in \\mathscr F$."}
+{"_id": "23368", "title": "Definition:Locally Euclidean Space/Complex", "text": "$M$ is a '''complex locally Euclidean space of dimension $d$''' {{iff}} each point in $M$ has an open neighbourhood homeomorphic to an open subset of complex Euclidean space $\\C^d$."}
+{"_id": "23369", "title": "Definition:Category with Products", "text": "Let $\\mathbf C$ be a metacategory. Then $\\mathbf C$ is said to '''have products''' or to be a '''(meta)category with products''' iff: :For all sets of objects $\\mathcal C \\subseteq \\mathbf C_0$, there is a product $\\displaystyle \\prod \\mathcal C$ for $\\mathcal C$. === Category with Binary Products === {{:Definition:Category with Products/Binary}} === Category with Finite Products === {{:Definition:Category with Products/Finite}}"}
+{"_id": "23370", "title": "Definition:Category with Products/Finite", "text": "Let $\\mathbf C$ be a metacategory. Then $\\mathbf C$ is said to '''have products''' or to be a '''(meta)category with products''' iff: :For all finite sets of objects $\\mathcal C \\subseteq \\mathbf C_0$, there is a product $\\displaystyle \\prod \\mathcal C$ for $\\mathcal C$."}
+{"_id": "23371", "title": "Definition:Category with Exponentials", "text": "Suppose $\\mathbf C$ has an exponential $C^B$ for all objects $B$ and $C$ of $\\mathbf C$. Then $\\mathbf C$ is called a '''category with exponentials'''."}
+{"_id": "23372", "title": "Definition:Cartesian Closed Category", "text": "Let $\\mathbf C$ be a metacategory. Then $\\mathbf C$ is called '''Cartesian closed''' {{iff}} $\\mathbf C$ is a metacategory with finite products and with exponentials."}
+{"_id": "23373", "title": "Definition:Topological Manifold", "text": "=== Differentiable Manifold === {{:Definition:Topological Manifold/Differentiable Manifold}} === Smooth Manifold === {{:Definition:Topological Manifold/Smooth Manifold}} === Complex Manifold === {{:Definition:Topological Manifold/Complex Manifold}}"}
+{"_id": "23374", "title": "Definition:Connected (Topology)/Topological Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a non-empty topological space."}
+{"_id": "23375", "title": "Definition:Connected (Topology)/Set", "text": "{{:Definition:Connected (Topology)/Set/Definition 1}}"}
+{"_id": "23376", "title": "Definition:Connected (Topology)/Points", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $a, b \\in S$. Then $a$ and $b$ are '''connected''' (in $T$) {{iff}} there exists a connected set in $T$ containing both $a$ and $b$."}
+{"_id": "23377", "title": "Definition:Disconnected (Topology)", "text": "=== Topological Space === {{Definition:Disconnected (Topology)/Topological Space}} === Subset of Topological Space === {{Definition:Disconnected (Topology)/Set}} === Points in Topological Space === {{:Definition:Disconnected (Topology)/Points}}"}
+{"_id": "23378", "title": "Definition:Degenerate Connected Set/Non-Degenerate", "text": "A '''non-degenerate connected set''' of $T$ is a connected set of $T$ containing more than one element."}
+{"_id": "23379", "title": "Definition:Connected (Topology)/Topological Space/Definition 1", "text": "$T$ is '''connected''' {{iff}} it admits no separation."}
+{"_id": "23380", "title": "Definition:Connected (Topology)/Topological Space/Definition 2", "text": "$T$ is '''connected''' {{iff}} it has no two disjoint nonempty closed sets whose union is $S$."}
+{"_id": "23381", "title": "Definition:Connected (Topology)/Topological Space/Definition 3", "text": "$T$ is '''connected''' {{iff}} its only subsets whose boundary is empty are $S$ and $\\varnothing$."}
+{"_id": "23382", "title": "Definition:Connected (Topology)/Topological Space/Definition 4", "text": "$T$ is '''connected''' {{iff}} its only clopen sets are $S$ and $\\O$."}
+{"_id": "23383", "title": "Definition:Connected (Topology)/Topological Space/Definition 5", "text": "$T$ is '''connected''' {{iff}} there are no two non-empty separated sets whose union is $S$."}
+{"_id": "23384", "title": "Definition:Connected (Topology)/Topological Space/Definition 6", "text": "$T$ is '''connected''' {{iff}} there exists no continuous surjection from $T$ onto a discrete two-point space."}
+{"_id": "23385", "title": "Definition:Exponential (Category Theory)/Evaluation", "text": "The morphism: :$\\epsilon: C^B \\times B \\to C$ associated to $C^B$ is called the '''evaluation morphism'''."}
+{"_id": "23386", "title": "Definition:Exponential (Category Theory)/Transpose", "text": "For a morphism $f: A \\times B \\to C$, the unique: :$\\tilde f: A \\to C^B$ provided by the UMP for $C^B$ is called the '''exponential transpose''' of $f$. For a morphism $g: A \\to C^B$, the morphism $\\bar g: A \\times B \\to C$ defined by: :$\\bar g = \\epsilon \\circ \\left({g \\times \\operatorname{id}_B}\\right)$ is also called the '''exponential transpose''' of $g$."}
+{"_id": "23387", "title": "Definition:Smooth Path/Complex", "text": "Let $\\left[{a \\,.\\,.\\, b}\\right]$ be a closed real interval. Let $\\gamma: \\left[{a \\,.\\,.\\, b}\\right] \\to \\C$ be a path in $\\C$. That is, $\\gamma$ is a continuous complex-valued function from $\\left[{a \\,.\\,.\\, b}\\right]$ to $\\C$. Define the real function $x : \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$ by: :$\\forall t \\in \\left[{a \\,.\\,.\\, b}\\right]: x \\left({t}\\right) = \\operatorname{Re} \\left({\\gamma \\left({t}\\right)}\\right)$ Define the real function $y: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$ by: :$\\forall t \\in \\left[{a \\,.\\,.\\, b}\\right]: y \\left({t}\\right) = \\operatorname{Im} \\left({\\gamma \\left({t}\\right)}\\right)$ where: : $\\operatorname{Re} \\left({\\gamma \\left({t}\\right)}\\right)$ denotes the real part of the complex number $\\gamma \\left({t}\\right)$ : $\\operatorname{Im} \\left({\\gamma \\left({t}\\right)}\\right)$ denotes the imaginary part of $\\gamma \\left({t}\\right)$. Then $\\gamma$ is a '''smooth path (in $\\C$)''' {{iff}}: :$(1): \\quad$ Both $x$ and $y$ are continuously differentiable :$(2): \\quad$ For all $t \\in \\left[{a \\,.\\,.\\, b}\\right]$, either $x' \\left({t}\\right) \\ne 0$ or $y' \\left({t}\\right) \\ne 0$."}
+{"_id": "23388", "title": "Definition:Contour/Complex Plane", "text": "Let $C_1, \\ldots, C_n$ be directed smooth curves in the complex plane $\\C$. For each $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$, let $C_i$ be parameterized by the smooth path $\\gamma_i: \\left[{a_i \\,.\\,.\\, b_i}\\right] \\to \\C$. For each $i \\in \\left\\{ {1, \\ldots, n - 1}\\right\\}$, let the endpoint of $\\gamma_i$ equal the start point of $\\gamma_{i + 1}$: :$\\gamma_i \\left({b_i}\\right) = \\gamma_{i + 1} \\left({a_{i + 1} }\\right)$ Then the finite sequence $\\left\\langle{C_1, \\ldots, C_n}\\right\\rangle$ is a '''contour'''."}
+{"_id": "23389", "title": "Definition:Connected (Topology)/Set/Definition 1", "text": "$H$ is a '''connected set of $T$''' {{iff}} it is not the union of any two non-empty separated sets of $T$."}
+{"_id": "23390", "title": "Definition:Connected (Topology)/Set/Definition 2", "text": "$H$ is a '''connected set of $T$''' {{iff}} it is not '''disconnected in $T$'''."}
+{"_id": "23391", "title": "Definition:Connected (Topology)/Set/Definition 3", "text": "$H$ is a '''connected set of $T$''' {{iff}}: :the topological subspace $\\struct {H, \\tau_H}$ of $T$ is a connected topological space."}
+{"_id": "23392", "title": "Definition:Contour/Closed/Complex Plane", "text": "$C$ is a '''closed contour''' {{iff}} the start point of $C$ is equal to the end point of $C$: :$\\gamma_1 \\left({a_1}\\right) = \\gamma_n \\left({b_n}\\right)$"}
+{"_id": "23393", "title": "Definition:Complex Riemann Integral", "text": "Let $\\mathbb I := \\closedint a b$ be a closed real interval. Let $f: \\mathbb I \\to \\C$ be a bounded complex function. Define the real function $x: \\mathbb I \\to \\R$ by: :$\\forall t \\in \\mathbb I: \\map x t = \\map \\Re {\\map f t}$ Define the real function $y: \\mathbb I \\to \\R$ by: :$\\forall t \\in \\mathbb I: \\map y t = \\map \\Im {\\map f t}$ Here: :$\\map \\Re {\\map f t}$ denotes the real part of the complex number $\\map f t$ :$\\map \\Im {\\map f t}$ denotes the imaginary part of $\\map f t$. Suppose that both $x$ and $y$ are Riemann integrable over $\\mathbb I$. Then the '''complex (Riemann) integral of $f$ over $\\mathbb I$''' is defined as: :$\\displaystyle \\int_a^b \\map f t \\rd t = \\int_a^b \\map \\Re {\\map f t} \\rd t + i \\int_a^b \\map \\Im {\\map f t} \\rd t$ $f$ is formally defined as '''(properly) complex integrable over $\\mathbb I$ in the sense of Riemann''', or '''(properly) complex Riemann integrable over $\\mathbb I$'''. More usually (and informally), we say: :'''$f$ is (Riemann) complex integrable over $\\mathbb I$.'''"}
+{"_id": "23394", "title": "Definition:Path (Topology)/Initial Point", "text": "The '''initial point''' of $\\gamma$ is $\\map \\gamma a$. That is, the path '''starts''' (or '''begins''') at $\\map \\gamma a$."}
+{"_id": "23395", "title": "Definition:Compatible Atlases", "text": "Let $M$ be a topological space. Let $\\mathscr F, \\mathscr G$ be $d$-dimensional atlases of class $C^k$ on $M$. === Definition 1 === {{:Definition:Compatible Atlases/Definition 1}} === Definition 2 === {{:Definition:Compatible Atlases/Definition 2}}"}
+{"_id": "23396", "title": "Definition:Path (Topology)/Final Point", "text": "The '''final point''' of $\\gamma$ is $\\map \\gamma b$. That is, the path '''ends''' (or '''finishes''') at $\\map \\gamma b$."}
+{"_id": "23397", "title": "Definition:Coordinate System/Coordinate Function", "text": "Let $\\left \\langle {a_n} \\right \\rangle$ be a coordinate system of a unitary $R$-module $G$. For each $x \\in G$ let $x_1, x_2, \\ldots, x_n$ be the coordinates of $x$ relative to $\\left \\langle {a_n} \\right \\rangle$. Then for $i = 1, \\ldots, n$ the mapping $f_i : G \\to R$ defined by $f_i \\left({x}\\right) = x_i$ is called the '''$i$-th coordinate function on $G$ relative to $\\left \\langle {a_n} \\right \\rangle$'''."}
+{"_id": "23398", "title": "Definition:Component (Topology)/Definition 1", "text": "From Connectedness of Points is Equivalence Relation, $\\sim$ is an equivalence relation. From the Fundamental Theorem on Equivalence Relations, the points in $T$ can be partitioned into equivalence classes. These equivalence classes are called the '''(connected) components''' of $T$. If $x \\in S$, then the '''component of $T$ containing $x$''' (that is, the set of points $y \\in S$ with $x \\sim y$) is denoted by $\\map {\\operatorname {Comp}_x} T$."}
+{"_id": "23399", "title": "Definition:Component (Topology)/Definition 2", "text": "The '''component of $T$ containing $x$''' is defined as: : $\\displaystyle \\operatorname{Comp}_x \\left({T}\\right) = \\bigcup \\left\\{{A \\subseteq S: x \\in A \\land A}\\right.$ is connected $\\left.\\right\\}$"}
+{"_id": "23400", "title": "Definition:Component (Topology)/Definition 3", "text": "The '''component of $T$ containing $x$''' is defined as: : the maximal connected set of $T$ that contains $x$."}
+{"_id": "23401", "title": "Definition:Closed Ball/Metric Space/Radius", "text": "The value $\\epsilon$ is referred to as the '''radius''' of $\\map { {B_\\epsilon}^-} a$."}
+{"_id": "23402", "title": "Definition:Continuous Mapping (Metric Space)/Point/Definition 1", "text": "'''$f$ is continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$)''' {{iff}}: :$\\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: \\forall x \\in A_1: \\map {d_1} {x, a} < \\delta \\implies \\map {d_2} {\\map f x, \\map f a} < \\epsilon$ where $\\R_{>0}$ denotes the set of all strictly positive real numbers."}
+{"_id": "23403", "title": "Definition:Continuous Mapping (Metric Space)/Point/Definition 2", "text": "$f$ is '''continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$)''' {{iff}}: :$(1): \\quad$ The limit of $f \\left({x}\\right)$ as $x \\to a$ exists :$(2): \\quad \\displaystyle \\lim_{x \\to a} f \\left({x}\\right) = f \\left({a}\\right)$."}
+{"_id": "23404", "title": "Definition:Continuous Mapping (Metric Space)/Point/Definition 3", "text": "$f$ is '''continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$)''' {{iff}}: :$\\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: f \\left[{B_\\delta \\left({a; d_1}\\right)}\\right] \\subseteq B_\\epsilon \\left({f \\left({a}\\right); d_2}\\right)$ where $B_\\epsilon \\left({f \\left({a}\\right); d_2}\\right)$ denotes the open $\\epsilon$-ball of $f \\left({a}\\right)$ with respect to the metric $d_2$, and similarly for $B_\\delta \\left({a; d_1}\\right)$."}
+{"_id": "23405", "title": "Definition:Continuous Mapping (Metric Space)/Space/Definition 1", "text": "$f$ is '''continuous from $\\left({A_1, d_1}\\right)$ to $\\left({A_2, d_2}\\right)$''' {{iff}} it is continuous at every point $x \\in A_1$."}
+{"_id": "23406", "title": "Definition:Limit of Function (Metric Space)/Epsilon-Delta Condition", "text": ":$\\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: 0 < d_1 \\left({x, c}\\right) < \\delta \\implies d_2 \\left({f \\left({x}\\right), L}\\right) < \\epsilon$ That is, for every real positive $\\epsilon$ there exists a real positive $\\delta$ such that ''every'' point in the domain of $f$ within $\\delta$ of $c$ has an image within $\\epsilon$ of some point $L$ in the codomain of $f$."}
+{"_id": "23407", "title": "Definition:Limit of Function (Metric Space)/Epsilon-Ball Condition", "text": ":$\\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: f \\left({B_\\delta \\left({c; d_1}\\right) \\setminus \\left\\{{c}\\right\\}}\\right) \\subseteq B_\\epsilon \\left({L; d_2}\\right)$. where: : $B_\\delta \\left({c; d_1}\\right) \\setminus \\left\\{{c}\\right\\}$ is the deleted $\\delta $-neighborhood of $c$ in $M_1$ : $B_\\epsilon \\left({L; d_2}\\right)$ is the open $\\epsilon$-ball of $L$ in $M_2$. That is, for every open $\\epsilon$-ball of $L$ in $M_2$, there exists a deleted $\\delta $-neighborhood of $c$ in $M_1$ whose image is a subset of that open $\\epsilon$-ball."}
+{"_id": "23408", "title": "Definition:Totally Bounded Metric Space/Definition 1", "text": "A metric space $M = \\struct {A, d}$ is '''totally bounded''' {{iff}}: :for every $\\epsilon \\in \\R_{>0}$ there exists a finite $\\epsilon$-net for $M$. That is, $M$ is '''totally bounded''' {{iff}}: :for every $\\epsilon \\in \\R_{>0}$ there exists a finite set of points $x_1, \\ldots, x_n \\in A$ such that: ::$\\displaystyle A = \\bigcup_{i \\mathop = 1}^n \\map {B_\\epsilon} {x_i}$ :where $\\map {B_\\epsilon} {x_i}$ denotes the open $\\epsilon$-ball of $x_i$. That is: $M$ is '''totally bounded''' {{iff}}, given any $\\epsilon \\in \\R_{>0}$, one can find a finite number of open $\\epsilon$-balls which cover $A$."}
+{"_id": "23409", "title": "Definition:Totally Bounded Metric Space/Definition 2", "text": "A metric space $M = \\struct {A, d}$ is '''totally bounded''' {{iff}}: :for every $\\epsilon \\in \\R_{>0}$ there exist finitely many points $x_0, \\dots, x_n \\in A$ such that: ::$\\displaystyle \\inf_{0 \\mathop \\le i \\mathop \\le n} \\map d {x_i, x} \\le \\epsilon$ :for all $x \\in A$."}
+{"_id": "23410", "title": "Definition:Uniform Convergence/Metric Space", "text": "Let $S$ be a set. Let $M = \\left({A, d}\\right)$ be a metric space. Let $\\left \\langle {f_n} \\right \\rangle$ be a sequence of mappings $f_n: S \\to A$. Let: : $\\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\R: \\forall n \\ge N, \\forall x \\in S: d \\left({f_n \\left({x}\\right), f \\left({x}\\right)}\\right) < \\epsilon$ Then '''$\\left \\langle {f_n} \\right \\rangle$ converges to $f$ uniformly on $S$ as $n \\to \\infty$'''."}
+{"_id": "23411", "title": "Definition:Uniform Convergence/Real Numbers", "text": "Let $\\sequence {f_n}$ be a sequence of real functions defined on $D \\subseteq \\R$. Let: :$\\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\R: \\forall n \\ge N, \\forall x \\in D: \\size {\\map {f_n} x - \\map f x} < \\epsilon$ That is:  :$\\displaystyle \\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\R: \\forall n \\ge N: \\sup_{x \\mathop \\in D} \\size {\\map {f_n} x - \\map f x} < \\epsilon$ Then '''$\\sequence {f_n}$ converges to $f$ uniformly on $D$ as $n \\to \\infty$'''."}
+{"_id": "23412", "title": "Definition:Real Function/Domain", "text": "Let $f: S \\to \\R$ be a real function. The domain of $f$ is the set $S$. It is frequently the case that $S$ is not explicitly specified. If this is so, then it is understood that the domain is to consist of ''all'' the values in $\\R$ for which the function is defined. This often needs to be determined as a separate exercise in itself, by investigating the nature of the function in question."}
+{"_id": "23413", "title": "Definition:Modulo Operation/Modulo Zero", "text": ":$\\forall x \\in \\R: x \\bmod 0 = x$ This can be considered as a '''special case''' of the modulo operation, but it is interesting to note that most of the results concerning the modulo operation still hold."}
+{"_id": "23414", "title": "Definition:Modulo Operation/Modulo One", "text": ":$x \\bmod 1 = x - \\left \\lfloor {x}\\right \\rfloor$ from which it follows directly that: :$x = \\left \\lfloor {x}\\right \\rfloor + \\left({x \\bmod 1}\\right)$"}
+{"_id": "23415", "title": "Definition:Symmetric Difference/Definition 1", "text": ":$S * T := \\paren {S \\setminus T} \\cup \\paren {T \\setminus S}$"}
+{"_id": "23416", "title": "Definition:Symmetric Difference/Definition 2", "text": ":$S * T = \\paren {S \\cup T} \\setminus \\paren {S \\cap T}$"}
+{"_id": "23417", "title": "Definition:Symmetric Difference/Definition 3", "text": ": $S * T = \\left({S \\cap \\overline T}\\right) \\cup \\left({\\overline S \\cap T}\\right)$"}
+{"_id": "23418", "title": "Definition:Symmetric Difference/Definition 4", "text": ":$S * T = \\paren {S \\cup T}\\cap \\paren {\\overline S \\cup \\overline T}$"}
+{"_id": "23419", "title": "Definition:Standard Deviation", "text": "Let $X$ be a random variable. Then the '''standard deviation of $X$''', written $\\sigma_X$ or $\\sigma$, is defined as the principal square root of the variance of $X$: :$\\sigma_X := \\sqrt {\\var X}$"}
+{"_id": "23420", "title": "Definition:Transfinite Ordinal", "text": "Let $\\alpha$ be an ordinal. Then $\\alpha$ is said to be '''transfinite''' {{iff}} it is an infinite set."}
+{"_id": "23421", "title": "Definition:Characteristic Function of Random Variable", "text": "Let $X$ be a random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. The '''characteristic function of $X$''' is the mapping $\\phi: \\R \\to \\C$ defined by: :$\\map \\phi t = \\expect {e^{i t X} }$ where: :$i$ is the imaginary unit :$\\expect \\cdot$ denotes expectation."}
+{"_id": "23422", "title": "Definition:Transfinite Sequence", "text": "Let $\\alpha$ be an infinite ordinal. Let $\\left({x_\\beta}\\right)_{\\beta \\mathop \\in \\alpha}$ be an $\\alpha$-indexed family. Then $\\left({x_\\beta}\\right)_{\\beta \\mathop \\in \\alpha}$ is called a '''transfinite sequence'''."}
+{"_id": "23423", "title": "Definition:Nest", "text": "Let $S$ be a set. Let $\\powerset S$ be its power set. Let $N \\subseteq \\powerset S$ be a subset of $\\powerset S$. Then $N$ is a '''nest''' {{iff}}: :$\\forall X, Y \\in N: X \\subseteq Y$ or $Y \\subseteq X$ === Class Theory === {{:Definition:Nest/Class Theory}}"}
+{"_id": "23424", "title": "Definition:Ideal (Order Theory)", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $I \\subseteq S$ be a non-empty subset of $S$. Then $I$ is an '''ideal''' of $S$ {{iff}} it is both a lower set and a directed set. That is, $I$ is an '''ideal''' {{iff}}: :$\\forall x \\in I, y \\in P: y \\preceq x \\implies y \\in I$ :$\\forall x, y \\in I: \\exists z \\in I: x \\preceq z$ and $y \\preceq z$ Category:Definitions/Order Theory qc6oyt63p8ea0goz322i0fy4l2z3f9j"}
+{"_id": "23425", "title": "Definition:Characteristic Function (Set Theory)/Set/Support", "text": "Let $S$ be a set Let $E \\subseteq S$ be a subset. Let $\\chi_E: S \\to \\set {0, 1}$ be the characteristic function of $E$. The '''support''' of $\\chi_E$, denoted $\\operatorname{supp} \\chi_E$, is the set $E$. That is: :$\\operatorname{supp} \\chi_E = \\set {x \\in S: \\map {\\chi_E} x = 1}$"}
+{"_id": "23426", "title": "Definition:Contour/Length/Complex Plane", "text": "Let $C$ be a '''contour''' in $C$ defined by the (finite) sequence $\\left\\langle{C_1, \\ldots, C_n}\\right\\rangle$ of directed smooth curves in $\\C$. Let $C_i$ be parameterized by the smooth path $\\gamma_i: \\left[{a_i\\,.\\,.\\,b_i}\\right] \\to \\C$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. The '''length''' of $C$ is defined as: :$\\displaystyle L \\left({C}\\right) := \\sum_{i \\mathop = 1}^n \\int_{a_i}^{b_i} \\left\\vert{\\gamma_i' \\left({t}\\right) }\\right\\vert \\rd t$"}
+{"_id": "23427", "title": "Definition:Two Ring", "text": "Let $\\struct {\\Z_2, +_2, \\times_2}$ be the ring of integers modulo $2$. Then $\\struct {\\Z_2, +_2, \\times_2}$ is referred to as the '''two ring'''."}
+{"_id": "23428", "title": "Definition:Idempotent Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. $R$ is an '''idempotent ring''' {{iff}} its ring product $\\circ$ is an idempotent operation."}
+{"_id": "23429", "title": "Definition:Boolean Ring Axioms", "text": "{{begin-axiom}} {{axiom | n = \\text A 0         | lc= Closure under addition         | q = \\forall a, b \\in R         | m = a * b \\in R }} {{axiom | n = \\text A 1         | q = \\forall a, b, c \\in R         | m = \\paren {a * b} * c = a * \\paren {b * c}         | lc= Associativity of addition }} {{axiom | n = \\text A 2         | q = \\forall a, b \\in R         | m = a * b = b * a         | lc= Commutativity of addition }} {{axiom | n = \\text A 3         | q = \\exists 0_R \\in R: \\forall a \\in R         | m = a * 0_R = a = 0_R * a         | lc= Identity element for addition: the zero }} {{axiom | n = \\text {AC} 2         | q = \\forall a \\in R         | m = a * a = 0_R         | lc= Characteristic 2 for addition: }} {{axiom | n = \\text M 0         | q = \\forall a, b \\in R         | m = a \\circ b \\in R         | lc= Closure under product }} {{axiom | n = \\text M 1         | q = \\forall a, b, c \\in R         | m = \\paren {a \\circ b} \\circ c = a \\circ \\paren {b \\circ c}         | lc= Associativity of product }} {{axiom | n = \\text M 2         | q = \\exists 1_R \\in R: \\forall a \\in R         | m = 1_R \\circ a = a = a \\circ 1_R         | lc= Identity element for product: the unity }} {{axiom | n = \\text {MI}         | q = \\forall a \\in R         | m = a \\circ a = a         | lc= Idempotence of product }} {{axiom | n = \\text D         | q = \\forall a, b, c \\in R         | m = a \\circ \\paren {b * c} = \\paren {a \\circ b} * \\paren {a \\circ c},         | lc= Product is distributive over addition }} {{axiom | m = \\paren {a * b} \\circ c = \\paren {a \\circ c} * \\paren {b \\circ c} }} {{end-axiom}} These criteria are called the '''Boolean ring axioms'''."}
+{"_id": "23430", "title": "Definition:Two-Valued Function", "text": "Let $2$ be the two ring. Let $f: X \\to 2$ be a mapping with codomain $2$. Then $f$ is called a '''$2$-valued function'''."}
+{"_id": "23431", "title": "Definition:Ring of Idempotents", "text": "Let $\\left({R, +, \\circ}\\right)$ be a commutative ring. Let $A$ be the set of all idempotent elements of $R$ under $\\circ$: :$A = \\left\\{{x \\in R: x \\circ x = x}\\right\\}$ Define a binary operation $\\circ$ on $A$ by: :$\\forall x, y \\in A: x \\oplus y := x + y - 2 x \\circ y$ Denote also with $\\circ$ the restriction of $\\circ$ to $A$. The algebraic structure $\\left({A, \\oplus, \\circ}\\right)$ is called the '''ring of idempotents''' of $R$. It is an idempotent ring, as shown on Ring of Idempotents is Idempotent Ring."}
+{"_id": "23432", "title": "Definition:Algebraic Element of Ring Extension", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\\struct {D, +, \\circ}$ be an integral subdomain of $R$. Let $x \\in R$. Then $x$ is '''algebraic over $D$''' {{iff}}: :$\\exists \\map f x$ over $D$ such that $\\map f x = 0_R$ where $\\map f x$ is a non-null polynomial in $x$ over $D$."}
+{"_id": "23433", "title": "Definition:Algebraic Element of Field Extension", "text": "Let $E / F$ be a field extension. Let $\\alpha \\in E$. === Definition 1 === {{:Definition:Algebraic Element of Field Extension/Definition 1}} === Definition 2 === {{:Definition:Algebraic Element of Field Extension/Definition 2}}"}
+{"_id": "23434", "title": "Definition:Algebraic Field Extension", "text": "A field extension $E / F$ is said to be '''algebraic''' {{iff}}: : $\\forall \\alpha \\in E: \\alpha$ is algebraic over $F$"}
+{"_id": "23435", "title": "Definition:Prime Ideal (Order Theory)", "text": "Let $I$ be an ideal in an ordered set $S$. Then $I$ is a '''prime ideal''' in $S$ {{iff}} $S \\setminus I$ is a filter."}
+{"_id": "23436", "title": "Definition:Distributive Lattice", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a lattice. Then $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ is '''distributive''' {{iff}} one (hence all) of the following equivalent statements holds: {{begin-axiom}} {{axiom | n = 1         | q = \\forall x, y, z \\in S         | m = x \\wedge \\left({y \\vee z}\\right) = \\left({x \\wedge y}\\right) \\vee \\left({x \\wedge z}\\right) }} {{axiom | n = 1'         | q = \\forall x, y, z \\in S         | m = \\left({x \\vee y}\\right) \\wedge z = \\left({x \\wedge z}\\right) \\vee \\left({y \\wedge z}\\right) }} {{axiom | n = 2         | q = \\forall x, y, z \\in S         | m = x \\vee \\left({y \\wedge z}\\right) = \\left({x \\vee y}\\right) \\wedge \\left({x \\vee z}\\right) }} {{axiom | n = 2'         | q = \\forall x, y, z \\in S         | m = \\left({x \\wedge y}\\right) \\vee z = \\left({x \\vee z}\\right) \\wedge \\left({y \\vee z}\\right) }} {{end-axiom}}"}
+{"_id": "23437", "title": "Definition:Stone Space", "text": "Let $B$ be a Boolean algebra. The '''Stone space of $B$''' is the topological space: :$\\map S B = \\struct {U, \\tau}$ where: :$U$ is the set of ultrafilters in $B$ :$\\tau$ is the topology generated by the basis consisting of all sets of the form: :$\\set {x \\in \\map S B: b \\in x}$ :for some $b \\in B$."}
+{"_id": "23438", "title": "Definition:Contour Integral/Complex", "text": "Let $C$ be a contour defined by a finite sequence $C_1, \\ldots, C_n$ of directed smooth curves. Let $C_i$ be parameterized by the smooth path $\\gamma_i: \\closedint {a_i} {b_i} \\to \\C$ for all $i \\in \\set {1, \\ldots, n}$. Let $f: \\Img C \\to \\C$ be a continuous complex function, where $\\Img C$ denotes the image of $C$. The '''contour integral of $f$ along $C$''' is defined by: :$\\displaystyle \\int_C \\map f z \\rd z = \\sum_{i \\mathop = 1}^n \\int_{a_i}^{b_i} \\map f {\\map {\\gamma_i} t} \\map {\\gamma_i'} t \\rd t$ From Contour Integral is Well-Defined, it follows that the complex Riemann integral on the right side is defined and is independent of the parameterizations of $C_1, \\ldots, C_n$. === Contour Integral along Closed Contour === {{:Definition:Contour Integral/Complex/Closed}}"}
+{"_id": "23439", "title": "Definition:Smooth Path/Closed/Complex Plane", "text": "Let $\\gamma$ be a smooth path in $\\C$. $\\gamma$ is a '''closed smooth path''' {{iff}} $\\gamma$ is a closed path. That is, {{iff}} $\\gamma \\left({a}\\right) = \\gamma \\left({b}\\right)$."}
+{"_id": "23440", "title": "Definition:Ordered Sum/General Definition", "text": "Let $S_1, S_2, \\ldots, S_n$ be tosets. Then we define $T_n$ as the ordered sum of $S_1, S_2, \\ldots, S_n$ as: :$\\forall n \\in \\N_{>0}: T_n = \\begin{cases} S_1 & : n = 1 \\\\ T_{n-1} + S_n & : n > 1 \\end{cases}$"}
+{"_id": "23442", "title": "Definition:Moore-Smith Sequence", "text": "Let $\\left({S, \\preceq}\\right)$ be a directed set. Let $T$ be a set. Let $f: S \\to T$ be a mapping. Then $f$ is a '''Moore-Smith sequence''' in $T$."}
+{"_id": "23443", "title": "Definition:Ordered Square", "text": "Let: : $S = \\left[{0 \\,.\\,.\\, 1}\\right] \\times \\left[{0 \\,.\\,.\\, 1}\\right]$ where $\\left[{0 \\,.\\,.\\, 1}\\right]$ is the closed real interval from $0$ to $1$. Let $\\preceq$ be the lexicographic ordering on $S$ induced by the usual ordering of $\\left[{0 \\,.\\,.\\, 1}\\right]$. Let $\\tau$ be the order topology on $S$ induced by $\\preceq$. Then $\\left({S, \\tau}\\right)$ is the '''ordered square'''."}
+{"_id": "23444", "title": "Definition:Cartesian Product of Relations", "text": "Let $\\left\\langle{S_i}\\right\\rangle_{i \\mathop \\in I}$ and $\\left\\langle{T_i}\\right\\rangle_{i \\mathop \\in I}$ be families of sets indexed by $I$. For each $i \\in I$, let $\\mathcal R_i \\subseteq S_i \\times T_i$ be a relation from $S_i$ to $T_i$. Let $S$ and $T$ be the Cartesian product of $\\left\\langle{S_i}\\right\\rangle_{i \\mathop \\in I}$ and $\\left\\langle{T_i}\\right\\rangle_{i \\mathop \\in I}$ respectively: :$\\displaystyle S = \\prod_{i \\mathop \\in I} S_i$ :$\\displaystyle T = \\prod_{i \\mathop \\in I} T_i$ Then the '''product''' of the relations $\\mathcal R_i$ is defined as the relation $\\mathcal R \\subseteq S \\times T$ such that: :$x \\mathrel{\\mathcal R} y \\iff \\forall i \\in I: x_i \\mathrel{\\mathcal R_i} y_i$"}
+{"_id": "23445", "title": "Definition:Time-Constructible Function", "text": "=== Definition 1 === {{:Definition:Time-Constructible Function/Definition 1}} === Definition 2 === {{:Definition:Time-Constructible Function/Definition 2}}"}
+{"_id": "23446", "title": "Definition:Boolean Algebra", "text": "=== Definition 1 === {{:Definition:Boolean Algebra/Definition 1}} === Definition 2 === {{:Definition:Boolean Algebra/Definition 2}} === Definition 3 === {{:Definition:Boolean Algebra/Definition 3}} The operations $\\vee$ and $\\wedge$ are called '''join''' and '''meet''', respectively. The identities $\\bot$ and $\\top$ are called '''bottom''' and '''top''', respectively. Also, $\\neg a$ is called the '''complement''' of $a$. The operation $\\neg$ is called '''complementation'''."}
+{"_id": "23447", "title": "Definition:Concatenation of Contours", "text": "Let $\\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ and $D$ be contours in $\\R^n$. Thus: :$C$ is a (finite) sequence of directed smooth curves $\\sequence {C_1, \\ldots, C_n}$ :$D$ is a (finite) sequence of directed smooth curves $\\sequence {D_1, \\ldots, D_m}$. Let $C_i$ be parameterized by the smooth path: :$\\gamma_i: \\closedint {a_i} {b_i} \\to \\R^n$ for all $i \\in \\set {1, \\ldots, n}$ Let $D_i$ be parameterized by the smooth path: :$\\sigma_i: \\closedint {c_i} {d_i} \\to \\R^n$ for all $i \\in \\set {1, \\ldots, m}$ Let $\\map {\\gamma_n} {b_n} = \\map {\\sigma_1} {c_1}$. Then the '''concatenation of the contours''' $C$ and $D$, denoted $C \\cup D$, is the contour defined by the (finite) sequence: :$\\sequence {C_1, \\ldots, C_n, D_1, \\ldots, D_m}$"}
+{"_id": "23448", "title": "Definition:Probability Density Function", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $X: \\Omega \\to \\R$ be a continuous random variable on $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\Omega_X = \\Img X$, the image of $X$. Then the '''probability density function''' of $X$ is the mapping $f_X: \\R \\to \\closedint 0 1$ defined as: :$\\forall x \\in \\R: \\map {f_X} x = \\begin {cases} \\displaystyle \\lim_{\\epsilon \\mathop \\to 0^+} \\frac {\\map \\Pr {x - \\frac \\epsilon 2 \\le X \\le x + \\frac \\epsilon 2} } \\epsilon & : x \\in \\Omega_X \\\\ 0 & : x \\notin \\Omega_X \\end {cases}$"}
+{"_id": "23449", "title": "Definition:Finite State Machine", "text": "A '''finite state machine''' is an ordered tuple: : $F = \\left({ S, A, I, \\Sigma, T }\\right)$ where: : $S$ is the (finite) set of states : $A \\subseteq S$ is the set of ''accepting'' states : $I \\in S$ is the ''initial state'' : $\\Sigma$ is the ''alphabet'' of symbols that can be fed into the machine : $T : \\left({ S \\times \\Sigma }\\right) \\rightarrow S$ is the ''transition function''. A finite state machine operates as follows: :$(1): \\quad$ At the beginning, the ''current state'' $s$ of the finite state machine is $I$. :$(2): \\quad$ One by one, the ''input'' (a sequence of symbols from $\\Sigma$) is fed into the machine. :$(3): \\quad$ After each input symbol $\\sigma$, the current state $s$ is set to the result of $T\\left({s, \\sigma}\\right)$. If, at the end of processing an input word $w$, $s \\in A$, the finite state machine is said to '''accept''' $w$, otherwise to '''reject''' it. The set of words $w$ ''accepted'' by the machine $F$ is called the '''accepted language''' $L\\left({F}\\right)$. Category:Definitions/Mathematical Logic 4y24bc1umm82j5akk79k7162mjwjpui"}
+{"_id": "23450", "title": "Definition:Regular Expression", "text": "A '''regular expression''' is an algebraic structure on an ''alphabet'' $\\Sigma$ defined as follows: * The empty-set regular expression, $\\varnothing$, is a regular expression. * The empty-word regular expression, $\\epsilon$, is a regular expression. * Every $\\sigma \\in \\Sigma$ is a regular expression. (These are called ''literals''.) * If $R_1$ and $R_2$ are regular expressions, $R_1 R_2$ is a regular expression (''concatenation''). * If $R_1$ and $R_2$ are regular expressions, $R_1 \\mid R_2$ is a regular expression (''alternation''). * If $R$ is a regular expression, $R^*$ is a regular expression (''Kleene star''). Every regular expression $R$ on an alphabet $\\Sigma$ defines a language $L \\left({R}\\right) \\subseteq \\Sigma^*$, where $\\Sigma^*$ is the set of all (finite-length) ''words'' (sequences) of symbols in $\\Sigma$: * $ L \\left({\\varnothing}\\right) = \\varnothing $ (the empty set). * $ L \\left({\\epsilon}\\right) = \\left\\{ { \\left[{}\\right] }\\right\\} $ (the set containing only the empty word). * If $R$ is a ''literal'' $\\sigma$, $ L \\left({R}\\right) = \\left\\{ { \\left[{ \\sigma }\\right] }\\right\\} $ (i.e., the set containing only the single-symbol word “$\\sigma$”). * If $R$ is a ''concatenation'' $R_1 R_2$, $ L \\left({R}\\right) = \\left\\{ { w_1 w_2 : w_1 \\in L \\left({R_1}\\right), w_2 \\in L \\left({R_2}\\right) }\\right\\} $, where $w_1 w_2$ is the concatenation of the words $w_1$ and $w_2$. * If $R$ is an ''alternation'' $R_1 \\mid R_2$, $ L \\left({R}\\right) = L \\left({R_1}\\right) \\cup L \\left({R_2}\\right)$. * If $R$ is a ''Kleene star'' $R_1^*$, $ L\\left({R}\\right) $ is the smallest set satisfying the following: ** $\\left[{}\\right] \\in L \\left({R}\\right) $ (the empty word is in the set); ** if $w_1 \\in L \\left({R}\\right)$ and $w_2 \\in L \\left({R_1}\\right)$, then $w_1 w_2 \\in L \\left({R}\\right)$. Category:Definitions/Formal Systems 5tdtcn3ny6g9fhsewv68p3ahwc0e47x"}
+{"_id": "23451", "title": "Definition:Directed Smooth Curve", "text": "Let $\\R^n$ be a real cartesian space of $n$ dimensions. Let $\\rho: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R^n$ be a smooth path in $\\R^n$. The '''directed smooth curve''' with '''parameterization''' $\\rho$ is defined as an equivalence class of smooth paths as follows: A smooth path $\\sigma: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R^n$ belongs to the equivalence class of $\\rho$ {{iff}}: : there exists a bijective differentiable strictly increasing real function: :: $\\phi: \\left[{c \\,.\\,.\\, d}\\right] \\to \\left[{a \\,.\\,.\\, b}\\right]$ : such that $\\sigma = \\rho \\circ \\phi$. </onlyinclude> It follows from Directed Smooth Curve Relation is Equivalence and Fundamental Theorem on Equivalence Relations that this does in fact define an equivalence class. If a '''directed smooth curve''' is only defined by a smooth path $\\rho$, then it is often denoted with the same symbol $\\rho$. === Parameterization === {{:Definition:Directed Smooth Curve/Parameterization}} === Endpoints === {{:Definition:Directed Smooth Curve/Endpoints}}"}
+{"_id": "23452", "title": "Definition:Smooth Path/Simple/Complex Plane", "text": "Let $\\gamma : \\left[{a \\,.\\,.\\, b}\\right] \\to \\C$ be a '''smooth path in $\\C$'''. $\\gamma$ is a '''simple smooth path''' {{iff}}: :$(1): \\quad \\gamma$ is injective on the half-open interval $\\left[{a \\,.\\,.\\, b}\\right)$ :$(2): \\quad \\forall t \\in \\left({a \\,.\\,.\\, b}\\right): \\gamma \\left({t}\\right) \\ne \\gamma \\left({b}\\right)$ That is, if $t_1, t_2 \\in \\left({a \\,.\\,.\\, b}\\right)$ with $t_1 \\ne t_2$, then $\\gamma \\left({a}\\right) \\ne \\gamma \\left({t_1}\\right) \\ne \\gamma \\left({t_2}\\right) \\ne \\gamma \\left({b}\\right)$."}
+{"_id": "23453", "title": "Definition:Arc-Connected/Points", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $a, b \\in S$ be such that there exists an arc from $a$ to $b$. That is, there exists a continuous injection $f: \\closedint 0 1 \\to S$ such that $\\map f 0 = a$ and $\\map f 1 = b$. Then $a$ and $b$ are '''arc-connected'''. It is also declared that any point $a$ is '''arc-connected''' to itself."}
+{"_id": "23454", "title": "Definition:Arc-Connected/Topological Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then $T$ is '''arc-connected''' {{iff}} every two points in $T$ are arc-connected in $T$. That is, $T$ is '''arc-connected''' {{iff}}: :for every $x, y \\in X, \\exists$ a continuous injection $f: \\closedint 0 1 \\to X$ such that $\\map f 0 = x$ and $\\map f 1 = y$."}
+{"_id": "23455", "title": "Definition:Arc-Connected/Subset", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $U \\subseteq S$ be a subset of $T$. Let $T\\,' = \\left({U, \\tau_U}\\right)$ be the subspace of $T$ induced by $U$. Then $U$ is '''arc-connected in $T$''' {{iff}} every two points in $U$ are arc-connected in $T\\,'$. That is, $U$ is '''arc-connected''' {{iff}}: :for every $x, y \\in U$, there exists a continuous injection $f: \\left[{0 \\,.\\,.\\, 1}\\right] \\to S$ such that $f \\left({0}\\right) = x$ and $f \\left({1}\\right) = y$."}
+{"_id": "23456", "title": "Definition:Mistake/Spello", "text": "A '''spello''' (abbreviation for '''spelling mistake''') is a mistake caused by inadequate spelling skills of the writer. These can sometimes be corrected by the writer having the self-doubt necessary to occasion the act of referring to a reliable manual, for example, a dictionary. A competent proof-reader (that is, one whose spelling skills are more comprehensive than the writer) is usually an adequate resource for reducing spelloes."}
+{"_id": "23457", "title": "Definition:Mistake/Typo", "text": "A '''typo''' (abbreviation for '''typographical error''') is a mistake arising from pressing the wrong button on the machine via which you are transcribing your thoughts onto a storage medium. These can be corrected by assiduous checking by the writer."}
+{"_id": "23458", "title": "Definition:Mistake/Homophone Horror", "text": "A '''homophone horror''' is a mistake caused by the use of a word which sounds the same as, but is spelled differently to, the word intended. A proficient proof-reader is needed to reduce instances of this sort of mistake."}
+{"_id": "23459", "title": "Definition:Directed Smooth Curve/Parameterization/Complex Plane", "text": "Let $C$ be a directed smooth curve in the complex plane $\\C$. Let $\\gamma: \\left[{a \\,.\\,.\\, b}\\right] \\to \\C$ be a smooth path in $\\C$. Then $\\gamma$ is a '''parameterization''' of $C$ {{iff}} $\\gamma$ is an element of the equivalence class that constitutes $C$."}
+{"_id": "23460", "title": "Definition:Directed Smooth Curve/Endpoints/Complex Plane", "text": "Let $C$ be a directed smooth curve in the complex plane $\\C$. Let $C$ be parameterized by a smooth path $\\gamma: \\left[{a \\,.\\,.\\, b}\\right] \\to \\C$. Then: : $\\gamma \\left({a}\\right)$ is the '''start point''' of $C$ : $\\gamma \\left({b}\\right)$ is the '''end point''' of $C$. Collectively, $\\gamma \\left({a}\\right)$ and $\\gamma \\left({b}\\right)$ are known as the '''endpoints''' of $\\rho$."}
+{"_id": "23461", "title": "Definition:Integral Domain/Definition 1", "text": "An '''integral domain''' $\\struct {D, +, \\circ}$ is: :a commutative ring which is non-null :with a unity :in which there are no (proper) zero divisors, that is: ::: $\\forall x, y \\in D: x \\circ y = 0_D \\implies x = 0_D \\text{ or } y = 0_D$ that is (from the Cancellation Law of Ring Product of Integral Domain) in which all non-zero elements are cancellable."}
+{"_id": "23462", "title": "Definition:Integral Domain/Definition 2", "text": "An '''integral domain''' $\\left({D, +, \\circ}\\right)$ is a commutative ring such that $\\left({D^*, \\circ}\\right)$ is a monoid, all of whose elements are cancellable. In this context, $D^*$ denotes the ring $D$ without zero: $D \\setminus \\left\\{{0_D}\\right\\}$."}
+{"_id": "23463", "title": "Definition:Integral Domain Axioms", "text": "An integral domain is an algebraic structure $\\struct {D, *, \\circ}$, on which are defined two binary operations $\\circ$ and $*$, which satisfy the following conditions: {{begin-axiom}} {{axiom | n = A0         | q = \\forall a, b \\in D         | m = a * b \\in D         | lc= Closure under addition }} {{axiom | n = A1         | q = \\forall a, b, c \\in D         | m = \\paren {a * b} * c = a * \\paren {b * c}         | lc= Associativity of addition }} {{axiom | n = A2         | q = \\forall a, b \\in D         | m = a * b = b * a         | lc= Commutativity of addition }} {{axiom | n = A3         | q = \\exists 0_D \\in D: \\forall a \\in D         | m = a * 0_D = a = 0_D * a         | lc= Identity element for addition: the zero }} {{axiom | n = A4         | q = \\forall a \\in D: \\exists a' \\in D         | m = a * a' = 0_D = a' * a         | lc= Inverse elements for addition: negative elements }} {{axiom | n = M0         | q = \\forall a, b \\in D         | m = a \\circ b \\in D         | lc= Closure under product }} {{axiom | n = M1         | q = \\forall a, b, c \\in D         | m = \\paren {a \\circ b} \\circ c = a \\circ \\paren {b \\circ c}         | lc= Associativity of product }} {{axiom | n = D         | q = \\forall a, b, c \\in D         | m = a \\circ \\paren {b * c} = \\paren {a \\circ b} * \\paren {a \\circ c}         | lc= Product is distributive over addition }} {{axiom | m = \\paren {a * b} \\circ c = \\paren {a \\circ c} * \\paren {b \\circ c} }} {{axiom | n = C         | q = \\forall a, b \\in D         | m = a \\circ b = b \\circ a         | lc= Product is commutative }} {{axiom | n = U         | q = \\exists 1_D \\in D: \\forall a \\in D         | m = a \\circ 1_D = a = 1_D \\circ a         | lc= Identity element for product: the unity }} {{axiom | n = ZD         | q = \\forall a, b \\in D         | m = a \\circ b = 0_D \\iff a = 0 \\lor b = 0         | lc= No proper zero divisors }} {{end-axiom}} These criteria are called the '''integral domain axioms'''."}
+{"_id": "23464", "title": "Definition:Triangular Matrix/Upper Triangular Matrix", "text": "An '''upper triangular matrix''' is a matrix in which all the lower triangular elements are zero. That is, all the non-zero elements are on the main diagonal or in the upper triangle. That is, $\\mathbf U$ is '''upper triangular''' {{iff}}: :$\\forall a_{ij} \\in \\mathbf U: i > j \\implies a_{ij} = 0$"}
+{"_id": "23465", "title": "Definition:Triangular Matrix/Lower Triangular Matrix", "text": "A '''lower triangular matrix''' is a matrix in which all the upper triangular elements are zero. That is, all the non-zero elements are on the main diagonal or in the lower triangle. That is, $\\mathbf L$ is '''lower triangular''' {{iff}}: :$\\forall a_{i j} \\in \\mathbf U: i < j \\implies a_{i j} = 0$"}
+{"_id": "23466", "title": "Definition:Contour/Image/Complex Plane", "text": "Let $C$ be a '''contour''' in $\\C$ defined by the (finite) sequence $\\left\\langle{C_1, \\ldots, C_n}\\right\\rangle$ of directed smooth curves in $\\C$. Let $C_i$ be parameterized by the smooth path $\\gamma_i: \\left[{a_i\\,.\\,.\\,b_i}\\right] \\to \\C$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. The '''image of $C$''' is defined as: :$\\displaystyle \\operatorname{Im} \\left({C}\\right) := \\bigcup_{i \\mathop = 1}^n \\operatorname{Im} \\left({\\gamma_i}\\right)$ where $\\operatorname{Im} \\left({\\gamma_i}\\right)$ denotes the image of $\\gamma_i$. If $\\operatorname{Im} \\left({C}\\right) \\subseteq D$, where $D$ is a subset of $\\C$, we say that $C$ is a '''contour in $D$'''."}
+{"_id": "23467", "title": "Definition:Contour/Parameterization/Complex Plane", "text": "Let $C_1, \\ldots, C_n$ be directed smooth curves in the complex plane $\\C$. Let $C_i$ be parameterized by the smooth path $\\gamma_i: \\left[{a_i \\,.\\,.\\, b_i}\\right] \\to \\C$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. Let $C$ be the contour defined by the finite sequence $C_1, \\ldots, C_n$. The '''parameterization of $C$''' is defined as the function $\\gamma: \\left[{a_1 \\,.\\,.\\, c_n}\\right] \\to \\C$ with: :$\\gamma \\restriction_{\\left[{c_i \\,.\\,.\\, c_{i + 1} }\\right] } \\left({t}\\right) = \\gamma_i \\left({t}\\right)$ where $\\displaystyle c_i = a_1 + \\sum_{j \\mathop = 1}^i b_j - \\sum_{j \\mathop = 1}^i a_j$ for $i \\in \\left\\{ {0, \\ldots, n}\\right\\}$. Here, $\\gamma \\restriction_{\\left[{c_i \\,.\\,.\\, c_{i + 1} }\\right] }$ denotes the restriction of $\\gamma$ to $\\left[{c_i \\,.\\,.\\, c_{i + 1} }\\right]$."}
+{"_id": "23468", "title": "Definition:Contour/Endpoints/Complex Plane", "text": "Let $C_1, \\ldots, C_n$ be directed smooth curves in $\\C$. Let $C_i$ be parameterized by the smooth path $\\gamma_i: \\left[{a_i \\,.\\,.\\, b_i}\\right] \\to \\C$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. Let $C$ be the contour defined by the finite sequence $C_1, \\ldots, C_n$. The '''start point''' of $C$ is $\\gamma_1 \\left({a_1}\\right)$. The '''end point''' of $C$ is $\\gamma_n \\left({b_n}\\right)$. Collectively, $\\gamma_1 \\left({a_1}\\right)$ and $\\gamma_n \\left({b_n}\\right)$ are referred to as the '''endpoints''' of $C$."}
+{"_id": "23469", "title": "Definition:Contour/Simple/Complex Plane", "text": "Let $C_1, \\ldots, C_n$ be directed smooth curves in the complex plane $\\C$. Let $C_i$ be parameterized by the smooth path $\\gamma_i: \\left[{a_i \\,.\\,.\\, b_i}\\right] \\to \\C$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. Let $C$ be the contour defined by the finite sequence $C_1, \\ldots, C_n$. $C$ is a '''simple contour''' {{iff}}: :$(1): \\quad$ For all $i,j \\in \\left\\{ {1, \\ldots, n}\\right\\}, t_1 \\in \\left[{a_i \\,.\\,.\\, b_i}\\right), t_2 \\in \\left[{a_j \\,.\\,.\\, b_j}\\right)$ with $t_1 \\ne t_2$, we have $\\gamma_i \\left({t_1}\\right) \\ne \\gamma_j \\left({t_2}\\right)$. :$(2): \\quad$ For all $k \\in \\left\\{ {1, \\ldots, n}\\right\\}, t \\in \\left[{a_k \\,.\\,.\\, b_k}\\right)$ where either $k \\ne 1$ or $t \\ne a_1$, we have $\\gamma_k \\left({t}\\right) \\ne \\gamma_n \\left({b_n}\\right)$."}
+{"_id": "23470", "title": "Definition:Quotient Epimorphism/Group", "text": "Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $G / N$ be the quotient group of $G$ by $N$. The mapping $q_N: G \\to G / N$ defined as: :$\\forall x \\in G: \\map {q_N} x = x N$ is known as the '''quotient (group) epimorphism''' from $G$ to $G / N$."}
+{"_id": "23471", "title": "Definition:Quotient Epimorphism/Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let $\\struct {R / J, +, \\circ}$ be the quotient ring defined by $J$. The mapping $\\phi: R \\to R / J$ given by: : $\\forall x \\in R: \\map \\phi x = x + J$ is known as the '''quotient (ring) epimorphism from $\\struct {R, +, \\circ}$ (on)to $\\struct {R / J, +, \\circ}$'''."}
+{"_id": "23472", "title": "Definition:Integrable Function/p-Integrable", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f \\in \\MM_{\\overline \\R}, f: X \\to \\overline \\R$ be a measurable function. Let $p \\ge 1$ be a real number. Then $f$ is said to be '''$p$-integrable in respect to $\\mu$''' {{iff}}: :$\\displaystyle \\int \\size f^p \\rd \\mu < +\\infty$ is $\\mu$-integrable."}
+{"_id": "23473", "title": "Definition:Derivative of Smooth Path/Complex Plane", "text": "Let $\\gamma: \\left[{a \\,.\\,.\\, b}\\right] \\to \\C$ be a '''smooth path in $\\C$'''. Define the real function $x : \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$ by: :$\\forall t \\in \\left[{a \\,.\\,.\\, b}\\right]: x \\left({t}\\right) = \\operatorname{Re} \\left({\\gamma \\left({t}\\right)}\\right)$ Define the real function $y: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$ by: :$\\forall t \\in \\left[{a \\,.\\,.\\, b}\\right]: y \\left({t}\\right) = \\operatorname{Im} \\left({\\gamma \\left({t}\\right)}\\right)$ where: : $\\operatorname{Re} \\left({\\gamma \\left({t}\\right)}\\right)$ denotes the real part of the complex number $\\gamma \\left({t}\\right)$ : $\\operatorname{Im} \\left({\\gamma \\left({t}\\right)}\\right)$ denotes the imaginary part of $\\gamma \\left({t}\\right)$. It follows from the definition of a  smooth path that both $x$ and $y$ are continuously differentiable. Let $x' \\left({t}\\right)$ and $y' \\left({t}\\right)$ denote the derivative of $x$ and $y$ {{WRT|Differentiation}} $t$. The '''derivative of $\\gamma$''' is the continuous complex function $\\gamma': \\left[{a \\,.\\,.\\, b}\\right] \\to \\C$ defined by: :$\\forall t \\in \\left[{a \\,.\\,.\\, b}\\right]: \\gamma' \\left({t}\\right) = x' \\left({t}\\right) + i y' \\left({t}\\right)$"}
+{"_id": "23474", "title": "Definition:Ring Direct Product", "text": "{{:Definition:Ring Direct Product/Finite Case}}"}
+{"_id": "23475", "title": "Definition:Reversed Contour", "text": "Let $\\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ be a contour in $\\R^n$. Then $C$ is defined as a concatenation of a finite sequence $C_1, \\ldots, C_n$ of directed smooth curves in $\\R^n$. The '''reversed contour of $C$''' is denoted $-C$ and is defined as the concatenation of the finite sequence: :$-C_n, -C_{n - 1}, \\ldots, -C_1$ where $-C_i$ is the reversed directed smooth curve of $C_i$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$."}
+{"_id": "23476", "title": "Definition:Reversed Directed Smooth Curve", "text": "Let $\\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ be a directed smooth curves in $\\R^n$. Let $C$ be parameterized by the smooth path $\\gamma: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R^n$. The '''reversed directed smooth curve of $C$''' is denoted $-C$ and is defined as the directed smooth curve that is parameterized by: :$\\rho: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R^n: \\rho \\left({t}\\right) = \\gamma \\left({a + b - t}\\right)$"}
+{"_id": "23477", "title": "Definition:Universal Net", "text": "Let $S$ be a non-empty set. Let $H$ be a set directed by $\\preceq$. Let $f: H \\to S$ be a net in $S$. Then $f$ is a '''universal net''' iff for each subset $A \\subseteq S$, $f$ is eventually in $A$ or $f$ is eventually in $S \\setminus A$. {{explain|\"eventually\"}} Category:Definitions/Set Theory Category:Definitions/Topology g6pdy53dz1lzccyidk0ptoihymj12jg"}
+{"_id": "23478", "title": "Definition:Subnet", "text": "Let $\\left({E, \\le}\\right)$ and $\\left({D,\\preceq}\\right)$ be directed sets. Let $A$ be a non-empty set. Let $T: E \\to A$ and $S: D \\to A$ be nets in $A$. Then $T$ is a '''subnet''' of $S$ iff there exists a cofinal mapping $N: E \\to D$ such that $T = S \\circ N$. That is, for each $m$ in $D$, there is an $n$ in $E$ such that for each $p \\in E$, $p \\ge n \\implies N \\left({p}\\right) \\succeq m$."}
+{"_id": "23479", "title": "Definition:Ordered Tuple as Ordered Set/Ordered Triple", "text": "The '''ordered triple''' $\\tuple {a, b, c}$ of elements $a$, $b$ and $c$ can be defined either as the ordered pair: :$\\tuple {a, \\tuple {b, c} }$ or as the ordered pair: :$\\tuple {\\tuple {a, b}, c}$ where $\\tuple {a, b}$ and $\\tuple {b, c}$ are themselves ordered pairs."}
+{"_id": "23480", "title": "Definition:Ordered Tuple as Ordered Set/Ordered Quadruple", "text": "The '''ordered quadruple''' $\\tuple {a, b, c, d}$ of elements $a$, $b$, $c$ and $d$ is defined either as the ordered pair: :$\\tuple {a, \\tuple {b, c, d} }$ or: :$\\tuple {\\tuple {a, b, c}, d}$ where $\\tuple {a, b, c}$ and $\\tuple {b, c, d}$ are themselves ordered triples."}
+{"_id": "23481", "title": "Definition:Ordered Tuple as Ordered Set/Ordered Tuple", "text": "The '''ordered tuple''' $\\tuple {a_1, a_2, \\ldots, a_n}$ of elements $a_1, a_2, \\ldots, a_n$ is defined as either the ordered pair: :$\\tuple {a_1, \\tuple {a_2, a_3, \\ldots, a_n} }$ or as the ordered pair: :$\\tuple {\\tuple {a_1, a_2, \\ldots, a_{n - 1} }, a_n}$ where $\\tuple {a_2, a_3, \\ldots, a_n}$ and $\\tuple {a_1, a_2, \\ldots, a_{n - 1} }$ are themselves ordered tuples."}
+{"_id": "23482", "title": "Definition:Relation/Relation as Ordered Pair", "text": "Some sources define a relation between $S$ and $T$ as an ordered pair: :$\\struct {S \\times T, \\map P {s, t} }$ where: :$S \\times T$ is the Cartesian product of $S$ and $T$ :$\\map P {s, t}$ is a propositional function on ordered pairs $\\tuple {s, t}$ of $S \\times T$. Note that this approach leaves the domain and codomain inadequately defined. This situation arises in the case that $S$ or $T$ are empty, whence it follows that $S \\times T$ is empty, but $T$ or $S$ are not themselves uniquely determined."}
+{"_id": "23483", "title": "Definition:Relation/Relation as Mapping", "text": "It is possible to define a relation as a mapping from the cartesian product $S \\times T$ to the set of truth values $\\set {\\text {true}, \\text {false} }$: :$\\RR: S \\times T \\to \\set {\\text {true}, \\text {false} }: \\forall \\tuple {s, t} \\in S \\times T: \\map \\RR {s, t} = \\begin{cases} \\text {true} & : \\tuple {s, t} \\in \\RR \\\\ \\text {false} & : \\tuple {s, t} \\notin \\RR \\end{cases}$ This is called the characteristic function of the relation. However, care needs to be taken that a mapping then cannot be defined as a special relation, as this would be circular."}
+{"_id": "23484", "title": "Definition:Meet (Order Theory)", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $a, b \\in S$, and suppose that their infimum $\\inf \\set {a, b}$ exists in $S$. Then $a \\wedge b$, the '''meet of $a$ and $b$''', is defined as: :$a \\wedge b = \\inf \\set {a, b}$ Expanding the definition of infimum, one sees that $c = a \\wedge b$ {{iff}}: :$c \\preceq a$ and $c \\preceq b$ and $\\forall s \\in S: s \\preceq a \\land s \\preceq b \\implies s \\preceq c$"}
+{"_id": "23485", "title": "Definition:Join (Order Theory)", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a, b \\in S$. Let their supremum $\\sup \\left\\{{a, b}\\right\\}$ exist in $S$. Then the '''join of $a$ and $b$''' is defined as: :$a \\vee b = \\sup \\left\\{{a, b}\\right\\}$ Expanding the definition of supremum, one sees that $c = a \\vee b$ {{iff}}: :$a \\preceq c$ and $b \\preceq c$ and $\\forall s \\in S: a \\preceq s \\land b \\preceq s \\implies c \\preceq s$"}
+{"_id": "23486", "title": "Definition:Join Semilattice", "text": "{{:Definition:Join Semilattice/Definition 1}}"}
+{"_id": "23487", "title": "Definition:Meet Semilattice", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Suppose that for all $a, b \\in S$: :$a \\wedge b \\in S$, where $a \\wedge b$ is the meet of $a$ and $b$. Then the ordered structure $\\struct {S, \\wedge, \\preceq}$ is called a '''meet semilattice'''."}
+{"_id": "23488", "title": "Definition:Semilattice", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Then $\\struct {S, \\circ}$ is called a '''semilattice''' {{iff}} $\\circ$ is a commutative and idempotent operation."}
+{"_id": "23489", "title": "Definition:Lattice", "text": "=== Definition 1 === {{:Definition:Lattice/Definition 1}} === Definition 2 === {{:Definition:Lattice/Definition 2}} === Definition 3 === {{:Definition:Lattice/Definition 3}}"}
+{"_id": "23490", "title": "Definition:Lattice/Definition 1", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Suppose that $S$ admits all finite non-empty suprema and finite non-empty infima. Denote with $\\vee$ and $\\wedge$ the join and meet operations on $S$, respectively. Then the ordered structure $\\struct {S, \\vee, \\wedge, \\preceq}$ is called a '''lattice'''."}
+{"_id": "23491", "title": "Definition:Lattice/Definition 2", "text": "Let $\\struct {S, \\vee, \\wedge, \\preceq}$ be an ordered structure. Then $\\struct {S, \\vee, \\wedge, \\preceq}$ is called a '''lattice''' {{iff}}: :$\\struct {S, \\vee, \\preceq}$ is a join semilattice :$\\struct {S, \\wedge, \\preceq}$ is a meet semilattice"}
+{"_id": "23492", "title": "Definition:Lattice/Definition 3", "text": "Let $\\struct {S, \\vee}$ and $\\struct {S, \\wedge}$ be semilattices on a set $S$. Suppose that $\\vee$ and $\\wedge$ satisfy the absorption laws, that is, for all $a, b \\in S$: :$a \\vee \\paren {a \\wedge b} = a$ :$a \\wedge \\paren {a \\vee b} = a$ Let $\\preceq$ be the ordering on $S$ defined by: :$\\forall a, b \\in S: a \\preceq b$ {{iff}} $a \\vee b = b$ as on Semilattice Induces Ordering. Then the ordered structure $\\struct {S, \\vee, \\wedge, \\preceq}$ is called a '''lattice'''."}
+{"_id": "23493", "title": "Definition:Bounded Lattice", "text": "=== Definition 1 === {{:Definition:Bounded Lattice/Definition 1}} === Definition 2 === {{:Definition:Bounded Lattice/Definition 2}} === Definition 3 === {{:Definition:Bounded Lattice/Definition 3}}"}
+{"_id": "23494", "title": "Definition:Bounded Lattice/Definition 1", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $S$ admit all finite suprema and finite infima. Let $\\vee$ and $\\wedge$ be the join and meet operations on $S$, respectively. Then the ordered structure $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ is a '''bounded lattice'''."}
+{"_id": "23495", "title": "Definition:Bounded Lattice/Definition 2", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a lattice. Let $\\vee$ and $\\wedge$ have identity elements $\\bot$ and $\\top$ respectively. Then $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ is a '''bounded lattice'''."}
+{"_id": "23497", "title": "Definition:Lowest Common Multiple/Integers", "text": "For all $a, b \\in \\Z: a b \\ne 0$, there exists a smallest $m \\in \\Z: m > 0$ such that $a \\divides m$ and $b \\divides m$. This $m$ is called the '''lowest common multiple of $a$ and $b$''', and denoted $\\lcm \\set {a, b}$."}
+{"_id": "23498", "title": "Definition:Lowest Common Multiple/Integral Domain", "text": "Let $D$ be an integral domain and let $a, b \\in A$ be nonzero. $l$ is the '''lowest common multiple''' of $a$ and $b$ {{iff}}: :$(1): \\quad$ both $a$ and $b$ divide $l$ :$(2): \\quad$ if $m$ is another element such that $a$ and $b$ divide $m$, then $l$ divides $m$."}
+{"_id": "23499", "title": "Definition:Operation/Binary Operation/Product", "text": "Let $z = x \\circ y$. Then $z$ is called the '''product''' of $x$ and $y$. This is an extension of the normal definition of product that is encountered in conventional arithmetic."}
+{"_id": "23500", "title": "Definition:Diophantine Equation/Linear Diophantine Equation", "text": "A '''linear Diophantine equation''' is a Diophantine equation in which all the arguments appear to no higher than the first degree. For example: :$ax + by + c = 0$ :$a_1 x_1 + a_2 x_2 + \\cdots + a_n x_n = b$"}
+{"_id": "23502", "title": "Definition:Absorption Law", "text": "Let $\\struct {S, \\circ, *}$ be an algebraic structure. Let both $\\circ$ and $*$ be commutative. Then '''$\\circ$ absorbs $*$''' {{iff}}: :$\\forall a, b \\in S: a \\circ \\paren {a * b} = a$ This equality is called the '''absorption law of $\\circ$ for $*$'''. Category:Definitions/Abstract Algebra 5obgfea09xjreoixndqhmz2wsjj44jp"}
+{"_id": "23503", "title": "Definition:Supremum of Set/Finite Supremum", "text": "If $T$ is finite, $\\sup T$ is called a '''finite supremum'''."}
+{"_id": "23504", "title": "Definition:Infimum of Set/Finite Infimum", "text": "If $T$ is finite, $\\inf T$ is called a '''finite infimum'''."}
+{"_id": "23505", "title": "Definition:Empty Supremum", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then the '''empty supremum''' is the supremum $\\sup \\O$."}
+{"_id": "23506", "title": "Definition:Empty Infimum", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then the '''empty infimum''' is the infimum $\\inf \\O$. By Infimum of Empty Set is Greatest Element, it exists {{iff}} $\\struct {S, \\preceq}$ has a greatest element."}
+{"_id": "23507", "title": "Definition:Dual Ordering", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $\\succeq$ be the inverse relation to $\\preceq$. That is, for all $a, b \\in S$: :$a \\succeq b$ {{iff}} $b \\preceq a$ Then $\\succeq$ is called the '''dual ordering of $\\preceq$'''. By Dual Ordering is Ordering, it is indeed an ordering. === Dual Ordered Set === {{:Definition:Dual Ordering/Dual Ordered Set}} === Notation for Inverse Ordering === {{:Definition:Inverse Relation/Ordering Notation}} === Notation for Inverse Strict Ordering === {{:Definition:Inverse Relation/Strict Ordering Notation}}"}
+{"_id": "23508", "title": "Definition:Dual Ordering/Dual Ordered Set", "text": "The ordered set $\\left({S, \\succeq}\\right)$ is called the '''dual ordered set''' (or just '''dual''') '''of $\\left({S, \\preceq}\\right)$'''."}
+{"_id": "23509", "title": "Definition:Conditional/Necessary Condition", "text": "Let $p \\implies q$ be a conditional statement. Then $q$ is a '''necessary condition''' for $p$. That is, if $p \\implies q$, then it is ''necessary'' that $q$ be true for $p$ to be true. This is because unless $q$ is true, $p$ can ''not'' be true."}
+{"_id": "23510", "title": "Definition:Conditional/Sufficient Condition", "text": "Let $p \\implies q$ be a conditional statement. Then $p$ is a '''sufficient condition''' for $q$. That is, if $p \\implies q$, then for $q$ to be true, it is ''sufficient'' to know that $p$ is true. This is because of the fact that if you know that $p$ is true, you know enough to know also that $q$ is true."}
+{"_id": "23512", "title": "Definition:Dual Statement (Order Theory)", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $\\succeq$ be the dual ordering to $\\preceq$. Let $\\Sigma$ be any statement pertaining to $\\left({S, \\preceq}\\right)$ (be it in natural language or a formal language). The '''dual statement of $\\Sigma$''', denoted $\\Sigma^*$, is the statement obtained from replacing every reference to $\\preceq$ in $\\Sigma$ with a reference to its dual $\\succeq$. This '''dual statement''' may then be turned into a statement about $\\preceq$ again by applying the equivalences on Dual Pairs (Order Theory)."}
+{"_id": "23513", "title": "Definition:Relation Conversely Compatible with Operation", "text": "Let $\\left({S, \\circ}\\right)$ be a closed algebraic structure. Let $\\mathcal R$ be a relation in $S$. Then $\\mathcal R$ is '''conversely compatible with $\\circ$''' {{iff}}: :$\\forall x, y, z \\in S: \\left({x \\circ z}\\right) \\mathrel {\\mathcal R} \\left({y \\circ z}\\right) \\implies x \\mathrel {\\mathcal R} y$ :$\\forall x, y, z \\in S: \\left({z \\circ x}\\right) \\mathrel {\\mathcal R} \\left({z \\circ y}\\right) \\implies x \\mathrel {\\mathcal R} y$ Category:Definitions/Compatible Relations kl5soecrnof9n332oxdlch8n9yz685r"}
+{"_id": "23514", "title": "Definition:Relation Strongly Compatible with Operation", "text": "Let $\\left({S, \\circ}\\right)$ be a closed algebraic structure. Let $\\mathcal R$ be a relation in $S$. Then $\\mathcal R$ is '''strongly compatible with $\\circ$''' {{iff}}: :$\\forall x, y, z \\in S: x \\mathrel {\\mathcal R} y \\iff \\left({x \\circ z}\\right) \\mathrel {\\mathcal R} \\left({y \\circ z}\\right)$ :$\\forall x, y, z \\in S: x \\mathrel {\\mathcal R} y \\iff \\left({z \\circ x}\\right) \\mathrel {\\mathcal R} \\left({z \\circ y}\\right)$. That is, {{iff}} $\\mathcal R$ is compatible with $\\circ$ and conversely compatible with $\\circ$."}
+{"_id": "23515", "title": "Definition:Union of Relations", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R_1$ and $\\mathcal R_2$ be relations on $S \\times T$. The '''union of $\\mathcal R_1$ and $\\mathcal R_2$''' is the relation $\\mathcal Q$ defined by: :$\\mathcal Q := \\mathcal R_1 \\cup \\mathcal R_2$ where $\\cup$ denotes set union. Explicitly, for $s \\in S$ and $t \\in T$, we have: :$s \\mathrel{\\mathcal Q} t$ {{iff}} $s \\mathrel{\\mathcal R_1} t$ or $s \\mathrel{\\mathcal R_2} t$ === General Definition === {{:Definition:Union of Relations/General Definition}}"}
+{"_id": "23516", "title": "Definition:Union of Relations/General Definition", "text": "Let $\\mathscr R$ be a collection of relations on $S \\times T$. The '''union of $\\mathscr R$''' is the relation $\\mathcal R$ defined by: :$\\mathcal R = \\displaystyle \\bigcup \\mathscr R$ where $\\bigcup$ denotes set union. Explicitly, for $s \\in S$ and $t \\in T$: :$s \\mathrel{\\mathcal R} t$ {{iff}} for some $\\mathcal Q \\in \\mathscr R$, $s \\mathrel{\\mathcal Q} t$"}
+{"_id": "23517", "title": "Definition:Intersection of Relations", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R_1$ and $\\mathcal R_2$ be relations on $S \\times T$. The '''intersection of $\\mathcal R_1$ and $\\mathcal R_2$''' is the relation $\\mathcal Q$ defined by: :$\\mathcal Q := \\mathcal R_1 \\cap \\mathcal R_2$ where $\\cap$ denotes set intersection. Explicitly, for $s \\in S$ and $t \\in T$, we have: :$s \\mathrel{\\mathcal Q} t$ iff both $s \\mathrel{\\mathcal R_1} t$ and $s \\mathrel{\\mathcal R_2} t$ === General Definition === {{:Definition:Intersection of Relations/General Definition}}"}
+{"_id": "23518", "title": "Definition:Intersection of Relations/General Definition", "text": "Let $\\mathscr R$ be a collection of relations on $S \\times T$. The '''intersection of $\\mathscr R$''' is the relation $\\mathcal R$ defined by: :$\\mathcal R = \\displaystyle \\bigcap \\mathscr R$ where $\\bigcap$ denotes set intersection. Explicitly, for $s \\in S$ and $t \\in T$: :$s \\mathrel{\\mathcal R} t$ {{iff}} for all $\\mathcal Q \\in \\mathscr R$, $s \\mathrel{\\mathcal Q} t$"}
+{"_id": "23519", "title": "Definition:Complement (Lattice Theory)", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded lattice. Denote by $\\bot$ and $\\top$ the bottom and top of $S$, respectively. Let $a \\in S$. Then $b \\in S$ is called a '''complement of $a$''' {{iff}}: :$b \\vee a = \\top$ :$b \\wedge a = \\bot$ If $a$ has a unique '''complement''', it is denoted by $\\neg a$. === Complemented Lattice === {{:Definition:Complemented Lattice}}"}
+{"_id": "23520", "title": "Definition:Finitely Generated Algebraic Structure", "text": "Let $\\struct {A, \\circ}$ be an algebraic structure. Let $\\struct {A, \\circ}$ have a generator which is finite. Then $\\struct {A, \\circ}$ is '''finitely generated'''."}
+{"_id": "23521", "title": "Definition:Topological Group/Definition 2", "text": "Let the mapping $\\psi: \\left({G, \\tau}\\right) \\times \\left({G, \\tau}\\right) \\to \\left({G, \\tau}\\right)$ be defined as: :$\\psi \\left({x, y}\\right) = x \\odot y^{-1}$ $\\left({G, \\odot, \\tau}\\right)$ is a '''topological group''' {{iff}}: : $\\psi$ is a continuous mapping where $\\left({G, \\tau}\\right) \\times \\left({G, \\tau}\\right)$ is considered as $G \\times G$ with the product topology."}
+{"_id": "23522", "title": "Definition:Topological Group/Definition 1", "text": "$\\left({G, \\odot, \\tau}\\right)$ is a '''topological group''' {{iff}}: :$(1): \\quad \\odot: \\left({G, \\tau}\\right) \\times \\left({G, \\tau}\\right) \\to \\left({G, \\tau}\\right)$ is a continuous mapping :$(2): \\quad \\phi: \\left({G, \\tau}\\right) \\to \\left({G, \\tau}\\right)$ such that $\\forall x \\in G: \\phi \\left({x}\\right) = x^{-1}$ is also a continuous mapping where $\\left({G, \\tau}\\right) \\times \\left({G, \\tau}\\right)$ is considered as $G \\times G$ with the product topology."}
+{"_id": "23523", "title": "Definition:Topological Ring", "text": "Let $\\struct{R, +, \\circ}$ be a ring. Let $\\tau$ be a topology over $R$. Then $\\struct {R, +, \\circ, \\tau}$ is a '''topological ring''' {{iff}}: :$(1): \\quad \\struct {R, +, \\tau}$ is a topological group :$(2): \\quad \\struct {R, \\circ,\\tau}$ is a topological semigroup."}
+{"_id": "23524", "title": "Definition:Topological Field", "text": "Let $\\struct {F, +, \\circ}$ be a field with zero $0_F$. Let $\\tau$ be a topology on $F$. Let $\\struct {F, +, \\circ, \\tau}$ be a topological division ring. Then $\\struct {F, +, \\circ, \\tau}$ is a '''topological field'''. That is, a topological field is a commutative topological division ring."}
+{"_id": "23525", "title": "Definition:Field Zero", "text": "Let $\\struct {F, +, \\times}$ be a field. The identity for field addition is called the '''field zero''' (of $\\struct {F, +, \\times}$). It is denoted $0_F$ (or just $0$ if there is no danger of ambiguity)."}
+{"_id": "23526", "title": "Definition:Ray (Order Theory)", "text": "Let $\\struct {S, \\preccurlyeq}$ be a totally ordered set. Let $\\prec$ be the reflexive reduction of $\\preccurlyeq$. Let $a \\in S$ be any point in $S$. === Open Ray === {{:Definition:Ray (Order Theory)/Open}} === Closed Ray === {{:Definition:Ray (Order Theory)/Closed}} === Upward-Pointing Ray === {{:Definition:Ray (Order Theory)/Upward-Pointing}} === Downward-Pointing Ray === {{:Definition:Ray (Order Theory)/Downward-Pointing}}"}
+{"_id": "23527", "title": "Definition:Complemented Lattice", "text": "Suppose that every $a \\in S$ admits a complement. Then $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ is called a '''complemented lattice'''."}
+{"_id": "23528", "title": "Definition:Top (Lattice Theory)", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a lattice. === Definition 1 === {{:Definition:Top (Lattice Theory)/Definition 1}} === Definition 2 === {{:Definition:Top (Lattice Theory)/Definition 2}}"}
+{"_id": "23529", "title": "Definition:Top (Lattice Theory)/Definition 1", "text": "Let $S$ admit a greatest element $\\top$. Then $\\top$ is called the '''top''' of $S$."}
+{"_id": "23530", "title": "Definition:Top (Lattice Theory)/Definition 2", "text": "Let $\\wedge$ have an identity element $\\top$. Then $\\top$ is called the '''top''' of $S$."}
+{"_id": "23531", "title": "Definition:Bottom (Lattice Theory)", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a lattice. === Definition 1 === {{:Definition:Bottom (Lattice Theory)/Definition 1}} === Definition 2 === {{:Definition:Bottom (Lattice Theory)/Definition 2}}"}
+{"_id": "23532", "title": "Definition:Bottom (Lattice Theory)/Definition 1", "text": "Let $S$ admit a smallest element $\\bot$. Then $\\bot$ is called the '''bottom''' of $S$."}
+{"_id": "23533", "title": "Definition:Bottom (Lattice Theory)/Definition 2", "text": "Let $\\vee$ have an identity element $\\bot$. Then $\\bot$ is called the '''bottom''' of $S$."}
+{"_id": "23534", "title": "Definition:Distance-Preserving Mapping", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces, pseudometric spaces, or quasimetric spaces. Let $\\phi: M_1 \\to M_2$ be a mapping such that: : $\\forall a, b \\in M_1: d_1 \\left({a, b}\\right) = d_2 \\left({\\phi \\left({a}\\right), \\phi \\left({b}\\right)}\\right)$ Then $\\phi$ is called a '''distance-preserving mapping'''."}
+{"_id": "23535", "title": "Definition:Connected Domain (Complex Analysis)", "text": "Let $D \\subseteq \\C$ be a subset of the set of complex numbers. Then $D$ is a '''connected domain''' {{iff}} $D$ is open and connected."}
+{"_id": "23536", "title": "Definition:Connected Domain (Complex Analysis)/Simply Connected Domain", "text": "Let $D \\subseteq \\C$ be a connected domain. Then $D$ is called a '''simply connected domain''' {{iff}} $D$ is simply connected. === Simply Connectedness Requirement ===  {{:Definition:Connected Domain (Complex Analysis)/Simply Connected Domain/Requirement}}"}
+{"_id": "23537", "title": "Definition:Topologist's Sine Curve", "text": "Let $G = \\set {\\tuple {x, \\sin \\dfrac 1 x} : 0 < x \\le 1}$, considered as a subset of the real number plane with the usual (Euclidean) topology $\\struct {\\R^2, \\tau_d}$. Then the set $G \\cup \\set {\\tuple {0, 0} }$ is called the '''Topologist's Sine Curve'''."}
+{"_id": "23538", "title": "Definition:Staircase Contour", "text": "Let $C$ be a contour that is a concatenation of the directed smooth curves $C_1, \\ldots, C_n$. For all $i \\in \\set {1, \\ldots, n}$, let it be possible for $C_i$ be parameterized by a smooth path $\\gamma_i: \\closedint 0 1 \\to \\C$ such that either: :$\\map {\\gamma_i} t = z_i + t r_i$ or :$\\map {\\gamma_i} t = z_i + i t r_i$ for some $z_i \\in \\C, r_i \\in \\R$ for all $t \\in \\closedint 0 1$. Then $C$ is called a '''staircase contour'''."}
+{"_id": "23539", "title": "Definition:Differentiable Mapping/Complex Function/Point", "text": "Let $D\\subset \\C$ be an open set. Let $f : D \\to \\C$ be a complex function. Let $z_0 \\in D$ be a point in $D$. Then $f$ is '''complex-differentiable''' at $z_0$ {{iff}} the limit: : $\\displaystyle \\lim_{h \\to 0} \\frac {f \\left({z_0+h}\\right) - f \\left({z_0}\\right)} h$ exists as a finite number."}
+{"_id": "23540", "title": "Definition:Differentiable Mapping/Complex Function/Region", "text": "Let $D \\subseteq \\C$ be an open set. Let $f: D \\to \\C$ be a complex function. Then $f$ is '''complex-differentiable in $D$''' {{iff}} $f$ is complex-differentiable at every point in $D$."}
+{"_id": "23541", "title": "Definition:Inequality", "text": "An '''inequality''' is a mathematical statement that two expressions relate in one of several conventional ways: : $a < b$ : $a \\le b$ : $a > b$ : $a \\ge b$ A statement of the form:  :$a \\ne b$ may or may not be considered an '''inequality'''."}
+{"_id": "23542", "title": "Definition:Convex Set (Vector Space)/Line Segment", "text": "The set: :$\\set {t x + \\paren {1 - t} y: t \\in \\closedint 0 1}$  is called the '''(straight) line segment joining $x$ and $y$'''. A convex set can thus be described as a set containing all '''straight line segments''' between its elements."}
+{"_id": "23544", "title": "Definition:Boolean Lattice", "text": "=== Definition 1 === {{:Definition:Boolean Lattice/Definition 1}} === Definition 2 === {{:Definition:Boolean Lattice/Definition 2}} === Definition 3 === {{:Definition:Boolean Lattice/Definition 3}}"}
+{"_id": "23545", "title": "Definition:Boolean Lattice/Definition 1", "text": "A '''Boolean lattice''' is a complemented distributive lattice."}
+{"_id": "23546", "title": "Definition:Boolean Lattice/Definition 2", "text": "An ordered structure $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ is a '''Boolean lattice''' {{iff}}: $(1): \\quad \\left({S, \\vee, \\wedge}\\right)$ is a Boolean algebra $(2): \\quad$ For all $a, b \\in S$: $a \\wedge b \\preceq a \\vee b$"}
+{"_id": "23547", "title": "Definition:Metalanguage/Formal Systems", "text": "In the context of formal systems, a '''metalanguage''' is a formal language used to specify another formal language."}
+{"_id": "23548", "title": "Definition:Metalanguage/Object Language", "text": "The '''object language''' of a metalanguage is the language described by that metalanguage."}
+{"_id": "23549", "title": "Definition:Metalanguage/Metasyntax", "text": "The syntax of a metalanguage is called a '''metasyntax''' of the object language of that metalanguage."}
+{"_id": "23550", "title": "Definition:Metalanguage/Metasymbol", "text": "A '''metasymbol''' is a symbol used in a metalanguage to represent an arbitrary collation in the object language."}
+{"_id": "23551", "title": "Definition:Theorem/Logic", "text": "A '''theorem''' in logic is a statement which can be shown to be the conclusion of a logical argument which depends on ''no'' premises except axioms. A sequent which denotes a theorem $\\phi$ is written $\\vdash \\phi$, indicating that there are no premises. In this context, $\\vdash$ is read as: :'''It is a theorem that ...'''"}
+{"_id": "23552", "title": "Definition:Theorem/Formal System", "text": "Let $\\mathcal L$ be a formal language. Let $\\mathscr P$ be a proof system for $\\mathcal L$. A '''theorem of $\\mathscr P$''' is a well-formed formula of $\\mathcal L$ which can be deduced from the axioms and axiom schemata of $\\mathscr P$ by means of its rules of inference. That a WFF $\\phi$ is a '''theorem''' of $\\mathscr P$ may be denoted as: :$\\vdash_{\\mathscr P} \\phi$"}
+{"_id": "23553", "title": "Definition:Theorem/Mathematics", "text": "The term '''theorem''' is used throughout the whole of mathematics to mean a statement which has been proved to be true from whichever axioms relevant to that particular branch. Note that statements which are taken as axioms in one branch of mathematics may be theorems in others."}
+{"_id": "23555", "title": "Definition:Distance/Sets/Real Numbers", "text": "Let $S, T$ be a subsets of the set of real numbers $\\R$. Let $x \\in \\R$ be a real number. The '''distance between $x$ and $S$''' is defined and annotated $\\displaystyle \\map d {x, S} = \\map {\\inf_{y \\mathop \\in S} } {\\map d {x, y} }$, where $\\map d {x, y}$ is the distance between $x$ and $y$. The '''distance between $S$ and $T$''' is defined and annotated $\\displaystyle \\map d {S, T} = \\map {\\inf_{\\substack {x \\mathop \\in S \\\\ y \\mathop \\in T} } } {\\map d {x, y} }$."}
+{"_id": "23556", "title": "Definition:Distance/Sets/Metric Spaces", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $x \\in A$. Let $S, T$ be subsets of $A$. The '''distance between $x$ and $S$''' is defined and annotated $\\displaystyle d \\left({x, S}\\right) = \\inf_{y \\mathop \\in S} \\left({d \\left({x, y}\\right)}\\right)$. The '''distance between $S$ and $T$''' is defined and annotated $\\displaystyle d \\left({S, T}\\right) = \\inf_{\\substack{x \\mathop \\in S \\\\ y \\mathop \\in T}} \\left({d \\left({x, y}\\right)}\\right)$."}
+{"_id": "23557", "title": "Definition:Bounded Lattice/Definition 3", "text": "Let $\\left({S, \\wedge, \\vee, \\preceq}\\right)$ be a lattice. Let $S$ be bounded in $\\left({S,\\preceq}\\right)$. Then $\\left({S, \\wedge, \\vee, \\preceq}\\right)$ is a '''bounded lattice'''."}
+{"_id": "23558", "title": "Definition:Uniquely Complemented Lattice", "text": "Let $\\left({S,\\wedge,\\vee,\\preceq}\\right)$ be a complemented lattice. If each element of $S$ has only one complement, then $\\left({S,\\wedge,\\vee,\\preceq}\\right)$ is a '''uniquely complemented lattice'''. That is, a uniquely complemented lattice is a lattice in which each element has exactly one complement."}
+{"_id": "23559", "title": "Definition:Boolean Algebra/Definition 3", "text": "A '''Boolean algebra''' is an algebraic structure $\\struct {S, \\vee, \\wedge}$ such that: {{begin-axiom}} {{axiom | n = \\text {BA} 0         | lc=         | t = $S$ is closed under both $\\vee$ and $\\wedge$ }} {{axiom | n = \\text {BA} 1         | lc=         | t = Both $\\vee$ and $\\wedge$ are commutative }} {{axiom | n = \\text {BA} 2         | lc=         | t = Both $\\vee$ and $\\wedge$ distribute over the other }} {{axiom | n = \\text {BA} 3         | lc=         | t = Both $\\vee$ and $\\wedge$ have identities $\\bot$ and $\\top$ respectively }} {{axiom | n = \\text {BA} 4         | lc=         | t = $\\forall a \\in S: \\exists \\neg a \\in S: a \\vee \\neg a = \\top, a \\wedge \\neg a = \\bot$ }} {{end-axiom}}"}
+{"_id": "23560", "title": "Definition:Boolean Algebra/Definition 1", "text": "{{:Definition:Boolean Algebra/Axioms/Definition 1}}"}
+{"_id": "23561", "title": "Definition:Degenerate Boolean Algebra", "text": "Let $\\struct {S, \\vee, \\wedge, \\neg}$ be a Boolean algebra. Then $\\struct {S, \\vee, \\wedge, \\neg}$ is said to be '''degenerate''' {{iff}} $S$ is a singleton."}
+{"_id": "23562", "title": "Definition:Exponential Function/Real/Inverse of Natural Logarithm", "text": "Consider the natural logarithm $\\ln x$, which is defined on the open interval $\\openint 0 {+\\infty}$. From Logarithm is Strictly Increasing: :$\\ln x$ is strictly increasing. From Inverse of Strictly Monotone Function: :the inverse of $\\ln x$ always exists. The inverse of the natural logarithm function is called the '''exponential function''', which is denoted as $\\exp$. Thus for $x \\in \\R$, we have: :$y = \\exp x \\iff x = \\ln y$"}
+{"_id": "23563", "title": "Definition:Exponential Function/Real/Extension of Rational Exponential", "text": "Let $e$ denote Euler's number. Let $f: \\Q \\to \\R$ denote the real-valued function defined as: :$f \\left({ x }\\right) = e^x$ That is, let $f \\left({ x }\\right)$ denote $e$ to the power of $x$, for rational $x$. Then $\\exp : \\R \\to \\R$ is defined to be the unique continuous extension of $f$ to $\\R$. $\\exp \\left({ x }\\right)$ is called the '''exponential of $x$'''."}
+{"_id": "23564", "title": "Definition:Exponential Function/Real/Limit of Sequence", "text": "The '''exponential function''' can be defined as the following limit of a sequence: :$\\exp x := \\displaystyle \\lim_{n \\mathop \\to +\\infty} \\paren {1 + \\frac x n}^n$"}
+{"_id": "23565", "title": "Definition:Exponential Function/Real/Sum of Series", "text": "The '''exponential function''' can be defined as a power series: :$\\exp x := \\displaystyle \\sum_{n \\mathop = 0}^\\infty \\frac {x^n} {n!}$"}
+{"_id": "23566", "title": "Definition:Exponential Function/Real/Differential Equation", "text": "The '''exponential function''' can be defined as the unique solution $y = \\map f x$ to the first order ODE: :$\\dfrac {\\d y} {\\d x} = y$ satisfying the initial condition $\\map f 0 = 1$."}
+{"_id": "23567", "title": "Definition:Logical Not/Truth Table", "text": "The characteristic truth table of the negation operator $\\neg p$ is as follows: :$\\begin{array}{|c||c|} \\hline p & \\neg p \\\\ \\hline \\F & \\T \\\\ \\T & \\F \\\\ \\hline \\end{array}$"}
+{"_id": "23568", "title": "Definition:Truth Value/Aristotelian Logic", "text": "In Aristotelian logic, a statement can be either true or false, and there is no undefined, in-between value. Whether it is true or false is called its '''truth value'''. Note that a statement's '''truth value''' may change depending on circumstances. Thus, the statement: :''It is currently raining on the grass outside my window'' has the truth value false, whereas it had the truth value true last week. The statement: :''I am listening to Shostakovich's 4th symphony'' is currently true, but that will last only for the next twenty minutes or so as I type. The '''truth values''' true and false are usually represented in one of two ways: :$\\T$ for true and $\\F$ for false; :$1$ for true and $0$ for false. There are advantages for both notations. In particular, the second lends itself to extending the discipline of logic into that of probability theory."}
+{"_id": "23569", "title": "Definition:Conjunction/Truth Table", "text": "The characteristic truth table of the logical conjunction operator $p \\land q$ is as follows: :$\\begin{array}{|cc||c|} \\hline p & q & p \\land q \\\\ \\hline \\F & \\F & \\F \\\\ \\F & \\T & \\F \\\\ \\T & \\F & \\F \\\\ \\T & \\T & \\T \\\\ \\hline \\end{array}$"}
+{"_id": "23570", "title": "Definition:Compactification", "text": "Let $\\left({X, \\tau_1}\\right)$ be a topological space. Let $\\left({Y, \\tau_2}\\right)$ be a compact space. Let $f: X \\to Y$ be a topological embedding. Let $f \\left({X}\\right)$ be everywhere dense in $Y$. Then either $f$ or $\\left({Y, \\tau_2}\\right)$ may be called a '''compactification''' of $\\left({X, \\tau_1}\\right)$."}
+{"_id": "23571", "title": "Definition:Disjunction/Truth Table", "text": "The characteristic truth table of the logical disjunction operator $p \\lor q$ is as follows: :$\\begin{array}{|cc||c|} \\hline p & q & p \\lor q \\\\ \\hline \\F & \\F & \\F \\\\ \\F & \\T & \\T \\\\ \\T & \\F & \\T \\\\ \\T & \\T & \\T \\\\ \\hline \\end{array}$"}
+{"_id": "23572", "title": "Definition:Biconditional/Truth Table", "text": "The characteristic truth table of the biconditional operator $p \\iff q$ is as follows: :$\\begin{array}{|cc||c|} \\hline p & q & p \\iff q \\\\ \\hline \\F & \\F & \\T \\\\ \\F & \\T & \\F \\\\ \\T & \\F & \\F \\\\ \\T & \\T & \\T \\\\ \\hline \\end{array}$"}
+{"_id": "23573", "title": "Definition:Exclusive Or/Truth Table", "text": "The characteristic truth table of the exclusive or operator $p \\oplus q$ is as follows: :$\\begin{array}{|cc||c|} \\hline p & q & p \\oplus q \\\\ \\hline \\F & \\F & \\F \\\\ \\F & \\T & \\T \\\\ \\T & \\F & \\T \\\\ \\T & \\T & \\F \\\\ \\hline \\end{array}$"}
+{"_id": "23574", "title": "Definition:Conditional/Truth Table", "text": "The characteristic truth table of the conditional (implication) operator $p \\implies q$ is as follows: :$\\begin{array}{|cc||c|} \\hline p & q & p \\implies q \\\\ \\hline \\F & \\F & \\T \\\\ \\F & \\T & \\T \\\\ \\T & \\F & \\F \\\\ \\T & \\T & \\T \\\\ \\hline \\end{array}$"}
+{"_id": "23575", "title": "Definition:Logical NAND/Truth Table", "text": "The characteristic truth table of the logical NAND operator $p \\uparrow q$ is as follows: :$\\begin{array}{|cc||c|} \\hline p & q & p \\uparrow q \\\\ \\hline F & F & T \\\\ F & T & T \\\\ T & F & T \\\\ T & T & F \\\\ \\hline \\end{array}$"}
+{"_id": "23576", "title": "Definition:Logical NOR/Truth Table", "text": "The characteristic truth table of the logical NOR operator $p \\downarrow q$ is as follows: :$\\begin{array}{|cc||c|} \\hline p & q & p \\downarrow q \\\\ \\hline F & F & T \\\\ F & T & F \\\\ T & F & F \\\\ T & T & F \\\\ \\hline \\end{array}$"}
+{"_id": "23577", "title": "Definition:Series/General", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroup. Let $\\left \\langle {a_n} \\right \\rangle$ be a sequence in $S$. Informally, a '''series''' is what results when an infinite product is taken of $\\left \\langle {a_n} \\right \\rangle$: :$\\displaystyle s := \\sum_{n \\mathop = 1}^\\infty a_n = a_1 \\circ a_2 \\circ a_3 \\circ \\cdots$ Formally, a '''series''' is a sequence in $S$."}
+{"_id": "23578", "title": "Definition:Series/Number Field", "text": "The '''series''' is what results when $\\sequence {a_n}$ is summed to infinity: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n = a_1 + a_2 + a_3 + \\cdots$"}
+{"_id": "23579", "title": "Definition:Series/Tail", "text": "Let $N \\in \\N$. The expression $\\displaystyle \\sum_{n \\mathop = N}^\\infty a_n$ is known as a '''tail''' of the '''series''' $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$."}
+{"_id": "23580", "title": "Definition:Conditional/Semantics of Conditional", "text": "$p \\implies q$ can be stated thus: * '''''If'' $p$ is true ''then'' $q$ is true.''' * '''$q$ is true ''if'' $p$ is true.''' * '''(The truth of) $p$ ''implies'' (the truth of) $q$.''' * '''(The truth of) $q$ ''is implied by'' (the truth of) $p$.''' * '''$q$ ''follows from'' $p$.''' * '''$p$ is true ''only if'' $q$ is true.''' The latter one may need some explanation. $p$ can be either true or false, as can $q$. But if $q$ is false, and $p \\implies q$, then $p$ can not be true. Therefore, $p$ can be true ''only if'' $q$ is also true, which leads us to our assertion. * '''$p$ is true ''therefore'' $q$ is true.''' * '''$p$ is true ''entails'' that $q$ is true.''' * '''$q$ is true ''because'' $p$ is true.''' * '''$p$ ''may'' be true ''unless'' $q$ is false.''' * '''''Given that'' $p$ is true, $q$ is true.''' * '''$q$ is true ''whenever'' $p$ is true.''' * '''$q$ is true ''provided that'' $p$ is true.''' * '''$q$ is true ''in case'' $p$ is true.''' * '''$q$ is true ''assuming that'' $p$ is true.''' * '''$q$ is true ''on the condition that'' $p$ is true.'''"}
+{"_id": "23581", "title": "Definition:Conditional/Language of Conditional/Weak", "text": "In a conditional $p \\implies q$, the statement $q$ is '''weaker''' than $p$."}
+{"_id": "23582", "title": "Definition:Conditional/Language of Conditional/Strong", "text": "In a conditional $p \\implies q$, the statement $p$ is '''stronger''' than $q$."}
+{"_id": "23583", "title": "Definition:Conditional/Superimplicant", "text": "In a conditional $p \\implies q$, the statement $p$ is '''superimplicant''' to $q$."}
+{"_id": "23584", "title": "Definition:Conditional/Subimplicant", "text": "In a conditional $p \\implies q$, the statement $q$ is '''subimplicant''' to $p$."}
+{"_id": "23585", "title": "Definition:Conditional/Language of Conditional", "text": "The conditional has been discussed at great length throughout the ages, and a whole language has evolved around it. For now, here are a few definitions: === Weak === {{:Definition:Conditional/Language of Conditional/Weak}} === Strong === {{:Definition:Conditional/Language of Conditional/Strong}} Thus we have the notion of certain theorems having a weak and a strong version. === Superimplicant === {{:Definition:Conditional/Superimplicant}} === Subimplicant === {{:Definition:Conditional/Subimplicant}} === Antecedent === {{:Definition:Conditional/Antecedent}} === Consequent === {{:Definition:Conditional/Consequent}} === Necessary Condition === {{:Definition:Conditional/Necessary Condition}} === Sufficient Condition === {{:Definition:Conditional/Sufficient Condition}}"}
+{"_id": "23586", "title": "Definition:Independent Statements", "text": "Let $p$ and $q$ be statements. Let it be the case that: :$(1): \\quad p$ and $q$ are not contrary :$(2): \\quad p$ and $q$ are not subcontrary :$(3): \\quad p$ is not superimplicant to $q$ :$(4): \\quad p$ is not subimplicant to $q$ :$(5): \\quad p$ and $q$ are not equivalent :$(6): \\quad p$ and $q$ are not contradictory. Then $p$ and $q$ are '''independent statements'''."}
+{"_id": "23587", "title": "Definition:Primitive (Calculus)/Complex", "text": "Let $F: D \\to \\C$ be a complex function which is complex-differentiable on a connected domain $D$. Let $f: D \\to \\C$ be a continuous complex function. Let: :$\\forall z \\in D: \\map {F'} z = \\map f z$ where $F'$ denotes the derivative of $F$ with respect to $z$. Then $F$ is '''a primitive of $f$''', and is denoted: :$\\displaystyle F = \\int \\map f z \\rd z$"}
+{"_id": "23588", "title": "Definition:Primitive (Calculus)/Real", "text": "Let $F$ be a real function which is continuous on the closed interval $\\closedint a b$ and differentiable on the open interval $\\openint a b$. Let $f$ be a real function which is continuous on the open interval $\\openint a b$. Let: :$\\forall x \\in \\openint a b: \\map {F'} x = \\map f x$ where $F'$ denotes the derivative of $F$ with respect to $x$. Then $F$ is '''a primitive of $f$''', and is denoted: :$\\displaystyle F = \\int \\map f x \\rd x$"}
+{"_id": "23589", "title": "Definition:Primitive (Calculus)/Arbitrary Constant", "text": "From the language in which it is couched, it is apparent that the primitive of a function may not be unique, otherwise we would be referring to $F$ as '''''the'' primitive''' of $f$. This point is made apparent in Primitives which Differ by Constant: if a function has a primitive, there is an infinite number of them, all differing by a constant. That is, if $F$ is a primitive for $f$, then so is $F + C$, where $C$ is a constant. This constant is known as an '''arbitrary constant'''."}
+{"_id": "23590", "title": "Definition:Primitive (Calculus)/Indefinite Integral", "text": "From the Fundamental Theorem of Calculus, it is apparent that to find the value of a definite integral for a function between two points, one can find the value of the primitive of the function at those points and subtract one from the other. Thus arises the notation: :$\\displaystyle \\int \\map f x \\rd x = \\map F x + C$ where $C$ is the arbitrary constant. In this context, the expression $\\displaystyle \\int \\map f x \\rd x$ is known as the '''indefinite integral''' of $f$."}
+{"_id": "23591", "title": "Definition:Circular Relation", "text": "$\\mathcal R$ is '''circular''' {{iff}}: :$\\left({x, y}\\right) \\in \\mathcal R \\land \\left({y, z}\\right) \\in \\mathcal R \\implies \\left({z, x}\\right) \\in \\mathcal R$"}
+{"_id": "23592", "title": "Definition:Exponential Function/Complex/Sum of Series", "text": "The '''exponential function''' can be defined as a (complex) power series: {{begin-eqn}} {{eqn | l = \\exp z       | r = \\sum_{n \\mathop = 0}^\\infty \\frac {z^n} {n!}       | c =  }} {{eqn | r = 1 + \\frac z {1!} + \\frac {z^2} {2!} + \\frac {z^3} {3!} + \\cdots + \\frac {z^n} {n!} + \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "23593", "title": "Definition:Exponential Function/Complex/Real Functions", "text": "The '''exponential function''' can be defined by the real exponential, sine and cosine functions: :$\\exp z := e^x \\paren {\\cos y + i \\sin y}$ where $z = x + i y$ with $x, y \\in \\R$. Here, $e^x$ denotes the real exponential function, which must be defined first."}
+{"_id": "23594", "title": "Definition:Exponential Function/Complex/Limit of Sequence", "text": "The '''exponential function''' can be defined as a limit of a sequence: :$\\displaystyle \\exp z := \\lim_{n \\to \\infty} \\left({1 + \\dfrac z n}\\right)^n$"}
+{"_id": "23595", "title": "Definition:Exponential Function/Complex/Differential Equation", "text": "The '''exponential function''' can be defined as the unique particular solution $y = \\map f z$ to the first order ODE: :$\\dfrac {\\d y} {\\d z} = y$ satisfying the initial condition $\\map f 0 = 1$. That is, the defining property of $\\exp$ is that it is its own derivative."}
+{"_id": "23596", "title": "Definition:Exponential Function/Complex", "text": "For all definitions of the '''complex exponential function''': : The domain of $\\exp$ is $\\C$. : The image of $\\exp$ is $\\C \\setminus \\left\\{ {0}\\right\\}$, as shown in Image of Complex Exponential Function. For $z \\in \\C$, the complex number $\\exp z$ is called the '''exponential of $z$'''. === As a Sum of a Series === {{:Definition:Exponential Function/Complex/Sum of Series}} === By Real Functions === {{:Definition:Exponential Function/Complex/Real Functions}} === As a Limit of a Sequence === {{:Definition:Exponential Function/Complex/Limit of Sequence}} === As the Solution of a Differential Equation === {{:Definition:Exponential Function/Complex/Differential Equation}}"}
+{"_id": "23597", "title": "Definition:Topological Division Ring", "text": "Let $\\left({R, +, \\circ}\\right)$ be a division ring with zero $0_R$. Let $\\tau$ be a topology on $R$. Let the mapping $\\phi: R \\setminus \\left\\{{0_R}\\right\\} \\to R$ be defined as: :$\\phi \\left({x}\\right) = x^{-1}$ for each $x \\in R \\setminus \\left\\{{0_R}\\right\\}$ Then $\\left({R,+,\\circ,\\tau}\\right)$ is a '''topological division ring''' {{iff}}: :$(1): \\quad \\left({R, +, \\circ, \\tau}\\right)$ is a topological ring :$(2): \\quad \\phi$ is a $\\tau'$-$\\tau$-continuous mapping, where $\\tau'$ is the $\\tau$-relative subspace topology on $R\\setminus\\left\\{{0_R}\\right\\}$."}
+{"_id": "23598", "title": "Definition:Injective on Objects", "text": "Let $\\mathbf C$ and $\\mathbf D$ be metacategories. Let $F: \\mathbf C \\to \\mathbf D$ be a functor. Then $F$ is said to be '''injective on objects''' iff for all objects $C_1, C_2$ of $\\mathbf C$: :$F C_1 = F C_2$ implies $C_1 = C_2$"}
+{"_id": "23599", "title": "Definition:Surjective on Objects", "text": "Let $\\mathbf C$ and $\\mathbf D$ be metacategories. Let $F: \\mathbf C \\to \\mathbf D$ be a functor. Then $F$ is said to be '''surjective on objects''' iff: :For every object $D$ of $\\mathbf D$, there is an object $C$ of $\\mathbf C$ such that $F C = D$."}
+{"_id": "23600", "title": "Definition:Injective on Morphisms", "text": "Let $\\mathbf C$ and $\\mathbf D$ be metacategories. Let $F: \\mathbf C \\to \\mathbf D$ be a functor. Then $F$ is said to be '''injective on morphisms''' iff for all morphisms $f, g$ of $\\mathbf C$: :$F f = F g$ implies $f = g$ Note that it is '''not''' required that $f$ and $g$ have equal domains or codomains."}
+{"_id": "23601", "title": "Definition:Surjective on Morphisms", "text": "Let $\\mathbf C$ and $\\mathbf D$ be metacategories. Let $F: \\mathbf C \\to \\mathbf D$ be a functor. Then $F$ is said to be '''surjective on morphisms''' iff: :For every morphism $g$ of $\\mathbf D$, there is a morphism $f$ of $\\mathbf C$ such that $F f = g$"}
+{"_id": "23602", "title": "Definition:Full Functor", "text": "Let $\\mathbf C$ and $\\mathbf D$ be metacategories. Let $F: \\mathbf C \\to \\mathbf D$ be a covariant functor. Then $F$ is '''full''' iff for all objects $C_1, C_2$ of $\\mathbf C$: :$F: \\operatorname{Hom}_{\\mathbf C} \\left({C_1, C_2}\\right) \\to \\operatorname{Hom}_{\\mathbf D} \\left({F C_1, F C_2}\\right), \\ f \\mapsto F f$ is a surjection. Here $\\operatorname{Hom}$ signifies a hom class."}
+{"_id": "23603", "title": "Definition:Faithful Functor", "text": "Let $\\mathbf C$ and $\\mathbf D$ be metacategories. Let $F: \\mathbf C \\to \\mathbf D$ be a functor. Then $F$ is '''faithful''' iff for all objects $C_1, C_2$ of $\\mathbf C$: :$F: \\operatorname{Hom}_{\\mathbf C} \\left({C_1, C_2}\\right) \\to \\operatorname{Hom}_{\\mathbf D} \\left({F C_1, F C_2}\\right), \\ f \\mapsto F f$ is an injection. Here $\\operatorname{Hom}$ signifies a hom class."}
+{"_id": "23604", "title": "Definition:Full Subcategory", "text": "Let $\\mathbf D$ be a metacategory, and let $\\mathbf C$ be a subcategory of $\\mathbf D$. Then $\\mathbf C$ is called a '''full subcategory''' iff the inclusion functor $i: \\mathbf C \\to \\mathbf D$ is full."}
+{"_id": "23605", "title": "Definition:Subcategory", "text": "=== Definition 1 === Let $\\mathbf C$ be a metacategory. A '''subcategory''' of $\\mathbf C$ is a metacategory with: {{begin-axiom}} {{axiom|lc= Objects:        |t = Any collection of objects from $\\mathbf C$ }} {{axiom|lc= Morphisms:        |t = Any collection of morphisms from $\\mathbf C$ }} {{axiom|lc= Composition:        |t = Inherited from $\\mathbf C$ }} {{axiom|lc= Identity morphisms:        |t = Inherited from $\\mathbf C$ }} {{end-axiom}} Colloquially, a '''subcategory''' of $\\mathbf C$ is a \"part of $\\mathbf C$ that is a category in its own right\". === Definition 2 === Let $\\mathbf C$ be a metacategory. A '''subcategory''' of $\\mathbf C$ is a monic functor $F: \\mathbf D \\to \\mathbf C$ to $\\mathbf C$"}
+{"_id": "23606", "title": "Definition:Jordan Arc", "text": "Let $\\tuple {x_1, y_1}, \\tuple {x_2, y_2} \\in \\R^2$. Let $f: \\closedint 0 1 \\to \\R^2$ be a path from $\\tuple {x_1, y_1}$ to $\\tuple {x_2, y_2}$. Then $f$ is a '''Jordan arc''' {{iff}} $f$ is an injection, except that we allow the possibility $\\tuple {x_1, y_1} = \\tuple {x_2, y_2}$."}
+{"_id": "23607", "title": "Definition:Jordan Curve", "text": "Let $f$ be a Jordan arc from $\\tuple {x_1, y_1}$ to $\\tuple {x_2, y_2}$. Then $f$ is a '''Jordan curve''' {{iff}} $\\tuple {x_1, y_1} = \\tuple {x_2, y_2}$."}
+{"_id": "23608", "title": "Definition:Orthogonal (Linear Algebra)/Real Vector Space", "text": "Let $\\mathbf u$, $\\mathbf v$ be vectors in $\\R^n$. Then $\\mathbf u$ and $\\mathbf v$ are said to be '''orthogonal''' {{iff}} their dot product is zero: :$\\mathbf u \\cdot \\mathbf v = 0$ As Dot Product is Inner Product, this is a special case of the definition of orthogonal vectors."}
+{"_id": "23609", "title": "Definition:Normal Subgroup/Definition 1", "text": ":$\\forall g \\in G: g \\circ N = N \\circ g$"}
+{"_id": "23610", "title": "Definition:Normal Subset", "text": "Let $\\left({G, \\circ}\\right)$ be a group. Let $S \\subseteq G$ be a general subset of $G$. Then $S$ is a '''normal subset of $G$''' {{iff}}: === Definition 1=== {{:Definition:Normal Subset/Definition 1}} === Definition 2=== {{:Definition:Normal Subset/Definition 2}} === Definition 3=== {{:Definition:Normal Subset/Definition 3}} === Definition 4=== {{:Definition:Normal Subset/Definition 4}} === Definition 5 === {{:Definition:Normal Subset/Definition 5}} === Definition 6 === {{:Definition:Normal Subset/Definition 6}} === Definition 7 === {{:Definition:Normal Subset/Definition 7}}"}
+{"_id": "23611", "title": "Definition:Normal Subset/Definition 1", "text": ":$\\forall g \\in G: g \\circ S = S \\circ g$"}
+{"_id": "23612", "title": "Definition:Normal Subset/Definition 2", "text": ":$\\forall g \\in G: g \\circ S \\circ g^{-1} = S$ or, equivalently: :$\\forall g \\in G: g^{-1} \\circ S \\circ g = S$"}
+{"_id": "23613", "title": "Definition:Normal Subset/Definition 3", "text": ": $\\forall g \\in G: g \\circ S \\circ g^{-1} \\subseteq S$ or, equivalently: : $\\forall g \\in G: g^{-1} \\circ S \\circ g \\subseteq S$"}
+{"_id": "23614", "title": "Definition:Normal Subset/Definition 4", "text": ": $\\forall g \\in G: S \\subseteq g \\circ S \\circ g^{-1}$ or, equivalently: : $\\forall g \\in G: S \\subseteq g^{-1} \\circ S \\circ g$"}
+{"_id": "23615", "title": "Definition:Normal Subgroup/Definition 2", "text": ": Every right coset of $N$ in $G$ is a left coset that is: : The right coset space of $N$ in $G$ equals its left coset space."}
+{"_id": "23616", "title": "Definition:Normal Subset/Definition 5", "text": ":$\\forall x, y \\in G: x \\circ y \\in S \\implies y \\circ x \\in S$"}
+{"_id": "23617", "title": "Definition:Normal Subgroup/Definition 3", "text": ":$\\forall g \\in G: g \\circ N \\circ g^{-1} \\subseteq N$ :$\\forall g \\in G: g^{-1} \\circ N \\circ g \\subseteq N$"}
+{"_id": "23618", "title": "Definition:Normal Subgroup/Definition 4", "text": ": $\\forall g \\in G: N \\subseteq g \\circ N \\circ g^{-1}$ : $\\forall g \\in G: N \\subseteq g^{-1} \\circ N \\circ g$"}
+{"_id": "23619", "title": "Definition:Normal Subgroup/Definition 5", "text": ":$\\forall g \\in G: g \\circ N \\circ g^{-1} = N$ :$\\forall g \\in G: g^{-1} \\circ N \\circ g = N$"}
+{"_id": "23620", "title": "Definition:Normal Subgroup/Definition 6", "text": ":$\\forall g \\in G: \\paren {n \\in N \\iff g \\circ n \\circ g^{-1} \\in N}$ :$\\forall g \\in G: \\paren {n \\in N \\iff g^{-1} \\circ n \\circ g \\in N}$"}
+{"_id": "23621", "title": "Definition:Normal Subgroup/Definition 7", "text": ":$N$ is a normal subset of $G$."}
+{"_id": "23622", "title": "Definition:Subset Product/Singleton", "text": "Let $A \\subseteq S$ be a subset of $S$. Then: :$(1): \\quad a \\circ S := \\set a \\circ S$ :$(2): \\quad S \\circ a := S \\circ \\set a$"}
+{"_id": "23623", "title": "Definition:Convex Angle", "text": "A '''convex angle''' is an angle that is larger than a zero angle and smaller than a straight angle. That is, the measurement of a '''convex angle''' is larger than $0 \\degrees$ and smaller than $180 \\degrees$. In radians, the measurement of a '''convex angle''' is larger than $0$ and smaller than $\\pi$."}
+{"_id": "23624", "title": "Definition:Inverse of Subset/Monoid", "text": "Let $\\struct {S, \\circ}$ be a monoid whose identity is $e_S$. Let $C \\subseteq S$ be the set of cancellable elements of $S$. Let $X \\subseteq C$. Then the '''inverse''' of the subset $X$ is defined as: :$X^{-1} = \\set {y \\in S: \\exists x \\in X: x \\circ y = e_S}$ That is, it is the set of all the inverses of all the elements of $X$."}
+{"_id": "23625", "title": "Definition:Inverse of Subset/Group", "text": "Let $\\struct {G, \\circ}$ be a group. Let $X \\subseteq G$. Then the '''inverse''' of the subset $X$ is defined as: :$X^{-1} = \\set {x \\in G: x^{-1} \\in X}$ or equivalently: :$X^{-1} = \\set {x^{-1}: x \\in X}$"}
+{"_id": "23626", "title": "Definition:Natural Deduction/Elementary Valid Argument Forms", "text": "In most treatments of PropLog various subsets of the following rules are treated as the axioms. Some of them are obvious. Others are more subtle. These rules are not all independent, in that it is possible to prove some of them using sequents constructed from combinations of others. However, when a set of proof rules is selected as the axioms for any particular treatment of this subject, those rules are usually selected carefully so that they ''are'' independent. === {{ProofRuleLink|Rule of Assumption}} === {{:Rule of Assumption/Proof Rule}} === {{ProofRuleLink|Rule of Conjunction}} === {{:Rule of Conjunction/Proof Rule}} === {{ProofRuleLink|Rule of Simplification}} === {{:Rule of Simplification/Proof Rule}} === {{ProofRuleLink|Rule of Addition}} === {{:Rule of Addition/Proof Rule}} === {{ProofRuleLink|Proof by Cases}} === {{:Proof by Cases/Proof Rule}} === {{ProofRuleLink|Modus Ponendo Ponens}} === {{:Modus Ponendo Ponens/Proof Rule}} === {{ProofRuleLink|Modus Tollendo Tollens}} === {{:Modus Tollendo Tollens/Proof Rule}} === {{ProofRuleLink|Modus Tollendo Ponens}} === {{:Modus Tollendo Ponens/Proof Rule}} === {{ProofRuleLink|Modus Ponendo Tollens}} === {{:Modus Ponendo Tollens/Proof Rule}} === {{ProofRuleLink|Rule of Implication}} === {{:Rule of Implication/Proof Rule}} === {{ProofRuleLink|Double Negation Introduction}} === {{:Double Negation Introduction/Proof Rule}} === {{ProofRuleLink|Double Negation Elimination}} === {{:Double Negation Elimination/Proof Rule}} === {{ProofRuleLink|Biconditional Introduction}} === {{:Biconditional Introduction/Proof Rule}} === {{ProofRuleLink|Biconditional Elimination}} === {{:Biconditional Elimination/Proof Rule}} === {{ProofRuleLink|Principle of Non-Contradiction}} === {{:Principle of Non-Contradiction/Proof Rule}} === {{ProofRuleLink|Proof by Contradiction}} === {{:Proof by Contradiction/Proof Rule}} === {{ProofRuleLink|Rule of Explosion}} === {{:Rule of Explosion/Proof Rule}} === {{ProofRuleLink|Law of Excluded Middle}} === {{:Law of Excluded Middle/Proof Rule}} === {{ProofRuleLink|Reductio ad Absurdum}} === {{:Reductio ad Absurdum/Proof Rule}}"}
+{"_id": "23627", "title": "Definition:Invariant Subset", "text": "Let $S$ be a set. Let $T \\subseteq S$ be a subset of $S$. Let $f: S \\to S$ be a self-map on $S$. Then $T$ is '''invariant under''' $f$ {{iff}}: :$\\forall t \\in T: \\map f t \\in T$"}
+{"_id": "23628", "title": "Definition:Inversion Mapping", "text": "Let $\\struct {G, \\circ}$ be a group. The '''inversion mapping''' on $G$ is the mapping $\\iota: G \\to G$ defined by: :$\\forall g \\in G: \\map \\iota g = g^{-1}$ That is, $\\iota$ assigns to an element of $G$ its inverse."}
+{"_id": "23629", "title": "Definition:Isomorphism of Categories", "text": "Let $\\mathbf C$ and $\\mathbf D$ be metacategories. Let $F: \\mathbf C \\to \\mathbf D$ be a functor. Then $F$ is an '''isomorphism (of categories)''' {{iff}} there exists a functor $G: \\mathbf C \\to \\mathbf D$ such that: :$G F: \\mathbf C \\to \\mathbf C$ is the identity functor $I_{\\mathbf C}$ :$F G: \\mathbf D \\to \\mathbf D$ is the identity functor $I_{\\mathbf D}$"}
+{"_id": "23630", "title": "Definition:Isomorphism of Categories/Isomorphic Categories", "text": "Let $F: \\mathbf C \\to \\mathbf D$ be an isomorphism of categories. Then $\\mathbf C$ and $\\mathbf D$ are said to be '''isomorphic''', and we write $\\mathbf C \\cong \\mathbf D$."}
+{"_id": "23631", "title": "Definition:Equivalence of Categories", "text": "Let $\\mathbf C$ and $\\mathbf D$ be metacategories. An '''equivalence of $\\mathbf C$ and $\\mathbf D$''' comprises: :Functors $F: \\mathbf C \\to \\mathbf D$ and $G: \\mathbf D \\to \\mathbf C$ :Natural isomorphisms $\\alpha: G F \\overset{\\sim}{\\longrightarrow} \\operatorname{id}_{\\mathbf C}$ and $\\beta: F G \\overset{\\sim}{\\longrightarrow} \\operatorname{id}_{\\mathbf D}$ === Equivalent Categories === {{:Definition:Equivalence of Categories/Equivalent Categories}}"}
+{"_id": "23632", "title": "Definition:Equivalence of Categories/Equivalent Categories", "text": "Let there exist an equivalence of categories between $\\mathbf C$ and $\\mathbf D$. Then $\\mathbf C$ and $\\mathbf D$ are said to be '''equivalent''', denoted $\\mathbf C \\simeq \\mathbf D$."}
+{"_id": "23633", "title": "Definition:Conjunction/Semantics of Conjunction", "text": "The '''conjunction''' is used to symbolise ''any'' statement in natural language such that two substatements are held to be true simultaneously. Thus it is also used to symbolise the concept of '''but''' as well as '''and'''. Thus $p \\land q$ can be also interpreted as: * '''$p$ ''and'' $q$''' * '''$p$ ''but'' $q$''' * '''$p$; ''however'', $q$''' * '''$p$; ''on the other hand'' $q$''' * '''''Not only'' $p$ ''but also'' $q$''' * '''''Despite'' $p$, $q$'''"}
+{"_id": "23634", "title": "Definition:Biconditional/Semantics of Biconditional", "text": "The concept of the biconditional has been defined such that $p \\iff q$ means: :'''If $p$ is true then $q$ is true, and if $q$ is true then $p$ is true.''' $p \\iff q$ can be considered as a ''shorthand'' to replace the use of the longer and more unwieldy expression involving two conditionals and a conjunction. If we refer to ways of expressing the conditional, we see that: * $q \\implies p$ can be interpreted as '''$p$ is true if $q$ is true''', and * $p \\implies q$ can be interpreted as '''$p$ is true only if $q$ is true'''. Thus we arrive at the usual way of reading '''$p \\iff q$''' which is: '''$p$ is true ''if and only if'' $q$ is true.''' This can also be said as: * '''The truth value of $p$ is ''equivalent'' to the truth value of $q$.''' * '''$p$ is ''equivalent'' to $q$.''' * '''$p$ and $q$ are ''equivalent''.''' * '''$p$ and $q$ are ''coimplicant''.''' * '''$p$ and $q$ are ''logically equivalent''.''' * '''$p$ and $q$ are ''materially equivalent''.''' * '''$p$ is true ''exactly when'' $q$ is true.''' * '''$p$ is true ''iff'' $q$ is true.''' This is another convenient and useful (if informal) shorthand which is catching on in the mathematical community."}
+{"_id": "23635", "title": "Definition:Free Set of Sets", "text": "Let $\\mathcal S$ be a set of sets. Then $\\mathcal S$ is a '''free set of sets''' {{iff}} the intersection of $\\mathcal S$ is empty. That is: :$\\displaystyle \\bigcap \\mathcal S = \\varnothing$"}
+{"_id": "23636", "title": "Definition:Gaussian Integral/One Variable", "text": "The '''Gaussian Integral''' (of one variable) is the following improper integral, considered as a real function: :$\\phi_1: \\R \\to \\R$: :$\\map {\\phi_1} x = \\displaystyle \\int_{\\mathop \\to -\\infty}^x \\frac 1 {\\sqrt {2 \\pi} } \\map \\exp {-\\frac {t^2} 2 } \\rd t$ where $\\exp$ is the real exponential function."}
+{"_id": "23637", "title": "Definition:Precise Refinement of Cover", "text": "Let: :$\\mathcal S = \\left\\{ {S_\\gamma: \\gamma \\in \\Gamma}\\right\\}$ be a cover of a set $X$. {{explain|What is $\\Gamma$? Is this an indexing set? If so, this needs to be noted.}} Also let: :$\\mathcal T = \\left\\{ {T_\\gamma: \\gamma \\in \\Gamma}\\right\\}$ be a cover of $X$. Then $\\mathcal T$ is a '''precise refinement''' of $\\mathcal S$ {{iff}}: :$\\forall \\gamma \\in \\Gamma: T_\\gamma \\subseteq S_\\gamma$ Category:Definitions/Covers oim53wijy4z95p55k1lpgihgipfd2lw"}
+{"_id": "23638", "title": "Definition:Fixed Set of Sets", "text": "Let $\\mathcal S$ be a set of sets. Then $\\mathcal S$ is a '''fixed set of sets''' {{iff}} the intersection of $\\mathcal S$ is non-empty. That is: :$\\displaystyle \\bigcap \\mathcal S \\ne \\varnothing$"}
+{"_id": "23639", "title": "Definition:Gaussian Integral/Two Variables", "text": "The '''Gaussian Integral''' (of two variables) is the following definite integral, considered as a real-valued function: :$\\phi_2: \\left\\{{\\left({a, b}\\right) \\in \\R^2: a \\le b}\\right\\} \\to \\R$:  :$\\phi_2 \\left({a, b}\\right) = \\displaystyle \\int_a^b \\frac 1 {\\sqrt{2 \\pi}} \\exp \\left({- \\frac {t^2} 2}\\right) \\ \\mathrm d t$ where $\\exp$ is the real exponential function."}
+{"_id": "23640", "title": "Definition:Preorder Category/Definition 1", "text": "Let $\\left({S, \\precsim}\\right)$ be a preordered set. One can interpret $\\left({S, \\precsim}\\right)$ as being a category, with: {{DefineCategory |ob = The elements of $S$ |mor = Precisely one morphism $a \\to b$ for every $a, b \\in S$ with $a \\precsim b$ }} More formally, we let the morphisms be the elements of the relation ${\\precsim} \\subseteq S \\times S$. Thus, $a \\to b$ in fact denotes the ordered pair $\\left({a, b}\\right)$. The category that so arises is called a '''preorder category'''."}
+{"_id": "23641", "title": "Definition:Preorder Category/Definition 2", "text": "Let $\\mathbf C$ be a metacategory. Then $\\mathbf C$ is a '''preorder category''' {{iff}} :For all objects $C, C'$ of $\\mathbf C$, there is at most one morphism $f: C \\to C'$"}
+{"_id": "23642", "title": "Definition:Order Category/Definition 1", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. One can interpret $\\left({S, \\preceq}\\right)$ as being a category, with: {{DefineCategory |ob = The elements of $S$ |mor = Precisely one morphism $a \\to b$ for every $a, b \\in S$ with $a \\preceq b$ }} More formally, we let the morphisms be the elements of the relation ${\\preceq} \\subseteq S \\times S$. Thus, $a \\to b$ in fact denotes the ordered pair $\\left({a, b}\\right)$. The category that so arises is called an '''order category'''."}
+{"_id": "23643", "title": "Definition:Order Category/Definition 2", "text": "Let $\\mathbf C$ be a metacategory. Then $\\mathbf C$ is an '''order category''' iff: :For all objects $C, C'$ of $\\mathbf C$, there is at most one morphism $f: C \\to C'$ :Whenever $f: C \\to C'$ is an isomorphism, $C = C'$ Thus, an '''order category''' is a skeletal preorder category."}
+{"_id": "23644", "title": "Definition:Biconditional/Semantics of Biconditional/Necessary and Sufficient", "text": "Let: :$p \\iff q$ where $\\iff$ denotes the biconditional operator. Then it can be said that '''$p$ is ''necessary and sufficient'' for $q$.''' This is a consequence of the definitions of necessary and sufficient conditions."}
+{"_id": "23645", "title": "Definition:Star Convex Set", "text": "Let $V$ be a vector space over $\\R$ or $\\C$. A subset $A \\subseteq V$ is said to be '''star convex''' iff there exists $a \\in A$ such that: :$\\forall x \\in A: \\forall t \\in \\left[{0 \\,.\\,.\\, 1}\\right]: t x + \\left({1 - t}\\right) a \\in A$ === Star Center === {{:Definition:Star Convex Set/Star Center}} A '''star convex set''' can thus be described as a set containing all line segments between the '''star center''' and an element of the set."}
+{"_id": "23647", "title": "Definition:Jordan Curve/Interior", "text": "It follows from the Jordan Curve Theorem that $\\R^2 \\setminus \\Img f$ is a union of two disjoint connected components, one of which is bounded. This bounded component is called the '''interior of $f$''', and is denoted as $\\Int f$."}
+{"_id": "23648", "title": "Definition:Jordan Curve/Exterior", "text": "It follows from the Jordan Curve Theorem that $\\R^2 \\setminus \\operatorname{Im} \\left({f}\\right)$ is a union of two disjoint connected components, one of which is unbounded. This unbounded component is called the '''exterior of $f$''', and is denoted as $\\operatorname{Ext} \\left({f}\\right)$."}
+{"_id": "23649", "title": "Definition:Crossing (Jordan Curve)", "text": "Let $P$ be a polygon embedded in $\\R^2$. Let $q \\in \\R^2 \\setminus \\partial P$. Let $\\mathbf v \\in R^2 \\setminus \\set {\\mathbf 0}$ be a non-zero vector. Let $\\LL = \\set {q + s \\mathbf v: s \\in \\R_{\\ge 0} }$ be a ray with start point $q$. Then $\\LL \\cap \\partial P$ consists of a finite number of line segments, where $\\partial P$ denotes the boundary of $P$. As two adjacent sides in $P$ do not form a straight angle by the definition of polygon, each line segment is either a single point or an entire side of $P$. Each of these line segments is called a '''crossing''' {{iff}} the line segment is one of these: :a single point which is not a vertex of $P$ :a single vertex of $P$, and its adjacent sides lie on opposite sides of $\\LL$ :a side $S$ of $P$, and the two sides adjacent to $S$ lie on opposite sides of $\\LL$. === Parity === {{:Definition:Crossing (Jordan Curve)/Parity}}"}
+{"_id": "23650", "title": "Definition:Crossing (Jordan Curve)/Parity", "text": "Let $\\map N q$ be the number of '''crossings''' between $\\LL$ and the boundary $\\partial P$ of $P$. Then the '''parity of $q$''' is defined as: :$\\map {\\operatorname{par} } q := \\map N q \\bmod 2$."}
+{"_id": "23651", "title": "Definition:Local Sub-Basis", "text": "Let $\\left({X, \\tau}\\right)$ be a topological space. Let $x \\in X$. Then a '''local sub-basis''' of $x$ relative to $\\tau$ is a neighborhood sub-basis of $x$ consisting of open sets."}
+{"_id": "23652", "title": "Definition:Neighborhood Sub-Basis", "text": "Let $\\struct {S, \\tau}$ be a topological space. Let $x \\in S$. Let $\\BB$ be a set of neighborhoods of $x$. Then $\\BB$ is a '''neighborhood sub-basis of $x$ relative to $\\tau$''' {{iff}}: :for each neighborhood $N$ of $x$, there exists a finite subset $K$ of $\\BB$ such that $\\bigcap K \\subseteq N$."}
+{"_id": "23653", "title": "Definition:Star Convex Set/Star Center", "text": "The point $a \\in A$ is called a '''star center of $A$'''."}
+{"_id": "23654", "title": "Definition:Sublattice", "text": "Let $\\left({L,\\wedge_L,\\vee_L,\\preceq_L}\\right)$ be a lattice. Let $S$ be a subset of $L$ and let $\\wedge_S$, $\\vee_S$, and $\\preceq_S$ be the restrictions to $S$ of $\\wedge_L$, $\\vee_L$, and $\\preceq_L$, respectively. Then $\\left({S,\\wedge_S,\\vee_S,\\preceq_S}\\right)$ is a '''sublattice''' of $\\left({L,\\wedge_L,\\vee_L,\\preceq_L}\\right)$ iff $S$ is closed under  $\\wedge_S$ and $\\vee_S$. If in addition $L$ is a bounded lattice and its top and bottom elements are in $S$, then $S$ is called a '''$0,1$-sublattice''' of $L$. If in addition $L$ is a complete lattice and for each subset $T$ of $S$, $\\sup_L T,\\inf_L T \\in S$, then $S$ is called a '''complete sublattice''' of $L$."}
+{"_id": "23655", "title": "Definition:Generalized Ordered Space", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. Let $\\tau$ be a topology for $S$. === Definition 1 === {{:Definition:Generalized Ordered Space/Definition 1}} === Definition 2 === {{:Definition:Generalized Ordered Space/Definition 2}} === Definition 3 === {{:Definition:Generalized Ordered Space/Definition 3}}"}
+{"_id": "23656", "title": "Definition:Generalized Ordered Space/Definition 2", "text": "$\\left({S, \\preceq, \\tau}\\right)$ is a '''generalized ordered space''' {{iff}}: :$(1): \\quad$ there exists a linearly ordered space $\\left({S', \\preceq', \\tau'}\\right)$ :$(2): \\quad$ there exists a mapping $\\phi: S \\to S'$ such that $\\phi$ is both an order embedding and a topological embedding."}
+{"_id": "23657", "title": "Definition:Generalized Ordered Space/Definition 1", "text": "$\\left({S, \\preceq, \\tau}\\right)$ is a '''generalized ordered space''' {{iff}}: :$(1): \\quad \\left({S, \\tau}\\right)$ is a Hausdorff space :$(2): \\quad$ there exists a basis for $\\left({S, \\tau}\\right)$ whose elements are convex in $S$."}
+{"_id": "23658", "title": "Definition:Argumentum ad Hominem", "text": "An '''argumentum ad hominem''' is a logical argument that concludes the falsehood of a statement because it is made by a particular person or group of people. It can be argued that it is reasonable to be suspicious of an argument if most of the assertions made by said person or group regarding the topic under discussion are false. However, such reasoning is still fallacious."}
+{"_id": "23659", "title": "Definition:Argumentum ad Baculum", "text": "An '''argumentum ad baculum''' is a logical argument that, rather than prove or present evidence for a claim, threatens any who dare argue with the person or group making the claim. The presence of such a threat is often a good reason to avoid ''stating'' an opposing view, but manifestly does not support the truth or falsity of the claim under consideration."}
+{"_id": "23660", "title": "Definition:Conjunction/General Definition", "text": "Let $p_1, p_2, \\ldots, p_n$ be statements. The '''conjunction''' of $p_1, p_2, \\ldots, p_n$ is defined as: :$\\displaystyle \\bigwedge_{i \\mathop = 1}^n \\ p_i = \\begin{cases} p_1 & : n = 1 \\\\ & \\\\ \\displaystyle \\left({\\bigwedge_{i \\mathop = 1}^{n-1} \\ p_i}\\right) \\land p_n & : n > 1 \\end{cases}$ That is: :$\\displaystyle \\bigwedge_{i \\mathop = 1}^n \\ p_i = p_1 \\land p_2 \\land \\cdots \\land p_{n-1} \\land p_n$ In terms of the set $P = \\left\\{{p_1, \\ldots, p_n}\\right\\}$, this can also be rendered: :$\\displaystyle \\bigwedge P$ and is referred to as the '''conjunction of $P$'''."}
+{"_id": "23661", "title": "Definition:Disjunction/General Definition", "text": "Let $p_1, p_2, \\ldots, p_n$ be statements. The '''disjunction''' of $p_1, p_2, \\ldots, p_n$ is defined as: :$\\displaystyle \\bigvee_{i \\mathop = 1}^n \\ p_i = \\begin{cases} p_1 & : n = 1 \\\\ & \\\\ \\displaystyle \\paren {\\bigvee_{i \\mathop = 1}^{n - 1} \\ p_i} \\lor p_n & : n > 1 \\end{cases}$ That is: :$\\displaystyle \\bigvee_{i \\mathop = 1}^n \\ p_i = p_1 \\lor p_2 \\lor \\cdots \\lor p_{n - 1} \\lor p_n$ In terms of the set $P = \\set {p_1, \\ldots, p_n}$ this can also be rendered: :$\\displaystyle \\bigvee P$ and is referred to as the '''disjunction of $P$'''."}
+{"_id": "23662", "title": "Definition:Countable Set/Countably Infinite", "text": "{{:Definition:Countable Set/Countably Infinite/Definition 1}}"}
+{"_id": "23663", "title": "Definition:Logical Not/Boolean Interpretation", "text": "The truth value of $\\neg \\mathbf A$ under a boolean interpretation $v$ is given by: :$\\map v {\\neg \\mathbf A} = \\begin{cases} \\T & : \\map v {\\mathbf A} = \\F \\\\ \\F & : \\map v {\\mathbf A} = \\T \\end{cases}$"}
+{"_id": "23664", "title": "Definition:Collation", "text": "A '''collation''' is a structured alignment with certain placeholders that underpins the construction of formal languages. These placeholders may be replaced by elements of an alphabet $\\mathcal A$ under consideration. A '''collation in $\\mathcal A$''' is one where all placeholders are replaced by symbols from $\\mathcal A$. For example, if we take $\\square$ to denote a placeholder, then $\\square\\square\\square\\square\\square$ represents the collation \"a word of length $5$\". We can see that then the word \"sheep\" is an instance of the '''collation''' \"a word of length $5$\" '''in the English alphabet''', as is \"axiom\". Typical examples of '''collations''' encountered in mathematics are words or structured graphics like labeled trees. {{transclude:Definition:Collation/Collation System   |section = def   |title = Collation System   |header = 3   |link = true }} {{transclude:Definition:Collation/Unique Readability|   |section = def   |title = Unique Readability   |header = 3   |link = true }}"}
+{"_id": "23666", "title": "Definition:Conditional/Boolean Interpretation", "text": "The truth value of $\\mathbf A \\implies \\mathbf B$ under a boolean interpretation $v$ is given by: :$\\map v {\\mathbf A \\implies \\mathbf B} = \\begin{cases} \\T & : \\map v {\\mathbf A} = \\F \\text{ or } \\map v {\\mathbf B} = \\T \\\\ \\F & : \\text{otherwise} \\end{cases}$ and the truth value of $\\mathbf A \\impliedby \\mathbf B$ under a boolean interpretation $v$ is given by: :$\\map v {\\mathbf A \\impliedby \\mathbf B} = \\begin{cases} \\T & : \\map v {\\mathbf A} = \\T \\text{ or } \\map v {\\mathbf B} = \\F \\\\ \\F & : \\text{otherwise} \\end{cases}$"}
+{"_id": "23667", "title": "Definition:Conjunction/Boolean Interpretation", "text": "The truth value of $\\mathbf A \\land \\mathbf B$ under a boolean interpretation $v$ is given by: :$\\map v {\\mathbf A \\land \\mathbf B} = \\begin{cases} \\T & : \\map v {\\mathbf A} = \\map v {\\mathbf B} = \\T \\\\ \\F & : \\text{otherwise} \\end{cases}$"}
+{"_id": "23668", "title": "Definition:Disjunction/Boolean Interpretation", "text": "The truth value of $\\mathbf A \\lor \\mathbf B$ under a boolean interpretation $v$ is given by: :$\\map v {\\mathbf A \\lor \\mathbf B} = \\begin{cases} \\F & : \\map v {\\mathbf A} = \\map v {\\mathbf B} = \\F \\\\ \\T & : \\text{otherwise} \\end{cases}$"}
+{"_id": "23669", "title": "Definition:Biconditional/Boolean Interpretation", "text": "The truth value of $\\mathbf A \\iff \\mathbf B$ under a boolean interpretation $v$ is given by: :$\\map v {\\mathbf A \\iff \\mathbf B} = \\begin{cases} \\T & : \\map v {\\mathbf A} = \\map v {\\mathbf B} \\\\ \\F & : \\text{otherwise} \\end{cases}$"}
+{"_id": "23670", "title": "Definition:Exclusive Or/Boolean Interpretation", "text": "The truth value of $\\mathbf A \\oplus \\mathbf B$ under a boolean interpretation $v$ is given by: :$v \\left({\\mathbf A \\oplus \\mathbf B}\\right) = \\begin{cases} F & : v \\left({\\mathbf A}\\right) = v \\left({\\mathbf B}\\right) \\\\ T & : \\text{otherwise} \\end{cases}$"}
+{"_id": "23671", "title": "Definition:Order Automorphism", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $\\phi: S \\to S$ be an order isomorphism from $\\left({S, \\preceq}\\right)$ to $\\left({S, \\preceq}\\right)$. Then $\\phi$ is an '''(order) automorphism'''."}
+{"_id": "23672", "title": "Definition:Logical NAND/Boolean Interpretation", "text": "The truth value of $\\mathbf A \\uparrow \\mathbf B$ under a boolean interpretation $v$ is given by: :$v \\left({\\mathbf A \\uparrow \\mathbf B}\\right) = \\begin{cases} T & : v \\left({\\mathbf A}\\right) = F \\text{ or } v \\left({\\mathbf B}\\right) = F\\\\ F & : \\text{otherwise} \\end{cases}$"}
+{"_id": "23673", "title": "Definition:Logical NOR/Boolean Interpretation", "text": "The truth value of $\\mathbf A \\downarrow \\mathbf B$ under a boolean interpretation $v$ is given by: :$v \\left({\\mathbf A \\downarrow \\mathbf B}\\right) = \\begin{cases} T & : v \\left({\\mathbf A}\\right) = v \\left({\\mathbf B}\\right) = F\\\\ F & : \\text{otherwise} \\end{cases}$"}
+{"_id": "23674", "title": "Definition:Dual Isomorphism (Order Theory)", "text": "Let $\\left({S, \\preceq_S}\\right)$ and $\\left({T, \\preceq_T}\\right)$ be ordered sets. Let $\\phi:S \\to T$ be a bijection. Then $\\phi$ is a '''dual isomorphism''' between $\\left({S, \\preceq_S}\\right)$ and $\\left({T, \\preceq_T}\\right)$ iff $\\phi$ and $\\phi^{-1}$ are decreasing mappings. If there is a dual isomorphism between $\\left({S, \\preceq_S}\\right)$ and $\\left({T, \\preceq_T}\\right)$, then $\\left({S, \\preceq_S}\\right)$ is '''dual''' to $\\left({T, \\preceq_T}\\right)$. Equivalently, $\\left({S, \\preceq_S}\\right)$ is '''dual''' to $\\left({T, \\preceq_T}\\right)$ iff $S$ with the dual ordering is isomorphic to $T$."}
+{"_id": "23675", "title": "Definition:Self-Dual (Order Theory)", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. If $S$ is dual to $S$, then $S$ is '''self-dual'''."}
+{"_id": "23676", "title": "Definition:Labeled Tree", "text": "A '''labeled tree''' is a finite rooted tree in which each node has attached to it a '''label'''. These '''labels''' may be any symbol or other object that one can think of, but usually consist of one letter or symbol. {{expand|A picture like [http://mathworld.wolfram.com/images/eps-gif/PrueferCode_700.gif this], exemplifying the below}} The parent function is usually depicted by placing the ancestor above the child and connecting them with a line. The various children of a node may also be considered to be in a particular order. This is usually depicted by placing them in a left-to-right order. Because of the intuitive clarity of these relations, '''labeled trees''' may be considered collations as basic as strings are. However, because of the space limitations that most printed media have to take into account, it is usual that one quickly resorts to representing trees by strings in one way or another. Category:Definitions/Formal Systems g30ri96v5v2842jfjrgnt4jzs1jw7iq"}
+{"_id": "23677", "title": "Definition:Upper Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $U \\subseteq S$."}
+{"_id": "23678", "title": "Definition:Lower Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $L \\subseteq S$. Then $L$ is a '''lower set''' in $S$ {{iff}}: : For all $l \\in L$ and $s \\in S$: if $s \\preceq l$ then $s \\in L$. That is, $L$ is a '''lower set''' {{iff}} it contains its own lower closure."}
+{"_id": "23679", "title": "Definition:Generalized Ordered Space/Definition 3", "text": "$\\left({S, \\preceq, \\tau}\\right)$ is a '''generalized ordered space''' {{iff}}: :$(1): \\quad \\left({S, \\tau}\\right)$ is a Hausdorff space :$(2): \\quad$ there exists a sub-basis for $\\left({S, \\tau}\\right)$ each of whose elements is an upper set or lower set in $S$."}
+{"_id": "23680", "title": "Definition:Singular Statement/Individuating Description", "text": "An '''individuating description''' is a predicate whose purpose is to uniquely identify a particular object."}
+{"_id": "23681", "title": "Definition:In-Order Traversal of Labeled Tree", "text": "Let $T$ be a binary labeled tree. '''In-order traversal''' of $T$ is an algorithm designed to obtain a string representation of $T$. The steps are as follows: === Variant 1 === {{:Definition:In-Order Traversal of Labeled Tree/Variant 1}} === Variant 2 === {{:Definition:In-Order Traversal of Labeled Tree/Variant 2}}"}
+{"_id": "23682", "title": "Definition:In-Order Traversal of Labeled Tree/Variant 1", "text": "$\\mathtt{Inorder} (T):$ * $n \\gets t$, where $t$ is the root node of $T$. * If $n$ is a leaf node, output the label of $n$, and stop. * Let $T_1$ and $T_2$ be the left and right subtrees of $T$. * If $n$ has only one child, skip this step. Output $\\mathtt{Inorder} (T_1)$. * Output the label of $n$. * Output $\\mathtt{Inorder} (T_2)$. * Stop."}
+{"_id": "23683", "title": "Definition:In-Order Traversal of Labeled Tree/Variant 2", "text": "$\\mathtt{Inorder} (T):$ * $n \\gets t$, where $t$ is the root node of $T$. * If $n$ is a leaf node, output the label of $n$, and stop. * Let $T_1$ and $T_2$ be the left and right subtrees of $T$. * Output a left bracket $($. * If $n$ has only one child, skip this step. Output $\\mathtt{Inorder} (T_1)$. * Output the label of $n$. * Output $\\mathtt{Inorder} (T_2)$. * Output a right bracket $)$. * Stop."}
+{"_id": "23684", "title": "Definition:Pre-Order Traversal of Labeled Tree", "text": "Let $T$ be a binary labeled tree. '''Pre-order traversal''' of $T$ is an algorithm designed to obtain a string representation of $T$. The steps are as follows: $\\mathtt{Preorder} (T):$ * $n \\gets t$, where $t$ is the root node of $T$. * Output the label of $n$. * If $n$ is a leaf node, stop. * Let $T_1$ and $T_2$ be the left and right subtrees of $T$. * If $n$ has only one child, skip this step. Output $\\mathtt{Preorder} (T_1)$. * Output $\\mathtt{Preorder} (T_2)$. * Stop. The resulting string will be in Polish notation."}
+{"_id": "23685", "title": "Definition:Polish Notation", "text": "'''Polish notation''' is a method of writing expressions without the need for parentheses. The main method in writing an expression in '''Polish notation''' is to output an operator $\\Box$, directly followed by the output of all its arguments $p, q, \\ldots$, yielding: :$\\Box p q \\ldots$ If we had another, operator $\\diamond$ taking only one argument, then also: :$\\Box {\\diamond} p {\\diamond} q \\ldots$ is in '''Polish notation'''. Note that for the applicability of '''Polish notation''', it is vital to be able to efficiently distinguish an argument from an operator, or a sequence of arguments. === Formal Definition === {{:Definition:Polish Notation/Formal Definition}} === Reverse Polish Notation === {{:Definition:Polish Notation/Reverse Polish Notation}} {{expand|Could benefit from being defined in BNF or otherwise formally, as an exercise in formal languages}}"}
+{"_id": "23688", "title": "Definition:Many-to-One Relation/Defined/Element", "text": "Let $f \\subseteq S \\times T$ be a many-to-one relation. <section notitle=\"true\" name=\"tc-defn\"> Let $s \\in S$. Then $f$ is '''defined at $s$''' {{iff}} $s \\in \\Dom f$, the domain of $f$. </section>"}
+{"_id": "23689", "title": "Definition:Many-to-One Relation/Defined/Set", "text": "Let $f \\subseteq S \\times T$ be a many-to-one relation. <section notitle=\"true\" name=\"tc-defn\"> Let $R \\subseteq S$. Then $f$ is '''defined on $R$''' {{iff}} it is defined at all $r \\in R$. Equivalently, {{iff}} $R \\subseteq \\Dom f$, the domain of $f$. </section>"}
+{"_id": "23690", "title": "Definition:Logical Not/Notational Variants", "text": "Various symbols are encountered that denote the concept of the logical not: {| border=\"1\" cellpadding=\"4\" cellspacing=\"0\" style=\"font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $\\neg p$ |   |   |- | $\\mathsf{NOT}\\ p$ |   |   |- | $\\sim p$ or $\\tilde p$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} | '''tilde''' or '''curl''' |- | $- p$ |  |   |- | $\\bar p$ |   | '''Bar $p$''' |- | $p'$ |   | '''$p$ prime''' or '''$p$ complement''' |- | $! p$ |   | '''Bang $p$''' |- | $\\operatorname{N} p$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |  |}"}
+{"_id": "23691", "title": "Definition:Deductive Method", "text": "'''The deductive method''' is a general term for the process of building a logical argument from a (usually) limited number of premises. Thus from a small number of statements which have been accepted as true, and a small number of rules of deduction, a theoretically unlimited number of further statements are determined to be true as a result."}
+{"_id": "23692", "title": "Definition:Conjunction/Notational Variants", "text": "Various symbols are encountered that denote the concept of logical conjunction: {| border=\"1\" cellpadding=\"4\" cellspacing=\"0\" style=\"font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \\land q$ |   | '''wedge''' |- | $p\\ \\mathsf{AND} \\ q$ |   |   |- | $p \\ . \\ q$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} | '''dot''' |- | $p \\ \\And \\ q$ |  | '''Ampersand''' |- | $\\operatorname K p q$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |  |}"}
+{"_id": "23693", "title": "Definition:Conditional/Notational Variants", "text": "Various symbols are encountered that denote the concept of the conditional: {| border=\"1\" cellpadding=\"4\" cellspacing=\"0\" style=\"font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \\implies q$ |  | '''Implies''' |- | $p \\to q$ |   | often used when space is limited |- | $p \\supset q$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} | '''hook''' or '''horseshoe''' |- | $p \\, \\mathop {-\\!\\!\\!<} q$ | {{AuthorRef|Charles Sanders Peirce}} | '''sign of illation''' |- | $\\operatorname C p q$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |  |} In mathematics, as opposed to works concerned purely with logic, it is usual to use \"$\\implies$\", as then it can be ensured that it is understood to mean exactly the same thing when we use it in the \"mathematical\" context. There are other uses in mathematics for the other symbols."}
+{"_id": "23694", "title": "Definition:Biconditional/Notational Variants", "text": "Various symbols are encountered that denote the concept of biconditionality: {| border=\"1\" cellpadding=\"4\" cellspacing=\"0\" style=\"font-weight:bold; text-align:center; |- ! Symbol  ! Origin  |- | $p \\iff q$ |   |- | $p\\ \\mathsf{EQ} \\ q$ |   |- | $p \\equiv q$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} |- | $p = q$ | |- | $p \\leftrightarrow q$ | |- | $\\operatorname E p q$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |} It is usual in mathematics to use $\\iff$, as there are other uses for the other symbols."}
+{"_id": "23695", "title": "Definition:Universal Quantifier/Notational Variants", "text": "Various symbols are encountered that denote the concept of universal quantifier: {| border=\"1\" cellpadding=\"4\" cellspacing=\"0\" style=\"font-weight:bold; text-align:center; |- ! Symbol  ! Origin  |- | $\\forall x$ | {{AuthorRef|Gerhard Gentzen}}: ''Untersuchungen über das logische Schließen'' (1935) |- | $\\paren x$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} |- | $\\Pi x$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |- | $\\wedge x$ or $\\bigwedge x$ |  |- | $\\ds \\operatorname{\\Large {\\textsf A} } \\limits_{x, y, \\dotsc}$ | {{BookReference|Introduction to Logic and to the Methodology of Deductive Sciences|1946|Alfred Tarski}} |}"}
+{"_id": "23696", "title": "Definition:Disjunction/Notational Variants", "text": "Various symbols are encountered that denote the concept of disjunction: {| border=\"1\" cellpadding=\"4\" cellspacing=\"0\" style=\"font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \\lor q$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} | '''vee''' or '''vel''' |- | $p\\ \\mathsf{OR} \\ q$ |   |   |- | $p + q$ |   |   |- | $\\operatorname A p q$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |   |}"}
+{"_id": "23697", "title": "Definition:Existential Quantifier/Notational Variants", "text": "Various symbols are encountered that denote the concept of existential quantifier: {| border=\"1\" cellpadding=\"4\" cellspacing=\"0\" style=\"font-weight:bold; text-align:center; |- ! Symbol  ! Origin  |- | $\\exists x$ | {{AuthorRef|Giuseppe Peano}}: ''Formulario Mathematico (2nd ed.)'' (1896) |- | $\\Sigma x$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |- | $\\lor x$ or $\\bigvee x$ |  |- | $\\ds \\operatorname {\\Large {\\textsf E} } \\limits_{x, y \\dotsc}$ | {{BookReference|Introduction to Logic and to the Methodology of Deductive Sciences|1946|Alfred Tarski}} |}"}
+{"_id": "23698", "title": "Definition:Exclusive Or/Notational Variants", "text": "Various symbols are encountered that denote the concept of exclusive or: {| border=\"1\" cellpadding=\"4\" cellspacing=\"0\" style=\"font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \\oplus q$ | | sometimes called '''o-plus''' |- | $p\\ \\mathsf{XOR} \\ q$ | | |- | $p + q$ | | |- | $p \\not \\Leftrightarrow q$ | | |- | $p \\not \\equiv q$ | | |- | $p \\ne q$ | | |- | $p \\ \\dot \\lor \\ q$ | | |- | $p \\ \\_ \\lor \\ q$ | | |}"}
+{"_id": "23699", "title": "Definition:Logical NAND/Notational Variants", "text": "Various symbols are encountered that denote the concept of logical NAND: {| border=\"1\" cellpadding=\"4\" cellspacing=\"0\" style=\"font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \\mid q$ | {{AuthorRef|Henry Maurice Sheffer|Henry Sheffer}} | Sheffer stroke |- | $p \\uparrow q$ |  | Also sometimes referred to as the '''Sheffer stroke''' |- | $p \\ \\mathsf{NAND} \\ q$ |  |   |- | $p / q$ |{{BookReference|Introduction to Symbolic Logic|1959|A.H. Basson|author2 = D.J. O'Connor}} | who refer to it as the '''stroke function''' |- | $p \\bar \\curlywedge q$ | {{AuthorRef|Charles Sanders Peirce}} | Modified Ampheck |} The all-uppercase rendition NAND originates from the digital electronics industry, where, because NAND is Functionally Complete, this operator has a high importance."}
+{"_id": "23700", "title": "Definition:Logical NOR/Notational Variants", "text": "Various symbols are encountered that denote the concept of logical NOR: {| border=\"1\" cellpadding=\"4\" cellspacing=\"0\" style=\"font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \\downarrow q$ | {{AuthorRef|Willard Van Orman Quine|Willard Quine}} | Quine arrow |- | $p \\ \\mathsf{NOR} \\ q$ |   |   |- | $p \\mathop \\bot q$ |  | |- | $p \\curlywedge q$ | {{AuthorRef|Charles Sanders Peirce}} | Ampheck |} The all-uppercase rendition NOR originates from the digital electronics industry, where, because NOR is Functionally Complete, this operator has a high importance."}
+{"_id": "23701", "title": "Definition:Axiom/Logic", "text": "An '''axiom''' in logic is a statement which is taken as '''self-evident'''. Note, however, that there has been disagreement for as long as there have been logicians and philosophers as to whether particular statements are true or not. For example, the Law of Excluded Middle is accepted as axiomatic by philosophers and logicians of the Aristotelian school but is denied by the intuitionist school."}
+{"_id": "23702", "title": "Definition:Axiom/Formal Systems", "text": "Let $\\mathcal L$ be a formal language. Part of defining a proof system $\\mathscr P$ for $\\mathcal L$ is to specify its '''axioms'''. An '''axiom of $\\mathscr P$''' is a well-formed formula of $\\mathcal L$ that $\\mathscr P$ approves of by definition."}
+{"_id": "23703", "title": "Definition:Axiom/Mathematics", "text": "The term '''axiom''' is used throughout the whole of mathematics to mean a statement which is accepted as true for that particular branch. Different fields of mathematics usually have different sets of statements which are considered as being '''axiomatic'''. So statements which are taken as axioms in one branch of mathematics may be theorems, or irrelevant, in others."}
+{"_id": "23705", "title": "Definition:Strictly Precede/Definition 2", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $a \\preceq b$ such that $a \\ne b$. Then '''$a$ strictly precedes $b$'''."}
+{"_id": "23706", "title": "Definition:Strictly Precede/Definition 1", "text": "Let $\\left({S, \\prec}\\right)$ be a strictly ordered set. Let $a, b \\in S$ and $a \\prec b$. Then '''$a$ strictly precedes $b$'''."}
+{"_id": "23707", "title": "Definition:Strictly Ordered Set", "text": "A '''strictly ordered set''' is a relational structure $\\left({S, \\prec}\\right)$ such that the relation $\\prec$ is an strict ordering."}
+{"_id": "23708", "title": "Definition:Strictly Well-Ordered Set", "text": "{{expand|Add a definition to the effect that a '''strictly ordered set''' is a relational structure where the relation is a strict well-ordering.}} Let $\\left({S, \\prec}\\right)$ be a strictly totally ordered set. Then $\\left({S, \\prec}\\right)$ is a '''strictly well-ordered set''' {{iff}} $\\prec$ is a foundational relation."}
+{"_id": "23709", "title": "Definition:Minimal/Relation", "text": "Let $\\left({S, \\mathcal R}\\right)$ be a relational structure. Let $T \\subseteq S$ be a subset of $S$. An element $x \\in T$ is an '''$\\mathcal R$-minimal element of $T$''' {{iff}}: :$\\forall y \\in T: y \\not \\mathrel {\\mathcal R} x$"}
+{"_id": "23710", "title": "Definition:Maximal/Relation", "text": "Let $\\left({S, \\mathcal R}\\right)$ be a relational structure. Let $T \\subseteq S$ be a subset of $S$. An element $x \\in T$ is an '''$\\mathcal R$-maximal element of $T$''' {{iff}}: :$\\forall y \\in T: x \\not \\mathrel {\\mathcal R} y$"}
+{"_id": "23711", "title": "Definition:Dual Order Embedding", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be ordered sets. A '''dual order embedding''' is a mapping $\\phi: S \\to T$ such that: :$\\forall x, y \\in S: x \\preceq_1 y \\iff \\phi \\left({y}\\right) \\preceq_2 \\phi \\left({x}\\right)$ That is: : if $\\phi$ is an order embedding of $\\left({S, \\preceq_1}\\right)$ into $\\left({T, \\succeq_2}\\right)$ where $\\succeq_2$ is the dual of $\\preceq_2$. Category:Definitions/Order Embeddings fyhwwnrx9b6ea2fmrab1iyhqq1jglui"}
+{"_id": "23715", "title": "Definition:Backus-Naur Form/Non-Terminal", "text": "Symbols that appear on the {{LHS}} of a specification in Backus-Naur form are called '''non-terminals.'''"}
+{"_id": "23716", "title": "Definition:Backus-Naur Form/Terminal", "text": "Symbols that never appear on the {{LHS}} of a specification in Backus-Naur form are called '''terminals.'''"}
+{"_id": "23717", "title": "Definition:Polish Notation/Reverse Polish Notation", "text": "For stack-based programming languages, '''reverse Polish language''' is a useful variant of Polish notation, because it naturally coincides with how the input is to be structured for the language. As the name suggests, a string $\\mathsf P$ is in '''reverse Polish language''' {{iff}} reversing it gives a string $\\tilde {\\mathsf P}$ in Polish notation. Thus the '''reverse Polish language''' equivalents of these examples of Polish notation: :$\\Box p q \\ldots$ :$\\Box {\\diamond} p {\\diamond} q \\ldots$ are: :$\\ldots q p \\Box$ :$\\ldots q {\\diamond} p {\\diamond} \\Box$"}
+{"_id": "23718", "title": "Definition:Top (Logic)", "text": "'''Top''' is a constant of propositional logic interpreted to mean the canonical, undoubted tautology whose truth nobody could possibly ever question. The symbol used is $\\top$."}
+{"_id": "23719", "title": "Definition:Bottom (Logic)", "text": "'''Bottom''' is a constant of propositional logic interpreted to mean the canonical, undoubted contradiction whose falsehood nobody could possibly ever question. The symbol used is $\\bot$."}
+{"_id": "23720", "title": "Definition:Top (Logic)/Boolean Interpretation", "text": "There is only one boolean interpretation for $\\top$: :$\\map v \\top = \\T$ where $\\T$ symbolises true."}
+{"_id": "23721", "title": "Definition:Bottom (Logic)/Boolean Interpretation", "text": "There is only one boolean interpretation for $\\bot$: :$\\map v \\bot = \\F$ where $\\F$ symbolises false."}
+{"_id": "23722", "title": "Definition:Order Topology/Definition 2", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. Define: :$\\map {\\Uparrow} S = \\set {s^\\succ: s \\in S}$ :$\\map {\\Downarrow} S = \\set {s^\\prec: s \\in S}$ where $s^\\succ$ and $s^\\prec$ denote the strict upper closure and strict lower closure of $s$, respectively. The '''order topology''' $\\tau$ on $S$ is the topology on $S$ generated by $\\map {\\Uparrow} S \\cup \\map {\\Downarrow} S$."}
+{"_id": "23723", "title": "Definition:Interval/Ordered Set/Open", "text": "The '''open interval between $a$ and $b$''' is the set: :$\\openint a b := a^\\succ \\cap b^\\prec = \\set {s \\in S: \\paren {a \\prec s} \\land \\paren {s \\prec b} }$ where: :$a^\\succ$ denotes the strict upper closure of $a$ :$b^\\prec$ denotes the strict lower closure of $b$."}
+{"_id": "23724", "title": "Definition:Interval/Ordered Set/Left Half-Open", "text": "The '''left half-open interval between $a$ and $b$''' is the set: :$\\hointl a b := a^\\succ \\cap b^\\preccurlyeq = \\set {s \\in S: \\paren {a \\prec s} \\land \\paren {s \\preccurlyeq b} }$ where: : $a^\\succ$ denotes the strict upper closure of $a$ : $b^\\preccurlyeq$ denotes the lower closure of $b$."}
+{"_id": "23725", "title": "Definition:Interval/Ordered Set/Right Half-Open", "text": "The '''right half-open interval between $a$ and $b$''' is the set: :$\\hointr a b := a^\\succcurlyeq \\cap b^\\prec = \\set {s \\in S: \\paren {a \\preccurlyeq s} \\land \\paren {s \\prec b} }$ where: :$a^\\succcurlyeq$ denotes the upper closure of $a$ :$b^\\prec$ denotes the strict lower closure of $b$."}
+{"_id": "23726", "title": "Definition:Interval/Ordered Set/Closed", "text": "The '''closed interval between $a$ and $b$''' is the set: :$\\closedint a b := a^\\succcurlyeq \\cap b^\\preccurlyeq = \\set {s \\in S: \\paren {a \\preccurlyeq s} \\land \\paren {s \\preccurlyeq b} }$ where: : $a^\\succcurlyeq$ denotes the upper closure of $a$ : $b^\\preccurlyeq$ denotes the lower closure of $b$."}
+{"_id": "23727", "title": "Definition:Interval/Ordered Set", "text": "Let $\\struct {S, \\preccurlyeq}$ be an ordered set. Let $a, b \\in S$. The '''intervals between $a$ and $b$''' are defined as follows: === Open Interval === {{:Definition:Interval/Ordered Set/Open}} === Left Half-Open Interval === {{:Definition:Interval/Ordered Set/Left Half-Open}} === Right Half-Open Interval === {{:Definition:Interval/Ordered Set/Right Half-Open}} === Closed Interval === {{:Definition:Interval/Ordered Set/Closed}}"}
+{"_id": "23728", "title": "Definition:Upper Closure/Set", "text": "Let $\\struct {S, \\preceq}$ be an ordered set or preordered set. Let $T \\subseteq S$. The '''upper closure of $T$ (in $S$)''' is defined as: :$T^\\succeq := \\bigcup \\set {t^\\succeq: t \\in T}$ where $t^\\succeq$ denotes the upper closure of $t$ in $S$. That is: :$T^\\succeq := \\set {u \\in S: \\exists t \\in T: t \\preceq u}$"}
+{"_id": "23729", "title": "Definition:Order Topology/Definition 1", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. Let $\\XX$ be the set of open rays in $S$. Let $\\tau$ be the topology on $S$ generated by $\\XX$. Then $\\tau$ is called the '''order topology''' on $S$."}
+{"_id": "23730", "title": "Definition:Lower Closure/Element", "text": "Let $\\struct {S, \\preccurlyeq}$ be an ordered set. Let $a \\in S$. The '''lower closure of $a$ (in $S$)''' is defined as: :$a^\\preccurlyeq := \\set {b \\in S: b \\preccurlyeq a}$ That is, $a^\\preccurlyeq$ is the set of all elements of $S$ that precede $a$."}
+{"_id": "23731", "title": "Definition:Lower Closure/Set", "text": "Let $\\struct {S, \\preccurlyeq}$ be an ordered set or preordered set. Let $T \\subseteq S$. The '''lower closure of $T$ (in $S$)''' is defined as: :$T^\\preccurlyeq := \\bigcup \\set {t^\\preccurlyeq: t \\in T}$ where $t^\\preccurlyeq$ is the lower closure of $t$. That is: :$T^\\preccurlyeq := \\set {l \\in S: \\exists t \\in T: l \\preccurlyeq t}$"}
+{"_id": "23733", "title": "Definition:Upper Set/Definition 2", "text": "$U$ is an '''upper set''' in $S$ {{iff}}: :$U^\\succeq \\subseteq U$ where $U^\\succeq$ is the upper closure of $U$."}
+{"_id": "23734", "title": "Definition:Upper Set/Definition 3", "text": "$U$ is an '''upper set''' in $S$ {{iff}}: :$U^\\succeq = U$ where $U^\\succeq$ is the upper closure of $U$."}
+{"_id": "23735", "title": "Definition:Closed Set/Closure Operator", "text": "Let $S$ be a set. Let $\\cl: \\powerset S \\to \\powerset S$ be a closure operator. Let $T \\subseteq S$ be a subset. === Definition 1 === The subset $T$ is '''closed''' (with respect to $\\cl$) {{iff}}: :$\\map \\cl T = T$ === Definition 2 === The subset $T$ is '''closed''' (with respect to $\\cl$) {{iff}} $T$ is in the image of $\\cl$: :$T \\in \\Img \\cl$"}
+{"_id": "23736", "title": "Definition:Kuratowski Closure Operator", "text": "=== Definition 1 === {{:Definition:Kuratowski Closure Operator/Definition 1}} === Definition 2 === {{:Definition:Kuratowski Closure Operator/Definition 2}}"}
+{"_id": "23737", "title": "Definition:Kuratowski Closure Operator/Definition 1", "text": "Let $S$ be a set. Let $\\cl: \\powerset S \\to \\powerset S$ be a mapping from the power set of $S$ to itself. Then $\\cl$ is a '''Kuratowski closure operator''' {{iff}} it satisfies the following '''Kuratowski closure axioms''' for all $A, B \\subseteq S$: {{begin-axiom}} {{axiom | n =1         | m = A \\subseteq \\map \\cl A         | rc= $\\cl$ is inflationary }} {{axiom | n = 2         | m = \\map \\cl {\\map \\cl A} = \\map \\cl A         | rc= $\\cl$ is idempotent }} {{axiom | n = 3         | m = \\map \\cl {A \\cup B} = \\map \\cl A \\cup \\map \\cl B         | rc= $\\cl$ preserves binary unions }} {{axiom | n = 4         | m = \\map \\cl \\O = \\O }} {{end-axiom}}"}
+{"_id": "23738", "title": "Definition:Kuratowski Closure Operator/Definition 2", "text": "Let $S$ be a set. Let $\\cl: \\powerset S \\to \\powerset S$ be a mapping from the power set of $S$ to itself. Then $\\cl$ is a '''Kuratowski closure operator''' {{iff}} it satisfies the following axioms for all $A, B \\subseteq X$: {{begin-axiom}} {{axiom | n = 1         | t = $\\cl$ is a closure operator }} {{axiom | n = 2         | t = $\\map \\cl {A \\cup B} = \\map \\cl A \\cup \\map \\cl B$         | rc= $\\cl$ preserves binary unions }} {{axiom | n = 3         | t = $\\map \\cl \\O = \\O$ }} {{end-axiom}}"}
+{"_id": "23739", "title": "Definition:Supremum of Mapping", "text": "Let $S$ be a set. Let $\\struct {T, \\preceq}$ be an ordered set. Let $f: S \\to T$ be a mapping from $S$ to $T$. Let $f \\sqbrk S$, the image of $f$, admit a supremum. Then the '''supremum''' of $f$ (on $S$) is defined by: :$\\displaystyle \\sup_{x \\mathop \\in S} \\map f x = \\sup f \\sqbrk S$ === Real-Valued Function === {{:Definition:Supremum of Mapping/Real-Valued Function}}"}
+{"_id": "23740", "title": "Definition:Projective Space/Over a Field", "text": "Let $V$ be a vector space over a field $K$ of dimension $n + 1 \\ge 1$.   Let $\\sim$ be the equivalence relation defined on the set $V \\setminus \\left\\{{0}\\right\\}$ by: :$x, y \\in V \\setminus \\left\\{{0}\\right\\}: x \\sim y \\iff \\exists \\lambda \\in K: x = \\lambda y$ The '''projective space''' associated to $V$ of dimension $n$ over $K$ is the quotient set $\\left({V \\setminus \\left\\{{0}\\right\\}}\\right) / \\sim$ and is denoted $\\mathbb P \\left({ V }\\right)$. If $V = K^{n+1}$ for $n \\ge 0$ a natural number, projective space is sometimes denoted $\\mathbb P\\left( K^{n+1} \\right) = \\mathbb P^n\\left( K \\right)$. This is because while $K^{n+1}$ is an $\\left(n+1\\right)$-dimensional vector space, the projective space $\\mathbb P\\left( K^{n+1} \\right)$ has dimension $n$. The notation $\\mathbb P\\left( K^{n+1} \\right) = K\\mathbb P^n$ is also in use."}
+{"_id": "23742", "title": "Definition:Universal Bounds", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $S$ have a smallest element, $\\bot$. Let $S$ have a greatest element, $\\top$. Then $\\bot$ and $\\top$ are the '''universal bounds''' of $S$."}
+{"_id": "23743", "title": "Definition:Reflexive Closure/Union with Diagonal", "text": "Let $\\mathcal R$ be a relation on a set $S$. The '''reflexive closure''' of $\\mathcal R$ is denoted $\\mathcal R^=$, and is defined as: :$\\mathcal R^= := \\mathcal R \\cup \\set {\\tuple {x, x}: x \\in S}$ That is: :$\\mathcal R^= := \\mathcal R \\cup \\Delta_S$ where $\\Delta_S$ is the diagonal relation on $S$."}
+{"_id": "23744", "title": "Definition:Reflexive Closure/Intersection of Reflexive Supersets", "text": "Let $\\mathcal R$ be a relation on a set $S$. Let $\\mathcal Q$ be the set of all reflexive relations on $S$ that contain $\\mathcal R$. The '''reflexive closure''' of $\\mathcal R$ is denoted $\\mathcal R^=$, and is defined as: :$\\mathcal R^= := \\bigcap \\mathcal Q$ That is: :$\\mathcal R^=$ is the intersection of all reflexive relations on $S$ containing $\\mathcal R$."}
+{"_id": "23745", "title": "Definition:Reflexive Closure/Smallest Reflexive Superset", "text": "Let $\\RR$ be a relation on a set $S$. The '''reflexive closure''' of $\\RR$ is defined as the smallest reflexive relation on $S$ that contains $\\RR$ as a subset. The '''reflexive closure''' of $\\RR$ is denoted $\\RR^=$."}
+{"_id": "23746", "title": "Definition:Transitive Closure (Relation Theory)/Smallest Transitive Superset", "text": "Let $\\RR$ be a relation on a set $S$. The '''transitive closure of $\\RR$''' is defined as the smallest transitive relation on $S$ which contains $\\RR$ as a subset."}
+{"_id": "23747", "title": "Definition:Transitive Closure (Relation Theory)/Intersection of Transitive Supersets", "text": "Let $\\mathcal R$ be a relation on a set $S$. The '''transitive closure''' of $\\mathcal R$ is defined as the intersection of all transitive relations on $S$ which contain $\\mathcal R$."}
+{"_id": "23748", "title": "Definition:Transitive Closure (Relation Theory)/Finite Chain", "text": "Let $\\mathcal R$ be a relation on a set or class $S$. The '''transitive closure''' of $\\mathcal R$ is the relation $\\mathcal R^+$ defined as follows: For $x, y \\in S$, $x \\mathrel {\\mathcal R^+} y$ {{iff}} for some $n \\in \\N_{>0}$ there exist $s_0, s_1, \\dots, s_n \\in S$ such that $s_0 = x$, $s_n = y$, and: {{begin-eqn}} {{eqn | l = s_0       | o = \\mathcal R       | r = s_1 }} {{eqn | l = s_1       | o = \\mathcal R       | r = s_2 }} {{eqn | o = \\vdots }} {{eqn | l = s_{n - 1}       | o = \\mathcal R       | r = s_n }} {{end-eqn}}"}
+{"_id": "23749", "title": "Definition:Transitive Closure (Relation Theory)/Union of Compositions", "text": "Let $\\mathcal R$ be a relation on a set $S$. Let: :$\\mathcal R^n := \\begin{cases} \\mathcal R & : n = 0 \\\\ \\mathcal R^{n-1} \\circ \\mathcal R & : n > 0 \\end{cases}$ where $\\circ$ denotes composition of relations. Finally, let: :$\\displaystyle \\mathcal R^+ = \\bigcup_{i \\mathop \\in \\N} \\mathcal R^i$ Then $\\mathcal R^+$ is called the '''transitive closure''' of $\\mathcal R$."}
+{"_id": "23750", "title": "Definition:Affine Frame", "text": "Let $\\mathcal E$ be an affine space with difference space $V$. === Definition 1 === {{:Definition:Affine Frame/Definition 1}} === Definition 2 === {{:Definition:Affine Frame/Definition 2}} Category:Definitions/Affine Geometry pz5qivyva9bgvrp3nv82l8s4yxut4l6"}
+{"_id": "23751", "title": "Definition:Coordinate System/Coordinates on Affine Space", "text": "Let $\\mathcal E$ be an affine space of dimension $n$ over a field $k$. Let $\\mathcal R = \\left({p_0, e_1, \\ldots, e_n}\\right)$ be an affine frame in $\\mathcal E$. Let $p \\in \\mathcal E$ be a point. Since Affine Coordinates are Well-Defined, there exists a unique ordered tuple $\\left({\\lambda_1, \\ldots, \\lambda_n}\\right) \\in k^n$ such that: :$\\displaystyle p = p_0 + \\sum_{i \\mathop = 1}^n \\lambda_i e_i$ The numbers $\\lambda_1, \\ldots, \\lambda_n$ are the '''coordinates''' of $p$ in the frame $\\mathcal R$."}
+{"_id": "23752", "title": "Definition:Barycenter", "text": "Let $\\mathcal E$ be an affine space over a field $k$. Let $p_1,\\ldots,p_n \\in \\mathcal E$ be points. Let $\\lambda_1,\\ldots,\\lambda_n \\in k$ such that $\\displaystyle \\sum_{i \\mathop = 1}^n \\lambda_i = 1$. The '''barycenter''' of $p_1,\\ldots,p_n$ with '''weights''' $\\lambda_1,\\ldots,\\lambda_n$ is the unique point $q$ of $\\mathcal E$ such that for every point $r \\in \\mathcal E$ :$\\displaystyle q = r + \\sum_{i \\mathop = 1}^n\\lambda_i \\vec{r p_i}$"}
+{"_id": "23753", "title": "Definition:Negatively Transitive Relation", "text": "Let $\\RR$ be a relation on a set $S$. Then $\\RR$ is '''negatively transitive''' {{iff}}: :$\\forall x, y, z \\in S: \\neg \\paren {x \\mathrel \\RR y} \\land \\neg \\paren {y \\mathrel \\RR z} \\implies \\neg \\paren {x \\mathrel \\RR z}$ By De Morgan's Laws, this can be given the alternative form: :$\\forall x, y, z \\in S: \\paren {x \\mathrel \\RR z} \\implies \\paren {x \\mathrel \\RR y} \\lor \\paren {y \\mathrel \\RR z}$"}
+{"_id": "23754", "title": "Definition:Reflexive Transitive Closure/Transitive Closure of Reflexive Closure", "text": "The '''reflexive transitive closure''' of $\\mathcal R$ is denoted $\\mathcal R^*$, and is defined as the transitive closure of the reflexive closure of $\\mathcal R$: :$\\mathcal R^* = \\left({\\mathcal R^=}\\right)^+$"}
+{"_id": "23755", "title": "Definition:Reflexive Transitive Closure/Reflexive Closure of Transitive Closure", "text": "The '''reflexive transitive closure''' of $\\mathcal R$ is denoted $\\mathcal R^*$, and is defined as the reflexive closure of the transitive closure of $\\mathcal R$: :$\\mathcal R^* = \\left({\\mathcal R^+}\\right)^=$"}
+{"_id": "23756", "title": "Definition:Reflexive Transitive Closure/Smallest Reflexive Transitive Superset", "text": "The '''reflexive transitive closure''' of $\\mathcal R$ is denoted $\\mathcal R^*$, and is defined as the smallest reflexive and transitive relation on $S$ which contains $\\mathcal R$."}
+{"_id": "23757", "title": "Definition:Closure Operator/Ordering", "text": "{{:Definition:Closure Operator/Ordering/Definition 1}}"}
+{"_id": "23758", "title": "Definition:Closed Element", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $\\operatorname{cl}$ be a closure operator on $S$. Let $x \\in S$. === Definition 1 === {{:Definition:Closed Element/Definition 1}} === Definition 2 === {{:Definition:Closed Element/Definition 2}}"}
+{"_id": "23759", "title": "Definition:Affine Subspace", "text": "Let $\\mathcal E$ be an affine space with tangent space $E$. Let $\\mathcal F \\subseteq \\mathcal E$ be a subset of $\\mathcal E$. Then $\\mathcal F$ is an '''affine subspace''' of $\\mathcal E$ {{iff}} there exists a point $p \\in \\mathcal E$ such that: :$F_p := \\set {q - p: q \\in \\mathcal F}$ is a vector subspace of the vector space $E$."}
+{"_id": "23760", "title": "Definition:Product of Affine Spaces", "text": "Let $\\mathcal E, \\mathcal F$ be affine spaces with difference spaces $E, F$ respectively. Let $\\mathcal G = \\mathcal E \\times \\mathcal F$ be the cartesian product of the sets $\\mathcal E,\\mathcal F$. Let $G = E \\times F$ be the direct product of the vector spaces $E, F$. Define sum and difference operations $+ : \\mathcal G \\times G \\to \\mathcal G$ and $- : \\mathcal G \\times \\mathcal G \\to G$ by, for all $p, p' \\in \\mathcal E$ and $q, q' \\in \\mathcal F$: :$\\left({p, q}\\right) + \\left({p', q'}\\right) := \\left({p + p', q + q'}\\right)$ :$\\left({p, q}\\right) - \\left({p', q'}\\right) := \\left({p - p', q - q'}\\right)$ Then the set $\\mathcal G$ together with the vector space $G$ and the operations $+, -$ is called the '''product''' of the affine spaces $\\mathcal E$ and $\\mathcal F$."}
+{"_id": "23761", "title": "Definition:Standard Affine Structure on Vector Space", "text": "Let $E$ be a vector space. Let $\\mathcal E$ be the underlying set of $E$. Let $+$ denote the addition operation $E \\times E \\to E$, viewed as a mapping $\\mathcal E \\times E \\to \\mathcal E$. Let $-$ denote the subtraction operation $E \\times E \\to E$, viewed as a mapping $\\mathcal E \\times \\mathcal E \\to E$. Then the set $\\mathcal E$, together with the vector space $E$ and the operations $+,-$, is called the '''standard affine structure''' on the vector space $E$."}
+{"_id": "23762", "title": "Definition:Vectorialization of Affine Space", "text": "Let $\\mathcal E$ be an affine space over a field $k$ with difference space $E$. Let $\\mathcal R = \\left({p_0, e_1, \\ldots, e_n}\\right)$ be an affine frame in $\\mathcal E$. Define a mapping $\\Theta_{\\mathcal R} : k^n \\to \\mathcal E$ by: :$\\displaystyle \\Theta_{\\mathcal R} \\left({\\lambda_1, \\ldots, \\lambda_n}\\right) = p_0 + \\sum_{i \\mathop = 1}^n \\lambda_i e_i$. By Affine Coordinates are Well-Defined, $\\Theta_{\\mathcal R}$ is a bijection. For any $\\mu \\in k$, $p, q \\in \\mathcal E$ let: :$\\mu \\cdot p = \\Theta_{\\mathcal R}\\left({ \\mu \\cdot \\Theta_{\\mathcal R}^{-1} \\left({p}\\right) }\\right)$ and: :$p + q = \\Theta_{\\mathcal R} \\left({ \\Theta_{\\mathcal R}^{-1} \\left({p}\\right) + \\Theta_{\\mathcal R}^{-1} \\left({q}\\right) }\\right)$ We call the set $\\mathcal E$, together with the operations $\\cdot,+$ the '''vectorialization of $\\mathcal E$ with origin $p_0$'''."}
+{"_id": "23763", "title": "Definition:Strict Ordering/Asymmetric and Transitive", "text": "Let $\\RR$ be a relation on a set $S$. Then $\\RR$ is a '''strict ordering (on $S$)''' {{iff}} the following two conditions hold: {{begin-axiom}} {{axiom | n = 1         | lc= Asymmetry         | q = \\forall a, b \\in S         | ml= a \\mathrel \\RR b         | mo= \\implies         | mr= \\neg \\paren {b \\mathrel \\RR a} }} {{axiom | n = 2         | lc= Transitivity         | q = \\forall a, b, c \\in S         | ml= \\paren {a \\mathrel \\RR b} \\land \\paren {b \\mathrel \\RR c}         | mo= \\implies         | mr= a \\mathrel \\RR c }} {{end-axiom}}"}
+{"_id": "23764", "title": "Definition:Strict Ordering/Antireflexive and Transitive", "text": "Let $\\mathcal R$ be a relation on a set $S$. Then $\\mathcal R$ is a '''strict ordering (on $S$)''' {{iff}} the following two conditions hold: {{begin-axiom}} {{axiom | n = 1         | lc= Antireflexivity         | q = \\forall a \\in S         | m = \\neg \\paren {a \\mathrel {\\mathcal R} a} }} {{axiom | n = 2         | lc= Transitivity         | q = \\forall a, b, c \\in S         | m = \\paren {a \\mathrel {\\mathcal R} b} \\land \\paren {b \\mathrel {\\mathcal R} c} \\implies a \\mathrel {\\mathcal R} c }} {{end-axiom}}"}
+{"_id": "23765", "title": "Definition:Closure Operator", "text": "=== Ordering === {{:Definition:Closure Operator/Ordering}} === Power Set === When the ordering in question is the subset relation on a power set, the definition can be expressed as follows: {{:Definition:Closure Operator/Power Set}}"}
+{"_id": "23766", "title": "Definition:Subband", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $T \\subseteq S$ such that $\\struct {T, \\circ {\\restriction_T} }$ is a band, where $\\circ {\\restriction_T}$ denotes the restriction of $\\circ$ to $T$. Then $\\struct {T, \\circ {\\restriction_T} }$ is a '''subband''' of $S$."}
+{"_id": "23767", "title": "Definition:Closure Operator/Ordering/Definition 1", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. A '''closure operator''' on $S$ is a mapping: :$\\cl: S \\to S$ which satisfies the following conditions for all elements $x, y \\in S$: {{begin-axiom}} {{axiom | lc= $\\cl$ is inflationary         | ml= x         | mo= \\preceq          | mr= \\map \\cl x }} {{axiom | lc= $\\cl$ is increasing         | ml= x \\preceq y         | mo= \\implies         | mr= \\map \\cl x \\preceq \\map \\cl y }} {{axiom | lc= $\\cl$ is idempotent         | ml= \\map \\cl {\\map \\cl x}          | mo= =         | mr= \\map \\cl x }} {{end-axiom}}"}
+{"_id": "23768", "title": "Definition:Closure Operator/Ordering/Definition 2", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. A '''closure operator''' on $S$ is a mapping: :$\\cl: S \\to S$ which satisfies the following condition for all elements $x, y \\in S$: :$x \\preceq \\map \\cl y \\iff \\map \\cl x \\preceq \\map \\cl y$"}
+{"_id": "23769", "title": "Definition:Closed Element/Definition 1", "text": "The element $x$ is a '''closed element of $S$ (with respect to $\\cl$)''' {{iff}} $x$ is a fixed point of $\\cl$: :$\\map \\cl x = x$"}
+{"_id": "23770", "title": "Definition:Closed Element/Definition 2", "text": "The element $x$ is a '''closed element of $S$ (with respect to $\\cl$)''' {{iff}} $x$ is in the image of $\\cl$: :$x \\in \\Img \\cl$"}
+{"_id": "23771", "title": "Definition:Univalent Relation", "text": "Let $\\RR$ be a relation on a set $S$. Then $\\RR$ is '''univalent''' {{iff}}: :$\\RR \\circ \\RR^{-1} \\subseteq \\Delta_S$ That is, $\\RR$ composed with its inverse $\\RR^{-1}$ is a subset of the diagonal relation on $S$."}
+{"_id": "23772", "title": "Definition:Dual Relation/Complement of Inverse", "text": "Let $\\RR \\subseteq S \\times T$ be a binary relation. Then the '''dual''' of $\\RR$ is denoted $\\RR^d$ and is defined as: :$\\RR^d := \\overline {\\paren {\\RR^{-1} } }$ where: :$\\RR^{-1}$ denotes the inverse of $\\RR$ :$\\overline {\\paren {\\RR^{-1} } }$ denotes the complement of the inverse of $\\RR$."}
+{"_id": "23773", "title": "Definition:Dual Relation/Inverse of Complement", "text": "Let $\\RR \\subseteq S \\times T$ be a binary relation. Then the '''dual''' of $\\RR$ is denoted $\\RR^d$ and is defined as: :$\\RR^d := \\paren {\\overline \\RR}^{-1}$ where: :$\\overline \\RR$ denotes the complement of $\\RR$ :$\\paren {\\overline \\RR}^{-1}$ denotes the inverse of the complement of $\\RR$."}
+{"_id": "23774", "title": "Definition:Dual Relation", "text": "=== Inverse of Complement === {{:Definition:Dual Relation/Inverse of Complement}} === Complement of Inverse === {{:Definition:Dual Relation/Complement of Inverse}}"}
+{"_id": "23776", "title": "Definition:Trivial Gradation", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring. Let $\\left({M, \\cdot, e}\\right)$ be a monoid. Define a gradation on $R$ as follows: :$\\displaystyle R_e = R$ :$\\displaystyle \\forall m \\in M \\setminus \\left\\{{e}\\right\\} : R_m = \\mathbf 0$ where $\\mathbf 0$ is the zero ring. This is called the '''trivial $M$-grading''' on $R$."}
+{"_id": "23777", "title": "Definition:Irreducible (Representation Theory)", "text": "=== Linear Representation === {{:Definition:Irreducible (Representation Theory)/Linear Representation}} === G-Module === {{:Definition:Irreducible (Representation Theory)/G-Module}}"}
+{"_id": "23778", "title": "Definition:Transcendental (Abstract Algebra)/Ring", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\\struct {D, +, \\circ}$ be an integral subdomain of $R$. Let $x \\in R$. Then $x$ is '''transcendental over $D$''' {{iff}}: :$\\displaystyle \\forall n \\in \\Z_{\\ge 0}: \\sum_{k \\mathop = 0}^n a_k \\circ x^k = 0_R \\implies \\forall k: 0 \\le k \\le n: a_k = 0_R$ That is, $x$ is '''transcendental over $D$''' {{iff}} the only way to express $0_R$ as a polynomial in $x$ over $D$ is by the null polynomial."}
+{"_id": "23779", "title": "Definition:Transcendental (Abstract Algebra)/Field Extension/Element", "text": "Let $E / F$ be a field extension. Let $\\alpha \\in E$. Then $\\alpha$ is '''transcendental over $F$''' {{iff}}: : $\\nexists f \\left({x}\\right) \\in F \\left[{x}\\right] \\setminus \\left\\{{0}\\right\\}: f \\left({\\alpha}\\right) = 0$ where $f \\left({x}\\right)$ denotes a polynomial in $x$ over $F$."}
+{"_id": "23780", "title": "Definition:Transcendental (Abstract Algebra)/Field Extension", "text": "A field extension $E / F$ is said to be '''transcendental''' {{iff}}: :$\\exists \\alpha \\in E: \\alpha$ is transcendental over $F$ That is, a field extension is '''transcendental''' {{iff}} it contains at least one transcendental element."}
+{"_id": "23781", "title": "Definition:Monomial of Free Commutative Monoid/Multiplication", "text": "The set of monomials over $\\family {X_j: j \\in J}$ has '''multiplication''' $\\circ$ defined by: :$\\ds \\paren {\\prod_{j \\mathop \\in J} X_j^{k_j} } \\circ \\paren {\\prod_{j \\mathop \\in J} X_j^{k_j'} } = \\paren {\\prod_{j \\mathop \\in J} X_j^{k_j + k_j'} }$ which using multiindex addition notation reads: :$\\mathbf X^k \\circ \\mathbf X^{k'} = \\mathbf X^{k + k'}$"}
+{"_id": "23782", "title": "Definition:Monomial of Free Commutative Monoid/Degree", "text": "The '''degree''' of a monomial is defined as: :$\\ds \\sum_{j \\mathop \\in J} k_j$ that is, the modulus of the corresponding multiindex."}
+{"_id": "23783", "title": "Definition:Degree of Polynomial/Null Polynomial", "text": "The null polynomial $0_R \\in S \\left[{X}\\right]$ does ''not'' have a degree."}
+{"_id": "23784", "title": "Definition:Degree of Polynomial/Zero", "text": "A polynomial $f \\in S \\sqbrk x$ in $x$ over $S$ is of '''degree zero''' {{iff}} $x$ is a non-zero element of $S$, that is, a constant polynomial."}
+{"_id": "23785", "title": "Definition:Multiplication of Polynomials", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\struct {S, +, \\circ}$ be a subring of $R$. Let $x \\in R$. Let: :$\\displaystyle f = \\sum_{j \\mathop = 0}^n a_j x^j$ :$\\displaystyle g = \\sum_{k \\mathop = 0}^n b_k x^k$ be polynomials in $x$ over $S$ such that $a_n \\ne 0$ and $b_m \\ne 0$. {{questionable|this does not need to be defined; it's just the product of ring elements}} The '''product of $f$ and $g$''' is defined as: :$\\displaystyle f g := \\sum_{l \\mathop = 0}^{m + n} c_l x^l$ where: :$\\displaystyle \\forall l \\in \\set {0, 1, \\ldots, m + n}:  c_l = \\sum_{\\substack {j \\mathop + k \\mathop = l \\\\ j, k \\mathop \\in \\Z}} a_j b_k$ === Polynomial Forms === {{:Definition:Multiplication of Polynomials/Polynomial Forms}} === Polynomials as Sequences === {{:Definition:Multiplication of Polynomials/Sequence}}"}
+{"_id": "23786", "title": "Definition:Ring of Polynomials in Ring Element", "text": "The subring of $R$ consisting of all the polynomials in $x$ over $D$ is called the '''ring of polynomials over $D$''' and is denoted $D \\sqbrk x$."}
+{"_id": "23787", "title": "Definition:Order Embedding/Definition 1", "text": "$\\phi$ is an '''order embedding of $S$ into $T$''' {{iff}}: :$\\forall x, y \\in S: x \\preceq_1 y \\iff \\map \\phi x \\preceq_2 \\map \\phi y$"}
+{"_id": "23788", "title": "Definition:Order Embedding/Definition 4", "text": "Let $T' = \\Img S$ be the image of $S$ under $\\phi$. $\\phi$ is an '''order embedding of $S$ into $T$''' {{iff}}: :the restriction of $\\phi$ to $S \\times T'$ is an order isomorphism between $\\struct {S, \\preceq_1}$ and $\\struct {T', \\preceq_2 \\restriction_{T' \\times T'} }$."}
+{"_id": "23789", "title": "Definition:Argumentum ad Passiones", "text": "An '''argumentum ad passiones''' is a logical argument that, rather than prove or present evidence for a claim, attempts to generate support for an idea by instilling either: : an emotional attachment to the argument being presented : a feeling of embarrassment at holding a position which is subject to ridicule or : a feeling of distaste for the argument being refuted. Such arguments are commonly seen along with argumentum ad hominem."}
+{"_id": "23790", "title": "Definition:Order Embedding/Definition 3", "text": "$\\phi$ is an '''order embedding of $S$ into $T$''' {{iff}} both of the following conditions hold: :$(1): \\quad \\phi$ is an injection :$(2): \\quad \\forall x, y \\in S: x \\prec_1 y \\iff \\map \\phi x \\prec_2 \\map \\phi y$"}
+{"_id": "23791", "title": "Definition:Null Polynomial/Ring", "text": "Let $\\left({R, +, \\times}\\right)$ be a ring. The zero $0_R$ of $R$ can be considered as being the '''null polynomial over $R$''' of any arbitrary element $x$ of $R$."}
+{"_id": "23792", "title": "Definition:Null Polynomial/Polynomial Form", "text": "Let $f = a_1 \\mathbf X^{k_1} + \\cdots + a_r \\mathbf X^{k_r}$ be a polynomial form over $R$ in the indeterminates $\\left\\{{X_j: j \\in J}\\right\\}$. For all $i = 1, 2, \\ldots, r$, let $a_i = 0$. Then $f$ is the '''null polynomial''' in the indeterminates $\\left\\{{X_j: j \\in J}\\right\\}$."}
+{"_id": "23793", "title": "Definition:Polynomial/Real Numbers", "text": "A '''polynomial (in $\\R$)''' is an expression of the form: :$\\displaystyle \\map P x = \\sum_{j \\mathop = 0}^n \\paren {a_j x^j} = a_0 + a_1 x + a_2 x^2 + \\cdots + a_{n - 1} x^{n - 1} + a_n x^n$ where: :$x \\in \\R$ :$a_0, \\ldots a_n \\in \\mathbb k$ where $\\mathbb k$ is one of the standard number sets $\\Z, \\Q, \\R$."}
+{"_id": "23794", "title": "Definition:Polynomial/Complex Numbers", "text": "A '''polynomial (in $\\C$)''' is an expression of the form: :$\\displaystyle P \\left({z}\\right) = \\sum_{j \\mathop = 0}^n \\left({a_j z^j}\\right) = a_0 + a_1 z + a_2 z^2 + \\cdots + a_{n-1} z^{n-1} + a_n z^n$ where: :$z \\in \\C$ :$a_0, \\ldots a_n \\in \\mathbb k$ where $\\mathbb k$ is one of the standard number sets $\\Z, \\Q, \\R, \\C$."}
+{"_id": "23795", "title": "Definition:Degree of Polynomial/Ring", "text": "{{questionable|ill-defined}} Let $\\left({R, +, \\circ}\\right)$ be a ring. Let $\\left({S, +, \\circ}\\right)$ be a subring of $R$. Let $x \\in R$. Let $\\displaystyle P = \\sum_{j \\mathop = 0}^n \\left({r_j \\circ x^j}\\right) = r_0 + r_1 \\circ x + \\cdots + r_n \\circ x^n$ be a polynomial in the element $x$ over $S$ such that $r_n \\ne 0$. Then the '''degree of $P$''' is $n$. The '''degree of $P$''' can be denoted $\\deg \\left({P}\\right)$ or $\\partial P$."}
+{"_id": "23796", "title": "Definition:Degree of Polynomial/Integral Domain", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity whose zero is $0_R$. Let $\\struct {D, +, \\circ}$ be an integral subdomain of $R$. Let $X \\in R$ be transcendental over $D$. Let $\\displaystyle f = \\sum_{j \\mathop = 0}^n \\paren {r_j \\circ X^j} = r_0 + r_1 X + \\cdots + r_n X^n$ be a polynomial over $D$ in $X$ such that $r_n \\ne 0$. Then the '''degree of $f$''' is $n$. The '''degree of $f$''' is denoted on {{ProofWiki}} by $\\map \\deg f$."}
+{"_id": "23797", "title": "Definition:Degree of Polynomial/Sequence/Field", "text": "Let $f = \\sequence {a_k} = \\tuple {a_0, a_1, a_2, \\ldots}$ be a polynomial over a field $F$. The '''degree of $f$''' is defined as the largest $n \\in \\Z$ such that $a_n \\ne 0$."}
+{"_id": "23798", "title": "Definition:Antisymmetric Quotient", "text": "Let $\\struct {S, \\precsim}$ be a preordered set. Let $\\sim$ be the equivalence relation on $S$ induced by $\\precsim$. Let $S / {\\sim}$ be the quotient set of $S$ by $\\sim$. Let $\\preceq$ be the relation on $S / {\\sim}$ defined by letting $P \\preceq Q$ {{iff}}: :$\\exists p \\in P: \\exists q \\in Q: p \\precsim q$ Then $\\struct {S / {\\sim}, \\preceq}$ is the '''antisymmetric quotient''' of $\\struct {S, \\precsim}$."}
+{"_id": "23801", "title": "Definition:Fuzzy Set", "text": "Let $S$ be a set. Let $\\left [{0 \\,.\\,.\\, 1}\\right]$ denotes the closed unit interval. Let $\\mu: S \\to \\left [{0 \\,.\\,.\\, 1}\\right]$ be a mapping. Then $\\left({S, \\mu}\\right)$ is a '''fuzzy set'''."}
+{"_id": "23802", "title": "Definition:Fuzzy Set/Domain", "text": "$S$ is the '''domain''' of $\\left({S, \\mu}\\right)$."}
+{"_id": "23803", "title": "Definition:Fuzzy Set/Membership Function", "text": "$\\mu$ is the '''membership function''' of $\\left({S, \\mu}\\right)$."}
+{"_id": "23804", "title": "Definition:Fuzzy Intersection", "text": "Let $\\textbf A = \\left({A, \\mu_A}\\right)$ and $\\textbf B = \\left({B, \\mu_B}\\right)$ be fuzzy sets. Then the '''(fuzzy) intersection''' of $\\textbf A$ and $\\textbf B$ is denoted $\\textbf A \\cap \\textbf B$. It is defined to be the fuzzy set such that: * The domain of $\\textbf A \\cap \\textbf B$ is $A \\cap B$. * The membership function of $\\textbf A \\cap \\textbf B$ is: ::$\\forall x \\in A \\cap B: \\mu_{A \\cap B}(x) = \\operatorname{min}\\left({\\mu_A(x), \\mu_B(x)}\\right)$ where $\\operatorname{min}$ is the min operation."}
+{"_id": "23805", "title": "Definition:Addition of Polynomials", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring. Let $\\left({S, +, \\circ}\\right)$ be a subring of $R$. For arbitrary $x \\in R$, let $S \\left[{x}\\right]$ be the set $S \\left[{x}\\right]$ be the set of polynomials in $x$ over $S$. Let  $p, q \\in S \\left[{x}\\right]$ be polynomials in $x$ over $S$: : $\\displaystyle p = \\sum_{k \\mathop = 0}^m a_k \\circ x^k$ : $\\displaystyle q = \\sum_{k \\mathop = 0}^n b_k \\circ x^k$ where: : $(1): \\quad a_k, b_k \\in S$ for all $k$ : $(2): \\quad m, n \\in \\Z_{\\ge 0}$. The operation '''polynomial addition''' is defined as: :$\\displaystyle p + q := \\sum_{k \\mathop = 0}^{\\max \\left({m, n}\\right)} \\left({a_k + b_k}\\right) x^k$ where: :$\\forall k \\in \\Z: k > m \\implies a_k = 0$ :$\\forall k \\in \\Z: k > n \\implies b_k = 0$ The expression $p + q$ is known as the '''sum''' of $p$ and $q$. === Polynomial Forms === {{:Definition:Addition of Polynomials/Polynomial Forms}} === Polynomials as Sequences === {{:Definition:Addition of Polynomials/Sequence}} Category:Definitions/Polynomial Theory c5gvtv5plr8rzi4rsqy8dwzxpc63f2i"}
+{"_id": "23806", "title": "Definition:Zero Divisor/Commutative Ring", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring. A '''zero divisor (in $R$)''' is an element $x \\in R$ such that: : $\\exists y \\in R^*: x \\circ y = 0_R$ where $R^*$ is defined as $R \\setminus \\set {0_R}$. The expression: :'''$x$ is a zero divisor''' can be written: : $x \\divides 0_R$"}
+{"_id": "23807", "title": "Definition:Leading Coefficient of Polynomial/Polynomial Form", "text": "Let $R$ be a commutative ring with unity. Let $f = a_0 + a_1 X + \\cdots + a_{r-1} X^{r-1} + a_r X^r$ be a polynomial form in the single indeterminate $X$ over $R$. Then the ring element $a_r$ is called the '''leading coefficient''' of $f$."}
+{"_id": "23808", "title": "Definition:Degree of Polynomial/Field", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0_F$. Let $\\struct {K, +, \\times}$ be a subfield of $F$. Let $x \\in F$. Let $\\displaystyle f = \\sum_{j \\mathop = 0}^n \\paren {a_j x^j} = a_0 + a_1 x + \\cdots + a_n x^n$ be a polynomial over $K$ in $x$ such that $a_n \\ne 0$. Then the '''degree of $f$''' is $n$. The '''degree of $f$''' can be denoted $\\map \\deg f$ or $\\partial f$."}
+{"_id": "23809", "title": "Definition:Degree of Polynomial/Sequence", "text": "=== Ring === {{:Definition:Degree of Polynomial/Sequence/Ring}} === Field === {{:Definition:Degree of Polynomial/Sequence/Field}}"}
+{"_id": "23810", "title": "Definition:Degree of Polynomial/Sequence/Ring", "text": "Let $f = \\left \\langle {a_k}\\right \\rangle = \\left({a_0, a_1, a_2, \\ldots}\\right)$ be a polynomial over a ring $R$. The '''degree of $f$''' is defined as the largest $n \\in \\Z$ such that $a_n \\ne 0$."}
+{"_id": "23811", "title": "Definition:Degree of Polynomial/Polynomial Form", "text": "Let $f = a_1 \\mathbf X^{k_1} + \\cdots + a_r \\mathbf X^{k_r}$ be a polynomial in the indeterminates $\\family {X_j: j \\in J}$ for some multiindices $k_1, \\ldots, k_r$. Let $f$ '''not''' be the null polynomial. Let $k = \\family {k_j}_{j \\mathop \\in J}$ be a multiindex. Let $\\ds \\size k = \\sum_{j \\mathop \\in J} k_j \\ge 0$ be the degree of the monomial $\\mathbf X^k$. The '''degree of $f$''' is the supremum: :$\\ds \\map \\deg f = \\max \\set {\\size {k_r}: i = 1, \\ldots, r}$"}
+{"_id": "23812", "title": "Definition:Coefficient of Polynomial/Polynomial Form", "text": "Let $R$ be a commutative ring with unity. Let $f = a_1 \\mathbf X^{k_1} + \\cdots + a_r \\mathbf X^{k_r}$ be a polynomial in the indeterminates $\\left\\{{X_j: j \\in J}\\right\\}$ over $R$. The ring elements $a_1, \\ldots, a_r$ are the '''coefficients''' of $f$."}
+{"_id": "23813", "title": "Definition:Monic Polynomial/Polynomial Form", "text": "Let $R$ be a commutative ring with unity $1_R$. Let $f = a_0 + a_1 X + \\cdots + a_{r-1} X^{r-1} + a_r X^r$ be a polynomial from in the single indeterminate $X$ over $R$. Then $f$ is '''monic''' if the leading coefficient of $f$ is $1_R$."}
+{"_id": "23814", "title": "Definition:Degree of Polynomial/Null Polynomial/Integral Domain", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity whose zero is $0_R$. Let $\\struct {D, +, \\circ}$ be an integral subdomain of $R$. For arbitrary $x \\in R$, let $D \\sqbrk x$ be the ring of polynomials in $x$ over $D$. The null polynomial $0_R \\in D \\sqbrk x$ does ''not'' have a degree."}
+{"_id": "23816", "title": "Definition:Inflationary Mapping/Subset", "text": "Let $C$ be a class. Let $f: C \\to C$ be a mapping from $C$ to $C$. Then $f$ is '''inflationary''' {{iff}}: :$x \\in C \\implies x \\subseteq \\map f x$ That is, {{iff}} for each $x \\in C$, $x$ is a subset of $\\map f x$."}
+{"_id": "23817", "title": "Definition:Subclass", "text": "Let $A$ and $B$ be classes. Then $A$ is a '''subclass''' of $B$, and we write $A \\subseteq B$, {{iff}}: :$\\forall x: \\paren {x \\in A \\implies x \\in B}$ where $x \\in A$ denotes that $x$ is an element of $A$."}
+{"_id": "23818", "title": "Definition:Inductive Class", "text": "Let $A$ be a class. Then $A$ is '''inductive''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | lc= $A$ contains the empty set:         | q =          | m = \\quad \\O \\in A }} {{axiom | n = 2         | lc= $A$ is closed under the successor mapping:         | q = \\forall x         | m = \\paren {x \\in A \\implies x^+ \\in A}         | rc= where $x^+$ is the successor of $x$ }} {{axiom | rc= That is, where $x^+ = x \\cup \\set x$ }} {{end-axiom}}"}
+{"_id": "23819", "title": "Definition:Polynomial Function/Real", "text": "Let $S \\subset \\R$ be a subset of the real numbers. === Definition 1 === {{Definition:Polynomial Function/Real/Definition 1}} === Definition 2 === {{Definition:Polynomial Function/Real/Definition 2}}"}
+{"_id": "23820", "title": "Definition:Polynomial", "text": "=== Real Numbers === {{Definition:Polynomial/Real Numbers}} === Complex Numbers === {{Definition:Polynomial/Complex Numbers}}"}
+{"_id": "23821", "title": "Definition:Polynomial Function/Complex", "text": "Let $S \\subset \\C$ be a subset of the complex numbers. === Definition 1 === {{Definition:Polynomial Function/Complex/Definition 1}} === Definition 2 === {{Definition:Polynomial Function/Complex/Definition 2}}"}
+{"_id": "23822", "title": "Definition:Argumentum ad Verecundiam", "text": "'''Argumentum ad verecundiam''', or '''argument from authority''', is a logical argument that infers something is true on the basis that a knowledgeable and well-reputed person has said it. An '''argument from authority''' may be valid or invalid, depending on the circumstances. If the trustworthy person is also an expert in the field, and is free to speak their mind about the matter, then the argument has some validity. Otherwise, it generally does not. That said, '''argumentum ad verecundiam''' can never serve as a (mathematical) proof, since no authority could ever be infallible. Thus, in the context of {{ProofWiki}}, such an argument should be considered fallacious."}
+{"_id": "23823", "title": "Definition:Eisenstein Integer", "text": "An '''Eisenstein integer''' is a complex number of the form :$a + b \\omega$ where $a$ and $b$ are both integers and: :$\\omega = e^{2 \\pi i / 3} = \\dfrac 1 2 \\paren {i \\sqrt 3 - 1}$ that is, the (complex) cube roots of unity. The set of all '''Eisenstein integers''' can be denoted $\\Z \\left[{\\omega}\\right]$: :$\\Z \\sqbrk \\omega = \\set {a + b \\omega: a, b \\in \\Z}$"}
+{"_id": "23824", "title": "Definition:Eisenstein Prime", "text": "Let $\\Z \\sqbrk \\omega$ be the ring of Eisenstein integers. Let $\\alpha \\in \\Z \\sqbrk \\omega$ be an Eisenstein integer. Then $\\alpha$ is an '''Eisenstein prime''' if $\\alpha$ is prime in $\\Z \\sqbrk \\omega$. {{NamedforDef|Ferdinand Gotthold Max Eisenstein|cat = Eisenstein}} Category:Definitions/Number Theory Category:Definitions/Eisenstein Integers ivyums544kha0szun3kswtcm6p9gxw2"}
+{"_id": "23825", "title": "Definition:Degree of Polynomial/Null Polynomial/Sequence", "text": "Let $f = \\sequence {a_k} = \\tuple {a_0, a_1, a_2, \\ldots}$ be a polynomial over a field $F$. Let $0_F$ be the zero of $F$. Let $a_0 = a_1 = a_2 = \\ldots = 0_F$. Then $f$ is a '''null polynomial over $F$'''."}
+{"_id": "23826", "title": "Definition:Monic Polynomial/Sequence", "text": "Let $f = \\sequence {a_k} = \\tuple {a_0, a_1, a_2, \\ldots}$ be a polynomial over a field $F$. Then $f$ is a '''monic polynomial''' {{iff}} its leading coefficient $a_n$ is $1$."}
+{"_id": "23827", "title": "Definition:Addition of Polynomials/Sequence", "text": "Let: :$f = \\sequence {a_k} = \\tuple {a_0, a_1, a_2, \\ldots}$ and: :$g = \\sequence {b_k} = \\tuple {b_0, b_1, b_2, \\ldots}$ be polynomials over a field $F$. Then the operation of '''(polynomial) addition''' is defined as: :$f + g := \\tuple {a_0 + b_0, a_1 + b_1, a_2 + b_2, \\ldots}$"}
+{"_id": "23828", "title": "Definition:Multiplication of Polynomials/Sequence", "text": "Let: : $f = \\sequence {a_k} = \\tuple {a_0, a_1, a_2, \\ldots}$ and: : $g = \\sequence {b_k} = \\tuple {b_0, b_1, b_2, \\ldots}$ be polynomials over a field $F$. Then the operation of '''(polynomial) multiplication''' is defined as: :$f g := \\tuple {c_0, c_1, c_2, \\ldots}$ where $\\displaystyle c_i = \\sum_{j \\mathop + k \\mathop = i} a_j b_k$"}
+{"_id": "23829", "title": "Definition:Standard Affine Space", "text": "Let $n \\geq 0$ be an integer. Let $k$ be a field. The '''standard affine space of dimension $n$''' over $k$ is the vector space $k^n$ together with the standard affine structure. Category:Definitions/Affine Geometry r6vqbj6ch7xapphyz93jf4n3uj6uxj8"}
+{"_id": "23830", "title": "Definition:Null Polynomial/Sequence", "text": "Let $f = \\sequence {a_k} = \\tuple {a_0, a_1, a_2, \\ldots}$ be a polynomial over a field $F$. Let $0_F$ be the zero of $F$. Let $a_0 = a_1 = a_2 = \\ldots = 0_F$. Then $f$ is known as the '''null polynomial'''."}
+{"_id": "23831", "title": "Definition:Irreducible Polynomial", "text": "{{Definition:Irreducible Polynomial/Definition 1}}"}
+{"_id": "23832", "title": "Definition:Operations on Polynomial Ring of Sequences", "text": "{{begin-axiom}} {{axiom | n = 1         | lc= '''Ring Addition:'''         | ml= \\sequence {r_0, r_1, r_2, \\ldots} \\oplus \\sequence {s_0, s_1, s_2, \\ldots}         | mo= =         | mr= \\sequence {r_0 + s_0, r_1 + s_1, r_2 + s_2, \\ldots}         | c =  }} {{axiom | n = 2         | lc= '''Ring Negative:'''         | ml= -\\sequence {r_0, r_1, r_2, \\ldots}         | mo= =         | mr= \\sequence {-r_0, -r_1, -r_2, \\ldots}         | c =  }} {{axiom | n = 3         | lc= '''Ring Product:'''         | ml= \\sequence {r_0, r_1, r_2, \\ldots} \\odot \\sequence {s_0, s_1, s_2, \\ldots}         | mo= =         | mr= \\sequence {t_0, t_1, t_2, \\ldots}         | rc= where $\\displaystyle t_i = \\sum_{j \\mathop + k \\mathop = i} r_j \\circ s_k$ }} {{end-axiom}}"}
+{"_id": "23833", "title": "Definition:Constant Polynomial", "text": "Let $R$ be a commutative ring with unity. Let $P \\in R \\left[{x}\\right]$ be a polynomial in one variable over $R$. === Definition 1 === {{Definition:Constant Polynomial/Definition 1}} === Definition 2 === {{Definition:Constant Polynomial/Definition 2}} === Definition 3 === {{Definition:Constant Polynomial/Definition 3}}"}
+{"_id": "23834", "title": "Definition:Greatest Set by Set Inclusion/Class Theory", "text": "Let $A$ be a class. Then a set $M$ is the '''greatest element''' of $A$ (with respect to the subset relation) {{iff}}: :$(1): \\quad M \\in A$ :$(2): \\quad \\forall S: \\paren {S \\in A \\implies S \\subseteq M}$"}
+{"_id": "23835", "title": "Definition:Relation of Set Inclusion", "text": "The '''relation of set inclusion''', written $\\subseteq$, is the class of all ordered pairs $\\left({x, y}\\right)$ such that $x$ is a subset of $y$."}
+{"_id": "23836", "title": "Definition:Substitution", "text": "'''Substitution''' is a word that denotes the process of replacement of a term in an expression or equation with another which has the same value. {{Disambiguation}} * Formal Systems: ** Definition:Substitution (Formal Systems) *** Definition:Substitution for Well-Formed Part *** Definition:Substitution for Letter **** Definition:Alphabetic Substitution *** Definition:Substitution of Term *** Definition:Substitution for Metasymbol * Mathematical Logic: ** Definition:Substitution (Mathematical Logic)"}
+{"_id": "23837", "title": "Definition:Well-Formed Part/Proper Well-Formed Part", "text": "Let $\\mathbf B$ be a well-formed part of $\\mathbf A$. Then $\\mathbf B$ is a '''proper well-formed part of $\\mathbf A$''' {{iff}} $\\mathbf B$ is not equal to $\\mathbf A$."}
+{"_id": "23838", "title": "Definition:Logical Not/Truth Function", "text": "The logical not connective defines the truth function $f^\\neg$ as follows: {{begin-eqn}} {{eqn | l = \\map {f^\\neg} \\F       | r = \\T }} {{eqn | l = \\map {f^\\neg} \\T       | r = \\F }} {{end-eqn}}"}
+{"_id": "23839", "title": "Definition:Conjunction/Truth Function", "text": "The conjunction connective defines the truth function $f^\\land$ as follows: {{begin-eqn}} {{eqn | l = \\map {f^\\land} {F, F}       | r = F }} {{eqn | l = \\map {f^\\land} {F, T}       | r = F }} {{eqn | l = \\map {f^\\land} {T, F}       | r = F }} {{eqn | l = \\map {f^\\land} {T, T}       | r = T }} {{end-eqn}}"}
+{"_id": "23840", "title": "Definition:Disjunction/Truth Function", "text": "The disjunction connective defines the truth function $f^\\lor$ as follows: {{begin-eqn}} {{eqn | l = f^\\lor \\left({F, F}\\right)       | r = F }} {{eqn | l = f^\\lor \\left({F, T}\\right)       | r = T }} {{eqn | l = f^\\lor \\left({T, F}\\right)       | r = T }} {{eqn | l = f^\\lor \\left({T, T}\\right)       | r = T }} {{end-eqn}}"}
+{"_id": "23841", "title": "Definition:Conditional/Truth Function", "text": "The conditional connective defines the truth function $f^\\to$ as follows: {{begin-eqn}} {{eqn | l = f^\\to \\left({F, F}\\right)       | r = T }} {{eqn | l = f^\\to \\left({F, T}\\right)       | r = T }} {{eqn | l = f^\\to \\left({T, F}\\right)       | r = F }} {{eqn | l = f^\\to \\left({T, T}\\right)       | r = T }} {{end-eqn}}"}
+{"_id": "23842", "title": "Definition:Biconditional/Truth Function", "text": "The biconditional connective defines the truth function $f^\\leftrightarrow$ as follows: {{begin-eqn}} {{eqn | l = f^\\leftrightarrow \\left({F, F}\\right)       | r = T }} {{eqn | l = f^\\leftrightarrow \\left({F, T}\\right)       | r = F }} {{eqn | l = f^\\leftrightarrow \\left({T, F}\\right)       | r = F }} {{eqn | l = f^\\leftrightarrow \\left({T, T}\\right)       | r = T }} {{end-eqn}}"}
+{"_id": "23843", "title": "Definition:Proper Relational Structure", "text": "Let $A$ be a set or class. Let $\\mathcal R$ be a relation on $A$. Then $(A, \\mathcal R)$ is a '''proper relational structure''' {{iff}}: : For each $a \\in A$, the preimage $\\mathcal R^{-1} \\left({a}\\right)$ of $a$ under $\\mathcal R$ is a set (or small class)."}
+{"_id": "23844", "title": "Definition:Transitive Closure (Set Theory)/Definition 2", "text": "Let $x$ be a set. For each natural number $n \\in \\N_{\\ge 0}$ let: : $\\bigcup^n x = \\underbrace{\\bigcup \\bigcup \\cdots \\bigcup}_n x$ Then the '''transitive closure''' of $x$ is the union of the sets: :$\\left\\{ {x}\\right\\}, x, \\bigcup x, \\bigcup^2 x, \\dots, \\bigcup^n x, \\dots$"}
+{"_id": "23845", "title": "Definition:Transitive Closure (Set Theory)/Definition 1", "text": "Let $x$ be a set. Then the '''transitive closure''' of $x$ is the smallest transitive superset of $x$."}
+{"_id": "23846", "title": "Definition:Positively Totally Ordered Semigroup", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a totally ordered semigroup. Then $\\left({S, \\circ, \\preceq}\\right)$ is a '''positively totally ordered semigroup''' iff for all $a, b \\in S$: : $a \\preceq a \\circ b$ : $b \\preceq a \\circ b$"}
+{"_id": "23847", "title": "Definition:Right Naturally Totally Ordered Semigroup", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a positively totally ordered semigroup. Then $\\left({S, \\circ, \\preceq}\\right)$ is a '''right naturally totally ordered semigroup''' iff for all $a, b \\in S$: : $a < b$ implies that for some $x \\in S$, $b = a \\circ x$."}
+{"_id": "23848", "title": "Definition:Left Naturally Totally Ordered Semigroup", "text": "Let $\\struct {S, \\circ, \\preceq}$ be a positively totally ordered semigroup. Then $\\struct {S, \\circ, \\preceq}$ is a '''left naturally totally ordered semigroup''' {{iff}} for all $a, b \\in S$: :$a < b$ implies that for some $y \\in S$, $b = x \\circ a$."}
+{"_id": "23849", "title": "Definition:Naturally Totally Ordered Semigroup", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a right naturally totally ordered semigroup and a left naturally totally ordered semigroup. Then $\\left({S, \\circ, \\preceq}\\right)$ is a '''naturally totally ordered semigroup'''."}
+{"_id": "23850", "title": "Definition:Ordinal/Definition 2", "text": "Let $A$ be a set. Then $A$ is an '''ordinal''' {{iff}} $A$ is: : transitive : epsilon-connected, that is: :: $\\forall x, y \\in A: x \\ne y \\implies x \\in y \\lor y \\in x$ : well-founded"}
+{"_id": "23851", "title": "Definition:Strongly Well-Founded Relation", "text": "Let $A$ be a class. Let $\\RR$ be a relation on $A$. Then $A$ is '''strongly well-founded''' {{iff}}: :whenever $B$ is a non-empty subclass of $A$, $B$ has an $\\RR$-minimal element."}
+{"_id": "23852", "title": "Definition:Well-Founded Class", "text": "Let $A$ be a class. Then $A$ is '''well-founded''' iff for every non-empty subclass $B \\subseteq A$: : $\\exists x \\in B: x \\cap B = \\varnothing$ {{MissingLinks|non-empty class}} Category:Definitions/Class Theory 3cuwtmdpfv67ljjzn1hhicw9nvskb8c"}
+{"_id": "23853", "title": "Definition:Top (Logic)/Truth Table", "text": "The characteristic truth table of the top constant $\\top$ of propositional logic is as follows: :$\\begin{array}{|c|} \\hline \\top \\\\ \\hline \\T \\\\ \\hline \\end{array}$"}
+{"_id": "23854", "title": "Definition:Bottom (Logic)/Truth Table", "text": "The characteristic truth table of the bottom constant $\\bot$ of propositional logic is as follows: :$\\begin{array}{|c|} \\hline \\bot \\\\ \\hline \\F \\\\ \\hline \\end{array}$"}
+{"_id": "23859", "title": "Definition:Polygon/Height", "text": "The '''height''' of a polygon is the length of a perpendicular from the base to the vertex most distant from the base. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book VI/4 - Height}}'' {{EuclidDefRefNocat|VI|4|Height}}"}
+{"_id": "23860", "title": "Definition:Linear Measure/Height", "text": "'''Height''', like depth, is used as a term for linear measure in a dimension perpendicular to both length and breadth. However, whereas depth has connotations of '''down''', '''height''' is used for distances '''up''' from the plane."}
+{"_id": "23862", "title": "Definition:Linear Measure/Breadth", "text": "'''Breadth''' is linear measure in a dimension perpendicular to length. In the context of a two-dimensional geometric figure, the '''breadth''' is in the plane of that figure. In a three-dimensional figure, the choice of which direction is referred to as '''breadth''' is often arbitrary."}
+{"_id": "23863", "title": "Definition:Dimension (Measurement)/Fundamental Dimensions", "text": "The SI-recommended '''fundamental dimensions''' are: :$\\mathsf M$: mass :$\\mathsf L$: length  :$\\mathsf T$: time :$\\Theta$: temperature :$\\mathsf I$: electric current :$\\mathsf N$: amount of substance :$\\mathsf J$: luminous intensity"}
+{"_id": "23864", "title": "Definition:Dimension (Measurement)/Units", "text": "Compare with units of measurement. This concept of '''dimension''' is more abstract than that of units, which are standard quantities of the particular dimension in question. === Examples === * Displacement has dimension $\\mathsf L$. * Velocity has dimension $\\mathsf {L T}^{-1}$ (change in displacement, that is length traveled, per unit of time). * Acceleration has dimension $\\mathsf {L T}^{-2}$ (change in velocity per unit of time). * Force has dimension $\\mathsf {M L T}^{-2}$ (mass times acceleration, from Newton's Second Law of Motion)."}
+{"_id": "23865", "title": "Definition:Linear Measure/Length", "text": "'''Length''' is linear measure taken in a particular direction. Usually, in multi-dimensional figures, the dimension in which the linear measure is greatest is referred to as '''length'''. It is the most widely used term for linear measure, as it is the standard term used when only one dimension is under consideration. '''Length''' is the fundamental notion of Euclidean geometry, never defined but regarded as an intuitive concept at the basis of every geometrical theorem."}
+{"_id": "23866", "title": "Definition:Linear Measure/Depth", "text": "'''Depth''' is linear measure in a dimension perpendicular to both length and breadth. The choice of '''depth''' is often arbitrary, although in two-dimensional diagrams of three-dimensional figures, '''depth''' is usually imagined as being the dimension perpendicular to the plane the figure is drawn in."}
+{"_id": "23867", "title": "Definition:Set/Implicit Set Definition/Infinite Set", "text": "If there is no end to the list of elements in the set, the ellipsis can be left open: :$S = \\set {1, 2, 3, \\ldots}$ which is taken to mean: :'''$S = $ the set containing $1, 2, 3, $ and so on for ever.'''"}
+{"_id": "23868", "title": "Definition:Endorelation/General Definition", "text": "An $n$-ary relation $\\RR$ on a cartesian space $S^n$ is an '''$n$-ary endorelation on $S$''': :$\\RR = \\struct {S, S, \\ldots, S, R}$ where $R \\subseteq S^n$."}
+{"_id": "23869", "title": "Definition:Well-Ordering/Definition 1", "text": "The ordering $\\preceq$ is a '''well-ordering''' on $S$ {{iff}} ''every'' non-empty subset of $S$ has a smallest element under $\\preceq$: :$\\forall T \\subseteq S, T \\ne \\O: \\exists a \\in T: \\forall x \\in T: a \\preceq x$"}
+{"_id": "23870", "title": "Definition:Well-Ordering/Definition 2", "text": "The ordering $\\preceq$ is a '''well-ordering''' on $S$ {{iff}} $\\preceq$ is a well-founded total ordering."}
+{"_id": "23871", "title": "Definition:Domain (Set Theory)/Relation/General Definition", "text": "Let $\\displaystyle \\prod_{i \\mathop = 1}^n S_i$ be the cartesian product of sets $S_1$ to $S_n$. Let $\\displaystyle \\mathcal R \\subseteq \\prod_{i \\mathop = 1}^n S_i$ be an $n$-ary relation on $\\displaystyle \\prod_{i \\mathop = 1}^n S_i$. The '''domain of $\\mathcal R$''' is the set defined as: :$\\displaystyle \\Dom {\\mathcal R} := \\set {\\tuple {s_1, s_2, \\ldots, s_{n - 1} } \\in \\prod_{i \\mathop = 1}^{n - 1} S_i: \\exists s_n \\in S_n: \\tuple {s_1, s_2, \\ldots, s_n} \\in \\mathcal R}$ The concept is usually encountered when $\\mathcal R$ is an endorelation on $S$: :$\\displaystyle \\Dom {\\mathcal R} := \\set {\\tuple {s_1, s_2, \\ldots, s_{n - 1} } \\in S^{n - 1}: \\exists s_n \\in S_n: \\tuple {s_1, s_2, \\ldots, s_n} \\in \\mathcal R}$"}
+{"_id": "23872", "title": "Definition:Image (Set Theory)/Relation/Relation/General Definition", "text": "Let $\\displaystyle \\prod_{i \\mathop = 1}^n S_i$ be the cartesian product of sets $S_1$ to $S_n$. Let $\\displaystyle \\mathcal R \\subseteq \\prod_{i \\mathop = 1}^n S_i$ be an $n$-ary relation on $\\displaystyle \\prod_{i \\mathop = 1}^n S_i$. The '''image of $\\mathcal R$''' is the set defined as: :$\\Img {\\mathcal R} := \\set {s_n \\in S_n: \\exists \\tuple {s_1, s_2, \\ldots, s_{n - 1} } \\in \\displaystyle \\prod_{i \\mathop = 1}^{n - 1} S_i: \\tuple {s_1, s_2, \\ldots, s_n} \\in \\mathcal R}$ The concept is usually encountered when $\\mathcal R$ is an endorelation on $S$: :$\\Img {\\mathcal R} := \\set {s_n \\in S: \\exists \\tuple {s_1, s_2, \\ldots, s_{n - 1} } \\in S^{n - 1}: \\tuple {s_1, s_2, \\ldots, s_n} \\in \\mathcal R}$"}
+{"_id": "23873", "title": "Definition:Strict Well-Ordering/Definition 1", "text": "Let $\\prec$ be a strict total ordering on a class $A$. Then $\\prec$ is a '''strict well-ordering''' on $A$ {{iff}} $\\prec$ is a foundational relation on $A$. That is, expressed symbolically: :${\\prec} \\mathrel{\\operatorname{We}} A \\iff \\left({\\prec \\operatorname{Or} A \\land {\\prec} \\mathrel{\\operatorname{Fr}} A}\\right)$"}
+{"_id": "23874", "title": "Definition:Strict Strong Well-Ordering", "text": "Let $A$ be a class. Let $\\mathcal R$ be a relation on $A$. Then $\\mathcal R$ is a '''strict strong well-ordering''' of $A$ {{iff}}: : $\\mathcal R$ connects $A$ : $\\mathcal R$ is strongly well-founded.  That is, whenever $B$ is a non-empty subclass of $A$, $B$ has an $\\mathcal R$-minimal element."}
+{"_id": "23875", "title": "Definition:Mapping/General Definition", "text": "Let $\\displaystyle \\prod_{i \\mathop = 1}^n S_i$ be the cartesian product of sets $S_1$ to $S_n$. Let $\\displaystyle \\RR \\subseteq \\prod_{i \\mathop = 1}^n S_i$ be an $n$-ary relation on $\\displaystyle \\prod_{i \\mathop = 1}^n S_i$. Then $\\RR$ is a '''mapping''' {{iff}}: :$\\displaystyle \\forall x := \\tuple {x_1, x_2, \\ldots, x_{n - 1} } \\in \\prod_{i \\mathop = 1}^{n - 1} S_i: \\forall y_1, y_2 \\in S_n: \\tuple {x, y_1} \\in \\RR \\land \\tuple {x, y_2} \\in \\RR \\implies y_1 = y_2$ and :$\\displaystyle \\forall x := \\tuple {x_1, x_2, \\ldots, x_{n - 1} } \\in \\prod_{i \\mathop = 1}^{n - 1} S_i: \\exists y \\in S_n: \\tuple {x, y} \\in \\RR$ Thus, a '''mapping''' is an $n$-ary relation which is: :Many-to-one :Left-total, that is, defined for all elements in the domain."}
+{"_id": "23876", "title": "Definition:Cartesian Product/Family of Sets/Definition 1", "text": "Let $I$ be an indexing set. Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of sets indexed by $I$. The '''Cartesian product of $\\family {S_i}_{i \\mathop \\in I}$''' is the set of all families $\\family {s_i}_{i \\mathop \\in I}$ with $s_i \\in S_i$ for each $i \\in I$. This can be denoted $\\displaystyle \\prod_{i \\mathop \\in I} S_i$ or, if $I$ is understood, $\\displaystyle \\prod_i S_i$."}
+{"_id": "23877", "title": "Definition:Cartesian Product/Family of Sets/Definition 2", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be an indexed family of sets. The '''Cartesian product of $\\family {S_i}_{i \\mathop \\in I}$''' is the set: :$\\displaystyle \\prod_{i \\mathop \\in I} S_i := \\set {f: \\paren {f: I \\to \\bigcup_{i \\mathop \\in I} S_i} \\land \\paren {\\forall i \\in I: \\paren {\\map f i \\in S_i} } }$ where $f$ denotes a mapping. When $S_i = S$ for all $i \\in I$, the expression is written: :$\\displaystyle S^I := \\set {f: \\paren {f: I \\to S} \\land \\paren {\\forall i \\in I: \\paren {\\map f i \\in S} } }$ which follows from Union is Idempotent: :$\\displaystyle \\bigcup_{i \\mathop \\in I} S = S$"}
+{"_id": "23878", "title": "Definition:Order Isomorphism/Wosets", "text": "Let $\\struct {S, \\preceq_1}$ and $\\struct {T, \\preceq_2}$ be well-ordered sets. Let $\\phi: S \\to T$ be a bijection such that $\\phi: S \\to T$ is order-preserving: :$\\forall x, y \\in S: x \\preceq_1 y \\implies \\map \\phi x \\preceq_2 \\map \\phi y$ Then $\\phi$ is an '''order isomorphism'''."}
+{"_id": "23879", "title": "Definition:Set-Like Relation", "text": "Let $\\mathcal R$ be a relation. Let $A$ be a class. Then $\\mathcal R$ is '''set-like''' on $A$ {{iff}}: :For all $x \\in A$, $\\left\\{{y \\in A: y \\mathrel{\\mathcal R} x}\\right\\}$ is a set."}
+{"_id": "23880", "title": "Definition:Boolean Algebra/Definition 2", "text": "{{:Definition:Boolean Algebra/Axioms/Definition 2}}"}
+{"_id": "23881", "title": "Definition:Strict Well-Ordering/Definition 2", "text": "Let $A$ be a set or class. Let $\\prec$ be a relation on $A$. Then $\\prec$ is a '''strict well-ordering''' of $A$ {{iff}}: : $\\prec$ connects $A$ : $\\prec$ is well-founded.  That is, whenever $b$ is a non-empty subset of $A$, $b$ has a $\\prec$-minimal element."}
+{"_id": "23882", "title": "Definition:Boolean Algebra/Axioms/Definition 1", "text": "A '''Boolean algebra''' is an algebraic system $\\struct {S, \\vee, \\wedge, \\neg}$, where $\\vee$ and $\\wedge$ are binary, and $\\neg$ is a unary operation. Furthermore, these operations are required to satisfy the following axioms: {{begin-axiom}} {{axiom | n = \\text {BA}_1 0         | lc=         | t = $S$ is closed under $\\vee$, $\\wedge$ and $\\neg$ }} {{axiom | n = \\text {BA}_1 1         | lc=         | t = Both $\\vee$ and $\\wedge$ are commutative }} {{axiom | n = \\text {BA}_1 2         | lc=         | t = Both $\\vee$ and $\\wedge$ distribute over the other }} {{axiom | n = \\text {BA}_1 3         | lc=         | t = Both $\\vee$ and $\\wedge$ have identities $\\bot$ and $\\top$ respectively }} {{axiom | n = \\text {BA}_1 4         | lc=         | t = $\\forall a \\in S: a \\vee \\neg a = \\top, a \\wedge \\neg a = \\bot$ }} {{end-axiom}}"}
+{"_id": "23883", "title": "Definition:Boolean Algebra/Axioms/Definition 2", "text": "A '''Boolean algebra''' is an algebraic system $\\struct {S, \\vee, \\wedge, \\neg}$, where $\\vee$ and $\\wedge$ are binary, and $\\neg$ is a unary operation. Furthermore, these operations are required to satisfy the following axioms: {{begin-axiom}} {{axiom | n = \\text {BA}_2 0         | lc= Closure:         | q = \\forall a, b \\in S         | m = a \\vee b \\in S }} {{axiom | m = a \\wedge b \\in S }} {{axiom | m = \\neg a \\in S }} {{axiom | n = \\text {BA}_2 1         | lc= Commutativity:         | q = \\forall a, b \\in S         | m = a \\vee b = b \\vee a }} {{axiom | m = a \\wedge b = b \\wedge a }} {{axiom | n = \\text {BA}_2 2         | lc= Associativity:         | q = \\forall a, b, c \\in S         | m = a \\vee \\paren {b \\vee c} = \\paren {a \\vee b} \\vee c }} {{axiom | m = a \\wedge \\paren {b \\wedge c} = \\paren {a \\wedge b} \\wedge c }} {{axiom | n = \\text {BA}_2 3         | lc= Absorption Laws:         | q = \\forall a, b \\in S         | m = \\paren {a \\wedge b} \\vee b = b }} {{axiom | m = \\paren {a \\vee b} \\wedge b = b }} {{axiom | n = \\text {BA}_2 4         | lc= Distributivity:         | q = \\forall a, b, c \\in S         | m = a \\wedge \\paren {b \\vee c} = \\paren {a \\wedge b} \\vee \\paren {a \\wedge c} }} {{axiom | m = a \\vee \\paren {b \\wedge c} = \\paren {a \\vee b} \\wedge \\paren {a \\vee c} }} {{axiom | n = \\text {BA}_2 5         | lc= Identity Elements:         | q = \\forall a, b \\in S         | m = \\paren {a \\wedge \\neg a} \\vee b = b }} {{axiom | m = \\paren {a \\vee \\neg a} \\wedge b = b }} {{end-axiom}}"}
+{"_id": "23884", "title": "Definition:Ordinal/Definition 1", "text": "Let $S$ be a set. Let $\\Epsilon \\! \\restriction_S$ be the restriction of the epsilon relation on $S$. Then $S$ is an '''ordinal''' {{iff}}: :$S$ is a transitive set :$\\Epsilon \\! \\restriction_S$ strictly well-orders $S$."}
+{"_id": "23885", "title": "Definition:Ordinal/Definition 3", "text": "An '''ordinal''' is a strictly well-ordered set $\\struct {S, \\prec}$ such that: :$\\forall a \\in S: S_a = a$ where $S_a$ is the initial segment of $S$ determined by $a$. From the definition of an initial segment, and Ordering on Ordinal is Subset Relation, we have that: :$S_a = \\set {x \\in S: x \\subsetneqq a}$ From Initial Segment of Ordinal is Ordinal we have that $S_a$ is itself an ordinal."}
+{"_id": "23886", "title": "Definition:Order Isomorphism/Definition 1", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be ordered sets. Let $\\phi: S \\to T$ be a bijection such that: : $\\phi: S \\to T$ is order-preserving : $\\phi^{-1}: T \\to S$ is order-preserving. Then $\\phi$ is an '''order isomorphism'''."}
+{"_id": "23887", "title": "Definition:Order Isomorphism/Definition 2", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be ordered sets. Let $\\phi: S \\to T$ be a surjective order embedding. Then $\\phi$ is an '''order isomorphism'''. That is, $\\phi$ is an '''order isomorphism''' {{iff}}: :$(1): \\quad \\phi$ is surjective :$(2): \\quad \\forall x, y \\in S: x \\preceq_1 y \\iff \\phi \\left({x}\\right) \\preceq_2 \\phi \\left({y}\\right)$"}
+{"_id": "23888", "title": "Definition:Order-Reflecting Mapping", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be ordered sets. Let $\\phi: S \\to T$ be a mapping. Then $\\phi$ is an '''order-reflecting mapping''' or '''reflects order''' {{iff}}: : $\\forall x, y \\in S: \\phi \\left({x}\\right) \\preceq_2 \\phi\\left({y}\\right) \\implies x \\preceq_1 y$"}
+{"_id": "23889", "title": "Definition:Probability Theory", "text": "'''Probability theory''' is the branch of mathematics which studies probability spaces."}
+{"_id": "23890", "title": "Definition:Event/Occurrence", "text": "Let the probability space of an experiment $\\EE$ be $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $A, B \\in \\Sigma$ be events, so that $A \\subseteq \\Omega$ and $B \\subseteq \\Omega$. Let the outcome of the experiment be $\\omega \\in \\Omega$. Then the following real-world interpretations of the '''occurrence''' of events can be determined: :If $\\omega \\in A$, then '''$A$ occurs'''. :If $\\omega \\notin A$, that is $\\omega \\in \\Omega \\setminus A$, then '''$A$ does not occur'''."}
+{"_id": "23891", "title": "Definition:Event Space/Discrete", "text": "Let $\\EE$ be an experiment. Let $\\Omega$ be a discrete sample space of $\\EE$. Then it is commonplace to take $\\Sigma$ to be the power set $\\powerset \\Omega$ of $\\Omega$, that is, the set of all possible subsets of $\\Omega$."}
+{"_id": "23892", "title": "Definition:Tangent Map", "text": "=== Differentable Mappings === {{:Definition:Tangent Map/Differentable Mappings}} === Affine Transformation === {{:Definition:Tangent Map/Affine Transformation}} Category:Definitions/Affine Geometry Category:Definitions/Differential Geometry pjf1hlk5fpsme5uopj9891idbe7tqn9"}
+{"_id": "23893", "title": "Definition:Probability Space/Discrete", "text": "Let $\\Omega$ be a discrete sample space. Then $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ is known as a '''discrete probability space'''."}
+{"_id": "23894", "title": "Definition:Probability Space/Continuous", "text": "Let $\\Omega$ be a continuum. Then $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ is known as a '''continuous probability space'''."}
+{"_id": "23895", "title": "Definition:Probability Function", "text": "Let $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ be a probability space. The probability measure $\\Pr$ on $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ can be considered as a function on elements of $\\Omega$ and $\\Sigma$."}
+{"_id": "23896", "title": "Definition:Strong Well-Ordering", "text": "Let $A$ be a class. Let $\\preceq$ be a relation on $A$. Then $\\preceq$ is a '''strong well-ordering''' iff: : $\\preceq$ is an ordering. : For each non-empty subclass $B \\subseteq A$, $B$ has a smallest element with respect to $\\preceq$."}
+{"_id": "23897", "title": "Definition:Associate/Integral Domain/Definition 1", "text": "'''$x$ is an associate of $y$ (in $D$)''' {{iff}} they are both divisors of each other. That is, $x$ and $y$ are '''associates (in $D$)''' {{iff}} $x \\divides y$ and $y \\divides x$."}
+{"_id": "23898", "title": "Definition:Associate/Commutative and Unitary Ring", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $x, y \\in R$. Then $x$ and $y$ are '''associates (in $R$)''' {{iff}} there exists a unit $u$ of $\\struct {R, +, \\circ}$ such that $u \\circ x = y$."}
+{"_id": "23899", "title": "Definition:Associate/Integral Domain/Definition 2", "text": "$x$ and $y$ are '''associates (in $D$)''' {{iff}}: :$\\ideal x = \\ideal y$ where $\\ideal x$ and $\\ideal y$ denote the ideals generated by $x$ and $y$ respectively."}
+{"_id": "23901", "title": "Definition:Independent Events/Definition 1", "text": "The events $A$ and $B$ are defined as '''independent (of each other)''' {{iff}} the occurrence of one of them does not affect the probability of the occurrence of the other one. Formally, $A$ is independent of $B$ {{iff}}: :$\\map \\Pr {A \\mid B} = \\map \\Pr A$ where $\\map \\Pr {A \\mid B}$ denotes the conditional probability of $A$ given $B$."}
+{"_id": "23902", "title": "Definition:Independent Events/Definition 2", "text": "The events $A$ and $B$ are defined as '''independent (of each other)''' {{iff}} the occurrence of both of them together has the same probability as the product of the probabilities of each of them occurring on their own. Formally, $A$ and $B$ are independent {{iff}}: :$\\map \\Pr {A \\cap B} = \\map \\Pr A \\map \\Pr B$"}
+{"_id": "23903", "title": "Definition:Independent Events/General Definition", "text": "Let $\\AA = \\family {A_i}_{i \\mathop \\in I}$ be an indexed family of events of $\\EE$. Then $\\AA$ is '''independent''' {{iff}}, for all finite subsets $J$ of $I$: :$\\displaystyle \\map \\Pr {\\bigcap_{i \\mathop \\in J} A_i} = \\prod_{i \\mathop \\in J} \\map \\Pr {A_i}$ That is, if the occurrence of any finite collection of $\\AA$ has the same probability as the product of each of those sets occurring individually. === Pairwise Independent === {{:Definition:Independent Events/Pairwise Independent}}"}
+{"_id": "23904", "title": "Definition:Independent Events/Pairwise Independent", "text": "Let $\\mathcal A = \\family {A_i}_{i \\mathop \\in I}$ be an indexed family of events of $\\EE$. Then $\\AA$ is '''pairwise independent''' {{iff}}: :$\\forall j, k \\in I: \\map \\Pr {A_j \\cap A_k} = \\map \\Pr {A_j} \\, \\map \\Pr {A_k}$ That is, if every pair of events of $\\EE$ are independent of each other. That is, $\\AA$ is '''pairwise independent''' {{iff}} the condition for general independence: :$\\displaystyle \\map \\Pr {\\bigcap_{i \\mathop \\in J} A_i} = \\prod_{i \\mathop \\in J} \\map \\Pr {A_i}$ holds whenever $\\card J = 2$."}
+{"_id": "23905", "title": "Definition:Independent Events/Dependent", "text": "If $A$ and $B$ are not independent, then they are '''dependent (on each other)''', and vice versa."}
+{"_id": "23906", "title": "Definition:Limit of Sequence of Events/Increasing", "text": "Let $\\left \\langle{A_n}\\right \\rangle_{n \\mathop \\in \\N}$ be an increasing sequence of events. Then the union: :$\\displaystyle A = \\bigcup_{i \\mathop \\in \\N} A_i$ of such a sequence is called the '''limit of the sequence $\\left \\langle{A_n}\\right \\rangle_{n \\in \\N}$'''."}
+{"_id": "23907", "title": "Definition:Limit of Sequence of Events/Decreasing", "text": "Let $\\left \\langle{A_n}\\right \\rangle_{n \\in \\N}$ be an decreasing sequence of events. Then the intersection: :$\\displaystyle A = \\bigcap_{i \\mathop \\in \\N} A_i$ of such a sequence is called the '''limit of the sequence $\\left \\langle{A_n}\\right \\rangle_{n \\in \\N}$'''."}
+{"_id": "23908", "title": "Definition:Probability Mass Function/Joint", "text": "Let $X: \\Omega \\to \\R$ and $Y: \\Omega \\to \\R$ both be discrete random variables on $\\struct {\\Omega, \\Sigma, \\Pr}$. Then the '''joint (probability) mass function''' of $X$ and $Y$ is the (real-valued) function $p_{X, Y}: \\R^2 \\to \\closedint 0 1$ defined as: :$\\forall \\tuple {x, y} \\in \\R^2: \\map {p_{X, Y} } {x, y} = \\begin {cases} \\map \\Pr {\\set {\\omega \\in \\Omega: \\map X \\omega = x \\land \\map Y \\omega = y} } & : x \\in \\Omega_X \\text { and } y \\in \\Omega_Y \\\\ 0 & : \\text {otherwise} \\end {cases}$ That is, $\\map {p_{X, Y} } {x, y}$ is the probability that the discrete random variable $X$ takes the value $x$ at the same time that the discrete random variable $Y$ takes the value $y$."}
+{"_id": "23910", "title": "Definition:Geometric Distribution/Shifted", "text": "$X$ has the '''shifted geometric distribution with parameter $p$''' {{iff}}: :$\\map X \\Omega = \\set {1, 2, \\ldots} = \\N_{>0}$ :$\\map \\Pr {X = k} = p \\paren {1 - p}^{k-1}$ where $0 < p < 1$. It is frequently seen as: :$\\map \\Pr {X = k} = q^{k - 1} p$ where $q = 1 - p$. It is written: :$X \\sim \\ShiftedGeometric p$"}
+{"_id": "23911", "title": "Definition:Negative Binomial Distribution/First Form", "text": "$X$ has the '''negative binomial distribution (of the first form) with parameters $n$ and $p$''' if: :$\\Img X = \\set {0, 1, 2, \\ldots}$ :$\\map \\Pr {X = k} = \\dbinom {n + k - 1} {n - 1} p^k \\paren {1 - p}^n$ where $0 < p < 1$. It is frequently seen as: :$\\map \\Pr {X = k} = \\dbinom {n + k - 1} {n - 1} p^k q^n$ where $q = 1 - p$."}
+{"_id": "23912", "title": "Definition:Negative Binomial Distribution/Second Form", "text": "$X$ has the '''negative binomial distribution (of the second form) with parameters $n$ and $p$''' if: :$\\Img X = \\set {n, n + 1, n + 2, \\dotsc}$ :$\\map \\Pr {X = k} = \\dbinom {k - 1} {n - 1} p^n \\paren {1 - p}^{k - n}$ where $0 < p < 1$. It is frequently seen as: :$\\map \\Pr {X = k} = \\dbinom {k - 1} {n - 1} q^{k - n} p^n $ where $q = 1 - p$."}
+{"_id": "23913", "title": "Definition:Variance/Discrete/Definition 1", "text": ":$\\var X := \\expect {\\paren {X - \\expect X}^2}$ That is: it is the expectation of the squares of the deviations from the expectation."}
+{"_id": "23914", "title": "Definition:Variance/Discrete/Definition 2", "text": ":$\\displaystyle \\var X := \\sum_{x \\mathop \\in \\Omega_X} \\paren {x - \\mu^2} \\map \\Pr {X = x}$ where: :$\\mu := \\expect X$ is the expectation of $X$ :$\\Omega_X$ is the image of $X$ :$\\map \\Pr {X = x}$ is the probability mass function of $X$."}
+{"_id": "23916", "title": "Definition:Countable Set/Definition 2", "text": "$S$ is '''countable''' {{iff}} it is finite or countably infinite."}
+{"_id": "23917", "title": "Definition:Countable Set/Definition 1", "text": "$S$ is '''countable''' {{iff}} there exists an injection: :$f: S \\to \\N$"}
+{"_id": "23918", "title": "Definition:Independent Random Variables/General Definition", "text": "Let $X = \\tuple {X_1, X_1, \\ldots, X_n}$ be an ordered tuple of random variables. Then $X$ is '''independent''' {{iff}}: :$\\displaystyle \\map \\Pr {X_1 = x_1, X_2 = x_2, \\ldots, X_n = x_n} = \\prod_{k \\mathop = 1}^n \\map \\Pr {X_k = x_k}$ for all $x = \\tuple {x_1, x_2, \\ldots, x_n} \\in \\R^n$."}
+{"_id": "23919", "title": "Definition:Independent Random Variables/Pairwise Independent", "text": "Let $X = \\tuple {X_1, X_1, \\ldots, X_n}$ be an ordered tuple of random variables. Then $X$ is '''pairwise independent''' {{iff}} $X_i$ and $X_j$ are independent (of each other) whenever $i \\ne j$."}
+{"_id": "23920", "title": "Definition:Independent Random Variables/Dependent", "text": "Let $X$ and $Y$ be random variables on $\\struct {\\Omega, \\Sigma, \\Pr}$. Then $X$ and $Y$ are defined as '''dependent (on each other)''' {{iff}} $X$ and $Y$ are not independent (of each other)."}
+{"_id": "23921", "title": "Definition:Euler's Number/Limit of Sequence", "text": "The sequence $\\sequence {x_n}$ defined as $x_n = \\paren {1 + \\dfrac 1 n}^n$ converges to a limit as $n$ increases without bound. That limit is called '''Euler's Number''' and is denoted $e$."}
+{"_id": "23922", "title": "Definition:Euler's Number/Limit of Series", "text": "The series $\\displaystyle \\sum_{n \\mathop = 0}^\\infty \\frac 1 {n!}$ converges to a limit. This limit is Euler's number $e$."}
+{"_id": "23923", "title": "Definition:Euler's Number/Base of Logarithm", "text": "The number $e$ can be defined as the number satisfied by: :$\\ln e = 1$ where $\\ln e$ denotes the natural logarithm of $e$. That $e$ is unique follows from Logarithm is Strictly Increasing."}
+{"_id": "23924", "title": "Definition:Euler's Number/Exponential Function", "text": "The number $e$ can be defined as the number satisfied by: :$e := \\exp 1 = e^1$ where $\\exp 1$ denotes the exponential function of $1$."}
+{"_id": "23925", "title": "Definition:Euler's Number/Decimal Expansion", "text": "The decimal expansion of Euler's number $e$ starts: :$2 \\cdotp 71828 \\, 18284 \\, 59045 \\, 23536 \\, 02874 \\, 71352 \\, 66249 \\, 77572 \\, 47093 \\, 69995 \\ldots$"}
+{"_id": "23926", "title": "Definition:Exponential Generating Function", "text": "Let $A = \\left \\langle {a_n}\\right \\rangle$ be a sequence in $\\R$. Then $\\displaystyle G_A \\left({z}\\right) = \\sum_{n \\mathop \\ge 0} \\frac {a_n} {n!} z^n$ is called the '''(exponential) generating function''' for the sequence $A$."}
+{"_id": "23927", "title": "Definition:Dismal Addition", "text": "'''Dismal addition''' is an operation defined on the natural numbers as follows. Let $a, b \\in \\N$ be expressed in decimal notation: :$\\displaystyle a = \\sum_{i \\mathop = 0}^n a_i 10^i = \\left[{a_n a_{n-1} \\ldots a_2 a_1 a_0}\\right]_{10}$ :$\\displaystyle b = \\sum_{j \\mathop = 0}^m b_j 10^j = \\left[{b_m b_{m-1} \\ldots b_2 b_1 b_0}\\right]_{10}$ The '''dismal sum''' of $a$ and $b$ is defined as: :$\\displaystyle a + b = \\sum_{k \\mathop = 0}^{\\max \\left\\{{m, n}\\right\\}} \\max \\left\\{{a_k, b_k}\\right\\} 10^k$ That is, for each pair of corresponding digits, the maximum is taken."}
+{"_id": "23928", "title": "Definition:Random Variable/Definition 1", "text": "Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $\\struct {X, \\Sigma'}$ be a measurable space. A '''random variable (on $\\struct {\\Omega, \\Sigma, \\Pr}$)''' is a $\\Sigma \\, / \\, \\Sigma'$-measurable mapping $f: \\Omega \\to X$."}
+{"_id": "23929", "title": "Definition:Random Variable/Definition 3", "text": ":$\\forall x \\in \\R: X^{-1} \\sqbrk {\\hointl {-\\infty} x} \\in \\Sigma$ where: :$\\hointl {-\\infty} x$ denotes the unbounded closed interval $\\set {y \\in \\R: y \\le x}$ :$X^{-1} \\sqbrk {\\hointl {-\\infty} x}$ denotes the preimage of $\\hointl {-\\infty} x$ under $X$."}
+{"_id": "23930", "title": "Definition:Random Variable/Definition 2", "text": "Let $\\EE$ be an experiment with a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. A '''random variable on $\\struct {\\Omega, \\Sigma, \\Pr}$''' is a mapping $X: \\Omega \\to \\R$ such that: :$\\forall x \\in \\R: \\set {\\omega \\in \\Omega: \\map X \\omega \\le x} \\in \\Sigma$"}
+{"_id": "23932", "title": "Definition:Tree (Graph Theory)/Node", "text": "The vertices of a tree are called its '''nodes'''."}
+{"_id": "23933", "title": "Definition:Tree (Graph Theory)/Infinite", "text": "A tree is '''infinite''' if it contains a (countably) infinite number of nodes."}
+{"_id": "23934", "title": "Definition:Tree (Graph Theory)/Finite", "text": "A tree is '''finite''' {{iff}} it contains a finite number of nodes."}
+{"_id": "23935", "title": "Definition:Ultrafilter on Set/Definition 4", "text": "Let $S$ be a non-empty set. Let $\\mathcal F$ be a non-empty set of subsets of $S$. Then $\\mathcal F$ is an '''ultrafilter''' on $S$ {{iff}} both of the following hold: : $\\mathcal F$ has the finite intersection property : For all $U \\subseteq S$, either $U \\in \\mathcal F$ or $U^c \\in \\mathcal F$ where $U^c$ is the complement of $U$ in $S$."}
+{"_id": "23936", "title": "Definition:Ultrafilter on Set/Definition 1", "text": "Let $S$ be a set. Let $\\FF \\subseteq \\powerset S$ be a filter on $S$. Then $\\FF$ is an '''ultrafilter (on $S$)''' {{iff}}: :there is no filter on $S$ which is strictly finer than $\\FF$ or equivalently, {{iff}}: :whenever $\\GG$ is a filter on $S$ and $\\FF \\subseteq \\mathcal G$ holds, then $\\FF = \\GG$."}
+{"_id": "23937", "title": "Definition:Finite Character", "text": "Let $S$ be a set. Let $\\mathcal F$ be a set of subsets of $S$. Then $\\mathcal F$ has '''finite character''' {{iff}} for each $A \\subseteq S$: : $A \\in \\mathcal F$ {{iff}} every finite subset of $A$ is in $\\mathcal F$."}
+{"_id": "23938", "title": "Definition:Rooted Tree/Ancestor Node/Proper", "text": "A '''proper ancestor node''' of $t$ is an ancestor node of $t$ that is not $t$ itself."}
+{"_id": "23939", "title": "Definition:Rooted Tree/Child Node", "text": "Let $T$ be a rooted tree with root $r_T$. Let $t$ be a node of $T$. The '''child nodes''' of $t$ are the elements of the set: :$\\left\\{{s \\in T: \\pi \\left({s}\\right) = t}\\right\\}$ where $\\pi \\left({s}\\right)$ denotes the parent mapping of $s$. That is, the '''children''' of $t$ are all the nodes of $T$ of which $t$ is the parent. === Grandchild Node === {{:Definition:Rooted Tree/Child Node/Grandchild Node}}"}
+{"_id": "23940", "title": "Definition:Rooted Tree/Proper Descendant", "text": "A '''proper descendant node''' of $t$ is a descendant of $t$ which is not $t$ itself."}
+{"_id": "23941", "title": "Definition:Rooted Tree/Descendant", "text": "Let $T$ be a rooted tree with root $r_T$. Let $t$ be a node of $T$. A '''descendant node''' $s$ of a $t$ is a node such that $t$ is in the path from $s$ to $r_T$. That is, the '''descendant nodes''' of $t$ are all the nodes of $T$ of which $t$ is an ancestor node. === Proper Descendant === {{:Definition:Rooted Tree/Proper Descendant}}"}
+{"_id": "23942", "title": "Definition:Rooted Tree/Sibling", "text": "Let $T$ be a rooted tree with root $r_T$. Two children of the same node of $T$ are called '''siblings'''. That is, '''siblings''' are nodes which both have the same parent."}
+{"_id": "23943", "title": "Definition:Rooted Tree/Branch/Infinite", "text": "Let $T$ be a rooted tree with root node $r_T$. Let $\\Gamma$ be a branch of $T$. Then $\\Gamma$ is infinite {{iff}} it has no leaf node at the end."}
+{"_id": "23944", "title": "Definition:Rooted Tree/Branch/Length", "text": "Let $T$ be a rooted tree with root node $r_T$. Let $\\Gamma$ be a finite branch of $T$. The '''length''' of $\\Gamma$ is defined as the number of ancestors of the leaf at the end of that branch."}
+{"_id": "23945", "title": "Definition:Rooted Tree/Branch", "text": "Let $T$ be a rooted tree with root node $r_T$. A subset $\\Gamma$ of $T$ is a '''branch''' {{iff}} all the following conditions hold: :$(1): \\quad$ The root node $r_T$ belongs to $\\Gamma$ :$(2): \\quad$ The parent of each node in $\\Gamma \\setminus \\left\\{{r_T}\\right\\}$ is in $\\Gamma$ :$(3): \\quad$ Each node in $\\Gamma$ either: ::$(a): \\quad$ is a leaf node of $T$ :or: ::$(b): \\quad$ has exactly one child node in $\\Gamma$."}
+{"_id": "23946", "title": "Definition:Rooted Tree/Root Node", "text": "Let $T$ be a rooted tree. The '''root node''' of $T$ is the node of $T$ which is distinguished from the others by being the ancestor node of every node of $T$."}
+{"_id": "23947", "title": "Definition:Finite Join Property", "text": "Let $(B, \\vee, \\wedge, \\neg, \\bot, \\top)$ be a Boolean algebra. Let $J \\subseteq B$. Then $J$ has the '''finite join property''' iff for any $x_1, x_2, \\dots, x_n \\in J$: : $x_1 \\vee x_2 \\vee \\dots \\vee x_n \\ne \\top$ Category:Definitions/Lattice Theory mp6xrnbd4t6fk1u28q618hjklixdisn"}
+{"_id": "23948", "title": "Definition:Finite Character/Mappings", "text": "Let $S$ and $T$ be sets. Let $\\mathcal F$ be a set of mappings from subsets of $S$ to $T$. That is, let $\\mathcal F$ be a set of partial mappings from $S$ to $T$. Then $\\mathcal F$ has '''finite character''' {{iff}} for each partial mapping $f \\subseteq S \\times T$: : $f \\in \\mathcal F$ {{iff}} for each finite subset $K$ of the domain of $f$, the restriction of $f$ to $K$ is in $\\mathcal F$."}
+{"_id": "23949", "title": "Definition:Rooted Tree/Branch/Finite", "text": "Let $T$ be a rooted tree with root node $r_T$. Let $\\Gamma$ be a branch of $T$. Then $\\Gamma$ is '''finite''' {{iff}} it has exactly one leaf node."}
+{"_id": "23951", "title": "Definition:Contradictory/Logical Formulas", "text": "Let $\\Delta$ be a set of propositional formulas. Then $\\Delta$ is '''contradictory''' if there exists some propositional formula $P$ such that $P \\in \\Delta$ and $\\neg P \\in \\Delta$."}
+{"_id": "23952", "title": "Definition:Lattice (Group Theory)", "text": "=== Definition 1 === {{:Definition:Lattice_(Group_Theory)/Definition_1}} === Definition 2 === {{:Definition:Lattice_(Group_Theory)/Definition_2}} Category:Definitions/Group Theory giz5748waqza4slf1bjgzp3rvd5si9i"}
+{"_id": "23953", "title": "Definition:Finished", "text": "=== Finished Set of WFFs of Propositional Logic === {{:Definition:Finished Set of WFFs of Propositional Logic}} === Finished Branch of Propositional Tableau === {{:Definition:Finished Branch of Propositional Tableau}} === Finished Propositional Tableau === {{:Definition:Finished Propositional Tableau}} Category:Definitions/Propositional Tableaus 7fyihzey4x8pypzxbqh54khqnw8ml2n"}
+{"_id": "23954", "title": "Definition:Null Sequence (Homological Algebra)", "text": "Let $\\left({R, +, \\cdot}\\right)$ be a ring. Let: :$(1): \\quad \\cdots \\longrightarrow M_i \\stackrel{d_i}{\\longrightarrow} M_{i+1} \\stackrel{d_{i+1}}{\\longrightarrow} M_{i+2} \\stackrel{d_{i+2}}{\\longrightarrow} \\cdots$ be a sequence of $R$-modules $M_i$ and $R$-module homomorphisms $d_i$. Then the sequence $(1)$ is '''null''' if $d_i \\circ d_{i+1} = 0$ for all $i$."}
+{"_id": "23955", "title": "Definition:Short Exact Sequence of Modules", "text": "Let $\\left({R, +, \\cdot}\\right)$ be a ring. Let: :$(1): \\quad \\cdots \\longrightarrow M_i \\stackrel{d_i}{\\longrightarrow} M_{i+1} \\stackrel{d_{i+1}}{\\longrightarrow} M_{i+2} \\stackrel{d_{i+2}}{\\longrightarrow} \\cdots$ be a sequence of $R$-modules $M_i$ and $R$-module homomorphisms $d_i$. Then the sequence $(1)$ is a '''short exact sequence''' if it exact, and is finite of the form ::$0 \\longrightarrow M_2 \\stackrel {d_2} {\\longrightarrow} M_3 \\stackrel {d_3} {\\longrightarrow} M_4 \\longrightarrow 0$"}
+{"_id": "23956", "title": "Definition:Harmonic Numbers/General Definition", "text": "Let $r \\in \\R_{>0}$. For $n \\in \\N_{> 0}$ the '''Harmonic numbers order $r$''' are defined as follows: :$\\displaystyle H_n^{\\paren r} = \\sum_{k \\mathop = 1}^n \\frac 1 {k^r}$"}
+{"_id": "23957", "title": "Definition:Homomorphism of Complexes", "text": "Let $\\left({R, +, \\cdot}\\right)$ be a ring. Let: :$M: \\quad \\cdots \\longrightarrow M_i \\stackrel {d_i} {\\longrightarrow} M_{i + 1} \\stackrel {d_{i + 1} } {\\longrightarrow} M_{i + 2} \\stackrel {d_{i + 2} } {\\longrightarrow} \\cdots$ and :$N: \\quad \\cdots \\longrightarrow N_i \\stackrel {d'_i} {\\longrightarrow} N_{i + 1} \\stackrel {d'_{i + 1} } {\\longrightarrow} N_{i + 2} \\stackrel {d'_{i + 2} } {\\longrightarrow} \\cdots$ be two differential complexes of $R$-modules. Let $\\phi = \\left\\{ {\\phi_i: i \\in \\Z}\\right\\}$ be a family of module homomorphisms $\\phi_i: M_i \\to N_i$. Then $\\phi$ is a '''homomorphism of complexes''' {{iff}} for each $i \\in \\Z$: :$\\phi_{i+1} \\circ d_i = \\phi_i \\circ d'_i$"}
+{"_id": "23958", "title": "Definition:Arborescence/Definition 1", "text": "$G$ is an '''arborescence of root $r$''' {{iff}}: :For each $v \\in V$ there is exactly one directed walk from $r$ to $v$."}
+{"_id": "23959", "title": "Definition:Arborescence/Definition 2", "text": "$G$ is an '''arborescence of root $r$''' {{iff}}: : $(1): \\quad$ $G$ is an orientation of a tree : $(2): \\quad$ For each $v \\in V$, $v$ is reachable from $r$."}
+{"_id": "23960", "title": "Definition:Arborescence", "text": "Let $G = \\left({V, A}\\right)$ be a directed graph. Let $r \\in V$. === Definition 1 === {{:Definition:Arborescence/Definition 1}} === Definition 2 === {{:Definition:Arborescence/Definition 2}} === Definition 3 === {{:Definition:Arborescence/Definition 3}}"}
+{"_id": "23961", "title": "Definition:Reachability Relation/Definition 1", "text": "Let $G = \\left({V, A}\\right)$ be a directed graph. Then the '''reachability relation''' of $G$ is the transitive closure of $A$."}
+{"_id": "23962", "title": "Definition:Reachability Relation/Definition 2", "text": "Let $G = \\left({V, A}\\right)$ be a directed graph. Let $\\mathcal R$ be the relation on $V$ defined by letting $x \\mathrel {\\mathcal R} y$ {{iff}} $y$ is reachable from $x$. That is, $x \\mathrel {\\mathcal R} y$ {{iff}} there exists a directed walk from $x$ to $y$. Then $\\mathcal R$ is the '''reachability relation''' of $G$."}
+{"_id": "23963", "title": "Definition:Reachability Relation", "text": "=== Definition 1 === {{:Definition:Reachability Relation/Definition 1}} === Definition 2 === {{:Definition:Reachability Relation/Definition 2}}"}
+{"_id": "23964", "title": "Definition:Reachable/Definition 1", "text": "Let $G = \\left({V, A}\\right)$ be a directed graph. Let $u, v \\in V$. Then $v$ is '''reachable''' from $u$ {{iff}} there exists a directed walk from $u$ to $v$."}
+{"_id": "23965", "title": "Definition:Reachable/Definition 2", "text": "Let $G = \\left({V, A}\\right)$ be a directed graph. Let $\\mathcal R$ be the reachability relation of $G$. That is, $\\mathcal R$ is the transitive closure of $A$. Let $u, v \\in V$. Then $v$ is '''reachable''' from $u$ {{iff}} $u \\mathrel {\\mathcal R} v$."}
+{"_id": "23966", "title": "Definition:Reachable", "text": "=== Definition 1 === {{:Definition:Reachable/Definition 1}} === Definition 2 === {{:Definition:Reachable/Definition 2}}"}
+{"_id": "23967", "title": "Definition:Directed Walk", "text": "Let $G = \\struct {V, A}$ be a directed graph. A '''directed walk''' in $G$ is a finite or infinite sequence $\\sequence {x_k}$ such that: :$\\forall k \\in \\N: k + 1 \\in \\Dom {\\sequence {x_k} }: \\tuple {x_k, x_{k + 1} } \\in A$"}
+{"_id": "23968", "title": "Definition:Arborescence/Definition 3", "text": "$G$ is an '''arborescence of root $r$''' {{iff}}: : $(1): \\quad$ Each vertex $v \\ne r$ is the final vertex of exactly one arc. : $(2): \\quad$ $r$ is not the final vertex of any arc. : $(3): \\quad$ For each $v \\in V$ such that $v \\ne r$ there is a directed walk from $r$ to $v$."}
+{"_id": "23969", "title": "Definition:Orientation (Graph Theory)", "text": "Let $G = \\struct {V, E}$ be a simple graph. Let $H = \\struct {V, A}$ be a directed graph. Then $H$ is an '''orientation''' of $G$ {{iff}} both of the following hold: :$G$ is a simple digraph. That is, $A$ is antisymmetric. :$\\forall x, y \\in V: \\paren {\\set {x, y} \\in E \\iff \\tuple {x, y} \\in A \\lor \\tuple {y, x} \\in A}$"}
+{"_id": "23971", "title": "Definition:Pseudocomplement", "text": "Let $(L, \\wedge, \\vee, \\preceq)$ be a lattice with smallest element $\\bot$. Let $x, x^* \\in L$. Then $x^*$ is the '''pseudocomplement''' of $x$ iff : $x^*$ is the greatest element of $L$ such that $x \\wedge x^* = \\bot$."}
+{"_id": "23972", "title": "Definition:Pseudocomplemented Lattice", "text": "Let $(L, \\wedge, \\vee, \\preceq)$ be a lattice with smallest element $\\bot$. Then $(L, \\wedge, \\vee, \\preceq)$ is a '''pseudocomplemented lattice''' iff each element $x$ of $L$ has a pseudocomplement. The pseudocomplement of $x$ is denoted $x^*$."}
+{"_id": "23973", "title": "Definition:Relative Pseudocomplement", "text": "Let $(L, \\wedge, \\vee, \\preceq)$ be a lattice. Let $x, y \\in L$. Then the '''relative pseudocomplement''' of $x$ with respect to $y$ is the greatest element $z \\in L$ such that $x \\wedge z \\preceq y$, if such an element exists. The relative pseudocomplement of $x$ with respect to $y$ is denoted $x \\to y$."}
+{"_id": "23974", "title": "Definition:Brouwerian Lattice", "text": "Let $\\left({L, \\wedge, \\vee, \\preceq}\\right)$ be a lattice. Then $\\left({L, \\wedge, \\vee, \\preceq}\\right)$ is a '''Brouwerian lattice''' {{iff}}: :for each $x, y \\in L$: $x$ has a relative pseudocomplement with respect to $y$. This pseudocomplement is denoted $x \\to y$."}
+{"_id": "23975", "title": "Definition:Heyting Algebra", "text": "Let $\\left({L, \\wedge, \\vee, \\preceq}\\right)$ be a lattice. Then $\\left({L, \\wedge, \\vee, \\preceq}\\right)$ is a '''Heyting algebra''' {{iff}}: : $(1): \\quad \\left({L, \\wedge, \\vee, \\preceq}\\right)$ is a Brouwerian lattice : $(2): \\quad L$ has a smallest element."}
+{"_id": "23976", "title": "Definition:Conditional/Notational Variants/Sign of Illation", "text": "The '''sign of illation''' $-\\!\\!\\!<$ is a notation invented by {{AuthorRef|Charles Sanders Peirce}} to denote the conditional operator. {{AuthorRef|Charles Sanders Peirce|Peirce}} derives $-\\!\\!\\!<$ as a variant of the sign $\\le$ for ''less than or equal to'', so as to denote that: : $A \\mathop {-\\!\\!\\!<} B$ represents the situation such that whenever a particular statement $A$ is true, then so is statement $B$."}
+{"_id": "23978", "title": "Definition:Infix Notation/Binary Relation", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a binary relation. When $\\left({s, t}\\right) \\in \\mathcal R$, we can write either: : $\\mathcal R \\left({s, t}\\right)$ or : $s \\mathop {\\mathcal R} t$ The notation $s \\mathop {\\mathcal R} t$ is known as '''infix notation'''."}
+{"_id": "23979", "title": "Definition:Set Partition/Definition 1", "text": "A '''partition''' of $S$ is a set of subsets $\\Bbb S$ of $S$ such that: :$(1): \\quad$ $\\Bbb S$ is pairwise disjoint: $\\forall S_1, S_2 \\in \\Bbb S: S_1 \\cap S_2 = \\O$ when $S_1 \\ne S_2$ :$(2): \\quad$ The union of $\\Bbb S$ forms the whole set $S$: $\\displaystyle \\bigcup \\Bbb S = S$ :$(3): \\quad$ None of the elements of $\\Bbb S$ is empty: $\\forall T \\in \\Bbb S: T \\ne \\O$."}
+{"_id": "23980", "title": "Definition:Set Partition/Definition 2", "text": "A '''partition''' of $S$ is a set of non-empty subsets $\\Bbb S$ of $S$ such that each element of $S$ lies in exactly one element of $\\Bbb S$."}
+{"_id": "23981", "title": "Definition:Cycle Decomposition", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\pi$ be a permutation on $S_n$. The '''cycle decomposition''' of $\\pi$ is the product of the disjoint cycles in which $\\pi$ can be expressed."}
+{"_id": "23982", "title": "Definition:Permutation on n Letters/Cycle Notation/Canonical Representation", "text": "The permutation: :$\\begin{pmatrix}   1 & 2 & 3 & 4 & 5 \\\\   2 & 1 & 4 & 3 & 5  \\end{pmatrix}$ can be expressed in '''cycle notation''' as: :$\\begin{pmatrix} 1 & 2 \\end{pmatrix} \\begin{pmatrix} 3 & 4 \\end{pmatrix}$ or as: :$\\begin{pmatrix} 3 & 4 \\end{pmatrix} \\begin{pmatrix} 5 \\end{pmatrix} \\begin{pmatrix} 1 & 2 \\end{pmatrix}$ or as: :$\\begin{pmatrix} 4 & 3 \\end{pmatrix} \\begin{pmatrix} 2 & 1 \\end{pmatrix}$ etc. However, only the first is conventional. This is known as the '''canonical representation'''."}
+{"_id": "23983", "title": "Definition:Cantor-Bendixson Rank", "text": "Let $\\struct {X, \\tau}$ be a topological space. Let $S \\subseteq X$. For each ordinal $\\alpha$, let $S^{\\paren \\alpha}$ be the $\\alpha$th Cantor-Bendixson derivative of $S$. Then the '''Cantor-Bendixson rank''' of $S$ is the least ordinal $\\alpha$ such that: :$S^{\\paren {\\alpha^+} } = S^{\\paren \\alpha}$ if such an ordinal exists. {{LinkWanted|Definition:Smallest Ordinal, to replace least ordinal}} {{NamedforDef|Georg Cantor|name2 = Ivar Otto Bendixson|cat = Cantor|cat2 = Bendixson}}"}
+{"_id": "23984", "title": "Definition:Sign of Permutation on n Letters", "text": "Let $n \\in \\N$ be a natural number. Let $N^*_{\\le n}$ denote the set of natural numbers $\\left\\{ {1, 2, \\ldots, n}\\right\\}$. Let $S_n$ denote the set of permutations on $N^*_{\\le n}$. Let $\\pi \\in S_n$ be a permutation of $N^*_{\\le n}$. Let $K$ be the cycle decomposition of $\\pi$. Let each cycle of $K$ be factored into transpositions. Let $k$ be the total number of transpositions that compose $K$. The '''sign of $\\pi$''' is defined as: :$\\forall \\pi \\in S_n: \\sgn \\paren \\pi = \\begin{cases} 1 & : k \\text{ even} \\\\ -1 & : k \\text{ odd} \\\\ \\end{cases}$"}
+{"_id": "23986", "title": "Definition:Set Equality/Definition 1", "text": "$S$ and $T$ are equal {{iff}} they have the same elements: :$S = T \\iff \\paren {\\forall x: x \\in S \\iff x \\in T}$"}
+{"_id": "23987", "title": "Definition:Set Equality/Definition 2", "text": "$S$ and $T$ are equal {{iff}} both: :$S$ is a subset of $T$ and :$T$ is a subset of $S$"}
+{"_id": "23988", "title": "Definition:Antisymmetric Relation/Definition 1", "text": "$\\RR$ is '''antisymmetric''' {{iff}}: :$\\tuple {x, y} \\in \\RR \\land \\tuple {y, x} \\in \\RR \\implies x = y$ that is: :$\\set {\\tuple {x, y}, \\tuple {y, x} } \\subseteq \\RR \\implies x = y$"}
+{"_id": "23989", "title": "Definition:Antisymmetric Relation/Definition 2", "text": "$\\mathcal R$ is '''antisymmetric''' {{iff}}: :$\\tuple {x, y} \\in \\mathcal R \\land x \\ne y \\implies \\tuple {y, x} \\notin \\mathcal R$"}
+{"_id": "23990", "title": "Definition:Direct Proof", "text": "A '''direct proof''' is an argument form which establishes the truth of a conclusion by assuming that the premises are true and demonstrating that the conclusion is necessarily true as a consequence."}
+{"_id": "23991", "title": "Definition:Graph (Graph Theory)/Edge/Join", "text": "Let $u$ and $v$ be vertices of $G$. Let $e = u v$ be an edge of $G$. Then $e$ '''joins''' the vertices $u$ and $v$."}
+{"_id": "23994", "title": "Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph", "text": "Let $G = \\struct {V, E}$ be an undirected graph. Two vertices $u, v \\in V$ of $G$ are '''adjacent''' if there exists an edge $e = \\set {u, v} \\in E$ of $G$ to which they are both incident."}
+{"_id": "23995", "title": "Definition:Adjacent (Graph Theory)/Vertices/Digraph", "text": "Let $G = \\struct {V, E}$ be a digraph. Two vertices $u, v \\in V$ of $G$ are '''adjacent''' if there exists an arc $e = \\left({u, v}\\right) \\in E$ of $G$ to which they are both incident."}
+{"_id": "23996", "title": "Definition:Adjacent (Graph Theory)/Vertices/Non-Adjacent", "text": "Let $G = \\struct {V, E}$ be a graph. Two vertices $u, v \\in V$ of $G$ are '''non-adjacent''' if they are not adjacent."}
+{"_id": "23997", "title": "Definition:Adjacent (Graph Theory)/Vertices", "text": "=== Undirected Graph === {{:Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph}} === Digraph === {{:Definition:Adjacent (Graph Theory)/Vertices/Digraph}}"}
+{"_id": "23998", "title": "Definition:Incident (Graph Theory)/Undirected Graph", "text": "Let $G = \\struct {V, E}$ be an undirected graph. Let $u, v \\in V$ be vertices of $G$. Let $e = \\set {u, v} \\in E$ be an edge of $G$: :320px Then: :$u$ and $v$ are each '''incident with $e$''' :$e$ is '''incident with $u$''' and '''incident with $v$'''."}
+{"_id": "23999", "title": "Definition:Incident (Graph Theory)/Digraph/Incident From", "text": ": $e$ is '''incident from $u$''' : $v$ is '''incident from $e$'''."}
+{"_id": "24000", "title": "Definition:Incident (Graph Theory)/Digraph/Incident To", "text": ": $e$ is '''incident to $v$''' : $u$ is '''incident to $e$'''."}
+{"_id": "24001", "title": "Definition:Incident (Graph Theory)/Digraph", "text": "Let $G = \\struct {V, E}$ be a digraph. Let $u, v \\in V$ be vertices of $G$. Let $e = \\tuple {u, v}$ be an arc that is directed from $u$ to $v$: :320px Then the following definitions are used: === Incident From === {{:Definition:Incident (Graph Theory)/Digraph/Incident From}} === Incident To === {{:Definition:Incident (Graph Theory)/Digraph/Incident To}}"}
+{"_id": "24002", "title": "Definition:Adjacent (Graph Theory)/Edges/Undirected Graph", "text": "Let $G = \\struct {V, E}$ be an undirected graph. Two edges $e_1, e_2 \\in E$ of $G$ '''adjacent''' if there exists a vertex $v \\in V$ to which they are both incident."}
+{"_id": "24003", "title": "Definition:Adjacent (Graph Theory)/Edges/Non-Adjacent", "text": "Let $G = \\struct {V, E}$ be a graph. Two edges $u, v \\in V$ of $G$ are '''non-adjacent''' if they are not adjacent."}
+{"_id": "24004", "title": "Definition:Adjacent (Graph Theory)/Edges/Digraph", "text": "Let $G = \\struct {V, E}$ be a digraph. Two arcs $e_1, e_2 \\in E$ of $G$ '''adjacent''' if there exists a vertex $v \\in V$ to which they are both incident."}
+{"_id": "24005", "title": "Definition:Adjacent (Graph Theory)/Edges", "text": "=== Undirected Graph === {{Definition:Adjacent (Graph Theory)/Edges/Undirected Graph}} === Digraph === {{Definition:Adjacent (Graph Theory)/Edges/Digraph}}"}
+{"_id": "24006", "title": "Definition:Adjacent (Graph Theory)/Faces", "text": "Let $G = \\left({V, E}\\right)$ be a planar graph. Two faces of $G$ are '''adjacent''' {{iff}} they are both incident to the same edge (or edges)."}
+{"_id": "24007", "title": "Definition:Directed Graph/Symmetric Digraph", "text": "Let $D = \\struct {V, E}$ be a digraph such that the relation $E$ in $D$ is symmetric. Then $D$ is called a '''symmetric digraph'''."}
+{"_id": "24008", "title": "Definition:Directed Graph/Arc/Endvertex", "text": "Let $D = \\struct {V, E}$ be a digraph. Let $e = u v$ be an arc of $D$, that is, $e \\in E$. The '''endvertices''' of $e$ are the vertices $u$ and $v$."}
+{"_id": "24009", "title": "Definition:Network/Undirected", "text": "An '''undirected network''' is a network whose underlying graph is an undirected graph: :450px"}
+{"_id": "24010", "title": "Definition:Underlying Graph", "text": "Let $N = \\left({V, E, f}\\right)$ be a network where $f: E \\to \\R$ is the mapping from the edge set $E$ to the real numbers $\\R$. The '''underlying graph''' of $N$ is the graph $G = \\left({V, E}\\right)$ consisting only of the vertex set $V$ and the edge set $E$. Category:Definitions/Graph Theory 3a2ultitf4nqj9xpxd1sawtxkt6hvuo"}
+{"_id": "24011", "title": "Definition:Network/Directed", "text": "A '''directed network''' is a network whose underlying graph is a digraph: :320px"}
+{"_id": "24012", "title": "Definition:Network/Loop-Network", "text": "A '''loop-network''' (directed or undirected) is a loop-graph together with a mapping which maps the edge set into the set $\\R$ of real numbers. That is, it is a '''network''' which may have loops."}
+{"_id": "24013", "title": "Definition:Network/Weight", "text": "Let $N = \\struct {V, E, w}$ be a network with weight function $w: E \\to \\R$. The values of the elements of $E$ under $w$ are known as the '''weights''' of the edges of $N$. The '''weights''' of a network $N$ can be depicted by writing the appropriate numbers next to the edges of the underlying graph of $N$."}
+{"_id": "24014", "title": "Definition:Network/Weight Function", "text": "Let $N = \\left({V, E, w}\\right)$ be a network. The mapping $w: E \\to \\R$ is known as the '''weight function''' of $N$."}
+{"_id": "24015", "title": "Definition:Multigraph/Simple Edge", "text": "Let $G = \\struct {V, E}$ be a multigraph. A '''simple edge''' is an edge $u v$ of $G$ which is the only edge of $G$ which is incident to both $u$ and $v$."}
+{"_id": "24016", "title": "Definition:Multigraph/Multiplicity", "text": "The '''multiplicity''' of a multigraph is the maximum multiplicity of its (multiple) edges."}
+{"_id": "24018", "title": "Definition:Loop-Graph/Loop-Multigraph", "text": "A '''loop-multigraph''' is a multigraph which is allowed to have loops: :380px"}
+{"_id": "24019", "title": "Definition:Graph (Graph Theory)/Notation", "text": "Let $G$ be a graph whose order is $p$ and whose size is $q$. Then $G$ can be referred to as a '''$\\tuple {p, q}$-graph'''. A wider category: a graph whose order is $n$ can be referred to as an '''$n$-graph'''."}
+{"_id": "24024", "title": "Definition:Path (Graph Theory)/Open", "text": "An '''open path''' is a path in which the first and last vertices are distinct."}
+{"_id": "24025", "title": "Definition:Walk (Graph Theory)/Open", "text": "An '''open walk''' is a walk whose first vertex and last vertex are distinct. That is, it is a walk which ends on a different vertex from the one where it starts."}
+{"_id": "24026", "title": "Definition:Connected (Graph Theory)/Vertices", "text": "Let $G$ be a graph. Two vertices $u, v \\in G$ are '''connected''' {{iff}} either: :$(1): \\quad u = v$ :$(2): \\quad u \\ne v$, and there exists a walk between them."}
+{"_id": "24027", "title": "Definition:Connected (Graph Theory)/Graph/Disconnected", "text": "Let $G$ be a graph. Then $G$ is '''disconnected''' {{iff}} it is not connected. That is, if there exists (at least) two vertices $u, v \\in G$ such that $u$ and $v$ are not connected."}
+{"_id": "24028", "title": "Definition:Connected (Graph Theory)/Graph", "text": "Let $G$ be a graph. Then $G$ is '''connected''' {{iff}} every pair of vertices in $G$ is connected."}
+{"_id": "24029", "title": "Definition:Cycle (Graph Theory)/Odd", "text": "An '''odd cycle''' is a cycle with odd length, that is, with an odd number of edges."}
+{"_id": "24030", "title": "Definition:Cycle (Graph Theory)/Even", "text": "An '''even cycle''' is a cycle with even length, that is, with an even number of edges."}
+{"_id": "24031", "title": "Definition:Edge Deletion", "text": "Let $G = \\struct {V, E}$ be an (undirected) graph. Let $F \\subseteq E$ be a set of edges of $G$. Then the '''graph obtained by deleting $F$ from $G$''', denoted by $G - F$, is the subgraph of $G$ containing the same vertices as $G$ but with all the elements of $F$ removed. That is: :$G - F = \\struct {V, E \\setminus F}$ Informally, $G - F$ is the graph obtained from $G$ by removing all edges in $F$. If $F$ is a singleton such that $F = \\set e$, then $G - F$ may be expressed $G - e$."}
+{"_id": "24032", "title": "Definition:Platonic Graph/Tetrahedron", "text": ":400px The graph of the regular tetrahedron is $3$-regular."}
+{"_id": "24033", "title": "Definition:Platonic Graph/Octahedron", "text": ":400px The graph of the regular octahedron is $4$-regular."}
+{"_id": "24034", "title": "Definition:Platonic Graph/Icosahedron", "text": ":400px The graph of the regular icosahedron is $5$-regular."}
+{"_id": "24035", "title": "Definition:Platonic Graph/Cube", "text": ":400px The graph of the cube is $3$-regular."}
+{"_id": "24036", "title": "Definition:Platonic Graph/Dodecahedron", "text": ":400px The graph of the regular dodecahedron is $3$-regular."}
+{"_id": "24037", "title": "Definition:Tree (Graph Theory)/Definition 1", "text": "A '''tree''' is a simple connected graph with no circuits."}
+{"_id": "24038", "title": "Definition:Tree (Graph Theory)/Definition 2", "text": "A '''tree''' is a simple connected graph with no cycles."}
+{"_id": "24039", "title": "Definition:Spanning Tree/Building-Up Method", "text": "Start with the edgeless graph $N$ whose vertices correspond with those of $G$. Select edges of $G$ one by one, such that no cycles are created, and add them to $N$. Continue till all vertices are included."}
+{"_id": "24040", "title": "Definition:Spanning Tree/Cutting-Down Method", "text": "Start with the graph $G$. Choose any cycle in $G$, and remove any one of its edges. By Condition for Edge to be Bridge, this will not disconnect $G$. Repeat this procedure till no cycles are left in $G$."}
+{"_id": "24041", "title": "Definition:Slope", "text": "=== Straight Line === {{:Definition:Slope/Straight Line}} === Curve === {{:Definition:Slope/Curve}}"}
+{"_id": "24042", "title": "Definition:Slope/Curve", "text": "Let $P$ be a point on a curve $C$. {{explain|Specify the domain -- it is apparent that a curve in a 2-dimensional space is probably to be inferred, but this needs to be made clear.}} The '''slope''' of $C$ at $P$ is defined as the slope of the tangent to $C$ at $P$."}
+{"_id": "24043", "title": "Definition:Slope/Straight Line", "text": "Let $L$ be a straight line. The '''slope''' of $L$ is defined as the change in $y$ divided by the change in $x$."}
+{"_id": "24044", "title": "Definition:Octahedron/Regular", "text": "A '''regular octahedron''' is an octahedron whose $8$ faces are all congruent equilateral triangles."}
+{"_id": "24045", "title": "Definition:Regular Polyhedron", "text": "A '''regular polyhedron''' is a polyhedron: :$(1): \\quad$ whose faces are congruent regular polygons :$(2): \\quad$ each of whose vertices is the common vertex of the same number of faces."}
+{"_id": "24046", "title": "Definition:Polyhedron", "text": "A '''polyhedron''' is a three-dimensional figure that is contained by polygonal faces."}
+{"_id": "24047", "title": "Definition:Platonic Solid", "text": "A '''platonic solid''' is a convex polyhedron: :$(1): \\quad$ whose faces are congruent regular polygons :$(2): \\quad$ each of whose vertices is the common vertex of the same number of faces."}
+{"_id": "24048", "title": "Definition:Prism", "text": ":500px {{EuclidDefinition|book = XI|def = 13|name = Prism}}"}
+{"_id": "24049", "title": "Definition:Cylinder", "text": "A '''cylinder''' is a solid made by rotating a rectangle along one of its sides."}
+{"_id": "24050", "title": "Definition:Kernel of Homomorphism of Differential Complexes", "text": "Let $\\left({R, +, \\cdot}\\right)$ be a ring. Let: :$M: \\quad \\cdots \\longrightarrow M_i \\stackrel{d_i}{\\longrightarrow} M_{i+1} \\stackrel{d_{i+1}}{\\longrightarrow} M_{i+2} \\stackrel{d_{i+2}}{\\longrightarrow} \\cdots$ and :$N: \\quad \\cdots \\longrightarrow N_i \\stackrel{d'_i}{\\longrightarrow} N_{i+1} \\stackrel{d'_{i+1}}{\\longrightarrow} N_{i+2} \\stackrel{d'_{i+2}}{\\longrightarrow} \\cdots$ be two differential complexes of $R$-modules. Let $\\phi = \\left\\{ \\phi_i : i \\in \\Z \\right\\}$ be a homomorphism $M \\to N$. For each $i \\in \\Z$ let $K_i$ be the kernel of $\\phi_i$. For each $i \\in \\Z$ let $f_i$ be the restriction of $d_i$ to $K_i$. Then the '''kernel''' of $\\phi$ is: :$\\ker \\phi : \\quad \\cdots \\longrightarrow K_i \\stackrel{f_i}{\\longrightarrow} K_{i+1} \\stackrel{f_{i+1}}{\\longrightarrow} K_{i+2} \\stackrel{f_{i+2}}{\\longrightarrow} \\cdots$"}
+{"_id": "24051", "title": "Definition:Angle/Rectilineal", "text": "{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book I/9 - Rectilineal Angle}}'' {{EuclidDefRefNocat|I|9|Rectilineal Angle}} Thus the distinction is made between straight-line angles and curved-line angles. Most of the time the fact that angles are rectilineal is taken for granted."}
+{"_id": "24052", "title": "Definition:Angle/Subtend", "text": "Let $AB$ be a line segment and $C$ be a point: :250px The line segment $AB$ is said to '''subtend''' the angle $\\angle ACB$."}
+{"_id": "24053", "title": "Definition:Angle/Adjacent", "text": "Two angles are '''adjacent''' if they have an intersecting line in common: :250px"}
+{"_id": "24054", "title": "Definition:Angle/Unit", "text": "The usual units of measurement for angle are as follows: === Degree === {{:Definition:Angle/Unit/Degree}} === Minute === {{:Definition:Angle/Unit/Minute}} === Second === {{:Definition:Angle/Unit/Second}} === Radian === {{:Definition:Angle/Unit/Radian}}"}
+{"_id": "24055", "title": "Definition:Addition/Integers", "text": "The '''addition operation''' in the domain of integers $\\Z$ is written $+$. We have that the set of integers is the Inverse Completion of Natural Numbers. Thus it follows that elements of $\\Z$ are the isomorphic images of the elements of equivalence classes of $\\N \\times \\N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same. Thus addition can be formally defined on $\\Z$ as the operation induced on those equivalence classes as specified in the definition of integers. That is, the integers being defined as all the difference congruence classes, integer addition can be defined directly as the operation induced by natural number addition on these congruence classes: :$\\forall \\tuple {a, b}, \\tuple {c, d} \\in \\N \\times \\N: \\eqclass {a, b} \\boxminus + \\eqclass {c, d} \\boxminus = \\eqclass {a + c, b + d} \\boxminus$"}
+{"_id": "24056", "title": "Definition:Addition/Real Modulo Addition", "text": "Let $z \\in \\R$. Let $\\R_z$ be the set of residue classes modulo $z$ of $\\R$. The addition operation is defined on $\\R_z$ as follows: :$\\eqclass a z +_z \\eqclass b z = \\eqclass {a + b} z$ This operation is called '''addition modulo $z$'''."}
+{"_id": "24057", "title": "Definition:Urn", "text": "An '''urn''' is a hypothetical container from which one may draw one of a number of objects (usually defined to be balls) of assorted colours. The idea is that one cannot see the ball one is withdrawing until it is out of the urn. Therefore it is not possible to know in advance what colour ball one is taking. The probability that a particular ball being withdrawn has a particular colour depends entirely on the distribution within the urn of the various colours of the balls within. This is used as an abstraction of a randomizing device in probability theory."}
+{"_id": "24058", "title": "Definition:Quadrilateral/Square", "text": "A '''square''' is a regular quadrilateral. That is, a regular polygon with $4$ sides. That is, a '''square''' is a plane figure with four sides all the same length and whose angles are all equal. :200px"}
+{"_id": "24059", "title": "Definition:Quadrilateral/Oblong", "text": "An '''oblong''' is a quadrilateral whose angles are all right angles, but whose sides are ''not'' all the same length: :400px"}
+{"_id": "24060", "title": "Definition:Quadrilateral/Rectangle/Containment", "text": "{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book II/1 - Containment of Rectangle}}'' {{EuclidDefRefNocat|II|1|Containment of Rectangle}}"}
+{"_id": "24061", "title": "Definition:Quadrilateral/Rectangle", "text": "A '''rectangle''' is a quadrilateral all of whose angles are equal to a right angle, and whose sides ''may or may not'' all be the same length."}
+{"_id": "24062", "title": "Definition:Quadrilateral/Parallelogram/Base", "text": "In a given parallelogram, one of the sides is distinguished as being the '''base'''. It is immaterial which is so chosen, but usual practice is that it is one of the two longer sides. In the parallelogram above, line $AB$ is considered to be the '''base'''."}
+{"_id": "24063", "title": "Definition:Quadrilateral/Parallelogram/Altitude", "text": "An '''altitude''' of a parallelogram is a line drawn perpendicular to its base, through one of its vertices to the side opposite to the base (which is extended if necessary). In the parallelogram above, line $DE$ is an '''altitude''' of the parallelogram $ABCD$. The term is also used for the length of such a line."}
+{"_id": "24064", "title": "Definition:Quadrilateral/Parallelogram", "text": "A '''parallelogram''' is a quadrilateral whose opposite sides are parallel to each other, and whose sides ''may or may not'' all be the same length.   :450px"}
+{"_id": "24065", "title": "Definition:Quadrilateral/Rhombus", "text": "A '''rhombus''' is a parallelogram whose sides are all the same length. :350px Its angles ''may or may not'' all be equal."}
+{"_id": "24066", "title": "Definition:Quadrilateral/Rhomboid", "text": "A '''rhomboid''' is a parallelogram whose sides are ''not'' all the same length. Its angles ''may or may not'' all be equal."}
+{"_id": "24067", "title": "Definition:Quadrilateral/Trapezoid", "text": "A '''trapezoid''' is a quadrilateral which has '''exactly one''' pair of sides parallel: :700px"}
+{"_id": "24068", "title": "Definition:Quadrilateral/Trapezium", "text": "A '''trapezium''' is a quadrilateral with no parallel sides. :300px"}
+{"_id": "24069", "title": "Definition:Quadrilateral/Kite", "text": "A '''kite''' is an irregular quadrilateral which has both pairs of adjacent sides equal. :300px"}
+{"_id": "24070", "title": "Definition:Quadrilateral/Dart", "text": "A '''dart''' is an irregular quadrilateral with a reflex angle. :500px"}
+{"_id": "24071", "title": "Definition:Boundary (Geometry)/Containment", "text": "A geometric figure is said to be '''contained''' by its boundary or boundaries."}
+{"_id": "24072", "title": "Definition:Polygon/Side", "text": "The line segments which make up a polygon are known as its '''sides'''. Thus, in the polygon above, the '''sides''' are identified as $a, b, c, d$ and $e$."}
+{"_id": "24073", "title": "Definition:Polygon/Vertex", "text": "A corner of a polygon is known as a '''vertex'''. Thus, in the polygon above, the vertices are $A, B, C, D$ and $E$."}
+{"_id": "24074", "title": "Definition:Polygon/Adjacent", "text": "=== Adjacent Side to Vertex === {{:Definition:Polygon/Adjacent/Side to Vertex}} === Adjacent Sides === {{:Definition:Polygon/Adjacent/Sides}} === Adjacent Vertex to Side === {{:Definition:Polygon/Adjacent/Vertex to Side}} === Adjacent Vertices === {{:Definition:Polygon/Adjacent/Vertices}}"}
+{"_id": "24075", "title": "Definition:Polygon/Opposite", "text": "When a polygon has an even number of sides, each side has an '''opposite side''', and each vertex likewise has an '''opposite vertex'''. When a polygon has an odd number of sides, each side has an '''opposite vertex'''. The '''opposite side''' (or '''opposite vertex''') to a given side (or '''vertex''') is that side (or vertex) which has the same number of sides between it and the side (or vertex) in question."}
+{"_id": "24076", "title": "Definition:Polygon/Internal Angle", "text": "The '''internal angle''' of a vertex of a polygon is the size of the angle between the sides adjacent to that vertex, as measured ''inside'' the polygon."}
+{"_id": "24077", "title": "Definition:Polygon/External Angle", "text": "Contrary to intuition, the '''external angle''' of a vertex of a polygon is ''not'' the size of the angle between the sides forming that vertex, as measured outside the polygon. An '''external angle''' is in fact an angle formed by one side of a polygon and a line produced from an adjacent side. :200px While $\\angle AFE$ is the internal angle of vertex $F$, the '''external angle''' of this vertex is $\\angle EFG$."}
+{"_id": "24078", "title": "Definition:Polygon/Base", "text": "For a given polygon, any one of its sides may be temporarily distinguished from the others, and referred to as the '''base'''. It is immaterial which is so chosen. The usual practice is that the polygon is drawn so that the '''base''' is made horizontal, and at the bottom."}
+{"_id": "24079", "title": "Definition:Polygon/Equilateral", "text": "An '''equilateral polygon''' is a polygon in which all the sides are the same length."}
+{"_id": "24080", "title": "Definition:Polygon/Equiangular", "text": "An '''equiangular polygon''' is a polygon in which all the vertices have the same angle."}
+{"_id": "24081", "title": "Definition:Polygon/Regular", "text": "A '''regular polygon''' is a polygon which is both equilateral and equiangular. That is, in which all the sides are the same length, and all the vertices have the same angle: :300px"}
+{"_id": "24082", "title": "Definition:Polygon/Multi-lateral", "text": "A '''multi-lateral polygon''' is a term to define a polygon with more than four sides."}
+{"_id": "24083", "title": "Definition:Triangle (Geometry)/Right-Angled/Hypotenuse", "text": "In a right-angled triangle, the opposite side to the right angle is known as the '''hypotenuse'''. In the above figure, the side labeled $b$ is the '''hypotenuse'''."}
+{"_id": "24084", "title": "Definition:Triangle (Geometry)/Right-Angled/Adjacent", "text": "In a right-angled triangle, for a given non-right angled vertex, the adjacent side which is ''not'' the hypotenuse is referred to as '''the adjacent'''. In the above figure: : the '''adjacent''' to vertex $A$ is side $c$ : the '''adjacent''' to vertex $C$ is side $a$."}
+{"_id": "24085", "title": "Definition:Geometric Figure/Plane Figure", "text": "A '''plane figure''' is a geometric figure embedded in the plane."}
+{"_id": "24086", "title": "Definition:Geometric Figure/Three-Dimensional Figure", "text": "A '''three-dimensional figure''' is a geometric figure which cannot be embedded in the plane, but which '''can''' be embedded in three-dimensional space."}
+{"_id": "24088", "title": "Definition:Geometric Figure/Diameter", "text": "The '''diameter''' of a geometric figure is the greatest length that can be formed between two opposite parallel straight lines that can be drawn tangent to its boundary."}
+{"_id": "24089", "title": "Definition:Triangle (Geometry)/Isosceles/Base", "text": "The base of an isosceles triangle is specifically defined to be the side which is a different length from the other two. In the above diagram, $BC$ is the '''base'''."}
+{"_id": "24090", "title": "Definition:Triangle (Geometry)/Isosceles/Base Angles", "text": "The two (equal) vertices adjacent to the base of an isosceles triangle are called the '''base angles'''. In the above diagram, $\\angle ABC$ and $\\angle ACB$ are the '''base angles'''."}
+{"_id": "24091", "title": "Definition:Triangle (Geometry)/Isosceles/Apex", "text": "The vertex opposite the base of an isosceles triangle is called the '''apex''' of the triangle. In the above diagram, $A$ is the '''apex'''."}
+{"_id": "24092", "title": "Definition:Triangle (Geometry)/Isosceles/Legs", "text": "The sides of an isosceles triangle which are adjacent to the apex are called the '''legs''' of the triangle. In the above diagram, $AB$ and $AC$ are the '''legs'''."}
+{"_id": "24093", "title": "Definition:Triangle (Geometry)/Adjacent", "text": "The two sides of a triangle which form a particular vertex are referred to as '''adjacent''' to that angle. Similarly, the two vertices of a triangle to which a particular side contributes are referred to as '''adjacent''' to that side."}
+{"_id": "24094", "title": "Definition:Triangle (Geometry)/Opposite", "text": "The side of a triangle which is ''not'' one of the sides adjacent to a particular vertex is referred to as its '''opposite'''. Thus, each vertex has an opposite side, and each side has an opposite vertex."}
+{"_id": "24095", "title": "Definition:Triangle (Geometry)/Base", "text": "For a given triangle, one of the sides can be distinguished as being the '''base'''. It is immaterial which is so chosen. The usual practice is that the triangle is drawn so that the '''base''' is made horizontal, and at the bottom. In the above diagram, it would be conventional for the side $AC$ to be identified as the '''base'''."}
+{"_id": "24096", "title": "Definition:Triangle (Geometry)/Apex", "text": "Having selected one side of a triangle to be the base, the opposite vertex to that base is called the '''apex'''. In the above diagram, if $AC$ is taken to be the base of $\\triangle ABC$, then $B$ is the '''apex'''."}
+{"_id": "24097", "title": "Definition:Triangle (Geometry)/Height", "text": "The '''height''' of a triangle is the length of a perpendicular from the apex to whichever side has been chosen as its base. That is, the length of the '''altitude''' so defined."}
+{"_id": "24098", "title": "Definition:Triangle (Geometry)/Nomenclature", "text": ":300px The vertices of a triangle are conventionally labeled $A, B, C$ (or with other uppercase letters), and the sides with lowercase letters corresponding to the opposite vertex, as above. In order to emphasize that a particular vertex being referred to is in fact a vertex, the symbol $\\angle$ is often placed by the letter corresponding to that vertex. Thus, for example: : $\\angle A$ is adjacent to sides $b$ and $c$ : Side $a$ is adjacent to $\\angle B$ and $\\angle C$ : $\\angle A$ is opposite side $a$ : Side $a$ is opposite $\\angle A$."}
+{"_id": "24099", "title": "Definition:Triangle (Geometry)/Equilateral", "text": "An '''equilateral triangle''' is a triangle in which all three sides are the same length: That is, a regular polygon with $3$ sides. :300px"}
+{"_id": "24100", "title": "Definition:Triangle (Geometry)/Scalene", "text": "A '''scalene triangle''' is a triangle in which all three sides are of different lengths. :500px"}
+{"_id": "24102", "title": "Definition:Triangle (Geometry)/Acute", "text": "An '''acute triangle''' is a triangle in which all three of the vertices are acute angles."}
+{"_id": "24103", "title": "Definition:Triangle (Geometry)/Obtuse", "text": "An '''obtuse triangle''' is a triangle in which one of the vertices is an obtuse angle."}
+{"_id": "24104", "title": "Definition:Quadrilateral/Trapezoid/Height", "text": "The '''height''' of a '''trapezoid''' is defined as the length of a line perpendicular to the bases. In the above diagram, the '''heights''' of the given trapezoids are indicated by the letter $h$."}
+{"_id": "24105", "title": "Definition:Icosahedron/Regular", "text": "A '''regular icosahedron''' is an icosahedron whose $20$ faces are all congruent equilateral triangles."}
+{"_id": "24106", "title": "Definition:Dodecahedron/Regular", "text": "A '''regular dodecahedron''' is a dodecahedron whose $12$ faces are all congruent regular pentagons."}
+{"_id": "24107", "title": "Definition:Tetrahedron/Regular", "text": "A '''regular tetrahedron''' is a tetrahedron whose $4$ faces are all congruent equilateral triangles."}
+{"_id": "24108", "title": "Definition:Plane Number", "text": "A '''plane number''' is the product of two (natural) numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book VII/16 - Plane Number}}'' {{EuclidDefRefNocat|VII|16|Plane Number}}"}
+{"_id": "24109", "title": "Definition:Solid Number", "text": "A '''solid number''' is the product of three (natural) numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book VII/17 - Solid Number}}'' {{EuclidDefRefNocat|VII|17|Solid Number}}"}
+{"_id": "24110", "title": "Definition:Incommensurable", "text": "Let $a, b \\in \\R_{>0}$ be (strictly) positive real numbers. $a$ and $b$ are '''incommensurable''' {{iff}} $\\dfrac a b$ is irrational. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X/1 - Commensurable}}'' {{EuclidDefRefNocat|X|1|Commensurable}}"}
+{"_id": "24111", "title": "Definition:Commensurable", "text": "Let $a, b \\in \\R_{>0}$ be (strictly) positive real numbers. $a$ and $b$ are '''commensurable''' iff $\\dfrac a b$ is rational. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X/1 - Commensurable}}'' {{EuclidDefRefNocat|X|1|Commensurable}}"}
+{"_id": "24112", "title": "Definition:Inclination", "text": "=== Straight Line to Plane === {{Definition:Inclination/Straight Line to Plane}} === Plane to Plane === {{Definition:Inclination/Plane to Plane}}"}
+{"_id": "24113", "title": "Definition:Similar Inclination", "text": "{{EuclidDefinition|book=XI|def=7|name=Similarly Inclined}} Category:Definitions/Angles Category:Definitions/Solid Geometry o1c4yyqx26loroltm7v20o0ja4d48u9"}
+{"_id": "24114", "title": "Definition:Sphere/Geometry/Axis", "text": "By definition, a sphere is made by turning a semicircle around a straight line. That straight line is called the '''axis of the sphere'''. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book XI/15 - Axis of Sphere}}'' {{EuclidDefRefNocat|XI|15|Axis of Sphere}}"}
+{"_id": "24115", "title": "Definition:Sphere/Geometry/Diameter", "text": "The '''diameter of a sphere''' is the length of any straight line drawn from a point on the surface to another point on the surface through the center."}
+{"_id": "24116", "title": "Definition:Cylinder/Axis", "text": "{{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book XI/22 - Axis of Cylinder}}'' {{EuclidDefRefNocat|XI|22|Axis of Cylinder}} In the above diagram, the '''axis''' of the cylinder $ACBEFD$ is the straight line $GH$."}
+{"_id": "24117", "title": "Definition:Cylinder/Base", "text": "{{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book XI/23 - Base of Cylinder}}'' {{EuclidDefRefNocat|XI|23|Base of Cylinder}} In the above diagram, the '''bases''' of the cylinder $ACBEDF$ are the faces $ABC$ and $DEF$."}
+{"_id": "24118", "title": "Definition:Cylinder/Similar Cylinders", "text": "Let $h_1$ and $h_2$ be the heights of two cylinders. Let $d_1$ and $d_2$ be the diameters of the bases of the two cylinders. Then the two cylinders are '''similar''' {{iff}}: :$\\dfrac {h_1} {h_2} = \\dfrac {d_1} {d_2}$ {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book XI/24 - Similar Cones and Cylinders}}'' {{EuclidDefRefNocat|XI|24|Similar Cones and Cylinders}}"}
+{"_id": "24119", "title": "Definition:Segment of Circle/Base", "text": "The '''base''' of a segment of a circle is the straight line forming one of the boundaries of the seqment. In the above diagram, $AB$ is the '''base''' of the highlighted segment."}
+{"_id": "24120", "title": "Definition:Segment of Circle/Angle of Segment", "text": "{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book III/7 - Angle of Segment}}'' {{EuclidDefRefNocat|III|7|Angle of Segment}} That is, it is the angle the base makes with the circumference where they meet. It can also be defined as the angle between the base and the tangent to the circle at the end of the base: :300px"}
+{"_id": "24123", "title": "Definition:Sector/Angle", "text": "The '''angle of a sector''' is the angle between the two radii which delimit the sector. In the above diagram, the '''angle of the sector''' $BAC$ is $\\theta$. The conjugate angle of $\\theta$ also forms a '''sector''', denoted $CAB$ (see above)."}
+{"_id": "24124", "title": "Definition:Inclination/Straight Line to Plane", "text": "{{EuclidDefinition|book = XI|def = 5|name = Inclination of Straight Line}}"}
+{"_id": "24125", "title": "Definition:Inclination/Plane to Plane", "text": "{{EuclidDefinition|book = XI|def = 6|name = Inclination of Plane}}"}
+{"_id": "24126", "title": "Definition:Pyramid", "text": ":400px Let $P$ be a polygon. Let $Q$ be a point not in the plane of $P$. From each vertex of $P$, let lines be drawn to $Q$. The polyhedron which is contained by $P$ and the triangles formed by the sides of $P$ and the lines to $Q$ is a '''pyramid'''."}
+{"_id": "24127", "title": "Definition:Right Circular Cone/Axis", "text": "Let $K$ be a right circular cone. Let point $A$ be the apex of $K$. Let point $O$ be the center of the base of $K$. Then the line $AO$ is the '''axis''' of $K$. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book XI/19 - Axis of Cone}}'' {{EuclidDefRefNocat|XI|19|Axis of Cone}}"}
+{"_id": "24128", "title": "Definition:Right Circular Cone/Base", "text": "Let $\\triangle AOB$ be a right-angled triangle such that $\\angle AOB$ is the right angle. Let $K$ be the right circular cone formed by the rotation of $\\triangle AOB$ around $OB$. Let $BC$ be the circle described by $B$. The '''base''' of $K$ is the plane surface enclosed by the circle $BC$. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book XI/20 - Base of Cone}}'' {{EuclidDefRefNocat|XI|20|Base of Cone}}"}
+{"_id": "24129", "title": "Definition:Right Circular Cone/Acute-Angled", "text": "Let $K$ be a right circular cone. Then $K$ is '''acute-angled''' {{iff}} the opening angle of $K$ is an acute angle."}
+{"_id": "24130", "title": "Definition:Right Circular Cone/Right-Angled", "text": "Let $K$ be a right circular cone. Then $K$ is '''right-angled''' {{iff}} the opening angle of $K$ is a right angle."}
+{"_id": "24131", "title": "Definition:Right Circular Cone/Obtuse-Angled", "text": "Let $K$ be a right circular cone. Then $K$ is '''obtuse-angled''' {{iff}} the opening angle of $K$ is an obtuse angle."}
+{"_id": "24132", "title": "Definition:Circle/Center", "text": "{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book I/16 - Center of Circle}}'' {{EuclidDefRefNocat|I|16|Center of Circle}} In the above diagram, the '''center''' is the point $A$."}
+{"_id": "24133", "title": "Definition:Right Circular Cone/Similar Cones", "text": "Let $h_1$ and $h_2$ be the lengths of the axes of two right circular cones. Let $d_1$ and $d_2$ be the lengths of the diameters of the bases of the two right circular cones. Then the two right circular cones are '''similar''' {{iff}}: :$\\dfrac {h_1} {h_2} = \\dfrac {d_1} {d_2}$ {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book XI/24 - Similar Cones and Cylinders}}'' {{EuclidDefRefNocat|XI|24|Similar Cones and Cylinders}}"}
+{"_id": "24134", "title": "Definition:Commensurable in Square", "text": "Let $a, b \\in \\R_{>0}$ be (strictly) positive real numbers. Then $a$ and $b$ are '''commensurable in square''' iff $\\left ({\\dfrac a b}\\right)^2$ is rational. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X/2 - Commensurable in Square}}'' {{EuclidDefRefNocat|X|2|Commensurable in Square}}"}
+{"_id": "24135", "title": "Definition:Incommensurable in Square", "text": "Let $a, b \\in \\R_{>0}$ be (strictly) positive real numbers. Then $a$ and $b$ are '''incommensurable in square''' iff $\\left ({\\dfrac a b}\\right)^2$ is irrational. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X/2 - Commensurable in Square}}'' {{EuclidDefRefNocat|X|2|Commensurable in Square}}"}
+{"_id": "24136", "title": "Definition:Binomial (Euclidean)", "text": "Let $a$ and $b$ be two (strictly) positive real numbers such that: : $(1): \\quad \\dfrac a b \\notin \\Q$ : $(2): \\quad \\left({\\dfrac a b}\\right)^2 \\in \\Q$ where $\\Q$ denotes the set of rational numbers. Then $a + b$ is a '''binomial'''. {{:Euclid:Proposition/X/36}} === First Binomial === {{Definition:Binomial (Euclidean)/First Binomial}} === Second Binomial === {{Definition:Binomial (Euclidean)/Second Binomial}} === Third Binomial === {{Definition:Binomial (Euclidean)/Third Binomial}} === Fourth Binomial === {{Definition:Binomial (Euclidean)/Fourth Binomial}} === Fifth Binomial === {{Definition:Binomial (Euclidean)/Fifth Binomial}} === Sixth Binomial === {{Definition:Binomial (Euclidean)/Sixth Binomial}}"}
+{"_id": "24137", "title": "Definition:Binomial (Euclidean)/First Binomial", "text": "Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''first binomial''' {{iff}}: : $(1): \\quad a \\in \\Q$ : $(2): \\quad \\dfrac {\\sqrt {a^2 - b^2} } a \\in \\Q$ where $\\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (II)/1 - First Binomial}}'' {{EuclidDefRefNocat|X (II)|1|First Binomial}}"}
+{"_id": "24138", "title": "Definition:Circle/Circumference", "text": "The '''circumference''' of a circle is the line that forms its boundary."}
+{"_id": "24139", "title": "Definition:Binomial (Euclidean)/Second Binomial", "text": "Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''second binomial''' {{iff}}: : $(1): \\quad b \\in \\Q$ : $(2): \\quad \\dfrac {\\sqrt {a^2 - b^2}} a \\in \\Q$ where $\\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (II)/2 - Second Binomial}}'' {{EuclidDefRefNocat|X (II)|2|Second Binomial}}"}
+{"_id": "24140", "title": "Definition:Binomial (Euclidean)/Third Binomial", "text": "Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''third binomial''' {{iff}}: : $(1): \\quad a \\notin \\Q$ : $(2): \\quad b \\notin \\Q$ : $(3): \\quad \\dfrac {\\sqrt {a^2 - b^2}} a \\in \\Q$ where $\\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (II)/3 - Third Binomial}}'' {{EuclidDefRefNocat|X (II)|3|Third Binomial}}"}
+{"_id": "24141", "title": "Definition:Binomial (Euclidean)/Fourth Binomial", "text": "Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''fourth binomial''' {{iff}}: : $(1): \\quad a \\in \\Q$ : $(2): \\quad \\dfrac {\\sqrt {a^2 - b^2}} a \\notin \\Q$ where $\\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (II)/4 - Fourth Binomial}}'' {{EuclidDefRefNocat|X (II)|4|Fourth Binomial}}"}
+{"_id": "24142", "title": "Definition:Binomial (Euclidean)/Fifth Binomial", "text": "Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''fifth binomial''' {{iff}}: : $(1): \\quad b \\in \\Q$ : $(2): \\quad \\dfrac {\\sqrt {a^2 - b^2}} a \\notin \\Q$ where $\\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (II)/5 - Fifth Binomial}}'' {{EuclidDefRefNocat|X (II)|5|Fifth Binomial}}"}
+{"_id": "24143", "title": "Definition:Binomial (Euclidean)/Sixth Binomial", "text": "Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''sixth binomial''' {{iff}}: : $(1): \\quad: a \\notin \\Q$ : $(2): \\quad: b \\notin \\Q$ : $(3): \\quad: \\dfrac {\\sqrt {a^2 - b^2}} a \\notin \\Q$ where $\\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (II)/6 - Sixth Binomial}}'' {{EuclidDefRefNocat|X (II)|6|Sixth Binomial}}"}
+{"_id": "24144", "title": "Definition:Circle/Diameter", "text": "{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book I/17 - Diameter of Circle}}'' {{EuclidDefRefNocat|I|17|Diameter of Circle}} In the above diagram, the line $CD$ is a '''diameter'''."}
+{"_id": "24145", "title": "Definition:Circle/Radius", "text": "A '''radius''' of a circle is a straight line segment whose endpoints are the center and the circumference of the circle. In the above diagram, the line $AB$ is a '''radius'''."}
+{"_id": "24146", "title": "Definition:Circle/Arc", "text": "An '''arc''' of a circle is any part of its circumference."}
+{"_id": "24147", "title": "Definition:Circle/Semicircle", "text": "{{EuclidDefinition|book=I|def=18|name=Semicircle}}"}
+{"_id": "24148", "title": "Definition:Circle/Chord", "text": "A '''chord''' of a circle is a straight line segment whose endpoints are on the circumference of the circle. In the diagram above, the lines $CD$ and $EF$ are both '''chords'''."}
+{"_id": "24149", "title": "Definition:Circle/Equality", "text": "{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book III/1 - Equal Circles}}'' {{EuclidDefRefNocat|III|1|Equal Circles}}"}
+{"_id": "24151", "title": "Definition:Line/Endpoint", "text": "Each of the points at either end of a line segment is called an '''endpoint''' of that line segment. Similarly, the point at which an infinite half-line terminates is called '''the endpoint''' of that line. {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book I/3 - Line Extremities}}'' {{EuclidDefRefNocat|I|3|Line Extremities}}"}
+{"_id": "24152", "title": "Definition:Apotome", "text": "Let $a, b \\in \\set {x \\in \\R_{>0} : x^2 \\in \\Q}$ be two rationally expressible numbers such that $a > b$. Then $a - b$ is an '''apotome''' {{iff}}: :$(1): \\quad \\dfrac a b \\notin \\Q$ :$(2): \\quad \\paren {\\dfrac a b}^2 \\in \\Q$ where $\\Q$ denotes the set of rational numbers. {{:Euclid:Proposition/X/73}}"}
+{"_id": "24153", "title": "Definition:Apotome/First Apotome", "text": "Let $a, b \\in \\set {x \\in \\R_{>0} : x^2 \\in \\Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''first apotome''' {{iff}}: :$(1): \\quad a \\in \\Q$ :$(2): \\quad \\dfrac {\\sqrt {a^2 - b^2}} a \\in \\Q$ where $\\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (III)/1 - First Apotome}}'' {{EuclidDefRefNocat|X (III)|1|First Apotome}}"}
+{"_id": "24154", "title": "Definition:Apotome/Second Apotome", "text": "Let $a, b \\in \\set {x \\in \\R_{>0} : x^2 \\in \\Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''second apotome''' {{iff}}: :$(1): \\quad b \\in \\Q$ :$(2): \\quad \\dfrac {\\sqrt {a^2 - b^2}} a \\in \\Q$ where $\\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (III)/2 - Second Apotome}}'' {{EuclidDefRefNocat|X (III)|2|Second Apotome}}"}
+{"_id": "24155", "title": "Definition:Apotome/Third Apotome", "text": "Let $a, b \\in \\set {x \\in \\R_{>0} : x^2 \\in \\Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''third apotome''' {{iff}}: :$(1): \\quad a \\notin \\Q$ :$(2): \\quad b \\notin \\Q$ :$(3): \\quad \\dfrac {\\sqrt {a^2 - b^2}} a \\in \\Q$ where $\\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (III)/3 - Third Apotome}}'' {{EuclidDefRefNocat|X (III)|3|Third Apotome}}"}
+{"_id": "24156", "title": "Definition:Line/Straight Line", "text": "{{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book I/4 - Straight Line}}'' {{EuclidDefRefNocat|I|4|Straight Line}}"}
+{"_id": "24157", "title": "Definition:Line/Curve", "text": "A '''curve''' is a line which may or may not be straight."}
+{"_id": "24158", "title": "Definition:Curve/Arc", "text": "An '''arc''' is a specific part of a curve."}
+{"_id": "24159", "title": "Definition:Apotome/Fourth Apotome", "text": "Let $a, b \\in \\set {x \\in \\R_{>0} : x^2 \\in \\Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''fourth apotome''' {{iff}}: :$(1): \\quad a \\in \\Q$ :$(2): \\quad \\dfrac {\\sqrt {a^2 - b^2}} a \\notin \\Q$ where $\\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (III)/4 - Fourth Apotome}}'' {{EuclidDefRefNocat|X (III)|4|Fourth Apotome}}"}
+{"_id": "24160", "title": "Definition:Apotome/Fifth Apotome", "text": "Let $a, b \\in \\set {x \\in \\R_{>0} : x^2 \\in \\Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''fifth apotome''' {{iff}}: :$(1): \\quad b \\in \\Q$ :$(2): \\quad \\dfrac {\\sqrt {a^2 - b^2}} a \\notin \\Q$ where $\\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (III)/5 - Fifth Apotome}}'' {{EuclidDefRefNocat|X (III)|5|Fifth Apotome}}"}
+{"_id": "24161", "title": "Definition:Apotome/Sixth Apotome", "text": "Let $a, b \\in \\set {x \\in \\R_{>0} : x^2 \\in \\Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''sixth apotome''' {{iff}}: :$(1): \\quad a \\notin \\Q$ :$(2): \\quad b \\notin \\Q$ :$(3): \\quad \\dfrac {\\sqrt {a^2 - b^2}} a \\notin \\Q$ where $\\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (III)/6 - Sixth Apotome}}'' {{EuclidDefRefNocat|X (III)|6|Sixth Apotome}}"}
+{"_id": "24162", "title": "Definition:Line/Straight Line Segment", "text": "A '''straight line segment''' is a line segment which is straight. {{EuclidSaid}} :''A straight line segment can be drawn joining any two points.'' ({{BookLink|The Elements|Euclid}}: Postulates: Euclid's Second Postulate)"}
+{"_id": "24163", "title": "Definition:Line/Segment", "text": "A '''line segment''' is any line (straight or not) which terminates at two points. === Straight Line Segment === {{:Definition:Line/Straight Line Segment}} === Endpoint === {{:Definition:Line/Endpoint}} === Midpoint === {{:Definition:Line/Midpoint}}"}
+{"_id": "24165", "title": "Definition:Line/Infinite Half-Line", "text": "An '''infinite half-line''' is a line which terminates at an endpoint at one end, but has no such endpoint at the other."}
+{"_id": "24167", "title": "Definition:Line/Equality", "text": "Two line segments are '''equal''' {{iff}} they have the same length."}
+{"_id": "24168", "title": "Definition:Incircle", "text": "Let $P$ be a polygon. Let $C$ be a circle which is inscribed within $P$. :300px Then $C$ is an '''incircle''' of $P$. === Incenter === {{:Definition:Incircle/Incenter}} === Inradius === {{:Definition:Incircle/Inradius}}"}
+{"_id": "24169", "title": "Definition:Circumcircle", "text": "Let $P$ be a polygon. Let $C$ be a circle which is circumscribed around $P$. :300px Then $C$ is a '''circumcircle''' of $P$."}
+{"_id": "24170", "title": "Definition:Incircle of Triangle/Incenter", "text": "The center of an incircle of a triangle is called an '''incenter of the triangle'''."}
+{"_id": "24171", "title": "Definition:Incircle of Triangle/Inradius", "text": "A radius of an incircle of a triangle is called an '''inradius of the triangle'''. In the above diagram, $r$ is an inradius."}
+{"_id": "24172", "title": "Definition:Circumcircle of Triangle/Circumcenter", "text": "The center of a circumcircle of a triangle is called a '''circumcenter of a triangle'''."}
+{"_id": "24173", "title": "Definition:Circumcircle of Triangle/Circumradius", "text": "A radius of a circumcircle of a triangle is called a '''circumradius of a triangle'''. In the above diagram, $R$ is a circumradius."}
+{"_id": "24174", "title": "Definition:Circumcenter", "text": "Let $P$ be a polygon. Let $P$ have a circumcircle $C$. :300px Then the center of $C$ is the '''circumcenter''' of $P$."}
+{"_id": "24175", "title": "Definition:Excircle of Triangle/Exradius", "text": "A radius of an excircle of a triangle is called an '''exradius of the triangle'''. In the above diagram, $r$ is an exradius."}
+{"_id": "24176", "title": "Definition:Excircle of Triangle/Excenter", "text": "The center of an excircle of a triangle is called an '''excenter of the triangle'''."}
+{"_id": "24177", "title": "Definition:Tangent/Geometry/Tangent Line", "text": "{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book III/2 - Tangent to Circle}}'' {{EuclidDefRefNocat|III|2|Tangent to Circle}} :300px In the above diagram, the line is '''tangent''' to the circle at the point $C$."}
+{"_id": "24178", "title": "Definition:Tangent/Geometry/Tangent Circles", "text": "{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book III/3 - Tangent Circles}}'' {{EuclidDefRefNocat|III|3|Tangent Circles}} :400px In the above diagram, the two circles are '''tangent''' to each other at the point $C$."}
+{"_id": "24179", "title": "Definition:Piecewise Continuous Function", "text": "Let $f$ be a real function defined on a closed interval $\\left[{a \\,.\\,.\\, b}\\right]$. $f$ is '''piecewise continuous''' {{iff}}: :there exists a finite subdivision $\\left\\{ {x_0, x_1, \\ldots, x_n}\\right\\}$ of $\\left[{a \\,.\\,.\\, b}\\right]$, where $x_0 = a$ and $x_n = b$, such that: ::for all $i \\in \\left\\{ {1, 2, \\ldots, n}\\right\\}$, $f$ is continuous on $\\left({x_{i − 1} \\,.\\,.\\, x_i}\\right)$."}
+{"_id": "24180", "title": "Definition:Piecewise Continuously Differentiable Function", "text": "Let $f$ be a real function defined on a closed interval $\\closedint a b$. === Definition 1 === {{:Definition:Piecewise Continuously Differentiable Function/Definition 1}} === Definition 2 === {{:Definition:Piecewise Continuously Differentiable Function/Definition 2}}"}
+{"_id": "24181", "title": "Definition:Cone/Double Napped Cone", "text": "A '''double napped cone''' is a cone where the lines joining the apex to the circumference of the base extend indefinitely in either dimension: :300px"}
+{"_id": "24182", "title": "Definition:Cone/Double Napped Cone/Nappe", "text": "Each of the halves of a double napped cone is a '''nappe'''."}
+{"_id": "24183", "title": "Definition:Plane Surface", "text": "{{EuclidDefinition|book = I|def = 7|name = Plane Surface}}"}
+{"_id": "24184", "title": "Definition:Plane Surface/The Plane", "text": "'''The plane''' is the term used for the general plane surface which is infinite in all directions."}
+{"_id": "24185", "title": "Definition:Cone/Apex", "text": "In the above diagram, the point $A$ is known as the '''apex''' of the cone."}
+{"_id": "24186", "title": "Definition:Cone/Base", "text": "The plane figure $PQR$ is called the '''base''' of the cone."}
+{"_id": "24187", "title": "Definition:Right Circular Cone/Opening Angle", "text": "Let $K$ be a right circular cone. Let point $A$ be the apex of $K$. Let $B$ and $C$ be the endpoints of a diameter of the base of $K$. Then the angle $\\angle BAC$ is the '''opening angle''' of $K$. In the above diagram, $\\phi$ is the '''opening angle''' of the right circular cone depicted."}
+{"_id": "24188", "title": "Definition:Conic Section", "text": "=== Intersection with Cone === {{:Definition:Conic Section/Intersection with Cone}} === Focus-Directrix Property === {{:Definition:Conic Section/Focus-Directrix Property}}"}
+{"_id": "24189", "title": "Definition:Ellipse", "text": "=== Intersection with Cone === {{:Definition:Conic Section/Intersection with Cone}} {{:Definition:Conic Section/Intersection with Cone/Ellipse}} === Focus-Directrix Property === {{:Definition:Ellipse/Focus-Directrix}} === Equidistance Property === {{:Definition:Ellipse/Equidistance}}"}
+{"_id": "24190", "title": "Definition:Parallel (Geometry)/Lines", "text": "{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book I/23 - Parallel Lines}}'' {{EuclidDefRefNocat|I|23|Parallel Lines}} The contemporary interpretation of the concept of parallelism declares that a straight line is parallel to itself."}
+{"_id": "24191", "title": "Definition:Parallel (Geometry)/Planes", "text": "Two planes are '''parallel''' {{iff}}, when produced indefinitely, do not intersect at any point. {{EuclidDefinition|book = XI|def = 8|name = Parallel Planes}} The contemporary interpretation of the concept of parallelism declares that a plane is parallel to itself."}
+{"_id": "24192", "title": "Definition:Right Circular Cone/Directrix", "text": "Let $K$ be a right circular cone. Let $B$ be the base of $K$. The circumference of $B$ is the '''directrix''' of $K$."}
+{"_id": "24193", "title": "Definition:Right Circular Cone/Generatrix", "text": "Let $K$ be a right circular cone. Let $A$ be the apex of $K$. Let $B$ be the base of $K$. Then a line joining the apex of $K$ to its directrix is a '''generatrix of $K$'''."}
+{"_id": "24194", "title": "Definition:Generatrix", "text": "A '''generatrix''' is an element of a set of straight lines which constitute a given surface."}
+{"_id": "24195", "title": "Definition:Conic Section/Intersection with Cone/Parabola", "text": ":400px Let $\\phi = \\theta$. Then $K$ is a parabola."}
+{"_id": "24196", "title": "Definition:Conic Section/Intersection with Cone/Circle", "text": ":400px Let $\\phi = \\dfrac \\pi 2 - \\theta$, thereby making $D$ perpendicular to the axis of $C$. Then $D$ and $B$ are parallel, and so $K$ is a circle."}
+{"_id": "24197", "title": "Definition:Conic Section/Intersection with Cone/Ellipse", "text": ":400px Let $\\theta < \\phi < \\dfrac \\pi 2 - \\theta$. That is, the angle between $D$ and the axis of $C$ is between that for which $K$ is a circle and that which $K$ is a parabola. Then $K$ is an ellipse."}
+{"_id": "24198", "title": "Definition:Conic Section/Intersection with Cone/Hyperbola", "text": ":400px Let $\\phi < \\theta$. Then $K$ is a hyperbola. Note that in this case $D$ intersects $C$ in two places: one for each nappe of $C$."}
+{"_id": "24199", "title": "Definition:Ellipse/Focus-Directrix", "text": "400px Let $D$ be a straight line. Let $F$ be a point. Let $\\epsilon \\in \\R: 0 < \\epsilon < 1$. Let $K$ be the locus of points $b$ such that the distance $p$ from $P$ to $D$ and the distance $q$ from $P$ to $F$ are related by the condition: :$\\epsilon \\, p = q$ Then $K$ is an '''ellipse'''."}
+{"_id": "24200", "title": "Definition:Ellipse/Equidistance", "text": ":400px Let $F_1$ and $F_2$ be two points in the plane. Let $d$ be a length greater than the distance between $F_1$ and $F_2$. Let $K$ be the locus of points $P$ which are subject to the condition: :$d_1 + d_2 = d$ where: :$d_1$ is the distance from $P$ to $F_1$ :$d_2$ is the distance from $P$ to $F_2$. Then $K$ is an '''ellipse'''. This property is known as the '''equidistance property'''."}
+{"_id": "24201", "title": "Definition:Line/Midpoint", "text": "Let $L = AB$ be a line segment whose endpoints are $A$ and $B$. Let $M$ be a point on $L$ such that the line segment $AM$ is equal to the line segment $MB$. Then $M$ is the '''midpoint''' of $L$."}
+{"_id": "24202", "title": "Definition:Ellipse/Major Axis", "text": "The '''major axis''' of $K$ is the line segment passing through both $F_1$ and $F_2$ whose endpoints are where it intersects $K$."}
+{"_id": "24203", "title": "Definition:Ellipse/Minor Axis", "text": "The '''minor axis''' of $K$ is the line segment through the center of $K$ perpendicular to the major axis of $K$ such that its endpoints are the points of intersection with $K$."}
+{"_id": "24204", "title": "Definition:Ellipse/Minor Axis/Semi-Minor Axis", "text": "A '''semi-minor axis''' of $K$ is either half of the minor axis of $K$ from its midpoint to its endpoint."}
+{"_id": "24205", "title": "Definition:Ellipse/Major Axis/Semi-Major Axis", "text": "A '''semi-major axis''' of $K$ is either half of the major axis of $K$ from its midpoint to its endpoint."}
+{"_id": "24206", "title": "Definition:Ellipse/Covertex", "text": "A '''covertex''' of $K$ is either one of the endpoints of the minor axis of $K$."}
+{"_id": "24207", "title": "Definition:Ellipse/Vertex", "text": "A '''vertex''' of $K$ is either one of the endpoints of the major axis of $K$."}
+{"_id": "24208", "title": "Definition:Cartesian Plane", "text": "324pxrightthumbA general point $Q = \\tuple {x, y}$ in the Cartesian plane The '''Cartesian plane''' is a Cartesian coordinate system of $2$ dimensions. Every point on the plane can be identified uniquely by means of an ordered pair of real coordinates $\\tuple {x, y}$, as follows: Identify one distinct point on the plane as the origin $O$. Select a point $P$ on the plane different from $O$. Construct an infinite straight line through $O$ and $P$ and call it the $x$-axis. Identify the $x$-axis with the real number line such that: :$0$ is identified with the origin $O$ :$1$ is identified with the point $P$ The orientation of the $x$-axis is determined by the relative positions of $O$ and $P$. It is conventional to locate $P$ to the right of $O$, so as to arrange that: :to the right of the origin, the numbers on the $x$-axis are positive :to the left of the origin, the numbers on the $x$-axis are negative. Construct an infinite straight line through $O$ perpendicular to the $x$-axis and call it the $y$-axis. Identify the point $P'$ on the $y$-axis such that $OP' = OP$. Identify the $y$-axis with the real number line such that: :$0$ is identified with the origin $O$ :$1$ is identified with the point $P'$ The orientation of the $y$-axis is determined by the position of $P'$ relative to $O$. It is conventional to locate $P'$ such that, if one were to imagine being positioned at $O$ and facing along the $x$-axis towards $P$, then $P'$ is on the left.  Hence with the conventional orientation of the $x$-axis as horizontal and increasing to the right: :going vertically \"up\" the page or screen from the origin, the numbers on the $y$-axis are positive :going vertically \"down\" the page or screen from the origin, the numbers on the $y$-axis are negative."}
+{"_id": "24209", "title": "Definition:Cartesian Coordinate System/X Coordinate", "text": "Let $x$ be the length of the line segment from the origin $O$ to the foot of the perpendicular from $Q$ to the $x$-axis. Then $x$ is known as the '''$x$ coordinate'''. If $Q$ is in the positive direction along the real number line that is the $x$-axis, then $x$ is positive. If $Q$ is in the negative direction along the real number line that is the $x$-axis, then $x$ is negative."}
+{"_id": "24210", "title": "Definition:Cartesian Coordinate System/Y Coordinate", "text": "Let $y$ be the length of the line segment from the origin $O$ to the foot of the perpendicular from $Q$ to the $y$-axis. Then $y$ is known as the '''$y$ coordinate'''. If $Q$ is in the positive direction along the real number line that is the $y$-axis, then $y$ is positive. If $Q$ is in the negative direction along the real number line that is the $y$-axis, then $y$ is negative."}
+{"_id": "24212", "title": "Definition:Cartesian Plane/Quadrants", "text": "For ease of reference, the cartesian plane is often divided into four quadrants by the axes: === First Quadrant === {{:Definition:First Quadrant}} === Second Quadrant === {{:Definition:Second Quadrant}} === Third Quadrant === {{:Definition:Third Quadrant}} === Fourth Quadrant === {{:Definition:Fourth Quadrant}} Note that the axes themselves are generally not considered to belong to any quadrant."}
+{"_id": "24213", "title": "Definition:Axis/Coordinate Axes", "text": "Consider a coordinate system. One of the reference lines of such a system is called an '''axis'''."}
+{"_id": "24214", "title": "Definition:Axis/X-Axis", "text": "In a cartesian coordinate system, the '''$x$-axis''' is the one usually depicted and visualised as going from left to right. It consists of all the points in the real vector space in question (usually either $\\R^2$ or $\\R^3$) at which all the elements of its coordinates but $x$ are zero."}
+{"_id": "24215", "title": "Definition:Axis/Y-Axis", "text": "In a cartesian coordinate system, the '''$y$-axis''' is the one usually depicted and visualised as going from \"bottom\" to \"top\" of the paper (or screen). It consists of all the points in the real vector space in question (usually either $\\R^2$ or $\\R^3$) at which all the elements of its coordinates but $y$ are zero."}
+{"_id": "24216", "title": "Definition:Axis/Z-Axis/Right-Hand Rule", "text": "The usual convention for the orientation of the $z$-axis is that of the right-hand rule: Let the coordinate axes be oriented as follows: :Let the $x$-axis increase from '''West''' to '''East'''. :Let the $y$-axis increase from '''South''' to '''North'''. Then the $z$-axis increases from '''below''' to '''above'''. If the $x$-axis and $y$-axis are aligned with a piece of paper or a screen aligned perpendicular to the line of sight, this translates into the following orientation: :Let the $x$-axis increase from '''left''' to '''right'''. :Let the $y$-axis increase from '''bottom''' to '''top'''. Then the $z$-axis increases from '''behind''' to '''in front''' (that is, from '''further away''' to '''closer in''')."}
+{"_id": "24217", "title": "Definition:Axis/Z-Axis", "text": "In a cartesian coordinate system, the '''$z$-axis''' is the axis passing through $x = 0, y = 0$ which is perpendicular to both the $x$-axis and the $y$-axis. It consists of all the points in the real vector space in question (usually $\\R^3$) at which all the elements of its coordinates but $z$ are zero."}
+{"_id": "24218", "title": "Definition:Formal Language", "text": "A '''formal language''' is a structure $\\mathcal L$ which comprises: : A set of symbols $\\mathcal A$ called the alphabet of $\\mathcal L$ : A collation system with the unique readability property for $\\mathcal A$ : A formal grammar that determines which collations belong to the formal language and which do not."}
+{"_id": "24220", "title": "Definition:Coordinate System/Coordinate/Element of Ordered Pair", "text": "Let $\\tuple {a, b}$ be an ordered pair. The following terminology is used: :$a$ is called the '''first coordinate''' :$b$ is called the '''second coordinate'''. This definition is compatible with the equivalent definition in the context of Cartesian coordinate systems."}
+{"_id": "24221", "title": "Definition:Polar Coordinates", "text": "'''Polar coordinates''' are a technique for unique identification of points on the plane. A distinct point $O$ is identified. === Pole === {{:Definition:Polar Coordinates/Pole}} === Polar Axis === {{:Definition:Polar Coordinates/Polar Axis}}"}
+{"_id": "24222", "title": "Definition:Polar Coordinates/Radial Coordinate", "text": "The length of $OP$ is called the '''radial coordinate''' of $P$, and usually labelled $r$."}
+{"_id": "24223", "title": "Definition:Polar Coordinates/Pole", "text": "The point $O$ is referred to as the '''pole''' of the polar coordinate plane."}
+{"_id": "24224", "title": "Definition:Polar Coordinates/Polar Axis", "text": "A ray is drawn from $O$, usually to the right, and referred to as the '''polar axis'''."}
+{"_id": "24225", "title": "Definition:Polar Coordinates/Angular Coordinate", "text": "The angle measured anticlockwise from the polar axis to $OP$ is called the '''angular coordinate''' of $P$, and usually labelled $\\theta$. If the angle is measured clockwise from the polar axis to $OP$, its value is considered negative."}
+{"_id": "24226", "title": "Definition:Meaningful Product", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroup. Let $a_1, \\ldots, a_n$ be a sequence of elements of $S$. Then we define a '''meaningful product''' of $a_1, \\ldots, a_n$ inductively as follows: If $n = 1$ then the only meaningful product is $a_1$. If $n > 1$ then a meaningful product is defined to be any product of the form: :$\\left({a_1 \\ldots a_m}\\right)\\left({a_{m+1} \\ldots a_n}\\right)$ where $m < n$ and $\\left({a_1 \\ldots a_m}\\right)$ and  $\\left({a_{m+1} \\ldots a_n}\\right)$ are meaningful products of $m$ and $n - m$ elements respectively."}
+{"_id": "24227", "title": "Definition:Standard n Product", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroup. Let $a_1, \\ldots, a_n$ be a sequence of elements over $S$. Then we denote the '''standard n product''' of $a_1, \\ldots, a_n$ as: :$\\displaystyle \\prod_{i \\mathop = 1}^n a_i$ We define it inductively as follows: If $n = 1$ then: :$\\displaystyle \\prod_{i \\mathop = 1}^1 a_i = a_1$ If $n > 1$ then: :$\\displaystyle \\prod_{i \\mathop = 1}^n a_i = \\left({\\displaystyle \\prod_{i \\mathop = 1}^{n-1} a_i}\\right) a_n$"}
+{"_id": "24229", "title": "Definition:Cartesian Plane/Quadrants/First", "text": ":400px Quadrant $\\text{I}: \\quad$ The area above the $x$-axis and to the right of the $y$-axis is called '''the first quadrant'''. That is, '''the first quadrant''' is where both the $x$ coordinate and the $y$ coordinate of a point are positive."}
+{"_id": "24230", "title": "Definition:Cartesian Plane/Quadrants/Third", "text": ":400px Quadrant $\\text{III}: \\quad$ The area below the $x$-axis and to the left of the $y$-axis is called '''the third quadrant'''. That is, '''the third quadrant''' is where both the $x$ coordinate and the $y$ coordinate of a point are negative."}
+{"_id": "24231", "title": "Definition:Cartesian Plane/Quadrants/Fourth", "text": ":400px Quadrant $\\text{IV}: \\quad$ The area below the $x$-axis and to the right of the $y$-axis is called '''the fourth quadrant'''. That is, '''the fourth quadrant''' is where the $x$ coordinate of a point is positive and the $y$ coordinate of a point is negative."}
+{"_id": "24232", "title": "Definition:Cartesian Plane/Quadrants/Second", "text": ":400px Quadrant $\\text{II}: \\quad$ The area above the $x$-axis and to the left of the $y$-axis is called '''the second quadrant'''. That is, '''the second quadrant''' is where the $x$ coordinate of a point is negative and the $y$ coordinate of a point is positive."}
+{"_id": "24233", "title": "Definition:Canonical Epimorphism", "text": "Let $m \\in \\Z$. Let $f:\\Z \\to \\Z_m$ be a mapping such that: :$\\forall n \\in \\Z: \\map f n = \\eqclass n m$ where: ::$\\Z_m$ denotes the integers modulo $m$. ::$\\eqclass n m$ denotes the residue class of $n$ modulo $m$. Then $f$ is referred to as the '''canonical epimorphism''' ( '''from $\\Z$ to $\\Z_m$'''). That this is an epimorphism is proved in Canonical Epimorphism is Epimorphism. {{expand|Note that this definition is unnecessarily specific - it is an instance of the concept as applied to the integers modulo $m$. It may be appropriate to link to the fuller definition as given in Definition:Quotient Epimorphism, or depending on the context in Hungerford it may be better to rewrite this as a proof that it is such a specific instance. OTOH that ought already to have been documented somewhere (search around) as this area of group theory has already been covered in considerable detail.<br>In fact I've found it: Ring Epimorphism from Integers to Integers Modulo m.}}"}
+{"_id": "24234", "title": "Definition:Tangent Function", "text": "=== Definition from Triangle === {{:Definition:Tangent Function/Definition from Triangle}} === Definition from Circle === {{:Definition:Tangent Function/Definition from Circle}} === Real Function === {{:Definition:Tangent Function/Real}} === Complex Function === {{:Definition:Tangent Function/Complex}}"}
+{"_id": "24235", "title": "Definition:Tangent Function/Real", "text": "Let $x \\in \\R$ be a real number. The real function $\\tan x$ is defined as: :$\\tan x = \\dfrac {\\sin x} {\\cos x}$ where: : $\\sin x$ is the sine of $x$ : $\\cos x$ is the cosine of $x$. The definition is valid for all $x \\in \\R$ such that $\\cos x \\ne 0$."}
+{"_id": "24236", "title": "Definition:Tangent Function/Complex", "text": "Let $z \\in \\C$ be a complex number. The complex function $\\tan z$ is defined as: :$\\tan z = \\dfrac {\\sin z} {\\cos z}$ where: : $\\sin z$ is the sine of $z$ : $\\cos z$ is the cosine of $z$. The definition is valid for all $z \\in \\C$ such that $\\cos z \\ne 0$."}
+{"_id": "24237", "title": "Definition:Cotangent/Real Function", "text": "Let $x \\in \\R$ be a real number. The real function $\\cot x$ is defined as: :$\\cot x = \\dfrac {\\cos x} {\\sin x} = \\dfrac 1 {\\tan x}$ where: : $\\sin x$ is the sine of $x$ : $\\cos x$ is the cosine of $x$ : $\\tan x$ is the tangent of $x$ The definition is valid for all $x \\in \\R$ such that $\\sin x \\ne 0$."}
+{"_id": "24238", "title": "Definition:Cotangent/Complex Function", "text": "Let $z \\in \\C$ be a complex number. The complex function $\\cot z$ is defined as: :$\\cot z = \\dfrac {\\cos z} {\\sin z} = \\dfrac 1 {\\tan z}$ where: : $\\sin z$ is the sine of $z$ : $\\cos z$ is the cosine of $z$ : $\\tan z$ is the tangent of $z$ The definition is valid for all $z \\in \\C$ such that $\\cos z \\ne 0$."}
+{"_id": "24239", "title": "Definition:Secant Function/Real", "text": "Let $x \\in \\R$ be a real number. The real function $\\sec x$ is defined as: :$\\sec x = \\dfrac 1 {\\cos x}$ where $\\cos x$ is the cosine of $x$. The definition is valid for all $x \\in \\R$ such that $\\cos x \\ne 0$."}
+{"_id": "24240", "title": "Definition:Secant Function/Complex", "text": "Let $z \\in \\C$ be a complex number. The complex function $\\sec z$ is defined as: :$\\sec z = \\dfrac 1 {\\cos z}$ where $\\cos z$ is the cosine of $z$. The definition is valid for all $z \\in \\C$ such that $\\cos z \\ne 0$."}
+{"_id": "24241", "title": "Definition:Cosecant/Real Function", "text": "Let $x \\in \\C$ be a real number. The real function $\\csc x$ is defined as: :$\\csc x = \\dfrac 1 {\\sin x}$ where $\\sin x$ is the sine of $x$. The definition is valid for all $x \\in \\R$ such that $\\sin x \\ne 0$."}
+{"_id": "24242", "title": "Definition:Cosecant/Complex Function", "text": "Let $z \\in \\C$ be a complex number. The complex function $\\csc z$ is defined as: :$\\csc z = \\dfrac 1 {\\sin z}$ where $\\sin z$ is the sine of $z$. The definition is valid for all $z \\in \\C$ such that $\\sin z \\ne 0$."}
+{"_id": "24243", "title": "Definition:Sine/Definition from Triangle", "text": ":400px In the above right triangle, we are concerned about the angle $\\theta$. The '''sine''' of $\\angle \\theta$ is defined as being $\\dfrac {\\text{Opposite}} {\\text{Hypotenuse}}$."}
+{"_id": "24244", "title": "Definition:Sine/Definition from Circle/First Quadrant", "text": "Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P = \\tuple {x, y}$ be the point on $C$ in the first quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $x$-axis. Then the '''sine''' of $\\theta$ is defined as the length of $AP$."}
+{"_id": "24245", "title": "Definition:Sine/Definition from Circle/Second Quadrant", "text": ":500px Let $P = \\tuple {x, y}$ be the point on $C$ in the second quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $x$-axis. Then the '''sine''' of $\\theta$ is defined as the length of $AP$."}
+{"_id": "24246", "title": "Definition:Sine/Definition from Circle/Third Quadrant", "text": ":500px Let $P = \\tuple {x, y}$ be the point on $C$ in the third quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $x$-axis. Then the '''sine''' of $\\theta$ is defined as the length of $AP$."}
+{"_id": "24247", "title": "Definition:Sine/Definition from Circle/Fourth Quadrant", "text": ":500px Let $P = \\tuple {x, y}$ be the point on $C$ in the fourth quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $x$-axis. Then the '''sine''' of $\\theta$ is defined as the length of $AP$."}
+{"_id": "24248", "title": "Definition:Sine/Definition from Circle", "text": "{{:Definition:Sine/Definition from Circle/First Quadrant}}"}
+{"_id": "24249", "title": "Definition:Right Angle/Perpendicular/Foot", "text": "The '''foot''' of a perpendicular is the point where it intersects the line to which it is at right angles. In the above diagram, the point $C$ is the '''foot''' of the perpendicular $CD$."}
+{"_id": "24250", "title": "Definition:Right Angle/Perpendicular", "text": "{{EuclidDefinition|book = I|def = 10|name = Right Angle}} :400px In the above diagram, the line $CD$ has been constructed so as to be a '''perpendicular''' to the line $AB$."}
+{"_id": "24251", "title": "Definition:Cosine/Definition from Triangle", "text": ":400px In the above right triangle, we are concerned about the angle $\\theta$. The '''cosine''' of $\\angle \\theta$ is defined as being $\\dfrac {\\text{Adjacent}} {\\text{Hypotenuse}}$."}
+{"_id": "24252", "title": "Definition:Cosine/Definition from Circle", "text": "{{:Definition:Cosine/Definition from Circle/First Quadrant}}"}
+{"_id": "24253", "title": "Definition:Cosine/Definition from Circle/First Quadrant", "text": "Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P = \\tuple {x, y}$ be the point on $C$ in the first quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $y$-axis. Then the '''cosine''' of $\\theta$ is defined as the length of $AP$."}
+{"_id": "24254", "title": "Definition:Cosine/Definition from Circle/Second Quadrant", "text": ":500px Let $P = \\tuple {x, y}$ be the point on $C$ in the second quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $y$-axis. Then the '''cosine''' of $\\theta$ is defined as the length of $AP$."}
+{"_id": "24255", "title": "Definition:Cosine/Definition from Circle/Third Quadrant", "text": ":500px Let $P = \\tuple {x, y}$ be the point on $C$ in the third quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $y$-axis. Then the '''cosine''' of $\\theta$ is defined as the length of $AP$."}
+{"_id": "24256", "title": "Definition:Cosine/Definition from Circle/Fourth Quadrant", "text": ":500px Let $P = \\tuple {x, y}$ be the point on $C$ in the fourth quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $y$-axis. Then the '''cosine''' of $\\theta$ is defined as the length of $AP$."}
+{"_id": "24257", "title": "Definition:Tangent Function/Definition from Triangle", "text": ":400px In the above right triangle, we are concerned about the angle $\\theta$. The '''tangent''' of $\\angle \\theta$ is defined as being $\\dfrac{\\text{Opposite}} {\\text{Adjacent}}$."}
+{"_id": "24258", "title": "Definition:Tangent Function/Definition from Circle", "text": "{{:Definition:Tangent Function/Definition from Circle/First Quadrant}}"}
+{"_id": "24259", "title": "Definition:Tangent Function/Definition from Circle/First Quadrant", "text": "Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the first quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \\tuple {1, 0}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''tangent''' of $\\theta$ is defined as the length of $AB$."}
+{"_id": "24260", "title": "Definition:Tangent Function/Definition from Circle/Second Quadrant", "text": "Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the second quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \\tuple {1, 0}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''tangent''' of $\\theta$ is defined as the length of $AB$."}
+{"_id": "24261", "title": "Definition:Tangent Function/Definition from Circle/Third Quadrant", "text": "Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the third quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \\tuple {1, 0}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''tangent''' of $\\theta$ is defined as the length of $AB$."}
+{"_id": "24262", "title": "Definition:Tangent Function/Definition from Circle/Fourth Quadrant", "text": "Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the fourth quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \\tuple {1, 0}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''tangent''' of $\\theta$ is defined as the length of $AB$."}
+{"_id": "24263", "title": "Definition:Cotangent/Definition from Triangle", "text": ":400px In the above right triangle, we are concerned about the angle $\\theta$. The '''cotangent''' of $\\angle \\theta$ is defined as being $\\dfrac {\\text{Adjacent}} {\\text{Opposite}}$."}
+{"_id": "24264", "title": "Definition:Cotangent/Definition from Circle", "text": "{{:Definition:Cotangent/Definition from Circle/First Quadrant}}"}
+{"_id": "24265", "title": "Definition:Cotangent/Definition from Circle/First Quadrant", "text": "Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the first quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \\tuple {0, 1}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''cotangent''' of $\\theta$ is defined as the length of $AB$."}
+{"_id": "24266", "title": "Definition:Cotangent/Definition from Circle/Second Quadrant", "text": "Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the second quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \\tuple {0, 1}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''cotangent''' of $\\theta$ is defined as the length of $AB$."}
+{"_id": "24267", "title": "Definition:Cotangent/Definition from Circle/Third Quadrant", "text": "Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the third quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \\tuple {0, 1}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''cotangent''' of $\\theta$ is defined as the length of $AB$."}
+{"_id": "24268", "title": "Definition:Cotangent/Definition from Circle/Fourth Quadrant", "text": "Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the fourth quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \\tuple {0, 1}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''cotangent''' of $\\theta$ is defined as the length of $AB$."}
+{"_id": "24269", "title": "Definition:Secant Function/Definition from Triangle", "text": ":400px In the above right triangle, we are concerned about the angle $\\theta$. The '''secant''' of $\\angle \\theta$ is defined as being $\\dfrac{\\text{Hypotenuse}} {\\text{Adjacent}}$."}
+{"_id": "24270", "title": "Definition:Cosecant/Definition from Triangle", "text": ":400px In the above right triangle, we are concerned about the angle $\\theta$. The '''cosecant''' of $\\angle \\theta$ is defined as being $\\dfrac {\\text {Hypotenuse}} {\\text {Opposite}}$."}
+{"_id": "24271", "title": "Definition:Secant Function/Definition from Circle", "text": "{{:Definition:Secant Function/Definition from Circle/First Quadrant}}"}
+{"_id": "24272", "title": "Definition:Cosecant/Definition from Circle", "text": "{{:Definition:Cosecant/Definition from Circle/First Quadrant}}"}
+{"_id": "24273", "title": "Definition:Secant Function/Definition from Circle/First Quadrant", "text": "Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the first quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \\tuple {1, 0}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''secant''' of $\\theta$ is defined as the length of $OB$."}
+{"_id": "24274", "title": "Definition:Secant Function/Definition from Circle/Second Quadrant", "text": ":500px Let $P$ be the point on $C$ in the second quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \\tuple {1, 0}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''secant''' of $\\theta$ is defined as the length of $OB$. As $OP$ needs to be produced in the opposite direction to $P$, the '''secant''' is therefore a negative number in the second quadrant."}
+{"_id": "24275", "title": "Definition:Secant Function/Definition from Circle/Third Quadrant", "text": ":500px Let $P$ be the point on $C$ in the third quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \\tuple {1, 0}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''secant''' of $\\theta$ is defined as the length of $OB$. As $OP$ needs to be produced in the opposite direction to $P$, the '''secant''' is therefore a negative number in the third quadrant."}
+{"_id": "24276", "title": "Definition:Secant Function/Definition from Circle/Fourth Quadrant", "text": ":500px Let $P$ be the point on $C$ in the fourth quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \\tuple {1, 0}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''secant''' of $\\theta$ is defined as the length of $OB$."}
+{"_id": "24277", "title": "Definition:Cosecant/Definition from Circle/First Quadrant", "text": "Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the first quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \\tuple {0, 1}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''cosecant''' of $\\theta$ is defined as the length of $OB$."}
+{"_id": "24278", "title": "Definition:Cosecant/Definition from Circle/Second Quadrant", "text": ":500px Let $P$ be the point on $C$ in the second quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \\tuple {0, 1}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''cosecant''' of $\\theta$ is defined as the length of $OB$."}
+{"_id": "24279", "title": "Definition:Cosecant/Definition from Circle/Third Quadrant", "text": ":500px Let $P$ be the point on $C$ in the third quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \\tuple {0, 1}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''secant''' of $\\theta$ is defined as the length of $OB$. As $OP$ needs to be produced in the opposite direction to $P$, the '''cosecant''' is therefore a negative number in the third quadrant."}
+{"_id": "24280", "title": "Definition:Cosecant/Definition from Circle/Fourth Quadrant", "text": ":500px Let $P$ be the point on $C$ in the fourth quadrant such that $\\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \\tuple {0, 1}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''secant''' of $\\theta$ is defined as the length of $OB$. As $OP$ needs to be produced in the opposite direction to $P$, the '''cosecant''' is therefore a negative number in the fourth quadrant."}
+{"_id": "24281", "title": "Definition:Inverse Secant/Real/Arcsecant", "text": "{{:Graph of Arcsecant Function}} From Shape of Secant Function, we have that $\\sec x$ is continuous and strictly increasing on the intervals $\\hointr 0 {\\dfrac \\pi 2}$ and $\\hointl {\\dfrac \\pi 2} \\pi$. From the same source, we also have that: :$\\sec x \\to + \\infty$ as $x \\to \\dfrac \\pi 2^-$ :$\\sec x \\to - \\infty$ as $x \\to \\dfrac \\pi 2^+$ Let $g: \\hointr 0 {\\dfrac \\pi 2} \\to \\hointr 1 \\to$ be the restriction of $\\sec x$ to $\\hointr 0 {\\dfrac \\pi 2}$. Let $h: \\hointl {\\dfrac \\pi 2} \\pi \\to \\hointl \\gets {-1}$ be the restriction of $\\sec x$ to $\\hointl {\\dfrac \\pi 2} \\pi$. Let $f: \\closedint 0 \\pi \\setminus \\dfrac \\pi 2 \\to \\R \\setminus \\openint {-1} 1$: :$\\map f x = \\begin{cases} \\map g x & : 0 \\le x < \\dfrac \\pi 2 \\\\ \\map h x & : \\dfrac \\pi 2 < x \\le \\pi \\end{cases}$ From Inverse of Strictly Monotone Function, $\\map g x$ admits an inverse function, which will be continuous and strictly increasing on $\\hointr 1 \\to$. From Inverse of Strictly Monotone Function, $\\map h x$ admits an inverse function, which will be continuous and strictly increasing on $\\hointl \\gets {-1}$. As both the domain and range of $g$ and $h$ are disjoint, it follows that: :$\\map {f^{-1} } x = \\begin{cases} \\map {g^{-1} } x & : x \\ge 1 \\\\ \\map {h^{-1} } x & : x \\le -1 \\end{cases}$ This function $f^{-1} \\left({x}\\right)$ is called '''arcsecant''' of $x$ and is written $\\arcsec x$. Thus: :The domain of $\\arcsec x$ is $\\R \\setminus \\openint {-1} 1$ :The image of $\\arcsec x$ is $\\closedint 0 \\pi \\setminus \\dfrac \\pi 2$."}
+{"_id": "24282", "title": "Definition:Inverse Cosecant/Real/Arccosecant", "text": "{{:Graph of Arccosecant Function}} From Shape of Cosecant Function, we have that $\\csc x$ is continuous and strictly decreasing on the intervals $\\hointr {-\\dfrac \\pi 2} 0$ and $\\hointl 0 {\\dfrac \\pi 2}$. From the same source, we also have that: :$\\csc x \\to + \\infty$ as $x \\to 0^+$ :$\\csc x \\to - \\infty$ as $x \\to 0^-$ Let $g: \\hointr {-\\dfrac \\pi 2} 0 \\to \\hointl {-\\infty} {-1}$ be the restriction of $\\csc x$ to $\\hointr {-\\dfrac \\pi 2} 0$. Let $h: \\hointl 0 {\\dfrac \\pi 2} \\to \\hointr 1 \\infty$ be the restriction of $\\csc x$ to $\\hointl 0 {\\dfrac \\pi 2}$. Let $f: \\closedint {-\\dfrac \\pi 2} {\\dfrac \\pi 2} \\setminus \\set 0 \\to \\R \\setminus \\openint {-1} 1$: :$\\map f x = \\begin{cases} \\map g x & : -\\dfrac \\pi 2 \\le x < 0 \\\\ \\map h x & : 0 < x \\le \\dfrac \\pi 2 \\end{cases}$ From Inverse of Strictly Monotone Function, $\\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\\hointl {-\\infty} {-1}$. From Inverse of Strictly Monotone Function, $\\map h x$ admits an inverse function, which will be continuous and strictly decreasing on $\\hointr 1 \\infty$. As both the domain and range of $g$ and $h$ are disjoint, it follows that: :$\\map {f^{-1} } x = \\begin{cases} \\map {g^{-1} } x & : x \\le -1 \\\\ \\map {h^{-1} } x & : x \\ge 1 \\end{cases}$ This function $\\map {f^{-1} } x$ is called '''arccosecant''' of $x$ and is written $\\arccsc x$. Thus: :The domain of $\\arccsc x$ is $\\R \\setminus \\openint {-1} 1$ :The image of $\\arccsc x$ is $\\closedint {-\\dfrac \\pi 2} {\\dfrac \\pi 2} \\setminus \\set 0$."}
+{"_id": "24283", "title": "Definition:Normal Set", "text": "Let $S$ be a set. Then $S$ is a '''normal set''' {{iff}}: :$S \\notin S$ That is, $S$ is not an element of itself."}
+{"_id": "24284", "title": "Definition:Abnormal Set", "text": "Let $S$ be a set. Then $S$ is an '''abnormal set''' {{iff}}: :$S \\in S$ That is, $S$ is an element of itself."}
+{"_id": "24285", "title": "Definition:Group of Rationals Modulo One", "text": "Define a relation $\\sim$ on $\\Q$ such that: :$\\forall p, q \\in \\Q: p \\sim q \\iff p - q \\in \\Z$ Then $\\left({\\Q / \\sim, +}\\right)$ is a group referred to as the '''group of rationals modulo one'''."}
+{"_id": "24287", "title": "Definition:Hyperbolic Function", "text": "There are six basic '''hyperbolic functions''', as follows: === Hyperbolic Sine === {{:Definition:Hyperbolic Sine}} === Hyperbolic Cosine === {{:Definition:Hyperbolic Cosine}} === Hyperbolic Tangent === {{:Definition:Hyperbolic Tangent}} === Hyperbolic Cotangent === {{:Definition:Hyperbolic Cotangent}} === Hyperbolic Secant === {{:Definition:Hyperbolic Secant}} === Hyperbolic Cosecant === {{:Definition:Hyperbolic Cosecant}}"}
+{"_id": "24288", "title": "Definition:Polar Coordinates/Polar Plane", "text": "A plane upon which a system of polar coordinates has been applied is known as a polar coordinate plane."}
+{"_id": "24289", "title": "Definition:Azimuth", "text": "=== Azimuth (Polar Coordinates) === Another name for the angular coordinate in a system of polar coordinates: {{:Definition:Polar Coordinates/Angular Coordinate}} === Azimuth (Astronomy) === {{:Definition:Azimuth (Astronomy)}} Category:Definitions/Polar Coordinates Category:Definitions/Astronomy 176b5pou9jmsdkjhrgkuoa577igox13"}
+{"_id": "24290", "title": "Definition:Irreducible Representation", "text": "Let $G$ be a group. Let $V$ be a $G$-representation. Then $V$ is '''irreducible''' iff the only subrepresentations are $V$ and $0$. Category:Definitions/Representation Theory 33aw8fk8gee83a3bx6abf6l2qjq0y39"}
+{"_id": "24291", "title": "Definition:Half-Integer", "text": "A '''half-integer''' is a number of the form: :$n + \\dfrac 1 2$ where $n \\in \\Z$. The set of all '''half-integers''' can be defined as: :$\\left\\{{n + \\dfrac 1 2: n \\in \\Z}\\right\\}$ and can be denoted as $\\Z + \\dfrac 1 2$."}
+{"_id": "24292", "title": "Definition:Muggle", "text": "A '''muggle''' is a person who is mathematically illiterate. Specifically, it refers to those who do not realise that they are indeed embarrassingly mathematically unsophisticated, and insist on peppering their speech with mathematical-sounding phrases, like ''Q.E.D'' (on believing they have won an argument) or ''I had $n$ pints of beer last night'' (being under the impression that $n$ means \"a big unspecified number\"). It may be argued that it is unfair and elitist to call such people with such a demeaning cognomen, but we stand by our declaration on the grounds that it is frequently mathematicians who are equally unfairly derided by many in common society as being socially inept, awkward and lacking in sophistication. Such displays of utter balderdash in the media are exemplified in such productions as ''{{WP|The_Big_Bang_Theory|The Big Bang Theory}}'', which, while attempting to make braininess accessible to the averagely intellectually endowed, end up propagating the very stereotype which it purports to debunk."}
+{"_id": "24293", "title": "Definition:Subset/Superset", "text": "If $S$ is a subset of $T$, then $T$ is a '''superset''' of $S$. This can be expressed by the notation $T \\supseteq S$. This can be interpreted as '''$T$ includes $S$''', or (more rarely) '''$T$ contains $S$'''. Thus $S \\subseteq T$ and $T \\supseteq S$ mean the same thing."}
+{"_id": "24294", "title": "Definition:Proper Subset/Proper Superset", "text": "If $S$ is a proper subset of $T$, then $T$ is a '''proper superset''' of $S$. This can be expressed by the notation $T \\supsetneqq S$. This can be interpreted as '''$T$ properly contains $S$'''."}
+{"_id": "24295", "title": "Definition:Set Union/Family of Sets/Subsets of General Set", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be an indexed family of subsets of a set $X$. Then the '''union''' of $\\family {S_i}$ is defined as: :$\\displaystyle \\bigcup_{i \\mathop \\in I} S_i := \\set {x \\in X: \\exists i \\in I: x \\in S_i}$"}
+{"_id": "24296", "title": "Definition:Set Union/Family of Sets/Universal Set", "text": "Let $\\mathbb U$ be a universal set. Let $I$ be an indexing set. Let $\\family {S_i}_{i \\mathop \\in I}$ be an indexed family of subsets of $\\mathbb U$. Then the '''union''' of $\\family {S_i}$ is defined and denoted as: :$\\displaystyle \\bigcup_{i \\mathop \\in I} S_i := \\set {x \\in \\mathbb U: \\exists i \\in I: x \\in S_i}$"}
+{"_id": "24297", "title": "Definition:Set Intersection/Family of Sets/Universal Set", "text": "Let $\\mathbb U$ be a universal set. Let $I$ be an indexing set. Let $\\family {S_i}_{i \\mathop \\in I}$ be an indexed family of subsets of $\\mathbb U$. Then the '''intersection''' of $\\family {S_i}$ is defined and denoted as: :$\\displaystyle \\bigcap_{i \\mathop \\in I} S_i := \\set {x \\in \\mathbb U: \\forall i \\in I: x \\in S_i}$"}
+{"_id": "24298", "title": "Definition:Set Intersection/Family of Sets/Subsets of General Set", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be an indexed family of subsets of a set $X$. Then the '''intersection''' of $\\family {S_i}$ is defined as: :$\\displaystyle \\bigcap_{i \\mathop \\in I} S_i := \\set {x \\in X: \\exists i \\in I: x \\in S_i}$"}
+{"_id": "24299", "title": "Definition:Bijection/Definition 1", "text": "A mapping $f: S \\to T$ is a '''bijection''' {{iff}} both: :$(1): \\quad f$ is an injection and: :$(2): \\quad f$ is a surjection."}
+{"_id": "24300", "title": "Definition:Bijection/Definition 2", "text": "A mapping $f: S \\to T$ is a '''bijection''' {{iff}}: :$f$ has both a left inverse and a right inverse."}
+{"_id": "24301", "title": "Definition:Bijection/Definition 3", "text": "A mapping $f: S \\to T$ is a '''bijection''' {{iff}}: :the inverse $f^{-1}$ of $f$ is a mapping from $T$ to $S$."}
+{"_id": "24302", "title": "Definition:Bijection/Definition 5", "text": "A relation $f \\subseteq S \\times T$ is a '''bijection''' {{iff}}: :$(1): \\quad$ for each $x \\in S$ there exists one and only one $y \\in T$ such that $\\tuple {x, y} \\in f$ :$(2): \\quad$ for each $y \\in T$ there exists one and only one $x \\in S$ such that $\\tuple {x, y} \\in f$."}
+{"_id": "24303", "title": "Definition:Bijection/Definition 4", "text": "A mapping $f \\subseteq S \\times T$ is a '''bijection''' {{iff}}: :for each $y \\in T$ there exists one and only one $x \\in S$ such that $\\tuple {x, y} \\in f$."}
+{"_id": "24305", "title": "Definition:Subtraction/Natural Numbers", "text": "Let $\\N$ be the set of natural numbers. Let $m, n \\in \\N$ such that $m \\le n$. Let $p \\in \\N$ such that $n = m + p$. Then we define the operation '''subtraction''' as: :$n - m = p$ The natural number $p$ is known as the '''difference''' between $m$ and $n$."}
+{"_id": "24306", "title": "Definition:Subtraction/Integers", "text": "The '''subtraction''' operation in the domain of integers $\\Z$ is written \"$-$\". As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of $\\Z$ are the isomorphic images of the elements of equivalence classes of $\\N \\times \\N$ where two tuples are equivalent if the difference between the two elements of each tuples is the same. Thus '''subtraction''' can be formally defined on $\\Z$ as the operation induced on those equivalence classes as specified in the definition of integers. It follows that: :$\\forall a, b, c, d \\in \\N: \\eqclass {\\tuple {a, b} } \\boxminus - \\eqclass {\\tuple {c, d} } \\boxminus = \\eqclass {\\tuple {a, b} } \\boxminus + \\tuple {-\\eqclass {\\tuple {c, d} } \\boxminus} = \\eqclass {\\tuple {a, b} } \\boxminus + \\eqclass {\\tuple {d, c} } \\boxminus$ Thus '''integer subtraction''' is defined between all pairs of integers, such that: :$\\forall x, y \\in \\Z: x - y = x + \\paren {-y}$"}
+{"_id": "24307", "title": "Definition:Addition/Naturally Ordered Semigroup", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a naturally ordered semigroup. The operation $\\circ$ in $\\left({S, \\circ, \\preceq}\\right)$ is called '''addition'''."}
+{"_id": "24308", "title": "Definition:Multiplication/Rational Numbers", "text": "The '''multiplication operation''' in the domain of rational numbers $\\Q$ is written $\\times$. Let $a = \\dfrac p q, b = \\dfrac r s$ where $p, q \\in \\Z, r, s \\in \\Z \\setminus \\set 0$. Then $a \\times b$ is defined as $\\dfrac p q \\times \\dfrac r s = \\dfrac {p \\times r} {q \\times s}$. This definition follows from the definition of and proof of existence of the field of quotients of any integral domain, of which the set of integers is one."}
+{"_id": "24309", "title": "Definition:Ring (Abstract Algebra)/Binding Priority", "text": "In order to simplify expressions involving both $+$ and $\\circ$, it is the convention that ring product has a higher precedence than ring addition: :$a \\circ b + c := \\paren {a \\circ b} + c$"}
+{"_id": "24310", "title": "Definition:Summation/Vacuous Summation", "text": "Take the summation: :$\\displaystyle \\sum _{\\Phi \\left({j}\\right)} a_j$ where $\\Phi \\left({j}\\right)$ is a propositional function of $j$. Suppose that there are no values of $j$ for which $\\Phi \\left({j}\\right)$ is true. Then $\\displaystyle \\sum_{\\Phi \\left({j}\\right)} a_j$ is defined as being $0$. This summation is called a '''vacuous summation'''. This is because: :$\\forall a: a + 0 = a$ where $a$ is a number. Hence for all $j$ for which $\\Phi \\left({j}\\right)$ is false, the sum is unaffected. This is most frequently seen in the form: :$\\displaystyle \\sum_{j \\mathop = m}^n a_j = 0$ where $m > n$. In this case, $j$ can not at the same time be both greater than or equal to $m$ and less than or equal to $n$. Some sources consider such a treatment as abuse of notation."}
+{"_id": "24311", "title": "Definition:Summation/Summand", "text": "The set of elements $\\set {a_j \\in S: 1 \\le j \\le n, \\map R j}$ is called the '''summand'''."}
+{"_id": "24312", "title": "Definition:Product Notation (Algebra)/Multiplicand", "text": "The set of elements $\\left\\{{a_j \\in S: 1 \\le j \\le n, R \\left({j}\\right)}\\right\\}$ is called the '''multiplicand'''."}
+{"_id": "24313", "title": "Definition:Product Notation (Algebra)/Vacuous Product", "text": "Take the composite expressed in product notation: :$\\displaystyle \\prod_{\\map R j} a_j$ where $\\map R j$ is a propositional function of $j$. Suppose that there are no values of $j$ for which $\\map R j$ is true. Then $\\displaystyle \\prod_{\\map R j} a_j$ is defined as being $1$. '''Beware:''' ''not'' zero. This composite is called a '''vacuous product'''. This is because: :$\\forall a: a \\times 1 = a$ where $a$ is a number. Hence for all $j$ for which $\\map R j$ is false, the product is unaffected. This is most frequently seen in the form: :$\\displaystyle \\prod_{j \\mathop = m}^n a_j = 1$ where $m > n$. In this case, $j$ can not at the same time be both greater than or equal to $m$ and less than or equal to $n$."}
+{"_id": "24314", "title": "Definition:Associate/Integers", "text": "Let $x, y \\in \\Z$. Then '''$x$ is an associate of $y$''' {{iff}} they are both divisors of each other. That is, $x$ and $y$ are '''associates''' {{iff}} $x \\divides y$ and $y \\divides x$."}
+{"_id": "24315", "title": "Definition:Composant/Continuum", "text": "$C$ is a '''composant''' of $H$ {{iff}}: :there exists some $p \\in H$ such that $C$ contains all points $x \\in S$ such that $x$ and $p$ are both contained in some proper subcontinua of $H$."}
+{"_id": "24316", "title": "Definition:Composant/Point", "text": "Let $p \\in H$. Then $C$ is the '''composant''' of $p$ {{iff}}: :$C$ is the union of all proper subcontinua of $H$ that contain $p$."}
+{"_id": "24317", "title": "Definition:Subcontinuum/Proper Subcontinuum", "text": "Let $K$ be a subcontinuum of $H$ such that $K \\ne H$. That is, let $K$ be a proper subset of $H$. Then $K$ is a '''proper subcontinuum''' of $H$."}
+{"_id": "24318", "title": "Definition:Degenerate Continuum/Non-Degenerate", "text": "A '''non-degenerate continuum''' of $T$ is a continuum in $T$ containing more than one element."}
+{"_id": "24319", "title": "Definition:Meager Space/Non-Meager", "text": "$A$ is '''non-meager in $T$''' {{iff}} it cannot be constructed as a countable union of subsets of $S$ which are nowhere dense in $T$. That is, $A$ is '''non-meager in $T$''' {{iff}} it is not meager in $T$."}
+{"_id": "24320", "title": "Definition:Coprime/Integers/Relatively Composite", "text": "If $\\gcd \\left\\{{a, b}\\right\\} > 1$, then $a$ and $b$ are '''relatively composite'''. That is, two integers are '''relatively composite''' if they are not coprime. {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book VII/14 - Relatively Composite}}'' {{EuclidDefRefNocat|VII|14|Relatively Composite}}"}
+{"_id": "24324", "title": "Definition:Prime Number/Definition 1", "text": "A '''prime number''' $p$ is a positive integer that has exactly two divisors which are themselves positive integers."}
+{"_id": "24325", "title": "Definition:Prime Number/Definition 2", "text": "Let $p$ be a positive integer. Then $p$ is a prime number {{iff}} $p$ has exactly four integral divisors: $\\pm 1$ and $\\pm p$."}
+{"_id": "24326", "title": "Definition:Prime Number/Definition 3", "text": "Let $p$ be a positive integer. Then $p$ is a prime number {{iff}}: : $\\tau \\left({p}\\right) = 2$ where $\\tau \\left({p}\\right)$ denotes the tau function of $p$."}
+{"_id": "24327", "title": "Definition:Pairwise Coprime", "text": "=== GCD Domain === {{:Definition:Pairwise Coprime/GCD Domain}} === Euclidean Domain === {{:Definition:Pairwise Coprime/Euclidean Domain}} === Integers === {{:Definition:Pairwise Coprime/Integers}} Category:Definitions/Abstract Algebra Category:Definitions/Coprimality 0sonl2zoaxr0t5bd1szcb4dwe1udned"}
+{"_id": "24328", "title": "Definition:Pairwise Coprime/Integers", "text": "A set of integers $S$ is '''pairwise coprime''' {{iff}}: :$\\forall x, y \\in S: x \\ne y \\implies x \\perp y$ where $x \\perp y$ denotes that $x$ and $y$ are coprime."}
+{"_id": "24329", "title": "Definition:Pairwise Coprime/GCD Domain", "text": "Let $\\struct {D, +, \\times}$ be a GCD domain. A subset $S \\subseteq D$ is '''pairwise coprime (in $D$)''' {{iff}}: :$\\forall x, y \\in S: x \\ne y \\implies x \\perp y$ where $x \\perp y$ denotes that $x$ and $y$ are coprime."}
+{"_id": "24330", "title": "Definition:Pairwise Coprime/Euclidean Domain", "text": "Let $\\struct {D, +, \\times}$ be a Euclidean domain. A subset $S \\subseteq D$ is '''pairwise coprime (in $D$)''' {{iff}}: :$\\forall x, y \\in S: x \\ne y \\implies x \\perp y$ where $x \\perp y$ denotes that $x$ and $y$ are coprime."}
+{"_id": "24332", "title": "Definition:Polynomial Congruence/Number of Solutions", "text": "Let $S = \\left\\{{b_1, b_2, \\ldots, b_n}\\right\\}$ be a complete set of residues modulo $n$. The '''number of solutions''' of $P \\left({x}\\right) \\equiv 0 \\pmod n$ is the number of integers $b \\in S$ for which $P \\left({b}\\right) \\equiv 0 \\pmod n$."}
+{"_id": "24335", "title": "Definition:Signature (Logic)", "text": "Let $\\mathcal L$ be a formal language. A choice of vocabulary for $\\mathcal L$ is called a '''signature''' for $\\mathcal L$. === Signature for Predicate Logic === {{:Definition:Signature (Logic)/Predicate Logic}}"}
+{"_id": "24336", "title": "Definition:Non-Archimedean/Norm (Vector Space)", "text": "A norm $\\norm {\\,\\cdot\\,} $ on a vector space $X$ is '''non-Archimedean''' {{iff}} $\\norm {\\, \\cdot \\,}$ satisfies the axiom: {{begin-axiom}} {{axiom | n = N4         | lc= Ultrametric Inequality:         | q = \\forall x, y \\in R         | ml= \\norm {x + y}         | mo= \\le         | mr= \\max \\set {\\norm x, \\norm y} }} {{end-axiom}}"}
+{"_id": "24337", "title": "Definition:Non-Archimedean/Norm (Vector Space)/Archimedean", "text": "A norm $\\norm {\\,\\cdot\\,} $ on a vector space $X$ is '''Archimedean''' {{iff}} it is not non-Archimedean."}
+{"_id": "24338", "title": "Definition:P-adic Norm/P-adic Metric", "text": "The '''$p$-adic metric''' on $\\Q$ is the metric induced by $\\norm{\\cdot}_p$: :$\\forall x, y \\in \\Q: \\map d {x, y} = \\norm{x - y}_p$"}
+{"_id": "24339", "title": "Definition:Non-Archimedean/Metric/Archimedean", "text": "A metric is '''Archimedean''' {{iff}} it is not non-Archimedean."}
+{"_id": "24340", "title": "Definition:Non-Archimedean/Metric", "text": "A metric $d$ on a metric space $X$ is '''non-Archimedean''' {{iff}}: :$\\map d {x, y} \\le \\max \\set {\\map d {x, z}, \\map d {y, z} }$ for all $x, y, z \\in X$."}
+{"_id": "24341", "title": "Definition:Max Operation/General Definition", "text": "Let $S^n$ be the cartesian $n$th power of $S$. The '''max operation''' is the $n$-ary operation on $\\struct {S, \\preceq}$ defined recursively as: :$\\forall x := \\family {x_i}_{1 \\mathop \\le i \\mathop \\le n} \\in S^n: \\map \\max x = \\begin{cases} x_1 & : n = 1 \\\\ \\map \\max {x_1, x_2} & : n = 2 \\\\ \\map \\max {\\map \\max {x_1, \\ldots, x_{n - 1} }, x_n} & : n > 2 \\\\ \\end{cases}$ where $\\map \\max {x, y}$ is the binary max operation on $S^2$."}
+{"_id": "24342", "title": "Definition:Language of Propositional Logic/Alphabet/Letter", "text": "The letters of $\\LL_0$, called '''propositional symbols''', can be any infinite collection $\\PP_0$ of arbitrary symbols. It is usual to specify them as a limited subset of the English alphabet with appropriate subscripts. A typical set of '''propositional symbols''' would be, for example: :$\\PP_0 = \\set {p_1, p_2, p_3, \\ldots, p_n, \\ldots}$"}
+{"_id": "24343", "title": "Definition:Language of Propositional Logic/Alphabet/Sign/Bracket", "text": "{{begin-eqn}} {{eqn | ll= \\bullet       | l = (       | o = :       | r = \\)the '''left bracket''' sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = )       | o = :       | r = \\)the '''right bracket''' sign\\(       | c =  }} {{end-eqn}}"}
+{"_id": "24344", "title": "Definition:Language of Propositional Logic/Alphabet/Sign/Connective", "text": "{{begin-eqn}} {{eqn | ll= \\bullet       | l = \\land       | o = :       | r = \\)the conjunction sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\lor       | o = :       | r = \\)the disjunction sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\implies       | o = :       | r = \\)the conditional sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\iff       | o = :       | r = \\)the biconditional sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\neg       | o = :       | r = \\)the negation sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\top       | o = :       | r = \\)the tautology sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\bot       | o = :       | r = \\)the contradiction sign\\(       | c =  }} {{end-eqn}} These comprise: * The nullary connectives $\\top$ and $\\bot$, representing the canonical tautology and contradiction, respectively * The unary connective $\\neg$, representing negation * The binary connectives $\\land, \\lor, \\implies$ and $\\iff$, representing, respectively, conjunction, disjunction, implication and biconditional."}
+{"_id": "24346", "title": "Definition:Language of Propositional Logic/Alphabet", "text": "The alphabet $\\mathcal A$ of the language of propositional logic $\\mathcal L_0$ is defined as follows: <section notitle=\"true\" name=\"tc\"> === Letters === {{:Definition:Language of Propositional Logic/Alphabet/Letter}} {{transclude:Definition:Language of Propositional Logic/Alphabet/Sign   |section = def   |title = Signs   |header = 3   |link = true   |increase = 1 }} </section>"}
+{"_id": "24347", "title": "Definition:Language of Propositional Logic/Formal Grammar/Backus-Naur Form", "text": "In Backus-Naur form, the formal grammar of the language of propositional logic takes the following form: {{begin-eqn}} {{eqn | l = \\)<tt><formula></tt>\\(       | o = \\)<tt>::=</tt>\\(       | r = p \\ \\mid \\ \\top \\ \\mid \\ \\bot       | c = where $p \\in \\PP_0$ is a letter }} {{eqn | l = \\)<tt><formula></tt>\\(       | o = \\)<tt>::=</tt>\\(       | r = \\neg\\) <tt><formula></tt>\\( }} {{eqn | l = \\)<tt><formula></tt>\\(       | o = \\)<tt>::=</tt>\\(       | r = (\\) <tt><formula></tt> <tt><op></tt> <tt><formula></tt> \\() }} {{eqn | l = \\)<tt><op></tt>\\(       | o = \\)<tt>::=</tt>\\(       | r = \\land \\ \\mid \\ \\lor \\ \\mid \\implies \\mid \\iff }} {{end-eqn}} Note that this is a top-down grammar: :we start with a metasymbol <tt><formula></tt> :progressively replace it with constructs containing other metasymbols and/or primitive symbols until finally we are left with a well-formed formula of $\\LL_0$ consisting of nothing but primitive symbols."}
+{"_id": "24348", "title": "Definition:Language of Propositional Logic/Formal Grammar/Bottom-Up Specification", "text": "The following rules of formation constitute a bottom-up grammar for the formation of well-formed formulas (WFFs) of the language of propositional logic $\\LL_0$. :Let $\\PP_0$ be the vocabulary of $\\LL_0$. :Let $\\mathrm {Op} = \\set {\\land, \\lor, \\implies, \\iff}$. The rules are: {| |- | $\\mathbf W: \\text {TF}$ || | $:$ | $\\top$ is a WFF, and $\\bot$ is a WFF. |- | $\\mathbf W: \\PP_0$ || | $:$ | If $p \\in \\PP_0$, then $p$ is a WFF. |- | $\\mathbf W: \\neg$ || | $:$ | If $\\mathbf A$ is a WFF, then $\\neg \\mathbf A$ is a WFF. |- | $\\mathbf W: \\text {Op}$ || | $:$ | If $\\mathbf A$ is a WFF and $\\mathbf B$ is a WFF and $\\circ \\in \\mathrm {Op}$, then $\\paren {\\mathbf A \\circ \\mathbf B}$ is a WFF. |} Any string which can not be created by means of the above rules is not a WFF."}
+{"_id": "24350", "title": "Definition:Language of Propositional Logic", "text": "In order to define $\\mathcal L_0$, it is necessary to specify: * An alphabet $\\mathcal A$ * A collation system with the unique readability property for $\\mathcal A$ * A formal grammar (which determines the WFFs of $\\mathcal L_0$)"}
+{"_id": "24351", "title": "Definition:Max Operation/General Definition/Real Numbers", "text": "The '''max operation''' on the real cartesian space $\\R^n$ is the real-valued function $\\max: \\R^n \\to \\R$ defined recursively as: :$\\forall x := \\family {x_i}_{1 \\mathop \\le i \\mathop \\le n} \\in \\R^n: \\map \\max x = \\begin{cases} x_1 & : n = 1 \\\\ \\map \\max {x_1, x_2} & : n = 2 \\\\ \\map \\max {\\map \\max {x_1, \\ldots, x_{n - 1} }, x_n} & : n > 2 \\\\ \\end{cases}$ where $\\map \\max {x, y}$ is the binary max operation on $\\R \\times \\R$."}
+{"_id": "24352", "title": "Definition:Cartesian Product/Cartesian Space/Real Cartesian Space", "text": "Let $n \\in \\N_{>0}$. Then $\\R^n$ is the cartesian product defined as follows: :$\\ds \\R^n = \\underbrace {\\R \\times \\R \\times \\cdots \\times \\R}_{\\text {$n$ times} } = \\prod_{k \\mathop = 1}^n \\R$ Similarly, $\\R^n$ can be defined as the set of all real $n$-tuples: :$\\R^n = \\set {\\tuple {x_1, x_2, \\ldots, x_n}: x_1, x_2, \\ldots, x_n \\in \\R}$"}
+{"_id": "24353", "title": "Definition:Language of Propositional Logic/Keisler-Robbin", "text": "There are many formal languages expressing propositional logic. The formal language used on {{ProofWiki}} is defined on Definition:Language of Propositional Logic. This page defines the formal language $\\LL_0$ used in: * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin}} Explanations are omitted as this is intended for reference use only. === Alphabet === ==== Letters ==== The letters used are a non-empty set of symbols $\\PP_0$. See the {{ProofWiki}} definition. ==== Signs ==== ===== Brackets ===== The '''brackets''' used are '''square brackets''': {{begin-eqn}} {{eqn | ll= \\bullet       | l = [       | o = :       | r = \\)the '''left bracket''' sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = ]       | o = :       | r = \\)the '''right bracket''' sign\\(       | c =  }} {{end-eqn}} See the {{ProofWiki}} definition. ===== Connectives ===== The following '''connectives''' are used: {{begin-eqn}} {{eqn | ll= \\bullet       | l = \\land       | o = :       | r = \\)the conjunction sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\lor       | o = :       | r = \\)the disjunction sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\implies       | o = :       | r = \\)the conditional sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\iff       | o = :       | r = \\)the biconditional sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\neg       | o = :       | r = \\)the negation sign\\(       | c =  }} {{end-eqn}} See the {{ProofWiki}} definition. === Collation System === The collation system used is that of words and concatenation. See the {{ProofWiki}} definition. === Formal Grammar === The following bottom-up formal grammar is used. Let $\\PP_0$ be the vocabulary of $\\LL_0$. Let $Op = \\set {\\land, \\lor, \\implies, \\iff}$. The rules are: {| |- | $\\mathbf W: \\PP_0$ || | $:$ | If $p \\in \\PP_0$, then $p$ is a WFF. |- | $\\mathbf W: \\neg$ || | $:$ | If $\\mathbf A$ is a WFF, then $\\neg \\mathbf A$ is a WFF. |- | $\\mathbf W: Op$ || | $:$ | If $\\mathbf A$ and $\\mathbf B$ are WFFs and $\\circ \\in Op$, then $\\sqbrk {\\mathbf A \\circ \\mathbf B}$ is a WFF. |} See the {{ProofWiki}} definition."}
+{"_id": "24354", "title": "Definition:Order of Pole/Simple Pole", "text": "Let the order of the pole at $x$ be $1$. Then $x$ is a '''simple pole'''."}
+{"_id": "24355", "title": "Definition:Order of Zero/Simple Zero", "text": "Let the order of the zero $x$ be $1$. Then $x$ is a '''simple zero'''."}
+{"_id": "24356", "title": "Definition:Isolated Singularity/Pole", "text": "Let $z_0$ be an isolated singularity of $f$. Then $z_0$ is a '''pole''' {{iff}}: :$\\displaystyle \\lim_{z \\mathop \\to z_0} \\cmod {\\map f z} \\to \\infty$"}
+{"_id": "24357", "title": "Definition:Characteristic of Ring/Definition 1", "text": "Let $n \\cdot x$ be defined as in Definition:Power of Element. The '''characteristic''' $\\Char R$ of $R$ is the smallest $n \\in \\Z, n > 0$ such that $n \\cdot 1_R = 0_R$. If there is no such $n$, then $\\Char R = 0$."}
+{"_id": "24358", "title": "Definition:Characteristic of Ring/Definition 2", "text": "Let $g: \\Z \\to R$ be the initial homomorphism, with $\\map g n = n \\cdot 1_R$. Let $\\ideal p$ be the principal ideal of $\\struct {\\Z, +, \\times}$ generated by $p$. The '''characteristic''' $\\Char R$ of $R$ is the positive integer $p \\in \\Z_{\\ge 0}$ such that $\\ideal p$ is the kernel of $g$."}
+{"_id": "24359", "title": "Definition:Characteristic of Ring/Definition 3", "text": "The '''characteristic of $R$''', denoted $\\Char R$, is defined as follows. Let $p$ be the order of $1_R$ in the additive group $\\struct {R, +}$ of $\\struct {R, +, \\circ}$. If $p \\in \\Z_{>0}$, then $\\Char R := p$. If $1_R$ is of infinite order, then $\\Char R := 0$."}
+{"_id": "24360", "title": "Definition:Monoid Axioms", "text": "A monoid is an algebraic structure $\\struct {S, \\circ, e_S}$ which satisfies the following properties: {{begin-axiom}} {{axiom | n = \\text S 0         | lc= Closure         | q = \\forall a, b \\in S         | m = a \\circ b \\in S }} {{axiom | n = \\text S 1         | lc= Associativity         | q = \\forall a, b, c \\in S         | m = a \\circ \\paren {b \\circ c} = \\paren {a \\circ b} \\circ c }} {{axiom | n = \\text S 2         | lc= Identity         | q = \\exists e_S \\in S: \\forall a \\in S         | m = e_S \\circ a = a = a \\circ e_S }} {{end-axiom}} The element $e_S$ is called the identity element."}
+{"_id": "24361", "title": "Definition:Unique Parsability", "text": "Let $\\mathcal L$ be a formal language. Let every WFF of $\\mathcal L$ result from a unique rule of formation. Then $\\mathcal L$ has '''unique parsability'''. === Bottom-Up Grammar === For a bottom-up grammar, a WFF $\\phi$ results from a unique rule of formation iff: :$\\phi$ results from applying the rule of formation $\\mathbf R$ to WFFs $\\phi_1, \\ldots \\phi_n$ :$\\phi$ results from applying the rule of formation $\\mathbf R'$ WFFs $\\psi_1, \\ldots \\psi_m$ imply that $\\mathbf R = \\mathbf R'$, $m = n$, and for $i = 1, \\ldots, n$, $\\phi_i = \\psi_i$. === Top-Down Grammar === For a top-down grammar, a WFF $\\phi$ results from a unique rule of formation iff: :$\\phi$ results from applying the rule of formation $\\mathbf R$, substituting the WFFs $\\phi_1, \\ldots, \\phi_n$ for the metasymbols of $\\mathbf R$ :$\\phi$ results from applying the rule of formation $\\mathbf R'$, substituting the WFFs $\\psi_1, \\ldots, \\psi_n$ for the metasymbols of $\\mathbf R'$ imply that $\\mathbf R = \\mathbf R'$, $m = n$, and for $i = 1, \\ldots, n$, $\\phi_i = \\psi_i$. Category:Definitions/Formal Systems qi4j9q92vrqo6xazzvorkgvractiw51"}
+{"_id": "24362", "title": "Definition:Bottom-Up Form of Top-Down Grammar", "text": "Let $\\mathcal L$ be a formal language. Let $\\mathcal L$ be given by a top-down formal grammar $\\mathcal T$. The '''bottom-up form of $\\mathcal T$''' is the formal grammar $\\mathcal B$ defined by declaring that: :Each letter of $\\mathcal L$ is a $\\mathcal B$-WFF and, for each rule of formation $\\mathbf R$ of $\\mathcal T$ of the form: :A metasymbol may be replaced by the collation $\\phi$ with metasymbols $\\phi_1, \\ldots, \\phi_n$ declaring that $\\mathcal B$ has the rule of formation $\\mathbf R_{\\mathcal B}$: :If $\\phi_1, \\ldots, \\phi_n$ are $\\mathcal B$-WFFs, so is $\\phi$."}
+{"_id": "24363", "title": "Definition:Semigroup Axioms", "text": "A semigroup is an algebraic structure $\\struct {S, \\circ}$ which satisfies the following properties: {{begin-axiom}} {{axiom | n = \\text S 0         | lc= Closure         | q = \\forall a, b \\in S         | m = a \\circ b \\in S }} {{axiom | n = \\text S 1         | lc= Associativity         | q = \\forall a, b, c \\in S         | m = a \\circ \\paren {b \\circ c} = \\paren  {a \\circ b} \\circ c }} {{end-axiom}}"}
+{"_id": "24364", "title": "Definition:Latin Square Property", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure. $\\left({S, \\circ}\\right)$ has the '''Latin square property''' {{iff}}: :$\\forall a \\in S$, the left and right regular representations $\\lambda_a$ and $\\rho_a$ are permutations on $S$. That is: :$\\forall a, b \\in S: \\exists ! x: x \\circ a = b$ :$\\forall a, b \\in S: \\exists ! y: a \\circ y = b$"}
+{"_id": "24365", "title": "Definition:Properties of Algebraic Structures of One Operation", "text": "The purpose of this page is to gather into one place the various types of algebraic structure of one binary operation and classify them according to the properties they hold. {| border=\"1\" !  ! Closure ! Associativity ! Identity ! Inverses ! Commutativity ! Latin Square Property |- | Magma | align=\"center\"| $\\checkmark$ | | | | | |- | Quasigroup | align=\"center\"| $\\checkmark$ | | | | | align=\"center\"| $\\checkmark$ |- | Algebra Loop | align=\"center\"| $\\checkmark$ | | align=\"center\"| $\\checkmark$ | | | align=\"center\"| $\\checkmark$ |- | Semigroup | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | | | | |- | Commutative Semigroup | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | | | align=\"center\"| $\\checkmark$ |  |- | Monoid | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | | | |- | Commutative Monoid | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | | align=\"center\"| $\\checkmark$ | |- | Group | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | | align=\"center\"| $(\\checkmark)$: see here |- | Abelian Group | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $(\\checkmark)$: see here |- |} A '''checkmark''' in brackets: $(\\checkmark)$ denotes that the property indicated can be derived from the others."}
+{"_id": "24366", "title": "Definition:Properties of Algebraic Structures of Two Operations", "text": "The purpose of this page is to gather into one place the various types of algebraic structure of two binary operations: $+, *$, and to classify them according to the properties they hold. The properties are defined by the following key: {| border=\"1\" | $A0$ | Closure of $+$ |- | $A1$ | Associativity of $+$ |- | $A2$ | Identity Element of $+$ |- | $A3$ | Inverses of $+$ |- | $A4$ | Commutativity of $+$ |- | $M0$ | Closure of $+$ |- | $M1$ | Associativity of $*$ |- | $M2$ | Identity Element of $*$ |- | $M3$ | Inverses of $*$ |- | $M4$ | Commutativity of $*$ |- | $D$ | Distributivity of $*$ over $+$ |- | $NZD$ | No (Proper) Zero Divisors |- | $AC2$ | Characteristic 2 for $+$ |- | $AZ$ | Zero Element of $+$ |- | $D'$ | Distributivity of $+$ over $*$ |- | $MI$ | Idempotence of $*$ |- | $MZ$ | Zero Element of $*$ |- |} {| border=\"1\" !  ! $A0$ ! $A1$ ! $A2$ ! $A3$ ! $A4$ ! $M0$ ! $M1$ ! $M2$ ! $M3$ ! $M4$ ! $D$  ! $NZD$ ! $AC2$  ! $AZ$ ! $MI$ ! $MZ$ ! $D'$ |- | Semiring | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ |  |  |  | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | | |  | align=\"center\"| $\\checkmark$ |  |  | |  | |  |- | Additive Semiring | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ |  |  | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | | |  | align=\"center\"| $\\checkmark$ |  |  | |  | |  |- | Rig | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ |  | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | | |  | align=\"center\"| $\\checkmark$ |  |  | |  | align=\"center\"| $\\checkmark$ |  |- | Ring without associativity | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ |  | | |  | align=\"center\"| $\\checkmark$ |  |  | |  | align=\"center\"| $(\\checkmark)$ (see here) |  |- | Ring | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | | |  | align=\"center\"| $\\checkmark$ |  |  | |  | align=\"center\"| $(\\checkmark)$ (see here) |  |- | Ring with Unity | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | |  | align=\"center\"| $\\checkmark$ |  |  | |  | align=\"center\"| $(\\checkmark)$ (see here) |  |- | Commutative Ring with Unity | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ |  |  | |  | align=\"center\"| $(\\checkmark)$ (see here) |  |- | Integral Domain | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ |  | |  | align=\"center\"| $(\\checkmark)$ (see here) |  |- | Division Ring | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ |  | |  | align=\"center\"| $(\\checkmark)$ (see here) |  |- | Field | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ |  | |  | align=\"center\"| $(\\checkmark)$ (see here) |  |- | Idempotent Ring | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | |  |  | align=\"center\"| $\\checkmark$ | | align=\"center\"| $\\checkmark$ |  | align=\"center\"| $\\checkmark$ |  |  |- | Boolean Ring | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ |  |  | align=\"center\"| $\\checkmark$ | | align=\"center\"| $\\checkmark$ |  | align=\"center\"| $\\checkmark$ |  |  |- | Boolean Algebra | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | | | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ | align=\"center\"| $\\checkmark$ |- |} A '''checkmark''' in brackets: $(\\checkmark)$ denotes that the property indicated can be derived from the others."}
+{"_id": "24367", "title": "Definition:Support of Mapping to Algebraic Structure/Real-Valued Function", "text": "Let $S$ be a set. Let $f: S \\to \\R$ be a real-valued function. The '''support of $f$''' is the set of elements $x$ of $S$ whose values under $f$ are non-zero. That is: :$\\operatorname{supp} \\left({f}\\right) := \\left\\{{x \\in S: f \\left({x}\\right) \\ne 0}\\right\\}$"}
+{"_id": "24368", "title": "Definition:Continuous Real-Valued Vector Function", "text": "Let $\\R^n$ be the cartesian $n$-space. Let $f: \\R^n \\to \\R$ be a real-valued function on $\\R^n$. Then '''$f$ is continuous on $\\R^n$''' iff: :$\\forall a \\in \\R^n: \\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: \\forall x \\in \\R^n: d \\left({x, a}\\right) < \\delta \\implies \\left|{f \\left({x}\\right) - f \\left({a}\\right)}\\right| < \\epsilon$ where $d \\left({x, a}\\right)$ is the distance function on $\\R^n$: :$\\displaystyle d: \\R^n \\to \\R: d \\left({x, y}\\right) := \\sqrt {\\left({\\sum_{i \\mathop = 1}^n \\left({x_i - y_i}\\right)}\\right)}$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({y_1, y_2, \\ldots, y_n}\\right)$ are general elements of $\\R^n$."}
+{"_id": "24369", "title": "Definition:Support of Continuous Mapping/Real-Valued", "text": "Let $f: \\R^n \\to \\R$ be a continuous real-valued function. The '''support of $f$''' is the closure of the set of elements $x$ of $\\R^n$ whose values under $f$ are non-zero. That is: :$\\operatorname{supp} \\left({f}\\right) = \\overline{\\left\\{{x \\in \\R^n: f \\left({x}\\right) \\ne 0}\\right\\}}$"}
+{"_id": "24370", "title": "Definition:Support of Distribution", "text": "Let $\\Omega \\subseteq \\R^n$ be an open set. {{definition wanted|The link goes to a definition of an open set which has been made only for the real number line. Needs to be expanded to include the general real cartesian space. This has already been defined in the context of Metric Spaces; it's just a matter of creating a page which connects the concepts of an open set on a real number line to that on a general space.}} Let $\\mathcal D \\left({\\Omega}\\right)$ be the space of continuous functions compactly supported in $\\Omega$. Let $T \\in \\mathcal D\\,' \\left({\\Omega}\\right)$ be a distribution. {{explain|$D\\,'$: presumably a derivative?}} The '''support''' $\\operatorname{supp} \\left({T}\\right) \\subseteq \\Omega$ of $T$ is defined by: : $\\displaystyle x \\notin \\operatorname{supp} \\left({T}\\right)$ {{iff}}: :: there exists an open neighborhood $U$ of $x$ such that: ::: for all $\\phi \\in \\mathcal D \\left({\\Omega}\\right)$ such that $\\operatorname{supp} \\left({\\phi}\\right) \\subseteq U$: :::: $T \\left({\\phi}\\right) = 0$"}
+{"_id": "24371", "title": "Definition:Partition of Unity (Topology)/Subordinate", "text": "Let $X$ be a topological space. Let $\\left\\{{\\phi_\\alpha : \\alpha \\in A}\\right\\}$ be a partition of unity. Let $\\mathcal B = \\left\\{{U_\\beta: \\beta \\in B}\\right\\}$ be an open cover of $X$. Suppose the set $\\left\\{{\\operatorname{supp} \\left({\\phi_\\alpha}\\right)^\\circ : \\alpha \\in A}\\right\\}$ of interiors of supports is a refinement of $\\mathcal B$ Then $\\mathcal A$ is said to be '''subordinate''' to the cover $\\mathcal B$."}
+{"_id": "24372", "title": "Definition:Literal", "text": "A '''literal''' is either: :an atom $p$ of propositional logic, that is, a statement, or :the negation $\\neg p$ of an atom $p$. In the language of propositional logic, these correspond to: :a letter $p$; :the WFF $\\neg p$, where $p$ is a letter. === Positive Literal === {{:Definition:Literal/Positive}} === Negative Literal === {{:Definition:Literal/Negative}}"}
+{"_id": "24373", "title": "Definition:Conjunctive Normal Form", "text": "A propositional formula $P$ is in '''conjunctive normal form''' {{iff}} it consists of a conjunction of: :$(1):\\quad$ disjunctions of literals and/or: :$(2):\\quad$ literals."}
+{"_id": "24374", "title": "Definition:Set of Truth Values", "text": "The '''set of truth values''' of propositional logic is the $2$-element set: :$\\Bbb B = \\set {\\T, \\F}$ of (Aristotelian) truth values."}
+{"_id": "24375", "title": "Definition:Continued Fraction/Finite", "text": "Let $n \\geq 0$ be a natural number. Informally, a '''finite continued fraction of length $n$''' in $F$ is an expression of the form: :$a_0 + \\cfrac 1 {a_1 + \\cfrac 1 {a_2 + \\cfrac 1 {\\ddots \\cfrac {} {a_{n-1} + \\cfrac 1 {a_n}} }}}$ where $a_0, a_1, a_2, \\ldots, a_n \\in F$. Formally, a '''finite continued fraction of length $n$''' in $F$ is a finite sequence, called '''sequence of partial quotients''', whose domain is the integer interval $\\left[0 \\,.\\,.\\, n\\right]$. A finite continued fraction should not be confused with its '''value''', when it exists."}
+{"_id": "24376", "title": "Definition:Continued Fraction/Infinite", "text": "Informally, an '''infinite continued fraction''' in $F$ is an expression of the form: :$a_0 + \\cfrac 1 {a_1 + \\cfrac 1 {a_2 + \\cfrac 1 {\\ddots \\cfrac {} {a_{n-1} + \\cfrac 1 {a_n + \\cfrac 1 {\\ddots}}} }}}$ where $a_0, a_1, a_2, \\ldots, a_n, \\ldots \\in F$. Formally, an '''infinite continued fraction''' in $F$ is a sequence, called '''sequence of partial quotients''', whose domain is $\\N_{\\geq 0}$. An infinite continued fraction should not be confused with its '''value''', when it exists."}
+{"_id": "24377", "title": "Definition:Partial Quotient", "text": "Let $(a_n)_{n\\geq0}$ be a continued fraction in $k$, either finite or infinite. Let $n \\geq 0$ be a natural number. The '''$n$th partial quotient''' is the $n$th term $a_n$."}
+{"_id": "24378", "title": "Definition:Continued Fraction/Simple/Finite", "text": "Let $n\\geq 0$ be a natural number. A '''simple finite continued fraction of length $n$''' is a finite continued fraction in $\\R$ of length $n$ whose partial quotients are integers that are strictly positive, except perhaps the first. That is, it is a finite sequence $a : \\left[0 \\,.\\,.\\, n\\right] \\to \\Z$ with $a_n > 0$ for $n >0$."}
+{"_id": "24379", "title": "Definition:Continued Fraction/Simple/Infinite", "text": "A '''simple infinite continued fraction''' is a infinite continued fraction in $\\R$ whose partial quotients are integers that are strictly positive, except perhaps the first. That is, it is a sequence $a : \\N_{\\geq 0} \\to \\Z$ with $a_n > 0$ for $n >0$."}
+{"_id": "24380", "title": "Definition:Continued Fraction/Simple", "text": "Let $\\R$ be the field of real numbers. === Simple Finite Continued Fraction === {{:Definition:Continued Fraction/Simple/Finite}} === Simple Infinite Continued Fraction === {{:Definition:Continued Fraction/Simple/Infinite}}"}
+{"_id": "24382", "title": "Definition:Value of Continued Fraction", "text": "Let $F$ be a field, such as the field of real numbers $\\R$. === Finite Continued Fraction === {{Definition:Value of Continued Fraction/Finite}} === Infinite Continued Fraction === {{:Definition:Value of Continued Fraction/Infinite}}"}
+{"_id": "24383", "title": "Definition:Convergent of Continued Fraction", "text": "Let $F$ be a field, such as the field of real numbers. Let $n \\in \\N \\cup \\{\\infty\\}$ be an extended natural number. Let $C = \\left[{a_0, a_1, a_2, \\ldots}\\right]$ be a continued fraction in $F$ of length $n$. Let $k \\leq n$ be a natural number. === Definition 1 === {{Definition:Convergent of Continued Fraction/Definition 1}} === Definition 2 === {{Definition:Convergent of Continued Fraction/Definition 2}}"}
+{"_id": "24384", "title": "Definition:Convergent of Continued Fraction/Even", "text": "The '''even convergents''' of $\\left[{a_0, a_1, a_2, \\ldots}\\right]$ are the convergents $C_0, C_2, C_4, \\ldots$, that is, those with an even subscript."}
+{"_id": "24385", "title": "Definition:Convergent of Continued Fraction/Odd", "text": "The '''odd convergents''' of $\\left[{a_0, a_1, a_2, \\ldots}\\right]$ are the convergents $C_1, C_3, C_5, \\ldots$, that is, those with an odd subscript."}
+{"_id": "24386", "title": "Definition:Numerators and Denominators of Continued Fraction", "text": "Let $F$ be a field. Let $n \\in \\N \\cup \\set \\infty$ be an extended natural number. Let $C = \\sqbrk {a_0, a_1, a_2, \\ldots}$ be a continued fraction in $F$ of length $n$. === Definition 1: recursive definition === The '''sequence of numerators''' of $C$ is the sequence $\\sequence {p_k}_{0 \\mathop \\le k \\mathop \\le n}$ that is recursively defined by: :$p_k = \\begin {cases}   a_0 & : k = 0 \\\\   a_1 a_0 + 1 & : k = 1 \\\\   a_k p_{k - 1} + p_{k - 2} & : k \\ge 2 \\end {cases}$ The '''sequence of denominators''' of $C$ is the sequence $\\sequence {q_k}_{0 \\mathop \\le k \\mathop \\le n}$ that is recursively defined by: :$q_k = \\begin {cases}   1 & : k = 0 \\\\   a_1 & : k = 1 \\\\   a_k q_{k - 1} + q_{k - 2} & : k \\ge 2 \\end {cases}$ === Definition 2: using matrix products === Let $k \\ge 0$, and let the indexed matrix product: :$\\displaystyle \\prod_{i \\mathop = 0}^k \\begin {pmatrix}   a_i & 1 \\\\   1 & 0 \\end {pmatrix} = \\begin {pmatrix}   x_{1 1}^{\\paren k} & x_{1 2}^{\\paren k} \\\\   x_{2 1}^{\\paren k} & x_{2 2}^{\\paren k} \\end {pmatrix}$ The $k$th '''numerator''' is $x_{1 1}^{\\paren k}$ and the $k$th '''denominator''' is $x_{2 1}^{\\paren k}$."}
+{"_id": "24387", "title": "Definition:Generalized Continued Fraction", "text": "Let $k$ be a field. Informally, a '''generalized continued fraction''' in $k$ is an expression of the form: :$b_0 + \\cfrac {a_1} {b_1 + \\cfrac {a_2} {b_2 + \\cfrac {a_3} {\\ddots \\cfrac {} {b_{n-1} + \\cfrac {a_n} {b_n + \\cfrac {a_{n+1}} {\\ddots}}} }}}$ Formally, a '''generalized continued fraction''' in $k$ is a pair of sequences $((b_n)_{n\\geq 0}, (a_n)_{n\\geq 1})$ in $k$, called '''sequence of partial denominators''' and '''sequence of partial numerators''' respectively. </onlyinclude>"}
+{"_id": "24388", "title": "Definition:Sheffer Operator", "text": "A '''Sheffer operator''' is a truth function which forms a functionally complete singleton set."}
+{"_id": "24389", "title": "Definition:Functionally Complete", "text": "Let $S$ be a set of truth functions. Then $S$ is '''functionally complete''' {{iff}} all possible truth functions are definable from $S$."}
+{"_id": "24390", "title": "Definition:Complement of Truth Function", "text": "Let $f: \\Bbb B^k \\to \\Bbb B$ be a truth function. The '''complement''' of $f$ is the function $f'$ defined by: :$f': \\Bbb B^k \\to \\Bbb B, f' \\left({p}\\right) = \\neg \\left({f \\left({p}\\right)}\\right)$"}
+{"_id": "24391", "title": "Definition:Support of Continuous Mapping", "text": "=== Continuous Real-Valued Function in $\\R^n$ === {{Definition:Support of Continuous Mapping/Real-Valued}} === General topological group === Let $X$ be a topological space. Let $G$ be a topological group with identity $e$. Let $f : X \\to G$ be a continuous mapping. The '''support of $f$''' is the closure of the set of elements of $X$ that do not map to $e$ under $f$: :$\\operatorname{supp} \\left({f}\\right) = \\overline{\\left\\{{x \\in X: f \\left({x}\\right) \\ne e}\\right\\}}$"}
+{"_id": "24392", "title": "Definition:Propositional Function", "text": "A '''propositional function''' $\\map P {x_1, x_2, \\ldots}$ is an operation which acts on the objects denoted by the object variables $x_1, x_2, \\ldots$ in a particular universe to return a truth value which depends on: :$(1): \\quad$ The values of $x_1, x_2, \\ldots$ :$(2): \\quad$ The nature of $P$. === Satisfaction === {{:Definition:Propositional Function/Satisfaction}}"}
+{"_id": "24393", "title": "Definition:Boolean Interpretation/Formula", "text": "Let $v: \\mathcal L_0 \\to \\left\\{{T, F}\\right\\}$ be a (partial) boolean interpretation. Let $\\phi$ be a WFF of propositional logic. Then $v$ is called a '''boolean interpretation for $\\phi$''' iff $v$ is defined at $\\phi$. Otherwise, $v$ is called a '''partial (boolean) interpretation for $\\phi$'''."}
+{"_id": "24394", "title": "Definition:Boolean Interpretation/Set of Formulas", "text": "Let $v: \\mathcal L_0 \\to \\left\\{{T, F}\\right\\}$ be a (partial) boolean interpretation. Let $\\mathcal F$ be a set of WFFs of $\\mathcal L_0$. Then $v$ is called a '''boolean interpretation for $\\mathcal F$''' iff $v$ is defined on $\\mathcal F$. Otherwise, $v$ is called a '''partial (boolean) interpretation for $\\mathcal F$'''."}
+{"_id": "24395", "title": "Definition:Boolean Interpretation/Truth Value", "text": "Let $\\phi$ be a WFF of propositional logic. Let $v: \\mathcal L_0 \\to \\left\\{{T, F}\\right\\}$ be a boolean interpretation of $\\phi$. The '''truth value of $\\phi$ under $v$''' is $v (\\phi)$."}
+{"_id": "24396", "title": "Definition:Legendre Polynomial", "text": "The '''Legendre polynomials''' are the solutions to Legendre's differential equation."}
+{"_id": "24397", "title": "Definition:Inverse Hyperbolic Sine/Complex", "text": "=== Definition 1 === {{:Definition:Inverse Hyperbolic Sine/Complex/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Sine/Complex/Definition 2}}"}
+{"_id": "24398", "title": "Definition:Variable/Propositional Logic", "text": "A '''statement variable''' is a variable which is used to stand for arbitrary and unspecified statements. For '''statement variables''', lowercase letters are usually used, e.g.: :$p, q, r, \\ldots{}$, etc. or lowercase Greek letters, e.g.: :$\\phi, \\psi, \\chi$ etc. The citing of a variable can be interpreted as an assertion that the statement represented by that symbol is true.  That is: :'''$p$''' means :'''$p \\text { is true}$'''"}
+{"_id": "24399", "title": "Definition:Scope (Logic)/Connective", "text": "Let $\\mathcal L_0$ be the language of propositional logic. Let $\\circ$ be a connective of $\\mathcal L_0$. Let $\\mathbf W$ be a well-formed formula of $\\mathcal L_0$. The '''scope''' of an occurrence of $\\circ$ in $\\mathbf W$ is the smallest well-formed part of $\\mathbf W$ containing this occurrence of $\\circ$."}
+{"_id": "24400", "title": "Definition:Abbreviation of WFFs of Propositional Logic", "text": "The WFFs of propositional logic can be made more readable by allowing them to be abbreviated. The resulting strings are not actually WFFs as such, but can be translated back uniquely into full WFFs. === Rules for Abbreviation of WFFs === {{:Definition:Abbreviation of WFFs of Propositional Logic/Rules}}"}
+{"_id": "24402", "title": "Definition:Abbreviation of WFFs of Propositional Logic/Standard Abbreviation", "text": "The string obtained by applying as many of the rules for abbreviation of WFFs  to a WFF $\\mathbf A$ as possible is known as the '''standard abbreviation of $\\mathbf A$'''."}
+{"_id": "24403", "title": "Definition:Binding Priority/Propositional Logic", "text": "The binding priority convention which is almost universally used for the connectives of propositional logic is: :$(1): \\quad \\neg$ binds more tightly than $\\lor$ and $\\land$ :$(2): \\quad \\lor$ and $\\land$ bind more tightly than $\\implies$ and $\\impliedby$ :$(3): \\quad \\implies$ and $\\impliedby$ bind more tightly than $\\iff$ Note that there is no overall convention defining which of $\\land$ and $\\lor$ bears a higher binding priority, and therefore we consider them to have equal priority. Because of this fact, ''unless specifically defined'', expressions such as $p \\land q \\lor r$ can not be interpreted unambiguously, and parenthesis ''must'' be used to determine the exact priorities which are to be used to interpret particular statements which may otherwise be ambiguous. Most sources do not recognise the use of $\\impliedby$ as a separate connective from $\\implies$, so the priority of one over the other is rarely a question."}
+{"_id": "24405", "title": "Definition:Main Connective/Propositional Logic/Definition 1", "text": "Let $\\mathbf C$ be a WFF of propositional logic. Let $\\circ$ be a binary connective. Then $\\circ$ is the '''main connective''' iff the scope of $\\circ$ is $\\mathbf C$."}
+{"_id": "24406", "title": "Definition:Main Connective/Propositional Logic/Definition 2", "text": "Let $\\mathbf C$ be a WFF of propositional logic such that: : $\\mathbf C = \\left({\\mathbf A \\circ \\mathbf B}\\right)$ where both $\\mathbf A$ and $\\mathbf B$ are both WFFs and $\\circ$ is a binary connective. Then $\\circ$ is the '''main connective''' of $\\mathbf C$. Alternatively, let $\\mathbf A$ be a WFF of propositional logic such that: : $\\mathbf A = \\neg \\mathbf B$ where $\\mathbf B$ is a WFF. Then $\\neg$ is the '''main connective''' of $\\mathbf A$."}
+{"_id": "24407", "title": "Definition:Main Connective/Propositional Logic/Definition 3", "text": "Let $T$ be a WFF of propositional logic in the labeled tree specification. Suppose $T$ has more than one node. Then the label of the root of $T$ is called the '''main connective''' of $T$."}
+{"_id": "24408", "title": "Definition:Classical Propositional Logic", "text": "'''Classical propositional logic''' is the branch of propositional logic based upon Aristotelian logic, whose philosophical tenets state that: :$(1): \\quad$ A statement must be either true or false, and there can be no middle value. :$(2): \\quad$ A statement cannot be both true and not true at the same time, that is, it may not contradict itself."}
+{"_id": "24409", "title": "Definition:Labeled Tree for Propositional Logic", "text": "A '''labeled tree for propositional logic''' is a system containing: * A rooted tree $T$; * A countable set $\\mathbf H$ of WFFs of propositional logic; * A WFF $\\Phi \\left({t}\\right)$ attached to each non-root node $t$ of $T$. Such a structure can be denoted $\\left({T, \\mathbf H, \\Phi}\\right)$. === Hypothesis Set === {{:Definition:Labeled Tree for Propositional Logic/Hypothesis Set}} === Attached === {{:Definition:Labeled Tree for Propositional Logic/Attached}} === Child WFF === {{:Definition:Labeled Tree for Propositional Logic/Child WFF}} === Ancestor WFF === {{:Definition:Labeled Tree for Propositional Logic/Ancestor WFF}} === Along a Branch === {{:Definition:Labeled Tree for Propositional Logic/Along a Branch}}"}
+{"_id": "24410", "title": "Definition:Labeled Tree for Propositional Logic/Attached", "text": "Let $t$ be a non-root node of $T$. Let $\\mathbf A$ be a WFF. Then '''$\\mathbf A$ is attached to $t$''' {{iff}} $\\mathbf A = \\Phi \\left({t}\\right)$."}
+{"_id": "24411", "title": "Definition:Labeled Tree for Propositional Logic/Child WFF", "text": "A WFF that is attached to a child of a node $t$ is called a '''child WFF of $t$'''."}
+{"_id": "24412", "title": "Definition:Labeled Tree for Propositional Logic/Ancestor WFF", "text": "A WFF that is attached to an ancestor node of a node $t$ is called an '''ancestor WFF of $t$'''."}
+{"_id": "24413", "title": "Definition:Labeled Tree for Propositional Logic/Along a Branch", "text": "Let $\\Gamma$ be a branch of $T$. Let $\\mathbf A$ be a WFF that is attached to a node $t \\in \\Gamma$. Then '''$\\mathbf A$ occurs along $\\Gamma$'''. This includes the case where $\\mathbf A \\in \\mathbf H$, that is, $\\mathbf A$ is attached to the root node."}
+{"_id": "24414", "title": "Definition:Intuitionistic Propositional Logic", "text": "The '''intuitionist''' school of mathematics is one which adopts the following philosophical position: :\"Although we may know that it is not the case that a statement $p$ is (provably) false, we don't necessarily know that it is (provably) true either.\" Thus the intuitionist school rejects the Law of the Excluded Middle. The classical school, by affirming that if a statement is not true it must be false, and if it is not false it must be true, accepts as an axiom that \"not not-true\" must mean \"true\". {{refactor|The below needs its own place (it's \"Natural Deduction for Intuitionistic Propositional Logic\")}} The system of '''intuitionistic propositional logic''' is, in consequence, based on the same axioms as that of classical propositional logic except for that disputed Law of the Excluded Middle."}
+{"_id": "24415", "title": "Definition:Language of Propositional Logic/Labeled Tree", "text": "There are many formal languages expressing propositional logic. The formal language used on {{ProofWiki}} is defined on Definition:Language of Propositional Logic. This page defines the formal language $\\LL_0$ used in: * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition}} Explanations are omitted as this is intended for reference use only. === Alphabet === ==== Letters ==== The letters used are an infinite set of symbols $\\PP_0$. This is the same as the {{ProofWiki}} definition. ==== Signs ==== ===== Connectives ===== The following '''connectives''' are used: {{begin-eqn}} {{eqn | ll= \\bullet       | l = \\neg       | o = :       | r = \\)the negation sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\lor       | o = :       | r = \\)the disjunction sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\land       | o = :       | r = \\)the conjunction sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\implies       | o = :       | r = \\)the conditional sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\iff       | o = :       | r = \\)the biconditional sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\oplus       | o = :       | r = \\)the exclusive or sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\downarrow       | o = :       | r = \\)the nor sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\uparrow       | o = :       | r = \\)the nand sign\\(       | c =  }} {{end-eqn}} See the {{ProofWiki}} definition. === Collation System === The collation system used is that of labeled trees and adding ancestors. {{explain|\"adding ancestors\" signifies combining labeled trees into a new one by creating a common ancestor for their roots}} See the {{ProofWiki}} definition. === Formal Grammar === The following bottom-up formal grammar is used. Let $\\PP_0$ be the vocabulary of $\\LL_0$. The WFFs of $\\LL_0$ are the smallest set $\\FF$ of labeled trees such that: {{begin-axiom}} {{axiom | n = T1         | t = For each letter $p \\in \\PP_0$, the labeled tree with one node, whose label is $p$, is in $\\FF$. }} {{axiom | n = T2         | t = For $T \\in \\FF$, the labeled tree obtained by adding an ancestor with label $\\neg$ to the root node of $T$, is again in $\\FF$. }} {{axiom | n = T3         | t = For $T_1 \\in \\FF$ and $T_2 \\in \\FF$ and a binary connective $\\mathsf B$, the labeled tree obtained by adding a common ancestor labeled $\\mathsf B$ of the root nodes of $T_1$ and $T_2$, is again in $\\FF$. }} {{end-axiom}} Graphically, this means one has the following means to construct WFFs: ::$\\begin{xy}\\xymatrix{  p & &  \\neg   \\ar@{-}[d] & & &  \\mathsf B   \\ar@{-}[ld]   \\ar@{-}[rd] \\\\ & &  <{\\sf WFF}> & &   <{\\sf WFF}> & &  <{\\sf WFF}> }\\end{xy}$ See the {{ProofWiki}} definition."}
+{"_id": "24416", "title": "Definition:Absolute Value/Ordered Integral Domain", "text": "Let $\\struct {D, +, \\times, \\le}$ be an ordered integral domain whose zero is $0_D$. Then for all $a \\in D$, the '''absolute value''' of $a$ is defined as: :$\\size a = \\begin{cases} a & : 0_D \\le a \\\\ -a & : a < 0_D \\end{cases}$"}
+{"_id": "24417", "title": "Definition:Truth Function", "text": "Let $\\mathbb B$ be the set of truth values, and let $k$ be a natural number. A mapping $f: \\mathbb B^k \\to \\mathbb B$ is called a '''truth function'''."}
+{"_id": "24419", "title": "Definition:Logical NAND/Truth Function", "text": "The NAND connective defines the truth function $f^\\uparrow$ as follows: {{begin-eqn}} {{eqn | l=f^\\uparrow \\left({F, F}\\right)       | r=T }} {{eqn | l=f^\\uparrow \\left({F, T}\\right)       | r=T }} {{eqn | l=f^\\uparrow \\left({T, F}\\right)       | r=T }} {{eqn | l=f^\\uparrow \\left({T, T}\\right)       | r=F }} {{end-eqn}}"}
+{"_id": "24420", "title": "Definition:Logical NOR/Truth Function", "text": "The NOR connective defines the truth function $f^\\downarrow$ as follows: {{begin-eqn}} {{eqn | l=f^\\downarrow \\left({F, F}\\right)       | r=T }} {{eqn | l=f^\\downarrow \\left({F, T}\\right)       | r=F }} {{eqn | l=f^\\downarrow \\left({T, F}\\right)       | r=F }} {{eqn | l=f^\\downarrow \\left({T, T}\\right)       | r=F }} {{end-eqn}}"}
+{"_id": "24421", "title": "Definition:Exclusive Or/Truth Function", "text": "The exclusive or connective defines the truth function $f^\\oplus$ as follows: {{begin-eqn}} {{eqn | l=f^\\oplus \\left({F, F}\\right)       | r=F }} {{eqn | l=f^\\oplus \\left({F, T}\\right)       | r=T }} {{eqn | l=f^\\oplus \\left({T, F}\\right)       | r=T }} {{eqn | l=f^\\oplus \\left({T, T}\\right)       | r=F }} {{end-eqn}}"}
+{"_id": "24422", "title": "Definition:Truth Table/Characteristic", "text": "Let $\\circledcirc$ be a logical connective. The '''characteristic truth table''' for $\\circledcirc$ is the truth table describing the truth function of $\\circledcirc$: :$\\begin{array}{|cc||c|} \\hline p & q & p \\circledcirc q \\\\ \\hline F & F & x \\\\ F & T & x \\\\ T & F & x \\\\ T & T & x \\\\ \\hline \\end{array}$ where $x$ is replaced by either $F$ or $T$ as appropriate on each row. The characteristic truth tables of the various logical connectives are listed below. === Logical Negation === {{:Definition:Logical Not/Truth Table}} === Logical Conjunction === {{:Definition:Conjunction/Truth Table}} === Logical Disjunction === {{:Definition:Disjunction/Truth Table}} === Biconditional === {{:Definition:Biconditional/Truth Table}} === Exclusive Disjunction === {{:Definition:Exclusive Or/Truth Table}} === Conditional === {{:Definition:Conditional/Truth Table}} === Logical NAND === {{:Definition:Logical NAND/Truth Table}} === Logical NOR === {{:Definition:Logical NOR/Truth Table}}"}
+{"_id": "24423", "title": "Definition:Generator of Monoid", "text": "Let $\\left({M, \\circ}\\right)$ be a monoid Let $S \\subseteq M$. Let $H$ be the smallest submonoid of $M$ such that $S \\subseteq H$. Then: * $S$ is a '''generator'''  of $\\left({H, \\circ}\\right)$ * $S$ '''generates''' $\\left({H, \\circ}\\right)$ * $\\left({H, \\circ}\\right)$ is the '''submonoid of $\\left({M, \\circ}\\right)$ generated by $S$'''. This is written $H = \\left\\langle {S} \\right\\rangle$."}
+{"_id": "24424", "title": "Definition:Boolean Interpretation", "text": "Let $\\LL_0$ be the language of propositional logic, with vocabulary $\\PP_0$. A '''boolean interpretation''' for $\\LL_0$ is a propositional function: :$v: \\PP_0 \\to \\set {\\T, \\F}$"}
+{"_id": "24425", "title": "Definition:Isomorphism (Abstract Algebra)/Monoid Isomorphism", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be monoids. Let $\\phi: S \\to T$ be a (monoid) homomorphism. Then $\\phi$ is a monoid isomorphism {{iff}} $\\phi$ is a bijection."}
+{"_id": "24427", "title": "Definition:Monoid Automorphism", "text": "Let $\\struct {S, \\circ}$ be a monoid. Let $\\phi: S \\to S$ be a (monoid) isomorphism from $S$ to itself. Then $\\phi$ is a monoid automorphism."}
+{"_id": "24429", "title": "Definition:Additive Group of Real Numbers", "text": "The '''additive group of real numbers''' $\\struct {\\R, +}$ is the set of real numbers under the operation of addition."}
+{"_id": "24430", "title": "Definition:Additive Group of Rational Numbers", "text": "The '''additive group of rational numbers''' $\\left({\\Q, +}\\right)$ is the set of rational numbers under the operation of addition."}
+{"_id": "24431", "title": "Definition:Multiplicative Group of Rational Numbers", "text": "The '''multiplicative group of rational numbers''' $\\left({\\Q_{\\ne 0}, \\times}\\right)$ is the set of rational numbers without zero under the operation of multiplication."}
+{"_id": "24432", "title": "Definition:Additive Group of Complex Numbers", "text": "The '''additive group of complex numbers''' $\\struct {\\C, +}$ is the set of complex numbers under the operation of addition."}
+{"_id": "24433", "title": "Definition:Multiplicative Group of Complex Numbers", "text": "The '''multiplicative group of complex numbers''' $\\struct {\\C_{\\ne 0}, \\times}$ is the set of complex numbers without zero under the operation of multiplication."}
+{"_id": "24434", "title": "Definition:Multiplicative Group of Real Numbers", "text": "The '''multiplicative group of real numbers''' $\\struct {\\R_{\\ne 0}, \\times}$ is the set of real numbers without zero under the operation of multiplication."}
+{"_id": "24435", "title": "Definition:Field of Complex Numbers", "text": "The '''field of complex numbers''' $\\struct {\\C, +, \\times}$ is the set of complex numbers under the two operations of addition and multiplication."}
+{"_id": "24436", "title": "Definition:Field of Real Numbers", "text": "The '''field of real numbers''' $\\struct {\\R, + \\times, \\le}$ is the set of real numbers under the two operations of addition and multiplication, with an ordering $\\le$ compatible with the ring structure of $\\R$.."}
+{"_id": "24437", "title": "Definition:Field of Rational Numbers", "text": "The '''field of rational numbers''' $\\struct {\\Q, + \\times, \\le}$ is the set of rational numbers under the two operations of addition and multiplication, with an ordering $\\le$ compatible with the ring structure of $\\Q$."}
+{"_id": "24438", "title": "Definition:Multiplicative Monoid of Integers Modulo m", "text": "Let $m \\in \\Z$ such that $m > 1$. The '''multiplicative monoid of integers modulo $m$''' $\\struct {\\Z_m, \\times_m}$ is the set of integers modulo $m$ under the operation of multiplication modulo $m$."}
+{"_id": "24439", "title": "Definition:Additive Group of Integers Modulo m", "text": "Let $m \\in \\Z$ such that $m > 1$. The '''additive group of integers modulo $m$''' $\\struct {\\Z_m, +_m}$ is the set of integers modulo $m$ under the operation of addition modulo $m$."}
+{"_id": "24440", "title": "Definition:Radical Extension", "text": "Let $L / F$ be a field extension. Then $L$ is a '''radical extension of $F$''' iff there exist $\\alpha_1, \\ldots, \\alpha_m \\in F$ and $n_1, \\ldots, n_2 \\in \\Z_{>0}$ such that: :$(1): \\quad L = K \\left[{\\alpha_1, \\ldots, \\alpha_m}\\right]$ :$(2): \\quad \\alpha_1^{n_1} \\in F$ :$(3): \\quad \\forall i \\in \\N_m: \\alpha_i^{n_i} \\in F \\left[{\\alpha_1, \\ldots, \\alpha_{i-1}}\\right]$ {{explain|link to definitions of $K \\left[{\\alpha_1, \\ldots, \\alpha_m}\\right]$ and $F \\left[{\\alpha_1, \\ldots, \\alpha_{i-1} }\\right]$}} Category:Definitions/Field Extensions mhhrn2hhw9k4rj8re7rflpi2nd0knsp"}
+{"_id": "24441", "title": "Definition:Additive Monoid of Natural Numbers", "text": "The '''additive monoid of natural numbers''' $\\left({\\N, +}\\right)$ is the set of natural numbers under the operation of addition."}
+{"_id": "24442", "title": "Definition:Language of Propositional Logic/Formal Grammar/WFF", "text": "Let $\\mathbf A$ be approved of by the formal grammar of propositional logic. Then $\\mathbf A$ is called a '''well-formed formula of propositional logic'''. Often, one abbreviates \"well-formed formula\", speaking of a '''WFF of propositional logic''' instead."}
+{"_id": "24443", "title": "Definition:Statement Form/Substatement", "text": "A '''substatement''' of a statement form $\\mathbf A$ is another statement form which occurs as a part of $\\mathbf A$."}
+{"_id": "24444", "title": "Definition:Finite Monoid", "text": "A '''finite monoid''' is a monoid whose underlying set is finite. Category:Definitions/Monoids 2fqcp6gr4u7mcakapzp30gtr2ap94ho"}
+{"_id": "24446", "title": "Definition:Additive Group of Integer Multiples", "text": "Let $n \\in \\Z_{>0}$. The '''additive group $\\left({n \\Z, +}\\right)$ of integer multiples of $n$'''  is the set of integer multiples of $n$ under the operation of addition."}
+{"_id": "24447", "title": "Definition:Root of Unity/Complex", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. The '''complex $n$th roots of unity''' are the elements of the set: :$U_n = \\set {z \\in \\C: z^n = 1}$"}
+{"_id": "24448", "title": "Definition:Root of Unity/Primitive", "text": "A '''primitive $n$th root of unity of $F$''' is an element $\\alpha \\in U_n$ such that: :$U_n = \\set {1, \\alpha, \\ldots, \\alpha^{n - 1} }$"}
+{"_id": "24449", "title": "Definition:Affine Group of One Dimension", "text": "Let $S$ be the set of mappings $f_{a b}: \\R \\to \\R$ defined as: :$S := \\left\\{{f_{a b}: x \\mapsto a x + b: a \\in \\R_{\\ne 0}, b \\in \\R}\\right\\}$ The algebraic structure $\\left({S, \\circ}\\right)$, where $\\circ$ denotes composition of mappings, is called the '''$1$-dimensional affine group on $\\R$''' and can be denoted $\\operatorname{Af}_1 \\left({\\R}\\right)$."}
+{"_id": "24450", "title": "Definition:Multiplicative Group of Positive Rational Numbers", "text": "The '''multiplicative group of positive rational numbers''' $\\left({\\Q_{> 0}, \\times}\\right)$ is the set of (strictly) positive rational numbers under the operation of multiplication."}
+{"_id": "24451", "title": "Definition:Multiplicative Group of Positive Real Numbers", "text": "The '''multiplicative group of positive real numbers''' $\\left({\\R_{> 0}, \\times}\\right)$ is the set of (strictly) positive real numbers under the operation of multiplication."}
+{"_id": "24452", "title": "Definition:Googol", "text": "A '''googol''' is defined to be $10^{100}$."}
+{"_id": "24453", "title": "Definition:Googolplex", "text": "A '''googolplex''' is defined to be $10$ to the power of a googol, that is $10^{10^{100}}$ or $10^{\\mathrm {googol}}$."}
+{"_id": "24454", "title": "Definition:Hyperbola", "text": "=== Intersection with Cone === {{:Definition:Conic Section/Intersection with Cone}} {{:Definition:Conic Section/Intersection with Cone/Hyperbola}} === Focus-Directrix Property === {{:Definition:Conic Section/Focus-Directrix Property/Hyperbola}} === Equidistance Property === {{:Definition:Hyperbola/Equidistance}}"}
+{"_id": "24455", "title": "Definition:Parabola", "text": "=== Intersection with Cone === {{:Definition:Conic Section/Intersection with Cone}} {{:Definition:Conic Section/Intersection with Cone/Parabola}} === Focus-Directrix Property === {{:Definition:Conic Section/Focus-Directrix Property/Parabola}}"}
+{"_id": "24456", "title": "Definition:Centralizer/Group Subset", "text": "Let $\\struct {G, \\circ}$ be a group. Let $S \\subseteq G$. The '''centralizer of $S$ (in $G$)''' is the set of elements of $G$ which commute with all $s \\in S$: :$\\map {C_G} S = \\set {x \\in G: \\forall s \\in S: x \\circ s = s \\circ x}$"}
+{"_id": "24457", "title": "Definition:Möbius Strip", "text": "A '''Möbius strip''' is a surface with boundary obtained by twisting a side of a rectangle by $180$ degrees, then joining the twisted side and the opposite side of it together. {{expand|An illustration of this process will aid understanding}}"}
+{"_id": "24458", "title": "Definition:Differentiable at Point", "text": "=== Real Function === {{:Definition:Differentiable Mapping/Real Function/Point}} === Complex Function === {{:Definition:Differentiable Mapping/Complex Function/Point}} === Real-Valued Function === {{:Definition:Differentiable Mapping/Real-Valued Function/Point}} === Between Manifolds === {{:Definition:Differentiable Mapping between Manifolds/Point}} Category:Definitions/Differential Calculus qkhwikxfxwgzmut1nsenhyu19q3cm2c"}
+{"_id": "24459", "title": "Definition:Faithful Group Action", "text": "Let $G$ be a group with identity $e$. Let $X$ be a set. Let $\\phi: G \\times X \\to X$ be a group action. === Definition 1 === {{:Definition:Faithful Group Action/Definition 1}} === Definition 2 === {{:Definition:Faithful Group Action/Definition 2}}"}
+{"_id": "24460", "title": "Definition:Transitive Group Action", "text": "Let $G$ be a group. Let $S$ be a set. Let $*: G \\times S \\to S$ be a group action. The group action is '''transitive''' {{iff}} for any $x, y \\in S$ there exists $g \\in G$ such that $g * x = y$. That is, {{iff}} for all $x \\in S$: :$\\Orb x = S$ where $\\Orb x$ denotes the orbit of $x \\in S$ under $*$."}
+{"_id": "24461", "title": "Definition:Trivial Group Action", "text": "Let $G$ be a group. Let $S$ be a set. Let $*: G \\times S \\to S$ be the group action defined as: :$\\forall \\tuple {g, s} \\in G \\times S: g * s = s$ Then $*$ is the '''trivial group action''' of $G$ on $S$."}
+{"_id": "24462", "title": "Definition:Group Action Axioms", "text": "{{begin-axiom}} {{axiom | n = \\text {GA} 1         | q = \\forall g, h \\in G, x \\in X         | m = g * \\paren {h * x} = \\paren {g \\circ h} * x }} {{axiom | n = \\text {GA} 2         | q = \\forall x \\in X         | m = e * x = x }} {{end-axiom}}"}
+{"_id": "24463", "title": "Definition:Algebra of Sets/Definition 1", "text": "Let $X$ be a set. Let $\\powerset X$ be the power set of $X$. Let $\\RR \\subseteq \\powerset X$ be a set of subsets of $X$. Then $\\RR$ is an '''algebra of sets over $X$''' {{iff}} the following conditions hold: {{begin-axiom}} {{axiom | n = \\text {AS} 1         | lc= Unit:         | m = X \\in \\RR }} {{axiom | n = \\text {AS} 2         | lc= Closure under Union:         | q = \\forall A, B \\in \\RR         | m = A \\cup B \\in \\RR }} {{axiom | n = \\text {AS} 3         | lc= Closure under Complement Relative to $X$:         | q = \\forall A \\in \\RR         | m = \\relcomp X A \\in \\RR }} {{end-axiom}}"}
+{"_id": "24464", "title": "Definition:Algebra of Sets/Definition 2", "text": "An '''algebra of sets''' is a ring of sets with a unit."}
+{"_id": "24466", "title": "Definition:Logical Argument/Finitary", "text": "A '''finitary argument''' is a logical argument which starts with a finite number of axioms, and can be translated into a finite number of statements."}
+{"_id": "24467", "title": "Definition:Deductive Argument", "text": "A '''deductive argument''' is one whose conclusion follows unshakeable from the premises."}
+{"_id": "24468", "title": "Definition:Argument Form", "text": "An '''argument form''' is a collation of symbols which contains statement variables such that: : when statements are used to replace statement variables (the same statement replacing the same statement variable throughout), the result is a logical argument. === Specific Form === {{:Definition:Argument Form/Specific Form}}"}
+{"_id": "24469", "title": "Definition:Argument Form/Specific Form", "text": "The '''specific form''' of a given logical argument is that argument form from which the logical argument results from replacing each distinct statement variable by a different simple statement."}
+{"_id": "24470", "title": "Definition:Statement Form/Specific Form", "text": "The '''specific form''' of a given statement is that propositional formula from which the statement form results from replacing each distinct statement variable by a different simple statement."}
+{"_id": "24471", "title": "Definition:Elementary Valid Argument", "text": "An '''elementary valid argument''' is a logical argument which is a substitution instance of an elementary valid argument form. The argument forms which are considered to be elementary valid argument forms will depend on the specific approach to propositional logic which is under discussion."}
+{"_id": "24472", "title": "Definition:Operation/Arity/Ternary", "text": "A '''ternary operation''' (or '''three-place operation''') is an operation which takes three operands. That is, its arity is $3$."}
+{"_id": "24473", "title": "Definition:Operation/Arity/Multiary", "text": "Certain types of operation have a variable number of operands. An operation which does not have a fixed arity is called '''multiary'''."}
+{"_id": "24474", "title": "Definition:Operation/Arity/Finitary", "text": "A '''finitary operation''' is an operation which takes a finite number of operands."}
+{"_id": "24475", "title": "Definition:Operation/Arity/Zero", "text": "It is possible to conceive of an operator which takes ''no'' operands. A constant can be considered as an operation which takes no operands. That is, it has an arity of zero."}
+{"_id": "24476", "title": "Definition:Variable/Value", "text": "A variable $x$ may be (temporarily, conceptually) identified with a particular object. If so, then that object is called the '''value''' of $x$."}
+{"_id": "24477", "title": "Definition:Propositional Function/Satisfaction", "text": "Let $\\map P {x_1, x_2, \\ldots, x_n}$ be an $n$-ary propositional function. If $a_1, a_2, \\ldots, a_n$ have values which make $\\map P {x_1, x_2, \\ldots, x_n}$ true, then the ordered tuple $\\tuple {a_1, a_2, \\ldots, a_n}$ '''satisfies''' $\\map P {x_1, x_2, \\ldots, x_n}$."}
+{"_id": "24479", "title": "Definition:Integer Division", "text": "Let $a, b \\in \\Z$ be integers such that $b \\ne 0$.. From the Division Theorem: :$\\exists_1 q, r \\in \\Z: a = q b + r, 0 \\le r < \\left|{b}\\right|$ where $q$ is the quotient and $r$ is the remainder. The process of finding $q$ and $r$ is known as '''(integer) division'''."}
+{"_id": "24480", "title": "Definition:Diagram (Graphical Technique)", "text": "A '''diagram''' is a graphical technique for illustrating a concept in picture form. It is generally considered that its use should be limited to that of an aid to understanding, and should not be used in order to prove something."}
+{"_id": "24481", "title": "Definition:Quantified Statement", "text": "A '''quantified statement''' is a simple statement in predicate logic whose subject is qualified by either the universal quantifier or the existential quantifier. That is, it is either a universal statement or an existential statement. Category:Definitions/Predicate Logic 8uoinmnj34wnioqjbfik4kc9hgljjp5"}
+{"_id": "24482", "title": "Definition:Square of Opposition", "text": "The '''square of opposition''' is a diagram whose purpose is to illustrate the relations between the various categories of quantified statement. ::$\\begin{xy} <-10em,0em>*+{\\forall x: \\map \\Phi x} = \"A\", <10em,0em>*+{\\forall x: \\neg \\map \\Phi x} = \"E\", <-10em,-20em>*+{\\exists x: \\map \\Phi x} = \"I\", <10em,-20em>*+{\\exists x: \\neg \\map \\Phi x} = \"O\", \"A\";\"E\" **@{-} ?<*@{<} ?>*@{>} ?*!/^.8em/{\\text{Contraries}}, \"A\";\"I\" **@{-} ?>*@{>} ?*!/^3.2em/{\\text{Subimplicant}}, \"A\";\"O\" **@{-} ?<*@{<} ?>*@{>} ?*!/^4em/{\\text{Contradictories}}, \"I\";\"E\" **@{-} ?<*@{<} ?>*@{>} ?*!/_4em/{\\text{Contradictories}}, \"I\";\"O\" **@{-} ?<*@{<} ?>*@{>} ?*!/_.8em/{\\text{Subcontraries}}, \"E\";\"O\" **@{-} ?>*@{>} ?*!/^-3.2em/{\\text{Subimplicant}}, \\end{xy}$ This therefore illustrates the relations: :''All $x$ have $\\Phi$'' is contrary to ''All $x$ do not have $\\Phi$'' :''All $x$ have $\\Phi$'' is contradictory to ''Some $x$ do not have $\\Phi$'' :''Some $x$ have $\\Phi$'' is contradictory to ''All $x$ do not have $\\Phi$'' :''Some $x$ have $\\Phi$'' is subimplicant to ''All $x$ have $\\Phi$'' :''Some $x$ do not have $\\Phi$'' is subimplicant to ''All $x$ do not have $\\Phi$'' :''Some $x$ have $\\Phi$'' is subcontrary to ''Some $x$ do not have $\\Phi$'' where $x$ is an object and $\\Phi$ is a property. === Vacuous Universe === ;Beware: Note that if the universe of discourse is empty, then the square of opposition no longer holds. Although $\\forall x: \\map \\Phi x$ is vacuously true for such an empty universe, $\\exists x: \\map \\Phi x$ is not. Thus $\\exists x: \\map \\Phi x$ is no longer subimplicant to $\\forall x: \\map \\Phi x$. Similarly, as $\\exists x: \\neg \\map \\Phi x$ is also false, it follows that $\\exists x: \\map \\Phi x$ and $\\exists x: \\neg \\map \\Phi x$ are no longer subcontrary."}
+{"_id": "24483", "title": "Definition:Predicate/Is of Identity", "text": "Consider the sentence: :'''Socrates is the philosopher who taught Plato.''' This could be reworded as: :'''The object named Socrates has the property of being the philosopher who taught Plato.''' However, the meaning that is ''really'' being conveyed here is that of: :'''The object named Socrates ''is the same object as'' the object which is the philosopher who taught Plato.''' In this context, '''is''' is not being used in the same way as the ''is'' of predication. When being used to indicate that one object is the same object as another object, '''is''' is called '''the ''is'' of identity'''. In this context, ''is'' means the same as equals."}
+{"_id": "24484", "title": "Definition:Predicate/Is of Predication", "text": "Consider the statement: :'''Socrates is a man.''' This means: :'''The object named Socrates has the property of being a man.''' Thus we see that '''is''' here means '''has the property of being'''. In this context, '''is''' here is called '''the ''is'' of predication'''."}
+{"_id": "24485", "title": "Definition:Universal Affirmative", "text": ":''Every $S$ is $P$'' where $S$ and $P$ are predicates."}
+{"_id": "24486", "title": "Definition:Universal Affirmative/Set Theory", "text": "The universal affirmative $\\forall x: \\map S x \\implies \\map P x$ can be expressed in set language as: :$\\set {x: \\map S x} \\subseteq \\set {x: \\map P x}$ or, more compactly: :$S \\subseteq P$"}
+{"_id": "24487", "title": "Definition:Universal Negative", "text": ":''No $S$ is $P$'' where $S$ and $P$ are predicates."}
+{"_id": "24488", "title": "Definition:Universal Negative/Set Theory", "text": "The universal negative $\\forall x: \\map S x \\implies \\neg \\map P x$ can be expressed in set language as: :$\\set {x: \\map S x} \\implies \\set {x: \\map P x} = \\O$ or, more compactly: :$S \\subseteq \\map \\complement P$"}
+{"_id": "24489", "title": "Definition:Particular Affirmative", "text": ":''Some $S$ is $P$'' where $S$ and $P$ are predicates."}
+{"_id": "24490", "title": "Definition:Particular Affirmative/Set Theory", "text": "The particular affirmative $\\exists x: \\map S x \\land \\map P x$ can be expressed in set language as: :$\\set {x: \\map S x} \\cap \\set {x: \\map P x} \\ne \\O$ or, more compactly: :$S \\cap P \\ne \\O$"}
+{"_id": "24491", "title": "Definition:Particular Negative/Set Theory", "text": "The particular negative $\\exists x: \\map S x \\land \\neg \\map P x$ can be expressed in set language as: :$\\set {x: \\map S x} \\cap \\set {x: \\neg \\map P x} \\ne \\O$ or, more compactly: :$S \\cap \\map \\complement P \\ne \\O$"}
+{"_id": "24492", "title": "Definition:Particular Negative", "text": ":''Some $S$ is not $P$'' where $S$ and $P$ are predicates."}
+{"_id": "24494", "title": "Definition:Categorical Syllogism", "text": "A '''categorical syllogism''' is a logical argument which is structured as follows: $(1): \\quad$ It has exactly two premises and one conclusion. : The first premise is usually referred to as the major premise. : The second premise is usually referred to as the minor premise. $(2): \\quad$ It concerns exactly three terms, which are usually denoted: {{begin-axiom}} {{axiom | q = P         | t = the primary term }} {{axiom | q = M         | t = the middle term }} {{axiom | q = S         | t = the secondary term }} {{end-axiom}} $(3): \\quad$ Each of the premises and conclusion is a categorical statement."}
+{"_id": "24495", "title": "Definition:Categorical Syllogism/Secondary Term", "text": "The '''secondary term''' of a categorical syllogism is the term that appears as the first predicate of the conclusion of the syllogism. It also appears once in one of the premises of the syllogism, traditionally the minor premise. It is usually denoted by $S$."}
+{"_id": "24496", "title": "Definition:Categorical Syllogism/Middle Term", "text": "The '''middle term''' of a categorical syllogism is the term that does not appear in the conclusion of the syllogism. It appears once in each of the premises of the syllogism. It is usually denoted by $M$."}
+{"_id": "24497", "title": "Definition:Categorical Syllogism/Primary Term", "text": "The '''primary term''' of a categorical syllogism is the term that appears as the second predicate of the conclusion of the syllogism. It also appears once in one of the premises of the syllogism, traditionally the major premise. It is usually denoted by $P$."}
+{"_id": "24498", "title": "Definition:Categorical Syllogism/Premises", "text": "There are two premises in a categorical syllogism: === Major Premise === {{:Definition:Categorical Syllogism/Premises/Major Premise}} === Minor Premise === {{:Definition:Categorical Syllogism/Premises/Minor Premise}}"}
+{"_id": "24499", "title": "Definition:Categorical Syllogism/Terms", "text": "There are three terms in a categorical syllogism: === Primary Term of Syllogism === {{:Definition:Categorical Syllogism/Primary Term}} === Middle Term of Syllogism === {{:Definition:Categorical Syllogism/Middle Term}} === Secondary Term of Syllogism === {{:Definition:Categorical Syllogism/Secondary Term}}"}
+{"_id": "24500", "title": "Definition:Categorical Syllogism/Premises/Major Premise", "text": "The '''major premise''' of a categorical syllogism is conventionally stated first. It is a categorical statement which expresses the logical relationship between the primary term and the middle term of the syllogism."}
+{"_id": "24501", "title": "Definition:Categorical Syllogism/Premises/Minor Premise", "text": "The '''minor premise''' of a categorical syllogism is conventionally stated second. It is a categorical statement which expresses the logical relationship between the secondary term and the middle term of the syllogism."}
+{"_id": "24502", "title": "Definition:Categorical Syllogism/Conclusion", "text": "The '''conclusion''' of a categorical syllogism is stated last, conventionally prefaced with the word ''Therefore''. It is a categorical statement which expresses the logical relationship between the primary term and the secondary term of the syllogism. Furthermore, this categorical statement is ''specifically'' of the form in which the secondary term occurs ''first'' and the primary term occurs ''second''."}
+{"_id": "24503", "title": "Definition:Figure of Categorical Syllogism", "text": "There are four patterns of categorical syllogism depending on the order of the terms in the major premise and minor premise. Let $\\mathbf \\Phi_1, \\mathbf \\Phi_2, \\mathbf \\Phi_3$ each be one of the categorical statements $\\mathbf A$, $\\mathbf E$, $\\mathbf I$ or $\\mathbf O$. Let $P$ denote the primary term, $S$ denote the secondary term and $M$ denote the middle term. The four possible '''figures''' are as follows: === $\\text I$ === {{:Definition:Figure of Categorical Syllogism/I}} === $\\text {II}$ === {{:Definition:Figure of Categorical Syllogism/II}} === $\\text {III}$ === {{:Definition:Figure of Categorical Syllogism/III}} === $\\text {IV}$ === {{:Definition:Figure of Categorical Syllogism/IV}}"}
+{"_id": "24504", "title": "Definition:Figure of Categorical Syllogism/I", "text": "{| |- | &nbsp; Major Premise: &nbsp; | align=\"right\" | $\\map {\\mathbf \\Phi_1} {M, P}$ |- | &nbsp; Minor Premise: &nbsp; | align=\"right\" | $\\map {\\mathbf \\Phi_2} {S, M}$ |- | &nbsp; Conclusion: &nbsp; | style=\"border-style: solid; border-width: 1px 0 0 0\"  align=\"right\" | $\\map {\\mathbf \\Phi_3} {S, P}$ |}"}
+{"_id": "24505", "title": "Definition:Figure of Categorical Syllogism/II", "text": "{| |- | &nbsp; Major Premise: &nbsp; | align=\"right\" | $\\map {\\mathbf \\Phi_1} {P, M}$ |- | &nbsp; Minor Premise: &nbsp; | align=\"right\" | $\\map {\\mathbf \\Phi_2 } {S, M}$ |- | &nbsp; Conclusion: &nbsp; | style=\"border-style: solid; border-width: 1px 0 0 0\"  align=\"right\" | $\\map {\\mathbf \\Phi_3} {S, P}$ |}"}
+{"_id": "24506", "title": "Definition:Figure of Categorical Syllogism/III", "text": "{| |- | &nbsp; Major Premise: &nbsp; | align=\"right\" | $\\map {\\mathbf \\Phi_1} {M, P}$ |- | &nbsp; Minor Premise: &nbsp; | align=\"right\" | $\\map {\\mathbf \\Phi_2} {M, S}$ |- | &nbsp; Conclusion: &nbsp; | style=\"border-style: solid; border-width: 1px 0 0 0\"  align=\"right\" | $\\map {\\mathbf \\Phi_3} {S, P}$ |}"}
+{"_id": "24507", "title": "Definition:Figure of Categorical Syllogism/IV", "text": "{| |- | &nbsp; Major Premise: &nbsp; | align=\"right\" | $\\map {\\mathbf \\Phi_1} {P, M}$ |- | &nbsp; Minor Premise: &nbsp; | align=\"right\" | $\\map {\\mathbf \\Phi_2} {M, S}$ |- | &nbsp; Conclusion: &nbsp; | style=\"border-style: solid; border-width: 1px 0 0 0\"  align=\"right\" | $\\map {\\mathbf \\Phi_3} {S, P}$ |}"}
+{"_id": "24508", "title": "Definition:Standard Instance of Categorical Syllogism", "text": "For each categorical syllogism, there are four figures: :$\\begin{array}{r|rl} \\text I & & \\\\ \\hline \\\\ \\text{Major Premise}: & \\mathbf \\Phi_1 & \\tuple {M, P} \\\\ \\text{Minor Premise}: & \\mathbf \\Phi_2 & \\tuple {S, M} \\\\ \\hline \\\\ \\text{Conclusion}: & \\mathbf \\Phi_3 & \\tuple {S, P} \\\\ \\end{array} \\qquad \\begin{array}{r|rl} \\text {II} & & \\\\ \\hline \\\\ \\text{Major Premise}: & \\mathbf \\Phi_1 & \\tuple {P, M} \\\\ \\text{Minor Premise}: & \\mathbf \\Phi_2 & \\tuple {S, M} \\\\ \\hline \\\\ \\text{Conclusion}: & \\mathbf \\Phi_3 & \\tuple {S, P} \\\\ \\end{array}$ :$\\begin{array}{r|rl} \\text {III} & & \\\\ \\hline \\\\ \\text{Major Premise}: & \\mathbf \\Phi_1 & \\tuple {M, P} \\\\ \\text{Minor Premise}: & \\mathbf \\Phi_2 & \\tuple {M, S} \\\\ \\hline \\\\ \\text{Conclusion}: & \\mathbf \\Phi_3 & \\tuple {S, P} \\\\ \\end{array} \\qquad \\begin{array}{r|rl} \\text {IV} & & \\\\ \\hline \\\\ \\text{Major Premise}: & \\mathbf \\Phi_1 & \\tuple {P, M} \\\\ \\text{Minor Premise}: & \\mathbf \\Phi_2 & \\tuple {M, S} \\\\ \\hline \\\\ \\text{Conclusion}: & \\mathbf \\Phi_3 & \\tuple {S, P} \\\\ \\end{array}$ where $\\mathbf \\Phi_1$, $\\mathbf \\Phi_2$ and $\\mathbf \\Phi_3$ each denote one of the categorical statements $\\mathbf A$, $\\mathbf E$, $\\mathbf I$ or $\\mathbf O$. A '''standard instance of a categorical syllogism''' is obtained by substituting $\\mathbf A$, $\\mathbf E$, $\\mathbf I$ or $\\mathbf O$ for each of $\\mathbf \\Phi_1$, $\\mathbf \\Phi_2$ and $\\mathbf \\Phi_3$ in one of the above figures. Not all of these '''standard instances''' are valid."}
+{"_id": "24509", "title": "Definition:Length of Curve", "text": "The '''length''' of a curve is defined as the limit of the length of a polygonal line inscribed within the curve as the maximum length of the chords which form that polygonal line tends to zero."}
+{"_id": "24510", "title": "Definition:Excluded Point Topology/Finite", "text": "Let $S$ be finite. Then $\\tau_{\\bar p}$ is a '''finite excluded point topology''', and $\\struct {S, \\tau_{\\bar p} }$ is a '''finite excluded point space'''."}
+{"_id": "24511", "title": "Definition:Excluded Point Topology/Countable", "text": "Let $S$ be countably infinite. Then $\\tau_{\\bar p}$ is a '''countable excluded point topology''', and $\\struct {S, \\tau_{\\bar p} }$ is a '''countable excluded point space'''."}
+{"_id": "24512", "title": "Definition:Excluded Point Topology/Uncountable", "text": "Let $S$ be uncountable. Then $\\tau_{\\bar p}$ is an '''uncountable excluded point topology''', and $\\struct {S, \\tau_{\\bar p} }$ is an '''uncountable excluded point space'''."}
+{"_id": "24513", "title": "Definition:Excluded Point Topology/Infinite", "text": "Let $S$ be infinite. Then $\\tau_{\\bar p}$ is an '''infinite excluded point topology''', and $\\left({S, \\tau_{\\bar p}}\\right)$ is a '''infinite excluded point space'''."}
+{"_id": "24514", "title": "Definition:Countable Complement Topology/Countable", "text": "It is possible to define the countable complement topology on a countable set set $S$, but as every subset of a countable set has a countable complement, it is clear that this is trivially equal to the discrete space. This is why the countable complement topology is usually understood to apply to uncountable sets only."}
+{"_id": "24515", "title": "Definition:Particular Point Topology/Finite", "text": "Let $S$ be finite. Then $\\tau_p$ is a '''finite particular point topology''', and $\\struct {S, \\tau_p}$ is a '''finite particular point space'''."}
+{"_id": "24516", "title": "Definition:Particular Point Topology/Infinite", "text": "Let $S$ be infinite. Then $\\tau_p$ is an '''infinite particular point topology''', and $\\left({S, \\tau_p}\\right)$ is an '''infinite particular point space'''."}
+{"_id": "24517", "title": "Definition:Particular Point Topology/Countable", "text": "Let $S$ be countably infinite. Then $\\tau_p$ is a '''countable particular point topology''', and $\\struct {S, \\tau_p}$ is a '''countable particular point space'''."}
+{"_id": "24518", "title": "Definition:Particular Point Topology/Uncountable", "text": "Let $S$ be uncountable. Then $\\tau_p$ is an '''uncountable particular point topology''', and $\\struct {S, \\tau_p}$ is an '''uncountable particular point space'''."}
+{"_id": "24519", "title": "Definition:Finite Complement Topology/Finite", "text": "It is possible to define the finite complement topology on a finite set $S$, but as every subset of a finite set has a finite complement, it is clear that this is trivially equal to the discrete space. This is why the finite complement topology is usually understood to apply to infinite sets only."}
+{"_id": "24520", "title": "Definition:Finite Complement Topology/Countable", "text": "Let $S$ be countably infinite. Then $\\tau$ is a '''finite complement topology on a countable space''', and $\\struct {S, \\tau}$ is a '''countable finite complement space'''."}
+{"_id": "24521", "title": "Definition:Finite Complement Topology/Uncountable", "text": "Let $S$ be uncountable. Then $\\tau$ is a '''finite complement topology on an uncountable space''', and $\\struct {S, \\tau}$ is a '''uncountable finite complement space'''."}
+{"_id": "24522", "title": "Definition:Fort Space/Countable", "text": "Let $S$ be countably infinite. Then $\\tau_p$ is a '''countable Fort topology''', and $\\struct {S, \\tau_p}$ is a '''countable Fort space'''."}
+{"_id": "24523", "title": "Definition:Fort Space/Uncountable", "text": "Let $S$ be uncountable. Then $\\tau_p$ is an '''uncountable Fort topology''', and $\\struct {S, \\tau_p}$ is an '''uncountable Fort space'''."}
+{"_id": "24524", "title": "Definition:Complete Metric Space/Definition 1", "text": "A metric space $M = \\struct {A, d}$ is '''complete''' {{iff}} every Cauchy sequence is convergent."}
+{"_id": "24525", "title": "Definition:Complete Metric Space/Definition 2", "text": "A metric space $M = \\struct {A, d}$ is '''complete''' {{iff}} the intersection of every nested sequence of closed balls whose radii tend to zero is non-empty."}
+{"_id": "24526", "title": "Definition:Affirmative Categorical Statement", "text": "An '''affirmative categorical statement''' is either one of: === Universal Affirmative === {{:Definition:Universal Affirmative}} === Particular Affirmative === {{:Definition:Particular Affirmative}}"}
+{"_id": "24527", "title": "Definition:Negative Categorical Statement", "text": "A '''negative categorical statement''' is either one of: === Universal Negative === {{:Definition:Universal Negative}} === Particular Negative === {{:Definition:Particular Negative}} Category:Definitions/Categorical Statements i4tlzj2k8juuoz2pt0zforyz6b8lyw6"}
+{"_id": "24528", "title": "Definition:Reduction to First Figure", "text": "'''Reduction to the first figure''' is a method for determining the validity of a categorical syllogism, as follows: :$(1): \\quad$ Certain \"self-evident\" patterns are identified in the first figure of the categorical syllogism. :$(2): \\quad$ Using various rules of categorical statements, the valid patterns of other figures of the categorical syllogism are deduced. There are two forms this procedure takes: === Direct Reduction === {{:Definition:Reduction to First Figure/Direct}} === Indirect Reduction === {{:Definition:Reduction to First Figure/Indirect}}"}
+{"_id": "24530", "title": "Definition:Reduction to First Figure/Indirect", "text": "'''Indirect reduction to the first figure''' is a form of the reduction to the first figure method for determining the validity of a categorical syllogism. This form of the method works backward by supposing that a pattern is not valid, and then deriving a contradiction against a known valid pattern in the first figure. Hence by Reductio ad Absurdum the pattern is deduced to be valid."}
+{"_id": "24531", "title": "Definition:Square of Opposition/Categorical Statements/Vacuous Terms", "text": "The traditional treatment of the categorical syllogism makes the assumption that no term is vacuous. However, from the point of view of the full predicate logic, this assumption may not be valid. Note that if $S$ is empty, then the square of opposition no longer holds. Although ''All $S$ are $P$'' is vacuously true for such an empty universe, ''Some $S$ are $P$'' is not. Thus ''Some $S$ are $P$'' is no longer subimplicant to ''All $S$ are $P$''. Similarly, as ''Some $S$ are not $P$'' is also false, it follows that ''All $S$ are $P$'' and ''Some $S$ are not $P$'' are no longer subcontrary."}
+{"_id": "24532", "title": "Definition:Categorical Syllogism/Shorthand", "text": "In order to specify the pattern of a categorical syllogism completely, it is necessary and sufficient to specify: :$(1) \\quad$ The figure ($\\text I$ to $\\text {IV}$) of the syllogism and :$(2) \\quad$ The types of the three categorical statements that compose the syllogism. Hence, for example, the following categorical syllogism, which is of the first figure: :$\\begin{array}{r|rl} \\text I & & \\\\ \\hline \\\\ \\text{Major Premise}: & \\mathbf E & \\left({M, P}\\right) \\\\ \\text{Minor Premise}: & \\mathbf I & \\left({S, M}\\right) \\\\ \\hline \\\\ \\text{Conclusion}: & \\mathbf O & \\left({S, P}\\right) \\\\ \\end{array}$ is specified completely by: :$\\text I: EIO$"}
+{"_id": "24533", "title": "Definition:Categorical Statement/Abbreviation", "text": "A '''categorical statement''' connecting $S$ and $P$ can be abbreviated as: :$\\mathbf{\\Phi} \\left({S, P}\\right)$ where $\\Phi$ is one of either $\\mathbf{A}$, $\\mathbf{E}$, $\\mathbf{I}$ or $\\mathbf{O}$, signifying Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative respectively. Thus: :$\\mathbf{A} \\left({S, P}\\right)$ denotes ''All $S$ are $P$'' :$\\mathbf{E} \\left({S, P}\\right)$ denotes ''No $S$ are $P$'' :$\\mathbf{I} \\left({S, P}\\right)$ denotes ''Some $S$ are $P$'' :$\\mathbf{O} \\left({S, P}\\right)$ denotes ''Some $S$ are not $P$''."}
+{"_id": "24534", "title": "Definition:Categorical Statement/Subject", "text": "The symbol $S$ can be referred to as the '''subject''' of $\\map {\\mathbf {\\Phi} } {S, P}$."}
+{"_id": "24535", "title": "Definition:Categorical Statement/Predicate", "text": "The symbol $P$ can be referred to as the '''predicate''' of $\\map {\\mathbf {\\Phi} } {S, P}$."}
+{"_id": "24536", "title": "Definition:Universal Categorical Statement", "text": "A '''universal categorical statement''' is either one of: === Universal Affirmative === {{:Definition:Universal Affirmative}} === Universal Negative === {{:Definition:Universal Negative}} Category:Definitions/Categorical Statements kd494xo7vtho2h5zy959svqc60zum76"}
+{"_id": "24537", "title": "Definition:Particular Categorical Statement", "text": "A '''particular categorical statement''' is either one of: === Particular Affirmative === {{:Definition:Particular Affirmative}} === Particular Negative === {{:Definition:Particular Negative}} Category:Definitions/Categorical Statements 9n19giae1i2ax8o8kalelftnxe1gifw"}
+{"_id": "24538", "title": "Definition:Distributed Term of Categorical Syllogism/Subject", "text": "Let $\\mathbf{\\Phi} \\left({S, P}\\right)$ be a categorical statement, expressed in abbreviated form. Let $\\mathbf{\\Phi}$ be a universal categorical statement. Then $S$ is described as being '''distributed'''."}
+{"_id": "24539", "title": "Definition:Distributed Term of Categorical Syllogism", "text": "A term in a syllogism is '''distributed''' if the categorical statement in which it occurs refers to the whole of the class designated by the term. === Distributed Subject === {{:Definition:Distributed Term of Categorical Syllogism/Subject}} === Distributed Predicate === {{:Definition:Distributed Term of Categorical Syllogism/Predicate}}"}
+{"_id": "24540", "title": "Definition:Distributed Term of Categorical Syllogism/Predicate", "text": "Let $\\mathbf{\\Phi} \\left({S, P}\\right)$ be a categorical statement, expressed in abbreviated form. Let $\\mathbf{\\Phi}$ be a negative categorical statement. Then $S$ is described as being '''distributed'''."}
+{"_id": "24541", "title": "Definition:Undistributed Term of Categorical Syllogism", "text": "A term in a syllogism is '''undistributed''' if the categorical statement in which it occurs does not refer to the whole of the class designated by the term. === Undistributed Subject === {{:Definition:Undistributed Term of Categorical Syllogism/Subject}} === Undistributed Predicate === {{:Definition:Undistributed Term of Categorical Syllogism/Predicate}}"}
+{"_id": "24542", "title": "Definition:Undistributed Term of Categorical Syllogism/Predicate", "text": "Let $\\mathbf{\\Phi} \\left({S, P}\\right)$ be a categorical statement, expressed in abbreviated form. Let $\\mathbf{\\Phi}$ be an affirmative categorical statement. Then $P$ is described as being '''undistributed'''."}
+{"_id": "24543", "title": "Definition:Undistributed Term of Categorical Syllogism/Subject", "text": "Let $\\mathbf{\\Phi} \\left({S, P}\\right)$ be a categorical statement, expressed in abbreviated form. Let $\\mathbf{\\Phi}$ be a particular categorical statement. Then $S$ is described as being '''undistributed'''."}
+{"_id": "24544", "title": "Definition:Distributed Term of Categorical Syllogism/Examples", "text": "=== Universal Affirmative === {{:Definition:Distributed Term of Categorical Syllogism/Examples/Universal Affirmative}} === Universal Negative === {{:Definition:Distributed Term of Categorical Syllogism/Examples/Universal Negative}} === Particular Affirmative === {{:Definition:Distributed Term of Categorical Syllogism/Examples/Particular Affirmative}} === Particular Negative === {{:Definition:Distributed Term of Categorical Syllogism/Examples/Particular Negative}} This can be tabulated as follows: :$\\begin{array}{rcl} \\mathbf{A} & (S, & P) \\\\ & d & u \\end{array} \\qquad \\begin{array}{rcl} \\mathbf{E} & (S, & P) \\\\ & d & d \\end{array}$ :$\\begin{array}{rcl} \\mathbf{I} & (S, & P) \\\\ & u & u \\end{array} \\qquad \\begin{array}{rcl} \\mathbf{O} & (S, & P) \\\\ & u & d \\end{array}$ where $d$ denotes a distributed term and $u$ denotes an undistributed term."}
+{"_id": "24549", "title": "Definition:Non-Trivial Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. A '''non-trivial subgroup''' of $\\struct {G, \\circ}$ is a subgroup $\\struct {H, \\circ}$ of $G$ such that $H \\ne \\set e$."}
+{"_id": "24550", "title": "Definition:Non-Trivial Element", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. A '''non-trivial element''' of $G$ is an element of $G$ distinct from $e$."}
+{"_id": "24551", "title": "Definition:Commutative/Set", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $X \\subseteq S$ be a subset of $S$ such that: :$\\forall a, b \\in X: a \\circ b = b \\circ a$ That is, every element of $X$ commutes with every other element. Then $X$ is a '''commuting set of elements''' of $S$."}
+{"_id": "24552", "title": "Definition:Proper Subgroup/Non-Trivial", "text": "Let $\\struct {H, \\circ}$ be a subgroup of $\\struct {G, \\circ}$ such that $\\set e \\subset H \\subset G$, that is: :$H \\ne \\set e$ :$H \\ne G$ Then $\\struct {H, \\circ}$ is a '''non-trivial proper subgroup of $\\struct {G, \\circ}$'''."}
+{"_id": "24553", "title": "Definition:Group Type", "text": "An equivalence class under group isomorphism is a '''(group) type'''."}
+{"_id": "24554", "title": "Definition:Real Function/Definition by Formula", "text": "A function $f: S \\to T$ can be considered as a '''formula''' which tells us how to determine what the value of $y \\in T$ is when we have selected a value for $x \\in S$. === As an Equation === {{:Definition:Real Function/Definition by Equation}}"}
+{"_id": "24555", "title": "Definition:Real Function/Definition by Equation", "text": "It is often convenient to refer to an equation or formula as though it were a function. What is meant is that the equation ''defines'' the function; that is, it specifies the rule by which we obtain the value of $y$ from a given $x$. === Square Function === For example, let $x, y \\in \\R$. {{:Definition:Real Square Function|Square Function}} We may express this as $y = x^2$, and use this equation to ''define'' this function. This may be conceived as: :For each $x \\in \\R$, the number $y \\in \\R$ assigned to it is that which we get by squaring $x$. Another useful notation is: :$\\forall x \\in \\R: x \\mapsto x^2$"}
+{"_id": "24556", "title": "Definition:Nu Function", "text": "The '''$\\nu$ function''' is the function $\\nu: \\Z_{>0} \\to \\Z_{>0}$ is defined as: :$\\forall n \\in \\Z_{>0}: \\map \\nu n = $ the number of types of group of order $n$"}
+{"_id": "24557", "title": "Definition:Ring of Sets/Definition 1", "text": "A '''ring of sets''' $\\RR$ is a system of sets with the following properties: {{begin-axiom}} {{axiom | n = \\text {RS} 1_1         | lc= Non-Empty:         | m = \\RR \\ne \\O }} {{axiom | n = \\text {RS} 2_1         | lc= Closure under Intersection:         | q = \\forall A, B \\in \\RR         | m = A \\cap B \\in \\RR }} {{axiom | n = \\text {RS} 3_1         | lc= Closure under Symmetric Difference:         | q = \\forall A, B \\in \\RR         | m = A * B \\in \\RR }} {{end-axiom}}"}
+{"_id": "24558", "title": "Definition:Ring of Sets/Definition 2", "text": "A '''ring of sets''' $\\RR$ is a system of sets with the following properties: {{begin-axiom}} {{axiom | n = \\text {RS} 1_2         | lc= Empty Set:         | m = \\O \\in \\RR }} {{axiom | n = \\text {RS} 2_2         | lc= Closure under Set Difference:         | q = \\forall A, B \\in \\RR         | m = A \\setminus B \\in \\RR }} {{axiom | n = \\text {RS} 3_2         | lc= Closure under Union:         | q = \\forall A, B \\in \\RR         | m = A \\cup B \\in \\RR }} {{end-axiom}}"}
+{"_id": "24559", "title": "Definition:Limit Inferior of Sequence of Sets/Definition 1", "text": "Let $\\sequence {E_n : n \\in \\N}$ be a sequence of sets. Then the '''limit inferior''' of $\\sequence {E_n : n \\in \\N}$, denoted $\\ds \\liminf_{n \\mathop \\to \\infty} E_n$, is defined as: {{begin-eqn}} {{eqn | l = \\liminf_{n \\mathop \\to \\infty} E_n       | o = :=       | r = \\bigcup_{n \\mathop = 0}^\\infty \\bigcap_{i \\mathop = n}^\\infty E_n }} {{eqn | r = \\paren {E_0 \\cap E_1 \\cap E_2 \\cap \\ldots} \\cup \\paren {E_1 \\cap E_2 \\cap E_3 \\cap \\ldots} \\cup \\cdots }} {{end-eqn}}"}
+{"_id": "24560", "title": "Definition:Limit Superior of Sequence of Sets/Definition 1", "text": "Let $\\sequence {E_n : n \\in \\N}$ be a sequence of sets. Then the '''limit superior''' of $\\sequence {E_n: n \\in \\N}$, denoted $\\ds \\limsup_{n \\mathop \\to \\infty} E_n$, is defined as: {{begin-eqn}} {{eqn | l = \\limsup_{n \\mathop \\to \\infty} E_n       | o = :=       | r = \\bigcap_{i \\mathop = 0}^\\infty \\bigcup_{n \\mathop = i}^\\infty E_n }} {{eqn | r = \\paren {E_0 \\cup E_1 \\cup E_2 \\cup \\ldots} \\cap \\paren {E_1 \\cup E_2 \\cup E_3 \\cup \\ldots} \\cap \\ldots }} {{end-eqn}}"}
+{"_id": "24561", "title": "Definition:Limit Superior of Sequence of Sets/Definition 2", "text": "Let $\\sequence {E_n: n \\in \\N}$ be a sequence of sets. Then the '''limit superior''' of $\\sequence {E_n: n \\in \\N}$, denoted $\\ds \\limsup_{n \\mathop \\to \\infty} E_n$, is defined as: :$\\ds \\limsup_{n \\mathop \\to \\infty} E_n = \\set {x : x \\in E_i \\text { for infinitely many } i}$"}
+{"_id": "24562", "title": "Definition:Limit Inferior of Sequence of Sets/Definition 2", "text": "Let $\\sequence {E_n : n \\in \\N}$ be a sequence of sets. Then the '''limit inferior''' of $\\sequence {E_n : n \\in \\N}$, denoted $\\ds \\liminf_{n \\mathop \\to \\infty} E_n$, is defined as: :$\\ds \\liminf_{n \\mathop \\to \\infty} E_n := \\set {x: x \\in E_i \\text { for all but finitely many } i}$"}
+{"_id": "24563", "title": "Definition:Ring of Sets/Definition 3", "text": "A '''ring of sets''' $\\RR$ is a system of sets with the following properties: {{begin-axiom}} {{axiom | n = \\text {RS} 1_3         | lc= Empty Set:         | m = \\varnothing \\in \\RR }} {{axiom | n = \\text {RS} 2_3         | lc= Closure under Set Difference:         | q = \\forall A, B \\in \\RR         | m = A \\setminus B \\in \\RR }} {{axiom | n = \\text {RS} 3_3         | lc= Closure under Disjoint Union:         | q = \\forall A, B \\in \\RR         | m = A \\cap B = \\O \\implies A \\cup B \\in \\RR }} {{end-axiom}}"}
+{"_id": "24564", "title": "Definition:Sigma-Ring/Definition 1", "text": "A '''$\\sigma$-ring''' is a ring of sets which is closed under countable unions. That is, a ring of sets $\\Sigma$ is a '''$\\sigma$-ring''' {{iff}}: :$\\ds A_1, A_2, \\ldots \\in \\Sigma \\implies \\bigcup_{n \\mathop = 1}^\\infty A_n \\in \\Sigma$"}
+{"_id": "24565", "title": "Definition:Sigma-Ring/Definition 2", "text": "A '''$\\sigma$-ring''' $\\Sigma$ is a system of sets with the following properties: {{begin-axiom}} {{axiom | n = \\text {SR} 1         | lc= Empty Set:         | m = \\O \\in \\Sigma }} {{axiom | n = \\text {SR} 2         | lc= Closure under Set Difference:         | q = \\forall A, B \\in \\Sigma         | m = A \\setminus B \\in \\Sigma }} {{axiom | n = \\text {SR} 3         | lc= Closure under Countable Unions:         | q = \\forall A_n \\in \\Sigma: n = 1, 2, \\ldots         | m = \\bigcup_{n \\mathop = 1}^\\infty A_n \\in \\Sigma }} {{end-axiom}}"}
+{"_id": "24566", "title": "Definition:Formal Semantics", "text": "A '''formal semantics''' for $\\mathcal L$ comprises: * A collection of objects called '''structures'''; * A notion of '''validity''' of $\\mathcal L$-WFFs in these structures. Often, a '''formal semantics''' provides these by using a lot of auxiliary definitions."}
+{"_id": "24567", "title": "Definition:Sigma-Algebra/Definition 4", "text": "Let $X$ be a set. A '''$\\sigma$-algebra''' $\\Sigma$ over $X$ is an algebra of sets which is closed under countable unions."}
+{"_id": "24568", "title": "Definition:Sigma-Algebra/Definition 1", "text": "Let $X$ be a set. A '''$\\sigma$-algebra''' $\\Sigma$ over $X$ is a system of subsets of $X$ with the following properties: {{begin-axiom}} {{axiom | n = \\text {SA} 1         | lc= Unit:         | m = X \\in \\Sigma }} {{axiom | n = \\text {SA} 2         | lc= Closure under Complement:         | q = \\forall A \\in \\Sigma         | m = \\relcomp X A \\in \\Sigma }} {{axiom | n = \\text {SA} 3         | lc= Closure under Countable Unions:         | q = \\forall A_n \\in \\Sigma: n = 1, 2, \\ldots         | m = \\bigcup_{n \\mathop = 1}^\\infty A_n \\in \\Sigma }} {{end-axiom}}"}
+{"_id": "24569", "title": "Definition:Logical Language", "text": "Let $\\mathcal L$ be a formal language used in symbolic logic. Then $\\mathcal L$ is called a '''logical language'''."}
+{"_id": "24570", "title": "Definition:Tautology", "text": "A '''tautology''' is a statement which is ''always true'', independently of any relevant circumstances that could theoretically influence its truth value. It is epitomised by the form: :$p \\implies p$ that is: :'''if $p$ is true then $p$ is true.''' An example of a \"relevant circumstance\" here is the truth value of $p$. The archetypal '''tautology''' is symbolised by $\\top$, and referred to as Top."}
+{"_id": "24571", "title": "Definition:Formal Semantics/Valid", "text": "Part of specifying a formal semantics $\\mathscr M$ for $\\LL$ is to define a notion of '''validity'''. Concretely, a precise meaning needs to be assigned to the phrase: :\"The $\\LL$-WFF $\\phi$ is '''valid''' in the $\\mathscr M$-structure $\\MM$.\" It can be expressed symbolically as: :$\\MM \\models_{\\mathscr M} \\phi$"}
+{"_id": "24572", "title": "Definition:Formal Semantics/Structure", "text": "Part of specifying a formal semantics $\\mathscr M$ for $\\LL$ is to specify '''structures''' $\\MM$ for $\\mathscr M$. A '''structure''' can in principle be any object one can think of. However, to get a useful formal semantics, the '''structures''' should support a meaningful definition of validity for the WFFs of $\\LL$. It is common that '''structures''' are sets, often endowed with a number of relations or functions."}
+{"_id": "24574", "title": "Definition:Model (Logic)", "text": "Let $\\mathscr M$ be a formal semantics for a logical language $\\LL$. Let $\\MM$ be a structure of $\\mathscr M$. === Model of Logical Formula === {{:Definition:Model (Logic)/Logical Formula}} === Model of Set of Logical Formulas === {{:Definition:Model (Logic)/Set of Logical Formulas}}"}
+{"_id": "24575", "title": "Definition:Sigma-Algebra/Definition 3", "text": "A '''$\\sigma$-algebra''' $\\Sigma$ is a $\\sigma$-ring with a unit."}
+{"_id": "24576", "title": "Definition:Boolean Interpretation/Formal Semantics", "text": "The boolean interpretations for $\\mathcal L_0$ can be interpreted as a formal semantics for $\\mathcal L_0$, which we denote by $\\mathrm{BI}$. The structures of $\\mathrm{BI}$ are the boolean interpretations. A WFF $\\phi$ is declared ($\\mathrm{BI}$-)valid in a boolean interpretation $v$ iff: :$v \\left({\\phi}\\right) = T$ Symbolically, this can be expressed as: :$v \\models_{\\mathrm{BI}} \\phi$"}
+{"_id": "24577", "title": "Definition:Model (Logic)/Logical Formula", "text": "Let $\\phi$ be a logical formula of $\\LL$. Then $\\MM$ is a '''model of $\\phi$''' {{iff}}: :$\\MM \\models_{\\mathscr M} \\phi$ that is, if $\\phi$ is valid in $\\MM$."}
+{"_id": "24578", "title": "Definition:Model (Logic)/Set of Logical Formulas", "text": "Let $\\FF$ be a set of logical formulas of $\\mathcal L$. Then $\\MM$ is a '''model of $\\FF$''' {{iff}}: :$\\MM \\models_{\\mathscr M} \\phi$ for every $\\phi \\in \\FF$ that is, if it is a model of every logical formula $\\phi \\in \\FF$."}
+{"_id": "24579", "title": "Definition:Sound Proof System", "text": "Let $\\LL$ be a logical language. Let $\\mathscr P$ be a proof system for $\\LL$, and let $\\mathscr M$ be a formal semantics for $\\LL$. Then $\\mathscr P$ is said to be '''sound for $\\mathscr M$''' {{iff}}: :Every $\\mathscr P$-theorem is an $\\mathscr M$-tautology. Symbolically, this can be expressed as the statement that, for every logical formula $\\phi$ of $\\LL$: :$\\vdash_{\\mathscr P} \\phi$ implies $\\models_{\\mathscr M} \\phi$ === Strongly Sound Proof System === {{:Definition:Sound Proof System/Strongly Sound}}"}
+{"_id": "24580", "title": "Definition:Complete Proof System", "text": "Let $\\LL$ be a logical language. Let $\\mathscr P$ be a proof system for $\\LL$, and let $\\mathscr M$ be a formal semantics for $\\LL$. Then $\\mathscr P$ is said to be '''complete for $\\mathscr M$''' {{iff}}: :Every $\\mathscr M$-tautology is a $\\mathscr P$-theorem. Symbolically, this can be expressed as the statement that, for every logical formula $\\phi$ of $\\LL$: :$\\models_{\\mathscr M} \\phi$ implies $\\vdash_{\\mathscr P} \\phi$ === Strongly Complete Proof System === {{:Definition:Complete Proof System/Strongly Complete}}"}
+{"_id": "24581", "title": "Definition:Unsatisfiable/Formula", "text": "A logical formula $\\phi$ of $\\LL$ is '''unsatisfiable for $\\mathscr M$''' {{iff}}: :$\\phi$ is valid in none of the structures of $\\mathscr M$ That is, for all structures $\\MM$ of $\\mathscr M$: :$\\MM \\not\\models_{\\mathscr M} \\phi$"}
+{"_id": "24582", "title": "Definition:Sigma-Ring/Definition 3", "text": "A '''$\\sigma$-ring''' $\\Sigma$ is a system of sets with the following properties: {{begin-axiom}} {{axiom | n = \\text {SR} 1'         | lc= Empty Set:         | m = \\O \\in \\Sigma }} {{axiom | n = \\text {SR} 2'         | lc= Closure under Set Difference:         | q = \\forall A, B \\in \\Sigma         | m = A \\setminus B \\in \\Sigma }} {{axiom | n = \\text {SR} 3'         | lc= Closure under Countable Disjoint Unions:         | q = \\forall A_n \\in \\Sigma: n = 1, 2, \\ldots         | m = \\bigsqcup_{n \\mathop = 1}^\\infty A_n \\in \\Sigma }} {{end-axiom}}"}
+{"_id": "24583", "title": "Definition:Sigma-Algebra/Definition 2", "text": "Let $X$ be a set. A '''$\\sigma$-algebra''' $\\Sigma$ over $X$ is a system of subsets of $X$ with the following properties: {{begin-axiom}} {{axiom | n = \\text {SA} 1'         | lc= Unit:         | m = X \\in \\Sigma }} {{axiom | n = \\text {SA} 2'         | lc= Closure under Complement:         | q = \\forall A \\in \\Sigma         | m = \\relcomp X A \\in \\Sigma }} {{axiom | n = \\text {SA} 3'         | lc= Closure under Countable Disjoint Unions:         | q = \\forall A_n \\in \\Sigma: n = 1, 2, \\ldots         | m = \\bigsqcup_{n \\mathop = 1}^\\infty A_n \\in \\Sigma }} {{end-axiom}}"}
+{"_id": "24584", "title": "Definition:Satisfiable/Formula", "text": "A logical formula $\\phi$ of $\\mathcal L$ is '''satisfiable for $\\mathscr M$''' iff: :$\\phi$ is valid in some structure $\\mathcal M$ of $\\mathscr M$ That is, there exists some structure $\\mathcal M$ of $\\mathscr M$ such that: :$\\mathcal M \\models_{\\mathscr M} \\phi$"}
+{"_id": "24585", "title": "Definition:Falsifiable/Formula", "text": "A logical formula $\\phi$ of $\\mathcal L$ is '''falsifiable for $\\mathscr M$''' iff: :$\\phi$ is not valid in some structure $\\mathcal M$ of $\\mathscr M$ That is, there exists some structure $\\mathcal M$ of $\\mathscr M$ such that: :$\\mathcal M \\not\\models_{\\mathscr M} \\phi$"}
+{"_id": "24586", "title": "Definition:Satisfiable", "text": "Let $\\mathcal L$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\mathcal L$. === Satisfiable Formula === {{:Definition:Satisfiable/Formula}} === Satisfiable Set of Formulas === {{:Definition:Satisfiable/Set of Formulas}} === Satisfiable for Boolean Interpretations === {{:Definition:Satisfiable/Boolean Interpretations}}"}
+{"_id": "24587", "title": "Definition:Satisfiable/Set of Formulas", "text": "A collection $\\mathcal F$ of logical formulas of $\\mathcal L$ is '''satisfiable for $\\mathscr M$''' iff: :There is some $\\mathscr M$-model $\\mathcal M$ of $\\mathcal F$ That is, there exists some structure $\\mathcal M$ of $\\mathscr M$ such that: :$\\mathcal M \\models_{\\mathscr M} \\mathcal F$"}
+{"_id": "24588", "title": "Definition:Falsifiable", "text": "Let $\\mathcal L$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\mathcal L$. === Falsifiable Formula === {{:Definition:Falsifiable/Formula}} === Falsifiable Set of Formulas === {{:Definition:Falsifiable/Set of Formulas}} === Falsifiable for Boolean Interpretations === {{:Definition:Falsifiable/Boolean Interpretations}}"}
+{"_id": "24589", "title": "Definition:Falsifiable/Set of Formulas", "text": "A collection $\\mathcal F$ of logical formulas of $\\mathcal L$ is '''falsifiable for $\\mathscr M$''' iff: :Some $\\mathscr M$-structure $\\mathcal M$ is not a model of $\\mathcal F$ That is, there exists some structure $\\mathcal M$ of $\\mathscr M$ such that: :$\\mathcal M \\not\\models_{\\mathscr M} \\mathcal F$"}
+{"_id": "24591", "title": "Definition:Unsatisfiable/Set of Formulas", "text": "A collection $\\FF$ of logical formulas of $\\LL$ is '''unsatisfiable for $\\mathscr M$''' {{iff}}: :There is no $\\mathscr M$-model $\\MM$ for $\\FF$ That is, for all structures $\\MM$ of $\\mathscr M$: :$\\MM \\not \\models_{\\mathscr M} \\FF$"}
+{"_id": "24592", "title": "Definition:Rooted Tree/Child Node/Grandchild Node", "text": "A child of a child node of a node $t$ can be referred to as a '''grandchild node''' of $t$. In terms of the parent mapping $\\pi$ of $T$, a '''grandchild node''' of $t$ is a node $s$ such that: :$\\pi \\left({\\pi \\left({s}\\right)}\\right) = t$"}
+{"_id": "24593", "title": "Definition:Propositional Tableau/Graphical Representation", "text": "The conditions by which a propositional tableau may be identified, respectively constructed, can be graphically represented like so: ::{| style=\"text-align: center\" |- | $\\begin{xy}\\xymatrix @R=1em @C=3em{  \\ar@{.}[d] & \\ar@{.}[d] & \\ar@{.}[d] & \\ar@{.}[d] \\\\  *++{\\neg \\neg \\mathbf A} \\ar@{.}[d] &  *++{\\mathbf A \\land \\mathbf B} \\ar@{.}[d] &  *++{\\neg \\left({\\mathbf A \\lor \\mathbf B}\\right)} \\ar@{.}[d] &  *++{\\neg \\left({\\mathbf A \\implies \\mathbf B}\\right)} \\ar@{.}[d] \\\\  *++{t} \\ar@{-}[d] &  *++{t} \\ar@{-}[d] &  *++{t} \\ar@{-}[d] &  *++{t} \\ar@{-}[d] \\\\  \\mathbf A &  \\mathbf A \\ar@{=}[d] &  \\neg \\mathbf A \\ar@{=}[d] &  \\mathbf A \\ar@{=}[d] \\\\   & \\mathbf B & \\neg \\mathbf B & \\neg \\mathbf B \\\\  \\boxed{\\neg\\neg} & \\boxed \\land & \\boxed{\\neg\\lor} & \\boxed{\\neg\\implies} }\\end{xy}$ |- | $\\begin{xy}\\xymatrix @R=1em @C=1em{  & \\ar@{.}[d] & & & & & & \\ar@{.}[d] & & & & & & \\ar@{.}[d] \\\\  & *++{\\mathbf A \\lor \\mathbf B} \\ar@{.}[d] &  & & & &  & *++{\\neg \\left({\\mathbf A \\land \\mathbf B}\\right)} \\ar@{.}[d] & & & & &  & *++{\\mathbf A \\implies \\mathbf B} \\ar@{.}[d] \\\\  & *++{t} \\ar@{-}[dl] \\ar@{-}[dr] & & & & &  & *++{t} \\ar@{-}[dl] \\ar@{-}[dr] & & & & &  & *++{t} \\ar@{-}[dl] \\ar@{-}[dr] \\\\  \\mathbf A & & \\mathbf B & & & &  \\neg \\mathbf A & & \\neg \\mathbf B & & & &  \\neg \\mathbf A & & \\mathbf B \\\\  & \\boxed{\\lor} & & & & & & \\boxed{\\neg \\land} & & & & & & \\boxed \\implies  }\\end{xy}$ |- | $\\begin{xy}\\xymatrix @R=1em @C=0.7em{  & \\ar@{.}[d] & & & & & & \\ar@{.}[d] \\\\  & *++{\\mathbf A \\iff \\mathbf B} \\ar@{.}[d] &  & & & &  & *++{\\neg \\left({\\mathbf A \\iff \\mathbf B}\\right)} \\ar@{.}[d] \\\\  & *++{t} \\ar@{-}[dl] \\ar@{-}[dr] & & & & &  & *++{t} \\ar@{-}[dl] \\ar@{-}[dr] \\\\  *++{\\mathbf A \\land \\mathbf B} & & *++{\\neg \\mathbf A \\land \\neg \\mathbf B} & & & &  *++{\\mathbf A \\land \\neg \\mathbf B} & & *++{\\neg \\mathbf A \\land \\mathbf B} \\\\  & \\boxed{\\iff} & & & & & & \\boxed{\\neg \\iff} }\\end{xy}$ |}"}
+{"_id": "24594", "title": "Definition:Projective Geometry", "text": "'''Projective Geometry''' is the field of geometry concerned with properties which are invariant under projective transformations."}
+{"_id": "24595", "title": "Definition:Analytic Geometry/Plane", "text": "'''Plane analytic geometry''' is the branch of analytic geometry that makes an algebraic study of the real number plane $\\R^2$."}
+{"_id": "24596", "title": "Definition:Analytic Geometry/Solid", "text": "'''Solid analytic geometry''' is the branch of analytic geometry that makes an algebraic study of the  real vector space $\\R^3$."}
+{"_id": "24597", "title": "Definition:Collinear Points", "text": "Three or more points which belong to the same line are '''collinear'''."}
+{"_id": "24598", "title": "Definition:Concurrent Lines", "text": "A set of $3$ or more (straight) lines through a single point are '''concurrent'''."}
+{"_id": "24599", "title": "Definition:Coplanar Points", "text": "Four or more points which belong to the same plane are '''coplanar'''."}
+{"_id": "24600", "title": "Definition:Collinear Planes", "text": "A set of planes which share the same (straight) line are '''collinear'''."}
+{"_id": "24601", "title": "Definition:Contradictory Branch", "text": "Let $T$ be a labeled tree for propositional logic. Let $\\Gamma$ be a branch of $T$. Then $\\Gamma$ is a '''contradictory branch''' iff, for some WFF of propositional logic $\\mathbf A$, both $\\mathbf A$ and $\\neg \\mathbf A$ occur along $\\Gamma$."}
+{"_id": "24602", "title": "Definition:Tableau Confutation", "text": "Let $\\mathbf H$ be a set of WFFs of propositional logic. A '''tableau confutation''' of $\\mathbf H$ is a propositional tableau $T$ with root $\\mathbf H$ such that every branch of $T$ is contradictory."}
+{"_id": "24603", "title": "Definition:Tableau Proof (Propositional Tableaus)", "text": "Let $\\mathbf H$ be a set of WFFs of propositional logic. Let $\\mathbf A$ be a WFF. A '''tableau proof''' of $\\mathbf A$ from $\\mathbf H$ is a tableau confutation of $\\mathbf H \\cup \\set {\\neg \\mathbf A}$. This definition also applies when $\\mathbf H = \\varnothing$. Then a '''tableau proof''' of $\\mathbf A$ is a tableau confutation of $\\set {\\neg \\mathbf A}$. If there exists a '''tableau proof''' of $\\mathbf A$ from $\\mathbf H$, one can write: :$\\mathbf H \\vdash_{\\mathrm{PT} } \\mathbf A$ Specifically, the notation: :$\\vdash_{\\mathrm{PT} } \\mathbf A$ means that there exists a '''tableau proof''' of $\\mathbf A$. === Proof System === {{:Definition:Tableau Proof (Propositional Tableaus)/Proof System}}"}
+{"_id": "24604", "title": "Definition:Consistent (Logic)", "text": "Let $\\LL$ be a logical language. Let $\\mathscr P$ be a proof system for $\\LL$. === Proof System === {{:Definition:Consistent (Logic)/Proof System}} === Set of Formulas === {{:Definition:Consistent (Logic)/Set of Formulas}}"}
+{"_id": "24605", "title": "Definition:Provable Consequence", "text": "Let $\\mathscr P$ be a proof system for a formal language $\\mathcal L$. Let $\\mathcal F$ be a collection of WFFs of $\\mathcal L$. Denote with $\\mathscr P \\left({\\mathcal F}\\right)$ the proof system obtained from $\\mathscr P$ by adding all the WFFs from $\\mathcal F$ as axioms. Let $\\phi$ be a theorem of $\\mathscr P \\left({\\mathcal F}\\right)$. Then $\\phi$ is called a '''provable consequence''' of $\\mathcal F$, and this is denoted as: :$\\mathcal F \\vdash_{\\mathscr P} \\phi$ Note in particular that for $\\mathcal F = \\varnothing$, this notation agrees with the notation for a $\\mathscr P$-theorem: :$\\vdash_{\\mathscr P} \\phi$"}
+{"_id": "24606", "title": "Definition:Monotone Sequence of Sets", "text": "Let $X$ be a set. Let $\\mathcal S \\subseteq \\powerset X$ be a collection of subsets of $X$. A '''monotone sequence of sets (in $\\mathcal S$)''' is a sequence $\\sequence {A_n}_{n \\mathop \\in \\N}$ in $\\mathcal S$, such that either: :$\\forall n \\in \\N: A_n \\subseteq A_{n + 1}$ or: :$\\forall n \\in \\N: A_n \\supseteq A_{n + 1}$ That is, such that $\\sequence  {A_n}_{n \\mathop \\in \\N}$ is either increasing or decreasing"}
+{"_id": "24607", "title": "Definition:Inconsistent (Logic)", "text": "Let $\\LL$ be a logical language. Let $\\mathscr P$ be a proof system for $\\LL$. A collection $\\FF$ of logical formulas is '''inconsistent for $\\mathscr P$''' {{iff}}: :For every logical formula $\\phi$, $\\FF \\vdash_{\\mathscr P} \\phi$. That is, ''every'' logical formula $\\phi$ is a provable consequence of $\\FF$."}
+{"_id": "24608", "title": "Definition:Proof System/Formal Proof", "text": "Let $\\phi$ be a WFF of $\\mathcal L$. A '''formal proof of $\\phi$''' in $\\mathscr P$ is a collection of axioms and rules of inference of $\\mathscr P$ that leads to the conclusion that $\\phi$ is a theorem of $\\mathscr P$. The term '''formal proof''' is also used to refer to specific presentations of such collections. For example, the term applies to tableau proofs in natural deduction."}
+{"_id": "24609", "title": "Definition:Tarski's Geometry", "text": "Tarski's geometry is an axiomatic treatment of geometry.  Unless specified otherwise, {{ProofWiki}} will use the term Tarski's geometry to be a formal systematic treatment of geometry containing only: :$(1):\\quad$ The language and axioms of first-order logic, and the disciplines preceding it :$(2):\\quad$ The undefined terms of Tarski's Geometry :$(3):\\quad$ Tarski's Axioms of Geometry."}
+{"_id": "24610", "title": "Definition:Boolean Lattice/Definition 3", "text": "A '''Boolean lattice''' is a bounded lattice $\\left({S, \\vee, \\wedge, \\preceq, \\bot, \\top}\\right)$ together with a unary operation $\\neg$ called '''complementation''', subject to: $(1): \\quad$ For all $a, b \\in S$, $a \\preceq \\neg b$ {{iff}} $a \\wedge b = \\bot$ $(2): \\quad$ For all $a \\in S$, $\\neg \\neg a = a$."}
+{"_id": "24611", "title": "Definition:Symmetric Relation/Definition 1", "text": "$\\RR$ is '''symmetric''' {{iff}}: :$\\tuple {x, y} \\in \\RR \\implies \\tuple {y, x} \\in \\RR$"}
+{"_id": "24612", "title": "Definition:Symmetric Relation/Definition 2", "text": "$\\RR$ is '''symmetric''' {{iff}} it equals its inverse: :$\\RR^{-1} = \\RR$"}
+{"_id": "24613", "title": "Definition:Reflexive Relation/Definition 1", "text": "$\\RR$ is '''reflexive''' {{iff}}: :$\\forall x \\in S: \\tuple {x, x} \\in \\RR$"}
+{"_id": "24614", "title": "Definition:Reflexive Relation/Definition 2", "text": "$\\RR$ is '''reflexive''' {{iff}} it is a superset of the diagonal relation:  :$\\Delta_S \\subseteq \\RR$"}
+{"_id": "24615", "title": "Definition:Transitive Relation/Definition 1", "text": "$\\RR$ is '''transitive''' {{iff}}: :$\\tuple {x, y} \\in \\RR \\land \\tuple {y, z} \\in \\RR \\implies \\tuple {x, z} \\in \\RR$ that is: :$\\set {\\tuple {x, y}, \\tuple {y, z} } \\subseteq \\RR \\implies \\tuple {x, z} \\in \\RR$"}
+{"_id": "24616", "title": "Definition:Transitive Relation/Definition 2", "text": "$\\RR$ is '''transitive''' {{iff}}: :$\\RR \\circ \\RR \\subseteq \\RR$ where $\\circ$ denotes composite relation."}
+{"_id": "24617", "title": "Definition:Field of Relation", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation. The '''field''' of $\\RR$ is defined as: :$\\map {\\operatorname {Field} } \\RR := \\set {x \\in S: \\exists t \\in T: \\tuple {x, t} \\in \\RR} \\cup \\set {x \\in T: \\exists s \\in S: \\tuple {s, x} \\in \\RR}$ That is, it is the union of the preimage of $\\RR$ with its image."}
+{"_id": "24618", "title": "Definition:Coreflexive Relation/Definition 1", "text": "$\\mathcal R$ is '''coreflexive''' {{iff}}: :$\\forall x, y \\in S: \\left({x, y}\\right) \\in \\mathcal R \\implies x = y$"}
+{"_id": "24619", "title": "Definition:Coreflexive Relation/Definition 2", "text": "$\\mathcal R$ is '''coreflexive''' {{iff}}: :$\\mathcal R \\subseteq \\Delta_S$ where $\\Delta_S$ is the diagonal relation."}
+{"_id": "24620", "title": "Definition:Finitely Branching", "text": "Let $T$ be a tree. Then $T$ is '''finitely branching''' {{iff}} $T$ is locally finite."}
+{"_id": "24621", "title": "Definition:Preordering/Definition 1", "text": "$\\RR$ is a '''preordering''' on $S$ {{iff}}: {{begin-axiom}} {{axiom | n = 1         | lc= $\\RR$ is reflexive         | q = \\forall a \\in S         | m = a \\mathrel \\RR a }} {{axiom | n = 2         | lc= $\\RR$ is transitive         | q = \\forall a, b, c \\in S         | m = a \\mathrel \\RR b \\land b \\mathrel \\RR c \\implies a \\mathrel \\RR c }} {{end-axiom}}"}
+{"_id": "24622", "title": "Definition:Preordering/Notation", "text": "Symbols used to denote a general preordering relation are usually variants on $\\lesssim$, $\\precsim$ or $\\precapprox$. A symbol for a preordering can be reversed, and the sense is likewise inverted: :$a \\precsim b \\iff b \\succsim a$ The notation $a \\sim b$ is defined as: :$a \\sim b$ {{iff}} $a \\precsim b$ and $b \\precsim a$ The notation $a \\prec b$ is defined as: :$a \\prec b$ {{iff}} $a \\precsim b$ and $a \\not \\sim b$"}
+{"_id": "24623", "title": "Definition:Preordering/Definition 2", "text": "$\\mathcal R$ is a '''preordering''' on $S$ {{iff}}: :$(1): \\quad \\mathcal R \\circ \\mathcal R = \\mathcal R$ :$(2): \\quad \\Delta_S \\subseteq \\mathcal R$ where: : $\\circ$ denotes relation composition : $\\Delta_S$ denotes the diagonal relation on $S$."}
+{"_id": "24624", "title": "Definition:Total Ordering/Definition 1", "text": "$\\RR$ is a '''total ordering''' on $S$ {{iff}}: :$(1): \\quad \\RR$ is an ordering on $S$ :$(2): \\quad \\RR$ is connected That is, $\\RR$ is an ordering with no non-comparable pairs: :$\\forall x, y \\in S: x \\mathop \\RR y \\lor y \\mathop \\RR x$"}
+{"_id": "24625", "title": "Definition:Total Ordering/Definition 2", "text": "$\\RR$ is a '''total ordering''' on $S$ {{iff}}: :$(1): \\quad \\RR \\circ \\RR = \\RR$ :$(2): \\quad \\RR \\cap \\RR^{-1} = \\Delta_S$ :$(3): \\quad \\RR \\cup \\RR^{-1} = S \\times S$"}
+{"_id": "24626", "title": "Definition:Ordering/Notation", "text": "Symbols used to denote a general ordering relation are usually variants on $\\preceq$, $\\le$ and so on. On {{ProofWiki}}, to denote a general ordering relation it is recommended to use $\\preceq$ and its variants: :$\\preccurlyeq$ :$\\curlyeqprec$ To denote the conventional ordering relation in the context of numbers, the symbol $\\le$ is to be used, or its variants: : $\\leqslant$ : $\\leqq$ : $\\eqslantless$ The symbol $\\subseteq$ is universally reserved for the subset relation. :$a \\preceq b$ can be read as: :'''$a$ precedes, or is the same as, $b$'''. Similarly: : $a \\preceq b$ can be read as: : '''$b$ succeeds, or is the same as, $a$'''. If, for two elements $a, b \\in S$, it is not the case that $a \\preceq b$, then the symbols $a \\npreceq b$ and $b \\nsucceq a$ can be used."}
+{"_id": "24627", "title": "Definition:Ordering/Definition 1", "text": "An '''ordering on $S$''' is a relation $\\RR$ on $S$ such that: {{begin-axiom}} {{axiom | n = 1         | lc= $\\RR$ is reflexive         | q = \\forall a \\in S         | m = a \\mathrel \\RR a }} {{axiom | n = 2         | lc= $\\RR$ is transitive         | q = \\forall a, b, c \\in S         | m = a \\mathrel \\RR b \\land b \\mathrel \\RR c \\implies a \\mathrel \\RR c }} {{axiom | n = 3         | lc= $\\RR$ is antisymmetric         | q = \\forall a \\in S         | m = a \\mathrel \\RR b \\land b \\mathrel \\RR a \\implies a = b }} {{end-axiom}}"}
+{"_id": "24628", "title": "Definition:Ordering/Definition 2", "text": "An '''ordering on $S$''' is a relation $\\RR$ on $S$ such that: :$(1): \\quad \\RR \\circ \\RR = \\RR$ :$(2): \\quad \\RR \\cap \\RR^{-1} = \\Delta_S$ where: :$\\circ$ denotes relation composition :$\\RR^{-1}$ denotes the inverse of $\\RR$ :$\\Delta_S$ denotes the diagonal relation on $S$."}
+{"_id": "24629", "title": "Definition:Operation/Binary Operation/Postfix Notation", "text": "Let $\\circ: S \\times T \\to U$ be a binary operation. The convention that places the symbol for the operation after the two operands: :$z = \\left ({x, y}\\right) \\circ$ is called '''postfix notation'''."}
+{"_id": "24630", "title": "Definition:Operation/Binary Operation/Prefix Notation", "text": "Let $\\circ: S \\times T \\to U$ be a binary operation. The convention that places the symbol for the operation before the two operands: :$z = \\circ \\left ({x, y}\\right)$ is called '''prefix notation'''."}
+{"_id": "24631", "title": "Definition:Surjection/Definition 1", "text": "$f: S \\to T$ is a '''surjection''' {{iff}}: :$\\forall y \\in T: \\exists x \\in \\Dom f: \\map f x = y$ That is, {{iff}} $f$ is right-total."}
+{"_id": "24632", "title": "Definition:Surjection/Definition 2", "text": "$f: S \\to T$ is a '''surjection''' {{iff}}: :$f \\sqbrk S = T$ or, in the language and notation of direct image mappings: :$\\map {f^\\to} S = T$ That is, $f$ is a '''surjection''' {{iff}} its image equals its codomain: :$\\Img f = \\Cdm f$"}
+{"_id": "24633", "title": "Definition:Ordered Tuple/Ordered Couples and Ordered Pairs", "text": "Notice the difference between ordered pairs and ordered couples. By definition, an ordered couple $\\tuple {a, b}$ is in fact the set $\\set {\\tuple {1, a}, \\tuple {2, b} }$, where each of $\\tuple {1, a}$ and $\\tuple {2, b}$ are ordered pairs. It is not possible to use the definition of ordered couple as the definition of ordered pair, as the latter is used to define a mapping, which is then used to define an ordered couple. However, in view of the Equality of Ordered Tuples, it is generally accepted that it is valid to use the notation $\\tuple {a, b}$ to mean both an ordered couple and an ordered pair. It is worth bearing this in mind, as there are times when it is important not to confuse them."}
+{"_id": "24634", "title": "Definition:Singleton Relation", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R \\subseteq S \\times T$ be a relation on $S \\times T$. Then $\\mathcal R$ is a singleton relation iff: :$\\exists_1 \\left({x, y}\\right) \\in S \\times T: \\left({x, y}\\right) \\in \\mathcal R$ That is, iff $\\mathcal R$ has exactly one element."}
+{"_id": "24635", "title": "Definition:Symmetric Relation/Definition 3", "text": "$\\mathcal R$ is '''symmetric''' {{iff}} it is a subset of its inverse: :$\\mathcal R \\subseteq \\mathcal R^{-1}$"}
+{"_id": "24636", "title": "Definition:Root of Unity Modulo m", "text": "Let $n \\in \\Z_{>0}$ be a positive integer. Let $m \\in \\Z_{>1}$ be a positive integer greater than one. Then $a$ is an '''$n^{th}$ root of unity modulo $m$''' {{iff}}: :$a^n \\equiv 1 \\pmod m$ Solving the following equation over the smallest integers modulo $m$: :$\\displaystyle \\left({a^n - 1}\\right) = \\left({a - 1}\\right) \\left({\\sum_{k \\mathop = 0}^{n-1} a^k}\\right) \\equiv 0 \\pmod m$ will produce the roots. Any root found will also have $a + k m$ as a solution, where $k \\in \\Z$ is any integer. Euler's Function $\\phi \\left({m}\\right)$ root of unity modulo $m$ is the set of all positive integers less than $m$. {{explain|The above statement needs tightening: what has been described \"$\\phi \\left({m}\\right)$ root of unity modulo $m$\" is not clearly defined and (while it possibly may be used to define a set by predicate, is does not appear itself actually to be a set. Difficult to tell.}}"}
+{"_id": "24637", "title": "Definition:Subtree (Graph Theory)", "text": "Let $T = \\left({V, E}\\right)$ be a tree. A '''subtree''' of $T$ is a subgraph of $T$ that is also a tree."}
+{"_id": "24638", "title": "Definition:Rooted Subtree", "text": "Let $\\left({T, r_T}\\right)$ be a rooted tree. A '''rooted subtree''' of $T$ is a rooted tree $\\left({S, r_S}\\right)$ such that: :$S$ is a subtree of $T$ :$r_S = r_T$ Note that the second condition implies that $r_T \\in S$."}
+{"_id": "24639", "title": "Definition:Extension of Rooted Tree", "text": "Let $\\left({T, r_T}\\right)$ and $\\left({S, r_S}\\right)$ be rooted trees. Then $\\left({S, r_S}\\right)$ is an '''extension of $T$''' {{iff}}: :$T$ is a subtree of $S$; :$r_S = r_T$. That is, {{iff}} $T$ is a rooted subtree of $S$. {{expand|Define using parent mapping}}"}
+{"_id": "24640", "title": "Definition:Extension of Branch of Rooted Tree", "text": "Let $T$ be a rooted tree, and let $\\Gamma$ be a branch of $T$. Let $S$ be an extension of $T$, and let $\\Gamma'$ be a branch of $S$. Then $\\Gamma'$ is an '''extension of $\\Gamma$''' {{iff}} $\\Gamma \\subseteq \\Gamma'$. Informally, a branch may be '''extended''' by successively adding children to its leaf node, one at a time."}
+{"_id": "24641", "title": "Definition:Extension of Propositional Tableau", "text": "Let $\\left({T, \\mathbf H, \\Phi}\\right)$ be a propositional tableau. === Definition 1 === {{:Definition:Extension of Propositional Tableau/Definition 1}} === Definition 2 === {{:Definition:Extension of Propositional Tableau/Definition 2}}"}
+{"_id": "24642", "title": "Definition:Convergent Series/Normed Vector Space", "text": "=== Definition 1 === {{:Definition:Convergent Series/Normed Vector Space/Definition 1}} === Definition 2 === {{:Definition:Convergent Series/Normed Vector Space/Definition 2}}"}
+{"_id": "24643", "title": "Definition:Convergent Series/Number Field", "text": "Let $S$ be one of the standard number fields $\\Q, \\R, \\C$. Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be a series in $S$. Let $\\sequence {s_N}$ be the sequence of partial sums of $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$. It follows that $\\sequence {s_N}$ can be treated as a sequence in the metric space $S$. If $s_N \\to s$ as $N \\to \\infty$, the series '''converges to the sum $s$''', and one writes $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n = s$. A series is said to be '''convergent''' {{iff}} it converges to some $s$."}
+{"_id": "24644", "title": "Definition:Extension of Branch of Propositional Tableau", "text": "Let $T$ be a propositional tableau, and let $\\Gamma$ be a branch of $T$. Let $S$ be an extension of $T$, and let $\\Gamma'$ be a branch of $S$. Then $\\Gamma'$ is an '''extension of $\\Gamma$''' iff $\\Gamma \\subseteq \\Gamma'$. Informally, a branch may be '''extended''' by successively adding children (with WFFs attached to them) to its leaf node, one at a time."}
+{"_id": "24645", "title": "Definition:Geometric Sequence/Common Ratio", "text": "The parameter: :$r \\in \\R: r \\ne 0$ is called the '''common ratio''' of $\\sequence {x_n}$."}
+{"_id": "24647", "title": "Definition:Geometric Series", "text": "Let $\\sequence {x_n}$ be a geometric sequence in $\\R$: :$x_n = a r^n$ for $n = 0, 1, 2, \\ldots$ Then the series defined as: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty x_n = a + a r + a r^2 + \\cdots + a r^n + \\cdots$ is a '''geometric series'''."}
+{"_id": "24648", "title": "Definition:Isolated Point (Metric Space)/Subset", "text": "Let $S \\subseteq A$ be a subset of $A$. $a \\in S$ is an '''isolated point''' of $S$ {{iff}} there exists an open $\\epsilon$-ball of $x$ in $M$ containing no points of $S$ other than $a$: :$\\exists \\epsilon \\in \\R_{>0}: \\map {B_\\epsilon} a \\cap S = \\set a$ That is: :$\\exists \\epsilon \\in \\R_{>0}: \\set {x \\in S: \\map d {x, a} < \\epsilon} = \\set a$"}
+{"_id": "24649", "title": "Definition:Isolated Point (Metric Space)/Space", "text": "$a \\in A$ is an '''isolated point''' of $M$ {{iff}} there exists an open $\\epsilon$-ball of $x$ containing no points other than $a$: :$\\exists \\epsilon \\in \\R_{>0}: \\map {B_\\epsilon} a = \\set a$ That is: :$\\exists \\epsilon \\in \\R_{>0}: \\set {x \\in A: \\map d {x, a} < \\epsilon} = \\set a$"}
+{"_id": "24650", "title": "Definition:Natural Logarithm/Positive Real", "text": "{{:Definition:Natural Logarithm/Positive Real/Definition 1}}"}
+{"_id": "24651", "title": "Definition:Natural Logarithm/Complex", "text": "{{:Definition:Natural Logarithm/Complex/Definition 1}}"}
+{"_id": "24652", "title": "Definition:General Logarithm/Common", "text": "Logarithms base $10$ are often referred to as common logarithms."}
+{"_id": "24653", "title": "Definition:Series/Sequence of Partial Sums", "text": "The sequence $\\sequence {s_N}$ defined as the indexed summation: :$\\displaystyle s_N: = \\sum_{n \\mathop = 1}^N a_n = a_1 + a_2 + a_3 + \\cdots + a_N$ is the '''sequence of partial sums''' of the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$."}
+{"_id": "24654", "title": "Definition:Series/General/Sequence of Partial Products", "text": "The sequence $\\left \\langle {s_N} \\right \\rangle$ defined as the indexed iterated operation: :$\\displaystyle s_N = \\sum_{n \\mathop = 1}^N a_n = a_1 \\circ a_2 \\circ \\cdots \\circ a_N$ is the '''sequence of partial products''' of the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$."}
+{"_id": "24655", "title": "Definition:Positive Series", "text": "Let $\\displaystyle s = \\sum_{n \\mathop = 1}^\\infty a_n$ be a series in the real numbers $\\R$. The series $s$ is a '''positive series''' {{iff}} either: :$\\forall n \\in \\N: a_n > 0$ or: :$\\forall n \\in \\N: a_n < 0$ That is, if all terms of $\\sequence {a_n}$ are either all (strictly) positive or (strictly) negative."}
+{"_id": "24656", "title": "Definition:Absolutely Convergent Series/General", "text": "Let $V$ be a normed vector space with norm $\\norm {\\, \\cdot \\,}$. Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be a series in $V$. Then the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ in $V$ is '''absolutely convergent''' {{iff}} $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\norm {a_n}$ is a convergent series in $\\R$."}
+{"_id": "24657", "title": "Definition:Absolutely Convergent Series/Real Numbers", "text": "Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be a series in the real number field $\\R$. Then $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ is '''absolutely convergent''' {{iff}}: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\size {a_n}$ is convergent where $\\size {a_n}$ denotes the absolute value of $a_n$."}
+{"_id": "24658", "title": "Definition:Alternating Series", "text": "Let $\\displaystyle s = \\sum_{n \\mathop = 1}^\\infty a_n$ be a series in the real numbers $\\R$. The series $s$ is an '''alternating series''' {{iff}} the terms of $\\sequence {a_n}$ alternate between positive and negative."}
+{"_id": "24659", "title": "Definition:Absolutely Convergent Series/Complex Numbers", "text": "Let $S = \\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be a series in the complex number field $\\C$. Then $S$ is '''absolutely convergent''' {{iff}}: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\cmod {a_n}$ is convergent where $\\cmod {a_n}$ denotes the complex modulus of $a_n$."}
+{"_id": "24660", "title": "Definition:Harmonic Series", "text": "The series defined as: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\frac 1 n = 1 + \\frac 1 2 + \\frac 1 3 + \\frac 1 4 + \\cdots$ is known as '''the harmonic series'''."}
+{"_id": "24661", "title": "Definition:Mercator's Constant", "text": "'''Mercator's constant''' is the real number: {{begin-eqn}} {{eqn | l = \\ln 2       | r = \\sum_{n \\mathop = 1}^\\infty \\frac {\\paren {-1}^\\paren {n - 1} } n       | c =  }} {{eqn | r = 1 - \\frac 1 2 + \\frac 1 3 - \\frac 1 4 + \\dotsb       | c =  }} {{eqn | r = 0 \\cdotp 69314 \\, 71805 \\, 59945 \\, 30941 \\, 72321 \\, 21458 \\, 17656 \\, 80755 \\, 00134 \\, 360 \\ldots \\ldots       | c =  }} {{end-eqn}}"}
+{"_id": "24662", "title": "Definition:Cauchy Product", "text": "Let $A := \\displaystyle \\sum_{n \\mathop = 0}^\\infty a_n$ and $B := \\displaystyle \\sum_{n \\mathop = 0}^\\infty b_n$ be two  series. The '''Cauchy product''' $C$ of $A$ and $B$ is defined as: :$\\displaystyle C := \\sum_{n \\mathop = 0}^\\infty c_n = \\sum_{n \\mathop = 0}^\\infty a_n \\cdot \\sum_{n \\mathop = 0}^\\infty b_n$ where: :$\\displaystyle \\forall n \\in \\N: c_n = \\sum_{k \\mathop = 0}^n a_k b_{n - k} = \\sum_{k \\mathop = 0}^n a_{n - k} b_k$ {{NamedforDef|Augustin Louis Cauchy|cat=Cauchy}}"}
+{"_id": "24663", "title": "Definition:Power Series/Coefficient", "text": "The terms: :$a_0, a_1, \\ldots, a_n, \\ldots$ are the '''coefficients''' of $\\displaystyle \\sum_{n \\mathop = 0}^\\infty a_n \\paren {x - \\xi}^n$."}
+{"_id": "24665", "title": "Definition:Maclaurin Series", "text": "Let $f$ be a real function which is smooth on the open interval $\\openint a b$. Then the '''Maclaurin series expansion of $f$''' is: :$\\displaystyle \\sum_{n \\mathop = 0}^\\infty \\frac {x^n} {n!} \\map {f^{\\paren n} } 0$ It is ''not'' necessarily the case that this power series is convergent with sum $\\map f x$. {{NamedforDef|Colin Maclaurin|cat = Maclaurin}}"}
+{"_id": "24666", "title": "Definition:Determinant/Matrix/Order 1", "text": ":$\\begin {vmatrix} a_{1 1} \\end {vmatrix} = a_{1 1}$ Thus the determinant of an order $1$ matrix is that element itself."}
+{"_id": "24667", "title": "Definition:Determinant/Matrix/Order 2", "text": "{{begin-eqn}} {{eqn | l = \\begin {vmatrix} a_{1 1} & a_{1 2} \\\\ a_{2 1} & a_{2 2} \\end{vmatrix}       | r = \\map \\sgn {1, 2} a_{1 1} a_{2 2} + \\map \\sgn {2, 1} a_{1 2} a_{2 1}       | c =  }} {{eqn | r = a_{1 1} a_{2 2} - a_{1 2} a_{2 1}       | c =  }} {{end-eqn}}"}
+{"_id": "24668", "title": "Definition:Determinant/Matrix/Order 3", "text": ":$\\map \\det {\\mathbf A} = \\begin {vmatrix} a_{1 1} & a_{1 2} & a_{1 3} \\\\ a_{2 1} & a_{2 2} & a_{2 3} \\\\ a_{3 1} & a_{3 2} & a_{3 3} \\end {vmatrix}$ Then: {{begin-eqn}} {{eqn | l = \\map \\det {\\mathbf A}       | r = a_{1 1} \\begin {vmatrix} a_{2 2} & a_{2 3} \\\\ a_{3 2} & a_{3 3} \\end {vmatrix} - a_{1 2} \\begin {vmatrix} a_{2 1} & a_{2 3} \\\\ a_{3 1} & a_{3 3} \\end {vmatrix} + a_{1 3} \\begin {vmatrix} a_{2 1} & a_{2 2} \\\\ a_{3 1} & a_{3 2} \\end{vmatrix}       | c =  }} {{eqn | r = \\map \\sgn {1, 2, 3} a_{1 1} a_{2 2}  a_{3 3}       | c =  }} {{eqn | o =        | ro= +       | r = \\map \\sgn {1, 3, 2} a_{1 1} a_{2 3}  a_{3 2}       | c =  }} {{eqn | o =        | ro= +       | r = \\map \\sgn {2, 1, 3} a_{1 2} a_{2 1}  a_{3 3}       | c =  }} {{eqn | o =        | ro= +       | r = \\map \\sgn {2, 3, 1} a_{1 2} a_{2 3}  a_{3 1}       | c =  }} {{eqn | o =        | ro= +       | r = \\map \\sgn {3, 1, 2} a_{1 3} a_{2 1}  a_{3 2}       | c =  }} {{eqn | o =        | ro= +       | r = \\map \\sgn {3, 2, 1} a_{1 3} a_{2 2}  a_{3 1}       | c =  }} {{eqn | r = a_{1 1} a_{2 2}  a_{3 3}       | c =  }} {{eqn | o =        | ro= -       | r = a_{1 1} a_{2 3}  a_{3 2}       | c =  }} {{eqn | o =        | ro= -       | r = a_{1 2} a_{2 1}  a_{3 3}       | c =  }} {{eqn | o =        | ro= +       | r = a_{1 2} a_{2 3}  a_{3 1}       | c =  }} {{eqn | o =        | ro= +       | r = a_{1 3} a_{2 1}  a_{3 2}       | c =  }} {{eqn | o =        | ro= -       | r = a_{1 3} a_{2 2}  a_{3 1}       | c =  }} {{end-eqn}} and thence in a single expression as: :$\\ds \\map \\det {\\mathbf A} = \\frac 1 6 \\sum_{i \\mathop = 1}^3 \\sum_{j \\mathop = 1}^3 \\sum_{k \\mathop = 1}^3 \\sum_{r \\mathop = 1}^3 \\sum_{s \\mathop = 1}^3 \\sum_{t \\mathop = 1}^3 \\map \\sgn {i, j, k} \\map \\sgn {r, s, t} a_{i r} a_{j s} a_{k t}$ where $\\map \\sgn {i, j, k}$ is the sign of the permutation $\\tuple {i, j, k}$ of the set $\\set {1, 2, 3}$. The values of the various instances of $\\map \\sgn {\\lambda_1, \\lambda_2, \\lambda_3}$ are obtained by applications of Parity of K-Cycle."}
+{"_id": "24669", "title": "Definition:Extension of Propositional Tableau/Definition 2", "text": "An '''extension of $T$''' is a propositional tableau $\\left({S, \\mathbf H', \\Phi'}\\right)$ such that: :$S$ is an extension of $T$; :$\\mathbf H = \\mathbf H'$; :$\\Phi'$ is an extension of $\\Phi$."}
+{"_id": "24670", "title": "Definition:Extension of Propositional Tableau/Definition 1", "text": "A tableau $T'$ is an '''extension of $T$''' if $T'$ can be obtained from $T$ by repeatedly adding nodes to the leaf nodes of $T$ (by means of the tableau extension rules)."}
+{"_id": "24671", "title": "Definition:Ordered Statistic", "text": "Let $X_1, X_2, X_3, \\ldots, X_N$ be a sequence of $N$ independent random variables chosen from an identical probability space. Let $X_{(1)},X_{(2)},X_{(3)},\\ldots,X_{(N)}$ be these measures re-arranged in increasing order. Then $X_{(i)}$ is said to be the '''$i$th ordered statistic'''. Category:Definitions/Probability Theory rzaj5o9lujxt7js071sccdi9t649l84"}
+{"_id": "24672", "title": "Definition:Exterior (Topology)/Notation", "text": "The exterior of $H$ can be denoted: :$\\map {\\mathrm {Ext} } H$ :$H^e$ The first is regarded by some as cumbersome, but has the advantage of being clear. $H^e$ is neat and compact, but has the disadvantage of being relatively obscure. On {{ProofWiki}}, the notation of choice is $H^e$."}
+{"_id": "24673", "title": "Definition:Exterior (Topology)/Definition 1", "text": "The '''exterior''' of $H$ is the complement of the closure of $H$ in $T$."}
+{"_id": "24674", "title": "Definition:Exterior (Topology)/Definition 2", "text": "The '''exterior''' of $H$ is the interior of the complement of $H$ in $T$."}
+{"_id": "24675", "title": "Definition:Enumeration/Finite", "text": "Let $X$ be a finite set of cardinality $n \\in \\N$. An '''enumeration''' of $X$ is a bijection $x: \\N_n \\to X$, where $\\N_n = \\set {1, \\ldots, n}$."}
+{"_id": "24676", "title": "Definition:Enumeration/Countably Infinite", "text": "Let $X$ be a countably infinite set. An '''enumeration''' of $X$ is a bijection $x: \\N \\to X$."}
+{"_id": "24677", "title": "Definition:Euclidean Plus Metric", "text": "Let $\\R$ be the set of real numbers. Let $\\set {r_i}$ be an enumeration of the rational numbers $\\Q$. The '''Euclidean plus metric''' $d: \\R \\times \\R \\to \\R$ is the metric defined as: :$\\map d {x, y} := \\set {x - y} + \\displaystyle \\sum_{i \\mathop = 1}^\\infty 2^{-i} \\map \\inf {1, \\size {\\max_{j \\mathop \\le i} \\frac 1 {\\size {x - r_j} } - \\max_{j \\mathop \\le i} \\frac 1 {\\size {y - r_j} } } }$ Thus $d$ adds to the Euclidean metric between $x$ and $y$ a contribution which measures the relative distances of $x$ and $y$ from $\\Q$."}
+{"_id": "24678", "title": "Definition:Tautology/Formal Semantics", "text": "Let $\\mathcal L$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\mathcal L$. A logical formula $\\phi$ of $\\mathcal L$ is a '''tautology for $\\mathscr M$''' {{iff}}: :$\\phi$ is valid in every structure $\\mathcal M$ of $\\mathscr M$ That $\\phi$ is a '''tautology for $\\mathscr M$''' can be denoted as: :$\\models_{\\mathscr M} \\phi$"}
+{"_id": "24679", "title": "Definition:Tautology/Formal Semantics/Boolean Interpretations", "text": "Let $\\mathbf A$ be a WFF of propositional logic. Then $\\mathbf A$ is called a '''tautology (for boolean interpretations)''' {{iff}}: :$\\map v {\\mathbf A} = T$ for every boolean interpretation $v$ of $\\mathbf A$. That $\\mathbf A$ is a '''tautology''' may be denoted as: :$\\models_{\\mathrm {BI} } \\mathbf A$"}
+{"_id": "24680", "title": "Definition:Contradiction", "text": "A '''contradiction''' is a statement which is ''always false'', independently of any relevant circumstances that could theoretically influence its truth value. This has the form: :$p \\land \\neg p$ or, equivalently: :$\\neg p \\land p$ that is: :'''$p$ is true and, at the same time, $p$ is not true.'''"}
+{"_id": "24681", "title": "Definition:Rational Number Space", "text": "Let $\\Q$ be the set of rational numbers. Let $d: \\Q \\times \\Q \\to \\R$ be the Euclidean metric on $\\Q$. Let $\\tau_d$ be the topology on $\\Q$ induced by $d$. Then $\\struct {\\Q, \\tau_d}$ is the '''rational number space'''."}
+{"_id": "24682", "title": "Definition:Irrational Number Space", "text": "Let $\\mathbb I := \\R \\setminus \\Q$ be the set of irrational numbers. Let $d: \\mathbb I \\times \\mathbb I \\to \\R$ be the Euclidean metric on $\\mathbb I$. Let $\\tau_d$ be the topology on $\\mathbb I$ induced by $d$. Then $\\struct {\\mathbb I, \\tau_d}$ is the '''irrational number space'''."}
+{"_id": "24683", "title": "Definition:Unsatisfiable/Boolean Interpretations", "text": "Let $\\mathbf A$ be a WFF of propositional logic. $\\mathbf A$ is called '''unsatisfiable (for boolean interpretations)''' {{iff}}: :$\\map v {\\mathbf A} = F$ for every boolean interpretation $v$ for $\\mathbf A$. In terms of validity, this can be rendered: :$v \\not \\models_{\\mathrm {BI} } \\mathbf A$ that is, $\\mathbf A$ is invalid in every boolean interpretation of $\\mathbf A$."}
+{"_id": "24684", "title": "Definition:Boolean Interpretation/Formal Semantics/Invalid", "text": "$\\phi$ is declared '''($\\mathrm{BI}$-)invalid''' in a boolean interpretation $v$ iff: :$v \\left({\\phi}\\right) = F$ Symbolically, this can be expressed as: :$v \\not\\models_{\\mathrm{BI}} \\phi$"}
+{"_id": "24685", "title": "Definition:Scattered Space/Definition 1", "text": "A topological space $T = \\struct {S, \\tau}$ is '''scattered''' {{iff}} it contains no non-empty subset which is dense-in-itself. That is, $T = \\struct {S, \\tau}$ is '''scattered''' {{iff}} every non-empty subset $H$ of $S$ contains at least one point which is isolated in $H$."}
+{"_id": "24686", "title": "Definition:Scattered Space/Also defined as", "text": "According to {{BookReference|Counterexamples in Topology|1978|Lynn Arthur Steen|author2 = J. Arthur Seebach, Jr.|ed = 2nd|edpage = Second Edition}}, a topological space $T$ is defined as '''scattered''': : ''... if it contains no non-empty dense-in-itself subsets; ...'' and it is immaterial whether those subsets are closed or not. On the other hand, [http://planetmath.org/encyclopedia/ScatteredSpace.html PlanetMath's definition] specifically requires that in order for a space to be classified as '''scattered''', only its closed subsets are required to contain one or more isolated points. There are few other reliable definitions to be found (the concept can be found neither on Wikipedia nor even MathWorld), and when the concept is used at all, the definitions go either way. However, it is apparent that the two definitions are equivalent, and so ultimately it does not matter which definition is used."}
+{"_id": "24687", "title": "Definition:Scattered Space/Definition 2", "text": "A topological space $T = \\left({S, \\tau}\\right)$ is '''scattered''' {{iff}} it contains no non-empty closed set which is dense-in-itself. That is, $T = \\left({S, \\tau}\\right)$ is '''scattered''' {{iff}} every non-empty closed set $H$ of $S$ contains at least one point which is isolated in $H$."}
+{"_id": "24688", "title": "Definition:Isolated Point (Topology)/Subset", "text": "{{:Definition:Isolated Point (Topology)/Subset/Definition 1}}"}
+{"_id": "24689", "title": "Definition:Isolated Point (Topology)/Space", "text": "$x \\in S$ is an '''isolated point of $T$''' {{iff}}: :$\\exists U \\in \\tau: U = \\set x$ That is, {{iff}} there exists an open set of $T$ containing no points of $S$ other than $x$."}
+{"_id": "24690", "title": "Definition:Integer Reciprocal Space", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line $\\R$ under the usual (Euclidean) topology $\\tau_d$. Let $A \\subseteq \\R$ be the set of all points on $\\R$ defined as: :$A := \\set {\\dfrac 1 n : n \\in \\Z_{>0} }$ That is: :$A := \\set {1, \\dfrac 1 2, \\dfrac 1 3, \\dfrac 1 4, \\ldots}$ Then $\\struct {A, \\tau_d}$ is the '''integer reciprocal space'''."}
+{"_id": "24691", "title": "Definition:Exponential Distribution", "text": "Let $X$ be a continuous random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Then $X$ has the '''exponential distribution with parameter $\\beta$''' {{iff}}: :$\\map X \\Omega = \\R_{\\ge 0}$ :$\\map \\Pr {X < x} = 1 - e^{-\\frac x \\beta}$ where $0 < \\beta$."}
+{"_id": "24692", "title": "Definition:Expectation/Continuous", "text": "Let $X$ be a continuous random variable over the probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $F = \\map \\Pr {X < x}$ be the cumulative probability function of $X$. The '''expectation of $X$''' is written $\\expect X$, and is defined over the probability measure as: :$\\expect X := \\displaystyle \\int_{x \\mathop \\in \\Omega} x \\rd F$ whenever the integral is absolutely convergent, i.e. when: :$\\displaystyle \\int_{x \\mathop \\in \\Omega} \\size x \\rd F < \\infty$"}
+{"_id": "24693", "title": "Definition:Logical Implication", "text": "In a valid argument, the premises '''logically imply''' the conclusion. If the truth of one statement $p$ can be shown in an argument directly to cause the meaning of another statement $q$ to be true, then $q$ follows from $p$ by '''logical implication'''. We may say: :'''$p$, therefore $q$''' and write $p \\vdash q$. :'''$q$, because $p$''' and write $q \\dashv p$. In symbolic logic, the concept of '''logical consequence''' occurs in the form of semantic consequence and provable consequence. In the context of proofs of a conventional mathematical nature on {{ProofWiki}}, the notation: :$p \\leadsto q$ is preferred, where $\\leadsto$ can be read as '''leads to'''. === Semantic Consequence === {{:Definition:Semantic Consequence}} === Provable Consequence === {{:Definition:Provable Consequence}}"}
+{"_id": "24694", "title": "Definition:Semantic Consequence/Boolean Interpretations", "text": "<section notitle=\"true\" name=\"tc\"> Let $\\mathcal F$ be a collection of WFFs of propositional logic. Then a WFF $\\mathbf A$ is a '''semantic consequence''' of $\\mathcal F$ iff: :$v \\models_{\\mathrm{BI}} \\mathcal F$ implies $v \\models_{\\mathrm{BI}} \\mathbf A$ where $\\models_{\\mathrm{BI}}$ is the models relation. </section> {{transclude:Definition:Semantic Consequence/Boolean Interpretations/Single Formula   |section = tc   |title = Semantic Consequence of Single Formula   |header = 3   |link = true   |increase = 1 }} === Notation === That $\\mathbf A$ is a '''semantic consequence''' of $\\mathcal F$ is denoted as: :$\\mathcal F \\models_{\\mathbf{BI}} \\mathbf A$"}
+{"_id": "24695", "title": "Definition:Union of Adjacent Open Intervals", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line $\\R$ under the usual (Euclidean) topology $\\tau_d$. Let $a, b, c \\in \\R$ where $a < b < c$. Let $A$ be the union of the two open intervals: :$A := \\openint a b \\cup \\openint b c$ Then $\\struct {A, \\tau_d}$ is the '''union of adjacent open intervals'''."}
+{"_id": "24696", "title": "Definition:Logical Equivalence", "text": "If two statements $p$ and $q$ are such that: :$p \\vdash q$, that is: $p$ therefore $q$ :$q \\vdash p$, that is: $q$ therefore $p$ then $p$ and $q$ are said to be '''(logically) equivalent'''. That is: :$p \\dashv \\vdash q$ means: :$p \\vdash q$ and $q \\vdash p$. Note that because the conclusion of an argument is a single statement, there can be only one statement on either side of the $\\dashv \\vdash$ sign. In symbolic logic, the notion of '''logical equivalence''' occurs in the form of provable equivalence and semantic equivalence. === Provable Equivalence === {{:Definition:Provable Equivalence}} === Semantic Equivalence === {{:Definition:Semantic Equivalence}}"}
+{"_id": "24697", "title": "Definition:Provable Equivalence", "text": "Let $\\mathcal P$ be a proof system for a formal language $\\mathcal L$. Let $\\phi, \\psi$ be $\\mathcal L$-WFFs. Then $\\phi$ and $\\psi$ are '''$\\mathscr P$-provably equivalent''' {{iff}}: :$\\phi \\vdash_{\\mathscr P} \\psi$ and $\\psi \\vdash_{\\mathscr P} \\phi$ that is, {{iff}} they are $\\mathscr P$-provable consequences of one another. The '''provable equivalence''' of $\\phi$ and $\\psi$ can be denoted by: :$\\phi \\dashv \\vdash_{\\mathscr P} \\psi$"}
+{"_id": "24698", "title": "Definition:Semantic Equivalence", "text": "Let $\\mathscr M$ be a formal semantics for a formal language $\\mathcal L$. Let $\\phi, \\psi$ be $\\mathcal L$-WFFs. Then $\\phi$ and $\\psi$ are $\\mathscr M$-'''semantically equivalent''' {{iff}}: :$\\phi \\models_{\\mathscr M} \\psi$ and $\\psi \\models_{\\mathscr M} \\phi$ that is, iff they are $\\mathscr M$-semantic consequences of one another."}
+{"_id": "24699", "title": "Definition:Semantic Equivalence/Boolean Interpretations", "text": "<section notitle=\"true\" name=\"tc\"> Let $\\mathbf A, \\mathbf B$ be WFFs of propositional logic. === Definition 1 === {{:Definition:Semantic Equivalence/Boolean Interpretations/Definition 1}} === Definition 2 === {{:Definition:Semantic Equivalence/Boolean Interpretations/Definition 2}} === Definition 3 === {{:Definition:Semantic Equivalence/Boolean Interpretations/Definition 3}} </section>"}
+{"_id": "24700", "title": "Definition:Semantic Consequence/Boolean Interpretations/Single Formula", "text": "<section notitle=\"true\" name=\"tc\"> Let $\\mathbf A, \\mathbf B$ be WFFs of propositional logic. === Definition 1 === {{:Definition:Semantic Consequence/Boolean Interpretations/Single Formula/Definition 1}} === Definition 2 === {{:Definition:Semantic Consequence/Boolean Interpretations/Single Formula/Definition 2}} </section> === Notation === That $\\mathbf A$ is a '''consequence of $\\mathbf B$ for boolean interpretations''' can be denoted as: :$\\mathbf B \\models_{\\mathrm{BI}} \\mathbf A$"}
+{"_id": "24701", "title": "Definition:Semantic Equivalence/Boolean Interpretations/Definition 1", "text": "Then $\\mathbf A$ and $\\mathbf B$ are '''equivalent for boolean interpretations''' {{iff}}: :$\\mathbf A \\models_{\\mathrm{BI}} \\mathbf B$ and $\\mathbf B \\models_{\\mathrm{BI}} \\mathbf A$ that is, each is a semantic consequence of the other. That is to say, $\\mathbf A$ and $\\mathbf B$ are '''equivalent''' {{iff}}: :$v \\left({\\mathbf A}\\right) = T$ {{iff}} $v \\left({\\mathbf B}\\right) = T$ for all boolean interpretations $v$."}
+{"_id": "24702", "title": "Definition:Semantic Equivalence/Boolean Interpretations/Definition 2", "text": "Then $\\mathbf A$ and $\\mathbf B$ are '''equivalent for boolean interpretations''' {{iff}}: :$\\map v {\\mathbf A} = \\map v {\\mathbf B}$ for all boolean interpretations $v$."}
+{"_id": "24703", "title": "Definition:Semantic Equivalence/Boolean Interpretations/Definition 3", "text": "Then $\\mathbf A$ and $\\mathbf B$ are '''equivalent for boolean interpretations''' {{iff}}: :$\\mathbf A \\iff \\mathbf B$ is a tautology where $\\iff$ is the biconditional connective."}
+{"_id": "24704", "title": "Definition:Semantic Consequence/Boolean Interpretations/Single Formula/Definition 1", "text": "Then $\\mathbf A$ is a '''semantic consequence''' of $\\mathbf B$ iff: :$v \\models_{\\mathrm{BI}} \\mathbf B$ implies $v \\models_{\\mathrm{BI}} \\mathbf A$ for all boolean interpretations $v$. Here, $\\models_{\\mathrm{BI}}$ is the models relation."}
+{"_id": "24705", "title": "Definition:Semantic Consequence/Boolean Interpretations/Single Formula/Definition 2", "text": "Then $\\mathbf A$ is a '''semantic consequence''' of $\\mathbf B$ {{iff}}: :$\\mathbf A \\implies \\mathbf B$ is a tautology where $\\implies$ is the conditional connective."}
+{"_id": "24706", "title": "Definition:Trace (Linear Algebra)/Matrix", "text": "Let $A = \\sqbrk a_n$ be a square matrix of order $n$. The '''trace''' of $A$ is: :$\\ds \\map \\tr A = \\sum_{i \\mathop = 1}^n a_{ii}$"}
+{"_id": "24707", "title": "Definition:Trace (Linear Algebra)/Linear Operator", "text": "Let $V$ be a vector space. Let $A: V \\to V$ be a linear operator of $V$. The '''trace''' of $A$ is the trace of the matrix of $A$ with respect to some basis."}
+{"_id": "24708", "title": "Definition:P-Element", "text": "Let $\\left({G, \\circ}\\right)$ be a group. Let $p$ be a prime number. Let $x \\in G$ be an element of $G$. Then $x$ is a '''$p$-element of $G$''' {{iff}} its order is a power of $p$."}
+{"_id": "24709", "title": "Definition:P-Group", "text": "A '''$p$-group''' is a group whose elements are all $p$-elements."}
+{"_id": "24710", "title": "Definition:Elementary Abelian P-Group", "text": "Let $G$ be an abelian group. Then $G$ is an '''elementary abelian $p$-group''' {{iff}} all its non-trivial elements are of order $p$."}
+{"_id": "24711", "title": "Definition:Abelian Group/Definition 1", "text": "An '''abelian group''' is a group $G$ where: : $\\forall a, b \\in G: a b = b a$ That is, every element in $G$ commutes with every other element in $G$."}
+{"_id": "24712", "title": "Definition:Abelian Group/Definition 2", "text": "An '''abelian group''' is a group $G$ {{iff}}: : $G = \\map Z G$ where $\\map Z G$ is the center of $G$."}
+{"_id": "24713", "title": "Definition:Normal Subset/Definition 6", "text": ":$\\map {N_G} S = G$ where $\\map {N_G} S$ denotes the normalizer of $S$ in $G$."}
+{"_id": "24714", "title": "Definition:Normal Subset/Definition 7", "text": ":$\\forall g \\in G: g \\circ S \\subseteq S \\circ g$ or: :$\\forall g \\in G: S \\circ g \\subseteq g \\circ S$"}
+{"_id": "24715", "title": "Definition:Normal Subgroup/Notation", "text": "The statement that $N$ is a normal subgroup of $G$ is represented symbolically as $N \\lhd G$. A normal subgroup is often represented by the letter $N$, as opposed to $H$ (which is used for a general subgroup which may or may not be normal). To use the notation introduced in the definition of the conjugate: :$N \\lhd G \\iff \\forall g \\in G: N^g = N$"}
+{"_id": "24716", "title": "Definition:Refinement of Normal Series/Proper Refinement", "text": "A '''proper refinement''' of a normal series is a refinement which is not equal to the original normal series. That is, it contains extra (normal) subgroups which are not present in the original normal series."}
+{"_id": "24717", "title": "Definition:Characteristic Subset", "text": "Let $G$ be a group. Let $S \\subseteq G$ be a subset of $G$ such that: :$\\forall \\phi \\in \\Aut G: \\map \\phi S = S$ where $\\Aut G$ is the automorphism group of $G$. Then $S$ is  '''characteristic (in $G$)''', or '''a characteristic subset of $G$'''."}
+{"_id": "24718", "title": "Definition:Minimal Subgroup", "text": "Let $G$ be a group. Let $M \\le G$ be a non-trivial subgroup of $G$. Then $M$ is a '''minimal subgroup of $G$''' {{iff}}: :For every subgroup $H$ of $G$, $H \\subseteq M$ means $H = M$ or $H = \\set e$. That is, if there is no subgroup of $M$, except $M$ and $\\set e$ itself, which is a subset of $M$."}
+{"_id": "24719", "title": "Definition:Asymptotics", "text": "Suppose the task to evaluate a certain number is sufficiently unwieldy that it becomes impractical to calculate it directly. In such cases it is frequently useful to have a completely different method to calculate an approximation to that number. Let the accuracy of this approximation improve as the number of operations involved in its calculation increases. The field of mathematics concerned with studying such techniques is called '''asymptotics'''."}
+{"_id": "24720", "title": "Definition:Primitive Triple", "text": "Let $x, y, z \\in \\Z_{>0}$ be (strictly) positive integers. Let $A, B, C \\in \\Z_{>0}$ be (strictly) positive integers that satisfy: :$A^x + B^y = C^z$ Let $\\left({A, B, C}\\right)$ be pairwise coprime. Then $\\left({A, B, C}\\right)$ is a '''primitive triple'''."}
+{"_id": "24721", "title": "Definition:Pythagorean Triple/Primitive/Canonical Form", "text": "Let $\\tuple {x, y, z}$ be a primitive Pythagorean triple. The convention for representing $\\tuple {x, y, z}$ as a (primitive) Pythagorean triple is that $x$ is the even element, while $y$ and $z$ are both odd. This is the '''canonical form''' of a Pythagorean triple."}
+{"_id": "24722", "title": "Definition:Pythagorean Triple/Primitive", "text": "Let $\\tuple {x, y, z}$ be a Pythagorean triple such that $x \\perp y$ (that is, $x$ and $y$ are coprime). Then $\\tuple {x, y, z}$ is a '''primitive Pythagorean triple'''."}
+{"_id": "24723", "title": "Definition:Perfect Set/Definition 1", "text": "A '''perfect set''' of a topological space $T = \\left({S, \\tau}\\right)$ is a subset $H \\subseteq S$ such that: : $H = H'$ where $H'$ is the derived set of $H$. That is, where: : every point of $H$ is a limit point of $H$ and : every limit point of $H$ is a point of $H$."}
+{"_id": "24724", "title": "Definition:Perfect Set/Definition 2", "text": "A '''perfect set''' of a topological space $T = \\left({S, \\tau}\\right)$ is a subset $H \\subseteq S$ such that: : $H$ is a closed set of $T$ : $H$ has no isolated points."}
+{"_id": "24725", "title": "Definition:Perfect Set/Definition 3", "text": "A '''perfect set''' of a topological space $T = \\struct {S, \\tau}$ is a subset $H \\subseteq S$ such that: :$H$ is dense-in-itself. :$H$ contains all its limit points."}
+{"_id": "24726", "title": "Definition:Conjugate Symmetric Mapping", "text": "Let $\\C$ be the field of complex numbers. Let $\\F$ be a subfield of $\\C$. Let $V$ be a vector space over $\\F$ Let $\\innerprod \\cdot \\cdot: V \\times V \\to \\mathbb F$ be a mapping. Then $\\innerprod \\cdot \\cdot: V \\times V \\to \\mathbb F$ is '''conjugate symmetric''' {{iff}}: :$\\forall x, y \\in V: \\quad \\innerprod x y = \\overline {\\innerprod y x}$ where $\\overline {\\innerprod y x}$ denotes the complex conjugate of $\\innerprod x y$."}
+{"_id": "24727", "title": "Definition:Symmetric Mapping (Linear Algebra)", "text": "Let $\\R$ be the field of real numbers. Let $\\F$ be a subfield of $\\R$. Let $V$ be a vector space over $\\F$ Let $\\innerprod \\cdot \\cdot: V \\times V \\to \\mathbb F$ be a mapping. Then $\\innerprod \\cdot \\cdot: V \\times V \\to \\mathbb F$ is '''symmetric''' {{iff}}: :$\\forall x, y \\in V: \\innerprod x y = \\innerprod y x$"}
+{"_id": "24728", "title": "Definition:Non-Negative Definite Mapping", "text": "Let $\\C$ be the field of complex numbers. Let $\\F$ be a subfield of $\\C$. Let $V$ be a vector space over $\\F$ Let $\\left \\langle {\\cdot, \\cdot} \\right \\rangle : V \\times V \\to \\mathbb F$ be a mapping. Then $\\left \\langle {\\cdot, \\cdot} \\right \\rangle : V \\times V \\to \\mathbb F$ is '''non-negative definite''' {{iff}}: :$\\forall x \\in V: \\quad \\left \\langle {x, x} \\right \\rangle \\in \\R_{\\geq 0}$ That is, the image of $\\left \\langle {x, x} \\right \\rangle$ is always a non-negative real number."}
+{"_id": "24729", "title": "Definition:Positiveness", "text": "Let $\\C$ be the field of complex numbers. Let $\\F$ be a subfield of $\\C$. Let $V$ be a vector space over $\\F$ Let $\\left \\langle {\\cdot, \\cdot} \\right \\rangle : V \\times V \\to \\mathbb F$ be a mapping. Then $\\left \\langle {\\cdot, \\cdot} \\right \\rangle : V \\times V \\to \\mathbb F$ is '''positive''' iff: :$\\forall x \\in V: \\quad \\left \\langle {x, x} \\right \\rangle = 0 \\implies x = \\mathbf 0_V$ where $\\mathbf 0_V$ denotes the zero vector of $V$."}
+{"_id": "24730", "title": "Definition:Prime Number/Definition 4", "text": "A '''prime number''' $p$ is an integer greater than $1$ that has no positive integer divisors other than $1$ and $p$."}
+{"_id": "24731", "title": "Definition:Partial Subtraction", "text": "'''Partial subtraction''' is the operation $\\dot - : \\N \\times \\N \\to \\N$ defined on the set of natural numbers $\\N$ as follows: :$\\forall a, b \\in \\N: a \\mathop {\\dot -} b = \\begin{cases} 0 & : a < b \\\\ a - b & : a \\ge b \\end{cases}$ where $a - b$ denotes natural number subtraction."}
+{"_id": "24732", "title": "Definition:Algebraically Closed Field/Definition 1", "text": ":The only algebraic field extension of $K$ is $K$ itself."}
+{"_id": "24733", "title": "Definition:Algebraically Closed Field/Definition 2", "text": ":Every irreducible polynomial $f$ over $K$ has degree $1$."}
+{"_id": "24734", "title": "Definition:Algebraically Closed Field/Definition 3", "text": ":Every polynomial $f$ over $K$ of strictly positive degree has a root in $K$."}
+{"_id": "24735", "title": "Definition:Hilbert 23", "text": "The '''Hilbert 23''' is a list of $23$ at-the-time unsolved problems in mathematics published by {{AuthorRef|David Hilbert}} during $1900$. They are as follows: === 1: The Continuum Hypothesis === {{:Definition:Hilbert 23/1}} === 2: Consistency of Axioms of Mathematics === {{:Definition:Hilbert 23/2}} === 3: Finite Dissection of Polyhedra === {{:Definition:Hilbert 23/3}} === 4: Construction of all Metrics where Lines are Geodesics === {{:Definition:Hilbert 23/4}} === 5: Whether Continuous Groups are Differential Groups === {{:Definition:Hilbert 23/5}} === 6: Axiomatize all of Physics === {{:Definition:Hilbert 23/6}} === 7: The Gelfond-Schneider Theorem === {{:Definition:Hilbert 23/7}} === 8a: The Riemann Hypothesis === {{:Definition:Hilbert 23/8a}} === 8b: The Goldbach Conjecture === {{:Definition:Hilbert 23/8b}} === 8c: The Twin Prime Conjecture === {{:Definition:Hilbert 23/8c}} === 9: General Reciprocity Theorem in Algebraic Number Field === {{:Definition:Hilbert 23/9}} === 10: Algorithm to determine whether Polynomial Diophantine Equation has Integer Solution === {{:Definition:Hilbert 23/10}} === 11: Quadratic Forms with Algebraic Numerical Coefficients === {{:Definition:Hilbert 23/11}} === 12: Extension of Kronecker-Weber Theorem to any base Number Field === {{:Definition:Hilbert 23/12}} === 13: Solution of 7th Degree Equations using Two Parameter Functions === {{:Definition:Hilbert 23/13}} === 14: Proof of Finiteness of certain Complete Systems of Functions === {{:Definition:Hilbert 23/14}} === 15: Rigorous foundation of Schubert's Enumerative Calculus === {{:Definition:Hilbert 23/15}} === 16: Topology of Algebraic Curves and Surfaces === {{:Definition:Hilbert 23/16}} === 17: Definite Rational Function as Quotient of Sums of Squares === {{:Definition:Hilbert 23/17}} === 18a: Existence of Non-Regular Space-Filling Polyhedron === {{:Definition:Hilbert 23/18a}} === 18b: Kepler's Conjecture (Densest Sphere Packing) === {{:Definition:Hilbert 23/18b}} === 19: Solutions of Lagrangian are Analytic === {{:Definition:Hilbert 23/19}} === 20: Existence of Solutions of Variational Problems with certain Boundary Conditions === {{:Definition:Hilbert 23/20}} === 21: Existence of Linear Differential Equation with prescribed Monodromic Group === {{:Definition:Hilbert 23/21}} === 22: Uniformization of Analytic Relations by means of Automorphic Functions === {{:Definition:Hilbert 23/22}} === 23: Further Development of the Calculus of Variations === {{:Definition:Hilbert 23/23}}"}
+{"_id": "24736", "title": "Definition:Millennium Problems", "text": "The '''Millennium problems''' are a collection of seven mathematical problems stated by the {{WP|Clay_Mathematics_Institute|Clay Mathematics Institute}} on $24$ May $2000$. Each carries a prize of $\\$1 \\, 000 \\, 000$ (US dollars). Six of them remain unsolved. They are as follows: # P versus NP # The Hodge Conjecture # The Poincaré Conjecture (the only one of the seven to be solved) # The Riemann Hypothesis # Yang-Mills Existence and Mass Gap # Navier-Stokes Existence and Smoothness # The Birch and Swinnerton-Dyer Conjecture"}
+{"_id": "24737", "title": "Definition:Almost Perfect Number", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. === Definition 1 === {{:Definition:Almost Perfect Number/Definition 1}} === Definition 2 === {{:Definition:Almost Perfect Number/Definition 2}} === Definition 3 === {{:Definition:Almost Perfect Number/Definition 3}} Thus an '''almost perfect number''' can be described as \"deficient, but only just\"."}
+{"_id": "24738", "title": "Definition:Quasiperfect Number", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. === Definition 1 === {{:Definition:Quasiperfect Number/Definition 1}} === Definition 2 === {{:Definition:Quasiperfect Number/Definition 2}} === Definition 3 === {{:Definition:Quasiperfect Number/Definition 3}}"}
+{"_id": "24739", "title": "Definition:Abundant Number/Definition 1", "text": "Let $A \\left({n}\\right)$ denote the abundance of $n$. $n$ is '''abundant''' {{iff}} $A \\left({n}\\right) > 0$."}
+{"_id": "24740", "title": "Definition:Abundant Number/Definition 2", "text": "Let $\\sigma \\left({n}\\right)$ be the sigma function of $n$. $n$ is '''abundant''' {{iff}} $\\dfrac {\\sigma \\left({n}\\right)} n > 2$."}
+{"_id": "24741", "title": "Definition:Deficient Number/Definition 1", "text": "Let $A \\left({n}\\right)$ denote the abundance of $n$. $n$ is '''deficient''' {{iff}} $A \\left({n}\\right) < 0$."}
+{"_id": "24742", "title": "Definition:Deficient Number/Definition 2", "text": "Let $\\sigma \\left({n}\\right)$ be the sigma function of $n$. $n$ is '''deficient''' {{iff}}: :$\\dfrac {\\sigma \\left({n}\\right)} n < 2$"}
+{"_id": "24743", "title": "Definition:Antisymmetric Matrix", "text": "Let $R$ be a ring. Let $\\mathbf A$ be a square matrix over $R$. $\\mathbf A$ is '''antisymmetric''' {{iff}}: :$\\mathbf A = - \\mathbf A^\\intercal$ where $\\mathbf A^\\intercal$ is the transpose of $\\mathbf A$."}
+{"_id": "24744", "title": "Definition:Orthogonal Matrix", "text": "Let $R$ be a ring with unity. Let $\\mathbf Q$ be an invertible square matrix over $R$. Then $\\mathbf Q$ is '''orthogonal''' {{iff}}: :$\\mathbf Q^{-1} = \\mathbf Q^\\intercal$ where: :$\\mathbf Q^{-1}$ is the inverse of $\\mathbf Q$ :$\\mathbf Q^\\intercal$ is the transpose of $\\mathbf Q$"}
+{"_id": "24745", "title": "Definition:Proper Orthogonal Matrix", "text": "Let $\\mathbf Q$ be an orthogonal matrix. Then $\\mathbf Q$ is a '''proper orthogonal matrix''' {{iff}}: :$\\map \\det {\\mathbf Q} = 1$ where $\\map \\det {\\mathbf Q}$ is the determinant of $\\mathbf Q$."}
+{"_id": "24746", "title": "Definition:Einstein Summation Convention", "text": "The '''Einstein summation convention''' is a notational device used in the manipulation of matrices and vectors, in particular square matrices in the context of physics and applied mathematics. If the same index occurs twice in a given expression involving matrices, then summation over that index is automatically assumed. Thus the summation sign can be omitted, and expressions can be written more compactly."}
+{"_id": "24747", "title": "Definition:Trace (Linear Algebra)/Matrix/Einstein Summation Convention", "text": "The trace of $A$, using the Einstein summation convention, is: :$\\map \\tr A = a_{ii}$"}
+{"_id": "24748", "title": "Definition:Determinant/Matrix/Order 3/Einstein Summation Convention", "text": "The determinant of a square matrix of order $3$ $\\mathbf A$ can be expressed using the Einstein summation convention as: :$\\map \\det {\\mathbf A} = \\dfrac 1 6 \\map \\sgn {i, j, k} \\map \\sgn {r, s, t} a_{i r} a_{j s} a_{k t}$ Note that there are $6$ indices which appear twice, and so $6$ summations are assumed."}
+{"_id": "24749", "title": "Definition:Matrix/Square Matrix/Order", "text": "Let $\\mathbf A$ be an $n \\times n$ square matrix. That is, let $\\mathbf A$ have $n$ rows (and by definition $n$ columns). Then the '''order''' of $\\mathbf A$ is defined as being $n$."}
+{"_id": "24750", "title": "Definition:Matrix Product (Conventional)/Einstein Summation Convention", "text": "The matrix product of $\\mathbf A$ and $\\mathbf B$ can be expressed using the Einstein summation convention as: Then: :$c_{i j} := a_{i k} \\circ b_{k j}$ The index which appears twice in the expressions on the {{RHS}} is the entry $k$, which is the one summated over."}
+{"_id": "24751", "title": "Definition:Dot Product/Definition 1", "text": "The '''dot product''' of $\\mathbf a$ and $\\mathbf b$ is defined as: :$\\ds \\mathbf a \\cdot \\mathbf b = a_1 b_1 + a_2 b_2 + \\cdots + a_n b_n = \\sum_{i \\mathop = 1}^n a_i b_i$ If the vectors are represented as column matrices: :$\\mathbf a = \\begin{bmatrix} a_1 \\\\ a_2 \\\\ \\vdots \\\\ a_n \\end{bmatrix} , \\mathbf b = \\begin{bmatrix} b_1 \\\\ b_2 \\\\ \\vdots \\\\ b_n \\end{bmatrix}$ we can express the dot product as: :$\\mathbf a \\cdot \\mathbf b = \\mathbf a^\\intercal \\mathbf b$ where: :$\\mathbf a^\\intercal = \\begin{bmatrix} a_1 & a_2 & \\cdots & a_n \\end{bmatrix}$ is the transpose of $\\mathbf a$ :the operation between the matrices is the matrix product."}
+{"_id": "24752", "title": "Definition:Dot Product/Definition 2", "text": "The '''dot product''' of $\\mathbf a$ and $\\mathbf b$ is defined as: :$\\mathbf a \\cdot \\mathbf b = \\norm {\\mathbf a} \\, \\norm {\\mathbf b} \\cos \\angle \\mathbf a, \\mathbf b$ where: :$\\norm {\\mathbf a}$ denotes the length of $\\mathbf a$ :$\\angle \\mathbf a, \\mathbf b$ is the angle between $\\mathbf a$ and $\\mathbf b$, taken to be between $0$ and $\\pi$."}
+{"_id": "24753", "title": "Definition:Left Module Axioms", "text": "{{begin-axiom}} {{axiom | n = \\text M 1         | lc= Scalar Multiplication (Left) Distributes over Module Addition         | q = \\forall \\lambda \\in R: \\forall x, y \\in G         | ml= \\lambda \\circ \\paren {x +_G y}         | mo= =         | mr= \\paren {\\lambda \\circ x} +_G \\paren {\\lambda \\circ y} }} {{axiom | n = \\text M 2         | lc= Scalar Multiplication (Right) Distributes over Scalar Addition         | q = \\forall \\lambda, \\mu \\in R: \\forall x \\in G         | ml= \\paren {\\lambda +_R \\mu} \\circ x         | mo= =         | mr= \\paren {\\lambda \\circ x} +_G \\paren {\\mu \\circ x} }} {{axiom | n = \\text M 3         | lc= Associativity of Scalar Multiplication         | q = \\forall \\lambda, \\mu \\in R: \\forall x \\in G         | ml= \\paren {\\lambda \\times_R \\mu} \\circ x         | mo= =         | mr= \\lambda \\circ \\paren {\\mu \\circ x} }} {{end-axiom}}"}
+{"_id": "24754", "title": "Definition:Unitary Module Axioms", "text": "{{begin-axiom}} {{axiom | n = \\text {UM} 1         | q = \\forall \\lambda \\in R: \\forall x, y \\in G         | m = \\lambda \\circ \\paren {x +_G y} = \\paren {\\lambda \\circ x} +_G \\paren {\\lambda \\circ y} }} {{axiom | n = \\text {UM} 2         | q = \\forall \\lambda, \\mu \\in R: \\forall x \\in G         | m = \\paren {\\lambda +_R \\mu} \\circ x = \\paren {\\lambda \\circ x} +_G \\paren {\\mu \\circ x} }} {{axiom | n = \\text {UM} 3         | q = \\forall \\lambda, \\mu \\in R: \\forall x \\in G         | m = \\paren {\\lambda \\times_R \\mu} \\circ x = \\lambda \\circ \\paren {\\mu \\circ x} }} {{axiom | n = \\text {UM} 4         | q = \\forall x \\in G         | m = 1_R \\circ x = x }} {{end-axiom}}"}
+{"_id": "24755", "title": "Definition:Simultaneous Equations/Solution", "text": "An ordered $n$-tuple $\\tuple {x_1, x_2, \\ldots, x_n}$ which satisfies each of the equations in a system of $m$ simultaneous equations in $n$ variables is called '''a solution''' of the system."}
+{"_id": "24756", "title": "Definition:Simultaneous Equations/Consistency", "text": "A system of simultaneous equations: :$\\forall i \\in \\set {1, 2, \\ldots m} : \\map {f_i} {x_1, x_2, \\ldots x_n} = \\beta_i$ that has at least one solution is '''consistent'''. If a system has no solutions, it is '''inconsistent'''."}
+{"_id": "24757", "title": "Definition:Echelon Matrix/Echelon Form/Non-Unity Variant", "text": "{{:Definition:Echelon Matrix/Echelon Form/Non-Unity Variant/Definition 1}}"}
+{"_id": "24758", "title": "Definition:Echelon Matrix/Reduced Echelon Form", "text": "The matrix $\\mathbf A$ is in '''reduced echelon form''' {{iff}}, in addition to being in echelon form, the leading $1$ in any non-zero row is the only non-zero element in the column in which that $1$ occurs. Such a matrix is called a '''reduced echelon matrix'''."}
+{"_id": "24759", "title": "Definition:Generalized Sum/Net Convergence", "text": "Let $\\left({g_n}\\right)_{n \\in \\N}$ be a sequence in $G$. The series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty g_n$ '''converges as a net''' or '''has net convergence''' {{iff}} the generalized sum $\\displaystyle \\sum \\left\\{{g_n: n \\in \\N}\\right\\}$ converges."}
+{"_id": "24760", "title": "Definition:Generalized Sum/Absolute Net Convergence", "text": "Let $V$ be a Banach space. Let $\\family {v_i}_{i \\mathop \\in I}$ be an indexed subset of $V$. Then $\\displaystyle \\sum \\set {v_i: i \\in I}$ '''converges absolutely''' {{iff}} $\\displaystyle \\sum \\set {\\norm {v_i}: i \\mathop \\in I}$ converges. This nomenclature is appropriate as we have Absolutely Convergent Generalized Sum Converges."}
+{"_id": "24761", "title": "Definition:Finite Cyclic Group", "text": "Let $\\struct {G, \\circ}$ be a cyclic group. Then $\\struct {G, \\circ}$ is a '''finite cyclic group''' {{iff}} it is a finite group."}
+{"_id": "24762", "title": "Definition:Sequential Continuity/Point", "text": "Let $x \\in X$. Then $f$ is '''sequentially continuous at $x$''' iff: :For every sequence $\\left\\langle{x_n}\\right\\rangle_{n \\ge 1}$ in $X$ which converges to $x$, the sequence $\\left\\langle{f \\left({x_n}\\right)}\\right\\rangle_{n \\ge 1}$ in $Y$ converges to $f \\left({x}\\right)$."}
+{"_id": "24764", "title": "Definition:Complex Number/Definition 1", "text": "A '''complex number''' is a number in the form $a + b i$ or $a + i b$ where: :$a$ and $b$ are real numbers :$i$ is a square root of $-1$, that is, $i = \\sqrt {-1}$."}
+{"_id": "24765", "title": "Definition:Complex Number/Definition 2", "text": "A '''complex number''' is an ordered pair $\\tuple {x, y}$ where $x, y \\in \\R$ are real numbers, on which the operations of addition and multiplication are defined as follows: === Complex Addition === {{:Definition:Complex Number/Definition 2/Addition}} === Complex Multiplication === {{:Definition:Complex Number/Definition 2/Multiplication}} === Scalar Product === {{:Definition:Complex Number/Definition 2/Scalar Product}}"}
+{"_id": "24766", "title": "Definition:Complex Number/Imaginary Unit", "text": "The entity $i := 0 + 1 i$ is known as the '''imaginary unit'''."}
+{"_id": "24767", "title": "Definition:Complex Number/Wholly Real", "text": "A complex number $z = a + i b$ is '''wholly real''' {{iff}} $b = 0$."}
+{"_id": "24768", "title": "Definition:Complex Number/Wholly Imaginary", "text": "A complex number $z = a + i b$ is '''wholly imaginary''' {{Iff}} $a = 0$."}
+{"_id": "24769", "title": "Definition:Complex Number/Complex Plane", "text": "Because a complex number can be expressed as an ordered pair, we can plot the number $x + i y$ on the real number plane $\\R^2$: :400px This representation is known as the '''complex plane'''. === Real Axis === {{:Definition:Complex Number/Complex Plane/Real Axis}} === Imaginary Axis === {{:Definition:Complex Number/Complex Plane/Imaginary Axis}}"}
+{"_id": "24770", "title": "Definition:Complex Number/Polar Form", "text": "For any complex number $z = x + i y \\ne 0$, let: {{begin-eqn}} {{eqn | l = r       | r = \\cmod z = \\sqrt {x^2 + y^2}       | c = the modulus of $z$, and }} {{eqn | l = \\theta       | r = \\arg z       | c = the argument of $z$ (the angle which $z$ yields with the real line) }} {{end-eqn}} where $x, y \\in \\R$. From the definition of $\\arg z$: :$(1): \\quad \\dfrac x r = \\cos \\theta$ :$(2): \\quad \\dfrac y r = \\sin \\theta$ which implies that: :$x = r \\cos \\theta$ :$y = r \\sin \\theta$ which in turn means that any number $z = x + i y \\ne 0$ can be written as: :$z = x + i y = r \\paren {\\cos \\theta + i \\sin \\theta}$ The pair $\\polar {r, \\theta}$ is called the '''polar form''' of the complex number $z \\ne 0$.  The number $z = 0 + 0 i$ is defined as $\\polar {0, 0}$."}
+{"_id": "24771", "title": "Definition:Complex Number/Complex Plane/Real Axis", "text": "Complex numbers of the form $\\tuple {x, 0}$, being wholly real, appear as points on the $x$-axis."}
+{"_id": "24772", "title": "Definition:Complex Number/Complex Plane/Imaginary Axis", "text": "Complex numbers of the form $\\tuple {0, y}$, being wholly imaginary, appear as points on the points on the $y$-axis. This line is known as the '''imaginary axis'''."}
+{"_id": "24773", "title": "Definition:Complex Number/Definition 2/Addition", "text": "Let $\\tuple {x_1, y_1}$ and $\\tuple {x_2, y_2}$ be complex numbers. Then $\\tuple {x_1, y_1} + \\tuple {x_2, y_2}$ is defined as: :$\\tuple {x_1, y_1} + \\tuple {x_2, y_2}:= \\tuple {x_1 + x_2, y_1 + y_2}$"}
+{"_id": "24774", "title": "Definition:Complex Number/Definition 2/Multiplication", "text": "Let $\\tuple {x_1, y_1}$ and $\\tuple {x_2, y_2}$ be complex numbers. Then $\\tuple {x_1, y_1} \\tuple {x_2, y_2}$ is defined as: :$\\tuple {x_1, y_1} \\tuple {x_2, y_2} := \\tuple {x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2}$"}
+{"_id": "24775", "title": "Definition:Argument of Complex Number/Principal Argument", "text": "Let $R$ be the principal range of the complex numbers $\\C$. The unique value of $\\theta$ in $R$ is known as the '''principal value of the argument''', or just '''principal argument''', of $z$. This is denoted $\\Arg z$. Note the capital $A$. The standard practice is for $R$ to be $\\hointl {-\\pi} \\pi$. This ensures that the '''principal argument''' is continuous on the real axis for positive numbers. Thus, if $z$ is represented in the complex plane, the '''principal argument''' $\\Arg z$ is intuitively defined as the angle which $z$ yields with the real ($y = 0$) axis."}
+{"_id": "24778", "title": "Definition:Pairwise Disjoint Events", "text": "Let $\\family {A_i}_{i \\mathop \\in I}$ be an indexed family of events in a probability space. Then $\\family {A_i}$ is '''pairwise disjoint''' {{iff}}: :$\\forall i, j \\in I, i \\ne j: A_i \\cap A_j = \\O$ That is, a set of events is '''pairwise disjoint''' {{iff}} every pair of them is disjoint."}
+{"_id": "24779", "title": "Definition:Complex Number/Polar Form/Exponential Form", "text": "From Euler's Formula: :$e^{i \\theta} = \\cos \\theta + i \\sin \\theta$ so $z$ can also be written in the form: :$z = r e^{i \\theta}$"}
+{"_id": "24780", "title": "Definition:Hyperbolic Cotangent", "text": "{{:Definition:Hyperbolic Cotangent/Definition 1}}"}
+{"_id": "24781", "title": "Definition:Hyperbolic Secant", "text": "{{:Definition:Hyperbolic Secant/Definition 1}}"}
+{"_id": "24782", "title": "Definition:Hyperbolic Cosecant", "text": "{{:Definition:Hyperbolic Cosecant/Definition 1}}"}
+{"_id": "24783", "title": "Definition:Hyperbolic Tangent/Definition 1", "text": "The '''hyperbolic tangent''' function is defined on the complex numbers as: :$\\tanh: X \\to \\C$: :$\\forall z \\in X: \\tanh z := \\dfrac {e^z - e^{-z} } {e^z + e^{-z} }$ where: : $X = \\set {z : z \\in \\C, \\ e^z + e^{-z} \\ne 0}$"}
+{"_id": "24784", "title": "Definition:Hyperbolic Tangent/Definition 2", "text": "The '''hyperbolic tangent''' function is defined on the complex numbers as: :$\\tanh: X \\to \\C$: :$\\forall z \\in X: \\tanh z := \\dfrac {\\sinh z} {\\cosh z}$ where: :$\\sinh$ is the hyperbolic sine :$\\cosh$ is the hyperbolic cosine :$X = \\set {z : z \\in \\C, \\ \\cosh z \\ne 0}$"}
+{"_id": "24785", "title": "Definition:Hyperbolic Tangent/Definition 3", "text": "The '''hyperbolic tangent''' function is defined on the complex numbers as: :$\\tanh: X \\to \\C$: :$\\forall z \\in X: \\tanh z := \\dfrac {e^{2 z} - 1} {e^{2 z} + 1}$ where: : $X = \\set {z: z \\in \\C, \\ e^{2 z} + 1 \\ne 0}$"}
+{"_id": "24786", "title": "Definition:Hyperbolic Cotangent/Definition 1", "text": "The '''hyperbolic cotangent''' function is defined on the complex numbers as: :$\\coth: X \\to \\C$: :$\\forall z \\in X: \\coth z := \\dfrac {e^z + e^{-z} } {e^z - e^{-z}}$ where: : $X = \\set {z : z \\in \\C, \\ e^z - e^{-z} \\ne 0}$"}
+{"_id": "24787", "title": "Definition:Hyperbolic Cotangent/Definition 2", "text": "The '''hyperbolic cotangent''' function is defined on the complex numbers as: :$\\coth: X \\to \\C$: :$\\forall z \\in X: \\coth z := \\dfrac {\\cosh z} {\\sinh z}$ where: :$\\sinh$ is the hyperbolic sine :$\\cosh$ is the hyperbolic cosine :$X = \\set {z : z \\in \\C, \\ \\sinh z \\ne 0}$"}
+{"_id": "24788", "title": "Definition:Hyperbolic Cotangent/Definition 3", "text": "The '''hyperbolic cotangent''' function is defined on the complex numbers as: :$\\coth: X \\to \\C$: :$\\forall z \\in X: \\coth z := \\dfrac {e^{2 z} + 1} {e^{2 z} - 1}$ where: :$X = \\set {z : z \\in \\C, \\ e^{2 z} - 1 \\ne 0}$"}
+{"_id": "24789", "title": "Definition:Hyperbolic Secant/Definition 1", "text": "The '''hyperbolic secant''' function is defined on the complex numbers as: :$\\sech: X \\to \\C$: :$\\forall z \\in X: \\sech z := \\dfrac 2 {e^z + e^{-z} }$ where: : $X = \\set {z: z \\in \\C, \\ e^z + e^{-z} \\ne 0}$"}
+{"_id": "24790", "title": "Definition:Hyperbolic Secant/Definition 2", "text": "The '''hyperbolic secant''' function is defined on the complex numbers as: :$\\sech: X \\to \\C$: :$\\forall z \\in X: \\sech z := \\dfrac 1 {\\cosh z}$ where: :$\\cosh$ is the hyperbolic cosine :$X = \\set {z: z \\in \\C, \\ \\cosh z \\ne 0}$"}
+{"_id": "24791", "title": "Definition:Hyperbolic Cosecant/Definition 1", "text": "The '''hyperbolic cosecant''' function is defined on the complex numbers as: :$\\csch: X \\to \\C$: :$\\forall z \\in X: \\csch z := \\dfrac 2 {e^z - e^{-z} }$ where: : $X = \\set {z: z \\in \\C, \\ e^z - e^{-z} \\ne 0}$"}
+{"_id": "24792", "title": "Definition:Hyperbolic Cosecant/Definition 2", "text": "The '''hyperbolic cosecant''' function is defined on the complex numbers as: :$\\csch: X \\to \\C$: :$\\forall z \\in X: \\csch z := \\dfrac 1 {\\sinh z}$ where: :$\\sinh$ is the hyperbolic sine :$X = \\set {z: z \\in \\C, \\ \\sinh z \\ne 0}$"}
+{"_id": "24793", "title": "Definition:Inverse Hyperbolic Function", "text": "Let $f: \\C \\to \\C$ be one of the hyperbolic functions on the set of complex numbers. The '''inverse hyperbolic function''' $f^{-1} \\subseteq \\C \\times \\C$ is actually a multifunction, as in general for a given $y \\in \\C$ there is more than one $x \\in \\C$ such that $y = f \\left({x}\\right)$. As with the inverse trigonometric functions, it is usual to restrict the codomain of the multifunction so as to allow $f^{-1}$ to be single-valued."}
+{"_id": "24794", "title": "Definition:Inverse Hyperbolic Sine/Real", "text": "=== Definition 1 === {{:Definition:Inverse Hyperbolic Sine/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Sine/Real/Definition 2}}"}
+{"_id": "24795", "title": "Definition:Inverse Hyperbolic Sine/Complex/Definition 1", "text": "The '''inverse hyperbolic sine''' is a multifunction defined as: :$\\forall z \\in \\C: \\sinh^{-1} \\left({z}\\right) := \\left\\{{w \\in \\C: z = \\sinh \\left({w}\\right)}\\right\\}$ where $\\sinh \\left({w}\\right)$ is the hyperbolic sine function."}
+{"_id": "24796", "title": "Definition:Inverse Hyperbolic Sine/Complex/Definition 2", "text": "The '''inverse hyperbolic sine''' is a multifunction defined as: :$\\forall z \\in \\C: \\sinh^{-1} \\left({z}\\right) := \\left\\{{\\ln \\left({z + \\sqrt{\\left|{z^2 + 1}\\right|} e^{\\left({i /  2}\\right) \\arg \\left({z^2 + 1}\\right)} }\\right) + 2 k \\pi i: k \\in \\Z}\\right\\}$ where: : $\\sqrt{\\left|{z^2 + 1}\\right|}$ denotes the positive square root of the complex modulus of $z^2 + 1$ : $\\arg \\left({z^2 + 1}\\right)$ denotes the argument of $z^2 + 1$ : $\\ln$ denotes the complex natural logarithm considered as a  multifunction."}
+{"_id": "24797", "title": "Definition:Inverse Hyperbolic Cosine/Complex", "text": "=== Definition 1 === {{:Definition:Inverse Hyperbolic Cosine/Complex/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cosine/Complex/Definition 2}}"}
+{"_id": "24798", "title": "Definition:Inverse Hyperbolic Tangent/Complex", "text": "Let $S$ be the subset of the complex plane: :$S = \\C \\setminus \\left\\{{-1 + 0 i, 1 + 0 i}\\right\\}$ === Definition 1 === {{:Definition:Inverse Hyperbolic Tangent/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Tangent/Definition 2}}"}
+{"_id": "24799", "title": "Definition:Inverse Hyperbolic Cotangent/Complex", "text": "Let $S$ be the subset of the complex plane: :$S = \\C \\setminus \\left\\{{-1 + 0 i, 1 + 0 i}\\right\\}$ === Definition 1 === {{:Definition:Inverse Hyperbolic Cotangent/Complex/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cotangent/Complex/Definition 2}}"}
+{"_id": "24800", "title": "Definition:Inverse Hyperbolic Secant/Complex", "text": "=== Definition 1 === {{:Definition:Inverse Hyperbolic Secant/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Secant/Definition 2}}"}
+{"_id": "24801", "title": "Definition:Inverse Hyperbolic Cosecant/Complex", "text": "=== Definition 1 === {{:Definition:Inverse Hyperbolic Cosecant/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cosecant/Definition 2}}"}
+{"_id": "24802", "title": "Definition:Inverse Hyperbolic Cosine/Complex/Definition 1", "text": "The '''inverse hyperbolic cosine''' is a multifunction defined as: :$\\forall z \\in \\C: \\cosh^{-1} \\left({z}\\right) := \\left\\{{w \\in \\C: z = \\cosh \\left({w}\\right)}\\right\\}$ where $\\cosh \\left({w}\\right)$ is the hyperbolic cosine function."}
+{"_id": "24803", "title": "Definition:Inverse Hyperbolic Tangent/Complex/Definition 1", "text": "The '''inverse hyperbolic tangent''' is a multifunction defined on $S$ as: :$\\forall z \\in S: \\tanh^{-1} \\paren z := \\set {w \\in \\C: z = \\tanh \\paren w}$ where $\\tanh \\paren w$ is the hyperbolic tangent function."}
+{"_id": "24804", "title": "Definition:Inverse Hyperbolic Cotangent/Complex/Definition 1", "text": "The '''inverse hyperbolic cotangent''' is a multifunction defined on $S$ as: :$\\forall z \\in S: \\coth^{-1} \\left({z}\\right) := \\left\\{{w \\in \\C: z = \\coth \\left({w}\\right)}\\right\\}$ where $\\coth \\left({w}\\right)$ is the hyperbolic cotangent function."}
+{"_id": "24805", "title": "Definition:Inverse Hyperbolic Secant/Complex/Definition 1", "text": "The '''inverse hyperbolic secant''' is a multifunction defined as: :$\\forall z \\in \\C_{\\ne 0}: \\operatorname{sech}^{-1} \\left({z}\\right) := \\left\\{{w \\in \\C: z = \\operatorname{sech} \\left({w}\\right)}\\right\\}$ where $\\operatorname{sech} \\left({w}\\right)$ is the hyperbolic secant function."}
+{"_id": "24806", "title": "Definition:Inverse Hyperbolic Cosecant/Complex/Definition 1", "text": "The '''inverse hyperbolic cosecant''' is a multifunction defined as: :$\\forall z \\in \\C_{\\ne 0}: \\map {\\csch^{-1} } z := \\set {w \\in \\C: z = \\map \\csch w}$ where $\\map \\csch w$ is the hyperbolic cosecant function."}
+{"_id": "24807", "title": "Definition:Inverse Hyperbolic Cosine/Complex/Definition 2", "text": "The '''inverse hyperbolic cosine''' is a multifunction defined as: :$\\forall z \\in \\C: \\cosh^{-1} \\left({z}\\right) := \\left\\{{\\ln \\left({z + \\sqrt{\\left|{z^2 - 1}\\right|} e^{\\left({i / 2}\\right) \\arg \\left({z^2 - 1}\\right)} }\\right) + 2 k \\pi i: k \\in \\Z}\\right\\}$ where: : $\\sqrt{\\left|{z^2 - 1}\\right|}$ denotes the positive square root of the complex modulus of $z^2 - 1$ : $\\arg \\left({z^2 - 1}\\right)$ denotes the argument of $z^2 - 1$ : $\\ln$ denotes the complex natural logarithm considered as a  multifunction."}
+{"_id": "24808", "title": "Definition:Inverse Hyperbolic Tangent/Complex/Definition 2", "text": "The '''inverse hyperbolic tangent''' is a multifunction defined on $S$ as: :$\\forall z \\in S: \\tanh^{-1} \\left({z}\\right) := \\left\\{{\\dfrac 1 2 \\ln \\left({\\dfrac {1 + z} {1 - z} }\\right) + k \\pi i: k \\in \\Z}\\right\\}$ where $\\ln$ denotes the complex natural logarithm considered as a  multifunction."}
+{"_id": "24809", "title": "Definition:Inverse Hyperbolic Cotangent/Complex/Definition 2", "text": "The '''inverse hyperbolic cotangent''' is a multifunction defined  on $S$ as: :$\\forall z \\in S: \\coth^{-1} \\left({z}\\right) := \\left\\{{\\dfrac 1 2 \\ln \\left({\\dfrac {z + 1} {z - 1} }\\right) + k \\pi i: k \\in \\Z}\\right\\}$ where $\\ln$ denotes the complex natural logarithm considered as a  multifunction."}
+{"_id": "24810", "title": "Definition:Inverse Hyperbolic Secant/Complex/Definition 2", "text": "The '''inverse hyperbolic secant''' is a multifunction defined as: :$\\forall z \\in \\C_{\\ne 0}: \\map {\\sech^{-1} } z := \\set {\\map \\ln {\\dfrac {1 + \\sqrt {\\size {1 - z^2} } e^{\\paren {i / 2} \\map \\arg {1 - z^2} } } z} + 2 k \\pi i: k \\in \\Z}$ where: :$\\sqrt {\\size {1 - z^2} }$ denotes the positive square root of the complex modulus of $1 - z^2$ :$\\map \\arg {1 - z^2}$ denotes the argument of $1 - z^2$ :$\\ln$ denotes the complex natural logarithm as a multifunction."}
+{"_id": "24811", "title": "Definition:Inverse Hyperbolic Cosecant/Complex/Definition 2", "text": "The '''inverse hyperbolic cosecant''' is a multifunction defined as: :$\\forall z \\in \\C_{\\ne 0}: \\map {\\csch^{-1} } z := \\set {\\map \\ln {\\dfrac {1 + \\sqrt {\\size {z^2 + 1} } e^{\\paren {i / 2} \\, \\map \\arg {z^2 + 1} } } z} + 2 k \\pi i: k \\in \\Z}$ where: :$\\sqrt {\\size {z^2 + 1} }$ denotes the positive square root of the complex modulus of $z^2 + 1$ :$\\map \\arg {z^2 + 1}$ denotes the argument of $z^2 + 1$ :$\\ln$ denotes the complex natural logarithm considered as a multifunction."}
+{"_id": "24812", "title": "Definition:Periodic Function/Period", "text": "The '''period''' of $f$ is the smallest value $\\cmod L \\in \\R_{\\ne 0}$ such that: : $\\forall x \\in X: \\map f x = \\map f {x + L}$ where $\\cmod L$ is the modulus of $L$."}
+{"_id": "24813", "title": "Definition:Periodic Function/Real", "text": "Let $f: \\R \\to \\R$ be a real function. Then $f$ is '''periodic''' {{iff}}: :$\\exists L \\in \\R_{\\ne 0}: \\forall x \\in \\R: \\map f x = \\map f {x + L}$"}
+{"_id": "24814", "title": "Definition:Periodic Function/Complex", "text": "Let $f: \\C \\to \\C$ be a complex function. Then $f$ is '''periodic''' {{iff}}: :$\\exists L \\in \\C_{\\ne 0}: \\forall x \\in \\C: f \\left({x}\\right) = f \\left({x + L}\\right)$"}
+{"_id": "24815", "title": "Definition:Inverse Sine", "text": "=== Real Numbers === {{:Definition:Inverse Sine/Real}} === Complex Plane === {{:Definition:Inverse Sine/Complex}}"}
+{"_id": "24816", "title": "Definition:Inverse Cosine", "text": "=== Real Numbers === {{:Definition:Inverse Cosine/Real}} === Complex Plane === {{:Definition:Inverse Cosine/Complex}}"}
+{"_id": "24817", "title": "Definition:Inverse Tangent", "text": "=== Real Numbers === {{:Definition:Inverse Tangent/Real}} === Complex Plane === {{:Definition:Inverse Tangent/Complex}}"}
+{"_id": "24818", "title": "Definition:Inverse Cosecant", "text": "=== Real Numbers === {{:Definition:Inverse Cosecant/Real}} === Complex Plane === {{:Definition:Inverse Cosecant/Complex}}"}
+{"_id": "24819", "title": "Definition:Inverse Secant", "text": "=== Real Numbers === {{:Definition:Inverse Secant/Real}} === Complex Plane === {{:Definition:Inverse Secant/Complex}}"}
+{"_id": "24820", "title": "Definition:Inverse Cotangent", "text": "=== Real Numbers === {{:Definition:Inverse Cotangent/Real}} === Complex Plane === {{:Definition:Inverse Cotangent/Complex}}"}
+{"_id": "24821", "title": "Definition:Quadratic Equation", "text": "A '''quadratic equation''' is a polynomial equation of the form: :$a x^2 + b x + c = 0$ such that $a \\ne 0$."}
+{"_id": "24822", "title": "Definition:Cubic Equation/Resolvent", "text": "Let: : $y = x + \\dfrac b {3 a}$ : $Q = \\dfrac {3 a c - b^2} {9 a^2}$ : $R = \\dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$ Let $y = u + v$ where $u v = -Q$. The '''resolvent equation''' of the cubic is given by: : $u^6 - 2 R u^3 - Q^3$"}
+{"_id": "24823", "title": "Definition:Division/Notation", "text": "The operation of division can be denoted as: :$a / b$, which is probably the most common in the general informal context :$\\dfrac a b$, which is the preferred style on {{ProofWiki}} :$a \\div b$, which is rarely seen outside grade school."}
+{"_id": "24824", "title": "Definition:Division/Rational Numbers", "text": "Let $\\struct {\\Q, +, \\times}$ be the field of rational numbers. The operation of '''division''' is defined on $\\Q$ as: :$\\forall a, b \\in \\Q \\setminus \\set 0: a / b := a \\times b^{-1}$ where $b^{-1}$ is the multiplicative inverse of $b$ in $\\Q$."}
+{"_id": "24825", "title": "Definition:Division/Real Numbers", "text": "Let $\\struct {\\R, +, \\times}$ be the field of real numbers. The operation of '''division''' is defined on $\\R$ as: :$\\forall a, b \\in \\R \\setminus \\set 0: a / b := a \\times b^{-1}$ where $b^{-1}$ is the multiplicative inverse of $b$ in $\\R$."}
+{"_id": "24826", "title": "Definition:Division/Complex Numbers", "text": "Let $\\struct {\\C, +, \\times}$ be the field of complex numbers. The operation of '''division''' is defined on $\\C$ as: :$\\forall a, b \\in \\C \\setminus \\set 0: \\dfrac a b := a \\times b^{-1}$ where $b^{-1}$ is the multiplicative inverse of $b$ in $\\C$."}
+{"_id": "24827", "title": "Definition:Division/Divisor", "text": "Let $c = a / b$ denote the division operation on two elements $a$ and $b$ of a field. The element $b$ is the '''divisor''' of $a$."}
+{"_id": "24828", "title": "Definition:Division/Dividend", "text": "Let $c = a / b$ denote the division operation on two elements $a$ and $b$ of a field. The element $a$ is the '''dividend''' of $b$."}
+{"_id": "24829", "title": "Definition:Division/Quotient", "text": "Let $c = a / b$ denote the division operation on two elements $a$ and $b$ of a field. The element $c$ is the '''quotient of $a$ (divided) by $b$'''."}
+{"_id": "24830", "title": "Definition:Division", "text": "Let $\\struct {F, +, \\times}$ be a field. Let the zero of $F$ be $0_F$. The operation of '''division''' is defined as: :$\\forall a, b \\in F \\setminus \\set {0_F}: a / b := a \\times b^{-1}$ where $b^{-1}$ is the multiplicative inverse of $b$."}
+{"_id": "24832", "title": "Definition:Irrational Number/Approximation", "text": "From its definition, it is not possible to express an irrational number precisely in terms of a fraction. From Decimal Expansion of Irrational Number neither Terminates nor Recurs, it is not possible to express it precisely by a decimal expansion either. However, it is possible to express it to an arbitrary level of precision. Let $x$ be an irrational number whose decimal expansion is $\\sqbrk {n.d_1 d_2 d_3 \\ldots}_{10}$. Then: :$\\displaystyle n + \\sum_{j \\mathop = 1}^k \\frac {d_j} {10^j} \\le x < n + \\sum_{j \\mathop = 1}^k \\frac {d_j} {10^j} + \\frac 1 {10^k}$ for all $k \\in \\Z: k \\ge 1$. Then all one needs to do is state that $x$ is expressed as '''accurate to $k$ decimal places'''."}
+{"_id": "24833", "title": "Definition:Rational Number/Fraction", "text": "By definition, a rational number is a number which can be expressed in the form: :$\\dfrac a b$ where $a$ and $b$ are integers. A '''fraction''' is a rational number such that, when expressed in canonical form $\\dfrac a b$ (that is, such that $a$ and $b$ are coprime), the denominator $b$ is not $1$."}
+{"_id": "24836", "title": "Definition:Subtraction/Complex Numbers", "text": "Let $\\struct {\\C, +, \\times}$ be the field of complex numbers. The operation of '''subtraction''' is defined on $\\C$ as: :$\\forall a, b \\in \\C: a - b := a + \\paren {-b}$ where $-b$ is the negative of $b$ in $\\C$."}
+{"_id": "24837", "title": "Definition:Negative/Complex Number", "text": "As the Complex Numbers cannot be Ordered Compatibly with Ring Structure, the concept of a '''negative complex number''', relative to a specified zero, is not defined. However, the '''negative''' of a complex number is defined as follows: Let $z = a + i b$ be a complex number. Then the '''negative of $z$''' is defined as: :$-z = -a - i b$"}
+{"_id": "24838", "title": "Definition:Cartesian Product/Coordinate", "text": "Let $\\displaystyle \\prod_{i \\mathop \\in I} S_i$ be a cartesian product. Let $j \\in I$, and let $s = \\sequence {s_i}_{i \\mathop \\in I} \\in \\displaystyle \\prod_{i \\mathop \\in I} S_i$. Then $s_j$ is called the '''$j$th coordinate of $s$'''. If the indexing set $I$ consists of ordinary numbers $1, 2, \\ldots, n$, one speaks about, for example, the '''first''', '''second''', or '''$n$th coordinate'''. For an element $\\tuple {s, t} \\in S \\times T$ of a binary cartesian product, $s$ is the '''first coordinate''', and $t$ is the '''second coordinate'''."}
+{"_id": "24839", "title": "Definition:Distance/Points/Complex Numbers", "text": "Let $x, y \\in \\C$ be complex numbers. Let $\\cmod {x - y}$ be the complex modulus of $x - y$. Then the function $d: \\C^2 \\to \\R$: :$\\map d {x, y} = \\cmod {x - y}$ is called the '''distance between $x$ and $y$'''."}
+{"_id": "24840", "title": "Definition:Argument of Complex Number/Principal Range", "text": "It is understood that the argument of a complex number $z$ is unique only up to multiples of $2 k \\pi$. With this understanding, we can limit the choice of what $\\theta$ can be for any given $z$ by requiring that $\\theta$ lie in some half open interval of length $2 \\pi$. The most usual of these are: :$\\hointr 0 {2 \\pi}$ :$\\hointl {-\\pi} \\pi$ but in theory any such interval may be used. This interval is known as the '''principal range'''."}
+{"_id": "24841", "title": "Definition:Substitution (Formal Systems)/Well-Formed Part", "text": "Let $\\FF$ be a formal language with alphabet $\\AA$. Let $\\mathbf B$ be a well-formed formula of $\\FF$. Let $\\mathbf A$ be a well-formed part of $\\mathbf B$. Let $\\mathbf A'$ be another well-formed formula. Then the '''substitution of $\\mathbf A'$ for $\\mathbf A$ in $\\mathbf B$''' is the collation resulting from $\\mathbf B$ by replacing all occurrences of $\\mathbf A$ in $\\mathbf B$ by $\\mathbf A'$. It is denoted as $\\map {\\mathbf B} {\\mathbf A' \\mathbin {//} \\mathbf A}$."}
+{"_id": "24843", "title": "Definition:Complex Number as Vector", "text": "Let $z = x + i y$ be a complex number. Then $z$ can be considered as a vector $OP$ in the complex plane such that: : its initial point is the origin : its terminal point $P$ is the point $\\tuple {x, y}$. Two vectors which have the same magnitude and direction, but different initial points, are considered '''equal'''."}
+{"_id": "24844", "title": "Definition:Spherical Representation of Complex Number", "text": "Let $\\PP$ be the complex plane. Let $\\mathbb S$ be the unit sphere which is tangent to $\\PP$ at $\\tuple {0, 0}$ (that is, where $z = 0$). Let the diameter of $\\mathbb S$ perpendicular to $\\PP$ through $\\tuple {0, 0}$ be $NS$ where $S$ is the point $\\tuple {0, 0}$. Let the point $N$ be referred to as the '''north pole''' of $\\mathbb S$ and $S$ be referred to as the '''south pole''' of $\\mathbb S$. Let $A$ be a point on $P$. Let the line $NA$ be constructed. :900px Then $NA$ passes through a point of $\\mathbb S$. Thus any complex number can be represented by a point on the surface of the unit sphere. The point $N$ on $\\mathbb S$ corresponds to the point at infinity. Thus any point on the surface of the unit sphere corresponds to a point on the extended complex plane."}
+{"_id": "24846", "title": "Definition:Complex Point at Infinity", "text": "The zero in the set of complex numbers $\\C$ has no inverse for multiplication. That is, the expression: :$\\dfrac 1 0$ has no meaning. The '''point at infinity''' is the element added to $\\C$ in order to allow $\\C$ to be closed under division: :$\\forall x, y \\in \\C: \\dfrac x y \\in \\C$ The set $\\C$ with that point added is known as the extended complex plane. Conceptually, it can be imagined as a point which is '''''at infinity''''' in '''''all''''' directions. It can also be considered as the $N$ point on the Riemann sphere which does not map to the complex plane. This point can be denoted $\\infty$."}
+{"_id": "24847", "title": "Definition:Definable Truth Function", "text": "Let $f: \\Bbb B^n \\to \\Bbb B$ be a truth function. Let $S$ be a set of truth functions. Then $f$ is '''definable from $S$''' iff there exist: :a truth function $g: \\Bbb B^m \\to \\Bbb B$, obtained by composition of truth functions from $S$ :an injection $i: \\Bbb B^n \\to \\Bbb B^m$ such that: :$f = g \\circ i$ {{expand|example}}"}
+{"_id": "24848", "title": "Definition:Satisfiable/Boolean Interpretations", "text": "Let $\\mathbf A$ be a WFF of propositional logic. $\\mathbf A$ is called '''satisfiable (for boolean interpretations)''' iff: :$v \\left({\\mathbf A}\\right) = T$ for some boolean interpretation $v$ for $\\mathbf A$. In terms of validity, this can be rendered: :$v \\models_{\\mathrm{BI}} \\mathbf A$ that is, $\\mathbf A$ is valid in the boolean interpretation $v$ of $\\mathbf A$."}
+{"_id": "24849", "title": "Definition:Falsifiable/Boolean Interpretations", "text": "Let $\\mathbf A$ be a WFF of propositional logic. $\\mathbf A$ is called '''falsifiable (for boolean interpretations)''' iff: :$v \\left({\\mathbf A}\\right) = F$ for some boolean interpretation $v$ for $\\mathbf A$. In terms of validity, this can be rendered: :$v \\not\\models_{\\mathrm{BI}} \\mathbf A$ that is, $\\mathbf A$ is invalid in the boolean interpretation $v$ of $\\mathbf A$."}
+{"_id": "24850", "title": "Definition:Extended Real Number Line/Definition 1", "text": "The '''extended real number line''' $\\overline \\R$ is defined as: :$\\overline \\R := \\R \\cup \\set {+\\infty, -\\infty}$ that is, the set of real numbers together with two auxiliary symbols: :$+\\infty$, '''positive infinity''' :$-\\infty$, '''negative infinity''' such that: :$\\forall x \\in \\R: x < +\\infty$ :$\\forall x \\in \\R: -\\infty < x$"}
+{"_id": "24851", "title": "Definition:Extended Real Number Line/Definition 2", "text": "The '''extended real number line''' $\\overline \\R$ is the order completion of the set of real numbers $\\R$. The greatest element of $\\overline \\R$ is often denoted by $+\\infty$ and its least element by $-\\infty$."}
+{"_id": "24852", "title": "Definition:Extended Complex Plane", "text": "The '''extended complex plane''' $\\overline \\C$ is defined as: :$\\overline \\C := \\C \\cup \\set \\infty$ that is, the set of complex numbers together with the point at infinity."}
+{"_id": "24853", "title": "Definition:Stereographic Projection", "text": "Let $\\PP$ be a the plane. Let $\\mathbb S$ be a sphere which is tangent to $\\PP$ at the origin $\\tuple {0, 0}$. Let the diameter of $\\mathbb S$ perpendicular to $\\PP$ through $\\tuple {0, 0}$ be $NS$ where $S$ is the point $\\tuple {0, 0}$. Let the point $N$ be referred to as the '''north pole''' of $\\mathbb S$ and $S$ be referred to as the '''south pole''' of $\\mathbb S$. Let $A$ be a point on $P$. Let the line $NA$ be constructed. :900px Then $NA$ passes through a point of $\\mathbb S$. Thus any point on $P$ can be represented by a point on $\\mathbb S$. With this construction, the point $N$ on $\\mathbb S$ maps to no point on $\\mathbb S$."}
+{"_id": "24854", "title": "Definition:Dot Product/Complex/Definition 2", "text": "The '''dot product''' of $z_1$ and $z_2$ is defined as: :$z_1 \\circ z_2 = \\cmod {z_1} \\, \\cmod{z_2} \\cos \\theta$ where: :$\\cmod {z_1}$ denotes the complex modulus of $z_1$ :$\\theta$ denotes the angle between $z_1$ and $z_2$."}
+{"_id": "24855", "title": "Definition:Dot Product/Complex/Definition 1", "text": "The '''dot product''' of $z_1$ and $z_2$ is defined as: :$z_1 \\circ z_2 = x_1 x_2 + y_1 y_2$"}
+{"_id": "24856", "title": "Definition:Dot Product/Complex", "text": "Let $z_1 := x_1 + i y_1$ and $z_2 := x_2 + i y_2$ be complex numbers. === Definition 1 === {{:Definition:Dot Product/Complex/Definition 1}} === Definition 2 === {{:Definition:Dot Product/Complex/Definition 2}} === Definition 3 === {{:Definition:Dot Product/Complex/Definition 3}} === Definition 4 === {{:Definition:Dot Product/Complex/Definition 4}}"}
+{"_id": "24857", "title": "Definition:Dot Product/Complex/Definition 3", "text": "The '''dot product''' of $z_1$ and $z_2$ is defined as: :$z_1 \\circ z_2 := \\map \\Re {\\overline {z_1} z_2}$ where: :$\\map \\Re z$ denotes the real part of a complex number $z$ :$\\overline {z_1}$ denotes the complex conjugate of $z_1$ :$\\overline {z_1} z_2$ denotes complex multiplication."}
+{"_id": "24858", "title": "Definition:Dot Product/Complex/Definition 4", "text": "The '''dot product''' of $z_1$ and $z_2$ is defined as: :$z_1 \\circ z_2 := \\dfrac {\\overline {z_1} z_2 + z_1 \\overline {z_2} } 2$ where: :$\\overline {z_1}$ denotes the complex conjugate of $z_1$ :$\\overline {z_1} z_2$ denotes complex multiplication."}
+{"_id": "24859", "title": "Definition:Vector Cross Product/Definition 1", "text": "The '''vector cross product''', denoted $\\mathbf a \\times \\mathbf b$, is defined as: :$\\mathbf a \\times \\mathbf b = \\begin{vmatrix} \\mathbf i & \\mathbf j & \\mathbf k\\\\ a_i & a_j & a_k \\\\ b_i & b_j & b_k \\\\ \\end{vmatrix}$ where $\\begin{vmatrix} \\ldots \\end{vmatrix}$ is interpreted as a determinant. More directly: : $\\mathbf a \\times \\mathbf b = \\mathbf i \\paren {a_j b_k - a_k b_j} - \\mathbf j \\paren {a_i b_k - a_k b_i} + \\mathbf k \\paren {a_i b_j - a_j b_i}$"}
+{"_id": "24860", "title": "Definition:Vector Cross Product/Definition 2", "text": "The '''vector cross product''', denoted $\\mathbf a \\times \\mathbf b$, is defined as: :$\\mathbf a \\times \\mathbf b = \\norm {\\mathbf a} \\, \\norm {\\mathbf b} \\sin \\theta \\hat {\\mathbf n}$ where: :$\\norm {\\mathbf a}$ denotes the length of $\\mathbf a$ :$\\theta$ denotes the angle from $\\mathbf a$ to $\\mathbf b$, measured in the positive direction :$\\hat {\\mathbf n}$ is the unit vector perpendicular to both $\\mathbf a$ and $\\mathbf b$ in the direction according to the right hand rule."}
+{"_id": "24861", "title": "Definition:Orthogonal (Linear Algebra)/Set", "text": "Let $S = \\set {u_1, \\ldots, u_n}$ be a subset of $V$. Then $S$ is an '''orthogonal set''' {{iff}} its elements are pairwise orthogonal: :$\\forall i \\ne j: \\innerprod {u_i}, {u_j} = 0$"}
+{"_id": "24862", "title": "Definition:Vector Cross Product/Complex", "text": "Let $z_1 := x_1 + i y_1$ and $z_2 := x_2 + i y_2$ be complex numbers. === Definition 1 === {{:Definition:Vector Cross Product/Complex/Definition 1}} === Definition 2 === {{:Definition:Vector Cross Product/Complex/Definition 2}} === Definition 3 === {{:Definition:Vector Cross Product/Complex/Definition 3}} === Definition 4 === {{:Definition:Vector Cross Product/Complex/Definition 4}}"}
+{"_id": "24863", "title": "Definition:Vector Cross Product/Complex/Definition 2", "text": "The '''cross product''' of $z_1$ and $z_2$ is defined as: :$z_1 \\times z_2 = \\cmod {z_1} \\, \\cmod {z_2} \\sin \\theta$ where: :$\\cmod {z_1}$ denotes the complex modulus of $z_1$ :$\\theta$ denotes the angle from $z_1$ to $z_2$, measured in the positive direction."}
+{"_id": "24864", "title": "Definition:Vector Cross Product/Complex/Definition 1", "text": "The '''cross product''' of $z_1$ and $z_2$ is defined as: :$z_1 \\times z_2 = x_1 y_2 - y_1 x_2$"}
+{"_id": "24865", "title": "Definition:Vector Cross Product/Complex/Definition 3", "text": "The '''cross product''' of $z_1$ and $z_2$ is defined as: :$z_1 \\times z_2 := \\map \\Im {\\overline {z_1} z_2}$ where: :$\\map \\Im z$ denotes the imaginary part of a complex number $z$ :$\\overline {z_1}$ denotes the complex conjugate of $z_1$ :$\\overline {z_1} z_2$ denotes complex multiplication."}
+{"_id": "24866", "title": "Definition:Vector Cross Product/Complex/Definition 4", "text": "The '''cross product''' of $z_1$ and $z_2$ is defined as: :$z_1 \\times z_2 := \\dfrac {\\overline {z_1} z_2 - z_1 \\overline {z_2}} {2 i}$ where: :$\\overline {z_1}$ denotes the complex conjugate of $z_1$ :$\\overline {z_1} z_2$ denotes complex multiplication."}
+{"_id": "24868", "title": "Definition:Bound of Real-Valued Function", "text": "Let $S$ be a set. Let $f: S \\to \\R$ be a real-valued function. Let $f$ be bounded. Then $B$ is a '''bound''' for $f$ {{iff}}: :$\\forall x \\in S: B \\ge \\size {\\map f x}$"}
+{"_id": "24869", "title": "Definition:Bounded Ordered Set/Unbounded", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. A subset $T \\subseteq S$ is '''unbounded (in $S$)''' {{iff}} it is not bounded."}
+{"_id": "24870", "title": "Definition:Bounded Above Set/Unbounded", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. A subset $T \\subseteq S$ is '''unbounded above (in $S$)''' {{iff}} it is not bounded above."}
+{"_id": "24873", "title": "Definition:Bounded Below Set/Unbounded", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. A subset $T \\subseteq S$ is '''unbounded below (in $S$)''' {{iff}} it is not bounded below."}
+{"_id": "24874", "title": "Definition:Bounded Mapping/Unbounded", "text": "$f$ is '''unbounded''' {{iff}} it is neither bounded above or bounded below."}
+{"_id": "24875", "title": "Definition:Bounded Sequence/Unbounded", "text": "A sequence which is not bounded is '''unbounded'''."}
+{"_id": "24878", "title": "Definition:Decision Procedure/Refutation Procedure", "text": "Given a decision procedure for satisfiability, one can craft a decision procedure for tautologies in the following way: Suppose one wanted to decide if a propositional formula $\\mathbf A$ is a tautology. Then apply the given procedure to decide if its negation $\\neg \\mathbf A$ is satisfiable. Now: :If $\\neg \\mathbf A$ is not satisfiable, then by Tautology iff Negation is Unsatisfiable, $\\mathbf A$ is a tautology. :If $\\neg \\mathbf A$ is satisfiable, then by Satisfiable iff Negation is Falsifiable, $\\mathbf A$ is falsifiable, so cannot be a tautology. Hence we have crafted a decision procedure for tautologies. Such a procedure is called a '''refutation procedure''', because it proceeds by refuting, i.e. proving unsatisfiability of the negation of the formula at hand."}
+{"_id": "24880", "title": "Definition:Bounded Below Mapping/Unbounded", "text": "Let $f: S \\to T$ be a mapping whose codomain is an ordered set $\\struct {T, \\preceq}$. Then $f$ is '''unbounded below (in $T \\ $)''' {{iff}} there exists no $L \\in S$ such that: :$\\forall x \\in S: L \\preceq \\map f x$"}
+{"_id": "24881", "title": "Definition:Upper Bound of Sequence", "text": "A special case of an upper bound of a mapping is an '''upper bound of a sequence''', where the domain of the mapping is $\\N$. Let $\\struct {T, \\preceq}$ be an ordered set. Let $\\sequence {x_n}$ be a sequence in $T$. Let $\\sequence {x_n}$ be bounded above in $T$ by $H \\in T$. Then $H$ is an '''upper bound of $\\sequence {x_n}$'''."}
+{"_id": "24882", "title": "Definition:Lower Bound of Sequence", "text": "A special case of a lower bound of a mapping is a '''lower bound of a sequence''', where the domain of the mapping is $\\N$. Let $\\struct {T, \\preceq}$ be an ordered set. Let $\\sequence {x_n}$ be a sequence in $T$. Let $\\sequence {x_n}$ be bounded below in $T$ by $L \\in T$. Then $L$ is a '''lower bound of $\\sequence {x_n}$'''."}
+{"_id": "24883", "title": "Definition:Upper Bound of Number", "text": "When considering the upper bound of a set of numbers, it is commonplace to ignore the set and instead refer just to the number itself. Thus the construction: :''The set of numbers which fulfil the propositional function $P \\left({n}\\right)$ is bounded above with the upper bound $N$'' would be reported as: :''The number $n$ such that $P \\left({n}\\right)$ has the upper bound $N$''. This construct obscures the details of what is actually being stated. Its use on {{ProofWiki}} is considered an abuse of notation and so discouraged. This also applies in the case where it is the upper bound of a mapping which is under discussion."}
+{"_id": "24885", "title": "Definition:Infimum of Mapping/Real-Valued Function", "text": "{{:Definition:Infimum of Mapping/Real-Valued Function/Definition 1}}"}
+{"_id": "24886", "title": "Definition:Bounded Mapping/Real-Valued/Definition 1", "text": "$f$ is '''bounded on $S$''' {{iff}}: :$f$ is bounded above on $S$ and also: :$f$ is bounded below on $S$."}
+{"_id": "24887", "title": "Definition:Bounded Mapping/Real-Valued/Definition 2", "text": "$f$ is '''bounded on $S$''' {{iff}}: :$\\exists K \\in \\R_{\\ge 0}: \\forall x \\in S: \\size {\\map f x} \\le K$ where $\\size {\\map f x}$ denotes the absolute value of $\\map f x$."}
+{"_id": "24888", "title": "Definition:Bounded Above Mapping/Real-Valued", "text": "Let $f: S \\to \\R$ be a real-valued function. $f$ is '''bounded above on $S$''' by the upper bound $H$ {{iff}}: :$\\forall x \\in S: \\map f x \\le H$"}
+{"_id": "24889", "title": "Definition:Bounded Above Mapping/Real-Valued/Unbounded", "text": "Let $f: S \\to \\R$ be a real-valued function. Then $f$ is '''unbounded above on $S$''' {{iff}} it is not bounded above on $S$: :$\\neg \\exists H \\in \\R: \\forall x \\in S: \\map f x \\le H$"}
+{"_id": "24890", "title": "Definition:Bounded Above Mapping/Unbounded", "text": "Let $f: S \\to T$ be a mapping whose codomain is an ordered set $\\struct {T, \\preceq}$. Then $f$ is '''unbounded above on $S$''' {{iff}} it is not bounded above on $S$: :$\\neg \\exists H \\in T: \\forall x \\in S: \\map f x \\preceq H$"}
+{"_id": "24891", "title": "Definition:Bounded Below Mapping/Real-Valued", "text": "Let $f: S \\to \\R$ be a real-valued function. Then $f$ is '''bounded below on $S$''' by the lower bound $L$ {{iff}}: :$\\forall x \\in S: L \\le \\map f x$"}
+{"_id": "24892", "title": "Definition:Bounded Below Mapping/Real-Valued/Unbounded Below", "text": "Let $f: S \\to \\R$ be a real-valued function. Then $f$ is '''unbounded below on $S$''' {{iff}} it is not bounded below on $S$: :$\\neg \\exists L \\in \\R: \\forall x \\in S: L \\le \\map f x$"}
+{"_id": "24893", "title": "Definition:Bound of Subset of Real Numbers", "text": "Let $S \\subseteq \\R$ be a subset of the real numbers $\\R$. Let $S$ be bounded. Then $K$ is a '''bound''' for $S$ {{iff}}: :$\\forall x \\in S: \\size x \\le K$"}
+{"_id": "24894", "title": "Definition:Bounded Ordered Set/Real Numbers/Definition 1", "text": "Let $T \\subseteq \\R$ be both bounded below and bounded above in $\\R$. Then $T$ is '''bounded in $\\R$'''."}
+{"_id": "24895", "title": "Definition:Bounded Ordered Set/Real Numbers/Definition 2", "text": "Let $T \\subseteq \\R$ be a subset of $\\R$ such that: :$\\exists K \\in \\R: \\forall x \\in T: \\size x \\le K$ where $\\size x$ denotes the absolute value of $x$. Then $T$ is '''bounded in $\\R$'''."}
+{"_id": "24896", "title": "Definition:Bounded Ordered Set/Real Numbers", "text": "{{:Definition:Bounded Ordered Set/Real Numbers/Definition 1}}"}
+{"_id": "24897", "title": "Definition:Upper Bound of Set/Real Numbers", "text": "Let $\\R$ be the set of real numbers. Let $T$ be a subset of $\\R$. An '''upper bound for $T$ (in $\\R$)''' is an element $M \\in \\R$ such that: :$\\forall t \\in T: t \\le M$ That is, $M$ is greater than or equal to every element of $T$."}
+{"_id": "24898", "title": "Definition:Bounded Above Set/Real Numbers", "text": "Let $\\R$ be the set of real numbers. A subset $T \\subseteq \\R$ is '''bounded above (in $\\R$)''' {{iff}} $T$ admits an upper bound (in $\\R$)."}
+{"_id": "24899", "title": "Definition:Bounded Below Set/Real Numbers", "text": "Let $\\R$ be the set of real numbers. A subset $T \\subseteq \\R$ is '''bounded below (in $\\R$)''' {{iff}} $T$ admits a lower bound (in $\\R$)."}
+{"_id": "24900", "title": "Definition:Lower Bound of Set/Real Numbers", "text": "Let $\\R$ be the set of real numbers. Let $T$ be a subset of $S$. A '''lower bound for $T$ (in $\\R$)''' is an element $m \\in \\R$ such that: :$\\forall t \\in T: m \\le t$"}
+{"_id": "24902", "title": "Definition:Bounded Below Mapping/Real-Valued/Unbounded", "text": "Let $f: S \\to \\R$ be a real-valued function. Then $f$ is '''unbounded below on $S$''' {{iff}} it is not bounded below on $S$: :$\\neg \\exists L \\in \\R: \\forall x \\in S: L \\le \\map f x$"}
+{"_id": "24903", "title": "Definition:Upper Bound of Mapping/Real-Valued", "text": "Let $f: S \\to \\R$ be a real-valued function. Let $f$ be bounded above in $\\R$ by $H \\in \\R$. Then $H$ is an '''upper bound of $f$'''."}
+{"_id": "24904", "title": "Definition:Lower Bound of Mapping/Real-Valued", "text": "Let $f: S \\to \\R$ be a real-valued function. Let $f$ be bounded below in $T$ by $L \\in T$. Then $L$ is a '''lower bound of $f$'''."}
+{"_id": "24905", "title": "Definition:Supremum of Mapping/Real-Valued Function/Definition 1", "text": "The '''supremum of $f$ on $S$''' is defined by: :$\\displaystyle \\sup_{x \\mathop \\in S} \\map f x := \\sup f \\sqbrk S$ where :$\\sup f \\sqbrk S$ is the supremum in $\\R$ of the image of $S$ under $f$."}
+{"_id": "24906", "title": "Definition:Supremum of Mapping/Real-Valued Function/Definition 2", "text": "The '''supremum of $f$ on $S$''' is defined as $\\displaystyle \\sup_{x \\mathop \\in S} \\map f x := K \\in \\R$ such that: :$(1): \\quad \\forall x \\in S: \\map f x \\le K$ :$(2): \\quad \\exists x \\in S: \\forall \\epsilon \\in \\R_{>0}: \\map f x > K - \\epsilon$"}
+{"_id": "24907", "title": "Definition:Infimum of Mapping/Real-Valued Function/Definition 1", "text": "The '''infimum of $f$ on $S$''' is defined by: :$\\displaystyle \\inf_{x \\mathop \\in S} \\map f x = \\inf f \\sqbrk S$ where :$\\inf f \\sqbrk S$ is the infimum in $\\R$ of the image of $S$ under $f$."}
+{"_id": "24908", "title": "Definition:Infimum of Mapping/Real-Valued Function/Definition 2", "text": "The '''infimum of $f$ on $S$''' is defined as $\\displaystyle \\inf_{x \\mathop \\in S} \\map f x := k \\in \\R$ such that: :$(1): \\quad \\forall x \\in S: k \\le \\map f x$ :$(2): \\quad \\forall \\epsilon \\in \\R_{>0}: \\exists x \\in S: \\map f x < k + \\epsilon$"}
+{"_id": "24909", "title": "Definition:Bounded Above Sequence/Real", "text": "Let $\\sequence {x_n}$ be a real sequence. Then $\\sequence {x_n}$ is '''bounded above''' {{iff}}: :$\\exists M \\in \\R: \\forall i \\in \\N: x_i \\le M$"}
+{"_id": "24910", "title": "Definition:Bounded Below Sequence/Real", "text": "Let $\\sequence {x_n}$ be a real sequence. Then $\\sequence {x_n}$ is '''bounded below''' {{iff}}: : $\\exists m \\in \\R: \\forall i \\in \\N: m \\le x_i$"}
+{"_id": "24911", "title": "Definition:Bounded Above Sequence/Real/Unbounded", "text": "$\\sequence {x_n}$ is '''unbounded above''' {{iff}} there exists no $M$ in $\\R$ such that: :$\\forall i \\in \\N: x_i \\le M$"}
+{"_id": "24912", "title": "Definition:Bounded Below Sequence/Real/Unbounded", "text": "$\\sequence {x_n}$ is '''unbounded below''' {{iff}} there exists no $m$ in $\\R$ such that: :$\\forall i \\in \\N: m \\le x_i$"}
+{"_id": "24913", "title": "Definition:Bounded Sequence/Real", "text": "Let $\\sequence {x_n}$ be a real sequence. Then $\\sequence {x_n}$ is '''bounded''' {{iff}} $\\exists m, M \\in \\R$ such that $\\forall i \\in \\N$: :$m \\le x_i$ :$x_i \\le M$"}
+{"_id": "24914", "title": "Definition:Bounded Sequence/Real/Unbounded", "text": "$\\sequence {x_n}$ is '''unbounded''' {{iff}} it is not bounded."}
+{"_id": "24915", "title": "Definition:Bounded Mapping/Real-Valued/Unbounded", "text": "$f$ is '''unbounded''' {{iff}} it is neither bounded above nor bounded below."}
+{"_id": "24916", "title": "Definition:Bounded Ordered Set/Real Numbers/Unbounded", "text": "$T \\subseteq \\R$ is '''unbounded (in $\\R$)''' {{iff}} it is not bounded."}
+{"_id": "24917", "title": "Definition:Bounded Above Set/Real Numbers/Unbounded", "text": "$T \\subseteq \\R$ is '''unbounded above (in $\\R$)''' {{iff}} it is not bounded above."}
+{"_id": "24918", "title": "Definition:Bounded Below Set/Real Numbers/Unbounded", "text": "$T \\subseteq \\R$ is '''unbounded below (in $\\R$)''' {{iff}} it is not bounded below."}
+{"_id": "24919", "title": "Definition:Lower Bound of Sequence/Real", "text": "Let $\\sequence {x_n}$ be a real sequence. Let $\\sequence {x_n}$ be bounded below in $T$ by $L \\in \\R$. Then $L$ is a '''lower bound of $\\sequence {x_n}$'''."}
+{"_id": "24920", "title": "Definition:Upper Bound of Sequence/Real", "text": "Let $\\sequence {x_n}$ be a real sequence. Let $\\sequence {x_n}$ be bounded above by $H \\in \\R$. Then $H$ is an '''upper bound of $\\sequence {x_n}$'''."}
+{"_id": "24921", "title": "Definition:Infimum of Set/Real Numbers", "text": "Let $T \\subseteq \\R$. A real number $c \\in \\R$ is the '''infimum of $T$ in $\\R$''' {{iff}}: :$(1): \\quad c$ is a lower bound of $T$ in $\\R$ :$(2): \\quad d \\le c$ for all lower bounds $d$ of $T$ in $\\R$."}
+{"_id": "24922", "title": "Definition:Supremum of Set/Real Numbers", "text": "Let $T \\subseteq \\R$ be a subset of the real numbers. A real number $c \\in \\R$ is the '''supremum of $T$ in $\\R$''' {{iff}}: :$(1): \\quad c$ is an upper bound of $T$ in $\\R$ :$(2): \\quad c \\le d$ for all upper bounds $d$ of $T$ in $\\R$."}
+{"_id": "24923", "title": "Definition:Bounded Metric Space/Complex", "text": "Let $D$ be a subset of the complex plane $\\C$. Then '''$D$ is bounded (in $\\C$)''' {{iff}} there exists $M \\in \\R$ such that: : $\\forall z \\in D: \\cmod z \\le M$"}
+{"_id": "24924", "title": "Definition:Bounded Metric Space/Complex/Unbounded", "text": "Let $D$ be a subset of the complex plane $\\C$. Then '''$D$ is unbounded (in $\\C$)''' {{iff}}: : $\\nexists M \\in \\R: \\forall z \\in D: \\cmod z \\le M$ That is, if $D$ is not bounded in $\\C$."}
+{"_id": "24925", "title": "Definition:Compact Space/Metric Space/Complex", "text": "Let $D$ be a subset of the complex plane $\\C$. Then $D$ is '''compact (in $\\C$)''' {{iff}}: : $D$ is closed in $\\C$ and : $D$ is bounded in $\\C$."}
+{"_id": "24926", "title": "Definition:Interior Point (Topology)/Definition 1", "text": "Let $h \\in H$. Then $h$ is an '''interior point''' of $H$ {{iff}}: :$h \\in H^\\circ$ where $H^\\circ$ denotes the interior of $H$."}
+{"_id": "24927", "title": "Definition:Interior Point (Topology)/Definition 2", "text": "Let $h \\in H$. $h$ is an '''interior point''' of $H$ {{iff}} $h$ has an open neighborhood $N_h$ such that $N_h \\subseteq H$."}
+{"_id": "24930", "title": "Definition:Theory", "text": "Let $\\mathcal L$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\mathcal L$. Let $\\mathcal F$ be a set of $\\mathcal L$-formulas. Then $\\mathcal F$ is an '''$\\mathcal L$-theory''' iff, for every $\\phi \\in \\mathcal L$: :$\\mathcal F \\models_{\\mathscr M} \\phi \\implies \\phi \\in \\mathcal F$ where $\\models_{\\mathscr M}$ denotes $\\mathscr M$-semantic consequence. === Theory of Set of Formulas === {{:Definition:Theory/Set of Formulas}}"}
+{"_id": "24931", "title": "Definition:Axiomatization", "text": "Let $\\mathcal L$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\mathcal L$. Let $\\mathcal F$ be an $\\mathcal L$-theory. An '''axiomatization of $\\mathcal F$''' is a subset $\\mathcal A \\subseteq \\mathcal F$ such that: :$\\mathcal F = \\left\\{{\\phi \\in \\mathcal L: \\mathcal A \\models_{\\mathscr M} \\phi}\\right\\}$ That is, all of $\\mathcal F$ is a semantic consequence of $\\mathcal A$. === Axiom === {{:Definition:Axiomatization/Axiom}}"}
+{"_id": "24932", "title": "Definition:Axiomatization/Axiom", "text": "Let $\\mathcal A$ be an axiomatization of $\\mathcal F$. Then a formula $\\phi \\in \\mathcal A$ is called an '''axiom'''."}
+{"_id": "24933", "title": "Definition:Finitely Axiomatizable", "text": "Let $\\mathcal L$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\mathcal L$. Let $\\mathcal F$ be an $\\mathcal L$-theory. Then $\\mathcal F$ is called '''finitely axiomatizable''' iff there exists a finite axiomatization $\\mathcal A$ of $\\mathcal F$."}
+{"_id": "24934", "title": "Definition:Closed Set/Metric Space/Definition 1", "text": "'''$H$ is closed (in $M$)''' {{iff}} its complement $A \\setminus H$ is open in $M$."}
+{"_id": "24935", "title": "Definition:Closed Set/Metric Space/Definition 2", "text": "'''$H$ is closed (in $M$)''' {{iff}} every limit point of $H$ is also a point of $H$."}
+{"_id": "24936", "title": "Definition:Closed Set/Topology", "text": "{{:Definition:Closed Set/Topology/Definition 1}}"}
+{"_id": "24937", "title": "Definition:Literal/Positive", "text": "A '''positive literal''' is an atom $p$ of propositional logic."}
+{"_id": "24938", "title": "Definition:Literal/Negative", "text": "A '''negative literal''' is the negation $\\neg p$ of an atom $p$ of propositional logic."}
+{"_id": "24939", "title": "Definition:Closed Set/Complex Analysis", "text": "Let $S \\subseteq \\C$ be a subset of the complex plane. '''$S$ is closed (in $\\C$)''' {{iff}} every limit point of $S$ is also a point of $S$. That is: {{iff}} $S$ contains all its limit points."}
+{"_id": "24940", "title": "Definition:Interior Point (Complex Analysis)", "text": "Let $S \\subseteq \\C$ be a subset of the complex plane. Let $z \\in S$. $z$ is an '''interior point''' of $S$ {{iff}} $z$ has an $\\epsilon$-neighborhood $\\map {N_\\epsilon} z$ such that $\\map {N_\\epsilon} z \\subseteq S$."}
+{"_id": "24941", "title": "Definition:Boundary Point (Complex Analysis)", "text": "Let $S \\subseteq \\C$ be a subset of the complex plane. Let $z_0 \\in \\C$. $z_0$ is a '''boundary point''' of $S$ {{iff}} every $\\epsilon$-neighborhood $\\map {N_\\epsilon} {z_0}$ of $z_0$ contains points of $\\C$ in $S$ and also points of $\\C$ which are not in $S$."}
+{"_id": "24942", "title": "Definition:Exterior Point (Complex Analysis)/Definition 1", "text": "$z_0$ is an '''exterior point''' of $S$ {{iff}} $z_0$ has an $\\epsilon$-neighborhood which is disjoint from $S$."}
+{"_id": "24943", "title": "Definition:Exterior Point (Complex Analysis)/Definition 2", "text": "$z_0$ is an '''exterior point''' of $S$ {{iff}}: :$z_0$ is not an interior point of $S$ and: :$z_0$ is not a boundary point of $S$."}
+{"_id": "24944", "title": "Definition:Exterior Point (Complex Analysis)", "text": "Let $S \\subseteq \\C$ be a subset of the complex plane. Let $z_0 \\in \\C$. === Definition 1 === {{:Definition:Exterior Point (Complex Analysis)/Definition 1}} === Definition 2 === {{:Definition:Exterior Point (Complex Analysis)/Definition 2}}"}
+{"_id": "24945", "title": "Definition:Open Set/Complex Analysis/Definition 1", "text": "Let $S \\subseteq \\C$ be a subset of the set of complex numbers. Let: : $\\forall z_0 \\in S: \\exists \\epsilon \\in \\R_{>0}: N_{\\epsilon} \\left({z_0}\\right) \\subseteq S$ where $N_{\\epsilon} \\left({z_0}\\right)$ is the $\\epsilon$-neighborhood of $z_0$ for $\\epsilon$. Then $S$ is an '''open set (of $\\C$)''', or '''open (in $\\C$)'''."}
+{"_id": "24946", "title": "Definition:Open Set/Complex Analysis/Definition 2", "text": "Let $S \\subseteq \\C$ be a subset of the set of complex numbers. Then $S$ is an '''open set (of $\\C$)''', or '''open (in $\\C$)''' {{iff}} every point of $S$ is an interior point."}
+{"_id": "24947", "title": "Definition:Region/Plane", "text": "A point set $R$ in the plane is a '''region''' {{iff}}: :$(1): \\quad$ Each point of $R$ is the center of a circle all of whose elements consist of points of $R$ :$(2): \\quad$ Each point of $R$ can be joined by a curve consisting entirely of points of $R$."}
+{"_id": "24948", "title": "Definition:Interior (Complex Analysis)", "text": "Let $H \\subseteq \\C$ be a subset of the complex plane. The '''interior of $H$''' is the subset of $H$ which consists of the interior points of $H$."}
+{"_id": "24949", "title": "Definition:Connected Set (Complex Analysis)", "text": "Let $D \\subseteq \\C$ be a subset of the set of complex numbers. === Definition 1 === {{:Definition:Connected Set (Complex Analysis)/Definition 1}} === Definition 2 === {{:Definition:Connected Set (Complex Analysis)/Definition 2}}"}
+{"_id": "24950", "title": "Definition:Open Region/Complex", "text": "Let $D \\subseteq \\C$ be a subset of the set of complex numbers. $D$ is an '''open region''' {{iff}} $D$ is: :$(1): \\quad$ An open set and :$(2): \\quad$ Connected."}
+{"_id": "24951", "title": "Definition:Open Region/Plane", "text": "An '''open region''' is a region without its boundary, i.e. the  interior of such a region."}
+{"_id": "24952", "title": "Definition:Closed Region", "text": "=== Complex Analysis === {{:Definition:Closed Region/Complex}} === Closed Region in the Plane === {{:Definition:Closed Region/Plane}} Category:Definitions/Complex Analysis Category:Definitions/Analytic Geometry bek3de33ql83xa4rml7jp62suzqxwta"}
+{"_id": "24953", "title": "Definition:Closed Region/Plane", "text": "A '''closed region''' is a region complete with its boundary."}
+{"_id": "24954", "title": "Definition:Closed Region/Complex", "text": "Let $D \\subseteq \\C$ be a subset of the set of complex numbers. $D$ is a '''closed region''' {{iff}} $D$ the closure of an open region."}
+{"_id": "24955", "title": "Definition:Region/Complex", "text": "Let $D \\subseteq \\C$ be a subset of the set of complex numbers. $D$ is a '''region of $\\C$''' {{iff}}: :$(1): \\quad$ $D$ is non-empty :$(2): \\quad$ $D$ is path-connected."}
+{"_id": "24956", "title": "Definition:Supremum of Sequence", "text": "A special case of a supremum of a mapping is a '''supremum of a sequence''', where the domain of the mapping is $\\N$. Let $\\struct {T, \\preceq}$ be an ordered set. Let $\\sequence {x_n}$ be a sequence in $T$. Let $\\set {x_n: n \\in \\N}$ admit a supremum. Then the '''supremum''' of $\\sequence {x_n}$) is defined as: :$\\displaystyle \\map \\sup {\\sequence {x_n} } = \\map \\sup {\\set {x_n: n \\in \\N} }$"}
+{"_id": "24957", "title": "Definition:Infimum of Sequence", "text": "A special case of an infimum of a mapping is an '''infimum of a sequence''', where the domain of the mapping is $\\N$. Let $\\struct {T, \\preceq}$ be an ordered set. Let $\\sequence {x_n}$ be a sequence in $T$. Let $\\set {x_n: n \\in \\N}$ admit an infimum. Then the '''infimum''' of $\\sequence {x_n}$) is defined as: :$\\displaystyle \\map \\inf {\\sequence {x_n} } = \\map \\inf {\\set {x_n: n \\in \\N} }$"}
+{"_id": "24958", "title": "Definition:Semantic Tableau", "text": "'''Semantic tableaus''' are the labeled trees which can be obtained in the course of executing the Semantic Tableau Algorithm. === Semantic Tableau Algorithm === {{:Semantic Tableau Algorithm}} === Completed Tableau === {{:Definition:Semantic Tableau/Completed|Completed Tableau}} === Closed Tableau === {{:Definition:Semantic Tableau/Closed|Closed Tableau}} === Open Tableau === {{:Definition:Semantic Tableau/Open|Open Tableau}}"}
+{"_id": "24959", "title": "Definition:Semantic Tableau/Completed", "text": "Let $T$ be a semantic tableau. Then $T$ is '''completed''' iff all of its leaves are marked."}
+{"_id": "24960", "title": "Definition:Semantic Tableau/Closed", "text": "Let $T$ be a semantic tableau. Then $T$ is '''closed''' iff all of its leaves are marked closed."}
+{"_id": "24961", "title": "Definition:Semantic Tableau/Open", "text": "Let $T$ be a semantic tableau. Then $T$ is '''open''' iff all of its leaves are marked open."}
+{"_id": "24962", "title": "Definition:Marked Leaf", "text": "Let $T$ be a semantic tableau. Let $t$ be a leaf of $T$. Then $t$ is '''marked''' iff it has occurred in '''Step 3''' of the Semantic Tableau Algorithm. Otherwise, it is called '''unmarked'''. === Marked Closed Leaf === {{:Definition:Marked Leaf/Closed}} === Marked Open Leaf === {{:Definition:Marked Leaf/Open}}"}
+{"_id": "24965", "title": "Definition:Alpha-Formula", "text": "Let $\\mathbf A$ be a WFF of propositional logic that is not a literal. Then $\\mathbf A$ is an '''$\\alpha$-formula''' {{iff}} either: :$\\mathbf A = \\neg \\neg \\mathbf A_1$ for some WFF $\\mathbf A_1$, or: :$\\mathbf A$ is semantically equivalent to a conjunction $\\mathbf A_1 \\land \\mathbf A_2$ for some WFFs $\\mathbf A_1, \\mathbf A_2$. === Table of $\\alpha$-Formulas === {{:Definition:Alpha-Formula/Table}}"}
+{"_id": "24966", "title": "Definition:Alpha-Formula/Table", "text": "From Classification of $\\alpha$-Formulas, we obtain the following table of $\\alpha$-formulas $\\mathbf A$ and corresponding $\\mathbf A_1$ and $\\mathbf A_2$: ::$\\begin{array}{ccc} \\hline \\mathbf A & \\mathbf A_1 & \\mathbf A_2\\\\ \\hline \\neg\\neg \\mathbf A_1 & \\mathbf A_1 & \\\\ \\mathbf A_1 \\land \\mathbf A_2 & \\mathbf A_1 & \\mathbf A_2 \\\\ \\neg \\left({\\mathbf A_1 \\lor \\mathbf A_2}\\right) & \\neg \\mathbf A_1 & \\neg \\mathbf A_2 \\\\ \\neg \\left({\\mathbf A_1 \\implies \\mathbf A_2}\\right) & \\mathbf A_1 & \\neg \\mathbf A_2 \\\\ \\neg \\left({\\mathbf A_1 \\mathbin \\uparrow \\mathbf A_2}\\right) & \\mathbf A_1 & \\mathbf A_2 \\\\ \\mathbf A_1 \\mathbin \\downarrow \\mathbf A_2 & \\neg \\mathbf A_1 & \\neg \\mathbf A_2 \\\\ \\mathbf A_1 \\iff \\mathbf A_2 & \\mathbf A_1 \\implies \\mathbf A_2 & \\mathbf A_2 \\implies \\mathbf A_1 \\\\ \\neg \\left({\\mathbf A_1 \\oplus \\mathbf A_2}\\right) & \\mathbf A_1 \\implies \\mathbf A_2 & \\mathbf A_2 \\implies \\mathbf A_1 \\\\ \\hline \\end{array}$"}
+{"_id": "24967", "title": "Definition:Beta-Formula/Table", "text": "From Classification of $\\beta$-Formulas, we obtain the following table of $\\beta$-formulas $\\mathbf B$ and corresponding $\\mathbf B_1$ and $\\mathbf B_2$: ::$\\begin{array}{ccc} \\hline \\mathbf B & \\mathbf B_1 & \\mathbf B_2\\\\ \\hline \\neg \\left({\\mathbf B_1 \\land \\mathbf B_2}\\right) & \\neg \\mathbf B_1 & \\neg \\mathbf B_2 \\\\ \\mathbf B_1 \\lor \\mathbf B_2 & \\mathbf B_1 & \\mathbf B_2 \\\\ \\mathbf B_1 \\implies \\mathbf B_2 & \\neg \\mathbf B_1 & \\mathbf B_2 \\\\ \\mathbf B_1 \\mathbin \\uparrow \\mathbf B_2 & \\neg \\mathbf B_1 & \\neg \\mathbf B_2 \\\\ \\neg \\left({\\mathbf B_1 \\mathbin \\downarrow \\mathbf B_2}\\right) & \\mathbf B_1 & \\mathbf B_2 \\\\ \\neg \\left({\\mathbf B_1 \\iff \\mathbf B_2}\\right) & \\neg \\left({\\mathbf B_1 \\implies \\mathbf B_2}\\right) & \\neg \\left({\\mathbf B_2 \\implies \\mathbf B_1}\\right) \\\\ \\mathbf B_1 \\oplus \\mathbf B_2 & \\neg \\left({\\mathbf B_1 \\implies \\mathbf B_2}\\right) & \\neg \\left({\\mathbf B_2 \\implies \\mathbf B_1}\\right) \\\\ \\hline \\end{array}$"}
+{"_id": "24968", "title": "Definition:Beta-Formula", "text": "Let $\\mathbf B$ be a WFF of propositional logic that is not a literal. Then $\\mathbf B$ is a '''$\\beta$-formula''' iff: :$\\mathbf B$ is semantically equivalent to a disjunction $\\mathbf B_1 \\lor \\mathbf B_2$ for some WFFs $\\mathbf B_1, \\mathbf B_2$. === Table of $\\beta$-Formulas === {{:Definition:Beta-Formula/Table}}"}
+{"_id": "24969", "title": "Definition:Triangle (Geometry)/Right-Angled/Opposite", "text": "In a right-angled triangle, for a given non-right angled vertex, the opposite side is referred to as '''the opposite'''. In the above figure: :the '''opposite''' to vertex $A$ is side $a$ :the '''opposite''' to vertex $C$ is side $c$."}
+{"_id": "24970", "title": "Definition:Theory/Set of Formulas", "text": "Let $\\mathcal F$ be a set of $\\mathcal L$-formulas. Then the '''$\\mathcal L$-theory of $\\mathcal F$''', denoted $T \\left({\\mathcal F}\\right)$ is the set: :$\\left\\{{\\phi \\in \\mathcal L: \\mathcal F \\models_{\\mathscr M} \\phi}\\right\\}$ where $\\models_{\\mathscr M}$ denotes $\\mathscr M$-semantic consequence."}
+{"_id": "24971", "title": "Definition:Lemniscate of Bernoulli", "text": "=== Geometric Definition === {{:Definition:Lemniscate of Bernoulli/Geometric Definition}} === Cartesian Definition === {{:Definition:Lemniscate of Bernoulli/Cartesian Definition}} === Polar Definition === {{:Definition:Lemniscate of Bernoulli/Polar Definition}} === Parametric Definition === {{:Definition:Lemniscate of Bernoulli/Parametric Definition}}"}
+{"_id": "24972", "title": "Definition:Gentzen Proof System", "text": "'''Gentzen proof systems''' are a class of proof systems for propositional and predicate logic. Their characteristics include: * The presence of few axioms and many rules of inference. * Use of formal, sequent-like notation involving the turnstile $\\vdash$. * Proofs whose structure can be viewed as rooted trees."}
+{"_id": "24973", "title": "Definition:Gentzen Proof System/Instance 1", "text": "<section notitle=\"true\" name=\"tc\"> This instance of a Gentzen proof system is used in: * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition}} Let $\\mathcal L$ be the language of propositional logic. The Gentzen system applies to sets of propositional formulae. The intuition behind the system is that a set $U$ represents its disjunction. $\\mathscr G$ has the following axioms and rules of inference: === Axioms === A set $U$ of propositional formulae is an axiom of $\\mathscr G$ iff $U$ contains a complementary pair of literals. The invocation of an axiom may be denoted by: :$\\vdash U$ === Rules of Inference === Let $U_1, U_2$ be sets of propositional formulae. $\\mathscr G$ has two rules of inference, the $\\alpha$-rule and the $\\beta$-rule:<ref>In {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition}}, the meanings of $\\alpha$-formula and $\\beta$-formula are interchanged at this point compared to their earlier appearances. This is done because of a later connection with semantic tableaus, but can serve only to confuse. Caution needs to be exercised when using this source, because of the context-dependent meaning of these two terms.</ref> ==== $\\alpha$-Rule ==== {{:Definition:Gentzen Proof System/Instance 1/Alpha-Rule}} ==== $\\beta$-Rule ==== {{:Definition:Gentzen Proof System/Instance 1/Beta-Rule}} Invocations of these rules in a proof can be denoted as: :$(\\alpha) \\dfrac {\\vdash U_1, \\mathbf A_1 \\hspace{3em} \\vdash U_2, \\mathbf A_2}{\\vdash U_1, U_2, \\mathbf A} \\hspace{3em}  (\\beta) \\dfrac {\\vdash U_1, \\mathbf B_1, \\mathbf B_2}{\\vdash U_1, \\mathbf B}$ This notation suppresses the set notation as a matter of convenience. </section> {{NamedforDef|Gerhard Karl Erich Gentzen|cat = Gentzen}}"}
+{"_id": "24974", "title": "Definition:Tableau Proof (Formal Systems)", "text": "A '''tableau proof''' for a proof system is a technique for presenting a logical argument in a straightforward, standard form. On {{ProofWiki}}, the proof system is usually natural deduction. A '''tableau proof''' is a sequence of lines specifying the order of premises, assumptions, inferences and conclusion in support of an argument. Each line of a '''tableau proof''' has a particular format. It consists of the following parts: * '''Line:''' The line number of the proof. This is a simple numbering from 1 upwards. * '''Pool:''' The list of all the lines containing the pool of assumptions for the formula introduced on this line. * '''Formula:''' The propositional formula introduced on this line. * '''Rule:''' The justification for introducing this line. This should be the rule of inference being used to derive this line. * '''Depends on:''' The lines (if any) upon which this line ''directly'' depends. For premises and assumptions, this field will be empty. Optionally, a comment may be added to explicitly point out possible intricacies. If any assumptions are discharged on a certain line, for the sake of clarity it is preferred that such be mentioned explicitly in a comment. At the end of a tableau proof, the only lines upon which the proof depends may be those which contain the premises."}
+{"_id": "24975", "title": "Definition:Tableau Proof (Formal Systems)/Length", "text": "The '''length''' of a tableau proof is the number of lines it has."}
+{"_id": "24976", "title": "Definition:Variable/Complex", "text": "A '''complex variable''' is a symbol which can stand for any one of a set of complex numbers."}
+{"_id": "24977", "title": "Definition:Independent Variable/Complex Function", "text": "Let $f: \\C \\to \\C$ be a complex function. Let $\\map f z = w$. Then $z$ is referred to as an '''independent variable (of $f$)'''."}
+{"_id": "24978", "title": "Definition:Dependent Variable/Complex Function", "text": "Let $f: \\C \\to \\C$ be a complex function. Let $\\map f z = w$. Then $w$ is referred to as the '''dependent variable (of $f$)'''."}
+{"_id": "24979", "title": "Definition:Left-Total Relation/Multifunction", "text": "In the field of complex analysis, a '''left-total relation''' is usually referred to as a '''multifunction'''."}
+{"_id": "24980", "title": "Definition:Left-Total Relation/Multifunction/Branch", "text": "Let $D \\subseteq \\C$ be a subset of the complex numbers. Let $f: D \\to \\C$ be a multifunction on $D$. Let $\\family {S_i}_{i \\mathop \\in I}$ be a partitioning of the codomain of $f$ such that: :$\\forall i \\in I: f \\restriction_{D \\times S_i}$ is a mapping. Then each $f \\restriction_{D \\times S_i}$ is a '''branch''' of $f$."}
+{"_id": "24981", "title": "Definition:Distinct/Plural", "text": "Two objects $x$ and $y$ are '''distinct''' {{iff}} $x \\ne y$. If $x$ and $y$ are '''distinct''', then that means they can be '''distinguished''', or '''identified as being different from each other'''."}
+{"_id": "24982", "title": "Definition:Distinct/Plural/Pairwise Distinct", "text": "A set of objects is '''pairwise distinct''' if each pair of elements of that set is distinct."}
+{"_id": "24983", "title": "Definition:Distinct/Singular", "text": "Let $x \\in S$ be an element of a set of objects $S$. $x$ is '''distinguished''' from the other elements of $S$ if is endowed with a property that the other elements of $S$ are specifically deemed not to possess. Such an element is identified as being '''distinct''' from the others."}
+{"_id": "24984", "title": "Definition:Distinct/Indistinguishable", "text": "Two objects are '''indistinguishable''' if they can not (in a particular context) be told apart from each other. So, two objects may be '''distinct''' but (at a given level) '''indistinguishable''', like identical twins."}
+{"_id": "24985", "title": "Definition:Left-Total Relation/Multifunction/Branch/Principal Branch", "text": "Let $D \\subseteq \\C$ be a subset of the complex numbers. Let $f: D \\to \\C$ be a multifunction on $D$. Let $\\sequence {S_i}_{i \\in I}$ be a partitioning of the codomain of $f$ into branches. It is usual to distinguish one such branch of $f$ from the others, and label it the '''principal branch''' of $f$."}
+{"_id": "24987", "title": "Definition:Real Function/Two Variables", "text": "Let $S, T \\subseteq \\R$ be subsets of the set of real numbers $\\R$. Let $f: S \\times T \\to \\R$ be a mapping. Then $f$ is defined as a '''(real) function of two (independent) variables'''.  The expression: :$z = \\map f {x, y}$ means: :(The dependent variable) $z$ is a function of (the independent variables) $x$ and $y$."}
+{"_id": "24988", "title": "Definition:Complex Transformation", "text": "A '''complex transformation''' is a mapping on the complex plane $f: \\C \\to \\C$ which is specifically ''not'' a multifunction. Let $z = x + i y$ be a complex variable. Let $w = u + i v = \\map f z$. Then $w$ can be expressed as: :$u + i v = \\map f {x + i y}$ such that: :$u = \\map u {x, y}$ and: :$v = \\map v {x, y}$ are real functions of two variables. Thus a point $P = \\tuple {x, y}$ in the complex plane is '''transformed''' to a point $P' = \\tuple {\\map u {x, y}, \\map v {x, y} }$ by $f$. Thus $P'$ is the image of $P$ under $f$."}
+{"_id": "24989", "title": "Definition:Curvilinear Coordinate System/Complex Plane", "text": "Let $u + i v = \\map f {x + i y}$ be a complex transformation. Let $P = \\tuple {x, y}$ be a point in the complex plane. Then $\\tuple {\\map u {x, y}, \\map v {x, y} }$ are the '''curvilinear coordinates of $P$ under $f$'''."}
+{"_id": "24990", "title": "Definition:Complex Curvilinear Coordinates/Coordinate Curves", "text": "Let $c_1$ and $c_2$ be constants. The curves: :$\\map u {x, y} = c_1$ :$\\map v {x, y} = c_2$ are the '''coordinate curves''' of $f$."}
+{"_id": "24991", "title": "Definition:Rational Function/Real", "text": "Let $P: \\R \\to \\R$ and $Q: \\R \\to \\R$ be polynomial functions on the set of real numbers. Let $S$ be the set $\\R$ from which all the roots of $Q$ have been removed. That is: : $S = \\R \\setminus \\left\\{{x \\in \\R: Q \\left({x}\\right) = 0}\\right\\}$. Then the equation $y = \\dfrac {P \\left({x}\\right)} {Q \\left({x}\\right)}$ defines a function from $S$ to $\\R$. Such a function is a '''rational function'''."}
+{"_id": "24992", "title": "Definition:Rational Function/Complex", "text": "Let $P: \\C \\to \\C$ and $Q: \\C \\to \\C$ be polynomial functions on the set of complex numbers. Let $S$ be the set $\\C$ from which all the roots of $Q$ have been removed. That is: :$S = \\C \\setminus \\set {z \\in \\C: \\map Q z = 0}$ Then the equation $y = \\dfrac {\\map P z} {\\map Q z}$ defines a function from $S$ to $\\C$. Such a function is a '''rational (algebraic) function'''."}
+{"_id": "24993", "title": "Definition:Power (Algebra)/Real Number/Complex", "text": "Let $x \\in \\R$ be a real number such that $x > 0$. Let $r \\in \\C$ be ''any'' complex number. Then we define $x^r$ as: :$x^r := \\map \\exp {r \\ln x}$ where $\\exp$ denotes the complex exponential function."}
+{"_id": "24994", "title": "Definition:Natural Logarithm/Complex/Definition 1", "text": "Let $z = r e^{i \\theta}$ be a complex number expressed in exponential form such that $z \\ne 0$. The '''complex natural logarithm''' of $z \\in \\C_{\\ne 0}$ is the multifunction defined as: :$\\map \\ln z := \\set {\\map \\ln r + i \\paren {\\theta + 2 k \\pi}: k \\in \\Z}$ where $\\map \\ln r$ is the natural logarithm of the (strictly) positive real number $r$."}
+{"_id": "24995", "title": "Definition:Natural Logarithm/Notation", "text": "The notation for the natural logarithm function is misleadingly inconsistent throughout the literature. It is written variously as: :$\\ln z$ :$\\log z$ :$\\Log z$ :$\\log_e z$ The first of these is commonly encountered, and is the preferred form on {{ProofWiki}}. However, many who consider themselves serious mathematicians believe this notation to be unsophisticated. The second and third are ambiguous (it doesn't tell you which base it is the logarithm of). While the fourth option is more verbose than the others, there is no confusion about exactly what is meant."}
+{"_id": "24996", "title": "Definition:Natural Logarithm/Complex/Definition 2", "text": "Let $z \\in \\C_{\\ne 0}$ be a non-zero complex number. The '''complex natural logarithm''' of $z$ is the multifunction defined as: :$\\map \\ln z := \\set {w \\in \\C: e^w = z}$"}
+{"_id": "24997", "title": "Definition:Natural Logarithm/Positive Real/Definition 1", "text": "Let $x \\in \\R$ be a real number such that $x > 0$. The '''(natural) logarithm''' of $x$ is defined as: :$\\displaystyle \\ln x := \\int_1^x \\frac {\\d t} t$"}
+{"_id": "24998", "title": "Definition:Natural Logarithm/Positive Real/Definition 2", "text": "Let $x \\in \\R$ be a real number such that $x > 0$. The '''(natural) logarithm''' of $x$ is defined as: :$\\ln x := y \\in \\R: e^y = x$ where $e$ is Euler's number."}
+{"_id": "24999", "title": "Definition:Gentzen Proof System/Instance 1/Alpha-Rule", "text": "$(\\alpha)$: For any $\\alpha$-formula $\\mathbf A$ and associated $\\mathbf A_1, \\mathbf A_2$ from the table of $\\alpha$-formulas: ::Given $U_1 \\cup \\left\\{{\\mathbf A_1}\\right\\}$ and $U_2 \\cup \\left\\{{\\mathbf A_2}\\right\\}$, one may infer $U_1 \\cup U_2 \\cup \\left\\{{\\mathbf A}\\right\\}$."}
+{"_id": "25000", "title": "Definition:Gentzen Proof System/Instance 1/Beta-Rule", "text": "$(\\beta)$: For any $\\beta$-formula $\\mathbf B$ and associated $\\mathbf B_1, \\mathbf B_2$ from the table of $\\beta$-formulas: :: Given $U_1 \\cup \\left\\{{\\mathbf B_1, \\mathbf B_2}\\right\\}$, one may infer $U_1 \\cup \\left\\{{\\mathbf B}\\right\\}$."}
+{"_id": "25001", "title": "Definition:Gentzen Proof System/Instance 1/Alpha-Rule/Notation", "text": "In a tableau proof, the $\\alpha$-rule can be used as follows: {{begin-axiom}} {{axiom|lc = '''Pool:'''        |t  = Empty. }} {{axiom|lc = '''Formula:'''         |t  = $U_1 \\cup U_2 \\cup \\left\\{ {\\mathbf A}\\right\\}$. }} {{axiom|lc = '''Description:'''         |t  = $\\alpha$-Rule. }} {{axiom|lc = '''Depends on:'''         |t  = The lines containing $U_1 \\cup \\left\\{ {\\mathbf A_1}\\right\\}$ and $U_2 \\cup \\left\\{ {\\mathbf A_2}\\right\\}$. }} {{axiom|lc = '''Abbreviation:'''         |t  = $\\alpha \\circ$, where $\\circ$ is the binary logical connective such that $\\mathbf A {{=}} \\mathbf A_1 \\circ \\mathbf A_2$ or $\\mathbf A {{=}} \\neg \\left({\\mathbf A_1 \\circ \\mathbf A_2}\\right)$, or $\\neg \\neg$ in the case that $\\mathbf A = \\neg \\neg \\mathbf A_1$. }} {{end-axiom}}"}
+{"_id": "25002", "title": "Definition:Gentzen Proof System/Instance 1/Beta-Rule/Notation", "text": "In a tableau proof, the $\\beta$-rule can be used as follows: {{begin-axiom}} {{axiom|lc = '''Pool:'''        |t  = Empty. }} {{axiom|lc = '''Formula:'''         |t  = $U_1 \\cup \\left\\{ {\\mathbf B}\\right\\}$. }} {{axiom|lc = '''Description:'''         |t  = $\\beta$-Rule. }} {{axiom|lc = '''Depends on:'''         |t  = The line containing $U_1 \\cup \\left\\{ {\\mathbf B_1, \\mathbf B_2}\\right\\}$. }} {{axiom|lc = '''Abbreviation:'''         |t  = $\\beta \\circ$, where $\\circ$ is the binary logical connective such that $\\mathbf B {{=}} \\mathbf B_1 \\circ \\mathbf B_2$ or $\\mathbf B {{=}} \\neg \\left({\\mathbf B_1 \\circ \\mathbf B_2}\\right)$. }} {{end-axiom}}"}
+{"_id": "25004", "title": "Definition:Natural Logarithm/Complex/Principal Branch", "text": "The principal branch of the complex natural logarithm is usually defined in one of two ways: :$\\map \\Ln z = \\map \\ln r + i \\theta$ for $\\theta \\in \\hointr 0 {2 \\pi}$ :$\\map \\Ln z = \\map \\ln r + i \\theta$ for $\\theta \\in \\hointl {-\\pi} \\pi$ It is important to specify which is in force during a particular exposition."}
+{"_id": "25006", "title": "Definition:Principle", "text": "A '''principle''' is a theorem which is deemed to be particularly important. The word has the same force about it as does '''law''', the two words being used interchangeably."}
+{"_id": "25007", "title": "Definition:General Logarithm/Positive Real", "text": "Let $x \\in \\R_{>0}$ be a strictly positive real number. Let $a \\in \\R_{>0}$ be a strictly positive real number such that $a \\ne 1$. The '''logarithm to the base $a$ of $x$''' is defined as: :$\\log_a x := y \\in \\R: a^y = x$ where $a^y = e^{y \\ln a}$ as defined in Powers of Real Numbers."}
+{"_id": "25008", "title": "Definition:General Logarithm/Complex", "text": "Let $z \\in \\C_{\\ne 0}$ be a non-zero complex number. Let $a \\in \\R_{>0}$ be a strictly positive real number such that $a \\ne 1$. The '''logarithm to the base $a$ of $z$''' is defined as: :$\\log_a z := \\set {y \\in \\C: a^y = z}$ where $a^y = e^{y \\ln a}$ as defined in Powers of Complex Numbers."}
+{"_id": "25009", "title": "Definition:Inverse Sine/Real/Arcsine", "text": "{{:Graph of Arcsine Function|Graph}} From Shape of Sine Function, we have that $\\sin x$ is continuous and strictly increasing on the interval $\\closedint {-\\dfrac \\pi 2} {\\dfrac \\pi 2}$. From Sine of Half-Integer Multiple of Pi: :$\\map \\sin {-\\dfrac {\\pi} 2} = -1$ and: :$\\sin \\dfrac {\\pi} 2 = 1$ Therefore, let $g: \\closedint {-\\dfrac \\pi 2} {\\dfrac \\pi 2} \\to \\closedint {-1} 1$ be the restriction of $\\sin x$ to $\\closedint {-\\dfrac \\pi 2} {\\dfrac \\pi 2}$. Thus from Inverse of Strictly Monotone Function, $g \\paren x$ admits an inverse function, which will be continuous and strictly increasing on $\\closedint {-1} 1$. This function is called '''arcsine of $x$''' and is written $\\arcsin x$. Thus: :The domain of $\\arcsin x$ is $\\closedint {-1} 1$ :The image of $\\arcsin x$ is $\\closedint {-\\dfrac \\pi 2} {\\dfrac \\pi 2}$."}
+{"_id": "25010", "title": "Definition:Inverse Sine/Real", "text": "Let $x \\in \\R$ be a real number such that $-1 \\le x \\le 1$. The '''inverse sine of $x$''' is the multifunction defined as: :$\\sin^{-1} \\left({x}\\right) := \\left\\{{y \\in \\R: \\sin \\left({y}\\right) = x}\\right\\}$ where $\\sin \\left({y}\\right)$ is the sine of $y$."}
+{"_id": "25011", "title": "Definition:Sine/Real Function", "text": "The real function $\\sin: \\R \\to \\R$ is defined as: {{begin-eqn}} {{eqn | l = \\sin x       | r = \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {x^{2 n + 1} } {\\paren {2 n + 1}!}       | c =  }} {{eqn | r = x - \\frac {x^3} {3!} + \\frac {x^5} {5!} - \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "25012", "title": "Definition:Sine/Complex Function", "text": "The complex function $\\sin: \\C \\to \\C$ is defined as: {{begin-eqn}} {{eqn | l = \\sin z       | r = \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {z^{2 n + 1 } } {\\paren {2 n + 1}!}       | c =  }} {{eqn | r = z - \\frac {z^3} {3!} + \\frac {z^5} {5!} - \\frac {z^7} {7!} + \\cdots + \\paren {-1}^n \\frac {z^{2 n + 1 } } {\\paren {2 n + 1}!} + \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "25013", "title": "Definition:Inverse Sine/Complex", "text": "{{:Definition:Inverse Sine/Complex/Definition 1}}"}
+{"_id": "25014", "title": "Definition:Hilbert Proof System", "text": "'''Hilbert proof systems''' are a class of proof systems for propositional and predicate logic. Their characteristics include: * The presence of many axioms; * Typically only Modus Ponendo Ponens as a rule of inference."}
+{"_id": "25015", "title": "Definition:Hilbert Proof System/Instance 1", "text": "<section notitle=\"true\" name=\"tc\"> This instance of a Hilbert proof system is used in: * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition}} Let $\\LL$ be the language of propositional logic. $\\mathscr H$ has the following axioms and rules of inference: === Axioms === Let $\\mathbf A, \\mathbf B, \\mathbf C$ be WFFs. The following three WFFs are axioms of $\\mathscr H$: {{begin-axiom}} {{axiom | lc= '''Axiom 1:'''         | m = \\mathbf A \\implies \\paren {\\mathbf B \\implies \\mathbf A} }} {{axiom | lc= '''Axiom 2:'''         | m = \\paren {\\mathbf A \\implies \\paren {\\mathbf B \\implies \\mathbf C} } \\implies \\paren {\\paren {\\mathbf A \\implies \\mathbf B} \\implies \\paren {\\mathbf A \\implies \\mathbf C} } }} {{axiom | lc= '''Axiom 3:'''         | m = \\paren {\\neg \\mathbf B \\implies \\neg \\mathbf A} \\implies \\paren {\\mathbf A \\implies \\mathbf B} }} {{end-axiom}} === Rules of Inference === The sole rule of inference is Modus Ponendo Ponens: :From $\\mathbf A$ and $\\mathbf A \\implies \\mathbf B$, one may infer $\\mathbf B$. </section> {{NamedforDef|David Hilbert}}"}
+{"_id": "25016", "title": "Definition:Derived Rule", "text": "Let $\\mathcal L$ be a formal language. Let $\\mathscr P$ be a proof system for $\\mathcal L$. A '''derived rule''' for $\\mathscr P$ is a rule of inference $R$ such that: :The proof system $\\mathscr P'$, obtained by adding $R$ to $\\mathscr P$, is equivalent to $\\mathscr P$. That is, $R$ is a '''derived rule''' iff it does not extend the collection of $\\mathscr P$-theorems."}
+{"_id": "25017", "title": "Definition:Cosine/Real Function", "text": "The real function $\\cos: \\R \\to \\R$ is defined as: {{begin-eqn}} {{eqn | l = \\cos x       | r = \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {x^{2 n} } {\\paren {2 n!} }       | c =  }} {{eqn | r = 1 - \\frac {x^2} {2!} + \\frac {x^4} {4!} - \\frac {x^6} {6!} + \\cdots + \\paren {-1}^n \\frac {x^{2 n} } {\\paren {2 n}!} + \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "25018", "title": "Definition:Cosine/Complex Function", "text": "The complex function $\\cos: \\C \\to \\C$ is defined as: {{begin-eqn}} {{eqn | l = \\cos z       | r = \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {z^{2 n} } {\\paren {2 n}!}       | c =  }} {{eqn | r = 1 - \\frac {z^2} {2!} + \\frac {z^4} {4!} - \\frac {z^6} {6!} + \\cdots + \\paren {-1}^n \\frac {z^{2 n} } {\\paren {2 n}!} + \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "25019", "title": "Definition:Inverse Sine/Complex/Definition 1", "text": "Let $z \\in \\C$ be a complex number. The '''inverse sine of $z$''' is the multifunction defined as: :$\\sin^{-1} \\paren z := \\set {w \\in \\C: \\sin \\paren w = z}$ where $\\sin \\paren w$ is the sine of $w$."}
+{"_id": "25020", "title": "Definition:Inverse Sine/Complex/Definition 2", "text": "Let $z \\in \\C$ be a complex number. The '''inverse sine of $z$''' is the multifunction defined as: :$\\sin^{-1} \\paren z := \\set {\\dfrac 1 i \\ln \\paren {i z + \\sqrt {\\cmod {1 - z^2} } \\exp \\paren {\\dfrac i 2 \\arg \\paren {1 - z^2} } } + 2 k \\pi: k \\in \\Z}$ where: : $\\sqrt {\\cmod {1 - z^2} }$ denotes the positive square root of the complex modulus of $1 - z^2$ : $\\arg \\paren {1 - z^2}$ denotes the argument of $1 - z^2$ : $\\ln$ is the complex natural logarithm considered as a  multifunction."}
+{"_id": "25021", "title": "Definition:Inverse Sine/Complex/Arcsine", "text": "The principal branch of the complex inverse sine function is defined as: :$\\map \\arcsin z = \\dfrac 1 i \\, \\map \\Ln {i z + \\sqrt {1 - z^2} }$ where: : $\\Ln$ denotes the principal branch of the complex natural logarithm : $\\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$."}
+{"_id": "25022", "title": "Definition:Inverse Cosine/Real", "text": "Let $x \\in \\R$ be a real number such that $-1 \\le x \\le 1$. The '''inverse cosine of $x$''' is the multifunction defined as: :$\\cos^{-1} \\left({x}\\right) := \\left\\{{y \\in \\R: \\cos \\left({y}\\right) = x}\\right\\}$ where $\\cos \\left({y}\\right)$ is the cosine of $y$."}
+{"_id": "25024", "title": "Definition:Inverse Cosine/Complex", "text": "{{:Definition:Inverse Cosine/Complex/Definition 1}}"}
+{"_id": "25025", "title": "Definition:Inverse Cosine/Complex/Definition 1", "text": "Let $z \\in \\C$ be a complex number. The '''inverse cosine of $z$''' is the multifunction defined as: :$\\cos^{-1} \\left({z}\\right) := \\left\\{{w \\in \\C: \\cos \\left({w}\\right) = z}\\right\\}$ where $\\cos \\left({w}\\right)$ is the cosine of $w$."}
+{"_id": "25026", "title": "Definition:Inverse Cosine/Complex/Definition 2", "text": "Let $z \\in \\C$ be a complex number. The '''inverse cosine of $z$''' is the multifunction defined as: :$\\cos^{-1} \\left({z}\\right) := \\left\\{{\\dfrac 1 i \\ln \\left({z + \\sqrt{\\left|{z^2 - 1}\\right|} e^{\\left({i / 2}\\right) \\arg \\left({z^2 - 1}\\right)} }\\right) + 2 k \\pi: k \\in \\Z}\\right\\}$ where: : $\\sqrt{\\left|{z^2 - 1}\\right|}$ denotes the positive square root of the complex modulus of $z^2 - 1$ : $\\arg \\left({z^2 - 1}\\right)$ denotes the argument of $z^2 - 1$ : $\\ln$ denotes the complex natural logarithm considered as a  multifunction."}
+{"_id": "25027", "title": "Definition:Inverse Cosine/Complex/Arccosine", "text": "The principal branch of the complex inverse cosine function is defined as: :$\\map \\arccos z = \\dfrac 1 i \\, \\map \\Ln {z + \\sqrt {z^2 - 1} }$ where: :$\\Ln$ denotes the principal branch of the complex natural logarithm :$\\sqrt {z^2 - 1}$ denotes the principal square root of $z^2 - 1$."}
+{"_id": "25028", "title": "Definition:Inverse Tangent/Real", "text": "Let $x \\in \\R$ be a real number. The '''inverse tangent of $x$''' is the multifunction defined as: :$\\tan^{-1} \\left({x}\\right) := \\left\\{{y \\in \\R: \\tan \\left({y}\\right) = x}\\right\\}$ where $\\tan \\left({y}\\right)$ is the tangent of $y$."}
+{"_id": "25030", "title": "Definition:Inverse Tangent/Complex", "text": "{{:Definition:Inverse Tangent/Complex/Definition 1}}"}
+{"_id": "25031", "title": "Definition:Inverse Tangent/Complex/Definition 1", "text": "The '''inverse tangent''' is a multifunction defined on $S$ as: :$\\forall z \\in S: \\tan^{-1} \\left({z}\\right) := \\left\\{{w \\in \\C: \\tan \\left({w}\\right) = z}\\right\\}$ where $\\tan \\left({w}\\right)$ is the tangent of $w$."}
+{"_id": "25032", "title": "Definition:Inverse Tangent/Complex/Definition 2", "text": "The '''inverse tangent''' is a multifunction defined on $S$ as: :$\\forall z \\in S: \\tan^{-1} \\paren z := \\set {\\dfrac 1 {2 i} \\ln \\paren {\\dfrac {i - z} {i + z} } + k \\pi: k \\in \\Z}$ where $\\ln$ denotes the complex natural logarithm as a multifunction."}
+{"_id": "25033", "title": "Definition:Inverse Tangent/Complex/Arctangent", "text": "The principal branch of the complex inverse tangent function is defined as: :$\\map \\arctan z := \\dfrac 1 {2 i} \\, \\map \\Ln {\\dfrac {i - z} {i + z} }$ where $\\Ln$ denotes the principal branch of the complex natural logarithm."}
+{"_id": "25035", "title": "Definition:Inverse Cotangent/Complex", "text": "{{:Definition:Inverse Cotangent/Complex/Definition 1}}"}
+{"_id": "25036", "title": "Definition:Inverse Cotangent/Complex/Definition 1", "text": "The '''inverse cotangent''' is a multifunction defined on $S$ as: :$\\forall z \\in S: \\cot^{-1} \\left({z}\\right) := \\left\\{{w \\in \\C: \\cot \\left({w}\\right) = z}\\right\\}$ where $\\cot \\left({w}\\right)$ is the cotangent of $w$."}
+{"_id": "25037", "title": "Definition:Inverse Cotangent/Complex/Definition 2", "text": "The '''inverse cotangent''' is a multifunction defined on $S$ as: :$\\forall z \\in S: \\cot^{-1} \\left({z}\\right) := \\left\\{{\\dfrac 1 {2 i} \\ln \\left({\\dfrac {z + i} {z - i}}\\right) + k \\pi: k \\in \\Z}\\right\\}$ where $\\ln$ denotes the complex natural logarithm as a multifunction."}
+{"_id": "25038", "title": "Definition:Inverse Cotangent/Complex/Arccotangent", "text": "The principal branch of the complex inverse cotangent function is defined as: :$\\map \\arccot z := \\dfrac 1 {2 i} \\, \\map \\Ln {\\dfrac {z + i} {z - i} }$ where $\\Ln$ denotes the principal branch of the complex natural logarithm."}
+{"_id": "25040", "title": "Definition:Inverse Secant/Real", "text": "Let $x \\in \\R$ be a real number such that $x \\le -1$ or $x \\ge 1$. The '''inverse secant of $x$''' is the multifunction defined as: :$\\map {\\sec^{-1} } x := \\set {y \\in \\R: \\map \\sec y = x}$ where $\\map \\sec y$ is the secant of $y$."}
+{"_id": "25041", "title": "Definition:Inverse Secant/Complex", "text": "{{:Definition:Inverse Secant/Complex/Definition 1}}"}
+{"_id": "25042", "title": "Definition:Inverse Secant/Complex/Definition 1", "text": "Let $z \\in \\C_{\\ne 0}$ be a non-zero complex number. The '''inverse secant of $z$''' is the multifunction defined as: :$\\map {\\sec^{-1} } z := \\set {w \\in \\C: \\map \\sec w = z}$ where $\\map \\sec w$ is the secant of $w$."}
+{"_id": "25043", "title": "Definition:Inverse Secant/Complex/Definition 2", "text": "Let $z \\in \\C_{\\ne 0}$ be a non-zero complex number. The '''inverse secant of $z$''' is the multifunction defined as: :$\\sec^{-1} z := \\set {\\dfrac 1 i \\map \\ln {\\dfrac {1 + \\sqrt {\\size {1 - z^2} } e^{\\paren {i / 2} \\map \\arg {1 - z^2} } } z} + 2 k \\pi: k \\in \\Z}$ where: :$\\sqrt {\\size {1 - z^2} }$ denotes the positive square root of the complex modulus of $1 - z^2$ :$\\map \\arg {1 - z^2}$ denotes the argument of $1 - z^2$ :$\\ln$ denotes the complex natural logarithm as a multifunction."}
+{"_id": "25044", "title": "Definition:Inverse Secant/Complex/Arcsecant", "text": "The principal branch of the complex inverse secant function is defined as: :$\\forall z \\in \\C_{\\ne 0}: \\map \\arcsec z := \\dfrac 1 i \\, \\map \\Ln {\\dfrac {1 + \\sqrt {1 - z^2} } z}$ where: :$\\Ln$ denotes the principal branch of the complex natural logarithm :$\\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$."}
+{"_id": "25048", "title": "Definition:Inverse Cosecant/Complex", "text": "{{:Definition:Inverse Cosecant/Complex/Definition 1}}"}
+{"_id": "25049", "title": "Definition:Inverse Cosecant/Complex/Definition 1", "text": "Let $z \\in \\C_{\\ne 0}$ be a non-zero complex number. The '''inverse cosecant of $z$''' is the multifunction defined as: :$\\csc^{-1} \\left({z}\\right) := \\left\\{{w \\in \\C: \\csc \\left({w}\\right) = z}\\right\\}$ where $\\csc \\left({w}\\right)$ is the cosecant of $w$."}
+{"_id": "25050", "title": "Definition:Inverse Cosecant/Complex/Definition 2", "text": "Let $z \\in \\C_{\\ne 0}$ be a non-zero complex number. The '''inverse cosecant of $z$''' is the multifunction defined as: :$\\csc^{-1} \\left({z}\\right) := \\left\\{{\\dfrac 1 i \\ln \\left({\\dfrac {i + \\sqrt{\\left|{z^2 - 1}\\right|} e^{\\left({i / 2}\\right) \\arg \\left({z^2 - 1}\\right)}} z}\\right) + 2 k \\pi: k \\in \\Z}\\right\\}$ where: : $\\sqrt{\\left|{z^2 - 1}\\right|}$ denotes the positive square root of the complex modulus of $z^2 - 1$ : $\\arg \\left({z^2 - 1}\\right)$ denotes the argument of $z^2 - 1$ : $\\ln$ denotes the complex natural logarithm considered as a  multifunction."}
+{"_id": "25051", "title": "Definition:Inverse Cosecant/Complex/Arccosecant", "text": "The principal branch of the complex inverse cosecant function is defined as: :$\\forall z \\in \\C_{\\ne 0}: \\map \\arccsc z := \\dfrac 1 i \\, \\map \\Ln {\\dfrac {i + \\sqrt {z^2 - 1} } z}$ where: :$\\Ln$ denotes the principal branch of the complex natural logarithm :$\\sqrt {z^2 - 1}$ denotes the principal square root of $z^2 - 1$."}
+{"_id": "25052", "title": "Definition:Inverse Hyperbolic Sine/Complex/Arcsine", "text": "The principal branch of the complex inverse hyperbolic sine function is defined as: :$\\forall z \\in \\C: \\map {\\Sinh^{-1} } z := \\map \\Ln {z + \\sqrt {z^2 + 1} }$ where: :$\\Ln$ denotes the principal branch of the complex natural logarithm :$\\sqrt {z^2 + 1}$ denotes the principal square root of $z^2 + 1$."}
+{"_id": "25054", "title": "Definition:Inverse Hyperbolic Sine/Arcsine", "text": "{{:Definition:Complex Hyperbolic Arcsine}}"}
+{"_id": "25055", "title": "Definition:Inverse Hyperbolic Sine/Real/Definition 2", "text": "The '''inverse hyperbolic sine''' $\\sinh^{-1}: \\R \\to \\R$ is a real function defined on $\\R$ as: :$\\forall x \\in \\R: \\map {\\sinh^{-1} } x := \\map \\ln {x + \\sqrt {x^2 + 1} }$ where: :$\\ln$ denotes the natural logarithm of a (strictly positive) real number :$\\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$."}
+{"_id": "25057", "title": "Definition:Inverse Hyperbolic Cosine/Real/Definition 1", "text": "The '''inverse hyperbolic cosine''' $\\cosh^{-1}: S \\to \\R$ is a real function defined on $S$ as: :$\\forall x \\in S: \\cosh^{-1} \\left({x}\\right) := y \\in \\R_{\\ge 0}: x = \\cosh \\left({y}\\right)$ where $\\cosh \\left({y}\\right)$ denotes the hyperbolic cosine function."}
+{"_id": "25058", "title": "Definition:Inverse Hyperbolic Cotangent/Real/Definition 1", "text": "The '''inverse hyperbolic cotangent''' $\\coth^{-1}: S \\to \\R$ is a real function defined on $S$ as: :$\\forall x \\in S: \\coth^{-1} x := y \\in \\R: x = \\coth y$ where $\\coth y$ denotes the hyperbolic cotangent function."}
+{"_id": "25059", "title": "Definition:Inverse Hyperbolic Secant/Real/Definition 1", "text": "The '''inverse hyperbolic secant''' $\\operatorname{sech}^{-1}: S \\to \\R$ is a real function defined on $S$ as: :$\\forall x \\in S: \\operatorname{sech}^{-1} \\left({x}\\right) := y \\in \\R_{\\ge 0}: x = \\operatorname{sech} \\left({y}\\right)$ where $\\operatorname{sech} \\left({y}\\right)$ denotes the hyperbolic secant function."}
+{"_id": "25060", "title": "Definition:Inverse Hyperbolic Cosecant/Real/Definition 1", "text": "The '''inverse hyperbolic cosecant''' $\\csch^{-1}: \\R_{\\ne 0} \\to \\R$ is a real function defined on the non-zero real numbers $\\R_{\\ne 0}$ as: :$\\forall x \\in \\R_{\\ne 0}: \\map {\\csch^{-1} } x := y \\in \\R: x = \\map \\csch y$ where $\\map \\csch y$ denotes the hyperbolic cosecant function of $y$."}
+{"_id": "25062", "title": "Definition:Inverse Hyperbolic Cosine/Complex/Arccosine", "text": "The principal branch of the complex inverse hyperbolic cosine function is defined as: :$\\forall z \\in \\C: \\map {\\Cosh^{-1} } z := \\map \\Ln {z + \\sqrt {z^2 - 1} }$ where: :$\\Ln$ denotes the principal branch of the complex natural logarithm :$\\sqrt {z^2 - 1}$ denotes the principal square root of $z^2 - 1$."}
+{"_id": "25063", "title": "Definition:Inverse Hyperbolic Cosecant/Complex/Arccosecant", "text": "The principal branch of the complex inverse hyperbolic cosecant function is defined as: :$\\forall z \\in \\C_{\\ne 0}: \\map {\\Csch^{-1} } z := \\map \\Ln {\\dfrac {1 + \\sqrt {z^2 + 1} } z}$ where: :$\\Ln$ denotes the principal branch of the complex natural logarithm :$\\sqrt {z^2 + 1}$ denotes the principal square root of $z^2 + 1$."}
+{"_id": "25064", "title": "Definition:Inverse Hyperbolic Secant/Complex/Arcsecant", "text": "The principal branch of the complex inverse hyperbolic secant function is defined as: :$\\forall z \\in \\C: \\map {\\Sech^{-1} } z := \\map \\Ln {\\dfrac {1 + \\sqrt{1 - z^2} } z}$ where: :$\\Ln$ denotes the principal branch of the complex natural logarithm :$\\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$."}
+{"_id": "25065", "title": "Definition:Inverse Hyperbolic Tangent/Complex/Arctangent", "text": "The principal branch of the complex inverse hyperbolic tangent function is defined as: :$\\forall z \\in \\C: \\map {\\Tanh^{-1} } z := \\dfrac 1 2 \\, \\map \\Ln {\\dfrac {1 + z} {1 - z} }$ where $\\Ln$ denotes the principal branch of the complex natural logarithm."}
+{"_id": "25066", "title": "Definition:Inverse Hyperbolic Cotangent/Complex/Arccotangent", "text": "The principal branch of the complex inverse hyperbolic cotangent function is defined as: :$\\forall z \\in \\C: \\map {\\Coth^{-1} } z := \\dfrac 1 2 \\, \\map \\Ln {\\dfrac {z + 1} {z - 1} }$ where $\\Ln$ denotes the principal branch of the complex natural logarithm."}
+{"_id": "25067", "title": "Definition:Inverse Hyperbolic Cosine/Real/Definition 2", "text": "The '''inverse hyperbolic cosine''' $\\cosh^{-1}: S \\to \\R$ is a real function defined on $S$ as: :$\\forall x \\in S: \\map {\\cosh^{-1} } x := \\map \\ln {x + \\sqrt {x^2 - 1} }$ where: :$\\sqrt {x^2 - 1}$ denotes the positive square root of $x^2 - 1$ :$\\ln$ denotes the natural logarithm of a (strictly positive) real number."}
+{"_id": "25069", "title": "Definition:Inverse Hyperbolic Tangent/Real/Definition 1", "text": "The '''inverse hyperbolic tangent''' $\\tanh^{-1}: S \\to \\R$ is a real function defined on $S$ as: :$\\forall x \\in S: \\tanh^{-1} \\left({x}\\right) := y \\in \\R: x = \\tanh \\left({y}\\right)$ where $\\tanh \\left({y}\\right)$ denotes the hyperbolic tangent function."}
+{"_id": "25070", "title": "Definition:Inverse Hyperbolic Sine/Real/Definition 1", "text": "The '''inverse hyperbolic sine''' $\\sinh^{-1}: \\R \\to \\R$ is a real function defined on $\\R$ as: :$\\forall x \\in \\R: \\map {\\sinh^{-1} } x := y \\in \\R: x = \\map \\sinh y$ where $\\map \\sinh y$ denotes the hyperbolic sine function."}
+{"_id": "25075", "title": "Definition:Inverse Hyperbolic Cosine/Real", "text": "=== Definition 1 === {{:Definition:Inverse Hyperbolic Cosine/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cosine/Real/Definition 2}}"}
+{"_id": "25076", "title": "Definition:Image (Set Theory)/Mapping/Mapping/Definition 1", "text": "The '''image''' of a mapping $f: S \\to T$ is the set: :$\\Img f = \\set {t \\in T: \\exists s \\in S: \\map f s = t}$ That is, it is the set of values taken by $f$."}
+{"_id": "25077", "title": "Definition:Image (Set Theory)/Mapping/Mapping/Definition 2", "text": "The '''image''' of a mapping $f: S \\to T$ is the set: :$\\Img f = f \\sqbrk S$ where $f \\sqbrk S$ is the image of $S$ under $f$."}
+{"_id": "25081", "title": "Definition:Inverse Hyperbolic Tangent/Real", "text": "Let $S$ denote the open real interval: : $S := \\left({-1 \\,.\\,.\\, 1}\\right)$ === Definition 1 === {{:Definition:Inverse Hyperbolic Tangent/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Tangent/Real/Definition 2}}"}
+{"_id": "25082", "title": "Definition:Inverse Hyperbolic Cotangent/Real", "text": "=== Definition 1 === {{:Definition:Inverse Hyperbolic Cotangent/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cotangent/Real/Definition 2}}"}
+{"_id": "25083", "title": "Definition:Inverse Hyperbolic Secant/Real", "text": "=== Definition 1 === {{:Definition:Inverse Hyperbolic Secant/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Secant/Real/Definition 2}}"}
+{"_id": "25084", "title": "Definition:Inverse Hyperbolic Cosecant/Real", "text": "=== Definition 1 === {{:Definition:Inverse Hyperbolic Cosecant/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cosecant/Real/Definition 2}}"}
+{"_id": "25085", "title": "Definition:Dominate (Set Theory)/Definition 1", "text": "Then '''$S$ is dominated by $T$''' {{iff}} there exists an injection from $S$ to $T$."}
+{"_id": "25086", "title": "Definition:Dominate (Set Theory)/Definition 2", "text": "Then '''$S$ is dominated by $T$''' {{iff}} $S$ is equivalent to some subset of $T$. That is, {{iff}} there exists a bijection $f: S \\to T'$ for some $T' \\subseteq T$."}
+{"_id": "25087", "title": "Definition:Dominate (Set Theory)/Strictly Dominate", "text": "'''$S$ is strictly dominated by set $T$''' {{iff}} $S \\preccurlyeq T$ but $\\neg T \\preccurlyeq S$. This can be written $S \\prec T$ or $S < T$."}
+{"_id": "25088", "title": "Definition:Exponentiation of Cardinals", "text": "Let $A$ and $B$ be sets. Let $\\mathbf a$ and $\\mathbf b$ be the cardinals associated respectively with $A$ and $B$. Then the '''exponentiation''' of $\\mathbf a$ to $\\mathbf b$ is defined as: :$\\mathbf a^{\\mathbf b} := \\operatorname{Card} \\left({A^B}\\right)$ where: :$A^B$ denotes the set of all mappings from $B$ to $A$ :$\\operatorname{Card} \\left({A^B}\\right)$ denotes the cardinal associated with $A^B$."}
+{"_id": "25089", "title": "Definition:Inverse Hyperbolic Tangent/Real/Definition 2", "text": "The '''inverse hyperbolic tangent''' $\\tanh^{-1}: S \\to \\R$ is a real function defined on $S$ as: :$\\forall x \\in S: \\map {\\tanh^{-1} } x := \\dfrac 1 2 \\map \\ln {\\dfrac {1 + x} {1 - x} }$ where $\\ln$ denotes the natural logarithm of a (strictly positive) real number."}
+{"_id": "25090", "title": "Definition:Inverse Hyperbolic Cotangent/Real/Definition 2", "text": "The '''inverse hyperbolic cotangent''' $\\coth^{-1}: S \\to \\R$ is a real function defined on $S$ as: :$\\forall x \\in S: \\coth^{-1} x := \\dfrac 1 2 \\map \\ln {\\dfrac {x + 1} {x - 1} }$ where $\\ln$ denotes the natural logarithm of a (strictly positive) real number."}
+{"_id": "25091", "title": "Definition:Inverse Hyperbolic Secant/Real/Definition 2", "text": "The '''inverse hyperbolic secant''' $\\sech^{-1}: S \\to \\R$ is a real function defined on $S$ as: :$\\forall x \\in S: \\map {\\sech^{-1} } x := \\map \\ln {\\dfrac {1 + \\sqrt {1 - x^2} } x}$ where: :$\\sqrt {1 - x^2}$ denotes the positive square root of $1 - x^2$ :$\\ln$ denotes the natural logarithm of a (strictly positive) real number."}
+{"_id": "25092", "title": "Definition:Inverse Hyperbolic Cosecant/Real/Definition 2", "text": "The '''inverse hyperbolic cosecant''' $\\csch^{-1}: \\R_{\\ne 0} \\to \\R$ is a real function defined on the non-zero real numbers $\\R_{\\ne 0}$ as: :$\\forall x \\in \\R_{\\ne 0}: \\map {\\csch^{-1} } x := \\map \\ln {\\dfrac 1 x + \\dfrac {\\sqrt {x^2 + 1} } {\\size x} }$ where: :$\\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$ :$\\ln$ denotes the natural logarithm of a (strictly positive) real number."}
+{"_id": "25093", "title": "Definition:Real Number/Axiomatic Definition", "text": "Let $\\struct {R, +, \\times, \\le}$ be a Dedekind complete ordered field. Then $R$ is called the '''(field of) real numbers'''. ==== Real Number Axioms ==== {{:Definition:Real Number/Axioms}}"}
+{"_id": "25094", "title": "Definition:Natural Numbers/Inductive Sets in Real Numbers", "text": "Let $\\R$ be the set of real numbers. Let $\\II$ be the set of all inductive sets defined as subsets of $\\R$. Then the '''natural numbers''' $\\N$ are defined as: :$\\N := \\displaystyle \\bigcap \\II$ where $\\displaystyle \\bigcap$ denotes intersection."}
+{"_id": "25096", "title": "Definition:Square Root/Positive Real", "text": "Let $x \\in \\R_{\\ge 0}$ be a positive real number. The '''square roots of $x$''' are the real numbers defined as: :$x^{\\paren {1 / 2} } := \\set {y \\in \\R: y^2 = x}$ where $x^{\\paren {1 / 2} }$ is the $2$nd root of $x$. The notation: :$y = \\pm \\sqrt x$ is usually encountered."}
+{"_id": "25097", "title": "Definition:Naturally Ordered Semigroup", "text": "The concept of a '''naturally ordered semigroup''' is intended to capture the behaviour of the natural numbers $\\N$, addition $+$ and the ordering $\\le$ as they pertain to $\\N$. === Naturally Ordered Semigroup Axioms === {{:Definition:Naturally Ordered Semigroup/Axioms}}"}
+{"_id": "25098", "title": "Definition:Square Root/Positive", "text": "The '''positive square root of $x$''' is the number defined as: :$+ \\sqrt x := y \\in \\R_{>0}: y^2 = x$"}
+{"_id": "25099", "title": "Definition:Square Root/Negative", "text": "The '''negative square root of $x$''' is the number defined as: :$- \\sqrt x := y \\in \\R_{<0}: y^2 = x$"}
+{"_id": "25100", "title": "Definition:Square Root/Negative Real", "text": "Let $x \\in \\R_{< 0}$ be a (strictly) negative real number. Then the '''square root of $x$''' is defined as: :$\\sqrt x = i \\paren {\\pm \\sqrt {-x} }$ where $i$ is the imaginary unit: :$i^2 = -1$"}
+{"_id": "25101", "title": "Definition:Square Root/Complex Number", "text": "{{:Definition:Square Root/Complex Number/Definition 1}}"}
+{"_id": "25102", "title": "Definition:Square Root/Complex Number/Definition 1", "text": "Let $z \\in \\C$ be a complex number expressed in polar form as $\\left \\langle{r, \\theta}\\right\\rangle = r \\left({\\cos \\theta + i \\sin \\theta}\\right)$. The '''square root of $z$''' is the $2$-valued multifunction: {{begin-eqn}} {{eqn | l = z^{1/2}       | r = \\left\\{ {\\sqrt r \\left({\\cos \\left({\\frac {\\theta + 2 k \\pi} 2}\\right) + i \\sin \\left({\\frac {\\theta + 2 k \\pi} 2}\\right) }\\right): k \\in \\left\\{ {0, 1}\\right\\} }\\right\\}       | c =  }} {{eqn | r = \\left\\{ {\\sqrt r \\left({\\cos \\left({\\frac \\theta 2 + k \\pi}\\right) + i \\sin \\left({\\frac \\theta 2 + k \\pi}\\right) }\\right): k \\in \\left\\{ {0, 1}\\right\\} }\\right\\}       | c =  }} {{end-eqn}} where $\\sqrt r$ denotes the positive square root of $r$."}
+{"_id": "25103", "title": "Definition:Square Root/Complex Number/Definition 2", "text": "Let $z \\in \\C$ be a complex number expressed in polar form as $\\left \\langle{r, \\theta}\\right\\rangle = r \\left({\\cos \\theta + i \\sin \\theta}\\right)$. The '''square root of $z$''' is the $2$-valued multifunction: :$z^{1/2} = \\left\\{ {\\pm \\sqrt r \\left({\\cos \\left({\\dfrac \\theta 2}\\right) + i \\sin \\left({\\dfrac \\theta 2}\\right) }\\right)}\\right\\}$ where $\\pm \\sqrt r$ denotes the positive and negative square roots of $r$."}
+{"_id": "25104", "title": "Definition:Square Root/Complex Number/Definition 3", "text": "Let $z \\in \\C$ be a complex number. The '''square root of $z$''' is the $2$-valued multifunction: :$z^{1/2} = \\left\\{ {\\sqrt {\\left\\vert{z}\\right\\vert} e^{\\left({i / 2}\\right) \\arg \\left({z}\\right)} }\\right\\}$ where: : $\\sqrt {\\left\\vert{z}\\right\\vert}$ denotes the positive square root of the complex modulus of $z$ : $\\arg \\left({z}\\right)$ denotes the argument of $z$ considered as a  multifunction."}
+{"_id": "25105", "title": "Definition:Square Root/Complex Number/Definition 4", "text": "Let $z \\in \\C$ be a complex number. The '''square root of $z$''' is the $2$-valued multifunction: :$z^{1/2} = \\left\\{ {w \\in \\C: w^2 = z}\\right\\}$"}
+{"_id": "25106", "title": "Definition:Gamma Function/Integral Form", "text": "The '''Gamma function''' $\\Gamma: \\C \\to \\C$ is defined, for the open right half-plane, as: :$\\displaystyle \\map \\Gamma z = \\map {\\MM \\set {e^{-t} } } z = \\int_0^{\\to \\infty} t^{z - 1} e^{-t} \\rd t$ where $\\MM$ is the Mellin transform. For all other values of $z$ except the non-positive integers, $\\map \\Gamma z$ is defined as: :$\\map \\Gamma {z + 1} = z \\, \\map \\Gamma z$"}
+{"_id": "25107", "title": "Definition:Gamma Function/Weierstrass Form", "text": "The '''{{AuthorRef|Karl Theodor Wilhelm Weierstrass|Weierstrass}} form''' of the '''Gamma function''' is: :$\\displaystyle \\frac 1 {\\map \\Gamma z} = z e^{\\gamma z} \\prod_{n \\mathop = 1}^\\infty \\paren {\\paren {1 + \\frac z n} e^{-z / n} }$ where $\\gamma$ is the Euler-Mascheroni constant. The '''Weierstrass form''' is valid for all $\\C$."}
+{"_id": "25108", "title": "Definition:Gamma Function/Euler Form", "text": "The '''{{AuthorRef|Leonhard Paul Euler|Euler}} form''' of the '''Gamma function''' is: :$\\displaystyle \\Gamma \\left({z}\\right) = \\frac 1 z \\prod_{n \\mathop = 1}^\\infty \\left({\\left({1 + \\frac 1 n}\\right)^z \\left({1 + \\frac z n}\\right)^{-1}}\\right) = \\lim_{m \\mathop \\to \\infty} \\frac {m^z m!} {z \\left({z + 1}\\right) \\left({z + 2}\\right) \\cdots \\left({z + m}\\right)}$ which is valid except for $z \\in \\left\\{{0, -1, -2, \\ldots}\\right\\}$."}
+{"_id": "25109", "title": "Definition:Square Root/Complex Number/Principal Square Root", "text": "Let $z \\in \\C$ be a complex number. Let $z^{1/2} = \\set {w \\in \\C: w^2 = z}$ be the square root of $z$. The '''principal square root''' of $z$ is the element $w$ of $z^{1/2}$ such that: :$\\begin{cases} \\map \\Im w > 0 : & \\map \\Im z \\ne 0 \\\\ \\map \\Re w \\ge 0 : & \\map \\Im z = 0 \\end{cases}$"}
+{"_id": "25110", "title": "Definition:Reduction Formula", "text": "Let $f: \\R \\times \\Z_{\\ge 0}: \\R$ be a Darboux integrable real-valued function on $\\R \\times \\N$. Let: :$\\displaystyle I_n = \\int \\map f {x, n} \\rd x$ for some $n \\in \\Z_{\\ge 0}$. A '''reduction formula''' is a recurrence relation of the form: :$\\displaystyle I_k = \\map g {\\displaystyle I_n}$ such that $k < n$. That is, it is a technique to '''reduce''' the integer parameter in the integrand in order to allow evaluation of the integral at, usually, $n = 0$ or $n = 1$. This technique relies upon the supposition that $\\map f {x, 0}$ or $\\map f {x, 1}$ can be integrated directly. Category:Definitions/Proof Techniques Category:Definitions/Integral Calculus 1kwzi5xzaqj9e47w5qfe5ui8vd20c7g"}
+{"_id": "25111", "title": "Definition:Zero Complement", "text": "Let $\\struct {S, \\circ, \\preceq}$ be a naturally ordered semigroup. Let $0$ be the zero of $S$. Let $S^* := \\relcomp S {\\set 0} = S \\setminus \\set 0$ be the complement of $\\set 0$ in $S$. Then $S^*$ is called the '''zero complement''' of $S$."}
+{"_id": "25112", "title": "Definition:Beta Function", "text": "{{:Definition:Beta Function/Definition 1}}"}
+{"_id": "25113", "title": "Definition:Subtraction/Naturally Ordered Semigroup", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a naturally ordered semigroup. Let $m, n \\in S$ such that $m \\preceq n$. By axiom $(NO3)$, there exists a $p \\in S$ such that: :$m + p = n$ This $p$ is the '''difference between $m$ and $n$''', and denoted $n - m$. The operation $-$, assigning to $m, n \\in S$ with $m \\preceq n$ their '''difference''' $n - m$ is called '''subtraction'''."}
+{"_id": "25114", "title": "Definition:Bernoulli Numbers", "text": "The '''Bernoulli numbers''' $B_n$ are a sequence of rational numbers defined by:  === Generating Function === {{:Definition:Bernoulli Numbers/Generating Function}} === Recurrence Relation === {{:Definition:Bernoulli Numbers/Recurrence Relation}}"}
+{"_id": "25115", "title": "Definition:Convex Real Function/Definition 1", "text": "$f$ is '''convex on $I$''' {{iff}}: :$\\forall x, y \\in I: \\forall \\alpha, \\beta \\in \\R_{>0}, \\alpha + \\beta = 1: f \\left({\\alpha x + \\beta y}\\right) \\le \\alpha f \\left({x}\\right) + \\beta f \\left({y}\\right)$"}
+{"_id": "25116", "title": "Definition:Convex Real Function/Definition 2", "text": "$f$ is '''convex on $I$''' {{iff}}: :$\\forall x_1, x_2, x_3 \\in I: x_1 < x_2 < x_3: \\dfrac {f \\left({x_2}\\right) - f \\left({x_1}\\right)} {x_2 - x_1} \\le \\dfrac {f \\left({x_3}\\right) - f \\left({x_2}\\right)} {x_3 - x_2}$"}
+{"_id": "25117", "title": "Definition:Convex Real Function/Definition 3", "text": "$f$ is '''convex on $I$''' {{iff}}: :$\\forall x_1, x_2, x_3 \\in I: x_1 < x_2 < x_3: \\dfrac {f \\left({x_2}\\right) - f \\left({x_1}\\right)} {x_2 - x_1} \\le \\dfrac {f \\left({x_3}\\right) - f \\left({x_1}\\right)} {x_3 - x_1}$"}
+{"_id": "25118", "title": "Definition:Concave Real Function/Definition 1", "text": "$f$ is '''concave on $I$''' {{iff}}: :$\\forall x, y \\in I: \\forall \\alpha, \\beta \\in \\R_{>0}, \\alpha + \\beta = 1: f \\left({\\alpha x + \\beta y}\\right) \\ge \\alpha f \\left({x}\\right) + \\beta f \\left({y}\\right)$"}
+{"_id": "25119", "title": "Definition:Concave Real Function/Definition 2", "text": "$f$ is '''concave on $I$''' {{iff}}: :$\\displaystyle \\forall x_1, x_2, x_3 \\in I: x_1 < x_2< x_3: \\frac {f \\left({x_2}\\right) - f \\left({x_1}\\right)} {x_2 - x_1} \\ge \\frac {f \\left({x_3}\\right) - f \\left({x_2}\\right)} {x_3 - x_2}$"}
+{"_id": "25120", "title": "Definition:Concave Real Function/Definition 3", "text": "$f$ is '''concave on $I$''' {{iff}}: :$\\forall x_1, x_2, x_3 \\in I: x_1 < x_2< x_3: \\dfrac {f \\left({x_2}\\right) - f \\left({x_1}\\right)} {x_2 - x_1} \\ge \\dfrac {f \\left({x_3}\\right) - f \\left({x_1}\\right)} {x_3 - x_1}$"}
+{"_id": "25121", "title": "Definition:Convex Real Function/Definition 1/Strictly", "text": "$f$ is '''strictly convex on $I$''' {{iff}}: :$\\forall x, y \\in I, x \\ne y: \\forall \\alpha, \\beta \\in \\R_{>0}, \\alpha + \\beta = 1: f \\left({\\alpha x + \\beta y}\\right) < \\alpha f \\left({x}\\right) + \\beta f \\left({y}\\right)$"}
+{"_id": "25122", "title": "Definition:Convex Real Function/Definition 2/Strictly", "text": "$f$ is '''strictly convex on $I$''' {{iff}}: :$\\forall x_1, x_2, x_3 \\in I: x_1 < x_2 < x_3: \\dfrac {f \\left({x_2}\\right) - f \\left({x_1}\\right)} {x_2 - x_1} < \\dfrac {f \\left({x_3}\\right) - f \\left({x_2}\\right)} {x_3 - x_2}$"}
+{"_id": "25123", "title": "Definition:Convex Real Function/Definition 3/Strictly", "text": "$f$ is '''strictly convex on $I$''' {{iff}}: :$\\forall x_1, x_2, x_3 \\in I: x_1 < x_2 < x_3: \\dfrac {f \\left({x_2}\\right) - f \\left({x_1}\\right)} {x_2 - x_1} < \\dfrac {f \\left({x_3}\\right) - f \\left({x_1}\\right)} {x_3 - x_1}$"}
+{"_id": "25124", "title": "Definition:Concave Real Function/Definition 1/Strictly", "text": "$f$ is '''strictly concave on $I$''' {{iff}}: :$\\forall x, y \\in I, x \\ne y: \\forall \\alpha, \\beta \\in \\R_{>0}, \\alpha + \\beta = 1: f \\left({\\alpha x + \\beta y}\\right) > \\alpha f \\left({x}\\right) + \\beta f \\left({y}\\right)$"}
+{"_id": "25125", "title": "Definition:Concave Real Function/Definition 2/Strictly", "text": "$f$ is '''strictly concave on $I$''' {{iff}}: :$\\forall x_1, x_2, x_3 \\in I: x_1 < x_2 < x_3: \\dfrac {f \\left({x_2}\\right) - f \\left({x_1}\\right)} {x_2 - x_1} > \\dfrac {f \\left({x_3}\\right) - f \\left({x_2}\\right)} {x_3 - x_2}$"}
+{"_id": "25126", "title": "Definition:Concave Real Function/Definition 3/Strictly", "text": "$f$ is '''strictly concave on $I$''' {{iff}}: :$\\forall x_1, x_2, x_3 \\in I: x_1 < x_2 < x_3: \\dfrac {f \\left({x_2}\\right) - f \\left({x_1}\\right)} {x_2 - x_1} > \\dfrac {f \\left({x_3}\\right) - f \\left({x_1}\\right)} {x_3 - x_1}$"}
+{"_id": "25127", "title": "Definition:Beta Function/Definition 1", "text": ":$\\displaystyle \\map \\Beta {x, y} := \\int_{\\mathop \\to 0}^{\\mathop \\to 1} t^{x - 1} \\paren {1 - t}^{y - 1} \\rd t$"}
+{"_id": "25128", "title": "Definition:Beta Function/Definition 2", "text": ":$\\displaystyle \\Beta \\left({x, y}\\right) := 2 \\int_0^{\\pi / 2} \\left({\\sin \\theta}\\right)^{2 x - 1} \\left({\\cos \\theta}\\right)^{2 y - 1} \\rd \\theta$"}
+{"_id": "25129", "title": "Definition:Beta Function/Definition 3", "text": ":$\\map \\Beta {x, y} := \\dfrac {\\map \\Gamma x \\, \\map \\Gamma y} {\\map \\Gamma {x + y} }$ where $\\Gamma$ is the Gamma function."}
+{"_id": "25130", "title": "Definition:Improper Integral/Half Open Interval", "text": "==== Open Above ==== {{:Definition:Improper Integral/Half Open Interval/Open Above}} ==== Open Below ==== {{:Definition:Improper Integral/Half Open Interval/Open Below}}"}
+{"_id": "25131", "title": "Definition:Improper Integral/Open Interval", "text": "Let $f$ be a real function which is continuous on the open interval $\\openint a b$. Then the improper integral of $f$ over $\\openint a b$ is defined as: :$\\displaystyle \\int_{\\mathop \\to a}^{\\mathop \\to b} \\map f t \\rd t := \\lim_{\\gamma \\mathop \\to a} \\int_\\gamma^c \\map f t \\rd t + \\lim_{\\gamma \\mathop \\to b} \\int_c^\\gamma \\map f t \\rd t$ for some $c \\in \\openint a b$."}
+{"_id": "25132", "title": "Definition:Improper Integral/Unbounded Closed Interval", "text": "==== Unbounded Above ==== {{:Definition:Improper Integral/Unbounded Closed Interval/Unbounded Above}} ==== Unbounded Below ==== {{:Definition:Improper Integral/Unbounded Closed Interval/Unbounded Below}}"}
+{"_id": "25133", "title": "Definition:Improper Integral/Unbounded Open Interval/Unbounded Above", "text": "Let $f$ be a real function which is continuous on the unbounded open interval $\\openint a {+\\infty}$. Then the improper integral of $f$ over $\\openint a {+\\infty}$ is defined as: :$\\displaystyle \\int_{\\mathop \\to a}^{\\mathop \\to +\\infty} \\map f t \\rd t := \\lim_{\\gamma \\mathop \\to a} \\int_\\gamma^c \\map f t \\rd t + \\lim_{\\gamma \\mathop \\to +\\infty} \\int_c^\\gamma \\map f t \\rd t$ for some $c \\in \\openint a {+\\infty}$."}
+{"_id": "25134", "title": "Definition:Improper Integral/Unbounded Open Interval/Unbounded Below", "text": "Let $f$ be a real function which is continuous on the unbounded open interval $\\openint {-\\infty} b$. Then the improper integral of $f$ over $\\openint {-\\infty} b$ is defined as: :$\\displaystyle \\int_{\\mathop \\to -\\infty}^{\\mathop \\to b} \\map f t \\rd t := \\lim_{\\gamma \\mathop \\to -\\infty} \\int_\\gamma^c \\map f t \\rd t + \\lim_{\\gamma \\mathop \\to b} \\int_c^\\gamma \\map f t \\rd t$ for some $c \\in \\openint {-\\infty} b$."}
+{"_id": "25135", "title": "Definition:Improper Integral/Unbounded Open Interval/Unbounded Above and Below", "text": "Let $f$ be a real function which is continuous everywhere. Then the improper integral of $f$ over $\\R$ is defined as: :$\\displaystyle \\int_{\\mathop \\to -\\infty}^{\\mathop \\to +\\infty} \\map f t \\rd t := \\lim_{\\gamma \\mathop \\to -\\infty} \\int_\\gamma^c \\map f t \\rd t + \\lim_{\\gamma \\mathop \\to +\\infty} \\int_c^\\gamma \\map f t \\rd t$ for some $c \\in \\R$. Usually $c$ is taken to be $0$ as this usually simplifies the evaluation of the expressions."}
+{"_id": "25136", "title": "Definition:Improper Integral/Unbounded Open Interval", "text": "==== Unbounded Above ==== {{:Definition:Improper Integral/Unbounded Open Interval/Unbounded Above}} ==== Unbounded Below ==== {{:Definition:Improper Integral/Unbounded Open Interval/Unbounded Below}} A specific and important instance of this occurs when the interval in question is the set of all real numbers: ==== Unbounded Above and Below ==== {{:Definition:Improper Integral/Unbounded Open Interval/Unbounded Above and Below}}"}
+{"_id": "25137", "title": "Definition:Improper Integral/Half Open Interval/Open Above", "text": "Let $f$ be a real function which is continuous on the half open interval $\\hointr a b$. Then the improper integral of $f$ over $\\hointr a b$ is defined as: :$\\displaystyle \\int_a^{\\mathop \\to b} \\map f t \\rd t := \\lim_{\\gamma \\mathop \\to b} \\int_a^\\gamma \\map f t \\rd t$"}
+{"_id": "25138", "title": "Definition:Improper Integral/Half Open Interval/Open Below", "text": "Let $f$ be a real function which is continuous on the half open interval $\\hointl a b$. Then the improper integral of $f$ over $\\hointl a b$ is defined as: :$\\displaystyle \\int_{\\mathop \\to a}^b \\map f t \\rd t := \\lim_{\\gamma \\mathop \\to a} \\int_\\gamma^b \\map f t \\rd t$"}
+{"_id": "25139", "title": "Definition:Improper Integral/Unbounded Closed Interval/Unbounded Above", "text": "Let $f$ be a real function which is continuous on the unbounded closed interval $\\hointr a {+\\infty}$. Then the improper integral of $f$ over $\\hointr a {+\\infty}$ is defined as: :$\\displaystyle \\int_a^{\\mathop \\to + \\infty} \\map f t \\rd t := \\lim_{\\gamma \\mathop \\to +\\infty} \\int_a^\\gamma \\map f t \\rd t$"}
+{"_id": "25140", "title": "Definition:Improper Integral/Unbounded Closed Interval/Unbounded Below", "text": "Let $f$ be a real function which is continuous on the unbounded closed interval $\\hointl {-\\infty} b$. Then the improper integral of $f$ over $\\hointl {-\\infty} b$ is defined as: :$\\displaystyle \\int_{\\mathop \\to -\\infty}^b \\map f t \\rd t := \\lim_{\\gamma \\mathop \\to -\\infty} \\int_\\gamma^b \\map f t \\rd t$"}
+{"_id": "25141", "title": "Definition:Positive Real Function", "text": "Let $I$ be a real interval. Let $f$ be a real function. Then $f$ is '''positive (on $I$)''' {{iff}}: :$\\forall x \\in I: f \\left({x}\\right) \\ge 0$"}
+{"_id": "25142", "title": "Definition:Negative Real Function", "text": "Let $I$ be a real interval. Let $f$ be a real function. Then $f$ is '''negative (on $I$)''' {{iff}}: :$\\forall x \\in I: \\map f x \\le 0$"}
+{"_id": "25143", "title": "Definition:Strictly Positive Real Function", "text": "Let $I$ be a real interval. Let $f$ be a real function. Then $f$ is '''strictly positive (on $I$)''' iff: :$\\forall x \\in I: f \\left({x}\\right) > 0$"}
+{"_id": "25144", "title": "Definition:Strictly Negative Real Function", "text": "Let $I$ be a real interval. Let $f$ be a real function. Then $f$ is '''strictly negative (on $I$)''' {{iff}}: :$\\forall x \\in I: \\map f x < 0$"}
+{"_id": "25147", "title": "Definition:Horizontal Line", "text": "=== Definition 1 === {{Definition:Horizontal Line/Definition 1}} === Definition 2 === {{Definition:Horizontal Line/Definition 2}} === Definition 3 === {{Definition:Horizontal Line/Definition 3}} File:Horizontal.png"}
+{"_id": "25148", "title": "Definition:Vertical Line", "text": "=== Definition 1 === {{:Definition:Vertical Line/Definition 1}} === Definition 2 === {{:Definition:Vertical Line/Definition 2}} :::File:Vertical.png"}
+{"_id": "25149", "title": "Definition:Laplace Transform", "text": "Let $f: \\R_{\\ge 0} \\to \\mathbb F$ be a function of a real variable $t$, where $\\mathbb F \\in \\set {\\R, \\C}$. The '''Laplace transform''' of $f$, denoted $\\laptrans f$ or $F$, is defined as: :$\\laptrans {\\map f t} = \\map F s = \\displaystyle \\int_0^{\\to +\\infty} e^{-s t} \\map f t \\rd t$ whenever this improper integral converges. If this improper integral does not converge, then $\\laptrans {\\map f t}$ does not exist."}
+{"_id": "25150", "title": "Definition:Euler Numbers", "text": "The '''Euler Numbers''' $E_n$ are a sequence of integers defined by the exponential generating function: :$\\displaystyle \\sech x = \\frac {2 e^x} {e^{2 x} + 1} = \\sum_{n \\mathop = 0}^\\infty \\frac {E_n x^n} {n!}$ where $\\size x < \\dfrac \\pi 2$."}
+{"_id": "25151", "title": "Definition:Catalan Number", "text": "The '''Catalan Numbers''' $C_n$ are a sequence of natural numbers defined by: :$C_n = \\dfrac 1 {n + 1} \\dbinom {2 n} n $"}
+{"_id": "25152", "title": "Definition:Population/Infinite", "text": "An '''infinite population''' is a population which is infinite."}
+{"_id": "25153", "title": "Definition:Population/Finite", "text": "A '''finite population''' is a population which is finite."}
+{"_id": "25154", "title": "Definition:Statistics", "text": "'''Statistics''' is the branch of science which studies the collection and analysis of data, and the presentation of organized inferences from that analysis. It is also concerned with the design of experiments and sample surveys, data reduction, data processing, and various other things. As such, it can be considered as a form of applied probability theory. It has been defined as '''the technology of the scientific method'''."}
+{"_id": "25155", "title": "Definition:Inductive Statistics", "text": "'''Inductive statistics''' is the phase of statistics which is concerned with the conditions under which conclusions about populations can be drawn from analysis of particular samples."}
+{"_id": "25156", "title": "Definition:Descriptive Statistics", "text": "'''Descriptive statistics''' is the phase of statistics whose task is to collect, analyze and summarize data about statistical samples, without drawing conclusions or inferences about the populations from which those samples are taken."}
+{"_id": "25157", "title": "Definition:Variable/Domain", "text": "The collection of all possible objects that a variable may refer to has to be specified. This collection is the '''domain''' of the variable."}
+{"_id": "25158", "title": "Definition:Variable/Continuous", "text": "A '''continuous variable''' is a variable which can take any value between two given values."}
+{"_id": "25159", "title": "Definition:Variable/Discrete", "text": "A '''discrete variable''' is a variable which is not continuous."}
+{"_id": "25160", "title": "Definition:Sample Statistic/Discrete", "text": "Data which can be described with a discrete variable are known as '''discrete data'''."}
+{"_id": "25161", "title": "Definition:Sample Statistic/Continuous", "text": "Data which can be described with a continuous variable are known as '''continuous data'''."}
+{"_id": "25162", "title": "Definition:Measurable Property/Measurement", "text": "'''Measurement''' is the process of determining the quantity of a measurable property. A '''measurement''' is reported as a (real) number multiplied by a unit of measurement for that quantity."}
+{"_id": "25163", "title": "Definition:Rounding", "text": "'''Rounding''' is the process of approximation of a value of a variable to a multiple of a given power of whatever number base one is working in (usually decimal). Let $n \\in \\Z$ be an integer. Let $x \\in \\R$ be a real number. Let $y \\in \\R$ such that: :$y = 10^n \\floor {\\dfrac x {10^n} + \\dfrac 1 2}$ or: :$y = 10^n \\ceiling {\\dfrac x {10^n} - \\dfrac 1 2}$ where $\\floor {\\, \\cdot \\,}$ denotes the floor function and $\\ceiling {\\, \\cdot \\,}$ denotes the ceiling function. Then $y$ is defined as '''$x$ rounded to the nearest $n$th power of $10$'''. Both of these definitions amount to the same thing, except for when $\\dfrac x {10^n}$ is exactly halfway between $\\floor {\\dfrac x {10^n} }$ and $\\ceiling {\\dfrac x {10^n} }$. How these instances is treated is known as the '''treatment of the half'''."}
+{"_id": "25164", "title": "Definition:Derivative/Higher Derivatives/Second Derivative", "text": "Let $f$ be a real function which is differentiable on an open interval $I$. Hence $f'$ is defined on $I$ as the derivative of $f$. Let $\\xi \\in I$ be a point in $I$. Let $f'$ be differentiable at the point $\\xi$. Then the '''second derivative''' $\\map {f''} \\xi$ is defined as: :$\\displaystyle f'' := \\lim_{x \\mathop \\to \\xi} \\dfrac {\\map {f'} x - \\map {f'} \\xi} {x - \\xi}$"}
+{"_id": "25165", "title": "Definition:Derivative/Higher Derivatives/Third Derivative", "text": "Let $f$ be a real function which is twice differentiable on an open interval $I$. Let $f''$ denote the second derivate. Then the '''third derivative''' $f'''$ is defined as: :$f''' := \\dfrac {\\d} {\\d x} f'' = \\map {\\dfrac {\\d} {\\d x} } {\\dfrac {\\d^2} {\\d x^2} f}$"}
+{"_id": "25166", "title": "Definition:Derivative/Higher Derivatives/Higher Order", "text": "The $n$th derivative of a function $y = \\map f x$ is defined as: :$\\map {f^{\\paren n} } x = \\dfrac {\\d^n y} {\\d x^n} := \\begin {cases} \\map {\\dfrac \\d {\\d x} } {\\dfrac {\\d^{n - 1} y} {\\d x^{n - 1} } } & : n > 0 \\\\ y & : n = 0 \\end {cases}$ assuming appropriate differentiability for a given $f^{\\paren {n - 1} }$."}
+{"_id": "25167", "title": "Definition:Derivative/Higher Derivatives/Zeroth Derivative", "text": "The '''zeroth derivative''' of a real function $f$ is defined as $f$ itself: :$f^{\\paren 0} := f$ where $f^{\\paren n}$ denotes the $n$th derivative of $f$."}
+{"_id": "25168", "title": "Definition:Dirichlet Eta Function", "text": "The '''Dirichlet $\\eta$ (eta) function''' $\\eta$ is the complex function defined on the half-plane $\\map \\Re s > 0$ as the series: :$\\displaystyle \\map \\eta s = \\sum_{n \\mathop = 1}^\\infty \\paren {-1}^{n - 1} n^{-s}$"}
+{"_id": "25169", "title": "Definition:Partial Derivative/Second Derivative", "text": "Let $\\map f {x, y}$ be a function of the two independent variables $x$ and $y$. The '''second partial derivatives of $f$ with respect to $x$ and $y$''' are defined and denoted by: {{begin-eqn}} {{eqn | n = 1       | l = \\dfrac {\\partial^2 f} {\\partial x^2}       | r = \\map {\\dfrac \\partial {\\partial x} } {\\dfrac {\\partial f} {\\partial x} }       | rr= =: \\map {f_{1 1} } {x, y}       | c =  }} {{eqn | n = 2       | l = \\dfrac {\\partial^2 f} {\\partial y^2}       | r = \\map {\\dfrac \\partial {\\partial y} } {\\dfrac {\\partial f} {\\partial y} }       | rr= =: \\map {f_{2 2} } {x, y}       | c =  }} {{eqn | n = 3       | l = \\quad \\dfrac {\\partial^2 f} {\\partial x \\partial y}       | r = \\map {\\dfrac \\partial {\\partial x} } {\\dfrac {\\partial f} {\\partial y} }       | rr= =: \\map {f_{2 1} } {x, y}       | c =  }} {{eqn | n = 4       | l = \\dfrac {\\partial^2 f} {\\partial y \\partial x}       | r = \\map {\\dfrac \\partial {\\partial y} } {\\dfrac {\\partial f} {\\partial x} }       | rr= =: \\map {f_{1 2} } {x, y}       | c =  }} {{end-eqn}}"}
+{"_id": "25170", "title": "Definition:Primitive (Calculus)/Integration", "text": "The process of finding a primitive for a function is known as '''integration'''."}
+{"_id": "25171", "title": "Definition:Bernoulli Numbers/Generating Function", "text": ":$\\displaystyle \\frac x {e^x - 1} = \\sum_{n \\mathop = 0}^\\infty \\frac {B_n x^n} {n!}$"}
+{"_id": "25172", "title": "Definition:Bernoulli Numbers/Recurrence Relation", "text": ":$B_n = \\begin{cases} 1 & : n = 0 \\\\ \\displaystyle - \\sum_{k \\mathop = 0}^{n - 1} \\binom n k \\frac {B_k} {n + 1 - k} & : n > 0 \\end{cases}$ or equivalently: :$B_n = \\begin{cases} 1 & : n = 0 \\\\ \\displaystyle - \\frac 1 {n+1} \\sum_{k \\mathop = 0}^{n - 1} \\binom {n+1} k B_k & : n > 0 \\end{cases}$"}
+{"_id": "25173", "title": "Definition:Logarithmic Integral/Eulerian", "text": "The '''Eulerian logarithmic integral''' is defined as: :$\\displaystyle \\map \\Li x = \\int_2^x \\frac {\\d t} {\\map \\ln t}$"}
+{"_id": "25174", "title": "Definition:Heaviside Step Function", "text": "Let $c \\ge 0$ be a constant real number. The '''Heaviside step function on $c$''' is the real function $u_c: \\R \\to \\R$ defined as: :$\\map {u_c} t := \\begin{cases} 1 & : t > c \\\\ 0 & : t < c \\end{cases}$ If $c = 0$, the subscript is often omitted: :$\\map u t := \\begin{cases} 1 & : t > 0 \\\\ 0 & : t < 0 \\end{cases}$"}
+{"_id": "25175", "title": "Definition:Doubly Periodic Function", "text": "Let $f: \\C \\to \\C$ be a complex function.  Then $f \\left({z}\\right)$ is a '''doubly-periodic function''' if there exist $\\omega_1, \\omega_2 \\in \\C$ such that: : $(1): \\quad \\omega_1, \\omega_2 \\ne 0$ : $(2): \\quad \\dfrac {\\omega_1} {\\omega_2} \\notin \\R$ : $(3): \\quad \\forall z \\in \\C: f \\left({z}\\right) = f \\left({z + \\omega_1}\\right) = f \\left({z + \\omega_2}\\right)$"}
+{"_id": "25176", "title": "Definition:Elliptic Function", "text": "Let $\\displaystyle y \\left({x}\\right) = \\int_0^x \\dfrac {\\d t} {\\sqrt {P \\left({t}\\right)} }$ be an elliptic integral, where $P \\left({t}\\right)$ is a polynomial of degree $3$ or $4$. Consider the inverse of $y \\left({x}\\right)$: :$x = \\phi \\left({y}\\right)$ Then $\\phi$ is an '''elliptic function'''."}
+{"_id": "25177", "title": "Definition:Quadratic Function", "text": "A '''quadratic function''' is an expression of the form: :$\\map Q x := a_0 + a_1 x + a_2 x^2$ where $a_0, a_1, a_2$ are constants."}
+{"_id": "25178", "title": "Definition:Exponential Order/Real Index", "text": "Let $e^{a t}$ be the exponential function, where $a \\in \\R$ is constant. Then $\\map f t$ is said to be of '''exponential order''' $a$, denoted $f \\in \\mathcal E_a$, {{iff}} there exist strictly positive real numbers $M, K$ such that: :$\\forall t \\ge M: \\size {\\map f t} < K e^{a t}$"}
+{"_id": "25179", "title": "Definition:Minimal Infinite Successor Set", "text": "=== Definition 1 === {{:Definition:Minimal Infinite Successor Set/Definition 1}} === Definition 2 === {{:Definition:Minimal Infinite Successor Set/Definition 2}} === Definition 3 === {{:Definition:Minimal Infinite Successor Set/Definition 3}}"}
+{"_id": "25180", "title": "Definition:Step Function", "text": "A real function $f: \\R \\to \\R$ is a '''step function''' {{iff}} it can be expressed as a finite linear combination of the form: :$\\map f x = \\lambda_1 \\chi_{\\mathbb I_1} + \\lambda_2 \\chi_{\\mathbb I_2} + \\cdots + \\lambda_n \\chi_{\\mathbb I_n}$ where: :$\\lambda_1, \\lambda_2, \\ldots, \\lambda_n$ are real constants :$\\mathbb I_1, \\mathbb I_2, \\ldots, \\mathbb I_n$ are open intervals, where these intervals partition $\\R$ (except for the endpoints) :$\\chi_{\\mathbb I_1}, \\chi_{\\mathbb I_2}, \\ldots, \\chi_{\\mathbb I_n}$ are characteristic functions of $\\mathbb I_1, \\mathbb I_2, \\ldots, \\mathbb I_n$."}
+{"_id": "25181", "title": "Definition:Limit Inferior of Sequence of Sets/Notation", "text": "The limit inferior of $E_n$ can also be seen denoted as: :$\\displaystyle {\\underline {\\lim} }_{n \\mathop \\to \\infty} \\ E_n$ but this notation is not used on {{ProofWiki}} because it does not render well. Some sources merely present this as: :$\\ds \\underline \\lim E_n$"}
+{"_id": "25182", "title": "Definition:Limit Superior of Sequence of Sets/Notation", "text": "The limit superior of $E_n$ can also be seen denoted as: :$\\ds \\overline {\\lim}_{n \\mathop \\to \\infty} \\ E_n$ but this notation is not used on {{ProofWiki}} because it does not render well. Some sources merely present this as: :$\\ds \\overline \\lim E_n$"}
+{"_id": "25183", "title": "Definition:Gamma Function/Graph", "text": "The graph of the Gamma function is illustrated here for real arguments. <!--thumbright400pxThe Gamma function: $\\Gamma \\left({z}\\right)$ (red solid line) and $\\dfrac 1 {\\Gamma\\left({z}\\right)}$ (blue broken line)--> :800pxThe Gamma function: $\\Gamma \\left({z}\\right)$ (red solid line) and $\\dfrac 1 {\\Gamma\\left({z}\\right)}$ (blue broken line) The Gamma function: :$\\Gamma \\left({z}\\right)$ (red solid line) :$\\dfrac 1 {\\Gamma \\left({z}\\right)}$ (blue broken line)"}
+{"_id": "25184", "title": "Definition:O Notation/Big-O Notation/Implied Constant", "text": "From the definition of the limit of a function, it can be seen that this is also equivalent to: :$\\exists c \\in \\R: c > 0, k \\ge 0: \\forall n > k, f \\left({n}\\right) \\le c g \\left({n}\\right)$ For some fixed $k$ (appropriate to the function under consideration) the infimum of such $c$ is called the '''implied constant'''."}
+{"_id": "25186", "title": "Definition:Arithmetic Series", "text": "An '''arithmetic series''' is a series whose underlying sequence is an arithmetic sequence: {{begin-eqn}} {{eqn | l = S_n       | r = \\sum_{k \\mathop = 0}^{n - 1} a + k d       | c =  }} {{eqn | r = a + \\paren {a + d} + \\paren {a + 2 d} + \\cdots + \\paren {a + \\paren {n - 1} d}       | c =  }} {{end-eqn}}"}
+{"_id": "25188", "title": "Definition:Arithmetic Sequence/Common Difference", "text": "The term $d$ is the '''common difference''' of $\\sequence {a_k}$."}
+{"_id": "25189", "title": "Definition:Arithmetic-Geometric Sequence", "text": "An '''arithmetic-geometric sequence''' is a sequence $\\sequence {a_k}$ in $\\R$ defined as: :$a_k = \\paren {a_0 + k d} r^k$ for $k = 0, 1, 2, \\ldots$ Thus its general form is: :$a_0, \\paren {a_0 + d} r, \\paren {a_0 + 2 d} r^2, \\paren {a_0 + 3 d} r^3, \\ldots$"}
+{"_id": "25190", "title": "Definition:Arithmetic-Geometric Series", "text": "An '''arithmetic-geometric series''' is a series whose underlying sequence is an arithmetic-geometric sequence: {{begin-eqn}} {{eqn | l = S_n       | r = \\sum_{k \\mathop = 0}^{n - 1} \\paren {a + k d} r^k       | c =  }} {{eqn | r = a + \\paren {a + d} r + \\paren {a + 2 d} r^2 + \\cdots + \\paren {a + \\paren {n - 1} d}r^{n-1}       | c =  }} {{end-eqn}}"}
+{"_id": "25191", "title": "Definition:Arithmetic Sequence/Last Term", "text": "The term $a_{n-1} = a_0 + \\paren {n - 1} d$ is the '''last term''' of $\\sequence {a_k}$."}
+{"_id": "25192", "title": "Definition:Geometric Mean/Mean Proportional", "text": "In the language of {{AuthorRef|Euclid}}, the geometric mean of two magnitudes is called the '''mean proportional'''. Thus the '''mean proportional''' of $a$ and $b$ is defined as that magnitude $c$ such that: :$a : c = c : b$ where $a : c$ denotes the ratio between $a$ and $c$."}
+{"_id": "25193", "title": "Definition:SI Units/Base Units", "text": "{| class=\"wikitable\" style=\"margin:1em auto 1em auto\" |+ SI base units |- !Name ! Unit symbol ! Dimension ! Symbol |- ! metre | $\\mathrm m$ | $\\mathsf L$: Length | $l$ |- ! kilogram | $\\mathrm {kg}$ | $\\mathsf M$: Mass | $m$ |- ! second | $\\mathrm s$ | $\\mathsf T$: Time | $t$ |- ! ampere | $\\mathrm A$ | $\\mathsf I$: Electric Current | $I$ |- ! kelvin | $\\mathrm K$ | $\\Theta$: Temperature | $T$ |- ! candela | $\\mathrm {cd}$ | $\\mathsf J$: Luminous Intensity | $I_v$ |- ! mole | $\\mathrm {mol}$ | $\\mathsf N$: Amount of Substance | $n$ |- |}"}
+{"_id": "25194", "title": "Definition:Bernoulli Numbers/Archaic Form", "text": "A different definition of the Bernoulli numbers can be found in older literature. Usually denoted with the symbol ${B_n}^*$, they are considered archaic, and will not be used on {{ProofWiki}}. === Definition 1 === {{:Definition:Bernoulli Numbers/Archaic Form/Definition 1}} === Definition 2 === {{:Definition:Bernoulli Numbers/Archaic Form/Definition 2}}"}
+{"_id": "25195", "title": "Definition:Planar Graph/Face", "text": "The '''faces''' of a planar graph are the areas which are surrounded by edges. In the above, the '''faces''' are $ABHC$, $CEGH$, $ACD$, $CDFE$ and $ADFEGHIHB$. === Incident === {{:Definition:Incident (Graph Theory)/Planar Graph}} === Adjacent === {{:Definition:Adjacent (Graph Theory)/Faces}} In the above diagram, $ABHC$ and $ACD$ are adjacent, but $ABHC$ and $CDFE$ are ''not'' adjacent."}
+{"_id": "25197", "title": "Definition:Tangent Map/Affine Transformation", "text": "Let $\\mathcal E$ and $\\mathcal F$ be affine spaces with difference spaces $E$ and $F$ respectively. Let $\\mathcal L : \\mathcal E \\to \\mathcal F$ be an affine transformation. The associated linear transformation $L: E \\to F$ of difference spaces is called the '''tangent map''' of $\\mathcal L$."}
+{"_id": "25198", "title": "Definition:Tangent Map/Differentable Mappings", "text": "{{stub}}"}
+{"_id": "25199", "title": "Definition:Cube", "text": "=== Geometry === {{:Definition:Cube/Geometry}} === Algebra === {{:Definition:Cube/Algebra}}"}
+{"_id": "25200", "title": "Definition:Cube/Algebra", "text": "The '''cube''' of a quantity is the third power of that quantity."}
+{"_id": "25201", "title": "Definition:Baire Space (Topology)/Definition 1", "text": "$T$ is a '''Baire space''' {{iff}} the union of any countable set of closed sets of $T$ whose interiors are empty also has an empty interior."}
+{"_id": "25202", "title": "Definition:Baire Space (Topology)/Definition 2", "text": "$T$ is a '''Baire space''' {{iff}} the intersection of any countable set of open sets of $T$ which are everywhere dense is everywhere dense."}
+{"_id": "25203", "title": "Definition:Baire Space (Topology)/Definition 3", "text": "$T$ is a '''Baire space''' {{iff}} the interior of the union of any countable set of closed sets of $T$ which are nowhere dense is empty."}
+{"_id": "25204", "title": "Definition:Baire Space (Topology)/Definition 4", "text": "$T$ is a '''Baire space''' {{iff}}, whenever the union of any countable set of closed sets of $T$ has an interior point, then one of those closed sets must have an interior point."}
+{"_id": "25206", "title": "Definition:Lychrel Number", "text": "A '''Lychrel number''' is a natural number which cannot form a palindromic number through repeated iteration of the reverse-and add process."}
+{"_id": "25207", "title": "Definition:Angle/Types", "text": "Angles can be divided into categories: === Zero Angle === {{:Definition:Zero Angle}} === Acute Angle === {{:Definition:Acute Angle}} === Right Angle === {{:Definition:Right Angle}} === Obtuse Angle === {{:Definition:Obtuse Angle}} === Straight Angle === {{:Definition:Straight Angle}} === Reflex Angle === {{:Definition:Reflex Angle}} === Full Angle === {{:Definition:Full Angle}} It is possible to consider angles outside the range $\\closedint {0 \\degrees} {360 \\degrees}$, that is, $\\closedint 0 {2 \\pi}$. However, in geometric contexts it is usually preferable to convert these to angles inside this range by adding or subtracting multiples of a full angle."}
+{"_id": "25208", "title": "Definition:Angle/Directed versus Undirected", "text": "The most basic definition of angle is an undirected angle on the interval $\\left[{0^\\circ  \\,.\\,.\\, 180^\\circ}\\right]$ or $\\left[{0 \\,.\\,.\\, \\pi}\\right]$. This definition is often insufficient, in cases such as the external angles of a polygon. Therefore, angles are most commonly defined in one of two ways: :$(1): \\quad$ Undirected angles on the interval $\\left[{0^\\circ \\,.\\,.\\, 360^\\circ}\\right]$ or $\\left[{0 \\,.\\,.\\, 2 \\pi}\\right]$. :$(2): \\quad$ Directed angles, with the positive direction being counterclockwise from a given line (or, if no line is specified, from the $x$-axis). :::This definition is more commonly found in applied mathematics, such as in surveying, navigation, or, more colloquially, in a $720^\\circ$ degree spin in skateboarding, skiing, etc."}
+{"_id": "25209", "title": "Definition:Vertical Angles", "text": "When two straight lines intersect, the angles opposite each other are called '''vertical angles''': :400px In the above diagram: :$\\alpha$ and $\\beta$ are '''vertical angles''' :$\\gamma$ and $\\delta$ are '''vertical angles'''."}
+{"_id": "25210", "title": "Definition:Smooth Mapping", "text": "Let $M, N$ be smooth manifolds.  Denote $m := \\dim M$ and $n := \\dim N$.  Let $\\phi: M \\to N$ be a mapping.  Then $\\phi$ is a '''smooth mapping''' {{iff}}: :for every chart $\\struct {U, \\kappa}$ on $M$ and every chart $\\struct {V, \\xi}$ on $N$ such that $V \\cap \\map \\phi U \\ne \\O$, the mapping: ::$\\displaystyle \\xi \\circ \\phi \\circ \\kappa^{-1}: \\map \\kappa U \\subseteq \\R^m \\to \\map \\xi {V \\cap \\map \\phi U} \\subseteq \\R^n$ :is smooth."}
+{"_id": "25211", "title": "Definition:Fiber Bundle", "text": "Let $M, E, F$ be topological spaces.  Let $\\pi: E \\to M$ be a continuous surjection.  Let $\\mathcal U := \\left\\{ {U_\\alpha \\subseteq M: \\alpha \\in I} \\right\\}$ be an open cover of $M$ with index set $I$.  Let $\\operatorname{pr}_{1, \\alpha} : U_\\alpha \\times F \\to U_\\alpha$ be the first projection on $U_\\alpha \\times F$. Let there exist homeomorphisms: :$\\chi_\\alpha : \\pi^{-1} \\left({U_\\alpha}\\right) \\to U_\\alpha \\times F$  such that for all $\\alpha \\in I$: :$\\pi {\\restriction}_{U_\\alpha} = \\operatorname{pr}_{1, \\alpha} \\mathop \\circ \\chi_\\alpha$ where $\\pi {\\restriction}_{U_\\alpha}$ is the restriction of $\\pi$ to $U_\\alpha \\in \\mathcal U$. Then the ordered tuple $\\left({E, M, \\pi, F}\\right)$ is called a '''fiber bundle over $M$'''."}
+{"_id": "25212", "title": "Definition:Transversal (Geometry)/Interior Angle", "text": "An '''interior angle''' of a transversal is an angle which is between the two lines cut by that transversal. In the above figure, the '''interior angles''' with respect to the transversal $EF$ are: :$\\angle AHJ$ :$\\angle CJH$ :$\\angle BHJ$ :$\\angle DJH$"}
+{"_id": "25213", "title": "Definition:Transversal (Geometry)/Alternate Angles", "text": "'''Alternate angles''' are interior angles of a transversal which are on opposite sides and different lines. In the above figure, the pairs of '''alternate angles''' with respect to the transversal $EF$ are: :$\\angle AHJ$ and $\\angle DJH$ :$\\angle CJH$ and $\\angle BHJ$"}
+{"_id": "25214", "title": "Definition:Transversal (Geometry)/Corresponding Angles", "text": "'''Corresponding angles''' are the angles in equivalent positions on the two lines cut by a transversal with respect to that transversal. In the above figure, the '''corresponding angles''' with respect to the transversal $EF$ are: :$\\angle AHJ$ and $\\angle CJF$ :$\\angle AHE$ and $\\angle CJH$ :$\\angle BHE$ and $\\angle DJH$ :$\\angle BHJ$ and $\\angle DJF$"}
+{"_id": "25215", "title": "Definition:Smooth Curve", "text": "Let $M$ be a smooth manifold.  Let $I$ be a open real interval, considered as a smooth manifold of dimension $1$. {{explain|We have various pages defining \"manifolds\" of various types (differentiable, smooth, complex), but not one defining a basic \"manifold\".  It can be assumed by inference that a manifold is \"a second-countable locally Euclidean space of finite integral dimension\" but this needs to be rigorously and unambiguously clarified by including that definition at the top level of the page Definition:Topological Manifold.}} Then a smooth mapping $\\gamma : I \\to M$ is called a '''smooth curve'''. Category:Definitions/Manifolds exaz9pzlswo05mgzfps2k4cek9jd44a"}
+{"_id": "25216", "title": "Definition:Proportion/Constant of Proportion", "text": "The constant $k$ is known as the '''constant of proportion'''."}
+{"_id": "25217", "title": "Definition:Coordinate Function", "text": "Let $M$ be a locally Euclidean space of dimension $n$.  Let $\\left({U, \\kappa}\\right)$ be a coordinate chart.  Let $\\operatorname{pr}_i: \\R^n \\to \\R$ be the $i$th projection.  Then the mapping $\\kappa_i$, defined as: :$\\kappa_i = \\operatorname{pr}_i \\mathop \\circ \\kappa: U \\to \\left({\\operatorname{pr}_i \\mathop \\circ \\kappa \\left({U}\\right)}\\right) \\subseteq \\R$ is called the '''$i$th coordinate function''' on $U$."}
+{"_id": "25218", "title": "Definition:Tangent Space", "text": "=== Real Submanifold === Let $M$ be a real submanifold of $\\R^n$ of dimension $d$. Let $p\\in M$. ==== Using Local Submersions ==== Let $U$ be a open neighborhood of $p$ in $\\R^n$ and $\\phi : U \\to \\R^{n-d}$ be a submersion such that: :$M \\cap U = \\phi^{-1}(0)$. The '''tangent space of $M$ at $p$''' is: :$T_pM = \\ker d\\phi(p)$ where $d\\phi(p)$ is the differential of $\\phi$ at $p$. ==== Using Local Embdeddings ==== Let $U$ be a open set of $\\R^d$ and $\\phi : U \\to \\R^{n}$ be an embedding such that: :$p\\in \\phi(U) \\subset M$ The '''tangent space of $M$ at $p$''' is: :$T_pM = \\operatorname{im} (d\\phi)(\\phi^{-1}(p))$ where $(d\\phi)(\\phi^{-1}(p))$ is the differential of $\\phi$ at $\\phi^{-1}(p)$. ==== Using Local Immersions ==== === Geometric Tangent Space === {{stub}} === Differentiable Manifold === There are various ways to construct the '''tangent space''' of a differentiable manifold. Let $M$ be a smooth manifold of dimension $m$. Let $p \\in M$.  ==== Abstract Definition ==== Let $A$ be an atlas of $M$. Let $B = \\{(U,\\phi) \\in A : p\\in U\\}$. The '''tangent space of $M$ at $p$''' is the real vector space :$\\displaystyle \\left(\\coprod_{b\\in B}\\{b\\} \\times \\R^m\\right)/\\sim$ where $\\sim$ is the equivalence relation defined by: :$(i,\\xi) \\sim (j,\\eta) \\iff d(\\phi_j \\circ \\phi_i^{-1})(\\phi_i(p)) (\\xi) = \\eta$ where $d(\\phi_j \\circ \\phi_i^{-1})(\\phi_i(p))$ is the differential of $\\phi_j \\circ \\phi_i^{-1}$ at $\\phi_i(p)$. Addition is defined by: :$[(i,\\xi)] + [(j,\\eta)] = \\left[\\left( i, d(\\phi_j \\circ \\phi_i^{-1})(\\phi_i(p)) (\\xi) + \\eta \\right)\\right]$ and scalar multiplication by: :$\\lambda \\cdot [(i,\\xi)] = [(i, \\lambda\\cdot\\xi)]$ ==== As a space of germs of curves ==== {{stub}} ==== As a space of derivations ==== The '''tangent space at $p$''', denoted by $T_p M$, is the set of all tangent vectors at $m$. {{refactor|make this explicit}}"}
+{"_id": "25219", "title": "Definition:Fiber Bundle/Total Space", "text": "The topological space $E$ is called the '''total space of $B$'''."}
+{"_id": "25220", "title": "Definition:Fiber Bundle/Base Space", "text": "The topological space $M$ is called the '''base space of $B$'''."}
+{"_id": "25221", "title": "Definition:Fiber Bundle/Base Point", "text": "A point $m \\in M$ is called a '''base point of $B$'''."}
+{"_id": "25222", "title": "Definition:Fiber Bundle/Bundle Projection", "text": "The continuous surjection $\\pi: E \\to M$ is called the '''bundle projection of $B$'''."}
+{"_id": "25223", "title": "Definition:Section (Topology)", "text": "Let $M, E$ be topological spaces.  Let $\\pi: E \\to M$ be a continuous surjection.  Let $I_M: M \\to M$ be the identity mapping on $M$.  Then a '''section''' of $E$ is a continuous mapping $s: M \\to E$ such that $\\pi \\circ s = I_M$."}
+{"_id": "25224", "title": "Definition:Local Trivialization", "text": "Let $\\left({E, M, \\pi, F}\\right)$ be a fiber bundle.  Let $\\operatorname{pr}_1: M \\times F \\to M$ be the first projection on $M \\times F$. By definition of fiber bundle, for every point $m \\in M$ there exists an open neighborhood $U$ of $m$ and a homeomorphism:  :$\\chi: \\pi^{-1} \\left({U}\\right) \\to U \\times F$  such that:  :$\\pi {\\restriction}_U = \\operatorname{pr}_1 \\mathop \\circ \\chi$ where $\\pi {\\restriction}_U$ is the restriction of $\\pi$ to $U$. Then the ordered pair $\\left({U, \\chi}\\right)$ is called a '''local trivialization''' of $E$ over $U$."}
+{"_id": "25225", "title": "Definition:Fiber Bundle/Model Fiber", "text": "The topological space $F$ is called the '''model fiber of $B$'''."}
+{"_id": "25226", "title": "Definition:Dimension (Topology)", "text": "=== Locally Euclidean Space === {{:Definition:Dimension (Topology)/Locally Euclidean Space}} === Hausdorff Dimension === {{:Definition:Dimension (Topology)/Hausdorff Dimension}} === Lebesgue Covering Dimension === {{:Definition:Dimension (Topology)/Lebesgue Covering Dimension}} === Inductive Dimension === {{:Definition:Dimension (Topology)/Inductive Dimension}} Category:Definitions/Topology clt3l1b93bpp2cr7znc3k39qnh3d28d"}
+{"_id": "25227", "title": "Definition:Transition Mapping", "text": "Let $B = \\left({E, M, \\pi, F}\\right)$ be a fiber bundle.  Let $\\left({U, \\chi}\\right)$, $\\left({V, \\xi}\\right)$ be two local trivializations with $U \\cap V \\ne \\varnothing$. Then the mapping: :$\\xi \\circ \\chi^{-1} : U \\cap V \\times F \\to U \\cap V \\times F$ is called a '''transition mapping''' from $\\left({U, \\chi}\\right)$ to $\\left({V, \\xi}\\right)$."}
+{"_id": "25228", "title": "Definition:Smooth Fiber Bundle", "text": "Let $E, M, F$ be smooth manifolds.  Let $\\pi$ be a smooth surjection.  Let $B = \\left({ E, M, \\pi, F }\\right)$ be a fiber bundle with a system of local trivializations $\\left\\{{ \\left({ U_\\alpha, \\chi_\\alpha }\\right) : \\alpha \\in I }\\right\\}$ such that $\\forall \\alpha \\in I$ :  :$\\chi_\\alpha : \\pi^{-1}\\left({ U_\\alpha }\\right) \\to U_\\alpha \\times F$ is smooth.  Then $B$ is called a '''smooth fiber bundle''' on $M$."}
+{"_id": "25229", "title": "Definition:Fiber Bundle/System of Local Trivializations", "text": "The set $\\left\\{ {\\left({U_\\alpha, \\chi_\\alpha}\\right): \\alpha \\in I}\\right\\}$ is called a '''system of local trivializations''' of $E$ on $M$."}
+{"_id": "25230", "title": "Definition:Vector Bundle", "text": "Let $B = \\struct {E, M , \\pi, F}$ be a fiber bundle.  Then $B$ is a '''vector bundle''' over $M$ if the model fiber $F$ is a topological vector space."}
+{"_id": "25231", "title": "Definition:Smooth Vector Bundle", "text": "Let $B = \\tuple {E, M, \\pi, F}$ be a smooth fiber bundle.  Let $F$ be a vector space.  Let vector addition and scalar multiplication on $F$ be smooth. Then $B$ is a '''smooth vector bundle''' over $M$."}
+{"_id": "25232", "title": "Definition:Borel Sigma-Algebra/Borel Set", "text": "The elements of $\\map {\\mathcal B} {S, \\tau}$ are called the '''Borel (measurable) sets''' of $\\struct {S, \\tau}$."}
+{"_id": "25233", "title": "Definition:Linear Measure/Height/Euclidean Definition", "text": "The '''height''' of a polygon is the linear measure going up the page."}
+{"_id": "25234", "title": "Definition:Even Integer/Even-Times Even", "text": "Let $n$ be an integer. Then $n$ is '''even-times even''' {{iff}} it has $4$ as a divisor. The first few non-negative '''even-times even''' numbers are: :$0, 4, 8, 12, 16, 20, \\ldots$"}
+{"_id": "25235", "title": "Definition:Even Integer/Even-Times Odd", "text": "Let $n$ be an integer. Then $n$ is '''even-times odd''' {{iff}} it has $2$ as a divisor and also an odd number. The first few non-negative '''even-times odd''' numbers are: :$2, 6, 10, 12, 14, 18, \\ldots$"}
+{"_id": "25236", "title": "Definition:Odd Integer/Odd-Times Odd", "text": "Let $n \\in \\Z$, i.e. let $n$ be an integer. === Definition 1 === {{:Definition:Odd Integer/Odd-Times Odd/Definition 1}} === Definition 2 === {{:Definition:Odd Integer/Odd-Times Odd/Definition 2}} === Sequence === {{:Definition:Odd Integer/Odd-Times Odd/Sequence}}"}
+{"_id": "25238", "title": "Definition:Plane Number/Side", "text": "The '''side''' of a plane number is one of the (natural) numbers which are its divisors."}
+{"_id": "25239", "title": "Definition:Solid Number/Side", "text": "The '''side''' of a solid number is one of the (natural) numbers which are its divisors."}
+{"_id": "25242", "title": "Definition:Side of Number", "text": "=== Side of Plane Number === {{:Definition:Side of Plane Number}} === Side of Solid Number === {{:Definition:Side of Solid Number}} Category:Definitions/Euclidean Number Theory fuwb3abn0ln4di6c41oov518khzuqsh"}
+{"_id": "25243", "title": "Definition:Solid Number/Similar Numbers", "text": "Let $m$ and $n$ be solid numbers. Let: : $m = p_1 \\times p_2 \\times p_3$ where $p_1 \\le p_2 \\le p_3$ : $n = q_1 \\times q_2 \\times q_3$ where $q_1 \\le q_2 \\le q_3$ Then $m$ and $n$ are '''similar''' iff: :$p_1 : q_1 = p_2 : q_2 = p_3 : q_3$"}
+{"_id": "25244", "title": "Definition:Similar Numbers", "text": "=== Similar Plane Numbers === {{:Definition:Plane Number/Similar Numbers}} === Similar Solid Numbers === {{:Definition:Solid Number/Similar Numbers}} {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book VII/21 - Similar Numbers}}'' {{EuclidDefRefNocat|VII|21|Similar Numbers}} Category:Definitions/Euclidean Number Theory bkaggf8650zkdak4px6fy31u10m43om"}
+{"_id": "25245", "title": "Definition:Plane Number/Similar Numbers", "text": "Let $m$ and $n$ be plane numbers. Let: :$m = p_1 \\times p_2$ where $p_1 \\le p_2$ :$n = q_1 \\times q_2$ where $q_1 \\le q_2$ Then $m$ and $n$ are '''similar''' {{iff}}: :$p_1 : q_1 = p_2 : q_2$"}
+{"_id": "25246", "title": "Definition:Parenthesization", "text": "Let $\\circ$ be a product defined on a set $S$. Let $a_1 \\circ a_2 \\circ \\cdots a_i \\circ \\cdots \\circ a_n$ denote a word in $S$ for some $n > 2$. To distinguish all possible products of $a_1, a_2, \\dotsc, a_n$, parentheses are inserted into the word. A set of parentheses applied on a product is called a '''parenthesization''' of that word."}
+{"_id": "25247", "title": "Definition:Binomial (Euclidean)/Order", "text": "The '''order''' of $a + b$ is the name of its classification into one of the six categories: first, second, third, fourth, fifth or sixth."}
+{"_id": "25249", "title": "Definition:Apotome/Order", "text": "The '''order''' of $a - b$ is the name of its classification into one of the six categories: first, second, third, fourth, fifth or sixth."}
+{"_id": "25250", "title": "Definition:Divisor (Algebra)/Real Number", "text": "Let $\\R$ be the set of real numbers. Let $x, y \\in \\R$. Then '''$x$ divides $y$''' is defined as: :$x \\divides y \\iff \\exists t \\in \\Z: y = t \\times x$ where $\\Z$ is the set of integers."}
+{"_id": "25251", "title": "Definition:Greatest Common Divisor/Real Numbers", "text": "Let $a, b \\in \\R$ be commensurable. Then there exists a greatest element $d \\in \\R_{>0}$ such that: : $d \\divides a$ : $d \\divides b$ where $d \\divides a$ denotes that $d$ is a divisor of $a$. This is called the '''greatest common divisor of $a$ and $b$''' and denoted $\\gcd \\set {a, b}$."}
+{"_id": "25252", "title": "Definition:Transversal (Geometry)/Exterior Angle", "text": "An '''exterior angle''' of a transversal is an angle which is not between the two lines cut by a transversal. In the above figure, the '''exterior angles''' with respect to the transversal $EF$ are: :$\\angle AHE$ :$\\angle CJF$ :$\\angle BHE$ :$\\angle DJF$"}
+{"_id": "25253", "title": "Definition:Angle/Containment", "text": "The two lines whose intersection forms an angle are said to '''contain''' that angle."}
+{"_id": "25254", "title": "Definition:Essentially Bounded Function", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $L^\\infty $ be the associated Lebesgue $\\infty$-space. Then any $f \\in L^\\infty$ is known as an '''essentially bounded function.'''"}
+{"_id": "25255", "title": "Definition:Supremum Seminorm", "text": "Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $f: X \\to \\R$ be a real-valued function. The '''supremum seminorm''' of $f$, commonly denoted as $\\left\\Vert{f}\\right\\Vert_\\infty$, is defined as: :$\\displaystyle \\left\\Vert{f}\\right\\Vert_\\infty = \\inf_{\\substack {N \\mathop \\in \\Sigma \\\\ \\mu \\left({N}\\right) \\mathop = 0}} \\sup \\ \\left\\{ {\\left\\vert{f \\left({x}\\right)}\\right\\vert: x \\notin N}\\right\\}$ Observe that for all $M > \\Vert f \\Vert_\\infty$, then: : $\\mu \\left({ \\left\\{ {x \\in X: \\left\\vert{f \\left({x}\\right)}\\right\\vert \\ge M}\\right\\} }\\right) = 0$ and that an essentially bounded function is almost everywhere equal to a bounded function. {{wtd|Move the above into their own proof and definition pages}}"}
+{"_id": "25256", "title": "Definition:Conjugate Exponents", "text": "Let $p, q \\in \\R_{\\ge 1}$. Then $p$ and $q$ are '''conjugate exponents''' iff: :$\\dfrac 1 p + \\dfrac 1 q = 1$ === General Definition === {{:Definition:Conjugate Exponents/General Definition}} Category:Definitions/Measure Theory 72hyj5t8tcetr3ftrhw01of3uv9fl6d"}
+{"_id": "25257", "title": "Definition:Conjugate Exponents/General Definition", "text": "Let $\\left\\{{p_1, p_2, \\ldots, p_n}\\right\\}$ be such that: : $\\forall i \\in \\left\\{{1, 2, \\ldots, n}\\right\\}: p_i \\in \\R_{\\ge 1}$ Then $\\left\\{{p_1, p_2, \\ldots, p_n}\\right\\}$ are '''conjugate exponents''' iff: :$\\displaystyle \\frac 1 {p_1} + \\frac 1 {p_2} + \\cdots + \\frac 1 {p_n} = \\sum_{i \\mathop = 1}^n \\frac 1 {p_i} = 1$"}
+{"_id": "25258", "title": "Definition:Inscribe/Circle in Polygon", "text": "A circle is '''inscribed in''' a polygon when it is tangent to each of the sides of that polygon: :300px"}
+{"_id": "25259", "title": "Definition:Inscribe/Polygon in Circle", "text": "A polygon is '''inscribed in''' a circle when each of its vertices lies on the circumference of the circle: :300px"}
+{"_id": "25260", "title": "Definition:Inscribe/Polygon in Polygon", "text": "A polygon is '''inscribed in''' another polygon when each of its vertices lies on the corresponding side of the other polygon."}
+{"_id": "25261", "title": "Definition:Circumscribe/Circle around Polygon", "text": "A circle is '''circumscribed around''' a polygon when its circumference passes through each of the vertices of that polygon: :300px"}
+{"_id": "25262", "title": "Definition:Circumscribe/Polygon around Circle", "text": "A polygon is '''circumscribed around''' a circle when each of its sides is tangent to the circumference of the circle: :300px"}
+{"_id": "25263", "title": "Definition:Circumscribe/Polygon around Polygon", "text": "A polygon is '''circumscribed around '''another polygon when each of its sides passes through each of the vertices of the other polygon."}
+{"_id": "25264", "title": "Definition:Leading Term", "text": "Let $P = a \\circ b$ be an expression. The term $a$ is known as the '''leading term''' of $P$."}
+{"_id": "25265", "title": "Definition:Following Term", "text": "Let $P = a \\circ b$ be an expression. The term $b$ is known as the '''following term''' of $P$."}
+{"_id": "25266", "title": "Definition:Unit (One)", "text": "A numerical quantity whose cardinality corresponds to the number $1$ (one) is called '''a unit'''. {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book VII/1 - Unit}}'' {{EuclidDefRefNocat|VII|1|Unit}}"}
+{"_id": "25267", "title": "Definition:Divisor (Algebra)/Real Number/Part", "text": "{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book V/1 - Part}}'' {{EuclidDefRefNocat|V|1|Part}}"}
+{"_id": "25268", "title": "Definition:Common Divisor/Real Numbers", "text": "Let $S$ be a finite set of real numbers, that is: :$S = \\set {x_1, x_2, \\ldots, x_n: \\forall k \\in \\N^*_n: x_k \\in \\R}$ Let $c \\in \\R$ such that $c$ divides all the elements of $S$, that is: :$\\forall x \\in S: c \\divides x$ Then $c$ is a '''common divisor''' of all the elements in $S$."}
+{"_id": "25269", "title": "Definition:Commensurable in Square Only", "text": "Let $a, b \\in \\R_{>0}$ be (strictly) positive real numbers. Then $a$ and $b$ are '''commensurable in square only''' iff: : $\\left ({\\dfrac a b}\\right)^2$ is rational. but: : $\\dfrac a b$ is irrational. That is, such that: : $a$ and $b$ are commensurable in square but: : $a$ and $b$ are incommensurable in length. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X/2 - Commensurable in Square}}'' {{EuclidDefRefNocat|X|2|Commensurable in Square}} and: :''{{Definition:Euclid's Definitions - Book X/1 - Commensurable}}'' {{EuclidDefRefNocat|X|1|Commensurable}}"}
+{"_id": "25270", "title": "Definition:Rational Line Segment", "text": "A '''rational line segment''' is a line segment the square of whose length is a rational number of units of area. In other words, a '''rational line segment''' is a line segment whose length belongs to the set $\\left\\{{x \\in \\R_{>0} : x^2 \\in \\Q}\\right\\}$. {{EuclidSaid}} : ''{{:Definition:Euclid's Definitions - Book X/3 - Rational Line Segment}}'' {{EuclidDefRefNocat|X|3|Rational Line Segment}}"}
+{"_id": "25271", "title": "Definition:Rational Area", "text": "A '''rational area''' is a geometric figure whose area is a rational number of units of area. {{EuclidSaid}} : ''{{:Definition:Euclid's Definitions - Book X/4 - Rational Area}}'' {{EuclidDefRefNocat|X|4|Rational Area}}"}
+{"_id": "25272", "title": "Definition:Irrational Line Segment", "text": "An '''irrational line segment''' is a line segment such that the square of whose length is an irrational number of units of area. {{EuclidSaid}} : ''{{:Definition:Euclid's Definitions - Book X/3 - Rational Line Segment}}'' {{EuclidDefRefNocat|X|3|Rational Line Segment}}"}
+{"_id": "25273", "title": "Definition:Medial", "text": "A '''medial''' is a strictly positive real number which is the mean proportional between two rational line segments which are commensurable in square only. Thus a magnitude $a \\in \\R_{>0}$ is '''medial''' {{iff}} $a$ is of the form: :$a = \\rho \\sqrt [4] k$ where: : $\\rho$ is a rational number : $k$ is a rational number whose square root is irrational. {{:Euclid:Proposition/X/21}}"}
+{"_id": "25274", "title": "Definition:Irrational Area", "text": "An '''irrational area''' is a geometric figure whose area is an irrational number of units of area. {{EuclidSaid}} : ''{{:Definition:Euclid's Definitions - Book X/4 - Rational Area}}'' {{EuclidDefRefNocat|X|4|Rational Area}}"}
+{"_id": "25275", "title": "Definition:Successor Mapping", "text": "Let $\\struct {P, s, 0}$ be a Peano structure. Then the mapping $s: P \\to P$ is called the '''successor mapping on $P$'''."}
+{"_id": "25276", "title": "Definition:Non-Successor Element", "text": "Let $\\struct {P, s, 0}$ be a Peano structure. Then the element $0 \\in P$ is called the '''non-successor element'''. This is justified by Axiom $(\\text P 4)$, which stipulates that $0$ is not in the image of the successor mapping $s$."}
+{"_id": "25277", "title": "Definition:Medial Area", "text": "A '''medial area''' is an area which is equal to the square on a medial straight line. Thus an area $A$ is '''medial''' {{iff}} it is of the form: :$A = \\rho^2 \\sqrt k$ where: : $\\rho$ is a rational number : $k$ is a rational number whose square root is irrational. {{:Euclid:Proposition/X/23/Porism}}"}
+{"_id": "25278", "title": "Definition:Bimedial/First Bimedial", "text": "Let $a, b \\in \\R_{>0}$ be in the forms: :$a = k^{1/4} \\rho$ :$b = k^{3/4} \\rho$ where: : $\\rho$ is a rational number : $k$ is a rational number whose square root is irrational. Then $a + b$ is a '''first bimedial'''. {{:Euclid:Proposition/X/37}}"}
+{"_id": "25279", "title": "Definition:Bimedial/Second Bimedial", "text": "Let $a, b \\in \\R_{>0}$ be in the forms: :$a = k^{1/4} \\rho$ :$b = \\dfrac {\\lambda^{1/2} \\rho} {k^{1/4} }$ where: : $\\rho$ is a rational number : $k$ is a rational number whose square root is irrational. : $\\lambda$ is a rational number whose square root is irrational. Then $a + b$ is a '''second bimedial'''. {{:Euclid:Proposition/X/38}}"}
+{"_id": "25281", "title": "Definition:Major (Euclidean)", "text": "Let $a, b \\in \\R_{>0}$ in the forms: :$a = \\dfrac \\rho {\\sqrt 2} \\sqrt {1 + \\dfrac k {\\sqrt {1 + k^2} } }$ :$b = \\dfrac \\rho {\\sqrt 2} \\sqrt {1 - \\dfrac k {\\sqrt {1 + k^2} } }$ where: : $\\rho$ is a rational number : $k$ is a rational number whose square root is irrational. Then $a + b$ is a '''major'''. {{:Euclid:Proposition/X/39}}"}
+{"_id": "25282", "title": "Definition:Side of Sum of Medial Areas", "text": "Let $a, b \\in \\R_{>0}$ be in the forms: :$a = \\dfrac {\\rho \\lambda^{1/4} } {\\sqrt 2} \\sqrt {1 + \\dfrac k {\\sqrt {1 + k^2} } }$ :$b = \\dfrac {\\rho \\lambda^{1/4} } {\\sqrt 2} \\sqrt {1 - \\dfrac k {\\sqrt {1 + k^2} } }$ where: : $\\rho$ is a rational number : $k$ is a rational number whose square root is irrational : $\\lambda$ is a rational number whose square root is irrational. Then $a + b$ is the '''side of the sum of (two) medial areas'''. {{:Euclid:Proposition/X/41}}"}
+{"_id": "25283", "title": "Definition:Side of Rational plus Medial Area", "text": "Let $a, b \\in \\R_{>0}$ be in the forms: :$a = \\dfrac \\rho {\\sqrt {2 \\left({1 + k^2}\\right)} } \\sqrt{\\sqrt {1 + k^2} + k}$ :$b = \\dfrac \\rho {\\sqrt {2 \\left({1 + k^2}\\right)} } \\sqrt{\\sqrt {1 + k^2} - k}$ where: : $\\rho$ is a rational number : $k$ is a rational number whose square root is irrational : $\\lambda$ is a rational number whose square root is irrational. Then $a + b$ is the '''side of the sum of (two) medial areas'''. {{:Euclid:Proposition/X/40}}"}
+{"_id": "25284", "title": "Definition:Binomial (Euclidean)/Term", "text": "The '''terms''' of $a + b$ are the elements $a$ and $b$."}
+{"_id": "25285", "title": "Definition:Bimedial", "text": "=== First Bimedial === {{:Definition:Bimedial/First Bimedial}} === Second Bimedial === {{:Definition:Bimedial/Second Bimedial}}"}
+{"_id": "25286", "title": "Definition:Bimedial/Order", "text": "The '''order''' of $a + b$ is the name of its classification into one of the two categories: first or second."}
+{"_id": "25287", "title": "Definition:Apotome of Medial/Second Apotome", "text": "Let $a, b \\in \\set {x \\in \\R_{>0} : x^2 \\in \\Q}$ be two rationally expressible numbers such that $a > b$ be in the forms: :$a = k^{1/4} \\rho$ :$b = \\dfrac {\\lambda^{1/2} \\rho} {k^{1/4} }$ where: :$\\rho$ is a rational number :$k$ is a rational number whose square root is irrational. :$\\lambda$ is a rational number whose square root is irrational. Then $a - b$ is a '''second apotome of a medial'''. {{:Euclid:Proposition/X/75}}"}
+{"_id": "25288", "title": "Definition:Apotome of Medial/First Apotome", "text": "Let $a, b \\in \\set {x \\in \\R_{>0} : x^2 \\in \\Q}$ be two rationally expressible numbers such that $a > b$ be in the forms: :$a = k^{1/4} \\rho$ :$b = k^{3/4} \\rho$ where: :$\\rho$ is a rational number :$k$ is a rational number whose square root is irrational. Then $a - b$ is a '''first apotome of a medial'''. {{:Euclid:Proposition/X/74}}"}
+{"_id": "25289", "title": "Definition:Apotome of Medial", "text": "=== First Apotome of Medial === {{:Definition:Apotome of Medial/First Apotome}} === Second Apotome of Medial === {{:Definition:Apotome of Medial/Second Apotome}}"}
+{"_id": "25290", "title": "Definition:Apotome of Medial/Order", "text": "The '''order''' of $a - b$ is the name of its classification into one of the two categories: first or second."}
+{"_id": "25292", "title": "Definition:Minor (Euclidean)", "text": "Let $a, b \\in \\R_{>0}$ in the forms: :$a = \\dfrac \\rho {\\sqrt 2} \\sqrt {1 + \\dfrac k {\\sqrt {1 + k^2} } }$ :$b = \\dfrac \\rho {\\sqrt 2} \\sqrt {1 - \\dfrac k {\\sqrt {1 + k^2} } }$ where: :$\\rho$ is a rational number :$k$ is a rational number whose square root is irrational. Then $a - b$ is a '''minor'''. {{:Euclid:Proposition/X/76}}"}
+{"_id": "25293", "title": "Definition:That which produces Medial Whole with Rational Area", "text": "Let $a, b \\in \\R_{>0}$ be in the forms: :$a = \\dfrac \\rho {\\sqrt {2 \\left({1 + k^2}\\right)} } \\sqrt{\\sqrt {1 + k^2} + k}$ :$b = \\dfrac \\rho {\\sqrt {2 \\left({1 + k^2}\\right)} } \\sqrt{\\sqrt {1 + k^2} - k}$ where: : $\\rho$ is a rational number : $k$ is a rational number whose square root is irrational : $\\lambda$ is a rational number whose square root is irrational. Then $a - b$ is '''that which produces a medial whole with a rational area'''. {{:Euclid:Proposition/X/77}}"}
+{"_id": "25294", "title": "Definition:That which produces Medial Whole with Medial Area", "text": "Let $a, b \\in \\R_{>0}$ be in the forms: :$a = \\dfrac {\\rho \\lambda^{1/4} } {\\sqrt 2} \\sqrt {1 + \\dfrac k {\\sqrt {1 + k^2} } }$ :$b = \\dfrac {\\rho \\lambda^{1/4} } {\\sqrt 2} \\sqrt {1 - \\dfrac k {\\sqrt {1 + k^2} } }$ where: : $\\rho$ is a rational number : $k$ is a rational number whose square root is irrational : $\\lambda$ is a rational number whose square root is irrational. Then $a - b$ is '''that which produces a medial whole with a medial area'''. {{:Euclid:Proposition/X/78}}"}
+{"_id": "25295", "title": "Definition:Apotome/Whole", "text": "The real number $a$ is called the '''whole''' of the apotome."}
+{"_id": "25296", "title": "Definition:Apotome/Annex", "text": "The real number $b$ is called the '''annex''' of the apotome."}
+{"_id": "25297", "title": "Definition:Apotome/Terms", "text": "The '''terms''' of $a - b$ are the elements $a$ and $b$. === Whole === {{Definition:Apotome/Whole}} === Annex === {{Definition:Apotome/Annex}}"}
+{"_id": "25298", "title": "Definition:Common Section", "text": "Let $A$ and $B$ be planes. The '''common section''' of $A$ and $B$ is the intersection of $A$ and $B$."}
+{"_id": "25299", "title": "Definition:Similar Planes", "text": "'''Similar planes''' are plane figures which are similar. Category:Definitions/Euclidean Geometry 6img9a3vyysc0t9nhmogs6mxs2ohfp6"}
+{"_id": "25300", "title": "Definition:Containment of Solid Figure", "text": "A solid geometric figure is said to be '''contained''' by the surfaces that form its boundary."}
+{"_id": "25301", "title": "Definition:Similar Solid Figures", "text": "{{EuclidDefinition|book=XI|def=9|name=Similar Solid Figures}} {{stub}} Category:Definitions/Solid Geometry g03otrtjq5ra8g4g45yp2bg50wx7o1a"}
+{"_id": "25302", "title": "Definition:Similar Equal Solid Figures", "text": "{{EuclidDefinition|book=XI|def=10|name=Similar Equal Solid Figures}} {{stub}} Category:Definitions/Solid Geometry 6jdhgei61uzyv7y7ax6v7tmsrgojcam"}
+{"_id": "25303", "title": "Definition:Solid Angle", "text": "{{EuclidDefinition|book=XI|def=11|name=Solid Angle}} === Containment of Solid Angle === {{:Definition:Solid Angle/Containment}} === Vertex of Solid Angle === {{:Definition:Solid Angle/Vertex}} Category:Definitions/Solid Geometry hyb7wr54ukev6qia0sl0294rd2x8opi"}
+{"_id": "25305", "title": "Definition:Plane of Reference", "text": "The '''plane of reference''' is a conceptual plane in a spatial context which is arbitrarily distinguished from the others. Category:Definitions/Solid Geometry obrrchumu5siktshe431w8i0996p6rj"}
+{"_id": "25306", "title": "Definition:Plane Surface/Side", "text": "From the definition of surface, it follows that a plane locally separates space into two '''sides'''. Thus the '''sides''' of a plane are the parts of that space into which the plane separates it."}
+{"_id": "25307", "title": "Definition:Solid Angle/Containment", "text": "The three plane angles which together form a solid angle are said to '''contain''' that solid angle."}
+{"_id": "25308", "title": "Definition:Parallelepiped", "text": "A '''parallelepiped''' is a polyhedron formed by three pairs of parallel planes: :500px In the above example, the pairs of parallel planes are: :$ABCD$ and $HGFE$ :$ADEH$ and $BCFG$ :$ABGH$ and $DCFE$"}
+{"_id": "25309", "title": "Definition:Geometric Figure/Three-Dimensional Figure/Face", "text": "The '''faces''' of a three-dimensional figure are the surfaces which form its extremities."}
+{"_id": "25310", "title": "Definition:Parallelepiped/Base", "text": "One of the faces of the parallelepiped is chosen arbitrarily, distinguished from the others and called the '''base of the parallelepiped'''. It is usual to choose the '''base''' to be the one which is conceptually on the bottom. In the above, $ABCD$ would conventionally be identified as being the '''base'''"}
+{"_id": "25311", "title": "Definition:Octahedron", "text": "An '''octahedron''' is a polyhedron which has $8$ faces."}
+{"_id": "25312", "title": "Definition:Tetrahedron", "text": "A '''tetrahedron''' is a polyhedron which has $4$ (triangular) faces."}
+{"_id": "25313", "title": "Definition:Dodecahedron", "text": "A '''dodecahedron''' is a polyhedron which has $12$ faces."}
+{"_id": "25314", "title": "Definition:Hexahedron", "text": "A '''hexahedron''' is a polyhedron which has $6$ faces. === Cube === {{:Definition:Hexahedron/Regular}}"}
+{"_id": "25315", "title": "Definition:Icosahedron", "text": "An '''icosahedron''' is a polyhedron which has $20$ faces."}
+{"_id": "25316", "title": "Definition:Polyhedron/Edge", "text": "The '''edges''' of a polyhedron are the sides of the polygons which constitute its faces."}
+{"_id": "25317", "title": "Definition:Polyhedron/Vertex", "text": "The '''vertices''' of a polyhedron are the vertices of the polygons which constitute its faces."}
+{"_id": "25318", "title": "Definition:Elevated", "text": "=== Elevated Point === {{:Definition:Elevated/Point}} === Elevated Line === {{:Definition:Elevated/Line}} Category:Definitions/Solid Geometry 347p9uj8vd4weqm5lcblp1pwasx7zp0"}
+{"_id": "25319", "title": "Definition:Angle/Vertex", "text": "The point at which the lines containing an angle meet is known as the '''vertex''' of that angle."}
+{"_id": "25320", "title": "Definition:Solid Angle/Vertex", "text": "The common vertex of the angles containing a solid angle is known as the '''vertex''' of that solid angle."}
+{"_id": "25321", "title": "Definition:Prism/Base", "text": "The '''bases''' of a prism are the two parallel polygons which form the faces at either end of the prism. In the above diagram, the faces $ABCDE$ and $FGHIJ$ are the '''bases''' of the prism."}
+{"_id": "25322", "title": "Definition:Pyramid/Base", "text": "The polygon of a pyramid to whose vertices the apex is joined is called the '''base''' of the pyramid. In the above diagram, $ABCDE$ is the '''base''' of the pyramid $ABCDEQ$."}
+{"_id": "25323", "title": "Definition:Pyramid/Apex", "text": "The vertex of a pyramid which is the common vertex of its triangular faces is called the '''apex''' of the pyramid. In the above diagram, $Q$ is the '''apex'''."}
+{"_id": "25324", "title": "Definition:Square Pyramid", "text": ":300px A '''square pyramid''' is a pyramid whose base is a square. Category:Definitions/Pyramids oxtzsa3zp635hqh0sj9c2aj1ryid8vh"}
+{"_id": "25325", "title": "Definition:Pyramid/Height", "text": "The '''height''' of a pyramid is the length of the perpendicular from the plane of the base to its apex. In the above diagram, $h$ is the height."}
+{"_id": "25326", "title": "Definition:Cylinder/Height", "text": "The '''height''' of a cylinder is the length of a line segment drawn perpendicular to the base and its opposite plane. In the above diagram, $h$ is the '''height''' of the cylinder $ACBDFE$."}
+{"_id": "25327", "title": "Definition:Height of Solid Figure", "text": "The '''height''' of a solid figure is the length of a perpendicular from the base to the point or points most distant from it. === Height of Parallelepiped === {{:Definition:Height of Parallelepiped}} === Height of Prism === :500px {{:Definition:Height of Prism}} === Height of Pyramid === {{:Definition:Height of Pyramid}} === Height of Cone === {{:Definition:Height of Cone}} === Height of Cylinder === {{:Definition:Height of Cylinder}} Category:Definitions/Solid Geometry 14tt10lxi71w6r2j1axuoo08vit7tmd"}
+{"_id": "25328", "title": "Definition:Base of Solid Figure", "text": "The '''base''' of a solid figure is one of its faces which has been distinguished from the others in some way. The solid figure is usually oriented so that the '''base''' is situated at the bottom. === Base of Parallelepiped === :500px {{:Definition:Base of Parallelepiped}} === Base of Pyramid === :400px {{:Definition:Base of Pyramid}} === Base of Prism === :500px {{:Definition:Base of Prism}} === Base of Cone === :300px {{:Definition:Base of Cone}} ==== Base of Right Circular Cone ==== :300px {{:Definition:Base of Right Circular Cone}} === Base of Cylinder === :300px {{:Definition:Base of Cylinder}}"}
+{"_id": "25329", "title": "Definition:Decagon", "text": "A '''decagon''' is a polygon with exactly $10$ sides. :300px"}
+{"_id": "25330", "title": "Definition:Geometric Sequence/Term", "text": "The elements: :$x_n$ for $n = 0, 1, 2, 3, \\ldots$ are the '''terms''' of $\\sequence {x_n}$."}
+{"_id": "25331", "title": "Definition:Proportion/Inverse", "text": "Two real variables $x$ and $y$ are '''inversely proportional''' {{iff}} their product is a constant: :$\\forall x, y \\in \\R: x \\propto \\dfrac 1 y \\iff \\exists k \\in \\R, k \\ne 0: x y = k$"}
+{"_id": "25332", "title": "Definition:Proportion/Joint", "text": "Two real variables $x$ and $y$ are '''jointly proportional''' to a third real variable $z$ {{iff}} the product of $x$ and $y$ is a constant multiple of $z$: :$\\forall x, y \\in \\R: x y \\propto z \\iff \\exists k \\in \\R, k \\ne 0: x y = k z$"}
+{"_id": "25333", "title": "Definition:Proportion/Continued", "text": "Four magnitudes $a, b, c, d$ are '''in continued proportion''' {{iff}} $a : b = b : c = c : d$."}
+{"_id": "25334", "title": "Definition:Linear Measure/Distance", "text": "The '''distance''' between two points $A$ and $B$ in space is defined as the length of a straight line that would be drawn from $A$ to $B$."}
+{"_id": "25335", "title": "Definition:Physical Constant", "text": "A '''physical constant''' is a physical quantity that is believed to be unchanging in both space and time."}
+{"_id": "25336", "title": "Definition:Recursive Sequence/Initial Terms", "text": "Let $S$ be a recursive sequence. In order for $S$ to be defined, it is necessary to define the '''initial term''' (or terms) explicitly. For example, in the Fibonacci sequence, the '''initial terms''' are defined as: :$F_0 = 0, F_1 = 1$"}
+{"_id": "25337", "title": "Definition:Proper Subset/Improper", "text": "$S$ is an '''improper subset''' of $T$ {{iff}} $S$ is a subset of $T$ but specifically ''not'' a proper subset of $T$. That is, either: :$S = T$ or: :$S = \\O$"}
+{"_id": "25338", "title": "Definition:Set Union/Family of Sets/Two Sets", "text": "Let $I = \\set {\\alpha, \\beta}$ be an indexing set containing exactly two elements. Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of sets indexed by $I$. From the definition of the union of $S_i$: :$\\displaystyle \\bigcup_{i \\mathop \\in I} S_i := \\set {x: \\exists i \\in I: x \\in S_i}$ it follows that: :$\\displaystyle \\bigcup \\set {S_\\alpha, S_\\beta} := S_\\alpha \\cup S_\\beta$"}
+{"_id": "25339", "title": "Definition:Set Intersection/Family of Sets/Two Sets", "text": "Let $I = \\set {\\alpha, \\beta}$ be an indexing set containing exactly two elements. Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of sets indexed by $I$. From the definition of the intersection of $S_i$: :$\\displaystyle \\bigcap_{i \\mathop \\in I} S_i := \\set {x: \\forall i \\in I: x \\in S_i}$ it follows that: :$\\displaystyle \\bigcap \\set {S_\\alpha, S_\\beta} := S_\\alpha \\cap S_\\beta$"}
+{"_id": "25340", "title": "Definition:Cartesian Product/Countable", "text": "Let $\\sequence {S_n}_{n \\mathop \\in \\N}$ be an infinite sequence of sets.  The '''cartesian product''' of $\\sequence {S_n}$ is defined as: :$\\displaystyle \\prod_{k \\mathop = 1}^\\infty S_k = \\set {\\tuple {x_1, x_2, \\ldots, x_n, \\ldots}: \\forall k \\in \\N: x_k \\in S_k}$ It defines the concept: :$S_1 \\times S_2 \\times \\cdots \\times S_n \\times \\cdots$ Thus $\\displaystyle \\prod_{k \\mathop = 1}^\\infty S_k$ is the set of all infinite sequences $\\tuple {x_1, x_2, \\ldots, x_n, \\ldots}$ with $x_k \\in S_k$."}
+{"_id": "25341", "title": "Definition:Composition of Mappings/General Definition", "text": "Let $f_1: S_1 \\to S_2, f_2: S_2 \\to S_3, \\ldots, f_n: S_n \\to S_{n + 1}$ be mappings such that the domain of $f_k$ is the same set as the codomain of $f_{k - 1}$. Then the '''composite of $f_1, f_2, \\ldots, f_n$''' is defined and denoted as: {{begin-eqn}} {{eqn | l = \\forall x \\in S_1: \\map {\\paren {f_n \\circ \\cdots \\circ f_2 \\circ f_1} } x       | o = :=       | r = \\begin {cases} \\map {f_1} x & : n = 1 \\\\ \\map {f_n} {\\map {\\paren {f_{n - 1} \\circ \\cdots \\circ f_2 \\circ f_1} } x} : & n > 1 \\end {cases}       | c =  }} {{eqn | r = \\map {f_n} {\\dotsm \\map {f_2} {\\map {f_1} x} \\dotsm}       | c =  }} {{end-eqn}}"}
+{"_id": "25342", "title": "Definition:Inverse Mapping/Definition 1", "text": "Let $f: S \\to T$ be a mapping. Let $f^{-1} \\subseteq T \\times S$ be the inverse of $f$: :$f^{-1} := \\set {\\tuple {t, s}: \\map f s = t}$ Let $f^{-1}$ itself be a mapping: :$\\forall y \\in T: \\tuple {y, x_1} \\in f^{-1} \\land \\tuple {y, x_2} \\in f^{-1} \\implies x_1 = x_2$ and :$\\forall y \\in T: \\exists x \\in S: \\tuple {y, x} \\in f$ Then $f^{-1}$ is called the '''inverse mapping of $f$'''."}
+{"_id": "25343", "title": "Definition:Chebyshev Distance", "text": "The '''Chebyshev distance''' on $A_1 \\times A_2$ is defined as: :$\\map {d_\\infty} {x, y} := \\max \\set {\\map {d_1} {x_1, y_1}, \\map {d_2} {x_2, y_2} }$ where $x = \\tuple {x_1, x_2}, y = \\tuple {y_1, y_2} \\in A_1 \\times A_2$."}
+{"_id": "25344", "title": "Definition:Taxicab Metric/General Definition", "text": "The '''taxicab metric''' on $\\displaystyle \\mathcal A = \\prod_{i \\mathop = 1}^n A_{i'}$ is defined as: : $\\displaystyle d_1 \\left({x, y}\\right) := \\sum_{i \\mathop = 1}^n d_{i'} \\left({x_i, y_i}\\right)$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({y_1, y_2, \\ldots, y_n}\\right) \\in \\mathcal A$."}
+{"_id": "25345", "title": "Definition:Chebyshev Distance/General Definition", "text": "The '''Chebyshev distance''' on $\\displaystyle \\mathcal A = \\prod_{i \\mathop = 1}^n A_i$ is defined as: :$\\displaystyle \\map {d_\\infty} {x, y} = \\max_{i \\mathop = 1}^n \\set {\\map {d_i} {x_i, y_i} }$ where $x = \\tuple {x_1, x_2, \\ldots, x_n}, y = \\tuple {y_1, y_2, \\ldots, y_n} \\in \\mathcal A$."}
+{"_id": "25346", "title": "Definition:Euclidean Metric", "text": "The '''Euclidean metric''' on $A_{1'} \\times A_{2'}$ is defined as: :$\\map {d_2} {x, y} := \\paren {\\paren {\\map {d_{1'} } {x_1, y_1} }^2 + \\paren {\\map {d_{2'} } {x_2, y_2} }^2}^{1/2}$ where $x = \\tuple {x_1, x_2}, y = \\tuple {y_1, y_2} \\in A_{1'} \\times A_{2'}$."}
+{"_id": "25347", "title": "Definition:Euclidean Metric/General Definition", "text": "The '''Euclidean metric''' on $\\displaystyle \\mathcal A = \\prod_{i \\mathop = 1}^n A_{i'}$ is defined as: : $\\displaystyle d_2 \\left({x, y}\\right) := \\left({\\sum_{i \\mathop = 1}^n \\left({d_{i'} \\left({x_i, y_i}\\right)}\\right)^2}\\right)^{\\frac 1 2}$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({y_1, y_2, \\ldots, y_n}\\right) \\in \\mathcal A$."}
+{"_id": "25348", "title": "Definition:Taxicab Metric/Real Vector Space", "text": "The '''taxicab metric''' on $\\R^n$ is defined as: : $\\displaystyle d_1 \\left({x, y}\\right) := \\sum_{i \\mathop = 1}^n \\left\\vert {x_i - y_i}\\right\\vert$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({y_1, y_2, \\ldots, y_n}\\right) \\in \\R^n$."}
+{"_id": "25349", "title": "Definition:Chebyshev Distance/Real Vector Space", "text": "The '''Chebyshev distance''' on $\\R^n$ is defined as: :$\\displaystyle \\map {d_\\infty} {x, y}:= \\max_{i \\mathop = 1}^n \\set {\\size {x_i - y_i} }$ where $x = \\tuple {x_1, x_2, \\ldots, x_n}, y = \\tuple {y_1, y_2, \\ldots, y_n} \\in \\R^n$."}
+{"_id": "25350", "title": "Definition:Standard Discrete Metric/Real Number Plane", "text": "The '''discrete metric''' on $\\R^2$ is defined as: : $\\displaystyle d_0 \\left({x, y}\\right) := \\begin{cases} 0 & : x = y \\\\ 1 & : \\exists i \\in \\left\\{{1, 2}\\right\\}: x_i \\ne y_i \\end{cases}$ where $x = \\left({x_1, x_2}\\right), y = \\left({y_1, y_2}\\right) \\in \\R^2$."}
+{"_id": "25351", "title": "Definition:Euclidean Metric/Real Number Plane", "text": "The '''Euclidean metric''' on $\\R^2$ is defined as: :$\\displaystyle \\map {d_2} {x, y} := \\sqrt {\\paren {x_1 - y_1}^2 + \\paren {x_2 - y_2}^2}$ where $x = \\tuple {x_1, x_2}, y = \\tuple {y_1, y_2} \\in \\R^2$."}
+{"_id": "25352", "title": "Definition:Taxicab Metric/Graphical Example", "text": "This diagram shows the open $\\epsilon$-ball of point $A$ in the $\\left({\\R^2, d_1}\\right)$ metric space where $d_1$ is the taxicab metric. :300px Note that $\\epsilon = \\epsilon_1 + \\epsilon_2$. Neither the boundary lines nor the extreme points are actually part of the open $\\epsilon$-ball."}
+{"_id": "25355", "title": "Definition:P-Product Metric/Real Vector Space", "text": "The '''$p$-product metric''' on $\\R^n$ is defined as: :$\\displaystyle d_p \\left({x, y}\\right) := \\left({\\sum_{i \\mathop = 1}^n \\left\\vert{x_i - y_i}\\right\\vert^p}\\right)^{\\frac 1 p}$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({y_1, y_2, \\ldots, y_n}\\right) \\in \\R^n$."}
+{"_id": "25356", "title": "Definition:Taxicab Metric/Real Number Plane", "text": "The '''taxicab metric''' on $\\R^2$ is defined as: :$\\map {d_1} {x, y} := \\size  {x_1 - y_1} + \\size {x_2 - y_2}$ where $x = \\tuple {x_1, x_2}$, $y = \\tuple {y_1, y_2} \\in \\R^2$."}
+{"_id": "25357", "title": "Definition:Chebyshev Distance/Real Number Plane", "text": "The '''Chebyshev distance''' on $\\R^2$ is defined as: :$\\map {d_\\infty} {x, y}:= \\max \\set {\\size {x_1 - y_1}, \\size {x_2 - y_2} }$ where $x = \\tuple {x_1, x_2}, y = \\tuple {y_1, y_2} \\in \\R^2$."}
+{"_id": "25358", "title": "Definition:Euclidean Metric/Ordinary Space", "text": "The '''Euclidean metric''' on $\\R^3$ is defined as: :$\\map {d_2} {x, y} := \\sqrt {\\paren {x_1 - y_1}^2 + \\paren {x_2 - y_2}^2 + \\paren {x_3 - y_3}^2}$ where $x = \\tuple {x_1, x_2, x_3}, y = \\tuple {y_1, y_2, y_3} \\in \\R^3$."}
+{"_id": "25359", "title": "Definition:Euclidean Metric/Complex Plane", "text": "The '''Euclidean metric''' on $\\C$ is defined as: :$\\displaystyle \\forall z_1, z_2 \\in \\C: \\map d {z_1, z_2} := \\size {z_1 - z_2}$ where $\\size {z_1 - z_2}$ denotes the modulus of $z_1 - z_2$."}
+{"_id": "25360", "title": "Definition:Euclidean Metric/Rational Number Plane", "text": "The '''Euclidean metric''' on $\\Q^2$ is defined as: :$\\displaystyle d_2 \\left({x, y}\\right) := \\sqrt{\\left({x_1 - y_1}\\right)^2 + \\left({x_2 - y_2}\\right)^2}$ where $x = \\left({x_1, x_2}\\right), y = \\left({y_1, y_2}\\right) \\in \\Q^2$."}
+{"_id": "25362", "title": "Definition:Continuous Real Function/Open Interval", "text": "Let $f$ be a real function defined on an open interval $\\openint a b$. Then $f$ is '''continuous on $\\openint a b$''' {{iff}} it is continuous at every point of $\\openint a b$."}
+{"_id": "25363", "title": "Definition:Continuous Real Function/Closed Interval/Definition 1", "text": "The function $f$ is '''continuous on $\\left[{a \\,.\\,.\\, b}\\right]$''' {{iff}} it is: :$(1): \\quad$ continuous at every point of $\\left({a \\,.\\,.\\, b}\\right)$ :$(2): \\quad$ continuous on the right at $a$ :$(3): \\quad$ continuous on the left at $b$. That is, if $f$ is to be continuous over the ''whole'' of a closed interval, it needs to be continuous at the end points. Because we only have \"access\" to the function on one side of each end point, all we can do is insist on continuity on the side of the end points on which the function is defined."}
+{"_id": "25364", "title": "Definition:Continuous Real Function/Half Open Interval", "text": "Let $f$ be a real function defined on a half open interval $\\hointl a b$. Then $f$ is '''continuous on $\\hointl a b$''' {{iff}} it is: :$(1): \\quad$ continuous at every point of $\\openint a b$ :$(2): \\quad$ continuous on the left at $b$. Let $f$ be a real function defined on a half open interval $\\hointr a b$. Then $f$ is '''continuous on $\\hointr a b$''' {{iff}} it is: :$(1): \\quad$ continuous at every point of $\\openint a b$ :$(2): \\quad$ continuous on the right at $a$."}
+{"_id": "25365", "title": "Definition:Supremum Metric/Bounded Real Functions on Interval", "text": "Let $\\left[{a \\,.\\,.\\, b}\\right] \\subseteq \\R$ be a closed real interval. Let $A$ be the set of all bounded real functions $f: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$. Let $d: A \\times A \\to \\R$ be the function defined as: :$\\displaystyle \\forall f, g \\in A: d \\left({f, g}\\right) := \\sup_{x \\mathop \\in \\left[{a \\,.\\,.\\, b}\\right]} \\left\\vert{f \\left({x}\\right) - g \\left({x}\\right)}\\right\\vert$ where $\\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$."}
+{"_id": "25366", "title": "Definition:Supremum Metric", "text": "Let $S$ be a set. Let $M = \\left({A', d'}\\right)$ be a metric space. Let $A$ be the set of all bounded mappings $f: S \\to M$. Let $d: A \\times A \\to \\R$ be the function defined as: :$\\displaystyle \\forall f, g \\in A: d \\left({f, g}\\right) := \\sup_{x \\mathop \\in S} d' \\left({f \\left({x}\\right), g \\left({x}\\right)}\\right)$ where $\\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$."}
+{"_id": "25367", "title": "Definition:Supremum Metric/Continuous Real Functions", "text": "Let $\\left[{a \\,.\\,.\\, b}\\right] \\subseteq \\R$ be a closed real interval. Let $A$ be the set of all continuous functions $f: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$. Let $d: A \\times A \\to \\R$ be the function defined as: :$\\displaystyle \\forall f, g \\in A: d \\left({f, g}\\right) := \\sup_{x \\mathop \\in \\left[{a \\,.\\,.\\, b}\\right]} \\left\\vert{f \\left({x}\\right) - g \\left({x}\\right)}\\right\\vert$ where $\\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$."}
+{"_id": "25368", "title": "Definition:L1 Metric", "text": "=== Closed Real Interval === {{:Definition:L1 Metric/Closed Real Interval}} Category:Definitions/L1 Metric aihu5b2srlf0jq3gj9kmoz3l6zyxaav"}
+{"_id": "25369", "title": "Definition:L1 Metric/Closed Real Interval", "text": "Let $S$ be the set of all real functions which are continuous on the closed interval $\\left[{a \\,.\\,.\\, b}\\right]$. Let the real-valued function $d: S \\times S \\to \\R$ be defined as: :$\\displaystyle \\forall f, g \\in S: d_1 \\left({f, g}\\right) := \\int_a^b \\left\\vert{f \\left({t}\\right) - g \\left({t}\\right)}\\right\\vert \\ \\mathrm d t$ Then $d_1$ is the '''$L^1$ metric''' on $\\left[{a \\,.\\,.\\, b}\\right]$."}
+{"_id": "25370", "title": "Definition:L2 Metric/Closed Real Interval", "text": "Let $S$ be the set of all real functions which are continuous on the closed interval $\\closedint a b$. Let the real-valued function $d: S \\times S \\to \\R$ be defined as: :$\\displaystyle \\forall f, g \\in S: \\map d {f, g} := \\paren {\\int_a^b \\paren {\\map f t - \\map g t}^2 \\rd t}^{\\frac 1 2}$ Then $d$ is the '''$L^2$ metric''' on $\\closedint a b$."}
+{"_id": "25371", "title": "Definition:L2 Metric", "text": "=== Closed Real Interval === {{:Definition:L2 Metric/Closed Real Interval}} Category:Definitions/L2 Metric omvah450xw3aqgxhw9r8qsidzeb3taw"}
+{"_id": "25372", "title": "Definition:Bounded Metric Space/Definition 1", "text": "'''$M'$ is bounded (in $M$)''' {{iff}}: :$\\exists a \\in A, K \\in \\R: \\forall x \\in B: \\map {d_B} {x, a} \\le K$ That is, there exists an element of $A$ within a finite distance of all elements of $B$."}
+{"_id": "25373", "title": "Definition:Bounded Metric Space/Definition 2", "text": "$M'$ is '''bounded''' {{iff}}: :$\\exists K \\in \\R: \\forall x, y \\in M': \\map {d_B} {x, y} \\le K$ That is, there exists a finite distance such that all pairs of elements of $B$ are within that distance."}
+{"_id": "25374", "title": "Definition:Hilbert Sequence Space", "text": "Let $d_2: A \\times A: \\to \\R$ be the real-valued function defined as: :$\\displaystyle \\forall x = \\sequence {x_i}, y = \\sequence {y_i} \\in A: \\map {d_2} {x, y} := \\paren {\\sum_{k \\mathop \\ge 0} \\paren {x_k - y_k}^2}^{\\frac 1 2}$ The metric space $\\struct {A, d_2}$ is the '''Hilbert sequence space on $\\R$''' and is denoted $\\ell^2$."}
+{"_id": "25375", "title": "Definition:Supremum Metric/Bounded Real-Valued Functions", "text": "Let $S$ be a set. Let $A$ be the set of all bounded real-valued functions $f: S \\to \\R$. Let $d: A \\times A \\to \\R$ be the function defined as: :$\\displaystyle \\forall f, g \\in A: d \\left({f, g}\\right) := \\sup_{x \\mathop \\in S} \\left\\vert{f \\left({x}\\right) - g \\left({x}\\right)}\\right\\vert$ where $\\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$."}
+{"_id": "25376", "title": "Definition:Supremum of Real Sequence", "text": "Let $\\sequence {x_n}$ be a real sequence. Let $\\set {x_n: n \\in \\N}$ admit a supremum. Then the '''supremum''' of $\\sequence {x_n}$) is defined as: :$\\displaystyle \\map \\sup {\\sequence {x_n} } = \\map \\sup {\\set {x_n: n \\in \\N} }$"}
+{"_id": "25377", "title": "Definition:Infimum of Real Sequence", "text": "Let $\\sequence {x_n}$ be a real sequence. Let $\\set {x_n: n \\in \\N}$ admit an infimum. Then the '''infimum''' of $\\sequence {x_n}$) is defined as: :$\\map \\inf {\\sequence {x_n} } = \\map \\inf {\\set {x_n: n \\in \\N} }$"}
+{"_id": "25378", "title": "Definition:Supremum Metric/Bounded Real Sequences", "text": "Let $A$ be the set of all bounded real sequences. Let $d: A \\times A \\to \\R$ be the function defined as: :$\\displaystyle \\forall \\left\\langle{x_i}\\right\\rangle, \\left\\langle{y_i}\\right\\rangle \\in A: d \\left({\\left\\langle{x_i}\\right\\rangle, \\left\\langle{y_i}\\right\\rangle}\\right) := \\sup_{n \\mathop \\in \\N} \\left\\vert{x_n - y_n}\\right\\vert$ where $\\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$."}
+{"_id": "25379", "title": "Definition:Supremum Metric/Bounded Continuous Mappings", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $A$ be the set of all continuous mappings $f: M_1 \\to M_2$ which are also bounded. Let $d: A \\times A \\to \\R$ be the function defined as: :$\\displaystyle \\forall f, g \\in A: d \\left({f, g}\\right) := \\sup_{x \\mathop \\in A_1} d_2 \\left({f \\left({x}\\right), g \\left({x}\\right)}\\right)$ where $\\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$."}
+{"_id": "25380", "title": "Definition:Supremum Metric/Differentiability Class", "text": "Let $\\left[{a \\,.\\,.\\, b}\\right] \\subseteq \\R$ be a closed real interval. Let $r \\in \\N$ be a natural number. Let $\\mathscr D^r \\left[{a \\,.\\,.\\, b}\\right]$ be the set of all continuous functions $f: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$ which are of differentiability class $r$. Let $d: \\mathscr D^r \\left[{a \\,.\\,.\\, b}\\right] \\times \\mathscr D^r \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$ be the function defined as: :$\\displaystyle \\forall f, g \\in \\mathscr D^r \\left[{a \\,.\\,.\\, b}\\right]: d \\left({f, g}\\right) := \\sup_{\\substack {x \\mathop \\in \\left[{a \\,.\\,.\\, b}\\right] \\\\ i \\in \\left\\{ {0, 1, 2, \\ldots, r}\\right\\} } } \\left\\vert{f^{\\left({i}\\right)} \\left({x}\\right) - g^{\\left({i}\\right)} \\left({x}\\right)}\\right\\vert$ where: : $f^{\\left({i}\\right)}$ denotes the $i$th derivative of $f$ : $f^{\\left({0}\\right)}$ denotes $f$ : $\\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $\\mathscr D^r \\left[{a \\,.\\,.\\, b}\\right]$."}
+{"_id": "25381", "title": "Definition:Lp Metric", "text": "=== Closed Real Interval === {{:Definition:Lp Metric/Closed Real Interval}} Category:Definitions/Lp Metrics 78tx38phmzq5alt9mzh28ohae55m5yg"}
+{"_id": "25382", "title": "Definition:Lp Metric/Closed Real Interval", "text": "Let $S$ be the set of all real functions which are continuous on the closed interval $\\left[{a \\,.\\,.\\, b}\\right]$. Let $p \\in \\R_{\\ge 1}$. Let the real-valued function $d: S \\times S \\to \\R$ be defined as: :$\\displaystyle \\forall f, g \\in S: d \\left({f, g}\\right) := \\left({\\int_a^b \\left\\vert{f \\left({t}\\right) - g \\left({t}\\right)}\\right\\vert^p \\ \\mathrm d t}\\right)^{\\frac 1 p}$ Then $d$ is the '''$L^p$ metric''' on $\\left[{a \\,.\\,.\\, b}\\right]$."}
+{"_id": "25383", "title": "Definition:P-Sequence Metric/Real Sequences", "text": "Let $d_p: A \\times A: \\to \\R$ be the real-valued function defined as: :$\\displaystyle \\forall x = \\left\\langle{x_i}\\right\\rangle, y = \\left\\langle{y_i}\\right\\rangle \\in A: d_p \\left({x, y}\\right) := \\left({\\sum_{k \\mathop \\ge 0} \\left\\vert{x_k - y_k}\\right\\vert^p}\\right)^{\\frac 1 p}$ The metric space $\\left({A, d_p}\\right)$ is the '''$p$-sequence space on $\\R$''' and is denoted $\\ell^p$."}
+{"_id": "25384", "title": "Definition:P-Sequence Metric", "text": "=== Real Sequences === {{:Definition:P-Sequence Metric/Real Sequences}} Category:Definitions/P-Sequence Metrics Category:Definitions/Examples of Metric Spaces m2zeg7e77318tab4lbkow9vm6dzc8el"}
+{"_id": "25385", "title": "Definition:P-adic Valuation/Integers", "text": "The '''$p$-adic valuation (on $\\Z$)''' is the mapping $\\nu_p^\\Z: \\Z \\to \\N \\cup \\set {+\\infty}$ defined as: :$\\map {\\nu_p^\\Z} n := \\begin {cases} +\\infty & : n = 0 \\\\ \\sup \\set {v \\in \\N: p^v \\divides n} & : n \\ne 0 \\end{cases}$ where: :$\\sup$ denotes supremum :$p^v \\divides n$ expresses that $p^v$ divides $n$."}
+{"_id": "25386", "title": "Definition:P-adic Valuation/Rational Numbers", "text": "Let the $p$-adic valuation on the integers $\\nu_p^\\Z$ be extended to $\\nu_p^\\Q: \\Q \\to \\Z \\cup \\left\\{{+\\infty}\\right\\}$ by: :$\\nu_p^\\Q \\left({\\dfrac a b}\\right) := \\nu_p^\\Z \\left({a}\\right) - \\nu_p^\\Z \\left({b}\\right)$ This mapping $\\nu_p^\\Q$ is called the '''$p$-adic valuation (on $\\Q$)''' and is usually denoted $\\nu_p: \\Q \\to \\Z \\cup \\left\\{{+\\infty}\\right\\}$."}
+{"_id": "25387", "title": "Definition:Valuation Axioms", "text": "{{begin-axiom}} {{axiom | n = \\text V 1         | q = \\forall a, b \\in R         | ml= \\map \\nu {a \\times b}         | mo= =         | mr= \\map \\nu a + \\map \\nu b }} {{axiom | n = \\text V 2         | q = \\forall a \\in R         | ml= \\map \\nu a = +\\infty         | mo= \\iff         | mr= a = 0_R         | rc= where $0_R$ is the ring zero }} {{axiom | n = \\text V 3         | q = \\forall a, b \\in R         | ml= \\map \\nu {a + b}         | mo= \\ge         | mr= \\min \\set {\\map \\nu a, \\map \\nu b} }} {{end-axiom}}"}
+{"_id": "25388", "title": "Definition:Multiplicative Norm Axioms", "text": "{{begin-axiom}} {{axiom | n = \\text N 1         | lc= Positive Definiteness:         | q = \\forall x \\in R         | ml= \\norm x = 0         | mo= \\iff         | mr= x = 0_R }} {{axiom | n = \\text N 2         | lc= Multiplicativity:         | q = \\forall x, y \\in R         | ml= \\norm {x \\circ y}         | mo= =         | mr= \\norm x \\times \\norm y }} {{axiom | n = \\text N 3         | lc= Triangle Inequality:         | q = \\forall x, y \\in R         | ml= \\norm {x + y}         | mo= \\le         | mr= \\norm x + \\norm y }} {{end-axiom}}"}
+{"_id": "25389", "title": "Definition:Norm Axioms (Vector Space)", "text": "{{begin-axiom}} {{axiom | n = \\text N 1         | lc= Positive definiteness:         | q = \\forall x \\in V         | ml= \\norm x = 0         | mo= \\iff         | mr= x = \\mathbf 0_V }} {{axiom | n = \\text N 2         | lc= Positive homogeneity:         | q = \\forall x \\in V, \\lambda \\in R         | ml= \\norm {\\lambda x}         | mo= =         | mr= \\norm {\\lambda}_R \\times \\norm x }} {{axiom | n = \\text N 3         | lc= Triangle inequality:         | q = \\forall x, y \\in V         | ml= \\norm {x + y}         | mo= \\le         | mr= \\norm x + \\norm y }} {{end-axiom}}"}
+{"_id": "25390", "title": "Definition:Topology Induced by Metric/Definition 1", "text": "The '''topology on the metric space $M = \\struct {A, d}$ induced by (the metric) $d$''' is defined as the set $\\tau$ of all open sets of $M$."}
+{"_id": "25391", "title": "Definition:Topology Induced by Metric/Definition 2", "text": "The '''topology on the metric space $M = \\struct {A, d}$ induced by (the metric) $d$''' is defined as the topology $\\tau$ generated by the basis consisting of the set of all open $\\epsilon$-balls in $M$."}
+{"_id": "25392", "title": "Definition:Closed Set/Real Analysis", "text": "=== Real Numbers === {{:Definition:Closed Set/Real Analysis/Real Numbers}} === Real Euclidean Space === {{:Definition:Closed Set/Real Analysis/Real Euclidean Space}}"}
+{"_id": "25393", "title": "Definition:System of Neighborhoods", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $a \\in A$. Let $\\mathcal N_a$ be the set of all neighborhoods of $a$ in $M$. Then $\\mathcal N_a$ is the '''system of neighborhoods''' of the point $a$."}
+{"_id": "25394", "title": "Definition:Continuous Mapping (Metric Space)/Point/Definition 4", "text": "$f$ is '''continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$)''' {{iff}}: :for each neighborhood $N'$ of $f \\left({a}\\right)$ in $M_2$ there exists a corresponding neighborhood $N$ of $a$ in $M_1$ such that $f \\left[{N}\\right] \\subseteq N'$."}
+{"_id": "25395", "title": "Definition:Basis for Neighborhood System", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $a \\in A$ be a point in $M$. Let $\\mathcal B_a$ be a set of neighborhoods of $a$ in $M$. Then $\\mathcal B_a$ is a '''basis for the neighborhood system at $a$''' iff: :$\\forall N_a \\subseteq M: \\exists B \\in \\mathcal B_a: B \\subseteq N_a$ where $N_a$ denotes a neighborhood of $a$ in $M$. That is, $\\mathcal B_a$ is a '''basis for the neighborhood system at $a$''' iff every neighborhood of $a$ contains an element of $\\mathcal B_a$ as a subset."}
+{"_id": "25396", "title": "Definition:Limit of Real Function/Limit at Infinity", "text": "=== Limit at (Positive) Infinity === {{:Definition:Limit of Real Function/Limit at Infinity/Positive}} === Limit at Negative Infinity === {{:Definition:Limit of Real Function/Limit at Infinity/Negative}}"}
+{"_id": "25397", "title": "Definition:Convergent Mapping/Metric Space", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $c$ be a limit point of $M_1$. Let $f: A_1 \\to A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ ''except possibly'' at $c$. Let $f \\left({x}\\right)$ tend to the limit $L$ as $x$ tends to $c$. Then $f$ '''converges to the limit $L$ as $x$ tends to $c$'''."}
+{"_id": "25398", "title": "Definition:Convergent Mapping/Complex Function", "text": "Let $f: \\C \\to \\C$ be a complex function defined everywhere on $\\C$ ''except possibly'' at $c$. Let $f \\left({z}\\right)$ tend to the limit $L$ as $z$ tends to $c$. Then $f$ '''converges to the limit $L$ as $z$ tends to $c$'''."}
+{"_id": "25399", "title": "Definition:Convergent Mapping/Real Function", "text": "Let $f: \\R \\to \\R$ be a real function defined everywhere on $A_1$ ''except possibly'' at $c$. Let $\\map f x$ tend to the limit $L$ as $x$ tends to $c$. Then $f$ '''converges to the limit $L$ as $x$ tends to $c$'''."}
+{"_id": "25400", "title": "Definition:Convergent Sequence/Metric Space/Definition 1", "text": "$\\sequence {x_k}$ '''converges to the limit $l \\in A$''' {{iff}}: :$\\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\R_{>0}: \\forall n \\in \\N: n > N \\implies \\map d {x_n, l} < \\epsilon$"}
+{"_id": "25401", "title": "Definition:Convergent Sequence/Metric Space/Definition 2", "text": "$\\sequence {x_k}$ '''converges to the limit $l \\in A$''' {{iff}}: :$\\forall \\epsilon > 0: \\exists N \\in \\R_{>0}: \\forall n \\in \\N: n > N \\implies x_n \\in \\map {B_\\epsilon} l$ where $\\map {B_\\epsilon} l$ is the open $\\epsilon$-ball of $l$."}
+{"_id": "25402", "title": "Definition:Convergent Sequence/Metric Space/Definition 3", "text": "$\\sequence {x_k}$ '''converges to the limit $l \\in A$''' {{iff}}: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map d {x_n, l} = 0$"}
+{"_id": "25403", "title": "Definition:Almost All/Set Theory", "text": "=== Countable === {{:Definition:Almost All/Set Theory/Countable}} === Uncountable === {{:Definition:Almost All/Set Theory/Uncountable}}"}
+{"_id": "25404", "title": "Definition:Limit of Sequence/Real Numbers", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let $\\sequence {x_n}$ converge to a value $l \\in \\R$. Then $l$ is a '''limit of $\\sequence {x_n}$ as $n$ tends to infinity'''."}
+{"_id": "25405", "title": "Definition:Convergent Sequence/Real Numbers", "text": "Let $\\sequence {x_k}$ be a sequence in $\\R$. The sequence $\\sequence {x_k}$ '''converges to the limit $l \\in \\R$''' {{iff}}: :$\\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\R_{>0}: n > N \\implies \\size {x_n - l} < \\epsilon$ where $\\size x$ denotes the absolute value of $x$."}
+{"_id": "25406", "title": "Definition:Limit of Sequence/Rational Numbers", "text": "Let $\\left \\langle {x_n} \\right \\rangle$ be a sequence in $\\Q$. Let $\\left \\langle {x_n} \\right \\rangle$ converge to a value $l \\in \\R$, where $\\R$ denotes the set of real numbers. Then $l$ is a '''limit of $\\left \\langle {x_n} \\right \\rangle$ as $n$ tends to infinity'''."}
+{"_id": "25407", "title": "Definition:Convergent Sequence/Rational Numbers", "text": "Let $\\sequence {x_k}$ be a sequence in $\\Q$. $\\sequence {x_k}$ '''converges to the limit $l \\in \\R$''' {{iff}}: :$\\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\R_{>0}: n > N \\implies \\size {x_n - l} < \\epsilon$ where $\\size x$ is the absolute value of $x$."}
+{"_id": "25408", "title": "Definition:Limit of Sequence/Complex Numbers", "text": "Let $\\sequence {z_n}$ be a sequence in $\\C$. Let $\\sequence {z_n}$ converge to a value $l \\in \\C$. Then $l$ is a '''limit of $\\sequence {z_n}$ as $n$ tends to infinity'''."}
+{"_id": "25409", "title": "Definition:Convergent Sequence/Complex Numbers/Definition 1", "text": "Let $\\sequence {z_k}$ be a sequence in $\\C$. $\\sequence {z_k}$ '''converges to the limit $c \\in \\C$''' {{iff}}: :$\\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\R: n > N \\implies \\cmod {z_n - c} < \\epsilon$ where $\\cmod z$ denotes the modulus of $z$."}
+{"_id": "25410", "title": "Definition:Rational Number/Fraction/Vulgar", "text": "A '''vulgar fraction''' is a rational number whose absolute value is less than $1$, expressed in the form $r = \\dfrac p q$, where $p$ and $q$ are integers."}
+{"_id": "25411", "title": "Definition:Rational Number/Fraction/Improper", "text": "An '''improper fraction''' is a rational number whose absolute value is greater than $1$, specifically when expressed in the form $r = \\dfrac p q$ where $p$ and $q$ are integers such that $p > q$."}
+{"_id": "25412", "title": "Definition:Rational Number/Fraction/Mixed Number", "text": "A '''mixed number''' is a rational number whose absolute value is greater than $1$, expressed in the form $r = n \\frac p q$ where: : $p$ and $q$ are integers such that $p < q$ : $r = n + \\dfrac p q$"}
+{"_id": "25413", "title": "Definition:Absolute Value/Definition 1", "text": "Let $x \\in \\R$ be a real number. The '''absolute value''' of $x$ is denoted $\\size x$, and is defined using the usual ordering on the real numbers as follows: :$\\size x = \\begin{cases} x & : x > 0 \\\\ 0 & : x = 0 \\\\ -x & : x < 0 \\end{cases}$"}
+{"_id": "25414", "title": "Definition:Absolute Value/Definition 2", "text": "Let $x \\in \\R$ be a real number. The '''absolute value''' of $x$ is denoted $\\size x$, and is defined as: :$\\size x = +\\sqrt {x^2}$ where $+\\sqrt {x^2}$ is the positive square root of $x^2$."}
+{"_id": "25415", "title": "Definition:Absolute Value/Number Classes", "text": "The absolute value function applies to the various number classes as follows: : Natural numbers $\\N$: All elements of $\\N$ are greater than or equal to zero, so the concept is irrelevant. : Integers $\\Z$: As defined here. : Rational numbers $\\Q$: As defined here. : Real numbers $\\R$: As defined here. : Complex numbers $\\C$: As $\\C$ is not an ordered set, the definition of the absolute value function based upon whether a complex number is greater than or less than zero cannot be applied. The notation $\\cmod z$, where $z \\in \\C$, is defined as the modulus of $z$ and has a different meaning."}
+{"_id": "25417", "title": "Definition:Open Ball/Center", "text": "In $\\map {B_\\epsilon} a$, the value $a$ is referred to as the '''center''' of the open $\\epsilon$-ball."}
+{"_id": "25418", "title": "Definition:Basis for Open Sets (Metric Space)", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $\\mathcal B$ be a set of open sets of $M$. Then $\\mathcal B$ is a '''basis for the open sets of $M$''' iff: :for each open set $U$ of $M$, $U$ is the union of sets of $\\mathcal B$."}
+{"_id": "25419", "title": "Definition:Closure (Metric Space)", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $H \\subseteq A$. Let $H'$ be the set of limit points of $H$. Let $H^i$ be the set of isolated points of $H$. The '''closure of $H$ (in $M$)''' is the union of all isolated points of $H$ and all limit points of $H$: :$H^- := H' \\cup H^i$"}
+{"_id": "25420", "title": "Definition:Closure (Topology)/Notation", "text": "The topological closure of $H$ is variously denoted: :$\\map \\cl H$ :$\\map {\\mathrm {Cl} } H$ :$\\overline H$ :$H^-$ Of these, it can be argued that $\\overline H$ has more problems with ambiguity than the others, as it is also frequently used for the set complement. $\\map \\cl H$ and $\\map {\\mathrm {Cl} } H$ are cumbersome, but they have the advantage of being clear. $H^-$ is neat and compact, but has the disadvantage of being relatively obscure. On {{ProofWiki}}, $H^-$ is notation of choice, although $\\map \\cl H$ can also be found in places."}
+{"_id": "25421", "title": "Definition:Hermitian Conjugate", "text": "Let $\\mathbf A = \\sqbrk \\alpha_{m n}$ be an $m \\times n$ matrix over the complex numbers $\\C$. Then the '''Hermitian conjugate''' of $\\mathbf A$ is denoted $\\mathbf A^\\dagger$ and is defined as: :$\\mathbf A^\\dagger = \\sqbrk \\beta_{n m}: \\forall i \\in \\set {1, 2, \\ldots, n}, j \\in \\set {1, 2, \\ldots, m}: \\beta_{i j} = \\overline {\\alpha_{j i} }$ where $\\overline {\\alpha_{j i} }$ denotes the complex conjugate of $\\alpha_{j i}$."}
+{"_id": "25422", "title": "Definition:Hermitian Matrix", "text": "Let $\\mathbf A$ be a square matrix over $\\C$. $\\mathbf A$ is '''Hermitian''' {{iff}}: :$\\mathbf A = \\mathbf A^\\dagger$ where $\\mathbf A^\\dagger$ is the Hermitian conjugate of $\\mathbf A$."}
+{"_id": "25423", "title": "Definition:Binomial Coefficient/Technical Note", "text": "The $\\LaTeX$ code to render the binomial coefficient $\\dbinom n k$ can be written in the following ways: :<code>\\dbinom n k</code> or: :<code>\\displaystyle {n \\choose k}</code> The <code>\\dbinom</code> form is preferred on {{ProofWiki}} because it is simpler. It is in fact an abbreviated form of <code>\\displaystyle \\binom n k</code>, which is the preferred construction when <code>\\displaystyle</code> is required for another entity in the expression. While the form <code>\\binom n k</code> is valid $\\LaTeX$ syntax, it renders the entity in the reduced size inline style: $\\binom n k$ which {{ProofWiki}} does not endorse. To render compound arguments, braces are needed to delimit the parameter when using <code>\\dbinom</code>, but (confusingly) not <code>\\choose</code>. For example, to render $\\dbinom {a + b} {c d}$ the following can be used: :<code><nowiki>\\dbinom {a + b} {c d}</nowiki></code> or: :<code><nowiki>\\displaystyle {a + b \\choose c d}</nowiki></code> $\\displaystyle {a + b \\choose c d}$ Again, for consistency across {{ProofWiki}}, the <code>\\dbinom</code> form is preferred."}
+{"_id": "25424", "title": "Definition:Unitary Matrix", "text": "Let $\\mathbf U$ be an invertible square matrix over the complex numbers $\\C$. Then $\\mathbf U$ is '''unitary''' {{iff}}: :$\\mathbf U^{-1} = \\mathbf U^\\dagger$ where: :$\\mathbf U^{-1}$ is the inverse of $\\mathbf U$ :$\\mathbf U^\\dagger$ is the Hermitian conjugate of $\\mathbf U$."}
+{"_id": "25425", "title": "Definition:Homeomorphism/Metric Spaces", "text": "=== Definition 1 === {{:Definition:Homeomorphism/Metric Spaces/Definition 1}} === Definition 2 === {{:Definition:Homeomorphism/Metric Spaces/Definition 2}} === Definition 3 === {{:Definition:Homeomorphism/Metric Spaces/Definition 3}} === Definition 4 === {{:Definition:Homeomorphism/Metric Spaces/Definition 4}}"}
+{"_id": "25426", "title": "Definition:Möbius Transformation", "text": "A '''Möbius transformation''' is a mapping $f: \\overline \\C \\to \\overline \\C$ of the form: :$\\map f z = \\dfrac {a z + b} {c z + d}$ where: :$\\overline C$ denotes the extended complex plane :$a, b, c, d \\in \\C$ such that $a d - b c \\ne 0$ We define: :$\\map f {-\\dfrac d c} = \\infty$ if $c \\ne 0$, and: :$\\map f \\infty = \\begin{cases} \\dfrac a c & : c \\ne 0 \\\\ \\infty & : c = 0 \\end{cases}$"}
+{"_id": "25427", "title": "Definition:Liouville Number", "text": "A real number $x$ is a '''Liouville number''' if for all $n \\in \\N$, there exist $p, q \\in \\Z$ (which may depend on $n$) with $q > 1$ such that: :$0 < \\size {x - \\dfrac p q} < \\dfrac 1 {q^n}$"}
+{"_id": "25428", "title": "Definition:Bell Number", "text": "The '''Bell Numbers''' $B_n$ are a sequence of natural numbers defined as the number of ways a set with $n$ elements can be partitioned."}
+{"_id": "25429", "title": "Definition:Catalan's Constant", "text": "'''Catalan's constant''' is the real number defined as: {{begin-eqn}} {{eqn | l = G       | r = \\map \\beta 2       | c =  }} {{eqn | r = \\sum_{n \\mathop = 0}^{\\infty} \\frac{\\paren {-1}^n} {\\paren {2 n + 1}^2}       | c =  }} {{eqn | r = \\frac 1 {1^2} - \\frac 1 {3^2} + \\frac 1 {5^2} - \\frac 1 {7^2} + \\cdots       | c =  }} {{end-eqn}} where $\\beta$ is the Dirichlet beta function. Its numerical value is approximately: :$G = 0 \\cdotp 91596 \\, 55941 \\, 77219 \\, 01505 \\, 46035 \\, 14932 \\, 38411 \\, 0774 \\ldots$"}
+{"_id": "25430", "title": "Definition:Lipschitz Equivalence/Metrics/Definition 1", "text": "Let $M_1 = \\struct {A, d_1}$ and $M_2 = \\struct {A, d_2}$ be metric spaces on the same underlying set $A$. Let $\\exists h, k \\in \\R_{>0}$ such that: : $\\forall x, y \\in A: h \\map {d_2} {x, y} \\le \\map {d_1} {x, y} \\le k \\map {d_2} {x, y}$ Then $d_1$ and $d_2$ are described as '''Lipschitz equivalent'''."}
+{"_id": "25431", "title": "Definition:Lipschitz Equivalence/Metrics/Definition 2", "text": "Let $M_1 = \\struct {A, d_1}$ and $M_2 = \\struct {A, d_2}$ be metric spaces on the same underlying set $A$. Let $\\exists K_1, K_2 \\in \\R_{>0}$ such that: : $(1): \\quad \\forall x, y \\in A: \\map {d_2} {x, y} \\le K_1 \\map {d_1} {x, y}$ : $(2): \\quad \\forall x, y \\in A: \\map {d_1} {x, y} \\le K_2 \\map {d_2} {x, y}$ Then $d_1$ and $d_2$ are described as '''Lipschitz equivalent'''."}
+{"_id": "25432", "title": "Definition:Homeomorphism/Metric Spaces/Definition 1", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f: A_1 \\to A_2$ be a bijection such that: : $f$ is continuous from $M_1$ to $M_2$ : $f^{-1}$ is continuous from $M_2$ to $M_1$. Then: : $f$ is a '''homeomorphism''' : $M_1$ and $M_2$ are '''homeomorphic'''."}
+{"_id": "25433", "title": "Definition:Homeomorphism/Metric Spaces/Definition 2", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f: A_1 \\to A_2$ be a bijection such that: : for all $U \\subseteq A_1$, $U$ is an open set of $M_1$ {{iff}} $f \\left[{U}\\right]$ is an open set of $M_2$. Then: : $f$ is a '''homeomorphism''' : $M_1$ and $M_2$ are '''homeomorphic'''."}
+{"_id": "25434", "title": "Definition:Homeomorphism/Metric Spaces/Definition 3", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f: A_1 \\to A_2$ be a bijection such that: : for all $V \\subseteq A_1$, $V$ is a closed set of $M_1$ {{iff}} $f \\left[{V}\\right]$ is a closed set of $M_2$. Then: : $f$ is a '''homeomorphism''' : $M_1$ and $M_2$ are '''homeomorphic'''."}
+{"_id": "25435", "title": "Definition:Homeomorphism/Metric Spaces/Definition 4", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $f: A_1 \\to A_2$ be a bijection such that: : for all $a \\in A_1$ and $N \\subseteq A_1$, $N$ is a neighborhood of $a$ {{iff}} $f \\left[{N}\\right]$ is a neighborhood of $f \\left({a}\\right)$. Then: : $f$ is a '''homeomorphism''' : $M_1$ and $M_2$ are '''homeomorphic'''."}
+{"_id": "25437", "title": "Definition:Cardinality of Continuum", "text": "The '''Cardinality of the Continuum''' is the cardinality of the set of real numbers, denoted $\\left \\lvert {\\R}\\right \\rvert$ or $\\mathfrak c$. {{refactor|State and prove this in a separate page}} It is an infinite cardinal number."}
+{"_id": "25438", "title": "Definition:Exact Sequence of Groups", "text": "Let $\\left({G, \\circ}\\right)$ be a group. Consider the sequence of groups $\\left\\langle{G_i}\\right\\rangle$ and group homomorphisms $\\phi_i$: :$\\displaystyle \\cdots \\stackrel{\\phi_{i-2}}{\\longrightarrow} G_{i-1} \\stackrel{\\phi_{i-1}}{\\longrightarrow} G_i \\stackrel{\\phi_i}{\\longrightarrow} G_{i+1} \\stackrel{\\phi_{i+1}}{\\longrightarrow} \\cdots$ $\\left\\langle{G_i}\\right\\rangle$ is  '''exact''' {{iff}}: :$\\forall i: \\operatorname{Im} \\left({\\phi_i}\\right) = \\ker \\left({\\phi_{i+1} }\\right)$ where: :$\\operatorname{Im} \\left({\\phi_i}\\right)$ denotes the image of $\\phi_i$ :$\\ker \\left({\\phi_{i+1} }\\right)$ denotes the kernel of $\\phi_{i+1}$."}
+{"_id": "25439", "title": "Definition:Short Exact Sequence of Groups", "text": "Let $\\left({G, \\cdot}\\right)$ be a group. An exact sequence of the form :$1 \\longrightarrow K \\stackrel{\\alpha}{\\longrightarrow} G \\stackrel{\\beta}{\\longrightarrow} H \\longrightarrow 1$ is called a '''short exact sequence''', where $1$ represents the trivial group. Category:Definitions/Group Theory 82r1xq04tv43yjqdjvkm62opeyg8m27"}
+{"_id": "25441", "title": "Definition:Dozen", "text": "A '''dozen''' is another name for a set whose cardinality is $12$."}
+{"_id": "25442", "title": "Definition:Gross", "text": "A '''gross''' is another name for a set whose cardinality is $12 \\times 12$, or $144$."}
+{"_id": "25443", "title": "Definition:Score", "text": "A '''score''' is another name for a set whose cardinality is $20$."}
+{"_id": "25444", "title": "Definition:Integral Sign", "text": "The symbol: :$\\displaystyle \\int \\ldots \\rd \\mu$ is called the '''integral sign'''. Note that there are two parts to this symbol, which embrace the function $f$ which is being integrated."}
+{"_id": "25445", "title": "Definition:Axis/Positive Direction", "text": "Consider a coordinate system whose axes are each aligned with an instance of the real number line $\\R$. The direction along an axis in which the corresponding elements of $\\R$ are increasing is called the '''positive direction'''."}
+{"_id": "25446", "title": "Definition:Smallest Set by Set Inclusion", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Let $\\TT \\subseteq \\powerset S$ be a subset of $\\powerset S$. Let $\\struct {\\TT, \\subseteq}$ be the ordered set formed on $\\TT$ by $\\subseteq$ considered as an ordering. Then $T \\in \\TT$ is the '''smallest set''' of $\\TT$ {{iff}} $T$ is the smallest element of $\\struct {\\TT, \\subseteq}$. That is: :$\\forall X \\in \\TT: T \\subseteq X$"}
+{"_id": "25447", "title": "Definition:Smallest Set by Set Inclusion/Class Theory", "text": "Let $A$ be a class. Then a set $m$ is the '''smallest element''' of $A$ (with respect to the inclusion relation) {{iff}}: :$(1): \\quad m \\in A$ :$(2): \\quad \\forall S: \\paren {S \\in A \\implies m \\subseteq S}$"}
+{"_id": "25448", "title": "Definition:Euclidean Space/Euclidean Topology/Real", "text": "Let $\\R^n$ be an $n$-dimensional real vector space. Let $M = \\struct {\\R^n, d}$ be a real Euclidean $n$-space. The topology $\\tau_d$ induced by the Euclidean metric $d$ is called the '''Euclidean topology'''."}
+{"_id": "25449", "title": "Definition:Euclidean Space/Euclidean Topology/Rational", "text": "Let $\\Q^n$ be an $n$-dimensional vector space of rational numbers. Let $M = \\left({\\Q^n, d}\\right)$ be a rational Euclidean $n$-space. The topology induced by the Euclidean metric $d$ is called the '''Euclidean topology'''."}
+{"_id": "25450", "title": "Definition:Euclidean Space/Euclidean Topology/Complex", "text": "Let $\\C$ be the complex plane. Let $M = \\left({\\C, d}\\right)$ be a complex Euclidean space. The topology induced by the Euclidean metric $d$ is called the '''Euclidean topology'''."}
+{"_id": "25451", "title": "Definition:Neighborhood Filter/Point", "text": "Let $x \\in S$. Let $\\mathcal N_x$ be the set of all neighborhoods of $x$ in $T$. Then $\\mathcal N_x$ is the '''neighborhood filter''' of $x$."}
+{"_id": "25452", "title": "Definition:Separated by Neighborhoods/Sets", "text": "{{:Definition:Separated by Neighborhoods/Sets/Neighborhoods}}"}
+{"_id": "25453", "title": "Definition:Separated by Neighborhoods/Points", "text": "{{:Definition:Separated by Neighborhoods/Points/Neighborhoods}}"}
+{"_id": "25454", "title": "Definition:Hausdorff Space/Definition 3", "text": "$\\left({S, \\tau}\\right)$ is a '''Hausdorff space''' or '''$T_2$ space''' {{iff}}: :$\\forall x, y \\in S, x \\ne y: \\exists N_x, N_y \\subseteq S: \\exists U, V \\in \\tau: x \\subseteq U \\subseteq N_x, y \\subseteq V \\subseteq N_y: N_x \\cap N_y = \\varnothing$ That is: : for any two distinct elements $x, y \\in S$ there exist disjoint neighborhoods $N_x, N_y \\subseteq S$ containing $x$ and $y$ respectively."}
+{"_id": "25456", "title": "Definition:Separated by Closed Neighborhoods/Points", "text": "Let $x, y \\in S$ such that: :$\\exists N_x, N_y \\subseteq S: \\exists U, V \\in \\tau: x \\subseteq U \\subseteq N_x, y \\subseteq V \\subseteq N_y: N_x^- \\cap N_y^- = \\O$ where $N_x^-$ and $N_y^-$ are the closures in $T$ of $N_x$ and $N_y$ respectively. That is, that $x$ and $y$ both have neighborhoods in $T$ whose closures are disjoint. Then $x$ and $y$ are described as '''separated by closed neighborhoods'''."}
+{"_id": "25457", "title": "Definition:Neighborhood Space", "text": "Let $S$ be a set. For each $x \\in S$, let there be given a set $\\mathcal N_x$ of subsets of $S$ which satisfy the neighborhood space axioms: {{:Axiom:Neighborhood Space Axioms}} The sets $\\mathcal N_x$ are the neighborhoods of $x$ in $S$. Let $\\mathcal N$ be the set of open sets of $S$: :$\\mathcal N = \\left\\{{U \\subseteq S: U}\\right.$ is a neighborhood of each of its elements $\\left.{}\\right\\}$ The set $S$ together with $\\mathcal N$ is called a '''neighborhood space''' and is denoted $\\left({S, \\mathcal N}\\right)$."}
+{"_id": "25458", "title": "Definition:Neighborhood Filter/Set", "text": "Let $A \\subseteq S$ such that $A \\ne \\O$. Let $\\NN_A$ be the set of all neighborhoods of $A$. Then $\\NN_A$ is the '''neighborhood filter''' of $A$."}
+{"_id": "25459", "title": "Definition:Neighborhood (Neighborhood Space)", "text": "Let $\\left({S, \\mathcal N}\\right)$ be a neighborhood space. Let $x \\in S$. Let $\\mathcal N_x$ satisfy the neighborhood space axioms. Then each element $N$ of $\\mathcal N_x$ is called a '''neighborhood''' of $x$."}
+{"_id": "25460", "title": "Definition:Open Set (Neighborhood Space)", "text": "Let $\\left({S, \\mathcal N}\\right)$ be a neighborhood space. Let $U \\subseteq S$ be a neighborhood of each of its elements. Then $U$ is an '''open set''' of $S$."}
+{"_id": "25461", "title": "Definition:Topological Space Induced by Neighborhood Space", "text": "Let $\\left({S, \\mathcal N}\\right)$ be a neighborhood space. Let $\\tau$ be the set of open sets of $\\left({S, \\mathcal N}\\right)$. Then $\\left({S, \\tau}\\right)$ is the '''topological space induced by $\\left({S, \\mathcal N}\\right)$'''."}
+{"_id": "25462", "title": "Definition:Combinatorics", "text": "'''Combinatorics''' is that branch of mathematics concerned with counting things. '''Combinatorial''' problems are so named because they are exercises in counting the number of combinations of various objects. It has been stated that it is the core of the discipline of discrete mathematics."}
+{"_id": "25463", "title": "Definition:Binomial Coefficient/Notation", "text": "The notation $\\dbinom n k$ for the binomial coefficient was introduced by {{AuthorRef|Andreas Freiherr von Ettingshausen}} in his $1826$ work {{BookLink|Die kombinatorische Analysis, als Vorbereitungslehre zum Studium der theoretischen höheren Mathematik|Andreas v. Ettingshausen}}. It appears to have become the ''de facto'' standard in recent years. As a result, $\\dbinom n k$ is frequently voiced '''the binomial coefficient $n$ over $k$'''. Other notations include: :$C \\left({n, k}\\right)$ :${}^n C_k$ :${}_n C_k$ :$C^n_k$ :${C_n}^k$ all of which can cause a certain degree of confusion."}
+{"_id": "25464", "title": "Definition:Binomial Coefficient/Integers/Definition 1", "text": "Let $n \\in \\Z_{\\ge 0}$ and $k \\in \\Z$. Then the '''binomial coefficient''' $\\dbinom n k$ is defined as: :$\\dbinom n k = \\begin{cases} \\dfrac {n!} {k! \\paren {n - k}!} & : 0 \\le k \\le n \\\\ & \\\\ 0 & : \\text { otherwise } \\end{cases}$ where $n!$ denotes the factorial of $n$."}
+{"_id": "25465", "title": "Definition:Binomial Coefficient/Historical Note", "text": "The binomial coefficients have been known about since at least the ancient Greeks and Romans, who were familiar with them for small values of $k$. See the historical note to Pascal's Triangle for further history."}
+{"_id": "25466", "title": "Definition:Binomial Coefficient/Integers/Definition 2", "text": "Let $n \\in \\Z_{\\ge 0}$ and $k \\in \\Z$. The number of different ways $k$ objects can be chosen (irrespective of order) from a set of $n$ objects is denoted: :$\\dbinom n k$ This number $\\dbinom n k$ is known as a '''binomial coefficient'''."}
+{"_id": "25467", "title": "Definition:Binomial Coefficient/Integers/Definition 3", "text": "Let $n \\in \\Z_{\\ge 0}$ and $k \\in \\Z$. Then the '''binomial coefficient''' $\\dbinom n k$ is defined as the coefficient of the term $a^k b^{n - k}$ in the expansion of $\\paren {a + b}^n$."}
+{"_id": "25468", "title": "Definition:Geometric Mean/Mean Proportional/General Definition", "text": "In the language of {{AuthorRef|Euclid}}, the terms of a (finite) geometric sequence of positive integers between (and not including) the first and last terms are called '''mean proportionals'''."}
+{"_id": "25469", "title": "Definition:Geometric Sequence of Integers in Lowest Terms", "text": "Let $G_n = \\sequence {a_j}_{0 \\mathop \\le j \\mathop \\le n}$ be a geometric sequence of integers. Let $r$ be the common ratio of $G_n$. Let $S$ be the set of all such geometric sequence: :$S = \\left\\{{G: G}\\right.$ is a geometric sequence of integers whose common ratio is $\\left.{r}\\right\\}$ Then $G_n$ is in '''lowest terms''' if the absolute values of the terms of $G_n$ are the smallest, term for term, of all the elements of $S$: :$\\forall Q = \\sequence {b_j}_{0 \\mathop \\le j \\mathop \\le n} \\in S: \\forall j \\in \\set {0, 1, \\ldots, n}: \\size {a_j} \\le \\size {b_j}$"}
+{"_id": "25470", "title": "Definition:Geometric Sequence/Finite", "text": "A '''finite geometric sequence''' is a geometric sequence with a finite number of terms ."}
+{"_id": "25471", "title": "Definition:Geometric Sequence/Finite/Extremes", "text": "Let $G_n = \\sequence {a_0, a_1, \\ldots, a_n}$ be a finite geometric sequence. The '''extremes''' of $G_n$ are the terms $a_0$ and $a_n$ of $G_n$."}
+{"_id": "25472", "title": "Definition:Geometric Sequence/Initial Term", "text": "Let $G = \\left\\langle{a_0, a_1, \\ldots}\\right\\rangle$ be a geometric sequence. The '''initial term''' of $G_n$ is the term $a_0$. The same definition applies to a finite geometric sequence $G_n = \\sequence {a_0, a_1, \\ldots, a_n}$."}
+{"_id": "25473", "title": "Definition:Geometric Sequence/Finite/Final Term", "text": "Let $G_n = \\sequence {a_0, a_1, \\ldots, a_n}$ be a finite geometric sequence. The '''final term''' of $G_n$ is the term $a_n$."}
+{"_id": "25474", "title": "Definition:Geometric Sequence/Integers", "text": "A '''geometric sequence of integers''' is a finite geometric sequence whose terms are all integers. Category:Definitions/Geometric Sequences o8n7wjhsvrr4hhufm061sjzlhrei5zf"}
+{"_id": "25475", "title": "Definition:Unit (One)/Naturally Ordered Semigroup", "text": "Let $\\struct {S, \\circ, \\preceq}$ be a naturally ordered semigroup. Let $S^*$ be the zero complement of $S$. By Zero Complement is Not Empty, $S^*$ is not empty. Therefore, by axiom $(NO4)$, it has a smallest element for $\\preceq$. This smallest element is called '''one''' and denoted $1$."}
+{"_id": "25476", "title": "Definition:Commensurable/Notation", "text": "There appears to be no universally acknowledged symbol to denote commensurability. {{AuthorRef|Thomas Little Heath|Thomas L. Heath}} in his edition of {{BookLink|Euclid: The Thirteen Books of The Elements: Volume 3|Sir Thomas L. Heath|ed = 2nd|edpage = Second Edition}} makes the following suggestions: :$(1): \\quad$ To denote that $A$ is commensurable or commensurable in length with $B$:  :::$A \\mathop{\\frown} B$ :$(2): \\quad$ To denote that $A$ is commensurable in square with $B$:  :::$A \\mathop{\\frown\\!\\!-} B$ :$(3): \\quad$ To denote that $A$ is incommensurable or incommensurable in length with $B$:  :::$A \\mathop{\\smile} B$ :$(4): \\quad$ To denote that $A$ is incommensurable in square with $B$:  :::$A \\mathop{\\smile\\!\\!-} B$ This convention may be used on {{ProofWiki}} if accompanied by a note which includes a link to this page."}
+{"_id": "25484", "title": "Definition:Apotome of Medial/Whole", "text": "The real number $a$ is called the '''whole''' of the apotome of a medial."}
+{"_id": "25485", "title": "Definition:Apotome of Medial/Annex", "text": "The real number $b$ is called the '''annex''' of the apotome of a medial."}
+{"_id": "25486", "title": "Definition:Apotome of Medial/Terms", "text": "The '''terms''' of $a - b$ are the elements $a$ and $b$. === Whole === {{Definition:Apotome of Medial/Whole}} === Annex === {{Definition:Apotome of Medial/Annex}}"}
+{"_id": "25487", "title": "Definition:Minor (Euclidean)/Whole", "text": "The real number $a$ is called the '''whole''' of the minor."}
+{"_id": "25488", "title": "Definition:Minor (Euclidean)/Annex", "text": "The real number $b$ is called the '''annex''' of the minor."}
+{"_id": "25489", "title": "Definition:Minor (Euclidean)/Terms", "text": "The '''terms''' of $a - b$ are the elements $a$ and $b$. === Whole === {{Definition:Minor (Euclidean)/Whole}} === Annex === {{Definition:Minor (Euclidean)/Annex}}"}
+{"_id": "25490", "title": "Definition:That which produces Medial Whole with Rational Area/Whole", "text": "The real number $a$ is called the '''whole''' of the straight line which produces with a rational area a medial whole."}
+{"_id": "25491", "title": "Definition:That which produces Medial Whole with Rational Area/Annex", "text": "The real number $b$ is called the '''annex''' of the straight line which produces with a rational area a medial whole."}
+{"_id": "25492", "title": "Definition:That which produces Medial Whole with Rational Area/Terms", "text": "The '''terms''' of $a - b$ are the elements $a$ and $b$. === Whole === {{Definition:That which produces Medial Whole with Rational Area/Whole}} === Annex === {{Definition:That which produces Medial Whole with Rational Area/Annex}}"}
+{"_id": "25493", "title": "Definition:That which produces Medial Whole with Medial Area/Whole", "text": "The real number $a$ is called the '''whole''' of the straight line which produces with a medial area a medial whole."}
+{"_id": "25494", "title": "Definition:That which produces Medial Whole with Medial Area/Annex", "text": "The real number $b$ is called the '''annex''' of the straight line which produces with a medial area a medial whole."}
+{"_id": "25495", "title": "Definition:That which produces Medial Whole with Medial Area/Terms", "text": "The '''terms''' of $a - b$ are the elements $a$ and $b$. === Whole === {{Definition:That which produces Medial Whole with Medial Area/Whole}} === Annex === {{Definition:That which produces Medial Whole with Medial Area/Annex}}"}
+{"_id": "25496", "title": "Definition:Piecewise Continuously Differentiable Function/Definition 2", "text": "$f$ is '''piecewise continuously differentiable''' {{iff}}: :there exists a finite subdivision $\\set {x_0, \\ldots, x_n}$ of $\\closedint a b$, $x_0 = a$ and $x_n = b$, such that: ::$f$ is continuously differentiable on $\\closedint {x_{i − 1} } {x_i}$, where the derivative at $x_{i − 1}$ understood as right-handed and the derivative at $x_i$ understood as left-handed, for every $i \\in \\set {1, \\ldots, n}$."}
+{"_id": "25497", "title": "Definition:Right Angle/Perpendicular/Plane", "text": "{{EuclidDefinition|book = XI|def = 3|name = Line at Right Angles to Plane}} :400px In the above diagram, the line $AB$ has been constructed so as to be a '''perpendicular''' to the plane containing the straight lines $CD$ and $EF$."}
+{"_id": "25498", "title": "Definition:Right Angle/Perpendicular/Plane to Plane", "text": "{{EuclidDefinition|book = XI|def = 4|name = Plane at Right Angles to Plane}} :400px In the above diagram, the two planes have been constructed so as to make lines perpendicular to their common section perpendicular to each other. Thus the two planes are perpendicular to each other."}
+{"_id": "25499", "title": "Definition:Pi/Definition 1", "text": "Take a circle in a plane whose circumference is $C$ and whose radius is $r$. Then $\\pi$ can be defined as $\\pi = \\dfrac C {2r}$."}
+{"_id": "25500", "title": "Definition:Pi/Definition 2", "text": "The real functions sine and cosine can be shown to be periodic. The number $\\pi$ is defined as the real number such that: :the period of both sine and cosine is $2 \\pi$."}
+{"_id": "25501", "title": "Definition:Piecewise Continuously Differentiable Function/Definition 1", "text": "$f$ is '''piecewise continuously differentiable''' {{iff}}: :$(1): \\quad f$ is continuous :$(2): \\quad$ there exists a finite subdivision $\\set {x_0, \\ldots, x_n}$ of $\\closedint a b$, $x_0 = a$ and $x_n = b$, such that: ::$(2.1): \\quad f$ is continuously differentiable on $\\openint {x_{i - 1}} {x_i}$ for every $i \\in \\set {1, \\ldots, n}$ ::$(2.2): \\quad$ the one-sided limits $\\displaystyle \\lim_{x \\mathop \\to {x_{i - 1}}^+} \\map {f'} x$ and $\\displaystyle \\lim_{x \\mathop \\to {x_i}^-} \\map {f'} x$ exist for every $i \\in \\set {1, \\ldots, n}$."}
+{"_id": "25502", "title": "Definition:Parallelepiped/Height", "text": ":400px The '''height''' of a parallelepiped is the length of the perpendicular from the plane of the base to the plane opposite. In the above diagram, $h$ is the '''height''' of the parallelepiped whose base is $AB$."}
+{"_id": "25503", "title": "Definition:Similar Situation", "text": "Two similar solid figures are said to be in a '''similar situation''' {{iff}} corresponding surfaces are similarly inclined and when corresponding edges are parallel."}
+{"_id": "25504", "title": "Definition:Solid Geometry", "text": "'''Solid geometry''' is the study of geometric figures in three dimensions. As such it is a misnomer, because the figures themselves under study are strictly speaking the surfaces of such figures."}
+{"_id": "25505", "title": "Definition:Group Theory", "text": "'''Group Theory''' is a branch of abstract algebra which studies groups and other related algebraic structures."}
+{"_id": "25506", "title": "Definition:Abstract Algebra", "text": "'''Abstract algebra''' is a branch of mathematics which studies algebraic structures and algebraic systems. It can be roughly described as the study of sets equipped with operations."}
+{"_id": "25507", "title": "Definition:Algebra (Mathematical Branch)", "text": "'''Algebra''' is the branch of mathematics which studies the techniques of manipulation of objects and expressions."}
+{"_id": "25508", "title": "Definition:Parallelepiped/Opposite Face", "text": "The '''opposite face''' of the face $F$ of a parallelepiped $P$ is the face of $P$ which is parallel to $F$. In the above example, the pairs of parallel planes are: :Face $ABCD$ is opposite $HGFE$ :Face $ADEH$ is opposite $BCFG$ :Face $ABGH$ is opposite $DCFE$"}
+{"_id": "25509", "title": "Definition:Prism/Height", "text": "The '''height''' of a prism is the length of the perpendicular between the bases of the prism. In the above diagram, the distance $h$ is the '''height''' of the prism $AJ$."}
+{"_id": "25510", "title": "Definition:Prism/Opposite", "text": "Let $P$ be a prism whose bases are $P_1$ and $P_2$. The '''opposite face''' of the base $P_1$ is the base $P_2$. In the above example, the base $ABCDE$ is opposite $FGHIJ$."}
+{"_id": "25511", "title": "Definition:Prism/Height/Euclidean Variant", "text": ":400px Although the height of a prism is generally understood to be the length of the perpendicular joining opposite faces, {{AuthorRef|Euclid}} was inconsistent in his usage in {{BookLink|The Elements|Euclid}}. In his {{EuclidPropLink|book = XI|prop = 39|title = Prisms of equal Height with Parallelogram and Triangle as Base}}, he defines the base of one prism as being one of the opposite parallel faces, but of the other he defines the base as being an arbitrary one of the parallelograms. Having defined the base in this manner, the '''height''' is then defined as being the height of one of the opposite parallel faces whose base is the edge which intersects the base so defined. Using this definition, the distance $h$ in the above diagram is the '''height''' of the prism $PQRSTU$."}
+{"_id": "25513", "title": "Definition:Prism/Lateral Face", "text": "The '''lateral faces''' of a prism are the parallelogramic faces which join the bases. In the above diagram, the faces $ABGF, BCHG, CDIH, DEJI$ and $EAFJ$ are the '''lateral faces''' of the prism."}
+{"_id": "25516", "title": "Definition:Triangular Prism", "text": ":250px A '''triangular prism''' is a prism whose bases are triangles. Category:Definitions/Prisms g9i3f6e84hppos7p0sans6sphy8prn2"}
+{"_id": "25517", "title": "Definition:Natural Numbers/Von Neumann Construction", "text": "Let $\\omega$ denote the minimal infinite successor set. The '''natural numbers''' can be defined as the elements of $\\omega$. Following Definition 2 of $\\omega$, this amounts to defining the '''natural numbers''' as the finite ordinals. In terms of the empty set $\\O$ and successor sets, we thus define: {{begin-eqn}} {{eqn | l = 0       | o = :=       | r = \\O = \\set {}       | c =  }} {{eqn | l = 1       | o = :=       | r = 0^+ = 0 \\cup \\set 0 = \\set 0       | c =  }} {{eqn | l = 2       | o = :=       | r = 1^+ = 1 \\cup \\set 1 = \\set {0, 1}       | c =  }} {{eqn | l = 3       | o = :=       | r = 2^+ = 2 \\cup \\set 2 = \\set {0, 1, 2}       | c =  }} {{eqn | o = \\vdots       | c =  }} {{eqn | l = n + 1       | o = :=       | r = n^+ = n \\cup \\set n       | c =  }} {{end-eqn}}"}
+{"_id": "25518", "title": "Definition:Piecewise Continuous Function/One-Sided Limits", "text": "$f$ is '''piecewise continuous with one-sided limits''' {{iff}}: :there exists a finite subdivision $\\set {x_0, x_1, \\ldots, x_n}$ of $\\closedint a b$, where $x_0 = a$ and $x_n = b$, such that, for all $i \\in \\set {1, 2, \\ldots, n}$: ::$(1): \\quad f$ is continuous on $\\openint {x_{i − 1} } {x_i}$ ::$(2): \\quad$ the one-sided limits $\\displaystyle \\lim_{x \\mathop \\to {x_{i − 1} }^+} \\map f x$ and $\\displaystyle \\lim_{x \\mathop \\to {x_i}^-} \\map f x$ exist."}
+{"_id": "25519", "title": "Definition:Piecewise Continuous Function/Bounded", "text": "$f$ is a '''bounded piecewise continuous function''' {{iff}}: :there exists a finite subdivision $\\set {x_0, x_1, \\ldots, x_n}$ of $\\closedint a b$, where $x_0 = a$ and $x_n = b$, such that: ::$(1): \\quad$ for all $i \\in \\set {1, 2, \\ldots, n}$, $f$ is continuous on $\\openint {x_{i − 1} } {x_i}$  ::$(2): \\quad$ $f$ is bounded on $\\closedint a b$."}
+{"_id": "25520", "title": "Definition:Axis of Solid Figure", "text": "=== Axis of Cone === :300px {{:Definition:Axis of Cone}} === Axis of Cylinder === :300px {{:Definition:Axis of Cylinder}} === Axis of Sphere === {{:Definition:Axis of Sphere}}"}
+{"_id": "25523", "title": "Definition:Apex", "text": "The '''apex''' of a geometric figure is the point which is distinguished from the others by dint of it being furthest away from its base. Not all figures have a discernible '''apex'''; for example, parallelograms, prisms and parallelepipeds do not. === Apex of Triangle === :300px {{:Definition:Apex of Triangle}} === Apex of Isosceles Triangle === :300px {{:Definition:Apex of Isosceles Triangle}} === Apex of Cone === :300px {{:Definition:Apex of Cone}} === Apex of Pyramid === :400px {{:Definition:Apex of Pyramid}}"}
+{"_id": "25524", "title": "Definition:Base of Geometric Figure", "text": "The '''base''' of a geometric figure is a specific part of that figure which is distinguished from the remainder of that figure and placed (actually or figuratively) at the '''bottom''' of a depiction or visualisation. In some cases the '''base''' is truly qualitiatively different from the rest of the figure. In other cases the '''base''' is selected arbitrarily as one of several parts of the figure which may equally well be so chosen."}
+{"_id": "25525", "title": "Definition:Quadrant", "text": "A '''quadrant''' is a sector of a circle whose angle is a right angle."}
+{"_id": "25526", "title": "Definition:Minimal Infinite Successor Set/Definition 3", "text": "{{Refactor|Based on Takeuti}} The '''minimal infinite successor set''' $\\omega$ is defined as: :$\\omega := \\set {x \\in \\On: \\paren {x \\cup \\set x} \\subseteq K_I}$ where: :$K_I$ is the class of all non-limit ordinals :$\\On$ is the class of all ordinals."}
+{"_id": "25527", "title": "Definition:Minimal Infinite Successor Set/Definition 1", "text": "Let $S$ be an infinite successor set. The '''minimal infinite successor set''' $\\omega$ is the infinite successor set given by: :$\\omega := \\displaystyle \\bigcap \\set {S' \\subseteq S: \\text{$S'$ is an infinite successor set} }$ that is, $\\omega$ is the intersection of every infinite successor set which is a subset of $S$."}
+{"_id": "25528", "title": "Definition:Equiangular Geometric Figures", "text": "Two geometric figures are '''equiangular (with each other)''' when the angles of each pair of their corresponding vertices are equal. Category:Definitions/Geometry 5fzaybsrt5vfpk414g7y44x8c7d6qw8"}
+{"_id": "25529", "title": "Definition:Tetrahedron/Base", "text": "One of the faces of a tetrahedron can be chosen arbitrarily, distinguished from the others and identified as the '''base''' of the tetrahedron. In the above diagram, $ABC$ is the '''base''' of the tetrahedron $ABCD$."}
+{"_id": "25530", "title": "Definition:Tetrahedron/Apex", "text": "Once the base of a tetrahedron has been identified, the vertex which does not lie on the base is called the '''apex''' of the tetrahedron. In the above diagram, given that the base of the tetrahedron $ABCD$ is the triangle $ABC$, the '''apex''' is $D$."}
+{"_id": "25531", "title": "Definition:Polyhedron/Face", "text": "The '''faces''' of a polyhedron are the polygons which contain it."}
+{"_id": "25532", "title": "Definition:Ludolphine Number", "text": "The '''Ludolphine number''' is the expression of the value of $\\pi$ to $35$ decimal places: :$3 \\cdotp 14159 \\, 26535 \\, 89793 \\, 23846 \\, 26433 \\, 83279 \\, 50288 \\ldots$"}
+{"_id": "25535", "title": "Definition:Sign of Number", "text": "The '''sign''' of a number is the symbol indicating whether it is: : positive, denoted by the symbol $+$ or: : negative, denoted by the symbol $-$. Hence a number's '''sign''' has evolved to define the fact of the number being positive or negative independently of the symbol itself. Thus: :the '''sign''' of $3.14159$ is positive and :the '''sign''' of $-75$ is negative."}
+{"_id": "25537", "title": "Definition:Piecewise Continuous Function/Improper Integrals", "text": "$f$ is '''piecewise continuous with improper integrals''' {{iff}}: :there exists a finite subdivision $\\left\\{{x_0, x_1, \\ldots, x_n}\\right\\}$ of $\\left[{a \\,.\\,.\\, b}\\right]$, where $x_0 = a$ and $x_n = b$, such that for all $i \\in \\left\\{ {1, 2, \\ldots, n}\\right\\}$: ::$(1): \\quad f$ is continuous on $\\left({x_{i − 1} \\,.\\,.\\, x_i}\\right)$  ::$(2): \\quad$ the improper integrals $\\displaystyle \\int_{ {x_{i - 1} }^+}^{ {x_i}^-} f \\left({x}\\right) \\rd x$ all exist."}
+{"_id": "25540", "title": "Definition:Commutative/Elements", "text": "Let $\\circ$ be a binary operation. Two elements $x, y$ are said to '''commute''' {{iff}}: :$x \\circ y = y \\circ x$"}
+{"_id": "25541", "title": "Definition:Commutative/Operation", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Then $\\circ$ is '''commutative on $S$''' {{iff}}: :$\\forall x, y \\in S: x \\circ y = y \\circ x$"}
+{"_id": "25542", "title": "Definition:Commutative Division Ring", "text": "A '''commutative division ring''' is a division ring $\\left({R, +, \\circ}\\right)$ in which the ring product $\\circ$ is commutative. Category:Definitions/Ring Theory awslszg1z8vsnyd5n03j4syolarfftl"}
+{"_id": "25543", "title": "Definition:Commutative/Algebraic Structure", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure whose operation $\\circ$ is a commutative operation. Then $\\left({S, \\circ}\\right)$ is a '''commutative (algebraic) structure'''."}
+{"_id": "25544", "title": "Definition:Operation/Operand", "text": "An '''operand''' is one of the objects on which an operation generates its new object."}
+{"_id": "25545", "title": "Definition:Arithmetic", "text": "'''Arithmetic''' is the branch of mathematics which concerns the manipulation of numbers, using the operations addition, subtraction, multiplication and division, and the taking of powers."}
+{"_id": "25547", "title": "Definition:Smallest Field", "text": "The '''smallest field''' is the set of integers modulo $2$ under modulo addition and modulo multiplication: :$\\struct {\\Z_2, +_2, \\times_2}$ This field has $2$ elements."}
+{"_id": "25548", "title": "Definition:Addition/Modulo Addition/Definition 1", "text": "Let $m \\in \\Z$ be an integer. Let $\\Z_m$ be the set of integers modulo $m$: :$\\Z_m = \\set {\\eqclass 0 m, \\eqclass 1 m, \\ldots, \\eqclass {m - 1} m}$ where $\\eqclass x m$ is the residue class of $x$ modulo $m$. The operation of '''addition modulo $m$''' is defined on $\\Z_m$ as: :$\\eqclass a m +_m \\eqclass b m = \\eqclass {a + b} m$"}
+{"_id": "25549", "title": "Definition:Addition/Modulo Addition/Definition 2", "text": "Let $m \\in \\Z$ be an integer. Let $\\Z_m$ be the set of integers modulo $m$: :$\\Z_m = \\left\\{{0, 1, \\ldots, m-1}\\right\\}$ The operation of '''addition modulo $m$''' is defined on $\\Z_m$ as: :$x +_m y$ equals the remainder after $x + y$ has been divided by $m$."}
+{"_id": "25550", "title": "Definition:Addition/Modulo Addition/Definition 3", "text": "Let $m \\in \\Z$ be an integer. Let $\\Z_m$ be the set of integers modulo $m$: :$\\Z_m = \\left\\{{0, 1, \\ldots, m-1}\\right\\}$ The operation of '''addition modulo $m$''' is defined on $\\Z_m$ as: :$x +_m y := x + y - j m$ where $j$ is the largest integer such that $j m \\le x + y$."}
+{"_id": "25551", "title": "Definition:Multiplication/Modulo Multiplication/Definition 1", "text": "Let $m \\in \\Z$ be an integer. Let $\\Z_m$ be the set of integers modulo $m$: :$\\Z_m = \\set {\\eqclass 0 m, \\eqclass 1 m, \\ldots, \\eqclass {m - 1} m}$ where $\\eqclass x m$ is the residue class of $x$ modulo $m$. The operation of '''multiplication modulo $m$''' is defined on $\\Z_m$ as: :$\\eqclass a m \\times_m \\eqclass b m = \\eqclass {a b} m$"}
+{"_id": "25552", "title": "Definition:Multiplication/Modulo Multiplication/Definition 3", "text": "Let $m \\in \\Z$ be an integer. Let $\\Z_m$ be the set of integers modulo $m$: :$\\Z_m = \\set {0, 1, \\ldots, m - 1}$ The operation of '''multiplication modulo $m$''' is defined on $\\Z_m$ as: :$x \\cdot_m y := x y - k m$ where $k$ is the largest integer such that $k m \\le x y$."}
+{"_id": "25553", "title": "Definition:Multiplication/Modulo Multiplication/Definition 2", "text": "Let $m \\in \\Z$ be an integer. Let $\\Z_m$ be the set of integers modulo $m$: :$\\Z_m = \\set {0, 1, \\ldots, m - 1}$ The operation of '''multiplication modulo $m$''' is defined on $\\Z_m$ as: :$x \\cdot_m y$ equals the remainder after $x y$ has been divided by $m$."}
+{"_id": "25554", "title": "Definition:Big Dipper Operation", "text": "Let $m, n \\in \\Z$ be integers such that $m \\ge 0, n > 0$. Let $+_{m, n}$ be the binary operation on $\\Z_{>0}$ defined as: :$\\forall a, b \\in \\Z_{>0}: a +_{m, n} b = \\begin{cases} a + b & : a + b < m \\\\ a + b - k n & : a + b \\ge m \\end{cases}$ where $k$ is the largest integer satisfying: :$m + k n \\le x + y$ The operation $+_{m, n}$ is known as the '''Big Dipper'''. :600px When the stars of the [http://en.wikipedia.org/wiki/Big_Dipper Big Dipper] are numbered as shown, the sequence: :$1, 1 +_{3, 4} 1, 1 +_{3, 4} 1 +_{3, 4} 1, \\ldots$ traces out those stars in the order: :first the handle: $\\text{Alkaid}, \\text{Mizar}, \\text{Alioth}$ then: :round the pan indefinitely: $\\text{Megrez}, \\text{Dubhe}, \\text{Merak}, \\text{Phecda}, \\text{Megrez}, \\ldots$ === Cayley Table === {{:Big Dipper Operation/Cayley Table}}"}
+{"_id": "25560", "title": "Definition:Parity Ring", "text": "The '''parity ring''' is the ring of two elements which defines the nature of the parity of integers under addition and multiplication: :$\\struct {\\set {\\text{even}, \\text{odd} }, +, \\times}$"}
+{"_id": "25561", "title": "Definition:Group of Gaussian Integer Units", "text": "Let $i$ be the imaginary unit: $i = \\sqrt {-1}$. Let $U_\\C$ be the set of complex numbers defined as: :$U_\\C = \\set {1, i, -1, -i}$ Let $\\times$ denote the operation of complex multiplication. The algebraic structure $\\struct {U_\\C, \\times}$ is the '''group of units of the ring of Gaussian integers'''."}
+{"_id": "25562", "title": "Definition:Ring of Gaussian Integers", "text": "The '''ring of Gaussian integers''' $\\struct {\\Z \\sqbrk i, +, \\times}$ is the algebraic structure formed from: : the set of Gaussian integers $\\Z \\sqbrk i$ : the operation of complex addition : the operation of complex multiplication."}
+{"_id": "25563", "title": "Definition:Finite Ordinal", "text": "Let $\\alpha$ be an ordinal. Then $\\alpha$ is said to be '''finite''' {{iff}} one of the following holds: :$\\alpha = \\O$ :$\\alpha = \\beta^+$ for some '''finite ordinal''' $\\beta$ where $\\O$ denotes the empty set, and $\\beta^+$ is the successor ordinal of $\\beta$."}
+{"_id": "25564", "title": "Definition:Minimal Infinite Successor Set/Definition 2", "text": "The '''minimal infinite successor set''' $\\omega$ is defined as the set of all finite ordinals: :$\\omega := \\set {\\alpha : \\text{$\\alpha$ is a finite ordinal} }$"}
+{"_id": "25565", "title": "Definition:Root of Unity/Complex/First", "text": "The root $e^{2 i \\pi / n}$ is known as the '''first (complex) $n$th root of unity'''."}
+{"_id": "25569", "title": "Definition:Finite Algebraic Structure", "text": "A '''finite algebraic structure''' is an algebraic structure whose underlying set is finite. Category:Definitions/Abstract Algebra c9ktkuy8penh33a3t2ls0vtwf8x35ln"}
+{"_id": "25572", "title": "Definition:Euclidean Relation/Left-Euclidean", "text": "$\\mathcal R$ is '''left-Euclidean''' {{iff}}: :$\\tuple {x, z} \\in \\mathcal R \\land \\tuple {y, z} \\in \\mathcal R \\implies \\tuple {x, y} \\in \\mathcal R$"}
+{"_id": "25574", "title": "Definition:Hasse Diagram/Examples", "text": "These are examples of Hasse diagrams : :350px $\\qquad$ 350px The diagram on the left illustrates the \"Divisor\" ordering on the set $S = \\left\\{{1, 2, 3, 4, 6, 8, 12, 24}\\right\\}$ where $S$ is the set of all elements of $\\N_{>0}$ which divide $24$. The diagram on the right illustrates the \"Subset\" relation on the power set $\\mathcal P \\left({S}\\right)$ where $S = \\left\\{{1, 2, 3}\\right\\}$."}
+{"_id": "25575", "title": "Definition:Smallest/Ordered Set/Subset", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T \\subseteq S$ be a subset of $S$. An element $x \\in T$ is '''the smallest element of $T$''' {{iff}}: :$\\forall y \\in T: x \\preceq \\restriction_T y$ where $\\preceq \\restriction_T$ denotes the restriction of $\\preceq$ to $T$."}
+{"_id": "25576", "title": "Definition:Inverse Relation/Ordering Notation", "text": "To denote the inverse of an ordering, the conventional technique is to reverse the symbol. Thus: :$\\succeq$ denotes $\\preceq^{-1}$ :$\\succcurlyeq$ denotes $\\preccurlyeq^{-1}$ :$\\curlyeqsucc$ denotes $\\curlyeqprec^{-1}$ and so: :$a \\preceq b \\iff b \\succeq a$ :$a \\preccurlyeq b \\iff b \\succcurlyeq a$ :$a \\curlyeqprec b \\iff b \\curlyeqsucc a$ Similarly for the standard symbols used to denote an ordering on numbers: :$\\ge$ denotes $\\le^{-1}$ :$\\geqslant$ denotes $\\leqslant^{-1}$ :$\\eqslantgtr$ denotes $\\eqslantless^{-1}$ and so on."}
+{"_id": "25577", "title": "Definition:Ordering/Size", "text": "An ordering can often be considered to be a comparison of the '''size''' of objects, perhaps in some intuitive sense. This is particularly applicable in the context of numbers. Thus the expression $A \\preceq B$ can in such contexts be interpreted as:  : '''$A$ is smaller than $B$''' : '''$A$ is less than $B$''' and $B \\preceq A$ can similarly be interpreted as:  : '''$A$ is larger than $B$''' : '''$A$ is greater than $B$''' In natural language, such terms are called '''[https://en.wikipedia.org/wiki/Comparative comparative adjectives]''', or just '''comparatives'''. Depending on the nature of the set being ordered, and depending on the nature of the ordering relation, this interpretation of an ordering as a comparison of size  may not be intellectually sustainable."}
+{"_id": "25578", "title": "Definition:Strict Ordering/Notation", "text": "Symbols used to denote a general strict ordering are usually variants on $\\prec$, $<$ and so on. On {{ProofWiki}}, to denote a general strict ordering it is recommended to use $\\prec$. To denote the conventional strict ordering in the context of numbers, the symbol $<$ is to be used. The symbol $\\subset$ is universally reserved for the (proper) subset relation. :$a \\prec b$ can be read as: :'''$a$ (strictly) precedes $b$'''. Similarly: : $a \\prec b$ can also be read as: : '''$b$ (strictly) succeeds $a$'''. If, for two elements $a, b \\in S$, it is not the case that $a \\prec b$, then the symbols $a \\nprec b$ and $b \\nsucc a$ can be used."}
+{"_id": "25579", "title": "Definition:Inverse Relation/Strict Ordering Notation", "text": "To denote the inverse of an strict ordering, the conventional technique is to reverse the symbol. Thus: :$\\succ$ denotes $\\prec^{-1}$ and so: :$a \\prec b \\iff b \\succ a$ Similarly for the standard symbol used to denote a strict ordering on numbers: :$>$ denotes $<^{-1}$ and so on."}
+{"_id": "25582", "title": "Definition:Lattice (Ordered Set)", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Then $\\left({S, \\preceq}\\right)$ is a '''lattice''' {{iff}}: :for all $x, y \\in S$, the subset $\\left\\{{x, y}\\right\\}$ admits both a supremum and an infimum."}
+{"_id": "25583", "title": "Definition:Order Embedding/Definition 2", "text": "$\\phi$ is an '''order embedding of $S$ into $T$''' {{iff}} both of the following conditions hold: :$(1): \\quad \\phi$ is an injection :$(2): \\quad \\forall x, y \\in S: x \\preceq_1 y \\iff \\map \\phi x \\preceq_2 \\map \\phi y$"}
+{"_id": "25584", "title": "Definition:Greatest/Ordered Set/Subset", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T \\subseteq S$ be a subset of $S$. An element $x \\in T$ is '''the greatest element of $T$''' {{iff}}: :$\\forall y \\in T: y \\preceq \\restriction_T x$ where $\\preceq \\restriction_T$ denotes the restriction of $\\preceq$ to $T$."}
+{"_id": "25586", "title": "Definition:Internal Direct Product/Decomposition", "text": "The set of algebraic substructures $\\left({S_1, \\circ {\\restriction_{S_1}}}\\right), \\left({S_2, \\circ {\\restriction_{S_2}}}\\right), \\ldots, \\left({S_n, \\circ {\\restriction_{S_n}}}\\right)$ whose direct product is isomorphic with $\\left({S, \\circ}\\right)$ is called a '''decomposition''' of $S$."}
+{"_id": "25587", "title": "Definition:Internal Group Direct Product/Decomposition", "text": "The set of subgroups $\\left({G_1, \\circ {\\restriction_{G_1}}}\\right), \\left({G_2, \\circ {\\restriction_{G_2}}}\\right), \\ldots, \\left({G_n, \\circ {\\restriction_{G_n}}}\\right)$ whose group direct product is isomorphic with $\\left({G, \\circ}\\right)$ is called a '''decomposition''' of $G$."}
+{"_id": "25588", "title": "Definition:Decomposable Group", "text": "Let $\\left({G, \\circ}\\right)$ be a group. Then $\\left({G, \\circ}\\right)$ is '''decomposable''' {{iff}} there exists a decomposition of $\\left({G, \\circ}\\right)$. That is, {{iff}} $\\left({G, \\circ}\\right)$ is the internal direct product of two (or more) proper subgroups of $G$. === Indecomposable === {{:Definition:Decomposable Group/Indecomposable}}"}
+{"_id": "25589", "title": "Definition:Decomposable Group/Indecomposable", "text": "$\\left({G, \\circ}\\right)$ is '''indecomposable''' {{iff}} it is not decomposable. That is, {{iff}} there does not exist a decomposition of $\\left({G, \\circ}\\right)$."}
+{"_id": "25590", "title": "Definition:Lexicographic Order/Ordinals", "text": "The '''lexicographic order''' is a relation on ordered pairs of ordinals denoted $\\operatorname{Le}$. $\\operatorname{Le}$ is the set of all ordered pairs $\\left({\\left({\\alpha, \\beta}\\right), \\left({\\gamma, \\delta}\\right)}\\right)$ such that: :$(1): \\quad$ Each $\\alpha, \\beta, \\gamma, \\delta$ is a member of the ordinal class :$(2): \\quad$ $\\alpha \\in \\gamma$ or $\\alpha = \\gamma \\land \\beta \\in \\delta$"}
+{"_id": "25591", "title": "Definition:Lexicographic Order/Tuples of Equal Length/Cartesian Space", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $n \\in \\N_{>0}$. Let $S^n$ be the cartesian $n$th power of $S$: :$S^n = \\underbrace{S \\times S \\times \\cdots \\times S}_{n \\text{ times} }$ The '''lexicographic order on $S^n$''' is the relation $\\preccurlyeq$ defined on $S^n$ as: :$\\left({x_1, x_2, \\ldots, x_n}\\right) \\preccurlyeq \\left({y_1, y_2, \\ldots, y_n}\\right)$ {{iff}}: ::$\\exists k: 1 \\le k \\le n: \\left({\\forall j: 1 \\le j < k: x_j = y_j}\\right) \\land \\left({x_k \\prec y_k}\\right)$ :or: ::$\\forall j: 1 \\le j \\le n: x_j = y_j$"}
+{"_id": "25592", "title": "Definition:Lexicographic Order/Tuples of Equal Length", "text": "Let $n \\in \\N_{>0}$. Let $\\left({S_1, \\preceq_1}\\right), \\left({S_2, \\preceq_2}\\right), \\ldots, \\left({S_n, \\preceq_n}\\right)$ be ordered sets. Let $\\displaystyle S = \\prod_{k \\mathop = 1}^n S_k = S_1 \\times S_2 \\times \\cdots \\times S_n$ be the Cartesian product of $S_1$ to $S_n$. The '''lexicographic order on $S$''' is the relation $\\preccurlyeq$ defined on $S$ as: :$\\left({x_1, x_2, \\ldots, x_n}\\right) \\preccurlyeq \\left({y_1, y_2, \\ldots, y_n}\\right)$ {{iff}}: ::$\\exists k: 1 \\le k \\le n: \\left({\\forall j: 1 \\le j < k: x_j = y_j}\\right) \\land \\left({x_k \\prec_k y_k}\\right)$ :or: ::$\\forall j: 1 \\le j \\le n: x_j = y_j$"}
+{"_id": "25593", "title": "Definition:Lexicographic Order/General Definition", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. For $n \\in \\N: n > 0$, we define $T_n$ as the set of all ordered $n$-tuples: :$\\left({x_1, x_2, \\ldots, x_n}\\right)$ of elements $x_j \\in S$. Let $\\displaystyle T = \\bigcup_{n \\mathop \\ge 1} T_n$. The '''lexicographic order on $T$''' is the relation $\\preccurlyeq$ defined on $T$ as: :$\\left({x_1, x_2, \\ldots, x_m}\\right) \\preccurlyeq \\left({y_1, y_2, \\ldots, y_n}\\right)$ {{iff}}: ::$\\exists k: 1 \\le k \\le \\min \\left({m, n}\\right): \\left({\\forall j: 1 \\le j < k: x_j = y_j}\\right) \\land x_k \\prec y_k$ :or: ::$m \\le n$ and $\\forall j: 1 \\le j \\le m: x_j = y_j$."}
+{"_id": "25594", "title": "Definition:Ordered Semigroup Axioms", "text": "An ordered semigroup is an algebraic structure $\\struct {S, \\circ, \\preceq}$ which satisfies the following properties: {{begin-axiom}} {{axiom | n = OS0         | lc= Closure         | q = \\forall a, b \\in S         | m = a \\circ b \\in S }} {{axiom | n = OS1         | lc= Associativity         | q = \\forall a, b, c \\in S         | m = a \\circ \\paren {b \\circ c} = \\paren {a \\circ b} \\circ c }} {{axiom | n = OS2         | lc= Compatibility of $\\preceq$ with $\\circ$         | q = \\forall a, b, c \\in S         | m = a \\preceq b \\implies \\paren {a \\circ c} \\preceq \\paren {b \\circ c} }} {{axiom | lc= where $\\preceq$ is an ordering         | m = a \\preceq b \\implies \\paren {c \\circ a} \\preceq \\paren {c \\circ b} }} {{end-axiom}}"}
+{"_id": "25595", "title": "Definition:Ordered Group Axioms", "text": "An ordered group is an algebraic structure $\\left({G, \\circ, \\preceq}\\right)$ which satisfies the following properties: {{begin-axiom}} {{axiom | n = OG0         | lc= Closure         | q = \\forall a, b \\in G         | m = a \\circ b \\in G }} {{axiom | n = OG1         | lc= Associativity         | q = \\forall a, b, c \\in G         | m = a \\circ \\left({b \\circ c}\\right) = \\left({a \\circ b}\\right) \\circ c }} {{axiom | n = OG2         | lc= Identity         | q = \\exists e \\in G: \\forall a \\in G         | m = e \\circ a = a = a \\circ e }} {{axiom | n = OG3         | lc= Inverse         | q = \\forall a \\in G: \\exists b \\in G         | m = a \\circ b = e = b \\circ a }} {{axiom | n = OG4         | lc= Compatibility of $\\preceq$ with $\\circ$         | q = \\forall a, b, c \\in G         | m = a \\preceq b \\implies \\left({a \\circ c}\\right) \\preceq \\left({b \\circ c}\\right) }} {{axiom | lc= where $\\preceq$ is an ordering         | m = a \\preceq b \\implies \\left({c \\circ a}\\right) \\preceq \\left({c \\circ b}\\right) }} {{end-axiom}}"}
+{"_id": "25596", "title": "Definition:Naturally Ordered Semigroup/Axioms", "text": "A '''naturally ordered semigroup''' is a (totally) ordered commutative semigroup $\\struct {S, \\circ, \\preceq}$ satisfying: {{begin-axiom}} {{axiom | n = \\text {NO} 1         | lc= $S$ is well-ordered by $\\preceq$         | q = \\forall T \\subseteq S         | m = T = \\varnothing \\lor \\exists m \\in T: \\forall n \\in T: m \\preceq n }} {{axiom | n = \\text {NO} 2         | lc= $\\circ$ is cancellable in $S$         | q = \\forall m, n, p \\in S         | m = m \\circ p = n \\circ p \\implies m = n }} {{axiom | m = p \\circ m = p \\circ n \\implies m = n }} {{axiom | n = \\text {NO} 3         | lc= Existence of product         | q = \\forall m, n \\in S         | m = m \\preceq n \\implies \\exists p \\in S: m \\circ p = n }} {{axiom | n = \\text {NO} 4         | lc= $S$ has at least two distinct elements         | q = \\exists m, n \\in S         | m = m \\ne n }} {{end-axiom}}"}
+{"_id": "25597", "title": "Definition:Initial Segment of Natural Numbers/Zero-Based", "text": "Let $n \\in \\N$ be a natural number. The initial segment of the natural numbers determined by $n$: :$\\set {0, 1, 2, \\ldots, n - 1}$ is denoted $\\N_{< n}$."}
+{"_id": "25598", "title": "Definition:Initial Segment of Natural Numbers/One-Based", "text": "The initial segment of the non-zero natural numbers determined by $n$: :$\\set {1, 2, 3, \\ldots, n}$ is denoted $\\N^*_{\\le n}$."}
+{"_id": "25599", "title": "Definition:Recursively Defined Mapping/Naturally Ordered Semigroup", "text": "Let $\\struct {S, \\circ, \\preceq}$ be a naturally ordered semigroup. Let $p \\in S$. Let $S' = \\set {x \\in S: p \\preceq x}$. Let $T$ be a set. Let $g: T \\to T$ be a mapping. Let $f: S' \\to T$ be the mapping defined as: :$\\forall n \\in S': \\map f x = \\begin{cases} a & : x = p \\\\ \\map g {\\map f n} & : x = n \\circ 1 \\end{cases}$ where $a \\in T$. Then $f$ is said to be '''recursively defined''' on $S'$."}
+{"_id": "25600", "title": "Definition:Recursively Defined Mapping", "text": "=== Natural Numbers === {{:Definition:Recursively Defined Mapping/Natural Numbers}}   === Peano Structure === {{:Definition:Recursively Defined Mapping/Peano Structure}} === Naturally Ordered Semigroup === {{:Definition:Recursively Defined Mapping/Naturally Ordered Semigroup}} === Minimal Infinite Successor Set === {{:Definition:Recursively Defined Mapping/Minimal Infinite Successor Set}}"}
+{"_id": "25601", "title": "Definition:Recursively Defined Mapping/Peano Structure", "text": "Let $\\left({P, 0, s}\\right)$ be a Peano structure. Let $T$ be a set. Let $g: T \\to T$ be a mapping. Let  $f: P \\to T$ be the mapping defined as: :$\\forall x \\in P: \\map f x = \\begin{cases}   a & : x = 0 \\\\   \\map g {\\map f n} & : x = \\map s n \\end{cases}$ where $a \\in T$. Then $f$ is said to be '''recursively defined''' on $P$."}
+{"_id": "25602", "title": "Definition:Recursively Defined Mapping/Minimal Infinite Successor Set", "text": "Let $\\omega$ be the minimal infinite successor set. Let $T$ be a set. Let $a \\in T$. Let $g: T \\to T$ be a mapping. Let $f: \\omega \\to T$ be the mapping defined as: :$\\forall x \\in \\omega: \\map f x = \\begin{cases} a & : x = 0 \\\\ \\map g {\\map f n} & : x = n^+ \\end{cases}$ where $n^+$ is the successor set of $n$. Then $f$ is said to be '''recursively defined''' on $\\omega$."}
+{"_id": "25603", "title": "Definition:Recursively Defined Mapping/Natural Numbers", "text": "Let $p \\in \\N$ be a natural number. Let $S = \\set {x \\in \\N: p \\le x}$. Let $T$ be a set. Let $g: T \\to T$ be a mapping. Let $f: S \\to T$ be the mapping defined as: :$\\forall x \\in S: \\map f x = \\begin{cases} a & : x = p \\\\ \\map g {\\map f n} & : x = n + 1 \\end{cases}$ for $a \\in T$. Then $f$ is said to be '''recursively defined''' on $S$."}
+{"_id": "25604", "title": "Definition:Power of Element/Magma with Identity", "text": "Let $\\left({S, \\circ}\\right)$ be a magma with an identity element $e$. Let $a \\in S$. Let the mapping $\\circ^n a: \\N \\to S$ be recursively defined as: :$\\forall n \\in S: \\circ^n a = \\begin{cases} e & : n = 0 \\\\ \\left({\\circ^r a}\\right) \\circ a & : n = r + 1 \\end{cases}$ The mapping $\\circ^n a$ is known as the '''$n$th power of $a$ (under $\\circ$)'''."}
+{"_id": "25605", "title": "Definition:Power of Element/Magma", "text": "Let $\\struct {S, \\circ}$ be a magma which has no identity element. Let $a \\in S$. Let the mapping $\\circ^n a: \\N_{>0} \\to S$ be recursively defined as: :$\\forall n \\in \\N_{>0}: \\circ^n a = \\begin{cases} a & : n = 1 \\\\ \\paren {\\circ^r a} \\circ a & : n = r + 1 \\end{cases}$ The mapping $\\circ^n a$ is known as the '''$n$th power of $a$ (under $\\circ$)'''."}
+{"_id": "25608", "title": "Definition:Commutative Semiring", "text": "Let $\\left({S, \\circ, *}\\right)$ be an additive semiring such that the distributor operation $*$ is a commutative operation. Then $\\left({S, \\circ}\\right)$ is a '''commutative semiring'''. Category:Definitions/Ring Theory 4b5tf6wyezwhoe4hpp66jxq652a2u2e"}
+{"_id": "25609", "title": "Definition:Addition/Natural Numbers", "text": "{{:Definition:Addition in Peano Structure}}"}
+{"_id": "25610", "title": "Definition:Ordered Tuple/Term", "text": "Let $\\sequence {a_k}_{k \\mathop \\in \\N^*_n}$ be an ordered tuple. The ordered pair $\\tuple {k, a_k}$ is called the '''$k$th term''' of the ordered tuple for each $k \\in \\N^*_n$."}
+{"_id": "25611", "title": "Definition:Sequence/Notation", "text": "The notation for a sequence is as follows. If $f: A \\to S$ is a sequence, then a symbol, for example \"$a$\", is chosen to represent elements of this sequence. Then for each $k \\in A$, $f \\left({k}\\right)$ is denoted $a_k$, and $f$ itself is denoted $\\left \\langle {a_k} \\right \\rangle_{k \\mathop \\in A}$. Other types of brackets may be encountered, for example: :$\\left({a_k}\\right)_{k \\mathop \\in A}$ :$\\left\\{{a_k}\\right\\}_{k \\mathop \\in A}$ The latter is discouraged because of the implication that the order of the terms does not matter. Any expression can be used to denote the domain of $f$ in place of $k \\in A$. For example: :$\\left \\langle {a_k} \\right \\rangle_{k \\mathop \\ge n}$ :$\\left \\langle {a_k} \\right \\rangle_{p \\mathop \\le k \\mathop \\le q}$ The sequence itself may be defined by a simple formula, and so for example: :$\\left \\langle {k^3} \\right \\rangle_{2 \\mathop \\le k \\mathop \\le 6}$ is the same as: :$\\left \\langle {a_k} \\right \\rangle_{2 \\mathop \\le k \\mathop \\le 6}$ where $a_k = k^3$ for all $k \\in \\left\\{{2, 3, \\ldots, 6}\\right\\}$. The set $A$ is usually taken to be the set of natural numbers $\\N = \\left\\{{0,1, 2, 3, \\ldots}\\right\\}$ or a subset. In particular, for a finite sequence, $A$ is usually $\\left\\{{0, 1, 2, \\ldots, n-1}\\right\\}$ or $\\left\\{{1, 2, 3, \\ldots, n}\\right\\}$. If this is the case, then it is usual to write $\\left \\langle {a_k} \\right \\rangle_{k \\mathop \\in A}$ as $\\left \\langle {a_k} \\right \\rangle$ or even as $\\left \\langle {a} \\right \\rangle$ if brevity and simplicity improve clarity. A finite sequence of length $n$ can be denoted: :$\\left({a_1, a_2, \\ldots, a_n}\\right)$ and by this notational convention the brackets are always round."}
+{"_id": "25612", "title": "Definition:Sequence/Equality", "text": "Let $f$ and $g$ be two sequences on the same set $A$: : $f = \\left\\langle{a_k}\\right\\rangle_{k \\mathop \\in A}$ : $g = \\left\\langle{b_k}\\right\\rangle_{k \\mathop \\in B}$ Then $f = g$ {{iff}}: : $A = B$ : $\\forall i \\in A: a_i = b_i$"}
+{"_id": "25613", "title": "Definition:Empty Set/Existence", "text": "Some authors have problems with the existence (or not) of the empty set: * {{BookReference|Sets and Groups|1965|J.A. Green}}: $\\S 1.3$: :: ''If $A, B$ are disjoint, then $A \\cap B$ is not really defined, because it has no elements. For this reason we introduce a conventional ''empty set'', denoted $\\O$, to be thought of as a 'set with no elements'. Of course this is a set only by courtesy, but it is convenient to allow $\\O$ the status of a set.'' * {{BookReference|The Theory of Groups|1968|Ian D. Macdonald}}: Appendix: :: ''The best attitude towards the empty set $\\O$ is, perhaps, to regard it as an interesting curiosity, a convenient fiction. To say that $x \\in \\O$ simply means that $x$ does not exist. Note that it is conveniently agreed that $\\O$ is a subset of every set, for elements of $\\O$ are supposed to possess every property.'' * {{BookReference|Topology|2000|James R. Munkres|ed = 2nd|edpage = Second Edition}}: $1$: Set Theory and Logic: $\\S 1$: Fundamental Concepts :: ''Now some students are bothered with the notion of an \"empty set\". \"How\", they say, \"can you have a set with nothing in it?\" ... The empty set is only a convention, and mathematics could very well get along without it. But it is a very convenient convention, for it saves us a good deal of awkwardness in stating theorems and proving them.'' Such a philosophical position is considered by many mathematicians to be a timid attitude harking back to the mediaeval distrust of zero. In any case, its convenience cannot be doubted: * {{BookReference|Lectures in Abstract Algebra|1951|Nathan Jacobson|volume = I|subtitle = Basic Concepts}}: Introduction $\\S 1$: Operations on Sets: :: ''One may regard [the vacuous set] as a zero element that is adjoined to the collection of \"real\" subsets.'' * {{BookReference|Modern Algebra|1965|Seth Warner}}: $\\S 1$: :: ''One practical advantage in admitting $\\O$ as a set is that we may wish to talk about a set without knowing ''a priori'' whether it has any members.'' * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth}}: $\\S 1$: :: ''The courtesy of regarding this as a set has several advantages ... In allowing $\\O$ the status of a set, we gain the advantage of being able to talk about a set without knowing at the outset whether or not it has any elements.'' Other sources allow the definition of the empty set, but because of the way natural numbers are defined, determine that it is neither finite nor infinite."}
+{"_id": "25614", "title": "Definition:Topologically Distinguishable/Indistinguishable", "text": "The two points $x$ and $y$ are '''topologically indistinguishable''' {{iff}} they are not '''topologically distinguishable'''. That is: :$\\forall U \\in \\tau: x \\in U \\iff y \\in U$"}
+{"_id": "25616", "title": "Definition:Strict Upper Closure/Element", "text": "Let $\\left({S, \\preccurlyeq}\\right)$ be an ordered set. Let $a \\in S$. The '''strict upper closure of $a$ (in $S$)''' is defined as: :$a^\\succ := \\left\\{{b \\in S: a \\preccurlyeq b \\land a \\ne b}\\right\\}$ or: :$a^\\succ := \\left\\{{b \\in S: a \\prec b}\\right\\}$ That is, $a^\\succ$ is the set of all elements of $S$ that strictly succeed $a$."}
+{"_id": "25617", "title": "Definition:Strict Upper Closure/Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set or preordered set. Let $T \\subseteq S$. The '''strict upper closure of $T$ (in $S$)''' is defined as: :$T^\\succ := \\bigcup \\left\\{{t^\\succ: t \\in T}\\right\\}$ where $t^\\succ$ denotes the strict upper closure of $t$ in $S$. That is: :$T^\\succ := \\left\\{ {u \\in S: \\exists t \\in T: t \\prec u}\\right\\}$"}
+{"_id": "25618", "title": "Definition:Strict Lower Closure/Element", "text": "Let $\\left({S, \\preccurlyeq}\\right)$ be an ordered set. Let $a \\in S$. The '''strict lower closure of $a$ (in $S$)''' is defined as: :$a^\\prec := \\left\\{{b \\in S: b \\preccurlyeq a \\land a \\ne b}\\right\\}$ or: :$a^\\prec := \\left\\{{b \\in S: b \\prec a}\\right\\}$ That is, $a^\\prec$ is the set of all elements of $S$ that strictly precede $a$."}
+{"_id": "25619", "title": "Definition:Strict Lower Closure/Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set or a preordered set. Let $T \\subseteq S$. The '''strict lower closure of $T$ (in $S$)''' is defined as: :$T^\\prec := \\displaystyle \\bigcup \\left\\{{t^\\prec: t \\in T}\\right\\}$ where $t^\\prec$ denotes the strict lower closure of $t$ in $S$. That is: :$T^\\prec := \\left\\{ {u \\in S: \\exists t \\in T: u \\prec t}\\right\\}$"}
+{"_id": "25620", "title": "Definition:Real Interval/Notation/Wirth", "text": "The notation used on this site to denote a real interval is a fairly recent innovation, and was introduced by {{AuthorRef|Niklaus Emil Wirth}}: {{begin-eqn}} {{eqn | l = \\openint a b       | o = :=       | r = \\set {x \\in \\R: a < x < b}       | c = Open Real Interval }} {{eqn | l = \\hointr a b       | o = :=       | r = \\set {x \\in \\R: a \\le x < b}       | c = Half-Open (to the right) Real Interval }} {{eqn | l = \\hointl a b       | o = :=       | r = \\set {x \\in \\R: a < x \\le b}       | c = Half-Open (to the left) Real Interval }} {{eqn | l = \\closedint a b       | o = :=       | r = \\set {x \\in \\R: a \\le x \\le b}       | c = Closed Real Interval }} {{end-eqn}} The term '''Wirth interval notation''' has consequently been coined by {{ProofWiki}}."}
+{"_id": "25621", "title": "Definition:Real Interval/Notation/Unbounded Intervals", "text": "Some authors (sensibly, perhaps) prefer not to use the $\\infty$ symbol and instead use $\\to$ and $\\gets$ for $+\\infty$ and $-\\infty$ repectively. In Wirth interval notation, such intervals are written as follows: {{begin-eqn}} {{eqn | l = \\hointr a \\to       | o = :=       | r = \\set {x \\in \\R: a \\le x} }} {{eqn | l = \\hointl \\gets a       | o = :=       | r = \\set {x \\in \\R: x \\le a} }} {{eqn | l = \\openint a \\to       | o = :=       | r = \\set {x \\in \\R: a < x} }} {{eqn | l = \\openint \\gets a       | o = :=       | r = \\set {x \\in \\R: x < a} }} {{eqn | l = \\openint \\gets \\to       | o = :=       | r = \\set {x \\in \\R} = \\R }} {{end-eqn}}"}
+{"_id": "25622", "title": "Definition:Strictly Totally Ordered Set", "text": "A '''strictly totally ordered set''' is a relational structure $\\left({S, \\prec}\\right)$ such that the relation $\\prec$ is a strict total ordering."}
+{"_id": "25623", "title": "Definition:Strictly Partially Ordered Set", "text": "A '''strictly partially ordered set''' is a relational structure $\\left({S, \\prec}\\right)$ such that the relation $\\prec$ is an strict partial ordering."}
+{"_id": "25624", "title": "Definition:Addition/Peano Structure", "text": "Let $\\struct {P, 0, s}$ be a Peano structure. The binary operation $+$ is defined on $P$ as follows: :$\\forall m, n \\in P: \\begin{cases} m + 0 & = m \\\\ m + \\map s n & = \\map s {m + n} \\end{cases}$ This operation is called '''addition'''."}
+{"_id": "25625", "title": "Definition:Real Number/Axioms", "text": "The properties of the field of real numbers $\\struct {R, +, \\times, \\le}$ are as follows: {{begin-axiom}} {{axiom | n = \\R \\text A 0         | lc= Closure under addition         | q = \\forall x, y \\in \\R         | m = x + y \\in \\R }} {{axiom | n = \\R \\text A 1         | lc= Associativity of addition         | q = \\forall x, y, z \\in \\R         | m = \\paren {x + y} + z = x + \\paren {y + z} }} {{axiom | n = \\R \\text A 2         | lc= Commutativity of addition         | q = \\forall x, y \\in \\R         | m = x + y = y + x }} {{axiom | n = \\R \\text A 3         | lc= Identity element for addition         | q = \\exists 0 \\in \\R: \\forall x \\in \\R         | m = x + 0 = x = 0 + x }} {{axiom | n = \\R \\text A 4         | lc= Inverse elements for addition         | q = \\forall x: \\exists \\paren {-x} \\in \\R         | m = x + \\paren {-x} = 0 = \\paren {-x} + x }} {{axiom | n = \\R \\text M 0         | lc= Closure under multiplication         | q = \\forall x, y \\in \\R         | m = x \\times y \\in \\R }} {{axiom | n = \\R \\text M 1         | lc= Associativity of multiplication         | q = \\forall x, y, z \\in \\R         | m = \\paren {x \\times y} \\times z = x \\times \\paren {y \\times z} }} {{axiom | n = \\R \\text M 2         | lc= Commutativity of multiplication         | q = \\forall x, y \\in \\R         | m = x \\times y = y \\times x }} {{axiom | n = \\R \\text M 3         | lc= Identity element for multiplication         | q = \\exists 1 \\in \\R, 1 \\ne 0: \\forall x \\in \\R         | m = x \\times 1 = x = 1 \\times x }} {{axiom | n = \\R \\text M 4         | lc= Inverse elements for multiplication         | q = \\forall x \\in \\R_{\\ne 0}: \\exists \\frac 1 x \\in \\R_{\\ne 0}         | m = x \\times \\frac 1 x = 1 = \\frac 1 x \\times x }} {{axiom | n = \\R \\text D         | lc= Multiplication is distributive over addition         | q = \\forall x, y, z \\in \\R         | m = x \\times \\paren {y + z} = \\paren {x \\times y} + \\paren {x \\times z} }} {{axiom | n = \\R \\text O 1         | lc= Usual ordering is compatible with addition         | q = \\forall x, y, z \\in \\R         | m = x > y \\implies x + z > y + z }} {{axiom | n = \\R \\text O 2         | lc= Usual ordering is compatible with multiplication         | q = \\forall x, y, z \\in \\R         | m = x > y, z > 0 \\implies x \\times z > y \\times z }} {{axiom | n = \\R \\text O 3         | lc= $\\struct {R, +, \\times, \\le}$ is Dedekind complete         | q =          | m =  }} {{end-axiom}} These are called the '''real number axioms'''."}
+{"_id": "25626", "title": "Definition:Indexing Set/Note on Terminology", "text": "It is a common approach to blur the distinction between an indexing function $x: I \\to S$ and the indexed family $\\family {x_i}_{i \\mathop \\in I}$ itself, and refer to the mapping as the indexed family. This approach is in accordance with the definition of a mapping as a relation defined as an as ordered triple $\\tuple {I, S, x}$, where the mapping is understood as being ''defined'' to include its domain and codomain. However, on {{ProofWiki}} the approach is taken to separate the concepts carefully such that an indexed family is defined as: :the set of terms of the indexed set together with: :the indexing function itself denoting the combination as $\\family {x_i}_{i \\mathop \\in I}$. The various approaches in the literature can be exemplified as follows. :''There are occasions when the range of a function is deemed to be more important than the function itself. When that is the case, both the terminology and the notation undergo radical alterations. Suppose, for instance, that $x$ is a function from a set $I$ to a set $X$. ... An element of the domain $I$ is called an '''index''', $I$ is called the '''index set''', the range of the function is called an '''indexed set''', the function itself is called a '''family''', and the value of the function $x$ at an index $i$, called a '''term''' of the family, is denoted by $x_i$. (This terminology is not absolutely established, but it is one of the standard choices among related slight variants...) An unacceptable but generally accepted way of communicating the notation and indicating the emphasis is to speak of a family $\\set {x_i}$ in $X$, or of a family $\\set {x_i}$ of whatever the elements of $X$ may be; when necessary, the index set $I$ is indicated by some such parenthetical expression as $\\paren {i \\in I}$. Thus, for instance, the phrase \"a family $\\set {A_i}$ of subsets of $X$\" is usually understood to refer to a function $A$, from some set $I$ of indices, into $\\powerset X$.'' :::{{BookReference|Naive Set Theory|1960|Paul R. Halmos}}: $\\S 9$: Families :''Occasionally, the special notation for sequences is also employed for functions that are not sequences. If $f$ is a function from $A$ into $E$, some letter or symbol is chosen, say \"$x$\", and $\\map f \\alpha$ is denoted by $x_\\alpha$ and $f$ itself by $\\paren {x_\\alpha}_{\\alpha \\mathop \\in A}$. When this notation is used, the domain $A$ of $f$ is called the set of '''indices''' of $\\paren {x_\\alpha}_{\\alpha \\mathop \\in A}$, and $\\paren {x_\\alpha}_{\\alpha \\mathop \\in A}$ is called a '''family of elements of $E$ indexed by $A$''' instead of a function from $A$ into $E$.'' :::{{BookReference|Modern Algebra|1965|Seth Warner}}: $\\S 18$: The Natural Numbers :''Let $I$ and $E$ be sets and let $f: I \\to E$ be a mapping, described by $i \\mapsto \\map f i$ for each $i \\in I$. We often find it convenient to write $x_i$ instead of $\\map f i$ and write the mapping as $\\paren {x_i}_{i \\mathop \\in I}$ which we shall call a '''family of elements of $E$ indexed by $I$'''. By abuse of language we refer to the $x_i$ as the '''elements of the family'''.'' :''...'' :''As we have already mentioned, many authors identify a mapping with its graph, thereby identifying the family $\\paren {x_i}_{i \\mathop \\in I}$ with the set $\\set {\\tuple {i, x_i}; i \\in I}$. In the case where the elements of the family are all distinct, some authors go even further and identify the mapping $\\paren {x_i}_{i \\mathop \\in I}$ with its image $\\set {x_i; i \\in I}$.'' :::{{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth}}: $\\S 6$. Indexed families; partitions; equivalence relations Some authors are specific about the types of objects to which this construction is applied: :''Let $\\mathcal A$ be a nonempty collection of sets. An '''indexing function''' for $\\mathcal A$ is a surjective function $f$ from some set $J$, called the '''index set''', to $\\mathcal A$. The collection $\\mathcal A$, together with the indexing function $f$, is called an '''indexed family of sets'''. Given $\\alpha \\in J$, we shall denote the set $\\map f \\alpha$ by the symbol $\\mathcal A_\\alpha$. And we shall denote the indexed family itself by the symbol $\\set {\\mathcal A_\\alpha}_{\\alpha \\mathop \\in J}$, which is read as \"the family of all $\\mathcal A_\\alpha$, as $\\alpha$ ranges over $J$.\"'' :::{{BookReference|Topology|2000|James R. Munkres|ed = 2nd|edpage = Second Edition}}: $\\S 5$: Cartesian Products"}
+{"_id": "25627", "title": "Definition:Indexing Set/Family", "text": "The image $\\Img x$, consisting of the terms $\\family {x_i}_{i \\mathop \\in I}$, along with the indexing function $x$ itself, is called a '''family of elements of $S$ indexed by $I$'''."}
+{"_id": "25628", "title": "Definition:Indexing Set/Notation", "text": "The family of elements $x$ of $S$ indexed by $I$ is often seen with one of the following notations: :$\\family {x_i}_{i \\mathop \\in I}$ :$\\paren {x_i}_{i \\mathop \\in I}$ :$\\set {x_i}_{i \\mathop \\in I}$ There is little consistency in the literature, but $\\paren {x_i}_{i \\mathop \\in I}$ is perhaps most common. The preferred notation on {{ProofWiki}} is $\\family {x_i}_{i \\mathop \\in I}$. The subscripted $i \\in I$ is often left out, if it is obvious in the particular context. Note the use of $x_i$ to denote the image of the index $i$ under the indexing function $x$. As $x$ is actually a mapping, one would expect the conventional notation $\\map x i$. However, this is generally not used, and $x_i$ is used instead."}
+{"_id": "25629", "title": "Definition:Indexing Set/Family of Subsets", "text": "Let $S$ be a set. Let $I$ be an indexing set. For each $i \\in I$, let $S_i$ be a corresponding subset of $S$. Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of subsets of $S$ indexed by $I$. Then $\\family {S_i}_{i \\mathop \\in I}$ is referred to as an '''indexed family of subsets (of $S$ by $I$)'''."}
+{"_id": "25630", "title": "Definition:Cartesian Product/Cartesian Space/Real Cartesian Space/Countable", "text": "The countable cartesian product defined as: :$\\displaystyle \\R^\\omega := \\R \\times \\R \\times \\cdots = \\prod_\\N \\R$ is called the '''countable-dimensional real cartesian space'''. Thus, $\\R^\\omega$ can be defined as the set of all real sequences: :$\\R^\\omega = \\left\\{{\\left({x_1, x_2, \\ldots}\\right): x_1, x_2, \\ldots \\in \\R}\\right\\}$"}
+{"_id": "25631", "title": "Definition:Composite Defined by Permutation", "text": "Let $\\oplus$ be an $n$-ary operation on a set $S$. Let $\\left\\langle{a_k}\\right\\rangle_{k \\mathop \\in A}$ be a sequence of $n$ terms of $S$. Let $\\sigma: A \\to A$ be a permutation of $A$. Then the '''composite of the ordered $n$-tuple defined by the sequence $\\left\\langle{a_{\\sigma \\left({k}\\right)}}\\right\\rangle_{k \\mathop \\in A}$''' is defined as: {{begin-eqn}} {{eqn | l = \\bigoplus_{k \\mathop \\in A} a_{\\sigma \\left({k}\\right)}       | r = \\begin{cases} a_{\\sigma \\left({1}\\right)} & : n = 1 \\\\ \\oplus \\left({a_{\\sigma \\left({1}\\right)}, a_{\\sigma \\left({2}\\right)}, \\ldots, a_{\\sigma \\left({m}\\right)} }\\right) \\oplus a_{\\sigma \\left({n + 1}\\right)} & : n = m + 1 \\end{cases}       | c =  }} {{eqn | r = a_{\\sigma \\left({1}\\right)} \\oplus a_{\\sigma \\left({2}\\right)} \\oplus \\cdots  \\oplus a_{\\sigma \\left({n}\\right)}       | c =  }} {{end-eqn}}"}
+{"_id": "25632", "title": "Definition:Power of Element/Semigroup", "text": "Let $\\struct {S, \\circ}$ be a semigroup which has no identity element. Let $a \\in S$. For $n \\in \\N_{>0}$, the $n$th power of $a$ (under $\\circ$) is defined as: :$\\circ^n a = \\begin{cases} a & : n = 1 \\\\ \\paren {\\circ^m a} \\circ a & : n = m + 1 \\end{cases}$ That is: :$a^n = \\underbrace {a \\circ a \\circ \\cdots \\circ a}_{n \\text{ copies of } a}$ which from the General Associativity Theorem is unambiguous."}
+{"_id": "25633", "title": "Definition:Incircle/Incenter", "text": "The center of an incircle of a polygon is called an '''incenter of the polygon'''."}
+{"_id": "25634", "title": "Definition:Incircle/Inradius", "text": "A radius of an incircle of a polygon is called an '''inradius of the polygon'''. In the above diagram, $OG$ is an inradius of the polygon $ABCDEF$."}
+{"_id": "25635", "title": "Definition:Independent Subgroups/Definition 2", "text": "The subgroups $H_1, H_2, \\ldots, H_n$ are '''independent''' {{iff}}: :$\\displaystyle \\forall k \\in \\set {2, 3, \\ldots, n}: \\paren {\\prod_{j \\mathop = 1}^{k - 1} H_j} \\cap H_k = \\set e$"}
+{"_id": "25636", "title": "Definition:Independent Subgroups/Definition 1", "text": "The subgroups $H_1, H_2, \\ldots, H_n$ are '''independent''' {{iff}}: :$\\displaystyle \\prod_{k \\mathop = 1}^n h_k = e \\iff \\forall k \\in \\set {1, 2, \\ldots, n}: h_k = e$ where $h_k \\in H_k$ for all $k \\in \\set {1, 2, \\ldots, n}$."}
+{"_id": "25637", "title": "Definition:Factorial/Historical Note", "text": "The symbol $!$ used on {{ProofWiki}} for the factorial, which is now universal, was introduced by {{AuthorRef|Christian Kramp}} in his $1808$ work {{BookLink|Élémens d'arithmétique universelle|C. Kramp}}. Before that, various symbols were used whose existence is now of less importance. Notations for $n!$ in history include the following: :$\\sqbrk n$ as used by {{AuthorRef|Leonhard Paul Euler|Euler}} :$\\mathop{\\Pi} n$ as used by {{AuthorRef|Carl Friedrich Gauss|Gauss}} :$\\left\\lvert {\\kern-1pt \\underline n} \\right.$ and $\\left. {\\underline n \\kern-1pt} \\right\\rvert$, once popular in England and Italy. {{AuthorRef|Augustus De Morgan}} declared his reservations about {{AuthorRef|Christian Kramp|Kramp}}'s notation thus: :''Amongst the worst barbarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation $n!$ ... which gives their pages the appearance of expressing admiration  that $2$, $3$, $4$, etc., should be found in mathematical results.'' ::::-- {{BookReference|A History of Mathematical Notations|1929|Florian Cajori|volume = 2}} The use of $n!$ for non-integer $n$ is uncommon, as the Gamma function tends to be used instead."}
+{"_id": "25638", "title": "Definition:Factorial/Multiindices", "text": "Let $\\alpha$ be a multiindex, indexed by a set $J$ such that for each $j \\in J$, $\\alpha_j \\geq 0$. Then we define: :$\\displaystyle\\alpha! = \\prod_{j \\mathop \\in J} \\alpha_j!$ where the factorial on the right is a factorial of natural numbers. Note that by definition, all by finitely many of the $\\alpha_j$ are zero, so the product over $J$ is convergent."}
+{"_id": "25639", "title": "Definition:Empty Topological Space", "text": "Let $\\varnothing$ denote the empty set. The topological space $\\left({\\varnothing, \\left\\{ {\\varnothing}\\right\\} }\\right)$ is called the '''empty topological space'''."}
+{"_id": "25640", "title": "Definition:Topology/Definition 1", "text": "Let $S$ be a set. A '''topology on $S$''' is a subset $\\tau \\subseteq \\powerset S$ of the power set of $S$ that satisfies the open set axioms: {{:Definition:Open Set Axioms}} If $\\tau$ is a '''topology''' on $S$, then $\\struct {S, \\tau}$ is called a topological space. The elements of $\\tau$ are called the open sets of $\\struct {S, \\tau}$."}
+{"_id": "25641", "title": "Definition:Topology/Definition 2", "text": "Let $S$ be a set. A '''topology on $S$''' is a subset $\\tau \\subseteq \\powerset S$ of the power set of $S$ that satisfies the following axioms: {{begin-axiom}} {{axiom | n = \\text O 1'         | t = The union of an arbitrary subset of $\\tau$ is an element of $\\tau$. }} {{axiom | n = \\text O 2'         | t = The intersection of any finite subset of $\\tau$ is an element of $\\tau$. }} {{end-axiom}}"}
+{"_id": "25642", "title": "Definition:Neighborhood (Topology)/Neighborhood defined as Open", "text": "Some authorities define a neighborhood of a set $A$ as what {{ProofWiki}} defines as an open neighborhood: :$N_A$ is a '''neighborhood of $A$''' {{iff}} $N_A$ is an open set of $T$ which itself contains $A$. That is, in order to be a neighborhood of $A$ in $T$, $N_A$ must not only be a subset of $T$, but also be an open set of $T$. However, this treatment is less common, and considered by many to be old-fashioned. When the term '''neighborhood''' is used on this site, it is assumed to be not necessarily open unless so specified."}
+{"_id": "25643", "title": "Definition:Basis (Topology)/Analytic Basis/Definition 1", "text": "Let $\\struct {S, \\tau}$ be a topological space. An '''analytic basis for $\\tau$''' is a subset $\\BB \\subseteq \\tau$ such that: :$\\displaystyle \\forall U \\in \\tau: \\exists \\AA \\subseteq \\BB: U = \\bigcup \\AA$ That is, such that for all $U \\in \\tau$, $U$ is a union of sets from $\\BB$."}
+{"_id": "25644", "title": "Definition:Basis (Topology)/Analytic Basis/Definition 2", "text": "Let $\\left({S, \\tau}\\right)$ be a topological space. Let $\\mathcal B \\subseteq \\tau$. Then $\\mathcal B$ is an analytic basis for $\\tau$ {{iff}}: :$\\forall U \\in \\tau: \\forall x \\in U: \\exists V \\in \\mathcal B: x \\in V \\subseteq U$"}
+{"_id": "25645", "title": "Definition:Isolated Point (Topology)/Subset/Definition 1", "text": "$x \\in H$ is an '''isolated point of $H$''' {{iff}}: :$\\exists U \\in \\tau: U \\cap H  = \\set x$ That is, {{iff}} there exists an open set of $T$ containing no points of $H$ other than $x$."}
+{"_id": "25646", "title": "Definition:Isolated Point (Topology)/Subset/Definition 2", "text": "$x \\in H$ is an '''isolated point of $H$''' {{iff}} $x$ is not a limit point of $H$. That is, {{iff}} $x$ is not in the derived set of $H$."}
+{"_id": "25647", "title": "Definition:Addition in Minimal Infinite Successor Set", "text": "Let $\\omega$ be the minimal infinite successor set. The binary operation $+$ is defined on $\\omega$ as follows: :$\\forall m,n \\in \\omega: \\begin{cases} m + 0 &= m \\\\ m + n^+ &= \\left({m + n}\\right)^+\\end{cases}$ where $m^+$ is the successor set of $m$. This operation is called '''addition'''."}
+{"_id": "25648", "title": "Definition:Addition for Natural Numbers in Real Numbers", "text": "Let $\\struct {\\R, +, \\times, \\le}$ be the field of real numbers. Let $\\N$ be the natural numbers in $\\R$. Then the restriction of $+$ to $\\N$ is called '''addition'''."}
+{"_id": "25649", "title": "Definition:Closed Set/Topology/Definition 1", "text": "'''$H$ is closed (in $T$)''' {{iff}} its complement $S \\setminus H$ is open in $T$.  That is, $H$ is '''closed''' {{iff}} $\\paren {S \\setminus H} \\in \\tau$. That is, {{iff}} $S \\setminus H$ is an element of the topology of $T$."}
+{"_id": "25650", "title": "Definition:Closed Set/Topology/Definition 2", "text": "'''$H$ is closed (in $T$)''' {{iff}} every limit point of $H$ is also a point of $H$. That is, by the definition of the derived set: :'''$H$ is closed (in $T$)''' {{iff}} $H' \\subseteq H$ where $H'$ denotes the derived set of $H$."}
+{"_id": "25651", "title": "Definition:Scaled Euclidean Metric", "text": "Let $\\R_{>0}$ be the set of strictly positive real numbers. Let $\\delta: \\R_{>0} \\times \\R_{>0} \\to \\R$ be the metric on $\\R_{>0}$ defined as: :$\\forall x, y \\in \\R_{>0}: \\map \\delta {x, y} = \\dfrac {\\size {x - y} } {x y}$ Then $\\delta$ is the '''scaled Euclidean metric''' on $\\R_{>0}$."}
+{"_id": "25652", "title": "Definition:Alexandroff Extension", "text": "Let $T = \\struct {S, \\tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \\cup \\set p$. Let $\\tau^*$ be the topology on $S^*$ defined such that $U \\subseteq S^*$ is open {{iff}}: :$U$ is an open set of $T$ or :$U$ is the complement in $T^*$ of a closed and compact subset of $T$. This topology is called the '''Alexandroff extension''' on $S$."}
+{"_id": "25653", "title": "Definition:Centroid of Triangle", "text": "Let $\\triangle ABC$ be a triangle. The '''centroid''' of $\\triangle ABC$ is where its three medians meet. :500px"}
+{"_id": "25654", "title": "Definition:One-Sided Limit of Real Function", "text": "A '''one-sided limit of real function''' is a right-hand limit of real function or a left-hand limit of real function: === Limit from Right === {{:Definition:Limit of Real Function/Right}} === Limit from Left === {{:Definition:Limit of Real Function/Left}} Category:Definitions/Real Analysis Category:Definitions/Limits k8m8gaomjoc9y8kcpjw3dzujindqdpf"}
+{"_id": "25656", "title": "Definition:Functor/Contravariant/Definition 1", "text": "Let $\\mathbf C$ and $\\mathbf D$ be metacategories. A '''contravariant functor''' $F : \\mathbf C \\to \\mathbf D$ consists of: * An '''object functor''' $F_0$ that assigns to each object $X$ of $\\mathbf C$ an object $FX$ of $\\mathbf D$. * An '''arrow functor''' $F_1$ that assigns to each arrow $f : X \\to Y$ of $\\mathbf C$ an arrow $Ff : FY \\to FX$ of $\\mathbf D$. These '''functors''' must satisfy, for any morphisms $X \\stackrel f \\longrightarrow Y \\stackrel g \\longrightarrow Z$ in $\\mathbf C$: :$\\map F {g \\circ f} = F f \\circ F g$ and: :$\\map F {\\operatorname {id}_X} = \\operatorname {id}_{F X}$ where: :$\\operatorname {id}_W$ denotes the identity arrow on an object $W$ and: :$\\circ$ is the composition of morphisms."}
+{"_id": "25657", "title": "Definition:Functor/Contravariant/Definition 2", "text": "Let $\\mathbf C$ and $\\mathbf D$ be metacategories. A '''contravariant functor''' $F : \\mathbf C \\to \\mathbf D$ is a covariant functor: :$F: \\mathbf C^{\\text{op}} \\to \\mathbf D$ where $\\mathbf C^{\\text{op}}$ is the dual category of $\\mathbf C$."}
+{"_id": "25659", "title": "Definition:Normal Extension/Definition 1", "text": "Let $L / K$ be a field extension. Then $L / K$ is a '''normal extension''' {{iff}}: : for every irreducible polynomial $f \\in K \\left[{x}\\right]$ with at least one root in $L$, $f$ splits completely in $L$."}
+{"_id": "25660", "title": "Definition:Normal Extension/Definition 2", "text": "Let $L / K$ be a field extension. Let $\\overline K$ be the algebraic closure of $K$. Let $\\operatorname{Gal} \\left({L / K}\\right)$ denote the set of embeddings of $L$ in $\\overline K$ which fix $K$ pointwise. Then $L/K$ is a '''normal extension''' {{iff}}: : $\\sigma \\left({L}\\right) = L$ for each $\\sigma \\in \\operatorname{Gal} \\left({L / K}\\right)$."}
+{"_id": "25661", "title": "Definition:Decomposable Element", "text": "Let $\\left({D, +, \\circ}\\right)$ be an integral domain. Let $x$ be a non-zero non-unit element of $D$. Let there exist a complete factorization of $x$ in $D$. Then $x$ is '''decomposable'''. Category:Definitions/Factorization 0zsw4x5af6xzpnef1uvc2tit86x6fwu"}
+{"_id": "25663", "title": "Definition:Zariski Topology/Affine Space", "text": "Let $k$ be a field. Let $\\mathbb A^n \\left({k}\\right) = k^n$ denote the standard affine space of dimension $n$ over $k$. The '''Zariski topology''' on $\\mathbb A^n \\left({k}\\right)$ is the topology on the direct product $k^n$ whose closed sets are the affine algebraic sets in $\\mathbb A^n \\left({k}\\right)$."}
+{"_id": "25664", "title": "Definition:Zariski Topology/Spectrum of Ring", "text": "Let $A$ be a commutative ring with unity. Let $\\operatorname{Spec} \\left({A}\\right)$ be the prime spectrum of $A$. The '''Zariski topology''' on $\\operatorname{Spec} A$ is the topology with closed sets the vanishing sets $V \\left({S}\\right)$ for $S \\subseteq A$."}
+{"_id": "25665", "title": "Definition:Logarithm/Base", "text": "Let $\\log_a$ denote the logarithm function on whatever domain: $\\R$ or $\\C$. The constant $a$ is known as the '''base''' of the logarithm."}
+{"_id": "25668", "title": "Definition:Tangent Vector/Definition 1", "text": "A '''tangent vector $X_m$ on $M$ at $m$''' is a linear transformation: :$X_m: C^\\infty \\left({V, \\R}\\right) \\to \\R$ which satisfies the Leibniz law: :$\\displaystyle X_m \\left({f g}\\right) = X_m \\left({f}\\right) \\, g \\left({m}\\right) + f \\left({m}\\right) \\, X_m \\left({g}\\right)$"}
+{"_id": "25669", "title": "Definition:Tangent Vector/Definition 2", "text": "Let $I$ be an open real interval with $0 \\in I$.  Let $\\gamma: I \\to M$ be a smooth curve with $\\gamma \\left({0}\\right) = m$.  Then a '''tangent vector''' $X_m$ at a point $m \\in M$ is a mapping  :$X_m: C^\\infty \\left({V, \\R}\\right) \\to \\R$ defined by: :$X_m \\left({f} \\right) := \\dfrac {\\mathrm d} {\\mathrm d \\tau} {\\restriction_0} \\, f \\circ \\gamma \\left({\\tau}\\right)$ for all $f \\in C^\\infty \\left({V, \\R}\\right)$."}
+{"_id": "25670", "title": "Definition:Inverse Laplace Transform", "text": "=== Definition 1 === {{:Definition:Inverse Laplace Transform/Definition 1}} === Definition 2 === {{:Definition:Inverse Laplace Transform/Definition 2}}"}
+{"_id": "25671", "title": "Definition:Inverse Laplace Transform/Definition 2", "text": "Let $\\map f s: S \\to \\R$ be a complex function, where $S \\subset \\R$. {{mistake|How can it be a complex function when both its domain and codomain are wholly real?}} The '''inverse Laplace transform''' of $f$, denoted $\\map F t: \\R \\to S$, is defined as: :$\\map F t = \\dfrac 1 {2 \\pi i} \\PV_{c \\mathop - i \\, \\infty}^{c \\mathop + i \\, \\infty} e^{s t} \\map f s \\rd s = \\frac 1 {2 \\pi i} \\lim_{T \\mathop \\to \\infty} \\int_{c \\mathop - i \\, T}^{c \\mathop + i \\, T} e^{s t} \\map f s \\rd s$ where: :$\\PV$ is the Cauchy principal value of the integral :$c$ is any real constant such that all the singular points of $\\map f s$ lie to the left of the line $\\map \\Re s = c$ in the complex $s$ plane."}
+{"_id": "25672", "title": "Definition:Inverse Laplace Transform/Definition 1", "text": "Let $f \\left({s}\\right) : S \\to \\R$ be a function, where $S \\subset \\R$. Let $F$ be another function such that $F$ is the Laplace transform of $f$. Then, $f$ is the '''inverse Laplace transform''' of $F$."}
+{"_id": "25673", "title": "Definition:Topological Surface", "text": "A '''surface''', in the context of topology, is a smooth topological manifold whose dimension is $2$. More briefly, a '''surface''' is a smooth $2$-manifold. Category:Definitions/Topology Category:Definitions/Algebraic Topology d03a75ochzeskx4zmadmgjvolmjlpax"}
+{"_id": "25675", "title": "Definition:Taylor Polynomial", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$ and $n + 1$ times differentiable on the open interval $\\openint a b$. Let $\\xi \\in \\openint a b$. The polynomial $T_n$ defined as:  :$\\displaystyle \\map {T_n} x = \\sum_{i \\mathop = 0}^i \\frac {\\paren {x - \\xi}^i} {i!} \\map {f^{\\paren i} } \\xi$ is known as the '''Taylor polynomial of degree $n$ for $f$ about $\\xi$'''. That is, a '''Taylor polynomial''' is a Taylor series taken for $n$ initial terms."}
+{"_id": "25676", "title": "Definition:Bessel Function", "text": "The '''Bessel functions''' are solutions to Bessel's equation: :$x^2 \\dfrac {\\d^2 y} {\\d x^2} + x \\dfrac {\\d y} {\\d x} + \\paren {x^2 - n^2} y = 0$ These solutions have two main classes: :the Bessel functions of the first kind $J_n$ and: :the Bessel functions of the second kind $Y_n$."}
+{"_id": "25677", "title": "Definition:Monster Group", "text": "A group $G$ is a '''Monster group''' and the largest sporadic simple group {{iff}} it has the order:   :$808017424794512875886459904961710757005754368000000000 = 2^{46}.3^{20}.5^9.7^6.11^2.13.17.19.23.29.31.41.47.59.71$"}
+{"_id": "25678", "title": "Definition:Dirac Delta Function", "text": "Let $\\epsilon \\in \\R_{>0}$ be a (strictly) positive real number. Consider the real function $F_\\epsilon: \\R \\to \\R$ defined as: :$\\map {F_\\epsilon} x := \\begin{cases} 0 & : x < 0 \\\\ \\dfrac 1 \\epsilon & : 0 \\le x \\le \\epsilon \\\\ 0 & : x > \\epsilon \\end{cases}$ The '''Dirac delta function''' is defined as: :$\\map \\delta x := \\displaystyle \\lim_{\\epsilon \\mathop \\to 0} \\map {F_\\epsilon} x$"}
+{"_id": "25679", "title": "Definition:Symmetric Set/Real Numbers", "text": "Let $\\R$ be the set of real numbers. Let $S \\subseteq \\R$ such that: :$\\forall x \\in S: -x \\in S$ That is, for every element in $S$, its negative is also in $S$. Then $S$ is a '''symmetric subset of $\\R$''', or (if $\\R$ is implicit) $S$ is a '''symmetric set'''."}
+{"_id": "25680", "title": "Definition:Additive Inverse/Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose ring addition operation is $+$. Let $a \\in R$ be any arbitrary element of $R$. The '''additive inverse''' of $a$ is its inverse under ring addition, denoted $-a$: :$a + \\paren {-a} = 0_R$ where $0_R$ is the zero of $R$."}
+{"_id": "25681", "title": "Definition:Additive Inverse/Number", "text": "Let $\\Bbb F$ be one of the standard number systems: $\\N$, $\\Z$, $\\Q$, $\\R$, $\\C$. Let $a \\in \\Bbb F$ be any arbitrary number. The '''additive inverse''' of $a$ is its inverse under addition, denoted $-a$: :$a + \\paren {-a} = 0$"}
+{"_id": "25682", "title": "Definition:Mellin Transform", "text": "Let $\\map f t: \\R_{\\ge 0} \\to \\C$ be a function of a real variable $t$. The '''Mellin Transform''' of $f$, denoted $\\map \\MM f$ or $\\phi$, is defined as: :$\\displaystyle \\map {\\MM \\set {\\map f t} } s = \\map \\phi s = \\int_0^{\\to +\\infty} t^{s - 1} \\map f t \\rd t$ wherever this improper integral exists. Here $\\map \\MM f$ is a complex function of the variable $s$."}
+{"_id": "25683", "title": "Definition:Quaternion Modulus", "text": "Let $\\mathbf x = a \\mathbf 1 + b \\mathbf i + c \\mathbf j + d \\mathbf k$ be a quaternion, where $a, b, c, d \\in \\R$. Then the '''(quaternion) modulus of''' $\\mathbf x$ is written as $\\vert \\mathbf x \\vert$ and is defined as: :$\\vert \\mathbf x \\vert := \\sqrt{a^2 + b^2 + c^2 + d^2}$ The '''quaternion modulus''' is a real-valued function, and as when appropriate is referred to as the '''quaternion modulus function'''."}
+{"_id": "25684", "title": "Definition:Quaternion Conjugate/Quaternion Conjugation", "text": "The operation of '''quaternion conjugation''' is the mapping: : $\\overline \\cdot: \\mathbb H \\to \\mathbb H: \\mathbf x \\mapsto \\overline{\\mathbf x}$. where $\\overline{\\mathbf x}$ is the quaternion conjugate of $x$. That is, it maps a quaternion to its quaternion conjugate."}
+{"_id": "25685", "title": "Definition:Natural Numbers/Axiomatization", "text": "=== Peano's Axioms === {{:Axiom:Peano's Axioms}} === Naturally Ordered Semigroup === {{:Definition:Naturally Ordered Semigroup}} === 1-Based Natural Numbers === The following axioms are intended to capture the behaviour of $\\N_{>0}$, the element $1 \\in \\N_{>0}$, and the operations $+$ and $\\times$ as they pertain to $\\N_{>0}$: {{:Axiom:Axiomatization of 1-Based Natural Numbers}}"}
+{"_id": "25686", "title": "Definition:Natural Numbers/Construction", "text": "=== Von Neumann Construction of Natural Numbers === {{:Definition:Natural Numbers/Von Neumann Construction}} === Inductive Set Definition for Natural Numbers === {{:Definition:Natural Numbers/Inductive Set Definition}} === Inductive Set Definition for Natural Numbers in Real Numbers === {{:Definition:Natural Numbers/Inductive Sets in Real Numbers}}"}
+{"_id": "25687", "title": "Definition:Ordering on Natural Numbers", "text": "=== Ordering on Peano Structure === {{:Definition:Ordering on Natural Numbers/Peano Structure}} === Ordering on Naturally Ordered Semigroup === {{:Definition:Ordering on Natural Numbers/Naturally Ordered Semigroup}} === Ordering on $1$-Based Natural Numbers === {{:Definition:Ordering on Natural Numbers/1-Based}} === Ordering on Minimal Infinite Successor Set === {{:Definition:Ordering on Natural Numbers/Minimal Infinite Successor Set}} === Ordering on Von Neumann Construction of Natural Numbers === {{:Definition:Ordering on Natural Numbers/Von Neumann Construction}} === Ordering on Natural Numbers in Real Numbers === {{:Definition:Ordering on Natural Numbers/Restriction of Real Numbers}}"}
+{"_id": "25688", "title": "Definition:Ordering on Natural Numbers/1-Based", "text": "Let $\\N_{>0}$ be the axiomatised $1$-based natural numbers. The '''strict ordering of $\\N_{>0}$''', denoted $<$, is defined as follows: :$\\forall a, b \\in \\N_{>0}: a < b \\iff \\exists c \\in \\N_{>0}: a + c = b$ The '''(weak) ordering of $\\N_{>0}$''', denoted $\\le$, is defined as: :$\\forall a, b \\in \\N_{>0}: a \\le b \\iff a = b \\lor a < b$"}
+{"_id": "25689", "title": "Definition:Ordering on Natural Numbers/Naturally Ordered Semigroup", "text": "Let $\\struct {S, \\circ, \\preceq}$ be a naturally ordered semigroup. The relation $\\preceq$ in $\\struct {S, \\circ, \\preceq}$ is called the '''ordering'''."}
+{"_id": "25690", "title": "Definition:Ordering on Natural Numbers/Peano Structure", "text": "Let $\\struct {P, 0, s}$ be a Peano structure. The '''ordering''' of $P$ is the relation $\\le$ defined by: :$\\forall m, n \\in P: m \\le n \\iff \\exists p \\in P: m + p = n$ where $+$ denotes addition in $\\struct {P, 0, s}$."}
+{"_id": "25691", "title": "Definition:Ordering on Natural Numbers/Minimal Infinite Successor Set", "text": "Let $\\omega$ be the minimal infinite successor set. The '''strict ordering''' of $\\omega$ is the relation $<$ defined by: :$\\forall m, n \\in \\omega: m < n \\iff m \\in n$ The '''(weak) ordering''' of $\\omega$ is the relation $\\le$ defined by: :$\\forall m, n \\in \\omega: m \\le n \\iff m < n \\lor m = n$"}
+{"_id": "25692", "title": "Definition:Ordering on Natural Numbers/Restriction of Real Numbers", "text": "Let $\\struct {\\R, +, \\times, \\le}$ be the field of real numbers. Let $\\N$ be the natural numbers in $\\R$. Then the '''ordering''' of $\\N$ is the restriction of $\\le$ to $\\N$."}
+{"_id": "25696", "title": "Definition:Addition on 1-Based Natural Numbers", "text": "Let $\\N_{>0}$ be the $1$-based natural numbers, axiomatized by: {{:Axiom:Axiomatization of 1-Based Natural Numbers}} The operation $+$ in this axiomatizaton is called '''addition'''."}
+{"_id": "25697", "title": "Definition:Octonion/Addition", "text": "The '''sum''' of two octonions $x=\\left({a, b}\\right)$ and $y=\\left({c, d}\\right)$ where $a, b, c, d \\in \\mathbb H$ are quaternions is defined as: :$x + y = \\left({a, b}\\right) + \\left({c, d}\\right) = \\left({a + c, b + d}\\right)$"}
+{"_id": "25698", "title": "Definition:Figurate Number", "text": "A '''figurate number''' is loosely defined as a (natural) number corresponding to a set of objects which can be arranged in a geometric pattern (in an arbitrary number of dimensions, but usually $2$ or $3$)."}
+{"_id": "25700", "title": "Definition:Periodic Continued Fraction/Cycle", "text": "The repeating block in a periodic (or purely periodic) continued fraction $F$ is called the '''cycle''' of $F$."}
+{"_id": "25702", "title": "Definition:Theory (Logic)/Complete", "text": "$T$ is '''complete''' {{iff}}: : for every $\\mathcal L$-sentence $\\phi$, either $T \\models \\phi$ or $T \\models \\neg \\phi$ where $T \\models \\phi$ denotes semantic entailment."}
+{"_id": "25703", "title": "Definition:Theory (Logic)/Maximal", "text": "$T$ is '''maximal''' {{iff}}: :for every $\\mathcal L$-sentence $\\phi$, either $\\phi \\in T$ or $\\neg \\phi \\in T$."}
+{"_id": "25704", "title": "Definition:Theory (Logic)/Structure", "text": "Let $\\mathcal M$ be an $\\mathcal L$-structure. The '''$\\mathcal L$-theory of $\\mathcal M$''' is the $\\mathcal L$-theory consisting of those $\\mathcal L$-sentences $\\phi$ such that: :$\\mathcal M \\models \\phi$ where $\\models$ denotes that $\\mathcal M$ is a model for $\\phi$. This theory can be denoted $\\operatorname{Th} \\left({\\mathcal M}\\right)$ when the language $\\mathcal L$ is understood."}
+{"_id": "25705", "title": "Definition:Stability (Model Theory)/Kappa-Stable Theory", "text": "$T$ is $\\kappa$-'''stable''' {{iff}}: : for all models $\\mathcal M$ of $T$ : for all subsets $A \\subseteq \\mathcal M$ of cardinality $\\kappa$ and: : for all $n \\in \\N$ the cardinality $\\left\\vert{ {S_n}^{\\mathcal M} \\left({A}\\right) }\\right\\vert$ of the set ${S_n}^{\\mathcal M} \\left({A}\\right)$ of complete $n$-types over $A$ is $\\kappa$."}
+{"_id": "25706", "title": "Definition:Stability (Model Theory)/Stable Theory", "text": "$T$ is '''stable''' {{iff}} it is $\\kappa$-stable for some $\\kappa \\ge \\aleph_0$."}
+{"_id": "25708", "title": "Definition:Stability (Model Theory)/Kappa-Stable Structure", "text": "Let $\\mathcal M$ be an $\\mathcal L$-structure. Let $\\operatorname {Th} \\left({\\mathcal M}\\right)$ be the $\\mathcal L$-theory of $\\mathcal M$. $\\mathcal M$ is '''$\\kappa$-stable''' if $\\operatorname{Th} \\left({\\mathcal M}\\right)$ is $\\kappa$-stable."}
+{"_id": "25709", "title": "Definition:Affine Monoid", "text": "An '''affine monoid''' is a monoid that is: : finitely generated and: : isomorphic to a submonoid of a free abelian group $\\Z^d$, for some $d \\in \\Z_{\\ge 0}$."}
+{"_id": "25710", "title": "Definition:Bounded Linear Transformation/Bounded Linear Operator", "text": "Let $H$ be a Hilbert space. Let $A: H \\to H$ be a linear operator. $A$ is a '''bounded linear operator''' {{iff}} :$\\exists c > 0: \\forall h \\in H: \\left\\Vert{A h}\\right\\Vert_H \\le c \\left\\Vert{h}\\right\\Vert_H$ That is, a '''bounded linear operator''' is a bounded linear transformation from a Hilbert space to itself."}
+{"_id": "25711", "title": "Definition:Arborescence/Root", "text": "The vertex $r$ of $G$ is known as the '''root of (the arborescence) $G$'''."}
+{"_id": "25712", "title": "Definition:Arborescence/Also defined as", "text": "Some sources, for example {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms||Donald E. Knuth}}, define an arborescence of root $r$ so as to reverse the orientation of $G$, so that the arcs are all directed toward the root rather than away from it."}
+{"_id": "25713", "title": "Definition:Arborescence/Also known as", "text": "An arborescence of root $r$ can be referred to as an an '''$r$-arborescence''', or just an '''arborescence'''. Some sources, for example {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms||Donald E. Knuth}}, call an arborescence an '''oriented tree'''."}
+{"_id": "25714", "title": "Definition:Interval/Ordered Set/Half-Open", "text": "Let $\\struct {S, \\preccurlyeq}$ be an ordered set. Let $a, b \\in S$. === Left Half-Open Interval === {{:Definition:Interval/Ordered Set/Left Half-Open}} === Right Half-Open Interval === {{:Definition:Interval/Ordered Set/Right Half-Open}}"}
+{"_id": "25715", "title": "Definition:Ray (Order Theory)/Open", "text": "The following sets are called '''open rays''' or '''open half-lines''': : $\\left\\{{x \\in S: a \\prec x}\\right\\}$ (the strict upper closure of $a$), denoted $a^\\succ$ : $\\left\\{{x \\in S: x \\prec a}\\right\\}$ (the strict lower closure of $a$), denoted $a^\\prec$."}
+{"_id": "25716", "title": "Definition:Ray (Order Theory)/Closed", "text": "The following sets are called '''closed rays''' or '''closed half-lines''': : $\\left\\{{x \\in S: a \\preccurlyeq x}\\right\\}$ (the upper closure of $a$), denoted $a^\\succcurlyeq$ : $\\left\\{{x \\in S: x \\preccurlyeq a}\\right\\}$ (the lower closure of $a$), denoted $a^\\preccurlyeq$."}
+{"_id": "25717", "title": "Definition:Ray (Order Theory)/Upward-Pointing", "text": "An '''upward-pointing ray''' is a ray which is bounded below: : an open ray $a^\\succ:= \\left\\{{x \\in S: a \\prec x}\\right\\}$ : a closed ray $a^\\succcurlyeq: \\left\\{{x \\in S: a \\preccurlyeq x}\\right\\}$"}
+{"_id": "25718", "title": "Definition:Ray (Order Theory)/Downward-Pointing", "text": "A '''downward-pointing ray''' is a ray which is bounded above: : an open ray $a^\\prec := \\left\\{{x \\in S: x \\prec a}\\right\\}$ : a closed ray $a^\\preccurlyeq : \\left\\{{x \\in S: x \\preccurlyeq a}\\right\\}$"}
+{"_id": "25719", "title": "Definition:Hereditarily Compact Space/Definition 1", "text": "$T$ is ''' hereditarily compact''' {{iff}} every subspace of $T$ is compact."}
+{"_id": "25720", "title": "Definition:Hereditarily Compact Space/Definition 2", "text": "$T$ is ''' hereditarily compact''' {{iff}}: : for each family $\\left\\langle{U_i}\\right\\rangle_{i \\mathop \\in I}$ of open sets of $T$, there exists a finite subset $J \\subset I$ such that: :::$\\displaystyle \\bigcup_{j \\mathop \\in J} U_j = \\bigcup_{i \\mathop \\in I} U_i$"}
+{"_id": "25721", "title": "Definition:Order of Structure/Infinite Structure", "text": "Let the underlying set $S$ of $\\struct {S, \\circ}$ be infinite. Then $\\struct {S, \\circ}$ is an '''infinite structure'''."}
+{"_id": "25722", "title": "Definition:Order of Structure/Finite Structure", "text": "Let the underlying set $S$ of $\\struct {S, \\circ}$ be finite. Then $\\struct {S, \\circ}$ a '''finite structure'''."}
+{"_id": "25723", "title": "Definition:Index of Subgroup/Finite", "text": "If $G / H$ is a finite set, then $\\index G H$ is '''finite''', and $H$ is '''of finite index''' in $G$."}
+{"_id": "25724", "title": "Definition:Index of Subgroup/Infinite", "text": "If $G / H$ is an infinite set, then $\\index G H$ is '''infinite''', and $H$ is '''of infinite index''' in $G$."}
+{"_id": "25728", "title": "Definition:Order of Group Element/Infinite", "text": "Let $G$ be a group whose identity is $e_G$. Let $x \\in G$ be an element of $G$. === Definition 1 === {{:Definition:Order of Group Element/Infinite/Definition 1}} === Definition 2 === {{:Definition:Order of Group Element/Infinite/Definition 2}} === Definition 3 === {{:Definition:Order of Group Element/Infinite/Definition 3}}"}
+{"_id": "25729", "title": "Definition:Order of Group Element/Finite", "text": "Let $G$ be a group whose identity is $e_G$. Let $x \\in G$ be an element of $G$. === Definition 1 === {{:Definition:Order of Group Element/Finite/Definition 1}} === Definition 2 === {{:Definition:Order of Group Element/Finite/Definition 2}}"}
+{"_id": "25730", "title": "Definition:Multiplication on 1-Based Natural Numbers", "text": "Let $\\N_{>0}$ be the $1$-based natural numbers, axiomatized by: {{:Axiom:Axiomatization of 1-Based Natural Numbers}} The operation $\\times$ in this axiomatization is called '''multiplication'''."}
+{"_id": "25731", "title": "Definition:Order of Group Element/Definition 1", "text": "The '''order of $x$ (in $G$)''', denoted $\\order x$, is the smallest $k \\in \\Z_{> 0}$ such that $x^k = e_G$."}
+{"_id": "25732", "title": "Definition:Order of Group Element/Definition 2", "text": "The '''order of $x$ (in $G$)''', denoted $\\order x$, is the order of the group generated by $x$: :$\\order x := \\order {\\gen x}$"}
+{"_id": "25733", "title": "Definition:Directed Graph/Arc/Endvertex/Final Vertex", "text": "The '''final vertex''' of $e$ is the endvertex $v$ which $e$ is incident to."}
+{"_id": "25734", "title": "Definition:Directed Graph/Arc/Endvertex/Initial Vertex", "text": "The '''initial vertex''' of $e$ is the endvertex $u$ which $e$ is incident from."}
+{"_id": "25735", "title": "Definition:Integer/Formal Definition/Notation", "text": "We have that $\\eqclass {\\tuple {a, b} } \\boxminus$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\\boxminus$. As this notation is cumbersome, it is commonplace though technically incorrect to streamline it to $\\eqclass {a, b} \\boxminus$, or $\\eqclass {a, b} {}$. This is generally considered acceptable, as long as it is made explicit as to the precise meaning of $\\eqclass {a, b} {}$ at the start of any exposition."}
+{"_id": "25736", "title": "Definition:Injection/Definition 1 a", "text": ":$\\forall x_1, x_2 \\in \\Dom f: x_1 \\ne x_2 \\implies \\map f {x_1} \\ne \\map f {x_2}$"}
+{"_id": "25737", "title": "Definition:Injection/Definition 1", "text": "A mapping $f$ is '''an injection''', or '''injective''' {{iff}}: :$\\forall x_1, x_2 \\in \\Dom f: \\map f {x_1} = \\map f {x_2} \\implies x_1 = x_2$ That is, an '''injection''' is a mapping such that the output uniquely determines its input."}
+{"_id": "25738", "title": "Definition:Injection/Definition 2", "text": "An '''injection''' is a relation which is both one-to-one and left-total."}
+{"_id": "25739", "title": "Definition:Injection/Definition 3", "text": "Let $f$ be a mapping. Then $f$ is an injection {{iff}}: :$f^{-1} {\\restriction_{\\Img f} }: \\Img f \\to \\Dom f$ is a mapping where $f^{-1} {\\restriction_{\\Img f} }$ is the restriction of the inverse of $f$ to the image set of $f$."}
+{"_id": "25740", "title": "Definition:Injection/Definition 4", "text": "Let $f$ be a mapping. $f$ is an '''injection''' {{iff}}: :$\\forall y \\in \\Img f: \\card {\\map {f^{-1} } y} = \\card {\\set {f^{-1} \\sqbrk {\\set y} } } = 1$ where: :$\\Img f$ denotes the image set of $f$ :$\\card {\\, \\cdot \\,}$ denotes the cardinality of a set :$\\map {f^{-1} } y$ is the preimage of $y$ :$f^{-1} \\sqbrk {\\set y}$ is the preimage of the subset $\\set y \\subseteq \\Img f$."}
+{"_id": "25741", "title": "Definition:Injection/Definition 5", "text": "Let $f: S \\to T$ be a mapping where $S \\ne \\O$. Then $f$ is an injection {{iff}}: :$\\exists g: T \\to S: g \\circ f = I_S$ where $g$ is a mapping. That is, {{iff}} $f$ has a left inverse."}
+{"_id": "25742", "title": "Definition:Injection/Definition 6", "text": "Let $f: S \\to T$ be a mapping where $S \\ne \\O$. Then $f$ is an '''injection''' {{iff}} $f$ is left cancellable: :$\\forall X: \\forall g_1, g_2: X \\to S: f \\circ g_1 = f \\circ g_2 \\implies g_1 = g_2$ where $g_1$ and $g_2$ are arbitrary mappings from an arbitrary set $X$ to the domain $S$ of $f$."}
+{"_id": "25744", "title": "Definition:Associative Operation", "text": "Let $S$ be a set. Let $\\circ : S \\times S \\to S$ be a binary operation. Then $\\circ$ is '''associative''' {{iff}}: :$\\forall x, y, z \\in S: \\paren {x \\circ y} \\circ z = x \\circ \\paren {y \\circ z}$"}
+{"_id": "25745", "title": "Definition:Set/Uniqueness of Elements", "text": "A set is '''uniquely determined''' by its elements. This means that the only thing that defines '''what a set is''' is '''what it contains'''. So, how you choose to '''list''' or '''define''' the contents makes '''no difference''' to what the contents actually '''are'''."}
+{"_id": "25746", "title": "Definition:Tally Marks", "text": "'''Tally marks''' are a system of recording (usually) small numbers. For each unit in a given natural number, a mark is made. Thus, for example, $13$ could be recorded $|||||||||||||$. It is usual for '''tally marks''' to be arranged in groups of $5$, by drawing every fifth line diagonally across the group of $4$ before it. Thus $13$ would then be recorded: :File:Tally5.pngFile:Tally5.pngFile:Tally3.png"}
+{"_id": "25748", "title": "Definition:Lebombo Bone", "text": "The '''Lebombo bone''' is a baboon's leg bone bearing $29$ notches which were carved into it between $33 \\ 000$ B.C.E. and $42 \\ 000$ B.C.E. It is the earliest known instance of a system of tally marks. It was found in {{WP|Border_Cave|Border Cave}} in {{WP|South_Africa|South Africa}} in the {{WP|Lebombo_Mountains|Lebombo mountains}}, near the border with {{WP|Swaziland|Swaziland}}. Various untestable hypotheses have been raised concerning this artefact, notably that it was created by an early African woman to count the days between her menses. However, as the bone appears to be broken at one end, it is possible that the count of $29$ is not the complete set of notches that were originally carved."}
+{"_id": "25749", "title": "Definition:Wolf Bone", "text": "The '''(Czechoslovakian) wolf bone''' is a wolf bone dating from approximately $30 \\ 000$ years ago, bearing $57$ marks which appear to have been made deliberately. It was discovered in $1937$ in {{WP|Czechoslovakia|Czechoslovakia}} at {{WP|Doln%C3%AD_V%C4%9Bstonice_(archaeology)|Dolní Věstonice}} in {{WP|Moravia|Moravia}}, by a team led by {{WP|Karel_Absolon|Karl Absolon}}. The markings are believed to be tally marks, and are arranged in $11$ groups of $5$, with $2$ left over. The untestable hypothesis has been made that the number corresponds to a double lunar period."}
+{"_id": "25750", "title": "Definition:Ishango Bone", "text": "The '''Ishango bone''' is a mathematical tool made from a leg bone of a baboon, dating from approximately $25 \\ 000$ years ago. It bears a system of markings which may be tally marks, but may in fact be for a more complex purpose. It was found in $1960$ by {{WP|Jean_de_Heinzelin_de_Braucourt|Jean de Heinzelin de Braucourt}} in the {{WP|Belgian_Congo|Belgian Congo}} (now the {{WP|Democratic_Republic_of_the_Congo|Democratic Republic of the Congo}}). It was discovered in the area of {{WP|Ishango|Ishango}} near the {{WP|Semliki_River|Semliki River}}. One row of marks contains the prime numbers between $10$ and $20$, that is, $11$, $13$, $17$ and $19$, which total $60$. Another row contains $9$, $11$, $19$ and $21$, which also total $60$. The third row contains marks which appear to represent a method of multiplication by means of a technique involving doubling and halving. While it is suggested that the markings form a calendar, it is also possible that the markings are coincidental, and perhaps even just put there to make the bone easier to hold."}
+{"_id": "25752", "title": "Definition:Cartesian Product/Cartesian Space/Family of Sets", "text": "Let $I$ be an indexing set. Let $\\family {S_i}_{i \\mathop \\in I}$ be an family of sets indexed by $I$. Let $\\displaystyle \\prod_{i \\mathop \\in I} S_i$ be the Cartesian product of $\\family {S_i}_{i \\mathop \\in I}$. Let $S$ be a set such that: :$\\forall i \\in I: S_i = S$ === Definition 1 === {{:Definition:Cartesian Product/Cartesian Space/Family of Sets/Definition 1}} === Definition 2 === {{:Definition:Cartesian Product/Cartesian Space/Family of Sets/Definition 2}}"}
+{"_id": "25753", "title": "Definition:Ordered Tuple/Definition 1", "text": "An '''ordered tuple (of length $n$)''' is a finite sequence whose domain is $\\N^*_n$."}
+{"_id": "25754", "title": "Definition:Ordered Tuple/Also known as", "text": "Some sources refer to an ordered tuple as a '''tuple'''. The term '''ordered $n$-tuple''' can sometimes be seen, particularly for specific instances of $n$.  Instead of writing '''2-tuple''', '''3-tuple''' and '''4-tuple''', the terms '''couple''', '''triple''' and '''quadruple''' are usually used. In the context of abstract algebra, the concept is encountered as '''(associative) word'''."}
+{"_id": "25755", "title": "Definition:Ordered Tuple/Notation", "text": "Notation for an ordered tuple varies throughout the literature. There are also specialised instances of an ordered tuple where the convention is to use angle brackets. However, it is common for an ordered tuple to be denoted: :$\\tuple {a_1, a_2, \\ldots, a_n}$ extending the notation for an ordered pair. For example: $\\tuple {6, 3, 3}$ is the ordered triple $f$ defined as: :$\\map f 1 = 6, \\map f 2 = 3, \\map f 3 = 3$ The notation: :$\\sequence {a_1, a_2, \\ldots, a_n}$ is recommended when use of round brackets would be ambiguous. Other notations which may be encountered are: :$\\sqbrk {a_1, a_2, \\ldots, a_n}$ :$\\set {a_1, a_2, \\ldots, a_n}$ but both of these are strongly discouraged: the square bracket format because there are rendering problems on this site, the latter because it is too easily confused with set notation. In order to further streamline notation, it is common to use the more compact $\\sequence {a_n}$ for $\\sequence {a_k}_{1 \\mathop \\le k \\mathop \\le n}$. Some sources, particularly in such fields as communication theory, where the elements of the domain of the ordered tuple is a specific set of symbols, use the notation $x_1 x_2 \\cdots x_n$ for $\\tuple {x_1, x_2, \\dotsc, x_n}$."}
+{"_id": "25756", "title": "Definition:Ordered Tuple/Definition 2", "text": "Let $\\family {S_i}_{i \\mathop \\in \\N_n}$ be a family of sets indexed by $\\N_n$. Let $\\displaystyle \\prod_{i \\mathop \\in \\N_n} S_i$ be the Cartesian product of $\\family {S_i}_{i \\mathop \\in \\N_n}$. An '''ordered tuple of length $n$''' of $\\family {S_i}$ is an element of $\\displaystyle \\prod_{i \\mathop \\in \\N_n} S_i$."}
+{"_id": "25757", "title": "Definition:Direct Image Mapping/Mapping", "text": "Let $f \\subseteq S \\times T$ be a mapping from $S$ to $T$. The '''direct image mapping''' of $f$ is the mapping $f^\\to: \\powerset S \\to \\powerset T$ that sends a subset $X \\subseteq S$ to its image under $f$: :$\\forall X \\in \\powerset S: \\map {f^\\to} X = \\begin {cases} \\set {t \\in T: \\exists s \\in X: \\map f s = t} & : X \\ne \\O \\\\ \\O & : X = \\O \\end {cases}$"}
+{"_id": "25758", "title": "Definition:Direct Image Mapping/Relation", "text": "Let $\\RR \\subseteq S \\times T$ be a relation on $S \\times T$. The '''direct image mapping''' of $\\RR$ is the mapping $\\RR^\\to: \\powerset S \\to \\powerset T$ that sends a subset $X \\subseteq T$ to its image under $\\RR$: :$\\forall X \\in \\powerset S: \\map {\\RR^\\to} X = \\begin {cases} \\set {t \\in T: \\exists s \\in X: \\tuple {s, t} \\in \\RR} & : X \\ne \\O \\\\ \\O & : X = \\O \\end {cases}$"}
+{"_id": "25759", "title": "Definition:Inverse Image Mapping", "text": "Let $S$ and $T$ be sets. Let $\\powerset S$ and $\\powerset T$ be their power sets. === Relation === Let $\\mathcal R \\subseteq S \\times T$ be a relation on $S \\times T$. {{Definition:Inverse Image Mapping/Relation/Definition 1}} === Mapping === Let $f: S \\to T$ be a mapping. {{Definition:Inverse Image Mapping/Mapping/Definition 1}}"}
+{"_id": "25760", "title": "Definition:Inverse Image Mapping/Relation", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation on $S \\times T$. === Definition 1 === {{:Definition:Inverse Image Mapping/Relation/Definition 1}} === Definition 2 === {{:Definition:Inverse Image Mapping/Relation/Definition 2}}"}
+{"_id": "25761", "title": "Definition:Inverse Image Mapping/Mapping", "text": "Let $f: S \\to T$ be a mapping. === Definition 1 === {{:Definition:Inverse Image Mapping/Mapping/Definition 1}} === Definition 2 === {{:Definition:Inverse Image Mapping/Mapping/Definition 2}}"}
+{"_id": "25762", "title": "Definition:Equivalence Relation/Also denoted as", "text": "When discussing equivalence relations, various notations are used for $\\tuple {x, y} \\in \\RR$. Examples are: :$x \\mathrel \\RR y$ :$x \\equiv \\map y \\RR$ :$x \\equiv y \\pmod \\RR$ and so on. Specialised equivalence relations generally have their own symbols, which can be defined as they are needed. Such symbols include: :$\\cong$, $\\equiv$, $\\sim$, $\\simeq$, $\\approx$"}
+{"_id": "25763", "title": "Definition:Equivalence Relation/Definition 2", "text": "$\\RR$ is an equivalence relation {{iff}}: :$\\Delta_S \\cup \\RR^{-1} \\cup \\RR \\circ \\RR \\subseteq \\RR$ where: :$\\Delta_S$ denotes the diagonal relation on $S$ :$\\RR^{-1}$ denotes the inverse relation :$\\circ$ denotes composition of relations"}
+{"_id": "25764", "title": "Definition:Equivalence Relation/Definition 1", "text": "Let $\\RR$ be: :$(1): \\quad$ reflexive :$(2): \\quad$ symmetric :$(3): \\quad$ transitive Then $\\RR$ is an '''equivalence relation''' on $S$."}
+{"_id": "25765", "title": "Definition:Minimal/Ordered Set/Definition 1", "text": "An element $x \\in T$ is a '''minimal element of $T$''' {{iff}}: :$\\forall y \\in T: y \\preceq x \\implies x = y$ That is, the only element of $T$ that $x$ '''succeeds or is equal to''' is itself."}
+{"_id": "25766", "title": "Definition:Minimal/Ordered Set/Definition 2", "text": "An element $x \\in T$ is a '''minimal element of $T$''' {{iff}}: :$\\neg \\exists y \\in T: y \\prec x$ where $y \\prec x$ denotes that $y \\preceq x \\land y \\ne x$. That is, {{iff}} $x$ has '''no strict predecessor'''."}
+{"_id": "25767", "title": "Definition:Minimal/Ordered Set/Comparison with Smallest Element", "text": "Compare the definition of minimal element with that of a smallest element. An element $x \\in T$ is '''the''' smallest element of $T$ {{iff}}: :$\\forall y \\in T: x \\preceq y$ That is, $x$ is comparable to, and precedes, or is equal to, every $y \\in T$. An element $x \\in T$ is '''a''' minimal element of $T$ {{iff}}: :$y \\preceq x \\implies x = y$ That is, $x$ precedes, or is equal to, every $y \\in T$ which ''is'' comparable to $x$. If ''all'' elements are comparable to $x$, then such a minimal element is indeed '''the smallest element'''. Note that when an ordered set is in fact a totally ordered set, the terms '''minimal element''' and '''smallest element''' are equivalent."}
+{"_id": "25768", "title": "Definition:Maximal/Ordered Set/Definition 1", "text": "An element $x \\in T$ is a '''maximal element of $T$''' {{iff}}: :$x \\preceq y \\implies x = y$ That is, the only element of $S$ that $x$ '''precedes or is equal to''' is itself."}
+{"_id": "25769", "title": "Definition:Maximal/Ordered Set/Definition 2", "text": "An element $x \\in T$ is a '''maximal element of $T$''' {{iff}}: :$\\neg \\exists y \\in T: x \\prec y$ where $x \\prec y$ denotes that $x \\preceq y \\land x \\ne y$. That is, {{iff}} $x$ has '''no strict successor'''."}
+{"_id": "25770", "title": "Definition:Maximal/Ordered Set/Comparison with Greatest Element", "text": "Compare the definition of maximal element with that of a greatest element. An element $x \\in T$ is '''the''' greatest element of $T$ {{iff}}: :$\\forall y \\in T: y \\preceq x$ That is, $x$ is comparable to, and succeeds, or is equal to, every $y \\in S$. An element $x \\in S$ is '''a''' maximal element of $T$ {{iff}}: :$x \\preceq y \\implies x = y$ That is, $x$ succeeds, or is equal to, every $y \\in S$ which ''is'' comparable to $x$. If ''all'' elements are comparable to $x$, then such a maximal element is indeed '''the greatest element'''. Note that when an ordered set is in fact a totally ordered set, the terms '''maximal element''' and '''greatest element''' are equivalent."}
+{"_id": "25771", "title": "Definition:Choice Function/Use of Axiom of Choice", "text": "The Axiom of Choice (abbreviated '''AoC''' or '''AC''') is the following statement: :''All $\\mathbb S$ as above have a choice function.'' It can be shown that the AoC it does not follow from the other usual axioms of set theory, and that it is relative consistent to these axioms (i.e., that AoC does not make the axiom system inconsistent, provided it was consistent without AoC). Note that for any given set $S \\in \\mathbb S$, one can select an element from it (without using AoC). AoC guarantees that there is a choice function, i.e., a function that \"simultaneously\" picks elements of all $S \\in \\mathbb S$.   AoC is needed to prove statements such as \"all countable unions of finite sets are countable\" (for many specific such unions this  can be shown without AoC), and AoC is equivalent to many other mathematical statements such as \"every vector space has a basis\"."}
+{"_id": "25772", "title": "Definition:Bessel Function/First Kind", "text": "A '''Bessel function of the first kind of order $n$''' is a Bessel function which is non-singular at the origin It is usually denoted $\\map {J_n} x$, where $x$ is the dependent variable of the instance of '''Bessel's equation''' to which $\\map {J_n} x$ forms a solution."}
+{"_id": "25773", "title": "Definition:Bessel Function/Second Kind", "text": "A '''Bessel function of the second kind of order $n$''' is a Bessel function which is singular at the origin. It is usually denoted $\\map {Y_n} x$, where $x$ is the dependent variable of the instance of '''Bessel's equation''' to which $\\map {Y_n} x$ forms a solution."}
+{"_id": "25774", "title": "Definition:Pythagoreans", "text": "The '''Pythagoreans''' were a semi-mystical cult which dated from around $550$ B.C.E., founded by {{AuthorRef|Pythagoras of Samos}}. Can claim to be the world's first university. It is feasible to suggest that their initial work in the field of geometry may have formed the basis of at least the first two books of {{ElementsLink}}. Attendees of the school were divided into two classes: :the '''Probationers''' (or '''listeners''') :the '''Pythagoreans'''. A student was a '''listener''' for $3$ years, after which he was allowed to be initiated into the class of '''Pythagoreans''', who were allowed to learn what was considered to be the deeper secrets. '''Pythagoreans''' were a closely-knit brotherhood, who held all their worldly goods in common, and were bound by oath never to reveal the secrets of the Founder. There exists a legend that one of the '''Pythagoreans''' was thrown overboard to drown after having revealed the existence of the regular dodecahedron. For some considerable time they dominated the political life of {{WP|Crotone|Croton}}, where they were based, but in $501$ B.C.E. there was a popular revolt in which a number of the leaders of the school were murdered. {{AuthorRef|Pythagoras of Samos|Pythagoras}} himself was murdered soon after. Some sources state that the reasons for this were based on the fact that their puritanical philosophy was at odds with the contemporary way of thinking. Others suggest that there was a reaction against their autocratic rule. Whatever the truth of the matter, between $501$ and about $460$ B.C.E. the political influence of the cult was destroyed. Its survivors scattered, many of them fleeing to {{WP|Thebes,_Egypt|Thebes}} in {{WP|Upper_Egypt|Upper Egypt}}, where they continued to exist more as a philosophical and mathematical society for another couple of centuries, secretive and ascetic to the end, publishing nothing, ascribing all their discoveries to the Master, {{AuthorRef|Pythagoras of Samos|Pythagoras}} himself. === Quadrivium === {{:Definition:Pythagoreans/Quadrivium}} === Trivium === {{:Definition:Pythagoreans/Trivium}}"}
+{"_id": "25775", "title": "Definition:Method of Exhaustion", "text": "The '''method of exhaustion''' is a technique for calculating an approximation to the physical extent of a geometric figure whose extremities are curved lines. To measure the area of a circle, for example, this is done by: : Circumscribing a polygon around the circle : Inscribing a polygon inside the circle : Measuring the areas of those polygons : Noting that the area of the circle is between those two : Increasing the number of sides on those polygons to make them closer and closer to the circle and each other. As the number of sides of the polygons increases, their areas become ever closer to each other."}
+{"_id": "25776", "title": "Definition:Markov Chain", "text": "Let $\\sequence {X_n}_{n \\mathop \\ge 0}$ be a sequence of random variables in a countable set $S$. Let $\\map \\Pr X$ denote the probability of the random variable $X$. Let $\\sequence {X_n}_{n \\mathop \\ge 0}$ satisfy the Markov property: :$\\map \\Pr {X_{n + 1} = i_{n + 1} \\mid X_0 = i_0, X_1 = i_1, \\ldots, X_n = i_n} = \\map \\Pr {X_{n + 1} = i_{n + 1} \\mid X_n = i_n}$ for all $n \\ge 0$ and all $i_0, i_1, \\ldots, i_{n + 1} \\in S$. Then $\\sequence {X_n}_{n \\mathop \\ge 0}$ is a '''Markov chain'''."}
+{"_id": "25777", "title": "Definition:Markov Chain/State Space", "text": "The set $S$ is called the '''state space''' of the Markov chain."}
+{"_id": "25779", "title": "Definition:Tool", "text": "A '''tool''' is a physical device used as an aid to a process."}
+{"_id": "25780", "title": "Definition:Straightedge", "text": "A '''straightedge''' is an ideal tool for constructing straight lines. A '''straightedge''' is of unlimited length, but has no markings on it, so it cannot be used for measurement."}
+{"_id": "25781", "title": "Definition:Compass", "text": "A '''compass''' is an ideal tool for drawing circles. Hence it can be used according to Euclid's third postulate to construct a circle using any two given points: :$(1): \\quad$ the center and: :$(2): \\quad$ an arbitrary point on the circumference."}
+{"_id": "25782", "title": "Definition:Compass and Straightedge Construction", "text": "A '''compass and straightedge construction''' is a technique of drawing geometric figures using only a straightedge and a compass. The operations available are: :using the straightedge to draw a straight line determined by two given points :using the compass to draw a circle whose center is at a given point and whose radius is the distance between two given points :finding the points of intersection between straight lines and circles."}
+{"_id": "25783", "title": "Definition:Pólya's Urn", "text": "'''Pólya's urn''' is an implementation of an urn such that, when a ball of a particular colour is taken out and examined, that ball is replaced and another of the same colour is added. {{NamedforDef|George Pólya|cat = Pólya}} Category:Definitions/Probability Theory ncbllm09u1lm8ewv2p4t2zx98p91h4n"}
+{"_id": "25784", "title": "Definition:Trisection", "text": "To '''trisect''' a finite geometrical object is to cut it into $3$ equal parts."}
+{"_id": "25785", "title": "Definition:Neusis Ruler", "text": "A '''neusis ruler''' is a straightedge on which two points have been marked with a specific fixed length between them."}
+{"_id": "25786", "title": "Definition:Neusis Construction", "text": "A '''neusis construction''' is a technique of drawing geometric figures using only a neusis ruler and a compass."}
+{"_id": "25787", "title": "Definition:Neusis Ruler/Diastema", "text": "On a neusis ruler, the '''diastema''' is the specific fixed length between the two marked points."}
+{"_id": "25788", "title": "Definition:Line/Curve/Plane", "text": "A '''plane curve''' is a curve which can be embedded in the plane."}
+{"_id": "25789", "title": "Definition:Conic Section/Focus-Directrix Property", "text": "A '''conic section''' is a plane curve which can be specified in terms of: :a given straight line $D$ known as the directrix :a given point $F$ known as a focus :a given constant $\\epsilon$ known as the eccentricity. Let $K$ be the locus of points $b$ such that the distance $p$ from $b$ to $D$ and the distance $q$ from $b$ to $F$ are related by the condition: :$(1): \\quad q = \\epsilon \\, p$ Then $K$ is a '''conic section'''. Equation $(1)$ is known as the '''focus-directrix property''' of $K$."}
+{"_id": "25790", "title": "Definition:Conic Section/Intersection with Cone", "text": ":600px Let $C$ be a double napped right circular cone whose base is $B$. Let $\\theta$ be half the opening angle of $C$. That is, let $\\theta$ be the angle between the axis of $C$ and a generatrix of $C$. Let a plane $D$ intersect $C$. Let $\\phi$ be the inclination of $D$ to the axis of $C$. Let $K$ be the set of points which forms the intersection of $C$ with $D$. Then $K$ is a '''conic section''', whose nature depends on $\\phi$."}
+{"_id": "25791", "title": "Definition:Parabola/Focus-Directrix", "text": ":300px Let $D$ be a straight line. Let $F$ be a point. Let $K$ be the locus of points $P$ such that the distance $p$ from $P$ to $D$ equals the distance $q$ from $P$ to $F$: :$p = q$ Then $K$ is a '''parabola'''."}
+{"_id": "25792", "title": "Definition:Hyperbola/Focus-Directrix", "text": ":300px Let $D$ be a straight line. Let $F_1$ be a point. Let $\\epsilon \\in \\R: \\epsilon > 1$. Let $K$ be the locus of points $P$ such that the distance $p$ from $P$ to $D$ and the distance $q$ from $P$ to $F_1$ are related by the condition: :$\\epsilon \\, p = q$ Then $K$ is a '''hyperbola'''."}
+{"_id": "25793", "title": "Definition:Conic Section/Directrix", "text": "The line $D$ is known as the '''directrix''' of the conic section."}
+{"_id": "25794", "title": "Definition:Conic Section/Focus", "text": "The point $F$ is known as the '''focus''' of the conic section."}
+{"_id": "25795", "title": "Definition:Conic Section/Eccentricity", "text": "The constant $\\epsilon$ is known as the '''eccentricity''' of the conic section."}
+{"_id": "25796", "title": "Definition:Ellipse/Focus", "text": "The point $F$ is known as the '''focus''' of the ellipse."}
+{"_id": "25797", "title": "Definition:Ellipse/Directrix", "text": "The line $D$ is known as the '''directrix''' of the ellipse."}
+{"_id": "25798", "title": "Definition:Ellipse/Eccentricity", "text": "The constant $\\epsilon$ is known as the '''eccentricity''' of the ellipse."}
+{"_id": "25799", "title": "Definition:Hyperbola/Focus", "text": "The point $F_1$ is known as a '''focus''' of the hyperbola. The symmetrically-positioned point $F_2$ is also a '''focus''' of the hyperbola."}
+{"_id": "25800", "title": "Definition:Hyperbola/Directrix", "text": "The line $D$ is known as the '''directrix''' of the hyperbola."}
+{"_id": "25801", "title": "Definition:Hyperbola/Eccentricity", "text": "The constant $\\epsilon$ is known as the '''eccentricity''' of the hyperbola."}
+{"_id": "25802", "title": "Definition:Parabola/Focus", "text": "The point $F$ is known as the '''focus''' of the parabola."}
+{"_id": "25803", "title": "Definition:Parabola/Directrix", "text": "The line $D$ is known as the '''directrix''' of the parabola."}
+{"_id": "25805", "title": "Definition:Greek Numerals", "text": "=== Attic System === {{:Definition:Greek Numerals/Attic System}} === Classical Period === {{:Definition:Greek Numerals/Classical Period}}"}
+{"_id": "25806", "title": "Definition:Greek Numerals/Attic System", "text": "The '''Greek numerals''' from the Attic period (c. 500 BCE to c. 300 BCE) are similar to the Roman system: {{begin-eqn}} {{eqn | l = \\textsf{I}:       | o =        | r = \\)one\\( }} {{eqn | l = \\Pi:       | o = \\)'''penta'''\\(:       | r = \\)five\\( }} {{eqn | l = \\Delta:       | o = \\)'''deka'''\\(:       | r = \\)ten\\( }} {{eqn | l = \\Eta:       | o = \\)'''hekaton'''\\(:       | r = \\)one hundred\\( }} {{eqn | l = \\Xi:       | o = \\)'''chilioi'''\\(:       | r = \\)one thousand\\( }} {{eqn | l = \\Mu:       | o = \\)'''myriad'''\\(:       | r = \\)ten thousand\\( }} {{end-eqn}} $\\Pi$ was later written in a different form, with a shorter right leg, and from there it appears to have evolved into $\\Gamma$. Numbers were combined in a similar way to Roman numbers, but with additive forms only. For example, $32718$ would have been written $\\Mu \\Mu \\Mu \\Xi \\Xi \\Eta \\Eta \\Eta \\Eta \\Eta \\Eta \\Eta \\Delta \\Pi \\textsf {III}$. Later evolutions introduced symbols for $50$, $500$ and $5000$, consisting of a tiny version of the appropriate power of $10$ nestled under the right hand (short) branch of the short-leg version of $\\Pi$. Such characters are difficult to render neatly, and as they are of limited importance, this has not been attempted on {{ProofWiki}}."}
+{"_id": "25807", "title": "Definition:Greek Numerals/Classical Period", "text": "The '''Greek numerals''' from the Classical period (c. $600$ BCE to c. $300$ BCE) are as follows: :$\\begin{array}{ccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\ \\alpha & \\beta & \\gamma & \\delta & \\epsilon & stigma & \\zeta & \\eta & \\theta \\\\ \\hline \\\\ 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 \\\\ \\iota & \\kappa & \\lambda & \\mu & \\nu & \\xi & \\omicron & \\pi & koppa \\\\ \\hline \\\\ 100 & 200 & 300 & 400 & 500 & 600 & 700 & 800 & 900 \\\\ \\rho & \\sigma & \\tau & \\upsilon & \\phi & \\chi & \\psi & \\omega & sampi \\\\ \\end{array}$ where $stigma$, $koppa$ and $sampi$ are characters that are not supported by MathJax. These were the letters of the everyday Greek alphabet, with extra ones added from the Phoenician alphabet. As it could be easy to confuse numbers with the letters of Greek words, a line was written over the numbers to distinguish them. Numbers bigger than $999$ could be written by putting a stroke in front of the symbols."}
+{"_id": "25808", "title": "Definition:Rational Number/Fraction/Denominator", "text": "The term $b$ is known as the '''denominator''' of $\\dfrac a b$."}
+{"_id": "25809", "title": "Definition:Rational Number/Fraction/Numerator", "text": "The term $a$ is known as the '''numerator''' of $\\dfrac a b$."}
+{"_id": "25812", "title": "Definition:Trigonometry", "text": "'''Trigonometry''' is the branch of mathematics which studies the relationships between the lengths of sides and angles of triangles."}
+{"_id": "25813", "title": "Definition:Spherical Geometry", "text": "'''Spherical geometry''' is the branch of mathematics which concerns spatial relationships on the surface of a sphere."}
+{"_id": "25814", "title": "Definition:Spherical Trigonometry", "text": "'''Spherical trigonometry''' is the branch of mathematics which concerns the measurement of spherical triangles and related figures on the surface of a sphere."}
+{"_id": "25815", "title": "Definition:Napier's Bones", "text": "rightNapier's Rods '''Napier's bones''' is a device for performing the operation of multiplication of numbers. The device consists of: : a set of '''rods''', one for each digit $0$ to $9$, containing the multiples of that number, again from $0$ to $9$, in a column, written as $2$ digits, divided by a diagonal line. : a '''board''' upon which the '''rods''' can be placed, labeled $1$ to $9$ down a column on the left hand side, each digit corresponding to the numbers on the '''rods''' The multiplicand is assembled by placing the rods for each of its digit next to each other on the '''board'''. The multiplier corresponds to the number down the left hand side of the '''board''' For each digit in the multiplicand, the numbers adjacent to each other on the row corresponding to the multiplier in each of the compartments formed by the diagonal lines aligned together are added together. The product is thus formed."}
+{"_id": "25816", "title": "Definition:Prosthaphaeresis", "text": "'''Prosthaphaeresis''' is the process whereby multiplication and division are performed by: :$(1): \\quad$ converting the numbers into a form where they can be added or subtracted :$(2): \\quad$ performing the addition or subtraction :$(3): \\quad$ converting the result back again, using the inverse process to step $(1)$."}
+{"_id": "25817", "title": "Definition:Prosthaphaeresis/Trigonometry", "text": "The trigonometric technique of prosthaphaeresis is based on the Prosthaphaeresis Formula for Sine plus Sine: :$\\dfrac {\\sin \\alpha + \\sin \\beta} 2 = \\map \\sin {\\dfrac {\\alpha + \\beta} 2} \\, \\map \\cos {\\dfrac {\\alpha - \\beta} 2}$"}
+{"_id": "25818", "title": "Definition:Napierian Logarithm", "text": "The '''Napierian logarithms''' are a system of logarithms where: :$\\log_{\\text {Nap} } 10 \\, 000 \\, 000 = 0$ :$\\log_{\\text {Nap} } 9 \\, 999 \\, 999 = 1$ Thus: :$\\log_{\\text {Nap} } 10^7 x y = \\log_{\\text {Nap} } x + \\log_{\\text{Nap} } y$"}
+{"_id": "25819", "title": "Definition:Prosthaphaeresis/Logarithms", "text": "The logarithmic technique of prosthaphaeresis is based on the rule of the Sum of Logarithms: :$\\log_b x y = \\log_b x + \\log_b y$ Any base $b$ can be used. The original tecnhique used Napierian logarithms where the base was $0.9999999$ In due course the common (base $10$) logarithm was used, as developed by {{AuthorRef|Henry Briggs}}, called Briggsian logarithms."}
+{"_id": "25820", "title": "Definition:Epicycle", "text": "An '''epicycle''' is the path described by an object moving in a uniform circular motion around a point which is itself moving in a uniform circular motion around another point. :400px That point may itself also be moving in a uniform circular motion around yet another point."}
+{"_id": "25821", "title": "Definition:Oblique Axes", "text": "'''Oblique axes''' are a pair of coordinate axes which are not at right angles to each other. :400px"}
+{"_id": "25822", "title": "Definition:Equation of Geometric Figure", "text": "Let there be a coordinate system. Let the variables $\\alpha, \\beta, \\gamma, \\ldots$ be used to identify points within that coordinate system. Let the variables be related to each other by means of an equation. Then the equations define the set of points which form a geometric figure. This equation is known as the '''equation of the geometric figure'''."}
+{"_id": "25823", "title": "Definition:Tensor Product of Modules", "text": "=== Commutative ring === Let $R$ be a commutative ring with unity. Let $M$ and $N$ be $R$-modules. === Definition 1 === Their '''tensor product''' is a pair $(M \\otimes_R N, \\theta)$ where: :$M \\otimes_R N$ is an $R$-module :$\\theta : M \\times N \\to M \\otimes_R N$ is an $R$-bilinear mapping satisfying the following '''universal property''': :For every pair $(P, \\omega)$ of an $R$-module and an $R$-bilinear mapping $\\omega : M \\times N \\to P$, there exists a unique $R$-module homomorphism $f : M \\otimes_R N \\to P$ with $\\omega = f \\circ \\theta$. === Definition 2 === Their '''tensor product''' is the pair $\\struct {M \\otimes_R N, \\theta}$, where: :$M \\otimes_R N$ is the quotient of the free $R$-module $R^{\\paren {M \\times N} }$ on the direct product $M \\times N$, by the submodule generated by the set of elements of the form: ::$\\tuple {\\lambda m_1 + m_2, n} - \\lambda \\tuple {m_1, n} - \\tuple {m_2, n}$ ::$\\tuple {m, \\lambda n_1 + n_2} - \\lambda \\tuple {m, n_1} - \\tuple {m, n_2}$ ::for $m, m_1, m_2 \\in M$, $n, n_1, n_2 \\in N$ and $\\lambda \\in R$, where we denote $tuple {m, n}$ for its image under the canonical mapping $M \\times N \\to R^{\\paren {M \\times N} }$. :$\\theta : M \\times N \\to M \\otimes_R N$ is the composition of the canonical mapping $M \\times N \\to R^{\\paren {M \\times N} }$ with the quotient module homomorphism $R^{\\paren {M \\times N} } \\to M \\otimes_R N$. === Noncommutative ring === Let $R$ be a ring. Let $M$ be a $R$-right module. Let $N$ be a $R$-left module. First construct a left module as a direct sum of all free left modules with a basis that is a single ordered pair in $M \\times N$ which is denoted $\\map R {m, n}$. :$T = \\displaystyle \\bigoplus_{s \\mathop \\in M \\mathop \\times N} R s$ That this is indeed a module is demonstrated in Tensor Product is Module. Next for all $m, m' \\in M$, $n, n' \\in N$ and $r \\in R$ we construct the following free left modules. :$L_{m, m', n}$ with a basis of $\\tuple {m + m', n}$, $\\tuple {m, n}$ and $\\tuple {m', n}$ :$R_{m, n, n'}$ with a basis of $\\tuple {m, n + n'}$, $\\tuple {m, n}$ and $\\tuple {m, n'}$ :$A_{r, m, n}$ with a basis of $r \\tuple {m, n}$ and $\\tuple {m r, n}$ :$B_{r, m, n}$ with a basis of $r \\tuple {m, n}$ and $\\tuple {m, r n}$ Let: :$D = \\displaystyle \\map {\\bigoplus_{r \\in R, n, n' \\in N, m, m' \\in M} } {L_{m, m', n} \\oplus R_{m, n, n'} \\oplus A_{r, m, n} \\oplus B_{r, m, n} }$ The '''tensor product''' $M \\otimes_R N$ is then our quotient module $T / D$."}
+{"_id": "25824", "title": "Definition:Folium of Descartes", "text": "=== Cartesian Form === {{:Definition:Folium of Descartes/Cartesian Form}} === Parametric Form === {{:Definition:Folium of Descartes/Parametric Form}} :500px"}
+{"_id": "25825", "title": "Definition:Archimedean Spiral", "text": "The '''Archimedean spiral''' is the locus of the equation expressed in Polar coordinates as: :$r = a \\theta$ :600px"}
+{"_id": "25826", "title": "Definition:Number Theory", "text": "'''Number theory''' is the branch of mathematics which studies the properties of the natural numbers."}
+{"_id": "25827", "title": "Definition:Algebraic Number Theory", "text": "'''Algebraic number theory''' is the branch of abstract algebra which studies structures in which the usual number fields are embedded. As such it can also be considered to be a branch of number theory."}
+{"_id": "25828", "title": "Definition:Topology (Mathematical Branch)", "text": "'''Topology''' is a geometry of transformations in which the only invariant is continuity. Some sources suggest that it can indeed be described simply as '''the study of continuity'''. As such, it is closely interwoven with the branch of '''analysis'''."}
+{"_id": "25829", "title": "Definition:Plane Geometry", "text": "'''Plane geometry''' is the study of geometric figures in two dimensions."}
+{"_id": "25830", "title": "Definition:Computer Science", "text": "'''Computer science''' is the branch of mathematics studying questions arising from the operation of [https://en.wikipedia.org/wiki/Computer digital computers]. As such it can be considered as an offshoot of both mathematical logic and discrete mathematics."}
+{"_id": "25831", "title": "Definition:Computability Theory", "text": "'''Computability theory''' is a branch of mathematical logic which concerns itself with the algorithmic implementation of mathematical proofs."}
+{"_id": "25832", "title": "Definition:Functional Analysis", "text": "'''Functional analysis''' is a branch of analysis, which studies vector spaces endowed a structure such as inner product, norm or topology."}
+{"_id": "25833", "title": "Definition:Analysis", "text": "'''Analysis''' is a branch of mathematics that studies continuous change. It subsumes the fields of calculus, differential equations, the calculus of variations, power series, Fourier series, real analysis and complex analysis. It is one of the main branches of mathematics, alongside geometry, number theory and algebra."}
+{"_id": "25834", "title": "Definition:Analysis/Real", "text": "'''Real analysis''' is a branch of mathematics that studies real functions."}
+{"_id": "25835", "title": "Definition:Analysis/Complex", "text": "'''Complex analysis''' is a branch of mathematics that studies complex functions."}
+{"_id": "25836", "title": "Definition:Ring Theory", "text": "'''Ring Theory''' is a branch of abstract algebra which studies rings and other related algebraic structures."}
+{"_id": "25837", "title": "Definition:Analytic Number Theory", "text": "'''Analytic number theory''' is a branch of number theory which uses tools from analysis to solve problems about the integers."}
+{"_id": "25839", "title": "Definition:Calculus/Differential", "text": "'''Differential calculus''' is a subfield of calculus which is concerned with the study of the rates at which quantities change."}
+{"_id": "25840", "title": "Definition:Calculus/Integral", "text": "'''Integral calculus''' is a subfield of calculus which is concerned with the study of the rates at which quantities accumulate. Equivalently, given the rate of change of a quantity '''integral calculus''' provides techniques of providing the quantity itself. The equivalence of the two uses are demonstrated in the Fundamental Theorem of Calculus. The technique is also frequently used for the purpose of calculating areas and volumes of curved geometric figures."}
+{"_id": "25841", "title": "Definition:Calculus", "text": "'''Calculus''' is the branch of mathematics which studies change."}
+{"_id": "25842", "title": "Definition:Mathematical Physics", "text": "'''Mathematical physics''' is the branch of mathematics concerned with the development of mathematical methods for application to problems in physics."}
+{"_id": "25843", "title": "Definition:Physics", "text": "'''Physics''' is the branch of science which studies basic concepts including motion, energy, force, and the fundamental nature of spacetime. The study of '''physics''' is generally based in mathematical concepts, especially those of calculus."}
+{"_id": "25844", "title": "Definition:Knot Theory", "text": "'''Knot theory''' is the branch of topology which studies the embedding of knots in $3$-dimensional space."}
+{"_id": "25845", "title": "Definition:Calculus of Variations", "text": "'''Calculus of variations''' is the subfield of analysis concerned maximizing or minimizing real functionals, which are mappings from a set of functions to the real numbers."}
+{"_id": "25846", "title": "Definition:Matrix Algebra", "text": "'''Matrix algebra''' is the subfield of algebra whose subjects of manipulation are matrices."}
+{"_id": "25847", "title": "Definition:Graph Theory", "text": "'''Graph theory''' is the branch of mathematics concerned with the structure and properties of graphs. As a (graph-theoretical) graph has the same conceptual definition as a relation, it follows that there is considerable overlap between the fields of graph theory and relation theory."}
+{"_id": "25848", "title": "Definition:Relation Theory", "text": "'''Relation theory''' is the subfield of set theory concerned with the properties of relations and relational structures. As a relation has the same conceptual definition as a (graph-theoretical) graph, it follows that there is considerable overlap between the fields of relation theory and graph theory."}
+{"_id": "25849", "title": "Definition:Mapping Theory", "text": "'''Mapping theory''' is the subfield of set theory concerned with the properties of mappings."}
+{"_id": "25850", "title": "Definition:Measure Theory", "text": "'''Measure theory''' is the subfield of analysis concerned with the properties of measures, particularly the Lebesgue measure."}
+{"_id": "25851", "title": "Definition:Linear Algebra", "text": "'''Linear algebra''' is the branch of algebra which studies vector spaces and linear transformations between them."}
+{"_id": "25852", "title": "Definition:Commutative Algebra (Mathematical Branch)", "text": "'''Commutative algebra''' is the branch of abstract algebra concerned with commutative and unitary rings."}
+{"_id": "25853", "title": "Definition:Hamiltonian Mechanics", "text": "'''Hamiltonian mechanics''' was a reformulation of classical mechanics using a more abstract mathematical framework. It laid the groundwork for quantum mechanics."}
+{"_id": "25854", "title": "Definition:Classical Mechanics", "text": "'''Classical mechanics''' is the study of the physical laws describing the motion of and forces on bodies under the action of a system of forces."}
+{"_id": "25856", "title": "Definition:Galois Theory", "text": "'''Galois theory''' is a subfield of abstract algebra which provides a connection between field theory and group theory."}
+{"_id": "25857", "title": "Definition:Field Theory", "text": "'''Field Theory''' is a branch of abstract algebra which studies fields and other related algebraic structures."}
+{"_id": "25858", "title": "Definition:Lattice Theory", "text": "'''Lattice theory''' is the branch of abstract algebra concerned with the structure and properties of lattices."}
+{"_id": "25859", "title": "Definition:Geometrical Mechanics", "text": "'''Geometrical mechanics''' is the branch of mathematics which uses techniques from geometry to solve problems in mechanics."}
+{"_id": "25860", "title": "Definition:Mechanics", "text": "'''Mechanics''' is the branch of applied mathematics concerned with the motion of and forces on bodies."}
+{"_id": "25861", "title": "Definition:Dimensional Analysis", "text": "'''Dimensional analysis''' is the branch of applied mathematics which studies the relationships between physical quantities by identifying their fundamental dimensions and units of measurement. As such it can also be considered to be a branch of number theory."}
+{"_id": "25862", "title": "Definition:Applied Mathematics", "text": "'''Applied mathematics''' is the branch of mathematics which concerns the solution of real world applications by the construction of mathematical models."}
+{"_id": "25863", "title": "Definition:Astronomy", "text": "'''Astronomy''' is the branch of physics which studies the features which can be seen in the sky that are outside the atmosphere of Earth. Certain of its mathematical aspects can be considered a subfield of solid geometry."}
+{"_id": "25865", "title": "Definition:Modulo Arithmetic", "text": "'''Modulo arithmetic''' is the branch of abstract algebra which studies the residue class of integers under a modulus. As such it can also be considered to be a branch of number theory."}
+{"_id": "25866", "title": "Definition:Homological Algebra", "text": "'''Homological algebra''' is the branch of algebra and topology which studies homology."}
+{"_id": "25867", "title": "Definition:Umbral Calculus", "text": "'''Umbral calculus''' is a branch of analysis which studies polynomial spaces by means of linear functionals. The modern approach concerns the study of Sheffer sequences"}
+{"_id": "25868", "title": "Definition:Order Theory", "text": "'''Order theory''' is the branch of relation theory which studies orderings."}
+{"_id": "25869", "title": "Definition:Category Theory", "text": "'''Category theory''' is the branch of abstract algebra which studies categories. It can be described as the '''theory of functors'''."}
+{"_id": "25871", "title": "Definition:Laws of Conservation", "text": "The '''laws of conservation''' are laws of physics that states that certain particular measurable properties of isolated physical systems do not change as the system evolves over time."}
+{"_id": "25872", "title": "Definition:Multiplication/Natural Numbers/Addition", "text": "Let $+$ denote addition. The binary operation $\\times$ is recursively defined on $\\N$ as follows: :$\\forall m, n \\in \\N: \\begin{cases} m \\times 0 & = 0 \\\\ m \\times \\left({n + 1}\\right) & = m \\times n + m \\end{cases}$ This operation is called '''multiplication'''. Equivalently, '''multiplication''' can be defined as: :$\\forall m, n \\in \\N: m \\times n := +^n m$  where $+^n m$ denotes the $n$th power of $m$ under $+$."}
+{"_id": "25873", "title": "Definition:Ideal Theory", "text": "'''Ideal Theory''' is a branch of abstract algebra which studies ideals of rings."}
+{"_id": "25875", "title": "Definition:Algebraic Topology", "text": "'''Algebraic Topology''' is a branch of topology which uses tools from abstract algebra to study topological spaces."}
+{"_id": "25876", "title": "Definition:Fractional Calculus", "text": "'''Fractional calculus''' is the branch of calculus which considers the concept of derivatives of fractional order."}
+{"_id": "25877", "title": "Definition:Approximation Theory", "text": "'''Approximation theory''' is a branch of numerical analysis and applied mathematics concerned with the approximation of a function over an interval can be approximated by a simpler function (usually polynomial)."}
+{"_id": "25878", "title": "Definition:Module Theory", "text": "'''Module Theory''' is the branch of abstract algebra which studies modules."}
+{"_id": "25879", "title": "Definition:Polynomial Theory", "text": "'''Polynomial Theory''' is a branch of abstract algebra which studies polynomials."}
+{"_id": "25881", "title": "Definition:Vector Algebra", "text": "'''Vector Algebra''' is the branch of mathematics which studies the algebra of vector spaces."}
+{"_id": "25884", "title": "Definition:Discrete Mathematics", "text": "'''Discrete mathematics''' is the branch of mathematics which studies mathematical structures that are discrete rather than continuous."}
+{"_id": "25885", "title": "Definition:Discrete (Word)", "text": "'''Discrete''' is a word used to define an object which is not continuous."}
+{"_id": "25886", "title": "Definition:Quantum Mechanics", "text": "'''Quantum mechanics''' is the branch of physics concerned with the behaviour of matter at the smallest scale accessible to observation."}
+{"_id": "25887", "title": "Definition:Ergodic Theory", "text": "'''Ergodic Theory''' is the study of dynamical systems with an invariant measure."}
+{"_id": "25889", "title": "Definition:Economics", "text": "'''Economics''' is the branch of applied mathematics which studies the movement of money."}
+{"_id": "25890", "title": "Definition:Mathematical Programming", "text": "'''Mathematical programming''' is the branch of applied mathematics which applies techniques from computer science to solve problems in economics."}
+{"_id": "25891", "title": "Definition:Game Theory", "text": "'''Game theory''' is the branch of discrete mathematics which studies mathematical models of conflict and cooperation between intelligent rational decision-makers. Problems in '''game theory''' are a special case of those of linear programming."}
+{"_id": "25892", "title": "Definition:Numerical Analysis", "text": "'''Numerical analysis''' is the branch of applied mathematics concerned with the processes by which certain kinds of numerical solutions to problems in (mainly) physics."}
+{"_id": "25893", "title": "Definition:Perturbation Theory", "text": "'''Perturbation Theory''' is a branch of applied mathematics which studies systems by investigating their behaviour when certain parameters are changed slghtly. That is, it is the study of the effects of small disturbances."}
+{"_id": "25894", "title": "Definition:Model Theory", "text": "'''Model theory''' is a sub-branch of mathematical logic which studies mathematical structures by means of formal languages."}
+{"_id": "25897", "title": "Definition:Fourier Analysis", "text": "'''Fourier analysis''' is a branch of analysis which seeks to express functions (usually periodic) by means of series of sines and cosines."}
+{"_id": "25898", "title": "Definition:Operator Theory", "text": "'''Operator theory''' is the study of linear operators on function spaces, beginning with differential operators and integral operators."}
+{"_id": "25899", "title": "Definition:Non-Euclidean Geometry", "text": "'''Non-Euclidean geometry''' is branch of geometry in which {{AuthorRef|Euclid}}'s fifth postulate does not hold."}
+{"_id": "25900", "title": "Definition:Ordered Geometry", "text": "'''Ordered geometry''' is a sub-branch of geometry which featured the concept of betweenness but, like projective geometry, omitting the basic notion of distance."}
+{"_id": "25901", "title": "Definition:Hyperbolic Geometry", "text": "'''Hyperbolic geometry''' is a branch of non-Euclidean geometry in which there more than one straight line can pass through a point parallel to another straight line."}
+{"_id": "25903", "title": "Definition:Algebraic Geometry", "text": "'''Algebraic geometry''' is the branch of geometry which studies objects in multi-dimensional space using the techniques of abstract algebra. In particular, techniques from commutative algebra are mainly used."}
+{"_id": "25904", "title": "Definition:Brill-Noether Theory", "text": "'''Brill-Noether Theory''' is the branch of algebraic geometry which studies special divisors."}
+{"_id": "25905", "title": "Definition:Tarski-Grothendieck Set Theory", "text": "'''Tarski-Grothendieck set theory''' is an implementation of axiomatic set theory which allows for the existence of a universe."}
+{"_id": "25906", "title": "Definition:Kummer Theory", "text": "'''Kummer Theory''' is a branch of abstract algebra and number theory which investigates certain types of field extensions."}
+{"_id": "25907", "title": "Definition:Tensor Theory", "text": "'''Tensor theory''' is the branch of linear algebra which studies tensors."}
+{"_id": "25908", "title": "Definition:Vector Analysis", "text": "'''Vector analysis''' is the branch of linear algebra concerned with differentiation and integration of vector spaces, primarily in Euclidean space of $3$-dimensional space."}
+{"_id": "25909", "title": "Definition:Fluid Mechanics", "text": "'''Fluid mechanics''' is the branch of mechanics concerned with the behaviour of fluids."}
+{"_id": "25912", "title": "Definition:Electromagnetism", "text": "'''Electromagnetism''' is the branch of physics concerned with the study of the combined phenomena of electricity and magnetism, and their interconnected nature."}
+{"_id": "25914", "title": "Definition:Partition Theory", "text": "'''Partition theory''' is the subfield of combinatorics which studies integer partitions."}
+{"_id": "25915", "title": "Definition:Proof Theory", "text": "'''Proof theory''' is the subfield of mathematical logic which studies proofs as mathematical objects, thus allowing them to be analysed mathematically."}
+{"_id": "25916", "title": "Definition:Fluid Dynamics", "text": "'''Fluid dynamics''' is the branch of fluid mechanics concerned with the behaviour of fluids in motion."}
+{"_id": "25917", "title": "Definition:Class Theory", "text": "'''Class theory''' is an extension of set theory which allows the creation of collections that are not sets by classes."}
+{"_id": "25918", "title": "Definition:Descriptive Set Theory", "text": "'''Descriptive set theory''' is a branch of set theory which studies certain classes subsets of the real number line and other Polish spaces."}
+{"_id": "25919", "title": "Definition:Recursion Theory", "text": "'''Recursion theory''' is a branch of computability theory which concerns itself with recursive structures."}
+{"_id": "25920", "title": "Definition:Inner Model Theory", "text": "'''Inner model theory''' is a branch of set theory and model theory which studies certain models of ZFC."}
+{"_id": "25921", "title": "Definition:Divisor (Algebra)/Notation", "text": "The conventional notation for '''$x$ is a divisor of $y$''' is \"$x \\mid y$\", but there is a growing trend to follow the notation \"$x \\divides y$\", as espoused by {{AuthorRef|Donald Ervin Knuth|Knuth}} etc. From {{BookReference|Concrete Mathematics: A Foundation for Computer Science||Ronald L. Graham|author2 = Donald E. Knuth|author3 = Oren Patashnik|ed = 2nd|edpage = Second Edition}}: :''The notation '$m \\mid n$' is actually much more common than '$m \\divides n$' in current mathematics literature. But vertical lines are overused -- for absolute values, set delimiters, conditional probabilities, etc. -- and backward slashes are underused. Moreover, '$m \\divides n$' gives an impression that $m$ is the denominator of an implied ratio. So we shall boldly let our divisibility symbol lean leftward.'' An unfortunate unwelcome side-effect of this notational convention is that to indicate non-divisibility, the conventional technique of implementing $/$ through the notation looks awkward with $\\divides$, so $\\not \\! \\backslash$ is eschewed in favour of $\\nmid$. Some sources use $\\ \\vert \\mkern -10mu {\\raise 3pt -} \\ $ or similar to denote non-divisibility."}
+{"_id": "25923", "title": "Definition:Cantor-Bendixson Derivative", "text": "Let $\\struct {X, \\tau}$ be a topological space. Let $S \\subseteq X$. Then for all ordinals $\\beta$, the $\\beta$th '''Cantor-Bendixson derivative''' of $S$ is defined by Transfinite Recursion thus: :$S^{\\paren \\beta} = \\begin {cases} S & : \\beta = 0 \\\\ \\paren {S^{\\paren \\alpha} }' & : \\beta = \\alpha^+ \\\\ \\displaystyle \\bigcap_{\\alpha \\mathop < \\lambda} S^{\\paren \\alpha} & : \\beta = \\lambda \\end{cases}$ where: :$\\paren {S^{\\paren \\alpha} }'$ is the derived set of $S^{\\paren \\alpha}$ :$\\lambda$ is a limit ordinal. {{NamedforDef|Georg Cantor|name2 = Ivar Otto Bendixson|cat = Cantor|cat2 = Bendixson}}"}
+{"_id": "25924", "title": "Definition:Directed Hamilton Cycle Problem/Function Version", "text": ":Given a directed graph $G$ with $n$ vertices, to find a Hamilton cycle in $G$."}
+{"_id": "25926", "title": "Definition:Directed Hamilton Cycle Problem", "text": "There are two versions of the '''Directed Hamilton Cycle Problem'''. === Function Version === {{:Definition:Directed Hamilton Cycle Problem/Function Version}} === Decision Version === {{:Definition:Directed Hamilton Cycle Problem/Decision Version}}"}
+{"_id": "25927", "title": "Definition:Finite Subset", "text": "Let $S$ be a set. Let $T$ be a subset of $S$ which is a finite set. Then $T$ is a '''finite subset''' of $S$. Category:Definitions/Subsets 18w2vinwyqq86kfnhdobc63kpxnfvps"}
+{"_id": "25928", "title": "Definition:Quotient Module", "text": "Let $M$ be an $R$-module and $M^\\ast$ be the underlying group. Let $N$ be a submodule of $M$ and $N^\\ast$ be the underlying subgroup. Let $a + N$ denote the coset of the quotient group $M^\\ast / N^\\ast$. Define the $R$-action on $M^\\ast / N^\\ast$ as: :$\\forall r \\in R, \\forall a \\in M / N: r \\left({a + N^\\ast}\\right) := r a + N^\\ast$ Then $(M^\\ast / N^\\ast,\\circ)$ is a '''quotient $R$-module'''."}
+{"_id": "25929", "title": "Definition:Right Module Axioms", "text": "{{begin-axiom}} {{axiom | n = \\text {RM} 1         | lc= Scalar Multiplication Right Distributes over Module Addition         | q = \\forall \\lambda \\in R: \\forall x, y \\in G         | ml= \\paren {x +_G y} \\circ \\lambda         | mo= =         | mr= \\paren {x \\circ \\lambda} +_G \\paren {y \\circ \\lambda} }} {{axiom | n = \\text {RM} 2         | lc= Scalar Multiplication Left Distributes over Scalar Addition         | q = \\forall \\lambda, \\mu \\in R: \\forall x \\in G         | ml= x \\circ \\paren {\\lambda +_R \\mu}         | mo= =         | mr= \\paren {x \\circ \\lambda} +_G \\paren {x\\circ \\mu} }} {{axiom | n = \\text {RM} 3         | lc= Associativity of Scalar Multiplication         | q = \\forall \\lambda, \\mu \\in R: \\forall x \\in G         | ml= x \\circ \\paren {\\lambda \\times_R \\mu}         | mo= =         | mr= \\paren {x \\circ \\lambda} \\circ \\mu }} {{end-axiom}}"}
+{"_id": "25930", "title": "Definition:Tensor Algebra", "text": "'''Tensor algebra''' is the branch of module theory which studies tensor products."}
+{"_id": "25931", "title": "Definition:Module Direct Product", "text": "=== Finite Case === {{:Definition:Module Direct Product/Finite Case}} === General Case === {{:Definition:Module Direct Product/General Case}}"}
+{"_id": "25932", "title": "Definition:Ramsey Theory", "text": "'''Ramsey theory''' is a branch of combinatorics that studies the conditions under which order must appear."}
+{"_id": "25934", "title": "Definition:Angle Inscribed in Circle", "text": ":300px Let $AB$ and $BC$ be two chords of a circle which meet at $B$. The angle $\\angle ABC$ is the '''angle inscribed at $B$ (with respect to $A$ and $C$)'''. Category:Definitions/Circles 62ne3b5n8dhn4vvvb4wya45hixd45f4"}
+{"_id": "25935", "title": "Definition:Path (Topology)/Endpoint", "text": "The initial point and final point of $\\gamma$ can be referred to as the '''endpoints of $\\gamma$'''"}
+{"_id": "25936", "title": "Definition:Infinite Path", "text": "An '''infinite path''' is a path of infinite length Category:Definitions/Graph Theory 9zbjdb17ulh1ua4rmn8i5zy8hutbh33"}
+{"_id": "25937", "title": "Definition:Supremum/Also known as", "text": "Particularly in the field of analysis, the supremum of a set $T$ is often referred to as the '''least upper bound of $T$''' and denoted $\\map {\\mathrm {lub} } T$ or $\\map {\\mathrm {l.u.b.} } T$. Some sources refer to the '''supremum of a set''' as the '''supremum ''on'' a set'''. Some sources refer to the '''supremum of a set''' as the '''join of the set''' and use the notation $\\bigvee S$. Some sources introduce the notation $\\displaystyle \\sup_{y \\mathop \\in S} y$, which may improve clarity in some circumstances. Some older sources, applying the concept to a (strictly) increasing real sequence, refer to a '''supremum''' as an '''upper limit'''."}
+{"_id": "25938", "title": "Definition:Supremum/Also defined as", "text": "Some sources refer to the supremum as being '''''the'' upper bound'''. Using this convention, any element greater than this is not considered to be an upper bound."}
+{"_id": "25940", "title": "Definition:Infimum/Also known as", "text": "Particularly in the field of analysis, the infimum of a set $T$ is often referred to as the '''greatest lower bound of $T$''' and denoted $\\map {\\mathrm {glb} } T$ or $\\map {\\mathrm {g.l.b.} } T$. Some sources refer to the '''infimum of a set''' as the '''infimum ''on'' a set'''. Some sources introduce the notation $\\displaystyle \\inf_{y \\mathop \\in S} y$, which may improve clarity in some circumstances. Some older sources, applying the concept to a (strictly) decreasing real sequence, refer to a '''infimum''' as a '''lower limit'''."}
+{"_id": "25941", "title": "Definition:Infimum/Also defined as", "text": "Some sources refer to the infimum as being '''''the'' lower bound'''. Using this convention, any element less than this is not considered to be a lower bound."}
+{"_id": "25944", "title": "Definition:Differential Equation/Solution", "text": "Let $\\Phi$ be a differential equation defined on a domain $D$. Let $\\phi$ be a function which satisfies $\\Phi$ on the whole of $D$. Then $\\phi$ is known as '''a solution''' of $\\Phi$. Note that, in general, there may be more than one '''solution''' to a given differential equation. On the other hand, there may be none at all. === General Solution === {{:Definition:Differential Equation/Solution/General Solution}} === Particular Solution === {{:Definition:Differential Equation/Solution/Particular Solution}}"}
+{"_id": "25945", "title": "Definition:Differential Equation/Solution/General Solution", "text": "The '''general solution''' of $\\Phi$ is the set of ''all'' functions $\\phi$ that satisfy $\\Phi$."}
+{"_id": "25946", "title": "Definition:Linear Representation/Group", "text": "Let $\\left({\\mathbb k, +, \\circ}\\right)$ be a field. Let $V$ be a vector space over $\\mathbb k$ of finite dimension. Let $\\operatorname {GL} \\left({V}\\right)$ be the general linear group of $V$. Let $\\left({G, \\cdot}\\right)$ be a finite group. A '''linear representation of $G$ on $V$''' is a group homomorphism $\\rho: G \\to \\operatorname {GL} \\left({V}\\right)$."}
+{"_id": "25947", "title": "Definition:Linear Representation/Algebra", "text": "Let $K$ be a field. Let $A$ be an associative unitary algebra over $K$. Then a '''(linear) representation of $A$''' is a vector space $V$ over $K$ equipped with a homomorphism of algebras: :$\\rho: A \\to \\map {\\operatorname {End} } V$ where $\\map {\\operatorname {End} } V$ is the endomorphism ring of $V$."}
+{"_id": "25948", "title": "Definition:Scalar Triple Product", "text": "Let $\\mathbf a$, $\\mathbf b$ and $\\mathbf c$ be vectors in a vector space $\\mathbf V$ of $3$ dimensions: {{begin-eqn}} {{eqn | l = \\mathbf a       | r = a_i \\mathbf i + a_j \\mathbf j + a_k \\mathbf k }} {{eqn | l = \\mathbf b       | r = b_i \\mathbf i + b_j \\mathbf j + b_k \\mathbf k }} {{eqn | l = \\mathbf c       | r = c_i \\mathbf i + c_j \\mathbf j + c_k \\mathbf k }} {{end-eqn}} where $\\tuple {\\mathbf i, \\mathbf j, \\mathbf k}$ is the standard ordered basis of $\\mathbf V$. The '''scalar triple product''', denoted as $\\mathbf a \\cdot \\paren {\\mathbf b \\times \\mathbf c}$, is defined as: :$\\mathbf a \\cdot \\paren {\\mathbf b \\times \\mathbf c} = \\begin {vmatrix} a_i & a_j & a_k \\\\ b_i & b_j & b_k \\\\ c_i & c_j & c_k \\\\ \\end {vmatrix}$"}
+{"_id": "25949", "title": "Definition:Singleton Graph", "text": "The '''singleton graph''' $N_1$ is the simple graph with one vertex: :60px"}
+{"_id": "25951", "title": "Definition:Complete Distributive Lattice", "text": "Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a lattice. Then $S$ is a '''complete distributive lattice''' {{iff}} :$(1): \\quad S$ is a complete lattice :$(2): \\quad S$ is a distributive lattice. Category:Definitions/Lattice Theory 19ry08xqa1fj2agz8s5bokuc36ohn0t"}
+{"_id": "25952", "title": "Definition:Cauchy Sequence/Complex Numbers", "text": "Let $\\sequence {z_n}$ be a sequence in $\\C$. Then $\\sequence {z_n}$ is a '''Cauchy sequence''' {{iff}}: : $\\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\N: \\forall m, n \\in \\N: m, n \\ge N: \\size {z_n - z_m} < \\epsilon$ where $\\size {z_n - z_m}$ denotes the complex modulus of $z_n - z_m$."}
+{"_id": "25953", "title": "Definition:Commutative and Unitary Ring/Axioms", "text": "A commutative and unitary ring is an algebraic structure $\\struct {R, *, \\circ}$, on which are defined two binary operations $\\circ$ and $*$, which satisfy the following conditions: {{begin-axiom}} {{axiom | n = \\text A 0         | q = \\forall a, b \\in R         | m = a * b \\in R         | lc= Closure under addition }} {{axiom | n = \\text A 1         | q = \\forall a, b, c \\in R         | m = \\paren {a * b} * c = a * \\paren {b * c}         | lc= Associativity of addition }} {{axiom | n = \\text A 2         | q = \\forall a, b \\in R         | m = a * b = b * a         | lc= Commutativity of addition }} {{axiom | n = \\text A 3         | q = \\exists 0_R \\in R: \\forall a \\in R         | m = a * 0_R = a = 0_R * a         | lc= Identity element for addition: the zero }} {{axiom | n = \\text A 4         | q = \\forall a \\in R: \\exists a' \\in R         | m = a * a' = 0_R = a' * a         | lc= Inverse elements for addition: negative elements }} {{axiom | n = \\text M 0         | q = \\forall a, b \\in R         | m = a \\circ b \\in R         | lc= Closure under product }} {{axiom | n = \\text M 1         | q = \\forall a, b, c \\in R         | m = \\paren {a \\circ b} \\circ c = a \\circ \\paren {b \\circ c}         | lc= Associativity of product }} {{axiom | n = \\text M 2         | q = \\forall a, b \\in R         | m = a \\circ b = b \\circ a         | lc= Commutativity of product }} {{axiom | n = \\text M 3         | q = \\exists 1_R \\in R: \\forall a \\in R         | m = a \\circ 1_R = a = 1_R \\circ a         | lc= Identity element for product: the unity }} {{axiom | n = \\text D         | q = \\forall a, b, c \\in R         | m = a \\circ \\paren {b * c} = \\paren {a \\circ b} * \\paren {a \\circ c}         | lc= Product is distributive over addition }} {{axiom | m = \\paren {a * b} \\circ c = \\paren {a \\circ c} * \\paren {b \\circ c} }} {{end-axiom}} These criteria are called the '''commutative and unitary ring axioms'''."}
+{"_id": "25954", "title": "Definition:Accumulation Point/Set", "text": "Let $x \\in S$. Then $x$ is an '''accumulation point''' of $A$ {{iff}}: :$x \\in \\cl {A \\setminus \\set x}$ where $\\operatorname{cl}$ denotes the (topological) closure of a set."}
+{"_id": "25955", "title": "Definition:Set Derivative", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A \\subseteq S$. The '''derivative''' of $A$ in $T$ is the set of all accumulation points of $A$. It is usually denoted on {{ProofWiki}} as $A'$."}
+{"_id": "25956", "title": "Definition:Pythagoras's Constant", "text": "'''Pythagoras's constant''' is sometimes used to mean $\\sqrt 2$, the square root of $2$. === Decimal Expansion === {{:Square Root of 2}} {{NamedforDef|Pythagoras of Samos|cat = Pythagoras}} Category:Number Theory m3ug0rekc6r54r0n4b62jnacyopnvvr"}
+{"_id": "25957", "title": "Definition:Existential Quantifier/Unique", "text": "The symbol $\\exists !$ denotes the existence of a unique object fulfilling a particular condition. :$\\exists ! x: \\map P x$ means: :'''There exists exactly one object $x$ such that $\\map P x$ holds''' or: :'''There exists one and only one $x$ such that $\\map P x$ holds'''. Formally: :$\\exists ! x: \\map P x \\dashv \\vdash \\exists x: \\map P x \\land \\forall y: \\map P y \\implies x = y$ In natural language, this means: ::''There exists exactly one $x$ with the property $P$'' : is logically equivalent to: ::''There exists an $x$ such that $x$ has the property $P$, and for every $y$, $y$ has the property $P$ only if $x$ and $y$ are the same object.''"}
+{"_id": "25958", "title": "Definition:Existential Quantifier/Exact", "text": "The symbol $\\exists_n$ denotes the existence of an exact number of objects fulfilling a particular condition. :$\\exists_n x: \\map P x$ means: :'''There exist exactly $n$ objects $x$ such that $\\map P x$ holds'''."}
+{"_id": "25960", "title": "Definition:Set Union/General Definition", "text": "Let $S$ be a collection, which could be either a set or a class. The '''union of $S$''' is: :$\\ds \\bigcup S := \\set {x: \\exists X \\in S: x \\in X}$ That is, the set of all elements of all elements of $S$ which are themselves sets."}
+{"_id": "25961", "title": "Definition:Ordered Pair/Informal Definition", "text": "An '''ordered pair''' is a two-element set together with an ordering. In other words, one of the elements ''is'' distinguished above the other - it comes first. Such a structure is written: :$\\tuple {a, b}$ and it means: : '''first $a$, then $b$'''."}
+{"_id": "25962", "title": "Definition:Ordered Pair/Kuratowski Formalization", "text": "The concept of an ordered pair can be formalized by the definition: :$\\tuple {a, b} := \\set {\\set a, \\set {a, b} }$ This formalization justifies the existence of ordered pairs in Zermelo-Fraenkel set theory."}
+{"_id": "25963", "title": "Definition:Inverse Mapping/Also defined as", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be an injection. Then its '''inverse mapping''' is the mapping $g$ such that: :its domain $\\Dom g$ equals the image $\\Img f$ of $f$ :$\\forall y \\in \\Img f: \\map f {\\map g y} = y$ Thus $f$ is seen to be a surjection by tacit use of Restriction of Mapping to Image is Surjection."}
+{"_id": "25964", "title": "Definition:Weight of Topological Space", "text": "Let $T$ be a topological space. Let $\\mathbb B$ be the set of all bases of $T$. === Definition 1 === {{:Definition:Weight of Topological Space/Definition 1}} === Definition 2 === {{:Definition:Weight of Topological Space/Definition 2}}"}
+{"_id": "25965", "title": "Definition:Operation Compatible with Equivalence", "text": "Let $F$ be a (unary) operation which can be applied to sets. Then $F$ is '''compatible with set equivalence''' {{iff}}: :$F \\left({A}\\right) = F \\left({B}\\right) \\iff A \\sim B$ where: :$A$ and $B$ are arbitrary sets :$\\sim$ denotes set equivalence."}
+{"_id": "25966", "title": "Definition:Weight of Topological Space/Definition 1", "text": "The '''weight''' of $T$ is defined as: :$\\displaystyle \\map w T := \\bigcap_{\\BB \\mathop \\in \\mathbb B} \\card \\BB$ where $\\card \\BB$ denotes the cardinality of $\\BB$."}
+{"_id": "25967", "title": "Definition:Weight of Topological Space/Definition 2", "text": "The '''weight''' of $T$ is the smallest cardinality of the elements of $\\mathbb B$: :$\\map w T := \\min \\set {\\card \\BB: \\BB \\in \\mathbb B}$"}
+{"_id": "25968", "title": "Definition:Character of Point in Topological Space", "text": "Let $T$ be a topological space. Let $x$ be a point of $T$. Let $\\mathbb B \\left({x}\\right)$ be the set of all local bases at $x$. The '''character of (the point) $x$ in $T$''' is the smallest cardinality of the elements of $\\mathbb B \\left({x}\\right)$: :$\\chi \\left({x,T}\\right) := \\min \\left\\{ {\\left\\vert{\\mathcal B}\\right\\vert: \\mathcal B \\in \\mathbb B \\left({x}\\right)}\\right\\}$"}
+{"_id": "25969", "title": "Definition:Character of Topological Space", "text": "Let $T$ be a topological space. The '''character of (the topological space) $T$''' is the supremum of the set of characters of the points of $T$: :$\\chi \\left({T}\\right) := \\sup \\left\\{ {\\chi \\left({x, T}\\right): x \\in T}\\right\\}$"}
+{"_id": "25971", "title": "Definition:Bound Variable/Examples/Universal Statement", "text": "In the universal statement: :$\\forall x: \\map P x$ the symbol $x$ is a '''bound variable'''. Thus, the meaning of $\\forall x: \\map P x$ does not change if $x$ is replaced by another symbol. That is, $\\forall x: \\map P x$ means the same thing as $\\forall y: \\map P y$ or $\\forall \\alpha: \\map P \\alpha$. And so on."}
+{"_id": "25972", "title": "Definition:Bound Variable/Examples/Existential Statement", "text": "In the existential statement: :$\\exists x: \\map P x$ the symbol $x$ is a '''bound variable'''. Thus, the meaning of $\\exists x: \\map P x$ does not change if $x$ is replaced by another symbol. That is, $\\exists x: \\map P x$ means the same thing as $\\exists y: \\map P y$ or $\\exists \\alpha: \\map P \\alpha$. And so on."}
+{"_id": "25973", "title": "Definition:Existential Statement/Conditional", "text": "A '''conditionally existential statement''' is an existential statement which states the existence of an object fulfilling a certain propositional function dependent upon the existence of certain other objects."}
+{"_id": "25975", "title": "Definition:Conditional/Formal Implication", "text": "'''Formal implication''' is a usage of an implication in which it is necessary for there to be a formal connection between the antecedent and the consequent in order for the implication to have any semantic meaning."}
+{"_id": "25976", "title": "Definition:Stoic School", "text": "The '''stoic school''' was a school of Greek philosophy which emphasized the importance of goodness and peace of mind gained from living a life of Virtue in accordance with Nature. The writings of its philosophers reveal the first beginnings of the discipline of propositional logic."}
+{"_id": "25977", "title": "Definition:Truth Table/Matrix Presentation", "text": "A two-element truth table can be presented in '''matrix form''': :{{:Definition:Conditional/Truth Table/Matrix Form}}$\\qquad${{:Definition:Conjunction/Truth Table/Matrix Form}}$\\qquad${{:Definition:Disjunction/Truth Table/Matrix Form}}$\\qquad${{:Definition:Biconditional/Truth Table/Matrix Form}} that is, in the form of a Cayley table. This, however, can be used only when the operation being displayed has two elements."}
+{"_id": "25982", "title": "Definition:Fundamental Truth Table", "text": "The '''fundamental truth tables''' are the characteristic truth tables for: : Logical Not : Conjunction : Disjunction : Conditional : Biconditional as follows: === Logical Not === {{:Definition:Logical Not/Truth Table}} === Conjunction === {{:Definition:Conjunction/Truth Table}} === Disjunction === {{:Definition:Disjunction/Truth Table}} === Conditional === {{:Definition:Conditional/Truth Table}} === Biconditional === {{:Definition:Biconditional/Truth Table}}"}
+{"_id": "25983", "title": "Definition:Derivative Truth Table", "text": "A '''derivative truth table''' is a truth table for a statement form which is not mapped by one of the fundamental truth tables: : Logical Not : Conjunction : Disjunction : Conditional : Biconditional"}
+{"_id": "25984", "title": "Definition:Truth Table/Row", "text": "A '''row''' of a truth table is one of the horizontal lines that consists of instances of the symbols $T$ and $F$. Each '''row''' contains the truth values of each of the boolean interpretations of the statement forms according to the propositional variables that comprise them. There are as many '''rows''' in a truth table as there are combinations of $T$ and $F$ for all the propositional variables that constitute the statement forms."}
+{"_id": "25985", "title": "Definition:Truth Table/Column", "text": "A '''row''' of a truth table is one of the vertical lines headed by a statement form presenting all the truth values that the statement form takes. Each entry in the '''column''' corresponds to one specific combination of truth values taken by the propositional variables that the statement form comprises."}
+{"_id": "25986", "title": "Definition:Conjugate Statements", "text": "Let $p$ and $q$ be statements. The conditional statement: :$p \\implies q$ its inverse: :$\\lnot p \\implies \\lnot q$ its converse: :$q \\implies p$ and its contrapositive: :$\\lnot q \\implies \\lnot p$ are all '''conjugate statements'''."}
+{"_id": "25987", "title": "Definition:Rubik's Cube", "text": "'''Rubik's cube''' is a toy consisting of a plastic cube divided into $27$ smaller cubes, interlocking at the center cube and allowing each face to rotate independently of the others. Each of the $6$ faces of the main cube is colored with one of six contrasting colors. Rotating the faces of the cube causes these $6$ faces to become mixed up. The object of the exercise is to restore the $6$ faces to their original single colors."}
+{"_id": "25989", "title": "Definition:Huntington Algebra", "text": "A '''Huntington algebra''' is an algebraic structure $\\struct {S, \\circ, *}$ such that: {{begin-axiom}} {{axiom | n = \\text {HA} 0         | lc=         | t =$S$ is closed under both $\\circ$ and $*$ }} {{axiom | n = \\text {HA} 1         | lc=         | t = Both $\\circ$ and $*$ are commutative }} {{axiom | n = \\text {HA} 2         | lc=         | t = Both $\\circ$ and $*$ distribute over the other }} {{axiom | n = \\text {HA} 3         | lc=         | t = Both $\\circ$ and $*$ have identities $e^\\circ$ and $e^*$ respectively, where $e^\\circ \\ne e^*$ }} {{axiom | n = \\text {HA} 4         | lc=         | t = $\\forall a \\in S: \\exists a' \\in S: a \\circ a' = e^*, a * a' = e^\\circ$ }} {{end-axiom}} The element $a'$ in $(\\text {HA} 4)$ is often called the '''complement''' of $a$. A '''Huntington algebra''' can also be considered as a mathematical system $\\set {S, O, A}$ where $O = \\set {\\circ, *}$ and $A$ consists of the set of axioms $(\\text {HA}  0)$ to $(\\text {HA}  4)$ as defined above."}
+{"_id": "25990", "title": "Definition:Cardinality/Finite", "text": "Let $S$ be a finite set. The '''cardinality''' $\\card S$ of $S$ is the '''number of elements in $S$'''. That is, if: :$S \\sim \\N_{< n}$ where: :$\\sim$ denotes set equivalence :$\\N_{<n}$ is the set of all natural numbers less than $n$ then we define: :$\\card S = n$"}
+{"_id": "25991", "title": "Definition:Rubik's Cube/Basic Moves", "text": "The set of '''basic moves''' of the Rubik's cube can be denoted: :$\\set {U, D, L, R, F, B}$ where $U$, $D$, $L$, $R$, $F$, $B$ are in Singmaster notation."}
+{"_id": "25993", "title": "Definition:Singmaster Notation", "text": "Let the cube be facing the user with the $U$ and $D$ faces horizontal and the $F$ face towards the user. In the following, '''clockwise''' denotes rotation of a layer of the cube in the direction of the clock '''as though that face were towards the user'''. :$U$: turns the up face one quarter turn clockwise. :$D$: turns the down face one quarter turn clockwise. :$L$: turns the left face one quarter turn clockwise. :$R$: turns the right face one quarter turn clockwise. :$F$: turns the front face one quarter turn clockwise. :$B$: turns the back face one quarter turn clockwise. Appending a $'$ symbol to one of the letters denotes a turn of that layer one quarter turn anticlockwise. Appending a $2$ to one of the letters denotes a half turn of that layer."}
+{"_id": "25994", "title": "Definition:Rubik's Cube/Facet", "text": "Each of the $9$ small squares into which the faces of Rubik's cube is subdivided is called a '''facet'''."}
+{"_id": "25995", "title": "Definition:Symmetric Difference/Definition 5", "text": ":$S * T := \\set {x: x \\in S \\oplus x \\in T}$"}
+{"_id": "25998", "title": "Definition:Rubik's Cube/Corner Facet", "text": "Of each of the $9$ facets of each face of Rubik's cube, the $4$ in the corners of the face are called the '''corner facets'''. The set of '''corner facets''' can be denoted $S_C$."}
+{"_id": "25999", "title": "Definition:Rubik's Cube/Edge Facet", "text": "Of each of the $9$ facets of each face of Rubik's cube, the $4$ on the edges of the face are called the '''edge facets'''. The set of '''edge facets''' can be denoted $S_E$."}
+{"_id": "26000", "title": "Definition:Rubik's Cube/Center Facet", "text": "Of each of the $9$ facets of each face of Rubik's cube, the ones in the centers of the faces are called the '''center facets'''. The set of '''center facets''' can be denoted $S_Z$."}
+{"_id": "26001", "title": "Definition:Trivial Partition/Singleton", "text": "The '''singleton partition''' on $S$ is defined as: :$\\mathcal P = \\left\\{{S}\\right\\}$ That is, it is a partition with only one component."}
+{"_id": "26002", "title": "Definition:Trivial Partition/Partition of Singletons", "text": "The '''partition of singletons''' on $S$ is defined as: :$\\mathcal P = \\set {\\set x: x \\in S}$ That is, it is a partition such that every component is a singleton."}
+{"_id": "26004", "title": "Definition:Prime Number/Definition 5", "text": "A '''prime number''' $p$ is an integer greater than $1$ that has no (positive) divisors less than itself other than $1$."}
+{"_id": "26005", "title": "Definition:Prime-Counting Function/Examples", "text": "The values of the prime-counting ($\\pi$) function for the first few integers are as follows: :{| border=\"1\" |- ! align=\"right\" style = \"padding: 2px 10px\" | $n$  ! align=\"right\" style = \"padding: 2px 10px\" | $\\map \\pi n$  |- | align=\"right\" style = \"padding: 2px 10px\" | $1$ | align=\"right\" style = \"padding: 2px 10px\" | $0$ |- | align=\"right\" style = \"padding: 2px 10px\" | $2$ | align=\"right\" style = \"padding: 2px 10px\" | $1$ |- | align=\"right\" style = \"padding: 2px 10px\" | $3$ | align=\"right\" style = \"padding: 2px 10px\" | $2$ |- | align=\"right\" style = \"padding: 2px 10px\" | $4$ | align=\"right\" style = \"padding: 2px 10px\" | $2$ |- | align=\"right\" style = \"padding: 2px 10px\" | $5$ | align=\"right\" style = \"padding: 2px 10px\" | $3$ |- | align=\"right\" style = \"padding: 2px 10px\" | $6$ | align=\"right\" style = \"padding: 2px 10px\" | $3$ |- | align=\"right\" style = \"padding: 2px 10px\" | $7$ | align=\"right\" style = \"padding: 2px 10px\" | $4$ |- | align=\"right\" style = \"padding: 2px 10px\" | $8$ | align=\"right\" style = \"padding: 2px 10px\" | $4$ |}"}
+{"_id": "26006", "title": "Definition:Prime-Counting Function/Examples/16", "text": "The value of the prime-counting ($\\pi$) function for $16$ is determined as follows. The prime numbers less than $16$ are: :$2, 3, 5, 7, 11, 13$ Hence: :$\\map \\pi {16} = 6$"}
+{"_id": "26008", "title": "Definition:Mapping/Definition 1", "text": "A '''mapping''' from $S$ to $T$ is a binary relation on $S \\times T$ which associates each element of $S$ with exactly one element of $T$."}
+{"_id": "26009", "title": "Definition:Mapping/Definition 2", "text": "A '''mapping $f$ from $S$ to $T$''', denoted $f: S \\to T$, is a relation $f = \\struct {S, T, G}$, where $G \\subseteq S \\times T$, such that: :$\\forall x \\in S: \\forall y_1, y_2 \\in T: \\tuple {x, y_1} \\in G \\land \\tuple {x, y_2} \\in G \\implies y_1 = y_2$ and :$\\forall x \\in S: \\exists y \\in T: \\tuple {x, y} \\in G$"}
+{"_id": "26010", "title": "Definition:Mapping/Definition 3", "text": "A '''mapping $f$ from $S$ to $T$''', denoted $f: S \\to T$, is a relation $f = \\struct {S, T, R}$, where $R \\subseteq S \\times T$, such that: :$\\forall \\tuple {x_1, y_1}, \\tuple {x_2, y_2} \\in f: y_1 \\ne y_2 \\implies x_1 \\ne x_2$ and :$\\forall x \\in S: \\exists y \\in T: \\tuple {x, y} \\in f$"}
+{"_id": "26011", "title": "Definition:Mapping/Definition 4", "text": "A '''mapping from $S$ to $T$''' is a relation on $S \\times T$ which is: :$(1): \\quad$ Many-to-one :$(2): \\quad$ Left-total, that is, defined for all elements in $S$."}
+{"_id": "26012", "title": "Definition:Countable Set/Also known as", "text": "When the terms '''denumerable''' and '''enumerable''' are encountered, they generally mean the same as countably infinite. Some modern pedagogues (for example {{AuthorRef|Vi Hart}} and {{AuthorRef|James Grime}}) use the term '''listable''', but this has yet to catch on."}
+{"_id": "26013", "title": "Definition:Countable Set/Countably Infinite/Cardinality", "text": "The cardinality of a countably infinite set is denoted by the symbol $\\aleph_0$ ('''aleph null''')."}
+{"_id": "26014", "title": "Definition:Countable Set/Countably Infinite/Definition 1", "text": "$S$ is '''countably infinite''' {{iff}} there exists a bijection: :$f: S \\to \\N$ where $\\N$ is the set of natural numbers."}
+{"_id": "26015", "title": "Definition:Countable Set/Countably Infinite/Definition 2", "text": "$S$ is '''countably infinite''' {{iff}} there exists a bijection: :$f: S \\to \\Z$ where $\\Z$ is the set of integers."}
+{"_id": "26016", "title": "Definition:Countable Set/Also defined as", "text": "Some sources define a countable set to be what is defined on {{ProofWiki}} as a countably infinite set. That is, they use '''countable''' to describe a set which has ''exactly the same'' cardinality as $\\N$. Thus under this criterion $X$ is said to be countable {{iff}} there exists a bijection from $X$ to $\\N$, that is, {{iff}} $X$ is equivalent to $\\N$. However, as the very concept of the term '''countable''' implies that a set '''can be counted''', which, plainly, a finite set can be, it is suggested that this interpretation may be counter-intuitive. Hence, on {{ProofWiki}}, the term countable set will be taken in the sense as to include the concept of finite set, and countably infinite will mean a countable set which is specifically ''not'' finite."}
+{"_id": "26017", "title": "Definition:Countable Set/Definition 3", "text": "$S$ is '''countable''' {{iff}} there exists a bijection between $S$ and a subset of $\\N$."}
+{"_id": "26018", "title": "Definition:Polish Notation/Formal Definition", "text": "Let $\\mathcal A$ be an alphabet. Let each $s \\in \\mathcal A$ be assigned a natural number called its arity. The formal grammar for Polish notation is given by the single bottom-up rule: ::If $s$ has arity $n$ and $\\phi_1, \\ldots, \\phi_n$ are well-formed formulas, then: :::$s \\phi_1 \\cdots \\phi_n$ ::is also a well-formed formula. Notably, in the case where $s$ has arity $0$, this is a vacuous truth, so any such $s$ constitutes a well-formed formula."}
+{"_id": "26019", "title": "Definition:Matrix Theory", "text": "'''Matrix theory''' is the field of mathematics which studies matrices."}
+{"_id": "26020", "title": "Definition:Unit Matrix/Examples/3 x 3", "text": "The '''$3 \\times 3$ unit matrix''' is as follows: :$\\mathbf I_3 = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ \\end{bmatrix}$"}
+{"_id": "26022", "title": "Definition:Matrix/Lower Triangular Elements", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be a matrix. The elements $a_{i j}: i > j$ are called the '''lower triangular elements'''."}
+{"_id": "26023", "title": "Definition:Matrix/Upper Triangular Elements", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be a matrix. The elements $a_{i j}: i < j$ are called the '''upper triangular elements'''."}
+{"_id": "26024", "title": "Definition:Permutation Matrix", "text": "A '''permutation matrix (of order $n$)''' is an $n \\times n$ square matrix with: : exactly one instance of $1$ in each row and column : $0$ elsewhere."}
+{"_id": "26025", "title": "Definition:Monomial Matrix", "text": "A '''monomial matrix (of order $n$)''' is an $n \\times n$ square matrix with: :exactly one element in each row and column which is not $0$ :$0$ elsewhere."}
+{"_id": "26026", "title": "Definition:(0, 1)-Matrix", "text": "A '''$\\tuple {0, 1}$-matrix''' is a matrix whose elements consist only of instances of $0$ and $1$."}
+{"_id": "26032", "title": "Definition:Submatrix/Notation", "text": "A submatrix of $\\mathbf A$ is denoted as follows. Let: : $\\left\\{ {a_1, a_2, \\ldots, a_r}\\right\\}$ be the indices of the $r$ selected rows : $\\left\\{ {b_1, b_2, \\ldots, b_s}\\right\\}$ be the indices of the $s$ selected columns where all of $a_1, \\ldots, a_r$ are between $1$ and $m$, and all of $b_1, \\ldots, b_s$ are between $1$ and $n$. Then the '''submatrix''' formed from rows $\\left\\{ {a_1, a_2, \\ldots, a_r}\\right\\}$ and columns $\\left\\{ {b_1, b_2, \\ldots, b_s}\\right\\}$ is denoted as: :$\\mathbf A \\left[{a_1, a_2, \\ldots, a_r; b_1, b_2, \\ldots, b_s}\\right]$ It is usual to specify the rows and columns in ascending numerical order."}
+{"_id": "26033", "title": "Definition:Submatrix/Notation/Order (m-1) x (n-1) Submatrix", "text": "Let a submatrix $\\mathbf B$ of $\\mathbf A$ be of order $\\left({m - 1}\\right) \\times \\left({n - 1}\\right)$. Then it is usual to denote $\\mathbf B$ by indicating the (single) row and column of $\\mathbf A$ which has been '''removed''', as follows: Let: : $a_j$ be the row of $\\mathbf A$ which is not included in $\\mathbf B$ : $b_k$ be the column of $\\mathbf A$ which is not included in $\\mathbf B$. Then the '''submatrix''' $\\mathbf B$ formed from the remaining rows and columns of $\\mathbf A$ can be denoted as: :$\\mathbf A \\left({a_j; b_k}\\right)$"}
+{"_id": "26034", "title": "Definition:Determinant/Matrix/Order", "text": "The '''order''' of a determinant is defined as the order of the square matrix on which it is defined."}
+{"_id": "26035", "title": "Definition:Minor of Determinant/Notation/Order n-1", "text": "Let a submatrix $\\mathbf B$ of $\\mathbf A$ be of order $n - 1$. Let: :$j$ be the row of $\\mathbf A$ which is not included in $\\mathbf B$ :$k$ be the column of $\\mathbf A$ which is not included in $\\mathbf B$. Thus, let $\\mathbf B := \\map {\\mathbf A} {j; k}$. Then $\\map \\det {\\mathbf B}$ can be denoted: :$D_{i j}$ That is, $D_{i j}$ is the minor of order $n - 1$ obtained from $D$ by deleting all the elements of row $i$ and column $j$."}
+{"_id": "26036", "title": "Definition:Minor of Determinant/Notation", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $D := \\map \\det {\\mathbf A}$ denote the determinant of $\\mathbf A$. Let: :$\\set {a_1, a_2, \\ldots, a_k}$ be the indices of the $k$ selected rows of $\\mathbf A$ :$\\set {b_1, b_2, \\ldots, b_k}$ be the indices of the $k$ selected columns of $\\mathbf A$ where all of $a_1, \\ldots, a_k$ and all of $b_1, \\ldots, b_k$ are between $1$ and $n$. Let: :$\\mathbf B := \\mathbf A \\sqbrk {a_1, a_2, \\ldots, a_k; b_1, b_2, \\ldots, b_k}$ be the '''submatrix''' formed from rows $\\set {a_1, a_2, \\ldots, a_k}$ and columns $\\set {b_1, b_2, \\ldots, b_k}$. The '''order-$k$ minor''' of $D$ formed from rows $r_1, r_2, \\ldots, r_k$ and columns $s_1, s_2, \\ldots, s_k$ can be denoted: :$\\map D {a_1, a_2, \\ldots, a_k \\mid b_1, b_2, \\ldots, b_k}$. Each element of $D$ is an order $1$ minor of $D$, and can be denoted: :$\\map D {a_i \\mid b_j}$"}
+{"_id": "26038", "title": "Definition:Cofactor/Element", "text": "Let $a_{r s}$ be an element of $D$. Let $D_{r s}$ be the determinant of order $n-1$ obtained from $D$ by deleting row $r$ and column $s$. Then the '''cofactor''' $A_{r s}$ of the element $a_{r s}$ is defined as: :$A_{r s} := \\paren {-1}^{r + s} D_{r s}$"}
+{"_id": "26039", "title": "Definition:Cofactor/Minor", "text": "Let $D \\left({r_1, r_2, \\ldots, r_k \\mid s_1, s_2, \\ldots, s_k}\\right)$ be a order-$k$ minor of $D$. Then the '''cofactor''' of $D \\left({r_1, r_2, \\ldots, r_k \\mid s_1, s_2, \\ldots, s_k}\\right)$ can be denoted: :$\\tilde D \\left({r_1, r_2, \\ldots, r_k \\mid s_1, s_2, \\ldots, s_k}\\right)$ and is defined as: :$\\tilde D \\left({r_1, r_2, \\ldots, r_k \\mid s_1, s_2, \\ldots, s_k}\\right) = \\left({-1}\\right)^t D \\left({r_{k+1}, r_{k+2}, \\ldots, r_n \\mid s_{k+1}, s_{k+2}, \\ldots, s_n}\\right)$ where: : $t = r_1 + r_2 + \\ldots + r_k + s_1 + s_2 + \\ldots s_k$ : $r_{k+1}, r_{k+2}, \\ldots, r_n$ are the numbers in $1, 2, \\ldots, n$ not in $\\left\\{{r_1, r_2, \\ldots, r_k}\\right\\}$ : $s_{k+1}, s_{k+2}, \\ldots, s_n$ are the numbers in $1, 2, \\ldots, n$ not in $\\left\\{{s_1, s_2, \\ldots, s_k}\\right\\}$ That is, the '''cofactor of a minor''' is the determinant formed from the rows and columns not in that minor, multiplied by the appropriate sign. When $k = 1$, this reduces to the cofactor of an element (as above). When $k = n$, the \"minor\" is in fact the whole determinant. For convenience its '''cofactor''' is defined as being $1$. Note that the '''cofactor''' of the '''cofactor''' of a minor is the minor itself (multiplied by the appropriate sign)."}
+{"_id": "26041", "title": "Definition:Non-Invertible Matrix/Definition 1", "text": "Let $\\mathbf A$ have no inverse. Then $\\mathbf A$ is referred to as '''non-invertible'''."}
+{"_id": "26042", "title": "Definition:Non-Invertible Matrix/Definition 2", "text": "Let the determinant of $\\mathbf A$ be equal to $0$. Then $\\mathbf A$ is referred to as '''non-invertible'''."}
+{"_id": "26043", "title": "Definition:Zero Vector/Euclidean Space", "text": "Let $\\struct {\\R^n, +, \\times}_\\R$ be a real vector space. The '''zero vector''' in $\\struct {\\R^n, +, \\times}_\\R$ is: :$\\mathbf 0_{n \\times 1} := \\begin {bmatrix} 0 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end {bmatrix}$ where $0 \\in \\R$."}
+{"_id": "26044", "title": "Definition:Congruence (Number Theory)/Integers/Remainder after Division", "text": "'''Congruence modulo $m$''' is defined as the relation $\\equiv \\pmod m$ on the set of all $a, b \\in \\Z$: :$a \\equiv b \\pmod m := \\set {\\tuple {a, b} \\in \\Z \\times \\Z: \\exists k \\in \\Z: a = b + k m}$ That is, such that $a$ and $b$ have the same remainder when divided by $m$."}
+{"_id": "26045", "title": "Definition:Congruence (Number Theory)/Remainder after Division", "text": "We define a relation $\\mathcal R_z$ on the set of all $x, y \\in \\R$: :$\\mathcal R_z := \\left\\{{\\left({x, y}\\right) \\in \\R \\times \\R: \\exists k \\in \\Z: x = y + k z}\\right\\}$ This relation is called '''congruence modulo $z$''', and the real number $z$ is called the '''modulus'''. When $\\left({x, y}\\right) \\in \\mathcal R_z$, we write: :$x \\equiv y \\pmod z$ and say: : '''$x$ is congruent to $y$ modulo $z$'''."}
+{"_id": "26046", "title": "Definition:Congruence (Number Theory)/Historical Note", "text": "The concept of '''congruence modulo an integer''' was first explored by {{AuthorRef|Carl Friedrich Gauss}}. He originated the notation $a \\equiv b \\pmod m$ in his work {{BookLink|Disquisitiones Arithmeticae|Carl Friedrich Gauss}}, published in $1801$."}
+{"_id": "26047", "title": "Definition:Equivalence Class/Notation", "text": "The notation used to denote an equivalence class varies throughout the literature, but is often some variant on the square bracket motif $\\eqclass x \\RR$. Other variants: * {{BookReference|Lectures in Abstract Algebra|1951|Nathan Jacobson|volume = I|subtitle = Basic Concepts}} uses $\\overline x$ for $\\eqclass x \\RR$. * {{BookReference|Introduction to Topology|1962|Bert Mendelson}} uses $\\map \\pi x$ for $\\eqclass x \\RR$. * {{BookReference|Sets and Groups|1965|J.A. Green}} uses $E_x$ for $\\eqclass x \\RR$. * {{BookReference|Modern Algebra|1965|Seth Warner}} uses $\\bigsqcup_\\RR \\mkern {-28 mu} {\\raise 1pt x} \\ \\ $ for $\\eqclass x \\RR$, which is even more challenging to render in our installed version of $\\LaTeX$ than $\\eqclass x \\RR$ itself. * {{BookReference|A Handbook of Terms used in Algebra and Analysis|1972|A.G. Howson}} uses $\\map {p_\\RR} x$. * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth}} uses $x / \\RR$ for $\\eqclass x \\RR$ (compare the notation for quotient set). This source also suggests $x_\\RR$ as a variant."}
+{"_id": "26048", "title": "Definition:Equivalence Class/Representative", "text": "Let $\\eqclass x \\RR$ be the equivalence class of $x$ under $\\RR$. Let $y \\in \\eqclass x \\RR$. Then $y$ is a '''representative of $\\eqclass x \\RR$."}
+{"_id": "26050", "title": "Definition:Word (Natural Language)", "text": "A '''word''' in natural language is intuitively understood as a sequence of sounds which expresses a concept. When written down, it appears as a sequence of letters, each one of which either is, or contributes to, a [https://en.wikipedia.org/wiki/Phoneme phoneme]."}
+{"_id": "26052", "title": "Definition:Scope of Occurrence", "text": "Let $\\FF$ be a formal language with alphabet $\\AA$. Let $\\mathbf A$ be a well-formed formula of $\\FF$. Let $a$ be a fixed occurrence of some element of $\\AA$ in $\\mathbf A$. Then the '''scope''' of $a$ is the smallest well-formed part of $\\mathbf A$ that contains $a$."}
+{"_id": "26053", "title": "Definition:Universal Closure of Well-Formed Formula", "text": "Let $\\mathcal L_1$ be the language of predicate logic. Let $\\mathbf A$ be a well-formed formula of $\\mathcal L_1$. A '''universal closure''' of $\\mathbf A$ is a sentence $\\mathbf B$ of $\\mathcal L_1$ of the form: :$\\forall x_1: \\cdots \\forall x_n: \\mathbf A$ By definition of sentence, this means that at least the variables occurring freely in $\\mathbf A$ should be quantified over."}
+{"_id": "26056", "title": "Definition:Conjunction/Truth Table/Number", "text": "The truth table number of the conjunction operator $p \\land q$ is as follows: Ascending order: :$0001$ or $\\F \\F \\F \\T$ Descending order: :$1000$ or $\\T \\F \\F \\F$"}
+{"_id": "26057", "title": "Definition:Truth Table/Number", "text": "The '''truth table number''' of the truth table of a statement form is a string consisting of the truth values in numerical form of the main connective. The order in which the '''truth table number''' is presented is usually in ascending or descending numerical order of the truth values of the simple statements contributing to that statement form, either: :Ascending order: $00, 01, 10, 11$, that is: $FF, FT, TF, TT$ or: :Descending order: $11, 10, 01, 00$, that is: $TT, TF, FT, FF$ Which format is being used needs to be specified. Hence the '''truth table number''' can be found by reading either '''up''' or '''down''' the column holding the main connective in the truth table of the statement form."}
+{"_id": "26058", "title": "Definition:Conditional/Truth Table/Number", "text": "The truth table number of the conjunction operator $p \\land q$ is as follows: Ascending order: :$1101$ or $\\T \\T \\F \\T$ Descending order: :$1011$ or $\\T \\F \\T \\T$"}
+{"_id": "26059", "title": "Definition:Disjunction/Truth Table/Number", "text": "The truth table number of the conjunction operator $p \\lor q$ is as follows: Ascending order: :$0111$ or $\\F \\T \\T \\T$ Descending order: :$1110$ or $\\T \\T \\T \\F$"}
+{"_id": "26060", "title": "Definition:Biconditional/Truth Table/Number", "text": "The truth table number of the biconditional operator $p \\iff q$ is as follows: Ascending and descending order: :$1001$ or $\\T \\F \\F \\T$"}
+{"_id": "26061", "title": "Definition:Exclusive Or/Truth Table/Number", "text": "The truth table number of the exclusive or operator $p \\oplus q$ is as follows: Ascending and descending order: :$0110$ or $\\F \\T \\T \\F$"}
+{"_id": "26062", "title": "Definition:Structure for Predicate Logic", "text": "Let $\\LL_1$ be the language of predicate logic. A '''structure $\\AA$ for $\\LL_1$''' comprises: :$(1): \\quad$ A non-empty set $A$; :$(2): \\quad$ For each function symbol $f$ of arity $n$, a mapping $f_\\AA: A^n \\to A$; :$(3): \\quad$ For each predicate symbol $p$ of arity $n$, a mapping $p_\\AA: A^n \\to \\Bbb B$ where $\\Bbb B$ denotes the set of truth values. $A$ is called the '''underlying set''' of $\\AA$. $f_\\AA$ and $p_\\AA$ are called the '''interpretations''' of $f$ and $p$ in $\\AA$, respectively."}
+{"_id": "26063", "title": "Definition:Assignment for Structure/Formula", "text": "Let $\\mathbf A$ be a well-formed formula of $\\mathcal L_1$. Denote with $V \\left({\\mathbf A}\\right)$ the variables which occur freely in $\\mathbf A$. An '''assignment for $\\mathbf A$ in $\\mathcal A$''' is a mapping $\\sigma$ with codomain $A$, whose domain is subject to the following condition: :$V \\left({\\mathbf A}\\right) \\subseteq \\operatorname{dom} \\left({\\sigma}\\right) \\subseteq \\mathrm{VAR}$ That is, the domain of $\\sigma$ contains only variables, and at least those with a free occurrence in $\\mathbf A$."}
+{"_id": "26064", "title": "Definition:Assignment for Structure/Term", "text": "Let $\\tau$ be a term of $\\mathcal L_1$. Denote with $V \\left({\\tau}\\right)$ the variables which occur in $\\tau$. An '''assignment for $\\tau$ in $\\mathcal A$''' is a mapping $\\sigma$ with codomain $A$, whose domain is subject to the following condition: :$V \\left({\\tau}\\right) \\subseteq \\operatorname{dom} \\left({\\sigma}\\right) \\subseteq \\mathrm{VAR}$ That is, the domain of $\\sigma$ contains only variables, and at least those which occur in $\\tau$."}
+{"_id": "26066", "title": "Definition:Language of Predicate Logic/Formal Grammar/Term", "text": "The '''terms''' of $\\LL_1$ are identified by the following bottom-up grammar: {{begin-axiom}} {{axiom | n = \\mathbf T \\ \\textrm {VAR}         | t = Any variable of $\\LL_1$ is a '''term'''; }} {{axiom | n = \\mathbf T \\ \\FF_n         | t = Given an $n$-ary function symbol $f \\in \\FF_n$ and '''terms''' $\\tau_1, \\ldots, \\tau_n$: :$\\map f {\\tau_1, \\ldots, \\tau_n}$ is also a '''term'''. }} {{end-axiom}}"}
+{"_id": "26067", "title": "Definition:Local Minimum in Set of Reals", "text": "Let $X$ be a subset of $\\R$, the set of all real numbers. Let $x$ be a real number, $x \\in \\R$. $x$ is '''local minimum in set''' $X$ {{iff}} :$x \\in X \\land \\exists y \\in \\R: y < x \\land \\left({y \\,.\\,.\\, x}\\right) \\cap X = \\varnothing$"}
+{"_id": "26068", "title": "Definition:Variable/Restricted", "text": "A '''restricted variable''' is a variable whose values are confined to some only of those of which it is capable."}
+{"_id": "26069", "title": "Definition:Variable/Unrestricted", "text": "An '''unrestricted variable''' is a variable whose values are not confined in any way to some only of those of which it is capable."}
+{"_id": "26070", "title": "Definition:Prime Number/Definition 6", "text": "Let $p \\in \\N$ be an integer such that $p \\ne 0$ and $p \\ne \\pm 1$. Then $p$ is a '''prime number''' {{iff}} :$\\forall a, b \\in \\Z: p \\divides a b \\implies p \\divides a$ or $p \\divides b$ where $\\divides$ means '''is a divisor of'''."}
+{"_id": "26071", "title": "Definition:Twin Primes", "text": "'''Twin primes''' are pairs of prime numbers whose difference is $2$."}
+{"_id": "26072", "title": "Definition:Quadratic Residue/Non-Residue", "text": "If there is no such integer $x$ such that $x^2 \\equiv a \\pmod p$, then $a$ is a '''quadratic non-residue of $p$'''."}
+{"_id": "26073", "title": "Definition:Quadratic Residue/Character", "text": "$a$ is either a quadratic residue or a quadratic non-residue of $p$. Whether it is or not is known as the '''quadratic character of $a$ modulo $p$'''."}
+{"_id": "26074", "title": "Definition:Heptagon", "text": "A '''heptagon''' is a polygon with exactly $7$ sides. :300px === Regular Heptagon === {{:Definition:Heptagon/Regular}}"}
+{"_id": "26075", "title": "Definition:Octagon", "text": "An '''octagon''' is a polygon with exactly $8$ sides. :300px === Regular Octagon === {{:Definition:Octagon/Regular}}"}
+{"_id": "26076", "title": "Definition:Nonagon", "text": "A '''nonagon''' is a polygon with exactly $9$ sides. :300px === Regular Nonagon === {{:Definition:Nonagon/Regular}}"}
+{"_id": "26077", "title": "Definition:Hendecagon", "text": "A '''hendecagon''' is a polygon with exactly $11$ sides. :300px === Regular Hendecagon === {{:Definition:Hendecagon/Regular}}"}
+{"_id": "26078", "title": "Definition:Dodecagon", "text": "A '''dodecagon''' is a polygon with exactly $12$ sides. :300px === Regular Dodecagon === {{:Definition:Dodecagon/Regular}}"}
+{"_id": "26079", "title": "Definition:Heptadecagon", "text": "A '''heptadecagon''' is a polygon with exactly $17$ sides. :300px === Regular Heptadecagon === {{:Definition:Heptadecagon/Regular}}"}
+{"_id": "26080", "title": "Definition:Pentagon/Regular", "text": "A '''regular pentagon''' is a pentagon which is both equilateral and equiangular. That is, a regular polygon with $5$ sides. That is, a pentagon in which all the sides are the same length, and all the vertices have the same angle: :300px"}
+{"_id": "26081", "title": "Definition:Hexagon/Regular", "text": "A '''regular hexagon''' is a hexagon which is both equilateral and equiangular. That is, a regular polygon with $6$ sides. That is, a hexagon in which all the sides are the same length, and all the vertices have the same angle: :300px"}
+{"_id": "26082", "title": "Definition:Heptagon/Regular", "text": "A '''regular heptagon''' is a heptagon which is both equilateral and equiangular. That is, a regular polygon with $7$ sides. That is, a heptagon in which all the sides are the same length, and all the vertices have the same angle: :300px"}
+{"_id": "26083", "title": "Definition:Octagon/Regular", "text": "A '''regular octagon''' is an octagon which is both equilateral and equiangular. That is, a regular polygon with $8$ sides. That is, an octagon in which all the sides are the same length, and all the vertices have the same angle: :300px"}
+{"_id": "26084", "title": "Definition:Nonagon/Regular", "text": "A '''regular nonagon''' is a nonagon which is both equilateral and equiangular. That is, a regular polygon with $9$ sides. That is, a nonagon in which all the sides are the same length, and all the vertices have the same angle: :300px"}
+{"_id": "26085", "title": "Definition:Decagon/Regular", "text": "A '''regular decagon''' is a decagon which is both equilateral and equiangular. That is, a regular polygon with $10$ sides. That is, a decagon in which all the sides are the same length, and all the vertices have the same angle: :300px"}
+{"_id": "26086", "title": "Definition:Hendecagon/Regular", "text": "A '''regular hendecagon''' is a hendecagon which is both equilateral and equiangular. That is, a regular polygon with $11$ sides. That is, a hendecagon in which all the sides are the same length, and all the vertices have the same angle: :300px"}
+{"_id": "26087", "title": "Definition:Heptadecagon/Regular", "text": "A '''regular heptadecagon''' is a heptadecagon which is both equilateral and equiangular. That is, a regular polygon with $17$ sides. That is, a heptadecagon in which all the sides are the same length, and all the vertices have the same angle: :300px"}
+{"_id": "26088", "title": "Definition:Dodecagon/Regular", "text": "A '''regular dodecagon''' is a dodecagon which is both equilateral and equiangular. That is, a regular polygon with $12$ sides. That is, a dodecagon in which all the sides are the same length, and all the vertices have the same angle: :300px"}
+{"_id": "26098", "title": "Definition:RSA Algorithm", "text": "The '''RSA algorithm''' is a technique of encoding a message such that the method of encoding can be made public without compromising the security. Let Alice be sending a message to Bob. Alice and Bob both agree on two large primes $p$ and $q$, each having at least $100$ digits. Let $M = p q$. $M$ can be made public if they so wish. Let $K = \\left({p - 1}\\right) \\left({q - 1}\\right)$. $K$ is to be kept secret. Let Alice choose some number $a$ such that $a \\perp K$. $a$ is made known to Bob, and may even be made public. Alice represents her message in a series of numbers in the range $0$ to $M$. Let $x$ be one such number. Alice calculates $y \\equiv x^a \\pmod M$. The sequence of numbers $y$ is sent to Bob as the coded message. Bob needs to know a number $b$ such that $a b \\equiv 1 \\pmod K$. This number needs to be kept secret. To decode $y$, Bob computes $y^b \\pmod M$. This works because of Fermat's Little Theorem: :$y^b \\equiv \\left({x^a}\\right)^b \\equiv x^1 = x \\pmod M$ The method works because: :$(1): \\quad$ There are efficient tests to find large primes :$(2): \\quad$ There are no known methods for finding the prime factors of large numbers efficiently. So making $p q$ public does not help $p$ and $q$ be found, and without those it is impossible to work out what $b$ is. {{NamedforDef|Ronald Linn Rivest|name2 = Adi Shamir|name3 = Leonard Max Adleman|cat = Rivest|cat2 = Shamir|cat3 = Adleman}}"}
+{"_id": "26099", "title": "Definition:Cryptography", "text": "'''Cryptography''' is the branch of mathematics which concerns the encoding of information for secrecy and security."}
+{"_id": "26100", "title": "Definition:Real Function/Multivariable", "text": "Let $f: S_1 \\times S_2 \\times \\cdots \\times S_n \\to \\R$ be a mapping where $S_1, S_2, \\ldots, S_n \\subseteq \\R$. Then $f$ is defined as a '''(real) function of $n$ (independent) variables'''.  The expression: :$y = \\map f {x_1, x_2, \\ldots, x_n}$ means: :(The dependent variable) $y$ is a function of (the independent variables) $x_1, x_2, \\ldots, x_n$."}
+{"_id": "26101", "title": "Definition:Derivative/Real Function/Derivative at Point/Definition 1", "text": "That is, suppose the limit $\\displaystyle \\lim_{x \\mathop \\to \\xi} \\frac {f \\left({x}\\right) - f \\left({\\xi}\\right)} {x - \\xi}$ exists. Then this limit is called the '''derivative of $f$ at the point $\\xi$'''."}
+{"_id": "26102", "title": "Definition:Derivative/Real Function/Derivative at Point/Definition 2", "text": "That is, suppose the limit $\\displaystyle \\lim_{h \\mathop \\to 0} \\frac {\\map f {\\xi + h} - \\map f \\xi} h$ exists. Then this limit is called the '''derivative of $f$ at the point $\\xi$'''."}
+{"_id": "26103", "title": "Definition:Secant Line", "text": "Let $f: \\R \\to \\R$ be a real function. Let the graph of $f$ be depicted on a Cartesian plane. :400px A '''secant''' to $f$ is a straight line which intersects the graph of $f$ in (at least) two points. In the above diagram, the '''secant''' is the line $AB$ in $\\color {blue} {\\text {blue} }$."}
+{"_id": "26104", "title": "Definition:Tangent/Analytic Geometry", "text": "Let $f: \\R \\to \\R$ be a real function. Let the graph of $f$ be depicted on a Cartesian plane. :400px Let $A = \\tuple {x, \\map f x}$ be a point on $G$. The '''tangent to $f$ at $A$''' is defined as: :$\\displaystyle \\lim_{h \\mathop \\to 0} \\frac {\\map f {x + h} - \\map f x} h$ Thus '''tangent to $f$ at $x$''' can be considered as the secant $AB$ to $G$ where: :$B = \\tuple {x + h, \\map f {x + h} }$ as $B$ gets closed and closer to $A$. By taking $h$ smaller and smaller, the secant approaches more and more closely the tangent to $G$ at $A$. Hence the '''tangent''' to $f$ is a straight line which intersects the graph of $f$ locally at a single point. :400px In the above diagram, the '''tangent''' is the straight line passing through $A$."}
+{"_id": "26105", "title": "Definition:Celestial Mechanics", "text": "'''Celestial mechanics''' is the branch of astronomy which studies the motion of celestial bodies such as planets, stars and comets."}
+{"_id": "26107", "title": "Definition:Orbit (Physics)", "text": "The '''orbit''' of body $A$ around body $B$ is the path taken by $A$ as it travels around $B$ under the influence of the force acting between $A$ and $B$. === Period of Orbit === {{:Definition:Orbit (Physics)/Period}}"}
+{"_id": "26108", "title": "Definition:Orbit (Physics)/Period", "text": "The '''period''' of an orbit of body $A$ around body $B$ is the length of time it takes for $A$ to travel once around $B$ and return to its original position."}
+{"_id": "26109", "title": "Definition:Astrophysics", "text": "'''Astrophysics''' is the branch of physics which studies the internal nature of celestial bodies as opposed to their positions and movements."}
+{"_id": "26110", "title": "Definition:Celestial Body", "text": "A '''celestial body''' is a more-or-less internally coherent body of matter that exists outside the Earth's atmosphere."}
+{"_id": "26111", "title": "Definition:Planet", "text": "A '''planet''' is a celestial body which orbits a star (or the remains of a star) that: :$(1): \\quad$ has enough mass so that its own gravitational field makes it approximately spherical :$(2): \\quad$ does not have enough mass to trigger thermonuclear fusion :$(3): \\quad$ has cleared its neighbouring region of planetesimals."}
+{"_id": "26112", "title": "Definition:Star (Physics)", "text": "A '''star''' is a celestial body that consists of a luminous sphere of plasma held together by its own gravitational field."}
+{"_id": "26114", "title": "Definition:Sun", "text": "The '''Sun''' is the star around which Earth (and the other planets of the solar system) is in orbit."}
+{"_id": "26115", "title": "Definition:Earth", "text": "'''Earth''' is the planet on which we live. It is the third planet of the solar system from the sun. Its orbit lies between those of Venus and Mars."}
+{"_id": "26116", "title": "Definition:Solar System", "text": "The '''solar system''' is the system of celestial bodies which are under the direct influence of the gravitational field of the sun."}
+{"_id": "26117", "title": "Definition:Mercury (Planet)", "text": "'''Mercury''' is the innermost planet of the solar system. Its orbit lies within that of Venus. Category:Definitions/Solar System plp5e2aklqfluk6yspee40yd9qjgu7b"}
+{"_id": "26118", "title": "Definition:Venus", "text": "'''Venus''' is the second planet of the solar system from the sun. Its orbit lies between those of Mercury and Earth. Category:Definitions/Solar System 24j4o8crdk8571medw9pcgt6ngffo04"}
+{"_id": "26119", "title": "Definition:Mars", "text": "'''Mars''' is the fourth planet of the solar system from the sun. Its orbit lies between those of Earth and Jupiter."}
+{"_id": "26120", "title": "Definition:Jupiter", "text": "'''Jupiter''' is the fifth planet of the solar system from the sun. Its orbit lies between those of Mars and Saturn."}
+{"_id": "26121", "title": "Definition:Saturn", "text": "'''Saturn''' is the sixth planet of the solar system from the sun. Its orbit lies between those of Jupiter and Uranus. Category:Definitions/Solar System 158qpaeeuovkq9yz648dncuu4nzyb1r"}
+{"_id": "26122", "title": "Definition:Uranus", "text": "'''Uranus''' is the seventh planet of the solar system from the sun. Its orbit lies between those of Saturn and Neptune. Category:Definitions/Solar System mavyjtjplsijg8xj0d8uvoilw3g6ur3"}
+{"_id": "26123", "title": "Definition:Neptune", "text": "'''Neptune''' is the eighth planet of the solar system from the sun. Its orbit lies mainly between those of Uranus and Pluto, although the latter comes closer to the sun than '''Neptune''' on part of its orbit."}
+{"_id": "26124", "title": "Definition:Pluto", "text": "'''Pluto''' is a dwarf planet of the solar system. Its orbit lies mostly outside that of Neptune. Category:Definitions/Solar System jljizc0dsjw0y2cdo2fyfxxb9rbpteb"}
+{"_id": "26125", "title": "Definition:Dwarf Planet", "text": "A '''dwarf planet''' is a celestial body which orbits a star (or the remains of a star) that: :$(1): \\quad$ has enough mass so that its own gravitational field makes it approximately spherical :$(2): \\quad$ does not have enough mass to trigger thermonuclear fusion :$(3): \\quad$ has not cleared its neighbouring region of planetesimals."}
+{"_id": "26126", "title": "Definition:Planetesimal", "text": "A '''planetesimal''' is a celestial body consisting of solid matter such that: :$(1): \\quad$ it is held together by its own gravitational field :$(2): \\quad$ it is not large enough for its gravitational field to significantly affect other celestial bodies nearby."}
+{"_id": "26127", "title": "Definition:Time/Dimension", "text": "'''Time''' is one of the fundamental dimensions of physics. In dimensional analysis it is assigned the symbol $T$ or $\\mathbf T$. It is a scalar quantity which can be mapped directly to the real number line."}
+{"_id": "26128", "title": "Definition:Time/Unit Time", "text": "The usual units of '''time''' are seconds, and compound units consisting of multiples of seconds. However, when discussing a general physical process it is convenient to discuss the general situation, in which case the term '''unit time''' is used. Thus the specific '''units''' are not mentioned."}
+{"_id": "26129", "title": "Definition:Time/Length", "text": "A quantity of '''time''' is usually referred to in natural language in the same terms as linear measure. Thus it is usual to refer to a '''length of time''' of an elapsed quantity of time."}
+{"_id": "26130", "title": "Definition:Time/Unit", "text": "The units of measurement of time are universal in all systems of measurement. === Second === {{:Definition:Time/Unit/Second}} Derived units of time include: === Minute === {{:Definition:Time/Unit/Minute}} === Hour === {{:Definition:Time/Unit/Hour}} === Day === {{:Definition:Time/Unit/Day}} There are longer time periods, all of which are more or less approximate: === Week === {{:Definition:Time/Unit/Week}} === Fortnight === {{:Definition:Time/Unit/Fortnight}} === Month === {{:Definition:Time/Unit/Month}} === Lunar Month === {{:Definition:Time/Unit/Lunar Month}} === Season === {{:Definition:Time/Unit/Season}} === Quarter-Year === {{:Definition:Time/Unit/Quarter-Year}} === Year === {{:Definition:Time/Unit/Year}} === Decade === {{:Definition:Time/Unit/Decade}} === Century === {{:Definition:Time/Unit/Century}} === Millennium === {{:Definition:Time/Unit/Millennium}} === Great Year of Plato === {{:Definition:Time/Unit/Great Year of Plato}}"}
+{"_id": "26131", "title": "Definition:Derived Unit", "text": "'''Derived units''' are units of measurement created by taking specific multiples or fractions of fundamental units."}
+{"_id": "26132", "title": "Definition:System of Measurement", "text": "A '''system of measurement''' is a set of fundamental units of measurement with which one can measure any measurable physical property."}
+{"_id": "26133", "title": "Definition:Time/Unit/Minute", "text": "The '''minute''' is a derived unit of time. :$1$ '''minute''' $= 60$ seconds."}
+{"_id": "26134", "title": "Definition:Time/Unit/Second/Symbol", "text": "The symbol for the '''second''' is $\\mathrm s$."}
+{"_id": "26135", "title": "Definition:Time/Unit/Minute/Symbol", "text": "The symbol for the '''minute''' is $\\mathrm {min}$."}
+{"_id": "26136", "title": "Definition:Time/Unit/Hour/Symbol", "text": "The symbol for the '''hour''' is $\\mathrm {hr}$."}
+{"_id": "26137", "title": "Definition:Time/Unit/Hour", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''hour''' }} {{eqn | r = 60       | c = minutes }} {{eqn | r = 60 \\times 60       | rr= = 3600       | c = seconds }} {{end-eqn}}"}
+{"_id": "26138", "title": "Definition:Time/Unit/Day", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''day''' }} {{eqn | r = 24       | c = hours }} {{eqn | r = 60 \\times 24       | rr= = 1440       | c = minutes }} {{eqn | r = 60 \\times 60 \\times 24       | rr= = 86\\, 400       | c = seconds }} {{end-eqn}}"}
+{"_id": "26139", "title": "Definition:Time/Unit/Day/Symbol", "text": "The symbol for the '''day''' is $\\mathrm {day}$ or $\\mathrm d$."}
+{"_id": "26140", "title": "Definition:Sidereal Day", "text": "A '''sidereal day''' is (approximately) the length of time it takes for Earth to rotate once on its axis relative to the stars. The mean duration of a '''sidereal day''' is: : $23$ hours, $56$ minutes and $4.0916$ seconds : $23.9344699$ hours : $0.99726958$ (solar) days. The duration of any given '''sidereal day''' is not constant because of various astronomical effects."}
+{"_id": "26142", "title": "Definition:Algorithm/Input", "text": "An '''algorithm''' has zero or more '''inputs'''. These are values supplied either: : before the algorithm starts : as the algorithm runs. These '''inputs''' are taken from specified sets of objects."}
+{"_id": "26143", "title": "Definition:Algorithm/Output", "text": "An '''algorithm''' has one or more '''outputs'''. These are values which are specifically determined by the inputs."}
+{"_id": "26144", "title": "Definition:Computational Problem", "text": "A '''computational problem''' is a mathematical object which represents a collection of questions that are to be solved by algorithmic techniques."}
+{"_id": "26145", "title": "Definition:Algorithm/Finiteness", "text": "An '''algorithm''' must '''terminate''' after a finite number of '''steps'''."}
+{"_id": "26146", "title": "Definition:Algorithm/Definiteness", "text": "Each '''step''' of an algorithm must be '''precisely defined'''."}
+{"_id": "26147", "title": "Definition:Algorithm/Effectiveness", "text": "An '''algorithm''' is supposed to be '''effective'''. That is, its '''operations''' must be basic enough to be able to be done '''exactly''' and in a finite length of time by, for example, somebody using pencil and paper."}
+{"_id": "26148", "title": "Definition:Algorithm/Analysis", "text": "There are two primary methods for analyzing '''algorithms''' formally: === Correctness === {{:Definition:Algorithm/Analysis/Correctness}} === Complexity === {{:Definition:Algorithm/Analysis/Complexity}}"}
+{"_id": "26150", "title": "Definition:Algorithm/Analysis/Complexity", "text": "An algorithm may be analysed in the context of its '''complexity'''. This is typically via measures of run-times using big-O, or big-omega, or big-theta notation. Complexity can be through 3 main types of analysis: # Average-Case Analysis, using a distribution of cases to find an average case of the algorithm run-time for analysis # Best-Case Analysis, using the best possible problem instance of the algorithm for analysis # Worst-Case Analysis,using the worst possible problem instance of the algorithm for analysis"}
+{"_id": "26151", "title": "Definition:Stationary", "text": "A body is '''stationary''' if its velocity (in a given frame of reference) is zero."}
+{"_id": "26152", "title": "Definition:Infinitesimal", "text": "An '''infinitesimal''' is a mathematical object $\\delta$ resembling a (real) number such that: :$(1): \\quad \\delta > 0$ :$(2): \\quad \\forall x \\in \\R_{>0}: \\delta < x$ That is, an '''infinitesimal''' is a (strictly) positive real number which is smaller than every other (strictly) positive real number."}
+{"_id": "26154", "title": "Definition:Solid of Revolution", "text": "Let $F$ be a plane figure. Let $L$ be a straight line in the same plane as $F$. Let $F$ be rotated one full turn about $L$ in $3$-dimensional space. The '''solid of revolution generated by $F$ around $L$''' is the solid figure whose boundaries are the paths taken by the sides of $F$ as they go round $L$."}
+{"_id": "26157", "title": "Definition:Differential Equation/Order", "text": "The '''order''' of a '''differential equation''' is defined as being the order of the highest order derivative that is present in the equation."}
+{"_id": "26158", "title": "Definition:Differential Equation/Ordinary", "text": "An '''ordinary differential equation''' (abbreviated '''O.D.E.''' or '''ODE''') is a '''differential equation''' which has exactly one independent variable. All the derivatives occurring in it are therefore ordinary. The general '''ODE''' of order $n$ is: :$\\map f {x, y, \\dfrac {\\d x} {\\d y}, \\dfrac {\\d^2 x} {\\d y^2}, \\ldots, \\dfrac {\\d^n x} {\\d y^n} } = 0$ or, using the prime notation: :$\\map f {x, y, y', y'', \\ldots, y^{\\paren n} } = 0$"}
+{"_id": "26159", "title": "Definition:Differential Equation/Partial", "text": "A '''partial differential equation''' (abbreviated '''P.D.E.''' or '''PDE''') is a '''differential equation''' which has: :one dependent variable :more than one independent variable. The derivatives occurring in it are therefore partial."}
+{"_id": "26161", "title": "Definition:Differential Equation/Linear", "text": "A '''linear differential equation''' is a '''differential equation''' where all dependent variables and their derivatives appear to the first power. Neither are products of dependent variables allowed."}
+{"_id": "26162", "title": "Definition:Differential Equation/Non-Linear", "text": "A '''non-linear differential equation''' is a '''differential equation''' which is not linear."}
+{"_id": "26163", "title": "Definition:Differential Equation/System", "text": "A '''system of differential equations''' is a set of simultaneous '''differential equations'''. The solutions for each of the differential equations are in general expected to be consistent."}
+{"_id": "26164", "title": "Definition:Differential Equation/System/Autonomous", "text": "A '''system of differential equations''' is '''autonomous''' if all of the '''differential equations''' which it comprises are themselves autonomous."}
+{"_id": "26165", "title": "Definition:Differential Equation/Explicit", "text": "A '''differential equation''' is called '''explicit''' if it can be written in the form: :$y^{\\left({n}\\right)} = f \\left({x, y, y', y'', \\dots, y^{\\left({n-1}\\right)}}\\right)$"}
+{"_id": "26167", "title": "Definition:Differential Equation/Autonomous", "text": "A '''differential equation''' is '''autonomous''' if none of the derivatives depend on the independent variable. The $n$th order '''autonomous differential equation''' takes the form: :$y^{\\paren n} = \\map f {y, y', y'', \\dots, y^{\\paren {n - 1} } }$"}
+{"_id": "26168", "title": "Definition:Legendre's Differential Equation", "text": ":$\\displaystyle \\paren {1 - x^2} \\frac {\\d^2 y} {\\d x^2} - 2 x \\frac {\\d y} {\\d x} + p \\paren {p + 1} y = 0$"}
+{"_id": "26169", "title": "Definition:Integral Curve", "text": "Let $\\map f {x, y}$ be a continuous real function within a rectangular region $R$ of the Cartesian plane. Consider the first order ODE: :$(1): \\quad \\dfrac {\\d y} {\\d x} = \\map f {x, y}$ :500px Let $P_0 = \\tuple {x_0, y_0}$ be a point in $R$. The number: :$\\paren {\\dfrac {\\d y} {\\d x} }_{P_0} = \\map f {x_0, y_0}$ determines the slope of a straight line passing through $P_0$. Let $P_1 = \\tuple {x_1, y_1}$ be a point in $R$ close to $P_0$. Then: :$\\paren {\\dfrac {\\d y} {\\d x} }_{P_1} = \\map f {x_1, y_1}$ determines the slope of a straight line passing through $P_1$. Let $P_2 = \\tuple {x_2, y_2}$ be a point in $R$ close to $P_1$. Then: :$\\paren {\\dfrac {\\d y} {\\d x} }_{P_2} = \\map f {x_2, y_2}$ determines the slope of a straight line passing through $P_2$. Continuing this process, we obtain a curve made of a sequence of straight line segments. As successive points $P_0, P_1, P_2, \\ldots$ are taken closer and closer to each other, the sequence of straight line segments $P_0 P_1 P_2 \\ldots$ approaches a smooth curve $C$ passing through an initial point $P_0$. By construction, for each point $\\tuple {x, y}$ on $C$, the slope of $C$ at $\\tuple {x, y}$ is given by $\\map f {x, y}$. Hence this curve is a solution to $(1)$. Starting from a different point, a different curve is obtained. Thus the general solution to $(1)$ takes the form of a set of curves. This set of curves are referred to collectively as '''integral curves'''."}
+{"_id": "26170", "title": "Definition:Parameter of Differential Equation", "text": "Let $f$ be a differential equation with general solution $F$. A '''parameter''' of $F$ is an arbitrary constant arising from the solving of a primitive during the course of obtaining the solution of $f$."}
+{"_id": "26172", "title": "Definition:Substance", "text": "A '''substance''' is a particular kind of matter with properties specific to it."}
+{"_id": "26173", "title": "Definition:First-Order Reaction", "text": "Suppose a body has a tendency to decompose spontaneously into smaller bodies at a rate independent of the presence of other bodies. Then the number of bodies that decompose in a single unit of time is proportional to the total number present. Such a reaction is called a '''first-order reaction'''."}
+{"_id": "26174", "title": "Definition:Matter", "text": "'''Matter''' is the stuff of which the physical universe is composed. At a certain level, can be considered as a form of energy. '''Matter''' is (to a first degree of approximation) composed of atoms."}
+{"_id": "26175", "title": "Definition:Chemical Element", "text": "A '''chemical element''' is a substance which is composed of atoms of one type. A '''chemical element''' is characterised uniquely by the number of protons it has in the nucleus of its atoms."}
+{"_id": "26176", "title": "Definition:Atom (Physics)", "text": "An '''atom''' (in the context of physics and chemistry) is the smallest piece of matter that can exist of a particular type of substance. '''Atoms''' can be subdivided into smaller particles, but then it ceases to be that substance."}
+{"_id": "26177", "title": "Definition:Radioactive Decay", "text": "'''Radioactive decay''' is a first-order reaction in which the atoms of a substance may spontaneously break apart, thereby turning into atoms of a different substance."}
+{"_id": "26178", "title": "Definition:Radioactive Decay/Half-Life", "text": "The '''half-life''' of a radioactive isotope is the time it takes for exactly half of an arbitrary quantity of that isotope to undergo radioactive decay. Thus it is the time it takes for exactly half of an arbitrary quantity of that isotope to remain."}
+{"_id": "26179", "title": "Definition:Radioactive Decay/Radioactive Element", "text": "A chemical element whose isotopes are ''all'' subject to '''radioactive decay''' is known as a '''radioactive element'''."}
+{"_id": "26181", "title": "Definition:First-Order Reaction/Rate Constant", "text": "Let $S$ be a substance which decomposes spontaneously in a '''first-order reaction'''. By First-Order Reaction, this is governed by the equation: :$x = x_0 e^{-k t}$ where: :$x$ is the quantity of $S$ at time $t$ :$x_0$ is the quantity of $S$ at time $t = 0$ :$k$ is a positive number. The number $k$ is called the '''rate constant'''. $k$ is constant for a particular substance, and different substances, in general, have different '''rate constants'''."}
+{"_id": "26182", "title": "Definition:Chemical Element/Isotope", "text": "'''Isotopes''' are instances of chemical elements which have the same number of protons in the nuclei of their atoms, but different numbers of neutrons. Thus it is usual to discuss '''an isotope''' of a particular element."}
+{"_id": "26183", "title": "Definition:Atom (Physics)/Nucleus", "text": "The '''nucleus''' of an atom is the relatively dense core at its center. It consists (to a first degree of approximation) of a number of protons and a similar number of neutrons."}
+{"_id": "26186", "title": "Definition:Elementary Particle", "text": "An '''elementary particle''' is one of the basic building blocks of matter. To a first degree of approximation, those building blocks can be considered to be protons, neutrons and electrons. Modern research in particle physics suggests the existence of smaller constituents of protons and neutrons, but at time of writing, electrons are believe to be truly, fundamentally indivisible. Category:Definitions/Particle Physics 2yejk9j01j4hb0trbc4tynec4cuny6z"}
+{"_id": "26187", "title": "Definition:Particle Physics", "text": "'''Particle physics''' is the branch of physics which studies the structure and interactions of the elementary particles of matter."}
+{"_id": "26189", "title": "Definition:Radioactive Decay/Radioactive Isotope", "text": "An isotope of a chemical element which is subject to '''radioactive decay''' is known as a '''radioactive isotope'''."}
+{"_id": "26190", "title": "Definition:Time/Unit/Year", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''year''' }} {{eqn | o = \\approx       | r = 365 \\cdotp 24219 \\, 878       | c = days }} {{eqn | o = \\approx       | r = 365 \\cdotp 24219 \\, 878 \\times 24 \\times 60 \\times 60       | c = seconds }} {{end-eqn}} That is: {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''year''' }} {{eqn | o = \\approx       | r = 365       | c = days, }} {{eqn | o =        | r = 5       | c = hours, }} {{eqn | o =        | r = 48       | c = minutes, and }} {{eqn | o =        | r = 45 \\cdotp 9747       | c = seconds }} {{end-eqn}} It is defined to be equal to the orbital period of Earth round the sun."}
+{"_id": "26191", "title": "Definition:Time/Unit/Year/Symbol", "text": "The symbol for the '''year''' is $\\mathrm {yr}$ or $\\mathrm y$. In some scientific contexts $\\mathrm a$ (for '''annus''', Latin for '''year''') is used."}
+{"_id": "26192", "title": "Definition:Time/Unit/Year/Calendar", "text": "A '''calendar year''' is the number of days between dates with the same designation in adjacent years. As a '''year''' is not an integral number of days, it is necessarily an approximation to the actual year. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''calendar year''' }} {{eqn | r = 12       | c = months usually, depending on the calendar used }} {{eqn | r = 365 \\text{ or } 366       | c = days, usually, but it depends on the calendar used }} {{end-eqn}}"}
+{"_id": "26193", "title": "Definition:Radioactive Decay/Stable", "text": "An isotope of a chemical element which is ''not'' subject to '''radioactive decay''' is described as '''stable'''."}
+{"_id": "26196", "title": "Definition:Density of Topological Space", "text": "Let $T$ be a topological space. The '''density''' of $T$ is the smallest cardinality of dense subset of $T$: :$d \\left({T}\\right) := \\min \\left\\{{\\left\\vert{A}\\right\\vert: A \\subseteq T \\land A}\\right.$ is dense$\\left.{}\\right\\}$"}
+{"_id": "26203", "title": "Definition:Chemical Compound", "text": "A '''(chemical) compound''' is a (chemical) substance that consists of molecules composed of more than one type of chemical element."}
+{"_id": "26204", "title": "Definition:Molecule", "text": "A '''molecule''' is an arrangement of atoms, possibly of different elements, which are held together in a configuration which requires the application of external energy to break apart. The '''molecule''' is the basic unit of a chemical compound."}
+{"_id": "26205", "title": "Definition:Chemical Reaction", "text": "A '''chemical reaction''' is a physical process in which the atoms  composing the molecules of the chemical substances involved change their configurations into different molecules. Category:Definitions/Chemistry Category:Definitions/Physics qoic8xfj9qwghpvwl74fqa2ik9qbkue"}
+{"_id": "26206", "title": "Definition:Second-Order Reaction", "text": "Let $\\AA$ and $\\BB$ be two chemical substances in solution which react together to form a compound $\\CC$. Let the reaction occur by means of the molecules of $\\AA$ and $\\BB$ colliding and interacting as a result. Then the rate of formation of $\\CC$ is proportional to the number of collisions in unit time. This in turn is jointly proportional to the quantities of $\\AA$ and $\\BB$ which have not yet transformed. Such a chemical reaction is called a '''second-order reaction'''."}
+{"_id": "26207", "title": "Definition:Thermodynamics", "text": "'''Thermodynamics''' is the branch of physics which studies the behaviour of heat."}
+{"_id": "26208", "title": "Definition:Heat", "text": "'''Heat''' is a form of energy which is held within a body in the form of the physical motion of the molecules or atoms of which that body is composed. Category:Definitions/Physics Category:Definitions/Thermodynamics dij5rrbowaoa9k8gdp108dyza63xjcl"}
+{"_id": "26209", "title": "Definition:Energy", "text": "'''Energy''' is a property of physical bodies which can be defined as the ability to do work. Like time, it is difficult to define exactly what '''energy''' actually ''is''."}
+{"_id": "26210", "title": "Definition:Linear Measure/Thickness", "text": "'''Thickness''', like breadth, is used as a term for linear measure in a dimension perpendicular to both length and depth. However, whereas breadth has connotations of '''across''', '''thickness''' is used for distances '''through''' the solid figure."}
+{"_id": "26211", "title": "Definition:Kinetic Energy", "text": "Let $B$ be a body. The '''kinetic energy''' of $B$ is defined as the energy appertaining to $B$ due to its motion. It is a scalar quantity."}
+{"_id": "26212", "title": "Definition:Potential Energy", "text": "Let $B$ be a body. The '''potential energy''' of $B$ is the energy appertaining to $B$ as a result of its position in a force field or because of the internal configuration of its components."}
+{"_id": "26213", "title": "Definition:Force Field", "text": "A '''force field''' is a field in which the physical quantity associated with every point is a force which applies to a body placed at that point. Category:Definitions/Physics lg50nsn2tmwne5pkz9xntjsy0t777br"}
+{"_id": "26214", "title": "Definition:Constant (Physics)", "text": "A physical quantity is defined as being '''constant''' if it does not change with the passage of time."}
+{"_id": "26215", "title": "Definition:Pendulum/Ideals", "text": "The ideals of a pendulum are as follows: :The rod is of zero thickness, infinite stiffness and zero mass. :The pivot is immovable and  has zero friction. :The bob is of zero volume. :The gravitation field is (usually) taken to be constant and uniform. :The medium in which the pendulum moves offers no resistive force, unless otherwise specifically stated."}
+{"_id": "26216", "title": "Definition:Pendulum/Bob", "text": "The point mass at the free end of the rod of a pendulum is called the '''(pendulum) bob'''."}
+{"_id": "26217", "title": "Definition:Pendulum", "text": "A '''pendulum''' is an ideal body consisting of two parts: :$(1): \\quad$ a rod attached at one end to a pivot :$(2): \\quad$ a point mass at the other end of the rod."}
+{"_id": "26219", "title": "Definition:Rod", "text": "A '''rod''' is a rigid straight body whose width and mass are approximated to zero."}
+{"_id": "26221", "title": "Definition:Escape Velocity", "text": "Let a body be propelled vertically upwards from the surface of a planet Let the body keep travelling away from the planet indefinitely. Let resistance from the atmosphere of the planet be considered as negligible. The smallest initial speed at which this is possible is known as the '''escape velocity''' of the planet."}
+{"_id": "26222", "title": "Definition:Terminal Velocity", "text": "Let a body travelling under a constant force through a medium which resists its progress. The '''terminal velocity''' of the body is the limiting speed through that medium for the given force."}
+{"_id": "26223", "title": "Definition:Speed/Also known as", "text": "Some authors erroneously or carelessly refer to the '''speed''' of a body as its velocity. But this is technically wrong if the author does not specify its direction as well as its magnitude."}
+{"_id": "26224", "title": "Definition:Sine/Real Function/Arch", "text": "Each section of the sine function between adjacent zeroes is called an '''arch''' of the sine function"}
+{"_id": "26230", "title": "Definition:Lemniscate of Bernoulli/Lobe", "text": "Each of the two loops that constitute the '''lemniscate''' can be referred to as a '''lobe''' of the lemniscate."}
+{"_id": "26248", "title": "Definition:Brachistochrone", "text": "Let a point $A$ be joined by a wire to a lower point $B$. Let the wire be allowed to be bent into whatever shape is required. Let a bead be released at $A$ to slide down without friction to $B$. The shape of the wire so that the bead takes least time to descend from $A$ to $B$ is called the '''brachistochrone'''."}
+{"_id": "26249", "title": "Definition:Wire", "text": "A '''wire''' is a rigid body modelled by a specific curve whose width and mass are approximated to zero."}
+{"_id": "26253", "title": "Definition:Cycloid/Generating Circle", "text": "The circle which rolls along the straight line is called the '''generating circle''' of the cycloid."}
+{"_id": "26254", "title": "Definition:Confocal Conics", "text": "Let $F$ be a set of conic sections. Then $F$ is a set of '''confocal conics''' {{iff}} they all share the same foci."}
+{"_id": "26255", "title": "Definition:Self-Orthogonal Trajectories", "text": "Let $\\map f {x, y, c}$ define a one-parameter family of curves $F$. Let the family of orthogonal trajectories of $F$ be $F$ itself. Then $\\map f {x, y, c}$ is '''self-orthogonal'''."}
+{"_id": "26256", "title": "Definition:One-Parameter Family of Curves/Parameter", "text": "The value $c$ is the '''parameter''' of $F$."}
+{"_id": "26257", "title": "Definition:T1/2 Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Then $T$ is a '''$T_{\\frac 1 2}$ space''' {{iff}}: :$\\forall A \\subseteq S: A'$ is closed where $A'$ denotes the derivative of $A$."}
+{"_id": "26258", "title": "Definition:Confocal Ellipses", "text": "Let $F$ be a set of ellipses. Then $F$ is a set of '''confocal ellipses''' {{iff}} they all share the same foci. Category:Definitions/Conic Sections l20rp6jtvnamympx7ddmlkq7prci5is"}
+{"_id": "26259", "title": "Definition:Confocal Hyperbolas", "text": "Let $F$ be a set of hyperbolas. Then $F$ is a set of '''confocal hyperbolas''' {{iff}} they all share the same foci. Category:Definitions/Conic Sections r8883h0y4x4pw2nxv9c8eyajujtws6h"}
+{"_id": "26266", "title": "Definition:Asymptote", "text": ":500pxrightthumbAsymptotes to $y = x + \\dfrac 1 x$ (blue): [[Definition:Y-Axis$y$-axis, and the line $x = y$ (red)]] Let $C$ be a curve in the plane. Let $L$ be a straight (infinite) line such that the distance between $C$ and $L$ approaches zero as they tend to infinity. Then $L$ is an '''asymptote to''' (or '''of''') $C$."}
+{"_id": "26268", "title": "Definition:Conic Section/Center", "text": "The '''center''' of a section is the point midway between the foci. === Center of Circle === The circle is usually treated differently. From Circle has Two Coincident Foci, the foci and the center are all the same point. :300px {{:Definition:Circle/Center}} === Center of Ellipse === Let $K$ be an ellipse. :500px {{:Definition:Ellipse/Center}} === Center of Hyperbola === Let $K$ be a hyperbola. :500px {{:Definition:Hyperbola/Center}} === Center of Parabola === {{:Definition:Parabola/Center}}"}
+{"_id": "26270", "title": "Definition:Conic Section/Reduced Form", "text": "Let $K$ be a conic section. Let $K$ be embedded in a cartesian plane such that: :$(1)$ the center is at the origin :$(2)$ the foci are at $\\left({\\pm c, 0}\\right)$ for some $c \\in \\R_{\\ge 0}$. This can be interpreted in the contexts of the specific classes of conic section as follows: === Reduced Form of Ellipse === {{:Definition:Conic Section/Reduced Form/Ellipse}} === Reduced Form of Hyperbola === {{:Definition:Conic Section/Reduced Form/Hyperbola}} === Reduced Form of Circle === {{:Definition:Conic Section/Reduced Form/Circle}} === Reduced Form of Parabola === {{:Definition:Conic Section/Reduced Form/Parabola}}"}
+{"_id": "26271", "title": "Definition:Parabola/Center", "text": "The parabola has no center. From Parabola has One Focus, the definition of the center as being the point midway between the foci, it follows that for a parabola the definition is inapplicable."}
+{"_id": "26273", "title": "Definition:Conic Section/Reduced Form/Ellipse", "text": "Let $K$ be an ellipse embedded in a cartesian plane. $K$ is in '''reduced form''' {{iff}}: :$(1)$ its major axis is aligned with the $x$-axis :$(2)$ its minor axis is aligned with the $y$-axis. :500px"}
+{"_id": "26274", "title": "Definition:Conic Section/Reduced Form/Hyperbola", "text": "Let $K$ be a hyperbola embedded in a cartesian plane. $K$ is in '''reduced form''' {{iff}}: :$(1)$ its major axis is aligned with the $x$-axis :$(2)$ its minor axis is aligned with the $y$-axis. :500px"}
+{"_id": "26275", "title": "Definition:Conic Section/Reduced Form/Circle", "text": "Let $K$ be a circle embedded in a cartesian plane. $K$ is in '''reduced form''' {{iff}} its center is located at the origin. :500px"}
+{"_id": "26276", "title": "Definition:Conic Section/Reduced Form/Parabola", "text": "Let $K$ be a parabola embedded in a cartesian plane. As a Parabola has no Center, it is not possible to define the reduced form of a parabola in the same way as for the other classes of conic section. Instead, $K$ is in '''reduced form''' {{iff}}: :$(1)$ its focus is at the point $\\tuple {c, 0}$ :$(2)$ its directrix is aligned with the line $x = -c$ for some $c \\in \\R_{> 0}$. :400px"}
+{"_id": "26277", "title": "Definition:Hyperbola/Equidistance", "text": "&nbsp; :400px Let $F_1$ and $F_2$ be two points in the plane. Let $d$ be a length less than the distance between $F_1$ and $F_2$. Let $K$ be the locus of points $P$ which are subject to the condition: :$\\size {d_1 - d_2} = d$ where: :$d_1$ is the distance from $P$ to $F_1$ :$d_2$ is the distance from $P$ to $F_2$ :$\\size {d_1 - d_2}$ denotes the absolute value of $d_1 - d_2$. Then $K$ is a '''hyperbola'''."}
+{"_id": "26278", "title": "Definition:Set of Condensation Points", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A$ be a subset of $S$. The '''set of condensation points''' of $A$ is the set of all condensation points of $A$, :$A^0 = \\left\\{{x \\in S:x}\\right.$ is condensation point of $\\left.{A}\\right\\}$"}
+{"_id": "26279", "title": "Definition:Homogeneous Function/Real Space/Degree", "text": "The integer $n$ is known as the '''degree''' of $f$."}
+{"_id": "26280", "title": "Definition:Homogeneous Function/Real Space", "text": "Let $f: \\R^2 \\to \\R$ be a real-valued function of two variables. $f \\left({x, y}\\right)$ is a '''homogeneous function of degree zero''' {{iff}}: :$\\exists n \\in \\Z: \\forall t \\in \\R: f \\left({t x, t y}\\right) = t^n f \\left({x, y}\\right)$"}
+{"_id": "26282", "title": "Definition:Homogeneous Function/Real Space/Zero Degree", "text": "$f \\left({x, y}\\right)$ is a '''homogeneous function of degree zero''' or '''of zero degree''' {{iff}}: :$\\forall t \\in \\R: f \\left({t x, t y}\\right) = t^0 f \\left({x, y}\\right) = f \\left({x, y}\\right)$"}
+{"_id": "26284", "title": "Definition:Linear Ordinary Differential Equation", "text": "A '''linear ordinary differential equation''' is an '''ordinary differential equation''' where all dependent variables and their derivatives appear to the first power. Neither are products of dependent variables allowed."}
+{"_id": "26285", "title": "Definition:Linear Second Order Ordinary Differential Equation", "text": "A '''linear second order ordinary differential equation''' is a differential equation which is in (or can be manipulated into) the form: :$\\dfrac {\\d^2 y} {\\d x^2} + \\map P x \\dfrac {\\d y} {\\d x} + \\map Q x y = \\map R x$ where, as is indicated by the notation, $\\map P x$, $\\map Q x$ and $\\map R x$ are functions of $x$ alone (or constants)."}
+{"_id": "26286", "title": "Definition:Niemytzki Plane", "text": "The '''Niemytzki plane''' is the topological space $T = \\struct {S, \\tau}$ defined as: {{begin-eqn}} {{eqn | l = S       | r = \\set {\\tuple {x, y} \\in \\R^2: y \\ge 0} }} {{eqn | l = \\map \\BB {x, y}       | r = \\set {\\map {B_r} {x, y} \\cap S: r > 0}       | c = if $x, y \\in \\R, y > 0$ }} {{eqn | l = \\map \\BB {x, 0}       | r = \\set {\\map {B_r} {x, r} \\cup \\set {\\tuple {x, 0} }: r > 0}       | c = if $x \\in \\R$ }} {{eqn | l = \\tau       | r = \\set {\\bigcup \\GG: \\GG \\subseteq \\bigcup_{\\tuple {x, y} \\mathop \\in S} \\map \\BB {x, y} } }} {{end-eqn}} where $\\map {B_r} {x, y}$ denotes the open $r$-ball of $\\tuple {x, y}$ in the $\\R^2$ Euclidean space."}
+{"_id": "26287", "title": "Definition:Neighborhood System", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. A '''neighborhood system''' is a family $\\family {\\NN_x}_{x \\mathop \\in S}$ indexed by points of $S$, such that $\\NN_x$ is a local basis at $x$ for $x \\in S$."}
+{"_id": "26289", "title": "Definition:Tractrix", "text": "Let $S$ be a cord situated as a (straight) line segment whose endpoints are $P$ and $T$. Let $T$ be dragged in a direction perpendicular to the straight line in which $S$ is aligned. The curve along which $P$ travels is known as a '''tractrix'''. :400px"}
+{"_id": "26290", "title": "Definition:Tractrix/Linguistic Note", "text": "The word '''tractrix''' derives from the Latin '''traho (trahere, traxi, tractum)''' meaning '''to pull''' or '''to drag'''. The plural is '''tractrices'''."}
+{"_id": "26292", "title": "Definition:Pursuit Curve", "text": "A '''pursuit curve''' is a curve constructed to model pursuers and pursuees. The '''pursuit curve''' is the curve traced by the pursuers. The pursuee is always on the pursuer's tangent."}
+{"_id": "26293", "title": "Definition:Rate of Change", "text": "The '''rate of change''' of a physical quantity is its (first) derivative with respect to time. Loosely speaking, it means '''how fast something changes''', with a wider scope than change in physical displacement. === Dimension === {{:Definition:Rate of Change/Dimension}}"}
+{"_id": "26294", "title": "Definition:Steady-State", "text": "=== First Order ODE === {{:Definition:Steady-State/First Order ODE}} === Second Order ODE === {{:Definition:Steady-State/Second Order ODE}}"}
+{"_id": "26295", "title": "Definition:Steady-State/Electronics", "text": "The electric current $I$ in $K$ is given by the equation: :$I = \\dfrac {E_0} R + \\left({I_0 - \\dfrac {E_0} R}\\right) e^{-R t / L}$ The term $\\dfrac {E_0} R$ is known as the '''steady-state''' part of $(1)$."}
+{"_id": "26296", "title": "Definition:Transient/Electronics", "text": "The electric current $I$ in $K$ is given by the equation: :$I = \\dfrac {E_0} R + \\left({I_0 - \\dfrac {E_0} R}\\right) e^{-R t / L}$ The term: :$\\left({I_0 - \\dfrac {E_0} R}\\right) e^{-R t / L}$ which dies away to zero over time, is known as the '''transient''' part."}
+{"_id": "26297", "title": "Definition:Transient", "text": "Consider the Decay Equation: :$\\dfrac {\\d y} {\\d x} = k \\paren {y_a - y}$ where: :$k \\in \\R: k > 0$ :$y = y_0$ at $x = 0$ which has the particular solution: :$(1): \\quad y = y_a + \\paren {y_0 - y_a} e^{-k x}$ The term: :$\\paren {y_0 - y_a} e^{-k x}$ which tends to zero with increasing $x$, is known as the '''transient''' part. === Electronics === The term is often seen in the context of the current in an $LR$ electric circuit $K$: {{:Definition:Transient/Electronics}}"}
+{"_id": "26298", "title": "Definition:Electronics", "text": "'''Electronics''' is the branch of electromagnetism which studies how to control electricity."}
+{"_id": "26299", "title": "Definition:Barrier", "text": "A complex function $\\varphi \\in \\map \\C {\\overline \\Omega}$ is a '''barrier''' for $\\Omega$ at $z \\in \\partial \\Omega$ {{iff}}: :$\\varphi$ is subharmonic :$\\map \\varphi z = 0$ :$\\varphi < 0$ on $\\partial \\Omega \\setminus \\set z$ {{explain|Define $\\map \\C {\\overline \\Omega}$}} {{explain|Define $\\Omega$}} {{explain|Define $\\partial \\Omega$}} {{explain|What is the domain and range of $\\varphi$?}} {{explain|If $\\varphi$ is a complex function, its range is also complex, and so cannot be ordered, so the meaning of $\\varphi < 0$ needs to be explained carefully}}"}
+{"_id": "26300", "title": "Definition:Surface of Revolution", "text": "Let $F$ be a plane figure. Let $L$ be a straight line in the same plane as $F$. Let $F$ be rotated one full turn about $L$ in $3$-dimensional space. The '''surface of revolution generated by $F$ around $L$''' is the surface of the solid of revolution generated by $F$ around $L$."}
+{"_id": "26301", "title": "Definition:Riccati Equation", "text": "The '''Riccati equation''' is the first order ordinary differential equation: :$y' = p \\left({x}\\right) + q \\left({x}\\right) y + r \\left({x}\\right) y^2$"}
+{"_id": "26302", "title": "Definition:Special Riccati Equation", "text": "The '''special Riccati equation''' is the first order ordinary differential equation: :$y' + b y^2 = c x^m$ {{NamedforDef|Jacopo Francesco Riccati|cat = Riccati}}"}
+{"_id": "26303", "title": "Definition:Burnout Velocity", "text": "Let $R$ be a rocket which is propelled by backwards expulsion of the exhaust products of the burning of its internal fuel supply. Let $R$ be fired straight up from the surface of a planet. The '''burnout velocity''' of $R$ is the velocity attained by $R$ at the instant when its fuel has become exhausted."}
+{"_id": "26304", "title": "Definition:Dynamics", "text": "'''Dynamics''' is the branch of the mechanics describing the motion of bodies under the influence of forces."}
+{"_id": "26305", "title": "Definition:Burnout Height", "text": "Let $R$ be a rocket which is propelled by backwards expulsion of the exhaust products of the burning of its internal fuel supply. Let $R$ be fired straight up from the surface of a planet. The '''burnout height''' of $R$ is the height attained by $R$ at the instant when its fuel has become exhausted."}
+{"_id": "26306", "title": "Definition:Homogeneous Linear Second Order ODE", "text": "A '''homogeneous linear second order ODE''' is a differential equation which is in (or can be manipulated into) the form: :$\\dfrac {\\d^2 y} {\\d x^2} + \\map P x \\dfrac {\\d y} {\\d x} + \\map Q x y = 0$ where, as is indicated by the notation, $\\map P x$ and $\\map Q x$ are functions of $x$ alone (or constants)."}
+{"_id": "26307", "title": "Definition:Nonhomogeneous Linear Second Order ODE", "text": "A '''nonhomogeneous linear second order ODE''' is a differential equation which is in (or can be manipulated into) the form: :$\\dfrac {\\d^2 y} {\\d x^2} + \\map P x \\dfrac {\\d y} {\\d x} + \\map Q x y = \\map R x$ where: :$\\map P x$, $\\map Q x$ and $\\map R x$ are functions of $x$ alone (or constants) :$\\map R x$ is not the zero constant function. That is, a '''nonhomogeneous linear second order ODE''' is a linear second order ODE which is not homogeneous."}
+{"_id": "26308", "title": "Definition:Linear Combination of Solutions to Homogeneous Linear 2nd Order ODE", "text": "Let $C_1$ and $C_2$ be real numbers. Let $\\map {y_1} x$ and $\\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE: :$(1): \\quad \\dfrac {\\d^2 y} {\\d x^2} + \\map P x \\dfrac {\\d y} {\\d x} + \\map Q x y = 0$ Then: :$C_1 \\, \\map {y_1} x + C_2 \\, \\map {y_2} x$ is known as a '''linear combination of particular solutions to $(1)$."}
+{"_id": "26309", "title": "Definition:Linearly Dependent Real Functions", "text": "Let $f \\left({x}\\right)$ and $g \\left({x}\\right)$ be real functions defined on a closed interval $\\left[{a \\,.\\,.\\, b}\\right]$. Let $f$ and $g$ be constant multiples of each other: :$\\exists c \\in \\R: \\forall x \\in \\left[{a \\,.\\,.\\, b}\\right]: f \\left({x}\\right) = c g \\left({x}\\right)$ or: :$\\exists c \\in \\R: \\forall x \\in \\left[{a \\,.\\,.\\, b}\\right]: g \\left({x}\\right) = c f \\left({x}\\right)$ Then $f$ and $g$ are '''linearly dependent'''."}
+{"_id": "26310", "title": "Definition:Linearly Independent Real Functions", "text": "Let $f \\left({x}\\right)$ and $g \\left({x}\\right)$ be real functions defined on a closed interval $\\left[{a \\,.\\,.\\, b}\\right]$. Let $f$ and $g$ be such that they are not linearly dependent. Then $f$ and $g$ are '''linearly independent'''."}
+{"_id": "26311", "title": "Definition:Wronskian/General Definition", "text": "Let $\\map {f_1} x, \\map {f_2} x, \\dotsc, \\map {f_n} x$ be real functions defined on a closed interval $\\closedint a b$. Let $f_1, f_2, \\ldots, f_n$ be $n - 1$ times differentiable on $\\closedint a b$. The '''Wronskian''' of $f_1, f_2, \\ldots, f_n$ on $\\closedint a b$ is defined as: :$\\map W {f_1, f_2, \\dotsc, f_n} = \\begin {vmatrix} \\map {f_1} x & \\map {f_2} x & \\cdots & \\map {f_n} x \\\\ \\map { {f_1}'} x & \\map { {f_2}'} x & \\cdots & \\map { {f_n}'} x \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\map { {f_1}^{\\paren {n - 1} } } x & \\map { {f_2}^{\\paren {n - 1} } } x & \\cdots & \\map { {f_n}^{\\paren {n - 1} } } x \\\\ \\end{vmatrix}$ where: :$\\begin{vmatrix} \\cdots \\end{vmatrix}$ denotes the determinant :$\\map { {f_1}^{\\paren {n - 1} } } x$ denotes the $n - 1$th derivative of $f_1$."}
+{"_id": "26312", "title": "Definition:Wronskian", "text": "Let $\\map f x$ and $\\map g x$ be real functions defined on a closed interval $\\closedint a b$. Let $f$ and $g$ be differentiable on $\\closedint a b$. The '''Wronskian''' of $f$ and $g$ is defined as: :$\\map W {f, g} = \\begin {vmatrix} \\map f x & \\map g x \\\\ \\map {f'} x & \\map {g'} x \\\\ \\end {vmatrix} = \\map f x \\, \\map {g'} x - \\map g x \\, \\map {f'} x$ === General Definition === {{:Definition:Wronskian/General Definition}}"}
+{"_id": "26313", "title": "Definition:Homogeneous Linear Second Order ODE with Constant Coefficients", "text": "A '''homogeneous linear second order ODE with constant coefficients''' is a second order ODE which can be manipulated into the form: :$y'' + p y' + q y = 0$ where $p$ and $q$ are real constants. Thus it is a homogeneous linear second order ODE: :$y'' + \\map P x y' + \\map Q x y = 0$ where $\\map P x$ and $\\map Q x$ are constant functions."}
+{"_id": "26314", "title": "Definition:Auxiliary Equation", "text": "Let: :$(1): \\quad y'' + p y' + q y = 0$ be a constant coefficient homogeneous linear second order ODE. The '''auxiliary equation''' of $(1)$ is the quadratic equation: :$m^2 + p m + q = 0$"}
+{"_id": "26315", "title": "Definition:Cauchy-Euler Equation", "text": "The linear second order ordinary differential equation: :$x^2 \\dfrac {\\d^2 y} {\\d x^2} + p x \\dfrac {\\d y} {\\d x} + q y = 0$ is the '''Cauchy-Euler equation'''."}
+{"_id": "26316", "title": "Definition:Cauchy-Euler Equation/General Form", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive integer. The linear ordinary differential equation: :$a_n x^n \\, \\map {y^{\\paren n} } x + \\dotsb + a_1 x \\, \\map {y'} x + a_0 \\, \\map y x = 0$ is the '''$n$th order Cauchy-Euler equation'''."}
+{"_id": "26317", "title": "Definition:Directed Subset", "text": "Let $\\struct {S, \\precsim}$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is a '''directed subset''' of $S$ {{iff}}: :$\\forall x, y \\in H: \\exists z \\in H: x \\precsim z$ and $y \\precsim z$"}
+{"_id": "26318", "title": "Definition:Filtered Subset", "text": "Let $\\left({S, \\precsim}\\right)$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is a '''filtered subset''' of $S$ {{iff}}: :$\\forall x, y \\in H: \\exists z \\in H: z \\precsim x$ and $z \\precsim y$"}
+{"_id": "26319", "title": "Definition:Spring/Ideal", "text": "An '''ideal spring''' obeys Hooke's Law: :$\\mathbf F = -k \\mathbf x$ where: : $\\mathbf F$ is the force caused by a displacement $\\mathbf x$ : $k$ is the constant of proportion. The negative sign indicates that the force pulls in the opposite direction to that of the displacement imposed."}
+{"_id": "26320", "title": "Definition:Spring/Force Constant", "text": "The constant of proportion: :$k \\in \\R_{>0}$ is known as the '''(spring) force contant'''. Its value depends on the particular spring under investigation. The SI unit of the '''force constant''' is newtons per metre: $\\mathrm N \\ \\mathrm m^{-1}$."}
+{"_id": "26321", "title": "Definition:Wall", "text": "A '''wall''' is an immovable vertical plane in space to which other physical objects may be attached."}
+{"_id": "26322", "title": "Definition:Cart", "text": "A '''cart''' is a body with a given mass which is free to move with zero friction in a single dimension, that is, back and forth. It is usually imagined as having wheels on which it can roll."}
+{"_id": "26323", "title": "Definition:Immovable", "text": "An '''immovable''' body is one whose position cannot be changed by applying a force to it. It can be considered as having an infinite mass and zero momentum."}
+{"_id": "26324", "title": "Definition:Spring/Equilibrium Position", "text": "The '''equilibrium position''' of a body $B$ attached to a spring $S$ is the position it occupies when $S$ is exerting no force upon $B$. For an ideal spring obeying Hooke's Law $\\mathbf F = -k \\mathbf x$, the '''equilibrium position''' is set to be the point $\\mathbf x = \\bszero$."}
+{"_id": "26325", "title": "Definition:Simple Harmonic Motion/Amplitude", "text": "The parameter $A$ is known as the '''amplitude''' of the motion."}
+{"_id": "26326", "title": "Definition:Simple Harmonic Motion", "text": "Consider a physical system $S$ whose motion can be expressed in the form of the following equation: :$x = A \\map \\sin {\\omega t + \\phi}$ where $A$ and $\\phi$ are constants. Then $S$ is in a state of '''simple harmonic motion'''. === Amplitude === {{:Definition:Simple Harmonic Motion/Amplitude}} === Period === {{:Definition:Simple Harmonic Motion/Period}} === Frequency === {{:Definition:Simple Harmonic Motion/Frequency}}"}
+{"_id": "26327", "title": "Definition:Simple Harmonic Motion/Period", "text": "The '''period''' $T$ of the motion of $S$ is the time required for one complete cycle: :$T = \\dfrac {2 \\pi} \\omega$"}
+{"_id": "26328", "title": "Definition:Simple Harmonic Motion/Frequency", "text": "The '''frequency''' $\\nu$ of the motion of $S$ is the number of complete cycles per unit time: :$\\nu = \\dfrac 1 T = \\dfrac \\omega {2 \\pi}$"}
+{"_id": "26329", "title": "Definition:Periodic Function/Frequency", "text": "The '''frequency''' $\\nu$ of $f$ is the reciprocal of the period $L$ of $f$: :$\\nu = \\dfrac 1 L$"}
+{"_id": "26330", "title": "Definition:Fluid", "text": "A '''fluid''' is a substance which has no defined shape, but which is free to flow according to external forces. === Ideal Fluid === {{:Definition:Fluid/Ideal}} Category:Definitions/Fluid Mechanics Category:Definitions/Ideals in Physics 36by7c8128e3a1nfmb1k1rs3pv2ei7t"}
+{"_id": "26331", "title": "Definition:Medium", "text": "A '''medium''' is a fluid $F$ through which a body $B$ can pass. Category:Definitions/Mechanics 49mnpwqf8fu8ddljeprdc05kuq7kusi"}
+{"_id": "26332", "title": "Definition:Damping Force", "text": "Consider a physical system consisting of a body $B$ passing through a medium $M$. In order for $B$ to move, it needs to push $M$ out of the way to make room for it. Hence $M$ exerts a force upon $B$ which opposes the motion of $B$. This force opposing the motion of $B$ is referred to as a '''damping force'''."}
+{"_id": "26333", "title": "Definition:Viscosity", "text": "The viscosity of a fluid is a measure of how large a damping force it exerts upon a body moving through it. {{stub|Establish its formal definition: how great the damping force against how large a body at what velocity. Fluid mechanics has not really been started on this site yet. I have a degree semester to infodump.}}"}
+{"_id": "26334", "title": "Definition:Fluid/Ideal", "text": "An '''ideal fluid''' is a fluid which delivers a damping force whose magnitude is proportional to the velocity of a body passing through it."}
+{"_id": "26335", "title": "Definition:Overdamped", "text": "Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \\quad \\dfrac {\\d^2 x} {\\d t^2} + 2 b \\dfrac {\\d x} {\\d t} + a^2 x = 0$ for $a, b \\in \\R_{>0}$. Let $b > a$, so that the solution of $(1)$ is in the form: :$x = C_1 e^{m_1 t} + C_2 e^{m_1 t}$ for $m_1, m_2 < 0$. Then $S$ is described as being '''overdamped'''. :500px"}
+{"_id": "26336", "title": "Definition:Critically Damped", "text": "Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \\quad \\dfrac {\\d^2 x} {\\d t^2} + 2 b \\dfrac {\\d x} {\\d t} + a^2 x = 0$ for $a, b \\in \\R_{>0}$. Let $b = a$, so that the solution of $(1)$ is in the form: :$x = C_1 e^{-a t} + C_2 t e^{-a t}$ Then $S$ is described as being '''critically damped'''. :500px"}
+{"_id": "26337", "title": "Definition:Underdamped", "text": "Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \\quad \\dfrac {\\d^2 x} {\\d t^2} + 2 b \\dfrac {\\d x} {\\d t} + a^2 x = 0$ for $a, b \\in \\R_{>0}$. Let $b < a$, so that the solution of $(1)$ is in the form: :$x = e^{-b t} \\paren {C_1 \\cos \\alpha t + C_2 \\sin \\alpha t}$ where $\\alpha = \\sqrt {a^2 - b^2}$. Then $S$ is described as being '''underdamped'''. :600px"}
+{"_id": "26338", "title": "Definition:Period of Underdamped Oscillation", "text": "Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \\quad \\dfrac {\\d^2 x} {\\d t^2} + 2 b \\dfrac {\\d x} {\\d t} + a^2 x = 0$ for $a, b \\in \\R_{>0}$. Let $b < a$, so as to make $S$ underdamped. :600px While the behaviour of $S$ is not strictly speaking periodic, its oscillations can be defined to have a \"period\" as follows: Let $T$ be the smallest value of $t$ such that: :$x = 0$ :$x' < 0$ Then $T$ is the '''period''' of the oscillation of $S$."}
+{"_id": "26339", "title": "Definition:Amplitude of Underdamped Oscillation", "text": "Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \\quad \\dfrac {\\d^2 x} {\\d t^2} + 2 b \\dfrac {\\d x} {\\d t} + a^2 x = 0$ for $a, b \\in \\R_{>0}$. Let $(1)$ be subject to the initial conditions: :$x = x_0$ at $t = 0$ :$x' = 0$ at $t = 0$ Let $b < a$, so as to make $S$ underdamped. :600px The '''amplitude''' of the oscillation of $S$ is defined as: :$A = \\dfrac {x_0 a} {\\sqrt {a^2 - b^2} } e^{-b t}$"}
+{"_id": "26340", "title": "Definition:Natural Frequency", "text": "Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \\quad \\dfrac {\\mathrm d^2 x} {\\mathrm d t^2} + 2 b \\dfrac {\\mathrm d x} {\\mathrm d t} + a^2 x = 0$ for $a, b \\in \\R_{>0}$. Let $b < a$, so as to make $S$ underdamped. :600px The '''natural frequency''' $\\nu$ of $S$ is defined as: :$\\nu = \\dfrac 1 T$ where $T$ is the period of the oscillation of $S$."}
+{"_id": "26341", "title": "Definition:Free Vibration", "text": "Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \\quad \\dfrac {\\d^2 x} {\\d t^2} + 2 b \\dfrac {\\d x} {\\d t} + a^2 x = 0$ for $a, b \\in \\R_{>0}$. Let $b < a$, so as to make $S$ underdamped. Then the motion of $S$ is referred to as '''free vibration''' of $S$. :600px"}
+{"_id": "26342", "title": "Definition:Steady-State/First Order ODE", "text": "Consider the Decay Equation: :$\\dfrac {\\d y} {\\d x} = k \\paren {y_a - y}$ where: :$k \\in \\R: k > 0$ :$y = y_0$ at $x = 0$ which has the particular solution: :$(1): \\quad y = y_a + \\paren {y_0 - y_a} e^{-k x}$ The term $y_a$ is known as the '''steady-state''' component of $(1)$."}
+{"_id": "26343", "title": "Definition:Steady-State/Second Order ODE", "text": "Consider the second order ODE: :$\\dfrac {\\mathrm d^2 y} {\\mathrm d x^2} + 2 b \\dfrac {\\mathrm d y} {\\mathrm d x} + a^2 x = K \\cos \\omega x$ where: :$K \\in \\R: k > 0$ :$a, b \\in \\R_{>0}: b < a$ which has the general solution: :$(1): \\quad y = e^{-b x} \\left({C_1 \\cos \\alpha x + C_2 \\sin \\alpha x}\\right) + \\dfrac K {\\sqrt {4 b^2 \\omega^2 + \\left({a^2 - \\omega^2}\\right)^2} } \\cos \\left({\\omega x - \\phi}\\right)$ where: :$\\alpha = \\sqrt {a^2 - b^2}$ :$\\phi = \\arctan \\left({\\dfrac {2 b \\omega} {a^2 - \\omega^2} }\\right)$ The term $\\dfrac K {\\sqrt {4 b^2 \\omega^2 + \\left({a^2 - \\omega^2}\\right)^2} } \\cos \\left({\\omega x - \\phi}\\right)$ is known as the '''steady-state''' component of $(1)$."}
+{"_id": "26344", "title": "Definition:Resonance", "text": "Consider a physical system $S$ whose behaviour is defined by the second order ODE: :$\\dfrac {\\d^2 y} {\\d x^2} + 2 b \\dfrac {\\d y} {\\d x} + a^2 x = K \\cos \\omega x$ where: :$K \\in \\R: k > 0$ :$a, b \\in \\R_{>0}: b < a$ which has the general solution: :$(1): \\quad y = e^{-b x} \\paren {C_1 \\cos \\alpha x + C_2 \\sin \\alpha x} + \\dfrac K {\\sqrt {4 b^2 \\omega^2 + \\paren {a^2 - \\omega^2}^2} } \\map \\cos {\\omega x - \\phi}$ where: :$\\alpha = \\sqrt {a^2 - b^2}$ :$\\phi = \\map \\arctan {\\dfrac {2 b \\omega} {a^2 - \\omega^2} }$ Let $\\omega$ be such that the amplitude of the steady-state component of $(1)$ is at a maximum. Then $S$ is said to be in '''resonance'''."}
+{"_id": "26345", "title": "Definition:Forced Vibration", "text": "Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \\quad \\dfrac {\\mathrm d^2 x} {\\mathrm d t^2} + 2 b \\dfrac {\\mathrm d x} {\\mathrm d t} + a^2 x = K \\cos \\omega x$ for $a, b \\in \\R_{>0}$. Let $b < a$, so as to make $S$ underdamped. Then the steady-state motion of $S$ is referred to as '''forced vibration''' of $S$. === Forcing Frequency === {{:Definition:Forced Vibration/Forcing Frequency}}"}
+{"_id": "26346", "title": "Definition:Forced Vibration/Forcing Frequency", "text": "The frequency: :$\\dfrac \\omega {2 \\pi}$ is known as the '''forcing frequency''' of $S$."}
+{"_id": "26347", "title": "Definition:Resonance Frequency", "text": "Consider a second order ODE in the form: :$\\dfrac {\\mathrm d^2 x} {\\mathrm d t^2} + 2 b \\dfrac {\\mathrm d x} {\\mathrm d t} + a^2 x = K \\cos \\omega x$ for $a, b \\in \\R_{>0}$. Let $b < a$, so as to make $S$ underdamped. Let the forcing frequency $\\omega$ be such that $S$ is in resonance. Then $\\omega$ is known as the '''resonance frequency'''."}
+{"_id": "26348", "title": "Definition:Logarithmic Decrement", "text": "Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \\quad \\dfrac {\\mathrm d^2 x} {\\mathrm d t^2} + 2 b \\dfrac {\\mathrm d x} {\\mathrm d t} + a^2 x = 0$ for $a, b \\in \\R_{>0}$. Let $b < a$, so as to make $S$ underdamped. :600px Let $T$ be the period of oscillation of $S$. Let $x_1$ and $x_2$ be successive local maxima of $x$. From Ratio of Successive Local Maxima for Underdamped Free Vibration: :$\\dfrac {x_1} {x_2} = e^{b T}$ The '''logarithmic decrement''' of $S$ is defined as: :$\\ln \\left({\\dfrac {x_1} {x_2} }\\right) = b T$"}
+{"_id": "26349", "title": "Definition:Central Force", "text": "Consider a particle $p$ of mass $m$ moving in the plane under the influence of a force $\\mathbf F$. Let the position of $p$ at time $t$ be given in polar coordinates as $\\left\\langle{r, \\theta}\\right\\rangle$. Let $\\mathbf F$ be expressed as: :$\\mathbf F = F_r \\mathbf u_r + F_\\theta \\mathbf u_\\theta$ where: :$\\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$ :$\\mathbf u_\\theta$ is the unit vector in the direction of the angular coordinate of $p$ :$F_r$ and $F_\\theta$ are the magnitudes of the components of $\\mathbf F$ in the directions of $\\mathbf u_r$ and $\\mathbf u_\\theta$ respectively. :600px Let $\\mathbf F$ have no component perpendicular to $\\mathbf u_r$. That is, such that $F_\\theta = 0$. Then $\\mathbf F$ is referred to as a '''central force'''."}
+{"_id": "26350", "title": "Definition:Polish Space", "text": "A '''Polish space''' is a topological space which is separable and metrizable, with a metric that makes it a complete metric space. Category:Definitions/Descriptive Set Theory Category:Definitions/Topology Category:Definitions/Metric Spaces gdlarwvzk7c4gy9pd4q2wo64zxxup1s"}
+{"_id": "26360", "title": "Definition:Tetractys", "text": "The '''tetractys''' is a central concept in the mystical beliefs of the Pythagoreans. It consists of $10$ points arranged in a triangle: :300px based on the notions that: :$1$ represents the point :$2$ represents the line :$3$ represents the surface :$4$ represents the solid :$1 + 2 + 3 + 4 = 10$ and therefore: :$10$ represents the universe."}
+{"_id": "26361", "title": "Definition:Numerology/Historical Note", "text": "The concept of '''numerology''' seems to have begun with {{AuthorRef|Pythagoras of Samos|Pythagoras}} and the Pythagorean school, continuing with the neo-Pythagoreans. {{AuthorRef|Plato}} adopted the ideas, and from there they were passed on through the emerging Christian tradition into the heart of the philosophy of Western civilization, where it still exerts a powerful influence."}
+{"_id": "26362", "title": "Definition:Numerology", "text": "'''Numerology''' is a philosophical belief structure, borrowed from the Pythagoreans, that considers numbers to possess mystical properties. At its heart is the pseudo-science of '''gematria''' or '''arithmology''': the process of translating linguistic text (in particular a name) into numerical form, and thence to interpret the significance of that text from the mystical power of that particular number. The techniques of that translation differ between practitioners, and in many cases the same practitioner may use more than one such technique, thereby allowing for multiple different numerical interpretations of any particular given text. Hence a popular criticism of numerology is to accuse practitioners of deciding what they want the answer to be, and then juggling the numbers in order to achieve that particular effect. The practice may have originated from the Greek numerals of the Classical period, in which the natural ordering of the letters of the Greek alphabet were assigned numerical values. It should not be necessary to place much emphasis on the fact that '''numerology''' is not a branch of mathematics."}
+{"_id": "26364", "title": "Definition:Plato's Academy", "text": "'''Plato's Academy''' was the second university in the Western world, after the Pythagorean school. It placed great emphasis on the study of mathematics."}
+{"_id": "26365", "title": "Definition:Mapping Preserves Supremum", "text": "Let $\\left({S_1, \\preceq_1}\\right)$, $\\left({S_2, \\preceq_2}\\right)$ be ordered sets. Let $f: S_1 \\to S_2$ be a mapping."}
+{"_id": "26366", "title": "Definition:Mapping Preserves Supremum/Subset", "text": "Let $F$ be a subset of $S_1$. $f$ '''preserves supremum of''' $F$ {{iff}} :$F$ admits a supremum in $\\left({S_1, \\preceq_1}\\right)$ implies :: $f^\\to \\left({F}\\right)$ admits a supremum in $\\left({S_2, \\preceq_2}\\right)$ and $\\sup \\left({f^\\to\\left({F}\\right)}\\right) = f \\left({\\sup F}\\right)$"}
+{"_id": "26367", "title": "Definition:Mapping Preserves Supremum/All", "text": "$f$ '''preserves all suprema''' {{iff}} :for every subset $F$ of $S_1$, $f$ preserves the supremum of $F$."}
+{"_id": "26368", "title": "Definition:Mapping Preserves Supremum/Join", "text": "$f$ '''preserves join''' {{iff}} :for every pair of elements $x, y$ of $S_1$, $f$ preserves the supremum of $\\left\\{ {x, y}\\right\\}$."}
+{"_id": "26369", "title": "Definition:Mapping Preserves Supremum/Finite", "text": "$f$ '''preserves finite suprema''' {{iff}} :for every finite subset $F$ of $S_1$, $f$ preserves the supremum of $F$."}
+{"_id": "26370", "title": "Definition:Mapping Preserves Supremum/Directed", "text": "$f$ '''preserves directed suprema''' {{iff}} :for every directed subset $F$ of $S_1$, $f$ preserves the supremum of $F$."}
+{"_id": "26371", "title": "Definition:Mapping Preserves Infimum/Subset", "text": "Let $F$ be a subset of $S_1$. $f$ '''preserves infimum of''' $F$ {{iff}} :$F$ admits a infimum in $\\left({S_1, \\preceq_1}\\right)$ implies :: $f^\\to \\left({F}\\right)$ admits an infimum in $\\left({S_2, \\preceq_2}\\right)$ and $\\inf \\left({f^\\to \\left({F}\\right)}\\right) = f \\left({\\inf F}\\right)$"}
+{"_id": "26372", "title": "Definition:Mapping Preserves Infimum/All", "text": "$f$ '''preserves all infima''' {{iff}} :for every subset $F$ of $S_1$, $f$ preserves the infimum of $F$."}
+{"_id": "26373", "title": "Definition:Mapping Preserves Infimum/Meet", "text": "$f$ '''preserves meet''' {{iff}} :for every pair of elements $x, y$ of $S_1$, $f$ preserves the infimum of $\\left\\{ {x, y}\\right\\}$."}
+{"_id": "26374", "title": "Definition:Mapping Preserves Infimum/Finite", "text": "$f$ '''preserves finite infima''' {{iff}} :for every finite subset $F$ of $S_1$, $f$ preserves the infimum of $F$."}
+{"_id": "26375", "title": "Definition:Mapping Preserves Infimum", "text": "Let $\\left({S_1, \\preceq_1}\\right)$, $\\left({S_2, \\preceq_2}\\right)$ be ordered sets. Let $f: S_1 \\to S_2$ be a mapping."}
+{"_id": "26376", "title": "Definition:Mapping Preserves Infimum/Filtered", "text": "$f$ '''preserves filtered infima''' {{iff}} :for every filtered subset $F$ of $S_1$, $f$ preserves the infimum of $F$."}
+{"_id": "26378", "title": "Definition:Conic Section/Focus-Directrix Property/Historical Note", "text": "The focus-directrix definition of a conic section was first documented by {{AuthorRef|Pappus of Alexandria}}. It appears in his {{BookLink|Collection|Pappus of Alexandria}}. As he was scrupulous in documenting his sources, and he gives none for this construction, it can be supposed that it originated with him."}
+{"_id": "26379", "title": "Definition:Conic Section/Intersection with Cone/Tilting Angle", "text": "The inclination $\\phi$ of the slicing plane with the axis of $C$ is known as the '''tilting angle''' of $D$ for $K$."}
+{"_id": "26380", "title": "Definition:Conic Section/Intersection with Cone/Slicing Plane", "text": "The plane $D$ is known as the '''slicing plane''' of $C$ for $K$."}
+{"_id": "26381", "title": "Definition:Vertical Line/Definition 1", "text": "A line is '''vertical''' {{iff}} it is aligned perpendicular to the horizon."}
+{"_id": "26382", "title": "Definition:Vertical Line/Definition 2", "text": "A line is '''vertical''' {{iff}} it is parallel to the path taken by a body released from rest in a constant gravitational field."}
+{"_id": "26383", "title": "Definition:Vertical", "text": "=== Vertical Line === {{:Definition:Vertical Line/Definition 2}} Hence the adjective '''vertically''', meaning '''in a vertical line'''. === Vertical Plane === {{:Definition:Vertical Plane}}"}
+{"_id": "26384", "title": "Definition:Vertical Plane", "text": "A plane is '''vertical''' {{iff}} it contains a vertical line."}
+{"_id": "26385", "title": "Definition:Horizontal", "text": "=== Horizontal Line === {{:Definition:Horizontal Line/Definition 2}} Hence the adjective '''horizontally''', meaning '''in a horizontal line'''. === Horizontal Plane === {{:Definition:Horizontal Plane}}"}
+{"_id": "26386", "title": "Definition:Horizontal Line/Definition 2", "text": "A line is '''horizontal''' {{iff}} it is aligned parallel to the horizon:"}
+{"_id": "26387", "title": "Definition:Horizontal Line/Definition 1", "text": "A line is '''horizontal''' {{iff}} it is perpendicular to a given vertical line."}
+{"_id": "26388", "title": "Definition:Horizontal Line/Definition 3", "text": "A line is '''horizontal''' {{iff}} it can be embedded within a horizontal plane."}
+{"_id": "26389", "title": "Definition:Horizontal Plane", "text": "A plane is '''horizontal''' {{iff}} it is perpendicular to a given vertical line."}
+{"_id": "26390", "title": "Definition:Conic Section/Focus-Directrix Property/Circle", "text": ":400px It is not possible to define the circle using the focus-directrix property. This is because as the eccentricity $e$ tends to $0$, the distance $p$ from $P$ to $D$ tends to infinity. Thus a circle can in a sense be considered to be a degenerate ellipse whose foci are at the same point, that is, the center of the circle."}
+{"_id": "26391", "title": "Definition:Circle/Focus", "text": "As a circle cannot be specified using the focus-directrix property, the focus cannot be defined using that technique. Consider an ellipse $K$ with eccentricity $e$. Let $e$ tend to zero. Then as $e$ becomes smaller, the foci of $K$ become closer together. In the limit, the foci coincide. Thus the focus of the circle can be understood as being its center."}
+{"_id": "26393", "title": "Definition:Acoustics", "text": "'''Acoustics''' is the branch of physics which studies the properties of sound."}
+{"_id": "26394", "title": "Definition:Optics", "text": "'''Optics''' is the branch of physics which studies the properties of light."}
+{"_id": "26395", "title": "Definition:Light", "text": "'''Light''' is electromagnetic radiation whose wavelength is between approximately $700 \\, \\text{nm}$ and $390 \\, \\text{nm}$. It is characterised by its ability to stimulate the visual senses of humans. Category:Definitions/Optics Category:Definitions/Physics 4rebzuewkr183shrxfw4160ir9ltfv2"}
+{"_id": "26396", "title": "Definition:Sound (Physics)", "text": "'''Sound''' is the vibration of an elastic medium whose frequency ranges between approximately $20 \\, \\text {Hz}$ and $20 \\, \\text {kHz}$. It is characterised by the ability of humans to detect it with their aural sensory systems."}
+{"_id": "26399", "title": "Definition:Filter in Ordered Set", "text": "Let $\\left({S, \\preceq}\\right)$ be a preordered set. Let $F$ be a subset of $S$. $F$ is '''filter''' in $\\left({S, \\preceq}\\right)$ {{iff}} :$F$ is non-empty filtered and upper."}
+{"_id": "26400", "title": "Definition:Ideal in Ordered Set", "text": "Let $\\left({S, \\preceq}\\right)$ be a preordered set. Let $I$ be a subset of $S$. $I$ is '''ideal''' in $\\left({S, \\preceq}\\right)$ {{iff}}: :$I$ is non-empty directed and lower."}
+{"_id": "26401", "title": "Definition:Logarithmic Spiral", "text": "The '''logarithmic spiral''' is the locus of the equation expressed in Polar coordinates as: :$r = a e^{b \\theta}$ :600px"}
+{"_id": "26402", "title": "Definition:Spiral", "text": "A '''spiral''' is a plane curve, or part of a plane curve, which can be expressed in polar coordinates in the form: :$r = \\map f \\theta$ where $f$ is either (strictly) increasing or (strictly) decreasing. Hence a '''spiral''' is a plane curve which emanates from a central point, getting progressively farther away (or closer in) as it revolves around that point."}
+{"_id": "26403", "title": "Definition:Gabriel's Horn", "text": "'''Gabriel's Horn''' is the name given to the solid of revolution formed by rotating about the $x$-axis the curve: :$y = \\dfrac 1 x$ :400px"}
+{"_id": "26404", "title": "Definition:Hydrostatics", "text": "'''Hydrostatics''' is the branch of physics concerned with the study of forces on fluids at rest."}
+{"_id": "26407", "title": "Definition:Tautochrone", "text": "Let $A$ and $B$ be points in space that are neither on the same horizontal line nor on the same vertical line. A '''tautochrone''' is the curve into which a wire is to be shaped so that a bead sliding down it (without friction) takes the same time to reach the lowest point independently of where on that wire it is released from rest."}
+{"_id": "26409", "title": "Definition:Evolute", "text": "Consider a curve $C$ embedded in a plane. The '''evolute''' of $C$ is the locus of the centers of curvature of each point on $C$."}
+{"_id": "26411", "title": "Definition:Involute", "text": "Consider a curve $C$ embedded in a plane. Imagine an ideal (zero thickness) cord $K$ wound round $C$. The '''involute''' of $C$ is the locus of the end of $K$ as it is unwound from $C$."}
+{"_id": "26423", "title": "Definition:Abscissa", "text": "Consider the graph $y = \\map f x$ of a real function $f$ embedded in a Cartesian plane. :500px The $x$ coordinate of a point $P = \\tuple {x, y}$ on $f$ is known as the '''abscissa''' of $P$."}
+{"_id": "26424", "title": "Definition:Ordinate", "text": "Consider the graph $y = \\map f x$ of a real function $f$ embedded in a Cartesian plane. :500px The $y$ coordinate of a point $P = \\tuple {x, y}$ on $f$ is known as the '''ordinate''' of $P$."}
+{"_id": "26433", "title": "Definition:Lemniscate of Bernoulli/Linguistic Note", "text": "The word '''lemniscate''' comes from the Latin word '''lemniscus''', which means '''pendant ribbon'''. The word may ultimately derive from the Latin '''lēmniscātus''', which means '''decorated with ribbons'''. This may in turn come from the ancient Greek island of '''Lemnos''' where ribbons were worn as decorations."}
+{"_id": "26444", "title": "Definition:Integer Partition/Part", "text": "In a '''partition''' of a (strictly) positive integer, one of the summands in that partition is referred to as a '''part'''."}
+{"_id": "26445", "title": "Definition:Integer Partition", "text": "A '''partition''' of a (strictly) positive integer $n$ is a way of writing $n$ as a sum of (strictly) positive integers."}
+{"_id": "26446", "title": "Definition:Integer Partition/Partition Function", "text": "The '''partition function''' $p: \\Z_{>0} \\to \\Z_{>0}$ is defined as: :$\\forall n \\in \\Z_{>0}: \\map p n =$ the number of partitions of the (strictly) positive integer $n$."}
+{"_id": "26448", "title": "Definition:Genus of Surface", "text": "Let $S$ be a surface. Let $G = \\left({V, E}\\right)$ be a graph which is embedded in $S$. Let $G$ be such that each of its faces is a simple closed curve. Let $\\chi \\left({G}\\right) = v - e + f = 2 - 2 p$ be the Euler characteristic of $G$ where: : $v = \\left|{V}\\right|$ is the number of vertices : $e = \\left|{E}\\right|$ is the number of edges : $f$ is the number of faces. Then $p$ is known as the '''genus''' of $S$."}
+{"_id": "26450", "title": "Definition:Plane Geometry/Historical Note", "text": "The first definitive work on Plane Geometry was {{ElementsLink}}."}
+{"_id": "26471", "title": "Definition:Hydrodynamics", "text": "'''Hydrodynamics''' is the branch of physics concerned with the study of fluids in motion."}
+{"_id": "26472", "title": "Definition:Analytical Mechanics", "text": "'''Analytical mechanics''' is a subfield of mathematical physics which uses techniques of analysis, in particular the calculus of variations, to solve problems in mechanics. Hence, instead of solving equations in vector quantities, it involves solutions of differential equations of scalar quantities."}
+{"_id": "26475", "title": "Definition:Metric System", "text": "The '''metric system''' is the colloquial term for the system of measurement based on the metre. Its main characteristic is that its units are constructed on a decimal system."}
+{"_id": "26476", "title": "Definition:Decimal System", "text": "A '''decimal system''' is a system of measurement in which the standard multiples and fractions of the units of measurement are powers of $10$."}
+{"_id": "26477", "title": "Definition:Up-Complete", "text": "Let $\\left({S, \\precsim}\\right)$ be a preordered set. Then $\\left({S, \\precsim}\\right)$ is '''up-complete''' {{iff}}: :every directed subset of $S$ admits a supremum in $\\left({S, \\precsim}\\right)$."}
+{"_id": "26479", "title": "Definition:Definite Integral/Integrand", "text": "In the expression for the definite integral: :$\\displaystyle \\int_a^b f \\left({x}\\right) \\, \\mathrm d x$ the function $f$ is called the '''integrand'''."}
+{"_id": "26480", "title": "Definition:Method of Least Squares (Approximation Theory)", "text": "Let there be a set of points $\\set {\\tuple {x_k, y_k}: k \\in \\set {1, 2, \\ldots, n} }$ plotted on a Cartesian $x y$ plane which correspond to measurements of a physical system. Let it be required that a straight line is to be fitted to the points. The '''method of least squares''' is a technique of producing a straight line of the form $y = m x + c$ such that: :the points $\\set {\\tuple {x_k', y_k'}: k \\in \\set {1, 2, \\ldots, n} }$ are on the line $y = m x + c$ :$\\forall k \\in \\set {1, 2, \\ldots, n}: y_k' = y_k$ :$\\displaystyle \\sum_n \\paren {x_k' = x_k}^2$ is minimised."}
+{"_id": "26482", "title": "Definition:Gaussian Distribution", "text": "Let $X$ be a continuous random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Then $X$ has a '''Gaussian distribution''' {{iff}} the probability density function of $X$ is: :$\\map {f_X} x = \\dfrac 1 {\\sigma \\sqrt {2 \\pi} } \\map \\exp {-\\dfrac {\\paren {x - \\mu}^2} {2 \\sigma^2} }$ for $\\mu \\in \\R, \\sigma \\in \\R_{> 0}$. This is written:  :$X \\sim \\Gaussian \\mu {\\sigma^2}$"}
+{"_id": "26483", "title": "Definition:Stirling Numbers/Karamata Notation", "text": "The notation $\\displaystyle {n \\brack k}$ and $\\displaystyle {n \\brace k}$ for  '''Stirling numbers''' is known as '''Karamata notation'''."}
+{"_id": "26484", "title": "Definition:Stirling Numbers of the First Kind/Notation", "text": "The notation $\\displaystyle {n \\brack k}$ for the unsigned Stirling numbers of the first kind is that proposed by {{AuthorRef|Jovan Karamata}} and publicised by {{AuthorRef|Donald Ervin Knuth|Donald E. Knuth}}. The notation $\\map s {n, k}$ for the signed Stirling numbers of the first kind is similar to variants of that sometimes given for the unsigned. Usage is inconsistent in the literature."}
+{"_id": "26485", "title": "Definition:Stirling Numbers of the First Kind/Unsigned", "text": "{{:Definition:Stirling Numbers of the First Kind/Unsigned/Definition 1}}"}
+{"_id": "26486", "title": "Definition:Stirling Numbers of the First Kind/Signed", "text": "{{:Definition:Stirling Numbers of the First Kind/Signed/Definition 1}}"}
+{"_id": "26487", "title": "Definition:Stirling Numbers of the Second Kind/Notation", "text": "The notation $\\displaystyle {n \\brace k}$ for Stirling numbers of the second kind is that proposed by {{AuthorRef|Jovan Karamata}} and publicised by {{AuthorRef|Donald Ervin Knuth|Donald E. Knuth}}. Other notations exist. Usage is inconsistent in the literature."}
+{"_id": "26488", "title": "Definition:Integer Sequence", "text": "An '''integer sequence''' is a sequence (usually infinite) whose codomain is the set of integers $\\Z$."}
+{"_id": "26489", "title": "Definition:Hypergeometric Series", "text": "A '''hypergeometric series''' is a power series: :$\\beta_0 + \\beta_1 z + \\beta_2 z^2 + \\dots = \\sum_{n \\mathop \\ge 0} \\beta_n z^n$ in which the ratio of successive coefficients is a rational function of $n$: :$\\dfrac {\\beta_{n + 1} } {\\beta_n} = \\dfrac {\\map A n} {\\map B n}$ where $\\map A n$ and $\\map B n$ are polynomials in $n$. {{expand|Needs to include a derivation of the notation ${}_p F_q \\tuple {a_1, \\ldots, a_p; b_1, \\ldots, b_q; z}$}} {{questionable|{{BookLink|The Penguin Dictionary of Mathematics|David Nelson|ed = 4th|edpage = Fourth Edition}} defines this where the coefficient $\\beta_n$ is specifically $\\dfrac {a^{\\overline n} b^{\\overline n} } {c^{\\overline n} n!}$}}"}
+{"_id": "26492", "title": "Definition:Geodesy", "text": "'''Geodesy''' is the branch of geometry which studies the shape of the surface of the Earth."}
+{"_id": "26493", "title": "Definition:First Fundamental Form", "text": "Let $S$ be a surface of a three-dimensional euclidean space. Let $p$ be a point of $S$ and $T_pS$ be the tangent space to $S$ at the point $p$. The '''first fundamental form''' is the bilinear form: :$\\operatorname I: T_p S \\times T_p S \\longrightarrow \\R$ induced from the dot product of $\\R^3$:  :$\\map {\\operatorname I} {x, y} = \\innerprod x y$ The '''first fundamental form''' is a way to calculate the length of a given line $C \\subset S$ or the area of a given bounded region $R \\subset S$. {{Disambiguate|Definition:Region}} {{expand}}"}
+{"_id": "26495", "title": "Definition:Gaussian Curvature", "text": "The '''Gaussian curvature''' $\\Kappa$ of a surface at a point is the product of the principal curvatures, $\\kappa_1$ and $\\kappa_2$, at the given point: :$\\Kappa = \\kappa_1 \\kappa_2$"}
+{"_id": "26496", "title": "Definition:Geodesic Curve", "text": "Let $A$ and $B$ be points on a surface $S$. A '''geodesic curve''' between $A$ and $B$ is the line between $A$ and $B$ completely embedded in $S$ whose length is the smallest of all lines between $A$ and $B$."}
+{"_id": "26498", "title": "Definition:Integral Curvature", "text": "Let $ABC$ be a geodesic triangle on a surface $S$. The '''integral curvature''' of $ABC$ is given by: :$\\displaystyle \\int_{ABC} \\Kappa \\, \\mathrm d a$ where $\\Kappa$ is the Gaussian curvature at each point on $ABC$."}
+{"_id": "26499", "title": "Definition:Geodesic Triangle", "text": "Let $A$, $B$ and $C$ be points on a surface $S$. Let $AB$, $BC$ and $CA$ be geodesic curves between $A$ and $B$, $B$ and $C$, and $C$ and $A$ respectively. The figure enclosed by $AB$, $BC$ and $CA$ is known as a '''geodesic triangle''', and can be denoted $ABC$."}
+{"_id": "26501", "title": "Definition:Biquadratic Residue", "text": "Let $p$ be an odd prime. Let $a \\in \\Z$ be an integer such that $a \\not \\equiv 0 \\pmod p$. Then $a$ is a '''biquadratic residue of $p$''' {{iff}} $x^4 \\equiv a \\pmod p$ has a solution. That is, {{iff}}: : $\\exists x \\in \\Z: x^4 \\equiv a \\pmod p$"}
+{"_id": "26504", "title": "Definition:Gaussian Prime", "text": "A '''Gaussian prime''' is a Gaussian integer which has exactly $4$ divisors which are themselves Gaussian integers."}
+{"_id": "26507", "title": "Definition:Harmonic Function", "text": "A '''harmonic function''' is a is a twice continuously differentiable function $f: U \\to \\R$ (where $U$ is an open set of $\\R^n$) which satisfies Laplace's equation: :$\\dfrac {\\partial^2 f} {\\partial {x_1}^2} + \\dfrac {\\partial^2 f} {\\partial {x_2}^2} + \\cdots + \\dfrac {\\partial^2 f} {\\partial {x_n}^2} = 0$ everywhere on $U$. This is usually written as: :$\\nabla^2 f = 0$ or: :$\\Delta f = 0$"}
+{"_id": "26508", "title": "Definition:Galois Connection", "text": "Let $\\struct {S, \\preceq}$ and $\\struct {T, \\precsim}$ be ordered sets. Let $g: S \\to T$, $d: T \\to S$ be mappings. Then $\\struct {g, d}$ is a '''Galois connection''' {{iff}}: :$g$ and $d$ are increasing mappings and ::$\\forall s \\in S, t \\in T: t \\precsim \\map g s \\iff \\map d t \\preceq s$ $g$ is '''upper adjoint''' and $d$ is '''lower adjoint''' of a Galois connection. {{definition wanted|monotone, antitone, closure operator, closed}} {{NamedforDef|Évariste Galois|cat = Galois}}"}
+{"_id": "26522", "title": "Definition:Taylor Series/Remainder", "text": "Consider the Taylor series expansion $\\map T {\\map f \\xi}$ of $f$ about the point $\\xi$: :$\\displaystyle \\sum_{n \\mathop = 0}^\\infty \\frac {\\paren {x - \\xi}^n} {n!} \\map {f^{\\paren n} } \\xi$ Let $\\map {T_n} {\\map f \\xi}$ be the Taylor polynomial: :$\\displaystyle \\sum_{n \\mathop = 0}^n \\frac {\\paren {x - \\xi}^n} {n!} \\map {f^{\\paren n} } \\xi$ for some $n \\in \\N$. The difference: :$\\displaystyle \\map {R_n} x = \\map f x - \\map {T_n} {\\map f \\xi} = \\int_\\xi^x \\map {f^{\\paren {n + 1} } } t \\dfrac {\\paren {x - t}^n} {n!} \\rd t$ is known as the '''remainder of $\\map T {\\map f \\xi}$''' at $x$."}
+{"_id": "26523", "title": "Definition:Taylor Series/Remainder/Lagrange Form", "text": "The '''Lagrange form''' of the remainder $R_n$ is given by: :$R_n = \\dfrac {\\map {f^{\\paren {n + 1} } } {x^*} } {\\paren {n + 1}!} \\paren {x - \\xi}^{n + 1}$ where $x^* \\in \\openint \\xi x$."}
+{"_id": "26524", "title": "Definition:Taylor Series/Remainder/Cauchy Form", "text": "The '''Cauchy form''' of the remainder $R_n$ is given by: :$R_n = \\dfrac {\\paren {x - \\eta}^n} {n!} \\paren {x - \\xi} \\map {f^{\\paren {n + 1} } } \\eta$ where $\\eta \\in \\closedint \\xi x$."}
+{"_id": "26528", "title": "Definition:Pentagonal Number/Examples", "text": "The first few pentagonal numbers are as follows: :700px === Sequence of Pentagonal Numbers === {{:Definition:Pentagonal Number/Sequence}}"}
+{"_id": "26530", "title": "Definition:Polygonal Number/Degenerate Case", "text": "Consider the polygonal number $P \\left({2, n}\\right)$ when $k = 2$. In this case, the polygon degenerates into a straight line, and the recurrence formula becomes: :$P \\left({2, n}\\right) = \\begin{cases} 0 & : n = 0 \\\\ P \\left({2, n-1}\\right) + 0 \\times \\left({n-1}\\right) + 1 & : n > 0 \\end{cases}$ Hence: :$P \\left({2, n}\\right) = P \\left({2, n-1}\\right) + 1$ and the sequence goes: :$0, 1, 2, 3, \\ldots$ which is of course the natural numbers."}
+{"_id": "26531", "title": "Definition:Triangular Number/Definition 2", "text": ":$\\displaystyle T_n = \\sum_{i \\mathop = 1}^n i = 1 + 2 + \\cdots + \\left({n-1}\\right) + n$"}
+{"_id": "26532", "title": "Definition:Triangular Number/Definition 1", "text": ":$T_n = \\begin{cases} 0 & : n = 0 \\\\ n + T_{n-1} & : n > 0 \\end{cases}$"}
+{"_id": "26533", "title": "Definition:Triangular Number/Definition 3", "text": ":$\\forall n \\in \\N: T_n = P \\left({3, n}\\right) = \\begin{cases} 0 & : n = 0 \\\\ P \\left({3, n - 1}\\right) + \\left({n - 1}\\right) + 1 & : n > 0 \\end{cases}$ where $P \\left({k, n}\\right)$ denotes the $k$-gonal numbers."}
+{"_id": "26534", "title": "Definition:Polygonal Number/Examples", "text": "=== Triangular Numbers === When $k = 3$, the recurrence relation is: {{:Definition:Triangular Number/Definition 3}} See Triangular Number. === Square Numbers === When $k = 4$, the recurrence relation is: {{:Definition:Square Number/Definition 4}} See Square Number. Square numbers are of course better known as: :$S_n = n^2$ === Pentagonal Numbers === When $k = 5$, the recurrence relation is: {{:Definition:Pentagonal Number/Definition 3}} See Pentagonal Number. === Hexagonal Numbers === When $k = 6$, the recurrence relation is: {{:Definition:Hexagonal Number/Definition 3}} See Hexagonal Number. === Heptagonal Numbers === When $k = 7$, the recurrence relation is: {{:Definition:Heptagonal Number/Definition 3}} See Heptagonal Number. === Octagonal Numbers === When $k = 8$, the recurrence relation is: {{:Definition:Octagonal Number/Definition 3}} See Octagonal Number."}
+{"_id": "26535", "title": "Definition:Pentagonal Number/Definition 1", "text": ":$P_n = \\begin{cases} 0 & : n = 0 \\\\ P_{n-1} + 3 n - 2 & : n > 0 \\end{cases}$"}
+{"_id": "26536", "title": "Definition:Pentagonal Number/Definition 2", "text": ":$\\displaystyle P_n = \\sum_{i \\mathop = 1}^n \\left({3 i - 2}\\right) = 1 + 4 + \\cdots + \\left({3 \\left({n - 1}\\right) - 2}\\right) + \\left({3 n - 2}\\right)$"}
+{"_id": "26537", "title": "Definition:Pentagonal Number/Definition 3", "text": ":$\\forall n \\in \\N: P_n = \\map P {5, n} = \\begin {cases} 0 & : n = 0 \\\\ \\map P {5, n - 1} + 3 \\paren {n - 1} + 1 & : n > 0 \\end {cases}$ where $\\map P {k, n}$ denotes the $k$-gonal numbers."}
+{"_id": "26538", "title": "Definition:Hexagonal Number/Definition 1", "text": ":$H_n = \\begin{cases} 0 & : n = 0 \\\\ H_{n-1} + 4 \\left({n-1}\\right) + 1 & : n > 0 \\end{cases}$"}
+{"_id": "26539", "title": "Definition:Hexagonal Number/Definition 2", "text": ":$\\displaystyle H_n = \\sum_{i \\mathop = 1}^n \\left({4 \\left({i - 1}\\right) + 1}\\right) = 1 + 5 + \\cdots + \\left({4 \\left({n-2}\\right) + 1}\\right) + \\left({4 \\left({n - 1}\\right) + 1}\\right)$"}
+{"_id": "26540", "title": "Definition:Hexagonal Number/Definition 3", "text": ":$\\forall n \\in \\N: H_n = P \\left({6, n}\\right) = \\begin{cases} 0 & : n = 0 \\\\ P \\left({6, n - 1}\\right) + 4 \\left({n - 1}\\right) + 1 & : n > 0 \\end{cases}$ where $P \\left({k, n}\\right)$ denotes the $k$-gonal numbers."}
+{"_id": "26541", "title": "Definition:Square Number/Definition 2", "text": ":$S_n = \\begin {cases} 0 & : n = 0 \\\\ S_{n - 1} + 2 n - 1 & : n > 0 \\end {cases}$"}
+{"_id": "26542", "title": "Definition:Square Number/Definition 1", "text": "An integer $n$ is classified as a '''square number''' {{iff}}: :$\\exists m \\in \\Z: n = m^2$ where $m^2$ denotes the integer square function. ==== Euclid's Definition ==== {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book VII/18 - Square Number}}'' {{EuclidDefRefNocat|VII|18|Square Number}}"}
+{"_id": "26543", "title": "Definition:Square Number/Definition 3", "text": ":$\\displaystyle S_n = \\sum_{i \\mathop = 1}^n \\paren {2 i - 1} = 1 + 3 + 5 + \\cdots + \\paren {2 n - 1}$"}
+{"_id": "26544", "title": "Definition:Square Number/Definition 4", "text": ":$\\forall n \\in \\N: S_n = \\map P {4, n} = \\begin{cases} 0 & : n = 0 \\\\ \\map P {4, n - 1} + 2 \\paren {n - 1} + 1 & : n > 0 \\end{cases}$ where $\\map P {k, n}$ denotes the $k$-gonal numbers."}
+{"_id": "26545", "title": "Definition:Triangular Number/Sequence", "text": "The sequence of triangular numbers, for $n \\in \\Z_{\\ge 0}$, begins: :$0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, \\ldots$"}
+{"_id": "26546", "title": "Definition:Pentagonal Number/Sequence", "text": "The sequence of pentagonal numbers, for $n \\in \\Z_{\\ge 0}$, begins: :$0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, \\ldots$"}
+{"_id": "26547", "title": "Definition:Hexagonal Number/Sequence", "text": "The sequence of hexagonal numbers, for $n \\in \\Z_{\\ge 0}$, begins: :$0, 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, \\ldots$"}
+{"_id": "26550", "title": "Definition:Odd Integer/Definition 1", "text": "An integer $n \\in \\Z$ is '''odd''' {{iff}} it is not divisible by $2$. That is, {{iff}} it is not even."}
+{"_id": "26551", "title": "Definition:Odd Integer/Definition 2", "text": "An integer $n \\in \\Z$ is '''odd''' {{iff}}: :$\\exists m \\in \\Z: n = 2 m + 1$"}
+{"_id": "26557", "title": "Definition:Convex Polyhedron", "text": "Let $P$ be a polyhedron. $P$ is a '''convex polyhedron''' {{iff}}: :For all points $A$ and $B$ located inside $P$, the line $AB$ is also inside $P$."}
+{"_id": "26559", "title": "Definition:Neusis Construction/Historical Note", "text": "Many of the constructions performed by {{AuthorRef|Archimedes of Syracuse|Archimedes}} were '''neusis constructions'''. Some sources suggest that {{AuthorRef|Oenopides of Chios}} (c. $440$ BCE) was the first to rank '''compass and straightedge constructions''' above '''neusis constructions''' in importance. It may have been {{AuthorRef|Hippocrates of Chios}} (c. $430$ BCE) who publicised the principle to use a '''compass and straightedge construction''' in preference to a '''neusis construction''' whenever possible. {{ElementsLink}} uses '''compass and straightedge constructions''' exclusively. Under {{AuthorRef|Plato}}, '''neusis''' was considered a last resort when all other methods of construction had failed. By the time of {{AuthorRef|Pappus of Alexandria}}, to use a '''neusis construction''' when other methods would work was considered a serious error."}
+{"_id": "26560", "title": "Definition:Center of Gravity", "text": "Let $B$ be a body. The '''center of gravity''' of $B$ is that point through which the resultant of the system of parallel forces formed by the weights of all the particles constituting $B$ passes for all positions of $B$."}
+{"_id": "26563", "title": "Definition:Archimedean Spiral/Archimedes' Definition", "text": ":''If a straight line of which one extremity remains fixed be made to revolve at a uniform rate in a plane until it returns to the position from which it started, and if, at the same time as the straight line revolves, a point moves at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane.''"}
+{"_id": "26568", "title": "Definition:Information Theory", "text": "'''Information theory''' is that branch of mathematics concerned with the flow of information."}
+{"_id": "26570", "title": "Definition:Special Function", "text": "'''Special functions''' are particular functions which have more or less established names and notations due to their importance in, for example,  analysis, functional analysis or physics. There is no general formal definition, but the list of mathematical functions contains functions which are commonly accepted as special."}
+{"_id": "26572", "title": "Definition:Stochastic Process", "text": "A '''stochastic process''' is a collection of random variables representing the evolution of some real-world system over time."}
+{"_id": "26573", "title": "Definition:Non-Linear System", "text": "A '''non-linear system''' is a system of differential equations which are non-linear."}
+{"_id": "26574", "title": "Definition:Catastrophe Theory", "text": "'''Catastrophe theory''' is a branch of bifurcation theory in the study of dynamical systems. It is also a particular special case of more general singularity theory in geometry."}
+{"_id": "26575", "title": "Definition:Chaos Theory", "text": "'''Chaos theory''' is the branch of non-linear systems that studies the behavior and condition of dynamical systems that are highly sensitive to initial conditions."}
+{"_id": "26576", "title": "Definition:Atomic Physics", "text": "'''Atomic physics''' is the branch of physics concerned with the structure and behaviour of the atom."}
+{"_id": "26577", "title": "Definition:Wave Mechanics", "text": "'''Wave mechanics''' is the branch of physics concerned with the physical behaviour of waves."}
+{"_id": "26580", "title": "Definition:Oscillation/Oscillation on Set", "text": "Let $A \\subseteq X$ be any non-empty subset $A$ of $X$. The '''oscillation of $f$ on (or over) $A$ with respect to $d$''', denoted $\\omega_f \\left({A; d}\\right)$, is defined as the diameter of $f \\left({A}\\right)$: :$\\displaystyle \\omega_f \\left({A; d}\\right) := \\operatorname{diam} \\left({f \\left({A}\\right)}\\right) = \\sup_{x,y \\mathop \\in A} d \\left({f \\left({x}\\right), f \\left({y}\\right)}\\right)$ where the supremum is taken in the extended real numbers $\\overline \\R$. The metric $d$ is often suppressed from the notation if it is clear from context, in which case one would simply write $\\omega_f \\left({A}\\right)$. Similarly, one would speak of the '''oscillation of $f$ on $A$''' in this case."}
+{"_id": "26581", "title": "Definition:Oscillation/Oscillation at Point", "text": "Let $x \\in X$. Let $X$ be a topological space. Denote with $\\mathcal N_x$ the set of neighborhoods of $x$. The '''oscillation of $f$ at $x$ with respect to $d$''', denoted by $\\omega_f \\left({x; d}\\right)$, is defined as: :$\\displaystyle \\omega_f \\left({x; d}\\right) := \\inf_{U \\mathop \\in \\mathcal N_x} \\omega_f \\left({U; d}\\right)$ where $\\omega_f \\left({U; d}\\right)$ denotes the oscillation of $f$ on $U$. The metric $d$ is often suppressed from the notation if it is clear from context, in which case one would simply write $\\omega_f \\left({x}\\right)$. Similarly, one would speak of the '''oscillation of $f$ at $x$''' in this case."}
+{"_id": "26594", "title": "Definition:Point of Inflection", "text": "Let $f$ be a real function which is differentiable on an interval $\\Bbb I \\subseteq \\R$. Let $\\xi \\in \\Bbb I$. === Definition 1 === {{:Definition:Point of Inflection/Definition 1}} === Definition 2 === {{:Definition:Point of Inflection/Definition 2}}"}
+{"_id": "26595", "title": "Definition:Point of Inflection/Definition 1", "text": "$f$ has a '''point of inflection at $\\xi$''' {{iff}} $\\xi$ is a point on $f$ at which $f$ changes from being concave to convex, or vice versa."}
+{"_id": "26596", "title": "Definition:Point of Inflection/Definition 2", "text": "$f$ has a '''point of inflection at $\\xi$''' {{iff}} the derivative $f'$ of $f$ has either a local maximum or a local minimum at $\\xi$."}
+{"_id": "26597", "title": "Definition:Ordering on Mappings", "text": "Let $S$ be a set. Let $\\left({T, \\preceq}\\right)$ be an ordered set. Let $f, g: S \\to T$ be mappings. Then '''ordering on mappings''' $f$ and $g$ denoted $f \\preceq g$ is defined by :$\\forall s \\in S: f\\left({s}\\right) \\preceq g\\left({s}\\right)$"}
+{"_id": "26598", "title": "Definition:Euler-Lagrange Equation", "text": "The '''Euler–Lagrange equation''' is an equation satisfied by a function $\\mathbf q$ of a real argument $t$, which is a stationary point of the functional: :$\\ds \\map S {\\mathbf q} = \\int_a^b \\map L {t, \\map {\\mathbf q} t, \\map {\\mathbf q'} t} \\rd t$ where: :$\\mathbf q$ is the function to be found: :$\\mathbf q: \\closedint a b \\subset \\R \\to X : t \\mapsto x = \\map {\\mathbf q} t$ such that: ::$\\mathbf q$ is differentiable ::$\\map {\\mathbf q} a = \\mathbf x_a$ ::$\\map {\\mathbf q} b = \\mathbf x_b$ :$\\mathbf q'$ is the derivative of $\\mathbf q$: ::$\\mathbf q': \\closedint a b \\to T_{\\map {\\mathbf q} t} X: t \\mapsto v = \\map {\\mathbf q'} t$ :$T_{\\map {\\mathbf q} t} X$ denotes the tangent space to $X$ at the point $\\map {\\mathbf q} t$ :$L$ is a real-valued function with continuous first partial derivatives: ::$L: \\closedint a b \\times T X \\to \\R: \\tuple {t, x, v} \\mapsto \\map L {t, x, v}$ where: :$T X$ is the tangent bundle of $X$ defined by: ::$T X = \\bigcup_{x \\mathop \\in X} \\set x \\times T_x X$ The ''Euler–Lagrange equation'', then, is given by: :$\\displaystyle \\map {L_x} {t, \\map {\\mathbf q} t, \\map {\\mathbf q'} t} - \\dfrac \\d {\\d t} \\map {L_v} {t, \\map {\\mathbf q} t, \\map {\\mathbf q'} t} = 0$ where: :$L_x$ and $L_v$ denote the partial derivatives of $L$ with respect to the second and third arguments respectively."}
+{"_id": "26600", "title": "Definition:Oscillation/Real Space/Oscillation at Point/Infimum", "text": "Let $\\mathcal N_x$ be the set of open subset neighborhoods of $x$. The '''oscillation of $f$ at $x$''' is defined as: :$\\displaystyle \\omega_f \\left({x}\\right) := \\inf_{U \\mathop \\in \\mathcal N_x} \\omega_f \\left({U \\cap X}\\right)$ where $\\omega_f \\left({U \\cap X}\\right)$ denotes the oscillation of $f$ on $U \\cap X$."}
+{"_id": "26601", "title": "Definition:Oscillation/Real Space/Oscillation at Point/Limit", "text": "The '''oscillation of $f$ at $x$''' is defined as: :$\\displaystyle \\omega_f \\left({x}\\right) := \\lim_{h \\to 0^+} \\omega_f \\left({\\left({x - h \\,.\\,.\\, x + h}\\right) \\cap X}\\right)$ where $\\omega_f \\left({\\left({x - h \\,.\\,.\\, x + h}\\right) \\cap X}\\right)$ denotes the oscillation of $f$ on $\\left({x - h \\,.\\,.\\, x + h}\\right) \\cap X$."}
+{"_id": "26602", "title": "Definition:Language of Predicate Logic", "text": "In order to define $\\LL_1$, it is necessary to specify: * An alphabet $\\AA$ * A collation system with the unique readability property for $\\AA$ * A formal grammar (which determines the WFFs of $\\LL_1$)"}
+{"_id": "26604", "title": "Definition:Abel's Integral Equation", "text": "'''Abel's integral equation''' is an integral equation whose purpose is to solve Abel's mechanical problem, which finds how long it will take a bead to slide down a wire. The purpose of '''Abel's integral equation''' is to find the shape of the curve into which the wire is bent in order to yield that result: Let $T \\left({y}\\right)$ be a function which specifies the total time of descent for a given starting height. :$\\displaystyle T \\left({y_0}\\right) = \\int_{y \\mathop = y_0}^{y \\mathop = 0} \\, \\mathrm d t = \\frac 1 {\\sqrt {2 g} } \\int_0^{y_0} \\frac 1 {\\sqrt {y_0 - y} } \\frac {\\mathrm d s} {\\mathrm d y} \\, \\mathrm d y$ where: :$y$ is the height of the bead at time $t$ :$y_0$ is the height from which the bead is released :$g$ is Acceleration Due to Gravity :$s \\left({y}\\right)$ is the distance along the curve as a function of height."}
+{"_id": "26610", "title": "Definition:Abelian Integral", "text": "An '''Abelian integral''' is a complex Riemann integral of the form :$\\displaystyle \\int_{z_0}^z R \\left({x, w}\\right) \\mathrm d x$ where $R \\left({x, w}\\right)$ is an arbitrary rational function of the two variables $x$ and $w$. These variables are related by the equation: :$F \\left({x, w}\\right) = 0$  where $F \\left({x, w}\\right)$ is an irreducible polynomial in $w$: :$F \\left ({x, w}\\right) \\equiv \\phi_n \\left({x}\\right) w^n + \\cdots + \\phi_1 \\left({x}\\right) w + \\phi_0 \\left({x}\\right)$ whose coefficients $\\phi_j \\left({x}\\right), j = 0, 1, \\ldots, n$ are rational functions of $x$.  {{NamedforDef|Niels Henrik Abel|cat = Abel}}"}
+{"_id": "26611", "title": "Definition:Abelian Function", "text": "An '''Abelian function''' is an inverse function of an Abelian integral. {{NamedforDef|Niels Henrik Abel|cat = Abel}}"}
+{"_id": "26612", "title": "Definition:Dirichlet Function", "text": "A '''Dirichlet function''' $D: \\R \\to \\R$ is a real function defined as: :$\\forall x \\in \\R: \\map D x = \\begin {cases} c & : x \\in \\Q \\\\ d & : x \\notin \\Q \\end {cases}$ for $c, d \\in \\R$ such that $c \\ne d$. The canonical example of this has $c = 1$ and $d = 0$. {{NamedforDef|Johann Peter Gustav Lejeune Dirichlet|cat = Dirichlet}}"}
+{"_id": "26614", "title": "Definition:Liouville's Constant", "text": "'''Liouville's constant''' is the real number defined as: {{begin-eqn}} {{eqn | l = \\sum_{n \\mathop \\ge 1} \\dfrac 1 {10^{n!} }       | r = \\frac 1 {10^1} + \\frac 1 {10^2} + \\frac 1 {10^6} +  \\frac 1 {10^{24} } + \\cdots       | c =  }} {{eqn | r = 0 \\cdotp 11000 \\, 10000 \\, 00000 \\, 00000 \\, 00010 \\, 00 \\ldots       | c =  }} {{end-eqn}} {{OEIS|A012245}}"}
+{"_id": "26621", "title": "Definition:Riemann Surface/Elliptic/Definition 2", "text": "A '''Riemann surface''' is '''elliptic''' {{iff}} it admits a metric of constant positive curvature."}
+{"_id": "26623", "title": "Definition:Riemann Surface/Elliptic", "text": "{{:Definition:Riemann Surface/Elliptic/Definition 1}}"}
+{"_id": "26624", "title": "Definition:Riemann Surface/Parabolic", "text": "{{:Definition:Riemann Surface/Parabolic/Definition 1}}"}
+{"_id": "26632", "title": "Definition:Riemannian Geometry (Mathematical Branch)", "text": "'''Riemannian geometry''' is the branch of differential geometry that studies Riemannian manifolds."}
+{"_id": "26634", "title": "Definition:Riemannian Manifold", "text": "A '''Riemannian manifold''' is a smooth manifold on the real space $\\R^n$ upon which a Riemannian metric has been imposed. === Dimension === {{:Definition:Riemannian Manifold/Dimension}} {{NamedforDef|Bernhard Riemann|cat = Riemann}}"}
+{"_id": "26636", "title": "Definition:Riemannian Metric", "text": "Consider a smooth manifold $\\mathcal M$ on the real space $\\R^n$. A '''Riemannian metric''' on $\\mathcal M$ is a metric $\\mathrm d s$ between nearby points $\\left({x_1, x_2, \\ldots, x_n}\\right)$ and $\\left({x_1 + \\mathrm d x_1, x_2 + \\mathrm d x_2, \\ldots, x_n + \\mathrm d x_n}\\right)$ by means of the quadratic differential form: :$\\displaystyle \\mathrm d s^2 = \\sum_{i, j \\mathop = 1}^n g_{i j} \\, \\mathrm d x_i \\, \\mathrm d x_j$ where each $g_{i j}$ is a suitable real-valued function of $x_1, \\ldots, x_n$. Different instances of $g_{i j}$ define different Riemannian geometries on the manifold under discussion. A manifold with such a '''Riemannian metric''' applied is known as a '''Riemannian manifold'''. {{NamedforDef|Bernhard Riemann|cat = Riemann}}"}
+{"_id": "26638", "title": "Definition:Riemann-Christoffel Tensor", "text": "A '''Riemann-Christoffel tensor''' is a tensor field which expresses the curvature of a Riemannian manifold. The '''Riemann-Christoffel tensor''' is given in terms of the Levi-Civita connection $\\nabla$ by: :$R \\left({u, v}\\right) w = \\nabla_u \\nabla_v w - \\nabla_v \\nabla_u w - \\nabla_{\\left[{u, v}\\right]} w$ where $\\left[{u, v}\\right]$ is the Lie bracket of vector fields. It measures the extent to which the metric tensor is not locally isometric to that of Euclidean space."}
+{"_id": "26640", "title": "Definition:Riemannian Manifold/Dimension", "text": "The '''dimension''' of the '''Riemannian manifold''' on $\\R^n$ is $n$."}
+{"_id": "26641", "title": "Definition:Riemannian Geometry", "text": "A '''Riemannian geometry''' is a Riemannian manifold $\\mathcal M$ together with the Riemannian metric that has been applied to $M$."}
+{"_id": "26643", "title": "Definition:Riemann Zeta Function/Critical Line", "text": "Let $s = \\sigma + i t$. The line defined by the equation $\\sigma = \\dfrac 1 2$ is known as the '''critical line'''. Hence the popular form of the statement of the Riemann Hypothesis: :''All the nontrivial zeroes of the Riemann zeta function lie on the '''critical line'''.''"}
+{"_id": "26644", "title": "Definition:Weierstrass Function", "text": "A '''Weierstrass function''' is a real function defined on a closed real interval $I$ which is: :$(1):\\quad$ continuous on $I$ :$(2):\\quad$ nowhere differentiable on $I$."}
+{"_id": "26650", "title": "Definition:Entire Function/Transcendental", "text": "Let $f$ be an '''entire function''' that has an essential singularity at $\\infty$. Then $f$ is a '''transcendental entire function'''."}
+{"_id": "26652", "title": "Definition:Dedekind Cut/Definition 1", "text": "A '''Dedekind cut''' of $\\struct {S, \\preceq}$ is a non-empty proper subset $L \\subsetneq S$ such that: :$(1): \\quad \\forall x \\in L: \\forall y \\in S: y \\prec x \\implies y \\in L$ ($L$ is a lower set in $S$) :$(2): \\quad \\forall x \\in L: \\exists y \\in L: x \\prec y$"}
+{"_id": "26653", "title": "Definition:Dedekind Cut/Definition 2", "text": "A '''Dedekind cut''' of $\\struct {S, \\preceq}$ is an ordered pair $\\tuple {L, R}$ such that: :$(1): \\quad \\set {L, R}$ is a partition of $S$. :$(2): \\quad L$ does not have a greatest element. :$(3): \\quad \\forall x \\in L: \\forall y \\in R: x \\prec y$."}
+{"_id": "26659", "title": "Definition:Cartesian Product of Ordered Sets", "text": "Let $\\left({S_1, \\preceq_1}\\right)$, $\\left({S_2, \\preceq_2}\\right)$ be ordered sets. '''Cartesian product of (ordered sets)''' $\\left({S_1, \\preceq_1}\\right)$ and $\\left({S_2, \\preceq_2}\\right)$ is a pair $\\left({S_1 \\times S_2, \\preceq}\\right)$ where :${\\preceq} \\subseteq \\left({S_1 \\times S_2}\\right) \\times \\left({S_1 \\times S_2}\\right)$ such that :$\\forall s_1, t_1 \\in S_1, s_2, t_2 \\in S_2: \\left({s_1,s_2}\\right) \\preceq \\left({t_1,t_2}\\right) \\iff s_1 \\preceq_1 t_1 \\land s_2 \\preceq_2 t_2$"}
+{"_id": "26661", "title": "Definition:Prime Number/Definition 7", "text": "A '''prime number''' $p$ is an integer greater than $1$ which cannot be written in the form: :$p = a b$ where $a$ and $b$ are both positive integers less than $p$."}
+{"_id": "26662", "title": "Definition:Perfect Number/Definition 1", "text": "A '''perfect number''' is a (strictly) positive integer equal to its aliquot sum."}
+{"_id": "26663", "title": "Definition:Perfect Number/Definition 2", "text": "A '''perfect number''' $n$ is a (strictly) positive integer such that: : $\\sigma \\left({n}\\right) = 2 n$ where $\\sigma: \\Z_{>0} \\to \\Z_{>0}$ is the sigma function."}
+{"_id": "26664", "title": "Definition:Momentum", "text": "=== Linear Momentum === {{:Definition:Linear Momentum}} === Angular Momentum === {{:Definition:Angular Momentum}}"}
+{"_id": "26665", "title": "Definition:Linear Momentum/Relativistic Model", "text": "A more accurate model for the linear momentum of a body is given by: :$\\mathbf p = \\gamma m \\mathbf v$ where $\\gamma$ is the Lorentz Factor: :$\\gamma = \\dfrac c {\\sqrt {c^2 - v^2} } = \\dfrac 1 {\\sqrt {1 - v^2 / c^2} }$ where: : $c$ is the speed of light in vacuum : $v$ is the magnitude of $\\mathbf v$: $v = \\size {\\mathbf v}$ It is clear $\\gamma \\approx 1$ (and thus that $\\mathbf p \\approx m \\mathbf v$) for values of $v$ much less than $c$."}
+{"_id": "26666", "title": "Definition:Thrust", "text": "The '''thrust''' of a rocket is defined as the product of the exhaust velocity and the rate at which its fuel is being consumed: :$\\mathbf t = \\mathbf b \\paren {\\dfrac {\\d m} {\\d t} }$ where: :$\\mathbf b$ is the exhaust velocity :$m$ is the mass of the rocket as a function of time $t$."}
+{"_id": "26671", "title": "Definition:Meet-Continuous Lattice", "text": "Let $\\left({S, \\preceq}\\right)$ be a meet semilattice. Then $\\left({S, \\preceq}\\right)$ is '''meet-continuous''' {{iff}} :$\\left({S, \\preceq}\\right)$ is up-complete and ::(MC): for every an element $x \\in S$ and a directed subset $D$ of $S$: $x \\wedge \\sup D = \\sup \\left\\{ {x \\wedge d: d \\in D}\\right\\}$"}
+{"_id": "26672", "title": "Definition:Twin Primes/Sequence", "text": "The sequence of twin primes begins: :$3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, \\ldots$"}
+{"_id": "26674", "title": "Definition:Prime-Counting Function/Approximations", "text": "A table of some of the values of the prime-counting ($\\pi$) function compared with $\\dfrac x {\\ln x}$ and the Eulerian logarithmic integral $\\displaystyle \\map \\Li x = \\int_2^x \\frac {\\d t} {\\map \\ln t}$: :{| border=\"1\" |- ! align=\"right\" style = \"padding: 2px 10px\" | $n$  ! align=\"right\" style = \"padding: 2px 10px\" | $\\map \\pi n$  ! align=\"right\" style = \"padding: 2px 10px\" | $\\dfrac x {\\ln x}$ ! align=\"right\" style = \"padding: 2px 10px\" | $\\map \\Li x$ |- | align=\"right\" style = \"padding: 2px 10px\" | $1 \\, 000$ | align=\"right\" style = \"padding: 2px 10px\" | $168$ | align=\"right\" style = \"padding: 2px 10px\" | $145$ | align=\"right\" style = \"padding: 2px 10px\" | $178$ |- | align=\"right\" style = \"padding: 2px 10px\" | $10 \\, 000$ | align=\"right\" style = \"padding: 2px 10px\" | $1 \\, 229$ | align=\"right\" style = \"padding: 2px 10px\" | $1 \\, 068$ | align=\"right\" style = \"padding: 2px 10px\" | $1 \\, 246$ |- | align=\"right\" style = \"padding: 2px 10px\" | $100 \\, 000$ | align=\"right\" style = \"padding: 2px 10px\" | $9 \\, 596$ | align=\"right\" style = \"padding: 2px 10px\" | $8 \\, 686$ | align=\"right\" style = \"padding: 2px 10px\" | $9 \\, 630$ |- | align=\"right\" style = \"padding: 2px 10px\" | $1 \\, 000 \\, 000$ | align=\"right\" style = \"padding: 2px 10px\" | $78 \\, 498$ | align=\"right\" style = \"padding: 2px 10px\" | $72 \\, 382$ | align=\"right\" style = \"padding: 2px 10px\" | $78 \\, 628$ |- | align=\"right\" style = \"padding: 2px 10px\" | $10 \\, 000 \\, 000$ | align=\"right\" style = \"padding: 2px 10px\" | $664 \\, 579$ | align=\"right\" style = \"padding: 2px 10px\" | $620 \\, 421$ | align=\"right\" style = \"padding: 2px 10px\" | $664 \\, 918$ |}"}
+{"_id": "26676", "title": "Definition:Rational Point in Plane", "text": "Consider the Cartesian plane $\\R^2$. The points $\\tuple {x, y} \\in \\R^2$ such that $x, y \\in \\Q$ are called the '''rational points (of $\\R^2$)'''."}
+{"_id": "26677", "title": "Definition:Linear Polynomial", "text": "A '''linear polynomial''' is a polynomial whose degree is $1$. Category:Definitions/Polynomial Theory 8ffuktvxhlxm4sq1yfifr3z2ugp5i91"}
+{"_id": "26678", "title": "Definition:Algebraic Number/Degree", "text": "Let $\\alpha$ be an algebraic number. By definition, $\\alpha$ is the root of at least one polynomial $P_n$ with rational coefficients. The '''degree''' of $\\alpha$ is the degree of the minimal polynomial $P_n$ whose coefficients are all in $\\Q$."}
+{"_id": "26681", "title": "Definition:Gelfond-Schneider Constant", "text": "The '''Gelfond-Schneider constant''' is defined as $2$ raised to the power of the square root of $2$: :$2^{\\sqrt 2}$ Its decimal expansion begins: :$2^{\\sqrt 2} \\approx 2 \\cdotp 66514 \\, 4$ {{OEIS|A007507}} {{NamedforDef|Alexander Osipovich Gelfond|name2 = Theodor Schneider|cat = Gelfond|cat2 = Schneider}}"}
+{"_id": "26683", "title": "Definition:Bernoulli Numbers/Sequence", "text": "The sequence of '''Bernoulli numbers''' begins: {{begin-eqn}} {{eqn | l = B_0       | r = 1       | c =  }} {{eqn | l = B_1       | ro= -       | r = \\dfrac 1 2       | c =  }} {{eqn | l = B_2       | r = \\dfrac 1 6       | c =  }} {{eqn | l = B_4       | ro= -       | r = \\dfrac 1 {30}       | c =  }} {{eqn | l = B_6       | r = \\dfrac 1 {42}       | c =  }} {{eqn | l = B_8       | ro= -       | r = \\dfrac 1 {30}       | c =  }} {{eqn | l = B_{10}       | r = \\dfrac 5 {66}       | c =  }} {{eqn | l = B_{12}       | ro= -       | r = \\dfrac {691} {2730}       | c =  }} {{end-eqn}} The odd index Bernoulli numbers, apart from $B_1$, are all equal to $0$."}
+{"_id": "26684", "title": "Definition:Partial Fractions Expansion", "text": "Let $\\map R x = \\dfrac {\\map P x} {\\map Q x}$ be a rational function, where $\\map P x$ and $\\map Q x$ are expressible as polynomial functions. Let $\\map Q x$ be expressible as: :$\\map Q x = \\displaystyle \\prod_{k \\mathop = 1}^n \\map {q_k} x$ where the $\\map {q_k} x$ are themselves polynomial functions of degree at least $1$. Let $\\map R x$ be expressible as: :$\\map R x = \\map r x \\displaystyle \\sum_{k \\mathop = 0}^n \\dfrac {\\map {p_k} x} {\\map {q_k} x}$ where: :$\\map r x$ is a polynomial function which may or may not be the null polynomial, or be of degree $0$ (that is, a constant) :each of the $\\map {p_k} x$ are polynomial functions :the degree of $\\map {p_k} x$ is strictly less than the degree of $\\map {q_k} x$ for all $k$. Then $\\map r x \\displaystyle \\sum_{k \\mathop = 0}^n \\dfrac {\\map {p_k} x} {\\map {q_k} x}$ is a '''partial fractions expansion''' of $\\map R x$."}
+{"_id": "26685", "title": "Definition:Hypocycloid", "text": "Let a circle $C_1$ roll around the inside of another (larger) circle $C_2$. The locus of a given point on the circumference of $C_1$ is known as a '''hypocycloid'''."}
+{"_id": "26686", "title": "Definition:Epicycloid", "text": "Let a circle $C_1$ roll around the outside of another circle $C_2$. The locus of a given point on the circumference of $C_1$ is known as an '''epicycloid'''."}
+{"_id": "26690", "title": "Definition:Astroid", "text": "An '''astroid''' is a hypocycloid with $4$ cusps. :400px"}
+{"_id": "26692", "title": "Definition:Circular Error of Pendulum", "text": "The period of a pendulum $P$ is dependent upon the angle from the vertical from which the bob is released. That period of $P$ is described by Motion of Pendulum as being: :$T = 2 \\sqrt {\\dfrac a g} \\, \\map K {\\map \\sin {\\dfrac \\alpha 2} }$ where: :$a$ is the length of the rod of $P$ :$\\map K k$ is the complete elliptic integral of the first kind. The '''circular error''' of $P$ is the difference between $T$ and its approximate period as described by period Approximate Motion of Pendulum: :$T' = 2 \\pi \\sqrt {\\dfrac a g}$"}
+{"_id": "26694", "title": "Definition:Tusi Couple", "text": "A '''Tusi couple''' is a hypocycloid with $2$ cusps. :400px"}
+{"_id": "26695", "title": "Definition:Hypocycloid/Generator", "text": "The circles $C_1$ and $C_2$ can be referred to as the '''generators''' of the hypocycloid. === Stator === {{Definition:Hypocycloid/Generator/Stator}} === Rotor === {{Definition:Hypocycloid/Generator/Rotor}}"}
+{"_id": "26696", "title": "Definition:Hypocycloid/Generator/Rotor", "text": "The circle $C_1$ can be referred to as the '''rotor''' of the hypocycloid."}
+{"_id": "26697", "title": "Definition:Hypocycloid/Generator/Stator", "text": "The circle $C_2$ can be referred to as the '''stator''' of the hypocycloid."}
+{"_id": "26698", "title": "Definition:Epicycloid/Generator", "text": "The circles $C_1$ and $C_2$ can be referred to as the '''generators''' of the epicycloid. === Stator === {{Definition:Epicycloid/Generator/Stator}} === Rotor === {{Definition:Epicycloid/Generator/Rotor}}"}
+{"_id": "26699", "title": "Definition:Epicycloid/Generator/Stator", "text": "The circle $C_2$ can be referred to as the '''stator''' of the epicycloid."}
+{"_id": "26700", "title": "Definition:Epicycloid/Generator/Rotor", "text": "The circle $C_1$ can be referred to as the '''rotor''' of the epicycloid."}
+{"_id": "26703", "title": "Definition:Deltoid", "text": "A '''deltoid''' is a hypocycloid with $3$ cusps. :400px"}
+{"_id": "26704", "title": "Definition:Oscillation/Real Space/Oscillation on Set", "text": "Let $A \\subseteq X$ be any non-empty subset $A$ of $X$. The '''oscillation of $f$ on (or over) $A$''' is defined as: :$\\displaystyle \\map {\\omega_f} A := \\sup_{x, y \\mathop \\in A} \\size {\\map f x - \\map f y}$ where the supremum is taken in the extended real numbers $\\overline \\R$."}
+{"_id": "26705", "title": "Definition:Oscillation/Real Space/Oscillation at Point", "text": "Let $X$ and $Y$ be real sets. Let $f: X \\to Y$ be a real function. <section notitle=\"true\" name=\"definition\"> Let $x \\in X$. === Definition 1 === {{:Definition:Oscillation/Real Space/Oscillation at Point/Infimum}} === Definition 2 === {{:Definition:Oscillation/Real Space/Oscillation at Point/Epsilon}} === Definition 3 === {{:Definition:Oscillation/Real Space/Oscillation at Point/Limit}} </section>"}
+{"_id": "26706", "title": "Definition:Oscillation/Real Space", "text": "Let $X$ and $Y$ be real sets. Let $f: X \\to Y$ be a real function. === Oscillation on a Set === {{:Definition:Oscillation/Real Space/Oscillation on Set}} {{transclude:Definition:Oscillation/Real Space/Oscillation at Point   |section  = definition   |increase = 1   |title    = Oscillation at a Point   |header   = 3   |link     = true }}"}
+{"_id": "26708", "title": "Definition:Nephroid", "text": "A '''nephroid''' is an epicycloid with $2$ cusps. :600px"}
+{"_id": "26710", "title": "Definition:Curvature", "text": "Let $C$ be a curve defined by a real function which is twice differentiable. The '''curvature''' of a $C$ is the reciprocal of the radius of the osculating circle to $C$ and is often denoted $\\kappa$ (Greek '''kappa''')."}
+{"_id": "26711", "title": "Definition:Radius of Curvature", "text": "The '''radius of curvature''' of a curve $C$ at a point $P$ is defined as the reciprocal of the absolute value of its curvature: :$\\rho = \\dfrac 1 {\\size k}$"}
+{"_id": "26712", "title": "Definition:Normal to Curve", "text": "Let $C$ be a curve embedded in the plane. The '''normal to $C$''' at a point $P$ is defined as the straight line which lies perpendicular to the tangent at $P$ and in the same plane as $P$."}
+{"_id": "26713", "title": "Definition:Center of Curvature", "text": "Let $C$ be a curve embedded in the plane. Let $N$ be the normal to $C$ at a point $P$. Let $\\rho$ be the radius of curvature of $C$ at $P$. Let $Q$ be the point on $N$ which is at a distance $\\rho$ from $P$ on the concave side of $C$. $Q$ is known as the '''center of curvature''' of $C$ at $P$."}
+{"_id": "26714", "title": "Definition:Curvature/Cartesian Form", "text": "Let $C$ be embedded in a cartesian plane. The '''curvature''' $\\kappa$ of $C$ at a point $P = \\tuple {x, y}$ is given by: :$\\kappa = \\dfrac {y''} {\\paren {1 + y'^2}^{3/2} }$ where: :$y' = \\dfrac {\\d y} {\\d x}$ is the derivative of $y$ {{WRT|Differentiation}} $x$ at $P$ :$y'' = \\dfrac {\\d^2 y} {\\d x^2}$ is the second derivative of $y$ {{WRT|Differentiation}} $x$ at $P$."}
+{"_id": "26715", "title": "Definition:Intrinsic Equation", "text": "Let $C$ be a curve. An '''intrinsic equation''' of $C$ is an equation that defines $C$ using a relation between the intrinsic properties of $C$. Thus an '''intrinsic equation''' defines the shape of $C$ without specifying its position relative to an arbitrarily defined coordinate system. === Natural Equation === {{:Definition:Intrinsic Equation/Natural Equation}} === Cesàro Equation === {{:Definition:Intrinsic Equation/Cesàro Equation}} === Whewell Equation === {{:Definition:Intrinsic Equation/Whewell Equation}}"}
+{"_id": "26716", "title": "Definition:Intrinsic Property", "text": "Let $C$ be a curve. The '''intrinsic properties''' of $C$ are those properties that do not depend on the nature of the embedding of $C$ in a coordinate system."}
+{"_id": "26717", "title": "Definition:Intrinsic Property/Examples", "text": "Examples of the intrinsic properties of a curve include: :Arc length $s$ :Curvature $\\kappa$ :Turning angle $\\psi$ :Torsion $\\tau$ :Radius of curvature $\\rho$"}
+{"_id": "26718", "title": "Definition:Turning Angle", "text": "Let $C$ be a curve. The '''turning angle''' $\\psi$ of $C$ at a point $P$ is defined by the differential equation: :$\\kappa = \\dfrac {\\d \\psi} {\\d s}$ where: :$\\kappa$ is the curvature at $P$ :$\\dfrac {\\d \\psi} {\\d s}$ is the derivative of $\\psi$ {{WRT|Differentiation}} the arc length $s$. That is, the '''turning angle''' is a measure of how sharply a curve changes direction. The '''turning angle''' is an intrinsic property of $C$."}
+{"_id": "26719", "title": "Definition:Intrinsic Equation/Whewell Equation", "text": "A '''Whewell equation''' is an intrinsic equation for a curve $C$ such that $C$ is expressed in terms of its intrinsic properties: :Arc length $s$ :Turning angle $\\psi$"}
+{"_id": "26720", "title": "Definition:Curvature/Whewell Form", "text": "The '''curvature''' $\\kappa$ of $C$ at a point $P$ can be expressed in the form of a Whewell equation as: :$\\kappa = \\dfrac {\\d \\psi} {\\d s}$ where: :$\\psi$ is the turning angle of $C$ :$s$ is the arc length of $C$."}
+{"_id": "26721", "title": "Definition:Curvature/Parametric Form", "text": "Let $C$ be embedded in a cartesian plane and defined by the parametric equations: :$\\begin{cases} x = \\map x t \\\\ y = \\map y t \\end{cases}$ The '''curvature''' $\\kappa$ of $C$ at a point $P = \\tuple {x, y}$ is given by: :$\\kappa = \\dfrac {x' y'' - y' x''} {\\tuple {x'^2 + y'^2}^{3/2} }$ where: :$x' = \\dfrac {\\d x} {\\d t}$ is the derivative of $x$ {{WRT|Differentiation}} $t$ at $P$ :$y' = \\dfrac {\\d y} {\\d t}$ is the derivative of $y$ {{WRT|Differentiation}} $t$ at $P$ :$x''$ and $y''$ are the second derivatives of $x$ and $y$ {{WRT|Differentiation}} $t$ at $P$."}
+{"_id": "26725", "title": "Definition:Astronomical Unit/Approximate Values", "text": "The '''astronomical unit''' is approximately $150 \\, 000 \\, 000 \\, \\mathrm {k m}$, or $93$ million (international) miles."}
+{"_id": "26726", "title": "Definition:Astronomical Unit", "text": "The '''astronomical unit''' is a derived unit of length. It is defined as being $149 \\, 597 \\, 870 \\, 700$ metres. The '''astronomical unit''' is the standard unit of measurement used by astronomers when discussing distances within the solar system. It is derived as the approximate mean distance from the Earth to the Sun. === Approximate Values === {{:Definition:Astronomical Unit/Approximate Values}} === Symbol === {{:Definition:Astronomical Unit/Symbol}}"}
+{"_id": "26733", "title": "Definition:Division Algebra/Definition 1", "text": "$\\left({A_F, \\oplus}\\right)$ is a '''division algebra''' {{iff}}: : $\\forall a, b \\in A_F, b \\ne \\mathbf 0_A: \\exists_1 x \\in A_F, y \\in A_F: a = b \\oplus x, a = y \\oplus b$"}
+{"_id": "26734", "title": "Definition:Division Algebra/Definition 2", "text": "$A$ is a division algebra {{iff}} it has no zero divisors: :$\\forall a, b \\in A_F: a \\oplus b = \\mathbf 0_A \\implies a = \\mathbf 0_A \\lor b = \\mathbf 0_A$"}
+{"_id": "26735", "title": "Definition:Conjugate Quaternion/Matrix Form", "text": "Let $\\mathbf x$ be a quaternion defined in matrix form as: :$\\mathbf x = \\begin{bmatrix} a + bi & c + di \\\\ -c + di & a - bi \\end{bmatrix}$ The conjugate quaternion of $\\mathbf x$ is defined as: :$\\overline {\\mathbf x} = \\begin{bmatrix} a - bi & -c - di \\\\ c - di & a + bi \\end{bmatrix}$"}
+{"_id": "26736", "title": "Definition:Conjugate Quaternion/Ordered Pair", "text": "Let $\\mathbf x$ be a quaternion defined as an ordered pair $\\left({a, b}\\right)$  of complex numbers. The conjugate quaternion of $\\mathbf x$ is defined as: :$\\overline {\\mathbf x} = \\overline {\\left({a, b}\\right)} = \\left({\\overline a, -b}\\right)$"}
+{"_id": "26737", "title": "Definition:Field Norm of Quaternion", "text": "Let $\\mathbf x = a \\mathbf 1 + b \\mathbf i + c \\mathbf j + d \\mathbf k$ be a quaternion. Let $\\overline {\\mathbf x}$ be the conjugate of $\\mathbf x$. The '''norm''' of $\\mathbf x$ is the real number defined as: :$n \\left({\\mathbf x}\\right) := \\left\\vert{\\mathbf x \\overline {\\mathbf x} }\\right\\vert = \\left\\vert{\\overline {\\mathbf x} \\mathbf x }\\right\\vert = a^2 + b^2 + c^2 + d^2$"}
+{"_id": "26738", "title": "Definition:Positive Definite (Ring)", "text": "Let $\\struct {R, +, \\times}$ be a ring whose zero is denoted $0_R$. Let $f: R \\to \\R$ be a (real-valued) function on $R$. Then $f$ is '''positive definite''' {{iff}}: :$\\forall x \\in R: \\begin {cases} \\map f x = 0 & : x = 0_R \\\\ \\map f x > 0 & : x \\ne 0_R \\end {cases}$ Category:Definitions/Ring Theory p3oa3bgzq9l393gfnrxy925ltk362pq"}
+{"_id": "26740", "title": "Definition:Nonassociative Algebra", "text": "Let $\\left({A_R, \\oplus}\\right)$ be an algebra over a ring. Then $\\left({A_R, \\oplus}\\right)$ is a nonassociative algebra {{iff}} it is not necessarily the case that $\\oplus$ is associative."}
+{"_id": "26741", "title": "Definition:Lie Algebra", "text": "Let $\\struct {A_R, \\oplus}$ be an algebra over a ring. Then $\\struct {A_R, \\oplus}$ is a Lie algebra {{iff}}: :$\\forall a \\in A_R: a^2 = 0$ :$\\forall a, b, c \\in A_R: a \\paren {b c} + b \\paren {c a} + c \\paren {a b} = 0$"}
+{"_id": "26742", "title": "Definition:Lie Group", "text": "A '''Lie group''' is a group which is also a smooth manifold."}
+{"_id": "26743", "title": "Definition:Cayley Algebra", "text": "The '''Cayley algebra''' is the nonassociative division algebra formed by the Octonions under octonion multiplication."}
+{"_id": "26745", "title": "Definition:Normal Family", "text": "Let $X = \\left({M_1, d_1}\\right)$ and $Y = \\left({M_2, d_2}\\right)$ be complete metric spaces. Let $\\mathcal F = \\left \\langle{f_i}\\right \\rangle_{i \\mathop \\in I}$ be a family of continuous mappings $f_i: X \\to Y$. Then $\\mathcal F$ is a '''normal family''' {{iff}}: : every sequence of mappings in $\\mathcal F$ contains a subsequence which converges uniformly on compact subsets of $X$ to a continuous function $f: X \\to Y$."}
+{"_id": "26747", "title": "Definition:Locally Bounded/Mapping", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $f$ be a mapping defined on $M$. Then $f$ is said to be '''locally bounded''' {{iff}}: :for all $x \\in A$, there is some neighbourhood $N$ of $x$ such that $f \\left[{N}\\right]$ is bounded."}
+{"_id": "26748", "title": "Definition:Uniformly Bounded", "text": "Let $X$ be a  set and let $Y = \\left({A, d}\\right)$ be a metric space. Let $\\mathcal F = \\left \\langle{f_i}\\right \\rangle_{i \\mathop \\in I}$ be a family of mappings $f_i: X \\to Y$. Then $\\mathcal F$ is said to be '''uniformly bounded''' if every mapping $f \\in \\mathcal F$ can be bounded by the same constant. That is, if there exists some $M \\in \\R$ such that: :$\\forall x, y \\in X, i \\in I : d \\left({f_i \\left({x}\\right), f_i \\left({y}\\right)}\\right) \\le M$ Category:Definitions/Metric Spaces Category:Definitions/Boundedness 9v1jbgvwtmrcsgrniksvg1s7rlg2f48"}
+{"_id": "26749", "title": "Definition:Locally Bounded/Family of Mappings", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $\\mathcal F = \\left \\langle{f_i}\\right \\rangle_{i \\mathop \\in I}$ be a family of mappings defined on $M$. Then $\\mathcal F$ is said to be '''locally bounded''' {{iff}}: : for all $x \\in A$, there is some neighbourhood $N$ of $x$ such that $\\mathcal F$ is uniformly bounded on $N$."}
+{"_id": "26750", "title": "Definition:Locally Bounded", "text": "=== Locally Bounded Mapping === {{:Definition:Locally Bounded/Mapping}} === Locally Bounded Family of Mappings === {{:Definition:Locally Bounded/Family of Mappings}} Category:Definitions/Metric Spaces Category:Definitions/Boundedness 2nyk2o70hbyhpuzn2alte3406z76g5w"}
+{"_id": "26752", "title": "Definition:Axiom/Also known as", "text": "An '''axiom''' is also known as a '''postulate'''. Among ancient Greek philosophers, the term '''axiom''' was used for a general truth that was common to everybody (see Euclid's \"common notions\"), while  '''postulate''' had a specific application to the subject under discussion. For most authors, the distinction is no longer used, and the terms are generally used interchangeably. This is the position of {{ProofWiki}}. However, some believe there is a difference significant enough to matter: :''... we shall use \"postulate\" instead of \"axiom\" hereafter, as \"axiom\" has a pernicious historical association of \"self-evident, necessary truth\", which \"postulate\" does not have; a postulate is an arbitrary assumption laid down by the mathematician himself and not by God Almighty.'' :::: -- {{BookReference|Men of Mathematics|1937|Eric Temple Bell}}: Chapter $\\text{II}$: Modern Minds in Ancient Bodies"}
+{"_id": "26753", "title": "Definition:Bounded Sequence/Complex", "text": "Let $\\sequence {z_n}$ be a complex sequence. Then $\\sequence {z_n}$ is '''bounded''' {{iff}}: :$\\exists M \\in \\R$ such that $\\forall i \\in \\N: \\cmod {z_i} \\le M$ where $\\cmod {z_i}$ denotes the complex modulus of $z_i$."}
+{"_id": "26754", "title": "Definition:Bounded Sequence/Complex/Unbounded", "text": "$\\sequence {z_n}$ is '''unbounded''' {{iff}} it is not bounded."}
+{"_id": "26756", "title": "Definition:Mechanical Curve", "text": "A '''mechanical curve''' is a curve which cannot be constructed using a compass and straightedge. Note that while it is not possible to construct such a curve in this manner, it may well be possible to construct as many points on it as required."}
+{"_id": "26757", "title": "Definition:Conic Section/Intersection with Cone/Degenerate Hyperbola", "text": "Let $\\phi < \\theta$, that is: so as to make $K$ a hyperbola. However, let $D$ pass through the apex of $C$. Then $K$ degenerates into a pair of intersecting straight lines."}
+{"_id": "26758", "title": "Definition:Geometric Curve", "text": "A '''geometric curve''' is a curve which can be constructed using a compass and straightedge."}
+{"_id": "26762", "title": "Definition:Number Theory/Also known as", "text": "Older sources refer to '''number theory''' variously as: :'''theory of numbers''' :'''the higher arithmetic''' :'''arithmetic'''. {{AuthorRef|Carl Friedrich Gauss}} himself used the term '''arithmetic''', after the manner of the ancient Greeks. '''Number theory''' can also be seen described by the fanciful name '''the Queen of Mathematics''' for various whimsical reasons."}
+{"_id": "26766", "title": "Definition:Ore Graph", "text": "Let $G = \\struct {V, E}$ be an undirected simple graph. Then $G$ is an '''Ore graph''' {{iff}}: :the sum of the degrees of every pair of non-adjacent vertices is greater than or equal to the order of $G$. That is, {{iff}}: :$\\forall u, v \\in V: \\set {u, v} \\notin E \\implies \\map {\\deg_G} u + \\map {\\deg_G} v \\ge \\card V$ {{NamedforDef|Øystein Ore|cat = Ore}}"}
+{"_id": "26767", "title": "Keith.U/Whatever/Definition:Exponential/Real", "text": "For all definitions of the '''real exponential function''': : The domain of $\\exp$ is $\\R$ : The codomain of $\\exp$ is $\\R_{>0}$ For $x \\in \\R$, the real number $\\exp x$ is called the '''exponential of $x$'''. === As a Sum of a Series === {{:Definition:Exponential Function/Real/Sum of Series}} === As a Limit of a Sequence === {{:Definition:Exponential Function/Real/Limit of Sequence}} === As an Extension of the Rational Exponential === {{:Definition:Exponential Function/Real/Extension of Rational Exponential}} === As the Inverse to the Natural Logarithm === {{:Definition:Exponential Function/Real/Inverse of Natural Logarithm}} === As the Solution of a Differential Equation === {{:Definition:Exponential Function/Real/Differential Equation}}"}
+{"_id": "26768", "title": "Definition:Exponential Function/Real", "text": "For all definitions of the '''real exponential function''': : The domain of $\\exp$ is $\\R$ : The codomain of $\\exp$ is $\\R_{>0}$ For $x \\in \\R$, the real number $\\exp x$ is called the '''exponential of $x$'''. === As a Sum of a Series === {{:Definition:Exponential Function/Real/Sum of Series}} === As a Limit of a Sequence === {{:Definition:Exponential Function/Real/Limit of Sequence}} === As an Extension of the Rational Exponential === {{:Definition:Exponential Function/Real/Extension of Rational Exponential}} === As the Inverse to the Natural Logarithm === {{:Definition:Exponential Function/Real/Inverse of Natural Logarithm}} === As the Solution of a Differential Equation === {{:Definition:Exponential Function/Real/Differential Equation}}"}
+{"_id": "26777", "title": "Definition:Boundary Value Problem", "text": "A '''boundary value problem''' is a differential equation for which a particular solution is to be found, such that the particular solution must satisfy certain boundary conditions."}
+{"_id": "26778", "title": "Definition:Boundary Condition", "text": "Let $\\Phi$ be a differential equation to which a particular solution is to be found. A '''boundary condition''' is an equation relating particular values of the dependent and independent variables which the particular solution to $\\Phi$ must satisfy. It is usual for such '''boundary conditions''' to correspond to the physical extremities of bodies, aspects of whose internal nature is being modelled by means of $\\Phi$."}
+{"_id": "26779", "title": "Definition:Subdivision (Graph Theory)", "text": "Let $G = \\struct {V, E}$ be a graph. === Edge Subdivision === {{:Definition:Subdivision (Graph Theory)/Edge}} === Graph Subdivision === {{:Definition:Subdivision (Graph Theory)/Graph}}"}
+{"_id": "26782", "title": "Definition:Isoperimetrical Problem", "text": "An '''isoperimetrical problem''' is a mathematical problem that asks what figure has the greatest area of all those figures which have a perimeter of a given length."}
+{"_id": "26783", "title": "Definition:Subdivision (Graph Theory)/Edge", "text": "The '''edge subdivision operation''' for an edge $\\set {u, v} \\in E$ is the deletion of $\\set {u, v}$ from $G$ and the addition of two edges $\\set {u, w}$ and $\\set {w, v}$ along with the new vertex $w$.  This operation generates a new graph $H$: :$H = \\struct {V \\cup \\set w, \\paren {E \\setminus \\set {u, v} } \\cup \\set {\\set {u, w}, \\set {w, v} } }$"}
+{"_id": "26784", "title": "Definition:Subdivision (Graph Theory)/Graph", "text": "A graph which has been derived from $G$ by a sequence of edge subdivision operations is called a '''subdivision of $G$'''."}
+{"_id": "26786", "title": "Definition:Algorism/Algorist", "text": "During the middle ages, an '''algorist''' was an arithmetician who used algorism, as opposed to an  abacist who calculated using an abacus. By the time of {{AuthorRef|Leonhard Paul Euler}}, the term had evolved to mean a mathematician who devised algorithms for more complicated operations than those of the classical algorithms."}
+{"_id": "26788", "title": "Definition:Abacism/Abacist", "text": "An '''abacist''' is an arithmetician who uses an abacus to do arithmetic, as opposed to an  algorist who calculates using algorism."}
+{"_id": "26789", "title": "Definition:Riesel Number", "text": "Let $k$ be an odd positive integer. Then $k$ is a '''Riesel number''' {{iff}}: :For all positive integers $n$, the number $k \\, 2^n - 1$ is composite."}
+{"_id": "26791", "title": "Definition:Real Interval/Unit Interval/Open", "text": "The open interval between $0$ and $1$ is referred to as the '''open unit interval'''. :$\\left({0 \\,.\\,.\\, 1}\\right) = \\left\\{ {x \\in \\R: 0 < x < 1}\\right\\}$"}
+{"_id": "26792", "title": "Definition:Real Interval/Unit Interval/Closed", "text": "The closed interval from $0$ to $1$ is denoted $\\mathbb I$ (or a variant) by some authors: :$\\mathbb I := \\closedint 0 1 = \\set {x \\in \\R: 0 \\le x \\le 1}$ This is often referred to as the '''closed unit interval'''."}
+{"_id": "26794", "title": "Definition:Power (Algebra)/Real Number/Definition 2", "text": "Let $f : \\Q \\to \\R$ be the real-valued function defined as: :$f \\left({ q }\\right) = x^q$ where $a^q$ denotes $a$ to the power of $q$. Then we define $x^r$ as the unique continuous extension of $f$ to $\\R$."}
+{"_id": "26795", "title": "Definition:Power (Algebra)/Real Number/Definition 1", "text": "We define $x^r$ as: :$x^r := \\map \\exp {r \\ln x}$ where $\\exp$ denotes the exponential function."}
+{"_id": "26796", "title": "Definition:Nonnumerical Analysis", "text": "'''Nonnumerical analysis''' is a branch of discrete mathematics that studies subjects emerging from the evolution of computer science, and includes such topics as: :information processing :analysis of algorithms and so on."}
+{"_id": "26800", "title": "Definition:Replacement Operation", "text": "The '''replacement operation''' on two variables $a$ and $b$ is denoted: :$a \\gets b$ and is interpreted as: :the value of variable $a$ is to be replaced by the current value of variable $b$, while leaving the value of variable $b$ the same."}
+{"_id": "26801", "title": "Definition:Exchange Operation", "text": "The '''exchange operation''' on two variables $a$ and $b$ is denoted: :$a \\leftrightarrow b$ and is interpreted as: :the value of variable $a$ is to be replaced by the current value of variable $b$ and at the same time: :the value of variable $b$ is to be replaced by the current value of variable $a$."}
+{"_id": "26802", "title": "Definition:Computational Method/Computational Sequence", "text": "Each $x \\in I$ defines a '''computational sequence''' $x_0, x_1, x_2, \\ldots$ as follows: : $x_0 = x$ : $\\forall k \\ge 0: x_{k+1} = f \\left({x_k}\\right)$"}
+{"_id": "26803", "title": "Definition:Algorithm/Step", "text": "An algorithm consists of a finite set of '''steps''', uniquely identified by means of a label, conventionally numeric. A '''step''' of an algorithm consists of: :an '''operation''' :an '''instruction''' as to what the algorithm is to do next, which will be one of the following: ::$(1): \\quad$ By default: to move onto the next '''step''' in the sequence ::$(2): \\quad$ Based on the result of a '''condition''', the specific '''step''' to perform next ::$(3): \\quad$ To '''terminate'''."}
+{"_id": "26804", "title": "Definition:Flow Chart", "text": "A '''flow chart''' is a graphical depiction of an algorithm in which the steps are depicted in the form of boxes connected together by arrows. Conventionally, the shape of the box representing a step is dependent upon the type of operation encapsulated within the step: :Rectangular for an action :A different shape, conventionally a diamond, for a condition. On {{ProofWiki}}, the preferred shape for condition boxes is rectangular with rounded corners.  This is to maximise ease and neatness of presentation: configuring a description inside a diamond shaped boxes in order for it to be aesthetically pleasing can be challenging and tedious. Also on {{ProofWiki}}, it is part of the accepted style to implement the start and end points of the algorithm using a box of a particular style, in this case with a double border."}
+{"_id": "26806", "title": "Definition:Algorithm/Action", "text": "In an algorithm, an '''action''' is part of a step of that algorithm. It is an operation which changes the state of the algorithm in some manner."}
+{"_id": "26808", "title": "Definition:Algorithm/Condition", "text": "In an algorithm, a '''condition''' is part of a step of that algorithm. It is an operation which determines the outcome of a decision as to which step to execute next,"}
+{"_id": "26809", "title": "Definition:Element is Way Below", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $x, y \\in S$. Then $x$ '''is way below''' $y$, denoted $x \\ll y$, {{iff}} :for every directed subset $D$ of $S$ ::if $D$ admits a supremum and $y \\preceq \\sup D$ ::then there exists $d \\in D$ such that $x \\preceq d$"}
+{"_id": "26810", "title": "Definition:Programming Language", "text": "A '''programming language''' is a special purpose language used to implement algorithms for a computer to execute."}
+{"_id": "26811", "title": "Definition:Program", "text": "A '''program''' is an algorithm which has been written for a computer to execute."}
+{"_id": "26815", "title": "Definition:Algorithm/Formal Specification", "text": "An '''algorithm''' can be implemented formally as a computational method $\\left({Q, I, \\Omega, f}\\right)$ as follows: Let $A$ be a finite set of symbols. Let $A^*$ be the set of all collations on $A$: :$\\left\\{ {x_1 x_2 \\cdots x_n: n \\ge 0, \\forall j: 1 \\le j \\le n: x_j \\in A}\\right\\}$ The states of the computation are encoded so as to be represented by elements of $A^*$. Let $N \\in \\Z_{\\ge 0}$. Let $Q$ be the set of all ordered pairs $\\left({\\sigma, j}\\right)$ where $\\sigma \\in A^*, j \\in \\Z: 0 \\le j \\le N$. Let $I \\subseteq Q$ such that $j = 0$. Let $\\Omega \\subseteq Q$ such that $j = N$. Let $\\theta, \\sigma \\in A^*$. Then $\\theta$ '''occurs in $\\sigma$''' {{iff}} $\\sigma$ has the form: :$\\alpha \\theta \\omega$ where $\\alpha, \\omega \\in A^*$. Let $f$ be a mapping of the following type: :$f \\left({\\left({\\sigma, j}\\right)}\\right) = \\begin{cases}\\left({\\sigma, a_j}\\right) : & \\sigma_j \\text { does not occur in } \\sigma \\\\ \\left({\\alpha \\theta_j \\omega, b_j}\\right) : & \\alpha \\text { is the shortest element of $A^*$ such that } \\sigma  = \\alpha \\theta_j \\omega\\end{cases}$ :$f \\left({\\left({\\sigma, N}\\right)}\\right) = \\left({\\sigma, N}\\right)$"}
+{"_id": "26819", "title": "Definition:Golden Mean/Geometrical Interpretation", "text": "Let $\\Box ADEB$ be a square. Let $\\Box ADFC$ be a rectangle such that: :$AC : AD = AD : BC$ where $AC : AD$ denotes the ratio of $AC$ to $AD$. :200px Then if you remove $\\Box ADEB$ from $\\Box ADFC$, the sides of the remaining rectangle have the same ratio as the sides of the original one. Thus if $AC = \\phi$ and $AD = 1$ we see that this reduces to: :$\\phi : 1 = 1 : \\phi - 1$ where $\\phi$ is the golden mean."}
+{"_id": "26820", "title": "Definition:Golden Mean/Definition 1", "text": "Let a line segment $AB$ be divided at $C$ such that: :$AB : AC = AC : BC$ Then the '''golden mean''' $\\phi$ is defined as: :$\\phi := \\dfrac {AB} {AC}$"}
+{"_id": "26823", "title": "Definition:Golden Mean/One Minus Golden Mean", "text": "The number: :$1 - \\phi$ is often denoted $\\hat \\phi$."}
+{"_id": "26824", "title": "Definition:Golden Mean/Definition 2", "text": "The '''golden mean''' is the unique positive real number $\\phi$ satisfying: :$\\phi = \\dfrac {1 + \\sqrt 5} 2$"}
+{"_id": "26825", "title": "Definition:Golden Mean/Definition 3", "text": "The '''golden mean''' is the unique positive real number $\\phi$ satisfying: :$\\phi = \\dfrac 1 {\\phi - 1}$"}
+{"_id": "26826", "title": "Definition:Basis Expansion/Positive Real Numbers", "text": "Let $x \\in \\R$ be a real number such that $x \\ge 0$. Let $b \\in \\N: b \\ge 2$. Let us define the recursive sequence: :$\\forall n \\in \\N: n \\ge 1: \\sequence {f_n} = \\begin {cases} b \\paren {x - \\floor x} & : n = 1 \\\\ b \\paren {f_{n - 1} - \\floor {f_{n - 1} } } & : n > 1 \\end{cases}$ Then we define: :$\\forall n \\in \\N: n \\ge 1: \\sequence {d_n} = \\floor {f_n}$ It follows from the method of construction and the definition of the floor function that: :$\\forall n: 0 \\le f_n < b$ and hence $\\forall n: 0 \\le d_n \\le b - 1$ :$\\forall n: f_n = 0 \\implies f_{n + 1} = 0$ and hence $d_{n + 1} = 0$. Hence we can express $x = \\floor x + \\displaystyle \\sum_{j \\mathop \\ge 1} \\frac {d_j} {b^j}$ as: :$\\sqbrk {s \\cdotp d_1 d_2 d_3 \\ldots}_b$ where: :$s = \\floor x$ :it is not the case that there exists $m \\in \\N$ such that $d_M = b - 1$ for all $M \\ge m$. (That is, the sequence of digits does not end with an infinite sequence of $b - 1$.) This is called the '''expansion of $x$ in base $b$'''. The generic term for such an expansion is a '''basis expansion'''. It follows from the Division Theorem that for a given $b$ and $x$ this expansion is unique."}
+{"_id": "26827", "title": "Definition:Basis Expansion/Negative Real Numbers", "text": "Let $x \\in \\R: x < 0$. We take the absolute value $y$ of $x$, that is: :$y = \\size x$ Then we take the '''expansion of $y$ in base $b$''': :$\\size {s . d_1 d_2 d_3 \\ldots}_b$ where $s = \\floor y$. Finally, the '''expansion of $x$ in base $b$''' is defined as: :$-\\sqbrk {s . d_1 d_2 d_3 \\ldots}_b$"}
+{"_id": "26828", "title": "Definition:Less Than (Real Numbers)", "text": "Let $\\R_{>0}$ denote the set of  strictly positive real numbers. Let $x, y \\in \\R$. Then we write $x < y$ {{iff}}: :$ y - x \\in \\R_{>0}$ and we say that $x$ is '''less than''' $y$."}
+{"_id": "26829", "title": "Definition:Strictly Positive/Real Number/Definition 2", "text": "The '''strictly positive real numbers''', written $R_{>0}$, is the subset of $\\R$ that satisfies the following: {{begin-axiom}} {{axiom | n = \\R_{>0} 1         | lc= Closure under addition         | q = \\forall x, y \\in \\R_{>0}         | m = x + y \\in \\R_{>0} }} {{axiom | n = \\R_{>0} 2         | lc= Closure under multiplication         | q = \\forall x, y \\in \\R_{>0}         | m = xy \\in \\R_{>0} }} {{axiom | n = \\R_{>0} 3         | lc=  Trichotomy         | q = \\forall x \\in \\R         | m = x \\in \\R_{>0} \\lor x = 0 \\lor -x \\in \\R_{>0} }} {{end-axiom}}"}
+{"_id": "26830", "title": "Definition:Strictly Positive/Real Number/Definition 1", "text": "The '''strictly positive real numbers''' are the set defined as: :$\\R_{>0} := \\set {x \\in \\R: x > 0}$ That is, all the real numbers that are strictly greater than zero."}
+{"_id": "26831", "title": "Definition:Decimal Expansion/Decimal Point", "text": "The dot that separates the integer part from the fractional part of $x$ is called the '''decimal point'''. That is, it is the radix point when used specifically for a base $10$ representation."}
+{"_id": "26833", "title": "Definition:Number Base/Integers", "text": "Let $n \\in \\Z$ be an integer. Let $b$ be any integer such that $b > 1$. By the Basis Representation Theorem, $n$ can be expressed uniquely in the form: :$\\displaystyle n = \\sum_{j \\mathop = 0}^m r_j b^j$ where: :$m$ is such that $b^m \\le n < b^{m + 1}$ :all the $r_j$ are such that $0 \\le r_j < b$. <noinclude> {{MissingLinks|The bounds on $n$ are not stated as part of the Basis Representation Theorem. Is there some other link to these bounds?}}</noinclude> <noinclude>{{Improve|The definition is incomplete as the Basis Representation Theorem is only stated for strictly positive integers}}</noinclude> The number $b$ is known as the '''number base''' to which $n$ is represented. $n$ is thus described as being '''(written) in base $b$'''. Thus we can write $\\displaystyle n = \\sum_{j \\mathop = 0}^m {r_j b^j}$ as: :$\\sqbrk {r_m r_{m - 1} \\ldots r_2 r_1 r_0}_b$ or, if the context is clear: :${r_m r_{m - 1} \\ldots r_2 r_1 r_0}_b$"}
+{"_id": "26834", "title": "Definition:Number Base/Real Numbers", "text": "Let $x \\in \\R$ be a real number such that $x \\ge 0$. Let $b \\in \\N: b \\ge 2$. See the definition of Basis Expansion for how we can express $x$ in the form: :$x = \\sqbrk {s \\cdotp d_1 d_2 d_3 \\ldots}_b$ Then we express $m$ as for integers, and arrive at: :$x = \\sqbrk {r_m r_{m - 1} \\ldots r_2 r_1 r_0 \\cdotp d_1 d_2 d_3 \\ldots}_b$ or, if the context is clear: :$r_m r_{m - 1} \\ldots r_2 r_1 r_0 \\cdotp d_1 d_2 d_3 \\ldots_b$"}
+{"_id": "26835", "title": "Definition:Number Base/Integer Part", "text": "In the basis expansion: :$x = \\left[{r_m r_{m-1} \\ldots r_2 r_1 r_0 . d_1 d_2 d_3 \\ldots}\\right]_b$ the part $r_m r_{m-1} \\ldots r_2 r_1 r_0$ is known as the '''integer part'''."}
+{"_id": "26836", "title": "Definition:Number Base/Fractional Part", "text": "In the basis expansion: :$x = \\sqbrk {r_m r_{m - 1} \\ldots r_2 r_1 r_0 . d_1 d_2 d_3 \\ldots}_b$ the part $.d_1 d_2 d_3 \\ldots$ is known as the '''fractional part'''."}
+{"_id": "26837", "title": "Definition:Number Base/Radix Point", "text": "In the basis expansion: :$x = \\sqbrk {r_m r_{m - 1} \\ldots r_2 r_1 r_0 \\cdotp d_1 d_2 d_3 \\ldots}_b$ the dot that separates the integer part from the fractional part is called the '''radix point'''."}
+{"_id": "26839", "title": "Definition:Natural Logarithm/Positive Real/Definition 3", "text": "Let $x \\in \\R$ be a real number such that $x > 0$. The '''(natural) logarithm''' of $x$ is defined as: :$ \\displaystyle \\ln x := \\lim_{n \\mathop \\to \\infty} n \\paren {\\sqrt [n] x - 1}$"}
+{"_id": "26841", "title": "Definition:General Logarithm/Binary", "text": "Logarithms base $2$ are becoming increasingly important in computer science. They are often referred to as '''binary logarithms'''."}
+{"_id": "26843", "title": "Definition:Pointwise Equicontinuous", "text": "Let $X = \\left({A, d}\\right)$ and $Y = \\left({B, \\rho}\\right)$ be metric spaces. Let $\\left\\langle{f_i}\\right\\rangle_{i \\mathop \\in I}$ be a family of mappings $f_i: X \\to Y$. Then $\\left\\langle{f_i}\\right\\rangle_{i \\mathop \\in I}$ is said to be '''pointwise equicontinuous''' at $x_0 \\in A$ {{iff}}: :$\\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: \\forall i \\in I: \\forall x \\in A: d \\left({x, x_0}\\right) < \\delta \\implies \\rho \\left({f_i \\left({x}\\right), f_i \\left({x_0}\\right)}\\right) < \\epsilon$"}
+{"_id": "26844", "title": "Definition:Uniformly Equicontinuous", "text": "Let $X = \\left({A, d}\\right)$ and $Y = \\left({B, \\rho}\\right)$ be metric spaces. Let $\\left\\langle{f_i}\\right\\rangle_{i \\mathop \\in I} $ be a family of mappings $f_i: X \\to Y$. Then $\\left\\langle{f_i}\\right\\rangle_{i \\mathop \\in I}$ is said to be '''uniformly equicontinuous''' on $S \\subseteq A$ {{iff}}: :$\\forall \\epsilon \\in \\R_{>0} : \\exists \\delta \\in \\R_{>0}: \\forall i \\in I: \\forall x, y \\in S : d \\left({x, y}\\right) < \\delta \\implies \\rho \\left({f_i \\left({x}\\right), f_i \\left({y}\\right) }\\right) < \\epsilon$"}
+{"_id": "26845", "title": "Definition:Discontinuity of the First Kind", "text": "Let $X$ be an open subset of $\\R$. Let $f: X \\to Y$ be a real function. Let $f$ be discontinuous at some point $c \\in X$. Then $c$ is called a '''discontinuity of the first kind''' of $f$ {{iff}}: :$\\displaystyle \\lim_{x \\mathop \\to c^-} \\map f x$ and $\\displaystyle \\lim_{x \\mathop \\to c^+} \\map f x$ exist where $\\displaystyle \\lim_{x \\mathop \\to c^-} \\map f x$ and $\\displaystyle \\lim_{x \\mathop \\to c^+} \\map f x$ denote the limit from the left and limit from the right at $c$ respectively."}
+{"_id": "26846", "title": "Definition:Jump Discontinuity", "text": "Let $X$ be an open subset of $\\R$. Let $f: X \\to Y$ be a real function. Let $f$ be discontinuous at some point $c \\in X$. Then $c$ is called a '''jump discontinuity''' of $f$ {{iff}}: :$\\displaystyle \\lim_{x \\mathop \\to c^-} \\map f x$ and $\\displaystyle \\lim_{x \\mathop \\to c^+} \\map f x$ exist and are not equal where $\\displaystyle \\lim_{x \\mathop \\to c^-} \\map f x$ and $\\displaystyle \\lim_{x \\mathop \\to c^+} \\map f x$ denote the limit from the left and limit from the right at $c$ respectively. === Jump === {{:Definition:Jump Discontinuity/Jump}}"}
+{"_id": "26848", "title": "Definition:Summation/Indexed", "text": "The composite is called the '''summation''' of $\\tuple {a_1, a_2, \\ldots, a_n}$, and is written: :$\\displaystyle \\sum_{j \\mathop = 1}^n a_j = \\paren {a_1 + a_2 + \\cdots + a_n}$"}
+{"_id": "26849", "title": "Definition:Summation/Inequality", "text": "The '''summation''' of $\\left({a_1, a_2, \\ldots, a_n}\\right)$ can be written: :$\\displaystyle \\sum_{1 \\mathop \\le j \\mathop \\le n} a_j = \\left({a_1 + a_2 + \\cdots + a_n}\\right)$"}
+{"_id": "26850", "title": "Definition:Summation/Propositional Function", "text": "Let $R \\left({j}\\right)$ be a propositional function of $j$. Then we can write the '''summation''' as: :$\\displaystyle \\sum_{R \\left({j}\\right)} a_j = \\text{ The sum of all $a_j$ such that $R \\left({j}\\right)$ holds}$. If more than one propositional function is written under the summation sign, they must ''all'' hold."}
+{"_id": "26851", "title": "Definition:Summation/Index Variable", "text": "Consider the '''summation''', in either of the three forms: :$\\displaystyle \\sum_{j \\mathop = 1}^n a_j \\qquad \\sum_{1 \\mathop \\le j \\mathop \\le n} a_j \\qquad \\sum_{\\map R j} a_j$ The variable $j$, an example of a bound variable, is known as the '''index variable''' of the summation."}
+{"_id": "26852", "title": "Definition:Summation/Infinite", "text": "Let an infinite number of values of $j$ satisfy the propositional function $R \\left({j}\\right)$. Then the precise meaning of $\\displaystyle \\sum_{R \\left({j}\\right)} a_j$ is: :$\\displaystyle \\sum_{R \\left({j}\\right)} a_j = \\left({\\lim_{n \\mathop \\to \\infty} \\sum_{\\substack {R \\left({j}\\right) \\\\ -n \\mathop \\le j \\mathop < 0}} a_j}\\right) + \\left({\\lim_{n \\mathop \\to \\infty} \\sum_{\\substack {R \\left({j}\\right) \\\\ 0 \\mathop \\le j \\mathop \\le n} } a_j}\\right)$ provided that both limits exist. If either limit ''does'' fail to exist, then the '''infinite summation''' does not exist."}
+{"_id": "26853", "title": "Definition:Real Interval/Bounded", "text": "Let $I$ be an interval. Let $I$ be either open, half-open or closed. Then $I$ is said to be a '''bounded (real) interval'''."}
+{"_id": "26854", "title": "Definition:Fiber of Truth", "text": "Let $P: X \\to \\set {\\mathrm T, \\mathrm F}$ be a propositional function defined on a domain $X$. The '''fiber of truth (under $P$)''' is the preimage, or '''fiber''', of $\\mathrm T$ under $P$: :$\\set {x \\in X: \\map P x = \\mathrm T}$ That is, the elements of $X$ whose image under $P$ is $\\mathrm T$."}
+{"_id": "26855", "title": "Definition:Way Below Closure", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $x \\in S$. The '''way below closure''' of $x$, denoted by $x^\\ll$, is defined by :$x^\\ll := \\set {y \\in S: y \\ll x}$ where $y \\ll x$ denotes that $y$ is way below $x$."}
+{"_id": "26856", "title": "Definition:Axiom of Approximation", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. '''Axiom of approximation''' says  :$\\forall x \\in S: x = \\sup \\left({x^\\ll}\\right)$ where $x^\\ll$ denotes the way below closure of $x$."}
+{"_id": "26857", "title": "Definition:Continuous Ordered Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. {{explain|$x^\\ll$}} Then $\\left({S, \\preceq}\\right)$ is '''continuous''' {{iff}} :(for all elements $x$ of $S$: $x^\\ll$ is directed) and :$\\left({S, \\preceq}\\right)$ is up-complete and satisfies axiom of approximation."}
+{"_id": "26859", "title": "Definition:Euler's Number/Base of Exponential", "text": "There is a number $x \\in \\R$ such that: :$\\displaystyle \\lim_{h \\to 0} \\frac{ x^{h} - 1 }{ h } = 1$ This number is called '''Euler's Number''' and is denoted $e$."}
+{"_id": "26860", "title": "Definition:Midpoint-Convex", "text": "Let $f$ be a real function defined on a real interval $I$. $f$ is '''midpoint-convex''' {{iff}}: :$\\forall x, y \\in I: f \\left({\\dfrac {x + y} 2}\\right) \\le \\dfrac {f \\left({x}\\right) + f \\left({y}\\right)} 2$"}
+{"_id": "26861", "title": "Definition:Midpoint-Concave", "text": "Let $f$ be a real function defined on a real interval $I$. $f$ is '''midpoint-concave''' {{iff}}: :$\\forall x, y \\in I: f \\left({\\dfrac {x + y} 2}\\right) \\ge \\dfrac {f \\left({x}\\right) + f \\left({y}\\right)} 2$"}
+{"_id": "26862", "title": "Definition:Strictly Midpoint-Convex", "text": "Let $f$ be a real function defined on a real interval $I$. $f$ is '''strictly midpoint-convex''' {{iff}}: :$\\forall x, y \\in I : f \\left({\\dfrac {x + y} 2}\\right) < \\dfrac {f \\left({x}\\right) + f \\left({y}\\right)} 2$"}
+{"_id": "26863", "title": "Definition:Strictly Midpoint-Concave", "text": "Let $f$ be a real function defined on a real interval $I$. $f$ is '''strictly midpoint-concave''' {{iff}}: :$\\forall x, y \\in I : f \\left({\\dfrac {x + y} 2}\\right) > \\dfrac {f \\left({x}\\right) + f \\left({y}\\right)} 2$"}
+{"_id": "26864", "title": "Definition:Summation/Propositional Function/Iverson's Convention", "text": "Let $\\displaystyle \\sum_{R \\left({j}\\right)} a_j$ be the summation over all $a_j$ such that $j$ satisfies $R$. This can also be expressed: :$\\displaystyle \\sum_{j \\mathop \\in \\Z} a_j \\left[{R \\left({j}\\right)}\\right]$ where $\\left[{R \\left({j}\\right)}\\right]$ is Iverson's convention."}
+{"_id": "26867", "title": "Definition:Product Notation (Algebra)/Index", "text": "The composite is called the '''product''' of $\\tuple {a_1, a_2, \\ldots, a_n}$, and is written: :$\\displaystyle \\prod_{j \\mathop = 1}^n a_j = \\paren {a_1 \\times a_2 \\times \\cdots \\times a_n}$"}
+{"_id": "26868", "title": "Definition:Product Notation (Algebra)/Inequality", "text": "The '''product''' of $\\left({a_1, a_2, \\ldots, a_n}\\right)$ can be written: :$\\displaystyle \\prod_{1 \\mathop \\le j \\mathop \\le n} a_j = \\left({a_1 \\times a_2 \\times \\cdots \\times a_n}\\right)$"}
+{"_id": "26869", "title": "Definition:Product Notation (Algebra)/Propositional Function", "text": "Let $R \\left({j}\\right)$ be a propositional function of $j$. Then we can write: :$\\displaystyle \\prod_{R \\left({j}\\right)} a_j = \\text{ The product of all } a_j \\text{ such that } R \\left({j}\\right) \\text{ holds}$. If more than one propositional function is written under the product sign, they must ''all'' hold."}
+{"_id": "26871", "title": "Definition:Product Notation (Algebra)/Infinite", "text": "Let an infinite number of values of $j$ satisfy the propositional function $\\map R j$. Then the precise meaning of $\\displaystyle \\prod_{\\map R j} a_j$ is: :$\\displaystyle \\prod_{\\map R j} a_j = \\paren {\\lim_{n \\mathop \\to \\infty} \\prod_{\\substack {\\map R j \\\\ -n \\mathop \\le j \\mathop < 0} } a_j} \\times \\paren {\\lim_{n \\mathop \\to \\infty} \\prod_{\\substack {\\map R j \\\\ 0 \\mathop \\le j \\mathop \\le n} } a_j}$ provided that both limits exist. If either limit ''does'' fail to exist, then the '''infinite product''' does not exist."}
+{"_id": "26872", "title": "Definition:Product Notation (Algebra)/Index Variable", "text": "Consider the '''product''', in either of the three forms: :$\\displaystyle \\prod_{j \\mathop = 1}^n a_j \\qquad \\prod_{1 \\mathop \\le j \\mathop \\le n} a_j \\qquad \\prod_{R \\left({j}\\right)} a_j$ The variable $j$, an example of a bound variable, is known as the '''index variable''' of the product."}
+{"_id": "26873", "title": "Definition:Summation/Inequality/Multiple Indices", "text": "Let $\\displaystyle \\sum_{0 \\mathop \\le j \\mathop \\le n} a_j$ denote the '''summation''' of $\\left({a_0, a_1, a_2, \\ldots, a_n}\\right)$. Summands with multiple indices can be denoted by propositional functions in several variables, for example: :$\\displaystyle \\sum_{0 \\mathop \\le i \\mathop \\le n} \\left({\\sum_{0 \\mathop \\le j \\mathop \\le n} a_{i j} }\\right) = \\sum_{0 \\mathop \\le i, j \\mathop \\le n} a_{i j}$ :$\\displaystyle \\sum_{0 \\mathop \\le i \\mathop \\le n} \\left({\\sum_{0 \\mathop \\le j \\mathop \\le i} a_{i j} }\\right) = \\sum_{0 \\mathop \\le j \\mathop \\le i \\mathop \\le n} a_{i j}$"}
+{"_id": "26874", "title": "Definition:Telescoping Series", "text": "A '''telescoping series''' is a series whose partial sums eventually only have a fixed number of terms after cancellation through algebraic manipulation."}
+{"_id": "26876", "title": "Definition:Norm of Subdivision", "text": "Let $\\closedint a b$ be a closed interval of the set of real numbers $\\R$. Let $\\Delta = \\set {x_0, x_1, x_2, \\ldots, x_{n - 1}, x_n}$ form a finite subdivision of $\\closedint a b$. Then the '''norm''' of $\\Delta$ is defined as: :$\\max \\set {x_1 - x_0, x_2 - x_1, \\ldots, x_n - x_{n - 1} }$ and is denoted $\\norm \\Delta$."}
+{"_id": "26877", "title": "Definition:Continuous Extension/Real Function", "text": "Let $A$, $B \\subseteq \\R$ be subsets of the real numbers such that $A \\subseteq B$. Let $f: A \\to \\R$ and $g: B \\to \\R$ be continuous real functions. Then $g$ is a '''continuous extension''' of $f$ {{iff}}: :$\\forall x \\in A : f \\left({x}\\right) = g \\left({x}\\right)$"}
+{"_id": "26878", "title": "Definition:Definite Integral/Riemann", "text": "Let $\\Delta$ be a finite subdivision of $\\closedint a b$, $\\Delta = \\set {x_0, \\ldots, x_n}$, $x_0 = a$ and $x_n = b$. Let there for $\\Delta$ be a corresponding sequence $C$ of sample points $c_i$, $C = \\tuple {c_1, \\ldots, c_n}$, where $c_i \\in \\closedint {x_{i - 1} } {x_i}$ for every $i \\in \\set {1, \\ldots, n}$. Let $\\map S {f; \\Delta, C}$ denote the Riemann sum of $f$ for the subdivision $\\Delta$ and the sample point sequence $C$. Then $f$ is said to be '''(properly) Riemann integrable''' on $\\closedint a b$ {{iff}}: :$\\exists L \\in \\R: \\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: \\forall$ finite subdivisions $\\Delta$ of $\\closedint a b: \\forall$ sample point sequences $C$ of $\\Delta: \\norm \\Delta < \\delta \\implies \\size {\\map S {f; \\Delta, C} - L} < \\epsilon$ where $\\norm \\Delta$ denotes the norm of $\\Delta$. The real number $L$ is called the '''Riemann integral''' of $f$ over $\\closedint a b$ and is denoted: :$\\displaystyle \\int_a^b \\map f x \\rd x$"}
+{"_id": "26879", "title": "Definition:Definite Integral/Darboux", "text": "Let $f$ be bounded on $\\closedint a b$. Suppose that: :$\\displaystyle \\underline {\\int_a^b} \\map f x \\rd x = \\overline {\\int_a^b} \\map f x \\rd x$ where $\\displaystyle \\underline {\\int_a^b}$ and $\\displaystyle \\overline {\\int_a^b}$ denote the lower integral and upper integral, respectively. Then the '''definite (Darboux) integral of $f$ over $\\closedint a b$''' is defined as: :$\\displaystyle \\int_a^b \\map f x \\rd x = \\underline {\\int_a^b} \\map f x \\rd x = \\overline {\\int_a^b} \\map f x \\rd x$ $f$ is formally defined as '''(properly) integrable over $\\closedint a b$ in the sense of Darboux''', or '''(properly) Darboux integrable over $\\closedint a b$'''. More usually (and informally), we say: :'''$f$ is (Darboux) integrable over $\\closedint a b$.'''"}
+{"_id": "26881", "title": "Definition:Auxiliary Relation", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let $\\mathcal R \\subseteq S \\times S$ be a relation on $S$. Then $\\mathcal R$ is an '''auxiliary relation''' {{iff}} {{begin-axiom}} {{axiom | n = 1         | q = \\forall x, y \\in S         | m = \\left({x, y}\\right) \\in \\mathcal R \\implies x \\preceq y }} {{axiom | n = 2         | q = \\forall x, y, z, u \\in S         | m = x \\preceq y \\land \\left({y, z}\\right) \\in \\mathcal R \\land z \\preceq u \\implies \\left({x, u}\\right) \\in \\mathcal R }} {{axiom | n = 3         | q = \\forall x, y, z \\in S         | m = \\left({x, z}\\right) \\in \\mathcal R \\land \\left({y, z}\\right) \\in \\mathcal R \\implies \\left({x \\vee y, z}\\right) \\in \\mathcal R }} {{axiom | n = 4         | q = \\forall x \\in S         | m = \\left({\\bot, x}\\right) \\in \\mathcal R }} {{end-axiom}}"}
+{"_id": "26882", "title": "Definition:Sign of Ordered Tuple", "text": "Let $n \\in \\N$ be a natural number such that $n > 1$. Let $\\tuple {x_1, x_2, \\ldots, x_n}$ be an ordered $n$-tuple of real numbers. Let $\\map {\\Delta_n} {x_1, x_2, \\ldots, x_n}$ be the product of differences of $\\tuple {x_1, x_2, \\ldots, x_n}$: :$\\displaystyle \\map {\\Delta_n} {x_1, x_2, \\ldots, x_n} = \\prod_{1 \\mathop \\le i \\mathop < j \\mathop \\le n} \\paren {x_i - x_j}$ The '''sign of $\\tuple {x_1, x_2, \\ldots, x_n}$''' is defined and denoted as: :$\\map \\epsilon {x_1, x_2, \\ldots, x_n} := \\map \\sgn {\\Delta_n}$ where $\\sgn$ denotes the signum function. That is: :$\\displaystyle \\map \\epsilon {x_1, x_2, \\ldots, x_n} := \\map \\sgn {\\prod_{1 \\mathop \\le i \\mathop < j \\mathop \\le n} \\paren {x_i - x_j} }$ where: :$\\map \\sgn \\pi := \\sqbrk {x > 0} - \\sqbrk {x < 0}$ :$\\sqbrk {x > 0}$ etc. is Iverson's convention."}
+{"_id": "26883", "title": "Definition:Floor Function/Definition 1", "text": "The '''floor function of $x$''' is defined as the supremum of the set of integers no greater than $x$: :$\\floor x := \\sup \\set {m \\in \\Z: m \\le x}$ where $\\le$ is the usual ordering on the real numbers."}
+{"_id": "26886", "title": "Definition:Floor Function/Notation", "text": "Before around $1970$, the usual symbol for the '''floor function''' was $\\sqbrk x$. The notation $\\floor x$ for the '''floor function''' is a relatively recent development. Compare the notation for the corresponding ceiling function, $\\ceiling x$, which in the context of discrete mathematics is used almost as much. Some sources use $\\map {\\mathrm {fl} } x$ for the '''floor function''' of $x$. However, this notation is clumsy, and will not be used on {{ProofWiki}}."}
+{"_id": "26887", "title": "Definition:Floor Function/Definition 3", "text": "The '''floor function of $x$''' is the unique integer $\\floor x$ such that: :$\\floor x \\le x < \\floor x + 1$"}
+{"_id": "26888", "title": "Definition:Ceiling Function/Notation", "text": "The notation $\\ceiling x$ for the '''ceiling function''' is a relatively recent development. Compare the notation $\\floor x$ for the corresponding  floor function."}
+{"_id": "26889", "title": "Definition:Language of Predicate Logic/Alphabet", "text": "The alphabet $\\AA$ of the language of predicate logic $\\LL_1$ is defined as follows: <section notitle=\"true\" name=\"tc\"> === Letters === The letters of $\\LL_1$ are separated in three classes: * variables; * predicate or relation symbols; * function symbols. Each of these three classes is handled differently by the formal grammar of predicate logic. ==== Variables ==== The variables constitute an infinite set $\\mathrm{VAR}$ of arbitrary symbols, for example: :$\\mathrm{VAR} = \\set {x, y, z, x_0, y_0, z_0, x_1, y_1, z_1, \\ldots}$ ==== Predicate Symbols ==== The predicate symbols are a collection of arbitrary symbols. Each of these symbols is considered to be endowed with an arity (a natural number $n \\in \\N$). We agree to write $\\PP$ for the set of predicate symbols, grouped by their arity: :$\\PP = \\set {\\PP_0, \\PP_1, \\PP_2, \\ldots, \\PP_k, \\ldots}$ The symbols in $\\PP_0$ are inherited from the language of propositional logic. For example, if $P \\in \\PP_5$ then $P$ is a quinternary predicate symbol. ==== Function Symbols ==== The function symbols are a collection (possibly empty) of arbitrary symbols. Each of these symbols is considered to be endowed with an arity (a natural number $n \\in \\N$). We agree to write $\\FF$ for the set of function symbols, grouped by their arity: :$\\FF = \\set {\\FF_0, \\FF_1, \\ldots, \\FF_k, \\ldots}$ The symbols in $\\FF_0$ are often called '''parameters''' or '''constants'''. Some sources write $\\KK$ for the collection of '''parameters'''. === Signs === The signs of $\\LL_1$ are an extension of the signs of propositional logic. They split in three classes: * connectives; * quantifiers; * punctuation. ==== Connectives ==== The '''connectives''' of $\\LL_1$ comprise: {{begin-axiom}} {{axiom | ml= \\land          | mo= :         | t = the conjunction sign }} {{axiom | ml= \\lor         | mo= :         | t = the disjunction sign }} {{axiom | ml= \\implies         | mo= :         | t = the conditional sign }} {{axiom | ml= \\iff         | mo= :         | t = the biconditional sign }} {{axiom | ml= \\neg         | mo= :         | t = the negation sign }} {{axiom | ml= \\top         | mo= :         | t = the top sign }} {{axiom | ml= \\bot         | mo= :         | t = the bottom sign }} {{end-axiom}} The symbols $\\land, \\lor, \\implies$ and $\\iff$ are called the '''binary connectives'''. The symbols $\\neg$ is called a '''unary connective'''. The symbols $\\top$ and $\\bot$ are called the '''nullary connectives'''. ==== Quantifiers ==== The '''quantifiers''' of $\\LL_1$ are: {{begin-axiom}} {{axiom | ml= \\exists         | mo= :         | t = the existential quantifier sign }} {{axiom | ml= \\forall         | mo= :         | t = the universal quantifier sign }} {{end-axiom}} ==== Punctuation ==== The punctuation symbols used in $\\LL_1$ are: {{begin-axiom}} {{axiom | ml= (         | mo= :         | t = the left parenthesis sign }} {{axiom | ml= )         | mo= :         | t = the right parenthesis sign }} {{axiom | ml= :         | mo= :         | t = the colon }} {{axiom | ml= ,         | mo= :         | t = the comma }} {{end-axiom}} </section>"}
+{"_id": "26890", "title": "Definition:Language of Predicate Logic/Formal Grammar", "text": "Part of specifying the language of propositional logic is to provide its formal grammar. <section notitle=\"true\" name=\"def\"> The following rules of formation constitute a bottom-up grammar for the language of predicate logic $\\LL_1$. The definition proceeds in two steps. First, we will define terms, and then well-formed formulas. === Terms === {{:Definition:Language of Predicate Logic/Formal Grammar/Term}} === Well-Formed Formulas === The WFFs of $\\LL_1$ are defined by the following bottom-up grammar: {{begin-axiom}} {{axiom | n = \\mathbf W ~ \\PP_n         | t = If $t_1, \\ldots, t_n$ are terms, and $p \\in \\PP_n$ is an $n$-ary predicate symbol, then $\\map p {t_1, t_2, \\ldots, t_n}$ is a WFF. }} {{axiom | n = \\mathbf W ~ \\neg         | t = If $\\mathbf A$ is a WFF, then $\\neg \\mathbf A$ is a WFF. }} {{axiom | n = \\mathbf W ~ \\lor, \\land, \\Rightarrow, \\Leftrightarrow         | t = If $\\mathbf A, \\mathbf B$ are WFFs and $\\circ$ is one of $\\lor, \\land, \\mathord \\implies, \\mathord \\iff$, then $\\paren {\\mathbf A \\circ \\mathbf B}$ is a WFF }} {{axiom | n = \\mathbf W ~ \\forall, \\exists         | t = If $\\mathbf A$ is a WFF and $x$ is a variable, then $\\paren {\\forall x: \\mathbf A}$ and $\\paren {\\exists x: \\mathbf A}$ are WFFs. }} {{end-axiom}} </section>"}
+{"_id": "26891", "title": "Definition:Relation Segment", "text": "Let $X$ be a set. Let $R$ be a relation on $X$. Let $x \\in X$. Then the '''$R$-segment of $x$''', denoted by $x^R$, is defined by: :$x^R := \\left\\{ {y \\in X: \\left({y, x}\\right) \\in R}\\right\\}$"}
+{"_id": "26892", "title": "Definition:Remainder/Real", "text": "The '''remainder of $x$ on division by $y$''' is defined as the value of $r$ in the expression: :$\\forall x, y \\in \\R, y \\ne 0: \\exists! q \\in \\Z, r \\in \\R: x = q y + r, 0 \\le r < \\left|{y}\\right|$ From the definition of the Modulo Operation: :$x \\bmod y := x - y \\left \\lfloor {\\dfrac x y}\\right \\rfloor$ it can be seen that the '''remainder of $x$ on division by $y$''' is defined as: : $r = x \\bmod y$"}
+{"_id": "26893", "title": "Definition:Quotient (Algebra)/Real", "text": "The '''quotient of $x$ on division by $y$''' is defined as the value of $q$ in the expression: :$\\forall x, y \\in \\R, y \\ne 0: \\exists! q \\in \\Z, r \\in \\R: x = q y + r, 0 \\le r < \\left|{y}\\right|$ From the definition of the Modulo Operation: :$x \\bmod y := x - y \\left \\lfloor {\\dfrac x y}\\right \\rfloor = r$ it can be seen that the '''quotient of $x$ on division by $y$''' is defined as: : $q = \\left \\lfloor {\\dfrac x y}\\right \\rfloor$"}
+{"_id": "26894", "title": "Definition:Increasing Mappings Satisfying Inclusion in Lower Closure", "text": "Let $R = \\left({S, \\preceq}\\right)$ be an ordered set. Let ${\\it Ids}\\left({R}\\right)$ be the set of all ideals in $R$. Let $L = \\left({ {\\it Ids}\\left({R}\\right), \\precsim}\\right)$ be an ordered set where $\\precsim \\mathop = \\subseteq\\restriction_{ {\\it Ids}\\left({R}\\right) \\times {\\it Ids}\\left({R}\\right)}$ Then '''ordered set $M$ of increasing mappings $f:R \\to L$ satisfying''' $\\forall x \\in S: f\\left({x}\\right) \\subseteq x^\\preceq$ is defined by :$M = \\left({F, \\preccurlyeq}\\right)$ where :$F = \\left\\{ {f: S \\to {\\it Ids}\\left({R}\\right): f}\\right.$ is increasing mapping $\\left.{\\land \\forall x \\in S: f\\left({x}\\right) \\subseteq x^\\preceq}\\right\\}$ and :$\\preccurlyeq$ is ordering on mappings generated by $\\precsim$ where $x^\\preceq$ denotes the lower closure of $x$."}
+{"_id": "26895", "title": "Definition:Coprime/Notation", "text": "Let $a$ and $b$ be coprime integers, that is, such that $\\gcd \\left\\{{a, b}\\right\\} = 1$. Then the notation $a \\perp b$ is preferred on {{ProofWiki}}. If $\\gcd \\left\\{{a, b}\\right\\} \\ne 1$, the notation $a \\not \\!\\! \\mathop{\\perp} b$ can be used."}
+{"_id": "26896", "title": "Definition:Modulo Subtraction", "text": "Let $m \\in \\Z$ be an integer. Let $\\Z_m$ be the set of integers modulo $m$: :$\\Z_m = \\set {\\eqclass 0 m, \\eqclass 1 m, \\ldots, \\eqclass {m - 1} m}$ where $\\eqclass x m$ is the residue class of $x$ modulo $m$. The operation of '''subtraction modulo $m$''' is defined on $\\Z_m$ as: :$\\eqclass a m -_m \\eqclass b m = \\eqclass {a - b} m$"}
+{"_id": "26897", "title": "Definition:Congruence (Number Theory)/Notation", "text": "The relation '''$x$ is congruent to $y$ modulo $z$''', usually denoted: :$x \\equiv y \\pmod z$ is also frequently seen denoted as: :$x \\equiv y \\ \\paren {\\mathop {\\operatorname{modulo} } z}$ Some (usually older) sources render it as: :$x \\equiv y \\ \\paren {\\mathop {\\operatorname{mod.} } z}$"}
+{"_id": "26899", "title": "Definition:Permutation/Ordered Selection/Notation", "text": "The number of $r$-permutations from a set of cardinality $n$ is denoted variously: :$P_{n r}$ :${}^r P_n$ :${}_r P_n$ :${}_n P_r$ (extra confusingly) There is little consistency in the literature). On {{ProofWiki}} the notation of choice is ${}^r P_n$."}
+{"_id": "26901", "title": "Definition:Factorial/Definition 1", "text": "The '''factorial of $n$''' is defined inductively as: :$n! = \\begin{cases} 1 & : n = 0 \\\\ n \\paren {n - 1}! & : n > 0 \\end{cases}$"}
+{"_id": "26902", "title": "Definition:Factorial/Definition 2", "text": "The '''factorial of $n$''' is defined as: {{begin-eqn}} {{eqn | l = n!       | r = \\prod_{k \\mathop = 1}^n k       | c =  }} {{eqn | r = 1 \\times 2 \\times \\cdots \\times \\paren {n - 1} \\times n       | c =  }} {{end-eqn}} where $\\displaystyle \\prod$ denotes product notation."}
+{"_id": "26904", "title": "Definition:Gamma Function/Hankel Form", "text": "The '''{{AuthorRef|Hermann Hankel|Hankel}} form''' of the '''Gamma function''' is: :$\\displaystyle \\frac 1 {\\Gamma \\left({z}\\right)} = \\dfrac 1 {2 \\pi i} \\oint_{\\mathcal H} \\frac {e^t \\, \\mathrm d t} {t^z}$ where $\\mathcal H$ is the contour starting at $-\\infty$, circling the origin in an anticlockwise direction, and returning to $-\\infty$. The '''Hankel form''' is valid for all $\\C$."}
+{"_id": "26907", "title": "Definition:Permanent", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. That is, let: : $\\mathbf A = \\begin {pmatrix}   a_{1 1} & a_{1 2} & \\cdots & a_{1 n} \\\\   a_{2 1} & a_{2 2} & \\cdots & a_{2 n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\   a_{n 1} & a_{n 2} & \\cdots & a_{n n} \\end {pmatrix}$ Let $\\lambda: \\N_{>0} \\to \\N_{>0}$ be a permutation on $\\N_{>0}$. Then the '''permanent''' of $\\mathbf A$ is defined as: :$\\displaystyle \\sum_{\\lambda} \\paren {\\prod_{k \\mathop = 1}^n a_{k \\map \\lambda k} } = \\sum_{\\lambda} a_{1 \\map \\lambda 1} a_{2 \\map \\lambda 2} \\cdots a_{n \\map \\lambda n}$ where: :the summation $\\displaystyle \\sum_\\lambda$ goes over all the $n!$ permutations of $\\set {1, 2, \\ldots, n}$."}
+{"_id": "26908", "title": "Definition:Constant Symbol", "text": "Let $\\LL$ be a formal language (for example, the language of predicate logic $\\LL_1$). A '''constant symbol''' is a letter of $\\LL$ used to describe a constant. The name '''constant symbol''' is a gesture to the reader to make clear what such a symbol should (intuitively) represent in the formal language $\\LL$."}
+{"_id": "26909", "title": "Definition:Value of Term under Assignment", "text": "Let $\\tau$ be a term in the language of predicate logic $\\mathcal L_1$. Let $\\mathcal A$ be an $\\mathcal L_1$-structure. Let $\\sigma$ be an assignment for $\\tau$ in $\\mathcal A$. Then the '''value of $\\tau$ under $\\sigma$''', denoted $\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau}\\right) } \\left[{\\sigma}\\right]$, is defined recursively by: :If $\\tau = x$, with $x \\in \\operatorname{dom} \\left({\\sigma}\\right)$ a variable in the domain of $\\sigma$: ::$\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({x}\\right) } \\left[{\\sigma}\\right]  :=  \\sigma \\left({x}\\right)$ :If $\\tau = f \\left({\\tau_1, \\ldots, \\tau_n}\\right)$ with $\\tau_i$ terms and $f \\in \\mathcal F_n$ an $n$-ary function symbol: ::$\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({f \\left({\\tau_1, \\ldots, \\tau_n}\\right) }\\right) } \\left[{\\sigma}\\right]  :=  f_{\\mathcal A} \\left({ \\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau_1}\\right) } \\left[{\\sigma}\\right], \\ldots, \\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau_n}\\right) } \\left[{\\sigma}\\right] }\\right)$ :where $f_{\\mathcal A}$ denotes the interpretation of $f$ in $\\mathcal A$. If $\\tau$ contains no variables, one writes $\\operatorname{val}_{\\mathcal A} \\left({\\tau}\\right)$ instead of $\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau}\\right) } \\left[{\\sigma}\\right]$."}
+{"_id": "26910", "title": "Definition:Value of Formula under Assignment", "text": "Let $\\mathbf A$ be a WFF in the language of predicate logic $\\mathcal L_1$. Let $\\mathcal A$ be an $\\mathcal L_1$-structure. Let $\\sigma$ be an assignment for $\\mathbf A$ in $\\mathcal A$. Then the '''value of $\\mathbf A$ under $\\sigma$''', denoted $\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\mathbf A}\\right) } \\left[{\\sigma}\\right]$, is defined recursively by: :If $\\mathbf A = p \\left({\\tau_1, \\ldots, \\tau_n}\\right)$ with $\\tau_i$ terms and $p \\in \\mathcal P_n$ an $n$-ary predicate symbol: ::$\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({p \\left({\\tau_1, \\ldots, \\tau_n}\\right) }\\right) } \\left[{\\sigma}\\right]  :=  p_{\\mathcal A} \\left({ \\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau_1}\\right) } \\left[{\\sigma}\\right], \\ldots, \\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau_n}\\right) } \\left[{\\sigma}\\right] }\\right)$ :where $p_{\\mathcal A}$ denotes the interpretation of $p$ in $\\mathcal A$ and $\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau_i}\\right) } \\left[{\\sigma}\\right]$ is the value of $\\tau_i$ under $\\sigma$. :If $\\mathbf A = \\neg \\mathbf B$ with $\\mathbf B$ a WFF: ::$\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\neg \\mathbf B}\\right) } \\left[{\\sigma}\\right]  :=  f^\\neg \\left({ \\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\mathbf B}\\right) } \\left[{\\sigma}\\right] }\\right)$ :where $f^\\neg$ denotes the truth function of $\\neg$. :If $\\mathbf A = \\left({\\mathbf B \\circ \\mathbf B'}\\right)$ with $\\mathbf B, \\mathbf B'$ WFFs and $\\circ$ one of $\\land, \\lor, \\implies, \\iff$: ::$\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\mathbf B \\circ \\mathbf B'}\\right) } \\left[{\\sigma}\\right]  :=  f^\\circ \\left({ \\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\mathbf B}\\right) } \\left[{\\sigma}\\right], \\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\mathbf B'}\\right) } \\left[{\\sigma}\\right] }\\right)$ :where $f^\\circ$ denotes the respective truth function of $\\circ$. :If $\\mathbf A = \\left({ \\exists x: \\mathbf B}\\right)$ with $x \\in \\mathrm{VAR}$ and $\\mathbf B$ a WFF: ::$\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\exists x: \\mathbf B}\\right) } \\left[{\\sigma}\\right]  :=  \\begin{cases}  T & \\text{if for some $a \\in A$, $\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\mathbf B}\\right) } \\left[{\\sigma + (x / a)}\\right] = T$} \\\\ F & \\text{otherwise} \\end{cases}$ :where $\\sigma + (x / a)$ is the extension of $\\sigma$ by mapping $x$ to $a$. :If $\\mathbf A = \\left({ \\forall x: \\mathbf B}\\right)$ with $x \\in \\mathrm{VAR}$ and $\\mathbf B$ a WFF: ::$\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\forall x: \\mathbf B}\\right) } \\left[{\\sigma}\\right]  :=  \\begin{cases}  T & \\text{if for all $a \\in A$, $\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\mathbf B}\\right) } \\left[{\\sigma + (x / a)}\\right] = T$} \\\\ F & \\text{otherwise} \\end{cases}$ :where $\\sigma + (x / a)$ is the extension of $\\sigma$ by mapping $x$ to $a$. === Sentence === {{:Definition:Value of Formula under Assignment/Sentence}}"}
+{"_id": "26911", "title": "Definition:Extension of Assignment", "text": "Let $\\mathcal A$ be a structure for predicate logic. Let $\\sigma$ be an assignment for $\\mathcal A$. Let $y \\in \\mathrm{VAR}$ be a variable. Let $a \\in A$ be arbitrary. Then the '''extension of $\\sigma$ by mapping $y$ to $a$''', denoted $\\sigma + (y / a)$, is defined by: :$\\forall x \\in \\operatorname{dom} \\left({\\sigma}\\right) \\cup \\left\\{{y}\\right\\}: \\left({\\sigma + (y / a)}\\right)\\left({x}\\right) := \\begin{cases} a & \\text{if $x = y$} \\\\ \\sigma \\left({x}\\right) & \\text{otherwise} \\end{cases}$ Note in particular the case where $y \\in \\operatorname{dom} \\left({\\sigma}\\right)$. If $\\sigma \\left({y}\\right) = a'$, say, then $\\sigma + (y / a)$ ''overwrites'' this value to become $a$ instead."}
+{"_id": "26912", "title": "Definition:Value of Formula under Assignment/Sentence", "text": "Let $\\mathbf A$ be a sentence in the language of predicate logic. The '''value of $\\mathbf A$ in $\\mathcal A$''', denoted $\\operatorname{val}_{\\mathcal A} \\left({\\mathbf A}\\right)$, is defined as: :$\\operatorname{val}_{\\mathcal A} \\left({\\mathbf A}\\right) := \\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\mathbf A}\\right) } \\left[{\\varnothing}\\right]$ where $\\varnothing$ is the empty mapping considered as an assignment for $\\mathbf A$ and $\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\mathbf A}\\right) } \\left[{\\varnothing}\\right]$ is the value of $\\mathbf A$ under $\\varnothing$."}
+{"_id": "26913", "title": "Definition:Assignment for Structure", "text": "Let $\\mathcal L_1$ be the language of predicate logic. Let $\\mathrm{VAR}$ be the collection of variables of $\\mathcal L_1$. Let $\\mathcal A$ be an $\\mathcal L_1$-structure on a set $A$. An '''assignment for $\\mathcal A$''' is a mapping $\\sigma$ such that: :the codomain of $\\sigma$ is $A$ :the domain of $\\sigma$ is a subset of $\\mathrm{VAR}$ That is, the domain of $\\sigma$ contains only variables, and maps them to elements of $A$. === Assignment for Term === {{:Definition:Assignment for Structure/Term}} === Assignment for Formula === {{:Definition:Assignment for Structure/Formula}}"}
+{"_id": "26914", "title": "Definition:Structure for Predicate Logic/Formal Semantics", "text": "Let $\\mathcal L_1$ be the language of predicate logic. <section notitle=\"true\" name=\"tc\"> {{transclude:Definition:Structure for Predicate Logic/Formal Semantics/Sentence  |section  = tc  |title    = Formal Semantics for Sentences  |header   = 3  |increase = 1  |link     = true }} {{transclude:Definition:Structure for Predicate Logic/Formal Semantics/Well-Formed Formula  |section  = tc  |title    = Formal Semantics for WFFs  |header   = 3  |increase = 1  |link     = true }} </section>"}
+{"_id": "26915", "title": "Definition:Semantic Consequence/Predicate Logic", "text": "<section notitle=\"true\" name=\"tc\"> Let $\\mathcal F$ be a collection of WFFs of predicate logic. Then a WFF $\\mathbf A$ is a '''semantic consequence''' of $\\mathcal F$ {{iff}}: :$\\mathcal A \\models_{\\mathrm{PL}} \\mathcal F$ implies $\\mathcal A \\models_{\\mathrm{PL}} \\mathbf A$ for all structures $\\mathcal A$, where $\\models_{\\mathrm{PL}}$ is the models relation. </section> === Notation === That $\\mathbf A$ is a '''semantic consequence''' of $\\mathcal F$ is denoted as: :$\\mathcal F \\models_{\\mathrm{PL}} \\mathbf A$"}
+{"_id": "26916", "title": "Definition:Structure for Predicate Logic/Formal Semantics/Sentence", "text": "Let $\\mathcal L_1$ be the language of predicate logic. <section notitle=\"true\" name=\"tc\"> The structures for $\\mathcal L_1$ can be interpreted as a formal semantics for $\\mathcal L_1$, which we denote by $\\mathrm{PL}$. For the purpose of this formal semantics, we consider only sentences instead of all WFFs. The structures of $\\mathrm{PL}$ are said structures for $\\mathcal L_1$. A sentence $\\mathbf A$ is declared ($\\mathrm{PL}$-)valid in a structure $\\mathcal A$ {{iff}}: :$\\operatorname{val}_{\\mathcal A} \\left({\\mathbf A}\\right) = T$ where $\\operatorname{val}_{\\mathcal A} \\left({\\mathbf A}\\right)$ is the value of $\\mathbf A$ in $\\mathcal A$. Symbolically, this can be expressed as: :$\\mathcal A \\models_{\\mathrm{PL}} \\mathbf A$ </section>"}
+{"_id": "26917", "title": "Definition:Structure for Predicate Logic/Formal Semantics/Well-Formed Formula", "text": "Let $\\mathcal L_1$ be the language of predicate logic. <section notitle=\"true\" name=\"tc\"> The structures for $\\mathcal L_1$ can be interpreted as a formal semantics for $\\mathcal L_1$, which we denote by $\\mathrm{PL_A}$. The structures of $\\mathrm{PL_A}$ are pairs $\\left({\\mathcal A, \\sigma}\\right)$, where: :$\\mathcal A$ is a structure for $\\mathcal L_1$ :$\\sigma$ is an assignment for $\\mathcal A$ A WFF $\\mathbf A$ is declared ($\\mathrm{PL_A}$-)valid in a structure $\\mathcal A$ {{iff}}: :$\\sigma$ is an assignment for $\\mathbf A$ :$\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\mathbf A}\\right) } \\left[{\\sigma}\\right] = T$ where $\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\mathbf A}\\right) } \\left[{\\sigma}\\right]$ is the value of $\\mathbf A$ under $\\sigma$. Symbolically, this can be expressed as one of the following: :$\\mathcal A, \\sigma \\models_{\\mathrm{PL_A}} \\mathbf A$ :$\\mathcal A \\models_{\\mathrm{PL_A}} \\mathbf A \\left[{\\sigma}\\right]$ </section>"}
+{"_id": "26918", "title": "Definition:Hypergeometric Function", "text": "A '''hypergeometric function''' is an infinite series power series defined as: :$\\displaystyle {}_m \\operatorname F_n \\left({ {a_1, \\ldots, a_m} \\atop {b_1, \\ldots, b_n} } \\, \\middle \\vert {\\, z}\\right) := \\sum_{k \\mathop \\ge 0} \\dfrac { {a_1}^{\\overline k} \\cdots {a_m}^{\\overline k} } { {b_1}^{\\overline k} \\cdots {b_n}^{\\overline k} } \\dfrac {z^k} {k!}$ where $x^{\\overline k}$ denotes the $k$th rising factorial power of $x$."}
+{"_id": "26919", "title": "Definition:Gaussian Binomial Coefficient", "text": "Let $q \\in \\R_{\\ne 1}$, $r \\in \\R$, $m \\in \\Z_{\\ge 0}$. The '''Gaussian binomial coefficient''' is an extension of the more conventional binomial coefficient as follows: {{begin-eqn}} {{eqn | l = \\binom r m_q       | o = :=       | r = \\prod_{k \\mathop = 0}^{m - 1} \\dfrac {1 - q^{r - k} } {1 - q^{k + 1} }       | c =  }} {{eqn | r = \\dfrac {\\paren {1 - q^r} \\paren {1 - q^{r - 1} } \\cdots \\paren {1 - q^{r - m + 1} } } {\\paren {1 - q^m} \\paren {1 - q^{m - 1} } \\cdots \\paren {1 - q^1} }       | c =  }} {{end-eqn}}"}
+{"_id": "26920", "title": "Definition:Multinomial Coefficient", "text": "Let $k_1, k_2, \\ldots, k_m \\in \\Z_{\\ge 0}$ be positive integers. The '''multinomial coefficient''' of $k_1, \\ldots, k_m$ is defined as: :$\\dbinom {k_1 + k_2 + \\cdots + k_m} {k_1, k_2, \\ldots, k_m} := \\dfrac {\\left({k_1 + k_2 + \\cdots + k_m}\\right)!} {k_1! \\, k_2! \\, \\ldots k_m!}$"}
+{"_id": "26921", "title": "Definition:Multinomial Coefficient/Trinomial", "text": "The '''trinomial coefficient''' of $k_1, k_2, k_3$ is a particular case of a multinomial coefficient, defined as: :$\\dbinom {k_1 + k_2 + k_3} {k_1, k_2, k_3} := \\dfrac {\\left({k_1 + k_2 + k_3}\\right)!} {k_1! \\, k_2! \\, k_3!}$"}
+{"_id": "26922", "title": "Definition:Complete Proof System/Strongly Complete", "text": "$\\mathscr P$ is '''strongly complete for $\\mathscr M$ {{iff}}: :Every $\\mathscr M$-semantic consequence is a $\\mathscr P$-provable consequence. Symbolically, this can be expressed as the statement that, for every collection $\\mathcal F$ of logical formulas, and every logical formula $\\phi$ of $\\mathcal L$: :$\\mathcal F \\models_{\\mathscr M} \\phi$ implies $\\mathcal F \\vdash_{\\mathscr P} \\phi$"}
+{"_id": "26923", "title": "Definition:Sound Proof System/Strongly Sound", "text": "$\\mathscr P$ is '''strongly sound for $\\mathscr M$''' iff: :Every $\\mathscr P$-provable consequence is an $\\mathscr M$-semantic consequence. Symbolically, this can be expressed as the statement that, for every collection of logical formulas $\\mathcal F$, and logical formula $\\phi$ of $\\mathcal L$: :$\\mathcal F \\vdash_{\\mathscr P} \\phi$ implies $\\mathcal F \\models_{\\mathscr M} \\phi$"}
+{"_id": "26924", "title": "Definition:Finitely Satisfiable", "text": "Let $\\mathcal L$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\mathcal L$. Let $\\mathcal F$ be a collection of logical formulas of $\\mathcal L$. Then $\\mathcal F$ is '''finitely satisfiable for $\\mathscr M$''' {{iff}}: :For each finite subset $\\mathcal F' \\subseteq \\mathcal F$, there is some $\\mathscr M$-model $\\mathcal M$ of $\\mathcal F'$ That is, for each such $\\mathcal F'$, there exists some structure $\\mathcal M$ of $\\mathscr M$ such that: :$\\mathcal M \\models_{\\mathscr M} \\mathcal F'$"}
+{"_id": "26925", "title": "Definition:Cardinality of Structure", "text": "Let $\\mathcal A$ be a first-order structure. Then the '''cardinality of $\\mathcal A$''', denoted $\\left\\vert{ \\mathcal A }\\right\\vert$, is defined as: :$\\left\\vert{ \\mathcal A }\\right\\vert := \\left\\vert{ A }\\right\\vert$ where $\\left\\vert{ A }\\right\\vert$ is the cardinality of the underlying set $A$."}
+{"_id": "26926", "title": "Definition:Semantic Equivalence/Predicate Logic", "text": "<section notitle=\"true\" name=\"tc\"> Let $\\mathbf A, \\mathbf B$ be WFFs of predicate logic. === Definition 1 === {{:Definition:Semantic Equivalence/Predicate Logic/Definition 1|Definition 1}} === Definition 2 === {{:Definition:Semantic Equivalence/Predicate Logic/Definition 2|Definition 2}} </section>"}
+{"_id": "26927", "title": "Definition:Semantic Equivalence/Predicate Logic/Definition 1", "text": "Then $\\mathbf A$ and $\\mathbf B$ are '''equivalent''' {{iff}}: :$\\mathbf A \\models_{\\mathrm{PL_A}} \\mathbf B$ and $\\mathbf B \\models_{\\mathrm{PL_A}} \\mathbf A$ that is, each is a semantic consequence of the other. That is to say, $\\mathbf A$ and $\\mathbf B$ are '''equivalent''' {{iff}}, for all structures $\\mathcal A$ and assignments $\\sigma$: :$\\mathcal A, \\sigma \\models_{\\mathrm{PL_A}} \\mathbf A$ iff $\\mathcal A, \\sigma \\models_{\\mathrm{PL_A}} \\mathbf B$ where $\\models_{\\mathrm{PL_A}}$ denotes the models relation."}
+{"_id": "26929", "title": "Definition:Tautology/Formal Semantics/Predicate Logic", "text": "Let $\\mathbf A$ be a WFF of predicate logic. Then $\\mathbf A$ is a '''tautology''' {{iff}}: :$\\mathcal A, \\sigma \\models_{\\mathrm{PL_A}} \\mathbf A$  for every structure $\\mathcal A$ and assignment $\\sigma$. That $\\mathbf A$ is a '''tautology''' can be denoted as: :$\\models_{\\mathrm{PL_A}} \\mathbf A$"}
+{"_id": "26930", "title": "Definition:Approximating Relation", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an ordered set. Let $\\mathcal R$ be a relation on $S$. Then $\\mathcal R$ is '''approximating relation''' on $S$ {{iff}} :$\\forall x \\in S: x = \\sup \\left({x^{\\mathcal R} }\\right)$ where $x^{\\mathcal R}$ denotes the $\\mathcal R$-segment of $x$."}
+{"_id": "26931", "title": "Definition:Stirling Numbers of the First Kind/Unsigned/Definition 2", "text": "'''Unsigned Stirling numbers of the first kind''' are defined as the polynomial coefficients $\\displaystyle {n \\brack k}$ which satisfy the equation: :$\\displaystyle x^{\\underline n} = \\sum_k \\left({-1}\\right)^{n - k} {n \\brack k} x^k$ where $x^{\\underline n}$ denotes the $n$th falling factorial of $x$."}
+{"_id": "26932", "title": "Definition:Stirling Numbers of the First Kind/Unsigned/Definition 1", "text": "'''Unsigned Stirling numbers of the first kind''' are defined recursively by: :$\\displaystyle {n \\brack k} := \\begin{cases} \\delta_{n k} & : k = 0 \\text { or } n = 0 \\\\ & \\\\ \\displaystyle {n - 1 \\brack k - 1} + \\paren {n - 1} {n - 1 \\brack k} & : \\text{otherwise} \\\\ \\end{cases}$"}
+{"_id": "26933", "title": "Definition:Stirling Numbers of the Second Kind/Definition 1", "text": "'''Stirling numbers of the second kind''' are defined recursively by: :$\\displaystyle {n \\brace k} := \\begin{cases} \\delta_{n k} & : k = 0 \\text{ or } n = 0 \\\\ & \\\\ \\displaystyle {n - 1 \\brace k - 1} + k {n - 1 \\brace k} & : \\text{otherwise} \\\\ \\end{cases}$"}
+{"_id": "26934", "title": "Definition:Stirling Numbers of the Second Kind/Definition 2", "text": "'''Stirling numbers  of the second kind''' are defined as the coefficients $\\displaystyle {n \\brace k}$ which satisfy the equation: :$\\displaystyle x^n = \\sum_k {n \\brace k} x^{\\underline k}$ where $x^{\\underline k}$ denotes the $k$th falling factorial of $x$."}
+{"_id": "26935", "title": "Definition:Stirling Numbers of the First Kind/Signed/Definition 1", "text": "'''Signed Stirling numbers of the first kind''' are defined recursively by: :$\\map s {n, k} := \\begin{cases} \\delta_{n k} & : k = 0 \\text{ or } n = 0 \\\\ \\map s {n - 1, k - 1} - \\paren {n - 1} \\map s {n - 1, k} & : \\text{otherwise} \\\\ \\end{cases}$"}
+{"_id": "26936", "title": "Definition:Stirling Numbers of the First Kind/Signed/Definition 2", "text": "'''Signed Stirling numbers of the first kind''' are defined as the polynomial coefficients $\\map s {n, k}$ which satisfy the equation: :$\\displaystyle x^{\\underline n} = \\sum_k \\map s {n, k} x^k$ where $x^{\\underline n}$ denotes the $n$th falling factorial of $x$."}
+{"_id": "26937", "title": "Definition:Relative Semantic Equivalence", "text": "Let $\\mathcal F$ be a theory in the language of predicate logic. === Relative Semantic Equivalence of WFFs === {{:Definition:Relative Semantic Equivalence/WFF}} === Relative Semantic Equivalence of Terms === {{:Definition:Relative Semantic Equivalence/Term}}"}
+{"_id": "26938", "title": "Definition:Relative Semantic Equivalence/WFF", "text": "Let $\\mathbf A, \\mathbf B$ be WFFs. Let $\\mathbf C$ be the universal closure of $\\mathbf A \\iff \\mathbf B$. Then $\\mathbf A$ and $\\mathbf B$ are '''semantically equivalent with respect to $\\mathcal F$''' {{iff}}: :$\\mathcal F \\models_{\\mathrm{PL}} \\mathbf C$ That is, iff $\\mathbf C$ is a semantic consequence of $\\mathcal F$."}
+{"_id": "26939", "title": "Definition:Relative Semantic Equivalence/Term", "text": "Let $\\tau_1, \\tau_2$ be terms. Then $\\tau_1$ and $\\tau_2$ are '''semantically equivalent with respect to $\\mathcal F$''' {{iff}}: :$\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau_1}\\right) } \\left[{\\sigma}\\right] = \\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau_2}\\right) } \\left[{\\sigma}\\right]$ for all models $\\mathcal A$ of $\\mathcal F$ and assignments $\\sigma$ for $\\tau_1,\\tau_2$ in $\\mathcal A$. Here $\\mathop{ \\operatorname{val}_{\\mathcal A} \\left({\\tau_1}\\right) } \\left[{\\sigma}\\right]$ denotes the value of $\\tau_1$ under $\\sigma$."}
+{"_id": "26940", "title": "Definition:Substitution (Formal Systems)/Term/In Term", "text": "Let $\\beta, \\tau$ be terms of predicate logic. Let $x \\in \\mathrm{VAR}$ be a variable. Let $\\beta \\left({x \\gets \\tau}\\right)$ be the term resulting from $\\beta$ by replacing all occurrences of $x$ by $\\tau$. Then $\\beta \\left({x \\gets \\tau}\\right)$ is called the '''substitution instance of $\\beta$ substituting $\\tau$ for $x$'''."}
+{"_id": "26941", "title": "Definition:Substitution (Formal Systems)/Term/In WFF", "text": "Let $\\mathbf A$ be a WFF of predicate logic. Let $\\tau$ be a term of predicate logic. Let $x \\in \\mathrm{VAR}$ be a variable. Let $\\mathbf A \\left({x \\gets \\tau}\\right)$ be the WFF resulting from $\\mathbf A$ by replacing all free occurrences of $x$ by $\\tau$. Then $\\mathbf A \\left({x \\gets \\tau}\\right)$ is called the '''substitution instance of $\\mathbf A$ substituting $\\tau$ for $x$'''."}
+{"_id": "26942", "title": "Definition:Additive Semiring/Axioms", "text": "An additive semiring is an algebraic structure $\\left({R, *, \\circ}\\right)$, on which are defined two binary operations $\\circ$ and $*$, which satisfy the following conditions: {{begin-axiom}} {{axiom | n = A0         | q = \\forall a, b \\in S         | m = a * b \\in S         | rc= Closure under $*$ }} {{axiom |  n = A1         |  q = \\forall a, b, c \\in S         | m = \\left({a * b}\\right) * c = a * \\left({b * c}\\right)         | rc= Associativity of $*$ }} {{axiom | n = A2         | q = \\forall a, b \\in S         | m = a * b = b * a         | rc= Commutativity of $*$ }} {{axiom | n = M0         | q = \\forall a, b \\in S         | m = a \\circ b \\in S         | rc= Closure under $\\circ$ }} {{axiom | n = M1         | q = \\forall a, b, c \\in S         | m = \\left({a \\circ b}\\right) \\circ c = a \\circ \\left({b \\circ c}\\right)         | rc= Associativity of $\\circ$ }} {{axiom | n = D         | q = \\forall a, b, c \\in S         | m = a \\circ \\left({b * c}\\right) = \\left({a \\circ b}\\right) * \\left({a \\circ c}\\right), \\left({a * b}\\right) \\circ c = \\left({a \\circ c}\\right) * \\left({a \\circ c}\\right)         | rc= $\\circ$ is distributive over $*$ }} {{end-eqn}} These criteria are called the '''additive semiring axioms'''."}
+{"_id": "26943", "title": "Definition:Join Semilattice/Definition 1", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Suppose that for all $a, b \\in S$: :$a \\vee b \\in S$ where $a \\vee b$ is the join of $a$ and $b$ with respect to $\\preceq$. Then the ordered structure $\\struct {S, \\vee, \\preceq}$ is called a '''join semilattice'''."}
+{"_id": "26944", "title": "Definition:Join Semilattice/Definition 2", "text": "Let $\\struct {S, \\vee}$ be a semilattice. Let $\\preceq$ be the ordering on $S$ defined by: :$a \\preceq b \\iff \\paren {a \\vee b} = b$ Then the ordered structure $\\struct {S, \\vee, \\preceq}$ is called a '''join semilattice'''."}
+{"_id": "26945", "title": "Definition:Parenthesization/Equivalent", "text": "Two parenthesizations of $a_1, \\ldots, a_n$ are '''equivalent''' {{iff}} the product defined by them yields the same result."}
+{"_id": "26946", "title": "Definition:Trivial Norm/Vector Space", "text": "Let $\\struct {K, +, \\circ}$ be a division ring endowed with the trivial norm. Let $V$ be a vector space over $K$, with zero $0_V$. Then the map $\\norm {\\cdot}: V \\to \\R_+ \\cup \\set 0$ given by: :$\\norm x = \\begin{cases}   0 & : \\text {if $x = 0_V$} \\\\   1 & : \\text {otherwise} \\end{cases}$ defines a norm on $V$, called the '''trivial norm'''."}
+{"_id": "26947", "title": "Definition:Trivial Norm/Division Ring", "text": "Let $\\struct {R, +, \\circ}$ be a division ring, and denote its zero by $0_R$. Then the map $\\norm {\\cdot}: R \\to \\R_{\\ge 0}$ given by: :$\\norm x = \\begin{cases}   0 & : \\text{if $x = 0_R$}\\\\   1 & : \\text{otherwise} \\end{cases}$ defines a norm on $R$, called the '''trivial norm'''."}
+{"_id": "26948", "title": "Definition:Signature (Logic)/Predicate Logic", "text": "Let $\\mathcal L_1$ be the language of predicate logic. Then a '''signature''' for $\\mathcal L_1$ is an explicit choice of the alphabet of $\\mathcal L_1$. That is to say, it amounts to choosing, for each $n \\in \\N$: :A collection $\\mathcal F_n$ of $n$-ary function symbols :A collection $\\mathcal P_n$ of $n$-ary relation symbols It is often conceptually enlightening to explicitly address the $0$-ary function symbols separately, as constant symbols."}
+{"_id": "26950", "title": "Definition:Variable/Also known as", "text": "When it occurs in a mathematical equation, a '''variable''' is often referred to as an '''unknown'''. In the specific context of elementary algebra, the ugly misnomer '''pronumeral''' is frequently found in Australia. This was introduced by extension of the concept of a {{WP|Pronoun|pronoun}}: a symbol that '''stands in''' for a '''numeral''', by which the term number is actually meant. Thankfully the term appears not to have caught on in general."}
+{"_id": "26953", "title": "Definition:Bipartite Graph/Partite Set", "text": "Let $G = \\left({A \\mid B, E}\\right)$ be a bipartite graph. Then $A$ and $B$ are called the '''partite sets''' of the bipartite graph."}
+{"_id": "26955", "title": "Definition:Conjugation (Abstract Algebra)/Conjugate", "text": "Let $a \\in A$. Then $C \\left({a}\\right)$ is called the '''conjugate''' of $a$."}
+{"_id": "26956", "title": "Definition:Gamma Function/Partial", "text": "Let $m \\in \\Z_{\\ge 0}$. The '''partial Gamma function at $m$''' is defined as: :$\\displaystyle \\Gamma_m \\left({z}\\right) := \\frac {m^z m!} {z \\left({z + 1}\\right) \\left({z + 2}\\right) \\cdots \\left({z + m}\\right)}$ which is valid except for $z \\in \\left\\{{0, -1, -2, \\ldots, -m}\\right\\}$."}
+{"_id": "26957", "title": "Definition:Neighborhood (Real Analysis)/Open Subset", "text": "Let $N_\\alpha$ be a subset of $\\R$ which contains (as a subset) an open real set which itself contains (as an element) $\\alpha$. Then $N_\\alpha$ is a '''neighborhood''' of $\\alpha$."}
+{"_id": "26958", "title": "Definition:Neighborhood (Real Analysis)", "text": "Let $\\alpha \\in \\R$ be a real number. === Open Subset Neighborhood === {{:Definition:Neighborhood (Real Analysis)/Open Subset}} === Epsilon-Neighborhood === {{:Definition:Neighborhood (Real Analysis)/Epsilon}}"}
+{"_id": "26959", "title": "Definition:Oscillation/Real Space/Oscillation at Point/Epsilon", "text": "The '''oscillation of $f$ at $x$''' is defined as: :$\\displaystyle \\omega_f \\left({x}\\right) := \\inf \\left\\{{\\omega_f \\left({\\left({x - \\epsilon \\,.\\,.\\, x + \\epsilon}\\right) \\cap X}\\right): \\epsilon \\in \\R_{>0}}\\right\\}$ where $\\omega_f \\left({\\left({x - \\epsilon \\,.\\,.\\, x + \\epsilon}\\right) \\cap X}\\right)$ denotes the oscillation of $f$ on $\\left({x - \\epsilon \\,.\\,.\\, x + \\epsilon}\\right) \\cap X$."}
+{"_id": "26960", "title": "Definition:Subsignature", "text": "Let $\\mathcal L, \\mathcal L'$ be signatures for the language of predicate logic. Then $\\mathcal L$ is said to be a '''subsignature''' of $\\mathcal L'$''', denoted $\\mathcal L \\subseteq \\mathcal L'$, {{iff}}, for each $n \\in \\N$: :$\\mathcal F_n \\left({\\mathcal L}\\right) \\subseteq \\mathcal F_n \\left({\\mathcal L'}\\right)$ :$\\mathcal P_n \\left({\\mathcal L}\\right) \\subseteq \\mathcal P_n \\left({\\mathcal L'}\\right)$ where $\\mathcal F_n$ denotes the collection of $n$-ary function symbols, and $\\mathcal P_n$ denotes the collection of $n$-ary predicate symbols. === Supersignature === {{:Definition:Subsignature/Supersignature}}"}
+{"_id": "26961", "title": "Definition:Subsignature/Supersignature", "text": "Let $\\mathcal L$ be a subsignature of $\\mathcal L'$. Then $\\mathcal L'$ is said to be a '''supersignature''' of $\\mathcal L$, denoted: :$\\mathcal L' \\supseteq \\mathcal L$"}
+{"_id": "26962", "title": "Definition:Lower Semicontinuous/Subset", "text": "Let $A \\subseteq S$, and $A \\ne \\varnothing$. The function $f$ is said to be '''lower semicontinuous on''' $A$ {{iff}} $f$ is lower semicontinuous at every $a \\in A$."}
+{"_id": "26963", "title": "Definition:Reduct of Structure", "text": "Let $\\mathcal L, \\mathcal L'$ be signatures of the language of predicate logic. Let $\\mathcal L$ be a subsignature of $\\mathcal L'$. Let $\\mathcal A, \\mathcal A'$ be structures for $\\mathcal L, \\mathcal L'$, respectively. Then $\\mathcal A$ is called the '''reduct of $\\mathcal A'$ to $\\mathcal L$''' {{iff}}: :For all function symbols $f$ of $\\mathcal L$, one has $f_{\\mathcal A'} = f_{\\mathcal A}$ :For all predicate symbols $p$ of $\\mathcal L$, one has $p_{\\mathcal A'} = p_{\\mathcal A}$ where $f_{\\mathcal A'}$ is the interpretation of the function symbol $f$ in the structure $\\mathcal A'$. Symbolically, one may write $\\mathcal A = \\mathcal A' \\restriction_{\\mathcal L}$."}
+{"_id": "26964", "title": "Definition:Expansion of Structure", "text": "Let $\\mathcal L, \\mathcal L'$ be signatures of the language of predicate logic. Let $\\mathcal L'$ be a supersignature of $\\mathcal L$. Let $\\mathcal A, \\mathcal A'$ be structures for $\\mathcal L, \\mathcal L'$, respectively. Then $\\mathcal A'$ is called an '''expansion of $\\mathcal A$ to $\\mathcal L'$''' {{iff}}: :For all function symbols $f$ of $\\mathcal L$, one has $f_{\\mathcal A'} = f_{\\mathcal A}$ :For all predicate symbols $p$ of $\\mathcal L$, one has $p_{\\mathcal A'} = p_{\\mathcal A}$ where $f_{\\mathcal A'}$ is the interpretation of the function symbol $f$ in the structure $\\mathcal A'$."}
+{"_id": "26965", "title": "Definition:Meet Irreducible", "text": "Let $\\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Let $g \\in S$. Then $g$ is '''meet irreducible''' {{iff}}: :$\\forall x, y \\in S: g = x \\wedge y \\implies g = x$ or $g = y$"}
+{"_id": "26966", "title": "Definition:Stencil", "text": "A quintuple $\\tuple {X, S_X, \\map S 0, \\Delta, \\delta}$ defines a '''stencil''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | lc= Index Domain:         | t = $X$ is discrete: $X \\subseteq \\Z^{\\size X}$ and of finite dimension: $\\size X \\in \\N$ }} {{axiom | n = 2         | lc= State Range:         | t = $S_X$ is a well-defined set }} {{axiom | n = 3         | lc= Initial State:         | t = $\\map S 0$ maps from index space to state space $\\map S 0: X \\to S_X$ }} {{axiom | n = 4         | lc= Neighbourhood Delta:         | t = $\\Delta$ is a vector of '''index offsets''': $\\Delta \\in \\paren {\\Z^{\\size X} }^{\\size \\Delta}$ }} {{axiom | n = 5         | lc= Transition Combinator:         | t = $\\delta$ is a mapping $\\delta: S_X^{\\size \\Delta} \\to S_X$ from neighbourhood states to states }} {{end-axiom}} {{refactor|Put update sweep and stencil evolution into their own pages}} By means of the '''update sweep''' $\\forall \\vec x \\in X: \\map {\\map S {n + 1} } {\\vec x} = \\displaystyle \\map \\delta {\\prod_{i \\mathop = 1}^{\\size \\Delta} \\map {\\map S n} {\\vec x + \\Delta_i} }$ this induces a '''stencil evolution''' $\\N \\ni n \\mapsto \\map S n \\in \\paren {X \\to S_X}$ from the '''initial state''' $\\map S 0$. Category:Definitions/Cellular Automata s4pak4bzdx71q0bp97qi89alappd492"}
+{"_id": "26967", "title": "Definition:Conway Life", "text": "A cellular evolution deriving from a '''stencil''' shaped as: :$(1): \\quad X := \\Z^2$ :$(2): \\quad S_X := 2 := \\set {0, 1}$ :$(3): \\quad \\map S 0 \\in \\paren {\\Z^2 \\to 2}$ :$(4): \\quad \\Delta := \\tuple {\\tuple {0, 0}, \\tuple {0, 1}, \\tuple {1, 1}, \\tuple {1, 0}, \\tuple {1, -1}, \\tuple {0, -1}, \\tuple {-1, -1}, \\tuple {-1, 0}, \\tuple {-1, 1} }$ :$(5): \\quad \\map \\delta {m, n_1, n_2, n_3, n_4, n_5, n_6, n_7, n_8} := \\tuple {m, \\displaystyle \\sum_{i \\mathop = 1}^8 n_i} \\in \\set {\\tuple {0, 3}, \\tuple {1, 2}, \\tuple {1, 3} } ? 1 : 0$ is called a '''Conway life'''."}
+{"_id": "26969", "title": "Definition:Legendre Symbol/Definition 1", "text": "The '''Legendre symbol''' $\\left({\\dfrac a p}\\right)$ is defined as: :$\\left({\\dfrac a p}\\right) := \\begin{cases} +1 & : a^{\\frac{(p - 1)} 2} \\bmod p = 1 \\\\ 0 & : a^{\\frac{(p - 1)} 2} \\bmod p = 0 \\\\ -1 & : a^{\\frac{(p - 1)} 2} \\bmod p = p - 1 \\end{cases}$"}
+{"_id": "26970", "title": "Definition:Legendre Symbol/Definition 2", "text": "The '''Legendre symbol''' $\\left({\\dfrac a p}\\right)$ is defined as: :$\\left({\\dfrac a p}\\right) := a^{\\frac{(p - 1)} 2} \\bmod p$"}
+{"_id": "26971", "title": "Definition:Termial", "text": "Let $n \\in \\Z_{> 0}$ be a positive integer. The '''termial''' of $n$ is denoted $n?$ and defined as: :$\\displaystyle n? = \\sum_{k \\mathop = 1}^n k = 1 + 2 + \\cdots + n$"}
+{"_id": "26973", "title": "Definition:Way Below Open", "text": "Let $\\left({S, \\preceq}\\right)$ a preordered set. Let $X$ be a subset of $S$. Then $X$ is '''way below open''' {{iff}}: :$\\forall x \\in X: \\exists y \\in X: y \\ll x$ where $\\ll$ denotes the way below relation."}
+{"_id": "26974", "title": "Definition:Way Above Closure", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $x \\in S$. The '''way above closure''' of $x$, denoted by $x^\\gg$, is defined by: :$x^\\gg := \\left\\{ {y \\in S: x \\ll y}\\right\\}$ where $\\ll$ denotes the way below relation."}
+{"_id": "26975", "title": "Definition:Termial/Real Numbers", "text": "Let $x \\in \\R$ be a real number. The '''termial''' of $x$ is denoted $x?$ and defined as: :$\\displaystyle x? = \\dfrac {x \\left({x + 1}\\right)} 2$"}
+{"_id": "26977", "title": "Definition:Operation/Binary Operation/Product/Right", "text": "Let $x$ and $y$ be elements which are operated on by a given operation $\\circ$. The '''right-hand product of $x$ by $y$''' is the product $x \\circ y$."}
+{"_id": "26978", "title": "Definition:Operation/Binary Operation/Product/Left", "text": "Let $x$ and $y$ be elements which are operated on by a given operation $\\circ$. The '''left-hand product of $x$ by $y$''' is the product $y \\circ x$."}
+{"_id": "26984", "title": "Definition:Special Orthogonal Group", "text": "Let $k$ be a field. The '''special ($n$th) orthogonal group (on $k$)''', denoted $\\map {\\mathrm {SO} } {n, k}$, is: :the set of all proper orthogonal order-$n$ square matrices over $k$ :under (conventional) matrix multiplication. That is: $\\map {\\mathrm {SO} } {n, k} = \\map {\\mathrm O} {n, k} \\cap \\SL {n, k}$"}
+{"_id": "26985", "title": "Definition:Finite Semigroup", "text": "A '''finite semigroup''' is a semigroup of finite order. That is, a semigroup $\\struct {S, \\circ}$ is a '''finite semigroup''' {{iff}} its underlying set $G$ is finite. That is, a '''finite semigroup''' is a semigroup with a finite number of elements."}
+{"_id": "26986", "title": "Definition:Inverse Row form of Cayley Table for Group", "text": "Let $\\struct {G, \\circ}$ be a finite group. The Cayley table for $\\struct {G, \\circ}$ can be presented in a form such that the rows are headed by the inverse elements of the elements which head the corresponding columns. This form is known as the '''inverse row form''' of the Cayley table for $\\struct {G, \\circ}$."}
+{"_id": "26987", "title": "Definition:Group Generated by Reciprocal of z and 1 minus z", "text": "Let: :$S = \\set {f_1, f_2, f_3, f_4, f_5, f_6}$ where $f_1, f_2, \\ldots, f_6$ are complex functions defined for all $z \\in \\C \\setminus \\set {0, 1}$ as: {{begin-eqn}} {{eqn | l = \\map {f_1} z       | r = z }} {{eqn | l = \\map {f_2} z       | r = \\dfrac 1 {1 - z} }} {{eqn | l = \\map {f_3} z       | r = \\dfrac {z - 1} z }} {{eqn | l = \\map {f_4} z       | r = \\dfrac 1 z }} {{eqn | l = \\map {f_5} z       | r = 1 - z }} {{eqn | l = \\map {f_6} z       | r = \\dfrac z {z - 1} }} {{end-eqn}} Let $\\circ$ denote composition of functions. Then $\\struct {S, \\circ}$ is the '''group generated by $\\dfrac 1 z$ and $1 - z$'''."}
+{"_id": "26988", "title": "Definition:Coprime Residue Class", "text": "Let $m \\in \\Z: m \\ge 1$. Let $a \\in \\Z$ such that: :$a \\perp m$ where $\\perp$ denotes that $a$ is prime to $m$. Let $\\eqclass a m$ be the residue class of $a$ (modulo $m$): :$\\set {x \\in \\Z: \\exists k \\in \\Z: x = a + k m}$ Then $\\eqclass a m$ is referred to as a '''coprime residue class'''."}
+{"_id": "26989", "title": "Definition:Reduced Residue System", "text": "The '''reduced residue system modulo $m$''', denoted $\\Z'_m$, is the set of all residue classes of $k$ (modulo $m$) which are prime to $m$: :$\\Z'_m = \\set {\\eqclass k m \\in \\Z_m: k \\perp m}$ Thus $\\Z'_m$ is the '''set of all coprime residue classes modulo $m$''': :$\\Z'_m = \\set {\\eqclass {a_1} m, \\eqclass {a_2} m, \\ldots, \\eqclass {a_{\\map \\phi m} } m}$ where: :$\\forall k: a_k \\perp m$ :$\\map \\phi m$ denotes the Euler phi function of $m$."}
+{"_id": "26990", "title": "Definition:Set of Residue Classes/Real Modulus", "text": "The quotient set of congruence modulo $z$ denoted $\\R_z$ is: :$\\R_z = \\dfrac \\R {\\mathcal R_z}$ Thus $\\R_z$ is the '''set of all residue classes modulo $z$'''. It follows from the Fundamental Theorem on Equivalence Relations that the quotient set $\\R_z$ of congruence modulo $z$ forms a partition of $\\R$."}
+{"_id": "26991", "title": "Definition:Order Generating", "text": "Let $\\left({S, \\preceq}\\right)$ be a preordered set. Let $X$ be a subset of $S$. Then $X$ is '''order generating''' {{iff}} :$\\forall x \\in S: x^\\succeq \\cap X$ admits an infimum and $x = \\inf \\left({x^\\succeq \\cap X}\\right)$"}
+{"_id": "26992", "title": "Definition:Finite Group/Axioms", "text": "A finite group is an algebraic structure $\\struct {G, \\circ}$ which satisfies the following four conditions: {{begin-axiom}} {{axiom | n = \\text {FG} 0         | lc= Closure         | q = \\forall a, b \\in G         | m = a \\circ b \\in G }} {{axiom | n = \\text {FG} 1         | lc= Associativity         | q = \\forall a, b, c \\in G         | m = a \\circ \\paren {b \\circ c} = \\paren {a \\circ b} \\circ c }} {{axiom | n = \\text {FG} 2         | lc= Finiteness         | q = \\exists n \\in \\N         | m = \\order G = n }} {{axiom | n = \\text {FG} 3         | lc= Cancellability         | q = \\forall a, b, c \\in G         | m = c \\circ a = c \\circ b \\implies a = b }} {{axiom | m = a \\circ c = b \\circ c \\implies a = b }} {{end-axiom}} These four stipulations are called the '''finite group axioms'''."}
+{"_id": "26993", "title": "Definition:Group of Rotation Matrices Order 4", "text": "Consider the algebraic structure $S$ of rotation matrices: :$R_4 = \\set {\\begin {bmatrix} 1 & 0 \\\\ 0 & 1 \\end {bmatrix}, \\begin {bmatrix} 0 & 1 \\\\ -1 & 0 \\end {bmatrix}, \\begin {bmatrix} -1 & 0 \\\\ 0 & -1 \\end {bmatrix}, \\begin {bmatrix} 0 & -1 \\\\ 1 & 0 \\end {bmatrix} }$ under the operation of (conventional) matrix multiplication. $R_4$ is the '''group of rotation matrices of order $4$'''."}
+{"_id": "26994", "title": "Definition:Additive Group of Integers Modulo 4", "text": "The '''additive group of integers modulo $4$''' $\\struct {\\Z_4, +_4}$ is the set of integers modulo $4$ under the operation of addition modulo $4$."}
+{"_id": "26995", "title": "Definition:Dimension of Vector Space/Vector", "text": "Informally, an element of an $n$-dimensional vector space is often referred to as an '''$n$-dimensional vector'''.  It must be understood that this is no more than a convenient shorthand. It is not the ''vector'' which possesses the dimensionality, but the ''space in which it is embedded''."}
+{"_id": "26996", "title": "Definition:Complex Root", "text": "Let $z \\in \\C$ be a complex number such that $z \\ne 0$. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $w \\in \\C$ such that: :$w^n = z$ Then $w$ is a '''(complex) $n$th root of $z$''', and we can write: :$w = z^{1 / n}$"}
+{"_id": "26997", "title": "Definition:Root of Unity/Complex/Primitive", "text": "A '''primitive (complex) $n$th root of unity''' is an element $\\alpha \\in U_n$ such that: :$U_n = \\set {1, \\alpha, \\alpha^2, \\ldots, \\alpha^{n - 1} }$"}
+{"_id": "27000", "title": "Definition:Prime Element (Order Theory)", "text": "Let $\\struct {S, \\wedge, \\preceq}$ be a meet semilattice. Let $p \\in S$. Then $p$ is '''a prime element (of $\\struct {S, \\wedge, \\preceq}$)''' {{iff}}: :$\\forall x, y \\in S: \\paren {x \\wedge y \\preceq p \\implies x \\preceq p \\text { or } y \\preceq p}$"}
+{"_id": "27001", "title": "Definition:Relation/Relation as Subset of Cartesian Product", "text": "Most treatments of set theory and relation theory define a '''relation on $S \\times T$''' to refer to just the truth set itself: :$\\RR \\subseteq S \\times T$ where: :$S \\times T$ is the Cartesian product of $S$ and $T$. Thus under this treatment, $\\RR$ is a set of ordered pairs, the first coordinate from $S$ and the second coordinate from $T$. This approach leaves the precise nature of $S$ and $T$ undefined."}
+{"_id": "27002", "title": "Definition:Airy's Equation", "text": "'''Airy's equation''' is the second order ODE: :$y'' = x y$"}
+{"_id": "27004", "title": "Definition:Game", "text": "In the context of game theory, a '''game''' is any social situation which involves more than one person. More specifically, a '''game''' is a problem of strategy between two or more parties in which all parties involved seek (usually) to maximise their outcome. A '''game''' $G$ can be specified by: :its players :the moves available to each player :the preference relation for each player: that is: :$G = \\sequence {N, \\sequence {A_i}, \\sequence {\\succsim_i} }$ $G$ can also be specified by: :its players :the moves available to each player :the payoff function for each player: that is: :$G = \\sequence {N, \\sequence {A_i}, \\sequence {u_i} }$"}
+{"_id": "27005", "title": "Definition:Game/Player", "text": "Each of the parties involved in a '''game''' are called '''players'''."}
+{"_id": "27006", "title": "Definition:Game/Strategy", "text": "A '''strategy''' is a complete plan of action that defines what a '''player''' will do under all circumstances in a  '''game'''."}
+{"_id": "27007", "title": "Definition:Game/Payoff", "text": "The '''payoff''' of a '''game''' is the reward or punishment made to a '''player''' at the end of the '''game''' as a result of combination of the various strategies employed."}
+{"_id": "27008", "title": "Definition:Game of Perfect Information", "text": "A '''game of perfect information''' is a game in which all players have complete information on the state of the game."}
+{"_id": "27010", "title": "Definition:Two-Person Game", "text": "A '''two-person game''' is a game in which there are (exactly) two players."}
+{"_id": "27011", "title": "Definition:Zero-Sum Game", "text": "A '''zero-sum game''' is a game in which the total of the payoffs to all players is zero."}
+{"_id": "27012", "title": "Definition:Equilibrium Point", "text": "An '''equilibrium point''' is a stable outcome of a game associated with a particular set of strategies."}
+{"_id": "27013", "title": "Definition:Zero-Sum Game/Examples", "text": "Examples of zero-sum game include: * [https://en.wikipedia.org/wiki/Poker Poker]: the total amount of money on the table is fixed -- what one person loses the other players win."}
+{"_id": "27014", "title": "Definition:Stable Outcome", "text": "An '''stable outcome''' is an end point of a game such that if a players were to use a new strategy, that player would experience a lower payoff."}
+{"_id": "27015", "title": "Definition:Two-Person Zero-Sum Game", "text": "A '''two-person zero-sum game''' is a two-person game with zero sum."}
+{"_id": "27016", "title": "Definition:Equilibrium Strategies", "text": "A system of strategies, one for each player, is '''in equilibrium''' {{iff}} they result in an equilibrium point. Such strategies are known as '''equilibrium strategies'''. In the singular, an '''equilibrium strategy''' is one that contributes to an equilibrium point."}
+{"_id": "27017", "title": "Definition:Linear Programming", "text": "'''Linear programming''' is the branch of mathematical programming which studies optimization of mathematical models whose requirements are represented by linear relationships."}
+{"_id": "27019", "title": "Definition:Matching Pennies", "text": "There are two players: $A$ and $B$. Each player puts down a coin, traditionally a [https://en.wikipedia.org/wiki/Penny penny], either head or tail up, without showing it to the other player. The coins are then uncovered. If they both show the same side, $A$ is deemed to have won, and he takes both coins. If they show different sides, $B$ is deemed to have won, and he takes both coins."}
+{"_id": "27020", "title": "Definition:Coin/Coin-Tossing", "text": "'''Coin-tossing''' is a technique to select randomly one of two options. The usual scenario is to resolve an issue between two parties. The most usual of these is to determine which of the two parties is to choose whether to take the first move, or otherwise determine the starting arrangement, in a game or sport. A coin is thrown into the air in such a manner that it turns over and over. This can be conveniently achieved by coiling the hand into a loose fist, balancing the coin on the first knuckle of the index finger, and then pinging the thumb up from under the index finger by a physical process derived from the science of catastrophe theory. The nail of the thumb will then impact smartly against the edge of the coin which projects over the edge of the index finger's knuckle, propelling it upwards in such a manner that, by an application of the physics of angular momentum, it will be caused to rotate more-or-less rapidly about a diameter. When it descends, the side which will remain uppermost will be dependent upon both the speed of rotation and the distance travelled, neither of which is easy to gauge precisely enough to make the process predictable. The descending coin will either be caught in the hand by the one flipping it (the \"tosser\"), to be immediately clasped to the back of the other hand, or allowed to fall directly to the ground. The other party (the \"caller\") is offered the chance of guessing which of the two sides of the coin is on top. The two sides are usually referred to (in the English-speaking world) as \"heads\" and \"tails\". The \"heads\" side tends to be the one that features the head of state of the nation to which the coin belongs, while the \"tails\" is the other side. Once the caller has made the call, the uppermost side of the coin is revealed. If the side matches that called by the caller, the caller has won, and is given the option of taking the choice as to the starting configuration of the game. Otherwise the tosser has won, and takes the option.<ref>The play ''{{WP|Rosencrantz_and_Guildenstern_Are_Dead|Rosencrantz and Guildenstern are Dead}}'' by {{WP|Tom_Stoppard|Tom Stoppard}} begins with an unfeasibly long sequence of \"heads\" calls in a game of '''coin-tossing'''.</ref>"}
+{"_id": "27021", "title": "Definition:Coin", "text": "A '''coin''' is a device for choosing a random number in the set $\\left\\{ {0, 1}\\right\\}$. It has been suggested that this device has other uses, but social anthropologists are divided as to what they might be. Perhaps they can resolve their differences by tossing a coin."}
+{"_id": "27023", "title": "Definition:Coin/Head", "text": "The '''head''' of a coin is the side which traditionally bears an image of the head of state of the nation to which the coin is issued."}
+{"_id": "27024", "title": "Definition:Coin/Tail", "text": "The '''tail''' of a coin is the opposite side to its head. It traditionally bears an image of cultural or historical significance to the nation to which the coin is issued."}
+{"_id": "27026", "title": "Definition:Game/Rules", "text": "A game is specified completely by its '''rules'''. They prescribe for each player a model of rational choice: {{:Definition:Model of Rational Choice}}"}
+{"_id": "27027", "title": "Definition:Move", "text": "A '''move''' is an action by a player in a game made according to the rules."}
+{"_id": "27028", "title": "Definition:Play", "text": "A '''play''' is a particular instance of the operation of a game."}
+{"_id": "27029", "title": "Definition:Blotto", "text": "'''Blotto''' is a class of game of which an example has the mechanics as follows: There are two players: '''General $A$''' and '''Colonel $B$(lotto)'''. $A$ commands $4$ companies. $B$ commands $5$ companies. $A$ can reach a town on either of two different roads. It is the aim of $A$ to reach a town, which can be achieved by travelling down either of two different roads. He can send each of his $4$ companies on whatever road he chooses. $B$ can order any of his $5$ companies to defend either of the two roads. $A$ wins if, on either road, he has (strictly) more companies than $B$."}
+{"_id": "27030", "title": "Definition:Card Game with Bluffing", "text": "There are two players: $A$ and $B$. First it is $A$'s move. $A$ receives one of two possible cards: either the '''high card''' or the '''low card'''. If he receives the '''high card''', he must bid $2$ credits. If he receives the '''low card''', he has two options, either: :$A (1): \\quad$ He may pay $1$ credit, and the play is complete. or :$A (2): \\quad$ He may bid $2$ credits. If $A$ has bid $2$ credits, it is $B$'s move. $B$ has two options, either: :$B (1): \\quad$ He may pay $1$ credit or :$B (2): \\quad$ He may challenge $A$'s bid. If $A$ had the '''high card''', $B$ must pay $2$ credits to $A$. If $A$ had the '''low card''', $A$ must pay $2$ credits to $B$. The play is complete."}
+{"_id": "27031", "title": "Definition:Normalization of Game", "text": "The operation which reduces the structure of a game to its strategies is known as '''normalizing''' the game."}
+{"_id": "27032", "title": "Definition:Extensive Form of Game", "text": "A game is described in its '''extensive form''' as follows: :$(1): \\quad$ Each successive move of each player is analysed :$(2): \\quad$ The pattern of information is also taken into account."}
+{"_id": "27033", "title": "Definition:Non-Zero-Sum Game/Examples", "text": "Examples of non-zero-sum games include: === Trade === Two parties trading: both may simultaneously gain by a transaction. It can be assumed that the buyer attaches a value to goods purchased which is higher than the price paid. At the same time, the price paid to the vendor can also be assumed to be higher than the value, to the vendor, of those goods. Thus, at the end of this transaction, both parties have a positive payoff. === Naval Warfare === The loss of a particular ship may cause a setback of considerably greater value to the enemy than the cost of the materiel used by the attacker to destroy it."}
+{"_id": "27034", "title": "Definition:Zero-Sum Game/Non-Zero", "text": "A '''non-zero-sum game''' is a game in which the total of the payoffs to all players is dependent upon the moves made to reach those payoffs."}
+{"_id": "27035", "title": "Definition:Multi-Person Game", "text": "A '''multi-person game''' is a game which has (strictly) more than two players."}
+{"_id": "27036", "title": "Definition:Cooperative Game", "text": "A '''cooperative game''' is a multi-person game in which the players may form coalitions outside the game."}
+{"_id": "27037", "title": "Definition:Coalition", "text": "A '''coalition''' is an agreement among players in a cooperative multi-person game such that they coordinate their strategies so as to confer an advantage to all players in that '''coalition'''."}
+{"_id": "27038", "title": "Definition:Non-Cooperative Game", "text": "A '''non-cooperative game''' is a game in which the players cannot form coalitions outside the game. Hence each player is concerned only with his / her own fortunes."}
+{"_id": "27039", "title": "Definition:Payoff Table", "text": "Let $G$ be a two-person game. A '''payoff table''' for $G$ is an array which specifies the payoff to each player for each strategy of both players. $G$ is completely defined by its '''payoff table'''. {{PayoffTable|table = $\\begin{array} {r {{|}} c {{|}} } & \\text{L} & \\text{R} \\\\ \\hline \\text{T} & w_1, w_2 & x_1, x_2 \\\\ \\hline \\text{B} & y_1, y_2 & z_1, z_2 \\\\ \\hline \\end{array}$}} The two numbers in the entry formed by row $r$ and column $c$ are the payoffs when the row player's moves is $r$ and the column player's moves is $c$. The first component given is the payoff to the row player. If the names of the players are $1$ and $2$, the convention is that the row player is player $1$ and the column player is player $2$. If the names of the players are $\\text{A}$ and $\\text{B}$, the convention is that the row player is player $\\text{A}$ and the column player is player $\\text{B}$."}
+{"_id": "27040", "title": "Definition:Maximising Player", "text": "Let $G$ be a two-person game whose players are identified with the symbols $A$ and $B$. It is the convention that the first of the players listed (in this case $A$) is distinguished from the other and referred to as the '''maximising player'''."}
+{"_id": "27041", "title": "Definition:Minimising Player", "text": "Let $G$ be a two-person game whose players are identified with the symbols $A$ and $B$. It is the convention that the second of the players listed (in this case $B$) is distinguished from the other and referred to as the '''minimising player'''."}
+{"_id": "27042", "title": "Definition:Distance to Nearest Integer Function/Definition 1", "text": ":$\\norm \\alpha:= \\min \\set {\\size {n - \\alpha}: n \\in \\Z}$"}
+{"_id": "27043", "title": "Definition:Distance to Nearest Integer Function/Definition 2", "text": ":$\\norm \\alpha:= \\min \\set {\\set \\alpha, 1 - \\set \\alpha}$ where $\\set \\alpha$ is the fractional part of $\\alpha$."}
+{"_id": "27044", "title": "Definition:Refinement of Cover/Finer Cover", "text": "Let $\\mathcal V$ be a refinement of $\\mathcal U$. Then $\\mathcal V$ is '''finer''' than $\\mathcal U$."}
+{"_id": "27045", "title": "Definition:Refinement of Cover/Coarser Cover", "text": "Let $\\mathcal V$ be a refinement of $\\mathcal U$. Then $\\mathcal U$ is '''coarser''' than $\\mathcal V$."}
+{"_id": "27046", "title": "Definition:Maximin Strategy", "text": "Let $G$ be a two-person game between players $A$ and $B$. A '''minimax strategy''' for $A$ is one in which: :$(1): \\quad A$ assumes that $B$ will make that move which makes $A$'s payoff (or average payoff, in the case of a random element) as small as possible. :$(2): \\quad A$ then makes the move which makes this smallest gain the largest possible. Hence it is the strategy that will '''maxi'''mise $A$'s '''min'''imum payoff. The reasoning is: :''I fear that my opponent will make the move which makes my payoff the smallest possible.  Hence I make my move so as to make this smallest payoff as large as possible.''"}
+{"_id": "27049", "title": "Definition:Payoff Table/Entry", "text": "Each of the values in a '''payoff table''' corresponding to the payoff for a combination of a move by each player is called an '''entry'''."}
+{"_id": "27050", "title": "Definition:Totally Pathwise Disconnected Space/Definition 1", "text": "A topological space $T = \\struct {S, \\tau}$ is '''totally pathwise disconnected''' {{iff}} all path components of $T$ are singletons."}
+{"_id": "27051", "title": "Definition:Saddle Point (Game Theory)", "text": "Let $G$ be a two-player zero-sum game. Let $G$ be defined by a payoff table $T$. Let $E$ be an entry in $T$ such that: :$E$ is the smallest entry in its row :$E$ is the largest entry in its column. Then $E$ is a '''saddle point''' of $G$."}
+{"_id": "27053", "title": "Definition:Specially Strictly Determined Game", "text": "Let $G$ be a two-person zero-sum game. Let $G$ have a saddle point. Then $G$ can be referred to as a '''specially strictly determined game'''."}
+{"_id": "27055", "title": "Definition:Eluding Game", "text": "'''The eluding game''' is a two-person game whose mechanics are as follows. The two players are $A$ and $B$. $A$ and $B$ each have three possible moves, the values: :$1, 2, 3$ If both choose the same number, $A$ pays to $B$ the amount of the chosen number. Otherwise, $A$ receives the amount of his own number from $B$. === Payoff Table === The payoff table of the '''eluding game''' is as follows: {{:Definition:Eluding Game/Payoff Table}}"}
+{"_id": "27057", "title": "Definition:Mixed Strategy", "text": "Let $G$ be a game. A '''mixed strategy''' for $G$ is a combination of different strategies for subsequent plays of $G$ which are played with particular frequencies."}
+{"_id": "27058", "title": "Definition:Pure Strategy", "text": "Let $G$ be a game. A '''pure strategy''' for $G$ is single strategy for consecutive plays of $G$ which is used exclusively. It can be seen that a '''pure strategy''' is a special case of a mixed strategy for which the frequencies of all strategies are zero except for one, which is used every time."}
+{"_id": "27059", "title": "Definition:Totally Pathwise Disconnected Space/Definition 2", "text": "A topological space $T = \\struct {S, \\tau}$ is '''totally pathwise disconnected''' {{iff}} the only continuous mappings from the closed unit interval $\\closedint 0 1$ to $T$ are constant mappings."}
+{"_id": "27060", "title": "Definition:Solution of Game", "text": "Let $G$ be a game. A '''solution''' of $G$ is a systematic description of the outcomes that may emerge in a family of games."}
+{"_id": "27062", "title": "Definition:Optimal Strategy", "text": "Let $G$ be a game. Let $P$ be a player of $G$. An '''optimal strategy''' for $P$ is a strategy that contributes towards a solution of $G$."}
+{"_id": "27063", "title": "Definition:Value of Game", "text": "Let $G$ be a game. The '''value''' of $G$ is the payoff resulting from a solution of $G$."}
+{"_id": "27064", "title": "Definition:Completely Mixed Game", "text": "Let $G$ be a game. Then $G$ is '''completely mixed''' {{iff}} all players use all their strategies in its optimal solution."}
+{"_id": "27065", "title": "Definition:Dominating Strategy", "text": "Let $G$ be a game. Let player $P$ have pure strategies $A_1$ and $A_2$ in $G$. Then $A_1$ '''dominates''' $A_2$ {{iff}}: :for any strategy of an opposing player, $A_1$ is at least as good as $A_2$ :for at least one strategy of an opposing player, $A_1$ is strictly better than $A_2$."}
+{"_id": "27066", "title": "Definition:Strategic Game", "text": "A '''strategic game''' is a game in which: :$(1): \\quad$ Each player chooses a strategy once per play :$(2): \\quad$ The move(s) of each player operates simultaneously.  That is, when choosing a move, each player has any information about the move that another player will make. It consists of: :A finite set $N$ of players :For each player $i \\in N$, a non-empty set $A_i$ of moves available to player $i$ :For each player $i \\in N$, a preference relation $\\succsim_i$ on $A = \\displaystyle \\prod_{j \\mathop \\in N} A_j$ of player $i$."}
+{"_id": "27067", "title": "Definition:Extensive Game", "text": "An '''extensive game''' is a game in which: :$(1): \\quad$ moves are made according to an order of events :$(2): \\quad$ A player can consider and reconsider a course of action at any time a decision needs to be made, throughout the play of a game."}
+{"_id": "27068", "title": "Definition:Game of Imperfect Information", "text": "A '''game of imperfect information''' is a game in which some information about other players is not known by all the players."}
+{"_id": "27070", "title": "Definition:Rational Player", "text": "The players of a game are generally assumed to be '''rational''' in the sense that: :$(1): \\quad$ Each player is fully aware of all possible choices that can be made :$(2): \\quad$ Each player may form expectations about any unknowns :$(3): \\quad$ Each player deliberately chooses an action based on a process of optimization."}
+{"_id": "27071", "title": "Definition:Model of Rational Choice", "text": ":$(1): \\quad$ A set $A$ of moves from which the player may choose :$(2): \\quad$ A set $C$ of consequences of each of those moves :$(3): \\quad$ A consequence function $g: A \\to C$ which maps a consequence to each action :$(4): \\quad$ A preference relation $\\succsim$, which is a total ordering on $C$."}
+{"_id": "27072", "title": "Definition:Consequence", "text": "A '''consequence''' is a state in a game which results from a move made by a player in that game made according to the rules."}
+{"_id": "27073", "title": "Definition:Consequence Function", "text": "Let $G$ be a game. Let $P$ be a player of $G$. Let $A$ be the set of moves available to $P$. Let $C$ be the set of consequences of those moves. A '''consequence function''' for $P$ is a mapping from the set $A$ to the set $C$: :$g: A \\to C$"}
+{"_id": "27074", "title": "Definition:Preference Relation", "text": "Let $G$ be a game. Let $P$ be a player of $G$. Let $C$ be the set $C$ of consequences of the moves available to $P$. A '''preference relation''' on $P$ is a total preordering $\\succsim$ on $C$ which ranks $C$ according to $P$'s eventual payoff."}
+{"_id": "27075", "title": "Definition:Utility Function", "text": "Let $G$ be a game. Let $P$ be a player of $G$. Let $C$ be the set $C$ of consequences of the moves available to $P$. A '''utility function''' on $C$ is a mapping from $C$ to the real numbers $\\R$ so as to define a preference relation on $C$: :$U: C \\to \\R$ by the condition: :$\\forall x, y \\in C: x \\succsim y \\iff U \\left({x}\\right) \\ge U \\left({y}\\right)$"}
+{"_id": "27076", "title": "Definition:Rational Decision-Maker", "text": "Let $G$ be a game. Let $P$ be a player of $G$. Let $A$ be the set of moves available to $P$. Let $C$ be the set $C$ of consequences of the moves available to $P$. Let $g: A \\to C$ be the consequence function on $A$. Let $B \\subseteq A$ be a the set of moves which are feasible. $P$ is a '''rational decision-maker''' {{iff}} $P$ chooses an element $a^* \\in B$ such that: :$\\forall a \\in a^*: g \\left({a^*}\\right) \\succsim g \\left({a}\\right)$ where $\\succsim$ is the preference relation on $C$. That is, that $P$ solves the problem: :$\\displaystyle \\max_{a \\mathop \\in B} U \\left({g \\left({a}\\right)}\\right)$ where $U$ is the utility function on $C$. It is assumed that $P$ uses the same preference relation for all $B \\subseteq A$."}
+{"_id": "27077", "title": "Definition:Steady State Interpretation", "text": "Let $G$ be a game. The '''steady state interpretation''' of the solution of $G$ treats $G$ as a model whose purpose is to explain some sort of regularity in a collection of similar situations."}
+{"_id": "27078", "title": "Definition:Deductive Interpretation", "text": "Let $G$ be a game. The '''deductive interpretation''' of the solution of $G$ treats $G$ in isolation, as a singular event, and, under the assumption that the players are rational, deduces the outcome of $G$ based on the rational model."}
+{"_id": "27079", "title": "Definition:Positive Real Vector Space", "text": "Let $\\R_{\\ge 0}$ be the set of positive real numbers. Then the $\\R_{\\ge 0}$-module $\\R_{\\ge 0}^n$ is called the '''positive real ($n$-dimensional) vector space'''."}
+{"_id": "27080", "title": "Definition:Absolute Real Vector Ordering", "text": "Let $x$ and $y$ be elements of the real vector space $\\R^n$. The '''absolute real vector ordering''' is the partial ordering $\\ge$ defined on the real vector space $\\R^n$ as: :$\\forall x, y \\in \\R^n: x \\ge y \\iff \\forall i \\in \\left\\{ {1, 2, \\ldots, n}\\right\\}: x_i \\ge y_i$"}
+{"_id": "27081", "title": "Definition:Absolute Real Vector Strict Ordering", "text": "Let $x$ and $y$ be elements of the real vector space $\\R^n$. The '''absolute real vector strict ordering''' is the strict partial ordering $\\ge$ defined on the real vector space $\\R^n$ as: :$\\forall x, y \\in \\R^n: x \\ge y \\iff \\forall i \\in \\left\\{ {1, 2, \\ldots, n}\\right\\}: x_i \\ge y_i$"}
+{"_id": "27082", "title": "Definition:Maximizers of Real Function", "text": "Let $f: X \\to \\R$ be a real-valued function. The '''set of maximisers''' of a $f$ are denoted: :$\\displaystyle \\arg \\max_{x \\mathop \\in X} f \\left({x}\\right)$ {{definition wanted|The notation is defined in my source work, but not the concept.}}"}
+{"_id": "27083", "title": "Definition:Profile", "text": "Let $G$ be a game. Let $N$ be the set of players of $G$. Let $V$ be a variable defining some aspect of each of the players. The family of values of $V$ is referred to as a '''profile''', denoted: :$\\family {x_i}_{i \\mathop \\in N}$ If the fact that $i \\in N$ is understood, then $\\family {x_i}$ can be used."}
+{"_id": "27084", "title": "Definition:Profile/Less Player", "text": "Let $x = \\left\\langle{x_i}\\right\\rangle_{i \\mathop \\in N}$ be a profile of $V$. Then $x_{-i}$ is used to denote $x$ for all players except for $i$: :$x_{-i} := \\left\\langle{x_j}\\right\\rangle_{j \\mathop \\in N \\setminus \\left\\{ {i}\\right\\} }$"}
+{"_id": "27085", "title": "Definition:Continuous Total Preordering", "text": "Let $S$ be a set. Let $\\precsim$ be a total preordering on $S$. Let $\\precsim$ be such that: :$a \\precsim b$ whenever there exist sequences $\\left\\langle{a^k}\\right\\rangle_k$ and $\\left\\langle{b^k}\\right\\rangle_k$ that converge to $a$ and $b$ respectively for which $a^k \\precsim b^k$ for all $k$. Then $\\precsim$ is '''continuous'''. {{explain|The above has been rendered more-or-less verbatim from the source cited, but it is not well explained. Its context appears to be important in the discipline of game theory.}}"}
+{"_id": "27086", "title": "Definition:Quasi-Concave Total Preordering", "text": "Let $R^n$ be a real vector space. Let $\\precsim$ be a total preordering on $\\R^n$. Let $\\precsim$ be such that: :for all $b \\in \\R^n$, the set $\\left\\{ {a \\in \\R^n: b \\precsim a}\\right\\}$ is convex. Then $\\precsim$ is '''quasi-concave'''."}
+{"_id": "27087", "title": "Definition:Strictly Quasi-Concave Total Preordering", "text": "Let $R^n$ be a real vector space. Let $\\precsim$ be a total preordering on $\\R^n$. Let $\\precsim$ be such that: :for all $b \\in \\R^n$, the set $\\left\\{ {a \\in \\R^n: b \\precsim a}\\right\\}$ is strictly convex. Then $\\precsim$ is '''strictly quasi-concave'''."}
+{"_id": "27088", "title": "Definition:Pareto Efficiency", "text": "Let $N$ be a finite set. Let $X \\subseteq \\R^N$ be a set. {{explain|what does $\\R^N$ mean in this context? This definition has been rendered verbatim from the source work and needs amplification.}} Then $x \\in X$ is '''Pareto efficient''' {{iff}} there exists no $y \\in X$ for which $x_i < y_i$ for all $i \\in N$. {{NamedforDef|Vilfredo Federico Damaso Pareto|cat = Pareto}}"}
+{"_id": "27089", "title": "Definition:Strong Pareto Efficiency", "text": "Let $N$ be a finite set. Let $X \\subseteq \\R^N$ be a set. {{explain|what does $\\R^N$ mean in this context? This definition has been rendered verbatim from the source work and needs amplification.}} Then $x \\in X$ is '''strongly Pareto efficient''' {{iff}} there exists no $y \\in X$ for which $x_i \\le y_i$ for all $i \\in N$ and for which $x_i < y_i$ for at least one $i \\in N$. {{NamedforDef|Vilfredo Federico Damaso Pareto|cat = Pareto}}"}
+{"_id": "27090", "title": "Definition:Limit of Vector-Valued Function/Definition 1", "text": "Let: :$\\mathbf r: t \\mapsto \\begin{bmatrix} f_1\\left({t}\\right) \\\\ f_2\\left({t}\\right) \\\\ \\vdots \\\\ f_n\\left({t}\\right) \\end{bmatrix}$ be a vector-valued function. The '''limit''' of $\\mathbf r$ as $t$ approaches $c$ is defined as follows: {{begin-eqn}} {{eqn | l = \\lim_{t \\to c} \\ \\mathbf r \\left({t}\\right)       | o = :=       | r = \\begin{bmatrix} \\lim_{t \\to c} \\ f_1\\left({t}\\right) \\\\ \\lim_{t \\to c} \\ f_2\\left({t}\\right) \\\\ \\vdots \\\\ \\lim_{t \\to c} \\ f_n\\left({t}\\right)\\end{bmatrix} }} {{end-eqn}} where each $\\lim$ on the RHS is a limit of a real function. The limit is defined to exist precisely when all the respective limits of the component functions exist."}
+{"_id": "27092", "title": "Definition:Finite Game", "text": "A '''finite game''' is a game in which the set $A_i$ of moves available to player $i$ is finite for each player $i$."}
+{"_id": "27093", "title": "Definition:Profile of Preference Relations", "text": "Let $G$ be a game. Let $P$ be a player of $G$. Let $C$ be the set $C$ of consequences of the moves available to $P$. The '''profile $\\left\\langle{\\succsim_i^*}\\right\\rangle$ of preference relations over $C$''' is defined as: :$\\forall a, b \\in A: \\left({a \\succsim_i b}\\right) \\iff \\left({g \\left({a}\\right) \\succsim_i^* g \\left({b}\\right)}\\right)$"}
+{"_id": "27094", "title": "Definition:Consequence Function with Probability", "text": "Let $G$ be a game. Let $P$ be a player of $G$. Let $A$ be the set of moves available to $P$. Let $C$ be the set of consequences of those moves. Let the consequences of those moves be affected by a random variable on a probability space $\\Omega$ whose realization is not known to the players before they make their moves. A '''consequence function''' for $P$ is a mapping from $A \\times \\Omega$ to $C$: :$g: A \\times \\Omega \\to C$ interpreted as that $g \\left({a, \\omega}\\right)$ is the consequence when the move is $a \\in A$ and the realization $\\omega$ of the random variable is $\\omega \\in \\Omega$."}
+{"_id": "27095", "title": "Definition:Lottery", "text": "A '''lottery''' is a game in which the consequence of each move is determined by the realization of a random variable. Let $X$ denote the set of prizes which a player may receive. Let $\\Omega$ denote the set of possible states. A '''lottery''' is a mapping $f: X \\times \\Omega \\to \\R_{\\ge 0}$ such that: :$\\forall t \\in \\Omega: \\displaystyle \\sum_{x \\mathop \\in X} f \\left({x, t}\\right) = 1$ === Probability Model === {{:Definition:Lottery/Probability Model}} === State-Variable Model === {{:Definition:Lottery/State-Variable Model}}"}
+{"_id": "27096", "title": "Definition:Lottery Induced by Preference Relation", "text": "Let $G$ be a game. Let $N$ be the set of players of $G$. Let $A$ be the set of moves available to player $i \\in N$. Let $C$ be the set of consequences of those moves. Let the consequences of those moves be affected by a random variable on a probability space $\\Omega$ whose realization is not known to the players before they make their moves. Let $g: A \\times \\Omega \\to C$ be the consequence function for player $i$. Then the '''lottery on $C$ induced by the profile of preference relations over $C$''' is defined by: :$\\forall a, b \\in A: \\left({a \\succsim_i b}\\right) \\iff \\left({g \\left({a, \\omega_a}\\right) \\succsim_i^* g \\left({b, \\omega_b}\\right)}\\right)$ where $\\succsim_i$ is the preference relation for player $i$."}
+{"_id": "27097", "title": "Definition:Payoff Function", "text": "Let $G$ be a game. Let $N$ be the set of players of $G$. Let $A$ be the set of moves available to player $i \\in N$. A '''payoff function''' on $A$ is a mapping $u_i$ from $A$ to the real numbers $\\R$: :$u_i: A \\to \\R$ defined by the condition: :$\\forall a, b \\in A: a \\succsim_i b \\iff u_i \\left({a}\\right) \\ge u_i \\left({b}\\right)$ where $\\succsim_i$ denotes the preference relation for player $i$."}
+{"_id": "27098", "title": "Definition:Payoff Table/Zero-Sum", "text": "Let $G$ be a two-person zero-sum game. A '''payoff table''' for $G$ is an array which specifies the payoff to (conventionally) the maximising player for each strategy of both players. As $G$ is zero-sum, there is no need to specify the payoff to the minimising player, as it will be the negative of the payoff to the maximising player. {{PayoffTable|table = $\\begin{array} {r {{|}} c {{|}} } & \\text{L} & \\text{R} \\\\ \\hline \\text{T} & w & x \\\\ \\hline \\text{B} & y & z \\\\ \\hline \\end{array}$}}"}
+{"_id": "27099", "title": "Definition:Consecutive Strategic Games", "text": "The interpretation of a strategic game $G$ can be that a particular play of $G$ is one of a sequence of such plays. In such a case, it is assumed that a player can form an expectation of the behaviour of other players through experience of previous plays."}
+{"_id": "27100", "title": "Definition:Nash Equilibrium", "text": "Let a strategic game $G$ be modelled by: :$G = \\stratgame N {A_i} {\\succsim_i}$ A '''Nash equilibrium''' of $G$ is a profile $a^* \\in A$ of moves which has the property that: :$\\forall i \\in N: \\forall a_i \\in A_i: \\tuple {a^*_{-i}, a^*_i} \\succsim_i \\tuple {a^*_{-i}, a_i}$ Thus, for $a^*$ to be a '''Nash equilibrium''', no player $i$ has a move yielding a preferable outcome to that when $a^*_i$ is chosen, given that every other player $j$ has chosen his own equilibrium move. That is, no player can profitably deviate, if no other player also deviates. {{NamedforDef|John Forbes Nash|cat = Nash}}"}
+{"_id": "27101", "title": "Definition:Best-Response Function", "text": "Let a strategic game $G$ be modelled by: :$G = \\left\\langle{N, \\left\\langle{A_i}\\right\\rangle, \\left\\langle{\\succsim_i}\\right\\rangle}\\right\\rangle$ Let $a^*$ be a Nash equilibrium of $G$: :$\\forall i \\in N: \\forall a_i \\in A_i: \\left({a^*_{-i}, a^*_i}\\right) \\succsim_i \\left({a^*_{-i}, a_i}\\right)$ For any $a_{-1} \\in A_{-i}$, let $B_i \\left({a_{-i} }\\right)$ be the set of player $i$'s best moves, defined as: :$B_i \\left({a_{-i} }\\right) = \\left\\{ {a_i \\in A_i: \\forall a'_i \\in A_i: \\left({a_{-i}, a_i}\\right) \\precsim_i  \\left({a_{-i}, a'_i}\\right)}\\right\\}$ Then $B_{-i}$ is known as the '''best-response function''' of player $i$."}
+{"_id": "27103", "title": "Definition:Bach or Stravinsky?", "text": "There are two players: $\\text A$lexis and $\\text B$everley. They wish to go out together to a musical concert, but $\\text A$ prefers Bach and $\\text B$ prefers Stravinsky. The key points are: :$\\text A$lexis and $\\text B$everley wish to coordinate their behaviour but: :they have conflicting interests."}
+{"_id": "27104", "title": "Definition:Coordination Game", "text": "There are two players: $\\text A$lexis and $\\text B$everley. They wish to go out together to a musical concert to experience either the music of Mozart or Mahler. Unaccountably, both $\\text A$ and $\\text B$ prefer Mozart.  (It takes all sorts to make a world.) The key points are: :$\\text A$lexis and $\\text B$everley wish to coordinate their behaviour but: :they have common interests."}
+{"_id": "27110", "title": "Definition:Strictly Competitive Game", "text": "A '''strictly competitive game''' is a game in which the interests of each player are diametrically opposed."}
+{"_id": "27111", "title": "Definition:Sealed-Bid Auction", "text": "A '''sealed-bid auction''' is a game in the following format: An object $F$ is to be assigned to a player in a set $\\left\\{ {1, 2, \\ldots, n}\\right\\}$ in exchange for a payment. The valuation of $F$ by player $i$ is $v_i$, and: :$v_1 > v_2 > \\cdots > v_n > 0$ Each player simultaneously submits a non-negative number, called a '''bid'''. $F$ is then given to the player with the lowest index among those who submit the highest bid, in exchange for a payment. === First Price Auction === {{:Definition:Sealed-Bid Auction/First Price}} === Second Price Auction === {{:Definition:Sealed-Bid Auction/Second Price}}"}
+{"_id": "27112", "title": "Definition:Sealed-Bid Auction/First Price", "text": "A '''first price auction''' is a sealed-bid auction in which the payment made by the winner is the price which is bid by that player."}
+{"_id": "27113", "title": "Definition:Sealed-Bid Auction/Second Price", "text": "A '''second price auction''' is a sealed-bid auction in which the payment made by the winner is the highest bid made by the players who did ''not'' win. So if only one player submits the highest bid, the price paid is the second highest bid."}
+{"_id": "27114", "title": "Definition:Weakly Dominant Move", "text": "Let $G$ be a game. Let $a$ be a move available to player $P$. Then $a$ is '''weakly dominant''' {{iff}} :for any move of an opposing player, the payoff of $a$ is not less than the payoff of any other move available to $P$."}
+{"_id": "27115", "title": "Definition:Intelligent Player", "text": "Let $G$ be a game. Let $P$ be a player in $G$. Then $P$ is described as '''intelligent''' {{iff}} $P$ knows everything about $G$ that there is to know, and can use that knowledge to make informed decisions about what move to make. In game theory, all players are assumed to be '''intelligent'''."}
+{"_id": "27116", "title": "Definition:Decision Theory", "text": "'''Decision theory''' is the study of the reasoning underlying an agent's choices."}
+{"_id": "27117", "title": "Definition:Bayesian Decision Theory", "text": "'''Bayesian decision theory''' is a branch of decision theory which is informed by Bayesian probability. It is a statistical system that tries to quantify the tradeoff between various decisions, making use of probabilities and costs."}
+{"_id": "27118", "title": "Definition:Subjective Probability Distribution", "text": "A '''subjective probability distribution''' is a probability distribution based on the beliefs of a rational decision-maker about all relevant unknown factors concerning a game. As new information becomes available, the '''subjective probabilities''' can then revised according to Bayes' Formula."}
+{"_id": "27119", "title": "Definition:Lottery/Probability Model", "text": "In a '''probability model''', a lottery is a probability distribution over a set of prizes."}
+{"_id": "27120", "title": "Definition:Lottery/State-Variable Model", "text": "In a '''state-variable model''', a lottery is defined as a mapping from a set of possible states into a set of prizes."}
+{"_id": "27121", "title": "Definition:Objective Unknown", "text": "An '''objective unknown''' is an event which has a well-defined objective probability of occurring. Two '''objective unknowns''' with the same probability are equivalent in the field of decision theory. An '''objective unknown''' is appropriately modelled by means of a probability model."}
+{"_id": "27123", "title": "Definition:Subjective Unknown", "text": "An '''subjective unknown''' is an event which does not have a well-defined objective probability of occurring. An '''subjective unknown''' is appropriately modelled by means of a state-variable model. === Examples === {{:Definition:Subjective Unknown/Examples}}"}
+{"_id": "27125", "title": "Definition:Prize", "text": "Let $G$ be a game. A prize in $G$ is a payoff whose utility value is strictly positive. The term is usually seen in the context of a lottery."}
+{"_id": "27126", "title": "Definition:State", "text": "Let $G$ be a game whose outcome is determined by the realization of a random variable $X$. Each of the possible values that can be taken by $X$ is known as a '''state''' of $G$."}
+{"_id": "27127", "title": "Definition:True State", "text": "Let $G$ be a game whose outcome is determined by the realization of a random variable $X$. The particular state of $G$ which $X$ actually takes is known as the '''true state (of the world)'''."}
+{"_id": "27128", "title": "Definition:Conditional Preference", "text": "Let $G$ be a lottery. Let $P$ be a player of $G$. Let $X$ denote the set of prizes which $P$ may receive. Let $\\Omega$ denote the set of possible states of $G$. Let $\\Xi$ be the event space of $G$. Let $L$ be the set of all plays of $G$. Let $f$ and $g$ be two lotteries in $G$. Let $S \\subseteq \\Xi$ be an event. A '''conditional preference''' is a preference relation $\\succsim_S$ such that: :$f \\succsim_S g$ {{iff}} $f$ would be at least as desirable to $P$ as $g$, if $P$ was aware that the true state of the world was $S$. That is, $f \\succsim_S g$ {{iff}} $P$ prefers $f$ to $g$ and he knows only that $S$ has occurred. The notation $a \\sim_S b$ is defined as: :$a \\sim_S b$ {{iff}} $a \\succsim_S b$ and $b \\succsim_S a$ The notation $a \\succ_S b$ is defined as: :$a \\succ_S b$ {{iff}} $a \\succsim_S b$ and $a \\not \\sim_S a$ When no conditioning event $S$ is mentioned, the notation $a \\succsim_\\Omega b$, $a \\succ_\\Omega b$ and $a \\sim_\\Omega b$ can be used, which mean the same as $a \\succsim b$, $a \\succ b$ and $a \\sim b$. {{handwaving|Myerson is as lax as all the other game theory writers when it comes to defining rigorous concepts. I am going to have to abandon this field of study until I really understand exactly what the underlying mathematical objects are.}}"}
+{"_id": "27129", "title": "Definition:Biquadrate", "text": "A '''biquadrate number''' (or just '''biquadrate''') is a number which can be expressed as the fourth power of an integer. The sequence of positive '''biquadrates''' begins: :$1, 16, 81, 256, 625, 1297, 2401, 4096, 6561, 10000, \\ldots$ {{OEIS|A000583}}"}
+{"_id": "27130", "title": "Definition:Egyptian Fraction", "text": "An '''Egyptian fraction''' is a fraction either: :whose numerator is $1$ or :which is $\\dfrac 2 3$"}
+{"_id": "27131", "title": "Definition:Primorial/Prime", "text": "Let $p_n$ be the $n$th prime number. Then the '''$n$th primorial''' $p_n \\#$ is defined as: :$\\displaystyle p_n \\# := \\prod_{k \\mathop = 1}^n p_k$ That is, $p_n \\#$ is the product of the first $n$ primes."}
+{"_id": "27132", "title": "Definition:Primorial/Positive Integer", "text": "Let $n$ be a positive integer. Then: : $\\displaystyle n\\# := \\prod_{i \\mathop = 1}^{\\pi \\left({n}\\right)} p_i = p_{\\pi \\left({n}\\right)}\\#$ where $\\pi \\left({n}\\right)$ is the prime counting function. That is, $n\\#$ is defined as the product of all primes less than or equal to $n$. Thus: :$n\\# = \\begin{cases} 1 & : n \\le 1 \\\\ n \\left({\\left({n - 1}\\right)\\#}\\right) & : n \\mbox { prime} \\\\ \\left({n - 1}\\right)\\# & : n \\mbox { composite} \\end{cases}$"}
+{"_id": "27134", "title": "Definition:Repunit", "text": "Let $b \\in \\Z_{>1}$ be an integer greater than $1$. Let a (positive) integer $n$, greater than $b$ be expressed in base $b$. $n$ is a '''repunit base $b$''' {{iff}} $n$ is a repdigit number base $b$ whose digits are $1$. That is, $n$ is a '''repunit base $b$''' {{iff}} all of the digits of $n$ are $1$. When $b$ is the usual base $10$, $n$ is merely referred to as a '''repunit'''."}
+{"_id": "27135", "title": "Definition:Unit Fraction", "text": "A '''unit fraction''' is a fraction whose numerator is $1$."}
+{"_id": "27136", "title": "Definition:Myriad", "text": "'''Myriad''' is an archaic word for the number $10 \\ 000$ (ten thousand)."}
+{"_id": "27137", "title": "Definition:Thousand", "text": "A '''thousand''' is $1000$: $10$ to the power of $3$: :$1000 = 10^3 = 10 \\times 10 \\times 10$"}
+{"_id": "27144", "title": "Definition:Almost All/Set Theory/Countable", "text": "Let $S$ be a countably infinite set. Let $P: S \\to \\left\\{ {\\text{true}, \\text{false} }\\right\\}$ be a property of $S$ such that: :$\\left\\{ {s \\in S: \\neg P \\left({s}\\right)}\\right\\}$ is finite. Then $P$ holds for '''almost all''' of the elements of $S$."}
+{"_id": "27146", "title": "Definition:Champernowne Constant", "text": "The '''Champernowne constant''' is the real number whose decimal expansion is formed by concatenating the positive integers in ascending order: :$C_{10} = 0 \\cdotp 12345 \\, 67891 \\, 01112 \\, 13141 \\, 51617 \\, 18192 \\, 02122 \\ldots$ {{OEIS|A033307}}"}
+{"_id": "27147", "title": "Definition:Absolutely Normal Real Number", "text": "A real number $r$ is '''absolutely normal''' if it is normal with respect to ''every'' number base $b$. That is, {{iff}} its basis expansion in every number base $b$ is such that: :no finite sequence of digits of $r$ of length $n$ occurs more frequently than any other such finite sequence of length $n$."}
+{"_id": "27148", "title": "Definition:Anomalous Cancellation", "text": "Let $r = \\dfrac a b$ be a fraction where $a$ and $b$ are integers expressed in conventional decimal notation. '''Anomalous cancellation''' is a phenomenon whereby deleting (that is, cancelling) common digits from the numerator $a$ and the denominator $b$ of $r$, the value of $r$ the fraction does not change."}
+{"_id": "27152", "title": "Definition:Newton-Mercator Series", "text": "Let $\\ln x$ denote the natural logarithm function. Then: {{begin-eqn}} {{eqn | l = \\map \\ln {1 + x}       | r = x - \\dfrac {x^2} 2 + \\dfrac {x^3} 3 - \\dfrac {x^4} 4 + \\cdots       | c =  }} {{eqn | r = \\sum_{n \\mathop = 1}^\\infty \\frac {\\paren {-1}^{n + 1} } n x^n       | c =  }} {{end-eqn}} The series converges to the natural logarithm (shifted by $1$) for $-1 < x \\le 1$. This is known as the '''Newton-Mercator series'''."}
+{"_id": "27154", "title": "Definition:Filter on Set/Filtered Set", "text": "Let $\\FF$ be a filter on $S$. Then $S$ is said to be '''filtered by $\\FF$''', or just a '''filtered set'''."}
+{"_id": "27156", "title": "Definition:Filter on Set/Trivial Filter", "text": "A filter $\\FF$ on $S$ by definition specifically does ''not'' include the empty set $\\O$. If a filter $\\FF$ ''were'' to include $\\O$, then from Empty Set is Subset of All Sets it would follow that ''every'' subset of $S$ would have to be in $\\FF$, and so $\\FF = \\powerset S$. Such a \"filter\" is called '''the trivial filter''' on $S$."}
+{"_id": "27157", "title": "Definition:Filter on Set/Definition 1", "text": "A '''filter on $S$''' (or '''filter of $S$''') is a set $\\FF \\subset \\powerset S$ which satisfies the following conditions: {{begin-axiom}} {{axiom | n = \\text F 1         | m = S \\in \\FF }} {{axiom | n = \\text F 2         | m = \\O \\notin \\FF }} {{axiom | n = \\text F 3         | m = U, V \\in \\FF \\implies U \\cap V \\in \\FF }} {{axiom | n = \\text F 4         | m = \\forall U \\in \\FF: U \\subseteq V \\subseteq S \\implies V \\in \\FF }} {{end-axiom}}"}
+{"_id": "27158", "title": "Definition:Filter on Set/Definition 2", "text": "A '''filter on $S$''' (or '''filter of $S$''') is a set $\\FF \\subset \\powerset S$ which satisfies the following conditions: {{begin-axiom}} {{axiom | n = \\text F 1         | m = S \\in \\FF }} {{axiom | n = \\text F 2         | m = \\O \\notin \\FF }} {{axiom | n = \\text F 3         | m = \\forall n \\in \\N: U_1, \\ldots, U_n \\in \\FF \\implies \\bigcap_{i \\mathop = 1}^n U_i \\in \\FF }} {{axiom | n = \\text F 4         | m = \\forall U \\in \\FF: U \\subseteq V \\subseteq S \\implies V \\in \\FF }} {{end-axiom}}"}
+{"_id": "27159", "title": "Definition:Filter/Proper Filter", "text": "Let $\\mathcal F$ be a filter on $\\left({S, \\preccurlyeq}\\right)$. Then $\\mathcal F$ is a '''proper filter''' on $S$ {{iff}} $\\mathcal F \\ne S$. That is, {{iff}} $\\mathcal F$ is a proper subset of $S$."}
+{"_id": "27161", "title": "Definition:Closed under Mapping/Arbitrary Product", "text": "Let $\\phi: X^I \\to T$ be a mapping or a partial mapping, taking $I$-indexed families as arguments. Denote with $\\Dom \\phi$ the domain of $\\phi$ (if $\\phi$ is a mapping, this is simply $X^I$). A set $S$ is '''closed under $\\phi$''' {{iff}}: :$\\forall \\family {s_i}_{i \\mathop \\in I} \\in S^I \\cap \\Dom \\phi: \\map \\phi {\\family {s_i}_{i \\mathop \\in I} } \\in S$ Phrased in terms of image of a mapping, this translates to: :$\\map \\phi {S^I \\cap \\Dom \\phi} \\subseteq S$ Thus, in words, $S$ is '''closed under $\\phi$''', {{iff}}: :Whenever $\\phi$ is defined for an $I$-indexed family from $S$, it maps that indexed family into $S$ again."}
+{"_id": "27162", "title": "Definition:Filter Basis/Equivalent Filter Bases", "text": "Two '''filter bases''' are '''equivalent''' {{iff}} they both generate the same filter."}
+{"_id": "27163", "title": "Definition:Unit Cube", "text": "A '''unit cube''' is a cube each of whose edges is one unit long."}
+{"_id": "27164", "title": "Definition:Cube Root/Real", "text": "Let $x \\in \\R_{\\ge 0}$ be a positive real number. The '''cube roots of $x$''' is the real number defined as: :$x^{\\paren {1 / 3} } := \\set {y \\in \\R: y^3 = x}$ where $x^{\\paren {1 / 3} }$ is the $3$rd root of $x$. The notation: :$y = \\sqrt [3] x$ is usually encountered."}
+{"_id": "27165", "title": "Definition:Archytas Curve", "text": "The '''Archytas curve''' is a curve formed as the intersection of a cone, a cylinder and a horn torus. {{DefinitionWanted}} {{NamedforDef|Archytas of Tarentum|cat = Archytas}}"}
+{"_id": "27166", "title": "Definition:Conchoid of Nicomedes", "text": "The '''conchoid of Nicomedes''' is the plane curve defined in Cartesian coordinates as: :$\\paren {x - a}^2 \\paren {x^2 + y^2} = b^2 x^2$ or in polar coordinates as: :$r = b + a \\sec \\theta$ for some real constants $a \\in \\R$, $b \\in \\R_{> 0}$. :600px The above diagram illustrates the '''conchoid of Nicomedes''' for $b = 1$ and various values of $a$ from $0$ to $3$."}
+{"_id": "27168", "title": "Definition:Cissoid of Diocles", "text": "Let $C$ be a a circle of radius $a$ with a distinguished point $O$ on its circumference. Let $L$ be the tangent to $C$ at the other end of the diameter of $C$ through $O$. Let $R$ be a point on the circumference of $C$. Let $OR$ be produced to meet $L$ at $S$. Let $P$ be the point on $OS$ such that $OP$ = $RS$. The '''cissoid of Diocles''' is the locus of points $P$ as $R$ travels around the circumference of $C$."}
+{"_id": "27170", "title": "Definition:Pell Numbers", "text": "The '''Pell numbers''' are a sequence $\\sequence {P_n}$ which is formally defined by the recurrence relation: :$P_n = \\begin{cases} 0 & : n = 0 \\\\ 1 & : n = 1 \\\\ 2 P_{n - 1} + P_{n - 2} & : \\text {otherwise}\\end{cases}$ The sequence of '''Pell numbers''' begins: :$0, 1, 2, 5, 12, 29, 70, 169, 408, 985, \\ldots$"}
+{"_id": "27171", "title": "Definition:Pell-Lucas Numbers", "text": "The '''Pell-Lucas numbers''' are a sequence $\\left \\langle {Q_n}\\right \\rangle$ which is formally defined by the recurrence relation: :$P_n = \\begin{cases} 2 & : n = 0 \\\\ 2 & : n = 1 \\\\ 2 P_{n -1} + P_{n - 2} & : \\text {otherwise}\\end{cases}$ The sequence of '''Pell-Lucas numbers''' begins: :$2, 2, 6, 14, 34, 82, 198, 478, 1154, \\ldots$ {{OEIS|A002203}}"}
+{"_id": "27172", "title": "Definition:Best Rational Approximation", "text": "Let $x \\in \\R$ be an (irrational) real number. The rational number $a = \\dfrac p q$ is a '''best rational approximation''' to $x$ {{iff}}: :$(1): \\quad a$ is in canonical form, that is $p$ is coprime to $q$: $p \\perp q$ :$(2): \\quad \\left\\vert{x - \\dfrac p q}\\right\\vert = \\min \\left\\{ {\\left\\vert{x - \\dfrac {p'} {q'} }\\right\\vert: q' \\le q}\\right\\}$ That is: :$\\left\\vert{x - \\dfrac p q}\\right\\vert$ is smaller than for any $\\dfrac {p'} {q'}$ where $q' \\le q$ where $\\left\\vert{x}\\right\\vert$ denotes the absolute value of $x$. === Sequence === {{definition wanted|sorted by increasing denominator}} Category:Definitions/Number Theory Category:Definitions/Real Analysis Category:Definitions/Continued Fractions k8dnfwqwqiwk8ba1vq2mz6snubc52ys"}
+{"_id": "27173", "title": "Definition:Prime Filter (Order Theory)", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $F$ be a filter in $\\left({S, \\preceq}\\right)$. $F$ is a '''prime filter''' {{iff}}: :$\\forall x, y \\in S: \\left({x \\vee y \\in F \\implies x \\in F \\lor y \\in F}\\right)$"}
+{"_id": "27174", "title": "Definition:Quadratic Irrational/Reduced", "text": "An irrational root $\\alpha$ of a quadratic equation with integer coefficients is a '''reduced quadratic irrational''' {{iff}} :$(1): \\quad \\alpha > 1$ :$(2): \\quad$ its conjugate $\\tilde{\\alpha}$ satisfies: ::::$-1 < \\tilde{\\alpha} < 0$"}
+{"_id": "27177", "title": "Definition:Power Tower Function", "text": "The '''power tower function''' is the real function $f: \\R \\to \\R$ defined on the closed interval $\\closedint {e^{-e} } {e^{1/e} }$ as: :$\\map f x = x^{x^{x^{x^{x^{\\cdot^{\\cdot^\\cdot} } } } } }$ where the tower of powers tends to infinity."}
+{"_id": "27178", "title": "Definition:Pentagram", "text": "A '''pentagram''' is a geometric figure formed by the line segments connecting alternate vertices of a regular pentagon: :400px"}
+{"_id": "27181", "title": "Definition:Golden Rectangle", "text": "A '''golden rectangle''' is a rectangle whose sides are in extreme and mean ratio: :500px Thus, in the above rectangle: :$AB = AD$ :$AB : BC = AC : AB$"}
+{"_id": "27182", "title": "Definition:Beatty Sequence", "text": "Let $x$ be an irrational number. The '''Beatty sequence on $x$''' is the integer sequence $\\BB_x$ defined as: :$\\BB_x := \\sequence{\\floor{n x} }_{n \\mathop \\in \\Z_{\\ge 0} }$"}
+{"_id": "27183", "title": "Definition:Beatty Sequence/Complementary", "text": "Let $\\mathcal B_x$ be the '''Beatty sequence''' on $x$. The '''complementary Beatty sequence on $x$''' is the integer sequence formed by the integers which are missing from $\\mathcal B_x$."}
+{"_id": "27184", "title": "Definition:Lower Wythoff Sequence", "text": "The '''lower Wythoff sequence''' is the Beatty sequence on the golden section $\\phi$. It starts: :$0, 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, \\ldots$"}
+{"_id": "27185", "title": "Definition:Ultrafilter (Order Theory)", "text": "Let $O = \\left({S, \\preceq}\\right)$ be an ordered set. Let $F$ be a filter in $O$. Then $F$ is '''ultrafilter''' (on $O$) {{iff}} :$F$ is proper subset of $S$ and :for all filter $G$ in $O$: $\\left({F \\subseteq G \\implies F = G \\lor G = S}\\right)$"}
+{"_id": "27186", "title": "Definition:Upper Wythoff Sequence", "text": "{{:Definition:Upper Wythoff Sequence/Definition 2}} The '''upper Wythoff sequence''' starts: :$0, 2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, \\ldots$"}
+{"_id": "27187", "title": "Definition:Upper Wythoff Sequence/Definition 1", "text": "The '''upper Wythoff sequence''' is the complementary Beatty sequence on the golden section $\\phi$."}
+{"_id": "27188", "title": "Definition:Upper Wythoff Sequence/Definition 2", "text": "The '''upper Wythoff sequence''' is the Beatty sequence on the square $\\phi^2$ of the golden section $\\phi$."}
+{"_id": "27189", "title": "Definition:Wythoff Pair", "text": "A '''Wythoff pair''' is an ordered pair of integers of the form: :$\\tuple {\\floor {k \\phi}, \\floor {k \\phi^2} }$ where: :$\\phi$ denotes the golden section: $\\phi = 1 \\cdotp 618 \\ldots$ :$\\floor x$ denotes the floor of $x$ :$k$ signifies a positive integer: $k \\in \\Z_{\\ge 0}$. Thus the coordinates of a '''Wythoff pair''' are corresponding terms of the lower and upper Wythoff sequences. {{NamedforDef|Willem Abraham Wythoff|cat = Wythoff}}"}
+{"_id": "27190", "title": "Definition:Wythoff Sequence", "text": "=== Lower Wythoff Sequence === {{:Definition:Lower Wythoff Sequence}} === Upper Wythoff Sequence === {{:Definition:Upper Wythoff Sequence}}"}
+{"_id": "27192", "title": "Definition:Finite Infima Set", "text": "Let $P = \\left({S, \\preceq}\\right)$ be an ordered set. Let $X$ be a subset of $S$. Then '''finite infima set''' of $X$, denoted $\\operatorname{fininfs}\\left({X}\\right)$, is defined by :$\\left\\{ {\\inf A: A \\in \\mathit{Fin}\\left({X}\\right) \\land A}\\right.$ admits an infimum$\\left.{}\\right\\}$ where $\\mathit{Fin}\\left({X}\\right)$ denotes the set of all finite subsets of $X$."}
+{"_id": "27193", "title": "Definition:Wythoff's Game", "text": "'''Wythoff's game''' is a two-person game whose rules are as follows: :$(1): \\quad$ The game starts with two piles of counters. :$(2): \\quad$ Each player takes it in turns to remove either: ::::A number of counters from one pile ::::A number of counters from both pile such that the number of counters removed from each pile must be equal. :$(3): \\quad$ The person who removes the last counter (or counters) is the winner. {{NamedforDef|Willem Abraham Wythoff|cat = Wythoff}}"}
+{"_id": "27194", "title": "Definition:Tribonacci Constant", "text": "The '''Tribonacci constant''' $\\eta$ is the one real root of the cubic: :$x^3 - x^2 - x - 1 = 0$ Its decimal expansion starts: :$\\eta = 1 \\cdotp 83928 \\, 67552 \\, 1416 \\ldots$ {{OEIS|A058265}}"}
+{"_id": "27196", "title": "Definition:Tribonacci Sequence/General", "text": "A '''general Tribonacci sequence''' is a sequence $\\left \\langle {u_n}\\right \\rangle$ which is formally defined recursively as: :$u_n = \\begin{cases} a & : n = 0 \\\\ b & : n = 1 \\\\ c & : n = 2 \\\\ u_{n - 1} + u_{n - 2} + u_{n - 3} & : n > 2 \\end{cases}$ where $a, b, c \\in \\Z$ are constants."}
+{"_id": "27197", "title": "Definition:Tribonacci Sequence", "text": "The '''Tribonacci sequence''' is a sequence $\\left \\langle {u_n}\\right \\rangle$ which is formally defined recursively as: :$u_n = \\begin{cases} 0 & : n = 0 \\\\ 0 & : n = 1 \\\\ 1 & : n = 2 \\\\ u_{n - 1} + u_{n - 2} + u_{n - 3} & : n > 2 \\end{cases}$"}
+{"_id": "27198", "title": "Definition:Brun's Constant", "text": "'''Brun's constant''' is the sum of the series consisting of the reciprocals of the twin primes: :$B_2 := \\paren {\\dfrac 1 3 + \\dfrac 1 5} + \\paren {\\dfrac 1 5 + \\dfrac 1 7} + \\paren {\\dfrac 1 {11} + \\dfrac 1 {13} } + \\paren {\\dfrac 1 {17} + \\dfrac 1 {19} } + \\paren {\\dfrac 1 {29} + \\dfrac 1 {31} } + \\cdots$ Its approxmiate decimal expansion is: :$B_2 \\approx 1 \\cdotp 90216 \\, 05831 \\, 04 \\ldots$ {{OEIS|A065421}} Estimates of its value are occasionally refined as further work is done to establish its nature."}
+{"_id": "27200", "title": "Definition:Bilateral Symmetry", "text": "Let $F$ be a geometric figure. Let there exists a reflection $R$ in the such that $\\map R F$ is a symmetry. Then $F$ has '''bilateral symmetry'''. === Axis of Symmetry === {{:Definition:Bilateral Symmetry/Axis}}"}
+{"_id": "27204", "title": "Definition:Metre/Square Metre", "text": "The '''square metre''' is the SI unit of area. The symbol for the '''square metre''' is $\\mathrm m^2$."}
+{"_id": "27206", "title": "Definition:Centimetre/Square Centimetre", "text": "The '''square centimetre''' is the CGS unit of area. The symbol for the '''square centimetre''' is $\\mathrm {cm}^2$ or (the informal and ugly) $\\mathrm {sq}. \\ \\mathrm {cm}$."}
+{"_id": "27209", "title": "Definition:SI Units/Derived", "text": "The units derived from the SI base units include the following: {{begin-axiom}} {{axiom | lc= Square metre:         | m = \\mathrm m^2 }} {{axiom | lc= Cubic metre:         | m = \\mathrm m^3 }} {{axiom | lc= Newton:         | m = \\mathrm {kg} \\, \\mathrm m \\, \\mathrm s^{-2} }} {{axiom | lc= Coulomb:         | m = \\mathrm A \\, \\mathrm s }} {{end-axiom}} {{stub}}"}
+{"_id": "27210", "title": "Definition:FPS/Base Units", "text": "{| class=\"wikitable\" style=\"margin:1em auto 1em auto\" |+ FPS base units |- !Name ! Unit symbol ! Dimension ! Symbol |- ! foot | $\\mathrm {ft}$ | Length | $l$ |- ! pound | $\\mathrm {lb}$ | Mass | $m$ |- ! second | $\\mathrm s$ | Time | $t$ |- |} === Foot === {{:Definition:FPS/Foot}} === Pound === {{:Definition:FPS/Pound}} === Second === {{:Definition:FPS/Second}}"}
+{"_id": "27211", "title": "Definition:FPS/Pound", "text": "The '''pound''' is the FPS unit of measurement of mass, which has a number of different standards. The most common standard is for it to be defined in kilograms to be exactly $0.453 \\, 592 \\, 37 \\ \\mathrm{kg}$."}
+{"_id": "27212", "title": "Definition:Imperial/Volume", "text": "The imperial units of volume are based on a binary system, in which each unit is a factor of $2$ larger than the next smaller unit. === Fluid Ounce === {{Definition:Imperial/Volume/Fluid Ounce}} === Gill === {{Definition:Imperial/Volume/Gill}} === Chopin === {{Definition:Imperial/Volume/Chopin}} === Pint === {{Definition:Imperial/Volume/Pint}} === Quart === {{Definition:Imperial/Volume/Quart}} === Pottle === {{Definition:Imperial/Volume/Pottle}} === Gallon === {{Definition:Imperial/Volume/Gallon}} === Peck === {{Definition:Imperial/Volume/Peck}} === Demi-Bushel === {{Definition:Imperial/Volume/Demi-Bushel}} === Bushel === {{Definition:Imperial/Volume/Bushel}} === Kilderkin === {{Definition:Imperial/Volume/Kilderkin}} === Barrel === {{Definition:Imperial/Volume/Barrel}} === Hogshead === {{Definition:Imperial/Volume/Hogshead}} === Pipe === {{Definition:Imperial/Volume/Pipe}} === Tun === {{Definition:Imperial/Volume/Tun}}"}
+{"_id": "27214", "title": "Definition:Imperial/Volume/Tun", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''tun''' }} {{eqn | r = 2       | c = pipes }} {{eqn | r = 1 \\cdotp 16379 \\, 904       | c = cubic metres }} {{eqn | r = 1 \\, 163 \\cdotp 79904       | c = litres }} {{end-eqn}}"}
+{"_id": "27215", "title": "Definition:Imperial/Volume/Pipe", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''pipe''' }} {{eqn | r = 2       | c = hogsheads }} {{eqn | r = 0 \\cdotp 58189 \\, 952       | c = cubic metres }} {{eqn | r = 581 \\cdotp 89952       | c = litres }} {{end-eqn}}"}
+{"_id": "27216", "title": "Definition:Imperial/Volume/Hogshead", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''hogshead''' }} {{eqn | r = 2       | c = barrels }} {{eqn | r = 0 \\cdotp 29094 \\, 976       | c = cubic metres }} {{eqn | r = 290 \\cdotp 94976       | c = litres }} {{end-eqn}}"}
+{"_id": "27217", "title": "Definition:Imperial/Volume/Barrel", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''barrel''' }} {{eqn | r = 2       | c = kilderkins }} {{eqn | r = 0 \\cdotp 14547 \\, 488       | c = cubic metres }} {{eqn | r = 145 \\cdotp 47488       | c = litres }} {{end-eqn}}"}
+{"_id": "27219", "title": "Definition:Litre", "text": "The '''litre''' is an SI unit of volume. It is defined as: {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''litre''' }} {{eqn | r = 1000       | c = cubic centimetres }} {{eqn | r = 10^{-3}       | c = cubic metres }} {{end-eqn}}"}
+{"_id": "27220", "title": "Definition:Imperial/Volume/Kilderkin", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''kilderkin''' }} {{eqn | r = 2       | c = bushels }} {{eqn | r = 0 \\cdotp 07273 \\, 744       | c = cubic metres }} {{eqn | r = 72 \\cdotp 73744       | c = litres }} {{end-eqn}}"}
+{"_id": "27221", "title": "Definition:Imperial/Volume/Bushel", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''bushel''' }} {{eqn | r = 2       | c = demi-bushels }} {{eqn | r = 0 \\cdotp 03636 \\, 872       | c = cubic metres }} {{eqn | r = 36 \\cdotp 36872       | c = litres }} {{end-eqn}}"}
+{"_id": "27222", "title": "Definition:Imperial/Volume/Demi-Bushel", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''demi-bushel''' }} {{eqn | r = 2       | c = pecks }} {{eqn | r = 18 \\cdotp 18436       | c = litres }} {{end-eqn}}"}
+{"_id": "27223", "title": "Definition:Imperial/Volume/Peck", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''peck''' }} {{eqn | r = 2       | c = (imperial) gallons }} {{eqn | r = 9 \\cdotp 09218       | c = litres }} {{end-eqn}}"}
+{"_id": "27224", "title": "Definition:Imperial/Volume/Gallon", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''gallon''' }} {{eqn | r = 2       | c = pottles }} {{eqn | r = 4 \\cdotp 54609       | c = litres }} {{eqn | r = 4 \\, 546 \\cdotp 09       | c = millilitres }} {{end-eqn}}"}
+{"_id": "27226", "title": "Definition:Imperial/Volume/Pottle", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''pottle''' }} {{eqn | r = 2       | c = quarts }} {{eqn | r = 2 \\cdotp 273045       | c = litres }} {{eqn | r = 2 \\, 273 \\cdotp 045       | c = millilitres }} {{end-eqn}}"}
+{"_id": "27227", "title": "Definition:Imperial/Volume/Quart", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''quart''' }} {{eqn | r = 2       | c = pints }} {{eqn | r = 1 \\cdotp 13652 25       | c = litres }} {{eqn | r = 1 \\, 136 \\cdotp 5225       | c = millilitres }} {{end-eqn}}"}
+{"_id": "27228", "title": "Definition:Imperial/Volume/Pint", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''pint''' }} {{eqn | r = 2       | c = chopins }} {{eqn | r = 0 \\cdotp 5682 \\, 6125       | c = litres }} {{eqn | r = 568 \\cdotp 26125       | c = millilitres }} {{end-eqn}}"}
+{"_id": "27229", "title": "Definition:Imperial/Volume/Chopin", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''chopin''' }} {{eqn | r = 2       | c = gills }} {{eqn | r = 0 \\cdotp 28413 \\, 0625       | c = litres }} {{eqn | r = 284 \\cdotp 13062 \\, 5       | c = millilitres }} {{end-eqn}}"}
+{"_id": "27230", "title": "Definition:Imperial/Volume/Gill", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''gill''' }} {{eqn | r = 5       | c = fluid ounces }} {{eqn | r = 0 \\cdotp 14206 \\, 53125       | c = litres }} {{eqn | r = 142 \\cdotp 06531 \\, 25       | c = millilitres }} {{end-eqn}}"}
+{"_id": "27233", "title": "Definition:Imperial/Volume/Fluid Ounce", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''fluid ounce''' }} {{eqn | r = \\dfrac 1 {20}       | c = pint }} {{eqn | r = 28 \\cdotp 41306 \\, 25       | c = millilitres }} {{end-eqn}}"}
+{"_id": "27237", "title": "Definition:Wallis's Number", "text": "'''Wallis's number''' is the real root of the cubic: :$x^3 - 2 x - 5 = 0$ Its approximate value is: :$2 \\cdotp 09455 \\, 1 \\ldots$ {{OEIS|A007493}}"}
+{"_id": "27238", "title": "Definition:Stirling's Constant", "text": "'''Stirling's constant''' is a name given to the square root of $2$ times $\\pi$ (pi). It is approximated by the decimal expansion: :$\\sqrt {2 \\pi} \\approx 2 \\cdotp 50662 \\, 82746 \\, 31000 \\, 5 \\ldots$ {{OEIS|A019727}} {{NamedforDef|James Stirling|cat = Stirling}}"}
+{"_id": "27241", "title": "Definition:Gelfond's Constant", "text": "'''Gelfond's constant''' is the number obtained by raising Euler's number $e$ to the power of $\\pi$ (pi): :$e^\\pi$"}
+{"_id": "27271", "title": "Definition:Time/Fourth Dimension", "text": "In the Theory of Relativity, it is convenient to consider time as another dimension in addition to, and treated similarly to, the $3$ dimensions of ordinary space."}
+{"_id": "27273", "title": "Definition:Quadratrix of Hippias/Definition 1", "text": "The '''quadratrix of Hippias''' is the plane curve defined in Cartesian coordinates as: :$y = x \\cot \\left({\\dfrac {\\pi x }{2 a} }\\right)$ for some real constant $a \\in \\R$. :600px The above diagram illustrates the '''quadratrix of Hippias'''."}
+{"_id": "27275", "title": "Definition:Quadratrix of Hippias", "text": "=== Definition 1 === {{:Definition:Quadratrix of Hippias/Definition 1}} === Definition 2 === {{:Definition:Quadratrix of Hippias/Definition 2}}"}
+{"_id": "27277", "title": "Definition:Orthocenter", "text": "Let $\\triangle ABC$ be a triangle. The '''orthocenter''' of $\\triangle ABC$ is where its three altitudes meet. :500px"}
+{"_id": "27278", "title": "Definition:Medial Triangle", "text": "Let $\\triangle ABC$ be a triangle. The '''medial triangle''' of $\\triangle ABC$ is the triangle formed by joining the midpoint of each of the sides of $\\triangle ABC$."}
+{"_id": "27279", "title": "Definition:Euler Line", "text": "Let $\\triangle ABC$ be a triangle which is not equilateral. The '''Euler line''' of $\\triangle ABC$ is the straight line through the orthocenter, centroid and circumcenter of $\\triangle ABC$. :700px In the above, the line $OGK$ is the '''Euler line''' of $\\triangle ABC$."}
+{"_id": "27280", "title": "Definition:Tessellation", "text": "A '''tessellation''' is a division of the plane into plane geometric figures, referred to as tiles. The division is made in such a way that the tiles are fitted together with no gaps and no overlaps."}
+{"_id": "27281", "title": "Definition:Tile", "text": "A '''tile''' is a plane geometric figure which forms part of a tessellation. Category:Definitions/Tessellations fpbdcx8a60ckfnpjfhai3x6htecvlwa"}
+{"_id": "27282", "title": "Definition:Regular Tessellation", "text": "A '''regular tessellation''' is a tessellation such that: :Its tiles consist of congruent regular polygons :No vertex of one tile may lie against the side of another tile."}
+{"_id": "27283", "title": "Definition:Magic Square", "text": "A '''magic square''' is an arrangement of $n^2$ numbers into an $n \\times n$ square array such that: :the sum of the elements of each row :the sum of the elements in each column :the sum of the elements along each diagonal are the same. When the elements are not specified, it is conventional for them to be the first $n^2$ (strictly) positive integers."}
+{"_id": "27284", "title": "Definition:Magic Square/Order", "text": "An $n \\times n$ magic square is called an '''order $n$ magic square'''."}
+{"_id": "27286", "title": "Definition:Coloring/Why Colors?", "text": "It is clear that the nature of the actual elements of a coloring $C$ is irrelevant. They are traditionally referred to as '''colors''' because this subfield of graph theory arose from considerations of the coloring of the faces of planar graphs such that adjacent faces have different '''colors'''. This was the origin of the famous Four Color Theorem."}
+{"_id": "27287", "title": "Definition:Coloring/Vertex Coloring", "text": "A '''vertex $k$-coloring''' of a simple graph $G = \\left({V, E}\\right)$ is defined as an assignment of one element from a set $C$ of $k$ colors to each vertex in $V$. That is, a '''vertex $k$-coloring''' of the graph $G = \\left({V, E}\\right)$ is a mapping $c: V \\to \\left\\{{1, 2, \\ldots k}\\right\\}$. A graph with such a coloring is called a '''vertex-colored graph'''."}
+{"_id": "27289", "title": "Definition:Continued Square Root", "text": "A '''continued square root''' is an expression of the form: :$a_0 + \\sqrt {a_1 + \\sqrt {a_2 + \\sqrt {a_3 + \\sqrt {\\cdots} } } }$ where $\\left\\langle{a_n}\\right\\rangle$ is an arbitrary sequence of numbers."}
+{"_id": "27290", "title": "Definition:Perfect Power", "text": "A '''perfect power''' is an integer which can be expressed in the form: :$a^k$ where both $a$ and $k$ are integers with $a \\ge 1$ and $k \\ge 2$. === Sequence === {{:Definition:Perfect Power/Sequence}}"}
+{"_id": "27291", "title": "Definition:Perfect Power/Sequence", "text": "The sequence of perfect powers begins: :$1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, \\ldots$"}
+{"_id": "27292", "title": "Definition:Zu Chongzhi Fraction", "text": "The '''Zu Chongzhi fraction''' is an exceptionally accurate approximation to $\\pi$ (pi): :$\\pi \\approx \\dfrac {355} {113}$ whose decimal expansion is: :$\\pi \\approx 3 \\cdotp 14159 \\, 292$ {{OEIS|A068079}}"}
+{"_id": "27296", "title": "Definition:Squarely Irrational Number", "text": "A number is '''squarely irrational''' if both the number itself and its square is irrational. Thus, the set of the '''squarely irrational numbers''' is $\\left\\{{x \\in \\R \\setminus \\Q : x^2 \\in \\R \\setminus \\Q}\\right\\}$."}
+{"_id": "27297", "title": "Definition:Rationally Expressible Number", "text": "A number is '''rationally expressible''' if its square is rational. Thus, the set of the '''rationally expressible numbers''' is $\\set {x \\in \\R : x^2 \\in \\Q}$."}
+{"_id": "27298", "title": "Definition:Square Root/Historical Note", "text": "It is suggested by some sources that the symbol $\\surd$ (a stylised '''r''' for '''radix''') for the '''square root''' may have originated with {{AuthorRef|René Descartes}}, but there is evidence that it may have been around a lot earlier than that."}
+{"_id": "27301", "title": "Definition:Philosophical Element", "text": "The Pythagoreans postulated that the Universe was composed of $4$ '''elements''': :Earth :Water :Air :Fire. Subsequently, the alchemical tradition proposed the existence of a fifth '''element''': quintessence, or aether. === Earth === {{:Definition:Philosophical Element/Earth}} === Water === {{:Definition:Philosophical Element/Water}} === Air === {{:Definition:Philosophical Element/Air}} === Fire === {{:Definition:Philosophical Element/Fire}} === Quintessence === {{:Definition:Philosophical Element/Quintessence}}"}
+{"_id": "27302", "title": "Definition:Philosophical Element/Earth", "text": "'''Earth''' was one of the original four elements that were postulated by the Pythagoreans to compose the Universe. All the '''earthy''' matter of the Universe was supposed to collect at its natural place, which was the centre of the Universe, that is, Earth itself."}
+{"_id": "27303", "title": "Definition:Philosophical Element/Air", "text": "'''Air''' was one of the original four elements that were postulated by the Pythagoreans to compose the Universe. It had its natural place around the rim of the sphere of water."}
+{"_id": "27304", "title": "Definition:Philosophical Element/Fire", "text": "'''Fire''' was one of the original four elements that were postulated by the Pythagoreans to compose the Universe. It had its natural place outside the sphere of air."}
+{"_id": "27305", "title": "Definition:Philosophical Element/Water", "text": "'''Water''' was one of the original four elements that were postulated by the Pythagoreans to compose the Universe. It had its natural place about the rim of the surface of Earth."}
+{"_id": "27307", "title": "Definition:Linearly Irrational Number", "text": "A number is '''linearly irrational''' if both the number itself is irrational but its square is rational. Thus, the set of the '''linearly irrational numbers''' is $\\left\\{{x \\in \\R \\setminus \\Q : x^2 \\in \\Q}\\right\\}$."}
+{"_id": "27308", "title": "Definition:Exponential Order", "text": "Let $f: \\R \\to \\F$ be a function, where $\\F \\in \\set {\\R, \\C}$. Let $f$ be continuous on the real interval $\\hointr 0 \\to$, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\\hointr 0 \\to$. {{explain|Establish whether it is \"finite subinterval\" that is needed here, or what we have already defined as \"Definition:Finite Subdivision\".}} Then $f$ is said to be of '''exponential order''', denoted $f \\in \\mathcal E$, {{iff}} it is of exponential order $a$ for some $a > 0$."}
+{"_id": "27313", "title": "Definition:Philosophical Element/Quintessence", "text": "'''Quintessence''' was the fifth element postulated by the Pythagoreans, and was supposed to be the material out of which the heavens were made. It was postuiated in order to explain how the heavenly bodies were in constant motion, and never came to rest like everything else that could be seen in the universe."}
+{"_id": "27314", "title": "Definition:Pythagoreans/Quadrivium", "text": "The '''quadrivium''' was the medieval name of the required course of study of the Pythagoreans, which had been adopted by the educational establishments in Europe. The required bodies of knowledge were divided into '''discrete''' and '''continuous''': Thus: :'''Arithmetic''': study of the absolute discrete :'''Music''': study of the relative discrete :'''Geometry''': study of the '''stable''' continuous :'''Astronomy''': study of the '''moving''' continuous."}
+{"_id": "27316", "title": "Definition:Pythagoreans/Trivium", "text": "The '''trivium''' was the medieval name of the supplementary course of study of the Pythagoreans, adopted by the educational establishments in Europe. These supplementary bodies of knowledge were: :'''Grammar''' :'''Rhetoric''' :'''Logic'''."}
+{"_id": "27317", "title": "Definition:Pythagoreans/Quadrivium/Linguistic Note", "text": "The word '''quadrivium''' is a Latin word meaning '''the four ways''', or '''place where four roads meet''' (that is, literally: '''crossroads''')."}
+{"_id": "27318", "title": "Definition:Pythagoreans/Trivium/Linguistic Note", "text": "The word '''trivium''' is a Latin word meaning '''the three ways''', or '''place where three roads meet'''. As the '''trivium''' was a preliminary course of study to the quadrivium, and thereby simpler, from this root the word '''trivial''' evolved, which has a meaning in accordance with '''insignificant''', '''self-evident''' and '''commonplace''', and the like."}
+{"_id": "27319", "title": "Definition:Dual Polyhedron", "text": "Let $P$ be a polyhedron. The '''dual polyhedron''' $D$ of $P$ is the polyhedron which can be constructed as follows: :$(1): \\quad$ The vertices of $D$ are the centroids of the faces of $P$. :$(2): \\quad$ For each edge of $P$ which is adjacent to two faces $F_1$ and $F_2$ of $P$, an edge of $D$ is constructed which is adjacent to the vertices of $D$ forming the centroids of $F_1$ and $F_2$."}
+{"_id": "27320", "title": "Definition:Brown Numbers", "text": "'''Brown numbers''' are pairs of integers $\\left({m, n}\\right)$ such that: :$n! + 1 = m^2$ There are only $3$ known such pairs: :$\\left({5, 4}\\right)$ :$\\left({11, 5}\\right)$ :$\\left({71, 7}\\right)$"}
+{"_id": "27322", "title": "Definition:Feigenbaum Constants", "text": "The '''Feigenbaum constants''' are a pair of real constants which arise in bifurcation theory. === First Feigenbaum Constant === {{:Definition:Feigenbaum Constants/First}} === Second Feigenbaum Constant === {{:Definition:Feigenbaum Constants/Second}} {{NamedforDef|Mitchell Jay Feigenbaum|cat = Feigenbaum}} Category:Definitions/Bifurcation Theory 0as984mc1pfs2fgfhoyhgnds31c9x4p"}
+{"_id": "27324", "title": "Definition:Feigenbaum Constants/First", "text": "The '''first Feigenbaum constant''' is the limiting ratio of each bifurcation interval to the next between every period doubling of a one-parameter mapping: :$x_{i + 1} = f \\left({x_i}\\right)$ where $f \\left({x}\\right)$ is a function parameterized by the bifurcation parameter $a$. It is given by the limit: :$\\displaystyle \\delta = \\lim_{n \\mathop \\to \\infty} \\dfrac{a_ {n - 1} - a_{n - 2} } {a_n - a_{n - 1} } = 4 \\cdotp 66920 \\, 16091 \\, 02990 \\, 67185 \\, 32038 \\, 20466 \\, 20161 \\, 72 \\ldots$ where $a_n$ are discrete values of $a$ at the $n$th period doubling. Its precise value appears to be a topic of research, as various resources quote it differently from the above, including the {{AuthorRef|David Wells}} $1997$ source work {{BookLink|Curious and Interesting Numbers|David Wells|ed = 2nd|edpage = Second Edition}}."}
+{"_id": "27325", "title": "Definition:Feigenbaum Constants/Second", "text": "The '''secondFeigenbaum constant''' $\\alpha$ is the ratio between the width of a tine and the width of one of its two subtines (except the tine closest to the fold). A negative sign is applied to $\\alpha$ when the ratio between the lower subtine and the width of the tine is measured. Its approximate value is given by: :$\\alpha \\approx 2 \\cdotp 50290 \\, 78750 \\, 95892 \\, 82228 \\, 39028 \\, 73218 \\, 21578 \\cdots$"}
+{"_id": "27326", "title": "Definition:Pythagorean Triangle", "text": "A '''Pythagorean triangle''' is a right triangle whose sides all have lengths which are integers."}
+{"_id": "27327", "title": "Definition:Automorphic Number", "text": "An '''automorphic number''' is a positive integer all of whose powers end in that number."}
+{"_id": "27329", "title": "Definition:Absolutely Convergent Integral", "text": "Let $I = \\displaystyle \\int_S \\map f x \\rd x$ be an improper integral, where $S$ is an unbounded closed interval. $I$ is '''absolutely convergent''' {{iff}} $\\displaystyle \\int_S \\size {\\map f x} \\rd x$ converges."}
+{"_id": "27332", "title": "Definition:Untouchable Number", "text": "An '''untouchable number''' is a (strictly) positive integer that is not the aliquot sum of any integer."}
+{"_id": "27336", "title": "Definition:Polytope", "text": "A '''polytope''' is an $n$-dimensional generalization of a polygon and polyhedron, for $n \\in \\Z_{>0}$."}
+{"_id": "27337", "title": "Definition:Simplex", "text": "A '''simplex''' is an $n$-dimensional generalization of a triangle and tetrahedron, for $n \\in \\Z_{>0}$. A $k$-simplex is a $k$-dimensional polytope which is the convex hull of its $k + 1$ vertices."}
+{"_id": "27338", "title": "Definition:Pentatope", "text": "A '''pentatope''' is a $4$-simplex: :400px"}
+{"_id": "27340", "title": "Definition:Triangle (Geometry)/Right-Angled/Legs", "text": "In a right-angled triangle, the two sides which are not the hypotenuse are referred to as its '''legs'''."}
+{"_id": "27341", "title": "Definition:Fibonacci Number/Negative", "text": "The definition of '''Fibonacci numbers''' for negative integers is an extension of the definition for positive integers: :$F_n = \\begin{cases} 0 & : n = 0 \\\\ 1 & : n = 1 \\\\ F_{n + 2} - F_{n + 1} & : n < 0 \\end{cases}$ for all $n \\in \\Z$."}
+{"_id": "27343", "title": "Definition:Extension of Sequence", "text": "As a sequence is, by definition, also a mapping, the definition of an '''extension of a sequence''' is the same as that for an extension of a mapping: Let: : $\\left \\langle {a_k} \\right \\rangle_{k \\mathop \\in A}$ be a sequence on $A$, where $A \\subseteq \\N$. : $\\left \\langle {b_k} \\right \\rangle_{k \\mathop \\in B}$ be a sequence on $B$, where $B \\subseteq \\N$. : $A \\subseteq B$ : $\\forall k \\in A: b_k = a_k$. Then $\\left \\langle {b_k} \\right \\rangle_{k \\mathop \\in B}$ '''extends''' or '''is an extension of''' $\\left \\langle {a_k} \\right \\rangle_{k \\mathop \\in A}$."}
+{"_id": "27344", "title": "Definition:Extension of Sequence/Negative Integers", "text": "A sequence on $\\N$ can be extended to the negative integers. Let $\\left \\langle {a_k} \\right \\rangle_{k \\mathop \\in \\N}$ and $\\left \\langle {b_k} \\right \\rangle_{k \\mathop \\in \\N}$ be sequences on $\\N$. Let $a_0 = b_0$. Let $c_k$ be defined as: :$\\forall k \\in \\Z: c_k = \\begin{cases} a_k & : k \\ge 0 \\\\ b_{-k} : & k \\le 0 \\end{cases}$ Then $\\left \\langle {c_k} \\right \\rangle_{k \\mathop \\in \\Z}$ '''extends''' (or '''is an extension of''') $\\left \\langle {a_k} \\right \\rangle_{k \\mathop \\in \\N}$ to the negative integers."}
+{"_id": "27345", "title": "Definition:Ansatz", "text": "An '''Ansatz''' is a guessed solution, or general form of a solution, to a mathematical problem. A mathematician making the '''Ansatz''' will then attempt to prove that the '''Ansatz''' is indeed a solution to the original problem. If he or she succeeds, the '''Ansatz''' is justified retroactively."}
+{"_id": "27346", "title": "Definition:Finite Suprema Set", "text": "Let $P = \\left({S, \\preceq}\\right)$ be an ordered set. Let $X$ be a subset of $S$. Then '''finite suprma set''' of $X$, denoted $\\operatorname{finsups}\\left({X}\\right)$, is defined by :$\\left\\{ {\\sup A: A \\in \\mathit{Fin}\\left({X}\\right) \\land A}\\right.$ admits a supremum$\\left.{}\\right\\}$ where $\\mathit{Fin}\\left({X}\\right)$ denotes the set of all finite subsets of $X$."}
+{"_id": "27348", "title": "Definition:Contour Integral/Complex/Closed", "text": "Let $C$ be a closed contour in $\\C$. Then the symbol $\\displaystyle \\oint$ is used for the '''contour integral''' on $C$. The definition remains the same: :$\\displaystyle \\oint_C f \\left({z}\\right) \\rd z := \\sum_{i \\mathop = 1}^n \\int_{a_i}^{b_i} f \\left({\\gamma_i \\left({t}\\right) }\\right) \\gamma_i' \\left({t}\\right) \\rd t$"}
+{"_id": "27350", "title": "Definition:Goldbach Decomposition", "text": "Let $n \\in \\Z_{>0}$ be a positive integer such that $n$ is even. A '''Goldbach decomposition''' for $n$ is a pair of primes $p_1, p_2$ such that: :$p_1 + p_2 = n$"}
+{"_id": "27351", "title": "Definition:Cauchy Principal Value", "text": "The '''Cauchy principal value''' is an extension of the concept of an improper integral when the latter might not exist. === Real Integral === {{:Definition:Cauchy Principal Value/Real Integral}} === Complex Integral === {{:Definition:Cauchy Principal Value/Complex Integral}} === Contour Integral === {{:Definition:Cauchy Principal Value/Contour Integral}}"}
+{"_id": "27353", "title": "Definition:Tesseract", "text": "A '''tesseract''' is the $4$-dimensional regular polytope corresponding to the cube: :500px {{stub}}"}
+{"_id": "27355", "title": "Definition:Steiner Tree", "text": "Let $G$ be a configuration of points in space. A '''Steiner tree''' is a network of lines joining the points of $G$ such that the total length of those lines is a minimum. {{NamedforDef|Jakob Steiner|cat = Steiner}}"}
+{"_id": "27356", "title": "Definition:Fermat Quotient", "text": "Let $a$ be a positive integer. Let $p$ be an odd prime. The '''Fermat quotient of $a$ with respect to $p$''' is defined as: :$\\map {q_p} a = \\dfrac {a^{p - 1} - 1} p$"}
+{"_id": "27357", "title": "Definition:Rational Diagonal", "text": "The number $7$ (seven) was referred to by the ancient Greeks as the '''rational diagonal''' of the $5 \\times 5$ square. This was on account of the fact that: :$5^2 + 5^2 = 50 \\approx 7^2 = 49$"}
+{"_id": "27358", "title": "Definition:Dissection", "text": "A '''dissection''' is a partition of a geometric figure into two or more components."}
+{"_id": "27360", "title": "Definition:Frieze", "text": "A '''frieze''' is a plane geometrical figure which has translational symmetry in one line only."}
+{"_id": "27361", "title": "Definition:Pseudoprime (Order Theory)", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be an up-complete lattice. Let $p \\in S$. Then $p$ is '''pseudoprime''' {{iff}} :there exists a prime ideal $P$ in $L$: $p = \\sup P$"}
+{"_id": "27363", "title": "Definition:Cyclotomic Polynomial", "text": "Let $n \\ge 1$ be a natural number. The '''$n$th cyclotomic polynomial''' is the polynomial :$\\displaystyle \\Phi_n \\paren x = \\prod_\\zeta \\paren {x - \\zeta}$ where the product runs over all primitive complex $n$th roots of unity, that is, those whose order is $n$."}
+{"_id": "27364", "title": "Definition:Fano Plane", "text": "The '''Fano plane''' is the finite projective plane of order $2$. It can be denoted: :$\\operatorname {PG} \\left({2, 2}\\right)$ where: :$\\operatorname {PG}$ stands for '''projective geometry''' :the first parameter $2$ specifies the dimension :the second parameter $2$ specifies the order. $\\operatorname {PG} \\left({2, 2}\\right)$ has the following properties: : It has $7$ points : It has $7$ lines : There are $3$ points on each line : There are $3$ lines through each point. :400px"}
+{"_id": "27365", "title": "Definition:Separable Polynomial", "text": "=== Definition 1 === {{:Definition:Separable Polynomial/Definition 1}} === Definition 2 === {{:Definition:Separable Polynomial/Definition 2}} === Definition 3 === {{:Definition:Separable Polynomial/Definition 3}}"}
+{"_id": "27366", "title": "Definition:Multiplicity (Polynomial)", "text": "Let $R$ be a commutative ring with unity. Let $P \\in R \\left[{X}\\right]$ be a nonzero polynomial. Let $a \\in R$ be a root of $P$. The '''multiplicity''' of $a$ in $P$ is the largest positive integer $n$ such that $\\left({X - a}\\right)^n$ divides $P \\left({X}\\right)$ in $R \\left[{X}\\right]$. {{refactor|Extract the following into a separate page, transcluded (perhaps an \"examples\" page)}} A '''double root''' is a root of multiplicity at least $2$. Category:Definitions/Polynomial Theory 744jj09nfz55cql8hm6ls1w0j0ryn7o"}
+{"_id": "27367", "title": "Definition:Knot Product", "text": "The '''product''' of two knots is informally defined as what you get when you tie those two knots together on the same string. {{stub|Yes I know this is sketchy and inadequate -- a lot more background is needed before this can be fleshed out and made rigorous.}}"}
+{"_id": "27368", "title": "Definition:Prime Knot", "text": "A '''prime knot''' is a knot which is not the product of two or more other knots."}
+{"_id": "27369", "title": "Definition:Monoid Ring", "text": "Let $R$ be a ring with unity. Let $\\left({G, *}\\right)$ be a monoid. Let $R^{\\left({G}\\right)}$ be the free $R$-module on $G$. Let $\\left\\{ {e_g: g \\in G}\\right\\}$ be its canonical basis. {{explain|Before the above notation can be properly understood, the precise nature of the canonical basis needs to be expanded so as to make the operations completely explicit.}} By Multilinear Mapping from Free Modules is Determined by Bases, there exists a unique bilinear map: :$\\circ: R^{\\left({G}\\right)} \\times R^{\\left({G}\\right)} \\to R^{\\left({G}\\right)}$ which satisfies: :$e_g \\circ e_h = e_{g \\mathop * h}$ Then $R \\left[{G}\\right] = \\left({R^{\\left({G}\\right)}, +, \\circ}\\right)$ is called the '''monoid ring of $G$ over $R$'''."}
+{"_id": "27370", "title": "Definition:Free Module on Set", "text": "Let $R$ be a ring. {{explain|For this concept to be more easily understood, it is suggested that the ring be defined using its full specification, i.e. complete with operators, $\\left({R, +, \\circ}\\right)$ and so on, and similarly that a symbol be used to make the module scalar product equally explicit.}} Let $I$ be an indexing set. The '''free $R$-module on $I$''' is the direct sum of $R$ as a module over itself: :$\\displaystyle R^{\\left({I}\\right)} := \\bigoplus_{i \\mathop \\in I} R$ of the family $I \\to \\{R\\}$ to the singleton $\\{R\\}$. === Canonical Basis === {{Definition:Free Module on Set/Canonical Basis}} === Canonical Mapping === {{Definition:Free Module on Set/Canonical Mapping}}"}
+{"_id": "27371", "title": "Definition:Multiplicative Relation", "text": "Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Let $\\mathcal R$ be a relation on $S$. Then $\\mathcal R$ is '''multiplicative (relation)''' {{iff}} :$\\forall a, x, y \\in S: \\left({\\left({a, x}\\right), \\left({a, y}\\right) \\in \\mathcal R \\implies \\left({a, x \\wedge y}\\right) \\in \\mathcal R}\\right)$"}
+{"_id": "27372", "title": "Definition:Polynomial Evaluation Homomorphism", "text": "Let $R, S$ be commutative rings with unity. Let $\\kappa : R \\to S$ be a unital ring homomorphism. === Single Indeterminate === {{:Definition:Polynomial Evaluation Homomorphism/Single Indeterminate}} === Multiple Indeterminates === {{:Definition:Polynomial Evaluation Homomorphism/Multiple Indeterminates}}"}
+{"_id": "27374", "title": "Definition:Free Module on Set/Canonical Basis", "text": "Let $R$ be a ring with unity."}
+{"_id": "27376", "title": "Definition:Homomorphism Ring between Modules", "text": "Let $R$ be a ring. Let $M, N$ be $R$-modules. The '''homomorphism ring between $M$ and $N$''' is defined as the set of all $R$-module homomorphisms from $M$ to $N$, denoted: :$\\map {\\operatorname {Hom}_R} {M, N}$ or just: :$\\map {\\operatorname {Hom} } {M, N}$ if the specific ring $R$ in question is clear from the context."}
+{"_id": "27377", "title": "Definition:Group Ring", "text": "Let $R$ be a ring. Let $\\struct {G, +}$ be a group. The '''group ring of $G$ over $R$''' is the monoid ring of $G$ over $R$."}
+{"_id": "27378", "title": "Definition:Power (Algebra)/Complex Number/Principal Branch", "text": "The principal branch of a complex number raised to a complex power is defined as: :$z^k = e^{k \\Ln z}$ where $\\Ln z$ is the principal branch of the natural logarithm."}
+{"_id": "27379", "title": "Definition:Power (Algebra)/Complex Number/Principal Branch/Positive Real Base", "text": "Let $t > 0$ be a real number and let $k$ be a complex number. The principal branch of a positive real number raised to a complex power is defined as: :$t^k = e^{k \\ln t}$ where $\\ln$ is the natural logarithm of a positive real number."}
+{"_id": "27380", "title": "Definition:Z-Module Associated with Abelian Group", "text": "Let $\\struct {G, *}$ be an abelian group with identity $e$. Let $\\struct {\\Z, +, \\times}$ be the ring of integers. === Definition 1 === {{:Definition:Z-Module Associated with Abelian Group/Definition 1}} === Definition 2 === {{:Definition:Z-Module Associated with Abelian Group/Definition 2}}"}
+{"_id": "27381", "title": "Definition:Octagonal Number", "text": "'''Octagonal numbers''' are those denumerating a collection of objects which can be arranged in the form of a regular octagon."}
+{"_id": "27382", "title": "Definition:Octagonal Number/Definition 1", "text": ":$O_n = \\begin{cases} 0 & : n = 0 \\\\ O_{n - 1} + 6 n - 5 & : n > 0 \\end{cases}$"}
+{"_id": "27383", "title": "Definition:Octagonal Number/Definition 2", "text": ":$\\displaystyle O_n = \\sum_{i \\mathop = 1}^n \\left({6 i - 5}\\right) = 1 + 7 + \\cdots + \\left({6 \\left({n - 1}\\right) - 5}\\right) + \\left({6 n - 5}\\right)$"}
+{"_id": "27384", "title": "Definition:Octagonal Number/Definition 3", "text": ":$\\forall n \\in \\N: O_n = P \\left({8, n}\\right) = \\begin{cases} 0 & : n = 0 \\\\ P \\left({8, n - 1}\\right) + 6 \\left({n - 1}\\right) + 1 & : n > 0 \\end{cases}$ where $P \\left({k, n}\\right)$ denotes the $k$-gonal numbers."}
+{"_id": "27386", "title": "Definition:Octagonal Number/Sequence", "text": "The sequence of octagonal numbers, for $n \\in \\Z_{\\ge 0}$, begins: :$0, 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, \\ldots$"}
+{"_id": "27387", "title": "Definition:Ring of Integers", "text": "The '''ring of integers''' $\\struct {\\Z, +, \\times}$ is the set of integers under the two operations of integer addition and integer multiplication."}
+{"_id": "27388", "title": "Definition:Perfect Magic Cube", "text": "A '''perfect magic cube''' is an arrangement of the first $n^3$ (strictly) positive integers into an $n \\times n \\times n$ cubic array such that: :the sum of the entries in each row in each of the $3$ dimensions :the sum of the entries along the main diagonal of each plane :the sum of the entries along the space diagonals are the same."}
+{"_id": "27389", "title": "Definition:Magic Cube/Order", "text": "An $n \\times n \\times n$ magic cube is called an '''order $n$ magic cube'''."}
+{"_id": "27390", "title": "Definition:Magic Cube", "text": "A '''magic cube''' is an arrangement of the first $n^3$ (strictly) positive integers into an $n \\times n \\times n$ cubic array such that: :the sum of the entries in each row in each of the $3$ dimensions :the sum of the entries along the space diagonals are the same. It is not guaranteed that the entries along each of the main diagonals of each plane also sum to the same constant."}
+{"_id": "27393", "title": "Definition:Deltahedron", "text": "A '''deltahedron''' is a polyhedron each of whose faces are equilateral triangles."}
+{"_id": "27397", "title": "Definition:Bipyramid", "text": "A '''bipyramid''' is a polyhedron formed by taking two pyramids with congruent bases and placing those bases together: :400px"}
+{"_id": "27398", "title": "Definition:Triangular Bipyramid", "text": "A '''triangular bipyramid''' is a polyhedron formed by taking two regular tetrahedra and placing their bases together: :300px The '''triangular bipyramid''' is an example of a deltahedron."}
+{"_id": "27399", "title": "Definition:Pentagonal Bipyramid", "text": "A '''pentagonal bipyramid''' is a polyhedron formed by taking two pyramids whose bases are regular pentagons and whose other faces are equilateral triangles, and placing their bases together: :300px The '''pentagonal bipyramid''' is an example of a deltahedron."}
+{"_id": "27400", "title": "Definition:Lucky Number", "text": "Start with the list of (strictly) positive integers: :$1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, \\ldots$ Remove every $2$nd number from $2$ onwards (that is, all the even integers): :$1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, \\ldots$ The $2$nd term is $3$. Starting from the $3$rd element in this list (which is $5$), remove every $3$rd element from what is left: :$1, 3, 7, 9, 13, 15, 19, 21, 25, \\ldots$ The $3$rd number is now $7$. Starting from the $7$th element in this list (which is $19$), remove every $7$th element from what is left: :$1, 3, 7, 9, 13, 15, 21, 25, \\ldots$ The numbers remaining are the '''lucky numbers'''."}
+{"_id": "27401", "title": "Definition:Lucky Number/Sequence", "text": "The sequence of lucky numbers begins: :$1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, \\ldots$"}
+{"_id": "27402", "title": "Definition:Kaprekar Number", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Suppose that $n^2$, when expressed in number base $b$, can be split into two parts that add up to $n$. Then $n$ is a '''Kaprekar number''' for base $b$."}
+{"_id": "27403", "title": "Definition:Kaprekar Number/Sequence", "text": "The sequence of Kaprekar numbers begins: :$1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, \\ldots$"}
+{"_id": "27404", "title": "Definition:Subfactorial", "text": "Let $n \\in \\Z_{\\ge0}$ be a (strictly) positive integer. The '''subfactorial''' of $n$ is defined and denoted as: {{begin-eqn}} {{eqn | l = !n       | o = :=       | r = n! \\sum_{k \\mathop = 0}^n \\frac {\\left({-1}\\right)^k} {k!}       | c =  }} {{eqn | r = n! \\left({1 - \\dfrac 1 {1!} + \\dfrac 1 {2!} - \\dfrac 1 {3!} + \\cdots + \\dfrac {\\left({-1}\\right)^n} {n!} }\\right)       | c =  }} {{end-eqn}} It arises as the number of derangements of $n$ distinct objects."}
+{"_id": "27406", "title": "Definition:Compact Subset of Lattice", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. '''Compact subset''' of $S$, denoted $K\\left({L}\\right)$, equals to the set of all compact elements of $S$."}
+{"_id": "27411", "title": "Definition:Compact Closure", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an ordered set. Let $x \\in S$. Then '''compact closure''' of $x$, denoted $x^{\\mathrm{compact}}$, is defined by :$x^{\\mathrm{compact}} := \\left\\{ {y \\in S: y \\preceq x \\land y}\\right.$ is compact$\\left.{}\\right\\}$"}
+{"_id": "27412", "title": "Definition:Axiom of K-Approximation", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. '''Axiom of K-approximation''' says  :$\\forall x \\in S: x = \\sup \\left({x^{\\mathrm{compact} } }\\right)$ where $x^{\\mathrm{compact} }$ denotes the compact closure of $x$."}
+{"_id": "27413", "title": "Definition:Algebraic Ordered Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Then $\\left({S, \\preceq}\\right)$ is '''algebraic''' {{iff}} :(for all elements $x$ of $S$: $x^{\\mathrm{compact} }$ is directed) and :$\\left({S, \\preceq}\\right)$ is up-complete and satisfies axiom of K-approximation where $x^{\\mathrm{compact} }$ denotes the compact closure of $x$."}
+{"_id": "27414", "title": "Definition:Asymmetric Relation/Definition 1", "text": "$\\RR$ is '''asymmetric''' {{iff}}: :$\\tuple {x, y} \\in \\RR \\implies \\tuple {y, x} \\notin \\RR$"}
+{"_id": "27415", "title": "Definition:Asymmetric Relation/Definition 2", "text": "$\\RR$ is '''asymmetric''' {{iff}} it and its inverse are disjoint: :$\\RR \\cap \\RR^{-1} = \\O$"}
+{"_id": "27417", "title": "Definition:Idempotence/Relation", "text": "Let $\\RR \\subseteq S \\times S$ be a relation on $S$. Then $\\RR$ is '''idempotent''' {{iff}}: :$\\RR \\circ \\RR = \\RR$ where $\\circ$ denotes composition of relations."}
+{"_id": "27418", "title": "Definition:Idempotence/Algebraic Structure", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure. Let $\\circ$ be an idempotent operation on $S$. Then $\\left({S, \\circ}\\right)$ is an '''idempotent algebraic structure'''."}
+{"_id": "27419", "title": "Definition:Meet Closed", "text": "Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Let $X$ be a subset of $S$. Then $X$ is '''meet closed''' {{iff}} :$\\forall x, y \\in X: x \\wedge y \\in X$"}
+{"_id": "27420", "title": "Definition:Arithmetic Ordered Set", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. $L$ is '''arithmetic''' {{iff}} :$L$ is algebraic and $K\\left({L}\\right)$ is meet closed where $K\\left({L}\\right)$ denotes the compact subset of $L$."}
+{"_id": "27421", "title": "Definition:Endomorphism Ring of Abelian Group", "text": "Let $\\struct {G, +}$ be an abelian group. Let $\\map {\\mathrm {End} } G$ be the set of endomorphisms of $G$. The '''endomorphism ring of $G$''' is the algebraic structure: :$\\struct {\\map {\\mathrm {End} } G, +, \\circ}$ where: :$\\circ$ denotes composition :$+$ denotes pointwise addition."}
+{"_id": "27422", "title": "Definition:Support of Permutation", "text": "Let $S$ be a set. Let $f$ be a permutation on $S$. The '''support''' of $f$ is the subset of moved elements: :$\\operatorname {supp} \\left({f}\\right) = \\left\\{ {x \\in S: f \\left({x}\\right) \\ne  x}\\right\\}$"}
+{"_id": "27423", "title": "Definition:Support of Element of Direct Product", "text": "Let $\\left({S_i, \\circ_i}\\right)_{i \\mathop \\in I}$ be a family of algebraic structures with identity. Let $\\displaystyle S = \\prod_{i \\mathop \\in I} S_i$ be their direct product. Let $e_i$ be an identity of $S_i$ for all $i \\in I$. Let $m = \\left({m_i}\\right)_{i \\mathop \\in I} \\in S$. The '''support''' of $m$ is defined as: :$\\operatorname {supp} \\left\\{ {i \\in I: m_i \\ne e_i}\\right\\}$ === Finite Support === The element is said to have '''finite support''' {{iff}} its support is a finite set."}
+{"_id": "27424", "title": "Definition:Liouville Function", "text": "Let $n \\in \\Z_{>0}$, that is, a strictly positive integer. The '''Liouville function''' is the function $\\lambda: \\Z_{>0} \\to \\Z_{>0}$ defined as: :$\\lambda \\left({n}\\right) = \\left({-1}\\right)^{\\Omega \\left({n}\\right)}$ where $\\Omega \\left({n}\\right)$ is the number of prime factors of $n$, counted with multiplicity."}
+{"_id": "27425", "title": "Definition:Module of All Mappings", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {M, +_M, \\circ}_R$ be an $R$-module. Let $S$ be a set. Let $M^S$ be the set of all mappings from $S$ to $M$. Let: :$+$ be the operation induced on $M^S$ by $+_M$ :$\\forall \\lambda \\in R: \\forall f \\in M^S: \\forall x \\in S: \\map {\\paren {\\lambda \\circ f} } x = \\lambda \\circ \\paren {\\map f x}$ Then $\\struct {M^S, +, \\circ}_R$ is the '''module of all mappings from $S$ to $M$'''."}
+{"_id": "27426", "title": "Definition:Module Direct Product/Finite Case", "text": "Let $R$ be a ring. Let $\\left({M_1, +_1, \\circ_1}\\right)_R, \\left({M_2, +_2, \\circ_2}\\right)_R, \\ldots, \\left({M_n, +_n, \\circ_n}\\right)_R$ be $R$-modules. Let: :$\\displaystyle M = \\prod_{k \\mathop = 1}^n M_k$ be the cartesian product of $M_1$ to $M_n$. Let: : $+$ be the operation induced on $M$ by the operations $+_1, +_2, \\ldots, +_n$ on $M_1, M_2, \\ldots, M_n$ : $\\circ$ be defined as $\\lambda \\circ \\left({x_1, x_2, \\ldots, x_n}\\right) = \\left({\\lambda \\circ_1 x_1, \\lambda \\circ_2 x_2, \\ldots, \\lambda \\circ_n x_n}\\right)$ In Finite Direct Product of Modules is Module, it is shown that $\\left\\langle{M, +, \\circ}\\right\\rangle$ is an $R$-module. The module $\\left({M, +, \\circ}\\right)_R$ is called the '''(external) direct product''' of $M_1$ to $M_n$."}
+{"_id": "27427", "title": "Definition:Module Direct Product/General Case", "text": "Let $R$ be a ring. Let $\\left\\{ \\left\\langle M_i,+_i,\\circ_i\\right\\rangle\\right\\}_{i \\in I}$ be a family of $R$-modules. {{explain|Expand the notation of the module to specify the operations of the ring.}} Let: :$M = \\displaystyle \\prod_{i \\mathop \\in I} M_i$ be the cartesian product of these modules. The operation $+$ induced on $M$ by $(+_i)_{i\\in I}$ is the operation defined by: :$\\left\\langle{a_i}\\right\\rangle_{i \\mathop \\in I} + \\left\\langle{b_i}\\right\\rangle_{i \\mathop \\in I} = \\left\\langle{a_i +_i b_i}\\right\\rangle_{i \\mathop \\in I}$ That is, the additive group of the module $M$ is the direct product of the groups $\\left\\{ \\left(M_i,+_i\\right)\\right\\}_{i \\in I}$. The $R$-action $\\circ$ induced on $M$ by $(\\circ_i)_{i\\in I}$ is the operation defined by: :$r \\circ \\left\\langle{m_i}\\right\\rangle_{i \\mathop \\in I} = \\left\\langle{r \\circ_i m_i}\\right\\rangle_{i \\mathop \\in I}$ In Direct Product of Modules is Module, it is shown that $\\left\\langle{M, +, \\circ}\\right\\rangle$ is an $R$-module. The module $\\left\\langle{M, +, \\circ}\\right\\rangle$  is called the '''(external) direct product''' of $\\left\\{ \\left\\langle M_i,+_i,\\circ_i\\right\\rangle\\right\\}_{i \\in I}$."}
+{"_id": "27428", "title": "Definition:Direct Product of Vector Spaces/Finite Case", "text": "Let $K$ be a field. Let $V_1, V_2, \\ldots, V_n$ be $K$-vector spaces. Let: :$\\displaystyle V = \\prod_{k \\mathop = 1}^n V_k$ be their cartesian product. Let: :$+$ be the operation induced on $V$ by the operations $+_1, +_2, \\ldots, +_n$ on $V_1, V_2, \\ldots, V_n$ :$\\circ$ be defined as $\\lambda \\circ \\tuple {x_1, x_2, \\ldots, x_n} := \\tuple {\\lambda \\circ x_1, \\lambda \\circ x_2, \\ldots, \\lambda \\circ x_n}$ $\\struct {V, +, \\circ}_K$ is called the '''direct product of $V_1, \\ldots, V_n$'''."}
+{"_id": "27429", "title": "Definition:Direct Product of Vector Spaces", "text": "Let $K$ be a field. Let $V, W$ be $K$-vector spaces. The '''direct product of $V$ and $W$''' is their module direct product. === Finite Case === {{:Definition:Direct Product of Vector Spaces/Finite Case}} === General Case === {{:Definition:Direct Product of Vector Spaces/General Case}}"}
+{"_id": "27430", "title": "Definition:Direct Product of Vector Spaces/General Case", "text": "Let $K$ be a field. Let $\\family {V_i, +_i, \\circ_i}_{i \\mathop \\in I}$ be a family of $K$-vector spaces. The '''(external) direct product''' of $\\family {V_i, +_i, \\circ_i}_{i \\mathop \\in I}$ is their module direct product."}
+{"_id": "27431", "title": "Definition:Module of Homomorphisms Between Modules", "text": "Let $R$ be a commutative ring. Let $M$ and $N$ be $R$-modules. Let $\\map {\\operatorname {Hom}_{R - \\operatorname {mod} } } {M, N}$ denote the set of $R$-module homomorphisms from $M$ to $N$. Let: :$+$ be the operation on $\\map {\\operatorname {Hom}_{R - \\operatorname {mod} } } {M, N}$ defined by $f + g: m \\mapsto \\map f m + \\map g m$ :$\\circ$ be defined as $\\lambda \\circ f: m \\mapsto \\lambda \\map f m$ Then $\\struct {\\map {\\operatorname {Hom}_{R - \\operatorname {mod} } } {M, N}, + , \\circ}$ is called the '''module of homomorphisms''' between $M$ and $N$."}
+{"_id": "27432", "title": "Definition:Dual Module", "text": "Let $R$ be a commutative ring. Let $M$ be an $R$-module. Then the '''dual module''' of $M$ is the module of homomorphisms $\\operatorname{Hom}_{R\\text{-mod}}(M,R)$. That is, the elements of the dual module are the linear forms on $M$. Category:Definitions/Abstract Algebra Category:Definitions/Module Theory ro54nx7d4h6z3no66m8rlngyeot2ku6"}
+{"_id": "27438", "title": "Definition:Order Complete Set/Definition 1", "text": "$\\struct {S, \\preceq}$ is '''order complete''' {{iff}}: :Each non-empty subset $H \\subseteq S$ which has an upper bound admits a supremum."}
+{"_id": "27439", "title": "Definition:Order Complete Set/Definition 2", "text": "$\\struct {S, \\preceq}$ is '''order complete''' {{iff}}: :Each non-empty subset $H \\subseteq S$ which has a lower bound admits an infimum."}
+{"_id": "27440", "title": "Definition:Order Complete Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set."}
+{"_id": "27443", "title": "Definition:Normal Subgroup/Also known as", "text": "It is usual to describe a '''normal subgroup of $G$''' as '''normal in $G$'''. Some sources refer to a '''normal subgroup''' as an '''invariant subgroup''' or a '''self-conjugate subgroup'''. This arises from Definition 6: {{:Definition:Normal Subgroup/Definition 6}} which is another way of stating that $N$ is '''normal''' {{iff}} $N$ stays the same under all inner automorphisms of $G$. Some sources use '''distinguished subgroup'''."}
+{"_id": "27444", "title": "Definition:Kepler-Poinsot Polyhedron", "text": "A '''Kepler-Poinsot polyhedron''' is any of the four regular star polyhedra."}
+{"_id": "27445", "title": "Definition:Altitude of Triangle/Foot", "text": "The point at which $h_a$ meets $BC$ is the '''foot of the altitude $h_a$'''."}
+{"_id": "27446", "title": "Definition:Feuerbach Circle", "text": "Let $\\triangle ABC$ be a triangle. The '''Feuerbach circle''' of $\\triangle ABC$ is the circle which passes through each of the $9$ points: : the feet of the altitudes of $\\triangle ABC$ : the midpoints of the sides of $\\triangle ABC$ : the midpoints of the lines from the vertices of $\\triangle ABC$ to the orthocenter of $\\triangle ABC$. :500px"}
+{"_id": "27448", "title": "Definition:Cosine Integral Function", "text": "The '''cosine integral function''' is the real function $\\Ci: \\R_{>0} \\to \\R$ defined as: :$\\map \\Ci x = \\displaystyle \\int_{t \\mathop = x}^{t \\mathop \\to +\\infty} \\frac {\\cos t} t \\rd t$"}
+{"_id": "27449", "title": "Definition:Magic Square/Magic Constant", "text": "The '''magic constant''' of a magic square is the number that each of the rows and columns adds up to."}
+{"_id": "27450", "title": "Definition:Field Extension/Complex", "text": "Let $\\left({F, +, \\times}\\right)$ be a subfield of $\\left({\\Bbb C, +, \\times}\\right)$, the field of complex numbers. Let $X_1, X_2, \\ldots, X_n$ be complex numbers, in or not in $F$. Then $F \\left({X_1, X_2, \\ldots, X_n}\\right)$ is the smallest field extension over $F$ containing $X_1, X_2, \\ldots, X_n$."}
+{"_id": "27451", "title": "Definition:Euclid Prime", "text": "A '''Euclid prime''' is a natural number which is both a Euclid number and a prime number."}
+{"_id": "27452", "title": "Definition:Lambert W Function/Principal Branch/Real Valued", "text": "The principal branch of the Lambert W function is the real function $W_0: \\hointr {-\\dfrac 1 e} \\to \\to \\hointr {-1} \\to$ such that: :$x = \\map {W_0} x e^{\\map {W_0} x}$"}
+{"_id": "27453", "title": "Definition:Lambert W Function", "text": "Let $e^z: \\C \\to \\C$ be the complex exponential function. Then $\\map W z$ is defined to be the multifunction that satisfies:  :$\\map W z e^{\\map W z} = z$"}
+{"_id": "27454", "title": "Definition:Omega Constant", "text": "The '''omega constant''' $\\Omega$ is the constant as the value of the principal branch of the Lambert W function at $1$: :$\\Omega \\, e^ \\Omega = 1$ That is, it is the root of the equation: :$x e^x = 1$ where $e$ denotes Euler's number."}
+{"_id": "27455", "title": "Definition:Euclid Prime/Sequence", "text": "The sequence of '''Euclid primes''' begins: :$2, 3, 7, 31, 211, 2311, 200 \\, 560 \\, 490 \\, 131, \\ldots$"}
+{"_id": "27457", "title": "Definition:Euclid Number/Sequence", "text": "The sequence of Euclid numbers begins as follows: {{begin-eqn}} {{eqn | ll= E_0       | l = = p_0\\# + 1       | r = 1 + 1       | rr= = 2       | c =  }} {{eqn | ll= E_1       | l = = p_1\\# + 1       | r = 2 + 1       | rr= = 3       | c =  }} {{eqn | ll= E_2       | l = = p_2\\# + 1       | r = 2 \\times 3 + 1       | rr= = 7       | c =  }} {{eqn | ll= E_3       | l = = p_3\\# + 1       | r = 2 \\times 3 \\times 5 + 1       | rr= = 31       | c =  }} {{eqn | ll= E_4       | l = = p_4\\# + 1       | r = 2 \\times 3 \\times 5 \\times 7 + 1       | rr= = 211       | c =  }} {{eqn | ll= E_5       | l = = p_5\\# + 1       | r = 2 \\times 3 \\times 5 \\times 7 \\times 11 + 1       | rr= = 2311       | c =  }} {{eqn | ll= E_6       | l = = p_6\\# + 1       | r = 2 \\times 3 \\times 5 \\times 7 \\times 11 \\times 13 + 1       | rr= = 30031       | c =  }} {{end-eqn}}"}
+{"_id": "27458", "title": "Definition:Compact Element", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $x \\in S$. Then $x$ is '''compact (element)''' {{iff}} $x \\ll x$ where $\\ll$ denotes the way below relation."}
+{"_id": "27459", "title": "Definition:Lambert W Function/Lower Branch", "text": "The '''lower branch'''  of the Lambert W function is the real function $W_{-1}: \\left[{-\\dfrac 1 e \\,.\\,.\\, 0}\\right) \\to \\left({\\gets \\,.\\,.\\, -1}\\right]$ such that: :$x = W_{-1} \\left({x}\\right) e^{W_{-1} \\left({x}\\right)}$"}
+{"_id": "27460", "title": "Definition:Foiaș Constant", "text": "=== First Foiaș Constant === {{:Definition:Foiaș Constant/First}} === Second Foiaș Constant === {{:Definition:Foiaș Constant/Second}}"}
+{"_id": "27461", "title": "Definition:Foiaș Constant/First/Decimal Expansion", "text": "The decimal expansion of the first Foiaș Constant starts: :$x_{\\infty} = 2 \\cdotp 29316 \\, 62874 \\, 11861 \\, 03150 \\, 80282 \\, 91250 \\, 80586 \\, 43722 \\, 57290 \\, 32712 \\, 12485 \\, 37 \\ldots$"}
+{"_id": "27462", "title": "Definition:Foiaș Constant/First", "text": "Let: :$x_{n+1} = \\left({1 + \\dfrac 1 {x_n} }\\right)^{x_n}$ for $n = 1, 2, 3, \\ldots$ The '''first Foiaș constant''' is the limit of $x_n$ as $n \\to \\infty$."}
+{"_id": "27463", "title": "Definition:Foiaș Constant/Second/Decimal Expansion", "text": "The decimal expansion of the second Foiaș Constant starts: :$\\alpha = 1 \\cdotp 18745 \\, 23511 \\, 26501 \\ldots$"}
+{"_id": "27464", "title": "Definition:Foiaș Constant/Second", "text": "Let $x_1 \\in \\R_{>0}$ be a (strictly) positive real number. Let: :$x_{n + 1} = \\left({1 + \\dfrac 1 {x_n} }\\right)^n$ for $n = 1, 2, 3, \\ldots$ The '''second Foiaș constant''' is defined as the unique real number $\\alpha$ such that if $x_1 = \\alpha$ then the sequence $\\left\\langle{x_{n + 1} }\\right\\rangle$ diverges to infinity."}
+{"_id": "27465", "title": "Definition:Pandigital Fraction", "text": "A '''pandigital fraction''' is a fraction where the digits $1$ to $9$ appear exactly once in the decimal expansions of the numerator and the denominator of the fraction combined."}
+{"_id": "27466", "title": "Definition:Tetrahedral Number", "text": "'''Tetrahedral numbers''' are those denumerating a collection of objects which can be arranged in the form of a regular tetrahedron."}
+{"_id": "27468", "title": "Definition:Polyhedral Number/Examples", "text": "=== Tetrahedral Number === {{:Definition:Tetrahedral Number}}"}
+{"_id": "27471", "title": "Definition:Noncototient", "text": "A '''noncototient''' is a positive integer $n$ such that: :$\\nexists m \\in \\Z_{>0}: m - \\phi \\left({m}\\right) = n$ where $\\phi \\left({m}\\right)$ denotes the Euler $\\phi$ function. That is, a '''noncototient''' is a positive integer which is not the cototient of any positive integer."}
+{"_id": "27472", "title": "Definition:Noncototient/Sequence", "text": "The sequence of '''noncototients''' begins: :$10, 26, 34, 50, 52, 58, 86, 100, 116, 122, 130, 134, 146, 154, 170, 172, 186, 202, \\ldots$"}
+{"_id": "27473", "title": "Definition:Cototient", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. The '''cototient''' of $n$ is defined as: :$n - \\phi \\left({n}\\right)$ where $\\phi \\left({n}\\right)$ denotes the Euler $\\phi$ function."}
+{"_id": "27475", "title": "Definition:Latin Square/Order", "text": "Let $\\mathbf L$ be an $n \\times n$ Latin square. The '''order''' of $\\mathbf L$ is $n$."}
+{"_id": "27478", "title": "Definition:Latin Square/Row", "text": "Let $\\mathbf L$ be a Latin square. The '''rows''' of $\\mathbf L$ are the lines of elements reading '''across''' the page."}
+{"_id": "27479", "title": "Definition:Latin Square/Column", "text": "Let $\\mathbf L$ be a Latin square. The '''columns''' of $\\mathbf L$ are the lines of elements reading '''down''' the page."}
+{"_id": "27483", "title": "Definition:Latin Square/Element", "text": "Let $\\mathbf L$ be a Latin square of order $n$. The individual $n \\times n$ symbols that go to form $\\mathbf L$ are known as the '''elements''' of $\\mathbf L$. The element at row $i$ and column $j$ is called '''element $\\left({i, j}\\right)$ of $\\mathbf L$''', and can be written $a_{i j}$, or $a_{i, j}$ if $i$ and $j$ are of more than one character. If the indices are still more complicated coefficients and further clarity is required, then the form $a \\left({i, j}\\right)$ can be used. Note that the first subscript determines the row, and the second the column, of the Latin square where the element is positioned."}
+{"_id": "27485", "title": "Definition:Orthogonal Latin Squares", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\mathbf A$ and $\\mathbf B$ be Latin squares of order $n$. Let $\\left[{a}\\right]_{i j}$ and $\\left[{b}\\right]_{i j}$ be the elements of $\\mathbf A$ and $\\mathbf B$ respectively whose indices are $i$ and $j$. Then $\\mathbf A$ and $\\mathbf B$ are '''orthogonal''' {{iff}}: :for all $i$ and $j$, the ordered pairs $\\left({\\left[{a}\\right]_{i j}, \\left[{b}\\right]_{i j}}\\right)$ are distinct. That is, {{iff}} $\\left({\\left[{a}\\right]_{i j}, \\left[{b}\\right]_{i j}}\\right)$ appears no more than once for every ordered pair $\\left({i, j}\\right)$. === Mutually Orthogonal === {{:Definition:Orthogonal Latin Squares/Mutually Orthogonal}}"}
+{"_id": "27486", "title": "Definition:Orthogonal Latin Squares/Mutually Orthogonal", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $S$ be a set of Latin squares of order $n$. Then the $S$ is '''mutually orthogonal''' {{iff}} every element of $S$ is orthogonal to every other element of $S$."}
+{"_id": "27487", "title": "Definition:Multiplicative Persistence", "text": "Let $n \\in \\N$ be a natural number. Let $n$ be expressed in decimal notation. Multiply the digits of $n$ together. Repeat with the answer, and again until a single digit remains. The number of steps it takes to reach $1$ digit is called the '''multiplicative persistence''' of $n$."}
+{"_id": "27490", "title": "Definition:Octave", "text": "An '''octave''' is a musical interval whose components have frequencies in the ratio $2 : 1$."}
+{"_id": "27491", "title": "Definition:Music", "text": "'''Music''' is the branch of study which creates aesthetically pleasing combinations of sounds."}
+{"_id": "27493", "title": "Definition:Fifth (Music)", "text": "A '''fifth''' is a musical interval whose components have frequencies in the ratio $3 : 2$."}
+{"_id": "27494", "title": "Definition:Fourth (Music)", "text": "A '''fourth''' is a musical interval whose components have frequencies in the ratio $4 : 3$."}
+{"_id": "27495", "title": "Definition:Tone (Music)", "text": "A '''tone''' is a musical interval whose components have frequencies in the ratio $9 : 8$."}
+{"_id": "27496", "title": "Definition:Well-Tempered Scale", "text": "A '''well-tempered scale''' is an octave which has been divided into $12$ equal intervals called semitones."}
+{"_id": "27497", "title": "Definition:Semitone", "text": "A '''semitone''' is a musical interval whose components have frequencies in the ratio $2^{1/12} : 1$."}
+{"_id": "27499", "title": "Definition:Imperial/Length", "text": "The imperial units of length are as follows: === Mil === {{Definition:Imperial/Length/Mil}} === Inch === {{Definition:Imperial/Length/Inch}} === Foot === {{Definition:Imperial/Length/Foot}} === Yard === {{Definition:Imperial/Length/Yard}} === Rod, Pole or Perch === {{Definition:Imperial/Length/Rod, Pole or Perch}} === Chain === {{Definition:Imperial/Length/Chain}} === Furlong === {{Definition:Imperial/Length/Furlong}} === International Mile === {{Definition:Imperial/Length/International Mile}} === League === {{Definition:Imperial/Length/League}}"}
+{"_id": "27500", "title": "Definition:Imperial/Length/Inch", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''inch''' }} {{eqn | r = 2 \\cdotp 54       | c = centimetres }} {{eqn | r = 0 \\cdotp 0254       | c = metres }} {{end-eqn}} This definition is exact, and is how the imperial unit of length is defined."}
+{"_id": "27501", "title": "Definition:Imperial/Length/Foot", "text": "The '''foot''' is the FPS base unit of length. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''foot''' }} {{eqn | r = 12       | c = inches }} {{eqn | r = 30.48       | c = centimetres }} {{eqn | r = 0 \\cdotp 3048       | c = metres }} {{end-eqn}}"}
+{"_id": "27502", "title": "Definition:Imperial/Length/Yard", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''yard''' }} {{eqn | r = 3       | c = feet }} {{eqn | r = 36       | c = inches }} {{eqn | r = 91 \\cdot 44       | c = centimetres }} {{eqn | r = 0 \\cdotp 9144       | c = metres }} {{end-eqn}}"}
+{"_id": "27503", "title": "Definition:Imperial/Length/Rod, Pole or Perch", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''rod, pole or perch''' }} {{eqn | r = 5 \\tfrac 1 2       | c = yards }} {{eqn | r = 16 \\tfrac 1 2       | c = feet }} {{eqn | r = 502 \\cdot 92       | c = centimetres }} {{eqn | r = 5 \\cdotp 0292       | c = metres }} {{end-eqn}}"}
+{"_id": "27505", "title": "Definition:Imperial/Length/Chain", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''chain''' }} {{eqn | r = 4       | c = rods, poles or perches }} {{eqn | r = 22       | c = yards }} {{eqn | r = 66       | c = feet }} {{eqn | r = 20 \\cdotp 1168       | c = metres }} {{end-eqn}}"}
+{"_id": "27506", "title": "Definition:Imperial/Length/Furlong", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''furlong''' }} {{eqn | r = 10       | c = chains }} {{eqn | r = 40       | c = rods, poles or perches }} {{eqn | r = 220       | c = yards }} {{eqn | r = 660       | c = feet }} {{eqn | r = 201 \\cdotp 168       | c = metres }} {{end-eqn}}"}
+{"_id": "27507", "title": "Definition:Imperial/Length/International Mile", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''(international) mile''' }} {{eqn | r = 8       | c = furlongs }} {{eqn | r = 80       | c = chains }} {{eqn | r = 320       | c = rods, poles or perches }} {{eqn | r = 1760       | c = yards }} {{eqn | r = 1609 \\cdotp 344       | c = metres (exactly) }} {{end-eqn}}"}
+{"_id": "27508", "title": "Definition:Imperial/Length/League", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''league''' }} {{eqn | o =        | r = 3       | c = (international) miles }} {{eqn | r = 5280       | c = yards }} {{eqn | o = \\approx       | r = 2       | c = leuca }} {{eqn | o = \\approx       | r = 3000       | c = paces }} {{eqn | r = 4828 \\cdotp 032       | c = metres (exactly) }} {{end-eqn}}"}
+{"_id": "27509", "title": "Definition:Palindromic Number", "text": "A '''palindromic number (base $b$)''' is a number which is a palindrome when expressed in number base $b$. That is, it reads the same reversed as it does forwards. When the base is not specified, it is commonplace to understand it as being $10$."}
+{"_id": "27510", "title": "Definition:Palindromic Prime", "text": "A '''palindromic prime''' is a prime number which is also a palindromic number."}
+{"_id": "27512", "title": "Definition:Palindrome", "text": "A '''palindrome''' is a sequence of objects which stays the same when reversed."}
+{"_id": "27513", "title": "Definition:Ulam Number", "text": "The sequence of '''Ulam numbers''' is defined as: :It begins $1, 2, 3$ :Each new term is the next to be uniquely the sum of $2$ previous terms in the sequence."}
+{"_id": "27514", "title": "Definition:Ulam Number/Sequence", "text": "The sequence of '''Ulam numbers''' begins as follows: :$1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, 48, 53, 57, 62, 69, \\ldots{}$"}
+{"_id": "27515", "title": "Definition:Lucas Number/Sequence", "text": "The '''Lucas sequence''' begins: :$2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, \\ldots$"}
+{"_id": "27516", "title": "Definition:Continuity/Functional", "text": "Let $S$ be a set of mappings. Let $y \\in S$ be a mapping. Let $J \\sqbrk y: S \\to \\R$ be a functional. Suppose:    :$\\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: \\size {y - y_0} < \\delta \\implies \\size {J \\sqbrk y - J \\sqbrk {y_0} } < \\epsilon$ Then $J \\sqbrk y$ is said to be a '''continuous functional''' and is continuous at the point $y_0 \\in S$."}
+{"_id": "27517", "title": "Definition:Functional/Real", "text": "Let $S$ be a set of mappings. Let $J: S \\to \\R$ be a mapping from $S$ to the real numbers $\\R$: :$\\forall y \\in S: \\exists x \\in \\R: J \\sqbrk y = x$ Then $J: S \\to \\R$ is known as a '''(real) functional''', denoted by $J \\sqbrk y$. That is, a '''(real) functional''' is a real-valued function whose arguments are themselves mappings."}
+{"_id": "27518", "title": "Definition:Ordered Set of All Mappings", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an ordered set. Let $X$ be a set. The '''ordered set of all mappings''' from $X$ to $L$ is defined by: :$L^X := \\left({S^X, \\precsim}\\right)$ where :$\\forall f, g \\in S^X: f \\precsim g \\iff f \\preceq g$ :$\\preceq$ denotes the ordering on mappings, :$S^X$ denotes the set of all mappings from $X$ into $S$."}
+{"_id": "27519", "title": "Definition:General Fibonacci Sequence", "text": "Let $r, s, t, u$ be numbers, usually integers but not necessarily so limited. Let $\\sequence {a_n}$ be the sequence defined as: :$a_n = \\begin{cases} r & : n = 0 \\\\ s & : n = 1 \\\\ t a_{n - 2} + u a_{n - 1} & : n > 1 \\end{cases}$ Then $\\sequence {a_n}$ is a '''general Fibonacci sequence'''."}
+{"_id": "27521", "title": "Definition:Lucas Number/Definition 1", "text": "The '''Lucas numbers''' are a sequence which is formally defined recursively as: :$L_n = \\begin{cases} 2 & : n = 0 \\\\ 1 & : n = 1 \\\\ L_{n - 1} + L_{n - 2} & : \\text{otherwise} \\end{cases}$"}
+{"_id": "27522", "title": "Definition:Lucas Number/Definition 2", "text": "The '''Lucas numbers''' are a sequence defined as: :$L_n = F_{n - 1} + F_{n + 1}$ where $F_k$ is the $k$th Fibonacci number."}
+{"_id": "27523", "title": "Definition:Increment/Functional", "text": "Let $S$ be a set of mappings. Let $y, h: \\R \\to \\R$ be real functions. Let $J \\sqbrk y: S \\to \\R$ be a functional defined on a normed vector space. Consider the following difference: :$ \\Delta J \\sqbrk {y; h} = J \\sqbrk {y + h} - J \\sqbrk y$ Then $\\Delta J \\sqbrk {y; h}$ is known as the '''increment of the functional $J$'''."}
+{"_id": "27524", "title": "Definition:Differentiable Functional", "text": "Let $S$ be a normed linear space of mappings. Let $y, h \\in S: \\R \\to \\R$ be real functions. Let $J \\sqbrk y$, $\\phi \\sqbrk {y; h}$ be functionals. Let $\\Delta J \\sqbrk {y; h}$ be an increment of the functional $J$ such that: :$\\Delta J \\sqbrk {y; h} = \\phi \\sqbrk {y;h} + \\epsilon \\norm h$ where $\\epsilon = \\epsilon \\sqbrk {y; h}$ is a functional, and $\\norm h$ is the norm of $S$. Suppose $\\phi \\sqbrk {y; h}$ is a linear {{WRT}} $h$ and: :$\\ds \\lim_{\\norm h \\mathop \\to 0} \\epsilon = 0$ Then the functional $J \\sqbrk y $ is said to be '''differentiable'''."}
+{"_id": "27525", "title": "Definition:Ordered Set of Increasing Mappings", "text": "Let $L = \\left({S_1, \\preceq_1}\\right)$, $K = \\left({S_2, \\preceq_2}\\right)$ be ordered sets. The '''ordered set of increasing mappings''' from $K$ into $L$ is ordered subset of $L^{S_2} = \\left({S_1^{S_2}, \\preceq'}\\right)$ and is defined by :$\\operatorname{Increasing}\\left({K, L}\\right) = \\left({X, \\precsim}\\right)$ where :$X = \\left\\{ {f:S_2 \\to S_1: f}\\right.$ is increasing mapping$\\left.\\right\\}$ :$\\mathord\\precsim = \\mathord\\preceq' \\cap \\left({X \\times X}\\right)$ :$L^{S_2}$ denotes the ordered set of all mappings from $S_2$ into $L$, :$S_1^{S_2}$ denotes the set of all mappings from $S_2$ into $S_1$. $\\operatorname{Increasing}\\left({K, L}\\right)$ as ordered subset of ordered set is ordered set by Ordered Subset of Ordered Set is Ordered Set."}
+{"_id": "27526", "title": "Definition:Prime Decomposition/Linguistic Note", "text": "The UK English spelling of '''prime factorization''' is '''prime factorisation'''."}
+{"_id": "27527", "title": "Definition:Prime Decomposition/Multiplicity", "text": "For each $p_j \\in \\left\\{ {p_1, p_2, \\ldots, p_r}\\right\\}$, its power $k_j$ is known as the '''multiplicity of $p_j$'''."}
+{"_id": "27528", "title": "Definition:Ordered Subset", "text": "Let $R = \\struct{S, \\preceq}$ be a relational structure. {{Questionable|A relational substructure of a relational structure is not necessarily ordered. So either: (a) $\\struct{S, \\preceq}$ needs to be ordered; (b) $\\struct{S', \\preceq'}$ needs to be ordered; or (c) the page needs to be renamed. The theorem Ordered Subset of Ordered Set is Ordered Set and many links to the page suggests (a) is needed.|Leigh.Samphier}} A '''ordered subset''' of $R$ is a relational structure $\\struct{S', \\preceq'}$ such that :$S'$ is a subset of $S$, :$\\mathord\\preceq' = \\mathord\\preceq \\cap \\paren{S' \\times S'}$ That is :$\\forall x, y \\in S': x \\preceq y \\iff x \\preceq' y$"}
+{"_id": "27529", "title": "Definition:Abundant Number/Definition 3", "text": "$n$ is '''abundant''' {{iff}} it is smaller than its aliquot sum."}
+{"_id": "27530", "title": "Definition:Deficient Number/Definition 3", "text": "$n$ is '''deficient''' {{iff}} it is greater than its aliquot sum."}
+{"_id": "27531", "title": "Definition:Ordered Set of Closure Operators", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an ordered set. The '''ordered set of closure operators''' of $L$ is ordered subset of $\\operatorname{Increasing} \\left({L, L}\\right) = \\left({X, \\preceq'}\\right)$ and is defined by :$\\operatorname{Closure}\\left({L}\\right) := \\left({Y, \\precsim}\\right)$ where :$Y = \\left\\{ {f:S \\to S: f}\\right.$ is closure operator$\\left.\\right\\}$ :$\\mathord\\precsim = \\mathord\\preceq' \\cap \\left({Y \\times Y}\\right)$ :$\\operatorname{Increasing} \\left({L, L}\\right)$  denotes the ordered set of increasing mappings from $L$ into $L$. $\\operatorname{Closure}\\left({L}\\right)$ as an ordered subset of an ordered set is an ordered set by Ordered Subset of Ordered Set is Ordered Set."}
+{"_id": "27532", "title": "Definition:Extremum/Functional", "text": "Let $S$ be a set of mappings. Let $J \\sqbrk y: S \\to \\R$ be a functional. Let $y, \\hat y: \\R \\to \\R$ be real functions. Suppose for $y = \\hat y \\paren x$ there exists a neighbourhood of the curve $y = \\hat y \\paren x$ such that the difference $J \\sqbrk y - J \\sqbrk {\\hat y}$ does not change its sign in this neighbourhood. Then $y = \\hat y$ is called a ('''relative''') '''extremum''' of the functional $J$. {{Help|Check if \"some neighbourhood\" could be more precisely defined |user = Julius}}"}
+{"_id": "27533", "title": "Definition:Superabundant Number", "text": "Let $n \\in \\Z_{>0}$ be a positive integer. Then $n$ is '''superabundant''' {{iff}}: :$\\forall m \\in \\Z_{>0}, m < n: \\dfrac {\\map \\sigma m} m < \\dfrac {\\map \\sigma n} n$ where $\\map \\sigma n$ is the $\\sigma$ function of $n$."}
+{"_id": "27535", "title": "Definition:Duodecimal Notation", "text": "'''Duodecimal notation''' is the technique of expressing numbers in base $12$."}
+{"_id": "27537", "title": "Definition:Imperial/Mass", "text": "There are three imperial systems of measurement of mass: * Avoirdupois * Apothecaries' weights * Troy === Grain === {{Definition:Imperial/Mass/Grain}} == Avoirdupois == {{:Definition:Avoirdupois}} == Avoirdupois: Short Variants == {{:Definition:Avoirdupois/Short}} == Apothecaries' Weights == {{:Definition:Apothecaries' Weights and Measures/Mass}} == Troy == {{:Definition:Troy}}"}
+{"_id": "27538", "title": "Definition:Avoirdupois/Ounce", "text": ":$1$ '''ounce avoirdupois''' $= 16$ drams $= 437 \\cdotp 5$ grains $= 28 \\cdotp 34952 \\, 3125$ grams. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''ounce avoirdupois''' }} {{eqn | r = 16       | c = drams }} {{eqn | r = 437 \\cdotp 5       | c = grains }} {{eqn | r = 28 \\cdotp 34952 \\, 3125       | c = grams }} {{end-eqn}} The gram equivalent is defined to be exact."}
+{"_id": "27539", "title": "Definition:Ounce/Linguistic Note", "text": "The word '''ounce''' derives from the Latin word '''uncia''', meaning '''$\\dfrac 1 {12}$ part''' (of an '''as'''). Despite the gradual migration to the metric system, the word '''ounce''' still lives on as a rhetorical flourish for '''something small''', for example: :''If you only had an '''ounce''' of sense you would understand that ...''"}
+{"_id": "27540", "title": "Definition:Apothecaries' Weights and Measures/Mass/Scruple", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''scruple''' }} {{eqn | r = 20       | c = grains }} {{eqn | r = 1 \\cdotp 3       | c = grams }} {{end-eqn}}"}
+{"_id": "27541", "title": "Definition:Ounce", "text": "=== Ounce Avoirdupois === {{:Definition:Avoirdupois/Ounce}} === Apothecaries' Ounce === {{:Definition:Apothecaries' Weights and Measures/Mass/Ounce}} === Troy Ounce === {{:Definition:Troy/Ounce}} === Uncia (Roman Ounce) === {{:Definition:Roman Weights and Measures/Mass/Uncia}}"}
+{"_id": "27543", "title": "Definition:Apothecaries' Weights and Measures", "text": "The '''apothecaries' (system of) weights and measures''' is a system of mass and volume units, derived from the Roman system. They were used by physicians and apothecaries for medical recipes, and also sometimes by scientists. Although in some cases bearing the same names as their imperial counterparts, the actual units themselves were subtly different in size."}
+{"_id": "27544", "title": "Definition:Apothecaries' Weights and Measures/Mass", "text": "=== Scruple === {{Definition:Apothecaries' Weights and Measures/Mass/Scruple}} === Drachm === {{Definition:Apothecaries' Weights and Measures/Mass/Drachm}} === Ounce === {{Definition:Apothecaries' Weights and Measures/Mass/Ounce}} === Pound === {{Definition:Apothecaries' Weights and Measures/Mass/Pound}}"}
+{"_id": "27545", "title": "Definition:System (Order Theory)", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an ordered set. The '''system''' of $L$ is an ordered subset of $L$."}
+{"_id": "27546", "title": "Definition:Infima Inheriting", "text": "Let $L = \\struct {S, \\preceq}$ be an ordered set. Let $R = \\struct {T, \\preceq'}$ be an ordered subset of $L$. Then $R$ '''inherits infima''' of $L$ {{iff}} :for all subsets $X$ of $T$ if $X$ admits an infimum of $L$, then ::$X$ admits an infimum of $R$ and $\\inf_R X = \\inf_L X$"}
+{"_id": "27547", "title": "Definition:Closure System", "text": "Let $L$ be an ordered set. Let $S$ be a system of $L$. Then $S$ is '''closure''' {{iff}} $S$ inherits infima."}
+{"_id": "27548", "title": "Definition:Ordered Set of Closure Systems", "text": "Let $L$ be an ordered set. The '''ordered set of closure systems''' of $L$ is a relational structure :$\\operatorname{ClSystems}\\left({L}\\right) = \\left({X, \\precsim}\\right)$ where :$X$ is the set of all closure systems of $L$, :dor all closure systems $S_1 = \\left({T_1, \\preceq_1}\\right), S_2 = \\left({T_2, \\preceq_2}\\right)$ of $L$: $S_1 \\precsim S_2 \\iff T_1 \\subseteq T_2$"}
+{"_id": "27549", "title": "Definition:Weak Extremum", "text": "Let $S$ be a set of mappings. Let $y, \\hat y \\in S: \\R \\to \\R$ be real functions. Let $J \\sqbrk y: S \\to \\R$ be a functional. Suppose there exists $\\epsilon \\in \\R_{> 0}$ such that for $\\norm {y - \\hat y}_1 < \\epsilon$ the expression $J \\sqbrk y - J \\sqbrk {\\hat y}$ has the same sign for all $y$. Here $\\norm{\\, \\cdot \\,}_1$ denotes the norm of in the space $C^1$. Then $y = \\hat y$ is a '''weak extremum''' of the functional $J \\sqbrk y$."}
+{"_id": "27551", "title": "Definition:Apothecaries' Weights and Measures/Mass/Ounce", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''apothecaries' ounce''' }} {{eqn | r = 8       | c = drachms }} {{eqn | r = 24       | c = scruples }} {{eqn | r = 480       | c = grains }} {{eqn | r = 1       | c = troy ounce }} {{eqn | r = 31 \\cdotp 1       | c = grams }} {{end-eqn}}"}
+{"_id": "27552", "title": "Definition:Troy/Ounce", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''troy ounce''' }} {{eqn | r = 20       | c = pennyweight }} {{eqn | r = 480       | c = grains }} {{eqn | r = 1       | c = apothecaries' ounce }} {{eqn | o = \\approx       | r = 31 \\cdotp 1       | c = grams }} {{end-eqn}}"}
+{"_id": "27553", "title": "Definition:Troy", "text": "=== Pennyweight === {{Definition:Troy/Pennyweight}} === Ounce === {{Definition:Troy/Ounce}} === Pound === {{Definition:Troy/Pound}} === Talent === {{Definition:Troy/Talent}}"}
+{"_id": "27556", "title": "Definition:Roman Weights and Measures/Mass/Uncia", "text": ":$1$ '''uncia''' $= \\dfrac 1 {12}$ libra $= 24$ scrupuli. Its value in modern units is estimated to be approximately $423$ grains or $27.4$ grams."}
+{"_id": "27557", "title": "Definition:Roman Weights and Measures", "text": "The '''Roman system of weights''' is a system of mass units originating from the time of the Roman empire. This system continued to be used throughout Europe, although it evolved in different directions according to use."}
+{"_id": "27558", "title": "Definition:Roman Weights and Measures/Mass", "text": "The Roman units of mass are as follows: === Calcus === {{Definition:Roman Weights and Measures/Mass/Calcus}} === Siliqua === {{Definition:Roman Weights and Measures/Mass/Siliqua}} === Scrupulus === {{Definition:Roman Weights and Measures/Mass/Scrupulus}} === Uncia === {{Definition:Roman Weights and Measures/Mass/Uncia}} === Libra === {{Definition:Roman Weights and Measures/Mass/Libra}}"}
+{"_id": "27561", "title": "Definition:Roman Weights and Measures/Mass/Scrupulus", "text": ":$1$ '''scrupulus''' $= \\dfrac 1 {24}$ uncia $= 8$ calci. Its value in modern units is estimated to be approximately $17.6$ grains, or approximately $0.14$ grams."}
+{"_id": "27564", "title": "Definition:Roman Weights and Measures/Mass/Calcus", "text": ":$1$ '''calcus''' $= \\dfrac 1 8$ scrupulus. Its value in modern units is estimated to be approximately $2.2$ grains, or approximately $0.14$ grams."}
+{"_id": "27566", "title": "Definition:Operator Generated by Ordered Subset", "text": "Let $L = \\left({X, \\precsim}\\right)$ be an ordered set. Let $S = \\left({T, \\preceq}\\right)$ be an ordered subset of $L$. The '''operator generated by ordered set''' $S$ is defined by :$\\forall x \\in X:\\operatorname{operator}\\left({S}\\right)\\left({x}\\right) := \\inf_L\\left({x^\\succeq \\cap T}\\right)$ where $x^\\succeq$ denotes the upper closure of $x$."}
+{"_id": "27567", "title": "Definition:Roman Weights and Measures/Mass/Siliqua", "text": ":$1$ '''siliqua''' $= \\dfrac 1 6$ scrupulus $= \\dfrac 1 {144}$ uncia. Its value in modern units is estimated to be approximately $2.9$ grains, or approximately $0.19$ grams. Its modern equivalent is the carat."}
+{"_id": "27570", "title": "Definition:Roman Weights and Measures/Mass/Libra", "text": ":$1$ '''libra''' $= 12$ unciae. Its value in modern units is estimated to be approximately $5 \\, 076$ grains or $329$ grams."}
+{"_id": "27572", "title": "Definition:Avoirdupois/Dram", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''dram''' }} {{eqn | r = 27 \\cdotp 34375       | c = grains }} {{eqn | r = \\dfrac 1 {16}       | c = ounce avoirdupois }} {{eqn | o = \\approx       | r = 1 \\cdotp 77       | c = grams }} {{end-eqn}}"}
+{"_id": "27574", "title": "Definition:Imperial/Mass/Grain", "text": "The '''grain''' is the imperial unit of mass which is used as the basis of all three of the imperial weight systems. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''grain''' }} {{eqn | r = \\dfrac 1 {437 \\cdot 5}       | c = ounce avoirdupois }} {{eqn | o = \\approx       | r = 64 \\cdotp 8       | c = milligrams }} {{end-eqn}}"}
+{"_id": "27575", "title": "Definition:Avoirdupois", "text": "=== Dram === {{Definition:Avoirdupois/Dram}} === Ounce === {{Definition:Avoirdupois/Ounce}} === Pound === {{Definition:Avoirdupois/Pound}} === Stone === {{Definition:Avoirdupois/Stone}} === Quarter === {{Definition:Avoirdupois/Quarter}} === Hundredweight === {{Definition:Avoirdupois/Hundredweight}} === Ton === {{Definition:Avoirdupois/Ton}}"}
+{"_id": "27577", "title": "Definition:Avoirdupois/Pound", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''pound avoirdupois''' }} {{eqn | r = 16       | c = ounces avoirdupois }} {{eqn | r = 7 \\, 000       | c = grains }} {{eqn | r = 453 \\cdotp 59237       | c = grams }} {{end-eqn}} The gram equivalent is defined to be exact."}
+{"_id": "27578", "title": "Definition:Avoirdupois/Stone", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''stone''' }} {{eqn | r = 14       | c = pounds avoirdupois }} {{eqn | r = 224       | c = ounces avoirdupois }} {{eqn | o = \\approx       | r = 6 \\, 350 \\cdotp 29318       | c = grams }} {{end-eqn}} The gram equivalent is exact."}
+{"_id": "27581", "title": "Definition:Avoirdupois/Ton", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''ton''' }} {{eqn | r = 20       | c = hundredweight }} {{eqn | r = 80       | c = quarters }} {{eqn | r = 160       | c = stone }} {{eqn | r = 2240       | c = pounds avoirdupois }} {{eqn | o = \\approx       | r = 1016       | c = kilograms }} {{end-eqn}}"}
+{"_id": "27582", "title": "Definition:Avoirdupois/Hundredweight", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''hundredweight''' }} {{eqn | r = 4       | c = quarters }} {{eqn | r = 8       | c = stone }} {{eqn | r = 112       | c = pounds avoirdupois }} {{eqn | o = \\approx       | r = 50 \\cdotp 8       | c = kilograms }} {{end-eqn}}"}
+{"_id": "27583", "title": "Definition:Avoirdupois/Quarter", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''quarter''' }} {{eqn | r = 2       | c = stone }} {{eqn | r = 28       | c = pounds avoirdupois }} {{eqn | o = \\approx       | r = 12 \\cdotp 7       | c = kilograms }} {{end-eqn}}"}
+{"_id": "27584", "title": "Definition:Avoirdupois/Short", "text": "=== Short Hundredweight === {{Definition:Avoirdupois/Short/Hundredweight}} === Short Ton === {{Definition:Avoirdupois/Short/Ton}}"}
+{"_id": "27585", "title": "Definition:Avoirdupois/Short/Hundredweight", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''short hundredweight''' }} {{eqn | r = 100       | c = pounds avoirdupois }} {{eqn | o = \\approx       | r = 45 \\cdotp 359       | c = kilograms }} {{end-eqn}}"}
+{"_id": "27586", "title": "Definition:Avoirdupois/Short/Ton", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''short ton''' }} {{eqn | r = 20       | c = short hundredweight }} {{eqn | r = 2000       | c = pounds avoirdupois }} {{eqn | o = \\approx       | r = 907 \\cdot 2       | c = kilograms }} {{end-eqn}}"}
+{"_id": "27593", "title": "Definition:Apothecaries' Weights and Measures/Mass/Drachm", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''drachm''' }} {{eqn | r = 3       | c = scruples }} {{eqn | r = 60       | c = grains }} {{eqn | r = 3 \\cdotp 89       | c = grams }} {{end-eqn}}"}
+{"_id": "27594", "title": "Definition:Apothecaries' Weights and Measures/Mass/Pound", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''apothecaries' pound''' }} {{eqn | r = 12       | c = apothecaries' ounces }} {{eqn | r = 5 \\, 760       | c = grains }} {{eqn | r = 373 \\cdotp 24       | c = grams }} {{eqn | r = 1       | c = troy pound }} {{end-eqn}}"}
+{"_id": "27596", "title": "Definition:Troy/Pennyweight", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''pennyweight''' }} {{eqn | r = 24       | c = grains }} {{eqn | o = \\approx       | r = 1 \\cdotp 56       | c = grams }} {{end-eqn}}"}
+{"_id": "27598", "title": "Definition:Troy/Pound", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''troy pound''' }} {{eqn | r = 12       | c = troy ounces }} {{eqn | r = 5 \\, 760       | c = grains }} {{eqn | r = 1       | c = apothecaries' pound }} {{eqn | o = \\approx       | r = 373 \\cdotp 24       | c = grams }} {{end-eqn}}"}
+{"_id": "27602", "title": "Definition:Pound/Symbol", "text": "The symbol for the pound is $\\text{lb}$. This derives from the libra (Roman pound) from which the pound evolved. It always needs to be distinguished from the pound sterling, whose symbol is $\\pounds$."}
+{"_id": "27603", "title": "Definition:Ounce/Symbol", "text": "The symbol for the ounce is $\\text{oz}$. This derives indirectly from the uncia (Roman ounce) via the $15$th century Italian '''onza'''."}
+{"_id": "27604", "title": "Definition:Avoirdupois/Quarter/Symbol", "text": "The symbol for the quarter is $\\text{qtr}$."}
+{"_id": "27606", "title": "Definition:Troy/Pennyweight/Symbol", "text": "The symbol for the pennyweight is $\\text{dwt}$. This derives from the Latin '''denarius''', which evolved into the (old) penny."}
+{"_id": "27609", "title": "Definition:Kilogram/Symbol", "text": "The symbol for the '''kilogram''' is $\\mathrm {kg}$."}
+{"_id": "27612", "title": "Definition:Milligram", "text": ":$1$ '''milligram''' is defined as $= \\dfrac 1 {1 \\, 000}$ of a gram."}
+{"_id": "27613", "title": "Definition:Milligram/Symbol", "text": "The symbol for the '''milligram''' is $\\mathrm {mg}$."}
+{"_id": "27615", "title": "Definition:Sexagesimal Notation", "text": "'''Sexagesimal notation''' is the technique of expressing numbers in base $60$."}
+{"_id": "27621", "title": "Definition:Long Hundred", "text": "A '''long hundred''' is another term for the number $120$."}
+{"_id": "27623", "title": "Definition:Short Hundred", "text": "A '''short hundred''' is another term for the number $100$."}
+{"_id": "27625", "title": "Definition:Rhombic Dodecahedron", "text": "The '''rhombic dodecahedron''' is a Catalan polyhedron with $12$ congruent faces each in the shape of a rhombus."}
+{"_id": "27626", "title": "Definition:Reversal", "text": "Let $m = \\sqbrk {a_n a_{n - 1} a_{n - 2} \\ldots a_2 a_1 a_0}$ be an integer expressed in base $10$. That is: :$m = \\displaystyle \\sum_{k \\mathop = 0}^n a_k 10^k$ Its '''reversal''' $m'$ is the integer created by writing the digits of $m$ in the opposite order: :$m' = \\sqbrk {a_0 a_1 a_2 \\ldots a_{n - 2} a_{n - 1} a_n}$ That is: :$m' = \\displaystyle \\sum_{k \\mathop = 0}^n a_{n - k} 10^k$"}
+{"_id": "27627", "title": "Definition:Emirp", "text": "An '''emirp''' is a prime number whose reversal is a different prime number."}
+{"_id": "27630", "title": "Definition:Variational Derivative", "text": "Let $\\map y x$ be a real function. Let $J = J \\sqbrk y$ be a functional dependent on $y$. Let $\\map h x$ be a real function, which differs from zero only in the neighbourhood of $x_0$. Consider an increment of functional $\\Delta J \\sqbrk {y; h}$. Denote the area between $\\map y x + \\map h x$ and $\\map y x$ (or, equivalently, between $\\map h x$ and x-axis) as $\\Delta \\sigma$. Let $\\Delta \\sigma \\to 0$ in such a way, that  $\\ds \\lim_{\\Delta \\sigma \\mathop \\to 0} \\map \\max {\\map h x} = 0$ and the length of interval where $\\map h x$ differs from 0 would go to 0. If the ratio $\\dfrac {\\Delta J \\sqbrk {y; h} } {\\Delta \\sigma}$ converges to a limit as $\\Delta \\sigma \\to 0$, then: $\\ds \\lim_{\\Delta \\sigma \\mathop \\to 0} \\frac {\\Delta J \\sqbrk {y; h} } {\\Delta \\sigma} = \\intlimits {\\frac {\\delta J} {\\delta y} } {x = x_0} {}$ where $\\intlimits {\\dfrac {\\delta J} {\\delta y} } {x = x_0} {}$ is called the '''variational derivative''' at the point $x = x_0$ for the function $y = \\map y x$. {{Stub|Check if there is a definition not using a special point $x_0$}}"}
+{"_id": "27631", "title": "Definition:Euler's Equation for Vanishing Variation", "text": "Let $\\map y x$ be a real function. Let $\\map F {x, y, z}$ be a real function belonging to $C^2$ {{WRT}} all its variables. Let $J \\sqbrk y$ be a functional of the form: :$\\displaystyle \\int_a^b \\map F {x, y, y'} \\rd x$ Then '''Euler's equation for vanishing variation''' is defined a differential equation, resulting from condition: :$\\displaystyle \\delta \\int_a^b \\map F {x, y, y'} \\rd x = 0$ In other words: :$\\displaystyle F_y - \\dfrac \\d {\\d x} F_{y'} = 0$ {{Stub|Check if conditions can be stricter, add special cases, examples, multidimensional and multiderivative forms}} {{NamedforDef|Leonhard Paul Euler|cat = Euler}}"}
+{"_id": "27632", "title": "Definition:Directed Suprema Inheriting", "text": "Let $L = \\left({X, \\preceq}\\right)$ be an ordered set. Let $S = \\left({Y, \\precsim}\\right)$ be an ordered subset of $L$. Then $S$ '''inherits directed suprema''' {{iff}} :for all directed subsets $A$ of $Y$: if $A$ admits a supremum in $L$, then $\\sup_L A \\in Y$"}
+{"_id": "27633", "title": "Definition:Ordered Set of Subalgebras", "text": "Let $L$ be an ordered set. The '''ordered set of subalgebras''' of $L$ is ordered subset of $\\operatorname{ClSystems}\\left({L}\\right)$ and is defined by :$\\operatorname{Subalgeras}\\left({L}\\right) := \\left({X, \\precsim}\\right)$ where :$X$ is the set of all directed suprema inheriting closure systems on $L$, :$\\operatorname{ClSystems}\\left({L}\\right)$ denotes the ordered set of closure systems on $L$."}
+{"_id": "27634", "title": "Definition:Lattice (Group Theory)/Definition 1", "text": "A (point) '''lattice''' is a discrete subgroup of $\\R^m$ under addition."}
+{"_id": "27635", "title": "Definition:Lattice (Group Theory)/Definition 2", "text": "Let $\\R^m$ be the $m$-dimensional real Euclidean space. Let $\\set {b_1, b_2, \\ldots, b_n}$ be a set of linearly independent vectors of $\\R^m$. A '''lattice in $\\R^m$''' is the set of all integer linear combinations of such vectors. That is: :$\\displaystyle \\map \\LL {b_1, b_2, \\ldots, b_n} = \\set {\\sum_{i \\mathop = 1}^n x_i b_i : x_i \\in \\Z}$"}
+{"_id": "27636", "title": "Definition:Coarser Subset (Order Theory)", "text": "Let $L = \\left({S, \\preceq}\\right)$ be a preordered set. Let $X, Y$ be subsets of $S$. Then $X$ is '''coarser (subset)''' than $Y$ {{iff}} :$\\forall x \\in X: \\exists y \\in Y: y \\preceq x$"}
+{"_id": "27637", "title": "Definition:Finer Subset (Order Theory)", "text": "Let $L = \\left({S, \\preceq}\\right)$ be a preordered set. Let $X, Y$ be subsets of $S$. Then $X$ is '''finer (subset)''' than $Y$ {{iff}} :$\\forall x \\in X: \\exists y \\in Y: x \\preceq y$"}
+{"_id": "27638", "title": "Definition:Archimedean Polyhedron", "text": "An '''Archimedean polyhedron''' is a convex polyhedron with the following properties: :$(1): \\quad$ Each of its faces is a regular polygon :$(2): \\quad$ It is vertex-transitive :$(3): \\quad$ The faces are not all congruent. :$(4): \\quad$ It is not a regular prism or a regular antiprism."}
+{"_id": "27640", "title": "Definition:Semiregular Polyhedron", "text": "A '''semiregular polyhedron''' is a polyhedron with the following properties: :$(1): \\quad$ Each of its faces is a regular polygon :$(2): \\quad$ It is vertex-transitive :$(3): \\quad$ The faces are not all congruent."}
+{"_id": "27641", "title": "Definition:Inaccessible by Directed Suprema", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an up-complete ordered set. Let $X$ be a subset of $S$. Then $X$ is '''inaccessible by directed suprema''' {{iff}} :for all directed subsets $D$ of $S$: $\\sup D \\in X \\implies X \\cap D \\ne \\varnothing$"}
+{"_id": "27642", "title": "Definition:Closed under Directed Suprema", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an up-complete ordered set. Let $X$ be a subset of $S$. Then $X$ is '''closed under directed suprema''' {{iff}} :for all directed subsets $D$ of $S$: $D \\subseteq X \\implies \\sup D \\in X$"}
+{"_id": "27643", "title": "Definition:Small Stellated Dodecahedron", "text": "The small stellated dodecahedron is a Kepler-Poinsot polyhedron composed of $12$ pentagrammic faces with $5$ pentagrams meeting at each vertex. {{stub|Picture needed for a start}}"}
+{"_id": "27646", "title": "Definition:Great Stellated Dodecahedron", "text": "The great stellated dodecahedron is a Kepler-Poinsot polyhedron composed of $12$ pentagrammic faces with $3$ pentagrams meeting at each vertex. {{stub|Picture needed for a start}}"}
+{"_id": "27649", "title": "Definition:Catalan Polyhedron", "text": "A '''Catalan polyhedron''' is a dual polyhedron of an Archimedean polyhedron."}
+{"_id": "27650", "title": "Definition:Contractible Space", "text": "=== Definition 1 === {{:Definition:Contractible Space/Definition 1}} === Definition 2 === {{:Definition:Contractible Space/Definition 2}}"}
+{"_id": "27651", "title": "Definition:Property (S)", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an up-complete ordered set. Let $X$ be a subset of $S$. Then $X$ has '''property (S)''' {{iff}} :for all directed subsets $D$ of $S$: $\\sup D \\in X \\implies \\exists y \\in D:\\forall x \\in D: y \\preceq x \\implies x \\in X$"}
+{"_id": "27652", "title": "Definition:Stellation", "text": "'''Stellation''' is the process of extending the extremities of a geometric figure in a specifically structured way, in order to form new figures from those extensions. The figure so generated is itself known as a '''stellation''' or a '''stellated''' original figure."}
+{"_id": "27653", "title": "Definition:Stellation/Polygon", "text": "A '''stellation''' of a polygon $P$ is a plane figure formed by extending the sides of $P$ until they meet."}
+{"_id": "27654", "title": "Definition:Stellation/Polyhedron", "text": "A '''stellation''' of a polyhedron $H$ is a solid figure formed by extending the faces of $H$ until they meet."}
+{"_id": "27656", "title": "Definition:Quadratic Gauss Sum", "text": "Let $p$ be an odd prime. Let $a$ be an integer. The '''quadratic Gauss sum''' is: :$\\displaystyle g \\left({a, p}\\right) = \\sum_{n \\mathop = 0}^{p - 1} e^{2 \\pi i a n^2 / p}$ Category:Definitions/Number Theory kuh39bp5zd1fa4xw5xigtf5htzh1m2i"}
+{"_id": "27657", "title": "Definition:Relational Structure with Topology", "text": "A triple $\\left({S, \\preceq, \\tau}\\right)$ is a '''relational structure with topology''' where :$\\preceq$ is a relation on $S$ and :$\\tau$ is a topology on $S$."}
+{"_id": "27658", "title": "Definition:Weierstrass Elementary Factor", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. The '''$n$th (Weierstrass) elementary factor''' is the function $E_n: \\C \\to \\C$ defined as: :$E_n \\left({z}\\right) = \\displaystyle \\begin{cases} 1 - z & : n = 0 \\\\ \\left({1 - z}\\right) \\exp \\left({z + \\dfrac {z^2} 2 + \\cdots + \\dfrac{z^n} n}\\right) & : \\text{otherwise}\\end{cases}$"}
+{"_id": "27660", "title": "Definition:Absolute Convergence of Product/General Definition/Definition 1", "text": "The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\paren{1 + a_n}$ is '''absolutely convergent''' {{iff}} $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\paren{1 + \\norm{a_n}}$ is convergent."}
+{"_id": "27661", "title": "Definition:Absolute Convergence of Product/Complex Numbers", "text": "Let $\\sequence {a_n}$ be a sequence in $\\C$. === Definition 1 === {{:Definition:Absolute Convergence of Product/Complex Numbers/Definition 1}} === Definition 2 === {{:Definition:Absolute Convergence of Product/Complex Numbers/Definition 2}} === Definition 3 === {{:Definition:Absolute Convergence of Product/Complex Numbers/Definition 3}}"}
+{"_id": "27663", "title": "Definition:Convergent Product/Number Field", "text": "Let $\\mathbb K$ be one of the standard number fields $\\Q, \\R, \\C$. === Nonzero Sequence === {{:Definition:Convergent Product/Number Field/Nonzero Sequence}} === Arbitrary Sequence === {{:Definition:Convergent Product/Number Field/Arbitrary Sequence}}"}
+{"_id": "27664", "title": "Definition:Scott Topology", "text": "Let $T = \\left({S, \\preceq, \\tau}\\right)$ be a relational structure with topology where $\\left({S, \\preceq}\\right)$ is an up-complete ordered set. Then $T$ has '''Scott topology''' {{iff}} :$\\tau$ is the set of all upper and inaccessible by directed suprema subsets of $S$. {{NamedforDef|Dana Stewart Scott|cat = Scott, Dana}}"}
+{"_id": "27665", "title": "Definition:Convergent Product/Arbitrary Field", "text": "Let $\\struct {\\mathbb K, \\norm {\\,\\cdot\\,} }$ be a valued field. === Nonzero Sequence === {{:Definition:Convergent Product/Arbitrary Field/Nonzero Sequence}} === Arbitrary Sequence === {{:Definition:Convergent Product/Arbitrary Field/Arbitrary Sequence}}"}
+{"_id": "27666", "title": "Definition:Divergent Product", "text": "An infinite product which is not convergent is '''divergent'''. === Divergence to zero === {{:Definition:Divergent Product/Divergence to Zero}}"}
+{"_id": "27667", "title": "Definition:Primitive Pythagorean Triangle", "text": "A '''primitive Pythagorean triangle''' is a Pythagorean triangle whose sides form a primitive Pythagorean triple."}
+{"_id": "27669", "title": "Definition:O Notation/Big-O Notation/Real/Point", "text": "Let $x_0 \\in \\R$. Let $f$ and $g$ be real-valued or complex-valued functions defined on a punctured neighborhood of $x_0$. The statement: :$f \\left({x}\\right) = \\mathcal O \\left({g \\left({x}\\right)}\\right)$ as $x \\to x_0$ is equivalent to: :$\\exists c \\in \\R: c \\ge 0: \\exists \\delta \\in \\R : \\delta > 0 : \\forall x \\in \\R : \\left({0 < \\left\\lvert{x - x_0}\\right\\rvert < \\delta \\implies \\left\\lvert{f \\left({x}\\right)}\\right\\rvert \\le c \\cdot \\left\\lvert{g \\left({x}\\right)}\\right\\rvert}\\right)$ That is: :$\\left\\lvert{f \\left({x}\\right)}\\right\\rvert \\le c \\cdot \\left\\lvert{g \\left({x}\\right)}\\right\\rvert$ for all $x$ in a punctured neighborhood of $x_0$."}
+{"_id": "27670", "title": "Definition:O Notation/Big-O Notation/Complex/Infinity", "text": "Let $f$ and $g$ be complex functions defined for all complex numbers whose modulus is sufficiently large. The statement: :$f(z) = \\mathcal O \\left({g(z)}\\right)$ as $|z|\\to\\infty$ is equivalent to: :$\\displaystyle \\exists c\\in \\R: c\\ge 0 : \\exists r_0 \\in \\R : \\forall z \\in \\C : (|z| \\geq r_0 \\implies |f(z)| \\leq c \\cdot |g(z)|)$ That is: :$|f(z)| \\leq c \\cdot |g(z)|$ for all $z$ in a neighborhood of infinity in $\\CC$."}
+{"_id": "27671", "title": "Definition:O Notation/Big-O Notation/Complex/Point", "text": "Let $z_0 \\in \\C$. Let $f$ and $g$ be complex functions defined on an punctured neighborhood of $z_0$. The statement: :$f(z) = \\mathcal O \\left({g(z)}\\right)$ as $z\\to z_0$ is equivalent to: :$\\displaystyle \\exists c \\in \\R : c \\ge 0 : \\exists \\delta \\in \\R : \\delta > 0 : \\forall z \\in \\C : (0<|z-z_0|<\\delta \\implies |f(z)| \\leq c \\cdot |g(z)|)$ That is: :$|f(z)| \\leq c \\cdot |g(z)|$ for all $z$ in a punctured neighborhood of $z_0$."}
+{"_id": "27672", "title": "Definition:O Notation/Big-O Notation/Real/Infinity", "text": "Let $f$ and $g$ be real-valued or complex-valued functions defined on a neighborhood of $+ \\infty$ in $\\R$. The statement: :$\\map f x = \\map {\\mathcal O} {\\map g x}$ as $x \\to \\infty$ is equivalent to: :$\\exists c \\in \\R: c \\ge 0: \\exists x_0 \\in \\R: \\forall x \\in \\R: \\paren {x \\ge x_0 \\implies \\size {\\map f x} \\le c \\cdot \\size {\\map g x} }$ That is: :$\\size {\\map f x} \\le c \\cdot \\size {\\map g x}$ for $x$ sufficiently large. This statement is voiced '''$f$ is big-O of $g$''' or simply '''$f$ is big-O $g$'''."}
+{"_id": "27673", "title": "Definition:Asymptotically Equal/Sequences", "text": "Let $\\left \\langle {a_n} \\right \\rangle$ and $\\left \\langle {b_n} \\right \\rangle$ be sequences in $\\R$. === Definition 1 === {{:Definition:Asymptotically Equal/Sequences/Definition 1}} === Definition 2 === {{:Definition:Asymptotically Equal/Sequences/Definition 2}} === Definition 3 === {{:Definition:Asymptotically Equal/Sequences/Definition 3}} This is denoted: $a_n \\sim b_n$."}
+{"_id": "27674", "title": "Definition:Asymptotically Equal/Functions", "text": "Let $f$ and $g$ real functions defined on $\\R$. Then: : '''$f$ is asymptotically equal to $g$''' {{iff}}: :$\\dfrac {f \\left({x}\\right)} {g \\left({x}\\right)} \\to 1$ as $x \\to +\\infty$. That is, the larger the $x$, the closer $f$ gets (relatively) to $g$."}
+{"_id": "27675", "title": "Definition:O Notation/Big-O Notation/General Definition/Point", "text": "Let $(X,\\tau)$ be a topological space. Let $V$ be a normed vector space over $\\R$ or $\\C$ with norm $\\left\\Vert{\\,\\cdot\\,}\\right\\Vert$ Let $x_0\\in X$. Let $f,g:X\\setminus\\{x_0\\}\\to V$ be functions. The statement :$f(x) = \\mathcal O \\left({g(x)}\\right)$ as $x\\to x_0$ is equivalent to: :$\\displaystyle \\exists c\\in \\R: c\\ge 0 : \\exists U\\in\\tau: x_0\\in U : \\forall x\\in U\\setminus\\{x_0\\} : \\Vert f(x)\\Vert \\leq c\\cdot\\Vert g(x)\\Vert$ That is: :$\\Vert f(x)\\Vert\\leq c\\cdot\\Vert g(x)\\Vert$ for all $x$ in a punctured neighborhood of $x_0$."}
+{"_id": "27676", "title": "Definition:Asymptotically Equal/General Definition/Point", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $V$ be a normed vector space over $\\R$ or $\\C$ with norm $\\left\\Vert{\\, \\cdot \\,}\\right\\Vert$. Let $f, g: S \\to V$ be mappings. Let $x_0 \\in X$. Then: : $f$ is '''asymptotically equal''' to $g$ as $x \\to x_0$ {{iff}}: : $f - g = o \\left({g}\\right)$ as $x \\to x_0$ where $o$ denotes little-O notation."}
+{"_id": "27677", "title": "Definition:O Notation/Little-O Notation/Real Functions", "text": "Let $f$ and $g$ be real-valued or complex-valued functions on a subset of $\\R$ containing all sufficiently large real numbers. === Definition 1 === {{:Definition:O Notation/Little-O Notation/Real Functions/Definition 1}} === Definition 2 === {{:Definition:O Notation/Little-O Notation/Real Functions/Definition 2}} This is denoted: :$f = o \\left({g}\\right)\\qquad(x\\to\\infty)$ </onlyinclude> This statement is voiced '''$f$ is little-o of $g$''' or simply '''$f$ is little-o $g$'''."}
+{"_id": "27678", "title": "Definition:Sequence of Partial Products", "text": "The sequence $\\sequence {p_N}$ defined as: :$\\displaystyle p_N: = \\prod_{n \\mathop = 1}^N a_n = a_1 \\cdot a_2 \\cdots a_N$ is the '''sequence of partial products''' of the infinite product $\\displaystyle p = \\prod_{n \\mathop = 1}^\\infty a_n$."}
+{"_id": "27679", "title": "Definition:O Notation/Little-O Notation/General Definition/Point", "text": "Let $X$ be a topological space. Let $V$ be a normed vector space over $\\R$ or $\\C$ with norm $\\left\\Vert{\\,\\cdot\\,}\\right\\Vert$ Let $f,g:X\\to V$ be functions. Let $x_0\\in X$. The statement :$f(x) = o \\left({g(x)}\\right)$ as $x\\to x_0$ is equivalent to the statement: :For all $\\epsilon>0$, there exists a neighborhood $U$ of $x_0$ such that $\\Vert f(x)\\Vert\\leq \\epsilon\\cdot\\Vert g(x)\\Vert$ for all $x\\in U$"}
+{"_id": "27680", "title": "Definition:Topological Lattice", "text": "A '''topological lattice''' is a relational structure with topology $\\left({S, \\preceq, \\tau}\\right)$ where $\\left({S, \\preceq}\\right)$ is a lattice and $\\left({S, \\tau}\\right)$ is a topological space."}
+{"_id": "27681", "title": "Definition:Generator for Primitive Pythagorean Triple", "text": "The '''generator''' for a primitive Pythagorean triple is an ordered pair: :$G = \\left({m, n}\\right)$ where $m, n \\in \\Z$ such that: : $m, n \\in \\Z_{>0}$ are (strictly) positive integers : $m \\perp n$, that is, $m$ and $n$ are coprime : $m$ and $n$ are of opposite parity : $m > n$. The primitive Pythagorean triple which has been '''generated''' by $G$ is: :$\\left({2 m n, m^2 - n^2, m^2 + n^2}\\right)$"}
+{"_id": "27682", "title": "Definition:Generator for Pythagorean Triple", "text": "Let $P = \\left\\{ {a, b, c}\\right\\}$ be a Pythagorean triple  A '''generator''' for $P$ is an ordered pair: :$G = \\left({m, n}\\right)$ where $m, n \\in \\Z$ such that: : $a = 2 m n$ : $b = m^2 - n^2$ : $c = m^2 + n^2$"}
+{"_id": "27684", "title": "Definition:Almost Isosceles Pythagorean Triangle", "text": "Let $P$ be a Pythagorean triangle whose legs are $a$ and $b$. Then $P$ is '''almost isosceles''' {{iff}} $\\size {a - b} = 1$ That is, if the legs of $P$ differ by $1$. === Sequence === {{:Generator for Almost Isosceles Pythagorean Triangle/Sequence}}"}
+{"_id": "27685", "title": "Definition:Neighborhood of Infinity (Real Analysis)", "text": "=== Neighborhood of Positive Infinity === {{:Definition:Neighborhood of Infinity (Real Analysis)/Positive Infinity}} === Neighborhood of Negative Infinity === {{:Definition:Neighborhood of Infinity (Real Analysis)/Negative Infinity}}"}
+{"_id": "27686", "title": "Definition:Neighborhood of Infinity (Real Analysis)/Positive Infinity", "text": "A '''neighborhood of $+\\infty$''' is a subset of the set of real numbers $\\R$ wich contains an interval $(a,+\\infty)$ for some $a\\in\\R$. That is, a subset which contains all sufficiently large real numbers."}
+{"_id": "27687", "title": "Definition:Neighborhood of Infinity (Real Analysis)/Negative Infinity", "text": "A '''neighborhood of $-\\infty$''' is a subset of the set of real numbers $\\R$ wich contains an interval $(-\\infty,a)$ for some $a\\in\\R$. That is, a subset which contains all sufficiently large negative real numbers."}
+{"_id": "27688", "title": "Definition:Nearest Integer Function", "text": "The '''nearest integer function''' is defined as: :$\\forall x \\in \\R: \\nint x = \\begin {cases} \\floor {x + \\dfrac 1 2} & : x \\notin 2 \\Z + \\dfrac 1 2 \\\\ x - \\dfrac 1 2 & : x \\in 2 \\Z + \\dfrac 1 2 \\end{cases}$ where $\\floor x$ is the floor function."}
+{"_id": "27689", "title": "Definition:Pythagorean Prime", "text": "A '''Pythagorean prime''' is a prime number of the form: :$p = 4 n + 1$ where $n \\in \\Z_{\\ge 0}$ is a positive integer. === Sequence === {{:Definition:Pythagorean Prime/Sequence}}"}
+{"_id": "27691", "title": "Definition:Non-Pythagorean Prime", "text": "A '''non-Pythagorean prime''' is an odd prime number of the form: :$p = 4 n + 3$ where $n \\in \\Z_{\\ge 0}$ is a positive integer. === Sequence === {{:Definition:Non-Pythagorean Prime/Sequence}}"}
+{"_id": "27693", "title": "Definition:O Notation/Big-O Notation/Sequence", "text": "Let $\\sequence {a_n}$ and $\\sequence {b_n}$ be sequences of real or complex numbers. '''$a_n$ is big-O of $b_n$''' {{iff}} :$\\exists c \\in \\R: c \\ge 0 : \\exists n_0 \\in \\N : \\paren {n \\ge n_0 \\implies \\size {a_n} \\le c \\cdot \\size {b_n} }$ That is: :$\\size {a_n} \\le c \\cdot \\size {b_n}$ for all sufficiently large $n$."}
+{"_id": "27694", "title": "Definition:O Notation/Big-O Notation/General Definition", "text": "=== Estimate at infinity === {{:Definition:O Notation/Big-O Notation/General Definition/Infinity}} === Point Estimate === {{:Definition:O Notation/Big-O Notation/General Definition/Point}}"}
+{"_id": "27695", "title": "Definition:Neighborhood of Infinity (Topology)", "text": "Let $X$ be a non-empty topological space. A '''neighborhood of infinity''' is a subset of $X$ which contains the complement of a closed and compact subset of $X$."}
+{"_id": "27696", "title": "Definition:Hamiltonian", "text": "Let $J \\sqbrk {\\dotsm y_i \\dotsm}$ be a functional of the form: :$\\displaystyle J \\sqbrk {\\dotsm y_i \\dotsm} = \\intlimits {\\int_{x_0}^{x_1} \\map F {x, \\cdots y_i \\dotsm, \\dotsm y_i \\dotsm} \\rd x} {i \\mathop = 1} {i \\mathop = n}$ Then the '''Hamilonian''' $H$ corresponding to $J \\sqbrk {\\dotsm y_i \\dotsm}$ is defined as :$\\displaystyle H = -F + \\sum_{i \\mathop = 1}^n y_i' F_{y_i'}$ {{NamedforDef|William Rowan Hamilton|cat = Hamilton}}"}
+{"_id": "27697", "title": "Definition:Canonical Variable", "text": "Let $\\map {\\mathbf y} x : \\R \\to \\R^n$, $n \\in \\N $ be a vector-valued function. Let $F: \\R^{2 n + 1} \\to \\R$ be a differentiable mapping. Let $J \\sqbrk {\\mathbf y}$ be a functional of the form: :$\\d J \\sqbrk {\\mathbf y} = \\int_{x_0}^{x_1} \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ Consider the (real) variables $x, \\mathbf y, \\mathbf y'$, and the mapping $F$. Make the following transformation: :$F_{y_i'} = p_i$ where $F_x$ denotes the partial derivative of $F$ {{WRT}} $x$. Let $H$ be the Hamiltonian corresponding to $J \\sqbrk {\\mathbf y}$. Then the new variables $x, \\mathbf y, \\mathbf p$, and the mapping $H$ corresponding to $J \\sqbrk {\\mathbf y}$ are called '''the canonical variables'''."}
+{"_id": "27699", "title": "Definition:Time/Unit/Week", "text": ":$1$ '''week''' $=7$ days."}
+{"_id": "27700", "title": "Definition:Time/Unit/Week/Symbol", "text": "The symbol for the '''week''' is $\\mathrm {week}$ or $\\mathrm {wk}$."}
+{"_id": "27701", "title": "Definition:Time/Unit/Fortnight", "text": ":$1$ '''fortnight''' $=2$ weeks $=14$ days."}
+{"_id": "27703", "title": "Definition:O Notation/Big-O Notation/Parameter", "text": "Let $X$ be a topological space. Let $V$ be a normed vector space over $\\R$ or $\\C$ with norm $\\norm {\\, \\cdot \\,}$ Let $x_0 \\in X \\cup \\set \\infty$. Let $A$ be a set. Let $X_\\alpha$ be a subset of $X$ for every $\\alpha \\in A$, which, if $x_0 \\ne \\infty$, contains $x_0$. Let $f_\\alpha : X_\\alpha \\setminus \\set {x_0} \\to V$ be a mapping for every $\\alpha \\in A$. Let $g : X \\to V$ be a mapping. The statement: :$f_\\alpha = \\map {\\OO_\\alpha} g$ as $x \\to x_0$ is equivalent to: :$\\forall \\alpha \\in A : f_\\alpha = \\map \\OO g$ as $x \\to x_0$ The $\\OO$-estimate is said to be '''independent''' of $\\alpha \\in A$ {{iff}}: :there exists a neighborhood $U$ of $x_0$ in $X$ such that: ::$\\exists c \\in \\R: c \\ge 0 : \\forall \\alpha \\in A : \\forall x \\in \\paren {U \\setminus \\set {x_0} } \\cap X_\\alpha : \\norm {\\map {f_\\alpha} x} \\le c \\cdot \\norm {\\map g x}$ That is, if the implied constant and implied neighborhood can be chosen the same for all $\\alpha \\in A$."}
+{"_id": "27705", "title": "Definition:O Notation/Big-O Notation/Real", "text": "=== Estimate at infinity === {{:Definition:O Notation/Big-O Notation/Real/Infinity}} === Point Estimate === {{:Definition:O Notation/Big-O Notation/Real/Point}}"}
+{"_id": "27706", "title": "Definition:O Notation/Big-O Notation/Complex", "text": "=== Estimate at infinity === {{:Definition:O Notation/Big-O Notation/Complex/Infinity}} === Point Estimate === {{:Definition:O Notation/Big-O Notation/Complex/Point}}"}
+{"_id": "27707", "title": "Definition:O Notation/Big-O Notation/General Definition/Infinity", "text": "Let $\\left({X, \\tau}\\right)$ be a topological space. Let $V$ be a normed vector space over $\\R$ or $\\C$ with norm $\\left\\Vert{\\,\\cdot\\,}\\right\\Vert$. Let $f, g : X \\to V$ be functions. The statement: :$f \\left({x}\\right) = \\mathcal O \\left({g \\left({x}\\right)}\\right)$ as $x \\to \\infty$ is equivalent to: :There exists a neighborhood of infinity $U \\subset X$ such that: ::$\\exists c \\in \\R: c \\ge 0: \\forall x \\in U: \\left\\Vert{f \\left({x}\\right)}\\right\\Vert \\le c \\cdot \\left\\Vert{g \\left({x}\\right)}\\right\\Vert$ That is: :$\\Vert f \\left({x}\\right) \\Vert \\le c \\cdot \\Vert g \\left({x}\\right) \\Vert$ for all $x$ in a neighborhood of infinity."}
+{"_id": "27708", "title": "Definition:Time/Unit/Month", "text": "The '''month''' is measured only approximately in duration, and is not used as a scientific measure. :$1$ '''month''' is between $28$ and $31$ days, depending on which '''month''' it is."}
+{"_id": "27710", "title": "Definition:Time/Unit/Season", "text": "The '''season''' is measured only approximately in duration, and is not used as a scientific measure. :$1$ '''season''' $= 3$ months, although exactly which months fall into which '''season''' depends on the locale and the culture."}
+{"_id": "27711", "title": "Definition:Neighborhood of Infinity (Complex Analysis)", "text": "A '''neighborhood of $\\infty$''' in $\\C$ is a subset of the set of complex numbers $\\C$ wich contains a set of the form $\\{z \\in \\C : |z| > r\\}$ for some $r\\in\\R$. That is, a subset which contains all complex numbers whose modulus is sufficiently large."}
+{"_id": "27713", "title": "Definition:Covolume of Lattice", "text": "Let $L$ be a lattice in $\\R^n$. Let $\\left({v_1, \\ldots, v_n}\\right)$ be an ordered basis for $L$. Let $v_i = \\left({v_{i 1}, \\ldots , v_{i n} }\\right)$ for $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. The '''covolume''' of $L$ is the determinant of the matrix: :$\\begin{bmatrix} v_{1 1} & v_{1 2} & \\cdots & v_{1 n} \\\\ v_{2 1} & v_{2 2} & \\cdots & v_{2 n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ v_{n 1} & v_{n 2} & \\cdots & v_{n n} \\\\ \\end{bmatrix}$"}
+{"_id": "27714", "title": "Definition:Integer Lattice", "text": "Let $n$ be a positive integer. The '''integer lattice''' in $\\R^n$ is the lattice $\\Z^n$. Category:Definitions/Linear Algebra Category:Definitions/Geometry of Numbers 01q0zkrsaqbk9hph0uhv482i84e1ows"}
+{"_id": "27715", "title": "Definition:Fundamental Domain (Lattice)", "text": "Let $L \\subset \\R^n$ be an integral lattice. Let $B = \\left({v_1, \\ldots, v_n}\\right)$ be an ordered basis for $L$. The '''fundamental domain''' of $L$ associated to $B$ is the set: :$\\displaystyle \\left\\{ {\\sum_{i \\mathop = 1}^n \\lambda_i v_i: \\forall i: 0 \\le \\lambda_i < 1}\\right\\}$ Category:Definitions/Linear Algebra Category:Definitions/Geometry of Numbers nh0m2th0ored6dymzrx9lt7pjpnoojz"}
+{"_id": "27716", "title": "Definition:Integral Lattice", "text": "Let $n$ be a positive integer. An '''integral lattice''' in $\\R^n$ is a subgroup that is isomorphic to the integer lattice $\\Z^n$."}
+{"_id": "27717", "title": "Definition:Square Pyramidal Number", "text": "'''Square pyramidal numbers''' are those denumerating a collection of objects which can be arranged in the form of a square pyramid."}
+{"_id": "27718", "title": "Definition:Discrete Group", "text": "A '''discrete group''' is a topological group whose topology is discrete. Category:Definitions/Topological Groups 2yt2c8qbqpt5dqn8tinr0cwz056jo7p"}
+{"_id": "27720", "title": "Definition:Discrete Subgroup", "text": "Let $G$ be a topological group. Let $H$ be a subgroup of $G$. Then $H$ is called a '''discrete subgroup''' if it is a discrete group for the induced topology. === Real Numbers === {{:Definition:Discrete Subgroup/Real Numbers}} Category:Definitions/Subgroups Category:Definitions/Topological Groups 84q4o7iejd52d7n595dv1qhcr5fxy1p"}
+{"_id": "27721", "title": "Definition:Limit Inferior of Moore-Smith Sequence", "text": "Let $\\left({S, \\preceq}\\right)$ be a directed set. Let $L = \\left({T, \\precsim}\\right)$ be a complete lattice. Let $N: S \\to T$ be a Moore-Smith sequence in $T$. Then '''limit inferior''' of $N$ is defined as follows: :$\\liminf N := \\sup_L \\left\\{ {\\inf_L \\left({N \\left[{\\preceq \\left({j}\\right)}\\right]}\\right): j \\in S}\\right\\}$ where :$\\preceq \\left({j}\\right)$ denotes the image of $j$ by $\\preceq$, :$N \\left[{\\preceq \\left({j}\\right)}\\right]$ denotes the image of $\\preceq \\left({j}\\right)$ under $N$."}
+{"_id": "27722", "title": "Definition:Discrete Subgroup/Real Numbers", "text": "Let $G$ be a subgroup of the additive group of real numbers. Then $G$ is '''discrete''' {{iff}}: :$\\forall g \\in G : \\exists \\epsilon > 0: \\openint {g - \\epsilon} {g + \\epsilon} \\cap G = \\set g$ That is, there exists a neighborhood of $g$ which contains no other elements of $G$."}
+{"_id": "27724", "title": "Definition:Everywhere Dense/Real Numbers", "text": "Let $S$ be a subset of the real numbers. Then $S$ is '''(everywhere) dense''' in $\\R$ {{iff}}: :$\\forall x \\in \\R : \\forall \\epsilon \\in \\R : \\epsilon > 0: \\exists s \\in S: x - \\epsilon < s < x + \\epsilon$. That is, {{iff}} in every neighborhood of every real number lies an element of $S$."}
+{"_id": "27726", "title": "Definition:Poisson Bracket", "text": "Let  :$\\map A {x, \\mathbf y, \\mathbf p}: \\R^{2n+1} \\to \\R$, :$\\map B {x, \\mathbf y, \\mathbf p}: \\R^{2n+1} \\to \\R$, $n \\in \\N$  be real functions, dependent on canonical variables. The '''Poisson Bracket''' of functions $A$ and $B$ is defined as: :$\\displaystyle \\sqbrk {A, B} := \\sum_{i \\mathop = 1}^n \\paren {\\frac {\\partial A} {\\partial y_i} \\frac {\\partial B} {\\partial p_i} - \\frac {\\partial B} {\\partial y_i} \\frac {\\partial A} {\\partial p_i} }$ where the notation $\\dfrac {\\partial A} {\\partial y_i}$ denotes partial differentiation. {{NamedforDef|Siméon-Denis Poisson|cat = Poisson}}"}
+{"_id": "27727", "title": "Definition:First Integral of System of Differential Equations", "text": "Let $S$ be a system of differential equations. Let $g$ be a function which satisfies $S$. Let $f$ be a function. {{explain|Establish the domain and codomain of these functions}} Let $f$ depend on variables (denoted by ordered tuples here) of $S$ independently as well as through $g$ and its derivatives: :$f = \\map f {\\sequence {x_i} _{1 \\mathop \\le i \\mathop \\le n}, \\sequence {g^{\\paren j} \\paren {\\sequence {x_i}_{1 \\mathop \\le i \\mathop \\le n} }_{0 \\mathop \\le j \\mathop \\le k} } }, \\quad {n, k} \\in \\N$ Suppose there exists $g$ such that $f$ is a constant. Then $f$ is the first integral of $S$."}
+{"_id": "27728", "title": "Definition:Proper Mapping", "text": "Let $X$ and $Y$ be topological spaces. A mapping $f: X \\to Y$ is '''proper''' {{iff}} for every compact subset $K \\subset Y$, its preimage $f^{-1} \\left({K}\\right)$ is also compact. {{explain|Here and elsewhere: is it worth replacing \"compact subset\" with \"compact subspace\" wherever it appears, as \"subset\" implies just the set of elements -- \"subspace\" makes it clear that the topology imposed on that subset is important. I don't know whether this flies against the prevailing convention, but it would help to make the language of this complicated area, where extreme clarity is required, more easily understood.}}"}
+{"_id": "27729", "title": "Definition:Generator Set of Filter", "text": "Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Let $F$ be a filter on $L$. The '''generator set''' $G$ of $F$ is defined as follows: :$F = \\left({\\operatorname{fininfs}\\left({G}\\right)}\\right)^\\succeq$ where :$\\operatorname{fininfs}\\left({G}\\right)$ denotes the finite infima set of $G$, :$G^\\succeq$ denotes the upper closure of $G$."}
+{"_id": "27730", "title": "Definition:Proper Group Action", "text": "Let $G$ be a topological group. Let $X$ be a topological space. A group action $\\phi: G \\times X \\to X$ is called '''proper''' {{iff}} $\\phi$ is a proper mapping. Here $G\\times X$ is equipped with the product topology."}
+{"_id": "27731", "title": "Definition:Continuous Group Action", "text": "Let $G$ be a topological group. Let $X$ be a topological space. A group action $\\phi: G \\times X \\to X$ is defined as '''continuous''' {{iff}} $\\phi$ is continuous."}
+{"_id": "27732", "title": "Definition:Loop (Topology)", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\gamma: \\closedint 0 1 \\to S$ be a path in $T$. $\\gamma$ is a '''loop (in $T$)''' {{iff}}: :$\\map \\gamma 0 = \\map \\gamma 1$ === Base Point === {{Definition:Loop (Topology)/Base Point}}"}
+{"_id": "27733", "title": "Definition:Concatenation of Paths", "text": "Let $X$ be a topological space. Let $f, g: \\left[{0 \\,.\\,.\\, 1}\\right] \\to X$ be paths. Suppose that $f \\left({1}\\right) = g \\left({0}\\right)$. The '''concatenation''' of $f$ and $g$ is the mapping $f * g: \\left[{0 \\,.\\,.\\, 1}\\right] \\to X$ defined by: :$\\displaystyle \\left({f * g}\\right) \\left({s}\\right) = \\begin{cases} f \\left({2 s}\\right) & : 0 \\le s \\le \\dfrac 1 2 \\\\ g \\left({2 s - 1}\\right) & : \\dfrac 1 2 \\le s \\le 1 \\end{cases}$"}
+{"_id": "27734", "title": "Definition:Loop (Topology)/Base Point", "text": "The '''base point''' of $\\gamma$ is $\\gamma \\left ({0}\\right)$."}
+{"_id": "27735", "title": "Definition:Homotopy/Path", "text": "Let $X$ be a topological space. Let $f, g: \\left[{0 \\,.\\,.\\, 1}\\right] \\to X$ be paths. We say that $f$ and $g$ are '''path-homotopic''' if they are homotopic relative to $\\left\\{ {0, 1}\\right\\}$."}
+{"_id": "27736", "title": "Definition:Homotopy Class/Path", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $f: \\left[{0 \\,.\\,.\\, 1}\\right] \\to S$ be a path in $T$. The '''homotopy class''' of the path $f$ is the homotopy class of $f$ relative to $\\left\\{ {0, 1}\\right\\}$. That is, the equivalence class of $f$ under the equivalence relation defined by path-homotopy."}
+{"_id": "27737", "title": "Definition:Nontotient", "text": "A '''nontotient''' is a positive even integer $n$ such that: :$\\nexists m \\in \\Z_{>0}: \\phi \\left({m}\\right) = n$ where $\\phi \\left({m}\\right)$ denotes the Euler $\\phi$ function. That is, a '''nontotient''' is a positive even integer which is not the totient of any positive integer."}
+{"_id": "27738", "title": "Definition:Compact-Open Topology", "text": "Let $X$ and $Y$ be topological spaces. Let $\\map \\CC {X, Y}$ be the set of continuous maps from $X$ to $Y$. For all compact subsets $K \\subset X$ and all open subsets $U \\subset Y$, let: :$\\map V {K, U} = \\set {f \\in \\map \\CC {X, Y}: f \\sqbrk K \\subset U}$ Let: :$\\BB = \\set {\\map V {K, U}: K \\subset X \\text{ compact}, U \\subset Y \\text{ open} }$ The '''compact-open topology''' on $\\map \\CC {X, Y}$ is the topology generated by $\\BB$."}
+{"_id": "27739", "title": "Definition:Multiplication of Homotopy Classes of Paths", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $\\alpha, \\beta$ be homotopy classes of paths in $T$. Let $f, g: \\left[{0 \\,.\\,.\\, 1}\\right] \\to S$ be representative paths for $\\alpha$ and $\\beta$ respectively. Let $f \\left({1}\\right) = g \\left({0}\\right)$. The '''product of the homotopy classes''' $\\alpha$ and $\\beta$ is the homotopy class of the concatenated path $f * g$."}
+{"_id": "27740", "title": "Definition:Legendre Transform", "text": "Let $\\map f x$ be a strictly convex real function. Let $p = \\map {f'} x$. Let $\\map {f^*} p = - \\map f{\\map x p} + p \\map x p$. {{explain|The above seems to imply that $x$ is a function.<br/>Yes it does, doesn't it. Exactly what it does mean is to be added to this page, and if necessary a new definition page is needed to specify it.}} The '''Legendre Transform on $x$ and $f$''' is the mapping of the variable and function pair: :$\\paren{x, \\map f x} \\to \\paren{p, \\map {f^*} p}$ {{expand|generalise, add properties and connect with Young's inequality<br/>But not in here, do it somewhere else.}} {{NamedforDef|Adrien-Marie Legendre|cat = Legendre}}"}
+{"_id": "27741", "title": "Definition:Maximal Normal Subgroup", "text": "Let $G$ be a group. Let $N \\le G$ be a proper normal subgroup. Then $N$ is a '''maximal normal subgroup of $G$''' iff: :For every normal subgroup $M$ of $G$, $N \\subseteq M \\subseteq G$ implies $N = M$ or $M = G$. That is, if there is no normal subgroup of $G$, except $N$ and $G$ itself, which contains $N$."}
+{"_id": "27742", "title": "Definition:Ascending Chain Condition", "text": "Let $\\left({P, \\leq}\\right)$ be an ordered set. Then $S$ is said to have the '''ascending chain condition (ACC)''' if every increasing sequence $x_1 \\leq x_2 \\leq x_3 \\leq \\cdots$ with $x_i \\in P$ eventually terminates: there is $n \\in \\N$ such that $x_n = x_{n+1} = \\cdots$. === ACC on submodules === {{:Definition:Ascending Chain Condition/Module}} === ACC on ideals === {{:Definition:Ascending Chain Condition/Ideals}} === ACC on principal ideals === {{:Definition:Ascending Chain Condition/Principal Ideals}}"}
+{"_id": "27743", "title": "Definition:Ascending Chain Condition/Ideals", "text": "Let $R$ be a commutative ring. Then $R$ is said to have the '''ascending chain condition on ideals''' {{iff}} every increasing sequence of ideals stabilizes."}
+{"_id": "27744", "title": "Definition:Nontotient/Sequence", "text": "The sequence of '''nontotients''' begins: :$14, 26, 34, 38, 50, 62, 68, 74, 76, 86, 90, 94, 98, 114, 118, 122, 124, 134, 142, 146, 152, 154, \\ldots$"}
+{"_id": "27745", "title": "Definition:Cofactor Matrix", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $A_{r s}$ denote the cofactor of the element whose indices are $\\tuple {r, s}$. The '''cofactor matrix''' of $\\mathbf A$ is the square matrix of order $n$: :$\\mathbf C = \\begin {bmatrix} A_{1 1} & A_{1 2} & \\cdots & A_{1 n} \\\\ A_{2 1} & A_{2 2} & \\cdots & A_{2 n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ A_{n 1} & A_{n 2} & \\cdots & A_{n n} \\\\ \\end {bmatrix}$"}
+{"_id": "27746", "title": "Definition:Fibonacci-Like Sequence", "text": "Let $A = \\left({a_0, a_1, \\ldots, a_{n - 1} }\\right)$ be an ordered tuple of numbers. The '''Fibonacci-like sequence formed from $A$''' is defined as: :$F_A \\left({k}\\right) = \\begin{cases} \\qquad \\qquad a_k & : 0 \\le k < n \\\\ & \\\\ \\displaystyle \\sum_{k - n \\mathop \\le j \\mathop < k} a_j & : k \\ge n \\end{cases}$ That is, apart from the first $n$  terms, every term is the sum of the previous $n$ terms. The main term can also be expressed as: :$F_A \\left({k}\\right) = 2 F_A \\left({k - 1}\\right) - F_A \\left({k - n}\\right)$"}
+{"_id": "27747", "title": "Definition:Keith Number", "text": "Let $K \\in \\Z_{>9}$ be an $n$-digit integer, where $n > 1$. Let $A = \\left({a_1, a_2, \\ldots, a_n}\\right)$ be the digits of $K$ in order. Let $F_A$ be the Fibonacci-like sequence on $A$. Then $K$ is a '''Keith number''' {{iff}} $K$ occurs somewhere in $F_A$."}
+{"_id": "27748", "title": "Definition:Keith Number/Sequence", "text": "The sequence of '''Keith numbers''' begins: :$14, 19, 28, 47, 61, 75, 197, 742, 1104, \\ldots$"}
+{"_id": "27750", "title": "Definition:Dorroh Extension", "text": "Let $R$ be a ring. We define two operations on the cartesian product $R \\times \\Z$ as: :$\\tuple {r, n} + \\tuple {s ,m} = \\tuple {r + s, n + m}$ :$\\tuple {r, n} \\cdot \\tuple {s, m} = \\tuple {r s + n s + m r, n m}$ The '''Dorroh extension''' of $R$ is the ring $\\struct {R \\times \\Z, +, \\cdot}$."}
+{"_id": "27751", "title": "Definition:Ramanujan-Nagell Equation", "text": "The '''Ramanujan-Nagell equation''' is the Diophantine equation: :$x^2 + 7 = 2^n$"}
+{"_id": "27754", "title": "Definition:Canonical Transformation", "text": "Let $\\paren{x,\\mathbf y,\\mathbf p,H}$ be canonical variables. Let $\\paren{x,\\mathbf Y,\\mathbf P,H^*}$ be another set of canonical variables. A mapping between these is a '''canonical transformation''' {{iff}}: :$\\dfrac {\\d y_i} {\\d x} = \\dfrac {\\partial H} {\\partial p_i},\\quad\\dfrac {\\d p_i} {\\d x}=-\\dfrac {\\partial H} {\\partial y_i}$ imply: :$\\dfrac {\\d Y_i} {\\d x}=\\dfrac {\\partial H^*} {\\partial P_i},\\quad\\dfrac {\\d P_i} {\\d x}=-\\dfrac {\\partial H^*} {\\partial Y_i}$"}
+{"_id": "27755", "title": "Definition:Irreducible Subset (Topology)", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $A$ be a subset of $S$. Then $A$ is '''irreducible (subset)''' {{iff}} :$A$ is non-empty and closed and for all closed subsets $B, C$ of $S$: $A = B \\cup C \\implies A = B$ or $A = C$"}
+{"_id": "27756", "title": "Definition:Invariant Functional under Transformation", "text": "Let $y_i$, $F$, $\\Phi$, $\\Psi$ be real functions. Let $\\mathbf y = \\sequence {y_i}_{1 \\mathop \\le i \\mathop \\le n}$. Let $\\ds J \\sqbrk {\\mathbf y} = \\int_{x_0}^{x_1} \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ be a functional. Let: :$X = \\map \\Phi {x, \\mathbf y, \\mathbf y'}$ :$\\mathbf Y = \\map {\\mathbf \\Psi} {x, \\mathbf y, \\mathbf y'}$ Let curve $\\gamma$ defined by:  :$\\mathbf y = \\map {\\mathbf y} x, \\quad x_0 \\le x \\le x_1$  be transformed into a curve $\\Gamma$ defined by: :$\\mathbf Y = \\map {\\mathbf Y} X, \\quad X_0 \\le X \\le X_1$ Then the functional $J \\sqbrk {\\mathbf y}$ is '''invariant''' under the given transformation {{iff}}: :$J \\sqbrk \\Gamma = J \\sqbrk \\gamma$ That is, {{iff}}: :$\\displaystyle \\int_{x_0}^{x_1} \\map F {x, \\mathbf y, \\mathbf y'} \\rd x = \\int_{X_0}^{X_1} \\map F {X, \\mathbf Y, \\mathbf Y'} \\rd X$"}
+{"_id": "27758", "title": "Definition:Dense (Order Theory)/Element", "text": "Let $x \\in S$. Then $x$ is '''dense''' {{iff}} :$\\forall y \\in S: y \\ne \\bot \\implies x \\wedge y \\ne \\bot$ where $\\bot$ denotes the smallest element in $L$."}
+{"_id": "27759", "title": "Definition:Dense (Order Theory)/Subset", "text": "Let $A$ be a subset of $S$. Then $A$ is '''dense''' {{iff}} it includes only dense elements. That means that iff $\\forall x \\in A: x$ is a dense element."}
+{"_id": "27760", "title": "Definition:Dense (Order Theory)", "text": "Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be a bounded below meet semilattice. === Dense Element === {{:Definition:Dense (Order Theory)/Element}} === Dense Subset === {{:Definition:Dense (Order Theory)/Subset}} Category:Definitions/Order Theory iuhfbmay1t7m60yyawdxgnb34uzgi9a"}
+{"_id": "27762", "title": "Definition:Almost Perfect Number/Definition 1", "text": "Let $A \\left({n}\\right)$ denote the abundance of $n$. $n$ is '''almost perfect''' {{iff}} $A \\left({n}\\right) = -1$."}
+{"_id": "27764", "title": "Definition:Almost Perfect Number/Definition 2", "text": "$n$ is '''almost perfect''' {{iff}}: :$\\sigma \\left({n}\\right) = 2 n - 1$ where $\\sigma \\left({n}\\right)$ denotes the $\\sigma$ function of $n$."}
+{"_id": "27765", "title": "Definition:Almost Perfect Number/Definition 3", "text": "$n$ is '''almost perfect''' {{iff}} it is exactly one greater than the sum of its aliquot parts."}
+{"_id": "27770", "title": "Definition:Geodetic Distance", "text": "Let $y_i$, $F$ be real functions. Let $\\mathbf y = \\sequence {y_i}_ {1 \\mathop \\le i \\mathop \\le n}$ be a vector.  Let: :$\\ds J \\sqbrk {\\mathbf y} = \\int_{x_0}^{x_1} \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ be a functional with only one extremal passing any two points: :$A = \\map A {x_0, \\mathbf y^0}$ :$B = \\map B {x_1, \\mathbf y^1}$ Suppose a curve $\\gamma$ is an extremal of $J$. {{explain|Extremum or extremal?}} Then: :$\\ds \\map S {x_0, x_1, \\mathbf y} = \\int_{x_0}^{x_1} \\map F {x, \\mathbf y, \\mathbf y'} \\big \\rvert_{\\gamma}\\rd x$ is called a '''geodetic distance''' between $A$ and $B$."}
+{"_id": "27771", "title": "Definition:Hamilton-Jacobi Equation", "text": "Let $\\map S {x_0, x_1, \\mathbf y} = \\map S {x, \\mathbf y}$ be the geodetic distance, where $x_0$ is fixed and $x_1 = x$. Let $H$ be Hamiltonian. The differential equation: :$\\displaystyle \\frac {\\partial S} {\\partial x} + \\map H {x, \\mathbf y, \\nabla_{\\mathbf y} S} = 0$ is known as the '''Hamilton-Jacobi Equation'''. {{NamedforDef|William Rowan Hamilton|name2 = Carl Gustav Jacob Jacobi|cat = Hamilton|cat2 = Jacobi}}"}
+{"_id": "27773", "title": "Definition:Quasiperfect Number/Definition 2", "text": "$n$ is '''quasiperfect''' {{iff}}: :$\\sigma \\left({n}\\right) = 2 n + 1$ where $\\sigma \\left({n}\\right)$ denotes the $\\sigma$ function of $n$."}
+{"_id": "27774", "title": "Definition:Quasiperfect Number/Definition 3", "text": "$n$ is '''quasiperfect''' {{iff}} it is exactly one less than the sum of its aliquot parts."}
+{"_id": "27775", "title": "Definition:Principal Ideal of Ordered Set/Definition 1", "text": "Then $I$ is '''principal (ideal)''' {{iff}} :$\\exists x \\in I: x$ is upper bound for $I$"}
+{"_id": "27776", "title": "Definition:Principal Ideal of Ordered Set/Definition 2", "text": "Then $I$ is '''principal (ideal)''' {{iff}} :$\\exists x \\in S: I = x^\\preceq$ where $x^\\preceq$ denotes the lower closure of $x$."}
+{"_id": "27777", "title": "Definition:Principal Ideal of Ordered Set", "text": "Let $\\left({S, \\preceq}\\right)$ be a preordered set. Let $I$ be an ideal in $S$. === Definition 1 === {{:Definition:Principal Ideal of Ordered Set/Definition 1}} === Definition 2 === {{:Definition:Principal Ideal of Ordered Set/Definition 2}}"}
+{"_id": "27778", "title": "Definition:Wallpaper Pattern", "text": "A '''wallpaper pattern''' is a plane geometrical figure which has translational symmetry in two non-parallel lines."}
+{"_id": "27779", "title": "Definition:Euler Lucky Number", "text": "=== Definition 1 === {{:Definition:Euler Lucky Number/Definition 1}} === Definition 2 === {{:Definition:Euler Lucky Number/Definition 2}}"}
+{"_id": "27780", "title": "Definition:Stern Prime", "text": "A '''Stern prime''' is a prime number which can not be represented in the form: :$2 a^2 + p$ where: :$a \\in \\Z_{>0}$ is a (strictly) positive integer :$p$ is a prime number. === Sequence === {{:Definition:Stern Prime/Sequence}}"}
+{"_id": "27782", "title": "Definition:Stern Number", "text": "A '''Stern number''' is an odd number which can not be represented in the form: :$2 a^2 + p$ where: :$a \\in \\Z_{>0}$ is a (strictly) positive integer :$p$ is a prime number."}
+{"_id": "27786", "title": "Definition:Centered Hexagonal Number", "text": "A '''centered hexagonal number''' is a figurate number which is represented by: :a point in the center :all other points surrounding the center at the points of a triangular lattice."}
+{"_id": "27788", "title": "Definition:Centered Hexagonal Number/Sequence", "text": "The sequence of centered hexagonal numbers, for $n \\in \\Z_{> 0}$, begins: :$1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, \\ldots$"}
+{"_id": "27789", "title": "Definition:Continuous Lattice Subframe", "text": "Let $L = \\left({X, \\preceq}\\right)$ be an ordered set. Let $S = \\left({Y, \\preceq'}\\right)$ be an ordered subset of $L$. Then $S$ is '''continuous lattice subframe''' of $L$ {{iff}} :$S$ inherits infima and directed suprema."}
+{"_id": "27790", "title": "Definition:Magic Hexagon", "text": "A magic hexagon is a centered hexagonal pattern of cells containing the natural numbers from $1$ to the number of cells such that the numbers in all the cells in a given straight line all add up to the same number."}
+{"_id": "27792", "title": "Definition:Magic Hexagon/Order", "text": "Let $H$ be an magic hexagon created on the $n$th centered hexagonal number. That is, let $H$ have $n$ cells along each side. Then $H$ is an '''order $n$ magic hexagon'''."}
+{"_id": "27796", "title": "Definition:Magic Hexagon/Line", "text": "A '''line''' of a magic hexagon is a set of consecutively adjacent cells, in any of the three directions parallel to the sides of the underlying hexagon."}
+{"_id": "27798", "title": "Definition:Semiperfect Number", "text": "A '''semiperfect number''' is a positive integer which is equal to the sum of some or all of its aliquot parts."}
+{"_id": "27800", "title": "Definition:Semiperfect Number/Sequence", "text": "The sequence of '''semiperfect numbers''' begins: :$6, 12, 18, 20, 24, 28, 30, \\ldots$"}
+{"_id": "27801", "title": "Definition:Vigesimal System", "text": "The '''vigesimal system''' is base $20$ notation. That is, every number $x \\in \\R$ is expressed in the form: :$\\displaystyle x = \\sum_{j \\mathop \\in \\Z} r_j 20^j$ where: :$\\forall j \\in \\Z: r_j \\in \\set {0, 1, \\ldots, 19}$"}
+{"_id": "27806", "title": "Definition:Bilinear Functional", "text": "Let $y_1$, $y_2$, $z$ be mappings, belonging to some normed linear space. {{explain|See talk page.}} Let $S$ be a set of ordered pairs $\\tuple {y_1, y_2}$. Let $B: S \\to \\R$ be a mapping defined as: :$\\forall \\tuple {y_1, y_2} \\in S: \\exists x \\in \\R: B \\sqbrk {y_1, y_2} = x$ {{explain|Establish the precise meaning of $B \\sqbrk {y_1, y_2}$, and see whether it actually might mean the same as $\\map B {y_1, y_2}$, in which case use the latter. If not, explain what it does mean.}} Let $B$ be linear {{WRT}} $y_1$ for fixed $y_2$, and linear {{WRT}} $y_2$ for fixed $y_1$: :$B \\sqbrk {\\alpha y_1 + \\beta z, y_2} = \\alpha B \\sqbrk {\\alpha y_1, y_2} + \\beta B \\sqbrk {z, y_2}$ :$B \\sqbrk {y_1, \\alpha y_2 + \\beta z} = \\alpha B \\sqbrk {y_1, y_2} + \\beta B \\sqbrk {y_1, z}$ where $\\alpha, \\beta \\in \\R$. Then $B: S \\to \\R$ is known as a '''bilinear functional''', denoted by $B \\sqbrk {y_1, y_2}$"}
+{"_id": "27807", "title": "Definition:Quadratic Functional", "text": "Let $B \\sqbrk {x, y}$ be a bilinear functional. Let $x = y$. Then the functional $A \\sqbrk x = B \\sqbrk {x, x}$ is called a '''quadratic functional'''."}
+{"_id": "27808", "title": "Definition:Bilinear Form", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. A '''bilinear form''' on $M$ is a bilinear mapping $b : M \\times M \\to R$."}
+{"_id": "27809", "title": "Definition:Twice Differentiable/Functional", "text": "Let $\\Delta J \\sqbrk {y; h}$ be an increment of a functional. Let: :$\\Delta J \\sqbrk {y; h} = \\phi_1 \\sqbrk {y; h} + \\phi_2 \\sqbrk {y; h} + \\epsilon \\size h^2$ where: :$\\phi_1 \\sqbrk {y; h}$ is a linear functional :$\\phi_2 \\sqbrk {y; h}$ is a quadratic functional {{WRT}} $h$ :$\\epsilon \\to 0$ as $\\size h \\to 0$. Then the functional $J\\sqbrk y$ is '''twice differentiable'''. {{refactor|Extract the following two into their own definition pages, appropriately named and transcluded.}} The linear part $\\phi_1$ is the '''first variation''', denoted: :$\\delta J \\sqbrk {y; h}$ $\\phi_2$ is called the '''second variation''' (or '''differential''') of a functional, and is denoted: :$\\delta^2 J \\sqbrk {y; h}$"}
+{"_id": "27810", "title": "Definition:Integer Square", "text": "An '''integer square''' is a square whose sides are integers. Category:Definitions/Geometry Category:Definitions/Recreational Mathematics 0jvtx0iavnfxizw8ornszqt2o2nd0t4"}
+{"_id": "27811", "title": "Definition:Perfect Square Dissection", "text": "A '''perfect square dissection''' is a dissection of an integer square into a number of smaller integer squares all of different sizes."}
+{"_id": "27812", "title": "Definition:Finite Projective Plane/Order 4", "text": "The finite projective plane of order $4$. It can be denoted: :$\\operatorname {PG} \\left({4, 4}\\right)$ where: :$\\operatorname {PG}$ stands for '''projective geometry''' :the first parameter $2$ specifies the dimension :the second parameter $2$ specifies the order. $\\operatorname {PG} \\left({4, 4}\\right)$ has the following properties: : It has $21$ points : It has $21$ lines : There are $5$ points on each line : There are $5$ lines through each point."}
+{"_id": "27813", "title": "Definition:Minimum Value of Functional", "text": "Let $S$ be a set of mappings. Let $y, \\hat y \\in S: \\R \\to \\R$ be real functions. Let $J \\sqbrk y: S \\to \\R$ be a functional. Let $J$ have a (relative) extremum for $y = \\hat y$. Let $J \\sqbrk y - J \\sqbrk {\\hat y} \\ge 0$ in the neighbourhood of $y = \\hat y$. {{Disambiguate|Definition:Neighborhood}} Then this extremum is called the '''minimum''' of the functional $J$."}
+{"_id": "27814", "title": "Definition:Maximum Value of Functional", "text": "Let $S$ be a set of mappings. Let $y, \\hat y \\in S: \\R \\to \\R$ be real functions. Let $J \\sqbrk y: S \\to \\R $ be a functional. Let $J$ have a (relative) extremum for $y = \\hat y$. Suppose, $J \\sqbrk y - J \\sqbrk {\\hat y} \\le 0$ in the neighbourhood of $y = \\hat y$. {{explain|Make sure the link to Definition:Neighborhood invokes the correct subpage -- if there is no such appropriate subpage, then write it}} Then this extremum is called the '''maximum''' of the functional $J$."}
+{"_id": "27815", "title": "Definition:Strongly Positive Quadratic Functional", "text": "Let $J\\sqbrk y$ be a quadratic functional {{WRT}} $y$, defined on normed linear space. Suppose there exists $k\\in\\R_{> 0}$ such that: :$J\\sqbrk y\\ge k\\size {y}^2$ Then the quadratic functional $J$ is '''strongly positive'''."}
+{"_id": "27816", "title": "Definition:Second Pentagonal Number/Definition 1", "text": "The '''second pentagonal numbers''' are the integers obtained by applying the Closed Form for Pentagonal Numbers to negative $n$: :$\\map {P'} n = \\dfrac {-n \\paren {-3 n - 1} } 2$"}
+{"_id": "27818", "title": "Definition:Second Pentagonal Number", "text": "The '''second pentagonal numbers''' are those denumerating a collection of objects which can be arranged in the form of a regular pentagon."}
+{"_id": "27820", "title": "Definition:Generalized Pentagonal Number", "text": "=== Definition 1 === {{:Definition:Generalized Pentagonal Number/Definition 1}} === Definition 2 === {{:Definition:Generalized Pentagonal Number/Definition 2}}"}
+{"_id": "27822", "title": "Definition:Generalized Pentagonal Number/Definition 1", "text": "Recall the sequence of pentagonal numbers: :$0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, \\ldots$ and the sequence of second pentagonal numbers: :$0, 2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, \\ldots$ The sequence of '''generalized pentagonal numbers''' consists of the elements of both of these sequences merged and arranged into ascending order."}
+{"_id": "27823", "title": "Definition:Generalized Pentagonal Number/Definition 2", "text": "The '''generalized pentagonal numbers''' are the integers obtained from the formula: :$GP_n = \\begin{cases} \\dfrac {m \\left({3 m + 1}\\right)} 2 & : n = 2 m \\\\ \\dfrac {m \\left({3 m - 1}\\right)} 2 & : n = 2 m - 1 \\end{cases}$ for $n = 0, 1, 2, \\ldots$"}
+{"_id": "27826", "title": "Definition:Conjugate Point", "text": "=== Definition 1 === {{:Definition:Conjugate Point/Definition 1}} === Definition 2 === {{:Definition:Conjugate Point/Definition 2}} === Definition 3 === {{:Definition:Conjugate Point/Definition 3}}"}
+{"_id": "27827", "title": "Definition:Factorial Number System", "text": "The '''factorial number system''' is a mixed radix positional numeral system in which the digit in the $i$th column from the right is of base $i$."}
+{"_id": "27829", "title": "Definition:Cullen Number", "text": "A '''Cullen number''' is a positive integer of the form: :$n \\times 2^n + 1$"}
+{"_id": "27831", "title": "Definition:Cullen Prime", "text": "A '''Cullen prime''' is a Cullen number: :$n \\times 2^n + 1$ which is also prime."}
+{"_id": "27833", "title": "Definition:Palindromic Number/Sequence", "text": "The sequence of '''palindromic integers''' in base 10 begins: :$0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, \\ldots$"}
+{"_id": "27834", "title": "Definition:Polygon/Chord", "text": "A '''chord''' of a polygon $P$ is a straight line connecting two non-adjacent vertices of $P$: :300px In the above diagram, $DF$ is a '''chord''' of polygon $ABCDEFG$."}
+{"_id": "27835", "title": "Definition:Convex Polygon", "text": "Let $P$ be a polygon. $P$ is a '''convex polygon''' {{iff}}: :For all points $A$ and $B$ located inside $P$, the line $AB$ is also inside $P$."}
+{"_id": "27836", "title": "Definition:Aliquot Sum", "text": "Let $n \\in \\Z$ be a positive integer. The '''aliquot sum''' of $n$ is defined as the sum of its aliquot parts."}
+{"_id": "27837", "title": "Definition:Sociable Chain", "text": "Let $m$ be a positive integer. Let $\\map s m$ be the aliquot sum of $m$. Define the sequence $\\sequence {a_k}$ recursively as: :$a_{k + 1} = \\begin{cases} m & : k = 0 \\\\ \\map s {a_k} & : k > 0 \\end{cases}$ A '''sociable chain''' is such a sequence $\\sequence {a_k}$ where: :$a_r = a_0$ for some $r > 0$."}
+{"_id": "27838", "title": "Definition:Sociable Chain/Order", "text": "The '''order''' of $a_k$ is the '''smallest''' $r \\in \\Z_{>0}$ such that :$a_r = a_0$"}
+{"_id": "27839", "title": "Definition:Sociable Number", "text": "A '''sociable number''' is a positive integer which occurs as a term in a sociable chain."}
+{"_id": "27840", "title": "Definition:Topological Group Homomorphism", "text": "Let $G$ and $H$ be topological groups. A mapping $\\phi:G \\to H$ is a '''homomorphism of topological groups''' if it is a continuous group homomorphism. Category:Definitions/Topological Groups Category:Definitions/Group Homomorphisms fj4jz7ytxgcrn863hjj99fqcearnq20"}
+{"_id": "27841", "title": "Definition:Quotient Space by Group Action", "text": "Let $G$ be a group acting on a topological space $X$. The '''quotient of $X$ by the action of $G$''', denoted $X/G$, is the set of orbits equipped with the quotient topology."}
+{"_id": "27842", "title": "Definition:Group Action by Homeomorphisms", "text": "Let $G$ be a group. Let $X$ be a topological space. Let $\\phi: G \\times X \\to X$ be a group action Then $G$ '''acts by homeomorphisms''' {{iff}} for all $g \\in G$, the mapping: :$\\phi_g : X \\to X : x \\mapsto \\phi \\left({g, x}\\right)$ is a homeomorphism."}
+{"_id": "27843", "title": "Definition:Locally Connected Space/Definition 1", "text": "A topological space $T = \\struct{S, \\tau}$ is '''locally connected''' {{iff}} each point of $T$ has a local basis consisting entirely of connected sets in $T$."}
+{"_id": "27844", "title": "Definition:Locally Connected Space/Definition 2", "text": "A topological space $T = \\struct{S, \\tau}$ is '''locally connected''' {{iff}} $T$ is weakly locally connected at each point of $T$."}
+{"_id": "27845", "title": "Definition:Locally Connected Space/Definition 3", "text": "A topological space $T = \\struct{S, \\tau}$ is '''locally connected''' {{iff}} it has a basis consisting of connected sets in $T$."}
+{"_id": "27846", "title": "Definition:Locally Path-Connected Space/Definition 3", "text": "A topological space $T = \\struct {S, \\tau}$ is a '''locally path-connected space''' {{iff}} it has a basis consisting of path-connected sets  in $T$."}
+{"_id": "27847", "title": "Definition:Locally Path-Connected Space/Definition 1", "text": "A topological space $T = \\struct{S, \\tau}$ is a '''locally path-connected space''' {{iff}} each point of $T$ has a local basis consisting of path-connected sets in $T$."}
+{"_id": "27848", "title": "Definition:Locally Path-Connected Space/Definition 2", "text": "A topological space $T = \\struct{S, \\tau}$ is a '''locally path-connected space''' {{iff}} each point of $T$ has a neighborhood basis consisting of path-connected sets in $T$."}
+{"_id": "27852", "title": "Definition:Locally Compact Hausdorff Space/Definition 1", "text": "$T$ is a '''locally compact Hausdorff space''' {{iff}} each point of $S$ has a compact neighborhood. That is, {{iff}} $T$ is weakly locally compact."}
+{"_id": "27853", "title": "Definition:Locally Compact Hausdorff Space/Definition 2", "text": "$T$ is a '''locally compact Hausdorff space''' {{iff}} each point has a neighborhood basis consisting of compact sets. That is, {{iff}} $T$ is locally compact (in the general sense)."}
+{"_id": "27854", "title": "Definition:Locally Compact Hausdorff Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a Hausdorff topological space. === Definition 1 === {{:Definition:Locally Compact Hausdorff Space/Definition 1}} === Definition 2 === {{:Definition:Locally Compact Hausdorff Space/Definition 2}}"}
+{"_id": "27855", "title": "Definition:Relatively Closed Set/Definition 1", "text": "'''$A$ is relatively closed in $B$''' {{iff}} $A$ is closed in the relative topology of $B$."}
+{"_id": "27856", "title": "Definition:Relatively Closed Set/Definition 2", "text": "'''$A$ is relatively closed in $B$''' {{iff}} there is a closed set $C \\subseteq S$ with $C \\cap B = A$."}
+{"_id": "27857", "title": "Definition:Zeroth", "text": "'''Zeroth''' is the adjectival form of the ordinal number zero. Category:Definitions/Zero 5jvx0jgag2e6s22dppm6xc4a6y772eg"}
+{"_id": "27858", "title": "Definition:Polynomial in Ring Element", "text": "Let $R$ be a commutative ring. Let $S$ be a subring with unity of $R$. Let $x\\in R$. === Definition 1 === {{Definition:Polynomial in Ring Element/Definition 1}} === Definition 2 === {{Definition:Polynomial in Ring Element/Definition 2}}"}
+{"_id": "27859", "title": "Definition:Polynomial over Ring as Sequence", "text": "A '''polynomial''' over $R$ can be defined as a sequence in $R$ whose domain is $\\N$ and which has finite support. That is, it is an element of the ring of sequences of finite support."}
+{"_id": "27860", "title": "Definition:Polynomial Ring/Monoid Ring on Natural Numbers", "text": "Let $\\N$ denote the additive monoid of natural numbers. Let $R \\left[{\\N}\\right]$ be the monoid ring of $\\N$ over $R$. The '''polynomial ring over $R$''' is the ordered triple $\\left({R \\left[{\\N}\\right], \\iota, X}\\right)$ where: :$X \\in R \\left[{\\N}\\right]$ is the standard basis element associated to $1\\in \\N$. :$\\iota : R \\to R \\left[{\\N}\\right]$ is the canonical mapping."}
+{"_id": "27861", "title": "Definition:Saturation (Equivalence Relation)", "text": "Let $\\sim$ be an equivalence relation on a set $S$. Let $T\\subset S$ be a subset. === Definition 1 === {{:Definition:Saturation (Equivalence Relation)/Definition 1}} === Definition 2 === {{:Definition:Saturation (Equivalence Relation)/Definition 2}} === Definition 3 === {{:Definition:Saturation (Equivalence Relation)/Definition 3}}"}
+{"_id": "27862", "title": "Definition:Saturation (Equivalence Relation)/Definition 1", "text": "The '''saturation of $T$''' is the set of all elements that are equivalent to some element in $T$: :$\\overline T = \\{s \\in S : \\exists t\\in T : s\\sim t\\}$"}
+{"_id": "27863", "title": "Definition:Saturation (Equivalence Relation)/Definition 3", "text": "The '''saturation of $T$''' is the preimage of its image under the quotient mapping: :$\\overline T = q^{-1} \\sqbrk {q \\sqbrk T}$"}
+{"_id": "27864", "title": "Definition:Saturation (Equivalence Relation)/Definition 2", "text": "The '''saturation of $T$''' is the union of the equivalence classes of its elements: :$\\displaystyle \\overline T = \\bigcup_{t \\mathop \\in T} \\eqclass t \\sim$"}
+{"_id": "27865", "title": "Definition:Saturated Set (Equivalence Relation)", "text": "Let $\\sim$ be an equivalence relation on a set $S$. Let $T\\subset S$ be a subset. === Definition 1 === {{:Definition:Saturated Set (Equivalence Relation)/Definition 1}} === Definition 2 === {{:Definition:Saturated Set (Equivalence Relation)/Definition 2}} === Definition 3 === {{:Definition:Saturated Set (Equivalence Relation)/Definition 3}}"}
+{"_id": "27866", "title": "Definition:Saturated Set (Equivalence Relation)/Definition 1", "text": "$T$ is '''saturated''' {{iff}} it equals its saturation: :$T = \\overline T$"}
+{"_id": "27867", "title": "Definition:Saturated Set (Equivalence Relation)/Definition 3", "text": "$T$ is '''saturated''' {{iff}} it is the preimage of some set under the quotient mapping: :$\\exists V \\subset S / \\sim \\; : T = q^{-1} \\left[{V}\\right]$"}
+{"_id": "27868", "title": "Definition:Saturated Set (Equivalence Relation)/Definition 2", "text": "$T$ is '''saturated''' {{iff}} it is a union of equivalence classes: :$\\displaystyle \\exists U \\subset S : T = \\bigcup_{u \\mathop \\in U} \\left[\\!\\left[{u}\\right]\\!\\right]$"}
+{"_id": "27869", "title": "Definition:Quotient Topology/Definition 1", "text": "Let $\\tau_\\RR$ be the identification topology on $S / \\RR$ by $q_\\RR$: :$\\tau_\\RR := \\set {U \\subseteq S / \\RR: q_\\RR^{-1} \\sqbrk U \\in \\tau}$ Then $\\tau_\\RR$ is the '''quotient topology on $S / \\RR$ by $q_\\RR$'''."}
+{"_id": "27871", "title": "Definition:Saturation (Group Action)", "text": "Let $G$ be a group acting on a set $X$. Let $S\\subset X$ be a subset. === Definition 1 === {{:Definition:Saturation (Group Action)/Definition 1}} === Definition 2 === {{:Definition:Saturation (Group Action)/Definition 2}}"}
+{"_id": "27872", "title": "Definition:Saturation (Group Action)/Definition 1", "text": "The '''saturation of $S$''' is its saturation by the equivalence relation induced by the action."}
+{"_id": "27873", "title": "Definition:Equivalence Relation Induced by Group Action", "text": "Let $G$ be a group. Let $X$ be a set. Let $\\phi : G\\times X \\to X$ be a group action. The '''equivalence relation on $X$ induced by (the action) $\\phi$''' is the relation $\\mathcal R_G$ defined as: :$x \\mathrel {\\mathcal R_G} y \\iff y \\in \\Orb x$ where: :$\\Orb x$ denotes the orbit of $x \\in X$. That is: :$x \\mathrel {\\mathcal R_G} y \\iff \\exists g \\in G: y = g*x$"}
+{"_id": "27874", "title": "Definition:Saturation (Group Action)/Definition 2", "text": "The '''saturation of $S$''' is the union of its images under the group action: :$\\overline S = \\displaystyle \\bigcup_{g \\mathop \\in G} g S$"}
+{"_id": "27875", "title": "Definition:Primitive Semiperfect Number", "text": "A '''primitive semiperfect number''' is a semiperfect number which is not a multiple of a smaller semiperfect number."}
+{"_id": "27876", "title": "Definition:Primitive Semiperfect Number/Sequence", "text": "The sequence of '''primitive semiperfect numbers''' begins: :$6, 20, 28, 88, 104, 272, 304, 350, 368, 464, 490, 496, 550, 572, \\ldots$"}
+{"_id": "27877", "title": "Definition:Primitive Abundant Number", "text": "A '''primitive abundant number''' is an abundant number whose aliquot parts are all deficient."}
+{"_id": "27878", "title": "Definition:Primitive Abundant Number/Sequence", "text": "The sequence of '''primitive abundant numbers''' begins: :$20, 70, 88, 104, 272, 304, 368, 464, 550, 572, 650, 748, 836, 945, 1184, 1312, \\ldots$"}
+{"_id": "27879", "title": "Definition:Ring of Polynomial Forms", "text": "Let  $R$ be a  commutative ring with unity. Let $I$ be a set Let $\\family {X_i: i \\in I}$ be an indexed set. Let $A = R \\sqbrk {\\family {X_i: i \\in I} }$ be the set of all polynomial forms over $R$ in $\\family {X_i: i \\in I}$. Let $+$ and $\\circ$ denote the standard addition and multiplication of polynomial forms. The '''ring of polynomial forms''' is the ordered triple  $\\struct {A, +, \\circ}$."}
+{"_id": "27880", "title": "Definition:Induced Homomorphism Between Fundamental Groups", "text": "Let $X,Y$ be topological spaces. Let $f:X\\to Y$ be a continuous map. Let $x_0\\in X$ and $y_0=f(x_0)\\in Y$. Let $\\pi_1(X,x_0)$ and $\\pi_1(Y,y_0)$ be their fundamental groups. The '''homomorphism induced by $f$''' is the group homomorphism $f_* : \\pi_1(X,x_0) \\to \\pi_1(Y,y_0)$ defined by: :$f_*([\\gamma]) = [f\\circ\\gamma]$"}
+{"_id": "27881", "title": "Definition:Topological Covering Map", "text": "Let $E$ and $B$ be topological spaces. Let $p:E\\to B$ be a continuous surjection. Then $p:E\\to B$ is a '''covering map''' if every $b\\in B$ has an open neighborhood whose preimage is a disjoint union of open sets such that the restriction of $p$ to each of them is a homeomorphism That is: :Every $b \\in B$ has an evenly covered open neighborhood."}
+{"_id": "27882", "title": "Definition:Evenly Covered", "text": "Let $E$ and $B$ be topological spaces. Let $p: E \\to B$ be a continuous surjection. An open set $U \\subset B$ is '''evenly covered by $p$''' if its preimage is a disjoint union of open sets such that the restriction of $p$ to each of them is a homeomorphism to $U$."}
+{"_id": "27884", "title": "Definition:Contractible Space/Definition 2", "text": "$X$ is called '''contractible''' {{iff}} it is homotopy equivalent to a point."}
+{"_id": "27885", "title": "Definition:Fourth Power", "text": "A '''fourth power''' is an integer which can be expressed as the $4$th power of an integer."}
+{"_id": "27886", "title": "Definition:Separable Polynomial/Definition 1", "text": "$P$ is '''separable''' {{iff}} its roots are distinct in an algebraic closure of $K$."}
+{"_id": "27887", "title": "Definition:Separable Polynomial/Definition 2", "text": "$P$ is '''separable''' {{iff}} it has no double roots in every field extension of $K$."}
+{"_id": "27888", "title": "Definition:Separable Polynomial/Definition 3", "text": "$P$ is '''separable''' {{iff}} it has $n$ distinct roots in every field extension where $P$ splits."}
+{"_id": "27889", "title": "Definition:Separable Element", "text": "Let $L/K$ be a field extension. Let $\\alpha\\in L$. Then $\\alpha$ is '''separable over $K$''' {{iff}} its minimal polynomial over $K$ is separable."}
+{"_id": "27890", "title": "Definition:Quadratic Mean", "text": "Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''quadratic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$Q_n := \\displaystyle \\sqrt {\\frac 1 n \\sum_{k \\mathop = 1}^n x_k^2}$"}
+{"_id": "27891", "title": "Definition:Automorphism Group", "text": "=== Group of Automorphisms === {{:Definition:Automorphism Group/Group}}"}
+{"_id": "27892", "title": "Definition:Galois Extension/Finite/Definition 2", "text": "$L/K$ is a '''Galois extension''' {{Iff}} it is normal and separable."}
+{"_id": "27893", "title": "Definition:Galois Extension/Finite/Definition 1", "text": "$L/K$ is a '''Galois extension''' {{Iff}} the fixed field of its automorphism group is $K$: :$\\operatorname{Fix}_L(\\operatorname{Gal}(L/K)) = K$"}
+{"_id": "27894", "title": "Definition:Frobenius Endomorphism", "text": "Let $p$ be a prime number. Let $R$ be a commutative ring with unity of characteristic $p$. The '''Frobenius endomorphism''' is the ring endomorphism $\\Frob: R \\to R$ defined as: :$\\forall x \\in R: \\map \\Frob x = x^p$"}
+{"_id": "27895", "title": "Definition:Perfect Field", "text": "Let $F$ be a field. === Definition 1 === {{:Definition:Perfect Field/Definition 1}} === Definition 2 === {{:Definition:Perfect Field/Definition 2}}"}
+{"_id": "27896", "title": "Definition:Giuga Number", "text": "Let $n$ be a positive integer. $n$ is a '''Giuga number''' {{iff}}: :$\\displaystyle \\sum_{p \\mathop \\divides n} \\frac 1 p - \\prod_{p \\mathop \\divides n} \\frac 1 p \\in \\Z_{\\ge 0}$"}
+{"_id": "27900", "title": "Definition:Area/Dimension", "text": "'''Area''' is a dimension of measurement of physics. The dimension of '''area''' is $L^2$: length squared."}
+{"_id": "27902", "title": "Definition:Imperial/Area", "text": "The imperial units of area are based on a system of units in which each unit is the square of a corresponding imperial length unit. The imperial units of area are as follows: === Square Inch === {{:Definition:Imperial/Area/Square Inch}} === Square Foot === {{:Definition:Imperial/Area/Square Foot}} === Square Yard === {{:Definition:Imperial/Area/Square Yard}} === Square Rod, Pole or Perch === {{:Definition:Imperial/Area/Square Rod, Pole or Perch}} === Square Chain === {{:Definition:Imperial/Area/Square Chain}} === Rood === {{:Definition:Imperial/Area/Rood}} === Acre === {{:Definition:Imperial/Area/Acre}} === Square Mile === {{:Definition:Imperial/Area/Square Mile}}"}
+{"_id": "27903", "title": "Definition:Imperial/Area/Square Rod, Pole or Perch", "text": "One '''square rod, pole or perch''' is equal to a square of side $1$ rod, pole or perch in length. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''square rod, pole or perch''' }} {{eqn | r = 30 \\dfrac 1 4 = \\paren {5 \\dfrac 1 2}^2       | c = square yards }} {{eqn | r = 272 \\dfrac 1 4 = \\paren {16 \\dfrac 1 2}^2       | c = square feet }} {{end-eqn}}"}
+{"_id": "27906", "title": "Definition:Jacobi's Equation of Functional", "text": "Let: :$(1): \\quad \\ds \\int_a^b \\map F {x, y, y'} \\rd x$ be a functional such that: :$\\map y a = A$ :$\\map y b = B$ Let: :$(2): \\quad \\displaystyle \\int_a^b \\paren {P h'^2 + Q h^2} \\rd x$ be a quadratic functional such that: :$P = \\dfrac 1 2 F_{y'y'}$ :$Q = \\dfrac 1 2 \\paren {F_{yy} - \\dfrac \\d {\\d x} F_{yy'} }$ Then Euler's equation of functional $(2)$: :$-\\map {\\dfrac \\d {\\d x} } {P h'} + Q h = 0$ is called '''Jacobi's Equation''' of functional $(1)$. {{NamedforDef|Carl Gustav Jacob Jacobi|cat = Jacobi}}"}
+{"_id": "27907", "title": "Definition:Happy Number", "text": "Let $n \\in \\Z_{>0}$ be a positive integer. Square the digits of $n$ and add the results. Repeat. If this process eventually reaches $1$, then $n$ is a '''happy number'''."}
+{"_id": "27910", "title": "Definition:Variational Equation of Differential Equation", "text": "For $n, i \\in \\N$, let: :$\\map F {x, \\sequence {\\map {y^{\\paren i} } x}_{0 \\mathop \\le i \\mathop \\le n} } = 0$ where $\\sequence {\\map {y^{\\paren i} } x}_{0 \\mathop \\le i \\mathop \\le n}$ is a sequence of derivatives of a function $y$, be a differential equation. Let $\\map y x, \\map g x$ be real functions which solve the given differential equation, such that :$\\map g x = \\map y x + \\map h x$ Then, neglecting $\\map {\\mathcal O} {h^2}$, the differential equation satisfied by $h$ is called the '''variational equation''' of the differential equation $F = 0$."}
+{"_id": "27911", "title": "Definition:Fahrenheit", "text": "'''Fahrenheit''' is a temperature scale. Its two reference points are: : $32 \\fahr$, which is set at the melting point of water. : $212 \\fahr$, which is set at the boiling point of water, as defined at sea level and standard atmospheric pressure. A temperature measured in '''Fahrenheit''' is often referred to as so many '''degrees Fahrenheit'''. === Symbol === {{:Definition:Fahrenheit/Symbol}} {{NamedforDef|Daniel Gabriel Fahrenheit|cat = Fahrenheit}}"}
+{"_id": "27912", "title": "Definition:Fahrenheit/Symbol", "text": "The symbol for the '''degree Fahrenheit''' is $\\fahr$."}
+{"_id": "27915", "title": "Definition:Temperature/Scales", "text": "There are several scales against which '''temperature''' is measured. Each one has two reference points. {| class=\"wikitable\" style=\"margin:1em auto 1em auto\" |+ Temperature Scales |- ! Name ! Unit symbol ! Absolute Zero ! Melting point of water ! Boiling point of water |- ! Celsius | ${}^\\circ \\mathrm C$ | $-273.15 \\ {}^\\circ \\mathrm C$ | $0 \\ {}^\\circ \\mathrm C$ | $99.9839 \\ {}^\\circ \\mathrm C$ |- ! Fahrenheit | ${}^\\circ \\mathrm F$ | $-459.67 \\ {}^\\circ \\mathrm F$ | $32 \\ {}^\\circ \\mathrm F$ | $211.9710 \\ {}^\\circ \\mathrm F$ |- ! Kelvin | $\\mathrm K$ | $0 \\ \\mathrm K$ | $273.15 \\ \\mathrm K$ | $373.1339 \\ \\mathrm K$ |- ! [http://en.wikipedia.org/wiki/Rankine_scale Rankine] | ${}^\\circ \\mathrm R$ | $0 \\ {}^\\circ \\mathrm R$ | $491.67 \\ {}^\\circ \\mathrm R$ | $671.641 \\ {}^\\circ \\mathrm R$ |- |} There are others. As a general rule, only Kelvin is used in physics nowadays. Celsius is usually used in the domestic context, for weather reporting and so on, in most nations, and sometimes seen in the teaching of physics, but usually at the most elementary levels in schools. Fahrenheit is still used as the official temperature scale only in the US and Belize, although can still be seen on occasion in the contexts of weather reporting and health monitoring in the UK. The [http://en.wikipedia.org/wiki/Rankine_scale Rankine scale] is used in a few specialist engineering applications in the US and Canada."}
+{"_id": "27917", "title": "Definition:Multiplicatively Perfect Number", "text": "Let $n \\in \\Z_{>0}$ be a positive integer. $n$ is '''multiplicatively perfect''' {{iff}} the product of the divisors of $n$ is equal to $n^2$."}
+{"_id": "27918", "title": "Definition:Order of Group Element/Definition 3", "text": "The '''order of $x$ (in $G$)''', denoted $\\left\\vert{x}\\right\\vert$, is the largest $k \\in \\Z_{\\gt 0}$ such that: :$\\forall i, j \\in \\Z: 0 \\le i < j < k \\implies x^i \\ne x^j$"}
+{"_id": "27919", "title": "Definition:Inclusion Ordered Set", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. $\\left({S, \\preceq}\\right)$ is '''inclusion ordered set''' {{iff}} :$\\mathord\\preceq = \\mathord\\subseteq \\cap \\left({S \\times S}\\right)$ That means, :$\\forall x, y \\in S: x \\preceq y \\iff x \\subseteq y$"}
+{"_id": "27920", "title": "Definition:Unit Square", "text": "A '''unit square''' is a square of side of unit length."}
+{"_id": "27921", "title": "Definition:Unit Length", "text": "A '''unit length''' is a straight line segment of arbitrary length can defined as having a length of $1$ ('''one'''). During the course of the exposition arising from the definition a '''unit length''' $L$, all subsequent lengths can then be expressed as multiples of $L$. Category:Definitions/Geometry 4z1h1wptocuc7qcfs0wp4yp6laxaf55"}
+{"_id": "27922", "title": "Definition:Polyomino", "text": "A '''polyomino''' is a configuration of $n$ unit squares, for a given (strictly) positive integer $n$, which are placed side by side with vertices touching, to form a plane figure."}
+{"_id": "27924", "title": "Definition:Monomino", "text": "A '''monomino''' is a $1$-omino: :100px That is, it is a polyomino consisting of exactly $1$ unit square."}
+{"_id": "27925", "title": "Definition:Domino", "text": "A '''domino''' is a $2$-omino: :200px That is, it is a polyomino consisting of exactly $2$ unit squares."}
+{"_id": "27928", "title": "Definition:Tromino", "text": "A '''tromino''' is a $3$-omino: :500px That is, it is a polyomino consisting of exactly $3$ unit squares."}
+{"_id": "27940", "title": "Definition:Pascal's Triangle/Column", "text": "Each of the vertical lines of numbers headed by a given $\\dbinom n m$ is known as the '''$m$th column''' of Pascal's triangle. Hence the leftmost '''column''', containing all $1$s, is identified as the '''zeroth column''', or '''column $0$'''."}
+{"_id": "27941", "title": "Definition:Pascal's Triangle/Row", "text": "Each of the horizontal lines of numbers corresponding to a given $n$ is known as the '''$n$th row''' of Pascal's triangle. Hence the top '''row''', containing a single $1$, is identified as the '''zeroth row''', or '''row $0$'''."}
+{"_id": "27942", "title": "Definition:Pascal's Triangle/Diagonal", "text": "The $n$th '''diagonal''' of Pascal's triangle consists of the entries $\\dbinom {n + m} m$ for $m \\ge 0$: :$\\dbinom n 0, \\dbinom {n + 1} 1, \\dbinom {n + 2} 2, \\dbinom {n + 3} 3, \\ldots$ Hence the '''diagonal''' leading down and to the right from $\\dbinom 0 0$, containing all $1$s, is identified as the '''zeroth diagonal''', or '''diagonal $0$'''."}
+{"_id": "27943", "title": "Definition:Pascal's Triangle/Lesser Diagonal", "text": "The $n$th '''lesser diagonal''' of Pascal's triangle consists of the entries $\\dbinom {n - m} m$ for $m \\ge 0$, leading up and to the right from the entry in row $n$ and column $0$: :$\\dbinom n 0, \\dbinom {n - 1} 1, \\dbinom {n - 2} 2, \\dbinom {n - 3} 3, \\ldots$"}
+{"_id": "27944", "title": "Definition:Pascal's Triangle/Order of Numbers", "text": "The entries in column $n$ can be referred to as '''numbers of the $n$th order (of Pascal's triangle)''', or '''$n$th order numbers'''."}
+{"_id": "27946", "title": "Definition:Pascal's Triangle/Edge Cells", "text": "The numbers in column $0$ and in diagonal $0$, containing all $1$s, are referred to as the '''edge cells'''."}
+{"_id": "27949", "title": "Definition:Simplicial Polytopic Number", "text": "The '''simplicial polytopic numbers''' are those denumerating a collection of objects which can be arranged in the form of a simplex."}
+{"_id": "27950", "title": "Definition:Harmonic Sequence", "text": "A '''harmonic sequence''' is a sequence $\\sequence {a_k}$ in $\\R$ defined as: :$h_k = \\dfrac 1 {a + k d}$ where: :$k \\in \\set {0, 1, 2, \\ldots}$ :$-\\dfrac a d \\notin \\set {0, 1, 2, \\ldots}$ Thus its general form is: :$\\dfrac 1 a, \\dfrac 1 {a + d}, \\dfrac 1 {a + 2 d}, \\dfrac 1 {a + 3 d}, \\ldots$"}
+{"_id": "27953", "title": "Definition:Leibniz Harmonic Triangle", "text": ":$\\begin{array}{r|rrrrrr} n & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline 0  & \\frac 1 1 \\\\ 1  & \\frac 1 2 & \\frac 1 2 \\\\ 2  & \\frac 1 3 & \\frac 1 6 &  \\frac 1 3 \\\\ 3  & \\frac 1 4 & \\frac 1 {12} & \\frac 1 {12} & \\frac 1 4 \\\\ 4  & \\frac 1 5 & \\frac 1 {20} & \\frac 1 {30} & \\frac 1 {20} & \\frac 1 5 \\\\ 5  & \\frac 1 6 & \\frac 1 {30} & \\frac 1 {60} & \\frac 1 {60} & \\frac 1 {30} & \\frac 1 6 \\\\ \\end{array}$"}
+{"_id": "27954", "title": "Definition:Leibniz Harmonic Triangle/Row", "text": "Each of the horizontal lines of numbers corresponding to a given $n$ is known as the '''$n$th row''' of Leibniz harmonic triangle. Hence the top '''row''', containing a single $1$, is identified as the '''zeroth row''', or '''row $0$'''."}
+{"_id": "27955", "title": "Definition:Leibniz Harmonic Triangle/Column", "text": "Each of the vertical lines of numbers is known as the '''$m$th column''' of Leibniz harmonic triangle. The leftmost '''column''', containing the reciprocals of the non-negative integers, is identified as the '''zeroth column''', or '''column $0$'''."}
+{"_id": "27956", "title": "Definition:Leibniz Harmonic Triangle/Diagonal", "text": "The $n$th '''diagonal''' of Leibniz harmonic triangle consists of the entries in row $n + m$ and column $m$ for $m \\ge 0$: :$\\left({n, 0}\\right), \\left({n + 1, 1}\\right), \\left({n + 2, 2}\\right), \\ldots$ Hence the '''diagonal''' leading down and to the right from $\\left({0, 0}\\right)$, containing  the reciprocals of the non-negative integers, is identified as the '''zeroth diagonal''', or '''diagonal $0$'''."}
+{"_id": "27957", "title": "Definition:Knot (Knot Theory)/Sphere Knot", "text": "A '''knotted $n$-sphere''' is a knotted embedding: : $\\phi: \\Bbb S^n \\to \\R^{n + 2}$"}
+{"_id": "27959", "title": "Definition:Knot (Knot Theory)/Elementary Knot", "text": "Circle knots can often be quite wild and unwieldy - most of modern knot theory concerns itself with a specific kind of knot. These knots are described as a finite set of points in $\\R^3$ called $\\left\\{{ x_1, x_2, \\dots, x_n }\\right\\}$, together with line segments from $x_i$ to $x_{i+1}$ and a line segment from $x_n$ to $x_1$. The union of all these line segments is clearly a circle knot, or an '''unknot''', an embedding of the circle which is homotopic to a circle. {{explain|Rewrite in a rigorous dictionary style so as to make it understandable independently of the parent page.}}"}
+{"_id": "27963", "title": "Definition:Centered Hexagonal Number/Linguistic Note", "text": "A '''centered hexagonal number''' is rendered in British English as '''centred hexagonal number''', on account of the British spelling of '''centre'''."}
+{"_id": "27965", "title": "Definition:Scott Sigma", "text": "Let $L = \\left({S, \\preceq}\\right)$ be an ordered set. '''Scott sigma''' $\\sigma\\left({L}\\right)$ is a set of subsets of $S$ and equals to :$\\left\\{ {U \\subseteq S: U}\\right.$ is upper and has property (S)$\\left.\\right\\}$ {{NamedforDef|Dana Stewart Scott|cat = Scott, Dana}}"}
+{"_id": "27966", "title": "Definition:Jointly Scott Continuous", "text": "Let $\\struct{S, \\preceq}$ be an ordered set. Let $f:S \\times S \\to S$ be a mapping. $f$ is '''jointly Scott continuous''' {{iff}} :for every relational structure with Scott topology $\\struct{S, \\preceq, \\tau}$ :for every topological space $T = \\struct{S, \\tau}$: $f$ is continuous as a mapping from $T \\times T$ into $T$. where $T \\times T$ denotes the product space. {{NamedforDef|Dana Stewart Scott|cat = Scott, Dana}}"}
+{"_id": "27967", "title": "Definition:Euler Lucky Number/Definition 1", "text": "The '''Euler lucky numbers''' are the prime numbers $p$ such that: :$n^2 + n + p$ is prime for $0 \\le n < p - 1$."}
+{"_id": "27968", "title": "Definition:Euler Lucky Number/Definition 2", "text": "The '''Euler lucky numbers''' are the prime numbers $p$ such that: :$n^2 - n + p$ is prime for $1 \\le n < p$."}
+{"_id": "27969", "title": "Definition:Twice Differentiable/Functional/Dependent on N functions", "text": "Let $\\Delta J \\sqbrk {\\mathbf y; \\mathbf h}$ be an increment of a functional, where $\\mathbf y = \\paren {\\sequence {y_i}_{1 \\mathop \\le i \\mathop \\le N} }$ is a vector. Let: :$\\Delta J \\sqbrk {\\mathbf y; \\mathbf h} = \\phi_1 \\sqbrk {\\mathbf y; \\mathbf h} + \\phi_2 \\sqbrk {\\mathbf y; \\mathbf h} + \\epsilon \\size {\\mathbf h}^2$ where: :$(1): \\quad \\phi_1 \\sqbrk {\\mathbf y; \\mathbf h}$ is a linear functional :$(2): \\quad \\phi_2 \\sqbrk {\\mathbf y; \\mathbf h}$ is a quadratic functional {{WRT}} $\\mathbf h$ :$(3): \\quad \\displaystyle \\size {\\mathbf h} = \\sum_{i \\mathop = 1}^N \\size {h_i}_1 = \\sum_{i \\mathop = 1}^N \\max_{a \\mathop \\le x \\mathop \\le b} \\set {\\size {\\map {h_i} x} + \\size {\\map {h_i'} x} }$ :$(4): \\quad \\epsilon \\to 0$ as $\\size {\\mathbf h} \\to 0$. Then the functional $J \\sqbrk {\\mathbf y}$ is '''twice differentiable'''. {{refactor|Extract the following two into their own definition pages, appropriately named and transcluded.}} The linear part $\\phi_1$ is the '''first variation''', denoted: :$\\delta J \\sqbrk {\\mathbf y; \\mathbf h}$ $\\phi_2$ is called the '''second variation''' (or '''differential''') of a functional, and is denoted: :$\\delta^2 J \\sqbrk {\\mathbf y; \\mathbf h}$"}
+{"_id": "27970", "title": "Definition:Class Mapping", "text": "Let $S$ and $T$ be classes. A '''class mapping $f$ from $S$ to $T$''', denoted $f: S \\to T$, is a class relation $f = \\mathcal R \\subseteq S \\times T$ such that: :$\\forall x \\in S: \\forall y_1, y_2 \\in T: \\left({x, y_1}\\right) \\in f \\land \\left({x, y_2}\\right) \\in f \\implies y_1 = y_2$ and :$\\forall x \\in S: \\exists y \\in T: \\left({x, y}\\right) \\in f$"}
+{"_id": "27971", "title": "Definition:Class Injection", "text": "Let $A$ and $B$ be classes. Let $f: A \\to B$ be a class mapping from $A$ to $B$. Then $f$ is said to be a '''class injection''' if and only if: :$\\forall x_1, x_2 \\in A: f \\left({x_1}\\right) = f \\left({x_2}\\right) \\implies x_1 = x_2$"}
+{"_id": "27972", "title": "Definition:Class Surjection", "text": "Let $A$ and $B$ be classes. Let $f: A \\to B$ be a class mapping from $A$ to $B$. Then $f$ is said to be a '''class surjection''' if and only if: :$\\forall y \\in B: \\exists x \\in A: f \\left({x}\\right) = y$"}
+{"_id": "27973", "title": "Definition:Class Bijection", "text": "Let $A$ and $B$ be classes. Let $f: A \\to B$ be a class mapping from $A$ to $B$. Then $f$ is said to be a '''class bijection''' if and only if: :$f$ is both a class injection and a class surjection. Category:Definitions/Class Mappings pddqb0wn23p9f221zl7n4nj8v1wwasw"}
+{"_id": "27975", "title": "Definition:Greatest Common Divisor/Polynomial Ring over Field", "text": "Let $F$ be a field. Let $P, Q, R \\in F \\left[{X}\\right]$ be polynomials. Then $R$ is '''the greatest common divisor''' of $P$ and $Q$ {{iff}} it is a monic greatest common divisor. This is denoted $\\gcd \\left({P, Q}\\right) = R$."}
+{"_id": "27976", "title": "Definition:Euclidean Number Theory", "text": "'''Euclidean number theory''' concerns aspects of number theory and real analysis as presented in {{ElementsLink}}. Particular attention is paid to certain properties of the irrational numbers."}
+{"_id": "27978", "title": "Definition:Göbel's Sequence", "text": "'''Göbel's sequence''' is the sequence defined recursively as: :$x_n = \\begin{cases} 1 & : n = 0 \\\\ \\displaystyle \\left({1 + \\sum_{k \\mathop = 0}^{n - 1} {x_k}^2}\\right) / n & : n > 0 \\end{cases}$"}
+{"_id": "27979", "title": "Definition:Göbel's Sequence/Sequence", "text": "'''Göbel's sequence''' begins: :$1, 2, 3, 5, 10, 28, 154, 3520, 1 \\, 551 \\, 880, 267 \\, 593 \\, 772 \\, 160, \\ldots$"}
+{"_id": "27980", "title": "Definition:Göbel's Sequence/Historical Note", "text": "Some sources link Göbel's sequence with the name of {{AuthorRef|Michael Somos}}, but it appears that the latter has a different sequence named for him."}
+{"_id": "27981", "title": "Definition:Göbel's Sequence/General/Examples/3", "text": "The '''$3$-Göbel sequence''' begins: :$1, 2, 5, 45, 22 \\, 815, 2 \\, 375 \\, 152 \\, 056 \\, 927, \\ldots$"}
+{"_id": "27984", "title": "Definition:Class Relation", "text": "Let $S$ and $T$ be classes. Let $S \\times T$ be the Cartesian Product of $S$ and $T$. Then a '''relation''' on $S \\times T$ is defined as a subclass of $S \\times T$: :$\\mathcal R \\subseteq S \\times T$ Category:Definitions/Class Theory igju4mpaz3swp08ql307vsp0po3to1b"}
+{"_id": "27985", "title": "Definition:Conjugate Point/Dependent on N Functions", "text": "Let $K$ be a functional such that: :$\\ds K \\sqbrk h = \\int_a^b \\paren {\\mathbf h'\\mathbf P \\mathbf h' + \\mathbf h \\mathbf Q \\mathbf h} \\rd x$ Consider Euler's equation related to the functional $K$: :$-\\map {\\dfrac \\d {\\d x} } {\\mathbf P \\mathbf h'} + \\mathbf Q \\mathbf h = 0$ where $\\mathbf P$ and $\\mathbf Q$ are symmetric matrices. Let the general solution to this equation be: :$\\set {\\mathbf h^{\\paren i} = \\paren {\\sequence {h_{ij} } }: i,j \\in \\N_{\\le N} }$ Let: :$\\exists j: \\forall k \\ne j: \\paren {\\map {\\mathbf h^{\\paren j} } a = 0} \\land \\paren {\\map {h_{j j}'} a = 1, h'_{j k} = 0}$ Let the determinant, built from $h_{ij}$, be such that: :$\\size {h_{i j} } \\paren {\\tilde a} = 0$ Here $i$ denotes rows, and $j$ denotes columns. Then $\\tilde a$ is said to be '''conjugate''' to point $a$ {{WRT}} the functional $K$."}
+{"_id": "27986", "title": "Definition:Derangement/Historical Note", "text": "The number of a derangements of a finite set was first investigated by {{AuthorRef|Nicolaus I Bernoulli}} and {{AuthorRef|Pierre Raymond de Montmort}} between about $1708$ and $1713$. They solved it at around the same time. The question is often couched in the idea of counting the number of ways of placing letters at random in envelopes such that nobody receives the correct letter."}
+{"_id": "27987", "title": "Definition:Kaprekar Triple", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Suppose that $n^3$, when expressed in number base $b$, can be split into three parts that add up to $n$. Then $n$ is a '''Kaprekar triple''' for base $b$."}
+{"_id": "27988", "title": "Definition:Kaprekar Triple/Sequence", "text": "The sequence of Kaprekar triples begins: :$1, 8, 45, 297, 2322, 2728, 4445, 4544, 4949, 5049, 5455, 5554, \\ldots$"}
+{"_id": "27990", "title": "Definition:Hexamorphic Number", "text": "A '''hexamorphic number''' is a hexagonal number $H_n$ whose decimal representation ends in $n$."}
+{"_id": "27991", "title": "Definition:Hexamorphic Number/Sequence", "text": "The sequence of hexamorphic numbers, for $n \\in \\Z_{\\ge 0}$, begins: {{begin-eqn}} {{eqn | l = H_1       | r = 1 }} {{eqn | l = H_5       | r = 45 }} {{eqn | l = H_6       | r = 66 }} {{eqn | l = H_{25}       | r = 1225 }} {{eqn | l = H_{26}       | r = 1326 }} {{eqn | l = H_{50}       | r = 4950 }} {{end-eqn}}"}
+{"_id": "27993", "title": "Definition:Uniform Absolute Convergence", "text": "Let $S$ be a set. Let $\\left({V, \\left\\lVert{\\, \\cdot \\,}\\right\\rVert}\\right)$ be a normed vector space. Let $\\left \\langle {f_n} \\right \\rangle$ be a sequence of mappings $f_n: S \\to V$. Then the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty f_n$ '''converges uniformly absolutely''' {{iff}} the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\left\\lVert{f_n}\\right\\rVert$ converges uniformly."}
+{"_id": "27994", "title": "Definition:Locally Uniform Convergence", "text": "=== General Definition === {{:Definition:Locally Uniform Convergence/General Definition}} === Complex Functions === {{:Definition:Locally Uniform Convergence/Complex Functions}} === Series === {{:Definition:Locally Uniform Convergence/Series}}"}
+{"_id": "27996", "title": "Definition:Locally Uniform Convergence/General Definition", "text": "Let $X$ be a topological space. Let $M$ be a metric space. Let $\\left\\langle{f_n}\\right\\rangle$ be a sequence of mappings $f_n: X \\to M$. Then $f_n$ '''converges locally uniformly''' to $f: X \\to M$ if every point of $X$ has a neighborhood on which $f_n$ converges uniformly to $f$."}
+{"_id": "27997", "title": "Definition:Compact Convergence", "text": "Let $X$ be a topological space. Let $M$ be a metric space. Let $\\left\\langle{f_n}\\right\\rangle$ be a sequence of mappings $f_n : X \\to M$. Let $f: X \\to M$ be a mapping. Then $f_n$ '''converges compactly''' to $f$ {{iff}} $f_n$ converges uniformly to $f$ on every compact subset of $X$."}
+{"_id": "27998", "title": "Definition:Amicable Pair", "text": "Let $m \\in \\Z_{>0}$ and $n \\in \\Z_{>0}$ be (strictly) positive integers. === Definition 1 === {{:Definition:Amicable Pair/Definition 1}} === Definition 2 === {{:Definition:Amicable Pair/Definition 2}} === Definition 3 === {{:Definition:Amicable Pair/Definition 3}}"}
+{"_id": "27999", "title": "Definition:Quasiamicable Numbers", "text": "Let $m \\in \\Z_{>0}$ and $n \\in \\Z_{>0}$ be (strictly) positive integers. === Definition 1 === {{:Definition:Quasiamicable Numbers/Definition 1}} === Definition 2 === {{:Definition:Quasiamicable Numbers/Definition 2}}"}
+{"_id": "28000", "title": "Definition:Uniform Convergence of Product/Compact Space", "text": "Let $X$ be a compact topological space. Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,}}$ be a complete valued field. Let $\\sequence{f_n}$ be a sequence of continuous mappings $f_n: X \\to \\mathbb K$. The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ '''converges uniformly''' {{Iff}} there exists $n_0 \\in \\N$ such that the sequence of partial products of $\\displaystyle \\prod_{n \\mathop = n_0}^\\infty f_n$ converges uniformly and is nonzero."}
+{"_id": "28001", "title": "Definition:Locally Uniform Absolute Convergence", "text": "Let $X$ be a topological space. Let $\\left({V, \\left\\lVert{\\, \\cdot \\,}\\right\\rVert}\\right)$ be a normed vector space. Let $\\left \\langle {f_n} \\right \\rangle$ be a sequence of mappings $f_n: X \\to V$. Then the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty f_n$ '''converges locally uniformly absolutely''' {{iff}} the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\left\\lVert{f_n}\\right\\rVert$ converges locally uniformly."}
+{"_id": "28002", "title": "Definition:Compact Absolute Convergence", "text": "Let $X$ be a topological space. Let $\\left({V, \\left\\lVert{\\, \\cdot \\,}\\right\\rVert}\\right)$ be a normed vector space. Let $\\left \\langle {f_n} \\right \\rangle$ be a sequence of mappings $f_n: X \\to V$. Then the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty f_n$ '''converges compactly (uniformly) absolutely''' {{iff}} the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\left\\lVert{f_n}\\right\\rVert$ converges compactly."}
+{"_id": "28003", "title": "Definition:Local Uniform Absolute Convergence of Product", "text": "Let $X$ be a topological space. Let $\\struct {\\mathbb K, \\norm {\\,\\cdot\\,} }$ be a valued field. Let $\\sequence {f_n}$ be a sequence of mappings $f_n: X \\to \\mathbb K$. Then the infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ '''converges locally uniformly absolutely''' {{iff}} every point of $X$ has a neighborhood on which it converges uniformly absolutely."}
+{"_id": "28005", "title": "Definition:Absolute Convergence of Product/General Definition", "text": "Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,} }$ be a valued field. Let $\\sequence {a_n}$ be a sequence in $\\mathbb K$. === Definition 1 === {{:Definition:Absolute Convergence of Product/General Definition/Definition 1}} === Definition 2 === {{:Definition:Absolute Convergence of Product/General Definition/Definition 2}}"}
+{"_id": "28007", "title": "Definition:Absolute Convergence of Product/General Definition/Definition 2", "text": "The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\paren{1 + a_n}$ is '''absolutely convergent''' {{iff}} the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ is absolutely convergent."}
+{"_id": "28008", "title": "Definition:Quasiamicable Numbers/Definition 1", "text": "$m$ and $n$ are '''quasiamicable numbers''' {{iff}}: :the sum of the proper divisors of $m$ is equal to $n$ and: :the sum of the proper divisors of $n$ is equal to $m$."}
+{"_id": "28009", "title": "Definition:Quasiamicable Numbers/Definition 2", "text": "$m$ and $n$ are '''quasiamicable numbers''' {{iff}}: :$\\sigma \\left({m}\\right) = \\sigma \\left({n}\\right) = m + n + 1$ where $\\sigma \\left({m}\\right)$ denotes the $\\sigma$ function."}
+{"_id": "28010", "title": "Definition:Absolute Convergence of Product/Complex Numbers/Definition 1", "text": "The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\left({1 + a_n}\\right)$ is '''absolutely convergent''' {{iff}} $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\left({1 + \\left\\vert{a_n}\\right\\vert}\\right)$ is convergent."}
+{"_id": "28011", "title": "Definition:Absolute Convergence of Product/Complex Numbers/Definition 2", "text": "The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\left({1 + a_n}\\right)$ is '''absolutely convergent''' {{iff}} the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ is absolutely convergent."}
+{"_id": "28012", "title": "Definition:Absolute Convergence of Product/Complex Numbers/Definition 3", "text": "The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\paren {1 + a_n}$ is '''absolutely convergent''' {{iff}} there exists $n_0 \\in \\N$ such that: :$a_n \\ne -1$ for $n > n_0$ :The series $\\displaystyle \\sum_{n \\mathop = n_0 + 1}^\\infty \\log \\paren {1 + a_n}$ is absolutely convergent where $\\log$ denotes the complex logarithm."}
+{"_id": "28013", "title": "Definition:Trimorphic Number", "text": "An '''trimorphic number''' is a positive integer whose cube ends in that number."}
+{"_id": "28015", "title": "Definition:Divergent Product/Divergence to Zero", "text": "If either: :there exist infinitely many $n \\in \\N$ with $a_n = 0$ :there exists $n_0 \\in \\N$ with $a_n \\ne 0$ for all $n > n_0$ and the sequence of partial products of $\\displaystyle \\prod_{n \\mathop = n_0 + 1}^\\infty a_n$ converges to $0$ the product '''diverges to $0$''', and we assign the value: :$\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n = 0$"}
+{"_id": "28016", "title": "Definition:Cauchy's Criterion for Products", "text": "Let $\\struct {\\mathbb K, \\norm {\\,\\cdot\\,}}$ be a valued field. Let $\\sequence {a_n}$ be a sequence of elements of $\\mathbb K$. The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ satisfies '''Cauchy's criterion''' {{Iff}}: :$\\displaystyle \\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\N: \\forall m, n \\in \\N: m\\geq n \\ge N: \\norm {\\prod_{k \\mathop = n}^m a_n - 1} < \\epsilon$"}
+{"_id": "28017", "title": "Definition:Cauchy Sequence/Cauchy Criterion", "text": "The '''Cauchy criterion''' is the condition: :For any (strictly) positive real number $\\epsilon \\in \\R_{>0}$, for a sufficiently large natural number $N \\in \\N$, the difference between the $m$th and $n$th terms of a Cauchy sequence, where $m, n \\ge N$, will be less than $\\epsilon$. Informally: :For any number you care to pick (however small), if you go out far enough into the sequence, past a certain point, the difference between any two terms in the sequence is less than the number you picked. Or to put it another way, the terms get arbitrarily close together the farther out you go."}
+{"_id": "28018", "title": "Definition:Uniform Absolute Convergence of Product/General Definition", "text": "Let $X$ be a set. Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,}}$ be a valued field. Let $\\sequence {f_n} $ be a sequence of bounded mappings $f_n: X \\to \\mathbb K$. === Definition 1 === {{:Definition:Uniform Absolute Convergence of Product/General Definition/Definition 1}} === Definition 2 === {{:Definition:Uniform Absolute Convergence of Product/General Definition/Definition 2}}"}
+{"_id": "28019", "title": "Definition:Uniform Absolute Convergence of Product/General Definition/Definition 1", "text": "The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\left({1 + f_n}\\right)$ '''converges uniformly absolutely''' {{iff}} the sequence of partial products of $\\displaystyle \\prod_{n \\mathop = 1}^\\infty(1+ \\norm{f_n})$ converges uniformly."}
+{"_id": "28020", "title": "Definition:Uniform Absolute Convergence of Product/General Definition/Definition 2", "text": "The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\left({1 + f_n}\\right)$ '''converges uniformly absolutely''' {{iff}} the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty f_n$ converges uniformly absolutely."}
+{"_id": "28021", "title": "Definition:Uniform Absolute Convergence of Product/Compact Space", "text": "{{refactor|integrate this in the other lists of definitions}} Let $X$ be a set. Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,}}$ be a valued field. Let $\\sequence {f_n} $ be a sequence of bounded mappings $f_n: X \\to \\mathbb K$. The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\left({1 + f_n}\\right)$ '''converges uniformly absolutely''' {{iff}} $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\left({1 + \\norm{f_n}}\\right)$ converges uniformly."}
+{"_id": "28022", "title": "Definition:Uniform Absolute Convergence of Product/Complex Functions", "text": "Let $X$ be a set. Let $\\left \\langle {f_n} \\right \\rangle$ be a sequence of bounded mappings $f_n: X \\to \\C$. === Definition 1 === {{:Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 1}} === Definition 2 === {{:Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 2}} === Definition 3 === {{:Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 3}}"}
+{"_id": "28023", "title": "Definition:Locally Uniform Convergence of Product", "text": "Let $T = \\left({S, \\tau}\\right)$ be a weakly locally compact topological space. Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,}}$ be a valued field. Let $\\left \\langle {f_n} \\right \\rangle$ be a sequence of locally bounded mappings $f_n: S \\to \\mathbb K$. === Definition 1 === {{:Definition:Locally Uniform Convergence of Product/Definition 1}} === Definition 2 === {{:Definition:Locally Uniform Convergence of Product/Definition 2}}"}
+{"_id": "28024", "title": "Definition:Locally Uniform Convergence of Product/Definition 1", "text": "The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ '''converges locally uniformly''' {{iff}} every point of $T$ has a compact neighborhood on which it converges uniformly."}
+{"_id": "28025", "title": "Definition:Locally Uniform Convergence of Product/Definition 2", "text": "The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ '''converges locally uniformly''' {{iff}} it converges uniformly on every compact subspace of $T$."}
+{"_id": "28026", "title": "Definition:Normed Division Ring", "text": "Let $\\struct {R, +, \\circ}$ be a division ring. Let $\\norm {\\,\\cdot\\,}$ be a norm on $R$. Then $\\struct {R, \\norm{\\,\\cdot\\,} }$ is a '''normed division ring'''."}
+{"_id": "28027", "title": "Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 1", "text": "The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\left({1 + f_n}\\right)$ '''converges uniformly absolutely''' {{iff}} the sequence of partial products of $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\left({1 + \\left\\vert{f_n}\\right\\vert}\\right)$ converges uniformly."}
+{"_id": "28028", "title": "Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 3", "text": "The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\left({1 + f_n}\\right)$ '''converges uniformly absolutely''' {{iff}} there exists $n_0 \\in \\N$ such that: :$(1): \\quad f_n \\left({x}\\right) \\ne -1$ for $n \\ge n_0$ and $x \\in X$ and: :$(2): \\quad$ The series $\\displaystyle \\sum_{n \\mathop = n_0}^\\infty \\log \\left({1 + f_n}\\right)$ is uniformly absolutely convergent."}
+{"_id": "28029", "title": "Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 2", "text": "The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\left({1 + f_n}\\right)$ '''converges uniformly absolutely''' {{iff}} the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty f_n$ converges uniformly absolutely."}
+{"_id": "28030", "title": "Definition:Locally Uniform Convergence/Series", "text": "Let $X$ be a topological space. Let $V$ be a normed vector space. Let $\\left\\langle{f_n}\\right\\rangle$ be a sequence of mappings $f_n:X\\to V$. Then the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty f_n$ '''converges locally uniformly''' {{iff}} the sequence of partial sums converges locally uniformly."}
+{"_id": "28032", "title": "Definition:Order of Entire Function/Definition 2", "text": "Let $f$ be not identically zero. The '''order''' $\\alpha \\in \\closedint 0 {+\\infty}$ of $f$ is the infimum of the $\\beta \\ge 0$ for which: :$\\displaystyle \\map \\ln {\\max_{\\size z \\mathop \\le R} \\size {\\map f z} } = \\map \\OO {R^\\beta}$ or $\\infty$ if no such $\\beta$ exists, where $\\OO$ denotes big-O notation The '''order''' of $0$ is $0$."}
+{"_id": "28033", "title": "Definition:Order of Entire Function/Definition 3", "text": "Let $f$ be non-constant. The '''order''' $\\alpha \\in \\closedint 0 {+\\infty}$ of $f$ is the limit superior: :$\\displaystyle \\limsup_{R \\mathop \\to \\infty} \\frac {\\displaystyle \\ln \\ln \\max_{\\cmod z \\le R} \\cmod f} {\\ln R}$ The '''order''' of a constant function is $0$."}
+{"_id": "28034", "title": "Definition:Order of Entire Function/Definition 1", "text": "The '''order''' $\\alpha \\in \\closedint 0 {+\\infty}$ of $f$ is the infimum of the $\\beta \\ge 0$ for which: :$\\map f z = \\map \\OO {\\map \\exp {\\size z^\\beta} }$ or $\\infty$ if no such $\\beta$ exists, where $\\OO$ denotes big-O notation."}
+{"_id": "28035", "title": "Definition:Jacobi's Equation of Functional/Dependent on N Functions", "text": "Let: :$\\ds \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ be a (real) functional, where $\\map {\\mathbf y} a = A$ and $\\map {\\mathbf y} b = B$. Let: :$\\ds \\int_a^b \\paren {\\mathbf h' \\mathbf P \\mathbf h' + \\mathbf h \\mathbf Q \\mathbf h} \\rd x$ be a quadratic functional, where: :$P_{ij} = \\dfrac 1 2 F_{y_i'y_j'}$ :$Q_{ij} = \\dfrac 1 2 \\paren {F_{y_i y_j} - \\dfrac \\d {\\d x} F_{y_i y_j'} }$ Then the Euler's equation of the latter functional: :$-\\map {\\dfrac \\d {\\d x} } {\\mathbf P \\mathbf h'} + \\mathbf Q \\mathbf h = \\mathbf 0$ is called '''Jacobi's Equation''' of the former functional. {{NamedforDef|Carl Gustav Jacob Jacobi|cat = Jacobi}}"}
+{"_id": "28037", "title": "Definition:Exponent of Convergence", "text": "Let $\\sequence {a_n}$ be a sequence of nonzero complex numbers. The '''exponent of convergence''' of $\\sequence {a_n}$ is the infimum of $\\tau \\ge 0$ for which the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\size {a_n}^{-\\tau}$ converges. The '''exponent of convergence''' of a finite sequence is $0$."}
+{"_id": "28038", "title": "Definition:Rank of Entire Function", "text": "Let $f: \\C \\to \\C$ be an entire function. Let $\\left\\langle{a_n}\\right\\rangle$ be the sequence of nonzero zeroes of $f$, repeated according to multiplicity. The '''rank''' of $f$ is the smallest integer $p\\geq0$ for which the series $\\displaystyle\\sum_{n\\mathop=1}^\\infty |a_n|^{-p-1}$ converges, or $\\infty$ if there is no such integer. If $f$ has finitely many zeroes, its '''rank''' is $0$."}
+{"_id": "28040", "title": "Definition:Hadamard's Canonical Factorization", "text": "Let $f: \\C \\to \\C$ be a nonzero entire function of finite rank $p \\in \\N$. Let $0$ be a zero of $f$ of multiplicity $m\\geq0$. Let $\\left\\langle{a_n}\\right\\rangle$ be the sequence of nonzero zeroes of $f$, repeated according to multiplicity. The '''canonical representation''' of $f$ is: :$\\displaystyle f \\left({z}\\right) = z^m e^{g \\left({z}\\right)} \\prod_{n \\mathop = 1}^\\infty E_p\\left({\\frac z {a_n} }\\right)$ where: :$g: \\C \\to \\C$ is an entire function :$E_p$ denotes the $p$th Weierstrass elementary factor. If $f$ has finitely many zeroes, the product is understood to be finite. {{NamedforDef|Jacques Salomon Hadamard|cat = Hadamard}}"}
+{"_id": "28042", "title": "Definition:Pentagonal Pyramidal Number", "text": "'''Pentagonal pyramidal numbers''' are those denumerating a collection of objects which can be arranged in the form of a pentagonal pyramid."}
+{"_id": "28043", "title": "Definition:Hexagonal Pyramidal Number", "text": "'''Hexagonal pyramidal numbers''' are those denumerating a collection of objects which can be arranged in the form of a hexagonal pyramid."}
+{"_id": "28045", "title": "Definition:Cubic Recurring Digital Invariant", "text": "A '''cubic recurring digital invariant''' is a recurring digital invariant of order $3$. === Sequence === {{:Definition:Cubic Recurring Digital Invariant/Sequence}}"}
+{"_id": "28046", "title": "Definition:Mutually Consistent Boundary Conditions", "text": "Let $\\map {\\mathbf y} x$, $\\map {\\boldsymbol \\psi} {\\mathbf y}$ be an N-dimensional vectors. Consider the system of differential equations: :$(1): \\quad \\mathbf y'' = \\map {\\mathbf f} {x, \\mathbf y, \\mathbf y'}$ Let derivatives of $\\mathbf y$ satisfy: :$\\bigintlimits {\\mathbf y'} {x \\mathop = x_1} {} = \\bigintlimits {\\map {\\boldsymbol \\psi^{\\paren 1} } {\\mathbf y} } {x \\mathop = x_1} {}$ :$\\bigintlimits {\\mathbf y'} {x \\mathop = x_2} {} = \\bigintlimits {\\map {\\boldsymbol \\psi^{\\paren 2} } {\\mathbf y} } {x \\mathop = x_2} {}$ If every solution of $(1)$ satisfying conditions at $x = x_1$ automatically satisfies conditions at $x = x_2$ (or vice versa), then these boundary conditions are called '''mutually consistent'''."}
+{"_id": "28048", "title": "Definition:Recurring Digital Invariant", "text": "Let $k \\in \\Z_{>0}$ be a positive integer. Let $f: \\Z_{>0} \\to \\Z_{>0}$ be the mapping defined as: :$\\forall m \\in \\Z_{>0}: f \\left({m}\\right) = $ the sum of the $k$th powers of the digits of $n$. Let $n_0 \\in \\Z_{>0}$ be a positive integer. Consider the sequence: :$s_n = \\begin{cases} n_0 & : n = 0 \\\\ f \\left({s_{n - 1} }\\right) & : n > 0 \\end{cases}$ If: :$\\exists r \\in \\N_{>0}: s_r = n_0$ then the smallest of the terms $n_0, n_1, \\ldots, n_r$ is a '''recurring digital invariant of order $k$'''."}
+{"_id": "28050", "title": "Definition:Separated Sets/Definition 2", "text": "$A$ and $B$ are '''separated (in $T$)''' {{iff}} there exist $U,V\\in\\tau$ with: :$A\\subset U$ and $U\\cap B = \\varnothing$ :$B\\subset V$ and $V\\cap A = \\varnothing$ where $\\varnothing$ denotes the empty set."}
+{"_id": "28051", "title": "Definition:Separated Sets/Definition 1", "text": "$A$ and $B$ are '''separated (in $T$)''' {{iff}}: :$A^- \\cap B = A \\cap B^- = \\O$ where: :$A^-$ denotes the closure of $A$ in $T$ :$\\O$ denotes the empty set."}
+{"_id": "28052", "title": "Definition:Logarithmic Mean Value", "text": "Let $f: \\N \\to \\C$ be an arithmetic function. The '''logarithmic mean value''' of $f$ is the limit: :$L \\left({f}\\right) = \\displaystyle \\lim_{x \\mathop \\to \\infty} \\frac 1 {\\ln x} \\sum_{n \\mathop \\le x} \\frac {f \\left({n}\\right)} n$ if it exists."}
+{"_id": "28053", "title": "Definition:Ordinary Mean Value", "text": "Let $f: \\N \\to \\C$ be an arithmetic function. The '''(ordinary) mean value''' of $f$ is the limit: :$M \\left({f}\\right) = \\displaystyle \\lim_{x \\mathop \\to \\infty} \\frac 1 x \\sum_{n \\mathop \\le x} f \\left({n}\\right)$ if it exists."}
+{"_id": "28054", "title": "Definition:Summatory Function", "text": "Let $a: \\N \\to \\C$ be an arithmetic function. Its '''summatory function''' is the mapping: :$A \\left({x}\\right) = \\displaystyle\\sum_{n \\mathop \\le x} a \\left({n}\\right)$ defined for $x \\in \\Z_{\\ge 0}$."}
+{"_id": "28055", "title": "Definition:Dirichlet Convolution/Definition 1", "text": "The '''Dirichlet convolution''' of $f$ and $g$ is the arithmetic function: :$\\displaystyle \\left({f * g}\\right) \\left({n}\\right) := \\sum_{d \\mathop \\backslash n} f \\left({d}\\right) g \\left({\\frac n d}\\right)$ where the summation runs over the set of positive divisors $d$ of $n$."}
+{"_id": "28056", "title": "Definition:Dirichlet Convolution/Definition 2", "text": "The '''Dirichlet convolution''' of $f$ and $g$ is the arithmetic function: :$\\displaystyle \\left({f * g}\\right) \\left({n}\\right) := \\sum_{a b \\mathop = n} f \\left({a}\\right) g \\left({b}\\right)$ where the summation runs over all pairs of positive integers $\\left({a, b}\\right)$ with $a b = n$."}
+{"_id": "28058", "title": "Definition:Pentatope Number", "text": "'''Pentatope numbers''' are those denumerating a collection of objects which can be arranged in $4$ dimensions in the form of a regular pentatope."}
+{"_id": "28059", "title": "Definition:Polytope Number", "text": "'''Polytope numbers''' are those denumerating a collection of objects which can be arranged in $4$ dimensions in the form of a regular polytope."}
+{"_id": "28060", "title": "Definition:Polytope Number/Examples", "text": "=== Pentatope Number === {{:Definition:Pentatope Number}}"}
+{"_id": "28062", "title": "Definition:Convergent Product/Number Field/Nonzero Sequence", "text": "Let $\\sequence{a_n}$ be a sequence of nonzero elements of $\\mathbb K$. The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ is '''convergent''' {{Iff}} its sequence of partial products converges to a nonzero limit $a\\in\\mathbb K\\setminus\\{0\\}$."}
+{"_id": "28063", "title": "Definition:Convergent Product/Number Field/Arbitrary Sequence", "text": "Let $\\sequence{a_n}$ be a sequence of elements of $\\mathbb K$. The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ is '''convergent''' {{Iff}}: * There exists $n_0\\in\\N$ such that the sequence of partial products of $\\displaystyle \\prod_{n \\mathop = n_0}^\\infty a_n$ converges to some $b\\in\\mathbb K\\setminus\\{0\\}$. The sequence of partial products of $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ is then convergent to some $a\\in\\mathbb K$."}
+{"_id": "28064", "title": "Definition:Convergent Product/Arbitrary Field/Arbitrary Sequence", "text": "Let $\\sequence {a_n}$ be a sequence of elements of $\\mathbb K$. The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ is '''convergent''' {{Iff}}: : There exists $n_0 \\in \\N$ such that the sequence of partial products of $\\displaystyle \\prod_{n \\mathop = n_0}^\\infty a_n$ converges to some $b \\in \\mathbb K \\setminus \\set 0$. The sequence of partial products of $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ is then convergent to some $a \\in \\mathbb K$."}
+{"_id": "28065", "title": "Definition:Convergent Product/Arbitrary Field/Nonzero Sequence", "text": "Let $\\sequence {a_n}$ be a sequence of nonzero elements of $\\mathbb K$. The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ is '''convergent''' {{Iff}} its sequence of partial products converges to a nonzero limit $a \\in \\mathbb K \\setminus \\set 0$."}
+{"_id": "28067", "title": "Definition:Group of Permutation Matrices", "text": "The '''group of permutation matrices''' of order $n$ is the set of all $n \\times n$ permutation matrices with multiplication of matrices. {{Disambiguate|Definition:Order}}"}
+{"_id": "28068", "title": "Definition:Change of Basis Matrix/Definition 1", "text": "The '''matrix of change of basis from $A$ to $B$''' is the matrix whose columns are the coordinate vectors of the elements of the '''new basis''' $\\left \\langle {b_n} \\right \\rangle$ relative to the '''original basis''' $\\left \\langle {a_n} \\right \\rangle$."}
+{"_id": "28069", "title": "Definition:Change of Basis Matrix/Definition 2", "text": "Let $I_G$ be the identity linear operator on $G$. Let $\\left[{I_G; \\left \\langle {a_n} \\right \\rangle, \\left \\langle {b_n} \\right \\rangle}\\right]$ be the matrix of $I_G$ relative to $\\left \\langle {b_n} \\right \\rangle$ and $\\left \\langle {a_n} \\right \\rangle$. Then $\\left[{I_G; \\left \\langle {a_n} \\right \\rangle, \\left \\langle {b_n} \\right \\rangle}\\right]$ is called the '''matrix corresponding to the change of basis from $\\left \\langle {a_n} \\right \\rangle$ to $\\left \\langle {b_n} \\right \\rangle$'''."}
+{"_id": "28070", "title": "Definition:Basis of Module", "text": "Let $R$ be a ring with unity. Let $\\left({G, +_G, \\circ}\\right)_R$ be a unitary $R$-module. === Definition 1 === {{:Definition:Basis of Module/Definition 1}} === Definition 2 === {{:Definition:Basis of Module/Definition 2}}"}
+{"_id": "28071", "title": "Definition:Basis of Vector Space/Definition 1", "text": "A '''basis of $G$''' is a linearly independent subset of $G$ which is a generator for $G$."}
+{"_id": "28072", "title": "Definition:Basis of Vector Space/Definition 2", "text": "A '''basis''' is a maximal linearly independent subset of $G$."}
+{"_id": "28073", "title": "Definition:Basis of Vector Space", "text": "Let $K$ be a division ring. Let $\\struct {G, +_G, \\circ}_R$ be a vector space over $K$. === Definition 1 === {{:Definition:Basis of Vector Space/Definition 1}} === Definition 2 === {{:Definition:Basis of Vector Space/Definition 2}}"}
+{"_id": "28075", "title": "Definition:Field of Directions", "text": "Consider the following system of differential equations: :$(1): \\quad \\mathbf y' = \\map {\\mathbf f} {x, \\mathbf y, \\mathbf y'}$ where $\\mathbf y$ is an $n$-dimensional vector. Let the boundary conditions be prescribed $\\forall x \\in \\closedint a b$: :$\\mathbf y' = \\map {\\boldsymbol \\psi} {x, \\mathbf y}$ Let these boundary conditions be consistent for all $x_1, x_2 \\in \\closedint a b$. Then the family of mutually consistent boundary conditions is called a '''field of directions''' for the given system $(1)$. That is, the first-order system is valid in an interval instead of a countable set of points."}
+{"_id": "28077", "title": "Definition:Angle/Unit/Minute", "text": "The '''minute (of arc)''' is a measurement of plane angles, symbolized by $'$. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''minute''' }} {{eqn | r = 60       | c = seconds }} {{eqn | r = \\dfrac 1 {60}       | c = degree of arc (by definition) }} {{end-eqn}}"}
+{"_id": "28078", "title": "Definition:Angle/Unit/Second", "text": "The '''second (of arc)''' is a measurement of plane angles, symbolized by $''$. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''second''' }} {{eqn | r = \\dfrac 1 {60}       | c = minute of arc (by definition) }} {{eqn | r = \\dfrac 1 {60 \\times 60} = \\dfrac 1 {3600}       | c = degree of arc }} {{end-eqn}}"}
+{"_id": "28081", "title": "Definition:Second", "text": "=== Second of Time === {{:Definition:Second of Time}} === Second of Arc === {{:Definition:Second of Arc}}"}
+{"_id": "28082", "title": "Definition:Minute", "text": "=== Minute of Time === {{:Definition:Minute of Time}} === Minute of Arc === {{:Definition:Minute of Arc}}"}
+{"_id": "28084", "title": "Definition:Splitting Polynomial", "text": "Let $K$ be a field. Let $P \\in K \\sqbrk X$ be a polynomial. === Definition 1 === The polynomial $P$ '''splits (completely)''' in $K$ {{iff}} it is a product of polynomials of degree $1$. === Definition 2 === The polynomial $P$ '''splits (completely)''' in $K$ {{iff}} every irreducible factor of $P$ has degree $1$."}
+{"_id": "28085", "title": "Definition:Self-Adjoint Boundary Conditions", "text": "Consider the functional $J \\sqbrk {\\mathbf y}$, such that: :$\\ds J \\sqbrk {\\mathbf y} = \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ Let the momenta of $J$ be: :$\\mathbf p = \\nabla_{\\mathbf y'} \\map F {x, \\mathbf y, \\mathbf y'}$ Let the following boundary conditions hold: :$\\map {\\mathbf y'} a = \\bigvalueat {\\map {\\boldsymbol \\psi} {\\mathbf y} } {x \\mathop = a}$ If: :$\\exists \\map g {x, \\mathbf y}: \\bigvalueat {\\map {\\mathbf p} {x, \\mathbf y, \\map {\\boldsymbol \\psi} {\\mathbf y} } } {x \\mathop = a} = \\bigvalueat {\\nabla_{\\mathbf y'} \\map g {x, \\mathbf y} } {x \\mathop = a}$ then the boundary conditions are called '''self-adjoint'''."}
+{"_id": "28086", "title": "Definition:Unitary Group", "text": "Let $n$ be a positive integer. The '''unitary group''' $\\operatorname{U}_n$ is the group of all $n\\times n$ unitary matrices. === Unitary Group of Hilbert Space === Let $H$ be a Hilbert space. Its '''unitary group''' is the group consisting of its unitary operators."}
+{"_id": "28087", "title": "Definition:Special Unitary Group", "text": "Let $n$ be a positive integer. The '''special unitary group''' $\\operatorname{SU}_n$ is the group of all $n\\times n$ unitary matrices with determinant $1$: :$\\operatorname{SU}_n = \\operatorname{U}_n \\cap \\operatorname{SL}_n(\\C)$"}
+{"_id": "28088", "title": "Definition:Standard Basis Matrix", "text": "Let $R$ be a ring with unity. Let $m, n \\ge 1$ be positive integers. Let $i, j \\in \\set {1, \\ldots, m} \\times \\set {1, \\ldots, n}$. The $\\tuple {i, j}$th '''standard basis matrix''' is the $m \\times n$ matrix which is $0$ everywhere except a $1$ at the $\\tuple {i, j}$th indices."}
+{"_id": "28089", "title": "Definition:Coordinate Vector", "text": "Let $R$ be a ring with unity. Let $M$ be a free $R$-module of dimension $n$. Let $B = \\left \\langle {b_k} \\right \\rangle_{1 \\mathop \\le k \\mathop \\le n}$ be an ordered basis of $M$. Let $x\\in M$. If $\\lambda_1,\\ldots,\\lambda_n\\in R$ are such that $x = \\displaystyle\\sum_{i=1}^n \\lambda_i b_i$, then $(\\lambda_1,\\ldots,\\lambda_n)^\\intercal \\in R^n$ is the '''coordinate vector''' of $x$ with respect to $B$. This can be denoted: $[x]_B$."}
+{"_id": "28090", "title": "Definition:Highly Composite Number", "text": "Let $n \\in \\Z_{>0}$ be a positive integer. Then $n$ is '''highly composite''' {{iff}}: :$\\forall m \\in \\Z_{>0}, m < n: \\map \\tau m < \\map \\tau n$ where $\\map \\tau n$ is the $\\tau$ function of $n$."}
+{"_id": "28091", "title": "Definition:Highly Composite Number/Sequence", "text": "The sequence of highly composite numbers begins: :$1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, \\ldots$"}
+{"_id": "28094", "title": "Definition:Internal Direct Sum of Modules", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $(M_i)_{i\\in I}$ be a family of submodules. === Definition 1 === {{:Definition:Internal Direct Sum of Modules/Definition 1}} === Definition 2 === {{:Definition:Internal Direct Sum of Modules/Definition 2}} === Definition 3 === {{:Definition:Internal Direct Sum of Modules/Definition 3}}"}
+{"_id": "28095", "title": "Definition:Seventh Power", "text": "A '''seventh power''' is an integer which can be expressed as the $7$th power of an integer."}
+{"_id": "28096", "title": "Definition:Internal Direct Sum of Modules/Definition 1", "text": "$M$ is the '''internal direct sum''' of $(M_i)_{i\\in I}$ {{Iff}} every $m\\in M$ can be written uniquely as a sum $\\sum m_i$ with each $m_i\\in M_i$."}
+{"_id": "28097", "title": "Definition:Internal Direct Sum of Modules/Definition 2", "text": "$M$ is the '''internal direct sum''' of $(M_i)_{i\\in I}$ {{Iff}}: * $\\displaystyle\\bigcup_{i\\in I}M_i$ generates $M$ * For all $i\\in I$, $M_i\\cap \\displaystyle\\sum_{j\\neq i} M_j = \\{0\\}$"}
+{"_id": "28098", "title": "Definition:Internal Direct Sum of Modules/Definition 3", "text": "Let $\\displaystyle \\bigoplus_{i \\mathop \\in I} M_i$ be the external direct sum of $\\left\\langle{M_i}\\right\\rangle_{i \\mathop \\in I}$. $M$ is the '''internal direct sum''' of $\\left\\langle{M_i}\\right\\rangle_{i \\mathop \\in I}$ {{iff}} the mapping given by Universal Property of Direct Sum of Modules is an isomorphism onto $M$."}
+{"_id": "28099", "title": "Definition:Basis of Module/Definition 1", "text": "A '''basis of $G$''' is a linearly independent subset of $G$ which is a generator for $G$."}
+{"_id": "28100", "title": "Definition:Basis of Module/Definition 2", "text": "Let $\\mathcal B = \\family {b_i}_{i \\mathop \\in I}$ be a family of elements of $M$. Let $\\Psi: R^{\\paren I} \\to M$ be the homomorphism given by Universal Property of Free Module on Set. Then $\\mathcal B$ is a '''basis''' {{Iff}} $\\Psi$ is an isomorphism."}
+{"_id": "28101", "title": "Definition:Relative Matrix of Bilinear Form", "text": "Let $R$ be a ring with unity. Let $M$ be a free $R$-module of finite dimension $n>0$. Let $\\mathcal B = \\left \\langle {b_m} \\right \\rangle$ be an ordered basis of $M$. Let $f : M\\times M\\to R$ be a bilinear form. The '''matrix of $f$ relative to $\\mathcal B$''' is the $n\\times n$ matrix $\\mathbf M_{f, \\mathcal B}$ where: :$\\displaystyle \\forall \\left({i, j}\\right) \\in \\left[{1 \\,.\\,.\\, n}\\right] \\times \\left[{1 \\,.\\,.\\, n}\\right] : (\\mathbf M_{f, \\mathcal B})_{ij} = f \\left({b_i, b_j}\\right)$"}
+{"_id": "28102", "title": "Definition:Mutually Consistent Boundary Conditions/wrt Functional", "text": "Let $J$ be a (real) functional, such that: :$\\displaystyle J = \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ where its Euler's equations are: :$\\nabla_{\\mathbf y'} F - \\dfrac \\d {\\d x} \\nabla_{\\mathbf y} F = 0$ Consider the following boundary conditions: :$\\bigvalueat {\\mathbf y} {x \\mathop = x_1} = \\bigvalueat {\\map {\\boldsymbol \\psi^{\\paren 1} } {\\mathbf y} } {x \\mathop = x_1}$ :$\\bigvalueat {\\mathbf y} {x \\mathop = x_2} = \\bigvalueat {\\map {\\boldsymbol \\psi^{\\paren 2} } {\\mathbf y} } {x \\mathop = x_2}$ If they are consistent {{WRT}} the Euler equations, then these boundary conditions are called '''mutually consistent''' {{WRT}} the functional $J$."}
+{"_id": "28103", "title": "Definition:Endomorphism Ring of Module", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. The '''endomorphism ring''' $\\operatorname{End}_R(M)$ is the ring of all endomorphisms of $M$ where: : Multiplication is composition of mappings : Addition is point-wise"}
+{"_id": "28104", "title": "Definition:Module Endomorphism", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. An '''module endomorphism''' of $M$ is a module homomorphism from $M$ to itself."}
+{"_id": "28105", "title": "Definition:Associated Quadratic Form", "text": "Let $\\mathbb K$ be a field. Let $V$ be a vector space over $\\mathbb K$. Let $b : V\\times V \\to \\mathbb K$ be a bilinear form. The '''quadratic form associated to $b$''' is the quadratic form: :$q : V \\to \\mathbb K : x \\mapsto b(x, x)$"}
+{"_id": "28106", "title": "Definition:Associated Bilinear Form", "text": "Let $\\mathbb K$ be a field of characteristic $\\operatorname{char} \\mathbb K\\neq2$. Let $V$ be a vector space over $\\mathbb K$. Let $q : V \\to \\mathbb K$ be a quadratic form. The '''bilinear form associated to $q$''' is the bilinear form: :$b : V \\times V \\to \\mathbb K : (v,w) \\mapsto \\frac12\\left( q(v+w) - q(v) - q(w) \\right)$"}
+{"_id": "28107", "title": "Definition:Matrix Congruence", "text": "Let $R$ be a commutative ring with unity. Let $n$ be a positive integer. Let $\\mathbf A$ and $\\mathbf B$ be square matrices of order $n$ over $R$. Then $\\mathbf A$ and $\\mathbf B$ are '''congruent''' {{Iff}} there exists an invertible matrix $\\mathbf P\\in R^{n\\times n}$ such that $\\mathbf B = \\mathbf P^\\intercal \\mathbf A \\mathbf P$."}
+{"_id": "28108", "title": "Definition:Symmetric Bilinear Form", "text": "Let $R$ be a ring Let $M$ be an $R$-module. Let $b: M \\times M \\to R$ be a bilinear form. Then $b$ is '''symmetric''' {{iff}}: :$\\forall v, w, \\in M: b \\left({v, w}\\right) = b \\left({w, v}\\right)$"}
+{"_id": "28109", "title": "Definition:Represented by Quadratic Form", "text": "Let $R$ be an integral domain. Let $M$ be an $R$-module. Let $q : M \\to R$ be a quadratic form. Let $a\\in R$. Then '''$q$ represents $a$ (over $R$)''' {{Iff}} there exists $x\\in M\\setminus\\{0\\}$ with $q(x) = a$."}
+{"_id": "28110", "title": "Definition:Isotropic Quadratic Form", "text": "Let $\\mathbb K$ be a field of characteristic $\\operatorname{char}\\mathbb K \\neq 2$. Let $V$ be a vector space over $\\mathbb K$. Let $q : V\\times V \\mapsto \\mathbb K$ be a quadratic form. Then $q$ is '''isotropic''' {{Iff}} it represents $0$. That is: $q(v) = 0$ for some $v\\in V\\setminus\\{0\\}$. === Anisotropic Quadratic Form === A quadratic form that is not isotropic is said to be '''anisotropic'''."}
+{"_id": "28111", "title": "Definition:Alternating Bilinear Form", "text": "Let $\\mathbb K$ be a field. Let $V$ be a vector space over $\\mathbb K$. Let $b$ be a bilinear form on $V$. Then $b$ is '''alternating''' {{iff}}: : $\\forall v \\in V: b \\left({v, v}\\right) = 0$"}
+{"_id": "28112", "title": "Definition:Reflexive Bilinear Form", "text": "Let $\\mathbb K$ be a field. Let $V$ be a vector space over $\\mathbb K$. Let $b$ be a bilinear form on $V$. Then $b$ is '''reflexive''' {{iff}}: : $\\forall v, w \\in V: b \\left({v, w}\\right) = 0 \\implies b \\left({w, v}\\right) = 0$"}
+{"_id": "28113", "title": "Definition:Apéry's Constant", "text": "'''Apéry's constant''' is the value of the infinite sum: :$\\map \\zeta 3 = \\displaystyle \\sum_{n \\mathop = 1}^\\infty \\frac 1 {n^3}$ where $\\zeta$ denotes the Riemann zeta function. Its approximate value is given by: :$\\map \\zeta 3 \\approx 1 \\cdotp 20205 \\, 69031 \\, 59594 \\, 28539 \\, 97381 \\, 61511 \\, 44999 \\, 07649 \\, 86292 \\ldots$"}
+{"_id": "28116", "title": "Definition:Discriminant of Bilinear Form", "text": "Let $\\mathbb K$ be a field. Let $V$ be a vector space over $\\mathbb K$ of finite dimension $n>0$. Let $b : V\\times V \\to \\mathbb K$ be a bilinear form on $V$. Let $A$ be the matrix of $b$ relative to an ordered basis of $V$. If $b$ is nondegenerate, its '''discriminant''' is the equivalence class of the determinant $\\det A$ in the quotient group $\\mathbb K^\\times / (\\mathbb K^\\times)^2$. If $b$ is degenerate, its '''discriminant''' is $0$."}
+{"_id": "28117", "title": "Definition:Degenerate Bilinear Form", "text": "Let $\\mathbb K$ be a field. Let $V$ be a vector space over $\\mathbb K$. Let $b : V\\times V \\to \\mathbb K$ be a bilinear form on $V$. Then $b$ is '''degenerate''' {{Iff}} there exists $v\\in V\\setminus\\{0\\}$ such that $b(v,u) = 0$ for all $u\\in V$. === Nondegenerate Bilinear Form === {{:Definition:Nondegenerate Bilinear Form}}"}
+{"_id": "28118", "title": "Definition:Nondegenerate Bilinear Form", "text": "A bilinear form on $V$ which is not degenerate is '''nondegenerate'''."}
+{"_id": "28119", "title": "Definition:Quadratic Space", "text": "Let $\\mathbb K$ be a field. A '''quadratic space''' over $\\mathbb K$ is a pair $\\left({V, q}\\right)$ where: :$V$ is a vector space over $\\mathbb K$ of finite dimension $n>0$ :$q$ is a quadratic form on $V$."}
+{"_id": "28120", "title": "Definition:Orthogonal (Bilinear Form)", "text": "Let $\\mathbb K$ be a field. Let $V$ be a vector space over $\\mathbb K$. Let $b: V \\times V \\to \\mathbb K$ be a reflexive bilinear form on $V$. Let $v,w\\in V$. Then $v$ and $w$ are '''orthogonal (with respect to $b$)''' {{iff}} $b \\left({v, w}\\right) = b \\left({w, v}\\right) = 0$ This is denoted: $v \\perp w$. === Orthogonal Subsets === {{:Definition:Orthogonal (Bilinear Form)/Subsets}} === Orthogonal Complement === {{:Definition:Orthogonal (Bilinear Form)/Orthogonal Complement}} === Radical === {{:Definition:Orthogonal (Bilinear Form)/Radical}}"}
+{"_id": "28121", "title": "Definition:Orthogonal (Bilinear Form)/Radical", "text": "The '''radical''' of $V$ is the orthogonal complement of $V$: :$\\operatorname{rad}(V) = V^\\perp$"}
+{"_id": "28122", "title": "Definition:Orthogonal (Bilinear Form)/Orthogonal Complement", "text": "Let $S\\subset V$ be a subset. The '''orthogonal complement of $S$ (with respect to $b$)''' is the set of all $v \\in V$ which are orthogonal to all $s \\in S$. This is denoted: $S^\\perp$. If $S = \\left\\{ {v}\\right\\}$ is a singleton, we also write $v^\\perp$."}
+{"_id": "28124", "title": "Definition:Relative Matrix of Quadratic Form", "text": "Let $\\mathbb K$ be a field of characteristic $\\operatorname{char}\\mathbb K \\neq2$. Let $V$ be a vector space over $\\mathbb K$ of finite dimension $n>0$. Let $\\mathcal B$ be an ordered basis of $V$. Let $q$ be a quadratic form on $V$. Its '''matrix  relative to $\\mathcal B$''' is the matrix of its associated bilinear form relative to $\\mathcal B$, denoted $\\mathbf M_{q, \\mathcal B}$. Category:Definitions/Quadratic Forms 4nmj6syc0tmb5ovh892qw2ag3haw907"}
+{"_id": "28125", "title": "Definition:Symplectic Basis", "text": "Let $\\mathbb K$ be a field. Let $\\struct {V, f}$ be a bilinear space over $\\mathbb K$ of finite dimension $2 n > 0$. Let $\\BB = \\tuple {b_1, c_1, \\ldots, b_n, c_n}$ be an ordered basis of $V$. Then $\\BB$ is '''symplectic''' {{iff}}: :$\\map f {b_i, b_j} = \\map f {c_i, c_j} = 0$ for all $i, j$ :$\\map f {b_i, c_j} = \\delta_{i j}$ for all $i, j$ where $\\delta$ denotes Kronecker delta. That is, {{iff}} the matrix of $f$ relative to $\\BB$ has the form: :$\\begin{pmatrix}  0 & 1 \\\\ -1 & 0 \\\\  & &  0 & 1 \\\\  & & -1 & 0 \\\\  & & & & \\ddots \\\\  & & & & &  0 & 1 \\\\   & & & & & -1 & 0 \\end{pmatrix}$"}
+{"_id": "28126", "title": "Definition:Bilinear Space", "text": "Let $\\mathbb K$ be a field. A '''bilinear space''' over $\\mathbb K$ is a pair $\\left({V, f}\\right)$ where: :$V$ be a vector space over $\\mathbb K$ of finite dimension $n > 0$ :$f$ is a bilinear form on $V$."}
+{"_id": "28127", "title": "Definition:Orthogonal Basis", "text": "=== Orthogonal Basis of Vector Space === {{:Definition:Orthogonal Basis of Vector Space}} === Orthogonal Basis of Inner Product Space === {{:Definition:Orthogonal Basis of Inner Product Space}} === Orthogonal Basis of Bilinear Space === {{:Definition:Orthogonal Basis of Bilinear Space}} Category:Definitions/Linear Algebra 5ccrihgddx0hdkbw28ybw4mk0efqf71"}
+{"_id": "28128", "title": "Definition:Orthogonal Basis/Bilinear Space", "text": "Let $\\mathbb K$ be a field. Let $\\struct {V, f}$ be a bilinear space over $\\mathbb K$ of finite dimension $n > 0$. Let $\\BB = \\tuple {b_1, \\ldots, b_n}$ be an ordered basis of $V$. Then $\\BB$ is '''orthogonal''' {{iff}}: :$\\map f {b_i, b_j} = 0$ for $i \\ne j$. That is, {{iff}} the matrix of $f$ relative to $\\mathcal B$ is diagonal."}
+{"_id": "28129", "title": "Definition:Internal Orthogonal Sum (Bilinear Space)", "text": "Let $\\mathbb K$ be a field. Let $\\left({V, f}\\right)$ be a reflexive bilinear space over $\\mathbb K$. Let $U, W \\subset V$ be subspaces of $V$. Then $V$ is the '''internal orthogonal (direct) sum''' of $U$ and $W$ {{iff}}: :$V = U \\oplus W$, that is, $V$ is the internal direct sum of $U$ and $W$ :$U \\perp W$, that is, $U$ and $W$ are orthogonal. This is denoted: $V = U\\oplus W$."}
+{"_id": "28130", "title": "Definition:Reflexive Bilinear Space", "text": "Let $\\mathbb K$ be a field. Let $\\left({V, f}\\right)$ be a bilinear space over $\\mathbb K$. Then $\\left({V, f}\\right)$ is '''reflexive''' {{iff}} $f$ is a reflexive bilinear form. Category:Definitions/Bilinear Forms hk94i1mcmdwfx8os8bp7i5ldr18shcq"}
+{"_id": "28131", "title": "Definition:Orthogonal (Bilinear Form)/Subsets", "text": "Let $S, T \\subset V$ be subsets. Then $S$ and $T$ are '''orthogonal''' {{iff}} for all $s\\in S$ and $t\\in T$, $s$ and $t$ are orthogonal: $s \\perp t$."}
+{"_id": "28132", "title": "Definition:Field of Quotients/Definition 2", "text": "A '''field of quotients''' of $D$ is a pair $\\struct {F, \\iota}$ such that: :$(1): \\quad F$ is a field :$(2): \\quad \\iota: D \\to F$ is a ring monomorphism  :$(3): \\quad$ If $K$ is a field with $\\iota \\sqbrk D \\subset K \\subset F$, then $K = F$. That is, the '''field of quotients''' of an integral domain $D$ is the smallest field containing $D$ as a subring."}
+{"_id": "28133", "title": "Definition:Field of Quotients/Definition 3", "text": "A '''field of quotients''' of $D$ is a pair $\\struct {F, \\iota}$ where: :$(1): \\quad$ $F$ is a field :$(2): \\quad$ $\\iota : D \\to F$ is a ring monomorphism :$(3): \\quad$ it satisfies the following universal property: ::::For every field $E$ and for every ring monomorphism $\\varphi: D \\to E$, there exists a unique field homomorphism $\\bar \\varphi: F \\to E$ such that $\\varphi = \\bar \\varphi \\circ \\iota$"}
+{"_id": "28134", "title": "Definition:Field of Quotients/Definition 1", "text": "A '''field of quotients''' of $D$ is a pair $\\struct {F, \\iota}$ where: :$(1): \\quad$ $F$ is a field :$(2): \\quad$ $\\iota : D \\to F$ is a ring monomorphism :$(3): \\quad \\forall z \\in F: \\exists x \\in D, y \\in D_{\\ne 0}: z = \\dfrac {\\map \\iota x} {\\map \\iota y}$"}
+{"_id": "28135", "title": "Definition:Field of Directions/Functional", "text": "Let $\\mathbf y$ be an $N$-dimensional vector. Let the functional $J$ be such that: :$\\ds J \\sqbrk {\\mathbf y} = \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x $ Let the following be a family of boundary conditions, presribed $\\forall x \\in \\closedint a b$: :$\\mathbf y' = \\map {\\boldsymbol \\psi} {x, \\mathbf y}$ Let these boundary conditions be self-adjoint and consistent $\\forall x_1, x_2 \\in \\closedint a b$. Then these boundary conditions are called '''field of directions''' of the functional $J$."}
+{"_id": "28136", "title": "Definition:Pluperfect Digital Invariant", "text": "Let $n \\in \\Z_{>0}$ be a positive integer. $n$ is a '''pluperfect digital invariant''' {{iff}} when expressed in decimal notation the $n$th powers of its digits add up to $n$."}
+{"_id": "28138", "title": "Definition:Inconsummate Number", "text": "Let $m \\in \\Z_{>0}$ be a positive integer. Let $s_{10}$ denote the digit sum base $10$  . $m$ is an '''inconsummate number''' {{iff}}: :$\\nexists n \\in \\Z_{>0}: n = m \\times s_{10} \\left({n}\\right)$ That is, {{iff}} there exists no positive integer $n \\in \\Z_{>0}$ such that $n$ equals $m$ multiplied by the digit sum of $n$."}
+{"_id": "28140", "title": "Definition:Kaprekar Mapping", "text": "The '''Kaprekar mapping''' is the arithmetic function $K: \\Z_{>0} \\to \\Z_{>0}$ defined on the positive integers  as follows: Let $n \\in \\Z_{>0}$ be expressed in some number base $b$ (where $b$ is usually $10$). Let $n'$ be the positive integer created by arranging the digits of $n$ into descending order of size. Let $n''$ be the positive integer created by arranging the digits of $n$ into ascending order of size. Then: :$K \\left({n}\\right) = n' - n''$ making sure to retain any leading zeroes to ensure that $K \\left({n}\\right)$ has the same number of digits as $n$."}
+{"_id": "28141", "title": "Definition:Kaprekar's Process", "text": "'''Kaprekar's process''' is the repeated application of the '''Kaprekar mapping''' to a given positive integer."}
+{"_id": "28142", "title": "Definition:Sixth Power", "text": "A '''sixth power''' is an integer which can be expressed as the $6$th power of an integer."}
+{"_id": "28144", "title": "Definition:Epsilon Relation/Restriction", "text": "Let $S$ be a set. The '''restriction of the epsilon relation''' on $S$ is defined as the endorelation $\\Epsilon {\\restriction_S} = \\left({S, S, \\in_S}\\right)$, where: :$\\in_S \\; := \\left\\{{\\left({x, y}\\right) \\in S \\times S: x \\in y}\\right\\}$"}
+{"_id": "28145", "title": "Definition:Weird Number", "text": "A '''weird number''' is an abundant number which is not at the same time semiperfect."}
+{"_id": "28147", "title": "Definition:Weird Number/Sequence", "text": "The sequence of weird numbers begins: :$70, 836, 4030, 5830, 7192, 7912, 9272, \\ldots$"}
+{"_id": "28148", "title": "Definition:Heptagonal Number", "text": "'''Heptagonal numbers''' are those denumerating a collection of objects which can be arranged in the form of a regular heptagon."}
+{"_id": "28149", "title": "Definition:Heptagonal Number/Definition 3", "text": ":$\\forall n \\in \\N: H_n = P \\left({7, n}\\right) = \\begin{cases} 0 & : n = 0 \\\\ P \\left({7, n - 1}\\right) + 5 \\left({n - 1}\\right) + 1 & : n > 0 \\end{cases}$ where $P \\left({k, n}\\right)$ denotes the $k$-gonal numbers."}
+{"_id": "28150", "title": "Definition:Heptagonal Number/Definition 1", "text": ":$H_n = \\begin{cases} 0 & : n = 0 \\\\ H_{n-1} + 5 n - 4 & : n > 0 \\end{cases}$"}
+{"_id": "28151", "title": "Definition:Heptagonal Number/Definition 2", "text": ":$\\displaystyle H_n = \\sum_{i \\mathop = 1}^n \\left({5 i - 4}\\right) = 1 + 6 + \\cdots + \\left({5 \\left({n - 1}\\right) - 4}\\right) + \\left({5 n - 4}\\right)$"}
+{"_id": "28153", "title": "Definition:Heptagonal Number/Sequence", "text": "The sequence of heptagonal numbers, for $n \\in \\Z_{\\ge 0}$, begins: :$0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, \\ldots$"}
+{"_id": "28154", "title": "Definition:Zero Digit", "text": "Let $x \\in \\R$ be a number. Let $b \\in \\Z$ such that $b > 1$ be a number base in which $x$ is represented. By the Basis Representation Theorem, $x$ can be expressed uniquely in the form: :$\\displaystyle x = \\sum_{j \\mathop \\in \\Z}^m r_j b^j$ Any instance of $r_j$ being equal to $0$ is known as a '''zero (digit)''' of $n$."}
+{"_id": "28155", "title": "Definition:Prime Gap", "text": "A '''prime gap''' is a difference between two consecutive prime numbers."}
+{"_id": "28157", "title": "Definition:Sophie Germain Prime/Sequence", "text": "The sequence of Sophie Germain primes begins: :$2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, \\ldots$"}
+{"_id": "28159", "title": "Definition:Reverse-and-Add", "text": "Let $m \\in \\Z_{>0}$ be a positive integer expressed in decimal notation. Let $r: \\Z_{>0} \\to \\Z_{>0}$ be the mapping defined as: :For all $n \\in \\Z_{>0}$: reverse the digits of $m$ and add the result to $m$. The mapping $r$ is the '''reverse-and-add operation'''."}
+{"_id": "28161", "title": "Definition:Central Field", "text": "Let $R$ be a simply connected region in $\\paren {n + 1}$-dimensional space. Let $\\tuple {x, \\mathbf y}$ be a point in $R$. Let $c = \\tuple {\\sequence {c_i}_{0 \\mathop \\le i \\mathop \\le n} }$ be a point lying outside of $R$. Let $J$ be a functional such that: :$\\ds J \\sqbrk {\\mathbf y} = \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ whose extremals $\\mathbf y$ are curves in $\\paren{n+1}$-dimensional space. {{explain|Extremum or extremal, or are they the same thing? If they are, then an \"also known as\" section needs to be placed on the page Definition:Extremum of Functional. If not, then the terminology needs to be specifically amended and/or corrected.}} Suppose that one and only one extremal of $J$ leaves $c$ and passes through $\\tuple {x, \\mathbf y}$, thereby for every point in $R$ defining a relation: :$(1): \\quad \\map {\\mathbf y'} x = \\map {\\boldsymbol \\psi} {x, \\mathbf y}$. Then the field of directions $(1)$ is called a '''central field'''."}
+{"_id": "28162", "title": "Definition:Time/Unit/Quarter-Year", "text": "The '''quarter-year''' is defined as: :$91$ days or :$13$ weeks of $7$ days each."}
+{"_id": "28163", "title": "Definition:Fermat Pseudoprime/Base 3", "text": "Let $q$ be a composite number such that $3^q \\equiv 3 \\pmod q$. Then $q$ is a '''Fermat pseudoprime to base $3$'''."}
+{"_id": "28164", "title": "Definition:Celsius", "text": "'''Celsius''' is a temperature scale. Its two reference points are: : $0 \\cels$, which is set at the melting point of water. : $100 \\cels$, which is set at the boiling point of water, as defined at sea level and standard atmospheric pressure. A temperature measured in '''Celsius''' is often referred to as so many '''degrees Celsius'''. === Symbol === {{:Definition:Celsius/Symbol}}"}
+{"_id": "28165", "title": "Definition:Celsius/Symbol", "text": "The symbol for the '''degree Celsius''' is $\\cels$."}
+{"_id": "28167", "title": "Definition:Array/Row", "text": "Let $\\mathbf A$ be an array. The '''rows''' of $\\mathbf A$ are the lines of elements reading '''across''' the page."}
+{"_id": "28170", "title": "Definition:Array/Element/Index", "text": "Let $\\mathbf A$ be an $m \\times n$ array. Let $a_{i j}$ be an element of $\\mathbf A$. Then the subscripts $i$ and $j$ are referred to as the '''indices''' (singular: '''index''') of $a_{i j}$."}
+{"_id": "28171", "title": "Definition:Array/Element", "text": "Let $\\mathbf A$ be an Array. The individual $n \\times n$ symbols that go to form $\\mathbf L$ are known as the '''elements''' of $\\mathbf L$. The '''element''' at row $i$ and column $j$ is called '''element $\\tuple {i, j}$ of $\\mathbf A$''', and can be written $a_{i j}$, or $a_{i, j}$ if $i$ and $j$ are of more than one character. If the indices are still more complicated coefficients and further clarity is required, then the form $a \\left({i, j}\\right)$ can be used. Note that the first subscript determines the row, and the second the column, of the array where the '''element''' is positioned."}
+{"_id": "28172", "title": "Definition:Array/Square", "text": "An array whose dimensions are equal is called a '''square array'''. That is, a '''square array''' is an array which has the same number of rows as it has columns. A '''square array''' $\\left[{a}\\right]_{n n}$ is usually denoted $\\left[{a}\\right]_n$."}
+{"_id": "28175", "title": "Definition:Prime Magic Square", "text": "A '''prime magic square''' is a magic square whose elements are all prime numbers, and may or may not also include the number $1$."}
+{"_id": "28176", "title": "Definition:Extremal Embedding in Field of Functional", "text": "Let $J$ be a functional such that: :$\\ds J \\sqbrk {\\mathbf y} = \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ Let $\\gamma$ be an extremal of $J$. {{explain|Extremum or extremal?}} Let $R$ be a simply connected open region which contains $\\gamma$ as a subset. {{explain|GF puts \"open\" in brackets; does it matter?}} Let a field of functional $J$ be defined at every point of $R$. Let one of the trajectories of the field be $\\gamma$. Then $\\gamma$ can be '''embedded''' in a field of functional $J$. {{explain|If $\\gamma$ \"can be\" embedded, is it correct to say that $\\gamma$ \"is\" embedded? If not, then is it more grammatically accurate to define $\\gamma$ as being \"embeddable\"? Otherwise it looks as though \"can be\" introduces a statement that needs to be proven.}}"}
+{"_id": "28177", "title": "Definition:Permutable Prime", "text": "A '''permutable prime''' is a prime number $p$ which has the property that all anagrams of $p$ are prime."}
+{"_id": "28178", "title": "Definition:Permutable Prime/Sequence", "text": "The sequence of '''permutable primes''' begins: :$2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, R_{19}, R_{23}, R_{317}, R_{1091}, \\ldots$ where $R_n$ denotes the repunit of $n$ digits."}
+{"_id": "28179", "title": "Definition:Anagram", "text": "Let $b \\in \\Z_{>1}$ be an integer greater than $1$. Let $m$ be an integer expressed in base $b$. An '''anagram base $b$''' of $m$ is an integer formed by the digits of $m$ written in a different order. When the number base of $m$ is not specified, base $10$ is assumed."}
+{"_id": "28180", "title": "Definition:Repunit Prime", "text": "Let $b \\in \\Z_{>1}$ be an integer greater than $1$. A '''repunit prime base $b$''' is a repunit base $b$ which is prime."}
+{"_id": "28185", "title": "Definition:Repunit Prime/Index", "text": "The '''index''' of a '''repunit prime''' is the number of digits it has. Thus $R_n$ denotes a '''repunit prime''' with $n$ digits."}
+{"_id": "28186", "title": "Definition:Repunit/Index", "text": "The '''index''' of a '''repunit''' is the number of digits it has. Thus a '''repunit''' with $n$ digits can be referred to as $R_n$."}
+{"_id": "28187", "title": "Definition:Fermat Set", "text": "The '''Fermat set''' is the set of integers: :$\\left\\{ {1, 3, 8, 120}\\right\\}$ It has the property of being a Diophantine quadruple."}
+{"_id": "28188", "title": "Definition:Diophantine m-Tuple/Quadruple", "text": "A '''Diophantine quadruple''' is a Diophantine $m$-tuple where $m = 4$. That is, a '''Diophantine quadruple''' is a set of $4$ positive integers such that the product of any $2$ of them plus $1$ is a square number."}
+{"_id": "28189", "title": "Definition:Diophantine m-Tuple", "text": "A '''Diophantine $m$-tuple''' set of $m$ positive integers such that the product of any $2$ of them plus $1$ is a square number."}
+{"_id": "28191", "title": "Definition:Diophantine m-Tuple/Quintuple", "text": "A '''Diophantine quintuple''' is a Diophantine $m$-tuple where $m = 5$. That is, a '''Diophantine quintuple''' is a set of $5$ positive integers such that the product of any $2$ of them plus $1$ is a square number."}
+{"_id": "28193", "title": "Definition:Triperfect Number", "text": "A '''triperfect number''' is a positive integer $n$ such that the sum of its divisors is equal to $3$ times $n$."}
+{"_id": "28194", "title": "Definition:Multiply Perfect Number", "text": "A '''multiply perfect number''' is a positive integer $n$ such that the sum of its divisors is equal to an integer multiple of $n$."}
+{"_id": "28199", "title": "Definition:Hilbert's Invariant Integral", "text": "Let $\\mathbf y$ be an $n$-dimensional vector. Let $H$ be Hamiltonian and $\\mathbf p$ momenta. Let $\\Gamma$ be a curve connecting points $\\tuple {x_0, \\map{\\mathbf y} {x_0} }$ and $\\tuple {x, \\mathbf y}$. Then the following line integral is known as '''Hilbert's Invariant Integral''': :$\\displaystyle \\map g {x, \\mathbf y} = \\int_\\Gamma \\paren {-H \\rd x + \\mathbf p \\rd \\mathbf y}$ {{NamedforDef|David Hilbert|cat = Hilbert}}"}
+{"_id": "28200", "title": "Definition:Weierstrass E-Function", "text": "Let $\\mathbf y, \\mathbf z, \\mathbf w : \\R \\to \\R^n$ be $n$-dimensional vector-valued functions. Let $\\mathbf y$ be such that $\\map {\\mathbf y} a = A$ and $\\map {\\mathbf y} b = B$. Let $F: \\R^{2 n + 1} \\to \\R$ be twice differentiable {{WRT|Differentiation}} its (independent) variables. Let $J$ be a functional such that: :$\\ds J \\sqbrk{\\mathbf y} = \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ where:  :$\\mathbf y' := \\dfrac {\\d \\mathbf y} {\\d x}$ is the derivative of a vector-valued function. === Definition 1 === {{:Definition:Weierstrass E-Function/Definition 1}} === Definition 2 === {{:Definition:Weierstrass E-Function/Definition 2}}"}
+{"_id": "28202", "title": "Definition:Multiply Perfect Number/Order", "text": "Let $n \\in \\Z_{>0}$ be a '''multiply perfect number''' such that the sum of its divisors is equal to $m \\times n$. Then $n$ is '''multiply perfect of order $m$'''."}
+{"_id": "28203", "title": "Definition:Wonderful Demlo Number", "text": "The '''Wonderful Demlo numbers''' are the positive integers defined as the coefficients of the powers of $x$ in the expansion of: :$\\dfrac {1 + 10 x} {\\left({1 - x}\\right) \\left({1 - 10 x}\\right) \\left({1 - 100 x}\\right)}$"}
+{"_id": "28204", "title": "Definition:Wonderful Demlo Number/Sequence", "text": "The sequence of '''Wonderful Demlo numbers''' begins: :$1, 121, 12 \\, 321, 1 \\, 234 \\, 321, 123 \\, 454 \\, 321, \\ldots$"}
+{"_id": "28206", "title": "Definition:Obstinate Number", "text": "An '''obstinate number''' is a positive odd integer which cannot be expressed as the sum of a power of $2$ and a prime."}
+{"_id": "28208", "title": "Definition:Ore Number", "text": "Let $n \\in \\Z_{>0}$ be a positive integer. $n$ is an '''Ore number''' {{iff}} the harmonic mean of its divisors is an integer."}
+{"_id": "28213", "title": "Definition:Woodall Number", "text": "A '''Woodall number''' is a positive integer of the form: :$n \\times 2^n - 1$"}
+{"_id": "28217", "title": "Definition:Primitive Prime Factor of Fibonacci Number", "text": "Let $F_n$ denote the $n$th Fibonacci number. A '''primitive prime factor of $F_n$''' is a prime number $p$ of $F_n$ such that: :$p \\divides F_n$ :$\\nexists k \\in \\Z_{>0}: k < n: p \\divides F_k$ where $a \\divides b$ denotes that $a$ is a divisor of $b$. That is, a prime factor of $F_n$ but of no smaller Fibonacci numbers."}
+{"_id": "28218", "title": "Definition:Primitive Prime Factor", "text": "Let $\\left\\langle{a_n}\\right\\rangle$ be an integer sequence. A '''primitive prime factor''' for a term $a_n$ is a prime number $p$ of $a_n$ such that: :$p \\divides a_n$ :$\\nexists k \\in \\Z_{>0}: k < n: p \\divides a_k$ where $a \\divides b$ denotes that $a$ is a divisor of $b$."}
+{"_id": "28219", "title": "Definition:Factorion", "text": "A '''factorion (base $b$)''' is a positive integer which, when expressed in base $b$, equals the sum of the factorials of its digits."}
+{"_id": "28222", "title": "Definition:Philosophy", "text": "'''Philosophy''' is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language. As there are considerable overlaps with mathematics, it is sometimes argued that mathematics can be considered a branch of '''philosophy'''. It is the controversial opinion of some mathematicians that philosophy is for people who are not smart enough for mathematics."}
+{"_id": "28223", "title": "Definition:Polynomial Ring/Sequences", "text": "Let $R^{\\left({\\N}\\right)}$ be the ring of sequences of finite support over $R$. Let $\\iota : R \\to R^{\\left({\\N}\\right)}$ be the mapping defined as: :$\\iota \\left({r}\\right) = \\left \\langle {r, 0, 0, \\ldots}\\right \\rangle$. Let $X$ be the sequence $\\left \\langle {0, 1, 0, \\ldots}\\right \\rangle$. The '''polynomial ring over $R$''' is the ordered triple $\\left({R^{\\left({\\N}\\right)}, \\iota, X}\\right)$."}
+{"_id": "28224", "title": "Definition:Highly Abundant Number", "text": "Let $n \\in \\Z_{>0}$ be a positive integer. Then $n$ is '''highly abundant''' {{iff}}: :$\\forall m \\in \\Z_{>0}, m < n: \\map \\sigma m < \\map \\sigma n$ where $\\map \\sigma n$ is the $\\sigma$ function of $n$."}
+{"_id": "28229", "title": "Definition:Heptagonal Pyramidal Number", "text": "'''Heptagonal pyramidal numbers''' are those denumerating a collection of objects which can be arranged in the form of a heptagonal pyramid."}
+{"_id": "28232", "title": "Definition:Lychrel Number/Candidate", "text": "No natural number has been '''proved''' to be a '''Lychrel number''' as of time of writing (June $2017$).  However, plenty of numbers have not shown themselves to terminate in a palindromic number, although in some cases millions of iterations have been tested. Hence a '''candidate Lychrel number''' is a natural number which '''is not known to''' form a palindromic number through repeated iteration of the reverse-and add process."}
+{"_id": "28236", "title": "Definition:Sequence/Minimizing/Functional", "text": "Let $y$ be a real mapping defined on a space $\\MM$. {{Stub|what is $\\MM$?}} Let $J \\sqbrk y$ be a functional such that: :$\\exists y \\in \\MM: J \\sqbrk y < \\infty$ :$\\ds \\exists \\mu > -\\infty: \\inf_y J \\sqbrk y = \\mu$ Let $\\sequence {y_n}$ be a sequence such that: :$\\ds \\lim_{n \\mathop \\to \\infty} J \\sqbrk {y_n} = \\mu$ Then the sequence $\\sequence {y_n}$ is called a '''minimizing sequence (of the functional $J \\sqbrk y$)'''."}
+{"_id": "28237", "title": "Definition:Sequence/Minimizing/Functional/Limit Minimizing Function of", "text": "Let $\\sequence {y_n}$ be a minimizing sequence of a functional $J$. Suppose: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} y_n = \\hat y$ and :$\\displaystyle \\lim_{n \\mathop \\to \\infty} J \\sqbrk {y_n} = J \\sqbrk {\\hat y}$ Then $\\hat y$ is the '''limit minimizing function''' of $J \\sqbrk {y_n}$ and $J \\sqbrk {\\hat y} = \\mu$."}
+{"_id": "28238", "title": "Definition:Ritz Method", "text": "Let $\\MM$ be a normed linear space. Let $J\\sqbrk y$ be a functional defined on space $\\MM$. Let $\\sequence{\\phi_n}$ be an infinite sequence of mappings in $\\MM$. Let $\\MM_n$ be an $n$-dimensional linear subspace of $\\MM$, spanned by first $n$ mapping of $\\sequence{\\phi_n}$. Let $\\eta_n = \\boldsymbol \\alpha \\boldsymbol \\phi$, where $\\boldsymbol \\alpha$ is a real $n$-dimensional vector. Minimise $J \\sqbrk {\\eta_n}$ {{WRT}} $\\boldsymbol\\alpha$. Then $J \\sqbrk {\\eta_n}$ is an approximate minimum of $J \\sqbrk y$, and is denoted by $\\mu_n$. This method is known as '''Ritz method'''."}
+{"_id": "28239", "title": "Definition:Multiplicative Magic Square", "text": "A '''multiplicative magic square''' is an arrangement of $n^2$ distinct numbers into an $n \\times n$ square array such that: :the product of the elements of each row :the product of the elements in each column :the product of the elements along each diagonal are the same."}
+{"_id": "28240", "title": "Definition:Multiplicative Magic Square/Order", "text": "An $n \\times n$ multiplicative magic square is called an '''order $n$ multiplicative magic square'''."}
+{"_id": "28243", "title": "Definition:Complete Ritz Sequence", "text": "Let $\\MM$ be a normed linear space. Let $\\sequence {\\phi_n}$ be a Ritz sequence in $\\MM$. Let $\\MM_n$ be an $n$-dimensional linear subspace of $\\MM$, spanned by the first $n$ mappings of $\\sequence {\\phi_n}$. Let $\\eta_n$ be of the form: :$\\eta_n = \\boldsymbol \\alpha \\boldsymbol \\phi$ where $\\boldsymbol \\alpha$ is an $n$-dimensional real vector. Suppose: :$\\forall y \\in \\MM: \\forall \\epsilon > 0: \\exists \\map n \\epsilon \\in \\N: \\exists \\eta_n \\in \\MM_n: \\size {\\eta_n - y} < \\epsilon$ {{explain|which norm is it?}} Then the sequence $\\sequence{\\phi_n}$ is called '''complete''' in $\\MM$. {{NamedforDef|Walther Ritz|cat = Ritz}}"}
+{"_id": "28244", "title": "Definition:Ritz Sequence", "text": "Let $\\sequence {\\phi_n}$ be an infinite sequence of mappings in a normed linear space, where: :$\\phi_n: \\R \\to \\R$ Let this sequence be constrained by the requirements defined in the definition of the Ritz method. {{explain|Specify what these are. For ultimate reusability, consider extracting those requirements into an appropriately-named and transcluded subpage of the above.}} Then the sequence $\\set {\\phi_n}$ can be called a '''Ritz Sequence'''."}
+{"_id": "28245", "title": "Definition:Plato's Geometrical Number", "text": "The actual value of what is generally known as '''Plato's geometrical number''' is uncertain. The passage in question from {{AuthorRef|Plato}}'s {{BookLink|Republic|Plato}} is obscure and difficult to interpret: :''But the number of a human creature is the first number in which root and square increases, having received three distances and four limits, of elements that make both like and unlike wax and wane, render all things conversable and rational with one another.'' There are two numbers which it is generally believed it could be: :$216$ :$12 \\, 960, \\, 000$ It is believed that the expression: :''three distances and four limits'' refers to cubing. It is further believed that the reference is to the area of the $3-4-5$ triangle, which is $6$. The passage is also deduced to contain a reference to $2 \\times 3$. It is also interpreted by other commentators as being $12 \\, 960 \\, 000$, which is $60^4$."}
+{"_id": "28247", "title": "Definition:Fedorov Group", "text": "A '''Fedorov group''' is the symmetry group of a $3$-dimensional configuration in space."}
+{"_id": "28248", "title": "Definition:Space Group", "text": "A '''space group''' is the symmetry group of a configuration in $n$-dimensional space."}
+{"_id": "28250", "title": "Definition:Wallpaper Group", "text": "A '''wallpaper group''' is the symmetry group of a $2$-dimensional configuration in the plane."}
+{"_id": "28252", "title": "Definition:Amicable Pair/Definition 1", "text": "$m$ and $n$ are an '''amicable pair''' {{iff}}: :the aliquot sum of $m$ is equal to $n$ and: :the aliquot sum of $n$ is equal to $m$."}
+{"_id": "28253", "title": "Definition:Amicable Pair/Definition 2", "text": "$m$ and $n$ are an '''amicable pair''' {{iff}}: :$\\map \\sigma m = \\map \\sigma n = m + n$ where $\\map \\sigma m$ denotes the $\\sigma$ function."}
+{"_id": "28254", "title": "Definition:Amicable Pair/Definition 3", "text": "$m$ and $n$ are an '''amicable pair''' {{iff}} they form a sociable chain of order $2$."}
+{"_id": "28256", "title": "Definition:Thabit Pair", "text": "A '''Thabit pair''' is an amicable pair formed by an application of Thabit's Rule: :$\\tuple {2^n a b, 2^n c}$ where $n$ be a positive integer such that: {{begin-eqn}} {{eqn | l = a       | r = 3 \\times 2^n - 1       | c =  }} {{eqn | l = b       | r = 3 \\times 2^{n - 1} - 1       | c =  }} {{eqn | l = c       | r = 9 \\times 2^{2 n - 1} - 1       | c =  }} {{end-eqn}} are all prime."}
+{"_id": "28257", "title": "Definition:Thabit Number", "text": "A '''Thabit number''' is an integer in the form: :$3 \\times 2^n - 1$ where $n \\in \\Z_{\\ge 0}$ is a non-negative integer."}
+{"_id": "28258", "title": "Definition:Thabit Number/Sequence", "text": "The sequence of Thabit numbers begins: :$2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, \\ldots$"}
+{"_id": "28259", "title": "Definition:Thabit Prime", "text": "A '''Thabit prime''' is a Thabit number which is prime."}
+{"_id": "28260", "title": "Definition:Thabit Prime/Sequence", "text": "The sequence of Thabit primes begins: :$2, 5, 11, 23, 47, 191, 383, 6143, 786 \\, 431, 51 \\, 539 \\, 607 \\, 551, \\ldots$"}
+{"_id": "28262", "title": "Definition:Amicable Triplet", "text": "Let $m_1, m_2, m_3 \\in \\Z_{>0}$ be (strictly) positive integers. === Definition 1 === {{:Definition:Amicable Triplet/Definition 1}} === Definition 2 === {{:Definition:Amicable Triplet/Definition 2}}"}
+{"_id": "28264", "title": "Definition:Amicable Triplet/Definition 1", "text": "$\\tuple {m_1, m_2, m_3}$ are an '''amicable triplet''' {{iff}} the aliquot sum of any one of them equals the sum of the other two: :the aliquot sum of $m_1$ is equal to $m_2 + m_3$ and: :the aliquot sum of $m_2$ is equal to $m_1 + m_3$ and: :the aliquot sum of $m_3$ is equal to $m_1 + m_2$"}
+{"_id": "28265", "title": "Definition:Amicable Triplet/Definition 2", "text": "$\\left({m_1, m_2, m_3}\\right)$ are an '''amicable triplet''' {{iff}}: :$\\sigma \\left({m_1}\\right) = \\sigma \\left({m_2}\\right) = \\sigma \\left({m_3}\\right) = m_1 + m_2 + m_3$ where $\\sigma \\left({m}\\right)$ denotes the $\\sigma$ function."}
+{"_id": "28266", "title": "Definition:Ljunggren Equation", "text": "The '''Ljunggren equation''' is the indeterminate Diophantine equation: :$x^2 + 1 = 2 y^4$"}
+{"_id": "28268", "title": "Definition:Sierpiński Number", "text": "There are two kinds of '''Sierpiński number''': === Sierpiński Numbers of the First Kind === {{:Definition:Sierpiński Number of the First Kind}} === Sierpiński Numbers of the Second Kind === {{:Definition:Sierpiński Number of the Second Kind}}"}
+{"_id": "28269", "title": "Definition:Sierpiński Number of the First Kind", "text": "The '''Sierpiński numbers of the first kind''' are the integers $S_n$ in the form: :$S_n := n^n + 1$ for all integers $n$."}
+{"_id": "28271", "title": "Definition:Sierpiński Number of the Second Kind/Sequence", "text": "The  sequence of known '''Sierpiński numbers of the second kind''' starts: :$78\\ 557, \\ 271\\ 129, \\ 271\\ 577, \\ 322\\ 523, \\ 327\\ 739, \\ 482\\ 719, \\ 575\\ 041, \\ 603\\ 713, \\ 903\\ 983, \\ 934\\ 909, \\ 965\\ 431, \\ \\ldots$"}
+{"_id": "28275", "title": "Definition:Sierpiński Number of the Second Kind/Historical Note", "text": "{{AuthorRef|Wacław Franciszek Sierpiński}} proved in $1960$ that there is an infinite number of Sierpiński numbers of the second kind."}
+{"_id": "28276", "title": "Definition:Balanced Prime", "text": "=== Definition 1 === {{:Definition:Balanced Prime/Definition 1}} === Definition 2 === {{:Definition:Balanced Prime/Definition 2}} === Definition 3 === {{:Definition:Balanced Prime/Definition 3}}"}
+{"_id": "28278", "title": "Definition:Balanced Prime/Definition 2", "text": "Let $\\paren {p_{n - 1}, p_n, p_{n + 1} }$ be a triplet of consecutive prime numbers. $p_n$ is a '''balanced prime''' {{iff}}: {{begin-eqn}} {{eqn | l = p_{n - 1} + d       | r = p_n       | c =  }} {{eqn | l = p_{n - 1} + 2 d       | r = p_{n + 1}       | c =  }} {{end-eqn}} for some $d \\in \\Z$."}
+{"_id": "28279", "title": "Definition:Balanced Prime/Definition 1", "text": "Let $\\left({p_{n - 1}, p_n, p_{n + 1} }\\right)$ be a triplet of consecutive prime numbers. $p_n$ is a '''balanced prime''' {{iff}}: :$p_n = \\dfrac {p_{n - 1} + p_{n + 1} } 2$"}
+{"_id": "28282", "title": "Definition:Aliquot Sequence", "text": "Let $m$ be a positive integer. Let $\\map s m$ be the aliquot sum of $m$. Define the sequence $\\sequence {a_k}$ recursively as: :$a_{k + 1} = \\begin{cases} m & : k = 0 \\\\ \\map s {a_k} & : k > 0 \\end{cases}$ The sequence $\\sequence {a_k}$ is known as an '''aliquot sequence'''."}
+{"_id": "28286", "title": "Definition:Eighth Power", "text": "A '''eighth power''' is an integer which can be expressed as the $8$th power of an integer."}
+{"_id": "28287", "title": "Definition:Superfactorial", "text": "The '''superfactorial of $n$''' is defined as: :$n\\$ = \\displaystyle \\prod_{k \\mathop = 1}^n k! = 1! \\times 2! \\times \\cdots \\times \\left({n - 1}\\right)! \\times n!$ where $k!$ denotes the factorial of $n$."}
+{"_id": "28289", "title": "Definition:Completely Irreducible", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $p \\in S$. An element $p$ is '''completely irreducible''' {{iff}} :$p^\\succeq \\setminus \\set p$ admits a minimum element where $p^\\succeq$ denotes the upper closure of $p$."}
+{"_id": "28290", "title": "Definition:Poulet Number", "text": "Let $q$ be a composite number such that $2^q \\equiv 2 \\pmod q$. Then $q$ is a '''Poulet number'''."}
+{"_id": "28296", "title": "Definition:Time/Unit/Lunar Month", "text": "The '''lunar month''' is defined as the length of time between $2$ new moons. :$1$ '''lunar month''' $\\approx 29 \\cdotp 530588$ days or: :$1$ '''lunar month''' $\\approx 29$ days, $12$ hours, $44$ minutes and $2 \\cdotp 8$ seconds."}
+{"_id": "28297", "title": "Definition:Calendar", "text": "A '''calendar''' is a technique to keep track of the passing of time throughout the course of a year. Its objective is to provide a more-or-less standard way of identifying the time of year by associating a name and number to each day in the year."}
+{"_id": "28298", "title": "Definition:Calendar/Julian/Year", "text": "A '''Julian year''' is the length of a year as defined using the '''Julian calendar'''. :$1$ '''Julian year''' $= \\begin{cases} 366 \\, \\text {days} & : 4 \\divides y \\\\ 365 \\, \\text {days} & : 4 \\nmid y \\end{cases}$ where: :$y$ denotes the number of the year :$4 \\divides y$ denotes that $y$ is divisible by $4$ :$4 \\nmid y$ denotes that $y$ is not divisible by $4$."}
+{"_id": "28299", "title": "Definition:Calendar/Julian", "text": "The '''Julian calendar''' is the calendar which was introduced by [https://en.wikipedia.org/wiki/Julius_Caesar Julius Caesar] in $45 \\, \\text{BCE}$. It divides the year into: :$365$ days for every $3$ out of $4$ years and: :$366$ days for every $1$ out of $4$ years, known as a leap year. The years themselves are given an index number, and are known by that number. A standard numbering system was introduced by [https://en.wikipedia.org/wiki/Dionysius_Exiguus Dionysus Exiguus]. He identified a particular year as being $525 \\, \\text{A.D.}$, where $\\text{A.D.}$ is an abbreviation for '''Anno Domini''', Latin for '''in the year of the Lord'''. The year $1$ was conventionally supposed to identify the year of the birth of [https://en.wikipedia.org/wiki/Jesus Jesus of Nazareth], although the accuracy of this has since been questioned. Years before $1 \\, \\text{A.D.}$ are counted backwards and assigned the label $\\text{B.C.}$. However, note that the year immediately prior to $1 \\, \\text{A.D.}$ is $1 \\, \\text{B.C.}$, not the intuitive '''year $0$''', a discrepancy that can cause confusion. Using this system of numbering, a leap year is identified by this number being divisible by $4$. The $365$ or $366$ days in the year are divided into $12$ approximately equal sections called months, which are assigned both names and index numbers: {{begin-eqn}} {{eqn | l = 1:       | o =        | c = January       | cc= $31$ days }} {{eqn | l = 2:       | o =        | c = February       | cc= $28$ days, or $29$ days in a leap year }} {{eqn | l = 3:       | o =        | c = March       | cc= $31$ days }} {{eqn | l = 4:       | o =        | c = April       | cc= $30$ days }} {{eqn | l = 5:       | o =        | c = May       | cc= $31$ days }} {{eqn | l = 6:       | o =        | c = June       | cc= $30$ days }} {{eqn | l = 7:       | o =        | c = July       | cc= $31$ days }} {{eqn | l = 8:       | o =        | c = August       | cc= $31$ days }} {{eqn | l = 9:       | o =        | c = September       | cc= $30$ days }} {{eqn | l = 10:       | o =        | c = October       | cc= $31$ days }} {{eqn | l = 11:       | o =        | c = November       | cc= $30$ days }} {{eqn | l = 12:       | o =        | c = December       | cc= $31$ days }} {{end-eqn}} Thus, for example, the day following the $31$st of January is the $1$st of February, and the $30$th of June is followed by the $1$st of July."}
+{"_id": "28300", "title": "Definition:Leap Year", "text": "The number of days in a year is not an integer, but lies somewhere between $365$ and $366$. However, it is convenient to assign an integer number of days to any given year. It is also desirable to ensure that a particular day of the year occurs at approximately the same time each year. In the Julian and Gregorian calendars: :some years are assigned $365$ days and: :some years are assigned $366$ days. The years which have $366$ days are called '''leap years'''."}
+{"_id": "28301", "title": "Definition:Calendar/Gregorian", "text": "The '''Gregorian calendar''' is the calendar which was introduced by [https://en.wikipedia.org/wiki/Pope_Gregory_XIII Pope Gregory XIII] in $1582 \\, \\text{CE}$. Over the course of the next few centuries it was gradually adopted by the various subcultures of [https://en.wikipedia.org/wiki/Western_culture Western civilization]. It was derived as a refinement of the Julian calendar to correct for the discrepancy between the Julian calendar and the tropical year. The years themselves are assigned the same numbers as their Julian counterparts. An evolving modern convention is to refer to $\\text{A.D.}$ and $\\text{B.C.}$ as $\\text{CE}$ and $\\text{BCE}$, for '''common era''' and '''before common era''' respectively. Like the Julian calendar, it divides the year into either $365$ days or $366$ days, according to the year number. The months, similarly, are kept the same as for the Julian calendar. The only difference is in the determination of which years are classified as leap years. Let $y$ be the year number. Then $y$ is a leap year {{iff}} :$y$ is divisible by $400$ or: :$y$ is divisible by $4$ and $y$ is not divisible by $100$. Thus, for example: :$2016$ was a leap year, because $2016$ is divisible by $4$ and not divisible by $100$. :$1900$ was '''not''' a leap year, because $1900$ is divisible by $100$ but not $400$. :$2000$ was a leap year, as $2000$ is divisible by $400$."}
+{"_id": "28303", "title": "Definition:Calendar/Gregorian/Year", "text": "A '''Gregorian year''', also known as the '''civil year''', is the length of a year as defined using the '''Gregorian calendar'''. :$1$ '''Gregorian year''' $= \\begin{cases} 366 \\, \\text {days} & : 400 \\divides y \\\\ 365 \\, \\text {days} & : 400 \\nmid  y \\text { and } 100 \\divides y \\\\ 366 \\, \\text {days} & : 100 \\nmid  y \\text { and } 4 \\divides y \\\\ 365 \\, \\text {days} & : 4 \\nmid y \\end{cases}$ where: :$y$ denotes the number of the year :$4 \\divides y$ denotes that $y$ is divisible by $4$ :$4 \\nmid y$ denotes that $y$ is not divisible by $4$."}
+{"_id": "28304", "title": "Definition:Calendar/Muslim", "text": "The '''Muslim calendar''' is a calendar whose period is based on $12$ full cycles of phases of the moon. Each Muslim year lasts either $354$ or $355$ days, depending on precisely when the relevant new moons happen The years are numbered from the year when [https://en.wikipedia.org/wiki/Muhammad Muhammad] and his followers migrated from [https://en.wikipedia.org/wiki/Mecca Mecca] to [https://en.wikipedia.org/wiki/Medina Medina] in $622 \\, \\text{A.D}$ by the Julian calendar. Years are denoted with the suffix $\\text{A.H.}$, meaning '''Anno Hegirae''', literally '''in the year of the Hejira'''. The start and end of the Muslim year are $10$ to $12$ days earlier every civil year. The $354$ or $355$ days in the Muslim year are divided into $12$ approximately equal sections called months, each either $29$ or $30$ days, depending on the phases of the moon. They are assigned both names and index numbers: {{begin-eqn}} {{eqn | l = 1:       | o =        | c = Muḥarram }} {{eqn | l = 2:       | o =        | c = Ṣafar }} {{eqn | l = 3:       | o =        | c = Rabī‘ al-awwal }} {{eqn | l = 4:       | o =        | c = Rabī‘ ath-thānī }} {{eqn | l = 5:       | o =        | c = Jumādá al-ūlá }} {{eqn | l = 6:       | o =        | c = Jumādá al-ākhirah }} {{eqn | l = 7:       | o =        | c = Rajab }} {{eqn | l = 8:       | o =        | c = Sha‘bān }} {{eqn | l = 9:       | o =        | c = Ramaḍān }} {{eqn | l = 10:       | o =        | c = Shawwāl }} {{eqn | l = 11:       | o =        | c = Dhū al-Qa‘dah }} {{eqn | l = 12:       | o =        | c = Dhū al-Ḥijjah }} {{end-eqn}}"}
+{"_id": "28305", "title": "Definition:Calendar/Muslim/Year", "text": "A '''Muslim year''', also known as the '''Islamic year''' or '''Hijra year''', is the length of a year as defined using the '''Muslim calendar'''. :$1$ '''Muslim year''' $= \\begin{cases} 355 \\, \\text {days} & : \\text {Leap year} \\\\ 354 \\, \\text {days} & : \\text {Non-leap year} \\end{cases}$"}
+{"_id": "28306", "title": "Definition:Calendar/Jewish/Year", "text": "A '''Jewish year''', also known as the '''Hebrew year''', is the length of a year as defined using the '''Jewish calendar'''. Its length is either $354$ or $355$ days, or longer, if an extra month has been inserted."}
+{"_id": "28307", "title": "Definition:Calendar/Jewish", "text": "The '''Jewish calendar''' is a calendar based on both the phases of the moon and on the orbital period of earth around the sun. The whose period is based on $12$ full cycles of phases of the moon. The basic Jewish year is $354$ or $355$ days of $12$ months, each alternately $29$ and $30$ days long. When the error between the Jewish year and tropical year becomes approximately $30$ days, a $13$th month is inerted into the year."}
+{"_id": "28308", "title": "Definition:Lucas-Carmichael Number", "text": "Let $n$ have the property that: :$p \\divides n \\implies \\paren {p + 1} \\divides \\paren {n + 1}$ where: :$p$ is prime :$p \\divides n$ denotes that $p$ is a divisor of $n$. Then $n$ is classified as a '''Lucas-Carmichael number'''."}
+{"_id": "28315", "title": "Definition:Complement of Subgroup", "text": "Let $G$ be a group with identity $e$. Let $H$ and $K$ be subgroups. Let $HK$ be their subset product and $H \\cap K$ their intersection. === Definition 1 === {{Definition:Complement of Subgroup/Definition 1}} === Definition 2 === {{Definition:Complement of Subgroup/Definition 2}}"}
+{"_id": "28316", "title": "Definition:Inner Semidirect Product", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $N$ be a normal subgroup of $G$. Let $H$ and $N$ be complementary. Then $G$ is the '''inner semidirect product''' of $N$ and $H$. This is denoted $G = N \\rtimes H$ or $G = H \\ltimes N$."}
+{"_id": "28317", "title": "Definition:Semidirect Product", "text": "Let $H$ and $N$ be groups. Let $\\Aut N$ denote the automorphism group of $N$. Let $\\phi: H \\to \\Aut N$ be a group homomorphism, that is, let $H$ act on $N$. The '''(outer) semidirect product of $N$ and $H$ with respect to $\\phi$''' is the cartesian product $N \\times H$ with the group operation defined as: :$\\tuple {n_1, h_1} \\circ \\tuple {n_2, h_2} = \\tuple {n_1 \\cdot \\map \\phi {h_1} \\paren {n_2}, h_1 \\cdot h_2}$ {{explain|What specifically is \"$\\cdot$\" here? This is an instance where it is important (in my belief) to be completely explicit as to what the operations are within the groups $H$ and $N$. It may also be clearer to refer to $G_1$ and $G_2$ because $N$ and $H$ have connotations of \"normal subgroup\" and \"subgroup\" respectively, and it is highly desirable to be consistent with other pages.}} It is denoted $N \\rtimes_\\phi H$."}
+{"_id": "28318", "title": "Definition:Continued Fraction/Expansion of Real Number", "text": "=== Irrational Number === Let $x$ be an irrational number. The '''continued fraction expansion of $x$''' is the simple continued fraction $\\paren {\\floor {\\alpha_n} }_{n \\ge 0}$ where $\\alpha_n$ is recursively defined as: :$\\alpha_n = \\displaystyle \\begin{cases} x & : n = 0 \\\\ \\dfrac 1 {\\fractpart {\\alpha_{n - 1} } } & : n \\ge 1 \\end{cases}$ where: :$\\floor {\\, \\cdot \\,}$ is the floor function :$\\fractpart {\\, \\cdot \\,}$ is the fractional part function. === Rational Number === Let $x$ be a rational number. The '''continued fraction expansion of $x$''' is found using the Euclidean Algorithm. {{explain|how}}"}
+{"_id": "28319", "title": "Definition:Derivation on Ring", "text": "Let $R$ be a ring. A '''derivation''' on $R$ is a group homomorphism $D$ of the additive group of $R$ which satsfies the Leibniz law: :$\\forall a, b \\in R: \\map D {a b} = \\map D a b + a \\map D b$"}
+{"_id": "28320", "title": "Definition:Differential Ring", "text": "A '''differential ring''' is a pair $\\struct {R, D}$ where: :$R$ is a ring :$D$ is a derivation on $R$."}
+{"_id": "28321", "title": "Definition:Differential Field", "text": "A '''differential field''' is a differential ring which is a field."}
+{"_id": "28322", "title": "Definition:Galois Extension/Finite/Definition 3", "text": "$L / K$ is a '''Galois extension''' {{iff}} the order of the automorphism group $\\Aut {L / K}$ equals the degree $\\index L K$: :$\\order {\\Aut {L / K} } = \\index L K$"}
+{"_id": "28323", "title": "Definition:Field Extension/Degree/Finite", "text": "$E / F$ is a '''finite field extension''' {{iff}} its degree $\\index E F$ is finite."}
+{"_id": "28324", "title": "Definition:Field Extension/Degree/Infinite", "text": "$E / F$ is an '''infinite field extension''' {{iff}} its degree $\\index E F$ is not finite."}
+{"_id": "28325", "title": "Definition:Field Norm/Definition 1", "text": "By Vector Space on Field Extension is Vector Space, $L$ is naturally a finite dimensional vector space over $K$. Let $\\theta_\\alpha$ be the linear operator: :$\\theta_\\alpha: L \\to L: \\beta \\mapsto \\alpha \\beta$ The '''field norm''' $\\map {N_{L/K} }\\alpha$ of $\\alpha$ is the determinant of $\\theta_\\alpha$."}
+{"_id": "28326", "title": "Definition:Field Norm/Definition 2", "text": "Let $L / K$ be Galois. By Finite Field Extension has Finite Galois Group, the Galois group $\\operatorname{Gal} \\left({L / K}\\right)$ is finite. The '''field norm''' $N_{L / K} \\left({\\alpha}\\right)$ of $\\alpha$ is $\\displaystyle \\prod_{\\sigma \\mathop \\in \\operatorname{Gal} \\left({L / K}\\right)} \\sigma \\left({\\alpha}\\right)$."}
+{"_id": "28327", "title": "Definition:Nilradical of Ring/Definition 1", "text": "The '''nilradical''' of $A$ is the subset consisting of all nilpotent elements of $A$."}
+{"_id": "28328", "title": "Definition:Nilradical of Ring/Definition 2", "text": "Let $\\Spec A$ denote the prime spectrum of $A$. The '''nilradical''' of $A$ is: :$\\displaystyle \\Nil A = \\bigcap_{\\mathfrak p \\mathop \\in \\Spec A} \\mathfrak p$ That is, it is the intersection of all prime ideals of $A$."}
+{"_id": "28329", "title": "Definition:Permutation Representation", "text": "Let $G$ be a group. Let $X$ be a set. Let $\\struct {\\map \\Gamma X, \\circ}$ be the symmetric group on $X$. A '''permutation representation''' of $G$ is a group homomorphism from $G$ to $\\struct {\\map \\Gamma X, \\circ}$. === Associated to Group Action === {{:Definition:Permutation Representation/Group Action}}"}
+{"_id": "28330", "title": "Definition:Faithful Group Action/Definition 1", "text": "$\\phi$ is '''faithful''' {{iff}} $e$ is the only element if $G$ which acts trivially: :$\\forall g \\in G: \\paren {\\forall x \\in X: g * x = x \\implies g = e}$"}
+{"_id": "28331", "title": "Definition:Faithful Group Action/Definition 2", "text": "$\\phi$ is '''faithful''' {{iff}} its permutation representation is injective."}
+{"_id": "28332", "title": "Definition:Free Group Action", "text": "Let $G$ be a group with identity $e$ acting on a set $X$. The group action is '''free''' {{iff}}: :$\\forall g \\in G: \\forall x \\in X : g * x = x \\implies g = e$"}
+{"_id": "28333", "title": "Definition:Kernel of Group Action", "text": "Let $G$ be a group with identity $e$. Let $X$ be a set. Let $* : G\\times X\\to X$ be a group action. === Definition 1 === {{Definition:Kernel of Group Action/Definition 1}} === Definition 2 === {{Definition:Kernel of Group Action/Definition 2}}"}
+{"_id": "28334", "title": "Definition:Kernel of Group Action/Definition 1", "text": "The '''kernel of the group action''' is the set: :$G_0 = \\set {g \\in G: \\forall x \\in X: g * x = x}$"}
+{"_id": "28335", "title": "Definition:Kernel of Group Action/Definition 2", "text": "The '''kernel of the group action''' is the kernel of its permutation representation."}
+{"_id": "28336", "title": "Definition:Permutation Representation/Group Action", "text": "Let $\\phi: G \\times X \\to X$ be a group action. Define for $g \\in G$ the mapping $\\phi_g : X \\to X$ by: :$\\map {\\phi_g} x = \\map \\phi {g, x}$ The '''permutation representation of $G$ associated to the group action''' is the group homomorphism $G \\to \\struct {\\map \\Gamma X, \\circ}$ which sends $g$ to $\\phi_g$."}
+{"_id": "28337", "title": "Definition:Group Action/Permutation Representation", "text": "Let $\\struct {\\map \\Gamma X, \\circ}$ be the symmetric group on $X$. Let $\\rho: G \\to \\struct {\\map \\Gamma X, \\circ}$ be a permutation representation. The '''group action of $G$ associated to the permutation representation $\\rho$''' is the group action $\\phi: G \\times X \\to X$ defined by: :$\\map \\phi {g, x} = \\map {\\rho_g} x$ where $\\rho_g : X \\to X$ is the permutation representation associated to $\\rho$ for $g \\in G$ by $\\map {\\rho_g} x = \\map \\phi {g, x}$."}
+{"_id": "28338", "title": "Definition:Field of Quotients/Definition 4", "text": "A '''field of quotients''' of $D$ is a pair $\\struct {F, \\iota}$ which is its total ring of fractions, that is, the localization of $D$ at the nonzero elements $D_{\\ne 0}$."}
+{"_id": "28339", "title": "Definition:Fixed Field", "text": "Let $F$ be a field. Let $G \\le \\Aut F$ be a subgroup of the automorphism group of $F$. The '''fixed field''' of $G$ is the set: :$\\Fix G = \\set {f \\in F : \\forall \\sigma \\in G : \\map \\sigma f = f}$"}
+{"_id": "28340", "title": "Definition:Group Action Induced on Subgroup", "text": "Let $G$ be a group. Let $X$ be a set. Let $\\phi : G \\times X \\to X$ be a group action. Let $H \\le G$ be a subgroup. The '''group action induced on $H$''' is the restriction of $\\phi$ to $H \\times X$. Equivalently, the '''group action induced on $H$''' is the group action associated to the permutation representation: :$\\rho \\circ \\iota : H \\to \\struct {\\map \\Gamma X, \\circ}$ where: :$\\iota : H \\to G$ is the inclusion homomorphism :$\\rho$ is the  permutation representation of $\\phi$ :$\\struct {\\map \\Gamma X, \\circ}$ is the symmetric group on $X$."}
+{"_id": "28341", "title": "Definition:Stable Under Group Action", "text": "Let $G$ be a group. Let $X$ be a set. Let $\\phi : G \\times X \\to X$ be a group action. Let $S \\subset X$ be a subset of $X$. Then $S$ is '''stable (under $\\phi$)''' {{iff}}: :$\\map \\phi {G \\times S} \\subset S$ Category:Definitions/Group Actions 80def9y5meto98n8cex3b88rvy86d4n"}
+{"_id": "28342", "title": "Definition:Group Action Induced on Stable Subset", "text": "Let $G$ be a group. Let $X$ be a set. Let $\\phi : G \\times X \\to X$ be a group action. Let $S \\subset X$ be a stable subset. The '''group action induced on $S$''' is the restriction of $\\phi$ to $G \\times S$."}
+{"_id": "28344", "title": "Definition:Complete Quotient", "text": "Let $x$ be an irrational number. The '''sequence of complete quotients''' is defined recursively by: :$\\alpha_0 = x$ :$\\alpha_{n+1} = \\displaystyle\\frac1{\\{\\alpha_n\\}}$ where $\\{\\cdot\\}$ denotes fractional part."}
+{"_id": "28345", "title": "Definition:Parametric Equation", "text": "Let $\\map \\RR {x_1, x_2, \\ldots, x_n}$ be a relation on the variables $x_1, x_2, \\ldots, x_n$. Let the truth set of $\\RR$ be definable as: :$\\forall k \\in \\N: 1 \\le k \\le n: x_k = \\map {\\phi_k} t$ where: :$t$ is a variable whose domain is to be defined :each of $\\phi_k$ is a mapping whose domain is the domain of $t$ and whose codomain is the domain of $x_k$. Then each of: :$x_k = \\map {\\phi_k} t$ is a '''parametric equation''' where $t$ is the '''parameter'''. The set: :$\\set {\\phi_k: 1 \\le k \\le n}$ is a '''set of parametric equations specifying $\\RR$'''. === $2$ Dimensions === {{:Definition:Parametric Equation/2 Dimensions}} Category:Definitions/Analysis Category:Definitions/Mapping Theory mske9rzz1wru8u49yo7w5lkbg7xrft2"}
+{"_id": "28346", "title": "Definition:Basis Expansion/Recurrence/Period", "text": "The '''period of recurrence''' is the number of digits in the recurring part after which it repeats itself."}
+{"_id": "28347", "title": "Definition:Force/Dimension", "text": "The dimension of measurement of '''force''' is $\\mathsf {M L T}^{-2}$. This arises from Newton's Second Law of Motion and its definition as a mass (of dimension $\\mathsf M$) multiplied by an acceleration (of dimension $\\mathsf {L T}^{-2}$)."}
+{"_id": "28348", "title": "Definition:Force/Unit", "text": "The units of measurement of '''force''' are as follows: === Newton === {{:Definition:Newton (Unit)}} === Dyne === {{:Definition:Dyne}} Thus we see: :$1 \\ \\mathrm N = 10^5 \\ \\mathrm {dyn}$"}
+{"_id": "28350", "title": "Definition:Electromotive Force/Dimension", "text": "The dimension of measurement of '''electromotive force''' is $\\mathsf {M L}^2 \\mathsf T^{-3} \\mathsf I^{-1}$. This derives from its definition as: :$\\dfrac {\\text{Work} } {\\text {Charge} }$"}
+{"_id": "28351", "title": "Definition:Electromotive Force/Unit", "text": "The unit of measurement of '''electromotive force'''  is the volt $\\mathrm V$: :$1 \\ \\mathrm V = 1 \\ \\mathrm J \\ \\mathrm C^{-1}$ that is, $1$ joule per coulomb."}
+{"_id": "28352", "title": "Definition:Electromotive Force", "text": "'''Electromotive force''' is a quantity that measures the source of potential energy in an electrical circuit. It is defined as the amount of work per unit electric charge."}
+{"_id": "28353", "title": "Definition:Inductance", "text": "'''Inductance''' is the property of an electrical conductor by which a change in electric current through it induces an electromotive force in both the conductor itself and also in any nearby conductors."}
+{"_id": "28354", "title": "Definition:Inductance/Dimension", "text": "The dimension of measurement of '''inductance''' is: :$\\mathsf {M L}^2 \\mathsf T^{-2} \\mathsf I^{-2}$ This derives from its definition as: :$v = L \\dfrac {\\mathrm d I} {\\mathrm d t}$"}
+{"_id": "28356", "title": "Definition:Dynamical System", "text": "A '''dynamical system''' is a non-linear system in which a function describes the time dependence of a point in a geometrical space."}
+{"_id": "28357", "title": "Definition:Resistance", "text": "'''Resistance''' is a measure of how difficult it is to pass an electric current through a conductor. It is defined as the electromotive force needed to push a unit current through that conductor."}
+{"_id": "28358", "title": "Definition:Resistance/Dimension", "text": "The dimension of measurement of '''resistance''' is: :$\\mathsf {M L}^2 \\mathsf T^{−3} \\mathsf I^{−2}$ This derives from its definition as: :$R = \\dfrac {\\text {EMF} } {\\text {current} }$"}
+{"_id": "28359", "title": "Definition:Resistance/Unit", "text": "The unit of measurement of '''resistance'''  is the ohm $\\Omega$: :$1 \\ \\Omega = 1 \\ \\mathrm{V} \\ \\mathrm{A}^{-1}$ that is, $1$ volt per ampere."}
+{"_id": "28360", "title": "Definition:Electric Current/Dimension", "text": "'''Electric current''' is one of the fundamental dimensions of physics. In dimensional analysis it is assigned the symbol $\\mathsf I$."}
+{"_id": "28362", "title": "Definition:Volume/Dimension", "text": "'''Volume''' is a dimension of measurement of physics. The dimension of measurement of '''volume''' is: :$L^3$ length cubed."}
+{"_id": "28363", "title": "Definition:Volume/Unit", "text": "The SI unit of '''volume''' is the cubic metre: : $\\mathrm m^3$ The CGS unit of '''volume''' is the cubic centimetre: :$\\mathrm {cm}^3$ or, less formally: :$\\mathrm {cc}$ Thus: :$1 \\ \\mathrm m^3 = 10^6 \\ \\mathrm {cm}^3 = 1 \\,000 \\,000 \\ \\mathrm {cc}$"}
+{"_id": "28365", "title": "Definition:Lucas Prime", "text": "A '''Lucas prime''' is a Lucas number which happens to be prime."}
+{"_id": "28367", "title": "Definition:Series (Electronics)", "text": "Two or more components in an electrical circuit are connected in '''series''' {{iff}} the current through them flows along a single path through the circuit. {{finish|Needs to be formalised, with the concept of an electrical circuit being mapped by means of a graph. This will need plenty of work.}} Category:Definitions/Electronics obfg4v861zaqqatagkvp8wsqve6x4g3"}
+{"_id": "28368", "title": "Definition:Anticlockwise", "text": "'''Anticlockwise''' is this direction: :400px"}
+{"_id": "28369", "title": "Definition:Clockwise", "text": "'''Clockwise''' is this direction: :400px"}
+{"_id": "28370", "title": "Definition:Rocket", "text": "A '''rocket''' is a device whose means of propulsion works by throwing material out of one end at high speed in order to take advantage of the Principle of Conservation of Momentum to move in the opposite direction. {{finish}} Category:Definitions/Rocket Science 8into5ob296rmdwo4ylnrumq3nrehv9"}
+{"_id": "28371", "title": "Definition:Cuboid", "text": "A '''cuboid''' is a parallelepiped whose faces are all rectangular. :400px"}
+{"_id": "28372", "title": "Definition:Moon", "text": "'''The moon''' is the single natural satellite of Earth."}
+{"_id": "28373", "title": "Definition:Integration With Respect To", "text": "Let $f$ be a function which is integrable on a domain $D$. Let $F$ be the primitive of $f$: :$\\displaystyle F = \\int \\map f x \\rd x$ The operation of evaluating the primitive of $f$ is referred to as '''integration with respect to $x$.'''"}
+{"_id": "28374", "title": "Definition:Spherical Triangle", "text": "A '''spherical triangle''' is a geometric figure described on the surface of a sphere which is bounded by the arcs of three great circles which intersect in $3$ vertices: :500px"}
+{"_id": "28375", "title": "Definition:Spherical Triangle/Side", "text": "The '''sides''' of a '''spherical triangle''' $T$ are the arcs of the $3$ great circles which form the boundaries of $T$ between the points at which they intersect."}
+{"_id": "28376", "title": "Definition:Linear Momentum/Dimension", "text": "The dimension of measurement of '''linear momentum''' is $\\mathsf {M L T}^{-1}$."}
+{"_id": "28377", "title": "Definition:Frequency (Physics)", "text": "Let $\\map f t$ be a periodic function of time $t$. The '''frequency''' of $f$ is the number of periods of $f$ that occur during a unit time interval."}
+{"_id": "28381", "title": "Definition:Zero Operator", "text": "Let $H$ be a Hilbert space. Let $A$ be a linear operator on $H$. Suppose that: :$\\forall h \\in H: A \\left({h}\\right) = 0$ Then $A$ is called a '''zero operator''', and is denoted as $\\mathbf 0$. Category:Definitions/Hilbert Spaces qijcp7dtvjflubcfb0t86jre0x7cua6"}
+{"_id": "28382", "title": "Definition:Space Diagonal", "text": "Consider a parallelepiped $P$. The '''space diagonals''' of $P$ are the line segments that can be drawn from each vertex $X$ of $P$ to the vertex $Y$ of $P$ which is not contained in any of the faces of $P$ which contain $X$. :400px In the above diagram, the '''space diagonals''' of the parallelepiped $ABCDEFGH$ are $AF$, $BE$, $CH$ and $DG$."}
+{"_id": "28383", "title": "Definition:Fermat-Torricelli Point", "text": "Let $ABC$ be a triangle. The '''Fermat Point''' is defined to be the point $P$ which minimizes the following quantity: :$AP + BP + CP$ which is the total distance from the three vertices to point $P$."}
+{"_id": "28384", "title": "Definition:Weakly Abnormal Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. === Definition 1 === {{:Definition:Weakly Abnormal Subgroup/Definition 1}} === Definition 2 === {{:Definition:Weakly Abnormal Subgroup/Definition 2}} === Definition 3 === {{:Definition:Weakly Abnormal Subgroup/Definition 3}}"}
+{"_id": "28385", "title": "Definition:Weakly Abnormal Subgroup/Definition 1", "text": "$H$ is '''weakly abnormal in $G$''' {{iff}}: :$\\forall g \\in G: g \\in H^{\\gen g}$ where $H^{\\gen g}$ denotes the smallest subgroup of $G$ containing $H$, generated by the conjugacy action by the cyclic subgroup of $G$ generated by $g$."}
+{"_id": "28388", "title": "Definition:Weakly Pronormal Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. === Definition 1 === {{:Definition:Weakly Pronormal Subgroup/Definition 1}} === Definition 2 === {{:Definition:Weakly Pronormal Subgroup/Definition 2}}"}
+{"_id": "28389", "title": "Definition:Trivial Vector Space", "text": "Let $V$ be a vector space. Then $V$ is said to be '''trivial''' {{iff}}: :$V = \\left\\{ {\\mathbf 0}\\right\\}$ where $\\mathbf 0$ denotes the zero vector."}
+{"_id": "28390", "title": "Definition:Weakly Pronormal Subgroup/Definition 1", "text": "$H$ is '''weakly pronormal in $G$''' {{iff}}: :$\\forall g \\in G: \\exists x \\in H^{\\gen g}: H^x = H^g$ where: :$H^{\\gen g}$ denotes the smallest subgroup of $G$ containing $H$, generated by the conjugacy action by the cyclic subgroup of $G$ generated by $g$ :$H^x$ denotes the conjugate of $H$ by $x$."}
+{"_id": "28392", "title": "Definition:Left Zero Divisor", "text": "A '''left zero divisor (in $R$)''' is an element $x \\in R$ such that: : $\\exists y \\in R^*: x \\circ y = 0_R$"}
+{"_id": "28393", "title": "Definition:Right Zero Divisor", "text": "A '''right zero divisor (in $R$)''' is an element $x \\in R$ such that: : $\\exists y \\in R^*: y \\circ x = 0_R$"}
+{"_id": "28394", "title": "Definition:Exhaust Velocity", "text": "Let $R$ be a rocket. The '''exhaust velocity''' of $R$ is the speed at which the propellant of $R$ is emitted out of the back end, to make $R$ travel forwards. Category:Definitions/Rocket Science b3t2yufyhfr3kiisnasodnx14rg98ov"}
+{"_id": "28395", "title": "Definition:Usual Ordering", "text": "Let $X$ be one of the sets of numbers: $\\N$, $\\Z$, $\\Q$, $\\R$. The '''usual ordering''' on $X$ is the conventional counting and measuring order on $X$ that is learned when one is initially introduced to numbers. === Natural Numbers === {{Definition:Ordering on Natural Numbers}} === Integers === {{Definition:Ordering on Integers}} {{expand|This needs to be formalised. The page is included as it is one of the most wanted redline links.}}"}
+{"_id": "28396", "title": "Definition:Bid", "text": "Let $A$ be an auction. A '''bid''' is a move by a player in $A$ which consists of the submission of a non-negative number. Such a '''bid''' is associated with a payment which is to be made by the winning player in exchange for the prize. Category:Definitions/Game Theory c6zmgj0zfanq5kdfbry41nbd4qjnkip"}
+{"_id": "28397", "title": "Definition:Fractal", "text": "A '''fractal''' is an object that has the property of self-similarity on all scales."}
+{"_id": "28398", "title": "Definition:Universal Cover", "text": "A '''universal cover''' is a covering space which is simply connected. {{explain|Context}} Category:Definitions/Topology ku749z3mkng85xpdpgoubu0l53k7f7p"}
+{"_id": "28399", "title": "Definition:Conjugacy Action", "text": "Let $\\struct {G, \\circ}$ be a group. The '''(left) conjugacy action''' of $G$ is the left group action $* : G \\times G \\to G$ defined as: :$\\forall g, x \\in G: g * x = g \\circ x \\circ g^{-1}$ The '''right conjugacy action''' of $G$ is the right group action $* : G \\times G \\to G$ defined as: :$\\forall x, g \\in G: x * g = g^{-1} \\circ x \\circ g$"}
+{"_id": "28400", "title": "Definition:Subinterval", "text": "Let $I$ be a real interval. A '''subinterval''' $J$ of $I$ is a real interval such that $J \\subseteq I$. {{Proofread|A purely intuitive definition which may or may not be accurate -- for example, is $\\left({a \\,.\\,.\\, b}\\right)$ strictly speaking a subinterval of $\\left[{a \\,.\\,.\\, b}\\right]$?}} Category:Definitions/Real Analysis kp8w81att27xxfvloskryzhnohdgq9m"}
+{"_id": "28401", "title": "Definition:Subdivision (Real Analysis)/Finite", "text": "Let $x_0, x_1, x_2, \\ldots, x_{n - 1}, x_n$ be points of $\\R$ such that: :$a = x_0 < x_1 < x_2 < \\cdots < x_{n - 1} < x_n = b$ Then $\\set {x_0, x_1, x_2, \\ldots, x_{n - 1}, x_n}$ form a '''finite subdivision of $\\closedint a b$'''."}
+{"_id": "28402", "title": "Definition:Subdivision (Real Analysis)/Infinite", "text": "Let $x_0, x_1, x_2, \\ldots$ be an infinite number of points of $\\R$ such that: :$a = x_0 < x_1 < x_2 < \\cdots < x_{n - 1} < \\ldots \\le b$ Then $\\set {x_0, x_1, x_2, \\ldots}$ forms an '''infinite subdivision of $\\closedint a b$'''."}
+{"_id": "28403", "title": "Definition:Orthonormal Basis", "text": "Let $\\mathbb F$ be a field. Let $V$ be a vector space over $\\mathbb F$. Let $S \\subset V$ be a subset of $V$. Then $S$ is an '''orthonormal basis''' {{iff}}: :$(1): \\quad S$ is a basis. :$(2): \\quad S$ is orthonormal. Category:Definitions/Linear Algebra rn1yswtz371sj368r4xkcfz5q37cxc5"}
+{"_id": "28404", "title": "Definition:Left Module", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {G, +_G}$ be an abelian group. A left module over $R$ is an $R$-algebraic structure $\\struct {G, +_G, \\circ}_R$ with one operation $\\circ$, the '''(left) ring action''', which satisfies the left module axioms: {{:Definition:Left Module Axioms}}"}
+{"_id": "28405", "title": "Definition:Right Module", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {G, +_G}$ be an abelian group. A right module over $R$ is an $R$-algebraic structure $\\struct {G, +_G, \\circ}_R$ with one operation $\\circ$, the '''(right) ring action''', which satisfies the right module axioms: {{:Definition:Right Module Axioms}}"}
+{"_id": "28408", "title": "Definition:O Notation/Little-O Notation/Sequence", "text": "Let $\\left \\langle {a_n} \\right \\rangle$ and $\\left \\langle {b_n} \\right \\rangle$ be sequences of real or complex numbers. === Definition 1 === {{:Definition:O Notation/Little-O Notation/Sequence/Definition 1}} === Definition 2 === {{:Definition:O Notation/Little-O Notation/Sequence/Definition 2}} This is denoted :$a_n = o \\left({b_n}\\right)$"}
+{"_id": "28409", "title": "Definition:O Notation/Little-O Notation/Sequence/Definition 1", "text": "Let $b_n\\neq0$ for all $n$. '''$a_n$ is little-O of $b_n$''' {{iff}} :$\\displaystyle \\lim_{n\\to\\infty}\\frac{a_n}{b_n} = 0$"}
+{"_id": "28410", "title": "Definition:O Notation/Little-O Notation/Sequence/Definition 2", "text": "'''$a_n$ is little-O of $b_n$''' {{iff}} :$\\forall \\epsilon \\in \\R: \\epsilon > 0 : \\exists n_0 \\in \\N : \\left({n \\ge n_0 \\implies \\left\\vert{a_n}\\right\\vert \\le \\epsilon \\cdot \\left\\vert{b_n}\\right\\vert}\\right)$ That is: :For all $\\epsilon>0$, $\\left\\vert{a_n}\\right\\vert \\le \\epsilon \\cdot \\left\\vert{b_n}\\right\\vert$ for all sufficiently large $n$."}
+{"_id": "28411", "title": "Definition:O Notation/Little-O Notation/Real Functions/Definition 1", "text": "Let $g(x)\\neq0$ for $x$ sufficiently large. '''$f$ is little-o of $g$''' as $x \\to \\infty$ {{Iff}}: :$\\displaystyle \\lim_{x \\to \\infty} \\ \\frac{f \\left({x}\\right)} {g \\left({x}\\right)} = 0$"}
+{"_id": "28412", "title": "Definition:O Notation/Little-O Notation/Real Functions/Definition 2", "text": "'''$f$ is little-o of $g$''' as $x \\to \\infty$ {{Iff}}: :$\\forall \\epsilon \\in \\R : \\epsilon > 0 : \\exists x_0 \\in \\R : \\forall x \\geq x_0 : |f(x)| \\leq \\epsilon \\cdot |g(x)|$"}
+{"_id": "28413", "title": "Definition:Ambivalent Group", "text": "Let $G$ be a group. Then $G$ is '''ambivalent''' {{iff}} every element of $G$ is conjugate to its inverse: :$\\forall g \\in G : \\exists h \\in G : h g h^{-1} = g^{-1}$"}
+{"_id": "28414", "title": "Definition:Real Group Element", "text": "Let $G$ be a group. Let $g \\in G$. Then $g$ is a '''real element (of $G$)''' {{iff}} it is conjugate to its inverse: :$\\exists h \\in G : hgh^{-1} = g^{-1}$"}
+{"_id": "28415", "title": "Definition:O Notation/Big-O Notation/Uniform", "text": "Let $X$ be a set. Let $V$ be a normed vector space over $\\R$ or $\\C$ with norm $\\norm {\\, \\cdot \\,}$. Let $f, g : X \\to V$ be mappings. Then '''$f$ is big $\\OO$ of $g$ uniformly''' {{iff}}: :$\\exists c > 0 : \\forall x \\in X : \\norm {\\map f x} \\le c \\cdot \\norm {\\map g x}$ This is denoted: :$f = \\map \\OO g$"}
+{"_id": "28416", "title": "Definition:Group Action/Right Group Action", "text": "A '''right group action''' is a mapping $\\phi: X \\times G \\to X$ such that: :$\\forall \\tuple {x, g} \\in X \\times G : x * g := \\map \\phi {x, g} \\in X$ in such a way that the right group action axioms are satisfied: {{:Definition:Right Group Action Axioms}}"}
+{"_id": "28417", "title": "Definition:Right Group Action Axioms", "text": "{{begin-axiom}} {{axiom | n = \\text {RGA} 1         | q = \\forall g, h \\in G, x \\in X         | m = \\paren {x * g} * h = x * \\paren {g \\circ h} }} {{axiom | n = \\text {RGA} 2         | q = \\forall x \\in X         | m =  x * e = x }} {{end-axiom}}"}
+{"_id": "28418", "title": "Definition:Conjugacy Action/Subgroups", "text": "Let $X$ be the set of all subgroups of $G$. The '''(left) conjugacy action on subgroups''' is the group action $* : G \\times X \\to X$: :$g * H = g \\circ H \\circ g^{-1}$ The '''right conjugacy action on subgroups''' is the group action $* : X \\times G \\to X$: :$H * g = g^{-1} \\circ H \\circ g$"}
+{"_id": "28419", "title": "Definition:Direct Sum of Module Homomorphisms", "text": "Let $R$ be a ring. Let $M,N,P,Q$ be $R$-modules. Let $M\\oplus N$ and $P\\oplus Q$ be their direct sum. Let $f : M \\to P$ and $g : N \\to Q$ be module homomorphisms. The '''direct sum''' of $f$ and $g$ is the module homomorphism $f\\oplus g : M\\oplus N \\to P\\oplus Q$ defined as: :$(f\\oplus g) (m, n) = ( f(m) , g(n) )$"}
+{"_id": "28420", "title": "Definition:Transitive Group Action/n-transitive", "text": "Let $n\\geq1$ be a natural number. The group action is '''$n$-transitive''' {{iff}} for any two ordered $n$-tuples $(x_1, \\ldots, x_n)$ and $(y_1, \\ldots, y_n)$ of pairwise distinct elements of $S$, there exists $g\\in G$ such that: :$\\forall i\\in \\{1, \\ldots, n\\} : g * x_i = y_i$"}
+{"_id": "28421", "title": "Definition:Generated Submodule", "text": "=== Definition 1 === {{:Definition:Generated Submodule/Definition 1}} === Definition 2 === {{:Definition:Generated Submodule/Definition 2}}"}
+{"_id": "28423", "title": "Definition:Wilson Prime/Sequence", "text": "The sequence of Wilson primes begins: :$5, 13, 563$ The next term in the sequence, if there is one, is greater than $2 \\times 10^{13}$."}
+{"_id": "28425", "title": "Definition:Field Norm of Complex Number", "text": "Let $z = a + i b$ be a complex number, where $a, b \\in \\R$. Then the '''field norm of $z$''' is written $\\map N z$ and is defined as: :$\\map N z := \\cmod \\alpha^2 = a^2 + b^2$ where $\\cmod \\alpha$ denotes the complex modulus of $\\alpha$."}
+{"_id": "28426", "title": "Definition:Value of Continued Fraction/Finite", "text": "Let $n\\geq0$ be a natural number. Let $(a_k)_{0 \\leq k \\leq n}$ be a finite continued fraction in $F$. Let $\\overline F = F \\cup \\{\\infty\\}$ be extended by infinity. === Definition 1 === The '''value''' $[a_0, a_1, \\ldots, a_n] \\in F \\cup \\{\\infty\\}$ is the right iteration of the binary operation: :$[\\cdot,\\cdot] :  F \\times \\overline F \\to \\overline F$: :$[a, b] = a + \\dfrac 1 b$. That is, it is recursively defined as: :$[a_0, \\ldots, a_n] = \\begin{cases}    a_0 & : n = 0 \\\\   a_0 + \\dfrac 1 {\\left[{a_1, \\ldots, a_n}\\right]} & : n > 0 \\\\ \\end{cases}$ or as: :$[a_0, \\ldots, a_n] = \\begin{cases}    a_0 & : n = 0 \\\\   \\left[a_0, \\ldots, a_{n-2}, a_{n-1} + \\dfrac 1 {a_n}\\right] & : n > 0 \\\\ \\end{cases}$ === Definition 2 === Let the matrix product: :$\\begin{pmatrix}a_0 & 1 \\\\ 1 & 0\\end{pmatrix}\\cdots\\begin{pmatrix}a_n & 1 \\\\ 1 & 0\\end{pmatrix} = \\begin{pmatrix}x_{11} & x_{12} \\\\ x_{21} & x_{22}\\end{pmatrix}$ The '''value''' of the finite continued fraction is $\\dfrac{x_{11}}{x_{21}}$"}
+{"_id": "28427", "title": "Definition:Value of Continued Fraction/Infinite", "text": "Let $\\struct {F, \\norm {\\,\\cdot\\,}}$ be a valued field. Let $C = (a_n)_{n\\geq 0}$ be a infinite continued fraction in $F$. Then $C$ '''converges''' to its '''value''' $x \\in F$ {{iff}} the following hold: #For all natural numbers $n \\in \\N_{\\geq 0}$, the $n$th denominator is nonzero #The sequence of convergents $\\sequence{C_n}_{n\\geq 0}$ converges to $x$."}
+{"_id": "28428", "title": "Definition:Maximal Ideal of Ring/Left", "text": "A left ideal $J$ of $R$ is a '''maximal left ideal''' {{iff}}: :$(1): \\quad J \\subsetneq R$ :$(2): \\quad$ There is no left ideal $K$ of $R$ such that $J \\subsetneq K \\subsetneq R$."}
+{"_id": "28429", "title": "Definition:Maximal Ideal of Ring/Right", "text": "A right ideal $J$ of $R$ is a '''maximal right ideal''' {{iff}}: :$(1): \\quad J \\subsetneq R$ :$(2): \\quad$ There is no right ideal $K$ of $R$ such that $J \\subsetneq K \\subsetneq R$."}
+{"_id": "28430", "title": "Definition:Residue Field of Local Ring", "text": "Let $R$ be a commutative local ring. Let $m$ be its maximal ideal. The '''residue field''' is the quotient ring $R/m$."}
+{"_id": "28432", "title": "Definition:Asymptotically Equal/Sequences/Definition 1", "text": "Let $b_n \\ne 0$ for all $n$. $\\sequence {a_n}$ is '''asymptotically equal''' to  $\\sequence {b_n}$ {{iff}}: :$\\displaystyle \\lim_{n \\to \\infty} \\dfrac {a_n} {b_n} \\to 1$"}
+{"_id": "28433", "title": "Definition:Asymptotically Equal/Sequences/Definition 2", "text": "$\\sequence {a_n}$ is '''asymptotically equal''' to  $\\sequence {b_n}$ {{iff}}: :$a_n - b_n = \\map o {b_n}$ where $o$ denotes little-o notation."}
+{"_id": "28434", "title": "Definition:Asymptotically Equal/Sequences/Definition 3", "text": "$\\sequence {a_n}$ is '''asymptotically equal''' to  $\\sequence {b_n}$ {{iff}}: :$a_n - b_n = \\map o {a_n}$ where $o$ denotes little-o notation."}
+{"_id": "28436", "title": "Definition:Differential/Functional", "text": "Let $J \\sqbrk y$ be a differentiable functional. Let $h$ be an increment of the independent variable $y$. Then the term linear {{WRT}} $h$ is called the '''differential''' of the functional $J$, and is denoted by $\\delta J \\sqbrk {y; h}$."}
+{"_id": "28438", "title": "Definition:Compatible Atlases/Definition 1", "text": "$\\mathscr F, \\mathscr G$ are '''$C^k$-compatible''' {{iff}} their union $\\mathscr F \\cup \\mathscr G$ is an atlas of class $C^k$."}
+{"_id": "28439", "title": "Definition:Compatible Atlases/Definition 2", "text": "$\\mathscr F$ and $\\mathscr G$ are '''$C^k$-compatible''' {{iff}} every pair of charts $\\struct {U, \\phi} \\in \\mathscr F$ and $\\struct {V, \\psi} \\in \\mathscr G$ are $C^k$-compatible."}
+{"_id": "28440", "title": "Definition:Atlas/Maximal Atlas", "text": "Let $A$ be a $d$-dimensional atlas of class $C^k$ of $M$. === Definition 1 === {{:Definition:Atlas/Maximal Atlas/Definition 1}} === Definition 2 === {{:Definition:Atlas/Maximal Atlas/Definition 2}} === Definition 3 === {{:Definition:Atlas/Maximal Atlas/Definition 3}}"}
+{"_id": "28441", "title": "Definition:Transition Mapping between Charts", "text": "Let $M$ be a topological space. Let $d$ be a natural number. Let $(U,\\phi)$ and $(V,\\psi)$ be $d$-dimensional charts of $M$. Let $U\\cap V \\neq\\emptyset$. The '''transition map from $\\phi$ to $\\psi$''' is the mapping: :$\\psi \\circ \\phi^{-1} : \\phi(U\\cap V) \\to \\psi(U\\cap V)$"}
+{"_id": "28442", "title": "Definition:Compatible Charts", "text": "Let $M$ be a topological space. Let $d$ be a natural number. Let $\\struct {U, \\phi}$ and $\\struct {V, \\psi}$ be $d$-dimensional charts of $M$. Then $\\struct {U, \\phi}$ and $\\struct {V, \\psi}$ are '''$C^k$-compatible''' {{iff}} their transition mapping: :$\\psi \\circ \\phi^{-1}: \\map \\phi {U \\cap V} \\to \\map \\psi {U \\cap V}$ is of class $C^k$. === Smoothly Compatible Charts === {{:Definition:Compatible Charts/Smooth}}"}
+{"_id": "28443", "title": "Definition:Compatible Charts/Smooth", "text": "$\\struct {U, \\phi}$ and $\\struct {V, \\psi}$ are '''smoothly compatible''' {{iff}} their transition mapping: :$\\psi \\circ \\phi^{-1} : \\map \\phi {U \\cap V} \\to \\map \\psi {U \\cap V}$ is of class $C^\\infty$."}
+{"_id": "28445", "title": "Definition:Chart Compatible with Atlas", "text": "Let $M$ be a topological space. Let $A$ be a $d$-dimensional $C^k$-atlas on $M$. Let $\\struct {U, \\phi}$ be a $d$-dimensional chart of $M$. Then $\\struct {U, \\phi}$ is '''$C^k$-compatible with $A$''' {{iff}} $\\struct {U, \\phi}$ is $C^k$-compatible with every chart of $A$. Category:Definitions/Manifolds 9eyy1syk3175jgb8zs3rnvcf1n4i0mw"}
+{"_id": "28446", "title": "Definition:Atlas/Maximal Atlas/Definition 1", "text": "$A$ is a '''maximal $C^k$-atlas''' of dimension $d$ {{iff}} $A$ is not strictly contained in another $C^k$-atlas."}
+{"_id": "28447", "title": "Definition:Atlas/Maximal Atlas/Definition 2", "text": "$A$ is a '''maximal $C^k$-atlas''' {{iff}} $A$ contains all charts of $M$ that are $C^k$-compatible with $A$."}
+{"_id": "28448", "title": "Definition:Atlas/Maximal Atlas/Definition 3", "text": "$A$ is a '''maximal $C^k$-atlas''' {{iff}} $A$ is a maximal element of some differentiable structure, partially ordered by inclusion. That is, a maximal element of some equivalence class of the set of atlases of class $\\mathcal C^k$ on $M$ under the equivalence relation of compatibility."}
+{"_id": "28449", "title": "Definition:Uniform Contraction Mapping", "text": "Let $M$ and $N$ be metric spaces. Let $f : M \\times N \\to M$ be a mapping. Then $f$ is a '''uniform contraction''' {{iff}} there exists $K<1$ such that for all $x,y\\in M$ and $t\\in N$: :$d(f(x,t), f(y,t)) \\leq K\\cdot d(x,y)$."}
+{"_id": "28450", "title": "Definition:Local Diffeomorphism", "text": "Let $n$ and $k$ be natural numbers. Let $U \\subset \\R^n$ be an open set. {{explain|Context of Open Set}} Let $f: U \\to \\R^n$ be a mapping. Then $f$ is a '''local $C^k$-diffeomorphism''' {{iff}} every $a \\in U$ has a open neighborhhood such that the restriction of $f$ to it is a $C^k$-diffeomorphism on its image. === Smooth Local Diffeomorphism === {{definition wanted}} Category:Definitions/Analysis th6qvtq1u3389gwzzwspm3q7qcdgvso"}
+{"_id": "28451", "title": "Definition:Open Set/Real Analysis/Real Numbers", "text": "Let $I \\subseteq \\R$ be a subset of the set of real numbers. Then $I$ is '''open (in $\\R$)''' {{iff}}: : $\\forall x_0 \\in I: \\exists \\epsilon \\in \\R_{>0}: \\openint {x_0 - \\epsilon} {x_0 + \\epsilon} \\subseteq I$ where $\\openint {x_0 - \\epsilon} {x_0 + \\epsilon}$ is an open interval. Note that $\\epsilon$ may depend on $x_0$."}
+{"_id": "28452", "title": "Definition:Open Set/Real Analysis/Real Euclidean Space", "text": "Let $n \\ge 1$ be a natural number. Let $U \\subseteq \\R^n$ be a subset. Then $U$ is '''open (in $\\R^n$)''' {{iff}}: :$\\forall x \\in U : \\exists R > 0: \\map B {x, R} \\subset U$ where $\\map B {x, R}$ denotes the open ball of radius $R$ centered at $x$."}
+{"_id": "28453", "title": "Definition:Smith Number", "text": "A '''Smith number''' is a composite number for which the sum of its digits is equal to the sum of the digits in its prime decomposition."}
+{"_id": "28456", "title": "Definition:Smith Brothers", "text": "'''Smith brothers''' are two consecutive integers which are both '''Smith numbers'''. === Sequence of Smith Brothers === {{:Definition:Smith Brothers/Sequence}}"}
+{"_id": "28458", "title": "Definition:Differentiable Mapping/Real Function/Point/Definition 1", "text": "$f$ is '''differentiable at the point $\\xi$''' {{iff}} the limit: :$\\displaystyle \\lim_{x \\mathop \\to \\xi} \\frac {\\map f x - \\map f \\xi} {x - \\xi}$ exists."}
+{"_id": "28460", "title": "Definition:Differentiable Mapping/Real Function/Point/Definition 2", "text": "$f$ is '''differentiable at the point $\\xi$''' {{iff}} the limit: :$\\displaystyle \\lim_{h \\mathop \\to 0} \\frac {\\map f {\\xi + h} - \\map f \\xi} h$ exists."}
+{"_id": "28461", "title": "Definition:Open Ball/Real Analysis", "text": "Let $n \\ge 1$ be a natural number. Let $\\R^n$ denote a real Euclidean space Let $\\left\\Vert{\\cdot}\\right\\Vert$ denote the Euclidean norm. Let $a \\in \\R^n$. Let $R > 0$ be a strictly positive real number. The '''open ball of center $a$ and radius $R$''' is the subset: :$B \\left({a, R}\\right) = \\left\\{ {x \\in \\R^n : \\left\\Vert{x - a}\\right\\Vert < R}\\right\\}$"}
+{"_id": "28462", "title": "Definition:Closed Rectangle", "text": "Let $n\\geq1$ be a natural number. Let $a_1, \\ldots, a_n, b_1, \\ldots, b_n$ be real numbers. The Cartesian product of closed intervals: :$\\displaystyle \\prod_{i \\mathop = 1}^n \\left[{a_i \\,.\\,.\\, b_i}\\right] = \\left[{a_1 \\,.\\,.\\, b_1}\\right] \\times \\cdots \\times \\left[{a_n \\,.\\,.\\, b_n}\\right] \\subseteq \\R^n$ is called a '''closed rectangle in $\\R^n$''' or '''closed $n$-rectangle'''. === Degenerate Case === In case $a_i > b_i$ for some $i$, the rectangle is taken to be the empty set $\\varnothing$. This is in accordance with the result Cartesian Product is Empty iff Factor is Empty for general Cartesian products."}
+{"_id": "28463", "title": "Definition:Closed Ball/Real Analysis", "text": "Let $n \\ge 1$ be a natural number. Let $\\R^n$ denote real Euclidean space Let $\\left\\Vert{\\, \\cdot \\,}\\right\\Vert$ denote the Euclidean norm. Let $a \\in \\R^n$. Let $R > 0$ be a strictly positive real number. The '''closed ball of center $a$ and radius $R$''' is the subset: :$B \\left({a, R}\\right) = \\left\\{ {x \\in \\R^n : \\left\\Vert{x - a}\\right\\Vert \\le R}\\right\\}$"}
+{"_id": "28464", "title": "Definition:Closed Set/Real Analysis/Real Numbers", "text": "Let $S \\subseteq \\R$ be a subset of the set of real numbers. Then $S$ is '''closed (in $\\R$)''' {{iff}} its complement $\\R \\setminus S$ is an open set."}
+{"_id": "28465", "title": "Definition:Closed Set/Real Analysis/Real Euclidean Space", "text": "Let $n\\geq1$ be a natural number. Let $S \\subseteq \\R^n$ be a subset. Then $S$ is '''closed (in $\\R^n$)''' {{iff}} its complement $\\R^n \\setminus S$ is an open set."}
+{"_id": "28466", "title": "Definition:Topology Generated by Synthetic Basis/Definition 1", "text": "The '''topology on $S$ generated by $\\BB$''' is defined as: :$\\tau = \\set{\\bigcup \\AA: \\AA \\subseteq \\BB}$"}
+{"_id": "28467", "title": "Definition:Topology Generated by Synthetic Basis/Definition 2", "text": "The '''topology on $S$ generated by $\\BB$''' is defined as: :$\\tau = \\set {U \\subseteq S: U = \\bigcup \\set {B \\in \\BB: B \\subseteq U}}$"}
+{"_id": "28468", "title": "Definition:Basis (Topology)/Synthetic Basis/Definition 1", "text": "A '''synthetic basis on $S$''' is a subset $\\mathcal B \\subseteq \\mathcal P \\left({S}\\right)$ of the power set of $S$ such that: {{begin-axiom}} {{axiom | n = B1         | t = $\\mathcal B$ is a cover for $S$ }} {{axiom | n = B2         | q = \\forall U, V \\in \\mathcal B         | t = $\\exists \\mathcal A \\subseteq \\mathcal B: U \\cap V = \\bigcup \\mathcal A$ }} {{end-axiom}} That is, the intersection of any pair of elements of $\\mathcal B$ is a union of sets of $\\mathcal B$."}
+{"_id": "28469", "title": "Definition:Basis (Topology)/Synthetic Basis/Definition 2", "text": "A '''synthetic basis on $S$''' is a subset $\\mathcal B \\subseteq \\mathcal P \\left({S}\\right)$ of the power set of $S$ such that: : $\\mathcal B$ is a cover for $S$ : $\\forall U, V \\in \\mathcal B: \\forall x \\in U \\cap V: \\exists W \\in \\mathcal B: x \\in W \\subseteq U \\cap V$"}
+{"_id": "28470", "title": "Definition:Topology Induced by Atlas", "text": "Let $X$ be a set. Let $A$ be an atlas on $X$. The '''topology induced by the atlas''' is the topology generated by the synthetic sub-basis: :$\\displaystyle\\bigcup_{(U,\\phi) \\in A }\\left\\{\\phi^{-1}(V) : V \\text{ is open in } U \\right\\} \\subset X$ Category:Definitions/Manifolds h4if4yplnmv9jzgj98we7uukzysh7oo"}
+{"_id": "28471", "title": "Definition:Generated Submodule/Definition 1", "text": "The '''submodule generated by $S$''' is the intersection of all submodules of $M$ containing $S$."}
+{"_id": "28472", "title": "Definition:Generated Submodule/Definition 2", "text": "The '''submodule generated by $S$''' is the set of all linear combinations of elements of $S$."}
+{"_id": "28473", "title": "Definition:Generator of Module/Definition 1", "text": "$S$ is a '''generator of $M$''' {{iff}} every element of $M$ is a linear combination of elements of $S$."}
+{"_id": "28474", "title": "Definition:Generator of Module/Definition 2", "text": "$S$ is a '''generator of $M$''' {{iff}} $M$ has no proper submodule containing $S$."}
+{"_id": "28475", "title": "Definition:Generator of Module/Definition 3", "text": "$S$ is a '''generator of $M$''' {{iff}} $M$ is the submodule generated by $S$."}
+{"_id": "28476", "title": "Definition:Generator of Vector Space", "text": "Let $K$ be a division ring. Let $\\mathbf V$ be a vector space over $K$. Let $S \\subseteq \\mathbf V$ be a subset of $\\mathbf V$. $S$ is a '''generator of $\\mathbf V$''' {{iff}} every element of $\\mathbf V$ is a linear combination of elements of $S$."}
+{"_id": "28477", "title": "Definition:Real Submanifold", "text": "Let $n,k\\geq1$ be natural numbers. Let $M\\subset\\R^n$ be a subset. === Definition Using Local Diffeomorphisms === $M$ is a '''real $C^k$-submanifold of dimension $d$''' of $\\R^n$ {{iff}} for all $p\\in M$ there exists a open neighborhood $U$ of $p$ in $\\R^n$ and a differentiable function $\\phi : U \\to \\R^n$ that is a $C^k$-diffeomorphism on its image, such that: :$\\phi(M \\cap U) = \\phi(U) \\cap (\\R^d\\times\\{0\\})$ === Definition Using Local Submersions === $M$ is a '''real $C^\\infty$-submanifold of dimension $d$''' of $\\R^n$ {{iff}} for all $p\\in M$ there exists a open neighborhood $U$ of $p$ in $\\R^n$ and a $C^\\infty$-submersion $\\phi : U \\to \\R^{n-d}$ such that: :$M \\cap U = \\phi^{-1}(0)$ === Definition Using Local Embeddings === $M$ is a '''real $C^\\infty$-submanifold of dimension $d$''' of $\\R^n$ {{iff}} for all $p\\in M$ there exists an open neighborhood $U$ in $\\R^d$ and a $C^\\infty$-embedding $\\phi : U \\to \\R^{n}$ such that: :$p \\in \\phi(U) \\subset M$"}
+{"_id": "28479", "title": "Definition:Embedding (Differential Geometry)", "text": "Let $m,n\\geq1$ be natural numbers. Let $U\\subset\\R^n$ be open. Let $f : U \\to \\R^m$ be a mapping. Then $f$ is a $C^k$-'''embedding''' {{iff}} $f$ is: * injective * a $C^k$-immersion * a homeomorphism on its image === Rank === The '''rank''' of an embedding is the rank of its differential at any point. === Smooth Embedding === {{stub}}"}
+{"_id": "28480", "title": "Definition:Open Neighborhood/Real Analysis", "text": "=== Real Numbers === {{:Definition:Open Neighborhood/Real Analysis/Real Numbers}} === Real Euclidean Space === {{:Definition:Open Neighborhood/Real Analysis/Real Euclidean Space}}"}
+{"_id": "28481", "title": "Definition:Open Neighborhood/Real Analysis/Real Numbers", "text": "Let $x\\in\\R$ be a real number. Let $I \\subseteq \\R$ be a subset. Then $I$ is an '''open neighborhood of $x$''' {{iff}} $I$ is open and $I$ is a neighborhood of $x$."}
+{"_id": "28483", "title": "Definition:Continuously Differentiable/Real Function", "text": "=== On an Open Interval === {{:Definition:Continuously Differentiable/Real Function/Open Interval}} === On an Open Set === {{:Definition:Continuously Differentiable/Real Function/Open Set}}"}
+{"_id": "28484", "title": "Definition:Continuously Differentiable/Real Function/Open Interval", "text": "Let $I\\subset\\R$ be an open interval. Then $f$ is '''continuously differentiable on $I$''' {{iff}} $f$ is differentiable on $I$ and its derivative is continuous on $I$."}
+{"_id": "28485", "title": "Definition:Continuously Differentiable/Real Function/Open Set", "text": "Let $I \\subset \\R$ be an open set. Then $f$ is '''continuously differentiable on $I$''' {{iff}} $f$ is differentiable on $I$ and its derivative is continuous on $I$."}
+{"_id": "28486", "title": "Definition:Continuously Differentiable/Real-Valued Function", "text": "=== In an Open Set === {{:Definition:Continuously Differentiable/Real-Valued Function/Open Set}}"}
+{"_id": "28487", "title": "Definition:Continuously Differentiable/Real-Valued Function/Open Set", "text": "Let $U$ be an open subset of $\\R^n$.  Let $f: U \\to \\R$ be a real-valued function. Then $f$ is '''continuously differentiable in the open set $U$''' {{iff}}: :$(1): \\quad f$ is differentiable in $U$. :$(2): \\quad$ the partial derivatives of $f$ are continuous in $U$."}
+{"_id": "28488", "title": "Definition:Continuously Differentiable/Vector-Valued Function", "text": "=== On an Open Set === {{:Definition:Continuously Differentiable/Vector-Valued Function/Open Set}}"}
+{"_id": "28489", "title": "Definition:Continuously Differentiable/Vector-Valued Function/Open Set", "text": "Let $U \\subset \\R^n$ be an open set. Let $f: U \\to \\R^m$ be a vector-valued function. Then $f$ is '''continuously differentiable in $U$''' {{iff}} $f$ is differentiable in $U$ and its partial derivatives are continuous in $U$."}
+{"_id": "28490", "title": "Definition:Holomorphic Function/Complex Plane", "text": "Let $U\\subset\\C$ be an open set. Let $f : U \\to \\C$ be a complex function. Then $f$ is '''holomorphic in $U$''' {{iff}} $f$ is differentiable at each point of $U$."}
+{"_id": "28491", "title": "Definition:Derivative/Complex Function/Point", "text": "Let $D\\subseteq \\C$ be an open set. Let $f : D \\to \\C$ be a complex function. Let $z_0 \\in D$ be a point in $D$. Let $f$ be complex-differentiable at the point $z_0$. That is, suppose the limit $\\displaystyle \\lim_{h \\to 0} \\ \\frac {f \\left({z_0 + h}\\right) - f \\left({z_0}\\right)} h$ exists. Then this limit is called the '''derivative of $f$ at the point $z_0$'''."}
+{"_id": "28492", "title": "Definition:Derivative/Complex Function/Open Set", "text": "Let $D\\subseteq \\C$ be an open set. Let $f : D \\to \\C$ be a complex function. Let $f$ be complex-differentiable in $D$. Then the '''derivative of $f$''' is the complex function  $f': D \\to \\C$ whose value at each point $z \\in D$ is the derivative $f' \\left({z}\\right)$: :$\\displaystyle \\forall z \\in D : f' \\left({z}\\right) := \\lim_{h \\mathop \\to 0} \\frac {f \\left({z + h}\\right) - f \\left({z}\\right)} h$"}
+{"_id": "28493", "title": "Definition:Differentiable Mapping/Vector-Valued Function/Point/Definition 1", "text": "$f$ is '''differentiable''' at $x \\in \\R^n$ {{iff}} there exists a linear transformation $T: \\R^n \\to \\R^m$ and a mapping $r : U \\to \\R^m$ such that: :$(1): \\quad \\map f {x + h} = \\map f x + \\map T h + \\map r h \\cdot \\norm h$ :$(2): \\quad \\displaystyle \\lim_{h \\mathop \\to 0} \\map r h = 0$"}
+{"_id": "28494", "title": "Definition:Differentiable Mapping/Vector-Valued Function/Point/Definition 2", "text": "$f$ is '''differentiable''' at $x \\in \\R^n$ {{iff}} for each real-valued function $f_j: j = 1, 2, \\ldots, m$: :$f_j: \\mathbb X \\to \\R$ is differentiable at $x$."}
+{"_id": "28495", "title": "Definition:Differential/Real Function/Point", "text": "Let $U \\subset \\R$ be an open set. Let $f: U \\to \\R$ be a real function. Let $f$ be differentiable at a point $x \\in U$. The '''differential of $f$ at $x$''' is the linear transformation $\\rd f \\left({x}\\right) : \\R \\to \\R$ defined as: :$\\rd f \\left({x}\\right) \\left({h}\\right) = f' \\left({x}\\right) \\cdot h$ where $f' \\left({x}\\right)$ is the derivative of $f$ at $x$."}
+{"_id": "28496", "title": "Definition:Differential/Real Function/Open Set", "text": "Let $U \\subset \\R$ be an open set. Let $f : U \\to \\R$ be a real function. Let $f$ be differentiable in $U$. The '''differential''' $\\rd f$ is the mapping $\\rd f : U \\to \\operatorname{Hom} \\left({\\R, \\R}\\right)$ defined as: :$\\left({\\mathrm d f}\\right) \\left({x}\\right) = \\rd f \\left({x}\\right)$ where: :$\\rd f \\left({x}\\right)$ is the differential of $f$ at $x$ :$\\operatorname{Hom} \\left({\\R, \\R}\\right)$ is the set of all linear transformations from $\\R$ to $\\R$."}
+{"_id": "28498", "title": "Definition:Zuckerman Number", "text": "A '''Zuckerman number''' is a positive integer which is divisible by the product of its digits."}
+{"_id": "28500", "title": "Definition:Partial Derivative/Real Analysis", "text": "{{:Definition:Partial Derivative/Real Analysis/Point}}"}
+{"_id": "28501", "title": "Definition:Partial Derivative/Real Analysis/Point", "text": "Let $U \\subset \\R^n$ be an open set. Let $f: U \\to \\R$ be a real-valued function. Let $a = \\tuple {a_1, a_2, \\ldots, a_n}^\\intercal \\in U$. Let $f$ be differentiable at $a$. Let $i \\in \\set {1, 2, \\ldots, n}$. ==== Definition 1 ==== {{:Definition:Partial Derivative/Real Analysis/Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Partial Derivative/Real Analysis/Point/Definition 2}}"}
+{"_id": "28502", "title": "Definition:Partial Derivative/Real Analysis/Open Set", "text": "Let $U\\subset\\R^n$ be an open set. Let $f : U \\to \\R$ be a real-valued function. Let $f$ be differentiable in $U$. The '''$i$th partial derivative (function) of $f$ with respect to $x_i$''' is the real-valued function which sends each $x\\in U$ to the $i$th partial derivative at $x$."}
+{"_id": "28503", "title": "Definition:Partial Derivative/Real Analysis/Point/Definition 1", "text": "The '''partial derivative of $f$ with respect to $x_i$ at $a$''' is denoted and defined as: :$\\map {\\dfrac {\\partial f} {\\partial x_i} } a := \\map {g_i'} {a_i}$ where: :$g_i$ is the real function defined as $\\map g {x_i} = \\map f {a_1, \\ldots, x_i, \\dots, a_n}$ :$\\map {g_i'} {a_i}$ is the derivative of $g$ at $a_i$."}
+{"_id": "28504", "title": "Definition:Partial Derivative/Real Analysis/Point/Definition 2", "text": "The '''$i$th partial derivative of $f$ at $a$''' is the limit: :$\\map {\\dfrac {\\partial f} {\\partial x_i} } a = \\displaystyle \\lim_{x_i \\mathop \\to a_i} \\frac {\\map f {a_1, a_2, \\ldots, x_i, \\ldots, a_n} - \\map f a} {x_i - a}$"}
+{"_id": "28506", "title": "Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 2", "text": "$f$ is '''differentiable at $x$''' {{iff}} there exists a linear transformation $T: \\R^n \\to \\R$ and a real-valued function $r: U \\setminus \\set x \\to \\R$ such that: :$(1):\\quad \\map f {x + h} = \\map f x + \\map T h + \\map r h \\cdot h$ :$(2):\\quad \\displaystyle \\lim_{h \\mathop \\to 0} \\map r h = 0$"}
+{"_id": "28507", "title": "Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 1", "text": "$f$ is '''differentiable at $x$''' {{iff}} there exist $\\alpha_1, \\ldots, \\alpha_n \\in \\R$ and a real-valued function $r: U \\setminus \\set x \\to \\R$ such that: :$(1):\\quad \\map f {x + h} = \\map f x + \\alpha_1 h_1 + \\cdots + \\alpha_n h_n + \\map r h\\cdot h$ :$(2):\\quad \\displaystyle \\lim_{h \\mathop \\to 0} \\map r h = 0$"}
+{"_id": "28510", "title": "Definition:Differential/Real-Valued Function/Point", "text": "Let $U \\subset \\R^n$ be an open set. Let $f: U \\to \\R$ be a real-valued function. Let $f$ be differentiable at a point $x \\in U$. :$\\displaystyle \\d f \\left({x; h}\\right) := \\sum_{i \\mathop = 1}^n \\frac {\\partial f \\left({x}\\right)} {\\partial x_i} h_i = \\frac {\\partial f \\left({x}\\right)} {\\partial x_1} h_1 + \\frac {\\partial f \\left({x}\\right)} {\\partial x_2} h_2 + \\cdots + \\frac {\\partial f \\left({x}\\right)} {\\partial x_n} h_n$ where: : $h = \\left({h_1, h_2, \\ldots, h_n}\\right) \\in \\R^n$ : $\\dfrac {\\partial f} {\\partial x_i}$ is the partial derivative of $f$ {{WRT|Differentiation}} $x_i$."}
+{"_id": "28511", "title": "Definition:Differential/Vector-Valued Function/Point", "text": "Let $U \\subset \\R^n$ be an open set. Let $f: U \\to \\R^m$ be a vector-valued function. Let $f$ be differentiable at a point $x \\in U$. The '''differential of $f$ at $x$''' is the linear transformation $\\d f \\left({x}\\right): \\R^n \\to \\R^m$ defined as: :$\\d f \\left({x}\\right) \\left({h}\\right) = J_f \\left({x}\\right) \\cdot h$ where: : $J_f \\left({x}\\right)$ is the Jacobian matrix of $f$ at $x$."}
+{"_id": "28512", "title": "Definition:Derivative/Vector-Valued Function/Point", "text": "Let $U \\subset \\R$ be an open set. Let $\\mathbf f \\left({x}\\right) = \\displaystyle \\sum_{k \\mathop = 1}^n f_k \\left({x}\\right) \\mathbf e_k: U \\to \\R^n$ be a vector-valued function. Let $\\mathbf f$ be differentiable at $u \\in U$. That is, let each $f_j$ be differentiable at $u \\in U$. The '''derivative of $\\mathbf f$ {{WRT|Differentiation}} $x$ at $u$''' is defined as :$\\dfrac {\\d \\mathbf f} {\\d x} \\left({u}\\right) = \\displaystyle \\sum_{k \\mathop = 1}^n \\dfrac {\\d f_k} {\\d x} \\left({u}\\right) \\mathbf e_k$ where $\\dfrac {\\d f_k} {\\d x} \\left({u}\\right)$ is the derivative of $f_k$ with respect to $x$ at $u$."}
+{"_id": "28513", "title": "Definition:Lipschitz Continuity/Point", "text": "Let $a\\in A$. $f$ is a '''Lipschitz continuous at $a$''' {{iff}} there exists a positive real number $K \\in \\R_{\\ge 0}$ such that: :$\\forall x \\in A: d\\,' \\left({f \\left({x}\\right), f \\left({a}\\right)}\\right) \\le K d \\left({x, a}\\right)$"}
+{"_id": "28514", "title": "Definition:Differentiable Mapping between Manifolds", "text": "Let $M$ and $N$ be differentiable manifolds. Let $f : M \\to N$ be continuous. === Definition 1 === $f$  is '''differentiable''' {{iff}} for every pair of charts $(U, \\phi)$ and $(V,\\psi)$ of $M$ and $N$: :$\\psi\\circ f\\circ \\phi^{-1} : \\phi ( U \\cap f^{-1}(V)) \\to \\psi(V)$ is differentiable. === Definition 2 === $f$  is '''differentiable''' {{iff}} $f$ is  differentiable at every point of $M$. === At a Point === {{:Definition:Differentiable Mapping between Manifolds/Point}}"}
+{"_id": "28515", "title": "Definition:Differentiable Mapping between Manifolds/Point", "text": "Let $M$ and $N$ be differentiable manifolds. Let $f: M \\to N$ be continuous. Let $p \\in M$. ==== Definition 1 ==== {{:Definition:Differentiable Mapping between Manifolds/Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Differentiable Mapping between Manifolds/Point/Definition 2}}"}
+{"_id": "28516", "title": "Definition:Differentiable Mapping between Manifolds/Point/Definition 1", "text": "$f$ is '''differentiable at $p$''' {{iff}} for every pair of charts $\\left({U, \\phi}\\right)$ and $\\left({V, \\psi}\\right)$ of $M$ and $N$ with $p \\in U$ and $f \\left({p}\\right) \\in V$: :$\\psi \\circ f \\circ \\phi^{-1}: \\phi \\left({U \\cap f^{-1} \\left({V}\\right)}\\right) \\to \\psi \\left({V}\\right)$ is differentiable at $\\phi \\left({p}\\right)$."}
+{"_id": "28517", "title": "Definition:Differentiable Mapping between Manifolds/Point/Definition 2", "text": "$f$ is '''differentiable at $p$''' {{iff}} there exists a pair of charts $\\left({U, \\phi}\\right)$ and $\\left({V, \\psi}\\right)$ of $M$ and $N$ with $p \\in U$ and $f \\left({p}\\right) \\in V$ such that: :$\\psi \\circ f \\circ \\phi^{-1}: \\phi \\left({U \\cap f^{-1} \\left({V}\\right)}\\right) \\to \\psi \\left({V}\\right)$ is differentiable at $\\phi \\left({p}\\right)$."}
+{"_id": "28518", "title": "Definition:Injective Space", "text": "Let $Z = \\left({S, \\tau_1}\\right)$ be a topological space. Then $Z$ is '''injective (space)''' {{iff}} :for all topological spaces $X = \\left({H, \\tau_2}\\right)$ :and for all continuous mappings $f:H \\to S$ :and for all topological spaces $Y = \\left({T, \\tau_3}\\right)$ such that $X$ is topological subspace of $Y$: :there exists a continuous mapping $g:T \\to S$: $g \\restriction H = f$"}
+{"_id": "28519", "title": "Definition:Wieferich Prime", "text": "A '''Wieferich prime''' is a prime number $p$ such that: :$p^2 \\divides 2^{p − 1} − 1$ where $\\divides$ denotes divisibility."}
+{"_id": "28520", "title": "Definition:Wieferich Prime/Sequence", "text": "The sequence of Wieferich primes begins: :$1093, 3511 \\ldots$ No other Wieferich primes exist below $4 \\cdotp 968543 \\times 10^{17}$."}
+{"_id": "28521", "title": "Definition:Fermat's Equation", "text": "'''Fermat's equation''' is the Diophantine equation: :$x^n + y^n = z^n$"}
+{"_id": "28526", "title": "Definition:Retraction (Topology)", "text": "Let $T_1 = \\left({S_1, \\tau_1}\\right)$ be a topological space. Let $T_2 = \\left({S_2, \\tau_2}\\right)$ be a topological subspace of $T_1$. It means that :$S_2 \\subseteq S_1$ Let $f: S_1 \\to S_2$ be a mapping. Then $f$ is '''retraction''' of $T_1$ {{iff}} :$\\forall s \\in S_2: f\\left({s}\\right) = s$"}
+{"_id": "28527", "title": "Definition:Retract (Topology)", "text": "Let $T_1 = \\left({S_1, \\tau_1}\\right)$ be a topological space. Let $T_2 = \\left({S_2, \\tau_2}\\right)$ be a topological subspace of $T_1$. Then $T_2$ is a '''retract''' of $T_1$ {{iff}} :there exists a continuous retraction $f: S_1 \\to S_2$ of $T_1$."}
+{"_id": "28528", "title": "Definition:Sphenic Number", "text": "A '''sphenic number''' is a (positive) integer which has exactly $3$ distinct prime factors all with multiplicity of $1$."}
+{"_id": "28531", "title": "Definition:Imperial/Volume/Cubic Inch", "text": ":$1$ '''cubic inch''' $= 16 \\cdotp 38706 \\, 4$ cubic centimetres $= 0 \\cdotp 01638 \\, 7064$ litres. This definition is exact, and is derived from the definition of the imperial unit of length as $2 \\cdotp 54$ centimetres."}
+{"_id": "28532", "title": "Definition:Imperial/Volume/Cubic Foot", "text": "The '''cubic foot''' is the FPS unit of measurement of volume. :$1$ '''cubic foot''' $= 12^3 = 1728$ cubic inches $= 28 \\cdotp 31684 \\, 6592$ litres $= 0 \\cdotp 02831 \\, 68465 \\, 92$ cubic metres."}
+{"_id": "28533", "title": "Definition:Taxicab Number", "text": "A '''taxicab number''' is a positive integer which can be expressed as the sum of $2$ cubes in $2$ different ways."}
+{"_id": "28539", "title": "Definition:Image of Topological Space", "text": "Let $T = \\left({S, \\tau}\\right)$ and $Q = \\left({X, \\tau'}\\right)$ be topological spaces. Let $f:S \\to X$ be a mapping. Then '''image (of the topological space $T$) of''' $f$ is equal to :$\\operatorname{Im}\\left({f}\\right) := Q_{f\\left[{S}\\right]} = \\left({f\\left[{S}\\right], \\tau'_{f\\left[{S}\\right]} }\\right)$ where $\\tau'_{f\\left[{S}\\right]}$ denotes the subspace topology on $f\\left[{S}\\right]$."}
+{"_id": "28540", "title": "Definition:Mile", "text": "There are a number of definitions for the '''mile'''. === International Mile === {{:Definition:Imperial/Length/International Mile}} === Nautical Mile === {{:Definition:Nautical Mile}} === Admiralty Mile === {{:Definition:Admiralty Mile}} Category:Definitions/Length tcd8q4x2mvlyelr8n81fgj5zhg8v9wt"}
+{"_id": "28541", "title": "Definition:Weak Retract (Topology)", "text": "Let $T_1 = \\left({S_1, \\tau_1}\\right)$ and $T_2 = \\left({S_2, \\tau_2}\\right)$ be topological spaces. Then $T_1$ is '''a weak retract of''' $T_2$ {{iff}} :there exists a continuous mapping $f:S_2 \\to S_2$: $f \\circ f = f$ and $\\operatorname{Im}\\left({f}\\right), T_1$ are homeomorphic."}
+{"_id": "28542", "title": "Definition:Lower Topology", "text": "Let $T = \\left({S, \\preceq, \\tau}\\right)$ be a relational structure with topology. Then $T$ has '''lower topology''' {{iff}} :$\\left\\{ {\\complement_S\\left({x^\\succeq}\\right): x \\in S}\\right\\}$ is sub-basis of $T$ where $x^\\succeq$ denotes the upper closure of $x$."}
+{"_id": "28545", "title": "Definition:Ball", "text": "=== Open Ball === {{:Definition:Open Ball}} === Closed Ball === {{:Definition:Closed Ball}} === Unit Ball === {{:Definition:Unit Ball}}"}
+{"_id": "28546", "title": "Definition:Homogeneous Cyclotomic Polynomial", "text": "Let $n\\geq1$ be a positive integer. The $n$th '''homogeneous cyclotomic polynomial''' is the homogenization of the $n$th cyclotomic polynomial. Category:Definitions/Cyclotomic Polynomials iqgnx6lwxx9ncuh3cywe3zn7j8718gw"}
+{"_id": "28548", "title": "Definition:Special Highly Composite Number", "text": "A '''special highly composite number''' is a highly composite number which is a divisor of all larger highly composite numbers."}
+{"_id": "28549", "title": "Definition:Special Highly Composite Number/Sequence", "text": "The sequence of special highly composite numbers consists of: :$1, 2, 6, 12, 60, 2520$ {{OEIS|A106037}}"}
+{"_id": "28551", "title": "Definition:Modified Kaprekar Mapping", "text": "The '''modified Kaprekar mapping''' is the arithmetic function $K: \\Z_{>0} \\to \\Z_{>0}$ defined on the positive integers  as follows: Let $n \\in \\Z_{>0}$ be expressed in some number base $b$ (where $b$ is usually $10$). Let $n'$ be the positive integer created by: :arranging the digits of $n$ into descending order of size then: :exchanging the last two digits. Let $n''$ be the positive integer created by: :arranging the digits of $n$ into ascending order of size then: :exchanging the first two digits. Then: :$\\map {K'} n = n' - n''$ making sure to retain any leading zeroes to ensure that $\\map K n$ has the same number of digits as $n$."}
+{"_id": "28553", "title": "Definition:Modified Kaprekar Process", "text": "The '''modified Kaprekar process''' is the repeated application of the '''modified Kaprekar mapping''' to a given positive integer."}
+{"_id": "28555", "title": "Definition:Central Tree", "text": "A tree $T$ is '''central''' {{iff}} it has a center."}
+{"_id": "28556", "title": "Definition:Bicentral Tree", "text": "A tree $T$ is '''bicentral''' {{iff}} it has a bicenter."}
+{"_id": "28557", "title": "Definition:Perfect Digit-to-Digit Invariant", "text": "A '''perfect digit-to-digit invariant''' is a number $n$ which is equal to the sum of the digits of $n$ each raised to the power of itself."}
+{"_id": "28560", "title": "Definition:Perfect Digital Invariant", "text": "Let $n \\in \\Z_{>0}$ be a positive integer. $n$ is a '''perfect digital invariant''' {{iff}} when expressed in decimal notation the $n$th powers of its digits add up to some positive integer $m$."}
+{"_id": "28562", "title": "Definition:Imperial/Area/Acre", "text": "One '''acre''' is equal to an oblong measuring $1$ chain by $1$ furlong. That is, measuring $4$ rods, poles or perches by $10$ chains. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''acre''' }} {{eqn | r = 4       | c = roods }} {{eqn | r = 10       | c = square chains }} {{eqn | r = 160 = 4 \\times 40       | c = square rods, poles or perches }} {{eqn | r = 4840 = 22 \\times 220       | c = square yards }} {{end-eqn}} :810px"}
+{"_id": "28565", "title": "Definition:Imperial/Area/Square Yard", "text": "One '''square yard''' is equal to a square of side $1$ yard in length. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''square yard''' }} {{eqn | r = 9 = 3^2       | c = square feet }} {{eqn | r = 1296 = 36^2       | c = square inches }} {{end-eqn}}"}
+{"_id": "28566", "title": "Definition:Imperial/Area/Square Inch", "text": "One '''square inch''' is equal to a square of side $1$ inch in length. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''square inch''' }} {{eqn | r = 6 \\cdotp 4516 = 2 \\cdotp 54^2       | c = square centimetres }} {{end-eqn}} The definition is exact."}
+{"_id": "28567", "title": "Definition:Imperial/Area/Square Foot", "text": "One '''square foot''' is equal to a square of side $1$ foot in length. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''square foot''' }} {{eqn | r = 144 = 12^2       | c = square inches }} {{end-eqn}}"}
+{"_id": "28570", "title": "Definition:Kaprekar's Constant", "text": "Let $n \\in \\Z_{>0}$ be a positive integer with $4$ digits, not all of which are equal. Let Kaprekar's process be applied to $n$. The number reached after no more than $8$ iterations is known as '''Kaprekar's constant''', and has the value of $6174$. {{NamedforDef|Dattathreya Ramchandra Kaprekar|cat = Kaprekar}}"}
+{"_id": "28573", "title": "Definition:Tri-Automorphic Number", "text": "A '''tri-automorphic number''' is a positive integer $n$ such that $3 n^2$ ends in a repetition of $n$."}
+{"_id": "28577", "title": "Definition:Support of Module", "text": "Let $A$ be a commutative ring with unity. Let $M$ be a unitary $A$-module. The '''support''' $\\operatorname{Supp}(M)$ of $M$ is the set of prime ideals $P$ of $A$ such that the localization of $M$ at $P$ is nonzero: :$\\operatorname{Supp}(M) = \\{P\\in \\operatorname{Spec}(A) : M_P\\neq0 \\}$ where $\\operatorname{Spec}(A)$ is the spectrum of $A$."}
+{"_id": "28580", "title": "Definition:Integral Ring Extension", "text": "Let $A$ be a commutative ring with unity. Let $R\\subset A$ be a subring. The ring extension $R \\subseteq A$ is said to be '''integral''' {{iff}} for all $a \\in A$, $a$ is integral over $R$."}
+{"_id": "28584", "title": "Definition:Support of Mapping to Algebraic Structure", "text": "=== Real-Valued Function on an Abstract Set === {{Definition:Support of Mapping to Algebraic Structure/Real-Valued Function}} === General Real-Valued Function in $\\R^n$ === === General Algebraic Structure === Let $(A, *)$ be an algebraic structure with an identity element $e$. Let $S$ be a set. Let $f : S \\to A$ be a mapping. The '''support of $f$''' is the set: :$\\operatorname{supp}(f) = \\{s \\in S : f(s) \\neq e\\}$ {{expand}}"}
+{"_id": "28588", "title": "Definition:Weakly Locally Compact Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Then $T$ is '''weakly locally compact''' {{iff}} every point of $S$ has a compact neighborhood."}
+{"_id": "28589", "title": "Definition:Noetherian Ring/Definition 1", "text": "A commutative ring with unity $A$ is '''Noetherian''' {{iff}} every ideal of $A$ is finitely generated."}
+{"_id": "28590", "title": "Definition:Noetherian Ring/Definition 2", "text": "A commutative ring with unity $A$ is '''Noetherian''' {{iff}} it satisfies the ascending chain condition on ideals."}
+{"_id": "28591", "title": "Definition:Noetherian Ring/Definition 3", "text": "A commutative ring with unity $A$ is '''Noetherian''' {{iff}} it satisfies the maximal condition on ideals."}
+{"_id": "28592", "title": "Definition:Noetherian Ring/Definition 4", "text": "A commutative ring with unity $A$ is '''Noetherian''' {{iff}} it is Noetherian as an $A$-module."}
+{"_id": "28593", "title": "Definition:Noetherian Module/Definition 1", "text": "$M$ is a '''Noetherian module''' {{iff}} every submodule of $M$ is finitely generated."}
+{"_id": "28594", "title": "Definition:Noetherian Module/Definition 2", "text": "$M$ is a '''Noetherian module''' {{iff}} it satisfies the ascending chain condition on submodules."}
+{"_id": "28595", "title": "Definition:Noetherian Module/Definition 3", "text": "$M$ is a '''Noetherian module''' {{iff}} it satisfies the maximal condition on submodules."}
+{"_id": "28597", "title": "Definition:Sylow p-Subgroup/Definition 1", "text": "Let $\\order G = k p^n$ where $p \\nmid k$. A '''Sylow $p$-subgroup''' is a $p$-subgroup of $G$ which has $p^n$ elements."}
+{"_id": "28598", "title": "Definition:Sylow p-Subgroup/Definition 2", "text": "A '''Sylow $p$-subgroup''' of $G$ is a '''maximal $p$-subgroup''' $P$ of $G$. In this context, maximality means that if $Q$ is a $p$-subgroup of $G$ and $P \\le Q$, then $P = Q$."}
+{"_id": "28601", "title": "Definition:Cover of Set/Subset", "text": "Let $A \\subseteq S$ be a subset. Let $\\mathcal C$ be a set of subsets of $S$. Then $\\mathcal C$ is a '''cover of $A$''' {{iff}} $A \\subseteq \\displaystyle \\bigcup \\mathcal C$, where $\\cup$ denotes union."}
+{"_id": "28602", "title": "Definition:Open Cover/Subset", "text": "Let $H$ be a subset of $S$. Let $\\CC$ be a cover of $H$. Then $\\CC$ is an '''open cover for $H$''' {{iff}}: :$\\CC \\subseteq \\tau$ That is, {{iff}} all the elements of $\\CC$ are open sets."}
+{"_id": "28603", "title": "Definition:Prime Number Race", "text": "The set of prime numbers $\\Bbb P$ may be partitioned into subsets according to a particular property. A '''prime number race''' is a comparison of the count of the number of prime numbers in each partition with increasing $p \\in \\Bbb P$."}
+{"_id": "28609", "title": "Definition:Pandigital Set", "text": "A set of integers is '''pandigital''' if it contains exactly $1$ instance of each digit. The digits may or may not contain zero, depending on context."}
+{"_id": "28610", "title": "Definition:Pandigital Set/Integer", "text": "The element of a singleton pandigital set is itself referred to as a '''pandigital integer''' or '''pandigital number'''."}
+{"_id": "28611", "title": "Definition:Ackermann-Péter Function", "text": "The '''Ackermann-Péter function''' $A: \\Z_{\\ge 0} \\times \\Z_{\\ge 0} \\to \\Z_{> 0}$ is an integer-valued function defined on the set of ordered pairs of positive integers as: :$A \\left({x, y}\\right) = \\begin{cases} y + 1 & : x = 0 \\\\ A \\left({x - 1, 1}\\right) & : x > 0, y = 0 \\\\ A \\left({x - 1, A \\left({x, y - 1}\\right)}\\right) & : \\text{otherwise} \\end{cases}$"}
+{"_id": "28613", "title": "Definition:Fermat Number/Naming Conventions", "text": "The '''Fermat number''' $F_0$ is often referred to as the ''' $1$st Fermat number''', so (confusingly) this convention dictates that $F_n$ is the '''$n + 1$th Fermat number'''. However, another convention is that $F_0$ can be referred to as the '''zeroth Fermat number''', thus bringing the appellation in line such that $F_n$ is the '''$n$th Fermat number'''. Both conventions are in place, sometimes in the same work. For example, {{AuthorRef|David Wells}}, in his {{BookLink|Curious and Interesting Numbers|David Wells|ed = 2nd|edpage = Second Edition}} of $1997$, refers to $5 = F_1$ in Section $5$ as the '''$2$nd Fermat number'''. However, in Section $257$ he defines $F_3 = 2^{2^3} + 1 = 257$ as the '''$3$rd Fermat number'''. Similarly, in Section $65,537$ he defines $F_4 = 2^{2^4} + 1 = 65 \\, 537$ as the '''$4$th Fermat number''', and so on. Both of these naming conventions is more or less clumsy. {{ProofWiki}} takes the position that the cat has to jump one way or the other, and so uses the second of these conventions: :$F_n$ is the '''$n$th Fermat number'''."}
+{"_id": "28614", "title": "Definition:Unitary Divisor", "text": "Let $n \\in \\Z$ be an integer. Let $d \\in \\Z$ be such that: :$d$ is a divisor of $n$ :$d$ and $\\dfrac n d$ are coprime. Then $d$ is a unitary divisor of $n$."}
+{"_id": "28615", "title": "Definition:Unitary Perfect Number", "text": "A '''unitary perfect number''' is a positive integer which equals the sum of its positive unitary divisors apart from itself."}
+{"_id": "28617", "title": "Definition:Prime Quadruple", "text": "A '''prime quadruple''' is a set of $4$ prime numbers of the form $n$, $n + 2$, $n + 6$, $n + 8$. That is, they are two pairs of twin primes separated by $4$."}
+{"_id": "28619", "title": "Definition:Propositional Tableau/Construction/Finite", "text": "The '''finite propositional tableaus''' are precisely those labeled trees singled out by the following bottom-up grammar: :{| style=\"border-spacing:20px;\" | $\\boxed{\\mathrm{Root}}$ | A labeled tree whose only node is its root node is a '''finite propositional tableau'''. |- | colspan=2 | For the following clauses, let $t$ be a leaf node of a '''finite propositional tableau''' $T$. |- | $\\boxed{\\neg \\neg}$ | If $\\neg \\neg \\mathbf A$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\\Phi \\left({s}\\right) = \\mathbf A$  is a '''finite propositional tableau'''. |- | $\\boxed \\land$ | If $\\mathbf A \\land \\mathbf B$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\\Phi \\left({s}\\right) = \\mathbf A$  :a child $r$ to $s$, with $\\Phi \\left({r}\\right) = \\mathbf B$ is a '''finite propositional tableau'''. |- | $\\boxed{\\neg \\land}$ | If $\\neg \\left({\\mathbf A \\land \\mathbf B}\\right)$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\\Phi \\left({s}\\right) = \\neg\\mathbf A$ :another child $s'$ to $t$, with $\\Phi \\left({s'}\\right) = \\neg\\mathbf B$ is a '''finite propositional tableau'''. |- | $\\boxed \\lor$ | If $\\mathbf A \\lor \\mathbf B$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\\Phi \\left({s}\\right) = \\mathbf A$ :another child $s'$ to $t$, with $\\Phi \\left({s'}\\right) = \\mathbf B$ is a '''finite propositional tableau'''. |- | $\\boxed{\\neg\\lor}$ | If $\\neg \\left({\\mathbf A \\lor \\mathbf B}\\right)$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\\Phi \\left({s}\\right) = \\neg\\mathbf A$  :a child $r$ to $s$, with $\\Phi \\left({r}\\right) = \\neg \\mathbf B$ is a '''finite propositional tableau'''. |- | $\\boxed \\implies$ | If $\\mathbf A \\implies \\mathbf B$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\\Phi \\left({s}\\right) = \\neg\\mathbf A$ :another child $s'$ to $t$, with $\\Phi \\left({s'}\\right) = \\mathbf B$ is a '''finite propositional tableau'''. |- | $\\boxed{\\neg\\implies}$ | If $\\neg \\left({\\mathbf A \\implies \\mathbf B}\\right)$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\\Phi \\left({s}\\right) = \\mathbf A$  :a child $r$ to $s$, with $\\Phi \\left({r}\\right) = \\neg \\mathbf B$ is a '''finite propositional tableau'''. |- | $\\boxed \\iff$ | If $\\mathbf A \\iff \\mathbf B$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\\Phi \\left({s}\\right) = \\mathbf A \\land \\mathbf B$  :another child $s'$ to $t$, with $\\Phi \\left({s'}\\right) = \\neg \\mathbf A \\land \\neg\\mathbf B$ is a '''finite propositional tableau'''. |- | $\\boxed{\\neg\\iff}$ | If $\\neg \\left({\\mathbf A \\iff \\mathbf B}\\right)$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\\Phi \\left({s}\\right) = \\mathbf A \\land \\neg \\mathbf B$  :another child $s'$ to $t$, with $\\Phi \\left({s'}\\right) = \\neg \\mathbf A \\land \\mathbf B$ is a '''finite propositional tableau'''. |} Note how the boxes give an indication of the ancestor WFF mentioned in the clause. These clauses together are called the '''tableau extension rules'''."}
+{"_id": "28620", "title": "Definition:Propositional Tableau/Construction/Infinite", "text": "An infinite labeled tree $\\left({T, \\mathbf H, \\Phi}\\right)$ is a '''propositional tableau''' {{iff}}: :There exists an exhausting sequence of sets $\\left({T_n}\\right)_{n \\in \\N}$ of $T$ such that for all $n \\in \\N$: ::$\\left({T_n, \\mathbf H, \\Phi \\restriction_{T_n}}\\right)$ :is a finite propositional tableau, where $\\Phi \\restriction_{T_n}$ denotes the restriction of $\\Phi$ to $T_n$."}
+{"_id": "28621", "title": "Definition:Cyclic Number", "text": "A '''cyclic number''' is an integer $n$ whose cyclic permutations of its digits are integer multiples or integer divisors of $n$."}
+{"_id": "28623", "title": "Definition:Sorted Number", "text": "A '''sorted number''' is an integer whose digits form a non-decreasing sequence, for example: :$1 \\, 123 \\, 334 \\, 899$"}
+{"_id": "28624", "title": "Definition:Rare Number", "text": "A '''rare number''' is a non-palindromic integer $n$ which has the property that $n + r$ and $n - r$ are both square, where $r$ is the reversal of $n$."}
+{"_id": "28629", "title": "Definition:Two-Sided Prime", "text": "A '''two-sided prime''' is a prime number which remains prime when: :any number of digits are removed from the left hand end and: :any number of digits are removed from the right hand end but, generally, not from both ends at once."}
+{"_id": "28632", "title": "Definition:Cunningham Chain", "text": "There are $2$ types of '''Cunningham chain''': === First Kind === {{:Definition:Cunningham Chain/First Kind}} === Second Kind === {{:Definition:Cunningham Chain/Second Kind}}"}
+{"_id": "28634", "title": "Definition:Undecidable Statement", "text": "An '''undecidable statement''' is a statement which, in the context of a given deductive apparatus, cannot be assigned a definitive truth value."}
+{"_id": "28635", "title": "Definition:Equivalence of Norms", "text": "Let $\\norm {\\,\\cdot\\,}_1$ and $\\norm {\\,\\cdot\\,}_2$ be norms on a vector space $V$. $\\norm {\\,\\cdot\\,}_1$ and $\\norm {\\,\\cdot\\,}_2$ are '''equivalent''' {{iff}} there exist real constants $c$ and $C$ such that: :$\\forall \\mathbf x \\in V: c \\norm {\\mathbf x}_1 \\le \\norm {\\mathbf x}_2 \\le C \\norm {\\mathbf x}_1$"}
+{"_id": "28636", "title": "Definition:Language of Propositional Logic/Basson-O'Connor", "text": "There are many formal languages expressing propositional logic. The formal language used on {{ProofWiki}} is defined on Definition:Language of Propositional Logic. This page defines the formal language $\\LL_0$ used in: * {{BookReference|Introduction to Symbolic Logic|1959|A.H. Basson|author2 = D.J. O'Connor|ed = 3rd|edpage = Third Edition}} Explanations are omitted as this is intended for reference use only. === Alphabet === ==== Letters ==== The letters used are a non-empty set of symbols $\\PP_0$. See the {{ProofWiki}} definition. ==== Signs ==== ===== Brackets ===== The '''brackets''' used are '''round brackets''': {{begin-eqn}} {{eqn | ll= \\bullet       | l = (       | o = :       | r = \\)the '''left bracket''' sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = )       | o = :       | r = \\)the '''right bracket''' sign\\(       | c =  }} {{end-eqn}} See the {{ProofWiki}} definition. ===== Connectives ===== The following '''connectives''' are used: {{begin-eqn}} {{eqn | ll= \\bullet       | l = .       | o = :       | r = \\)the conjunction sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\lor       | o = :       | r = \\)the disjunction sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\supset       | o = :       | r = \\)the conditional sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\equiv       | o = :       | r = \\)the biconditional sign\\(       | c =  }} {{eqn | ll= \\bullet       | l = \\sim       | o = :       | r = \\)the negation sign\\(       | c =  }} {{end-eqn}} See the {{ProofWiki}} definition. === Collation System === The collation system used is that of words and concatenation. See the {{ProofWiki}} definition. === Formal Grammar === The following bottom-up formal grammar is used. Let $\\PP_0$ be the vocabulary of $\\LL_0$. Let $Op = \\left\\{{., \\lor, \\supset, \\equiv}\\right\\}$. The rules are: {| |- | $\\mathbf W: \\PP_0$ || | $:$ | If $p \\in \\PP_0$, then $p$ is a WFF. |- | $\\mathbf W: \\neg$ || | $:$ | If $\\mathbf A$ is a WFF, then $\\sim \\mathbf A$ is a WFF. |- | $\\mathbf W: Op$ || | $:$ | If $\\mathbf A$ and $\\mathbf B$ are WFFs and $\\circ \\in Op$, then $\\paren {\\mathbf A \\circ \\mathbf B}$ is a WFF. |} See the {{ProofWiki}} definition."}
+{"_id": "28637", "title": "Definition:Pea Pattern/Sequential", "text": "For the '''sequential pea pattern''', the concatenation into $p_n$ is in sequential order of how the first instance of the distinct digits appear in $p_{n - 1}$."}
+{"_id": "28638", "title": "Definition:Pea Pattern", "text": "The first term $p_0$ is an arbitrary integer. $p_n$ is formed from $p_{n - 1}$ as follows. First, the distinct digits in $p_{n - 1}$ are counted, and the number of each is noted. The count of each distinct digit is concatenated with an instance of the digit itself. Then those concatenations are themselves concatenated into $p_n$ according to a predetermined order."}
+{"_id": "28639", "title": "Definition:Pea Pattern/Ascending", "text": "For the '''ascending pea pattern''', the concatenation into $p_n$ is in ascending order of the distinct digits of $p_{n - 1}$."}
+{"_id": "28640", "title": "Definition:Pea Pattern/Descending", "text": "For the '''descending pea pattern''', the concatenation into $p_n$ is in descending order of the distinct digits of $p_{n - 1}$."}
+{"_id": "28645", "title": "Definition:Time/Unit/Great Year of Plato", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''great year of Plato''' }} {{eqn | r = 12 \\, 960 \\, 000       | c = days }} {{eqn | o = \\approx       | r = 36 \\, 000       | c = years (of $360$ days) }} {{end-eqn}}"}
+{"_id": "28647", "title": "Definition:Counterexample", "text": "Let $X$ be the universal statement: :$\\forall x \\in S: \\map P x$ That is: :''For all the elements $x$ of a given set $S$, the property $P$ holds.'' Such a statement may or may not be true. Let $Y$ be the existential statement: :$\\exists y \\in S: \\neg \\map P y$ That is: :''There exists at least one element $y$ of the set $S$ such that the property $P$ does ''not'' hold.'' It follows immediately by De Morgan's laws that if $Y$ is true, then $X$ must be false. Such a statement $Y$ is referred to as a '''counterexample to $X$'''."}
+{"_id": "28648", "title": "Definition:Substitution (Formal Systems)", "text": "=== Substitution for Well-Formed Part === {{:Definition:Substitution (Formal Systems)/Well-Formed Part}} === Substitution for Letter === {{:Definition:Substitution (Formal Systems)/Letter}} === Substitution of Term for Variable === {{:Definition:Substitution (Formal Systems)/Term}} === Substitution for Metasymbol === {{:Definition:Substitution (Formal Systems)/Metasymbol}}"}
+{"_id": "28649", "title": "Definition:Substitution (Formal Systems)/Letter", "text": "Let $\\mathbf B$ be a well-formed formula of $\\FF$. Let $p$ be a letter of $\\FF$. Let $\\mathbf A$ be another well-formed formula. Then the '''substitution of $\\mathbf A$ for $p$ in $\\mathbf B$''' is the collation resulting from $\\mathbf B$ by replacing all occurrences of $p$ in $\\mathbf B$ by $\\mathbf A$. It is denoted as $\\map {\\mathbf B} {\\mathbf A \\mathbin {//} p}$."}
+{"_id": "28651", "title": "Definition:Substitution (Formal Systems)/Term", "text": "{{:Definition:Substitution (Formal Systems)/Term/In WFF}}"}
+{"_id": "28652", "title": "Definition:Substitution (Formal Systems)/Metasymbol", "text": "Let $S_1$ be a statement form. Let $p$ be a metasymbol which occurs one or more times in $S_1$. Let $T$ be a statement. Let $S_2$ be the string formed by replacing ''every'' occurrence of $p$ in $S_1$ with $T$. Then $S_2$ results from the '''substitution of $p$ by $T$ in $S_1$'''. $S_2$ is called a '''substitution instance''' of $S_1$."}
+{"_id": "28653", "title": "Definition:Right-Truncatable Prime", "text": "A '''right-truncatable prime''' is a prime number which remains prime when any number of digits are removed from the right hand end."}
+{"_id": "28656", "title": "Definition:Hilbert Proof System/Instance 2", "text": "<section notitle=\"true\" name=\"tc\"> This instance of a Hilbert proof system is used in: * {{BookReference|Introduction to Symbolic Logic|1959|A.H. Basson|author2 = D.J. O'Connor|ed = 3rd|edpage = Third Edition}} Let $\\mathcal L$ be the language of propositional logic. $\\mathscr H$ has the following axioms and rules of inference: === Axioms === Let $p, q, r$ be propositional variables. Then the following WFFs are axioms of $\\mathscr H$: {{begin-axiom}} {{axiom|n = A1        |lc = Rule of Idempotence        |m = (p \\lor p) \\implies p }} {{axiom|n = A2        |lc = Rule of Addition        |m = q \\implies (p \\lor q) }} {{axiom|n = A3        |lc = Rule of Commutation        |m = (p \\lor q) \\implies (q \\lor p) }} {{axiom|n = A4        |lc = Factor Principle        |m = (q \\implies r) \\implies \\left({ (p \\lor q) \\implies (p \\lor r)}\\right) }} {{end-axiom}} === Rules of Inference === ==== $RST \\, 1$: Rule of Uniform Substitution ==== {{refactor|This rule should have its own dedicated page like the others. But separate from the Rule of Substitution, which is too vague at this point (might be a subpage though). Also review source flow afterwards}} Any WFF $\\mathbf A$ may be substituted for any propositional variable $p$ in a $\\mathscr H_2$-theorem $\\mathbf B$. The resulting theorem can be denoted $\\mathbf B \\paren{ \\mathbf A \\mathbin{//} p }$. See the Rule of Substitution. ==== $RST \\, 2$: Rule of Substitution by Definition ==== The following expressions are regarded definitional abbreviations: {{begin-axiom}} {{axiom|n = 1        |lc = Conjunction        |ml = \\mathbf A \\land \\mathbf B        |mo = {{=}}_{\\text{def} }        |mr = \\neg \\left({ \\neg \\mathbf A \\lor \\neg \\mathbf B }\\right) }} {{axiom|n = 2        |lc = Conditional        |ml = \\mathbf A \\implies \\mathbf B        |mo = {{=}}_{\\text{def} }        |mr = \\neg \\mathbf A \\lor \\mathbf B }} {{axiom|n = 3        |lc = Biconditional        |ml = \\mathbf A \\iff \\mathbf B        |mo = {{=}}_{\\text{def} }        |mr = (\\mathbf A \\implies \\mathbf B) \\land (\\mathbf B \\implies \\mathbf A) }} {{end-axiom}} ==== $RST \\, 3$: Rule of Detachment ==== If $\\mathbf A \\implies \\mathbf B$ and $\\mathbf A$ are theorems of $\\mathscr H$, then so is $\\mathbf B$. That is, Modus Ponendo Ponens. ==== $RST \\, 4$: Rule of Adjunction ==== If $\\mathbf A$ and $\\mathbf B$ are theorems of $\\mathscr H$, then so is $\\mathbf A \\land \\mathbf B$. That is, the Rule of Conjunction. (This rule can be proved from the other three and so is only a convenience.) </section>"}
+{"_id": "28657", "title": "Definition:Hardy-Ramanujan Number", "text": "The $n$th '''Hardy-Ramanujan number''' $\\operatorname {Ta} \\left({n}\\right)$ is the smallest positive integer which can be expressed as the sum of $2$ cubes in $n$ different ways."}
+{"_id": "28658", "title": "Definition:Cube Number/Sequence", "text": "The sequence of positive cube numbers begins: :$1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, \\ldots$"}
+{"_id": "28662", "title": "Definition:Summation/Finite Support", "text": "Let $G$ be an abelian group. Let $S$ be a set. Let $f: S \\to G$ be a mapping. Let the support $\\operatorname{Supp} \\left({f}\\right)$ be finite. Let $g$ be the restriction of $f$ to $\\operatorname{Supp}f$. The '''summation of $f$ over $S$''', denoted $\\displaystyle \\sum_{s \\mathop \\in S} f \\left({s}\\right)$, is the summation over the finite set $\\operatorname{Supp} \\left({f}\\right)$ of $g$: :$\\displaystyle \\sum_{s \\mathop \\in S} f \\left({s}\\right) = \\sum_{s \\mathop \\in \\operatorname {Supp} \\left({f}\\right)} g \\left({s}\\right)$"}
+{"_id": "28663", "title": "Definition:Independent Proof System", "text": "Let $\\mathcal L$ be a logical language. Let $\\mathscr P$ be a proof system. Then $\\mathscr P$ is '''independent''' if it is not possible to derive one axiom or rule of inference of $\\mathscr P$ from the others. {{stub|Deliberately vague, as the context in which this definition was originally couched was equally vague. Needs to be made rigorous, or merged with pages building a more comprehensive treatment.}}"}
+{"_id": "28666", "title": "Definition:Sum of Finite Set", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $S \\subset \\mathbb A$ be a finite subset. The '''sum of $S$''' is the summation of the identity mapping on $S$ over $S$: :$\\displaystyle \\sum S = \\sum_{s \\mathop \\in S} s$ Category:Definitions/Summations 2qv4z6kzqhyodq18cepnvy97xmbqzvb"}
+{"_id": "28668", "title": "Definition:Unital Magma", "text": "A '''unital magma''' is a magma that has an identity element."}
+{"_id": "28669", "title": "Definition:Commutative Magma", "text": "A '''commutative magma''' is a magma whose operation is commutative."}
+{"_id": "28670", "title": "Definition:N-Ary Operation Induced by Binary Operation", "text": "Let $\\struct {G, \\oplus}$ be a magma. Let $n \\ge 1$ be a natural number. Let $G^n$ be the $n$th cartesian power of $G$. The '''$n$-ary operation induced by $\\oplus$''' is the $n$-ary operation $\\oplus_n: G^n \\to G$ defined as: :$\\map {\\oplus_n} f = \\displaystyle \\bigoplus_{i \\mathop = 1}^n \\map f i$ where $\\bigoplus$ denotes indexed iteration of $f$ from $1$ to $n$."}
+{"_id": "28671", "title": "Definition:Constant", "text": "A '''constant''' is a name for an object (usually a number, but the concept has wider applications) which ''does not change'' during the context of a logical or mathematical argument. A constant can be considered as an operator which takes no operands. A constant can also be considered as a variable whose domain is a singleton."}
+{"_id": "28672", "title": "Definition:Polynomial Ring/Indeterminate", "text": "==== Single indeterminate ==== Let $\\left({S, \\iota, X}\\right)$ be a polynomial ring over $R$. The '''indeterminate''' of $\\left({S, \\iota, X}\\right)$ is the term $X$. ==== Multiple Indeterminates ==== Let $I$ be a set. Let $\\left({S, \\iota, f}\\right)$ be a polynomial ring over $R$ in $I$ '''indeterminates'''. The '''indeterminates''' of $\\left({S, \\iota, f}\\right)$ are the elements of the image of the family $f$."}
+{"_id": "28673", "title": "Definition:Polynomial Ring/Universal Property", "text": "A '''polynomial ring over $R$''' is a pointed $R$-algebra $(S, \\iota, X)$ that satisfies the following universal property: :For every pointed $R$-algebra $(A, \\kappa, a)$ there exists a unique pointed algebra homomorphism $h : S\\to A$, called evaluation homomorphism. This is known as the '''universal property of a polynomial ring'''."}
+{"_id": "28675", "title": "Definition:Polynomial Algebra", "text": "Let $R$ be a ring. Let $\\Z \\sqbrk X$ be the polynomial ring over the ring of integers $\\Z$. The '''polynomial algebra in one indeterminate''' over $R$ is the tensor product $R \\otimes_\\Z \\Z \\sqbrk X$. {{expand}}"}
+{"_id": "28676", "title": "Definition:Unital Associative Commutative Algebra", "text": "Let $R$ be a commutative ring with unity. === Definition 1 === {{:Definition:Unital Associative Commutative Algebra/Definition 1}} === Definition 2 === {{:Definition:Unital Associative Commutative Algebra/Definition 2}} == Also known as == A '''unital associative commutative algebra''' over $R$ is also known as an '''$R$-algebra''', as is a general algebra over $R$. == Equivalence of Definitions == While, strictly speaking, the above definitions do define different objects, they are equivalent in the following sense: * An algebra $\\left({A, *}\\right)$ over $R$ that is unital, associative and commutative and whose underlying module is unitary, is identified with the ring under $R$ equal to its underlying ring together with its canonical mapping $R \\to A$. The algebra is '''viewed as a ring'''. * A ring under $R$, $(A, f)$, is identified with the algebra defined by $f$. The algebra is '''viewed as an algebra'''. For the detailed statements, see Equivalence of Definitions of Unital Associative Commutative Algebra."}
+{"_id": "28677", "title": "Definition:Unital Ring Homomorphism", "text": "Let $R$ and $S$ be rings with unities $1_R$ and $1_S$. A '''unital ring homomorphism''' is a ring homomorphism $f: R \\to S$ such that $\\map f {1_R} = 1_S$."}
+{"_id": "28678", "title": "Definition:Unital Associative Commutative Algebra/Definition 1", "text": "A '''unital associative commutative algebra''' over $R$ is an algebra $\\left({A, *}\\right)$ over $R$ that is unital, associative and commutative and whose underlying module is unitary."}
+{"_id": "28679", "title": "Definition:Unital Associative Commutative Algebra/Definition 2", "text": "A '''unital associative commutative algebra''' over $R$ is a ring under $A$, that is, an ordered pair $(A, f)$ where: :$A$ is a commutative ring with unity :$f : R \\to A$ is a unital ring homomorphism <!--whose image is contained in the center of $A$-->"}
+{"_id": "28680", "title": "Definition:Algebra Defined by Ring Homomorphism", "text": "Let $R$ be a commutative ring. Let $\\struct {S, +, *}$ be a ring. Let $f : R \\to S$ be a ring homomorphism. Let the image of $f$ be a subset of the center of $S$. Let $S_R$ be the module defined by the ring homomorphism $f$. The '''algebra defined by $f$''' is the algebra over $R$ defined as: :$\\struct {S_R, *}$"}
+{"_id": "28681", "title": "Definition:Underlying Ring of Associative Algebra", "text": "Let $R$ be a commutative ring. Let $(A, *)$ be an associative algebra over $R$. Let $A = (M, +, \\circ)$ be the underlying module of $(A, *)$. The '''underlying ring of $(A, *)$''' is the ring $(A, +, *)$."}
+{"_id": "28682", "title": "Definition:Pointed Algebra over Ring", "text": "Let $R$ be a commutative ring. A '''pointed algebra over $R$''' is an ordered pair $(A, a)$ where: :$A$ is an algebra over $R$ :$a$ is an element of $A$"}
+{"_id": "28683", "title": "Definition:Polynomial Ring/Embedding", "text": "Let the ordered triple $(S, \\iota, X)$ be a polynomial ring over $R$ in one indeterminate $X$. The unital ring homomorphism $\\iota$ is called the '''canonical embedding into the polynomial ring'''. === Multiple Indeterminates === Let $I$ be a set. Let $(S, \\iota, X)$ be a polynomial ring over $R$ in $I$ indeterminates. The unital ring homomorphism $\\iota$ is called the '''canonical embedding into the polynomial ring'''."}
+{"_id": "28684", "title": "Definition:Monoid Ring/Canonical Mapping", "text": "Let $e_1$ be the canonical basis element. The '''canonical mapping to $R \\left[{G}\\right]$''' is the mapping $R \\to R \\left[{G}\\right]$ which sends $r$ to $r * e_1$."}
+{"_id": "28685", "title": "Definition:Unital Subalgebra/Definition 1", "text": "The subalgebra $\\left({B_R, *}\\right)$ is a '''unital subalgebra''' of $A_R$ {{iff}} $1_A \\in B$."}
+{"_id": "28686", "title": "Definition:Unital Subalgebra/Definition 2", "text": "The subalgebra $\\left({B_R, *}\\right)$ is a '''unital subalgebra''' of $A_R$ {{iff}}: :$(1) \\quad B_R$ is unital :$(2) \\quad$Its unit is $1_A$. That is, a '''unital subalgebra''' of $A_R$ must not only have a unit, but that unit must also be the same unit as that of $A_R$."}
+{"_id": "28687", "title": "Definition:Field of Rational Fractions", "text": "Let $R$ be an integral domain."}
+{"_id": "28688", "title": "Definition:Continuous Mapping (Topology)/Point/Open Sets", "text": "The mapping $f$ is '''continuous at (the point) $x$''' (with respect to the topologies $\\tau_1$ and $\\tau_2$) {{iff}}: :For every neighborhood $N$ of $\\map f x$ in $T_2$, there exists a neighborhood $M$ of $x$ in $T_1$ such that $f \\sqbrk M \\subseteq N$."}
+{"_id": "28689", "title": "Definition:Continuous Mapping (Topology)/Point/Filters", "text": "The mapping $f$ is '''continuous at (the point) $x$''' {{iff}} for any filter $\\mathcal F$ on $T_1$ that converges to $x$, the corresponding image filter $f \\left({\\mathcal F}\\right)$ converges to $f \\left({x}\\right)$."}
+{"_id": "28690", "title": "Definition:Continuous Mapping (Topology)/Everywhere/Pointwise", "text": "The mapping $f$ is '''continuous everywhere''' (or simply '''continuous''') {{iff}} $f$ is continuous at every point $x \\in S_1$."}
+{"_id": "28691", "title": "Definition:Open Neighborhood/Topology", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $A \\subseteq S$ be a subset of $S$. Let $N_A$ be a neighborhood of $A$. If $N_A \\in \\tau$, that is, if $N_A$ is itself open in $T$, then $N_A$ is called an '''open neighborhood'''."}
+{"_id": "28692", "title": "Definition:Continuous Real Function/Everywhere", "text": "Let $f: \\R \\to \\R$ be a real function. Then $f$ is '''everywhere continuous''' {{iff}} $f$ is continuous at every point in $\\R$."}
+{"_id": "28693", "title": "Definition:Continuous Real Function/Closed Interval", "text": "Let $f$ be a real function defined on a closed interval $\\left[{a \\,.\\,.\\, b}\\right]$. ==== Definition 1 ==== {{Definition:Continuous Real Function/Closed Interval/Definition 1}} ==== Definition 2 ==== {{Definition:Continuous Real Function/Closed Interval/Definition 2}}"}
+{"_id": "28694", "title": "Definition:Continuous Real Function/Closed Interval/Definition 2", "text": "The function $f$ is '''continuous on $\\left[{a \\,.\\,.\\, b}\\right]$''' {{iff}} it is continuous at every point of $\\left[{a \\,.\\,.\\, b}\\right]$."}
+{"_id": "28695", "title": "Definition:Indexed Iterated Binary Operation", "text": "Let $\\struct {G, *}$ be a magma. Let $a, b \\in \\Z$ be integers. Let $\\closedint a b$ be the integer interval between $a$ and $b$. Let $f: \\closedint a b \\to G$ be a mapping. The '''indexed iteration of $*$ of $f$ from $a$ to $b$''' is recursively defined and denoted: :$\\ds \\prod_{k \\mathop = a}^b \\map f k = \\begin {cases} \\map f a & : b = a \\\\ \\paren {\\ds \\prod_{k \\mathop = a}^{b - 1} \\map f k} * \\map f b & : b > a \\end {cases}$ For each ordered $n$-tuple $\\tuple {a_1, a_2, \\ldots, a_n} \\in S^n$, the '''composite''' of $\\tuple {a_1, a_2, \\ldots, a_n}$ for $\\oplus$ is the value at $\\tuple {a_1, a_2, \\ldots, a_n}$ of the $n$-ary operation defined by $\\oplus$. This '''composite''' is recursively defined and denoted: {{begin-eqn}} {{eqn | l = \\bigoplus_{k \\mathop = 1}^n a_k       | r = \\map {\\oplus_n} {a_1, a_2, \\ldots, a_n}       | c =  }} {{eqn | r = \\begin {cases} a & : n = 1 \\\\ \\map {\\oplus_m} {a_1, \\ldots, a_m} \\oplus a_{m + 1} & : n = m + 1 \\end {cases}       | c =  }} {{eqn | r = \\paren {\\paren {\\cdots \\paren {\\paren {a_1 \\oplus a_2} \\oplus a_3} \\oplus \\cdots} \\oplus a_{n - 1} } \\oplus a_n       | c =  }} {{end-eqn}}"}
+{"_id": "28696", "title": "Definition:Iterated Binary Operation over Finite Set", "text": "Let $\\struct {G, *}$ be a commutative semigroup. Let $S$ be a finite non-empty set. Let $f: S \\to G$ be a mapping. Let $n \\in \\N$ be the cardinality of $S$. Let $g: \\N_{<n} \\to S$ be a bijection, where $\\N_{<n}$ is an initial segment of the natural numbers. The '''iteration of $*$ of $f$ over $S$''', denoted $\\displaystyle \\prod_{s \\mathop \\in S} \\map f s$, is the indexed iteration of $*$ of the composition $f \\circ g$ over $\\N_{<n}$: :$\\displaystyle \\prod_{s \\mathop \\in S} \\map f s = \\displaystyle \\prod_{i \\mathop = 0}^{n - 1} \\map f {\\map g i}$"}
+{"_id": "28697", "title": "Definition:Iterated Binary Operation over Set with Finite Support", "text": "Let $\\struct {G, *}$ be a commutative monoid. Let $S$ be a set. Let $f: S \\to G$ be a mapping. Let the support $\\operatorname {Supp} f$ be finite. The '''iteration of $*$ of $f$ over $S$''', denoted $\\displaystyle \\prod_{s \\mathop \\in S} \\map f s$, is the iteration over the finite set $\\operatorname{Supp} f$ of $f$: :$\\displaystyle \\prod_{s \\mathop \\in S} \\map f s = \\prod_{s \\mathop \\in \\operatorname {Supp} f} \\map f s$"}
+{"_id": "28698", "title": "Definition:Ring of Sequences of Finite Support", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $P \\sqbrk R$ be the set of all sequences in $R$ whose domain is $\\N$: :$P \\sqbrk R = \\set {\\sequence {r_0, r_1, r_2, \\ldots} }$ and whose support is finite. Let the operations $\\oplus$ and $\\odot$ on $P \\sqbrk R$ be defined as follows: {{:Definition:Operations on Polynomial Ring of Sequences}} The '''ring of sequences of finite support over $R$''' is the ring $\\struct {P \\sqbrk R, \\oplus, \\odot}$."}
+{"_id": "28699", "title": "Definition:Consistent (Logic)/Proof System", "text": "Then $\\mathscr P$ is '''consistent''' {{iff}}: :There exists a logical formula $\\phi$ such that $\\not \\vdash_{\\mathscr P} \\phi$ That is, some logical formula $\\phi$ is '''not''' a theorem of $\\mathscr P$."}
+{"_id": "28701", "title": "Definition:Consistent (Logic)/Proof System/Propositional Logic/Definition 1", "text": "Then $\\mathscr P$ is '''consistent''' {{iff}}: :There exists a logical formula $\\phi$ such that $\\not \\vdash_{\\mathscr P} \\phi$ That is, some logical formula $\\phi$ is '''not''' a theorem of $\\mathscr P$."}
+{"_id": "28702", "title": "Definition:Consistent (Logic)/Proof System/Propositional Logic/Definition 2", "text": "Suppose that in $\\mathscr P$, the Rule of Explosion (Variant 3) holds. Then $\\mathscr P$ is '''consistent''' {{iff}}: :For every logical formula $\\phi$, not ''both'' of $\\phi$ and $\\neg \\phi$ are theorems of $\\mathscr P$"}
+{"_id": "28703", "title": "Definition:Consistent (Logic)/Set of Formulas", "text": "Let $\\FF$ be a collection of logical formulas. Then $\\FF$ is '''consistent for $\\mathscr P$''' {{iff}}: :There exists a logical formula $\\phi$ such that $\\FF \\nvdash_{\\mathscr P} \\phi$. That is, some logical formula $\\phi$ is '''not''' a provable consequence of $\\FF$."}
+{"_id": "28704", "title": "Definition:Left-Truncatable Prime", "text": "A '''left-truncatable prime''' is a prime number which remains prime when any number of digits are removed from the left hand end. Zeroes are excluded, in order to eliminate, for example, prime numbers of the form $10^n + 3$ for arbitrarily large $n$."}
+{"_id": "28705", "title": "Definition:Irreducible Component", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. A subspace $Y \\subset T$ is an '''irreducible component of $T$''' {{iff}}: : $Y$ is irreducible : $Y$ is not a proper subset of an irreducible subspace of $T$. That is, {{iff}} $Y$ is maximal in the ordered set of irreducible subspaces of $T$, ordered by inclusion."}
+{"_id": "28706", "title": "Definition:Irreducible Space/Definition 1", "text": "A topological space $T = \\struct {S, \\tau}$ is '''irreducible''' {{iff}} there exists no cover of $T$ by two proper closed sets of $T$."}
+{"_id": "28707", "title": "Definition:Irreducible Space/Definition 2", "text": "A topological space $T = \\struct {S, \\tau}$ is '''irreducible''' {{iff}} there is no finite cover of $T$ by proper closed sets of $T$."}
+{"_id": "28708", "title": "Definition:Irreducible Space/Definition 3", "text": "A topological space $T = \\struct {S, \\tau}$ is '''irreducible''' {{iff}} every two non-empty open sets of $T$ have non-empty intersection: :$\\forall U, V \\in \\tau: U, V \\ne \\O \\implies U \\cap V \\ne \\O$"}
+{"_id": "28709", "title": "Definition:Irreducible Space/Definition 4", "text": "A topological space $T = \\struct {S, \\tau}$ is '''irreducible''' {{iff}} every non-empty open set of $T$ is (everywhere) dense in $T$."}
+{"_id": "28710", "title": "Definition:Irreducible Space/Definition 5", "text": "A topological space $T = \\struct {S, \\tau}$ is '''irreducible''' {{iff}} the interior of every proper closed set of $T$ is empty."}
+{"_id": "28711", "title": "Definition:Krull Dimension", "text": "=== Krull Dimension of a Ring === {{Definition:Krull Dimension of Ring}} === Krull Dimension of a Topological Space === {{Definition:Krull Dimension of Topological Space}} {{NamedforDef|Wolfgang Krull|cat = Krull}} Category:Definitions/Algebraic Geometry t98gwppos4spvq3h1086xm9c80ofd9a"}
+{"_id": "28712", "title": "Definition:Length of Chain", "text": "Let $T$ be a chain in $S$. Let $T$ be finite and non-empty. The '''length''' of the chain $T$ is its cardinality minus $1$."}
+{"_id": "28714", "title": "Definition:Krull Dimension of Topological Space", "text": "Let $T$ be a topological space. Its '''Krull dimension''' $\\map {\\dim_{\\mathrm {Krull} } } T$ is the supremum of lengths of chains of closed irreducible sets of $T$, ordered by inclusion. Thus, the '''Krull dimension''' is $\\infty$ if there exist arbitrarily long chains."}
+{"_id": "28716", "title": "Definition:Noetherian Topological Space", "text": "=== Definition 1 === {{:Definition:Noetherian Topological Space/Definition 1}} === Definition 2 === {{:Definition:Noetherian Topological Space/Definition 2}}"}
+{"_id": "28717", "title": "Definition:Set Ordered by Inclusion", "text": "Let $S$ be a set. Then '''$S$ ordered by inclusion''' is the ordered set $\\struct {S, \\subseteq}$, where $\\subseteq$ is the inclusion relation on $S$."}
+{"_id": "28719", "title": "Definition:Total Ring of Fractions", "text": "Let $A$ be a commutative ring with unity. The '''total ring of fractions''' of $A$ is the localization of $A$ at the set of its regular elements."}
+{"_id": "28720", "title": "Definition:Regular Element of Ring", "text": "Let $A$ be a commutative ring with unity. Let $a\\in A$. Then $a$ is '''regular''' {{iff}} it is not a zero divisor."}
+{"_id": "28722", "title": "Definition:Monomial of Polynomial Ring", "text": "=== Multiple Variables === {{Definition:Monomial of Polynomial Ring/Multiple Variables}}"}
+{"_id": "28723", "title": "Definition:Pointed Set", "text": "A '''pointed set''' is an ordered pair $\\struct {A, a}$ where: :$A$ is a set :$a \\in A$ is an element."}
+{"_id": "28724", "title": "Definition:Pointed Mapping", "text": "Let $\\struct {A, a}$ and $\\struct {B, b}$ be pointed sets. A '''pointed mapping''' $f: \\struct {A, a} \\to \\struct {B, b}$ is a mapping $f : A \\to B$ such that $\\map f a = b$."}
+{"_id": "28726", "title": "Definition:Multiple Pointed Set", "text": "Let $I$ be a set. An '''$I$-pointed set''' is an ordered pair $\\left({A, f}\\right)$ where: :$A$ is a set :$f : I \\to A$ is a mapping"}
+{"_id": "28727", "title": "Definition:Multiple Pointed Algebra", "text": "Let $R$ be a commutative ring with unity. Let $I$ be a set. An '''$I$-pointed algebra over $R$''' is an ordered pair $(A, f)$ where: :$A$ is an algebra over $R$ :$f : I \\to A$ is a mapping"}
+{"_id": "28728", "title": "Definition:Square Pyramorphic Number", "text": "A '''square pyramorphic number''' is a square pyramidal number $P_n$ whose decimal representation ends in $n$."}
+{"_id": "28729", "title": "Definition:Square Pyramorphic Number/Sequence", "text": "The sequence of square pyramorphic numbers, for $n \\in \\Z_{\\ge 0}$, begins: {{begin-eqn}} {{eqn | l = P_1       | r = 1 }} {{eqn | l = P_5       | r = 55 }} {{eqn | l = P_{25}       | r = 5525 }} {{eqn | l = P_{40}       | r = 22 \\, 140 }} {{eqn | l = P_{65}       | r = 93 \\, 665 }} {{eqn | l = P_{80}       | r = 1 \\, 043 \\, 280 }} {{end-eqn}}"}
+{"_id": "28731", "title": "Definition:Additive Arithmetic Function", "text": "Let $f : \\N \\to \\C$ be an arithmetic function. Then $f$ is '''additive''' {{iff}}: :$m \\perp n \\implies \\map f {m n} = \\map f m + \\map f n$"}
+{"_id": "28733", "title": "Definition:Multiplicative Function on UFD", "text": "Let $R$ be a unique factorization domain. Let $f : R \\to \\C$ be a complex-valued function. Then $f$ is '''multiplicative''' {{iff}}: :For all coprime $x, y\\in R$: $f \\left({x y}\\right) = f \\left({x}\\right) f \\left({y}\\right)$"}
+{"_id": "28734", "title": "Definition:Constructed Semantics", "text": "Let $\\mathcal L$ be a formal language. A '''constructed semantics''' for $\\mathcal L$ is a formal semantics which is invented ''solely'' for proving a property about $\\mathcal L$ or other entities related to $\\mathcal L$."}
+{"_id": "28735", "title": "Definition:Additive Function on UFD", "text": "Let $R$ be a unique factorization domain. Let $f : R \\to \\C$ be a complex-valued function. Then $f$ is '''additive''' {{iff}}: :For all coprime $x,y\\in R$, $f(xy) = f(x) + f(y)$"}
+{"_id": "28736", "title": "Definition:Multiplicative Arithmetic Function", "text": "Let $f : \\N \\to \\C$ be an arithmetic function. Then $f$ is '''multiplicative''' {{iff}}: :$m \\perp n \\implies \\map f {m n} = \\map f m \\map f n$ where $m \\perp n$ denotes that $m$ is coprime to $n$."}
+{"_id": "28737", "title": "Definition:Constructed Semantics/Instance 1", "text": "Let $\\mathcal L_0$ be the language of propositional logic. The constructed semantics $\\mathscr C_1$ for $\\mathcal L_0$ is used for the following results: * Hilbert Proof System Instance 2 is Consistent === Structures === Define the structures of $\\mathscr C_1$ as mappings $v$ by the Principle of Recursive Definition, as follows. Let $\\mathcal P_0$ be the vocabulary of $\\mathcal L_0$. Let a mapping $v: \\mathcal P_0 \\to \\{ 1, 2 \\}$ be given. Next, regard the following as definitional abbreviations: {{begin-axiom}} {{axiom | n  = 1         | lc = Conjunction         | ml = \\mathbf A \\land \\mathbf B         | mo = {{=}}_{\\text{def} }         | mr = \\neg \\left({ \\neg \\mathbf A \\lor \\neg \\mathbf B }\\right) }} {{axiom | n  = 2         | lc = Conditional         | ml = \\mathbf A \\implies \\mathbf B         | mo = {{=}}_{\\text{def} }         | mr = \\neg \\mathbf A \\lor \\mathbf B }} {{axiom | n  = 3         | lc = Biconditional         | ml = \\mathbf A \\iff \\mathbf B         | mo = {{=}}_{\\text{def} }         | mr = \\left({\\mathbf A \\implies \\mathbf B}\\right) \\land \\left({\\mathbf B \\implies \\mathbf A}\\right) }} {{end-axiom}} It only remains to define $v \\left({ \\neg \\phi }\\right)$ and $v \\left({ \\phi \\lor \\psi}\\right)$ recursively, by: {{begin-eqn}} {{eqn | l = v \\left({\\neg \\phi }\\right)       | o = :=       | r = \\begin{cases} 1 & : \\text{if $v \\left({\\phi}\\right) = 2$} \\\\ 2 & : \\text{if $v \\left({\\phi}\\right) = 1$}\\end{cases} }} {{eqn | l = v \\left({ \\phi \\lor \\psi }\\right)       | o = :=       | r = \\begin{cases} 1 & : \\text{if $v \\left({\\phi}\\right) = v \\left({\\psi}\\right) = 1$} \\\\ 2 & : \\text{otherwise}\\end{cases} }} {{end-eqn}} === Validity === Define validity in $\\mathscr C_1$ by declaring: :$\\models_{\\mathscr C_1} \\phi$ {{iff}} $v \\left({\\phi}\\right) = 2$"}
+{"_id": "28740", "title": "Definition:Floor Function/Definition 2", "text": "The '''floor function of $x$''', denoted $\\floor x$, is defined as the greatest element of the set of integers: :$\\set {m \\in \\Z: m \\le x}$ where $\\le$ is the usual ordering on the real numbers."}
+{"_id": "28741", "title": "Definition:Ceiling Function/Definition 1", "text": "The '''ceiling function of $x$''' is defined as the infimum of the set of integers no smaller than $x$: :$\\ceiling x := \\inf \\set {m \\in \\Z: x \\le m}$ where $\\le$ is the usual ordering on the real numbers."}
+{"_id": "28742", "title": "Definition:Ceiling Function/Definition 2", "text": "The '''ceiling function of $x$''', denoted $\\ceiling x$, is defined as the smallest element of the set of integers: :$\\set {m \\in \\Z: x \\le m}$ where $\\le$ is the usual ordering on the real numbers."}
+{"_id": "28743", "title": "Definition:Ceiling Function/Definition 3", "text": "The '''ceiling function of $x$''' is the unique integer $\\ceiling x$ such that: :$\\ceiling x - 1 < x \\le \\ceiling x$"}
+{"_id": "28745", "title": "Definition:Functor Category", "text": "Let $C$ and $D$ be categories. The '''functor category''' $\\operatorname{Funct}(C, D)$ is the category with: {{DefineCategory |ob = covariant functors $C \\to D$ |mor = natural transformations |comp = vertical composition of natural transformations |id = identity natural transformations }} {{definition wanted|the contravariant case}}"}
+{"_id": "28747", "title": "Definition:Generated Subgroup", "text": "Let $G$ be a group. Let $S \\subset G$ be a subset. === Definition 1 === {{Definition:Generated Subgroup/Definition 1}} === Definition 2 === {{Definition:Generated Subgroup/Definition 2}} === Definition 3 === {{Definition:Generated Subgroup/Definition 3}}"}
+{"_id": "28748", "title": "Definition:Generated Subgroup/Definition 1", "text": "The '''subgroup generated by $S$''' is the smallest subgroup containing $S$."}
+{"_id": "28749", "title": "Definition:Generated Subgroup/Definition 3", "text": "Let $S^{-1}$ be the set of inverses of $S$. The '''subgroup generated by $S$''' is the set of words on the union $S \\cup S^{-1}$."}
+{"_id": "28750", "title": "Definition:Generator of Ring", "text": "Let $R$ be a ring. Let $S \\subset R$ be a subset. Then $S$ is a '''generator''' of $R$ {{iff}} $R$ is the subring generated by $S$."}
+{"_id": "28751", "title": "Definition:Generator of Ring Extension", "text": "Let $f : A \\to B$ be a ring extension of commutative rings with unity. Let $S \\subset B$ be a subset. Then $S$ is a '''generator of $B$ over $A$''' {{iff}} $B$ is the ring extension generated by $S$."}
+{"_id": "28753", "title": "Definition:Generated Normal Subgroup", "text": "Let $G$ be a group. Let $S \\subset G$ be a subset. === Definition 1 === {{Definition:Generated Normal Subgroup/Definition 1}} === Definition 2 === {{Definition:Generated Normal Subgroup/Definition 2}} === Definition 3 === {{Definition:Generated Normal Subgroup/Definition 3}}"}
+{"_id": "28754", "title": "Definition:Generated Normal Subgroup/Definition 1", "text": "The '''normal subgroup generated by $S$''', denoted $\\gen {S^G}$,  is the intersection of all normal subgroups of $G$ containing $S$."}
+{"_id": "28755", "title": "Definition:Generated Normal Subgroup/Definition 2", "text": "The '''normal subgroup generated by $S$''', denoted $\\gen {S^G}$, is the subgroup generated by the set of conjugates of $S$: :$S^G = \\set {g^{−1}sg: g \\in G, s \\in S}$"}
+{"_id": "28757", "title": "Definition:Generator of Ideal of Ring", "text": "Let $R$ be a commutative ring.  Let $I \\subset R$ be an ideal. Let $S \\subset I$ be a subset. Then $S$ is a '''generator''' of $I$ {{iff}} $I$ is the ideal generated by $S$."}
+{"_id": "28772", "title": "Definition:Riemann Zeta Function/Zero/Nontrivial", "text": "The '''nontrivial zeroes''' of the Riemann zeta function $\\zeta$ are the zeroes of $\\zeta$ that are not trivial."}
+{"_id": "28773", "title": "Definition:Formal Derivative of Dirichlet Series", "text": "Let $(a_n) : \\N \\to \\C$ be an arithmetic function. Let $F(s)=\\displaystyle \\sum_{n\\mathop=1}^\\infty\\frac{a_n}{n^s}$ be the corresponding formal Dirichlet series. The '''formal derivative''' of $F(s)$ is the formal Dirichlet series: :$\\displaystyle \\sum_{n\\mathop=1}^\\infty\\frac{a_n\\cdot \\log n}{n^s}$"}
+{"_id": "28774", "title": "Definition:Abscissa of Convergence", "text": "Let $f \\left({s}\\right)$ be a Dirichlet series The '''abscissa of convergence''' of $f$ is the extended real number $\\sigma_0 \\in \\overline \\R$ defined by: :$\\sigma_0 = \\displaystyle \\inf \\left\\{ {\\operatorname{Re} \\left({s}\\right) : s \\in \\C, f \\left({s}\\right) \\text{converges} }\\right\\}$ where $\\inf \\varnothing = +\\infty$."}
+{"_id": "28775", "title": "Definition:Dirichlet Inverse of Arithmetic Function", "text": "Let $f : \\N \\to \\C$ be an arithmetic function. Let $\\varepsilon$ be the identity arithmetic function. A '''Dirichlet inverse''' of $f$ is an arithmetic function $g$ such that $g*f = \\varepsilon$, where $*$ denotes Dirichlet convolution. That is, an inverse of $f$ in the ring of arithmetic functions."}
+{"_id": "28776", "title": "Definition:Formal Product of Dirichlet Series", "text": "Let $f,g : \\N \\to \\C$ be arithmetic functions. Let $F,G$ be their formal Dirichlet series. The '''formal product''' of $F$ and $G$ is the formal Dirichlet series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\frac{(f*g)(n)}{n^s}$ where $*$ denotes Dirichlet convolution."}
+{"_id": "28777", "title": "Definition:Ring of Arithmetic Functions", "text": "Let $\\AA$ be the set of all arithmetic functions. Let $*$ denote Dirichlet convolution, and $+$ the pointwise sum of functions. The '''ring of arithmetic functions''' is the ring $\\struct {\\AA, +, *}$."}
+{"_id": "28779", "title": "Definition:Consistent (Logic)/Set of Formulas/Propositional Logic/Definition 1", "text": "Then $\\FF$ is '''consistent for $\\mathscr P$''' {{iff}}: :There exists a logical formula $\\phi$ such that $\\FF \\not \\vdash_{\\mathscr P} \\phi$ That is, some logical formula $\\phi$ is '''not''' a $\\mathscr P$-provable consequence of $\\FF$."}
+{"_id": "28780", "title": "Definition:Consistent (Logic)/Set of Formulas/Propositional Logic/Definition 2", "text": "Suppose that in $\\mathscr P$, the Rule of Explosion (Variant 3) holds. Then $\\FF$ is '''consistent for $\\mathscr P$''' {{iff}}: :For every logical formula $\\phi$, not ''both'' of $\\phi$ and $\\neg \\phi$ are $\\mathscr P$-provable consequences of $\\FF$"}
+{"_id": "28782", "title": "Definition:Pole of Complex Function", "text": "Let $U \\subset \\C$ be an open subset. Let $f : U \\to \\C$ be a holomorphic function. Let $p\\in \\C$ be an isolated singularity of $f$. === Definition 1 === The point $p$ is a '''pole''' of $f$ {{iff}} the Laurent expansion of $f$ around $p$ has the form: :$f(z) = \\displaystyle \\sum_{k = -n}^\\infty a_k(z-p)^k$ === Definition 2 === The point $p$ is a '''pole''' of $f$ {{iff}} there exists a natural number $m>0$ such that: :$\\displaystyle\\lim_{z \\to p}(z-p)^m f(z) \\in \\C\\setminus\\{0\\}$ === Definition 3 === The point $p$ is a '''pole''' of $f$ {{iff}} the improper limit: :$\\displaystyle\\lim_{z\\to p}|f(z)| = \\infty$."}
+{"_id": "28783", "title": "Definition:Complex Disk", "text": "Let $a \\in \\C$ be a complex number. Let $R>0$ be a real number. === Open disk === {{Definition:Complex Disk/Open}} === Closed disk === {{Definition:Complex Disk/Closed}}"}
+{"_id": "28784", "title": "Definition:Complex Disk/Open", "text": "The '''open (complex) disk of center $a$ and radius $R$''' is the set: :$\\map B {a, R} = \\set {z \\in \\C: \\cmod {z - a} < R}$ where $\\cmod {\\, \\cdot \\,}$ denotes complex modulus."}
+{"_id": "28785", "title": "Definition:Complex Disk/Closed", "text": "The '''closed (complex) disk of center $a$ and radius $R$''' is the set: :$\\map B {a, R} = \\set {z \\in \\C: \\cmod {z - a} \\le R}$ where $\\cmod {\\, \\cdot \\,}$ denotes complex modulus."}
+{"_id": "28786", "title": "Definition:Unit Ball", "text": "Let $V$ be a normed vector space with norm $\\norm {\\, \\cdot \\,}$. The '''closed unit ball''' of $V$, denoted $\\operatorname {ball} V$, is the set: :$\\set {v \\in V: \\norm v_V \\mathop \\le 1}$"}
+{"_id": "28787", "title": "Definition:Design", "text": "A '''design''' is an ordered pair $\\tuple {X, \\mathcal B}$ where: :$X$ is a non-empty finite set of points :$\\mathcal B$ is a multiset of blocks. Category:Definitions/Design Theory 37g472ap1lcm782m0j6gjg3gk9wulrl"}
+{"_id": "28788", "title": "Definition:Point (Design Theory)", "text": "Let $X$ be non-empty finite set. If $X$ is to be analyzed in the context of Design Theory, then its elements are referred to as '''points'''. In application, the '''points''' in $X$ are often variables. Category:Definitions/Design Theory 9d5dhk8qpe9vg081c8xwfxi2o0ravo0"}
+{"_id": "28789", "title": "Definition:Block", "text": "Let $X$ be a set of points. Let $\\mathcal B$ be a multiset of subsets of $X$. Then the elements of $\\mathcal B$ are called '''blocks'''. Category:Definitions/Design Theory 1p8idw86cxu109u4kfn62pj0gl2shmq"}
+{"_id": "28791", "title": "Definition:Real Interval/Definition 2", "text": "A '''real interval''' is a subset of $\\R$ that is one of the following real interval types: * closed bounded interval * open bounded interval * left half-open bounded interval * right half-open bounded interval * closed and bounded on the right, also known as a closed unbounded below real interval * open and bounded on the right, also known as an open unbounded below real interval * closed and bounded on the left, also known as a closed unbounded above real interval * open and bounded on the left, also known as an open unbounded above real interval * unbounded interval without endpoints"}
+{"_id": "28792", "title": "Definition:Real Interval Types", "text": "It is usual to define intervals in terms of inequalities. These are in the form of a pair of brackets, either round or square, enclosing the two endpoints of the interval separated by two dots. Whether the bracket at either end is round or square depends on whether the end point is '''inside''' or '''outside''' the interval, as specified in the following. Let $a, b \\in \\R$ be real numbers. === Bounded Intervals === ==== Open Interval ==== {{:Definition:Real Interval/Open}} ==== Half-Open Interval ==== {{:Definition:Real Interval/Half-Open}} ==== Closed Interval ==== {{:Definition:Real Interval/Closed}} ==== Bounded Interval ==== {{:Definition:Real Interval/Bounded}} === Unbounded Intervals === {{:Definition:Real Interval/Unbounded}} === Other Intervals === ==== Empty Interval ==== {{:Definition:Real Interval/Empty}} ==== Singleton Interval ==== {{:Definition:Real Interval/Singleton}} ==== Unit Interval ==== {{:Definition:Real Interval/Unit Interval}}"}
+{"_id": "28794", "title": "Definition:Real Interval/Half-Open/Right", "text": "The '''right half-open (real) interval''' from $a$ to $b$ is the subset: :$\\hointr a b := \\set {x \\in \\R: a \\le x < b}$"}
+{"_id": "28795", "title": "Definition:Real Interval/Half-Open/Left", "text": "The '''left half-open (real) interval''' from $a$ to $b$ is the subset: :$\\hointl a b := \\set {x \\in \\R: a < x \\le b}$"}
+{"_id": "28796", "title": "Definition:Compact Real Interval", "text": "A '''compact real interval''' is a closed interval of the form: :$[a\\,.\\,.\\,b]$ with $a,b\\in \\R$."}
+{"_id": "28798", "title": "Definition:Removable Discontinuity of Real Function", "text": "Let $A \\subseteq \\R$ be a subset of the real numbers. Let $f: A \\to \\R$ be a real function. Let $f$ be discontinuous at $a\\in A$. === Definition 1  === {{Definition:Removable Discontinuity of Real Function/Definition 1}} === Definition 2  === {{Definition:Removable Discontinuity of Real Function/Definition 2}}"}
+{"_id": "28799", "title": "Definition:Removable Discontinuity of Real Function/Definition 1", "text": "The point $a$ is a '''removable discontinuity''' of $f$ {{iff}} the limit $\\displaystyle \\lim_{x \\mathop \\to a} \\map f x$ exists."}
+{"_id": "28800", "title": "Definition:Removable Discontinuity of Real Function/Definition 2", "text": "The point $a$ is a '''removable discontinuity''' of $f$ {{iff}} there exists $b \\in \\R$ such that the function $f_b$ defined by: :$\\map {f_b} x = \\begin {cases} \\map f x &: x \\ne a \\\\ b &: x = a \\end {cases}$ is continuous at $a$."}
+{"_id": "28801", "title": "Definition:Discontinuous Mapping/Real Function/Point", "text": "Let $A \\subseteq \\R$ be a subset of the real numbers. Let $f : A \\to \\R$ be a real function. Let $a\\in A$. Then $f$ is '''discontinuous''' at $a$ {{iff}} $f$ is not continuous at $a$."}
+{"_id": "28802", "title": "Definition:Discontinuous Mapping/Real Function", "text": "=== At a Point === {{Definition:Discontinuous Mapping/Real Function/Point}}"}
+{"_id": "28803", "title": "Definition:Discontinuous Mapping/Topological Space/Point", "text": "Let $T_1 = \\left({A_1, \\tau_1}\\right)$ and $T_2 = \\left({A_2, \\tau_2}\\right)$ be topological spaces. Let $f: A_1 \\to A_2$ $x \\in T_1$ be a mapping from $A_1$ to $A_2$. Then by definition $f$ is continuous at $x$ if for every neighborhood $N$ of $f \\left({x}\\right)$ there exists a neighborhood $M$ of $x$ such that $f \\left({M}\\right) \\subseteq N$. Therefore, $f$ is discontinuous at $x$ if for some neighbourhood $N$ of $f \\left({x}\\right)$ and every neighbourhood $M$ of $x$, $f \\left({M}\\right) \\nsubseteq N$. The point $x$ is called a '''discontinuity of $f$'''."}
+{"_id": "28805", "title": "Definition:Polynomial Function/Real/Definition 1", "text": "A '''real polynomial function''' on $S$ is a function $f: S \\to \\R$ for which there exist: :a natural number $n\\in \\N$ :real numbers $a_0, \\ldots, a_n \\in \\R$ such that for all $x \\in S$: :$\\map f x = \\displaystyle \\sum_{k \\mathop = 0}^n a_k x^k$ where $\\sum$ denotes indexed summation."}
+{"_id": "28806", "title": "Definition:Polynomial Function/Complex/Definition 1", "text": "A '''complex polynomial function''' on $S$ is a function $f : S \\to \\C$ for which there exist: * a natural number $n\\in \\N$ * complex numbers $a_0, \\ldots, a_n \\in \\C$ such that for all $z \\in S$: :$\\map f z = \\displaystyle \\sum_{k \\mathop = 0}^n a_k z^k$ where $\\displaystyle \\sum$ denotes indexed summation."}
+{"_id": "28807", "title": "Definition:Polynomial Function/Complex/Definition 2", "text": "Let $\\C \\sqbrk X$ be the polynomial ring in one variable over $\\C$. Let $\\C^S$ be the ring of mappings from $S$ to $\\C$. Let $\\iota \\in \\C^S$ denote the inclusion $S \\hookrightarrow \\C$. A '''complex polynomial function''' on $S$ is a function $f: S \\to \\C$ which is in the image of the evaluation homomorphism $\\C \\sqbrk X \\to \\C^S$ at $\\iota$."}
+{"_id": "28808", "title": "Definition:Polynomial Function/Real/Definition 2", "text": "Let $\\R \\sqbrk X$ be the polynomial ring in one variable over $\\R$. Let $\\R^S$ be the ring of mappings from $S$ to $\\R$. Let $\\iota \\in \\R^S$ denote the inclusion $S \\hookrightarrow \\R$. A '''real polynomial function''' on $S$ is a function $f: S \\to \\R$ which is in the image of the evaluation homomorphism $\\R \\sqbrk X \\to \\R^S$ at $\\iota$."}
+{"_id": "28809", "title": "Definition:Polynomial Function/Ring", "text": "Let $R$ be a commutative ring with unity. Let $S \\subset R$ be a subset. === Definition 1 === {{Definition:Polynomial Function/Ring/Definition 1}} === Definition 2 === {{Definition:Polynomial Function/Ring/Definition 2}}"}
+{"_id": "28810", "title": "Definition:Polynomial Function/Ring/Definition 1", "text": "A '''polynomial function''' on $S$ is a mapping $f : S \\to R$ for which there exist: : a natural number $n \\in \\N$ : $a_0, \\ldots, a_n \\in R$ such that for all $x\\in S$: :$\\map f x = \\displaystyle \\sum_{k \\mathop = 0}^n a_k x^k$ where $\\sum$ denotes indexed summation."}
+{"_id": "28811", "title": "Definition:Polynomial Function/Ring/Definition 2", "text": "Let $R \\sqbrk X$ be the polynomial ring in one variable over $R$. Let $R^S$ be the ring of mappings from $S$ to $R$. Let $\\iota \\in R^S$ denote the inclusion $S \\hookrightarrow R$. A '''polynomial function''' on $S$ is a mapping $f : S \\to R$ which is in the image of the evaluation homomorphism $R \\sqbrk X \\to R^S$ at $\\iota$."}
+{"_id": "28812", "title": "Definition:Polynomial in Ring Element/Definition 1", "text": "A '''polynomial in $x$ over $S$''' is an element $y \\in R$ for which there exist: :a natural number $n \\in \\N$ :$a_0, \\ldots, a_n \\in S$ such that: :$y = \\displaystyle \\sum_{k \\mathop = 0}^n a_k x^k$ where: :$\\displaystyle \\sum$ denotes indexed summation :$x^k$ denotes the $k$th power of $x$"}
+{"_id": "28813", "title": "Definition:Polynomial in Ring Element/Definition 2", "text": "Let $S[X]$ be the polynomial ring in one variable over $S$. A '''polynomial in $x$ over $S$''' is an element that is in the image of the evaluation homomorphism $S[X]\\to R$ at $x$."}
+{"_id": "28815", "title": "Definition:Monomial of Polynomial Ring/One Variable", "text": "Let $R \\sqbrk X$ be a polynomial ring over $R$ in one indeterminate $X$. A '''monomial''' of $R \\sqbrk X$ is an element that is a power of $X$."}
+{"_id": "28816", "title": "Definition:Monomial of Polynomial Ring/Multiple Variables", "text": "Let $I$ be a set. Let $R \\sqbrk {\\family {X_i}_{i \\mathop \\in I} }$ be a polynomial ring in $I$ variables $\\family {X_i}_{i \\mathop \\in I}$. Let $y \\in R \\sqbrk {\\family {X_i}_{i \\mathop \\in I} }$. A '''monomial''' of $R \\sqbrk {\\family {X_i}_{i \\mathop \\in I} }$ is an element that is a product of variables; specifically: === Definition 1 === {{Definition:Monomial of Polynomial Ring/Multiple Variables/Definition 1}} === Definition 2 === {{Definition:Monomial of Polynomial Ring/Multiple Variables/Definition 2}}"}
+{"_id": "28817", "title": "Definition:Discontinuous Mapping/Topological Space", "text": "=== At a Point === {{Definition:Discontinuous Mapping/Topological Space/Point}}"}
+{"_id": "28818", "title": "Definition:Strong Fibonacci Pseudoprime", "text": "A '''strong Fibonacci pseudoprime''' is a Carmichael number which also satisfies one of the following conditions: === Type I === {{:Definition:Strong Fibonacci Pseudoprime/Type I}} === Type II === {{:Definition:Strong Fibonacci Pseudoprime/Type II}}"}
+{"_id": "28819", "title": "Definition:Strong Fibonacci Pseudoprime/Type I", "text": "A '''strong Fibonacci pseudoprime of type I''' is a Carmichael number $N = \\displaystyle \\prod p_i$ such that an even number of the prime factors $p_i$ are of the form $4 m - 1$ where: {{begin-eqn}} {{eqn | n = 1       | l = 2 \\paren {p_i + 1}       | o = \\divides       | r = \\paren {N - 1}       | c = for those $p_i$ of the form $4 m - 1$ }} {{eqn | n = 2       | l = \\paren {p_i + 1}       | o = \\divides       | r = \\paren {N \\pm 1}       | c = for those $p_i$ of the form $4 m + 1$ }} {{end-eqn}}"}
+{"_id": "28820", "title": "Definition:Strong Fibonacci Pseudoprime/Type II", "text": "A '''strong Fibonacci pseudoprime of type II''' is a Carmichael number $N = \\displaystyle \\prod p_i$ such that an odd number of the prime factors $p_i$ are of the form $4 m - 1$ where: :$2 \\paren {p_i + 1} \\divides \\paren {N - p_i}$ for all $p_i$"}
+{"_id": "28821", "title": "Definition:Cunningham Chain/First Kind", "text": "A '''Cunningham chain of the first kind''' is a (finite) sequence $\\left({p_1, p_2, \\ldots, p_n}\\right)$ such that: :$(1): \\quad \\forall i \\in \\left\\{ {1, 2, \\ldots, n - 1}\\right\\}: p_{i + 1} = 2 p_i + 1$ :$(2): \\quad p_i$ is prime for all $i \\in \\left\\{ {1, 2, \\ldots, n - 1}\\right\\}$ :$(3): \\quad n$ is not prime such that $2 n + 1 = p_1$ :$(4): \\quad 2 p_n + 1$ is not prime. Thus: : each term except the last is a Sophie Germain prime : each term except the first is a safe prime."}
+{"_id": "28824", "title": "Definition:Aurifeuillian Factorization", "text": "An '''Aurifeuillian factorization''' is an operation to find the prime factors of integers of various forms based upon the identity: :$a^2 + b^2 = \\left({a + \\sqrt {2a b} + b}\\right) \\left({a - \\sqrt{2ab} + b}\\right)$"}
+{"_id": "28828", "title": "Definition:Polydivisible Number", "text": "=== Definition 1 === {{:Definition:Polydivisible Number/Definition 1}} === Definition 2 === {{:Definition:Polydivisible Number/Definition 2}} === Sequence of Polydivisible Numbers === {{:Definition:Polydivisible Number/Sequence}}"}
+{"_id": "28835", "title": "Definition:Fischer-Griess Monster", "text": "The '''Fischer-Griess Monster''' is the automorphism group of the Griess algebra."}
+{"_id": "28838", "title": "Definition:Vigintillion", "text": "=== Short Scale === {{:Definition:Vigintillion/Short Scale}} === Long Scale === {{:Definition:Vigintillion/Long Scale}}"}
+{"_id": "28840", "title": "Definition:Number-Naming System", "text": "There are various '''number-naming systems''' for naming large numbers (that is: greater than $1 \\, 000 \\, 000$). === Short Scale === {{:Definition:Number-Naming System/Short Scale}} === Long Scale === {{:Definition:Number-Naming System/Long Scale}}"}
+{"_id": "28841", "title": "Definition:Number-Naming System/Short Scale", "text": "The '''short scale system''' is the number-naming system which uses: :the word '''million''' for $10^6 = 1 \\, 000 \\, 000$ :the Latin-derived prefixes '''bi-''', '''tri-''', '''quadri-''', '''quint-''', etc. for each further multiple of $1 \\, 000$, appended to the root '''-(i)llion''', corresponding to the indices $2$, $3$, $4$, $5$, $\\ldots$ Thus: {{begin-axiom}} {{axiom | lc= one billion:         | m = = 1 \\, 000 \\, 000 \\, 000         | mm= = 10^9 = 10^{2 \\times 3 + 3} }} {{axiom | lc= one trillion         | m = = 1 \\, 000 \\, 000 \\, 000 \\, 000         | mm= = 10^{12} = 10^{3 \\times 3 + 3} }} {{axiom | lc= one quadrillion         | m = = 1 \\, 000 \\, 000 \\, 000 \\, 000 \\, 000         | mm= = 10^{15} = 10^{4 \\times 3 + 3} }} {{axiom | lc= one quintillion         | m = = 1 \\, 000 \\, 000 \\, 000 \\, 000 \\, 000 \\, 000         | mm= = 10^{18} = 10^{5 \\times 3 + 3} }} {{end-axiom}} Thus '''one $n$-illion''' equals $1000 \\times 10^{3 n}$ or $10^{3 n + 3}$"}
+{"_id": "28842", "title": "Definition:Number-Naming System/Long Scale", "text": "The '''long scale system''' is the number-naming system which uses: :the word '''million''' for $10^6 = 1 \\, 000 \\, 000$ :the Latin-derived prefixes '''bi-''', '''tri-''', '''quadri-''', '''quint-''', etc. for each further multiple of $1 \\, 000 \\, 000$, appended to the root '''-(i)llion''', corresponding to the indices $2$, $3$, $4$, $5$, $\\ldots$ Thus: {{begin-axiom}} {{axiom | lc= one billion:         | m = = 1 \\, 000 \\, 000 \\, 000 \\, 000         | mm= = 10^{12} = 10^{2 \\times 6} }} {{axiom | lc= one trillion         | m = = 1 \\, 000 \\, 000 \\, 000 \\, 000 \\, 000 \\, 000         | mm= = 10^{18} = 10^{3 \\times 6} }} {{axiom | lc= one quadrillion         | m = = 1 \\, 000 \\, 000 \\, 000 \\, 000 \\, 000 \\, 000 \\, 000 \\, 000         | mm= = 10^{24} = 10^{4 \\times 6} }} {{axiom | lc= one quintillion         | m = = 1 \\, 000 \\, 000 \\, 000 \\, 000 \\, 000 \\, 000 \\, 000 \\, 000 \\, 000 \\, 000         | mm= = 10^{30} = 10^{5 \\times 6} }} {{end-axiom}} Thus '''one $n$-illion''' equals $10^{6 n}$. Additional terms are occasionally found to fill some of the gaps, but these are rare nowadays: {{begin-axiom}} {{axiom | lc= one milliard:         | m = = 1 \\, 000 \\, 000 \\, 000         | mm= = 10^9 }} {{axiom | lc= one billiard         | m = = 1 \\, 000 \\, 000 \\, 000 \\, 000 \\, 000         | mm= = 10^{15} }} {{end-axiom}}"}
+{"_id": "28844", "title": "Definition:Vigintillion/Also known as", "text": "Some sources use the word '''vigillion''' for '''vigintillion'''."}
+{"_id": "28845", "title": "Definition:Vigintillion/Short Scale", "text": "'''Vigintillion''' is a name for $10^{63}$ in the short scale system: :'''One vigintillion''' $= 10^{3 \\times 20 + 3}$"}
+{"_id": "28847", "title": "Definition:Vigintillion/Long Scale", "text": "'''Vigintillion''' is a name for $10^{120}$ in the long scale system: :'''One vigintillion''' $= 10^{6 \\times 20}$"}
+{"_id": "28848", "title": "Definition:Multiindex Power Notation", "text": "Let $M$ be a commutative monoid. Let $I$ be a set. Let $\\mathbf x = (x_i)_{i\\in I}$ be a family of elements of $M$. Let $\\mathbf a = (a_i)_{i\\in I}$ be a family of natural numbers of finite support. We denote $\\mathbf x^{\\mathbf a} = \\displaystyle \\prod_{i \\in I} x_i^{a_i}$ where: * $x_i^{a_i}$ is the $a_i$th power of $x_i$ * $\\prod$ denotes product with finite support Category:Definitions/Polynomial Theory bfvaevrkwgbfdcr1qv3ob8rjtqyee8y"}
+{"_id": "28849", "title": "Definition:Restriction of Scalars of Module", "text": "Let $R$ and $S$ be commutative rings with unity. Let $f : R \\to S$ be a ring homomorphism. Let $M$ be an $S$-module with ring representation $\\rho$. The '''restriction of scalars''' of $M$ to $R$ via $f$ is the $R$-module with ring representation $\\rho \\circ f$."}
+{"_id": "28851", "title": "Definition:Centillion", "text": "=== Short Scale === {{:Definition:Centillion/Short Scale}} === Long Scale === {{:Definition:Centillion/Long Scale}}"}
+{"_id": "28852", "title": "Definition:Centillion/Short Scale", "text": "'''Centillion''' is a name for $10^{303}$ in the short scale system: :'''One centillion''' $= 10^{3 \\times 100 + 3}$"}
+{"_id": "28854", "title": "Definition:Milli-Millillion/Short Scale", "text": "'''Milli-millillion''' is a name for $10^{3 \\, 000 \\, 003}$ in the short scale system: :'''One milli-millillion''' $= 10^{3 \\times 1 \\, 000 \\, 000 + 3}$"}
+{"_id": "28855", "title": "Definition:Module Structure of Ring", "text": "Let $R$ be a commutative ring with unity. Then '''$R$ as an $R$-module''' is the $R$-module $R$ with ring action: :$R\\times R \\to R$ equal to the ring product of $R$."}
+{"_id": "28859", "title": "Definition:Koti", "text": "'''Koti''' is a Jaina word for a '''hundred-hundred-thousand''' ($10 \\, 000 \\, 000$)."}
+{"_id": "28860", "title": "Definition:Pakoti", "text": "'''Pakoti''' is a Jaina word for a koti multiplied by a koti: that is: :$10 \\, 000 \\, 000 \\times 10 \\, 000 \\, 000$"}
+{"_id": "28861", "title": "Definition:Asankhyeya", "text": "'''Asankhyeya''' is a Jaina word for a koti to the power of $20$: that is: :$10 \\, 000 \\, 000^{20} = 10^{140}$"}
+{"_id": "28862", "title": "Definition:Module Structure of Polynomial Ring", "text": "Let $R$ be a commutative ring with unity. Let $R \\sqbrk X$ be a polynomial ring in one variable $X$ over $R$. The '''$R$-module structure''' on $R \\sqbrk X$ is the module structure as an algebra over $R$."}
+{"_id": "28863", "title": "Definition:Ring Representation", "text": "Let $R$ be a ring. Let $M$ be an abelian group. A '''ring representation''' of $R$ on $M$ is a ring homomorphism from $R$ to the endomorphism ring $\\map {\\operatorname {End} } M$. === Unital Ring Representation === {{:Definition:Ring Representation/Unital}}"}
+{"_id": "28864", "title": "Definition:Constant Polynomial/Definition 1", "text": "The polynomial $P$ is a '''constant polynomial''' {{iff}} its coefficients of $x^k$ are zero for $k \\ge 1$."}
+{"_id": "28865", "title": "Definition:Constant Polynomial/Definition 2", "text": "The polynomial $P$ is a '''constant polynomial''' {{iff}} $P$ is either the zero polynomial or has degree $0$."}
+{"_id": "28866", "title": "Definition:Constant Polynomial/Definition 3", "text": "The polynomial $P$ is a '''constant polynomial''' {{iff}} it is in the image of the canonical embedding $R \\to R \\left[{x}\\right]$."}
+{"_id": "28867", "title": "Definition:Minimal Polynomial/Definition 1", "text": "The '''minimal polynomial''' of $\\alpha$ over $K$ is the unique monic polynomial $f \\in K \\sqbrk x$ of smallest degree such that $\\map f \\alpha = 0$."}
+{"_id": "28868", "title": "Definition:Minimal Polynomial/Definition 2", "text": "The '''minimal polynomial''' of $\\alpha$ over $K$ is the unique irreducible, monic polynomial $f \\in K \\left[{x}\\right]$ such that $f \\left({\\alpha}\\right) = 0$."}
+{"_id": "28869", "title": "Definition:Minimal Polynomial/Definition 3", "text": "The '''minimal polynomial''' of $\\alpha$ over $K$ is the unique monic polynomial $f \\in K \\left[{x}\\right]$ that generates the kernel of the evaluation homomorphism $K \\left[{x}\\right] \\to L$ at $\\alpha$. That is, such that for all $g \\in K \\left[{x}\\right]$: :$g \\left({\\alpha}\\right) = 0$ {{iff}} $f$ divides $g$."}
+{"_id": "28870", "title": "Definition:Integral Element of Algebra", "text": "Let $A$ be a commutative ring with unity. Let $f : A \\to B$ be a commutative $A$-algebra. Let $b\\in B$. === Definition 1 === {{Definition:Integral Element of Algebra/Definition 1}} === Definition 2 === {{Definition:Integral Element of Algebra/Definition 2}} === Definition 3 === {{Definition:Integral Element of Algebra/Definition 3}} === Definition 4 === {{Definition:Integral Element of Algebra/Definition 4}}"}
+{"_id": "28872", "title": "Definition:Integral Element of Algebra/Definition 1", "text": "The element $b$ is '''integral''' over $A$ {{iff}} it is a root of a monic polynomial in $A[x]$."}
+{"_id": "28873", "title": "Definition:Integral Element of Algebra/Definition 2", "text": "The element $b$ is '''integral''' over $A$ {{iff}} the generated subalgebra $A[b]$ is a finitely generated module over $A$."}
+{"_id": "28874", "title": "Definition:Integral Element of Algebra/Definition 3", "text": "The element $b$ is '''integral''' over $A$ {{iff}} the generated subalgebra $A[b]$ is contained in a subalgebra $C\\leq B$ which is a finitely generated module over $A$."}
+{"_id": "28875", "title": "Definition:Integral Element of Algebra/Definition 4", "text": "The element $b$ is '''integral''' over $A$ {{iff}} there exists a [[Definition:Faithful Module|faithful $A[b]$-module]] whose restriction of scalars to $A$ is finitely generated."}
+{"_id": "28876", "title": "Definition:Presheaf on Topological Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $\\mathbf C$ be a category. === Definition 1 === {{:Definition:Presheaf on Topological Space/Definition 1}} === Definition 2 === {{:Definition:Presheaf on Topological Space/Definition 2}}"}
+{"_id": "28878", "title": "Definition:Sheaf on Topological Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\mathbf C$ be a category. === Definition 1 === {{:Definition:Sheaf on Topological Space/Definition 1}} === Definition 2 === {{:Definition:Sheaf on Topological Space/Definition 2}} === Definition 3 === {{:Definition:Sheaf on Topological Space/Definition 3}}"}
+{"_id": "28879", "title": "Definition:Morphism of Presheaves on Topological Space", "text": "Let $X$ be a topological space. Let $\\mathbf C$ be a category. Let $\\mathcal F$ and $\\mathcal G$ be $\\mathbf C$-valued presheaves on $X$. === Definition 1 === {{definition wanted|using restrictions}} === Definition 2 === A '''morphism of presheaves''' from $\\mathcal F$ to $\\mathcal G$ is a natural transformation $\\mathcal F \\to \\mathcal G$."}
+{"_id": "28880", "title": "Definition:Abelian Sheaf", "text": "Let $X$ be a topological space. === Definition 1 === An '''abelian sheaf''' on $X$ is a sheaf on $X$ with values in the category of abelian groups. === Definition 2 === An '''abelian sheaf''' on $X$ is a sheaf of $\\underline{\\mathbb{Z}}$-modules on $X$, where $\\underline{\\mathbb{Z}}$ is the constant sheaf with value the ring of integers $\\Z$."}
+{"_id": "28881", "title": "Definition:Stalk of Presheaf", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $\\mathbf C$ be a category which has all small inductive limits. Let $\\mathcal F$ be a $\\mathbf C$-valued presheaf on $T$. Let $x \\in S$. The '''stalk''' $\\mathcal F_x$ of $\\mathcal F$ at $x$ is the inductive limit: :$\\displaystyle\\varinjlim_{U \\ni x} \\mathcal F \\left({U}\\right)$ over all open $U \\subseteq S$ containing $x$."}
+{"_id": "28882", "title": "Definition:Morphism of Sheaves", "text": "Let $X$ be a topological space. Let $\\mathbf C$ be a category. Let $\\mathcal F$ and $\\mathcal G$ be $\\mathbf C$-valued sheaves on $X$. A '''sheaf morphism''' from $\\mathcal F$ to $\\mathcal G$ is a presheaf morphism $\\mathcal F \\to\\mathcal G$."}
+{"_id": "28883", "title": "Definition:Category of Sheaves on Topological Space", "text": "Let $X$ be a topological space.  Let $\\mathbf C$ be a category. The '''category of $\\mathbf C$-valued sheaves on $X$''' is the category with: {{DefineCategory |ob = $\\mathbf C$-valued sheaves on $X$ |mor = morphisms of presheaves |id = identity morphisms on presheaves |comp = composition of morphisms of presheaves }}"}
+{"_id": "28884", "title": "Definition:Category of Presheaves on Topological Space", "text": "Let $X$ be a topological space.  Let $\\mathbf C$ be a category. The '''category of $\\mathbf C$-valued presheaves on $X$''' is the category with: {{DefineCategory |ob = $\\mathbf C$-valued presheaves on $X$ |mor = morphisms of presheaves |id = identity morphisms on presheaves |comp = composition of morphisms of presheaves }}"}
+{"_id": "28885", "title": "Definition:Ringed Space", "text": "A '''ringed space''' is a pair $\\struct {X, \\OO_X}$ where: :$X$ is a topological space :$\\OO_X$ is a sheaf of rings with unity on $X$."}
+{"_id": "28887", "title": "Definition:Locally Ringed Space", "text": "A '''locally ringed space''' is a ringed space $\\struct {X, \\OO_X}$ such that the stalks of the structure sheaf $\\OO_X$ are local rings."}
+{"_id": "28888", "title": "Definition:Natural Transformation/Covariant Functors", "text": "Let $F, G : \\mathbf C \\to \\mathbf D$ be covariant functors. A '''natural transformation''' $\\eta$ from $F$ to $G$ is a mapping on $\\mathbf C$ such that: :$(1): \\quad$ For all $x \\in \\mathbf C$, $\\eta_x$ is a morphism from $\\map F x$ to $\\map G x$. :$(2): \\quad$ For all $x, y \\in C$ and morphism $f: x \\to y$, the following diagram commutes: ::$\\xymatrix{ \\map F x \\ar[d]^{\\eta_x} \\ar[r]^{\\map F f} & \\map F y \\ar[d]^{\\eta_y} \\\\ \\map G x \\ar[r]^{\\map G f}                 & \\map G y}$"}
+{"_id": "28889", "title": "Definition:Natural Transformation/Contravariant Functors", "text": "Let $F, G: \\mathbf C \\to \\mathbf D$ be contravariant functors. A '''natural transformation''' $\\eta$ from $F$ to $G$ is a mapping on $\\mathbf C$ such that: :$(1): \\quad$ For all $x \\in \\mathbf C$, $\\eta_x$ is a morphism from $\\map F x$ to $\\map G x$. :$(2): \\quad$ For all $x, y \\in C$ and morphism $f: x \\to y$, the following diagram commutes: ::$\\xymatrix{ \\map F x \\ar[d]^{\\eta_x} & \\map F y \\ar[d]^{\\eta_y} \\ar[l]^{\\map F f}  \\\\ \\map G x                 & \\map G y \\ar[l]^{\\map G f} }$"}
+{"_id": "28890", "title": "Definition:Natural Transformation", "text": "Let $\\mathbf C$ and $\\mathbf D$ be categories. === Covariant Functors === {{Definition:Natural Transformation/Covariant Functors}} === Contravariant Functors === {{Definition:Natural Transformation/Contravariant Functors}}"}
+{"_id": "28891", "title": "Definition:Natural Isomorphism", "text": "Let $\\mathbf C$ and $\\mathbf D$ be categories. === Covariant Functors === {{Definition:Natural Isomorphism between Covariant Functors}} === Contravariant Functors === {{Definition:Natural Isomorphism between Contravariant Functors}}"}
+{"_id": "28892", "title": "Definition:Natural Isomorphism between Covariant Functors", "text": "Let $F, G : \\mathbf C \\to \\mathbf D$ be covariant functors. === Definition 1 === A '''natural isomorphism''' from $F$ to $G$ is a natural transformation $\\eta : F \\to G$ such that for all $x\\in \\mathbf C$, $\\eta_x : F(x) \\to G(x)$ is an isomorphism. === Definition 2 === A '''natural isomorphism''' from $F$ to $G$ is an isomorphism in the functor category $\\operatorname{Funct}(\\mathbf C, \\mathbf D)$, that is, a natural transformation $\\eta : F \\to G$ for which there exists a natural transformation $\\xi : G \\to F$ such that the compositions $\\xi \\circ \\eta = 1_F$ and $\\eta \\circ \\xi = 1_G$ are identity natural transformations."}
+{"_id": "28894", "title": "Definition:Hom Functor", "text": "Let $\\mathbf{Set}$ be the category of sets. Let $\\mathbf C$ be a locally small category. === Hom Bifunctor === {{Definition:Hom Bifunctor}} === Covariant Hom Functor === {{Definition:Covariant Hom Functor}} === Contravariant Hom Functor === {{Definition:Contravariant Hom Functor}} {{definition wanted|for enriched categories}}"}
+{"_id": "28895", "title": "Definition:Covariant Hom Functor", "text": "Let $C \\in \\mathbf C_0$ be an object of $\\mathbf C$. The '''covariant hom functor based at $C$''', $\\operatorname{Hom}_{\\mathbf C} \\left({C, \\cdot}\\right): \\mathbf C \\to \\mathbf{Set}$, is the covariant functor defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \\operatorname{Hom}_{\\mathbf C} \\left({C, D}\\right) = \\operatorname{Hom}_{\\mathbf C} \\left({C, D}\\right) }} {{axiom|lc= Morphism functor:        |m = \\operatorname{Hom}_{\\mathbf C} \\left({C, f}\\right): \\operatorname{Hom}_{\\mathbf C} \\left({C, A}\\right) \\to \\operatorname{Hom}_{\\mathbf C} \\left({C, B}\\right), g \\mapsto f \\circ g        |rc= for $f: A \\to B$ }} {{end-axiom}} where $\\operatorname{Hom}_{\\mathbf C} \\left({C, D}\\right)$ denotes a hom set. Thus, the morphism functor is defined to be postcomposition."}
+{"_id": "28896", "title": "Definition:Hom Bifunctor", "text": "The '''hom bifunctor''' on $\\mathbf C$ is the covariant functor $\\map {\\operatorname {Hom} } {-, -} : \\mathbf C^{\\operatorname {op} } \\times \\mathbf C \\to \\mathbf {Set}$ from the product with the opposite category to the category of sets such that: :$(1): \\quad \\map {\\operatorname {Hom} } {a, b}$ is the hom class :$(2): \\quad$ If $\\tuple {f^{\\operatorname {op} }, g}: \\tuple {a, b} \\to \\tuple {c, d}$ is a morphism, $\\map {\\operatorname {Hom} } {f^{\\operatorname{op} }, g}: \\map {\\operatorname {Hom} } {a, b} \\to \\map {\\operatorname {Hom} } {c, d}$ is $f_* \\circ g^*$, the postcomposition with $g$ composed with the precomposition with $f$."}
+{"_id": "28897", "title": "Definition:Lehmer's Electromechanical Sieve", "text": "rightLehmer's Electromechanical Sieve '''Lehmer's electromechanical sieve''' is a device which was invented for the purpose of solving number theory problems, such as: : determining the prime decomposition of integers : determining whether an integer is prime. It consists of: : a mechanical assemblage of gearwheels : photo detector : an electronic amplifier. It is currently located at the [http://www.computerhistory.org/ Computer History Museum], where its lot number is $\\text X 85.82$. {{NamedforDef|Derrick Henry Lehmer|cat = Lehmer D H}}"}
+{"_id": "28898", "title": "Definition:Category of Abelian Groups", "text": "The '''category of abelian groups''' is the category $\\mathbf{Ab}$ with: {{DefineCategory |ob = abelian groups |mor = group homomorphisms |id = identity mappings |comp = composition of mappings }}"}
+{"_id": "28899", "title": "Definition:Category of Open Sets", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. === Definition 1 === {{:Definition:Category of Open Sets/Definition 1}} === Definition 2 === {{:Definition:Category of Open Sets/Definition 2}}"}
+{"_id": "28900", "title": "Definition:Direct Image of Presheaf", "text": "Let $\\mathbf C$ be a category. Let $T_1 = \\left({S_1, \\tau_1}\\right)$ and $T_2 = \\left({S_2, \\tau_2}\\right)$ be topological spaces. Let $f : T_1 \\to T_2$ be continuous. Let $\\mathcal F$ be a $\\mathbf C$-valued presheaf on $T_1$. === Definition 1 === {{:Definition:Direct Image of Presheaf/Definition 1}} === Definition 2 === {{:Definition:Direct Image of Presheaf/Definition 2}}"}
+{"_id": "28901", "title": "Definition:Direct Image of Sheaf", "text": "Let $X$ be a topological space. Let $\\mathbf C$ be a category. Let $\\mathcal F$ be a $\\mathbf C$-valued sheaf on $X$. The '''direct image''' of $\\mathcal F$ is the direct image of the presheaf $\\mathcal F$."}
+{"_id": "28902", "title": "Definition:Étalé Space of Presheaf", "text": "Let $X$ be a topological space. Let $\\mathbf C$ be a category. Let $\\mathcal F$ be a $\\mathbf C$-valued presheaf on $X$. The '''étalé space''' of $\\mathcal F$ is the pair $\\struct {\\map {\\operatorname {\\acute Et} } {\\mathcal F}, \\pi}$ where: :$\\map {\\operatorname {\\acute Et} } {\\mathcal F}$ is the disjoint union $\\displaystyle \\bigsqcup_{x \\mathop \\in X} \\mathcal F_x$ of stalks of $\\mathcal F$ :$\\pi: \\map {\\operatorname {\\acute Et} } {\\mathcal F} \\to X$ is the canonical projection. === Topology on Étalé Space === {{Definition:Topology on Étalé Space of Presheaf}}"}
+{"_id": "28904", "title": "Definition:Associated Section of Étalé Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\mathcal F$ be a presheaf of sets. Let $\\map {\\operatorname {\\acute Et} } {\\mathcal F}$ be its étalé space. Let $U \\subseteq S$ be open in $T$. Let $s \\in \\map {\\mathcal F} U$ be a section. The '''associated section''' of $\\map {\\operatorname {\\acute Et} } {\\mathcal F}$ is the mapping: :$\\overline s : U \\to \\map {\\operatorname {\\acute Et} } {\\mathcal F} : x \\mapsto \\tuple {x, s_x}$ where $s_x$ is the image in the stalk at $x$."}
+{"_id": "28905", "title": "Definition:Topology on Étalé Space of Presheaf", "text": "The '''topology on $\\map {\\operatorname {\\acute Et} } {\\mathcal F}$''' is the final topology with respect to the sections associated to elements of $\\map {\\mathcal F} U$ with $U \\subseteq S$ open."}
+{"_id": "28906", "title": "Definition:Trapdoor Function", "text": "A '''trapdoor function''' is a mapping $f$ which is: :for a given $x$ in the domain of $f$ is easy to calculate  :for a given $y$ in the image of $f$ is hard to calculate unless a specific piece of information $t$ is supplied, in which case it is easy. That piece of information is known as the '''trapdoor'''."}
+{"_id": "28908", "title": "Definition:Section of Étalé Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $\\mathcal F$ be a presheaf of sets on $T$. Let $\\left({\\operatorname{\\acute Et} \\left({\\mathcal F}\\right), \\pi}\\right)$ be its étalé space. Let $U \\subseteq S$ be open. === Definition 1 === {{:Definition:Section of Étalé Space/Definition 1}} === Definition 2 === {{:Definition:Section of Étalé Space/Definition 2}}"}
+{"_id": "28909", "title": "Definition:Sheafification", "text": "Let $X$ be a topological space.  Let $\\mathbf C$ be a category. Let $\\mathcal F$ be a $\\mathbf C$-valued presheaf on $X$. === Definition 1 === {{definition wanted|as a representation of a functor}} === Definition 2 === Let $\\mathcal F$ be a presheaf of sets. The '''sheafification''' of $\\mathcal F$ is the sheaf of continuous sections of its étalé space $\\map {\\operatorname {\\acute Et} } {\\mathcal F}$."}
+{"_id": "28910", "title": "Definition:Inverse Image Presheaf", "text": "Let $T_1 = \\left({S_1, \\tau_1}\\right)$ and $T_2 = \\left({S_2, \\tau_2}\\right)$ be topological spaces. Let $f: T_1 \\to T_2$ be continuous. Let $\\mathbf C$ be a category which has all small inductive limits. Let $\\mathcal F$ be a $\\mathbf C$-valued presheaf on $T_2$. The '''inverse image presheaf''' of $\\mathcal F$ via $f$ is the presheaf $f^{-1}_{\\operatorname{Psh}} \\mathcal F$ on $T_1$ with: : $\\left({f^{-1}_{\\operatorname{Psh}} \\mathcal F}\\right) \\left({U}\\right) = \\displaystyle \\varinjlim_{V \\supseteq f \\left({U}\\right)} \\mathcal F \\left({V}\\right)$ where the inductive limit goes over open $V \\subseteq Y$ : $\\operatorname{res}^U_W$ is the induced map on the inductive limit of the subset $\\left\\{ {V: V \\supseteq f \\left({U}\\right)}\\right\\} \\subseteq \\left\\{ {V : V \\supseteq f \\left({W}\\right)}\\right\\}$"}
+{"_id": "28911", "title": "Definition:Unique up to Unique Isomorphism", "text": "Let $\\mathbf C$ be a category. Let $S \\subseteq \\operatorname{Ob}(\\mathbf C)$ be a subclass of its objects. The class $S$ is '''unique up to unique isomorphism''' {{iff}} for all objects $s,t \\in S$ there is a unique isomorphism from $s$ to $t$."}
+{"_id": "28912", "title": "Definition:Zero Object", "text": "Let $\\mathbf C$ be a category. A '''zero object''' of $\\mathbf C$ is an object which is initial and terminal."}
+{"_id": "28914", "title": "Definition:RSA 129", "text": "'''RSA $129$''' is the name given to the semiprime: :$114 \\, 381 \\, 625 \\, 757 \\, 888 \\, 867 \\, 669 \\, 235 \\, 779 \\, 976 \\, 146 \\, 612 \\, 010 \\, 218 \\, 296 \\, 721 \\, 242 \\, 362 \\, 562 \\, 561 \\, 842 \\, 935 \\, 706 \\, 935 \\, 245 \\, 733 \\, 897 \\, 830 \\, 597 \\, 123 \\, 563 \\, 958 \\, 705 \\, 058 \\, 989 \\, 075 \\, 147 \\, 599 \\, 290 \\, 026 \\, 879 \\, 543 \\, 541$ Its factors are: :$3 \\, 490 \\, 529 \\, 510 \\, 847 \\, 650 \\, 949 \\, 147 \\, 849 \\, 619 \\, 903 \\, 898 \\, 133 \\, 417 \\, 764 \\, 638 \\, 493 \\, 387 \\, 843 \\, 990 \\, 820 \\, 577$ and: :$32 \\, 769 \\, 132 \\, 993 \\, 266 \\, 709 \\, 549 \\, 961 \\, 988 \\, 190 \\, 834 \\, 461 \\, 413 \\, 177 \\, 642 \\, 967 \\, 992 \\, 942 \\, 539 \\, 798 \\, 288 \\, 533$"}
+{"_id": "28915", "title": "Definition:Tychonoff Topology/Natural Sub-Basis", "text": "The '''natural sub-basis on $X$''' is defined as: :$\\mathcal S = \\set {\\pr_i^{-1} \\sqbrk U: i \\in I, \\, U \\in \\tau_i}$"}
+{"_id": "28916", "title": "Definition:Tychonoff Topology/Natural Basis", "text": "The '''natural basis on $X$''' is defined as the basis generated by $\\SS$."}
+{"_id": "28919", "title": "Definition:RSA 130", "text": "'''RSA $130$''' is the name given to the semiprime: :$1 \\, 807 \\, 082 \\, 088 \\, 687 \\, 404 \\, 805 \\, 951 \\, 656 \\, 164 \\, 405 \\, 905 \\, 566 \\, 278 \\, 102 \\, 516 \\, 769 \\, 401 \\, 349 \\, 170 \\, 127 \\, 021 \\, 450 \\, 056 \\, 662 \\, 540 \\, 244 \\, 048 \\, 387 \\, 341 \\, 127 \\, 590 \\, 812 \\, 303 \\, 371 \\, 781 \\, 887 \\, 966 \\, 563 \\, 182 \\, 013 \\, 214 \\, 880 \\, 557$ Its factors are: :$39 \\, 685 \\, 999 \\, 459 \\, 597 \\, 454 \\, 290 \\, 161 \\, 126 \\, 162 \\, 883 \\, 786 \\, 067 \\, 576 \\, 449 \\, 112 \\, 810 \\, 064 \\, 832 \\, 555 \\, 157 \\, 243$ and: :$45 \\, 534 \\, 498 \\, 646 \\, 735 \\, 972 \\, 188 \\, 403 \\, 686 \\, 897 \\, 274 \\, 408 \\, 864 \\, 356 \\, 301 \\, 263 \\, 205 \\, 069 \\, 600 \\, 999 \\, 044 \\, 599$"}
+{"_id": "28920", "title": "Definition:Mersenne Prime/Index", "text": "The '''index''' of the '''Mersenne prime''' $M_p = 2^p - 1$ is the (prime) number $p$."}
+{"_id": "28921", "title": "Definition:Direct Limit", "text": "Let $\\mathbf C$ be a category. Let $I$ be an upward directed set. Let $\\mathbf I$ be its order category. Let $D : \\mathbf I \\to \\mathbf C$ be a diagram. The '''direct limit''' of $D$ is the colimit of the diagram $D$."}
+{"_id": "28922", "title": "Definition:Contravariant Hom Functor", "text": "Let $C \\in \\mathbf C_0$ be an object of $\\mathbf C$. The '''contravariant hom functor based at $C$''': : $\\operatorname{Hom}_{\\mathbf C} \\left({\\cdot, C}\\right): \\mathbf C \\to \\mathbf{Set}$ is the covariant functor defined by: {{begin-axiom}} {{axiom | lc= Object functor:         | m = \\operatorname{Hom}_{\\mathbf C} \\left({B, C}\\right) = \\operatorname{Hom}_{\\mathbf C} \\left({B, C}\\right) }} {{axiom | lc= Morphism functor:         | m = \\operatorname{Hom}_{\\mathbf C} \\left({f, C}\\right): \\operatorname{Hom}_{\\mathbf C} \\left({B, C}\\right) \\to \\operatorname{Hom}_{\\mathbf C} \\left({A, C}\\right), g \\mapsto g \\circ f         | rc= for $f: A \\to B$ }} {{end-axiom}} where $\\operatorname{Hom}_{\\mathbf C} \\left({B, C}\\right)$ denotes a hom set. Thus, the morphism functor is defined to be precomposition."}
+{"_id": "28923", "title": "Definition:Representable Functor", "text": "Let $\\mathbf C$ be a locally small category. Let $\\mathbf{Set}$ be the category of sets. Let $F : \\mathbf C \\to \\mathbf{Set}$ be a covariant functor. Then $F$ is '''representable''' if there exists an object $a\\in \\mathbf C$ such that $F$ is naturally isomorphic to the covariant hom functor $\\operatorname{Hom}(a,-)$. {{DefinitionWanted|contravariant}}"}
+{"_id": "28924", "title": "Definition:Yoneda Functor/Yoneda Embedding", "text": "The '''Yoneda embedding''' of $C$ is the covariant functor $h_- : C \\to \\left[{C^{\\operatorname{op}}, \\mathbf{Set} }\\right]$ which sends: : an object $X\\in C$ to the contravariant hom-functor $h_X = \\operatorname{Hom} \\left({-, X}\\right)$ : a morphism $f : X \\to Y$ to the postcomposition natural transformation $h_f : \\operatorname{Hom} \\left({-, X}\\right) \\to \\operatorname{Hom} \\left({-, Y}\\right)$"}
+{"_id": "28925", "title": "Definition:Yoneda Functor/Contravariant", "text": "The '''contravariant Yoneda functor''' of $C$ is the contravariant functor $h^- : C \\to \\left[{C, \\mathbf{Set} }\\right]$ which sends : an object $X \\in C$ to the covariant hom-functor $h^X = \\operatorname{Hom} \\left({X, -}\\right)$ : a morphism $f : X \\to Y$ to the precomposition natural transformation $h^f : \\operatorname{Hom} \\left({Y, -}\\right) \\to \\operatorname{Hom} \\left({X, -}\\right) : g \\mapsto g \\circ f$"}
+{"_id": "28926", "title": "Definition:Yoneda Functor", "text": "Let $C$ be a locally small category. Let $C^{\\operatorname{op}}$ be its opposite category. Let $\\mathbf{Set}$ be the category of sets. Let $\\left[{C^{\\operatorname{op}}, \\mathbf{Set} }\\right]$ be the functor category between them. === Yoneda Embedding === {{Definition:Yoneda Embedding}} === Contravariant Yoneda Functor === {{Definition:Contravariant Yoneda Functor}}"}
+{"_id": "28927", "title": "Definition:Whiskering", "text": "Let $B,C,D,E$ be categories. Let $F, G : C \\to D$ be covariant functors. Let $\\eta : F \\to G$ be a natural transformation. === Natural Transformation followed by Functor === Let $H : D \\to E$ be a covariant functor. The '''right whiskering''' of $H$ and $\\eta$ is the natural transformation $H\\eta : H \\circ F \\to H \\circ G$ between compositions of functors defined by $(H\\eta)_A = H(\\eta_A)$ for $A\\in C$. === Functor followed by Natural Transformation === Let $K : B \\to C$ be a covariant functor. The '''left whiskering''' of $\\eta$ and $K$ is the natural transformation $\\eta K : F \\circ K \\to G \\circ K$ between compositions of functors defined by $(\\eta K)_A = \\eta_{K(A)}$ for $A\\in B$."}
+{"_id": "28928", "title": "Definition:Functor Evaluation Bifunctor", "text": "Let $C$ and $D$ be categories. Let $\\operatorname{Funct}(C, D)$ be their covariant functor category. Let $\\operatorname{Funct}(C, D) \\times C$ be the product category. The '''evaluation bifunctor''' $\\operatorname{ev} : \\operatorname{Funct}(C, D) \\times C \\to D$ is the covariant functor that sends: *an object $(F, a)$ to $F(a)$ *a morphism $(\\eta, f) : (F, a) \\to (G, b)$ to $G(f) \\circ \\eta_a = \\eta_b \\circ F(f)$"}
+{"_id": "28929", "title": "Definition:Vertical Composition of Natural Transformations", "text": "Let $C$ and $D$ be categories. Let $F,G,H : C \\to D$ be covariant functors. Let $\\eta : F \\to G$ and $\\xi : G \\to H$ be natural transformations. The '''vertical composition''' of $\\eta$ and $\\xi$ is the natural transformation $\\xi \\circ \\eta : F \\Rightarrow H$ with $(\\xi \\circ \\eta)_A = \\xi_A \\circ \\eta_A$ for $A\\in C$."}
+{"_id": "28930", "title": "Definition:Titanic Prime", "text": "A '''titanic prime''' is a prime number with $1000$ digits or more."}
+{"_id": "28931", "title": "Definition:Representation of Functor", "text": "Let $\\mathbf C$ be a locally small category. Let $\\mathbf{Set}$ be the category of sets. Let $F : \\mathbf C \\to \\mathbf{Set}$ be a covariant functor. A '''representation''' of $F$ is a pair $(A, \\eta)$ where $\\eta : \\operatorname{Hom}(A, -) \\to F$ is a natural isomorphism with the covariant hom functor of $A$."}
+{"_id": "28932", "title": "Definition:Fully Faithful Functor", "text": "Let $C$ and $D$ be categories. Let $F : C \\to D$ be a covariant functor. The functor $F$ is '''fully faithful''' {{iff}} it is full and faithful. Category:Definitions/Category Theory i96p1iro3gt54jq9tgt3qy5ryxnud99"}
+{"_id": "28933", "title": "Definition:Embedding of Categories", "text": "Let $C$ and $D$ be categories. Let $F : C \\to D$ be a functor. === Definition 1 === The functor $F$ is an '''embedding''' {{iff}} it is: * injective on objects * faithful === Definition 2 === The functor $F$ is an '''embedding''' {{iff}} it is injective on morphisms. === Definition 3 === The functor $F$ is an '''embedding''' {{iff}} it is a monomorphisms in the category of categories."}
+{"_id": "28934", "title": "Definition:Group Category", "text": "Let $G$ be a group. The '''group category''' of $G$ is the monoid category of $G$. Category:Definitions/Examples of Categories 4574r9kgtovfd1xa7xh0kz05tjwo9q9"}
+{"_id": "28935", "title": "Definition:Primorial Prime", "text": "A '''primorial prime''' is a prime number $p$ such that $p + 1$ is a primorial. That is, a prime number which is $1$ less than a primorial."}
+{"_id": "28937", "title": "Definition:Covariant Power Set Functor", "text": "The '''(covariant) power set functor''' $\\mathcal P : \\mathbf{Set} \\to \\mathbf{Set}$ is the covariant functor which sends: * An object $x$ to its power set $\\mathcal P(x)$. * A morphism $f : x \\to y$ to the direct image mapping $\\mathcal P(f) : \\mathcal P(x) \\to \\mathcal P(y)$."}
+{"_id": "28938", "title": "Definition:Contravariant Power Set Functor", "text": "The '''contravariant power set functor''' $\\overline{\\mathcal P} : \\mathbf{Set} \\to \\mathbf{Set}$ is the contravariant functor which sends: * An object $x$ to its power set $\\mathcal P(x)$. * A morphism $f : x \\to y$ to the inverse image mapping $\\overline{\\mathcal P}(f) : \\mathcal P(y) \\to \\mathcal P(x)$."}
+{"_id": "28939", "title": "Definition:Power Set Functor", "text": "=== Covariant === {{Definition:Covariant Power Set Functor}} === Contravariant === {{Definition:Contravariant Power Set Functor}} {{definition wanted|Create the analogous definition for relations.}} Category:Definitions/Examples of Functors qzkvcpd1lr99266stgimt8nmreynnz1"}
+{"_id": "28941", "title": "Definition:Gigantic Prime", "text": "A '''gigantic prime''' is a prime number with $10 \\, 000$ digits or more."}
+{"_id": "28942", "title": "Definition:Mapping Induced on Powerset", "text": "The '''mapping induced on a powerset''' can be either of two concepts: == Direct Image Mapping == {{:Definition:Direct Image Mapping}} == Inverse Image Mapping == {{:Definition:Inverse Image Mapping}}"}
+{"_id": "28943", "title": "Definition:Proth Prime", "text": "A '''Proth prime''' is a Proth number which is also a prime."}
+{"_id": "28944", "title": "Definition:Proth Number", "text": "A '''Proth number''' is a natural number of the form: :$N = k \\times 2^n + 1$ where: :$k$ is an odd integer :$n$ is a positive integer :$2^n > k$ The condition is included, otherwise every odd integer greater than $1$ would be a '''Proth number'''."}
+{"_id": "28952", "title": "Definition:Hyperoperation", "text": "=== Hyperoperation Sequence === {{:Definition:Hyperoperation/Sequence}} === $n$th Hyperoperation === {{:Definition:Hyperoperation/Nth Hyperoperation}}"}
+{"_id": "28953", "title": "Definition:Hyperoperation/Sequence", "text": "The '''hyperoperation sequence''' is the sequence $\\left\\langle{H_n}\\right\\rangle$ of binary operations $H_n : \\Z_{\\ge 0} \\times \\Z_{\\ge 0} \\to \\Z_{\\ge 0}$, defined as: $\\forall n, x, y \\in \\Z_{\\ge 0}: H_n \\left({x, y}\\right) = \\begin{cases} y + 1 & : n = 0 \\\\ x & : n = 1, y = 0 \\\\ 0 & : n = 2, y = 0 \\\\ 1 & : n > 2, y = 0 \\\\ H_{n - 1} \\left({x, H_n \\left({x, y - 1}\\right)}\\right) & : n > 0, y > 0 \\end{cases}$"}
+{"_id": "28954", "title": "Definition:Hyperoperation/Nth Hyperoperation", "text": "The $n$th term of $\\left\\langle{H_n}\\right\\rangle$, which is the binary operation $H_n : \\Z_{\\ge 0} \\times \\Z_{\\ge 0} \\to \\Z_{\\ge 0}$, is known as the '''$n$th hyperoperation'''."}
+{"_id": "28955", "title": "Definition:Power (Algebra)/Integer/Knuth Notation", "text": "In certain contexts in number theory, the symbol $\\uparrow$ is used to denote the (usually) integer power operation: :$x \\uparrow y := x^y$ This notation is usually referred to as '''Knuth (uparrow) notation'''."}
+{"_id": "28957", "title": "Definition:Gigaplex", "text": "A '''gigaplex''' is defined to be $10$ to the power of a (short scale) billion, that is $10^{1 \\, 000 \\, 000 \\, 000}$."}
+{"_id": "28958", "title": "Definition:Billion", "text": "=== Short Scale === {{:Definition:Billion/Short Scale}} === Long Scale === {{:Definition:Billion/Long Scale}}"}
+{"_id": "28959", "title": "Definition:Billion/Short Scale", "text": "'''Billion''' is a name for $10^9$ in the short scale system: :'''One billion''' $= 10^{3 \\times 2 + 3}$"}
+{"_id": "28960", "title": "Definition:Billion/Long Scale", "text": "'''Billion''' is a name for $10^{12}$ in the long scale system: :'''One billion''' $= 10^{6 \\times 2}$"}
+{"_id": "28962", "title": "Definition:Skewes' Number", "text": "'''Skewes' number''' is: :$10^{10^{10^{34} } }$ In Knuth notation this can be presented as: :$10 \\uparrow \\paren {10 \\uparrow \\paren {10 \\uparrow 34} }$"}
+{"_id": "28964", "title": "Definition:Graham's Number", "text": "Let $3 \\uparrow 3$ denote Knuth's notation for powers: :$3 \\uparrow 3 := 3^3$ Further, let: :$3 \\uparrow \\uparrow 3 := 3 \\uparrow \\paren {3 \\uparrow 3} = 3^{\\paren {3^3} }$ and so define: :$3 \\underbrace {\\uparrow \\uparrow \\ldots \\uparrow \\uparrow}_n 3 := 3 \\underbrace {\\uparrow \\uparrow \\ldots \\uparrow \\uparrow}_{n - 1} \\paren {3 \\underbrace {\\uparrow \\uparrow \\ldots \\uparrow \\uparrow}_{n - 1} 3}$ Thus, for example: :$3 \\uparrow \\uparrow \\uparrow \\uparrow 3 = 3 \\uparrow \\uparrow \\uparrow \\paren {3 \\uparrow \\uparrow \\uparrow 3}$ Let: : $n_1 := 3 \\underbrace {\\uparrow \\uparrow \\ldots \\uparrow \\uparrow}_{3 \\uparrow \\uparrow \\uparrow \\uparrow 3} 3$ That is, a total of $3 \\uparrow \\uparrow \\uparrow \\uparrow 3$ instances of $\\uparrow$. Similarly, let: :$n_2 := 3 \\underbrace {\\uparrow \\uparrow \\ldots \\uparrow \\uparrow}_{n_1} 3$ In general for $m \\ge 2$: :$n_m := 3 \\underbrace {\\uparrow \\uparrow \\ldots \\uparrow \\uparrow}_{n_{m - 1} } 3$ and so specifically: :$n_{63} := 3 \\underbrace {\\uparrow \\uparrow \\ldots \\uparrow \\uparrow}_{n_{62} } 3$ It is this $n_{63}$ which is defined as '''Graham's number'''. {{NamedforDef|Ronald Lewis Graham|cat = Graham}}"}
+{"_id": "28975", "title": "Definition:Metrizable Topology/Linguistic Note", "text": "The UK English spelling of '''metrizable''' is '''metrisable''', but it is rarely found."}
+{"_id": "28976", "title": "Definition:Noetherian Topological Space/Definition 1", "text": "A topological space $T = \\left({S, \\tau}\\right)$ is '''Noetherian''' {{iff}} its set of closed sets, ordered by inclusion, satisfies the descending chain condition."}
+{"_id": "28977", "title": "Definition:Noetherian Topological Space/Definition 2", "text": "A topological space $T = \\left({S, \\tau}\\right)$ is '''Noetherian''' {{iff}} its set of open sets, ordered by inclusion, satisfies the ascending chain condition."}
+{"_id": "28979", "title": "Definition:Everywhere Dense/Definition 1", "text": "The subset $H$ is '''(everywhere) dense in $T$''' {{iff}}: :$H^- = S$ where $H^-$ is the closure of $H$."}
+{"_id": "28980", "title": "Definition:Everywhere Dense/Definition 2", "text": "The subset $H$ is '''(everywhere) dense in $T$''' {{iff}} the intersection of $H$ with every open subset of $T$ is non-empty: :$\\forall U \\in \\tau: H \\cap U \\ne \\O$"}
+{"_id": "28981", "title": "Definition:Product Space (Topology)/Factor Space", "text": "Each of the topological spaces $\\struct{S_i, \\tau_i}$ are called the '''factors''' of $\\struct{S, \\tau}$, and can be referred to as '''factor spaces'''."}
+{"_id": "28983", "title": "Definition:Finer Filter on Set", "text": "Let $\\FF \\subseteq \\FF'$. Then  $\\FF'$ is '''finer''' than $\\FF$."}
+{"_id": "28984", "title": "Definition:Coarser Filter on Set", "text": "Let $\\FF \\subseteq \\FF'$. Then $\\FF$ is '''coarser''' than $\\FF$."}
+{"_id": "28985", "title": "Definition:Finer Filter on Set/Strictly Finer", "text": "Let $\\FF \\subset \\FF'$, that is, $\\FF \\subseteq \\FF'$ but $\\FF \\ne \\FF'$. Then $\\FF'$ is '''strictly finer''' than $\\FF$."}
+{"_id": "28987", "title": "Definition:Comparable Filters on Set", "text": "Let $S$ be a set. Let $\\powerset S$ be the power set of $S$. Let $\\FF, \\FF' \\subset \\powerset S$ be two filters on $S$. $\\FF$ and $\\FF'$ are '''comparable''' {{iff}} one is finer (or coarser) than the other."}
+{"_id": "28988", "title": "Definition:Ultrafilter on Set/Definition 2", "text": "Let $S$ be a set. Let $\\FF \\subseteq \\powerset S$ be a filter on $S$. Then $\\FF$ is an '''ultrafilter (on $S$)''' {{iff}}: :for every $A \\subseteq S$ and $B \\subseteq S$ such that $A \\cap B = \\O$ and $A \\cup B \\in \\FF$, either $A \\in \\FF$ or $B \\in \\FF$."}
+{"_id": "28989", "title": "Definition:Ultrafilter on Set/Definition 3", "text": "Let $S$ be a set. Let $\\FF \\subseteq \\powerset S$ be a filter on $S$. Then $\\FF$ is an '''ultrafilter (on $S$)''' {{iff}}: :for every $A \\subseteq S$, either $A \\in \\FF$ or $\\relcomp S A \\in \\FF$ where $\\relcomp S A$ is the relative complement of $A$ in $S$, that is, $S \\setminus A$."}
+{"_id": "28990", "title": "Definition:Ring under Ring", "text": "Let $A$ be a commutative ring with unity. A '''ring under $A$''' is a pair $(B, f)$ where: * $B$ is a commutative ring with unity * $f : A \\to B$ is a unital ring homomorphism"}
+{"_id": "28992", "title": "Definition:Principal Ultrafilter/Nonprincipal", "text": "Let $\\FF \\subset \\powerset S$ be an ultrafilter on $S$ which does not have a cluster point. Then $\\FF$ is a '''nonprincipal ultrafilter'''  on $S$."}
+{"_id": "28993", "title": "Definition:Limit Point/Filter Basis/Definition 1", "text": "A point $x \\in S$ is called a '''limit point of $\\BB$''' {{iff}} $\\FF$ converges on $x$. $\\BB$ is likewise said to converge on $x$."}
+{"_id": "28994", "title": "Definition:Limit Point/Filter Basis/Definition 2", "text": "A point $x \\in S$ is called a '''limit point of $\\BB$''' {{iff}} every neighborhood of $x$ contains a set of $\\BB$."}
+{"_id": "29002", "title": "Definition:Determinant", "text": "=== Determinant of Matrix === {{:Definition:Determinant of Matrix}} === Determinant of Linear Operator === {{:Definition:Determinant of Linear Operator}}"}
+{"_id": "29004", "title": "Definition:Compact Space/Topology/Definition 1", "text": "A topological space $T = \\struct {S, \\tau}$ is '''compact''' {{iff}} every open cover for $S$ has a finite subcover."}
+{"_id": "29005", "title": "Definition:Compact Space/Topology/Definition 2", "text": "A topological space $T = \\struct {S, \\tau}$ is '''compact''' {{iff}} it satisfies the Finite Intersection Axiom."}
+{"_id": "29006", "title": "Definition:Compact Space/Topology/Definition 3", "text": "A topological space $T = \\struct {S, \\tau}$ is '''compact''' {{iff}} $\\tau$ has a sub-basis $\\BB$ such that: :from every cover of $S$ by elements of $\\BB$, a finite subcover of $S$ can be selected."}
+{"_id": "29007", "title": "Definition:Compact Space/Topology/Definition 4", "text": "A topological space $T = \\left({S, \\tau}\\right)$ is '''compact''' {{iff}} every filter on $S$ has a limit point in $S$."}
+{"_id": "29009", "title": "Definition:Countably Compact Space/Definition 1", "text": "A topological space $T = \\struct {S, \\tau}$ is '''countably compact''' {{iff}}: :every countable open cover of $T$ has a finite subcover."}
+{"_id": "29010", "title": "Definition:Countably Compact Space/Definition 2", "text": "A topological space $T = \\struct {S, \\tau}$ is '''countably compact''' {{iff}}: :every countable set of closed sets of $T$ whose intersection is empty has a finite subset whose intersection is empty. That is, $T$ satisfies the countable finite intersection axiom."}
+{"_id": "29011", "title": "Definition:Countably Compact Space/Definition 3", "text": "A topological space $T = \\struct {S, \\tau}$ is '''countably compact''' {{iff}}: :every infinite sequence in $S$ has an accumulation point in $S$."}
+{"_id": "29013", "title": "Definition:Countably Compact Space/Definition 5", "text": "A topological space $T = \\struct {S, \\tau}$ is '''countably compact''' {{iff}}: :every infinite subset of $S$ has an $\\omega$-accumulation point in $S$."}
+{"_id": "29015", "title": "Definition:Normed Algebra", "text": "Let $R$ be a division ring with norm $\\norm {\\,\\cdot\\,}_R$. A '''normed algebra''' over $R$ is a pair $\\struct {A, \\norm{\\,\\cdot\\,} }$ where: :$A$ is a algebra over $R$ :$ \\norm{\\,\\cdot\\,} $ is a norm on $A$."}
+{"_id": "29016", "title": "Definition:Normed Ring", "text": "Let $\\norm {\\, \\cdot \\,}$ be a norm on $R$. Then $\\struct {R, \\norm {\\, \\cdot \\,} }$ is a '''normed ring'''."}
+{"_id": "29017", "title": "Definition:Convergent Product/Normed Algebra", "text": "Let $\\mathbb K$ be a division ring with norm $\\norm{\\,\\cdot\\,}_{\\mathbb K}$. Let $\\struct{ A, \\norm{\\,\\cdot\\,} }$ be an associative normed unital algebra over $\\mathbb K$. Let $\\sequence{a_n}$ be a sequence in $A$. === Definition 1 === The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ is '''convergent''' {{Iff}} there exists $n_0\\in\\N$ such that: # $a_n$ is invertible for $n \\geq n_0$ # the sequence of partial products of $\\displaystyle \\prod_{n \\mathop = n_0}^\\infty a_n$ converges to some invertible $b\\in A^\\times$. === Definition 2: for complete algebras === Let $\\struct{ A, \\norm{\\,\\cdot\\,} }$ be complete. The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ is '''convergent''' {{Iff}} there exists $n_0\\in\\N$ such that: : the sequence of partial products of $\\displaystyle \\prod_{n \\mathop = n_0}^\\infty a_n$ converges to some invertible $a\\in A^\\times$."}
+{"_id": "29018", "title": "Definition:Vectorization of Matrix", "text": "Let $S$ be a set. Let $m, n \\ge 1$ be natural numbers. Let $A = \\sqbrk {a_{i j} }$ be a $m \\times n$ matrix over $S$. === Definition 1 === The '''vectorization''' of $A$ is the $m n \\times 1$ column matrix: :$\\map {\\operatorname {vec} } A = \\sqbrk {a_{11}, \\ldots, a_{m1}, a_{12}, \\ldots, a_{m2}, \\ldots, a_{1n}, \\ldots, a_{mn} }^\\intercal$ informally obtained by stacking the columns of $A$. That is: :$\\map {\\operatorname {vec} } A_k = a_{\\floor {k/m}, k \\bmod m}$ where: :$\\floor {\\, \\cdot \\,}$ is the floor function :$\\bmod$ is the modulo operation. === Definition 2 === Let $R$ be a ring with unity. Let $A$ be an $m\\times n$ matrix over $R$. The '''vectorization''' of $A$ is its coordinate vector with respect to the standard matrix basis."}
+{"_id": "29019", "title": "Definition:Block Matrix", "text": "Let $S$ be a set. Let $m, n \\ge 1$ be positive integers. Let $A = \\sqbrk {A_{i j} }$ be an $m \\times n$ matrix of matrices over $S$. Let for every $i \\in \\set {1, \\ldots, m}$, the elements of the $i$th row of $A$ have equal height $m_i$. Let for every $j \\in \\set {1,\\ldots, n}$ the elements of the $j$th column of $A$ have equal width $n_i$. Define $M = \\displaystyle \\sum_{i \\mathop = 1}^m m_i$ and $N = \\displaystyle \\sum_{i \\mathop = 1}^n n_i$ as indexed summations. Let more generally $M_k = \\displaystyle \\sum_{i \\mathop = 1}^k m_i$ and $N_l = \\displaystyle \\sum_{i \\mathop = 1}^l n_i$ for $k \\in \\set {0, \\ldots, m}$ and $l \\in \\set {0, \\ldots, n}$. Then the '''block matrix''' of $A$ is the $M \\times N$ matrix $\\sqbrk {b_{i j} }$ over $S$ defined as the union of the mappings: :$b_{i j} = \\sqbrk {A_{kl} }_{i - M_{k - 1}, j - N_{l - 1} }$ on $\\set {M_{k - 1}, \\ldots, M_k} \\times \\set {N_{l - 1}, \\ldots, N_l}$ for $k \\in \\set {1, \\ldots, m}$ and $ l \\in \\set {1, \\ldots, n}$."}
+{"_id": "29020", "title": "Definition:Standard Matrix Basis", "text": "Let $R$ be a ring with unity. Let $m,n\\geq1$ be positive integers. Let $i, j \\in \\left\\{ {1, \\ldots, m}\\right\\} \\times \\{ 1, \\ldots, n\\}$. The '''standard matrix basis''' of $m\\times n$ matrices over $R$ is the ordered basis of standard basis matrices ordered by the colexicographic order on $\\left\\{ {1, \\ldots, m}\\right\\} \\times \\{ 1, \\ldots, n\\}$."}
+{"_id": "29021", "title": "Definition:Sigma-Locally Compact Space", "text": "$T$ is '''$\\sigma$-locally compact''' {{iff}}: : $T$ is $\\sigma$-compact : $T$ is locally compact That is, $T$ is '''$\\sigma$-locally compact''' {{iff}}: : $T$ is the union of countably many compact subspaces : every point of $S$ has a local basis $\\mathcal B$ such that all elements of $\\mathcal B$ are  compact."}
+{"_id": "29022", "title": "Definition:Normed Unital Algebra", "text": "Let $R$ be a division ring with norm $\\norm {\\,\\cdot\\,}_R$. A '''normed unital algebra''' over $R$ is a pair $\\struct {A, \\norm {\\,\\cdot\\,} }$ where: :$A$ is a unital algebra over $R$ :$\\norm {\\,\\cdot\\,}$ is a norm on $A$."}
+{"_id": "29024", "title": "Definition:Strongly Locally Compact Space/Definition 1", "text": "The space $T$ is '''strongly locally compact''' {{iff}}: :every point of $S$ is contained in an open set whose closure is compact."}
+{"_id": "29025", "title": "Definition:Strongly Locally Compact Space/Definition 2", "text": "The space $T$ is '''strongly locally compact''' {{iff}}: :every point has a closed compact neighborhood. That is: :every point of $S$ is contained in an open set which is contained in a closed compact subspace."}
+{"_id": "29027", "title": "Definition:Ringed Space/Structure Sheaf", "text": "Let $\\left({X, \\mathcal O_X}\\right)$ be a ringed space. The '''structure sheaf''' of $\\left({X, \\mathcal O_X}\\right)$ is the term $\\mathcal O_X$."}
+{"_id": "29028", "title": "Definition:Category of Rings", "text": "The '''category of rings''', denoted $\\mathbf{Ring}$, is the category with: {{DefineCategory | ob = rings | mor = ring homomorphisms |id = identity mappings |comp = composition of mappings }} === Category of rings with unity === {{Definition:Category of Rings with Unity}}"}
+{"_id": "29029", "title": "Definition:Category of Rings with Unity", "text": "The '''category of rings with unity''' is the category with: {{DefineCategory | ob = rings with unity | mor = unital ring homomorphisms |id = identity mappings |comp = composition of mappings }}"}
+{"_id": "29030", "title": "Definition:Morphism of Ringed Spaces", "text": "Let $(X, \\mathcal O_X)$ and $(Y, \\mathcal O_Y)$ be ringed spaces. === Definition 1 === A '''morphism of ringed spaces''' from $(X, \\mathcal O_X)$ to $(Y, \\mathcal O_Y)$ is a pair $(f, f^\\sharp)$ where: * $f : X \\to Y$ is continuous * $f^\\sharp : \\mathcal O_Y \\to f_* \\mathcal O_X$ is a morphism of sheaves to the direct image sheaf of $\\mathcal O_X$ via $f$ === Definition 2 === A '''morphism of ringed spaces''' from $(X, \\mathcal O_X)$ to $(Y, \\mathcal O_Y)$ is a pair $(f, f^\\sharp)$ where: * $f : X \\to Y$ is continuous * $f^\\sharp : f^{-1}\\mathcal O_Y \\to \\mathcal O_X$ is a morphism of sheaves from the inverse image sheaf of $\\mathcal O_Y$ via $f$ === Definition 3 === {{definition wanted|a description using open subsets, between the two above}}"}
+{"_id": "29031", "title": "Definition:Restriction of Presheaf to Open Set", "text": "Let $C$ be a category. Let $X$ be a topological space. Let $\\mathcal F$ be a $C$-valued presheaf on $X$. Let $U \\subset X$ be open. The '''restriction''' of $\\mathcal F$ to $U$ is the restriction of the contravariant functor $\\mathcal F$ to the subcategory $U$."}
+{"_id": "29032", "title": "Definition:Scheme", "text": "A '''scheme''' is a locally ringed space $\\struct {X, \\OO_X}$ such that every point $x \\in X$ has an open neighborhood $U$ such that the restriction of $\\struct {X, \\OO_X}$ to $U$ is an affine scheme."}
+{"_id": "29033", "title": "Definition:Affine Scheme", "text": "An '''affine scheme''' is a ringed space which is isomorphic to the spectrum of a commutative ring with unity."}
+{"_id": "29034", "title": "Definition:Sum of Ideals of Ring", "text": "Let $R$ be a ring. === Two ideals === Let $I$ and $J$ be ideals of $R$. Their '''sum''' is the ideal equal to their subset sum: :$I + J  = \\set {i + j : i \\in I \\land j \\in J}$ === Multiple Ideals === {{definition wanted|1) explicit description, and 2) using the iterated induced operation on subsets}}"}
+{"_id": "29035", "title": "Definition:Unique up to Unique Morphism", "text": "Let $\\mathbf C$ be a category. Let $S \\subseteq \\operatorname{Ob}(\\mathbf C)$ be a subclass of its objects. === Definition 1 === The class $S$ is '''unique up to unique morphism''' {{iff}} for all objects $s,t \\in S$ there is a unique morphism from $s$ to $t$. === Definition 2 === The class $S$ is '''unique up to unique morphism''' {{iff}} for all objects $s,t \\in S$ there is a unique morphism from $s$ to $t$, and it is an isomorphism."}
+{"_id": "29036", "title": "Definition:Set of Literals", "text": "Let $S$ be a set. A '''set of literals''' on $S$ is a triple $\\struct {S^\\pm, \\iota, \\theta}$ where: :$S^\\pm$ is a set :$\\iota : S \\to S^\\pm$ is a mapping, the '''canonical injection''' :$\\theta : S^\\pm \\to S^\\pm$ is an involution without fixed points, the '''inversion mapping''', and we also denote $\\map \\theta s = s^{-1}$ such that $S^\\pm = \\iota \\sqbrk S \\sqcup \\theta \\sqbrk {\\iota \\sqbrk S}$ is the disjoint union of the image of $S$ under $\\iota$ and its image under $\\theta$. Explicitly, $S^\\pm$ can be defined as follows. Let $S^\\pm = S \\sqcup S = S \\times \\set 0 \\cup S \\times \\set 1$ be the disjoint union of $S$ with $S$. Let $\\iota: S \\to S^\\pm$ be the canonical mapping: :$s \\mapsto \\tuple {s, 0}$ Let $\\theta : S^\\pm \\to S^\\pm$ be the mapping: :$\\tuple {s, i} \\mapsto \\tuple {s, 1 - i}$ {{tidy|The above phrasing is not up to par}}"}
+{"_id": "29037", "title": "Definition:Reduced Group Word on Set", "text": "Let $S$ be a set. Let $n \\ge 0$ be a natural number. Let $w = w_1 \\cdots w_i \\cdots w_n$ be a group word on $S$ of length $n$. Then $w$ is '''reduced''' {{iff}} $w_i \\ne w_{i + 1}^{-1}$ for all $i \\in \\set {1, \\ldots, n - 1}$"}
+{"_id": "29038", "title": "Definition:Elementary Reduction of Group Word on Set", "text": "Let $X$ be a set. Let $v$ and $w$ be group words on $X$. Let $n$ be the length of $v$. Then $w$ is an '''elementary reduction''' of $v$ {{iff}}: :$w$ has length $n - 2$ :There exists $k \\in \\set {1, \\ldots, n - 1}$ such that: ::$v_k = v_{k + 1}^{-1}$ ::$w_i = \\begin {cases} v_i & : i < k \\\\ v_{i + 2} & : i > k + 1 \\end {cases}$ This is denoted $v \\overset 1 \\longrightarrow w$."}
+{"_id": "29039", "title": "Definition:Reduction of Group Word on Set", "text": "Let $X$ be a set. Let $w$ be a group word on $X$. A '''reduction''' of $w$ is a finite sequence of group words $\\tuple {w^{\\paren 0}, \\ldots, w^{\\paren n} }$ such that: :$w^{\\paren 0} = w$ :$w_{i + 1}$ is an elementary reduction of $w_i$, for all $i \\in \\set {0, \\ldots, n - 1}$ :$w^{\\paren n}$ is reduced"}
+{"_id": "29040", "title": "Definition:Reduced Form of Group Word", "text": "Let $X$ be a set. Let $w$ be a group word on $X$. The '''reduced form''' $\\map {\\operatorname {red} } w$ of $w$ is the unique reduced word for which there exists a reduction: :$w = w^{\\paren 0} \\to w^{\\paren 1} \\to \\cdots \\to w^{\\paren n} = \\map {\\operatorname {red} } w$"}
+{"_id": "29041", "title": "Definition:Composition of Reduced Group Words", "text": "Let $X$ be a set. Let $v$ and $w$ be reduced group words on $X$. The '''composition''' of $v$ and $w$ is the reduced form of their concatenation: :$v \\cdot w = \\operatorname{red}(vw)$"}
+{"_id": "29042", "title": "Definition:Group of Reduced Group Words", "text": "Let $S$ be a set. The '''group of reduced group words''' on $S$ is the group $\\left({F_S, \\circ}\\right)$ where: :$F_S$ is the set of reduced group words on $S$ :$\\circ$ denotes composition of reduced group words."}
+{"_id": "29043", "title": "Definition:Free Group on Set", "text": "Let $X$ be a set. A '''free group''' on $X$ is a certain $X$-pointed group, that is, a pair $(F, \\iota)$ where: :$F$ is a group :$\\iota : X \\to F$ is a mapping that can be defined as follows: === Definition 1: by universal property === A '''free group''' on $X$ is an $X$-pointed group $(F, \\iota)$ that satisfies the following universal property: :For every $X$-pointed group $(G, \\kappa)$ there exists a unique group homomorphism $\\phi : F \\to G$ such that $\\phi \\circ \\iota = \\kappa$, that is, a morphism of pointed groups $F \\to G$. === Definition 2: As the group of reduced group words === The '''free group''' on $X$ is the pair $(F, \\iota)$ where: :$F$ is the group of reduced group words on $X$ :$\\iota : X \\to F$ is the canonical injection"}
+{"_id": "29044", "title": "Definition:Group Defined by Presentation", "text": "Let $S$ be a set. Let $R$ be a set of group words on $S$. Let $F_S$ be the group of reduced group words on $S$. Let $\\operatorname{red} \\left({R}\\right)$ be the set of reduced forms of elements of $R$. Let $N$ be the normal subgroup generated by $\\operatorname{red} \\left({R}\\right)$ in $F$. The '''group defined by the presentation''' $\\langle S \\mid R \\rangle$ is the quotient group $F_S / N$."}
+{"_id": "29045", "title": "Definition:Group Word on Set", "text": "Let $S$ be a set. A '''group word''' on $S$ is an ordered tuple on the set of literals $S^\\pm$ of $S$."}
+{"_id": "29046", "title": "Definition:Rank of Free Group", "text": "Let $F$ be a free group. The '''rank''' of $F$ is the dimension of its abelianization as a module over $\\Z$."}
+{"_id": "29047", "title": "Definition:Commutator Subgroup", "text": "Let $G$ be a group. Its '''commutator subgroup''' $\\sqbrk {G, G}$ is the subgroup generated by all commutators. === Higher derived subgroups === Let $n \\ge 0$ be a natural number. The $n$th '''derived subgroup''' $G^{\\paren n}$ is recursively defined as: :$G^{\\paren n} = \\begin{cases} G & : n = 0 \\\\ \\sqbrk {G^{\\paren {n - 1} }, G^{\\paren {n - 1} } } & : n \\ge 1 \\end{cases}$"}
+{"_id": "29048", "title": "Definition:Abelianization of Group", "text": "Let $G$ be a group. Its '''abelianization''' is the quotient by its commutator subgroup: :$G^{\\mathrm {ab} } = G / \\sqbrk {G, G}$"}
+{"_id": "29049", "title": "Definition:Cartesian Closed Set", "text": "Let $S$ be a set. Then $S$ is '''cartesian closed''' {{iff}} the cartesian product $S \\times S$ is a subset of $S$."}
+{"_id": "29051", "title": "Definition:Group Relation on Set", "text": "Let $X$ be a set. A '''group relation''' on $X$ is a pair $(u, v)$ where $u$ and $v$ are group words on $X$. A '''group relation''' $(u, v)$ is also denoted $u=v$."}
+{"_id": "29054", "title": "Definition:Generator of Algebraic Structure", "text": "Let $\\left({A, \\circ}\\right)$ be an algebraic structure. Let $G \\subset A$ be a subset. === Definition 1 === The subset $G$ is a '''generator''' of $A$ {{iff}} $A$ is the algebraic substructure generated by $G$. === Definition 2 === The subset $G$ is a '''generator''' of $A$ {{iff}}: * $\\forall x, y \\in G: x \\circ y \\in A$; * $\\forall z \\in A: \\exists x, y \\in W \\left({G}\\right): z = x \\circ y$ where $W \\left({G}\\right)$ is the set of words of $G$. That is, every element in $A$ can be formed as the product of a finite number of elements of $G$. If $G$ is such a set, then we can write $A = \\left \\langle {G}\\right \\rangle$."}
+{"_id": "29055", "title": "Definition:Cartesian Product/Uncountable", "text": "Let $I$ be an indexing set with uncountable cardinality. Let $\\family {S_\\alpha}_{\\alpha \\mathop \\in I}$ be a family of sets indexed by $I$. The '''cartesian product''' of $\\family {S_\\alpha}$ is denoted: :$\\displaystyle \\prod_{\\alpha \\mathop \\in I} S_\\alpha$"}
+{"_id": "29059", "title": "Definition:Cauchy Sequence in Topological Group", "text": "Let $G$ be a topological group with identity element $e$. Let $\\sequence {a_n}$ be a sequence in $G$. Then $\\sequence {a_n}$ is a '''Cauchy sequence''' {{iff}} for every neighborhood $U$ of $e$ there exists $N \\in \\N$ such that for all $m, n \\ge N : a_n a_m^{-1} \\in U$."}
+{"_id": "29060", "title": "Definition:Semiring of Natural Numbers", "text": "The '''semiring of natural numbers''' is the semiring $\\struct {\\N, +, \\times}$ where: :$\\N$ is the set of natural numbers :$+$ is the addition :$\\times$ is the multiplication"}
+{"_id": "29061", "title": "Definition:Localization of Ring at Prime Ideal", "text": "Let $A$ be a commutative ring with unity. Let $\\mathfrak p$ be a prime ideal of $A$. The '''localization of $A$ at $\\mathfrak p$''' is the localization of $A$ at the complement $A\\setminus\\mathfrak p$: :$A_{\\mathfrak p} = (A\\setminus\\mathfrak p)^{-1}A$"}
+{"_id": "29062", "title": "Definition:Localization of Ring at Element", "text": "Let $A$ be a commutative ring with unity. Let $f\\in A$ be an element. The '''localization of $A$ at $f$''' is the localization of $A$ at the set of powers $\\{1, f, f^2, \\ldots\\}$: :$A_f = (\\{1, f, f^2, \\ldots\\})^{-1}A$"}
+{"_id": "29063", "title": "Definition:Saturation of Multiplicatively Closed Subset of Ring", "text": "Let $A$ be a commutative ring with unity. Let $S \\subset A$ be a multiplicatively closed subset. === Definition 1 === The '''saturation''' of $S$ is the smallest saturated multiplicatively closed subset of $A$ containing $S$. That is, it is the intersection of all saturated multiplicatively closed subsets containing $S$. {{circularStructure|this is a general issue with definitions of examples of closure operators}} === Definition 2 === The '''saturation''' of $S$ is the set of divisors of elements of $S$. === Definition 3 === The '''saturation''' of $S$ is the set of elements whose image in the localization $A_S$ is a unit of $A$. === Definition 4 === The '''saturation''' of $S$ is the complement relative to $A$ of the union of prime ideals that are disjoint from $S$: :$\\map {\\operatorname {Sat} } S = A \\setminus \\displaystyle \\bigcup \\set {\\mathfrak p \\in \\operatorname{Spec} A: \\mathfrak p \\cap S = \\O}$"}
+{"_id": "29064", "title": "Definition:Closure of Set under Closure Operator", "text": "Let $S$ be a set. Let $\\operatorname{cl}$ be a closure operator on $S$. Let $T \\subseteq S$ be a subset of $S$. The '''closure''' of $T$ is its image $\\operatorname{cl}(T)$."}
+{"_id": "29065", "title": "Definition:Weakly Locally Connected at Point", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $x \\in S$. === Definition 1 === {{Definition:Weakly Locally Connected at Point/Definition 1}} === Definition 2 === {{Definition:Weakly Locally Connected at Point/Definition 2}}"}
+{"_id": "29067", "title": "Definition:Zero Morphism via Zero Object", "text": "Let $C$ be a category with has a zero object $0$. Let $a,b\\in C$ be objects. The '''zero morphism $0$''' from $a$ to $b$ is the composition of the unique morphism $a \\to 0$ and the unique morphism $0 \\to b$: :$0 : a \\to 0 \\to b$"}
+{"_id": "29068", "title": "Definition:Pointed Category", "text": "Let $C$ be a category. Then $C$ is '''pointed''' {{iff}} $C$ has a zero object. Category:Definitions/Category Theory 8fy4dm6wwomuet3up4qrlrjhr06rgaj"}
+{"_id": "29069", "title": "Definition:Universal Object", "text": "Let $C$ be a category. A '''universal object''' of $C$ is an object that is initial or terminal."}
+{"_id": "29070", "title": "Definition:Disconnected (Topology)/Topological Space", "text": "A topological space $T$ is '''disconnected''' {{iff}} it is not connected."}
+{"_id": "29071", "title": "Definition:Kernel (Category Theory)", "text": "Let $C$ be a category. Let $f : A \\to B $ be a morphism. === Definition 1: for categories with initial objects === Let $C$ have an initial object $0$. A '''kernel''' of $f$ is a morphism $\\map \\ker f \\to A$ which is a pullback of the unique morphism $0 \\to B$ via $f$ to $A$. === Definition 2: for categories with zero objects === Let $C$ have an zero object $0$. A '''kernel''' of $f$ is the equalizer of $f$ and the zero morphism $0 : A \\to B$."}
+{"_id": "29072", "title": "Definition:Generated Field Extension", "text": "Let $E / F$ be a field extension. Let $S \\subset E$ be a subset of $E$. === Definition 1 === {{:Definition:Generated Field Extension/Definition 1}} === Definition 2 === {{:Definition:Generated Field Extension/Definition 2}}"}
+{"_id": "29073", "title": "Definition:Non-Associative Ring", "text": "A '''non-associative ring''' is a ringoid $\\struct {R, +, \\times}$ such that $\\struct {R, +}$ is an abelian group."}
+{"_id": "29074", "title": "Definition:Nonassociative Ring with Unity", "text": "A '''non-associative ring with unity''' is a non-associative ring $(R, +, \\times)$ such that $(R, \\times)$ is a unitary magma."}
+{"_id": "29075", "title": "Definition:Group Action of GL(2,Q) on Irrational Numbers", "text": "Let $\\GL {2, \\Q}$ be the general linear group on the field of rational numbers. Let $\\R \\setminus \\Q$ be the set of irrational numbers. The '''standard group action''' of $\\GL {2, \\Q}$ on $\\R \\setminus \\Q$ is the group action: :$\\GL {2, \\Q} \\times \\paren {\\R \\setminus \\Q} \\to \\R \\setminus \\Q$: :$\\paren {\\begin {pmatrix} a & b \\\\ c & d \\end {pmatrix}, x} \\mapsto \\dfrac {a x + b} {c x + d}$"}
+{"_id": "29076", "title": "Definition:Disconnected (Topology)/Set", "text": "{{:Definition:Disconnected (Topology)/Set/Definition 1}}"}
+{"_id": "29077", "title": "Definition:Disconnected (Topology)/Points", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $a, b \\in S$. Then $a$ and $b$ are '''disconnected (in $T$)''' {{iff}} they are not '''connected (in $T$)'''."}
+{"_id": "29078", "title": "Definition:Local Dimension of Topological Space", "text": "Let $X$ be a topological space. Let $x \\in X$. The '''local dimension of $X$ at $x$''' is the supremum of lengths of chains of closed irreducible sets of $T$ containing $x$, ordered by inclusion. Thus, the '''Krull dimension''' is $\\infty$ {{iff}} there exist arbitrarily long chains containing $x$."}
+{"_id": "29079", "title": "Definition:Krull Dimension of Scheme", "text": "Let $\\struct {X, \\mathcal O_X}$ be a scheme. Its '''Krull dimension''' is the Krull dimension of $X$."}
+{"_id": "29080", "title": "Definition:Disconnected (Topology)/Set/Definition 1", "text": "$H$ is a '''disconnected set of $T$''' {{iff}} it is not a '''connected set of $T$'''."}
+{"_id": "29081", "title": "Definition:Disconnected (Topology)/Set/Definition 2", "text": "$H$ is a '''disconnected set of $T$''' {{iff}} there exist open sets $U$ and $V$ of $T$ such that: : $H \\subseteq U \\cup V$ : $H \\cap U \\cap V = \\O$ : $U \\cap H \\ne \\O$ and: : $V \\cap H \\ne \\O$"}
+{"_id": "29082", "title": "Definition:Local Krull Dimension of Scheme", "text": "Let $\\struct {X, \\mathcal O_X}$ be a scheme. Let $x \\in X$. The '''local Krull dimension at $x$''' is the local dimension of $X$ at $x$."}
+{"_id": "29083", "title": "Definition:Vanishing Set of Subset of Ring", "text": "Let $A$ be a commutative ring with unity. Let $S \\subseteq A$ be a subset. The '''vanishing set''' of $S$ is the set of prime ideals of $A$ containing $S$: :$V(S) = \\{\\mathfrak p \\in \\operatorname{Spec}(A) : \\mathfrak p \\supseteq S \\}$"}
+{"_id": "29084", "title": "Definition:Structure Sheaf of Spectrum of Ring", "text": "Let $A$ be a commutative ring with unity. Let $\\struct {\\Spec A, \\tau}$ be its spectrum with Zariski topology $\\tau$ === Definition 1 === Note that Principal Open Subsets form Basis of Zariski Topology on Prime Spectrum. We define the structure sheaf of $\\Spec A$ to be the sheaf induced by a sheaf on this basis defined as follows: :For $f \\in A$, $\\map \\OO {\\map X f}$ is the localization of $A$ at $f$ :For $f, g \\in A$ with $\\map X f \\supset \\map X g$, the restriction is the induced homomorphism of $A$-algebras $A_f \\to A_g$. === Definition 2 === {{definition wanted|the étalé space point of view}} === Definition 3 === {{definition wanted|using explicit formulas}}"}
+{"_id": "29085", "title": "Definition:Irreducible Scheme", "text": "Let $\\struct {X, \\mathcal O_X}$ be a scheme. Then $\\struct {X, \\mathcal O_X}$ is '''irreducible''' {{iff}} $X$ is an irreducible space."}
+{"_id": "29086", "title": "Definition:Local Ring Homomorphism", "text": "Let $(A, \\mathfrak m)$ and $(B, \\mathfrak n)$ be commutative local rings. Let $f : A \\to B$ be a unital ring homomorphism. === Definition 1 === {{Definition:Local Ring Homomorphism/Definition 1}} === Definition 2 === {{Definition:Local Ring Homomorphism/Definition 2}} === Definition 3 === {{Definition:Local Ring Homomorphism/Definition 3}}"}
+{"_id": "29087", "title": "Definition:Quasicompact Scheme", "text": "Let $\\struct {X, \\mathcal O_X}$ be a scheme. Then $X$ is '''quasicompact''' {{iff}} $X$ is a compact space."}
+{"_id": "29088", "title": "Definition:Preadditive Category", "text": "A '''preadditive category''' is a monoidally enriched category over the monoidal category of abelian groups $\\mathbf {A b}$. That is, a category such that: :its hom sets are abelian groups and where: :composition is bilinear."}
+{"_id": "29089", "title": "Definition:Sequence of Partial Quotients", "text": "Let $F$ be a field. Let $n \\in \\N\\cup\\{\\infty\\}$ be an extended natural number. Let $C$ be a continued fraction in $F$ of length $n$. The '''sequence of partial quotients''' of $C$ is just $C$ itself. That is, a continued fraction equals its '''sequence of partial quotients'''."}
+{"_id": "29090", "title": "Definition:Degenerate Connected Set/Topological Space", "text": "$T$ is a '''degenerate connected space''' {{iff}} it contains exactly one element."}
+{"_id": "29091", "title": "Definition:Canonical Form", "text": "A '''canonical form''' of a mathematical object is a standard way of presenting that object as a mathematical expression."}
+{"_id": "29093", "title": "Definition:Ordered Set of Natural Numbers", "text": "The '''ordered set of natural numbers''' is the ordered set $(\\N, \\leq)$ where: :$\\N$ is the set of natural numbers :$\\leq$ is the ordering on the natural numbers"}
+{"_id": "29094", "title": "Definition:Extended Natural Numbers", "text": "Let $\\N$ be the set of natural numbers. === Definition 1 === The '''extended natural numbers''' are the elements of the ordered set $\\struct {\\N \\cup \\set \\infty, \\le}$, where :$\\infty$ is a new element, called '''infinity''' :$\\le$ is the ordering on the extended natural numbers === Definition 2 === The '''extended natural numbers''' are the order completion of the ordered set of natural numbers."}
+{"_id": "29095", "title": "Definition:Length of Continued Fraction", "text": "Let $k$ be a field. Let $C$ be a continued fraction in $k$, either finite or infinite. The '''length''' of $C$ is an extended natural number equal to: *$\\infty$ if $C$ is an infinite continued fraction. *$n$ if $C$ is a finite continued fraction with domain the integer interval $\\left[0 \\,.\\,.\\, n\\right]$."}
+{"_id": "29096", "title": "Definition:Nonzero Ring Element", "text": "Let $R$ be a ring. A '''nonzero element''' of $R$ is an element that is not its ring zero."}
+{"_id": "29097", "title": "Definition:Divisor-Finite Monoid", "text": "Let $(M, *)$ be a monoid. Then $M$ is '''divisor-finite''' {{iff}} for all $m \\in M$ the set: :$\\{(x, y) \\in M^2 : x*y = m \\}$ is finite."}
+{"_id": "29098", "title": "Definition:Convolution of Mappings on Divisor-Finite Monoid", "text": "Let $\\struct {M, \\cdot}$ be a divisor-finite monoid. Let $\\struct {R, +, \\times}$ be a non-associative ring. Let $f, g : M \\to R$ be mappings. The '''convolution''' of $f$ and $g$ is the mapping $f * g: M \\to R$ defined as: :$\\forall m \\in M: \\map {\\paren {f * g} } m := \\displaystyle \\sum_{x y \\mathop = m} \\map f x \\times \\map g y$ where the summation is over the finite set $\\set {\\tuple {x, y} \\in M^2: x y = m}$."}
+{"_id": "29099", "title": "Definition:Big Monoid Ring", "text": "Let $(M,\\cdot)$ be a divisor-finite monoid. Let $(R, +, \\times)$ be an additive semiring. The '''big monoid ring''' of $R$ over $M$ is the ringoid $(R^M, +, *)$ where: :$R^M$ be the set of all mappings $M \\to R$ :$+$ is the pointwise operation induced by $+$ :$*$ denotes convolution of mappings"}
+{"_id": "29101", "title": "Definition:Generic Point of Topological Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $x \\in S$ be an element of $S$. === Definition 1 === The point $x$ is a '''generic point of $T$''' {{iff}} the closure of the singleton $\\left\\{{x}\\right\\}$ is $S$. === Definition 2 === The point $x$ is a '''generic point of $T$''' {{iff}} $x$ is contained in every non-empty open subset of $T$."}
+{"_id": "29103", "title": "Definition:Sober Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Then $T$ is a '''sober space''' {{iff}} each closed irreducible subspace of $T$ has a unique generic point. Category:Definitions/Topology iiihtfrup7ghe0k7t25kuwvjw1yyw1s"}
+{"_id": "29104", "title": "Definition:Ring of Formal Power Series", "text": "Let $R$ be a commutative ring with unity. === One variable === A '''ring of formal power series''' over $R$ is a pointed algebra over $R$, that is, an ordered triple $\\left({RX, \\iota, X}\\right)$ where: :$RX$ is a commutative ring with unity :$\\iota : R \\to RX$ is a unital ring homomorphism, called canonical embedding :$X$ is an element of $RX$, called indeterminate that may be defined as follows: Let $\\N$ be the additive monoid of natural numbers. Let $R\\N$ be the big monoid ring of $R$ over $\\N$. Let $\\iota : R \\to R\\N$ be the embedding. Let $X \\in R\\N$ be the mapping $X : \\N \\to R$ defined by $X(n) = 1$ if $n=1$ and $X(n) = 0$ otherwise. The '''ring of formal power series''' over $R$ is the ordered triple $\\left({R\\N, \\iota, X}\\right)$ === Multiple variables === {{definition wanted}}"}
+{"_id": "29105", "title": "Definition:Ring of Formal Laurent Series", "text": "A '''ring of formal Laurent series in one variable''' over $R$ is a pointed algebra over $R$, that is, an ordered triple $\\tuple {\\map R {\\paren X}, \\iota, X}$ where: :$\\map R {\\paren X}$ is a commutative ring with unity :$\\iota: R \\to \\map R {\\paren X}$ is a unital ring homomorphism, called canonical embedding :$X$ is an element of $\\map R {\\paren X}$, called variable that may be defined as follows: {{explain|Research that double parenthesis notation and investigate its validity and specific meaning}} Let $\\tuple {R \\sqbrk {\\sqbrk X}, \\kappa, X}$ be a ring of formal power series in one variable over $R$. Let $\\tuple {\\map R {\\paren X}, \\lambda}$ be the localization of $R \\sqbrk {\\sqbrk X}$ at $X$. The '''ring of formal Laurent series''' over $R$ is the ordered triple $\\tuple {\\map R {\\paren X}, \\lambda \\circ \\kappa, \\map \\lambda X}$."}
+{"_id": "29106", "title": "Definition:Ring of Formal Dirichlet Series", "text": "The '''ring of formal Dirichlet series''' is the topological ring $(\\mathcal A, \\tau)$ where: :$\\mathcal A$ is the ring of arithmetic functions :$\\tau$ is the topology on formal dirichlet series Category:Definitions/Dirichlet Series l5z5hd2tvlwjh8bdwmnpwvrr0dx7dw2"}
+{"_id": "29107", "title": "Definition:Field of Formal Laurent Series", "text": "Let $k$ be a field. The ring of formal Laurent series $k((X))$ over $k$ in a variable $X$ is also called the '''field of formal Laurent series'''."}
+{"_id": "29108", "title": "Definition:Continued Fraction Expansion of Laurent Series", "text": "Let $k$ be a field. Let $k((t^{-1}))$ be the field of formal Laurent series in the variable $t^{-1}$. === Irrational Laurent series === Let $f \\in k((t^{-1}))$ be an irrational formal Laurent series. The '''continued fraction expansion''' of $f$ is the infinite continued fraction $(\\lfloor \\alpha_n \\rfloor)_{n\\geq0}$ where $\\alpha_n$ is recursively defined as: :$\\alpha_n = \\displaystyle \\begin{cases} f & : n = 0 \\\\  \\dfrac 1 {f - \\lfloor f \\rfloor} & : n \\geq 1 \\end{cases}$ where $\\lfloor \\cdot \\rfloor$ denotes the polynomial part. === Rational Laurent series === {{definition wanted}}"}
+{"_id": "29110", "title": "Definition:Irreducible Space/Definition 6", "text": "A topological space $T = \\struct {S, \\tau}$ is '''irreducible''' {{iff}} the closure of every non-empty open set is the whole space: :$\\forall U \\in \\tau: U^- = S$"}
+{"_id": "29111", "title": "Definition:Ultraconnected Space/Definition 1", "text": "A topological space $T = \\struct {S, \\tau}$ is '''ultraconnected''' {{iff}} no two non-empty closed sets are disjoint."}
+{"_id": "29112", "title": "Definition:Ultraconnected Space/Definition 2", "text": "A topological space $T = \\struct {S, \\tau}$ is '''ultraconnected''' {{iff}} the closures of every distinct pair of elements of $S$ are not disjoint: :$\\forall x, y \\in S: \\set x^- \\cap \\set y^- \\ne \\O$"}
+{"_id": "29114", "title": "Definition:Irreducible Space/Definition 7", "text": "A topological space $T = \\struct {S, \\tau}$ is '''irreducible''' {{iff}} every open set of $T$ is connected."}
+{"_id": "29115", "title": "Definition:Locally Connected Space/Definition 4", "text": "A topological space $T = \\struct {S, \\tau}$ is '''locally connected''' {{iff}} the components of the open sets of $T$ are also open in $T$."}
+{"_id": "29116", "title": "Definition:Convergent of Continued Fraction/Definition 1", "text": "The '''$k$th convergent''' $C_k$ of $C$ is the value of the finite continued fraction: :$C_k = \\left[{a_0, a_1, \\ldots, a_k}\\right]$"}
+{"_id": "29117", "title": "Definition:Convergent of Continued Fraction/Definition 2", "text": "The '''$k$th convergent''' $C_k$ of $C$ is the quotient of the $k$th numerator $p_k$ by the $k$th denominator $q_k$: :$C_k = \\dfrac{p_k}{q_k}$"}
+{"_id": "29118", "title": "Definition:New Element", "text": "Let $S$ be a set. A '''new element for $S$''' is a set that is not an element of $S$. By Set does not Contain Itself, it may be defined as $S$ itself."}
+{"_id": "29119", "title": "Definition:Locally Path-Connected Space/Definition 4", "text": "A topological space $T = \\struct{S, \\tau}$ is a '''locally path-connected space''' {{iff}} the path components of open sets of $T$ are also open in $T$."}
+{"_id": "29120", "title": "Definition:Weakly Locally Connected at Point/Definition 1", "text": "The space $T$ is '''weakly locally connected at $x$''' {{iff}} $x$ has a neighborhood basis consisting of connected sets."}
+{"_id": "29121", "title": "Definition:Weakly Locally Connected at Point/Definition 2", "text": "The space $T$ is '''weakly locally connected at $x$''' {{iff}} every open neighborhood $U$ of $x$ contains an open neighborhood $V$ such that every two points of $V$ lie in some connected subset of $U$."}
+{"_id": "29122", "title": "Definition:Fundamental Group Functor", "text": "Let $\\mathbf{Top}^*$ be the category of pointed topological spaces. Let $\\mathbf{Grp}$ be the category of groups. The '''fundamental group functor''' is the covariant functor $\\pi_1 : \\mathbf{Top}^* \\to \\mathbf{Grp}$ with: {{DefineFunctor |ob = A pointed topological space $(X, x_0)$ is sent to its fundamental group. |mor = A continuous pointed mapping $f$ is sent to its induced homomorphism of fundamental groups $\\pi_1(f) = f_*$. }}"}
+{"_id": "29123", "title": "Definition:Pointed Topological Space", "text": "A '''pointed topological space''' is an ordered pair $\\left({T, x_0}\\right)$ where: :$T$ is a topological space :$x_0$ is a point of $T$."}
+{"_id": "29124", "title": "Definition:Discrete Category on Set", "text": "Let $S$ be a set. The '''discrete category on $S$''' is the discrete category $\\mathbf{Dis} \\left({S}\\right)$ with: {{DefineCategory |ob = elements of $S$ |mor = only identity mappings |comp = composition of mappings }}"}
+{"_id": "29125", "title": "Definition:Graded Object", "text": "Let $C$ be a category. Let $S$ be a set. Let $\\mathbf{Dis}(S)$ be its discrete category. An '''$S$-graded object''' of $C$ is a covariant functor $\\mathbf{Dis}(S) \\to C$."}
+{"_id": "29126", "title": "Definition:Binary Biproduct", "text": "Let $A$ be a category. Let $a_1, a_2$ be objects of $A$. === Definition 1 === A '''biproduct''' of $a_1$ and $a_2$ is an ordered tuple $\\tuple {a_1 \\oplus a_2, p_1, p_2, i_1, i_2}$ such that: :$\\tuple {a_1 \\oplus a_2, p_1, p_2}$ is a binary product :$\\tuple {a_1 \\oplus a_2, i_1, i_2}$ is a binary coproduct === Definition 2: for categories with zero morphisms === Let $A$ be a category with zero morphisms. Then $a_1$ and $a_2$ are said to '''have a biproduct''' {{iff}}: :they have a coproduct $\\tuple {a_1 \\sqcup a_2, j_1, j_2}$ and a product $\\tuple {a_1 \\times a_2, q_1, q_2}$ :the canonical mapping $r: a_1 \\sqcup a_2 \\to a_1 \\times a_2$ is an isomorphism in which case: :$\\tuple {a_1 \\sqcup a_2, j_1, j_2, q_1 \\circ r, q_2 \\circ r}$ :$\\tuple {a_1 \\times a_2, r \\circ j_1, r \\circ j_2, q_1, q_2}$ are '''biproducts''' of $a_1$ and $a_2$. === Definition 3: for preadditive categories === Let $A$ be a preadditive category. A '''biproduct''' of $a_1$ and $a_2$ is an ordered tuple $\\tuple {a_1 \\oplus a_2, p_1, p_2, i_1, i_2}$ where: :$a \\oplus a_2$ is an object of $A$ :$i_1 : a_1 \\to a_1 \\oplus a_2$ :$i_2 : a_1 \\to a_1 \\oplus a_2$ :$p_1 : a_1 \\oplus a_2 \\to a_2$ :$p_2 : a_1 \\oplus a_2 \\to a_2$ are morphisms such that: :$p_1 \\circ i_1 = 1_{a_1}$ :$p_2 \\circ i_2 = 1_{a_2}$ :$i_1 \\circ p_1 + i_2 \\circ p_2 = 1_{a_1 \\oplus a_2}$ where $1$ denotes the identity morphism. === Definition 4: for preadditive categories === Let $A$ be a preadditive category. A '''biproduct''' of $a_1$ and $a_2$ is an ordered tuple $\\tuple {a_1 \\oplus a_2, p_1, p_2, i_1, i_2}$ where: :$a \\oplus a_2$ is an object of $A$ :$i_1 : a_1 \\to a_1 \\oplus a_2$ :$i_2 : a_1 \\to a_1 \\oplus a_2$ :$p_1 : a_1 \\oplus a_2 \\to a_2$ :$p_2 : a_1 \\oplus a_2 \\to a_2$ are morphisms such that: :$p_1 \\circ i_1 = 1_{a_1}$ :$p_2 \\circ i_2 = 1_{a_2}$ :$p_1 \\circ i_2 = 0_{a_1}$ :$p_2 \\circ i_1 = 0_{a_2}$ :$i_1 \\circ p_1 + i_2 \\circ p_2 = 1_{a_1 \\oplus a_2}$ where: :$1$ denotes the identity morphism :$0$ denotes the zero morphism."}
+{"_id": "29127", "title": "Definition:Zero Morphism in Preadditive Category", "text": "Let $A$ be a preadditive category. Let $a,b$ be objects of $A$. The '''zero morphism''' $0 : a \\to b$ is the identity element of their hom set $\\operatorname{Hom}(a, b)$."}
+{"_id": "29128", "title": "Definition:Additive Category", "text": "Let $A$ be a preadditive category. === Definition 1: binary biproducts === The category $A$ is an '''additive category''' {{iff}}: * it has a zero object * it has all binary biproducts. === Definition 2: binary products === The category $A$ is an '''additive category''' {{iff}}: * it has a zero object * it has all binary products. === Definition 3: binary coproducts === The category $A$ is an '''additive category''' {{iff}}: * it has a zero object * it has all binary coproducts. === Definition 4: finite biproducts === The category $A$ is an '''additive category''' {{iff}} it has all finite biproducts. === Definition 5: finite products === The category $A$ is an '''additive category''' {{iff}} it has all finite products. === Definition 6: finite coproducts === The category $A$ is an '''additive category''' {{iff}} it has all finite coproducts."}
+{"_id": "29129", "title": "Definition:Canonical Mapping from Coproduct to Product", "text": "Let $C$ be a category with zero morphisms. Let $a, b$ be objects of $C$. Assume they have a coproduct $(a \\sqcup b, i_1, i_2)$ and a product $(a \\times b, p_1, p_2)$. === Definition 1 === Let: :$j_1 : a \\to a \\times b$ be the unique morphism such that: :*$p_1 \\circ j_1 = 1 : a \\to a$ :*$p_2 \\circ j_1 = 0 : a \\to b$ :$j_2 : b \\to a \\times b$ be the unique morphism such that: :*$p_1 \\circ j_2 = 0 : b \\to a$ :*$p_2 \\circ j_2 = 1 : b \\to b$ The '''canonical mapping''' from $a \\sqcup b$ to $a \\times b$ is the unique morphism $r : a \\sqcup b \\to a \\times b$ such that: :$r \\circ i_1 = j_1$ :$r \\circ i_2 = j_2$ === Definition 2 === Let: :$q_1 : a \\sqcup b \\to a$ be the unique morphism such that: :*$q_1 \\circ j_1 = 1 : a \\to a$ :*$q_1 \\circ j_2 = 0 : b \\to a$ :$q_2 : a \\sqcup b \\to b$ be the unique morphism such that: :*$q_2 \\circ j_1 = 0 : a \\to a$ :*$q_2 \\circ j_2 = 1 : b \\to a$ The '''canonical mapping''' from $a \\sqcup b$ to $a \\times b$ is the unique morphism $r : a \\sqcup b \\to a \\times b$ such that: :$p_1 \\circ r = q_1$ :$p_2 \\circ r = q_2$ {{definition wanted|for arbitrarily many objects}}"}
+{"_id": "29130", "title": "Definition:Totally Separated Space/Definition 1", "text": "A topological space $T = \\struct {S, \\tau}$ is '''totally separated''' {{iff}}: :For every $x, y \\in S: x \\ne y$ there exists a separation $U \\mid V$ of $T$ such that $x \\in U, y \\in V$."}
+{"_id": "29131", "title": "Definition:Totally Separated Space/Definition 2", "text": "A topological space $T = \\struct {S, \\tau}$ is '''totally separated''' {{iff}} each of its quasicomponents is a singleton set."}
+{"_id": "29132", "title": "Definition:Pre-Abelian Category", "text": "=== Definition 1 === A '''pre-abelian category''' is an additive category in which every morphism has a kernel and a cokernel. === Definition 2 === A '''pre-abelian category''' is an additive category with all finite limits and finite colimits."}
+{"_id": "29133", "title": "Definition:Abelian Category", "text": "=== Definition 1 === An '''abelian category''' is a pre-abelian category in which: * every monomorphism is a kernel * every epimorphism is a cokernel === Definition 2 === An '''abelian category''' is a pre-abelian category in which: * every monomorphism is the kernel of its cokernel * every epimorphism is the cokernel of its kernel === Definition 3 === An '''abelian category''' is a pre-abelian category in which for every morphism $f$, the canonical morphism from its coimage to its image $\\operatorname{coim}(f) \\to \\operatorname{im}(f)$ is an isomorphism."}
+{"_id": "29134", "title": "Definition:Extremally Disconnected Space/Definition 1", "text": "A $T_2$ (Hausdorff) topological space $T = \\struct {S, \\tau}$ is '''extremally disconnected''' {{iff}} the closure of every open set of $T$ is open."}
+{"_id": "29135", "title": "Definition:Extremally Disconnected Space/Definition 3", "text": "A $T_2$ (Hausdorff) topological space $T = \\struct {S, \\tau}$ is '''extremally disconnected''' {{iff}} the closures of every pair of open sets which are disjoint are also disjoint."}
+{"_id": "29136", "title": "Definition:Extremally Disconnected Space/Definition 2", "text": "A $T_2$ (Hausdorff) topological space $T = \\struct {S, \\tau}$ is '''extremally disconnected''' {{iff}} the interior of every closed set of $T$ is closed."}
+{"_id": "29140", "title": "Definition:Isometry (Metric Spaces)/Definition 1", "text": "Let $M_1 = \\tuple {A_1, d_1}$ and $M_2 = \\tuple {A_2, d_2}$ be metric spaces or pseudometric spaces. Let $\\phi: A_1 \\to A_2$ be a bijection such that: :$\\forall a, b \\in A_1: \\map {d_1} {a, b} = \\map {d_2} {\\map \\phi a, \\map \\phi b}$ Then $\\phi$ is called an '''isometry'''. That is, an '''isometry''' is a distance-preserving bijection."}
+{"_id": "29141", "title": "Definition:Isometry (Metric Spaces)/Definition 2", "text": "Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces or pseudometric spaces. :$M_1$ and $M_2$ are '''isometric''' {{iff}} there exist inverse mappings $\\phi: A_1 \\to A_2$ and $\\phi^{-1}: A_2 \\to A_1$ such that: ::$\\forall a, b \\in A_1: \\map {d_1} {a, b} = \\map {d_2} {\\map \\phi a, \\map \\phi b}$ :and: ::$\\forall u, v \\in A_2: \\map {d_2} {u, v} = \\map {d_1} {\\map {\\phi^{-1} } u, \\map {\\phi^{-1} } v}$"}
+{"_id": "29142", "title": "Definition:Isometry (Metric Spaces)/Into", "text": "Let $\\phi: A_1 \\to A_2$ be an injection such that: : $\\forall a, b \\in A_1: d_1 \\left({a, b}\\right) = d_2 \\left({\\phi \\left({a}\\right), \\phi \\left({b}\\right)}\\right)$ Then $\\phi$ is called an '''isometry (from $M_1$) ''into'' $M_2$'''."}
+{"_id": "29150", "title": "Definition:Presheaf on Topological Space/Definition 1", "text": "A $\\mathbf C$-valued '''presheaf''' on $T$ is a pair $\\left({\\mathcal F, \\operatorname{res} }\\right)$ where: : $\\mathcal F$ is a mapping on $\\tau$ whose image consists of objects of $\\mathbf C$ : $\\operatorname{res}$ is a mapping on $\\left\\{ {\\left({U, V}\\right) \\in \\tau^2: U \\supseteq V}\\right\\}$ such that for all $U, V, W \\in \\tau$ with $U \\supseteq V \\supseteq W$: :: $\\operatorname{res}_V^U$ is a morphism from $\\mathcal F \\left({U}\\right)$ to $\\mathcal F \\left({V}\\right)$ :: $\\operatorname{res}_U^U = \\operatorname{id}_{\\mathcal F \\left({U}\\right)}$, the identity morphism on $\\mathcal F \\left({U}\\right)$ :: $\\operatorname{res}_V^U \\circ \\operatorname{res}_W^V = \\operatorname{res}_W^U$, where $\\circ$ is the composition in $\\mathbf C$"}
+{"_id": "29151", "title": "Definition:Presheaf on Topological Space/Definition 2", "text": "Let $\\tau$ be the category of open sets of $T$. A $\\mathbf C$-valued '''presheaf''' on $T$ is a contravariant functor $\\tau \\to \\mathbf C$."}
+{"_id": "29152", "title": "Definition:Section of Étalé Space/Definition 1", "text": "A '''section''' of $\\map {\\operatorname {\\acute Et} } {\\mathcal F}$ on $U$ is a continuous mapping $s: U \\to \\map {\\operatorname {\\acute Et} } {\\mathcal F}$ such that: :$\\pi \\circ s = I_U$ where $I_U$ is the identity mapping on $U$."}
+{"_id": "29153", "title": "Definition:Section of Étalé Space/Definition 2", "text": "A '''section''' of $\\map {\\operatorname {\\acute Et} } {\\mathcal F}$ on $U$ is a mapping $s: U \\to \\map {\\operatorname {\\acute Et} } {\\mathcal F}$ such that: :for all $x \\in U$ there exists an open neighborhhood $V$ of $x$ in $U$ such that the restriction of $s$ to $V$ is the section associated to some $t \\in \\map {\\mathcal F} V$."}
+{"_id": "29154", "title": "Definition:Direct Image of Presheaf/Definition 1", "text": "The '''direct image''' of $\\mathcal F$ via $f$ is the $\\mathbf C$-valued presheaf $f_* \\mathcal F$ on $T_2$ with: : $f_* \\mathcal F \\left({V}\\right) = \\mathcal F \\left({f^{-1} \\left({V}\\right)}\\right)$ for all open $V \\subset T_2$ : Restrictions $\\operatorname{res}^U_V = \\operatorname{res}^{f^{-1} \\left({U}\\right)}_{f^{-1} \\left({V}\\right)}$"}
+{"_id": "29155", "title": "Definition:Direct Image of Presheaf/Definition 2", "text": "The '''direct image''' of $\\mathcal F$ via $f$ is the $\\mathbf C$-valued presheaf $f_*\\mathcal F$ that is the composition $\\mathcal F \\circ \\operatorname{Open} \\left({f}\\right)$, where $\\operatorname{Open}$ is the open subsets functor."}
+{"_id": "29156", "title": "Definition:Sheaf on Topological Space/Definition 1", "text": "A $\\mathbf C$-valued '''sheaf''' $\\mathcal F$ on $T$ is a $\\mathbf C$-valued presheaf such that for all open $U \\subseteq S$ and all open covers $\\left\\langle{U_i}\\right\\rangle_{i \\mathop \\in I}$ of $U$: :$\\left({\\mathcal F \\left({U}\\right), \\left({\\operatorname{res}^U_{U_i} }\\right)_{i \\mathop \\in I} }\\right)$ is the limit of the restriction of $\\mathcal F$ to $\\left\\{ {U}\\right\\} \\cup \\left\\{ {U_i: i \\in I}\\right\\} \\cup \\left\\{ {U_i \\cap U_j : \\left({i, j}\\right) \\in I^2}\\right\\}$"}
+{"_id": "29157", "title": "Definition:Sheaf on Topological Space/Definition 2", "text": "Let $\\mathbf C$ be a complete category. {{definition wanted|using equalizers}}"}
+{"_id": "29158", "title": "Definition:Sheaf on Topological Space/Definition 3", "text": "Let $\\mathbf C$ be a complete abelian category. A $\\mathbf C$-valued '''sheaf''' $\\mathcal F$ on $T$ is a $\\mathbf C$-valued presheaf such that for all open $U \\subset S$ and all open covers $\\left\\langle{U_i}\\right\\rangle_{i \\mathop \\in I}$ of $U$ the sequence: :$0 \\to \\mathcal F \\left({U}\\right) \\to \\displaystyle \\prod_{i \\mathop \\in I} \\mathcal F \\left({U_i}\\right) \\to \\prod_{\\left({i ,j}\\right) \\mathop \\in I^2} \\mathcal F \\left({U_i \\cap U_j}\\right)$ is exact. {{explain|what the maps are}} {{definition wanted|using Definition:étalé Space of Presheaf, another via Definition:Sheaf of Sets}}"}
+{"_id": "29159", "title": "Definition:Edge Cut", "text": "Let $G$ be a graph. An '''edge cut''' of $G$ is a set of edges $W \\subseteq \\map E G$ such that the edge deletion $G \\setminus W$ is disconnected."}
+{"_id": "29160", "title": "Definition:Particular Point", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. A '''particular point of $T$''' is an element $p \\in S$ which is distinguished in some manner from the other elements of $S$."}
+{"_id": "29161", "title": "Definition:Euclidean Space/Euclidean Topology/Real Number Line", "text": "Let $\\R$ denote the real number line. Let $d: \\R \\times \\R \\to \\R$ denote the Euclidean metric on $\\R$. Let $\\tau_d$ denote the topology on $\\R$ induced by $d$. The topology $\\tau_d$ induced by $d$ is called the '''Euclidean topology'''. Hence $\\struct {\\R, \\tau_d}$ is referred to as the '''real number line with the Euclidean topology'''."}
+{"_id": "29162", "title": "Definition:Euclidean Space/Euclidean Topology/Real Number Plane", "text": "Let $\\R^n$ be an $n$-dimensional real vector space. Let $M = \\left({\\R^2, d}\\right)$ be a real Euclidean space of $2$ dimensions. The topology $\\tau_d$ induced by the Euclidean metric $d$ is called the '''Euclidean topology'''. The space $\\left({\\R^2, \\tau_d}\\right)$ is known as the '''(real) Euclidean plane'''."}
+{"_id": "29163", "title": "Definition:Fréchet Space (Functional Analysis)", "text": "Let $\\R^\\omega$ denote the countable-dimensional real Cartesian space. Let: :$x := \\family {x_i}_{i \\mathop \\in \\N} = \\tuple {x_0, x_1, x_2, \\ldots}$ and: :$y := \\family {y_i}_{i \\mathop \\in \\N} = \\tuple {y_0, y_1, y_2, \\ldots}$ denote arbitrary elements of $\\R^\\omega$. Let the distance function $d: \\R^\\omega \\times \\R^\\omega \\to \\R$ be applied to $\\R^\\omega$ as: :$\\forall x, y \\in \\R^\\omega: \\map d {x, y} = \\displaystyle \\sum_{i \\mathop \\in \\N} \\dfrac {2^{-i} \\size {x_i - y_i} } {1 + \\size {x_i - y_i} }$ The distance function $d$ is referred to as the '''Fréchet (product) metric'''. The resulting metric space $\\struct {\\R^\\omega, d}$ is then referred to as the '''Fréchet (metric) space'''."}
+{"_id": "29165", "title": "Definition:Hilbert Cube", "text": "The '''Hilbert cube''' $\\struct {I^\\omega, d_2}$ is the subspace of the Hilbert sequence space $I^\\omega$ defined as: :$\\displaystyle I^\\omega = \\prod_{k \\mathop \\in \\N} \\closedint 0 {\\dfrac 1 k}$ under the same metric as that of the Hilbert sequence space: :$\\displaystyle \\forall x = \\sequence {x_i}, y = \\sequence {y_i} \\in I^\\omega: \\map {d_2} {x, y} := \\paren {\\sum_{k \\mathop \\ge 0} \\paren {x_k - y_k}^2}^{\\frac 1 2}$"}
+{"_id": "29166", "title": "Definition:Convex Set (Order Theory)/Definition 1", "text": "A subset $A$ of an ordered set $\\left({S, \\preceq}\\right)$ is '''convex (in $S$)''' {{iff}}: :$\\forall x, y \\in A: \\forall z \\in S: x \\preceq z \\preceq y \\implies z \\in A$"}
+{"_id": "29167", "title": "Definition:Convex Set (Order Theory)/Definition 2", "text": "A subset $A$ of an ordered set $\\struct {S, \\preceq}$ is '''convex (in $S$)''' {{iff}}: :$\\forall x, y \\in A: \\forall z \\in S: x \\prec z \\prec y \\implies z \\in A$"}
+{"_id": "29168", "title": "Definition:Interval/Open Interval", "text": "Let $\\left({S, \\preceq}\\right)$ be a totally ordered set. Let $a, b \\in S$. === Open Interval on Ordered Set === {{:Definition:Open Interval}} === Open Real Interval === In the context of the real number line $\\R$: {{:Definition:Open Real Interval}}"}
+{"_id": "29169", "title": "Definition:Interval/Closed Interval", "text": "Let $\\left({S, \\preceq}\\right)$ be a totally ordered set. Let $a, b \\in S$. === Closed Interval on Ordered Set === {{:Definition:Closed Interval}} === Closed Real Interval === In the context of the real number line $\\R$: {{:Definition:Closed Real Interval}}"}
+{"_id": "29170", "title": "Definition:Interval/Half-Open Interval", "text": "Let $\\left({S, \\preceq}\\right)$ be a totally ordered set. Let $a, b \\in S$. === Left Half-Open Interval === {{:Definition:Interval/Ordered Set/Left Half-Open}} === Right Half-Open Interval === {{:Definition:Interval/Ordered Set/Right Half-Open}} === Half-Open Real Interval === In the context of the real number line $\\R$: {{:Definition:Half-Open Real Interval}}"}
+{"_id": "29171", "title": "Definition:Convex Component", "text": "Let $\\struct {S, \\preccurlyeq}$ be an ordered set. The maximally convex sets into which $S$ is partitioned are the '''convex components''' of $S$."}
+{"_id": "29172", "title": "Definition:Interval/Ordered Set/Endpoint", "text": "The elements $a, b \\in S$ are known as the '''endpoints''' (or '''end points''') of the interval. $a$ is sometimes called the '''left hand endpoint''' and $b$ the '''right hand end point''' of the interval."}
+{"_id": "29173", "title": "Definition:Ordinal Space", "text": "Let $\\Gamma$ be a limit ordinal. === Open Ordinal Space === {{:Definition:Ordinal Space/Open}} === Closed Ordinal Space === {{:Definition:Ordinal Space/Closed}}"}
+{"_id": "29174", "title": "Definition:Uncountable Ordinal", "text": "Let $\\alpha$ be an ordinal. Then $\\alpha$ is said to be '''uncountable''' {{iff}} it is an uncountable set."}
+{"_id": "29175", "title": "Definition:Ordinal Space/Closed", "text": "The '''closed ordinal space on $\\Gamma$''' is the set $\\closedint 0 \\Gamma$ of all ordinal numbers less than or equal to $\\Gamma$, together with the order topology."}
+{"_id": "29176", "title": "Definition:Ordinal Space/Open", "text": "The '''open ordinal space on $\\Gamma$''' is the set $\\hointr 0 \\Gamma$ of all ordinal numbers (strictly) less than $\\Gamma$, together with the order topology."}
+{"_id": "29177", "title": "Definition:Ordinal Space/Open/Countable", "text": "Let $\\Omega$ denote the first uncountable ordinal. The '''countable open ordinal space on $\\Gamma$''' is a particular case of an open ordinal space $\\hointr 0 \\Gamma$ where $\\Gamma < \\Omega$."}
+{"_id": "29178", "title": "Definition:Ordinal Space/Closed/Uncountable", "text": "Let $\\Omega$ denote the first uncountable ordinal. The '''uncountable closed ordinal space on $\\Omega$''' is the particular case of a closed ordinal space $\\closedint 0 \\Gamma$ where $\\Gamma = \\Omega$."}
+{"_id": "29179", "title": "Definition:Ordinal Space/Closed/Countable", "text": "Let $\\Omega$ denote the first uncountable ordinal. The '''countable closed ordinal space on $\\Gamma$''' is a particular case of an closed ordinal space $\\closedint 0 \\Gamma$ where $\\Gamma < \\Omega$."}
+{"_id": "29180", "title": "Definition:Ordinal Space/Open/Uncountable", "text": "Let $\\Omega$ denote the first uncountable ordinal. The '''uncountable open ordinal space on $\\Omega$''' is the particular case of an open ordinal space $\\hointr 0 \\Gamma$ where $\\Gamma = \\Omega$."}
+{"_id": "29181", "title": "Definition:Characteristic Polynomial of Element of Algebra", "text": "Let $A$ be a commutative ring with unity. Let $B$ be an algebra over $A$ such that $B$ is a finite-dimensional free module over $A$. Let $b \\in B$. The '''characteristic polynomial''' of $b$ is the characteristic polynomial of the regular representation $\\lambda_b : B \\to B$ over $A$."}
+{"_id": "29182", "title": "Definition:Trace of Element of Algebra over Ring", "text": "Let $A$ be a commutative ring with unity. Let $B$ be an algebra over $A$ such that $B$ is a finite-dimensional free module over $A$. Let $b \\in B$. The '''trace''' $\\operatorname{Tr}_{B/A}(b)$ of $b$ is the trace of the regular representation $\\lambda_b : B \\to B$ over $A$."}
+{"_id": "29183", "title": "Definition:Norm of Element of Algebra over Ring", "text": "Let $A$ be a commutative ring with unity. Let $B$ be an algebra over $A$ such that $B$ is a finite-dimensional free module  over $A$. Let $b \\in B$. The '''trace''' $N_{B/A}(b)$ of $b$ is the determinant of the regular representation $\\lambda_b : B \\to B$ over $A$."}
+{"_id": "29184", "title": "Definition:Characteristic Polynomial of Matrix", "text": "Let $R$ be a commutative ring with unity. Let $M$ be a square matrix over $R$ of order $n > 0$. Let $I_n$ be the $n\\times n$ identity matrix. Let $R[x]$ be the polynomial ring in one variable over $R$. The '''characteristic polynomial''' of $M$ is the determinant of a matrix over $R[x]$: :$p_M (x) = \\operatorname{det}(xI - M)$."}
+{"_id": "29187", "title": "Definition:Torsion Element of Module", "text": "Let $R$ be a commutative ring with unity. Let $M$ be a unitary module over $R$. Let $m \\in M$. Then $m$ is a '''torsion element''' {{iff}} there exists a regular element $a \\in R$ with $a m = 0$."}
+{"_id": "29188", "title": "Definition:Torsion Submodule", "text": "Let $R$ be a commutative ring with unity. Let $M$ be a unitary module over $R$. The '''torsion submodule''' $T(M)$ of $M$ is the submodule of all torsion elements of $M$."}
+{"_id": "29189", "title": "Definition:Torsion Module", "text": "Let $R$ be a commutative ring with unity. Let $M$ be a unitary module over $R$. Then $M$ is a '''torsion module''' {{iff}} every element of $M$ is of torsion, that is, $M$ equals its torsion submodule $T(M)$."}
+{"_id": "29190", "title": "Definition:Zariski Topology on Maximal Spectrum of Ring", "text": "Let $A$ be a commutative ring with unity. Let $\\operatorname{MaxSpec}(A)$ be its maximal spectrum. === Definition 1 === The '''Zariski topology''' on $\\operatorname{MaxSpec}(A)$ is the topology with as closed sets the maximal zero loci. === Definition 2 === The '''Zariski topology''' on $\\operatorname{MaxSpec}(A)$ is the subspace topology induced by the Zariski topology on the spectrum $\\operatorname{Spec} A$."}
+{"_id": "29191", "title": "Definition:Torsion Subgroup", "text": "Let $G$ be an abelian group. Its '''torsion subgroup''' $\\map T G$ is the subgroup of all torsion elements."}
+{"_id": "29192", "title": "Definition:Torsion Group", "text": "Let $G$ be a group. Then $G$ is a '''torsion group''' {{iff}} all its elements are torsion elements."}
+{"_id": "29195", "title": "Definition:Purely Inseparable Field Extension", "text": "Let $E/F$ be an algebraic field extension. === Definition 1 === The extension $E/F$ is '''purely inseparable''' {{iff}} every element $\\alpha \\in E \\setminus F$ is inseparable. === Definition 2 === Let $F$ have positive characteristic $p$. The extension $E/F$ is '''purely inseparable''' {{iff}} for each $\\alpha \\in E$ there exists $n \\in \\N$ such that $\\alpha^{p^n} \\in F$. === Definition 3 === Let $F$ have positive characteristic $p$. The extension $E/F$ is '''purely inseparable''' {{iff}} each element of $E$ has a minimal polynomial of the form $X^{p^n} - a$."}
+{"_id": "29196", "title": "Definition:Initial Homomorphism from Integers to Ring with Unity", "text": "Let $\\Z$ be the ring of integers. Let $R$ be a ring with unity. The '''initial homomorphism''' $\\Z \\to R$ is the unital ring homomorphism that sends $n \\in \\Z$ to the $n$th power of $1$ in $R$: :$ n \\mapsto n \\cdot 1$."}
+{"_id": "29197", "title": "Definition:Positive Characteristic", "text": "Let $R$ be a ring with unity. Then $R$ has '''positive characteristic''' {{iff}} its characteristic is not $0$."}
+{"_id": "29198", "title": "Definition:Inseparable Element of Field Extension", "text": "Let $E/F$ be a field extension. Let $\\alpha \\in E$ be algebraic over $F$. Then $\\alpha$ is '''inseparable''' {{iff}} it is not separable."}
+{"_id": "29199", "title": "Definition:Relative Separable Closure", "text": "Let $E/F$ be a field extension. The '''(relative) separable closure''' of $E$ in $F$ is the intermediate field consisting of all separable elements of $E$ over $F$."}
+{"_id": "29200", "title": "Definition:Inseparable Field Extension", "text": "Let $E/F$ be an algebraic field extension. Then $E/F$ is '''inseparable''' {{iff}} it is not separable."}
+{"_id": "29201", "title": "Definition:Separable Closure", "text": "Let $K$ be a field. A '''separable closure''' of $K$ is a separably closed algebraic field extension of $F$. It may be defined as the relative separable closure of $F$ in its algebraic closure."}
+{"_id": "29202", "title": "Definition:Separably Closed Field", "text": "Let $K$ be a field. Then $K$ is '''separably closed''' {{iff}} === Definition 1 === :The only separable field extension of $K$ is $K$ itself. === Definition 2 === :Every separable irreducible polynomial over $K$ has degree $1$. === Definition 3 === :Every separable polynomial over $K$ of strictly positive degree has a root in $K$."}
+{"_id": "29203", "title": "Definition:Separable Degree", "text": "Let $E / F$ be a field extension. Let $S \\subseteq E$ be the separable closure of $F$ in $E$. The '''separable degree''' $\\index E F_{\\operatorname {sep} }$ of $E / F$ is the degree $\\index S F$."}
+{"_id": "29204", "title": "Definition:Inseparable Degree", "text": "Let $E/F$ be a field extension. Let $S \\subseteq E$ be the separable closure of $F$ in $E$. The '''inseparable degree''' $[E : F]_i$ of $E/F$ is the degree $[E : S]$."}
+{"_id": "29205", "title": "Definition:Absolute Galois Group", "text": "Let $K$ be a field. === Definition 1 === {{:Definition:Absolute Galois Group/Definition 1}} === Definition 2 === {{:Definition:Absolute Galois Group/Definition 2}}"}
+{"_id": "29206", "title": "Definition:Profinite Group", "text": "=== Definition 1 === A '''profinite group''' is a topological group that is isomorphic to a small inverse limit of finite discrete groups, with the inverse limit topology. === Definition 2 === A '''profinite group''' is a topological group that is compact, Hausdorff and totally disconnected."}
+{"_id": "29207", "title": "Definition:Complete Order Topology", "text": "Let $\\struct {S, \\preceq}$ be a complete ordered set. Let $\\tau$ be the order topology on $\\struct {S, \\preceq}$. Then $\\tau$ is a '''complete order topology'''. Hence $\\struct {S, \\preceq, \\tau}$ is a '''complete order space'''."}
+{"_id": "29210", "title": "Definition:Integral Transform", "text": "Let $p$ be a variable whose domain is a subset of the set of real numbers $\\R$. Let $\\closedint a b$ be a closed real interval for some $a, b \\in \\R: a \\le b$. Let $f: \\closedint a b \\to \\R$ be a real function defined on the domain $\\closedint a b$. Let $\\map K {p, x}$ be a real-valued function defined for all $p$ in its domain and all $x \\in \\closedint a b$. Let $\\map f x \\map K {p, x}$ be integrable {{WRT|Integration}} $x$ for all $p$ in its domain and all $x \\in \\closedint a b$. Consider the real function $\\map F p$ defined as: :$\\map F p = \\displaystyle \\int_a^b \\map f x \\map K {p, x} \\rd x$ Then $\\map F p$ is an '''integral transform''' of $\\map f x$."}
+{"_id": "29211", "title": "Definition:Integral Transform/Kernel", "text": "The function $K \\left({p, x}\\right)$ is the '''kernel''' of $F \\left({p}\\right)$."}
+{"_id": "29212", "title": "Definition:Integral Transform/Operator", "text": "This can be written in the form: :$F = \\map T f$ where $T$ is interpreted as the (unitary) operator meaning: :''Multiply this by $\\map K {p, x}$ and integrate {{WRT|Integration}} $x$ between the limits $a$ and $b$.'' Thus $T$ transforms the function $\\map f x$ into its image $\\map F p$, which is itself another real function."}
+{"_id": "29213", "title": "Definition:Integral Transform/Image Space", "text": "The domain of $p$ is known as the '''image space''' of $T$."}
+{"_id": "29214", "title": "Definition:Inverse Integral Operator", "text": "Let $T: f \\to F$ be an integral operator on a function $f$. Let there be a (unitary) operator $T^{-1}: F \\to f$ such that for a given $F \\left({p}\\right)$ there exists a unique $f \\left({x}\\right)$ such that $f = T \\left({f}\\right)$. Then $T^{-1}$ is the '''inverse integral operator''' of $T$."}
+{"_id": "29215", "title": "Definition:Inversion Theorem", "text": "Let $F \\left({p}\\right)$ be an integral transform: :$F \\left({p}\\right) = \\displaystyle \\int_a^b f \\left({x}\\right) K \\left({p, x}\\right) \\, \\mathrm d x$ Let $T$ be the integral operator associated with $F \\left({p}\\right)$: :$F = t \\left({f}\\right)$ An '''inversion theorem''' is a specification for an inverse integral operator $T^{-1}$ of the form $f = T^{-1} \\left({F}\\right)$ such that: :$f \\left({x}\\right) = \\displaystyle \\int_\\alpha^\\beta F \\left({p}\\right) H \\left({x, p}\\right) \\, \\mathrm d p$ should such an $H \\left({x, p}\\right)$ exist. It is not necessarily the case that it does exist.."}
+{"_id": "29216", "title": "Definition:Trigonometric Series", "text": "A '''trigonometric series''' is a series of the type: :$\\map S x = \\dfrac {a_0} 2 + \\displaystyle \\sum_{n \\mathop = 1}^\\infty \\paren {a_n \\cos n x + b_n \\sin n x}$ where: :the domain of $x$ is the set of real numbers $\\R$ :the coefficients $a_0, a_1, a_2, \\ldots, a_n, \\ldots; b_1, b_2, \\ldots, b_n, \\ldots$ are real numbers independent of $x$."}
+{"_id": "29217", "title": "Definition:Trigonometric Series/Complex Form", "text": "A '''trigonometric series''' can be expressed in a form using complex functions as follows: :$\\map S x = \\displaystyle \\sum_{n \\mathop = -\\infty}^\\infty c_n e^{i n x}$ where: :the domain of $x$ is the set of real numbers $\\R$ :the coefficients $\\ldots, c_{-n}, \\ldots, c_{-2}, c_{-1}, c_0, c_1, c_2, \\ldots, c_n, \\ldots$ are real numbers independent of $x$ :$c_{-n} = \\overline {c_n}$ where $\\overline {c_n}$ is the complex conjugate of $c_n$."}
+{"_id": "29218", "title": "Definition:Fourier Series/Fourier Coefficient/Range 2 Pi", "text": "The constants: : $a_0, a_1, a_2, \\ldots, a_n, \\ldots; b_1, b_2, \\ldots, b_n, \\ldots$ are the '''Fourier coefficients''' of $f$."}
+{"_id": "29219", "title": "Definition:Dirichlet Conditions", "text": "{{begin-axiom}} {{axiom | n = \\text D 1         | t = $\\quad f$ is absolutely integrable }} {{axiom | n = \\text D 2         | t = $\\quad f$ has a finite number of local maxima and local minima }} {{axiom | n = \\text D 3         | t = $\\quad f$ has a finite number of discontinuities, all of them finite }} {{end-axiom}}"}
+{"_id": "29222", "title": "Definition:Absolutely Integrable Function", "text": "Let $f$ be a real function. $\\map f x$ is '''absolutely integrable''' on $S \\subseteq \\R$ {{iff}} its absolute value of $f$ is an integrable function on $S$. That is, {{iff}} the definite integral of the absolute value of $f$ over any interval $\\openint \\alpha \\beta \\subseteq S$ is bounded. Category:Definitions/Integral Calculus 35eag25292vx0cl2p32luts6g380kx9"}
+{"_id": "29223", "title": "Definition:Fourier Series/Fourier Coefficient", "text": "The constants: : $a_0, a_1, a_2, \\ldots, a_n, \\ldots; b_1, b_2, \\ldots, b_n, \\ldots$ are the '''Fourier coefficients''' of $f$."}
+{"_id": "29224", "title": "Definition:Spence's Function", "text": "'''Spence's function''', also known as the '''dilogarithm''', is a special case of the polylogarithm, defined for $z \\in \\C$ by the integral:  :$\\displaystyle \\operatorname {Li}_2 \\paren z = -\\int_0^z \\frac {\\Ln \\paren {1 - t} } t \\rd t$ where: :$\\displaystyle \\int_0^z$ is an integral across the straight line in the complex plane connecting $0$ and $z$.  :$\\Ln$ is the principal branch of the complex natural logarithm."}
+{"_id": "29225", "title": "Definition:Half-Range Fourier Sine Series", "text": "=== Formulation 1 === {{:Definition:Half-Range Fourier Sine Series/Formulation 1}} === Formulation 2 === {{:Definition:Half-Range Fourier Sine Series/Formulation 2}}"}
+{"_id": "29226", "title": "Definition:Half-Range Fourier Cosine Series", "text": "=== Formulation 1 === {{:Definition:Half-Range Fourier Cosine Series/Formulation 1}} === Formulation 2 === {{:Definition:Half-Range Fourier Cosine Series/Formulation 2}}"}
+{"_id": "29227", "title": "Definition:Fourier Series/Formulation 1", "text": "Let $\\alpha \\in \\R$ be a real number. Let $\\lambda \\in \\R_{>0}$ be a strictly positive real number. Let $f: \\R \\to \\R$ be a function such that $\\displaystyle \\int_\\alpha^{\\alpha + 2 \\lambda} \\map f x \\rd x$ converges absolutely. Let: {{begin-eqn}} {{eqn | l = a_n       | r = \\dfrac 1 \\lambda \\int_\\alpha^{\\alpha + 2 \\lambda} \\map f x \\cos \\frac {n \\pi x} \\lambda \\rd x }} {{eqn | l = b_n       | r = \\dfrac 1 \\lambda \\int_\\alpha^{\\alpha + 2 \\lambda} \\map f x \\sin \\frac {n \\pi x} \\lambda \\rd x }} {{end-eqn}} Then: :$\\displaystyle \\frac {a_0} 2 + \\sum_{n \\mathop = 1}^\\infty \\paren {a_n \\cos \\frac {n \\pi x} \\lambda + b_n \\sin \\frac {n \\pi x} \\lambda}$ is the '''Fourier Series''' for $f$."}
+{"_id": "29228", "title": "Definition:Fourier Series/Formulation 2", "text": "Let $a, b \\in \\R$ be real numbers. Let $f: \\R \\to \\R$ be a function such that $\\displaystyle \\int_a^b \\map f x \\rd x$ converges absolutely. Let: {{begin-eqn}} {{eqn | l = A_m       | r = \\dfrac 2 {b - a} \\int_a^b \\map f x \\cos \\frac {2 m \\pi \\paren {x - a} } {b - a} \\rd x }} {{eqn | l = B_m       | r = \\dfrac 2 {b - a} \\int_a^b \\map f x \\sin \\frac {2 m \\pi \\paren {x - a} } {b - a} \\rd x }} {{end-eqn}} Then: :$\\displaystyle \\frac {A_0} 2 + \\sum_{m \\mathop = 1}^\\infty \\paren {A_m \\cos \\frac {2 m \\pi \\paren {x - a} } {b - a} + B_m \\sin \\frac {2 m \\pi \\paren {x - a} } {b - a} }$ is the '''Fourier Series''' for $f$."}
+{"_id": "29229", "title": "Definition:Half-Range Fourier Sine Series/Formulation 2", "text": "Let $\\map f x$ be a real function defined on the interval $\\openint a b$. Then the ''' half-range Fourier sine series''' of $\\map f x$ over $\\openint a b$ is the series: :$\\displaystyle \\map f x \\sim \\sum_{m \\mathop = 1}^\\infty B_m \\sin \\frac {m \\pi \\paren {x - a} } {b - a}$ where for all $n \\in \\Z_{> 0}$: :$B_m = \\displaystyle \\frac 2 {b - a} \\int_a^b \\map f x \\sin\\frac {m \\pi \\paren {x - a} } {b - a} \\rd x$"}
+{"_id": "29230", "title": "Definition:Half-Range Fourier Cosine Series/Formulation 2", "text": "Let $\\map f x$ be a real function defined on the interval $\\openint a b$. Then the ''' half-range Fourier cosine series''' of $\\map f x$ over $\\openint a b$ is the series: :$\\displaystyle \\map f x \\sim \\frac {A_0} 2 + \\sum_{m \\mathop = 1}^\\infty A_m \\cos \\frac {m \\pi \\paren {x - a} } {b - a}$ where for all $m \\in \\Z_{\\ge 0}$: :$A_m = \\displaystyle \\frac 2 {b - a} \\int_a^b \\map f x \\cos \\frac {m \\pi \\paren {x - a} } {b - a} \\rd x$"}
+{"_id": "29231", "title": "Definition:Orthonormal Set of Real Functions", "text": "Let $I$ be an indexing set. Let $S := \\family {\\map {\\phi_i} x}_{i \\mathop \\in I}$ be an indexed family of real functions all of which are integrable over the interval $\\openint a b$. Let $S$ have the property that: :$\\forall m, n \\in I: \\displaystyle \\int_a^b \\map {\\phi_m} x \\map {\\phi_n} x \\rd x = \\delta_{m n}$ where $\\delta_{m n}$ denotes the Kronecker delta. Then $S$ is defined as '''orthonormal'''."}
+{"_id": "29232", "title": "Definition:Complete Orthonormal Set of Real Functions", "text": "Let $I$ be an indexing set. Let $S := \\family {\\map {\\phi_i} x}_{i \\mathop \\in I}$ be an orthonormal set of real functions over the interval $\\openint a b$. Let $S$ have the property that: :$\\forall n \\in I: \\displaystyle \\int_a^b \\map \\psi x \\map {\\phi_n} x \\rd x = 0 \\implies \\map \\psi x \\equiv 0$ for any real function $\\psi$ integrable over the interval $\\openint a b$. Then $S$ is a '''complete orthonormal set''' of real functions."}
+{"_id": "29233", "title": "Definition:Divisor of Polynomial", "text": "Let $D$ be an integral domain. Let $D \\sqbrk x$ be the polynomial ring in one variable over $D$. Let $f, g \\in D \\sqbrk x$ be polynomials. Then: :'''$f$ divides $g$''' :'''$f$ is a divisor of $g$''' :'''$f$ is a factor of $g$''' :'''$g$ is divisible by $f$''' {{iff}}: :$\\exists h \\in D \\sqbrk x : g = f h$ This is denoted: :$f \\divides g$ === Notation === {{:Definition:Divisor Notation}}"}
+{"_id": "29234", "title": "Definition:Monomial of Polynomial Ring/Multiple Variables/Definition 1", "text": "The element $y$ is a '''monomial''' of $R \\sqbrk {\\family {X_i}_{i \\mathop \\in I} }$ {{iff}} there exists a mapping $a: I \\to \\N$ with finite support such that: :$y = \\ds \\prod_{i \\mathop \\in I} X_i^{a_i}$ where: :$\\prod$ denotes the product with finite support over $I$ :$X_i^{a_i}$ denotes the $a_i$th power of $X_i$."}
+{"_id": "29235", "title": "Definition:Monomial of Polynomial Ring/Multiple Variables/Definition 2", "text": "The element $y$ is a '''monomial''' of $R \\sqbrk {\\family {X_i}_{i \\mathop \\in I} }$ {{iff}} there exists a finite set $S$ and a mapping $f: S \\to \\set {X_i : i \\in I}$ such that it equals :$y = \\ds \\prod_{s \\mathop \\in S} \\map f s$ where $\\prod$ denotes the product over the finite set $S$."}
+{"_id": "29236", "title": "Definition:Inclusion-Preserving Mapping", "text": "Let $A$ and $B$ be sets. Let $f : A \\to B$ be a mapping. Then $f$ is '''inclusion-preserving''' {{iff}} for every two sets $a_1, a_2 \\in A$: :$a_1 \\subseteq a_2 \\implies f(a_1) \\subseteq f(a_2)$"}
+{"_id": "29237", "title": "Definition:Topologically Nilpotent Ring Element", "text": "Let $A$ be a Hausdorff topological ring. Let $a \\in A$. Then $a$ is '''topologically nilpotent''' {{iff}} the sequence of powers $a^n$ converges to $0$."}
+{"_id": "29238", "title": "Definition:Induced Mapping on Spectra of Rings", "text": "Let $A$ and $B$ be commutative rings with unity. Let $f : A \\to B$ be a ring homomorphism. The '''induced mapping on spectra by $f$''' is the mapping $f^* : \\operatorname{Spec} B \\to \\operatorname{Spec} A$ between their spectra with: :$f^* (\\mathfrak p) = f^{-1}(\\mathfrak p)$, the preimage of a prime ideal $\\mathfrak p \\in \\operatorname{Spec} B$. === Induced morphism of locally ringed spaces === {{definition wanted}}"}
+{"_id": "29239", "title": "Definition:Saturation of Ideal by Multiplicatively Closed Subset", "text": "Let $A$ be a commutative ring with unity. Let $\\mathfrak a \\subseteq A$ be an ideal. Let $S \\subseteq A$ be a multiplicatively closed subset. === Definition 1 === The '''saturation''' of $\\mathfrak a$ by $S$ is the ideal: :$\\{ a \\in A : \\exists s \\in S : as \\in \\mathfrak a\\}$ === Definition 2 === Let $A \\overset \\iota \\to A_S$ be the localization of $A$ at $S$. The '''saturation''' of $\\mathfrak a$ is the preimage of its image under the ring homomorphism $\\iota : A \\to A_S$: :$\\iota^{-1}(\\iota(\\mathfrak a))$"}
+{"_id": "29240", "title": "Definition:Topological Closed Embedding", "text": "Let $X$ and $Y$ be topological spaces. Let $f : X \\to Y$ be a mapping. Then $f$ is a '''(topological) closed embedding''' {{iff}}: *$f$ is a topological embedding *its image $f(X)$ is closed in $Y$"}
+{"_id": "29241", "title": "Definition:Topological Open Embedding", "text": "Let $X$ and $Y$ be topological spaces. Let $f : X \\to Y$ be a mapping. Then $f$ is a '''(topological) open embedding''' {{iff}}: *$f$ is a topological embedding *its image $f(X)$ is open in $Y$"}
+{"_id": "29242", "title": "Definition:Unit Ideal of Ring", "text": "Let $R$ be a ring. Its '''unit ideal''' is the ideal $R$."}
+{"_id": "29243", "title": "Definition:Semi-Local Ring", "text": "Let $A$ be a commutative ring with unity. Then $A$ is '''semi-local''' {{iff}} $A$ has a finite number of maximal ideals."}
+{"_id": "29244", "title": "Definition:Local Ring/Commutative", "text": "Let $A$ be a commutative ring with unity. === Definition 1 === {{Definition:Local Ring/Commutative/Definition 1}} === Definition 2 === {{Definition:Local Ring/Commutative/Definition 2}}"}
+{"_id": "29245", "title": "Definition:Localization of Module", "text": "Let $A$ be a commutative ring with unity. Let $S \\subseteq A$ be a multiplicatively closed subset. Let $A_S$ be the localization of $A$ at $S$. Let $M$ be an $A$-module. A '''localization of $M$ at $S$''' is a pair $(M_S, \\iota)$ where: *$M_S$ is an $A_S$-module *$\\iota : M \\to M_S$ is the '''localization map''', an $A$-module homomorphism to the restriction of scalars of $M_S$ to $A$ Such that: :For every $A_S$-module $N$ and $A$-module homomorphism $f: M \\to \\operatorname{res}_A^{A_S}N$ to the restriction of scalars to $A$, there exists a unique $A_S$-module homomorphism $g : M_S \\to N$ such that $f = g \\circ \\iota$. That is, the precomposition mapping between modules of homomorphisms: :$\\operatorname{Hom}_{A_S} (M_S, N) \\overset {\\iota^*} \\to \\operatorname{Hom}_A(M, N)$ is a bijection."}
+{"_id": "29246", "title": "Definition:Precomposition Mapping", "text": "Let $A, B, C$ be sets. Let $\\operatorname{Hom}(B, C)$ denote the set of all mappings from $B$ to $C$. Let $f : A \\to B$ be a mapping. The '''precomposition mapping''' $f^* : \\operatorname{Hom}(B, C) \\to \\operatorname{Hom}(A, C)$ is the mapping that sends a mapping $g : B \\to C$ to the precomposition $g \\circ f$ with $f$."}
+{"_id": "29248", "title": "Definition:Vanishing Ideal of Set of Prime Ideals", "text": "Let $A$ be a commutative ring with unity. Let $V \\subseteq \\operatorname{Spec} A$ be a set of prime ideals of $A$. Its '''vanishing ideal''' is its intersection, the set of elements of $A$ that are in each $\\mathfrak p \\in V$: :$I(V) = \\bigcap V$"}
+{"_id": "29249", "title": "Definition:Vanishing Ideal of Subset of Affine Space", "text": "Let $k$ be a field. Let $n \\ge 0$ be a natural number. Let $k \\sqbrk {X_1, \\ldots, X_n}$ be the polynomial ring in $n$ variables over $k$. Let $S \\subseteq \\mathbb A^n_k$ be a subset of the standard affine space over $k$. Its '''vanishing ideal''' is the ideal: :$\\map I S = \\set {f \\in k \\sqbrk {X_1, \\ldots, X_n} : \\forall x \\in S : \\map f x = 0}$"}
+{"_id": "29250", "title": "Definition:Divisor (Algebra)/Terminology", "text": "Let $x \\divides y$ denote that '''$x$ divides $y$'''. Then the following terminology can be used:  : $x$ is a '''divisor''' of $y$ : $y$ is a '''multiple''' of $x$ : $y$ is '''divisible by''' $x$. In the field of Euclidean geometry, in particular: : $x$ '''measures''' $y$. To indicate that $x$ does ''not'' divide $y$, we write $x \\nmid y$."}
+{"_id": "29251", "title": "Definition:Divisor (Algebra)/Gaussian Integer", "text": "Let $\\struct {\\Z \\left[{i}\\right], +, \\times}$ be the ring of Gaussian integers. Let $x, y \\in \\Z \\left[{i}\\right]$. Then '''$x$ divides $y$''' is defined as: :$x \\divides y \\iff \\exists t \\in \\Z \\left[{i}\\right]: y = t \\times x$"}
+{"_id": "29252", "title": "Definition:Spectrum of Ring Functor", "text": "Let $\\mathbf{Ring}$ be the category of commutative rings with unity. === To topological spaces === Let $\\mathbf{Top}$ be the category of topological spaces. The '''spectrum functor''' $\\Spec {} : \\mathbf{Ring} \\to \\mathbf{Top}$ is the contravariant functor with: {{DefineFunctor |ob = $\\Spec A$ is the spectrum of a ring $A$ |mor = If $f : A \\to B$ is a ring homomorphism, $\\Spec f : \\Spec B \\to \\Spec A$ is the  induced map on spectra }} {{definition wanted|to schemes}} Category:Definitions/Examples of Functors Category:Definitions/Commutative Algebra Category:Definitions/Algebraic Geometry szewf05u9zadxzq1453qaeva2p9bwhz"}
+{"_id": "29253", "title": "Definition:Ring Antirepresentation", "text": "Let $R$ be a ring. Let $M$ be an abelian group. A '''ring antirepresentation''' of $R$ on $M$ is a ring antihomomorphism to the endomorphism ring of $M$."}
+{"_id": "29254", "title": "Definition:Ring Action Defined by Ring Representation", "text": "Let $R$ be a ring. Let $M$ be an abelian group. Let $\\rho : R \\to \\map {\\operatorname {End} } M$ be a ring representation. The '''associated (left) ring action''' is the linear ring action: :$R \\times M \\to M$: :$\\tuple {r, m} \\mapsto \\map {\\map \\rho r} m$"}
+{"_id": "29255", "title": "Definition:Ring Representation/Unital", "text": "Let $R$ be a ring with unity. Let $M$ be an abelian group. A '''unital ring representation''' of $R$ on $M$ is a ring representation $R \\to \\map {\\operatorname {End} } M$ which is unital. That is, it is a unital ring homomorphism from $R$ to the endomorphism ring $\\map {\\operatorname {End} } M$."}
+{"_id": "29256", "title": "Definition:Closed Set Axioms", "text": "Let $S$ be a set. The '''closed set axioms''' are the conditions under which a subset $F \\subseteq \\mathcal P \\left({S}\\right)$ of the power set of $S$ consists of the closed sets of a topology on $S$: {{begin-axiom}} {{axiom | n = C1         | t = The intersection of an arbitrary subset of $F$ is an element of $F$. }} {{axiom | n = C2         | t = The union of any two elements of $F$ is an element of $F$. }} {{axiom | n = C3         | t = $\\varnothing$ is an element of $F$. }} {{end-axiom}}"}
+{"_id": "29257", "title": "Definition:Topology Defined by Closed Sets", "text": "Let $S$ be a set. Let $F \\subseteq \\mathcal P(S)$ be a subset of its power set satisfying the closed set axioms. The '''topology defined by $F$''' is the topology whose open sets are the complements of elements of $F$: :$\\tau = \\{U \\subseteq S : S \\setminus U \\in F\\}$"}
+{"_id": "29258", "title": "Definition:Two-Sided Linear Combination in Ring", "text": "Let $R$ be a ring. Let $\\family {x_i}_{i \\mathop \\in I}$ be a family of elements of $R$. A '''two-sided linear combination''' of the family is an element of the form: :$\\displaystyle \\sum_{i \\mathop \\in I} a_i x_i b_i$ where: :$\\family {a_i}_{i \\mathop \\in I}$ and $\\family {b_i}_{i \\mathop \\in I}$ are families in $R$ of finite support :$\\sum$ denotes summation with finite support"}
+{"_id": "29259", "title": "Definition:Prime Ideal of Number Field", "text": "Let $K$ be a number field. Let $\\mathcal O_K$ be its ring of integers. Let $\\mathfrak p \\subseteq \\mathcal O_K$ be an ideal. Then $\\mathfrak p$ is a '''prime ideal''' {{iff}} it is not the unit ideal $(1)$ and $\\mathfrak p$ has no divisors other than $\\mathfrak p$ and $(1)$."}
+{"_id": "29260", "title": "Definition:Rational Integer", "text": "Let $K$ be a number field. A '''rational integer''' of $K$ is an integer of $K$, that is, an element of $\\mathbb Z \\subseteq K$."}
+{"_id": "29261", "title": "Definition:Dedekind Domain", "text": "=== Definition 1 === {{Definition:Dedekind Domain/Definition 1}} === Definition 2 === {{Definition:Dedekind Domain/Definition 2}} === Definition 3 === {{Definition:Dedekind Domain/Definition 3}} === Definition 4 === {{Definition:Dedekind Domain/Definition 4}} === Definition 5 === {{Definition:Dedekind Domain/Definition 5}} === Definition 6 === {{Definition:Dedekind Domain/Definition 6}}"}
+{"_id": "29262", "title": "Definition:Integrally Closed Integral Domain", "text": "Let $R$ be an integral domain. Then $R$ is '''integrally closed''' {{iff}} it is integrally closed in its field of fractions."}
+{"_id": "29263", "title": "Definition:Integrally Closed in Ring Extension", "text": "Let $\\phi : A \\hookrightarrow B$ be a ring extension. Let $C$ be the integral closure of $A$ in $B$. Then $A$ is '''integrally closed''' in $B$ {{iff}} $C = \\phi(A)$."}
+{"_id": "29264", "title": "Definition:Fractional Ideal", "text": "Let $R$ be an integral domain with field of fractions $K$. A '''fractional ideal''' of $R$ is a subset of $K$ that is a product of an ideal $I$ with the inverse $\\lambda^{-1}$ of an element of $R$: :$\\lambda^{-1} I = \\left\\{ {\\lambda^{-1} a : a \\in I}\\right\\}$"}
+{"_id": "29265", "title": "Definition:Dedekind Domain/Definition 1", "text": "A '''Dedekind domain''' is an integral domain in which every nonzero proper ideal has a prime ideal factorization that is unique up to permutation of the factors. {{explain|unique up to permutation}}"}
+{"_id": "29266", "title": "Definition:Dedekind Domain/Definition 2", "text": "A '''Dedekind domain''' is an integral domain of which every nonzero fractional ideal is invertible."}
+{"_id": "29267", "title": "Definition:Dedekind Domain/Definition 3", "text": "A '''Dedekind domain''' is a noetherian domain of dimension $1$ that is integrally closed."}
+{"_id": "29268", "title": "Definition:Dedekind Domain/Definition 4", "text": "A '''Dedekind domain''' is a noetherian domain of dimension $1$ in which every primary ideal is the power of a prime ideal."}
+{"_id": "29269", "title": "Definition:Dedekind Domain/Definition 5", "text": "A '''Dedekind domain''' is a noetherian domain $A$ of dimension $1$ such that for every maximal ideal $\\mathfrak p$, the localization $A_{\\mathfrak p}$ is a discrete valuation ring."}
+{"_id": "29270", "title": "Definition:Dedekind Domain/Definition 6", "text": "A '''Dedekind domain''' is a Krull domain of dimension $1$."}
+{"_id": "29272", "title": "Definition:Polynomial over Ring/Multiple Variables", "text": "Let $I$ be a set. A '''polynomial over $I$ in one variable''' is an element of a polynomial ring in $I$ variables over $R$. Thus: :''Let $P \\in R \\left[{\\left\\langle{X_i}\\right\\rangle_{i \\mathop \\in I} }\\right]$ be a polynomial'' is a short way of saying: : Let $R \\left[{\\left\\langle{X_i}\\right\\rangle_{i \\mathop \\in I} }\\right]$ be a polynomial ring in $I$ variables over $R$, call its family of variables $\\left\\langle{X_i}\\right\\rangle_{i \\mathop \\in I}$, and let $P$ be an element of this ring."}
+{"_id": "29273", "title": "Definition:Digamma Function", "text": "The '''digamma function''', $\\psi$, is defined, for $z \\in \\C \\setminus \\Z_{\\le 0}$, by the logarithmic derivative of the gamma function:  :$\\map \\psi z = \\dfrac {\\map {\\Gamma'} z} {\\map \\Gamma z}$ where $\\Gamma$ is the gamma function, and $\\Gamma'$ denotes its derivative."}
+{"_id": "29274", "title": "Definition:Generated Ideal of Ring/Definition 1", "text": "The '''ideal generated by $S$''' is the smallest ideal of $R$ containing $S$, that is, the intersection of all ideals containing $S$."}
+{"_id": "29275", "title": "Definition:Generated Ideal of Ring/Definition 2", "text": "Let $R$ be a commutative ring with unity."}
+{"_id": "29277", "title": "Definition:Product of Fractional Ideals", "text": "Let $R$ be an integral domain with fraction field $K$. Let $I, J \\subseteq K$ be fractional ideals of $R$. The '''product of $I$ and $J$''' is the set of summations: :$\\left\\{ \\displaystyle\\sum_{i = 1}^r a_i b_i : a_i \\in I, b_i \\in J, r \\in \\N \\right\\}$"}
+{"_id": "29278", "title": "Definition:Invertible Fractional Ideal", "text": "Let $R$ be an integral domain with fraction field $K$. Let $I\\subseteq K$ be a fractional ideal of $R$. Then $I$ is '''invertible''' {{iff}} there exists a fractional ideal $J\\subseteq K$ such that their product is the unit ideal of $R$: :$I J = \\ideal 1$"}
+{"_id": "29279", "title": "Definition:Minimal Condition/Ordered Set", "text": "Let $(P, \\leq)$ be an ordered set. Then $P$ satisfies the '''minimal condition''' {{iff}} it is '''well-founded''': :Every non-empty subset has a minimal element."}
+{"_id": "29280", "title": "Definition:Coprime Ideals", "text": "Let $A$ be a commutative ring with unity. Let $I, J \\subseteq A$ be ideals. Then $I$ and $J$ are '''coprime''' {{iff}} their sum is the unit ideal: :$I + J = A$."}
+{"_id": "29281", "title": "Definition:Unit Ideal", "text": "Let $A$ be a commutative ring with unity. Its '''unit ideal''' is $A$ itself."}
+{"_id": "29282", "title": "Definition:Saturated Multiplicatively Closed Subset of Ring", "text": "Let $A$ be a commutative ring with unity. Let $S \\subseteq A$ be a multiplicatively closed subset. Then $S$ is '''saturated''' {{iff}} it equals its saturation, that is: :$x, y \\in A, xy \\in S \\implies x,y \\in S$ Category:Definitions/Localization of Rings eoer31a87qrcjz3ej5uao1tm83mcoov"}
+{"_id": "29283", "title": "Definition:Contraction of Ideal", "text": "Let $A$ and $B$ be commutative ring with unity. Let $f : A \\to B$ be a ring homomorphism. Let $\\mathfrak b \\subseteq B$ be an ideal. The '''contraction''' of $\\mathfrak b$ by $f$ is its preimage under $f$: :$\\mathfrak b^c = f^{-1}(\\mathfrak b)$"}
+{"_id": "29284", "title": "Definition:Canonical Homomorphism from Ring to Unital Algebra", "text": "Let $A$ be a commutative ring with unity. Let $B$ be an unital algebra over $A$ with unit $1$ whose underlying module is unitary. The '''canonical homomorphism''' $A \\to B$ is the algebra homomorphism that sends $a \\in A$ to $a \\cdot 1$."}
+{"_id": "29285", "title": "Definition:Algebra Homomorphism", "text": "Let $R$ be a commutative ring. Let $(A, *)$ and $(B, \\times)$ be algebras over $R$. An '''algebra homomorphism''' $f : A \\to B$ is a module homomorphism such that: :$\\forall a, b \\in A : f(a * b) = f(a) \\times f(b)$"}
+{"_id": "29286", "title": "Definition:Underlying Module of Algebra", "text": "Let $R$ be a commutative ring. Let $(A, *)$ be an algebra over $R$. Its '''underlying module''' is the $R$-module $A$."}
+{"_id": "29287", "title": "Definition:Unital Algebra Homomorphism", "text": "Let $R$ be a commutative ring. Let $\\left({A, *}\\right)$ and $\\left({B, \\times}\\right)$ be unital algebras over $R$ with units $1_A$ and $1_B$. A '''unital algebra homomorphism''' $f : A \\to B$ is a algebra homomorphism such that $f \\left({1_A}\\right) = 1_B$. Category:Definitions/Algebras 7qphmf7ukyuspdrvr52lwuc1qes6ebc"}
+{"_id": "29288", "title": "Definition:Riemann P-symbol", "text": "The '''Riemann P-symbol''', written:  :$\\displaystyle \\map f z = \\operatorname P \\set {\\begin{matrix} a & b & c \\\\ \\alpha & \\beta & \\gamma & z \\\\ \\alpha' & \\beta' & \\gamma' \\end{matrix} }$ denotes the solutions to the hypergeometric differential equation:  {{begin-eqn}} {{eqn | o =        | r = \\frac {\\d^2 f} {\\d z^2} }} {{eqn | o =        | ro= +       | r = \\paren {\\frac {1 - \\alpha - \\alpha'} {z - a} + \\frac {1 - \\beta - \\beta'} {z - b} + \\frac {1 - \\gamma - \\gamma'} {z - c} } \\frac {\\d f} {\\d z} }} {{eqn | o =        | ro= +       | r = \\paren {\\frac {\\alpha \\alpha' \\paren {a - b} \\paren {a - c} } {z - a} + \\frac {\\beta \\beta' \\paren {b - c} \\paren {b - a} } {z - b} + \\frac {\\gamma \\gamma' \\paren {c - a} \\paren {c - b} } {z - c} } \\frac f {\\paren {z - a} \\paren {z - b} \\paren {z - c} } }} {{eqn | r = 0 }} {{end-eqn}} where: :$\\alpha + \\alpha' + \\beta + \\beta' + \\gamma + \\gamma' = 1$"}
+{"_id": "29289", "title": "Definition:Induced Homomorphism between Localizations of Ring", "text": "Let $A$ be a commutative ring with unity. Let $S, T \\subseteq A$ be multiplicatively closed subsets. Let $S$ be a subset of the saturation of $T$. The '''induced homomorphism''' between localizations $A_S \\to A_T$ is the unique $A$-algebra homomorphism between them."}
+{"_id": "29290", "title": "Definition:Principal Open Subset of Spectrum", "text": "Let $A$ be a commutative ring with unity. Let $f \\in A$. The '''principal open subset''' determined by $f$ of the spectrum $\\operatorname{Spec} A$ is the complement of the vanishing set $V \\left({f}\\right)$: :$D \\left({f}\\right) = \\operatorname{Spec} A - V \\left({f}\\right)$ That is, it is the set of prime ideals $\\mathfrak p \\subseteq A$ with $f \\notin \\mathfrak p$."}
+{"_id": "29291", "title": "Definition:Unital Associative Commutative Algebra Homomorphism", "text": "Let $A$ be a commutative ring with unity. Let $B$ and $C$ be $A$-algebras. === Definition 1 === Let $B$ and $C$ be viewed as rings under $A$, say $(B, f)$ and $(C, g)$. An '''$A$-algebra homomorphism''' $h : B \\to C$ is a morphism of rings under $A$. That is, a unital ring homomorphism $h$ such that $g = h \\circ f$: :$\\xymatrix{ A \\ar[d]_f \\ar[r]^{g} & C\\\\ B \\ar[ru]_{h} }$ === Definition 2 === Let $B$ and $C$ be viewed as unital algebras over $A$. An '''$A$-algebra homomorphism''' $h : B \\to C$ is a unital algebra homomorphism."}
+{"_id": "29292", "title": "Definition:Radical of Ideal of Ring/Definition 1", "text": "The '''radical of $I$''' is the ideal of elements of which some power is in $I$: :$\\map {\\operatorname {Rad} } I := \\set {a \\in A: \\exists n \\in \\N : a^n \\in I}$"}
+{"_id": "29293", "title": "Definition:Radical of Ideal of Ring/Definition 2", "text": "Let $A / I$ be the quotient ring. Let $\\Nil {A / I}$ be its nilradical. Let $\\pi: A \\to A / I$ be the quotient mapping. The '''radical''' of $I$ is the preimage of $\\Nil {A / I}$ under $\\pi$: :$\\map {\\operatorname {Rad} } I = \\pi^{-1} \\sqbrk {\\Nil {A / I} }$"}
+{"_id": "29294", "title": "Definition:Finitely Generated Algebra", "text": "Let $A$ be a commutative ring. Let $B$ be an $A$-algebra. Then $B$ is '''finitely generated''' {{iff}} $B$ has a generator which is finite. Category:Definitions/Algebras qq40jwih7e3gzrirnshtqgb4dep6ajp"}
+{"_id": "29295", "title": "Definition:Characteristic Polynomial of Linear Operator", "text": "Let $A$ be a commutative ring with unity. Let $M$ be a free module over $A$ of finite rank $n > 0$. Let $\\phi : M \\to M$ be a linear operator. === Definition 1 === The '''characteristic polynomial''' of $\\phi$ is the characteristic polynomial of the relative matrix of $\\phi$ with respect to a basis of $M$. === Definition 2 === Let $A[x]$ be the polynomial ring in one variable over $A$. Let $\\operatorname{id}$ be the identity mapping on $M$. Let $M \\otimes_A A[x]$ be the extension of scalars of $M$ to $A[x]$. The '''characteristic polynomial''' of $\\phi$ is the determinant of the linear operator $x\\operatorname{id} - \\phi$ on $M \\otimes_A A[x]$."}
+{"_id": "29296", "title": "Definition:Induced Mapping on Maximal Spectra", "text": "Let $k$ be a field. Let $A$ and $B$ be finitely generated $k$-algebras. Let $f : A \\to B$ be a $k$-algebra homomorphism. The '''induced mapping on spectra by $f$''' is the mapping $f^* : \\operatorname{Max} B \\to \\operatorname{Max} A$ between their maximal spectra with: :$f^* (\\mathfrak m) = f^{-1}(\\mathfrak m)$, the preimage of a maximal ideal $\\mathfrak m \\in \\operatorname{Max} B$. === Induced morphism of locally ringed spaces === {{definition wanted}}"}
+{"_id": "29297", "title": "Definition:Finite Algebra over Ring", "text": "Let $A$ be a commutative ring with unity. Let $B$ be an $A$-algebra. Then $B$ is '''finite''' over $A$ {{iff}} its underlying module is finitely generated."}
+{"_id": "29298", "title": "Definition:Module Defined by Ring Homomorphism", "text": "Let $A$ and $B$ be a rings. Let $f: A \\to B$ be a ring homomorphism. === Definition 1 === The '''left $A$-module structure''' of $B$ via $f$ is the module with left ring action: :$A \\times B \\to B$ :$ \\tuple {a, b} \\mapsto \\map f a \\cdot b$ === Definition 2 === The '''left $A$-module structure''' of $B$ via $f$ is the restriction of scalars of the $B$-module structure of $B$. === Definition 3 === Let $\\lambda: B \\to \\map {\\operatorname {End} } B$ be its left regular ring representation. The '''left $A$-module structure''' of $B$ via $f$ is the module with ring representation the composition $\\lambda \\circ f$."}
+{"_id": "29299", "title": "Definition:Weierstrass's Elliptic Function", "text": "'''Weierstrass's Elliptic Function''' is an elliptic function, given for all complex $z$ (except for $z \\in \\set {2 m \\omega_1 + 2 n \\omega_2: \\tuple {n, m} \\in \\Z^2}$ where the function has double poles, by Poles of Weierstrass's Elliptic Function) by:  :$\\displaystyle \\map \\wp {z; \\omega_1, \\omega_2} = \\frac 1 {z^2} + {\\sum_{\\tuple {n, m} \\mathop \\in \\Z^2 \\setminus \\tuple {0, 0} } } \\paren {\\frac 1 {\\paren {z - 2 m \\omega_1 - 2 n \\omega_2}^2} - \\frac 1 {\\paren {2 m \\omega_1 + 2 n \\omega_2}^2} }$ where $\\omega_1$ and $\\omega_2$ are non-zero complex constants with $\\dfrac {\\omega_1} {\\omega_2}$ having a positive imaginary part."}
+{"_id": "29301", "title": "Definition:Right-Hand Derivative/Real Function", "text": "Let $f: \\R \\to \\R$ be a real function. The '''right-hand derivative''' of $f$ is defined as the right-hand limit: :$\\displaystyle \\map {f'_+} x = \\lim_{h \\mathop \\to 0^+} \\frac {\\map f {x + h} - \\map f x} h$ If the '''right-hand derivative''' exists, then $f$ is said to be '''right-hand differentiable''' at $x$."}
+{"_id": "29302", "title": "Definition:Left-Hand Derivative/Real Function", "text": "Let $f: \\R \\to \\R$ be a real function. The '''left-hand derivative''' of $f$ is defined as the left-hand limit: :$\\displaystyle \\map {f'_-} x = \\lim_{h \\mathop \\to 0^-} \\frac {\\map f {x + h} - \\map f x} h$ If the '''left-hand derivative''' exists, then $f$ is said to be '''left-hand differentiable''' at $x$."}
+{"_id": "29303", "title": "Definition:Ideal Quotient", "text": "Let $A$ be a commutative ring with unity. Let $\\mathfrak a, \\mathfrak b \\subseteq A$ be ideals of $A$. Their '''ideal quotient''' is the ideal consisting of elements whose product with $\\mathfrak b$ is a subset of $\\mathfrak a$: :$\\ideal {\\mathfrak a : \\mathfrak b} := \\set {x \\in A : x \\mathfrak b \\subseteq \\mathfrak a}$"}
+{"_id": "29304", "title": "Definition:Annihilator of Ideal of Ring", "text": "Let $A$ be a commutative ring with unity. Let $I \\subseteq A$ be an ideal. === Definition 1 === The '''annihilator''' of $I$ is the ideal consisting of the elements $a \\in A$ such that $a \\cdot x = 0$ for all $x \\in I$, where $0 \\in A$ is its zero. === Definition 2 === The '''annihilator''' of $I$ is the ideal quotient $\\ideal {0 : I}$, where $0$ is the zero ideal."}
+{"_id": "29305", "title": "Definition:Extension of Ideal", "text": "Let $A$ and $B$ be commutative ring with unity. Let $f : A \\to B$ be a ring homomorphism. Let $\\mathfrak a \\subseteq A$ be an ideal. The '''extension''' of $\\mathfrak a$ by $f$ is the ideal generated by its image under $f$: :$\\mathfrak a^e = \\left\\langle f(\\mathfrak a) \\right\\rangle$"}
+{"_id": "29306", "title": "Definition:Irreducible Polynomial/Definition 1", "text": "Let $R$ be an integral domain. An '''irreducible polynomial''' over $R$ is an irreducible element of the polynomial ring $R \\left[{X}\\right]$."}
+{"_id": "29307", "title": "Definition:Irreducible Polynomial/Definition 2", "text": "Let $K$ be a field. An '''irreducible polynomial''' over $K$ is a nonconstant polynomial over $K$ that is not the product of two polynomials of smaller degree."}
+{"_id": "29308", "title": "Definition:Irreducible Polynomial/Definition 3", "text": "Let $K$ be a field. An '''irreducible polynomial''' over $K$ is a polynomial over $K$ that is not the product of two nonconstant polynomials."}
+{"_id": "29309", "title": "Definition:Linear Ring Action", "text": "Let $R$ be a ring. Let $M$ be an abelian group. === Left Ring Action === {{:Definition:Linear Ring Action/Left}} === Right Ring Action === {{:Definition:Linear Ring Action/Right}}"}
+{"_id": "29310", "title": "Definition:Gradation on Abelian Group", "text": "Let $G$ be an abelian group. Let $\\Delta$ be a set. A '''gradation of type $\\Delta$''' on $G$ is a family of subgroups $\\family {G_\\lambda}_{\\lambda \\in \\Delta}$ of which $G$ is the internal direct sum."}
+{"_id": "29311", "title": "Definition:Graded Abelian Group", "text": "Let $\\Delta$ be a set. A '''graded abelian group of type $\\Delta$''' is a pair $\\struct {G, f}$ where: :$G$ is an abelian group :$f$ is a gradation on $G$ indexed by $\\Delta$, the '''set of degrees'''."}
+{"_id": "29312", "title": "Definition:N-Graded Ring", "text": "Let $\\N$ be the set of natural numbers. An '''$\\N$-graded ring''' or '''positively $\\Z$-graded ring''' is a graded ring of type the additive monoid of natural numbers. That is, it is a pair $\\struct {R, f}$ where: :$R$ is a ring :$f$ is a sequence $\\sequence {R_n}_{n \\mathop \\in \\N}$ of subgroups of the additive group of $R$, of which it is the internal direct sum, and such that: ::$\\forall x \\in R_n, y \\in R_m: x y \\in R_{m + n}$"}
+{"_id": "29313", "title": "Definition:Z-Graded Ring", "text": "Let $\\Z$ be the set of integers. A '''$\\Z$-graded ring''' is a graded ring of type the additive group of integers. That is, it is a pair $\\struct {R, f}$ where: :$R$ is a ring :$f$ is a family $\\family {R_n}_{n \\mathop \\in \\Z}$ of subgroups of the additive group of $R$, of which it is the internal direct sum, and such that: ::$\\forall x \\in R_n, y \\in R_m: x y \\in R_{m + n}$"}
+{"_id": "29314", "title": "Definition:Opposite Ring", "text": "Let $\\struct {R, +, \\times}$ be a ring. Let $* : R \\times R \\to R$ be the binary operation on $S$ defined by: :$\\forall x, y \\in S: x * y = y \\times x$ The '''opposite ring''' of $R$ is the algebraic structure $\\struct {R, +, *}$."}
+{"_id": "29315", "title": "Definition:Ring Representation Defined by Ring Action", "text": "Let $R$ be a ring. Let $M$ be an abelian group. Let $\\phi : R \\times M \\to M$ be a left linear ring action. The '''associated ring representation''' is the ring representation $\\rho: R \\to \\map {\\operatorname {End} } M$ with: :$\\map {\\map \\rho r} m = \\map \\phi {r, m}$"}
+{"_id": "29316", "title": "Definition:Category of Modules", "text": "Let $R$ be a commutative ring. The '''category of $R$-modules''' is the category $\\mathbf{R-Mod}$ with: {{DefineCategory |ob = $R$-modules |mor = $R$-module homomorphisms |id = identity mappings |comp = composition of mappings }} === Category of Unitary Modules === {{Definition:Category of Unitary Modules}}"}
+{"_id": "29317", "title": "Definition:Underlying Group of Module", "text": "Let $R$ be a ring. Let $\\struct {M, +, \\times}$ be a left module or right module over $R$. Its '''underlying (abelian) group''' is the abelian group $\\struct {M, +}$."}
+{"_id": "29318", "title": "Definition:Biadditive Mapping", "text": "Let $M, N, P$ be abelian groups. Let $M \\times N$ be the cartesian product. A '''biadditive mapping''' $f : M \\times N \\to P$ is a mapping such that: :$\\forall m_1, m_2 \\in M : \\forall n \\in N: \\map f {m_1 + m_2, n} = \\map f {m_1, n} + \\map f {m_2, n}$ :$\\forall m \\in M: \\forall n_1, n_2 \\in N: \\map f {m, n_1 + n_2} = \\map f {m, n_1} + \\map f {m, n_2}$"}
+{"_id": "29319", "title": "Definition:Cartesian Square of Set", "text": "Let $S$ be a set. Its '''cartesian square''' is the cartesian power $S^2$. Category:Definitions/Cartesian Product i2nh881gpp361x4qw0f83uv8um0ek3x"}
+{"_id": "29320", "title": "Definition:R-Balanced Mapping", "text": "Let $R$ be a ring. Let $M$ be a right $R$-module and $N$ be a left $R$-module. Let $M \\times N$ be their cartesian product. Let $P$ be an abelian group. An '''$R$-balanced mapping''' $f : M \\times N \\to P$ is a biadditive mapping with: :$\\forall m \\in M: \\forall n \\in N: \\forall r \\in R: \\map f {m \\cdot r, n} = \\map f {m, r \\cdot n}$"}
+{"_id": "29321", "title": "Definition:Tensor Product of Abelian Groups", "text": "Let $A$ and $B$ be abelian groups. === Definition 1: by universal property === Their '''tensor product''' is a pair $(A \\otimes B, \\theta)$ where: * $A \\otimes B$ is an abelian group * $\\theta : A \\times B \\to A \\otimes B$ is a biadditive mapping such that, for every pair $(C, \\omega)$ where: * $C$ is an abelian group * $\\omega : A \\times B \\to C$ is a biadditive mapping there exists a unique group homomorphism $g : A \\otimes B \\to C$ with $\\omega = g \\circ \\theta$. === Definition 2: construction === Their '''tensor product''' is the pair $(A \\otimes B, \\theta)$ where: * $A \\otimes B$ is the quotient of the free abelian group $\\Z^{(A \\times B)}$ on the cartesian product $A \\times B$ by the subgroup generated by the elements of the form: ** $(a_1 + a_2, b) - (a_1, b) - (a_2, b)$ ** $(a, b_1 + b_2) - (a, b_1) - (a, b_2)$ *:for $a, a_1, a_2 \\in A$, $b, b_1, b_2 \\in B$, where we denote $(a, b)$ for its image under the canonical mapping $A \\times B \\to \\Z^{(A\\times B)}$. * $\\theta : A \\times B \\to A \\otimes B$ is the composition of the canonical mapping $A \\times B \\to \\Z^{(A\\times B)}$ with the quotient group epimorphism $\\Z^{(A\\times B)} \\to A \\otimes B$."}
+{"_id": "29322", "title": "Definition:Right Ring Action Defined by Ring Antirepresentation", "text": "Let $R$ be a ring. Let $M$ be an abelian group. Let $\\rho: R \\to \\map {\\operatorname {End} } M$ be a ring antirepresentation. The '''associated right ring action''' is the right linear ring action: :$M \\times R \\to M$: :$\\tuple {m, r} \\mapsto \\map \\rho r \\paren m$"}
+{"_id": "29323", "title": "Definition:Tensor Product of Modules as Abelian Group", "text": "Let $R$ be a ring with unity. Let $M$ be a unitary right module and $N$ a unitary left module over $R$. === Definition 1: by universal property === Their '''tensor product''' is a pair $(M \\otimes_R N, \\theta)$ where: * $M \\otimes_R N$ is an abelian group * $\\theta : M \\times N \\to M \\otimes_R N$ is an $R$-balanced mapping satisfying the following '''universal property''': :For every pair $(P, \\omega)$ of an abelian group and an $R$-balanced mapping $\\omega : M \\times N \\to P$, there exists a unique group homomorphism $f : M \\otimes_R N \\to P$ with $\\omega = f \\circ \\theta$. === Definition 2: direct construction === Their '''tensor product''' is the pair $(M \\otimes_R N, \\theta)$, where: *$M \\otimes_R N$ is the quotient group of the free $R$-module $R^{(M\\times N)}$ on the direct product $M \\times N$, by the subgroup generated by the set of elements of the form: ** $(m_1 + m_2, n) - \\lambda (m_1, n) - (m_2, n)$ ** $(m, n_1 + n_2) - \\lambda (m, n_1) - (m, n_2)$ ** $(m \\cdot \\lambda, n) - (m, \\lambda \\cdot n)$ *:for $m, m_1, m_2 \\in M$, $n, n_1, n_2 \\in N$ and $\\lambda \\in R$, where we denote $(m, n)$ for its image under the canonical mapping $M \\times N \\to R^{(M\\times N)}$. * $\\theta : M \\times N \\to M \\otimes_R N$ is the composition of the canonical mapping $M \\times N \\to R^{(M\\times N)}$ with the quotient module epimorphism $R^{(M\\times N)} \\to M \\otimes_R N$. === Definition 3: construction via abelian groups === Let $(M \\otimes_\\Z N, u)$ be the tensor product of their underlying abelian groups. The '''tensor product''' of $M$ and $N$ is the the pair $(M \\otimes_R N, \\theta)$, where: * $M \\otimes_R N$ is the quotient of $M \\otimes_\\Z N$ by the subgroup generated by the elements of the form: *:$u(m \\cdot \\lambda, n) - u(m, \\lambda \\cdot n)$ * $\\theta : M \\times N \\to M \\otimes_R N$ is the composition of $u$ with the quotient group epimorphism $M \\otimes_\\Z N \\to M \\otimes_R N$."}
+{"_id": "29324", "title": "Definition:Free Abelian Group on Set", "text": "Let $\\Z$ be the additive group of integers. Let $S$ be a set. The '''free abelian group on $S$''' is the pair $(\\Z^{(S)}, \\iota)$ where: * $\\Z^{(S)}$ is the direct sum of $S$ copies of $\\Z$. That is, of the indexed family $S \\to \\{\\Z\\}$ * $\\iota : S \\to \\Z^{(S)}$ is the '''canonical mapping''', which sends $s$ to the mapping $\\delta_{st} \\in \\Z^{(S)}$, where $\\delta$ denotes Kronecker delta."}
+{"_id": "29325", "title": "Definition:Direct Sum of Groups", "text": "Let $I$ be a indexing set. Let $(G_i)_{i \\in I}$ be a family of groups. The '''direct sum''' of $(G_i)_{i \\in I}$ is the subgroup of their direct product consisting of mappings of finite support."}
+{"_id": "29326", "title": "Definition:Category of Right Modules", "text": "Let $R$ be a ring. The '''category of right $R$-modules''' is the category $\\mathbf {Mod-R}$ with: {{DefineCategory | ob = right modules over $R$ | mor = right $R$-module homomorphisms | id = identity mappings | comp = composition of mappings }}"}
+{"_id": "29327", "title": "Definition:Category of Left Modules", "text": "Let $R$ be a ring. The '''category of left $R$-modules''' is the category $\\mathbf {R-Mod}$ with: {{DefineCategory | ob = left modules over $R$ | mor = left $R$-module homomorphisms | id = identity mappings | comp = composition of mappings }}"}
+{"_id": "29328", "title": "Definition:Category of Unitary Modules", "text": "Let $R$ be a commutative ring with unity. The '''category of unitary $R$-modules''' is the category $\\mathbf{R-Mod}$ with: {{DefineCategory |ob = unitary $R$-modules |mor = $R$-module homomorphisms |id = identity mappings |comp = composition of mappings }}"}
+{"_id": "29329", "title": "Definition:Forgetful Functor from Modules to Abelian Groups", "text": "Let $R$ be a ring. Let $\\mathbf C$ be the category of left modules or category of left modules over $R$. Let $\\mathbf{Ab}$ be the category of abelian groups. The '''forgetful functor''' $\\mathbf C \\to \\mathbf{Ab}$ is the covariant functor with {{DefineFunctor | ob = sends a left module or right module to its underlying abelian group. | mor = sends a mapping to itself }}"}
+{"_id": "29330", "title": "Definition:Associated Z-Module Functor", "text": "Let $\\mathbf{Ab}$ be the category of abelian groups. Let $\\Z$ be the ring of integers and $\\mathbf{Z-Mod}$ the category of unitary modules over $\\Z$. The '''associated Z-module functor''' $\\mathbf{Ab} \\to \\mathbf{Z-Mod}$ is the covariant functor with {{DefineFunctor |ob = sends an abelian group to its associated Z-module |mor = sends a mapping to itself }}"}
+{"_id": "29332", "title": "Definition:Jacobi Theta Function", "text": "Let $\\tau$ be a complex constant with a positive imaginary part.  Let $q = e^{i \\pi \\tau}$.  Then the '''Jacobi Theta functions''' are defined for all complex $z$ by: === First Type === {{:Definition:Jacobi Theta Function/First Type}} === Second Type === {{:Definition:Jacobi Theta Function/Second Type}} === Third Type === {{:Definition:Jacobi Theta Function/Third Type}} === Fourth Type === {{:Definition:Jacobi Theta Function/Fourth Type}}"}
+{"_id": "29333", "title": "Definition:Jacobi Theta Function/First Type", "text": ":$\\displaystyle \\vartheta_1 \\left({z, q}\\right) = 2 \\sum_{n \\mathop = 0}^\\infty \\left({-1}\\right)^n q^{\\left({n + \\frac 1 2}\\right)^2} \\sin \\left({2 n + 1}\\right) z$"}
+{"_id": "29334", "title": "Definition:Jacobi Theta Function/Second Type", "text": ":$\\displaystyle \\vartheta_2 \\left({z, q}\\right) = 2 \\sum_{n \\mathop = 0}^\\infty q^{\\left({n + \\frac 1 2}\\right)^2} \\cos \\left({2 n + 1}\\right) z$"}
+{"_id": "29335", "title": "Definition:Jacobi Theta Function/Third Type", "text": ":$\\displaystyle \\vartheta_3 \\left({z, q}\\right) = 1 + 2 \\sum_{n \\mathop = 0}^\\infty q^{n^2} \\cos 2 n z$"}
+{"_id": "29336", "title": "Definition:Jacobi Theta Function/Fourth Type", "text": ":$\\displaystyle \\vartheta_4 \\left({z, q}\\right) = 1 + 2 \\sum_{n \\mathop = 0}^\\infty \\left({-1}\\right)^n q^{n^2} \\cos 2 n z$"}
+{"_id": "29337", "title": "Definition:Leibniz Law", "text": "Let $R$ be a ring. Let $A$ be a $R$-algebra. Then a function $f : A \\to A$ is said to be satisfying '''Leibniz Law''' if the following holds: :$\\forall a, b \\in A: \\map f {a b} = \\map f a b + a \\map f b$ Category:Definitions/Differential Algebra bvtjsn4vemtu8t3g5euveqfulwyq6mi"}
+{"_id": "29338", "title": "Definition:Compatible Module Structures", "text": "Let $A$ and $B$ be rings. Let $\\struct {M, +}$ be an abelian group. Let $* : A \\times M \\to M$ and $\\circledast: B \\times M \\to M$ be left or right linear ring actions so that: :$(1): \\quad \\struct {M, +, *}$ is a left or right module over $A$ :$(2): \\quad \\struct {M, +, \\circledast}$ is a left or right module over $B$ === Definition 1 === The module structures are '''compatible''' {{iff}} for all $a \\in A$, $b \\in B$, the homotheties $h_a$ and $h_b$ commute. That is, for all $m \\in M$, $a \\in A$, $b \\in B$: :$a * \\paren {b \\circledast m} = b \\circledast \\paren {a * m}$ === Definition 2 === The module structures are '''compatible''' {{iff}} for all $a \\in A$, the homothety $h_a : M \\to M$ is an endomorphism of the $B$-module $M$. That is, {{iff}} the image of the ring representation $A \\to \\map {\\operatorname {End} } M$ is contained in the endomorphism ring $\\map {\\operatorname {End}_B } M$. === Definition 3 === The module structures are '''compatible''' {{iff}} for all $b \\in A$, the homothety $h_b : M \\to M$ is an endomorphism of the $A$-module $M$. That is, {{iff}} the image of the ring representation $B \\to \\map {\\operatorname {End} } M$ is contained in the endomorphism ring $\\map {\\operatorname {End}_A} M$."}
+{"_id": "29339", "title": "Definition:Invariant Mapping Under Equivalence Relation", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R$ be an equivalence relation on $S$. Let $f: S \\to T$ be a mapping. Then $f$ is '''invariant under $\\mathcal R$''' {{iff}}: :$x \\mathrel {\\mathcal R} y \\implies f \\left({x}\\right) = f \\left({y}\\right) $"}
+{"_id": "29340", "title": "Definition:Complement of Subgroup/Definition 1", "text": "$K$ is a '''complement''' of $H$ {{iff}}: : $G = H K$ and $H \\cap K = \\set e$"}
+{"_id": "29341", "title": "Definition:Complement of Subgroup/Definition 2", "text": "$K$ is a '''complement''' of $H$ {{iff}}: :$G = K H$ and $H \\cap K = \\set e$"}
+{"_id": "29342", "title": "Definition:Finer Equivalence Relation", "text": "Let $X$ be a set. Let $\\equiv$ and $\\sim$ be equivalence relations on $X$. Then $\\equiv$ is '''finer''' than $\\sim$ {{iff}}: :$\\forall x, y \\in X : x \\equiv y \\implies x \\sim y$"}
+{"_id": "29343", "title": "Definition:Finer Topology/Definition 1", "text": "$\\tau_1$ is '''finer''' than $\\tau_2$ {{iff}} $\\tau_1 \\supseteq \\tau_2$."}
+{"_id": "29345", "title": "Definition:Gaussian Hypergeometric Function", "text": "The '''Gaussian Hypergeometric Function''' is a hypergeometric function, given for $\\size z < 1$ by: :$\\displaystyle {}_2 \\map {F_1} {a, b; c; z} = \\sum_{n \\mathop = 0}^\\infty \\dfrac { a^{\\overline n} b^{\\overline n} } { c^{\\overline n} } \\dfrac {z^n} {n!}$ where $x^{\\overline n}$ denotes the $n$th rising factorial power of $x$."}
+{"_id": "29346", "title": "Definition:Local Ring/Commutative/Definition 1", "text": "The ring $A$ is '''local''' {{iff}} it has a unique maximal ideal."}
+{"_id": "29348", "title": "Definition:Local Ring/Noncommutative", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity. === Definition 1 === {{Definition:Local Ring/Noncommutative/Definition 1}} === Definition 2 === {{Definition:Local Ring/Noncommutative/Definition 2}} === Definition 3 === {{Definition:Local Ring/Noncommutative/Definition 3}} === Definition 4 === $R$ is a '''local ring''' {{iff}} it is nontrivial and the sum of any two non-units is a non-unit. {{refactor}} * The zero does not equal the unity, and for all $a \\in R$, either $a$ or $1 + \\paren {-a}$ is a unit. * If the summation $\\displaystyle \\sum_{i \\mathop = 1}^n a_i$ is a unit, then some of the $a_i$ are also units (in particular the empty sum is not a unit)."}
+{"_id": "29349", "title": "Definition:Local Ring/Noncommutative/Definition 1", "text": "$R$ is a '''local ring''' {{iff}} it has a unique maximal left ideal."}
+{"_id": "29350", "title": "Definition:Local Ring/Noncommutative/Definition 2", "text": "$R$ is a '''local ring''' {{iff}} it has a unique maximal right ideal."}
+{"_id": "29351", "title": "Definition:Local Ring/Noncommutative/Definition 3", "text": "Let $\\operatorname {rad} R$ be its Jacobson radical. Then $R$ is a '''local ring''' {{iff}} the quotient ring $R / \\operatorname{rad} R$ is a division ring."}
+{"_id": "29352", "title": "Definition:Local Ring Homomorphism/Definition 1", "text": "The homomorphism $f$ is '''local''' {{iff}} the image $f(\\mathfrak m) \\subseteq \\mathfrak n$."}
+{"_id": "29353", "title": "Definition:Local Ring Homomorphism/Definition 2", "text": "The homomorphism $f$ is '''local''' {{iff}} the preimage $f^{-1}(\\mathfrak n) \\supseteq \\mathfrak m$."}
+{"_id": "29354", "title": "Definition:Local Ring Homomorphism/Definition 3", "text": "The homomorphism $f$ is '''local''' {{iff}} the preimage $f^{-1}(\\mathfrak n) = \\mathfrak m$."}
+{"_id": "29355", "title": "Definition:Tensor Product of Abelian Groups/Family", "text": "Let $I$ be an indexing set. Let $\\family {G_i}_{i \\mathop \\in I}$ be a family of abelian groups. Let $G = \\displaystyle \\prod_{i \\mathop \\in I} G_i$ be their direct product. === Definition 1: by universal property === {{Definition:Tensor Product of Abelian Groups/Family/Definition 1}} === Definition 2: construction === {{Definition:Tensor Product of Abelian Groups/Family/Definition 2}}"}
+{"_id": "29356", "title": "Definition:Free Module on Set/Canonical Mapping", "text": "The '''canonical mapping''' $I \\to R^{(I)}$ is the mapping that sends $i \\in I$ to the $i$th standard basis element $e_i$."}
+{"_id": "29357", "title": "Definition:Tensor Product of Abelian Groups/Family/Definition 1", "text": "Their '''tensor product''' is a pair $\\left( \\displaystyle \\bigotimes_{i \\in I} G_i, \\theta \\right)$ where: * $\\displaystyle \\bigotimes_{i \\in I} G_i$ is an abelian group * $\\theta : G \\to \\displaystyle\\bigotimes_{i \\in I} G_i$ is a multiadditive mapping such that, for every pair $(C, \\omega)$ where: * $C$ is an abelian group * $\\omega : G \\to C$ is a multiadditive mapping there exists a unique group homomorphism $g : \\displaystyle \\bigotimes_{i \\in I} G_i \\to C$ with $\\omega = g \\circ \\theta$. :$\\xymatrix{ G \\ar[d]_\\theta \\ar[r]^\\omega & C\\\\ \\displaystyle \\bigotimes_{i \\in I} G_i \\ar@{.>}[ru]_g }$"}
+{"_id": "29358", "title": "Definition:Tensor Product of Abelian Groups/Family/Definition 2", "text": "Their '''tensor product''' is the pair: :$\\tuple {\\displaystyle \\bigotimes_{i \\mathop \\in I} G_i, \\theta}$ where: *$\\displaystyle \\bigotimes_{i \\in I} G_i$ is the quotient of the free abelian group $\\Z \\sqbrk G$ on $G$, by the subgroup generated by the elements of the form $\\tuple {x + y, \\family {z_i}_{i \\mathop \\ne j} } - \\tuple {x, \\family {z_i}_{i \\mathop \\ne j} } - \\tuple {y, \\family {z_i}_{i \\mathop \\ne j} }$ *:for $j \\in I$, $x, y \\in G_j$, $\\family {z_i}_{i \\mathop \\ne j} \\in \\displaystyle \\prod_{i \\mathop \\ne j} G_i$, where we denote $\\tuple {x, \\family {z_i}_{i \\mathop \\ne j} }$ for: *:*the family in $G$ whose $j$th term is $x$ and whose $i$th term is $z_i$, for $i \\ne j$ *:*its image under the canonical mapping $G \\to \\Z \\sqbrk G$. * $\\theta : G \\to \\displaystyle \\bigotimes_{i \\mathop \\in I} G_i$ is the composition of the canonical mapping $G \\to \\Z \\sqbrk G$ with the quotient group epimorphism $\\Z \\sqbrk G \\to \\displaystyle \\bigotimes_{i \\mathop \\in I} G_i$: *:$G \\hookrightarrow \\Z \\sqbrk G \\twoheadrightarrow \\displaystyle \\bigotimes_{i \\mathop \\in I} G_i$"}
+{"_id": "29359", "title": "Definition:Hurwitz Zeta Function", "text": "The '''Hurwitz zeta function''' is a generalisation of the Riemann zeta function. It is defined for $\\map \\Re s > 1$ and $\\map \\Re q > 0$, by:  :$\\displaystyle \\map \\zeta {s, q} = \\sum_{n \\mathop = 0}^\\infty \\frac 1 {\\paren {n + q}^s}$"}
+{"_id": "29360", "title": "Definition:Ordered Tuple/Empty", "text": "Let $S$ be a set. The '''empty ordered tuple on $S$''' is the empty mapping: :$\\varnothing \\to S$ from the empty set $\\varnothing$ to $S$."}
+{"_id": "29361", "title": "Definition:Ordered Tuple/On Set", "text": "Let $S$ be a set. Let $s: \\N^*_n \\to S$ be an ordered tuple. Then $s$ is called an '''ordered tuple on $S$''', its codomain."}
+{"_id": "29362", "title": "Definition:Weierstrass E-Function/Definition 1", "text": "The following mapping is known as the '''Weierstrass E-Function''' of $J \\sqbrk {\\mathbf y}$: :$\\map E {x, \\mathbf y, \\mathbf z, \\mathbf w} = \\map F {x, \\mathbf y, \\mathbf w} - \\map F {x, \\mathbf y, \\mathbf z} + \\paren {\\mathbf w - \\mathbf z} F_{\\mathbf y'} \\paren {x, \\mathbf y, \\mathbf z}$"}
+{"_id": "29363", "title": "Definition:Weierstrass E-Function/Definition 2", "text": "Let $\\theta \\in \\R: 0 < \\theta < 1$. The following mapping is known as the '''Weierstrass E-Function of $J \\sqbrk {\\mathbf y}$''': :$\\ds \\map E {x, \\mathbf y, \\mathbf z, \\mathbf w} = \\frac 1 2 \\sum_{i, k \\mathop = 1}^n \\paren {w_i - z_i} \\paren {w_k - z_k} F_{y_i' y_k'} \\paren {x, \\mathbf y, \\mathbf z + \\theta \\paren {\\mathbf w - \\mathbf z} }$"}
+{"_id": "29364", "title": "Definition:Euler Numbers/Sequence", "text": "The sequence of Euler numbers begins: {{begin-eqn}} {{eqn | l = E_0       | r = 1       | c =  }} {{eqn | l = E_2       | r = -1       | c =  }} {{eqn | l = E_4       | r = 5       | c =  }} {{eqn | l = E_6       | r = -61       | c =  }} {{eqn | l = E_8       | r = 1385       | c =  }} {{eqn | l = E_{10}       | r = -50 \\, 521       | c =  }} {{eqn | l = E_{12}       | r = 2 \\, 702 \\, 765       | c =  }} {{eqn | l = E_{14}       | r = -199 \\, 360 \\, 981       | c =  }} {{eqn | l = E_{16}       | r = 19 \\, 391 \\, 512 \\, 145       | c =  }} {{eqn | l = E_{18}       | r = -2 \\, 404 \\, 879 \\, 675 \\, 441       | c =  }} {{eqn | l = E_{20}       | r = 370 \\, 371 \\, 188 \\, 237 \\, 525       | c =  }} {{eqn | l = E_{22}       | r = -69 \\, 348 \\, 874 \\, 393 \\, 137 \\, 901       | c =  }} {{eqn | l = E_{24}       | r = 15 \\, 514 \\, 534 \\, 163 \\, 557 \\, 086 \\, 905       | c =  }} {{end-eqn}} Odd index Euler numbers are all $0$."}
+{"_id": "29365", "title": "Definition:Million", "text": "'''Million''' is a name for $10^6$. :$1 \\, 000 \\, 000 = 1000 \\times 1000$"}
+{"_id": "29366", "title": "Definition:Variance", "text": "=== Discrete Random Variable === {{:Definition:Variance of Discrete Random Variable}} === Continuous Random Variable === {{:Definition:Variance of Continuous Random Variable}}"}
+{"_id": "29367", "title": "Definition:Variance/Continuous", "text": "Let $X$ be a continuous random variable.  Then the '''variance of $X$''', written $\\var X$, is a measure of how much the values of $X$ varies from the expectation $\\expect X$, and is defined as: :$\\var X := \\expect {\\paren {X - \\expect X}^2}$ That is, the expectation of the squares of the deviations from the expectation."}
+{"_id": "29368", "title": "Definition:Taylor Series/Two Variables", "text": "Let $f: \\R^2 \\to \\R$ be a real-valued function of $2$ variables which is smooth on the open rectangle $\\left({a \\,.\\,.\\, b}\\right) \\times \\left({c \\,.\\,.\\, d}\\right)$. Let $\\left({\\xi, \\zeta}\\right) \\in \\left({a \\,.\\,.\\, b}\\right) \\times \\left({c \\,.\\,.\\, d}\\right)$. Then the '''Taylor series expansion of $f$''' about $\\left({\\xi, \\zeta}\\right)$ is: {{begin-eqn}} {{eqn | l = f \\left({x, y}\\right)       | r = f \\left({\\xi, \\zeta}\\right) + \\left({x - \\xi}\\right) f_x \\left({\\xi, \\zeta}\\right) + \\left({y - \\zeta}\\right) f_y \\left({\\xi, \\zeta}\\right)       | c =  }} {{eqn | o =        | ro= +       | r = \\frac 1 {2!} \\left({\\left({x - \\xi}\\right)^2 f_{xx} \\left({\\xi, \\zeta}\\right) + 2 \\left({x - \\xi}\\right) \\left({y - \\zeta}\\right) f_{xy} \\left({\\xi, \\zeta}\\right) + \\left({y - \\zeta}\\right)^2 f_{yy} \\left({\\xi, \\zeta}\\right) }\\right)       | c =  }} {{eqn | o =        | ro= +       | r = \\cdots       | c =  }} {{end-eqn}} where $f_x \\left({\\xi, \\zeta}\\right)$, $f_y \\left({\\xi, \\zeta}\\right)$ denote partial derivatives {{WRT|Differentiation}} $x, y, \\ldots$ evaluated at $x = \\xi$, $y = \\zeta$."}
+{"_id": "29369", "title": "Definition:Uniform Distribution/Continuous", "text": "Let $X$ be a continuous random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $a, b \\in \\R$ such that $a < b$. $X$ is said to be '''uniformly distributed''' on the closed real interval $\\closedint a b$ {{iff}} it has probability density function:  :$\\map {f_X} x = \\begin{cases} \\dfrac 1 {b - a} & a \\le x \\le b \\\\ 0 & \\text{otherwise} \\end{cases}$  This is written:  :$X \\sim \\ContinuousUniform a b$"}
+{"_id": "29370", "title": "Definition:Median of Continuous Random Variable", "text": "Let $X$ be a continuous random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $X$ have probability density function $f_X$.  A '''median''' of $X$ is defined as a real number $m_X$ such that:  :$\\displaystyle \\map \\Pr {X < m_X} = \\int_{-\\infty}^{m_X} \\map {f_X} x \\rd x = \\frac 1 2$"}
+{"_id": "29372", "title": "Definition:Mode of Continuous Random Variable", "text": "Let $X$ be a continuous random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $X$ have probability density function $f_X$.  We call $M$ a '''mode''' of $X$ if $f_X$ attains its (global) maximum at $M$."}
+{"_id": "29373", "title": "Definition:Gamma Distribution", "text": "Let $X$ be a continuous random variable on a probability space $\\left({\\Omega, \\Sigma, \\Pr}\\right)$. Let $\\operatorname{Im} \\left({X}\\right) = \\R_{\\ge 0}$. $X$ is said to have a '''Gamma distribution''' if it has probability density function:  :$\\displaystyle f_X\\left({x}\\right) = \\frac{ \\beta^\\alpha x^{\\alpha - 1} e^{-\\beta x} } {\\Gamma \\left({\\alpha}\\right)}$ for $\\alpha, \\beta > 0$, where $\\Gamma$ is the Gamma function. This is written:  :$X \\sim \\Gamma \\left({\\alpha, \\beta}\\right)$"}
+{"_id": "29374", "title": "Definition:Bernoulli Numbers/Archaic Form/Definition 1", "text": "{{begin-eqn}} {{eqn | l = \\frac x {e^x - 1}       | r = 1 - \\frac x 2 + \\sum_{n \\mathop = 1}^\\infty \\left({-1}\\right)^{n - 1} \\frac{B_n^* x^{2 n} } {\\left({2 n}\\right)!}       | c =  }} {{eqn | r = 1 - \\frac x 2 + \\frac{B_1^* x^2} {2!} - \\frac{B_2^* x^4} {4!} + \\frac{B_3^* x^6} {6!} - \\cdots       | c =  }} {{end-eqn}} for $x \\in \\R$ such that $\\left\\lvert{x}\\right\\rvert < 2 \\pi$"}
+{"_id": "29375", "title": "Definition:Bernoulli Numbers/Archaic Form/Sequence", "text": "The sequence of old style Bernoulli numbers begins: {{begin-eqn}} {{eqn | l = B_1^*       | r = \\dfrac 1 6       | c =  }} {{eqn | l = B_2^*       | r = \\dfrac 1 {30}       | c =  }} {{eqn | l = B_3^*       | r = \\dfrac 1 {42}       | c =  }} {{eqn | l = B_4^*       | r = \\dfrac 1 {30}       | c =  }} {{eqn | l = B_5^*       | r = \\dfrac 5 {66}       | c =  }} {{eqn | l = B_6^*       | r = \\dfrac {691} {2730}       | c =  }} {{eqn | l = B_7^*       | r = \\dfrac 7 6       | c =  }} {{eqn | l = B_8^*       | r = \\dfrac {3617} {510}       | c =  }} {{eqn | l = B_9^*       | r = \\dfrac {43 \\, 867} {798}       | c =  }} {{eqn | l = B_{10}^*       | r = \\dfrac {174 \\, 611} {330}       | c =  }} {{eqn | l = B_{11}^*       | r = \\dfrac {854 \\, 513} {138}       | c =  }} {{eqn | l = B_{12}^*       | r = \\dfrac {236 \\, 364 \\, 091} {2730}       | c =  }} {{end-eqn}}"}
+{"_id": "29376", "title": "Definition:Bernoulli Numbers/Archaic Form/Definition 2", "text": ":$\\displaystyle 1 - \\frac x 2 \\cot \\frac x 2 = \\sum_{n \\mathop = 1}^\\infty \\frac{B_n^* x^{2 n} } {\\left({2 n}\\right)!}$ {{begin-eqn}} {{eqn | l = 1 - \\frac x 2 \\cot \\frac x 2       | r = \\sum_{n \\mathop = 1}^\\infty \\frac{B_n^* x^{2 n} } {\\left({2 n}\\right)!}       | c =  }} {{eqn | r = \\frac{B_1^* x^2} {2!} + \\frac{B_2^* x^4} {4!} + \\frac{B_3^* x^6} {6!} + \\cdots       | c =  }} {{end-eqn}} for $x \\in \\R$ such that $\\left\\lvert{x}\\right\\rvert < \\pi$"}
+{"_id": "29377", "title": "Definition:Euler Numbers/Alternative Form", "text": "An alternative form of the Euler numbers can often be found. Usually denoted with the symbol $E_n^*$, they are generally not used on {{ProofWiki}}. === Definition 1 === {{:Definition:Euler Numbers/Alternative Form/Definition 1}} === Definition 2 === {{:Definition:Euler Numbers/Alternative Form/Definition 2}}"}
+{"_id": "29378", "title": "Definition:Euler Numbers/Alternative Form/Definition 1", "text": "{{begin-eqn}} {{eqn | l = \\sech x       | r = 1 + \\sum_{n \\mathop = 1}^\\infty \\paren {-1}^n \\frac {E_n^* x^{2 n} } {\\paren {2 n}!}       | c =  }} {{eqn | r = 1 - \\frac {E_1^* x^2} {2!} + \\frac {E_2^* x^4} {4!} - \\frac {E_3^* x^6} {6!} + \\cdots       | c =  }} {{end-eqn}} where $\\size x < \\dfrac \\pi 2 $."}
+{"_id": "29379", "title": "Definition:Euler Numbers/Alternative Form/Definition 2", "text": "{{begin-eqn}} {{eqn | l = \\sec x       | r = 1 + \\sum_{n \\mathop = 1}^\\infty \\frac {E^*_n x^{2 n} } {\\paren {2 n}!}       | c =  }} {{eqn | r = 1 + \\frac {E_1^* x^2} {2!} + \\frac {E_2^* x^4} {4!} + \\frac {E_3^* x^6} {6!} + \\cdots       | c =  }} {{end-eqn}} where $\\size x < \\dfrac \\pi 2$."}
+{"_id": "29380", "title": "Definition:Euler Numbers/Alternative Form/Sequence", "text": "The sequence of the alternative form of Euler numbers begins: {{begin-eqn}} {{eqn | l = E_1^*       | r = 1       | c =  }} {{eqn | l = E_2^*       | r = 5       | c =  }} {{eqn | l = E_3^*       | r = 61       | c =  }} {{eqn | l = E_4^*       | r = 1385       | c =  }} {{eqn | l = E_5^*       | r = 50 \\, 521       | c =  }} {{eqn | l = E_6^*       | r = 2 \\, 702 \\, 765       | c =  }} {{eqn | l = E_7^*       | r = 199 \\, 360 \\, 981       | c =  }} {{eqn | l = E_8^*       | r = 19 \\, 391 \\, 512 \\, 145       | c =  }} {{eqn | l = E_9^*       | r = 2 \\, 404 \\, 879 \\, 675 \\, 441       | c =  }} {{eqn | l = E_{10}^*       | r = 370 \\, 371 \\, 188 \\, 237 \\, 525       | c =  }} {{eqn | l = E_{11}^*       | r = 69 \\, 348 \\, 874 \\, 393 \\, 137 \\, 901       | c =  }} {{eqn | l = E_{12}^*       | r = 15 \\, 514 \\, 534 \\, 163 \\, 557 \\, 086 \\, 905       | c =  }} {{end-eqn}}"}
+{"_id": "29381", "title": "Definition:Moment Generating Function", "text": "Let $X$ be a random variable.  Then the '''moment generating function''' of $X$, $M_X$, is defined as:  :$\\map {M_X} t = \\expect {e^{t X} }$ for all $t$ such that this expectation exists."}
+{"_id": "29382", "title": "Definition:Velocity/Dimension", "text": "The dimension of measurement of '''velocity''' is $\\mathsf {L T}^{-1}$."}
+{"_id": "29383", "title": "Definition:Velocity/Units", "text": "=== SI === {{:Definition:Velocity/Units/SI}} === CGS === {{:Definition:Velocity/Units/CGS}} === FPS === {{:Definition:Velocity/Units/FPS}}"}
+{"_id": "29385", "title": "Definition:Vector Quantity/Component", "text": "Let $\\mathbf a$ be a vector quantity embedded in an $n$-dimensional Cartesian coordinate system $C_n$. Let $\\mathbf a$ be represented with its initial point at the origin of $C_n$. Let $\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n$ be the unit vectors in the positive direction of the coordinate axes of $C_n$. Then: :$\\mathbf a = a_1 \\mathbf e_1 + a_2 \\mathbf e_2 + \\cdots + a_3 \\mathbf e_n$ where: :$a_1 \\mathbf e_1, a_2 \\mathbf e_2, \\ldots, a_3 \\mathbf e_n$ are the '''component vectors''' of $\\mathbf a$ in the directions of $\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n$  :$a_1, a_2, \\ldots, a_3$ are the '''components''' of $\\mathbf a$ in the directions of $\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n$. The number of '''components''' in $\\mathbf a$ is determined by the number of dimensions in the Cartesian coordinate system of its frame of reference."}
+{"_id": "29386", "title": "Definition:Differential Entropy", "text": "'''Differential entropy''' extends the concept of entropy to continuous random variables. Let $X$ be a continuous random variable. Let $X$ have probability density function $f_X$.  Then the '''differential entropy''' of $X$, $\\map h X$ measured in nats, is given by:  :$\\ds \\map h X = -\\int_{-\\infty}^\\infty \\map {f_X} x \\ln \\map {f_X} x \\rd x$ Where $\\map {f_X} x = 0$, we take $\\map {f_X} x \\ln \\map {f_X} x = 0$ by convention."}
+{"_id": "29387", "title": "Definition:Vector Calculus", "text": "'''Vector calculus''' is the branch of linear algebra concerned with the application of the techniques of calculus to vector spaces."}
+{"_id": "29388", "title": "Definition:Del Operator", "text": "Let $\\mathbf V$ be a vector space of $n$ dimensions. Let $\\tuple {\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}$ be the standard ordered basis of $\\mathbf V$. The '''del operator''' is a unary operator on $\\mathbf V$ defined as: :$\\nabla := \\displaystyle \\sum_{k \\mathop = 1}^n \\mathbf e_k \\dfrac \\partial {\\partial x_k}$ where $\\mathbf v = \\displaystyle \\sum_{k \\mathop = 0}^n x_k \\mathbf e_k$ is an arbitrary vector of $\\mathbf V$."}
+{"_id": "29392", "title": "Definition:Gradient Operator", "text": "Let $\\struct {M, g}$ be a Riemannian manifold equiped with a metric $g$. Let $f \\in \\map {\\CC^\\infty} M$ be a smooth mapping on $M$. The '''gradient''' of $f$ is defined as: {{begin-eqn}} {{eqn | l = \\grad f        | o = :=        | r = \\nabla f }} {{eqn | r = g^{-1} \\d_{\\d R} f }} {{end-eqn}} where $\\d_{\\d R}$ is de Rham differential."}
+{"_id": "29393", "title": "Definition:Beta Distribution", "text": "Let $X$ be a continuous random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\Img X = \\closedint 0 1$. $X$ is said to have a '''Beta distribution''' {{iff}} it has probability density function:  :$\\map {f_X} X = \\dfrac {x^{\\alpha - 1} \\paren {1 - x}^{\\beta - 1} } {\\map \\Beta {\\alpha, \\beta} }$ for $\\alpha, \\beta > 0$, where $\\Beta$ denotes the Beta function. This is written:  :$X \\sim \\BetaDist \\alpha \\beta$"}
+{"_id": "29394", "title": "Definition:Divergence Operator", "text": "Let $\\struct {M, g}$ be a Riemannian manifold equiped with a metric $g$. Let $\\mathbf X : \\map {\\CC^\\infty} M \\to \\map {\\CC^\\infty} M$ be a smooth vector field. The '''divergence''' of $\\mathbf X$ is defined as: {{begin-eqn}} {{eqn | l = \\operatorname {div} \\mathbf X        | o = :=        | r = \\nabla \\cdot \\mathbf X }} {{eqn | r = \\star^{−1}_g \\d_{\\d R} \\star_g \\map g {\\mathbf X} }} {{end-eqn}} where: :$\\star_g$ is the Hodge star operator of $\\struct {M, g}$ :$\\d_{\\d R}$ is de Rham differential."}
+{"_id": "29395", "title": "Definition:Divergence Operator/Real Cartesian Space", "text": "Let $\\R^n \\left({x_1, x_2, \\ldots, x_n}\\right)$ denote the real Cartesian space of $n$ dimensions. Let $\\left({\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}\\right)$ be the standard ordered basis on $\\R^n$. Let $\\mathbf f = \\left({f_1 \\left({\\mathbf x}\\right), f_2 \\left({\\mathbf x}\\right), \\ldots, f_n \\left({\\mathbf x}\\right)}\\right): \\R^n \\to \\R^n$ be a vector-valued function on $\\R^n$. Let the partial derivative of $\\mathbf f$ with respect to $x_k$ exist for all $f_k$. The '''divergence of $\\mathbf f$''' is defined as: {{begin-eqn}} {{eqn | l = \\operatorname {div} \\mathbf f       | o = :=       | r = \\nabla \\cdot \\mathbf f }} {{eqn | r = \\left({\\sum_{k \\mathop = 1}^n \\mathbf e_k \\dfrac \\partial {\\partial u_k} }\\right) \\cdot \\mathbf f       | c = {{Defof|Del Operator}} }} {{eqn | r = \\sum_{k \\mathop = 1}^n \\dfrac {\\partial f_k} {\\partial x_k}       | c =  }} {{end-eqn}}"}
+{"_id": "29396", "title": "Definition:Curl Operator", "text": "Let $\\map {\\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.. Let $\\tuple {\\mathbf i, \\mathbf j, \\mathbf k}$ be the standard ordered basis on $\\R^3$. Let $\\mathbf f := \\tuple {\\map {f_x} {\\mathbf x}, \\map {f_y} {\\mathbf x}, \\map {f_z} {\\mathbf x} }: \\R^3 \\to \\R^3$ be a vector-valued function on $\\R^3$. The '''curl''' of $\\mathbf f$ is defined as: {{begin-eqn}} {{eqn | l = \\curl \\mathbf f       | o = :=       | r = \\nabla \\times \\mathbf f       | c = where $\\nabla$ denotes the del operator }} {{eqn | r = \\paren {\\mathbf i \\dfrac \\partial {\\partial x} + \\mathbf j \\dfrac \\partial {\\partial y} + \\mathbf k \\dfrac \\partial {\\partial z} } \\times \\paren {f_x \\mathbf i + f_y \\mathbf j + f_z \\mathbf k}       | c = {{Defof|Del Operator}} }} {{eqn | r = \\begin {vmatrix} \\mathbf i & \\mathbf j & \\mathbf k \\\\ \\dfrac \\partial {\\partial x} & \\dfrac \\partial {\\partial y} & \\dfrac \\partial {\\partial z} \\\\ f_x & f_y & f_z \\end{vmatrix}       | c = {{Defof|Vector Cross Product}} }} {{eqn | r = \\paren {\\dfrac {\\partial f_z} {\\partial y} - \\dfrac {\\partial f_y} {\\partial z} } \\mathbf i + \\paren {\\dfrac {\\partial f_x} {\\partial z} - \\dfrac {\\partial f_z} {\\partial x} } \\mathbf j + \\paren {\\dfrac {\\partial f_y} {\\partial x} - \\dfrac {\\partial f_x} {\\partial y} } \\mathbf k       | c =  }} {{end-eqn}}"}
+{"_id": "29397", "title": "Definition:Laplacian", "text": "=== Real-Valued Function === {{:Definition:Laplacian/Real-Valued Function}} === Vector-Valued Function === {{:Definition:Laplacian/Vector-Valued Function}}"}
+{"_id": "29398", "title": "Definition:Laplacian/Real-Valued Function", "text": "Let $\\R^n$ denote the real Cartesian space of $n$ dimensions. Let $\\map f {x_1, x_2, \\ldots, x_n}$ denote a real-valued function on $\\R^n$. Let $\\tuple {\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}$ be the standard ordered basis on $\\R^n$. Let the partial derivative of $f$ with respect to $x_k$ exist for all $x_k$. The '''Laplacian of $f$''' is defined as: {{begin-eqn}} {{eqn | l = \\nabla^2 f       | o = :=       | r = \\nabla \\cdot \\left({\\nabla f}\\right) }} {{eqn | r = \\paren {\\sum_{k \\mathop = 1}^n \\mathbf e_k \\dfrac \\partial {\\partial x_k} } \\cdot \\paren {\\sum_{k \\mathop = 1}^n \\dfrac {\\partial f} {\\partial x_k} \\mathbf e_k}       | c = {{Defof|Del Operator}} }} {{eqn | r = \\sum_{k \\mathop = 1}^n \\dfrac {\\partial^2 f} {\\partial {x_k}^2}       | c =  }} {{end-eqn}}"}
+{"_id": "29399", "title": "Definition:Laplacian/Vector-Valued Function", "text": "Let $\\R^n \\left({x_1, x_2, \\ldots, x_n}\\right)$ denote the real Cartesian space of $n$ dimensions. Let $\\left({\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}\\right)$ be the standard ordered basis on $\\mathbf V$. Let $\\mathbf f = \\left({f_1 \\left({\\mathbf x}\\right), f_2 \\left({\\mathbf x}\\right), \\ldots, f_n \\left({\\mathbf x}\\right)}\\right): \\mathbf V \\to \\mathbf V$ be a vector-valued function on $\\mathbf V$. Let the partial derivative of $\\mathbf f$ with respect to $x_k$ exist for all $f_k$. The '''Laplacian of $\\mathbf f$''' is defined as: {{begin-eqn}} {{eqn | l = \\nabla^2 \\mathbf f       | o = :=       | r = \\left({\\sum_{k \\mathop = 1}^n \\dfrac {\\partial^2 \\mathbf f} {\\partial {x_k}^2} }\\right) }} {{end-eqn}}"}
+{"_id": "29400", "title": "Definition:Biharmonic Operator", "text": "Let $\\R^n$ denote the real Cartesian space of $n$ dimensions. Let $f \\left({x_1, x_2, \\ldots, x_n}\\right)$ denote a real-valued function on $\\R^n$. Let $\\left({\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}\\right)$ be the standard ordered basis on $\\R^n$. Let the partial derivative of $f$ with respect to $x_k$ exist for all $x_k$. The '''biharmonic operator on $f$''' is defined as: {{begin-eqn}} {{eqn | l = \\nabla^4 f       | o = :=       | r = \\nabla^2 \\left({\\nabla^2 f}\\right) }} {{eqn | r = \\left({\\sum_{k \\mathop = 1}^n \\dfrac {\\partial^2} {\\partial x_k^2} }\\right) \\left({\\sum_{k \\mathop = 1}^n \\dfrac {\\partial^2} {\\partial x_k^2} }\\right) f       | c = {{Defof|Del Operator}} }} {{end-eqn}}"}
+{"_id": "29401", "title": "Definition:Primitive (Calculus)/Vector-Valued Function", "text": "Let $U \\subset \\R$ be an open set in $\\R$. Let $\\mathbf f: U \\to \\R^n$ be a vector-valued function on $U$: :$\\forall x \\in U: \\map {\\mathbf f} x = \\displaystyle \\sum_{k \\mathop = 1}^n \\map {f_k} x \\mathbf e_k$ where: :$f_1, f_2, \\ldots, f_n$ are real functions from $U$ to $\\R$ :$\\tuple {e_1, e_2, \\ldots, e_k}$ denotes the standard ordered basis on $\\R^n$. Let $\\mathbf f$ be differentiable on $U$. Let $\\map {\\mathbf g} x := \\dfrac \\d {\\d x} \\map {\\mathbf f} x$ be the derivative of $\\mathbf f$ {{WRT|Differentiation}} $x$. The '''primitive of $\\mathbf g$ {{WRT|Integration}} $x$''' is defined as: :$\\displaystyle \\int \\map {\\mathbf g} x \\rd x := \\map {\\mathbf f} x + \\mathbf c$ where $\\mathbf c$ is an arbitrary constant vector."}
+{"_id": "29402", "title": "Definition:Definite Integral of Vector-Valued Function", "text": "Let $I \\:= \\closedint a b \\subset \\R$ be a closed real interval. Let $\\mathbf f: I \\to \\R^n$ be a vector-valued function on $I$: :$\\forall x \\in I: \\map {\\mathbf f} x = \\displaystyle \\sum_{k \\mathop = 1}^n \\map {f_k} x \\mathbf e_k$ where: :$f_1, f_2, \\ldots, f_n$ are real functions from $U$ to $\\R$ :$\\tuple {e_1, e_2, \\ldots, e_k}$ denotes the standard ordered basis on $\\R^n$. Let $\\mathbf f$ be differentiable on $I$. Let $\\map {\\mathbf g} x := \\dfrac \\d {\\d x} \\map {\\mathbf f} x$ be the derivative of $\\mathbf f$ {{WRT|Differentiation}} $x$. The '''definite integral of $\\mathbf g$ {{WRT|Integration}} $x$ from $a$ to $b$''' is defined as: :$\\displaystyle \\int_a^b \\map {\\mathbf g} x \\rd x := \\map {\\mathbf f} b - \\map {\\mathbf f} a$"}
+{"_id": "29403", "title": "Definition:Derivative of Smooth Path", "text": "=== Real Cartesian Space === {{:Definition:Derivative of Smooth Path/Real Cartesian Space}} === Complex Plane === {{:Definition:Derivative of Smooth Path/Complex Plane}} Category:Definitions/Vector Analysis q44bxnogmof4cosxrf20xduw0w1a9l2"}
+{"_id": "29404", "title": "Definition:Smooth Path", "text": "=== Real Cartesian Space === {{:Definition:Smooth Path/Real Cartesian Space}} === Complex Analysis === {{:Definition:Smooth Path/Complex}} Category:Definitions/Vector Analysis ghfzofsi4chsdwv7pk61lzmt4d9zrpl"}
+{"_id": "29405", "title": "Definition:Smooth Path/Real Cartesian Space", "text": "Let $\\R^n$ be a real cartesian space of $n$ dimensions. Let $\\left[{a \\,.\\,.\\, b}\\right]$ be a closed real interval. Let $\\rho: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R^n$ be a path in $\\R^n$. That is, let $\\rho$ be a continuous real-valued function from $\\left[{a \\,.\\,.\\, b}\\right]$ to $\\R^n$. For each $k \\in \\left\\{ {1, 2, \\ldots, n}\\right\\}$, define the real function $\\rho_k: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$ by: :$\\forall t \\in \\left[{a \\,.\\,.\\, b}\\right]: \\rho_k \\left({t}\\right) = \\pr_k \\left({\\rho \\left({t}\\right)}\\right)$ where $\\pr_k$ denotes the $k$th projection from the image $\\operatorname{Im} \\left({\\rho}\\right)$ of $\\rho$ to $\\R$. Then $\\rho$ is a '''smooth path (in $\\R^n$)''' {{iff}}: :$(1): \\quad$ all of $\\pr_k$ are continuously differentiable :$(2): \\quad$ for all $t \\in \\left[{a \\,.\\,.\\, b}\\right]$, at least one $\\rho_k' \\left({t}\\right) \\ne 0$, where $\\rho_k'$ denotes the derivative of $\\rho_k$ {{WRT|Differentiation}} $t$."}
+{"_id": "29406", "title": "Definition:Smooth Path/Closed/Real Cartesian Space", "text": "Let $\\rho: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R^n$ be a '''smooth path in $\\R^n$'''. $\\rho$ is a '''closed smooth path''' {{iff}} $\\rho$ is a closed path. That is, {{iff}} $\\rho \\left({a}\\right) = \\rho \\left({b}\\right)$."}
+{"_id": "29407", "title": "Definition:Smooth Path/Simple/Real Cartesian Space", "text": "Let $\\rho: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R^n$ be a '''smooth path in $\\R^n$'''. $\\rho$ is a '''simple smooth path (in $\\R^n$)''' {{iff}}: :$(1): \\quad \\rho$ is injective on the half-open interval $\\left[{a \\,.\\,.\\, b}\\right)$ :$(2): \\quad \\forall t \\in \\left({a \\,.\\,.\\, b}\\right): \\rho \\left({t}\\right) \\ne \\rho \\left({b}\\right)$ That is, {{iff}} $t_1, t_2 \\in \\left({a \\,.\\,.\\, b}\\right)$ with $t_1 \\ne t_2$, then $\\gamma \\left({a}\\right) \\ne \\gamma \\left({t_1}\\right) \\ne \\gamma \\left({t_2}\\right) \\ne \\gamma \\left({b}\\right)$."}
+{"_id": "29408", "title": "Definition:Derivative of Smooth Path/Real Cartesian Space", "text": "Let $\\R^n$ be a real cartesian space of $n$ dimensions. Let $\\left[{a \\,.\\,.\\, b}\\right]$ be a closed real interval. Let $\\rho: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R^n$ be a '''smooth path in $\\R^n$'''. For each $k \\in \\left\\{ {1, 2, \\ldots, n}\\right\\}$, define the real function $\\rho_k: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$ by: :$\\forall t \\in \\left[{a \\,.\\,.\\, b}\\right]: \\rho_k \\left({t}\\right) = \\pr_k \\left({\\rho \\left({t}\\right)}\\right)$ where $\\pr_k$ denotes the $k$th projection from the image $\\operatorname{Im} \\left({\\rho}\\right)$ of $\\rho$ to $\\R$. It follows from the definition of a  smooth path that $\\rho_k$ is continuously differentiable for all $k$. Let $\\rho_k' \\left({t}\\right)$ denote the derivative of $\\rho_k$ {{WRT|Differentiation}} $t$. The '''derivative of $\\rho$''' is the continuous vector-valued function $\\rho': \\left[{a \\,.\\,.\\, b}\\right] \\to \\R^n$ defined by: :$\\forall t \\in \\left[{a \\,.\\,.\\, b}\\right]: \\rho' \\left({t}\\right) = \\displaystyle \\sum_{k \\mathop = 1}^n \\rho_k' \\left({t}\\right) \\mathbf e_k$ where $\\left({\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}\\right)$ denotes the standard ordered basis of $\\R^n$."}
+{"_id": "29409", "title": "Definition:Contour", "text": "Let $\\R^n$ be a real cartesian space of $n$ dimensions. Let $C_1, \\ldots, C_n$ be directed smooth curves in $\\R^n$. For each $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$, let $C_i$ be parameterized by the smooth path $\\rho_i: \\left[{a_i \\,.\\,.\\, b_i}\\right] \\to \\R^n$. For each  $i \\in \\left\\{ {1, \\ldots, n-1}\\right\\}$, let the end point of $\\rho_i$ equal the start point of $\\rho_{i + 1}$: :$\\rho_i \\left({b_i}\\right) = \\rho_{i + 1} \\left({a_{i + 1} }\\right)$ Then the finite sequence $\\left\\langle{C_1, \\ldots, C_n}\\right\\rangle$ is called a '''contour''' (in $\\R^n$)."}
+{"_id": "29410", "title": "Definition:Image of Contour", "text": "=== Real Cartesian Space === {{:Definition:Image of Contour (Real Cartesian Space)}} === Complex Plane === {{:Definition:Image of Contour (Complex Plane)}} Category:Definitions/Vector Analysis 7uaz6viadxp9y8eryj3rfwpmsdr83wu"}
+{"_id": "29411", "title": "Definition:Student's t-Distribution", "text": "Let $X$ be a continuous random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\Img X = \\R$. $X$ is said to have a '''$t$-distribution''' with $k$ degrees of freedom {{iff}} it has probability density function:  :$\\map {f_X} x = \\dfrac {\\map \\Gamma {\\frac {k + 1} 2} } {\\sqrt {\\pi k} \\map \\Gamma {\\frac k 2} } \\paren {1 + \\dfrac {x^2} k}^{-\\frac {k + 1} 2}$ for some $k \\in \\R_{> 0}$. This is written:  :$X \\sim \\StudentT k$"}
+{"_id": "29412", "title": "Definition:Estimator", "text": "Let $X_1, X_2, \\ldots, X_n$ be random variables.  Let the joint distribution of $X_1, X_2, \\ldots, X_n$ be indexed by a population parameter $\\theta$.  Let $\\delta: \\R^n \\to \\R$ be a real-valued function.  The random variable $\\hat \\theta = \\map \\delta {X_1, X_2, \\ldots, X_n}$ is called an '''estimator''' of $\\theta$.  A particular realization of $\\hat \\theta$ is called an '''estimate''' of $\\theta$."}
+{"_id": "29414", "title": "Definition:Directed Smooth Curve/Complex Plane", "text": "Let $\\gamma : \\left[{ a \\,.\\,.\\, b }\\right] \\to \\C$ be a smooth path in $\\C$. The '''directed smooth curve''' with '''parameterization''' $\\gamma$ is defined as an equivalence class of smooth paths as follows: A smooth path $\\sigma: \\left[{ a \\,.\\,.\\, b }\\right] \\to \\C$ belongs to the equivalence class of $\\gamma$ {{iff}}: : there exists a bijective differentiable strictly increasing real function: :: $\\phi: \\left[{c \\,.\\,.\\, d}\\right] \\to \\left[{a \\,.\\,.\\, b}\\right]$ : such that $\\sigma = \\gamma \\circ \\phi$."}
+{"_id": "29415", "title": "Definition:Directed Smooth Curve/Parameterization", "text": "Let $\\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ be a directed smooth curve in $\\R^n$. Let $\\rho: \\left[{a \\,.\\,.\\, b}\\right] \\to \\C$ be a smooth path in $\\R^n$. Then $\\rho$ is a '''parameterization''' of $C$ {{iff}} $\\rho$ is an element of the equivalence class that constitutes $C$."}
+{"_id": "29417", "title": "Definition:Parameterization", "text": "=== Parameterization of Directed Smooth Curve === {{:Definition:Parameterization of Directed Smooth Curve}} === Parameterization of Contour === {{:Definition:Parameterization of Contour}}"}
+{"_id": "29418", "title": "Definition:Directed Smooth Curve/Endpoints", "text": "Let $C$ be parameterized by a smooth path $\\rho: \\left[{a \\,.\\,.\\, b}\\right] \\to \\C$. Then: : $\\rho \\left({a}\\right)$ is the '''start point''' of $C$ : $\\rho \\left({b}\\right)$ is the '''end point''' of $C$. Collectively, $\\rho \\left({a}\\right)$ and $\\rho \\left({b}\\right)$ are known as the '''endpoints''' of $\\rho$."}
+{"_id": "29419", "title": "Definition:Bias of Estimator", "text": "Let $\\theta$ be a population parameter of some statistical model.  Let $\\mathbf X$ be a random sample from this population. Let $\\delta$ be an estimator of $\\theta$.  The '''bias''' of $\\delta$ is defined as:  :$\\map {\\operatorname{bias} } \\delta = \\expect {\\map \\delta {\\mathbf X} } - \\theta$"}
+{"_id": "29420", "title": "Definition:Contour/Parameterization", "text": "Let $C_1, \\ldots, C_n$ be directed smooth curves in $\\R^n$. Let $C_i$ be parameterized by the smooth path $\\rho_i: \\left[{a_i \\,.\\,.\\, b_i}\\right] \\to \\R^n$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. Let $C$ be the contour defined by the finite sequence $C_1, \\ldots, C_n$. The '''parameterization of $C$''' is defined as the function $\\rho: \\left[{a_1 \\,.\\,.\\, c_n}\\right] \\to \\R^n$ with: :$\\rho \\restriction_{\\left[{c_i \\,.\\,.\\, c_{i + 1} }\\right] } \\left({t}\\right) = \\rho_i \\left({t}\\right)$ where: : $\\displaystyle c_i = a_1 + \\sum_{j \\mathop = 1}^i b_j - \\sum_{j \\mathop = 1}^i a_j$ for $i \\in \\left\\{ {0, \\ldots, n}\\right\\}$ : $\\rho \\restriction_{\\left[{c_i \\,.\\,.\\, c_{i + 1} }\\right] }$ denotes the restriction of $\\rho$ to $\\left[{c_i \\,.\\,.\\, c_{i + 1} }\\right]$."}
+{"_id": "29421", "title": "Definition:Contour/Closed", "text": "Let $C$ be the '''contour''' in $\\R^n$ defined by the (finite) sequence $\\left\\langle{C_1, \\ldots, C_n}\\right\\rangle$ of directed smooth curves in $\\R^n$. Let $C_i$ be parameterized by the smooth path $\\rho_i: \\left[{a_i \\,.\\,.\\, b_i}\\right] \\to \\R^n$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. $C$ is a '''closed contour''' {{iff}} the start point of $C$ is equal to the end point of $C$: :$\\rho_1 \\left({a_1}\\right) = \\rho_n \\left({b_n}\\right)$"}
+{"_id": "29422", "title": "Definition:Contour/Simple", "text": "Let $C_1, \\ldots, C_n$ be directed smooth curves in $\\R^n$. Let $C_i$ be parameterized by the smooth path $\\rho_i: \\left[{a_i \\,.\\,.\\, b_i}\\right] \\to \\R^n$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. Let $C$ be the contour in $\\R^n$ defined by the finite sequence $C_1, \\ldots, C_n$. $C$ is a '''simple contour''' {{iff}}: :$(1): \\quad$ For all $i, j \\in \\left\\{ {1, \\ldots, n}\\right\\}, t_1 \\in \\left[{a_i \\,.\\,.\\, b_i}\\right), t_2 \\in \\left[{a_j \\,.\\,.\\, b_j}\\right)$ with $t_1 \\ne t_2$, we have $\\rho_i \\left({t_1}\\right) \\ne \\rho_j \\left({t_2}\\right)$ :$(2): \\quad$ For all $k \\in \\left\\{ {1, \\ldots, n}\\right\\}, t \\in \\left[{a_k \\,.\\,.\\, b_k}\\right)$ where either $k \\ne 1$ or $t \\ne a_1$, we have $\\rho_k \\left({t}\\right) \\ne \\rho_n \\left({b_n}\\right)$."}
+{"_id": "29423", "title": "Definition:Contour/Length", "text": "Let $C$ be a '''contour''' in $C$ defined by the (finite) sequence $\\left\\langle{C_1, \\ldots, C_n}\\right\\rangle$ of directed smooth curves in $\\R^n$. Let $C_i$ be parameterized by the smooth path $\\rho_i: \\left[{a_i \\,.\\,.\\, b_i}\\right] \\to \\R^n$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. The '''length''' of $C$ is defined as: :$\\displaystyle L \\left({C}\\right) := \\sum_{i \\mathop = 1}^n \\int_{a_i}^{b_i} \\left\\vert{\\rho_i' \\left({t}\\right) }\\right\\vert \\rd t$"}
+{"_id": "29424", "title": "Definition:Contour/Image", "text": "Let $C$ be a '''contour''' in $\\R^n$ defined by the (finite) sequence $\\left\\langle{C_1, \\ldots, C_n}\\right\\rangle$ of directed smooth curves in $\\R^n$. Let $C_i$ be parameterized by the smooth path $\\rho_i: \\left[{a_i \\,.\\,.\\, b_i}\\right] \\to \\R^n$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. The '''image of $C$''' is defined as: :$\\displaystyle \\operatorname{Im} \\left({C}\\right) := \\bigcup_{i \\mathop = 1}^n \\operatorname{Im} \\left({\\rho_i}\\right)$ where $\\operatorname{Im} \\left({\\rho_i}\\right)$ denotes the image of $\\rho_i$."}
+{"_id": "29425", "title": "Definition:Contour/Endpoints", "text": "Let $C_1, \\ldots, C_n$ be directed smooth curves in $\\R^n$. Let $C_i$ be parameterized by the smooth path $\\rho_i: \\left[{a_i \\,.\\,.\\, b_i}\\right] \\to \\R^n$ for all $i \\in \\left\\{ {1, \\ldots, n}\\right\\}$. Let $C$ be the contour defined by the finite sequence $C_1, \\ldots, C_n$. The '''start point''' of $C$ is $\\rho_1 \\left({a_1}\\right)$. The '''end point''' of $C$ is $\\rho_n \\left({b_n}\\right)$. Collectively, $\\rho_1 \\left({a_1}\\right)$ and $\\rho_n \\left({b_n}\\right)$ are referred to as the '''endpoints''' of $C$."}
+{"_id": "29426", "title": "Definition:Concatenation of Contours/Complex Plane", "text": "Let $C$ and $D$ be contours in the complex plane $\\C$. Thus: :$C$ is a (finite) sequence of directed smooth curves $\\sequence {C_1, \\ldots, C_n}$ :$D$ is a (finite) sequence of directed smooth curves $\\sequence {D_1, \\ldots, D_m}$. Let $C_i$ be parameterized by the smooth path: :$\\gamma_i: \\closedint {a_i} {b_i} \\to \\C$ for all $i \\in \\set {1, \\ldots, n}$ Let $D_i$ be parameterized by the smooth path: :$\\sigma_i: \\closedint {c_i} {d_i} \\to \\C$ for all $i \\in \\set {1, \\ldots, m}$ Let $\\map {\\gamma_n} {b_n} = \\map {\\sigma_1} {c_1}$. Then the '''concatenation of the contours''' $C$ and $D$, denoted $C \\cup D$, is the contour defined by the (finite) sequence: :$\\sequence {C_1, \\ldots, C_n, D_1, \\ldots, D_m}$"}
+{"_id": "29427", "title": "Definition:Pointwise Operation/Real-Valued Functions/Multiary", "text": "For ease of notation, write $\\left[{S \\to \\R}\\right]$ for $\\R^S$. Let $I$ be some index set. Let $\\oplus^I: \\R^I \\to \\R$ be an $I$-ary operation on $\\R$. Then $\\oplus^I: \\left[{S \\to \\R}\\right]^I \\to \\left[{S \\to \\R}\\right]$, referred to as '''pointwise $\\oplus^I$''', is defined as: :$\\forall \\left({f_i}\\right)_{i \\mathop \\in I} \\in \\left[{S \\to \\R}\\right]^I: \\forall s \\in S: \\left({\\oplus^I \\left({f_i}\\right)_{i \\in I} }\\right) \\left({s}\\right) := \\oplus^I \\left({f_i \\left({s}\\right) }\\right)_{i \\in I}$"}
+{"_id": "29433", "title": "Definition:Confidence Interval", "text": "Let $\\theta$ be a population parameter of some population. Let $\\mathbf X$ be a random sample from this population. Let $I = \\openint {\\map f {\\mathbf X} } {\\map g {\\mathbf X} }$ for some real-valued functions $f$, $g$. $I$ is said to be a '''$100\\gamma \\%$ confidence interval''' for $\\theta$ if:  :$\\map \\Pr {\\theta \\in I} = \\gamma$ where $0 < \\gamma < 1$."}
+{"_id": "29435", "title": "Definition:Random Vector", "text": "Let $X_1, X_2, \\ldots, X_n$ be random variables on a probability space $\\left({\\Omega, \\Sigma, \\Pr}\\right)$. Then the vector $\\mathbf X = \\left({X_1, X_2, \\ldots, X_n}\\right)$ is referred to as a '''random vector'''."}
+{"_id": "29436", "title": "Definition:Mean Squared Error of Estimator", "text": "Let $\\theta$ be a population parameter of some population. Let $\\mathbf X$ be a random sample from this population. Let $\\hat \\theta$ be an estimator of $\\theta$.  The '''mean squared error''' of $\\hat \\theta$ is defined by: :$\\map {\\operatorname {MSE} } {\\hat \\theta} = \\expect {\\paren {\\map {\\hat \\theta} {\\mathbf X} - \\theta}^2} $"}
+{"_id": "29439", "title": "Definition:Density (Physics)/Area", "text": "The '''area density''' of a two-dimensional body is its mass per unit area."}
+{"_id": "29440", "title": "Definition:Density (Physics)/Linear", "text": "The '''linear density''' of a one-dimensional body is its mass per unit length."}
+{"_id": "29443", "title": "Definition:Expectation of Random Vector", "text": "Let $X_1, X_2, \\ldots, X_n$ be random variables on a probability space $\\left({\\Omega, \\Sigma, \\Pr}\\right)$. Let $\\mathbf X = \\left({X_1, X_2, \\ldots, X_n}\\right)$ be a random vector.  Then the expected value of $\\mathbf X$, $\\mathbb E \\left[{\\mathbf X}\\right]$, is defined by:  :$\\mathbb E \\left[{\\mathbf X}\\right] = \\left({\\mathbb E \\left[{X_1}\\right], \\mathbb E \\left[{X_2}\\right], \\ldots, \\mathbb E \\left[{X_n}\\right]}\\right)$"}
+{"_id": "29446", "title": "Definition:Field of Rational Functions", "text": "Let $K$ be a field. Let $K \\sqbrk x$ be the integral domain of polynomial forms on $K$. Let $\\map K x$ be the set of rational functions on $K$: :$\\map K x := \\set {\\forall f \\in K \\sqbrk x, g \\in K \\sqbrk x^*: \\dfrac {\\map f x} {\\map g x} }$ where $K \\sqbrk x^* = K \\sqbrk x \\setminus \\set {\\text {the null polynomial} }$. Then $\\map K x$ is the '''field of rational functions''' on $K$."}
+{"_id": "29449", "title": "Definition:Sylow p-Subgroup/Definition 3", "text": "Let $n$ be the largest integer such that: :$p^n \\divides \\order G$ where $\\divides$ denotes divisibility. A '''Sylow $p$-subgroup''' is a $p$-subgroup of $G$ which has $p^n$ elements."}
+{"_id": "29450", "title": "Definition:Cross-Covariance Matrix", "text": "Let $\\mathbf X = \\left({X_1, X_2, \\ldots, X_n}\\right)^T$ and $\\mathbf Y = \\left({Y_1, Y_2, \\ldots, Y_m}\\right)^T$ be random vectors.   Then the '''cross-covariance matrix''' of $\\mathbf X$ and $\\mathbf Y$ is defined by:  :$\\operatorname{cov} \\left({\\mathbf X, \\mathbf Y}\\right) = \\mathbb E \\left[{\\left({\\mathbf X - \\mathbb E \\left[{\\mathbf X}\\right]}\\right) \\left({\\mathbf Y - \\mathbb E \\left[{\\mathbf Y}\\right]}\\right)^T}\\right]$ where this expectation exists."}
+{"_id": "29453", "title": "Definition:Doubleton/Class Theory", "text": "Let $a$ and $b$ be sets. The class $\\set {a, b}$ is a '''doubleton (class)'''. It is defined as the class of all $x$ such that $x = a$ or $x = b$: :$\\set {a, b} = \\set {x: x = a \\lor x = b: a \\ne b}$"}
+{"_id": "29454", "title": "Definition:Variance of Random Vector", "text": "Let $\\mathbf X$ be a random vector. The '''variance''' of $\\mathbf X$ is defined by:  :$\\operatorname{var} \\left({\\mathbf X}\\right) = \\operatorname{cov} \\left({\\mathbf X, \\mathbf X}\\right)$ where $\\operatorname{cov}$ is the cross-covariance matrix."}
+{"_id": "29455", "title": "Definition:Even Permutation", "text": "$\\rho$ is an '''even permutation''' {{iff}}: :$\\map \\sgn \\rho = 1$"}
+{"_id": "29456", "title": "Definition:Odd Permutation", "text": "$\\rho$ is an '''odd permutation''' {{iff}}: :$\\map \\sgn \\rho = -1$"}
+{"_id": "29457", "title": "Definition:Zero Residue Class", "text": "Let $m \\in \\Z$. Let $\\mathcal R_m$ be the congruence relation modulo $m$ on the set of all $a, b \\in \\Z$: :$\\mathcal R_m = \\set {\\tuple {a, b} \\in \\Z \\times \\Z: \\exists k \\in \\Z: a = b + k m}$ Let $\\eqclass 0 m$ be the residue class of $0$ (modulo $m$): :$\\eqclass 0 m = \\set {x \\in \\Z: \\exists k \\in \\Z: x = k m}$ Then $\\eqclass 0 m$ is known as the '''zero residue class (modulo $m$)'''."}
+{"_id": "29462", "title": "Definition:Binomial Coefficient/Complex Numbers", "text": "Let $z, w \\in \\C$. Then $\\dbinom z w$ is defined as: :$\\dbinom z w := \\displaystyle \\lim_{\\zeta \\mathop \\to z} \\lim_{\\omega \\mathop \\to w} \\dfrac {\\Gamma \\left({\\zeta + 1}\\right)} {\\Gamma \\left({\\omega + 1}\\right) \\Gamma \\left({\\zeta - \\omega + 1}\\right)}$ where $\\Gamma$ denotes the Gamma function. When $z$ is a negative integer and $w$ is not an integer, $\\dbinom z w$ is infinite."}
+{"_id": "29464", "title": "Definition:Power (Algebra)/Real Number/Definition 3", "text": "First let $x > 1$. Let $r$ be expressed by its decimal expansion: :$r = n \\cdotp d_1 d_2 d_3 \\ldots$ For $k \\in \\Z_{> 0}$, let $\\psi_1, \\psi_2 \\in \\Q$ be rational numbers defined as: {{begin-eqn}} {{eqn | l = \\psi_1       | r = n + \\sum_{j \\mathop = 1}^k \\frac {d_1} {10^k} = n + \\frac {d_1} {10} + \\cdots + \\frac {d_k} {10^k} }} {{eqn | l = \\psi_2       | r = \\psi_1 + \\dfrac 1 {10^k} }} {{end-eqn}} Then $x^r$ is defined as the (strictly) positive real number $\\xi$ defined as: :$\\displaystyle \\lim_{k \\mathop \\to \\infty} x^{\\psi_1} \\le \\xi \\le x^{\\psi_2}$ In this context, $x^{\\psi_1}, x^{\\psi_2}$ denote $x$ to the rational powers $\\psi_1$ and $\\psi_2$. Next let $x < 1$. Then $x^r$ is defined as: :$x^r := \\left({\\dfrac 1 x}\\right)^{-r}$ Finally, when $x = 1$: :$x^r = 1$"}
+{"_id": "29465", "title": "Definition:Power (Algebra)/Real Number/Definition 3/Binary Expansion", "text": "First let $x > 1$. Let $r$ be expressed in binary notation: :$r = n \\cdotp d_1 d_2 d_3 \\ldots$ where $d_1, d_2, d_3 \\ldots$ are in $\\left\\{ {0, 1}\\right\\}$. For $k \\in \\Z_{> 0}$, let $\\psi_1, \\psi_2 \\in \\Q$ be rational numbers defined as: {{begin-eqn}} {{eqn | l = \\psi_1       | r = n + \\sum_{j \\mathop = 1}^k \\frac {d_1} {2^k} = n + \\frac {d_1} 2 + \\cdots + \\frac {d_k} {2^k} }} {{eqn | l = \\psi_2       | r = \\psi_1 + \\dfrac 1 {2^k} }} {{end-eqn}} Then $x^r$ is defined as the (strictly) positive real number $\\xi$ defined as: :$\\displaystyle \\lim_{k \\mathop \\to \\infty} x^{\\psi_1} \\le \\xi \\le x^{\\psi_2}$ In this context, $x^{\\psi_1}, x^{\\psi_2}$ denote $x$ to the rational powers $\\psi_1$ and $\\psi_2$. Next let $x < 1$. Then $x^r$ is defined as: :$x^r := \\left({\\dfrac 1 x}\\right)^{-r}$ Finally, when $x = 1$: :$x^r = 1$"}
+{"_id": "29468", "title": "Definition:General Logarithm/Common/Historical Note", "text": "'''Common logarithms''' were developed by {{AuthorRef|Henry Briggs}}, as a direct offshoot of the work of {{AuthorRef|John Napier}}. After seeing the tables that {{AuthorRef|John Napier|Napier}} published, {{AuthorRef|Henry Briggs|Briggs}} consulted {{AuthorRef|John Napier|Napier}}, and suggested defining them differently, using base $10$. In $1617$, {{AuthorRef|Henry Briggs|Briggs}} published a set of tables of logarithms of the first $1000$ positive integers. In $1624$, he published tables of logarithms which included $30 \\, 000$ logarithms going up to $14$ decimal places. Before the advent of cheap means of electronic calculation, '''common logarithms''' were widely used as a technique for performing multiplication."}
+{"_id": "29469", "title": "Definition:Bit", "text": "A '''bit''' is a '''binary digit'''. Category:Definitions/Computer Science 7exngkynvfq4qe904b9xsblx6x5yu1k"}
+{"_id": "29472", "title": "Definition:Product Notation (Algebra)/Propositional Function/Iverson's Convention", "text": "Let $\\displaystyle \\prod_{R \\left({j}\\right)} a_j$ be the product over all $a_j$ such that $j$ satisfies $R$. This can also be expressed: :$\\displaystyle \\prod_{j \\mathop \\in \\Z} a_j^{\\left[{R \\left({j}\\right)}\\right]}$ where $\\left[{R \\left({j}\\right)}\\right]$ is Iverson's convention."}
+{"_id": "29476", "title": "Definition:Uncountable Sum", "text": "Let $X$ be an uncountable set. Let $f: X \\to \\left[{0 \\,.\\,.\\, +\\infty}\\right]$ be an extended real-valued function. The '''uncountable sum''' of $f$ over $X$ is defined to be the supremum of the finite sums: :$\\displaystyle \\sum_{x \\mathop \\in X} f \\left({x}\\right) := \\sup \\left\\{ {\\sum_{x \\mathop \\in F} f \\left({x}\\right): F \\subseteq X, F \\text{ finite} }\\right\\}$"}
+{"_id": "29477", "title": "Definition:Supremum of Set/Real Numbers/Propositional Function", "text": "Let $\\family {a_j}_{j \\mathop \\in I}$ be a family of elements of the real numbers $\\R$ indexed by $I$. Let $\\map R j$ be a propositional function of $j \\in I$. Then we can define the '''supremum of $\\family {a_j}_{j \\mathop \\in I}$''' as: :$\\displaystyle \\sup_{\\map R j} a_j := \\text{ the supremum of all $a_j$ such that $\\map R j$ holds}$ If more than one propositional function is written under the supremum sign, they must ''all'' hold."}
+{"_id": "29478", "title": "Definition:Supremum of Set/Real Numbers/Propositional Function/Finite Range", "text": "Let the fiber of truth of $\\map R j$ be finite. Then the '''supremum of $\\family {a_j}_{j \\mathop \\in I}$''' can be expressed as: :$\\displaystyle \\max_{\\map R j} a_j = \\text{ the maxmum of all $a_j$ such that $\\map R j$ holds}$ and can be referred to as the '''maximum''' of $\\family {a_j}_{j \\mathop \\in I}$."}
+{"_id": "29479", "title": "Definition:Supremum of Set/Real Numbers/Propositional Function/Vacuous Supremum", "text": "Take the indexed supremum: :$\\displaystyle \\sup _{\\Phi \\left({j}\\right)} a_j$ where $\\Phi \\left({j}\\right)$ is a propositional function of $j$. Suppose that there are no values of $j$ for which $\\Phi \\left({j}\\right)$ is true. Then $\\displaystyle \\sup_{\\Phi \\left({j}\\right)} a_j$ is defined as being $-\\infty$. This supremum is called a '''vacuous supremum'''. This is because: :$\\forall a \\in \\R: \\sup \\left\\{ {a, -\\infty}\\right\\} = a$ Hence for all $j$ for which $\\Phi \\left({j}\\right)$ is false, the supremum is unaffected. In this context $-\\infty$ is considered as '''minus infinity''', the hypothetical quantity that has the property: : $\\forall n \\in \\Z: -\\infty< n$"}
+{"_id": "29481", "title": "Definition:Rounding/Integer", "text": "Let $y \\in \\R$ such that: :$y = \\floor {x + \\dfrac 1 2}$ Then $y$ is defined as '''$x$ rounded to the nearest integer'''."}
+{"_id": "29482", "title": "Definition:Rounding/Treatment of Half", "text": "Consider the situation when $\\dfrac x {10^n} + \\dfrac 1 2$ is an integer. That is, $\\dfrac x {10^n}$ is exactly midway between the two integers $\\dfrac x {10^n} - \\dfrac 1 2$ and $\\dfrac x {10^n} + \\dfrac 1 2$. Recall that the general philosophy of the process of rounding is to find the closest approximation to $x$ to a given power of $10$. Thus there are two equally valid such approximations: :$\\dfrac x {10^n} - \\dfrac 1 2$ and $\\dfrac x {10^n} + \\dfrac 1 2$ between which $\\dfrac x {10^n}$ is exactly midway. There are a number of conventions which determine which is to be used."}
+{"_id": "29483", "title": "Definition:General Dirichlet Series", "text": "Let $a_n$ be a sequence in $\\C$. Let $\\sequence {\\lambda_n}$ be a strictly increasing sequence of non-negative real numbers whose limit is infinity. A '''general Dirichlet series of type $\\lambda_n$''' is a complex function $f: \\C \\to \\C$ defined by the series: :$\\displaystyle \\map f s = \\sum_{n \\mathop = 1}^\\infty a_n e^{-\\lambda_n s}$"}
+{"_id": "29485", "title": "Definition:Sufficiently Small", "text": "Let $P$ be a property of real numbers. Then: : '''$P \\left({x}\\right)$ holds for all sufficiently small $x$''' {{iff}}: :$\\exists \\epsilon \\in \\R: \\forall x \\in \\R: \\left\\lvert{x}\\right\\rvert \\le \\epsilon: P \\left({x}\\right)$ That is, {{iff}}: :''There exists a real number $\\epsilon$ such that for every (real) number not more than $\\epsilon$ in in absolute value, the property $P$ holds.'' It is not necessarily the case, for a given property $P$ about which such a statement is made, that the value of $\\epsilon$ actually needs to be known, merely that such a value can be demonstrated to exist."}
+{"_id": "29487", "title": "Definition:Extended Real Sequence", "text": "An '''extended real sequence''' is a sequence (usually infinite) whose codomain is the extended real number line."}
+{"_id": "29488", "title": "Definition:Limit Superior of Extended Real Sequence", "text": "Let $\\left \\langle {x_n} \\right \\rangle$ be an extended real sequence. The '''limit superior''' of $\\left \\langle {x_n} \\right \\rangle$ is defined as: :$\\displaystyle \\limsup x_n : = \\inf_{k \\mathop \\ge 1} \\left({\\sup_{n \\mathop \\ge k} x_n}\\right)$"}
+{"_id": "29489", "title": "Definition:Limit Inferior of Extended Real Sequence", "text": "Let $\\left \\langle {x_n} \\right \\rangle$ be an extended real sequence. The '''limit inferior''' of $\\left \\langle {x_n} \\right \\rangle$ is defined as: :$\\displaystyle \\liminf x_n : = \\sup_{k \\mathop \\ge 1} \\left({\\inf_{n \\mathop \\ge k} x_n}\\right)$"}
+{"_id": "29490", "title": "Definition:N-Cube (Euclidean Space)", "text": "Let $R, c \\in \\R$ be real numbers with $R > 0$. Let $\\struct {\\R^n, d}$ be a Euclidean $n$-Space equipped with the usual metric $d$. An '''$n$-cube''' is a subset of $\\struct {\\R^n, d}$ defined as the cartesian product of closed real intervals of the form: :$\\displaystyle \\prod_{i \\mathop = 1}^n \\closedint {c - R} {c + R}_i$ where $\\closedint {c - R} {c + R}_i$ is an interval in the $i$th factor of $\\R^n$. The $n$-cube can be concisely expressed as: :$\\closedint {c - R} {c + R}^n$ in contexts where the indices of the product are unimportant."}
+{"_id": "29491", "title": "Definition:Diameter of Subset of Metric Space", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $S \\subseteq A$ be subset of $A$. Then the '''diameter''' of $S$ is the extended real number defined by: : $\\operatorname {diam} \\left({S}\\right) := \\begin{cases} \\sup \\left\\{ {d \\left({x, y}\\right): x, y \\in S }\\right\\}, \\ \\text{if this quantity is finite} \\\\ +\\infty, \\ \\text{otherwise} \\end{cases}$ In the finite case, this is, by the definition of the supremum, the smallest real number $D$ such that any two points of $S$ are at most a distance $D$ apart."}
+{"_id": "29492", "title": "Definition:Arbitrarily Small", "text": "Let $P$ be a property of real numbers. We say that '''$P \\left({x}\\right)$ holds for arbitrarily small $\\epsilon$''' (or '''there exist arbitrarily small $x$ such that $P \\left({x}\\right)$ holds''') {{iff}}: :$\\forall \\epsilon \\in \\R_{> 0}: \\exists x \\in \\R: \\left\\lvert{x}\\right\\rvert \\le \\epsilon: P \\left({x}\\right)$ That is: :''For any real number $a$, there exists a (real) number not more than $a$ such that the property $P$ holds.'' or, more informally and intuitively: :''However small a number you can think of, there will be an even smaller one for which $P$ still holds.''"}
+{"_id": "29494", "title": "Definition:Half-Plane of Convergence", "text": "Let $f \\left({s}\\right)$ be a (general) Dirichlet series. The '''half-plane of convergence of $f$''' is the set: :$\\left\\{ {s: s \\in \\C, \\operatorname{Re} \\left({s}\\right) > \\sigma_0}\\right\\}$ where $\\sigma_0$ is abscissa of convergence of $f$."}
+{"_id": "29496", "title": "Definition:Stirling Numbers of the First Kind/Unsigned/Complex Numbers", "text": "{{AuthorRef|Donald E. Knuth}}, in his {{BookLink|The Art of Computer Programming: Volume 1: Fundamental Algorithms|Donald E. Knuth|ed = 3rd|edpage = Third Edition}} of $1997$, suggests an extension of the '''unsigned Stirling numbers of the first kind''' $\\displaystyle {r \\brack r - m}$ to the real and complex numbers. However, beyond stating that such a number is a polynomial in $r$ of degree $2 m$, and providing a few examples, he goes no further than that, and the details of this extension are unclear."}
+{"_id": "29497", "title": "Definition:Stirling Numbers of the Second Kind/Complex Numbers", "text": "{{AuthorRef|Donald E. Knuth}}, in his {{BookLink|The Art of Computer Programming: Volume 1: Fundamental Algorithms|Donald E. Knuth|ed = 3rd|edpage = Third Edition}} of $1997$, suggests an extension of the '''Stirling numbers of the second kind''' $\\displaystyle {r \\brace r - m}$ to the real and complex numbers. However, beyond stating that such a number is a polynomial in $r$ of degree $2 m$, and providing a few examples, he goes no further than that, and the details of this extension are unclear."}
+{"_id": "29498", "title": "Definition:Minimal Uncountable Well-Ordered Set", "text": "Let $\\Omega$ be an uncountable well-ordered set. Then $\\Omega$ is the '''minimal uncountable well-ordered set''' if every initial segment in $\\Omega$ is countable."}
+{"_id": "29500", "title": "Definition:Combinatorial Number System", "text": "The '''combinatorial number system''' is a system for representing a positive integer $m$ by a sequence of digits which are the upper coefficient of a sequence of $n$ binomial coefficients for some $n \\in \\Z_{>0}$: :$m := k_1 k_2 k_3 \\ldots k_n$ where: :$m = \\dbinom {k_1} 1 + \\dbinom {k_2} 2 + \\dbinom {k_3} 3 + \\cdots + \\dbinom {k_n} n$ :$0 \\le k_1 < k_2 < \\cdots < k_n$"}
+{"_id": "29502", "title": "Definition:Combination with Repetition", "text": "Let $S$ be a (finite) set with $n$ elements. A '''$k$ combination of $S$ with repetition''' is a multiset with $k$ elements selected from $S$."}
+{"_id": "29504", "title": "Definition:Bernoulli Polynomial", "text": "The '''Bernoulli polynomials''' $B_n \\left({x}\\right)$ are the terms of the sequence of polynomials defined as:  :$\\displaystyle B_n \\left({x}\\right) = \\sum_{k \\mathop = 0}^n \\binom n k B_{n - k} x^k$ where $B_k$ are the Bernoulli numbers."}
+{"_id": "29511", "title": "Definition:Midline of Triangle", "text": "Let $\\triangle ABC$ be a triangle. Let $D$ and $E$ be the midpoints of any two of the sides of $\\triangle ABC$. The line $DE$ is a '''midline of $\\triangle ABC$'''. 400px"}
+{"_id": "29512", "title": "Definition:Second Order Fibonacci Number", "text": "The '''second order Fibonacci numbers''' are a sequence $\\left \\langle {\\mathcal F_n}\\right \\rangle$ of integers which is formally defined recursively for all $n \\in \\Z_{\\ge 0}$ as: :$\\mathcal F_n = \\begin{cases} 0 & : n = 0 \\\\ 1 & : n = 1 \\\\ \\mathcal F_{n - 1} + \\mathcal F_{n - 2} + F_{n - 2} & : \\text{otherwise} \\end{cases}$ where $F_{n - 2}$ denotes the $n - 2$th Fibonacci number."}
+{"_id": "29513", "title": "Definition:Fibonomial Coefficient", "text": "Let $n \\in \\Z_{\\ge 0}$ and $k \\in \\Z$. Then the '''Fibonomial coefficient''' $\\dbinom n k$ is defined as: :$\\dbinom n k_\\mathcal F = \\begin{cases} 0 & : n < 0, n > k \\\\ 1 & : n \\ge 0, k = 0 \\\\ \\dfrac {F_n F_{n - 1} \\cdots F_{n - k + 1} } {F_k F_{k - 1} \\cdots F_1} = \\displaystyle \\prod_{j \\mathop = 1}^k \\dfrac {F_{n - k + j} } {F_j} & : \\text{otherwise} \\end{cases}$ where $F_n$ denotes the $n$th Fibonacci number."}
+{"_id": "29516", "title": "Definition:Tower in Set", "text": "Let $X$ be a set. Let $T$ be any non-empty subset of $X$ Let $c$ be a fixed choice function on the non-empty subsets $T$ of $X$. Let $\\preccurlyeq$ be a well-ordering on $T$. The well-ordered set $\\left({T,\\preccurlyeq}\\right)$ is a '''tower in $X$''' {{iff}}, for all $t \\in T$: :$t = c \\left({X \\setminus S_t \\left({T}\\right)}\\right)$ where $S_t \\left({T}\\right)$ is the initial segment of $T$ determined by $t$."}
+{"_id": "29517", "title": "Definition:Zeckendorf Representation", "text": "'''Zeckendorf representation''' is a system for representing a positive integer $m$ by a sequence of digits which are the indices of a sequence of $r$ Fibonacci numbers: :$n := k_1 k_2 k_3 \\ldots k_r$ where: :$n = F_{k_1} + F_{k_2} + F_{k_3} + \\cdots + F_{k_r}$ :$k_1 \\gg k_2 \\gg k_3 \\gg \\cdots \\gg k_r \\gg 0$ where $n \\gg k$ denotes that $n \\ge k + 2$."}
+{"_id": "29518", "title": "Definition:Golden Mean Number System", "text": "The '''golden mean number system''' is a system for representing a non-negative real number $x$ by a sequence of zeroes and  ones using the golden mean $\\phi$ as a number base."}
+{"_id": "29520", "title": "Definition:Proper Subtower in Set", "text": "Let $X$ be a set. Let $\\left({T, \\preccurlyeq}\\right)$ be a tower in $X$. Then $\\left({T, \\preccurlyeq}\\right)$ is a '''proper subtower in $X$''' {{iff}}: : $T$ is a proper subset of some set $T' \\subseteq X$ and: : $\\left({T', \\preccurlyeq}\\right)$ is a tower in $X$."}
+{"_id": "29522", "title": "Definition:Golden Mean Number System/Simplification", "text": "Let $x \\in \\R_{\\ge 0}$ have a representation which includes the string $011$, say: :$x = p011q$ where $p$ and $q$ are strings in $\\left\\{ {0, 1}\\right\\}$. From 100 in Golden Mean Number System is Equivalent to 011, $x$ can also be written as: :$x = p100q$ The expression $p100q$ is a '''simplification''' of $p011q$."}
+{"_id": "29524", "title": "Definition:String/Infinite", "text": "A '''string''' $S$ in $\\mathcal A$ is an '''infinite string''' {{iff}} the sequence of symbols of which it is composed is infinite."}
+{"_id": "29525", "title": "Definition:String/Finite", "text": "A '''string''' $S$ in $\\mathcal A$ is a '''finite string''' {{iff}} the sequence of symbols of which it is composed is finite."}
+{"_id": "29528", "title": "Definition:Fibonacci String", "text": "Consider the alphabet $\\left\\{ {\\text {a}, \\text {b} }\\right\\}$. For all $n \\in \\Z_{>0}$, let $S_n$ be the (finite) string formed as: :$S_n = \\begin{cases} \\text {a} & : n = 1 \\\\ \\text {b} & : n = 2 \\\\ S_{n - 1} S_{n - 2} & : n > 2 \\end{cases}$ where $S_{n - 1} S_{n - 2}$ denotes that $S_{n - 1}$ and $S_{n - 2}$ are concatenated. The terms of the sequence $\\left\\langle{S_n}\\right\\rangle$ are '''Fibonacci strings'''."}
+{"_id": "29531", "title": "Definition:Fibonacci Nim", "text": "'''Fibonacci nim''' is a two-person game whose rules are as follows: :$(1): \\quad$ The game starts with one pile of $n$ counters. :$(2): \\quad$ The first player removes a number of counters $c_1$ such that $1 \\le c_1 < n$. :$(3): \\quad$ Each player takes it in turns to remove $c_n$ counters such that $1 \\le c_n \\le 2 c_{n - 1}$ where $c_{n - 1}$ is the number of counters taken in the other player's prevous move. :$(3): \\quad$ The person who removes the last counter (or counters) is the winner."}
+{"_id": "29534", "title": "Definition:Generating Function/Parameter/Also denoted as", "text": "The symbol used for the '''parameter''' varies. $x$ is often used instead. In the field of probability theory $s$ tends to be the symbol of choice. Some authors use $\\zeta$."}
+{"_id": "29538", "title": "Definition:Linearly Recurrent Sequence", "text": "A '''linearly recurrent sequence''' is a sequence which can be defined by a recurrence relation of the form: :$a_n = c_1 a_{n - 1} + c_2 a_{n - 2} + \\cdots + c_m a_{n - m}$ with appropriate initial values for $a_1, a_2, \\ldots, a_{n - 1}$."}
+{"_id": "29539", "title": "Definition:Generating Function/Extraction of Coefficient", "text": "Let $\\map G z$ be the generating function for the sequence $S = \\sequence {a_n}$. The '''coefficient of $z^n$ extracted from $\\map G z$''' is the $n$th term of $S$, and can be denoted: :$\\sqbrk {z^n} \\map G z := a_n$"}
+{"_id": "29540", "title": "Definition:Elementary Symmetric Function", "text": "Let $a, b \\in \\Z$ be integers such that $b \\ge a$. Let $U$ be a set of $n = b - a + 1$ numbers $\\set {x_a, x_{a + 1}, \\ldots, x_b}$. Let $m \\in \\Z_{>0}$ be a (strictly) positive integer. An '''elementary symmetric function of degree $m$''' is a polynomial which can be defined by the formula: {{begin-eqn}} {{eqn | l = \\map {e_m} U       | r = \\sum_{a \\mathop \\le j_1 \\mathop < j_2 \\mathop < \\mathop \\cdots \\mathop < j_m \\mathop \\le b} \\paren {\\prod_{i \\mathop = 1}^m x_{j_i} }       | c =  }} {{eqn | r = \\sum_{a \\mathop \\le j_1 \\mathop < j_2 \\mathop < \\mathop \\cdots \\mathop < j_m \\mathop \\le b} x_{j_1} x_{j_2} \\cdots x_{j_m}       | c =  }} {{end-eqn}} That is, it is the sum of all products of $m$ distinct elements of $\\set {x_a, x_{a + 1}, \\dotsc, x_b}$."}
+{"_id": "29541", "title": "Definition:Power Sum", "text": "Let $S$ be a finite set of numbers. Let $p \\in \\R$ be a real number. Then the '''$p$th power sum''' of $S$ is defined as: :$S_p = \\displaystyle \\sum_{x \\mathop \\in S} x^p$ That is, it is the summation of the $p$th powers of all the elements of $S$."}
+{"_id": "29543", "title": "Definition:Sequence/Doubly Subscripted", "text": "A '''doubly subscripted sequence''' is a mapping whose domain is a subset of the cartesian product $\\N \\times \\N$ of the set of natural numbers $\\N$ with itself. It can be seen that a '''doubly subscripted sequence''' is an instance of a family of elements indexed by $\\N^2$. A '''doubly subscripted sequence''' can be denoted $\\left\\langle{a_{m n} }\\right\\rangle_{m, \\, n \\mathop \\ge 0}$"}
+{"_id": "29544", "title": "Definition:Generating Function/Doubly Subscripted Sequence", "text": "Let $A = \\left \\langle {a_{m, n} }\\right \\rangle$ be a doubly subscripted sequence in $\\R$ for $m, n \\in \\Z_{\\ge 0}$. Then $\\displaystyle G_A \\left({w, z}\\right) = \\sum_{m, \\, n \\mathop \\ge 0} a_{m n} w^m z^n$ is called the '''generating function''' for the sequence $A$."}
+{"_id": "29548", "title": "Definition:Complex Arithmetic", "text": "'''Complex arithmetic''' is the branch of arithmetic which concerns the manipulation of complex numbers"}
+{"_id": "29549", "title": "Definition:Conjugate Pair", "text": "Let $z \\in \\C$ be a complex number. Let $\\overline z$ be the complex conjugate of $z$. Then $z$ and $\\overline z$ are a '''conjugate pair'''. Category:Definitions/Complex Conjugates 6kg0eyhxfetdrg3qh1bqlt7grgic968"}
+{"_id": "29550", "title": "Definition:Complex Algebra", "text": "'''Complex algebra''' is the branch of algebra which specifically involves complex arithmetic. Thus it is the branch of mathematics which studies the techniques of manipulation of expressions in the domain of complex numbers."}
+{"_id": "29551", "title": "Definition:Half-Plane", "text": "Let $P$ denote the plane. Let $L$ denote an infinite straight line in $P$. Then each of the areas of $P$ on either side of $L$ is a '''half-plane'''."}
+{"_id": "29554", "title": "Definition:Circle of Apollonius", "text": "Let $A, B$ be distinct points in the plane. Let $\\lambda \\in \\R_{>0}$ be a strictly positive real number. Let $X$ be the locus of points in the plane such that: :$X A = \\map \\lambda {X B}$ :400px Then $X$ is a '''circle of Apollonius'''."}
+{"_id": "29555", "title": "Definition:Root of Unity/Complex/Order", "text": "Let $z \\in U_n$. The '''order''' of $z$ is the smallest $p \\in \\Z_{> 0}$ such that: :$z^p = 1$"}
+{"_id": "29556", "title": "Definition:Cauchy Sequence/Normed Vector Space", "text": "Let $\\struct {V, \\norm {\\,\\cdot\\,} }$ be a normed vector space. Let $\\sequence {x_n}$ be a sequence in $V$. Then $\\sequence {x_n}$ is a '''Cauchy sequence''' {{iff}}: : $\\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\N: \\forall m, n \\in \\N: m, n \\ge N: \\norm {x_n - x_m} < \\epsilon$"}
+{"_id": "29557", "title": "Definition:Cauchy Sequence/Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} } $ be a normed division ring Let $\\sequence {x_n}$ be a sequence in $R$. Then $\\sequence {x_n} $ is a '''Cauchy sequence in the norm $\\norm {\\, \\cdot \\,}$''' {{iff}}: :$\\sequence {x_n}$ is a cauchy sequence in the metric induced by the norm $\\norm {\\, \\cdot \\,}$ That is: : $\\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\N: \\forall m, n \\in \\N: m, n \\ge N: \\norm {x_n - x_m} < \\epsilon$"}
+{"_id": "29558", "title": "Definition:Sextic Equation", "text": "Let $f \\paren x = a x^6 + b x^5 + c x^4 + d x^3 + e x^2 + f x + g$ be a polynomial function over a field $\\mathbb k$ of degree $6$. Then the equation $f \\paren x = 0$ is the general '''sextic equation''' over $\\mathbb k$."}
+{"_id": "29560", "title": "Definition:Convergent Sequence/Complex Numbers", "text": "=== Definition 1 === {{:Definition:Convergent Sequence/Complex Numbers/Definition 1}} === Definition 2 === {{:Definition:Convergent Sequence/Complex Numbers/Definition 2}}"}
+{"_id": "29561", "title": "Definition:Convergent Sequence/Complex Numbers/Definition 2", "text": "Let $\\sequence {z_k} = \\sequence {x_k + i y_k}$ be a sequence in $\\C$. $\\sequence {z_k}$ '''converges to the limit $c = a + i b$''' {{iff}} both: :$\\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\R: n > N \\implies \\size {x_n - a} < \\epsilon \\text { and } \\size {y_n - b} < \\epsilon$ where $\\size {x_n - a}$ denotes the absolute value of $x_n - a$."}
+{"_id": "29562", "title": "Definition:Series/Complex", "text": "A '''complex series''' $S_n$ is the limit to infinity of the sequence of partial sums of a complex sequence $\\sequence {a_n}$: {{begin-eqn}} {{eqn | l = S_n       | r = \\lim_{N \\mathop \\to \\infty} \\sum_{n \\mathop = 1}^N a_n       | c =  }} {{eqn | r = \\sum_{n \\mathop = 1}^\\infty a_n       | c =  }} {{eqn | r = a_1 + a_2 + a_3 + \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "29563", "title": "Definition:Bounded Mapping/Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,}}$ be a normed division ring. Let $f: S \\to R$ be a mapping from $S$ into $R$. Then $f$ is '''bounded''' {{iff}} the real-valued function $\\norm {\\,\\cdot\\,} \\circ f: S \\to \\R$ is bounded, where $\\norm {\\,\\cdot\\,} \\circ f$ is the composite of $\\norm {\\,\\cdot\\,}$ and $f$. That is, $f$ is '''bounded''' if there is a constant $K \\in \\R_{\\ge 0}$ such that $\\norm{f \\paren {s}} \\le K$ for all $s \\in S$."}
+{"_id": "29564", "title": "Definition:Bounded Sequence/Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\sequence {x_n}$ be a sequence in $R$. Then $\\sequence {x_n}$ is '''bounded''' {{iff}}: :$\\exists K \\in \\R$ such that $\\forall n \\in \\N: \\norm {x_n} \\le K$"}
+{"_id": "29565", "title": "Definition:Bounded Sequence/Normed Division Ring/Unbounded", "text": "$\\sequence {x_n}$ is '''unbounded''' {{iff}} it is not bounded."}
+{"_id": "29566", "title": "Definition:Bounded Sequence/Metric Space", "text": "Let $M$ be a metric space. Let $\\sequence {x_n}$ be a sequence in $M$. Then $\\sequence {x_n}$ is a '''bounded sequence''' {{iff}} $\\sequence {x_n}$ is bounded in $M$. That is: :$\\exists K \\in \\R: \\forall n, m \\in \\N: \\map d {x_n, x_m} \\le K$"}
+{"_id": "29567", "title": "Definition:Bounded Sequence/Metric Space/Unbounded", "text": "$\\sequence {x_n}$ is '''unbounded''' {{iff}} it is not bounded."}
+{"_id": "29568", "title": "Definition:Metric Induced by Norm on Division Ring", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Then the '''induced metric''' or the '''metric induced by $\\norm {\\,\\cdot\\,}$''' is the map $d: R \\times R \\to \\R_{\\ge 0}$ defined as: :$d \\paren {x, y} = \\norm {x - y}$"}
+{"_id": "29569", "title": "Definition:Series/Real", "text": "A '''real series''' $S_n$ is the limit to infinity of the sequence of partial sums of a real sequence $\\sequence {a_n}$: {{begin-eqn}} {{eqn | l = S_n       | r = \\lim_{N \\mathop \\to \\infty} \\sum_{n \\mathop = 1}^N a_n       | c =  }} {{eqn | r = \\sum_{n \\mathop = 1}^\\infty a_n       | c =  }} {{eqn | r = a_1 + a_2 + a_3 + \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "29573", "title": "Definition:Non-Archimedean/Norm (Division Ring)", "text": "=== Definition 1 === {{:Definition:Non-Archimedean/Norm (Division Ring)/Definition 1}}"}
+{"_id": "29574", "title": "Definition:Non-Archimedean/Norm (Division Ring)/Archimedean", "text": "A norm $\\norm {\\, \\cdot \\,}$ on a division ring $R$ is '''Archimedean''' {{iff}} it is not non-Archimedean."}
+{"_id": "29584", "title": "Definition:Bell Number/Sequence", "text": "The sequence of Bell numbers begins: :$(1), 1, 2, 5, 15, 52, 203, 877, 4140, 21 \\, 147, \\ldots$ where the first $(1)$ is the zeroth '''Bell number''' $B_0$."}
+{"_id": "29588", "title": "Definition:Divisor (Algebra)/Also known as", "text": "A '''divisor''' can also be referred to as a '''factor'''."}
+{"_id": "29589", "title": "Definition:Greatest Common Divisor/Also known as", "text": "The '''greatest common divisor''' is often seen abbreviated as '''GCD''', '''gcd''' or '''g.c.d.''' Some sources write $\\gcd \\set {a, b}$ as $\\tuple {a, b}$, but this notation can cause confusion with ordered pairs. The notation $\\map \\gcd {a, b}$ is frequently seen, but the set notation, although a little more cumbersome, can be argued to be preferable. The '''greatest common divisor''' is also known as the '''highest common factor''', or '''greatest common factor'''. '''Highest common factor''' when it occurs, is usually abbreviated as '''HCF''', '''hcf''' or '''h.c.f.''' It is written $\\hcf \\set {a, b}$ or $\\map \\hcf {a, b}$. The archaic term '''greatest common measure''' can also be found, mainly in such as {{ElementsLink}}."}
+{"_id": "29591", "title": "Definition:Ring of Sequences", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Given the natural numbers $\\N$, the '''ring of sequences over $R$''' is the ring of mappings $\\struct {R^\\N, +', \\circ'}$ where: :$(1): \\quad R^\\N$ is the set of sequences in $R$ :$(2): \\quad +'$ and $\\circ'$ are the (pointwise) operations induced by $+$ and $\\circ$."}
+{"_id": "29592", "title": "Definition:Ring of Cauchy Sequences", "text": "Let $\\struct {R, +, \\circ, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\struct {R^\\N, +', \\circ'}$ be the ring of sequences over $R$. Let $\\CC$ be the set of Cauchy sequences on $R$. The '''ring of Cauchy sequences over $R$''' is the subring $\\struct {\\CC, +', \\circ'}$ of $R^\\N$ with unity. The (pointwise) ring operations on the '''ring of Cauchy sequences over $R$''' are defined as: :$\\forall \\sequence {x_n}, \\sequence {y_n} \\in R^\\N: \\sequence {x_n} +' \\sequence {y_n} = \\sequence {x_n + y_n}$ :$\\forall \\sequence {x_n}, \\sequence {y_n} \\in R^\\N: \\sequence {x_n} \\circ' \\sequence {y_n} = \\sequence {x_n \\circ y_n}$ The zero of the '''ring of Cauchy sequences''' is the sequence $\\tuple {0, 0, 0, \\dots}$, where $0$ is the zero in $R$. The unity of the '''ring of Cauchy sequences''' is the sequence $\\tuple {1, 1, 1, \\dots}$, where $1$ is the unity of $R$. By Corollary to Cauchy Sequences form Ring with Unity, if $R$ is a valued field then the '''ring of Cauchy sequences over $R$''' is a commutative ring with unity."}
+{"_id": "29596", "title": "Definition:Image/Also denoted as", "text": "The notation $\\Img f$ to denote the image of a mapping $f$ is specific to {{ProofWiki}}. The usual notation is $\\map {\\mathrm {Im} } f$ or a variant, but this is too easily confused with $\\map \\Im z$, the imaginary part of a complex number. Hence the non-standard usage $\\Img f$. Some sources use $f \\sqbrk S$, where $S$ is the domain of $f$. Others just use $\\map f S$, but that notation is deprecated on {{ProofWiki}} so as not to confuse it with the notation for the image of an element."}
+{"_id": "29597", "title": "Definition:Inverse Image Mapping/Relation/Definition 1", "text": "The '''inverse image mapping''' of $\\mathcal R$ is the mapping $\\mathcal R^\\gets: \\powerset T \\to \\powerset S$ that sends a subset $Y \\subseteq T$ to its preimage $\\map {\\mathcal R^{-1} } Y$ under $\\mathcal R$: :$\\forall Y \\in \\powerset T: \\map {\\mathcal R^\\gets} Y = \\begin {cases} \\set {s \\in S: \\exists t \\in Y: \\tuple {t, s} \\in \\mathcal R^{-1} } & : \\Img {\\mathcal R} \\cap Y \\ne \\O \\\\ \\O & : \\Img {\\mathcal R} \\cap Y = \\O \\end {cases}$"}
+{"_id": "29598", "title": "Definition:Inverse Image Mapping/Relation/Definition 2", "text": "The '''inverse image mapping''' of $\\mathcal R$ is the direct image mapping of the inverse $\\mathcal R^{-1}$ of $\\mathcal R$: :$\\mathcal R^\\gets = \\paren {\\mathcal R^{-1} }^\\to: \\powerset T \\to \\powerset S$ That is: :$\\forall Y \\in \\powerset T: \\map {\\mathcal R^\\gets} Y = \\set {s \\in S: \\exists t \\in Y: \\tuple {t, s} \\in \\mathcal R^{-1} }$"}
+{"_id": "29601", "title": "Definition:Inverse Image Mapping/Mapping/Definition 1", "text": "The '''inverse image mapping''' of $f$ is the mapping $f^\\gets: \\powerset T \\to \\powerset S$ that sends a subset $Y \\subseteq T$ to its preimage $f^{-1} \\paren T$ under $f$: :$\\forall Y \\in \\powerset T: \\map {f^\\gets} Y = \\begin {cases} \\set {s \\in S: \\exists t \\in Y: \\map f s = t} & : \\Img f \\cap Y \\ne \\O \\\\ \\O & : \\Img f \\cap Y = \\O \\end {cases}$"}
+{"_id": "29602", "title": "Definition:Inverse Image Mapping/Mapping/Definition 2", "text": "The '''inverse image mapping''' of $f$ is the direct image mapping of the inverse $f^{-1}$ of $f$: :$f^\\gets = \\paren {f^{-1} }^\\to: \\powerset T \\to \\powerset S$: That is: :$\\forall Y \\in \\powerset T: \\map {f^\\gets} Y = \\set {s \\in S: \\exists t \\in Y: \\map f s = t}$"}
+{"_id": "29603", "title": "Definition:Mapping/Also known as", "text": "Words which are often used to mean the same thing as '''mapping''' include: :'''transformation''' (particularly in the context of self-maps) :'''operator''' or '''operation''' :'''function''' (usually in the context of numbers) :'''map''' (but this term is discouraged, as the term is also used by some writers for '''planar graph'''). Some sources introduce the concept with informal words such as '''rule''' or '''idea''' or '''mathematical notion'''. Sources which define a '''mapping (function)''' to be only a many-to-one relation refer to a '''mapping (function)''' as a '''total mapping (total function)'''. Some use the term '''single-valued relation'''. Sources which go into analysis of multifunctions often refer to a conventional '''mapping''' as: :a '''single-valued mapping''' or '''single-valued function''' :a '''many-to-one mapping''', '''many-to-one transformation''', or '''many-to-one correspondence''', and so on. The wording can vary, for example: '''many-one''' can be seen for '''many-to-one'''. A '''mapping $f$ from $S$ to $T$''' is also described as a '''mapping on $S$ into $T$'''."}
+{"_id": "29604", "title": "Definition:Image (Set Theory)/Mapping/Element/Also known as", "text": "The image of an element $s$ under a mapping $f$ is also called the '''functional value''', or '''value''', of $f$ at $s$. The terminology: :'''$f$ maps $s$ to $\\map f s$''' :'''$f$ assigns the value $\\map f s$ to $s$''' :'''$f$ carries $s$ into $\\map f s$ can be found. The modifier '''by $f$''' can also be used for '''under $f$'''. Thus, for example, the '''image of $s$ by $f$''' means the same as the '''image of $s$ under $f$'''. In the context of computability theory, the following terms are frequently found: If $\\tuple {x, y} \\in f$, then $y$ is often called the '''output''' of $f$ for '''input''' $x$, or simply, the '''output of $f$ at $x$'''."}
+{"_id": "29610", "title": "Definition:Surjection/Also known as", "text": "The phrase '''$f$ is surjective''' is often used for '''$f$ is a surjection'''. Authors who prefer to limit the jargon of mathematics tend to use the term '''an onto mapping''' for '''a surjection''', and '''onto''' for '''surjective'''. A mapping which is '''not surjective''' is thence described as '''into'''. A surjection $f$ from $S$ to $T$ is sometimes denoted: :$f: S \\twoheadrightarrow T$ to emphasize surjectivity. {{LatexFor|for = f: S \\twoheadrightarrow T}}"}
+{"_id": "29611", "title": "Definition:Constructed Semantics/Instance 2", "text": "Let $\\mathcal L_0$ be the language of propositional logic. The constructed semantics $\\mathscr C_2$ for $\\mathcal L_0$ is used for the following results: * Hilbert Proof System Instance 2 Independence Results: Independence of $(A1)$ === Structures === Define the structures of $\\mathscr C_2$ as mappings $v$ by the Principle of Recursive Definition, as follows. Let $\\mathcal P_0$ be the vocabulary of $\\mathcal L_0$. Let a mapping $v: \\mathcal P_0 \\to \\{ 0, 1, 2 \\}$ be given. Next, regard the following as definitional abbreviations: {{begin-axiom}} {{axiom | n  = 1         | lc = Conjunction         | ml = \\mathbf A \\land \\mathbf B         | mo = {{=}}_{\\text{def} }         | mr = \\neg \\left({ \\neg \\mathbf A \\lor \\neg \\mathbf B }\\right) }} {{axiom | n  = 2         | lc = Conditional         | ml = \\mathbf A \\implies \\mathbf B         | mo = {{=}}_{\\text{def} }         | mr = \\neg \\mathbf A \\lor \\mathbf B }} {{axiom | n  = 3         | lc = Biconditional         | ml = \\mathbf A \\iff \\mathbf B         | mo = {{=}}_{\\text{def} }         | mr = \\left({\\mathbf A \\implies \\mathbf B}\\right) \\land \\left({\\mathbf B \\implies \\mathbf A}\\right) }} {{end-axiom}} It only remains to define $v \\left({ \\neg \\phi }\\right)$ and $v \\left({ \\phi \\lor \\psi}\\right)$ recursively, by: {{begin-eqn}} {{eqn | l = v \\left({\\neg \\phi }\\right)       | o = :=       | r = \\begin{cases} 1 & : \\text{if $v \\left({\\phi}\\right) = 0$} \\\\ 0 & : \\text{if $v \\left({\\phi}\\right) = 1$} \\\\ 2 & : \\text{if $v \\left({\\phi}\\right) = 2$}\\end{cases} }} {{eqn | l = v \\left({ \\phi \\lor \\psi }\\right)       | o = :=       | r = \\begin{array}{c{{!}}ccc} \\phi \\lor \\psi & 0 & 1 & 2 \\\\ \\hline 0 & 0 & 0 & 0 \\\\ 1 & 0 & 1 & 2 \\\\ 2 & 0 & 2 & 0 \\\\ \\end{array} }} {{end-eqn}} === Validity === Define validity in $\\mathscr C_2$ by declaring: :$\\models_{\\mathscr C_2} \\phi$ {{iff}} $v \\left({\\phi}\\right) = 0$"}
+{"_id": "29612", "title": "Definition:Constructed Semantics/Instance 3", "text": "Let $\\mathcal L_0$ be the language of propositional logic. The constructed semantics $\\mathscr C_3$ for $\\mathcal L_0$ is used for the following results: * Hilbert Proof System Instance 2 Independence Results: Independence of $(A2)$ === Structures === Define the structures of $\\mathscr C_3$ as mappings $v$ by the Principle of Recursive Definition, as follows. Let $\\mathcal P_0$ be the vocabulary of $\\mathcal L_0$. Let a mapping $v: \\mathcal P_0 \\to \\{ 0, 1, 2 \\}$ be given. Next, regard the following as definitional abbreviations: {{begin-axiom}} {{axiom | n  = 1         | lc = Conjunction         | ml = \\mathbf A \\land \\mathbf B         | mo = {{=}}_{\\text{def} }         | mr = \\neg \\left({ \\neg \\mathbf A \\lor \\neg \\mathbf B }\\right) }} {{axiom | n  = 2         | lc = Conditional         | ml = \\mathbf A \\implies \\mathbf B         | mo = {{=}}_{\\text{def} }         | mr = \\neg \\mathbf A \\lor \\mathbf B }} {{axiom | n  = 3         | lc = Biconditional         | ml = \\mathbf A \\iff \\mathbf B         | mo = {{=}}_{\\text{def} }         | mr = \\left({\\mathbf A \\implies \\mathbf B}\\right) \\land \\left({\\mathbf B \\implies \\mathbf A}\\right) }} {{end-axiom}} It only remains to define $v \\left({ \\neg \\phi }\\right)$ and $v \\left({ \\phi \\lor \\psi}\\right)$ recursively, by: {{begin-eqn}} {{eqn | l = v \\left({\\neg \\phi }\\right)       | o = :=       | r = \\begin{cases} 2 & : \\text{if $v \\left({\\phi}\\right) = 0$} \\\\ 1 & : \\text{if $v \\left({\\phi}\\right) = 1$} \\\\ 0 & : \\text{if $v \\left({\\phi}\\right) = 2$}\\end{cases} }} {{eqn | l = v \\left({ \\phi \\lor \\psi }\\right)       | o = :=       | r = \\begin{array}{c{{!}}ccc} \\phi \\lor \\psi & 0 & 1 & 2 \\\\ \\hline 0 & 0 & 0 & 0 \\\\ 1 & 0 & 1 & 2 \\\\ 2 & 0 & 2 & 2 \\\\ \\end{array} }} {{end-eqn}} === Validity === Define validity in $\\mathscr C_3$ by declaring: :$\\models_{\\mathscr C_3} \\phi$ {{iff}} $v \\left({\\phi}\\right) \\in \\set{ 0, 1 }$"}
+{"_id": "29613", "title": "Definition:Constructed Semantics/Instance 4", "text": "Let $\\mathcal L_0$ be the language of propositional logic. The constructed semantics $\\mathscr C_4$ for $\\mathcal L_0$ is used for the following results: * Hilbert Proof System Instance 2 Independence Results: Independence of $(A3)$ === Structures === Define the structures of $\\mathscr C_4$ as mappings $v$ by the Principle of Recursive Definition, as follows. Let $\\mathcal P_0$ be the vocabulary of $\\mathcal L_0$. Let a mapping $v: \\mathcal P_0 \\to \\{ 0, 1, 2, 3 \\}$ be given. Next, regard the following as definitional abbreviations: {{begin-axiom}} {{axiom | n  = 1         | lc = Conjunction         | ml = \\mathbf A \\land \\mathbf B         | mo = {{=}}_{\\text{def} }         | mr = \\neg \\left({ \\neg \\mathbf A \\lor \\neg \\mathbf B }\\right) }} {{axiom | n  = 2         | lc = Conditional         | ml = \\mathbf A \\implies \\mathbf B         | mo = {{=}}_{\\text{def} }         | mr = \\neg \\mathbf A \\lor \\mathbf B }} {{axiom | n  = 3         | lc = Biconditional         | ml = \\mathbf A \\iff \\mathbf B         | mo = {{=}}_{\\text{def} }         | mr = \\left({\\mathbf A \\implies \\mathbf B}\\right) \\land \\left({\\mathbf B \\implies \\mathbf A}\\right) }} {{end-axiom}} It only remains to define $v \\left({ \\neg \\phi }\\right)$ and $v \\left({ \\phi \\lor \\psi}\\right)$ recursively, by: {{begin-eqn}} {{eqn | l = v \\left({\\neg \\phi }\\right)       | o = :=       | r = \\begin{cases} 1 & : \\text{if $v \\left({\\phi}\\right) = 0$} \\\\ 0 & : \\text{if $v \\left({\\phi}\\right) = 1$} \\\\ 0 & : \\text{if $v \\left({\\phi}\\right) = 2$} \\\\ 2 & : \\text{if $v \\left({\\phi}\\right) = 3$}\\end{cases} }} {{eqn | l = v \\left({ \\phi \\lor \\psi }\\right)       | o = :=       | r = \\begin{array}{c{{!}}cccc} \\phi \\lor \\psi & 0 & 1 & 2 & 3\\\\ \\hline 0 & 0 & 0 & 0 & 0 \\\\ 1 & 0 & 1 & 2 & 3 \\\\ 2 & 0 & 2 & 2 & 0 \\\\ 3 & 0 & 3 & 3 & 3 \\end{array} }} {{end-eqn}} === Validity === Define validity in $\\mathscr C_4$ by declaring: :$\\models_{\\mathscr C_4} \\phi$ {{iff}} $v \\left({\\phi}\\right) = 0$"}
+{"_id": "29614", "title": "Definition:Cauchy Functional Equation", "text": "Let $f: \\R \\to \\R$ be a real function. Then the '''Cauchy functional equation''' is: :$\\forall x, y \\in \\R: f \\paren {x + y} = f \\paren x + f \\paren y$"}
+{"_id": "29615", "title": "Definition:Constructed Semantics/Instance 5", "text": "Let $\\mathcal L_0$ be the language of propositional logic. The constructed semantics $\\mathscr C_5$ for $\\mathcal L_0$ is used for the following results: * Hilbert Proof System Instance 2 Independence Results: Independence of $(A4)$ === Structures === Define the structures of $\\mathscr C_5$ as mappings $v$ by the Principle of Recursive Definition, as follows. Let $\\mathcal P_0$ be the vocabulary of $\\mathcal L_0$. Let a mapping $v: \\mathcal P_0 \\to \\{ 0, 1, 2, 3 \\}$ be given. Next, regard the following as definitional abbreviations: {{begin-axiom}} {{axiom | n  = 1         | lc = Conjunction         | ml = \\mathbf A \\land \\mathbf B         | mo = {{=}}_{\\text{def} }         | mr = \\neg \\left({ \\neg \\mathbf A \\lor \\neg \\mathbf B }\\right) }} {{axiom | n  = 2         | lc = Conditional         | ml = \\mathbf A \\implies \\mathbf B         | mo = {{=}}_{\\text{def} }         | mr = \\neg \\mathbf A \\lor \\mathbf B }} {{axiom | n  = 3         | lc = Biconditional         | ml = \\mathbf A \\iff \\mathbf B         | mo = {{=}}_{\\text{def} }         | mr = \\left({\\mathbf A \\implies \\mathbf B}\\right) \\land \\left({\\mathbf B \\implies \\mathbf A}\\right) }} {{end-axiom}} It only remains to define $v \\left({ \\neg \\phi }\\right)$ and $v \\left({ \\phi \\lor \\psi}\\right)$ recursively, by: {{begin-eqn}} {{eqn | l = v \\left({\\neg \\phi }\\right)       | o = :=       | r = \\begin{cases} 1 & : \\text{if $v \\left({\\phi}\\right) = 0$} \\\\ 0 & : \\text{if $v \\left({\\phi}\\right) = 1$} \\\\ 0 & : \\text{if $v \\left({\\phi}\\right) = 2$} \\\\ 2 & : \\text{if $v \\left({\\phi}\\right) = 3$}\\end{cases} }} {{eqn | l = v \\left({ \\phi \\lor \\psi }\\right)       | o = :=       | r = \\begin{array}{c{{!}}cccc} \\phi \\lor \\psi & 0 & 1 & 2 & 3\\\\ \\hline 0 & 0 & 0 & 0 & 0 \\\\ 1 & 0 & 1 & 2 & 3 \\\\ 2 & 0 & 2 & 2 & 0 \\\\ 3 & 0 & 3 & 0 & 3 \\end{array} }} {{end-eqn}} === Validity === Define validity in $\\mathscr C_5$ by declaring: :$\\models_{\\mathscr C_5} \\phi$ {{iff}} $v \\left({\\phi}\\right) = 0$"}
+{"_id": "29616", "title": "Definition:Subtraction/Real Numbers", "text": "Let $\\struct {\\R, +, \\times}$ be the field of real numbers. The operation of '''subtraction''' is defined on $\\R$ as: :$\\forall a, b \\in \\R: a - b := a + \\paren {-b}$ where $-b$ is the negative of $b$ in $\\R$."}
+{"_id": "29617", "title": "Definition:Subtraction/Rational Numbers", "text": "Let $\\struct {\\Q, +, \\times}$ be the field of rational numbers. The operation of '''subtraction''' is defined on $\\Q$ as: :$\\forall a, b \\in \\Q: a - b := a + \\paren {-b}$ where $-b$ is the negative of $b$ in $\\Q$."}
+{"_id": "29620", "title": "Definition:Self-Inverse Element/Definition 1", "text": "$x$ is a '''self-inverse element of $\\struct {S, \\circ}$''' {{iff}} $x \\circ x = e$."}
+{"_id": "29621", "title": "Definition:Self-Inverse Element/Definition 2", "text": "$x$ is a '''self-inverse element of $\\struct {S, \\circ}$''' {{iff}}: :$x$ is invertible and: :$x = x^{-1}$, where $x^{-1}$ is the inverse of $x$."}
+{"_id": "29623", "title": "Definition:Symmetric Group/n Letters", "text": "Let $S_n$ denote the set of permutations on $n$ letters. Let $\\struct {S_n, \\circ}$ denote the symmetric group on $S_n$. Then $\\struct {S_n, \\circ}$ is referred to as the '''symmetric group on $n$ letters'''."}
+{"_id": "29624", "title": "Definition:Symmetric Group/Notation", "text": "In order not to make notation for operations on a '''symmetric group''' overly cumbersome, '''product notation''' is usually used for mapping composition. Thus $\\pi \\circ \\rho$ is written $\\pi \\rho$. Also, for the same reason, rather than using $I_{S_n}$ for the identity mapping, the symbol $e$ is usually used."}
+{"_id": "29626", "title": "Definition:Symmetric Group/Also known as", "text": "In view of the isomorphism between symmetric groups on sets of the same cardinality, the terminology '''symmetric group of degree $n$''' is often used when the nature of the underlying set is immaterial. Some sources use the term '''$n$th symmetric group'''. These terms will sometimes be used on {{ProofWiki}}. Some sources refer to the '''symmetric group''' on a set as the '''full symmetric group (on $S$)'''. Others use the term '''complete symmetric group'''. Similarly, the '''symmetric group on $n$ letters''' can be found referred to as the '''full symmetric group on $n$ letters'''. The term '''(full) symmetric group on $n$ objects''' can be found for both the general '''symmetric group''' and the '''symmetric group on $n$ letters''' Some sources use the notation $S \\paren A$ to denote the set of permutations on a given set $A$, and thence $S \\paren A$ to denote the '''symmetric group''' on $A$. In line with this, the notation $S \\paren n$ is often used for $S_n$ to denote the '''symmetric group on $n$ letters'''. Others use $\\mathcal S_n$ or some such variant. The notation $\\operatorname {Sym} \\paren n$ for $S_n$ can also be found. Some older sources denote the '''symmetric group on $A$''' as $\\mathfrak S_A$. Such sources consequently denote the '''symmetric group on $n$ letters''' as $\\mathfrak S_n$. However, this ''fraktur'' font is rarely used nowadays as it is cumbersome to reproduce and awkward to read. Be careful not to refer to $\\struct {\\Gamma \\paren S, \\circ}$ for $\\card S = n$ or $S_n$ as the '''symmetric group of order $n$''', as the order of these groups is not $n$ but $n!$, from Order of Symmetric Group. === Isomorphism between Symmetric Groups === {{:Definition:Symmetric Group/Isomorphism}}"}
+{"_id": "29627", "title": "Definition:Rotation (Geometry)", "text": "A '''rotation''' in the context of Euclidean geometry is an isometry from a Euclidean Space $\\R^n$ as follows. A '''rotation''' is defined usually for either: :$n = 2$, representing the plane or: :$n = 3$, representing ordinary space. === Rotation in the Plane === {{:Definition:Rotation (Geometry)/Plane}} === Rotation in Space === {{:Definition:Rotation (Geometry)/Space}}"}
+{"_id": "29628", "title": "Definition:Rotation (Geometry)/Plane", "text": "A '''rotation''' $r_\\alpha$ in the plane is an isometry on the Euclidean Space $\\Gamma = \\R^2$ as follows. Let $O$ be a distinguished point in $\\Gamma$, which has the property that: :$\\map {r_\\alpha} O = O$ That is, $O$ maps to itself. Let $P \\in \\Gamma$ such that $P \\ne O$. Let $OP$ be joined by a straight line. Let a straight line $OP'$ be constructed such that: :$(1): \\quad OP' = OP$ :$(2): \\angle POP' = \\alpha$ such that $OP \\to OP'$ is in the anticlockwise direction: :300px Then: :$\\map {r_\\alpha} P = P'$ Thus $r_\\alpha$ is a '''rotation (in the plane) of (angle) $\\alpha$ about (the axis) $O$'''."}
+{"_id": "29629", "title": "Definition:Rotation (Geometry)/Space", "text": "A '''rotation''' $r_\\theta$ in space is an isometry on the Euclidean Space $\\Gamma = \\R^3$ as follows. Let $AB$ be a distinguished straight line in $\\Gamma$, which has the property that: :$\\forall P \\in AB: \\map {r_\\theta} P = P$ That is, all points on $AB$ map to themselves. Let $P \\in \\Gamma$ such that $P \\notin AB$. Let a straight line be constructed from $P$ to $O$ on $AB$ such that $OP$ is perpendicular to $AB$. Let a straight line $OP'$ be constructed perpendicular to $AB$ such that: :$(1): \\quad OP' = OP$ :$(2): \\quad \\angle POP' = \\theta$ such that $OP \\to OP'$ is in the anticlockwise direction: :400px Then: :$\\map {r_\\theta} P = P'$ Thus $r_\\theta$ is a '''rotation (in space) of (angle) $\\theta$ about (the axis) $O$'''."}
+{"_id": "29630", "title": "Definition:Rotation (Geometry)/Axis", "text": "Let $r_\\theta$ be a rotation in the Euclidean Space $\\Gamma = \\R^n$. The set $A$ of points in $\\Gamma$ such that: :$\\forall P \\in A: \\map {r_\\theta} P = P$ is called the '''axis of rotation''' of $r_\\theta$."}
+{"_id": "29632", "title": "Definition:Reflection (Geometry)", "text": "A '''reflection''' in the context of Euclidean geometry is an isometry from a Euclidean Space $\\R^n$ as follows. A '''reflection''' is defined usually for either: :$n = 2$, representing the plane or: :$n = 3$, representing ordinary space. === Reflection in the Plane === {{:Definition:Reflection (Geometry)/Plane}} === Reflection in Space === {{:Definition:Reflection (Geometry)/Space}} === Point Reflection in Space === {{:Definition:Reflection (Geometry)/Point}}"}
+{"_id": "29633", "title": "Definition:Reflection (Geometry)/Plane", "text": "A '''reflection''' $\\phi_{AB}$ in the plane is an isometry on the Euclidean Space $\\Gamma = \\R^2$ as follows. Let $AB$ be a distinguished straight line in $\\Gamma$, which has the property that: :$\\forall P \\in AB: \\map {\\phi_{AB} } P = P$ That is, every point on $AB$ maps to itself. Let $P \\in \\Gamma$ such that $P \\notin AB$. Let a straight line be constructed from $P$ to $O$ on $AB$ such that $OP$ is perpendicular to $AB$. Let $PO$ be produced to $P'$ such that $OP = OP'$. :400px Then: :$\\map {\\phi_{AB} } P = P'$ Thus $\\phi_{AB}$ is a '''reflection (in the plane) in (the axis of reflection) $AB$'''."}
+{"_id": "29640", "title": "Definition:Translation Mapping/Abelian Group", "text": "Let $\\struct {G, +}$ be an abelian group. Let $g \\in G$. Then '''translation by $g$''' is the mapping $\\tau_g: G \\to G$ defined by: :$\\forall h \\in G: \\map {\\tau_g} h = h + \\paren {-g}$ where $-g$ is the inverse of $g$ with respect to $+$ in $G$."}
+{"_id": "29642", "title": "Definition:Translation Mapping/Euclidean Space", "text": "A '''translation''' $\\tau_\\mathbf x$ is an isometry on the Euclidean Space $\\Gamma = \\R^n$ defined as: :$\\forall \\mathbf y \\in \\R^n: \\map {\\tau_\\mathbf x} {\\mathbf y} = \\mathbf y - \\mathbf x$ where $\\mathbf x$ is a vector in $\\R^n$."}
+{"_id": "29645", "title": "Definition:Aleph Number", "text": "The '''aleph numbers''' are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. The cardinality of the natural numbers is $\\aleph_0$. The next larger cardinality is $\\aleph_1$. Then $\\aleph_2$ and so on."}
+{"_id": "29647", "title": "Definition:Order of Group Element/Also known as", "text": "Some sources refer to the '''order''' of an element of a group as its '''period'''."}
+{"_id": "29648", "title": "Definition:Order of Group Element/Also denoted as", "text": "The '''order''' of an element $x$ in a group is sometimes seen as $\\map o x$. Some sources render it as $\\map {\\operatorname {Ord} } x$."}
+{"_id": "29649", "title": "Definition:Cyclic Group/Notation", "text": "A '''cyclic group''' with $n$ elements is often denoted $C_n$. Some sources use the notation $\\sqbrk g$ or $\\gen g$ to denote the '''cyclic group''' generated by $g$. From Integers Modulo m under Addition form Cyclic Group, $\\struct {\\Z_m, +_m}$ is a '''cyclic group'''. Thus $\\struct {\\Z_m, +_m}$ often taken as the archetypal example of a '''cyclic group''', and the notation $\\Z_m$ is used. This is justified as, from Cyclic Groups of Same Order are Isomorphic, $\\Z_m$ is isomorphic to $C_m$. In certain contexts $\\Z_m$ is particularly useful, as it allows results about '''cyclic groups''' to be demonstrated using number theoretical techniques."}
+{"_id": "29650", "title": "Definition:Cyclic Group/Definition 1", "text": "The group $G$ is '''cyclic''' {{iff}} every element of $G$ can be expressed as the power of one element of $G$: :$\\exists g \\in G: \\forall h \\in G: h = g^n$ for some $n \\in \\Z$."}
+{"_id": "29651", "title": "Definition:Cyclic Group/Definition 2", "text": "The group $G$ is '''cyclic''' {{iff}} it is generated by one element $g \\in G$: :$G = \\gen g$"}
+{"_id": "29652", "title": "Definition:Generator of Group", "text": "Let $\\struct {G, \\circ}$ be a group. Let $S \\subseteq G$. Then '''$S$ is a generator of $G$''', denoted $G = \\gen S$, {{iff}} $G$ is the subgroup generated by $S$."}
+{"_id": "29653", "title": "Definition:Cyclic Group/Generator", "text": "Let $a \\in G$ be an element of $G$ such that $\\gen a = G$. Then $a$ is '''a generator of $G$'''."}
+{"_id": "29654", "title": "Definition:Multiplicative Group of Reduced Residues", "text": "Let $m \\in \\Z_{> 0}$ be a (strictly) positive integer. Let $\\Z'_m$ denote the reduced residue system modulo $m$. Consider the algebraic structure: :$\\struct {\\Z'_m, \\times_m}$ where $\\times_m$ denotes multiplication modulo $m$. Then $\\struct {\\Z'_m, \\times_m}$ is referred to as '''the multiplicative group of reduced residues modulo $m$'''."}
+{"_id": "29657", "title": "Definition:Completion (Normed Division Ring)", "text": "Let $\\struct {R_1, \\norm {\\, \\cdot \\,}_1}$ and $\\struct {R_2, \\norm {\\, \\cdot \\,}_2}$ be normed division rings. Let $M_1 = \\struct {R_1, d_1}$ and $M_2 = \\struct {R_2, d_2}$ be the metric spaces where $d_1: R_1 \\times R_1 \\to \\R_{\\ge 0}$ and $d_2: R_2 \\times R_2 \\to \\R_{\\ge 0}$ are the metrics induced by $\\norm {\\, \\cdot \\,}_1$ and $\\norm {\\, \\cdot \\,}_2$ respectively. Then '''$\\struct {R_2, \\norm {\\, \\cdot \\,}_2}$ is a completion of $\\struct {R_1, \\norm {\\, \\cdot \\,}_1}$''' {{iff}}: :$(1): \\quad$ there exists a distance-preserving ring monomorphism $\\phi: R_1 \\to R_2$ :$(2): \\quad M_2$ is a metric completion of $\\map \\phi {M_1}$. That is, '''$\\struct {R_2, \\norm{\\,\\cdot\\,}_2}$ is a completion of $\\struct {R_1,\\norm{\\,\\cdot\\,}_1}$''' {{iff}}: :$(a): \\quad M_2$ is a complete metric space :$(b): \\quad$ there exists a distance-preserving ring monomorphism $\\phi: R_1 \\to R_2$ :$(c): \\quad \\map \\phi {R_1}$ is a dense subspace in $M_2$."}
+{"_id": "29664", "title": "Definition:Existential Quantifier/Unique/Also denoted as", "text": "The symbol $\\exists_1$ is also found for the same concept, being an instance of the exact existential quantifier $\\exists_n$. Some sources, for example {{BookReference|Axiomatic Set Theory|1972|Patrick Suppes}}, use $\\operatorname E !$, which is idiosyncratic, considering the use in the same source of $\\exists$ for the general existential quantifier."}
+{"_id": "29665", "title": "Definition:Order of Group Element/Infinite/Definition 1", "text": "$x$ '''is of infinite order''', or '''has infinite order''' {{iff}} there exists no $k \\in \\Z_{> 0}$ such that $x^k = e_G$: :$\\order x = \\infty$"}
+{"_id": "29666", "title": "Definition:Order of Group Element/Infinite/Definition 2", "text": "$x$ '''is of infinite order''', or '''has infinite order''' {{iff}} the powers $x, x^2, x^3, \\ldots$ of $x$ are all distinct: :$\\order x = \\infty$"}
+{"_id": "29667", "title": "Definition:Order of Group Element/Finite/Also known as", "text": "An '''element of finite order''' of $G$ is also known as a '''torsion element''' of $G$."}
+{"_id": "29668", "title": "Definition:Order of Group Element/Finite/Definition 1", "text": "$x$ '''is of finite order''', or '''has finite order''' {{iff}} there exists $k \\in \\Z_{> 0}$ such that $x^k = e_G$."}
+{"_id": "29669", "title": "Definition:Order of Group Element/Finite/Definition 2", "text": "$x$ '''is of finite order''', or '''has finite order''' {{iff}} there exist $m, n \\in \\Z_{> 0}$ such that $m \\ne n$ but $x^m = x^n$."}
+{"_id": "29670", "title": "Definition:Complete Normed Division Ring", "text": "A normed division ring $\\struct {R, \\norm {\\, \\cdot \\,} }$ is '''complete''' {{iff}} the metric space $\\struct {R, d}$ is a complete metric space where $d$ is the metric induced by the norm $\\norm {\\, \\cdot \\,}$. That is, a normed division ring $\\struct {R, \\norm {\\, \\cdot \\,} }$ is '''complete''' {{iff}} every Cauchy sequence is convergent."}
+{"_id": "29671", "title": "Definition:Order of Group Element/Infinite/Definition 3", "text": "$x$ '''is of infinite order''', or '''has infinite order''' {{iff}} the group $\\gen x$ generated by $x$ is of infinite order. :$\\order x = \\infty \\iff \\order {\\gen x} = \\infty$"}
+{"_id": "29672", "title": "Definition:Cardinality/Also known as", "text": "Some authors prefer the term '''order''' instead of '''cardinality''', particularly in the context of finite sets. {{AuthorRef|Georg Cantor}} used the term '''power''' and equated it with the term '''cardinal number''', using the notation $\\overline {\\overline M}$ for the '''cardinality''' of $M$. Some sources cut through all the complicated language and call it the '''size'''. Some sources use $\\map {\\#} S$ (or a variant) to denote '''set cardinality'''. This notation has its advantages in certain contexts, and is used on occasion on this website. Others use $\\map C S$, but this is easy to confuse with other uses of the same or similar notation. A clear but relatively verbose variant is $\\Card \\paren S$ or $\\operatorname{card} \\paren S$. {{BookReference|Introductory Real Analysis|1968|A.N. Kolmogorov|author2 = S.V. Fomin}} use $\\map m A$ for the '''power''' of the set $A$. Further notations are $\\map n A$ and $\\overline A$."}
+{"_id": "29673", "title": "Definition:Cardinality/Infinite", "text": "Let $S$ be an infinite set. The '''cardinality''' $\\card S$ of $S$ can be indicated as: :$\\card S = \\infty$ However, it needs to be noted that this just means that the cardinality of $S$ cannot be assigned a number $n \\in \\N$. It means that $\\card S$ is ''at least'' $\\aleph_0$ (aleph null)."}
+{"_id": "29675", "title": "Definition:Field of Integers Modulo Prime", "text": "Let $p \\in \\Bbb P$ be a prime number. Let $\\Z_p$ be the set of integers modulo $p$. Let $+_p$ and $\\times_p$ denote addition modulo $p$ and multiplication modulo $p$ respectively. The algebraic structure $\\struct {\\Z_p, +_p, \\times_p}$ is '''the field of integers modulo $p$'''."}
+{"_id": "29677", "title": "Definition:Conjugate (Group Theory)/Element/Also known as", "text": "Some sources refer to the '''conjugate''' of $x$ as the '''transform''' of $x$. Some sources refer to '''conjugacy''' as '''conjugation'''."}
+{"_id": "29678", "title": "Definition:Conjugate (Group Theory)/Element/Also defined as", "text": "Some sources define the '''conjugate of $x$ by $a$ in $G$''' as: :$x \\sim y \\iff \\exists a \\in G: x \\circ a = a \\circ y$ or: :$x \\sim y \\iff \\exists a \\in G: a^{-1} \\circ x \\circ a = y$"}
+{"_id": "29679", "title": "Definition:Conjugate (Group Theory)/Element/Definition 1", "text": "The  '''conjugacy relation''' $\\sim$ is defined on $G$ as: :$\\forall \\tuple {x, y} \\in G \\times G: x \\sim y \\iff \\exists a \\in G: a \\circ x = y \\circ a$"}
+{"_id": "29680", "title": "Definition:Conjugate (Group Theory)/Element/Definition 2", "text": "The  '''conjugacy relation''' $\\sim$ is defined on $G$ as: :$\\forall \\tuple {x, y} \\in G \\times G: x \\sim y \\iff \\exists a \\in G: a \\circ x \\circ a^{-1} = y$"}
+{"_id": "29682", "title": "Definition:Ring (Abstract Algebra)/Historical Note", "text": "According to {{AuthorRef|Ian Stewart}}, in his {{BookLink|Galois Theory|Ian Stewart|ed = 3rd|edpage = Third Edition}} of $2004$, the ring axioms were first formulated by {{AuthorRef|Heinrich Martin Weber}} in $1893$."}
+{"_id": "29684", "title": "Definition:Norm/Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is denoted $0_R$. A '''(submultiplicative) norm''' on $R$ is a mapping from $R$ to the non-negative reals: {{:Definition:Submultiplicative Norm on Ring}}"}
+{"_id": "29685", "title": "Definition:Norm/Division Ring", "text": "Let $\\struct {R, +, \\circ}$ be a division ring whose zero is denoted $0_R$. A '''(multiplicative) norm''' on $R$ is a mapping from $R$ to the non-negative reals: {{:Definition:Multiplicative Norm on Ring}}"}
+{"_id": "29686", "title": "Definition:Norm/Vector Space", "text": "Let $\\struct {R, +, \\circ}$ be a division ring with norm $\\norm {\\,\\cdot\\,}_R$. Let $V$ be a vector space over $R$, with zero $0_V$. A '''norm''' on $V$ is a map from $V$ to the nonnegative reals: :$\\norm{\\,\\cdot\\,}: V \\to \\R_{\\ge 0}$ satisfying the '''(vector space) norm axioms''': {{Definition:Norm Axioms (Vector Space)}}"}
+{"_id": "29687", "title": "Definition:Norm/Algebra", "text": "Let $R$ be a division ring with norm $\\norm {\\,\\cdot\\,}_R$. Let $A$ be an algebra over $R$. A '''norm''' on $A$ is a vector space norm $\\norm{\\,\\cdot\\,}: A \\to \\R_{\\ge 0}$ on $A$ as a vector space such that: :for all $a, b \\in A: \\norm {a b} \\le \\norm a \\cdot \\norm b$"}
+{"_id": "29688", "title": "Definition:Norm/Unital Algebra", "text": "Let $A$ be an unital algebra over $R$ with unit $e$. A '''norm''' on $A$ is an algebra norm $\\norm{\\,\\cdot\\,}: A \\to \\R_{\\ge 0}$ such that: :$\\norm e = 1$"}
+{"_id": "29689", "title": "Definition:Norm/Bounded Linear Transformation", "text": "Let $H, K$ be Hilbert spaces, and let $A: H \\to K$ be a bounded linear transformation. Then the '''norm''' of $A$, denoted $\\norm A$, is the real number defined by: :$(1): \\quad \\norm A = \\sup \\set {\\norm {A h}_K: \\norm h_H \\le 1}$ :$(2): \\quad \\norm A = \\sup \\set {\\dfrac {\\norm {A h}_K} {\\norm h_H}: h \\in H, h \\ne \\mathbf 0_H}$ :$(3): \\quad \\norm A = \\sup \\set {\\norm {A h}_K: \\norm h_H \\le 1}$ :$(4): \\quad \\norm A = \\inf \\set {c > 0: \\forall h \\in H: \\norm {A h}_K \\le c \\norm h_H}$ These definitions are equivalent, as proved in Equivalence of Definitions of Norm of Linear Transformation."}
+{"_id": "29690", "title": "Definition:Norm/Bounded Linear Functional/Definition 1", "text": "The '''norm''' of $L$ is the real number defined as the supremum: :$\\norm L = \\sup \\set {\\size {L h}: \\norm h_H \\le 1}$"}
+{"_id": "29691", "title": "Definition:Norm/Bounded Linear Functional/Definition 2", "text": "The '''norm''' of $L$ is the supremum: :$\\norm{L} = \\sup \\set{\\left|{Lh}\\right|: \\norm{h}_H = 1}$"}
+{"_id": "29692", "title": "Definition:Norm/Bounded Linear Functional/Definition 3", "text": "The '''norm''' of $L$ is the supremum: :$\\norm L = \\displaystyle \\sup \\set {\\frac {\\size {L h} } {\\norm h _H}: h \\in H, h \\ne \\bszero_H}$"}
+{"_id": "29693", "title": "Definition:Norm/Bounded Linear Functional/Definition 4", "text": "The '''norm''' of $L$ is the infimum: :$\\norm{L} = \\inf \\set{c > 0: \\forall h \\in H: \\left|{Lh}\\right| \\le c \\norm{h}_H}$"}
+{"_id": "29694", "title": "Definition:Norm/Bounded Linear Functional", "text": "Let $H$ be a Hilbert space, and let $L$ be a bounded linear functional on $H$. === Definition 1 === {{:Definition:Norm/Bounded Linear Functional/Definition 1}} === Definition 2 === {{:Definition:Norm/Bounded Linear Functional/Definition 2}} === Definition 3 === {{:Definition:Norm/Bounded Linear Functional/Definition 3}} === Definition 4 === {{:Definition:Norm/Bounded Linear Functional/Definition 4}} {{refactor|These ought to go into their own pages, appropriately linked}} As a consequence of definition $(4)$, we have for all $h \\in H$ that $\\size {L h} \\le \\norm L \\norm h$. As $L$ is bounded, it is assured that $\\norm L < \\infty$."}
+{"_id": "29696", "title": "Definition:Group Direct Product/Finite Product", "text": "Let $\\struct {G_1, \\circ_1}, \\struct {G_2, \\circ_2}, \\ldots, \\struct {G_n, \\circ_n}$ be groups. Let $\\displaystyle G = \\prod_{k \\mathop = 1}^n G_k$ be their cartesian product. Let $\\circ$ be the operation defined on $G$ as: :$\\circ := \\tuple {g_1, g_2, \\ldots, g_n} \\circ \\tuple {h_1, h_2, \\ldots, h_n} = \\tuple {g_1 \\circ_1 h_1, g_2 \\circ_2 h_2, \\ldots, g_n \\circ_n h_n}$ for all ordered $n$-tuples in $G$. The group $\\struct {G, \\circ}$ is called the '''(external) direct product''' of $\\struct {G_1, \\circ_1}, \\struct {G_2, \\circ_2}, \\ldots, \\struct {G_n, \\circ_n}$."}
+{"_id": "29698", "title": "Definition:Bilateral Symmetry/Axis", "text": "Let $F$ be a plane geometric figure. Let $\\map R F$ be a reflection in an axis of reflection $AB$ which is a symmetry. Then $AB$ is referred to as an '''axis of symmetry''' of $F$."}
+{"_id": "29703", "title": "Definition:Multiplicative Group of Complex Roots of Unity", "text": "Let $n \\in \\Z$ be an integer such that $n > 0$. Let $U_n := \\set {z \\in \\C: z^n = 1}$ denote the set of complex $n$th roots of unity. Let $\\struct {U_n, \\times}$ be the algebraic structure formed by $U_n$ under complex multiplication. Then $\\struct {U_n, \\times}$ is the '''multiplicative group of complex $n$th roots of unity'''."}
+{"_id": "29704", "title": "Definition:Isolated Zero", "text": "Let $f: \\C \\to \\C$ be a complex function. Let $\\map f {z_0} = 0$ for some $z_0 \\in \\C$.  We say $z_0$ is an '''isolated zero''' of $f$ if there exists an open ball $B$ containing $z_0$, such that $\\map f w \\ne 0$ for all $w \\in B \\setminus \\set {z_0}$."}
+{"_id": "29705", "title": "Definition:Conjugacy Action/Subsets", "text": "Let $\\powerset G$ be the power set of $G$. The '''(left) conjugacy action on subsets''' is the group action $* : G \\times \\powerset G \\to \\powerset G$: :$g * S = g \\circ S \\circ g^{-1}$ The '''right conjugacy action on subsets''' is the group action $* : \\powerset G \\times G \\to \\powerset G$: :$S * g = g^{-1} \\circ S \\circ g$"}
+{"_id": "29706", "title": "Definition:Subgroup Action", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\struct {H, \\circ}$ be a subgroup of $G$. Let $*: H \\times G \\to G$ be the operation defined as: :$\\forall h \\in H, g \\in G: h * g := h \\circ g$ This is the '''subgroup action''' of $H$ on $G$."}
+{"_id": "29707", "title": "Definition:Norm/Ring/Multiplicative", "text": ":$\\norm {\\,\\cdot\\,}: R \\to \\R_{\\ge 0}$ satisfying the '''(ring) multiplicative norm axioms''': {{:Definition:Multiplicative Norm Axioms}}"}
+{"_id": "29708", "title": "Definition:Norm/Ring/Submultiplicative", "text": ":$\\norm {\\,\\cdot\\,}: R \\to \\R_{\\ge 0}$ satisfying the '''(ring) submultiplicative norm axioms''': {{:Definition:Submultiplicative Norm Axioms}}"}
+{"_id": "29709", "title": "Definition:Isometric Isomorphism", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,}_R}$ and $\\struct {S, \\norm {\\,\\cdot\\,}_S}$ be normed division rings. Let $d_R$ and $d_S$ be the metric induced by the norms $\\norm {\\,\\cdot\\,}_R$ and $\\norm {\\,\\cdot\\,}_S$ respectively. Let $\\phi:R \\to S$ be a bijection such that: :$(1): \\quad \\phi: \\struct {R, d_R} \\to \\struct {S, d_S}$ is an isometry :$(2): \\quad \\phi: R \\to S$ is a ring isomorphism. Then $\\phi$ is called an '''isometric isomorphism'''. The normed division rings $\\struct {R, \\norm {\\,\\cdot\\,}_R}$ and $\\struct {S, \\norm {\\,\\cdot\\,}_S}$ are said to be '''isometrically isomorphic'''. Category:Definitions/Normed Division Rings 64o9alx30idd000xrp4c8v9wwl428ll"}
+{"_id": "29711", "title": "Definition:Cartesian Product/Cartesian Space/Two Dimensions", "text": "The '''cartesian $2$nd power of $S$''' is: :$S^2 = S \\times S = \\set {\\tuple {x_1, x_2}: x_1, x_2 \\in S}$ The set $S^2$ called a '''cartesian space of $2$ dimensions'''."}
+{"_id": "29715", "title": "Definition:Bijection/Also known as", "text": "The terms :'''biunique correspondence''' :'''bijective correspondence''' are sometimes seen for '''bijection'''. Authors who prefer to limit the jargon of mathematics tend to use the term '''one-one and onto mapping''' for '''bijection'''. If a '''bijection''' exists between two sets $S$ and $T$, then $S$ and $T$ are said to be '''in one-to-one correspondence'''. Occasionally you will see the term '''set isomorphism''', but the term '''isomorphism''' is usually reserved for mathematical structures of greater complexity than a set. Some authors, developing the concept of '''inverse mapping''' independently from that of the '''bijection''', call such a mapping '''invertible'''. The symbol $f: S \\leftrightarrow T$ is sometimes seen to denote that $f$ is a '''bijection''' from $S$ to $T$. Also seen sometimes is the notation $f: S \\cong T$ or $S \\stackrel f \\cong T$ but this is cumbersome and the symbol has already got several uses. == Technical Note == {{:Definition:Bijection/Technical Note}}"}
+{"_id": "29719", "title": "Definition:Normed Division Subring", "text": "Let $\\struct {R, +, \\circ, \\norm {\\, \\cdot \\,} }$ be a normed division ring. A '''normed division subring''' of $\\struct {R, +, \\circ, \\norm {\\, \\cdot \\,} }$ is a subset $S$ of $R$ such that $\\struct{S, +_S, \\circ_S, \\norm{\\, \\cdot \\,}_S}$ is a normed division ring where: :$(1) \\quad +_S$ is the binary operation $+$ restricted to $S \\times S$ :$(2) \\quad \\circ_S$ is the binary operation $\\circ$ restricted to $S \\times S$ :$(3) \\quad \\norm {\\, \\cdot \\,}_S$ is the norm $\\norm {\\, \\cdot \\,}$ restricted to $S$. Category:Definitions/Normed Division Rings qkji2ugsu8opfzad3kzqdsibjpr37iy"}
+{"_id": "29720", "title": "Definition:Square/Mapping", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $f: S \\to S$ be the mapping from $S$ to $S$ defined as: :$\\forall x \\in S: \\map f x := x \\circ x$ This is usually denoted $x^2$: :$x^2 := x \\circ x$"}
+{"_id": "29721", "title": "Definition:Square/Mapping/Element", "text": "A '''square (element of $S$)''' is an element $y$ of $S$ for which: :$\\exists x \\in S: y = x^2$ Such a $y = x^2$ is referred to as '''the square of $x$'''."}
+{"_id": "29722", "title": "Definition:Square/Function/Definition 1", "text": "The '''square (function) on $\\F$''' is the mapping $f: \\F \\to \\F$ defined as: :$\\forall x \\in \\F: \\map f x = x \\times x$ where $\\times$ denotes multiplication."}
+{"_id": "29723", "title": "Definition:Square/Function", "text": "Let $\\F$ denote one of the standard classes of numbers: $\\N$, $\\Z$, $\\Q$, $\\R$, $\\C$. === Definition 1 === {{:Definition:Square/Function/Definition 1}} === Definition 2 === {{:Definition:Square/Function/Definition 2}}"}
+{"_id": "29724", "title": "Definition:Square/Function/Definition 2", "text": "The '''square (function) on $\\F$''' is the mapping $f: \\F \\to \\F$ defined as: :$\\forall x \\in \\F: \\map f x = x^2$ where $x^2$ denotes the $2$nd power of $x$."}
+{"_id": "29725", "title": "Definition:Square/Function/Integer", "text": "The '''(integer) square function''' is the integer function $f: \\Z \\to \\Z$ defined as: :$\\forall x \\in \\Z: \\map f x = x^2$"}
+{"_id": "29726", "title": "Definition:Ultrametric Space", "text": "An '''ultrametric space''' is a metric space $\\struct {X, d}$ where $d$ is non-Archimedean."}
+{"_id": "29727", "title": "Definition:Unique up to Isomorphism", "text": "Let $\\mathbf C$ be a category. Let $S \\subseteq \\operatorname{Ob}(\\mathbf C)$ be a subclass of its objects. The class $S$ is '''unique up to isomorphism''' {{iff}} for all objects $s,t \\in S$ there is a isomorphism from $s$ to $t$."}
+{"_id": "29728", "title": "Definition:Power of Element/Also defined as", "text": "Some treatments do not define the '''power of an element''' for a magma whose operation is non-associative."}
+{"_id": "29729", "title": "Definition:Power Associativity", "text": "Let $\\struct {S, \\circ}$ be a magma. $\\struct {S, \\circ}$ is '''power associative''' {{iff}}: :$\\forall a \\in S: a \\circ \\paren {a \\circ a} = \\paren {a \\circ a} \\circ a$ Category:Definitions/Powers (Abstract Algebra) dyk1d393oyc3d4lh49p5uynq81ugw4t"}
+{"_id": "29731", "title": "Definition:Finitely Generated Group", "text": "Let $G$ be a group. $G$ is '''finitely generated''' {{iff}} $G$ has a generator which is finite."}
+{"_id": "29732", "title": "Definition:Infinite Cyclic Group/Definition 1", "text": "An '''infinite cyclic group''' is a cyclic group $G$ such that: :$\\forall n \\in \\Z_{> 0}: n > 0 \\implies \\nexists a \\in G, a \\ne e: a^n = e$"}
+{"_id": "29733", "title": "Definition:Infinite Cyclic Group/Definition 2", "text": "An '''infinite cyclic group''' is a cyclic group $G$ such that: :$\\forall a \\in G, a \\ne e: \\forall m, n \\in \\Z: m \\ne n \\implies a^m \\ne a^n$ where $e$ is the identity element of $G$. That is, such that all the powers of $a$ are distinct."}
+{"_id": "29734", "title": "Definition:Subset Product Action/Left", "text": "The '''(left) subset product action''' of $G$ is the group action $*: G \\times \\powerset G \\to \\powerset G$: :$\\forall g \\in G, S \\in \\powerset G: g * S = g \\circ S$"}
+{"_id": "29735", "title": "Definition:Subset Product Action", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\HH$ be the set of subgroups of $G$. === Left Subset Product Action === {{:Definition:Subset Product Action/Left}} === Right Subset Product Action === {{:Definition:Subset Product Action/Right}}"}
+{"_id": "29736", "title": "Definition:Subset Product Action/Right", "text": "The '''(right) subset product action''' of $G$ is the group action $*: G \\times \\powerset G \\to \\powerset G$: :$\\forall g \\in G, S \\in \\powerset G: g * S = S \\circ g$"}
+{"_id": "29738", "title": "Definition:Generated Subgroup/Definition 2", "text": "The '''subgroup generated by $S$''' is the intersection of all subgroups of $G$ containing $S$."}
+{"_id": "29739", "title": "Definition:Group Action on Coset Space", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. The mapping $*: G \\times G / H \\to G / H$ by the rule: :$\\forall g \\in G, \\forall g' H \\in G / H: g * \\paren {g' H} := \\paren {g g'} H$ is the '''group action on the (left) coset space''' $G / H$."}
+{"_id": "29741", "title": "Definition:Trivial Quotient Group", "text": "Let $G$ be a group. The quotient group: :$G / \\set {e_G} \\cong G$ where: :$\\cong$ denotes group isomorphism :$e_G$ denotes the identity element of $G$. is known as the '''trivial quotient group'''."}
+{"_id": "29742", "title": "Definition:Inner Automorphism Group", "text": "Let $G$ be a group. The '''inner automorphism group''' of $G$ is the algebraic structure: :$\\struct {\\Inn G, \\circ}$ where: :$\\Inn G$ is the '''set of inner automorphisms''' of $G$ :$\\circ$ denotes composition of mappings."}
+{"_id": "29746", "title": "Definition:Zeraoulia Function", "text": "The '''Zeraoulia function''' is a new special function see <ref>E. W. Ng and M. Geller, “A table of integrals of the error functions,” Journal of Research of the National Bureau of Standards, vol. 73B, pp. 1–20, 1969 </ref> , <ref> E. W. Ng and M. Geller, “A table of integrals of the error functions,” Journal of Research of the National Bureau of Standards, vol. 73B, pp. 1–20, 1969. View at Google Scholar · View at MathSciNet</ref>   proposed by Zeraoulia Rafik in 17/02/2017 and have been studied by Zeraoulia Rafik,Alvaro Humberto Salas  Davide L.Ocampo  and published in  International Journal of Mathematics and Mathematical Sciences , <ref> Zeraoulia Rafik, Alvaro H. Salas, and David L. Ocampo, “A New Special Function and Its Application in Probability,” International Journal of Mathematics and Mathematical Sciences, vol. 2018, Article ID 5146794, 12 pages, 2018. https://doi.org/10.1155/2018/5146794</ref>,  it behave like more than error function and it is defined as : :$\\displaystyle \\map T a = \\int_0^a \\paren {e^{-x^2} }^{\\map \\erf x} \\rd x$ Mathematica gives for the first $100$ digits: :$\\map T \\infty = 0 \\cdotp 9721069927691785931510778754423911755542721833855699009722910408441888759958220033410678218401258734$ with $\\map T 0 = 0$"}
+{"_id": "29747", "title": "Definition:Generator of Subgroup/Definition by Predicate", "text": "A generator of a subgroup can be defined by a '''predicate'''. For example: :$\\gen {x \\in G: x^2 = e}$ defines the subgroup of $G$ generated by the elements of $G$ of order $2$."}
+{"_id": "29748", "title": "Definition:Generated Normal Subgroup/Definition 3", "text": "The '''normal subgroup generated by $S$''', denoted $\\gen {S^G}$,  is the smallest normal subgroup of $G$ containing $S$: :$\\gen {S^G} = \\gen {x G x^{-1}: x \\in G}$"}
+{"_id": "29749", "title": "Definition:Nu Function/Sequence", "text": "The sequence of values of '''$\\nu$ function''': $\\map \\nu n$ for $n = 1, 2, 3, \\ldots$ begins: :$1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, \\ldots$"}
+{"_id": "29750", "title": "Definition:Extension Problem", "text": "Let $H$ and $K$ be groups. The '''extension problem''' is to determine all groups $G$ such that: :$K \\trianglelefteq G$ :$G / K \\cong H$ where: :$K \\trianglelefteq G$ denotes that $K$ is a normal subgroup of $G$ :$G / K$ denotes the quotient group of $G$ by $K$ :$\\cong$ denotes group isomorphism."}
+{"_id": "29753", "title": "Definition:Involution (Mapping)/Definition 1", "text": "$f: A \\to A$ is an '''involution''' precisely when: :$\\forall x \\in A: \\map f {\\map f x} = x$ That is: :$f \\circ f = I_A$ where $I_A$ denotes the identity mapping on $A$."}
+{"_id": "29755", "title": "Definition:Involution (Mapping)/Definition 3", "text": "Let $f: A \\to A$ be a mapping on $A$. Then $f$ is an involution {{iff}} $f$ is both a bijection and a symmetric relation. That is, {{iff}} $f$ is a bijection such that $f = f^{-1}$."}
+{"_id": "29756", "title": "Definition:Self-Inverse Element/Also known as", "text": "The definition of a '''self-inverse element''' is usually made in the context of a group. Some sources refer to such an element as an '''involution'''."}
+{"_id": "29757", "title": "Definition:Characterization Theorem", "text": "Let $E$ be a non-abelian finite simple group. Let $u \\in E$ be a self-inverse element of $E$. Let $H = \\map {C_E} u$ be the centralizer of $u$ in $E$. Let $G$ be a finite simple group with a self-inverse element $t$ such that $H \\cong \\map {C_G} t$. A '''characterization theorem''' is a theorem that proves there is only one such group type $G$. That is, that $G \\cong E$ necessarily."}
+{"_id": "29759", "title": "Definition:Group of Units/Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity. Then the set $U_R$ of units of $\\struct {R, +, \\circ}$ is called the '''group of units''' of $\\struct {R, +, \\circ}$. This can be denoted explicitly as $\\struct {U_R, \\circ}$."}
+{"_id": "29761", "title": "Definition:Chess", "text": "'''Chess''' is a game for two players. It is played using '''pieces''' placed on a '''chessboard''', where the players take turns to move a piece. A piece is '''captured''' if an opposing piece moves onto the same space. The opposing piece is '''attacking'''. The captured piece is then removed from the board, unless it is a king. It is forbidden to capture your own piece. If a piece could be captured in one move by any piece, it is '''threatened'''. A piece cannot move over another piece, unless it is a knight, or it is castling. If a king is captured, the player who captured it wins immediately. A king is put in '''check''' if it is threatened by any piece. This is '''checkmate''' if the player who owns the king cannot prevent the king from being captured using one move. The only other way to win is if a player indicates that he resigns, traditionally by knocking over his own king."}
+{"_id": "29766", "title": "Definition:Odd Integer/Odd-Times Odd/Sequence", "text": "The sequence of '''odd-times odd integers''' begins: :$9, 15, 21, 25, 27, \\ldots$"}
+{"_id": "29773", "title": "Definition:Completely Metrizable Topology", "text": "Let $\\struct {S, \\tau}$ be a topological space. The space $\\struct {S, \\tau}$ is said to be '''completely metrizable''' {{iff}} there exists a metric $d$ such that: :$\\struct {S, d}$ is a complete metric space and: :$\\tau$ is the topological space induced by the metric $d$. == Also see == {{refactor|Trivial, but the following statement needs to be justified in its own page}} All completely metrizable topologies are in particular metrizable topologies."}
+{"_id": "29776", "title": "Definition:Limit of Sequence/Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\sequence {x_n} $ be a sequence in $R$. <section notitle=\"true\" name=\"LimitDefinition\"> Let $\\sequence {x_n}$ converge to $x \\in R$. Then $x$ is a '''limit of $\\sequence {x_n}$ as $n$ tends to infinity''' which is usually written: :$\\displaystyle x = \\lim_{n \\mathop \\to \\infty} x_n$ </section>"}
+{"_id": "29778", "title": "Definition:Null Sequence/Analysis", "text": "=== Complex Numbers === {{:Definition:Null Sequence/Complex Numbers}} === Real Numbers === {{:Definition:Null Sequence/Real Numbers}} === Rational Numbers === {{:Definition:Null Sequence/Rational Numbers}}"}
+{"_id": "29779", "title": "Definition:Null Sequence/Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring with zero $0_R$. Let $\\sequence {x_n}$ be a sequence in $R$ which converges to the limit $0_R$: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = 0_R$ Then $\\sequence {x_n}$ is called a '''null sequence'''."}
+{"_id": "29780", "title": "Definition:Null Sequence", "text": "=== Normed Division Ring === {{:Definition:Null Sequence/Normed Division Ring}}"}
+{"_id": "29781", "title": "Definition:Fermat Pseudoprime/Base 4", "text": "Let $q$ be a composite number such that $4^q \\equiv 4 \\pmod q$. Then $q$ is a '''Fermat pseudoprime to base $4$'''."}
+{"_id": "29783", "title": "Definition:Fermat Pseudoprime/Base 5", "text": "Let $q$ be a composite number such that $5^q \\equiv 5 \\pmod q$. Then $q$ is a '''Fermat pseudoprime to base $5$'''."}
+{"_id": "29786", "title": "Definition:Null Sequence/Complex Numbers", "text": "Let $\\sequence {z_n}$ be a sequence in $\\C$ which converges to a limit of $0$: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} z_n = 0$ Then $\\sequence {z_n}$ is called a '''(complex) null sequence'''."}
+{"_id": "29787", "title": "Definition:Null Sequence/Real Numbers", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$ which converges to a limit of $0$: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = 0$ Then $\\sequence {x_n}$ is called a '''(real) null sequence'''."}
+{"_id": "29788", "title": "Definition:Null Sequence/Rational Numbers", "text": "Let $\\sequence {x_n}$ be a sequence in $\\Q$ which converges to a limit of $0$: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = 0$ Then $\\sequence {x_n}$ is called a '''(rational) null sequence'''."}
+{"_id": "29789", "title": "Definition:Convergent Sequence", "text": "=== Topological Space === {{:Definition:Convergent Sequence/Topology}} === Metric Space === {{:Definition:Convergent Sequence/Metric Space}} For other equivalent definitions of a '''convergent sequence''' in a Metric Space see: Definition:Convergent Sequence in Metric Space === Normed Division Ring === {{:Definition:Convergent Sequence/Normed Division Ring}}"}
+{"_id": "29790", "title": "Definition:Convergent Sequence/Metric Space", "text": "Let $M = \\left({A, d}\\right)$ be a metric space or a pseudometric space. Let $\\sequence {x_k}$ be a sequence in $A$."}
+{"_id": "29791", "title": "Definition:Convergent Sequence/Normed Division Ring", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\sequence {x_n} $ be a sequence in $R$."}
+{"_id": "29792", "title": "Definition:Icosahedron/Truncated", "text": "A '''truncated icosahedron''' is an icosahedron whose $12$ vertices are truncated such that one third of each edge is cut off at each of both ends."}
+{"_id": "29794", "title": "Definition:Floating Point Representation", "text": "'''Floating point''' representation is a technique whereby numbers are held in a computer in standard scientific notation."}
+{"_id": "29795", "title": "Definition:Scientific Notation", "text": "'''Scientific notation''' is a technique for representing approximations to (usually large) numbers by presenting them in the form: :$n \\approx m \\times 10^e$ where: :$m$ is a rational number such that $1 \\le m < 10$, expressed in decimal notation :$e$ is an integer."}
+{"_id": "29799", "title": "Definition:Megaflop", "text": "A '''megaflop''' is a measure of calculation speed of a computer. :$1$ '''megaflop''' $= 1$ million floating point operations per second."}
+{"_id": "29800", "title": "Definition:Gigaflop", "text": "A '''gigaflop''' is a measure of calculation speed of a computer. {{begin-eqn}} {{eqn | o = =       | r = 1        | c = '''gigaflop''' }} {{eqn | r = 1000       | c = megaflops }} {{eqn | r = 1 \\times 10^9       | c = floating point operations per second }} {{end-eqn}}"}
+{"_id": "29803", "title": "Definition:Closed Disk", "text": "Let $V$ be a vectorspace and $x \\in V$, $r \\in \\R$. Then the '''closed disk''' $\\bar{D}$ centered at $x$ with radius $r$ is the set: {{begin-eqn}} $\\bar{D}(x, r) := \\{v \\in V | \\| x - v \\| \\le r\\}$ {{end-eqn}}"}
+{"_id": "29804", "title": "Definition:Group Presentation/Relation", "text": "The standard form of a '''relation''' in a '''group presentation''' is: :$w = e$ where $w$ is a word in the group."}
+{"_id": "29805", "title": "Definition:Topologically Equivalent Metrics/Definition 1", "text": "$d_1$ and $d_2$ are '''topologically equivalent''' {{iff}}: :For all metric spaces $\\left({B, d}\\right)$ and $\\left({C, d'}\\right)$: :For all mappings $f: B \\to A$ and $g: A \\to C$: :: $(1): \\quad f$ is $\\left({d, d_1}\\right)$-continuous {{iff}} $f$ is $\\left({d, d_2}\\right)$-continuous :: $(2): \\quad g$ is $\\left({d_1, d'}\\right)$-continuous {{iff}} $g$ is $\\left({d_2, d'}\\right)$-continuous."}
+{"_id": "29806", "title": "Definition:Topologically Equivalent Metrics/Definition 2", "text": "$d_1$ and $d_2$ are '''topologically equivalent''' {{iff}}: :$U \\subseteq A$ is $d_1$-open $\\iff$ $U \\subseteq A$ is $d_2$-open."}
+{"_id": "29807", "title": "Definition:Lipschitz Equivalence/Terminology", "text": "Despite the close connection with the concept of Lipschitz continuity, this concept is rarely seen in mainstream mathematics, and appears not to have a well-established name. The name '''Lipschitz equivalence''' appears in {{BookReference|Introduction to Metric and Topological Spaces|1975|W.A. Sutherland}}: :''There does not appear to be a standard name for this; the name we use is reasonably appropriate ...''"}
+{"_id": "29808", "title": "Definition:Equivalent Division Ring Norms", "text": "Let $R$ be a division ring. Let $\\norm{\\,\\cdot\\,}_1: R \\to \\R_{\\ge 0}$ and $\\norm{\\,\\cdot\\,}_2: R \\to \\R_{\\ge 0}$ be norms on $R$. Let $d_1$ and $d_2$ be the metrics induced by the norms $\\norm{\\,\\cdot\\,}_1$ and $\\norm{\\,\\cdot\\,}_2$ respectively. === Topologically Equivalent === {{:Definition:Equivalent Division Ring Norms/Topologically Equivalent}} === Convergently Equivalent === {{Transclude:Definition:Equivalent Division Ring Norms/Convergently Equivalent|section=Definition}} === Null Sequence Equivalent === {{:Definition:Equivalent Division Ring Norms/Null Sequence Equivalent}} === Open Unit Ball Equivalent === {{:Definition:Equivalent Division Ring Norms/Open Unit Ball Equivalent}} === Norm is the Power of the Other Norm === {{:Definition:Equivalent Division Ring Norms/Norm is Power of Other Norm}} === Cauchy Sequence Equivalent === {{Transclude:Definition:Equivalent Division Ring Norms/Cauchy Sequence Equivalent|section=Definition}}"}
+{"_id": "29809", "title": "Definition:Equivalent Division Ring Norms/Topologically Equivalent", "text": "$\\norm {\\, \\cdot \\,}_1$ and $\\norm {\\, \\cdot \\,}_2$ are '''equivalent''' {{iff}} $d_1$ and $d_2$ are topologically equivalent metrics."}
+{"_id": "29810", "title": "Definition:Equivalent Division Ring Norms/Convergently Equivalent", "text": "$\\norm {\\,\\cdot\\,}_1$ and $\\norm {\\,\\cdot\\,}_2$ are '''equivalent''' {{iff}} for all sequences $\\sequence {x_n}$ in $R$: :$\\sequence {x_n}$ converges to $l$ in $\\norm{\\,\\cdot\\,}_1 \\iff \\sequence {x_n}$ converges to $l$ in $\\norm {\\,\\cdot\\,}_2$"}
+{"_id": "29811", "title": "Definition:Equivalent Division Ring Norms/Null Sequence Equivalent", "text": "$\\norm {\\,\\cdot\\,}_1$ and $\\norm {\\,\\cdot\\,}_2$ are '''equivalent''' {{iff}} for all sequences $\\sequence {x_n}$ in $R$: :$\\sequence {x_n}$ is a null sequence in $\\norm{\\,\\cdot\\,}_1 \\iff \\sequence {x_n}$ is a null sequence in $\\norm {\\,\\cdot\\,}_2$"}
+{"_id": "29812", "title": "Definition:Equivalent Division Ring Norms/Open Unit Ball Equivalent", "text": "$\\norm {\\,\\cdot\\,}_1$ and $\\norm {\\,\\cdot\\,}_2$ are '''equivalent''' {{iff}} $\\forall x \\in R: \\norm x_1 < 1 \\iff \\norm x_2 < 1$"}
+{"_id": "29813", "title": "Definition:Equivalent Division Ring Norms/Norm is Power of Other Norm", "text": "$\\norm{\\,\\cdot\\,}_1$ and $\\norm{\\,\\cdot\\,}_2$ are '''equivalent''' {{iff}} $\\exists \\alpha \\in \\R_{\\gt 0}: \\forall x \\in R: \\norm{x}_1 = \\norm{x}_2^\\alpha$"}
+{"_id": "29814", "title": "Definition:Equivalent Division Ring Norms/Cauchy Sequence Equivalent", "text": "$\\norm {\\,\\cdot\\,}_1$ and $\\norm {\\,\\cdot\\,}_2$ are '''equivalent''' {{iff}} for all sequences $\\sequence {x_n}$ in $R$: :$\\sequence {x_n}$ is a Cauchy sequence in $\\norm {\\,\\cdot\\,}_1 \\iff \\sequence {x_n}$ is a Cauchy sequence in $\\norm{\\,\\cdot\\,}_2$"}
+{"_id": "29816", "title": "Definition:International Standard Book Number", "text": "An '''international standard book number''' is an index number used to uniquely identify a published {{WP|Book|book}}. === ISBN-$10$ === {{:Definition:International Standard Book Number/ISBN-10}} === ISBN-$13$ === {{:Definition:International Standard Book Number/ISBN-13}}"}
+{"_id": "29817", "title": "Definition:International Standard Book Number/ISBN-10", "text": "The '''ISBN-$10$''' format consists of a string of $9$ digits followed by a check digit in the set $\\set {0, 1, \\ldots, 9, X}$. The check digit $d$ is calculated as follows. Let $c$ be calculated as: :$c = \\displaystyle \\sum_{k \\mathop = 1}^9 k d_k \\pmod {11}$ where $d_k$ denotes the $k$th digit in the '''ISBN-$10$'''. Then the check digit $d$ is defined as: :$d = \\begin{cases} c & : c \\le 9 \\\\ X &: c = 10 \\end{cases}$"}
+{"_id": "29818", "title": "Definition:International Standard Book Number/ISBN-13", "text": "The '''ISBN-$13$''' format consists of a string of $12$ digits followed by a check digit. The check digit $d$ is calculated as: :$d = \\displaystyle \\sum_{k \\mathop = 1}^{12} k d_k \\pmod {10}$"}
+{"_id": "29820", "title": "Definition:International Standard Book Number/Linguistic Note", "text": "When referring to an '''international standard book number''' by its abbreviation '''ISBN''', beware not to vocalise it as '''\"ISBN number\"'''. The word '''number''' is already included in the abbreviation '''ISBN''', so does not need to be repeated."}
+{"_id": "29821", "title": "Definition:Check Digit", "text": "A '''check digit''' is a digit or other symbol used as an aid to identifying whether a mistake has been made transmitting a codeword. It is calculated from the symbols in the codeword itself, and appended to the codeword as an extra symbol. If a mistake is made transmitting the codeword, the '''check digit''' will then be incompatible with the rest of the contents of the codeword, and reveal that a transmission error is present."}
+{"_id": "29822", "title": "Definition:Error-Correcting Code", "text": "An '''error-correcting code''' is a technique whereby a transmitted codeword contains extra characters whose purpose is to detect whether the codeword has been corrupted during transmission."}
+{"_id": "29823", "title": "Definition:Linear Code", "text": "Let $p$ be a prime number. Let $\\Z_p$ be the set of residue classes modulo $p$. Let $\\map V {n, p}$ denote the set of sequences of length $n$ of elements of $\\Z_p$. A '''linear $\\tuple {n, k}$-code''' is a $k$-dimensional subspace $C$ of $\\map V {n, p}$ considered as a vector space over $\\Z_p$ of $n$ dimensions."}
+{"_id": "29824", "title": "Definition:Addition of Codewords in Linear Code", "text": "Let $\\map V {n, p}$ denote the linear $\\tuple {n, n}$-code modulo $p$. The operation of '''addition''' on $\\map V {n, p}$ is defined as follows. Let $A$ and $B$ be elements of $\\map V {n, p}$, that is, sequences of length $n$ of residue classes modulo $p$. Let $a_k$ and $b_k$ denote the $k$th term of $A$ and $B$ respectively. Then $C = A + B$ is the sequence of length $n$ of residue classes modulo $p$ whose $k$th term $c_k$ is defined as: :$c_k : = a_k +_p b_k$ where $+_p$ denotes the operation of addition modulo $p$."}
+{"_id": "29826", "title": "Definition:Linear Code/Master Code", "text": "$\\map V {n, p}$ is itself a '''linear $\\tuple {n, n}$-code''', and is referred to on {{ProofWiki}} as a '''master code''' (for a '''linear $\\tuple {n, k}$-code''' over $\\Z_p$)."}
+{"_id": "29827", "title": "Definition:Multiple of Codeword in Linear Code", "text": "Let $\\map V {n, p}$ denote the linear $\\tuple {n, n}$-code modulo $p$. The operation of '''multiplication''' on $\\map V {n, p}$ is defined as follows. Let $A$ be an elements of $\\map V {n, p}$, that is, a sequence of length $n$ of residue classes modulo $p$. Let $a_k$ denote the $k$th term of $A$. Let $c \\in \\Z_p$ be an element of the set $\\Z_p$ of residue classes modulo $p$. Then $c A$ is the sequence of length $n$ of residue classes modulo $p$ whose $k$th term $c_k$ is defined as: :$c_k : = c \\times_p a_k$ where $\\times_p$ denotes the operation of multiplication modulo $p$."}
+{"_id": "29828", "title": "Definition:Codeword", "text": "=== Codeword of Linear Code === {{:Definition:Linear Code/Codeword}}"}
+{"_id": "29829", "title": "Definition:Linear Code/Codeword", "text": "The elements of $\\map V {n, p}$ are referred to as '''codewords'''. A '''codeword''' is usually written without punctuation, so that, for example: :$\\tuple {\\eqclass 1 p, \\eqclass 0 p, \\eqclass 1 p}$ is presented as: :$101$"}
+{"_id": "29830", "title": "Definition:Weight of Linear Codeword", "text": "Let $C$ be a codeword of a linear code. The '''weight''' of $C$ is the number of non-zero terms of $C$."}
+{"_id": "29831", "title": "Definition:Distance between Linear Codewords", "text": "Let $u$ and $v$ be two codewords of a linear code. The '''distance''' between $u$ and $v$ is the number of corresponding terms at which $u$ and $v$ are different."}
+{"_id": "29832", "title": "Definition:Minimum Distance of Linear Code", "text": "Let $C$ be a linear code whose master code is $\\map V {n, p}$. The '''minimum distance $d$ of $C$''' is defined as: :$\\map d C := \\displaystyle \\min_{u, v \\mathop \\in C: u \\mathop \\ne v} \\set {\\map d {u, v} }$ where $\\map d {u, v}$ denotes the distance between $u$ and $v$."}
+{"_id": "29834", "title": "Definition:Standard Generator Matrix for Linear Code", "text": "Let $n, k \\in \\Z_{>0}$ be strictly positive integers such that $n > k$. Let $p$ be a prime number. Let $\\Z_p$ denote the set of residue classes modulo $p$. A '''(standard) generator matrix''' $G$ over $\\Z_p$ is a $k \\times n$ matrix such that: :The elements of $G$ are elements of $\\Z_p$ :The first $k$ columns form the $k \\times k$ identity matrix."}
+{"_id": "29835", "title": "Definition:Golay Ternary Code", "text": "The '''Golay ternary code''' is the linear $\\tuple {11, 6}$ code over $\\Z_3$ whose standard generator matrix $G$ is given by: :$G := \\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 1 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 2 & 1 & 0 & 1 & 2 \\\\ 0 & 0 & 0 & 1 & 0 & 0 & 2 & 2 & 1 & 0 & 1 \\\\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 & 1 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1  \\end{pmatrix}$"}
+{"_id": "29836", "title": "Definition:Transmission Error", "text": "Let $c$ be a transmitted codeword of a message sent over a transmission line $T$. Let $c'$ be the corresponding word received across $T$. Each term of $c'$ may or may not match the corresponding term of $c$. Each term of $c'$ which does ''not'' match the corresponding term of $c$ is referred to as an instance of a '''transmission error'''. Category:Definitions/Information Theory hx64fvv2g0bag2ytqohht7jnyphtbmj"}
+{"_id": "29837", "title": "Definition:Transmitted Codeword", "text": "Let $M$ be a message sent over a transmission line $T$. Let $M$ consist of a number of codewords. Each of the codewords comprising $M$, as it is before it has been transmitted across $T$, is referred to as a '''transmitted codeword'''."}
+{"_id": "29838", "title": "Definition:Received Word", "text": "Let $M$ be a message sent over a transmission line $T$. Let $M$ consist of a number of codewords. Each of the codewords comprising $M$, as it is after it has been transmitted across $T$, is referred to as a '''received word'''. Note that if a '''received word''' $c'$ does not match its corresponding transmitted codeword $c$, then it is possible that $c'$ is not actually a codeword. Category:Definitions/Information Theory 6t0pb7pjlbka8zk4gzbsktw21q1p1eb"}
+{"_id": "29839", "title": "Definition:Trillion", "text": "=== Short Scale === {{:Definition:Trillion/Short Scale}} === Long Scale === {{:Definition:Trillion/Long Scale}}"}
+{"_id": "29840", "title": "Definition:Trillion/Short Scale", "text": "'''Trillion''' is a name for $10^{12}$ in the short scale system: :'''One trillion''' $= 10^{3 \\times 3 + 3}$"}
+{"_id": "29841", "title": "Definition:Trillion/Long Scale", "text": "'''Trillion''' is a name for $10^{18}$ in the long scale system: :'''One trillion''' $= 10^{6 \\times 3}$"}
+{"_id": "29842", "title": "Definition:Quadrillion", "text": "=== Short Scale === {{:Definition:Quadrillion/Short Scale}} === Long Scale === {{:Definition:Quadrillion/Long Scale}}"}
+{"_id": "29843", "title": "Definition:Quadrillion/Long Scale", "text": "'''Quadrillion''' is a name for $10^{24}$ in the long scale system: :'''One quadrillion''' $= 10^{6 \\times 4}$"}
+{"_id": "29844", "title": "Definition:Quadrillion/Short Scale", "text": "'''Quadrillion''' is a name for $10^{15}$ in the short scale system: :'''One quadrillion''' $= 10^{3 \\times 4 + 3}$"}
+{"_id": "29846", "title": "Definition:Quintillion/Short Scale", "text": "'''Quintillion''' is a name for $10^{18}$ in the short scale system: :'''One quintillion''' $= 10^{3 \\times 5 + 3}$"}
+{"_id": "29847", "title": "Definition:Quintillion/Long Scale", "text": "'''Quintillion''' is a name for $10^{30}$ in the long scale system: :'''One quintillion''' $= 10^{6 \\times 5}$"}
+{"_id": "29848", "title": "Definition:Milliard", "text": "'''Milliard''' is an archaic name for $10^9$."}
+{"_id": "29849", "title": "Definition:Billiard", "text": "'''Billiard''' is an archaic name for $10^{15}$."}
+{"_id": "29850", "title": "Definition:Zero Codeword", "text": "A '''zero codeword''' is a codeword whose terms are all zero. Category:Definitions/Information Theory eppxi3htxndryrm50ptdbnt07osvxvj"}
+{"_id": "29851", "title": "Definition:Coset Decoding Table", "text": "A '''coset decoding table''' is a technique for decoding a linear $\\tuple {n, k}$ code. Let $C$ be a linear $\\tuple {n, k}$ code whose master code is $\\map V {n, p}$. Let $T$ be an array constructed as follows: :The first row consists of the codewords of $C$, starting with the zero codeword first. {{explain|It has not yet been rigorously proved that $C$ does actually contain the zero codeword.}} :Each subsequent row is a left coset of $C$. :The entries of the first column of $T$ are the coset representatives, now called '''coset leaders'''. :The $r$th coset leader is allocated by choosing any element of $\\map V {n, p}$ of minimum weight which is not already included in the first $r - 1$ rows. Then $T$ is a '''coset decoding table'''. Note that it may not always be easy to find the $r$th coset leader."}
+{"_id": "29852", "title": "Definition:Standard Parity Check Matrix", "text": "Let $C$ be a linear $\\tuple {n, k}$ code whose master code is $\\map V {n, p}$. Let $G$ be the $k \\times n$ standard generator matrix of $C$: :$G = \\paren {\\begin{array} {c|c} \\mathbf I_k & \\mathbf A \\end{array} }$ where: :$\\mathbf I$ denotes the identity matrix of order $k$ :$\\mathbf A$ denotes some $k \\times \\paren {n - k}$ matrix. The '''(standard) parity check matrix associated with $G$''' is the $\\paren {n - k} \\times n$ matrix: :$P = \\paren {\\begin{array} {c|c} -\\mathbf A^\\intercal & \\mathbf I_{n - k} \\end{array} }$ where: :$\\mathbf A^\\intercal$ denotes the transpose of $\\mathbf A$ :$\\mathbf I_{n - k}$ denotes the identity matrix of order $n - k$ :the $-$ sign before $\\mathbf A^\\intercal$ denotes that each of the elements of $\\mathbf A$ is to be replaced with its inverse element in $\\Z_p$, the additive group of integers modulo $p$."}
+{"_id": "29853", "title": "Definition:Trivial Norm/Division Ring/Nontrivial", "text": "A norm $\\norm {\\, \\cdot \\,}$ on a division ring $R$ is '''nontrivial''' {{iff}} it is not trivial."}
+{"_id": "29858", "title": "Definition:Antihomomorphism/Group Antihomomorphism", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be groups. Then $\\phi: S \\to T$ is a '''group antihomomorphism''' {{iff}}: :$\\forall x, y \\in S:\\map \\phi {x \\circ y} = \\map \\phi y * \\map \\phi x$"}
+{"_id": "29859", "title": "Definition:Antihomomorphism/Field Antihomomorphism", "text": "Let $\\struct {R, +, \\circ}$ and $\\struct {S, \\oplus, *}$ be fields. Then a ring antihomomorphism $\\phi: \\struct {R, +, \\circ} \\to \\struct {S, \\oplus, *}$ is called a '''field antihomomorphism'''."}
+{"_id": "29860", "title": "Definition:Absolute Galois Group/Definition 1", "text": "The '''absolute Galois group''' of $K$ is the Galois group $\\Gal {K^{\\operatorname{sep} } \\mid K}$ of its separable closure."}
+{"_id": "29861", "title": "Definition:Absolute Galois Group/Definition 2", "text": "The '''absolute Galois group''' of $K$ is the automorphism group $\\Aut {\\overline K \\mid K}$ of its algebraic closure."}
+{"_id": "29862", "title": "Definition:Acceleration/Dimension", "text": "'''Acceleration''' has dimension $\\mathsf {L T}^{-2}$."}
+{"_id": "29863", "title": "Definition:Acceleration/Units", "text": "=== SI === {{:Definition:Acceleration/Units/SI}} === CGS === {{:Definition:Acceleration/Units/CGS}} === FPS === {{:Definition:Acceleration/Units/FPS}}"}
+{"_id": "29872", "title": "Definition:Group Action/Left Group Action", "text": "A '''(left) group action''' is an operation $\\phi: G \\times X \\to X$ such that: :$\\forall \\tuple {g, x} \\in G \\times X: g * x := \\map \\phi {g, x} \\in X$ in such a way that the group action axioms are satisfied: {{:Definition:Group Action Axioms}}"}
+{"_id": "29874", "title": "Definition:Orbit (Group Theory)/Definition 1", "text": "The '''orbit''' of an element $x \\in X$ is defined as: :$\\Orb x := \\set {y \\in X: \\exists g \\in G: y = g * x}$ where $*$ denotes the group action. That is, $\\Orb x = G * x$. Thus the '''orbit''' of an element is all its possible destinations under the group action."}
+{"_id": "29875", "title": "Definition:Orbit (Group Theory)/Definition 2", "text": "Let $\\mathcal R$ be the relation on $X$ defined as: :$\\forall x, y \\in X: x \\mathrel {\\mathcal R} y \\iff \\exists g \\in G: y = g * x$ where $*$ denotes the group action. From Group Action Induces Equivalence Relation, $\\mathcal R$ is an equivalence relation. The '''orbit''' of $x$, denoted $\\Orb x$, is the equivalence class of $x$ under $\\mathcal R$."}
+{"_id": "29879", "title": "Definition:Gazillion", "text": "'''Gazillion''' is a rhetorical flourish to mean '''a very large number''', generally understood to be of the order of an unspecified power of $10$."}
+{"_id": "29880", "title": "Definition:Internal Group Direct Product/Definition 1", "text": "The group $\\struct {G, \\circ}$ is the '''internal group direct product of $H_1$ and $H_2$''' {{iff}} the mapping: :$C: H_1 \\times H_2 \\to G: \\map C {h_1, h_2} = h_1 \\circ h_2$ is a group isomorphism from the (group) direct product $\\struct {H_1, \\circ {\\restriction_{H_1} } } \\times \\struct {H_2, \\circ {\\restriction_{H_2} } }$ onto $\\struct {G, \\circ}$."}
+{"_id": "29881", "title": "Definition:Internal Group Direct Product/Definition 2", "text": "The group $\\struct {G, \\circ}$ is the '''internal group direct product of $H_1$ and $H_2$''' {{iff}}: :$(1): \\quad \\struct {H_1, \\circ {\\restriction_{H_1} } }$ and $\\struct {H_2, \\circ {\\restriction_{H_2} } }$ are both normal subgroups of $\\struct {G, \\circ}$ :$(2): \\quad$ every element of $G$ can be expressed uniquely in the form: ::::$g = h_1 \\circ h_2$"}
+{"_id": "29883", "title": "Definition:Internal Group Direct Product/General Definition/Definition 1", "text": "The group $\\struct {G, \\circ}$ is the '''internal group direct product of $\\sequence {H_n}$''' {{iff}} the mapping: :$\\displaystyle C: \\prod_{k \\mathop = 1}^n H_k \\to G: \\map C {h_1, \\ldots, h_n} = \\prod_{k \\mathop = 1}^n h_k$ is a group isomorphism from the group direct product $\\struct {H_1, \\circ {\\restriction_{H_1} } } \\times \\cdots \\times \\struct {H_n, \\circ {\\restriction_{H_n} } }$ onto $\\struct {G, \\circ}$."}
+{"_id": "29884", "title": "Definition:Internal Group Direct Product/General Definition/Definition 2", "text": ":$(1): \\quad$ Each $H_1, H_2, \\ldots, H_n$ is a normal subgroup of $G$ :$(2): \\quad$ Each element $g$ of $G$ can be expressed uniquely in the form: ::::$g = h_1 \\circ h_2 \\circ \\cdots \\circ h_n$ :::where $h_i \\in H_i$ for all $i \\in \\set {1, 2, \\ldots, n}$."}
+{"_id": "29885", "title": "Definition:Internal Group Direct Product/Definition 3", "text": "The group $\\struct {G, \\circ}$ is the '''internal group direct product of $H_1$ and $H_2$''' {{iff}}: :$(1): \\quad \\struct {H_1, \\circ {\\restriction_{H_1} } }$ and $\\struct {H_2, \\circ {\\restriction_{H_2} } }$ are both normal subgroups of $\\struct {G, \\circ}$ :$(2): \\quad G$ is the subset product of $H_1$ and $H_2$, that is: $G = H_1 \\circ H_2$ :$(3): \\quad$ $H_1 \\cap H_2 = \\set e$ where $e$ is the identity element of $G$."}
+{"_id": "29888", "title": "Definition:Even Integer/Definition 1", "text": "An integer $n \\in \\Z$ is '''even''' {{iff}} it is divisible by $2$."}
+{"_id": "29889", "title": "Definition:Even Integer/Definition 3", "text": "An integer $n \\in \\Z$ is '''even''' {{iff}}: :$x \\equiv 0 \\pmod 2$ where the notation denotes congruence modulo $2$."}
+{"_id": "29890", "title": "Definition:Even Integer/Definition 2", "text": "An integer $n \\in \\Z$ is '''even''' {{iff}} it is of the form: :$n = 2 r$ where $r \\in \\Z$ is an integer."}
+{"_id": "29893", "title": "Definition:Odd Integer/Definition 3", "text": "An integer $n \\in \\Z$ is '''odd''' {{iff}}: :$x \\equiv 1 \\pmod 2$ where the notation denotes congruence modulo $2$."}
+{"_id": "29894", "title": "Definition:Balanced Ternary Representation", "text": "Let $n \\in \\Z$ be an integer. '''Balanced ternary representation''' is the unique representation of $n$ in the form: :$\\sqbrk {r_m r_{m - 1} \\ldots r_2 r_1 r_0}$ such that: :$\\displaystyle n = \\sum_{j \\mathop = 0}^m r_j 3^j$ where: :$m \\in \\Z_{>0}$ is a strictly positive integer such that $3^m \\le \\size n < 3^{m + 1}$ :all the $r_j$ are such that $r_j \\in \\set {\\underline 1, 0, 1}$, where $\\underline 1 := -1$."}
+{"_id": "29895", "title": "Definition:Integral Ideal", "text": "Let $J \\subseteq \\Z$ be a non-empty subset of the set of integers. Let $J$ fulfil the following conditions: :$(1): \\quad n, m \\in J \\implies m + n \\in J, m - n \\in J$ :$(2): \\quad n \\in J, r \\in \\Z \\implies r n \\in J$ Then $J$ is an '''integral ideal'''."}
+{"_id": "29896", "title": "Definition:Greatest Common Divisor/Integers/Definition 1", "text": "The '''greatest common divisor of $a$ and $b$''' is defined as: :the largest $d \\in \\Z_{>0}$ such that $d \\divides a$ and $d \\divides b$"}
+{"_id": "29897", "title": "Definition:Greatest Common Divisor/Integers/Definition 2", "text": "The '''greatest common divisor of $a$ and $b$''' is defined as the (strictly) positive integer $d \\in \\Z_{>0}$ such that: :$(1): \\quad d \\divides a \\land d \\divides b$ :$(2): \\quad c \\divides a \\land c \\divides b \\implies c \\divides d$"}
+{"_id": "29898", "title": "Definition:Lowest Common Multiple/Also known as", "text": "The '''lowest common multiple''' is also known as the '''least common multiple'''. It is usually abbreviated '''LCM''', '''lcm''' or '''l.c.m.''' The notation $\\lcm \\set {a, b}$ can be found written as $\\sqbrk {a, b}$. This usage is not recommended as it can cause confusion."}
+{"_id": "29899", "title": "Definition:Open Ball/Normed Division Ring", "text": "Let $\\struct{R, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Let $a \\in R$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. The '''open $\\epsilon$-ball of $a$ in $\\struct {R, \\norm {\\,\\cdot\\,} }$''' is defined as: :$\\map {B_\\epsilon} a = \\set {x \\in R: \\norm{x - a} < \\epsilon}$ If it is necessary to show the norm itself, then the notation $\\map {B_\\epsilon} {a; \\norm {\\,\\cdot\\,} }$ can be used."}
+{"_id": "29900", "title": "Definition:Open Ball/Normed Division Ring/Radius", "text": "In $\\map {B_\\epsilon} a$, the value $\\epsilon$ is referred to as the '''radius''' of the open $\\epsilon$-ball."}
+{"_id": "29901", "title": "Definition:Open Ball/Normed Division Ring/Center", "text": "In $\\map {B_\\epsilon} a$, the value $a$ is referred to as the '''center''' of the open $\\epsilon$-ball."}
+{"_id": "29902", "title": "Definition:Closed Ball/Normed Division Ring", "text": "Let $\\struct{R, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Let $a \\in R$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. The '''closed $\\epsilon$-ball of $a$ in $\\struct {R, \\norm {\\,\\cdot\\,} }$''' is defined as: :$\\map { {B_\\epsilon}^-} a = \\set {x \\in R: \\norm {x - a} \\le \\epsilon}$ If it is necessary to show the norm itself, then the notation $\\map { {B_\\epsilon}^-} {a; \\norm {\\,\\cdot\\,} }$ can be used."}
+{"_id": "29903", "title": "Definition:Closed Ball/Normed Division Ring/Radius", "text": "In $\\map { {B_\\epsilon}^-} a$, the value $\\epsilon$ is referred to as the '''radius''' of the closed $\\epsilon$-ball."}
+{"_id": "29904", "title": "Definition:Closed Ball/Normed Division Ring/Center", "text": "In $\\map { {B_\\epsilon}^-} a$, the value $a$ is referred to as the '''center''' of the closed $\\epsilon$-ball."}
+{"_id": "29905", "title": "Definition:Lattice Point", "text": "Let $\\CC$ be a cartesian coordinate system of $n$ dimensions. Let $P = \\tuple {a_1, a_2, \\ldots, a_n}$ be a point in $\\CC$. Then $P$ is a '''lattice point''' of $\\CC$ {{iff}} $a_1, a_2, \\ldots, a_n$ are all integers. === Rational Lattice Point === {{:Definition:Lattice Point/Rational}}"}
+{"_id": "29906", "title": "Definition:Intercept", "text": "Let $\\mathcal C$ be a curve embedded in a cartesian plane. An '''intercept''' of $\\mathcal C$ is a point where $\\mathcal C$ intersects one of the coordinate axes. In particular: === $x$-Intercept === {{:Definition:Intercept/X-Intercept}} === $y$-Intercept === {{:Definition:Intercept/Y-Intercept}}"}
+{"_id": "29907", "title": "Definition:Intercept/X-Intercept", "text": "An '''$x$-intercept''' of $\\mathcal C$ is a point where $\\mathcal C$ intersects the $x$-axis."}
+{"_id": "29908", "title": "Definition:Intercept/Y-Intercept", "text": "A '''$y$-intercept''' of $\\mathcal C$ is a point where $\\mathcal C$ intersects the $y$-axis."}
+{"_id": "29909", "title": "Definition:Perpendicular Distance", "text": "Let $\\mathcal L$ be a straight line. Let $P$ be a point not on $\\mathcal L$. Let a perpendicular be dropped from $P$ to $\\mathcal L$, intersecting $\\mathcal L$ at $A$. The '''perpendicular distance''' from $P$ to $\\mathcal L$ is defined as being the length of the line $AP$. :300px"}
+{"_id": "29910", "title": "Definition:Lowest Common Multiple/Integers/General Definition", "text": "Let $S = \\set {a_1, a_2, \\ldots, a_n} \\subseteq \\Z$ such that $\\displaystyle \\prod_{a \\mathop \\in S} a = 0$ (that is, all elements of $S$ are non-zero). Then the '''lowest common multiple''' of $S$: :$\\lcm \\paren S = \\lcm \\set {a_1, a_2, \\ldots, a_n}$ is defined as the smallest $m \\in \\Z_{>0}$ such that: :$\\forall x \\in S: x \\divides m$"}
+{"_id": "29911", "title": "Definition:Sphere/Metric Space", "text": "Let $M = \\struct{A, d}$ be a metric space or pseudometric space. Let $a \\in A$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. The '''$\\epsilon$-sphere of $a$ in $M$''' is defined as: :$\\map {S_\\epsilon} a = \\set {x \\in A: \\map d {x, a} = \\epsilon}$"}
+{"_id": "29912", "title": "Definition:Sphere/Metric Space/Center", "text": "In $S_\\epsilon \\paren{a}$, the value $a$ is referred to as the '''center''' of the $\\epsilon$-sphere."}
+{"_id": "29913", "title": "Definition:Sphere/Metric Space/Radius", "text": "In $S_\\epsilon \\paren{a}$, the value $\\epsilon$ is referred to as the '''radius''' of the $\\epsilon$-sphere."}
+{"_id": "29914", "title": "Definition:Sphere/Normed Division Ring", "text": "Let $\\struct{R, \\norm{\\,\\cdot\\,}}$ be a normed division ring. Let $a \\in R$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. The '''$\\epsilon$-sphere of $a$ in $\\struct{R, \\norm{\\,\\cdot\\,}}$''' is defined as: :$S_\\epsilon \\paren{a} = \\set {x \\in R: \\norm{x - a} = \\epsilon}$"}
+{"_id": "29915", "title": "Definition:Sphere/Normed Division Ring/Radius", "text": "In $\\map {S_\\epsilon} a$, the value $\\epsilon$ is referred to as the '''radius''' of the $\\epsilon$-sphere."}
+{"_id": "29916", "title": "Definition:Sphere/Normed Division Ring/Center", "text": "In $\\map {S_\\epsilon} a$, the value $a$ is referred to as the '''center''' of the $\\epsilon$-sphere."}
+{"_id": "29917", "title": "Definition:Stellation/Polygon/Regular", "text": "A '''regular stellated polygon''' is a stellated polygon which is both equilateral and equiangular. That is, in which all the sides are the same length, and all the vertices have the same angle: :300px"}
+{"_id": "29918", "title": "Definition:Triskaidecagon", "text": "A '''triskaidecagon''' is a polygon with exactly $13$ sides. :300px === Regular Triskaidecagon === {{:Definition:Triskaidecagon/Regular}}"}
+{"_id": "29922", "title": "Definition:Topology Induced by Division Ring Norm", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring whose norm is $\\norm {\\,\\cdot\\,}$. The '''topology on $\\struct {R, \\norm {\\,\\cdot\\,} }$ induced by $\\norm {\\,\\cdot\\,}$''' is defined as the topology $\\tau$ generated by the basis consisting of the set of all open balls of $\\struct {R, \\norm {\\,\\cdot\\,} }$. Let $d$ be the metric induced by the norm $\\norm {\\,\\cdot\\,}$. By the definition of the metric induced by a norm on division ring it follows that the '''topology on the normed division ring $\\struct{R, \\norm{\\,\\cdot\\,}}$ induced by (the norm) $\\norm{\\,\\cdot\\,}$''' is the topology induced by the metric d."}
+{"_id": "29923", "title": "Definition:Complete Residue System", "text": "Let $m \\in \\Z_{\\ne 0}$ be a non-zero integer. Let $S := \\set {r_1, r_2, \\dotsb, r_s}$ be a set of integers with the properties that: :$(1): \\quad i \\ne j \\implies r_i \\not \\equiv r_j \\pmod m$ :$(2): \\quad \\forall n \\in \\Z: \\exists r_i \\in S: n \\equiv r_i \\pmod m$ Then $S$ is a '''complete residue system modulo $m$'''."}
+{"_id": "29924", "title": "Definition:Deck of Cards", "text": "A '''deck of cards''' is a set of pieces of (traditionally) cardboard with a design on one side uniquely identifying the card, and on the other side a single design which is exactly the same for all cards in the '''deck'''. === Card === {{:Definition:Deck of Cards/Card}} === Suit === {{:Definition:Deck of Cards/Suit}}"}
+{"_id": "29925", "title": "Definition:Deck of Cards/Card", "text": "Each of the elements of a deck of cards is a '''card'''."}
+{"_id": "29926", "title": "Definition:Deck of Cards/Suit", "text": "The cards in a deck of cards are partitioned into a number of '''suits'''. The traditional '''deck of cards''' that is usually discussed in the context of probability theory is divided into $4$ '''suits''' of $13$ cards each. There may also be a small number (usually between $1$ and $3$) of extra cards, usually called '''jokers''', which do not belong to any '''suit'''."}
+{"_id": "29927", "title": "Definition:Shuffle", "text": "A '''shuffle''' is an operation on a deck of cards which changes the order of the cards in the deck. Category:Definitions/Cards qljox4ecvgily80liwz5ry5pm6nkhr4"}
+{"_id": "29928", "title": "Definition:Order of Cards", "text": "Let $D$ be a deck of cards. The cards in the deck can be considered to be stacked uniformly, usually all in the same orientation, with a card on the top and a card on the bottom. The '''order''' of the cards is the total ordering $\\prec$ defined as: :$a \\prec b$ {{iff}} $a$ is somewhere underneath $b$ Category:Definitions/Cards kn84y5p05gjai83ol0n83qrxiewm53o"}
+{"_id": "29929", "title": "Definition:Modified Perfect Faro Shuffle", "text": "A '''modified perfect Faro shuffle''' is a shuffle performed on a deck of cards $D$ of size $2 n$. Cut $D$ into $2$ piles $D_1$ (the top half) and $D_2$ (the bottom half) of exactly $n$ cards each. The shuffled deck $D'$ is assembled by alternating the cards in $D$: :The cards in positions $2, 4, 6, \\ldots, 2 n$ of $D'$ are to be occupied by the cards $1, 2, \\ldots, n$ of $D_1$ in the same order :The cards in positions $1, 3, 5, \\ldots, 2 n - 1$ of $D'$ are to be occupied by the cards $1, 2, \\ldots, n$ of $D_2$ in the same order. Thus a card in position $k$ in $D$ ends up in position $2 k \\bmod \\paren {2 n + 1}$ in $D'$."}
+{"_id": "29931", "title": "Definition:Fifth Root/Real", "text": "Let $x \\in \\R_{\\ge 0}$ be a positive real number. The '''fifth root of $x$''' is the real number defined as: :$x^{\\paren {1 / 5} } := \\set {y \\in \\R: y^5 = x}$ where $x^{\\paren {1 / 5} }$ is the $5$th root of $x$. The notation: :$y = \\sqrt [5] x$ is usually encountered."}
+{"_id": "29934", "title": "Definition:Affine Space/Weyl's Axioms", "text": "Let $K$ be a field. Let $\\struct{V, +_V, \\circ}$ be a vector space over $K$. Let $\\EE$ be a set on which a mapping is defined: :$- : \\EE \\times \\EE \\to V$ satisfying the following associativity conditions: {{begin-axiom}} {{axiom | n = \\text W 1         | q = \\forall p \\in \\EE: \\forall v \\in V: \\exists ! q \\in \\EE         | m = v = q - p }} {{axiom | n = \\text W 2         | q = \\forall p, q, r \\in \\EE         | m = \\paren{r - q} +_V \\paren{q - p} = r - p }} {{end-axiom}} Then the ordered pair $\\tuple {\\EE, -}$ is an '''affine space'''."}
+{"_id": "29935", "title": "Definition:Affine Space/Group Action", "text": "Let $K$ be a field. Let $\\left({V, +_V, \\circ}\\right)$ be a vector space over $K$. Let $\\mathcal E$ be a set. Let $\\phi: \\mathcal E \\times V \\to \\mathcal E$ be a free and transitive group action of $\\struct{V, +_V}$ on $\\mathcal E$. Then the ordered pair $\\tuple{\\mathcal E, \\phi}$ is an '''affine space'''."}
+{"_id": "29936", "title": "Definition:Affine Space/Associativity Axioms", "text": "Let $K$ be a field. Let $\\struct {V, +_V, \\circ}$ be a vector space over $K$. Let $\\EE$ be a set on which two mappings are defined: :$+ : \\EE \\times V \\to \\EE$ :$- : \\EE \\times \\EE \\to V$ satisfying the following associativity conditions: {{begin-axiom}} {{axiom | n = \\text A 1         | q = \\forall p, q \\in \\EE         | m = p + \\paren {q - p} = q }} {{axiom | n = \\text A 2         | q = \\forall p \\in \\EE: \\forall u, v \\in V         | m = \\paren {p + u} + v = p + \\paren {u +_V v} }} {{axiom | n = \\text A 3         | q = \\forall p, q \\in \\EE: \\forall u \\in V         | m = \\paren {p - q} +_V u = \\paren {p + u} - q }} {{end-axiom}} Then the ordered triple $\\struct {\\EE, +, -}$ is an '''affine space'''."}
+{"_id": "29937", "title": "Definition:Tangent Space (Affine Geometry)", "text": "Let $\\tuple {\\EE, +, -}$ be an affine space. Let $V$ be the vector space that is the codomain of $-$. Then $V$ is the '''tangent space''' of $\\EE$."}
+{"_id": "29938", "title": "Definition:Vector (Affine Geometry)", "text": "Let $\\EE$ be an affine space. Let $V$ be the tangent space of $\\EE$. Any element $v$ of $V$ is called a '''vector'''."}
+{"_id": "29940", "title": "Definition:Subtraction (Affine Geometry)", "text": "Let $\\tuple {\\EE, +, -}$ be an affine space. Then the mapping $-$ is called '''affine subtraction'''."}
+{"_id": "29941", "title": "Definition:Addition (Affine Geometry)", "text": "Let $\\tuple {\\EE, +, -}$ be an affine space. Then the mapping $+$ is called '''affine addition'''."}
+{"_id": "29942", "title": "Definition:Addition/Sum", "text": "Let $a + b$ denote the operation of addition on two objects $a$ and $b$. Then the result $a + b$ is referred to as the '''sum''' of $a$ and $b$."}
+{"_id": "29944", "title": "Definition:Multiplication/Notation", "text": "There are several variants of the notation for '''multiplication''': :$n \\times m$ This is usually used when numbers are under consideration, for example: $3 \\times 5 = 15$. However, it can be used in the context of algebra where extra clarity is needed. :$n m$ This is most common in algebra, but not with numbers, as it is difficult to make it obvious where one number ends and the next number begins. :$\\paren n \\paren m$ This form can be used for either symbols denoting variables or numbers, for example: $\\paren 3 \\paren 4 = 12$. :$n \\cdot m$ or $n . m$ These have their uses in algebra, but the dot has the danger of being confused with the decimal point when used for numbers. :$n * m$ This notation specifically evolved in the field of computer science, but can occasionally be seen encroaching into mathematics. Its use is not recommended, as it can be confused with other operations that use the same or similar notation, for example convolution."}
+{"_id": "29945", "title": "Definition:Multiplication/Multiplicand", "text": "Let $a \\times b$ denote the operation of multiplication on two objects. The object $b$ is known as the '''multiplicand of $a$'''. That is, it is the object which is to be multiplied by the multiplier."}
+{"_id": "29946", "title": "Definition:Multiplication/Multiplier", "text": "Let $a \\times b$ denote the operation of multiplication on two objects. The object $a$ is known as the '''multiplier of $b$'''. That is, it is the object which is to multiply the multiplicand."}
+{"_id": "29947", "title": "Definition:Multiplication/Product", "text": "Let $a \\times b$ denote the operation of multiplication on two objects $a$ and $b$. Then the result $a \\times b$ is referred to as the '''product''' of $a$ and $b$."}
+{"_id": "29950", "title": "Definition:Real Number/Real Number Line/Origin", "text": "The point representing the number $0$ (zero) is referred to as the '''origin'''."}
+{"_id": "29952", "title": "Definition:Variable/Real", "text": "A '''real variable''' is a symbol which can stand for any one of a set of real numbers."}
+{"_id": "29954", "title": "Definition:Complex Number/Definition 2/Scalar Product", "text": "Let $\\tuple {x, y}$ be a complex numbers. Let $m \\in \\R$ be a real number. Then $m \\tuple {x, y}$ is defined as: :$m \\tuple {x, y} := \\tuple {m x, m y}$"}
+{"_id": "29955", "title": "Definition:Complex Conjugate Coordinates", "text": "Let $P$ be a point in the complex plane. $P$ may be located using '''complex conjugate coordinates''' $\\tuple {z, \\overline z}$ based on: {{begin-eqn}} {{eqn | l = x       | r = \\dfrac {z + \\overline z} 2       | c = Sum of Complex Number with Conjugate }} {{eqn | l = y       | r = \\dfrac {z - \\overline z} {2 i}       | c = Difference of Complex Number with Conjugate }} {{end-eqn}} where $P = \\tuple {x, y}$ is expressed in Cartesian coordinates."}
+{"_id": "29956", "title": "Definition:Connected Set (Complex Analysis)/Definition 1", "text": "$D$ is '''connected''' {{iff}} every pair of points in $D$ can be joined by a staircase contour."}
+{"_id": "29957", "title": "Definition:Connected Set (Complex Analysis)/Definition 2", "text": "$D$ is '''connected''' {{iff}} every pair of points in $D$ can be joined by a polygonal path all points of which are in $D$."}
+{"_id": "29958", "title": "Definition:Polygonal Path", "text": "Let $\\C$ denote the complex plane. A '''polygonal path''' in $\\C$ is a contour in $\\C$ which is composed of a finite number of straight line segments."}
+{"_id": "29959", "title": "Definition:Valuation Ring Induced by Non-Archimedean Norm", "text": "Let $\\struct {R, \\norm{\\,\\cdot\\,}}$ be a non-Archimedean normed division ring with zero $0_R$. The '''valuation ring induced by the non-Archimedean norm''' $\\norm{\\,\\cdot\\,}$ is the set: :$\\mathcal O = \\set {x \\in R: \\norm x \\le 1}$ That is, the '''valuation ring induced by the non-Archimedean norm''' $\\norm{\\,\\cdot\\,}$ is the closed ball ${B_1}^- \\paren {0_R}$."}
+{"_id": "29960", "title": "Definition:Valuation Ideal Induced by Non-Archimedean Norm", "text": "Let $\\struct {R, \\norm{\\,\\cdot\\,}}$ be a non-Archimedean normed division ring with zero $0_R$. The '''valuation ideal induced by the non-Archimedean norm''' $\\norm{\\,\\cdot\\,}$ is the set: :$\\mathcal P = \\set{x \\in R: \\norm{x} \\lt 1}$ That is, the '''valuation ideal induced by the non-Archimedean norm''' $\\norm{\\,\\cdot\\,}$ is the open ball $B_1 \\paren {0_R}$."}
+{"_id": "29961", "title": "Definition:Residue Division Ring Induced by Non-Archimedean Norm", "text": "Let $\\struct {R, \\norm {\\,\\cdot\\,}}$ be a non-Archimedean normed division ring. Let $\\OO$ be the valuation ring induced by the non-Archimedean norm $\\norm {\\,\\cdot\\,}$. Let $\\PP$ be the valuation ideal induced by the non-Archimedean norm $\\norm {\\,\\cdot\\,}$. The '''residue division ring induced by the norm''' $\\norm {\\,\\cdot\\,}$ is the quotient ring $\\OO / \\PP$. If $R$ is a field then the quotient ring $\\OO / \\PP$ is called the '''residue field induced by the norm''' $\\norm {\\,\\cdot\\,}$."}
+{"_id": "29962", "title": "Definition:P-adic Integers", "text": "Let $\\norm {\\,\\cdot\\,}_p$ be the $p$-adic norm on the $p$-adic numbers $\\Q_p$ for some prime $p$. The set of '''$p$-adic integers''', denoted $\\Z_p$, is the valuation ring induced by $\\norm {\\,\\cdot\\,}_p$, that is: :$\\Z_p = \\set {x \\in \\Q_p: \\norm x_p \\le 1}$"}
+{"_id": "29963", "title": "Definition:Coincident Straight Lines", "text": "Two straight lines are '''coincident''' {{iff}} they lie on top of each other. That is, two straight lines $L_1$ and $L_2$ are '''coincident''' {{iff}}: :$L_1 \\cap L_2 = L_1 = L_2$. Category:Definitions/Geometry qmrzpxwjm4xris3ziy02ig6puschszu"}
+{"_id": "29964", "title": "Definition:Equilibrant", "text": "Let $\\mathbf F_1, \\mathbf F_2, \\ldots, \\mathbf F_n$ be a set of $n$ forces acting on a particle $B$ at a point $P$ in space. The '''equilibrant''' of $\\mathbf F_1, \\mathbf F_2, \\ldots, \\mathbf F_n$ is defined as the force which is needed to prevent $B$ from moving."}
+{"_id": "29965", "title": "Definition:Lemniscate of Bernoulli/Major Axis", "text": "The line $P_1 P_2$ is the '''major axis''' of the lemniscate."}
+{"_id": "29966", "title": "Definition:Lemniscate of Bernoulli/Major Semiaxis", "text": "Each of the lines $O P_1$ and $O P_2$ is a '''major semiaxis''' of the lemniscate."}
+{"_id": "29969", "title": "Definition:Division Ring/Definition 2", "text": ":Every non-zero element of $R$ is a unit."}
+{"_id": "29971", "title": "Definition:Ring of Quaternions", "text": "The '''ring of quaternions''' $\\struct {\\H, +, \\times}$ is the set of quaternions under the two operations of quaternion addition and quaternion multiplication."}
+{"_id": "29972", "title": "Definition:Subfield/Field", "text": "Let $\\struct {F, +, \\circ}$ be a field. Let $K$ be a subset of $F$ such that $\\struct {K, +, \\circ}$ is also a field. Then $\\struct {K, +, \\circ}$ is a '''subfield''' of $\\struct {F, +, \\circ}$."}
+{"_id": "29973", "title": "Definition:Subfield/Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity. Let $K$ be a subset of $R$ such that $\\struct {K, +, \\circ}$ is a field. Then $\\struct {K, +, \\circ}$ is a '''subfield''' of $\\struct {R, +, \\circ}$."}
+{"_id": "29974", "title": "Definition:Subfield/Proper Subfield", "text": "Let $\\struct {K, +, \\circ}$ be a subfield of $\\struct {F, +, \\circ}$. Then $\\struct {K, +, \\circ}$ is a '''proper subfield''' of $\\struct {F, +, \\circ}$ {{iff}} $K \\ne F$. That is, $\\struct {K, +, \\circ}$ is a '''proper subfield''' of $\\struct {F, +, \\circ}$ {{iff}}: :$(1): \\quad \\struct {K, +, \\circ}$ is a subfield of $\\struct {F, +, \\circ}$ :$(2): \\quad K$ is a proper subset of $F$."}
+{"_id": "29975", "title": "Definition:Periodic Function/Periodic Element", "text": "Let $L \\in X_{\\ne 0}$. Then $L$ is a '''periodic element''' of $f$ {{iff}}: :$\\forall x \\in X: \\map f x = \\map f {x + L}$"}
+{"_id": "29976", "title": "Definition:Trivial Factorization/Non-Trivial Factorization", "text": "A factorization in $\\struct {D, +, \\circ}$ of the form $x = z \\circ y$, where neither $y$ nor $z$ is a unit of $D$, is called a '''non-trivial factorization'''."}
+{"_id": "29977", "title": "Definition:Associate/Integral Domain/Definition 3", "text": "$x$ and $y$ are '''associates (in $D$)''' {{iff}} there exists a unit $u$ of $\\struct {D, +, \\circ}$ such that: :$y = u \\circ x$ and consequently: :$x = u^{-1} \\circ y$ That is, {{iff}} $x$ and $y$ are unit multiples of each other."}
+{"_id": "29991", "title": "Definition:Cauchy Distribution", "text": "Let $X$ be a continuous random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\Img X = \\R$.  $X$ is said to have a '''Cauchy distribution''' if it has probability density function: :$\\map {f_X} x = \\dfrac 1 {\\pi \\gamma \\paren {1 + \\paren {\\frac {x - x_0} \\gamma}^2} }$ for some $\\gamma > 0$.  This is written:  :$X \\sim \\Cauchy {x_0} \\gamma$"}
+{"_id": "29993", "title": "Definition:Antiperiodic Function", "text": "=== Antiperiodic Real Function === {{:Definition:Antiperiodic Function/Real}} === Antiperiodic Complex Function === {{:Definition:Antiperiodic Function/Complex}}"}
+{"_id": "29996", "title": "Definition:Antiperiodic Function/Antiperiodic Element", "text": "Let $L \\in X_{\\ne 0}$. Then $L$ is an '''anti-periodic element''' of $f$ {{iff}}: : $\\forall x \\in X: - \\map f x = \\map f {x + L}$"}
+{"_id": "29997", "title": "Definition:Antiperiodic Function/Antiperiod", "text": "The '''antiperiod''' of $f$ is the smallest value $\\cmod L \\in \\R_{\\ne 0}$ such that: : $\\forall x \\in X: - \\map f x = \\map f {x + L}$ where $\\cmod L$ is the modulus of $L$."}
+{"_id": "29998", "title": "Definition:Skewness", "text": "'''Skewness''' is a measure of the asymmetry of a probability distribution about its mean. Let $X$ be a random variable with mean $\\mu$ and standard deviation $\\sigma$. Then the '''skewness''' of $X$, usually denoted $\\gamma_1$, is defined as:  :$\\gamma_1 = \\expect {\\paren {\\dfrac {X - \\mu} \\sigma}^3}$ where $\\expect X$ denotes the expectation of $X$."}
+{"_id": "29999", "title": "Definition:Uniform Distribution", "text": "=== Discrete Uniform Distribution === {{:Definition:Discrete Uniform Distribution}} === Continuous Uniform Distribution === {{:Definition:Continuous Uniform Distribution}}"}
+{"_id": "30000", "title": "Definition:Prime Ideal of Ring/Commutative and Unitary Ring/Definition 1", "text": "A '''prime ideal''' of $R$ is a proper ideal $P$ such that: :$\\forall a, b \\in R : a \\circ b \\in P \\implies a \\in P$ or $b \\in P$"}
+{"_id": "30001", "title": "Definition:Prime Ideal of Ring/Commutative and Unitary Ring/Definition 2", "text": "A '''prime ideal''' of $R$ is a proper ideal $P$ of $R$ such that: :$I \\circ J \\subseteq P \\implies I \\subseteq P \\text { or } J \\subseteq P$ for all ideals $I$ and $J$ of $R$."}
+{"_id": "30002", "title": "Definition:Prime Ideal of Ring/Commutative and Unitary Ring", "text": "=== Definition 1 === {{:Definition:Prime Ideal of Commutative and Unitary Ring/Definition 1}} === Definition 2 === {{:Definition:Prime Ideal of Commutative and Unitary Ring/Definition 2}} === Definition 3 === {{:Definition:Prime Ideal of Commutative and Unitary Ring/Definition 3}}"}
+{"_id": "30003", "title": "Definition:Prime Ideal of Ring/Commutative and Unitary Ring/Definition 3", "text": "A '''prime ideal''' of $R$ is a proper ideal $P$ of $R$ such that: :the complement $R \\setminus P$ of $P$ in $R$ is closed under the ring product $\\circ$."}
+{"_id": "30004", "title": "Definition:Moment (Probability Theory)", "text": "Let $X$ be a random variable on some probability space.  Let $a$ be a real number. Then the '''$n$th moment of $X$ about $a$''', usually denoted $\\map {\\mu_n} a$, is defined as:  :$\\map {\\mu_n} a = \\expect {\\paren {X - a}^n}$ where $\\expect X$ denotes the expectation of $X$."}
+{"_id": "30005", "title": "Definition:Raw Moment", "text": "Let $X$ be a random variable on some probability space.  Then the '''$n$th raw moment of $X$''', usually denoted $\\mu'_n$, is defined as:  :$\\mu'_n = \\expect {X^n}$ where $\\expect X$ denotes the expectation of $X$. That is, the $n$th raw moment of $X$ is its $n$th moment about $0$.  The first raw moment of $X$ is the mean of $X$, and is instead usually denoted $\\mu$."}
+{"_id": "30006", "title": "Definition:Central Moment", "text": "Let $X$ be a random variable on some probability space with mean $\\mu$.  Then the '''$n$th central moment of $X$''', usually denoted $\\mu_n$, is defined as:  :$\\mu_n = \\expect {\\paren {X - \\mu}^n}$ where $\\expect X$ denotes the expectation of $X$. That is, the $n$th central moment of $X$ is its $n$th moment about $\\mu$.  The second central moment of $X$ is the variance of $X$, and is instead usually denoted $\\sigma^2$."}
+{"_id": "30007", "title": "Definition:Norm Axioms", "text": "=== Multiplicative Norm Axioms === Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Let $\\norm {\\, \\cdot \\,}: R \\to \\R_{\\ge 0}$ be a multiplicative norm on $R$. The '''multiplicative norm axioms''' are the conditions on $\\norm {\\, \\cdot \\,}$ which are satisfied for all elements of $R$ in order for $\\norm {\\, \\cdot \\,}$ to be a multiplicative norm: {{:Definition:Multiplicative Norm Axioms}} When the concept of '''norm axioms''' is raised without qualification, it is usually the case that '''multiplicative norm axioms''' are under discussion. === Submultiplicative Norm Axioms === Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Let $\\norm {\\, \\cdot \\,}: R \\to \\R_{\\ge 0}$ be a submultiplicative norm on $R$. The '''submultiplicative norm axioms''' are the conditions on $\\norm {\\, \\cdot \\,}$ which are satisfied for all elements of $R$ in order for $\\norm {\\, \\cdot \\,}$ to b a submultiplicative norm: {{:Definition:Submultiplicative Norm Axioms}} === Norm Axioms (Vector Space) === Let $\\struct {R, +, \\circ}$ be a division ring with norm $\\norm {\\,\\cdot\\,}_R$. Let $V$ be a vector space over $R$, with zero $\\mathbf 0_V$. Let $\\norm {\\,\\cdot\\,}: V \\to \\R_{\\ge 0}$ be a norm on $V$. The '''norm axioms''' are the following conditions on $\\norm {\\,\\cdot\\,}$ which define $\\norm {\\,\\cdot\\,}$ as being a norm: {{:Definition:Norm Axioms (Vector Space)}} Category:Definitions/Norm Theory gyi8v5yi0lz255ux3g3vc1fwvcw6qr1"}
+{"_id": "30008", "title": "Definition:Submultiplicative Norm Axioms", "text": "{{begin-axiom}} {{axiom | n = \\text N 1         | lc= Positive Definiteness:         | q = \\forall x \\in R         | ml= \\norm x = 0         | mo= \\iff         | mr= x = 0_R }} {{axiom | n = \\text N 2         | lc= Submultiplicativity:         | q = \\forall x, y \\in R         | ml= \\norm {x \\circ y}         | mo= \\le         | mr= \\norm x \\times \\norm y }} {{axiom | n = \\text N 3         | lc= Triangle Inequality:         | q = \\forall x, y \\in R         | ml= \\norm {x + y}         | mo= \\le         | mr= \\norm x + \\norm y }} {{end-axiom}}"}
+{"_id": "30010", "title": "Definition:Multiplicative Function on Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $f: R \\to \\R$ be a (real-valued) function on $R$. $f$ is a '''multiplicative function on $R$''' {{iff}}: :$\\forall x, y \\in R: \\map f {x \\circ y} = \\map f x \\times \\map f y$"}
+{"_id": "30011", "title": "Definition:Standardized Moment", "text": "Let $X$ be a random variable on some probability space with standard deviation $\\sigma$.  Then the '''$n$th standardized moment of $X$''', usually denoted $\\alpha_n$, is defined as:  :$\\alpha_n = \\dfrac {\\mu_n} {\\sigma^n}$ where $\\mu_n$ is the $n$th central moment of $X$. The third standardized moment of $X$ is the skewness of $X$, and is instead usually denoted $\\gamma_1$.  The fourth standardized moment of $X$ is the kurtosis of $X$."}
+{"_id": "30012", "title": "Definition:Cyclotomic Ring", "text": "Let $\\Z \\sqbrk {i \\sqrt n}$ be the set $\\set {a + i b \\sqrt n: a, b \\in \\Z}$. The algebraic structure $\\struct {\\Z \\sqbrk {i \\sqrt n}, +, \\times}$ is the '''$n$th cyclotomic ring'''."}
+{"_id": "30013", "title": "Definition:Kurtosis", "text": "Let $X$ be a random variable with mean $\\mu$ and standard deviation $\\sigma$. Then the '''kurtosis''' of $X$, usually denoted $\\alpha_4$, is defined as:  :$\\alpha_4 = \\expect {\\paren {\\dfrac {X - \\mu} \\sigma}^4}$ where $\\expect X$ denotes the expectation of $X$. That is, the '''kurtosis''' of $X$ is the fourth standardized moment of $X$."}
+{"_id": "30014", "title": "Definition:Constant Term of Polynomial", "text": "Let $R$ be a commutative ring with unity. Let $P \\in R \\sqbrk X$ be a nonzero polynomial over $R$: :$\\displaystyle f = \\sum_{k \\mathop = 0}^n a_k \\circ x^k$ where $n$ is the degree of $P$. The '''constant term''' of $P$ is the coefficient of $a_0$."}
+{"_id": "30015", "title": "Definition:Ring of Eisenstein Integers", "text": "The '''ring of Eisenstein integers''' $\\struct {\\Z \\sqbrk \\omega, +, \\times}$ is the algebraic structure formed from: :the set of Eisenstein integers $\\Z \\sqbrk \\omega$, where $\\omega = e^{2 \\pi i / 3}$ :the operation of complex addition :the operation of complex multiplication."}
+{"_id": "30016", "title": "Definition:Ring of Square Matrices", "text": "Let $R$ be a ring. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\map {\\mathcal M_R} n$ denote the $n \\times n$ matrix space over $R$. Let $+$ denote the operation of matrix entrywise addition. Let $\\times$ be (temporarily) used to denote the operation of conventional matrix multiplication.  The algebraic structure: :$\\struct {\\map {\\mathcal M_R} n, +, \\times}$ is known as the '''ring of square matrices of order $n$ over $R$'''"}
+{"_id": "30021", "title": "Definition:Additive Group", "text": "=== Additive Group of Integers === {{:Definition:Additive Group of Integers}} === Additive Group of Integer Multiples === {{:Definition:Additive Group of Integer Multiples}} === Additive Group of Integers Modulo $m$ === {{:Definition:Additive Group of Integers Modulo m}} === Additive Group of Rational Numbers === {{:Definition:Additive Group of Rational Numbers}} === Additive Group of Real Numbers === {{:Definition:Additive Group of Real Numbers}} === Additive Group of Complex Numbers === {{:Definition:Additive Group of Complex Numbers}}"}
+{"_id": "30022", "title": "Definition:Excess Kurtosis", "text": "'''Excess kurtosis''' is defined as the difference between the kurtosis of a particular probability distribution and that of the Gaussian distribution. Let $X$ be a random variable with kurtosis $\\alpha_4$.  By Kurtosis of Gaussian Distribution, the kurtosis of a Gaussian distribution is equal to $3$.  So the '''excess kurtosis''' of $X$, usually denoted $\\gamma_2$, is given by:  :$\\gamma_2 = \\alpha_4 - 3$"}
+{"_id": "30023", "title": "Definition:Die/Historical Note", "text": "Dice have been around for thousands of years. An early reference to the design of a die can be found in {{AuthorRef|W.R. Paton}}'s $1918$ translation of {{BookLink|The Greek Anthology Book XIV|W.R. Paton}}: :''The numbers on a die run so: six one, five two, three four.''"}
+{"_id": "30024", "title": "Definition:Ordered Integral Domain/Definition 1", "text": "An '''ordered integral domain''' is an integral domain $\\struct {D, +, \\times}$ which has a strict positivity property $P$: {{:Definition:Strict Positivity Property}}"}
+{"_id": "30025", "title": "Definition:Ordered Integral Domain/Definition 2", "text": "An '''ordered integral domain''' is an ordered ring $\\struct {D, +, \\times, \\le}$ which is also an integral domain. That is, it is an integral domain with an ordering $\\le$ compatible with the ring structure of $\\struct {D, +, \\times}$: {{begin-axiom}} {{axiom | n = \\text {OID} 1         | lc= $\\le$ is compatible with ring addition:         | q = \\forall a, b, c \\in D         | ml= a \\le b         | mo= \\implies         | mr= \\paren {a + c} \\le \\paren {b + c} }} {{axiom | n = \\text {OID} 2         | lc= Strict positivity is closed under ring product:         | q = \\forall a, b \\in D         | ml= 0_D \\le a, 0_D \\le b         | mo= \\implies         | mr= 0_D \\le a \\times b }} {{end-axiom}}"}
+{"_id": "30026", "title": "Definition:Positivity Property", "text": "{{begin-axiom}} {{axiom | n = \\text P 1         | m = P + P \\subseteq P }} {{axiom | n = \\text P 2         | m = P \\cap \\paren {-P} = \\set {0_R} }} {{axiom | n = \\text P 3         | m = P \\circ P \\subseteq P }} {{end-axiom}}"}
+{"_id": "30028", "title": "Definition:Total Ordering Induced by Strict Positivity Property", "text": "Let $\\struct {D, +, \\times, \\le}$ be an ordered integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $P: D \\to \\set {\\mathrm T, \\mathrm F}$ denote the strict positivity property: {{:Definition:Strict Positivity Property}} Then the total ordering $\\le$ compatible with the ring structure of $D$ is called the '''(total) ordering induced by (the strict positivity property) $P$'''."}
+{"_id": "30029", "title": "Definition:Dottie Number", "text": "The '''Dottie number''' $\\d$ is defined as the unique fixed point of the cosine function: :$\\cos \\rd = \\d$"}
+{"_id": "30030", "title": "Definition:Strict Ordering on Integers", "text": "=== Definition 1 === {{:Definition:Strict Ordering on Integers/Definition 1}} === Definition 2 === {{:Definition:Strict Ordering on Integers/Definition 2}}"}
+{"_id": "30031", "title": "Definition:Well-Ordered Integral Domain/Definition 1", "text": "$\\struct {D, +, \\times \\le}$ is a '''well-ordered integral domain''' {{iff}} the ordering $\\le$ is a well-ordering on the set $P$ of (strictly) positive elements of $D$."}
+{"_id": "30032", "title": "Definition:Well-Ordered Integral Domain/Definition 2", "text": "$\\struct {D, +, \\times \\le}$ is a '''well-ordered integral domain''' {{iff}} every subset $S$ of the set $P$ of (strictly) positive elements of $D$ has a minimal element: :$\\forall S \\subseteq D_{\\ge 0_D}: \\exists x \\in S: \\forall a \\in S: x \\le a$ where $D_{\\ge 0_D}$ denotes all the elements $d \\in D$ such that $\\map P d$."}
+{"_id": "30046", "title": "Definition:P-adic Number/P-adic Norm Completion of Rational Numbers", "text": "The '''p-adic numbers''', denoted $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$, is the unique (up to isometric isomorphism) non-Archimedean valued field that completes $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$. Each element of $\\Q_p$ is called a '''$p$-adic number'''."}
+{"_id": "30047", "title": "Definition:P-adic Number/Quotient of Cauchy Sequences in P-adic Norm", "text": "Let $\\CC$ be the commutative ring of Cauchy sequences over $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$. Let $\\NN$ be the set of null sequences in $\\struct {\\Q, \\norm {\\,\\cdot\\,}_p}$. Let $\\Q_p$ denote the quotient ring $\\CC \\, \\big / \\NN$. Let $\\norm {\\, \\cdot \\,}_p:\\Q_p \\to \\R_{\\ge 0}$ be the norm on the quotient ring $\\Q_p$ defined by: :$\\displaystyle \\forall \\sequence {x_n} + \\NN: \\norm {\\sequence {x_n} + \\NN }_p = \\lim_{n \\mathop \\to \\infty} \\norm{x_n}_p$ By Corollary to Quotient Ring of Cauchy Sequences is Normed Division Ring, then $\\struct {\\Q_p, \\norm {\\, \\cdot \\,}_p}$ is a valued field. Each left coset $\\sequence {x_n} + \\NN \\in \\CC \\, \\big / \\NN$ is called a '''$p$-adic number'''. The '''p-adic numbers''' is the valued field $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$."}
+{"_id": "30054", "title": "Definition:Multiple/Integral Domain", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain. Let $x, y \\in D$. Let $x$ be a divisor of $y$. Then $y$ is a '''multiple''' of $x$."}
+{"_id": "30055", "title": "Definition:Multiple/Integer", "text": "Let $\\Z$ denote the set of integers. Let $x, y \\in \\Z$. Let $x$ be a divisor of $y$. Then $y$ is a '''multiple''' of $x$."}
+{"_id": "30067", "title": "Definition:Unit of Ring/Definition 1", "text": "An element $x \\in R$ is a '''unit of $\\struct {R, +, \\circ}$''' {{iff}} $x$ is invertible under $\\circ$. That is, a '''unit''' of $R$ is an element of $R$ which has an inverse. :$\\exists y \\in R: x \\circ y = 1_R = y \\circ x$"}
+{"_id": "30068", "title": "Definition:Unit of Ring/Definition 2", "text": "An element $x \\in R$ is a '''unit of $\\struct {R, +, \\circ}$''' {{iff}} $x$ is divisor of $1_R$."}
+{"_id": "30071", "title": "Definition:Proper Divisor/Integer", "text": "Let $\\struct {\\Z, +, \\times}$ be the ring of integers. Let $x, y \\in \\Z$. Then '''$x$ divides $y$''' is defined as: :$x \\divides y \\iff \\exists t \\in \\Z: y = t \\times x$ Then $x$ is a '''proper divisor''' of $y$ {{iff}}: :$(1): \\quad x \\divides y$ :$(2): \\quad \\size x \\ne \\size y$ :$(3): \\quad x \\ne \\pm 1$ That is: :$(1): \\quad x$ is a divisor of $y$ :$(2): \\quad x$ and $y$ are not equal in absolute value :$(3): \\quad x$ is not equal to either $1$ or $-1$."}
+{"_id": "30078", "title": "Definition:Content of Polynomial/Integer", "text": "Let $f \\in \\Z \\sqbrk X$ be a polynomial with integer coefficients. Then the '''content''' of $f$, denoted $\\cont f$, is the greatest common divisor of the coefficients of $f$."}
+{"_id": "30079", "title": "Definition:Content of Polynomial/Rational", "text": "Let $f \\in \\Q \\sqbrk X$ be a polynomial with rational coefficients. The '''content''' of $f$ is defined as: :$\\cont f := \\dfrac {\\cont {n f} } n$ where $n \\in \\N$ is such that $n f \\in \\Z \\sqbrk X$."}
+{"_id": "30080", "title": "Definition:Content of Polynomial/GCD Domain", "text": "Let $D$ be a GCD domain. Let $K$ be the field of quotients of $D$. Let $f \\in K \\sqbrk X$ be a polynomial. Let $a \\in D$ be such that $a f \\in D \\sqbrk X$. Let $d$ be the greatest common divisor of the coefficients of $a f$. Then we define the '''content''' of $f$ to be: :$\\cont f := \\dfrac d a$"}
+{"_id": "30082", "title": "Definition:Z-Module Associated with Abelian Group/Definition 1", "text": "The '''$\\Z$-module associated with $G$''' is the $\\Z$-module $\\struct {G, *, \\circ}$ with ring action: :$\\circ: \\Z \\times G \\to G$: :$\\tuple {n, x} \\mapsto *^n x$ where $*^n x$ is the $n$th power of $x$."}
+{"_id": "30083", "title": "Definition:Z-Module Associated with Abelian Group/Definition 2", "text": "The '''$\\Z$-module associated with $G$''' is the $\\Z$-module on $G$ with ring representation $\\Z \\to \\map {\\operatorname {End} } G$ equal to the initial homomorphism."}
+{"_id": "30084", "title": "Definition:Linear Ring Action/Left", "text": "A '''(left) linear ring action''' of $R$ on $M$ is a mapping from the cartesian product $\\circ : R \\times M \\to M$ such that: {{begin-axiom}} {{axiom | n = 1         | q = \\forall \\lambda \\in R: \\forall m, n \\in M         | ml= \\lambda \\circ \\paren {m + n}         | mo= =         | mr= \\paren {\\lambda \\circ m} + \\paren {\\lambda \\circ n} }} {{axiom | n = 2         | q = \\forall \\lambda, \\mu \\in R: \\forall m \\in M         | ml= \\paren {\\lambda + \\mu} \\circ m         | mo= =         | mr= \\paren {\\lambda \\circ m} + \\paren {\\mu \\circ m} }} {{axiom | n = 3         | q = \\forall \\lambda, \\mu \\in R: \\forall m \\in M         | ml= \\paren {\\lambda \\mu} \\circ m         | mo= =         | mr= \\lambda \\circ \\paren {\\mu \\circ m} }} {{end-axiom}}"}
+{"_id": "30085", "title": "Definition:Linear Ring Action/Right", "text": "A '''right linear ring action''' of $R$ on $M$ is a mapping from the cartesian product $\\circ : M \\times R \\to M$ such that: {{begin-axiom}} {{axiom | n = 1         | q = \\forall \\lambda \\in R: \\forall m, n \\in M         | ml= \\paren {m + n} \\circ \\lambda         | mo= =         | mr= \\paren {m \\circ \\lambda} + \\paren {n \\circ \\lambda} }} {{axiom | n = 2         | q = \\forall \\lambda, \\mu \\in R: \\forall m \\in M         | ml= m \\circ \\paren {\\lambda + \\mu}         | mo= =         | mr= \\paren {m \\circ \\lambda} + \\paren {m \\circ \\mu} }} {{axiom | n = 3         | q = \\forall \\lambda, \\mu \\in R: \\forall m \\in M         | ml= m \\circ \\paren {\\lambda\\mu}         | mo= =         | mr= \\paren {m \\circ \\lambda} \\circ \\mu }} {{end-axiom}}"}
+{"_id": "30090", "title": "Definition:Closed Statement", "text": "Let $P$ be a statement. $P$ is a '''closed statement''' {{iff}} $P$ contains only bound occurrences of any variables that may appear in it. That is, such that it contains no free occurrences of variables."}
+{"_id": "30091", "title": "Definition:Open Statement", "text": "Let $P$ be a statement. $P$ is an '''open statement''' {{iff}} $P$ contains at least one free occurrence of a variables that appears in it."}
+{"_id": "30094", "title": "Definition:Generated Subspace", "text": "=== Definition 1 === {{:Definition:Generated Subspace/Definition 1}} === Definition 2 === {{:Definition:Generated Subspace/Definition 2}}"}
+{"_id": "30095", "title": "Definition:Generated Subspace/Definition 1", "text": "The '''subspace generated by $S$''' is the intersection of all subspaces of $\\mathbf V$ containing $S$."}
+{"_id": "30096", "title": "Definition:Generated Subspace/Definition 2", "text": "The '''subspace generated by $S$''' is the set of all linear combinations of elements of $S$."}
+{"_id": "30097", "title": "Definition:Dimension of Vector Space/Definition 2", "text": "The '''dimension of $V$''' is the maximum cardinality of a linearly independent subset of $V$."}
+{"_id": "30098", "title": "Definition:Dimension of Vector Space/Definition 1", "text": "The '''dimension of $V$''' is the number of vectors in a basis for $V$."}
+{"_id": "30100", "title": "Definition:Sheldon Prime", "text": "A '''Sheldon prime''' is a prime number which satisfies both the product property and the mirror property."}
+{"_id": "30101", "title": "Definition:Product Property of Primes", "text": "Let $p_n$ denote the $n$th prime number. $p_n$ satisfies the '''product property''' if the product of the digits of its base $10$ representation equals $n$."}
+{"_id": "30102", "title": "Definition:Mirror Property of Primes", "text": "Let $p_n$ denote the $n$th prime number. Let $\\map {\\operatorname {rev} } {p_n}$ denote the reversal of $p_n$. $p_n$ satisfies the '''mirror property''' {{iff}}: :$(1): \\quad \\map {\\operatorname {rev} } {p_n}$ is also a prime number :$(2): \\quad$ the product of the digits of the base $10$ representation of $\\map {\\operatorname {rev} } {p_n}$ equals $\\map {\\operatorname {rev} } n$."}
+{"_id": "30103", "title": "Definition:Smallest Field containing Subfield and Complex Number", "text": "Let $F$ be a field. Let $\\theta \\in \\C$ be a complex number. Let $S$ be the intersection of all fields such that: :$S \\subseteq F$ :$\\theta \\in F$ Then $S$ is denoted $\\map F \\theta$ and referred to as the '''smallest field containing $F$ and $\\theta$'''. {{help|I encounter this definition in {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham}} without having much idea of the context. The element $\\theta$ is specifically defined as being a complex number. At the moment I have not got a clue as to why the specific nature of $\\theta$, but I will go along with it as defined in Clapham. If the context needs to be expanded, that can happen in due course.}} === General Definition === {{:Definition:Smallest Field containing Subfield and Complex Number/General Definition}}"}
+{"_id": "30104", "title": "Definition:Smallest Field containing Subfield and Complex Number/General Definition", "text": "Let $F$ be a field. Let $\\theta_1, \\theta_2, \\ldots, \\theta_n \\in \\C$ be complex numbers. Let $S$ be the intersection of all fields such that: :$S \\subseteq F$ :$\\theta_1, \\theta_2, \\ldots, \\theta_n \\in F$ Then $S$ is denoted $\\map F {\\theta_1, \\theta_2, \\ldots, \\theta_n}$ and referred to as the '''smallest field containing $F$ and $\\theta_1, \\theta_2, \\ldots, \\theta_n$'''."}
+{"_id": "30105", "title": "Definition:Algebraic Number over Field", "text": "Let $F$ be a field. Let $z$ be a complex number. $z$ is '''algebraic over $F$''' {{iff}} $z$ is a root of a polynomial with coefficients in $F$."}
+{"_id": "30107", "title": "Definition:Generated Field Extension/Definition 1", "text": "The '''field extension $F \\sqbrk S$ generated by $S$''' is the smallest subfield extension of $E$ containing $S$, that is, the intersection of all subfields of $E$ containing $S$ and $F$. Thus $S$ is a '''generator''' of $F \\sqbrk S$ {{iff}} $F \\sqbrk S$ has no proper subfield extension containing $S$."}
+{"_id": "30109", "title": "Definition:Simple Algebraic Field Extension", "text": "Let $F$ be a field. Let $z$ be a complex number which is algebraic over $F$. Then the smallest field $\\map F z$ containing $F$ and $z$ is known as a '''simple algebraic (field) extension''' of $F$."}
+{"_id": "30110", "title": "Definition:Algebraic Number over Field/Degree", "text": "Let $F$ be a field. Let $z \\in \\C$ be algebraic over $F$. The '''degree''' of $\\alpha$ is the degree of the minimal polynomial $\\map m x$ whose coefficients are all in $F$."}
+{"_id": "30111", "title": "Definition:Algebraic Element of Field Extension/Definition 1", "text": "$\\alpha$ is '''algebraic over $F$''' {{iff}} it is a root of some nonzero polynomial over $F$: :$\\exists f \\in F \\sqbrk X \\setminus \\set 0: \\map f \\alpha = 0$ where $F \\sqbrk X$ denotes the ring of polynomial forms in $X$."}
+{"_id": "30112", "title": "Definition:Algebraic Element of Field Extension/Definition 2", "text": "$\\alpha$ is '''algebraic over $F$''' {{iff}} the evaluation homomorphism $F \\sqbrk X \\to K$ at $\\alpha$ is not injective."}
+{"_id": "30113", "title": "Definition:Polynomial Evaluation Homomorphism/Single Indeterminate", "text": "Let $\\struct {R \\sqbrk X, \\iota, X}$ be a polynomial ring in one variable over $R$. Let $s\\in S$. A ring homomorphism $h : R \\sqbrk X \\to S$ is called an '''evaluation in $s$''' {{iff}}: :$\\map h X = s$ :$h \\circ \\iota = \\kappa$ where $\\circ$ denotes composition of mappings."}
+{"_id": "30114", "title": "Definition:Polynomial Evaluation Homomorphism/Multiple Indeterminates", "text": "Let $\\family {s_i}_{i \\mathop \\in I}$ be an indexed family of elements of $S$. Let $R \\sqbrk {\\set {X_i: i \\mathop \\in I} }$ be a polynomial ring over $R$. A ring homomorphism $g: R \\sqbrk {\\set {X_i: i \\mathop \\in I} } \\to S$ is called an '''evaluation at $\\family {s_i}_{i \\mathop \\in I}$''' {{iff}}: :$\\forall r \\in R : \\map g r = \\map \\kappa r$ :$\\forall j \\in J : \\map g {X_j} = s_j$"}
+{"_id": "30116", "title": "Definition:Constructible Point in Plane", "text": "Let $\\mathcal C$ be a Cartesian coordinate plane. Let $S$ be a set of points in $\\mathcal C$. Let $P$ be a point in $\\mathcal C$. Let there exist a compass and straightedge construction for $P$ from a line segment $AB$, where $A, B \\in S$ Then $P$ is defined as '''constructible from $S$'''."}
+{"_id": "30117", "title": "Definition:Coherent Sequence", "text": "Let $p$ be a prime number. Let $\\sequence {\\alpha_n}$ be an integer sequence such that: :$(1): \\quad \\forall n \\in \\N: 0 \\le \\alpha_n \\le p^{n + 1} - 1$ :$(2): \\quad \\forall n \\in \\N: \\alpha_{n + 1} \\equiv \\alpha_n \\pmod {p^{n + 1}}$ The sequence $\\sequence {\\alpha_n}$ is said to be a '''coherent sequence'''. If it is necessary to emphasize the choice of prime $p$ then the sequence $\\sequence {\\alpha_n}$ is said to be a '''$p$-adically coherent sequence'''."}
+{"_id": "30118", "title": "Definition:P-adic Norm/P-adic Numbers/P-adic Metric", "text": "The '''$p$-adic metric''' on $\\Q_p$ is the metric induced by $\\norm{\\cdot}_p$: :$\\forall x, y \\in \\Q_p: \\map d {x, y} = \\norm{x - y}_p$"}
+{"_id": "30119", "title": "Definition:P-adic Norm/P-adic Numbers", "text": "The norm $\\norm {\\,\\cdot\\,}_p$ on $\\Q_p$ is called the '''$p$-adic norm''' on $\\Q_p$."}
+{"_id": "30122", "title": "Definition:Irreducible Element of Ring/Definition 1", "text": "$x$ is defined as '''irreducible''' {{iff}} it has no non-trivial factorization in $D$. That is, {{iff}} $x$ cannot be written as a product of two non-units."}
+{"_id": "30123", "title": "Definition:Irreducible Element of Ring/Definition 2", "text": "$x$ is defined as '''irreducible''' {{iff}} the only divisors of $x$ are its associates and the units of $D$. That is, {{iff}} $x$ has no proper divisors."}
+{"_id": "30125", "title": "Definition:Laplace Transform/Restriction to Reals", "text": "Although the definition of the '''Laplace transform''' has $s$ be a complex variable, sometimes the restriction of $\\map {\\laptrans f} s$ to wholly real $s$ is sufficient to solve a particular differential equation.  Therefore, elementary textbooks introducing the Laplace transform will often write something like the following: :''... where we assume at present that the parameter $s$ is real. Later it will be found useful to consider $s$ complex.'' ::::-- {{BookReference|Theory and Problems of Laplace Transforms|1965|Murray R. Spiegel}}: Chapter $1$: The Laplace Transform: Definition of the Laplace Transform :''A profound understanding of the workings of the Laplace transform requires considering it to be a so-called analytic function of a complex variable, but in most of this book we shall assume that the variable $s$ is real.'' ::::-- {{BookReference|Fourier Analysis and its Applications|2003|Anders Vretblad}}: $\\S 3.1$"}
+{"_id": "30126", "title": "Definition:Laplace Transform/Graphical Interpretation", "text": "Define $\\gamma$ as the integrand of $\\displaystyle \\int_0^{\\to +\\infty} e^{-s t} \\map f t \\rd t$ as a function of $\\map f t$, $s$, and $t$: :$\\map \\gamma {\\map f t, t; s} = \\map f t \\, e^{-s t}$ For any particular function $f$, holding $s$ fixed, the integrand of the Laplace Transform $\\kappa$ can be interpreted as a contour. That is, for a given function $f$ and a particular complex number $s_0$ held constant: :$\\map \\kappa t: \\R_{\\ge 0} \\to \\C$ :$\\map \\kappa t = \\map \\gamma {\\map f t, t; s_0}$ is a parameterization of a contour."}
+{"_id": "30129", "title": "Definition:Laplace Transform/Discontinuity at Zero", "text": "Let $f: \\R_{> 0} \\to \\mathbb F$ be a function of a real variable $t$, where $\\mathbb F \\in \\set {\\R, \\C}$. Let $f$ be discontinuous or not defined at $t = 0$. Then the '''Laplace transform''' of $f$  is defined as: :$\\displaystyle \\laptrans {\\map f t} = \\map F s = \\int_{0^+}^{\\to +\\infty} e^{-s t} \\map f t \\rd t = \\lim_{\\epsilon \\mathop \\to 0^+} \\int_\\epsilon^{\\to +\\infty} e^{-s t} \\map f t \\rd t$ whenever this improper integral converges. If this improper integral does not converge, then $\\laptrans {\\map f t}$ does not exist."}
+{"_id": "30132", "title": "Definition:Cauchy Principal Value/Real Integral", "text": "Let $f: \\R \\to \\R$ be a real function which is piecewise continuous everywhere. Then the '''Cauchy principal value of $\\displaystyle \\int f$''' is defined as: :$\\PV_{-\\infty}^{+\\infty} \\map f t \\rd t := \\lim_{R \\mathop \\to +\\infty} \\int_{-R}^R \\map f t \\rd t$ where $\\displaystyle \\int_{-R}^R \\map f t \\rd t$ is a Riemann integral."}
+{"_id": "30137", "title": "Definition:Jump Discontinuity/Jump", "text": "Let $X$ be an open subset of $\\R$. Let $f: X \\to Y$ be a real function. Let $f$ be discontinuous at some point $c \\in X$ such that $c$ is a '''jump discontinuity''' of $f$. The '''jump''' at $c$ is defined as: :$\\displaystyle \\lim_{x \\mathop \\to c^+} \\map f x - \\lim_{x \\mathop \\to c^-} \\map f x$"}
+{"_id": "30138", "title": "Definition:Bessel Function/Order", "text": "The parameter $n$ is known as the '''order''' of the Bessel function."}
+{"_id": "30140", "title": "Definition:Modified Bessel Function", "text": "The '''modified Bessel functions''' are solutions to Bessel's modified equation: :$x^2 \\dfrac {\\d^2 y} {\\d x^2} + x \\dfrac {\\d y} {\\d x} - \\paren {x^2 + n^2} y = 0$ These solutions have two main classes: :the modified Bessel functions of the first kind $I_n$ and: :the modified Bessel functions of the second kind $K_n$."}
+{"_id": "30141", "title": "Definition:Modified Bessel Function/Order", "text": "The parameter $n$ is known as the '''order''' of the modified Bessel function."}
+{"_id": "30142", "title": "Definition:Modified Bessel Function/First Kind", "text": "A '''modified Bessel function of the first kind of order $n$''' is a modified Bessel function which is non-singular at the origin. It is usually denoted $\\map {I_n} x$, where $x$ is the dependent variable of the instance of '''Bessel's modified equation''' to which $\\map {I_n} x$ forms a solution."}
+{"_id": "30143", "title": "Definition:Modified Bessel Function/Second Kind", "text": "A '''modified Bessel function of the second kind of order $n$''' is a modified Bessel function which is singular at the origin. It is usually denoted $\\map {K_n} x$, where $x$ is the dependent variable of the instance of '''Bessel's modified equation''' to which $\\map {K_n} x$ forms a solution."}
+{"_id": "30144", "title": "Definition:Error Function", "text": "The '''error function''' is the following improper integral, considered as a real function $\\erf : \\R \\to \\R$: :$\\map {\\erf} x = \\displaystyle \\dfrac 2 {\\sqrt \\pi} \\int_0^x \\map \\exp {-t^2} \\rd t$ where $\\exp$ is the real exponential function."}
+{"_id": "30145", "title": "Definition:Complementary Error Function", "text": "The '''complementary error function''' is the real function $\\erfc: \\R \\to \\R$: {{begin-eqn}} {{eqn | l = \\map {\\erfc} x       | r = 1 - \\map \\erf x       | c = where $\\erf$ denotes the Error Function }} {{eqn | r = 1 - \\dfrac 2 {\\sqrt \\pi} \\int_0^x \\map \\exp {-t^2} \\rd t       | c = where $\\exp$ denotes the Real Exponential Function }} {{eqn | r = \\dfrac 2 {\\sqrt \\pi} \\int_x^\\infty \\map \\exp {-t^2} \\rd t       | c =  }} {{end-eqn}}"}
+{"_id": "30146", "title": "Definition:Exponential Integral Function", "text": "The '''exponential integral function''' is the real function $\\Ei: \\R_{> 0} \\to \\R$ defined as: :$\\map \\Ei x = \\displaystyle \\int_{t \\mathop = x}^{t \\mathop \\to +\\infty} \\frac {e^{-t} } t \\rd t$"}
+{"_id": "30147", "title": "Definition:Null Function", "text": "Let $\\mathcal N: \\R \\to \\R$ be a real function such that: :$\\forall x \\in \\R_{>0}: \\displaystyle \\int_0^x \\map {\\mathcal N} t \\rd t = 0$ Then $\\mathcal N$ is a '''null function'''."}
+{"_id": "30149", "title": "Definition:Half Wave Rectified Sine Curve", "text": "The half wave rectified sine curve is the real function $f: \\R \\to \\R$ defined as: :$\\forall t \\in \\R: \\map f t = \\begin {cases} \\sin t & : 2 n \\pi \\le t \\le \\paren {2 n + 1} \\pi \\\\ 0 & : \\paren {2 n + 1} \\pi \\le t \\le \\paren {2 n + 2} \\pi \\end {cases}$ for all integers $n$. === Graph of Half Wave Rectified Sine Curve === {{:Definition:Half Wave Rectified Sine Curve/Graph}}"}
+{"_id": "30150", "title": "Definition:Convolution Integral", "text": "Let $f$ and $g$ be real functions which are integrable. The '''convolution integral''' of $f$ and $g$ is defined as: :$\\displaystyle \\map f t * \\map g t := \\int_{-\\infty}^\\infty \\map f u \\map g {t - u} \\rd u$"}
+{"_id": "30157", "title": "Definition:Convergent Sequence/Real Numbers/Graphical Illustration", "text": "The following diagram illustrates the first few terms of a convergent real sequence. :510px For the value of  $\\epsilon$ given, a suitable value of $N$ such that: :$n > N \\implies \\size {x_n - l} < \\epsilon$ is $6$."}
+{"_id": "30158", "title": "Definition:Unbounded Divergent Sequence/Real Sequence/Positive Infinity", "text": "$\\sequence {x_n}$ '''diverges to $+\\infty$''' {{iff}}: :$\\forall H \\in \\R_{>0}: \\exists N: \\forall n > N: x_n > H$ That is, whatever positive real number $H$ you choose, for sufficiently large $n$, $x_n$ will exceed $H$. We write: :$x_n \\to +\\infty$ as $n \\to \\infty$ or: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n \\to +\\infty$"}
+{"_id": "30159", "title": "Definition:Unbounded Divergent Sequence/Real Sequence/Negative Infinity", "text": "$\\sequence {x_n}$ '''diverges to $-\\infty$''' {{iff}}: :$\\forall H \\in \\R_{>0}: \\exists N: \\forall n > N: x_n < -H$ That is, whatever positive real number $H$ you choose, for sufficiently large $n$, $x_n$ will be less than $-H$. We write: :$x_n \\to -\\infty$ as $n \\to \\infty$ or: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n \\to -\\infty$"}
+{"_id": "30160", "title": "Definition:Unbounded Divergent Sequence/Real Sequence/Infinity", "text": "$\\sequence {x_n}$ '''diverges to $\\infty$''' {{iff}}: :$\\forall H > 0: \\exists N: \\forall n > N: \\size {x_n} > H$"}
+{"_id": "30165", "title": "Definition:Local Basis/Local Basis for Open Sets", "text": "A '''local basis''' at $x$ is a set $\\BB$ of open neighborhoods of $x$ such that: :$\\forall U \\in \\tau: x \\in U \\implies \\exists H \\in \\BB: H \\subseteq U$ That is, such that every open neighborhood of $x$ also contains some set in $\\BB$."}
+{"_id": "30166", "title": "Definition:Local Basis/Neighborhood Basis of Open Sets", "text": "A '''local basis''' at $x$ is a set $\\BB$ of open neighborhoods of $x$ such that every neighborhood of $x$ contains a set in $\\BB$. That is, a '''local basis''' at $x$ is a neighborhood basis of $x$ consisting of open sets."}
+{"_id": "30169", "title": "Definition:Limit Superior/Definition 1", "text": "Let $L$ be the set of all real numbers which are the limit of some subsequence of $\\sequence {x_n}$. From Existence of Maximum and Minimum of Bounded Sequence, $L$ has a maximum. This maximum is called the '''limit superior'''. It can be denoted: :$\\displaystyle \\map {\\limsup_{n \\mathop \\to \\infty} } {x_n} = \\overline l$"}
+{"_id": "30170", "title": "Definition:Limit Superior/Definition 2", "text": "The '''limit superior of $\\sequence {x_n}$''' is defined and denoted as: :$\\displaystyle \\map {\\limsup_{n \\mathop \\to \\infty} } {x_n} = \\inf \\set {\\sup_{m \\mathop \\ge n} x_m: n \\in \\N}$"}
+{"_id": "30171", "title": "Definition:Limit Inferior/Definition 1", "text": "Let $L$ be the set of all real numbers which are the limit of some subsequence of $\\sequence {x_n}$. From Existence of Maximum and Minimum of Bounded Sequence, $L$ has a minimum. This minimum is called the '''limit inferior'''. It can be denoted: :$\\displaystyle \\map {\\liminf_{n \\mathop \\to \\infty} } {x_n} = \\underline l$"}
+{"_id": "30172", "title": "Definition:Limit Inferior/Definition 2", "text": "The '''limit inferior of $\\sequence {x_n}$''' is defined and denoted as: :$\\displaystyle \\map {\\liminf_{n \\mathop \\to \\infty} } {x_n} = \\sup \\set {\\inf_{m \\mathop \\ge n} x_m: n \\in \\N}$"}
+{"_id": "30173", "title": "Definition:Chi-Squared Distribution", "text": "Let $X$ be a continuous random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\Img X = \\hointr 0 \\infty$.  Let $r$ be a strictly positive integer. $X$ is said to have a '''chi-squared distribution''' with $r$ degrees of freedom {{iff}} it has probability density function: :$\\displaystyle \\map {f_X} x = \\dfrac 1 {2^{r / 2} \\map \\Gamma {r / 2} } x^{\\paren {r / 2} - 1} e^{- x / 2}$ where $\\Gamma$ denotes the gamma function. This is written:  :$X \\sim \\chi^2_r$"}
+{"_id": "30174", "title": "Definition:Operations Research", "text": "'''Operations research''' is the branch of applied mathematics which studies the management of human organizations."}
+{"_id": "30175", "title": "Definition:Chi Distribution", "text": "Let $X$ be a continuous random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\Img X = \\hointr 0 \\infty$.  Let $r$ be a strictly positive integer. $X$ is said to have a '''chi distribution''' with $r$ degrees of freedom {{iff}} it has probability density function: :$\\displaystyle \\map {f_X} x = \\dfrac 1 {2^{\\paren {r / 2} - 1} \\map \\Gamma {r / 2} } x^{r - 1} e^{- x^2 / 2}$ where $\\Gamma$ denotes the gamma function. This is written:  :$X \\sim \\chi_r$"}
+{"_id": "30176", "title": "Definition:F-Distribution", "text": "Let $X$ be a continuous random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\Img X = \\hointr 0 \\infty$.  Let $n, m$ be strictly positive integers. $X$ is said to have an '''F-distribution''' with $\\tuple {n, m}$ degrees of freedom {{iff}} it has probability density function: :$\\displaystyle \\map {f_X} x = \\frac {m^{m / 2} n^{n / 2} x^{\\paren {n / 2} - 1} } {\\paren {m + n x}^{\\paren {n + m} / 2} \\map \\Beta {n / 2, m / 2} }$ where $\\Beta$ denotes the beta function.  This is written:  :$X \\sim F_{n, m}$"}
+{"_id": "30177", "title": "Definition:Constant/Linguistic Note", "text": "The word '''constant''' can be used either as a {{WP|Noun|noun}}: :''Let $c$ be a constant between $0$ and $1$'' or as an {{WP|Adjective|adjective}}: :''Let $c$ be a constant real number between $0$ and $1$'' Which is intended can usually be deduced from the context."}
+{"_id": "30178", "title": "Definition:Chess/Chessboard", "text": "A '''chessboard''' is an array (usually square) of alternating dark and light squares (conventionally referred to as '''black''' and '''white''', even though they may well not be those actual colours). {{ChessDiagram| | |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   | }} This may be referred to as an $m \\times n$ '''chessboard''', where: :$m$ is the number of ranks :$n$ is the number of files. For the conventional '''chessboard''', as depicted above, $m = n = 8$. === Rank === {{:Definition:Chess/Chessboard/Rank}} === File === {{:Definition:Chess/Chessboard/File}} === Square === {{:Definition:Chess/Chessboard/Square}}"}
+{"_id": "30179", "title": "Definition:Hypothesis (Inductive Statistics)", "text": "A '''hypothesis''' is a statement about a population parameter."}
+{"_id": "30180", "title": "Definition:Chess/Chessboard/Rank", "text": "A '''row''' of a '''chessboard''' is known as a '''rank'''. It is identified by '''number''': :$1$ is the '''rank''' nearest white :The '''ranks''' are numbered sequentially to the '''rank''' nearest black."}
+{"_id": "30181", "title": "Definition:Chess/Player", "text": "The two players in a game of chess are identified by the colour of the pieces they are using. The sets of pieces are made from materials of two contrasting colours, conventionally referred to as '''black''' and '''white'''. Most everyday chess sets actually ''are'' '''black''' and '''white'''. === White === {{:Definition:Chess/Player/White}} === Black === {{:Definition:Chess/Player/Black}} A diagram of a chessboard is conventionally positioned on a page such that '''white''' is at the bottom, playing upwards towards '''black''', and '''black''' is at the top, playing downwards towards '''white'''."}
+{"_id": "30182", "title": "Definition:Chess/Player/White", "text": "The '''chess player''' who is using the lighter-coloured pieces is conventionally referred to as '''white'''."}
+{"_id": "30183", "title": "Definition:Chess/Player/Black", "text": "The '''chess player''' who is using the darker-coloured pieces is conventionally referred to as '''black'''."}
+{"_id": "30184", "title": "Definition:Chess/Chessboard/File", "text": "A '''column''' of a '''chessboard''' is known as a '''file'''. It is identified by '''letter''' :$\\text a$ is the '''file''' on the edge of the board to the left of white (that is, to the right of black) :The '''files''' are lettered sequentially up to the '''file''' on the edge of the board to the left of black (that is, to the right of white)."}
+{"_id": "30185", "title": "Definition:Chess/Chessboard/Square", "text": "The individual spaces of the chessboard are referred to as its '''squares'''. They are conventionally identified by a letter-number pair, where: :the letter identifies its file :the number identifies its rank."}
+{"_id": "30187", "title": "Definition:Chess/Piece/King", "text": "The '''king''' File:Chess klt45.svg can be moved one square in any direction, including diagonally. It cannot be moved into a position where it is in check."}
+{"_id": "30189", "title": "Definition:Chess/Rules/Castling", "text": "'''Castling''' is a special move where, in one move, the king moves two squares towards a rook, and the rook jumps over the king and lands on the next square the other side of the king. This can only be done if: :$(1): \\quad$ The king and rook are both in their initial positions. :$(2): \\quad$ The squares in between the king and rook are empty. :$(3): \\quad$ The king is not in check, and neither of the $2$ squares it moves through are under attack."}
+{"_id": "30190", "title": "Definition:Chess/Piece/Queen", "text": "The '''queen''' File:Chess qlt45.svg can be moved any number of squares in any of $8$ directions: forwards, backwards, left, right, and the $4$ diagonals. It cannot be moved past a square occupied by another piece."}
+{"_id": "30195", "title": "Definition:Chess/Draw/Insufficient Pieces", "text": "If neither player has sufficient pieces to force checkmate, then the game of chess is '''drawn'''."}
+{"_id": "30197", "title": "Definition:Chess/Rules/Under Attack", "text": "If a chess piece $A$ may legally move to a square which is occupied by a piece $B$ of the opposite colour, $B$ is said to be '''under attack''' by $A$. In the same way, any square to which $A$ may legally move may also be said to be '''under attack'''."}
+{"_id": "30198", "title": "Definition:Chess/Rules/In Check", "text": "Uniquely among the chess pieces the king may not legally move into a square which is under attack by a piece of the opposite colour. That is, a king may not be moved so as to be placed under attack. If player $a$ moves a piece so as to place the king of player $b$ under attack, then player $b$ is said to be '''in check'''. Player $a$ is then obliged to say '''\"Check.\"''' Player $b$ must make a move so that his or her king is no longer under attack. Such a move by player $b$ is known as '''getting out of check'''."}
+{"_id": "30199", "title": "Definition:Chess/Rules/Checkmate", "text": "If player $a$ has put player $b$'s king in check, but player $b$ has no legal move that will get him or her out of check, then player $a$ has won. This situation is known as '''checkmate'''. Player $a$ is said to have '''checkmated''' player $b$."}
+{"_id": "30200", "title": "Definition:Chess/Rules", "text": "With the exception of the knight, no chess piece may move past a square which is occupied by another piece. === Under Attack === {{:Definition:Chess/Rules/Under Attack}} === Capture === {{:Definition:Chess/Rules/Capture}} === Pawn === {{:Definition:Chess/Rules/Capture/Pawn}} === Castling === {{:Definition:Chess/Rules/Castling}} === In Check === {{:Definition:Chess/Rules/In Check}} === Checkmate === {{:Definition:Chess/Rules/Checkmate}}"}
+{"_id": "30201", "title": "Definition:Chess/Piece/Bishop", "text": "The '''bishop''' File:Chess blt45.svg can be moved any number of squares in any of the $4$ diagonal directions. It cannot be moved past a square occupied by another piece."}
+{"_id": "30203", "title": "Definition:Chess/Piece/Rook", "text": "The '''rook''' File:Chess rlt45.svg can be moved any number of squares in any of the $4$ orthogonal directions: up, down, left or right. It cannot be moved past a square occupied by another piece."}
+{"_id": "30204", "title": "Definition:Chess/Piece/Knight", "text": "The '''knight''' File:Chess nlt45.svg can be moved two squares in any orthogonal direction (up, down, left or right), and then one square at right angles to that direction. It passes over intervening pieces and does not stop until reaching the end of its move. It cannot be moved onto a square occupied by another piece of the same colour."}
+{"_id": "30205", "title": "Definition:Chess/Pawn/Promotion", "text": "If a pawn moves onto the square on the final rank, it is then exchanged for any other type of piece which replaces the pawn on that square. This is called '''promotion'''. The player of that pawn decides which piece to replace it with, which may be neither another pawn nor a king. If it is promoted to a queen (which is usual, the queen being the most powerful piece), this operation is colloquially referred to as '''queening'''."}
+{"_id": "30206", "title": "Definition:Chess/Pawn", "text": "A '''pawn''' File:Chess plt45.svg may move in the following modes. $(1): \\quad$ It may move one square towards the opposing player, {{iff}} that square is empty. $(2): \\quad$ If it has not yet been moved, and {{iff}} both of the squares is empty, it may move $2$ squares forward instead of $1$. Some authorities do not classify the pawn as a piece. === Promotion === {{:Definition:Chess/Pawn/Promotion}}"}
+{"_id": "30207", "title": "Definition:Chess/Rules/Capture/Pawn/Normal", "text": "If one of the $2$ diagonally adjacent squares towards the opposing player is occupied by an opposing piece, the pawn may move into that square and capture that piece."}
+{"_id": "30208", "title": "Definition:Chess/Rules/Capture/Pawn/En Passant", "text": "Let pawn $a$ be on the player's $5$th rank. Let pawn $b$ be of the opposite colour to pawn $a$. Let pawn $b$ be on its starting position, on one of the files adjacent to the one occupied by pawn $a$. Let pawn $b$ move forward $2$ spaces, in the process crossing over one of the squares which is under attack from pawn $a$. Then pawn $a$, '''on its next move only''', may move into that square crossed over by pawn $b$, and capture pawn $b$ \"while it is passing\". This mode of capture is known as '''capture ''en passant'''''."}
+{"_id": "30210", "title": "Definition:Chess/Rules/Capture/Pawn", "text": "A capture in chess by a pawn works completely differently from that by any other piece. It may happen in one of $2$ circumstances. ==== Normal Pawn Capture ==== {{:Definition:Chess/Rules/Capture/Pawn/Normal}} ==== En Passant Pawn Capture ==== {{:Definition:Chess/Rules/Capture/Pawn/En Passant}}"}
+{"_id": "30211", "title": "Definition:Chess/Piece", "text": "There are a number of different kinds of '''chess piece'''. They are distinguished by their modes of movement, as follows: === King === {{:Definition:Chess/Piece/King}} === Queen === {{:Definition:Chess/Piece/Queen}} === Bishop === {{:Definition:Chess/Piece/Bishop}} === Knight === {{:Definition:Chess/Piece/Knight}} === Rook === {{:Definition:Chess/Piece/Rook}} === Pawn === {{:Definition:Chess/Pawn}}"}
+{"_id": "30212", "title": "Definition:Chess/Starting Position", "text": "The '''starting position''' of a game of chess is: {{ChessDiagram| | |rd|nd|bd|qd|kd|bd|nd|rd |pd|pd|pd|pd|pd|pd|pd|pd |  |  |  |  |  |  |  |    |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |pl|pl|pl|pl|pl|pl|pl|pl |rl|nl|bl|ql|kl|bl|nl|rl | }} Note the following: :$(1): \\quad$ The top left square is white :$(2): \\quad$ The queens start on squares of their own colours :$(3): \\quad$ White is always first to move."}
+{"_id": "30213", "title": "Definition:Chess/Move", "text": "Each turn in a game of chess consists of transferring a piece from one square to another, according to the rules. Such a transfer is known as a '''move'''. There is one exception: the operation of '''castling''' involves the moving of $2$ pieces."}
+{"_id": "30215", "title": "Definition:Retrograde Analysis", "text": "'''Retrograde analysis''' is a technique used to determine the sequence of chess moves which were played that led to a given position."}
+{"_id": "30216", "title": "Definition:Hypothesis Test", "text": "A '''hypothesis test''' is a rule that specifies, for a null hypothesis $H_0$ and alternative hypothesis $H_1$:  * For which sample values the decision is made to accept $H_0$. * For which sample values $H_0$ is rejected and $H_1$ is accepted."}
+{"_id": "30217", "title": "Definition:Erlang Distribution", "text": "Let $X$ be a continuous random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\Img X = \\hointr 0 \\infty$.  Let $k$ be a strictly positive integer. Let $\\lambda$ be a strictly positive real number. $X$ is said to have an '''Erlang distribution''' with parameters $k$ and $\\lambda$ {{iff}} it has probability density function: :$\\map {f_X} x = \\dfrac {\\lambda^k x^{k - 1} e^{- \\lambda x} } {\\map \\Gamma k}$ where $\\Gamma$ denotes the gamma function. This is written:  :$X \\sim \\map {\\operatorname {Erlang} } {k, \\lambda}$"}
+{"_id": "30218", "title": "Definition:Laplace Distribution", "text": "Let $X$ be a continuous random variable on a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\Img X = \\R$.  Let $\\mu$ be a real number.  Let $b$ be a strictly positive real number. $X$ is said to have a '''Laplace distribution''' with parameters $\\mu$ and $b$ {{iff}} it has probability density function: :$\\map {f_X} x = \\dfrac 1 {2 b} \\map \\exp {- \\dfrac {\\size {x - \\mu} } b}$ This is written:  :$X \\sim \\map {\\operatorname {Laplace} } {\\mu, b}$"}
+{"_id": "30220", "title": "Definition:Bijection/Technical Note", "text": "{{LatexFor|for = f: S \\leftrightarrow T}} {{LatexFor|for = f: S \\cong T}} {{LatexFor|for = S \\stackrel f \\cong T}}"}
+{"_id": "30221", "title": "Definition:Simple Hypothesis", "text": "Let $\\theta$ be a population parameter of some population.  Let the parameter space of $\\theta$ be $\\Omega$.  Let $\\Omega_0$ and $\\Omega_1$ be disjoint subsets of $\\Omega$ such that $\\Omega_0 \\cup \\Omega_1 = \\Omega$.  Consider the hypotheses:  :$H_0: \\theta \\in \\Omega_0$ :$H_1: \\theta \\in \\Omega_1$ We call $H_i$, for $i \\in \\set {0, 1}$, a '''simple hypothesis''' if $\\Omega_i$ contains only a single element."}
+{"_id": "30222", "title": "Definition:Composite Hypothesis", "text": "Let $\\theta$ be a population parameter of some population.  Let the parameter space of $\\theta$ be $\\Omega$.  Let $\\Omega_0$ and $\\Omega_1$ be disjoint subsets of $\\Omega$ such that $\\Omega_0 \\cup \\Omega_1 = \\Omega$.  Consider the hypotheses:  :$H_0: \\theta \\in \\Omega_0$ :$H_1: \\theta \\in \\Omega_1$ We call $H_i$, for $i \\in \\set {0, 1}$, a '''composite hypothesis''' if $\\Omega_i$ contains more than a single element."}
+{"_id": "30223", "title": "Definition:Critical Region", "text": "Let $\\theta$ be a population parameter of some population $P$.  Let $\\Omega$ be the parameter space of $\\theta$.  Let $\\mathbf X$ be a random sample from $P$.  Let $T = \\map f {\\mathbf X}$ be a sample statistic. Let $\\delta$ be a test procedure of the form:  :reject $H_0$ if $T \\in C$ for some null hypothesis $H_0$ and some $C \\subset \\Omega$. We refer to $C$ as the '''critical region''' of $\\delta$."}
+{"_id": "30224", "title": "Definition:Test Statistic", "text": "Let $\\theta$ be a population parameter of some population $P$.  Let $\\Omega$ be the parameter space of $\\theta$.  Let $\\mathbf X$ be a random sample from $P$.  Let $T = \\map f {\\mathbf X}$ be a sample statistic. Let $\\delta$ be a test procedure of the form:  :reject $H_0$ if $T \\in C$ for some null hypothesis $H_0$ and some $C \\subset \\Omega$. We refer to $T$ as the '''test statistic''' of $\\delta$."}
+{"_id": "30225", "title": "Definition:Power Function", "text": "Let $\\theta$ be a population parameter of some population. Let $\\Omega$ be the parameter space of $\\theta$.    Let $\\delta$ be a test procedure for hypotheses about the value of $\\theta$.  Let $C$ be the critical region of $\\delta$.  Let $T$ be the test statistic of $\\delta$.  The '''power function''' of $\\delta$, written $\\map \\pi {\\theta_0}$, is defined by: :$\\map \\pi {\\theta_0} = \\map \\Pr {T \\in C \\mid \\theta = \\theta_0 }$ for all $\\theta_0 \\in \\Omega$."}
+{"_id": "30226", "title": "Definition:Parameter Space", "text": "Let $\\theta$ be a population parameter of some population. The '''parameter space''' of $\\theta$, typically denoted $\\Omega$, is the domain of $\\theta$. That is, it is the set of all possible values that $\\theta$ can take."}
+{"_id": "30227", "title": "Definition:Type I Error", "text": "A '''type I error''' is the rejection of a true null hypothesis as the result of a test procedure."}
+{"_id": "30228", "title": "Definition:Type II Error", "text": "A '''type II error''' is the failure to reject a false null hypothesis as the result of a test procedure."}
+{"_id": "30229", "title": "Definition:Size (Inductive Statistics)", "text": "Let $\\theta$ be a population parameter of some population. Let $\\Omega$ be the parameter space of $\\theta$.  Let $\\Omega_0$ and $\\Omega_1$ be disjoint subsets of $\\Omega$ such that $\\Omega_0 \\cup \\Omega_1 = \\Omega$. Let $\\delta$ be a test procedure of the hypotheses: :$H_0: \\theta \\in \\Omega_0$ :$H_1: \\theta \\in \\Omega_1$ Let $\\pi$ be the power function of $\\delta$.  The '''size''' of $\\delta$, usually denoted $\\alpha$, is defined as:  :$\\displaystyle \\alpha = \\sup_{\\theta \\in \\Omega_0} \\map \\pi \\theta$"}
+{"_id": "30230", "title": "Definition:Covariance", "text": "Let $X$ and $Y$ be random variables.  Let $\\mu_X = \\expect X$ and $\\mu_Y = \\expect Y$, the expectations of $X$ and $Y$ respectively, exist and be finite. Then the '''covariance''' of $X$ and $Y$ is defined by:  :$\\cov {X, Y} = \\expect {\\paren {X - \\mu_X} \\paren {Y - \\mu_Y} }$ where this expectation exists."}
+{"_id": "30231", "title": "Definition:Pearson Correlation Coefficient", "text": "Let $X$ and $Y$ be random variables.  Let the variances of $X$ and $Y$ exist and be finite. Then the '''Pearson correlation coefficient''' of $X$ and $Y$, typically denoted $\\map \\rho {X, Y}$, is defined by:  :$\\map \\rho {X, Y} = \\dfrac {\\cov {X, Y} } {\\sqrt {\\var X \\, \\var Y} }$ where $\\cov {X, Y}$ is the covariance of $X$ and $Y$."}
+{"_id": "30232", "title": "Definition:Transversality Conditions", "text": "Let $\\map {\\mathbf y} x$ be a differentiable vector-valued function. Let $J \\sqbrk {\\mathbf y}$ be a functional of the following form: :$\\ds J \\sqbrk {\\mathbf y} = \\int_{P_1}^{P_2} \\map F {x, \\mathbf y, \\mathbf y', \\ldots} \\rd x$ where $P_1$, $P_2$ are points on given differentiable manifolds $M_1$ and $M_2$. Suppose we are looking for $\\mathbf y$ extremizing $J$. The system of equations to be solved consists of differential Euler equations and algebraic equations at both endpoints. Then the set of all algebraic equations at both endpoints are called '''transversality conditions'''."}
+{"_id": "30234", "title": "Definition:Unbiased Estimator", "text": "Let $\\theta$ be a population parameter of some statistical model. Let $\\delta$ be an estimator of $\\theta$. We call $\\delta$ an '''unbiased estimator''' if its bias is equal to $0$ regardless of the true value of $\\theta$."}
+{"_id": "30235", "title": "Definition:Hardy-Littlewood Maximal Function", "text": "{{MissingLinks|throughout}} {{explain|presumably the \"Hardy-Littlewood Maximal Function\" is that function which is returned by the Hardy-Littlewood maximal operator? More precision needed here.}} The Hardy-Littlewood maximal operator takes a locally integrable function $f: \\R^d \\to \\R$  and returns another function $M f$ that, at each point $x \\in \\R^d$ gives the maximum average value that $f$ can have on balls centered at that point.  More precisely, :$\\displaystyle \\map {M f} x := \\sup_{r \\mathop > 0} \\frac 1 {\\size {\\map B {x, r} } } \\int_{\\map B {x, r} } \\size {\\map f y} \\rd y$ where: :$\\map B {x, r}$  is the ball of radius $r$ centered at $x$ :$\\size E$ denotes the Lebesgue measure of $E \\subset \\R^d$."}
+{"_id": "30236", "title": "Definition:Precisely One Function", "text": "Let $p_1, p_2, \\ldots, p_n$ be statements. The '''precisely one function''' is the propositional function $\\map P {p_1, p_2, \\ldots, p_n}$ defined as: :$\\map P {p_1, p_2, \\ldots, p_n}$ is true {{iff}} '''precisely one''' of $p_1, p_2, \\ldots, p_n$ is true."}
+{"_id": "30238", "title": "Definition:Differential Equation/Total", "text": "A '''total differential equation''' is a '''differential equation''' which contains: :more than one dependent variable :one independent variable which may or may not appear explicitly in that differential equation."}
+{"_id": "30239", "title": "Definition:Differential Equation/Degree", "text": "Let $f$ be a '''differential equation''' which can be expressed as a polynomial in all the derivatives involved. The '''degree''' of $f$ is defined as being the power to which the derivative of the highest order is raised. By default, if not specifically mentioned, the '''degree''' of a '''differential equation''' is assumed to be $1$."}
+{"_id": "30241", "title": "Definition:Alexandrian School", "text": "The '''Alexandrian School''' was active in the city of {{WP|Alexandria|Alexandria}} for the approximate millennium leading up to the city's destruction in $641$ C.E. It stood as the cultural hub of the Greek sphere of influence. The {{WP|Musaeum|Alexandrian Musaeum}} stood at the centre of this hub, and has been argued as being the archetypal forerunner of the modern University. It had a library which, at its peak, is believed to have contained over $700 \\, 000$ volumes. After the fall of {{WP|Alexandria|Alexandria}}, most of its scholars migrated to {{WP|Constantinople|Constantinople}}, preserving the ancient Greek mathematical learning, where it remained for some $800$ more years until the {{WP|Renaissance|Renaissance}}."}
+{"_id": "30243", "title": "Definition:Convolution of Probability Distributions", "text": "Let $X$ and $Y$ be independent random variables.  Let $Z = X + Y$.  The probability distribution of $Z$ is called the '''convolution''' of the probability distributions of $X$ and $Y$."}
+{"_id": "30244", "title": "Definition:Convergence in Probability", "text": "Let $\\sequence {X_n}_{n \\ge 1}$ be a sequence of random variables. Let $X$ be a random variable.  We say that $\\sequence {X_n}$ '''converges in probability''' to $X$ if:  :$\\displaystyle \\lim_{n \\to \\infty} \\map \\Pr {\\size {X_n - X} < \\varepsilon} = 1$ for all real $\\varepsilon > 0$. This is written:  :$X_n \\xrightarrow p X$"}
+{"_id": "30245", "title": "Definition:Convergence in Distribution", "text": "Let $\\sequence {X_n}_{n \\ge 1}$ be a sequence of random variables. Let $X$ be a random variable.  We say that $\\sequence {X_n}$ '''converges in distribution''' to $X$ if:  :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map \\Pr {X_n \\le x} = \\map \\Pr {X \\le x}$ for all $x$ for which the map $x \\mapsto \\map \\Pr {X \\le x}$ is continuous. This is written:  :$X_n \\xrightarrow d X$"}
+{"_id": "30246", "title": "Definition:Random Sample", "text": "Let $X_i$ be a random variable with $\\Img {X_i} = \\Omega$, for all $1 \\le i \\le n$. Let $F_i$ be the cumulative distribution function of $X_i$ for all $1 \\le i \\le n$.  We say that $X_1, X_2, \\ldots, X_n$ form a '''random sample''' of size $n$ if:  :$X_i$ and $X_j$ are independent if $i \\ne j$ :$\\map {F_1} x = \\map {F_i} x$ for all $x \\in \\Omega$ for all $1 \\le i, j \\le n$. If $X_1, X_2, \\ldots, X_n$ form a '''random sample''', they are said to be '''independent and identically distributed''', commonly abbreviated '''i.i.d'''."}
+{"_id": "30247", "title": "Definition:Almost Sure Convergence", "text": "Let $\\sequence {X_n}_{n \\ge 1}$ be a sequence of random variables. Let $X$ be a random variable.  We say that $\\sequence {X_n}$ '''almost surely converges''' to $X$ if:  :$\\displaystyle \\map \\Pr {\\lim_{n \\mathop \\to \\infty} \\size {X_n - X} < \\varepsilon} = 1$ for all real $\\varepsilon > 0$. This is written: :$X_n \\xrightarrow {\\text {a.s.}} X$"}
+{"_id": "30248", "title": "Definition:Path Component/Equivalence Class", "text": "Let $\\sim$ be the equivalence relation on $T$ defined as: :$x \\sim y \\iff x$ and $y$ are path-connected. The equivalence classes of $\\sim$ are called the '''path components of $T$'''. If $x \\in T$, then the '''path component of $T$''' containing $x$ (that is, the set of points $y \\in T$ with $x \\sim y$) can be denoted by $\\map {\\operatorname{PC}_x} T$."}
+{"_id": "30249", "title": "Definition:Path Component/Union of Path-Connected Sets", "text": "The '''path component of $T$ containing $x$''' is defined as: : $\\displaystyle \\map {\\operatorname{PC}_x} T = \\bigcup \\left\\{{A \\subseteq S: x \\in A \\land A}\\right.$ is path-connected $\\left.\\right\\}$"}
+{"_id": "30250", "title": "Definition:Path Component/Maximal Path-Connected Set", "text": "The '''path component of $T$ containing $x$''' is defined as: :the maximal path-connected set of $T$ that contains $x$."}
+{"_id": "30251", "title": "Definition:Sufficient Statistic", "text": "Let $X_1, X_2, \\ldots, X_n$ form a random sample from a population indexed by a parameter $\\theta$. Let $T$ be a sample statistic. Let $I = \\Img {\\map T {X_1, X_2, \\ldots, X_n} }$. Let $D$ be the conditional joint distribution of $X_1, X_2, \\ldots, X_n$ given $T = t$ and $\\theta$.   We call $T$ a '''sufficient statistic''' for $\\theta$ if $D$ is independent of the value of $\\theta$ for all $t \\in I$."}
+{"_id": "30253", "title": "Definition:Error/Absolute", "text": "The '''absolute error''' $\\varepsilon$ is the difference between $x$ and a $X$, and can be defined in one of three ways: {{begin-eqn}} {{eqn | n = 1       | l = \\varepsilon       | o = :=       | r = X - x }} {{eqn | n = 2       | l = \\varepsilon       | o = :=       | r = x - X }} {{eqn | n = 3       | l = \\varepsilon       | o = :=       | r = \\size {X - x} }} {{end-eqn}} where $\\size {X - x}$ denotes the absolute value of $X - x$. Different sources use different conventions."}
+{"_id": "30254", "title": "Definition:Error/Relative", "text": "The '''relative error''' in $x$ is defined as: :$\\dfrac {\\size {X - x} } X$ :where $\\size {X - x}$ denotes the absolute value of $X - x$."}
+{"_id": "30256", "title": "Definition:Abstraction", "text": "'''Abstraction''' is the process of making a general statement from a collection of particular statements so as to summarise a property that applies to the subject of those particular statements."}
+{"_id": "30257", "title": "Definition:Imperial/Area/Square Mile", "text": "One '''square mile''' is equal to a square whose side measures $1$ mile. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''square mile''' }} {{eqn | r = 640       | c = acres }} {{end-eqn}}"}
+{"_id": "30258", "title": "Definition:Hectare", "text": "One '''hectare''' is equal to a square whose side measures $100$ metres. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''hectare''' }} {{eqn | r = 100       | c = ares }} {{eqn | r = 10 \\, 000       | c = square metres }} {{eqn | o = \\approx       | r = 0.4       | c = acres }} {{end-eqn}}"}
+{"_id": "30259", "title": "Definition:Angle/Adjacent/Also defined as", "text": "Some sources give that '''adjacent angles''' are the angles on a straight line at the intersection of that line and another. Under such a definition, $\\angle \\alpha$ and $\\angle \\beta$ are '''adjacent''' in the diagram below: :250px Such '''adjacent angles''' are seen to be supplementary."}
+{"_id": "30261", "title": "Definition:The Algebra of Sets", "text": "Let $E$ be a universal set. The power set $\\powerset E$, together with: :the binary operations union $\\cup$ and intersection $\\cap$ :the unary operation complement $\\complement$ is referred to as '''the algebra of sets''' on $E$."}
+{"_id": "30262", "title": "Definition:Absolute Geometry", "text": "'''Absolute geometry''' is the study of '''Euclidean geometry''' without the parallel postulate."}
+{"_id": "30263", "title": "Definition:Normal Real Number", "text": "A real number $r$ is '''normal''' with respect to a number base $b$ {{iff}} its basis expansion in number base $b$ is such that: :no finite sequence of digits of $r$ of length $n$ occurs more frequently than any other such finite sequence of length $n$."}
+{"_id": "30268", "title": "Definition:Disjunction/Disjunct", "text": "The substatements $p$ and $q$ are known as the '''disjuncts''', or the '''members of the disjunction'''."}
+{"_id": "30274", "title": "Definition:Dirichlet Beta Function", "text": "The '''Dirichlet beta function''' $\\beta$ is the complex function defined on the half-plane $\\map \\Re s > 0$ by the series:  :$\\displaystyle \\map \\beta s = \\sum_{n \\mathop = 0}^\\infty \\frac {\\paren {-1}^n} {\\paren {2 n + 1}^s}$"}
+{"_id": "30275", "title": "Definition:Convergent Sequence/Metric Space/Definition 4", "text": "$\\sequence {x_k}$ '''converges to the limit $l \\in A$''' {{iff}}: :for every $\\epsilon \\in \\R{>0}$, the open $\\epsilon$-ball about $l$ contains all but finitely many of the $p_n$."}
+{"_id": "30276", "title": "Definition:Alexandroff Extension of Real Number Line", "text": "The '''Alexandroff extension of the real number line''' $\\R^*$ is defined as: :$\\R^* := \\R \\cup \\set \\infty$ that is, the set of real numbers together with an element $\\infty$ which is not in $\\R$."}
+{"_id": "30277", "title": "Definition:Möbius Transformation/Real Numbers", "text": "A '''Möbius transformation''' is a mapping $f: \\R^* \\to \\R^*$ of the form: :$\\map f x = \\dfrac {a x + b} {c x + d}$ where: :$\\R^*$ denotes the Alexandroff extension of the real number line :$a, b, c, d \\in \\R$ such that $a d - b c \\ne 0$ We define: :$\\map f {-\\dfrac d c} = \\infty$ if $c \\ne 0$, and: :$\\map f \\infty = \\begin{cases} \\dfrac a c & : c \\ne 0 \\\\ \\infty & : c = 0 \\end{cases}$"}
+{"_id": "30282", "title": "Definition:P-adic Expansion", "text": "Let $p$ be a prime number. A '''$p$-adic expansion''' is a power series in the rational numbers $\\Q$ of the form: :$\\displaystyle \\sum_{n \\mathop = m}^\\infty d_n p^n$ where: :$m \\in \\Z_{\\le 0}$ :$\\forall n \\in \\Z_{\\ge m}: d_n \\in \\N \\mathop {\\text{ and } } 0 \\le d_n < p$ :$m < 0 \\implies d_m \\ne 0$"}
+{"_id": "30285", "title": "Definition:Composition of Mappings/Definition 2", "text": "The '''composite of $f_1$ and $f_2$''' is defined and denoted as: :$f_2 \\circ f_1 := \\set {\\tuple {x, z} \\in S_1 \\times S_3: \\tuple {\\map {f_1} x, z} \\in f_2}$"}
+{"_id": "30286", "title": "Definition:Composition of Mappings/Definition 1", "text": "The '''composite mapping''' $f_2 \\circ f_1$ is defined as: :$\\forall x \\in S_1: \\map {\\paren {f_2 \\circ f_1} } x := \\map {f_2} {\\map {f_1} x}$"}
+{"_id": "30287", "title": "Definition:Generated Ring Extension", "text": "Let $S$ be a commutative rings with unity. Let $R$ be a subring of $S$ with unity such that the unity of $R$ is the unity of $S$. That is, $S$ is a ring extension of $R$. Let $T \\subseteq S$ be a subset of $S$."}
+{"_id": "30288", "title": "Definition:Generated Ring Extension/Smallest Subring", "text": "The '''ring extension $R \\sqbrk T$ generated by $T$''' is the smallest subring of $S$ containing $T$ and $R$, that is, the intersection of all subrings of $S$ containing $T$ and $R$. Thus $T$ is a '''generator''' of $R \\sqbrk T$ {{iff}} $R \\sqbrk T$ has no proper subring containing $T$ and $R$."}
+{"_id": "30289", "title": "Definition:Generated Ring Extension/Evaluation of Polynomial Ring", "text": "Let $R \\sqbrk {\\set {X_t} }$ be the polynomial ring in $T$ variables $X_t$. Let $\\operatorname {ev} : R \\sqbrk {\\set {X_t} } \\to S$ be the evaluation homomorphism associated with the inclusion $T \\hookrightarrow S$. The '''ring extension $R \\sqbrk T$ generated by $T$''' is $\\map {\\operatorname {Img}} {\\operatorname {ev}}$, the image  of $\\operatorname {ev}$. $T$ is said to be a '''generator''' of $R \\sqbrk T$."}
+{"_id": "30290", "title": "Definition:Composition of Mappings/Commutative Diagram", "text": "The concept of '''composition of mappings''' can be illustrated by means of a commutative diagram. This diagram illustrates the specific example of $f_2 \\circ f_1$: ::$\\begin{xy}\\xymatrix@+1em{  S_1   \\ar[r]^*+{f_1}   \\ar@{-->}[rd]_*[l]+{f_2 \\mathop \\circ f_1} &  S_2   \\ar[d]^*+{f_2} \\\\ &  S_3 }\\end{xy}$"}
+{"_id": "30291", "title": "Definition:Composition of Mappings/Binary Operation", "text": "Let $\\sqbrk {S \\to S}$ be the set of all mappings from a set $S$ to itself. Then the concept of '''composite mapping''' defines a binary operation on $\\sqbrk {S \\to S}$: :$\\forall f, g \\in \\sqbrk {S \\to S}: g \\circ f = \\set {\\tuple {s, t}: s \\in S, \\tuple {f \\paren s, t} \\in g} \\in \\sqbrk {S \\to S}$ Thus, for every pair $\\tuple {f, g}$ of mappings in $\\sqbrk {S \\to S}$, the composition $g \\circ f$ is another element of $\\sqbrk {S \\to S}$."}
+{"_id": "30292", "title": "Definition:Composition of Mappings/Warning", "text": "Let $f_1: S_1 \\to S_2$ and $f_2: S_2 \\to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$. If $\\Dom {f_2} \\ne \\Cdm {f_1}$, then the '''composite mapping''' $f_2 \\circ f_1$ is '''not defined'''. This definition is directly analogous to that of composition of relations owing to the fact that a mapping is a special kind of relation."}
+{"_id": "30293", "title": "Definition:Composition of Mappings/Also known as", "text": "In the context of analysis, this is often found referred to as a '''function of a function''', which (according to some sources) makes set theorists wince, as it is technically defined as a '''function on the codomain of a function'''. Some sources call $f_2 \\circ f_1$ the '''resultant of $f_1$ and $f_2$''' or the '''product of $f_1$ and $f_2$'''. Some authors write $f_2 \\circ f_1$ as $f_2 f_1$. Some use the notation $f_2 \\cdot f_1$ or $f_2 . f_1$. Some use the notation $f_2 \\bigcirc f_1$. Others, particularly in books having ties with computer science, write $f_1; f_2$ or $f_1 f_2$ (note the reversal of order), which is read as '''(apply) $f_1$, then $f_2$'''."}
+{"_id": "30294", "title": "Definition:Composition of Mappings/Definition 3", "text": "The '''composite of $f_1$ and $f_2$''' is defined and denoted as: :$f_2 \\circ f_1 := \\set {\\tuple {x, z} \\in S_1 \\times S_3: \\exists y \\in S_2: \\map {f_1} x = y \\land \\map {f_2} y = z}$"}
+{"_id": "30295", "title": "Definition:Cartesian Product/Cartesian Space/Family of Sets/Definition 1", "text": "The '''Cartesian space of $S$ indexed by $I$''' is the set of all families $\\family {s_i}_{i \\mathop \\in I}$ with $s_i \\in S$ for each $i \\in I$: :$S_I := \\displaystyle \\prod_I S = \\set {\\family {s_i}_{i \\mathop \\in I}: s_i \\in S}$"}
+{"_id": "30296", "title": "Definition:Cartesian Product/Cartesian Space/Family of Sets/Definition 2", "text": "The '''Cartesian space of $S$ indexed by $I$''' is defined and denoted as: :$\\displaystyle S^I := \\set {f: \\paren {f: I \\to S} \\land \\paren {\\forall i \\in I: \\paren {\\map f i \\in S} } }$"}
+{"_id": "30297", "title": "Definition:Definition/If or Iff", "text": "It is a standard convention, when making a definition in mathematics, to use '''if''' to introduce the definiens, when in fact the intent is generally '''iff''', that is: '''{{iff}}'''. This convention is specifically '''not followed''' on {{ProofWiki}}, where the mandatory style is to use '''{{iff}}'''."}
+{"_id": "30300", "title": "Definition:Characteristic Property of Set", "text": "Let $S$ be a set which is specified by means of such a propositional function: :$S = \\set {a: \\map P a}$ The open statement $\\map P x$ is known as a '''characteristic property''' of $S$."}
+{"_id": "30303", "title": "Definition:Fresnel Sine Integral Function", "text": "The '''Fresnel sine integral function''' is the real function $\\R \\to \\R$ defined by:  :$\\displaystyle \\map {\\operatorname S} x = \\sqrt {\\frac 2 \\pi} \\int_0^x \\sin u^2 \\rd u$"}
+{"_id": "30304", "title": "Definition:Fresnel Cosine Integral Function", "text": "The '''Fresnel cosine integral function''' is the real function $\\R \\to \\R$ defined by:  :$\\displaystyle \\map {\\operatorname C} x = \\sqrt {\\frac 2 \\pi} \\int_0^x \\cos u^2 \\rd u$"}
+{"_id": "30305", "title": "Definition:Ring of Mappings", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $S$ be a set. Let $R^S$ be the set of all mappings from $S$ to $R$. The '''ring of mappings''' from $S$ to $R$ is the algebraic structure $\\struct {R^S, +', \\circ'}$ where $+'$ and $\\circ'$ are the (pointwise) operations induced on $R^S$ by $+$ and $\\circ$."}
+{"_id": "30306", "title": "Definition:Hypergeometric Differential Equation", "text": "A '''hypergeometric differential equation''' is a second order ODE of the form: :$\\displaystyle x \\paren {1 - x} \\frac {\\d^2 y} {\\d x^2} + \\paren {c - \\paren {a + b + 1} x} \\frac {\\d y} {\\d x} - a b y = 0$"}
+{"_id": "30307", "title": "Definition:Legendre's Associated Differential Equation", "text": "'''Legendre's associated differential equation''' is a second order ODE of the form: :$\\displaystyle \\paren {1 - x^2} \\frac {\\d^2 y} {\\d x^2} - 2 x \\frac {\\d y} {\\d x} + \\paren {n \\paren {n + 1} - \\frac {m^2} {1 - x^2} } y = 0$"}
+{"_id": "30309", "title": "Definition:Ring of Mappings/Zero", "text": "The zero of the ring of mappings is the constant mapping $f_0 : S \\to R$ defined by: :$\\quad \\forall s \\in S : \\map {f_0} x = 0$ where $0$ is the zero in $R$"}
+{"_id": "30310", "title": "Definition:Ring of Mappings/Additive Inverse", "text": "The additive inverse in the ring of mappings is defined by: :$\\forall f \\in R^S : -f \\in R^S : \\forall s \\in S : \\map {\\paren {-f} } x = -\\map f x$"}
+{"_id": "30311", "title": "Definition:Ring of Mappings/Unity", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose unity is $1$."}
+{"_id": "30312", "title": "Definition:Ring of Mappings/Units", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity $1$."}
+{"_id": "30313", "title": "Definition:Ring of Mappings/Commutativity", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring."}
+{"_id": "30314", "title": "Definition:Coproduct/Sets", "text": "Let $S_1$ and $S_2$ be sets. A '''coproduct''' $\\struct {C, i_1, i_2}$ of $S_1$ and $S_2$ comprises a set $C$ together with mappings $i_1: S_1 \\to C$, $i_2: S_2 \\to C$ such that: :for all sets $X$ and mappings $f_1: S_1 \\to X$ and $f_2: S_2 \\to X$: ::there exists a unique mapping $h: C \\to X$ such that: :::$h \\circ i_1 = f_1$ :::$h \\circ i_2 = f_2$ Hence: ::$\\begin{xy} \\xymatrix@L+2mu@+1em{ & C \\ar@{-->}[dd]_*{h} & \\\\ S_1 \\ar[ru]^*{i_1} \\ar[rd]_*{f_1} & & S_2 \\ar[lu]_*{i_2} \\ar[ld]^*{f_2} \\\\ & X & }\\end{xy}$ :is a commutative diagram."}
+{"_id": "30315", "title": "Definition:Ring of Mappings/Pointwise Addition", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $S$ be a set. Let $\\struct {R^S, +', \\circ'}$ be the ring of mappings from $S$ to $R$. <section notitle=true name=PointwiseAddition> The pointwise operation $+'$ induced by $+$ on the ring of mappings from $S$ to $R$ is called '''pointwise addition''' and is defined as: :$\\forall f, g \\in R^S: f +’ g \\in R^S :$ :::$\\forall s \\in S : \\map {\\paren {f +’ g}} x = \\map f x + \\map g x$ </section>"}
+{"_id": "30316", "title": "Definition:Ring of Mappings/Pointwise Multiplication", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $S$ be a set. Let $\\struct {R^S, +', \\circ'}$ be the ring of mappings from $S$ to $R$. <section notitle=true name=PointwiseMultiplication> The pointwise operation $\\circ'$ induced by $\\circ$ on the ring of mappings from $S$ to $R$ is called '''pointwise multiplication''' and is defined as: :$\\forall f, g \\in R^S: f \\circ’ g \\in R^S :$ :::$\\forall s \\in S : \\map {\\paren {f \\circ’ g}} x = \\map f x \\circ \\map g x$ </section>"}
+{"_id": "30317", "title": "Definition:Ring of Sequences/Pointwise Addition", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\struct {R^\\N, +', \\circ'}$ be the ring of sequences over $R$. <section notitle=true name=PointwiseAddition> The pointwise operation $+'$ induced by $+$ on the ring of sequences is called '''pointwise addition''' and is defined as: :$\\forall \\sequence {x_n}, \\sequence {y_n} \\in R^\\N: \\sequence {x_n} +' \\sequence {y_n} = \\sequence {x_n + y_n}$ </section>"}
+{"_id": "30318", "title": "Definition:Ring of Sequences/Pointwise Multiplication", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\struct {R^\\N, +', \\circ'}$ be the ring of sequences over $R$. <section notitle=true name=PointwiseMultiplication> The pointwise operation $\\circ'$ induced by $\\circ$ on the ring of sequences is called '''pointwise multiplication''' and is defined as: :$\\forall \\sequence {x_n}, \\sequence {y_n} \\in R^{\\N}: \\sequence {x_n} \\circ' \\sequence {y_n} = \\sequence {x_n \\circ y_n}$ </section>"}
+{"_id": "30319", "title": "Definition:Ring of Sequences/Zero", "text": "The zero of the ring of sequences is the constant sequence $\\tuple {0, 0, 0, \\dots}$, where $0$ is the zero of $R$."}
+{"_id": "30320", "title": "Definition:Ring of Sequences/Unity", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity $1$."}
+{"_id": "30321", "title": "Definition:Ring of Sequences/Commutativity", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring."}
+{"_id": "30322", "title": "Definition:Ring of Sequences/Additive Inverse", "text": "The additive inverse in the ring of sequences is defined by: :$\\forall \\sequence {x_n} \\in R^\\N: -\\sequence {x_n} = \\sequence {-x_n}$"}
+{"_id": "30323", "title": "Definition:Ring of Sequences/Units", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity $1$."}
+{"_id": "30329", "title": "Definition:Euclidean Number", "text": "A '''Euclid number''' is a perfect number which is even."}
+{"_id": "30332", "title": "Definition:Friction/Coefficient", "text": "Let $B$ be a body at rest on a plane surface $S$ on which friction acts. Let $N$ be the normal reaction of $S$ on $B$. Let a force be applied to $B$ parallel to $S$. === Coefficient of Static Friction === {{:Definition:Friction/Coefficient/Static}} === Coefficient of Kinetic Friction === {{:Definition:Friction/Coefficient/Kinetic}} These '''coefficients of friction''' depend upon the materials out of which $B$ and $S$ are made. It is usual for $\\mu_s$ to be greater than $\\mu_k$."}
+{"_id": "30333", "title": "Definition:Friction/Coefficient/Static", "text": "Let $F$ be the magnitude of that force in the limiting case when $B$ is just about to move. Then the '''coefficient of static friction''' is defined and denoted: :$\\mu_s = \\dfrac F N$"}
+{"_id": "30334", "title": "Definition:Friction/Coefficient/Kinetic", "text": "Let $F$ be the magnitude of that force needed to keep $B$ moving at a constant velocity. Then the '''coefficient of kinetic friction''' is defined and denoted: :$\\mu_k = \\dfrac F N$"}
+{"_id": "30340", "title": "Definition:Bimodule", "text": "Let $\\struct {R, +_R, \\times_R}$ and $\\struct {S, +_S, \\times_S}$ be rings. Let $\\struct {G, +_G}$ be an abelian group. Let $\\circ_R : R \\times G \\to G$ and $\\circ_S : G \\times S \\to G$ be binary operations such that: :$(1): \\quad \\struct {G, +_G, \\circ_R}$ is a left module :$(2): \\quad \\struct {G, +_G, \\circ_S}$ is a right module :$(3): \\quad \\forall \\lambda \\in R: \\forall \\mu \\in S: \\forall x \\in G: \\paren {\\lambda \\circ_R x} \\circ_S \\mu = \\lambda \\circ_R \\paren {x \\circ_S \\mu}$ Then $\\struct {G, +_G, \\circ_R, \\circ_S}$ is a '''bimodule over $\\tuple {R, S}$'''. If $\\struct {S, +_S, \\times_S} = \\struct {R, +_R, \\times_R}$ then a '''bimodule over $\\tuple {R, R}$''' is simply called a '''bimodule over $R$'''"}
+{"_id": "30348", "title": "Definition:Weight (Physics)/Warning", "text": "There is a certain amount of confusion in the common mind between weight and mass. The latter is usually determined by measuring its weight. But while the mass of a body is (under normal circumstances) constant, its weight varies according to its position relative to the gravitational field it is in, and so is not a constant property of that body. However, under usual terrestrial conditions the gravitational field is more or less constant (any differences being detectable only by instruments). This means that the weight and mass of a body are commonly considered \"the same\". Thus a weighing machine, while indicating the mass of a body, does so by measuring its weight."}
+{"_id": "30349", "title": "Definition:Weight (Physics)/Dimension", "text": "The dimension of '''weight''' is $M L T^{-2}$: mass times acceleration, that is, a force."}
+{"_id": "30351", "title": "Definition:Whole Number", "text": "The phrase '''whole number''' is an ambiguous term which could mean either: :an integer :a natural number :a strictly positive integer. Hence it is recommended that a specific instance of the above is used instead."}
+{"_id": "30353", "title": "Definition:Are", "text": "One '''are''' is equal to a square whose side measures $10$ metres. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''are''' }} {{eqn | r = 100       | c = square metres }} {{eqn | o = \\approx       | r = 119 \\cdot 60       | c = square yards }} {{end-eqn}}"}
+{"_id": "30354", "title": "Definition:Finite Difference Operator/Forward Difference", "text": "The '''forward difference operator''' on $f$ is defined as: :$\\map {\\Delta f} x := \\map f {x + 1} - \\map f x$"}
+{"_id": "30355", "title": "Definition:Finite Difference Operator/Backward Difference", "text": "The '''backward difference operator''' on $f$ is defined as: :$\\map {\\nabla f} x := \\map f x - \\map f {x - 1}$"}
+{"_id": "30359", "title": "Definition:Bernoulli Trial/Bernoulli Variable", "text": "Let $X$ be a discrete random variable whose sample space is $\\Omega$ in such a Bernoulli trial. Then $X$ is known as a '''Bernoulli variable'''."}
+{"_id": "30360", "title": "Definition:Borromean Rings", "text": "The '''Borromean rings''' is an arrangement of $3$ interlinked rings such that no $2$ of the rings are themselves linked. That is, if one of them is removed, then the other $2$ are unlinked. :600px"}
+{"_id": "30365", "title": "Definition:Basis Representation", "text": "Let $b \\in \\Z$ be an integer such that $b > 1$. Let $n \\in \\Z$ be an integer such that $n \\ne 0$. The '''representation of $n$ to the base $b$''' is the unique string of digits: :$\\pm \\sqbrk {r_m r_{m - 1} \\ldots r_2 r_1 r_0}_b$ where $\\pm$ is: :the negative sign $-$ {{iff}} $n < 0$ :the positive sign $+$ (or omitted) {{iff}} $n > 0$ If $n = 0$, then $n$ is represented simply as $0$."}
+{"_id": "30366", "title": "Definition:Lagrangian Mechanics", "text": "'''Lagrangian Mechanics''' is a reformulation of mechanics, where the main object of study is the Lagrangian function."}
+{"_id": "30367", "title": "Definition:Action of Physical System", "text": "Let $S$ be an integral functional. Suppose that equations of motion of a physical system are to be derived from $S$ by the Principle of Stationary Action. Then $S$ is the '''action''' of a '''physical system'''."}
+{"_id": "30368", "title": "Definition:Lagrangian", "text": "Let $P$ be a physical system composed of $n \\in \\N$ particles. Let the real variable $t$ be the time of $P$. $\\forall i \\le n$ let $\\map { {\\mathbf x}_i} t$ be the position of the $i$th particle. Suppose, the action $S$ of $P$ is of the following form: :$\\displaystyle S = \\int_{t_1}^{t_2} L \\rd t$ where $L$ is a mapping of (possibly) $t$, $\\map {{\\mathbf x}_i} t$ and their derivatives. Then $L$ is the '''Lagrangian''' of $P$. {{NamedforDef|Joseph Louis Lagrange|cat = Lagrange}} Category:Definitions/Lagrangian Mechanics Category:Definitions/Physics Category:Definitions/Applied Mathematics k0yf3wthx912zivh4vfjf544vgbcowt"}
+{"_id": "30369", "title": "Definition:Free Group/Definition 1", "text": "A group $G$ is a '''free group''' {{iff}} it is isomorphic to the free group on some set."}
+{"_id": "30370", "title": "Definition:Free Group/Definition 2", "text": "A group $G$ is a '''free group''' {{iff}} it has a presentation of the form $\\gen S$, where $S$ is a set. That is, it has a presentation without relators. In this context, '''free''' means '''free of non-trivial relations'''."}
+{"_id": "30371", "title": "Definition:Standard Lagrangian", "text": "Let $P$ be a physical system composed of countable number of particles. Let $L$ be a Lagrangian of $P$. Let: :$L = T - V$ where $T$ is the kinetic energy and $V$ the potential energy of $P$. Then $L$ is the '''Standard Lagrangian''' of $P$. {{NamedforDef|Joseph Louis Lagrange|cat = Lagrange}}"}
+{"_id": "30372", "title": "Definition:Kinetic Energy of Classical Particle", "text": "Let $\\MM$ be an $n$-dimensional Euclidean manifold. Let $P$ be a particle with an inertial mass $m_i$. Let $t$ be the time variable of $P$. Suppose the position of $P$ is a real differentiable $n$-dimensional vector-valued mapping $\\mathbf x = \\map {\\mathbf x} t$. Then the '''kinetic energy''' of a '''classical particle''' $P$ is: :$T = \\dfrac {m_i} 2 \\paren {\\dfrac {\\d \\mathbf x} {\\d t} }^2$ where $\\paren {\\dfrac {\\d \\mathbf x} {\\d t} }^2$ is the dot product of the vector $\\dfrac {\\d \\mathbf x} {\\d t}$ with itself."}
+{"_id": "30374", "title": "Definition:Pedal Triangle", "text": "=== Pedal Triangle of Point with respect to Triangle === {{:Definition:Pedal Triangle/Point}} === Orthic (Pedal) Triangle === {{:Definition:Orthic Triangle}} {{:Definition:Orthic Triangle/Also known as}}"}
+{"_id": "30375", "title": "Definition:Orthic Triangle", "text": "Let $\\triangle ABC$ be a triangle. Let $\\triangle DEF$ be the triangle formed by the feet of the altitudes $AD$, $BC$ and $ED$ of $\\triangle ABC$. :400px $\\triangle DEF$ is known as the '''orthic triangle of $\\triangle ABC$'''. That is, the '''orthic triangle of $\\triangle ABC$''' is the pedal triangle of its orthocenter."}
+{"_id": "30376", "title": "Definition:Pedal Triangle/Point", "text": "Let $\\triangle ABC$ be a triangle. Let $P$ be a point in the plane of $\\triangle ABC$. Let $PD$, $PE$ and $PF$ be perpendiculars dropped from $P$ to $BC$, $AC$ and $AB$ respectively. Let $\\triangle DEF$ be the triangle formed by the feet of the perpendiculars $PD$, $PE$ and $PF$. :400px $\\qquad$ 400px $\\triangle DEF$ is known as the '''pedal triangle of $P$ with respect to $\\triangle ABC$'''."}
+{"_id": "30381", "title": "Definition:Total Energy of Particles", "text": "Let $P$ be a physical system composed of a countable number of particles. Suppose the kinetic and potential energy of $P$ is respectively $T$ and $U$. Denote: :$E = T + U$ Then $E$ is known as the '''total energy''' of $P$."}
+{"_id": "30391", "title": "Definition:Annulus (Geometry)", "text": "An '''annulus''' is a plane figure whose boundary consists of $2$ concentric circles: :400px In the above diagram, the '''annulus''' is the colored area."}
+{"_id": "30392", "title": "Definition:Annulus (Geometry)/Center", "text": "The '''center''' of an '''annulus''' is the common center of the $2$ concentric circles that form its boundary. In the above diagram, the '''center''' is the point $O$."}
+{"_id": "30396", "title": "Definition:Annulus (Geometry)/Inner Radius", "text": "The '''inner radius''' of an '''annulus''' is the radius of the smaller of the $2$ concentric circles that form its boundary. In the above diagram, the '''inner radius''' is denoted $r$."}
+{"_id": "30397", "title": "Definition:Annulus (Geometry)/Outer Radius", "text": "The '''outer radius''' of an '''annulus''' is the radius of the larger of the $2$ concentric circles that form its boundary. In the above diagram, the '''outer radius''' is denoted $R$."}
+{"_id": "30399", "title": "Definition:Power of Element/Ring", "text": "Let $\\struct {R, \\circ, +}$ be a ring. Let $r \\in R$. Let $n \\in \\Z_{>0}$ be the set of strictly positive integers. The '''$n$th power of $r$ in $R$''' is defined as the $n$th power of $r$ with respect to the semigroup $\\struct {R, \\circ}$: :$\\forall n \\in \\Z_{>0}: r^n = \\begin {cases} r & : n = 1 \\\\ r^{n - 1} \\circ r & : n > 1  \\end {cases}$ If $R$ is a ring with unity where $1_R$ is that unity, the definition extends to $n \\in \\Z_{\\ge 0}$: :$\\forall n \\in \\Z_{\\ge 0}: r^n = \\begin {cases} 1_R & : n = 0 \\\\ r^{n - 1} \\circ r & : n > 0 \\end {cases}$"}
+{"_id": "30400", "title": "Definition:Half-Range Fourier Series", "text": "=== Half-Range Fourier Cosine Series === {{:Definition:Half-Range Fourier Cosine Series/Formulation 1}} === Half-Range Fourier Sine Series === {{:Definition:Half-Range Fourier Sine Series/Formulation 1}}"}
+{"_id": "30401", "title": "Definition:Half-Plane/Edge", "text": "Let $H$ denote one of the half-planes into which $L$ divides $P$. Then $L$ is called the '''edge''' of $H$."}
+{"_id": "30402", "title": "Definition:Complex Half-Plane", "text": "Let $\\C$ denote the complex plane. A '''complex half-plane''' is a '''half-plane''' of $\\C$ whose edge is an infinite straight line of $\\C$. Category:Definitions/Complex Analysis 72g45o95hyx7v1pet7o7s9denayhta5"}
+{"_id": "30403", "title": "Definition:New Moon", "text": "The '''new moon''' is the astronomical phenomenon of the Moon in its orbit coming to its closest point to the Sun: :800px Note that the above diagram is not to scale. At this point in its orbit, the Moon is between Earth and the Sun. Thus the hemisphere of the Moon which is illuminated by the Sun is facing directly away from Earth. Thus for an observer of the Moon from Earth, unless an actual solar eclipse is in progress, at this point it cannot be seen, because: :$(1): \\quad$ It is dark :$(2): \\quad$ The Sun is more or less behind it, outshining everything else in the sky. Category:Definitions/Astronomy ts9gid2sqz6h6f9db2q271okr3mkbfs"}
+{"_id": "30407", "title": "Definition:Zero Mapping", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N,\\Z,\\Q,\\R,\\C$. Let $S$ be a set. Let $f_0: S \\to \\mathbb A$ denote the constant mapping: :$\\forall x \\in S: \\map {f_0} x = 0$ Then $f_0$ is referred to as the '''zero mapping'''."}
+{"_id": "30408", "title": "Definition:Non-Trivial Ring", "text": "A '''non-trivial ring''' is a ring which is not trivial. That is, a ring $R$ such that: :$\\exists x, y \\in R: x \\circ y \\ne 0_R$ where $0_R$ denotes the zero of $R$."}
+{"_id": "30409", "title": "Definition:Convergent Sequence/P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\sequence {x_n} $ be a sequence in $\\Q_p$."}
+{"_id": "30410", "title": "Definition:Phase of Moon", "text": "The '''phase of the Moon''' is its apparent shape when viewed in the night sky of Earth. This shape is dependent upon the position of the Moon with respect to the Sun and Earth. :800px When the Moon is between the Sun and Earth, the sunlit side faces more or less away from Earth, and so less than half of it can be seen lit up. When Earth is between the Sun and the Moon, the sunlit side faces more or less away towards Earth, and so more than half of it can be seen lit up. When the Moon is directly between the Sun and Earth, the sunlit side is facing completely away from Earth, and in normal conditions cannot be seen. This is the '''phase''' known as '''new moon'''. When Earth is directly between the Sun and the Moon, the sunlit side is facing completely towards Earth, and in normal conditions appears as a brign disk. This is the '''phase''' known as '''full moon'''. At the point half way between new moon and full moon, the '''phase''' is known as '''first quarter'''. At the point half way between full moon and new moon, the '''phase''' is known as '''last quarter'''. Category:Definitions/Astronomy ixuonqcy1kirih2w49f13upg59nu45c"}
+{"_id": "30411", "title": "Definition:Safe Prime/Sequence", "text": "The sequence of safe primes begins: :$5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, \\ldots$"}
+{"_id": "30413", "title": "Definition:Metric System/Scaling Prefixes/yocto-", "text": "'''yocto-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-24}$. === Symbol === {{:Symbols:Y/yocto-}}"}
+{"_id": "30414", "title": "Definition:Metric System/Scaling Prefixes/zepto-", "text": "'''zepto-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-21}$. === Symbol === {{:Symbols:Z/zepto-}}"}
+{"_id": "30415", "title": "Definition:Metric System/Scaling Prefixes/atto-", "text": "'''atto-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-18}$. === Symbol === {{:Symbols:A/atto-}}"}
+{"_id": "30416", "title": "Definition:Metric System/Scaling Prefixes/femto-", "text": "'''femto-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-15}$. === Symbol === {{:Symbols:F/femto-}}"}
+{"_id": "30417", "title": "Definition:Metric System/Scaling Prefixes/pico-", "text": "'''pico-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-12}$. === Symbol === {{:Symbols:P/pico-}}"}
+{"_id": "30418", "title": "Definition:Metric System/Scaling Prefixes/nano-", "text": "'''nano-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-9}$. === Symbol === {{:Symbols:N/nano-}}"}
+{"_id": "30419", "title": "Definition:Metric System/Scaling Prefixes/micro-", "text": "'''micro-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-6}$. === Symbol === {{:Symbols:Greek/Mu/micro-}}"}
+{"_id": "30420", "title": "Definition:Metric System/Scaling Prefixes/milli-", "text": "'''milli-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-3}$. === Symbol === {{:Symbols:M/milli-}}"}
+{"_id": "30421", "title": "Definition:Metric System/Scaling Prefixes/centi-", "text": "'''centi-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-2}$. === Symbol === {{:Symbols:C/centi-}}"}
+{"_id": "30422", "title": "Definition:Metric System/Scaling Prefixes/deci-", "text": "'''deci-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-1}$. === Symbol === {{:Symbols:D/deci-}}"}
+{"_id": "30423", "title": "Definition:Harmonic Potential Energy", "text": "Let $P$ be a physical particle. Let its position $\\map x t$ be a real function, where $t$ is time. Let $k > 0$. Then the potential energy of the form :$\\map U x = \\frac 1 2 k x^2$ is called the '''harmonic potential energy'''."}
+{"_id": "30424", "title": "Definition:Metric System/Scaling Prefixes/deka-", "text": "'''deka-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^1$. === Symbol === {{:Symbols:D/deka-}}"}
+{"_id": "30425", "title": "Definition:Harmonic Oscillator", "text": "Let $P$ be a physical particle. Let the potential energy of $P$ be that of the harmonic potential. Then $P$ is called the '''harmonic oscillator'''."}
+{"_id": "30426", "title": "Definition:Metric System/Scaling Prefixes/hecto-", "text": "'''hecto-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^2$. === Symbol === {{:Symbols:H/hecto-}}"}
+{"_id": "30427", "title": "Definition:Metric System/Scaling Prefixes/kilo-", "text": "'''kilo-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^3$. === Symbol === {{:Symbols:K/kilo-}}"}
+{"_id": "30428", "title": "Definition:Metric System/Scaling Prefixes/mega-", "text": "'''mega-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^6$. === Symbol === {{:Symbols:M/mega-}}"}
+{"_id": "30429", "title": "Definition:Metric System/Scaling Prefixes/giga-", "text": "'''giga-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^9$. === Symbol === {{:Symbols:G/giga-}}"}
+{"_id": "30430", "title": "Definition:Metric System/Scaling Prefixes/tera-", "text": "'''tera-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{12}$. === Symbol === {{:Symbols:T/tera-}}"}
+{"_id": "30431", "title": "Definition:Metric System/Scaling Prefixes/peta-", "text": "'''peta-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{15}$. === Symbol === {{:Symbols:P/peta-}}"}
+{"_id": "30432", "title": "Definition:Metric System/Scaling Prefixes/exa-", "text": "'''exa-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{18}$. === Symbol === {{:Symbols:E/exa-}}"}
+{"_id": "30433", "title": "Definition:Metric System/Scaling Prefixes/zetta-", "text": "'''zetta-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{21}$. === Symbol === {{:Symbols:Z/zetta-}}"}
+{"_id": "30434", "title": "Definition:Metric System/Scaling Prefixes/yotta-", "text": "'''yotta-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{24}$. === Symbol === {{:Symbols:Y/yotta-}}"}
+{"_id": "30435", "title": "Definition:Convergent Sequence/P-adic Numbers/Definition 1", "text": "The sequence $\\sequence {x_n}$ '''converges to $x \\in \\Q_p$''' {{iff}}: :$\\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\R_{>0}: \\forall n \\in \\N: n > N \\implies \\norm {x_n - x}_p < \\epsilon$"}
+{"_id": "30436", "title": "Definition:Convergent Sequence/P-adic Numbers/Definition 3", "text": "The sequence $\\sequence {x_n}$ '''converges to $x \\in \\Q_p$''' {{iff}}: :$\\sequence {x_n}$ converges to $x$ in the $p$-adic metric"}
+{"_id": "30437", "title": "Definition:Convergent Sequence/P-adic Numbers/Definition 4", "text": "The sequence $\\sequence {x_n}$ '''converges to $x \\in \\Q_p$''' {{iff}}: :the real sequence $\\sequence {\\norm {x_n - x}_p }$ converges to $0$ in the reals $\\R$"}
+{"_id": "30438", "title": "Definition:Limit of Sequence/P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\sequence {x_n} $ be a sequence in $\\Q_p$. <section notitle=\"true\" name=\"LimitDefinition\"> Let $\\sequence {x_n}$ converge to $x \\in \\Q_p$  Then $x$ is a '''limit of $\\sequence {x_n}$ as $n$ tends to infinity''' which is usually written: :$\\displaystyle x = \\lim_{n \\mathop \\to \\infty} x_n$ </section>"}
+{"_id": "30439", "title": "Definition:Convergent Sequence/P-adic Numbers/Definition 2", "text": "The sequence $\\sequence {x_n}$ '''converges to $x \\in \\Q_p$''' {{iff}}: :$\\sequence {x_n}$ converges to $x$ in the $p$-adic norm"}
+{"_id": "30441", "title": "Definition:Convergent Sequence/Normed Division Ring/Definition 1", "text": "The sequence $\\sequence {x_n}$ '''converges to $x \\in R$ in the norm $\\norm {\\, \\cdot \\,}$''' {{iff}}: :$\\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\R_{>0}: \\forall n \\in \\N: n > N \\implies \\norm {x_n - x} < \\epsilon$"}
+{"_id": "30442", "title": "Definition:Convergent Sequence/Normed Division Ring/Definition 2", "text": "The sequence $\\sequence {x_n}$ '''converges to $x \\in R$ in the norm $\\norm {\\, \\cdot \\,}$''' {{iff}}: :$\\sequence {x_n}$ converges to $x$ in the metric induced by the norm $\\norm {\\, \\cdot \\,}$"}
+{"_id": "30443", "title": "Definition:Convergent Sequence/Normed Division Ring/Definition 3", "text": "The sequence $\\sequence {x_n}$ '''converges to $x \\in R$ in the norm $\\norm {\\, \\cdot \\,}$''' {{iff}}: :the real sequence $\\sequence {\\norm {x_n - x} }$ converges to $0$ in the reals $\\R$"}
+{"_id": "30444", "title": "Definition:Rounding/Treatment of Half/Round Up", "text": "The '''round up''' convention is that the '''larger''' of those two integers is used: :$y = 10^n \\floor {\\dfrac x {10^n} + \\dfrac 1 2}$"}
+{"_id": "30445", "title": "Definition:Rounding/Treatment of Half/Round Down", "text": "The '''round down''' convention is that the '''smaller''' of those two integers is used: :$y = 10^n \\ceiling {\\dfrac x {10^n} - \\dfrac 1 2}$"}
+{"_id": "30446", "title": "Definition:Rounding/Treatment of Half/Round to Even", "text": "The '''round to even''' convention is that the '''nearest even integer''' to $\\dfrac x {10^n}$ is used: :$y = \\begin {cases} 10^n \\floor {\\dfrac x {10^n} + \\dfrac 1 2} & : \\text {$\\floor {\\dfrac x {10^n} + \\dfrac 1 2}$ even} \\\\ 10^n \\ceiling {\\dfrac x {10^n} - \\dfrac 1 2} & : \\text {$\\floor {\\dfrac x {10^n} + \\dfrac 1 2}$ odd} \\end {cases}$"}
+{"_id": "30447", "title": "Definition:Cumulative Rounding Error", "text": "Let $S$ be a set of continuous data which is to be rounded to the nearest $n$th power of $10$. The '''cumulative rounding error''' of $S$ is defined as: :$R = \\displaystyle \\sum_{x \\mathop \\in S} \\paren {x - \\overline x}$ where $\\overline x$ denotes the rounded value of a given $x$."}
+{"_id": "30449", "title": "Definition:Scientific Notation/Exponent", "text": "The number $e$ is known as the '''exponent'''."}
+{"_id": "30451", "title": "Definition:Significant Figures", "text": "Let $n$ be a number expressed in decimal notation. The number of digits to which $n$ is rounded, apart from any digits needed to locate the decimal point, are referred to as the '''significant figures''' of $n$."}
+{"_id": "30452", "title": "Definition:Graph (Statistics)", "text": "In the context of statistics, a '''graph''' is a word that loosely means a pictorial or diagrammatic presentation of a set of data."}
+{"_id": "30453", "title": "Definition:Bar Graph", "text": "A '''bar graph''' is a form of graph which consists of a finite set of (usually) vertical  bars whose length determines the statistic being communicated."}
+{"_id": "30455", "title": "Definition:Pie Graph", "text": "A '''pie graph''' is a form of graph which consists of a circle divided into sectors whose areas represent the proportion of the corresponding statistic relative to the whole."}
+{"_id": "30456", "title": "Definition:Pictograph", "text": "A '''pictograph''' is a variant of a bar graph whose  bars consist of a number of icons or other pictorial symbols arranged in a (usually) line."}
+{"_id": "30457", "title": "Definition:Inequality/Member", "text": "Let $\\RR$ be an inequality. Let $a \\mathrel \\RR b$. Then both $a$ and $b$ are referred to as '''members''' of the inequality $\\RR$."}
+{"_id": "30458", "title": "Definition:General Logarithm/Common/Notation for Negative Logarithm", "text": "Let $n \\in \\R$ be a real number such that $0 < n < 1$. Let $n$ be presented (possibly approximated) in scientific notation as: :$a \\times 10^{-d}$ where $d \\in \\Z_{>0}$ is a (strictly) positive integer. Let $\\log_{10} n$ denote the common logarithm of $n$. Then it is the standard convention to express $\\log_{10} n$ in the form: :$\\log_{10} n = \\overline d \\cdotp m$ where $m := \\log_{10} a$ is the mantissa of $\\log_{10} n$."}
+{"_id": "30459", "title": "Definition:General Logarithm/Common/Mantissa", "text": "Let $\\log_{10} n$ be expressed in the form: :$\\log_{10} n = \\begin {cases} c \\cdotp m & : d \\ge 0 \\\\ \\overline c \\cdotp m & : d < 0 \\end {cases}$ where: : $c = \\size d$ is the absolute value of $d$ : $m := \\log_{10} a$ $\\log_{10} a$ is the '''mantissa''' of $\\log_{10} n$."}
+{"_id": "30460", "title": "Definition:General Logarithm/Common/Characteristic", "text": "$c$ is the '''characteristic''' of $\\log_{10} n$."}
+{"_id": "30461", "title": "Definition:Antilogarithm", "text": "Let $x \\in \\R_{>0}$ be a strictly positive real number. Let $b \\in \\R_{>1}$ be a real number which is greater than $1$. Let $y = \\log_b x$ be the logarithm of $x$ base $b$. Then the '''antilogarithm of $y$ base $b$''' is: :$\\operatorname {alog}_b y = x$"}
+{"_id": "30462", "title": "Definition:Sample Statistic/Raw", "text": "'''Raw data''' are collected data which have not been organized or analysed in any way."}
+{"_id": "30464", "title": "Definition:Spherical Astronomy", "text": "'''Spherical astronomy''' is a branch of astronomy which studies the angular positions of the celestial bodies, with no concern for their distance from Earth. Its mathematical aspects can be considered a subfield of spherical geometry."}
+{"_id": "30465", "title": "Definition:Great Circle", "text": "Let $S$ be a sphere. A '''great circle''' of $S$ is defined as the intersection of $S$ with a plane passing through the center of $S$. :400px"}
+{"_id": "30466", "title": "Definition:Small Circle", "text": "Let $S$ be a sphere. A '''small circle''' on $S$ is defined as the intersection of $S$ with a plane which does not pass through the center of $S$. :400px"}
+{"_id": "30467", "title": "Definition:Pole of Great Circle", "text": "Let $C$ be a great circle of $S$. Let $AB$ be the diameter of $S$ situated perpendicular to the plane of $C$. :700px The points $A$ and $B$, where the diameter intersects $S$, are the '''poles''' of the great circle $C$."}
+{"_id": "30468", "title": "Definition:Pole of Circle", "text": "Let $S$ be a sphere whose center is $O$. Let $AB$ be a diameter of $S$, such that $A$ and $B$ are the points of intersection $S$ Let $C$ be a (small) circle of $S$ embedded in a plane perpendicular to $AB$.   :700px The points $A$ and $B$ are the '''poles''' of the circle $C$. === Special Case: Pole of Great Circle === {{:Definition:Pole of Great Circle}}"}
+{"_id": "30470", "title": "Definition:Geographical Equator", "text": "The '''(geographical) equator''' is the great circle described on the surface of Earth whose plane is perpendicular to Earth's axis of rotation. :500px"}
+{"_id": "30471", "title": "Definition:Earth's Poles", "text": "Let it be assumed to be established that planet Earth rotates in space about its axis. The '''poles''' of Earth are the endpoints of Earth's axis, where it intersects the surface of Earth. That is, the '''poles''' of Earth are the poles of Earth's equator. :500px"}
+{"_id": "30472", "title": "Definition:Earth's Poles/North", "text": "Let an observer $A$ be situated on Earth's equator. Let $A$ be facing the direction of Earth's rotation in space about its axis. Then the '''North Pole''' is the particular one of Earth's poles which is on the left of $A$."}
+{"_id": "30473", "title": "Definition:Earth's Poles/South", "text": "Let an observer $A$ be situated on Earth's equator. Let $A$ be facing the direction of Earth's rotation in space about its axis. Then the '''South Pole''' is the particular one of Earth's poles which is on the right of $A$."}
+{"_id": "30474", "title": "Definition:Earth's Axis", "text": "Let it be assumed to be established that planet Earth rotates in space about a more-or-less fixed diameter $D$. Then $D$ is known as '''Earth's axis'''. :500px"}
+{"_id": "30476", "title": "Definition:Closed Ball/P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. The '''closed $\\epsilon$-ball of $a$ in $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p }$''' is defined as: :$\\map { {B_\\epsilon}^-} a = \\set {x \\in R: \\norm {x - a}_p \\le \\epsilon}$"}
+{"_id": "30477", "title": "Definition:Closed Ball/P-adic Numbers/Radius", "text": "In $\\map { {B_\\epsilon}^-} a$, the value $\\epsilon$ is referred to as the '''radius''' of the closed $\\epsilon$-ball."}
+{"_id": "30478", "title": "Definition:Closed Ball/P-adic Numbers/Center", "text": "In $\\map { {B_\\epsilon}^-} a$, the value $a$ is referred to as the '''center''' of the closed $\\epsilon$-ball."}
+{"_id": "30479", "title": "Definition:Open Ball/P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in R$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. The '''open $\\epsilon$-ball of $a$ in $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$''' is defined as: :$\\map {B_\\epsilon} a = \\set {x \\in \\Q_p: \\norm{x - a}_p < \\epsilon}$"}
+{"_id": "30480", "title": "Definition:Open Ball/P-adic Numbers/Radius", "text": "In $\\map {B_\\epsilon} a$, the value $\\epsilon$ is referred to as the '''radius''' of the open $\\epsilon$-ball."}
+{"_id": "30481", "title": "Definition:Open Ball/P-adic Numbers/Center", "text": "In $\\map {B_\\epsilon} a$, the value $a$ is referred to as the '''center''' of the open $\\epsilon$-ball."}
+{"_id": "30483", "title": "Definition:Meridian (Terrestrial)", "text": "A '''(terrestrial) meridian''' is a semi-great circle on Earth's surface whose endpoints are Earth's poles. :600px"}
+{"_id": "30485", "title": "Definition:Meridian (Terrestrial)/Principal", "text": "The '''principal meridian''' is the meridian of Earth which passes through the {{WP|Royal_Observatory,_Greenwich|Royal Observatory}} in {{WP|Greenwich|Greenwich}}, {{WP|Greater_London|Greater London}}, {{WP|England|England}}, which is used as the internationally-recognised $0 \\degrees$ reference longitude meridian."}
+{"_id": "30486", "title": "Definition:North (Terrestrial)", "text": "'''North''' is the direction on (or near) Earth's surface along the meridian directly towards the North Pole."}
+{"_id": "30487", "title": "Definition:South (Terrestrial)", "text": "'''South''' is the direction on (or near) Earth's surface along the meridian directly towards the South Pole."}
+{"_id": "30488", "title": "Definition:East (Terrestrial)", "text": "'''East''' is the direction on (or near) Earth's surface along the small circle in the direction of Earth's rotation in space about its axis."}
+{"_id": "30489", "title": "Definition:West (Terrestrial)", "text": "'''West''' is the direction on (or near) Earth's surface along the small circle in the opposite direction from Earth's rotation in space about its axis."}
+{"_id": "30490", "title": "Definition:Cardinal Directions (Terrestrial)", "text": "The $4$ '''cardinal directions''' are the directions on (or near) Earth's surface defined as follows: === East === {{:Definition:East (Terrestrial)}} === West === {{:Definition:West (Terrestrial)}} === North === {{:Definition:North (Terrestrial)}} === South === {{:Definition:South (Terrestrial)}} :600px"}
+{"_id": "30491", "title": "Definition:Cardinal Directions", "text": "The $4$ cardinal directions: :North :South :East :West are the basis of navigation and geographical coordinates."}
+{"_id": "30492", "title": "Definition:East", "text": "=== East (Terrestrial) === {{:Definition:East (Terrestrial)}} === East (Celestial) === {{:Definition:East (Celestial)}}"}
+{"_id": "30493", "title": "Definition:West", "text": "=== West (Terrestrial) === {{:Definition:West (Terrestrial)}} === West (Celestial) === {{:Definition:West (Celestial)}}"}
+{"_id": "30494", "title": "Definition:South", "text": "=== South (Terrestrial) === {{:Definition:South (Terrestrial)}} === South (Celestial) === {{:Definition:South (Celestial)}}"}
+{"_id": "30495", "title": "Definition:North", "text": "=== North (Terrestrial) === {{:Definition:North (Terrestrial)}} === North (Celestial) === {{:Definition:North (Celestial)}}"}
+{"_id": "30496", "title": "Definition:Intercardinal Directions", "text": "The $4$ '''intercardinal directions''' are the directions on (or near) Earth's surface halfway between the cardinal directions: === Northeast === {{:Definition:Northeast}} === Northwest === {{:Definition:Northwest}} === Southeast === {{:Definition:Southeast}} === Southwest === {{:Definition:Southwest}}"}
+{"_id": "30497", "title": "Definition:Northeast", "text": "'''Northeast''' is the direction on (or near) Earth's surface halfway between '''north''' and '''east'''."}
+{"_id": "30501", "title": "Definition:Pole of Arc", "text": "Let $S$ be a sphere whose center is $O$. Let $C$ be the arc of a circle on $S$, either a small circle or a great circle. The '''poles''' of $C$ are the poles of the circle on $S$ of which $C$ is a part."}
+{"_id": "30502", "title": "Definition:Spherical Angle", "text": "Let $S$ be a sphere. Let $PQ$ be a diameter of $S$. Let $PAQ$ and $PBQ$ be great circles on $S$ both passing through the points $P$ and $Q$. Let tangents $S$ and $T$ to $PAQ$ and $PBQ$ respectively be drawn through $P$. :500px Then the angle between $S$ and $T$ is known as the '''spherical angle''' between $PAQ$ and $PBQ$. Thus a '''spherical angle''' is defined with respect only to $2$ great circles."}
+{"_id": "30503", "title": "Definition:Hemisphere", "text": "Let $S$ be a sphere. Let $S$ be bisected by a plane which passes through the center of $S$. Each of the halves of $S$ into which $S$ is divided is called a '''hemisphere'''."}
+{"_id": "30504", "title": "Definition:Spherical Triangle/Side/Length", "text": "Let $ABC$ be a spherical triangle on a sphere $S$. Let $AB$ be a side of $ABC$. The '''length''' of $AB$ is defined as the angle subtended at the center of $S$ by $AB$."}
+{"_id": "30506", "title": "Definition:Sphere/P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $a \\in \\Q_p$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. The '''$\\epsilon$-sphere of $a$ in $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$''' is defined as: :$\\map {S_\\epsilon} a = \\set {x \\in \\Q_p: \\norm {x - a} = \\epsilon}$"}
+{"_id": "30507", "title": "Definition:Sphere/P-adic Numbers/Radius", "text": "In $\\map {S_\\epsilon} a$, the value $\\epsilon$ is referred to as the '''radius''' of the $\\epsilon$-sphere."}
+{"_id": "30508", "title": "Definition:Sphere/P-adic Numbers/Center", "text": "In $\\map {S_\\epsilon} a$, the value $a$ is referred to as the '''center''' of the $\\epsilon$-sphere."}
+{"_id": "30510", "title": "Definition:Earth/Shape/First Approximation", "text": "To a first approximation, the shape of '''Earth''' can be considered to be spherical."}
+{"_id": "30512", "title": "Definition:Earth/Shape/Second Approximation", "text": "To a second approximation, '''Earth''' is an oblate spheroid."}
+{"_id": "30513", "title": "Definition:Spheroid/Oblate", "text": "An '''oblate spheroid''' is an ellipsoid which is the solid of revolution formed by rotating an ellipse about its minor axis."}
+{"_id": "30514", "title": "Definition:Spheroid/Prolate", "text": "A '''prolate spheroid''' is an ellipsoid which is the solid of revolution formed by rotating an ellipse about its major axis."}
+{"_id": "30515", "title": "Definition:Spheroid", "text": "A '''spheroid''' is an ellipsoid which is the solid of revolution formed by rotating an ellipse about one of its axes. === Oblate Spheroid === {{:Definition:Oblate Spheroid}} === Prolate Spheroid === {{:Definition:Prolate Spheroid}}"}
+{"_id": "30516", "title": "Definition:Ellipsoid", "text": "An '''ellipsoid''' is a solid figure whose plane sections are circles or ellipses."}
+{"_id": "30517", "title": "Definition:Ellipse/Axis", "text": "Let $K$ be an ellipse whose foci are at $F_1$ and $F_2$. === Major Axis === {{:Definition:Ellipse/Major Axis}} === Minor Axis === {{:Definition:Ellipse/Minor Axis}} Category:Definitions/Ellipses j11v3s7kuz7fer16mgjjoejkzdqyncn"}
+{"_id": "30518", "title": "Definition:Longitude (Terrestrial)", "text": "Let $J$ be a point on Earth's surface that is not one of the two poles $N$ and $S$. Let $\\bigcirc NJS$ be a meridian passing through $J$, whose endpoints are by definition $N$ and $S$. The '''longitude''' of $J$, and of the meridian $\\bigcirc NJS$ itself, is the (spherical) angle that $\\bigcirc NJS$ makes with the principal meridian $\\bigcirc NKS$. :500px If $\\bigcirc NJS$ is on the eastern hemisphere, the '''longitude''' is defined as '''longitude $n \\degrees$ east''', where $n \\degrees$ denotes $n$ degrees (of arc), written $n \\degrees \\, \\mathrm E$. If $\\bigcirc NJS$ is on the western hemisphere, the '''longitude''' is defined as '''longitude $n \\degrees$ west''', written $n \\degrees \\, \\mathrm W$. If $\\bigcirc NJS$ is the principal meridian itself, the '''longitude''' is defined as '''$0 \\degrees$ longitude'''. If $\\bigcirc NJS$ is the other half of the great circle that contains the principal meridian, the '''longitude''' is defined as '''$180 \\degrees$ longitude'''."}
+{"_id": "30520", "title": "Definition:Longitude", "text": "=== Terrestrial Longitude === {{:Definition:Longitude (Terrestrial)}} === Celestial Longitude === {{:Definition:Longitude (Celestial)}} === Galactic Longitude === {{:Definition:Longitude (Galactic)}}"}
+{"_id": "30521", "title": "Definition:Latitude (Terrestrial)", "text": "Let $J$ be a point on Earth's surface that is not one of the two poles $N$ and $S$. Let $\\bigcirc NJS$ be a meridian passing through $J$, whose endpoints are by definition $N$ and $S$. Let $\\bigcirc NJS$ pass through the equator at $L$. The '''latitude''' of $J$ is the (spherical) angle $\\sphericalangle LOJ$ , where $O$ is the center of Earth. :500px If $J$ is in the northern hemisphere of Earth, the '''latitude''' is defined as '''latitude $n \\degrees$ north''', where $n \\degrees$ denotes $n$ degrees (of arc), written $n \\degrees \\, \\mathrm N$. If $J$ is in the southern hemisphere of Earth, the '''latitude''' is defined as '''latitude $n \\degrees$ south''', written $n \\degrees \\, \\mathrm S$. At the North Pole, the '''latitude''' is $90 \\degrees \\, \\mathrm N$. At the South Pole, the '''latitude''' is $90 \\degrees \\, \\mathrm S$."}
+{"_id": "30522", "title": "Definition:Semi-Great Circle", "text": "Let $S$ be a sphere. A '''semi-great circle''' on $S$ is a semicircle consisting of half of a great circle on $S$."}
+{"_id": "30523", "title": "Definition:Northern Hemisphere", "text": "The '''northern hemisphere''' of Earth is the hemisphere between the equator and the North Pole. Points in the northern hemisphere have latitude between $0 \\degrees \\, \\mathrm N$ (the equator itself) and $90 \\degrees \\, \\mathrm N$ (the North Pole)."}
+{"_id": "30524", "title": "Definition:Southern Hemisphere", "text": "The '''southern hemisphere''' of Earth is the hemisphere between the equator and the South Pole. Points in the southern hemisphere have latitude between $0 \\degrees \\, \\mathrm S$ (the equator itself) and $90 \\degrees \\, \\mathrm S$ (the South Pole)."}
+{"_id": "30525", "title": "Definition:Eastern Hemisphere", "text": "The '''eastern hemisphere''' of Earth is the hemisphere immediately to the east of the principal meridian. Points in the eastern hemisphere have longitude between $0 \\degrees \\, \\mathrm E$ (the principal meridian itself) and $180 \\degrees \\, \\mathrm E$."}
+{"_id": "30526", "title": "Definition:Western Hemisphere", "text": "The '''western hemisphere''' of Earth is the hemisphere immediately to the west of the principal meridian. Points in the western hemisphere have longitude between $0 \\degrees \\, \\mathrm W$ (the principal meridian itself) and $180 \\degrees \\, \\mathrm W$."}
+{"_id": "30527", "title": "Definition:Latitude", "text": "=== Terrestrial Latitude === {{:Definition:Latitude (Terrestrial)}} === Celestial Latitude === {{:Definition:Latitude (Celestial)}} Category:Definitions/Geographical Coordinates Category:Definitions/Spherical Astronomy jbsqzvied2ezel4hr69qms34of08u5l"}
+{"_id": "30528", "title": "Definition:Terrestrial Hemisphere", "text": "=== Northern Hemisphere === {{:Definition:Northern Hemisphere}} === Southern Hemisphere === {{:Definition:Southern Hemisphere}} === Eastern Hemisphere === {{:Definition:Eastern Hemisphere}} === Western Hemisphere === {{:Definition:Western Hemisphere}} Category:Definitions/Earth alp6k1su22746wvaev2c4nuli2fp2mv"}
+{"_id": "30529", "title": "Definition:Colatitude (Terrestrial)", "text": "Let $J$ be a point on Earth's surface that is not one of the two poles $N$ and $S$. Let $\\phi$ denote the latitude of $J$. The '''colatitude''' of $J$ is the (spherical) angle $90 \\degrees - \\phi$, that is: :if $J$ is in the northern hemisphere of Earth, the '''colatitude''' is the (spherical) angle $\\sphericalangle NOJ$ :if $J$ is in the southern hemisphere of Earth, the '''colatitude''' is the (spherical) angle $\\sphericalangle SOJ$. :500px"}
+{"_id": "30530", "title": "Definition:Parallel of Latitude", "text": "Let $J$ be a point on Earth's surface that is not one of the two poles $N$ and $S$. Let $C$ be the small circle through $J$ parallel to the plane containing the equator. Then $C$ is known as the '''parallel of latitude''' of $J$. Thus $C$ consists of all the points on Earth's surface which have the same latitude as $J$. :500px"}
+{"_id": "30534", "title": "Definition:Imperial/Length/Mil", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''mil''' }} {{eqn | r = \\dfrac 1 {1000}       | c = inch }} {{eqn | r = 25 \\cdotp 4       | c = microns }} {{eqn | r = 0 \\cdotp 0254       | c = millimetres }} {{eqn | r = 2 \\cdotp 54 \\times 10^{-5}       | c = metres }} {{end-eqn}}"}
+{"_id": "30536", "title": "Definition:Metric System/Length", "text": "Various metric units of length which have specific names are as follows: === Micrometre === {{Definition:Metric System/Length/Micrometre}} === Millimetre === {{Definition:Metric System/Length/Millimetre}} === Centimetre === {{Definition:Metric System/Length/Centimetre}} === Decimetre === {{Definition:Metric System/Length/Decimetre}} === Metre === {{Definition:Metric System/Length/Metre}} === Hectometre === {{Definition:Metric System/Length/Hectometre}} === Kilometre === {{Definition:Metric System/Length/Kilometre}}"}
+{"_id": "30543", "title": "Definition:Non-Archimedean Norm Axioms", "text": "{{begin-axiom}} {{axiom | n = \\text N 1         | lc= Positive Definiteness:         | q = \\forall x \\in R         | ml= \\norm x = 0         | mo= \\iff         | mr= x = 0_R }} {{axiom | n = \\text N 2         | lc= Multiplicativity:         | q = \\forall x, y \\in R         | ml= \\norm {x \\circ y}         | mo= =         | mr= \\norm x \\times \\norm y }} {{axiom | n = \\text N 4         | lc= Ultrametric Inequality:         | q = \\forall x, y \\in R         | ml= \\norm {x + y}         | mo= \\le         | mr= \\max \\set {\\norm x, \\norm y} }} {{end-axiom}}"}
+{"_id": "30544", "title": "Definition:Non-Archimedean/Norm (Division Ring)/Definition 1", "text": "A norm $\\norm {\\, \\cdot \\,}$ on $R$ is '''non-Archimedean''' {{iff}} $\\norm {\\, \\cdot \\,}$ satisfies the axiom: {{begin-axiom}} {{axiom | n = \\text N 4         | lc= Ultrametric Inequality:         | q = \\forall x, y \\in R         | ml= \\norm {x + y}         | mo= \\le         | mr= \\max \\set {\\norm x, \\norm y} }} {{end-axiom}}"}
+{"_id": "30545", "title": "Definition:Non-Archimedean/Norm (Division Ring)/Definition 2", "text": "A '''non-Archimedean norm''' on $R$ is a mapping from $R$ to the non-negative reals: :$\\norm {\\, \\cdot \\,}: R \\to \\R_{\\ge 0}$ satisfying the non-Archimedean norm axioms: {{:Definition:Non-Archimedean Norm Axioms}}"}
+{"_id": "30546", "title": "Definition:Kilometre/Symbol", "text": "The symbol for the '''kilometre''' is $\\mathrm {km}$: :$\\mathrm k$ for kilo :$\\mathrm m$ for metre."}
+{"_id": "30548", "title": "Definition:Nautical Mile", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''nautical mile''' }} {{eqn | r = 1852       | c = metres (by definition) }} {{eqn | r = 10       | c = cables }} {{eqn | o = \\approx       | r = 6076       | c = feet }} {{eqn | o = \\approx       | r = 1 \\cdotp 151       | c = (international) mile }} {{eqn | o = \\approx       | r = 0 \\cdotp 999 \\, 363       | c = admiralty (UK nautical) mile }} {{end-eqn}}"}
+{"_id": "30549", "title": "Definition:Admiralty Mile", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''admiralty mile''' }} {{eqn | r = 6080       | c = feet (by definition) }} {{eqn | r = 1853 \\cdotp 184       | c = metres }} {{eqn | o = \\approx       | r = 1 \\cdotp 1515       | c = (international) mile }} {{end-eqn}}"}
+{"_id": "30551", "title": "Definition:Inner Angle of Spherical Triangle", "text": "Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Consider the $4$ consecutive parts of $\\triangle ABC$: :$B, a, C, b$ Then $C$ is known as the '''inner angle''' of $B, a, C, b$"}
+{"_id": "30552", "title": "Definition:Inner Side of Spherical Triangle", "text": "Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Consider the $4$ consecutive parts of $\\triangle ABC$: :$B, a, C, b$ Then $a$ is known as the '''inner side''' of $B, a, C, b$"}
+{"_id": "30554", "title": "Definition:Quadrantal Triangle", "text": "Let $\\triangle ABC$ be a spherical triangle. Let one of the sides of $\\triangle ABC$ be a right angle: $\\dfrac \\pi 2$. :500px Then $\\triangle ABC$ is known as a '''quadrantal (spherical) triangle'''."}
+{"_id": "30555", "title": "Definition:Right Spherical Triangle", "text": "Let $\\triangle ABC$ be a spherical triangle. Let one of the angles of $\\triangle ABC$ be a right angle: $\\dfrac \\pi 2$. :500px Then $\\triangle ABC$ is known as a '''right spherical triangle'''."}
+{"_id": "30556", "title": "Definition:Spherical Excess of Spherical Triangle", "text": "Let $\\triangle ABC$ be a spherical triangle. Let $E$ be the sum of the angles of $\\triangle ABC$. The '''spherical excess''' of $\\triangle ABC$ is how much $E$ is greater than $\\pi$ radians."}
+{"_id": "30557", "title": "Definition:Polar Triangle", "text": "Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let $A'$, $B'$ and $C'$ be the poles of the sides $BC$, $AC$ and $AB$ respectively which are in the same hemisphere as the points $A$, $B$ and $C$ respectively. :400px Then the spherical triangle $\\triangle A'B'C'$ is the '''polar triangle''' of $\\triangle ABC$."}
+{"_id": "30558", "title": "Definition:Length of Arc of Great Circle", "text": "Let $E$ be a great circle on a sphere $S$ whose center is $O$. Let $AB$ be an arc of $E$. The '''length''' of $AB$ is defined as the angle subtended at $O$ by $AB$."}
+{"_id": "30559", "title": "Definition:Haversine", "text": "The '''haversine''' of an angle is defined as: {{begin-eqn}} {{eqn | l = \\hav \\theta       | r = \\dfrac 1 2 \\paren {1 - \\cos \\theta}       | c =  }} {{eqn | r = \\dfrac 1 2 \\vers \\theta       | c = where $\\vers \\theta$ denotes the versed sine of $\\theta$ }} {{eqn | r = \\sin^2 \\dfrac {\\theta} 2       | c = Double Angle Formula for Cosine: Corollary 2 }} {{end-eqn}}"}
+{"_id": "30561", "title": "Definition:Versed Sine", "text": "The '''versed sine''' of an angle is defined as: :$\\vers \\theta = 1 - \\cos \\theta$"}
+{"_id": "30565", "title": "Definition:Knot (Unit of Measurement)", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''knot''' }} {{eqn | r = 1 \\, 852       | c = metres per hour (exactly) }} {{eqn | o = \\approx       | r = 0 \\cdotp 51444       | c = metres per second }} {{eqn | o = \\approx       | r = 1 \\cdotp 15078       | c = (international) miles per hour }} {{end-eqn}}"}
+{"_id": "30566", "title": "Definition:Zenith", "text": "Let an observer $O$ on the surface of Earth be assumed to be the center of the celestial sphere $C$. The '''zenith''' $Z$ is the point on $C$ which is vertically overhead."}
+{"_id": "30567", "title": "Definition:Celestial Sphere", "text": "The '''celestial sphere''' is an imaginary sphere, an indeterminate distance from Earth, which intersects the straight lines from the celestial bodies to the observer. :480px It is then convenient to identify the directions of the celestial bodies with their position on the '''celestial sphere'''."}
+{"_id": "30568", "title": "Definition:Celestial", "text": "In the context of astronomy, the word '''celestial''' is an adjective whose meaning is loosely '''of the sky'''. It is used to define bodies and phenomena vaguely defined as being outside Earth's atmosphere, hence does not apply to meteorological phenomena. It is often used as an antithesis for the word '''terrestrial''', which means '''of Earth'''."}
+{"_id": "30570", "title": "Definition:Terrestrial", "text": "The word '''terrestrial''' is an adjective whose meaning is loosely '''of Earth'''. It is usually used in the context of astronomy and geodesy to refer to concepts which need to be distinguished from their celestial counterparts."}
+{"_id": "30572", "title": "Definition:Celestial Sphere/Observer", "text": "In the context of astronomy, in particular spherical astronomy, the '''observer''' is considered to be a reference point located at the center of the celestial sphere. In practice, of course, an actual '''observer''' is physically located on the surface of Earth. However, the celestial sphere is considered to be so much larger than the radius of Earth that the '''observer''' is considered to be at the center of Earth."}
+{"_id": "30574", "title": "Definition:Celestial Horizon", "text": "The '''celestial horizon''' is the great circle on the celestial sphere whose plane is perpendicular to the line from the zenith to the observer. Thus the '''celestial horizon''' is the edge of the visible part of the celestial sphere. The rest of it is hidden by Earth itself. :480px"}
+{"_id": "30577", "title": "Definition:Visible Hemisphere", "text": "The '''visible hemisphere''' is the hemisphere of the celestial sphere between the celestial horizon and the zenith. That is, it is the half of the sky that can be seen from Earth at an arbitrary time. The other hemisphere of the celestial sphere is hidden from an observer by Earth itself. :480px"}
+{"_id": "30578", "title": "Definition:Vertical Circle", "text": "Let $X$ be the position of a star (or other celestial body) on the celestial sphere. The '''vertical circle''' through $X$ is defined as the great circle passing through both $X$ and the zenith $Z$."}
+{"_id": "30579", "title": "Definition:Celestial Altitude", "text": "Let $X$ be a point on the celestial sphere. The '''(celestial) altitude''' of $X$ is defined as the angle subtended by the the arc of the vertical circle through $X$ between the celestial horizon and $X$ itself."}
+{"_id": "30580", "title": "Definition:Zenith Distance", "text": "Let $X$ be the position of a star (or other celestial body) on the celestial sphere. The '''zenith distance''' of $X$ is defined as the angle subtended by the the arc of the vertical circle through $X$ between $X$ and the zenith."}
+{"_id": "30581", "title": "Definition:Parallel of Altitude", "text": "Let $X$ be a point on the celestial sphere. Let $LXM$ be a small circle through $X$ which is parallel to the celestial horizon. Then $LXM$ is a '''parallel of altitude''', and is such that all celestial bodies on it have the same altitude."}
+{"_id": "30582", "title": "Definition:North Celestial Pole", "text": "Let $O$ be a celestial observer situated at the surface of Earth in the Northern hemisphere. Let $OP$ be a straight line parallel to Earth's axis. Let $P$ be the point in the visible hemisphere at which $OP$ meets the celestial sphere. Then $P$ is the '''north celestial pole'''. That is, the '''north celestial pole''' is the pole of the celestial equator which is the zenith of the North Pole."}
+{"_id": "30583", "title": "Definition:Polaris", "text": "'''Polaris''' is a star whose position in the celestial sphere is very nearly exactly at the north celestial pole. Hence it appears not to move through the sky during the course of a night."}
+{"_id": "30584", "title": "Definition:Principal Vertical Circle", "text": "The vertical circle on the celestial sphere which passes through the north celestial pole (and south celestial pole) is called the '''principal vertical circle'''. :480px"}
+{"_id": "30585", "title": "Definition:North Point of Horizon", "text": "The point on the celestial horizon through which the principal vertical circle passes is called the '''north point of the horizon'''."}
+{"_id": "30586", "title": "Definition:South Point of Horizon", "text": "The point on the celestial horizon exactly opposite the '''north point of the horizon''' is called the '''south point of the horizon'''."}
+{"_id": "30587", "title": "Definition:West Point of Horizon", "text": "Let an observer $O$ be facing the '''north point of the horizon''' $N$, facing directly away from the '''south point of the horizon''' $S$. Let the celestial equator meet the celestial horizon at the points $E$ and $W$. Let $W$ be the point on the left of $O$. This is called the '''west point of the horizon'''."}
+{"_id": "30588", "title": "Definition:East Point of Horizon", "text": "Let an observer $O$ be facing the '''north point of the horizon''' $N$, facing directly away from the '''south point of the horizon''' $S$. Let the celestial equator meet the celestial horizon at the points $E$ and $W$. Let $E$ be the point on the right of $O$. This is called the '''east point of the horizon'''."}
+{"_id": "30589", "title": "Definition:Cardinal Points (Celestial)", "text": "The '''cardinal points''' of the celestial sphere are the '''north point''', the '''sourth point''', the '''east point''' and the '''west point''': === North Point of Horizon === {{:Definition:North Point of Horizon}} === South Point of Horizon === {{:Definition:South Point of Horizon}} === East Point of Horizon === {{:Definition:East Point of Horizon}} === West Point of Horizon === {{:Definition:West Point of Horizon}}"}
+{"_id": "30590", "title": "Definition:Azimuth (Astronomy)", "text": "Let $X$ be a point on the celestial sphere. The spherical angle between the principal vertical circle and the vertical circle on which $X$ lies is the '''azimuth''' of $X$. The '''azimuth''' is usually measured in degrees, $0 \\degrees$ to $180 \\degrees$ either west or east, depending on whether $X$ lies on the eastern or western hemisphere of the celestial sphere."}
+{"_id": "30591", "title": "Definition:Prime Vertical", "text": "The vertical circle on the celestial sphere which passes through both the east point and the west point is called the '''prime vertical'''."}
+{"_id": "30592", "title": "Definition:Celestial Equator", "text": "Consider the celestial sphere with observer $O$. Let $P$ be the north celestial pole. The great circle whose plane is perpendicular to $OP$ is known as the '''celestial equator'''."}
+{"_id": "30593", "title": "Definition:Nadir", "text": "Consider the celestial sphere $C$ with an observer $O$ at the center. Let $Z$ be the zenith of $C$ Let  the straight line $ZO$ be produced from $O$ to meet $C$ at $R$. The point $R$ is known as the '''nadir''' of $C$."}
+{"_id": "30596", "title": "Definition:South Celestial Pole", "text": "Let $O$ be a celestial observer situated at the surface of Earth in the Southern hemisphere. Let $OQ$ be a straight line parallel to Earth's axis. Let $Q$ be the point in the visible hemisphere at which $OQ$ meets the celestial sphere. Then $Q$ is the '''south celestial pole'''. That is, the '''south celestial pole''' is the pole of the celestial equator which is the zenith of the South Pole."}
+{"_id": "30597", "title": "Definition:Celestial Axis", "text": "Consider the celestial sphere with observer $O$. The '''celestial axis''' is the line which joins the north celestial pole and the south celestial pole."}
+{"_id": "30598", "title": "Definition:Celestial Pole", "text": "Consider the celestial sphere $C$. The '''celestial poles''' of $C$ are the north celestial pole and the south celestial pole: === North Celestial Pole === {{:Definition:North Celestial Pole}} === South Celestial Pole === {{:Definition:South Celestial Pole}}"}
+{"_id": "30599", "title": "Definition:Declination", "text": "Consider the celestial sphere $C$ with observer $O$. Let $P$ and $Q$ be the north celestial pole and south celestial pole respectively. Let $X$ be a point on $C$. Let $PXQ$ be the vertical circle through $X$. Let $D$ be the point where the celestial equator intersects $PXQ$. The length of the arc $DX$ of $PXQ$ is known as the '''declination''' of $X$."}
+{"_id": "30600", "title": "Definition:Declination/North", "text": "If $X$ is in the northern celestial hemisphere, $DX$ is '''north declination'''."}
+{"_id": "30601", "title": "Definition:Declination/South", "text": "If $X$ is in the southern celestial hemisphere, $DX$ is '''south declination'''."}
+{"_id": "30606", "title": "Definition:North Polar Distance", "text": "Consider the celestial sphere $C$ with observer $O$. Let $X$ be a point on $C$. Let $X$ have declination of $\\delta$. The '''north polar distance''' of $X$, denoted $\\operatorname {N.P.D.} x$, is defined as: :$\\operatorname {N.P.D.} x = 90 \\degrees - \\delta$ That is, the '''north polar distance''' of $X$ length of the arc from the north celestial pole $P$ along the vertical circle through $X$. :500px"}
+{"_id": "30608", "title": "Definition:Observer's Meridian", "text": "Consider the celestial sphere whose observer is $O$. Let $P$ and $Q$ be the north celestial pole and south celestial pole respectively. Let $Z$ be the zenith. Let $S$ be the south point of the horizon. The '''observer's meridian''' is the semi-great circle $PZSQ$."}
+{"_id": "30614", "title": "Definition:Babylonian Number System", "text": "The number system as used in the {{WP|First_Babylonian_Dynasty|Old Babylonian empire}} was a positional numeral system where the number base was a combination of decimal (base $10$) and sexagesimal (base $60$). The characters were written in {{WP|Cuneiform|cuneiform}} by a combination of: :a thin vertical wedge shape, to indicate the digit $1$ :a fat horizontal wedge shape, to indicate the digit $10$ arranged in groups to indicate the digits $2$ to $9$ and $20$ to $50$. :600px At $59$ the pattern stops, and the number $60$ is represented by the digit $1$ once again. Thus these groupings were placed side by side: :the rightmost grouping would indicate a number from $1$ to $59$ :the one to the left of that would indicate a number from $60 \\times 1$ to $60 \\times 59$ and so on, each grouping further to the left indicating another multiplication by $60$ For fractional numbers there was no actual radix point. Instead, the distinction was inferred by context. The fact that they had no symbol to indicate the zero digit means that this was not a true positional numeral system as such. For informal everyday arithmetic, they used a decimal system which was the decimal part of the full sexagesimal system."}
+{"_id": "30615", "title": "Definition:Number System", "text": "A '''number system''' is a technique for representing numbers."}
+{"_id": "30616", "title": "Definition:Numeral System", "text": "A '''numeral system''' is: :a set of symbols that is used to represent a specific subset of the set of numbers (usually natural numbers), referred to as numerals :a set of rules which define how to combine the numerals so as to be able to express other numbers."}
+{"_id": "30617", "title": "Definition:Numeral", "text": "A '''numeral''' is a symbol which is used to identify a particular number (usually a natural number). A '''numeral''' is one of a small subset of the numbers. Other numbers can be created by combining '''numerals''' using specific well-defined rules."}
+{"_id": "30618", "title": "Definition:Positional Numeral System", "text": "A '''positional number system''' is a number system with the following properties: :It has a set of numerals which represent a subset of the numbers. :The number being represented is written as a string of these numerals, which represent a different value according to their position in the numerals. The design of the '''positional number system''' is such that all numbers can be represented by such a string, which may or may not be infinite in length."}
+{"_id": "30624", "title": "Definition:Musical Interval", "text": "A '''musical interval''' is a ratio of frequencies of musical notes which is recognised by convention as being particularly aesthetically pleasing or meaningful. === Octave === {{:Definition:Octave}} === Fifth === {{:Definition:Fifth (Music)}} === Fourth === {{:Definition:Fourth (Music)}} === Tone === {{:Definition:Tone (Music)}} === Semitone === {{:Definition:Semitone}}"}
+{"_id": "30626", "title": "Definition:Neusis", "text": "=== Neusis Ruler === {{:Definition:Neusis Ruler}} === Neusis Construction === {{:Definition:Neusis Construction}}"}
+{"_id": "30629", "title": "Definition:Decimal Expansion/Historical Note", "text": "The idea of representing fractional values by extending the decimal notation to the right appears to have been invented by {{AuthorRef|Simon Stevin}}, who published the influential book {{BookLink|De Thiende|Simon Stevin}}. The idea was borrowed from the Babylonian number system, but streamlined to base $10$ from the cumbersome sexagesimal. However, his notation was cumbersome: he would write, for example, $25 \\bigcirc \\! \\! \\! \\! \\! \\! 0 \\ \\, 3 \\bigcirc \\! \\! \\! \\! \\! \\! 1 \\ \\, 7 \\bigcirc \\! \\! \\! \\! \\! \\! 2 \\ \\, 9 \\bigcirc \\! \\! \\! \\! \\! \\! 3$ for what we now give as $25 \\cdotp 379$. {{AuthorRef|John Napier}}, in the early $17$th century, appears to have been the first into print with the contemporary notation, although {{AuthorRef|Walter William Rouse Ball}} suggests that credit for this ought to be due to {{AuthorRef|Henry Briggs}}. It was not until a century later, however, that the decimal point came into general use."}
+{"_id": "30632", "title": "Definition:Trigonometry/Linguistic Note", "text": "The word '''trigonometry''' derives from the Greek for '''measurement of triangles'''."}
+{"_id": "30638", "title": "Definition:Epicycle/Historical Note", "text": "{{AuthorRef|Claudius Ptolemy}}'s {{BookLink|Almagest|Claudius Ptolemy}} modelled the orbits of the planets of the solar system as a system of epicycles."}
+{"_id": "30640", "title": "Definition:Sextant", "text": "A '''sextant''' is a device for measuring the celestial altitude of a celestial body. It consists of a telescope and a system of adjustable reflectors for presenting the image of the celestial body to the eye so as to be next to an image of the horizon. The resulting reading on the scale of the '''sextant''' gives an indication of the altitude of the body."}
+{"_id": "30642", "title": "Definition:Astrolabe", "text": "An '''astrolabe''' is a device for measuring the celestial altitude of a celestial body. It consists of a series of disks on which are shown the positions of prominent stars. These are aligned with their positions on what can be seen of the celestial sphere, and hence obtain a reading of their altitudes."}
+{"_id": "30645", "title": "Definition:GPS", "text": "'''GPS''', or '''Global Positioning System''', is a technique using signals from satellites to identify the geographical coordinates of an appropriately configured receiver."}
+{"_id": "30646", "title": "Definition:Noon", "text": "'''Noon''' is the point in time during the course of a day that the Sun reaches its highest celestial altitude. Category:Definitions/Navigation Category:Definitions/Geography Category:Definitions/Astronomy pj5i99obzjmns4nlyj39e74w9laz4xn"}
+{"_id": "30650", "title": "Definition:Unknown", "text": "An '''unknown''' is an indeterminate variable in a system of equations or inequalities whose value the solver is invited to determine."}
+{"_id": "30651", "title": "Definition:Diophantine Problem", "text": "A '''Diophantine problem''' is a system of Diophantine equations which has more unknowns than it has equations."}
+{"_id": "30655", "title": "Definition:Binary Tuple", "text": "A '''binary tuple''' is an ordered tuple whose domain is the set $\\set {0, 1}$. That is, it is an ordered $n$-tuple $\\tuple {x_1, x_2, \\ldots, x_n}$ such that each $x_i$ is either $0$ or $1$. The set of all binary $n$-tuples can often be seen denoted $V_n$: :$V_n := \\set {\\tuple {x_1, x_2, \\ldots, x_n}: \\forall i \\in \\set {1, 2, \\ldots n}: x_i \\in \\set {0, 1} }$"}
+{"_id": "30656", "title": "Definition:Set of Strings of Given Length", "text": "Let $n \\in \\N$ be a natural number. Let $\\Sigma$ be an alphabet. The set of all strings from $\\Sigma$ of length $n$ is denoted $\\Sigma^{\\paren n}$."}
+{"_id": "30657", "title": "Definition:Set of Finite Strings", "text": "Let $\\Sigma$ be an alphabet. The set of all finite strings from $\\Sigma$ is denoted $\\Sigma^*$."}
+{"_id": "30658", "title": "Definition:Symmetric Mapping", "text": "=== Mapping Theory === {{:Definition:Symmetric Mapping (Mapping Theory)}} === Linear Algebra === {{:Definition:Symmetric Mapping (Linear Algebra)}} Category:Definitions/Mapping Theory Category:Definitions/Linear Algebra g3wpcxdc1eztpzylzbb1npci2h88ha7"}
+{"_id": "30659", "title": "Definition:Symmetric Mapping (Mapping Theory)", "text": "Let $n \\in \\N$ be a natural number. Let $S^n$ be an $n$-dimensional cartesian space on a set $S$. Let $f: S^n \\to T$ be a mapping from $S^n$ to a set $T$. Then $f$ is a '''symmetric mapping''' {{iff}}: :$\\map f {x_1, x_2, \\dotsc, x_n} = \\map f {x_{\\map \\pi 1}, x_{\\map \\pi 2}, \\dotsc x_{\\map \\pi n} }$ for all permutations $\\pi$ on $\\set {1, 2, \\dotsc n}$. That is, a '''symmetric mapping''' is a mapping defined on a cartesian space whose values are preserved under permutation of its arguments."}
+{"_id": "30660", "title": "Definition:Uncertainty", "text": "Let $X$ be a random variable. Let $X$ take a finite number of values with probabilities $p_1, p_2, \\dotsc, p_n$. The '''uncertainty''' of $X$ is defined to be: :$\\map H X = \\displaystyle -\\sum_k p_k \\lg p_k$ where: :$\\lg$ denotes logarithm base $2$ :the summation is over those $k$ where $p_k > 0$."}
+{"_id": "30661", "title": "Definition:Chebyshev-Sylvester Constant", "text": "The '''Chebyshev-Sylvester constant''' is a mathematical constant $\\alpha$ which was shown by {{AuthorRef|Pafnuty Lvovich Chebyshev}} and {{AuthorRef|James Joseph Sylvester}} that for sufficiently large $x$, there exists at least one prime number $p$ satisfying: :$x < p < \\paren {1 + \\alpha} x$ The number $\\alpha$ was demonstrated to be $0 \\cdotp 092 \\ldots$ In $1896$, {{AuthorRef|Jacques Salomon Hadamard}} and {{AuthorRef|Charles de la Vallée Poussin}} independently proved the Prime Number Theorem. This demonstrated that the above inequality is true for all $\\alpha > 0$ for sufficiently large $x$,. Hence this constant is now only of historical interest. {{NamedforDef|Pafnuty Lvovich Chebyshev|name2 = James Joseph Sylvester|cat = Chebyshev|cat2 = Sylvester}}"}
+{"_id": "30662", "title": "Definition:Du Bois-Reymond Constants", "text": "The '''du Bois-Reymond constants''' are the constants $C_n$ where: :$C_n = \\displaystyle \\int_0^\\infty \\size {\\map {\\dfrac \\d {\\d t} } {\\dfrac {\\sin t} t}^n} \\rd t - 1$"}
+{"_id": "30663", "title": "Definition:Copeland-Erdős Constant", "text": "The '''Copeland-Erdős constant''' is the real number whose decimal expansion is formed by concatenating the prime numbers in ascending order: :$C_{10} = 0 \\cdotp 23571 \\, 11317 \\, 19232 \\, 93137 \\, 4143 \\ldots$ {{OEIS|A033308}}"}
+{"_id": "30664", "title": "Definition:Koebe Constant", "text": "The Koebe Quarter Theorem states: {{:Koebe Quarter Theorem}} That constant $\\dfrac 1 4$ by which $\\size {\\map {f'} 0}$ is bounded below can be found referred to as the '''Koebe constant'''. {{NamedforDef|Paul Koebe|cat = Koebe}}"}
+{"_id": "30665", "title": "Definition:Meissel-Mertens Constant", "text": "Consider the expression: :$\\displaystyle M = \\map {\\lim_{n \\mathop \\to \\infty} } {\\sum_{\\substack {p \\mathop \\le n \\\\ \\text {$p$ prime} } } \\dfrac 1 p - \\ln \\ln p}$ Then: :$M \\approx 0 \\cdotp 26149 \\, 72128 \\, 47642 \\, 78375 \\, 54268 \\, 38608 \\, 69585 \\, 90516 \\ldots$ {{OEIS|A077761}} The constant $M$ is known as the '''Meissel-Mertens Constant'''."}
+{"_id": "30666", "title": "Definition:Matroid Theory", "text": "'''Matroid Theory''' is the branch of mathematics which concerns the role which matroids play in disparate branches of combinatorial theory and algebra such as graph theory, lattice theory, combinatorial optimization, and linear algebra."}
+{"_id": "30667", "title": "Definition:Kakeya's Constant", "text": "'''Kakeya's constant''' is defined as the area of the smallest simple convex domain in which one can put a line segment of length $1$ which will coincide with itself when rotated $180 \\degrees$: :$K = \\dfrac {\\paren {5 - 2 \\sqrt 2} \\pi} {24} \\approx 0 \\cdotp 28425 \\, 82246 \\ldots$ {{OEIS|A093823}} {{expand|Needs considerable work done here by someone who understands exactly what is going on here. The case of the equilateral triangle is well known; so is the case of the Perron tree; I also remember a piece by Martin Gardner on the subject which demonstrates that a star-shaped area derived from the deltoid can be made arbitrarily small; and so on. Exactly what is meant here by simple convex domain needs rigorous clarification.}}"}
+{"_id": "30668", "title": "Definition:Matroid", "text": "Let $M = \\struct{S,\\mathscr I}$ be an independence system."}
+{"_id": "30669", "title": "Definition:Matroid Axioms", "text": "<onlyinclude> Let $S$ be a finite set. Let $\\mathscr I$ be a set of subsets of $S$. The '''matroid axioms''' are the conditions on $S$ and $\\mathscr I$ in order for the ordered pair $\\struct {S, \\mathscr I}$ to be a matroid:  === Axioms 1 === {{:Definition:Matroid Axioms/Axioms 1}} === Axioms 2 === {{:Definition:Matroid Axioms/Axioms 2}} === Axioms 3 === {{:Definition:Matroid Axioms/Axioms 3}} === Axioms 4 === {{:Definition:Matroid Axioms/Axioms 4}}"}
+{"_id": "30670", "title": "Definition:Matroid/Independent Set", "text": "An element of $\\mathscr I$ is called an '''independent set''' of $M$."}
+{"_id": "30671", "title": "Definition:Matroid/Dependent Set", "text": "A subset of $S$ that is not an element of $\\mathscr I$ is called a '''dependent set''' of $M$."}
+{"_id": "30672", "title": "Definition:Base of Matroid", "text": "A '''base''' of $M$ is a maximal independent subset of $S$."}
+{"_id": "30673", "title": "Definition:Rank Function (Matroid)", "text": "The '''rank function''' of $M$ is the mapping $\\rho : \\powerset S \\to \\Z$ from the power set of $S$ into the integers defined by: :$\\forall A \\subseteq S : \\map \\rho A = \\max \\set {\\size X : X \\subseteq A \\land X \\in \\mathscr I}$ where $\\size A$ denotes the cardinality of $A$."}
+{"_id": "30674", "title": "Definition:Rank (Matroid)", "text": "The '''rank''' of $M$, denoted by $\\map \\rho M$, is the image of $S$ under $\\rho$, that is: :$\\map \\rho M = \\map \\rho S$."}
+{"_id": "30675", "title": "Definition:Flat (Matroid)", "text": "A subset $A \\subseteq S$ is a '''flat''' of $M$ {{iff}}: :$\\forall x \\in S \\setminus A : \\map \\rho {A \\cup \\set x} = \\map \\rho A + 1$"}
+{"_id": "30676", "title": "Definition:Depends Relation (Matroid)", "text": "For $x \\in S$ and $A \\subseteq S$, $x$ is said to '''depend''' on $A$, denoted as $x \\sim A$, {{iff}}: :$\\map \\rho {A \\cup \\set x} = \\map \\rho A$"}
+{"_id": "30677", "title": "Definition:Closure Operator (Matroid)", "text": "The '''closure operator''' of the matroid $M$ is the mapping $\\sigma : \\powerset S \\to \\powerset S$ defined by: :$\\map \\sigma A$ is the set of elements of $S$ which depend on $A$"}
+{"_id": "30678", "title": "Definition:Spanning Subset (Matroid)", "text": "A subset $X \\subseteq S$ is '''spanning''' in $M$ {{iff}}: :there exists a base $B$ contained in $X$"}
+{"_id": "30679", "title": "Definition:Circuit (Matroid)", "text": "A '''circuit''' of $M$ is a minimal dependent subset of $S$."}
+{"_id": "30680", "title": "Definition:Base Axiom (Matroid)", "text": "{{begin-axiom}} {{axiom | n = \\text B 1         | q = \\forall B_1, B_2 \\in \\mathscr B         | mr= x \\in B_1 \\setminus B_2 \\implies \\exists y \\in B_2 \\setminus B_1 : \\paren {B_1 \\cup \\set y} \\setminus \\set x \\in \\mathscr B }} {{end-axiom}}"}
+{"_id": "30681", "title": "Definition:Rank Axioms (Matroid)/Definition 1", "text": "{{begin-axiom}} {{axiom | n = \\text R 1         | mr= \\map \\rho \\O = 0 }} {{axiom | n = \\text R 2         | q = \\forall X \\in \\powerset S \\land y \\in S         | mr= \\map \\rho X \\le \\map \\rho {X \\cup \\set y} \\le \\map \\rho X + 1 }} {{axiom | n = \\text R 3         | q = \\forall X \\in \\powerset S \\land y, z \\in S         | mr= \\map \\rho {X \\cup \\set y} = \\map \\rho {X \\cup \\set z} = \\map \\rho X \\implies \\map \\rho {X \\cup \\set{y,z} } = \\map \\rho X }} {{end-axiom}}"}
+{"_id": "30682", "title": "Definition:Rank Axioms (Matroid)/Definition 2", "text": "{{begin-axiom}} {{axiom | n = \\text R 1'         | q = \\forall X \\in \\powerset S         | mr = 0 \\le \\map \\rho X \\le \\size X }} {{axiom | n = \\text R 2'         | q = \\forall X, Y \\in \\powerset S         | mr = X \\subseteq Y \\implies \\map \\rho X \\le \\map \\rho Y }} {{axiom | n = \\text R 3'         | q = \\forall X, Y \\in \\powerset S         | mr = \\map \\rho {X \\cup Y} + \\map \\rho {X \\cap Y} \\le \\map \\rho X + \\map \\rho Y }} {{end-axiom}}"}
+{"_id": "30683", "title": "Definition:Rank Axioms (Matroid)", "text": "Let $S$ be a finite set. Let $\\rho : \\powerset S \\to \\Z$ be a mapping from the power set of $S$ into the integers === Definition 1 === $\\rho$ is said to satisfy the '''rank axioms''' {{iff}}: {{:Definition:Rank Axioms (Matroid)/Definition 1}} === Definition 2 === $\\rho$ is said to satisfy the '''rank axioms''' {{iff}}: {{:Definition:Rank Axioms (Matroid)/Definition 2}} Category:Definitions/Matroid Theory hchpgizbnsrgwofz4fawrvgwwsrjclf"}
+{"_id": "30687", "title": "Definition:Isomorphism (Matroid)", "text": "Let $M_1 = \\struct{S_1, \\mathscr I_1}$ and $M_2 = \\struct{S_2, \\mathscr I_2}$ be matroids. Let $f : S_1 \\to S_2$ be a bijection. Then $f$ is called an '''isomorphism''' if: :$\\forall X \\subseteq S : X \\in \\mathscr I_1 \\iff \\map f X \\in \\mathscr I_2$ If $f$ is an '''isomorphism''' then $M_1$ is said to be '''isomorphic''' to $M_2$."}
+{"_id": "30688", "title": "Definition:Bloch's Constant", "text": "Recall Bloch's Theorem: {{:Bloch's Theorem}} The lower bound of $B$ is known as '''Bloch's constant'''."}
+{"_id": "30689", "title": "Definition:Landau's Constant", "text": "Recall Landau's Theorem: {{:Landau's Theorem}} The constant of $L$ is known as '''Landau's constant'''."}
+{"_id": "30691", "title": "Definition:Uniform Matroid", "text": "Let $S$ be a finite set of cardinality $n$. Let $\\mathscr I_{k,n}$ be the set of all subsets of $S$ of cardinality less than or equal to $k$.  Then the ordered pair $\\struct {S, \\mathscr I_{k, n} }$ is called the '''uniform matroid of rank $k$''' and is denoted $U_{k,n}$."}
+{"_id": "30692", "title": "Definition:Free Matroid", "text": "Let $S$ be a finite set. Let $\\mathscr I = \\powerset S$ be the power set of $S$. That is, let $\\mathscr I$ be the set of all subsets of $S$: :$\\mathscr I := \\set {X: X \\subseteq S}$ Then the ordered pair $\\struct{S, \\mathscr I}$ is called the '''free matroid of $S$'''."}
+{"_id": "30693", "title": "Definition:Matroid Induced by Linear Independence/Vector Space", "text": "Let $V$ be a vector space. Let $S$ be a finite subset of $V$. Let $\\mathscr I$ be the set of linearly independent subsets of $S$. Then the ordered pair $\\struct{S, \\mathscr I}$ is called a '''matroid induced on $S$ by linear independence in $V$'''."}
+{"_id": "30694", "title": "Definition:Cycle Matroid", "text": "Let $G$ be a graph. Let $E$ be the edge set of $G$. Let $\\mathscr I$ be the set of edge sets of subgraphs of $G$ that contain no cycles. Then the ordered pair $\\struct{E, \\mathscr I}$ is called the '''cycle matroid''' of the graph $G$."}
+{"_id": "30696", "title": "Definition:Riemann Zeta Function/Zero", "text": "The '''zeroes''' of the Riemann zeta function are, according to the conventional definition of the zero of a (complex) function, the points $s \\in \\C$ such that: :$\\map \\zeta s = 0$ === Trivial Zeroes === {{:Definition:Riemann Zeta Function/Zero/Trivial}} === Nontrivial Zeroes === {{:Definition:Riemann Zeta Function/Zero/Nontrivial}}"}
+{"_id": "30697", "title": "Definition:Riemann Zeta Function/Zero/Trivial", "text": "The '''trivial zeroes''' of the Riemann zeta function $\\zeta$ are the strictly negative even integers : :$\\set {n \\in \\Z: n = -2 \\times k: k \\in \\N_{\\ne 0} } = \\set {-2, -4, -6, \\ldots}$"}
+{"_id": "30698", "title": "Definition:Brocard Points", "text": ":700px Let $\\triangle ABC$ be a triangle such that $A \\to B \\to C \\to A$ travels anticlockwise around $\\triangle ABC$. === First Brocard Point === {{:Definition:Brocard Points/First}} === Second Brocard Point === {{:Definition:Brocard Points/Second}}"}
+{"_id": "30699", "title": "Definition:Brocard Points/First", "text": "Let the point $P$ be constructed such that: :$\\angle PAB = \\angle PBC = \\angle PCA$ Then $P$ is the '''first Brocard point''' of $\\triangle ABC$."}
+{"_id": "30700", "title": "Definition:Brocard Points/Second", "text": "Let the point $Q$ be constructed such that: :$\\angle QAC = \\angle QCB = \\angle QBA$ Then $Q$ is the '''second Brocard point''' of $\\triangle ABC$."}
+{"_id": "30701", "title": "Definition:Brocard Angle", "text": ":700px Let $\\triangle ABC$ be a triangle such that $A \\to B \\to C \\to A$ travels anticlockwise around $\\triangle ABC$. Let $P$ and $Q$ be the first and second Brocard points of $\\triangle ABC$ respectively. Let: {{begin-eqn}} {{eqn | l = \\omega       | r = \\angle PAB = \\angle PBC = \\angle PCA }} {{eqn | l = \\omega'       | r = \\angle QAC = \\angle QCB = \\angle QBA }} {{end-eqn}} From Brocard Angle is Unique: :$\\omega = \\omega'$ This angle $\\omega = \\omega'$ is the '''Brocard angle''' of $\\triangle ABC$."}
+{"_id": "30702", "title": "Definition:Lehmer's Constant", "text": "Let the sequence $\\sequence {b_k}_{k \\mathop \\in \\N}$ be defined by the recurrence relation: {{begin-eqn}} {{eqn | l = b_k       | r = \\begin {cases} 0 & : k = 0 \\\\ {b_{k - 1} }^2 + b_{k - 1} + 1 & : \\text {otherwise} \\end{cases} }} {{eqn | r = \\sequence {0, 1, 3, 13, 183, 33973 \\ldots} }} {{end-eqn}} {{OEIS|A002065}} '''Lehmer's constant''' is the real number $\\xi$ defined as: {{begin-eqn}} {{eqn | l = \\xi       | r = \\map \\cot {\\sum_{k \\mathop = 0}^\\infty \\paren {-1}^k \\arccot b_k} }} {{eqn | r = \\map \\cot {\\arccot 0 - \\arccot 1 + \\arccot 3 - \\arccot 13 + \\arccot 183 - \\arccot 33973 + \\dotsb} }} {{eqn | o = \\approx       | r = 0 \\cdotp 59263 \\, 27182 \\ldots }} {{end-eqn}} {{OEIS|A030125}}"}
+{"_id": "30704", "title": "Definition:Vectorial Matroid", "text": "Let $V$ be a vector space. Let $S$ be a finite subset of $V$. Let $\\struct{S, \\mathscr I}$ be the matroid induced by linear independence in $V$ on $S$. From Matroid Induced by Linear Independence in Vector Space is Matroid, $\\struct{S, \\mathscr I}$ is a matroid. Then any matroid isomorphic to $\\struct{S, \\mathscr I}$ is called a '''vectorial matroid'''."}
+{"_id": "30705", "title": "Definition:Matroid Induced by Algebraic Independence", "text": "Let $L / K$ be a field extension. Let $S \\subseteq L$ be a finite subset of $L$. Let $\\mathscr I$ be the set of algebraically independent subsets of $S$. Then $\\struct {S, \\mathscr I}$ is called the '''matroid induced by algebraic independence over $K$ on $S$'''."}
+{"_id": "30706", "title": "Definition:Algebraic Matroid", "text": "Let $L / K$ be a field extension. Let $S \\subseteq L$ be a finite subset of $L$. Let $\\struct{S, \\mathscr I}$ be the matroid induced by Algebraic Independence over $K$ on $S$. From Matroid Induced by Algebraic Independence is Matroid, $\\struct{S, \\mathscr I}$ is a matroid. Then any matroid isomorphic to $\\struct{S, \\mathscr I}$ is called an '''algebraic matroid'''."}
+{"_id": "30707", "title": "Definition:Affinely Dependent", "text": "Let $\\R^n$ be the $n$-dimensional real Euclidean space. Let $S = \\set{x_1, \\dots, x_r}$ be a finite subset of $\\R^n$. An element $x \\in \\R^n$ is '''affinely dependent''' on $S$ if there exist real numbers $\\set{\\lambda_i: 1 \\le i \\le r}$ such that: :$(1) \\quad x = \\displaystyle \\sum_{i = 1}^r \\lambda_i x_i$  :$(2) \\quad \\displaystyle \\sum_{i = 1}^r \\lambda_i = 1$"}
+{"_id": "30708", "title": "Definition:Affinely Dependent/Independent", "text": "Let $\\R^n$ be the $n$-dimensional real Euclidean space. Let $X = \\set{x_1, \\dots, x_r}$ be a finite subset of $\\R^n$. The subset $X$ is '''affinely independent''' if no element $x \\in X$ is affinely dependent on $X \\setminus \\set x$"}
+{"_id": "30709", "title": "Definition:Euler-Gompertz Constant", "text": "=== Integral Form === {{:Definition:Euler-Gompertz Constant/Integral Form}} === As a Continued Fraction === {{:Definition:Euler-Gompertz Constant/Continued Fraction}}"}
+{"_id": "30710", "title": "Definition:Torsion-Free Group", "text": "Let $G$ be a group. Then $G$ is a '''torsion-free group''' {{iff}} the only torsion element of $G$ is the group identity element."}
+{"_id": "30711", "title": "Definition:Matroid Induced by Affine Independence", "text": "Let $\\R^n$ be the $n$-dimensional real Euclidean space. Let $S = \\set{x_1, \\dots, x_r}$ be a finite subset of $\\R^n$. Let $\\mathscr I$ be the set of affinely independent subsets of $S$. Then $\\struct{S, \\mathscr I}$ is called the '''matroid induced by affine independence on $S$'''."}
+{"_id": "30712", "title": "Definition:Matroid Induced by Linear Independence/Abelian Group", "text": "Let $\\struct{G, +}$ be a torsion-free Abelian group. Let $\\struct{G, +, \\times}$ be the $\\Z$-module associated with $G$. Let $S$ be a finite subset of $G$. Let $\\mathscr I$ be the set of linearly independent subsets of $S$. Then the ordered pair $\\struct{S, \\mathscr I}$ is called the '''matroid induced by linear independence in $G$ on $S$'''."}
+{"_id": "30717", "title": "Definition:Euler-Gompertz Constant/Integral Form", "text": "The '''Euler-Gompertz constant''' is the real number $G$ defined as: :$G = \\displaystyle \\int_0^\\infty \\dfrac {e^{-u} } {1 + u} \\rd u$"}
+{"_id": "30719", "title": "Definition:Twin Primes Constant", "text": "The '''twin primes constant''' is the real number: {{begin-eqn}} {{eqn | l = \\Pi_2       | o = :=       | r = \\displaystyle \\prod_{\\substack {p \\mathop \\ge 3 \\\\ \\text {$p$ prime} } } \\paren {1 - \\dfrac 1 {\\paren {p - 1}^2} } }} {{eqn | o = \\approx       | r = 0 \\cdotp 66016 \\, 18       | c =  }} {{end-eqn}} {{OEIS|A005597}}"}
+{"_id": "30720", "title": "Definition:Robbins Constant", "text": "The '''Robbins constant''' $R$ is defined as the mean distance $D$ between $2$ points chosen at random from the interior of a unit cube: {{begin-eqn}} {{eqn | l = R       | r = \\frac {4 + 17 \\sqrt 2 - 6 \\sqrt3 - 7 \\pi} {105} + \\frac {\\map \\ln {1 + \\sqrt 2 } } 5 + \\frac {2 \\, \\map \\ln {2 + \\sqrt 3} } 5 }} {{eqn | o = \\approx       | r = 0 \\cdotp 66170 \\, 71822 \\, 67176 \\, 23515 \\, 582 \\ldots       | c =  }} {{end-eqn}} {{OEIS|A073012}}"}
+{"_id": "30722", "title": "Definition:Matroid Induced by Linear Independence", "text": "=== Vector Space === {{:Definition:Matroid Induced by Linear Independence/Vector Space}} === Abelian Group === {{:Definition:Matroid Induced by Linear Independence/Abelian Group}} Category:Examples of Matroids ggj6ymjjxk3r856h4pw8v4ka19ao229"}
+{"_id": "30723", "title": "Definition:Loop (Matroid)", "text": "A '''loop''' of $M$ is an element $x$ of $S$ such that $\\set x$ is a dependent subset of $S$. That is, $x \\in S$ is a '''loop''' {{iff}} $\\set x \\not \\in \\mathscr I$."}
+{"_id": "30724", "title": "Definition:Parallel (Matroid)", "text": "Two elements $x, y \\in S$ are said to be '''parallel''' in $M$ {{iff}} they are not loops but $\\set {x, y}$ is a dependent subset of $S$. That is, $x, y \\in S$ are '''parallel''' {{iff}}: :$\\set x, \\set y \\in \\mathscr I$ and $\\set {x, y} \\notin \\mathscr I$."}
+{"_id": "30726", "title": "Definition:Central Binomial Coefficient", "text": "Let $n \\in \\N$ be a natural number. The binomial coefficient: :$\\dbinom {2 n} n$ is known as a '''central binomial coefficient'''."}
+{"_id": "30727", "title": "Definition:Landau-Ramanujan Constant", "text": "{{begin-eqn}} {{eqn | l = k       | r = \\sqrt {\\dfrac 1 2 \\displaystyle \\prod_{\\substack {r \\mathop = 4 n \\mathop + 3 \\\\ \\text {$r$ prime} } } \\paren {1 - \\dfrac 1 {r^2} }^{-1} } }} {{eqn | o = \\approx       | r = 0 \\cdotp 76422 \\, 3653 \\ldots }} {{end-eqn}}"}
+{"_id": "30728", "title": "Definition:Stolarsky-Harborth Constant", "text": "The '''Stolarsky-Harborth constant''' is the lower bound for the number $\\beta$ defined as: :$\\beta > \\dfrac {P_n} {n^{\\lg 3} }$ where: :$P_n$ is the number of odd elements in the first $n$ rows of Pascal's triangle :$\\lg 3$ denotes the logarithm base $2$ of $3$. Its value is given by: :$\\beta \\approx 0 \\cdotp 81255 \\, 65590 \\, 160063 \\, 8769 \\ldots$ {{OEIS|A077464}}"}
+{"_id": "30729", "title": "Definition:Lehmer's Polynomial", "text": "'''Lehmer's polynomial''' is the expression: :$x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$"}
+{"_id": "30730", "title": "Definition:Salem Number", "text": "A '''Salem number''' is a real algebraic integer $\\alpha$ such that: :$\\alpha > 1$ : its conjugate roots all have absolute value no greater than $1$, at least one of which has absolute value exactly $1$."}
+{"_id": "30732", "title": "Definition:Wilbraham-Gibbs Constant", "text": "The '''Wilbraham-Gibbs constant''' is the real number defined as: {{begin-eqn}} {{eqn | l = G'       | r = \\int_0^\\pi \\dfrac {\\sin u} u \\rd u       | c =  }} {{eqn | o = \\approx       | r = 1 \\cdotp 85193 \\, 7052 \\ldots       | c =  }} {{end-eqn}} {{OEIS|036792}}"}
+{"_id": "30734", "title": "Definition:Covertex", "text": "=== Covertex of Ellipse === Let $K$ be an ellipse. {{:Definition:Covertex of Ellipse}} === Covertex of Hyperbola === Let $K$ be a hyperbola. {{:Definition:Covertex of Hyperbola}}"}
+{"_id": "30735", "title": "Definition:Kepler's Equation", "text": "Consider a system consisting of two bodies $B_1$ and $B_2$ in elliptical orbit around each other. Let $e$ denote the eccentricity of that orbit. '''Kepler's equation''' is the equation that describes the relation between the mean anomaly and eccentric anomaly of $B_1$ with respect to $B_2$: :$M = E - e \\sin E$ where: :$M$ is the mean anomaly :$E$ is the eccentric anomaly. {{NamedforDef|Johannes Kepler|cat = Kepler}}"}
+{"_id": "30737", "title": "Definition:Lemniscate of Bernoulli/Geometric Definition", "text": "Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$. The '''lemniscate of Bernoulli''' is the locus of points $M$ in the plane such that: :$P_1 M \\times P_2 M = a^2$"}
+{"_id": "30738", "title": "Definition:Lemniscate of Bernoulli/Cartesian Definition", "text": "The '''lemniscate of Bernoulli''' is the curve defined by the Cartesian equation: :$\\paren {x^2 + y^2}^2 = 2 a^2 \\paren {x^2 - y^2}$"}
+{"_id": "30739", "title": "Definition:Lemniscate of Bernoulli/Polar Definition", "text": "The '''lemniscate of Bernoulli''' is the curve defined by the polar equation: :$r^2 = 2 a^2 \\cos 2 \\theta$"}
+{"_id": "30740", "title": "Definition:Lemniscate of Bernoulli/Parametric Definition", "text": "The '''lemniscate of Bernoulli''' is the curve defined by the parametric equation: :$\\begin{cases} x = \\dfrac {a \\sqrt 2 \\cos t} {\\sin^2 t + 1} \\\\ y = \\dfrac {a \\sqrt 2 \\cos t \\sin t} {\\sin^2 t + 1} \\end{cases}$"}
+{"_id": "30741", "title": "Definition:Lemniscate of Bernoulli/Cartesian Definition/Also defined as", "text": "Some sources define the Cartesian equation for the '''lemniscate of Bernoulli''' as: :$\\paren {x^2 + y^2}^2 = a^2 \\paren {x^2 - y^2}$ which is the same but for a scale factor: :620px"}
+{"_id": "30742", "title": "Definition:Lemniscate of Bernoulli/Polar Definition/Also defined as", "text": "Some sources define the polar equation for the '''lemniscate of Bernoulli''' as: :$r^2 = a^2 \\cos 2 \\theta$ which is the same but for a scale factor: :620px"}
+{"_id": "30744", "title": "Definition:Twin Primes Constant/Also defined as", "text": "{{AuthorRef|François Le Lionnais}} and {{AuthorRef|Jean Brette}}, in their {{BookLink|Les Nombres Remarquables|François Le Lionnais}} of $1983$, define the '''twin primes constant''' as: {{begin-eqn}} {{eqn | l = \\Pi_2       | o = :=       | r = 2 \\displaystyle \\prod_{\\substack {p \\mathop \\ge 3 \\\\ \\text {$p$ prime} } } \\paren {1 - \\dfrac 1 {\\paren {p - 1}^2} } }} {{eqn | o = \\approx       | r = 1 \\cdotp 32032 \\, 36316 \\ldots       | c =  }} {{end-eqn}}"}
+{"_id": "30745", "title": "Definition:Pisot-Vijayaraghavan Number", "text": "Let $\\alpha$ be a real algebraic integer greater than $1$. Let $\\alpha$ be such that the absolute values of its Galois conjugates are all less than $1$. Then $\\alpha$ is a '''Pisot-Vijayaraghavan number'''."}
+{"_id": "30748", "title": "Definition:Plastic Constant", "text": "The '''plastic constant''' is the real root of the cubic: :$x^3 - x - 1 = 0$ {{explain|There are $3$ roots of this equation -- needs to be established exactly which one we are talking about here.}} Its value is approximately: :$P = 1 \\cdotp 32471 \\, 795 \\ldots$ {{OEIS|A060006}}"}
+{"_id": "30749", "title": "Definition:Hermite Constant", "text": "Let $n \\in \\N$ be a natural number. Let $L$ be an integer lattice in Euclidean space $R^n$ unit covolume. That is: :$\\map {\\operatorname {vol} } {\\dfrac {R^n} L} = 1$ Let $\\map {\\lambda_1} L$ denote the least length of a nonzero element of $L$. Let $\\sqrt {\\gamma_n}$ be the maximum of $\\map {\\lambda_1} L$ over all such integer lattices $L$. The '''Hermite constant of dimension $n$''' is the constant $\\gamma_n$. {{NamedforDef|Charles Hermite|cat = Hermite}}"}
+{"_id": "30751", "title": "Definition:Landau's Problems", "text": "'''Landau's problems''' are a set of $4$ (still) unsolved problems about prime numbers listed by {{AuthorRef|Edmund Georg Hermann Landau}} in an address at the $1912$ International Congress of Mathematicians. They are as follows: === Goldbach's Conjecture === {{:Goldbach's Conjecture}} === Twin Prime Conjecture === {{:Twin Prime Conjecture}} === Legendre's Conjecture === {{:Legendre's Conjecture}} === Is there an Infinite Number of Primes of Form $n^2 + 1$? === {{:Is there an Infinite Number of Primes of Form n^2 + 1?}}"}
+{"_id": "30754", "title": "Definition:Polar Equation", "text": "A '''polar equation''' is an equation defining the locus of a set of points in the polar coordinate plane. Such an equation is generally presented in terms of the variables: :$r$: the radial coordinate :$\\theta$: the angular coordinate"}
+{"_id": "30755", "title": "Definition:Cartesian Equation", "text": "A '''Cartesian equation''' is an equation defining the locus of a set of points in a Cartesian space. Such an equation is generally presented in terms of the variables: :$x$ and $y$ for a $2$-dimensional space :$x$, $y$ and $z$ for a $3$-dimensional space :$x_1, x_2, \\ldots, x_n$ for a general $n$-dimensional space"}
+{"_id": "30758", "title": "Definition:Lemniscate of Bernoulli/Focus", "text": "Each of the two points $P_1$ and $P_2$ can be referred to as a '''focus''' of the lemniscate."}
+{"_id": "30759", "title": "Definition:Cardioid", "text": "A '''cardioid''' is an epicycloid with $1$ cusp. :500px"}
+{"_id": "30760", "title": "Definition:Trifolium/Cosine Form", "text": "The '''trifolium''' can be defined by the polar equation in the form: :$r = a \\cos 3 \\theta$ where: :$0 \\le \\theta \\le \\pi$ :400px"}
+{"_id": "30761", "title": "Definition:Trifolium/Sine Form", "text": "The '''trifolium''' can be defined by the polar equation in the form: :$r = a \\sin 3 \\theta$ where: :$0 \\le \\theta \\le \\pi$ :400px"}
+{"_id": "30764", "title": "Definition:Rhodonea Curve", "text": "A '''rhodonea curve''' is a curve defined by either one of the polar equations: {{begin-eqn}} {{eqn | l = r       | r = a \\cos n \\theta }} {{eqn | l = r       | r = a \\sin n \\theta }} {{end-eqn}}"}
+{"_id": "30767", "title": "Definition:Rhodonea Curve/Petal", "text": "Each of the loops of a '''rhodonea curve''' is referred to as a '''petal'''."}
+{"_id": "30769", "title": "Definition:Quadrifolium/Cosine Form", "text": "The '''quadrifolium''' can be defined by the polar equation in the form: :$r = a \\cos 2 \\theta$ where: :$0 \\le \\theta \\le 2 \\pi$ :400px"}
+{"_id": "30771", "title": "Definition:Quadrifolium/Sine Form", "text": "The '''quadrifolium''' can be defined by the polar equation in the form: :$r = a \\sin 2 \\theta$ where: :$0 \\le \\theta \\le 2 \\pi$ :400px"}
+{"_id": "30774", "title": "Definition:Trochoid", "text": "Consider a circle $C$ rolling without slipping along a straight line. Consider a point $P$ on the line of a radius of $C$ at a distance $b$ from the center of $C$. The curve traced out by $P$ is called a '''trochoid'''."}
+{"_id": "30775", "title": "Definition:Trochoid/Curtate", "text": "Let $b < a$. The curve traced out by $P$ is called a '''curtate trochoid'''. :500px"}
+{"_id": "30776", "title": "Definition:Trochoid/Prolate", "text": "Let $b > a$. The curve traced out by $P$ is called a '''prolate trochoid'''. :500px"}
+{"_id": "30777", "title": "Definition:Folium of Descartes/Cartesian Form", "text": "The '''folium of Descartes''' is the locus of the equation: :$x^3 + y^3 - 3 a x y = 0$"}
+{"_id": "30778", "title": "Definition:Omega Constant/Decimal Expansion", "text": "The decimal expansion of the omega constant $\\Omega$ starts: :$0 \\cdotp 56714 \\, 32904 \\, 097838 \\, 72999 \\, 96866 \\, 22 \\ldots$"}
+{"_id": "30779", "title": "Definition:Folium of Descartes/Parametric Form", "text": "The '''folium of Descartes''' is the locus of the equation given in parametric form as: :$\\begin {cases} x = \\dfrac {3 a t} {1 + t^3} \\\\ y = \\dfrac {3 a t^2} {1 + t^3} \\end {cases}$"}
+{"_id": "30780", "title": "Definition:Ovals of Cassini", "text": "Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$. The '''ovals of Cassini''' are the loci of points $M$ in the plane such that: :$P_1 M \\times P_2 M = b^2$ for a real constant $b$."}
+{"_id": "30781", "title": "Definition:Ovals of Cassini/Shape", "text": "When $b > a$, $M$ is in one continuous piece, either oval or bone-shaped. When $b < a$, $M$ is in two separate pieces, each surrounding one of the foci of $M$. When $b = a$, $M$ degenerates into the lemniscate of Bernoulli."}
+{"_id": "30782", "title": "Definition:Ovals of Cassini/Focus", "text": "Each of the two points $P_1$ and $P_2$ can be referred to as a '''focus''' of the ovals."}
+{"_id": "30784", "title": "Definition:Sigma-Algebra Generated by Collection of Subsets/Generator", "text": "One says that $\\GG$ is a '''generator''' for $\\map \\sigma {\\GG}$. Also, elements $G$ of $\\GG$ may be called '''generators'''."}
+{"_id": "30785", "title": "Definition:Sigma-Algebra Generated by Collection of Subsets/Definition 1", "text": "The '''$\\sigma$-algebra generated by $\\GG$''', $\\map \\sigma \\GG$, is the smallest $\\sigma$-algebra on $X$ that contains $\\GG$. That is, $\\map \\sigma \\GG$ is subject to: :$(1): \\quad \\GG \\subseteq \\map \\sigma \\GG$ :$(2): \\quad \\GG \\subseteq \\Sigma \\implies \\map \\sigma \\GG \\subseteq \\Sigma$ for any $\\sigma$-algebra $\\Sigma$ on $X$"}
+{"_id": "30786", "title": "Definition:Sigma-Algebra Generated by Collection of Subsets/Definition 2", "text": "The '''$\\sigma$-algebra generated by $\\GG$''', $\\map \\sigma \\GG$, is the intersection of all $\\sigma$-algebras on $X$ that contain $\\GG$."}
+{"_id": "30787", "title": "Definition:Limaçon of Pascal", "text": "Let $C$ be a circle of diameter $a$ with a distinguished point $O$ on the circumference. Let $OQ$ be a chord of $C$. The '''limaçons of Pascal''' are the loci of points $P$ in the plane such that: :$PQ = b$ where: :$OPQ$ is a straight line :$b$ is a real constant."}
+{"_id": "30788", "title": "Definition:Limaçon of Pascal/Shape", "text": "Let $L$ denote a limaçon of Pascal. Depending on the value of $b$, the shape of $L$ is as follows: :For $b \\ge 2 a$, $L$ is wholly convex. :For $a < b < 2 a$, $L$ has a concavity. :For $b = a$, $L$ degenerates to a cardioid. :For $0 < b < a$, $L$ has a loop inside its generating circle. :For $b = \\dfrac a 2$, the internal loop of $L$ passes through the center of the generating circle. :For $b = 0$, $L$ degenerates to a circle. :For $b < 0$, $L$ is the same curve as for $-b$. :200px 160px 125px 108px 96px 90px 74px 70px"}
+{"_id": "30793", "title": "Definition:Differential Equation/Solution/Particular Solution", "text": "Let $S$ denote the solution set of $\\Phi$. A '''particular solution''' of $\\Phi$ is the element of $S$, or subset of $S$, which satisfies a particular boundary condition of $\\Phi$."}
+{"_id": "30794", "title": "Definition:Convergent Sequence/Normed Vector Space", "text": "Let $\\tuple {X, \\norm {\\,\\cdot \\,}}$ be a normed vector space. Let $\\sequence {x_n}_{n \\mathop \\in \\N}$ be a sequence in $X$. Let $L \\in X$.  The sequence $\\sequence {x_n}_{n \\mathop \\in \\N}$ '''converges to the limit $L \\in X$''' {{iff}}: :$\\forall \\epsilon \\in \\R_{>0}: \\exists N \\in \\N: \\forall n \\in \\N: n > N \\implies \\norm {x_n - L} < \\epsilon$"}
+{"_id": "30795", "title": "Definition:Compact Space/Normed Vector Space", "text": "Let $\\struct {X, \\norm {\\,\\cdot\\,} }$ be a normed vector space.  Let $K \\subseteq X$. Then $K$ is '''compact''' {{iff}} every sequence in $K$ has a convergent subsequence with limit $L \\in K$. That is, if: :$\\sequence {x_n}_{n \\mathop \\in \\N} :\\forall n \\in \\N : x_n \\in K \\implies \\exists \\sequence {x_{n_k} }_{k \\mathop \\in \\N} : \\exists L \\in K: \\displaystyle  \\lim_{k \\mathop \\to \\infty} x_{n_k} = L$"}
+{"_id": "30799", "title": "Definition:Producer of Dedekind Cut", "text": "Let $\\struct {S, \\preceq}$ be a totally ordered set. Let $S' \\subseteq S$. Let $\\tuple {L, R}$ be a Dedekind cut of $S'$. An $\\alpha \\in S$ is referred to as a '''producer''' of $\\tuple {L, R}$ {{iff}}: :$l \\preceq \\alpha$ for all $l \\in L$ :$\\alpha \\preceq r$ for all $r \\in R$."}
+{"_id": "30800", "title": "Definition:Linear First Order Ordinary Differential Equation/Constant Coefficients", "text": "A '''linear first order ordinary differential equation with constant coefficients''' is a linear first order ordinary differential equation which is in (or can be manipulated into) the form: :$\\dfrac {\\d y} {\\d x} + a y = \\map Q x$ where: :$\\map Q x$ is a function of $x$ :$a$ is a constant."}
+{"_id": "30801", "title": "Definition:Complementary Function of Linear First Order ODE With Constant Coefficients", "text": "Consider the linear first order ODE with constant coefficients: :$(1): \\quad \\dfrac {\\d y} {\\d x} + a y = \\map Q x$ The general solution to the reduced equation: :$\\dfrac {\\d y} {\\d x} + a y = 0$ is the '''complementary function''' of $(1)$. From First Order ODE: $\\dfrac {\\d y} {\\d x} = k y$, the '''complementary function''' of $(1)$ is $C e^{-a x}$."}
+{"_id": "30802", "title": "Definition:Reduced Equation of Linear ODE with Constant Coefficients/First Order", "text": "Consider the linear first order ODE with constant coefficients: :$(1): \\quad \\dfrac {\\d y} {\\d x} + a y = \\map Q x$ The equation: :$\\dfrac {\\d y} {\\d x} + a y = 0$ is the '''reduced equation''' of $(1)$."}
+{"_id": "30803", "title": "Definition:Reduced Equation of Linear ODE with Constant Coefficients", "text": "Consider the linear $n$th order ODE with constant coefficients: :$(1): \\quad \\displaystyle \\sum_{k \\mathop = 0}^n a_k \\dfrac {\\d^k y} {d x^k} = \\map R x$ The equation: :$\\displaystyle \\sum_{k \\mathop = 0}^n a_k \\dfrac {\\d^k y} {d x^k} = 0$ is the '''reduced equation''' of $(1)$. === First Order Linear ODE === {{:Definition:Reduced Equation of Linear ODE with Constant Coefficients/First Order}} === Second Order Linear ODE === {{:Definition:Reduced Equation of Linear ODE with Constant Coefficients/Second Order}}"}
+{"_id": "30804", "title": "Definition:Reduced Equation of Linear ODE with Constant Coefficients/Second Order", "text": "Consider the linear second order ODE with constant coefficients: :$(1): \\quad \\dfrac {\\d^2 y} {\\d x^2} + p \\dfrac {\\d y} {\\d x} + q y = \\map R x$ The equation: :$\\dfrac {\\d^2 y} {\\d x^2} + p \\dfrac {\\d y} {\\d x} + q y = 0$ is the '''reduced equation''' of $(1)$."}
+{"_id": "30805", "title": "Definition:Linear Second Order ODE with Constant Coefficients", "text": "A '''linear second order ODE with constant coefficients''' is a second order ODE which can be manipulated into the form: :$y'' + p y' + q y = \\map R x$ where: :$p$ and $q$ are real constants :$\\map R x$ is a function of $x$. Thus it is a linear second order ODE: :$y'' + \\map P x y' + \\map Q x y = \\map R x$ where $\\map P x$ and $\\map Q x$ are constant functions."}
+{"_id": "30806", "title": "Definition:Open Ball/Normed Vector Space", "text": "Let $\\struct{X, \\norm {\\,\\cdot\\,} }$ be a normed vector space. Let $x \\in X$. Let $\\epsilon \\in \\R_{>0}$ be a strictly positive real number. The '''open $\\epsilon$-ball of $x$ in $\\struct {X, \\norm {\\,\\cdot\\,} }$''' is defined as: :$\\map {B_\\epsilon} x = \\set {y \\in X: \\norm{x - y} < \\epsilon}$"}
+{"_id": "30807", "title": "Definition:Open Set/Normed Vector Space", "text": "Let $V = \\struct{X, \\norm {\\,\\cdot\\,} }$ be a normed vector space. Let $U \\subseteq X$. Then $U$ is an '''open set in $V$''' {{iff}}: :$\\forall x \\in U: \\exists \\epsilon \\in \\R_{>0}: \\map {B_\\epsilon} x \\subseteq U$ where $\\map {B_\\epsilon} x$ is the open $\\epsilon$-ball of $x$."}
+{"_id": "30808", "title": "Definition:Closed Set/Normed Vector Space", "text": "Let $V = \\struct {X, \\norm {\\,\\cdot\\,} }$ be a normed vector space. Let $F \\subset X$. ==== Definition 1 ==== {{:Definition:Closed Set/Normed Vector Space/Definition 1}} ==== Definition 2 ==== {{:Definition:Closed Set/Normed Vector Space/Definition 2}}"}
+{"_id": "30809", "title": "Definition:P-adic Unit", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\Z_p$ denote the $p$-adic integers. The set of '''$p$-adic units''', denoted $\\Z_p^\\times$, is the set of invertible elements of $\\Z_p$. From P-adic Unit has Norm Equal to One, the set of '''$p$-adic units''' is: :$\\Z_p^\\times = \\set {x \\in \\Q_p: \\norm x_p = 1}$"}
+{"_id": "30814", "title": "Definition:Linear nth Order ODE with Constant Coefficients", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. A '''linear $n$th order ODE with constant coefficient''' is an ordinary differential equation of order $n$ which can be manipulated into the form: :$\\displaystyle \\sum_{k \\mathop = 0}^n a_k \\dfrac {\\d^k y} {d x^k} = \\map R x$ where $a_k$ is a real constant for all $0 \\le k \\le n$. That is: :$a_n \\dfrac {\\d^n y} {d x^n} + a_{n - 1} \\dfrac {\\d^{n - 1} y} {d x^{n - 1} } + \\dotsb + a_1 \\dfrac {\\d y} {d x} + a_0 y = \\map R x$"}
+{"_id": "30817", "title": "Definition:Lie Theory", "text": "'''Lie theory''' is the study of Lie groups and Lie algebras."}
+{"_id": "30819", "title": "Definition:Induced Norm on Quotient of Cauchy Sequences", "text": "Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a normed division ring. Let $\\mathcal C$ be the ring of Cauchy sequences over $R$ Let $\\mathcal N$ be the set of null sequences. For all $\\sequence {x_n} \\in \\mathcal C$, let $\\eqclass {x_n} {}$ denote the left coset $\\sequence {x_n} + \\mathcal N$ Let $\\norm {\\, \\cdot \\,}_1: \\mathcal C \\,\\big / \\mathcal N \\to \\R_{\\ge 0}$ be defined by: :$\\displaystyle \\forall \\eqclass {x_n} {} \\in \\mathcal C \\,\\big / \\mathcal N: \\norm {\\eqclass {x_n} {} }_1 = \\lim_{n \\mathop \\to \\infty} \\norm {x_n}$ $\\norm {\\, \\cdot \\,}_1$ is called the '''induced norm on the quotient ring of Cauchy sequences'''."}
+{"_id": "30821", "title": "Definition:Linear Differential Operator", "text": "A '''linear differential operator''' is a differential operator $\\mathscr L$ with the property that: :$\\map {\\mathscr L} {\\alpha \\phi_1 + \\beta \\phi_2} = \\alpha \\map {\\mathscr L} {\\phi_1} + \\beta \\map {\\mathscr L} {\\phi_2}$ Thus if $\\phi_1$ and $\\phi_2$ are solutions to the differential equation $\\map {\\mathscr L} {\\phi_i} = 0$, then so is any linear combination of $\\phi_1$ and $\\phi_2$."}
+{"_id": "30822", "title": "Definition:Linear Integral Operator", "text": "A '''linear integral operator''' is a integral operator $\\mathscr L$ with the property that: :$\\map {\\mathscr L} {\\alpha \\phi_1 + \\beta \\phi_2} = \\alpha \\map {\\mathscr L} {\\phi_1} + \\beta \\map {\\mathscr L} {\\phi_2}$ Thus if $\\phi_1$ and $\\phi_2$ are solutions to the integral equation $\\map {\\mathscr L} {\\phi_i} = 0$, then so is any linear combination of $\\phi_1$ and $\\phi_2$."}
+{"_id": "30823", "title": "Definition:Set/Distinction between Element and Set", "text": "It is important to distinguish between an element, for example $a$, and a singleton containing it, that is, $\\set a$. That is $a$ and $\\set a$ are ''not'' the same thing. While it is true that: :$a \\in \\set a$ it is not true that: :$a = \\set a$ neither is it true that: :$a \\in a$"}
+{"_id": "30828", "title": "Definition:Experiment/Informal Definition", "text": "An '''experiment''' is defined as: : ''a course of action whose consequence is not predetermined.''"}
+{"_id": "30829", "title": "Definition:Experiment/Formal Definition", "text": "An '''experiment''', which can conveniently be denoted $\\EE$, is a measure space $\\struct {\\Omega, \\Sigma, \\Pr}$ such that $\\map \\Pr \\Omega = 1$."}
+{"_id": "30830", "title": "Definition:Sample Space/Discrete", "text": "If $\\Omega$ is a countable set, whether finite or infinite, then it is known as a '''discrete sample space'''."}
+{"_id": "30831", "title": "Definition:Event/Simple Event", "text": "A '''simple event''' in $\\EE$ is an event in $\\EE$ which consists of exactly $1$ elementary event. That is, it is a singleton subset of the sample space $\\Omega$ of $\\EE$."}
+{"_id": "30832", "title": "Definition:Complementary Event", "text": "Let the probability space of an experiment $\\EE$ be $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $A \\in \\Sigma$ be an event in $\\EE$. The '''complementary event''' to $A$ is defined as $\\relcomp \\Omega A$. That is, it is the subset of the sample space of $\\EE$ consisting of all the elementary events of $\\EE$ that are not in $A$."}
+{"_id": "30833", "title": "Definition:Trivial Event", "text": "Let the probability space of an experiment $\\EE$ be $\\struct {\\Omega, \\Sigma, \\Pr}$. The empty set $\\O$ is a subset of $\\Sigma$, by Empty Set is Subset of All Sets, and so is an event in $\\EE$. It can be referred to as the '''the trivial event''' of $\\EE$."}
+{"_id": "30837", "title": "Definition:Probability Measure/Definition 1", "text": "Let $\\EE$ be defined as a measure space $\\struct {\\Omega, \\Sigma, \\Pr}$. Then $\\Pr$ is a measure on $\\EE$ such that $\\map \\Pr \\Omega = 1$."}
+{"_id": "30838", "title": "Definition:Probability Measure/Definition 2", "text": "Let $\\Omega$ be the sample space on $\\EE$. Let $\\Sigma$ be the event space of $\\mathcal E$. A '''probability measure on $\\EE$''' is a mapping $\\Pr: \\Sigma \\to \\R$ which fulfils the Kolmogorov axioms: {{:Axiom:Kolmogorov Axioms}}"}
+{"_id": "30839", "title": "Definition:Probability Measure/Definition 3", "text": "Let $\\Omega$ be the sample space on $\\EE$. Let $\\Sigma$ be the event space of $\\mathcal E$. A '''probability measure on $\\EE$''' is a mapping $\\Pr: \\Sigma \\to \\R$ which fulfils the following axioms: {{begin-axiom}} {{axiom | n = \\text I         | q = \\forall A \\in \\Sigma         | ml= \\map \\Pr A         | mo= \\ge         | mr= 0 }} {{axiom | n = \\text {II}         | ml= \\map \\Pr \\Omega         | mo= =         | mr= 1 }} {{axiom | n = \\text {III}         | q = \\forall A \\in \\Sigma         | ml= \\map \\Pr A         | mo= =         | mr= \\sum_{\\bigcup \\set e \\mathop = A} \\map \\Pr {\\set e}         | rc= where $e$ denotes the elementary events of $\\EE$ }} {{end-axiom}}"}
+{"_id": "30840", "title": "Definition:Probability Measure/Definition 4", "text": "Let $\\Omega$ be the sample space on $\\EE$. Let $\\Sigma$ be the event space of $\\mathcal E$. A '''probability measure on $\\EE$''' is an additive function $\\Pr: \\Sigma \\to \\R$ which fulfils the following axioms: {{begin-axiom}} {{axiom | n = 1         | q = \\forall A, B \\in \\Sigma: A \\cap B = \\O         | ml= \\map \\Pr {A \\cup B}         | mo= =         | mr= \\map \\Pr A + \\map \\Pr B }} {{axiom | n = 2         | ml= \\map \\Pr \\Omega         | mo= =         | mr= 1 }} {{end-axiom}}"}
+{"_id": "30842", "title": "Definition:Decimal", "text": "=== Decimal System === {{:Definition:Decimal System}} === Decimal Notation === {{:Definition:Decimal Notation}} === Decimal Expansion === {{:Definition:Decimal Expansion}} === Decimal Point === {{:Definition:Decimal Point|}} === Decimal Place === {{:Definition:Decimal Place}} === Decimal Part === Erroneously used to mean fractional part: {{:Definition:Decimal Part}} Category:Definitions/Number Systems cfaggi6xrg4msoheotg59i1d97p96lw"}
+{"_id": "30843", "title": "Definition:Decimal Expansion/Decimal Place", "text": "Let the '''decimal expansion''' of $x$ be: :$x = \\sqbrk {s \\cdotp d_1 d_2 d_3 \\ldots}_{10}$ Then $d_k$ is defined as being the digit in the $k$th '''decimal place'''."}
+{"_id": "30845", "title": "Definition:Number Names", "text": "This page gathers together instances of names of numbers in various languages and contexts."}
+{"_id": "30848", "title": "Definition:Number Names/Welsh", "text": "These are the names of the numbers in Welsh: {{begin-axiom}} {{axiom | n = 1         | lc = '''un''' }} {{axiom | n = 2         | lc = '''dau''', '''dwy''' }} {{axiom | n = 3         | lc = '''toi''', '''tair''' }} {{axiom | n = 4         | lc = '''pedwar''', '''pedair''' }} {{axiom | n = 5         | lc = '''pump''' (or '''pum''') }} {{axiom | n = 6         | lc = '''chwech''' (or '''chwe''') }} {{axiom | n = 7         | lc = '''saith''' }} {{axiom | n = 8         | lc = '''wyth''' }} {{axiom | n = 9         | lc = '''naw''' }} {{axiom | n = 10         | lc = '''deg''' }} {{axiom | n = 11         | lc = '''un ar ddeg''' }} {{axiom | n = 12         | lc = '''deuddeg''' }} {{axiom | n = 13         | lc = '''toi ar ddeg''', '''tair ar ddeg''' }} {{axiom | n = 14         | lc = '''pedwar ar ddeg''', '''pedair ar ddeg''' }} {{axiom | n = 15         | lc = '''pymtheg''' }} {{axiom | n = 16         | lc = '''un ar bymtheg''' }} {{axiom | n = 17         | lc = '''dau ar bymtheg''', '''dwy ar bymtheg''' }} {{axiom | n = 18         | lc = '''deunaw''' }} {{axiom | n = 19         | lc = '''pedwar ar bymtheg''', '''pedair ar bymtheg''' }} {{axiom | n = 20         | lc = '''ugain''' }} {{axiom | n = 30         | lc = '''deg ar hugain''' }} {{axiom | n = 40         | lc = '''dugain''' }} {{axiom | n = 50         | lc = '''deg ar deugain''' or '''hanner cant''' }} {{axiom | n = 60         | lc = '''trigain''' }} {{axiom | n = 70         | lc = '''trigain a deg''' or '''deg ar thrigain''' }} {{axiom | n = 80         | lc = '''pedwar ugain''' }} {{axiom | n = 90         | lc = '''pedwar ugain a deg''' or '''deg a phedwar ugain''' }} {{axiom | n = 100         | lc = '''cant''' }} {{end-axiom}} The different forms for $2$, $3$, $4$ and their derivatives are for masculine and feminine. The different forms for $5$ and $6$ are elisions for use in front of certain letters. The different forms for $50$, $70$ and $90$ are separate ways of saying the numbers. Note: :there are no distinct words for $11$ and $12$, indicating that there has been less influence of duodecimal systems on Welsh :there are clear indications of base $5$ and vigesimal influences as well as decimal :the interesting forms for $18$ (expressed as $2 \\times 9$) and $50$ (expressed both as $10 + 40$ and $\\dfrac 1 2 \\times 100$)."}
+{"_id": "30851", "title": "Definition:Continuous Real Function/Point/Definition by Epsilon-Delta", "text": "Then '''$f$ is continuous at $x$''' {{iff}} the limit $\\displaystyle \\lim_{y \\mathop \\to x} \\map f y$ exists and: :$\\displaystyle \\lim_{y \\mathop \\to x} \\, \\map f y = \\map f x$"}
+{"_id": "30852", "title": "Definition:Continuous Real Function/Point/Definition by Neighborhood", "text": "Then '''$f$ is continuous at $x$''' {{iff}} the limit $\\displaystyle \\lim_{y \\mathop \\to x} \\map f y$ exists and: :$\\displaystyle \\lim_{y \\mathop \\to x} \\, \\map f y = \\map f x$ :for every $\\epsilon$-neighborhood $N_\\epsilon$ of $\\map f x$ in $\\R$, there exists a $\\delta$-neighborhood $N_\\delta$ of $x$ in $A$ such that $\\map f x \\in N_\\epsilon$ whenever $x \\in N_\\delta$."}
+{"_id": "30853", "title": "Definition:Approximation", "text": "An '''approximation''' is an estimate of a quantity. It is usually the case that there exists some knowledge about the accuracy of the estimate. The notation: :$a \\approx b$ indicates that $b$ is an '''approximation''' to $a$."}
+{"_id": "30855", "title": "Definition:Cut (Analysis)", "text": "Let $\\alpha \\subset \\Q$ be a subset of the set of rational numbers $\\Q$ which has the following properties: :$(1): \\quad \\alpha \\ne \\O$ and $\\alpha \\ne \\Q$, that is: $\\alpha$ contains at least one rational number but not all rational numbers :$(2): \\quad$ If $p \\in \\Q$ and $q \\in \\Q$ such that $q < p$, then $q \\in \\Q$ :$(3): \\quad \\alpha$ does not contain a greatest element. Then $\\alpha$ is called a '''cut'''."}
+{"_id": "30856", "title": "Definition:Cut (Analysis)/Lower Number", "text": "Let $\\alpha$ be a cut. Let $p \\in \\alpha$. Then $p$ is referred to as a '''lower number''' of $\\alpha$."}
+{"_id": "30857", "title": "Definition:Cut (Analysis)/Upper Number", "text": "Let $\\alpha$ be a cut. Let $q \\in \\Q$ such that $q \\notin \\alpha$. Then $p$ is referred to as an '''upper number''' of $\\alpha$."}
+{"_id": "30858", "title": "Definition:Cut (Analysis)/Rational", "text": "Let $r \\in \\Q$ be a rational number. Let $\\alpha$ be the cut consisting of all rational numbers $p$ such that $p < r$. Then $\\alpha$ is referred to as a '''rational cut'''."}
+{"_id": "30859", "title": "Definition:Equality of Cuts", "text": "Let $\\alpha$ and $\\beta$ be cuts such that: :$p \\in \\alpha \\implies p \\in \\beta$ :$q \\in \\beta \\implies q \\in \\alpha$ Then $\\alpha$ and $\\beta$ are '''equal'''."}
+{"_id": "30860", "title": "Definition:Ordering of Cuts/Strict", "text": ":$\\alpha$ '''is less than ''' $\\beta$, denoted $\\alpha < \\beta$ {{iff}}: :there exists a rational number $p \\in \\Q$ such that $p \\in \\alpha$ but $p \\notin \\beta$."}
+{"_id": "30861", "title": "Definition:Ordering of Cuts", "text": "Let $\\alpha$ and $\\beta$ be cuts. $\\alpha$ and $\\beta$ conventionally have the following ordering imposed on them, as follows: :$\\alpha$ '''is less than or equal to''' $\\beta$, denoted $\\alpha \\le \\beta$ {{iff}}: :$\\alpha = \\beta$ or $\\alpha < \\beta$ where $<$ denotes the strict ordering relation between $\\alpha$ and $\\beta$. This can also be expressed as $\\beta \\ge \\alpha$. === Strict Ordering === {{:Definition:Ordering of Cuts/Strict}}"}
+{"_id": "30862", "title": "Definition:Positive Cut/Strictly Positive", "text": "Let $\\alpha > 0^*$. Then $\\alpha$ can be referred to as a '''strictly positive cut'''."}
+{"_id": "30863", "title": "Definition:Positive Cut", "text": "Let $\\alpha$ be a cut. Let $0^*$ be the rational cut associated with the (rational) number $0$. Let $\\alpha \\ge 0^*$. Then $\\alpha$ can be referred to as a '''positive cut'''. === Strictly Positive Cut === {{:Definition:Positive Cut/Strictly Positive}}"}
+{"_id": "30864", "title": "Definition:Negative Cut", "text": "Let $\\alpha$ be a cut. Let $0^*$ be the rational cut associated with the (rational) number $0$. Let $\\alpha \\le 0^*$. Then $\\alpha$ can be referred to as a '''negative cut'''. === Strictly Negative Cut === {{:Definition:Negative Cut/Strictly Negative}}"}
+{"_id": "30865", "title": "Definition:Negative Cut/Strictly Negative", "text": "Let $\\alpha < 0^*$. Then $\\alpha$ can be referred to as a '''strictly negative cut'''."}
+{"_id": "30866", "title": "Definition:Addition of Cuts", "text": "Let $\\alpha$ and $\\beta$ be cuts. Let the operation of '''addition''' be defined on $\\alpha$ and $\\beta$ as: :$\\gamma := \\alpha + \\beta$ where $\\gamma$ is the set of all rational numbers $r$ such that: :$\\exists p \\in \\alpha, q \\in \\beta: r = p + q$ In this context, $\\gamma$ is known as the '''sum of $\\alpha$ and $\\beta$'''."}
+{"_id": "30867", "title": "Definition:Negative of Cut", "text": "Let $\\alpha$ be a cut. Let $0^*$ be the rational cut associated with the (rational) number $0$: :$0^* = \\set {r \\in \\Q: r < 0}$ Let $\\beta$ be the unique cut such that: :$\\alpha + \\beta = 0^*$ where $+$ denotes the operation of addition of cuts. Then $\\beta$ is referred to as the '''negative''' of $\\alpha$. It is usually denoted $-\\alpha$."}
+{"_id": "30869", "title": "Definition:Subtraction of Cuts", "text": "Let $\\alpha$ and $\\beta$ be cuts. Let $\\beta - \\alpha$ denote the operation on $\\alpha$ and $\\beta$ defined as $\\beta + \\paren {-\\alpha}$. Then the operator $-$ is known as '''subtraction''', and $\\beta - \\alpha$ is itself called the '''difference''' between $\\beta$ and $\\alpha$."}
+{"_id": "30870", "title": "Definition:Multiplication of Positive Cuts", "text": "Let $0^*$ denote the rational cut associated with the (rational) number $0$. Let $\\alpha$ and $\\beta$ be positive cuts, that is, cuts such that $\\alpha \\ge 0^*$ and $\\beta \\ge 0^*$, where $\\ge$ denotes the ordering on cuts. Let the operation of '''multiplication''' be defined on $\\alpha$ and $\\beta$ as: :$\\gamma := \\alpha \\beta$ where $\\gamma$ is the set of all rational numbers $r$ such that either: :$r < 0$ or :$\\exists p \\in \\alpha, q \\in \\beta: r = p q$ where $p \\ge 0$ and $q \\ge 0$. In this context, $\\gamma$ is known as the '''product of $\\alpha$ and $\\beta$'''."}
+{"_id": "30871", "title": "Definition:Multiplication of Cuts", "text": "Let $0^*$ denote the rational cut associated with the (rational) number $0$. Let $\\alpha$ and $\\beta$ be cuts. The operation of '''multiplication''' is defined on $\\alpha$ and $\\beta$ as: :$\\alpha \\beta := \\begin {cases} \\size \\alpha \\, \\size \\beta & : \\alpha \\ge 0^*, \\beta \\ge 0^* \\\\ -\\paren {\\size \\alpha \\, \\size \\beta} & : \\alpha < 0^*, \\beta \\ge 0^* \\\\ -\\paren {\\size \\alpha \\, \\size \\beta} & : \\alpha \\ge 0^*, \\beta < 0^* \\\\ \\size \\alpha \\, \\size \\beta & : \\alpha < 0^*, \\beta < 0^* \\end {cases}$ where: :$\\size \\alpha$ denotes the absolute value of $\\alpha$ :$\\size \\alpha \\, \\size \\beta$ is defined as in Multiplication of Positive Cuts :$\\ge$ denotes the ordering on cuts. In this context, $\\alpha \\beta$ is known as the '''product of $\\alpha$ and $\\beta$'''."}
+{"_id": "30872", "title": "Definition:Absolute Value of Cut", "text": "Let $\\alpha$ be a cut. The '''absolute value of $\\alpha$''' is denoted and defined as: :$\\size \\alpha = \\begin {cases} \\alpha & : \\alpha \\ge 0^* \\\\ -\\alpha & : \\alpha < 0^* \\end {cases}$ where: :$0^*$ denotes the rational cut associated with the (rational) number $0$ :$\\ge$ denotes the ordering on cuts."}
+{"_id": "30874", "title": "Definition:Ordered Field Isomorphism", "text": "Let $\\struct {S, +, \\circ, \\preceq}$ and $\\struct {T, \\oplus, *, \\preccurlyeq}$ be ordered fields. An '''ordered field isomorphism''' from $\\struct {S, +, \\circ, \\preceq}$ to $\\struct {T, \\oplus, *, \\preccurlyeq}$ is a mapping $\\phi: S \\to T$ that is both: :$(1): \\quad$ An ordered group isomorphism from the ordered group $\\struct {S, +, \\preceq}$ to the ordered group $\\struct {T, \\oplus, \\preccurlyeq}$ :$(2): \\quad$ A group isomorphism from the group $\\struct {S_{\\ne 0}, \\circ}$ to the semigroup $\\struct {T_{\\ne 0}, *}$ where $S_{\\ne 0}$ and $T_{\\ne 0}$ denote the sets $S$ and $T$ without the zeros of $S$ and $T$ respectively."}
+{"_id": "30876", "title": "Definition:Real Number/Also known as", "text": "When the term '''number''' is used in general discourse, it is often tacitly understood as meaning '''real number'''. They are sometimes referred to in the pedagogical context as '''ordinary numbers''', so as to distinguish them from '''complex numbers''' However, depending on the context, the word '''number''' may also be taken to mean '''integer''' or '''natural number'''. Hence it is wise to be specific."}
+{"_id": "30877", "title": "Definition:Real Number/Number Line Definition", "text": "A '''real number''' is defined as a '''number''' which is identified with a point on the '''real number line'''. ==== Real Number Line ==== {{:Definition:Real Number/Real Number Line}}"}
+{"_id": "30878", "title": "Definition:Real Number/Cauchy Sequences", "text": "Consider the set of rational numbers $\\Q$. For any two Cauchy sequences of rational numbers $X = \\sequence {x_n}, Y = \\sequence {y_n}$, define an equivalence relation between the two as: :$X \\equiv Y \\iff \\forall \\epsilon \\in \\Q_{>0}: \\exists n \\in \\N: \\forall i, j > n: \\size {x_i - y_j} < \\epsilon$ A '''real number''' is an equivalence class $\\eqclass {\\sequence {x_n} } {}$ of Cauchy sequences of rational numbers."}
+{"_id": "30880", "title": "Definition:Real Number/Digit Sequence", "text": "Let $b \\in \\N_{>1}$ be a given natural number which is greater than  $1$. The set of '''real numbers''' can be expressed as the set of all sequences of digits: :$z = \\sqbrk {a_n a_{n - 1} \\dotsm a_2 a_1 a_0 \\cdotp d_1 d_2 \\dotsm d_{m - 1} d_m d_{m + 1} \\dotsm}$ such that: :$0 \\le a_j < b$ and $0 \\le d_k < b$ for all $j$ and $k$ :$\\displaystyle z = \\sum_{0 \\mathop \\le j \\le n} a_j b^j + \\sum_{k \\mathop \\ge 1} d_k b^{-k}$ It is usual for $b$ to be $10$."}
+{"_id": "30882", "title": "Definition:Final Topology/Definition 1", "text": "The '''final topology on $X$ with respect to $\\family {f_i}_{i \\mathop \\in I}$''' is defined as: :$\\tau = \\set{U \\subseteq X: \\forall i \\in I: \\map {f_i^{-1}} U \\in \\tau_i} \\subseteq \\powerset X$"}
+{"_id": "30883", "title": "Definition:Final Topology/Definition 2", "text": "Let $\\tau$ be the finest topology on $X$ such that each $f_i: Y_i \\to X$ is $\\tuple{\\tau_i, \\tau}$-continuous. Then $\\tau$ is known as the '''final topology on $X$ with respect to $\\family{f_i}_{i \\mathop \\in I}$'''."}
+{"_id": "30886", "title": "Definition:Subfamily", "text": "Let $I$ and $S$ be sets. Let $x: I \\to S$ be an indexing function for $S$. Let $\\family {x_i}_{i \\mathop \\in I}$ be a family indexed by $I$. Let $J \\subseteq I$ be a subset of $I$. The family $\\family {x_j}_{j \\mathop \\in J}$ indexed by $J$ is known as a '''subfamily''' of $\\family {x_i}_{i \\mathop \\in I}$."}
+{"_id": "30887", "title": "Definition:Accumulation Point", "text": "Let $\\paren { S, \\tau }$ be a topological space. Let $A \\subseteq S$. === Accumulation Point of Sequence === {{:Definition:Accumulation Point/Sequence}} === Accumulation Point of Set === {{:Definition:Accumulation Point/Set}}"}
+{"_id": "30888", "title": "Definition:Strictly Smaller Set", "text": "Let $S$ and $T$ be sets. $S$ is defined as being '''strictly smaller than''' $T$ {{iff}}: :$(1): \\quad$ There exists a bijection from $S$ to a subset of $T$ :$(2): \\quad$ There exists no such bijection from $T$ to a subset of $S$. '''$S$ is strictly smaller than $T$''' can be denoted: :$S < T$"}
+{"_id": "30889", "title": "Definition:Smaller Set", "text": "Let $S$ and $T$ be sets. $S$ is defined as being '''smaller than''' $T$ {{iff}} there exists a bijection from $S$ to a subset of $T$. '''$S$ is smaller than $T$''' can be denoted: :$S \\le T$"}
+{"_id": "30890", "title": "Definition:Strictly Larger Set", "text": "Let $S$ and $T$ be sets. $S$ is defined as being '''strictly larger than''' $T$ {{iff}}: :$(1): \\quad$ There exists a bijection from $T$ to a subset of $S$ :$(2): \\quad$ There exists no such bijection from $S$ to a subset of $T$. '''$S$ is strictly larger than $T$''' can be denoted: :$S > T$"}
+{"_id": "30891", "title": "Definition:Larger Set", "text": "Let $S$ and $T$ be sets. $S$ is defined as being ''' larger than''' $T$ {{iff}} there exists a bijection from $T$ to a subset of $S$. '''$S$ is larger than $T$''' can be denoted: :$S \\ge T$"}
+{"_id": "30892", "title": "Definition:Comparable Sets/Cardinality", "text": "Let $S$ and $T$ be sets. Then $S$ and $T$ are '''comparable (in size)''' {{iff}} either: :$S$ can be put into one-to-one correspondence with a subset of $T$ or: :$T$ can be put into one-to-one correspondence with a subset of $S$ or both. That is, if either $S$ is smaller than $T$ or $T$ is smaller than $S$."}
+{"_id": "30893", "title": "Definition:First Order Logic with Identity", "text": "A system of '''first order logic with identity''' is a system of predicate logic with the following elements in its alphabet: ==== Connectives ==== {{begin-axiom}} {{axiom | ml= \\land          | mo= :         | t = the conjunction sign }} {{axiom | ml= \\lor         | mo= :         | t = the disjunction sign }} {{axiom | ml= \\implies         | mo= :         | t = the conditional sign }} {{axiom | ml= \\iff         | mo= :         | t = the biconditional sign }} {{axiom | ml= \\neg         | mo= :         | t = the negation sign }} {{end-axiom}} ==== Quantifiers ==== {{begin-axiom}} {{axiom | ml= \\exists         | mo= :         | t = the existential quantifier sign }} {{axiom | ml= \\forall         | mo= :         | t = the universal quantifier sign }} {{end-axiom}} ==== Identity ==== {{begin-axiom}} {{axiom | ml= =         | mo= :         | t = the equality sign }} {{end-axiom}}"}
+{"_id": "30894", "title": "Definition:Frege Set Theory", "text": "The '''Frege system of set theory''' is a system of axiomatic set theory which has as its sole axiom the comprehension principle: {{:Axiom:Comprehension Principle}} In support of this, the various logical axioms supporting predicate logic also hold."}
+{"_id": "30895", "title": "Definition:Zermelo Set Theory", "text": "'''Zermelo set theory''' is a system of axiomatic set theory. Its basis consists of a system of Aristotelian logic, appropriately axiomatised, together with the following axioms: === The Axiom of Extension === {{:Axiom:Axiom of Extension/Set Theory}} === The Axiom of the Empty Set === {{:Axiom:Axiom of Empty Set/Set Theory}} === The Axiom of Pairing === {{:Axiom:Axiom of Pairing/Set Theory}} === The Axiom of Specification === {{:Axiom:Axiom of Specification/Set Theory}} === The Axiom of Unions === {{:Axiom:Axiom of Unions/Set Theory}} === The Axiom of Powers === {{:Axiom:Axiom of Powers/Set Theory}} === The Axiom of Infinity === {{:Axiom:Axiom of Infinity}}"}
+{"_id": "30897", "title": "Definition:Zermelo-Fraenkel-Skolem Set Theory", "text": "'''Zermelo-Fraenkel-Skolem set theory''' is a system of axiomatic set theory. It consists of Zermelo-Fraenkel set theory with the inclusion of a system of first order formulas to specify the existence of properties of sets. {{NamedforDef|Ernst Friedrich Ferdinand Zermelo|name2 = Abraham Halevi Fraenkel|name3 = Thoralf Albert Skolem|cat = Zermelo|cat2 = Fraenkel|cat3 = Skolem}}"}
+{"_id": "30899", "title": "Definition:Morse-Kelley Set Theory", "text": "'''Morse-Kelley set theory''' is a system of axiomatic set theory. It is a stronger form of Zermelo-Fraenkel-Skolem set theory which allows not only for first order formulas to specify the existence of properties of sets, but also defines properties by quantifying over properties as well as over sets. {{stub}} {{NamedforDef|Anthony Perry Morse|name2 = John Leroy Kelley|cat = Morse|cat2 = Kelley}}"}
+{"_id": "30901", "title": "Definition:Von Neumann-Bernays-Gödel Set Theory", "text": "'''Von Neumann-Bernays-Gödel set theory''' is a system of axiomatic set theory. Its main feature is that it classifies collections of objects into: :sets, whose construction is strictly controlled and: :classes, which have fewer restrictions on how they may be generated. All sets are classes, but not all classes are sets."}
+{"_id": "30903", "title": "Definition:Model of Class-Set Theory", "text": "Let $V$ be a collection of objects. Then $V$ is a '''model of class-set theory''' {{iff}} it satisfies the axioms of Von Neumann-Bernays-Gödel set theory. Thus:  :The elements of $V$ are then known as sets :The subcollections of $V$ are known as classes."}
+{"_id": "30904", "title": "Definition:Collection", "text": "A '''collection''' is an aggregation of objects whose nature is arbitrary. The precise nature and structure of the '''collection''' itself is left unspecified. Hence the word '''collection''' can be conveniently used for an aggregation which may be either a '''set''' or a '''class''', and such that whatever is being said about it applies to both."}
+{"_id": "30905", "title": "Definition:Element/Notation", "text": "The symbol universally used in modern mainstream mathematics to mean '''$x$ is an element of $S$''' is: :$x \\in S$ Similarly, $x \\notin S$ means '''$x$ is ''not'' an element of $S$'''. The symbol can be reversed: :$S \\ni x$ means '''$S$ has $x$ as an element''', that is, $x$ is an element of $S$ but this is rarely seen. Some texts (usually older ones) use $x \\mathop {\\overline \\in} S$ or $x \\mathop {\\in'} S$ instead of $x \\notin S$."}
+{"_id": "30906", "title": "Definition:Element/Also known as", "text": "The term '''member''' is sometimes used as a synonym for element (probably more for the sake of linguistic variation than anything else). In the contexts of geometry and topology, '''elements''' of a set are often called '''points''', in particular when they ''are'' (geometric) points. $x \\in S$ can also be read as: * '''$x$ is in $S$''' * '''$x$ belongs to $S$''' * '''$S$ includes $x$''' * '''$x$ is included in $S$''' * '''$S$ contains $x$''' However, '''beware''' of this latter usage: '''$S$ contains $x$''' can also be interpreted as '''$x$ is a subset of $S$'''. Such is the scope for misinterpretation that it is '''mandatory''' that further explanation is added to make it clear whether you mean subset or element."}
+{"_id": "30907", "title": "Definition:Element/Class", "text": "Let $S$ be a class. An '''element of $S$''' is a member of $S$."}
+{"_id": "30908", "title": "Definition:Class Equality/Definition 1", "text": "$A$ and $B$ are '''equal''', denoted $A = B$, {{iff}}: :$\\forall x: \\paren {x \\in A \\iff x \\in B}$ where $\\in$ denotes class membership."}
+{"_id": "30909", "title": "Definition:Class Equality/Definition 2", "text": "$A$ and $B$ are '''equal''', denoted $A = B$, {{iff}}: :$A \\subseteq B$ and $B \\subseteq A$ where $\\subseteq$ denotes the subclass relation."}
+{"_id": "30912", "title": "Definition:First-Order Property of Sets", "text": "A '''first-order property of sets''' is a property defined by a logical formula whose domain is entirely over sets. Thus the quantifiers $\\forall x$ and $\\exists x$ are valid when $x$ is a set, but $\\forall A$ and $\\exists A$ are ''not'' valid when $A$ is a class. Hence we allow, for example: :$\\map \\phi {A_1, A_2, \\ldots, A_n, x}$ to be a logical formula whose variables $A_1, A_2, \\ldots, A_n$ represent classes, and are all free, and $x$ is the only free variable representing a set."}
+{"_id": "30913", "title": "Definition:Swelled Class", "text": "Let $A$ denote a class. Then $A$ is a '''swelled class''' {{iff}} every subclass of every element of $A$ is also an element of $A$."}
+{"_id": "30914", "title": "Definition:Supercomplete Class", "text": "Let $A$ denote a class. Then $A$ is a '''supercomplete class''' {{iff}} both: :$A$ is transitive and: :$A$ is swelled."}
+{"_id": "30915", "title": "Definition:Von Neumann-Bernays-Gödel Axioms", "text": "=== The Axiom of Extension === {{:Axiom:Axiom of Extension/Class Theory}} === The Axiom of Specification === {{:Axiom:Axiom of Specification/Class Theory}} {{finish}}"}
+{"_id": "30916", "title": "Definition:Basic Universe", "text": "A '''basic universe''' $V$ is a universal class which satisfies the following axioms: {{:Definition:Basic Universe Axioms}}"}
+{"_id": "30917", "title": "Definition:Basic Universe Axioms", "text": "=== $\\text A 1$: Axiom of Transitivity === {{:Axiom:Axiom of Transitivity}} === $\\text A 2$: Axiom of Swelledness === {{:Axiom:Axiom of Swelledness}} === $\\text A 3$: Axiom of the Empty Set === {{:Axiom:Axiom of Empty Set/Class Theory}} === $\\text A 4$: Axiom of Pairing === {{:Axiom:Axiom of Pairing/Class Theory}} === $\\text A 5$: Axiom of Unions === {{:Axiom:Axiom of Unions/Class Theory}} === $\\text A 6$: Axiom of Powers === {{:Axiom:Axiom of Powers/Class Theory}}"}
+{"_id": "30918", "title": "Definition:Zermelo-Fraenkel Universe", "text": "A '''Zermelo-Fraenkel universe''' is a basic universe which also satisfies the axiom of infinity and the axiom of substitution: {{:Definition:Basic Universe Axioms}} === $\\text A 7$: Axiom of Infinity === {{:Axiom:Axiom of Infinity/Class Theory}} === $\\text A 8$: Axiom of Substitution === {{:Axiom:Axiom of Substitution}}"}
+{"_id": "30919", "title": "Definition:Zermelo Universe", "text": "A '''Zermelo universe''' is a basic universe which also satisfies the axiom of infinity: {{:Definition:Basic Universe Axioms}} === $\\text A 7$: Axiom of Infinity === {{:Axiom:Axiom of Infinity/Class Theory}}"}
+{"_id": "30921", "title": "Definition:Empty Class (Class Theory)", "text": "A class is defined as being '''empty''' {{iff}} it has no elements. That is: :$\\forall x: x \\notin A$ or: :$\\neg \\exists x: x \\in A$ The '''empty class''' is usually denoted $\\O$ or $\\emptyset$."}
+{"_id": "30922", "title": "Definition:Singleton Class", "text": "Let $a$ be a set. The class $\\set a$ is a '''singleton (class)'''. It can be referred to as '''singleton $a$'''."}
+{"_id": "30924", "title": "Definition:Ordered Pair/Wiener Formalization", "text": "The concept of an ordered pair can be formalized by the definition: :$\\tuple {a, b} := \\set {\\set {\\O, \\set a}, \\set {\\set b} }$"}
+{"_id": "30926", "title": "Definition:Ordered Pair/Empty Set Formalization", "text": "The concept of an ordered pair can be formalized by the definition: :$\\tuple {a, b} := \\set {\\set {\\O, a}, \\set {\\set \\O, b} }$"}
+{"_id": "30928", "title": "Definition:Set Intersection/Class", "text": "Let $A$ be a class. The '''intersection of $A$''' is: :$\\displaystyle \\bigcap A := \\set {x: \\forall y \\in A: x \\in y}$ That is, the class of all elements which belong to all the elements of $A$."}
+{"_id": "30930", "title": "Definition:Class Union", "text": "Let $A$ and $B$ be two classes. The '''(class) union''' $A \\cup B$ of $A$ and $B$ is defined as the class of all sets $x$ such that either $x \\in A$ or $x \\in B$ or both: :$x \\in A \\cup B \\iff x \\in A \\lor x \\in A$ or: :$A \\cup B = \\set {x: x \\in A \\lor x \\in B}$"}
+{"_id": "30931", "title": "Definition:Class Intersection", "text": "Let $A$ and $B$ be two classes. The '''(class) intersection''' $A \\cap B$ of $A$ and $B$ is defined as the class of all sets $x$ such that $x \\in A$ and $x \\in B$: :$x \\in A \\cap B \\iff x \\in A \\land x \\in A$ or: :$A \\cap B = \\set {x: x \\in A \\land x \\in B}$"}
+{"_id": "30934", "title": "Definition:Power Set/Class Theory", "text": "The '''power set''' of a set $S$ is the class of all the subsets of $S$: :$\\powerset S := \\set {T: T \\subseteq S}$"}
+{"_id": "30939", "title": "Definition:Polygon/Adjacent/Side to Vertex", "text": "Each vertex of a polygon is formed by the intersection of two sides. The two sides that form a particular vertex are referred to as the '''adjacents''' of that vertex, or described as '''adjacent to''' that vertex."}
+{"_id": "30940", "title": "Definition:Polygon/Adjacent/Vertex to Side", "text": "Each side of a polygon intersects two other sides, and so is terminated at either endpoint by two vertices. The two vertices that terminate a particular side are referred to as the '''adjacents''' of that side, or described as '''adjacent to''' that side."}
+{"_id": "30941", "title": "Definition:Polygon/Adjacent/Vertices", "text": "Those two vertices are described as '''adjacent to each other'''."}
+{"_id": "30942", "title": "Definition:Polygon/Adjacent/Sides", "text": "Two sides of a polygon that meet at the same vertex are '''adjacent''' to each other."}
+{"_id": "30944", "title": "Definition:Diophantine Analysis", "text": "'''Diophantine analysis''' is the branch of number theory which is concerned with the solution of Diophantine equations."}
+{"_id": "30946", "title": "Definition:Generatrix/Also known as", "text": "A '''generatrix''' of a surface can also be referred to as a '''generator''' or '''generator element'''."}
+{"_id": "30950", "title": "Definition:Vector Product", "text": "=== Dot Product === {{:Definition:Dot Product}} === Cross Product === {{:Definition:Vector Cross Product}} === Outer Product === {{:Definition:Outer Product}}"}
+{"_id": "30951", "title": "Definition:Stochastic Calculus", "text": "'''Stochastic calculus''' is a generalization of calculus to the subject of stochastic processes."}
+{"_id": "30953", "title": "Definition:QEF", "text": "The initials of '''Quod Erat Faciendum''', which is {{WP|Latin|Latin}} for '''which was to be done'''. These initials were traditionally added to the end of a geometric construction."}
+{"_id": "30954", "title": "Definition:QEI", "text": "The initials of '''Quod Erat Inveniendum''', which is {{WP|Latin|Latin}} for '''which was to be found'''. These initials were traditionally added after the completion of a calculation (either arithmetic or algebraic)."}
+{"_id": "30955", "title": "Definition:Quintal Notation", "text": "'''Quintal notation''' is the technique of expressing numbers in base $5$. That is, every number $x \\in \\R$ is expressed in the form: :$\\displaystyle x = \\sum_{j \\mathop \\in \\Z} r_j 5^j$ where: :$\\forall j \\in \\Z: r_j \\in \\set {0, 1, 2, 3, 4}$"}
+{"_id": "30956", "title": "Definition:Random", "text": "=== Random Error === {{:Definition:Random Error}} === Random Number === {{:Definition:Random Number}} === Random Sample === {{:Definition:Random Sample}} === Random Selection === {{:Definition:Random Selection}} === Random Variable === {{:Definition:Random Variable}} === Random Vector === {{:Definition:Random Vector}} === Random Walk === {{:Definition:Random Walk}}"}
+{"_id": "30957", "title": "Definition:Decade", "text": "A '''decade''' is a set or ordered tuple of $10$ numbers. Its most common usage is as a period of time: === Decade (Time Period) === {{:Definition:Decade (Time)}}"}
+{"_id": "30959", "title": "Definition:Parastrophe", "text": "=== $(1-3)$ Parastrophe === {{:Definition:(1-3) Parastrophe}} === $(2-3)$ Parastrophe === {{:Definition:(2-3) Parastrophe}} === $(1-2)$ Parastrophe === {{:Definition:(1-2) Parastrophe}}"}
+{"_id": "30960", "title": "Definition:Continuous Mapping (Normed Vector Space)", "text": "Let $M_1 = \\struct{X_1, \\norm {\\,\\cdot\\,}_{X_1} }$ and $M_2 = \\struct{X_2, \\norm {\\,\\cdot\\,}_{X_2} }$ be normed vector spaces. Let $f: X_1 \\to X_2$ be a mapping from $X_1$ to $X_2$. Let $a \\in X_1$ be a point in $X_1$. === Continuous at a Point === {{Definition:Continuous Mapping (Normed Vector Space)/Point}} === Continuous on a Space === {{Definition:Continuous Mapping (Normed Vector Space)/Space}}"}
+{"_id": "30961", "title": "Definition:Continuous Mapping (Normed Vector Space)/Point", "text": "{{Definition:Continuous Mapping (Normed Vector Space)/Point/Definition 1}}"}
+{"_id": "30962", "title": "Definition:Continuous Mapping (Normed Vector Space)/Point/Definition 1", "text": "'''$f$ is continuous at (the point) $a$ (with respect to the norms $\\norm {\\,\\cdot\\,}_{X_1}$ and $\\norm {\\,\\cdot\\,}_{X_2}$)''' {{iff}}: :$\\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: \\forall x \\in X_1: \\norm {x - a}_{X_1} < \\delta \\implies \\norm {\\map f x - \\map f a}_{X_2} < \\epsilon$ where $\\R_{>0}$ denotes the set of all strictly positive real numbers."}
+{"_id": "30963", "title": "Definition:Continuous Mapping (Normed Vector Space)/Space", "text": "{{Definition:Continuous Mapping (Normed Vector Space)/Space/Definition 1}}"}
+{"_id": "30964", "title": "Definition:Continuous Mapping (Normed Vector Space)/Space/Definition 1", "text": "$f$ is '''continuous from $\\struct{X_1, \\norm {\\,\\cdot\\,}_{X_1} }$ to $\\struct{X_2, \\norm {\\,\\cdot\\,}_{X_2} }$''' {{iff}} it is continuous at every point $x \\in X_1$."}
+{"_id": "30966", "title": "Definition:Statics", "text": "'''Statics''' is the branch of mechanics investigating the behaviour of forces of bodies at rest."}
+{"_id": "30968", "title": "Definition:Limit of Function (Normed Vector Space)/Epsilon-Delta Condition", "text": ":$\\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: 0 < \\norm {x - c}_{X_1} < \\delta \\implies \\norm {\\map f x - L}_{X_2} < \\epsilon$ That is, for every real positive $\\epsilon$ there exists a real positive $\\delta$ such that ''every'' point in the domain of $f$ within $\\delta$ of $c$ has an image within $\\epsilon$ of some point $L$ in the codomain of $f$."}
+{"_id": "30970", "title": "Definition:Limit of Function (Normed Vector Space)/Epsilon-Ball Condition", "text": ":$\\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: \\map f {B_\\delta \\left({c; \\norm {\\,\\cdot\\,}_{X_1}}\\right) \\setminus \\set c} \\subseteq \\map {B_\\epsilon} {L; \\norm {\\,\\cdot\\,}_{X_2}}$. where: : $\\map {B_\\delta} {c; \\norm {\\,\\cdot\\,}_{X_1}} \\setminus \\set c$ is the deleted $\\delta $-neighborhood of $c$ in $M_1$ : $B_\\epsilon \\left({L; \\norm {\\,\\cdot\\,}_{X_2}}\\right)$ is the open $\\epsilon$-ball of $L$ in $M_2$. That is, for every open $\\epsilon$-ball of $L$ in $M_2$, there exists a deleted $\\delta $-neighborhood of $c$ in $M_1$ whose image is a subset of that open $\\epsilon$-ball."}
+{"_id": "30971", "title": "Definition:Body/Rigid", "text": "A '''rigid body''' is a body $B$ with the property that the distance between any two points of $B$ remains constant regardless of any external forces."}
+{"_id": "30973", "title": "Definition:Motion", "text": "A body is defined as being '''in motion''' if it is not stationary with respect to the given frame of reference. Thus '''motion''' is the state of '''changing position'''."}
+{"_id": "30974", "title": "Definition:Ceteris Paribus", "text": "'''Ceteris paribus''' is a Latin phrase meaning '''all other things being equal'''. It is also translated as '''in the absence of any other change''' (in the conditions of a theorem, for example). It is often used in logical arguments in natural language when expressing a general comparison between two effects, and is a plea to the hearer to suspend counterarguments based on differences in the general situations."}
+{"_id": "30975", "title": "Definition:Natural Place", "text": "According to the ancient Greek philosophers, everything had a position in the universe where it was most comfortable. That position was referred to as its '''natural place'''. Hence, when displaced from its '''natural place''', it would eagerly try to return to that '''natural place'''. Hence: :a stone, being made of earth, when released from a hand holding it in midair, will push downwards in order to return as soon as possible to the centre of the universe, that is, Earth. :tongues of fire will shoot upwards, so as to rise to its '''natural place''' above the sphere of air :bubbles of air will rise up through water to its '''natural place''' above the sphere of water."}
+{"_id": "30978", "title": "Definition:Vacuum", "text": "A '''vacuum''' is the physical state of being completely empty of matter. It is a physical ideal, and has never been (and probably can never be) fully realised."}
+{"_id": "30980", "title": "Definition:Inclined Plane", "text": "An '''inclined plane''' is a plane surface which is neither horizontal nor vertical."}
+{"_id": "30990", "title": "Definition:Gravity", "text": "'''Gravity''' is the tendency of bodies with mass to attract each other."}
+{"_id": "30996", "title": "Definition:Gravity/Gravitational Force", "text": "The '''gravitational force''' on a body $B$ is the force which is exerted on $B$ as a result of the gravitational field whose influence it is under."}
+{"_id": "30997", "title": "Definition:Limit Point/Normed Vector Space", "text": "Let $M = \\struct {X, \\norm {\\, \\cdot \\,} }$ be a normed vector space. Let $Y \\subseteq X$ be a subset of $X$. === Limit Point of Set === {{:Definition:Limit Point/Normed Vector Space/Set}} === Limit Point of Sequence === {{:Definition:Limit Point/Normed Vector Space/Sequence}}"}
+{"_id": "30998", "title": "Definition:Displacement/Dimension", "text": "The dimension of '''displacement''' is length $\\mathsf L$."}
+{"_id": "31004", "title": "Definition:Mass/Units", "text": "The units of measurement of '''mass''' are as follows: * The SI unit of '''mass''' is the kilogram $\\mathrm {kg}$. * The CGS unit of '''mass''' is the gram $\\mathrm g$. * The FPS unit of '''mass''' is the pound $\\mathrm {lb}$. Thus: :$1 \\ \\mathrm {kg} = 10^3 \\ \\mathrm g$ :$1 \\ \\mathrm {lb} = 0.453 \\, 592 \\, 37 \\ \\mathrm{kg}$"}
+{"_id": "31005", "title": "Definition:Gram/Symbol", "text": "The symbol for the '''gram''' is $\\mathrm g$."}
+{"_id": "31006", "title": "Definition:CGS/Base Units", "text": "{| class=\"wikitable\" style=\"margin:1em auto 1em auto\" |+ CGS base units |- !Name ! Unit symbol ! Dimension ! Symbol |- ! centimetre | $\\mathrm{cm}$ | Length | $l$ |- ! gram | $\\mathrm g$ | Mass | $m$ |- ! second | $\\mathrm s$ | Time | $t$ |- |}"}
+{"_id": "31007", "title": "Definition:CGS/Derived Units/Dyne", "text": "The '''dyne''' is the '''CGS''' unit of force: :$1 \\ \\mathrm {dyn} = 1 \\ \\mathrm g \\ \\mathrm{cm} \\ \\mathrm s^{-2}$"}
+{"_id": "31010", "title": "Definition:MKS/Derived Units", "text": "=== Newton === {{:Definition:Newton (Unit)}}"}
+{"_id": "31013", "title": "Definition:Expectation/Discrete", "text": "Let $X$ be a discrete random variable. The '''expectation of $X$''' is written $\\expect X$, and is defined as: :$\\expect X := \\displaystyle \\sum_{x \\mathop \\in \\image X} x \\, \\map \\Pr {X = x}$ whenever the sum is absolutely convergent, that is, when: :$\\displaystyle \\sum_{x \\mathop \\in \\image X} \\size {x \\, \\map \\Pr {X = x} } < \\infty$"}
+{"_id": "31014", "title": "Definition:Limit Point/Normed Vector Space/Sequence", "text": "Let $L \\in Y$. Let $\\sequence {x_n}_{n \\mathop \\in \\N}$ be a sequence in $Y \\setminus \\set L$. Let $\\sequence {x_n}_{n \\mathop \\in \\N}$ converge to $L$.  Then $L$ is a '''limit of  $\\sequence {x_n}_{n \\mathop \\in \\N}$ as $n$ tends to infinity''' which is usually written: :$\\displaystyle L = \\lim_{n \\mathop \\to \\infty} x_n$"}
+{"_id": "31015", "title": "Definition:Artificial Intelligence", "text": "'''Artificial intelligence''' is a branch of computer science which studies intelligent behavior, and attempts to model it via computing techniques."}
+{"_id": "31017", "title": "Definition:Deduction", "text": "'''Deduction''' is the process in logic which derives new true statements from a set of premises, themselves assumed true."}
+{"_id": "31020", "title": "Definition:Temporal Logic", "text": "'''Temporal logic''' is a subcategory of modal logic which introduces time, by defining the concepts: :sometimes :always."}
+{"_id": "31021", "title": "Definition:Limit Point/Normed Vector Space/Set", "text": "Let $\\alpha \\in X$.  Then $\\alpha$ is a '''limit point of $Y$''' {{iff}} ''every'' deleted $\\epsilon$-neighborhood $\\map {B_\\epsilon} \\alpha \\setminus \\set \\alpha$ of $\\alpha$ contains a point in $Y$: :$\\forall \\epsilon \\in \\R_{>0}: \\map {B_\\epsilon} \\alpha \\setminus \\set \\alpha \\cap Y \\ne \\O$ that is: :$\\forall \\epsilon \\in \\R_{>0}: \\set {x \\in Y: 0 < \\norm {x - \\alpha} < \\epsilon} \\ne \\O$ Note that $\\alpha$ does not have to be an element of $A$ to be a '''limit point'''."}
+{"_id": "31024", "title": "Definition:Abel Summation Method", "text": "{{Help|It is difficult finding a concise and complete definition of exactly what the Abel Summation Method actually is. All and any advice as to how to implement this adequately is requested of anyone. This is what is said in the Spring encyclopedia on the page \"Abel summation method\":}} The series: :$\\displaystyle \\sum a_n$ can be summed by the Abel method ($A$-method) to the number $S$ if, for any real $x$ such that $0 < x < 1$, the series: :$\\displaystyle \\sum_{k \\mathop = 0}^\\infty a_k x^k$ is convergent and: :$\\displaystyle \\lim_{x \\mathop \\to 1^-} \\sum_{k \\mathop =0}^\\infty a_k x^k = S$ {{help|This is what we have on Wikipedia page {{WP|Divergent_series|Divergent series}}: }} :$\\displaystyle \\map f x = \\sum_{n \\mathop = 0}^\\infty a_n e^{-n x} = \\sum_{n \\mathop = 0}^\\infty a_n z^n$ where $z = \\map \\exp {−x}$. Then the limit of $\\map f x$ as $x$ approaches $0$ through positive reals is the limit of the power series for $\\map f z$ as $z$ approaches $1$ from below through positive reals. The '''Abel sum''' $\\map A s$ is defined as: :$\\displaystyle \\map A s = \\lim_{z \\mathop \\to 1^-} \\sum_{n \\mathop = 0}^\\infty a_n z^n$ {{NamedforDef|Niels Henrik Abel|cat = Abel}}"}
+{"_id": "31025", "title": "Definition:Above", "text": "In the context of real numbers, '''above''' means '''greater than'''."}
+{"_id": "31027", "title": "Definition:Maximum Value of Real Function", "text": "=== Absolute Maximum === {{:Definition:Maximum Value of Real Function/Absolute}} === Local Maximum === {{:Definition:Maximum Value of Real Function/Local}}"}
+{"_id": "31028", "title": "Definition:Minimum Value of Real Function", "text": "=== Absolute Minimum === {{:Definition:Minimum Value of Real Function/Absolute}} === Local Minimum === {{:Definition:Minimum Value of Real Function/Local}}"}
+{"_id": "31031", "title": "Definition:Topology Generated by Synthetic Basis/Definition 3", "text": "The '''topology on $S$ generated by $\\BB$''' is defined as: :$\\tau = \\set {U \\subseteq S: \\forall x \\in U: \\exists B \\in \\BB: x \\in B \\subseteq U}$"}
+{"_id": "31035", "title": "Definition:Compact Space/Normed Vector Space/Subspace", "text": "Let $M = \\struct{X, \\norm {\\,\\cdot\\,}}$ be a normed vector space. Let $K \\subseteq X$ be a subset of $X$. The normed vector subspace $M_K = \\struct {K, \\norm {\\,\\cdot\\,}_K}$ is '''compact in $M$''' {{iff}} $M_K$ is itself a compact normed vector space."}
+{"_id": "31041", "title": "Definition:Relation/Notation", "text": "If $\\tuple {x, y}$ is an ordered pair such that $\\tuple {x, y} \\in \\RR$, we use the notation: :$s \\mathrel \\RR t$ or: :$\\map \\RR {s, t}$ and can say: :'''$s$ bears $\\RR$ to $t$''' :'''$s$ stands in the relation $\\RR$ to $t$''' If $\\tuple {s, t} \\notin \\RR$, we can write: $s \\not \\mathrel \\RR t$, that is, by drawing a line through the relation symbol. See Complement of Relation."}
+{"_id": "31042", "title": "Definition:Relation/Class Theory", "text": "Let $V$ be a basic universe. In the context of class theory, a '''relation''' $\\RR$ is a subclass of the Cartesian product $V \\times V$."}
+{"_id": "31044", "title": "Definition:Braces", "text": "'''Braces''' is the name given to the pair of '''curly brackets''': :$\\set {\\cdots}$ These are conventionally used mostly for: :$(1): \\quad$ Set delimiters, to membership of sets and classes. :$(2): \\quad$ the fractional part of a real number: ::::$\\fractpart x := x - \\floor x$ :::where $\\floor x$ denotes the floor of $x$. :$(3): \\quad$ Karamata notation for the Stirling numbers of the second kind: ::::$\\displaystyle {n \\brace k}$ On {{ProofWiki}}, which implements the $\\LaTeX$ mathematical markup language, '''braces''' are also used to delimit arguments to $\\LaTeX$ commands. Category:Definitions/Language Definitions mo8a908hi9dpxipe6mapqibiyhtquki"}
+{"_id": "31046", "title": "Definition:Non-Limit Ordinal", "text": "An ordinal $\\lambda$ is a '''non-limit ordinal''' {{iff}} it is not a limit ordinal. That is, either the zero ordinal or a successor ordinal."}
+{"_id": "31047", "title": "Definition:Natural Numbers/Inductive Set Definition", "text": "Let $x$ be a set which is an element of every inductive set. Then $x$ is a natural number."}
+{"_id": "31048", "title": "Definition:Distinguished Element", "text": "Let $S$ be a set. A '''distinguished element of $S$''' is an element which is distinguished in some manner, by the imposition of some property or properties on $S$, from the other elements of $S$."}
+{"_id": "31049", "title": "Definition:Inductive Set", "text": "Let $S$ be a set of sets. Then $S$ is '''inductive''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | lc= $S$ contains the empty set:         | q =          | m = \\quad \\O \\in S }} {{axiom | n = 2         | lc= $S$ is closed under the successor mapping:         | q = \\forall x         | m = \\paren {x \\in S \\implies x^+ \\in S}         | rc= where $x^+$ is the successor of $x$ }} {{axiom | rc= That is, where $x^+ = x \\cup \\set x$ }} {{end-axiom}}"}
+{"_id": "31051", "title": "Definition:Roster Notation", "text": "'''Roster notation''' is the technique of specifying the elements in a set by listing them between a pair of braces $\\set{}$. === Explicit Definition === {{:Definition:Explicit Set Definition}} If there are many elements in a set, then it becomes tedious and impractical to list them all in one big long explicit definition. Fortunately, however, there are other techniques for listing sets. === Implicit Definition === {{:Definition:Implicit Set Definition}} Category:Definitions/Set Theory j6vc5f7i8000i4wxyug8pnf47osksua"}
+{"_id": "31052", "title": "Definition:Inductive Class/General", "text": "Let $A$ be a class. Let $g: A \\to A$ be a mapping on $A$. Then $A$ is '''inductive under $g$''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | lc= $A$ contains the empty set:         | q =          | m = \\quad \\O \\in A }} {{axiom | n = 2         | lc= $A$ is closed under $g$:         | q = \\forall x         | m = \\paren {x \\in A \\implies \\map g x \\in A} }} {{end-axiom}}"}
+{"_id": "31053", "title": "Definition:Minimally Inductive Class under General Mapping/Definition 1", "text": "$A$ is '''minimally inductive under $g$''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | t = $A$ is inductive under $g$ }} {{axiom | n = 2         | t = No proper subclass of $A$ is inductive under $g$. }} {{end-axiom}}"}
+{"_id": "31054", "title": "Definition:Minimally Inductive Class under General Mapping/Definition 2", "text": "$A$ is '''minimally inductive under $g$''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | t = $A$ is inductive under $g$ }} {{axiom | n = 2         | t = Every subclass of $A$ which is inductive under $g$ contains all the elements of $A$. }} {{end-axiom}}"}
+{"_id": "31055", "title": "Definition:Minimally Inductive Class under General Mapping", "text": "Let $A$ be a class. Let $g: A \\to A$ be a mapping. === Definition 1 === {{:Definition:Minimally Inductive Class under General Mapping/Definition 1}} === Definition 2 === {{:Definition:Minimally Inductive Class under General Mapping/Definition 2}} === Definition 3 === {{:Definition:Minimally Inductive Class under General Mapping/Definition 3}}"}
+{"_id": "31061", "title": "Definition:Left Normal Element of Relation", "text": "Let $A$ be a class. Let $\\RR$ be a relation on $A$. An element $x$ of $A$ is '''left normal''' with respect to $\\RR$ {{iff}}: :$\\forall y \\in A: \\map \\RR {x, y}$ holds."}
+{"_id": "31062", "title": "Definition:Right Normal Element of Relation", "text": "Let $A$ be a class. Let $\\RR$ be a relation on $A$. An element $x$ of $A$ is '''right normal''' with respect to $\\RR$ {{iff}}: :$\\forall y \\in A: \\map \\RR {y, x}$ holds."}
+{"_id": "31064", "title": "Definition:Comparable Sets", "text": "=== Comparable in Size === {{:Definition:Comparable Sets/Cardinality}} === Comparable by Subset Ordering === {{:Definition:Comparable Sets/Subset Ordering}}"}
+{"_id": "31065", "title": "Definition:Comparable Sets/Subset Ordering", "text": "Let $S$ and $T$ be sets. Then $S$ and $T$ are '''comparable (with respect to the subset ordering)''' {{iff}} either: :$S \\subseteq T$ or: :$T \\subseteq S$ or both."}
+{"_id": "31066", "title": "Definition:Nest/Class Theory", "text": "Let $C$ be a class. $C$ is a '''nest''' {{iff}}: :$\\forall x, y \\in C: x \\subseteq y$ or $y \\subseteq x$"}
+{"_id": "31068", "title": "Definition:Well-Ordered Class under Inclusion", "text": "Let $A$ be a class which is also a nest. Let $A$ have the property that every non-empty subclass of $A$ has a smallest element under the inclusion relation. Then $A$ is said to be '''well-ordered under inclusion'''."}
+{"_id": "31069", "title": "Definition:Bounded Class", "text": "Let $B$ be a class. === Bounded by Set === {{:Definition:Bounded Class/Bounded by Set}} === Bounded Subset of Class === {{:Definition:Bounded Class/Bounded Subset of Class}}"}
+{"_id": "31071", "title": "Definition:Bounded Class/Bounded Subset of Class", "text": "Let $B$ be a subclass of a class $A$. Then $B$ is a '''bounded subset of $A$''' {{iff}}: :there exists a set $x \\in A$ such that $B$ is '''bounded by $x$'''"}
+{"_id": "31072", "title": "Definition:Absolute Difference", "text": "Let $a$ and $b$ be real numbers. The '''absolute difference''' between $a$ and $b$ is defined and denoted as: :$\\size {a - b}$ where $\\size {\\, \\cdot \\,}$ is the absolute value function."}
+{"_id": "31078", "title": "Definition:Arbitrary Constant", "text": "An '''arbitrary constant''' is a symbol used to represent an object which is neither a specific number nor a variable. It is used to represent a general object (usually a number, but not necessarily) whose value can be assigned when the expression is instantiated. === In the context of Calculus === {{:Definition:Arbitrary Constant (Calculus)}}"}
+{"_id": "31079", "title": "Definition:Congruence (Number Theory)/Modulus", "text": "The number $m$ in this congruence is known as the '''modulus''' of the congruence."}
+{"_id": "31087", "title": "Definition:Cubic (Geometry)", "text": "'''Cubic''' is an adjective which means '''in the shape of a cube'''."}
+{"_id": "31096", "title": "Definition:Interior (Topology)/Definition 3", "text": "The '''interior''' of $H$ is the set of all interior points of $H$."}
+{"_id": "31097", "title": "Definition:Interior Point (Topology)/Definition 3", "text": "Let $h \\in H$. $h$ is an '''interior point''' of $H$ {{iff}} $h$ is an element of an open set of the subspace topology of $H$."}
+{"_id": "31098", "title": "Definition:Weighted Mean/Normalized", "text": "Let the weights be normalized. Then the '''weighted mean''' of $S$ can be expressed in the form: :$\\displaystyle \\bar x := \\sum_{i \\mathop = 1}^n \\map W {x_i} x_i$ as by definition of normalized weight function all the weights add up to $1$."}
+{"_id": "31102", "title": "Definition:Operation/Operator", "text": "An '''operator''' is a symbol used to identify an '''operation'''."}
+{"_id": "31104", "title": "Definition:Rate of Change/Dimension", "text": "Let a physical quantity $P$ have dimension $\\mathsf D$. Then the '''rate of change''' of $P$ has dimension $\\mathsf {D T}^{-1}$."}
+{"_id": "31105", "title": "Definition:Section (Geometry)", "text": "=== Section of Line by Line === {{:Definition:Section of Line by Line}} === Definition:Section of Solid by Plane === {{:Definition:Section of Solid by Plane}} Category:Definitions/Geometry 10vp9dywstfn2857na6umnnx396075o"}
+{"_id": "31110", "title": "Definition:Absolute Number", "text": "An '''absolute number''' is a number in an expression which has a single value. It is either expressed using actual figures, in an agreed number system, or by a symbol which is understood to represent that specific number. === Examples === :$2$, $\\sqrt 5$, $1.976$, $\\dfrac 2 3$, $\\pi$ (pi), $e$ (Euler's number)"}
+{"_id": "31111", "title": "Definition:Long Period Prime", "text": "A '''long period prime''' is a prime number $p$ whose reciprocal has the maximum period $p - 1$."}
+{"_id": "31112", "title": "Definition:Minimally Closed Class", "text": "Let $A$ be a class. Let $g: A \\to A$ be a mapping. === Definition 1 === {{:Definition:Minimally Closed Class/Definition 1}} === Definition 2 === {{:Definition:Minimally Closed Class/Definition 2}}"}
+{"_id": "31114", "title": "Definition:Minimally Closed Class/Definition 1", "text": "$A$ is '''minimally closed under $g$ with respect to $b$''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | t = $A$ is closed under $g$ }} {{axiom | n = 2         | t = There exists $b \\in A$ such that no proper subclass of $A$ containing $b$ is closed under $g$. }} {{end-axiom}}"}
+{"_id": "31115", "title": "Definition:Minimally Closed Class/Definition 2", "text": "$A$ is '''minimally closed under $g$ with respect to $b$''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | t = $A$ is closed under $g$ }} {{axiom | n = 2         | t = There exists $b \\in A$ such that every subclass of $A$ containing $b$ which is closed under $g$ contains all the elements of $A$. }} {{end-axiom}}"}
+{"_id": "31116", "title": "Definition:Minimally Inductive Class under General Mapping/Definition 3", "text": "$A$ is '''minimally inductive under $g$''' {{iff}} $A$ is minimally closed under $g$ with respect to $\\O$."}
+{"_id": "31121", "title": "Definition:Abstract Space", "text": "An '''abstract space''' is: :a set of objects together with: :a set of axioms which define operations on and relations between those objects. === Metric Space === {{:Definition:Metric Space}} === Topological Space === {{:Definition:Topological Space}} === Vector Space === {{:Definition:Vector Space}}"}
+{"_id": "31122", "title": "Definition:Rectangle Function", "text": "The '''rectangle function''' is the real function $\\Pi: \\R \\to \\R$ defined as: :$\\forall x \\in \\R: \\map \\Pi x := \\begin {cases} 1 : & \\size x \\le \\dfrac 1 2 \\\\ 0 : & \\size x > \\dfrac 1 2 \\end {cases}$"}
+{"_id": "31123", "title": "Definition:Triangle Function", "text": "The '''triangle function''' is the real function $\\Lambda: \\R \\to \\R$ defined as: :$\\forall x \\in \\R: \\map \\Lambda x := \\begin {cases} 1 - \\size x : & \\size x \\le 1 \\\\ 0 : & \\size x > 1 \\end {cases}$ where $\\size x$ denotes the absolute value function."}
+{"_id": "31124", "title": "Definition:Rectangle Function/Graph", "text": "The graph of the rectangle function is illustrated below: :400px"}
+{"_id": "31125", "title": "Definition:Triangle Function/Graph", "text": "The graph of the triangle function is illustrated below: :400px"}
+{"_id": "31127", "title": "Definition:Heaviside Step Function/Graph", "text": "The graph of the Heaviside step function is illustrated below: :600px"}
+{"_id": "31128", "title": "Definition:Signum Function/Graph", "text": "The graph of the signum function is illustrated below: :400px"}
+{"_id": "31129", "title": "Definition:Dirac Delta Function/Graph", "text": "The graph of the construction of the Dirac delta function is illustrated below: :400px In the limit, the graph can be approximated as follows, where it is understood that the blue arrow represents a ray from $0$ up the $y$-axis: :400px"}
+{"_id": "31130", "title": "Definition:Lituus", "text": "The '''lituus''' is the locus of the equation expressed in Polar coordinates as: :$r^2 \\theta = a^2$ :600px"}
+{"_id": "31132", "title": "Definition:Reciprocal Spiral", "text": "The '''reciprocal spiral''' is the locus of the equation expressed in Polar coordinates as: :$r = \\dfrac a \\theta$ :600px"}
+{"_id": "31133", "title": "Definition:Fermat's Spiral", "text": "'''Fermat's spiral''' is the locus of the equation expressed in Polar coordinates as: :$r^2 = \\dfrac a \\theta$ :600px"}
+{"_id": "31135", "title": "Definition:Cornu Spiral", "text": "The '''Cornu spiral''' is the locus $C$ of the equation expressed in intrinsic coordinates as: :$s = a^2 \\kappa$ where: :$s$ denotes the length of arc at a point of $C$ from the origin :$\\kappa$ denotes the curvature of $C$ at that point. :600px"}
+{"_id": "31137", "title": "Definition:Sampling Function", "text": "The '''sampling function''' is the real function $\\operatorname {III}: \\R \\to \\R$ defined as: :$\\forall x \\in \\R: \\map {\\operatorname {III} } x := \\displaystyle \\sum_{n \\mathop \\in \\Z} \\map \\delta {x - n}$ where $\\delta$ denotes the Dirac delta function."}
+{"_id": "31138", "title": "Definition:Sampling Function/Graph", "text": "The graph of the sampling function is illustrated below: :600px It is to be understood that the blue arrows represent rays from the $x$-axis for constant $n \\in \\Z$."}
+{"_id": "31139", "title": "Definition:Even Impulse Pair Function", "text": "The '''even impulse pair function''' is the real function $\\operatorname {II}: \\R \\to \\R$ defined as: :$\\forall x \\in \\R: \\map {\\operatorname {II} } x := \\dfrac 1 2 \\map \\delta {x + \\dfrac 1 2} + \\dfrac 1 2 \\map \\delta {x - \\dfrac 1 2}$ where $\\delta$ denotes the Dirac delta function."}
+{"_id": "31140", "title": "Definition:Even Impulse Pair Function/Graph", "text": "The graph of the even impulse pair function is illustrated below: :400px It is to be understood that the blue arrows represent rays from the $x$-axis for constant $n \\in \\set {-\\dfrac 1 2, \\dfrac 1 2}$."}
+{"_id": "31141", "title": "Definition:Odd Impulse Pair Function", "text": "The '''odd impulse pair function''' is the real function $\\operatorname {I_I}: \\R \\to \\R$ defined as: :$\\forall x \\in \\R: \\map {\\operatorname {I_I} } x := \\dfrac 1 2 \\map \\delta {x + \\dfrac 1 2} - \\dfrac 1 2 \\map \\delta {x - \\dfrac 1 2}$ where $\\delta$ denotes the Dirac delta function."}
+{"_id": "31142", "title": "Definition:Odd Impulse Pair Function/Graph", "text": "The graph of the odd impulse pair function is illustrated below: :300px It is to be understood that the blue arrows represent rays from the $x$-axis for constant $n \\in \\set {-\\dfrac 1 2, \\dfrac 1 2}$."}
+{"_id": "31143", "title": "Definition:Filtering Function", "text": "The '''filtering function''' is the real function $\\operatorname {sinc}: \\R \\to \\R$ defined as: :$\\forall x \\in \\R: \\map {\\operatorname {sinc} } x := \\dfrac {\\sin \\pi x} {\\pi x}$ where $\\sin$ denotes the (real) sine function."}
+{"_id": "31144", "title": "Definition:Filtering Function/Graph", "text": "The graph of the filtering function is illustrated below: :500px"}
+{"_id": "31145", "title": "Definition:Convolution Integral/Positive Real Domain", "text": "Let $f$ and $g$ be supported on the positive real numbers $\\R_{\\ge 0}$ only. The '''convolution integral''' of $f$ and $g$ may be defined as: :$\\displaystyle \\map f t * \\map g t := \\int_0^t \\map f u \\map g {t - u} \\rd u$"}
+{"_id": "31146", "title": "Definition:Convolution of Real Sequences", "text": "Let $\\sequence f$ and $\\sequence g$ be real sequences. The '''convolution''' of $f$ and $g$ is defined as: :$\\displaystyle \\sequence {f_i} * \\sequence {g_i} := \\sum_{j \\mathop \\in \\Z_{\\ge 0} } f_i g_{i - j}$"}
+{"_id": "31148", "title": "Definition:Even Impulse Pair Function/2 Dimensional", "text": "Let $\\operatorname {II}: \\R \\to \\R$ denote the '''even impulse pair function'''. The $2$-dimensional form of $\\operatorname {II}$ is defined and denoted: :$\\forall x, y \\in \\R: \\map {\\operatorname { {}^2 II} } {x, y} := \\map {\\operatorname {II} } x \\map {\\operatorname {II} } y$"}
+{"_id": "31149", "title": "Definition:Dirac Delta Function/2 Dimensional", "text": "Let $\\delta: \\R \\to \\R$ denote the '''Dirac delta function'''. The $2$-dimensional form of $\\delta$ is defined and denoted: :$\\forall x, y \\in \\R: \\map { {}^2 \\delta} {x, y} := \\map \\delta x \\map \\delta y$"}
+{"_id": "31150", "title": "Definition:Sampling Function/2 Dimensional", "text": "Let $\\operatorname {III}: \\R \\to \\R$ denote the '''sampling function'''. The $2$-dimensional form of $\\operatorname {III}$ is defined and denoted: :$\\forall x, y \\in \\R: \\map {\\operatorname { {}^2 III} } {x, y} := \\map {\\operatorname {III} } x \\map {\\operatorname {III} } y$"}
+{"_id": "31151", "title": "Definition:Filtering Function/2 Dimensional", "text": "Let $\\operatorname {sinc}: \\R \\to \\R$ denote the '''filtering function'''. The $2$-dimensional form of $\\operatorname {sinc}$ is defined and denoted: :$\\forall x, y \\in \\R: \\map {\\operatorname { {}^2 sinc} } {x, y} := \\map {\\operatorname {sinc} } x \\map {\\operatorname {sinc} } y$"}
+{"_id": "31152", "title": "Definition:Sub-Gaussian Distribution", "text": "The distribution of a random variable $X$ with expectation $\\mu = \\expect X$ is called '''sub-Gaussian''' if there exists a $\\sigma \\in \\R_{>0}$ such that: :$\\expect {e^{\\lambda \\paren {X - \\mu} } } \\le e^{\\sigma^2 \\lambda^2 / 2}$ for all $\\lambda \\in \\R$. Category:Definitions/Probability Distributions 51lpgxg6t77wxtvk5u4xkl7z40jpaa9"}
+{"_id": "31155", "title": "Definition:Sampling Function/Linguistic Note", "text": "The name '''shah''' for the '''sampling function''' derives from its similarity in shape and appearance to the Russian '''Ш''', whose name is itself pronounced '''shah'''."}
+{"_id": "31161", "title": "Definition:Fourier Transform/Real Function", "text": "{{:Definition:Fourier Transform/Real Function/Formulation 1}}"}
+{"_id": "31162", "title": "Definition:Sub-Exponential Distribution", "text": "The distribution of a random variable $X$ with expectation $\\mu = \\expect X$ is called '''sub-exponential''' if there exists $\\nu\\in \\R_{> 0}, \\alpha \\in \\R_{\\ge 0}$ such that: :$\\expect {e^{\\lambda \\paren {X - \\mu} } } \\le e^{\\nu^2 \\lambda^2 / 2}$ for all $\\size \\lambda < \\dfrac 1 \\alpha$. {{explain|Where does $\\alpha$ come into it?}}"}
+{"_id": "31171", "title": "Definition:Progression", "text": "The word '''progression''' is a term used for either '''sequence''' or '''series''' in the following specific contexts. Because its usage is in general ambiguous, it is deprecated on {{ProofWiki}}."}
+{"_id": "31172", "title": "Definition:Arithmetic Progression", "text": "The term '''arithmetic progression''' is used to mean one of the following: === Arithmetic Sequence === {{:Definition:Arithmetic Sequence}} === Arithmetic Series === {{:Definition:Arithmetic Series}}"}
+{"_id": "31173", "title": "Definition:Geometric Progression", "text": "The term '''geometric progression''' is used to mean one of the following: === Geometric Sequence === {{:Definition:Geometric Sequence}} === Geometric Series === {{:Definition:Geometric Series}}"}
+{"_id": "31174", "title": "Definition:Arithmetic-Geometric Progression", "text": "The term '''arithmetic-geometric progression''' is used to mean one of the following: === Arithmetic-Geometric Sequence === {{:Definition:Arithmetic-Geometric Sequence}} === Arithmetic-Geometric Series === {{:Definition:Arithmetic-Geometric Series}}"}
+{"_id": "31175", "title": "Definition:Harmonic Progression", "text": "The term '''harmonic progression''' is used to mean one of the following: === Harmonic Sequence === {{:Definition:Harmonic Sequence}} === Harmonic Series === {{:Definition:Harmonic Series}}"}
+{"_id": "31176", "title": "Definition:Harmonic Series/General", "text": "Let $\\sequence {x_n}$ be a sequence of numbers such that $\\sequence {\\size {x_n} }$ is a harmonic sequence. Then the series defined as: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty x_n$ is a '''harmonic series'''."}
+{"_id": "31177", "title": "Definition:Square Wave/Points of Discontinuity", "text": "The points $\\alpha + n \\lambda$, for $n \\in \\Z$, are jump discontinuities. The values $\\map S {\\alpha + n \\lambda}$ may or may not be explicitly defined. It is a common approach to include either endpoint of the intervals from $\\alpha$ to $\\alpha + \\lambda$, and from $\\alpha - \\lambda$ to $\\alpha$, in order to ensure that the domain of $S$ is simply defined, for example: :$\\forall x \\in \\R: \\map S x = \\begin {cases} \\delta + \\gamma & : x \\in \\hointr \\alpha {\\alpha + \\lambda} \\\\ \\delta - \\gamma & : x \\in \\hointr {\\alpha - \\lambda} \\alpha \\\\ \\map S {x + 2 \\lambda} & : x < \\alpha - \\lambda \\\\ \\map S {x - \\lambda l} & : x \\ge \\alpha + \\lambda \\end {cases}$ Another approach is to make $\\map S {\\alpha + n \\lambda} = \\delta$ for all $n \\in \\Z$. The precise treatment of the discontinuities is often irrelevant or immaterial."}
+{"_id": "31178", "title": "Definition:Square Wave", "text": "A '''square wave''' is a periodic real function $S: \\R \\to \\R$ defined as follows: :$\\forall x \\in \\R: \\map S x = \\begin {cases} \\delta + \\gamma & : x \\in \\openint \\alpha {\\alpha + \\lambda} \\\\ \\delta - \\gamma & : x \\in \\openint {\\alpha - \\lambda} \\alpha \\\\ \\map S {x + 2 \\lambda} & : x < \\alpha - \\lambda \\\\ \\map S {x - 2 \\lambda} & : x > \\alpha + \\lambda \\end {cases}$ where: :$\\alpha$, $\\lambda$, $\\gamma$ and $\\delta$ are given real constants."}
+{"_id": "31179", "title": "Definition:Square Wave/Graph", "text": "The graph of $S$ is given below: :800px"}
+{"_id": "31180", "title": "Definition:Given", "text": "A '''given''' is an object whose properties and existence are stipulated at the beginning of an exposition."}
+{"_id": "31181", "title": "Definition:Finite Discontinuity", "text": "Let $A \\subseteq \\R$ be a subset of the real numbers. Let $f: A \\to \\R$ be a real function. Let $f$ be discontinuous at $a \\in A$. The point $a$ is a '''finite discontinuity''' of $f$ {{iff}} either: :$(1): \\quad a$ is a removable discontinuity or: :$(2): \\quad a$ is a jump discontinuity. Category:Definitions/Continuity 9hs3d2ni3o3srwdp37y6gv50znysm6k"}
+{"_id": "31182", "title": "Definition:Triangle Wave", "text": "A '''triangle wave''' is a periodic real function $T: \\R \\to \\R$ defined as follows: :$\\forall x \\in \\R: \\map T x = \\begin {cases} \\size x & : x \\in \\closedint {-\\lambda} \\lambda \\\\ \\map T {x + 2 \\lambda} & : x < -\\lambda \\\\ \\map T {x - 2 \\lambda} & : x > +\\lambda \\end {cases}$ where: :$\\lambda$ is a given real constant :$\\size x$ denotes the absolute value of $x$."}
+{"_id": "31183", "title": "Definition:Triangle Wave/Graph", "text": "The graph of $T$ is given below: :1000px"}
+{"_id": "31184", "title": "Definition:Sawtooth Wave", "text": "A '''sawtooth wave''' is a periodic real function $S: \\R \\to \\R$ defined as follows: :$\\forall x \\in \\R: \\map S x = \\begin {cases} x & : x \\in \\openint {-\\lambda} \\lambda \\\\ \\map S {x + 2 \\lambda} & : x < -\\lambda \\\\ \\map S {x - 2 \\lambda} & : x > +\\lambda \\end {cases}$ where $\\lambda$ is a given real constant"}
+{"_id": "31185", "title": "Definition:Sawtooth Wave/Points of Discontinuity", "text": "The points $\\paren {2 r + 1} \\lambda$, for $r \\in \\Z$, are jump discontinuities. The values $\\map S {\\paren {2 r + 1} \\lambda}$ may or may not be explicitly defined. It is a common approach to include one of the endpoints of the interval from $-\\lambda$ to $\\lambda$, in order to ensure that the domain of $S$ is simply defined. For the sawtooth wave for example: :$\\forall x \\in \\R: \\map S x = \\begin {cases} x & : x \\in \\hointr {-\\lambda} \\lambda \\\\ \\map S {x + 2 \\lambda} & : x < -\\lambda \\\\ \\map S {x - 2 \\lambda} & : x \\ge +\\lambda \\end {cases}$ and, for the inverse sawtooth wave: :$\\forall x \\in \\R: \\map S x = \\begin {cases} -x & : x \\in \\hointr {-\\lambda} \\lambda \\\\ \\map S {x + 2 \\lambda} & : x < -\\lambda \\\\ \\map S {x - 2 \\lambda} & : x \\ge +\\lambda \\end {cases}$ Another approach is to make $\\map S {\\paren {2 r + 1} \\lambda} = 0$ for all $r \\in \\Z$. The precise treatment of the discontinuities is often irrelevant or immaterial."}
+{"_id": "31186", "title": "Definition:Sawtooth Wave/Graph", "text": "The graph of $S$ is given below: :800px"}
+{"_id": "31188", "title": "Definition:Sawtooth Wave/Inverse/Graph", "text": "The graph of $S$ is given below: :800px"}
+{"_id": "31189", "title": "Definition:Set/Point Set", "text": "A '''set''' whose elements are all (geometric) points is often called a '''point set'''. In particular, the Cartesian plane and complex plane can each be seen referred to as a '''two-dimensional point set'''."}
+{"_id": "31190", "title": "Definition:Definite Integral/Limits of Integration/Upper Limit", "text": "The limit $b$ is referred to as the '''upper limit''' of the integral'''."}
+{"_id": "31191", "title": "Definition:Definite Integral/Limits of Integration/Lower Limit", "text": "The limit $a$ is referred to as the '''lower limit''' of the integral'''."}
+{"_id": "31192", "title": "Definition:Fourier Series/Range 2 Pi", "text": "Let $\\alpha \\in \\R$ be a real number. Let $f: \\R \\to \\R$ be a function such that $\\displaystyle \\int_\\alpha^{\\alpha + 2 \\pi} \\map f x \\rd x$ converges absolutely. Let: {{begin-eqn}} {{eqn | l = a_n       | r = \\dfrac 1 \\pi \\int_\\alpha^{\\alpha + 2 \\pi} \\map f x \\cos n x \\rd x }} {{eqn | l = b_n       | r = \\dfrac 1 \\pi \\int_\\alpha^{\\alpha + 2 \\pi} \\map f x \\sin n x \\rd x }} {{end-eqn}} Then: :$\\dfrac {a_0} 2 + \\displaystyle \\sum_{n \\mathop = 1}^\\infty \\paren {a_n \\cos n x + b_n \\sin n x}$ is called the '''Fourier Series''' for $f$."}
+{"_id": "31194", "title": "Definition:Half-Range Fourier Cosine Series/Formulation 1", "text": "Let $\\map f x$ be a real function defined on the interval $\\openint 0 \\lambda$. Then the ''' half-range Fourier cosine series''' of $\\map f x$ over $\\openint 0 \\lambda$ is the series: :$\\map f x \\sim \\dfrac {a_0} 2 + \\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n \\cos \\frac {n \\pi x} \\lambda$ where for all $n \\in \\Z_{\\ge 0}$: :$a_n = \\displaystyle \\frac 2 \\lambda \\int_0^\\lambda \\map f x \\cos \\frac {n \\pi x} \\lambda \\rd x$"}
+{"_id": "31197", "title": "Definition:Half-Range Fourier Sine Series/Formulation 1", "text": "Let $\\map f x$ be a real function defined on the interval $\\openint 0 \\lambda$. Then the '''half-range Fourier sine series''' of $\\map f x$ over $\\openint 0 \\lambda$ is the series: :$\\map f x \\sim \\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n \\sin \\frac {n \\pi x} \\lambda$ where for all $n \\in \\Z_{> 0}$: :$b_n = \\displaystyle \\frac 2 \\lambda \\int_0^\\lambda \\map f x \\sin \\frac {n \\pi x} \\lambda \\rd x$"}
+{"_id": "31198", "title": "Definition:Half-Range Fourier Series/Range Pi", "text": "=== Half-Range Fourier Cosine Series === {{:Definition:Half-Range Fourier Cosine Series/Range Pi}} === Half-Range Fourier Sine Series === {{:Definition:Half-Range Fourier Sine Series/Range Pi}}"}
+{"_id": "31200", "title": "Definition:Half-Range Fourier Cosine Series/Range Pi", "text": "Let $\\map f x$ be a real function defined on the interval $\\openint 0 \\pi$. Then the ''' half-range Fourier cosine series''' of $\\map f x$ over $\\openint 0 \\pi$ is the series: :$\\map f x \\sim \\dfrac {a_0} 2 + \\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n \\cos n x$ where for all $n \\in \\Z_{\\ge 0}$: :$a_n = \\displaystyle \\frac 2 \\pi \\int_0^\\pi \\map f x \\cos n x \\rd x$"}
+{"_id": "31201", "title": "Definition:Half-Range Fourier Sine Series/Range Pi", "text": "Let $\\map f x$ be a real function defined on the interval $\\openint 0 \\pi$. Then the ''' half-range Fourier sine series''' of $\\map f x$ over $\\openint 0 \\pi$ is the series: :$\\map f x \\sim \\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n \\sin n x$ where for all $n \\in \\Z_{> 0}$: :$b_n = \\displaystyle \\frac 2 \\pi \\int_0^\\pi \\map f x \\sin n x \\rd x$"}
+{"_id": "31203", "title": "Definition:Everywhere Dense/Normed Vector Space", "text": "Let $M = \\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $Y \\subseteq X$ be a subset of $X$. Suppose: :$\\forall x \\in X : \\forall \\epsilon \\in \\R : \\epsilon > 0 : \\exists y \\in Y : \\norm {x - y} < \\epsilon$ Then $Y$ is '''(everywhere) dense''' in $X$."}
+{"_id": "31204", "title": "Definition:Separable Space/Normed Vector Space", "text": "Let $M = \\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $Y \\subseteq X$ be a subset of $X$. Let $Y$ be dense in $X$. Let $Y$ be a countable set: :$Y = \\set {y_1, y_2, \\ldots}$ Suppose: :$\\forall x \\in X : \\forall \\epsilon \\in \\R : \\epsilon > 0 : \\exists y_{n \\mathop \\in \\N} \\in Y : \\norm {y_n - x} < \\epsilon$ Then $X$ is '''separable'''."}
+{"_id": "31207", "title": "Definition:Conjugate Point/Definition 3", "text": "Let $y = \\map y x$ and $y = \\map {\\tilde y} x$ be extremal functions. Let: :$M = \\paren {a, \\map y a}$ :$\\tilde M = \\paren {\\tilde a, \\map y {\\tilde a} }$ Let both $y = \\map y x$ and $y = \\map {\\tilde y} x$ pass through the point $M$. Let  :$\\displaystyle \\lim_{\\norm {\\map y x - \\map {\\tilde y} x}_{1, \\infty} \\to 0} \\sqbrk {\\paren {x, \\map y x}: \\map y x - \\map {\\tilde y} x = 0} = \\tilde M$ In other words, let $\\tilde M$ be the limit points of intersection of $y = \\map y x$ and $y = \\map {\\tilde y} x$ as $\\norm {\\map y x - \\map {\\tilde y} x}_{1, \\infty} \\to 0$. {{explain|the notation $\\norm {\\map y x - \\map {\\tilde y} x}_{1, \\infty}$ with particular regard to the subscript This is most likely the standard norm of Sobolev space, also know as np-norm. Original source never calls it by word; only by special symbols}} Then $\\tilde M$ is '''conjugate''' to $M$."}
+{"_id": "31208", "title": "Definition:Separated by Neighborhoods/Sets/Neighborhoods", "text": "Let $A, B \\subseteq S$ such that: :$\\exists N_A, N_B \\subseteq S: \\exists U, V \\in \\tau: A \\subseteq U \\subseteq N_A, B \\subseteq V \\subseteq N_B: N_A \\cap N_B = \\O$ That is, that $A$ and $B$ both have neighborhoods in $T$ which are disjoint. Then $A$ and $B$ are described as '''separated by neighborhoods'''."}
+{"_id": "31209", "title": "Definition:Separated by Neighborhoods/Points/Neighborhoods", "text": "Let $x, y \\in S$ such that: :$\\exists N_x, N_y \\subseteq S: \\exists U, V \\in \\tau: x \\in U \\subseteq N_x, y \\in V \\subseteq N_y: N_x \\cap N_y = \\O$ That is, that $x$ and $y$ both have neighborhoods in $T$ which are disjoint. Then $x$ and $y$ are described as '''separated by neighborhoods'''."}
+{"_id": "31210", "title": "Definition:Separated by Neighborhoods/Points/Open Sets", "text": "Let $x, y \\in S$ such that: :$\\exists U, V \\in \\tau: x \\in U, y \\in V: U \\cap V = \\O$ That is, that $x$ and $y$ both have open neighborhoods in $T$ which are disjoint. Then $x$ and $y$ are described as '''separated by (open) neighborhoods'''."}
+{"_id": "31211", "title": "Definition:Separated by Neighborhoods/Sets/Open Sets", "text": "Let $A, B \\subseteq S$ such that: :$\\exists U, V \\in \\tau: A \\subseteq U, B \\subseteq V: U \\cap V = \\O$ That is, that $A$ and $B$ both have open neighborhoods in $T$ which are disjoint. Then $A$ and $B$ are described as '''separated by (open) neighborhoods'''."}
+{"_id": "31212", "title": "Definition:Positive Definite Functional", "text": "Let $f : S \\to T$ be a mapping. Let $f \\in \\MM$, where $\\MM$ stands for some function space. Let $F \\sqbrk f : \\MM \\to \\R$ be a real-valued functional. Suppose: :$\\forall f \\in \\MM: F \\sqbrk f > 0$ Then $F$ is a '''positive-definite functional'''. Category:Definitions/Functional Analysis kdrak7sbetlc0p4syfabm44plyku69w"}
+{"_id": "31213", "title": "Definition:Earthworm Sequence", "text": "The '''earthworm sequence''' is the integer sequence $\\sequence {h_n}$ defined as follows. Let $h: \\Z \\to \\Z$ be the mapping defined as: :$\\forall a \\in \\Z: \\map h a = 2 a \\pmod {100}$ Then: :$h_n = \\begin {cases} x: x \\in \\set {10, 11, \\dotsc, 99} & : n = 0 \\\\ \\map h {n - 1} & : n > 0 \\end {cases}$ That is: :$h_0$ is an arbitrary integer between $10$ and $99$ :$h_n$ is the result of multiplying the previous term in the sequence by $2$ and keeping only the last two digits."}
+{"_id": "31215", "title": "Definition:Bounded Sequence/Normed Vector Space", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,} }$ be a normed vector space. Let $\\sequence {x_n}$ be a sequence in $X$. Then $\\sequence {x_n}$ is '''bounded''' {{iff}}: :$\\exists K \\in \\R$ such that $\\forall n \\in \\N: \\norm {x_n} \\le K$"}
+{"_id": "31216", "title": "Definition:Bounded Sequence/Normed Vector Space/Unbounded", "text": "$\\sequence {x_n}$ is '''unbounded''' {{iff}} it is not bounded."}
+{"_id": "31217", "title": "Definition:Quality of Integer Triple", "text": "Let $\\tuple {a, b, c}$ be an ordered triple of (strictly) positive integers. The '''quality''' of $\\tuple {a, b, c}$ is defined and denoted as: :$\\map q {a, b, c} := \\dfrac {\\ln c} {\\map \\ln {\\map {\\operatorname {rad} } {abc} } }$ where $\\operatorname {rad}$ denotes the radical of an integer. Category:Definitions/Number Theory rp83swjw9ou6vt9w5luv52on617nlyg"}
+{"_id": "31218", "title": "Definition:Continuous Function Space", "text": "Let $X, Y$ be topological spaces. Let $f : X \\to Y$ be a continuous mapping. Then the set of all such mappings $f$ is known as '''continuous function space''' and is denoted by $\\CC \\paren {X,Y}$: :$\\CC \\paren {X, Y} = \\set {f : X \\to Y}$"}
+{"_id": "31219", "title": "Definition:Bourbaki Dangerous Bend Symbol", "text": "The '''Bourbaki dangerous bend symbol''' is a symbol invented by the {{AuthorRef|Nicolas Bourbaki|Bourbaki group}} to warn the reader that there exists the serious possibility of making a common mistake while interpreting or utilising the material presented. It was originally presented by {{AuthorRef|Nicolas Bourbaki|Bourbaki}}, as the pre-$1949$ French traffic road sign: :20px Variants exist, notably {{AuthorRef|Donald Ervin Knuth}}'s version: :40px {{NamedforDef|Nicolas Bourbaki|cat = Bourbaki}}"}
+{"_id": "31220", "title": "Definition:Continuous Real-Valued Function Space", "text": "Let $X$ be a topological space. Let $f : X \\to \\R$ be a continuous real valued mapping. Then the set of all such mappings $f$ is known as '''continuous real-valued function space''' and is denoted by $\\map \\CC X$: :$\\map \\CC X := \\map \\CC {X, \\R} = \\set {f : X \\to \\R}$"}
+{"_id": "31221", "title": "Definition:Space of Continuous on Closed Interval Real-Valued Functions", "text": "Let $f : \\closedint a b \\to \\R$ be a continuous real valued function. Then the set of all such mappings $f$ is known as '''continuous on closed interval real-valued function space''' and is denoted by $\\CC \\closedint a b$: :$\\CC \\closedint a b := \\CC \\paren {\\closedint a b, \\R} = \\set {f : \\closedint a b \\to \\R}$"}
+{"_id": "31222", "title": "Definition:Space of Continuous Functions of Differentiability Class k", "text": "Let $X, Y$ be normed vector spaces. Let $f : X \\to Y$ be a mapping of a differentiability class $k \\in \\N_{>0}$ in the sense of Frechet. Then the set of all such mappings $f$ is known as '''continuous function space of differentiability class k''' and is denoted by $\\CC^k \\paren {X,Y}$: :$\\CC^k \\paren {X, Y} = \\set {f : X \\to Y}$"}
+{"_id": "31223", "title": "Definition:Real-Valued Periodic Function Space", "text": "Let $I = \\closedint 0 T$ be a closed real interval. Let $\\map \\CC I$ be the space of real-valued functions, continuous on  $I$. Let the elements of $\\map \\CC I$ be real periodic functions with the period $T$:  :$f \\in \\map \\CC I: \\map f T = \\map f 0$ Then the set of all such mappings $f$ is known as '''real-valued periodic (over $I$) function space''' and is denoted by $\\map {\\CC_{per}} I$: :$\\map {\\CC_{per}} I:= \\CC_{per} \\paren {I, \\R} = \\set {f : I \\to \\R : \\map f T = \\map f 0}$"}
+{"_id": "31224", "title": "Definition:Space of Real Polynomials of Degree n", "text": "Let $\\map \\CC \\R$ be the space of real-valued continuous functions. Let $f \\in \\map \\CC \\R$ be a polynomial over real numbers of degree not greater than $n$. Then the set of all such mappings $f$ is known as '''space of real polynomials of degree $\\le n$''' and is denoted by $\\PP_n \\paren {\\R}$: :$\\PP_n \\paren {\\R} := \\set {f : \\R \\to \\R : \\map \\deg f \\le n}$"}
+{"_id": "31225", "title": "Definition:Space of Real Polynomials", "text": "Let $\\PP_n$ be the space of real polynomials of degree not greater than n. Then the union of all such spaces is called the '''space of real polynomials''' and is denoted by $\\PP$: :$\\displaystyle \\PP := \\bigcup _{n \\mathop = 0}^{\\infty}\\PP_n$"}
+{"_id": "31229", "title": "Definition:Jaina Mathematics", "text": "'''Jainism''' is a religion and philosophy which was founded in India around the $6$th century BCE. '''Jaina mathematics''' is mathematics done by those following Jainism. This has been practised on the Indian subcontinent from the founding of Jainism up to modern times."}
+{"_id": "31230", "title": "Definition:Complete Lattice/Definition 1", "text": "Let $\\struct {S, \\preceq}$ be a lattice. Then $\\struct {S, \\preceq}$ is a '''complete lattice''' {{iff}}: :$\\forall T \\subseteq S: T$ admits both a supremum and an infimum."}
+{"_id": "31231", "title": "Definition:Completely Hausdorff Space/Definition 1", "text": "$\\struct {S, \\tau}$ is a '''completely Hausdorff space''' or '''$T_{2 \\frac 1 2}$ space''' {{iff}}: :$\\forall x, y \\in S, x \\ne y: \\exists U, V \\in \\tau: x \\in U, y \\in V: U^- \\cap V^- = \\O$  That is, for any two distinct elements $x, y \\in S$ there exist open sets $U, V \\in \\tau$ containing $x$ and $y$ respectively whose closures are disjoint."}
+{"_id": "31232", "title": "Definition:Completely Hausdorff Space/Definition 2", "text": "$\\struct {S, \\tau}$ is a '''completely Hausdorff space''' or '''$T_{2 \\frac 1 2}$ space''' {{iff}}: :$\\forall x, y \\in S, x \\ne y : \\exists N_x, N_y \\subseteq S: \\exists U, V \\in \\tau: x \\subseteq U \\subseteq N_x, y \\subseteq V \\subseteq N_y: N_x^- \\cap N_y^- = \\O$ That is: :$\\struct {S, \\tau}$ is a '''$T_{2 \\frac 1 2}$ space''' {{iff}} every two points in $S$ are separated by closed neighborhoods."}
+{"_id": "31233", "title": "Definition:Polyomino/Free", "text": "A '''free polyomino''' is a polyomino which is not distinguished from its image under a reflection. That is, it is considered '''free''' of the plane in which it is embedded, and can be \"lifted up and turned over\"."}
+{"_id": "31236", "title": "Definition:Ordering on Natural Numbers/Von Neumann Construction", "text": "Let $\\omega$ denote the set of natural numbers as defined by the von Neumann construction. The '''strict ordering''' of $\\omega$ is the relation $<$ defined by: :$\\forall m, n \\in \\omega: m < n \\iff m \\subsetneq n$ The '''(weak) ordering''' of $\\omega$ is the relation $\\le$ defined by: :$\\forall m, n \\in \\omega: m \\le n \\iff m \\subseteq n$"}
+{"_id": "31237", "title": "Definition:Cardinality of Finite Class/Definition 1", "text": "Let $A$ be such that: :there exists a bijection $\\phi$ from $A$ to $n$ where $n$ is a natural number as defined by the von Neumann construction. Then $A$ has '''cardinality $n$'''."}
+{"_id": "31238", "title": "Definition:Cardinality of Finite Class/Definition 2", "text": "Let $A$ be such that: :there exists a bijection $\\phi$ from $A$ to the set $\\set {1, 2, \\dotsc, n} = n^+ \\setminus \\set 0$ where: :$n$ is a natural number as defined by the von Neumann construction :$n^+$ is the successor of $n$. Then $A$ has '''cardinality $n$'''."}
+{"_id": "31240", "title": "Definition:Finite Class", "text": "Let $A$ be a class. Let there exist a natural number $n$ as defined by the von Neumann construction such that there exists a bijection $\\phi$ from $A$ to $n$. Then $A$ is a '''finite class'''."}
+{"_id": "31241", "title": "Definition:Infinite Class", "text": "Let $A$ be a class. $A$ is an '''infinite class''' {{iff}} $A$ is not a finite class. </onlyinclude>"}
+{"_id": "31245", "title": "Definition:Statistic", "text": "A '''statistic''' is broadly defined as '''a quantity which is calculated from sample data."}
+{"_id": "31246", "title": "Definition:Dispersion (Statistics)", "text": "Let $S$ be a sample of a population in the context of statistics. The '''dispersion''' of $S$ is a general term meaning how much the data describing the sample are spread out. The word can also be applied to a random variable."}
+{"_id": "31247", "title": "Definition:Denumerable Class", "text": "Let $A$ be a class. $A$ is '''denumerable''' {{iff}} there exists a bijection: :$f: S \\to \\N$ where $\\N$ is the set of natural numbers."}
+{"_id": "31248", "title": "Definition:Non-Denumerable Class", "text": "A class which is neither denumerable nor finite is described as '''non-denumerable'''."}
+{"_id": "31251", "title": "Definition:Machine (Mechanics)", "text": "A '''machine''', in the context of mechanics, is a physical artefact which takes energy in a particular form and converts it into energy in another form for a specific purpose."}
+{"_id": "31252", "title": "Definition:Mach Number", "text": "'''Mach number''' is a unit of measurement of speed of a body $B$ in a fluid $F$. It is defined as the ratio of the speed of $B$ to the speed of sound in $F$. {{NamedforDef|Ernst Waldfried Josef Wenzel Mach|cat = Mach}}"}
+{"_id": "31255", "title": "Definition:Supersonic", "text": "The speed of a body $B$ is defined as '''supersonic''' {{iff}} its Mach number is greater than $1$. That is, if $B$ is moving faster than the speed of sound in the fluid in which it is moving."}
+{"_id": "31257", "title": "Definition:Rest", "text": "=== At Rest === The term '''at rest''' is a synonym for '''stationary''': {{:Definition:At Rest}} === Rest Mass === {{:Definition:Rest Mass}} Category:Definitions/Physics 91a1fkbupc2a1maip7g8fnbkrtui9l7"}
+{"_id": "31259", "title": "Definition:Electric Charge/Dimension", "text": "The dimension of measurement of '''electric charge''' is $\\mathsf {I T}$."}
+{"_id": "31260", "title": "Definition:Electric Charge/Units", "text": "The SI unit of '''electric charge''' is the coulomb $\\mathrm C$."}
+{"_id": "31262", "title": "Definition:Electric Field Strength/Units", "text": "The SI unit for '''electric field strength''' can be given either as: :volt per metre: $\\mathrm V / \\mathrm m$ :newton per coulomb: $\\mathrm N / \\mathrm C$"}
+{"_id": "31263", "title": "Definition:Electric Field Strength/Dimension", "text": "'''Electric field strength''' has the dimension $\\mathsf {M L T}^{-3} \\mathsf I^{-1}$."}
+{"_id": "31264", "title": "Definition:Electric Field Strength", "text": "'''Electric field strength''' is the measure of the intensity of an electric field."}
+{"_id": "31265", "title": "Definition:Electrostatics", "text": "'''Electrostatics''' is the branch of physics concerned with the study of stationary electrically charged bodies, and the forces and fields they give rise to."}
+{"_id": "31268", "title": "Definition:Electric Charge/Sign", "text": "The '''sign''' of an '''electric charge''' can be one of $2$ types: === Positive Electric Charge === {{:Definition:Electric Charge/Sign/Positive}} === Negative Electric Charge === {{:Definition:Electric Charge/Sign/Negative}}"}
+{"_id": "31269", "title": "Definition:Electric Charge/Sign/Like", "text": "If $2$ electric charges are of the same sign, they are referred to as being '''like (electric) charges'''."}
+{"_id": "31270", "title": "Definition:Electric Charge/Sign/Unlike", "text": "If $2$ electric charges are of opposite sign, they are referred to as being '''unlike (electric) charges'''."}
+{"_id": "31271", "title": "Definition:Electric Charge/Sign/Positive", "text": "A '''positive electric charge''' is an electric charge which is of the same sign as the electric charge on a proton. When it is necessary to assign a value to a '''positive electric charge''', a $+$ (plus) sign is used, and the value assigned is a positive number."}
+{"_id": "31272", "title": "Definition:Electric Charge/Sign/Negative", "text": "A '''negative electric charge''' is an electric charge which is of the same sign as the electric charge on an electron. When it is necessary to assign a value to a '''negative electric charge''', a $-$ (minus) sign is used, and the value assigned is a negative number."}
+{"_id": "31273", "title": "Definition:Electric Charge/Sign/Convention", "text": "It appears at first glance that the definition of the sign of an electric charge is both arbitrary and circular. That is: :a positive electric charge is the type of electric charge that can be found on a proton while: :a proton is defined as carrying a positive electric charge. Early in the development of the theory of electrostatics, it was noticed that there were two types of electric charge. What they were called was at that point arbitrary; one type was called '''positive''', and the other type was called '''negative'''. When the proton and electron were discovered, the signs of their electric charges were identified, and subsequently used to ''define'' the sign of an arbitrary electric charge. Hence the fact that a proton is positive and an electron is negative is simply the result of a convention."}
+{"_id": "31277", "title": "Definition:Coulomb (Unit)/Base Units", "text": "The SI base units of the '''coulomb''' are: :$\\mathrm C := \\mathrm A \\, \\mathrm s$ where: :$\\mathrm A$ denotes amperes :$\\mathrm s$ denotes seconds (of time)."}
+{"_id": "31280", "title": "Definition:Bounded Mapping/Real-Valued/Definition 3", "text": "$f$ is '''bounded on $S$''' {{iff}}: :$\\exists a, b \\in \\R_{\\ge 0}: \\forall x \\in S: \\map f x \\in \\closedint a b$ where $\\closedint a b$ denotes the (closed) real interval from $a$ to $b$."}
+{"_id": "31282", "title": "Definition:Negation Function", "text": "The '''negation function''' is the function defined on the various standard number systems as follows: === Integer Negation Function === {{:Definition:Negation Function/Integer}} === Rational Negation Function === {{:Definition:Negation Function/Rational}} === Real Negation Function === {{:Definition:Negation Function/Real}} === Complex Negation Function === {{:Definition:Negation Function/Complex}}"}
+{"_id": "31283", "title": "Definition:Negation Function/Integer", "text": "The negation function $h: \\Z \\to \\Z$ is defined on the set of integers as: :$\\forall n \\in \\Z: \\map h n = -n$"}
+{"_id": "31287", "title": "Definition:Union Mapping/Finite Set", "text": "Let $S = \\set {f_1, f_2, \\ldots, f_n}$ denote a finite set of mappings. The '''union mapping''' $f$ of $S$ is defined when: :$\\forall i, j \\in \\set {1, 2, \\ldots, n}: f_i$ and $f_j$ are combinable and is defined as: :$\\forall x \\in \\displaystyle \\bigcup \\set {\\Dom {f_i}: i \\in \\set {1, 2, \\ldots, n} } x \\in \\Dom {f_i} \\implies f = \\map {f_i} x$"}
+{"_id": "31288", "title": "Definition:Union Mapping/Family", "text": "Let $I$ be an indexing set. Let $F = \\family {f_i}_{i \\mathop \\in I}$ be a family of mappings indexed by $I$ The '''union mapping''' $f$ of $F$ is defined when: :$\\forall i, j \\in I: f_i$ and $f_j$ are combinable and is defined as: :$\\forall x \\in \\displaystyle \\bigcup \\set {\\Dom {f_i}: i \\in I} x \\in \\Dom {f_i} \\implies f = \\map {f_i} x$"}
+{"_id": "31290", "title": "Definition:Set Product/Projection", "text": "The mappings $\\phi_i$ are the '''projections''' of $P$."}
+{"_id": "31291", "title": "Definition:Group Product/Group Law", "text": "The operation $\\circ$ can be referred to as the '''group law'''."}
+{"_id": "31292", "title": "Definition:Group Product/Product Element", "text": "Let $a, b \\in G$ such that $ = a \\circ b$. Then $g$ is known as the '''product of $a$ and $b$'''."}
+{"_id": "31293", "title": "Definition:Closed under Inversion", "text": "Let $\\struct {G, \\circ}$ be an group with an identity element $e$. Let $H \\subseteq G$ be a subset of $G$. Then $H$ is '''closed under inversion''' {{iff}}: :$\\forall h \\in H: h^{-1} \\in H$ That is, {{iff}} the inverse of every element of $H$ is itself in $H$."}
+{"_id": "31295", "title": "Definition:Torus Group", "text": "The '''torus group''' is the direct product of the circle group $C$ with itself: :$T := \\struct {K, \\times} \\times \\struct {K, \\times}$ where: :$K = \\set {z \\in \\C: \\cmod z = 1}$"}
+{"_id": "31296", "title": "Definition:Closed Set/Normed Vector Space/Definition 1", "text": "'''$F$ is closed in $V$''' {{iff}} its complement $X \\setminus F$ is open in $V$."}
+{"_id": "31297", "title": "Definition:Closed Set/Normed Vector Space/Definition 2", "text": "'''$F$ is closed (in $V$)''' {{iff}} every limit point of $F$ is also a point of $F$. That is: {{iff}} $F$ contains all its limit points."}
+{"_id": "31298", "title": "Definition:Simple Graph/Formal Definition", "text": "Let $V$ be a set. Let $\\RR$ be an endorelation on $V$ which is antireflexive and symmetric. Let $E$ be the set whose elements of the form: :$\\set {\\tuple {v_a, v_b}, \\tuple {v_b, v_a} }$. where $\\tuple {v_a, v_b}$ and $\\tuple {v_b, v_a}$ are elements of $\\RR$  A '''simple graph''' is an ordered pair $G = \\struct {V, E}$, where $V$ and $E$ are defined as above. $V$ is called the vertex set. $E$ is called the edge set."}
+{"_id": "31301", "title": "Definition:Directed Graph/Formal Definition", "text": "A '''directed graph''' or '''digraph''' $D$ is a non-empty set $V$ together with an antireflexive relation $E$ on $V$. The elements of $E$ are the arcs."}
+{"_id": "31303", "title": "Definition:Directed Graph/Simple Digraph", "text": "If the relation $E$ in $D$ is also specifically asymmetric, then $D$ is called a '''simple digraph'''. That is, in a '''simple digraph''' there are no pairs of arcs (like there are between $v_1$ and $v_4$ in the diagram above) which go in both directions between two vertices."}
+{"_id": "31304", "title": "Definition:Star Graph", "text": "The '''star graph of order $n$''', denoted $S_n$ is a simple graph with $n$ vertices with the following properties: :One distinguished vertex is of degree $n - 1$ :The remaining vertices are all of degree $1$ and are adjacent only to the distinguished vertex. :700px"}
+{"_id": "31306", "title": "Definition:Loop-Graph/Formal Definition", "text": "A '''loop-graph''' $G$ is a non-empty set $V$ together with a symmetric relation $E$ on $G$. Thus it can be seen that a loop-graph is a simple graph with the stipulation that the relation $E$ does not need to be antireflexive."}
+{"_id": "31307", "title": "Definition:Bounded Normed Vector Space", "text": "Let $M = \\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $M' = \\struct {Y, \\norm {\\, \\cdot \\,}_Y}$ be a normed vector subspace of $M$. === Definition 1 === {{:Definition:Bounded Normed Vector Space/Definition 1}} === Definition 2 === {{:Definition:Bounded Normed Vector Space/Definition 2}}"}
+{"_id": "31308", "title": "Definition:Bounded Normed Vector Space/Definition 1", "text": "'''$M'$ is bounded (in $M$)''' {{iff}}: :$\\exists x \\in X, C \\in \\R: \\forall y \\in Y: \\norm {x - y} \\le C$"}
+{"_id": "31309", "title": "Definition:Bounded Normed Vector Space/Definition 2", "text": "'''$M'$ is bounded (in $M$)''' {{iff}}: :$\\exists \\epsilon \\in \\R_{>0} : \\exists x \\in X : Y \\subseteq \\map {B_\\epsilon^-} x$ where $\\map {B_\\epsilon^-} x$ is a closed ball in $M$."}
+{"_id": "31310", "title": "Definition:Loop-Graph/Loop-Digraph/Formal Definition", "text": "A '''loop-digraph''' $D$ is a non-empty set $V$ together with a relation $E$ on $D$. Thus it can be seen that a '''loop-digraph''' is a directed graph with the stipulation that the relation $E$ does not need to be antireflexive."}
+{"_id": "31311", "title": "Definition:Loop-Graph/Loop-Digraph/Loop", "text": "Such an arc is called a loop."}
+{"_id": "31317", "title": "Definition:Radiometric Dating", "text": "'''Radiometric dating''' is a technique whose purpose is to work out the age $T$ of a physical object $B$. The pieces of information are known: :$(1): \\quad$ The ratio $r_0$ of a radioactive isotope $E_R$ of a chemical element $E$ to its stable isotope $E_S$ in $B$ at the time it came into being :$(2): \\quad$ The ratio $r$ of $E_R$ to $E_S$ at the time now :$(3): \\quad$ The half-life of the radioactive isotope $E_R$ It is known from scientific investigation what $r_0$ is when a physical object is created. It is also known from scientific investigation what the rate of radioactive decay of $E_R$ is. Hence it can be worked out by use of the First-Order Reaction how long it would take for the ratio of $E_R$ to $E_S$ to reach its current ratio $r$."}
+{"_id": "31318", "title": "Definition:Radiometric Dating/Radiocarbon Dating", "text": "'''Radiocarbon dating''' is a specific application of '''radiometric dating''' which is used to determine how long a piece of organic matter has been dead. Because of bombardment by cosmic rays, the ratio of (radioactive) carbon-14 to (stable) carbon-12 in the atmosphere of Earth is fairly constant. This ratio is known. The ratio of carbon-14 to carbon-12 in a living organism is the same as it is in Earth's atmosphere, by biological respiration. However, when the organism dies, it no longer respires, and the carbon in its body stays where it was at the time of its death. As time passes, the carbon-14 decays to nitrogen-14 via the process of beta decay. The half-life of carbon-14 is known to be $5700 \\pm 40$ years."}
+{"_id": "31330", "title": "Definition:P-adic Number/Quotient of Cauchy Sequences in P-adic Norm/Representative", "text": "Each Cauchy sequence $\\sequence {y_n}$ of the left coset $\\sequence {x_n} + \\NN$ is called a '''representative''' of the $p$-adic number $\\sequence {x_n} + \\NN$."}
+{"_id": "31333", "title": "Definition:Real Function/Definition 1", "text": "Let $S \\subseteq \\R$ be a subset of the set of real numbers $\\R$. Suppose that, for each value of the independent variable $x$ of $S$, there exists a corresponding value of the dependent variable $y$. Then the dependent variable $y$ is a '''(real) function''' the independent variable $x$."}
+{"_id": "31334", "title": "Definition:Real Function/Range", "text": "Let $f: S \\to \\R$ be a real function. The range of $f$ is the set of values that the dependent variable can take."}
+{"_id": "31335", "title": "Definition:Real Function/Definition 2", "text": "A '''(real) function''' is correspondence between a domain set $D$ and a set range set $R$ that assigns to each element of $D$ a unique element of $R$."}
+{"_id": "31336", "title": "Definition:Implicit Function/General", "text": "Let: :$f: \\R^{n + 1} \\to \\R, \\tuple {x_1, x_2, \\ldots, x_n, z} \\mapsto \\map f {x_1, x_2, \\ldots, x_n, z}$ be a real-valued function on $\\R^{n + 1}$, where: :$\\tuple {x_1, x_2, \\ldots, x_n} \\in \\R^n, z \\in \\R$ Let a relation between $x_1, x_2, \\ldots, x_n$ and $z$ be expressed in the form: :$\\map f {x_1, x_2, \\ldots, x_n, z} = 0$ defined on some subset $S \\subseteq \\R^n$. If there exists a function $g: S \\to \\R$ such that: :$\\forall \\tuple {x_1, x_2, \\ldots, x_n} \\in S: z = \\map g {x_1, x_2, \\ldots, x_n} \\iff \\map f {x_1, x_2, \\ldots, x_n, z} = 0$ then the relation $\\map f {x_1, x_2, \\ldots, x_n, z} = 0$ defines $z$ as an '''implicitly defined function''' of $x_1, x_2, \\ldots, x_n$."}
+{"_id": "31337", "title": "Definition:Real Function/Two Variables/Substitution for y", "text": "Let $f: S \\times T \\to \\R$ be a (real) function of two variables: :$z = \\map f {x, y}$ Then: :$\\map f {x, a}$ means the real function of $x$ obtained by replacing the independent variable $y$ with $a$. In this context, $a$ can be: :a real constant such that $a \\in T$ :a real function $\\map g x$ whose range is a subset of $T$."}
+{"_id": "31338", "title": "Definition:Bounded Region of Plane", "text": "Let $D \\subseteq \\R^2$ be a subset of the plane. $D$ is '''bounded''' {{iff}} there exists a circle in the plane which completely encloses $D$."}
+{"_id": "31339", "title": "Definition:Semilattice Homomorphism", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be semilattices. Let $\\phi: S \\to T$ be a mapping such that $\\circ$ has the morphism property under $\\phi$. That is, $\\forall a, b \\in S$: :$\\map \\phi {a \\circ b} = \\map \\phi a * \\map \\phi b$ Then $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ is a semilattice homomorphism."}
+{"_id": "31340", "title": "Definition:Subsemilattice", "text": "Let $\\struct {R, \\circ}$ be an algebraic structure with a binary operation. A '''subsemilattice of $\\struct {R, \\circ}$''' is a subset $S$ of $R$ such that $\\struct {S, \\circ_S}$ is a semilattice."}
+{"_id": "31341", "title": "Definition:Atom (Physics)/Diameter", "text": "The diameter of an '''atom''' (considered approximately spherical) is between about $1$ and $5$ angstroms."}
+{"_id": "31342", "title": "Definition:Atom (Physics)/Classical", "text": "The classical model of the '''atom''' is as a hard spherical body which is electrically neutral. In this model, '''atoms''' exert no forces on one another until they are brought into contact, at which point they cannot be brought any closer together, no matter how hard they are pushed to do so."}
+{"_id": "31346", "title": "Definition:Gram-Equivalent", "text": "The '''gram-equivalent''' of a chemical element $E$ in a particular compound is the ratio of the mass of $E$ to another, with the mass of hydrogen taken as $1$."}
+{"_id": "31347", "title": "Definition:Ion", "text": "An '''ion''' is an atom or a fragment of a molecule which has an electric charge on it through having either more or fewer electrons than protons."}
+{"_id": "31348", "title": "Definition:Faraday (Unit of Charge)", "text": "The '''Faraday''' is defined as the amount of electric charge needed to release one gram-equivalent of a chemical element. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''Faraday''' }} {{eqn | r = 96 \\, 485       | c = coulombs per mole: $\\mathrm C \\, \\mathrm {mol}^{-1}$ }} {{end-eqn}} </onlyinclude> === Symbol === {{:Definition:Faraday (Unit of Charge)/Symbol}} {{NamedforDef|Michael Faraday|cat = Faraday}}"}
+{"_id": "31354", "title": "Definition:Water", "text": "'''Water''' is the common term for the chemical substance whose molecules are composed of $2$ atoms of hydrogen and $1$ atom of oxygen. It can be expressed using the chemical formula $\\mathrm H_2 \\mathrm O$."}
+{"_id": "31355", "title": "Definition:Homogeneous (Physics)/Warning", "text": "Just to specify that a body ''is made of the same substance throughout'' is '''not''' an adequate definition of '''homogeneous'''. For example, a column of air in the atmosphere is denser at the bottom than at the top. The fact that it is \"all made of air\" is one thing, but the air at the bottom can be distinguished from that higher up because the densities are different. Thus a column of air is '''not''' '''homogeneous'''."}
+{"_id": "31358", "title": "Definition:Molecular Weight", "text": "Let $S$ be a substance. Let $S$ be homogeneous. The '''molecular weight''' of $S$ is defined as: :$W_S = m_S \\times \\dfrac {12} {m_C}$ where: :$m_S$ is the (arithmetic) mean mass of one molecule of $S$ :$m_C$ is the mass of one molecule of carbon-12. It is understood that the atoms that compose the molecules of $S$ may be of different isotopes according to their relative natural concentrations at the source of wherever $S$ came from. Hence the use of the (arithmetic) mean mass in the definition of $m_S$."}
+{"_id": "31360", "title": "Definition:Avogadro's Number", "text": "'''Avogadro's number''' is defined as the number of particles in a mole. Its value is defined as: :$6 \\cdotp 02214 076 \\times 10^{23}$ and is usually denoted either $N_A$ or $L$. Thus it is the constant of proportion for converting the mass of one mole of a substance to the (arithmetic) mean mass of one molecule of that substance. {{NamedforDef|Lorenzo Romano Amedeo Carlo Avogadro|cat = Avogadro}}"}
+{"_id": "31362", "title": "Definition:Fiber of Truth/Solution", "text": "Let $P: X \\to \\set {\\T, \\F}$ be a propositional function defined on a domain $X$. Let $S = \\set {x \\in X: \\map P x = \\T}$ be the '''fiber of truth (under $P$)'''. Then an element of $S$ is known as a '''solution''' of $P$."}
+{"_id": "31364", "title": "Definition:Partial Derivative/Value at Point", "text": "Let $\\map f {x_1, x_2, \\ldots, x_n}$ be a real function of $n$ variables Let $f_i = \\dfrac {\\partial f} {\\partial x_i}$ be the partial derivative of $f$ {{WRT|Differentiation}} $x_i$. Then the value of $f_i$ at $x = \\tuple {a_1, a_2, \\ldots, a_n}$ can be denoted: :$\\valueat {\\dfrac {\\partial f} {\\partial x_i} } {x_1 \\mathop = a_1, x_2 \\mathop = a_2, \\mathop \\ldots, x_n \\mathop = a_n}$ or: :$\\valueat {\\dfrac {\\partial f} {\\partial x_i} } {a_1, a_2, \\mathop \\ldots, a_n}$ or: :$\\map {f_i} {a_1, a_2, \\mathop \\ldots, a_n}$ and so on. Hence we can express: :$\\map {f_i} {a_1, a_2, \\mathop \\ldots, a_n} = \\valueat {\\dfrac \\partial {\\partial x_i} \\map f {a_1, a_2, \\mathop \\ldots, a_{i - 1}, x_i, a_{i + i}, \\mathop \\ldots, a_n} } {x_i \\mathop = a_i}$ according to what may be needed as appropriate."}
+{"_id": "31365", "title": "Definition:Closure (Topology)/Definition 6", "text": "The '''closure of $H$ (in $T$)''', denoted $H^-$, is the set of all adherent points of $H$."}
+{"_id": "31367", "title": "Definition:Partial Derivative/Higher Derivative/Third Derivative", "text": "Let $u = \\map f {x, y, z}$ be a function of the $3$ independent variables $x$, $y$ and $z$. The following is an example of one of the $3$rd derivatives of $f$: :$\\dfrac {\\partial^3 u} {\\partial z^2 \\partial y} := \\map {\\dfrac \\partial {\\partial z} } {\\dfrac {\\partial^2 u} {\\partial z \\partial y} } =: \\map {f_{2 3 3} } {x, y, z}$"}
+{"_id": "31368", "title": "Definition:Partial Derivative/Higher Derivative/Fourth Derivative", "text": "Let $u = \\map f {x, y, z}$ be a function of the $3$ independent variables $x$, $y$ and $z$. The following is an example of one of the $4$th derivatives of $f$: :$\\dfrac {\\partial^4 u} { \\partial x \\partial y \\partial z^2} := \\map {\\dfrac \\partial {\\partial x} } {\\dfrac {\\partial^3 u} {\\partial y \\partial z^2} } =: \\map {f_{3 3 2 1} } {x, y, z}$"}
+{"_id": "31369", "title": "Definition:Grandi's Series", "text": "'''Grandi's series''' is the divergent series: :$\\displaystyle \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n$ {{NamedforDef|Luigi Guido Grandi|cat = Grandi}} Category:Definitions/Convergence Category:Definitions/Series 2gl0szzqyunwl361wc70zcwwk5gczpy"}
+{"_id": "31370", "title": "Definition:Partial Derivative/Order", "text": "$u = \\map f {x_1, x_2, \\ldots, x_n}$ be a function of the $n$ independent variables $x_1, x_2, \\ldots, x_n$. The '''order''' of a partial derivative of $u$ is the '''number of times it has been (partially) differentiated''' by at least one of $x_1, x_2, \\ldots, x_n$. For example: :a second partial derivative of $u$ is of '''second order''', or '''order $2$''' :a third partial derivative of $u$ is of '''third order''', or '''order $3$''' and so on."}
+{"_id": "31371", "title": "Definition:Event/Occurrence/Impossibility", "text": "Let $A \\in \\Sigma$ be an event of $\\EE$ whose probability of occurring is equal to $0$. Then $A$ is described as '''impossible'''. That is, it is an '''impossibility''' for $A$ to occur."}
+{"_id": "31372", "title": "Definition:Event/Occurrence/Certainty", "text": "Let $A \\in \\Sigma$ be an event of $\\EE$ whose probability of occurring is equal to $1$. Then $A$ is described as '''certain'''. That is, it is a '''certainty''' that $A$ occurs."}
+{"_id": "31373", "title": "Definition:Event/Occurrence/Equality", "text": "Let $A, B \\in \\Sigma$ be events of $\\EE$ such that $A = B$. Then: :the occurrence of $A$ inevitably brings about the occurrence of $B$ and: :the occurrence of $B$ inevitably brings about the occurrence of $A$."}
+{"_id": "31374", "title": "Definition:Event/Occurrence/Intersection", "text": "Let $\\omega \\in A \\cap B$, where $A \\cap B$ denotes the intersection of $A$ and $B$. Then '''both $A$ and $B$ occur'''."}
+{"_id": "31375", "title": "Definition:Event/Occurrence/Union", "text": "Let $\\omega \\in A \\cup B$, where $A \\cup B$ denotes the union of $A$ and $B$. Then '''either $A$ or $B$ occur'''."}
+{"_id": "31376", "title": "Definition:Event/Occurrence/Difference", "text": "Let $\\omega \\in A \\setminus B$, where $A \\setminus B$ denotes the difference of $A$ and $B$. Then '''$A$ occurs but $B$ does not occur'''."}
+{"_id": "31377", "title": "Definition:Event/Occurrence/Symmetric Difference", "text": "Let $\\omega \\in A * B$, where $A * B$ denotes the symmetric difference of $A$ and $B$. Then '''either $A$ occurs or $B$ occurs, but not both'''."}
+{"_id": "31378", "title": "Definition:Uniform Cauchy Criterion", "text": "Let $S \\subseteq \\mathbb R$. Let $\\sequence {f_n}$ be a sequence of real functions $S \\to \\R$.  We say that $\\sequence {f_n}$ satisfies the '''uniform Cauchy criterion''' or is '''uniformly Cauchy''' on $S$ if for all $\\varepsilon \\in \\R_{> 0}$, there exists $N \\in \\N$ such that:  :$\\size {\\map {f_n} x - \\map {f_m} x} < \\varepsilon$ for all $x \\in S$ and $n, m > N$. By Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent, this criterion gives a necessary and sufficient condition for a sequence of real functions to be uniformly convergent."}
+{"_id": "31379", "title": "Definition:Complete Set of Events", "text": "Let $I$ be an indexing set. Let $\\family {A_i}_{i \\mathop \\in I}$ be a family of events in a probability space indexed by $I$. $\\family {A_i}_{i \\mathop \\in I}$ is a '''complete set of events''' {{iff}}: :$\\displaystyle \\map \\Pr {\\bigcup_{i \\mathop \\in I} A_i} = 1$"}
+{"_id": "31381", "title": "Definition:Adherent Point/Definition 3", "text": "A point $x \\in S$ is an '''adherent point of $A$''' {{iff}} every neighborhood $N$ of $x$ satisfies: :$A \\cap N \\ne \\O$"}
+{"_id": "31382", "title": "Definition:Boundary (Topology)/Definition 1", "text": "The '''boundary of $H$''' consists of all the points in the closure of $H$ which are not in the interior of $H$. Thus, the '''boundary of $H$''' is defined as: :$\\partial H := H^- \\setminus H^\\circ$ where $H^-$ denotes the closure and $H^\\circ$ the interior of $H$."}
+{"_id": "31383", "title": "Definition:Boundary (Topology)/Definition 2", "text": "$x \\in S$ is a '''boundary point''' of $H$ if every neighborhood $N$ of $x$ satisfies: :$H \\cap N \\ne \\O$ and :$\\overline H \\cap N \\ne \\O$ where $\\overline H$ is the complement of $H$ in $S$. The '''boundary of $H$''' consists of all the '''boundary points''' of $H$."}
+{"_id": "31385", "title": "Definition:Set Meeting Set", "text": "Let $\\family {S_i}_{i \\mathop \\in I}$ be an family of sets indexed by some  indexing set $I$. The sets in $\\family {S_i}$ are said to '''meet''' {{iff}} their intersection is not empty. That is, {{iff}}: :$\\displaystyle \\bigcap_{i \\mathop \\in I} \\family {S_i} \\ne \\O$ That is, {{iff}} $\\family {S_i}_{i \\mathop \\in I}$ is not disjoint."}
+{"_id": "31386", "title": "Definition:Disjoint Sets/Family", "text": "Let $I$ be an indexing set. Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of sets indexed by $I$. Then $\\family {S_i}_{i \\mathop \\in I}$ is '''disjoint''' {{iff}} their intersection is empty: :$\\displaystyle \\bigcap_{i \\mathop \\in I} S_i = \\O$"}
+{"_id": "31388", "title": "Definition:Negative Vector", "text": "The inverse of a vector $\\mathbf x \\in G$ is usually denoted $-\\mathbf x$, and called the '''negative of $\\mathbf x$'''."}
+{"_id": "31389", "title": "Definition:Vector Subtraction", "text": "Let $\\struct {F, +_F, \\times_F}$ be a field. Let $\\struct {G, +_G}$ be an abelian group. Let $V := \\struct {G, +_G, \\circ}_R$ be the corresponding '''vector space over $F$'''. Let $\\mathbf x$ and $\\mathbf y$ be vectors of $V$. Then the operation of '''(vector) subtraction''' on $\\mathbf x$ and $\\mathbf y$ is defined as: :$\\mathbf x - \\mathbf y := \\mathbf x + \\paren {-\\mathbf y}$ where $-\\mathbf y$ is the negative of $\\mathbf y$. The $+$ on the {{RHS}} is vector addition. === Arrow Representation === {{:Definition:Vector Subtraction/Arrow Representation}}"}
+{"_id": "31390", "title": "Definition:Bounded Variation", "text": "Let $a, b$ be real numbers with $a < b$. Let $f : \\closedint a b \\to \\R$ be a real function.  For each finite subdivision $P$ of $\\closedint a b$, write:  :$P = \\set {x_0, x_1, \\ldots, x_n}$ with: :$a = x_0 < x_1 < x_2 < \\cdots < x_{n - 1} < x_n = b$ Also write: :$\\displaystyle \\map {V_f} P = \\sum_{i \\mathop = 1}^n \\size {\\map f {x_i} - \\map f {x_{i - 1} } }$ We say $f$ is of '''bounded variation''' if there exists a $M \\in \\R$ such that: :$\\map {V_f} P \\le M$ for all finite subdivisions $P$."}
+{"_id": "31392", "title": "Definition:Total Variation", "text": "Let $a, b$ be real numbers with $a < b$. Let $f : \\closedint a b \\to \\R$ be a function of bounded variation. Let $X$ be the set of finite subdivisions of $\\closedint a b$. For each $P \\in X$, write:  :$P = \\set {x_0, x_1, \\ldots, x_{\\size P - 1} }$ with: :$a = x_0 < x_1 < x_2 < \\cdots < x_{\\size P - 2} < x_{\\size P - 1} = b$ Also write: :$\\displaystyle \\map {V_f} P = \\sum_{i \\mathop = 1}^{\\size P - 1} \\size {\\map f {x_i} - \\map f {x_{i - 1} } }$ We define the '''total variation''' $V_f$ of $f$ on $\\closedint a b$ by:  :$\\displaystyle V_f = \\sup_{P \\in X} \\paren {\\map {V_f} P}$ This supremum is finite as, since $f$ is of bounded variation, there exists $M \\in \\R$ with: :$\\map {V_f} P \\le M$ for all $P \\in X$."}
+{"_id": "31393", "title": "Definition:Truncated Real-Valued Function", "text": "Let $f: \\R \\to \\R$ be a real function. Let $m \\in \\Z_{>0}$ be a (positive) integer constant. Let $f_m: \\R \\to \\R$ be the real function defined as: :$\\forall x \\in \\Dom f: \\map {f_m} x = \\begin {cases} \\map f x & : \\map f x \\le m \\\\ m & : \\map f x > m \\end {cases}$ Then $\\map {f_m} x$ is called '''(the function) $f$ truncated by $m$'''."}
+{"_id": "31394", "title": "Definition:Integrable Function/Unbounded Above", "text": "Let $f: \\R \\to \\R$ be a positive real function. Let $f$ be unbounded above on the open interval $\\openint a b$. Let $f_m$ denote the function $f$ truncated by $m$ for $m \\in \\Z_{>0}$. Suppose that $f_m$ is Darboux integrable on $\\openint a b$ for all $m \\in \\Z_{>0}$. Suppose also that the following limit exists: :$\\displaystyle \\lim_{m \\mathop \\to \\infty} \\int_a^b \\map {f_m} x \\rd x$ Then $f$ is '''integrable on $\\openint a b$''' and can be expressed in the conventional notation of the Darboux integral: :$\\displaystyle \\int_a^b \\map f x \\rd x$"}
+{"_id": "31395", "title": "Definition:Integrable Function/Unbounded", "text": "Let $f: \\R \\to \\R$ be a real function. Let $f$ be unbounded on the open interval $\\openint a b$. Let: :$f^+$ denote the positive part of $f$ :$f^-$ denote the negative part of $f$. Let $f^+$ and $-f^-$ both be integrable on $\\openint a b$. Then $f$ is '''integrable on $\\openint a b$ and its (definite) integral is understood to be: :$\\displaystyle \\int_a^b \\map f x \\rd x := \\int_a^b \\map {f^+} x \\rd x - \\int_a^b \\paren {-\\map {f^-} x} \\rd x$"}
+{"_id": "31396", "title": "Definition:Integrable Function/Measure Space", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f \\in \\MM_{\\overline \\R}, f: X \\to \\overline \\R$ be a measurable function. Then $f$ is said to be '''$\\mu$-integrable''' {{iff}}: :$\\displaystyle \\int f^+ \\rd \\mu < +\\infty$ and :$\\displaystyle \\int f^- \\rd \\mu < +\\infty$ where $f^+$, $f^-$ are the positive and negative parts of $f$, respectively. The integral signs denote $\\mu$-integration of positive measurable functions."}
+{"_id": "31399", "title": "Definition:Baire Category", "text": "The '''Baire category''' of a topological space $T$ is a way of specifying how dense $T$ is: === First Category (Meager) === {{:Definition:Meager Space}} === Second Category (Non-Meager) === {{:Definition:Non-Meager Space}} {{NamedforDef|René-Louis Baire|cat = Baire}}"}
+{"_id": "31400", "title": "Definition:Homeomorphism/Terminology", "text": "Let a '''homeomorphism''' exist between $T_\\alpha$ and $T_\\beta$. Then $T_\\alpha$ and $T_\\beta$ are said to be '''homeomorphic'''. The symbolism $T_\\alpha \\sim T_\\beta$ is often seen to denote that $T_\\alpha$ is '''homeomorphic''' to $T_\\beta$."}
+{"_id": "31401", "title": "Definition:Filter Basis/Definition 1", "text": "Let $\\BB \\subset \\powerset S$ such that $\\O \\notin \\BB$ and $\\BB \\ne \\O$. Then $\\FF := \\set {V \\subseteq S: \\exists U \\in \\BB: U \\subseteq V}$ is a filter on $S$ {{iff}}: :$\\forall V_1, V_2 \\in \\BB: \\exists U \\in \\BB: U \\subseteq V_1 \\cap V_2$ Such a $\\BB$ is called a '''filter basis''' of $\\FF$."}
+{"_id": "31402", "title": "Definition:Filter Basis/Generated Filter", "text": "$\\FF$ is said to be '''generated by $\\BB$'''."}
+{"_id": "31403", "title": "Definition:Filter Basis/Definition 2", "text": "Let $\\BB$ be a subset of a filter $\\FF$ on $S$ such that $\\BB \\ne \\O$. Then $\\BB$ is a '''filter basis''' of $\\FF$ {{iff}}: :$\\forall U \\in \\FF: \\exists V \\in \\BB: V \\subseteq U$"}
+{"_id": "31404", "title": "Definition:Fleenor-Heronian Triangle", "text": "A '''Fleenor-Heronian triangle''' is a '''Heronian triangle''' whose sides have lengths form a set of $3$ consecutive integers."}
+{"_id": "31405", "title": "Definition:Fleenor-Heronian Triangle/Sequence", "text": "The sequence of Fleenor-Heronian triangles begins: :$\\paren {1, 2, 3}$ :$\\paren {3, 4, 5}$ :$\\paren {13, 14, 15}$ :$\\paren {51, 52, 53}$ :$\\paren {193, 194, 195}$ :$\\paren {723, 724, 725}$ :$\\paren {2701, 2702, 2703}$ :$\\paren {10 \\, 083, 10 \\, 084, 10 \\, 085}$ :$\\paren {37 \\, 633, 37 \\, 634, 37 \\, 635}$"}
+{"_id": "31413", "title": "Definition:Countable Topological Space", "text": "A '''countable topological space''' is a '''topological space''' whose underlying set is '''countably infinite'''."}
+{"_id": "31414", "title": "Definition:Uncountable Topological Space", "text": "An '''uncountable topological space''' is a '''topological space''' whose underlying set is '''uncountable'''."}
+{"_id": "31416", "title": "Definition:Finite Topological Space", "text": "A '''finite topological space''' is a '''topological space''' whose underlying set is '''finite'''."}
+{"_id": "31418", "title": "Definition:Uncountable Discrete Ordinal Space", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\hointr 0 \\Omega$ denote the open ordinal space on $\\Omega$. Let $S$ be the set of points of $\\hointr 0 \\Omega$ of the form $\\alpha + 1$, where $\\alpha$ is a limit ordinal. Let $\\tau$ be the subspace topology induced by the order topology on $\\hointr 0 \\Omega$. $\\struct {S, \\tau}$ is known as the '''uncountable discrete ordinal space'''."}
+{"_id": "31419", "title": "Definition:Long Line", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\hointr 0 \\Omega$ denote the open ordinal space on $\\Omega$. Let $L$ be the set constructed as follows. Between each ordinal $\\alpha \\in \\hointr 0 \\Omega$ and its successor $\\alpha + 1$, let a copy of the open (real) unit interval $\\openint 0 1$ be inserted. Let a total ordering $\\preccurlyeq$ be applied to $L$ according to the betweenness described above. Let the order topology $\\tau$ be applied to the ordered structure $\\struct {L, \\preccurlyeq}$. The resulting topological space $\\struct {L, \\preccurlyeq, \\tau}$ is called the '''long line'''. Informally it can be seen that $L$ is of the form: :$0, \\openint 0 1, 1, \\openint 0 1, 2, \\openint 0 1, \\ldots, \\openint 0 1, \\alpha, \\openint 0 1, \\alpha + 1, \\openint 0 1, \\ldots, \\openint 0 1, \\Omega - 1, \\openint 0 1$"}
+{"_id": "31420", "title": "Definition:Extended Long Line", "text": "Let $\\Omega$ denote the first uncountable ordinal. Let $\\closedint 0 \\Omega$ denote the open ordinal space on $\\Omega$. Let $L^*$ be the set constructed as follows. Between each ordinal $\\alpha \\in \\closedint 0 \\Omega$ and its successor $\\alpha + 1$, let a copy of the open (real) unit interval $\\openint 0 1$ be inserted. Let a total ordering $\\preccurlyeq$ be applied to $L^*$ according to the betweenness described above. Let the order topology $\\tau$ be applied to the ordered structure $\\struct {L^*, \\preccurlyeq}$. The resulting topological space $\\struct {L^*, \\preccurlyeq, \\tau}$ is called the '''extended long line'''. Informally it can be seen that $L$ is of the form: :$0, \\openint 0 1, 1, \\openint 0 1, 2, \\openint 0 1, \\ldots, \\openint 0 1, \\alpha, \\openint 0 1, \\alpha + 1, \\openint 0 1, \\ldots, \\openint 0 1, \\Omega - 1, \\openint 0 1, \\Omega$"}
+{"_id": "31421", "title": "Definition:Altered Long Line", "text": "Let $\\struct {L, \\preccurlyeq, \\tau}$ denote the '''long line'''. Let $p$ be a point which is not an element of $L$. Consider the set $S := L \\cup \\set p$. Let $T = \\struct {S, \\tau_p}$ be the topological space whose topology $\\tau_p$ is defined as follows: The open sets of $T$ are the open sets of $L$ together with those generated by the following neighborhoods of $P$: :$\\map {U_s} p := \\set p \\cup \\set {\\displaystyle \\bigcup_{\\alpha \\mathop = \\beta}^\\Omega \\openint \\alpha {\\alpha + 1}: 1 \\le \\beta < \\Omega}$ where $\\Omega$ denotes the first uncountable ordinal. $\\map {U_s} p$ is therefore a right-hand ray without the ordinals. The topological space $T = \\struct {S, \\tau_p}$ is known as the '''altered long line'''."}
+{"_id": "31422", "title": "Definition:Altered Long Line/Greatest Element", "text": "$p$ is the greatest element of $S = L \\cup \\set p$."}
+{"_id": "31423", "title": "Definition:Unit Square under Lexicographic Ordering", "text": "Let $S$ be the unit square in the (real) Cartesian plane: {{begin-eqn}} {{eqn | l = S       | r = \\set {\\tuple {x, y}: x, y \\in \\R, 0 \\le x \\le 1, 0 \\le y \\le 1}       | c =  }} {{eqn | r = \\closedint 0 1 \\times \\closedint 0 1       | c = where $\\closedint 0 1$ is the closed unit interval }} {{end-eqn}} Let $\\preccurlyeq$ be the lexicographical ordering applied to $S$: :$\\forall \\tuple {x_1, y_1}, \\tuple {x_2, y_2} \\in S: \\tuple {x_1, y_1} \\preccurlyeq \\tuple {x_2, y_2} \\iff \\begin {cases} x_1 < x_2 \\\\ x_1 = x_2 \\land y_1 \\le y_2 \\end {cases}$ Let $\\tau$ be the order topology be applied to the ordered structure $\\struct {S, \\preccurlyeq}$. Then the topological space $\\struct {S, \\preccurlyeq, \\tau}$ is known as the '''unit square under lexicographical ordering'''."}
+{"_id": "31425", "title": "Definition:Right Order Topology", "text": "Let $\\struct {S, \\preccurlyeq}$ be a totally ordered set. Let $\\tau$ be the topology on $S$ generated by the basis sets of the form: :$S_a = \\set {x: a \\prec x}$ for $a \\in S$. Then the topological space $\\struct {S, \\preccurlyeq, \\tau}$ is known as the '''right order topology on $S$'''."}
+{"_id": "31426", "title": "Definition:Left Order Topology", "text": "Let $\\struct {S, \\preccurlyeq}$ be a totally ordered set. Let $\\tau$ be the topology on $S$ generated by the basis sets of the form: :$S_a = \\set {x: x \\prec a}$ for $a \\in S$. Then the topological space $\\struct {S, \\preccurlyeq, \\tau}$ is known as the '''left order topology on $S$'''."}
+{"_id": "31429", "title": "Definition:Nested Interval Topology", "text": "Let $S = \\openint 0 1$ denote the open real interval: :$\\openint 0 1 = \\set {x \\in \\R: 0 < x < 1}$ Let $\\tau$ be the topology defined on $\\openint 0 1$ by defining the open sets $U_n$ as: :$\\forall n \\in \\N_{>0}: U_n := \\openint 0 {1 - \\dfrac 1 n}$ together with $\\O$ and $S$ itself. Then $\\tau$ is referred to as the '''nested interval topology'''. The topological space $T = \\struct {S, \\tau}$ is referred to as the '''nested interval space'''."}
+{"_id": "31430", "title": "Definition:Overlapping Interval Topology", "text": "Let $S = \\closedint {-1} 1$ denote the open real interval: :$\\closedint {-1} 1 = \\set {x \\in \\R: -1 \\le x \\le 1}$ Let $\\BB$ be the set: :$\\BB = \\set {\\hointl a 1: -1 < a < 0} \\cup \\set {\\hointr {-1} b: 0 < b < 1}$ where: :$\\hointl a 1$ is the half-open interval $\\set {x \\in \\R: a < x \\le 1}$. :$\\hointr {-1} b$ is the half-open interval $\\set {x \\in \\R: -1 \\le x < b}$. Then $\\BB$ is the basis for a topology $\\tau$ on $\\R$. Thus the sets of the form $\\openint a b$ such that $a < 0 < b$ are open sets in $S$. $\\tau$ is referred to as the '''overlapping interval topology'''. The topological space $T = \\struct {S, \\tau}$ is referred to as the '''overlapping interval space'''."}
+{"_id": "31432", "title": "Definition:Hjalmar Ekdal Topology", "text": "Let $\\tau$ be the topology defined on the (strictly) positive integers $\\Z_{>0}$ as follows: A subset $H$ of $\\Z_{>0}$ is in $\\tau$ {{iff}} $H$ contains the (immediate) successor of every odd integer that is also in $H$: :$\\tau := \\set {H \\subseteq \\Z_{>0}: 2 n - 1 \\in H \\implies 2 n \\in H}$ Then $\\tau$ is referred to as the '''Hjalmar Ekdal topology'''. The topological space $T = \\struct {\\Z_{>0}, \\tau}$ is referred to as the '''Hjalmar Ekdal space'''."}
+{"_id": "31433", "title": "Definition:Prime Ideal Topology", "text": "Let $S$ be the set of all prime ideals $P$ of the integers $\\Z$. Let $\\BB$ be the set of all sets $V_x$ defined as: :$V_x = \\set {P \\in S: x \\notin P}$ for all $x \\in \\Z_{>0}$. Then $\\BB$ is the basis for a topology $\\tau$ on $S$. $\\tau$ is referred to as the '''prime ideal topology'''. The topological space $T = \\struct {S, \\tau}$ is referred to as the '''prime ideal space'''."}
+{"_id": "31434", "title": "Definition:Divisor Topology", "text": "Let $S = \\set {x \\in \\Z: x \\ge 2}$ denote the set of integers greater than $1$. Let $\\BB$ be the set of all sets $U_n$ defined for all $n \\ge 2$ as: :$U_n = \\set {x \\in \\Z_{>0}: x \\divides n}$ where $\\divides$ denotes the divisor relation. Then $\\BB$ is the basis for a topology $\\tau$ on $S$. Then $\\tau$ is referred to as the '''divisor topology'''. The topological space $T = \\struct {S, \\tau}$ is referred to as the '''divisor space'''."}
+{"_id": "31435", "title": "Definition:Evenly Spaced Integer Topology", "text": "Let $\\Z$ denote the set of integers. Let $\\BB$ be the set of sets defined as: :$\\BB = \\set {a + k \\Z: a, k \\in \\Z, k \\ne 0}$ where $a + k \\Z := \\set {a + k \\lambda: \\lambda \\in \\Z}$. Then $\\BB$ is the basis for a topology $\\tau$ on $S$. Then $\\tau$ is referred to as the '''evenly spaced integer topology'''. The topological space $T = \\struct {S, \\tau}$ is referred to as the '''evenly spaced integer space'''."}
+{"_id": "31436", "title": "Definition:P-adic Topology", "text": "Let $\\Z$ denote the set of integers. Let $p \\in \\Z$ be a fixed prime number. Let $\\BB$ be the set of all sets $\\map {U_\\alpha} n$ defined as: :$\\map {U_\\alpha} n = \\set {n + \\lambda p^\\alpha: \\lambda \\in \\Z}$ Then $\\BB$ is the basis for a topology $\\tau$ on $S$. $\\tau$ is referred to as the '''$p$-adic topology (on $\\Z$)'''. The topological space $T = \\struct {S, \\tau}$ is referred to as the '''$p$-adic (topological) space'''."}
+{"_id": "31437", "title": "Definition:Relatively Prime Integer Topology", "text": "Let $\\Z_{>0}$ denote the set of (strictly) positive integers. Let $\\BB$ be the set of sets $\\set {\\map {U_a} b: a, b \\in \\Z_{>0} }$ where: :$\\map {U_a} b = \\set {b + n a \\in \\Z_{>0}: \\gcd \\set {a, b} = 1}$ where $\\gcd \\set {a, b}$ denotes the greatest common divisor of $a$ and $b$. Then $\\BB$ is the basis for a topology $\\tau$ on $\\Z_{>0}$. $\\tau$ is then referred to as the '''relatively prime integer topology'''. The topological space $T = \\struct {\\Z_{>0}, \\tau}$ is referred to as the '''relatively prime integer space'''."}
+{"_id": "31441", "title": "Definition:Smirnov's Deleted Sequence Topology", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $A$ denote the set defined as: :$A := \\set {\\dfrac 1 n: n \\in \\Z_{>0} }$ Let $\\tau$ be the topology defined as: :$\\tau = \\set {H: \\exists U \\in \\tau_d, B \\subseteq A: H = U \\setminus B}$ That is, $\\tau$ consists of the open sets of $\\struct {\\R, \\tau_d}$ which have had any number of the set of the reciprocals of the positive integers removed. $\\tau$ is then referred to as '''Smirnov's deleted sequence topology on $\\R$'''."}
+{"_id": "31444", "title": "Definition:Indiscrete Extension of Reals", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $D$ be an everywhere dense subset of $\\struct {\\R, \\tau_d}$ with an everywhere dense complement in $\\R$. Let $\\BB$ be the set of sets: :$\\BB := \\set {H: \\exists U \\in \\tau_d: H = U \\cap D}$ Let $\\tau^*$ be the topology generated from $\\tau_d$ by the addition of all sets of $\\BB$. :$\\tau^* = \\tau_d \\cup \\BB$ $\\tau^*$ is then referred to as an '''indiscrete extension of $\\R$'''."}
+{"_id": "31445", "title": "Definition:Pointed Extension of Reals", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $D$ be an everywhere dense subset of $\\struct {\\R, \\tau_d}$ with an everywhere dense complement in $\\R$. Let $\\BB$ be the set of sets defined as: :$\\BB = \\set {\\set x \\cup \\paren {U \\cap D}: x \\in U \\in \\tau_d}$ Let $\\tau'$ be the topology generated from $\\BB$. $\\tau'$ is referred to as a '''pointed extension of $\\R$'''."}
+{"_id": "31446", "title": "Definition:Indiscrete Extension of Reals/Rational", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\\Q$ denote the set of rational numbers. Let $\\tau^*$ be the indiscrete extension of $\\struct {\\R, \\tau_d}$: :$\\tau^* = \\tau_d \\cup \\set {H: \\exists U \\in \\tau_d: H = U \\cap \\Q}$ $\\tau^*$ is then referred to as the '''indiscrete rational extension of $\\R$'''."}
+{"_id": "31448", "title": "Definition:Pointed Extension of Reals/Irrational", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\\Bbb I := \\R \\setminus \\Q$ denote the set of irrational numbers. Let $\\BB$ be the set of sets defined as: :$\\BB = \\set {\\set x \\cup \\paren {U \\cap \\Bbb I}: x \\in U \\in \\tau_d}$ Let $\\tau'$ be the topology generated from $\\BB$. $\\tau'$ is referred to as '''pointed irrational extension of $\\R$'''."}
+{"_id": "31449", "title": "Definition:Pointed Extension of Reals/Rational", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\\Q$ denote the set of rational numbers. Let $\\BB$ be the set of sets defined as: :$\\BB = \\set {\\set x \\cup \\paren {U \\cap \\Q}: x \\in U \\in \\tau_d}$ Let $\\tau'$ be the topology generated from $\\BB$. $\\tau'$ is referred to as the '''pointed rational extension of $\\R$'''."}
+{"_id": "31450", "title": "Definition:Discrete Extension of Reals", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $D$ be an everywhere dense subset of $\\struct {\\R, \\tau_d}$ with an everywhere dense complement in $\\R$. Let $\\BB$ be the set of sets defined as: :$\\BB = \\tau_d \\cup \\set {\\set x: x \\in D}$ Let $\\tau*$ be the topology generated from $\\BB$. $\\tau^*$ is referred to as a '''discrete extension of $\\R$'''."}
+{"_id": "31451", "title": "Definition:Discrete Extension of Reals/Rational", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\\Q$ denote the set of rational numbers. Let $\\BB$ be the set of sets defined as: :$\\BB = \\tau_d \\cup \\set {\\set x: x \\in \\Q}$ Let $\\tau*$ be the topology generated from $\\BB$. $\\tau^*$ is referred to as the '''discrete rational extension of $\\R$'''."}
+{"_id": "31452", "title": "Definition:Discrete Extension of Reals/Irrational", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\\Bbb I := \\R \\setminus \\Q$ denote the set of irrational numbers. Let $\\BB$ be the set of sets defined as: :$\\BB = \\tau_d \\cup \\set {\\set x: x \\in \\Bbb I}$ Let $\\tau*$ be the topology generated from $\\BB$. $\\tau^*$ is referred to as the '''discrete irrational extension of $\\R$'''."}
+{"_id": "31455", "title": "Definition:Telophase Topology", "text": "Let $S = \\closedint 0 1 \\cup \\set {1^*}$ where: :$\\closedint 0 1$ is the closed unit interval $\\set {x \\in \\R: 0 \\le x \\le 1}$ :$1^*$ is a second right hand endpoint of $\\closedint 0 1$. Let $\\BB$ be a local basis defined as: :$\\BB = \\set {\\openint a 1 \\cup \\set {1^*}: a \\in \\closedint 0 1}$ Let $\\tau$ be the topology generated from $\\BB$. $\\tau$ is referred to as the '''telophase topology'''."}
+{"_id": "31457", "title": "Definition:Simple Matroid", "text": "Let $M = \\struct{S, \\mathscr I}$ be a matroid. $M$ is called a '''simple matroid''' if it has no loops or parallel elements."}
+{"_id": "31460", "title": "Definition:Irrational Slope Topology", "text": "Let $S = \\set {\\tuple {x, y}: y \\ge 0, x, y \\in \\Q}$. Let $\\theta \\in \\R \\setminus \\Q$ be some fixed irrational number. :500pxrightthumb$\\epsilon$-neighborhood $\\map {N_\\epsilon} {x, y}$: intervals are of [[Definition:Length of Real Intervallength $\\epsilon$]] Let $\\BB$ be the set of $\\epsilon$-neighborhoods of the form: :$\\map {N_\\epsilon} {x, y} = \\set {\\tuple {x, y} } \\cup \\map {B_\\epsilon} {x + \\dfrac y \\theta} \\cup \\map {B_\\epsilon} {x - \\dfrac y \\theta}$ where: :$\\map {B_\\epsilon} \\zeta := \\set {r \\in \\Q: \\size {r - \\zeta} < \\epsilon}$ that is, the open $\\epsilon$-ball at $\\zeta$ in $\\Q^2$. Hence each $\\map {N_\\epsilon} {x, y}$ consists of: :the singleton $\\set {\\tuple {x, y} }$ together with: :two open intervals of length $\\epsilon$ in the rational numbers whose midpoints are at the irrational points $x \\pm \\dfrac y \\theta$ The lines joining $\\tuple {x, y}$ to $\\tuple {x \\pm \\dfrac y \\theta, 0}$ have slope $\\pm \\theta$ and are not part of $\\map {N_\\epsilon} {x, y}$. Let $\\tau$ be the topology generated from $\\BB$. $\\tau$ is referred to as the '''irrational slope topology'''."}
+{"_id": "31461", "title": "Definition:Deleted Diameter Topology", "text": "Let $\\struct {\\R^2, \\tau_d}$ be the real number plane with the usual (Euclidean) topology induced by the Euclidean metric $d$. Let $\\BB$ be the sub-basis of $\\tau_d$ defined as the set of all open balls of $\\struct {\\R^2, d}$ with the horizontal diameters apart from the centers excluded. Let $\\sigma$ be the topology generated from $\\BB$. $\\sigma$ is referred to as the '''deleted diameter topology'''."}
+{"_id": "31462", "title": "Definition:Deleted Radius Topology", "text": "Let $\\struct {\\R^2, \\tau_d}$ be the real number plane with the usual (Euclidean) topology induced by the Euclidean metric $d$. Let $\\BB$ be the sub-basis of $\\tau_d$ defined as the set of all open balls of $\\struct {\\R^2, d}$ with the horizontal radii to the right of the centers excluded. Let $\\tau$ be the topology generated from $\\BB$. $\\tau$ is referred to as the '''deleted radius topology'''."}
+{"_id": "31465", "title": "Definition:Half-Plane/Left", "text": "=== Open Left Half-Plane === {{:Definition:Half-Plane/Open/Left}} === Closed Left Half-Plane === {{:Definition:Half-Plane/Closed/Left}}"}
+{"_id": "31466", "title": "Definition:Half-Plane/Right", "text": "=== Open Right Half-Plane === {{:Definition:Half-Plane/Open/Right}} === Closed Right Half-Plane === {{:Definition:Half-Plane/Closed/Right}}"}
+{"_id": "31467", "title": "Definition:Half-Plane/Upper", "text": "=== Open Upper Half-Plane === {{:Definition:Half-Plane/Open/Upper}} === Closed Upper Half-Plane === {{:Definition:Half-Plane/Closed/Upper}}"}
+{"_id": "31468", "title": "Definition:Half-Plane/Lower", "text": "=== Open Lower Half-Plane === {{:Definition:Half-Plane/Open/Lower}} === Closed Lower Half-Plane === {{:Definition:Half-Plane/Closed/Lower}}"}
+{"_id": "31469", "title": "Definition:Half-Plane/Open/Left", "text": "The '''open left half-plane''' $H_{\\text {OL} }$ is the area of $P$ on the left of $L$. That is, where $x < 0$: :$H_{\\text {OL} } := \\set {\\tuple {x, y}: x \\in \\R_{<0} }$"}
+{"_id": "31470", "title": "Definition:Half-Plane/Closed/Left", "text": "The '''closed left half-plane''' $H_{\\text {CL} }$ is the area of $P$ on the left of and including $L$. That is, where $x \\le 0$: :$H_{\\text {CL} } := \\set {\\tuple {x, y}: x \\in \\R_{\\le 0} }$"}
+{"_id": "31471", "title": "Definition:Half-Plane/Closed/Right", "text": "The '''closed right half-plane''' $H_{\\text {CR} }$ is the area of $P$ on the right of and including $L$. That is, where $x \\ge 0$: :$H_{\\text {CR} } := \\set {\\tuple {x, y}: x \\in \\R_{\\ge 0} }$"}
+{"_id": "31472", "title": "Definition:Half-Plane/Closed/Upper", "text": "The '''closed upper half-plane''' $H_{\\text {CU} }$ is the area of $P$ above and including $L$. That is, where $y \\ge 0$: :$H_{\\text {CU} } := \\set {\\tuple {x, y}: y \\in \\R_{\\ge 0} }$"}
+{"_id": "31473", "title": "Definition:Half-Plane/Closed/Lower", "text": "The '''closed lower half-plane''' $H_{\\text {CL} }$ is the area of $P$ below and including $L$. That is, where $y \\le 0$: :$H_{\\text {CL} } := \\set {\\tuple {x, y}: y \\in \\R_{\\le 0} }$"}
+{"_id": "31474", "title": "Definition:Half-Plane/Open/Lower", "text": "The '''open lower half-plane''' $H_{\\text {OL} }$ is the area of $P$ below $L$. That is, where $y < 0$: :$H_{\\text {OL} } := \\set {\\tuple {x, y}: y \\in \\R_{< 0} }$"}
+{"_id": "31475", "title": "Definition:Half-Plane/Open/Upper", "text": "The '''open upper half-plane''' $H_{\\text {OU} }$ is the area of $P$ above $L$. That is, where $y > 0$: :$H_{\\text {OU} } := \\set {\\tuple {x, y}: y \\in \\R_{> 0} }$"}
+{"_id": "31476", "title": "Definition:Half-Plane/Open/Right", "text": "The '''open right half-plane''' $H_{\\text {OR} }$ is the area of $P$ on the right of $L$. That is, where $x > 0$: :$H_{\\text {OR} } := \\set {\\tuple {x, y}: x \\in \\R_{>0} }$"}
+{"_id": "31477", "title": "Definition:Half-Disc Topology", "text": "Let $P = \\set {\\tuple {x, y}: x \\in \\R, y \\in \\R_{>0} }$ be the open upper half-plane.  Let $\\struct {P, \\tau_d}$ be the open upper half-plane with the Euclidean topology. Let $L$ denote the $x$-axis Let $\\BB$ be the set of sets of the form: :$\\set x \\cup \\paren {U \\cap P}$ where: :$x \\in L$ :$U$ is a Euclidean neighborhood of $x$. Let $\\tau^*$ be the topology generated from $\\BB$. $\\tau^*$ is referred to as the '''half-disc topology'''."}
+{"_id": "31478", "title": "Definition:Irregular Lattice Topology", "text": "Let $S$ be the subset of the lattice points of the Cartesian plane defined as: :$S := \\set {\\tuple {i, j}: i, j \\in \\Z_{>0} } \\cup \\set {\\tuple {i, 0}: i \\in \\Z_{\\ge 0} }$ Let each point of $S$ be defined as being open. Let each point of $S$ of the form $\\tuple {i, 0}$ such that $i \\ne 0$ have as a local basis sets $U_n$ of the form: :$\\map {U_n} {i, 0} := \\set {\\tuple {i, k}: k = 0 \\text { or } k \\ge n}$ Let the point $\\tuple {0, 0}$ have as a local basis a set $V_n$ of the form: :$V_n := \\set {\\tuple {i, k}: i = k = 0 \\text { or } i, k \\ge n}$ Let $\\tau$ be the topology generated from all these $U_n$ and $V_n$. $\\tau$ is referred to as the '''irregular lattice topology'''."}
+{"_id": "31479", "title": "Definition:Lattice Point/Rational", "text": "$P$ is a '''rational lattice point''' of $\\CC$ {{iff}} $a_1, a_2, \\ldots, a_n$ are all rational numbers."}
+{"_id": "31480", "title": "Definition:Arens Square", "text": "Let $A$ be the subset of the unit square defined as: :$A := \\set {\\tuple {i, j}: 0 < i < 1, 0 < j < 1, i, j \\in \\Q} \\setminus \\set {\\tuple {i, j}: i = \\dfrac 1 2}$ That is: :$A := \\openint 0 1^2 \\cup \\Q \\times \\Q \\setminus \\set {\\tuple {\\dfrac 1 2}, j: j \\in \\Q}$ That is, the set of rational points in the interior of the unit square from which are excluded the points whose $x$ coordinates are equal to $\\dfrac 1 2$. :rightthumb300pxArens square with example local bases marked Let $S$ be the set defined as: :$S = A \\cup \\set {\\tuple {0, 0} } \\cup \\set {\\tuple {1, 0} } \\cup \\set {\\tuple {\\dfrac 1 2, r \\sqrt 2}: r \\in \\Q, 0 < r \\sqrt 2 < 1}$ Let $\\BB$ be the basis for a topology generated on $S$ be defined by granting: :to each point of $A$ the local basis of open sets inherited by $A$ from the Euclidean topology on the unit square; :to the other points of $S$ the following local bases: {{begin-eqn}} {{eqn | l = \\map {U_n} {0, 0}       | o = :=       | r = \\set {\\tuple {x, y}: 0 < x < \\dfrac 1 4, 0 < y < \\dfrac 1 n} \\cup \\set {\\tuple {0, 0} } }} {{eqn | l = \\map {U_n} {1, 0}       | o = :=       | r = \\set {\\tuple {x, y}: \\dfrac 3 4 < x < 1, 0 < y < \\dfrac 1 n} \\cup \\set {\\tuple {1, 0} } }} {{eqn | l = \\map {U_n} {\\tuple {\\dfrac 1 2, r \\sqrt 2} }       | o = :=       | r = \\set {\\tuple {x, y}: \\dfrac 1 4 < x < \\dfrac 3 4, \\size {y - r \\sqrt 2} < \\dfrac 1 n} }} {{end-eqn}} Let $\\tau$ be the topology generated from $\\BB$. $\\struct {S, \\tau}$ is referred to as the '''Arens square'''."}
+{"_id": "31481", "title": "Definition:Simplified Arens Square", "text": "Let $A$ be the set of points in the interior of the unit square: :$A := \\set {\\tuple {i, j}: 0 < i < 1, 0 < j < 1, i, j \\in \\R} = \\openint 0 1^2$ :rightthumb300pxSimplified Arens square with example local bases marked Let $S$ be the set defined as: :$S = A \\cup \\set {\\tuple {0, 0} } \\cup \\set {\\tuple {1, 0} }$ Let $\\BB$ be the basis for a topology generated on $S$ be defined by granting: :to each point of $A$ the local basis of open sets inherited by $A$ from the Euclidean topology on the unit square; :to the other points of $S$ the following local bases: {{begin-eqn}} {{eqn | l = \\map {U_n} {0, 0}       | o = :=       | r = \\set {\\tuple {x, y}: 0 < x < \\dfrac 1 2, 0 < y < \\dfrac 1 n} \\cup \\set {\\tuple {0, 0} } }} {{eqn | l = \\map {U_m} {1, 0}       | o = :=       | r = \\set {\\tuple {x, y}: \\dfrac 1 2 < x < 1, 0 < y < \\dfrac 1 m} \\cup \\set {\\tuple {1, 0} } }} {{end-eqn}} Let $\\tau$ be the topology generated from $\\BB$. $\\struct {S, \\tau}$ is referred to as the '''simplified Arens square'''."}
+{"_id": "31482", "title": "Definition:Metrizable Tangent Disc Topology", "text": "Let $A$ be a countable subset of the $x$-axis in the Cartesian plane $\\R^2$. Let $P$ denote the open upper half-plane in $\\R^2$. Let $S := A \\cup P$. Let $\\struct {S, \\tau}$ be the topological subspace of the Niemytzki plane. $\\struct {S, \\tau}$ is referred to as the '''metrizable tangent disc topology'''."}
+{"_id": "31483", "title": "Definition:Sorgenfrey's Half-Open Square Topology", "text": "Let $A = \\struct {\\R, \\sigma}$ be a Sorgenfrey line. Let $S = A \\times A$ denote the product space of $A$ with itself, whose topology $\\tau$ is such that: :the neighborhood of a point $\\tuple {p, q}$ is a rectangle in $\\R^2$ of the form: ::$\\set {\\set {x, y}: p \\le x < p + \\epsilon_1, q \\le y < q + \\epsilon_2}$ :for some $\\epsilon_1, \\epsilon_2 \\in \\R_{>0}$. $\\struct {S, \\tau}$ is referred to as '''Sorgenfrey's half-open square topology'''."}
+{"_id": "31484", "title": "Definition:Michael's Product Topology", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\\Bbb I := \\R \\setminus \\Q$ denote the set of irrational numbers. Let $\\struct {S, \\sigma} := \\struct {\\R, \\tau^*} \\times \\struct {\\Bbb I, \\tau'}$, where: :$\\tau^*$ is the discrete irrational extension of $\\tau_d$ by $\\Bbb I$ :$\\tau'$ is the subspace topology on $\\Bbb I$ induced by $\\tau_d$. $\\struct {S, \\sigma}$ is referred to as '''Michael's product topology'''."}
+{"_id": "31486", "title": "Definition:Tychonoff Plank", "text": "Let $\\omega$ be the first transfinite ordinal. Let $\\Omega$ be the first uncountable ordinal. Let $\\closedint 0 \\Omega$ and $\\closedint 0 \\omega$ be closed ordinal spaces which have both been given the interval topology. The '''Tychonoff plank''' is the topological space defined as: :$T = \\closedint 0 \\Omega \\times \\closedint 0 \\omega$"}
+{"_id": "31487", "title": "Definition:Tychonoff Plank/Deleted", "text": "Let $S = \\closedint 0 \\Omega$ and $\\closedint 0 \\omega$ be closed ordinal spaces which have both been given the interval topology. Hence let $T = \\struct {S, \\tau}$ denote the Tychonoff plank. The '''deleted Tychonoff plank''' is the topological subspace defined as: :$T_\\infty = \\struct {S \\setminus \\set {\\tuple {\\Omega, \\omega} }, \\tau}$"}
+{"_id": "31488", "title": "Definition:Alexandroff Plank", "text": ":rightthumb600px Let $\\Omega$ be the first uncountable ordinal. Let the closed ordinal space $\\closedint 0 \\Omega$ be given the interval topology. Let the closed real interval $\\closedint {-1} 1$ be given the interval topology. Let $\\struct {S, \\tau}$ be the product space of $\\closedint 0 \\Omega$ with $\\closedint {-1} 1$ Consider the point $p := \\tuple {\\Omega, 0} \\in S$ Let $\\sigma$ be the expansion of $\\tau$ generated by adding to $\\tau$ the sets of the form: :$\\map U {\\alpha, n} := \\set p \\cup \\hointl \\alpha \\Omega \\times \\openint 0 {\\dfrac 1 n}$ Then the topological space $T = \\struct {S, \\sigma}$ is known as the '''Alexandroff plank'''."}
+{"_id": "31489", "title": "Definition:Dieudonné Plank", "text": ":rightthumb600px Let $\\omega$ be the first transfinite ordinal. Let $\\Omega$ be the first uncountable ordinal. Let $S = \\paren {\\closedint 0 \\Omega \\times \\closedint 0 \\omega} \\setminus \\set {\\tuple {\\Omega, \\omega} }$ be the underlying set of the deleted Tychonoff plank. Let the topology $\\tau$ be generated by declaring as open: :each point of $\\hointr 0 \\Omega \\times \\hointr 0 \\omega$ together with the sets defined as: {{begin-eqn}} {{eqn | l = \\map {U_\\alpha} \\beta       | r = \\set {\\tuple {\\beta, \\gamma}: \\alpha < \\gamma \\le \\omega} }} {{eqn | l = \\map {V_\\alpha} \\beta       | r = \\set {\\tuple {\\gamma, \\beta}: \\alpha < \\gamma \\le \\Omega} }} {{end-eqn}} The topological space $\\struct {S, \\tau}$ is known as the '''Dieudonné plank'''."}
+{"_id": "31490", "title": "Definition:Tychonoff Corkscrew", "text": "Let $\\omega$ be the first transfinite ordinal. Let $\\Omega$ be the first uncountable ordinal. For each ordinal $\\alpha$, let $A_\\alpha$ denote the totally ordered set defined as: :$A_\\alpha := \\set {-0, -1, -2, \\ldots, \\alpha, \\ldots, 2, 1, 0}$ with the order topology. {{explain|What is the meaning of the above ordering?}} :thumbright600pxOne instance of the plane $P^*$ Let $P$ be the product space defined as: :$P := A_\\Omega \\times A_\\omega$ Let $P^*$ be the topological subspace of $P$: :$P^* := P \\setminus \\set {\\tuple {\\Omega, \\omega} }$ Consider an infinite stack of copies of $P^*$. Let a rectangular corkscrew lattice $C$ be formed such that it spirals in both directions by: :$(1): \\quad$ cutting each $P^*$ immediately below the positive $A_\\Omega$ axis $\\tuple {\\Omega, \\omega}$ to $\\tuple {0, \\omega}$ :$(2): \\quad$ joining the fourth quadrant of one $P^*$ to the first quadrant of the $P^*$ immediately below it, along the positive $A_\\Omega$ axis. Let $\\sequence {\\map { {A_\\Omega}^+} i}_{i \\mathop = -\\infty}^{+ \\infty}$ be the family of positive $A_\\Omega$ axes. Then $i$ will be referred to as the '''level''' of $A_\\Omega$. By convention, the points of $C$ which lie above $\\map { {A_\\Omega}^+} i$ are at a level '''greater''' than $i$. If $x$ is such a point, this is denoted $\\map L x > i$. We create $S$ by adding to $C$ two distinguished points $a^+$ and $a^-$ which can be considered as being infinity points at the top and bottom of the axis of the corkscrew. Basis neighborhoods of $a^+$ consist of all points of $S$ which lie above a certain level. Basis neighborhoods of $a^-$ consist of all points of $S$ which lie below a certain level. The topological space $\\struct {S, \\tau}$ so generated is referred to as the '''Tychonoff corkscrew'''."}
+{"_id": "31491", "title": "Definition:Tychonoff Corkscrew/Deleted", "text": "Let $T = \\struct {S, \\tau}$ denote the Tychonoff corkscrew. Let $a^-$ denote the distinguished point of $S$ considered as being the infinity point at the bottom of $T$. The '''deleted Tychonoff corkscrew''' is the topological subspace defined as: :$T_\\infty = \\struct {S \\setminus \\set {a^-}, \\tau}$"}
+{"_id": "31494", "title": "Definition:Thomas's Plank/Definition 1", "text": "Let $L_n$ be lines embedded in the Cartesian plane $\\R^2$ defined as: :$\\forall n \\in \\N: L_n = \\begin {cases} \\set {\\tuple {x, 0}: x \\in \\openint 0 1} & : n = 0 \\\\ \\set {\\tuple {x, \\dfrac 1 n}: x \\in \\hointr 0 1} & : n > 0 \\end {cases}$ Let $S = \\displaystyle \\bigcup {n \\mathop \\in \\N}$. Let a topology $\\tau$ be applied to $S$, defined as follows: :For $n \\ge 1$, each point of $L_n$ except for $\\tuple {0, \\dfrac 1 n}$ is open. :neighborhood bases of $\\tuple {0, \\dfrac 1 n}$ are subsets of $L_n$ with finite complements. :neighborhood bases of $\\tuple {x, 0}$ are the sets $\\map {U_i} {x, 0}$ defined as: ::$\\map {U_i} {x, 0} := \\set {\\tuple{x, \\dfrac 1 n}: n > i}$ '''Thomas's plank''' is the topological space $\\struct {S, \\tau}$."}
+{"_id": "31496", "title": "Definition:Thomas's Plank/Definition 2", "text": "Let $L := \\openint 0 1$ denote the open unit interval. Let $S_1 := L \\cup \\set p$ denote the Alexandroff extension of $L$. Let $S_2 := \\Z_{>0} \\cup \\set q$ denote the Alexandroff extension of the (strictly) positive integers $\\Z_{>0}$. Let $\\struct {S, \\tau} := \\paren {S_1 \\times S_2} \\setminus \\set {\\tuple {p, q} }$ be the subspace of the product space $S_1 \\times S_2$ with $\\set {\\tuple {p, q} }$ removed. '''Thomas's plank''' is the topological space $\\struct {S, \\tau}$."}
+{"_id": "31497", "title": "Definition:Thomas's Corkscrew", "text": "Let copies of Thomas's plank be used to construct a corkscrew in the same manner as the Tychonoff plank is used to construct the Tychonoff corkscrew. '''Thomas's corkscrew''' is the topological space that results."}
+{"_id": "31498", "title": "Definition:Parallel Line Topology/Strong", "text": "Let $\\BB$ be the set of sets of the form: {{begin-eqn}} {{eqn | l = \\map V {a, b}       | r = \\set {\\paren {x, 1}: a \\le x < b} }} {{eqn | l = \\map U {a, b}       | r = \\set {\\paren {x, 0}: a < x \\le b} \\cup \\set {\\paren {x, 1}: a < x \\le b} }} {{end-eqn}} that is: :the left half-open real intervals on $B$ and: :the right half-open real intervals on $A$ together with the interior of their projection onto $B$. $\\BB$ is then taken to be the basis for a topology $\\sigma$ on $S$. Thus $\\sigma$ is referred to as the '''strong parallel line topology'''. The topological space $T = \\struct {S, \\sigma}$ is referred to as the '''strong parallel line space'''."}
+{"_id": "31499", "title": "Definition:Parallel Line Topology/Weak", "text": "Let $\\BB$ be the set of sets of the form: {{begin-eqn}} {{eqn | l = \\map U {a, b}       | r = \\set {\\paren {x, 0}: a < x \\le b} \\cup \\set {\\paren {x, 1}: a < x \\le b} }} {{eqn | l = \\map W {a, b}       | r = \\set {\\paren {x, 0}: a < x < b} \\cup \\set {\\paren {x, 1}: a \\le x < b} }} {{end-eqn}} that is: :the left half-open real intervals on $B$ together with the interior of their projection onto $A$ and: :the right half-open real intervals on $A$ together with the interior of their projection onto $B$. $\\BB$ is then taken to be the basis for a topology $\\tau$ on $S$. Thus $\\tau$ is referred to as the '''weak parallel line topology'''. The topological space $T = \\struct {S, \\tau}$ is referred to as the '''weak parallel line space'''."}
+{"_id": "31501", "title": "Definition:Concentric Circle Topology", "text": ":rightthumb280pxOpen Set of $T$ Let $C_1$ and $C_2$ be concentric circles in the Cartesian plane $\\R^2$ such that $C_1$ is inside $C_2$. Let $S = C_1 \\cup C_2$. Let $\\BB$ be the set of sets consisting of: :all singleton sets of $C_2$ :all open intervals on $C_1$ each together with its projection from the center of the circles onto $C_2$ except for the midpoint. $\\BB$ is then taken to be the sub-basis for a topology $\\tau$ on $S$. Thus $\\tau$ is referred to as the '''concentric circle topology'''. The topological space $T = \\struct {S, \\tau}$ is referred to as the '''concentric circle space'''."}
+{"_id": "31502", "title": "Definition:Appert Space", "text": "Let $\\Z_{>0}$ denote the (strictly) positive integers. For a given subset $H$ of $\\Z_{>0}$ and element $n$ of $\\Z_{>0}$, let $\\map N {n, H}$ denote the number of integers in $E$ which are less than or equal to $n$: :$\\forall n \\in \\Z_{>0}, H \\subseteq \\Z_{>0}: \\map N {n, H} = \\card {\\set {x \\in H: x \\le n} }$ Let $\\tau$ be the topology defined as: :$1 \\notin H \\implies H \\in \\tau$ :$1 \\in H$ and $\\displaystyle \\lim_{n \\mathop \\to \\infty} \\dfrac {\\map N {n, H} } n = 1 \\implies H \\in \\tau$ for $H \\subseteq \\Z_{>0}$. $\\tau$ is referred to as the '''Appert topology'''. The topological space $T = \\struct {S, \\tau}$ is referred to as the '''Appert space'''."}
+{"_id": "31503", "title": "Definition:Maximal Compact Topology", "text": "Let $X = \\Z_{>0} \\times \\Z_{>0}$ be the set of all lattice points $\\tuple {i, j}$ of the Cartesian plane where $i$ and $j$ are both (strictly) positive integers. Let $S = X \\cup \\set {x, y}$ where $x$ and $y$ are two new elements of $X$. Let $\\tau$ be the topology defined on $S$ as follows: :Each lattice point of $S$ is an open point. :The open neighborhoods of $x$ are of the form $S \\setminus A$ where $A$ is any set of lattice points with at most finitely many points on each row :The open neighborhoods of $y$ are of the form $S \\setminus B$ where $B$ is any set of lattice points selected from at most finitely many rows. $\\tau$ is referred to as the '''maximal compact topology'''. The topological space $T = \\struct {S, \\tau}$ is referred to as the '''maximal compact space'''."}
+{"_id": "31504", "title": "Definition:Minimal Hausdorff Topology", "text": ":thumbright420pxExamples of [[Definition:Neighborhood Basisneighborhood bases of $a$ and $-a$]] Let $\\omega$ be the first transfinite ordinal. Let $\\struct {A, \\tau_1}$ denote the totally ordered set defined as: :$A := \\set {1, 2, 3, \\ldots, \\omega, \\ldots, -3, -2, -1}$ with the interval topology $\\tau_1$. Let $\\struct {\\Z_{>0}, \\tau_2}$ denote the (strictly) positive integers with the discrete topology $\\tau_2$. Let $S$ be defined as: :$S = A \\times \\Z_{>0} \\cup \\set {a, -a}$ where $a$ and $-a$ are new elements of $S$. Let $\\tau$ be the topology defined on $S$ as: :the product topology on $\\struct {A, \\tau_1} \\times \\struct {\\Z_{>0}, \\tau_2}$ together with neighborhood bases of $a$ and $-a$ defined as: {{begin-eqn}} {{eqn | l = \\map { {M_n}^+} a       | o = :=       | r = \\set a \\cup \\set {\\tuple {i, j}: i < \\omega, j > n} }} {{eqn | l = \\map { {M_n}^-} {-a}       | o = :=       | r = \\set {-a} \\cup \\set {\\tuple {i, j}: i > \\omega, j > n} }} {{end-eqn}} $\\tau$ is referred to as the '''minimal Hausdorff topology'''. The topological space $T = \\struct {S, \\tau}$ is referred to as the '''minimal Hausdorff space'''."}
+{"_id": "31505", "title": "Definition:Complex Vector Space", "text": "Let $\\C$ be the set of complex numbers. Then the $\\C$-module $\\C^n$ is called the '''complex ($n$-dimensional) vector space'''."}
+{"_id": "31506", "title": "Definition:Linearly Independent/Set/Complex Vector Space", "text": "Let $\\tuple {\\C^n, +, \\cdot}_\\C$ be a complex vector space. Let $S \\subseteq \\C^n$. Then $S$ is a '''linearly independent set of complex vectors''' if every finite sequence of distinct terms in $S$ is a linearly independent sequence. That is, such that: :$\\displaystyle \\forall \\set {\\lambda_k: 1 \\le k \\le n} \\subseteq \\C: \\sum_{k \\mathop = 1}^n \\lambda_k \\mathbf v_k = \\mathbf 0 \\implies \\lambda_1 = \\lambda_2 = \\cdots = \\lambda_n = 0$ where $\\mathbf v_1, \\mathbf v_2, \\ldots, \\mathbf v_n$ are distinct elements of $S$."}
+{"_id": "31507", "title": "Definition:Linearly Dependent/Set/Complex Vector Space", "text": "Let $\\struct {\\C^n, +, \\cdot}_\\C$ be a complex vector space. Let $S \\subseteq \\C^n$. Then $S$ is a '''linearly dependent set''' if there exists a sequence of distinct terms in $S$ which is a linearly dependent sequence. That is, such that: :$\\displaystyle \\exists \\set {\\lambda_k: 1 \\le k \\le n} \\subseteq \\C: \\sum_{k \\mathop = 1}^n \\lambda_k \\mathbf v_k = \\mathbf 0$ where $\\set {\\mathbf v_1, \\mathbf v_2, \\ldots, \\mathbf v_n} \\subseteq S$, and such that at least one of $\\lambda_k$ is not equal to $0$."}
+{"_id": "31508", "title": "Definition:Pointwise Operation/Induced Structure", "text": "The algebraic structure $\\struct {T^S, \\oplus}$ is called the '''algebraic structure on $T^S$ induced by $\\circ$'''."}
+{"_id": "31509", "title": "Definition:Pointwise Addition of Linear Operators", "text": "Let $V$ be a vector space. Let $\\map \\LL V$ denote the set of linear operators on $V$. :$+: \\map \\LL V \\times \\map \\LL V \\to \\map \\LL V: \\forall S, T \\in \\map \\LL V:$ ::$\\forall u \\in V: \\map {\\paren {S + T} } u := \\map S u + \\map T u$ where $+$ on the {{RHS}} is vector addition."}
+{"_id": "31510", "title": "Definition:Pointwise Addition on Complex Vector Space", "text": "Let $\\C^n$ be a complex vector space. Let $S$ and $T$ be linear operators on $\\C^n$. Then the '''pointwise sum of $S$ and $T$''' is defined as: :$S + T: \\C^n \\to \\C^n:$ ::$\\forall u \\in \\C^n: \\map {\\paren {S + T} } u := \\map S u + \\map T u$ where $+$ on the {{RHS}} is complex vector addition."}
+{"_id": "31511", "title": "Definition:Matrix Space/Real", "text": "Let $\\R$ denote the set of real numbers. The '''$m \\times n$ matrix space over $\\R$''' is referred to as the '''real matrix space''', and can be denoted $\\map {\\MM_\\R} {m, n}$."}
+{"_id": "31512", "title": "Definition:Matrix Space/Complex", "text": "Let $\\C$ denote the set of complex numbers. The '''$m \\times n$ matrix space over $\\C$''' is referred to as the '''complex matrix space''', and can be denoted $\\map {\\MM_\\C} {m, n}$."}
+{"_id": "31513", "title": "Definition:Ones Matrix/Square", "text": "A '''square ones matrix of order $n$''' is a square matrix of order $n$ all of whose elements are $1$. It is often denoted $\\mathbf J_n$."}
+{"_id": "31517", "title": "Definition:Matrix Entrywise Addition/Ring", "text": "Let $\\struct {R, +, \\cdot}$ be a ring. Let $\\map {\\MM_R} {m, n}$ be a $m \\times n$ matrix space over $R$. Let $\\mathbf A, \\mathbf B \\in \\map {\\MM_R} {m, n}$. Then the '''matrix entrywise sum of $\\mathbf A$ and $\\mathbf B$''' is written $\\mathbf A + \\mathbf B$, and is defined as follows. Let $\\mathbf A + \\mathbf B = \\mathbf C = \\sqbrk c_{m n}$. Then: :$\\forall i \\in \\closedint 1 m, j \\in \\closedint 1 n: c_{i j} = a_{i j} + b_{i j}$ Thus $\\sqbrk c_{m n}$ is the $m \\times n$ matrix whose entries are made by performing the (ring) addition operation $+$ on corresponding entries of $\\mathbf A$ and $\\mathbf B$."}
+{"_id": "31518", "title": "Definition:Limit of Sequence/Normed Vector Space", "text": "Let $L \\in X$. Let $\\sequence {x_n}_{n \\mathop \\in \\N}$ be a sequence in $X$. Let $\\sequence {x_n}_{n \\mathop \\in \\N}$ converge to $L$.  Then $L$ is a '''limit of  $\\sequence {x_n}_{n \\mathop \\in \\N}$ as $n$ tends to infinity''' which is usually written: :$\\displaystyle L = \\lim_{n \\mathop \\to \\infty} x_n$"}
+{"_id": "31519", "title": "Definition:Zero Matrix/Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Let $\\map {\\MM_R} {m, n}$ be an $m \\times n$ matrix space over $R$. The '''zero matrix of $\\map {\\MM_R} {m, n}$''', denoted $\\mathbf 0_R$, is the $m \\times n$ matrix whose elements are all $0_R$, and can be written $\\sqbrk {0_R}_{m n}$."}
+{"_id": "31520", "title": "Definition:Complex Matrix", "text": "A '''complex matrix''' is a matrix whose elements are all complex numbers."}
+{"_id": "31521", "title": "Definition:Zero Matrix", "text": "Let $\\Bbb F$ be one of the standard number system $\\N$, $\\Z$, $\\Q$, $\\R$ and $\\C$. Let $\\map \\MM {m, n}$ be an $m \\times n$ matrix space over $\\Bbb F$. The '''zero matrix of $\\map \\MM {m, n}$''', denoted $\\mathbf 0$, is the $m \\times n$ matrix whose elements are all zero, and can be written $\\sqbrk 0_{m n}$."}
+{"_id": "31523", "title": "Definition:Matrix Scalar Product/Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\mathbf A = \\sqbrk a_{m n}$ be an $m \\times n$ matrix over $\\struct {R, +, \\circ}$. Let $\\lambda \\in R$ be any element of $R$. The '''scalar product of $\\lambda$ and $\\mathbf A$''' is defined as follows. Let $\\lambda \\circ \\mathbf A = \\mathbf C$. Then: :$\\forall i \\in \\closedint 1 m, j \\in \\closedint 1 n: c_{i j} = \\lambda \\circ a_{i j}$ Thus $\\sqbrk c_{m n}$ is the $m \\times n$ matrix composed of the product of $\\lambda$ with the corresponding elements of $\\mathbf A$."}
+{"_id": "31524", "title": "Definition:Negative Matrix/Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\map {\\MM_R} {m, n}$ denote the $m \\times n$ matrix space over $\\struct {R, +, \\circ}$. Let $\\mathbf A = \\sqbrk a_{m n}$ be an element of $\\map {\\MM_R} {m, n}$. Then the '''negative (matrix) of $\\mathbf A$''' is denoted and defined as: :$-\\mathbf A := \\sqbrk {-a}_{m n}$ where $-a$ is the ring negative of $a$."}
+{"_id": "31525", "title": "Definition:Inverse Matrix/Right", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be a matrix of order $m \\times n$. Let $\\mathbf B = \\sqbrk b_{n m}$ be a matrix of order $n \\times m$ such that: :$\\mathbf A \\mathbf B = I_m$ where $I_m$ denotes the unit matrix of order $m$. Then $\\mathbf B$ is known as a '''right inverse (matrix)''' of $\\mathbf A$."}
+{"_id": "31526", "title": "Definition:Inverse Matrix/Left", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be a matrix of order $m \\times n$. Let $\\mathbf B = \\sqbrk b_{n m}$ be a matrix of order $n \\times m$ such that: :$\\mathbf B \\mathbf A = I_n$ where $I_n$ denotes the unit matrix of order $n$. Then $\\mathbf B$ is known as a '''left inverse (matrix)''' of $\\mathbf A$."}
+{"_id": "31527", "title": "Definition:Row Operation", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be an $m \\times n$ matrix over a field $K$. A '''row operation''' on $\\mathbf A$ is a sequence of '''elementary row operations''' performed on $\\mathbf A$ in turn."}
+{"_id": "31531", "title": "Definition:Echelon Matrix/Echelon Form/Non-Unity Variant/Definition 1", "text": "$\\mathbf A$ is in '''non-unity echelon form''' {{iff}}: :$(1): \\quad$ Each row (except perhaps row $1$) starts with a sequence of zeroes :$(2): \\quad$ Except when for row $k$ and row $k + 1$ are zero rows, the number of zeroes in this initial sequence in row $k + 1$ is strictly greater than the number of zeroes in this initial sequence in row $k$ :$(3): \\quad$ The non-zero rows appear before any zero rows."}
+{"_id": "31532", "title": "Definition:Echelon Matrix/Echelon Form/Non-Unity Variant/Definition 2", "text": "$\\mathbf A$ is in '''non-unity echelon form''' {{iff}} it contains no adjacent rows of the form: :$\\begin {pmatrix} 0 & 0 & \\cdots & 0 & x_1 & x_2 & \\cdots \\\\ 0 & 0 & \\cdots & 0 & y_1 & y_2 & \\cdots \\\\ \\end {pmatrix}$ where: :$(1): \\quad y_1 \\ne 0$ :$(2): \\quad x_1$ can be any value at all, including $0$."}
+{"_id": "31533", "title": "Definition:Echelon Matrix/Echelon Form", "text": "$\\mathbf A$ is in '''echelon form''' {{iff}}: :$(1): \\quad$ The leading coefficient in each non-zero row is $1$ :$(2): \\quad$ The leading $1$ in any non-zero row occurs to the right of the leading $1$ in any previous row :$(3): \\quad$ The non-zero rows appear before any zero rows."}
+{"_id": "31535", "title": "Definition:Transposition/Adjacent", "text": "Let $S_n$ denote the symmetric group on $n$ letters. An '''adjacent transposition''' is a transposition that exchanges two consecutive integers $j$ and $j + 1$, where $1 \\le j < n$. In cycle notation, they are denoted: :$\\begin {pmatrix} j & j + 1 \\end {pmatrix}$"}
+{"_id": "31537", "title": "Definition:Rank/Matrix/Definition 1", "text": "Let $K$ be a field. Let $\\mathbf A$ be an $m \\times n$ matrix over $K$. Then the '''rank''' of $\\mathbf A$, denoted $\\map \\rho {\\mathbf A}$, is the dimension of the subspace of $K^m$ generated by the columns of $\\mathbf A$. That is, it is the dimension of the column space of $\\mathbf A$."}
+{"_id": "31538", "title": "Definition:Gaussian Elimination", "text": "Let $\\mathbf A$ be a matrix over a field $K$. Let $\\mathbf E$ be a reduced echelon matrix which is row equivalent to $\\mathbf A$. The '''Gaussian elimination method''' is a technique for converting $\\mathbf A$ into $\\mathbf E$ by means of a sequence of elementary row operations."}
+{"_id": "31539", "title": "Definition:Simultaneous Linear Equations/Matrix Representation/Matrix of Coefficients", "text": "The matrix $\\mathbf A$ is known as the '''matrix of coeffficients''' of the system."}
+{"_id": "31540", "title": "Definition:Taxicab Norm", "text": "Let $\\mathbf v = \\tuple {v_1, v_2, \\ldots, v_n}$ be a vector in $\\R^n$. The '''taxicab norm''' of $\\mathbf v$ is defined as: :$\\displaystyle \\norm {\\mathbf v}_1 = \\sum_{k \\mathop = 1}^n \\size {v_k}$"}
+{"_id": "31541", "title": "Definition:Simultaneous Linear Equations/Matrix Representation/Augmented Matrix", "text": "Let $\\begin {bmatrix} \\mathbf A & \\mathbf b \\end {bmatrix}$ be the block matrix formed from $\\mathbf A$ and $\\mathbf b$. Then $\\begin {bmatrix} \\mathbf A & \\mathbf b \\end {bmatrix}$ is known as the '''augmented matrix''' of the system."}
+{"_id": "31542", "title": "Definition:Augmented Matrix", "text": "Let $\\mathbf A$ be a matrix of order $n \\times m$. Let $\\mathbf B$ be a matrix of order $n \\times k$. The '''augmented matrix''' of $\\mathbf A$ and $\\mathbf B$ is the block matrix $\\begin {pmatrix} \\mathbf A & \\mathbf B \\end {pmatrix}$ of order $n \\times \\paren {m + k}$. === Augmented Matrix of Simultaneous Equations === {{:Definition:Simultaneous Linear Equations/Matrix Representation/Augmented Matrix}}"}
+{"_id": "31543", "title": "Definition:Simultaneous Equations/Linear Equations/Solution", "text": "Let $\\tuple {x_1, x_2, \\ldots, x_n}$ satisfy each of the equations in $\\displaystyle \\sum_{j \\mathop = 1}^n \\alpha_{i j} x_j = \\beta_i$. Then $\\tuple {x_1, x_2, \\ldots, x_n}$ is referred to as a '''solution to the '''system of simultaneous linear equations'''"}
+{"_id": "31544", "title": "Definition:Equivalent Systems of Simultaneous Linear Equations", "text": "Let $S_1$ and $S_2$ be two systems of simultaneous linear equations. Then $S_1$ and $S_2$ are '''equivalent''' {{iff}}: :every solution to $S_1$ is also a solution to $S_2$ and: :every solution to $S_2$ is also a solution to $S_1$."}
+{"_id": "31545", "title": "Definition:Homogeneous Simultaneous Linear Equations", "text": "A '''system of simultaneous linear equations''': :$\\displaystyle \\forall i \\in \\set {1, 2, \\ldots, m}: \\sum_{j \\mathop = 1}^n \\alpha_{i j} x_j = \\beta_i$ is referred to as '''homogeneous''' {{iff}}: :$\\forall i \\in \\set {1, 2, \\ldots, m}: \\beta_i = 0$ That is: {{begin-eqn}} {{eqn | r = 0       | l = \\alpha_{1 1} x_1 + \\alpha_{1 2} x_2 + \\cdots + \\alpha_{1 n} x_n }} {{eqn | r = 0       | l = \\alpha_{2 1} x_1 + \\alpha_{2 2} x_2 + \\cdots + \\alpha_{2 n} x_n }} {{eqn | o = \\cdots}} {{eqn | r = 0       | l = \\alpha_{m 1} x_1 + \\alpha_{m 2} x_2 + \\cdots + \\alpha_{m n} x_n }} {{end-eqn}}"}
+{"_id": "31546", "title": "Definition:Consistent Simultaneous Linear Equations", "text": "A '''system of simultaneous linear equations''': :$\\displaystyle \\forall i \\in \\set {1, 2, \\ldots, m}: \\sum_{j \\mathop = 1}^n \\alpha_{i j} x_j = \\beta_i$ is referred to as '''consistent''' {{iff}} it has at least one solution."}
+{"_id": "31547", "title": "Definition:Trivial Solution to Homogeneous Simultaneous Linear Equations", "text": "Let $S$ be a '''system of homogeneous simultaneous linear equations''': :$\\displaystyle \\forall i \\in \\set {1, 2, \\ldots, m}: \\sum_{j \\mathop = 1}^n \\alpha_{i j} x_j = 0$ The solution: :$\\tuple {x_1, x_2, \\ldots, x_n}$ such that: :$\\forall j \\in \\set {1, 2, \\ldots, n}: x_j = 0$ is known as the '''trivial solution''' to $S$."}
+{"_id": "31549", "title": "Definition:Scalar", "text": "=== R-Algebraic Structure === Let $\\struct {R, +_R, \\times_R}$ be the scalar ring of an $R$-algebraic structure $\\struct {S, *_1, *_2, \\ldots, *_n, \\circ}_R$. {{:Definition:Scalar/R-Algebraic Structure}} === Module === Let $\\struct {R, +_R, \\times_R}$ be the scalar ring of a module $\\struct {G, +_G, \\circ}_R$. {{:Definition:Scalar/Module}} === Vector Space === Let $\\struct {K, +_K, \\times_K}$ be the scalar field of a vector space $\\struct {G, +_G, \\circ}_K$. {{:Definition:Scalar/Vector Space}} === Scalar (Matrix Theory) === {{:Definition:Scalar (Matrix Theory)}} === Scalar Quantity === {{:Definition:Scalar Quantity}}"}
+{"_id": "31550", "title": "Definition:Scalar (Matrix Theory)", "text": "Let $\\map \\MM {m, n}$ be a matrix space of order $m \\times n$ over an underlying structure $R$. The elements of $R$ are referred to as '''scalars'''."}
+{"_id": "31551", "title": "Definition:Matrix Scalar Product/Scalar", "text": "The element $\\lambda$ of the underlying structure of $\\map \\MM {m, n}$ is known as a '''scalar'''."}
+{"_id": "31552", "title": "Definition:Equality of Matrices", "text": "Let $\\mathbf A$ and $\\mathbf B$ be matrices over an underlying structure $R$. Then $\\mathbf A$ is equal to $\\mathbf B$ {{iff}}: :$(1): \\quad$ the order of $\\mathbf A$ equals the order of $\\mathbf B$: $m \\times n$, say :$(2): \\quad$ for all $i \\in \\set {1, 2, \\ldots, m}$ and $j \\in \\set {1, 2, \\ldots, n}$, we have that $a_{i j} = b_{i j}$."}
+{"_id": "31554", "title": "Definition:Matrix Entrywise Addition/General", "text": "Let $\\mathbf A_1, \\mathbf A_2, \\ldots, \\mathbf A_k$ be matrices all of order of $m \\times n$. Then the '''matrix entrywise sum of $\\mathbf A_1, \\mathbf A_2, \\ldots, \\mathbf A_k$''' is written $\\mathbf A_1 + \\mathbf A_2 + \\ldots + \\mathbf A_k$, and is defined as follows: Then: :$\\forall i \\in \\closedint 1 m, j \\in \\closedint 1 n: K_{i j} = \\paren {a_1}_{i j} + \\paren {a_2}_{i j} + \\cdots + \\paren {a_k}_{i j}$ where $\\paren {a_l}_{i j}$ is the element of $\\mathbf A_l$ whose indices are $\\tuple {i, j}$. Thus $\\mathbf K = \\sqbrk k_{m n}$ is the $m \\times n$ matrix whose entries are made by performing the adding corresponding entries of $\\mathbf A_1, \\mathbf A_2, \\ldots, \\mathbf A_k$. This can be expressed in summation form as: :$\\displaystyle K = \\sum_{j \\mathop = 1}^k \\mathbf A_j$"}
+{"_id": "31555", "title": "Definition:Rank/Matrix/Definition 2", "text": "Let $K$ be a field. Let $\\mathbf A$ be an $m \\times n$ matrix over $K$. Let $\\mathbf A$ be converted to echelon form $\\mathbf B$. Let $\\mathbf B$ have exactly $k$ non-zero rows. Then the '''rank''' of $\\mathbf A$, denoted $\\map \\rho {\\mathbf A}$, is $k$."}
+{"_id": "31558", "title": "Definition:Perfect Field/Definition 1", "text": "$F$ is a '''perfect field ''' {{iff}} $F$ has no inseparable extensions."}
+{"_id": "31559", "title": "Definition:Perfect Field/Definition 2", "text": "$F$ is a '''perfect field''' {{iff}} one of the following holds: :$\\Char F = 0$ :$\\Char F = p$ with $p$ prime and $\\Frob$ is an automorphism of $F$ where: :$\\Char F$ denotes the characteristic of $F$ :$\\Frob$ denotes the Frobenius endomorphism on $F$"}
+{"_id": "31560", "title": "Definition:Elementary Operation/Column", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be an $m \\times n$ matrix over a field $K$. The '''elementary column operations''' on $\\mathbf A$ are operations which act upon the columns of $\\mathbf A$ as follows. For some $i, j \\in \\closedint 1 n: i \\ne j$: {{begin-axiom}} {{axiom | n = \\text {ECO} 1         | t = For some $\\lambda \\in K_{\\ne 0}$, multiply column $i$ by $\\lambda$         | m = \\kappa_i \\to \\lambda \\kappa_i }} {{axiom | n = \\text {ECO} 2         | t = For some $\\lambda \\in K$, add $\\lambda$ times column $j$ to column $i$         | m = \\kappa_i \\to \\kappa_i + \\lambda \\kappa_j }} {{axiom | n = \\text {ECO} 3         | t = Interchange columns $i$ and $j$         | m = \\kappa_i \\leftrightarrow \\kappa_j }} {{end-axiom}}"}
+{"_id": "31561", "title": "Definition:Column Operation", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be an $m \\times n$ matrix over a field $K$. A '''column operation''' on $\\mathbf A$ is a sequence of '''elementary column operations''' performed on $\\mathbf A$ in turn."}
+{"_id": "31562", "title": "Definition:Column Equivalence", "text": "Two matrices $\\mathbf A = \\sqbrk a_{m n}, \\mathbf B = \\sqbrk b_{m n}$ are '''column equivalent''' if one can be obtained from the other by a finite sequence of elementary column operations. This relationship can be denoted $\\mathbf A \\sim \\mathbf B$."}
+{"_id": "31565", "title": "Definition:Determinant/Matrix/In Full", "text": "When written out in full, the determinant of $\\mathbf A$ is denoted: :$\\map \\det {\\mathbf A} = \\begin {vmatrix} a_{1 1} & a_{1 2} & \\cdots & a_{1 n} \\\\ a_{2 1} & a_{2 2} & \\cdots & a_{2 n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ a_{n 1} & a_{n 2} & \\cdots & a_{n n} \\\\ \\end {vmatrix}$"}
+{"_id": "31566", "title": "Definition:Determinant/Matrix/Definition 1", "text": "Let $\\lambda: \\N_{> 0} \\to \\N_{> 0}$ be a permutation on $\\N_{>0}$. The '''determinant''' of $\\mathbf A$ is defined as: :$\\displaystyle \\map \\det {\\mathbf A} := \\sum_{\\lambda} \\paren {\\map \\sgn \\lambda \\prod_{k \\mathop = 1}^n a_{k \\map \\lambda k} } = \\sum_\\lambda \\map \\sgn \\lambda a_{1 \\map \\lambda 1} a_{2 \\map \\lambda 2} \\cdots a_{n \\map \\lambda n}$ where: :the summation $\\displaystyle \\sum_\\lambda$ goes over all the $n!$ permutations of $\\set {1, 2, \\ldots, n}$ :$\\map \\sgn \\lambda$ is the sign of the permutation $\\lambda$."}
+{"_id": "31567", "title": "Definition:Determinant/Matrix/Definition 2", "text": "The '''determinant''' of $\\mathbf A$ is defined as follows: For $n = 1$, the order $1$ determinant is defined as: {{:Definition:Determinant/Matrix/Order 1}} For $n > 1$, the determinant of order $n$ is defined recursively as: :$\\displaystyle \\map \\det {\\mathbf A} := \\begin {vmatrix} a_{1 1} & a_{1 2} & a_{1 3} & \\cdots & a_{1 n} \\\\ a_{2 1} & a_{2 2} & a_{2 3} & \\cdots & a_{2 n} \\\\ a_{3 1} & a_{3 2} & a_{3 3} & \\cdots & a_{3 n} \\\\  \\vdots &  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ a_{n 1} & a_{n 2} & a_{n 3} & \\cdots & a_{n n} \\\\ \\end {vmatrix} = a_{1 1} \\begin {vmatrix} a_{2 2} & a_{2 3} & \\cdots & a_{2 n} \\\\ a_{3 2} & a_{3 3} & \\cdots & a_{3 n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ a_{n 2} & a_{n 3} & \\cdots & a_{n n} \\\\ \\end {vmatrix} - a_{1 2} \\begin {vmatrix} a_{2 1} & a_{2 3} & \\cdots & a_{2 n} \\\\ a_{3 1} & a_{3 3} & \\cdots & a_{3 n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ a_{n 1} & a_{n 3} & \\cdots & a_{n n} \\\\ \\end {vmatrix} + \\cdots + \\paren {-1}^{n + 1} a_{1 n} \\begin {vmatrix} a_{2 1} & a_{2 2} & \\cdots & a_{2, n - 1} \\\\ a_{3 1} & a_{3 3} & \\cdots & a_{3, n - 1} \\\\  \\vdots &  \\vdots & \\ddots &       \\vdots \\\\ a_{n 1} & a_{n 3} & \\cdots & a_{n, n - 1} \\\\ \\end {vmatrix}$"}
+{"_id": "31568", "title": "Definition:Polygamma Function", "text": "The $n$th '''polygamma function''', $\\psi_n$, is defined, for $z \\in \\C \\setminus \\Z_{\\le 0}$, by the $n$th derivative of the digamma function:  :$\\map {\\psi_n} z = \\dfrac {\\d^n} {\\d z^n} \\map \\psi z$ where $\\psi$ is the digamma function."}
+{"_id": "31570", "title": "Definition:Elementary Matrix/Row Operation", "text": "Let $\\mathbf E$ be a unit matrix on which exactly one elementary row operation $e$ has been performed. Then $\\mathbf E$ is called the '''elementary row matrix''' for the elementary row operation $e$."}
+{"_id": "31571", "title": "Definition:Elementary Matrix/Column Operation", "text": "Let $\\mathbf E$ be a unit matrix on which exactly one elementary column operation $e$ has been performed. Then $\\mathbf E$ is called the '''elementary column matrix''' for the elementary column operation $e$."}
+{"_id": "31573", "title": "Definition:Elementary Matrix Operation", "text": "=== Elementary Row Operation === {{:Definition:Elementary Row Operation}} === Elementary Column Operation === {{:Definition:Elementary Column Operation}}"}
+{"_id": "31574", "title": "Definition:Elementary Operation/Row", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be an $m \\times n$ matrix over a field $K$. The '''elementary row operations''' on $\\mathbf A$ are operations which act upon the rows of $\\mathbf A$ as follows. For some $i, j \\in \\closedint 1 m: i \\ne j$: {{begin-axiom}} {{axiom | n = \\text {ERO} 1         | t = For some $\\lambda \\in K_{\\ne 0}$, multiply row $i$ by $\\lambda$         | m = r_i \\to \\lambda r_i }} {{axiom | n = \\text {ERO} 2         | t = For some $\\lambda \\in K$, add $\\lambda$ times row $j$ to row $i$         | m = r_i \\to r_i + \\lambda r_j }} {{axiom | n = \\text {ERO} 3         | t = Exchange rows $i$ and $j$         | m = r_i \\leftrightarrow r_j }} {{end-axiom}}"}
+{"_id": "31575", "title": "Definition:Matrix Product (Conventional)/Pre-Multiplication", "text": "Let $\\mathbf A \\mathbf B$ be the product of $\\mathbf A$ with $\\mathbf B$. Then $\\mathbf B$ is '''pre-multiplied''' by $\\mathbf A$."}
+{"_id": "31576", "title": "Definition:Matrix Product (Conventional)/Post-Multiplication", "text": "Let $\\mathbf A \\mathbf B$ be the product of $\\mathbf A$ with $\\mathbf B$. Then $\\mathbf A$ is '''post-multiplied''' by $\\mathbf B$."}
+{"_id": "31577", "title": "Definition:Distance/Points/Normed Vector Space", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $x, y \\in X$ . Then the function $\\norm {\\, \\cdot \\,} : X \\times X \\to \\R$: :$\\map d {x, y} = \\norm {x - y}$ is called the '''distance between $x$ and $y$'''."}
+{"_id": "31578", "title": "Definition:Inverse of Elementary Row Operation", "text": "Let $e$ be an elementary row operation which transforms a matrix $\\mathbf A$ to another matrix $\\mathbf B$. Let $e'$ be an elementary row operation which transforms $\\mathbf B$ back to $\\mathbf A$. Then $e'$ is the '''inverse of the elementary row operation $e$'''."}
+{"_id": "31579", "title": "Definition:Inverse of Elementary Column Operation", "text": "Let $e$ be an elementary column operation which transforms a matrix $\\mathbf A$ to another matrix $\\mathbf B$. Let $e'$ be an elementary column operation which transforms $\\mathbf B$ back to $\\mathbf A$. Then $e'$ is the '''inverse of the elementary column operation $e$'''."}
+{"_id": "31581", "title": "Definition:Extended Weight Function", "text": "Let $S$ be a set. Let $\\mathscr F$ be the set of all finite subsets of $S$. Let $w: S \\to \\R$ be a weight function. The '''extended weight function''' of $w$ is the function $w^+: \\mathscr F \\to \\R$ defined by: :$\\forall A \\in \\mathscr F : \\map {w^+} A = \\displaystyle \\sum_{a \\mathop \\in A} \\map w a$"}
+{"_id": "31582", "title": "Definition:Matrix of Minors", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $a_{i j}$ denote the element whose indices are $\\tuple {i, j}$: :$\\mathbf A = \\begin {pmatrix} a_{1 1} & a_{1 2} & \\cdots & a_{1 n} \\\\ a_{2 1} & a_{2 2} & \\cdots & a_{2 n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ a_{n 1} & a_{n 2} & \\cdots & a_{n n} \\\\ \\end {pmatrix}$ For each $a_{i j}$, let $b_{i j}$ denote the minor of $a_{i j}$: :$b_{i j} = \\begin {vmatrix} a_{1 1} & a_{1 2} & \\cdots & a_{1, i - 1} & a_{1, i + 1} & \\cdots & \\cdots & a_{1 n} \\\\ a_{2 1} & a_{2 2} & \\cdots & a_{2, i - 1} & a_{2, i + 1} & \\cdots & \\cdots & a_{2 n} \\\\  \\vdots &  \\vdots & \\ddots & \\vdots &  \\vdots & \\ddots & \\vdots \\\\ a_{j - 1, 1} & a_{j - 1, 2} & \\cdots & a_{j - 1, i - 1} & a_{j - 1, i + 1} & \\cdots & \\cdots & a_{j - 1, n} \\\\ a_{j + 1, 1} & a_{j + 1, 2} & \\cdots & a_{j + 1, i - 1} & a_{j + 1, i + 1} & \\cdots & \\cdots & a_{j + 1, n} \\\\  \\vdots &  \\vdots & \\ddots & \\vdots &  \\vdots & \\ddots & \\vdots \\\\ a_{n 1} & a_{n 2} & \\cdots & a_{n, i - 1} & a_{n, i + 1} & \\cdots & \\cdots & a_{n n} \\\\ \\end {vmatrix}$ Then $\\mathbf B = \\sqbrk b_n$ is called the '''matrix of minors''' of $\\mathbf A$. :$\\mathbf B = \\begin {pmatrix} b_{1 1} & b_{1 2} & \\cdots & b_{1 n} \\\\ b_{2 1} & b_{2 2} & \\cdots & b_{2 n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ b_{n 1} & b_{n 2} & \\cdots & b_{n n} \\\\ \\end {pmatrix}$"}
+{"_id": "31585", "title": "Definition:Pythagorean Quadrilateral", "text": "A '''Pythagorean quadrilateral''' is a convex quadrilateral whose diagonals intersect at right angles, and is formed by fitting four right triangles with integer-valued sides. The interest is finding solutions where no two triangles are similar. The smallest Pythagorean quadrilateral is formed by the triangles with side lengths: :$\\tuple {25, 60, 65}, \\tuple {91, 60, 109}, \\tuple {91, 312, 325}, \\tuple {25, 312, 313}$ The smallest primitive Pythagorean quadrilateral, where each Pythagorean triple is primitive is: :$\\tuple {28435, 20292, 34933}, \\tuple {284795, 20292, 285517}, \\tuple {284795, 181908, 337933}, \\tuple {28435, 181908, 184117}$ The smallest anti-primitive Pythagorean quadrilateral, where no Pythagorean triples are primitive is: :$\\tuple {1209, 6188, 6305}, \\tuple {10659, 6188, 12325}, \\tuple {10659, 23560, 25859}, \\tuple {1209, 23560, 23591}$ with common divisors: :$13, 17, 19, 31$"}
+{"_id": "31586", "title": "Definition:EPORN", "text": "An '''EPORN''' is a natural number which can be expressed as the product of a number and its reversal in two different ways."}
+{"_id": "31590", "title": "Definition:Dudeney Number", "text": "A '''Dudeney number''' is a natural number which is equal to the sum of the digits of its cube when expressed in decimal notation."}
+{"_id": "31591", "title": "Definition:Supremum Norm/Continuous on Closed Interval Real-Valued Function", "text": "Let $I = \\closedint a b$ be a closed real interval. Let $\\map \\CC I$ be the space of real-valued functions, continuous on $I$. Let $f \\in \\map \\CC I$. Let $\\size {\\, \\cdot \\,}$ denote the absolute value. Suppose $\\sup$ denotes the supremum of real-valued functions. Then the '''supremum norm''' over $\\map \\CC I$ is defined as :$\\displaystyle \\norm {f}_\\infty := \\sup_{x \\mathop \\in I} \\size {\\map f x}$"}
+{"_id": "31593", "title": "Definition:Think of a Number", "text": "A '''think of a number''' puzzle is usually in the form of a game between two players. Player '''A''' asks player '''B''' to: :''Think of a number'' perhaps with constraints. Let this number be referred to as $n$. Player '''A''' asks player '''B''' to perform certain arithmetical manipulations on $n$. As a result, player '''B''' is left with another number, which we will refer to as $m$. The game now goes one of $2$ ways: :$(1): \\quad$ Player '''A''' announces: ::::''The number you have been left with is $m$.'' :$(2): \\quad$ Player '''A''' asks what $m$ is, and on learning what it is, instantaneously replies: ::::''The number you first thought of was $n$.''"}
+{"_id": "31594", "title": "Definition:Ancient Egyptian Papyrus", "text": "Most of our knowledge of the mathematics of {{WP|Ancient_Egypt|ancient Egypt}} has come to us through a number of documents written on {{WP|Papyrus|papyrus}}."}
+{"_id": "31596", "title": "Definition:Babylonian Mathematics", "text": "'''Babylonian mathematics''' is the name given to the mathematics of the {{WP|First_Babylonian_Dynasty|ancient Babylon}}. Its content was arithmetical and algebraical, and considerably in advance of the mathematics of the approximately contemporaneous {{WP|Ancient_Egypt|ancient Egypt}}."}
+{"_id": "31597", "title": "Definition:Loculus of Archimedes", "text": "The '''loculus of Archimedes''' is a dissection of the square into $14$ pieces: :500px"}
+{"_id": "31600", "title": "Definition:Rusty Compass", "text": "A '''rusty compass''' is (despite what its name may make it seem) an ideal tool for drawing circles. It differs from a normal compass in that, once it has been opened, it is too stiff and rusty to be changed. Hence it can draw circles with only one radius."}
+{"_id": "31601", "title": "Definition:Historic Measures", "text": "Various historic systems of measurement that are no longer in common use are gathered here."}
+{"_id": "31602", "title": "Definition:Historic Measures/Length", "text": "Various historic systems of measurement of length that are no longer in common use are gathered here."}
+{"_id": "31604", "title": "Definition:Historic Measures/Length/Leuca", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''leuca''' }} {{eqn | r = 1500       | c = paces }} {{eqn | r = 7500       | c = feet }} {{eqn | r = 90 \\, 000       | c = inches }} {{end-eqn}}"}
+{"_id": "31612", "title": "Definition:Heap Problem", "text": "A '''heap problem''' is one where: :an unknown $x$ is established :$x$ is expressed as the sum of one or more fractions of $x$ and one or more other whole number. The object is to find $x$."}
+{"_id": "31616", "title": "Definition:Sterling", "text": "'''Sterling''' is the system of monetary units that England (later Britain and the United Kingdom) inherited from the Roman system. Until $1971$ its main units were pounds, shillings and pence (that is, \"pennies\"). From $15$ February $1971$ its main units have been pounds and (new) pence."}
+{"_id": "31617", "title": "Definition:Money", "text": "'''Money''' is a measure of purchasing power."}
+{"_id": "31629", "title": "Definition:Penny", "text": "A '''penny''' is one of two coins of the Sterling monetary system: === New Penny === {{:Definition:Sterling/Penny/New}} === Old Penny === {{:Definition:Sterling/Penny/Old}}"}
+{"_id": "31634", "title": "Definition:Donkey and Mule Problem", "text": "A '''Donkey and Mule problem''' is of the form: :$2$ participants are comparing their quantity of goods they own relative to the other. :Participant $1$ suggests transferring some of the goods of one of them to the other, and remarks on their relative quantities after that exchange. :Participant $2$ then suggests the same (or a similar) thing, but transferring the goods in the other direction, and again remarks on their relative quantities after that exchange. From these remarks, $2$ simultaneous linear equations can be set up and (assuming the question is well-crafted) solved. The name comes from the original setting of this problem concerning a '''donkey''' and a '''mule''' each carrying a number of sacks."}
+{"_id": "31639", "title": "Definition:Troy/Talent", "text": "{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''talent''' }} {{eqn | r = 75       | c = troy pounds }} {{eqn | r = 900       | c = troy ounces }} {{end-eqn}}"}
+{"_id": "31641", "title": "Definition:Box-Jenkins Model", "text": "The '''Box-Jenkins model''' is a very general mathematical model for time series analysis in forecasting and prediction. === ARMA (Autoregressive Moving Average) === {{:Definition:Box-Jenkins Model/ARMA}} === ARIMA (Autoregressive Integrated Moving Average) === {{:Definition:Box-Jenkins Model/ARIMA}}"}
+{"_id": "31644", "title": "Definition:Recreational Chess", "text": "'''Recreational chess''' is a subgenre of recreational mathematics which explores the geometry of the chessboard and the moves of its pieces outside of the context of an actual game of chess itself. The name amusingly belies the fact chess is itself a recreation."}
+{"_id": "31649", "title": "Definition:Heronian Triangle/Definition 2", "text": "A '''Heronian triangle''' is a triangle whose side lengths and altitudes are all rational numbers."}
+{"_id": "31650", "title": "Definition:Proper Heronian Triangle", "text": "A '''proper Heronian triangle''' is a Heronian triangle which is specifically not right-angled."}
+{"_id": "31651", "title": "Definition:Integer Heronian Triangle", "text": "An '''integer Heronian triangle''' is a Heronian triangle whose sides are integers."}
+{"_id": "31652", "title": "Definition:Independence System", "text": "Let $S$ be a finite set. Let $\\mathscr F$ be a set of subsets of $S$ satisfying the independence system axioms: {{:Definition:Independence System Axioms}} The ordered pair $I = \\struct{S, \\mathscr F}$ is called an '''independence system on $S$''', or simply an '''independence system''' when the context is obvious."}
+{"_id": "31653", "title": "Definition:Independence System Axioms", "text": "{{begin-axiom}} {{axiom | n = \\text I 1         | mr= \\O \\in \\mathscr F  }} {{axiom | n = \\text I 2         | q = \\forall X \\in \\mathscr F: \\forall Y \\subseteq S         | mr= Y \\subseteq X \\implies Y \\in \\mathscr F }} {{end-axiom}}"}
+{"_id": "31656", "title": "Definition:Matroid Axioms/Axioms 1", "text": "{{begin-axiom}} {{axiom | n = \\text I 1         | mr= \\O \\in \\mathscr I  }} {{axiom | n = \\text I 2         | q = \\forall X \\in \\mathscr I: \\forall Y \\subseteq S         | mr= Y \\subseteq X \\implies Y \\in \\mathscr I }} {{axiom | n = \\text I 3         | q = \\forall U, V \\in \\mathscr I         | mr= \\size V < \\size U \\implies \\exists x \\in U \\setminus V : V \\cup \\set x \\in \\mathscr I }} {{end-axiom}}"}
+{"_id": "31657", "title": "Definition:Matroid Axioms/Axioms 2", "text": "{{begin-axiom}} {{axiom | n = \\text I 1         | mr= \\O \\in \\mathscr I  }} {{axiom | n = \\text I 2         | q = \\forall X \\in \\mathscr I: \\forall Y \\subseteq S         | mr= Y \\subseteq X \\implies Y \\in \\mathscr I }} {{axiom | n = \\text I 3'         | q = \\forall U, V \\in \\mathscr I         | mr= \\size U = \\size V + 1 \\implies \\exists x \\in U \\setminus V : V \\cup \\set x \\in \\mathscr I }} {{end-axiom}}"}
+{"_id": "31660", "title": "Definition:Matroid Axioms/Axioms 3", "text": "{{begin-axiom}} {{axiom | n = \\text I 1         | mr= \\O \\in \\mathscr I  }} {{axiom | n = \\text I 2         | q = \\forall X \\in \\mathscr I: \\forall Y \\subseteq S         | mr= Y \\subseteq X \\implies Y \\in \\mathscr I }} {{axiom | n = \\text I 3''         | q = \\forall U, V \\in \\mathscr I         | mr= \\size V < \\size U \\implies \\exists Z \\subseteq U \\setminus V : \\paren{V \\cup Z \\in \\mathscr I} \\land \\paren{\\size{V \\cup Z} = \\size U} }} {{end-axiom}}"}
+{"_id": "31661", "title": "Definition:Matroid Axioms/Axioms 4", "text": "{{begin-axiom}} {{axiom | n = \\text I 1         | mr= \\O \\in \\mathscr I  }} {{axiom | n = \\text I 2         | q = \\forall X \\in \\mathscr I: \\forall Y \\subseteq S         | mr= Y \\subseteq X \\implies Y \\in \\mathscr I }} {{axiom | n = \\text I 3'''         | q = \\forall A \\subseteq S         | mr= \\text{ all maximal subsets } Y \\subseteq A \\text{ with } Y \\in \\mathscr I \\text{ have the same cardinality} }} {{end-axiom}}"}
+{"_id": "31662", "title": "Definition:P-adic Norm/Definition 1", "text": "Let $p \\in \\N$ be a prime. Let $\\nu_p: \\Q \\to \\Z \\cup \\set {+\\infty}$ be the $p$-adic valuation on $\\Q$. The '''$p$-adic norm''' on $\\Q$ is the mapping $\\norm {\\,\\cdot\\,}_p: \\Q \\to \\R_{\\ge 0}$ defined as: :$\\forall q \\in \\Q: \\norm q_p := \\begin{cases} 0 & : q = 0 \\\\ p^{-\\map {\\nu_p} q} & : q \\ne 0 \\end{cases}$"}
+{"_id": "31663", "title": "Definition:P-adic Norm/Definition 2", "text": "Let $p \\in \\N$ be a prime. Let $k, m, n \\in \\Z : p \\nmid m, n$. Let $\\displaystyle r := p^k \\frac m n$. The '''$p$-adic norm''' on $\\Q$ is the mapping $\\norm {\\,\\cdot\\,}_p: \\Q \\to \\R_{\\ge 0}$ defined as: :$\\forall r \\in \\Q: \\norm r_p := \\begin{cases} 0 & : r = 0 \\\\ p^{-k} & : r \\ne 0 \\end{cases}$"}
+{"_id": "31665", "title": "Definition:Meridian", "text": "=== Terrestrial Meridian === {{:Definition:Meridian (Terrestrial)}} === Celestial Meridian === {{:Definition:Celestial Meridian}} Category:Definitions/Geography Category:Definitions/Spherical Astronomy mg2cgyg0lbb4qelfbv45cd272nq6nwq"}
+{"_id": "31667", "title": "Definition:Geography Puzzle", "text": "A '''geography puzzle''' is a puzzle whose solution is dependent upon some particular aspect of geography or geodesy."}
+{"_id": "31668", "title": "Definition:Chess/Chessboard/Square/White", "text": "A '''white square''' of a chessboard is one of the squares which is of the lighter colour."}
+{"_id": "31669", "title": "Definition:Chess/Chessboard/Square/Black", "text": "A '''black square''' of a chessboard is one of the squares which is of the darker colour."}
+{"_id": "31670", "title": "Definition:Formation of Ordinary Differential Equation by Elimination", "text": "Let $\\map f {x, y, C_1, C_2, \\ldots, C_n} = 0$ be an equation: :whose dependent variable is $y$ :whose independent variable is $x$ :$C_1, C_2, \\ldots, C_n$ are constants which are deemed to be arbitrary. A '''differential equation''' may be '''formed''' from $f$ by: :differentiating $n$ times {{WRT|Differentiation}} $x$ to obtain $n$ equations in $x$ and $\\dfrac {\\d^k y} {\\d x^k}$, for $k \\in \\set {1, 2, \\ldots, n}$ :'''eliminating''' $C_k$ from these $n$ equations, for $k \\in \\set {1, 2, \\ldots, n}$."}
+{"_id": "31671", "title": "Definition:Real Number/Equality", "text": "Two real numbers are defined as being '''equal''' {{iff}} they correspond to the same point on the real number line."}
+{"_id": "31672", "title": "Definition:Nonzero Integer", "text": "A '''nonzero integer''' is an element of the set: :$\\Z \\setminus \\set 0$ where: :$\\Z$ denotes the set of integers :$\\setminus$ denotes set difference. Thus: :$\\Z \\setminus \\set 0 = \\set {\\ldots, -3, -2, -1, 1, 2, 3, \\ldots}$ and can be denoted $\\Z_{\\ne 0}$."}
+{"_id": "31673", "title": "Definition:Basis Expansion/Termination", "text": "Let the basis expansion of $x$ in base $b$ be: :$\\sqbrk {s \\cdotp d_1 d_2 d_3 \\ldots}_b$ Let it be the case that: :$\\exists m \\in \\N: \\forall k \\ge m: d_k = 0$ That is, every digit of $x$ in base $b$ after a certain point is zero. Then $x$ is said to '''terminate'''."}
+{"_id": "31674", "title": "Definition:Basis Expansion/Recurrence", "text": "Let the basis expansion of $x$ in base $b$ be: :$\\sqbrk {s \\cdotp d_1 d_2 d_3 \\ldots}_b$ Let there be a finite sequence of $p$ digits of $x$: :$\\tuple {d_{r + 1} d_{r + 1} \\ldots d_{r + p} }$ such that for all $k \\in \\Z_{\\ge 0}$ and for all $j \\in \\set {1, 2, \\ldots, p}$: :$d_{r + j + k p} = d_{r + j}$ where $p$ is the smallest $p$ to have this property. That is, let $x$ be of the form: :$\\sqbrk {s \\cdotp d_1 d_2 d_3 \\ldots d_r d_{r + 1} d_{r + 2} \\ldots d_{r + p} d_{r + 1} d_{r + 2} \\ldots d_{r + p} d_{r + 1} d_{r + 2} \\ldots d_{r + p} d_{r + 1} \\ldots}_b$ That is, $\\tuple {d_{r + 1} d_{r + 2} \\ldots d_{r + p} }$ repeats from then on, or '''recurs'''. Then $x$ is said to '''recur'''."}
+{"_id": "31675", "title": "Definition:Basis Expansion/Recurrence/Notation", "text": "Let the basis expansion of $x$ in base $b$ be: :$\\sqbrk {s \\cdotp d_1 d_2 d_3 \\ldots}_b$ such that $x$ is recurring. Let the non-recurring part of $x$ be: :$\\sqbrk {s \\cdotp d_1 d_2 d_3 \\ldots d_r}_b$ Let the recurring part of $x$ be: :$\\sqbrk {\\ldots d_{r + 1} d_{r + 2} \\ldots d_{r + p} \\ldots}_b$ Then $x$ is denoted: :$x = s.d_1 d_2 d_3 \\ldots d_r \\dot d_{r + 1} d_{r + 2} \\ldots \\dot d_{r + p}$ That is, a dot is placed over the first and last digit of the first instance of the recurring part."}
+{"_id": "31676", "title": "Definition:Basis Expansion/Recurrence/Non-Recurring Part", "text": "Let the basis expansion of $x$ in base $b$ be recurring: :$\\sqbrk {s \\cdotp d_1 d_2 d_3 \\ldots d_r d_{r + 1} d_{r + 2} \\ldots d_{r + p} d_{r + 1} d_{r + 2} \\ldots d_{r + p } d_{r + 1} d_{r + 2} \\ldots d_{r + p} d_{r + 1} \\ldots}_b$ The '''non-recurring part''' of $x$  is: :$\\sqbrk {s \\cdotp d_1 d_2 d_3 \\ldots d_r}$"}
+{"_id": "31678", "title": "Definition:Closed Ball/Metric Space", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $a \\in A$. Let $\\epsilon \\in \\R_{>0}$ be a positive real number. The '''closed $\\epsilon$-ball of $a$ in $M$''' is defined as: :$\\map { {B_\\epsilon}^-} a := \\set {x \\in A: \\map d {x, a} \\le \\epsilon}$ where $B^-$ recalls the notation of topological closure. If it is necessary to show the metric itself, then the notation $\\map { {B_\\epsilon}^-} {a; d}$ can be used."}
+{"_id": "31679", "title": "Definition:Multiplicative Inverse/Number", "text": "Let $\\Bbb F$ be one of the standard number fields: $\\Q$, $\\R$, $\\C$. Let $a \\in \\Bbb F$ be any arbitrary number. The '''multiplicative inverse''' of $a$ is its inverse under addition and can be denoted: $a^{-1}$, $\\dfrac 1 a$, $1 / a$, and so on. :$a \\times a^{-1} = 1$"}
+{"_id": "31685", "title": "Definition:Instant of Time", "text": "An '''instant''' is a specific point in time. Category:Definitions/Time gqd2hk6fo21a7y0qbew88gqj2dafg19"}
+{"_id": "31686", "title": "Definition:Time Series", "text": "A '''time series''' is a sequence of observations taken at a sequence of instants of time."}
+{"_id": "31689", "title": "Definition:Time Series Analysis", "text": "'''Time series analysis''' is the discipline of analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Its main activity is concerned with techniques for determining the nature of the dependence of adjacent observations of such time series."}
+{"_id": "31690", "title": "Definition:Observation", "text": "An '''observation''' is an item of raw data that has been obtained as the result of measurement or counting. Hence it is a particular value taken by a variable in statistics."}
+{"_id": "31691", "title": "Definition:Torsion Element", "text": "=== Torsion Element of Group === Let $G$ be a group. {{:Definition:Torsion Element of Group}} === Torsion Element of Module === {{:Definition:Torsion Element of Module}} Category:Definitions/Abstract Algebra 49qe6r916v0d0x19ufjdzfby1gb0xwo"}
+{"_id": "31693", "title": "Definition:Time Series/Adjacent Observations", "text": "Two observations of a '''time series''' are '''adjacent''' {{iff}} the index of one of them is the immediate predecessor of the other (and the other is the immediate successor of the one)."}
+{"_id": "31694", "title": "Definition:Time Series/Dependence of Adjacent Observations", "text": "A defining characteristic of a '''time series''' is that every pair of adjacent observations is dependent."}
+{"_id": "31695", "title": "Definition:Time Series/Timestamp", "text": "A '''timestamp''' of an '''observation''' $x$ of a '''time series''' is the instant with which $x$ is associated."}
+{"_id": "31696", "title": "Definition:Forecasting", "text": "In the context of the discipline of '''time series analysis''', '''forecasting''' is the estimation of future observations of a given time series based on knowledge of current and past such observations."}
+{"_id": "31698", "title": "Definition:Time Series/Current Value", "text": "The '''current value''' of a '''time series''' $T$ is an '''observation''' $x$ of $T$ whose timestamp is the '''current time'''. That is, it is the most recent observation."}
+{"_id": "31700", "title": "Definition:Past", "text": "The '''past''' is the portion of time previous to the present. That is, the '''past''' is what has already happened."}
+{"_id": "31701", "title": "Definition:Present", "text": "The '''present''' is the time at this point '''now'''."}
+{"_id": "31702", "title": "Definition:Future", "text": "The '''future''' is the portion of time subsequent to the present. That is, the '''future''' is what has not happened yet."}
+{"_id": "31703", "title": "Definition:Transfer Function (Time Series Analysis)", "text": "A '''transfer function''', in the context of time series analysis, is a function of time which theoretically models the future output for each possible input."}
+{"_id": "31704", "title": "Definition:Intervention Event", "text": "An '''intervention event''' is an event whose occurrence is not characteristic of a particular time series such that it affects the behaviour of that time series. As a result, forecasting future values based upon the past values is likely to be compromised."}
+{"_id": "31705", "title": "Definition:Control Scheme", "text": "A '''control scheme''' is a system whereby potential deviations of future values from a desired target may be compensated for, by adjustment of input values."}
+{"_id": "31706", "title": "Definition:Lead Time", "text": "When forecasting on a time series, the '''lead time''' is the future time over which forecasts are needed. The '''lead time''' depends upon the problem for which the forecasting is being performed."}
+{"_id": "31707", "title": "Definition:Time Series/Discrete", "text": "A '''discrete time series''' is such that the timestamps of the observations occur at well-defined instants, separated one from another by a time interval."}
+{"_id": "31708", "title": "Definition:Time Series/Equispaced", "text": "A '''time series''' is '''equispaced''' {{iff}} the time intervals between the timestamps is equal for all pairs of adjacent observations."}
+{"_id": "31709", "title": "Definition:Time Series/Origin", "text": "The '''origin''' of a '''time series''' is an arbitrary timestamp of an observation which is chosen in order that all other timestamps can be measured from that '''origin'''."}
+{"_id": "31710", "title": "Definition:Forecast Function", "text": "Let $T$ be a time series. Let $S$ denote the range of $T$. Let $L$ denote the set of lead times of $T$. Let $t$ denote the origin of $T$. A '''forecast function''' $\\hat z_t : L \\to S$ is a function which provides forecasts of the future values of $T$ at the lead times $L$. The objective is to make the mean squares of the deviations $z_{t + l} - \\map {\\hat z_t} l$ as small as possible for each lead time $l$. In the above: :$\\map {\\hat z_t} l$ denotes the forecast value of the observation at the timestamp of lead time $l$ :$z_{t + l}$ denotes the actual value of the observation at the timestamp of $l$."}
+{"_id": "31711", "title": "Definition:Time Series/Forecast Value", "text": "A '''forecast value''' of a time series $T$ is an estimate of a future value at some lead time $t + l$."}
+{"_id": "31712", "title": "Definition:Time Series/Actual Value", "text": "An '''actual value''' of a time series $T$ is the result of a measurement of an observation at some time $t$. The term is usually made in reference to a forecast value made after its lead time has elapsed, and its timestamp is now the current time."}
+{"_id": "31713", "title": "Definition:Hyperbolic Sine/Real", "text": "The '''real hyperbolic sine''' function is defined on the real numbers as: :$\\sinh: \\R \\to \\R$: :$\\forall x \\in \\R: \\sinh x := \\dfrac {e^x - e^{-x} } 2$"}
+{"_id": "31714", "title": "Definition:Hyperbolic Cosine/Real", "text": "The '''real hyperbolic cosine''' function is defined on the real numbers as: :$\\cosh: \\R \\to \\R$: :$\\forall x \\in \\R: \\cosh x := \\dfrac {e^x + e^{-x} } 2$"}
+{"_id": "31717", "title": "Definition:Real Hyperbolic Tangent/Definition 2", "text": "The '''real hyperbolic tangent''' function is defined on the real numbers as: :$\\tanh: \\R \\to \\R$: :$\\forall x \\in \\R: \\tanh x := \\dfrac {\\sinh x} {\\cosh x}$ where: :$\\sinh$ is the real hyperbolic sine :$\\cosh$ is the real hyperbolic cosine"}
+{"_id": "31720", "title": "Definition:Real Hyperbolic Tangent/Definition 1", "text": "The '''real hyperbolic tangent''' function is defined on the real numbers as: :$\\tanh: \\R \\to \\R$: :$\\forall x \\in \\R: \\tanh x := \\dfrac {e^z - e^{-x} } {e^z + e^{-x} }$"}
+{"_id": "31721", "title": "Definition:Real Hyperbolic Cotangent/Definition 1", "text": "The '''real hyperbolic cotangent''' function is defined on the real numbers as: :$\\coth: \\R_{\\ne 0} \\to \\R$: :$\\forall x \\in \\R_{\\ne 0}: \\coth x := \\dfrac {e^x + e^{-x} } {e^x - e^{-x} }$ where it is noted that at $x = 0$: :$e^x - e^{-x} = 0$ and so $\\coth x$ is not defined at that point."}
+{"_id": "31722", "title": "Definition:Real Hyperbolic Cotangent/Definition 2", "text": "The '''real hyperbolic cotangent''' function is defined on the real numbers as: :$\\coth: \\R_{\\ne 0} \\to \\R$: :$\\forall x \\in \\R_{\\ne 0}: \\coth x := \\dfrac {\\cosh x} {\\sinh x}$ where: :$\\sinh$ is the real hyperbolic sine :$\\cosh$ is the real hyperbolic cosine It is noted that at $x = 0$ we have that $\\sinh x = 0$, and so $\\coth x$ is not defined at that point."}
+{"_id": "31728", "title": "Definition:Real Hyperbolic Cosecant/Definition 2", "text": "The '''real hyperbolic cosecant''' function is defined on the real numbers as: :$\\csch: \\R_{\\ne 0} \\to \\C$: :$\\forall x \\in \\R_{\\ne 0}: \\csch x := \\dfrac 1 {\\sinh x}$ where $\\sinh$ is the real hyperbolic sine. It is noted that at $x = 0$ we have that $\\sinh x = 0$, and so $\\csch x$ is not defined at that point."}
+{"_id": "31729", "title": "Definition:Real Hyperbolic Cosecant/Definition 1", "text": "The '''real hyperbolic cosecant''' function is defined on the real numbers as: :$\\csch: \\R_{\\ne 0} \\to \\R$: :$\\forall x \\in \\R_{\\ne 0}: \\csch x := \\dfrac 2 {e^x - e^{-x} }$ where it is noted that at $x = 0$: :$e^x - e^{-x} = 0$ and so $\\csch x$ is not defined at that point."}
+{"_id": "31730", "title": "Definition:Mean Square", "text": "Let $S$ be a set of numbers. The '''mean square''' of $S$ is the arithmetic mean of the squares of the elements of $S$: :$s^2 = \\dfrac 1 n \\displaystyle \\sum_{i \\mathop = 1}^n {x_i}^2$ where $S = \\set {x_1, x_2, \\ldots, x_n}$."}
+{"_id": "31731", "title": "Definition:Deviation", "text": "Let $S$ be a set of observations. Let $x \\in S$. The '''deviation''' of $x$ is the difference between $x$ and some other value, whose nature depends upon the context."}
+{"_id": "31732", "title": "Definition:Deviation from Forecast", "text": "Let $T$ be a time series. Let $S$ denote the range of $T$. Let $L$ denote the set of lead times of $T$. Let $\\hat z_t$ be a forecast function on $L$. Let $\\map {\\hat z_t} l$ denote the forecast value of the observation at the timestamp of lead time $l$. Let $z_{t + l}$ denote the actual value of the observation at the timestamp of $l$. The '''deviation (from forecast)''' is the difference between $\\map {\\hat z_t} l$ and $z_{t + l}$: :$\\Delta_l := z_{t + l} - \\map {\\hat z_t} l$"}
+{"_id": "31734", "title": "Definition:Probability Limit/Upper", "text": "Let $\\Delta_p$ be a probability limit for $\\map {\\hat z_t} l$. The '''upper probability limit''' of $\\map {\\hat z_t} l$ is the value $\\map {\\hat z_t} l + \\Delta_p$."}
+{"_id": "31735", "title": "Definition:Probability Limit", "text": "Let $T$ be a time series. Let $S$ denote the range of $T$. Let $L$ denote the set of lead times of $T$. Let $\\hat z_t$ be a forecast function on $L$. Let $\\map {\\hat z_t} l$ denote the forecast value of the observation at the timestamp of lead time $l$. A '''probability limit''', for a given probability $p$, is the deviation $\\Delta_p$, either positive or negative, from $\\map {\\hat z_t} l$ such that the probability that the actual value lies within $\\Delta_p$ of $\\map {\\hat z_t} l$ is greater than $p$. That is, the actual value of the time series at $l$, when it occurs, will be within those '''probability limits''' within that stated probability $p$."}
+{"_id": "31736", "title": "Definition:Probability Limit/Lower", "text": "Let $\\Delta_p$ be a probability limit for $\\map {\\hat z_t} l$. The '''lower probability limit''' of $\\map {\\hat z_t} l$ is the value $\\map {\\hat z_t} l - \\Delta_p$."}
+{"_id": "31737", "title": "Definition:Stochastic Model", "text": "A '''stochastic model''' is a mathematical model which incorporates stochastic (random) processes in its definition."}
+{"_id": "31738", "title": "Definition:Measure of Central Tendency", "text": "A '''measure of central tendency''' is a central or typical value for a probability distribution or set of sample data."}
+{"_id": "31740", "title": "Definition:Median (Statistics)", "text": "Let $S$ be a set of quantitative data. Let $S$ be arranged in order of size. The '''median''' is the element of $S$ that is in the '''middle''' of that ordered set. Suppose there are an odd number of element of $S$ such that $S$ has cardinality $2 n - 1$. The '''median''' of $S$ in that case is the $n$th element of $S$. Suppose there are an even number of element of $S$ such that $S$ has cardinality $2 n$. Then the '''middle''' of $S$ is not well-defined, and so the '''median''' of $S$ in that case is the arithmetic mean of the $n$th and $n + 1$th elements of $S$."}
+{"_id": "31741", "title": "Definition:Mode", "text": "Let $S$ be a set of quantitative data. The '''mode''' of $S$ is the element of $S$ which occurs most often in $S$. If there is more than one such element of $S$ which occurs equally most often, it is then understood that each of these is a '''mode'''. If there is no element of $S$ which occurs more often (in the extreme case, all elements are equal) then $S$ has '''no mode'''"}
+{"_id": "31742", "title": "Definition:Least Significant Digit", "text": "Let $b \\in \\Z: b \\ge 2$ be a number base Let $n$ be a number which is reported to $r$ significant figures, to base $b$, that is: :$n = d_1 \\times b^k + d_2 \\times b^{k - 1} + \\dotsb + d_{r - 1} \\times b^{k - r + 2} + d_r \\times b^{k - r + 1}$ where: :$d_1, d_2, \\dotsc, d_r$ are the significant figures of $n$ :$b^k$ is the largest power of $b$ less than or equal to $n$. Then the digit $d_r$ is known as the '''least significant digit''' of $n$. Note that the usual situation is when $b = 10$, but in the field of computer science, binary is usual."}
+{"_id": "31743", "title": "Definition:Most Significant Digit", "text": "Let $b \\in \\Z: b \\ge 2$ be a number base Let $n$ be a number which is reported to $r$ significant figures, to base $b$, that is: :$n = d_1 \\times b^k + d_2 \\times b^{k - 1} + \\dotsb + d_{r - 1} \\times b^{k - r + 2} + d_r \\times b^{k - r + 1}$ where: :$d_1, d_2, \\dotsc, d_r$ are the significant figures of $n$ :$b^k$ is the largest power of $b$ less than or equal to $n$. Then the digit $d_1$ is known as the '''most significant digit''' of $n$. Note that the usual situation is when $b = 10$, but in the field of computer science, binary is usual."}
+{"_id": "31744", "title": "Definition:Left (Direction)", "text": "The direction '''left''' is '''that way''': :$\\gets$"}
+{"_id": "31745", "title": "Definition:Right (Direction)", "text": "The direction '''right''' is '''that way''': :$\\to$"}
+{"_id": "31749", "title": "Definition:Time Series/Continuous", "text": "A '''continuous time series''' is one in which the set of timestamps of the observations forms a continuous function."}
+{"_id": "31750", "title": "Definition:Backward Shift Operator", "text": "Let $T = \\sequence {z_t}$ be a discrete time series. The '''backward shift operator''' $B$ is defined as: :$\\forall t: \\map B {z_t} = z_{t - 1}$"}
+{"_id": "31751", "title": "Definition:Forward Shift Operator", "text": "Let $T = \\sequence {z_t}$ be a discrete time series. The '''forward shift operator''' $F$ is defined as: :$\\forall t: \\map F {z_t} = z_{t + 1}$ === Iterated === {{:Definition:Forward Shift Operator/Iterated}}"}
+{"_id": "31752", "title": "Definition:Forward Shift Operator/Iterated", "text": "$F$ can be '''iterated''' on $\\sequence {z_t}$ as follows: :$\\map {F^m} {z_t} := z_{t + m}$"}
+{"_id": "31753", "title": "Definition:Limit of Real Function/Limit at Infinity/Positive", "text": "$L$ is the '''limit of $f$ at infinity''' {{iff}}: :$\\forall \\epsilon \\in \\R_{>0}: \\exists c \\in \\R: \\forall x > c : \\size {\\map f x - L} < \\epsilon$ This is denoted as: :$\\ds \\lim_{x \\mathop \\to \\infty} \\map f x = L$"}
+{"_id": "31754", "title": "Definition:Limit of Real Function/Limit at Infinity/Negative", "text": "$L$ is the '''limit of $f$ at minus infinity''' {{iff}}: :$\\forall \\epsilon \\in \\R_{>0}: \\exists c \\in \\R: \\forall x < c: \\size {\\map f x - L} < \\epsilon$ This is denoted as: :$\\displaystyle \\lim_{x \\mathop \\to - \\infty} \\map f x = L$"}
+{"_id": "31755", "title": "Definition:Limit of Real Function/Limit at Infinity/Positive/Increasing Without Bound", "text": "Suppose that: :$\\forall M \\in \\R_{>0}: \\exists N \\in \\R_{>0}: \\forall x > N : \\map f x > M$ for $M$ sufficiently large. Then we write: :$\\displaystyle \\lim_{x \\mathop \\to +\\infty} \\map f x = +\\infty$ or :$\\map f x \\to +\\infty$ as $x \\to +\\infty$ That is, $\\map f x$ can be made arbitrarily large by making $x$ sufficiently large. This is voiced: :'''$\\map f x$ increases without bound as $x$ increases without bound'''. or: :'''$\\map f x$ tends to (plus) infinity as $x$ tends to (plus) infinity'''."}
+{"_id": "31756", "title": "Definition:Limit of Real Function/Limit at Infinity/Negative/Increasing Without Bound", "text": "Suppose that: :$\\forall M \\in \\R_{>0}: \\exists N \\in \\R_{<0}: \\forall x < N : \\map f x > M$ for $M$ sufficiently large. Then we write: :$\\displaystyle \\lim_{x \\mathop \\to -\\infty} \\map f x = +\\infty$ or :$\\map f x \\to +\\infty$ as $x \\to -\\infty$ This is voiced: :'''$\\map f x$ increases without bound as $x$ decreases without bound'''. or: :'''$\\map f x$ tends to (plus) infinity as $x$ tends to minus infinity'''."}
+{"_id": "31757", "title": "Definition:Limit of Real Function/Limit at Infinity/Positive/Decreasing Without Bound", "text": "Suppose that: :$\\forall M \\in \\R_{<0}: \\exists N \\in \\R_{>0}: x > N \\implies \\map f x < M$ for $M$ sufficiently large in magnitude. Then we write: :$\\displaystyle \\lim_{x \\mathop \\to +\\infty} \\map f x = -\\infty$ or :$\\map f x \\to -\\infty$ as $x \\to +\\infty$ That is, $-\\map f x$ can be made arbitrarily large by making $x$ sufficiently large. This is voiced: :'''$\\map f x$ decreases without bound as $x$ increases without bound'''. or: :'''$\\map f x$ tends to minus infinity as $x$ tends to (plus) infinity'''."}
+{"_id": "31758", "title": "Definition:Limit of Real Function/Limit at Infinity/Negative/Decreasing Without Bound", "text": "Suppose that: :$\\forall M \\in \\R_{<0}: \\exists N \\in \\R_{<0}: x < N \\implies \\map f x < M$ for $M$ sufficiently large in magnitude. Then we write: :$\\displaystyle \\lim_{x \\mathop \\to -\\infty} \\map f x = +\\infty$ or :$\\map f x \\to +\\infty \\ \\text{as} \\ x \\to -\\infty$ This is voiced: :'''$\\map f x$ decreases without bound as $x$ decreases without bound'''. or: :'''$\\map f x$ tends to minus infinity as $x$ tends to minus infinity'''."}
+{"_id": "31759", "title": "Definition:Deterministic Model", "text": "A '''deterministic model''' is a mathematical model based on physical laws which are possible to be calculated exactly (or nearly exactly) for known inputs."}
+{"_id": "31762", "title": "Definition:Stationary Model", "text": "A '''stationary model''' is a stochastic model for describing a time series which assumes that the underlying stochastic process remains in equilibrium about a constant mean level. That is, it is a stochastic model with an underlying stochastic process which is itself stationary."}
+{"_id": "31763", "title": "Definition:Statistical Equilibrium", "text": "Let $S$ be a stochastic process. Suppose that the observations of the time series to which $S$ gives rise have a constant mean level. Then $S$ is said to be in '''(statistical) equilibrium'''."}
+{"_id": "31764", "title": "Definition:Stationary Stochastic Process", "text": "A '''stationary stochastic process''' is stochastic process which remains in equilibrium about a constant mean level."}
+{"_id": "31765", "title": "Definition:Constant Mean Level", "text": "Let $S$ be a stochastic process giving rise to a time series $T$. $T$ has a '''constant mean level''' {{iff}} the mean of $S$ does not change as time progresses."}
+{"_id": "31766", "title": "Definition:Non-Stationary Stochastic Process", "text": "A '''non-stationary stochastic process''' is a stochastic process which does not remain in equilibrium about a constant mean level. That is, it is a stochastic process which is not a '''stationary stochastic process'''."}
+{"_id": "31767", "title": "Definition:Non-Stationary Model", "text": "A '''non-stationary model''' is a stochastic model for describing a time series which assumes that the underlying stochastic process does not remain in equilibrium about a constant mean level. That is, it is a stochastic model with an underlying stochastic process which is itself non-stationary."}
+{"_id": "31768", "title": "Definition:White Noise Process", "text": "Let $S$ be a stochastic process consisting of a time series $\\map {z_r} t$ such that: :the terms of the sequence $\\sequence {z_r}$ are independent random variables :the terms of $\\sequence {z_r}$ are governed by a normal distribution with zero expectation and a given variance $\\sigma^2$. Then $S$ is known as a '''white noise process'''."}
+{"_id": "31769", "title": "Definition:Independent Shocks", "text": "Let $M$ be a stochastic model which describes a time series in which adjacent observations are highly dependent. Then $M$ may be able to be modelled by a time series whose elements are of the form: :$\\map z t = \\map {z_d} t + \\map {z_r} t$ where: :$\\map {z_d} t$ has a deterministic model :$\\map {z_r} t$ has a stochastic model which consists of a sequence of independent random variables from a specified probability distribution (usually a white noise process). The terms of the sequence $\\sequence {z_r}$ are known as '''independent shocks'''."}
+{"_id": "31770", "title": "Definition:Linear Filter", "text": "Let $S$ be a stationary stochastic process governed by a white noise process: :$\\map z t = \\mu + a_t$ where: :$\\mu$ is a constant mean level :$a_t$ is an independent shock at timestamp $t$. A '''linear filter''' takes the terms of $S$, and uses a weight function $\\psi$ to apply a weighted sum of the past values so that: {{begin-eqn}} {{eqn | l = \\map z t       | r = \\mu + a_t + \\psi_1 a_{t - 1} + \\psi_2 a_{t - 2} + \\cdots       | c =  }} {{eqn | r = \\mu + \\map \\psi B a_t       | c =  }} {{end-eqn}} where $B$ denotes the backward shift operator, hence: :$\\map \\psi B := 1 + \\psi_1 B + \\psi_2 B^2 + \\cdots$ === Transfer Function === {{:Definition:Linear Filter/Transfer Function}} === Stable === {{:Definition:Linear Filter/Stable}}"}
+{"_id": "31771", "title": "Definition:Linear Filter/Transfer Function", "text": "The operator: :$\\map \\psi B := 1 + \\psi_1 B + \\psi_2 B^2 + \\cdots$ is the transfer function of $L$."}
+{"_id": "31773", "title": "Definition:Ferrier's Prime", "text": "'''Ferrier's prime''' is the prime number: :$\\dfrac {2^{148} + 1} {17} = 20 \\, 988 \\, 936 \\, 657 \\, 440 \\, 586 \\,486 \\, 151 \\, 264 \\, 256 \\, 610 \\, 222 \\, 593 \\, 863 \\, 921$ {{NamedforDef|Aimé Ferrier|cat = Ferrier}}"}
+{"_id": "31775", "title": "Definition:Weighted Sum", "text": "Let $S = \\sequence {x_1, x_2, \\ldots, x_n}$ be a sequence of real numbers. Let $\\map W x$ be a weight function to be applied to the terms of $S$. The '''weighted sum''' of $S$ is defined as: :$\\bar x := \\displaystyle \\sum_{i \\mathop = 1}^n \\map W {x_i} x_i$ This means that elements of $S$ with a larger weight contribute more to the '''weighted sum''' than those with a smaller weight."}
+{"_id": "31776", "title": "Definition:Linear Filter/Stable", "text": "Consider the sequence $\\sequence {\\psi_k}$ formed by the weight function $\\psi$ of $L$. Suppose that: :$\\displaystyle \\sum_k \\size {\\psi_k} < \\infty$ Then $L$ is said to be '''stable''', and the model for $S$ is stationary. Hence $\\mu$ is the mean about which $S$ varies."}
+{"_id": "31777", "title": "Definition:Autoregressive Model", "text": "Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \\dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \\dotsc$ Let $\\tilde z_t, \\tilde z_{t - 1}, \\tilde z_{t - 2}, \\dotsc$ be deviations from a constant mean level $\\mu$: :$\\tilde z_t = z_t - \\mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \\dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \\dotsc$ Let $M$ be a model where the current value of $S$ is expressed as a finite linear aggregate of the past values along with a shock: :$\\tilde z_t = \\phi_1 \\tilde z_{t - 1} + \\phi_2 \\tilde z_{t - 2} + \\dotsb + \\phi_p \\tilde z_{t - p} + a_t$ $M$ is known as an '''autoregressive (AR) process of order $p$'''."}
+{"_id": "31778", "title": "Definition:Metrical Geometry", "text": "'''Metrical geometry''' is the study of points, lines, surfaces and volumes in real space, and is based on the concepts of distance and angle. It is what is usually understood in the everyday world as '''geometry''' proper."}
+{"_id": "31780", "title": "Definition:Regression Model", "text": "Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \\dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \\dotsc$ Let $\\tilde z_t, \\tilde z_{t - 1}, \\tilde z_{t - 2}, \\dotsc$ be deviations from a constant mean level $\\mu$: :$\\tilde z_t = z_t - \\mu$ Let $M$ be a linear filter model defined as :$\\tilde z_t = \\phi_1 \\tilde x_1 + \\phi_2 \\tilde x_2 + \\dotsb + \\phi_p \\tilde x_p + a$ which relates a dependent variable $z$ to a set of independent variables $x_1, x_2, \\dotsc, x_p$ plus an error term $a$. $M$ is known as a '''regression model'''."}
+{"_id": "31781", "title": "Definition:Autoregressive Model/Autoregressive Operator", "text": "Let $\\map \\phi B$ be defined as: :$\\map \\phi B = 1 - \\phi_1 B - \\phi_2 B^2 - \\dotsb - \\phi_p B^p$ where $B$ denotes the backward shift operator. Then $\\map \\phi B$ is referred to as the '''autoregressive operator'''. Hence the '''autoregessive model''' can be written in the following compact manner: :$\\map \\phi B \\tilde z_t = a_t$"}
+{"_id": "31782", "title": "Definition:Autoregressive Model/Parameter", "text": "The '''parameters''' of $M$ consist of: :the constant mean level $\\mu$ :the variance $\\sigma_a^2$ of the underlying (usually white noise) process of the independent shock $a_t$ :the coefficients $\\phi_1$ to $\\phi_p$."}
+{"_id": "31784", "title": "Definition:Vector Space/Definition 1", "text": "Let $\\struct {K, +_K, \\times_K}$ be a field. Let $\\struct {G, +_G}$ be an abelian group. Let $\\struct {G, +_G, \\circ}_K$ be a unitary $K$-module. Then $\\struct {G, +_G, \\circ}_K$ is a '''vector space over $K$''' or a '''$K$-vector space'''. That is, a '''vector space''' is a unitary module whose scalar ring is a field."}
+{"_id": "31785", "title": "Definition:Vector Space/Division Ring", "text": "Let $\\struct {K, +_K, \\times_K}$ be a division ring. Let $\\struct {G, +_G}$ be an abelian group. Let $\\struct {G, +_G, \\circ}_K$ be a unitary $K$-module. Then $\\struct {G, +_G, \\circ}_K$ is a '''vector space over $K$''' or a '''$K$-vector space'''. That is, a vector space is a unitary module whose scalar ring is a division ring."}
+{"_id": "31786", "title": "Definition:Vector Space/Definition 2", "text": "Let $\\struct {K, +_K, \\times_K}$ be a field whose unity is $1_K$. Let $\\struct {G, +_G}$ be an abelian group. Let $\\struct {\\map {\\mathrm {End} } G, +, \\circ}$ be the endomorphism ring of $\\struct {G, +_G}$ such that $I_G$ is the identity mapping. Let $\\cdot: \\struct {K, +_K, \\times_K} \\to \\struct {\\map {\\mathrm {End} } G, +, \\circ}$ be a ring homomorphism from $K$ to $\\map {\\mathrm {End} } G$ which maps $1_K$ to $I_G$. Then $\\struct {G, +_G, \\cdot, K}$ is a '''vector space over $K$''' or a '''$K$-vector space'''."}
+{"_id": "31790", "title": "Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 2", "text": "{{begin-axiom}} {{axiom | n = \\text B 2         | q = \\forall B_1, B_2 \\in \\mathscr B         | mr= x \\in B_1 \\setminus B_2 \\implies \\exists y \\in B_2 \\setminus B_1 : \\paren {B_1 \\cup \\set y} \\setminus \\set x \\in \\mathscr B }} {{end-axiom}}"}
+{"_id": "31791", "title": "Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1", "text": "{{begin-axiom}} {{axiom | n = \\text B 1         | q = \\forall B_1, B_2 \\in \\mathscr B         | mr= x \\in B_1 \\setminus B_2 \\implies \\exists y \\in B_2 \\setminus B_1 : \\paren {B_1 \\setminus \\set x} \\cup \\set y \\in \\mathscr B }} {{end-axiom}}"}
+{"_id": "31792", "title": "Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)", "text": "Let $S$ be a finite set. Let $\\mathscr B$ be a non-empty set of subsets of $S$. === Definition 1 === $\\mathscr B$ is said to satisfy the base axiom {{iff}}: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1}} === Definition 2 === $\\mathscr B$ is said to satisfy the base axiom {{iff}}: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 2}} === Definition 3 === $\\mathscr B$ is said to satisfy the base axiom {{iff}}: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 3}} === Definition 4 === $\\mathscr B$ is said to satisfy the base axiom {{iff}}: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 4}} === Definition 5 === $\\mathscr B$ is said to satisfy the base axiom {{iff}}: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 5}} === Definition 6 === $\\mathscr B$ is said to satisfy the base axiom {{iff}}: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 6}} === Definition 7 === $\\mathscr B$ is said to satisfy the base axiom {{iff}}: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 7}}"}
+{"_id": "31793", "title": "Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 3", "text": "{{begin-axiom}} {{axiom | n = \\text B 3         | q = \\forall B_1, B_2 \\in \\mathscr B         | mr= \\exists \\text{ a bijection } \\pi : B_1 \\setminus B_2 \\to B_2 \\setminus B_1 : \\forall x \\in B_1 \\setminus B_2 : \\paren {B_1 \\setminus \\set x } \\cup \\set {\\map \\pi x} \\in \\mathscr B }} {{end-axiom}}"}
+{"_id": "31794", "title": "Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 4", "text": "{{begin-axiom}} {{axiom | n = \\text B 4         | q = \\forall B_1, B_2 \\in \\mathscr B         | mr= x \\in B_1 \\setminus B_2 \\implies \\exists y \\in B_2 \\setminus B_1 : \\paren {B_1 \\setminus \\set x} \\cup \\set y, \\paren {B_2 \\setminus \\set y} \\cup \\set x \\in \\mathscr B }} {{end-axiom}}"}
+{"_id": "31795", "title": "Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 5", "text": "{{begin-axiom}} {{axiom | n = \\text B 5         | q = \\forall B_1, B_2 \\in \\mathscr B         | mr= x \\in B_1 \\setminus B_2 \\implies \\exists y \\in B_2 \\setminus B_1 : \\paren {B_2 \\setminus \\set y}  \\cup \\set x \\in \\mathscr B }} {{end-axiom}}"}
+{"_id": "31796", "title": "Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 7", "text": "{{begin-axiom}} {{axiom | n = \\text B 7         | q = \\forall B_1, B_2 \\in \\mathscr B         | mr= \\exists \\text{ a bijection } \\pi : B_1 \\setminus B_2 \\to B_2 \\setminus B_1 : \\forall x \\in B_1 \\setminus B_2 : \\paren {B_2 \\setminus \\set {\\map \\pi x} } \\cup \\set x \\in \\mathscr B }} {{end-axiom}}"}
+{"_id": "31799", "title": "Definition:Image of Subset under Mapping/Definition 2", "text": "The '''image of $X$ under $f$''' is the element of the codomain of the direct image mapping $f^\\to: \\powerset S \\to \\powerset T$ of $f$: :$\\forall X \\in \\powerset S: \\map {f^\\to} X := \\set {t \\in T: \\exists s \\in X: \\map f s = t}$"}
+{"_id": "31800", "title": "Definition:Preimage of Subset under Mapping/Definition 1", "text": "The '''preimage of $Y$ under $f$''' is defined as: :$f^{-1} \\sqbrk Y := \\set {s \\in S: \\exists t \\in Y: \\map f s = t}$ That is, the '''preimage of $Y$ under $f$''' is the image of $Y$ under $f^{-1}$, where $f^{-1}$ can be considered as a relation."}
+{"_id": "31801", "title": "Definition:Preimage of Subset under Mapping/Definition 2", "text": "The '''preimage of $Y$ under $f$''' can be seen to be an element of the codomain of the inverse image mapping $f^\\gets: \\powerset T \\to \\powerset S$ of $f$: :$\\forall Y \\in \\powerset T: \\map {f^\\gets} Y := \\set {s \\in S: \\exists t \\in Y: \\map f s = t}$ Thus: :$\\forall Y \\subseteq T: f^{-1} \\sqbrk Y = \\map {f^\\gets} Y$"}
+{"_id": "31804", "title": "Definition:Moving Average Model", "text": "Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \\dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \\dotsc$ Let $\\tilde z_t$ be the deviation from a constant mean level $\\mu$: :$\\tilde z_t = z_t - \\mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \\dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \\dotsc$ Let $M$ be a model where the current value of $\\tilde z_t$ is expressed as a finite linear aggregate of the shocks: :$\\tilde z_t = a_t - \\theta_1 a_{t - 1} - \\theta_2 a_{t - 2} - \\dotsb - \\theta_q a_{t - q}$ $M$ is known as a '''moving average (MA) process of order $q$'''."}
+{"_id": "31806", "title": "Definition:Moving Average Model/Moving Average Operator", "text": "Let $\\map \\theta B$ be defined as: :$\\map \\theta B = 1 - \\theta_1 B - \\theta_2 B^2 - \\dotsb - \\theta_q B^q$ where $B$ denotes the backward shift operator. Then $\\map \\theta B$ is referred to as the '''moving average operator'''. Hence the '''moving average model''' can be written in the following compact manner: :$\\tilde z_t = \\map \\theta B a_t$"}
+{"_id": "31807", "title": "Definition:Moving Average Model/Parameter", "text": "The '''parameters''' of $M$ consist of: :the constant mean level $\\mu$ :the variance $\\sigma_a^2$ of the underlying (usually white noise) process of the independent shocks $a_t$ :the coefficients $\\theta_1$ to $\\theta_q$."}
+{"_id": "31808", "title": "Definition:Box-Jenkins Model/ARMA", "text": "Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \\dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \\dotsc$ Let $\\tilde z_t, \\tilde z_{t - 1}, \\tilde z_{t - 2}, \\dotsc$ be deviations from a constant mean level $\\mu$: :$\\tilde z_t = z_t - \\mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \\dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \\dotsc$ Let $M$ be a model where the current value of $\\tilde z_t$ is expressed as a combination of a finite linear aggregate of the past values along with a finite linear aggregate of the shocks: :$\\tilde z_t = \\phi_1 \\tilde z_{t - 1} + \\phi_2 \\tilde z_{t - 2} + \\dotsb + \\phi_p \\tilde z_{t - p} + a_t - \\theta_1 a_{t - 1} - \\theta_2 a_{t - 2} - \\dotsb - \\theta_q a_{t - q}$ $M$ is known as a '''mixed autoregressive (order $p$), moving average (order $q$) process''', usually referred as an '''ARMA process'''."}
+{"_id": "31809", "title": "Definition:ARMA Model/ARMA Operator", "text": "Using the autoregressive operator: :$\\map \\phi B = 1 - \\phi_1 B - \\phi_2 B^2 - \\dotsb - \\phi_p B^p$ and the moving average operator: :$\\map \\theta B = 1 - \\theta_1 B - \\theta_2 B^2 - \\dotsb - \\theta_q B^q$ the '''ARMA model''' can be written in the following compact manner: :$\\map \\phi B \\tilde z_t = \\map \\theta B a_t$ where $B$ denotes the backward shift operator. Hence: :$\\tilde z_t = \\map {\\phi^{-1} } B \\map \\theta B a_t$"}
+{"_id": "31810", "title": "Definition:ARMA Model/Parameter", "text": "The '''parameters''' of $M$ consist of: :the constant mean level $\\mu$ :the variance $\\sigma_a^2$ of the underlying (usually white noise) process of the independent shocks $a_t$ :the coefficients $\\phi_1$ to $\\phi_p$ :the coefficients $\\theta_1$ to $\\theta_q$."}
+{"_id": "31811", "title": "Definition:Scalar Ring/Module", "text": "Let $\\struct {G, +_G, \\circ}_R$ be a module, where: :$\\struct {R, +_R, \\times_R}$ is a ring :$\\struct {G, +_G}$ is an abelian group :$\\circ: R \\times G \\to G$ is a binary operation. Then the ring $\\struct {R, +_R, \\times_R}$ is called the scalar ring of $\\struct {G, +_G, \\circ}_R$."}
+{"_id": "31812", "title": "Definition:Scalar Ring/Unitary Module", "text": "Let $\\struct {G, +_G, \\circ}_R$ be a module, where: :$\\struct {R, +_R, \\times_R}$ is a ring with unity :$\\struct {G, +_G}$ is an abelian group :$\\circ: R \\times G \\to G$ is a binary operation. Then the ring $\\struct {R, +_R, \\times_R}$ is called the scalar ring of $\\struct {G, +_G, \\circ}_R$."}
+{"_id": "31813", "title": "Definition:Scalar Ring/Vector Space over Division Ring", "text": "Let $\\struct {G, +_G, \\circ}_K$ be a vector space over a division ring, where: :$\\struct {K, +_K, \\times_K}$ is a division ring :$\\struct {G, +_G}$ is an abelian group $\\struct {G, +_G}$ :$\\circ: K \\times G \\to G$ is a binary operation. Then the division ring $\\struct {K, +_K, \\times_K}$ is called the '''scalar division ring''' of $\\struct {G, +_G, \\circ}_K$, or just '''scalar ring'''."}
+{"_id": "31814", "title": "Definition:Box-Jenkins Model/ARIMA", "text": "Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \\dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \\dotsc$ Let $a_t, a_{t - 1}, a_{t - 2}, \\dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \\dotsc$ Let: :$w_t = \\nabla^d z_t$ where $\\nabla^d$ denotes the $d$th iteration of the backward difference operator. Let $M$ be a model where the current value of $w_t$ is expressed as a combination of a finite linear aggregate of the past values along with a finite linear aggregate of the shocks: :$w_t = \\phi_1 w_{t - 1} + \\phi_2 w_{t - 2} + \\dotsb + \\phi_p w_{t - p} + a_t - \\theta_1 a_{t - 1} - \\theta_2 a_{t - 2} - \\dotsb - \\theta_q a_{t - q}$ $M$ is known as an '''autoregressive integrated moving average (ARIMA) process of order $p$, $d$, $q$'''."}
+{"_id": "31815", "title": "Definition:ARIMA Model/ARIMA Operator", "text": "Using the autoregressive operator: :$\\map \\phi B = 1 - \\phi_1 B - \\phi_2 B^2 - \\dotsb - \\phi_p B^p$ and the moving average operator: :$\\map \\theta B = 1 - \\theta_1 B - \\theta_2 B^2 - \\dotsb - \\theta_q B^q$ the '''ARIMA model''' can be written in the following compact manner: :$\\map \\phi B w_t = \\map \\theta B a_t$ where $B$ denotes the backward shift operator. Hence: :$\\map \\varphi B z_t = \\map \\phi B \\paren {1 - B}^d z_t = \\map \\theta B a_t$ where: :$\\map \\varphi B = \\map \\phi B \\paren {1 - B}^d$"}
+{"_id": "31819", "title": "Definition:Scalar/Module", "text": "The elements of the scalar ring $\\struct {R, +_R, \\times_R}$ are called '''scalars'''."}
+{"_id": "31820", "title": "Definition:Scalar/Vector Space", "text": "The elements of the scalar field $\\struct {K, +_K, \\times_K}$ are called '''scalars'''."}
+{"_id": "31822", "title": "Definition:Vector Field", "text": "If a vector quantity $\\mathbf v$ can be associated with every point in a given space, then $\\mathbf v$ is said to be a '''vector field'''."}
+{"_id": "31826", "title": "Definition:Density (Physics)/Dimension", "text": "The dimension of '''density''' is $\\mathsf {M L}^{-3}$: mass per unit volume."}
+{"_id": "31827", "title": "Definition:Density (Physics)/Units", "text": "* The SI units of '''density''' are $\\mathrm {kg} \\, \\mathrm m^{-3}$ (kilograms per cubic metre). * The CGS units of '''density''' are $\\mathrm g \\, \\mathrm{cm}^{-3}$ or, less formally: $\\mathrm g / \\mathrm {cc}$ (grams per cubic centimetre). Thus: :$1 \\, \\mathrm g \\, \\mathrm {cm}^{-3} = 1000 \\, \\mathrm {kg} \\, \\mathrm m^{-3}$"}
+{"_id": "31828", "title": "Definition:Density (Physics)/Linear/Symbol", "text": "The usual symbol used to denote '''linear density''' is $\\mu$ (Greek letter '''mu'''). Sometimes $\\lambda$ (Greek letter '''lambda''') is also used."}
+{"_id": "31836", "title": "Definition:Mistake/Whoosho", "text": "A '''whoosho''' is a mistake caused by a complete failure to understand a concept vital to the accurate communication of an idea. It is also used for a failure to understand or recognise a joke. The word comes from the popular meme of how '''whoosh''' is supposedly the sound of whatever it is going over the person's head. :''\"That went so far over my head I didn't even hear it go whoosh.\"''"}
+{"_id": "31837", "title": "Definition:Current Time", "text": "The '''current time''' in the context of time series analysis is the timestamp of the most recent observation. By the very nature of time itself, the '''current time''' is always going to be some non-zero instant in the past."}
+{"_id": "31841", "title": "Definition:Successive Values of Time Series", "text": "Let $T$ be a time series whose observations are $\\map z {\\tau_1}, \\map z {\\tau_2}, \\dotsb, \\map z {\\tau_t}, \\dotsb$ at an unbroken sequence of timestamps $\\tau_1, \\tau_2, \\dotsb, \\tau_t, \\dotsb$. When we have $N$ such '''successive''' observations, without a missing value, we write: :$z_1, z_2, \\dotsb, z_t, \\dotsb, z_N$ and it is understood that the subscripts correspond directly to the timestamps. === Equispaced Time Series === {{:Definition:Successive Values of Time Series/Equispaced}}"}
+{"_id": "31842", "title": "Definition:Successive Values of Time Series/Equispaced", "text": "Let $T$ be equispaced with time interval $h$ between adjacent observations.  Then $N$ '''successive observations''', written as: :$z_1, z_2, \\dotsb, z_t, \\dotsb, z_N$ occur at timestamps: :$\\tau_0 + h, \\tau_0 + 2 h, \\dotsb, \\tau_0 + t h, \\dotsb, \\tau_0 + N h$ Hence we can refer to the observations at timestamp $\\tau_0 + t h$ as $z_t$."}
+{"_id": "31843", "title": "Definition:Discrete Time Series/Sampling", "text": "A '''discrete time series''' can be obtained by taking observations at predetermined instants from a comtinuous time series of measurements of the process in question."}
+{"_id": "31844", "title": "Definition:Discrete Time Series/Accumulation", "text": "A '''discrete time series''' can be obtained by collecting a varying quantity over a period of time. The observations are the measurements of the quantities at the end of each period."}
+{"_id": "31846", "title": "Definition:Space of Bounded Sequences", "text": "The '''space of bounded sequences''', denoted $\\ell^\\infty$ is defined as: :$\\displaystyle \\ell^\\infty := \\set{\\sequence{z_n}_{n \\mathop \\in \\N} \\in \\C^\\N: \\sup_{n \\mathop \\in \\N} \\size {z_n} < \\infty}$ As such, $\\ell^\\infty$ is a subspace of $\\C^\\N$, the space of all complex sequences. {{stub}}"}
+{"_id": "31847", "title": "Definition:Field Negative", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0_F$. Let $x \\in F$. The inverse of $x$ with respect to the addition operation $+$ in the additive group $\\struct {F, +}$ of $F$ is referred to as the '''(field) negative''' of $x$ and is denoted $-x$. That is, the '''(field) negative''' of $x$ is the element $-x$ of $F$ such that: :$x + \\paren {-x} = 0_F$"}
+{"_id": "31850", "title": "Definition:Realization of Stochastic Process", "text": "Let $S$ be a stochastic process. Let $T$ be a time series of observations of $S$ which has been acquired as $S$ evolves, according to the underlying probability distribution of $S$. Then $T$ is referred to as a '''realization''' of $S$. Thus we can regard the observation $z_t$ at some timestamp $t$, for example $t = 25$, as the '''realization''' of a random variable with probability mass function $\\map p {z_t}$. Similarly the observations $z_{t_1}$ and $z_{t_2}$ at times $t_1$ and $t_2$ can be regarded as the '''realizations''' of two random variables with joint probability mass function $\\map p {z_{t_1} }$ and $\\map p {z_{t_2} }$. Similarly, the observations making an equispaced time series can be described by an $N$-dimensional random variable $\\tuple {z_1, z_2, \\dotsc, z_N}$ with associated probability mass function $\\map p {z_1, z_2, \\dotsc, z_N}$."}
+{"_id": "31851", "title": "Definition:Strictly Stationary Stochastic Process", "text": "Let $S$ be a stochastic process giving rise to a time series $T$. $S$ is a '''strictly stationary stochastic process''' if its properties are unaffected by a change of origin of $T$."}
+{"_id": "31852", "title": "Definition:Mean of Stochastic Process", "text": "Let $S$ be a stochastic process giving rise to a time series $T$. The '''mean''' of $S$ over an interval $Q$ is the arithmetic mean of the observations of $T$ over all the timestamps within $Q$."}
+{"_id": "31853", "title": "Definition:Variance of Stochastic Process", "text": "Let $S$ be a stationary stochastic process giving rise to a time series $T$. The '''variance''' of $S$ is calculated as: :$\\sigma_z^2 = \\expect {\\paren {z_t - \\mu}^2} = \\displaystyle \\int_{-\\infty}^\\infty \\paren {z - \\mu}^2 \\map p z \\rd z$ where $\\map p z$ is the (constant) probability mass function of $S$. It is a measure of the spread about the constant mean level $\\mu$."}
+{"_id": "31854", "title": "Definition:Histogram", "text": "A '''histogram''' is a form of bar graph used to represent the frequency distribution grouped by class intervals."}
+{"_id": "31855", "title": "Definition:Sample Mean of Stochastic Process", "text": "Let $S$ be a stochastic process giving rise to a time series $T$. The '''sample mean''' of $S$ over a set of $N$ successive values $\\set {z_1, z_2, \\dotsb, z_N}$ is defined as: :$\\overline z := \\dfrac 1 N \\displaystyle \\sum_{t \\mathop = 1}^N z_t$"}
+{"_id": "31856", "title": "Definition:Sample Variance of Stochastic Process", "text": "Let $S$ be a stochastic process giving rise to a time series $T$. The '''sample variance''' of $S$ over a set of $N$ successive values $\\set {z_1, z_2, \\dotsb, z_N}$ is defined as: :$\\hat \\sigma_z^2 := \\dfrac 1 N \\displaystyle \\sum_{t \\mathop = 1}^N \\paren {z_t - \\overline z}^2$ where $\\overline z$ denotes the sample mean of $S$ over $\\set {z_1, z_2, \\dotsb, z_N}$."}
+{"_id": "31857", "title": "Definition:Scatter Diagram", "text": "A '''scatter diagram''' is a technique of graphical representation of the distribution of ordered pairs of random variables. The ordered pairs are treated as points in a Cartesian plane: the first coordinate along the $x$-axis and the second coordinate along the $y$-axis."}
+{"_id": "31858", "title": "Definition:Lag", "text": "Let $T$ be a time series. A '''lag''' is a constant time interval between two timestamps of $T$. Thus, for every observation $z_t$ of $T$, a pair of observations can be created with a given '''lag''' $k$ between them: :$\\tuple {z_t, z_{t + k} }$"}
+{"_id": "31859", "title": "Definition:Field (Abstract Algebra)/Addition", "text": "The distributand $+$ of a field $\\struct {F, +, \\times}$ is referred to as '''field addition''', or just '''addition'''."}
+{"_id": "31860", "title": "Definition:Field (Abstract Algebra)/Product", "text": "The distributive operation $\\times$ in $\\struct {F, +, \\times}$ is known as the '''(field) product'''."}
+{"_id": "31862", "title": "Definition:Additive Group of Field", "text": "The group $\\struct {F, +}$ is known as the '''additive group of $F$'''."}
+{"_id": "31863", "title": "Definition:Additive Inverse", "text": "=== Additive Inverse in Ring === {{:Definition:Additive Inverse/Ring}} === Additive Inverse in Field === {{:Definition:Additive Inverse/Field}} === Additive Inverse of Number === The concept is often encountered in the context of numbers: {{:Definition:Additive Inverse/Number}}"}
+{"_id": "31864", "title": "Definition:Additive Inverse/Field", "text": "Let $\\struct {F, +, \\times}$ be a field whose addition operation is $+$. Let $a \\in R$ be any arbitrary element of $F$. The '''additive inverse''' of $a$ is its inverse under addition, denoted $-a$: :$a + \\paren {-a} = 0_F$ where $0_F$ is the zero of $R$."}
+{"_id": "31865", "title": "Definition:Unity (Abstract Algebra)", "text": "=== Unity of Ring === {{:Definition:Unity (Abstract Algebra)/Ring}} === Unity of Field === {{:Definition:Unity (Abstract Algebra)/Field}}"}
+{"_id": "31866", "title": "Definition:Unity (Abstract Algebra)/Field", "text": "Let $\\struct {F, +, \\times}$ be a field. The identity of the multiplicative group $\\struct {F, \\times}$ is referred to as '''the unity of the field $\\struct {F, +, \\times}$'''. It is (usually) denoted $1_F$, where the subscript denotes the particular field to which $1_F$ belongs (or often $1$ if there is no danger of ambiguity)."}
+{"_id": "31867", "title": "Definition:Space of Convergent Sequences", "text": "The '''space of convergent sequences''', denoted $c$ is defined as: :$\\displaystyle c := \\set{\\sequence{z_n}_{n \\in \\N} \\in \\C^\\N : \\exists L \\in \\C : \\forall \\epsilon \\in \\R_{>0} : \\exists N \\in \\R: n > N \\implies \\cmod {z_n - L} < \\epsilon}$ As such, $c$ is a subspace of $\\C^\\N$, the space of all complex sequences. {{stub}}"}
+{"_id": "31868", "title": "Definition:Scalar Addition", "text": "=== Scalar Addition on Module === {{:Definition:Scalar Addition/Module}} === Scalar Addition on Vector Space === {{:Definition:Scalar Addition/Vector Space}}"}
+{"_id": "31869", "title": "Definition:Scalar Addition/Vector Space", "text": "Let $\\struct {K, +_K, \\times_K}$ be a field. Let $\\struct {G, +_G}$ be an abelian group. Let $V := \\struct {G, +_G, \\circ}_K$ be the corresponding '''vector space over $K$'''. The field addition operation $+_K$ on $V$ is known as '''scalar addition''' on $V$."}
+{"_id": "31870", "title": "Definition:Scalar Addition/Module", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {G, +_G}$ be an abelian group. Let $M := \\struct {G, +_G, \\circ}_R$ be the corresponding '''module over $R$''' (either a left module or a right module). The ring addition operation $+_R$ on $M$ is known as '''scalar addition''' on $M$."}
+{"_id": "31872", "title": "Definition:Vector Addition/Vector Space", "text": "Let $\\struct {F, +_F, \\times_F}$ be a field. Let $\\struct {G, +_G}$ be an abelian group. Let $V := \\struct {G, +_G, \\circ}_R$ be the corresponding '''vector space over $F$'''. The group operation $+_G$ on $V$ is known as '''vector addition''' on $V$."}
+{"_id": "31873", "title": "Definition:Scalar Multiplication/Module", "text": "Let $\\struct {G, +_G, \\circ}_R$ be an module (either a left module or a right module or both), where: :$\\struct {R, +_R, \\times_R}$ is a ring :$\\struct {G, +_G}$ is an abelian group. The operation $\\circ: R \\times G \\to G$ is called '''scalar multiplication'''."}
+{"_id": "31874", "title": "Definition:Scalar Multiplication/Vector Space", "text": "Let $\\struct {G, +_G, \\circ}_K$ be an vector space, where: :$\\struct {K, +_K, \\times_K}$ is a field :$\\struct {G, +_G}$ is an abelian group. The operation $\\circ: K \\times G \\to G$ is called '''scalar multiplication'''."}
+{"_id": "31877", "title": "Definition:Negative of Element", "text": "=== Ring Negative === {{:Definition:Ring Negative}} === Field Negative === As a field is also a ring, the same definition can be used: {{:Definition:Field Negative}} Category:Definitions/Ring Theory Category:Definitions/Field Theory itnu454mo7keuscqageynxn280saaus"}
+{"_id": "31879", "title": "Definition:Space of Zero-Limit Sequences", "text": "The '''space of zero-limit sequences''', denoted $c$ is defined as: :$\\displaystyle c_0 := \\set{\\sequence{z_n}_{n \\in \\N} \\in \\C^\\N : \\forall \\epsilon \\in \\R_{>0} : \\exists N \\in \\R_{> 0}: n > N \\implies \\cmod {z_n} < \\epsilon}$ As such, $c_0$ is a subspace of $\\C^\\N$, the space of all complex sequences. {{stub}}"}
+{"_id": "31880", "title": "Definition:Space of Almost-Zero Sequences", "text": "The '''space of almost-zero sequences''', denoted $c_{00}$ is defined as: :$\\displaystyle c_{00} := \\set{\\sequence{z_n}_{n \\in \\N} \\in \\C^\\N : \\exists N \\in \\R_{> 0}: n > N \\implies z_n =0}$ As such, $c_{00}$ is a subspace of $\\C^\\N$, the space of all complex sequences. {{stub}}"}
+{"_id": "31883", "title": "Definition:C^k Norm", "text": "Let $f \\in \\map {\\CC^k} {\\R}$ be a function of differentiability class $k$. Then '''$C^k$ norm''' is defined as: :$\\displaystyle \\norm {f}_{\\map {C^k} \\R} := \\sum_{i \\mathop = 0}^k \\sup_{x \\in \\R} \\size {\\map {f^{\\paren i}} x} = \\sum_{i \\mathop = 0}^k \\norm{f^{\\paren i} }_\\infty$ where $f^{\\paren i}$ denotes the $i$-th derivative {{WRT}} $x$, and $\\norm {\\, \\cdot \\,}_\\infty$ denotes the supremum norm."}
+{"_id": "31884", "title": "Definition:Autocovariance", "text": "Let $S$ be a stochastic process giving rise to a time series $T$. The '''autocovariance''' of $S$ at lag $k$ is defined as: :$\\gamma_k := \\cov {z_t, z_{t + k} } = \\expect {\\paren {z_t - \\mu} \\paren {z_{t - k} - \\mu} }$ where: :$z_t$ is the observation at time $t$ :$\\mu$ is the mean of $S$ :$\\expect \\cdot$ is the expectation."}
+{"_id": "31885", "title": "Definition:Autocorrelation", "text": "Let $S$ be a stochastic process giving rise to a time series $T$. The '''autocorrelation''' of $S$ at lag $k$ is defined as: :$\\rho_k := \\dfrac {\\expect {\\paren {z_t - \\mu} \\paren {z_{t + k} - \\mu} } } {\\sqrt {\\expect {\\paren {z_t - \\mu}^2} \\expect {\\paren {z_{t + k} - \\mu}^2} } }$ where: :$z_t$ is the observation at time $t$ :$\\mu$ is the mean of $S$ :$\\expect \\cdot$ is the expectation."}
+{"_id": "31886", "title": "Definition:Autocovariance Matrix", "text": "Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\\sequence {s_n} = \\tuple {z_1, z_2, \\dotsb, z_n}$ The '''autocovariance matrix''' associated with $S$ for $\\sequence {s_n}$ is: :$\\boldsymbol \\Gamma_n = \\begin {pmatrix} \\gamma_0 & \\gamma_1 & \\gamma_2 & \\cdots & \\gamma_{n - 1} \\\\ \\gamma_1 & \\gamma_0 & \\gamma_1 & \\cdots & \\gamma_{n - 2} \\\\ \\gamma_2 & \\gamma_1 & \\gamma_0 & \\cdots & \\gamma_{n - 3} \\\\ \\vdots   & \\vdots   & \\vdots   & \\ddots & \\vdots \\\\ \\gamma_{n - 1} & \\gamma_{n - 2} & \\gamma_{n - 3} & \\cdots & \\gamma_0 \\end {pmatrix}$ where $\\gamma_k$ is the autocovariance of $S$ at lag $k$. That is, such that: :$\\sqbrk {\\Gamma_n}_{i j} = \\gamma_{\\size {i - j} }$"}
+{"_id": "31887", "title": "Definition:Autocorrelation Matrix", "text": "Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\\sequence {s_n} = \\tuple {z_1, z_2, \\dotsb, z_n}$ The '''autocorrelation matrix''' associated with $S$ for $\\sequence {s_n}$ is: :$\\mathbf P_n = \\begin {pmatrix} 1 & \\rho_1 & \\rho_2 & \\cdots & \\rho_{n - 1} \\\\ \\rho_1 & 1 & \\rho_1 & \\cdots & \\rho_{n - 2} \\\\ \\rho_2 & \\rho_1 & 1 & \\cdots & \\rho_{n - 3} \\\\ \\vdots   & \\vdots   & \\vdots   & \\ddots & \\vdots \\\\ \\rho_{n - 1} & \\rho_{n - 2} & \\rho_{n - 3} & \\cdots & 1 \\end {pmatrix}$ where $\\rho_k$ is the autocorrelation of $S$ at lag $k$. That is, such that: :$\\sqbrk {P_n}_{i j} = \\rho_{\\size {i - j} }$"}
+{"_id": "31888", "title": "Definition:Positive Definite Matrix", "text": "Let $\\mathbf A$ be a square matrix of order $n$. $\\mathbf A$ is '''positive definite''' {{iff}}: :$(1): \\quad \\mathbf A$ is symmetric :$(2): \\quad$ for all column matrices $\\mathbf x$ of order $n$, $\\mathbf x^\\intercal \\mathbf A \\mathbf x$ is strictly positive."}
+{"_id": "31889", "title": "Definition:Matrix/Order/Column", "text": "Let $\\mathbf A$ be an $n \\times 1$ column matrix. Then the '''order''' of $\\mathbf A$ is defined as being $n$."}
+{"_id": "31891", "title": "Definition:Induced Norm", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,}_X}$ be a normed vector space. Let $Y \\subseteq X$ be a subspace. Then the '''induced norm (on $Y$)''' is defined as the restriction of $\\norm {\\, \\cdot \\,}_X$ to $Y$. :$\\norm {\\, \\cdot \\,}_Y := \\norm {\\, \\cdot \\,}_X {\\restriction_Y}$"}
+{"_id": "31892", "title": "Definition:Wholly Real/Abbreviated Notation", "text": "Let $z = a + i b$ be a complex number such that $b = 0$. That is, let $z$ be wholly real: $z = a + 0 i$, or $\\tuple {a, 0}$ Despite the fact that $z$ is still a complex number, it is commonplace to use the same notation as if it were a real number, and hence say $z = a$. While it is in theory important to distinguish between a real number and its corresponding wholly real complex number, in practice it makes little difference."}
+{"_id": "31894", "title": "Definition:Pure Mathematics", "text": "'''Pure mathematics''' is the branch of mathematics which is not concerned with real-world applications."}
+{"_id": "31902", "title": "Definition:Parallel (Geometry)/Line to Plane", "text": "Let $L$ be a straight line. Let $P$ be a plane. Then $L$ and $P$ are '''parallel''' {{iff}}, when produced indefinitely, they do not intersect at any point."}
+{"_id": "31903", "title": "Definition:Method of Least Squares", "text": "=== Approximation Theory === {{:Definition:Method of Least Squares (Approximation Theory)}} === Statistics === {{:Definition:Method of Least Squares (Statistics)}} Category:Definitions/Approximation Theory Category:Definitions/Statistics 51987t69f5t5rza6b9j271im7jtcjbs"}
+{"_id": "31904", "title": "Definition:Cardinality/Natural Numbers", "text": "When the natural numbers are defined as von Neumann construction of natural numbers, the cardinality function can be viewed as the identity mapping on $\\N$. That is: :$\\forall n \\in N: \\card n := n$"}
+{"_id": "31909", "title": "Definition:Spline", "text": "=== Draftsman's tool === {{:Definition:Spline (Tool)}} === Spline Function === {{:Definition:Spline Function}}"}
+{"_id": "31910", "title": "Definition:Spline (Tool)", "text": ":400px A '''spline''' is a drawing tool which consists of a long strip of uniformly flexible material fixed in position at a number of points whose tension creates a smooth curve passing through those points. Its purpose is to transferring that curve to another material."}
+{"_id": "31911", "title": "Definition:Spline Function", "text": "Let $\\closedint a b$ be a closed real interval. Let $T : = \\set {a = t_0, t_1, t_2, \\ldots, t_{n - 1}, t_n = b}$ form a subdivision of $\\closedint a b$. Let $S: \\closedint a b \\to \\R$ be a continuous function on $\\closedint a b$ whose values on $t_0, t_1, \\ldots, t_n$ are known. On each of the intervals $\\closedint {t_k} {t_{k + 1} }$, let $P_k: \\closedint {t_k} {t_{k + 1} }: \\R$ be a polynomial function such that: :for $t$ on each of $t_k < t < t_{k + 1}$: $\\map S t = \\map {P_k} t$ The function $S: \\closedint a b \\to \\R$ is known as a '''spline function''' on $T$."}
+{"_id": "31912", "title": "Definition:Spline Function/Knot", "text": "The points $T := \\set {t_0, t_1, t_2, \\ldots, t_{n - 1}, t_n}$ of $S$ are known as the '''knots'''."}
+{"_id": "31913", "title": "Definition:Spline Function/Knot/Knot Vector", "text": "The ordered $n + 1$-tuple $\\mathbf t := \\tuple {t_0, t_1, t_2, \\ldots, t_{n - 1}, t_n}$ of $S$ is known as the '''knot vector'''."}
+{"_id": "31914", "title": "Definition:Spline Function/Uniform", "text": "$S$ is a '''uniform spline''' {{iff}} $T$ is a normal subdivision. That is, {{iff}} the knots of $S$ are equally spaced."}
+{"_id": "31915", "title": "Definition:Spline Function/Degree", "text": "The '''degree''' of $S$ is the maximum degree of the polynomials $P_k$ fitted between $t_k$ and $t_{k + 1}$."}
+{"_id": "31916", "title": "Definition:Spline Function/Order", "text": "Some sources, instead of referring to the '''degree''' of a spline, use the '''order'''. Let the maximum degree of the polynomials $P_k$ fitted between $t_k$ and $t_{k + 1}$ be $n$. The '''order''' of $S$ is then $n + 1$."}
+{"_id": "31917", "title": "Definition:Fundamental Circuit (Matroid)", "text": "Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $B$ be a base of $M$. Let $x \\in S \\setminus B$. The '''fundamental circuit of $x$ in the base B''', denoted $\\map C {x, B}$, is the unique circuit such that: :$x \\in \\map C {x, B} \\subseteq B \\cup \\set x$"}
+{"_id": "31918", "title": "Definition:Set/Uniqueness of Elements/Multiple Specification", "text": "For a given '''set''', an object is either '''in''' the set or '''not in''' the set. So, if an element is in a set, then it is in the set '''only once''', however many times it may appear in the definition of the set. Thus, the set $\\set {1, 2, 2, 3, 3, 4}$ is the same set as $\\set {1, 2, 3, 4}$. $2$ and $3$ are in the set, and '''listing''' them twice makes no difference to the set's '''contents'''. Like the membership of a club, if you're in, you're in -- however many membership cards you have to prove it."}
+{"_id": "31919", "title": "Definition:Set/Uniqueness of Elements/Order of Listing", "text": "It makes no difference in what order the elements of a set are specified. This means that the sets $S = \\set {1, 2, 3, 4}$ and $T = \\set {3, 4, 2, 1}$ are the ''same set''."}
+{"_id": "31920", "title": "Definition:Set/Uniqueness of Elements/Equality of Sets", "text": "By definition of '''set equality''' {{:Definition:Set Equality/Definition 1}} So, to take the club membership analogy, if two clubs had exactly the same members, the clubs would be considered as ''the same club'', although they may be given different '''names'''. This follows from the definition of '''equals''' given above. Note that there '''are''' mathematical constructs which '''do''' take into account both (or either of) the order in which the elements appear, and the number of times they appear, but these are ''not'' '''sets''' as such."}
+{"_id": "31922", "title": "Definition:Parabola/Vertex", "text": "The '''vertex''' of $P$ is the point where the axis intersects $P$."}
+{"_id": "31923", "title": "Definition:Parabola/Axis", "text": "The '''axis''' of $P$ is the straight line passing through the focus of $P$ perpendicular to the directrix $D$."}
+{"_id": "31926", "title": "Definition:Age (Time)", "text": "The '''age''' of a physical objectis defined as the period of time over which it has been in existence."}
+{"_id": "31938", "title": "Definition:Classical Algorithm", "text": "The '''classical algorithms''' are the conventional techniques for performing arithmetic on numbers with more than $1$ digit."}
+{"_id": "31940", "title": "Definition:Multi-Digit Integer", "text": "Let $b \\in \\N_{>1}$ be a natural number greater than $1$. An '''$n$-digit integer base $b$''' is an integer which has no more than $n$ digits when expressed in base $b$. That is, it is strictly less than $b^n$."}
+{"_id": "31941", "title": "Definition:Classical Algorithm/Primitive Operation", "text": "The '''primitive operations''' of the '''classical algorithms''' are the following operations which are performed on '''two''' $1$-digit integers $x$ and $y$: === Addition === {{:Definition:Classical Algorithm/Primitive Addition}} === Subtraction === {{:Definition:Classical Algorithm/Primitive Subtraction}} === Multiplication === {{:Definition:Classical Algorithm/Primitive Multiplication}} === Division === {{:Definition:Classical Algorithm/Primitive Division}}"}
+{"_id": "31942", "title": "Definition:Carry Digit", "text": "A '''carry digit''' is a digit that appears as a result of the operation of one of the primitive operations of the classical algorithms which is to be applied to a further primitive operation in some manner appropriate to the classical algorithm in question."}
+{"_id": "31943", "title": "Definition:Classical Algorithm/Primitive Addition", "text": "Let $x$ and $y$ be $1$-digit integers. Let $z$ be a '''carry digit''' such that either $z = 0$ or $z = 1$. Addition of $x$, $y$ and $z$ gives a $1$-digit '''sum''' $s$ and a $1$-digit '''carry''' $c$ such that: {{begin-eqn}} {{eqn | l = s       | r = \\paren {x + y + z} \\pmod b }} {{eqn | l = c       | r = \\floor {\\dfrac {x + y + z} b} }} {{end-eqn}}"}
+{"_id": "31944", "title": "Definition:Classical Algorithm/Primitive Subtraction", "text": "Let $x$ and $y$ be $1$-digit integers. Let $z$ be a '''carry digit''' such that either $z = 0$ or $z = -1$. Subtraction of $y$ from $x$ with $z$ gives a $1$-digit '''difference''' $d$ and a $1$-digit '''carry''' $c$ such that: {{begin-eqn}} {{eqn | l = d       | r = \\paren {x - y + z} \\pmod b }} {{eqn | l = c       | r = \\floor {\\dfrac {x - y + z} b} }} {{end-eqn}}"}
+{"_id": "31946", "title": "Definition:Classical Algorithm/Primitive Multiplication", "text": "Multiplication of '''two''' $1$-digit integers $x$ and $y$ gives a $1$-digit '''product''' $p$ and a $1$-digit '''carry''' $c$ such that: {{begin-eqn}} {{eqn | l = p       | r = x \\times y \\pmod b }} {{eqn | o =  }} {{eqn | l = c       | r = \\dfrac {x \\times y - p} b }} {{end-eqn}}"}
+{"_id": "31947", "title": "Definition:Classical Algorithm/Primitive Division", "text": "Let $y$ be a $1$-digit integer. Let $x$ be a $2$-digit integer $x_1 b + x_2$, where $x_1$ and $x_2$ are $1$-digit integers. Suppose $x_1 \\ge y$. Then division of $x_1$ by $y$ gives a $1$-digit '''quotient''' $q$ and a $1$-digit '''remainder''' $r$, which is used as a '''carry''', such that: {{begin-eqn}} {{eqn | l = x_1       | r = q \\times y + r }} {{end-eqn}} Suppose $x_1 < y$. Then division of $x = x_1 b + x_2$ by $y$ gives a $1$-digit '''quotient''' $q$ and a $1$-digit '''remainder''' $r$, which is used as a '''carry''', such that: {{begin-eqn}} {{eqn | l = x       | r = q \\times y + r }} {{end-eqn}}"}
+{"_id": "31948", "title": "Definition:Classical Algorithm/Primitive Addition/Base 10 Addition Table", "text": "The primitive addition operation for conventional base $10$ arithmetic of two $1$-digit integers can be presented as a pair of operation tables as follows: :$\\begin{array}{c|cccccccccc} s & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\ \\hline 0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\ 1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 0 \\\\ 2 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 0 & 1 \\\\ 3 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 0 & 1 & 2 \\\\ 4 & 4 & 5 & 6 & 7 & 8 & 9 & 0 & 1 & 2 & 3 \\\\ 5 & 5 & 6 & 7 & 8 & 9 & 0 & 1 & 2 & 3 & 4  \\\\ 6 & 6 & 7 & 8 & 9 & 0 & 1 & 2 & 3 & 4 & 5  \\\\ 7 & 7 & 8 & 9 & 0 & 1 & 2 & 3 & 4 & 5 & 6  \\\\ 8 & 8 & 9 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7  \\\\ 9 & 9 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8  \\\\ \\end{array} \\qquad \\begin{array}{c|cccccccccc} c & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\ \\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\\\ 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\\\ 4 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\\\ 5 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\\\ 6 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 7 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 8 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 9 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ \\end{array}$"}
+{"_id": "31949", "title": "Definition:Classical Algorithm/Primitive Subtraction/Base 10 Subtraction Table", "text": "The primitive subtraction operation for conventional base $10$ arithmetic on two $1$-digit integers can be presented as a pair of operation tables as follows: :$\\begin{array}{c|cccccccccc} d & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\ \\hline 0 & 0 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \\\\ 1 & 1 & 0 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 \\\\ 2 & 2 & 1 & 0 & 9 & 8 & 7 & 6 & 5 & 4 & 3 \\\\ 3 & 3 & 2 & 1 & 0 & 9 & 8 & 7 & 6 & 5 & 4 \\\\ 4 & 4 & 3 & 2 & 1 & 0 & 9 & 8 & 7 & 6 & 5 \\\\ 5 & 5 & 4 & 3 & 2 & 1 & 0 & 9 & 8 & 7 & 6  \\\\ 6 & 6 & 5 & 4 & 3 & 2 & 1 & 0 & 9 & 8 & 7  \\\\ 7 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 & 9 & 8  \\\\ 8 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 & 9  \\\\ 9 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0  \\\\ \\end{array} \\qquad \\begin{array}{c|cccccccccc} c & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\ \\hline 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 2 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 3 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 4 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\\\ 5 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\\\ 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\\\ 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\\\ 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ \\end{array}$"}
+{"_id": "31950", "title": "Definition:Classical Algorithm/Primitive Multiplication/Base 10 Multiplication Table", "text": "The primitive multiplication operation for conventional base $10$ arithmetic of two $1$-digit integers can be presented as a pair of operation tables as follows: :$\\begin{array}{c|cccccccccc} p & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\ \\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\ 2 & 0 & 2 & 4 & 6 & 8 & 0 & 2 & 4 & 6 & 8 \\\\ 3 & 0 & 3 & 6 & 9 & 2 & 5 & 8 & 1 & 4 & 7 \\\\ 4 & 0 & 4 & 8 & 2 & 6 & 0 & 4 & 8 & 2 & 6 \\\\ 5 & 0 & 5 & 0 & 5 & 0 & 5 & 0 & 5 & 0 & 5  \\\\ 6 & 0 & 6 & 2 & 8 & 4 & 0 & 6 & 2 & 8 & 4  \\\\ 7 & 0 & 7 & 4 & 1 & 8 & 5 & 2 & 9 & 6 & 3  \\\\ 8 & 0 & 8 & 6 & 4 & 2 & 0 & 8 & 6 & 4 & 2  \\\\ 9 & 0 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1  \\\\ \\end{array} \\qquad \\begin{array}{c|cccccccccc} c & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\ \\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 2 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\\\ 3 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 2 \\\\ 4 & 0 & 0 & 0 & 1 & 1 & 2 & 2 & 2 & 3 & 3 \\\\ 5 & 0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 & 4 & 4 \\\\ 6 & 0 & 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 \\\\ 7 & 0 & 0 & 1 & 2 & 2 & 3 & 4 & 4 & 5 & 6 \\\\ 8 & 0 & 0 & 1 & 2 & 3 & 4 & 4 & 5 & 6 & 7 \\\\ 9 & 0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\ \\end{array}$"}
+{"_id": "31951", "title": "Definition:Classical Algorithm/Addition", "text": "Let $u = \\sqbrk {u_{n - 1} u_{n - 2} \\dotsm u_1 u_0}_b$ and $v = \\sqbrk {v_{n - 1} v_{n - 2} \\dotsm v_1 v_0}_b$ be $n$-digit integers. The '''classical addition algorithm''' forms their $n + 1$-digit sum $u + v$: :$w = \\sqbrk {w_n w_{n - 1} \\dotsm w_1 w_0}_b$ where $w_n$ is either $0$ or $1$. The steps are: :$(\\text A 1): \\quad$ Set $j = 0$, $k = 0$. ::::$j$ is used to run through all the digit positions ::::$k$ keeps track of the carry digit between each step. :$(\\text A 2): \\quad$ Calculate digit $j$: ::::Calculate $\\begin {cases} s = \\paren {u_j + v_j + k} \\pmod b \\\\ c = \\floor {\\dfrac {u_j + v_j + k} b} \\end {cases}$ using the primitive addition. ::::Set $w_j$ to $s$. ::::Set $k$ to $c$. :$(\\text A 3): \\quad$ Add $1$ to $j$, using conventional integer addition. ::::If $j < n$, return to $(\\text A 2)$. ::::Otherwise, set $w_n$ equal to $k$ and exit."}
+{"_id": "31952", "title": "Definition:Classical Algorithm/Subtraction", "text": "Let $u = \\sqbrk {u_{n - 1} u_{n - 2} \\dotsm u_1 u_0}_b$ and $v = \\sqbrk {v_{n - 1} v_{n - 2} \\dotsm v_1 v_0}_b$ be $n$-digit integers. The '''classical subtraction algorithm''' forms their $n$-digit difference $u - v$: :$w = \\sqbrk {w_n w_{n - 1} \\dotsm w_1 w_0}_b$ where $w_n$ is either $0$ or $1$. The steps are: :$(\\text S 1): \\quad$ Set $j = 0$, $k = 0$. ::::$j$ is used to run through all the digit positions ::::$k$ keeps track of the carry digit between each step. :$(\\text S 2): \\quad$ Calculate digit $j$: ::::Calculate $\\begin {cases} d = \\paren {u_j + v_j - k} \\pmod b \\\\ c = \\floor {\\dfrac {u_j - v_j + k} b} \\end {cases}$ using the primitive subtraction. ::::Set $w_j$ to $d$. ::::Set $k$ to $c$. :$(\\text S 3): \\quad$ Add $1$ to $j$, using conventional integer addition. ::::If $j < n$, return to $(\\text S 2)$. ::::Otherwise exit."}
+{"_id": "31953", "title": "Definition:Classical Algorithm/Multiplication", "text": "Let $u = \\sqbrk {u_{m - 1} u_{m - 2} \\dotsm u_1 u_0}_b$ and $v = \\sqbrk {v_{n - 1} v_{n - 2} \\dotsm v_1 v_0}_b$ be $m$-digit and $n$-digit integers respectively. The '''classical multiplication algorithm''' forms their $m + n$-digit product $u v$: :$w = \\sqbrk {w_{m + n - 1} w_{m + n - 2} \\dotsm w_1 w_0}_b$ The steps are: :$(\\text M 1): \\quad$ Initialise: ::::Set $w_{m - 1}, w_{m - 2}, \\dotsc, w_1, w_0 := 0$ ::::Set $j := 0$ :$(\\text M 2): \\quad$ Is $v_j = 0$? ::::If so, go to Step $(\\text M 6)$. :$(\\text M 3): \\quad$ Initialise loop: ::::Set $i := 0$ ::::Set $k := 0$ :$(\\text M 4): \\quad$ Multiply and Add: ::::Set $t := u_i \\times v_j + w_{i + j} + k$ ::::Set $w_{i + j} := t \\pmod b$ ::::Set $k := \\floor {\\dfrac t b}$ :$(\\text M 5): \\quad$ Loop on $i$: ::::Increase $i$ by $1$. ::::If $i < m$, go back to Step $(\\text M 4)$. ::::Otherwise, set $w_{j + m} := k$ :$(\\text M 6): \\quad$ Loop on $j$: ::::Increase $j$ by $1$. ::::If $j < n$, go back to Step $(\\text M 2)$. ::::Otherwise exit."}
+{"_id": "31954", "title": "Definition:Classical Algorithm/Division", "text": "Let $u = \\sqbrk {u_{m + n - 1} u_{m - 2} \\dotsm u_1 u_0}_b$ and $v = \\sqbrk {v_{n - 1} v_{n - 2} \\dotsm v_1 v_0}_b$ be $m + n$-digit and $n$-digit integers respectively, where $v_{n - 1} \\ne 0$ and $n > 1$. The '''classical division algorithm''' forms: :their $m + 1$-digit quotient $\\floor {\\dfrac u v} = q = \\sqbrk {q_m q_{m - 1} \\dotsm q_1 q_0}_b$ and: :their $n$-digit remainder $u \\pmod v = r = \\sqbrk {r_{n - 1} \\dotsm r_1 r_0}_b$ The steps are: :$(\\text D 1): \\quad$ Normalize: ::::Set: {{begin-eqn}} {{eqn | l = d       | o = :=       | r = \\floor {\\dfrac b {v_{n - 1} - 1} } }} {{eqn | l = \\sqbrk {u_{m + n} u_{m - 1} \\dotsm u_1 u_0}_b       | o = :=       | r = \\sqbrk {u_{m + n - 1} u_{m - 2} \\dotsm u_1 u_0}_b \\times d }} {{eqn | l = \\sqbrk {v_{n - 1} \\dotsm v_1 v_0}_b       | o = :=       | r = \\sqbrk {v_{n - 1} \\dotsm v_1 v_0}_b \\times d }} {{end-eqn}} ::::Note that if $d = 1$, all that needs to be done is to set $u_{m + n} := 0$. :$(\\text D 2): \\quad$ Initialise $j$: ::::Set $j := m$ :$(\\text D 3): \\quad$ Calculate the quotient $\\hat q$ and remainder $\\hat r$: ::::Set $\\hat q := \\floor {\\dfrac {u_{j + n} b + u_{j + n - 1} } {v_{n - 1} } }$ ::::Set $\\hat r := \\paren {u_{j + n} b + u_{j + n - 1} } \\pmod {v_{n - 1} }$ ::::Test whether $\\hat q = b$ or $\\hat q v_{n - 1} > b \\hat r + u_{j + n - 2}$. ::::If so, decrease $\\hat q$ by $1$ and increase $\\hat r$ by $v_{n - 1}$. ::::If $\\hat r < b$ repeat this test. :$(\\text D 4): \\quad$ Multiply and Subtract: ::::Replace $\\sqbrk {u_{j + n} u_{j + n - 1} \\dotsm u_j}_b$ by $\\sqbrk {u_{j + n} u_{j + n - 1} \\dotsm u_j}_b - \\hat q \\sqbrk {0 v_{n - 1} v_{n - 2} \\dotsm v_1 v_0}_b$ ::::If the result of this step is negative, add $b_{n + 1}$ to it. :$(\\text D 5): \\quad$ Test remainder: ::::Set $q_j := \\hat q$. ::::If the result of '''Step $(\\text D 4)$''' was negative, go to '''Step $(\\text D 6)$''' ::::Otherwise go to '''Step $(\\text D 7)$''' :$(\\text D 6): \\quad$ Add back: ::::Decrease $q_j$ by $1$ ::::Add $\\sqbrk {0 v_{n - 1} v_{n - 2} \\dotsm v_1 v_0}_b$ to $\\sqbrk {u_{j + n} u_{j + n - 1} \\dotsm u_j}_b$ (ignoring the carry) :$(\\text D 7): \\quad$ Loop on $j$: ::::Decrease $j$ by $1$ ::::If $j \\ge 0$ go back to '''Step $(\\text D 3)$''' :$(\\text D 8): \\quad$ Unnormalise: ::::The required quotient is $\\sqbrk {q_m q_{m - 1} \\dotsm q_1 q_0}_b$ ::::The required remainder is obtained by dividing $\\sqbrk {u_{n - 1} u_{m - 2} \\dotsm u_1 u_0}_b$ by $d$."}
+{"_id": "31955", "title": "Definition:Unit Disk", "text": "A '''unit disk''' is a disk whose radius is equal to $1$. Category:Definitions/Metric Spaces dbjkoi549op8ywxtt93i4k4cig1yzxz"}
+{"_id": "31956", "title": "Definition:Disk/Radius", "text": "Let $\\map B {a, r}$ be a disk. The '''radius''' of $\\map B {a, r}$ is the parameter $r$."}
+{"_id": "31957", "title": "Definition:Disk/Center", "text": "Let $\\map B {a, r}$ be a disk. The '''center''' of $\\map B {a, r}$ is the parameter $a$."}
+{"_id": "31958", "title": "Definition:Transcendental Function", "text": "A '''transcendental function''' is an analytic function which is not an algebraic function. That is, it cannot be expressed as a polynomial equation. === Defined by Integral === {{:Definition:Transcendental Function/Defined by Integral}} === Defined by Differential Equation === {{:Definition:Transcendental Function/Defined by Differential Equation}}"}
+{"_id": "31959", "title": "Definition:Transcendental Function/Also defined as", "text": "Some sources define a '''transcendental function''' as a real function or complex function which is not an elementary function. However, the distinction between what is and is not an elementary function is more or less arbitrary, consisting of both algebraic functions and those derived from the exponential function, which itself is not algebraic. The current school of thought appears to be that this definition: \"not an elementary function\" is actually considered to be erroneous. However, the distinction is not considered particularly important nowadays. As long as it is made clear which definition is being used at the time, that would be adequate."}
+{"_id": "31960", "title": "Definition:Algebraic Function", "text": "=== Real Algebraic Function === {{:Definition:Algebraic Function/Real}} === Complex Algebraic Function === {{:Definition:Algebraic Function/Complex}}"}
+{"_id": "31961", "title": "Definition:Algebraic Function/Real", "text": "Let $y$ be a solution to the polynomial equation: :$\\map {p_0} x + \\map {p_1} x y + \\dotsb + \\map {p_{n - 1} } x y^{n - 1} + \\map {p_n} x y^n = 0$ where $\\map {p_0} x \\ne 0, \\map {p_1} x, \\dotsc, \\map {p_n} x$ are real polynomial functions in $x$. Then $y = \\map f x$ is a '''(real) algebraic function''':"}
+{"_id": "31962", "title": "Definition:Algebraic Function/Complex", "text": "Let $w$ be a solution to the polynomial equation: :$\\map {p_0} z + \\map {p_1} z w + \\dotsb + \\map {p_{n - 1} } z w^{n - 1} + \\map {p_n} z w^n = 0$ where $\\map {p_0} z \\ne 0, \\map {p_1} z, \\dotsc, \\map {p_n} z$ are complex polynomial functions in $z$. Then $w = \\map f z$ is a '''(complex) algebraic function''':"}
+{"_id": "31963", "title": "Definition:Perpendicular Projection", "text": "Let $\\PP$ denote the plane. Let $L$ denote a straight line in $\\PP$. For all $p \\in \\PP$, let $K_p$ denote the straight line through $P$ perpendicular to $L$. Let $p_L$ denote the point on $L$ where $K_p$ intersects $L$. Let $\\pi_L: \\PP \\to L$ denote the mapping defined as: :$\\forall p \\in \\PP: \\map {\\pi_L} p = p_L$ That is, $\\pi_L$ sends every point $p$ in $\\PP$ to the foot of the perpendicular from $p$ to $L$. $\\pi_L$ is called the '''perpendicular projection of the plane onto $L$'''."}
+{"_id": "31965", "title": "Definition:Ordering on Integers/Definition 1", "text": "The integers are ordered on the relation $\\le$ as follows: :$\\forall x, y \\in \\Z: x \\le y$ {{iff}}: :$\\exists c \\in P: x + c = y$ where $P$ is the set of positive integers. That is, $x$ is '''less than or equal''' to $y$ {{iff}} $y - x$ is non-negative."}
+{"_id": "31966", "title": "Definition:Strict Ordering on Integers/Definition 2", "text": "The integers are strictly ordered on the relation $<$ as follows: Let $x$ and $y$ be defined as from the formal definition of integers: :$x = \\eqclass {x_1, x_2} {}$ and $y = \\eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \\in \\N$. Then: :$x < y \\iff x_1 + y_2 < x_2 + y_1$ where: :$+$ denotes natural number addition :$a < b$ denotes natural number ordering $a \\le b$ such that $a \\ne b$."}
+{"_id": "31967", "title": "Definition:Ordering on Integers/Definition 2", "text": "The integers are ordered on the relation $\\le$ as follows: Let $x$ and $y$ be defined as from the formal definition of integers: :$x = \\eqclass {x_1, x_2} {}$ and $y = \\eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \\in \\N$. Then: :$x < y \\iff x_1 + y_2 \\le x_2 + y_1$ where: :$+$ denotes natural number addition :$\\le$ denotes natural number ordering."}
+{"_id": "31974", "title": "Definition:Imperial/Area/Rood", "text": "One '''rood''' is equal to an oblong measuring $1$ chain by $2 \\frac 1 2$ chains. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''rood''' }} {{eqn | r = 2 \\tfrac 1 2       | c = square chains }} {{eqn | r = 40       | c = square rods, poles or perches }} {{eqn | r = 1210       | c = square yards }} {{eqn | r = 10 \\, 890       | c = square feet }} {{end-eqn}} :400px"}
+{"_id": "31976", "title": "Definition:Parabola/Latus Rectum", "text": "The '''latus rectum''' of $P$ is the chord of $P$ passing through the focus of $P$ parallel to the directrix $D$."}
+{"_id": "31978", "title": "Definition:Work", "text": "Let $P$ be a particle whose position vector at time $t$ is $\\mathbf r$. Let a force applied to $P$ be represented by the vector $\\mathbf F$. Suppose that, during the time interval $\\delta t$, $P$ moves from $\\mathbf r$ to $\\mathbf r + \\delta \\mathbf r$. The '''work done''' by $\\mathbf F$ during $\\delta t$ is defined to be: :$\\delta W = \\mathbf F \\cdot \\delta \\mathbf r$ where $\\cdot$ denotes the dot product."}
+{"_id": "31979", "title": "Definition:Cartesian Product/Cartesian Space/Three Dimensions", "text": "The '''cartesian $3$rd power of $S$''' is: :$S^3 = S \\times S \\times S = \\set {\\tuple {x_1, x_2, x_3}: x_1, x_2, x_3 \\in S}$ The set $S^3$ called a '''cartesian space of $3$ dimensions'''."}
+{"_id": "31980", "title": "Definition:Cartesian 3-Space", "text": "The '''Cartesian $3$-space''' is a Cartesian coordinate system of $3$ dimensions. === Definition by Axes === {{:Definition:Cartesian 3-Space/Definition by Axes}} === Definition by Planes === {{:Definition:Cartesian 3-Space/Definition by Planes}}"}
+{"_id": "31982", "title": "Definition:Cartesian Coordinate System/Z Coordinate", "text": "Let $z$ be the length of the line segment from the origin $O$ to the foot of the perpendicular from $Q$ to the $z$-axis. Then $z$ is known as the '''$z$ coordinate'''. If $Q$ is in the positive direction along the real number line that is the $z$-axis, then $z$ is positive. If $Q$ is in the negative direction along the real number line that is the $z$-axis, then $z$ is negative."}
+{"_id": "31983", "title": "Definition:Rectangular Coordinate System", "text": "A '''rectangular coordinate system''' is a '''Cartesian coordinate system''' in which each of the coordinate axes are perpendicular to each other."}
+{"_id": "31984", "title": "Definition:Oblique Coordinate System", "text": "An '''oblique coordinate system''' is a coordinate system in which the position of a point is determined by its relation to a set of straight coordinate axes which are oblique. :400px"}
+{"_id": "31986", "title": "Definition:X-Y Plane", "text": "The '''$x$-$y$ plane''' is the Cartesian plane embedded in Cartesian $3$-space which contains the $x$-axis and the $y$-axis. It consists of all the points in $S$ such that $z = 0$."}
+{"_id": "31987", "title": "Definition:Y-Z Plane", "text": "The '''$y$-$z$ plane''' is the Cartesian plane embedded in Cartesian $3$-space which contains the $y$-axis and the $z$-axis. It consists of all the points in $S$ such that $x = 0$."}
+{"_id": "31988", "title": "Definition:X-Z Plane", "text": "The '''$x$-$z$ plane''' is the Cartesian plane embedded in Cartesian $3$-space which contains the $x$-axis and the $z$-axis. It consists of all the points in $S$ such that $x = 0$."}
+{"_id": "31990", "title": "Definition:Cartesian 3-Space/Definition by Axes", "text": "600pxrightthumbA general point in Cartesian $3$-Space Every point in ordinary $3$-space can be identified uniquely by means of an ordered triple of real coordinates  $\\tuple {x, y, z}$, as follows: Construct a Cartesian plane, with origin $O$ and axes identified as the $x$-axis and $y$-axis. Recall the identification of the point $P$ with the coordinate pair $\\tuple {1, 0}$ in the $x$-$y$ plane. Construct an infinite straight line through $O$ perpendicular to both the $x$-axis and the$y$-axis and call it the $z$-axis. Identify the point $P''$ on the $z$-axis such that $OP'' = OP$. Identify the $z$-axis with the real number line such that: :$0$ is identified with the origin $O$ :$1$ is identified with the point $P$ The orientation of the $z$-axis is determined by the position of $P''$ relative to $O$. It is conventional to locate $P''$ as follows. Imagine being positioned, standing on the $x$-$y$ plane at $O$, and facing along the $x$-axis towards $P$, with $P'$ on the left.  Then $P''$ is then one unit ''above'' the $x$-$y$ plane. Let the $x$-$y$ plane be identified with the plane of the page or screen. The orientation of the $z$-axis is then: :coming vertically \"out of\" the page or screen from the origin, the numbers on the $z$-axis are positive :going vertically \"into\" the page or screen from the origin, the numbers on the $z$-axis are negative."}
+{"_id": "31995", "title": "Definition:Decimal Expansion/Size Less than 1", "text": "A number $x$ such that $\\size x < 0$ has a units digit which is zero. Such a number may be expressed either with or without the zero, for example: :$0 \\cdotp 568$ or: :$\\cdotp 568$ While both are commonplace, the form with the zero is less prone to the mistake where decimal point is missed when reading it."}
+{"_id": "31996", "title": "Definition:Cartesian 3-Space/Definition by Planes", "text": "600pxrightthumb Every point in ordinary $3$-space can be identified uniquely by means of an ordered triple of real coordinates  $\\tuple {x, y, z}$, as follows: Identify one distinct point in space as the origin $O$. Let $3$ distinct planes be constructed through $O$ such that all are perpendicular. Each pair of these $3$ planes intersect in a straight line that passes through $O$. Let $X$, $Y$ and $Z$ be points, other than $O$, one on each of these $3$ lines of intersection. Then the lines $OX$, $OY$ and $OZ$ are named the $x$-axis, $y$-axis and $z$-axis respectively. Select a point $P$ on the $x$-axis different from $O$. Let $P$ be identified with the coordinate pair $\\tuple {1, 0}$ in the $x$-$y$ plane. Identify the point $P'$ on the $y$-axis such that $OP' = OP$. Identify the point $P''$ on the $z$-axis such that $OP'' = OP$. The orientation of the $z$-axis is determined by the position of $P''$ relative to $O$. It is conventional to locate $P''$ as follows. Imagine being positioned, standing on the $x$-$y$ plane at $O$, and facing along the $x$-axis towards $P$, with $P'$ on the left.  Then $P''$ is then one unit ''above'' the $x$-$y$ plane. Let the $x$-$y$ plane be identified with the plane of the page or screen. The orientation of the $z$-axis is then: :coming vertically \"out of\" the page or screen from the origin, the numbers on the $z$-axis are positive :going vertically \"into\" the page or screen from the origin, the numbers on the $z$-axis are negative."}
+{"_id": "31997", "title": "Definition:Wave Profile", "text": "Let $\\phi$ be a disturbance which is propagated along the $x$-axis with velocity $c$. At $t = 0$, let $\\phi = \\map f x$. Then $\\map f x$ is the '''wave profile''' of $\\phi$."}
+{"_id": "32000", "title": "Definition:Switching Circuit", "text": "A '''switching circuit''' is a mechanical, electronic or electromechanical system for controlling the route of information or resources. The study of '''switching circuits''' can be considered a field of applied logic."}
+{"_id": "32001", "title": "Definition:Electric Potential/Dimension", "text": "The dimension of measurement of '''electric potential''' is $\\mathsf M \\mathsf L^2 \\mathsf T^{−3} \\mathsf I^{−1}$."}
+{"_id": "32002", "title": "Definition:Electric Potential/Units", "text": "The SI unit of '''electric potential''' is the volt $\\mathrm V$."}
+{"_id": "32003", "title": "Definition:Electric Potential", "text": "An '''electric potential''' is the amount of work needed to move a unit of electric charge from a given reference point to a specific point in an electric field without producing an acceleration. The reference point is usually either Earth or a point at infinity, although any point can be used."}
+{"_id": "32004", "title": "Definition:Scalar Algebra", "text": "'''Scalar algebra''' is the application of the rules of conventional '''algebra''' in order to manipulate '''scalar quantities'''."}
+{"_id": "32007", "title": "Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 6", "text": "{{begin-axiom}} {{axiom | n = \\text B 6         | q = \\forall B_1, B_2 \\in \\mathscr B         | mr= x \\in B_1 \\setminus B_2 \\implies \\exists y \\in B_2 \\setminus B_1 : \\paren {B_2 \\cup \\set x} \\setminus \\set y \\in \\mathscr B }} {{end-axiom}}"}
+{"_id": "32019", "title": "Definition:Cartesian 3-Space/Coordinate Planes", "text": "Consider the '''Cartesian $3$-space''' defined by $3$ distinct perpendicular planes through the origin $O$. These $3$ planes are known as the '''coordinate planes''' of the Cartesian $3$-space."}
+{"_id": "32021", "title": "Definition:Electromagnetic Induction", "text": "'''Electromagnetic induction''' is the phenomenon whereby an electromotive force is generated across an electrical conductor in a changing magnetic field."}
+{"_id": "32022", "title": "Definition:Electric Field", "text": "An '''electric field''' is a vector field that associates to each point in space the (electrostatic or Coulomb) force per unit of charge exerted on an infinitesimal positive test charge at rest at that point."}
+{"_id": "32023", "title": "Definition:Magnetic Flux Density", "text": "'''Magnetic flux density''' is  the force exerted on a charged particle perpendicular to its velocity through a magnetic field. Hence it is a measure of the intensity of that magnetic field."}
+{"_id": "32025", "title": "Definition:Magnetic Flux Density/Units", "text": "The SI unit for '''magnetic flux density''' is the tesla $\\mathrm T$: :$1 \\ \\mathrm T = 1 \\ \\mathrm {kg} \\, \\mathrm s^{-2} \\mathrm A^{-1}$ that is: :$1 \\ \\mathrm T = 1 \\ \\mathrm N \\, \\mathrm m^{-1} \\mathrm A^{-1}$"}
+{"_id": "32026", "title": "Definition:Magnetic Flux Density/Dimension", "text": "'''Magnetic flux density''' has the dimension $\\mathsf {M T}^{-2} \\mathsf I^{-1}$."}
+{"_id": "32027", "title": "Definition:Rate/Dimension", "text": "A '''rate''' has dimension $\\mathsf T^{-1}$."}
+{"_id": "32028", "title": "Definition:Charged Particle", "text": "A '''charged particle''' is an ideal particle which bears an electric charge. Category:Definitions/Electrostatics Category:Definitions/Electromagnetism Category:Definitions/Physics gopofl6hh9cqhltggjsiuv9s01nhivp"}
+{"_id": "32029", "title": "Definition:Elementary Charge", "text": "The '''elementary charge''' is the value of the electric charge on a proton. It is defined as ''exactly'' $1.60217 \\, 6634 \\times 10^{−19}$ coulombs. === Symbol === {{:Definition:Elementary Charge/Symbol}}"}
+{"_id": "32031", "title": "Definition:Tesla (Unit)/Base Units", "text": "The SI base units of the '''tesla''' are: :$\\mathrm T := \\mathrm {kg} \\, \\mathrm s^{-2} \\mathrm A^{-1}$ where: :$\\mathrm {kg}$ denotes kilogram :$\\mathrm A$ denotes amperes :$\\mathrm s$ denotes seconds (of time)."}
+{"_id": "32032", "title": "Definition:Volume Charge Density", "text": "The '''volume charge density''' quantity of electric charge per unit volume, at any given point in that volume."}
+{"_id": "32034", "title": "Definition:Volume Charge Density/Dimension", "text": "The dimension of '''volume charge density''' is $\\mathsf {I T L}^{-3}$: electric charge per unit volume."}
+{"_id": "32035", "title": "Definition:Volume Charge Density/Units", "text": "The SI units of '''volume charge density''' are $\\mathrm C \\, \\mathrm m^{-3}$ (coulombs per cubic metre)."}
+{"_id": "32036", "title": "Definition:Electric Displacement Field", "text": "The '''electric displacement field''' is a vector quantity defined as the electric field strength multiplied by the permittivity of the medium through which it passes: :$\\mathbf D = \\varepsilon \\mathbf E$ where: :$\\varepsilon$ is the permittivity of the medium :$\\mathbf E$ is the electric field strength"}
+{"_id": "32040", "title": "Definition:Magnetic Field", "text": "A '''magnetic field''' is a vector field that gives rise to a force on a moving electric charge perpendicular to both the velocity of the charge and the direction of the '''magnetic field'''."}
+{"_id": "32041", "title": "Definition:Magnetic Field Strength", "text": "'''Magnetic field strength''' is the measure of the intensity of a magnetic field. It can be expressed as: :$\\mathbf H = \\dfrac {\\mathbf B} \\mu$ where: :$\\mathbf B$ is the magnetic flux density :$\\mu$ is the permeability of the medium."}
+{"_id": "32044", "title": "Definition:Magnetic Field Strength/Units", "text": "The SI unit for '''magnetic field strength''' is given as amperes per metre: :$\\mathrm A \\, \\mathrm m^{-1}$"}
+{"_id": "32045", "title": "Definition:Electrostatic Force", "text": "An '''electrostatic force''' is the force exerted on an electric charge caused by an electric field."}
+{"_id": "32047", "title": "Definition:Schrödinger Wave Equation", "text": "=== Time Independent Version === {{:Definition:Schrödinger Wave Equation/Time Independent}} {{NamedforDef|Erwin Rudolf Josef Alexander Schrödinger|cat = Schrödinger}} Category:Definitions/Quantum Mechanics 8vncojomzji3ry5cp1wj6to7krdhtjz"}
+{"_id": "32048", "title": "Definition:Schrödinger Wave Equation/Time Independent", "text": ":$-\\dfrac {\\hbar^2} {2 m} \\nabla^2 \\psi + V \\psi = E \\psi$ where: :$\\psi$ is the wave function whose nature is to be determined :$V$ denotes the potential energy (usually positional) :$E$ denotes the total energy of the system :$m$ denotes the mass of the particle whose motion is described by $\\psi$ :$\\hbar$ is Planck's constant divided by $2 \\pi$:  ::$\\hbar = \\dfrac h {2 \\pi}$"}
+{"_id": "32049", "title": "Definition:Angular Momentum", "text": "The '''angular momentum''' of a body about a point $P$ is its moment of inertia about $P$ multiplied by its angular velocity about $P$. Angular momentum is a vector quantity."}
+{"_id": "32050", "title": "Definition:Angular Momentum/Dimension", "text": "The dimension of measurement of '''angular momentum''' is $\\mathsf {M L}^2 \\mathsf T^{-1}$."}
+{"_id": "32051", "title": "Definition:Vector Quantity/Arrow Representation", "text": "A '''vector quantity''' $\\mathbf v$ is often represented diagramatically in the form of an '''arrow''' such that: :its length is proportional to the magnitude of $\\mathbf v$ :its direction corresponds to the direction of $\\mathbf v$. The head of the arrow then indicates the positive sense of the direction of $\\mathbf v$. It can be rendered on the page like so: :300px It is important to note that a vector quantity, when represented in this form, is not in general fixed in space. All that is being indicated using such a notation is its magnitude and direction, and not, in general, a point at which it acts."}
+{"_id": "32054", "title": "Definition:Vector Sum/Triangle Law", "text": "Let $\\mathbf u$ and $\\mathbf v$ be represented by arrows embedded in the plane such that: :$\\mathbf u$ is represented by $\\vec {AB}$ :$\\mathbf v$ is represented by $\\vec {BC}$ that is, so that the initial point of $\\mathbf v$ is identified with the terminal point of $\\mathbf u$. :400px Then their '''(vector) sum''' $\\mathbf u + \\mathbf v$ is represented by the arrow $\\vec {AC}$."}
+{"_id": "32055", "title": "Definition:Terminal Point of Vector", "text": "Let $\\mathbf u$ be a vector quantity represented by an arrow $\\vec {AB}$ embedded in the plane: :300px The point $B$ is the '''terminal point''' of $\\mathbf u$."}
+{"_id": "32056", "title": "Definition:Initial Point of Vector", "text": "Let $\\mathbf u$ be a vector quantity represented by an arrow $\\vec {AB}$ embedded in the plane: :300px The point $A$ is the '''initial point''' of $\\mathbf u$."}
+{"_id": "32061", "title": "Definition:Vector Sum/Component Definition", "text": "Let $\\mathbf u$ and $\\mathbf v$ be represented by their components considered to be embedded in a real $n$-space: {{begin-eqn}} {{eqn | l = \\mathbf u       | r = \\tuple {u_1, u_2, \\ldots, u_n} }} {{eqn | l = \\mathbf v       | r = \\tuple {v_1, v_2, \\ldots, v_n} }} {{end-eqn}} Then the '''(vector) sum''' of $\\mathbf u$ and $\\mathbf v$ is defined as: :$\\mathbf u + \\mathbf v := \\tuple {u_1 + v_1, u_2 + v_2, \\ldots, u_n + v_n}$ Note that the $+$ on the {{RHS}} is conventional addition of numbers, while the $+$ on the {{LHS}} takes on a different meaning. The distinction is implied by which operands are involved."}
+{"_id": "32062", "title": "Definition:Vector Length/Arrow Representation", "text": "Let $\\mathbf v$ be a vector quantity represented as an arrow in a real vector space $\\R^n$. The '''length''' of $\\mathbf v$ is the length of the line segment representing $\\mathbf v$ in $\\R^n$."}
+{"_id": "32063", "title": "Definition:Deceleration", "text": "'''Deceleration''' is the term used in natural language to mean '''negative acceleration'''. Thus, while '''acceleration''' is used to mean '''speeding up''', '''deceleration''' is used to mean '''slowing down'''. As in the applied mathematical context, '''acceleration''' is used to mean '''any''' change in velocity, including a change in direction while moving at a constant speed, the term '''deceleration''' is generally not used in mathematics."}
+{"_id": "32065", "title": "Definition:Rectilinear Coordinate System", "text": "A coordinate system whose coordinate axes are straight lines is called a system of '''rectilinear coordinate system'''."}
+{"_id": "32068", "title": "Definition:Curvilinear Coordinate System", "text": "A coordinate system such that at least one of the coordinate axes is a curved line is called a system of '''curvilinear coordinates'''."}
+{"_id": "32069", "title": "Definition:Curved Line", "text": "A '''curved line''' is a curve which is specifically not a straight line."}
+{"_id": "32072", "title": "Definition:Line/Curve/Space", "text": "A '''space curve''' is a curve which cannot be embedded in the plane, but can be embedded in $3$-space."}
+{"_id": "32073", "title": "Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)", "text": "Let $S$ be a finite set. Let $\\mathscr C$ be a non-empty set of subsets of $S$. === Definition 1 === {{:Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 1}} === Definition 2 === {{:Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 2}}"}
+{"_id": "32074", "title": "Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 1", "text": "{{begin-axiom}} {{axiom | n = C1         | q = \\forall C_1, C_2 \\in \\mathscr C         | mr = C_1 \\neq C_2 \\implies C_1 \\not \\subseteq C_2 }} {{axiom | n = C2         | q = \\forall C_1, C_2 \\in \\mathscr C         | mr = C_1 \\neq C_2 \\land z \\in C_1 \\cap C_2 \\implies \\exists C_3 \\in \\mathscr C : C_3 \\subseteq \\paren{C_1 \\cup C_2} \\setminus \\set z }} {{end-axiom}}"}
+{"_id": "32075", "title": "Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 2", "text": "{{begin-axiom}} {{axiom | n = C1         | q = \\forall C_1, C_2 \\in \\mathscr C         | mr = C_1 \\neq C_2 \\implies C_1 \\not \\subseteq C_2 }} {{axiom | n = C2'         | q = \\forall C_1, C_2 \\in \\mathscr C         | mr = C_1 \\neq C_2 \\land z \\in C_1 \\cap C_2 \\land w \\in C_1 \\setminus C_2 \\implies \\exists C_3 \\in \\mathscr C : y \\in C_3 \\subseteq \\paren{C_1 \\cup C_2} \\setminus \\set z }} {{end-axiom}}"}
+{"_id": "32076", "title": "Definition:Curvilinear Coordinate System/Cartesian Representation", "text": "The relation between curvilinear coordinates and Cartesian coordinates can be expressed as: {{begin-eqn}} {{eqn | l = x       | r = \\map x {q_1, q_2, q_3} }} {{eqn | l = y       | r = \\map y {q_1, q_2, q_3} }} {{eqn | l = z       | r = \\map z {q_1, q_2, q_3} }} {{end-eqn}} where: :$\\tuple {x, y, z}$ denotes the Cartesian coordinates :$\\tuple {q_1, q_2, q_3}$ denotes their curvilinear equivalents."}
+{"_id": "32077", "title": "Definition:Orthogonal Curvilinear Coordinates", "text": "Let $\\tuple {q_1, q_2, q_3}$ denote a set of curvilinear coordinates. Let the relation between those curvilinear coordinates and Cartesian coordinates be expressed as: {{begin-eqn}} {{eqn | l = x       | r = \\map x {q_1, q_2, q_3} }} {{eqn | l = y       | r = \\map y {q_1, q_2, q_3} }} {{eqn | l = z       | r = \\map z {q_1, q_2, q_3} }} {{end-eqn}} where $\\tuple {x, y, z}$ denotes the Cartesian coordinates. Let these equations have the property that: :$\\dfrac {\\partial x} {\\partial q_i} \\dfrac {\\partial x} {\\partial q_j} + \\dfrac {\\partial y} {\\partial q_i} \\dfrac {\\partial y} {\\partial q_j} + \\dfrac {\\partial z} {\\partial q_i} \\dfrac {\\partial z} {\\partial q_j} = 0$ wherever $i \\ne j$. Then $\\tuple {q_1, q_2, q_3}$ are '''orthogonal curvilinear coordinates'''."}
+{"_id": "32078", "title": "Definition:Functional", "text": "=== Real Functional === {{:Definition:Real Functional}} Category:Definitions/Functionals Category:Definitions/Calculus of Variations 1krmqjtyftdlzscrs2kbxvkcqf0rfq7"}
+{"_id": "32079", "title": "Definition:Permutation Theory", "text": "'''Permutation theory''' is a branch of abstract algebra which studies the properties of permutations."}
+{"_id": "32080", "title": "Definition:Landau Symbols", "text": "The '''Landau symbols''' are the $\\OO$ and $\\mathcal o$ that are used in the definition of big-$\\OO$ notation and little-$\\mathcal o$ notation: === Big-$\\OO$ Notation === {{:Definition:O Notation/Big-O Notation/Real/Infinity}} === Little-$\\mathcal o$ Notation === {{:Definition:O Notation/Little-O Notation/Real Functions/Definition 1}} {{NamedforDef|Edmund Georg Hermann Landau|cat = Landau}}"}
+{"_id": "32081", "title": "Definition:Vector Subtraction/Arrow Representation", "text": "Let $\\mathbf u$ and $\\mathbf v$ be represented by arrows embedded in the plane such that: :$\\mathbf u$ is represented by $\\vec {AB}$ :$\\mathbf v$ is represented by $\\vec {AC}$ that is, so that the initial point of $\\mathbf v$ is identified with the initial point of $\\mathbf u$. :300px Then their '''(vector) difference''' $\\mathbf u - \\mathbf v$ is represented by the arrow $\\vec {CB}$."}
+{"_id": "32082", "title": "Definition:Orientation of Coordinate Axes", "text": "The '''orientation''' of a coordinate system is the disposition of the coordinate axes relative to each other."}
+{"_id": "32083", "title": "Definition:Orientation of Coordinate Axes/Cartesian Plane", "text": "There are $2$ different orientations of a Cartesian plane: :Right-hand Cartesian plane320px $\\qquad \\qquad \\qquad$ Left-hand Cartesian plane320px"}
+{"_id": "32085", "title": "Definition:Orientation of Coordinate Axes/Cartesian Plane/Right-Handed", "text": "A Cartesian plane is defined as being '''right-handed''' if it has the following property: Let a '''right hand''' be placed, with palm uppermost, such that the thumb points along the $x$-axis in the positive direction, such that the thumb and index finger are at right-angles to each other. Then the index finger is pointed along the $y$-axis in the positive direction. :320px"}
+{"_id": "32086", "title": "Definition:Orientation of Coordinate Axes/Cartesian Plane/Left-Handed", "text": "A Cartesian plane is defined as being '''left-handed''' if it has the following property: Let a '''left hand''' be placed, with palm uppermost, such that the thumb points along the $x$-axis in the positive direction, such that the thumb and index finger are at right-angles to each other. Then the index finger is pointed along the $y$-axis in the positive direction. :320px"}
+{"_id": "32087", "title": "Definition:Orientation of Coordinate Axes/Cartesian 3-Space", "text": "There are $2$ different orientations of a Cartesian $3$-space: :450px $\\qquad \\qquad \\qquad$ 320px"}
+{"_id": "32088", "title": "Definition:Orientation of Coordinate Axes/Cartesian 3-Space/Right-Handed", "text": "A Cartesian $3$-Space is defined as being '''right-handed''' if it has the following property: Let a '''right hand''' be placed such that: :the thumb and index finger are at right-angles to each other :the $3$rd finger is at right-angles to the thumb and index finger, upwards from the palm :the thumb points along the $x$-axis in the positive direction :the index finger points along the $x$-axis in the positive direction. Then the $3$rd finger is pointed along the $z$-axis in the positive direction. :450px"}
+{"_id": "32089", "title": "Definition:Orientation of Coordinate Axes/Cartesian 3-Space/Left-Handed", "text": "A Cartesian $3$-Space is defined as being '''left-handed''' if it has the following property: Let a '''left hand''' be placed such that: :the thumb and index finger are at right-angles to each other :the $3$rd finger is at right-angles to the thumb and index finger, upwards from the palm :the thumb points along the $x$-axis in the positive direction :the index finger points along the $x$-axis in the positive direction. Then the $3$rd finger is pointed along the $z$-axis in the positive direction. :320px"}
+{"_id": "32090", "title": "Definition:Pairwise Perpendicular", "text": "Let $S = \\set {L_1, L_2, \\ldots, L_n}$ be a set of lines in a Euclidean space. Then the elements of $S$ are '''pairwise perpendicular''' {{iff}} :$\\forall i, j \\in \\set {1, 2, \\ldots, n}, i \\ne j:$ $L_i$ and $L_j$ are perpendicular to each other."}
+{"_id": "32091", "title": "Definition:Orthogonal Coordinate System", "text": "An '''orthogonal coordinate system''' is a coordinate system in which the coordinate axes are pairwise perpendicular."}
+{"_id": "32096", "title": "Definition:Smallest Natural Number", "text": "Let $S \\subseteq \\N$ be a subset of the natural numbers $\\N$. The '''smallest''' element $m$ of $S$ is defined as: :$\\forall n \\in S: m \\le n$ That is, it is the minimal element of $S$ under the usual ordering."}
+{"_id": "32098", "title": "Definition:Energy/Unit", "text": "The unit of measurement of '''energy'''  is the joule $\\mathrm J$: :$1 \\ \\mathrm J = 1 \\ \\mathrm N \\ \\mathrm m$ where: :$\\mathrm N$ denotes the newton :$\\mathrm m$ denotes the metre."}
+{"_id": "32099", "title": "Definition:Joule (Unit)", "text": "The '''joule''' is the SI unit of energy. It is defined as being: :the '''energy''' transferred to (or work done on) a body when a force of $1$ newton acts on that body in the direction of the force's motion through a distance of one metre :the '''energy''' dissipated as heat when an electric current of one ampere passes through a resistance of one ohm for one second."}
+{"_id": "32100", "title": "Definition:Joule (Unit)/Base Units", "text": "The SI base units of the '''joule''' are: :$\\mathrm J := \\mathrm {kg} \\, \\mathrm m^2 \\mathrm s^{-2}$ where: :$\\mathrm {kg}$ denotes kilograms :$\\mathrm m$ denotes metres :$\\mathrm s$ denotes seconds (of time)."}
+{"_id": "32101", "title": "Definition:Power (Physics)", "text": "'''Power''' the amount of energy transferred or converted per unit time. '''Power''' is a scalar quantity."}
+{"_id": "32103", "title": "Definition:Power (Physics)/Dimension", "text": "The dimension of measurement of '''power''' is $\\mathsf {M L}^2 \\mathsf T^{-3}$. This derives from its definition as: :$\\dfrac {\\text{Energy} } {\\text {Time} }$"}
+{"_id": "32104", "title": "Definition:Power (Physics)/Unit", "text": "The unit of measurement of '''power'''  is the watt $\\mathrm W$: :$1 \\ \\mathrm W = 1 \\ \\mathrm J \\ \\mathrm s^{-1}$ where: :$\\mathrm J$ denotes the joule :$\\mathrm s$ denotes the second."}
+{"_id": "32105", "title": "Definition:Watt (Unit)", "text": "The '''watt''' is the SI unit of power. It is defined as being: :the rate at which work is done when the velocity of a body is held constant at $1$ metre per second against a constant opposing force of $1$ newton :the rate at which work is performed when an electric current of $1$ ampere flows across an electrical potential difference of $1$ volt."}
+{"_id": "32108", "title": "Definition:Volt", "text": "The '''volt''' is the SI unit of electromotive force. It is defined as being the difference in '''electric potential between two points of an electrical conductor: :when an electric current of $1$ ampere dissipates $1$ watt of power between those points :that will impart $1$ joule of energy per coulomb of electric charge that passes through it."}
+{"_id": "32109", "title": "Definition:Volt/Base Units", "text": "The SI base units of the '''volt''' are: :$\\mathrm W := \\mathrm {kg} \\, \\mathrm m^2 \\mathrm s^{-3} \\mathrm A^{-1}$ where: :$\\mathrm {kg}$ denotes kilograms :$\\mathrm m$ denotes metres :$\\mathrm s$ denotes seconds (of time). :$\\mathrm A$ denotes ampere."}
+{"_id": "32110", "title": "Definition:Ohm (Unit)", "text": "The '''ohm''' is the SI unit of electrical resistance. It is defined as being the electrical resistance between two points of an electrical conductor when a constant electric potential of $1$ volt, applied to these points, produces in the conductor an electric current of $1$ ampere, the  conductor not being the seat of any electromotive force."}
+{"_id": "32111", "title": "Definition:Ohm (Unit)/Base Units", "text": "The SI base units of the '''ohm''' are: :$\\mathrm W := \\mathrm {kg} \\, \\mathrm m^2 \\mathrm s^{-3} \\mathrm A^{-2}$ where: :$\\mathrm {kg}$ denotes kilograms :$\\mathrm m$ denotes metres :$\\mathrm s$ denotes seconds (of time). :$\\mathrm A$ denotes ampere."}
+{"_id": "32118", "title": "Definition:Strong Nuclear Force", "text": "The '''strong nuclear force''' is a force which acts in the range of the order of $10^{-15} \\ \\mathrm m$. It is what holds protons and neutrons together in the nucleus of an atom. Beyond the range of approximately the diameter of a large atom its effect is diminished. This may be the reason why stable isotopes of large atoms are rare."}
+{"_id": "32120", "title": "Definition:Elementary Charge/Symbol", "text": "The symbol used to denote the '''elementary charge''' is usually $\\E$ or $e$. The preferred symbol on {{ProofWiki}} is $\\E$."}
+{"_id": "32124", "title": "Definition:Zero Vector/Vector Quantity", "text": "A '''vector quantity''' whose magnitude is zero is referred to as a '''zero vector'''."}
+{"_id": "32125", "title": "Definition:Scalar Multiplication/Vector Quantity", "text": "Let $\\mathbf a$ be a vector quantity. Let $m$ be a scalar quantity. The operation of scalar multiplication by $m$ of $\\mathbf a$ is denoted $m \\mathbf a$ and defined such that: :the magnitude of $m \\mathbf a$ is equal to $m$ times the magnitude of $\\mathbf a$: ::$\\size {m \\mathbf a} = m \\size {\\mathbf a}$ :the direction of $m \\mathbf a$ is the same as the direction of $\\mathbf a$."}
+{"_id": "32126", "title": "Definition:Vector Quantity/Component/Cartesian Plane", "text": "Let $\\mathbf a$ be a vector quantity embedded in a Cartesian plane $P$. Let $\\mathbf a$ be represented with its initial point at the origin of $P$. Let $\\mathbf i$ and $\\mathbf j$ be the unit vectors in the positive direction of the $x$-axis and $y$-axis. Then: :$\\mathbf a = x \\mathbf i + y \\mathbf j$ where: :$x \\mathbf i$ and $y \\mathbf j$ are the '''component vectors''' of $\\mathbf a$ in the $\\mathbf i$ and $\\mathbf j$ directions :$x$ and $y$ are the '''components''' of $\\mathbf a$ in the $\\mathbf i$ and $\\mathbf j$ directions."}
+{"_id": "32127", "title": "Definition:Vector Quantity/Component/Cartesian 3-Space", "text": "Let $\\mathbf a$ be a vector quantity embedded in Cartesian $3$-space $S$. Let $\\mathbf a$ be represented with its initial point at the origin of $S$. Let $\\mathbf i$, $\\mathbf j$ and $\\mathbf k$ be the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively. Then: :$\\mathbf a = x \\mathbf i + y \\mathbf j + z \\mathbf k$ where: :$x \\mathbf i$, $y \\mathbf j$ and $z \\mathbf k$ are the '''component vectors''' of $\\mathbf a$ in the $\\mathbf i, \\mathbf j, \\mathbf k$ directions :$x$, $y$ and $z$ are the '''components''' of $\\mathbf a$ in the $\\mathbf i$, $\\mathbf j$ and $\\mathbf k$ directions. It is usual to arrange that the coordinate axes form a right-handed Cartesian $3$-space."}
+{"_id": "32128", "title": "Definition:Variational Principle", "text": "A '''variational principle''' is a statement defining a condition that is required to be fulfilled in order to solve a problem in the calculus of variations."}
+{"_id": "32129", "title": "Definition:Vectorial Mechanics", "text": "'''Vectorial mechanics''' is an approach to problems in mechanics which handles quantities as vector quantities and uses vector algebra and vector calculus for their solution."}
+{"_id": "32131", "title": "Definition:Kinematics", "text": "'''Kinematics''' is the branch of the mechanics describing the motion of bodies without considering the forces that may affect this motion."}
+{"_id": "32132", "title": "Definition:Like Vector Quantities", "text": "Let $\\mathbf a$ and $\\mathbf b$ be vector quantities. Then $\\mathbf a$ and $\\mathbf b$ are known as '''like vector quantities''' {{iff}} they have the same direction."}
+{"_id": "32133", "title": "Definition:Negative of Vector Quantity", "text": "Let $\\mathbf a$ be a vector quantities. The '''negative''' $-\\mathbf a$ of $\\mathbf a$ is the vector quantity which has the same magnitude but the opposite direction."}
+{"_id": "32134", "title": "Definition:Scalar Division/Vector Quantity", "text": "Let $\\mathbf a$ be a vector quantity. Let $m$ be a scalar quantity. The operation of scalar division by $m$ of $\\mathbf a$ is defined as: :$\\dfrac {\\mathbf a} m = \\dfrac 1 m \\size {\\mathbf a}$ that is, scalar multiplication by $\\dfrac 1 m$ of $\\mathbf a$."}
+{"_id": "32135", "title": "Definition:Scalar Division", "text": "=== Vector Quantity === {{:Definition:Scalar Division/Vector Quantity}}"}
+{"_id": "32136", "title": "Definition:Scalar Multiplication by Negative Scalar", "text": "Let $\\mathbf a$ be a vector quantity. Let $-m$ be a negative scalar quantity. The operation of scalar multiplication by $-m$ of $\\mathbf a$ is defined as: :$-m \\mathbf a := \\paren {-m} \\mathbf a = m \\paren {-\\mathbf a}$ that is, the negative of the scalar product of $m$ with $\\mathbf a$."}
+{"_id": "32137", "title": "Definition:Direct Product Norm", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,}}$ and $\\struct {Y, \\norm {\\, \\cdot \\,}}$ be normed vector spaces. Let $V = X \\times Y$ be a direct product of vector spaces $X$ and $Y$ together with induced component-wise operations. Let $\\tuple {x, y} \\in V$, $x \\in X$ and $y \\in Y$. Then the '''direct product norm''' on $V$ is defined as: :$\\norm {\\tuple {x, y} } := \\map \\max {\\norm x, \\norm y}$ where $\\max$ denotes the max operation."}
+{"_id": "32138", "title": "Definition:Coplanar Vectors", "text": "Three or more vectors are '''coplanar''' if they can be embedded in the same plane."}
+{"_id": "32139", "title": "Definition:Coplanar Vectors/Non-Coplanar", "text": "Three or more vectors are '''non-coplanar''' if they are not '''coplanar'''. That is if they cannot be embedded in the same plane."}
+{"_id": "32140", "title": "Definition:Direction Cosine", "text": "Let $\\mathbf a$ be a vector quantity embedded in a Cartesian $3$-space. Let the angles which $\\mathbf a$ makes with the $x$-axis, $y$-axis and $z$-axis be $\\alpha$, $\\beta$ and $\\gamma$ respectively. Then the '''direction cosines''' of $\\mathbf a$ are $\\cos \\alpha$, $\\cos \\beta$ and $\\cos \\gamma$, defined individually such that: :$\\cos \\alpha$ is the '''direction cosine of $\\mathbf a$ with respect to the $x$-axis''' :$\\cos \\beta$ is the '''direction cosine of $\\mathbf a$ with respect to the $y$-axis''' :$\\cos \\gamma$ is the '''direction cosine of $\\mathbf a$ with respect to the $z$-axis'''."}
+{"_id": "32142", "title": "Definition:Birkhoff-James Orthogonality", "text": "Let $\\struct {V, \\norm {\\,\\cdot\\,} }$ be a normed linear space. Let $x, y \\in V$. Then $x$ and $y$ are '''Birkhoff-James orthogonal''' {{iff}}  :$\\norm {x + \\lambda y} \\ge \\norm y$ for all scalars $\\lambda$. This is denoted: :$x \\perp_B y$"}
+{"_id": "32144", "title": "Definition:Orthogonal Basis of Vector Space", "text": "Let $V$ be a vector space. Let $\\BB = \\tuple {\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}$ be a basis of $V$. Then $\\BB$ is an '''orthogonal basis''' {{iff}} $\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n$ are pairwise perpendicular."}
+{"_id": "32145", "title": "Definition:Orthonormal Basis of Vector Space", "text": "Let $V$ be a vector space. Let $\\BB = \\tuple {\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}$ be a basis of $V$. Then $\\BB$ is an '''orthonormal basis of $V$''' {{iff}}: :$(1): \\quad \\tuple {\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n}$ is an orthogonal basis of $V$ :$(2): \\quad \\norm {\\mathbf e_1} = \\norm {\\mathbf e_2} = \\cdots = \\norm {\\mathbf e_1} = 1$"}
+{"_id": "32150", "title": "Definition:Vector Quantity/Component/Einstein Summation Convention", "text": "Let $\\mathbf a$ be a vector quantity. $\\mathbf a$ can be expressed in component form using the Einstein summation convention as: :$\\mathbf a = a_i \\mathbf e_i$"}
+{"_id": "32151", "title": "Definition:Dot Product/Einstein Summation Convention", "text": "Let $\\mathbf a$ and $\\mathbf b$ be vector quantities. The dot product of $\\mathbf a$ and $\\mathbf b$ can be expressed as: {{begin-eqn}} {{eqn | l = \\mathbf a \\cdot \\mathbf b        | o = :=       | r = a_i b_j \\delta_{i j}       | c =  }} {{eqn | r = a_i b_i       | c =  }} {{eqn | r = a_j b_j       | c =  }} {{end-eqn}}"}
+{"_id": "32154", "title": "Positive Integer Greater than 1 has Prime Divisor", "text": "Positive Integer Greater than 1 has Prime Divisor 0 130 373934 291994 2018-10-29T23:01:52Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Every positive integer greater than $1$ has at least one divisor which is prime. </onlyinclude> == Proof 1 == {{:Positive Integer Greater than 1 has Prime Divisor/Proof 1}} == Proof 2 (not using FTA) == {{:Positive Integer Greater than 1 has Prime Divisor/Proof 2}} == Sources == * {{BookReference|Elements of Abstract Algebra|1971|Allan Clark|prev = Definition:Prime Number/Sequence|next = Euclid's Theorem}}: Chapter $1$: Properties of the Natural Numbers: $\\S 22 \\alpha$ Category:Prime Numbers Category:Positive Integer Greater than 1 has Prime Divisor 2oc0nobqc3qlfa2d3tcqy9rvhai5fqi"}
+{"_id": "32155", "title": "Axiom:Axiom of Extension", "text": "Axiom:Axiom of Extension 100 149 444633 444632 2020-01-23T20:49:07Z Prime.mover 59 wikitext text/x-wiki == Axiom == The '''axiom of extension''' is the fundamental definition of the nature of a collection: it is completely determined by its elements. <onlyinclude> === Set Theory === {{:Axiom:Axiom of Extension/Set Theory}} === Class Theory === The '''axiom of extension''' in the context of class theory has the same form: {{:Axiom:Axiom of Extension/Class Theory}}</onlyinclude> == Also known as == {{:Axiom:Axiom of Extension/Also known as}} == Also see == * Definition:Set Equality * Definition:Equals == Linguistic Note == {{:Axiom:Axiom of Extension/Linguistic Note}} Extension Extension Extension nyvyz3hodadhs77097a0v1mjkjr01o3"}
+{"_id": "32156", "title": "Axiom:Axiom of Empty Set", "text": "Axiom:Axiom of Empty Set 100 151 449936 449902 2020-02-18T23:25:43Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> The '''axiom of the empty set''' posits the existence of a set which has no elements. Depending on whether this axiom is declared in the context of set theory or class theory, it exists in different forms. === Set Theory === {{:Axiom:Axiom of Empty Set/Set Theory}} === Class Theory === In class theory, the existence of the empty class is not axiomatic, as it has been derived from previous axioms. Hence the '''axiom of the empty set''' takes this form: {{:Axiom:Axiom of Empty Set/Class Theory}} where $V$ denotes the basic universe. </onlyinclude> == Also known as == {{:Axiom:Axiom of Empty Set/Also known as}} == Also see == * Definition:Zermelo-Fraenkel Axioms * Definition:Empty Set * Empty Set is Unique Empty Set Empty Set Category:Definitions/Basic Universe qrgtgfa41x2x4dt8v416smyu66epkeq"}
+{"_id": "32157", "title": "Axiom:Axiom of Pairing", "text": "Axiom:Axiom of Pairing 100 152 444595 444583 2020-01-23T20:27:31Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> === Set Theory === {{:Axiom:Axiom of Pairing/Set Theory}} === Class Theory === {{:Axiom:Axiom of Pairing/Class Theory}}</onlyinclude> == Also known as == {{:Axiom:Axiom of Pairing/Also known as}} == Also see == * Definition:Doubleton * Definition:Ordered Pair {{LinkToCategory|Axiom of Pairing|the axiom of pairing}} Pairing Pairing Pairing cuzikepuwbqucmjwfidhnaql1cq6qna"}
+{"_id": "32158", "title": "Axiom:Axiom of Specification", "text": "Axiom:Axiom of Specification 100 153 493722 453375 2020-10-10T15:04:29Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true. Because we cannot quantify over functions, we need an axiom for every condition we can express. Therefore, this axiom is sometimes called an axiom ''schema'', as we introduce a lot of similar axioms. This axiom schema can be formally stated as follows: === Set Theory === {{:Axiom:Axiom of Specification/Set Theory}} === Class Theory === The '''axiom of specification''' in the context of class theory has a similar form: {{:Axiom:Axiom of Specification/Class Theory}}</onlyinclude> == Also known as == {{:Axiom:Axiom of Specification/Also known as}} == Also see == * Axiom:Comprehension Principle -- do not confuse that with this {{refactor|Include the following in its own theorem page, and present that deduction formally}} The '''axiom of specification''' can be deduced from the Axiom of Replacement. == Historical Note == {{:Axiom:Axiom of Specification/Historical Note}} {{Languages|Axiom of specification}} {{language|German|Aussonderungsaxiom|lit = axiom of segregation}} {{end-languages}} == Sources == * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin|prev = Definition:Class (Class Theory)|next = Definition:Finite Set}}: Appendix $\\text A$: Sets and Functions: $\\text{A}.1$: Sets Specification Specification Specification 2mhp95orxrykoudc2fopbmzs46tzk0a"}
+{"_id": "32159", "title": "Axiom:Axiom of Unions", "text": "Axiom:Axiom of Unions 100 158 445003 444875 2020-01-27T23:08:19Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> === Set Theory === {{:Axiom:Axiom of Unions/Set Theory}} === Class Theory === {{:Axiom:Axiom of Unions/Class Theory}}</onlyinclude> == Also known as == {{:Axiom:Axiom of Unions/Also known as}} Unions Unions Unions 614u0u5kgomeahmakt1wp1tl5beot67"}
+{"_id": "32160", "title": "Axiom:Axiom of Powers", "text": "Axiom:Axiom of Powers 100 159 445020 444039 2020-01-27T23:25:39Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> === Set Theory === {{:Axiom:Axiom of Powers/Set Theory}} === Class Theory === {{:Axiom:Axiom of Powers/Class Theory}}</onlyinclude> == Also known as == {{:Axiom:Axiom of Powers/Also known as}} == Also see == * Definition:Power Set Powers Powers Powers ehj2nolsk55rwqbwequ79o8o70oaewj"}
+{"_id": "32161", "title": "Axiom:Axiom of Infinity", "text": "Axiom:Axiom of Infinity 100 160 449889 449832 2020-02-18T22:40:16Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> === Set Theory === {{:Axiom:Axiom of Infinity/Set Theory}} === Class Theory === {{:Axiom:Axiom of Infinity/Class Theory}}</onlyinclude> Infinity Infinity Infinity 9gjyglkd40zx3kol5ox0syoks4l30ng"}
+{"_id": "32162", "title": "Axiom:Axiom of Replacement", "text": "Axiom:Axiom of Replacement 100 163 444040 443865 2020-01-18T01:52:31Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> For any function $f$ and subset $S$ of the domain of $f$, there is a set containing the image $\\map f S$. More formally, let us express this as follows: Let $\\map P {y, z}$ be a propositional function, which determines a function. That is, we have: :$\\forall y: \\exists x: \\forall z: \\paren {\\map P {y, z} \\iff x = z}$. Then we state as an axiom: :$\\forall w: \\exists x: \\forall y: \\paren {y \\in w \\implies \\paren {\\forall z: \\paren {\\map P {y, z} \\implies z \\in x} } }$ </onlyinclude> == Also presented as == The two above statements may be combined into a single (somewhat unwieldy) expression: :$\\paren {\\forall y: \\exists x: \\forall z: \\paren {\\map P {y, z} \\implies x = z} } \\implies \\forall w: \\exists x: \\forall y: \\paren {y \\in w \\implies \\forall z: \\paren {\\map P {y, z} \\implies z \\in x} }$ == Also known as == The '''axiom of replacement''' is also known as the '''axiom of substitution'''. == Historical Note == {{:Axiom:Axiom of Replacement/Historical Note}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Zermelo Set Theory/Historical Note|next = Axiom:Axiom of Replacement/Historical Note}}: Chapter $1$: General Background: $\\S 9$ Zermelo set theory * {{MathWorld|Zermelo-Fraenkel Axioms|Zermelo-FraenkelAxioms}} Replacement cb6yglqcwu8vyjbv8t8n9hr3wisjls5"}
+{"_id": "32163", "title": "Axiom:Axiom of Foundation", "text": "Axiom:Axiom of Foundation 100 164 444036 444035 2020-01-18T01:50:48Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> For all non-empty sets, there is an element of the set that shares no element with the set. That is: :$\\forall S: \\paren {\\paren {\\exists x: x \\in S} \\implies \\exists y \\in S: \\forall z \\in S: \\neg \\paren {z \\in y} }$ The antecedent states that $S$ is not empty. </onlyinclude> == Also defined as == It can also be stated as: :For every non-empty set $S$, there exists an element $x \\in S$ such that $x$ and $S$ are disjoint. :A set contains no infinitely descending (membership) sequence. :A set contains a (membership) minimal element. :The membership relation is a foundational relation on any non-empty set. == Also known as == The '''axiom of foundation''' is also known as the '''axiom of regularity'''. == Sources == * {{MathWorld|Zermelo-Fraenkel Axioms|Zermelo-FraenkelAxioms}} Foundation tj1k4ptei585ohnkgatal3y7fl9um99"}
+{"_id": "32164", "title": "Axiom:Axiom of Choice", "text": "Axiom:Axiom of Choice 100 165 462986 462981 2020-04-19T09:47:14Z Prime.mover 59 wikitext text/x-wiki == Axiom == === Formulation 1 === <onlyinclude>{{:Axiom:Axiom of Choice/Formulation 1}}</onlyinclude> === Formulation 2 === {{:Axiom:Axiom of Choice/Formulation 2}} === Formulation 3 === {{:Axiom:Axiom of Choice/Formulation 3}} == Comment == Although it seems intuitively obvious (\"surely you can just pick an element?\"), when it comes to infinite sets of sets this axiom leads to non-intuitive results, notably the famous Banach-Tarski Paradox. For this reason, the '''Axiom of Choice''' (often abbreviated '''AoC''' or '''AC''') is often treated separately from the rest of the Zermelo-Fraenkel Axioms. Set theory based on the Zermelo-Fraenkel axioms is referred to ZF, while that based on the Z-F axioms including the '''AoC''' is referred to as ZFC. == Additional forms == The following are equivalent, in ZF, to the '''Axiom of Choice''': * Zorn's Lemma * Kuratowski's Lemma * Hausdorff Maximal Principle * Tukey's Lemma * Tychonoff's Theorem * Kelley's Theorem * Vector Space has Basis == Historical Note == {{:Axiom:Axiom of Choice/Historical Note}} == Also see == * Equivalence of Versions of Axiom of Choice {{LinkToCategory|Axiom of Choice|the Axiom of Choice}} == Sources == * {{BookReference|Topology: An Introduction with Application to Topological Groups|1967|George McCarty|prev = Definition:Surjection/Definition 2|next = Definition:Commutative Diagram}}: Chapter $\\text{I}$: Sets and Functions: Composition of Functions * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin|prev = Surjection iff Right Inverse/Proof 1|next = Definition:Inverse Mapping/Definition 2}}: Appendix $\\text{A}.7$: Inverses * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Continuum Hypothesis is Independent of ZFC/Historical Note|next = Axiom:Axiom of Choice/Historical Note}}: Chapter $1$: General Background: $\\S 5$ The continuum problem * {{MathWorld|Zermelo-Fraenkel Axioms|Zermelo-FraenkelAxioms}} Choice Choice Choice Category:Philosophical Positions ohexnx43nsaw0l6fvvv4e34lo39rmrx"}
+{"_id": "32165", "title": "Axiom:Comprehension Principle", "text": "Axiom:Comprehension Principle 100 170 490600 454411 2020-09-25T06:24:01Z Prime.mover 59 wikitext text/x-wiki == Definition == The '''comprehension principle''' states: <onlyinclude> :Given any property $P$, there exists a unique set which consists of all and only those objects which have property $P$: ::$\\set {x: \\map P x}$ </onlyinclude> From the definition of a set: :''A set is any aggregation of objects, called elements, which can be precisely defined in some way or other.'' == Also known as == The '''comprehension principle''' can also be referred to as: :the '''abstraction principle''' :the '''axiom of abstraction''' :the '''unlimited abstraction principle''' == Also see == * Axiom:Axiom of Comprehension -- do not confuse that with this * Definition:Set Definition by Predicate == Historical Note == {{:Axiom:Comprehension Principle/Historical Note}} == Sources == * {{BookReference|Axiomatic Set Theory|1972|Patrick Suppes|ed = 2nd|edpage = Second Edition|prev = Definition:Free Variable|next = Axiom:Comprehension Principle/Historical Note}}: $\\S 1.3$ Axiom Schema of Abstraction and Russell's Paradox * {{BookReference|Introduction to Graph Theory|1993|Richard J. Trudeau|prev = Square Root of 2 is Irrational/Classic Proof|next = Russell's Paradox}}: $2$. Graphs: Paradox * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Abstraction|next = Definition:Abstract Space|entry = abstraction|index = 2}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Abstraction|next = Definition:Abstract Space|entry = abstraction|index = 2}} * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Frege Set Theory|next = Definition:Set Definition by Predicate}}: Chapter $1$: General Background: $\\S 7$ Frege set theory Category:Axioms/Frege Set Theory 6hwmglfowxv66erjlmffiztpsgsz0zp"}
+{"_id": "32166", "title": "Rule of Conjunction", "text": "Rule of Conjunction 0 266 244265 244259 2016-01-15T17:27:36Z Prime.mover 59 wikitext text/x-wiki == Sequent == The '''rule of conjunction''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Rule of Conjunction/Proof Rule}} === Sequent Form === {{:Rule of Conjunction/Sequent Form}} == Explanation == {{:Rule of Conjunction/Explanation}} Thus a conjunction is added to a sequent. == Also known as == {{:Rule of Conjunction/Also known as}} == Also see == * Rule of Simplification Category:Conjunction Category:Rule of Conjunction h7ohkmvyd9hjrf7hwtrhsis8p4uaiba"}
+{"_id": "32167", "title": "Rule of Simplification", "text": "Rule of Simplification 0 267 244269 244247 2016-01-15T17:31:54Z Prime.mover 59 wikitext text/x-wiki == Sequent == The '''rule of simplification'''  is a valid deduction sequent in propositional logic. === Proof Rule === {{:Rule of Simplification/Proof Rule}} === Sequent Form === {{:Rule of Simplification/Sequent Form}} == Explanation == {{:Rule of Simplification/Explanation}} == Also known as == {{:Rule of Simplification/Also known as}} == Also see == * Rule of Conjunction Category:Rule of Simplification Category:Conjunction d5y2fu6s06nxhtrbuhm14bqvrmbmcym"}
+{"_id": "32168", "title": "Rule of Addition", "text": "Rule of Addition 0 268 244636 244253 2016-01-16T14:22:07Z Prime.mover 59 wikitext text/x-wiki == Sequent == The '''rule of addition'''  is a valid deduction sequent in propositional logic. === Proof Rule === {{:Rule of Addition/Proof Rule}} === Sequent Form === {{:Rule of Addition/Sequent Form}} == Explanation == {{:Rule of Addition/Explanation}} == Also known as == {{:Rule of Addition/Also known as}} == Also see == * Proof by Cases Category:Disjunction Category:Rule of Addition k35h9024bd9cfpt88usbkj6vldgzcaj"}
+{"_id": "32169", "title": "Proof by Cases", "text": "Proof by Cases 0 269 244744 244633 2016-01-17T06:23:15Z Prime.mover 59 wikitext text/x-wiki == Sequent == '''Proof by cases''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Proof by Cases/Proof Rule}} === Sequent Form === {{:Proof by Cases/Sequent Form}} == Variants == The following forms can be used as variants of this theorem: === Formulation 1 === {{:Proof by Cases/Formulation 1}} === Formulation 2 === {{:Proof by Cases/Formulation 2}} === Formulation 3 === {{:Proof by Cases/Formulation 3}} == Explanation == {{:Proof by Cases/Explanation}} Thus a disjunction is '''eliminated''' from a sequent. == Also known as == {{:Proof by Cases/Also known as}} == Also see == * Rule of Addition * Principle of Dilemma Category:Proof by Cases Category:Disjunction fow9jm911mpvt1vg4uw0ptluhlxxo4f"}
+{"_id": "32170", "title": "Modus Ponendo Ponens", "text": "Modus Ponendo Ponens 0 272 493613 467111 2020-10-10T12:25:12Z Prime.mover 59 wikitext text/x-wiki == Proof Rule == The '''modus ponendo ponens''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Modus Ponendo Ponens/Proof Rule}} === Sequent Form === {{:Modus Ponendo Ponens/Sequent Form}} == Variants == The following forms can be used as variants of this theorem: === Variant 1 === {{:Modus Ponendo Ponens/Variant 1}} === Variant 2 === {{:Modus Ponendo Ponens/Variant 2}} === Variant 3 === {{:Modus Ponendo Ponens/Variant 3}} == Also known as == {{:Modus Ponendo Ponens/Also known as}} == Linguistic Note == {{:Modus Ponendo Ponens/Linguistic Note}} == Also see == The following are related argument forms: * Modus Ponendo Tollens * Modus Tollendo Ponens * Modus Tollendo Tollens == Sources == * {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell|volume = 1|prev = Law of Identity/Formulation 2|next = Modus Ponendo Ponens/Also known as}}: Chapter $\\text{I}$: Preliminary Explanations of Ideas and Notations * {{BookReference|Symbolic Logic|1973|Irving M. Copi|ed = 4th|edpage = Fourth Edition|prev = Method of Truth Tables|next = Modus Tollendo Tollens}}: $2.3$: Argument Forms and Truth Tables * {{BookReference|Algebra Volume 1|1982|P.M. Cohn|edpage = Second Edition|ed = 2nd|prev = Definition:Direct Proof|next = Definition:Distinction between Logical Implication and Conditional}}: Chapter $1$: Sets and mappings: $\\S 1.1$: The need for logic * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin|prev = Four Color Theorem for Finite Maps implies Four Color Theorem for Infinite Maps|next = Reductio ad Absurdum/Variant 2}}: $\\S 1.12$: Valid Arguments * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Rigidity Modulus|next = Modus Tollendo Ponens|entry = ''modus ponens''}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Modulus Sign|next = Modus Tollendo Ponens|entry = ''modus ponens''}} Category:Implication Category:Modus Ponendo Ponens akrpifwojcsk0lbtfid8j6h49tf4cwq"}
+{"_id": "32171", "title": "Rule of Implication", "text": "Rule of Implication 0 273 493647 354568 2020-10-10T12:39:50Z Prime.mover 59 wikitext text/x-wiki == Proof Rule == The '''rule of implication''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Rule of Implication/Proof Rule}} === Sequent Form === {{:Rule of Implication/Sequent Form}} == Explanation == {{:Rule of Implication/Explanation}} == Also known as == {{:Rule of Implication/Also known as}} == Sources == * {{BookReference|Symbolic Logic|1973|Irving M. Copi|ed = 4th|edpage = Fourth Edition|prev = Implication Equivalent to Negation of Conjunction with Negative/Formulation 1|next = Rule of Transposition/Formulation 2}}: $2.4$: Statement Forms Category:Implication Category:Rule of Implication oaxbv4hmjqs1uazzych5wwy0o0b6lvx"}
+{"_id": "32172", "title": "Proof by Contradiction", "text": "Proof by Contradiction 0 275 452511 409742 2020-03-05T07:34:23Z Prime.mover 59 wikitext text/x-wiki == Proof Rule == '''Proof by contradiction''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Proof by Contradiction/Proof Rule}} === Sequent Form === {{:Proof by Contradiction/Sequent Form}} == Explanation == {{:Proof by Contradiction/Explanation}} == Also known as == {{:Proof by Contradiction/Also known as}} == Variants == The following forms can be used as variants of this theorem: === Variant 1 === {{:Proof by Contradiction/Variant 1}} === Variant 2 === {{:Proof by Contradiction/Variant 2}} === Variant 3 === {{:Proof by Contradiction/Variant 3}} == Also see == * Reductio ad Absurdum * Law of Excluded Middle == Sources == * {{BookReference|Adventures in Group Theory|2008|David Joyner|ed = 2nd|edpage = Second Edition|prev = Definition:Contrapositive Statement|next = Reductio ad Absurdum}}: Chapter $1$: Elementary, my dear Watson: $\\S 1.1$: You have a logical mind if... * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Proof|next = Definition:Proper Divisor|entry = proof by contradiction}} Category:Proof Techniques Category:Negation Category:Contradiction Category:Proof by Contradiction 93pb8g7tez7ee41xr05zohr36av3p2t"}
+{"_id": "32173", "title": "Rule of Explosion", "text": "Rule of Explosion 0 278 323536 245018 2017-10-28T16:11:09Z Lord Farin 560 wikitext text/x-wiki == Proof Rule == The '''Rule of Explosion''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Rule of Explosion/Proof Rule}} === Sequent Form === {{:Rule of Explosion/Sequent Form}} == Explanation == {{:Rule of Explosion/Explanation}} == Variants == The following can be used as variants of this theorem: === Variant 1 === {{:Rule of Explosion/Variant 1}} === Variant 2 === {{:Rule of Explosion/Variant 2}} === Variant 3 === {{:Rule of Explosion/Variant 3}} == Also known as == {{:Rule of Explosion/Also known as}} == Also see == * False Statement implies Every Statement Category:Contradiction Category:Rule of Explosion 76pn95w4d19csj56l3czdm6kli77sae"}
+{"_id": "32174", "title": "Law of Excluded Middle", "text": "Law of Excluded Middle 0 279 490603 448675 2020-09-25T06:27:59Z Prime.mover 59 wikitext text/x-wiki == Proof Rule == The '''law of (the) excluded middle''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Law of Excluded Middle/Proof Rule}} === Sequent Form === {{:Law of Excluded Middle/Sequent Form}} == Explanation == {{:Law of Excluded Middle/Explanation}} == Also known as == {{:Law of Excluded Middle/Also known as}} == Also see == * Principle of Non-Contradiction == Sources == * {{BookReference|Algebra Volume 1|1982|P.M. Cohn|edpage = Second Edition|ed = 2nd|prev = Definition:Biconditional|next = Definition:Tautology (Boolean Interpretations)}}: Chapter $1$: Sets and mappings: $\\S 1.1$: The need for logic * {{BookReference|Logic for Mathematicians|1988|Alan G. Hamilton|ed = 2nd|edpage = Second Edition|prev = Definition:Unsatisfiable (Boolean Interpretations)|next = Principle of Non-Contradiction}}: $\\S 1$: Informal statement calculus: $\\S 1.2$: Truth functions and truth tables: Example $1.6 \\ \\text{(a)}$ * {{BookReference|Mathematical Logic for Computer Science|1993|M. Ben-Ari|prev = Definition:Intuitionistic Logic|next = Definition:Temporal Logic}}: Chapter $1$: Introduction: $\\S 1.4$: Non-standard logics * {{BookReference|Introduction to Graph Theory|1993|Richard J. Trudeau|prev = Russell's Paradox|next = Definition:Theory of Types}}: $2$. Graphs: Paradox Category:Proof Rules Category:Philosophical Positions Category:Disjunction Category:Tautology Category:Law of Excluded Middle i05cf2wzoalh8whklj6dam1xq18gyg7"}
+{"_id": "32175", "title": "Modus Tollendo Tollens", "text": "Modus Tollendo Tollens 0 304 493614 466200 2020-10-10T12:25:25Z Prime.mover 59 wikitext text/x-wiki == Proof Rule == The '''modus tollendo tollens''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Modus Tollendo Tollens/Proof Rule}} === Sequent Form === {{:Modus Tollendo Tollens/Sequent Form}} == Explanation == {{:Modus Tollendo Tollens/Explanation}} == Also known as == {{:Modus Tollendo Tollens/Also known as}} == Also see == The following are related argument forms: * Modus Ponendo Ponens * Modus Ponendo Tollens * Modus Tollendo Ponens The Rule of Transposition is conceptually similar, and can be derived from the MTT by a simple application of the Rule of Implication. These are classic fallacies: * Affirming the Consequent * Denying the Antecedent == Linguistic Note == {{:Modus Tollendo Tollens/Linguistic Note}} == Sources == * {{BookReference|Symbolic Logic|1973|Irving M. Copi|ed = 4th|edpage = Fourth Edition|prev = Modus Ponendo Ponens|next = Hypothetical Syllogism/Formulation 1}}: $2.3$: Argument Forms and Truth Tables * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Modus Tollendo Ponens|next = Definition:Mole|entry = ''modus tollens''}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Modus Tollendo Ponens|next = Definition:Mole|entry = ''modus tollens''}} Category:Implication Category:Negation Category:Modus Tollendo Tollens e6awiocdjyyx9qiw71r61kc22yic59x"}
+{"_id": "32176", "title": "Tautology and Contradiction", "text": "Tautology and Contradiction 0 310 241510 238677 2015-12-24T17:23:17Z Prime.mover 59 wikitext text/x-wiki == Theorems == === Contradiction is Negation of Tautology === {{:Contradiction is Negation of Tautology}} === Tautology is Negation of Contradiction === {{:Tautology is Negation of Contradiction}} === Conjunction with Tautology === {{:Conjunction with Tautology}} === Disjunction with Tautology === {{:Disjunction with Tautology}} === Conjunction with Contradiction === {{:Conjunction with Contradiction}} === Disjunction with Contradiction === {{:Disjunction with Contradiction}} == Comment == Note that the proofs of: * $\\neg \\bot \\vdash \\top$ * $\\neg \\top \\vdash \\bot$ * $p \\vdash p \\land \\top$ * $p \\lor \\top \\vdash \\top$ rely (directly or indirectly) upon Law of Excluded Middle - and it can be seen that they are just another way of stating that truth. The propositions: :''If it's not false, it must be true'' and :''If it's not true, it must be false'' are indeed valid ''only'' in the context where there are only two truth values. From the intuitionistic perspective, these results do not hold. == Sources == * {{BookReference|Elementary Logic|1980|D.J. O'Connor|author2 = Betty Powell|prev = Definition:Contingent Statement|next = Definition:Truth Value/Aristotelian Logic}}: $\\S \\text{I}: 3$: Logical Constants $(2)$ Category:Tautology Category:Contradiction eesax7f146b3vge7b2n6rksqx6ogou5"}
+{"_id": "32177", "title": "Reductio ad Absurdum", "text": "Reductio ad Absurdum 0 312 415595 410772 2019-07-31T22:20:05Z Prime.mover 59 wikitext text/x-wiki == Proof Rule == The '''reductio ad absurdum''' is a valid deduction sequent in propositional logic: === Proof Rule === {{:Reductio ad Absurdum/Proof Rule}} === Sequent Form === {{:Reductio ad Absurdum/Sequent Form}} == Explanation == {{:Reductio ad Absurdum/Explanation}} == Variants == The following forms can be used as variants of this theorem: === Variant 1 === {{:Reductio ad Absurdum/Variant 1}} === Variant 2 === {{:Reductio ad Absurdum/Variant 2}} {{LEM}} == Also see == * Clavius's Law * Proof by Contradiction, often treated as another aspect of the same thing. From the point of view of purely classical logic, this is acceptable. However, in the context of intuitionistic logic, it is '''essential''' to bear in mind that only the Proof by Contradiction is valid. This is because Proof by Contradiction starts with a positive assumption $\\phi$. As a result, it does not depend on the Law of Excluded Middle. == Linguistic Note == {{:Reductio ad Absurdum/Linguistic Note}} == Sources == * {{BookReference|A Handbook of Terms used in Algebra and Analysis|1972|A.G. Howson|prev = Definition:Consistent Proof System|next = Definition:Independent Proof System}}: $\\S 1$: Some mathematical language: Axiom systems * {{BookReference|Elementary Logic|1980|D.J. O'Connor|author2 = Betty Powell|prev = Biconditional as Disjunction of Conjunctions/Formulation 2/Proof 2|next = Definition:Natural Deduction}}: $\\S \\text{I}: 13$: A Short Cut to Truth-tables * {{BookReference|Algebra Volume 1|1982|P.M. Cohn|edpage = Second Edition|ed = 2nd|prev = Definition:Converse Statement|next = Square Root of 2 is Irrational/Classic Proof}}: Chapter $1$: Sets and mappings: $\\S 1.1$: The need for logic * {{BookReference|Adventures in Group Theory|2008|David Joyner|ed = 2nd|edpage = Second Edition|prev = Proof by Contradiction|next = Definition:Conjunction/Truth Table}}: Chapter $1$: Elementary, my dear Watson: $\\S 1.1$: You have a logical mind if... Category:Proof Rules Category:Negation Category:Reductio ad Absurdum 600liiy6mll0k1hdz49i5h42u7nz15y"}
+{"_id": "32178", "title": "Conjunction and Implication", "text": "Conjunction and Implication 0 314 163059 163057 2013-09-28T10:33:29Z Lord Farin 560 wikitext text/x-wiki == Theorems == === Conjunction Equivalent to Negation of Implication of Negative === {{:Conjunction Equivalent to Negation of Implication of Negative}} === Implication Equivalent to Negation of Conjunction with Negative === {{:Implication Equivalent to Negation of Conjunction with Negative}} === Conjunction with Negative Equivalent to Negation of Implication === {{:Conjunction with Negative Equivalent to Negation of Implication}} === Modus Ponendo Tollens === {{:Modus Ponendo Tollens/Variant}} == Law of Excluded Middle == Note that the Modus Ponendo Tollens: :$\\neg \\left({p \\land q}\\right) \\dashv \\vdash p \\implies \\neg q$ can be proved in both directions without resorting to Law of Excluded Middle. All the others: * $p \\land q \\vdash \\neg \\left({p \\implies \\neg q}\\right)$ * $p \\implies q \\vdash \\neg \\left({p \\land \\neg q}\\right)$ * $p \\land \\neg q \\vdash \\neg \\left({p \\implies q}\\right)$ are not reversible in intuitionistic logic. == Also see == * Negation of Conditional implies Antecedent * Negation of Conditional implies Negation of Consequent Category:Conjunction Category:Implication dm3bild2uri1op8gxw84msinlx7bzb6"}
+{"_id": "32179", "title": "Disjunction and Implication", "text": "Disjunction and Implication 0 315 244713 230588 2016-01-16T22:21:32Z Prime.mover 59 wikitext text/x-wiki {{refactor|in progress}} == Theorems == === Modus Tollendo Ponens === {{:Modus Tollendo Ponens/Variant}} === Rule of Material Implication === {{:Rule of Material Implication}} Both of the above come in negative forms: {{begin-eqn}} {{eqn | l = \\neg \\left({p \\implies q}\\right)       | o = \\dashv \\vdash       | r = \\neg \\left({\\neg p \\lor q}\\right)       | c =  }} {{eqn | l = \\neg \\left({\\neg p \\implies q}\\right)       | o = \\dashv \\vdash       | r = \\neg \\left({p \\lor q}\\right)       | c =  }} {{end-eqn}} Disjunction is definable through implication: : $p \\lor q \\dashv \\vdash \\left({p \\implies q}\\right) \\implies q$ === Alternative rendition === They can alternatively be rendered as: {{begin-eqn}} {{eqn | ll= \\vdash       | l = \\left({\\neg \\left({p \\implies q}\\right)}\\right)       | o = \\iff       | r = \\left({\\neg \\left({\\neg p \\lor q}\\right)}\\right)       | c =  }} {{eqn | ll= \\vdash       | l = \\left({\\neg \\left({\\neg p \\implies q}\\right)}\\right)       | o = \\iff       | r = \\left({\\neg \\left({p \\lor q}\\right)}\\right)       | c =  }} {{eqn | ll= \\vdash       | l = \\left({p \\lor q}\\right)       | o = \\iff       | r = \\left({\\left({p \\implies q}\\right) \\implies q}\\right)       | c =  }} {{end-eqn}} They can be seen to be logically equivalent to the forms above. == Proof == {{BeginTableau|\\neg \\left({p \\implies q}\\right) \\vdash \\neg \\left({\\neg p \\lor q}\\right)}} {{Premise|1|\\neg \\left({p \\implies q}\\right)}} {{Assumption|2|\\neg p \\lor q|Assume the opposite of what is to be proved ...}} {{SequentIntro|3|2|p \\implies q|2|Rule of Material Implication}} {{NonContradiction|4|1, 2|3|1| ... demonstrating a contradiction}} {{Contradiction|5|1|\\neg \\left({\\neg p \\lor q}\\right)|2|4}} {{EndTableau}} {{qed}} {{BeginTableau|\\neg \\left({\\neg p \\lor q}\\right) \\vdash \\neg \\left({p \\implies q}\\right)}} {{Premise|1|\\neg \\left({\\neg p \\lor q}\\right)}} {{Assumption|2|p \\implies q|Assume the opposite of what is to be proved ...}} {{SequentIntro|3|2|\\neg p \\lor q|2|Rule of Material Implication}} {{NonContradiction|4|1, 2|3|1| ... demonstrating a contradiction}} {{Contradiction|5|1|\\neg \\left({p \\implies q}\\right)|2|4}} {{EndTableau}} {{qed}} {{LEM|Rule of Material Implication}} {{BeginTableau|\\neg \\left({\\neg p \\implies q}\\right) \\vdash \\neg \\left({p \\lor q}\\right)}} {{Premise|1|\\neg \\left({\\neg p \\implies q}\\right)}} {{Assumption|2|p \\lor q|Assume the opposite of what is to be proved ...}} {{SequentIntro|3|2|\\neg p \\implies q|2|Modus Tollendo Ponens}} {{NonContradiction|4|1, 2|3|1| ... demonstrating a contradiction}} {{Contradiction|5|1|\\neg \\left({p \\lor q}\\right)|2|4}} {{EndTableau}} {{qed}} {{BeginTableau|\\neg \\left({p \\lor q}\\right) \\vdash \\neg \\left({\\neg p \\implies q}\\right)}} {{Premise|1|\\neg \\left({p \\lor q}\\right)}} {{Assumption|2|\\neg p \\implies q|Assume the opposite of what is to be proved ...}} {{DeMorgan|3|1|\\neg p \\land \\neg q|1|Conjunction of Negations}} {{Simplification|4|1|\\neg p|3|1}} {{ModusPonens|5|1, 2|q|2|4|... from the assumption}} {{Simplification|6|1|\\neg q|3|2}} {{NonContradiction|7|1, 2|5|6| ... demonstrating a contradiction}} {{Contradiction|8|1|\\neg \\left({\\neg p \\implies q}\\right)|2|7}} {{EndTableau}} {{qed}} {{BeginTableau|p \\lor q \\vdash (p \\implies q) \\implies q}} {{Premise|1|p \\lor q}} {{Assumption|2|p \\implies q}} {{Assumption|3|p}} {{ModusPonens|4|2, 3|q|2|3}} {{Assumption|5|q}} {{ProofByCases|6|2|r|1|3|4|5|5}} {{Implication|7|1|\\left({p \\implies q}\\right) \\implies q|2|6}} {{EndTableau}} {{qed}} {{Proofread}} === Comment === Note that this: * $\\neg \\left({\\neg p \\implies q}\\right) \\dashv \\vdash \\neg \\left({p \\lor q}\\right)$ can be proved in both directions without resorting to the Law of Excluded Middle. All the others: * $p \\lor q \\vdash \\neg p \\implies q$ * $\\neg p \\lor q \\vdash p \\implies q$ * $\\neg \\left({p \\implies q}\\right) \\vdash \\neg \\left({\\neg p \\lor q}\\right)$ are not reversible in intuitionistic logic. == Proof by Truth Table == We apply the Method of Truth Tables to the propositions in turn. As can be seen by inspection, in all cases the truth values under the main connectives match for all boolean interpretations. $\\begin{array}{|cccc||ccccc|} \\hline \\neg & (p & \\lor & q) & \\neg & (\\neg & p & \\implies & q) \\\\ \\hline T & F & F & F & T & T & F & F & F \\\\ F & F & T & T & F & T & F & T & T \\\\ F & T & T & F & F & F & T & T & F \\\\ F & T & T & T & F & F & T & T & T \\\\ \\hline \\end{array}$ {{qed}} $\\begin{array}{|cccc||ccccc|} \\hline \\neg & (p & \\implies & q) & \\neg & (\\neg & p & \\lor & q) \\\\ \\hline F & F & T & F & F & T & F & T & F \\\\ F & F & T & T & F & T & F & T & T \\\\ T & T & F & F & T & F & T & F & F \\\\ F & T & T & T & F & F & T & T & T \\\\ \\hline \\end{array}$ {{qed}} Category:Disjunction Category:Implication g1zern52jt3d4o747h8wjo1ii9qzyfs"}
+{"_id": "32180", "title": "Set Difference and Intersection form Partition/Corollary 2", "text": "Set Difference and Intersection form Partition/Corollary 2 0 473 418202 418200 2019-08-15T06:02:43Z Prime.mover 59 wikitext text/x-wiki == Corollary to Set Difference and Intersection form Partition == <onlyinclude> Let $\\O \\subsetneqq T \\subsetneqq S$. Then: :$\\set {T, \\relcomp S T}$ is a partition of $S$. </onlyinclude> == Proof == First we note that: :$\\O \\subsetneqq T \\implies T \\ne \\O$ :$T \\subsetneqq S \\implies T \\ne S$ from the definition of proper subset. From the definition of relative complement, we have: :$\\relcomp S T = S \\setminus T$. It follows from Set Difference with Self is Empty Set that: :$T \\ne S \\iff \\relcomp S T \\ne \\O$ From Intersection with Relative Complement is Empty: :$T \\cap \\relcomp S T = \\O$ That is, $T$ and $\\relcomp S T$ are disjoint. From Union with Relative Complement: :$T \\cup \\relcomp S T = S$ That is, the union of $T$ and $\\relcomp S T$ forms the whole set $S$. Thus all the conditions for a partition are satisfied. {{qed}} == Sources == * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth|prev = Definition:Set Partition/Definition 1|next = Definition:Endorelation}}: $\\S 6$. Indexed families; partitions; equivalence relations: Example $6.4$ Category:Relative Complement ec8skoyuz4jxzrs02ej5wqxkc325k68"}
+{"_id": "32181", "title": "Empty Intersection iff Subset of Complement", "text": "Empty Intersection iff Subset of Complement 0 511 416793 414063 2019-08-05T21:57:18Z Prime.mover 59 wikitext text/x-wiki == Corollary to Intersection with Complement is Empty iff Subset == <onlyinclude> :$S \\cap T = \\O \\iff S \\subseteq \\relcomp {} T$ </onlyinclude> where: :$S \\cap T$ denotes the intersection of $S$ and $T$ :$\\O$ denotes the empty set :$\\complement$ denotes set complement :$\\subseteq$ denotes subset. === Corollary === {{:Empty Intersection iff Subset of Complement/Corollary}} == Proof 1 == {{:Empty Intersection iff Subset of Complement/Proof 1}} == Proof 2 == {{:Empty Intersection iff Subset of Complement/Proof 2}} == Sources == * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth|prev = Complement Union with Superset is Universe/Corollary|next = Intersection is Empty and Union is Universe if Sets are Complementary}}: $\\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $11 \\ \\text{(b)}$ * {{BookReference|Introduction to Topology|1975|Bert Mendelson|ed = 3rd|edpage = Third Edition|prev = Intersection with Subset is Subset|next = Complement Union with Superset is Universe/Corollary}}: Chapter $1$: Theory of Sets: $\\S 3$: Set Operations: Union, Intersection and Complement: Exercise $1 \\ \\text{(c)}$ Category:Set Complement Category:Set Intersection Category:Empty Set Category:Subsets Category:Empty Intersection iff Subset of Complement 6sgo51puejhk70xmyjzrunwdfh8skgf"}
+{"_id": "32182", "title": "Image is Subset of Codomain/Corollary 1", "text": "Image is Subset of Codomain/Corollary 1 0 548 397277 397276 2019-03-25T20:03:12Z Prime.mover 59 wikitext text/x-wiki == Corollary to Image is Subset of Codomain == <onlyinclude> Let $\\mathcal R = S \\times T$ be a relation. The image of $\\mathcal R$ is a subset of the codomain of $\\mathcal R$: :$\\Img {\\mathcal R} \\subseteq T$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\Img {\\mathcal R}       | r = \\mathcal R \\sqbrk {\\Dom {\\mathcal R} }       | c = {{Defof|Image of Relation}} }} {{eqn | l = \\Dom {\\mathcal R}       | o = \\subseteq       | r = \\Dom {\\mathcal R}       | c = Set is Subset of Itself }} {{eqn | ll= \\leadsto       | l = \\Img {\\mathcal R}       | o = \\subseteq       | r = T       | c = Image is Subset of Codomain }} {{end-eqn}} {{qed}} Category:Relation Theory 29kiylqq2rnxxkgg2fzbgbqg50epl7y"}
+{"_id": "32183", "title": "Sundry Coset Results", "text": "Sundry Coset Results 0 1196 372738 372734 2018-10-23T07:57:00Z Prime.mover 59 wikitext text/x-wiki {{rename}} {{refactor|Once extension works with TOC, use it to fix headings}} == Theorems == Let $G$ be a group and let $H$ be a subgroup of $G$. Let $x, y \\in G$. Let: : $x H$ denote the left coset of $H$ by $x$; : $H y$ denote the right coset of $H$ by $y$. Then the following results apply: === Element in Coset iff Product with Inverse in Subgroup === {{:Element in Coset iff Product with Inverse in Subgroup}} === Cosets are Equal iff Product with Inverse in Subgroup === {{:Cosets are Equal iff Product with Inverse in Subgroup}} === Cosets are Equal iff Element in Other Coset === {{:Cosets are Equal iff Element in Other Coset}} === Coset Equals Subgroup iff Element in Subgroup === {{:Coset Equals Subgroup iff Element in Subgroup}} === Elements in Same Coset iff Product with Inverse in Subgroup === {{:Elements in Same Coset iff Product with Inverse in Subgroup}} === Regular Representation on Subgroup is Bijection to Coset === {{:Regular Representation on Subgroup is Bijection to Coset}} Category:Cosets dw55ecqly0ukrqdzu7y7en29tx3o5fn"}
+{"_id": "32184", "title": "Strictly Increasing Sequence on Ordered Set", "text": "Strictly Increasing Sequence on Ordered Set 0 1374 366577 366574 2018-09-20T12:36:57Z KarlFrei 3474 added category wikitext text/x-wiki == Lemma == Let $\\left({S, \\preceq}\\right)$ be a totally ordered set. Let $\\left \\langle {r_k} \\right \\rangle_{p \\mathop \\le k \\mathop \\le q}$ be a finite sequence of elements of $\\left({S, \\preceq}\\right)$. Then $\\left \\langle {r_k} \\right \\rangle_{p \\mathop \\le k \\mathop \\le q}$ is strictly increasing {{iff}}: :$\\forall k \\in \\left[{p + 1 \\,.\\,.\\, q}\\right]: r_{k - 1} \\prec r_k$ == Proof == Let $\\left \\langle {r_k} \\right \\rangle_{p \\mathop \\le k \\mathop \\le q}$ be strictly increasing. Because $\\forall k \\in \\N_{>0}: k - 1 < k$, it follows directly that: : $\\forall k \\in \\left[{p + 1 \\,.\\,.\\, q}\\right]: r_{k - 1} \\prec r_k$ For the other direction, we use a Proof by Contraposition. To that end, suppose $\\left \\langle {r_k} \\right \\rangle_{p \\mathop \\le k \\mathop \\le q}$ is ''not'' strictly increasing. Let $K$ be the set of all $k \\in \\left[{p \\,.\\,.\\, q}\\right]$ such that: : $\\exists j \\in \\left[{p \\,.\\,.\\, q}\\right]$ such that $j < k$ and $r_k \\preceq r_j$ The set $K$ is not empty because $\\left \\langle {r_k} \\right \\rangle_{p \\mathop \\le k \\mathop \\le q}$ is not strictly increasing. As $K \\subset \\N$ and the latter is well-ordered, then so is $K$. Thus $K$ has a minimal element $m$. Thus there exists $j \\in \\left[{p \\,.\\,.\\, q}\\right]$ such that: : $j < m$ and: : $r_m \\preceq r_j$ Because $m - 1 < m$: :$j \\le m - 1$ and so: :$m - 1 \\notin K$ So: : $r_m \\preceq r_j\\prec r_{m-1}.$ Since orderings are transitive, it follows : $r_m \\preceq r_{m-1}.$ From Rule of Transposition it follows that :$\\forall k \\in \\left[{p + 1 \\,.\\,.\\, q}\\right]: r_{k - 1} \\prec r_k\\implies \\left \\langle {r_k} \\right \\rangle_{p \\mathop \\le k \\mathop \\le q}$ is strictly increasing. The result follows. {{Qed}} == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev=Strictly Increasing Sequence induces Partition|next=Fundamental Principle of Counting}}: $\\S 18$: Lemma for Theorem $18.4$ Category:Sequences Category:Proofs by Contraposition mkquczmokcibfdu9m0vax0lkugixbos"}
+{"_id": "32185", "title": "Euclidean Algorithm", "text": "Euclidean Algorithm 0 1527 493847 445819 2020-10-11T06:57:17Z Prime.mover 59 wikitext text/x-wiki == Algorithm == <onlyinclude> The '''Euclidean algorithm''' is a method for finding the greatest common divisor (GCD) of two integers $a$ and $b$. Let $a, b \\in \\Z$ and $a \\ne 0 \\lor b \\ne 0$. The steps are: :$(1): \\quad$ Start with $\\tuple {a, b}$ such that $\\size a \\ge \\size b$. If $b = 0$ then the task is complete and the GCD is $a$. :$(2): \\quad$ If $b \\ne 0$ then you take the remainder $r$ of $\\dfrac a b$. :$(3): \\quad$ Set $a \\gets b, b \\gets r$ (and thus $\\size a \\ge \\size b$ again). :$(4): \\quad$ Repeat these steps until $b = 0$. Thus the GCD of $a$ and $b$ is the value of the variable $a$ after the termination of the algorithm. </onlyinclude> {{:Euclid:Proposition/VII/2}} == Proof 1 == {{:Euclidean Algorithm/Proof 1}} == Euclid's Proof == {{:Euclidean Algorithm/Euclid's Proof}} == Demonstration == {{:Euclidean Algorithm/Demonstration}} == Algorithmic Nature == {{:Euclidean Algorithm/Algorithmic Nature}} == Formal Implementation == {{:Euclidean Algorithm/Formal Implementation}} == Constructing an Integer Combination == {{:Euclidean Algorithm/Construction of Integer Combination}} == Also known as == The '''Euclidean algorithm''' is also known as '''Euclid's algorithm''' or the '''Euclidean division algorithm'''. == Examples == {{:Euclidean Algorithm/Examples}} == Also see == * Rational Numbers and SFCFs are Equivalent for an application of the '''Euclidean algorithm''' in a slightly different context. {{Euclid Note|2|VII}} {{Namedfor|Euclid|cat = Euclid}} == Sources == * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Bézout's Lemma/Proof 2|next = Definition:Prime Number/Definition 2}}: $\\S 3.11$ * {{BookReference|Number Theory|1971|George E. Andrews|prev = Euclidean Algorithm/Examples/341 and 527|next = Euclidean Algorithm/Demonstration}}: $\\text {2-2}$ Divisibility: Theorem $\\text {2-2}$ * {{BookReference|Elements of Abstract Algebra|1971|Allan Clark|prev = GCD and LCM Distribute Over Each Other|next = Euclidean Algorithm/Euclid's Proof}}: Chapter $1$: Properties of the Natural Numbers: $\\S 23 \\zeta$ * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Prime not Divisor implies Coprime|next = Euclidean Algorithm/Demonstration}}: $\\S 12$: Highest common factors and Euclid's algorithm * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Abel-Ruffini Theorem/Historical Note|next = Lamé's Theorem/Historical Note}}: $5$ * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Definition:Euclidean|next = Definition:Euclidean Construction|entry = Euclidean algorithm}} * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Definition:Perfect Number|next = Axiom:Euclid's Postulates|entry = Euclid's algorithm''' or '''Euclidean algorithm}} * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Definition:Greatest Common Divisor/Integers/Definition 2|next = Euclidean Algorithm/Demonstration}}: $\\text{A}.1$: Number theory * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Definition:Algorithm/Linguistic Note|next = Definition:Greatest Common Divisor/Integers/Definition 1}}: $\\S 1.1$: Algorithms: Algorithm $\\text{E}$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Abel-Ruffini Theorem/Historical Note|next = Lamé's Theorem/Historical Note}}: $5$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Mathematician:Euclid|next = Definition:Euclidean Construction|entry = Euclidean algorithm}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Mathematician:Euclid|next = Lamé's Theorem|entry = Euclidean algorithm}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Mathematician:Euclid|next = Definition:Euclidean Construction|entry = Euclidean Algorithm}} Category:Euclidean Algorithm Category:Number Theory Category:Greatest Common Divisor 7sgt63ld7znycn4u25fcsh8r7s54l1w"}
+{"_id": "32186", "title": "Real Numbers are Close Packed", "text": "Real Numbers are Close Packed 0 1643 411495 410825 2019-07-03T20:54:04Z Prime.mover 59 wikitext text/x-wiki == Corollary to Between two Real Numbers exists Rational Number == <onlyinclude> :$\\forall a, b \\in \\R: a < b \\implies \\paren {\\exists c \\in \\R: a < c < b}$ </onlyinclude> That is, the set of real numbers is close packed. == Proof 1 == {{:Real Numbers are Close Packed/Proof 1}} == Proof 2 == {{:Real Numbers are Close Packed/Proof 2}} == Sources == * {{BookReference|Algebra Volume 1|1982|P.M. Cohn|edpage = Second Edition|ed = 2nd|prev = Quantifier/Examples/Existence of Multiplicative Identity|next = Equivocation of Nothing}}: Chapter $1$: Sets and mappings: $\\S 1.1$: The need for logic: Exercise $(8)$ Category:Real Analysis Category:Real Numbers are Close Packed 6dpt6r61ivdbhad6ncibxrh7qi3z9gn"}
+{"_id": "32187", "title": "Group Action on Prime Power Order Subset", "text": "Group Action on Prime Power Order Subset 0 1995 404172 404159 2019-05-04T12:52:20Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $G$ be a finite group. Let $\\mathbb S = \\set {S \\subseteq G: \\card S = p^n}$ where $p$ is prime. That is, the set of all subsets of $G$ whose cardinality is the power of a prime number. Let $G$ act on $\\mathbb S$ by the group action defined in Group Action on Sets with k Elements: :$\\forall S \\in \\mathbb S: g * S = g S = \\set {x \\in G: x = g s: s \\in S}$. Then: === Stabilizer is $p$-Subgroup === {{:Group Action on Prime Power Order Subset/Stabilizer is p-Subgroup}} === Stabilizer of Maximal Power Order Subset === {{:Group Action on Prime Power Order Subset/Stabilizer of Maximal Power Order Subset}} Category:Examples of Group Actions Category:Group Action on Prime Power Order Subset Category:Group Action on Sets with k Elements kmqpneic98hd4d379weg4zt375fwfzw"}
+{"_id": "32188", "title": "Orbits of Group Action on Sets with Power of Prime Size", "text": "Orbits of Group Action on Sets with Power of Prime Size 0 2003 404166 404165 2019-05-04T12:47:52Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $G$ be a finite group such that $\\order G = k p^n$ where $p \\nmid k$. Let $\\mathbb S = \\set {S \\subseteq G: \\order S = p^n}$ Let $G$ act on $\\mathbb S$ by the group action defined in Group Action on Sets with k Elements: :$\\forall S \\in \\mathbb S: g * S = g S = \\set {x \\in G: x = g s: s \\in S}$ Then: === Orbit Length === {{:Orbits of Group Action on Sets with Power of Prime Size/Orbit Length}} === Orbit whose Length is not Divisible by $p$ === {{:Orbits of Group Action on Sets with Power of Prime Size/Orbit whose Length is not Divisible by p}} === Orbit whose Length is Divisible by $p$ === {{:Orbits of Group Action on Sets with Power of Prime Size/Orbit whose Length is Divisible by p}} Category:Group Action on Prime Power Order Subset Category:Orbits of Group Action on Sets with Power of Prime Size ljy7325x3umbtw6he4d6nyl1dh3jr9e"}
+{"_id": "32189", "title": "Axiom:Euclid's First Postulate", "text": "Axiom:Euclid's First Postulate 100 2125 439989 425102 2019-12-17T21:27:44Z Prime.mover 59 wikitext text/x-wiki == Postulate == <onlyinclude> A straight line segment can be drawn joining any two points. {{EuclidPostulateStatement|First|To draw a straight line from any point to any point.}} </onlyinclude> == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 1|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Definition:Euclid's Definitions - Book I/23 - Parallel Lines|next = Axiom:Euclid's Second Postulate}}: Book $\\text{I}$. Postulates * {{BookReference|Probability Theory|1965|A.M. Arthurs|prev = Definition:Straight Line|next = Definition:Axiom}}: Chapter $2$: Probability and Discrete Sample Spaces: $2.1$ Introduction * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Axiom:Euclid's Postulates|next = Axiom:Euclid's Second Postulate|entry = Euclid's axioms}} Euclid Postulate 1 n70ocyqo261qg5q7e6kw8y1t8m51qtw"}
+{"_id": "32190", "title": "Axiom:Euclid's Second Postulate", "text": "Axiom:Euclid's Second Postulate 100 2126 425103 196201 2019-09-13T16:23:29Z Prime.mover 59 wikitext text/x-wiki == Postulate == <onlyinclude> A straight line segment can be extended indefinitely to form a straight line. {{EuclidPostulateStatement|Second|To produce a finite straight line continuously in a straight line.}} </onlyinclude> === Production === {{:Axiom:Euclid's Second Postulate/Production}} == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 1|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Axiom:Euclid's First Postulate|next = Axiom:Euclid's Third Postulate}}: Book $\\text{I}$. Postulates * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Axiom:Euclid's First Postulate|next = Axiom:Euclid's Third Postulate|entry = Euclid's axioms}} Euclid Postulate 2 e9p10asae0xjolxihapf4tieho2npm3"}
+{"_id": "32191", "title": "Axiom:Euclid's Third Postulate", "text": "Axiom:Euclid's Third Postulate 100 2127 425104 196204 2019-09-13T16:24:18Z Prime.mover 59 wikitext text/x-wiki == Postulate == <onlyinclude> Given any line segment, a circle can be drawn using the segment as the radius with one endpoint as the center. {{EuclidPostulateStatement|Third|To describe a circle with any centre  and distance.}} </onlyinclude> == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 1|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Axiom:Euclid's Second Postulate|next = Axiom:Euclid's Fourth Postulate}}: Book $\\text{I}$. Postulates * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Axiom:Euclid's Second Postulate|next = Axiom:Euclid's Fourth Postulate|entry = Euclid's axioms}} Euclid Postulate 3 6s9apddjmowqegm9i6175yvq7plt7fd"}
+{"_id": "32192", "title": "Axiom:Euclid's Fourth Postulate", "text": "Axiom:Euclid's Fourth Postulate 100 2128 430025 425105 2019-10-08T12:22:27Z Prime.mover 59 wikitext text/x-wiki == Postulate == <onlyinclude> All right angles are congruent. {{EuclidPostulateStatement|Fourth|That all right angles are equal to one another.}} </onlyinclude> == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 1|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Axiom:Euclid's Third Postulate|next = Axiom:Euclid's Fifth Postulate}}: Book $\\text{I}$. Postulates * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Axiom:Euclid's Third Postulate|next = Axiom:Euclid's Fifth Postulate|entry = Euclid's axioms}} * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Axiom:Euclid's Common Notion 1|next = Definition:Axiom}}: Chapter $2$: The Logic of Shape: Euclid Euclid Postulate 4 gu9wlfhcdl178dovsnycj6mdlqgh6sq"}
+{"_id": "32193", "title": "Axiom:Euclid's Fifth Postulate", "text": "Axiom:Euclid's Fifth Postulate 100 2129 425109 425107 2019-09-13T16:40:36Z Prime.mover 59 wikitext text/x-wiki == {{AuthorRef|Euclid}}'s Statement == <onlyinclude> {{EuclidPostulateStatement|Fifth|If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.}} </onlyinclude> There are many equivalent ways to state this postulate.  See below for a selection of them. == Parallel Postulate == {{:Axiom:Parallel Postulate}} == Playfair's Axiom == {{:Axiom:Playfair's Axiom}} == Other equivalent statements == Many further attempts have been made to formulate equivalent definitions of this axiom, often with a view to finding a proof which relies on the other four axioms. (Such attempts have been universally doomed to failure.) Here are a few examples, in approximate chronological order: === {{AuthorRef|Proclus Lycaeus|Proclus}} === :If a straight line intersects one of two parallels, it will intersect the other also. Straight lines parallel to the same straight line are parallel to one another. === {{AuthorRef|Posidonius of Apameia|Posidonius}} and {{AuthorRef|Geminus of Rhodes|Geminus}} === :There exist straight lines everywhere equidistant from one another. (This can be compared with {{AuthorRef|Proclus Lycaeus|Proclus}}' tacit assumption that \"Parallels remain, throughout their length, at a finite distance from one another.\") === {{AuthorRef|Adrien-Marie Legendre|Legendre}} === :There exists a triangle in which the sum of the three angles is equal to two right angles. === {{AuthorRef|John Wallis|Wallis}}, {{AuthorRef|Lazare Nicolas Marguerite Carnot|Carnot}}, {{AuthorRef|Pierre-Simon de Laplace|Laplace}} === :Given any figure, there exists a figure similar to it of any size we please. {{AuthorRef|Giovanni Girolamo Saccheri|Saccheri}} points out that it is necessary only to postulate that: :There exist two unequal triangles with equal angles. === {{AuthorRef|Adrien-Marie Legendre|Legendre}} (again) === :Through any point within an angle less than two-thirds of a right angle a straight line can always be drawn which meets both sides of the angle. === {{AuthorRef|Johann Friedrich Lorenz|Lorenz}} === :Every straight line through a point within an angle must meet one of the sides of the angle.<ref>{{AuthorRef|Johann Friedrich Lorenz}}: ''Grundriss der reinen und angewandten Mathematik'', 1791.</ref> === {{AuthorRef|Adrien-Marie Legendre|Legendre}} and {{AuthorRef|Farkas Wolfgang Bolyai|W. Bolyai}} === :Given any three points not in a straight line, there exists a circle passing through them. === {{AuthorRef|Carl Friedrich Gauss|Gauss}} === :If I could prove that a rectilineal triangle is possible the content of which is greater than any given area, I am in a position to prove perfectly rigorously the whole of geometry.<ref>{{AuthorRef|Carl Friedrich Gauss|Gauss}}, in a letter to {{AuthorRef|Farkas Wolfgang Bolyai|W. Bolyai}}, 1799.</ref> === {{AuthorRef|Julius Worpitzky|Worpitzky}} === :There exists no triangle in which every angle is as small as we please. === {{AuthorRef|Alexis Claude Clairaut|Clairault}} === :If in a quadrilateral three angles are right angles, the fourth angle is a right angle also. (1741) === {{AuthorRef|Giuseppe Veronese|Veronese}} === :If two straight lines are parallel, they are figures opposite to (or the reflex of) one another with respect to the middle points of all their transversal segments.<ref>{{AuthorRef|Giuseppe Veronese|Veronese}}: ''Elementi'', 1904.</ref> === {{AuthorRef|G. Ingrami|Ingrami}} === :Two parallel straight lines intercept, on every transversal which passes through the middle point of a segment included within them, another segment the middle point of which is the middle point of the first.<ref>{{AuthorRef|G. Ingrami|Ingrami}}: ''Elementi'', 1904.</ref> == Historical Note == {{:Axiom:Euclid's Fifth Postulate/Historical Note}} == Also see == * {{TarskiAxiomLink|axiom = Euclid's Axiom|pageName = Euclid's Fifth Postulate}} == References == <references/> == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 1|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Mathematician:Dionysius Lardner|next = Mathematician:Proclus Lycaeus}}: Preface to the Second Edition * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 1|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Axiom:Euclid's Fourth Postulate|next = Axiom:Euclid's Common Notion 1}}: Book $\\text{I}$. Postulates * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Axiom:Euclid's Fourth Postulate|next = Axiom:Playfair's Axiom|entry = Euclid's axioms}} Euclid Postulate 5 cmj46mqk4dtm5m3ls5f1eonb6lckqmo"}
+{"_id": "32194", "title": "Axiom:Euclid's Common Notions", "text": "Axiom:Euclid's Common Notions 100 2634 445822 430022 2020-02-02T23:04:56Z Prime.mover 59 wikitext text/x-wiki == Common Notions == <onlyinclude> This is a set of axiomatic statements that appear at the start of Book $\\text{I}$ of {{ElementsLink}}. === Common Notion 1 === {{:Axiom:Euclid's Common Notion 1}} === Common Notion 2 === {{:Axiom:Euclid's Common Notion 2}} === Common Notion 3 === {{:Axiom:Euclid's Common Notion 3}} === Common Notion 4 === {{:Axiom:Euclid's Common Notion 4}} === Common Notion 5 === {{:Axiom:Euclid's Common Notion 5}}</onlyinclude> == Historical Note == {{:Axiom:Euclid's Common Notions/Historical Note}} == References == <references/> == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 1|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Axiom:Euclid's Postulates|next = Axiom:Playfair's Axiom}}: Book $\\text{I}$. Common Notions * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Axiom:Euclid's Postulates|next = Definition:Axiom}}: Chapter $\\text {A}.4$: Euclid (flourished ca. $300$ B.C.) * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Euclidean Geometry|next = Axiom:Euclid's Postulates|entry = Euclidean geometry}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Euclidean Geometry|next = Axiom:Euclid's Postulates|entry = Euclidean geometry}} * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Definition:Euclid's Definitions|next = Axiom:Euclid's Postulates}}: Chapter $2$: The Logic of Shape: Euclid * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Axiom:Euclid's Postulates|next = Mathematician:Eudoxus of Cnidus|entry = Euclid's axioms}} Category:Euclid's Common Notions ey6aqgq3tenjaykk0290pw8jxo2pi4r"}
+{"_id": "32195", "title": "Closure of Subset of Metric Space by Convergent Sequence", "text": "Closure of Subset of Metric Space by Convergent Sequence 0 3396 445080 334490 2020-01-28T21:42:27Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $M$ be a metric space. Let $C \\subseteq M$. Let $x \\in M$. Let $\\map \\cl C$ denote the closure of $C$ in $T$. Then $x \\in \\map \\cl C$ {{iff}} there exists a sequence $\\sequence {x_n}$ in $C$ which converges to $x$. == Proof == === Necessary Condition === Suppose there exists a sequence $\\sequence {x_n}$ in $C$ which converges to $x$. Let $\\epsilon > 0$. Then by definition: :$\\exists N \\in \\N: \\forall n > N: x_n \\in \\map {B_\\epsilon} x$ where $\\map {B_\\epsilon} x$ is the open $\\epsilon$-ball of $x$ in $M$. Since $\\forall n: x_n \\in C$, it follows that: :$\\forall \\epsilon > 0: \\map {B_\\epsilon} x \\cap C \\ne \\O$ Hence $x \\in \\map \\cl C$. {{qed|lemma}} === Sufficient Condition === Now suppose $x \\in \\map \\cl C$. By definition of closure: :$\\forall n \\in \\N: \\exists x_n \\in C \\cap \\map {B_{1 / n} } x$ Thus clearly $\\sequence {x_n}$ converges to $x$. {{handwaving|Lose the weasel word \"clearly\"}} {{qed}} == Sources == * {{BookReference|Introduction to Metric and Topological Spaces|1975|W.A. Sutherland|prev = Definition:Sequentially Compact Space|next = Definition:Cauchy Sequence (Metric Space)}}: $7.2$: Sequential compactness: Lemma $7.2.2$ Category:Metric Spaces Category:Sequences ipkattwqz1m3rifaregy8oby7lpkqwq"}
+{"_id": "32196", "title": "Series Law for Extremal Length/Rho is Well Defined", "text": "Series Law for Extremal Length/Rho is Well Defined 0 3435 435002 240506 2019-11-12T16:46:21Z Prime.mover 59 wikitext text/x-wiki {{MissingLinks|throughout}} == Lemma for Series Law for Extremal Length == {{explain|The details of what is to be proved needs to be stated here, in order that this page be appropriately self-contained.}} == Proof == To see that $\\rho$ is a well-defined metric, we need to check that it transforms correctly when changing local coordinates. Let $z = \\map z t$ and $w = \\map w t$ be charts on the Riemann surface $X$. Let $\\map {\\rho_1^z} t$ and $\\map {\\rho_1^w} t$ be the coefficient functions when $\\rho_1$ is expressed in the local coordinates $z$ and $w$, respectively. We use the analogous notation for $\\rho_2$ and $\\rho$. Since $\\rho_j$ is a metric for $j \\in \\set {1, 2}$, we have: :$\\map {\\rho_j^w} t = \\map {\\rho_j^z} t \\cdot \\size {\\dfrac {\\d z} {\\d w} }$ where $\\dfrac {\\d z} {\\d w}$ denotes, the derivative of the coordinate change $z \\circ w^{-1}$. Thus we have: {{begin-eqn}} {{eqn | l = \\map {\\rho^w} t       | r = \\max \\set {\\map {\\rho_1^w} t, \\map {\\rho_2^w} t}       | c =  }} {{eqn | r = \\max \\set {\\map {\\rho_1^z} t, \\map {\\rho_2^z} t} \\cdot \\size {\\frac {\\d z} {\\d w} }       | c =  }} {{eqn | r = \\map {\\rho^z} t \\cdot \\size {\\dfrac {\\d z} {\\d w} }       | c =  }} {{end-eqn}} This means that $\\rho$ transforms correctly and is a metric, as desired. {{qed}} bp2yqxpiygxf9e7xgb07wg13fk5vska"}
+{"_id": "32197", "title": "Goldbach Conjecture", "text": "Goldbach Conjecture 0 3618 492809 474006 2020-10-06T06:23:02Z Prime.mover 59 wikitext text/x-wiki == Conjecture == <onlyinclude> Every even integer greater than $2$ is the sum of two primes. </onlyinclude> === Marginal Conjecture === {{:Goldbach Conjecture/Marginal}} {{Hilbert23|8b}} == Landau's Problems == This is the $1$st of Landau's problems. == Also see == * Goldbach's Weak Conjecture * Goldbach Conjecture implies Goldbach's Marginal Conjecture {{Namedfor|Christian Goldbach|cat = Goldbach}} == Historical Note == {{:Goldbach Conjecture/Historical Note}} == Sources == * {{BookReference|Algebra Volume 1|1982|P.M. Cohn|edpage = Second Edition|ed = 2nd|prev = Proof by Counterexample|next = Definition:Sufficient Condition}}: Chapter $1$: Sets and mappings: $\\S 1.1$: The need for logic * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Tamref's Last Theorem/Mistake|next = Definition:Binary Notation}}: $2$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = 5^x + 12^y equals 13^z has Unique Solution|next = Definition:Binary Notation}}: $2$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Gödel's Incompleteness Theorems/Historical Note|next = Goldbach Conjecture/Historical Note|entry = Goldbach's conjecture}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Gödel's Incompleteness Theorems/Historical Note|next = Goldbach Conjecture/Historical Note|entry = Goldbach's conjecture}} * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Goldbach Conjecture/Marginal/Historical Note|next = Goldbach Conjecture/Historical Note}}: Chapter $7$: Patterns in Numbers: Fermat Category:Unproven Hypotheses Category:Prime Numbers Category:Goldbach Conjecture Category:Landau's Problems huazswegt2nronbgpmj6n6cxmtr4p3g"}
+{"_id": "32198", "title": "Gauss's Lemma (Number Theory)", "text": "Gauss's Lemma (Number Theory) 0 3718 357332 267473 2018-05-27T08:56:10Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $p$ be an odd prime. Let $a \\in \\Z: a \\not \\equiv 0 \\pmod p$. Let $S = \\left\\{{a, 2 a, 3 a, \\ldots, \\dfrac {p - 1} 2 a}\\right\\}$. Let $n$ denote the number of elements of $S$ whose least positive residue modulo $p$ is greater than $\\dfrac p 2$. Then: :$\\left({\\dfrac a p}\\right) = \\left({-1}\\right)^n$ where $\\left({\\dfrac a p}\\right)$ is the Legendre symbol. == Proof == First note that no two elements of $s$ are congruent modulo $p$: Let $r a \\equiv s a \\pmod p$ for some $r, s$ such that $1 \\le s \\le r \\le \\dfrac {p-1} 2$. As $a \\perp p$, we have that $r a \\equiv s a \\implies r \\equiv s \\pmod p$ from Cancellability of Congruences. Now $r \\equiv s \\pmod p$ can happen only when $r = s$ as $0 \\le r - s \\le \\dfrac {p-1} 2$. Next, no element of $S$ is congruent modulo $p$ to $0$. This is because $r a \\equiv 0 \\pmod p$ when $a \\not \\equiv 0$ requires $r \\equiv 0$ which doesn't happen. Now, we create $S'$ from $S$ by replacing each element of $S$ by its least positive residue modulo $p$. Arranging $S'$ into increasing order, we get: :$S' = \\left\\{{b_1, b_2, \\ldots, b_m, c_1, c_2, \\ldots, c_n}\\right\\}$ where $b_m < \\dfrac p 2 < c_1$ and $m + n = \\dfrac {p-1}2$. As $p$ is an odd prime, $\\dfrac p 2$ is not an integer so neither $b_m$ nor $c_1$ can be equal to $\\dfrac p 2$. Now we let $S'' = \\left\\{{b_1, b_2, \\ldots, b_m, p-c_1, p-c_2, \\ldots, p-c_n}\\right\\}$ We need to show that $S'' = \\left\\{{1, 2, 3, \\ldots, \\dfrac {p-1} 2}\\right\\}$. Now: * all the elements of $S''$ are positive, as all the $c_j < p$ * the largest one is either $b_m$ or $p - c_1$ * there are $\\dfrac {p-1} 2$ of them * they are all less than $\\dfrac p 2$. So $S''$ contains $\\dfrac {p-1} 2$ positive integers all less than $\\dfrac p 2$. So all we need to do is show that they are distinct. As all the elements of $S'$ are distinct, the $m$ elements $b_1, b_2, \\ldots, b_m$ are distinct. Similarly, so are the $n$ elements $p-c_1, p-c_2, \\ldots, p-c_n$. So, we suppose $b_i = p - c_j$ for some $i, j$ and try to derive a contradiction. So: :$p = b_i + c_j \\equiv r a + s a \\pmod p$ for some $r, s$ such that $1 \\le r, s \\le \\dfrac {p-1} 2$. But from: :$p \\equiv \\left({r+s}\\right) a \\pmod p$ and $a \\not \\equiv 0 \\pmod p$ we apply Euclid's Lemma to get $r + s \\equiv 0 \\pmod p$. But this is impossible since $2 \\le r + s \\le p-1$. This establishes that $S'' = \\left\\{{1, 2, 3, \\ldots, \\dfrac {p-1} 2}\\right\\}$. Now, we multiply all the elements of $S''$ together: {{begin-eqn}} {{eqn | l = \\left({\\frac {p - 1} 2}\\right)!       | r = b_1 b_2 \\cdots b_m \\left({p - c_1}\\right) \\left({p - c_2}\\right) \\cdots \\left({p - c_n}\\right)       | c =  }} {{eqn | o = \\equiv       | r = b_1 b_2 \\cdots b_m \\left({-c_1}\\right) \\left({-c_2}\\right) \\cdots \\left({-c_n}\\right)       | rr= \\pmod p       | c =  }} {{eqn | o = \\equiv       | r = \\left({-1}\\right)^n b_1 b_2 \\cdots b_m c_1 c_2 \\cdots c_n       | rr= \\pmod p       | c =  }} {{eqn | o = \\equiv       | r = \\left({-1}\\right)^n a \\times 2 a \\times 3 a \\times \\cdots \\frac{p - 1} 2 a       | rr= \\pmod p       | c =  }} {{eqn | o = \\equiv       | r = \\left({-1}\\right)^n a^{\\left({\\frac {p - 1} 2}\\right)} \\times \\left({\\frac {p - 1} 2}\\right)!       | rr=\\pmod p       | c = as the elements in $S'$ are congruent, in some order, to those in $S$ }} {{end-eqn}} Now, from GCD with Prime: :$\\gcd \\left\\{ {p, \\left({\\dfrac {p - 1} 2}\\right)!}\\right\\} = 1$ So from Cancellability of Congruences the term $\\left({\\dfrac {p - 1} 2}\\right)!$ can be cancelled from both sides of the above congruence: :$1 \\equiv \\left({-1}\\right)^n a^{\\left({\\frac {p - 1} 2}\\right)} \\pmod p$ Finally, from the definition of the Legendre symbol: :$\\left({\\dfrac a p}\\right) \\equiv a^{\\left({\\frac {p - 1} 2}\\right)} \\pmod p$ Multiplying both sides of the congruence by $\\left({-1}\\right)^n$: :$\\left({\\dfrac a p}\\right) = \\left({-1}\\right)^n$ {{qed}} {{Namedfor|Carl Friedrich Gauss|cat = Gauss}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Definition:Legendre Symbol/Definition 1|next = Second Supplement to Law of Quadratic Reciprocity}}: $\\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $47 \\ \\text {a)}$ Category:Legendre Symbol s2sbf32tuemp74ygoq89qclfms0pkzr"}
+{"_id": "32199", "title": "Euclid's Lemma for Prime Divisors", "text": "Euclid's Lemma for Prime Divisors 0 3733 462866 462838 2020-04-18T20:11:09Z Prime.mover 59 wikitext text/x-wiki {{Previous POTW|2 April 2009|10 April 2009}} == Lemma == <onlyinclude> Let $p$ be a prime number. Let $a$ and $b$ be integers such that: : $p \\divides a b$ where $\\divides$ means '''is a divisor of'''. Then $p \\divides a$ or $p \\divides b$. </onlyinclude> === General Result === {{:Euclid's Lemma for Prime Divisors/General Result}} === Corollary === {{:Euclid's Lemma for Prime Divisors/Corollary}} == Proof 1 == {{:Euclid's Lemma for Prime Divisors/Proof 1}} == Proof 2 == {{:Euclid's Lemma for Prime Divisors/Proof 2}} == Proof 3 == {{:Euclid's Lemma for Prime Divisors/Proof 3}} == Also presented as == Some sources present this as: Let $p$ be a prime number. Let $a$ and $b$ be integers such that: :$a b \\equiv 0 \\pmod p$ Then either $a \\equiv 0 \\pmod p$ or $b \\equiv 0 \\pmod p$. == Also known as == Some sources give the name of this as '''Euclid's first theorem'''. {{Namedfor|Euclid}} == Also see == Some sources use this property to '''define''' a prime number. == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Modulo Arithmetic/Examples/Residue of 2^512 Modulo 5|next = Converse of Euclid's Lemma does not Hold}}: Chapter $2$. Equivalence Relations: Exercise $7$ * {{BookReference |An Introduction to the Theory of Numbers|1979|G.H. Hardy|author2 = E.M. Wright|ed = 5th|edpage = Fifth Edition|prev = Fundamental Theorem of Arithmetic|next = Euclid's Lemma for Prime Divisors/General Result}}: $\\text I$: The Series of Primes: $1.3$ Statement of the fundamental theorem of arithmetic: Theorem $3$ * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Expression for Integer as Product of Primes is Unique|next = Definition:Irrational Number/Historical Note}}: Chapter $\\text {B}.16$: The Sequence of Primes: Problem $1 \\ \\text{(b)}$ Category:Prime Numbers Category:Euclid's Lemma Category:Euclid's Lemma for Prime Divisors pjrpdvhh81ve5wvdnyzmx4xl0yh6vyz"}
+{"_id": "32200", "title": "Continued Fraction Algorithm", "text": "Continued Fraction Algorithm 0 3829 470623 470622 2020-05-26T06:30:25Z Prime.mover 59 wikitext text/x-wiki == Algorithm == <onlyinclude> The '''Continued Fraction Algorithm''' is a method for finding the continued fraction expansion for any irrational number to as many partial quotients as desired. Let $x_0$ be the irrational number in question. The steps are: {{begin-eqn}} {{eqn | n = 1       | l = k       | r = 0       | c = initialise }} {{eqn | n = 2       | l = a_k       | r = \\floor {x_k}       | c = the $k$th partial quotient (that is, $a_k$) is the integer part of $x_k$ }} {{eqn | n = 3       | l = x_{k + 1}       | r = \\frac 1 {x_k - a_k}       | c = the subsequent term is the reciprocal of the fractional part of $x_k$ }} {{eqn | n = 4       | l = k       | r = k + 1       | c = increase $k$ by $1$ then go to step $(2)$ }} {{end-eqn}} Then $x_0 = \\sqbrk {a_0, a_1, a_2, \\ldots}$ is the required continued fraction expansion. </onlyinclude> == Examples == {{:Continued Fraction Algorithm/Examples}} == Proof 1 == {{:Continued Fraction Algorithm/Proof 1}} Category:Continued Fractions ijqk8sif36y0gd1bh8diivul5x2dpsa"}
+{"_id": "32201", "title": "Pell's Equation", "text": "Pell's Equation 0 3848 446491 431415 2020-02-05T10:52:20Z Prime.mover 59 wikitext text/x-wiki == Definition == The Diophantine equation: :$x^2 - n y^2 = 1$ is known as '''Pell's equation'''. == Solution == Let the continued fraction of $\\sqrt n$ have a cycle whose length is $s$: :$\\sqrt n = \\sqbrk {a_1 \\sequence {a_2, a_3, \\ldots, a_{s + 1} } }$ Let $a_n = \\dfrac {p_n} {q_n}$ be a convergent of $\\sqrt n$. Then: :${p_{r s} }^2 - n {q_{r s} }^2 = \\paren {-1}^{r s}$ for $r = 1, 2, 3, \\ldots$ and all solutions of: :$x^2 - n y^2 = \\pm 1$ are given in this way. == Proof == First note that if $x = p, y = q$ is a positive solution of $x^2 - n y^2 = 1$ then $\\dfrac p q$ is a convergent of $\\sqrt n$. The continued fraction of $\\sqrt n$ is periodic from Continued Fraction Expansion of Irrational Square Root and of the form: :$\\sqbrk {a \\sequence {b_1, b_2, \\ldots, b_{m - 1}, b_m, b_{m - 1}, \\ldots, b_2, b_1, 2 a} }$ or  :$\\sqbrk {a \\sequence {b_1, b_2, \\ldots, b_{m - 1}, b_m, b_m, b_{m - 1}, \\ldots, b_2, b_1, 2 a} }$ For each $r \\ge 1$ we can write $\\sqrt n$ as the (non-simple) finite continued fraction: :$\\sqrt n = \\sqbrk {a \\sequence {b_1, b_2, \\ldots, b_2, b_1, 2 a, b_1, b_2, \\ldots, b_2, b_1, x} }$ which has a total of $r s + 1$ partial quotients. The last element $x$ is of course not an integer. What we do have, though, is: {{begin-eqn}} {{eqn | l = x       | r = \\sqbrk {\\sequence {2 a, b_1, b_2, \\ldots, b_2, b_1} }       | c =  }} {{eqn | r = a + \\sqbrk {a, \\sequence {b_1, b_2, \\ldots, b_2, b_1, 2 a} }       | c =  }} {{eqn | r = a + \\sqrt n       | c =  }} {{end-eqn}} The final three convergents in the above FCF are: :$\\dfrac {p_{r s - 1} } {q_{r s - 1} }, \\quad \\dfrac {p_{r s} } {q_{r s} }, \\quad \\dfrac {x p_{r s} + p_{r s - 1} } {x q_{r s} + q_{r s - 1} }$ The last one of these equals $\\sqrt n$ itself. So: :$\\sqrt n \\paren {x q_{r s} + q_{r s - 1} } = \\paren {x p_{r s} + p_{r s - 1} }$ Substituting $a + \\sqrt n$ for $x$, we get: :$\\sqrt n \\paren {\\paren {a + \\sqrt n} q_{r s} + q_{r s - 1} } = \\paren {\\paren {a + \\sqrt n} p_{r s} + p_{r s - 1} }$ This simplifies to: :$\\sqrt n \\paren {a q_{r s} + q_{r s - 1} - p_{r s} } = a p_{r s} + p_{r s - 1} - n q_{r s}$ The {{RHS}} of this is an integer while the {{LHS}} is $\\sqrt n$ times an integer. Since $\\sqrt n$ is irrational, the only way that can happen is if both sides equal zero. This gives us: {{begin-eqn}} {{eqn | n = 1       | l = a q_{r s} + q_{r s - 1}       | r = p_{r s} }} {{eqn | n = 2       | l = a p_{r s} + p_{r s - 1}       | r = n q_{r s} }} {{end-eqn}} Multiplying $(1)$ by $p_{r s}$, $(2)$ by $q_{r s}$ and then subtracting: :$p_{r s}^2 - n q_{r s}^2 = p_{r s} q_{r s - 1} - p_{r s - 1} q_{r s}$ By Difference between Adjacent Convergents of Simple Continued Fraction, the {{RHS}} of this is $\\paren {-1}^{r s}$. When the cycle length $s$ of the continued fraction of $\\sqrt n$ is even, we have $\\paren {-1}^{r s} = 1$. Hence $x = p_{r s}, y = q_{r s}$ is a solution to Pell's Equation for each $r \\ge 1$. When $s$ is odd, though: : $x = p_{r s}, y = q_{r s}$ is a solution of $x^2 - n y^2 = -1$ when $r$ is odd : $x = p_{r s}, y = q_{r s}$ is a solution of $x^2 - n y^2 = 1$ when $r$ is even. {{qed}} == Sequence == {{:Pell's Equation/Sequence}} == Examples == {{:Pell's Equation/Examples}} == Also known as == A particular instance of '''Pell's equation''' can also be seen referred to as '''a Pellian equation'''. {{Namedfor|John Pell|cat = Pell}} == Historical Note == {{:Pell's Equation/Historical Note}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Best Rational Approximations to Root 2 generate Pythagorean Triples|next = Sprague's Property of Root 2}}: $1 \\cdotp 41421 \\, 35623 \\, 73095 \\, 04880 \\, 16887 \\, 24209 \\, 69807 \\, 85697 \\ldots$ * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = 4,729,494|next = Pell's Equation/Historical Note}}: $4,729,494$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Best Rational Approximations to Root 2 generate Pythagorean Triples|next = Square Root of 2 as Sum of Egyptian Fractions}}: $1 \\cdotp 41421 \\, 35623 \\, 73095 \\, 04880 \\, 16887 \\, 24209 \\, 69807 \\, 85697 \\ldots$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Definition:Emirp/Sequence|next = Pell's Equation/Examples/13}}: $13$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = 4,729,494|next = Pell's Equation/Historical Note}}: $4,729,494$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Mathematician:Charles Sanders Peirce|next = Definition:Pencil|entry = Pell's equation}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Mathematician:Charles Sanders Peirce|next = Definition:Pencil|entry = Pell's equation}} * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Mathematician:Pierre de Fermat|next = Mathematician:Carl Friedrich Gauss}}: Chapter $7$: Patterns in Numbers: Fermat * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Mathematician:John Pell|next = Definition:Pendulum|entry = Pell's equation}} * {{MathWorld|Pell Equation|PellEquation}} Category:Diophantine Equations Category:Pell's Equation rh0e3rxucadso4h7vp3zw8sc7ymemco"}
+{"_id": "32202", "title": "Clear Registers Program", "text": "Clear Registers Program 0 3990 51940 14186 2011-04-13T21:48:52Z Prime.mover 59 wikitext text/x-wiki == URM Program == Let $a, b \\in \\N$ be natural numbers such that $0 < a$. Then we define the URM program $Z \\left({a, b}\\right)$ to be: {| |- ! align=\"right\" | Line !! ! align=\"left\" | Command |- | align=\"right\" | $1$ || | align=\"left\" | $Z \\left({a}\\right)$ |- | align=\"right\" | $2$ || | align=\"left\" | $Z \\left({a + 1}\\right)$ |- | align=\"right\" | $3$ || | align=\"left\" | $Z \\left({a + 2}\\right)$ |- | align=\"right\" | $\\vdots$ || | align=\"left\" | $\\vdots$ |- | align=\"right\" | $b - a + 1$ || | align=\"left\" | $Z \\left({b}\\right)$ |} This program clears (that is, sets to $0$) all the registers from $R_a$ through to $R_b$. If $a > b$ then $Z \\left({a, b}\\right)$ is the null URM program. The length of $Z \\left({a, b}\\right)$ is: :$\\lambda \\left({Z \\left({a, b}\\right)}\\right) = \\begin{cases} 0 & : a > b \\\\ b - a + 1 & : a \\le b \\end{cases}$ == Proof == Self-evident. {{qed}} Category:URM Programs n08c640hq7dvmd9vt1l4oe1uij6w3bn"}
+{"_id": "32203", "title": "Biconditional Properties", "text": "Biconditional Properties 0 4483 126281 126278 2013-01-13T13:21:40Z Prime.mover 59 wikitext text/x-wiki == Theorems == === Biconditional is Commutative === {{:Biconditional is Commutative}} === Biconditional is Associative === {{:Biconditional is Associative}} === Biconditional is Reflexive === {{:Biconditional is Reflexive}} Category:Biconditional bt4u4kllobeob13f69cagqquif51ykv"}
+{"_id": "32204", "title": "Method of Truth Tables", "text": "Method of Truth Tables 0 4495 493612 492247 2020-10-10T12:24:57Z Prime.mover 59 wikitext text/x-wiki == Proof Technique == The '''method of truth tables''' is a technique for determining the validity of propositional formulas with respect to boolean interpretations. In particular, for discerning if a propositional formula is a tautology for boolean interpretations. To start with, we establish the characteristic truth tables of the various logical connectives. We write one row for each boolean interpretation of the set of variables that we are concerned with. From Count of Rows of Truth Table this amounts to $2^n$ rows if there are $n$ variables. There are therefore two rows in the truth table for the only non-trivial unary connective: $\\begin{array}{|c||c|} \\hline p & \\neg p \\\\ \\hline F & T \\\\ T & F \\\\ \\hline \\end{array}$ ... and four rows in the truth tables for the binary connectives (the usual subset of which is given below): $\\begin{array}{|cc||c|c|c|c|c|c|c|c|} \\hline p & q & p \\land q  & p \\lor q & p \\implies q & p \\iff q & p \\impliedby q & p \\oplus q & p \\uparrow q & p \\downarrow q \\\\ \\hline F & F & F & F & T & T & T & F & T & T \\\\ F & T & F & T & T & F & F & T & T & F \\\\ T & F & F & T & F & F & T & T & T & F \\\\ T & T & T & T & T & T & T & F & F & F \\\\ \\hline \\end{array}$ There are various sorts of proof this technique can be put to, as follows. These will be illustrated by various examples. == Proof of Tautology == {{:Method of Truth Tables/Proof of Tautology}} == Proof of Interderivability == {{:Method of Truth Tables/Proof of Interderivability}} == Proof of Logical Implication == {{:Method of Truth Tables/Proof of Logical Implication}} == Indirect Technique == {{:Method of Truth Tables/Indirect Technique}} == Notational Convenience == It is not actually necessary to include the truth values of the variables themselves (as we have done in the leftmost columns). One is equally justified to write this: $\\begin{array}{ccccccc} ((p & \\implies & q) & \\implies & p) & \\implies & p \\\\ \\hline F & T & F & F & F & T & F \\\\ F & T & T & F & F & T & F \\\\ T & F & F & T & T & T & T \\\\ T & T & T & T & T & T & T \\\\ \\end{array}$ and it serves just as well. However, it can help to clarify the derivation, as well as making the truth table easier to construct, if they ''are'' included. It's a matter of personal taste. == Also known as == The '''method of truth tables''' is also sometimes referred to as the '''method of matrices'''. However, it can be argued that the term '''truth table''' is more specific than '''matrix''' and hence preferable. == Comment == Note that solution by truth table is valid only for Aristotelian logic, as it takes for granted the Law of Excluded Middle and the Principle of Non-Contradiction. Within that context, it is a completely mechanical procedure and about as exciting as a strip-tease artist who starts the performance naked. == Sources == * {{BookReference|Introduction to Logic and to the Methodology of Deductive Sciences|1946|Alfred Tarski|ed = 2nd|edpage = Second Edition|prev = Hypothetical Syllogism|next = Definition:Logical Not/Notational Variants}}: $\\S \\text{II}.13$: Symbolism of sentential calculus * {{BookReference|Introduction to Symbolic Logic|1959|A.H. Basson|author2 = D.J. O'Connor|ed = 3rd|edpage = Third Edition|prev = Definition:Therefore|next = Method of Truth Tables/Proof of Tautology}}: $\\S 3.3$: The Construction and Application of Truth-Tables * {{BookReference|Logic: Techniques of Formal Reasoning|1964|Donald Kalish|author2 = Richard Montague|prev = Definition:Truth Table|next = Definition:Proper Name}}: $\\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\\S 6$ * {{BookReference|Beginning Logic|1965|E.J. Lemmon|prev = Definition:Biconditional/Truth Table/Matrix Form|next = Method of Truth Tables/Proof of Tautology}}: $\\S 2.3$: Truth-Tables * {{BookReference|Symbolic Logic|1973|Irving M. Copi|ed = 4th|edpage = Fourth Edition|prev = Modus Tollendo Ponens/Sequent Form/Case 1/Proof 2|next = Modus Ponendo Ponens}}: $2.3$: Argument Forms and Truth Tables * {{BookReference|Elementary Logic|1980|D.J. O'Connor|author2 = Betty Powell|prev = Definition:Conditional/Truth Table|next = Method of Truth Tables/Proof of Tautology}}: $\\S \\text{I}: 4$: Using the Definitions * {{BookReference|Logic for Mathematicians|1988|Alan G. Hamilton|ed = 2nd|edpage = Second Edition|prev = Bottom-Up Specification of Propositional Logic/Examples/Example 1|next = Rule of Material Implication/Formulation 1/Proof}}: $\\S 1$: Informal statement calculus: $\\S 1.2$: Truth functions and truth tables * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin|prev = Definition:Truth Table|next = Definition:Tautology/Formal Semantics}}: $\\S 1.6$: Truth Tables and Tautologies * {{BookReference|Logic in Computer Science: Modelling and reasoning about systems|2000|Michael R.A. Huth|author2 = Mark D. Ryan|prev = Count of Rows of Truth Table|next = Definition:Boolean Interpretation}}: $\\S 1.4.1$: The meaning of logical connectives Category:Proof Techniques Category:Truth Tables gsj9vhqlxuhxgg48vdazq5xccjnroqr"}
+{"_id": "32205", "title": "Finished Set Lemma", "text": "Finished Set Lemma 0 4617 360477 320261 2018-07-14T09:28:51Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $\\Delta$ be a finished set of WFFs of propositional logic. Then any model of the set of basic WFFs in $\\Delta$ is a model of $\\Delta$. === Corollary === {{:Finished Set Lemma/Corollary}} == Proof == Let $\\mathcal M$ be a model of the set of basic WFFs in $\\Delta$. We need to show that $\\mathcal M \\models \\Delta$. That is, that $\\mathcal M \\models \\mathbf C$ for each $\\mathbf C \\in \\Delta$. Now, let $R \\left({n}\\right)$ be a propositional function on the set of natural numbers $\\N$ such that: :$R \\left({n}\\right) = T$ iff: for every WFF $\\mathbf C$, if $\\mathbf C \\in \\Delta$ and $\\mathbf C$ has length at most $n$, then $\\mathcal M \\models \\mathbf C$. It is clear that $R \\left({0}\\right), R \\left({1}\\right), R \\left({2}\\right)$ are true because every WFF length 2 or less is basic, and $\\mathcal M$ models every basic WFF in $\\Delta$. So, assume $R \\left({k}\\right)$ is true for some $k \\in \\N$. Suppose $\\mathbf C$ has length at most $k+1$ and belongs to $\\Delta$. By examining each of the cases in the definition of finished set, we see that since $\\mathcal M$ models every WFF in $\\Delta$ of length at most $k$, then $\\mathcal M$ models $\\mathbf C$. Thus $R \\left({k+1}\\right)$ is true. Thus by strong induction, $R \\left({n}\\right)$ is true for all $n \\in \\N$. Hence the result. {{qed}} == Sources == * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin|prev = Finished Set Lemma/Corollary|next = Completeness Theorem for Propositional Tableaus and Boolean Interpretations}}: $\\S 1.9$: Finished Sets: Lemma $1.9.1$ Category:Propositional Tableaus Category:Boolean Interpretations 7zy3lgsjgzkgyb9ht1nctbm3oyc0rf0"}
+{"_id": "32206", "title": "Finite Main Lemma of Propositional Tableaus", "text": "Finite Main Lemma of Propositional Tableaus 0 4618 320147 319775 2017-10-08T18:35:01Z Lord Farin 560 wikitext text/x-wiki {{mergeto|Main Lemma of Propositional Tableaus}} == Lemma == Let $\\mathbf H$ be a finite set of WFFs of propositional logic. Either $\\mathbf H$ has a tableau confutation or $\\mathbf H$ has a model. == Proof == Let $\\mathbf H$ be a finite set of WFFs of propositional logic which does not have a tableau confutation. By the Tableau Extension Lemma, the tableau which consists only of a root node with hypothesis set $\\mathbf H$ can be extended into a finite finished tableau $T$. The tableau $T$ still has root $\\mathbf H$. Since $T$ is not a confutation, it has a finished branch $\\Gamma$. By the Corollary to the Finished Branch Lemma, $\\Gamma$ has a model, $\\mathcal M$, say. In particular, $\\mathcal M$ is a model of $\\mathbf H$ as required. {{qed}} == Comment == From Tableau Confutation implies Unsatisfiable, we already know that $\\mathbf H$ can not have both a tableau confutation ''and'' a model. This result gives us that $\\mathbf H$ has a tableau confutation iff $\\mathbf H$ does not have a model. == Sources == * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2=Joel Robbin|prev=Completeness Theorem for Propositional Tableaus and Boolean Interpretations|next=Definition:Finished Branch of Propositional Tableau}}: $\\S 1.10$: Completeness: Lemma $1.10.1$ Category:Propositional Tableaus i42my72x1ywpf40t08rxv3d9x03p0kr"}
+{"_id": "32207", "title": "Tableau Extension Lemma", "text": "Tableau Extension Lemma 0 4626 360478 320137 2018-07-14T09:29:24Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $T$ be a finite propositional tableau. Let its hypothesis set $\\mathbf H$ be finite. Then $T$ can be extended to a finished finite propositional tableau $T'$ which also has hypothesis set $\\mathbf H$. === General Statement === {{:Tableau Extension Lemma/General Statement}} == Proof == Let $\\mathbf A$ be a WFF of propositional logic at a node $t$ such that: * $\\mathbf A$ is not a basic WFF; * There is a non-contradictory branch through $t$ on which $\\mathbf A$ is not used. Such a WFF $\\mathbf A$ will be called '''unused'''. Note that a tableau is finished iff there are no unused WFFs in the tableau. Let $u \\left({T}\\right)$ be the length of the longest unused WFF in $T$. If $T$ is finished, we set $u \\left({T}\\right) = 0$. Since there are only finitely many WFFs in $T$, the number $u \\left({T}\\right)$ must exist. Let $R \\left({n}\\right)$ be the proposition that every finite propositional tableau with hypothesis set $\\mathbf H$ and with $u \\left({T}\\right) < n$ can be extended to a finite finished tableau. That is, $R \\left({n}\\right)$ is an assertion that this lemma is true whenever $u \\left({T}\\right) < n$. The statement $R \\left({1}\\right)$ is true, because a tableau $T$ such that $u \\left({T}\\right) < 1$ is already finished. So, let us assume the truth of $R \\left({k}\\right)$. Suppose that $u \\left({T}\\right) < k+1$. We extend $T$ to a new tableau $T'$ by using every unused WFF $\\mathbf A$ in $T$ once on every non-contradictory branch through $\\mathbf A$. Each of the unused WFFs in $T$ is used in the new tableau $T'$. Also, each new WFF which was added when forming $T'$ has a length less than $u \\left({T}\\right)$, because the added WFFs are always shorter than the used WFFs. So $u \\left({T'}\\right) < u \\left({T}\\right) < k + 1$, so $u \\left({T'}\\right) < k$. So, by the induction hypothesis $R \\left({k}\\right)$, there is a finite finished extension $T''$ of $T'$. $T''$ is also a finished extension of $T$. Hence $R \\left({k+1}\\right)$, and the result follows by strong induction. {{qed}} == Sources == * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin|prev = Definition:Extension of Propositional Tableau/Definition 1|next = Extended Completeness Theorem for Propositional Tableaus and Boolean Interpretations}}: $\\S 1.10$: Completeness: Lemma $1.10.2$ Category:Propositional Tableaus 0ojqs4qlskowxutkv14mp49d96ibfip"}
+{"_id": "32208", "title": "Countable Hypothesis Set has Finished Tableau", "text": "Countable Hypothesis Set has Finished Tableau 0 4630 319779 227105 2017-10-04T21:10:26Z Lord Farin 560 wikitext text/x-wiki == Lemma == Let $\\mathbf H$ be a countable set of WFFs of propositional logic. Then there exists a finished tableau whose root node is $\\mathbf H$. == Proof == The Tableau Extension Lemma shows that each finite hypothesis set $\\mathbf H$ is the root of some finished tableau. It remains, then, to show that the result still applies when $\\mathbf H = \\left\\{{\\mathbf A_1, \\ldots, \\mathbf A_n, \\ldots}\\right\\}$ is countably infinite. Let $\\mathbf H_n = \\left\\{{\\mathbf A_1, \\ldots, \\mathbf A_n}\\right\\} \\subset \\mathbf H$. We will say that a finite tableau $T_n$ with root $\\mathbf H$ is '''finished for $\\mathbf H_n$''' if the tableau $T'_n$ is finished where: : $T'_n$ is the same as $T_n$ except: : $T'_n$ has the root $\\mathbf H_n$ instead of $\\mathbf H$. We can use the Tableau Extension Lemma countably many times, and get a sequence of finite tableaus $T_0, T_1, \\ldots, T_n, \\ldots$ such that: : $T_0$ has only a root node; : For each $n > 0$, $T_n$ is an extension of $T_{n-1}$ such that $T_n$ is '''finished for $\\mathbf H_n$'''; : For each $n > 0$, $T_n$ has the property that no contradictory branch $\\Gamma$ of $T_{n-1}$ gets extended when forming $T_n$. Now, let $T$ be the union $\\displaystyle T = \\bigcup_{k \\mathop = 0}^\\infty T_k$. Let $\\Gamma$ be a branch of $T$. Suppose $\\Gamma$ is contradictory, with complementary pair $\\mathbf A, \\neg \\mathbf A$. Then $\\exists n \\in \\N$ such that both $\\mathbf A$ and $\\neg \\mathbf A$ are in $T_n$. Then $\\Gamma \\cap T_n$ is already a contradictory branch of $T_n$. So, by our method of construction of $T$, this branch $\\Gamma \\cap T_n$ is never extended past stage $n$, so $\\Gamma = \\Gamma \\cap T_n$ and $\\Gamma$ is finite. On the other hand, suppose $\\Gamma$ is not contradictory. Then the construction ensures that $\\Gamma$ is a finished branch. So $T$ is a finished tableau whose root node is $\\mathbf H$. {{qed}} == Sources == * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin|prev = Main Lemma of Propositional Tableaus|next = König's Tree Lemma/Proof 1}}: $\\S 1.11$: Compactness: Lemma $1.11.2$ Category:Propositional Tableaus klz60bypzjrzi04gxv4n71qk3hi6os3"}
+{"_id": "32209", "title": "Main Lemma of Propositional Tableaus", "text": "Main Lemma of Propositional Tableaus 0 4639 320139 320121 2017-10-08T18:30:09Z Lord Farin 560 wikitext text/x-wiki == Lemma == Let $\\mathbf H$ be a countable set of WFFs of propositional logic. Either $\\mathbf H$ has a tableau confutation or $\\mathbf H$ has a model. == Proof == If $\\mathbf H$ is finite, then the Finite Main Lemma of Propositional Tableaus applies. So, assume that $\\mathbf H$ is countably infinite. Suppose $\\mathbf H$ does not have a tableau confutation. By Countable Hypothesis Set has Finished Tableau, there is a finished tableau $T$ with hypothesis set $\\mathbf H$. By Finished Propositional Tableau has Finished Branch or is Confutation, as $T$ is (by hypothesis) not a confutation, it must have a finished branch; call it $\\Gamma$. By the Finished Set Lemma, the set of WFFs of propositional logic on $\\Gamma$ has a model $\\mathcal M$. Finally, because all the WFFs in the hypothesis set occur on $\\Gamma$, $\\mathcal M \\models \\mathbf H$. {{qed}} == Sources == * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2=Joel Robbin|prev=Extended Completeness Theorem for Propositional Tableaus and Boolean Interpretations|next=Countable Hypothesis Set has Finished Tableau}}: $\\S 1.11$: Compactness: Lemma $1.11.1$ Category:Propositional Tableaus 2mrm61u2wv571x6yzfbqp8a4sb3wg80"}
+{"_id": "32210", "title": "Polynomial Long Division", "text": "Polynomial Long Division 0 4873 240053 141276 2015-12-06T10:08:51Z Prime.mover 59 wikitext text/x-wiki == Technique == Let $P_n \\left({x}\\right)$ be a polynomial in $x$ of degree $n$. Let $Q_m \\left({x}\\right)$ be a polynomial in $x$ of degree $m$ where $m \\le n$. Then $P_n \\left({x}\\right)$ can be expressed in the form: :$P_n \\left({x}\\right) \\equiv Q_m \\left({x}\\right) D_{n-m} \\left({x}\\right) + R_k \\left({x}\\right)$ {{explain|what is $\\equiv$ being used for here?}} where: :$D_{n-m} \\left({x}\\right)$ is a polynomial in $x$ of degree $n-m$ :$R_k \\left({x}\\right)$ is a polynomial in $x$ of degree $k$, where $k < m$, or may be null. Hence we can define $\\dfrac {P_n \\left({x}\\right)} {Q_m \\left({x}\\right)}$: :$\\dfrac {P_n \\left({x}\\right)} {Q_m \\left({x}\\right)} = D_{n - m} \\left({x}\\right) + \\dfrac {R_k \\left({x}\\right)} {Q_m \\left({x}\\right)}$ The polynomial $R_k \\left({x}\\right)$ is called the '''remainder'''. The procedure for working out what $D_{n - m} \\left({x}\\right)$ and $R_k \\left({x}\\right)$ are is called '''(polynomial) long division'''. {{explain|Establish the precise versions of polynomial that are invoked via the links on this page.}} == Proof == Let $\\displaystyle P_n \\left({x}\\right) = \\sum_{j \\mathop = 0}^n p_j x^j$. Let $\\displaystyle Q_m \\left({x}\\right) = \\sum_{j \\mathop = 0}^m q_j x^j$. First calculate $\\displaystyle Q'_m \\left({x}\\right) = Q_m \\left({x}\\right) \\times \\frac {p_n} {q_m} x^{n-m}$. This gives: {{begin-eqn}} {{eqn | l = Q'_m \\left({x}\\right)       | r = \\sum_{j \\mathop = 0}^m \\frac {p_n q_j} {q_m} x^{n - m + j}       | c =  }} {{eqn | r = \\sum_{j \\mathop = n-m}^n \\frac {p_n q_{j - n + m} } {q_m} x^j       | c =  }} {{eqn | r = p_n x^n + \\sum_{j \\mathop = n - m}^{n - 1} \\frac {p_n q_{j - n + m} } {q_m} x^j       | c =  }} {{end-eqn}} Then evaluate: :$P'_{n - 1} \\left({x}\\right) = P_n \\left({x}\\right) - Q'_m \\left({x}\\right)$ which (after some algebra) works out as: :$\\displaystyle P_n \\left({x}\\right) - Q'_m \\left({x}\\right) = \\sum_{j \\mathop = n - m}^{n - 1} \\frac {p_n q_{j - n + m} } {q_m} x^j + \\sum_{j \\mathop = 0}^{n - m - 1} p_j x^j$ So we see that $P_n \\left({x}\\right) - Q'_m \\left({x}\\right)$ is a polynomial in $x$ of degree $n-1$. Let $\\dfrac {p_n} {q_m} = d_{n - m}$. Hence we have: :$P_n \\left({x}\\right) = d_{n - m} x^{n - m} Q_m \\left({x}\\right) + P'_{n-1} \\left({x}\\right)$ We can express $P'_{n-1} \\left({x}\\right)$ as: :$\\displaystyle P'_{n-1} \\left({x}\\right) = \\sum_{j \\mathop = 0}^{n - 1} p'_j x^j$ Repeat the above by subtracting $\\displaystyle \\frac {p'_{n - 1} } {q_m} x^{n - m - 1} Q_m \\left({x}\\right)$ from $P'_{n - 1} \\left({x}\\right)$, and letting $\\dfrac {p'_{n - 1} } {q_m} = d_{n - m - 1}$. Hence: :$P'_{n - 1} \\left({x}\\right) = d_{n - m - 1} x^{n - m - 1} Q_m \\left({x}\\right) + P''_{n - 2} \\left({x}\\right)$ The process can be repeated $n-m$ times. It can be seen that after the last stage, we have: :$P_n \\left({x}\\right) = D_{n-m} \\left({x}\\right) Q_m \\left({x}\\right) + R_k \\left({x}\\right)$ where: : $\\displaystyle D_{n-m} \\left({x}\\right) = \\sum_{j \\mathop = 0}^{n-m} d_j x^j$ : $R_k \\left({x}\\right)$ is a polynomial of degree at most $m - 1$. {{qed}} Category:Proof Techniques 5wfxx82wynr2yb1wbb73q5ul4ef8e5a"}
+{"_id": "32211", "title": "Prüfer Sequence from Labeled Tree", "text": "Prüfer Sequence from Labeled Tree 0 5179 477053 477052 2020-07-04T15:53:07Z Prime.mover 59 wikitext text/x-wiki == Algorithm == Given a finite labeled tree, it is possible to generate a Prüfer sequence corresponding to that tree. Let $T$ be a labeled tree of order $n$, where the labels are assigned the values $1$ to $n$. :'''Step 1''': If there are two (or less) nodes in $T$, then '''stop'''. Otherwise, continue on to '''step 2'''. :'''Step 2''': Find all the nodes of $T$ of degree $1$. There are bound to be some, from Finite Tree has Leaf Nodes. Choose the one $v$ with the lowest label. :'''Step 3''': Look at the node $v'$ adjacent to $v$, and assign the label of $v\\,'$ to the first available element of the Prüfer sequence being generated. :'''Step 4''': Remove the node $v$ and its incident edge. This leaves a smaller tree $T'$. Go back to '''step 1'''. The above constitutes an algorithm, for the following reasons: === Finiteness === For each iteration through the algorithm, '''step 4''' is executed, which reduces the number of nodes by $1$. Therefore, after $n-2$ iterations, at '''step 1''' there will be $2$ nodes left, and the algorithm will stop. === Definiteness === :'''Step 1''': There are either more than $2$ nodes in a tree or there are $2$ or less. :'''Step 2''': There are bound to be some nodes of degree $1$, from Finite Tree has Leaf Nodes. As integers are totally ordered, it is always possible to find the lowest label. :'''Step 3''': As the node $v$ is of degree $1$, there is a unique node $v'$ to which it is adjacent. (Note that this node will not ''also'' have degree $1$, for then $v v'$ would be a tree of order $2$, and we have established from step $1$ that this is not the case.) :'''Step 4''': The node and edge to be removed are unique and specified precisely, as this is a tree we are talking about. === Inputs === The input to this algorithm is the tree $T$. === Outputs === The output to this algorithm is the Prüfer sequence $\\tuple {\\mathbf a_1, \\mathbf a_2, \\ldots, \\mathbf a_{n - 2} }$. === Effective === Each step of the algorithm is basic enough to be done exactly and in a finite length of time. == Example == Let $T$ be the following labeled tree: :300px This tree has $8$ nodes, so the corresponding Prüfer sequence will have $6$ elements. === Iteration 1 === : '''Step 1''': There are $8$ nodes, so continue to '''step 2'''. : '''Step 2''': The nodes of degree $1$ are $8, 2, 6, 4, 3$. Of these, $2$ is the lowest. : '''Step 3''': $2$ is adjacent to $1$, so add $\\mathbf 1$ to the Prüfer sequence. : '''Step 4''': Removing node $2$ leaves the following tree: :300px At this stage, the Prüfer sequence is $\\tuple {\\mathbf 1}$. === Iteration 2 === : '''Step 1''': There are $7$ nodes, so continue to '''step 2'''. : '''Step 2''': The nodes of degree $1$ are $8, 6, 4, 3$. Of these, $3$ is the lowest. : '''Step 3''': $3$ is adjacent to $7$, so add $\\mathbf 7$ to the Prüfer sequence. : '''Step 4''': Removing node $3$ leaves the following tree: :300px At this stage, the Prüfer sequence is $\\tuple {\\mathbf 1, \\mathbf 7}$. === Iteration 3 === : '''Step 1''': There are $6$ nodes, so continue to '''step 2'''. : '''Step 2''': The nodes of degree $1$ are $8, 6, 4$. Of these, $4$ is the lowest. : '''Step 3''': $4$ is adjacent to $5$, so add $\\mathbf 5$ to the Prüfer sequence. : '''Step 4''': Removing node $4$ leaves the following tree: :250px At this stage, the Prüfer sequence is $\\tuple {\\mathbf 1, \\mathbf 7, \\mathbf 5}$. === Iteration 4 === : '''Step 1''': There are $5$ nodes, so continue to '''step 2'''. : '''Step 2''': The nodes of degree $1$ are $8, 6, 5$. Of these, $5$ is the lowest. : '''Step 3''': $5$ is adjacent to $7$, so add $\\mathbf 7$ to the Prüfer sequence. : '''Step 4''': Removing node $5$ leaves the following tree: :200px At this stage, the Prüfer sequence is $\\tuple {\\mathbf 1, \\mathbf 7, \\mathbf 5, \\mathbf 7}$. === Iteration 5 === : '''Step 1''': There are $4$ nodes, so continue to '''step 2'''. : '''Step 2''': The nodes of degree $1$ are $8, 6$. Of these, $6$ is the lowest. : '''Step 3''': $6$ is adjacent to $7$, so add $\\mathbf 7$ to the Prüfer sequence. : '''Step 4''': Removing node $6$ leaves the following tree: :200px At this stage, the Prüfer sequence is $\\tuple {\\mathbf 1, \\mathbf 7, \\mathbf 5, \\mathbf 7, \\mathbf 7}$. === Iteration 6 === : '''Step 1''': There are $3$ nodes, so continue to '''step 2'''. : '''Step 2''': The nodes of degree $1$ are $8, 7$. Of these, $7$ is the lowest. : '''Step 3''': $7$ is adjacent to $1$, so add $\\mathbf 1$ to the Prüfer sequence. : '''Step 4''': Removing node $7$ leaves the following tree: :150px At this stage, the Prüfer sequence is $\\tuple {\\mathbf 1, \\mathbf 7, \\mathbf 5, \\mathbf 7, \\mathbf 7, \\mathbf 1}$. === Iteration 7 === : '''Step 1''': There are $2$ nodes, so '''stop'''. The Prüfer sequence is $\\tuple {\\mathbf 1, \\mathbf 7, \\mathbf 5, \\mathbf 7, \\mathbf 7, \\mathbf 1}$. == Also see == Compare with the example given in Labeled Tree from Prüfer Sequence. These two results are pulled together in Bijection between Prüfer Sequences and Labeled Trees. Category:Tree Theory Category:Combinatorics ksok75784rgp5zboypq2ednespi5ojl"}
+{"_id": "32212", "title": "Labeled Tree from Prüfer Sequence", "text": "Labeled Tree from Prüfer Sequence 0 5191 477013 159217 2020-07-04T12:20:13Z Prime.mover 59 wikitext text/x-wiki == Algorithm == Given a Prüfer sequence, it is possible to generate a finite labeled tree corresponding to that sequence. Let $P = \\tuple {\\mathbf a_1, \\mathbf a_2, \\ldots, \\mathbf a_{n - 2} }$ be a Prüfer sequence. This will be called '''the sequence'''. It is assumed the sequence is not empty. :'''Step 1''': Draw the $n$ nodes of the tree we are to generate, and label them from $1$ to $n$. This will be called '''the tree'''. :'''Step 2''': Make a list of all the integers $\\left({1, 2, \\ldots, n}\\right)$. This will be called '''the list'''. :'''Step 3''': If there are two numbers left in '''the list''', connect them with an edge and then '''stop'''. Otherwise, continue on to '''step 4'''. :'''Step 4''': Find the smallest number in '''the list''' which is not in '''the sequence''', and also the first number in the '''the sequence'''. Add an edge to '''the tree''' connecting the nodes whose labels correspond to those numbers. :'''Step 5''': Delete the first of those numbers from '''the list''' and the second from '''the sequence'''. This leaves a smaller '''list''' and a shorter '''sequence'''. Then return to '''step 3'''. The above constitutes an algorithm, for the following reasons: === Finiteness === For each iteration through the algorithm, '''step 5''' is executed, which reduces the size of '''the list''' by $1$. Therefore, after $n-2$ iterations, at '''step 1''' there will be $2$ numbers left in '''the list''', and the algorithm will stop. === Definiteness === :'''Steps 1 and 2''': Trivially definite. :'''Step 3''': We are starting with a non-empty Prüfer sequence of length $n-2$, so '''the list''' must originally contain at least $3$ elements. As the number of elements in '''the list''' decreases by $1$ each iteration (see '''step 5'''), eventually there is bound to be just two elements in '''the list'''. :'''Step 4''': As there are more elements in '''the list''' than there are in '''the sequence''', by the Pigeonhole Principle there has to be at least one number in '''the list''' that is not in '''the sequence'''. :'''Step 5''': Trivially definite. === Inputs === The input to this algorithm is the Prüfer sequence $\\tuple {\\mathbf a_1, \\mathbf a_2, \\ldots, \\mathbf a_{n - 2} }$. === Outputs === The output to this algorithm is the tree $T$. The fact that $T$ is in fact a tree follows from the fact that: * $T$ has $n$ nodes and (from the method of construction) $n - 1$ edges; * Each new edge connects two as yet unconnected parts of $T$, so every edge is a bridge. Therefore there are no cycles in $T$, from Condition for Edge to be Bridge. So $T$ is a tree from Equivalent Definitions for Finite Tree. === Effective === Each step of the algorithm is basic enough to be done exactly and in a finite length of time. == Example == Let the starting Prüfer sequence be $\\tuple {\\mathbf 1, \\mathbf 7, \\mathbf 5, \\mathbf 7, \\mathbf 7, \\mathbf 1}$. :'''Step 1''': '''The sequence''' is length $6$, so the tree will have $8$ nodes: :250px :'''Step 2''': We generate '''the list''': $\\tuple {1, 2, 3, 4, 5, 6, 7, 8}$. === Iteration 1 === :'''Step 3''': There are $8$ elements in '''the list''', so we move on to '''step 4'''. :'''Step 4''': The smallest number in '''the list''' which is not in '''the sequence''' is $2$, and the first number in '''the sequence''' is $1$. We join $1$ and $2$, to obtain this graph: :250px :'''Step 5''': We delete $2$ from '''the list''' to obtain $\\tuple {1, 3, 4, 5, 6, 7, 8}$, and $1$ from the start of '''the sequence''' to obtain $\\tuple {\\mathbf 7, \\mathbf 5, \\mathbf 7, \\mathbf 7, \\mathbf 1}$. === Iteration 2 === :'''Step 3''': There are $7$ elements in '''the list''', so we move on to '''step 4'''. :'''Step 4''': The smallest number in '''the list''' which is not in '''the sequence''' is $3$, and the first number in '''the sequence''' is $7$. We join $3$ and $7$, to obtain this graph: :250px :'''Step 5''': We delete $3$ from '''the list''' to obtain $\\tuple {1, 4, 5, 6, 7, 8}$, and $7$ from the start of '''the sequence''' to obtain $\\tuple {\\mathbf 5, \\mathbf 7, \\mathbf 7, \\mathbf 1}$. === Iteration 3 === :'''Step 3''': There are $6$ elements in '''the list''', so we move on to '''step 4'''. :'''Step 4''': The smallest number in '''the list''' which is not in '''the sequence''' is $4$, and the first number in '''the sequence''' is $5$. We join $4$ and $5$, to obtain this graph: :250px :'''Step 5''': We delete $4$ from '''the list''' to obtain $\\tuple {1, 5, 6, 7, 8}$, and $5$ from the start of '''the sequence''' to obtain $\\tuple {\\mathbf 7, \\mathbf 7, \\mathbf 1}$. === Iteration 4 === :'''Step 3''': There are $5$ elements in '''the list''', so we move on to '''step 4'''. :'''Step 4''': The smallest number in '''the list''' which is not in '''the sequence''' is $5$, and the first number in '''the sequence''' is $7$. We join $5$ and $7$, to obtain this graph: :250px :'''Step 5''': We delete $5$ from '''the list''' to obtain $\\left({1, 6, 7, 8}\\right)$, and $7$ from the start of '''the sequence''' to obtain $\\tuple {\\mathbf 7, \\mathbf 1}$. === Iteration 5 === :'''Step 3''': There are $4$ elements in '''the list''', so we move on to '''step 4'''. :'''Step 4''': The smallest number in '''the list''' which is not in '''the sequence''' is $6$, and the first number in '''the sequence''' is $7$. We join $6$ and $7$, to obtain this graph: :250px :'''Step 5''': We delete $6$ from '''the list''' to obtain $\\tuple {1, 7, 8}$, and $7$ from the start of '''the sequence''' to obtain $\\tuple {\\mathbf 1}$. === Iteration 6 === :'''Step 3''': There are $3$ elements in '''the list''', so we move on to '''step 4'''. :'''Step 4''': The smallest number in '''the list''' which is not in '''the sequence''' is $7$, and the first number in '''the sequence''' is $1$. We join $7$ and $1$, to obtain this graph: :250px :'''Step 5''': We delete $7$ from '''the list''' to obtain $\\tuple {1, 8}$, and $1$ from the start of '''the sequence''', which is at this point empty. === Iteration 7 === :'''Step 3''': There are $2$ elements in '''the list''': $\\tuple {1, 8}$, so we join them to obtain this graph: :250px Then we '''stop'''. The algorithm has terminated, and the tree is complete. Rearranging the positions of the nodes, we can draw it like this: :300px == Also see == Compare with the example given in Prüfer Sequence from Labeled Tree. These two results are pulled together in Bijection between Prüfer Sequences and Labeled Trees. Category:Tree Theory Category:Combinatorics 8pwuhc94aw63lbxf1frlf230hfxfxx5"}
+{"_id": "32213", "title": "Kruskal's Algorithm", "text": "Kruskal's Algorithm 0 5215 498169 476926 2020-11-08T22:51:12Z Prime.mover 59 wikitext text/x-wiki == Algorithm == The purpose of this algorithm is to produce a minimum spanning tree for any given weighted graph $G$. :'''Step 1''': Start with the edgeless graph $T$ whose vertices correspond with those of $G$. :'''Step 2''': Choose an edge $e$ of $G$ such that: ::'''a)''': Adding $e$ to $T$ would not make a cycle in $T$; ::'''b)''': $e$ has the minimum weight of all the edges remaining in $G$ that fulfil the condition in '''a)'''. :'''Step 3''': Add $e$ to $T$. :'''Step 4''': If $T$ spans $G$, '''stop'''. Otherwise, go to '''Step 2'''. The above constitutes an algorithm, for the following reasons: === Finiteness === For each iteration through the algorithm, '''step 3''' is executed, which increases the number of edges in $T$ by 1. As a tree with $n$ nodes has $n-1$ edges, the algorithm will terminate after $n-1$ iterations. === Definiteness === :'''Step 1''': Trivially definite. :'''Step 2''': As the edges of a graph can be arranged in order of weight, there is a definite edge (or set of edges) with minimal weight. It is straightforward to select an edge $e$ which does not make a cycle in $T$, by ensuring that at least one end of $e$ is incident to a vertex which has not so far been connected into $T$. :'''Step 3''': Trivially definite. :'''Step 4''': It is straightforward to determine whether all the vertices are connected. === Inputs === The input to this algorithm is the weighted graph $G$. === Outputs === The output to this algorithm is the minimum spanning tree $T$. === Effective === Each step of the algorithm is basic enough to be done exactly and in a finite length of time. == Note == It is clear that this is a greedy algorithm: at each stage the minimum possible weight is chosen, without any analysis as to whether there may be a combination of larger weights which may produce a smaller-weight spanning tree. For this reason, it is sometimes called '''Kruskal's greedy algorithm'''. In this case, the greedy algorithm ''does'' produce the minimum spanning tree. {{Namedfor|Joseph Bernard Kruskal|cat = Kruskal}} == Also see == * Kruskal's Algorithm produces Minimum Spanning Tree * Prim's Algorithm == Sources == * {{BookReference|Introductory Graph Theory|1977|Gary Chartrand|prev = Definition:Minimum Spanning Tree|next = Kruskal's Algorithm produces Minimum Spanning Tree}}: $\\S 4.1$: The Minimal Connector Problem: An Introduction to Trees * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Kronecker Delta/Also denoted as|next = Kruskal-Wallis Test|entry = Kruskal's algorithm}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Kronecker Delta/Also denoted as|next = Kruskal-Wallis Test|entry = Kruskal's algorithm}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Kronecker's Lemma|next = Kruskal-Wallis Test|entry = Kruskal's algorithm}} Category:Graph Theory Category:Greedy Algorithms itasf7blibtyywvutl5n7lo5b7377po"}
+{"_id": "32214", "title": "Prim's Algorithm", "text": "Prim's Algorithm 0 5221 476927 476864 2020-07-04T08:51:51Z Prime.mover 59 wikitext text/x-wiki == Algorithm == The purpose of this algorithm is to produce a minimum spanning tree for any given weighted graph $G$. :'''Step 1''': Choose any vertex of $G$, and add it to $T$. :'''Step 2''': Add an edge of minimum weight $e$ to join a vertex in $T$ to one not in $T$. :'''Step 3''': If $T$ spans $G$, '''stop'''. Otherwise, go to '''Step 2'''. == Proof == The above constitutes an algorithm, for the following reasons: === Finiteness === For each iteration through the algorithm, '''step 2''' is executed, which increases the number of edges in $T$ by 1. As a tree with $n$ nodes has $n-1$ edges, the algorithm will terminate after $n-1$ iterations. === Definiteness === :'''Step 1''': Trivially definite. :'''Step 2''': As the edges connecting $T$ to the remaining vertices can be arranged in order of weight, there is a definite edge (or set of edges) with minimal weight. :'''Step 3''': It is straightforward to determine whether all the vertices are connected. === Inputs === The input to this algorithm is the weighted graph $G$. === Outputs === The output to this algorithm is the minimum spanning tree $T$. === Effective === Each step of the algorithm is basic enough to be done exactly and in a finite length of time. == Also known as == It is clear that this is a greedy algorithm: at each stage the minimum possible weight is chosen, without any analysis as to whether there may be a combination of larger weights which may produce a smaller-weight spanning tree. For this reason, it is sometimes called '''Prim's greedy algorithm'''. In this case, the greedy algorithm ''does'' produce the minimum spanning tree. {{Namedfor|Robert Clay Prim|cat = Prim}} == Historical Note == {{:Prim's Algorithm/Historical Note}} == Also see == * Prim's Algorithm produces Minimum Spanning Tree * Kruskal's Algorithm == Sources == * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Primitive Polynomial (Ring Theory)|next = Definition:Principal Axes|entry = Prim's algorithm}} Category:Graph Theory Category:Greedy Algorithms tmbdpk4ake83t3myj8o2i55xaz3gy9j"}
+{"_id": "32215", "title": "Continuum Hypothesis", "text": "Continuum Hypothesis 0 5242 450153 443656 2020-02-20T13:39:06Z Prime.mover 59 wikitext text/x-wiki == Hypothesis == <onlyinclude> There is no set whose cardinality is strictly between that of the integers and the real numbers. </onlyinclude> Symbolically, the '''Continuum Hypothesis''' asserts that $\\aleph_1 = \\mathfrak c$. {{MissingLinks|$\\aleph_1$}} === Generalized Continuum Hypothesis === {{:Generalized Continuum Hypothesis}} {{Hilbert23|1}} == Historical Note == {{:Continuum Hypothesis/Historical Note}} == Sources == * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin|prev = Rational Numbers are Countably Infinite|next = Definition:Left Inverse Mapping}}: Appendix $\\text{A}.6$: Cardinality * {{BookReference|A Handbook of Terms used in Algebra and Analysis|1972|A.G. Howson|prev = Definition:Countable Set/Countably Infinite/Definition 1|next = Definition:Integer}}: $\\S 4$: Number systems $\\text{I}$: A set-theoretic approach * {{BookReference|Introduction to Boolean Algebras|2008|Paul Halmos|author2 = Steven Givant|prev = Definition:Cardinal Number}}: Appendix $\\text{A}$: Set Theory: Cardinal Numbers * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Power Set of Natural Numbers is Cardinality of Continuum|next = Generalized Continuum Hypothesis}}: Chapter $1$: General Background: $\\S 5$ The continuum problem Category:Set Theory Category:Named Theorems Category:Open Questions gb2vdnxv9ysggtkor55vcgrx47i5ryl"}
+{"_id": "32216", "title": "Division by Zero", "text": "Division by Zero 0 5340 488685 484214 2020-09-17T20:30:42Z Prime.mover 59 wikitext text/x-wiki == Mistake == Let $x$ and $y$ be numbers such that $y = 0$. The quantity $\\dfrac x y$ is '''undefined'''. It is a common mistake to forget this fact when evaluating formulas. == Also see == * L'Hôpital's Rule: while $\\dfrac 0 0$ is undefined, the limit of a rational function whose denominator approaches $0$ is not necessarily undefined. == Historical Note == {{:Division by Zero/Historical Note}} == Sources == * {{BookReference|Calculus|1967|Michael Spivak|prev = Commutative Law of Multiplication|next = Multiplication by Non-Zero Numbers is Cancellable}}: Part $\\text I$: Prologue: Chapter $1$: Basic Properties of Numbers: $(\\text P 8)$ * {{BookReference|Theory and Problems of Statistics|1972|Murray R. Spiegel|author2 = R.W. Boxer|ed = SI|edpage = SI Edition|prev = Definition:Right Hand Side|next = Definition:Simultaneous Equations}}: Chapter $1$: Equations * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Definition:Real Number Axioms|next = Definition:Infinity}}: $\\S 1$: Real Numbers: $\\S 1.3$: Arithmetic * {{KhanAcademySecure|algebra/x2f8bb11595b61c86:foundation-algebra/x2f8bb11595b61c86:division-zero/v/why-dividing-by-zero-is-undefined}} Category:Fallacies and Mistakes Category:Real Division 9o5mmpxqxqvjt6iuoqqofmq4xoqwlck"}
+{"_id": "32217", "title": "Abel's Lemma", "text": "Abel's Lemma 0 5969 454399 412780 2020-03-14T13:19:15Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $\\sequence a$ and $\\sequence b$ be sequences in an arbitrary ring $R$. === Formulation 1 === {{:Abel's Lemma/Formulation 1}} === Formulation 2 === {{:Abel's Lemma/Formulation 2}} == Also known as == {{:Abel's Lemma/Also known as}} {{Namedfor|Niels Henrik Abel|cat = Abel}} Category:Algebra Category:Finite Calculus Category:Proof Techniques Category:Abel's Lemma gxlhdof52p9s2dlo33wuk8ggllq2zd8"}
+{"_id": "32218", "title": "Pairwise Disjoint Subsets in Semiring Part of Partition", "text": "Pairwise Disjoint Subsets in Semiring Part of Partition 0 6093 170249 46671 2013-12-01T03:19:00Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $\\mathbb S$ be a semiring of sets. Let $A, A_1, A_2, \\ldots, A_n$ all belong to $\\mathbb S$. Let $A_1, A_2, \\ldots, A_n$ all be pairwise disjoint subsets of $A$. Then there exists a finite expansion of $A$: :$\\displaystyle \\exists s \\ge n: A = \\bigcup_{k \\mathop = 1}^s A_k$ with $A_1, \\ldots, A_n$ as its first $n$ elements, such that: : $(1): \\quad \\forall k, 1 \\le k \\le s: A_k \\in \\mathbb S$ : $(2): \\quad \\forall k, l, 1 \\le k \\le s, 1 \\le l \\le s: k \\ne l \\implies A_k \\cap A_l = \\varnothing$. That is, the nature of a semiring is such that every collection of pairwise disjoint subsets of a given set $A$ of that semiring is part of a larger collection of pairwise disjoint subsets of $A$ which forms a complete partition of $A$. == Proof == By the definition of a semiring of sets, the lemma holds for $n = 1$. Now we suppose that the lemma holds for $n = m$, and we attempt to show it consequently holds for $n = m+1$. So, let $A_1, A_2, \\ldots, A_m, A_{m+1}$ all be pairwise disjoint subsets of $A$. By hypothesis: :$A = A_1 \\cup A_2 \\cup \\cdots \\cup A_m \\cup B_1 \\cup \\cdots \\cup B_p$ where $A_1, A_2, \\ldots, A_m, B_1, \\ldots, B_p$ are pairwise disjoint subsets of $A$, all belonging to $\\mathbb S$. Let $B_{q1} = A_{m+1} \\cap B_q$. By the definition of a semiring of sets: :$B_q = B_{q1} \\cup \\cdots \\cup B_{q r_q}$ where all the $B_{qj}$ are pairwise disjoint subsets of $B_q$, all belonging to $\\mathbb S$. But then we see that: :$\\displaystyle A = A_1 \\cup A_2 \\cup \\cdots \\cup A_m \\cup A_{m+1} \\cup \\bigcup_{q \\mathop = 1}^p \\left({\\bigcup_{j \\mathop = 2}^{r_q} B_{qj}}\\right)$ and so the lemma is true for $m+1$. The result follows by induction. {{qed}} Category:Semirings of Sets 4n6l469lda9nu2qd9ceqytlwls70idh"}
+{"_id": "32219", "title": "Axiom:Kolmogorov Axioms", "text": "Axiom:Kolmogorov Axioms 100 6103 496131 496130 2020-10-23T22:51:46Z Prime.mover 59 wikitext text/x-wiki == Definition == Let $\\EE$ be an experiment. Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability measure on $\\EE$. Then $\\EE$ can be defined as being a measure space $\\struct {\\Omega, \\Sigma, \\Pr}$, such that $\\map \\Pr \\Omega = 1$. Thus $\\Pr$ satisfies the '''Kolmogorov axioms''': == Axioms == <onlyinclude> {{begin-axiom}} {{axiom | n = 1         | q = \\forall A \\in \\Sigma         | ml= 0         | mo= \\le         | mr= \\map \\Pr A \\le 1         | rc= The probability of an event occurring is a real number between $0$ and $1$ }} {{axiom | n = 2         | ml= \\map \\Pr \\Omega         | mo= =         | mr= 1         | rc= The probability of some elementary event occurring in the sample space is $1$ }} {{axiom | n = 3         | ml= \\map \\Pr {\\bigcup_{i \\mathop \\ge 1} A_i}         | mo= =         | mr= \\sum_{i \\mathop \\ge 1} \\map \\Pr {A_i}         | rc= where $\\set {A_1, A_2, \\ldots}$ is a countable (possibly countably infinite) set of pairwise disjoint events }} {{axiom | rc= That is, the probability of any one of countably many pairwise disjoint events occurring }} {{axiom | rc= is the sum of the probabilities of the occurrence of each of the individual events }} {{end-axiom}} </onlyinclude> == Also defined as == Some sources include: :$\\map \\Pr \\O = 0$ but this is strictly speaking not axiomatic as it can be deduced from the other axioms. == Also see == * Elementary Properties of Probability Measure * Probability of Union of Disjoint Events is Sum of Individual Probabilities * Definition:Measure Space: the '''Kolmogorov axioms''' follow directly from the fact that $\\struct {\\Omega, \\Sigma, \\Pr}$ is an example of such. {{NamedforAxiom|Andrey Nikolaevich Kolmogorov|cat = Kolmogorov}} == Sources == * {{BookReference|Probability: An Introduction|1986|Geoffrey Grimmett|author2 = Dominic Welsh|prev = Definition:Probability Measure/Definition 2|next = Definition:Countable Set/Definition 3}}: $1$: Events and probabilities: $1.3$: Probabilities Category:Axioms/Probability Theory 9jfz5uval44btpxxwqib2ffcx34f5gw"}
+{"_id": "32220", "title": "Proof by Counterexample", "text": "Proof by Counterexample 0 6686 410774 405754 2019-06-28T18:44:11Z Prime.mover 59 wikitext text/x-wiki == Proof Technique == Consider the definition of a counterexample: {{:Definition:Counterexample}} Proving, or disproving, a statement in the form of $X$ by establishing the truth or falsehood of a statement in the form of $Y$ is known as the technique of '''proof by counterexample'''. {{Languages|Counterexample}} {{Language|German|Gegenbeispiel}} {{End-languages}} == Sources == * {{BookReference|Undergraduate Topology|1971|Robert H. Kasriel|prev = Definition:Counterexample|next = Definition:Union of Set of Sets}}: $\\S 1.7$: Counterexamples * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Definition:Counterexample|next = Definition:Conditional}}: $\\S 3$: Statements and conditions; quantifiers * {{BookReference|Algebra Volume 1|1982|P.M. Cohn|edpage = Second Edition|ed = 2nd|prev = Square Root of 2 is Irrational/Classic Proof|next = Goldbach's Conjecture}}: Chapter $1$: Sets and mappings: $\\S 1.1$: The need for logic Category:Proof Techniques Category:Logic tuc80mry1q1ge4pc5ehgiuxi5rkpajh"}
+{"_id": "32221", "title": "Talk:Surjection Induced by Powerset is Induced by Surjection", "text": "Talk:Surjection Induced by Powerset is Induced by Surjection 1 6966 403435 217992 2019-04-30T08:58:03Z Prime.mover 59 wikitext text/x-wiki Is there a trivial reason that I'm just missing to explain \"For the second, it can be seen that neither $\\left\\{{y_1}\\right\\}$ nor $\\left\\{{y_2}\\right\\}$ can be in $\\mathrm{Rng} \\left({f_{\\mathcal{R}} \\left({\\mathcal{P} \\left({S}\\right)}\\right)}\\right)$\"?  --Cynic  (talk) 03:28, 17 July 2010 (UTC) :My thinking was: ::\"Because $\\left({x, y_1}\\right) \\in f$ and $\\left({x, y_2}\\right) \\in f$, it follows that $\\left\\{{y_1, y_2}\\right\\} \\in \\mathrm{Rng} \\left({f_{\\mathcal{R}} \\left({\\mathcal{P} \\left({S}\\right)}\\right)}\\right)$. :: \"But as $\\left\\{{y_1}\\right\\}$ is not the image of $x$ ...\" : Except the more I look at this, the more rubbish it looks. Why can't $\\exists x_1 \\in S: \\left({x_1, y_1}\\right) \\in f$. : Either the result doesn't hold or I need another way to justify it. I've studied a lot since I wrote this result up, I may be able to take a step back and rethink it. --Prime.mover 05:39, 17 July 2010 (UTC) 18lntpn6bx3a4e7dsf17w5bh983mzer"}
+{"_id": "32222", "title": "Modus Ponendo Tollens", "text": "Modus Ponendo Tollens 0 6991 244569 244568 2016-01-16T13:25:20Z Prime.mover 59 wikitext text/x-wiki == Modus Ponendo Tollens == The '''modus ponendo tollens''' is a valid deduction sequent in propositional logic: === Proof Rule === {{:Modus Ponendo Tollens/Proof Rule}} == Variants == The following forms can be used as variants of this theorem: === Variant === {{:Modus Ponendo Tollens/Variant}} == Explanation == {{:Modus Ponendo Tollens/Explanation}} == Linguistic Note == {{:Modus Ponendo Tollens/Linguistic Note}} == Also see == The following are related argument forms: * Modus Ponendo Ponens * Modus Tollendo Ponens * Modus Tollendo Tollens Category:Conjunction Category:Negation Category:Modus Ponendo Tollens mjbrmbzhpqjtf2emif4b8qxlkrnc3qr"}
+{"_id": "32223", "title": "Modus Tollendo Ponens", "text": "Modus Tollendo Ponens 0 6992 466201 462043 2020-05-06T06:33:14Z Prime.mover 59 wikitext text/x-wiki == Sequent == The '''modus tollendo ponens''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Modus Tollendo Ponens/Proof Rule}} === Sequent Form === {{:Modus Tollendo Ponens/Sequent Form}} == Variants == The following forms can be used as variants of this theorem: === Variant === {{:Modus Tollendo Ponens/Variant}} Note that the form: {{:Modus Tollendo Ponens/Variant/Formulation 1/Reverse Implication}} requires Law of Excluded Middle. Therefore it is not valid in intuitionistic logic. == Explanation == {{:Modus Tollendo Ponens/Explanation}} == Also known as == {{:Modus Tollendo Ponens/Also known as}} == Also see == The following are related argument forms: * Modus Ponendo Ponens * Modus Ponendo Tollens * Modus Tollendo Tollens == Linguistic Note == {{:Modus Tollendo Ponens/Linguistic Note}} == Sources == * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Definition:Disjunctive Normal Form|next = Definition:Ball|entry = disjunctive syllogism}} * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Modus Ponendo Ponens|next = Modus Tollendo Tollens|entry = ''modus tollendo ponens''}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Modus Ponendo Ponens|next = Modus Tollendo Tollens|entry = ''modus tollendo ponens''}} Category:Disjunction Category:Negation Category:Modus Tollendo Ponens 017gfwoo0yla3186sm4z7ejx993e3ok"}
+{"_id": "32224", "title": "Axiom:Playfair's Axiom", "text": "Axiom:Playfair's Axiom 100 7470 425120 236994 2019-09-13T17:03:37Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> :Exactly one straight line can be drawn through any point not on a given line parallel to the given straight line in a plane. Or:  :Given any straight line and a point not on it, there exists one and only one line which passes through this point and does not intersect the first line no matter how far they are extended. :This unique line is defined as being parallel to the original line in question. Or: :Two straight lines which intersect one another cannot both be parallel to one and the same straight line. </onlyinclude> == Comment == This is a frequently seen alternative presentation of Euclid's Fifth Postulate. It can easily seen to be equivalent to that given by {{AuthorRef|Euclid}}, but it can be argued that it is easier to understand. {{NamedforAxiom|John Playfair|cat = Playfair}} However, he did not originate it, merely published it. When he did so, he credited others, specifically {{AuthorRef|William Ludlam}}, for having used it earlier. == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 1|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Axiom:Euclid's Common Notions|next = Axiom:Euclid's Common Notions/Historical Note}}: Book $\\text{I}$. Notes on Postulate $5$ * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Axiom:Euclid's Fifth Postulate|next = Axiom:Eudoxus' Axiom|entry = Euclid's axioms}} Category:Euclid's Fifth Postulate osaymmlkj8dywhyqbm780ttcwea7yuv"}
+{"_id": "32225", "title": "Axiom:Parallel Postulate", "text": "Axiom:Parallel Postulate 100 7472 430027 295117 2019-10-08T12:23:33Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> :If two straight lines are drawn which intersect a third in such a way that the sum of the measures of the two interior angles on one side is less than the sum of the measures of two right angles, then the two lines must intersect each other on that side if extended far enough. </onlyinclude> == Also see == * Axiom:Euclid's Fifth Postulate: This is the name and formulation by which Euclid's Fifth Postulate is usually known. As can be seen, its wording in its modern format gives it an intent very similar to {{AuthorRef|Euclid}}'s. == Sources == * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Definition:Axiom|next = Isosceles Triangle has Two Equal Angles}}: Chapter $2$: The Logic of Shape: Euclid Category:Euclid's Fifth Postulate 2kzf2g76z8xc4fmnjs4vzx50eanrhnt"}
+{"_id": "32226", "title": "Newton's Laws of Motion/Third Law", "text": "Newton's Laws of Motion/Third Law 0 8131 488527 448467 2020-09-17T05:11:17Z Prime.mover 59 wikitext text/x-wiki == Physical Law == '''Newton's third law of motion''' is one of three physical laws that forms the basis for classical mechanics. === Statement of Law === <onlyinclude> :To every force there is always an equal and opposite force. That is, the forces of two bodies on each other are always equal and are directed in opposite directions. </onlyinclude> As {{AuthorRef|Isaac Newton}} himself put it: :''Whenever one body exerts a force on a second body, the second body exerts a force on the first body. These forces are equal in magnitude and opposite in direction.'' == Also known as == This law is also referred to as the '''law of action and reaction'''. It is also often referred to as just '''Newton's third law'''. == Also see == * Newton's First Law of Motion * Newton's Second Law of Motion {{Namedfor|Isaac Newton}} == Sources == * {{BookReference|Men of Mathematics|1937|Eric Temple Bell|prev = Newton's Second Law of Motion|next = Definition:Rate of Change}}: Chapter $\\text{VI}$: On the Seashore * {{BookReference|Classical Mechanics|1965|J.W. Leech|ed = 2nd|edpage = Second Edition|prev = Newton's Second Law of Motion|next = Definition:Force}}: Chapter $\\text {I}$: Introduction: $(3)$ * {{BookReference|Understanding Physics|1966|Isaac Asimov|prev = Definition:Force/Unit|next = Newton's Law of Universal Gravitation/Historical Note}}: $\\text {I}$: Motion, Sound and Heat: Chapter $3$: The Laws of Motion: Action and Reaction * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Newton's Second Law of Motion|next = Newton's Method|entry = Newton's laws of motion}} Category:Newton's Laws of Motion f8htgrjffg7efsxw47gb5iij11ydk9c"}
+{"_id": "32227", "title": "Newton's Laws of Motion/First Law", "text": "Newton's Laws of Motion/First Law 0 8136 493529 488525 2020-10-10T10:20:55Z Prime.mover 59 wikitext text/x-wiki == Physical Law == '''Newton's first law of motion''' is one of three physical laws that forms the basis for classical mechanics. === Statement of Law === <onlyinclude> :Every body remains in a state of constant velocity unless it is acted upon by an external unbalanced force.   This of course includes it being stationary, that is, with a constant velocity of zero. </onlyinclude> This tendency is sometimes called inertia. As {{AuthorRef|Isaac Newton}} himself put it: :''A body remains at rest or, if already in motion, remains in uniform motion with constant speed in a straight line, unless it is acted on by an unbalanced external force.'' == Also known as == This law is also sometimes called the '''principle of inertia'''. It is also often referred to as just '''Newton's first law'''. == Also see == * Newton's Second Law of Motion * Newton's Third Law of Motion It is worth pointing out that '''Newton's first law of motion''' is in fact a special case of Newton's Second Law of Motion, putting $\\mathbf F = 0$. {{Namedfor|Isaac Newton}} == Sources == * {{BookReference|Men of Mathematics|1937|Eric Temple Bell|prev = Newton's Laws of Motion|next = Newton's Second Law of Motion}}: Chapter $\\text{VI}$: On the Seashore * {{BookReference|Classical Mechanics|1965|J.W. Leech|ed = 2nd|edpage = Second Edition|prev = Definition:Mass|next = Newton's Second Law of Motion}}: Chapter $\\text {I}$: Introduction: $(1)$ * {{BookReference|Understanding Physics|1966|Isaac Asimov|prev = Newton's Laws of Motion|next = Definition:Stationary}}: $\\text {I}$: Motion, Sound and Heat: Chapter $3$: The Laws of Motion: Inertia * {{BookReference|Tensor Calculus and Relativity|1975|Derek F. Lawden|ed = 3rd|edpage = Third Edition|next = Definition:Frame of Reference}}: Chapter $1$ Special Principle of Relativity. Lorentz Transformations: $1$. Newton's laws of motion * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Newton's Laws of Motion|next = Newton's Second Law of Motion|entry = Newton's laws of motion}} Category:Newton's Laws of Motion cdmovc4vas7r9fenkwfb9xet9pau00e"}
+{"_id": "32228", "title": "Newton's Laws of Motion/Second Law", "text": "Newton's Laws of Motion/Second Law 0 8137 493976 488526 2020-10-11T11:25:21Z Prime.mover 59 wikitext text/x-wiki == Physical Law == '''Newton's second law of motion''' is one of three physical laws that forms the basis for classical mechanics. === Statement of Law === <onlyinclude> :The total force applied on a body is equal to the derivative with respect to time of the linear momentum of the body: :$\\mathbf F = \\map {\\dfrac \\d {\\d t} } {m \\bsv}$ </onlyinclude> As {{AuthorRef|Isaac Newton}} himself put it: :''The acceleration produced by a particular force acting on a body is directly proportional to the magnitude of the force and inversely proportional to the mass of the body.'' == Also known as == This law is also often referred to as just '''Newton's second law'''. == Also see == * At velocities near the speed of light, see Einstein's Law of Motion. * Newton's First Law of Motion * Newton's Third Law of Motion {{Namedfor|Isaac Newton}} == Sources == * {{BookReference|Men of Mathematics|1937|Eric Temple Bell|prev = Newton's First Law of Motion|next = Newton's Third Law of Motion}}: Chapter $\\text{VI}$: On the Seashore * {{BookReference|Classical Mechanics|1965|J.W. Leech|ed = 2nd|edpage = Second Edition|prev = Newton's First Law of Motion|next = Newton's Third Law of Motion}}: Chapter $\\text {I}$: Introduction: $(2)$ * {{BookReference|Understanding Physics|1966|Isaac Asimov|prev = Definition:Mass|next = Definition:Gram}}: $\\text {I}$: Motion, Sound and Heat: Chapter $3$: The Laws of Motion: Mass * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Definition:Derivative|next = Definition:Free Fall}}: $1$: The Nature of Differential Equations: $\\S 1$: Introduction: $(1)$ * {{BookReference|Engineering Mathematics|1977|A.J.M. Spencer|volume = I|prev = Definition:Differential Equation|next = Definition:Ordinary Differential Equation}}: Chapter $1$ Ordinary Differential Equations: $1.1$ Introduction * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Volume of Sphere/Proof by Archimedes|next = Definition:Linear Momentum}}: Chapter $\\text {B}.7$: A Simple Approach to $E = M c^2$ * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Einstein's Mass-Energy Equation|next = Motion of Rocket in Outer Space/Proof 2}}: Chapter $\\text {B}.8$: Rocket Propulsion in Outer Space * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Volume of Solid of Revolution/Historical Note|next = Definition:Velocity}}: Chapter $8$: The System of the World: Newton * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Speed of Light|next = Definition:Newton (Unit)|entry = constant|subentry = in physical laws}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Newton's First Law of Motion|next = Newton's Third Law of Motion|entry = Newton's laws of motion}} Category:Newton's Laws of Motion ft6ilao12vjwlk0aei8sr6bfcckfhik"}
+{"_id": "32229", "title": "Newton's Law of Cooling", "text": "Newton's Law of Cooling 0 8240 486172 486171 2020-09-07T21:29:24Z Prime.mover 59 wikitext text/x-wiki == Physical Law == The rate at which a hot body loses heat is proportional to the difference in temperature between it and its surroundings. {{Namedfor|Isaac Newton|cat = Newton}} == Historical Note == {{:Newton's Law of Cooling/Historical Note}} == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Law of Mass Action|next = Newton's Law of Cooling/Historical Note}}: $1$: The Nature of Differential Equations: $\\S 4$: Growth, Decay and Chemical Reactions: Problem $5$ Category:Thermodynamics grksipow1gt13we3qj3atxf1kot5yc9"}
+{"_id": "32230", "title": "Acceleration Due to Gravity", "text": "Acceleration Due to Gravity 0 8242 448198 448180 2020-02-11T21:54:54Z Prime.mover 59 wikitext text/x-wiki == Physical Law == A body $B$ in a uniform gravitational field experiences a force which gives rise to a constant acceleration independent of the mass of the body. If the force due to the gravitational field is the '''only''' force on the body, it is said to be in free fall. == Derivation == This law can be derived from Newton's Law of Universal Gravitation. Let the mass of $B$ be $m$. Let the mass of the body $P$ which gives rise to the gravitational field be $M$. Then the force on $B$ is given by: :$F = G \\dfrac {M m} {r^2}$ where: :$G$ is the gravitational constant :$r$ is the distance between the centers of gravity of $B$ and $P$. The assumption is that $M$ is orders of magnitude greater than $m$, and $r$ is also several orders of magnitude greater than the displacements observed on $B$ in the local frame. Then: :$F = m \\dfrac {G M} {r^2}$ But from Newton's Second Law of Motion: :$F = m a$ where $a$ is the magnitude of the acceleration which would be imparted to the body if no other force were acting on it. Hence: :$a = \\dfrac {G M} {r^2}$ which is, as we said, independent of the mass of the object. {{qed}} == Gravity on Earth == {{:Acceleration Due to Gravity/Earth's Surface|Gravity on Earth}} == Historical Note == {{:Acceleration Due to Gravity/Historical Note}} == Sources == * {{BookReference|Understanding Physics|1966|Isaac Asimov|prev = Definition:Constant of Proportion|next = Definition:Gravity}}: $\\text {I}$: Motion, Sound and Heat: Chapter $2$: Falling Bodies: Free Fall Category:Mechanics Category:Gravity 5h2pl882vzxxfjnk5n71dggi0box3la"}
+{"_id": "32231", "title": "Newton's Law of Universal Gravitation", "text": "Newton's Law of Universal Gravitation 0 8245 462330 462318 2020-04-16T13:17:44Z Prime.mover 59 wikitext text/x-wiki == Physical Law == Let $a$ and $b$ be particles with mass $m_a$ and $m_b$ respectively. Then $a$ and $b$ exert a force upon each other whose magnitude and direction are given by '''Newton's law of universal gravitation''': :$\\mathbf F_{a b} \\propto \\dfrac {m_a m_b {\\mathbf r_{b a} } } {r^3}$ where: :$\\mathbf F_{a b}$ is the force exerted on $b$ by the electric charge on $a$ :$\\mathbf r_{b a}$ is the displacement vector from $b$ to $a$ :$r$ is the distance between $a$ and $b$. Thus it is seen that the direction of $\\mathbf F_{a b}$ is specifically '''towards''' $a$. By exchanging $a$ and $b$ in the above, it is seen that $b$ exerts the same force on $a$ as $a$ does on $b$, but in the opposite direction, that is, towards $b$. === Gravitational Constant === {{:Definition:Gravitational Constant}} Thus the equation becomes: :$\\mathbf F_{a b} = \\dfrac {G m_a m_b \\mathbf r_{b a} } {r^3}$ == Also presented as == :$\\mathbf F_{a b} \\propto \\dfrac {m_a m_b \\hat {\\mathbf r}_{b a} } {r^2}$ where $\\hat {\\mathbf r}_{a b}$ is the unit vector in the direction from $b$ to $a$. == Also known as == '''Newton's Law of Universal Gravitation''' is also known as just '''Newton's Law of Gravitation'''. Some sources refer to it as the '''inverse square law of gravitation'''. {{Namedfor|Isaac Newton|cat = Newton}} == Historical Note == {{:Newton's Law of Universal Gravitation/Historical Note}} == Sources == * {{BookReference|Men of Mathematics|1937|Eric Temple Bell|prev = Kepler's Third Law of Planetary Motion|next = Newton's Laws of Motion}}: Chapter $\\text{VI}$: On the Seashore * {{BookReference|Understanding Physics|1966|Isaac Asimov|prev = Mathematician:Frederick William Herschel|next = Definition:Gravitational Constant}}: $\\text {I}$: Motion, Sound and Heat: Chapter $4$: Gravitation: The Gravitational Constant * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Kepler's Laws of Planetary Motion/Historical Note|next = Definition:Gravitational Constant}}: $\\S 3.21$: Newton's Law of Gravitation: $(11)$ * {{BookReference|Electromagnetism|1990|I.S. Grant|author2 = W.R. Phillips|ed = 2nd|edpage = Second Edition|next = Coulomb's Law of Electrostatics}}: Chapter $1$: Force and energy in electrostatics * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Kepler's Laws of Planetary Motion|next = Book:Isaac Newton/Philosophiae Naturalis Principia Mathematica}}: Chapter $\\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Acceleration is Second Derivative of Displacement with respect to Time|next = Newton's Law of Universal Gravitation/Historical Note}}: Chapter $8$: The System of the World: Newton * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Inverse for Complex Multiplication|next = Definition:Inverse Trigonometric Function|entry = inverse square law of gravitation}} Category:Gravity feriqxzt5bc0wgv9sdr31pc7kdlycd7"}
+{"_id": "32232", "title": "Fermat's Principle of Least Time", "text": "Fermat's Principle of Least Time 0 8267 486203 448998 2020-09-07T21:50:19Z Prime.mover 59 wikitext text/x-wiki == Physical Law == === Original formulation === The path taken by a ray of light from one point to another is the one that can be traversed in the least time. === Modern version === The optical path length must be stationary, that is, be either a minimum, maximum or a point of inflection. == Proof == This follows from the Huygens-Fresnel Principle. {{proof wanted}} == Also see == * Snell-Descartes Law {{Namedfor|Pierre de Fermat|cat = Fermat}} == Historical Note == {{:Fermat's Principle of Least Time/Historical Note}} == Sources == * {{BookReference|Men of Mathematics|1937|Eric Temple Bell|prev = Definition:Euler-Lagrange Equation/Historical Note|next = Fermat's Principle of Least Time/Historical Note}}: Chapter $\\text{IV}$: The Prince of Amateurs * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Mathematician:Willebrord van Royen Snell|next = Brachistochrone is Cycloid/Proof 1}}: $1$: The Nature of Differential Equations: $\\S 6$: The Brachistochrone. Fermat and the Bernoullis * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Heron's Principle of Reflection|next = Fermat's Principle of Least Time/Historical Note}}: Chapter $\\text {A}.7$: Heron (first century A.D.) * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Snell-Descartes Law/Historical Note|next = Definition:Folium of Descartes/Historical Note}}: Chapter $\\text {A}.13$: Fermat ({{DateRange|1601|1665}}) Category:Optics 23ym34cgy2myo527vebsvhnb1brob0o"}
+{"_id": "32233", "title": "Snell-Descartes Law", "text": "Snell-Descartes Law 0 8272 486201 409925 2020-09-07T21:49:27Z Prime.mover 59 wikitext text/x-wiki == Physical Law == Consider a ray of light crossing the threshold between two media. Let its speed: :in medium $1$ be $v_1$ :in medium $2$ be $v_2$. Let it meet the threshold at: :an angle $\\alpha_1$ from the vertical in medium 1 :an angle $\\alpha_2$ from the vertical in medium 2. Then the '''Snell-Descartes law''' states that: :$\\dfrac {\\sin \\alpha_1} {v_1} = \\dfrac {\\sin \\alpha_2} {v_2}$ == Proof == The '''Snell-Descartes law''' can be derived from Fermat's Principle of Least Time as follows: Let the ray of light travel from $A$ to $P$ in the medium $1$. Then let it travel from $P$ to $B$ in medium $2$. :350px The total time $T$ required for that journey is: :$T = \\dfrac {\\sqrt {a^2 + x^2} } {v_1} + \\dfrac {\\sqrt {b^2 + \\paren {c - x}^2} } {v_2}$ from the geometry of the above diagram. From Fermat's Principle of Least Time, this time will be a minimum. From Derivative at Maximum or Minimum, we need: :$\\dfrac {\\d T} {\\d x} = 0$. So: :$\\dfrac x {v_1 \\sqrt {a^2 + x^2} } = \\dfrac {c - x} {v_2 \\sqrt {b^2 + \\paren {c - x}^2} }$ which leads directly to: :$\\dfrac {\\sin \\alpha_1} {v_1} = \\dfrac {\\sin \\alpha_2} {v_2}$ by definition of sine. {{qed}} == Also presented as == This law can also be seen expressed as: :$\\dfrac {\\sin \\alpha_1} {\\sin \\alpha_2} = \\dfrac {v_1} {v_2}$ {{Namedfor|Willebrord van Royen Snell|name2 = René Descartes|cat = Snell|cat2 = Descartes}} == Historical Note == {{:Snell-Descartes Law/Historical Note}} == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Definition:Brachistochrone|next = Mathematician:Willebrord van Royen Snell}}: $1$: The Nature of Differential Equations: $\\S 6$: The Brachistochrone. Fermat and the Bernoullis * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Largest Rectangle with Given Perimeter is Square/Historical Note|next = Snell-Descartes Law/Historical Note}}: Chapter $\\text {A}.13$: Fermat ({{DateRange|1601|1665}}) * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Length of Arc of Nephroid|next = Brachistochrone is Cycloid/Proof 1}}: Chapter $\\text {B}.22$: Bernoulli's Solution of the Brachistochrone Problem Category:Optics ljffje6b8d5mp6djnxrc1st4h40ggeq"}
+{"_id": "32234", "title": "Einstein's Mass-Velocity Equation", "text": "Einstein's Mass-Velocity Equation 0 8348 256262 256255 2016-05-19T06:05:06Z Prime.mover 59 wikitext text/x-wiki == Physical Law == The mass $m$ of a body is not constant. It varies with the body's velocity, according to the equation: : $m = \\dfrac {m_0} {\\sqrt{1 - \\dfrac {v^2} {c^2} } }$ where: : $v$ is the magnitude of the velocity of the body : $c$ is the speed of light in vacuum : $m_0$ is the rest mass of the body. The value $m$ is known as the '''relativistic mass''' of the body. The factor $\\dfrac 1 {\\sqrt{1 - \\dfrac {v^2} {c^2} } }$ is known as the Lorentz Factor. == Proof == {{Proofread|grammar}} Imagine a comet that flies towards a planet on which you are resting. The comet's velocity $u$ towards the planet is much smaller than the speed of light. Now imagine the impact caused by the comet striking the planet as a deformation of the planet. That impact can be seen as proportional to the momentum of the comet which is: : $p = m u$ where: : $p$ is the magnitude of the comet's momentum : $m$ is the comet's rest mass : $u$ is the magnitude of the comet's velocity. If someone else watches the crash from a space ship passing by with a relativistic velocity (for example $v = 0.7c$) he will find that the comet appears to move more slowly than it does from your stationary perspective on the planet. This is due to the time dilation, which is given by: : $\\Delta t' = \\dfrac {\\Delta t} {\\sqrt{1 - \\dfrac {v^2} {c^2}}}$ where: : $\\Delta t'$ is the time interval measured from the space ship : $\\Delta t$ is the time interval measured in the inertial system containing planet and comet : $v$ is the magnitude of the space ship's velocity : $c$ is the speed of light in vacuum. Because the time measured in the space ship is less, the comet will appear to need more time to cover a certain distance. Thus, its velocity seems smaller from the perspective of the space ship (Note: The space ship's trajectory be perpendicular to the comet's trajectory towards the planet, so there is no length contraction parallel to the trajectory of the comet). The comet's velocity measured from the planet is: : $u = \\dfrac {\\mathrm d s} {\\mathrm d t}$ where the comet's velocity measured from the space ship is: : $u' = \\dfrac {\\mathrm d s} {\\mathrm d t'}$ and as we know from the time dilation, the term for $u'$ is thus: : $u' = u \\sqrt{1 - \\frac {v^2}{c^2}}$ The observer in the space ship will nevertheless find out that the impact is equal to the one observed by the resting person. That means that the comet's momentum doesn't change, no matter from what inertial system you measure it. That can only be possible, if -- seen from the space ship -- the comet's mass increases, as its velocity decreases. The comet's momentum from the perspective of the space ship is: : $p' = m' u'$ where: : $p'$ is the magnitude of the comet's momentum measured from the inertial system of the space ship : $m'$ is the comet's relativistic mass measured from the inertial system of the space ship : $u'$ is the magnitude of the comet's velocity measured from the inertial system of the space ship. And because the measured momentums from both observers are the same, you can write: : $p = p'$  : $m u = m' u'$ : $m' = m \\dfrac u {u'}$ : $m' = m \\dfrac u {u \\sqrt{1 - \\frac {v^2} {c^2}}}$ : $m' = \\dfrac m {\\sqrt{1 - \\frac {v^2} {c^2}}}$ {{qed}} {{Namedfor|Albert Einstein|cat = Einstein}} == Historical Note == {{:Einstein's Mass-Velocity Equation/Historical Note}} {{MissingLinks}}  == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Definition:Burnout Height|next = Einstein's Mass-Energy Equation}}: Miscellaneous Problems for Chapter $2$: Problem $32$ * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Definition:Linear Momentum|next = Einstein's Law of Motion}}: Chapter $\\text {B}.7$: A Simple Approach to $E = M c^2$ Category:Physics 8rli0ucu43dqg2cj99uo08s38lsth5f"}
+{"_id": "32235", "title": "Einstein's Law of Motion", "text": "Einstein's Law of Motion 0 8349 447608 258812 2020-02-09T00:54:06Z Prime.mover 59 wikitext text/x-wiki == Physical Law == The force and acceleration on a body of constant rest mass are related by the equation: : $\\mathbf F = \\dfrac {m_0 \\mathbf a} {\\left({1 - \\dfrac{v^2}{c^2}}\\right)^{\\tfrac 3 2}}$ where: : $\\mathbf F$ is the force on the body : $\\mathbf a$ is the acceleration induced on the body : $v$ is the magnitude of the velocity of the body : $c$ is the speed of light : $m_0$ is the rest mass of the body. == Proof == Into Newton's Second Law of Motion: : $\\mathbf F = \\dfrac {\\mathrm d}{\\mathrm d t} \\left({m \\mathbf v}\\right)$ we substitute Einstein's Mass-Velocity Equation: : $m = \\dfrac {m_0} {\\sqrt {1 - \\dfrac {v^2} {c^2}}}$ to obtain: : $\\mathbf F = \\dfrac {\\mathrm d} {\\mathrm d t} \\left({\\dfrac {m_0 \\mathbf v}{\\sqrt{1 - \\dfrac {v^2}{c^2}}}}\\right)$ Then we perform the differentiation {{WRT|Differentiation}} time: {{begin-eqn}} {{eqn | l = \\frac{\\mathrm d}{\\mathrm d t} \\left({\\frac {\\mathbf v}{\\sqrt{1 - \\dfrac {v^2}{c^2} } } }\\right)       | r = \\frac{\\mathrm d}{\\mathrm d v} \\left({\\frac {\\mathbf v}{\\sqrt{1 - \\dfrac {v^2}{c^2} } } }\\right) \\frac{\\mathrm d v}{\\mathrm d t}       | c = Chain Rule for Derivatives }} {{eqn | r = \\mathbf a \\left({\\frac {\\sqrt{1 - \\dfrac {v^2}{c^2} } - \\dfrac v 2 \\dfrac 1 {\\sqrt{1 - \\dfrac {v^2}{c^2} } } \\dfrac{-2 v}{c^2} } {1 - \\dfrac {v^2}{c^2} } }\\right)       | c = Chain Rule for Derivatives, Quotient Rule, etc. }} {{eqn | r = \\mathbf a \\left({\\frac {c^2 \\left({1 - \\dfrac {v^2}{c^2} }\\right) + v^2} {c^2 \\left({1 - \\dfrac {v^2}{c^2} }\\right)^{3/2} } }\\right)       | c =  }} {{eqn | r = \\mathbf a \\left({\\frac 1 {\\left({1 - \\dfrac {v^2}{c^2} }\\right)^{3/2} } }\\right)       | c =  }} {{end-eqn}} Thus we arrive at the form: : $\\mathbf F = \\dfrac {m_0 \\mathbf a} {\\left({1 - \\dfrac{v^2}{c^2}}\\right)^{\\tfrac 3 2}}$ {{qed}} == Comment == Thus we see that at low velocities (i.e. much less than that of light), the well-known equation $\\mathbf F = m \\mathbf a$ holds to a high degree of accuracy. {{namedfor|Albert Einstein|cat = Einstein}} == Sources == * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Einstein's Mass-Velocity Equation|next = Einstein's Mass-Energy Equation}}: Chapter $\\text {B}.7$: A Simple Approach to $E = M c^2$ Category:Physics mxvf0tynbofgs7y6tmuxz6slcyrlg7i"}
+{"_id": "32236", "title": "Hooke's Law", "text": "Hooke's Law 0 8451 496104 451938 2020-10-23T20:49:30Z Prime.mover 59 wikitext text/x-wiki == Physical Law == '''Hooke's Law''' applies to an ideal spring: :$\\mathbf F = -k \\mathbf x$ where: :$\\mathbf F$ is the force caused by a displacement $\\mathbf x$ :$k$ is the spring force constant. The negative sign indicates that the force pulls in the opposite direction to that of the displacement imposed. :''The strain is proportional to the stress.'' == Also see == * Dimension of Spring Force Constant: the dimension of $k$ is $\\mathsf {M T}^{-2}$. == Sources == * {{BookReference|Understanding Physics|1966|Isaac Asimov|prev = Definition:Elasticity|next = Definition:Elastic Limit}}: $\\text {I}$: Motion, Sound and Heat: Chapter $4$: Gravitation: The Gravitational Constant * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Mathematician:Robert Hooke|next = Definition:Hopf Bifurcation|entry = Hooke's law}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Mathematician:Robert Hooke|next = Definition:Hopf Bifurcation|entry = Hooke's law}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Homotopy|next = Mathematician:Guillaume de l'Hôpital|entry = Hooke's law}} Category:Physics lfeol0b5fc6vyjku946eh2tcrrzgpon"}
+{"_id": "32237", "title": "Leibniz's Formula for Pi/Lemma", "text": "Leibniz's Formula for Pi/Lemma 0 8459 456449 258605 2020-03-19T17:45:58Z Prime.mover 59 wikitext text/x-wiki == Lemma == {{begin-eqn}} {{eqn | l = \\frac 1 {1 + t^2}       | r = 1 - t^2 + t^4 - t^6 + \\cdots + t^{4 n} - \\frac {t^{4 n + 2} } {1 + t^2}       | c =  }} {{eqn | r = \\paren {\\sum_{k \\mathop = 0}^{2 n} \\paren {-1}^k t^{2 k} } - \\frac {t^{4 n + 2} } {1 + t^2}       | c =  }} {{end-eqn}} This holds for all real $t \\in \\R$. == Proof == {{begin-eqn}} {{eqn | l = \\frac {1 - \\paren {-t^2}^{2 n + 1} } {1 - \\paren {-t^2} }       | r = \\sum_{k \\mathop = 0}^{2 n} \\paren {-t^2}^k       | c = Sum of Geometric Sequence }} {{eqn | ll= \\leadsto       | l = \\frac {1 + \\paren {t^2}^{2 n + 1} } {1 + t^2}       | r = 1 - t^2 + t^4 - t^6 + \\cdots + t^{4 n}       | c =  }} {{eqn | ll= \\leadsto       | l = \\frac 1 {1 + t^2} + \\frac {t^{4 n + 2} } {1 + t^2}       | r = 1 - t^2 + t^4 - t^6 + \\cdots + t^{4 n}       | c =  }} {{eqn | ll= \\leadsto       | l = \\frac 1 {1 + t^2}       | r = 1 - t^2 + t^4 - t^6 + \\cdots + t^{4 n} - \\frac {t^{4 n + 2} } {1 + t^2}       | c =  }} {{end-eqn}} From Square of Real Number is Non-Negative, we have that: :$t^2 \\ge 0$ for all real $t$. So $- t^2 \\le 0$ and so $- t^2 \\ne 1$. So the conditions of Sum of Geometric Sequence are satisfied, and so the above argument holds for all real $t$. {{qed}} Category:Analysis p4j6tw6gbc96p47fxnnxya9iq3srmwp"}
+{"_id": "32238", "title": "Stabilizer is Subgroup/Corollary", "text": "Stabilizer is Subgroup/Corollary 0 8606 479172 378345 2020-07-22T06:48:35Z Prime.mover 59 Prime.mover moved page Stabilizer is Subgroup/Corollary 2 to Stabilizer is Subgroup/Corollary wikitext text/x-wiki == Corollary to Stabilizer is Subgroup == <onlyinclude> Let $G$ be a group whose identity is $e$. Let $G$ act on a set $X$. Let $x \\in X$. Then: :$\\forall g, h \\in G: g * x = h * x \\iff g^{-1} h \\in \\Stab x$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = h * x       | r = g * x       | c =  }} {{eqn | ll= \\leadstoandfrom       | l = g^{-1} * \\paren {h * x}       | r = g^{-1} * \\paren {g * x}       | c =  }} {{eqn | ll= \\leadstoandfrom       | l = \\paren {g^{-1} h} * x       | r = \\paren {g^{-1} g} * x       | c =  }} {{eqn | ll= \\leadstoandfrom       | l = \\paren {g^{-1} h} * x       | r = e * x       | c =  }} {{eqn | ll= \\leadstoandfrom       | l = \\paren {g^{-1} h} * x       | r = x       | c =  }} {{eqn | ll= \\leadstoandfrom       | l = g^{-1} h       | o = \\in       | r = \\Stab x       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Definition:Transitive Group Action|next = Orbit-Stabilizer Theorem/Proof 2}}: $\\S 6.5$. Orbits: Lemma $\\text {(ii)}$ Category:Stabilizer is Subgroup bdt6cxy88ib67897zz493mozynn8f9a"}
+{"_id": "32239", "title": "Upper Bound is Lower Bound for Inverse Ordering", "text": "Upper Bound is Lower Bound for Inverse Ordering 0 8810 212639 212638 2015-04-07T12:51:49Z Prime.mover 59 wikitext text/x-wiki {{refactor|Not trivial. Result may need rephrasing using iff instead}} == Definition == Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T \\subseteq S$. Let $M$ be an upper bound for $\\left({T, \\preceq}\\right)$. Let $\\succeq$ be the dual ordering of $\\preceq$. Then $M$ is a lower bound for $\\left({T, \\succeq}\\right)$. == Proof == Let $M$ be an upper bound for $\\left({T, \\preceq}\\right)$. That is: :$\\forall a \\in T: a \\preceq M$ By definition of dual ordering, it follows that: :$\\forall a \\in T: M \\succeq a$ That is, $M$ is a lower bound for $\\left({T, \\succeq}\\right)$.  {{qed}} {{expand|Proof via duality}} == Also see == * Lower Bound is Upper Bound for Inverse Ordering Category:Order Theory d06yyjlwxyk51wcwiot068yvvtw6ecq"}
+{"_id": "32240", "title": "User talk:Linus44", "text": "User talk:Linus44 3 9010 346671 346666 2018-03-11T09:36:20Z Prime.mover 59 wikitext text/x-wiki ==== Welcome ==== Welcome to ProofWiki! Since you're new, you may want to check out the general help page. It's the best first stop to see how things are done (next to reading proofs, of course!). Please feel free to contribute to whichever area of mathematics interests you, either by adding new proofs, or fixing up existing ones. If you have any questions please feel free to contact me, or post your question on the questions page. Here are some useful pages to help get you started: * Community Portal - To see what needs to be done, and keep up to date with the community. * Recent Changes - To keep up with what's new, and what's being added. * Main Page talk - This is where most of the main discussions regarding the direction of the site take place. If you have any ideas, please share them! Cheers, prime.mover (talk) == Properties of Degree == We're obviously on precisely the same page here (literally!) as this was the next proof (after tidying up after the infrastructure update) that I was going to post! You're not working your way through Hartley and Hawkes yourself, are you?  Keep up the damn fine work, bro - and bear with me when I reformat stuff into our (so far unwritten) house style. --prime mover 00:48, 8 February 2011 (CST) : Just read your comment (later deleted) on polynomials: if you have a strategy for axiomatic foundation of Galois theory, then feel free to go for it. You've noticed there's a lot of groundwork done already - but the author of most of this (ahem, me) has little background in it except what's been self-taught, so feel free to rewrite or amend if stuff is insufficiently precise or accurate. And if you have an idea of how to turn into crystal the amorphous sludge that currently constitutes the section on polynomials, then go for it. --prime mover 00:59, 8 February 2011 (CST) == The big picture == You're obviously focused here - I'll let you get on with it as you definitely seem to be in the zone. You just happened to turn up as we're in the middle of changing the infrastructure to use a new LaTeX package MathJax. Some of the old commands no longer work like <nowiki>\\or</nowiki> and <nowiki>\\and</nowiki> - I'm replacing them with <nowiki>\\lor</nowiki> and <nowiki>\\land</nowiki> as I go - and display text formatting insists on centre justifying, which doesn't work too well here. It'll shake itself down in time, no doubt, but in the meantime expect plenty of pages to not look as good as they should. IF you see any then feel free to try and fix them, and if you have problems let me know and I'll see what I can do. --prime mover 00:22, 9 February 2011 (CST) : ... Just pick it up as you go along, you're all right.--prime mover 00:46, 9 February 2011 (CST) == Comments, then ... == You said \"comments welcome, okay then ... Nice job. In the groove. There's a few stylistic and \"house standard\" issues which you may want to take on board (not to worry if not, it'll get changed). * Good to separate each entity in LaTeX with a space - a rule of thumb is put a space between each variable and before each backslash starting a command. It's not important, but it makes it readable and allows for a saner line breaking. * More tediously, anything parenthesised ought to be between $\\left({ left and right }\\right)$ delimiters. Not sure if this is relevant any more actually, but with MediaWiki in particular this enforced the browser to space things neatly, even in cases where what's in the middle is a single letter. E.g. $f \\left({x}\\right)$ not $f(x)$. This needs to be waived, of course, if the brackets involved are in different latex strings. * You've noticed that the delimiter of choice is now dollar not < math >. The latter renders still, but interestingly not in transclusions (see Trigonometric Identities for an example of lots of transclusions). There is an ongoing exercise to convert all math delimiters to dollars. * Finally (this time round), the writing style here is (barring the occasional page or two in symbolic logic written by someone who didn't get it) \"short and sweet.\" : Long sentences are likely to be split up into shorter ones. : Each sentence needs to go on a separate line.  : The result is that it is easy to follow. : The above fact continues to hold even when the proof is tortuously complex. * Other minor stylistic quibbles are relatively inconsequential: display equations aren't punctuated with a full stop (that's an Ian Stewartism), lines before a display equation end with a colon: :because this is a display equation and punctuation does not go inside a latex object (despite what they say on wikipedia). That is: :Hence the result for $f(x)$. not: :Hence the result for $f(x).$ And yes I know I've broken more than one of the above rules in the above. Keep on trucking.--prime mover 13:34, 11 February 2011 (CST) == Numbering equations and referring back == Like this: :$\\label{einstein} e = m c^2$ Then you can refer back to \\eqref{einstein} like this. This is new with the MathJax package. I'm not sure how limited we are on how such equations may be formatted. I'm not a fan of centre justification on a wiki page as it doesn't lend itself to the eye when the sentences are short. I haven't found a way to define the default for display equations to default to left justification. I believe you can do it in standard LaTeX documents (long time since I wrote one) so it ought to be doable here. What I tend to do in the meantime is do it manually: :$(1) \\quad e = m c^2$ Then you can refer back to $(1)$ like this. --prime mover 02:13, 12 February 2011 (CST) == Help page == Wow! Nice one. That was something I was going to get round to doing one time. Saves me a pain! Thx. --prime mover 02:18, 12 February 2011 (CST) ... in fact it encouraged me to work on it. --prime mover 04:11, 12 February 2011 (CST) == Pointwise operations == We already have a page for this. I think it's in the Abstract Algebra category. Might take some hunting down - sorry, been called away for domestic reasons - I'll get back to you on it. --prime mover 06:21, 13 February 2011 (CST) : You wrote: \"... the definition of induced structure is quite robust so it covers addition (although interestingly not [I think] multiplication) of polynomial forms as well ...\" I don't see why not, if you treat the concept of $\\oplus$ as being either multiplication or addition. Can we continue this discussion in the talk page of Definition:Induced Structure? I need to understand what you're thinking. --prime mover 08:35, 13 February 2011 (CST) ::... Goodness. It'll take me a while to digest all that lot ... bear with me, I'm only able to bear a subset of my braincells at the moment. I never dreamed it would have been so involved. --prime mover 09:56, 13 February 2011 (CST) == Defining operators == If you need to define an operator in $\\LaTeX$, rather than use \\text{op} for example, use \\operatorname{op} instead. There's a reason for this: apparently it gets rendered differently so as to be more appropriate to being an operator (more space afterwards perhaps) in certain browsers. Or something. It's what we've been doing up till now. Probably no big deal, it's just a nicety. We may find that \\operatorname isn't what's supposed to be used now anyway, it's probably \\mathop now - but I'm not in a position to experiment. --prime mover 00:45, 17 February 2011 (CST) : Sure thing, it appears that \\mathop is for turning a composition of symbols into a single operator, e.g. ::$\\displaystyle \\mathop{\\bigoplus \\bigotimes}_{1\\leq i< j \\leq n} (ij)$ : while \\operatorname is for operators with textual names, hence makes a better standard of \\id etc. At least I think those their intended purposes. Linus44 13:39, 17 February 2011 (CST) :::backslashed your LaTeX for yez --prime mover 14:53, 17 February 2011 (CST) == Macros == Replied on my page: --prime mover 15:07, 20 February 2011 (CST) == Internal links == I'm interested as to your techniques for including internal links in your articles because I see they have the underscores in them. Not a big problem (can make it more tedious to produce a form for reading because then it forces you to format it) - but I'm curous as to what your technique is that makes this easier than writing the link without the underscores. --prime mover 00:25, 23 February 2011 (CST) == ... unrelatedly == While the Principle of Mathematical Induction implies Well-Ordering Principle, as proved on Equivalence of Well-Ordering Principle and Induction, it is not the case that the Well-Ordering Principle implies the Axiom of Choice. The latter is implied by the Well-Ordering Theorem (not documented yet), which states that any set can have an ordering under which that set is a well-ordered set. But that latter is NOT the Well-Ordering Principle.  Good question. --prime mover 00:41, 23 February 2011 (CST) : Ah, my book has the WOP as \"every set can be well ordered\". I was wondering why no-one had told me about that incredible set of equivalences. Thanks! --Linus44 00:48, 23 February 2011 (CST) :: There is a word in the WOP defn on this site warning not to confuse the WOP with the WOT - perhaps it needs a further word about how the WOT is also sometimes called the WOP! Scary biscuits ... --prime mover 01:57, 23 February 2011 (CST) == Listen here wolfchild == ... I speak to you of the science of mythology ... :-) Good catch. --prime mover 14:43, 23 February 2011 (CST) == big thankx == for helping with this tedious migration. Your effort is appreciated. --prime mover 15:14, 3 March 2011 (CST) : S'alright, I just figured out external editing in Chrome; the equations load about 10 times quicker than they did in Firefox so it's quite relaxing to do now --Linus44 23:50, 3 March 2011 (CST) == Sorting definition pages == When you categorise a new definition page, plz remember to add <nowiki>|{{SUBPAGENAME}}</nowiki> to the definitions category - otherwise the category page will sort it on \"Definition:...\" and therefore put it under D where it ought to be sorted as the subpage name. (Note to self: check this is stated in the edit help pages.) --prime mover 17:35, 4 March 2011 (CST) : Ah yes sorry, I had picked up on that....then forgotten again --Linus44 01:46, 5 March 2011 (CST) :: ... no big deal, we can correct them as we find them. :: Underneath the edit pane there's a \"MediaWiki Functions\" box which you can use to put the template for a category, definition category, etc. in place without having to type it in. Just click on the appropriate blue thing and a copy appears where your cursor is. Might be worth getting the habit then it's just muscle memory. --prime mover 03:23, 5 March 2011 (CST) == Minor changes == Something to beware of (I'm always doing it myself!) is hitting the \"minor edit\" check box when you don't want to, and you especially don't want to do it with a new proof. In the latter case what happens is that the proof does not get logged as the \"Latest proof\" (on front page) or added to the list of \"recently added proofs\" (see Community portal), and you lose the kudos. --prime mover 17:02, 27 March 2011 (CDT) : Ah yes, had it marking as minor by default, now changed. --Linus44 17:08, 27 March 2011 (CDT) == Properties of Gamma Function == Just a heads up: what I did with Properties of Gamma Function was split it up into 4 separate pages linked by transclusion. It's the same technique as I used on Properties of Binomial Coefficients, q.v. for comparison. The individual results all merit a page of their own (particularly the ones with names on them) and besides they'll probably end up being too big for comfort. Big pages don't work too well on this wiki. --prime mover 15:59, 29 March 2011 (CDT) : Works for me --Linus44 09:36, 30 March 2011 (CDT) == Does not divide == Using nmid. See Definition:Divisor Notation where the notation is discussed. It's not a perfect solution but using slash to negate a backslash makes something ugly. --prime mover 01:34, 3 May 2011 (CDT) == Using newcommand == I see you're experimenting with newcommand (fair play to you) but just a heads-up: I've found it doesn't work too well when transcluding pages with it on. I also think it doesn't work when the newcommand isn't right at the top of the page (had to change a Spec of yours back to an operatorname a couple days ago) and I haven't got to finding out why this is. Be aware you may get rendering issues. If we can solve these problems (I confess they're not high on my list ATM), then I wonder whether it might be feasible to set up some \"standard\" abbrevs for some stuff, a) because it will streamline certain pages, and b) because it will then give us a basis for a \"uniform\" symbology. One thing I've been meaning to get round to (but there's always exciting results to document instead) is a list of standard symbols (as part of \"house style\") like using $\\varnothing$ for the empty set rather than $\\emptyset$, and $\\backslash$ for \"divides\" and not \"\\mid\" and so on. Using a standard header file with \"newcommand\" commands in may give us a headstart on that experiment.--prime mover 16:10, 6 May 2011 (CDT) : Yeah, the newcommand stuff was just because I prefered $\\operatorname{Max}\\:\\operatorname{Spec}$ to $\\operatorname{MaxSpec}$, but the former takes forever to type. : Uniform notation would have it's advantages, but then it'd be good to have a more obvious link on each page saying \"The notation used is explained here\", otherwise there's a risk of being difficult to read for anyone unfamiliar with these pages, especially with stuff like $\\backslash$, where $\\mid$ is more commonly used. --Linus44 07:22, 7 May 2011 (CDT) :: My approach to this is always, when using a piece of notation which is possibly non-standard or unusual, to add \"... where $\\backslash$ denotes divides\" or whatever. How do I know whether it's non-standard enough to merit this? Someone complains they can't understand it. :-) --prime mover 08:14, 7 May 2011 (CDT) == Welcome back == I thought you'd gone forever! --prime mover (talk) 05:51, 28 September 2012 (UTC) :I've been making do with the internet on my phone for ages, since I never used it for much more than emails.  I'm not sure how often I'll be on here, the spare time I do have is usually a much needed break from mathematics, but we'll see. --Linus44 (talk) 16:41, 28 September 2012 (UTC) == Hold your horses, bro' == The recent edit you made to Definition:Differential has completely obscured the basic simplicity that it originally was and cast it into language which is inaccessible to undergraduates. We're trying not to lose the basic understandability here, and besides, the citation at the bottom of the page no longer has anything to do with what's on the page. You might want to take some time to refamiliarise yourself with the house style and the general philosophy of this site - the idea is for it to be more of a dictionary than an encyclopedia. In particular, I believe that what we might want to do with this page is to write the more complicated stuff in a separate page and transclude it. --prime mover (talk) 19:26, 28 September 2012 (UTC) : This is a point. It might be good to split the definition up into sections of successive generality, with the $\\R \\to \\R$ setting at the top. I'll rearrange it after I've eaten some chocolate bread.  : BTW, is there a page for vector spaces of the same dimension are isomorphic? There seemed to be enough linear algebra that it ought to be there, but I couldn't find it. --Linus44 (talk) 20:26, 28 September 2012 (UTC) == SourceReview template == Please add an invocation of Template:SourceReview at the bottom of any pages which you have refactored, like for the Gauss's Lemma page. It allows someone coming after to note the fact that the process flow of the works that are invoked can be amended as appropriate. --prime mover (talk) 06:23, 19 March 2013 (UTC) : ok --Linus44 (talk) 11:11, 19 March 2013 (UTC) ==  Fundamental Theorem of Algebra == Thanks for polishing up my proof! :-) It was my first attempt to write a proof here, so I'm really grateful for your corrections and additions. --BFeuerbacher (talk) 20:36, 16 April 2013 (UTC) :No worries, and thanks for your contribution. My changes were just cosmetic stuff and adding links to the results used, see here for an explanation of the changes. --Linus44 (talk) 22:23, 16 April 2013 (UTC) == Question about Definition:Partition of Unity (Topology) == I have addressed some of the unclarities in this page, but there is still an outstanding question about what is specifically meant by \"support\" - there are two definitions on that page which appear to be mutually incompatible. Are you in a position to be able to go back to this page and see whether (a) it is still accurate, and (b) it needs improvement? I gather you may be more familiar with this area of maths than I am - and besides, you wrote the original (presumably in support of the Stokes' Theorem page) ... --prime mover (talk) 07:52, 15 September 2013 (UTC) m21g5cp0m2ex8l1na3mzsx5hyu2kgk0"}
+{"_id": "32241", "title": "User:Linus44", "text": "User:Linus44 2 9084 455719 221265 2020-03-18T21:30:25Z Prime.mover 59 wikitext text/x-wiki Things to do: == Definition:Differential == Let $(E, \\| \\cdot \\|_E)$, $(F, \\| \\cdot \\|_F)$ be normed vector spaces. Let $U \\subseteq E$ be an open set. Let $f : U \\to F$ be a mapping. Let $a \\in U$ be an element of $U$. Then $f$ is '''differentiable''' at $a$ if there exists a continuous and linear map $df_a \\in \\mathcal L(E,F)$ such that :$\\displaystyle \\lim_{h \\to 0} \\| f(a+h) - f(a) - df_a \\cdot h \\|_F \\| h \\|_E^{-1} = 0$ Then $df_a$ is called the '''differental''' or the '''tangent map''' of $f$ at $a$. We say that $f$ is '''continuously differentiable''' if: :$\\displaystyle df : (U, \\| \\cdot \\|_E) \\to \\mathcal (L(E,F),\\| \\cdot \\|_{L(E,F)})$ :$\\displaystyle \\ : a \\mapsto df_a$ is continuous. If $E = \\R^n$, this is true iff the first order partial derivatives of $f$ exist and are continuous. == Induced polynomial homomorphism == Even this needs serious thought if it's to be any good. == Permutations == Definition:Cyclic Permutation $k$ well defined. Add canonicality. Incorrect: Definition:Permutation on n Letters/Cycle Notation permutation/cycle confusion? Also $\\rho$ should be $\\pi$ for consistency. Then here: Equality of Cycles == Rings, properties, equivalent definitions == GCD domain: *Equivalent Definitions of a GCD Domain *Properties of GCD Domains Bézout domain *Equivalent Definitions of a Bézout domain *Properties of Bézout Domains Definition:Unique Factorization Domain: *Equivalent Definitions of a Unique Factorization Domain *Properties of Unique Factorization Domains etc...needs organizing into something more standardized == The rest == *Vinogradov's Theorem: Pick a method of proof and think of a good structure for it. *Figure out conditions for the \"function-form epimorphism\" to have trivial kernel (see Epimorphism from Polynomial Forms to Polynomial Functions and especially Equality of Polynomials) === Proof of prime number theorem and Siegel Walfiz === See also stuff on Dirichlet's Theorem Poisson Summation Formula Definition:Schwarz Function Harmonic Properties of Schwarz Functions Hadamard Factorisation Theorem Jensen's Formula Definition:Completed Riemann Zeta Function Estimation Lemma Uniform Limit of Analytic Functions is Analytic Poles of Gamma Function Properties of Gamma Function Stirling's Formula for Gamma Function Definition:Order of Entire Function Completed Riemann Zeta Function has Order One Product Equation for Riemann Zeta Function Zeros of Functions of Finite Order Poles of Riemann Zeta Function Riemann Zeta Has No Zeros With Real Part One Unsymmetric Functional Equation for Riemann Zeta Function === Dirichlet: Finished, but check for errors === Analytic Continuation of Dirichlet L-Function L-Function does not Vanish at One Logarithm of Dirichlet L-Functions Dirichlet's Theorem on Arithmetic Sequences Dirichlet L-Function from Trivial Character Convergence of Dirichlet Series with Bounded Coefficients Convergence of Dirichlet Series with Bounded Partial Sums Definition:Dirichlet L-function Definition:Dirichlet Character Definition:Completed Dirichlet L-Function Functional Equation for Dirichlet L-Functions Orthogonality Relations for Characters === Unfinished Pages === Completeness Criterion (Metric Spaces) Open Sets in Real Number Line Equivalence of Definitions of Riemann Zeta Function Definition:Category Hardy-Littlewood Circle Method Vinogradov Circle Method Characterisation of Totally Ordered Field Properties of Totally Ordered Field Gauss's Lemma (Ring Theory) 8s9j00q06ec1pixsutawbecgaerh4p4"}
+{"_id": "32242", "title": "Euclid's Lemma for Irreducible Elements", "text": "Euclid's Lemma for Irreducible Elements 0 9515 402085 352815 2019-04-22T07:55:42Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $\\struct {D, +, \\times}$ be a Euclidean domain whose unity is $1$. Let $p$ be an irreducible element of $D$. Let $a, b \\in D$ such that: :$p \\divides a \\times b$ where $\\divides$ means '''is a divisor of'''. Then $p \\divides a$ or $p \\divides b$. === General Result === {{:Euclid's Lemma for Irreducible Elements/General Result}} == Proof == Let $p \\divides a \\times b$. Suppose $p \\nmid a$. Then from the definition of irreducible: : $p \\perp a$ Thus from Euclid's Lemma for Euclidean Domains: :$p \\divides b$ Similarly, if $p \\nmid b$: :$p \\divides a$ So: :$p \\divides a b \\implies p \\divides a$ or: :$p \\divides b$ as we needed to show. {{qed}} {{Namedfor|Euclid}} == Also see == * Euclid's Lemma for Prime Divisors, for the usual statement of this result, which is this lemma as applied specifically to the integers. == Sources == * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Euclid's Lemma for Euclidean Domains|next = Euclid's Lemma for Irreducible Elements/General Result}}: Chapter $6$: Polynomials and Euclidean Rings: $\\S 29$. Irreducible elements: Theorem $56 \\ \\text{(i)}$ Category:Euclidean Domains Category:Euclid's Lemma 4ejlquo28pgy35i9ww9c5y30e0814ov"}
+{"_id": "32243", "title": "Continuous Image of Separable Space is Separable", "text": "Continuous Image of Separable Space is Separable 0 9604 469520 468928 2020-05-21T13:49:26Z Prime.mover 59 wikitext text/x-wiki == Definition == Let $T_1 = \\struct {S_1, \\tau_1}$ and $T_2 = \\struct {S_2, \\tau_2}$ be topological spaces. Let $f: T_1 \\to T_2$ be a continuous mapping. If $T_1$ is separable, then so is the image $f \\sqbrk {T_1}$. That is, separability is a continuous invariant. == Proof == By definition, $T_1 = \\struct {S_1, \\tau_1}$ is separable {{iff}} there exists a countable subset $D \\subset S_1$ which is (everywhere) dense in $T_1$. We need to show that if there exists a continuous mapping $f: T_1 \\to T_2$, then $f \\sqbrk {T_1}$ is also separable. That is, that there exists a countable subset of $f \\sqbrk {S_1}$ which is dense in $T_2$. Let $x_2$ be any point in the image $f \\sqbrk {S_1}$ of $S_1$ under $f$. Let $U \\in \\tau_2$ be an arbitrary open set such that $x_2 \\in U$. By definition of image set, there exists some $x_1 \\in S_1$ with $\\map f {x_1} = x_2$. Since $f$ is continuous, $f^{-1} \\sqbrk U$ is open in $T_1$. By definition of preimage, $x_1$ is in this set. We are given that $D$ is a countable subset of $S_1$ which is dense in $T_1$. By definition of dense, $D \\cap f^{-1} \\sqbrk U \\ne \\O$ Thus there exists some $d \\in D$ such that $d \\in f^{-1} \\sqbrk U$. Therefore $\\map f d \\in U$. Since $U$ was arbitrary, it follows that $f \\sqbrk D$ is dense in $T_2$. By Image of Countable Set under Mapping is Countable, $f \\sqbrk D$ is countable. Hence $T_2$ is separable. {{qed}} == Sources == * {{BookReference|Counterexamples in Topology|1978|Lynn Arthur Steen|author2 = J. Arthur Seebach, Jr.|ed = 2nd|edpage = Second Edition|prev = Definition:Closed Invariant|next = Continuous Image of Compact Space is Compact}}: Part $\\text I$: Basic Definitions: Section $1$: General Introduction: Functions * {{BookReference|Counterexamples in Topology|1978|Lynn Arthur Steen|author2 = J. Arthur Seebach, Jr.|ed = 2nd|edpage = Second Edition|prev = Lindelöf Property is Preserved under Continuous Surjection|next = Compactness Properties Preserved under Projection Mapping}}: Part $\\text I$: Basic Definitions: Section $3$: Compactness: Invariance Properties Category:Separable Spaces Category:Continuous Invariants o526ot6ldkjyrmhw1zn4p9o4efhl777"}
+{"_id": "32244", "title": "Urysohn's Lemma", "text": "Urysohn's Lemma 0 9642 469196 469183 2020-05-20T06:59:31Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $T = \\struct {S, \\tau}$ be a $T_4$ topological space. Let $A, B \\subseteq S$ be closed sets of $T$ such that $A \\cap B = \\O$. Then there exists an Urysohn function for $A$ and $B$. == Proof == Let $T = \\struct {S, \\tau}$ be a $T_4$ space. Let $A, B \\subseteq S$ be closed sets of $T$ such that $A \\cap B = \\O$. Let $P = \\Q \\cap \\closedint 0 1$ where $\\closedint 0 1$ is the closed unit interval. $\\Q$ is countable, therefore so is $P$. === Creation of Domain === We are going to construct a set $\\Bbb U \\subseteq \\tau$ of open sets with $P$ as an indexing set: :$\\Bbb U = \\set {U_i: i \\in P}$ such that: :$\\forall p, q \\in P: p < q \\implies {U_p}^- \\subseteq U_q$ where ${U_p}^-$ denotes the set closure of $U_p$. We define $U_p$ by induction, as follows. List the elements of $P$ in the form of an infinite sequence $\\sequence z$. Let $z_0 = 1, z_1 = 0$. In general, let $P_n$ denote the set consisting of the first $n$ elements of $\\sequence z$. Let $\\map \\PP n$ be the proposition: :$U_p$ is defined for all $p \\in P_n$, and: ::$(1): \\quad \\forall p, q \\in P_n: p < q \\implies {U_p}^- \\subseteq U_q$ ==== Basis for the Induction ==== As $S$ is a $T_4$ space we have that: :$\\forall A, B \\in \\map \\complement \\tau, A \\cap B = \\O: \\exists U_1, V \\in \\tau: A \\subseteq U_1, B \\subseteq V$  We have that $A \\subseteq U_1$ is a closed set of $S$. We define $U_1 = S \\setminus B$ where $S \\setminus B$ denotes the complement of $B$ in $S$. As $S$ is $T_4$, we can choose an open set $U_0 \\in \\tau$ such that $A \\subseteq U_0$ and ${U_0}^- \\subseteq U_1$. Thus $\\map \\PP 1$ is shown to hold. This is our basis for the induction. ==== Induction Hypothesis ==== This is our induction hypothesis: Let $\\map \\PP k$ be the proposition: :$U_p$ is defined for all $p \\in P_k$, and: :: $(1): \\quad \\forall p, q \\in P_k: p < q \\implies {U_p}^- \\subseteq U_q$ We want to show that if $\\map \\PP k$ holds, then: :$U_p$ is defined for all $p \\in P_{k + 1}$, and: :: $(1): \\quad \\forall p, q \\in P_{k + 1}: p < q \\implies {U_p}^- \\subseteq U_q$ ==== Induction Step ==== This is our induction step: Let $r = z_{k + 1}$ be the next element in $\\sequence z$. Consider $P_{k + 1} = P_k \\cup \\set r$. It is a finite subset of the closed unit interval $\\closedint 0 1$. We consider the usual $<$ ordering on $P_{k + 1}$, which is a subset of $\\closedint 0 1$ which in turn is a subset of $\\R$.  From Finite Non-Empty Subset of Totally Ordered Set has Smallest and Greatest Elements, $P_{k + 1}$ has both a minimal element $m$ and a maximal element $M$. From Predecessor and Successor of Finite Toset, every element other than $m$ and $M$ has an immediate predecessor and immediate successor. We already know that $z_1 = 0$ is the minimal element and $z_0 = 1$ is the maximal element of $P_{k + 1}$. So $r$ must be neither of these. Thus: :$r$ has an immediate predecessor $p$ :$r$ has an immediate successor $q$ in $P_{k + 1}$. The sets $U_p$ and $U_q$ are already defined by the inductive hypothesis. As $T$ is a $T_4$ space, there exists an open set $U_r \\subseteq \\tau$ such that: :${U_p}^- \\subseteq U_r$ :${U_r}^- \\subseteq U_q$ We now show that $(1)$ holds for every pair of elements in $P_{k + 1}$. If both elements are in $P_n$, then $(1)$ is true by the inductive hypothesis. If one is $r$ and the other is $s \\in P_k$, then: :$s < p \\implies {U_s}^- \\subseteq {U_p}^- \\subseteq U_r$ and: :$s \\ge q \\implies U_r \\subseteq {U_q}^- \\subseteq {U_s}^-$ Thus $(1)$ holds for every pair of elements in $P_{k + 1}$. Therefore by induction, $U_p$ is defined for all $p \\in P$. We have defined $U_p$ for all rational numbers in $\\closedint 0 1$. We now extend this definition to every rational $p$ by defining: :$U_p = \\begin{cases} \\O & : p < 0 \\\\ S & : p > 1 \\end{cases}$ It is easily checked that $(1)$ still holds. === Definition of Function === Let $x \\in S$. Define $\\map \\Q x = \\set {p: x \\in U_p}$. This set contains no rational number less than $0$ and contains every rational number greater than $1$ by the definition of $U_p$ for $p < 0$ and $p > 1$. Thus $\\map \\Q x$ is bounded below and its infimum is an element in $\\closedint 0 1$. We define: :$\\map f x = \\map \\inf {\\map \\Q x}$ Now we need to show that $f$ satisfies the conditions of Urysohn's Lemma. Let $x \\in A$. Then $x \\in U_p$ for all $p \\ge 0$, so $\\map \\Q x$ equals the set $\\Q_+$ of all non-negative rationals. Hence $\\map f x = 0$. Let $x \\in B$. Then $x \\notin U_p$ for $p \\le 1$ so $\\map \\Q x$ equals the set of all the rationals greater than $1$. Hence $\\map f x = 1$. === Continuity of Function === To show that $f$ is continuous, we first prove two smaller results: :$(a): \\quad x \\in {U_r}^- \\implies \\map f x \\le r$ We have: $x \\in {U_r}^- \\implies \\forall s > r: x \\in U_s$ so $\\map \\Q x$ contains all rationals greater than $r$. Thus $\\map f x \\le r$ by definition of $f$. {{qed|lemma}} :$(b): \\quad x \\notin U_r \\implies \\map f x \\ge r$ We have: $x \\notin U_r \\implies \\forall s < r: x \\notin U_s$ so $\\map \\Q x$ contains no rational less than $r$. Thus $\\map f x \\ge r$. {{qed|lemma}} Let $x_0 \\in S$ and let $\\openint c d$ be an open real interval containing $\\map f x$. We will find a neighborhood $U$ of $x_0$ such that $\\map f U \\subseteq \\openint c d$. Choose $p, q \\in \\Q$ such that: :$c < p < \\map f {x_0} < q < d$ Let $U = U_q \\setminus {U_p}^-$. Then since $\\map f {x_0} < q$, we have that $(b)$ implies vacuously that $x \\in U_q$. Since $\\map f {x_0} > p$, $(a)$ implies that $x_0 \\notin U_p$. Hence $x_0 \\in U$ and thus $U$ is a neighborhood of $x_0$ because $U$ is open. Finally, let $x \\in U$. Then $x \\in U_q \\subseteq {U_q}^-$, so $\\map f x \\le q$ by $(a)$. Also, $x \\notin {U_p}^-$ so $x \\notin U_p$ and $\\map f x \\ge p$ by $(b)$. Thus: $\\map f x \\in \\closedint p q \\subseteq \\openint c d$ Therefore $f$ is continuous. {{qed}} == Also see == * Urysohn's Lemma Converse {{Namedfor|Pavel Samuilovich Urysohn|cat = Urysohn}} == Sources == * {{BookReference|Counterexamples in Topology|1978|Lynn Arthur Steen|author2 = J. Arthur Seebach, Jr.|ed = 2nd|edpage = Second Edition|prev = Definition:Urysohn Function|next = Urysohn's Lemma Converse}}: Part $\\text I$: Basic Definitions: Section $2$: Separation Axioms: Completely Regular Spaces * {{planetmath|url = proofofurysohnslemma|title = Urysohn's Lemma}}  Category:T4 Spaces Category:Closed Sets Category:Urysohn Functions lk4wrekvug3s7ehj3mfaidaplglyvka"}
+{"_id": "32245", "title": "Urysohn's Lemma Converse", "text": "Urysohn's Lemma Converse 0 9643 469195 469190 2020-05-20T06:58:26Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $T = \\struct {S, \\tau}$ be a topological space. Let there exist an Urysohn function for any two $A, B \\subseteq S$ which are closed sets in $T$ such that $A \\cap B = \\varnothing$. Then $T = \\struct {S, \\tau}$ is a $T_4$ space. == Proof == Let $A$ and $B$ be arbitrary closed sets of $T$ and let $f$ be an Urysohn function for $A$ and $B$. Let $C = \\hointr 0 {\\dfrac 1 4}$ and $D = \\hointl {\\dfrac 3 4} 1$. Then $C$ and $D$ are open in $\\closedint 0 1$. {{explain|There's a link available somewhere for the above fact.}} Hence, $\\map {f^{-1} } C$ and $\\map {f^{-1} } D$ are open in $T$. Furthermore, by definition of Urysohn function, $A \\subset \\map {f^{-1} } C$ and $B \\subset \\map {f^{-1} } D$. Also, from Preimage of Intersection under Mapping: :$C \\cap D = \\O \\implies \\map {f^{-1} } C \\cap \\map {f^{-1} } D = \\O$ Therefore, $T$ is a $T_4$ space. {{qed}} == Also see == * Urysohn's Lemma {{Namedfor|Pavel Samuilovich Urysohn|cat = Urysohn}} == Sources == * {{BookReference|Counterexamples in Topology|1978|Lynn Arthur Steen|author2 = J. Arthur Seebach, Jr.|ed = 2nd|edpage = Second Edition|prev = Urysohn's Lemma|next = Existence of Urysohn Function does not guarantee Normal Space}}: Part $\\text I$: Basic Definitions: Section $2$: Separation Axioms: Completely Regular Spaces Category:Closed Sets Category:T4 Spaces Category:Urysohn Functions l7l3u68ugbucfzt8ihtrr5vbjl3ux0o"}
+{"_id": "32246", "title": "Localization of Ring Exists/Lemma 1", "text": "Localization of Ring Exists/Lemma 1 0 9941 324545 306587 2017-11-04T13:26:20Z Barto 3079 wikitext text/x-wiki == Lemma for Localization of Ring Exists == Let $A$ be a commutative ring with unity. Let $S \\subseteq A$ be a multiplicatively closed subset with $0 \\notin S$. Let $\\sim$ be the relation defined on the Cartesian product $A \\times S$ by: :$\\left({a, s}\\right) \\sim \\left({b, t}\\right) \\iff \\exists u \\in S: a t u = b s u$ <onlyinclude> The relation $\\sim$ is an equivalence relation. </onlyinclude> == Proof ==  Since by definition $1 \\in S$ and $a s = a s$ for all $a \\in A, s \\in S$, it follows that $\\sim$ is reflexive. Also $\\sim$ is clearly symmetric because if $a t u = b s u$ then $b s u = a t u$ by symmetry of $=$. Lastly suppose that: : $\\left({a, s}\\right) \\sim \\left({b, t}\\right)$ and: : $\\left({b, t}\\right) \\sim \\left({c, u}\\right)$ Then there are $v, w \\in S$ such that: :$a t v = b s v$ :$b u w = c t w$ Therefore: :$a u t v w = b u w s v = c t w s v$ and so: : $a u \\left({t v w}\\right) = c s \\left({t v w}\\right)$ Since $S$ is multiplicatively closed: : $\\left({a, s}\\right) \\sim \\left({c, u}\\right)$ Thus $\\sim$ is transitive. {{qed}} Category:Commutative Algebra ize69zt3moayarm2v8m8ebbc6c41yg0"}
+{"_id": "32247", "title": "Localization of Ring Exists/Lemma 2", "text": "Localization of Ring Exists/Lemma 2 0 9943 306586 306585 2017-07-26T15:07:22Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> The operations $+$ and $\\cdot$ are well defined on $A_S$. </onlyinclude> {{explain|Put this into the context in which it is relevant so as to make the page standalone.}} == Proof == Let: :$a / s = c / u$ :$b / t = d / v$ be two sets of representatives for two distinct equivalence classes in $A_S$. We have $w, z \\in S$ such that: :$\\left({a u - c s}\\right) w = 0$ and :$\\left({b v - d t}\\right) z = 0$ Therefore: :$z w \\left[{\\left({a u - c s}\\right) w - \\left({b v - d t}\\right) z}\\right] = 0$ and: :$z w \\left[{\\left({a t + b s}\\right) u v - \\left({c v + d u}\\right)s t}\\right]$ So: :$\\left({a t + b s, s t}\\right) \\sim \\left({c v + d u, u v}\\right)$ That is: :$\\dfrac {a t + b s} {s t} = \\dfrac {c v + d u} {u v}$ For multiplication, with $z, w$ as above we have: :$\\left({a b u v - c d s t}\\right)z w = 0$ So: :$\\left({a b, s t}\\right) \\sim \\left({d c, u v}\\right)$ and: :$\\dfrac {a b} {s t} = \\dfrac {d c} {u v}$ {{qed}} Category:Commutative Algebra 6789zb5nsajxmkfe19f7fngtth7kwyv"}
+{"_id": "32248", "title": "Localization of Ring Exists/Lemma 3", "text": "Localization of Ring Exists/Lemma 3 0 9944 305246 305232 2017-07-16T13:25:21Z Prime.mover 59 wikitext text/x-wiki {{refactor|The definitions of the entities need to be included here as they are on the main page.}} == Lemma for Localization of Ring Exists == <onlyinclude> The map $h \\left({a / s}\\right) = g \\left({a}\\right) g \\left({s}\\right)^{-1}$ is a well defined ring homomorphism $A_S \\to B$. </onlyinclude> == Proof == Let $a / s = b / t \\in A_S$. Then there is $u \\in S$ such that $a t u = b s u$. Therefore, by the homomorphism property, and the fact that $g \\left[{S}\\right] \\subseteq B^\\times$: :$g \\left({a}\\right) g \\left({t}\\right) = g \\left({b}\\right) g \\left({s}\\right) \\to g \\left({a}\\right) g \\left({s}\\right)^{-1} = g \\left({b}\\right) g \\left({t}\\right)^{-1}$ We have that $g$ is a ring homomorphism. As $B$ is a commutative ring it follows that $h$ is also a ring homomorphism. {{qed}} Category:Commutative Algebra hjse67aa67h9mlbociitmtd58rzeita"}
+{"_id": "32249", "title": "Equivalence Relation on Cauchy Sequences", "text": "Equivalence Relation on Cauchy Sequences 0 10209 441888 394769 2020-01-01T14:38:38Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $\\struct {X, d}$ be a metric space. Let $\\CC \\sqbrk X$ denote the set of all Cauchy sequences of elements from $X$. Let a relation $\\sim$ be defined on $\\CC \\sqbrk X$ by: :$\\displaystyle \\sequence {x_n} \\sim \\sequence {y_n}  \\iff \\lim_{n \\mathop \\to \\infty} \\map d {x_n, y_n} = 0$ Then $\\sim$ is an equivalence relation on $\\CC \\sqbrk X$. == Proof == We must show that $\\sim$ is :reflexive, :symmetric and :transitive on $\\CC \\sqbrk X$. To this end, let $\\sequence {x_n}, \\sequence {y_n}, \\sequence {z_n} \\in \\CC \\sqbrk X$ be arbitrary. For each $n \\in \\N$ we have that $\\map d {x_n, x_n} = 0$ by metric space axiom '''M1'''. Therefore $\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map d {x_n, x_n} = 0$. This shows that $\\sequence {x_n} \\sim \\sequence {x_n}$. Thus $\\sim$ is reflexive. {{qed|lemma}} By metric space axiom '''M3''', $\\map d {x_n, y_n} = \\map d {y_n, x_n}$ for each $n \\in \\N$. Therefore: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map d {x_n, y_n} = \\lim_{n \\mathop \\to \\infty} \\map d {y_n, x_n}$ So $\\sequence {x_n} \\sim \\sequence {y_n}$ implies that $\\sequence {y_n} \\sim \\sequence {x_n}$. Thus $\\sim$ is symmetric. {{qed|lemma}} Finally, by metric space axiom '''M2''', $\\map d {x_n, z_n} \\le \\map d {x_n, y_n} + \\map d {y_n, z_n}$ for each $n \\in \\N$. Therefore, by the sum rule for limits of sequences: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map d {x_n, z_n} \\le \\lim_{n \\mathop \\to \\infty} \\map d {x_n, y_n} + \\lim_{n \\mathop \\to \\infty} \\map d {y_n, z_n}$ Thus $\\sequence {x_n} \\sim \\sequence {y_n}$ and $\\sequence {y_n} \\sim \\sequence {z_n}$ together imply that $\\sequence {x_n} \\sim \\sequence {z_n}$. Thus $\\sim$ is transitive. {{qed|lemma}} So $\\sim$ is shown to be reflexive, symmetric and transitive, and therefore an equivalence relation on $\\CC \\sqbrk X$. {{qed}} Category:Cauchy Sequences oyhdg80u6w2apmebp8s0go65tceb2dc"}
+{"_id": "32250", "title": "Field Adjoined Algebraic Elements is Algebraic", "text": "Field Adjoined Algebraic Elements is Algebraic 0 10490 315920 109939 2017-09-08T08:04:46Z Barto 3079 wikitext text/x-wiki == Definition == Let $L / K$ be a field extension and $S \\subseteq L$ a subset. If each $x \\in S$ is algebraic over $K$ then $K \\left({S}\\right)$ is algebraic over $K$. == Proof == Let $S \\subseteq L$ be arbitrary, and $x \\in K \\left({S}\\right)$. By Field Adjoined Set $x \\in K \\left({S}\\right)$ {{iff}} $x \\in K \\left({\\alpha_1, \\ldots, \\alpha_n}\\right)$ for some $\\alpha_1, \\ldots, \\alpha_n \\in S$. We have that $K \\left({\\alpha_1, \\ldots, \\alpha_n}\\right) / K$ is finite by Finitely Generated Algebraic Extension is Finite. Moreover a Finite Field Extension is Algebraic. Therefore $x$ is algebraic over $K$. {{qed}} Category:Field Extensions mwdjy617ad88plf9n1bhuk3irb02o9g"}
+{"_id": "32251", "title": "Vinogradov's Theorem/Minor Arcs/Lemma 1", "text": "Vinogradov's Theorem/Minor Arcs/Lemma 1 0 10754 456325 391779 2020-03-19T11:11:42Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> For $\\beta \\in \\R$, define: :$\\norm \\beta := \\min \\set {\\size {n - \\beta}: n \\in \\Z}$ Then: :$\\displaystyle \\forall \\alpha \\in \\R: \\size {\\sum_{k \\mathop = N_1}^{N_2} \\map e {\\alpha k} } \\le \\min \\set {N_2 - N_1, \\frac 1 {2 \\norm \\alpha} }$ </onlyinclude> == Proof == The bound $N_2 - N_1$ is trivial: Since $\\size {\\map e {\\alpha k} } = 1$ for all $k$, by the Triangle Inequality: :$\\displaystyle \\size {\\sum_{k \\mathop = N_1}^{N_2} \\map e {\\alpha k} } \\le \\sum_{k \\mathop = N_1}^{N_2} 1 = N_2 - N_1$ To show the second bound we evaluate the sum as a geometric series. We have: :$\\map e {\\alpha k} = \\map e \\alpha^k$ So by Sum of Geometric Sequence: {{begin-eqn}} {{eqn | l = \\size {\\sum_{k \\mathop = N_1}^{N_2} \\map e {\\alpha k} }       | r = \\size {\\frac {\\map e {\\alpha \\paren {N_2 + 1} } - \\map e {\\alpha N_1} } {\\map e \\alpha - 1} } }} {{eqn | r = \\frac 2 {\\size {\\map e \\alpha - 1} }       | o = \\le }} {{end-eqn}} By the polar form of a complex number: :$\\map e \\alpha - 1 = \\map e {\\alpha / 2} \\paren {\\map e {\\alpha / 2} - \\map e {-\\alpha / 2} }$ and: :$\\map e {\\alpha / 2} - \\map e {-\\alpha / 2} = \\map \\exp {\\pi i \\alpha} - \\map \\exp {-\\pi i \\alpha} = 2 i \\map \\sin {\\pi \\alpha}$ Therefore: :$\\displaystyle \\size {\\sum_{k \\mathop = N_1}^{N_2} \\map e {\\alpha k} } \\le \\frac 1 {\\size {\\map \\sin {\\pi \\alpha} } }$ We know that the sine function is concave on $\\closedint 0 {\\pi / 2}$. So: :$\\map \\sin {\\pi \\alpha} \\ge 2 \\alpha$ for $\\alpha \\in \\closedint 0 {1/2}$. By definition there exists $n \\in \\Z$ such that: :$\\alpha = n + \\norm \\alpha$ and: :$\\norm \\alpha \\in \\closedint 0 {\\pi/2}$. So: {{begin-eqn}} {{eqn | l = \\size {\\map \\sin {\\pi \\alpha} }       | r = \\size {\\sin {\\pi \\norm \\alpha} }       | c = Sine and Cosine are Periodic on Reals }} {{eqn | r = \\map \\sin {\\pi \\norm \\alpha}       | c = as $\\sin x > 0$ for $x \\in \\closedint 0 {\\pi/2}$ }} {{eqn | o = \\ge       | r = 2 \\norm \\alpha       | c =  }} {{end-eqn}} This completes the proof. {{qed}} Category:Vinogradov's Theorem 3zgphhzfwhljp1fbsyanz1v0j0gr0um"}
+{"_id": "32252", "title": "User:Yohji kinomoto", "text": "User:Yohji kinomoto 2 10774 56649 2011-06-08T20:58:56Z Yohji kinomoto 404 Created page with \"'''<big><big>日本への栄光</big>></big>'''\" wikitext text/x-wiki '''<big><big>日本への栄光</big>></big>''' jekumnbz28ww4r0l5dep1423smbmjhk"}
+{"_id": "32253", "title": "Vinogradov's Theorem/Major Arcs/Lemma 1", "text": "Vinogradov's Theorem/Major Arcs/Lemma 1 0 10842 458354 296435 2020-03-29T20:58:04Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $\\phi$ be the Euler $\\phi$ function. Let $\\mu$ be the Möbius function. Let $c_q$ be the Ramanujan sum modulo $q$. Let $P, N \\ge 1$. {{Explain|what are $P$ and $N$: real, integer, ...?}} Let: :$\\displaystyle \\map {\\SS_P} N := \\sum_{q \\mathop \\le P} \\frac {\\map \\mu q \\map {c_q} N} {\\map \\phi q^3}$ :$\\displaystyle \\map \\SS N := \\lim_{P \\mathop \\to \\infty} \\map {\\SS_P} N$ Then: :$\\map \\SS N = \\map {\\SS_P} N + \\map \\OO {P^{\\epsilon -1} }$ and $\\SS$ has the Euler product: :$\\displaystyle \\map \\SS N = \\prod_{p \\mathop \\nmid N} \\paren {1 + \\frac 1 {\\paren {p - 1}^3} } \\prod_{p \\mathop \\divides N} \\paren {1 - \\frac 1 {\\paren {p - 1}^2} }$ where: :$p \\nmid N$ denotes that $p$ is not a divisor of $N$ :$p \\divides N$ denotes that $p$ is a divisor of $N$. </onlyinclude> == Proof == We have: {{begin-eqn}} {{eqn | l = \\size {\\map {c_q} N}       | o = \\le        | r = \\sum_{\\substack {1 \\mathop \\le a \\mathop \\le q \\\\ \\map \\gcd {a, q} \\mathop = 1} } 1       | c = Triangle Inequality }} {{eqn | r = \\map \\phi q       | c = Definition of Euler $\\phi$ Function }} {{end-eqn}} Trivially we have $\\size {\\map \\mu q} \\le 1$. Therefore: {{begin-eqn}} {{eqn | l = \\size {\\map \\SS N - \\map {\\SS_P} N}       | r = \\sum_{q \\mathop > P} \\frac {\\size {\\map \\mu q \\map {c_q} N} } {\\map \\phi q^3}       | c =  }} {{eqn | o = \\le       | r = \\sum_{q \\mathop > P} \\frac 1 {\\map \\phi q^2}       | c =  }} {{eqn | o = \\le       | r = \\sum_{q \\mathop > P} q^{\\epsilon - 2}       | c = for large $P$ and $\\epsilon \\in \\R_{>0}$, by Asymptotic Growth of Euler Phi Function }} {{eqn | o = \\le       | r = P^{\\delta - 1} \\sum_{q \\mathop > P} q^{\\epsilon - \\delta - 1}       | c = for any $\\delta > \\epsilon$ }} {{end-eqn}} Therefore by Convergence of Powers of Reciprocals: :$\\map \\SS N = \\map {\\SS_P} N + \\map \\OO {P^{\\epsilon - 1} }$ as claimed. By Euler Product, and because $\\map \\mu {p^k} = 0$ for $k > 1$: :$\\displaystyle \\map \\SS N = \\prod_p \\paren {1 + \\frac {\\map \\mu q \\map {c_q} N} {\\map \\phi q^3} }$ Now for a prime $p$ we have: :$\\map \\mu p = -1$ :$\\map \\phi p = p - 1$ We also have Kluyver's Formula for Ramanujan's Sum: :$\\displaystyle \\map {c_p} n = \\sum_{d \\mathop \\divides \\map \\gcd {p, n} } \\rd \\map \\mu {\\frac p d}$ Let $p \\divides N$. Then: $\\map \\gcd {p, N} = p$ which gives: :$\\map {c_p} N = p \\map \\mu 1 + \\map \\mu p = p - 1$ Let $p \\nmid N$. Then: :$\\map \\gcd {p, N} = 1$ and so: :$\\map {c_p} N = -1$ Therefore: :$\\displaystyle \\map \\SS N = \\prod_{p \\mathop \\divides N} \\paren {1 - \\frac 1 {\\paren {p - 1}^2} } \\prod_{p \\mathop \\nmid N} \\paren {1 + \\frac 1 {\\paren {p - 1}^3} }$ This completes the proof. {{qed}} Category:Vinogradov's Theorem 01jbgz0ddezyfq6pa4c347tpvo0r7x9"}
+{"_id": "32254", "title": "Axiom:Axiom of Limitation of Size", "text": "Axiom:Axiom of Limitation of Size 100 10934 444291 443991 2020-01-20T15:30:46Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> For any class $\\CC$, a set $x$ such that $x = \\CC$ exists {{iff}} there is no bijection between $\\CC$ and the universe. </onlyinclude> {{explain|Check whether \"universal class\" is meant here}} Limitation of Size 7zgunwpct6381rvcoas04h8zhnakky0"}
+{"_id": "32255", "title": "Axiom:Class Comprehension Schema", "text": "Axiom:Class Comprehension Schema 100 10935 443994 443988 2020-01-18T01:05:48Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> For any formula $\\phi$ containing no quantifiers over classes, there is a class $A$ such that: :$\\forall x: \\paren {x \\in A \\iff \\map \\phi x}$ </onlyinclude> Class Comprehension Schema bkn58cjsj43c7fdu63cibngfneffbfv"}
+{"_id": "32256", "title": "Bijection between Specific Elements", "text": "Bijection between Specific Elements 0 11017 491812 57625 2020-09-30T21:52:08Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $A$ and $B$ be sets. Let $f: A \\leftrightarrow B$ be a bijection. Then given $a \\in A$ and $b \\in B$, there exists another bijection $g: A \\leftrightarrow B$ such that $\\map g a = b$. == Proof == Let $\\map f a = b'$. Since $f$ is surjective, there is $a' \\in A$ such that $\\map f {a'} = b$. We define the mapping $g: A \\to B$ as: :$\\map g x = \\begin {cases} b & : x = a \\\\ b' & : x = a' \\\\ \\map f x & : x \\in A \\setminus \\set {a, a'} \\end {cases}$ === $g$ is Surjective === Let $y \\in B \\setminus \\set {b, b'}$. Then: :$\\exists x \\in A: \\map f x = y$ and so by definition of $g$: :$\\exists x \\in A: \\map g x = y$ Otherwise: :$y = b \\implies b = \\map g a$ :$y = b' \\implies b = \\map g {a'}$ and so $y$ is seen to be surjective. === $g$ is Injective === Suppose $\\map g x = \\map g {x'}$. If $\\map g x, \\map g {x'} \\in B \\setminus \\set {b, b'}$ then $\\map f x = \\map f {x'}$. As $f$ is bijective and so injective it follows that $x = x'$. If $\\map g x, \\map g {x'} \\in \\set {b, b'}$ then either: :$\\map g x = \\map g {x'} = b$ and so $x = x' = a$ or :$\\map g x = \\map g {x'} = b'$ and so $x = x' = a'$ For the other two cases: :$\\map g x \\in \\set {b, b'}, \\map g {x'} \\notin \\set {b, b'}$ :$\\map g x \\notin \\set {b, b'}, \\map g {x'} \\in \\set {b, b'}$ it is clear that $\\map g x \\ne \\map g {x'}$. So $g$ has been shown to be injective. === $g$ is Bijective === So $g$ is both injective and surjective, and so by definition a bijection. {{qed}} == Comment == What this means is that given any two equivalent sets, it is always possible to find a bijection that connects any given pair of elements, one in one set and one in the other. Category:Bijections 4yekzc58r9rc2mkv5ztzgdnsq73zy6d"}
+{"_id": "32257", "title": "Tietze Extension Theorem/Lemma", "text": "Tietze Extension Theorem/Lemma 0 11212 338392 335952 2018-01-18T20:28:48Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $T = \\left({S, \\tau}\\right)$ be a topological space which is normal. Let $A \\subseteq S$ be closed in $T$. <onlyinclude> Let $f: A \\to \\R$ be a continuous mapping such that $\\left|{f \\left({x}\\right)}\\right| \\le 1$. Then there exists a continuous mapping $g: S \\to \\R$ such that: : $\\forall x \\in S: \\left|{g \\left({x}\\right)}\\right| \\le \\dfrac 1 3$ : $\\forall x \\in A: \\left|{f \\left({x}\\right) − g \\left({x}\\right)}\\right| \\le \\dfrac 2 3$ </onlyinclude> == Proof == The sets $f^{−1} \\left({\\left({−\\infty \\,.\\,.\\, −\\dfrac 1 3}\\right]}\\right)$ and $f^{−1} \\left({\\left[{\\dfrac 1 3 \\,.\\,.\\, \\infty}\\right)}\\right)$ are disjoint and closed in $A$. Since $A$ is closed, they are also closed in $S$. We have that $S$ is normal. Then by: : Urysohn's Lemma and: : the fact that $\\left[{0 \\,.\\,.\\, 1}\\right]$ is homeomorphic to $\\left[{−\\dfrac 1 3 \\,.\\,.\\, \\dfrac 1 3}\\right]$ there exists a continuous mapping $g: S \\to \\left[{−\\dfrac 1 3 \\,.\\,.\\, \\dfrac 1 3}\\right]$ such that: :$g \\left({f^{-1} \\left({ \\left({-\\infty \\,.\\,.\\, -\\dfrac 1 3}\\right]}\\right)}\\right) = -\\dfrac 1 3$ and :$g \\left({f^{-1} \\left({ \\left[{\\dfrac 1 3 \\,.\\,.\\, \\infty}\\right)}\\right)}\\right) = \\dfrac 1 3$ Thus: : $\\forall x \\in S: \\left|{g \\left({x}\\right)}\\right| \\le \\dfrac 1 3$ Now if $−1 \\le f \\left({x}\\right) \\le −\\dfrac 1 3$, then: :$g \\left({x}\\right) = −\\dfrac 1 3$ and thus: :$\\left|{f \\left({x}\\right) − g \\left({x}\\right)}\\right| \\le \\dfrac 2 3$ Similarly if $\\dfrac 1 3 \\le f \\left({x}\\right) \\le 1$, then: :$g \\left({x}\\right) = \\dfrac 1 3$ and thus: :$\\left|{f \\left({x}\\right) − g \\left({x}\\right)}\\right| \\le \\dfrac 2 3$ Finally, for $\\left|{f \\left({x}\\right)}\\right| \\le \\dfrac 1 3$ we have that: :$\\left|{g \\left({x}\\right)}\\right| \\le \\dfrac 1 3$ and so: : $\\left|{f \\left({x}\\right) − g \\left({x}\\right)}\\right| \\le \\dfrac 2 3$ Hence, for all $x \\in S$: : $\\left|{f \\left({x}\\right) − g \\left({x}\\right)}\\right| \\le \\dfrac 2 3$ {{qed}} {{MissingLinks|Some of the statements above need to be justified.}} {{Namedfor|Heinrich Franz Friedrich Tietze}} == Sources == {{Planetmath|id = 5566|title = Tietze extension theorem}} Category:Normal Spaces Category:Continuous Mappings 3fm1tg8pgstmtgqce1rqqi08u7oq6bi"}
+{"_id": "32258", "title": "Finitely Many Reduced Associated Quadratic Irrationals", "text": "Finitely Many Reduced Associated Quadratic Irrationals 0 11491 273258 178390 2016-10-16T10:23:46Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $n \\in \\Z: n > 0$. There are finitely many reduced quadratic irrationals which are associated to $n$. == Proof == <!-- This was taken from Continued Fraction Expansion of Irrational Square Root and may require some tinkering. --> By definition, we can write an arbitrary reduced irrational as $\\alpha = \\dfrac{\\sqrt n + P} Q$. By definition, $\\alpha > 1$ and its conjugate $\\tilde{\\alpha} > -1$. So we know $\\alpha + \\tilde{\\alpha} = \\dfrac {2P} Q > 0$. Hence with the assumption $Q > 0$ we have $P > 0$. Since $\\tilde{\\alpha} < 0$ we also have $P < \\sqrt n$. Also, since $\\alpha > 1$ by assumption, we have $Q < P + \\sqrt n < 2 \\sqrt n$. Thus there are finitely many choices for both $P$ and $Q$, forcing finitely many reduced quadratic irrationals associated to a fixed $n$. This amount is strictly bounded above by $2n$. {{explain|Why the word \"strictly\" is needed here - a quantity is bounded above or it's not. Also, clarify \"amount\".}} {{qed}} Category:Quadratic Irrationals cpzhwfzhio3upcvlwexgn8wi5mko3hk"}
+{"_id": "32259", "title": "Expansion of Associated Reduced Quadratic Irrational", "text": "Expansion of Associated Reduced Quadratic Irrational 0 11492 273250 59859 2016-10-16T09:59:47Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $\\alpha$ be a reduced quadratic irrational which is associated to $n$. Let $\\alpha$ be transformed into its integer part and fractional part via: :$\\alpha = \\left \\lfloor {\\alpha} \\right \\rfloor + \\dfrac 1{\\alpha'}$ Then the resulting quadratic irrational $\\alpha'$ is also reduced and associated to $n$. == Proof == Let $\\alpha = \\dfrac{\\sqrt n + P} Q$ and $X = \\left \\lfloor {\\alpha} \\right \\rfloor$ Then we have: :$\\dfrac 1 {\\alpha'} = \\dfrac {\\sqrt n - \\left({Q X - P}\\right)} Q$ Because $\\sqrt n$ is irrational, we must have: :$\\dfrac 1 {\\alpha'} > 0$ Because $\\dfrac 1 {\\alpha'}$ is the fractional part: :$0 < \\dfrac 1 {\\alpha'} < 1 \\implies \\alpha' > 1$ Transforming: {{begin-eqn}} {{eqn | l = \\alpha'       | r = \\frac Q {\\sqrt n - \\left({Q X - P}\\right)} \\cdot \\frac {\\sqrt n + \\left({Q X - P}\\right)} {\\sqrt n + \\left({Q X - P}\\right)}       | c =  }} {{eqn | r = \\frac {\\sqrt n + \\left({Q X - P}\\right)} {\\frac 1 Q \\left({n - \\left({Q X - P}\\right)^2}\\right)}       | c =  }} {{end-eqn}} we have: :$P' = Q X - P$ and: :$Q' = \\dfrac 1 Q \\left({n - \\left({Q X - P}\\right)^2}\\right)$ To show $Q' \\in \\Z$: We have that: :$n - \\left({Q X - P}\\right)^2 \\equiv n - P^2 \\pmod Q$ Because $\\alpha$ is associated to $n$, $Q$ must divide this quantity. Hence $Q'$ is an integer as defined. Because $X = \\left \\lfloor{\\dfrac{\\sqrt n + P} Q}\\right \\rfloor$ is an integer and $\\alpha$ is irrational: :$X < \\dfrac {\\sqrt n + P} Q$ Hence: :$P' = QX - P < \\sqrt n $ forcing: :$\\tilde{\\alpha}' < 0$ Because $\\alpha > 1$: :$X \\ge 1 \\iff 0 \\le X - 1$ Thus: {{begin-eqn}} {{eqn | l = \\tilde \\alpha = \\frac {P - \\sqrt n} Q       | o = <       | r = 0 \\le X - 1       | c =  }} {{eqn | ll= \\leadsto       | l = Q       | o = <       | r = \\sqrt n + \\left({Q X - P}\\right)       | c =  }} {{eqn | ll= \\leadsto       | l = Q \\left({\\sqrt n - \\left({Q X - P}\\right)}\\right)       | o = <       | r = n - \\left({Q X - P}\\right)^2       | c =  }} {{eqn | ll= \\leadsto       | l = -\\tilde \\alpha' = \\frac {\\sqrt n - \\left({Q X - P}\\right)} {\\frac 1 Q \\left({n - \\left({Q X - P}\\right)^2}\\right)}       | o = <       | r = 1       | c =  }} {{end-eqn}} Hence $\\tilde \\alpha' > -1$ and $\\alpha'$ is reduced. Because $Q' = \\dfrac 1 Q \\left({n - \\left({P'}\\right)^2}\\right)$: :$n - \\left({P'}\\right)^2 \\equiv Q Q' \\equiv 0 \\pmod {Q'}$ Hence $\\alpha'$ is associated to $n$. Thus $\\alpha'$ is both reduced and associated to $n$. {{qed}} Category:Continued Fractions Category:Quadratic Irrationals h0c2qxftxwlwpmwvydczgxr5s5adwr5"}
+{"_id": "32260", "title": "Axiom:Area Axioms", "text": "Axiom:Area Axioms 100 11793 196396 195134 2014-10-12T07:51:29Z Prime.mover 59 wikitext text/x-wiki == Axioms of Area == <onlyinclude> When discussing areas of surfaces, we usually take the following statements to be axiomatic: :$(1) \\quad$ The area of a square with a side of length one unit is defined to be one square unit. :$(2) \\quad$ Let a surface be divided into a finite number of smaller non-overlapping surfaces. ::Let the smaller surfaces cover the entire larger surface. ::Then the sum of the areas of the smaller surfaces equals the area of the larger surface. :$(3) \\quad$ Equal surfaces have equal areas. </onlyinclude> Category:Axioms/Geometry thsl5ryu5w2se2a3funl2fpd44tmf2y"}
+{"_id": "32261", "title": "Gelfond-Schneider Theorem/Lemma 1", "text": "Gelfond-Schneider Theorem/Lemma 1 0 11991 326204 175307 2017-11-11T10:19:20Z Barto 3079 \"work to do\" template wikitext text/x-wiki {{wtd|Links to Polynomial related results need to be resolved.}} == Lemma == <onlyinclude> Let $a_1 \\left({t}\\right), \\ldots, a_n \\left({t}\\right)$ be non-zero polynomials in $\\R \\left[{t}\\right]$ of degrees $d_1, \\ldots, d_n$ respectively. Let $w_1, \\ldots, w_n$ be pairwise distinct real numbers. Then: :$\\displaystyle F \\left({t}\\right) = \\sum_{j \\mathop = 1}^n a_j \\left({t}\\right) e^{w_j t}$ has at most $d_1 + \\cdots + d_n + n − 1$ real roots (counting multiplicities). </onlyinclude> == Proof == By multiplying through by $e^{−w_n t}$ if necessary, we may suppose that $w_n = 0$ and that otherwise $w_j \\ne 0$. Let $k = d_1 + \\cdots + d_n + n$. We use strong induction on $k$. If $k = 1$, then $n = 1$ and $d_1 = 0$, and the lemma easily follows. Let $l \\ge 2$ be such that the lemma holds whenever $k < l$. Suppose $k = l$. Let $N$ be the number of real roots of $F \\left({t}\\right)$. By Rolle's Theorem, the number of real roots of $F' \\left({t}\\right)$ is at least $N − 1$. On the other hand: :$\\displaystyle F' \\left({t}\\right) = \\sum_{j=1}^n b_j \\left({t}\\right) e^{w_j t}$ where: :$b_j \\left({t}\\right) = a'_j \\left({t}\\right) + w_j a_j \\left({t}\\right)$ Note that for $1 \\le j \\le n − 1$, we have that $b_j \\left({t}\\right)$ has degree exactly $d_j$. Also, since $w_n = 0$, either: : there are only $n−1$ non-zero polynomials $b_j \\left({t}\\right)$ in the expression for $F' \\left({t}\\right)$ above or: : there are $n$ such polynomials and the degree of $b_n \\left({t}\\right)$ is one less than the degree of $a_n \\left({t}\\right)$. We get from the induction hypothesis that $F' \\left({t}\\right)$ has at most $d_1 + \\cdots + d_n + n − 2$ real roots. Hence, $N − 1 \\le d_1 + \\cdots + d_n + n − 2$, and the result follows. {{qed|lemma}} Category:Transcendental Number Theory jxhani6jbvr3c03ovy9x4lyig5il3x4"}
+{"_id": "32262", "title": "Gelfond-Schneider Theorem/Lemma 2", "text": "Gelfond-Schneider Theorem/Lemma 2 0 11999 371662 371651 2018-10-17T20:15:31Z Prime.mover 59 wikitext text/x-wiki {{MissingLinks}} == Lemma == <onlyinclude> Let $\\map f z$ be an analytic function in the disk $D \\subseteq \\C: D = \\set {z : \\size z < R}$ for some real number $R > 0$. Let $f$ also be continuous on the closure of $D$, that is, on $D^- = \\set {z : \\size z \\le R}$. Then: :$\\forall z \\in D^-: \\size {\\map f z} \\le \\size f_R$, where we set $\\size f_R = \\max_{\\size z = R} \\size {\\map f z}$. </onlyinclude> This lemma is essentially a restatement of the maximum modulus principle for analytic functions. == Proof == Since $D^-$ is compact, the continuous function $\\size f$ has a maximum point $z_0 \\in D^-$, meaning that $\\size {\\map f {z_0} } = \\max_{D^-} \\size f$. If $z_0$ belongs to the open disk $D$, then the maximum modulus principle implies that $f$ is constant, in which case the statement is trivial. If instead $z_0$ belongs to the boundary $D^- \\setminus D$, then $\\size {z_0} = R$ and so $\\size {\\map f {z_0} } \\le \\size f_R$. Since $\\size {\\map f z} \\le \\size {\\map f {z_0} }$ for any $z \\in D^-$, the statement follows. {{qed|lemma}} Category:Transcendental Number Theory 6201s4jahub0y0hw17zwn71uddz54op"}
+{"_id": "32263", "title": "Gelfond-Schneider Theorem/Lemma 3", "text": "Gelfond-Schneider Theorem/Lemma 3 0 12001 371650 350834 2018-10-17T15:24:47Z Kappaenne 3501 wikitext text/x-wiki == Lemma == <onlyinclude> Let $r$ and $R$ be two real numbers such that $0 < r \\le R$. Let $f_1 \\left({z}\\right), f_2 \\left({z}\\right), \\ldots, f_L \\left({z}\\right)$ be: : analytic in $D \\subseteq \\C: D = \\left\\{{z : \\left|{z}\\right| < R}\\right\\}$ : continuous on the closure $D$, that is, $D^- = \\left\\{{z : \\left \\vert{z}\\right \\vert \\le R}\\right\\}$. Let $\\zeta_1, \\ldots, \\zeta_L$ be complex numbers such that: :$\\forall j \\in \\left\\{{1, 2, \\ldots, L}\\right\\}: \\left|{\\zeta_j}\\right| \\le r$ Then the determinant: :$\\Delta = \\det \\begin{bmatrix} f_1 \\left({\\zeta_1}\\right) & \\cdots & f_L \\left({\\zeta_1}\\right) \\\\ \\vdots & \\ddots & \\vdots \\\\ f_1 \\left({\\zeta_L}\\right) & \\cdots & f_L \\left({\\zeta_L}\\right) \\end{bmatrix}$ satisfies: :$\\displaystyle \\left|{\\Delta}\\right| \\le \\left({\\frac R r}\\right)^{−L \\left({L−1}\\right) / 2} L! \\prod_{\\lambda \\mathop = 1}^L \\left|{f_λ}\\right|_R$ </onlyinclude> == Proof == Let $h \\left({z}\\right)$ be the determinant of the $L \\times L$ matrix $\\left[{f_j \\left({\\zeta_i z}\\right)}\\right]$. Then $h \\left({z}\\right)$ is: : analytic in $D' = \\left\\{{z : \\left \\vert {z}\\right \\vert < R/r}\\right\\}$ : continuous on $D'^- = \\left\\{{z : \\left \\vert{z}\\right \\vert \\le R/r}\\right\\}$. Let $M = L \\left({L − 1}\\right) / 2$, and write: :$\\displaystyle f_j \\left({\\zeta_i z}\\right) = \\sum_{k \\mathop = 0}^{M−1} b_k \\left({j}\\right) \\zeta_i^k z^k + z^M g_{i,j} \\left({z}\\right)$ where: :$b_k \\left({j}\\right) \\in \\C$ for each $k$ :$g_{i,j} \\left({z}\\right)$ is analytic in $D'$ :$g_{i,j} \\left({z}\\right)$ is  continuous on $D'^-$ By evaluating along the columns it is seen that the determinant is linear in its columns. So we can view $h \\left({z}\\right)$ as $z^M$ times an analytic function plus terms involving the factor: :$z^{n_1 + \\cdots + n_L} \\det \\left[{\\zeta_i^{n_j} }\\right]$ where the $n_j$ denote non-negative integers. Observe that the determinant in this last expression is zero if the $n_j$ are not distinct. Therefore, the non-zero terms of this form satisfy: :$n_1 + n_2 + \\cdots + n_L \\ge 0 + 1 + \\cdots + \\left({L − 1}\\right) = \\dfrac {L \\left({L − 1}\\right)} 2$ Hence, we deduce that $h \\left({z}\\right)$ is divisible by $z^M$. More precisely, $\\dfrac {h \\left({z}\\right)} {z^M}$ is analytic in $D'$ and continuous on $D'^-$. Therefore, by Gelfond-Schneider Theorem: Lemma 2, for any $w \\in D'$: :$\\displaystyle \\left|{\\frac {h \\left({w}\\right)} {w^M}}\\right| \\le \\left|{\\frac {h \\left({z}\\right)} {z^M}}\\right|_{R/r} = \\left({\\frac r R}\\right)^M \\left|{h \\left({z}\\right)}\\right|_{R/r}$ For $\\left|{z}\\right| = R/r$, we get that $\\left|{\\zeta_i z}\\right| \\le R$. We create a bound for $\\left|{h \\left({z}\\right)}\\right|_{R/r}$ by multiplying the number of terms in $\\det \\left[{f_j \\left({\\zeta_i z}\\right)}\\right]$ by an obvious upper bound on the maximum such term. Thus: :$\\displaystyle \\left|{h \\left({z}\\right)}\\right|_{R/r} \\le L! \\prod_{\\lambda \\mathop = 1}^L \\left|{f_λ}\\right|_R$ Observe that $\\left|{\\Delta}\\right| = \\left|{h \\left({1}\\right)}\\right|$ and $1 \\le R/r$. We deduce that :$\\displaystyle \\left|{\\Delta}\\right| \\le \\left({\\frac r R}\\right)^M \\left|{h \\left({z}\\right)}\\right|_{R/r} \\le \\left({\\frac r R}\\right)^M L! \\prod_{\\lambda \\mathop = 1}^L \\left|{f_λ}\\right|_R$ giving the desired conclusion. {{qed|lemma}} Category:Transcendental Number Theory 173dtdvf0myyodnpyklcyk1gss1eemh"}
+{"_id": "32264", "title": "Gelfond-Schneider Theorem/Lemma 4", "text": "Gelfond-Schneider Theorem/Lemma 4 0 12002 490684 349511 2020-09-25T15:47:45Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let: :$\\Delta = \\det \\sqbrk {\\alpha_{i, j} }_{L \\times L}$ where the $\\alpha_{i, j}$ are algebraic numbers. Suppose that $T$ is a positive (rational) integer for which $T \\alpha_{i, j}$ is an algebraic integer for every $i, j \\in \\set {1, 2, \\ldots, L}$. Also, suppose that $\\Delta \\ne 0$. Then there is a conjugate of $\\Delta$ with absolute value $\\ge T^{−L}$. </onlyinclude> == Proof == Observe that $T^L \\Delta$ is an algebraic integer so that one of its conjugates has absolute value $\\ge 1$. The result follows. {{qed}} {{Proofread|There are plenty of definitions on this page which we may not have completely accurate.}} Category:Transcendental Number Theory mjaqjter8d19jybn9fgakn1m2tff5bt"}
+{"_id": "32265", "title": "Frobenius's Theorem/Lemma 1", "text": "Frobenius's Theorem/Lemma 1 0 12012 489385 146286 2020-09-20T10:35:06Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $\\struct {A, \\oplus}$ be a quadratic real algebra. Then: :$(1): \\quad U = \\set {u \\in A \\setminus \\R: u^2 \\in \\R} \\cup \\set 0$ is a linear subspace of $A$ :$(2): \\quad \\forall u, v \\in U: u v + v u \\in \\R$ :$(3): \\quad A = \\R \\oplus U$ :$(4): \\quad$ If $A$ is also a division algebra, then every nonzero $u \\in U$ can be written as $u = \\alpha v$ with $\\alpha \\in \\R$ and $v^2 = -1$. </onlyinclude> == Proof == === Proof of First Assertion === $U$ is closed under scalar multiplication. We have to show that $u, v \\in U$ implies $u + v \\in U$. Let $\\set {u, v, 1}$ be a linearly dependent set. Then there exist constants $a, b \\in \\R$ such that $u = av + b$. Then $u^2 = a^2 v^2 + 2 a b v + b^2$. Since $u^2$, $a^2 v^2$, and $b^2$ are all real, it follows that $2 a b v \\in \\R$, that is, $v \\in \\R$. Since $0$ is the only real element of $U$, it follows that $v = 0$. Reversing $u$ and $v$ in the preceding argument shows that also $u = 0$, so that $u + v = 0 \\in U$. Now let $\\set {u, v, 1}$ be a linearly independent set. We have: :$\\paren {u + v}^2 + \\paren {u - v}^2 = 2 u^2 + 2 v^2 \\in \\R$ On the other hand, as $A$ is quadratic there exist $\\lambda, \\mu \\in \\R$ such that: :$\\paren {u + v}^2 - \\lambda \\paren {u + v} \\in \\R$ :$\\paren {u - v}^2 - \\mu \\paren {u - v} \\in R$ Hence: :$\\lambda \\paren {u + v} + \\mu \\paren {u - v} \\in \\R$ However, we have that $\\set {u, v, 1}$ is linearly independent. Therefore $\\lambda + \\mu = \\lambda - \\mu = 0$, and so $\\lambda = \\mu = 0$. This proves that $u \\pm v \\in U$. Thus $U$ is indeed a subspace of $A$. {{qed|lemma}} === Proof of Second Assertion === Accordingly, from the result of $(1)$: :$\\forall u, v \\in U: u v + v u = \\paren {u + v}^2 - u^2 - v^2 \\in \\R$. {{qed|lemma}} === Proof of Third Assertion === Let $a \\in A \\setminus \\R$. Then: :$\\exists \\nu \\in \\R: a^2 - \\nu a \\in \\R$ Therefore, if we set :$u = a - \\dfrac \\nu 2 \\in U$ then $u^2 = a^2 - \\nu a + \\nu^2/4 \\in \\R$, so :$a = \\dfrac \\nu 2 + u \\in \\R \\oplus U$ which proves the assertion. {{qed|lemma}} === Proof of Fourth Assertion === This follows directly from the definition of division algebra. Also, we have that $u^2 \\in \\R$. Suppose $u^2 = 0$. Since $u$ is nonzero and since $A$ is a division algebra, there exists an inverse $u^{-1}$ such that $u u^{-1} = 1$. But then $u = u 1 = u u u^{-1} = 0 \\cdot u^{-1} = 0$, which cannot be since $u$ was assumed to be nonzero. Now suppose $u^2 > 0$. Then there exists an $\\alpha \\in \\R$ such that $\\alpha^2 = u^2$. But then $\\paren {u - \\alpha} \\paren {u + \\alpha} = u^2 - \\alpha^2$ would be $0$. Since $A$ is assumed to be a division algebra, this would imply that either $u = \\alpha$ or $u = -\\alpha$, which is impossible since $u \\in U$. So $u^2  < 0$ and so $u^2 = -\\alpha^2$ with $0 \\ne \\alpha \\in \\R$. Thus, $v = \\alpha^{-1} u$ is a desired element. {{qed}} {{MissingLinks}} Category:Abstract Algebra f6o1q5guzy610v4t64l15321xx6akoh"}
+{"_id": "32266", "title": "Frobenius's Theorem/Lemma 2", "text": "Frobenius's Theorem/Lemma 2 0 12013 146288 79502 2013-05-06T11:49:08Z Nearyan 2167 /* Proof */ wikitext text/x-wiki == Lemma == <onlyinclude> Let $A$ be a quadratic real division algebra. Let: :$U = \\left\\{{u \\in A \\setminus \\R: u^2 \\in \\R}\\right\\} \\cup \\left\\{{0}\\right\\}$ where $\\setminus$ denotes set difference. Suppose $e_1, \\ldots, e_k \\in U$ are such that: : $\\forall i \\le k: e_i^2 = -1$ : $\\forall i, j \\le k, i \\ne j: e_i e_j = -e_j e_i$ If $U$ is not equal to the linear span of $e_1, \\ldots, e_k$, then there exists $e_{k+1} \\in U$ such that: : $e_{k+1}^2 = -1$ : $\\forall i \\le k: e_i e_{k+1} = -e_{k+1} e_i$ </onlyinclude> == Proof == From Lemma 1, $U$ is a linear subspace of $A$ such that: :$\\forall u, v \\in U: u v + v u \\in \\R$ :$A = \\R \\oplus U$ Pick $u \\in U$ that is not contained in the linear span of $e_1, \\ldots, e_k$. By Lemma 1, we can set $\\alpha_i = \\dfrac 1 2 \\left({u e_i + e_i u}\\right) \\in \\R$. Note that $v = u + \\alpha_1 e_1 + \\cdots + \\alpha_k e_k$ satisfies $e_i v = v e_i$ for all $i \\le k$. Let $e_{k+1}$ be a scalar multiple of $v$ such that $e_{k+1}^2 = -1$ (whose existence follows from Lemma 1: Assertion 4). Then $e_{k+1}$ has all desired properties. {{qed|lemma}} Category:Abstract Algebra 94euoen9b0d870ivfbxtg45pqb9x1y4"}
+{"_id": "32267", "title": "Singleton Equality", "text": "Singleton Equality 0 12160 444460 444459 2020-01-22T06:14:56Z Prime.mover 59 wikitext text/x-wiki {{refactor|Two results here.}} == Theorems == Let $x$ and $y$ be sets. Then: :$\\set x \\subseteq \\set y \\iff x = y$ :$\\set x = \\set y \\iff x = y$ == Proof == {{begin-eqn}} {{eqn | o =        | rr= \\set x \\subseteq \\set y       | c =  }} {{eqn | o = \\leadstoandfrom       | r = \\forall z:       | rr= \\paren {z \\in \\set x \\implies z \\in \\set y}       | c = {{Defof|Subset}} }} {{eqn | o = \\leadstoandfrom       | r = \\forall z:       | rr= \\paren {z = x \\implies z = y}       | c = {{Defof|Singleton}} }} {{eqn | o = \\leadstoandfrom       | rr= x = y       | c = Equality implies Substitution }} {{end-eqn}} {{qed|lemma}} Then: {{begin-eqn}} {{eqn | l = x       | r = y       | c =  }} {{eqn | ll= \\leadsto       | l = \\set x       | r = \\set y       | c = Substitutivity of Equality }} {{eqn | ll= \\leadsto       | l = \\set x       | o = \\subseteq       | r = \\set y       | c =  }} {{eqn | ll= \\leadsto       | l = x       | r = y       | c = by the first part }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Set Theory and Its Logic|1963|Willard Van Orman Quine}}: $\\S 7.7$ Category:Set Theory Category:Subsets Category:Singletons 2wb5g0fjhg6xwbwjpofkvpe1yurrsse"}
+{"_id": "32268", "title": "Cardinals are Totally Ordered", "text": "Cardinals are Totally Ordered 0 12226 178891 123437 2014-03-07T07:11:21Z Prime.mover 59 wikitext text/x-wiki == Corollary to Zermelo's Theorem == Every set of cardinals is totally ordered under $\\le$. == Proof == Let $S$ be a set of cardinals. From Zermelo's theorem, $S$ is well-ordered. Let $a, b \\in S$. Consider the subset $X = \\left\\{{a, b}\\right\\}$ of $S$. Since $S$ is well-ordered, $\\inf \\left({X}\\right)$ exists and belongs to $X$. So either $\\inf \\left({X}\\right) = a$ or $\\inf \\left({X}\\right) = b$. By definition of infimum, we have that $\\inf \\left({X}\\right) \\le a$ and $\\inf \\left({X}\\right) \\le b$. It follows that either $a \\le b$ or $b \\le a$. So, by definition, $S$ is totally ordered under $\\le$. {{qed}} {{AoC|Zermelo's Theorem (Set Theory)}} == Sources == * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth|prev=Zermelo's Theorem (Set Theory)|next=Definition:Zero (Cardinal)}}: $\\S 8$: Theorem $8.3$: Corollary Category:Cardinals Category:Total Orderings tld7afth5rd0ezdwoarguq1q910izr8"}
+{"_id": "32269", "title": "Idempotent Operators", "text": "Idempotent Operators 0 12233 375015 374551 2018-11-06T09:31:44Z Leigh.Samphier 3031 wikitext text/x-wiki {{rename|something more descriptive}} Let $H$ be a Hilbert space $H$. A linear operator $P: H \\to H$ is called '''idempotent''' if :$P^2 = P$ or equivalently :$P x = x$ for $x \\in \\Rng P$. An idempotent operator is called a '''projector''' or '''orthogonal projector''' if :$\\forall x \\in H: P x - x \\perp \\Rng P$. Especially in the context of linear algebra, many texts refer to ''all'' idempotent operators as \"projectors\" and use the same definition as above only for \"orthogonal projectors.\" In such texts, idempotent operators that are ''not'' orthogonal projectors may be called '''oblique projectors'''. Orthogonal projectors are extremely important in applied linear algebra and spectral theory. They can be characterized in several additional ways: * An idempotent operator is an orthogonal projector {{iff}} it is self-adjoint. * An idempotent operator is an orthogonal projector {{iff}} its norm is 1 (proof). {{refactor|The following to be done:<br/>1: Idempotent needs to be given its own definition page in this context, as do Projector and Orthogonal Projector<br/>2: The \"immediate consequence\" above needs to be given its own proof page<br>3: A definition needs to be raised for \"linear idempotent operator\" unless its meaning is as straightforward as being a \"linear operator\" which is also an \"idempotent operator\" - and even then it might be worth giving it a page if the concept occurs frequently.<br/>It is also worth reviewing the meaning of $\\Rng P$ in this context, as \"range\" is ambiguously understood.}} {{MissingLinks}} Category:Hilbert Spaces lk84c9j9tuw0l9gubhntjfzjm8dh55d"}
+{"_id": "32270", "title": "Squeeze Theorem for Real Sequences/Corollary", "text": "Squeeze Theorem for Real Sequences/Corollary 0 12410 408180 408168 2019-06-15T13:22:04Z Prime.mover 59 wikitext text/x-wiki == Corollary to Squeeze Theorem for Real Sequences == <onlyinclude> Let $\\sequence {y_n}$ be a sequence in $\\R$ which is null, that is: :$y_n \\to 0$ as $n \\to \\infty$ Let: :$\\forall n \\in \\N: \\size {x_n - l} \\le y_n$ Then $x_n \\to l$ as $n \\to \\infty$. </onlyinclude> == Proof == From Negative of Absolute Value: Corollary 2: :$\\size {x_n - l} \\le y_n \\iff l - y_n \\le x_n \\le l + y_n$ From the Difference Rule for Real Sequences: :$l - y_n \\to l$ as $n \\to \\infty$ and from the Sum Rule for Real Sequences: :$l + y_n \\to l$ as $n \\to \\infty$ So by the Squeeze Theorem for Real Sequences, $x_n \\to l$ as $n \\to \\infty$. {{qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Squeeze Theorem for Real Sequences|next = Sequence of Powers of Number less than One/Necessary Condition/Proof 1}}: $\\S 4$: Convergent Sequences: $\\S 4.11$: Corollary Category:Squeeze Theorem nsvfni1ehlrolqwsh5fywnkicxecjfk"}
+{"_id": "32271", "title": "Taylor's Theorem/One Variable", "text": "Taylor's Theorem/One Variable 0 12521 484044 446910 2020-08-30T05:16:42Z Prime.mover 59 wikitext text/x-wiki == Theorem == <onlyinclude> {{:Taylor's Theorem/One Variable/Statement of Theorem}} </onlyinclude> == Proof == === Integral Version === {{:Taylor's Theorem/One Variable/Integral Version}} === Proof using Cauchy Mean Value Theorem === An alternative proof, which holds under milder technical assumptions on the function $f$, can be supplied using the Cauchy Mean Value Theorem. {{:Taylor's Theorem/One Variable/Proof by Cauchy Mean Value Theorem}} === Proof using Rolle's Theorem directly === Yet another proof  for Lagrange Form of the Remainder can be constructed applying Rolle's theorem directly $n$ times; this proof might be easier to visualize geometrically. {{:Taylor's Theorem/One Variable/Proof by Rolle's Theorem}} == Also see == * Mean Value Theorem: Taylor's Theorem when $n = 0$ {{Namedfor|Brook Taylor}} == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Mathematician:Brook Taylor|next = Definition:Taylor Series|entry = Taylor's theorem}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Mathematician:Brook Taylor|next = Definition:Taylor Series|entry = Taylor's theorem}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Taylor Series|next = Mathematician:Pafnuty Lvovich Chebyshev|entry = Taylor's Theorem}} Category:Taylor's Theorem oys9195a6vm91w58n0uh1qgkx8y1of4"}
+{"_id": "32272", "title": "Derivative of Exponential Function/Corollary 1", "text": "Derivative of Exponential Function/Corollary 1 0 12582 490881 485755 2020-09-26T09:47:50Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Exponential Function == Let $\\exp x$ be the exponential function. <onlyinclude> Let $c \\in \\R$. Then: :$\\map {\\dfrac \\d {\\d x} } {\\map \\exp {c x} } = c \\map \\exp {c x}$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\map {\\dfrac \\d {\\d x} } {\\map \\exp {c x} }       | r = c \\map {\\dfrac \\d {\\map \\d {c x} } } {\\map \\exp {c x} }       | c = Derivative of Function of Constant Multiple }} {{eqn | r = c \\map \\exp {c x}       | c = Derivative of Exponential Function }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Integration|1944|R.P. Gillespie|ed = 2nd|edpage = Second Edition|prev = Primitive of Reciprocal|next = Primitive of Exponential of a x}}: Chapter $\\text {II}$: Integration of Elementary Functions: $\\S 7$. Standard Integrals: $3$. Category:Derivatives involving Exponential Function 84gqrvoa9b6562l37zobp4zp7jf6n4m"}
+{"_id": "32273", "title": "Derivative of General Exponential Function", "text": "Derivative of General Exponential Function 0 12583 485952 485756 2020-09-06T21:07:05Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Exponential Function == <onlyinclude> Let $a \\in \\R: a > 0$. Let $a^x$ be $a$ to the power of $x$. Then: :$\\map {\\dfrac \\d {\\d x} } {a^x} = a^x \\ln a$ </onlyinclude> == Proof 1 == {{:Derivative of General Exponential Function/Proof 1}} == Proof 2 == {{:Derivative of General Exponential Function/Proof 2}} Category:Derivatives involving Exponential Function pox8ott2n50d2p1ayac7snucrugq5xc"}
+{"_id": "32274", "title": "Quaternion Group/Complex Matrices", "text": "Quaternion Group/Complex Matrices 0 12762 379946 379139 2018-12-03T18:26:28Z Prime.mover 59 wikitext text/x-wiki == Representation of Quaternion Group == <onlyinclude> Let $\\mathbf 1, \\mathbf i, \\mathbf j, \\mathbf k$ denote the following four elements of the matrix space $\\map {\\mathcal M_\\C} 2$: :$\\mathbf 1 = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} \\qquad \\mathbf i = \\begin{bmatrix} i & 0 \\\\ 0 & -i \\end{bmatrix} \\qquad \\mathbf j = \\begin{bmatrix} 0 & 1 \\\\ -1 & 0 \\end{bmatrix} \\qquad \\mathbf k = \\begin{bmatrix} 0 & i \\\\ i & 0 \\end{bmatrix}$ where $\\C$ is the set of complex numbers. The set: :$\\Dic 2 = \\set {\\mathbf 1, -\\mathbf 1, \\mathbf i, -\\mathbf i, \\mathbf j, -\\mathbf j, \\mathbf k, -\\mathbf k}$ under the operation of conventional matrix multiplication, forms the '''quaternion group''': </onlyinclude> === Cayley Table === Its Cayley table is given by: {{:Quaternion Group/Complex Matrices/Cayley Table}} == Also see == * Quaternions Defined by Matrices where it is shown that these have the appropriate properties. In Matrix Form of Quaternion it is shown that a general element $\\mathbf x$ of $\\mathbb H$ has the form: :$\\mathbf x = \\begin{bmatrix} a + b i & c + d i \\\\ -c + d i & a - bi \\end{bmatrix}$ == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Group of Gaussian Integer Units/Cayley Table|next = Quaternions Defined by Matrices}}: $\\S 34$. Examples of groups: $(6) \\ \\text{(ii)}$ Category:Quaternion Group raxbuehbo9ts4rsu3mbper98sgofdv0"}
+{"_id": "32275", "title": "Axiom:Leibniz's Law", "text": "Axiom:Leibniz's Law 100 12959 425237 425236 2019-09-14T13:23:03Z Prime.mover 59 wikitext text/x-wiki == Axiom == Let $=$ represent the relation of equality and let $P$ be an arbitrary property. Then: :$x = y \\dashv \\vdash \\map P x \\iff \\map P y$ for all $P$ in the universe of discourse. That is, two objects $x$ and $y$ are equal {{iff}} $x$ has every property $y$ has, and $y$ has every property $x$ has. === Application to Equality of Sets === {{:Leibniz's Law for Sets}} == Also see == * Axiom:Axioms of Equality * Definition:Equals {{NamedforAxiom|Gottfried Wilhelm von Leibniz|cat = Leibniz}} == Historical Note == {{:Axiom:Leibniz's Law/Historical Note}} == Sources == * {{BookReference|Introduction to Logic and to the Methodology of Deductive Sciences|1946|Alfred Tarski|ed = 2nd|edpage = Second Edition}}: $\\S 3.17$ * {{BookReference|A Handbook of Terms used in Algebra and Analysis|1972|A.G. Howson|prev = Equality is Transitive|next = Definition:Proof System}}: $\\S 1$: Some mathematical language: Equality: $\\text{(d)}$ * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Leibniz's Alternating Series Test|next = Leibniz's Law for Sets|entry = Leibniz's law|index = 1}} Category:Axioms/Logic tlr9rhzko3h8021an1sh8r376yx3qpz"}
+{"_id": "32276", "title": "Axiom:Axiom of Dependent Choice", "text": "Axiom:Axiom of Dependent Choice 100 13124 453399 444071 2020-03-08T13:00:27Z Prime.mover 59 wikitext text/x-wiki == Axiom == === Left-Total Form === {{:Axiom:Axiom of Dependent Choice/Left-Total}} === Right-Total Form === {{:Axiom:Axiom of Dependent Choice/Right-Total}} == Also known as == Some sources call this the '''Axiom of Dependent Choices''', reflecting the infinitely many choices made. This axiom can be abbreviated '''ADC''' or simply '''DC'''. == Also see == * This axiom is a weaker form of the axiom of choice, as shown in Axiom of Choice Implies Axiom of Dependent Choice. * This axiom is also a stronger form of the axiom of countable choice, as shown in Axiom of Dependent Choice Implies Axiom of Countable Choice. * Dependent Choice (Fixed First Element) shows that it is possible to choose any element of the set to be the first element of the sequence. Dependent Choice Dependent Choice 007arhkjbrui5rbabctehqgx8t5u1m0"}
+{"_id": "32277", "title": "Cartesian Product of Intersections/Corollary 1", "text": "Cartesian Product of Intersections/Corollary 1 0 13257 417118 375824 2019-08-07T20:31:54Z Prime.mover 59 wikitext text/x-wiki == Corollary to Cartesian Product of Intersections == <onlyinclude> :$A \\times \\paren {B \\cap C} = \\paren {A \\times B} \\cap \\paren {A \\times C}$ </onlyinclude> == Proof == Take the result Cartesian Product of Intersections: :$\\paren {S_1 \\cap S_2} \\times \\paren {T_1 \\cap T_2} = \\paren {S_1 \\times T_1} \\cap \\paren {S_2 \\times T_2}$ Put $S_1 = S_2 = A, T_1 = B, T_2 = C$: {{begin-eqn}} {{eqn | l = A \\times \\paren {B \\cap C}       | r = \\paren {A \\cap A} \\times \\paren {B \\cap C}       | c = Intersection is Idempotent }} {{eqn | r = \\paren {A \\times B} \\cap \\paren {A \\times C}       | c = Cartesian Product of Intersections }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Intersection Distributes over Symmetric Difference|next = Cartesian Product of Unions/Corollary}}: Chapter $1$. Sets: Exercise $8 \\ \\text{(i)}$ * {{BookReference|Undergraduate Topology|1971|Robert H. Kasriel|prev = Cartesian Product is Anticommutative|next = Cartesian Product of Subsets/Corollary 1}}: $\\S 1.9$: Cartesian Product: Theorem $9.2$ * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth|prev = Definition:Unbounded Open Real Interval|next = Cartesian Product Distributes over Union}}: $\\S 3$. Ordered pairs; cartesian product sets: Exercise $3 \\ (1)$ Category:Cartesian Product of Intersections ky822jywuu7e08ht1awdjabx97s3xsd"}
+{"_id": "32278", "title": "Image of Set Difference under Relation/Corollary 1", "text": "Image of Set Difference under Relation/Corollary 1 0 13584 495690 226802 2020-10-21T07:52:33Z Prime.mover 59 wikitext text/x-wiki == Corollary to Image of Set Difference under Relation == <onlyinclude> Let $\\RR \\subseteq S \\times T$ be a relation. Let $A \\subseteq B \\subseteq S$. Then: :$\\relcomp {\\RR \\sqbrk B} {\\RR \\sqbrk A} \\subseteq \\RR \\sqbrk {\\relcomp B A}$ where: :$\\RR \\sqbrk B$ denotes the image of $B$ under $\\RR$ :$\\complement$ (in this context) denotes relative complement. </onlyinclude> == Proof == We have that $A \\subseteq B$. Then by definition of relative complement: :$\\relcomp B A = B \\setminus A$ :$\\relcomp {\\RR \\sqbrk B} {\\RR \\sqbrk A} = \\RR \\sqbrk B \\setminus \\RR \\sqbrk A$ Hence, when $A \\subseteq B$: :$\\relcomp {\\RR \\sqbrk B} {\\RR \\sqbrk A} \\subseteq \\RR \\sqbrk {\\relcomp B A}$ means exactly the same thing as: :$\\RR \\sqbrk B \\setminus \\RR \\sqbrk A \\subseteq \\RR \\sqbrk {B \\setminus A}$ {{qed}} Category:Image of Set Difference under Relation 5058rrwlmkf3j5x4um5uteu6svor7p0"}
+{"_id": "32279", "title": "Image of Set Difference under Relation/Corollary 2", "text": "Image of Set Difference under Relation/Corollary 2 0 13585 495692 495691 2020-10-21T07:54:37Z Prime.mover 59 wikitext text/x-wiki == Corollary to Image of Set Difference under Relation == <onlyinclude> Let $\\RR \\subseteq S \\times T$ be a relation. Let $A$ be a subset of $S$. Then: :$\\relcomp {\\Img \\RR} {\\RR \\sqbrk A} \\subseteq \\RR \\sqbrk {\\relcomp S A}$ where: :$\\Img \\RR$ denotes the image of $\\RR$ :$\\RR \\sqbrk A$ denotes the image of $A$ under $\\RR$. </onlyinclude> == Proof == By definition of the image of $\\RR$: :$\\Img \\RR = \\RR \\sqbrk S$ So, when $B = S$ in Image of Set Difference under Relation: Corollary 1: :$\\relcomp {\\Img \\RR} {\\RR \\sqbrk A} = \\relcomp {\\RR \\sqbrk S} {\\RR \\sqbrk A}$ Hence: :$\\relcomp {\\Img \\RR} {\\RR \\sqbrk A} \\subseteq \\RR \\sqbrk {\\relcomp S A}$ means exactly the same thing as: :$\\relcomp {\\RR \\sqbrk S} {\\RR \\sqbrk A} \\subseteq \\RR \\sqbrk {\\relcomp S A}$ that is: :$\\RR \\sqbrk S \\setminus \\RR \\sqbrk A \\subseteq \\RR \\sqbrk {S \\setminus A}$ {{Qed}} Category:Image of Set Difference under Relation t36jcol89y1tjockt9oh052ij7wfcrh"}
+{"_id": "32280", "title": "Zassenhaus Lemma", "text": "Zassenhaus Lemma 0 13788 404925 360091 2019-05-15T21:41:09Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $G$ be a group. Let $H_1$ and $H_2$ be subgroups of $G$. Let: :$N_1 \\lhd H_1$ :$N_2 \\lhd H_2$ where $\\lhd$ denotes the relation of being a normal subgroup. Then: :$\\dfrac {N_1 \\paren {H_1 \\cap H_2} } {N_1 \\paren {H_1 \\cap N_2} } \\cong \\dfrac {H_1 \\cap H_2} {\\paren {H_1 \\cap N_2} \\paren {N_1 \\cap H_2} } \\cong \\dfrac {N_2 \\paren {H_1 \\cap H_2} } {N_2 \\paren {N_1 \\cap H_2} }$ where: :$N_1 \\paren {H_1 \\cap H_2}$, and so on, denotes subset product :$\\cong$ denotes group isomorphism. == Proof == Because of symmetry, only the first of the isomorphisms needs to be proved. === Proof of Normality === In order for the expressions to make sense, the expressions on the bottom of the fractions need to be normal subgroups. That is, we need to show that: :$(1): \\quad N_1 \\paren {H_1 \\cap N_2} \\lhd N_1 \\paren {H_1 \\cap H_2}$ :$(2): \\quad \\paren {H_1 \\cap N_2} \\paren {N_1 \\cap H_2} \\lhd H_1 \\cap H_2$ {{finish}} === Proof of Isomorphism === Let: :$H = H_1 \\cap H_2$ :$N = N_1 \\paren {H_1 \\cap N_2}$ Then: {{begin-eqn}} {{eqn | l = \\frac {N_1 \\paren {H_1 \\cap N_2} \\paren {H_1 \\cap H_2} } {N_1 \\paren {H_1 \\cap N_2} }       | r = \\frac {N H} N       | c =  }} {{eqn | o = \\cong       | r = \\frac H {H \\cap N}       | c = Second Isomorphism Theorem }} {{eqn | r = \\frac {H_1 \\cap H_2} {H_1 \\cap H_2 \\cap N_1 \\paren {H_1 \\cap N_2} }       | c =  }} {{end-eqn}} From Intersection with Subgroup Product of Superset: :If $X, Y, Z$ are subgroups of a group $\\struct {G, \\circ}$, and $Y \\subseteq X$, then: ::$X \\cap \\paren {Y \\circ Z} = Y \\circ \\paren {X \\cap Z}$ Thus: {{begin-eqn}} {{eqn | l = N_1 \\paren {H_1 \\cap N_2} \\paren {H_1 \\cap H_2}       | r = N_1 \\paren {H_1 \\cap H_2}       | c = by taking $X = H_1, Y = H_1 \\cap N_2, Z = H_2$ }} {{eqn | l = H_1 \\cap H_2 \\cap N_1 \\paren {H_1 \\cap N_2}       | r = \\paren {H_1 \\cap N_2} \\paren {N_1 \\cap H_2}       | c = by taking $X = H_1 \\cap H_2, Y = H_1 \\cap N_2, Z = N_1$ }} {{end-eqn}} Hence the result. {{qed}} == Also known as == This lemma is also known as: :the '''butterfly lemma''', so named because the Hasse diagram of the various subgroups involved can be drawn to resemble a butterfly :either the '''third isomorphism theorem''' or the '''fourth isomorphism theorem''', which names are not recommended because of the lack of any consistent naming convention for the Isomorphism Theorems in the literature. {{Namedfor|Hans Julius Zassenhaus|cat = Zassenhaus}} == Sources == * {{BookReference|Elements of Abstract Algebra|1971|Allan Clark|prev = Second Isomorphism Theorem for Groups|next = Definition:Normal Series}}: Chapter $2$: Group Homomorphism and Isomorphism: $\\S 70$. The Third Isomorphism Theorem Category:Group Isomorphisms Category:Normal Subgroups c393a6nodonyl3uuq3tba352zktj110"}
+{"_id": "32281", "title": "Axiom:Pasch's Axiom", "text": "Axiom:Pasch's Axiom 100 14315 230283 158844 2015-10-29T23:20:44Z Prime.mover 59 wikitext text/x-wiki {{refactor|These axioms state the same truth, but in different language. A disambiguation page is for documenting multiple uses of a term which are completely different. This paqe needs to be converted into a parent page with transclusions, and then a separate page which proves the logical equivalence of these two axioms.}} {{disambig}} * Pasch's Axiom in Euclidean Geometry * Pasch's Axiom in Tarski's Geometry Category:Axioms/Geometry 4tpavwovpquem1a758myy9mzy4c53ry"}
+{"_id": "32282", "title": "Taylor's Theorem/One Variable/Integral Version", "text": "Taylor's Theorem/One Variable/Integral Version 0 14611 483449 362648 2020-08-28T10:12:49Z Prime.mover 59 wikitext text/x-wiki == Theorem == {{:Taylor's Theorem/One Variable/Statement of Theorem}} == Proof == <onlyinclude> This proof requires $f^{\\paren n}$ to be absolutely continuous on $\\closedint a x$, so that the Fundamental Theorem of Calculus holds. Except at the end when the Mean Value Theorem is invoked, differentiability of $f^{\\paren n}$ need not be assumed, since absolute continuity implies: :differentiability almost everywhere :the validity of the Fundamental Theorem of Calculus provided the integrals involved are understood as Lebesgue integrals. Consequently, the integral form of the remainder holds with this particular weakening of the assumptions on $f$. We first prove Taylor's Theorem with the integral remainder term. The Fundamental Theorem of Calculus states that: :$\\displaystyle \\int_a^x \\map {f'} t \\rd t = \\map f x - \\map f a$ which can be rearranged to: :$\\displaystyle \\map f x = \\map f a + \\int_a^x \\map {f'} t \\rd t$ Now we can see that an application of Integration by Parts yields: {{begin-eqn}} {{eqn | l = \\map f x       | r = \\map f a + x \\map {f'} x - a \\map {f'} a - \\int_a^x t \\map {f''} t \\rd t       | c = $u = \\map {f'} t$ and $\\d v = \\d t$ }} {{eqn | r = \\map f a + \\int_a^x x \\map {f''} t \\rd t + x \\map {f'} a - a \\map {f'} a - \\int_a^x t \\map {f''} t \\rd t       | c = $\\displaystyle \\int_a^x x \\map {f''} t \\rd t = x \\map {f'} x - x \\map {f'} a$ }} {{eqn | r = \\map f a + \\paren {x - a} \\map {f'} a + \\int_a^x \\paren {x - t} \\map {f''} t \\rd t       | c = factoring out some common terms }} {{end-eqn}} Another application yields: :$\\displaystyle \\map f x = \\map f a + \\paren {x - a} \\map {f'} a + \\frac 1 2 \\paren {x - a}^2 \\map {f''} a + \\frac 1 2 \\int_a^x \\paren {x - t}^2 \\map {f'''} t \\rd t$ By repeating this process, we may derive Taylor's theorem for higher values of $n$. This can be formalized by applying the technique of Principle of Mathematical Induction. So, suppose that Taylor's theorem holds for a $n$, that is, suppose that: {{begin-eqn}} {{eqn | l = \\map f x       | r = \\map f a       | c =  }} {{eqn | o = +       | r = \\frac {\\map {f'} a} {1!} \\paren {x - a}       | c =  }} {{eqn | o = +       | r = \\cdots       | c =  }} {{eqn | o = +       | r = \\frac {\\map {f^{\\paren n} } a} {n!} \\paren {x - a}^n       | c =  }} {{eqn | o = +       | r = \\int_a^x \\frac {\\map {f^{\\paren {n + 1} } } t} {n!} \\paren {x - t}^n \\rd t       | c = $*$ }} {{end-eqn}} We can rewrite the integral using Integration by Parts. A primitive of $\\paren {x - t}^n$ as a function of $t$ is given by $\\dfrac {-\\paren {x - t}^{n + 1} } {n + 1}$. So: {{begin-eqn}} {{eqn | o =        | r = \\int_a^x \\map {\\frac {f^{\\paren {n + 1} } } t} {n!} \\paren {x - t}^n \\rd t       | c =  }} {{eqn | r = -\\intlimits {\\map {\\frac {f^{\\paren {n + 1} } } t} {\\paren {n + 1} n!} \\paren {x - t}^{n + 1} } a x + \\int_a^x \\frac {\\map {f^{\\paren {n + 2} } } t} {\\paren {n + 1} n!} \\paren {x - t}^{n + 1} \\rd t       | c =  }} {{eqn | r = \\frac {\\map {f^{\\paren {n + 1} } } a} {\\paren {n + 1}!} \\paren {x - a}^{n + 1} + \\int_a^x \\frac {\\map {f^{\\paren {n + 2} } } t} {\\paren {n + 1}!} \\paren {x - t}^{n + 1} \\rd t       | c =  }} {{end-eqn}} The last integral can be solved immediately, which leads to :$R_n = \\dfrac {\\map {f^{\\paren {n + 1} } } \\xi} {\\paren {n + 1}!} \\paren {x - a}^{n + 1}$ {{qed}} </onlyinclude> {{Namedfor|Brook Taylor}} Category:Taylor's Theorem 6vf2wc06etjrisnr8v6d25ktn1upas4"}
+{"_id": "32283", "title": "Taylor's Theorem/One Variable/Proof by Cauchy Mean Value Theorem", "text": "Taylor's Theorem/One Variable/Proof by Cauchy Mean Value Theorem 0 14612 484062 483450 2020-08-30T06:10:15Z Prime.mover 59 wikitext text/x-wiki == Theorem == {{:Taylor's Theorem/One Variable/Statement of Theorem}} == Proof == <onlyinclude> Let $G$ be a real-valued function continuous on $\\closedint a x$ and differentiable with non-vanishing derivative on $\\openint a x$. Let: :$\\map F t = \\map f t + \\dfrac {\\map {f'} t} {1!} \\paren {x - t} + \\dotsb + \\dfrac {\\map {f^{\\paren n} } t} {n!} \\paren {x - t}^n$ By the Cauchy Mean Value Theorem: :$(1): \\quad \\dfrac {\\map {F'} x} {\\map {G'} \\xi} = \\dfrac {\\map F x - \\map F a} {\\map G x - \\map G a}$ for some $\\xi \\in \\openint a x$. Note that the numerator: :$\\map F x - \\map F a = R_n$ is the remainder of the Taylor polynomial for $\\map f x$. On the other hand, computing $\\map {F'} \\xi$: :$\\map {F'} \\xi = \\map {f'} \\xi - \\map {f'} \\xi + \\dfrac {\\map {f''} \\xi} {1!} \\paren {x - \\xi} - \\dfrac {\\map {f''} \\xi} {1!} \\paren {x - \\xi} + \\dotsb + \\dfrac {\\map {f^{\\paren {n + 1} } } t} {n!} \\paren {x - \\xi}^n = \\dfrac {\\map {f^{\\paren {n + 1} } } \\xi} {n!} \\paren {x - \\xi}^n$ Putting these two facts together and rearranging the terms of $(1)$ yields: :$R_n = \\dfrac {\\map {f^{\\paren {n + 1} } } \\xi} {n!} \\paren {x - \\xi}^n \\dfrac {\\map G x - \\map G a} {\\map {G'} \\xi}$ which was to be shown. {{refactor|Find a way of neatly handling the two different remainder forms}} Note that the Lagrange Form of the Remainder comes from taking $\\map G t = \\paren {x - t}^{n + 1}$ and the given Cauchy Form of the Remainder comes from taking $\\map G t = t - a$. {{qed}} </onlyinclude> {{Namedfor|Brook Taylor}} Category:Taylor's Theorem bp8i1weofd8nv7m8rgfrqubreirojaz"}
+{"_id": "32284", "title": "Taylor's Theorem/One Variable/Proof by Rolle's Theorem", "text": "Taylor's Theorem/One Variable/Proof by Rolle's Theorem 0 14613 484136 483451 2020-08-30T11:45:11Z Prime.mover 59 wikitext text/x-wiki == Theorem == {{:Taylor's Theorem/One Variable/Statement of Theorem}} == Proof == <onlyinclude> Let the function $g$ be defined as: :$\\map g t = \\map {R_n} t - \\dfrac {\\paren {t - a}^{n + 1} } {\\paren {x - a}^{n + 1} } \\map {R_n} x$ Then: :$\\map {g^{\\paren k} } a = 0$ for $k = 0, \\dotsc, n$, and $\\map g x = 0$. Apply Rolle's Theorem successively to $g, g', \\dotsc, g^{\\paren n}$. Then there exist: :$\\xi_1, \\ldots, \\xi_{n + 1}$ between $a$ and $x$ such that: :$\\map {g'} {\\xi_1} = 0, \\map {g''} {\\xi_2} = 0, \\ldots, \\map {g^{\\paren {n + 1} } } {\\xi_{n + 1} } = 0$ Let $\\xi = \\xi_{n + 1}$. Then: :$0 = \\map {g^{\\paren {n + 1} } } \\xi = \\map {f^{\\paren {n + 1} } } \\xi - \\dfrac {\\paren {n + 1}!} {\\paren {x - a}^{n + 1} } \\map {R_n} x$ and the formula for $\\map {R_n} x$ follows. {{qed}} </onlyinclude> {{Namedfor|Brook Taylor}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Taylor's Theorem/One Variable/Statement of Theorem/Also presented as|next = Definition:Increasing Real Function}}: $\\S 11.10$ Category:Taylor's Theorem 5ca6mx3oeb2lvib8nmi92h6dw3zvvwg"}
+{"_id": "32285", "title": "Matrix Inverse Algorithm", "text": "Matrix Inverse Algorithm 0 14899 476345 476210 2020-06-29T13:42:09Z Prime.mover 59 wikitext text/x-wiki == Algorithm == <onlyinclude> The '''matrix inverse algorithm''' is an algorithm which either: :$(1): \\quad$ converts a matrix into its inverse, if it exists or: :$(2): \\quad$ determines that such an inverse does not exist. Let $\\mathbf A$ be the $n \\times n$ square matrix in question. Let $\\mathbf I$ be the unit matrix of order $n$. :'''Step $0$''': Create the augmented matrix $\\begin {bmatrix} \\mathbf A & \\mathbf I \\end {bmatrix}$. :'''Step $1$''': Perform elementary row operations until $\\begin {bmatrix} \\mathbf A & \\mathbf I \\end {bmatrix}$ is in reduced echelon form. By Matrix is Row Equivalent to Reduced Echelon Matrix, this is possible. By Reduced Echelon Matrix is Unique, this process is well-defined. Call this new augmented matrix $\\begin {bmatrix} \\mathbf H & \\mathbf C \\end {bmatrix}$. :'''Step $2$''':  ::If $\\mathbf H = \\mathbf I$, then take $\\mathbf C = \\mathbf A^{-1}$. ::If $\\mathbf H \\ne \\mathbf I$, $\\mathbf A$ is not invertible. </onlyinclude> == Proof == Suppose $\\begin {bmatrix} \\mathbf A & \\mathbf I \\end {bmatrix}$ can be transformed into an upper triangular matrix with no zeroes on its main diagonal. Then from Identity Matrix from Upper Triangular Matrix, it can further be transformed into $\\begin {bmatrix} \\mathbf H & \\mathbf C \\end {bmatrix}$ where $\\mathbf H = \\mathbf I$. From Row Operation is Equivalent to Pre-Multiplication by Product of Elementary Matrices, the row operation to convert $\\begin {bmatrix} \\mathbf A & \\mathbf I \\end {bmatrix}$ to $\\begin {bmatrix} \\mathbf H & \\mathbf C \\end {bmatrix}$ is equivalent to the matrix product of the elementary row matrices corresponding to the sequence of elementary row operations that compose that row operation. Let $\\mathbf R$ be that matrix corresponding to that row operation. Because $\\mathbf H = \\mathbf I$, it follows that: :$\\mathbf R \\mathbf A = \\mathbf I$ and so $\\mathbf R$ is the inverse of $\\mathbf A$. That is: :$\\mathbf R = \\mathbf A^{-1}$ We also have, from the action of $\\mathbf R$ on the {{RHS}} of the augmented matrix $\\begin {bmatrix} \\mathbf A & \\mathbf I \\end {bmatrix}$ that: :$\\mathbf R \\mathbf I = \\mathbf C$ and so: :$\\mathbf C = \\mathbf R = \\mathbf A^{-1}$ {{qed|lemma}} Now suppose that $\\mathbf H \\ne \\mathbf I$. {{finish|It remains to be shown that in such a case $\\mathbf A$ is not invertible.}} == Examples == {{:Matrix Inverse Algorithm/Examples}} == Sources == * {{BookReference|Linear Algebra|1998|Richard Kaye|author2 = Robert Wilson|prev = Definition:Augmented Matrix|next = Matrix Inverse Algorithm/Examples/Arbitrary Matrix 1}}: Part $\\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations Category:Inverse Matrices Category:Matrix Inverse Algorithm 258v7l48cdgzlyr3lvydeos4hc8ha87"}
+{"_id": "32286", "title": "Convergence of Generalized Sum of Complex Numbers/Corollary", "text": "Convergence of Generalized Sum of Complex Numbers/Corollary 0 14938 178094 178091 2014-03-02T17:20:16Z Prime.mover 59 wikitext text/x-wiki == Corollary to Convergence of Generalized Sum of Complex Numbers == Let $\\left({z_i}\\right)_{i \\in I}$ be an $I$-indexed family of complex numbers. That is, let $z_i \\in \\C$ for all $i \\in I$. <onlyinclude> Suppose that $\\displaystyle \\sum \\left\\{{ z_i: i \\in I }\\right\\}$ converges to $z \\in \\C$. Then $\\displaystyle \\sum \\left\\{{ \\overline{z_i}: i \\in I }\\right\\}$ converges to $\\overline z$, where $\\overline z$ denotes the complex conjugate of $z$. </onlyinclude> Here, the $\\sum$ denote generalized sums. == Proof == Using Convergence of Generalized Sum of Complex Numbers, one has: :$\\displaystyle \\sum \\left\\{{\\operatorname{Re} z_i : i \\in I}\\right\\} = \\operatorname{Re} z$ :$\\displaystyle \\sum \\left\\{{\\operatorname{Im} z_i : i \\in I}\\right\\} = \\operatorname{Im} z$ Now, observe that, from the definition of complex conjugate: {{begin-eqn}} {{eqn|l = \\overline z      |r = \\operatorname{Re} z - i \\operatorname{Im} z }} {{eqn|r = \\sum \\left\\{ {\\operatorname{Re} z_i : i \\in I}\\right\\} - i \\sum \\left\\{ {\\operatorname{Im} z_i : i \\in I}\\right\\} }} {{eqn|r = \\sum \\left\\{ {\\operatorname{Re} z_j - i \\operatorname{Im} z_j : j \\in I}\\right\\}      |c = Generalized Sum is Linear }} {{eqn|r = \\sum \\left\\{ {\\overline{z_j}: j \\in I}\\right\\} }} {{end-eqn}} {{qed}} Category:Generalized Sums Category:Complex Numbers mtmfc9a1rt4ibp1ms8sld6zo6yverm5"}
+{"_id": "32287", "title": "Bijection iff Left and Right Inverse/Corollary", "text": "Bijection iff Left and Right Inverse/Corollary 0 14953 447362 368399 2020-02-08T12:26:59Z Prime.mover 59 wikitext text/x-wiki == Corollary to Bijection iff Left and Right Inverse == <onlyinclude> Let $f: S \\to T$ and $g: T \\to S$ be mappings such that: : $g \\circ f = I_S$ : $f \\circ g = I_T$ Then both $f$ and $g$ are bijections. </onlyinclude> == Proof  == Suppose we have such mappings $f$ and $g$ with the given properties. From Bijection iff Left and Right Inverse, we have that $f$ is a bijection, by considering $g = g_1$ and $g = g_2$. It directly follows that by setting $g = f, f = g_1, f = g_2$, the result Bijection iff Left and Right Inverse can be used the other way about. {{qed}} == Sources == * {{BookReference|Lectures in Abstract Algebra|1951|Nathan Jacobson|volume = I|subtitle = Basic Concepts|prev = Composite of Bijection with Inverse is Identity Mapping|next = Definition:Binary Operation}}: Introduction $\\S 2$: Product sets, mappings * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Composite of Bijection with Inverse is Identity Mapping|next = Composite of Bijections is Bijection/Proof 1}}: $\\S 25.1$: Some further results and examples on mappings Category:Bijections Category:Inverse Mappings 09zl59d76hre8zvavs3lkuhcvyw266n"}
+{"_id": "32288", "title": "Exponential of Sum/Real Numbers/Lemma", "text": "Exponential of Sum/Real Numbers/Lemma 0 14995 396492 396466 2019-03-21T07:02:29Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $x, y \\in \\R$. Let $n \\in \\N_{> 0}$ such that $n > -\\paren {x + y}$. Then: :$1 + \\dfrac {x + y} n + \\dfrac {x y} {n^2} = \\paren {1 + \\dfrac {x + y} n} \\paren {1 + \\dfrac {\\paren {\\frac {x y} {n + x + y} } } n}$ {{qed|lemma}} </onlyinclude> == Proof == As $n \\in \\N_{> 0}$ we have that $n \\ne 0$ and so the fractions in the expressions are defined. {{begin-eqn}} {{eqn | l = 1 + \\frac {x + y} n + \\frac {x y} {n^2}       | r = \\frac {\\paren {1 + \\frac {x + y} n} \\paren {1 + \\frac {x + y} n + \\frac {x y} {n^2} } } {1 + \\frac {x + y} n}        | c = multiplying and dividing by $1 + \\dfrac {x + y} n$ }} {{eqn | r = \\paren {1 + \\frac {x + y} n} \\frac {\\paren {1 + \\frac {x + y} n + \\frac {x y} {n^2} } } {\\paren {1 + \\frac {x + y} n} }        | c = extracting a factor }} {{eqn | r = \\paren {1 + \\frac {x + y} n} \\paren {\\frac {n^2 + n x + n y + x y} {n^2 + n x + n y} }       | c = multiplying top and bottom by $n^2$ }} {{eqn | r = \\paren {1 + \\frac {x + y} n} \\paren {1 + \\frac {x y} {n^2 + n x + n y} }       | c = Polynomial Long Division }} {{eqn | r = \\paren {1 + \\frac {x + y} n} \\paren {1 + \\frac {\\paren {\\frac {x y} {n + x + y} } } n}       | c = dividing top and bottom by $n + x + y$ }} {{end-eqn}} That final step is justified, as we have that $n > -\\paren {x + y}$ and so $n + x + y \\ne 0$. {{qed}} == Also see == This lemma is used in: :Proof 2 of (Real) Exponential of Sum :Proof 3 of (Real) Exponential of Sum :$\\map \\exp {x + y} = \\paren {\\exp x} \\paren {\\exp y}$ for real $x$ and $y$. Category:Exponential of Sum mkhqezzsgxcvr5s3a3ntuqluxryficm"}
+{"_id": "32289", "title": "Natural Logarithm of 2 is Greater than One Half", "text": "Natural Logarithm of 2 is Greater than One Half 0 15131 356186 267867 2018-05-19T09:36:22Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> :$\\ln 2 \\ge \\dfrac 1 2$ where $\\ln$ denotes the natural logarithm function. </onlyinclude> == Proof 1 == {{:Natural Logarithm of 2 is Greater than One Half/Proof 1}} == Proof 2 == {{:Natural Logarithm of 2 is Greater than One Half/Proof 2}} == Also see == * Logarithm Tends to Infinity == Sources == * {{BookReference|Calculus|2005|Roland E. Larson|author2 = Robert P. Hostetler|author3 = Bruce H. Edwards|ed = 8th|edpage = Eighth Edition}}: Appendix $A$: ''Properties of the Natural Logarithmic Function'' Category:Natural Logarithms Category:Inequalities Category:Natural Logarithm of 2 is Greater than One Half nmpmqpkbmuc8l8xfz3lo3jsxua2ki78"}
+{"_id": "32290", "title": "Schur's Lemma (Representation Theory)/Corollary", "text": "Schur's Lemma (Representation Theory)/Corollary 0 15357 497538 316079 2020-11-02T06:59:29Z Prime.mover 59 wikitext text/x-wiki {{MissingLinks}} == Corollary to Schur's Lemma == <onlyinclude> Let $\\struct {G, \\cdot}$ be a finite group. Let $\\struct {V, \\phi}$ be a $G$-module. Let the underlying field $k$ of $V$ be algebraically closed. Let: :$\\map {\\mathrm {End}_G} V := \\leftset {f: V \\to V: f}$ is a homomorphism of $G$-modules$\\rightset {}$ Then: :$\\map {\\mathrm {End}_G} V$ is a field, with the same structure as $k$. </onlyinclude> == Proof == Denote the identity mapping on $V$ as $I_V: V \\to V$. If $f = 0$, since $0\\in k$ it can be written $f = 0 I_V$. {{explain|Not clear what the above line means.}} Let $f$ be an automorphism. We have that $k$ is algebraically closed. Therefore the characteristic polynomial of $f$ is complete reducible in $k \\sqbrk x$. Hence $f$ has all eigenvalue in $k$. Let $\\lambda \\in k$ be an eigenvalue of $f$. Consider the endomorphism: :$f - \\lambda I_V: V \\to V$ Because $\\lambda$ is an eigenvalue: :$\\map \\ker {f - \\lambda I_V} \\ne \\set 0$ From Schur's Lemma: :$f = \\lambda I_V$ :$\\paren {\\lambda I_V} \\circ \\paren {\\mu I_V} = \\paren {\\lambda \\mu} I_V$ :$\\lambda I_V + \\paren {-\\mu I_V} = \\paren {\\lambda - \\mu} I_V$ From Subring Test: :$\\map {\\mathrm {End}_G} V$ is a subring of the ring endomorphisms of $V$ as an abelian group. Let $\\phi: \\map {\\mathrm {End}_G} V \\to k$ be defined as: :$\\map \\phi {\\lambda I_V} = \\lambda$ Then: :$\\map \\phi {\\lambda I_V + \\mu I_V} = \\lambda + \\mu = \\map \\phi {\\lambda I_V} + \\map \\phi {\\mu I_V}$ :$\\map \\phi {\\paren {\\lambda I_V} \\circ \\paren {\\mu I_V} } = \\lambda \\mu = \\map \\phi {\\lambda I_V} \\map \\phi {\\mu I_V}$ Hence $\\phi$ is a ring isomorphism. But since $k$ is a field it is a field isomorphism. {{qed}}   {{proofread}} Category:Representation Theory 0iluo82z6mhpuey50216wc2typ8r5hj"}
+{"_id": "32291", "title": "De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection/Corollary", "text": "De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection/Corollary 0 15601 352420 225769 2018-05-01T20:45:53Z Prime.mover 59 wikitext text/x-wiki == Corollary to De Morgan's Laws: Difference with Intersection == Let $S, T_1, T_2$ be sets. <onlyinclude> Suppose that $T_1 \\subseteq S$. Then: :$S \\setminus \\left({T_1 \\cap T_2}\\right) = \\left({S \\setminus T_1}\\right) \\cup \\left({T_1 \\setminus T_2}\\right)$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = S \\setminus \\left({T_1 \\cap T_2}\\right)       | r = \\left({S \\setminus T_1}\\right) \\cup \\left({S \\setminus T_2}\\right)       | c = De Morgan's Laws: Difference with Intersection }} {{eqn | r = \\left({S \\setminus T_1}\\right) \\cup \\left({\\left({\\left({S \\setminus T_1}\\right) \\cup \\left({S \\cap T_1}\\right)}\\right) \\setminus T_2}\\right)       | c = Set Difference Union Intersection }} {{eqn | r = \\left({S \\setminus T_1}\\right) \\cup \\left({\\left({S \\setminus T_1}\\right) \\setminus T_2}\\right) \\cup \\left({\\left({S \\cap T_1}\\right) \\setminus T_2}\\right)       | c = Set Difference is Right Distributive over Union }} {{eqn | r = \\left({S \\setminus T_1}\\right) \\cup \\left({\\left({S \\setminus T_1}\\right) \\setminus T_2}\\right) \\cup \\left({T_1 \\setminus T_2}\\right)       | c = Intersection with Subset is Subset: $T_1 \\subseteq S$ }} {{eqn | r = \\left({S \\setminus T_1}\\right) \\cup \\left({T_1 \\setminus T_2}\\right)       | c = Set Difference Union First Set is First Set }} {{end-eqn}} {{qed}} Category:Set Difference Category:De Morgan's Laws scwakh2xporyr40vaai5w15c3j18g9n"}
+{"_id": "32292", "title": "Image of Preimage under Mapping/Corollary", "text": "Image of Preimage under Mapping/Corollary 0 15653 417770 414514 2019-08-11T11:14:31Z Prime.mover 59 wikitext text/x-wiki == Corollary to Image of Preimage under Mapping == Let $f: S \\to T$ be a mapping. Then: <onlyinclude> :$B \\subseteq \\Img S \\implies \\paren {f \\circ f^{-1} } \\sqbrk B = B$ </onlyinclude> == Proof == From Image of Subset under Relation is Subset of Image/Corollary 3 we have: :$B \\subseteq \\Img S \\implies f^{-1} \\sqbrk B \\subseteq f^{-1} \\sqbrk {\\Img S}$ and from Intersection with Subset is Subset we have: :$f^{-1} \\sqbrk B \\subseteq f^{-1} \\sqbrk {\\Img S} \\implies f^{-1} \\sqbrk B \\cap f^{-1} \\sqbrk {\\Img S} = f^{-1} \\sqbrk B$ Hence the result. {{qed}} == Sources == * {{BookReference|Rings, Modules and Linear Algebra|1970|B. Hartley|author2 = T.O. Hawkes|prev = Definition:Direct Image Mapping of Mapping|next = Image of Subset under Relation is Subset of Image/Corollary 2}}: $\\S 2.2$: Homomorphisms: $\\text{(i)}$ Category:Preimages under Mappings evr46sk5tjwomo75w6b2tyk4gkdm20i"}
+{"_id": "32293", "title": "Axiom:Axiom of Countable Choice", "text": "Axiom:Axiom of Countable Choice 100 15701 453398 444063 2020-03-08T13:00:08Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> === Form 1 === {{:Axiom:Axiom of Countable Choice/Form 1}} === Form 2 === {{:Axiom:Axiom of Countable Choice/Form 2}} </onlyinclude> == Also known as == This axiom can be abbreviated $\\mathrm{ACC}$, $\\mathrm{CC}$, $\\mathrm{AC}_\\omega$, or $\\mathrm{AC}_\\N$. == Also see == * Equivalence of Forms of Axiom of Countable Choice * This axiom is a weaker form of the axiom of dependent choice, as shown in Axiom of Dependent Choice Implies Axiom of Countable Choice. Countable Choice Countable Choice 4g8kezgx135n752bu4a8j7r5ij9wckk"}
+{"_id": "32294", "title": "Kernel of Linear Transformation contains Zero Vector", "text": "Kernel of Linear Transformation contains Zero Vector 0 16446 417034 349654 2019-08-07T12:48:10Z Prime.mover 59 wikitext text/x-wiki == Corollary to Linear Transformation Maps Zero Vector to Zero Vector == Let $\\mathbf V$ be a vector space, with zero $\\mathbf 0$. Likewise let $\\mathbf V'$ be another vector space, with zero $\\mathbf 0'$. Let $T: \\mathbf V \\to \\mathbf V'$ be a linear transformation. Then: <onlyinclude> :$\\mathbf 0 \\in \\map \\ker T$ where $\\map \\ker T$ is the kernel of $T$. </onlyinclude> == Proof == Follows from Linear Transformation Maps Zero Vector to Zero Vector and the definition of kernel. {{qed}} Category:Linear Transformations Category:Linear Transformation Maps Zero Vector to Zero Vector orb8mqtbdb8a78fe6cg9tz5otd2n0qi"}
+{"_id": "32295", "title": "Fermat's Little Theorem/Corollary 1", "text": "Fermat's Little Theorem/Corollary 1 0 16621 486213 445854 2020-09-07T21:56:28Z Prime.mover 59 wikitext text/x-wiki == Corollary to Fermat's Little Theorem == <onlyinclude> If $p$ is a prime number, then $n^p \\equiv n \\pmod p$. </onlyinclude> == Proof 1 == {{:Fermat's Little Theorem/Corollary 1/Proof 1}} == Proof 2 == {{:Fermat's Little Theorem/Corollary 1/Proof 2}} == Also known as == Some sources call this '''Fermat's Little Theorem''', and from it derive that theorem as a corollary. == Also reported as == This result can also be reported as: :If $p$ is a prime number, then $n^p - n$ is divisible by $p$. {{Namedfor|Pierre de Fermat}} == Historical Note == {{:Fermat's Little Theorem/Historical Note}} == Sources == * {{BookReference|Men of Mathematics|1937|Eric Temple Bell|prev = Construction of Regular Prime p-Gon Exists iff p is Fermat Prime/Historical Note|next = Fermat's Little Theorem/Historical Note}}: Chapter $\\text{IV}$: The Prince of Amateurs * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Fermat's Little Theorem|next = Prime Group has no Proper Subgroups}}: Exercise $25.6$ * {{BookReference|Number Theory|1971|George E. Andrews|prev = Binomial Theorem/Integral Index|next = Alternating Summation of Binomial Coefficient of Summation of Binomial Coefficient of Sequence}}: $\\text {3-1}$ Permutations and Combinations: Exercise $11$ * {{BookReference|Number Theory|1971|George E. Andrews|prev = Congruence Modulo Integer/Examples/11 equiv -1 mod 12|next = Wilson's Theorem}}: $\\text {4-1}$ Basic Properties of Congruences: Example $\\text {4-2}$ * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Fermat's Two Squares Theorem|next = Fermat's Last Theorem}}: $1$: The Nature of Differential Equations: $\\S 6$: The Brachistochrone. Fermat and the Bernoullis * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Chinese Remainder Theorem|next = Fermat's Little Theorem/Proof 1}}: $\\S 1.2.4$: Integer Functions and Elementary Number Theory: Theorem $\\text{F}$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Fermat's Little Theorem|next = Fermat's Little Theorem/Historical Note|entry = Fermat's theorem}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Fermat's Little Theorem|next = Fermat's Little Theorem/Historical Note|entry = Fermat's theorem}} * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Lagrange's Four Square Theorem|next = Fermat's Last Theorem}}: Chapter $7$: Patterns in Numbers: Fermat * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Fermat's Little Theorem|next = Feuerbach's Theorem|entry = Fermat's Little Theorem}} Category:Fermat's Little Theorem 9s54cx1hptsxhfr3264m8jfyymogv7s"}
+{"_id": "32296", "title": "Generated Sigma-Algebra by Generated Monotone Class/Corollary", "text": "Generated Sigma-Algebra by Generated Monotone Class/Corollary 0 16680 179757 169487 2014-03-11T16:49:28Z Prime.mover 59 wikitext text/x-wiki == Corollary to Generated Sigma-Algebra by Generated Monotone Class == <onlyinclude> Let $X$ be a set, and let $\\mathcal G \\subseteq \\mathcal P \\left({X}\\right)$ be a nonempty collection of subsets of $X$. Define $\\complement_X \\left({\\mathcal G}\\right)$ by: :$\\complement_X \\left({\\mathcal G}\\right) := \\left\\{{\\complement_X \\left({A}\\right): A \\in \\mathcal G}\\right\\}$ Then: :$\\sigma \\left({\\mathcal G}\\right) = \\mathfrak m \\left({\\mathcal G \\cup \\complement_X \\left({\\mathcal G}\\right)}\\right)$ </onlyinclude> Here, $\\mathfrak m$ denotes generated monotone class, and $\\sigma$ denotes generated $\\sigma$-algebra. == Proof == From Set is Subset of Union: : $\\mathcal G \\subseteq \\mathcal G \\cup \\complement_X \\left({\\mathcal G}\\right)$ Further, as $\\sigma \\left({\\mathcal G}\\right)$ is a $\\sigma$-algebra: :$S \\in \\sigma \\left({\\mathcal G}\\right) \\implies \\complement_X \\left({X}\\right) = X \\setminus S \\in \\sigma \\left({\\mathcal G}\\right)$ from Set Difference as Intersection with Relative Complement. Since $\\mathcal G \\subseteq \\sigma \\left({\\mathcal G}\\right)$: :$\\mathcal G \\cup \\complement_X \\left({\\mathcal G}\\right) \\subseteq \\sigma \\left({\\mathcal G}\\right)$ By Condition on Equality of Generated Sigma-Algebras: :$\\sigma \\left({\\mathcal G}\\right) = \\sigma \\left({\\mathcal G \\cup \\complement_X \\left({\\mathcal G}\\right)}\\right)$ Applying Generated Sigma-Algebra by Generated Monotone Class: :$\\sigma \\left({\\mathcal G}\\right) = \\mathfrak m \\left({\\mathcal G \\cup \\complement_X \\left({\\mathcal G}\\right)}\\right)$ {{qed}} Category:Sigma-Algebras Category:Monotone Classes 034okkbyk5kjaao4x9mh0k72o8wyjjb"}
+{"_id": "32297", "title": "Nth Derivative of Nth Power", "text": "Nth Derivative of Nth Power 0 16748 484121 466592 2020-08-30T10:52:22Z Prime.mover 59 wikitext text/x-wiki == Corollary to Nth Derivative of Mth Power == Let $n \\in \\Z$ be an integer such that $n \\ge 0$. The $n$th derivative of $x^n$ {{WRT|Differentiation}} $x$ is: <onlyinclude> :$\\dfrac {\\d^n} {\\d x^n} x^n = n!$ where $n!$ denotes $n$ factorial. </onlyinclude> == Proof == From Nth Derivative of Mth Power, we have: :$\\dfrac {\\d^n} {\\d x^n} x^m = \\begin {cases} m^\\underline n \\, x^{m - n} & : n \\le m \\\\ 0 & : n > m \\end {cases}$ where $m^\\underline n$ denotes the falling factorial. Putting $m = n$: :$\\dfrac {\\d^n} {\\d x^n} x^n = n^\\underline n$ where from the definition of the falling factorial: :$n^\\underline n = n!$ Hence the result. {{Qed}} Category:Derivatives n5qs4fn8tymcfoj9djk165f81mdu141"}
+{"_id": "32298", "title": "Unique Linear Transformation Between Vector Spaces", "text": "Unique Linear Transformation Between Vector Spaces 0 16781 147340 93014 2013-05-21T00:14:28Z Dfeuer 1672 Add to vector spaces category wikitext text/x-wiki == Corollary to Unique Linear Transformation Between Modules == <onlyinclude> Let $G$ be a finite-dimensional $K$-vector space. Let $H$ be a $K$-vector space (not necessarily finite-dimensional). Let $\\left \\langle {a_n} \\right \\rangle$ be a linearly independent sequence of vectors of $G$. Let $\\left \\langle {b_n} \\right \\rangle$ be a sequence of vectors of $H$. Then there is a unique linear transformation $\\phi: G \\to H$ satisfying $\\forall k \\in \\left[{1 \\,.\\,.\\, n}\\right]: \\phi \\left({a_k}\\right) = b_k$ </onlyinclude> == Proof == From Generator of Vector Space Contains Basis, $\\left\\{{a_1, \\ldots, a_m}\\right\\}$ is contained in a basis for $G$. The result then follows from Unique Linear Transformation Between Modules. {{Qed}} == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev=Unique Linear Transformation Between Modules|next=Definition:Rank of Linear Transformation}}: $\\S 28$: Theorem $28.4$: Corollary Category:Linear Transformations Category:Vector Spaces 0ke7kn6ysxeidzaioq8u2u93x6fr11x"}
+{"_id": "32299", "title": "Linear Transformations Isomorphic to Matrix Space/Corollary", "text": "Linear Transformations Isomorphic to Matrix Space/Corollary 0 16835 399279 182109 2019-04-06T22:05:54Z Prime.mover 59 wikitext text/x-wiki === Corollary to Linear Transformations Isomorphic to Matrix Space === <onlyinclude> Let $R$ be a commutative ring with unity. Let $M: \\struct {\\map {\\mathcal L_R} G, +, \\circ} \\to \\struct {\\map {\\mathcal M_R} n, +, \\times}$ be defined as: :$\\forall u \\in \\map {\\mathcal L_R} G: \\map M u = \\sqbrk {u; \\sequence {a_n} }$ Then $M$ is an isomorphism. </onlyinclude> == Proof == Follows directly from Linear Transformations Isomorphic to Matrix Space. {{qed}} {{Proofread}} == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Definition:Order of Square Matrix|next = Ring of Square Matrices over Commutative Ring with Unity}}: $\\S 29$: Theorem $29.2$ Category:Linear Algebra Category:Matrix Algebra 5fse8w10os2p8180ahpxdbqva6jv3ib"}
+{"_id": "32300", "title": "Fundamental Theorem of Calculus/First Part/Corollary", "text": "Fundamental Theorem of Calculus/First Part/Corollary 0 16873 486124 362632 2020-09-07T20:28:01Z Prime.mover 59 wikitext text/x-wiki == Corollary to Fundamental Theorem of Calculus (First Part) == Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$. Let $F$ be a real function which is defined on $\\closedint a b$ by: :$\\displaystyle \\map F x = \\int_a^x \\map f t \\rd t$ Then: <onlyinclude> :$\\displaystyle \\frac \\d {\\d x} \\int_a^x \\map f t \\rd t = \\map f x$ </onlyinclude> == Proof == Follows from the Fundamental Theorem of Calculus (First Part) and the definition of primitive. {{qed}} == Sources == * {{BookReference|Calculus|2005|Roland E. Larson|author2 = Robert P. Hostetler|author3 = Bruce H. Edwards|ed = 8th|edpage = Eighth Edition}}: $\\S 4.4$ Category:Fundamental Theorem of Calculus e7l3fggbx4owixdxnxj43t6fskbolvm"}
+{"_id": "32301", "title": "Ring of Integers Modulo Prime is Integral Domain", "text": "Ring of Integers Modulo Prime is Integral Domain 0 16893 399547 399546 2019-04-09T06:50:31Z Prime.mover 59 wikitext text/x-wiki == Corollary to Ring of Integers Modulo Prime is Field == <onlyinclude> Let $m \\in \\Z: m \\ge 2$. Let $\\struct {\\Z_m, +, \\times}$ be the ring of integers modulo $m$. Then: :$m$ is prime {{iff}}: :$\\struct {\\Z_m, +, \\times}$ is an integral domain. </onlyinclude> == Proof 1 == {{:Ring of Integers Modulo Prime is Integral Domain/Proof 1}} == Proof 2 == {{:Ring of Integers Modulo Prime is Integral Domain/Proof 2}} Category:Ring of Integers Modulo m Category:Integral Domains Category:Ring of Integers Modulo Prime is Field Category:Ring of Integers Modulo Prime is Integral Domain 7uqzb2473za599um7rmo9jc9r5orpp7"}
+{"_id": "32302", "title": "Reduced Residue System under Multiplication forms Abelian Group/Corollary", "text": "Reduced Residue System under Multiplication forms Abelian Group/Corollary 0 16903 374438 374436 2018-11-01T07:47:11Z Prime.mover 59 wikitext text/x-wiki == Corollary of Reduced Residue System under Multiplication forms Abelian Group == <onlyinclude> Let $p$ be a prime number. Let $\\Z_p$ be the set of integers modulo $p$. Let $\\struct {\\Z'_p, \\times}$ denote the multiplicative group of reduced residues modulo $p$. Then $\\struct {\\Z'_p, \\times}$ is an abelian group. </onlyinclude> == Proof == Suppose $p \\in \\Z$ be a prime number. From the definition of reduced residue system modulo $p$, as $p$ is prime, $\\Z'_p$ becomes: :$\\set {\\eqclass 1 p, \\eqclass 2 p, \\ldots, \\eqclass {p - 1} p}$ This is precisely $\\Z_p \\setminus \\set {\\eqclass 0 p}$ which is what we wanted to show. The result follows from Reduced Residue System under Multiplication forms Abelian Group. {{qed}} == Sources == * {{BookReference|The Theory of Groups|1968|Ian D. Macdonald|prev = Modulo Multiplication is Well-Defined/Proof 1|next = Modulo Multiplication is Associative}}: $\\S 1$: Some examples of groups: Example $1.11$ * {{BookReference|Elements of Abstract Algebra|1971|Allan Clark|prev = Multiplicative Inverse in Monoid of Integers Modulo m|next = Definition:Multiplicative Group of Reduced Residues}}: Chapter $2$: Examples of Group Structure: $\\S 34$ * {{BookReference|Algebra|1974|Thomas W. Hungerford|prev = Reduced Residue System under Multiplication forms Abelian Group/Proof 2|next = Definition:Group of Rationals Modulo One}}: $\\text{I}$: Groups: $\\S 1$: Semigroups, Monoids and Groups * {{BookReference|Algebra|1974|Thomas W. Hungerford|prev = Klein Four-Group/Cayley Table|next = Modulo One is Congruence Relation on Rational Numbers}}: $\\text{I}$: Groups: $\\S 1$: Semigroups, Monoids and Groups: Exercise $7$ * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Integers Modulo m under Addition form Abelian Group|next = Definition:Symmetric Group}}: $\\S 34$. Examples of groups: $(3)$ Category:Reduced Residue System under Multiplication forms Abelian Group 1o7zkplb21p4hmrgrkf9o439mnrtkva"}
+{"_id": "32303", "title": "Product with Ring Negative/Corollary", "text": "Product with Ring Negative/Corollary 0 16907 399490 346892 2019-04-08T21:27:04Z Prime.mover 59 wikitext text/x-wiki == Corollary to Product with Ring Negative == <onlyinclude> Let $\\struct {R, +, \\circ}$ be a ring with unity $1_R$. Then: :$\\forall x \\in R: \\paren {-1_R} \\circ x = -x$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\paren {-1_R} \\circ x       | r = -\\paren {1_R \\circ x}       | c = Product with Ring Negative }} {{eqn | r = -x       | c = {{Defof|Unity of Ring}} }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Product of Ring Negatives|next = Cancellation Law of Ring Product of Integral Domain}}: Chapter $1$: Integral Domains: $\\S 4$. Elementary Properties: Theorem $2 \\ \\text{(vi)}$ Category:Ring Theory Category:Rings with Unity mxg1scb688xhderm1zwzc5b3x88sxs9"}
+{"_id": "32304", "title": "Ring Homomorphism Preserves Subrings/Corollary", "text": "Ring Homomorphism Preserves Subrings/Corollary 0 16919 398070 397449 2019-03-29T08:17:53Z Prime.mover 59 wikitext text/x-wiki == Corollary to Ring Homomorphism Preserves Subrings == Let $\\struct {R_1, +_1, \\circ_1}$ and $\\struct {R_2, +_2, \\circ_2}$ be rings. <onlyinclude> The image of a ring homomorphism $\\phi: R_1 \\to R_2$ is a subring of $R_2$. </onlyinclude> == Proof == From Null Ring and Ring Itself Subrings, $R_1$ is a subring of itself. The result then follows from Ring Homomorphism Preserves Subrings. {{qed}} == Sources == * {{BookReference|Rings, Modules and Linear Algebra|1970|B. Hartley|author2 = T.O. Hawkes|prev = Kernel is Trivial iff Monomorphism/Ring|next = Definition:Coset}}: $\\S 2.2$: Homomorphisms: Lemma $2.6 \\ \\text{(ii)}$ * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Ring Homomorphism Preserves Subrings/Proof 3|next = Kernel of Ring Homomorphism is Subring}}: $\\S 57.3$ Ring homomorphisms: $\\text{(i)}$ Category:Ring Homomorphisms Category:Subrings Category:Ring Homomorphism Preserves Subrings i0g4hpd4lq79nx8j6qjddrzqtpndsfh"}
+{"_id": "32305", "title": "Handshake Lemma/Corollary", "text": "Handshake Lemma/Corollary 0 16955 482233 482230 2020-08-17T21:01:36Z Prime.mover 59 wikitext text/x-wiki == Corollary to the Handshake Lemma == Let $G$ be a $\\tuple {p, q}$-graph, which may be a multigraph or a loop-graph, or both. <onlyinclude> The number of odd vertices in $G$ is even. </onlyinclude> == Proof == Let $G$ be a $\\tuple {p, q}$-graph. Consider the sum of the degrees of its vertices: :$\\displaystyle K = \\sum_{v \\mathop \\in V} \\map {\\deg_G} v$ From the Handshake Lemma: :$K = 2 \\card E$ which is an even integer. Subtracting from $K$ the degrees of all even vertices, we are left with the sum of all degrees of odd vertices in $V$. That is: :$\\displaystyle \\paren {\\sum_{v \\mathop \\in V} \\map {\\deg_G} v} - \\paren {\\sum_{v \\mathop \\in V : \\map {\\deg_G} v \\mathop = 2 k} \\map {\\deg_G} v} = \\paren {\\sum_{v \\mathop \\in V : \\map {\\deg_G} v \\mathop = 2 k + 1} \\map {\\deg_G} v}$ This must still be an even number, as it is equal to the difference of two even numbers. Because this is a sum of exclusively odd terms, there must be an even number of such terms for the sum on the {{RHS}} to be even. Hence the number of odd vertices in $G$ must be even. {{qed}} == Sources == * {{BookReference|Introductory Graph Theory|1977|Gary Chartrand|prev = Definition:Odd Vertex (Graph Theory)|next = Definition:Regular Graph}}: Chapter $2$: Elementary Concepts of Graph Theory: $\\S 2.1$: The Degree of a Vertex: Theorem $2.2$ * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Definition:Eulerian Circuit|next = Characteristics of Eulerian Graph}}: Chapter $\\text {A}.21$: Euler ({{DateRange|1707|1783}}) * {{BookReference|Curious and Interesting Puzzles|1992|David Wells|prev = Definition:Odd Vertex (Graph Theory)|next = Orthogonal Latin Squares of Order 6 do not Exist/Historical Note}}: The Bridges of Königsberg: $134$ Category:Handshake Lemma ih6kbddv63eqeggf5ra0uqq495xnrue"}
+{"_id": "32306", "title": "Image of Composite Mapping/Corollary", "text": "Image of Composite Mapping/Corollary 0 16980 416117 226853 2019-08-03T07:46:11Z Prime.mover 59 wikitext text/x-wiki == Corollary to Image of Composite Mapping == <onlyinclude> Let $f: S \\to T$ and $g: R \\to S$ be mappings. Then: :$\\Img {f \\circ g} \\subseteq \\Img f$ </onlyinclude> where: :$f \\circ g$ denotes composition of $g$ and $f$ :$\\Img f$ denotes image of $f$. == Proof == From Image of Composite Mapping, it holds that: :$\\Img {f \\circ g} = f \\sqbrk {\\Img g}$ where $f \\sqbrk {\\, \\cdot \\,}$ denotes image of subset. By definition of composite mapping: :$\\Img g \\subseteq \\Dom f$ where $\\Dom f$ denotes the domain of $f$. Now Image of Subset under Relation is Subset of Image: Corollary 2 yields: :$\\Img {f \\circ g} \\subseteq \\Img f$ {{qed}} == Sources == * {{BookReference|Undergraduate Topology|1971|Robert H. Kasriel|prev = Domain of Composite Mapping|next = Composition of Mappings is Associative}}: $\\S 1.14$: Composition of Functions Category:Composite Mappings g66e40owxmfalc7v3i6ikwsrwrp9x7k"}
+{"_id": "32307", "title": "Relation between Two Ordinals/Corollary", "text": "Relation between Two Ordinals/Corollary 0 17233 144451 144447 2013-04-18T21:36:33Z Prime.mover 59 wikitext text/x-wiki == Corollary to Relation between Two Ordinals == <onlyinclude> Let $S$ and $T$ be ordinals. If $S \\ne T$, then either $S$ is an initial segment of $T$, or vice versa. </onlyinclude> == Proof 1 == {{:Relation between Two Ordinals/Corollary/Proof 1}} == Proof 2 == {{:Relation between Two Ordinals/Corollary/Proof 2}} == Proof 3 == {{:Relation between Two Ordinals/Corollary/Proof 3}} Category:Ordinals 7fyjylzo2mfuc548txqu06h1pl3bg7e"}
+{"_id": "32308", "title": "Complement Union with Superset is Universe/Corollary", "text": "Complement Union with Superset is Universe/Corollary 0 17270 416790 416716 2019-08-05T20:59:45Z Prime.mover 59 wikitext text/x-wiki == Corollary to Complement Union with Superset is Universe == <onlyinclude> :$S \\cup T = \\mathbb U \\iff \\map \\complement S \\subseteq T$ </onlyinclude> where: :$S \\subseteq T$ denotes that $S$ is a subset of $T$ :$S \\cup T$ denotes the union of $S$ and $T$ :$\\complement$ denotes set complement :$\\mathbb U$ denotes the universal set. == Proof == Let $X = \\map \\complement S$. Then: {{begin-eqn}} {{eqn | l = X \\subseteq T       | o = \\iff       | r = \\map \\complement X \\cup T = \\mathbb U       | c = Complement Union with Superset is Universe }} {{eqn | ll= \\leadsto       | l = \\map \\complement S \\subseteq T       | o = \\iff       | r = \\map \\complement {\\map \\complement S} \\cup T = \\mathbb U       | c = substituting $X = \\map \\complement S$ }} {{eqn | ll= \\leadsto       | l = \\map \\complement S \\subseteq T       | o = \\iff       | r = S \\cup T = \\mathbb U       | c = Complement of Complement }} {{end-eqn}} {{qed}} == Also see == * Empty Intersection iff Subset of Complement == Sources == * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth|prev = Cardinality of Set Union/Examples/Student Subjects/Mathematics and Chemistry|next = Empty Intersection iff Subset of Complement}}: $\\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $11 \\ \\text{(a)}$ * {{BookReference|Introduction to Topology|1975|Bert Mendelson|ed = 3rd|edpage = Third Edition|prev = Empty Intersection iff Subset of Complement|next = Set Complement inverts Subsets}}: Chapter $1$: Theory of Sets: $\\S 3$: Set Operations: Union, Intersection and Complement: Exercise $1 \\ \\text{(d)}$ Category:Subsets Category:Set Union Category:Set Complement Category:Empty Set qe36vxibbova8xblwt4zz2947z63op2"}
+{"_id": "32309", "title": "Image of Subset under Relation is Subset of Image/Corollary 1", "text": "Image of Subset under Relation is Subset of Image/Corollary 1 0 17281 484938 226706 2020-09-03T12:31:06Z Prime.mover 59 wikitext text/x-wiki == Corollary to Image of Subset under Relation is Subset of Image == Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation from $S$ to $T$. <onlyinclude> Let $C, D \\subseteq T$. Then: :$C \\subseteq D \\implies \\RR^{-1} \\sqbrk C \\subseteq \\RR^{-1} \\sqbrk D$ where $\\RR^{-1} \\sqbrk C$ is the preimage of $C$ under $\\RR$. </onlyinclude> == Proof == We have that $\\RR^{-1}$ is itself a relation, by definition of inverse relation. The result follows directly from Image of Subset under Relation is Subset of Image. {{qed}} Category:Subsets Category:Relation Theory Category:Mapping Theory i5ia4t6iycfmfoqbm1uamvevrgw1yjp"}
+{"_id": "32310", "title": "Image of Subset under Mapping is Subset of Image", "text": "Image of Subset under Mapping is Subset of Image 0 17282 417838 417397 2019-08-11T14:24:15Z Prime.mover 59 wikitext text/x-wiki == Corollary to Image of Subset under Relation is Subset of Image == Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping from $S$ to $T$. <onlyinclude> Let $A, B \\subseteq S$ such that $A \\subseteq B$. Then the image of $A$ is a subset of the image of $B$: :$A \\subseteq B \\implies f \\sqbrk A \\subseteq f \\sqbrk B$ This can be expressed in the language and notation of direct image mappings as: :$\\forall A, B \\in \\powerset S: A \\subseteq B \\implies \\map {f^\\to} A \\subseteq \\map {f^\\to} B$ </onlyinclude> == Proof == As $f: S \\to T$ is a mapping, it is also a relation, and thus: :$f \\subseteq S \\times T$ The result follows directly from Image of Subset under Relation is Subset of Image. {{qed}} == Sources == * {{BookReference|Rings, Modules and Linear Algebra|1970|B. Hartley|author2 = T.O. Hawkes|prev = Image of Preimage under Mapping/Corollary|next = Image of Subset under Relation is Subset of Image/Corollary 3}}: $\\S 2.2$: Homomorphisms: $\\text{(ii)}$ * {{BookReference|Undergraduate Topology|1971|Robert H. Kasriel|prev = Mapping Images are Disjoint only if Domains are Disjoint|next = Image of Union under Mapping}}: $\\S 1.10$: Functions: Exercise $5 \\ \\text{(c)}$ * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth|prev = Direct Image Mapping of Mapping is Empty iff Argument is Empty|next = Image of Subset under Relation is Subset of Image/Corollary 3}}: $\\S 5$. Induced mappings; composition; injections; surjections; bijections: Theorem $5.1: \\ \\text{(i)}$ * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Definition:Surjection/Also known as|next = Image of Intersection under Mapping/Proof 1}}: $\\S 21.3$: The image of a subset of the domain; surjections * {{BookReference|Topology|2000|James R. Munkres|ed = 2nd|edpage = Second Edition|prev = Preimage of Set Difference under Mapping|next = Image of Union under Mapping}}: $1$: Set Theory and Logic: $\\S 2$: Functions: Exercise $2.2 \\ \\text{(e)}$ Category:Subsets Category:Mapping Theory ovvbvaj2tmbfp9jxrpv24wvoupdglyq"}
+{"_id": "32311", "title": "Image of Subset under Relation is Subset of Image/Corollary 3", "text": "Image of Subset under Relation is Subset of Image/Corollary 3 0 17283 417837 417445 2019-08-11T14:24:08Z Prime.mover 59 wikitext text/x-wiki == Corollary of Image of Subset under Relation is Subset of Image == Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping from $S$ to $T$. <onlyinclude> Let $C, D \\subseteq T$. Then: :$C \\subseteq D \\implies f^{-1} \\sqbrk C \\subseteq f^{-1} \\sqbrk D$ This can be expressed in the language and notation of inverse image mappings as: :$\\forall C, D \\in \\powerset T: C \\subseteq D \\implies \\map {f^\\gets} C \\subseteq \\map {f^\\gets} D$ </onlyinclude> == Proof == As $f: S \\to T$ is a mapping, it is also a relation, and thus so is its inverse: :$f^{-1} \\subseteq T \\times S$ The result follows directly from Image of Subset under Relation is Subset of Image. {{qed}} == Sources == * {{BookReference|Rings, Modules and Linear Algebra|1970|B. Hartley|author2 = T.O. Hawkes|prev = Image of Subset under Relation is Subset of Image/Corollary 2|next = Fourth Isomorphism Theorem}}: $\\S 2.2$: Homomorphisms: $\\text{(ii)}$ * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth|prev = Image of Subset under Relation is Subset of Image/Corollary 2|next = Image of Union under Mapping/Proof 1}}: $\\S 5$. Induced mappings; composition; injections; surjections; bijections: Theorem $5.1: \\ \\text{(j)}$ * {{BookReference|Topology|2000|James R. Munkres|ed = 2nd|edpage = Second Edition|prev = Image of Preimage of Subset under Surjection equals Subset|next = Preimage of Union under Mapping}}: $1$: Set Theory and Logic: $\\S 2$: Functions: Exercise $2.2 \\ \\text{(a)}$ Category:Subsets Category:Preimages under Mappings 5xj29ze8j94smaws01ws6c6xapyr8bu"}
+{"_id": "32312", "title": "Subset of Preimage under Relation is Preimage of Subset/Corollary", "text": "Subset of Preimage under Relation is Preimage of Subset/Corollary 0 17289 417845 417454 2019-08-11T14:38:44Z Prime.mover 59 wikitext text/x-wiki == Corollary to Subset of Preimage under Relation is Preimage of Subset == Let $f: S \\to T$ be a mapping. Let $X \\subseteq S, Y \\subseteq T$. Then: <onlyinclude> :$X \\subseteq f^{-1} \\sqbrk Y \\iff f \\sqbrk X \\subseteq Y$ This can be expressed in the language and notation of direct image mappings and inverse image mappings as: :$\\forall X \\in \\powerset S, Y \\in \\powerset T: X \\subseteq \\map {f^\\gets} Y \\iff \\map {f^\\to} X \\subseteq Y$ </onlyinclude> == Proof == Let $f: S \\to T$ be a mapping. As a mapping is also a relation, it follows that $f$ is a relation and so: :$X \\subseteq f^{-1} \\sqbrk Y \\iff f \\sqbrk X \\subseteq Y$ holds on the strength of Subset of Preimage under Relation is Preimage of Subset. {{qed}} == Sources == * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth|prev = Complement of Preimage equals Preimage of Complement|next = Definition:Composition of Mappings/Definition 1}}: $\\S 5$. Induced mappings; composition; injections; surjections; bijections: Theorem $5.1 \\ \\text{(v)}$ Category:Subsets Category:Preimages under Mappings j3spfa22m9v2t52ky5c0280xviuz2f1"}
+{"_id": "32313", "title": "Monotone Convergence Theorem", "text": "Monotone Convergence Theorem 0 17378 497022 97090 2020-10-30T13:59:00Z Prime.mover 59 wikitext text/x-wiki {{Disambiguation}} * Monotone Convergence Theorem (Real Analysis) * Monotone Convergence Theorem (Measure Theory) mbc419yt8xmiqu5qcn58fczbv7yeyti"}
+{"_id": "32314", "title": "Cardinality of Cartesian Product/Corollary", "text": "Cardinality of Cartesian Product/Corollary 0 17491 376852 342139 2018-11-15T22:21:46Z Prime.mover 59 wikitext text/x-wiki == Corollary to Cardinality of Cartesian Product == Let $S \\times T$ be the cartesian product of two sets $S$ and $T$ which are both finite. Then: <onlyinclude> :$\\card {S \\times T} = \\card {T \\times S}$ </onlyinclude> where $\\card {S \\times T}$ denotes the cardinality of $S \\times T$. == Proof 1 == {{:Cardinality of Cartesian Product/Corollary/Proof 1}} == Proof 2 == {{:Cardinality of Cartesian Product/Corollary/Proof 2}} == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Surjection from Finite Set to Itself is Permutation|next = Definition:Binary Operation}}: Chapter $3$. Mappings: Exercise $9 \\ \\text {(i)}$ * {{BookReference|Topology|2000|James R. Munkres|ed = 2nd|edpage = Second Edition|prev = Definition:Countable-Dimensional Real Cartesian Space|next = Definition:Finite Set}}: $1$: Set Theory and Logic: $\\S 5$: Cartesian Products: Exercise $1$ Category:Cardinality of Cartesian Product jget1nbvchrxhkc9v1ziu3iypml1nxj"}
+{"_id": "32315", "title": "Modulo Subtraction is Well-Defined", "text": "Modulo Subtraction is Well-Defined 0 17673 451161 392592 2020-02-28T16:57:46Z Prime.mover 59 wikitext text/x-wiki == Corollary to Modulo Addition is Well-Defined == Let $m \\in \\Z$ be an integer. Let $\\Z_m$ be the set of integers modulo $m$. The modulo subtraction operation on $\\Z_m$, defined by the rule: <onlyinclude> :$\\eqclass a m -_m \\eqclass b m = \\eqclass {a - b} m$ is a well-defined operation. </onlyinclude> That is: :If $a \\equiv b \\pmod m$ and $x \\equiv y \\pmod m$, then $a - x \\equiv b - y \\pmod m$. == Proof == We have: {{begin-eqn}} {{eqn | l = \\eqclass a m -_m \\eqclass b m       | r = \\eqclass {a - b} m       | c =  }} {{eqn | r = \\eqclass {a + \\paren {-b} } m       | c =  }} {{eqn | r = \\eqclass a m +_m \\eqclass {-b} m       | c =  }} {{end-eqn}} The result follows from the fact that Modulo Addition is Well-Defined for all integers. {{qed}} == Examples == {{:Modulo Subtraction is Well-Defined/Examples}} == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Modulo Addition is Well-Defined/Proof 2|next = Modulo Multiplication is Well-Defined}}: $\\S 2.6$. Algebra of congruences: $\\text{(ii)}$ * {{BookReference|Number Theory|1971|George E. Andrews|prev = Modulo Addition is Well-Defined/Proof 1|next = Modulo Multiplication is Well-Defined/Proof 1}}: $\\text {4-1}$ Basic Properties of Congruences: Theorem $\\text {4-2}$ * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Modulo Addition is Well-Defined|next = Modulo Multiplication is Well-Defined}}: $\\S 14.3 \\ \\text {(ii)}$: Congruence modulo $m$ ($m \\in \\N$) * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Modulo Addition is Well-Defined|next = Modulo Multiplication is Well-Defined}}: $\\S 1.2.4$: Integer Functions and Elementary Number Theory: Law $\\text{A}$ * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Modulo Addition is Well-Defined|next = Modulo Multiplication is Well-Defined|entry = congruence}} Category:Modulo Subtraction Category:Modulo Addition Category:Modulo Addition is Well-Defined k8f18w2wfijpyrj6un3079cm8tje8hj"}
+{"_id": "32316", "title": "Order Isomorphism between Ordinals and Proper Class/Lemma", "text": "Order Isomorphism between Ordinals and Proper Class/Lemma 0 17766 335895 297355 2018-01-11T20:45:11Z Barto 3079 notation for image clarified wikitext text/x-wiki {{MissingLinks|one or two}} == Lemma for Order Isomorphism between Ordinals and Proper Class == <onlyinclude> Suppose the following conditions are met: Let $A$ be a class. We allow $A$ to be a proper class or a set. Let $\\left({A, \\prec}\\right)$ be a strict well-ordering. Let every $\\prec$-initial segment be a set, not a proper class. Let $\\operatorname{Im} \\left({x}\\right)$ denote the image of a subclass $x$. Let $G$ equal the class of all ordered pairs $\\left({x, y}\\right)$ satisfying: ::$y \\in \\left({A \\setminus \\operatorname{Im} \\left({x}\\right)}\\right)$ ::The initial segment $A_y$ of $\\left({A, \\prec}\\right)$ is a subset of $\\operatorname{Im} \\left({x}\\right)$ Let $F$ be a mapping with a domain of $\\operatorname{On}$. Let $F$ also satisfy: :$F \\left({x}\\right) = G \\left({F \\restriction x}\\right)$ Then: : $G$ is a mapping : $G \\left({x}\\right) \\in \\left({A \\setminus \\operatorname{Im} \\left({x}\\right)}\\right) \\iff \\left({A \\setminus \\operatorname{Im} \\left({x}\\right)}\\right) \\ne \\varnothing$ </onlyinclude> Note that only the first four conditions need hold: we may construct classes $F$ and $G$ satisfying the other conditions using transfinite recursion. {{explain|Far too woolly. If it is important to the statement of the theorem, then it needs to be expounded properly. If it is not, then it needs to be put somewhere else, either in a separate section, or as a separate proof, or be deleted.}} == Proof == {{begin-eqn}} {{eqn | o =        | r = \\left({\\left({x, y}\\right) \\in G \\land \\left({x, z}\\right) \\in G}\\right)       | c =  }} {{eqn | o = \\implies        | r = \\left({y \\in \\left({A \\setminus \\operatorname{Im} \\left({x}\\right)}\\right) \\land z \\in \\left({A \\setminus \\operatorname{Im} \\left({x}\\right)}\\right)}\\right)       | c = Definition of $G$ }} {{eqn | o = \\implies       | r = \\left({y \\notin A_z \\land y \\notin A_y}\\right)       | c = $A_y$ is disjoint with $\\left({A \\setminus \\operatorname{Im} \\left({x}\\right)}\\right)$.  Same with $A_z$. }} {{eqn | o = \\implies       | r = \\left({y \\not \\prec z \\land z \\not \\prec y}\\right)       | c = Definition of initial segment }} {{eqn | o = \\implies       | r = y = z       | c = $\\prec$ is a strict well-ordering }} {{end-eqn}} Therefore, we may conclude, that $G$ is a single-valued relation and therefore a mapping. For the second part: {{begin-eqn}} {{eqn | o =        | r = \\left({A \\setminus \\operatorname{Im} \\left({x}\\right)}\\right) \\ne \\varnothing       | c =  }} {{eqn | n = 1       | o = \\implies       | r = \\exists y \\in \\left({A \\setminus \\operatorname{Im} \\left({x}\\right)}\\right): \\left({A \\cap A_y}\\right) \\setminus \\operatorname{Im} \\left({x}\\right) = \\varnothing       | c = Proper Well-Ordering Determines Smallest Elements }} {{eqn | o = \\implies       | r = G \\left({x}\\right) = y       | c = Conditions are satisfied for $\\left({x, y}\\right) \\in G$.  Follows from first part. }} {{eqn | o = \\implies       | r = G \\left({x}\\right) \\in \\left({A \\setminus \\operatorname{Im} \\left({x}\\right)}\\right)       | c = equation $(1)$, $y \\in \\left({A \\setminus \\operatorname{Im} \\left({x}\\right)}\\right)$ }} {{end-eqn}} Furthermore: :$G \\left({x}\\right) \\in \\left({A \\setminus \\operatorname{Im} \\left({x}\\right)}\\right) \\implies \\left({A \\setminus \\operatorname{Im} \\left({x}\\right)}\\right) \\ne \\varnothing$ by the definition of nonempty. {{qed}} == Also see == * Transfinite Recursion * Condition for Injective Mapping on Ordinals * Maximal Injective Mapping from Ordinals to a Set == Sources == * {{BookReference|Introduction to Axiomatic Set Theory|1971|Gaisi Takeuti|author2 = Wilson M. Zaring}}: $\\S 7.48$ Category:Ordinals Category:Order Theory jcc4dvmpmjxol5bxzw6uuc74ozq9vsq"}
+{"_id": "32317", "title": "Single Instruction URM Programs/Identity Function", "text": "Single Instruction URM Programs/Identity Function 0 17866 99431 2012-07-29T21:49:10Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> The identity function $I_\\N: \\N \\to \\N$ defined as: :$\\forall n \\in \\N: I_\\N \\l...\" wikitext text/x-wiki == Theorem == <onlyinclude> The identity function $I_\\N: \\N \\to \\N$ defined as: :$\\forall n \\in \\N: I_\\N \\left({n}\\right) = n$ </onlyinclude> is URM computable by a single-instruction URM program. == Proof == Any of the following URM programs compute the identity function: {| |- ! align=\"right\" | Line !! ! align=\"left\" | Command |- | align=\"right\" | $1$ || | align=\"left\" | $Z \\left({m}\\right)$ |} ... where $m \\ne 1$. This sets the value $0$ into $R_m$ and then stops. {| |- ! align=\"right\" | Line !! ! align=\"left\" | Command |- | align=\"right\" | $1$ || | align=\"left\" | $S \\left({m}\\right)$ |} ... where $m \\ne 1$. The input $n$ is in $R_1$ when the program starts. The program adds $1$ to $r_m$ and then stops. {| |- ! align=\"right\" | Line !! ! align=\"left\" | Command |- | align=\"right\" | $1$ || | align=\"left\" | $C \\left({j, m}\\right)$ |} ... where $m \\ne 1$. The input $n$ is in $R_1$ when the program starts. The program copies $r_j$ to $r_m$ and then stops. {| |- ! align=\"right\" | Line !! ! align=\"left\" | Command |- | align=\"right\" | $1$ || | align=\"left\" | $C \\left({1, 1}\\right)$ |} The input $n$ is in $R_1$ when the program starts. The program copies $r_1$ to $r_1$, i.e. to itself, and then stops. In none of these programs is $R_1$ affected, and so no change is effected to the input, which is returned as the output unchanged. Hence they all compute the identity function. {{qed}} Note that the latter program, consisting entirely of $C \\left({1, 1}\\right)$, is a direct implementation of the projection function program $\\operatorname{pr}^1_1: \\N \\to \\N$. This latter function is the usual way of implementing the identity function as it is well-defined and obvious, and guaranteed to have no other side-effects when embedded in a larger program. Category:URM Programs Category:Primitive Recursive Functions qtu82uqpzhzzsxzp19zei4qyxl8tw20"}
+{"_id": "32318", "title": "Cancellability of Congruences/Corollary 1", "text": "Cancellability of Congruences/Corollary 1 0 17943 392725 392713 2019-02-18T22:04:24Z Prime.mover 59 wikitext text/x-wiki == Corollary to Cancellability of Congruences == <onlyinclude> Let $c$ and $n$ be coprime integers, that is: :$c \\perp n$ Then: :$c a \\equiv c b \\pmod n \\implies a \\equiv b \\pmod n$ </onlyinclude> where $\\equiv$ denotes congruence. == Proof 1 == {{:Cancellability of Congruences/Corollary 1/Proof 1}} == Proof 2 == {{:Cancellability of Congruences/Corollary 1/Proof 2}} == Proof 3 == {{:Cancellability of Congruences/Corollary 1/Proof 3}} Category:Cancellability of Congruences 6ulnth22u6ya1pzd0h89ftd3c7ohsry"}
+{"_id": "32319", "title": "Euler Phi Function of Integer/Corollary", "text": "Euler Phi Function of Integer/Corollary 0 18047 491187 342514 2020-09-27T13:00:53Z Prime.mover 59 wikitext text/x-wiki == Corollary to Euler Phi Function of Integer == Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $p$ be a prime number. Let $\\map \\phi n$ denote the Euler $\\phi$ function of $n$. <onlyinclude> Let $p$ be a divisor of $n$. Then $p - 1$ is a divisor of $\\phi \\left({n}\\right)$. </onlyinclude> == Proof == From Euler Phi Function of Integer: :$\\displaystyle \\map \\phi n = n \\prod_{p \\mathop \\divides n} \\paren {1 - \\frac 1 p}$ Let $n$ be expressed as its prime decomposition: :$n = p_1^{k_1} p_2^{k_2} \\ldots p_r^{k_r}, p_1 < p_2 < \\ldots < p_r$ Then: {{begin-eqn}} {{eqn | l = \\phi \\left({n}\\right)       | r = \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i} \\paren {1 - \\dfrac 1 {p_i} }       | c =  }} {{eqn | r = \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i} \\paren {\\dfrac {p_i - 1} {p_i} }       | c =  }} {{eqn | r = \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i - 1} \\paren {p_i - 1}       | c =  }} {{eqn | r = \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i - 1} \\paren {p_i - 1}       | c =  }} {{end-eqn}} Hence the result. {{qed}} Category:Euler Phi Function 1dua6m2t9769x60o0hprendhv8zk4de"}
+{"_id": "32320", "title": "Euler Phi Function of Prime Power/Corollary", "text": "Euler Phi Function of Prime Power/Corollary 0 18049 371220 102037 2018-10-14T12:28:50Z Prime.mover 59 wikitext text/x-wiki == Corollary to Euler Phi Function of Prime Power == Let $\\phi: \\Z_{>0} \\to \\Z_{>0}$ be the Euler $\\phi$ function. Then: <onlyinclude> :$\\map \\phi {2^k} = 2^{k-1}$ </onlyinclude> == Proof == We have that: :$\\displaystyle 1 - \\frac 1 2 = \\frac {2 - 1} 2 = \\frac 1 2$ It follows from Euler Phi Function of Prime Power: :$\\map \\phi {2^k} = \\paren {\\dfrac 1 2} 2^k = 2^{k - 1}$ {{qed}} Category:Euler Phi Function Category:Prime Numbers kcxcsg07owdm84w44f3o5tfdjisapfd"}
+{"_id": "32321", "title": "Abelian Group Factored by Prime/Corollary", "text": "Abelian Group Factored by Prime/Corollary 0 18066 389112 389034 2019-01-22T21:08:48Z Prime.mover 59 wikitext text/x-wiki == Corollary to Abelian Group Factored by Prime == <onlyinclude> Any finite abelian group $G$ can be factored as follows: Let $\\order G = \\displaystyle \\prod_{i \\mathop = 1}^k p_i^{n_i}$ be the prime factorisation of the order of $G$. Then we have $G = \\displaystyle \\prod_{i \\mathop = 1}^k H_i$, where $H_i = \\set {x \\in G : x^{p_i^{n_i} } = e}$. This factorisation is unique up to ordering of the factors. </onlyinclude> == Proof == Let $\\displaystyle \\prod_{i \\mathop = 1}^k p_i^{n_i}$ be the prime factorisation of $\\order G$. We proceed by induction on $k$. === Basis for the induction === For $n = 1$, the statement is trivial. === Induction Hypothesis === Now we assume the theorem is true for abelian groups whose order has $k - 1$ distinct prime factors. === Induction Step === Apply Abelian Group Factored by Prime to $G$ and $p_1$.  By definition, $H = H_1$. Also, the resulting $K$ has $\\order K = \\displaystyle \\prod_{i \\mathop = 2}^k p_i^{n_i}$. Therefore, it satisfies the induction hypothesis. It follows that $G = H_1 \\times \\displaystyle \\prod_{i \\mathop = 2}^k H_i$. From: :Subgroup of Abelian Group is Normal :the definition of Sylow $p$-subgroup all the $H_i$ are normal Sylow $p$-subgroups. From Sylow $p$-Subgroup is Unique iff Normal, the factorisation is unique up to ordering of the factors. {{qed}} Category:Abelian Groups shkkcod24tx5x3qpry41iu7ssb6xpeg"}
+{"_id": "32322", "title": "Measure is Subadditive/Corollary", "text": "Measure is Subadditive/Corollary 0 18086 100774 2012-08-06T07:40:12Z Lord Farin 560 Created page with \"== Theorem ==  Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. <onlyinclude> Let $E_1, \\ldots, E_n \\in \\Sigma$.   Then:  :$\\displaystyle \\...\" wikitext text/x-wiki == Theorem == Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. <onlyinclude> Let $E_1, \\ldots, E_n \\in \\Sigma$. Then: :$\\displaystyle \\mu \\left({\\bigcup_{k \\mathop = 1}^n E_k}\\right) \\le \\sum_{k \\mathop = 1}^n \\mu \\left({E_k}\\right)$. </onlyinclude> == Proof == We have Measure is Subadditive. The result follows by an application of Finite Union of Sets in Subadditive Function. {{qed}} Category:Measure Theory 4yu4a6sg5yzpmyk0x79qp2m3gzyxi8v"}
+{"_id": "32323", "title": "Composite Functor is Functor", "text": "Composite Functor is Functor 0 18201 439677 102954 2019-12-16T08:44:17Z Prime.mover 59 wikitext text/x-wiki == Definition == Let $\\mathbf C, \\mathbf D$ and $\\mathbf E$ be metacategories. Let $F: \\mathbf C \\to \\mathbf D$ and $G: \\mathbf D \\to \\mathbf E$ be covariant functors. Let $GF: \\mathbf C \\to \\mathbf E$ be the composition of $G$ with $F$. Then $G F$ is also a covariant functor. == Proof == Let $f, g$ be morphisms of $\\mathbf C$ such that $g \\circ f$ is defined. Then: {{begin-eqn}} {{eqn | l = \\map {G F} {g \\circ f}       | r = \\map G {\\map F {g \\circ f} }       | c = {{Defof|Composition of Functors}} }} {{eqn | r = \\map G {F g \\circ F f}       | c = $F$ is a Covariant Functor }} {{eqn | r = \\map G {F g} \\circ \\map G {F f}       | c = $G$ is a functor }} {{eqn | r = G F g \\circ G F f       | c = {{Defof|Composition of Functors}} }} {{end-eqn}} Also, for any object $C$ of $\\mathbf C$: {{begin-eqn}} {{eqn | l = G F I_C       | r = \\map G {F I_C}       | c = {{Defof|Composition of Functors}} }} {{eqn | r = G I_{F C}       | c = $F$ is a Covariant Functor }} {{eqn | r = I_{\\map G {F C} }       | c = $G$ is a Covariant Functor }} {{eqn | r = I_{G F C}       | c = {{Defof|Composition of Functors}} }} {{end-eqn}} Hence $G F$ is shown to be a covariant functor. {{qed}} Category:Category Theory d5uraaxxfmvvnznnmua2jo3e8jfd94v"}
+{"_id": "32324", "title": "Transitive Set is Proper Subset of Ordinal iff Element of Ordinal/Corollary", "text": "Transitive Set is Proper Subset of Ordinal iff Element of Ordinal/Corollary 0 18556 175087 107940 2014-01-26T01:40:13Z Prime.mover 59 wikitext text/x-wiki == Corollary to Transitive Set is Proper Subset of Ordinal iff Element of Ordinal == <onlyinclude> Let $A$ and $B$ be ordinals. Then: :$A \\subsetneq B \\iff A \\in B$ </onlyinclude> == Proof == We have that an ordinal is transitive. The result follows directly from Transitive Set is Proper Subset of Ordinal iff Element of Ordinal. {{qed}} == Sources == * {{BookReference|Introduction to Axiomatic Set Theory|1971|Gaisi Takeuti|author2=Wilson M. Zaring}}: $\\S 7.8$ Category:Ordinals qcqrzek318h7uxv2h5ih4a7edxf8ns9"}
+{"_id": "32325", "title": "Absolute Value of Integer is not less than Divisors/Corollary", "text": "Absolute Value of Integer is not less than Divisors/Corollary 0 18700 497629 401283 2020-11-02T22:58:51Z Prime.mover 59 wikitext text/x-wiki == Corollary to Absolute Value of Integer is not less than Divisors == <onlyinclude> Let $a, b \\in \\Z_{>0}$ be (strictly) positive integers. Let $a \\divides b$. Then: :$a \\le b$ </onlyinclude> == Proof == Follows directly from Absolute Value of Integer is not less than Divisors. {{qed}} == Sources == * {{BookReference|Elements of Abstract Algebra|1966|Richard A. Dean|prev = Definition:Integer|next = Definition:Divisor of Integer}}: $\\S 0.1$. Arithmetic: Theorem $1$ * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Divisor Relation is Antisymmetric/Corollary/Proof 2|next = Division Theorem/Positive Divisor/Existence/Proof 3}}: Chapter $3$: The Integers: $\\S 10$. Divisibility: Theorem $17 \\ \\text{(iii)}$ Category:Number Theory Category:Divisors 5u0zsn3k7vf61zx6tz78aj2617jkl42"}
+{"_id": "32326", "title": "Euclid's Lemma for Prime Divisors/General Result", "text": "Euclid's Lemma for Prime Divisors/General Result 0 18713 462840 462839 2020-04-18T17:06:14Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $p$ be a prime number. Let $\\displaystyle n = \\prod_{i \\mathop = 1}^r a_i$. Then if $p$ divides $n$, it follows that $p$ divides $a_i$ for some $i$ such that $1 \\le i \\le r$. That is: :$p \\divides a_1 a_2 \\ldots a_n \\implies p \\divides a_1 \\lor p \\divides a_2 \\lor \\cdots \\lor p \\divides a_n$ </onlyinclude> == Proof 1 == {{:Euclid's Lemma for Prime Divisors/General Result/Proof 1}} == Proof 2 == {{:Euclid's Lemma for Prime Divisors/General Result/Proof 2}} == Proof 3 == {{:Euclid's Lemma for Prime Divisors/General Result/Proof 3}} {{Namedfor|Euclid}} == Sources == * {{BookReference |An Introduction to the Theory of Numbers|1979|G.H. Hardy|author2 = E.M. Wright|ed = 5th|edpage = Fifth Edition|prev = Euclid's Lemma for Prime Divisors|next = Expression for Integer as Product of Primes is Unique/Proof 1}}: $\\text I$: The Series of Primes: $1.3$ Statement of the fundamental theorem of arithmetic Category:Euclid's Lemma for Prime Divisors qcekxkp6nvapbtae89pe99sdjj27qw2"}
+{"_id": "32327", "title": "Euclid's Lemma for Prime Divisors/General Result/Proof 2", "text": "Euclid's Lemma for Prime Divisors/General Result/Proof 2 0 18721 490838 401317 2020-09-25T22:13:55Z Prime.mover 59 wikitext text/x-wiki == Lemma == {{:Euclid's Lemma for Prime Divisors/General Result}} == Proof == <onlyinclude> Proof by induction: For all $r \\in \\N_{>0}$, let $\\map P r$ be the proposition: :$\\displaystyle p \\divides \\prod_{i \\mathop = 1}^r a_i \\implies \\exists i \\in \\closedint 1 r: p \\divides a_i$ $\\map P 1$ is true, as this just says $p \\divides a_1 \\implies p \\divides a_1$. === Basis for the Induction === $\\map P 2$ is the case: :$p \\divides a_1 a_2 \\implies p \\divides a_2$ or $p \\divides a_2$ which is proved in Euclid's Lemma for Prime Divisors. This is our basis for the induction. === Induction Hypothesis === Now we need to show that, if $\\map P k$ is true, where $k \\ge 1$, then it logically follows that $\\map P {k + 1}$ is true. So this is our induction hypothesis: :$\\displaystyle p \\divides \\prod_{i \\mathop = 1}^k a_i \\implies \\exists i \\in \\closedint 1 k: p \\divides a_i$ Then we need to show: :$\\displaystyle p \\divides \\prod_{i \\mathop = 1}^{k + 1} a_i \\implies \\exists i \\in \\closedint 1 {k + 1}: p \\divides a_i$ === Induction Step === This is our induction step: {{begin-eqn}} {{eqn | l = p       | o = \\divides       | r = a_1 a_2 \\ldots a_{k + 1}       | c =  }} {{eqn | ll= \\leadsto       | l = p       | o = \\divides       | r = \\paren {a_1 a_2 \\ldots a_k} \\paren {a_{k + 1} }       | c = }} {{eqn | ll= \\leadsto       | l = p       | o = \\divides       | r = a_1 a_2 \\ldots a_k \\lor p \\divides a_{k + 1}       | c = Basis for the Induction }} {{eqn | ll= \\leadsto       | l = p       | o = \\divides       | r = a_1 \\lor p \\divides a_2 \\lor \\ldots \\lor p \\divides a_k \\lor p \\divides a_{k + 1}       | c = Induction Hypothesis }} {{end-eqn}} So $\\map P k \\implies \\map P {k + 1}$ and the result follows by the Principle of Mathematical Induction. Therefore: :$\\displaystyle \\forall r \\in \\N: p \\divides \\prod_{i \\mathop = 1}^r a_i \\implies \\exists i \\in \\closedint 1 r: p \\divides a_i$ {{qed}} </onlyinclude> {{Namedfor|Euclid}} == Sources == * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Euclid's Lemma for Prime Divisors/Proof 2|next = Fundamental Theorem of Arithmetic}}: Chapter $3$: The Integers: $\\S 12$. Primes: Theorem $21 \\ \\text{(ii)}$ * {{BookReference|Number Theory|1971|George E. Andrews|prev = Euclid's Lemma for Prime Divisors/Proof 2|next = Euclidean Algorithm/Examples/527 and 765}}: $\\text {2-2}$ Divisibility: Corollary $\\text {2-4}$ * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Euclid's Lemma for Prime Divisors/Proof 2|next = Fundamental Theorem of Arithmetic}}: $\\S 12.6$: Highest common factors and Euclid's algorithm * {{BookReference|Computability and Unsolvability|1982|Martin Davis|ed = 2nd|edpage = Second Edition|prev = Euclid's Lemma for Prime Divisors/Proof 2|next = Fundamental Theorem of Arithmetic}}: Appendix $1$: Some Results from the Elementary Theory of Numbers: Corollary $9$ Category:Euclid's Lemma for Prime Divisors Category:Proofs by Induction j804jinwhibh8z6fiajssv98bjlprim"}
+{"_id": "32328", "title": "Euclid's Lemma for Irreducible Elements/General Result", "text": "Euclid's Lemma for Irreducible Elements/General Result 0 18722 402087 402086 2019-04-22T08:02:44Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $\\struct {D, +, \\times}$ be a Euclidean domain whose unity is $1$. <onlyinclude> Let $p$ be an irreducible element of $D$. Let $n \\in D$ such that: :$\\displaystyle n = \\prod_{i \\mathop = 1}^r a_i$ where $a_i \\in D$ for all $i: 1 \\le i \\le r$. If $p$ divides $n$, then $p$ divides $a_i$ for some $i$. That is: :$p \\divides a_1 a_2 \\ldots a_n \\implies p \\divides a_1 \\lor p \\divides a_2 \\lor \\cdots \\lor p \\divides a_n$ </onlyinclude> == Proof == Proof by induction: For all $r \\in \\N_{>0}$, let $\\map P r$ be the proposition: :$\\displaystyle p \\divides \\prod_{i \\mathop = 1}^r a_i \\implies \\exists i \\in \\closedint 1 r: p \\divides a_i$ $\\map P 1$ is true, as this just says $p \\divides a_1 \\implies p \\divides a_1$. === Basis for the Induction === $\\map P 2$ is the case: :$p \\divides a_1 a_2 \\implies p \\divides a_2$ or $p \\divides a_2$ which is proved in Euclid's Lemma for Irreducible Elements. This is our basis for the induction. === Induction Hypothesis === Now we need to show that, if $\\map P k$ is true, where $k \\ge 1$, then it logically follows that $\\map P {k + 1}$ is true. So this is our induction hypothesis: :$\\displaystyle p \\divides \\prod_{i \\mathop = 1}^k a_i \\implies \\exists i \\in \\closedint 1 k: p \\divides a_i$ Then we need to show: :$\\displaystyle p \\divides \\prod_{i \\mathop = 1}^{k + 1} a_i \\implies \\exists i \\in \\closedint 1 {k + 1}: p \\divides a_i$ === Induction Step === This is our induction step: {{begin-eqn}} {{eqn | l = p       | o = \\divides       | r = a_1 a_2 \\ldots a_{k + 1}       | c =  }} {{eqn | ll= \\leadsto       | l = p       | o = \\divides       | r = \\paren {a_1 a_2 \\ldots a_k} \\paren {a_{k + 1} }       | c =  }} {{eqn | ll= \\leadsto       | l = p       | o = \\divides       | r = a_1 a_2 \\ldots a_k \\lor p \\divides a_{k + 1}       | c = Basis for the Induction }} {{eqn | ll= \\leadsto       | l = p       | o = \\divides       | r = a_1 \\lor p \\divides a_2 \\lor \\ldots \\lor p \\divides a_k \\lor p \\divides a_{k + 1}       | c = Induction Hypothesis }} {{end-eqn}} So $\\map P k \\implies \\map P {k + 1}$ and the result follows by the Principle of Mathematical Induction. Therefore: :$\\displaystyle \\forall r \\in \\N: p \\divides \\prod_{i \\mathop = 1}^r a_i \\implies \\exists i \\in \\closedint 1 r: p \\divides a_i$ {{qed}} {{Namedfor|Euclid}} == Also see == * Euclid's Lemma for Prime Divisors, for the usual statement of this result, which is this lemma as applied specifically to the integers. == Sources == * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Euclid's Lemma for Irreducible Elements|next = Polynomial Forms over Field form Principal Ideal Domain/Corollary 2}}: Chapter $6$: Polynomials and Euclidean Rings: $\\S 29$. Irreducible elements: Theorem $56 \\ \\text{(ii)}$ Category:Euclidean Domains Category:Euclid's Lemma 2sd682kw559rrgpxie3yybdzq0nfccp"}
+{"_id": "32329", "title": "Common Divisor Divides Integer Combination", "text": "Common Divisor Divides Integer Combination 0 18747 493818 462684 2020-10-10T21:57:07Z Prime.mover 59 wikitext text/x-wiki == Corollary to Common Divisor in Integral Domain Divides Linear Combination == <onlyinclude> Let $c$ be a common divisor of two integers $a$ and $b$. That is: :$a, b, c \\in \\Z: c \\divides a \\land c \\divides b$ Then $c$ divides any integer combination of $a$ and $b$: :$\\forall p, q \\in \\Z: c \\divides \\paren {p a + q b}$ </onlyinclude> === Corollary === {{:Common Divisor Divides Integer Combination/Corollary}} === General Result === {{:Common Divisor Divides Integer Combination/General Result}} == Proof 1 == {{:Common Divisor Divides Integer Combination/Proof 1}} == Proof 2 == {{:Common Divisor Divides Integer Combination/Proof 2}} == Sources == * {{BookReference |An Introduction to the Theory of Numbers|1979|G.H. Hardy|author2 = E.M. Wright|ed = 5th|edpage = Fifth Edition|prev = Multiple of Divisor Divides Multiple|next = Definition:Prime Number/Definition 4}}: $\\text I$: The Series of Primes: $1.1$ Divisibility of integers * {{BookReference|Elementary Number Theory|1980|David M. Burton|ed = revised|edpage = Revised Printing|prev = Absolute Value of Integer is not less than Divisors|next = Common Divisor Divides Integer Combination/General Result}}: Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Theorem $2 \\text{-} 2 \\ (7)$ * {{BookReference|A Course in Number Theory|1994|H.E. Rose|ed = 2nd|edpage = Second Edition|prev = Divisor Relation is Transitive|next = Definition:Greatest Common Divisor/Integers/Definition 2}}: $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization: $\\text {(v)}$ Category:Divisors Category:Integer Combinations Category:Common Divisor Divides Integer Combination b2qsir1pvol6l0ql0sfpxscdcek3kax"}
+{"_id": "32330", "title": "Divisor Relation is Antisymmetric/Corollary/Proof 2", "text": "Divisor Relation is Antisymmetric/Corollary/Proof 2 0 18762 447447 411262 2020-02-08T13:30:51Z Prime.mover 59 wikitext text/x-wiki == Corollary to Divisor Relation is Antisymmetric == {{:Divisor Relation is Antisymmetric/Corollary}} == Proof == <onlyinclude> Let $a \\divides b$ and $b \\divides a$. Then by definition of divisor: :$\\exists c, d \\in \\Z: a c = b, b d = a$ Thus: {{begin-eqn}} {{eqn | l = a c d       | r = a       | c =  }} {{eqn | ll= \\leadsto       | l = c d       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = d       | r = \\pm 1       | c = Divisors of One }} {{eqn | ll= \\leadsto       | l = a       | r = \\pm b       | c = as $a = b d$ }} {{end-eqn}} {{qed}} </onlyinclude> == Sources == * {{BookReference|Lectures in Abstract Algebra|1951|Nathan Jacobson|volume = I|subtitle = Basic Concepts|prev = Divisor Relation is Transitive|next = Definition:Greatest Common Divisor/Integers/Definition 2}}: Introduction $\\S 6$: The division process in $I$ * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Divisors of One|next = Absolute Value of Integer is not less than Divisors/Corollary}}: Chapter $3$: The Integers: $\\S 10$. Divisibility: Theorem $17 \\ \\text{(ii)}$ * {{BookReference|Algebra Volume 1|1982|P.M. Cohn|edpage = Second Edition|ed = 2nd|prev = Divisor Divides Multiple|next = Definition:Associate of Integer}}: Chapter $2$: Integers and natural numbers: $\\S 2.2$: Divisibility and factorization in $\\mathbf Z$ Category:Divisor Relation is Antisymmetric d5drfre7at4ibdme38oh5qjbo896q6d"}
+{"_id": "32331", "title": "Polynomial Forms over Field form Principal Ideal Domain/Corollary 3", "text": "Polynomial Forms over Field form Principal Ideal Domain/Corollary 3 0 18766 489212 488790 2020-09-19T13:30:43Z Prime.mover 59 wikitext text/x-wiki == Corollary to Polynomial Forms over Field form Principal Ideal Domain == Let $\\struct {F, +, \\circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $X$ be transcendental over $F$. Let $F \\sqbrk X$ be the ring of polynomials in $X$ over $F$. Then <onlyinclude>$F \\sqbrk X$ is a unique factorization domain. </onlyinclude> == Proof == We have the result Principal Ideal Domain is Unique Factorization Domain. The result then follows from Polynomial Forms over Field form Principal Ideal Domain. {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Polynomial Forms over Field form Principal Ideal Domain/Proof 1|next = Polynomial Forms over Field form Principal Ideal Domain/Corollary 1}}: $\\S 65.2$ Some properties of $F \\sqbrk X$, where $F$ is a field Category:Polynomial Forms over Field form Principal Ideal Domain Category:Unique Factorization Domains eh1gxv09ue7w3wdhy4cflj5ey11daxz"}
+{"_id": "32332", "title": "Polynomial Forms over Field form Principal Ideal Domain/Corollary 1", "text": "Polynomial Forms over Field form Principal Ideal Domain/Corollary 1 0 18767 489213 488791 2020-09-19T13:30:54Z Prime.mover 59 wikitext text/x-wiki == Corollary to Polynomial Forms over Field form Principal Ideal Domain == Let $\\struct {F, +, \\circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $X$ be transcendental over $F$. Let $F \\sqbrk X$ be the ring of polynomials in $X$ over $F$. <onlyinclude> Let $f$ be an irreducible element of $F \\sqbrk X$. Then $F \\sqbrk X / \\ideal f$ is a field, where $\\ideal f$ denotes the ideal generated by $f$. </onlyinclude> == Proof == It follows from Principal Ideal of Principal Ideal Domain is of Irreducible Element iff Maximal that $\\ideal f$ is maximal for irreducible $f$. Therefore by Maximal Ideal iff Quotient Ring is Field, $F \\sqbrk X / \\ideal f$ is a field. {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Polynomial Forms over Field form Unique Factorization Domain|next = Field of Quotients of Ring of Polynomial Forms on Reals that yields Complex Numbers}}: $\\S 65.3$ Some properties of $F \\sqbrk X$, where $F$ is a field Category:Polynomial Forms over Field form Principal Ideal Domain klnder8eo8jiru8ocohfjwk67o1dh9c"}
+{"_id": "32333", "title": "No Arithmetic Sequence of 4 Primes with Common Difference 2", "text": "No Arithmetic Sequence of 4 Primes with Common Difference 2 0 18783 456429 456424 2020-03-19T17:27:22Z Prime.mover 59 wikitext text/x-wiki == Definition == There exist no $n \\in \\Z_{>0}$ such that $n, n + 2, n + 4, n + 6$ are all prime. === Corollary === {{:No Arithmetic Sequence of 4 Primes with Common Difference 2/Corollary}} == Proof == {{AimForCont}} $S$ is a set of $4$ prime numbers of the form $n, n + 2, n + 4, n + 6$. $S$ must contain as a subset a prime triplet. From Prime Triplet is Unique, the only one of these is $\\set {3, 5, 7}$. The only sets of the form $\\set {n, n + 2, n + 4, n + 6}$ containing $\\set {3, 5, 7}$ are: :$(1): \\quad \\set {1, 3, 5, 7}$: as $1$ is by convention not a prime, then this is not $S$. :$(2): \\quad \\set {3, 5, 7, 9}$: as $9 = 3 \\times 3$ is not a prime, then this is not $S$. There are no more possible $\\set {n, n + 2, n + 4, n + 6}$ all prime. Hence, by Proof by Contradiction, $S$ does not exist. {{qed}} Category:Prime Numbers Category:Arithmetic Sequences meyav5fl4qmca6v01w3blr8myf3737d"}
+{"_id": "32334", "title": "Cardinal Number Less than Ordinal/Corollary", "text": "Cardinal Number Less than Ordinal/Corollary 0 18870 113041 105265 2012-10-28T08:27:31Z Prime.mover 59 wikitext text/x-wiki == Corollary to Cardinal Number Less than Ordinal == <onlyinclude> Let $x$ be an ordinal. Let $\\left|{x}\\right|$ denote the cardinal number of $x$. Then: :$\\left|{x}\\right| \\le x$ </onlyinclude> == Proof == By Set Equivalence is Equivalence Relation: : $x \\sim x$ By Cardinal Number Less than Ordinal: : $\\left|{x}\\right| \\le x$ {{qed}} == Sources == * {{BookReference|Introduction to Axiomatic Set Theory|1971|Gaisi Takeuti|author2=Wilson M. Zaring}}: $\\S 10.13$ Category:Cardinals Category:Ordinals lluo5cfjxrroelppyxemowssnp4ensp"}
+{"_id": "32335", "title": "Set of Words Generates Group/Corollary", "text": "Set of Words Generates Group/Corollary 0 19302 440598 107132 2019-12-23T11:30:02Z Prime.mover 59 wikitext text/x-wiki == Corollary to Set of Words Generates Group == Let $G$ be a group. Let $T \\subseteq G$. Let $\\map W T$ be the set of words of $T$. <onlyinclude> If $T$ is closed under taking inverses, then $\\map W T$ is a subgroup of $G$. </onlyinclude> == Proof == This follows directly from Set of Words Generates Group and the fact that $T$ has the same properties as $\\hat S$ in that result. {{qed}} Category:Group Theory ojhkc24jnwmrt2q5h30afhh2bmg1sw2"}
+{"_id": "32336", "title": "Subset Product within Semigroup is Associative/Corollary", "text": "Subset Product within Semigroup is Associative/Corollary 0 19550 378263 371306 2018-11-24T16:28:47Z Prime.mover 59 wikitext text/x-wiki == Corollary to Subset Product within Semigroup is Associative == <onlyinclude> Let $\\struct {S, \\circ}$ be a semigroup. Then: {{begin-eqn}} {{eqn | l = x \\paren {y S}       | r = \\paren {x y} S }} {{eqn | l = x \\paren {S y}       | r = \\paren {x S} y }} {{eqn | l = \\paren {S x} y       | r = S \\paren {x y} }} {{end-eqn}} </onlyinclude> == Proof == From the definition of Subset Product with Singleton: {{begin-eqn}} {{eqn | l = x \\paren {y S}       | r = \\set x \\paren {\\set y S} }} {{eqn | l = x \\paren {S y}       | r = \\set x \\paren {S \\set y} }} {{eqn | l = \\paren {S x} y       | r = \\paren {S \\set x} \\set y }} {{end-eqn}} The result then follows directly from Subset Product within Semigroup is Associative. {{qed}} == Also see == * Subset Product within Commutative Structure is Commutative == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Definition:Subset Product with Singleton|next = Definition:Right Coset}}: $\\S 6.1$. The quotient sets of a subgroup Category:Subset Products 428at3duvbkpwh0be1wa186nd0yyl9q"}
+{"_id": "32337", "title": "Partition Topology is T4", "text": "Partition Topology is T4 0 19748 470070 469187 2020-05-23T21:52:16Z Prime.mover 59 wikitext text/x-wiki == Corollary to Partition Topology is T5 == Let $S$ be a set and let $\\PP$ be a partition on $S$ which is not the (trivial) partition of singletons. Let $T = \\struct {S, \\tau}$ be the partition space whose basis is $\\PP$. Then: <onlyinclude> :$T$ is a $T_4$ space. </onlyinclude> == Proof == We have that the Partition Topology is $T_5$. We also have that a $T_5$ Space is $T_4$ Space. The result follows. {{qed}} == Sources == * {{BookReference|Counterexamples in Topology|1978|Lynn Arthur Steen|author2 = J. Arthur Seebach, Jr.|ed = 2nd|edpage = Second Edition|prev = Partition Topology is T3 1/2|next = Partition Topology is T5}}: Part $\\text {II}$: Counterexamples: $5$. Partition Topology: $2$ Category:T4 Spaces Category:Partition Topology krnrmpuxydyeriig2u8po6xswjdg64a"}
+{"_id": "32338", "title": "Pointwise Addition on Real-Valued Functions is Commutative", "text": "Pointwise Addition on Real-Valued Functions is Commutative 0 19758 473610 472906 2020-06-12T08:19:56Z Prime.mover 59 wikitext text/x-wiki == Definition == Let $S$ be a set. <onlyinclude> Let $f, g: S \\to \\R$ be real-valued functions. Let $f + g: S \\to \\R$ denote the pointwise sum of $f$ and $g$. Then: :$f + g = g + f$ </onlyinclude> That is, pointwise addition of real-valued functions is commutative. == Proof == {{begin-eqn}} {{eqn | lo= \\forall x \\in S:       | l = \\map {\\paren {f + g} } x       | r = \\map f x + \\map g x       | c = {{Defof|Pointwise Addition of Real-Valued Functions}} }} {{eqn | r = \\map g x + \\map f x       | c = Real Addition is Commutative }} {{eqn | r = \\map {\\paren {g + f} } x       | c = {{Defof|Pointwise Addition of Real-Valued Functions}} }} {{end-eqn}} {{qed}} Category:Pointwise Addition Category:Real Addition Category:Commutativity g2yuv1giptnoh4uoh5b2ierg1xnwcbi"}
+{"_id": "32339", "title": "Compact Subspace of Hausdorff Space is Closed/Corollary", "text": "Compact Subspace of Hausdorff Space is Closed/Corollary 0 19761 310455 179959 2017-08-12T06:50:28Z Prime.mover 59 wikitext text/x-wiki == Corollary to Compact Subspace of Hausdorff Space is Closed == <onlyinclude> A finite subspace of a Hausdorff space is closed. </onlyinclude> == Proof == Follows directly from: : Finite Topological Space is Compact and: : Compact Subspace of Hausdorff Space is Closed. {{qed}} Category:Finite Topological Spaces Category:Compact Subspace of Hausdorff Space is Closed qlgcadopcukk0um0hgwvfipvbwlswp4"}
+{"_id": "32340", "title": "Brahmagupta-Fibonacci Identity/Extension", "text": "Brahmagupta-Fibonacci Identity/Extension 0 19835 396426 396423 2019-03-20T22:11:06Z Prime.mover 59 wikitext text/x-wiki == Extension to Brahmagupta-Fibonacci Identity == <onlyinclude> Let $a_1, a_2, \\ldots, a_n, b_1, b_2, \\ldots, b_n$ be integers. Then: :$\\displaystyle \\prod_{j \\mathop = 1}^n \\paren { {a_j}^2 + {b_j}^2} = c^2 + d^2$ where $c, d \\in \\Z$. That is: the product of any number of sums of two squares is also a sum of two squares. </onlyinclude> === General Version === {{:Brahmagupta-Fibonacci Identity/Extension/General}} == Proof 1 == {{:Brahmagupta-Fibonacci Identity/Extension/Proof 1}} == Proof 2 == {{:Brahmagupta-Fibonacci Identity/Extension/Proof 2}} == Proof 3 == {{:Brahmagupta-Fibonacci Identity/Extension/Proof 3}} Category:Brahmagupta-Fibonacci Identity jw9pxxhqrn182b29d5c2czrn1rvtyhc"}
+{"_id": "32341", "title": "Sum of Squared Deviations from Mean/Corollary 1", "text": "Sum of Squared Deviations from Mean/Corollary 1 0 19995 488868 348299 2020-09-18T12:17:29Z Prime.mover 59 wikitext text/x-wiki == Theorem  == Let $x_1, x_2, \\ldots, x_n$ be real data about some quantitative variable. Let $\\overline x$ be the arithmetic mean of the above data. Then:  <onlyinclude> :$\\displaystyle \\sum_{i \\mathop = 1}^n \\paren {x_i - \\overline x}^2 = \\sum_{i \\mathop = 1}^n x_i^2 - n \\overline x^2$ </onlyinclude>  == Proof == {{begin-eqn}} {{eqn | l = \\sum_{i \\mathop = 1}^n \\paren {x_i - \\overline x}^2       | r = \\sum_{i \\mathop = 1}^n \\paren {x_i^2 - \\overline x^2}       | c = Sum of Squared Deviations from Mean }} {{eqn | r = \\sum_{i \\mathop = 1}^n x_i^2 - \\sum_{i \\mathop = 1}^n \\overline x^2       | c = Summation is Linear }} {{eqn | r = \\sum_{i \\mathop = 1}^n x_i^2 - n \\overline x^2       | c = Sum of Identical Terms }} {{end-eqn}} {{qed}} Category:Sum of Squared Deviations from Mean d80mi4gqu49iwugeua4n833grnqgns6"}
+{"_id": "32342", "title": "Limit of Sequence to Zero Distance Point/Corollary 1", "text": "Limit of Sequence to Zero Distance Point/Corollary 1 0 20153 408658 408654 2019-06-16T22:26:39Z Prime.mover 59 wikitext text/x-wiki == Corollary to Limit of Sequence to Zero Distance Point == Let $S$ be a non-empty subset of $\\R$. Let the distance $\\map d {\\xi, S} = 0$ for some $\\xi \\in \\R$. <onlyinclude> If $S$ is bounded above, then there exists a sequence $\\sequence {x_n}$ in $S$ such that $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = \\sup S$. </onlyinclude> == Proof == Let  $\\xi = \\sup S$. Then from Distance from Subset of Real Numbers: :$\\map d {\\xi, S} = 0$ The result then follows directly from Limit of Sequence to Zero Distance Point. Note that the terms of this sequence do not necessarily have to be distinct. {{Qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Limit of Sequence to Zero Distance Point|next = Limit of Sequence to Zero Distance Point/Corollary 2}}: $\\S 4$: Convergent Sequences: Exercise $\\S 4.29 \\ (6)$ Category:Limit of Sequence to Zero Distance Point rjfax6iujwpafbe3bcdtct9a2b86yvn"}
+{"_id": "32343", "title": "Derivative of Function of Constant Multiple/Corollary", "text": "Derivative of Function of Constant Multiple/Corollary 0 20203 447516 447515 2020-02-08T19:18:17Z Prime.mover 59 wikitext text/x-wiki == Corollary of Derivative of Function of Constant Multiple == Let $f$ be a real function which is differentiable on $\\R$. <onlyinclude> Let $a, b \\in \\R$ be constants. Then: :$\\map {D_x} {\\map f {a x + b} } = a \\, \\map {D_{a x + b} } {\\map f {a x + b} }$ </onlyinclude> == Proof == First it is shown that $\\map {D_x} {a x + b} = a$: {{begin-eqn}} {{eqn | n = 1       | l = \\map {D_x} {a x + b}       | r = \\map {D_x} {a x} + \\map {D_x} b       | c = Sum Rule for Derivatives }} {{eqn | r = a + 0       | c = Derivative of Function of Constant Multiple and Derivative of Constant }} {{eqn | r = a       | c =  }} {{end-eqn}} Next: {{begin-eqn}} {{eqn | l = \\map {D_x} {\\map f {a x + b} }       | r = \\map {D_x} {a x + b} \\, \\map {D_{a x + b} } {\\map f {a x + b} }       | c = Chain Rule for Derivatives }} {{eqn | r = a \\, \\map {D_{a x + b} } {\\map f {a x + b} }       | c = from $(1)$ }} {{end-eqn}} {{qed}} Category:Differential Calculus aj264wkpsc0lsl0e6edhp2m9ze8ee0l"}
+{"_id": "32344", "title": "Open Real Interval is Open Set/Corollary", "text": "Open Real Interval is Open Set/Corollary 0 20317 467826 202377 2020-05-14T06:21:02Z Prime.mover 59 wikitext text/x-wiki == Corollary to Open Real Interval is Open Set == <onlyinclude> Let $\\R$ be the real number line considered as an Euclidean space. Let $A := \\openint a \\infty \\subset \\R$ be an open interval of $\\R$. Let $B := \\openint {-\\infty} b \\subset \\R$ be an open interval of $\\R$. Then both $A$ and $B$ are open sets of $\\R$. </onlyinclude> == Proof == From Open Real Interval is Open Set we have that for any $c \\in \\openint a b$ there exists an open $\\epsilon$-ball of $c$ lying wholly within $\\openint a b$. When either of $a \\to -\\infty$ or $b \\to \\infty$ the result still holds. The result follows by definition of open set. {{qed}} Category:Real Intervals Category:Open Sets 8gsilbpdhc1wyoz828748s8ia5w373g"}
+{"_id": "32345", "title": "Negative of Absolute Value/Corollary 1", "text": "Negative of Absolute Value/Corollary 1 0 20525 407044 407015 2019-06-08T13:50:48Z Prime.mover 59 wikitext text/x-wiki == Corollary to Negative of Absolute Value == Let $x, y \\in \\R$ be a real numbers. Let $\\size x$ be the absolute value of $x$. Then: <onlyinclude> :$\\size x < y \\iff -y < x < y$ </onlyinclude> == Proof == === Necessary Condition === Let $\\size x < y$. Then from Negative of Absolute Value: :$x \\le \\size x$ and: :$\\size x \\ge -x$ So $x < y$ and $-x < y$, and so $x > -y$ from Ordering of Inverses in Ordered Monoid. It follows that $-y < x < y$. {{qed|lemma}} === Sufficient Condition === Let $-y < x < y$. Then $x < y$ and $-x < y$. For all $x$: :$\\size x = x$ or: :$\\size x = -x$ Thus it follows that $\\size x < y$. {{qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Reverse Triangle Inequality/Real and Complex Fields/Examples/6 - (-1)|next = Reverse Triangle Inequality/Real and Complex Fields/Proof 2}}: $\\S 1$: Real Numbers: Exercise $\\S 1.20 \\ (1)$ Category:Negative of Absolute Value 0hochhv3flq6ud3fsdquw2ik490b1n6"}
+{"_id": "32346", "title": "Continuous Image of Connected Space is Connected/Corollary 1", "text": "Continuous Image of Connected Space is Connected/Corollary 1 0 20580 337657 328301 2018-01-16T23:21:45Z Prime.mover 59 wikitext text/x-wiki == Corollary to Continuous Image of Connected Space is Connected == <onlyinclude> Connectedness is a topological property. </onlyinclude> == Proof == Follows directly from Continuous Image of Connected Space is Connected and the definition of topological property. {{qed}} == Sources == * {{BookReference|Introduction to Metric and Topological Spaces|1975|W.A. Sutherland|prev = Continuous Image of Connected Space is Connected/Proof 1|next = Continuous Image of Connected Space is Connected/Corollary 2}}: $6.2$: Connectedness: Corollary $6.2.11$ Category:Connected Spaces Category:Continuous Image of Connected Space is Connected o4lnwd6b381lp6ql3phnjk8i8c7rscv"}
+{"_id": "32347", "title": "Continuous Image of Connected Space is Connected/Corollary 2", "text": "Continuous Image of Connected Space is Connected/Corollary 2 0 20581 337658 328302 2018-01-16T23:22:29Z Prime.mover 59 wikitext text/x-wiki == Corollary to Continuous Image of Connected Space is Connected == <onlyinclude> Let $T$ be a connected topological space. Let $f: T \\to \\R$ be a continuous real-valued mapping. Then $f \\left({T}\\right)$ is a real interval. </onlyinclude> == Proof == From Continuous Image of Connected Space is Connected, the continuous image of the connected space $T$ is connected. The result follows from Subset of Real Numbers is Interval iff Connected. {{qed}} == Sources == * {{BookReference|Introduction to Metric and Topological Spaces|1975|W.A. Sutherland|prev = Continuous Image of Connected Space is Connected/Corollary 1|next = Continuous Image of Connected Space is Connected/Corollary 3}}: $6.2$: Connectedness: Corollary $6.2.13$ Category:Connected Spaces Category:Continuous Mappings Category:Continuous Functions Category:Analysis Category:Continuous Image of Connected Space is Connected lyx1ep9b71ma9m3njstfgccykewvbcz"}
+{"_id": "32348", "title": "Image of Countable Set under Mapping is Countable", "text": "Image of Countable Set under Mapping is Countable 0 20635 336022 189918 2018-01-12T08:25:39Z Barto 3079 stable link to image of mapping wikitext text/x-wiki == Definition == Let $S$ be a countable set. Let $T$ be a set. Let $f: S \\to T$ be a mapping. Then the image of $f$ is countable. == Proof == Let $A$ be the image of $f$. Let $g: S \\to A$ be the restriction of $f$ to the cartesian product $S \\times A$. By Surjection by Restriction of Codomain, $g$ is a surjection. By Surjection from Natural Numbers iff Countable, there exists a surjection $\\phi: \\N \\to S$. Since the composition of surjections is a surjection, $g \\circ \\phi: \\N \\to A$ is a surjection. Hence, by Surjection from Natural Numbers iff Countable, it follows that $A$ is countable. {{qed}} Category:Countable Sets Category:Mapping Theory qqh9cn9h9q8i9ky1p5fbcwqrkel1phv"}
+{"_id": "32349", "title": "Infinite Sequence in Countably Compact Space has Accumulation Point", "text": "Infinite Sequence in Countably Compact Space has Accumulation Point 0 20668 471376 352748 2020-05-29T17:31:18Z Prime.mover 59 wikitext text/x-wiki == Corollary to Countably Infinite Set in Countably Compact Space has Omega-Accumulation Point == Let $T = \\struct {S, \\tau}$ be a countably compact topological space. <onlyinclude> Let $\\sequence {x_n}_{n \\mathop \\in \\N}$ be an infinite sequence in $S$. Then $\\sequence {x_n}$ has an accumulation point in $T$. </onlyinclude> == Proof == Let $A \\subseteq S$ be the range of $\\sequence {x_n}$: :$A = \\set {x_n: n \\in \\N}$ If $A$ is finite, then consider the equality: :$\\displaystyle \\N = \\bigcup_{y \\mathop \\in A} \\set {n \\in \\N: x_n = y}$ Therefore, there exists a $y \\in A$ such that $\\set {n \\in \\N: x_n = y}$ is an infinite set. Hence, $y$ is an accumulation point of $\\sequence {x_n}$. Otherwise, $A$ is countably infinite. Then $A$ has an $\\omega$-accumulation point in $T$. It follows that $\\sequence {x_n}$ has an accumulation point in $T$. {{explain|Link to a result that states that if $T$ has an $\\omega$-accumulation point, then $\\sequence {x_n}$ has an accumulation point}} {{qed}} == Sources == * {{BookReference|Counterexamples in Topology|1978|Lynn Arthur Steen|author2 = J. Arthur Seebach, Jr.|ed = 2nd|edpage = Second Edition|prev = Definition:Countable Finite Intersection Axiom|next = Countably Compact Space satisfies Countable Finite Intersection Axiom}}: Part $\\text I$: Basic Definitions: Section $3$: Compactness: Global Compactness Properties Category:Countably Compact Spaces Category:Sequences Category:Accumulation Points dzx7hnopemrgv97o7xm8894azbpua0x"}
+{"_id": "32350", "title": "L'Hôpital's Rule/Corollary 1", "text": "L'Hôpital's Rule/Corollary 1 0 20803 480849 480847 2020-08-04T05:21:44Z Prime.mover 59 wikitext text/x-wiki == Corollary to L'Hôpital's Rule == Let $f$ and $g$ be real functions which are continuous on the closed interval $\\closedint a b$ and differentiable on the open interval $\\openint a b$. Suppose that $\\forall x \\in \\openint a b: \\map {g^{\\prime} } x \\ne 0$. <onlyinclude> Suppose that $\\exists c \\in \\openint a b: \\map f c = \\map g c = 0$. Then: :$\\displaystyle \\lim_{x \\mathop \\to c} \\frac {\\map f x} {\\map g x} = \\lim_{x \\mathop \\to c} \\frac {\\map {f^{\\prime} } x} {\\map {g^{\\prime} } x}$ provided that the second limit exists. </onlyinclude> == Proof == This follows directly from the definition of limit. If $\\displaystyle \\lim_{x \\mathop \\to c} \\frac {\\map {f^{\\prime} } x} {\\map {g^{\\prime} } x}$ exists, it follows that: :$\\displaystyle \\lim_{x \\mathop \\to c} \\frac {\\map {f^{\\prime} } x} {\\map {g^{\\prime} } x} = \\lim_{x \\to c^+} \\frac {\\map {f^{\\prime} } x} {\\map {g^{\\prime} } x}$ That is, if there exists such a limit, it is also a limit from the right. {{qed}} {{Namedfor|Guillaume de l'Hôpital|cat = L'Hôpital}} == Historical Note == {{:L'Hôpital's Rule/Historical Note}} Category:Differential Calculus Category:Limits of Functions Category:L'Hôpital's Rule g3ve7t2qrgnf63pv1732omz1jws77lb"}
+{"_id": "32351", "title": "L'Hôpital's Rule/Corollary 2", "text": "L'Hôpital's Rule/Corollary 2 0 20804 480850 408110 2020-08-04T05:32:14Z Prime.mover 59 wikitext text/x-wiki == Corollary to L'Hôpital's Rule == Let $f$ and $g$ be real functions which are continuous on the closed interval $\\closedint a b$ and differentiable on the open interval $\\openint a b$. Suppose that $\\forall x \\in \\openint a b: \\map {g'} x \\ne 0$. <onlyinclude> Suppose that $\\map f x \\to \\infty$ and $\\map g x \\to \\infty$ as $x \\to a^+$. Then: :$\\displaystyle \\lim_{x \\mathop \\to a^+} \\frac {\\map f x} {\\map g x} = \\lim_{x \\mathop \\to a^+} \\frac {\\map {f'} x} {\\map {g'} x}$ provided that the second limit exists. </onlyinclude> == Proof == Let $\\displaystyle \\lim_{x \\mathop \\to a^+} \\frac {\\map {f'} x} {\\map {g'} x} = L$. Let $\\sequence {x_n}$ be a sequence such that: :$\\quad x_n \\in \\openint a b$ for all $n \\in \\N$ and $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = a$ From Intermediate Value Theorem for Derivatives and the definition of limit of real function, it follows that: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map g {x_n} = \\infty$ and $\\sequence {\\map g {x_n} }$ is strictly increasing. Consider the range $\\closedint {x_{n - 1} } {x_n} \\subset \\openint a b$ where $n \\ge 2$. By Cauchy Mean Value Theorem, there exists $c_n \\in \\openint {x_{n - 1} } {x_n}$ such that: :$\\displaystyle \\frac {\\map f {x_n} - \\map f {x_{n - 1} } } {\\map g {x_n} - \\map g {x_{n - 1} } } = \\frac {\\map {f'} {c_n} } {\\map {g'} {c_n} }$ From the above and the Squeeze Theorem for Real Sequences: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} c_n = a$ and: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\frac {\\map f {x_n} - \\map f {x_{n - 1} } } {\\map g {x_n} - \\map g {x_{n - 1} } } = \\lim_{n \\mathop \\to \\infty} \\frac {\\map {f'} {c_n} } {\\map {g'} {c_n} } = L$ So, by Stolz-Cesàro Theorem: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\frac {\\map f {x_n} } {\\map g {x_n} } = L$ From the definition of limit of real function, we deduce that: :$\\displaystyle \\lim_{x \\mathop \\to a^+} \\frac {\\map f x} {\\map g x} = L = \\lim_{x \\mathop \\to a^+} \\frac {\\map {f'} x} {\\map {g'} x}$ {{qed}} == Remarks == * The proof does not actually use the assumption $\\displaystyle \\lim_{x \\mathop \\to a^+} \\map f x = \\infty$. {{refactor|Add these as separate corollaries}} * Cases $x \\to b^-$, $x \\to \\pm \\infty$ and $\\map g x \\to -\\infty$ can be proven similarly. {{Namedfor|Guillaume de l'Hôpital|cat = L'Hôpital}} == Historical Note == {{:L'Hôpital's Rule/Historical Note}} Category:Differential Calculus Category:Limits of Functions Category:L'Hôpital's Rule gbkfejy6y0xc3hs9j3bdnnnzzzp7mxh"}
+{"_id": "32352", "title": "Law of Inverses (Modulo Arithmetic)/Corollary 1", "text": "Law of Inverses (Modulo Arithmetic)/Corollary 1 0 20969 392611 264171 2019-02-17T23:13:43Z Prime.mover 59 wikitext text/x-wiki == Corollary to Law of Inverses (Modulo Arithmetic) == <onlyinclude> Let $m, n \\in \\Z$ such that: :$m \\perp n$ that is, such that $m$ and $n$ are coprime. Then: :$\\exists n' \\in \\Z: n n' \\equiv 1 \\pmod m$ </onlyinclude> == Proof == By definition of coprime: :$m \\perp n \\iff \\gcd \\set {m, n} = 1$ The result follows directly from Law of Inverses (Modulo Arithmetic). {{qed}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Chinese Remainder Theorem|next = Common Factor Cancelling in Congruence/Corollary 1}}: $\\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $19$ Category:Modulo Arithmetic jmay40sjb275qvretmgl0auqlkzbo3u"}
+{"_id": "32353", "title": "Continuous Image of Compact Space is Compact/Corollary 2", "text": "Continuous Image of Compact Space is Compact/Corollary 2 0 21155 316453 179573 2017-09-09T06:59:47Z Barto 3079 wikitext text/x-wiki == Corollary to Continuous Image of Compact Space is Compact == <onlyinclude> A continuous mapping from a compact topological space to a metric space is bounded. </onlyinclude> == Proof == Follows from Continuous Image of Compact Space is Compact and Compact Subspace of Metric Space is Bounded. {{Qed}} == Sources == * {{BookReference|Principles of Mathematical Analysis|1953|Walter Rudin|prev = Continuous Image of Compact Space is Compact|next = Continuous Image of Compact Space is Compact/Corollary 3/Proof 1}}: $4.15$ * {{BookReference|Introduction to Metric and Topological Spaces|1975|W.A. Sutherland|prev=Continuous Image of Compact Space is Compact/Corollary 1|next=Continuous Image of Compact Space is Compact/Corollary 3/Proof 1}}: $5.5$: Continuous maps on compact spaces: Corollary $5.5.3$ Category:Compact Spaces Category:Continuous Invariants fmt1at4n9zvce3nub5xzkjzspun1yyd"}
+{"_id": "32354", "title": "Continuous Image of Compact Space is Compact/Corollary 3", "text": "Continuous Image of Compact Space is Compact/Corollary 3 0 21156 316616 316495 2017-09-09T16:43:02Z Barto 3079 wikitext text/x-wiki == Corollary to Continuous Image of Compact Space is Compact == <onlyinclude> Let $S$ be a compact topological space. Let $f: S \\to \\R$ be a continuous real-valued function. Then $f$ attains its bounds on $S$. </onlyinclude> == Proof 1 == {{:Continuous Image of Compact Space is Compact/Corollary 3/Proof 1}} == Proof 2 == {{:Continuous Image of Compact Space is Compact/Corollary 3/Proof 2}} Category:Compact Spaces Category:Continuous Invariants mpyx4tzrb8si8eo1v0otk1yu41sdea9"}
+{"_id": "32355", "title": "Area of Triangle in Terms of Two Sides and Angle", "text": "Area of Triangle in Terms of Two Sides and Angle 0 21377 451001 393586 2020-02-28T10:20:41Z Prime.mover 59 wikitext text/x-wiki == Corollary to Area of Triangle in Terms of Side and Altitude == <onlyinclude> The area of a triangle $ABC$ is given by: :$\\dfrac 1 2 a b \\sin C$ where: : $a, b$ are two of the sides : $C$ is the angle of the vertex opposite its other side $c$. </onlyinclude> == Proof 1 == {{:Area of Triangle in Terms of Two Sides and Angle/Proof 1}} == Proof 2 == {{:Area of Triangle in Terms of Two Sides and Angle/Proof 2}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Area of Triangle in Terms of Side and Altitude|next = Heron's Formula}}: $\\S 4$: Geometric Formulas: $4.5$ * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Area of Triangle in Terms of Side and Altitude|next = Law of Sines|entry = triangle|subentry = i}} Category:Areas of Triangles Category:Area of Triangle in Terms of Two Sides and Angle p0pjtycjchc38qyryusgp6rc6wthb2i"}
+{"_id": "32356", "title": "Derivative of Cosine Function/Corollary", "text": "Derivative of Cosine Function/Corollary 0 21389 485861 310524 2020-09-06T19:34:05Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Cosine Function == <onlyinclude> :$\\map {\\dfrac \\d {\\d x} } {\\cos a x} = -a \\sin a x$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\map {\\dfrac \\d {\\d x} } {\\cos x}       | r = -\\sin x       | c = Derivative of $\\cos x$ }} {{eqn | ll= \\leadsto       | l = \\map {\\dfrac \\d {\\d x} } {\\cos a x}       | r = -a \\sin a x       | c = Derivative of Function of Constant Multiple }} {{end-eqn}} {{qed}} == Also see == * Derivative of $\\sin a x$ * Derivative of $\\tan a x$ * Derivative of $\\cot a x$ * Derivative of $\\sec a x$ * Derivative of $\\csc a x$ Category:Derivative of Cosine Function ghumvt1h1tgyvdqixu8iuxia17moyc0"}
+{"_id": "32357", "title": "Derivative of Sine Function/Corollary", "text": "Derivative of Sine Function/Corollary 0 21393 485860 310492 2020-09-06T19:33:06Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Sine Function == <onlyinclude> :$\\map {\\dfrac \\d {\\d x} } {\\sin a x} = a \\cos a x$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\map {\\dfrac \\d {\\d x} } {\\sin x}       | r = \\cos x       | c = Derivative of $\\sin x$ }} {{eqn | ll= \\leadsto       | l = \\map {\\dfrac \\d {\\d x} } {\\sin a x}       | r = a \\cos a x       | c = Derivative of Function of Constant Multiple }} {{end-eqn}} {{qed}} == Also see == * Derivative of $\\cos a x$ * Derivative of $\\tan a x$ * Derivative of $\\cot a x$ * Derivative of $\\sec a x$ * Derivative of $\\csc a x$ Category:Derivative of Sine Function spsr35wfgis0woq4ajfod9z23jfa51r"}
+{"_id": "32358", "title": "Cosine of Sum/Corollary", "text": "Cosine of Sum/Corollary 0 21397 451129 396172 2020-02-28T16:07:44Z Prime.mover 59 wikitext text/x-wiki == Corollary to Cosine of Sum == <onlyinclude> :$\\map \\cos {a - b} = \\cos a \\cos b + \\sin a \\sin b$ </onlyinclude> where $\\sin$ denotes the sine and $\\cos$ denotes the cosine. == Proof == {{begin-eqn}} {{eqn | l = \\map \\cos {a - b}       | r = \\cos a \\, \\map \\cos {-b} - \\sin a \\, \\map \\sin {-b}       | c = Cosine of Sum }} {{eqn | r = \\cos a \\cos b - \\sin a \\, \\map \\sin {-b}       | c = Cosine Function is Even }} {{eqn | r = \\cos a \\cos b + \\sin a \\sin b       | c = Sine Function is Odd }} {{end-eqn}} {{qed}} == Historical Note == {{:Cosine of Sum/Historical Note}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Sine of Difference|next = Sum of Squares of Sine and Cosine/Proof 1}}: $\\S 4.5$. The Functions $e^z$, $\\cos z$, $\\sin z$ * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Cosine of Sum|next = Tangent of Sum}}: $\\S 5$: Trigonometric Functions: $5.35$ * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Cosine of Sum|next = Tangent of Sum}}: $2$: Functions, Limits and Continuity: The Elementary Functions: $4$ * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Cosine of Sum/Mistake|next = Tangent of Sum|entry = compound angle formulae|subentry = in trigonometry}} Category:Cosine Function Category:Cosine of Sum Category:Compound Angle Formulas 3oleic61i3pc7bknsxjwtqj59ky1jkt"}
+{"_id": "32359", "title": "Union of Connected Sets with Non-Empty Intersections is Connected/Corollary", "text": "Union of Connected Sets with Non-Empty Intersections is Connected/Corollary 0 21403 339341 339340 2018-01-21T23:04:22Z Prime.mover 59 wikitext text/x-wiki == Corollary to Union of Connected Sets with Non-Empty Intersections is Connected == <onlyinclude> Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $I$ be an indexing set. Let $\\mathcal A = \\left \\langle{A_\\alpha}\\right \\rangle_{\\alpha \\mathop \\in I}$ be an indexed family of subsets of $S$, all connected in $T$. Let $B$ be a connected set of $T$ such that: : $\\forall C \\in \\mathcal A: B \\cap C \\ne \\varnothing$ Then $\\displaystyle B \\cup \\bigcup \\mathcal A$ is connected. </onlyinclude> == Proof 1 == {{:Union of Connected Sets with Non-Empty Intersections is Connected/Corollary/Proof 1}} == Proof 2 == {{:Union of Connected Sets with Non-Empty Intersections is Connected/Corollary/Proof 2}} == Proof 3 == {{:Union of Connected Sets with Non-Empty Intersections is Connected/Corollary/Proof 3}} Category:Connected Spaces Category:Union of Connected Sets with Non-Empty Intersections is Connected nmd29npt3fbj0q9eijt52aifpflhnlp"}
+{"_id": "32360", "title": "Sine of Sum/Corollary", "text": "Sine of Sum/Corollary 0 21561 451122 396168 2020-02-28T15:55:07Z Prime.mover 59 wikitext text/x-wiki == Corollary to Sine of Sum == <onlyinclude> :$\\map \\sin {a - b} = \\sin a \\cos b - \\cos a \\sin b$ </onlyinclude> where $\\sin$ denotes the sine and $\\cos$ denotes the cosine. == Proof == {{begin-eqn}} {{eqn | l = \\map \\sin {a - b}       | r = \\sin a \\, \\map \\cos {-b} + \\cos a \\, \\map \\sin {-b}       | c = Sine of Sum }} {{eqn | r = \\sin a \\cos b + \\cos a \\, \\map \\sin {-b}       | c = Cosine Function is Even }} {{eqn | r = \\sin a \\cos b - \\cos a \\sin b       | c = Sine Function is Odd }} {{end-eqn}} {{qed}} == Historical Note == {{:Sine of Sum/Historical Note}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Cosine of Sum|next = Cosine of Difference}}: $\\S 4.5$. The Functions $e^z$, $\\cos z$, $\\sin z$ * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Sine of Sum|next = Cosine of Sum}}: $\\S 5$: Trigonometric Functions: $5.34$ * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Sine of Sum|next = Cosine of Sum}}: $2$: Functions, Limits and Continuity: The Elementary Functions: $4$ * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Sine of Sum|next = Cosine of Sum|entry = compound angle formulae|subentry = in trigonometry}} Category:Sine Function Category:Sine of Sum Category:Compound Angle Formulas ooshkhgan2izsxahq8ukue2ch9xey7z"}
+{"_id": "32361", "title": "Ideal of Unit is Whole Ring/Corollary", "text": "Ideal of Unit is Whole Ring/Corollary 0 21944 401535 398664 2019-04-19T22:29:39Z Prime.mover 59 wikitext text/x-wiki == Corollary to Ideal of Unit is Whole Ring == <onlyinclude> Let $\\struct {R, +, \\circ}$ be a ring with unity. Let $J$ be an ideal of $R$. If $J$ contains the unity of $R$, then $J = R$. </onlyinclude> == Proof == Follows directly from Ideal of Unit is Whole Ring and Unity is Unit. {{qed}} == Sources == * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Ideals of Ring of Integers Modulo m/Examples/Order 12|next = Field has 2 Ideals}}: Chapter $5$: Rings: $\\S 21$. Ideals: Theorem $35$ * {{BookReference|Rings, Modules and Linear Algebra|1970|B. Hartley|author2 = T.O. Hawkes|prev = Subring of Integers is Ideal|next = Ideals of Field}}: $\\S 2$: Exercise $1$ * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Ideal of Unit is Whole Ring|next = Field has 2 Ideals}}: $\\S 58.2$ Ideals Category:Ideal Theory ci3cu5ftvwct6sxi4zy7je8l1f71n6g"}
+{"_id": "32362", "title": "Sum of Geometric Sequence/Corollary 1", "text": "Sum of Geometric Sequence/Corollary 1 0 22078 456670 456638 2020-03-20T13:05:38Z Prime.mover 59 wikitext text/x-wiki == Corollary to Sum of Geometric Sequence == <onlyinclude> Let $a, a r, a r^2, \\ldots, a r^{n-1}$ be a geometric sequence. Then: :$\\displaystyle \\sum_{j \\mathop = 0}^{n - 1} a r^j = \\frac {a \\paren {r^n - 1} } {r - 1}$ </onlyinclude> {{:Euclid:Proposition/IX/35}} == Proof 1 == {{:Sum of Geometric Sequence/Corollary 1/Proof 1}} == Proof 2 == {{:Sum of Geometric Sequence/Corollary 1/Proof 2}} == Also presented as == This result is also seen presented as: :$\\displaystyle \\sum_{j \\mathop = 0}^{n - 1} a r^j = \\frac {a \\paren {1 - r^n} } {1 - r}$ which is usually more manageable when $r < 1$. {{Euclid Note|35|IX}} == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 2|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd|next = Theorem of Even Perfect Numbers/Sufficient Condition}}: Book $\\text{IX}$. Propositions * {{BookReference|Handbook of Mathematical Functions|1964|Milton Abramowitz|author2 = Irene A. Stegun|prev = Sum of Arithmetic Sequence|next = Sum of Infinite Geometric Sequence/Corollary 2}}: $3.1.10$: Sum of Geometric Progression to $n$ Terms * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Definition:Geometric Series|next = Sum of Infinite Geometric Sequence/Corollary 2}}: $\\S 19$: Geometric Series: $19.4$ * {{BookReference|Elementary Number Theory|1980|David M. Burton|ed = revised|edpage = Revised Printing|prev = Sum of Sequence of Cubes/Proof by Induction|next = Difference of Two Powers/Proof by Induction}}: Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction: Problems $1.1$: $2$ * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Definition:Geometric Series|next = Sum of Infinite Geometric Sequence/Corollary 2|entry = geometric series}} * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Common Ratio of Geometric Sequence|next = Definition:Geometric Series|entry = geometric progression (geometric sequence)}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Common Ratio of Geometric Sequence|next = Definition:Geometric Series|entry = geometric progression (geometric sequence)}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Geometric Series|next = Sum of Infinite Geometric Sequence/Corollary 2|entry = geometric series}} Category:Geometric Sequences Category:Sum of Geometric Sequence 2zf6hscfcls8ov9wr6ci34j8piu17c5"}
+{"_id": "32363", "title": "Upper and Lower Bounds of Integral/Corollary", "text": "Upper and Lower Bounds of Integral/Corollary 0 22213 495020 348011 2020-10-16T15:07:45Z Prime.mover 59 wikitext text/x-wiki == Corollary to Upper and Lower Bounds of Integral == <onlyinclude> Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$. Suppose that $\\forall t \\in \\closedint a b: \\size {\\map f t} < \\kappa$. Then: :$\\ds \\forall \\xi, x \\in \\closedint a b: \\size {\\int_x^\\xi \\map f t \\rd t} < \\kappa \\size {x - \\xi}$ </onlyinclude> == Proof == Follows directly from Upper and Lower Bounds of Integral. {{Qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Integral of Constant/Definite|next = Definite Integral of Function plus Constant}}: $\\S 13.6$ Category:Integral Calculus 2w7ztever04t4na0aazjqtgs2l5xw6y"}
+{"_id": "32364", "title": "Double Negation/Double Negation Introduction", "text": "Double Negation/Double Negation Introduction 0 22278 244793 244787 2016-01-17T09:07:26Z Prime.mover 59 wikitext text/x-wiki == Proof Rule == <onlyinclude> The rule of '''double negation introduction''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Double Negation/Double Negation Introduction/Proof Rule}} === Sequent Form === {{:Double Negation/Double Negation Introduction/Sequent Form}} </onlyinclude> Category:Proof Rules Category:Double Negation clgtfyvaav8oiviqjve6uup56dbwesn"}
+{"_id": "32365", "title": "User:Dfeuer/CRG2", "text": "User:Dfeuer/CRG2 2 22614 125790 2013-01-11T19:57:12Z Dfeuer 1672 Dfeuer moved page User:Dfeuer/CRG2 to Properties of Relation Compatible with Group Operation/CRG2 wikitext text/x-wiki #REDIRECT Properties of Relation Compatible with Group Operation/CRG2 brgxg8uewnap1es2qrv2q92h4j7pmhw"}
+{"_id": "32366", "title": "Biconditional is Commutative/Formulation 1", "text": "Biconditional is Commutative/Formulation 1 0 22685 209536 177784 2015-03-20T20:06:29Z Prime.mover 59 wikitext text/x-wiki == Theorems == The biconditional operator is commutative: <onlyinclude> :$p \\iff q \\dashv \\vdash q \\iff p$ </onlyinclude> == Proof 1 == {{:Biconditional is Commutative/Formulation 1/Proof 1}} == Proof 2 == {{:Biconditional is Commutative/Formulation 1/Proof 2}} == Proof 3 == {{:Biconditional is Commutative/Formulation 1/Proof 3}} == Sources == * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition|prev=Rule of Commutation/Conjunction/Formulation 1|next=Exclusive Or is Commutative}}: $\\S 2.3.3$ Category:Biconditional is Commutative 1025jdfsxybvnmq4dfp3kj2vo92qpsg"}
+{"_id": "32367", "title": "De Morgan's Laws (Logic)/Disjunction of Negations", "text": "De Morgan's Laws (Logic)/Disjunction of Negations 0 22816 445727 445716 2020-02-02T11:35:06Z Prime.mover 59 wikitext text/x-wiki == Theorems == ==== Formulation 1 ==== <onlyinclude>{{:De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 1}}</onlyinclude> ==== Formulation 2 ==== {{:De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 2}} == Also see == * De Morgan's Laws (Set Theory) for a set theoretic application. * De Morgan's Laws (Predicate Logic) for a predicate logic application. == Sources == * {{BookReference|Undergraduate Topology|1971|Robert H. Kasriel|prev = Conjunction implies Disjunction|next = Definition:Subset}}: $\\S 1.2$: Some Remarks on the Use of the Connectives ''and'', ''or'', ''implies'': Exercise $4$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = De Morgan's Laws (Set Theory)/Set Complement/Complement of Union|next = De Morgan's Laws (Logic)/Conjunction of Negations|entry = De Morgan's Laws}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = De Morgan's Laws (Set Theory)/Set Complement/Complement of Union|next = De Morgan's Laws (Logic)/Conjunction of Negations|entry = De Morgan's Laws}} Category:De Morgan's Laws (Logic) e3ja0teled439hfeurhls927e8duis0"}
+{"_id": "32368", "title": "Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1/Forward Implication", "text": "Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1/Forward Implication 0 22866 244691 242036 2016-01-16T19:37:03Z Prime.mover 59 wikitext text/x-wiki == Definition == <onlyinclude> :$p \\land \\left({q \\lor r}\\right) \\vdash \\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)$ </onlyinclude> == Proof == {{BeginTableau|p \\land \\left({q \\lor r}\\right) \\vdash \\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)}} {{Premise|1|p \\land \\left({q \\lor r}\\right)}} {{Simplification|2|1|p|1|1}} {{Simplification|3|1|q \\lor r|1|2}} {{Assumption|4|q}} {{Conjunction|5|1, 4|p \\land q|2|4}} {{Addition|6|1, 4|\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)|5|1}} {{Assumption|7|r}} {{Conjunction|8|1, 7|p \\land r|2|7}} {{Addition|9|1, 7|\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)|8|2}} {{ProofByCases|10|1|\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)|3|4|6|7|9}} {{EndTableau}} {{qed}} == Sources == * {{BookReference|Logic in Computer Science: Modelling and reasoning about systems|2000|Michael R.A. Huth|author2 = Mark D. Ryan|prev = Rule of Association/Disjunction/Formulation 1/Proof 1|next = Rule of Simplification/Sequent Form/Formulation 2/Proof 1/Form 1}}: $\\S 1.2.1$: Rules for natural deduction: Example $1.18$ Category:Rule of Distribution i1s2qhget7rr6g4ove1z8elkj2itrrs"}
+{"_id": "32369", "title": "Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1/Reverse Implication", "text": "Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1/Reverse Implication 0 22867 244692 143371 2016-01-16T19:37:35Z Prime.mover 59 wikitext text/x-wiki == Definition == <onlyinclude> :$\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right) \\vdash p \\land \\left({q \\lor r}\\right)$ </onlyinclude> == Proof == {{BeginTableau|\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right) \\vdash p \\land \\left({q \\lor r}\\right)}} {{Premise|1|\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)}} {{Assumption|2|p \\land q}} {{Simplification|3|2|p|2|1}} {{Assumption|4|p \\land r}} {{Simplification|5|4|p|2|1}} {{ProofByCases|6|1|p|1|2|3|4|5}} {{Assumption|7|p \\land q}} {{Simplification|8|7|q|7|2}} {{Addition|9|7|q \\lor r|7|1}} {{Assumption|10|p \\land r}} {{Simplification|11|10|r|10|2}} {{Addition|12|10|q \\lor r|11|2}} {{ProofByCases|13|1|q \\lor r|1|7|9|10|12}} {{Conjunction|14|1|p \\land \\left({q \\lor r}\\right)|6|13}} {{EndTableau}} {{qed}} == Sources == * {{BookReference|Logic in Computer Science: Modelling and reasoning about systems|2000|Michael R.A. Huth|author2 = Mark D. Ryan|prev = Absorption Laws (Logic)/Disjunction Absorbs Conjunction/Forward Implication|next = Definition:Contradiction}}: $\\S 1.2.1$: Rules for natural deduction: Exercises $1.4: \\ 2 \\ \\text{(q)}$ Category:Rule of Distribution 33gp2h9yg0azjmaudagosx4p0xyigwf"}
+{"_id": "32370", "title": "Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1", "text": "Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1 0 22868 427188 181061 2019-09-22T22:23:15Z Prime.mover 59 wikitext text/x-wiki == Definition == The conjunction operator is left distributive over the disjunction operator: <onlyinclude> :$p \\land \\paren {q \\lor r} \\dashv \\vdash \\paren {p \\land q} \\lor \\paren {p \\land r}$ </onlyinclude> This can be expressed as two separate theorems: === Forward Implication === {{:Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1/Forward Implication}} === Reverse Implication === {{:Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1/Reverse Implication}} == Proof == {{:Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1/Proof}} == Sources == * {{BookReference|Beginning Logic|1965|E.J. Lemmon|prev = Rule of Idempotence/Conjunction/Formulation 1|next = Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 1}}: $\\S 1.5$: Further Proofs: Résumé of Rules: Exercise $1 \\ \\text{(c)}$ * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed = 3rd|edpage = Third Edition|prev = Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 1|next = Rule of Material Equivalence/Formulation 1}}: $\\S 2.3.3$ * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed = 3rd|edpage = Third Edition|prev = Completeness Theorem for Semantic Tableaus|next = De Morgan's Laws (Logic)/Disjunction/Formulation 1}}: $\\S 2.10$: Exercise $2.3$ Category:Rule of Distribution 8qqk1syg44309nyzyukv8z0hqsksq5t"}
+{"_id": "32371", "title": "Conjunction Equivalent to Negation of Implication of Negative", "text": "Conjunction Equivalent to Negation of Implication of Negative 0 22964 133377 127713 2013-02-08T09:57:52Z Prime.mover 59 wikitext text/x-wiki == Theorems == <onlyinclude> ==== Formulation 1 ==== {{:Conjunction Equivalent to Negation of Implication of Negative/Formulation 1}} ==== Formulation 2 ==== {{:Conjunction Equivalent to Negation of Implication of Negative/Formulation 2}}</onlyinclude> Category:Implication Category:Conjunction heid7i5x21oznmqxorxkb728hkr9c3g"}
+{"_id": "32372", "title": "Operations of Boolean Algebra are Idempotent", "text": "Operations of Boolean Algebra are Idempotent 0 23151 445333 422670 2020-01-31T07:44:36Z Prime.mover 59 wikitext text/x-wiki == Definition == Let $\\left({S, \\vee, \\wedge}\\right)$ be a Boolean algebra. Then: :$\\forall x \\in S: x \\wedge x = x = x \\vee x$ That is, both $\\vee$ and $\\wedge$ are idempotent operations. == Proof == Let $x \\in S$. Then: {{begin-eqn}} {{eqn | l = x       | r = x \\vee \\bot       | c = as $\\bot$ is the identity of $\\vee$ }} {{eqn | r = x \\vee \\left({x \\wedge \\neg x}\\right)       | c = as $x \\wedge \\neg x = \\bot$ }} {{eqn | r = \\left({x \\vee x}\\right) \\wedge \\left({x \\vee \\neg x}\\right)       | c = both $\\vee$ and $*$ distribute over the other }} {{eqn | r = \\left({x \\vee x}\\right) \\wedge \\top       | c = as $x \\vee \\neg x = \\top$ }} {{eqn | r = x \\vee x       | c =  }} {{end-eqn}} So $x = x \\vee x$. {{qed|lemma}} The result $x = x \\wedge x$ follows from Duality Principle (Boolean Algebras). {{qed}} == Sources == * {{BookReference|Abstract Algebra|1964|W.E. Deskins|prev = Idempotent Elements for Integer Multiplication|next = Duality Principle (Boolean Algebras)}}: $\\S 1.5$: Theorem $1.13$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Duality Principle (Boolean Algebras)|next = Operations of Boolean Algebra are Associative|entry = Boolean algebra}} * {{BookReference|Introduction to Boolean Algebras|2008|Paul Halmos|author2 = Steven Givant|prev = Complement of Complement (Boolean Algebras)|next = De Morgan's Laws (Boolean Algebras)}}: $\\S 2$: Exercise $2$ * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Duality Principle (Boolean Algebras)|next = Operations of Boolean Algebra are Associative|entry = Boolean algebra}} Category:Boolean Algebras bkdgh8ozqpsvrh2sqakisbtqe2b6c0p"}
+{"_id": "32373", "title": "Method of Truth Tables/Proof of Interderivability", "text": "Method of Truth Tables/Proof of Interderivability 0 23237 170634 170350 2013-12-05T14:20:28Z Lord Farin 560 wikitext text/x-wiki == Proof Technique == <onlyinclude> Suppose we have two propositional formulas $P$ and $Q$ and we wish to see whether $P \\dashv \\vdash Q$ or not. :Example: Let $P$ be $p \\uparrow q$ and let $Q$ be $\\neg \\left({p \\land q}\\right)$. There are two things we can do. === Express two statements as a single WFF === We express $P \\dashv \\vdash Q$ as a single WFF $\\left({P \\iff Q}\\right)$ and perform the method of truth tables on that. Demonstrating this with the example given: $\\begin{array}{cc||cccccccc} p & q & (p & \\uparrow & q) & \\iff & \\neg & (p & \\land & q) \\\\ \\hline F & F & F & T & F & T & T & F & F & F \\\\ F & T & F & T & T & T & T & F & F & T \\\\ T & F & T & T & F & T & T & T & F & F \\\\ T & T & T & F & T & T & F & T & T & T \\\\ \\end{array}$ As can be seen, the column under the main connective $\\iff$ of $\\left({P \\iff Q}\\right)$ is all $T$, so $\\left({\\left({p \\uparrow q}\\right) \\iff \\neg \\left({p \\land q}\\right)}\\right)$ is a tautology. Hence from Equivalences are Interderivable, $\\left({\\left({p \\uparrow q}\\right) \\dashv \\vdash \\neg \\left({p \\land q}\\right)}\\right)$ and the two formulas are interderivable. {{Qed}} === Compare two WFFs in the same table === Alternatively, we can place the two WFFs side by side in the same truth table: $\\begin{array}{cc||ccc||cccc} p & q & (p & \\uparrow & q) & \\neg & (p & \\land & q) \\\\ \\hline F & F & F & T & F & T & F & F & F \\\\ F & T & F & T & T & T & F & F & T \\\\ T & F & T & T & F & T & T & F & F \\\\ T & T & T & F & T & F & T & T & T \\\\ \\end{array}$ This time, we need to make sure that the truth values in the columns under the main connectives of both formulae are the same. Note that this is exactly the same as putting a $\\iff$ between the two and making a WFF out of the pair of them. {{Qed}} </onlyinclude> Category:Proof Techniques Category:Truth Tables 4bt34xg0u08wexqqnryeeedvw2pws0i"}
+{"_id": "32374", "title": "Derivative of Complex Power Series/Proof 2/Lemma", "text": "Derivative of Complex Power Series/Proof 2/Lemma 0 23330 364935 364934 2018-09-08T15:39:47Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $n \\in N_{\\ge 1}$. Then: <onlyinclude> :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {\\dfrac {n \\paren {n - 1} } 2}^{1/n} = 1$ </onlyinclude> == Proof == Choose any $\\alpha > 1$.  It follows from the ratio test that: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\dfrac 1 {\\alpha^n} \\frac {n \\paren {n - 1} } 2 = 0$ Therefore, for all sufficiently large $n$: :$\\dfrac {n \\paren {n - 1} } 2 \\le \\alpha^n$ and so: :$\\paren {\\dfrac {n \\paren {n - 1} } 2}^{1/n} \\le \\alpha$ It follows that: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {\\dfrac {n \\paren {n - 1} } 2}^{1/n} \\le \\alpha$ Since $\\alpha > 1$ was arbitrary, we can conclude that: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {\\dfrac {n \\paren {n - 1} } 2}^{1/n} \\le 1$ It is clear that the following holds for sufficiently large $n$: :$\\displaystyle \\paren {\\dfrac {n \\paren {n - 1} } 2}^{1/n} \\ge 1^{1/n} = 1$ Therefore: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {\\dfrac {n \\paren {n - 1} } 2}^{1/n} = 1$ {{qed}} Category:Derivative of Complex Power Series ju1d8vthg9el62cjkrqfy99uuv857dv"}
+{"_id": "32375", "title": "Expansion Theorem for Determinants/Corollary", "text": "Expansion Theorem for Determinants/Corollary 0 23349 476037 350836 2020-06-26T07:39:26Z Prime.mover 59 wikitext text/x-wiki == Corollary to Expansion Theorem for Determinants == Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $D = \\map \\det {\\mathbf A}$ be the determinant of $\\mathbf A$. Let $a_{p q}$ be an element of $\\mathbf A$. Let $A_{p q}$ be the cofactor of $a_{p q}$ in $D$. <onlyinclude> Let $\\delta_{rs}$ be the Kronecker delta. Then: :$(1): \\quad \\displaystyle \\forall r \\in \\closedint 1 n: \\sum_{k \\mathop = 1}^n a_{r k} A_{s k} = \\delta_{r s} D$ :$(2): \\quad \\displaystyle \\forall r \\in \\closedint 1 n: \\sum_{k \\mathop = 1}^n a_{k r} A_{k s} = \\delta_{r s} D$ That is, if you multiply each element of a row or column by the cofactor of ''another'' row or column, the sum of those products is zero. </onlyinclude> == Proof == Let $D'$ be the determinant obtained by replacing row $s$ with row $r$. Let $a'_{ij}$ be an element of $D'$. Then: :$a'_{ij} = \\begin {cases} a_{ij} & : i \\ne s \\\\ a_{rj} & : i = s \\end {cases}$ Let the cofactor of $a'_{ij}$ in $D'$ be denoted by $A'_{i j}$. Then: :$\\forall k \\in \\closedint 1 n: A'_{s k} = A_{s k}$ So by the Expansion Theorem for Determinants: :$\\displaystyle D' = \\sum_{k \\mathop = 1}^n a'_{s k} A'_{s k}$ But the $r$th and $s$th rows are identical. So by Square Matrix with Duplicate Rows has Zero Determinant: :$D' = 0$ Hence the result. The result for columns follows from Determinant of Transpose.  {{Qed}} Category:Determinants ov756fsll8it09frfmi3vb9s4wfgvvi"}
+{"_id": "32376", "title": "Reductio ad Absurdum/Sequent Form", "text": "Reductio ad Absurdum/Sequent Form 0 23556 245156 245131 2016-01-18T06:13:29Z Prime.mover 59 wikitext text/x-wiki == Proof Rule == <onlyinclude> The Reductio ad Absurdum can be symbolised by the sequent: : $\\left({\\neg p \\vdash \\bot}\\right) \\vdash p$ </onlyinclude> == Proof == {{BeginTableau|\\left({\\neg p \\vdash \\bot}\\right) \\vdash p}} {{Premise|1|\\neg p \\vdash \\bot}} {{Assumption|2|\\neg p}} {{SequentIntro|3|1|\\bot|1|the premise itself}} {{Contradiction|4|1|\\neg \\neg p|2|3}} {{DoubleNegElimination|5|1|p|4}} {{EndTableau}} {{Qed}} {{LEM|Double Negation Elimination}} == Sources == * {{BookReference|Logic in Computer Science: Modelling and reasoning about systems|2000|Michael R.A. Huth|author2 = Mark D. Ryan|prev = Reductio ad Absurdum/Proof Rule|next = Law of Excluded Middle/Proof Rule}}: $\\S 1.2.2$: Derived rules Category:Reductio ad Absurdum 4tjhlvqthkkgl3dr0ipn67izelvddny"}
+{"_id": "32377", "title": "Sum of Squares of Sine and Cosine/Corollary 1", "text": "Sum of Squares of Sine and Cosine/Corollary 1 0 23659 455235 451537 2020-03-17T15:29:42Z Prime.mover 59 wikitext text/x-wiki == Corollary to Sum of Squares of Sine and Cosine == For all $x \\in \\C$: <onlyinclude> :$\\sec^2 x - \\tan^2 x = 1 \\quad \\text {(when $\\cos x \\ne 0$)}$ </onlyinclude> where $\\sec$, $\\tan$ and $\\cos$ are secant, tangent and cosine respectively. == Proof  == When $\\cos x \\ne 0$: {{begin-eqn}} {{eqn | l = \\cos^2 x + \\sin^2 x       | r = 1       | c = Sum of Squares of Sine and Cosine }} {{eqn | ll= \\leadsto       | l = 1 + \\frac {\\sin^2 x} {\\cos^2 x}       | r = \\frac 1 {\\cos^2 x}       | c = dividing both sides by $\\cos^2 x$, as $\\cos x \\ne 0$ }} {{eqn | ll= \\leadsto       | l = 1 + \\tan^2 x       | r = \\sec^2 x       | c = {{Defof|Tangent Function}} and {{Defof|Secant Function}} }} {{eqn | ll= \\leadsto       | l = \\sec^2 x - \\tan^2 x       | r = 1       | c = rearranging }} {{end-eqn}} {{qed}} == Also defined as == This result can also be reported as: :$\\sec^2 x = 1 + \\tan^2 x \\quad \\text {(when $\\cos x \\ne 0$)}$ or: :$\\tan^2 x = \\sec^2 x - 1 \\quad \\text {(when $\\cos x \\ne 0$)}$ == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Sum of Squares of Sine and Cosine|next = Difference of Squares of Cosecant and Cotangent}}: $\\S 5$: Trigonometric Functions: $5.20$ * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Sum of Squares of Sine and Cosine/Proof 5|next = Difference of Squares of Cosecant and Cotangent}}: $2$: Functions, Limits and Continuity: The Elementary Functions: $4$ Category:Tangent Function Category:Secant Function Category:Sum of Squares of Sine and Cosine 80bwpr4yeift0whx7blgrh8b0n16tpo"}
+{"_id": "32378", "title": "Biconditional Introduction", "text": "Biconditional Introduction 0 23917 244863 244862 2016-01-17T12:07:31Z Prime.mover 59 wikitext text/x-wiki == Proof Rule == The rule of '''biconditional introduction''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Biconditional Introduction/Proof Rule}} === Sequent Form === {{:Biconditional Introduction/Sequent Form}} == Also known as == {{:Biconditional Introduction/Also known as}} == Also see == * Biconditional Elimination Category:Biconditional Category:Biconditional Introduction Category:Implication akjh8pd2536ep8297vske6irfv9su7t"}
+{"_id": "32379", "title": "Modus Ponendo Tollens/Variant", "text": "Modus Ponendo Tollens/Variant 0 23962 134035 133706 2013-02-10T19:17:44Z Prime.mover 59 wikitext text/x-wiki == Theorems == <onlyinclude> ==== Formulation 1 ==== {{:Modus Ponendo Tollens/Variant/Formulation 1}} ==== Formulation 2 ==== {{:Modus Ponendo Tollens/Variant/Formulation 2}}</onlyinclude> Category:Modus Ponendo Tollens othpbszkidaufqqjci9i6wkrwnle9bb"}
+{"_id": "32380", "title": "Jordan Polygon Parity Lemma", "text": "Jordan Polygon Parity Lemma 0 24068 443062 443061 2020-01-09T07:54:42Z Prime.mover 59 wikitext text/x-wiki == Definition == Let $P$ be a polygon embedded in $\\R^2$. Let $U \\subseteq \\R^2 \\setminus \\partial P$ be a path-connected subset of $\\R^2 \\setminus \\partial P$. Let $q \\in U$, and let $\\mathbf v \\in \\R^2 \\setminus \\set {\\mathbf 0}$ be a non-zero vector. Let $\\LL = \\set {q + s \\mathbf v: s \\in \\R_{\\ge 0} }$ be a ray with start point $q$. Then: :$(1): \\quad$ The parity $\\map {\\operatorname{par} } q$ is independent of the choice of $v$. :$(2): \\quad$ All points in $U$ have the same parity. == Proof == Define $g: \\R \\to \\R^2$ by $\\map g \\theta = \\tuple {\\cos \\theta, \\sin \\theta}$. Define $\\LL_\\theta = \\set {q + s \\map g \\theta: s \\in \\R_{\\ge 0} }$ as a ray with start point $q$. Note that the image of $g$ is equal to the unit circle. Then: :$\\dfrac v {\\norm {\\mathbf v} } = \\map g {\\theta'}$ for some $\\theta' \\in \\R$, where $\\norm {\\mathbf v}$ denotes the Euclidean norm of $v$. Then $\\LL = \\LL_{\\theta'}$, as $q + s \\mathbf v = q + \\paren {s \\norm {\\mathbf v} } \\, \\map g {\\theta'}$. $\\LL_\\theta \\cap \\partial P$ consists of a finite number of line segments, some of which are crossings. Let $\\map {N_\\theta} q \\in \\N$ be the number of crossings of $\\LL_\\theta$. As the value of $\\theta$ increases, the value of $\\map {N_\\theta} q \\in \\N$ only changes when $\\LL_\\theta$ crosses a vertex of $P$. If $\\theta_0 \\in \\R$ is a value for which $\\LL_{\\theta_0}$ crosses exactly one vertex $A$ or side $S$, there are three possibilities: :$\\displaystyle \\map {N_{\\theta_0} } q = \\lim_{\\theta \\mathop \\to \\theta_0} \\map {N_\\theta} q$, if $A$ or $S$ is not part of a crossing. :$\\displaystyle \\map {N_{\\theta_0} } q = \\lim_{\\theta \\mathop \\to \\theta_0^-} \\map {N_\\theta} q = \\lim_{\\theta \\mathop \\to \\theta_0^+} \\map {N_\\theta} q - 2$, if $A$ or $S$ is part of a crossing, and $\\LL_{\\theta_0}$ intersects both lines adjacent to $A$ or $S$ for some $\\theta > \\theta_0$. :$\\displaystyle \\map {N_{\\theta_0} } q = \\lim_{\\theta \\mathop \\to \\theta_0^-} \\map {N_\\theta} q - 2 = \\lim_{\\theta \\mathop \\to \\theta_0^+} \\map {N_\\theta} q$, if $A$ or $S$ is part of a crossing, and $\\LL_{\\theta_0}$ intersects both lines adjacent to $A$ or $S$ for some $\\theta < \\theta_0$. Here, $\\displaystyle \\lim_{\\theta \\mathop \\to \\theta_0^-} $ denotes a limit from the left, and $\\displaystyle \\lim_{\\theta \\mathop \\to \\theta_0^+} $ denotes a limit from the right. :File:JordanPolygon2.png The figure shows the change of $\\map {N_\\theta} q$ each time $\\LL_\\theta$ intersects a vertex.  The change of $\\map {N_\\theta} q$ is for increasing values of $\\theta$, which corresponds to a counterclockwise rotation of $\\LL_\\theta$ around $q$. {{explain|Prove the above statement.}} If $\\LL_{\\theta_0}$ crosses more than one vertex, $\\displaystyle N_{\\theta_0} \\left({q}\\right)$ may change be a larger number, but always by a multiple of $2$. Hence, $\\map {N_\\theta} q \\bmod 2$ is independent of $\\theta$. Thus we define the parity of $q$ as $\\map {\\operatorname{par} } q := \\map {N_\\theta} q \\bmod 2$. Since $\\LL = \\LL_{\\theta'}$ for some $\\theta' \\in \\R$, it follows that $\\map {\\operatorname{par} } q$ can also be defined as the number of crossings modulo $2$ between $\\LL$ and $\\partial P$. Let $\\sigma: \\closedint 0 1 \\to \\R^2 \\setminus \\partial P$ be any path in $U$ with initial point $q$. For fixed $\\theta \\in \\R$ and $t \\in \\closedint 0 1$, define $\\LL_{\\theta, t} = \\set {\\map \\sigma t + s \\map g \\theta: s \\in \\R_{\\ge 0} }$ as a ray with start point $\\map \\sigma t$. A similar argument to the one above shows that for all $t_0 \\in \\closedint 0 1$: :$\\displaystyle \\lim_{t \\mathop \\to t_0} \\map {N_\\theta} {\\map \\sigma t + s \\map g \\theta} \\equiv \\map {N_\\theta} {\\map \\sigma {t_0} + s \\, \\map g \\theta} \\pmod 2$ So all points in $\\Img \\sigma$ have the same parity. This shows that all points in $U$ have the same parity. {{qed}} {{Namedfor|Marie Ennemond Camille Jordan|cat = Jordan}} == Sources == {{wtd|Replace this with a link to a hard-copy; links to web pages have been known to be ephemeral. See house style: Help:Sources.}} : [http://compgeom.cs.uiuc.edu/~jeffe/teaching/comptop/2009/notes/jordan-polygon-theorem.pdf Jeff Ericsson: Computational Topology] Category:Topology 3unndunkt1gy2dx7jajj737x1hyk4k6"}
+{"_id": "32381", "title": "Jordan Polygon Theorem/Lemma 1", "text": "Jordan Polygon Theorem/Lemma 1 0 24070 472015 451015 2020-06-02T19:15:55Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $P$ be a polygon embedded in $\\R^2$. Denote the boundary of $P$ as $\\partial P$. <onlyinclude> $\\R^2 \\setminus \\partial P$ is the union of at most two disjoint path-connected sets. </onlyinclude> == Proof == The polygon $P$ has $n$ sides, where $n \\in \\N$. Denote the vertices of $P$ as $A_1, \\ldots, A_n$. Denote the sides of $P$ as $S_1, \\ldots, S_n$, such that each vertex $A_i$ has adjacent sides $S_{i - 1}$ and $S_i$. We use the conventions that $S_0 = S_n$, and $A_{n + 1} = A_1$. Let $\\displaystyle \\delta_i = \\map d {S_i, \\bigcup_{j \\mathop = 1, \\ldots, n: \\, \\size {i - j} > 1 } S_j}$ be the Euclidean distance between a side $S_i$ and all sides not adjacent to $S_i$. From Distance between Closed Sets in Euclidean Space: :$\\delta_i > 0$ Put $\\displaystyle \\delta = \\min_{i \\mathop = 1, \\ldots, n} \\delta_i$. From Boundary of Polygon is Jordan Curve, it follows that $\\partial P$ is equal to the image of a Jordan curve: :$\\gamma: \\closedint 0 1 \\to \\R^2$ that is a concatenation of $n$ paths: :$\\gamma_1, \\ldots, \\gamma_n$ Each $\\gamma_i$ is a line segment that joins its initial point $A_i$ and its final point $A_{i + 1}$. Therefore the image of $\\gamma_i$ is equal to the side $S_i$. Let $\\mathbf v_i = \\dfrac {A_{i + 1} - A_i} {\\map d {A_{i + 1}, A_i} }$ be the direction vector of $\\gamma_i$ with norm $\\norm {\\mathbf v_i} = 1$. Let $\\mathbf w_i$ be the vector $v_i$ rotated $\\dfrac \\pi 2$ radians counterclockwise, so $\\norm {\\mathbf w_i} = 1$. {{explain|Invoke the fact that rotation is an isometry (still to be covered on {{ProofWiki}}).}} For any $\\epsilon \\in \\openint 0 {\\dfrac \\delta 2}$, we intend to construct two Jordan curves $\\sigma, \\overline \\sigma$ such that: :$\\Img \\sigma \\cup \\Img {\\overline \\sigma} = \\set {q \\in \\R^2: \\map d {q, \\partial P} = \\epsilon}$ rightthumbIllustration of the [[Definition:Jordan CurveJordan curves $\\sigma, \\overline \\sigma$ which both have distance $\\epsilon$ to $\\partial P$.<br/>Note the rounded corners of the curves that occur in place of non-convex angles.]] For $i \\in \\set {1, \\ldots, n}$, initially define the $\\sigma_i$ as the line segment that joins its initial point $A_i + \\epsilon \\mathbf w_i$ with its final point $A_{i + 1} + \\epsilon \\mathbf w_i$. Suppose $\\sigma_i$ and $\\sigma_{i + 1}$ intersect at some point $p_{i + 1} \\in R^2$. Then re-define the two line segments so that: :the final point of $\\sigma_i$ becomes $p_{i + 1}$ and: :the initial point of $\\sigma_{i + 1}$ becomes $p_{i + 1}$. Then define a path $\\rho_i$ as the constant function $\\map {\\rho_i} t = p_{i + 1}$. Otherwise, define a path $\\rho_i$ with initial point $A_{i + 1} + \\epsilon \\mathbf w_i$ and final point $A_{i + 1} + \\epsilon \\mathbf w_{i + 1}$, such that: :the image of $\\rho_i$ is part of the circumference of the circle with center $A_{i + 1}$ and radius $\\epsilon$. Define the path $\\sigma: \\closedint 0 1 \\to \\R^2$ as the concatenation: :$\\sigma = \\sigma_1 * \\rho_1 * \\sigma_2 * \\rho_2 * \\ldots \\sigma_n * \\rho_n$ Then $\\sigma$ is a closed path, as $\\sigma_1$ has initial point $A_1$ equal to the final point of $\\rho_n$. Each of the paths $\\sigma_i$ and $\\rho_i$ is injective. For all $i, j \\in \\set {1, \\ldots, n}$, $\\sigma_i$ only intersects $\\rho_{i - 1}$ and $\\rho_i$ in their endpoints. Also, $\\sigma_i$ can only possibly intersect $\\sigma_{i - 1}$ in its endpoint. In that case the path $\\rho_{i - 1}$ is constant. Similarly, $\\sigma_i$ can only possibly intersect $\\sigma_{i + 1}$ in its endpoint. In that case the path $\\rho_{i + 1}$ is constant. For $\\size {i - j} > 1$: :let $x_i \\in \\Img {\\sigma_i} \\cup \\Img {\\rho_i}$ :let $x_j \\in \\Img {\\sigma_j} \\cup \\Img {\\rho_j}$. Let $p_i, p_j \\in \\partial P$ be two points such that: :$\\map d {x_i, p_i} = \\map d {x_j, p_j} = \\epsilon$ Suppose $x_i \\in \\Img {\\sigma_i}$. Then we can put: :$p_i = x_i - \\epsilon \\mathbf w_i$ Suppose $x_i \\in \\Img {\\rho_i}$. Then we can put $p_i = A_{i + 1}$. Then $x_i \\ne x_j$, as: {{begin-eqn}} {{eqn | l = \\map d {x_i, x_j}       | r = \\map d {x_i, x_j} + \\map d {x_i, p_i} + \\map d {x_j, p_j} - 2 \\epsilon }} {{eqn | o = \\ge       | r = \\map d {p_i, p_j} - 2 \\epsilon       | c = Triangle Inequality for Vectors in Euclidean Space }} {{eqn | o = >       | r = \\delta - 2 \\epsilon       | c = as $p_i \\in S_i, p_j \\in S_j$, and $\\map d {S_i, S_j} \\ge \\delta$ }} {{eqn | r = 0 }} {{end-eqn}} It follows that $\\sigma_i$ and $\\rho_i$ do not intersect $\\sigma_j$ and $\\rho_j$. It follows that $\\sigma$ is a Jordan curve. Now for $i \\in \\set {1, \\ldots, n}$, initially define the $\\overline \\sigma_i$ as the line segment that joins its initial point $A_i - \\epsilon \\mathbf w_i$ with its final point $A_{i + 1} - \\epsilon \\mathbf w_i$. Proceed to define $\\overline \\rho_i$ and re-define $\\overline \\sigma_i$ similarly to what was done with $\\sigma_i$ and $\\rho_i$. Finally, define $\\overline \\sigma$ as the concatenation: :$\\overline \\sigma = \\overline \\sigma_1 * \\overline \\rho_1 * \\overline \\sigma_2 * \\overline \\rho_2 * \\ldots \\overline \\sigma_n * \\overline \\rho_n$ It follows that $\\overline \\sigma$ is a Jordan curve as above. We now show that $\\sigma$ and $\\overline \\sigma$ do not intersect. First, $\\sigma_i$ and $\\overline \\sigma_i$ do not intersect, as they are both line segments parallel to $S_i$ with a distance of $2 \\epsilon$. Second, $\\rho_i$ and $\\overline \\rho_i$ do not intersect, as one path, say $\\rho_i$, is constant with image equal to the crossing point of $\\sigma_{i - 1}$ and $\\sigma_i$.  The other path, say $\\overline \\rho_i$, is part of the circumference of the circle that joins the final point of $\\overline \\sigma_{i - 1}$ with the initial point of $\\overline \\sigma_i$. Third, $\\sigma_i$ and $\\overline \\sigma_{i + 1}$ do not intersect, since neither line segment crosses $\\partial P$, which would be necessary if the two line segment should intersect. Similarly, $\\sigma_i$ and $\\overline \\sigma_{i - 1}$ do not intersect. For all other combinations of $i, j \\in \\set {1, \\ldots, n}$, $\\sigma_i$ and $\\rho_i$ do not intersect with $\\overline \\sigma_j$ and $\\overline \\rho_j$.  This follows as: :$\\map d {\\sigma_i, S_i} = \\map d {\\rho_i, S_i} = \\map d {\\rho_i, S_{i - 1} } = \\epsilon$ and as $S_i$, or $S_{i - 1}$, are not adjacent sides to $S_j$, we have $\\map d {S_i, S_j} > 2 \\epsilon$. Now we can use the Triangle Inequality for Vectors in Euclidean Space as above, to prove that the paths do not intersect. {{qed|lemma}} Finally, let $q_1, q_2, q_3 \\in R^2 \\setminus \\partial P$.  Let: :$\\displaystyle \\epsilon = \\min \\set {\\map d {q_1, \\partial P} / 2, \\map d {q_2, \\partial P} / 2, \\map d {q_3, \\partial P} / 2, \\delta}$ For all $i \\in \\set {1, 2, 3}$, draw a line segment $\\LL_i$ joining $q_i$ with any point on the boundary $\\partial P$. As $\\map d {q_i, \\partial P} > \\epsilon$, it follows from the Intermediate Value Theorem that there is a point $x_i$ on $\\LL_i$ such that $\\map d {x_i, \\partial P} = \\epsilon$. Then we have either: :$x_i \\in \\Img \\sigma$ or: :$x_i \\in \\Img {\\overline \\sigma}$ when the Jordan curves $\\sigma$ and $\\overline \\sigma$ gets defined from the new value of $\\epsilon$. It follows that at least two out of three of the points $q_1, q_2, q_3$ is path-connected to the same Jordan curve. As a Jordan curve is a path, it follows that at least two of the points can be connected by a path. Hence, $R^2 \\setminus \\partial P$ is the union of at most two disjoint path-connected sets. {{qed}} == Sources == {{SourceReview|Revisit this link}} [http://compgeom.cs.uiuc.edu/~jeffe/teaching/comptop/2009/notes/jordan-polygon-theorem.pdf Jeff Ericsson: Computational Topology]   Category:Topology n10ulxyauewjglujhx6xsj8jzklkux3"}
+{"_id": "32382", "title": "Rule of Transposition/Variant 1/Formulation 2/Forward Implication/Proof", "text": "Rule of Transposition/Variant 1/Formulation 2/Forward Implication/Proof 0 24245 135343 2013-02-15T21:32:04Z Prime.mover 59 Created page with \"== Theorem ==  : $\\vdash \\left({p \\implies \\neg q}\\right) \\implies \\left({q \\implies \\neg p}\\right)$   == Proof == <onlyinclude> {{BeginTableau|\\vdash \\left({p \\implies \\neg q...\" wikitext text/x-wiki == Theorem == : $\\vdash \\left({p \\implies \\neg q}\\right) \\implies \\left({q \\implies \\neg p}\\right)$ == Proof == <onlyinclude> {{BeginTableau|\\vdash \\left({p \\implies \\neg q}\\right) \\implies \\left({q \\implies \\neg p}\\right)}} {{Assumption|1|p \\implies \\neg q}} {{Assumption|2|q}} {{DoubleNegIntro|3|2|\\neg \\neg q|2}} {{ModusTollens|4|1, 2|\\neg p|1|3}} {{Implication|5|1|q \\implies \\neg p|2|4}} {{Implication|6||\\left({p \\implies \\neg q}\\right) \\implies \\left({q \\implies \\neg p}\\right)|1|5}} {{EndTableau}} {{Qed}} </onlyinclude> Category:Rule of Transposition 9o85lre25t88tpxflgvogpdc7zv4we3"}
+{"_id": "32383", "title": "Rule of Transposition/Variant 1/Formulation 2/Reverse Implication/Proof", "text": "Rule of Transposition/Variant 1/Formulation 2/Reverse Implication/Proof 0 24246 135345 2013-02-15T21:33:30Z Prime.mover 59 Created page with \"== Theorem ==  : $\\vdash \\left({q \\implies \\neg p}\\right) \\implies \\left({p \\implies \\neg q}\\right)$   == Proof == <onlyinclude> {{BeginTableau|\\vdash \\left({q \\implies \\neg p...\" wikitext text/x-wiki == Theorem == : $\\vdash \\left({q \\implies \\neg p}\\right) \\implies \\left({p \\implies \\neg q}\\right)$ == Proof == <onlyinclude> {{BeginTableau|\\vdash \\left({q \\implies \\neg p}\\right) \\implies \\left({p \\implies \\neg q}\\right)}} {{Assumption|1|q \\implies \\neg p}} {{Assumption|2|p}} {{DoubleNegIntro|3|2|\\neg \\neg p|2}} {{ModusTollens|4|1, 2|\\neg q|1|3}} {{Implication|5|1|p \\implies \\neg q|2|4}} {{Implication|6||\\left({q \\implies \\neg p}\\right) \\implies \\left({p \\implies \\neg q}\\right)|1|5}} {{EndTableau}} {{Qed}} </onlyinclude> Category:Rule of Transposition oqcax6kqn2a2wky8z2vjzwr4rafw2ok"}
+{"_id": "32384", "title": "Rule of Transposition/Variant 1/Formulation 1/Forward Implication/Proof", "text": "Rule of Transposition/Variant 1/Formulation 1/Forward Implication/Proof 0 24254 135394 2013-02-15T22:18:32Z Prime.mover 59 Created page with \"== Theorem ==  :$p \\implies \\neg q \\vdash q \\implies \\neg p$   == Proof == <onlyinclude> {{BeginTableau|p \\implies \\neg q \\vdash q \\implies \\neg p}} {{Premise|1|p \\implies \\ne...\" wikitext text/x-wiki == Theorem == :$p \\implies \\neg q \\vdash q \\implies \\neg p$ == Proof == <onlyinclude> {{BeginTableau|p \\implies \\neg q \\vdash q \\implies \\neg p}} {{Premise|1|p \\implies \\neg q}} {{Assumption|2|q}} {{DoubleNegIntro|3|2|\\neg \\neg q|2}} {{ModusTollens|4|1, 2|\\neg p|1|3}} {{Implication|5|1|q \\implies \\neg p|2|4}} {{EndTableau}} {{Qed}} </onlyinclude> Category:Rule of Transposition an10xkw6ifgwgb8m81kh2nyi2phlsik"}
+{"_id": "32385", "title": "Rule of Transposition/Variant 1/Formulation 1/Reverse Implication/Proof", "text": "Rule of Transposition/Variant 1/Formulation 1/Reverse Implication/Proof 0 24256 135396 2013-02-15T22:21:31Z Prime.mover 59 Created page with \"== Theorem ==  : $q \\implies \\neg p \\vdash p \\implies \\neg q$   == Proof == <onlyinclude> {{BeginTableau|q \\implies \\neg p \\vdash p \\implies \\neg q}} {{Premise|1|q \\implies \\n...\" wikitext text/x-wiki == Theorem == : $q \\implies \\neg p \\vdash p \\implies \\neg q$ == Proof == <onlyinclude> {{BeginTableau|q \\implies \\neg p \\vdash p \\implies \\neg q}} {{Premise|1|q \\implies \\neg p}} {{Assumption|2|p}} {{DoubleNegIntro|3|2|\\neg \\neg p|2}} {{ModusTollens|4|1, 2|\\neg q|1|3}} {{Implication|5|1|p \\implies \\neg q|2|4}} {{EndTableau}} {{Qed}} </onlyinclude> Category:Rule of Transposition sybrt3850ifoizy9sqtgr37btaieihr"}
+{"_id": "32386", "title": "Axiom:Axiom of Dependent Choice/Right-Total", "text": "Axiom:Axiom of Dependent Choice/Right-Total 100 24480 444068 415825 2020-01-18T02:07:40Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> Let $\\mathcal R$ be a binary relation on a non-empty set $S$. Suppose that: :$\\forall a \\in S: \\exists b \\in S: b \\mathrel {\\mathcal R} a$ that is, that $\\mathcal R$ is a right-total relation. The '''axiom of dependent choice''' states that there exists a sequence $\\sequence {x_n}_{n \\mathop \\in \\N}$ in $S$ such that: :$\\forall n \\in \\N: x_{n + 1} \\mathrel {\\mathcal R} x_n$ </onlyinclude> Dependent Choice 55o04uhf17seubw32jq7zib415n7yjq"}
+{"_id": "32387", "title": "Transitive Closure is Closure Operator", "text": "Transitive Closure is Closure Operator 0 24660 138579 138437 2013-03-03T23:47:46Z Dfeuer 1672 wikitext text/x-wiki == Definition == Let $S$ be a set. Let $\\mathscr A$ be the set of all endorelations on $S$. Then the transitive closure operator is a closure operator on $\\mathscr A$. == Proof == Let $\\mathcal Q$ be the set of transitive relations on $S$. By Intersection of Transitive Relations is Transitive, the intersection of any subset of $\\mathcal Q$ is in $\\mathcal Q$. Recall the definition of transitive closure as the intersection of transitive supersets: :The transitive closure of a relation $\\mathcal R$ on $S$ is the intersection of elements of $\\mathcal Q$ that contain $S$. From Closure Operator from Closed Sets we conclude that transitive closure is a closure operator. {{qed}} Category:Transitive Closures Category:Closure Operators r8q0evdsyz24wk5t8or2growohwwxyq"}
+{"_id": "32388", "title": "Set Union Preserves Subsets/Corollary", "text": "Set Union Preserves Subsets/Corollary 0 24716 413984 413982 2019-07-20T22:28:07Z Prime.mover 59 wikitext text/x-wiki == Corollary to Set Union Preserves Subsets == <onlyinclude> Let $A, B, S$ be sets. Then: :$A \\subseteq B \\implies A \\cup S \\subseteq B \\cup S$ :$A \\subseteq B \\implies S \\cup A \\subseteq S \\cup B$ </onlyinclude> == Proof 1 == {{:Set Union Preserves Subsets/Corollary/Proof 1}} == Proof 2 == {{:Set Union Preserves Subsets/Corollary/Proof 2}} == Sources == * {{BookReference|Undergraduate Topology|1971|Robert H. Kasriel|prev = Intersection with Subset is Subset|next = Union with Empty Set}}: $\\S 1.6$: Set Identities and Other Set Relations: Exercise $1 \\ \\text{(d)}$ Category:Set Union Preserves Subsets 7cieht9076f142282ozekvgadqvxu95"}
+{"_id": "32389", "title": "Set Union Preserves Subsets/Corollary/Proof 2", "text": "Set Union Preserves Subsets/Corollary/Proof 2 0 24718 413987 352645 2019-07-20T22:30:10Z Prime.mover 59 wikitext text/x-wiki == Corollary to Set Union Preserves Subsets == {{:Set Union Preserves Subsets/Corollary}} == Proof == <onlyinclude> Let $A$, $B$, and $S$ be sets. Let $A \\subseteq B$. Let $x \\in A \\cup S$. By the definition of union: :$x \\in A$ or $x \\in S$ Suppose $x \\in A$. Then by the definition of subset: :$x \\in B$ Thus by the definition of union: :$x \\in B \\cup S$ Suppose instead that $x \\in S$. Then by the definition of union: :$x \\in B \\cup S$ Thus for all $x \\in A \\cup S$: :$x \\in B \\cup S$ The second result follows from Union is Commutative. {{qed}} </onlyinclude> Category:Set Union Preserves Subsets ckr3kx3z8dmfi2hlkiqc2nw13p5kamh"}
+{"_id": "32390", "title": "Zero Simple Staircase Integral Condition for Primitive/Lemma", "text": "Zero Simple Staircase Integral Condition for Primitive/Lemma 0 25235 442794 351912 2020-01-07T13:20:36Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $f: D \\to \\C$ be a continuous complex function, where $D$ is a connected domain. Let $C$ be a closed staircase contour in $D$. Then there exists a contour $C'$ such that: :$\\displaystyle \\oint_C \\map f z \\rd z = \\oint_{C'} \\map f z \\rd z$ This contour $C'$ has the property that for all $k \\in \\set {1, \\ldots, n - 1}$, the intersection of the images of $C_k$ and $C_{k + 1}$ is equal to their common end point $\\map {\\gamma_k} {b_k}$. </onlyinclude> == Proof == The proof is by induction over $n \\in \\N$, the number of directed smooth curves that $C$ is a concatenation of. === Basis for the Induction === As induction basis, if $C$ consists of just one directed smooth curve, we can simply put $C' = C$. Then, we have to check a statement for all $k \\in \\O$, so it is trivially true. === Induction Hypothesis === As induction hypothesis, suppose that if $C$ is a concatenation of $n$ directed smooth curves, we can find a contour $C'$ with the properties above. === Induction Step === In the induction step, let $C$ be a concatenation of $n + 1$ directed smooth curves. Suppose that there exists $k \\in \\set {1, \\ldots, n}$ such that the intersection of the images of $C_k$ and $C_{k + 1}$ contain more elements than their common end point $\\map {\\gamma_k} {b_k}$, otherwise we can put $C' = C$. Then, one of the two line segments must contain the other line segment, so: :$\\Img {C_k} \\cap \\Img {C_{k + 1} } = \\gamma_k \\sqbrk {\\closedint {a'} {b_k} } \\cup \\gamma_{k + 1} \\sqbrk {\\closedint {a_{k + 1} } {b'} }$ where either $a' = a_k$, or $b' = b_{k + 1}$. Define $C_k'$ as the contour that is parameterized as a line segment that goes from $\\map {\\gamma_k} {a_k}$ to $\\map {\\gamma_k} {a'}$, and then to $\\map {\\gamma_{k + 1} } {b_{k + 1} }$. Then, the image of $C_k'$ is equal to the part of the line segments that does not overlap. Define $\\tilde C_k$ as the contour with the parameterization $\\gamma_k \\restriction_{\\closedint {a'} {b_k} }$, and $\\tilde C_{k + 1}$ as the contour with the parameterization $\\gamma_{k + 1} \\restriction_{\\closedint {a_{k + 1} } {b'} }$. Here, $\\gamma_k \\restriction_{\\closedint {a'} {b_k} }$ denotes the restriction of $\\gamma_k$ to $\\closedint {a'} {b_k}$. Now $\\tilde C_k$ is the reversed contour of $\\tilde C_{k + 1}$, so: {{begin-eqn}} {{eqn | l = \\oint_C \\map f z \\rd z       | r = \\oint_{C_1 \\cup \\ldots \\cup C_{k - 1} } \\map f z \\rd z + \\oint_{C_k} \\map f z \\rd z + \\oint_{C_{k + 1} } \\map f z \\rd z + \\oint_{C_{k + 2} \\cup \\ldots \\cup C_{n + 1} } \\map f z \\rd z       | c = Contour Integral of Concatenation of Contours }} {{eqn | r = \\oint_{C_1 \\cup \\ldots \\cup C_{k - 1} } \\map f z \\rd z + \\oint_{C_k'} \\map f z \\rd z + \\oint_{\\tilde C_k} \\map f z \\rd z + \\oint_{\\tilde C_{k + 1} } \\map f z \\rd z + \\oint_{C_{k + 2} \\cup \\ldots \\cup C_{n + 1} } \\map f z \\rd z       | c = Contour Integral of Concatenation of Contours }} {{eqn | r = \\oint_{C_1 \\cup \\ldots \\cup C_{k - 1} } \\map f z \\rd z + \\oint_{C_k'} \\map f z \\rd z + \\oint_{\\tilde C_k} \\map f z \\rd z - \\oint_{\\tilde C_k} \\map f z \\rd z + \\oint_{C_{k + 2} \\cup \\ldots \\cup C_{n + 1} } \\map f z \\rd z       | c = Contour Integral along Reversed Contour }} {{eqn | r = \\oint_{C_1 \\cup \\ldots \\cup C_{k - 1} \\cup \\tilde C_k \\cup C_{k + 2} \\ldots \\cup C_{n + 1} } \\map f z \\rd z       | c = Contour Integral of Concatenation of Contours }} {{end-eqn}} which shows that the contour integral of $f$ along $C$ is equal to the contour integral of $f$ along $C'' = C_1 \\cup \\ldots \\cup C_{k - 1} \\cup \\tilde C_k \\cup C_{k + 2} \\ldots \\cup C_{n + 1}$. This new contour $C''$ is a staircase contour which is a concatenation of $n$ directed smooth curves, so by the induction hypothesis, there exists a contour $C'$ such that: :$\\displaystyle \\oint_{C'} \\map f z \\rd z = \\oint_{C''} \\map f z \\rd z$ where $C'$ has the desired property. {{qed}} Category:Complex Analysis j4it516u0i2u375g2cjnypku1rnupg1"}
+{"_id": "32391", "title": "Polynomial Factor Theorem/Corollary", "text": "Polynomial Factor Theorem/Corollary 0 25240 442610 407962 2020-01-06T20:43:45Z Prime.mover 59 wikitext text/x-wiki == Corollary to Polynomial Factor Theorem == <onlyinclude> Let $\\map P x$ be a polynomial in $x$ over the real numbers $\\R$ of degree $n$. Suppose there exists $\\xi \\in \\R: \\map P \\xi = 0$. Then $\\map P x = \\paren {x - \\xi} \\map Q x$, where $\\map Q x$ is a polynomial of degree $n - 1$. Hence, if $\\xi_1, \\xi_2, \\dotsc, \\xi_n \\in \\R$ such that all are different, and $\\map P {\\xi_1} = \\map P {\\xi_2} = \\dotsb = \\map P {\\xi_n} = 0$, then: :$\\displaystyle \\map P x = k \\prod_{j \\mathop = 1}^n \\paren {x - \\xi_j}$ where $k \\in \\R$. </onlyinclude> === Complex Number form === {{:Polynomial Factor Theorem/Corollary/Complex Numbers}} == Proof == Recall that Real Numbers form Field. The result then follows from the Polynomial Factor Theorem. {{qed}} == Also known as == Some sources refer to this (and its more general form) as the Factor Theorem, but there are multiple theorems with this name. == Sources == * {{BookReference|A Course in Pure Mathematics|1960|Margaret M. Gow|prev = Polynomial Remainder Theorem|next = }}: Chapter $1$: Polynomials; The Remainder and Factor Theorems; Undetermined Coefficients; Partial Fractions: $1.2$. The remainder and factor theorems * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Definition:Polynomial Function/Real/Definition 1|next = Definition:Factorial/Definition 2}}: $\\S 3$: Natural Numbers: Exercise $\\S 3.11 \\ (3)$ Category:Polynomial Theory kpusbhnxnnvoa2h5gxzlnh0uyxc72py"}
+{"_id": "32392", "title": "Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 2/Forward Implication", "text": "Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 2/Forward Implication 0 25526 141832 2013-03-24T17:45:43Z Dfeuer 1672 Created page with \"== Theorem == <onlyinclude> : $\\vdash \\left({p \\land \\left({q \\lor r}\\right)}\\right) \\implies \\left({\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)}\\right)$ </onlyincl...\" wikitext text/x-wiki == Theorem == <onlyinclude> : $\\vdash \\left({p \\land \\left({q \\lor r}\\right)}\\right) \\implies \\left({\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)}\\right)$ </onlyinclude> == Proof == {{BeginTableau|\\vdash \\left({p \\land \\left({q \\lor r}\\right)}\\right) \\implies \\left({\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)}\\right)}} {{Assumption|1|p \\land \\left({q \\lor r}\\right)}} {{SequentIntro|2|1|\\left({\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)}\\right)|1|Conjunction is Left Distributive over Disjunction: Formulation 1}} {{Implication|3||\\left({p \\land \\left({q \\lor r}\\right)}\\right) \\implies \\left({\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)}\\right)|1|2}} {{EndTableau}} {{qed}} Category:Rule of Distribution 44o08ry9xd4hrjz9tdnxcsck49f143v"}
+{"_id": "32393", "title": "Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 2/Reverse Implication", "text": "Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 2/Reverse Implication 0 25536 141890 2013-03-25T04:36:09Z Dfeuer 1672 Created page with \"== Theorem == <onlyinclude> :$\\vdash \\left({\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)}\\right) \\implies \\left({p \\land \\left({q \\lor r}\\right)}\\right)$ </onlyinclu...\" wikitext text/x-wiki == Theorem == <onlyinclude> :$\\vdash \\left({\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)}\\right) \\implies \\left({p \\land \\left({q \\lor r}\\right)}\\right)$ </onlyinclude> == Proof == {{BeginTableau|\\vdash \\left({\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)}\\right) \\implies \\left({p \\land \\left({q \\lor r}\\right)}\\right)}} {{Assumption|1|\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)}} {{SequentIntro|2|1|p \\land \\left({q \\lor r}\\right)|1|Conjunction is Left Distributive over Disjunction: Formulation 1}} {{Implication|3||\\left({\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)}\\right) \\implies \\left({p \\land \\left({q \\lor r}\\right)}\\right)|1|2}} {{EndTableau}} {{qed}} Category:Rule of Distribution g9i1e7hay4zpyt39iurxetwxljvt9lm"}
+{"_id": "32394", "title": "Degree of Product of Polynomials over Ring/Corollary 1", "text": "Degree of Product of Polynomials over Ring/Corollary 1 0 25647 348160 348158 2018-03-21T18:38:32Z Barto 3079 terminology: ring of polynomials wikitext text/x-wiki {{mergeto|Degree of Product of Polynomials over Integral Domain}} == Corollary to Degree of Product of Polynomials over Ring == Let $\\left({R, +, \\circ}\\right)$ be a ring with unity whose zero is $0_R$. Let $R \\left[{X}\\right]$ be the ring of polynomials over $R$ in the indeterminate $X$. For $f \\in R \\left[{X}\\right]$ let $\\deg \\left({f}\\right)$ be the degree of $f$. <onlyinclude> Let $R$ have no proper zero divisors. Then: :$\\forall f, g \\in R \\left[{X}\\right]: \\deg \\left({f g}\\right) = \\deg \\left({f}\\right) + \\deg \\left({g}\\right)$ </onlyinclude> == Proof == Let the leading coefficient of: : $f \\left({X}\\right)$ be $a_n$ : $g \\left({X}\\right)$ be $b_m$. From Degree of Product of Polynomials over Ring: :$\\deg \\left({f g}\\right) \\le \\deg \\left({f}\\right) + \\deg \\left({g}\\right)$ From the definition of polynomial multiplication: : $f \\left({X}\\right) g \\left({X}\\right) = a_n b_m X^{n+m} + \\cdots + a_0 b_0$ As $\\left({R, +, \\circ}\\right)$ has no proper zero divisors the $X^{n+m}$ term can not equal $0_R$ and so: :$\\deg \\left({f g}\\right) = \\deg \\left({f}\\right) + \\deg \\left({g}\\right)$ {{qed}} Category:Degree of Product of Polynomials over Ring 2t0ozz3b84kx5izs33ctam5bmqtt3xj"}
+{"_id": "32395", "title": "Union of Bijections with Disjoint Domains and Codomains is Bijection/Corollary", "text": "Union of Bijections with Disjoint Domains and Codomains is Bijection/Corollary 0 25665 200333 142500 2014-11-11T22:31:48Z Prime.mover 59 wikitext text/x-wiki == Corollary to Union of Bijections with Disjoint Domains and Codomains is Bijection == <onlyinclude> Let $A$, $B$, $C$, and $D$ be sets or classes. Let $A \\cap B = C \\cap D = \\varnothing$. Let $f: A \\to C$ and $g: D \\to B$ be bijections. Then $f \\cup g^{-1}: A \\cup B \\to C \\cup D$ is also a bijection. </onlyinclude> == Proof == By definition of bijection, $g^{-1}: B \\to D$ is a bijection. Hence the result by Union of Bijections with Disjoint Domains and Codomains is Bijection. {{qed}} Category:Bijections re730kt8jf0uk7g2td02l1aasv0v3z6"}
+{"_id": "32396", "title": "Set Difference and Intersection form Partition/Corollary 1", "text": "Set Difference and Intersection form Partition/Corollary 1 0 26175 496211 418193 2020-10-24T16:16:55Z Prime.mover 59 wikitext text/x-wiki == Corollary to Set Difference and Intersection form Partition == <onlyinclude> Let $S$ and $T$ be sets such that: :$S \\setminus T \\ne \\O$ :$T \\setminus S \\ne \\O$ :$S \\cap T \\ne \\O$ Then $S \\setminus T$, $T \\setminus S$ and $S \\cap T$ form a partition of $S \\cup T$, the union of $S$ and $T$. </onlyinclude> == Proof == From Set Difference and Intersection form Partition: :$S \\setminus T$ and $S \\cap T$ form a partition of $S$ :$T \\setminus S$ and $S \\cap T$ form a partition of $T$ From Set Difference is Disjoint with Reverse: :$\\paren {S \\setminus T} \\cap \\paren {T \\setminus S} = \\O$ So: :$S \\cup T = \\paren {S \\setminus T} \\cup \\paren {S \\cap T} \\cup \\paren {T \\setminus S} \\cup \\paren {S \\cap T}$ and the result follows. {{qed}} Category:Set Difference Category:Set Intersection Category:Set Union 05cyy0o0qirvwojqxolxdkmyfrur5lo"}
+{"_id": "32397", "title": "Inclusion-Exclusion Principle/Corollary", "text": "Inclusion-Exclusion Principle/Corollary 0 26197 390368 352624 2019-01-28T13:36:09Z Prime.mover 59 wikitext text/x-wiki == Corollary to Inclusion-Exclusion Principle == <onlyinclude> Let $\\mathcal S$ be an algebra of sets. Let $A_1, A_2, \\ldots, A_n$ be finite sets which are pairwise disjoint. Let $f: \\mathcal S \\to \\R$ be an additive function. Then: :$\\displaystyle \\map f {\\bigcup_{i \\mathop = 1}^n A_i} = \\sum_{i \\mathop = 1}^n \\map f {A_i}$ </onlyinclude> == Proof == As $A_1, A_2, \\ldots, A_n$ are pairwise disjoint, their intersections are all empty. Then Inclusion-Exclusion Principle holds. However, from Cardinality of Empty Set, all the terms apart from the first vanish. {{qed}} == Comment == This result is usually quoted in the context of combinatorics, where $f$ is the cardinality function. It is also seen in the context of probability theory, in which $f$ is taken to be a probability measure. Category:Set Systems Category:Set Union Category:Probability Theory Category:Combinatorics srw2dqr9ylgcyhz7g4qkyn3o0o2ya7m"}
+{"_id": "32398", "title": "Independent Events are Independent of Complement/Corollary", "text": "Independent Events are Independent of Complement/Corollary 0 26223 486487 189570 2020-09-08T14:58:16Z Prime.mover 59 wikitext text/x-wiki == Corollary to Independent Events are Independent of Complement == Let $A$ and $B$ be events in a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. <onlyinclude> $A$ and $B$ are independent {{iff}} $\\Omega \\setminus A$ and $\\Omega \\setminus B$ are independent. </onlyinclude> == Proof == Let $A$ and $B$ be independent. Then from Independent Events are Independent of Complement, $A$ and $\\Omega \\setminus B$ are independent. Setting $A' = \\Omega \\setminus B$ and $B' = A$, we see clearly that $A'$ and $B'$ are independent. So from the main result, $A'$ and $\\Omega \\setminus B'$ are independent. That is, $\\Omega \\setminus B$ and $\\Omega \\setminus A$ are independent. The \"only if\" part of the result follows directly from Relative Complement of Relative Complement and another application of this result. {{qed}} Category:Independent Events 5uhcjyyox39poiunckkjtnlx3vu6haf"}
+{"_id": "32399", "title": "Condition for Independence from Product of Expectations/Corollary", "text": "Condition for Independence from Product of Expectations/Corollary 0 26304 475737 475727 2020-06-24T05:16:15Z Prime.mover 59 wikitext text/x-wiki == Corollary to Condition for Independence from Product of Expectations == Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. <onlyinclude> Let $X$ and $Y$ be independent discrete random variables on $\\struct {\\Omega, \\Sigma, \\Pr}$. Then: :$\\expect {X Y} = \\expect X \\expect Y$ assuming the latter expectations exist. </onlyinclude> === General Result === {{:Condition for Independence from Product of Expectations/Corollary/General Result}} == Proof == From Condition for Independence from Product of Expectations, setting both $g$ and $h$ to the identity functions: :$\\forall x \\in \\R: \\map g x = x$ :$\\forall y \\in \\R: \\map h y = y$ It follows directly that if $X$ and $Y$ are independent, then: :$\\expect {X Y} = \\expect X \\expect Y$ assuming the latter expectations exist. {{qed}} == Note on Converse == Note that the converse of the corollary does not necessarily hold. {{:Condition for Independence from Product of Expectations/Corollary/Converse}} == Sources == * {{BookReference|Probability: An Introduction|1986|Geoffrey Grimmett|author2 = Dominic Welsh|prev = Condition for Independence of Discrete Random Variables|next = Condition for Independence from Product of Expectations}}: $\\S 3.3$: Independence of discrete random variables: Theorem $3 \\text{C}$ Category:Expectation 9235x95dydyt0aurp8hvfi3pfu4dzhk"}
+{"_id": "32400", "title": "Condition for Independence from Product of Expectations/Corollary/General Result", "text": "Condition for Independence from Product of Expectations/Corollary/General Result 0 26306 481558 475740 2020-08-13T10:30:02Z RandomUndergrad 3904 wikitext text/x-wiki == Corollary to Condition for Independence from Product of Expectations == Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. <onlyinclude> Let $X_1, X_2, \\ldots, X_n$ be independent discrete random variables. Then: :$\\displaystyle \\expect {\\prod_{k \\mathop = 1}^n {X_k} } = \\prod_{k \\mathop = 1}^n \\expect {X_k}$ assuming the latter expectations exist. </onlyinclude> == Proof == Proof by induction: For all $n \\in \\Z_{> 0}$, let $\\map P n$ be the proposition: :$\\displaystyle \\expect {\\prod_{k \\mathop = 1}^n {X_k} } = \\prod_{k \\mathop = 1}^n \\expect {X_k}$ === Basis for the Induction === $\\map P 1$ is the case: :$\\displaystyle \\expect {\\prod_{k \\mathop = 1}^1 {X_k} } = \\expect {X_1} = \\prod_{k \\mathop = 1}^1 \\expect {X_k}$ Thus $\\map P 1$ is seen to hold. This is the basis for the induction. === Induction Hypothesis === Now it needs to be shown that if $\\map P r$ is true, where $r \\ge 1$, then it logically follows that $\\map P {r + 1}$ is true. So this is the induction hypothesis: :$\\displaystyle \\expect {\\prod_{k \\mathop = 1}^r {X_k} } = \\prod_{k \\mathop = 1}^r \\expect {X_k}$ from which it is to be shown that: :$\\displaystyle \\expect {\\prod_{k \\mathop = 1}^{r + 1} {X_k} } = \\prod_{k \\mathop = 1}^{r + 1} \\expect {X_k}$ === Induction Step === This is the induction step: We have: {{begin-eqn}} {{eqn | l = \\expect {\\prod_{k \\mathop = 1}^{r + 1} {X_k} }       | r = \\expect {X_{r + 1} \\prod_{k \\mathop = 1}^r {X_k} } }} {{eqn | r = \\expect {X_{r + 1} } \\expect {\\prod_{k \\mathop = 1}^r {X_k} }       | c = Corollary to Condition for Independence from Product of Expectations }} {{eqn | r = \\expect {X_{r + 1} } \\prod_{k \\mathop = 1}^r \\expect {X_k}       | c = Induction Hypothesis }} {{eqn | r = \\prod_{k \\mathop = 1}^{r + 1} \\expect {X_k} }} {{end-eqn}} So $\\map P r \\implies \\map P {r + 1}$ and thus it follows by the Principle of Mathematical Induction that: :$\\displaystyle \\forall n \\in \\Z_{> 0}: \\expect {\\prod_{k \\mathop = 1}^n {X_k} } = \\prod_{k \\mathop = 1}^n \\expect {X_k}$ {{qed}} Category:Expectation hl637f6rigaldm19zwxt3j8q8ql2s68"}
+{"_id": "32401", "title": "Sum of Infinite Geometric Sequence/Corollary 1", "text": "Sum of Infinite Geometric Sequence/Corollary 1 0 26315 456204 456125 2020-03-19T09:16:53Z Prime.mover 59 wikitext text/x-wiki == Corollary to Sum of Infinite Geometric Sequence == Let $S$ be a standard number field, i.e. $\\Q$, $\\R$ or $\\C$. Let $z \\in S$. Let $\\size z < 1$, where $\\size z$ denotes: :the absolute value of $z$, for real and rational $z$ :the complex modulus of $z$ for complex $z$. Then: <onlyinclude> :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty z^n = \\frac z {1 - z}$ </onlyinclude> == Proof 1 == {{:Sum of Infinite Geometric Sequence/Corollary 1/Proof 1}} == Proof 2 == {{:Sum of Infinite Geometric Sequence/Corollary 1/Proof 2}} Category:Sum of Infinite Geometric Sequence 18ydlbwfe2ktt3eos2413uldw1w9zgn"}
+{"_id": "32402", "title": "Axiom:Axiom of Countable Choice for Finite Sets", "text": "Axiom:Axiom of Countable Choice for Finite Sets 100 26471 453393 453350 2020-03-08T12:52:18Z Prime.mover 59 wikitext text/x-wiki == Axiom == Let $S$ be a countable set of non-empty finite sets. Then $S$ has a choice function. == Also see == This axiom is a weakening of the Axiom of Countable Choice and the Axiom of Choice for Finite Sets. == Sources == {{wtd|Have source; will add.}} Category:Axioms/Axiom of Choice 4zeewvmxrtth0ae6bvrkg2eqzxjginl"}
+{"_id": "32403", "title": "Axiom:Axiom of Countable Choice/Form 2", "text": "Axiom:Axiom of Countable Choice/Form 2 100 26473 444057 444053 2020-01-18T02:01:49Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> Let $S$ be a countable set of non-empty sets. Then $S$ has a choice function. </onlyinclude> == Also see == * Equivalence of Forms of Axiom of Countable Choice == Sources == {{NoSources}} Countable Choice 9gilr1ghka7rq545viyvdgkfh6xvnkl"}
+{"_id": "32404", "title": "Axiom:Axiom of Countable Choice/Form 1", "text": "Axiom:Axiom of Countable Choice/Form 1 100 26474 444056 444051 2020-01-18T02:01:38Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> Let $\\sequence {S_n}_{n \\mathop \\in \\N}$ be a sequence of non-empty sets. The '''axiom of countable choice''' states that there exists a sequence: :$\\sequence {x_n}_{n \\mathop \\in \\N}$ such that $x_n \\in S_n$ for all $n \\in \\N$. </onlyinclude> == Also see == * Equivalence of Forms of Axiom of Countable Choice Countable Choice 5571sm8cd5w30rp4kupm6da3uthox0z"}
+{"_id": "32405", "title": "Between two Rational Numbers exists Irrational Number/Lemma 1", "text": "Between two Rational Numbers exists Irrational Number/Lemma 1 0 26902 369152 369112 2018-10-05T17:33:27Z Prime.mover 59 wikitext text/x-wiki == Lemma for Between two Rational Numbers exists Irrational Number == <onlyinclude> Let $\\alpha \\in \\Q \\setminus \\set 0$ and $\\beta \\in \\R \\setminus \\Q$.  Then:  :$\\alpha \\cdot \\beta \\in \\R \\setminus \\Q$ </onlyinclude> == Proof == {{AimForCont}} $\\alpha \\cdot \\beta \\in \\Q$. By the definition of rational numbers: :$\\exists n, m, p, q \\in \\Z: \\alpha = \\dfrac n m$ :$\\alpha \\cdot \\beta = \\dfrac p q$ Thus: :$\\beta = \\dfrac p q \\cdot \\dfrac 1 \\alpha = \\dfrac p q \\cdot \\dfrac m n$ By Rational Multiplication is Closed, we have $\\beta \\in \\Q$, which contradicts the statement that $\\beta \\in \\R \\setminus \\Q$. Therefore  $\\alpha \\cdot \\beta \\in \\R \\setminus \\Q$. {{qed}} Category:Between two Rational Numbers exists Irrational Number 35l02mdb33hx8y6ktvgn22juytppbwp"}
+{"_id": "32406", "title": "Between two Rational Numbers exists Irrational Number/Lemma 2", "text": "Between two Rational Numbers exists Irrational Number/Lemma 2 0 26903 483038 369153 2020-08-26T21:58:49Z Prime.mover 59 wikitext text/x-wiki == Lemma for Between two Rational Numbers exists Irrational Number == <onlyinclude> Let $\\alpha \\in \\Q$ and $\\beta \\in \\R \\setminus \\Q$.  Then:  :$\\alpha + \\beta \\in \\R \\setminus \\Q$ </onlyinclude> == Proof == {{AimForCont}} $\\alpha + \\beta \\in \\Q$. By the definition of rational numbers: :$\\exists p, q \\in \\Z: \\alpha + \\beta = \\dfrac p q$ Thus: :$\\beta = \\dfrac p q - \\alpha$ By Rational Subtraction is Closed, we have $\\beta \\in \\Q$, which contradicts the statement that $\\beta \\in \\R \\setminus \\Q$. Therefore: :$\\alpha + \\beta \\in \\R \\setminus \\Q$ {{qed}} Category:Between two Rational Numbers exists Irrational Number 7pc2qb6kr0efk8crs9joxmhmq28bb48"}
+{"_id": "32407", "title": "Mersenne Prime/Current Status", "text": "Mersenne Prime/Current Status 0 26909 484342 446311 2020-08-31T11:04:21Z Prime.mover 59 wikitext text/x-wiki == Currently known Mersenne Primes == The list of all known Mersenne primes is as follows: <onlyinclude> {{Begin-Mersenne-Primes}} {{Mersenne-Prime |  1 | 2 | 3 | 1 | Known to {{AuthorRef|Euclid}} | }} {{Mersenne-Prime |  2 | 3 | 7 | 1 | Known to {{AuthorRef|Euclid}} | }} {{Mersenne-Prime |  3 | 5 | 31 | 2 | Known to {{AuthorRef|Euclid}} | }} {{Mersenne-Prime |  4 | 7 | 127 | 3 | Known to {{AuthorRef|Euclid}} | }} {{Mersenne-Prime |  5 | 13 | 8191 | 4 | 1456 | }} {{Mersenne-Prime |  6 | 17 | 131 \\, 071 | 6 | 1588 | {{AuthorRef|Pietro Antonio Cataldi}} }} {{Mersenne-Prime |  7 | 19 | 524 \\, 287 | 6 | 1588 | {{AuthorRef|Pietro Antonio Cataldi}} }} {{Mersenne-Prime |  8 | 31 | 2 \\, 147 \\, 483 \\, 647 | 10 | 1772 | {{AuthorRef|Leonhard Paul Euler|}} }} {{Mersenne-Prime |  9 | 61 | 2 \\cdotp 305 \\times 10^{18} | 19 | 1883 | {{AuthorRef|Ivan Mikheevich Pervushin}} }} {{Mersenne-Prime | 10 | 89 | 6 \\cdotp 189 \\times 10^{26} | 27 | 1911 | {{AuthorRef|R.E. Powers}} }} {{Mersenne-Prime | 11 | 107 | 1 \\cdotp 622 \\times 10^{32} | 33 | 1914 | {{AuthorRef|R.E. Powers}} }} {{Mersenne-Prime | 12 | 127 | 1 \\cdotp 701 \\times 10^{38} | 39 | 1876 | {{AuthorRef|François Édouard Anatole Lucas}} }} {{Mersenne-Prime | 13 | 521 | 6 \\cdotp 865 \\times 10^{156} | 157 | 30 Jan 1952 | {{AuthorRef|Raphael Mitchel Robinson}} }} {{Mersenne-Prime | 14 | 607 | 5 \\cdotp 311 \\times 10^{182} | 183 | 30 Jan 1952 | {{AuthorRef|Raphael Mitchel Robinson}} }} {{Mersenne-Prime | 15 | 1279 | 1 \\cdotp 041 \\times 10^{385} | 386 | 25 Jun 1952 | {{AuthorRef|Raphael Mitchel Robinson}} }} {{Mersenne-Prime | 16 | 2203 | 1 \\cdotp 476 \\times 10^{663} | 664 | 7 Oct 1952 | {{AuthorRef|Raphael Mitchel Robinson}} }} {{Mersenne-Prime | 17 | 2281 | 4 \\cdotp 461 \\times 10^{686} | 687 | 9 Oct 1952 | {{AuthorRef|Raphael Mitchel Robinson}} }} {{Mersenne-Prime | 18 | 3217 | 2 \\cdotp 591 \\times 10^{968} | 969 | 8 Sept 1957 | {{AuthorRef|Hans Ivar Riesel}} }} {{Mersenne-Prime | 19 | 4253 | 1 \\cdotp 908 \\times 10^{1280} | 1281 | 3 Nov 1961 | {{AuthorRef|Alexander Hurwitz}} }} {{Mersenne-Prime | 20 | 4423 | 2 \\cdotp 855 \\times 10^{1331} | 1332 | 3 Nov 1961 | {{AuthorRef|Alexander Hurwitz}} }} {{Mersenne-Prime | 21 | 9689 | 4 \\cdotp 782 \\times 10^{2916} | 2917 | 11 May 1963 | {{AuthorRef|Donald Bruce Gillies}} }} {{Mersenne-Prime | 22 | 9941 | 3 \\cdotp 461 \\times 10^{2992} | 2993 | 16 May 1963 | {{AuthorRef|Donald Bruce Gillies}} }} {{Mersenne-Prime | 23 | 11 \\, 213 | 2 \\cdotp 814 \\times 10^{3375} | 3376 | 2 Jun 1963 | {{AuthorRef|Donald Bruce Gillies}} }} {{Mersenne-Prime | 24 | 19 \\, 937 | 4 \\cdotp 315 \\times 10^{6001} | 6002 | 4 Mar 1971 | {{AuthorRef|Bryant Tuckerman}} }} {{Mersenne-Prime | 25 | 21 \\, 701 | 4 \\cdotp 487 \\times 10^{6532} | 6533 | 30 Oct 1978 | {{AuthorRef|Landon Curt Noll}} and {{AuthorRef|Ariel Nickel}} }} {{Mersenne-Prime | 26 | 23 \\, 209 | 4 \\cdotp 029 \\times 10^{6986} | 6987 | 9 Feb 1979 | {{AuthorRef|Landon Curt Noll}} }} {{Mersenne-Prime | 27 | 44 \\, 497 | 8 \\cdotp 545 \\times 10^{13 \\, 394} | 13 \\, 395 | 8 Apr 1979 | {{AuthorRef|Harry Lewis Nelson}} and {{AuthorRef|David Slowinski}} }} {{Mersenne-Prime | 28 | 86 \\, 243 | 5 \\cdotp 369 \\times 10^{25 \\, 961} | 25 \\, 962 | 25 Sept 1982 | {{AuthorRef|David Slowinski}} }} {{Mersenne-Prime | 29 | 110 \\, 503 | 5 \\cdotp 219 \\times 10^{33 \\, 264} | 33 \\, 265 | 28 Jan 1988 | {{AuthorRef|Walt Colquitt}} and {{AuthorRef|Luke Welsh}} }} {{Mersenne-Prime | 30 | 132 \\, 049 | 5 \\cdotp 127 \\times 10^{39 \\, 750} | 39 \\, 751 | 19 Sept 1983 | {{AuthorRef|David Slowinski}} }} {{Mersenne-Prime | 31 | 216 \\, 091 | 7 \\cdotp 461 \\times 10^{65 \\, 049} | 65 \\, 050 | 1 Sept 1985 | {{AuthorRef|David Slowinski}} }} {{Mersenne-Prime | 32 | 756 \\, 839 | 1 \\cdotp 741 \\times 10^{227 \\, 831} | 227 \\, 832 | 19 Feb 1992 | {{AuthorRef|David Slowinski}} and {{AuthorRef|Paul Gage}} }} {{Mersenne-Prime | 33 | 859 \\, 433 | 1 \\cdotp 295 \\times 10^{258 \\, 715} | 258 \\, 716 | 4 Jan 1994 | {{AuthorRef|David Slowinski}} and {{AuthorRef|Paul Gage}} }} {{Mersenne-Prime | 34 | 1 \\, 257 \\, 787 | 4 \\cdotp 122 \\times 10^{378 \\, 631} | 378 \\, 632 | 3 Sept 1996 | {{AuthorRef|David Slowinski}} and {{AuthorRef|Paul Gage}} }} {{Mersenne-Prime | 35 | 1 \\, 398 \\, 269 | 8 \\cdotp 147 \\times 10^{420 \\, 920} | 420 \\, 921 | 13 Nov 1996 | GIMPS / {{AuthorRef|Joel Armengaud}} }} {{Mersenne-Prime | 36 | 2 \\, 976 \\, 221 | 6 \\cdotp 233 \\times 10^{895 \\, 931} | 895 \\, 932 | 24 Aug 1997 | GIMPS / {{AuthorRef|Gordon Spence}} }} {{Mersenne-Prime | 37 | 3 \\, 021 \\, 377 | 1 \\cdotp 274 \\times 10^{909 \\, 525} | 909 \\, 526 | 27 Jan 1998 | GIMPS / {{AuthorRef|Roland Clarkson}} }} {{Mersenne-Prime | 38 | 6 \\, 972 \\, 593 | 4 \\cdotp 371 \\times 10^{2 \\, 098 \\, 959} | 2 \\, 098 \\, 960 | 1 Jun 1999 | GIMPS / {{AuthorRef|Nayan Hajratwala}} }} {{Mersenne-Prime | 39 | 13 \\, 466 \\, 917 | 9 \\cdotp 249 \\times 10^{4 \\, 053 \\, 945} | 4 \\, 053 \\, 946 | 14 Nov 2001 | GIMPS / {{AuthorRef|Michael Cameron}} }} {{Mersenne-Prime | 40 | 20 \\, 996 \\, 011 | 1 \\cdotp 260 \\times 10^{6 \\, 320 \\, 429} | 6 \\, 320 \\, 430 | 17 Nov 2003 | GIMPS / {{AuthorRef|Michael Shafer}} }} {{Mersenne-Prime | 41 | 24 \\, 036 \\, 583 | 2 \\cdotp 994 \\times 10^{7 \\, 235 \\, 732} | 7 \\, 235 \\, 733 | 15 May 2004 | GIMPS / {{AuthorRef|Josh Findley}} }} {{Mersenne-Prime | 42 | 25 \\, 964 \\, 951 | 1 \\cdotp 222 \\times 10^{7 \\, 816 \\, 229} | 7 \\, 816 \\, 230 | 18 Feb 2005 | GIMPS / {{AuthorRef|Martin Nowak}} }} {{Mersenne-Prime | 43 | 30 \\, 402 \\, 457 | 3 \\cdotp 154 \\times 10^{9 \\, 152 \\, 051} | 9 \\, 152 \\, 052 | 15 Dec 2005 | GIMPS / {{AuthorRef|Curtis Cooper}} and {{AuthorRef|Steven Boone}} }} {{Mersenne-Prime | 44 | 32 \\, 582 \\, 657 | 1 \\cdotp 246 \\times 10^{9 \\, 808 \\, 358} | 9 \\, 808 \\, 358 | 4 Sept 2006 | GIMPS / {{AuthorRef|Curtis Cooper}} and {{AuthorRef|Steven Boone}} }} {{Mersenne-Prime | 45 | 37 \\, 156 \\, 667 | 2 \\cdotp 023 \\times 10^{11 \\, 185 \\, 271} | 11 \\, 185 \\, 272 | 6 Sept 2008 | GIMPS / {{AuthorRef|Hans-Michael Elvenich}} }} {{Mersenne-Prime | 46 | 42 \\, 643 \\, 801 | 1 \\cdotp 699 \\times 10^{12 \\, 837 \\, 063} | 12 \\, 837 \\, 064 | 12 Apr 2009 | GIMPS / {{AuthorRef|Odd Magnar Strindmo}} }} {{Mersenne-Prime | 47 | 43 \\, 112 \\, 609 | 3 \\cdotp 165 \\times 10^{12 \\, 978 \\, 188} | 12 \\, 978 \\, 189 | 23 Aug 2008 | GIMPS / {{AuthorRef|Edson Smith}} }} {{Mersenne-Prime |  | 57 \\, 885 \\, 161 | 5 \\cdotp 818 \\times 10^{17 \\, 425 \\, 169} | 17 \\, 425 \\, 170 | 25 Jan 2013 | GIMPS / {{AuthorRef|Curtis Cooper}} }} {{Mersenne-Prime |  | 74 \\, 207 \\, 281 | 3 \\cdotp 003 \\times 10^{22 \\, 338 \\, 617} | 22 \\, 338 \\, 618 | 07 Jan 2016 | GIMPS / {{AuthorRef|Curtis Cooper}} }} {{Mersenne-Prime |  | 77 \\, 232 \\, 917 | 4 \\cdotp 673 \\times 10^{23 \\, 249 \\, 424} | 23 \\, 249 \\, 425 | 26 Dec 2017 | GIMPS / {{AuthorRef|Jon Pace}} }} {{Mersenne-Prime |  | 82 \\, 589 \\, 933 | 1 \\cdotp 488 \\times 10^{24 \\, 862 \\, 047} | 24 \\, 862 \\, 048 | 07 Dec 2018 | GIMPS / {{AuthorRef|Patrick Laroche}} }} {{End-Mersenne-Primes}} Note that the index numbers of Mersenne primes after no. $47$ are uncertain, as there may still be undiscovered Mersenne primes between the $47$th and $51$st. Not all numbers in that range have been explored yet. </onlyinclude> == Also see == {{OEIS|A000668}} {{OEIS|A000043|order = index}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Barlow's Prediction|next = Do Mersenne Primes Go On For Ever?}}: $28$ * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Definition:Mersenne Prime/Historical Note|next = Mathematician:David Slowinski}}: Chapter $\\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Barlow's Prediction|next = Do Mersenne Primes Go On For Ever?}}: $28$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Mersenne Prime/Historical Note|next = Theorem of Even Perfect Numbers|entry = Mersenne numbers}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Mersenne Prime/Historical Note|next = Theorem of Even Perfect Numbers|entry = Mersenne numbers}} * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Definition:Mersenne Prime/Historical Note|next = Mathematician:Diophantus of Alexandria}}: Chapter $7$: Patterns in Numbers: Euclid * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Mersenne Prime|next = 1,257,787/Historical Note|entry = Mersenne prime}} Category:Mersenne Primes regie24mutkxbdw88rl7k3c450bvvcg"}
+{"_id": "32408", "title": "Divisibility by 9/Corollary", "text": "Divisibility by 9/Corollary 0 27285 380675 376067 2018-12-08T15:51:59Z Prime.mover 59 wikitext text/x-wiki == Corollary to Divisibility by 9 == <onlyinclude> A number expressed in decimal notation is divisible by $3$ {{iff}} the sum of its digits is divisible by $3$. That is: :$N = \\sqbrk {a_0 a_1 a_2 \\ldots a_n}_{10} = a_0 + a_1 10 + a_2 10^2 + \\cdots + a_n 10^n$ is divisible by $3$  {{iff}}: :$a_0 + a_1 + \\ldots + a_n$ is divisible by $3$. </onlyinclude> == Proof == From Divisibility by 9 we have that: :$N = \\sqbrk {a_0 a_1 a_2 \\ldots a_n}_{10} = a_0 + a_1 10 + a_2 10^2 + \\cdots + a_n 10^n$ is divisible by $3^2$  {{iff}}: :$a_0 + a_1 + \\ldots + a_n$ is divisible by $3^2$. So: {{begin-eqn}} {{eqn | l = \\paren {a_0 + a_1 10 + a_2 10^2 + \\cdots + a_n 10^n}       | o = \\equiv       | r = \\paren {a_0 + a_1 + a_2 + \\cdots + a_n}       | rr= \\pmod {3^2}       | c =  }} {{eqn | ll= \\leadstoandfrom       | l = \\paren {a_0 + a_1 10 + a_2 10^2 + \\cdots + a_n 10^n}       | o = \\equiv       | r = \\paren {a_0 + a_1 + a_2 + \\cdots + a_n}       | rr= \\pmod 3       | c = Congruence by Divisor of Modulus }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Divisibility by 9/Proof 1|next = Binomial Theorem/Integral Index}}: $\\S 2.6$. Algebra of congruences: Example $41$ * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Integer as Sum of Three Squares|next = Vinogradov's Theorem/Corollary 2}}: $3$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Integer as Sum of Three Squares|next = Vinogradov's Theorem/Corollary 2}}: $3$ Category:Divisibility Tests Category:3 c6f43kipzt6rfbqz66egca9gwvlk9ua"}
+{"_id": "32409", "title": "Square Root/Examples/2", "text": "Square Root/Examples/2 0 27339 447779 435327 2020-02-09T15:40:28Z Prime.mover 59 wikitext text/x-wiki == Decimal Expansion == <onlyinclude> The decimal expansion of the square root of $2$ starts: :$\\sqrt 2 \\approx 1 \\cdotp 41421 \\, 35623 \\, 73095 \\, 04880 \\, 16887 \\, 24209 \\, 69807 \\, 85697 \\ldots$ </onlyinclude> {{OEIS|A002193}} == Also known as == :$\\sqrt 2$ is sometimes known as '''{{AuthorRef|Pythagoras of Samos|Pythagoras}}'s constant''', for {{AuthorRef|Pythagoras of Samos}}. == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Definition:Euler's Number/Limit of Sequence|next = Square Root/Examples/3}}: $\\S 1$: Special Constants: $1.3$ * {{BookReference|Theory and Problems of Differential and Integral Calculus|1972|Frank Ayres, Jr.|author2 = J.C. Ault|ed = SI|edpage = SI Edition|prev = Definition:Real Number|next = Definition:Pi/Decimal Expansion}}: Chapter $1$: Variables and Functions: The Set of Real Numbers * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Square Root of 2 is Irrational|next = Pi is Irrational}}: $1$: Complex Numbers: The Real Number System: $4$ * {{BookReference|Les Nombres Remarquables|1983|François Le Lionnais|author2 = Jean Brette|prev = Number of Primes of Form n^2 + 1/Historical Note|next = Length of Diagonal of Unit Square}}: $1,41421 35623 73 \\ldots$ * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Doubling the Cube/Cissoid of Diocles/Historical Note|next = Length of Diagonal of Unit Square}}: $1 \\cdotp 41421 \\, 35623 \\, 73095 \\, 04880 \\, 16887 \\, 24209 \\, 69807 \\, 85697 \\ldots$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Doubling the Cube/Cissoid of Diocles/Historical Note|next = Length of Diagonal of Unit Square}}: $1 \\cdotp 41421 \\, 35623 \\, 73095 \\, 04880 \\, 16887 \\, 24209 \\, 69807 \\, 85697 \\ldots$ Category:Square Root of 2 Category:Examples of Square Roots Category:Square Root of 2 is Irrational paqwv8ns78wqocfd6f2v6mctkqevyn0"}
+{"_id": "32410", "title": "Square Root/Examples/5", "text": "Square Root/Examples/5 0 27342 433313 393211 2019-11-01T09:07:52Z Prime.mover 59 wikitext text/x-wiki == Decimal Expansion == <onlyinclude> The decimal expansion of $\\sqrt 5$ starts: :$\\sqrt 5 \\approx 2 \\cdotp 23606 \\, 79774 \\, 99789 \\, 6964 \\ldots$ </onlyinclude> {{OEIS|A002163}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Square Root of 3|next = Cube Root of 2}}: $\\S 1$: Special Constants: $1.5$ * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Definition:Wallis's Number/Historical Note|next = Natural Logarithm of 10}}: $2 \\cdotp 236 \\, 067 \\ldots$ Category:Examples of Square Roots oeqnocm2i451v28jlvpueviyb2j3u6a"}
+{"_id": "32411", "title": "Axiom:Euclid's Second Postulate/Production", "text": "Axiom:Euclid's Second Postulate/Production 100 27568 446523 424515 2020-02-05T12:19:33Z Prime.mover 59 wikitext text/x-wiki == Definition == <onlyinclude> The process of extending a straight line segment is called '''production'''. Such a straight line segment is said to be '''produced'''. The line segment can be extended in either direction, thus producing an infinite half-line, or in both directions, thus producing an infinite straight line. </onlyinclude> {{EuclidPostulateStatement|Second|To produce a finite straight line continuously in a straight line.}} == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Mathematician:Proclus Lycaeus|next = Definition:Product|entry = produce}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Mathematician:Proclus Lycaeus|next = Definition:Producer's Risk|entry = produce}} Category:Definitions/Geometry r4zwvubgslksrb9h401ckzxm1rfn7yx"}
+{"_id": "32412", "title": "Sine and Cosine are Periodic on Reals/Corollary", "text": "Sine and Cosine are Periodic on Reals/Corollary 0 27870 441347 400163 2019-12-29T08:52:42Z Prime.mover 59 wikitext text/x-wiki {{mergeto|Cosine of Angle plus Straight Angle]] and [[Sine of Angle plus Straight Angle}} == Corollary to Sine and Cosine are Periodic on Reals == <onlyinclude> :$\\map \\cos {x + \\pi} = -\\cos x$ :$\\map \\sin {x + \\pi} = -\\sin x$ :$\\cos x$ is strictly positive on the interval $\\openint {-\\dfrac \\pi 2} {\\dfrac \\pi 2}$ and strictly negative on the interval $\\openint {\\dfrac \\pi 2} {\\dfrac {3 \\pi} 2}$ :$\\sin x$ is strictly positive on the interval $\\openint 0 \\pi$ and strictly negative on the interval $\\openint \\pi {2 \\pi}$ </onlyinclude> == Proof == From Sine and Cosine are Periodic on Reals: :$\\map \\sin {x + \\dfrac \\pi 2} = \\cos x$ :$\\map \\cos {x + \\dfrac \\pi 2} = -\\sin x$ Thus: :$\\map \\sin {x + \\pi} = \\map \\cos {x + \\dfrac \\pi 2} = -\\sin x$ :$\\map \\cos {x + \\pi} = -\\map \\sin {x + \\dfrac \\pi 2} = -\\cos x$ It follows directly that: :$\\forall x \\in \\closedint {-\\dfrac \\pi 2} {\\dfrac \\pi 2}: \\cos x \\ge 0$ Hence: :$\\forall x \\in \\closedint {\\dfrac \\pi 2} {\\dfrac {3 \\pi} 2}: \\cos x \\le 0$ The result for $\\sin x$ follows similarly, or we can use: :$\\map \\sin {x + \\dfrac \\pi 2} = \\cos x$ {{qed}} Category:Sine Function Category:Cosine Function qgvtqrzg1h4bt2cizdu44meq4qeausj"}
+{"_id": "32413", "title": "Sum of Squares of Sine and Cosine/Corollary 2", "text": "Sum of Squares of Sine and Cosine/Corollary 2 0 28010 396161 310516 2019-03-18T21:07:09Z Prime.mover 59 wikitext text/x-wiki == Corollary to Sum of Squares of Sine and Cosine == For all $x \\in \\C$: <onlyinclude> :$\\csc^2 x - \\cot^2 x = 1 \\quad \\text {(when $\\sin x \\ne 0$)}$ </onlyinclude> where $\\csc$, $\\cot$ and $\\sin$ are cosecant, cotangent and sine respectively. == Proof  == When $\\sin x \\ne 0$: {{begin-eqn}} {{eqn | l = \\sin^2 x + \\cos^2 x       | r = 1       | c = Sum of Squares of Sine and Cosine }} {{eqn | ll= \\leadsto       | l = 1 + \\frac {\\cos^2 x} {\\sin^2 x}       | r = \\frac 1 {\\sin^2 x}       | c = dividing both sides by $\\sin^2 x$, as $\\sin x \\ne 0$ }} {{eqn | ll= \\leadsto       | l = 1 + \\cot^2 x       | r = \\csc^2 x       | c = {{Defof|Cotangent}} and {{Defof|Cosecant}} }} {{eqn | ll= \\leadsto       | l = \\csc^2 x - \\cot^2 x       | r = 1       | c = rearranging }} {{end-eqn}} {{qed}} == Also defined as == This result can also be reported as: :$\\csc^2 x = 1 + \\cot^2 x \\quad \\text{(when $\\sin x \\ne 0$)}$ or: :$\\cot^2 x = \\csc^2 x - 1 \\quad \\text{(when $\\sin x \\ne 0$)}$ == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Difference of Squares of Secant and Tangent|next = Shape of Sine Function}}: $\\S 5$: Trigonometric Functions: $5.21$ * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Difference of Squares of Secant and Tangent|next = Sine Function is Odd}}: $2$: Functions, Limits and Continuity: The Elementary Functions: $4$ Category:Cosecant Function Category:Cotangent Function Category:Sum of Squares of Sine and Cosine 9itjzv9n2qejn4wh2opxqmb342lp1o1"}
+{"_id": "32414", "title": "Kernel of Linear Transformation is Orthocomplement of Range of Adjoint/Corollary", "text": "Kernel of Linear Transformation is Orthocomplement of Range of Adjoint/Corollary 0 28057 477161 414675 2020-07-06T05:42:36Z Prime.mover 59 wikitext text/x-wiki == Corollary to Kernel of Linear Transformation is Orthocomplement of Range of Adjoint == Let $H$ be a Hilbert space. <onlyinclude> Let $A \\in \\map B H$ be a normal operator. Then: :$\\ker A = \\paren {\\Rng A}^\\perp$ </onlyinclude> where: :$A^*$ denotes the adjoint of $A$ :$\\ker A$ is the kernel of $A$ :$\\Rng A$ is the range of $A$ :$\\perp$ signifies orthocomplementation == Proof == {{begin-eqn}} {{eqn | l = \\ker A       | r = \\paren {\\Rng {A^*} }^\\perp       | c = Kernel of Linear Transformation is Orthocomplement of Range of Adjoint }} {{eqn | ll= \\leadsto       | l = \\ker A^*       | r = \\paren {\\Rng {A^*} }^\\perp       | c = Kernel of Normal Operator is Kernel of Adjoint }} {{eqn | ll= \\leadsto       | l = \\ker A^{**}       | r = \\paren {\\Rng {A^{**} } }^\\perp       | c = substituting $A^*$ for $A$ }} {{eqn | ll= \\leadsto       | l = \\ker A       | r = \\paren {\\Rng A}^\\perp       | c = Double Adjoint is Itself }} {{end-eqn}} {{qed}} Category:Adjoints Category:Linear Transformations on Hilbert Spaces jcdx00qq1bfueds6rc7iciu6120mx4t"}
+{"_id": "32415", "title": "Bisection of Angle in Cartesian Plane/Corollary", "text": "Bisection of Angle in Cartesian Plane/Corollary 0 28065 429534 154560 2019-10-04T22:39:12Z Prime.mover 59 wikitext text/x-wiki == Corollary to Bisection of Angle in Cartesian Plane == Let $\\theta$ be the angular coordinate of a point $P$ in a polar coordinate plane. Let $QOR$ be a straight line that bisects the angle $\\theta$. <onlyinclude> If $\\theta$ is in quadrant I or quadrant II, then the angular coordinates of $Q$ and $R$ are in quadrant I and quadrant III. If $\\theta$ is in quadrant III or quadrant IV, then the angular coordinates of $Q$ and $R$ are in quadrant II and quadrant IV. </onlyinclude> == Proof == :500px From Bisection of Angle in Cartesian Plane, the angular coordinates of $Q$ and $R$ are $\\dfrac \\theta 2$ and $\\pi + \\dfrac \\theta 2$. {{WLOG}}, let $\\angle Q = \\dfrac \\theta 2$ and $\\angle R = \\pi + \\dfrac \\theta 2$. Let $\\theta$ be in quadrant I or quadrant II. Then $0 < \\theta < \\pi$. Dividing each term in the inequality by $2$: :$0 < \\dfrac \\theta 2 < \\dfrac \\pi 2$ and so $Q$ lies in quadrant I. Adding $\\pi$ to each expression in the inequality: :$\\pi < \\pi + \\dfrac \\theta 2 < \\dfrac {3 \\pi} 2$ and so $R$ lies in quadrant I. {{qed|lemma}} Let $\\theta$ be in quadrant III or quadrant IV. Then $\\pi < \\theta < 2 \\pi$. Dividing each term in the inequality by $2$: :$\\dfrac \\pi 2 < \\dfrac \\theta 2 < \\pi$ and so $Q$ lies in quadrant II. Adding $\\pi$ to each expression in the inequality: :$\\dfrac {3 \\pi} 2 < \\pi + \\dfrac \\theta 2 < 2 \\pi$ and so $R$ lies in quadrant IV. {{qed}} Category:Analytic Geometry 5baopujzi1710fuimqg1baff7pfmldl"}
+{"_id": "32416", "title": "Shape of Cosecant Function/Graph", "text": "Shape of Cosecant Function/Graph 0 28287 155043 2013-08-05T21:22:52Z Prime.mover 59 Created page with \"== Theorem ==  The graph of the cosecant function appears as:  <onlyinclude> :800px </onlyinclude...\" wikitext text/x-wiki == Theorem == The graph of the cosecant function appears as: <onlyinclude> :800px </onlyinclude> == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev=Graph of Secant Function|next=Sine Function is Odd}}: $\\S 5$: Trigonometric Functions: $5.27$ Category:Cosecant Function 7wxbvz76udx7omx41aox29v4y85hk3f"}
+{"_id": "32417", "title": "Tangent of Sum/Corollary", "text": "Tangent of Sum/Corollary 0 28295 451135 396176 2020-02-28T16:13:31Z Prime.mover 59 wikitext text/x-wiki == Corollary to Tangent of Sum == <onlyinclude> :$\\map \\tan {a - b} = \\dfrac {\\tan a - \\tan b} {1 + \\tan a \\tan b}$ </onlyinclude> where $\\tan$ is tangent. == Proof == {{begin-eqn}} {{eqn | l = \\map \\tan {a - b}       | r = \\frac {\\tan a + \\map \\tan {-b} } {1 - \\tan a \\, \\map \\tan {-b} }       | c = Tangent of Sum }} {{eqn | r = \\frac {\\tan a - \\tan b} {1 + \\tan a \\tan b}       | c = Tangent Function is Odd }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Tangent of Sum|next = Cotangent of Sum}}: $\\S 5$: Trigonometric Functions: $5.36$ * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Tangent of Sum|next = Definition:Hyperbolic Sine}}: $2$: Functions, Limits and Continuity: The Elementary Functions: $4$ * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Tangent of Sum|next = Prosthaphaeresis Formulas|entry = compound angle formulae|subentry = in trigonometry}} Category:Tangent Function Category:Compound Angle Formulas mefu39i4ew9frgauq5j67uwkbdxd4v2"}
+{"_id": "32418", "title": "Cotangent of Sum/Corollary", "text": "Cotangent of Sum/Corollary 0 28297 405404 405403 2019-05-21T06:22:40Z Prime.mover 59 wikitext text/x-wiki == Corollary to Cotangent of Sum == <onlyinclude> :$\\map \\cot {a - b} = \\dfrac {\\cot a \\cot b + 1} {\\cot b - \\cot a}$ </onlyinclude> where $\\cot $ is cotangent. == Proof == {{begin-eqn}} {{eqn | l = \\map \\cot {a - b}       | r = \\frac {\\cot a \\, \\map \\cot {-b} - 1} {\\cot a + \\map \\cot {-b} }       | c = Cotangent of Sum }} {{eqn | r = \\frac {-\\cot a \\cot b - 1} {\\cot a - \\cot b}       | c = Cotangent Function is Odd }} {{eqn | r = \\frac {\\cot a \\cot b + 1} {\\cot b - \\cot a}       | c = multiplying numerator and denominator by $-1$ and rearranging }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Cotangent of Sum|next = Trigonometric Functions of Negative Angle}}: $\\S 5$: Trigonometric Functions: $5.37$ Category:Cotangent Function 5k1x25al0ub4ob134b1ahvttpe3llqk"}
+{"_id": "32419", "title": "Double Angle Formulas/Cosine/Corollary 1", "text": "Double Angle Formulas/Cosine/Corollary 1 0 28372 428443 363776 2019-09-29T10:11:48Z Prime.mover 59 wikitext text/x-wiki == Corollary to Double Angle Formula for Cosine  == <onlyinclude> :$\\cos 2 \\theta = 2 \\cos^2 \\theta - 1$ </onlyinclude> where $\\cos$ denotes cosine. == Proof  == {{begin-eqn}} {{eqn | l = \\cos 2 \\theta       | r = \\cos^2 \\theta - \\sin^2 \\theta       | c = Double Angle Formula for Cosine }} {{eqn | r = \\cos^2 \\theta - \\paren {1 - \\cos^2 \\theta}       | c = Sum of Squares of Sine and Cosine }} {{eqn | r = 2 \\cos^2 \\theta - 1 }} {{end-eqn}} {{qed}} == Also see == * Square of Cosine: $\\cos^2 \\theta = \\dfrac {1 + \\cos 2 \\theta} 2$ == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Double Angle Formulas/Cosine/Corollary 2|next = Double Angle Formula for Tangent}}: $\\S 5$: Trigonometric Functions: $5.39$ Category:Cosine Function Category:Double Angle Formula for Cosine d84haju1i992an6yk6pc6k6hgfn61ig"}
+{"_id": "32420", "title": "Double Angle Formulas/Cosine/Corollary 2", "text": "Double Angle Formulas/Cosine/Corollary 2 0 28373 428428 314404 2019-09-29T09:44:53Z Prime.mover 59 wikitext text/x-wiki == Corollary to Double Angle Formula for Cosine == <onlyinclude> :$\\cos 2 \\theta = 1 - 2 \\sin^2 \\theta$ </onlyinclude> where $\\cos$ and $\\sin$ denote cosine and sine respectively. == Proof == {{begin-eqn}} {{eqn | l = \\cos 2 \\theta       | r = \\cos^2 \\theta - \\sin^2 \\theta       | c = Double Angle Formula for Cosine }} {{eqn | r = \\paren {1 - \\sin^2 \\theta} - \\sin^2 \\theta       | c = Sum of Squares of Sine and Cosine }} {{eqn | r = 1 - 2 \\sin^2 \\theta }} {{end-eqn}} {{qed}} == Also see == * Square of Sine: $\\sin^2 \\theta = \\dfrac {1 - \\cos 2 \\theta} 2$ == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Double Angle Formula for Cosine|next = Double Angle Formulas/Cosine/Corollary 1}}: $\\S 5$: Trigonometric Functions: $5.39$ Category:Cosine Function Category:Double Angle Formula for Cosine 8jjxrseaf5s53xdzmyowd2wikpxx2v1"}
+{"_id": "32421", "title": "Double Angle Formulas/Sine/Corollary", "text": "Double Angle Formulas/Sine/Corollary 0 28375 452789 419747 2020-03-05T21:49:19Z Prime.mover 59 wikitext text/x-wiki == Corollary to Double Angle Formula for Sine == <onlyinclude> :$\\sin 2 \\theta = \\dfrac {2 \\tan \\theta} {1 + \\tan^2 \\theta}$ </onlyinclude> where $\\sin$ and $\\tan$ denote sine and tangent respectively. == Proof == {{begin-eqn}} {{eqn | l = \\sin 2 \\theta       | r = 2 \\sin \\theta \\cos \\theta       | c = Double Angle Formula for Sine }} {{eqn | r = 2 \\sin \\theta \\cos \\theta \\frac {\\cos \\theta} {\\cos \\theta}       | c =  }} {{eqn | r = 2 \\tan \\theta \\cos^2 \\theta       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \\frac {2 \\tan \\theta} {\\sec^2 \\theta}       | c = Secant is Reciprocal of Cosine }} {{eqn | r = \\frac {2 \\tan \\theta} {1 + \\tan^2 \\theta}       | c = Difference of Squares of Secant and Tangent }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Half Angle Formula for Sine|next = Tangent Half-Angle Substitution for Cosine|entry = half-angle formula}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Tangent Half-Angle Substitution Formulas|next = Tangent Half-Angle Substitution for Cosine|entry = $t$-formulae}} Category:Sine Function Category:Tangent Function Category:Double Angle Formula for Sine c0yrzrs3aqrnzj45c2snqu25i1eukvq"}
+{"_id": "32422", "title": "Tangent Half-Angle Substitution for Cosine", "text": "Tangent Half-Angle Substitution for Cosine 0 28376 452788 419753 2020-03-05T21:48:24Z Prime.mover 59 wikitext text/x-wiki == Corollary to Double Angle Formula for Cosine == <onlyinclude> :$\\cos 2 \\theta = \\dfrac {1 - \\tan^2 \\theta} {1 + \\tan^2 \\theta}$ </onlyinclude> where $\\cos$ and $\\tan$ denote cosine and tangent respectively. == Proof == {{begin-eqn}} {{eqn | l = \\cos 2 \\theta       | r = \\cos^2 \\theta - \\sin^2 \\theta       | c = Double Angle Formula for Cosine }} {{eqn | r = \\paren {\\cos^2 \\theta - \\sin^2 \\theta} \\frac {\\cos^2 \\theta}{\\cos^2 \\theta}       | c =  }} {{eqn | r = \\paren {1 - \\tan^2 \\theta} \\cos^2 \\theta       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \\frac {1 - \\tan^2 \\theta} {\\sec^2 \\theta}       | c = Secant is Reciprocal of Cosine }} {{eqn | r = \\frac {1 - \\tan^2 \\theta} {1 + \\tan^2 \\theta}       | c = Difference of Squares of Secant and Tangent }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Tangent Half-Angle Substitution for Sine|next = Double Angle Formula for Tangent|entry = half-angle formula}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Tangent Half-Angle Substitution for Sine|next = Mathematician:Thales of Miletus|entry = $t$-formulae}} Category:Cosine Function Category:Tangent Function Category:Double Angle Formula for Cosine cmell4vk6px0vbhnzeyqdcz456yc505"}
+{"_id": "32423", "title": "Quintuple Angle Formulas", "text": "Quintuple Angle Formulas 0 28391 155470 2013-08-07T19:50:49Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> === Quintuple Angle Formula for Sine === {{:Quintuple Angle Formulas/Sine}}  === Quintuple Angle Formulas/Cosin...\" wikitext text/x-wiki == Theorem == <onlyinclude> === Quintuple Angle Formula for Sine === {{:Quintuple Angle Formulas/Sine}} === Quintuple Angle Formula for Cosine === {{:Quintuple Angle Formulas/Cosine}} === Quintuple Angle Formula for Tangent === {{:Quintuple Angle Formulas/Tangent}} </onlyinclude> where $\\sin, \\cos, \\tan$ denote sine, cosine and tangent respectively. Category:Trigonometric Identities nok0n01h6jkfbx2i6u2owqr41zd7l9b"}
+{"_id": "32424", "title": "Sum of Geometric Sequence/Proof 4/Lemma", "text": "Sum of Geometric Sequence/Proof 4/Lemma 0 28621 456250 456109 2020-03-19T10:04:09Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $n \\in \\N_{>0}$. Then: :$\\displaystyle \\paren {1 - x} \\sum_{i \\mathop = 0}^{n - 1} x^i = 1 - x^n$ </onlyinclude> == Proof == Proof by induction on $n$: For all $n \\in \\N_{> 0}$, let $\\map P n$ be the proposition: :$\\displaystyle \\paren {1 - x} \\sum_{i \\mathop = 0}^{n - 1} x^i = 1 - x^n$ === Basis for the Induction === $\\map P 1$ is the case: {{begin-eqn}} {{eqn | l = \\paren {1 - x} \\sum_{i \\mathop = 1}^{1 - 1} x^i       | r = 1 \\paren {1 - x} }} {{eqn | r = 1 - x }} {{eqn | r = 1 - x^1 }} {{end-eqn}} This is our basis for the induction. === Induction Hypothesis === Now we need to show that, if $\\map P k$ is true, where $k \\ge 2$, then it logically follows that $\\map P {k + 1}$ is true. So this is our induction hypothesis: :$\\displaystyle \\paren {1 - x} \\sum_{i \\mathop = 0}^{k - 1} x^i = 1 - x^k$ Then we need to show: :$\\displaystyle \\paren {1 - x} \\sum_{i \\mathop = 0}^k x^i = 1 - x^{k + 1}$ === Induction Step === This is our induction step: {{begin-eqn}} {{eqn | l = \\paren {1 - x} \\sum_{i \\mathop = 0}^k x^i       | r = \\paren {1 - x} \\paren {x^k + \\sum_{i \\mathop = 0}^{k - 1} x^i} }} {{eqn | r = x^k - x^{k + 1} + \\paren {1 - x} \\sum_{i \\mathop = 0}^{k - 1} x^i }} {{eqn | r = x^k - x^{k + 1} + 1 - x^k       | c = from the induction hypothesis }} {{eqn | r = 1 - x^{k + 1}       | c = gathering terms }} {{end-eqn}} So $\\map P k \\implies \\map P {k + 1}$ and the result follows by the Principle of Mathematical Induction. Therefore: :$\\displaystyle \\paren {1 - x} \\sum_{i \\mathop = 0}^{n - 1} x^i = 1 - x^n$ {{qed}} Category:Sum of Geometric Sequence r79zvqlj9xu8id3oy0keckf5iabeyqf"}
+{"_id": "32425", "title": "General Distributivity Theorem/Lemma 1", "text": "General Distributivity Theorem/Lemma 1 0 28638 353098 353078 2018-05-02T21:19:51Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $\\left({R, \\circ, *}\\right)$ be a ringoid. Then for every sequence $\\left \\langle {a_k} \\right \\rangle_{1 \\mathop \\le k \\mathop \\le n}$ of elements of $R$, and for every $b \\in R$: :$\\displaystyle \\left({\\sum_{j \\mathop = 1}^n a_j}\\right) * b = \\sum_{j \\mathop = 1}^n \\left({a_j * b}\\right)$ </onlyinclude> where: :$\\displaystyle \\sum_{j \\mathop = 1}^n a_j$ is the composite $a_1 \\circ a_2 \\circ \\cdots \\circ a_n$ :$n$ is a strictly positive integer: $n \\in \\Z_{> 0}$. == Proof == The proof proceeds by the Principle of Mathematical Induction. Recall that as $\\left({R, \\circ, *}\\right)$ is a ringoid, $*$ is distributive over $\\circ$: :$\\forall a, b, c \\in S: \\left({a \\circ b}\\right) * c = \\left({a * c}\\right) \\circ \\left({b * c}\\right)$ For all $n \\in \\Z_{> 0}$, let $P \\left({n}\\right)$ be the proposition: :$\\displaystyle \\left({\\sum_{j \\mathop = 1}^n a_j}\\right) * b = \\sum_{j \\mathop = 1}^n \\left({a_j * b}\\right)$ === Basis for the Induction === $P(1)$ is true, as this just says: :$a_1 * b = a_1 * b$ $P(2)$ is the case: {{begin-eqn}} {{eqn | l = \\left({\\sum_{j \\mathop = 1}^2 a_j}\\right) * b       | r = \\left({a_1 \\circ a_2}\\right) * b       | c = Definition of Composite }} {{eqn | r = \\left({a_1 * b}\\right) \\circ \\left({a_2 * b}\\right)       | c = $*$ is distributive over $\\circ$ as $\\left({R, \\circ, *}\\right)$ is a ringoid }} {{eqn | r = \\sum_{j \\mathop = 1}^2 \\left({a_j * b}\\right)       | c = Definition of Composite }} {{end-eqn}} This is our basis for the induction. === Induction Hypothesis === Now we need to show that, if $P \\left({k}\\right)$ is true, where $k \\ge 2$, then it logically follows that $P \\left({k+1}\\right)$ is true. So this is our induction hypothesis: :$\\displaystyle \\left({\\sum_{j \\mathop = 1}^k a_j}\\right) * b = \\sum_{j \\mathop = 1}^k \\left({a_j * b}\\right)$ Then we need to show: :$\\displaystyle \\left({\\sum_{j \\mathop = 1}^{k + 1} a_j}\\right) * b = \\sum_{j \\mathop = 1}^{k + 1} \\left({a_j * b}\\right)$ === Induction Step === This is our induction step: {{begin-eqn}} {{eqn | l = \\left({\\sum_{j \\mathop = 1}^{k + 1} a_j}\\right) * b       | r = \\left({\\left({\\sum_{j \\mathop = 1}^k a_j}\\right) \\circ a_{k+1} }\\right) * b       | c = Definition of Composite }} {{eqn | r = \\left({\\left({\\sum_{j \\mathop = 1}^k a_j}\\right) * b}\\right) \\circ \\left({a_{k+1} * b}\\right)       | c = Basis for the Induction }} {{eqn | r = \\left({\\sum_{j \\mathop = 1}^k \\left({a_j * b}\\right)}\\right) \\circ \\left({a_{k+1} * b}\\right)       | c = Induction Hypothesis }} {{eqn | r = \\sum_{j \\mathop = 1}^{k + 1} \\left({a_j * b}\\right)       | c = Associativity of $\\circ$ in $\\left({R, \\circ, *}\\right)$ }} {{end-eqn}} So $P \\left({k}\\right) \\implies P \\left({k+1}\\right)$ and the result follows by the Principle of Mathematical Induction. Therefore: :$\\displaystyle \\forall n \\in \\Z_{> 0}: \\left({\\sum_{j \\mathop = 1}^k a_j}\\right) * b = \\sum_{j \\mathop = 1}^k \\left({a_j * b}\\right)$ {{qed}} == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev = General Commutativity Theorem|next = General Distributivity Theorem/Lemma 2}}: $\\S 18$: Theorem $18.8$ Category:General Distributivity Theorem qr2k266ad5c6kwabp79q8cn5pv5s9j5"}
+{"_id": "32426", "title": "General Distributivity Theorem/Lemma 2", "text": "General Distributivity Theorem/Lemma 2 0 28639 411216 401085 2019-07-02T07:24:13Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $\\struct {R, \\circ, *}$ be a ringoid. Then for every sequence $\\sequence {a_k}_{1 \\mathop \\le k \\mathop \\le n}$ of elements of $R$, and for every $b \\in R$: :$\\displaystyle b * \\paren {\\sum_{j \\mathop = 1}^n a_j} = \\sum_{j \\mathop = 1}^n \\paren {b * a_j}$ </onlyinclude> where: :$\\displaystyle \\sum_{j \\mathop = 1}^n a_j$ is the summation $a_1 \\circ a_2 \\circ \\cdots \\circ a_n$ :$n$ is a strictly positive integer: $n \\in \\Z_{> 0}$. == Proof == The proof proceeds by the Principle of Mathematical Induction. Recall that as $\\struct {R, \\circ, *}$ is a ringoid, $*$ is distributive over $\\circ$: :$\\forall a, b, c \\in R: a * \\paren {b \\circ c} = \\paren {a * b} \\circ \\paren {a * c}$ For all $n \\in \\Z_{> 0}$, let $\\map P n$ be the proposition: :$\\displaystyle b * \\paren {\\sum_{j \\mathop = 1}^n a_j} = \\sum_{j \\mathop = 1}^n \\paren {b * a_j}$ We have that $\\struct {R, \\circ, *}$ is a ringoid, and so: :$\\forall a, b, c \\in R: a * \\paren {b \\circ c} = \\paren {a * b} \\circ \\paren {a * c}$ === Basis for the Induction === $\\map P 1$ is true, as this just says: :$b * a_1 = b * a_1$ $\\map P 2$ is the case: {{begin-eqn}} {{eqn | l = b * \\paren {\\sum_{j \\mathop = 1}^2 a_j}       | r = b * \\paren {a_1 \\circ a_2}       | c = {{Defof|Composite (Abstract Algebra)|Composite}} }} {{eqn | r = \\paren {b * a_1} \\circ \\paren {b * a_2}       | c = $*$ is distributive over $\\circ$ as $\\paren {R, \\circ, *}$ is a ringoid }} {{eqn | r = \\sum_{j \\mathop = 1}^2 \\paren {b * a_j}       | c = {{Defof|Composite (Abstract Algebra)|Composite}} }} {{end-eqn}} This is our basis for the induction. === Induction Hypothesis === Now we need to show that, if $\\map P k$ is true, where $k \\ge 2$, then it logically follows that $\\map P {k + 1}$ is true. So this is our induction hypothesis: :$\\displaystyle b * \\paren {\\sum_{j \\mathop = 1}^k a_j} = \\sum_{j \\mathop = 1}^k \\paren {b * a_j}$ Then we need to show: :$\\displaystyle b * \\paren {\\sum_{j \\mathop = 1}^{k + 1} a_j} = \\sum_{j \\mathop = 1}^{k + 1} \\paren {b * a_j}$ === Induction Step === This is our induction step: {{begin-eqn}} {{eqn | l = b * \\paren {\\sum_{j \\mathop = 1}^{k + 1} a_j}       | r = b * \\paren {\\paren {\\sum_{j \\mathop = 1}^k a_j} \\circ a_{k + 1} }       | c =  }} {{eqn | r = \\paren {b * \\paren {\\sum_{j \\mathop = 1}^k a_j} } \\circ \\paren {b * a_{k + 1} }       | c = Basis for the Induction }} {{eqn | r = \\paren {\\sum_{j \\mathop = 1}^k \\paren {a_j * b} } \\circ \\paren {b * a_{k + 1} }       | c = Induction Hypothesis }} {{eqn | r = \\sum_{j \\mathop = 1}^{k + 1} \\paren {b * a_j}       | c = Associativity of $\\circ$ in $\\paren {R, \\circ, *}$ }} {{end-eqn}} So $\\map P k \\implies \\map P {k + 1}$ and the result follows by the Principle of Mathematical Induction. Therefore: :$\\displaystyle \\forall n \\in \\Z_{> 0}: b * \\paren {\\sum_{j \\mathop = 1}^k a_j} = \\sum_{j \\mathop = 1}^k \\paren {b * a_j}$ {{qed}} == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev = General Distributivity Theorem/Lemma 1|next = Definition:Indexing Set}}: $\\S 18$: Theorem $18.8$ * {{BookReference|Algebra Volume 1|1982|P.M. Cohn|edpage = Second Edition|ed = 2nd|prev = Principle of Mathematical Induction/Zero-Based|next = General Distributivity Theorem}}: Chapter $2$: Integers and natural numbers: $\\S 2.1$: The integers: Exercise $2$ Category:General Distributivity Theorem h7knb9uy7erih2y17j9y5ug27s476gt"}
+{"_id": "32427", "title": "Integer Multiplication Distributes over Addition/Corollary", "text": "Integer Multiplication Distributes over Addition/Corollary 0 28652 473551 353134 2020-06-12T07:06:00Z Prime.mover 59 wikitext text/x-wiki == Corollary to Integer Multiplication Distributes over Addition == <onlyinclude> The operation of multiplication on the set of integers $\\Z$ is distributive over subtraction: : $\\forall x, y, z \\in \\Z: x \\times \\left({y - z}\\right) = \\left({x \\times y}\\right) - \\left({x \\times z}\\right)$ : $\\forall x, y, z \\in \\Z: \\left({y - z}\\right) \\times x = \\left({y \\times x}\\right) - \\left({z \\times x}\\right)$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = x \\times \\left({y - z}\\right)       | r = x \\times \\left({y + \\left({- z}\\right)}\\right)       | c = {{Defof|Integer Subtraction}} }} {{eqn | r = x \\times y + x \\times \\left({- z}\\right)       | c =  }} {{eqn | r = x \\times y + \\left({- \\left({x \\times z}\\right)}\\right)       | c = Product with Ring Negative }} {{eqn | r = x \\times y - x \\times z       | c = {{Defof|Integer Subtraction}} }} {{end-eqn}} {{qed|lemma}} {{begin-eqn}} {{eqn | l = \\left({y - z}\\right) \\times x       | r = x \\times \\left({y - z}\\right)       | c = Integer Multiplication is Commutative }} {{eqn | r = x \\times y - x \\times z       | c = from above }} {{eqn | r = y \\times z - z \\times x       | c = Integer Multiplication is Commutative }} {{end-eqn}} {{qed}} Category:Integer Multiplication Category:Subtraction Category:Distributive Operations 57cvjbx161m0pv7sdlm5h1a05u5icz9"}
+{"_id": "32428", "title": "Divisors of Product of Coprime Integers/Corollary", "text": "Divisors of Product of Coprime Integers/Corollary 0 28770 437926 265344 2019-12-06T16:57:40Z Prime.mover 59 wikitext text/x-wiki == Corollary to Divisors of Product of Coprime Integers == <onlyinclude> Let $p$ be a prime. Let $p \\divides b c$, where $b \\perp c$. Then $p \\divides b$ or $p \\divides c$, but not both. </onlyinclude> == Proof == From the main result, $p = r s$, where $r \\divides b$ and $s \\divides c$. But as $p$ is prime, either: :$r = 1$ and $s = p$, or: :$r = p$ and $s = 1$. So $p \\divides b$ or $p \\divides c$. But $p$ can not divide both, as $b \\perp c$. {{qed}} Category:Coprime Integers Category:Prime Numbers jl8ilgejcoz1uv9o814h5jm23hh6oew"}
+{"_id": "32429", "title": "Common Factor Cancelling in Congruence/Corollary 1", "text": "Common Factor Cancelling in Congruence/Corollary 1 0 28800 392620 392607 2019-02-17T23:23:31Z Prime.mover 59 wikitext text/x-wiki == Corollary to Common Factor Cancelling in Congruence == Let $a, b, x, y, m \\in \\Z$. Let: :$a x \\equiv b y \\pmod m$ and $a \\equiv b \\pmod m$ where $a \\equiv b \\pmod m$ denotes that $a$ is congruent modulo $m$ to $b$. <onlyinclude> If $a$ is coprime to $m$, then: :$x \\equiv y \\pmod m$ </onlyinclude> == Proof == Let $a \\perp m$. Then by definition of coprime: :$\\gcd \\set {a, m} = 1$ The result follows immediately from Common Factor Cancelling in Congruence. {{qed}} == Warning == {{:Common Factor Cancelling in Congruence/Corollary 1/Warning}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Modulo Multiplication is Well-Defined|next = Congruence by Product of Moduli}}: $\\S 1.2.4$: Integer Functions and Elementary Number Theory: Law $\\text{B}$ * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Law of Inverses (Modulo Arithmetic)/Corollary 1|next = Fundamental Theorem of Arithmetic}}: $\\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $20$ Category:Common Factor Cancelling in Congruence 5l5a3qhfgplllpmlq3j5e75o0h2xfm6"}
+{"_id": "32430", "title": "Sequence of Powers of Reciprocals is Null Sequence/Corollary", "text": "Sequence of Powers of Reciprocals is Null Sequence/Corollary 0 28878 408133 408132 2019-06-15T12:41:08Z Prime.mover 59 wikitext text/x-wiki == Corollary to Sequence of Powers of Reciprocals is Null Sequence == <onlyinclude> Let $\\sequence {x_n}$ be the sequence in $\\R$ defined as: :$x_n = \\dfrac 1 n$ Then $\\sequence {x_n}$ is a null sequence. </onlyinclude> == Proof == $n = n^1$ from the definition of power. As $1 \\in \\Q_{>0}$ the result follows from Sequence of Powers of Reciprocals is Null Sequence. {{qed}} Category:Sequence of Powers of Reciprocals is Null Sequence 7yp45kvd2ebzco481calh7q6exmwqpe"}
+{"_id": "32431", "title": "Integer Less One divides Power Less One/Corollary", "text": "Integer Less One divides Power Less One/Corollary 0 29499 454188 164538 2020-03-13T09:02:32Z Prime.mover 59 wikitext text/x-wiki == Corollary to Integer Less One divides Power Less One == <onlyinclude> Let $m, n, q \\in \\Z_{>0}$. Let: :$m \\divides n$ where $\\divides$ denotes divisibility. Then: :$\\paren {q^m - 1} \\divides \\paren {q^n - 1}$ </onlyinclude> === Converse to Corollary === {{:Integer Less One divides Power Less One/Corollary/Converse}} == Proof == By hypothesis: :$m \\divides n$ By definition of divisibility: :$\\exists k \\in \\Z: k m = n$ Thus: :$q^n = q^{k m} = \\paren {q^m}^k$ {{explain|Find the exponent combination laws to justify the above}} Then by Integer Less One divides Power Less One: :$\\paren {q^m - 1} \\divides \\paren {\\paren {q^m}^k - 1}$ Hence the result. {{qed}} == Sources == * {{BookReference|Algebra Volume 1|1982|P.M. Cohn|edpage = Second Edition|ed = 2nd|prev = Integer Less One divides Power Less One|next = Integer Less One divides Power Less One/Corollary/Converse}}: $\\S 2.4$: The rational numbers and some finite fields: Further Exercises $5$ Category:Number Theory b69gx4718d7m09c0l3o10ufnr355dm6"}
+{"_id": "32432", "title": "Axiom:Axiom of Choice/Formulation 1", "text": "Axiom:Axiom of Choice/Formulation 1 100 30420 491830 491825 2020-10-01T07:04:09Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> For every set of non-empty sets, it is possible to provide a mechanism for choosing one element of each element of the set. :$\\displaystyle \\forall s: \\paren {\\O \\notin s \\implies \\exists \\paren {f: s \\to \\bigcup s}: \\forall t \\in s: \\map f t \\in t}$  That is, one can always create a choice function for selecting one element from each element of the set. </onlyinclude> == Also see == * Equivalence of Versions of Axiom of Choice {{LinkToCategory|Axiom of Choice|the Axiom of Choice}} == Sources == * {{BookReference|Naive Set Theory|1960|Paul R. Halmos|prev = Axiom:Axiom of Choice/Formulation 2|next = Equivalence of Versions of Axiom of Choice/Formulation 2 implies Formulation 1}}: $\\S 15$: The Axiom of Choice * {{BookReference|Point Set Topology|1964|Steven A. Gaal|prev = Definition:Choice Function|next = Axiom:Axiom of Choice/Historical Note}}: Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents * {{BookReference|The Axiom of Choice|1973|Thomas J. Jech|next = Definition:Choice Function}}: $1.1$ The Axiom of Choice Choice 34dmh7pd26iip1t65a6xykrb6fqsnr2"}
+{"_id": "32433", "title": "Axiom:Axiom of Choice/Formulation 2", "text": "Axiom:Axiom of Choice/Formulation 2 100 30421 444060 444022 2020-01-18T02:02:50Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> Let $\\family {X_i}_{i \\mathop \\in I}$ be an indexed family of sets all of which are non-empty, indexed by $I$ which is also non-empty. Then there exists an indexed family $\\family {x_i}_{i \\mathop \\in I}$ such that: :$\\forall i \\in I: x_i \\in X_i$ That is, the Cartesian product of a non-empty family of sets which are non-empty is itself non-empty. </onlyinclude> == Also see == * Equivalence of Versions of Axiom of Choice {{LinkToCategory|Axiom of Choice|the Axiom of Choice}} == Sources == * {{BookReference|Naive Set Theory|1960|Paul R. Halmos|prev = Definition:Lexicographic Order|next = Axiom:Axiom of Choice/Formulation 1}}: $\\S 15$: The Axiom of Choice * {{BookReference|Point Set Topology|1964|Steven A. Gaal|prev = Well-Ordering Theorem|next = Equivalence of Versions of Axiom of Choice}}: Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents Choice s59iex6vzqt7v2z3rdl05apu6t2kaw0"}
+{"_id": "32434", "title": "Axiom:Axiom of Choice/Formulation 3", "text": "Axiom:Axiom of Choice/Formulation 3 100 30422 450952 446516 2020-02-27T23:48:41Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> Let $\\SS$ be a set of non-empty pairwise disjoint sets. Then there is a set $C$ such that for all $S \\in \\SS$, $C \\cap S$ has exactly one element. Symbolically: :$\\forall s: \\paren {\\paren {\\O \\notin s \\land \\forall t, u \\in s: t = u \\lor t \\cap u = \\O} \\implies \\exists c: \\forall t \\in s: \\exists x: t \\cap c = \\set x}$ </onlyinclude> == Also see == * Equivalence of Versions of Axiom of Choice {{LinkToCategory|Axiom of Choice|the Axiom of Choice}} == Sources == * {{citation|date = 1908|title = Neuer Beweis für die Möglichkeit einer Wohlordnung|trans = A new proof of the possibility of well-ordering|journal = Mathematische Annalen|abbr = Math. Ann.|volume = 65|startpage = 107|endpage = 128|author = Ernst Zermelo}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Axiomatic System|next = Definition:Axis|entry = axiom of choice}} Choice aa2hjgpjv581bfh9pq3d6z8ccjnxs4g"}
+{"_id": "32435", "title": "Image of Empty Set is Empty Set/Corollary", "text": "Image of Empty Set is Empty Set/Corollary 0 30445 487397 374945 2020-09-13T12:44:23Z Prime.mover 59 wikitext text/x-wiki == Corollary of Image of Empty Set is Empty Set == <onlyinclude> Let $f: S \\to T$ be a mapping. The image of the empty set is the empty set: :$f \\sqbrk \\O = \\O$ </onlyinclude> == Proof == By definition, a mapping is a relation. Thus Image of Empty Set is Empty Set applies. {{qed}} == Sources == * {{BookReference|Undergraduate Topology|1971|Robert H. Kasriel|prev = Definition:Image of Subset under Mapping/Definition 1|next = Image of Domain of Mapping is Image Set}}: $\\S 1.10$: Functions: Remark $10.8 \\ \\text{(a)}$ Category:Mapping Theory Category:Empty Set 7xavt25fvkht9z61ptq1ge391e36cn5"}
+{"_id": "32436", "title": "Axiom:Propositions of Incidence/Line in Plane", "text": "Axiom:Propositions of Incidence/Line in Plane 100 30593 486759 181304 2020-09-10T07:22:08Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> A line defined by any two distinct points in a plane lies entirely within that plane. </onlyinclude> == Sources == * {{BookReference|Projective Geometry|1952|T. Ewan Faulkner|ed = 2nd|edpage = Second Edition|prev = Axiom:Propositions of Incidence/Plane|next = Plane contains Infinite Number of Lines}}: Chapter $1$: Introduction: The Propositions of Incidence: $1.2$: The projective method: The propositions of incidence Category:Axioms/Projective Geometry 0vbntmbzl40iteejpqipw7xojoqesbg"}
+{"_id": "32437", "title": "Axiom:Propositions of Incidence/Plane and Line", "text": "Axiom:Propositions of Incidence/Plane and Line 100 30599 486768 181300 2020-09-10T07:33:20Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> A plane and a (straight) line which does not lie in that plane have exactly one point in common. </onlyinclude> That is, a point is completely determined by any two of the infinite number of distinct (straight) lines which intersect at that point. == Sources == * {{BookReference|Projective Geometry|1952|T. Ewan Faulkner|ed = 2nd|edpage = Second Edition|prev = Definition:Coplanar Points|next = Two Planes have Line in Common}}: Chapter $1$: Introduction: The Propositions of Incidence: $1.2$: The projective method: The propositions of incidence Category:Axioms/Projective Geometry h68vlrk0phnfre2d5wb5p5w1pn0frga"}
+{"_id": "32438", "title": "Primitive of Reciprocal/Corollary 2", "text": "Primitive of Reciprocal/Corollary 2 0 30760 407575 387660 2019-06-12T19:52:43Z Prime.mover 59 wikitext text/x-wiki {{mergeto|Derivative of Natural Logarithm Function}} {{refactor|What would probably be best would be for this to be a corollary of the above page.}} == Corollary to Primitive of Reciprocal == <onlyinclude> :$\\displaystyle \\frac {\\d} {\\d x} \\ln \\size x = \\frac 1 x$ for $x \\ne 0$. </onlyinclude> == Proof == Follows directly from Primitive of Reciprocal. {{qed}} == Sources == * {{BookReference|Calculus|2005|Roland E. Larson|author2 = Robert P. Hostetler|author3 = Bruce H. Edwards|ed = 8th|edpage = Eighth Edition}}: $\\S 5.1$ Category:Primitive of Reciprocal 6jdjjj4dmtd0kkhoz4e9550qkatoy1z"}
+{"_id": "32439", "title": "Comparison Test/Corollary", "text": "Comparison Test/Corollary 0 30788 261392 172357 2016-07-02T10:40:21Z Prime.mover 59 wikitext text/x-wiki == Corollary to Comparison Test == <onlyinclude> Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n$ be a convergent series of positive real numbers. Let $\\left \\langle {a_n} \\right \\rangle$ be a sequence in sequence $\\R$ or sequence in $\\C$. Let $H \\in \\R$. Let $\\exists M: \\forall n > M: \\left\\vert {a_n}\\right\\vert \\le H b_n$. Then the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ converges. </onlyinclude> == Proof == Let $\\epsilon > 0$. Then $\\dfrac \\epsilon H > 0$. As $\\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n$ converges, its tail tends to zero. So: : $\\displaystyle \\exists N: \\forall n > N: \\sum_{k \\mathop = n+1}^\\infty b_k < \\frac \\epsilon H$ Let $\\left \\langle s_n \\right \\rangle$ be the sequence of partial sums of $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$. Then $\\forall n > m > \\max \\left\\{{M, N}\\right\\}$: {{begin-eqn}} {{eqn | l = \\left\\vert{s_n - s_m}\\right\\vert       | r = \\left\\vert{\\left({a_1 + a_2 + \\cdots + a_n}\\right) - \\left({a_1 + a_2 + \\cdots + a_m}\\right)}\\right\\vert       | c =  }} {{eqn | r = \\left\\vert{a_{m+1} + a_{m+2} + \\cdots + a_n}\\right\\vert       | c =  }} {{eqn | o = \\le       | r = \\left\\vert { a_{m+1} } \\right\\vert + \\left\\vert { a_{m+2} } \\right\\vert + \\cdots + \\left\\vert { a_n } \\right\\vert       | c = Triangle Inequality }} {{eqn | o = \\le       | r = H b_{m+1} + H b_{m+2} + \\cdots + H b_n       | c =  }} {{eqn | o = \\le       | r = H \\sum_{k \\mathop = n+1}^\\infty b_k }} {{eqn | o = <       | r = H \\frac \\epsilon H }} {{eqn | r = \\epsilon       | c =  }} {{end-eqn}} So $\\left \\langle s_n \\right \\rangle$ is a Cauchy sequence and the result follows from: : Real Number Line is Complete Metric Space or: : Complex Plane is Complete Metric Space. {{qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Comparison Test|next = Ratio Test}}: $\\S 6.15$ Category:Series Category:Convergence Tests Category:Named Theorems l0v7nb5242yzp18zkm48ik0e8dsm1q6"}
+{"_id": "32440", "title": "Sum of Sequence of Products of Consecutive Reciprocals/Corollary", "text": "Sum of Sequence of Products of Consecutive Reciprocals/Corollary 0 30789 492870 406446 2020-10-06T09:45:56Z Prime.mover 59 wikitext text/x-wiki == Corollary to Sum of Sequence of Products of Consecutive Reciprocals == <onlyinclude> :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\sum_{j \\mathop = 1}^n \\frac 1 {j \\paren {j + 1} } = 1$ </onlyinclude> == Proof == From Sum of Sequence of Products of Consecutive Reciprocals: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\sum_{j \\mathop = 1}^n \\frac 1 {j \\paren {j + 1} } = \\frac n {n + 1} = 1 - \\frac 1 {n + 1}$ We have that: :$\\dfrac 1 {n + 1} < \\dfrac 1 n$ and that $\\sequence {\\dfrac 1 n}$ is a basic null sequence. Thus by the Squeeze Theorem: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\sum_{j \\mathop = 1}^n \\frac 1 {j \\paren {j + 1} } = 1$ {{qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Sum of Sequence of Products of Consecutive Reciprocals/Proof 2|next = Harmonic Series is Divergent/Proof 1}}: $\\S 6.3$ * {{BookReference|Special Functions of Mathematics for Engineers|1992|Larry C. Andrews|ed = 2nd|edpage = Second Edition|prev = Definition:Divergent Series|next = Divergent Sequence may be Bounded}}: $\\S 1.2$: Infinite Series of Constants: Example $1$ Category:Limits of Series Category:Sum of Sequence of Products of Consecutive Reciprocals pjh79wjetqqj9wjlalfrvf0z5lnkxm0"}
+{"_id": "32441", "title": "Newton's Method", "text": "Newton's Method 0 30799 421737 421732 2019-08-29T07:05:45Z Prime.mover 59 wikitext text/x-wiki {{MissingLinks}} == Proof Technique == '''Newton's method''' is a method of solving an equation expressed as a real function for which there may be no convenient closed form solution. The derivative of the function has to be known in order to use '''Newton's method'''. Let the equation to be solved be of the form: :$y = \\map f x$ Let the value of $x$ be required for a given $y$. Then an iterative improvement on an initial guess is of the form :$x_2 = x_1 - \\dfrac {\\map f {x_1} - y} {\\map {f'} {x_1} }$ where $\\map {f'} {x_1}$ is the derivative of $f$ {{WRT|Differentiation}} $x$ evaluated at $x_1$. == Proof == The function $\\map f x$ can be expanded using Taylor's Theorem: :$\\map f {x_2} = \\map f {x_1} + \\map {f'} {x_1} \\paren {x_2 - x_1} + \\dfrac 1 2 \\map {f''} {x_1} \\paren {x_2 - x_1}^2 + \\dotsb$ As $x_2$ gets closer to $x_1$, this series can be truncated to: :$\\map f {x_2} = \\map f {x_1} + \\map {f'} {x_1} \\paren {x_2 - x_1}$ {{explain|Not sure but that assumptions about convergence need to be made?}} Let $x_\\infty$ be the exact solution where: :$\\map f {x_\\infty} = y$ Let $\\epsilon \\in \\R_{>0}$ be the difference from the new estimate to the solution: :$x_\\infty = x_2 + \\epsilon$ Then the function expanded around the new estimate is: :$y = \\map f {x_2} + \\map {f'} {x_2} \\epsilon$ Solving for $x_2$ produces: :$x_2 = x_1 - \\dfrac {\\map f {x_1} - y} {\\map {f'} {x_1} } - \\epsilon \\dfrac {\\map {f'} {x_2} } {\\map {f'} {x_1} }$ For $\\epsilon$ small enough the final term can be neglected: :$x_2 = x_1 - \\dfrac {\\map f {x_1} - y} {\\map {f'} {x_1} }$ {{explain|Greater linguistic and mathematical precision may be needed in the above statement.}} {{qed}} == Example == {{:Newton's Method/Example}} {{Namedfor|Isaac Newton|cat = Newton}} == Sources == {{SourceReview|Expressed in the below in a slightly different form. Need to check whether it is equivalent to the one given here.}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Newton's Third Law of Motion|next = Mathematician:Jerzy Neyman|entry = Newton's method}} Category:Calculus Category:Numerical Analysis Category:Newton's Method 6zkz90kzd5dpfo2n27n7pnf026sstc8"}
+{"_id": "32442", "title": "Negated Upper Index of Binomial Coefficient/Corollary 1", "text": "Negated Upper Index of Binomial Coefficient/Corollary 1 0 30863 492913 265088 2020-10-06T10:05:26Z Prime.mover 59 wikitext text/x-wiki == Corollary to Negated Upper Index of Binomial Coefficient == <onlyinclude> Let $r \\in \\R, k \\in \\Z$. Then: :$\\dbinom {-r} k = \\paren {-1}^k \\dbinom {r + k - 1} k$ where $\\dbinom {-r} k$ is a binomial coefficient. </onlyinclude> == Proof 1 == {{:Negated Upper Index of Binomial Coefficient/Corollary 1/Proof 1}} == Proof 2 == {{:Negated Upper Index of Binomial Coefficient/Corollary 1/Proof 2}} Category:Binomial Coefficients Category:Negated Upper Index of Binomial Coefficient rn42o7kdzo196kyc77gdmuvaiakf4vk"}
+{"_id": "32443", "title": "Set of Integer Combinations includes Zero", "text": "Set of Integer Combinations includes Zero 0 31285 490828 175187 2020-09-25T21:35:34Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $a, b \\in \\Z$ be integers. Let $S = \\set {a x + b y: x, y \\in \\Z}$ be the set of integer combinations of $a$ and $b$. Then $0 \\in S$. == Proof == By setting $x = 0$ and $y = 0$: :$a \\cdot 0 + b \\cdot 0 = 0$ {{qed}} == Sources == * {{BookReference|Computability and Unsolvability|1982|Martin Davis|ed = 2nd|edpage = Second Edition|prev = Definition:Coprime Integers|next = Set of Integer Combinations includes those Integers}}: Appendix $1$: Some Results from the Elementary Theory of Numbers: Lemma $1$ Category:Integer Combinations jg8bt0yar901ccdz6zqpjclfj6rdoyw"}
+{"_id": "32444", "title": "Euler's Formula/Real Domain/Corollary", "text": "Euler's Formula/Real Domain/Corollary 0 31506 365724 365148 2018-09-14T22:06:31Z Prime.mover 59 wikitext text/x-wiki == Corollary to Euler's Formula: Real Domain == Let $\\theta \\in \\R$ be a real number. Then: <onlyinclude> :$e^{-i \\theta} = \\cos \\theta - i \\sin \\theta$ </onlyinclude> where: : $e^{-i \\theta}$ denotes the complex exponential function : $\\cos \\theta$ denotes the real cosine function : $\\sin \\theta$ denotes the real sine function : $i$ denotes the imaginary unit. == Proof == {{begin-eqn}} {{eqn | l = e^{-i \\theta}       | r = \\cos \\paren {-\\theta} + i \\sin \\paren {-\\theta}       | c = Euler's Formula }} {{eqn | r = \\cos \\theta + i \\sin \\paren {-\\theta}       | c = Cosine Function is Even }} {{eqn | r = \\cos \\theta - i \\sin \\theta       | c = Sine Function is Odd }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Definition:Exponential Form of Complex Number|next = Cosine Exponential Formulation/Real Domain/Proof 3}}: $\\S 2$. Geometrical Representations * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Euler's Formula/Real Domain|next = Sine Exponential Formulation/Real Domain}}: $\\S 7$: Relationship between Exponential and Trigonometric Functions: $7.16$ Category:Euler's Formula oo9fbxooo0c061wwaf5h54uu2rztlkg"}
+{"_id": "32445", "title": "Hyperbolic Sine of Sum/Corollary", "text": "Hyperbolic Sine of Sum/Corollary 0 31573 493512 396199 2020-10-10T09:03:49Z Prime.mover 59 wikitext text/x-wiki == Corollary of Hyperbolic Sine of Sum == <onlyinclude> :$\\map \\sinh {a - b} = \\sinh a \\cosh b - \\cosh a \\sinh b$ </onlyinclude> where $\\sinh$ denotes the hyperbolic sine and $\\cosh$ denotes the hyperbolic cosine. == Proof == {{begin-eqn}} {{eqn | l = \\map \\sinh {a - b}       | r = \\sinh a \\map \\cosh {-b} + \\cosh a \\map \\sinh {-b}       | c = Hyperbolic Sine of Sum }} {{eqn | r = \\sinh a \\cosh b - \\cosh a \\sinh b       | c = Hyperbolic Cosine Function is Even and Hyperbolic Sine Function is Odd }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Hyperbolic Sine of Sum|next = Hyperbolic Cosine of Sum}}: $\\S 8$: Hyperbolic Functions: $8.20$: Addition Formulas * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Hyperbolic Sine of Sum|next = Hyperbolic Cosine of Sum}}: $2$: Functions, Limits and Continuity: The Elementary Functions: $5$ Category:Hyperbolic Sine Function e5cgmmolsrp6zpkc8l6envukvzfavbg"}
+{"_id": "32446", "title": "Hyperbolic Cosine of Sum/Corollary", "text": "Hyperbolic Cosine of Sum/Corollary 0 31576 495816 396201 2020-10-21T21:29:31Z Prime.mover 59 wikitext text/x-wiki == Corollary of Hyperbolic Cosine of Sum == <onlyinclude> :$\\map \\cosh {a - b} = \\cosh a \\cosh b - \\sinh a \\sinh b$ </onlyinclude> where $\\sinh$ denotes the hyperbolic sine and $\\cosh$ denotes the hyperbolic cosine. == Proof == {{begin-eqn}} {{eqn | l = \\map \\cosh {a - b}       | r = \\cosh a \\map \\cosh {-b} + \\sinh a \\map \\sinh {-b}       | c = Hyperbolic Sine of Sum }} {{eqn | r = \\cosh a \\cosh b - \\sinh a \\sinh b       | c = Hyperbolic Cosine Function is Even and Hyperbolic Sine Function is Odd }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Hyperbolic Cosine of Sum|next = Hyperbolic Tangent of Sum}}: $\\S 8$: Hyperbolic Functions: $8.21$: Addition Formulas * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Hyperbolic Cosine of Sum|next = Hyperbolic Tangent of Sum}}: $2$: Functions, Limits and Continuity: The Elementary Functions: $5$ Category:Hyperbolic Cosine Function miqq02aaj5mfsudrc7w3i8yhxecz1bh"}
+{"_id": "32447", "title": "Double Angle Formulas/Hyperbolic Cosine/Corollary 1", "text": "Double Angle Formulas/Hyperbolic Cosine/Corollary 1 0 31585 191810 176650 2014-08-19T05:52:09Z Prime.mover 59 wikitext text/x-wiki == Corollary to Double Angle Formula for Hyperbolic Cosine == <onlyinclude> : $\\cosh 2 x = 2 \\cosh^2 x - 1$ </onlyinclude> where $\\cosh$ denotes hyperbolic cosine. == Proof == {{begin-eqn}} {{eqn|l = \\cosh 2 x      |r = \\cosh^2 x + \\sinh^2 x      |c = Double Angle Formula for Hyperbolic Cosine }} {{eqn|r = \\cosh^2 x + \\left({\\cosh^2 x - 1}\\right)      |c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn|r = 2 \\ \\cosh^2 x - 1 }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev=Double Angle Formula for Hyperbolic Cosine|next=Double Angle Formulas/Hyperbolic Cosine/Corollary 2}}: $\\S 8$: Hyperbolic Functions: $8.25$: Double Angle Formulas Category:Hyperbolic Cosine Function rtooyulhdl9gowv7kklh74yqkpzyv4l"}
+{"_id": "32448", "title": "Double Angle Formulas/Hyperbolic Cosine/Corollary 2", "text": "Double Angle Formulas/Hyperbolic Cosine/Corollary 2 0 31586 191463 176652 2014-08-13T20:16:07Z Prime.mover 59 wikitext text/x-wiki == Corollary to Double Angle Formula for Hyperbolic Cosine == <onlyinclude> : $\\cosh 2 x = 1 + 2 \\sinh^2 x$ </onlyinclude> where $\\cosh$ and $\\sinh$ denote hyperbolic cosine and hyperbolic sine respectively. == Proof == {{begin-eqn}} {{eqn|l = \\cosh 2 x      |r = \\cosh^2 x + \\sinh^2 x      |c = Double Angle Formula for Hyperbolic Cosine }} {{eqn|r = \\left({1 + \\sinh^2 x}\\right) + \\sinh^2 \\theta      |c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn|r = 1 + 2 \\sinh^2 x }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev=Double Angle Formulas/Hyperbolic Cosine/Corollary 1|next=Double Angle Formula for Hyperbolic Tangent}}: $\\S 8$: Hyperbolic Functions: $8.25$: Double Angle Formulas Category:Hyperbolic Cosine Function mcdukzs104yk3brgaticzbby3hyrfy6"}
+{"_id": "32449", "title": "Complex Multiplication as Geometrical Transformation/Corollary", "text": "Complex Multiplication as Geometrical Transformation/Corollary 0 32293 394698 362679 2019-03-04T20:27:08Z Prime.mover 59 wikitext text/x-wiki == Corollary to Complex Multiplication as Geometrical Transformation == <onlyinclude> Let $z = \\polar {r, \\theta}$ be a complex number expressed in polar form. Let $z$ be represented on the complex plane $\\C$ in vector form. The effect of multiplying $z$ by $e^{i \\alpha}$ is to rotate it about the origin of $\\C$ by $\\alpha$ in the positive direction </onlyinclude> == Proof == From Complex Multiplication as Geometrical Transformation, the effect of multiplying a complex number by $r e^{i \\alpha}$ is: :to rotate it about the origin of $\\C$ by $\\alpha$ in the positive direction :to multiply its modulus by $r$. In this instance $r = 1$. Hence the result. {{qed}} == Sources == * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Power Reduction Formulas/Cosine to 4th/Proof 2|next = Periodicity of Complex Exponential Function}}: $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $24$ Category:Complex Multiplication as Geometrical Transformation cjjo033pavj4kfjh0i8k5yqyeu4im4e"}
+{"_id": "32450", "title": "Semantic Tableau Algorithm", "text": "Semantic Tableau Algorithm 0 32295 181022 180972 2014-03-26T11:54:06Z Lord Farin 560 wikitext text/x-wiki == Algorithm == <onlyinclude> Let $\\mathbf A$ be a WFF of propositional logic. The purpose of this algorithm is to create a completed semantic tableau $T$ for $\\mathbf A$. :'''Step 1''': Start with a labeled tree $T$ comprising only a root node labeled with the singleton $\\left\\{{\\mathbf A}\\right\\}$. :'''Step 2''': Choose a leaf node $t$ of $T$ that has not yet been marked. Let $U \\left({t}\\right)$ be the set that labels $t$. :'''Step 3''': If $U \\left({t}\\right)$ is a set of literals, mark $t$ as follows: ::'''a)''': If $U \\left({t}\\right)$ contains a complementary pair, mark it closed, $\\times$, and go to '''Step 2'''. ::'''b)''': Otherwise, mark it open, $\\odot$, and go to '''Step 2'''. :'''Step 4''': Choose a formula $\\mathbf B \\in U \\left({t}\\right)$ that is not a literal. Classify $\\mathbf B$ as an $\\alpha$-formula or a $\\beta$-formula, and determine $\\mathbf B_1, \\mathbf B_2$ accordingly. ::'''a)''': If $\\mathbf B$ is an $\\alpha$-formula, add a child $t'$ to $t$, labeled $U \\left({t'}\\right) = \\left({U \\left({t}\\right) \\setminus \\left\\{{\\mathbf B}\\right\\}}\\right) \\cup \\left\\{{\\mathbf B_1, \\mathbf B_2}\\right\\}$. ::'''b)''': If $\\mathbf B$ is a $\\beta$-formula, add two children $t', t''$ to $t$ with labels: ::::$U \\left({t'}\\right) = \\left({U \\left({t}\\right) \\setminus \\left\\{{\\mathbf B}\\right\\}}\\right) \\cup \\left\\{{\\mathbf B_1}\\right\\}$ ::::$U \\left({t''}\\right) = \\left({U \\left({t}\\right) \\setminus \\left\\{{\\mathbf B}\\right\\}}\\right) \\cup \\left\\{{\\mathbf B_2}\\right\\}$ :'''Step 5''': Go to '''Step 2''' if there remain leaves that are not marked. Otherwise, '''stop'''. </onlyinclude> === Finiteness === The finiteness of this algorithm is demonstrated on Semantic Tableau Algorithm Terminates. {{expand}} === Heuristics === {{:Semantic Tableau Algorithm/Heuristics}} == Sources == * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition|prev=Definition:Beta-Formula|next=Definition:Marked Leaf}}: $\\S 2.6.2$: Algorithm $2.64$ Category:Propositional Logic bkeg0ynhcyu8zstorn9rkmzkyf9ulgn"}
+{"_id": "32451", "title": "Quadruple Angle Formulas/Sine/Corollary 1", "text": "Quadruple Angle Formulas/Sine/Corollary 1 0 32365 395682 395680 2019-03-15T21:52:47Z Prime.mover 59 wikitext text/x-wiki == Corollary to Quadruple Angle Formula for Sine == <onlyinclude> For all $\\theta$ such that $\\theta \\ne 0, \\pm \\pi, \\pm 2 \\pi \\ldots$ : $\\dfrac {\\sin 4 \\theta} {\\sin \\theta} = 8 \\cos^3 \\theta - 4 \\cos \\theta$ where $\\sin$ denotes sine and $\\cos$ denotes cosine. </onlyinclude> == Proof == First note that when $\\theta = 0, \\pm \\pi, \\pm 2 \\pi \\ldots$: :$\\sin \\theta = 0$ so $\\dfrac {\\sin 4 \\theta} {\\sin \\theta}$ is undefined. Therefore for the rest of the proof it is assumed that $\\theta \\ne 0, \\pm \\pi, \\pm 2 \\pi \\ldots$ {{begin-eqn}} {{eqn | l = \\sin 4 \\theta       | r = 4 \\sin \\theta \\cos \\theta - 8 \\sin^3 \\theta \\cos \\theta       | c = Quadruple Angle Formula for Sine }} {{eqn | ll= \\leadsto       | l = \\dfrac {\\sin 4 \\theta} {\\sin \\theta}       | r = 4 \\cos \\theta - 8 \\paren {1 - \\cos^2 \\theta} \\cos \\theta       | c = Sum of Squares of Sine and Cosine }} {{eqn | r = 8 \\cos^3 \\theta - 4 \\cos \\theta       | c = multiplying out and gathering terms }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Cosine of 72 Degrees|next = Quadruple Angle Formulas/Sine/Corollary 2}}: $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $93 \\ \\text{(a)}$ ::(although see Quadruple Angle Formulas/Sine/Mistake for analysis of an error in that work) Category:Quadruple Angle Formula for Sine nvp3xlkcgwbsgfy6s6kjwe4gbhsohh1"}
+{"_id": "32452", "title": "Soundness and Completeness of Semantic Tableaus/Corollary 2", "text": "Soundness and Completeness of Semantic Tableaus/Corollary 2 0 32422 424258 181030 2019-09-09T13:48:45Z Prime.mover 59 wikitext text/x-wiki == Corollary to Soundness and Completeness of Semantic Tableaus == Let $\\mathbf A$ be a WFF of propositional logic. Then <onlyinclude>$\\mathbf A$ is a tautology {{iff}} $\\neg \\mathbf A$ has a closed tableau. </onlyinclude> == Proof == By Tautology iff Negation is Unsatisfiable, $\\mathbf A$ is a tautology {{iff}} $\\neg \\mathbf A$ is unsatisfiable. By the Soundness and Completeness of Semantic Tableaus, this amounts to the existence of a closed tableau for $\\neg \\mathbf A$. {{qed}} == Sources == * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed = 3rd|edpage = Third Edition|prev = Soundness and Completeness of Semantic Tableaus/Corollary 1|next = Semantic Tableau Algorithm is Decision Procedure for Tautologies}}: $\\S 2.7$: Corollary $2.69$ Category:Propositional Logic jd2m34fes6e32zzz1jlk10v28ul1oz3"}
+{"_id": "32453", "title": "Exponential of Sum/Real Numbers/Corollary", "text": "Exponential of Sum/Real Numbers/Corollary 0 32532 396500 396464 2019-03-21T07:17:34Z Prime.mover 59 wikitext text/x-wiki == Corollary to Exponential of Sum: Real Numbers == <onlyinclude> Let $x, y \\in \\R$ be real numbers. Let $\\exp x$ be the exponential of $x$. Then: :$\\map \\exp {x - y} = \\dfrac {\\exp x} {\\exp y}$ </onlyinclude> == Proof == By Exponential of Sum: Real Numbers: :$\\map \\exp {x - y} = \\exp x \\, \\map \\exp {-y}$ {{improve|No real counterpart exists to link to}} By Reciprocal of Complex Exponential: :$\\dfrac 1 {\\exp y} = \\map \\exp {-y}$ Combining these two, we obtain the result: :$\\map \\exp {x - y} = \\dfrac {\\exp x} {\\exp y}$ {{qed}} Category:Exponential of Sum mxtd2q3sd9p8g2l9a9hrt0wwaz5otr9"}
+{"_id": "32454", "title": "True Statement is implied by Every Statement/Formulation 2/Proof 2", "text": "True Statement is implied by Every Statement/Formulation 2/Proof 2 0 32742 182453 2014-04-07T11:54:46Z Lord Farin 560 Created page with \"== Theorem ==  : $\\vdash q \\implies \\left({p \\implies q}\\right)$   == Proof == <onlyinclude> We apply the Method of Truth Tables.  As can be seen by inspection, the Defi...\" wikitext text/x-wiki == Theorem == : $\\vdash q \\implies \\left({p \\implies q}\\right)$ == Proof == <onlyinclude> We apply the Method of Truth Tables. As can be seen by inspection, the truth value under the main connective, the first instance of $\\implies$, is $T$ for each boolean interpretation. $\\begin{array}{|ccccc|} \\hline q & \\implies & ( p & \\implies & q ) \\\\ \\hline F & T & T & F & F \\\\ F & T & F & T & F \\\\ T & T & T & T & T \\\\ T & T & F & T & T \\\\ \\hline \\end{array}$ {{qed}} </onlyinclude> Category:True Statement is implied by Every Statement Category:Truth Table Proofs 3dihouk51t4v9u4gxoo2ky98iaarh25"}
+{"_id": "32455", "title": "Axiom:Peano's Axioms", "text": "Axiom:Peano's Axioms 100 32855 488567 488564 2020-09-17T07:44:22Z Prime.mover 59 wikitext text/x-wiki == Axioms == '''Peano's Axioms''' are a set of properties which can be used to serve as a basis for logical deduction of the properties of the natural numbers. <onlyinclude> '''Peano's Axioms''' are intended to reflect the intuition behind $\\N$, the mapping $s: \\N \\to \\N: \\map s n = n + 1$ and $0$ as an element of $\\N$. Let there be given a set $P$, a mapping $s: P \\to P$, and a distinguished element $0$. Historically, the existence of $s$ and the existence of $0$ were considered the first two of '''Peano's Axioms''': {{begin-axiom}} {{axiom | n = \\text P 1         | q =          | m = 0 \\in P         | t = $0$ is an element of $P$ }} {{axiom | n = \\text P 2         | q = \\forall n \\in P         | m = \\map s n \\in P         | t = For all $n \\in P$, its successor $\\map s n$ is also in $P$ }} {{end-axiom}} The other three are as follows: </onlyinclude> === Formulation 1 === <onlyinclude>{{:Axiom:Peano's Axioms/Formulation 1}}</onlyinclude> === Formulation 2 === {{:Axiom:Peano's Axioms/Formulation 2}} == Terminology == === Successor Mapping === {{:Definition:Successor Mapping}} === Non-Successor Element === {{:Definition:Non-Successor Element}} === Peano Structure === Such a set $P$, together with the successor mapping $s$ and non-successor element $0$ as defined above, is known as a '''Peano structure'''. == Also presented as == Some sources present the axioms in a different order. For example, {{BookReference|Number Theory|1964|J. Hunter}} places the induction axiom $(\\text P 5)$ as axiom $(3)$ and moves $(\\text P 3)$ and $(\\text P 4)$ down to be $(4)$ and $(5)$ respectively. == Also defined as == {{:Axiom:Peano's Axioms/Also defined as}} {{expand|subpage with Peano's original exposition/axioms}} == Also known as == {{:Axiom:Peano's Axioms/Also known as}} == Also see == * Equivalence of Formulations of Peano's Axioms * Minimal Infinite Successor Set forms Peano Structure * Principle of Mathematical Induction for Peano Structure {{LinkToCategory|Peano's Axioms|Peano's axioms}} {{NamedforAxiom|Giuseppe Peano|author2 = Richard Dedekind|cat = Peano|cat2 = Dedekind}} == Historical Note == {{:Axiom:Peano's Axioms/Historical Note}} == Sources == * {{BookReference|Numbers, Sets and Axioms|1982|Alan G. Hamilton|prev = Successor Mapping on Natural Numbers has no Fixed Element|next = Natural Number Multiplication is Commutative}}: $\\S 1$: Numbers: $1.1$ Natural Numbers and Integers: Examples $1.1 \\ \\text {(e)}$ Category:Axioms/Number Theory Category:Axioms/Abstract Algebra Category:Axioms/Peano's Axioms h7wez3z3cfirxfptwzhjfbnjv7y8428"}
+{"_id": "32456", "title": "Stirling's Formula/Proof 2/Lemma 1", "text": "Stirling's Formula/Proof 2/Lemma 1 0 32930 349891 183945 2018-04-06T17:49:48Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $f \\left({x}\\right)$ be the real function defined on the open interval $\\left({-1 \\,.\\,.\\, 1}\\right)$ as: :$f \\left({x}\\right) := \\dfrac 1 {2 x} \\ln \\left({\\dfrac {1 + x} {1 - x} }\\right) - 1$ Then: :$\\displaystyle f \\left({x}\\right) = \\sum_{k \\mathop = 1}^\\infty \\dfrac {x^{2 n} } {2n + 1}$ == Proof == {{begin-eqn}} {{eqn | l = f \\left({x}\\right)       | o = :=       | r = \\frac 1 {2 x} \\ln \\left({\\frac {1 + x} {1 - x} }\\right) - 1       | c = for $\\left\\vert{x}\\right\\vert < 1$ }} {{eqn | r = \\frac 1 {2 x} \\left({\\ln \\left({1 + x}\\right) - \\ln \\left({1 - x}\\right)}\\right) - 1       | c = Difference of Logarithms }} {{eqn | r = \\frac 1 {2 x} \\left({\\sum_{k \\mathop = 1}^\\infty \\left({\\left({-1}\\right)^{k-1} \\frac {x^k} k}\\right) - \\sum_{k \\mathop = 1}^\\infty \\left({-\\frac {x^k} k}\\right)}\\right) - 1       | c = Power Series Expansion for $\\ln \\left({1 + x}\\right)$ }} {{eqn | r = \\sum_{k \\mathop = 1}^\\infty \\frac {x^{2n} } {2n + 1}       | c = simplifying }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev=Stirling's Formula/Proof 2|next=Stirling's Formula/Proof 2/Lemma 2}}: $\\S 17.2$ Category:Stirling's Formula n3vjrngd7oz373utxaven1ltvu78x5c"}
+{"_id": "32457", "title": "Stirling's Formula/Proof 2/Lemma 3", "text": "Stirling's Formula/Proof 2/Lemma 3 0 32936 456533 357722 2020-03-19T22:52:07Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $\\sequence {d_n}$ be the sequence defined as: :$d_n = \\map \\ln {n!} - \\paren {n + \\dfrac 1 2} \\ln n + n$ Then the sequence: :$\\sequence {d_n - \\dfrac 1 {12 n} }$ is increasing. == Proof == We have: {{begin-eqn}} {{eqn | l = d_n - d_{n + 1}       | r = \\map \\ln {n!} - \\paren {n + \\frac 1 2} \\ln n + n       | c =  }} {{eqn | o =        | ro= -       | r = \\paren {\\map \\ln {\\paren {n + 1}!} - \\paren {n + 1 + \\frac 1 2} \\map \\ln {n + 1} + n + 1}       | c =  }} {{eqn | r = -\\map \\ln {n + 1} - \\paren {n + \\frac 1 2} \\ln n + \\paren {n + \\frac 3 2} \\map \\ln {n + 1} - 1       | c = (as $\\map \\ln {\\paren {n + 1}!} = \\map \\ln {n + 1} + \\map \\ln {n!}$) }} {{eqn | r = \\paren {n + \\frac 1 2} \\map \\ln {\\frac {n + 1} n} - 1       | c = }} {{eqn | n = 1       | r = \\frac {2n + 1} 2 \\map \\ln {\\frac {1 + \\paren {2 n + 1}^{-1} } {1 - \\paren {2 n + 1}^{-1} } } - 1       | c = }} {{end-eqn}} Let: :$\\map f x := \\dfrac 1 {2 x} \\map \\ln {\\dfrac {1 + x} {1 - x} } - 1$ for $\\size x < 1$. Then: {{begin-eqn}} {{eqn | l = \\map f x       | r = \\sum_{k \\mathop = 1}^\\infty \\frac {x^{2 n} } {2 n + 1}       | c = Lemma 1 }} {{eqn | o = <       | r = \\frac {x^2} 3 \\sum_{k \\mathop = 0}^\\infty x^{2 n}       | c =  }} {{eqn | n = 1       | o = <       | r = \\frac {x^2} {3 \\paren {1 - x^2} }        | c = Sum of Infinite Geometric Sequence }} {{end-eqn}} As $-1 < \\dfrac 1 {2 n + 1} < 1$ it can be substituted for $x$ in $(1)$: {{begin-eqn}} {{eqn | l = d_n - d_{n - 1}       | o = \\le       | r = \\frac 1 {3 \\paren {\\paren {2 n + 1}^2 - 1} }       | c = simplifying }} {{eqn | r = \\frac 1 {12 n} - \\frac 1 {12 \\paren {n + 1} }       | c =  }} {{eqn | ll= \\leadsto       | l = d_n - \\frac 1 {12 n}        | o = \\le       | r = d_{n-1} - \\frac 1 {12 \\paren {n + 1} } }} {{end-eqn}} Thus the sequence: :$\\sequence {d_n - \\dfrac 1 {12 n} }$ is increasing. {{qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Stirling's Formula/Proof 2/Lemma 2|next = Definition:Integral Form of Gamma Function}}: $\\S 17.2$ Category:Stirling's Formula fwgijw7rkjuf1ymytkgtu8tr1ss7f48"}
+{"_id": "32458", "title": "Definite Integral of Even Function/Corollary", "text": "Definite Integral of Even Function/Corollary 0 33081 457202 457198 2020-03-23T11:40:02Z Prime.mover 59 wikitext text/x-wiki == Corollary to Definite Integral of Even Function == <onlyinclude> Let $f$ be an even function with a primitive on the open interval $\\openint {-a} a$, where $a > 0$. Then the improper integral of $f$ on $\\openint {-a} a$ is: :$\\displaystyle \\int_{\\mathop \\to -a}^{\\mathop \\to a} \\map f x \\rd x = 2 \\int_0^{\\mathop \\to a} \\map f x \\rd x$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\int_{\\mathop \\to -a}^{\\mathop \\to a} \\map f x \\rd x       | r = \\lim_{y \\to a} \\int_{-y}^y \\map f x \\rd x       | c = {{Defof|Improper Integral over Open Interval}} }} {{eqn | r = \\lim_{y \\mathop \\to a} 2 \\int_0^y \\map f x \\rd x       | c = Definite Integral of Even Function }} {{eqn | r = 2 \\lim_{y \\mathop \\to a} \\int_0^y \\map f x \\rd x       | c = Multiple Rule for Limits of Functions }} {{eqn | r = 2 \\int_0^{\\mathop \\to a} \\map f x \\rd x       | c = {{Defof|Improper Integral over Open Interval}} }} {{end-eqn}} {{qed}} Category:Integral Calculus Category:Even Functions Category:Improper Integrals k0k6hi21scfqdada75v9gd7i54hsbqj"}
+{"_id": "32459", "title": "Negative of Absolute Value/Corollary 2", "text": "Negative of Absolute Value/Corollary 2 0 33091 459712 459711 2020-04-06T12:21:05Z Prime.mover 59 wikitext text/x-wiki == Corollary to Negative of Absolute Value == Let $x, y \\in \\R$ be a real numbers. Let $\\size x$ be the absolute value of $x$. Then: <onlyinclude> :$\\size x \\le y \\iff -y \\le x \\le y$ that is: :$\\size x \\le y \\iff \\begin {cases} x & \\le y \\\\ -x & \\le y \\end {cases}$ </onlyinclude> == Proof == === Necessary Condition === Let $\\size x \\le y$. If $\\size x < y$ then from Corollary 1: :$-y < x < y$ Thus: :$-y \\le x \\le y$ Otherwise, if $\\size x = y$ then either $x = y$ or $-x = y$. Hence the result. {{qed|lemma}} === Sufficient Condition === Let $-y \\le x \\le y$. If $-y < x < y$ then from Corollary 1: :$\\size x < y$ Hence: :$\\size x \\le y$ Otherwise, if either $-y = x$ or $x = y$ then: :$\\size x = y$ Hence the result. {{qed}} Category:Negative of Absolute Value ak12cdl05u79k5whelfxgro09qcjliy"}
+{"_id": "32460", "title": "Derivative of Identity Function/Corollary", "text": "Derivative of Identity Function/Corollary 0 33458 466596 274640 2020-05-08T07:57:01Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Identity Function == <onlyinclude> :$\\map {\\dfrac {\\d} {\\d x} } {c x} = c$ where $c$ is a constant. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\map {\\frac \\d {\\d x} } {c x}       | r = c \\frac \\d {\\d x} x       | c = Derivative of Constant Multiple }} {{eqn | r = c \\times 1       | c = Derivative of Identity Function }} {{eqn | r = c       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Derivative of Constant|next = Power Rule for Derivatives/Corollary}}: $\\S 13$: General Rules of Differentiation: $13.3$ Category:Derivatives Category:Identity Mappings el09hmuomqzx0kmw0u8p3o825e3wlx9"}
+{"_id": "32461", "title": "Derivative of Constant Multiple/Real/Corollary", "text": "Derivative of Constant Multiple/Real/Corollary 0 33461 485979 351652 2020-09-06T22:20:46Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Constant Multiple == Let $f$ be a real function which is differentiable on $\\R$. Let $c \\in \\R$ be a constant. Then: <onlyinclude> :$\\map {\\dfrac {\\d^n} {\\d x^n} } {c \\map f x} = c \\map {\\dfrac {\\d^n} {\\d x^n} } {\\map f x}$ </onlyinclude> == Proof == {{refactor|Expand this into the formal {{ProofWiki}} structure for induction proofs.}} By induction: the base case is for $n = 1$ and is proved in Derivative of Constant Multiple. Now consider $\\map {\\dfrac {\\d^{k + 1} } {\\d x^{k + 1} } } {c \\map f x}$, assuming the induction hypothesis $\\map {\\dfrac {\\d^k} {\\d x^k} } {c \\map f x} = c \\map {\\dfrac {\\d^k} {\\d x^k} } {\\map f x}$: {{begin-eqn}} {{eqn | l = \\map {\\dfrac {\\d^{k + 1} } {\\d x^{k + 1} } } {c \\map f x}       | r = \\map {\\dfrac \\d {\\d x} } {\\map {\\dfrac {\\d^k} {\\d x^k} } {c \\map f x} }       | c = {{Defof|Higher Derivative}} }} {{eqn | r = \\map {\\dfrac \\d {\\d x} } {c \\map {\\dfrac {\\d^k} {\\d x^k} } {\\map f x} }       | c = Induction Hypothesis }} {{eqn | r = c \\map {\\dfrac \\d {\\d x} } {\\map {\\dfrac {\\d^k} {\\d x^k} } {\\map f x} }       | c = Basis for the Induction }} {{eqn | r = c \\map {\\dfrac {\\d^{k + 1} } {\\d x^{k + 1} } } {\\map f x}       | c = {{Defof|Higher Derivative}} }} {{end-eqn}} Hence the result by induction. {{qed}} Category:Differential Calculus 6hfezikkcl9kwfjgeu7oorglrv9re5v"}
+{"_id": "32462", "title": "Axiom:Axiomatization of 1-Based Natural Numbers", "text": "Axiom:Axiomatization of 1-Based Natural Numbers 100 33519 493106 400696 2020-10-07T21:59:07Z Prime.mover 59 wikitext text/x-wiki {{rename|Hardly elegant}} == Axioms == The following axioms are intended to capture the behaviour of the ($1$-based) natural numbers $\\N_{>0}$, the element $1 \\in \\N_{>0}$, and the operations of addition $+$ and multiplication $\\times$ as they pertain to $\\N_{>0}$: <onlyinclude> {{begin-axiom}} {{axiom | n = \\text A         | q = \\exists_1 1 \\in \\N_{> 0}         | m = a \\times 1 = a = 1 \\times a }} {{axiom | n = \\text B         | q = \\forall a, b \\in \\N_{> 0}         | m = a \\times \\paren {b + 1} = \\paren {a \\times b} + a }} {{axiom | n = \\text C         | q = \\forall a, b \\in \\N_{> 0}         | m = a + \\paren {b + 1} = \\paren {a + b} + 1 }} {{axiom | n = \\text D         | q = \\forall a \\in \\N_{> 0}, a \\ne 1         | m = \\exists_1 b \\in \\N_{> 0}: a = b + 1 }} {{axiom | n = \\text E         | q = \\forall a, b \\in \\N_{> 0}         | m = \\)Exactly one of these three holds:\\( }} {{axiom | m = a = b \\lor \\paren {\\exists x \\in \\N_{> 0}: a + x = b} \\lor \\paren {\\exists y \\in \\N_{> 0}: a = b + y} }} {{axiom | n = \\text F         | q = \\forall A \\subseteq \\N_{> 0}         | m = \\paren {1 \\in A \\land \\paren {z \\in A \\implies z + 1 \\in A} } \\implies A = \\N_{> 0} }} {{end-axiom}} </onlyinclude> == Note == The above axiom schema specifies the old-fashioned definition of the natural numbers as: :$\\text{The set of natural numbers} = \\set {1, 2, 3, \\ldots}$ as opposed to the more modern approach which defines them as: :$\\text{The set of natural numbers} = \\set {0, 1, 2, 3, \\ldots}$ In order to eliminate confusion, on {{ProofWiki}} the set $\\set {1, 2, 3, \\ldots}$ will be denoted as $\\N_{> 0}$ or $\\N_{\\ne 0}$ or $\\N_{\\ge 1}$. When $\\N$ is used, $\\N = \\set {0, 1, 2, 3, \\ldots}$ is to be understood. == Sources == * {{BookReference|Abstract Algebra|1964|W.E. Deskins|prev = Definition:Natural Numbers|next = Principle of Mathematical Induction/Predicate/One-Based/Proof 2}}: $\\S 2.1$: Definition $2.1$ Category:Axioms/Number Theory Category:Axioms/Abstract Algebra 52o5lgnwu3yvkvn2zs2xjk7ww3xqi3g"}
+{"_id": "32463", "title": "Primitive of Square of Cosine Function/Corollary", "text": "Primitive of Square of Cosine Function/Corollary 0 33567 450907 350620 2020-02-27T16:25:10Z Prime.mover 59 wikitext text/x-wiki == Corollary to Primitive of Square of Cosine Function == <onlyinclude> :$\\displaystyle \\int \\cos^2 x \\rd x = \\frac {x + \\sin x \\cos x} 2 + C$ where $C$ is an arbitrary constant. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\int \\sin^2 x \\rd x       | r = \\frac x 2 + \\frac {\\sin 2 x} 4 + C       | c = Primitive of Square of Cosine Function }} {{eqn | r = \\frac x 2 + \\frac {2 \\sin x \\cos x} 4 + C       | c = Double Angle Formula for Sine }} {{eqn | r = \\frac {x + \\sin x \\cos x} 2 + C       | c =  }} {{end-eqn}} {{Qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Primitive of Square of Cosine Function|next = Primitive of Product of Secant and Tangent}}: $\\S 14$: General Rules of Integration: $14.22$ Category:Primitive of Square of Cosine Function 98map9s8trcnvivvlb9kw3pk4xr5ylq"}
+{"_id": "32464", "title": "Primitive of Square of Hyperbolic Sine Function/Corollary", "text": "Primitive of Square of Hyperbolic Sine Function/Corollary 0 33613 450896 386012 2020-02-27T16:06:34Z Prime.mover 59 wikitext text/x-wiki == Corollary to Primitive of Square of Hyperbolic Sine Function == <onlyinclude> :$\\displaystyle \\int \\sinh^2 x \\rd x = \\frac {\\sinh x \\cosh x - x} 2 + C$ where $C$ is an arbitrary constant. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\int \\sinh^2 x \\rd x       | r = \\frac {\\sinh 2 x} 4 - \\frac x 2 + C       | c = Primitive of Square of Hyperbolic Sine Function }} {{eqn | r = \\frac {2 \\sinh x \\cosh x} 4 - \\frac x 2 + C       | c = Double Angle Formula for Hyperbolic Sine }} {{eqn | r = \\frac {\\sinh x \\cosh x - x} 2 + C       | c = rearranging }} {{end-eqn}} {{Qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Primitive of Square of Hyperbolic Sine Function|next = Primitive of Square of Hyperbolic Cosine Function}}: $\\S 14$: General Rules of Integration: $14.35$ Category:Primitives of Hyperbolic Functions Category:Hyperbolic Sine Function 3yt56sthwj4k60tsifyp5769ossyy77"}
+{"_id": "32465", "title": "Primitive of Square of Hyperbolic Cosine Function/Corollary", "text": "Primitive of Square of Hyperbolic Cosine Function/Corollary 0 33615 450895 187106 2020-02-27T16:06:15Z Prime.mover 59 wikitext text/x-wiki == Corollary to Primitive of Square of Hyperbolic Cosine Function == <onlyinclude> :$\\displaystyle \\int \\cosh^2 x \\rd x = \\frac {\\sinh x \\cosh x + x} 2 + C$ where $C$ is an arbitrary constant. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\int \\cosh^2 x \\rd x       | r = \\frac {\\sinh 2 x} 4 + \\frac x 2 + C       | c = Primitive of Square of Hyperbolic Cosine Function }} {{eqn | r = \\frac {2 \\sinh x \\cosh x} 4 + \\frac x 2 + C       | c = Double Angle Formula for Hyperbolic Sine }} {{eqn | r = \\frac {\\sinh x \\cosh x + x} 2 + C       | c = rearranging }} {{end-eqn}} {{Qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Primitive of Square of Hyperbolic Cosine Function|next = Primitive of Product of Hyperbolic Secant and Tangent}}: $\\S 14$: General Rules of Integration: $14.36$ Category:Primitives of Hyperbolic Functions Category:Hyperbolic Cosine Function rl2pzj3o9d8bfq71uvr5eota4bng132"}
+{"_id": "32466", "title": "Derivative of Arcsine Function/Corollary", "text": "Derivative of Arcsine Function/Corollary 0 33685 490935 490933 2020-09-26T10:56:08Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Arcsine Function == Let $a \\in \\R$ be a constant Let $x \\in \\R$ be a real number such that $x^2 < a^2$. Let $\\map \\arcsin {\\dfrac x a}$ be the arcsine of $\\dfrac x a$. Then: <onlyinclude> :$\\dfrac {\\map \\d {\\map \\arcsin {\\frac x a} } } {\\d x} = \\dfrac 1 {\\sqrt {a^2 - x^2} }$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\frac {\\map \\d {\\map \\arcsin {\\frac x a} } } {\\d x}       | r = \\frac 1 a \\frac 1 {\\sqrt {1 - \\paren {\\frac x a}^2} }       | c = Derivative of Arcsine Function and Derivative of Function of Constant Multiple\t }} {{eqn | r = \\frac 1 a \\frac 1 {\\sqrt {\\frac {a^2 - x^2} {a^2} } }       | c =  }} {{eqn | r = \\frac 1 a \\frac a {\\sqrt {a^2 - x^2} }       | c =  }} {{eqn | r = \\frac 1 {\\sqrt {a^2 - x^2} }       | c =  }} {{end-eqn}} {{qed}} == Also see == * Derivative of $\\arccos \\dfrac x a$ * Derivative of $\\arctan \\dfrac x a$ * Derivative of $\\arccot \\dfrac x a$ * Derivative of $\\arcsec \\dfrac x a$ * Derivative of $\\arccsc \\dfrac x a$ == Sources == * {{BookReference|Integration|1944|R.P. Gillespie|ed = 2nd|edpage = Second Edition|prev = Primitive of Square of Hyperbolic Cosecant Function|next = Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form}}: Chapter $\\text {II}$: Integration of Elementary Functions: $\\S 7$. Standard Integrals: $12$. Category:Derivatives of Inverse Trigonometric Functions Category:Arcsine Function bxzuxbn8fq0adzva1h8iktngpzrewj6"}
+{"_id": "32467", "title": "Derivative of Arccosine Function/Corollary", "text": "Derivative of Arccosine Function/Corollary 0 33688 490936 415995 2020-09-26T10:56:54Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Arccosine Function == Let $a \\in \\R$ be a constant Let $x \\in \\R$ be a real number such that $x^2 < a^2$. Let $\\map \\arccos {\\dfrac x a}$ be the arccosine of $\\dfrac x a$. Then: <onlyinclude> :$\\dfrac {\\map \\d {\\map \\arccos {\\frac x a} } } {\\d x} = \\dfrac {-1} {\\sqrt {a^2 - x^2} }$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\frac {\\map \\d {\\map \\arccos {\\frac x a} } } {\\d x}       | r = \\frac 1 a \\frac {-1} {\\sqrt {1 - \\paren {\\frac x a}^2} }       | c = Derivative of Arccosine Function and Derivative of Function of Constant Multiple\t }} {{eqn | r = \\frac 1 a \\frac {-1} {\\sqrt {\\frac {a^2 - x^2} {a^2} } }       | c =  }} {{eqn | r = \\frac 1 a \\frac {-a} {\\sqrt {a^2 - x^2} }       | c =  }} {{eqn | r = \\frac {-1} {\\sqrt {a^2 - x^2} }       | c =  }} {{end-eqn}} {{qed}} == Also see == * Derivative of $\\arcsin \\dfrac x a$ * Derivative of $\\arctan \\dfrac x a$ * Derivative of $\\arccot \\dfrac x a$ * Derivative of $\\arcsec \\dfrac x a$ * Derivative of $\\arccsc \\dfrac x a$ == Sources == * {{BookReference|Integration|1944|R.P. Gillespie|ed = 2nd|edpage = Second Edition|prev = Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form|next = Primitive of Reciprocal of Root of a squared minus x squared/Arccosine Form}}: Chapter $\\text {II}$: Integration of Elementary Functions: $\\S 7$. Standard Integrals: $12$. Category:Derivatives of Inverse Trigonometric Functions Category:Arccosine Function ihf059sfr876p92yzwofnydsnwomu26"}
+{"_id": "32468", "title": "Derivative of Arctangent Function/Corollary", "text": "Derivative of Arctangent Function/Corollary 0 33691 490938 345418 2020-09-26T11:03:31Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Arctangent Function == Let $x \\in \\R$. Let $\\map \\arctan {\\dfrac x a}$ denote the arctangent of $\\dfrac x a$. Then: <onlyinclude> :$\\dfrac {\\map \\d {\\map \\arctan {\\frac x a} } } {\\d x} = \\dfrac a {a^2 + x^2}$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\frac {\\map \\d {\\map \\arctan {\\frac x a} } } {\\d x}       | r = \\frac 1 a \\frac 1 {1 + \\paren {\\frac x a}^2}       | c = Derivative of Arctangent Function and Derivative of Function of Constant Multiple\t }} {{eqn | r = \\frac 1 a \\frac 1 {\\frac {a^2 + x^2} {a^2} }       | c =  }} {{eqn | r = \\frac 1 a \\frac {a^2} {a^2 + x^2}       | c =  }} {{eqn | r = \\frac a {a^2 + x^2}       | c =  }} {{end-eqn}} {{qed}} == Also defined as == This result can also be reported as: :$\\dfrac {\\map \\d {\\map \\arctan {\\frac x a} } } {\\d x} = \\dfrac a {x^2 + a^2}$ == Also see == * Derivative of $\\arcsin \\dfrac x a$ * Derivative of $\\arccos \\dfrac x a$ * Derivative of $\\arccot \\dfrac x a$ * Derivative of $\\arcsec \\dfrac x a$ * Derivative of $\\arccsc \\dfrac x a$ == Sources == * {{BookReference|Integration|1944|R.P. Gillespie|ed = 2nd|edpage = Second Edition|prev = Primitive of Reciprocal of Root of a squared minus x squared/Arccosine Form|next = Primitive of Reciprocal of x squared plus a squared/Arctangent Form}}: Chapter $\\text {II}$: Integration of Elementary Functions: $\\S 7$. Standard Integrals: $13$. Category:Derivative of Arctangent Function 9m2dw8z5a9aarqiw97351fdcd94r5o0"}
+{"_id": "32469", "title": "Derivative of Arccotangent Function/Corollary", "text": "Derivative of Arccotangent Function/Corollary 0 33693 345414 190993 2018-03-02T17:59:22Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Arccotangent Function == Let $x \\in \\R$. Let $\\operatorname{arccot} \\left({\\dfrac x a}\\right)$ be the arccotangent of $\\dfrac x a$. Then: <onlyinclude> :$\\dfrac {\\mathrm d \\left({\\operatorname{arccot} \\left({\\frac x a}\\right) }\\right)} {\\mathrm d x} = \\dfrac {-a} {a^2 + x^2}$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\frac {\\mathrm d \\left({\\operatorname{arccot} \\left({\\frac x a}\\right)}\\right)} {\\mathrm d x}       | r = \\frac 1 a \\frac {-1} {1 + \\left({\\frac x a}\\right)^2}       | c = Derivative of Arccotangent Function and Derivative of Function of Constant Multiple\t }} {{eqn | r = \\frac 1 a \\frac {-1} {\\frac {a^2 + x^2} {a^2} }       | c =  }} {{eqn | r = \\frac 1 a \\frac {-a^2} {a^2 + x^2}       | c =  }} {{eqn | r = \\frac {-a} {a^2 + x^2}       | c =  }} {{end-eqn}} {{qed}} == Also defined as == This result can also be reported as: :$\\dfrac {\\mathrm d \\left({\\operatorname{arccot} \\left({\\frac x a}\\right) }\\right)} {\\mathrm d x} = \\dfrac {-a} {x^2 + a^2}$ == Also see == * Derivative of $\\arcsin \\dfrac x a$ * Derivative of $\\arccos \\dfrac x a$ * Derivative of $\\arctan \\dfrac x a$ * Derivative of $\\operatorname{arcsec} \\dfrac x a$ * Derivative of $\\operatorname{arccsc} \\dfrac x a$ Category:Derivative of Arccotangent Function 158l17r4ursuysne6d8w58ey3dgfzqc"}
+{"_id": "32470", "title": "Derivative of Arcsecant Function/Corollary 1", "text": "Derivative of Arcsecant Function/Corollary 1 0 33697 485936 345409 2020-09-06T20:52:05Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Arcsecant Function == Let $x \\in \\R$. Let $\\map \\arcsec {\\dfrac x a}$ be the arcsecant of $\\dfrac x a$. Then: <onlyinclude> :$\\dfrac {\\map \\d {\\map \\arcsec {\\frac x a} } } {\\d x} = \\dfrac a {\\size x \\sqrt {x^2 - a^2} } = \\begin {cases} \\dfrac a {x \\sqrt {x^2 - a^2} } & : 0 < \\arcsec \\dfrac x a < \\dfrac \\pi 2 \\ (\\text {that is: $x > a$})  \\\\ \\dfrac {-a} {x \\sqrt {x^2 - a^2} } & : \\dfrac \\pi 2 < \\arcsec \\dfrac x a < \\pi \\ (\\text {that is: $x < -a$}) \\\\ \\end{cases}$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\frac {\\map \\d {\\map \\arcsec {\\frac x a} } } {\\d x}       | r = \\frac 1 a \\frac 1 {\\size {\\frac x a} \\sqrt {\\paren {\\frac x a}^2 - 1} }       | c = Derivative of Arcsecant Function and Derivative of Function of Constant Multiple\t }} {{eqn | r = \\frac 1 a \\frac 1 {\\size {\\frac x a} \\frac {\\sqrt {x^2 - a^2} } a}       | c =  }} {{eqn | r = \\frac 1 a \\frac {a^2} {\\size x {\\sqrt {x^2 - a^2} } }       | c =  }} {{eqn | r = \\frac a {\\size x {\\sqrt {x^2 - a^2} } }       | c =  }} {{end-eqn}} {{qed|lemma}} Similarly: {{begin-eqn}} {{eqn | l = \\frac {\\map \\d {\\map \\arcsec {\\frac x a} } } {\\d x}       | r = \\begin {cases} \\dfrac 1 a \\dfrac {+1} {\\frac x a \\sqrt {\\paren {\\frac x a}^2 - 1} } & : 0 < \\arcsec \\dfrac x a < \\dfrac \\pi 2 \\ (\\text {that is: $\\dfrac x a > 1$}) \\\\ \\dfrac 1 a \\dfrac {-1} {\\frac x a \\sqrt {\\paren {\\frac x a}^2 - 1} } & : \\dfrac \\pi 2 < \\arcsec \\dfrac x a < \\pi \\ (\\text {that is: $\\dfrac x a < -1$}) \\\\ \\end{cases}       | c = Derivative of Arcsecant Function <br/>and Derivative of Function of Constant Multiple\t }} {{eqn | r = \\begin {cases} \\dfrac a {x \\sqrt {x^2 - a^2} } & : 0 < \\arcsec \\dfrac x a < \\dfrac \\pi 2 \\\\ \\dfrac {-a} {x \\sqrt {x^2 - a^2} } & : \\dfrac \\pi 2 < \\arcsec \\dfrac x a < \\pi \\\\ \\end{cases}       | c = simplifying as above }} {{end-eqn}} {{qed}} == Also see == * Derivative of $\\arcsin \\dfrac x a$ * Derivative of $\\arccos \\dfrac x a$ * Derivative of $\\arctan \\dfrac x a$ * Derivative of $\\arccot \\dfrac x a$ * Derivative of $\\arccsc \\dfrac x a$ Category:Derivative of Arcsecant Function g9q7rhstacx1hjaz0jzh26nr4utcfp3"}
+{"_id": "32471", "title": "Derivative of Arccosecant Function/Corollary", "text": "Derivative of Arccosecant Function/Corollary 0 33700 415975 190995 2019-08-02T19:58:00Z Caliburn 3218 wikitext text/x-wiki == Corollary to Derivative of Arccosecant Function == Let $x \\in \\R$. Let $\\operatorname{arccsc} \\left({\\dfrac x a}\\right)$ be the arccosecant of $\\dfrac x a$. Then: <onlyinclude> :$\\dfrac {\\mathrm d \\left({\\operatorname{arccsc} \\left({\\frac x a}\\right) }\\right)} {\\mathrm d x} = \\dfrac {-a} {\\left\\vert{x}\\right\\vert {\\sqrt {x^2 - a^2} } } = \\begin{cases} \\dfrac {-a} {x \\sqrt {x^2 - a^2} } & : 0 < \\operatorname{arccsc} \\dfrac x a < \\dfrac \\pi 2 \\\\ \\dfrac a {x \\sqrt {x^2 - a^2} } & : -\\dfrac \\pi 2 < \\operatorname{arccsc} \\dfrac x a < 0 \\\\ \\end{cases}$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\frac {\\mathrm d \\left({\\operatorname{arccsc} \\left({\\frac x a}\\right)}\\right)} {\\mathrm d x}       | r = \\frac 1 a \\frac {-1} {\\left\\vert{\\frac x a}\\right\\vert \\sqrt {\\left({\\frac x a}\\right)^2 - 1} }       | c = Derivative of Arccosecant Function and Derivative of Function of Constant Multiple\t }} {{eqn | r = \\frac 1 a \\frac {-1} {\\left\\vert{\\frac x a}\\right\\vert \\frac {\\sqrt {x^2 - a^2} } a}       | c =  }} {{eqn | r = \\frac 1 a \\frac {-a^2} {\\left\\vert{x}\\right\\vert {\\sqrt {x^2 - a^2} } }       | c =  }} {{eqn | r = \\frac {-a} {\\left\\vert{x}\\right\\vert {\\sqrt {x^2 - a^2} } }       | c =  }} {{end-eqn}} {{qed|lemma}} Similarly: {{begin-eqn}} {{eqn | l = \\frac {\\mathrm d \\left({\\operatorname{arccsc} \\left({\\frac x a}\\right)}\\right)} {\\mathrm d x}       | r = \\begin{cases} \\dfrac 1 a \\dfrac {-1} {\\frac x a \\sqrt {\\left({\\frac x a}\\right)^2 - 1} } & : 0 < \\operatorname{arccsc} \\dfrac x a < \\dfrac \\pi 2 \\\\ \\dfrac 1 a \\dfrac {+1} {\\frac x a \\sqrt {\\left({\\frac x a}\\right)^2 - 1} } & : -\\dfrac \\pi 2 < \\operatorname{arccsc} \\dfrac x a < 0 \\\\ \\end{cases}       | c = Derivative of Arccosecant Function <br/>and Derivative of Function of Constant Multiple\t }} {{eqn | r = \\begin{cases} \\dfrac {-a} {x \\sqrt {x^2 - a^2} } & : 0 < \\operatorname{arccsc} \\dfrac x a < \\dfrac \\pi 2 \\\\ \\dfrac a {x \\sqrt {x^2 - a^2} } & : -\\dfrac \\pi 2 < \\operatorname{arccsc} \\dfrac x a < 0 \\\\ \\end{cases}       | c = simplifying as above }} {{end-eqn}} {{qed}} == Also see == * Derivative of $\\arcsin \\dfrac x a$ * Derivative of $\\arccos \\dfrac x a$ * Derivative of $\\arctan \\dfrac x a$ * Derivative of $\\operatorname{arccot} \\dfrac x a$ * Derivative of $\\operatorname{arcsec} \\dfrac x a$ Category:Derivatives of Inverse Trigonometric Functions Category:Arccosecant Function rha9282refmkf3hxesaj45e2azyhcus"}
+{"_id": "32472", "title": "Primitive of Reciprocal of x cubed by a x + b squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x cubed by a x + b squared/Partial Fraction Expansion 0 33732 407613 220626 2019-06-12T20:43:55Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of $\\dfrac 1 {x^3 \\left({a x + b}\\right)^2}$ == :$\\dfrac 1 {x^3 \\left({a x + b}\\right)^2} \\equiv \\dfrac {3 a^2} {b^4 x} - \\dfrac {2 a} {b^3 x^2} + \\dfrac 1 {b^2 x^3} - \\dfrac {3 a^3} {b^4 \\left({a x + b}\\right)} - \\dfrac {a^3} {b^3 \\left({a x + b}\\right)^2}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x^3 \\left({a x + b}\\right)^2}       | o = \\equiv       | r = \\dfrac A x + \\dfrac B {x^2} + \\dfrac C {x^3} + \\dfrac D {a x + b} + \\dfrac E {\\left({a x + b}\\right)^2}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = 1       | o = \\equiv       | r = A x^2 \\left({a x + b}\\right)^2 + B x \\left({a x + b}\\right)^2 + C \\left({a x + b}\\right)^2 + D x^3 \\left({a x + b}\\right) + E x^3       | c = multiplying through by $x^3 \\left({a x + b}\\right)^2$ }} {{eqn | o = \\equiv       | r = A a^2 x^4 + 2 A a b x^3 + A b^2 x^2       | c = multiplying everything out }} {{eqn | o =        | ro= +       | r = B a^2 x^3 + 2 B a b x^2 + B b^2 x       | c = (tedious though this is, it helps to }} {{eqn | o =        | ro= +       | r = C a^2 x^2 + 2 C a b x + C b^2       | c = identify the equal indices) }} {{eqn | o =        | ro= +       | r = D a x^4 + D b x^3 + E x^3       | c =  }} {{end-eqn}} Setting $a x + b = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = a x + b       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac b a       | c =  }} {{eqn | ll= \\implies       | l = E \\left({-\\frac b a}\\right)^3       | r = 1       | c = substituting for $x$ in $(1)$: terms in $a x + b$ are all $0$ }} {{eqn | ll= \\implies       | l = E       | r = -\\frac {a^3} {b^3}       | c =  }} {{end-eqn}} Equating constants in $(1)$: {{begin-eqn}} {{eqn | l = 1       | r = C b^2       | c =  }} {{eqn | n = 2       | ll= \\implies       | l = C       | r = \\frac 1 {b^2}       | c =  }} {{end-eqn}} Equating $1$st powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = 2 C a b + B b^2       | c =  }} {{eqn | ll= \\implies       | l = \\frac {2 a b} {b^2}       | r = -B b^2       | c = subtituting for $C$ from $(2)$ }} {{eqn | n = 3       | ll= \\implies       | l = B       | r = -\\frac {2 a} {b^3}       | c =  }} {{end-eqn}} Equating $2$nd powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A b^2 + 2 B a b + C a^2       | c =  }} {{eqn | r = A b^2 + 2 \\left({-\\frac {2 a} {b^3} }\\right) a b + \\left({\\frac 1 {b^2} }\\right) a^2       | c = substituting for $B$ and $C$ from $(2)$ and $(3)$ }} {{eqn | ll= \\implies       | l = A b^2       | r = \\frac {4 a^2} {b^2} - \\frac {a^2} {b^2}       | c = rearranging }} {{eqn | n = 4       | ll= \\implies       | l = A       | r = \\frac {3 a^2} {b^4}       | c = simplifying }} {{end-eqn}} Equating $4$th powers of $x$: {{begin-eqn}} {{eqn | l = 0       | r = A a^2 + D a       | c =  }} {{eqn | ll= \\implies       | l = D       | r = \\frac {3 a^3} {b^4}       | c = substituting for $A$ from $(3)$ and simplifying }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac {3 a^2} {b^4} }} {{eqn | l = B       | r = -\\frac {2 a} {b^3} }} {{eqn | l = C       | r = \\frac 1 {b^2} }} {{eqn | l = D       | r = \\frac {3 a^3} {b^4} }} {{eqn | l = E       | r = -\\frac {a^3} {b^3} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a x + b/Lemmata hk1k80vrcy5vr3kj71xhdbac62iz99x"}
+{"_id": "32473", "title": "Primitive of Reciprocal of x squared by a x + b squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x squared by a x + b squared/Partial Fraction Expansion 0 33733 407620 220627 2019-06-12T20:46:01Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of $\\dfrac 1 {x^2 \\left({a x + b}\\right)^2}$ == :$\\dfrac 1 {x^2 \\left({a x + b}\\right)^2} \\equiv -\\dfrac {2 a} {b^3 x} + \\dfrac 1 {b^2 x^2} + \\dfrac {2 a^2} {b^3 \\left({a x + b}\\right)} + \\dfrac {a^2} {b^2 \\left({a x + b}\\right)^2}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x^2 \\left({a x + b}\\right)^2}       | o = \\equiv       | r = \\dfrac A x + \\dfrac B {x^2} + \\dfrac C {a x + b} + \\dfrac D {\\left({a x + b}\\right)^2}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = 1       | o = \\equiv       | r = A x \\left({a x + b}\\right)^2 + B \\left({a x + b}\\right)^2 + C x^2 \\left({a x + b}\\right) + D x^2       | c = multiplying through by $x^2 \\left({a x + b}\\right)^2$ }} {{eqn | o = \\equiv       | r = A a^2 x^3 + 2 A a b x^2 + A b^2 x       | c = multiplying everything out }} {{eqn | o =        | ro= +       | r = B a^2 x^2 + 2 B a b x + B b^2       | c = (tedious though this is, it helps to }} {{eqn | o =        | ro= +       | r = C a x^3 + C b x^2 + D x^2       | c = identify the equal indices) }} {{end-eqn}} Setting $a x + b = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = a x + b       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac b a       | c =  }} {{eqn | ll= \\implies       | l = D \\left({-\\frac b a}\\right)^2       | r = 1       | c = substituting for $x$ in $(1)$: terms in $a x + b$ are all $0$ }} {{eqn | ll= \\implies       | l = D       | r = \\frac {a^2} {b^2}       | c =  }} {{end-eqn}} Equating constants in $(1)$: {{begin-eqn}} {{eqn | l = 1       | r = B b^2       | c =  }} {{eqn | n = 2       | ll= \\implies       | l = B       | r = \\frac 1 {b^2}       | c =  }} {{end-eqn}} Equating $1$st powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A b^2 + 2 B a b       | c =  }} {{eqn | ll= \\implies       | l = A b^2       | r = -\\frac {2 a b} {b^2}       | c = subtituting for $B$ from $(2)$ }} {{eqn | n = 3       | ll= \\implies       | l = A       | r = -\\frac {2 a} {b^3}       | c =  }} {{end-eqn}} Equating $3$rd powers of $x$: {{begin-eqn}} {{eqn | l = 0       | r = A a^2 + C a       | c =  }} {{eqn | ll= \\implies       | l = C       | r = \\frac {2 a^2} {b^3}       | c = substituting for $A$ from $(3)$ and simplifying }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = -\\frac {2 a} {b^3} }} {{eqn | l = B       | r = \\frac 1 {b^2} }} {{eqn | l = C       | r = \\frac {2 a^2} {b^3} }} {{eqn | l = D       | r = \\frac {a^2} {b^2} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a x + b/Lemmata 7t9zwmc4yk753nn430cy797l47zy5ct"}
+{"_id": "32474", "title": "Primitive of Reciprocal of x by a x + b squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x by a x + b squared/Partial Fraction Expansion 0 33734 495810 407598 2020-10-21T20:53:17Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of $\\dfrac 1 {x \\paren {a x + b}^2}$ == :$\\dfrac 1 {x \\paren {a x + b}^2} \\equiv \\dfrac 1 {b^2 x} - \\dfrac a {b^2 \\paren {a x + b} } - \\dfrac a {b \\paren {a x + b}^2}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x \\paren {a x + b}^2}       | o = \\equiv       | r = \\dfrac A x + \\dfrac B {a x + b} + \\dfrac C {\\paren {a x + b}^2}       | c =  }} {{eqn | n = 1       | ll= \\leadsto       | l = 1       | o = \\equiv       | r = A \\paren {a x + b}^2 + B x \\paren {a x + b} + C x       | c = multiplying through by $x \\paren {a x + b}^2$ }} {{eqn | o = \\equiv       | r = A a^2 x^2 + 2 A a b x + A b^2 + B a x^2 + B b x + C x       | c = multiplying everything out }} {{end-eqn}} Setting $a x + b = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = a x + b       | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = x       | r = -\\frac b a       | c =  }} {{eqn | ll= \\leadsto       | l = C \\paren {-\\frac b a}       | r = 1       | c = substituting for $x$ in $(1)$: terms in $a x + b$ are all $0$ }} {{eqn | ll= \\leadsto       | l = C       | r = -\\frac a b       | c =  }} {{end-eqn}} Equating constants in $(1)$: {{begin-eqn}} {{eqn | l = 1       | r = A b^2       | c =  }} {{eqn | n = 2       | ll= \\leadsto       | l = A       | r = \\frac 1 {b^2}       | c =  }} {{end-eqn}} Equating $2$nd powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A a^2 + B a       | c =  }} {{eqn | ll= \\leadsto       | l = B       | r = -\\frac a {b^2}       | c = substituting for $A$ from $(2)$ and simplifying }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac 1 {b^2} }} {{eqn | l = B       | r = -\\frac a {b^2} }} {{eqn | l = C       | r = -\\frac a b }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a x + b/Lemmata rg82b461fpapczoti63aa49viuxbwoj"}
+{"_id": "32475", "title": "Primitive of Reciprocal of x cubed by a x + b/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x cubed by a x + b/Partial Fraction Expansion 0 33735 407614 220629 2019-06-12T20:44:13Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of $\\dfrac 1 {x^3 \\left({a x + b}\\right)}$ == :$\\dfrac 1 {x^3 \\left({a x + b}\\right)} \\equiv \\dfrac {a^2} {b^3 x} + \\dfrac {-a} {b^2 x^2} + \\dfrac 1 {b x^3} + \\dfrac {-a^3} {b^3 \\left({a x + b}\\right)}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x^3 \\left({a x + b}\\right)}       | o = \\equiv       | r = \\dfrac A x + \\dfrac B {x^2} + \\dfrac C {x^3} + \\dfrac D {a x + b}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = 1       | o = \\equiv       | r = A x^2 \\left({a x + b}\\right) + B x \\left({a x + b}\\right) + C \\left({a x + b}\\right) + D x^3       | c = multiplying through by $x^3 \\left({a x + b}\\right)$ }} {{eqn | o = \\equiv       | r = A a x^3 + A b x^2 + B a x^2 + B b x + C a x + C b + D x^3       | c = multiplying everything out }} {{end-eqn}} Setting $a x + b = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = a x + b       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac b a       | c =  }} {{eqn | ll= \\implies       | l = D \\left({-\\frac b a}\\right)^3       | r = 1       | c = substituting for $x$ in $(1)$: terms in $a x + b$ are all $0$ }} {{eqn | n = 2       | ll= \\implies       | l = D       | r = -\\frac {a^3} {b^3}       | c =  }} {{end-eqn}} Equating constants in $(1)$: {{begin-eqn}} {{eqn | l = 1       | r = C b       | c =  }} {{eqn | ll= \\implies       | l = C       | r = \\frac 1 b       | c =  }} {{end-eqn}} Equating $3$rd powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A a + D       | c =  }} {{eqn | n = 3       | ll= \\implies       | l = A       | r = \\frac {a^2} {b^3}       | c = substituting for $D$ from $(2)$ and simplifying }} {{end-eqn}} Equating $2$nd powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A b + B a       | c =  }} {{eqn | r = \\frac {a^2} {b^3} b + B a       | c = substituting for $A$ from $(3)$ }} {{eqn | ll= \\implies       | l = B       | r = -\\frac a {b^2}       | c = rearranging and simplifying }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac {a^2} {b^3} }} {{eqn | l = B       | r = -\\frac a {b^2} }} {{eqn | l = C       | r = \\frac 1 b }} {{eqn | l = D       | r = -\\frac {a^3} {b^3} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a x + b/Lemmata f4ynmdmwy0z1el2feperqi60dv7rab3"}
+{"_id": "32476", "title": "Primitive of Reciprocal of x squared by a x + b/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x squared by a x + b/Partial Fraction Expansion 0 33736 407621 220630 2019-06-12T20:46:15Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of $\\dfrac 1 {x^2 \\left({a x + b}\\right)}$ == :$\\dfrac 1 {x^2 \\left({a x + b}\\right)} \\equiv -\\dfrac a {b^2 x} + \\dfrac 1 {b x^2} + \\dfrac {a^2} {b^2 \\left({a x + b}\\right)}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x^2 \\left({a x + b}\\right)}       | o = \\equiv       | r = \\dfrac A x + \\dfrac B {x^2} + \\dfrac C {a x + b}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = 1       | o = \\equiv       | r = A x \\left({a x + b}\\right) + B \\left({a x + b}\\right) + C x^2       | c = multiplying through by $x^2 \\left({a x + b}\\right)$ }} {{eqn | o = \\equiv       | r = A a x^2 + A b x + B a x + B b + C x^2       | c = multiplying everything out }} {{end-eqn}} Setting $a x + b = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = a x + b       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac b a       | c =  }} {{eqn | ll= \\implies       | l = C \\left({-\\frac b a}\\right)^2       | r = 1       | c = substituting for $x$ in $(1)$: terms in $a x + b$ are all $0$ }} {{eqn | n = 2       | ll= \\implies       | l = C       | r = \\frac {a^2} {b^2}       | c =  }} {{end-eqn}} Equating constants in $(1)$: {{begin-eqn}} {{eqn | l = 1       | r = B b       | c =  }} {{eqn | ll= \\implies       | l = B       | r = \\frac 1 b       | c =  }} {{end-eqn}} Equating $2$nd powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A a + C       | c =  }} {{eqn | ll= \\implies       | l = A       | r = -\\frac a {b^2}       | c = substituting for $C$ from $(2)$ and rearranging }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = -\\frac a {b^2} }} {{eqn | l = B       | r = \\frac 1 b }} {{eqn | l = C       | r = \\frac {a^2} {b^2} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a x + b/Lemmata fagh8fgr6gihgoev0j5vk8a4t178et7"}
+{"_id": "32477", "title": "Primitive of Reciprocal of x by a x + b/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x by a x + b/Partial Fraction Expansion 0 33737 407599 220631 2019-06-12T20:38:04Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of $\\dfrac 1 {x \\left({a x + b}\\right)}$ == :$\\dfrac 1 {x \\left({a x + b}\\right)} \\equiv \\dfrac 1 {b x} - \\dfrac a {b \\left({a x + b}\\right)}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x \\left({a x + b}\\right)}       | o = \\equiv       | r = \\dfrac A x + \\dfrac B {a x + b}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = 1       | o = \\equiv       | r = A \\left({a x + b}\\right) + B x       | c = multiplying through by $x \\left({a x + b}\\right)$ }} {{eqn | o = \\equiv       | r = A a x + A b + B x       | c = multiplying out }} {{end-eqn}} Setting $a x + b = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = a x + b       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac b a       | c =  }} {{eqn | ll= \\implies       | l = B \\left({-\\frac b a}\\right)       | r = 1       | c = substituting for $x$ in $(1)$: terms in $a x + b$ are all $0$ }} {{eqn | ll= \\implies       | l = B       | r = -\\frac a b       | c =  }} {{end-eqn}} Equating constants in $(1)$: {{begin-eqn}} {{eqn | l = 1       | r = A b       | c =  }} {{eqn | ll= \\implies       | l = A       | r = \\frac 1 b       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac 1 b }} {{eqn | l = B       | r = -\\frac a b }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a x + b/Lemmata ker6cbgwr67q36i17hcd1890g5lvgt6"}
+{"_id": "32478", "title": "Primitive of Reciprocal of x by a x + b cubed/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x by a x + b cubed/Partial Fraction Expansion 0 33745 407596 220653 2019-06-12T20:36:51Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of $\\dfrac 1 {x \\paren {a x + b}^3}$ == :$\\dfrac 1 {x \\paren {a x + b}^3} \\equiv \\dfrac 1 {b^3 x} - \\dfrac a {b^3 \\paren {a x + b} } - \\dfrac a {b^2 \\paren {a x + b}^2} - \\dfrac a {b \\paren {a x + b}^3}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x \\paren {a x + b}^3}       | o = \\equiv       | r = \\dfrac A x + \\dfrac B {a x + b} + \\dfrac C {\\paren {a x + b}^2} + \\dfrac D {\\paren {a x + b}^3}       | c =  }} {{eqn | n = 1       | ll= \\leadsto       | l = 1       | o = \\equiv       | r = A \\paren {a x + b}^3 + B x \\paren {a x + b}^2 + C x \\paren {a x + b} + D x       | c = multiplying through by $x \\paren {a x + b}^3$ }} {{eqn | o = \\equiv       | r = A a^3 x^3 + 3 A a^2 b x^2 + 3 A a b^2 x + A b^3       | c = multiplying everything out }} {{eqn | o =        | ro= +       | r = B a^2 x^3 + 2 B a b x^2 + B b^2 x + C a x^2 + C b x + D x       | c =  }} {{end-eqn}} Setting $a x + b = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = a x + b       | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = x       | r = -\\frac b a       | c =  }} {{eqn | ll= \\leadsto       | l = D \\paren {-\\frac b a}       | r = 1       | c = substituting for $x$ in $(1)$: terms in $a x + b$ are all $0$ }} {{eqn | ll= \\leadsto       | l = D       | r = -\\frac a b       | c =  }} {{end-eqn}} Equating constants in $(1)$: {{begin-eqn}} {{eqn | l = 1       | r = A b^3       | c =  }} {{eqn | n = 2       | ll= \\leadsto       | l = A       | r = \\frac 1 {b^3}       | c =  }} {{end-eqn}} Equating $3$rd powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A a^3 + B a^2       | c =  }} {{eqn | ll= \\leadsto       | l = B a^2       | r = -\\frac {a^3} {b^3}       | c = substituting for $A$ from $(2)$ }} {{eqn | n = 3       | ll= \\leadsto       | l = B       | r = -\\frac a {b^3}       | c = simplifying }} {{end-eqn}} Equating $2$nd powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = 3 A a^2 b + 2 B a b + C a       | c =  }} {{eqn | ll= \\leadsto       | l = C       | r = -3 A a b - 2 B b       | c = rerranging }} {{eqn | r = -3 a b \\frac 1 {b^3} + 2 b \\frac a {b^3}       | c = substituting for $A$ from $(2)$ and $B$ from $(3)$ }} {{eqn | r = -3 \\frac a {b^2} + 2 \\frac a {b^2}       | c = simplifying }} {{eqn | r = -\\frac a {b^2}       | c = simplifying }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac 1 {b^3} }} {{eqn | l = B       | r = -\\frac a {b^3} }} {{eqn | l = C       | r = -\\frac a {b^2} }} {{eqn | l = D       | r = -\\frac a b }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a x + b/Lemmata 9gna4h51gj3p4fkmtvxmmptx7xo65o7"}
+{"_id": "32479", "title": "Primitive of Reciprocal of x squared by a x + b cubed/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x squared by a x + b cubed/Partial Fraction Expansion 0 33747 407618 220652 2019-06-12T20:45:31Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of $\\dfrac 1 {x^2 \\left({a x + b}\\right)^3}$ == :$\\dfrac 1 {x^2 \\left({a x + b}\\right)^3} \\equiv \\dfrac {-3 a} {b^4 x} + \\dfrac 1 {b^3 x^2} + \\dfrac {3 a^2} {b^4 \\left({a x + b}\\right)} + \\dfrac {2 a^2} {b^3 \\left({a x + b}\\right)^2} + \\dfrac {a^2} {b^2 \\left({a x + b}\\right)^3}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x^2 \\left({a x + b}\\right)^3}       | o = \\equiv       | r = \\dfrac A x + \\dfrac B {x^2} + \\dfrac C {a x + b} + \\dfrac D {\\left({a x + b}\\right)^2} + \\dfrac E {\\left({a x + b}\\right)^3}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = 1       | o = \\equiv       | r = A x \\left({a x + b}\\right)^3 + B \\left({a x + b}\\right)^3 + C x^2 \\left({a x + b}\\right)^2 + D x^2 \\left({a x + b}\\right) + E x^2       | c = multiplying through by $x^2 \\left({a x + b}\\right)^3$ }} {{eqn | o = \\equiv       | r = A a^3 x^4 + 3 A a^2 b x^3 + 3 A a b^2 x^2 + A b^3 x       | c = multiplying everything out }} {{eqn | o =        | ro= +       | r = B a^3 x^3 + 3 B a^2 b x^2 + 3 B a b^2 x + B b^3       | c =  }} {{eqn | o =        | ro= +       | r = C a^2 x^4 + 2 C a b x^3 + C b^2 x^2 + D a x^3 + D b x^2 + E x^2       | c =  }} {{end-eqn}} Setting $a x + b = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = a x + b       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac b a       | c =  }} {{eqn | ll= \\implies       | l = E \\left({-\\frac b a}\\right)^2       | r = 1       | c = substituting for $x$ in $(1)$: terms in $a x + b$ are all $0$ }} {{eqn | ll= \\implies       | l = E       | r = \\frac {a^2} {b^2}       | c =  }} {{end-eqn}} Equating constants in $(1)$: {{begin-eqn}} {{eqn | l = 1       | r = B b^3       | c =  }} {{eqn | n = 2       | ll= \\implies       | l = B       | r = \\frac 1 {b^3}       | c =  }} {{end-eqn}} Equating $1$st powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A b^3 + 3 B a b^2       | c =  }} {{eqn | ll= \\implies       | l = A b^3       | r = -3 \\frac 1 {b^3} a b^2       | c = substituting for $B$ from $(2)$ }} {{eqn | n = 3       | ll= \\implies       | l = A       | r = \\frac {-3 a} {b^4}       | c = simplifying }} {{end-eqn}} Equating $4$th powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A a^3 + C a^2       | c =  }} {{eqn | ll= \\implies       | l = C a^2       | r = \\frac {3 a} {b^4} a^3       | c = substituting for $A$ from $(3)$ }} {{eqn | n = 4       | ll= \\implies       | l = C       | r = \\frac {3 a^2} {b^4}       | c = simplifying }} {{end-eqn}} Equating $3$rd powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = 3 A a^2 b + B a^3 + 2 C a b + D a       | c =  }} {{eqn | ll= \\implies       | l = D       | r = -3 A a b - B a^2 - 2 C b       | c = rearranging }} {{eqn | r = -3 a b \\frac {-3 a} {b^4} - \\frac 1 {b^3} a^2 - 2 b \\frac {3 a^2} {b^4}       | c = substituting for $A$ from $(3)$, $B$ from $(2)$ and $C$ from $(4)$ }} {{eqn | r = \\frac {9 a^2} {b^3} - \\frac {a^2} {b^3} - \\frac {6 a^2} {b^3}       | c = simplifying }} {{eqn | r = \\frac {2 a^2} {b^3}       | c = simplifying }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac {-3 a} {b^4} }} {{eqn | l = B       | r = \\frac 1 {b^3} }} {{eqn | l = C       | r = \\frac {3 a^2} {b^4} }} {{eqn | l = D       | r = \\frac {2 a^2} {b^3} }} {{eqn | l = E       | r = \\frac {a^2} {b^2} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a x + b/Lemmata 8x5pzzoxvxf4n4dse5oqpjokx8vk2yr"}
+{"_id": "32480", "title": "Primitive of Reciprocal of x cubed by a x + b cubed/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x cubed by a x + b cubed/Partial Fraction Expansion 0 33755 407611 220619 2019-06-12T20:43:25Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of $\\dfrac 1 {x^3 \\left({a x + b}\\right)^3}$ == :$\\dfrac 1 {x^3 \\left({a x + b}\\right)^3} \\equiv \\dfrac {6 a^2} {b^5 x} - \\dfrac {3 a} {b^4 x^2} + \\dfrac 1 {b^3 x^3} - \\dfrac {6 a^3} {b^5 \\left({a x + b}\\right)} -\\dfrac {3 a^3} {b^4 \\left({a x + b}\\right)^2} - \\dfrac {a^3} {b^3 \\left({a x + b}\\right)^3}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x^3 \\left({a x + b}\\right)^3}       | o = \\equiv       | r = \\dfrac A x + \\dfrac B {x^2} + \\dfrac C {x^3} + \\dfrac D {a x + b} + \\dfrac E {\\left({a x + b}\\right)^2} + \\dfrac F {\\left({a x + b}\\right)^3}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = 1       | o = \\equiv       | r = A x^2 \\left({a x + b}\\right)^3 + B x \\left({a x + b}\\right)^3 + C \\left({a x + b}\\right)^3 + D x^3 \\left({a x + b}\\right)^2 + E x^2 \\left({a x + b}\\right) + F x^2       | c = multiplying through by $x^2 \\left({a x + b}\\right)^3$ }} {{eqn | o = \\equiv       | r = A a^3 x^5 + 3 A a^2 b x^4 + 3 A a b^2 x^3 + A b^3 x^2       | c = multiplying everything out }} {{eqn | o =        | ro= +       | r = B a^3 x^4 + 3 B a^2 b x^3 + 3 B a b^2 x^2 + B b^3 x       | c =  }} {{eqn | o =        | ro= +       | r = C a^3 x^3 + 3 C a^2 b x^2 + 3 C a b^2 x + C b^3       | c =  }} {{eqn | o =        | ro= +       | r = D a^2 x^5 + 2 D a b x^4 + D b^2 x^3 + E a x^4 + E b x^3 + F x^3       | c =  }} {{end-eqn}} Setting $a x + b = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = a x + b       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac b a       | c =  }} {{eqn | ll= \\implies       | l = F \\left({-\\frac b a}\\right)^3       | r = 1       | c = substituting for $x$ in $(1)$: terms in $a x + b$ are all $0$ }} {{eqn | ll= \\implies       | l = F       | r = -\\frac {a^3} {b^3}       | c =  }} {{end-eqn}} Equating constants in $(1)$: {{begin-eqn}} {{eqn | l = 1       | r = C b^3       | c =  }} {{eqn | n = 2       | ll= \\implies       | l = C       | r = \\frac 1 {b^3}       | c =  }} {{end-eqn}} Equating $1$st powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = B b^3 + 3 C a b^2       | c =  }} {{eqn | ll= \\implies       | l = B b^3       | r = -3 \\frac 1 {b^3} a b^2       | c = substituting for $C$ from $(2)$ }} {{eqn | n = 3       | ll= \\implies       | l = B       | r = \\frac {-3 a} {b^4}       | c = simplifying }} {{end-eqn}} Equating $2$nd powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A b^3 + 3 B a b^2 + 3 C a^2 b       | c =  }} {{eqn | ll= \\implies       | l = A b^3       | r = -3 \\frac {-3 a} {b^4} a b^2 - 3 \\frac 1 {b^3} a^2 b       | c = substituting for $C$ from $(2)$ and $B$ from $(3)$ }} {{eqn | ll= \\implies       | l = A       | r = \\frac {9 a^2} {b^5} - 3 \\frac {a^2} {b^5}       | c = simplifying }} {{eqn | n = 4       | r = \\frac {6 a^2} {b^5}       | c = simplifying }} {{end-eqn}} Equating $5$th powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A a^3 + D a^2       | c =  }} {{eqn | ll= \\implies       | l = D a^2       | r = -\\frac {6 a^2} {b^5} a^3       | c = substituting for $A$ from $(4)$ }} {{eqn | n = 5       | ll= \\implies       | l = D       | r = -\\frac {6 a^3} {b^5}       | c = simplifying }} {{end-eqn}} Equating $4$th powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = 3 A a^2 b + B a^3 + 2 D a b + E a       | c =  }} {{eqn | ll= \\implies       | l = E       | r = -3 A a b - B a^2 - 2 D b       | c = rearranging }} {{eqn | r = -3 a b \\frac {6 a^2} {b^5} - \\frac {-3 a} {b^4} a^2 - 2 \\frac {-6 a^3} {b^5} b       | c = substituting for $A$ from $(4)$, $B$ from $(3)$ and $D$ from $(5)$ }} {{eqn | r = -\\frac {18 a^3} {b^4} + \\frac {3 a^3} {b^4} + \\frac {12 a^3} {b^4}       | c = simplifying }} {{eqn | r = -\\frac {3 a^3} {b^4}       | c = simplifying }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac {6 a^2} {b^5} }} {{eqn | l = B       | r = -\\frac {3 a} {b^4} }} {{eqn | l = C       | r = \\frac 1 {b^3} }} {{eqn | l = D       | r = -\\frac {6 a^3} {b^5} }} {{eqn | l = E       | r = -\\frac {3 a^3} {b^4} }} {{eqn | l = F       | r = -\\frac {a^3} {b^3} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a x + b/Lemmata t9cz268t2m64gf2f1xex6swv2eu44e9"}
+{"_id": "32481", "title": "Primitive of Root of a x + b over Power of x/Formulation 1", "text": "Primitive of Root of a x + b over Power of x/Formulation 1 0 33874 188090 2014-05-24T06:10:32Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> :$\\displaystyle \\int \\frac {\\sqrt{a x + b} } {x^m} \\ \\mathrm d x = -\\frac {\\sqrt{a x + b} } {\\left({m - 1}\\right) x^{m-1} } + \\frac a {2 \\left({m -...\" wikitext text/x-wiki == Theorem == <onlyinclude> :$\\displaystyle \\int \\frac {\\sqrt{a x + b} } {x^m} \\ \\mathrm d x = -\\frac {\\sqrt{a x + b} } {\\left({m - 1}\\right) x^{m-1} } + \\frac a {2 \\left({m - 1}\\right)} \\int \\frac {\\mathrm d x} {x^{m - 1} \\sqrt{a x + b} }$ </onlyinclude> == Proof == Let: {{begin-eqn}} {{eqn | l = u       | r = \\sqrt{a x + b}       | c =  }} {{eqn | ll= \\implies       | l = \\frac {\\mathrm d u}{\\mathrm d x}       | r = \\frac a {2 \\sqrt{a x + b} }       | c = Power Rule for Derivatives etc. }} {{eqn | l = v       | r = \\frac {-1} {\\left({m - 1}\\right) x^{m - 1} }       | c =  }} {{eqn | ll= \\implies       | l = \\frac {\\mathrm d v}{\\mathrm d x}       | r = \\frac 1 {x^m}       | c = Power Rule for Derivatives }} {{end-eqn}} From Integration by Parts: : $\\displaystyle \\int u \\frac {\\mathrm d v} {\\mathrm d x} \\ \\mathrm d x = u v - \\int v \\ \\frac {\\mathrm d u} {\\mathrm d x} \\ \\mathrm d x$ from which: {{begin-eqn}} {{eqn | l = \\int \\frac {\\sqrt{a x + b} } {x^m} \\ \\mathrm d x       | r = \\int \\sqrt{a x + b} \\frac 1 {x^m} \\ \\mathrm d x       | c =  }} {{eqn | r = \\sqrt{a x + b} \\frac {-1} {\\left({m - 1}\\right) x^{m - 1} } - \\int \\frac {-1} {\\left({m - 1}\\right) x^{m - 1} } \\frac a {2 \\sqrt{a x + b} } \\ \\mathrm d x       | c =  }} {{eqn | r = -\\frac {\\sqrt{a x + b} } {\\left({m - 1}\\right) x^{m-1} } + \\frac a {2 \\left({m - 1}\\right)} \\int \\frac {\\mathrm d x} {x^{m - 1} \\sqrt{a x + b} }       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev=Primitive of Power of x by Root of a x + b|next=Primitive of Root of a x + b over Power of x/Formulation 2}}: $\\S 14$: Integrals involving $\\sqrt{a x + b}$: $14.97$ Category:Primitives involving Root of a x + b pk9gve2i0ecocasv4ubaegm50wsvp19"}
+{"_id": "32482", "title": "Primitive of Reciprocal of a x + b by p x + q/Partial Fraction Expansion", "text": "Primitive of Reciprocal of a x + b by p x + q/Partial Fraction Expansion 0 33886 188115 188113 2014-05-24T19:15:15Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of a x + b by p x + q == :$\\dfrac 1 {\\left({a x + b}\\right) \\left({p x + q}\\right)} \\equiv \\dfrac {-a} {\\left({b p - a q}\\right) \\left({a x + b}\\right)} + \\dfrac p {\\left({b p - a q}\\right) \\left({p x + q}\\right)}$ == Proof == {{begin-eqn}} {{eqn | l = \\frac 1 {\\left({a x + b}\\right) \\left({p x + q}\\right)}       | o = \\equiv       | r = \\frac A {a x + b} + \\frac B {p x + q}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = 1       | o = \\equiv       | r = A \\left({p x + q}\\right) + B \\left({a x + b}\\right)       | c = multiplying through by $\\left({a x + b}\\right) \\left({p x + q}\\right)$ }} {{end-eqn}} Setting $p x + q = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = p x + q       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac q p       | c =  }} {{eqn | ll= \\implies       | l = B \\left({a \\left({-\\frac q p}\\right) + b}\\right)       | r = 1       | c = substituting for $x$ in $(1)$: term in $p x + q$ is $0$ }} {{eqn | ll= \\implies       | l = B \\left({\\frac {b p - a q} p}\\right)       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = B       | r = \\frac p {b p - a q}       | c =  }} {{end-eqn}} Setting $a x + b = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = a x + b       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac b a       | c =  }} {{eqn | ll= \\implies       | l = A \\left({p \\left({-\\frac b a}\\right) + q}\\right)       | r = 1       | c = substituting for $x$ in $(1)$: term in $a x + b$ is $0$ }} {{eqn | ll= \\implies       | l = A \\left({\\frac {-b p + a q} a}\\right)       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = A       | r = \\frac {-a} {b p - a q}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac {-a} {p b - a q} }} {{eqn | l = B       | r = \\frac p {b p - a q} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a x + b and p x + q/Lemmata 5971lztczzaln1zy4hpeeazsdflryak"}
+{"_id": "32483", "title": "Primitive of x over a x + b by p x + q/Partial Fraction Expansion", "text": "Primitive of x over a x + b by p x + q/Partial Fraction Expansion 0 33889 191849 188120 2014-08-19T22:16:01Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of x over a x + b by p x + q == :$\\dfrac x {\\left({a x + b}\\right) \\left({p x + q}\\right)} \\equiv \\dfrac b {\\left({b p - a q}\\right) \\left({a x + b}\\right)} - \\dfrac q {\\left({b p - a q}\\right) \\left({p x + q}\\right)}$ == Proof == {{begin-eqn}} {{eqn | l = \\frac x {\\left({a x + b}\\right) \\left({p x + q}\\right)}       | o = \\equiv       | r = \\frac A {a x + b} + \\frac B {p x + q}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = x       | o = \\equiv       | r = A \\left({p x + q}\\right) + B \\left({a x + b}\\right)       | c = multiplying through by $\\left({a x + b}\\right) \\left({p x + q}\\right)$ }} {{end-eqn}} Setting $p x + q = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = p x + q       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac q p       | c =  }} {{eqn | ll= \\implies       | l = B \\left({a \\left({-\\frac q p}\\right) + b}\\right)       | r = -\\frac q p       | c = substituting for $x$ in $(1)$: term in $p x + q$ is $0$ }} {{eqn | ll= \\implies       | l = B       | r = \\frac {-q} {b p - a q}       | c =  }} {{end-eqn}} Setting $a x + b = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = a x + b       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac b a       | c =  }} {{eqn | ll= \\implies       | l = A \\left({p \\left({-\\frac b a}\\right) + q}\\right)       | r = -\\frac b a       | c = substituting for $x$ in $(1)$: term in $a x + b$ is $0$ }} {{eqn | ll= \\implies       | l = A       | r = \\frac b {b p - a q}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac b {p b - a q} }} {{eqn | l = B       | r = \\frac {-q} {b p - a q} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a x + b and p x + q/Lemmata 1u26x3kg8tj7eax9vec04ldj16pfvot"}
+{"_id": "32484", "title": "Primitive of Reciprocal of a x + b squared by p x + q/Partial Fraction Expansion", "text": "Primitive of Reciprocal of a x + b squared by p x + q/Partial Fraction Expansion 0 33891 188129 188126 2014-05-24T21:28:47Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of a x + b squared by p x + q == :$\\dfrac 1 {\\left({a x + b}\\right)^2 \\left({p x + q}\\right)} \\equiv \\dfrac 1 {b p - a q} \\left({\\dfrac {-a p} {\\left({b p - a q}\\right) \\left({a x + b}\\right)} + \\dfrac {-a} {\\left({a x + b}\\right)^2} + \\dfrac {p^2} {\\left({b p - a q}\\right) \\left({p x + q}\\right)} }\\right)$ == Proof == {{begin-eqn}} {{eqn | l = \\frac 1 {\\left({a x + b}\\right)^2 \\left({p x + q}\\right)}       | o = \\equiv       | r = \\frac A {a x + b} + \\frac B {\\left({a x + b}\\right)^2} + \\frac C {p x + q}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = 1       | o = \\equiv       | r = A \\left({a x + b}\\right) \\left({p x + q}\\right) + B \\left({p x + q}\\right) + C \\left({a x + b}\\right)^2       | c = multiplying through by $\\left({a x + b}\\right)^2 \\left({p x + q}\\right)$ }} {{end-eqn}} Setting $a x + b = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = a x + b       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac b a       | c =  }} {{eqn | ll= \\implies       | l = B \\left({p \\left({-\\frac b a}\\right) + q}\\right)       | r = 1       | c = substituting for $x$ in $(1)$: term in $a x + b$ is $0$ }} {{eqn | ll= \\implies       | l = B       | r = \\frac {-a} {b p - a q}       | c =  }} {{end-eqn}} Setting $p x + q = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = p x + q       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac q p       | c =  }} {{eqn | ll= \\implies       | l = C \\left({a \\left({-\\frac q p}\\right) + b}\\right)^2       | r = 1       | c = substituting for $x$ in $(1)$: term in $a x + b$ is $0$ }} {{eqn | n = 2       | ll= \\implies       | l = C       | r = \\frac {p^2} {\\left({b p - a q}\\right)^2}       | c =  }} {{end-eqn}} Equating $2$nd powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A a p + C a^2       | c =  }} {{eqn | ll= \\implies       | l = A       | r = -C \\frac a p       | c =  }} {{eqn | ll= \\implies       | l = A       | r = -\\frac {p^2} {\\left({b p - a q}\\right)^2} \\frac a p       | c = substituting for $C$ from $(2)$ }} {{eqn | ll= \\implies       | l = A       | r = \\frac {-a p} {\\left({b p - a q}\\right)^2}       | c = simplifying }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac {-a p} {\\left({b p - a q}\\right)^2} }} {{eqn | l = B       | r = \\frac {-a} {b p - a q} }} {{eqn | l = C       | r = \\frac {p^2} {\\left({b p - a q}\\right)^2} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a x + b and p x + q/Lemmata t465uyeblrnmm6m8k04s7fdd575abd6"}
+{"_id": "32485", "title": "Primitive of x over a x + b squared by p x + q/Partial Fraction Expansion", "text": "Primitive of x over a x + b squared by p x + q/Partial Fraction Expansion 0 33893 188130 2014-05-24T21:29:54Z Prime.mover 59 Created page with \"== Lemma for Primitive of x over a x + b squared by p x + q ==  :$\\dfrac x {\\left({a x + b}\\right)^2 \\left({p x + q}\\right)} \\equiv \\dfrac 1 {b p - a q} \\left({\\dfrac {a q...\" wikitext text/x-wiki == Lemma for Primitive of x over a x + b squared by p x + q == :$\\dfrac x {\\left({a x + b}\\right)^2 \\left({p x + q}\\right)} \\equiv \\dfrac 1 {b p - a q} \\left({\\dfrac {a q} {\\left({b p - a q}\\right) \\left({a x + b}\\right)} + \\dfrac b {\\left({a x + b}\\right)^2} - \\dfrac {p q} {\\left({b p - a q}\\right) \\left({p x + q}\\right)} }\\right)$ == Proof == {{begin-eqn}} {{eqn | l = \\frac x {\\left({a x + b}\\right)^2 \\left({p x + q}\\right)}       | o = \\equiv       | r = \\frac A {a x + b} + \\frac B {\\left({a x + b}\\right)^2} + \\frac C {p x + q}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = x       | o = \\equiv       | r = A \\left({a x + b}\\right) \\left({p x + q}\\right) + B \\left({p x + q}\\right) + C \\left({a x + b}\\right)^2       | c = multiplying through by $\\left({a x + b}\\right)^2 \\left({p x + q}\\right)$ }} {{end-eqn}} Setting $a x + b = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = a x + b       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac b a       | c =  }} {{eqn | ll= \\implies       | l = B \\left({p \\left({-\\frac b a}\\right) + q}\\right)       | r = -\\frac b a       | c = substituting for $x$ in $(1)$: term in $a x + b$ is $0$ }} {{eqn | ll= \\implies       | l = B       | r = \\frac b {b p - a q}       | c =  }} {{end-eqn}} Setting $p x + q = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = p x + q       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac q p       | c =  }} {{eqn | ll= \\implies       | l = C \\left({a \\left({-\\frac q p}\\right) + b}\\right)^2       | r = -\\frac q p       | c = substituting for $x$ in $(1)$: term in $a x + b$ is $0$ }} {{eqn | n = 2       | ll= \\implies       | l = C       | r = \\frac {-p q} {\\left({b p - a q}\\right)^2}       | c =  }} {{end-eqn}} Equating $2$nd powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A a p + C a^2       | c =  }} {{eqn | ll= \\implies       | l = A       | r = -C \\frac a p       | c =  }} {{eqn | ll= \\implies       | l = A       | r = -\\frac {p q} {\\left({b p - a q}\\right)^2} \\frac a p       | c = substituting for $C$ from $(2)$ }} {{eqn | ll= \\implies       | l = A       | r = \\frac {a q} {\\left({b p - a q}\\right)^2}       | c = simplifying }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac {a q} {\\left({b p - a q}\\right)^2} }} {{eqn | l = B       | r = \\frac b {b p - a q} }} {{eqn | l = C       | r = \\frac {-p q} {\\left({b p - a q}\\right)^2} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a x + b and p x + q/Lemmata 1fsrcgxs6v6e1zjqvc301kuubv2lk1b"}
+{"_id": "32486", "title": "Primitive of x squared over a x + b squared by p x + q/Partial Fraction Expansion", "text": "Primitive of x squared over a x + b squared by p x + q/Partial Fraction Expansion 0 33896 221069 219845 2015-06-12T06:48:52Z Kc kennylau 2331 wikitext text/x-wiki == Lemma for Primitive of $\\dfrac {x^2} {\\left({a x + b}\\right)^2 \\left({p x + q}\\right)}$ == :$\\dfrac {x^2} {\\left({a x + b}\\right)^2 \\left({p x + q}\\right)} \\equiv \\dfrac {b \\left({b p - 2 a q}\\right)} {a \\left({b p - a q}\\right)^2 \\left({a x + b}\\right)} + \\dfrac {-b^2} {a \\left({b p - a q}\\right) \\left({a x + b}\\right)^2} + \\dfrac {q^2} {\\left({b p - a q}\\right)^2 \\left({p x + q}\\right)}$ == Proof == {{begin-eqn}} {{eqn | l = \\frac {x^2} {\\left({a x + b}\\right)^2 \\left({p x + q}\\right)}       | o = \\equiv       | r = \\frac A {a x + b} + \\frac B {\\left({a x + b}\\right)^2} + \\frac C {p x + q}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = x^2       | o = \\equiv       | r = A \\left({a x + b}\\right) \\left({p x + q}\\right) + B \\left({p x + q}\\right) + C \\left({a x + b}\\right)^2       | c = multiplying through by $\\left({a x + b}\\right)^2 \\left({p x + q}\\right)$ }} {{end-eqn}} Setting $a x + b = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = a x + b       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac b a       | c =  }} {{eqn | ll= \\implies       | l = B \\left({p \\left({-\\frac b a}\\right) + q}\\right)       | r = \\frac {b^2} {a^2}       | c = substituting for $x$ in $(1)$: term in $a x + b$ is $0$ }} {{eqn | ll= \\implies       | l = B       | r = \\frac {-b^2} {a \\left({b p - a q}\\right)}       | c =  }} {{end-eqn}} Setting $p x + q = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = p x + q       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = x       | r = -\\frac q p       | c =  }} {{eqn | ll= \\implies       | l = C \\left({a \\left({-\\frac q p}\\right) + b}\\right)^2       | r = \\frac {q^2} {p^2}       | c = substituting for $x$ in $(1)$: term in $a x + b$ is $0$ }} {{eqn | n = 2       | ll= \\implies       | l = C       | r = \\frac {q^2} {\\left({b p - a q}\\right)^2}       | c =  }} {{end-eqn}} Equating $2$nd powers of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 1       | r = A a p + C a^2       | c =  }} {{eqn | ll= \\implies       | l = A a p       | r = 1 - \\frac {q^2} {\\left({b p - a q}\\right)^2} a^2       | c = substituting for $C$ from $(2)$ }} {{eqn | ll= \\implies       | l = A       | r = \\frac 1 {a p} - \\frac {a q^2} {p \\left({b p - a q}\\right)^2}       | c =  }} {{eqn | ll= \\implies       | l = A       | r = \\frac {\\left({b p - a q}\\right)^2 - a^2 q^2} {a p \\left({b p - a q}\\right)^2}       | c = common denominator }} {{eqn | ll= \\implies       | l = A       | r = \\frac {b \\left({b p - 2 a q}\\right)} {a \\left({b p - a q}\\right)^2}       | c = simplifying }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac {b \\left({b p - 2 a q}\\right)} {a \\left({b p - a q}\\right)^2} }} {{eqn | l = B       | r = \\frac {-b^2} {a \\left({b p - a q}\\right)} }} {{eqn | l = C       | r = \\frac {q^2} {\\left({b p - a q}\\right)^2} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a x + b and p x + q/Lemmata 48bk2wpymjpu6yecpmawqt7e6w9pq00"}
+{"_id": "32487", "title": "Primitive of Reciprocal of x by x squared plus a squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x by x squared plus a squared/Partial Fraction Expansion 0 33949 454169 188334 2020-03-12T23:23:24Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x \\paren {x^2 + a^2}$ == :$\\dfrac 1 {x \\paren {x^2 + a^2} } \\equiv \\dfrac 1 {a^2 x} - \\dfrac x {a^2 \\paren {x^2 + a^2} }$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x \\paren {x^2 + a^2} }       | o = \\equiv       | r = \\dfrac A x + \\dfrac {B x + C} {x^2 + a^2}       | c =  }} {{eqn | n = 1       | ll= \\leadsto       | l = 1       | o = \\equiv       | r = A \\paren {x^2 + a^2} + B x^2 + C x       | c = multiplying through by $x \\paren {x^2 + a^2}$ }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = A a^2       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = A       | r = \\frac 1 {a^2}       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A + B       | c =  }} {{eqn | ll= \\leadsto       | l = B       | r = -\\frac 1 {a^2}       | c =  }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = C       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac 1 {a^2} }} {{eqn | l = B       | r = -\\frac 1 {a^2} }} {{eqn | l = C       | r = 0 }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving x squared plus a squared/Lemmata 4c3dqw2x7sri1z1phrcjhev095ulb8w"}
+{"_id": "32488", "title": "Primitive of Reciprocal of x squared by x squared plus a squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x squared by x squared plus a squared/Partial Fraction Expansion 0 33952 188332 2014-05-30T17:22:09Z Prime.mover 59 Created page with \"== Lemma for Primitive of Reciprocal of $x^2 \\left({x^2 + a^2}\\right)$ ==  :$\\dfrac 1 {x^2 \\left({x^2 + a^...\" wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x^2 \\left({x^2 + a^2}\\right)$ == :$\\dfrac 1 {x^2 \\left({x^2 + a^2}\\right)} \\equiv \\dfrac 1 {a^2 x^2} - \\dfrac 1 {a^2 \\left({x^2 + a^2}\\right)}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x^2 \\left({x^2 + a^2}\\right)}       | o = \\equiv       | r = \\dfrac A x + \\dfrac B {x^2} + \\dfrac {C x + D} {x^2 + a^2}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = 1       | o = \\equiv       | r = A x \\left({x^2 + a^2}\\right) + B \\left({x^2 + a^2}\\right) + C x^3 + D x^2       | c = multiplying through by $x^2 \\left({x^2 + a^2}\\right)$ }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = B a^2       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = B       | r = \\frac 1 {a^2}       | c =  }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A a^2       | c =  }} {{eqn | ll= \\implies       | l = A       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^3$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A + C       | c =  }} {{eqn | ll= \\implies       | l = C       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = B + D       | c =  }} {{eqn | ll= \\implies       | l = D       | r = -\\frac 1 {a^2}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = 0 }} {{eqn | l = B       | r = \\frac 1 {a^2} }} {{eqn | l = C       | r = 0 }} {{eqn | l = D       | r = -\\frac 1 {a^2} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving x squared plus a squared/Lemmata fvx9pp1l8t2jnv8imw5utyyvu26gvf6"}
+{"_id": "32489", "title": "Primitive of Reciprocal of x cubed by x squared plus a squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x cubed by x squared plus a squared/Partial Fraction Expansion 0 33954 391500 188336 2019-02-05T14:13:36Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x^3 \\paren {x^2 + a^2}$ == :$\\dfrac 1 {x^3 \\paren {x^2 + a^2} } \\equiv \\dfrac 1 {a^2 x^3} - \\dfrac 1 {a^4 x} + \\dfrac x {a^4 \\paren {x^2 + a^2} }$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x^3 \\paren {x^2 + a^2} }       | o = \\equiv       | r = \\dfrac A x + \\dfrac B {x^2} + \\dfrac C {x^3} + \\dfrac {D x + E} {x^2 + a^2}       | c =  }} {{eqn | n = 1       | ll= \\leadsto       | l = 1       | o = \\equiv       | r = A x^2 \\paren {x^2 + a^2} + B x \\paren {x^2 + a^2} + C \\paren {x^2 + a^2} + D x^4 + E x^3       | c = multiplying through by $x^3 \\paren {x^2 + a^2}$ }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = C a^2       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = C       | r = \\frac 1 {a^2}       | c =  }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = B a^2       | c =  }} {{eqn | ll= \\leadsto       | l = B       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A a^2 + C       | c =  }} {{eqn | ll= \\leadsto       | l = A       | r = -\\frac 1 {a^4}       | c =  }} {{end-eqn}} Equating coefficients of $x^3$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = B + E       | c =  }} {{eqn | ll= \\leadsto       | l = E       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^4$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A + D       | c =  }} {{eqn | ll= \\leadsto       | l = D       | r = \\frac 1 {a^4}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = -\\frac 1 {a^4} }} {{eqn | l = B       | r = 0 }} {{eqn | l = C       | r = \\frac 1 {a^2} }} {{eqn | l = D       | r = \\frac 1 {a^4} }} {{eqn | l = E       | r = 0 }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving x squared plus a squared/Lemmata 5wrkm4i3eax5r7zwncgbtftiu335t7m"}
+{"_id": "32490", "title": "Primitive of Reciprocal of x by x squared plus a squared squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x by x squared plus a squared squared/Partial Fraction Expansion 0 33960 188356 2014-05-31T06:41:14Z Prime.mover 59 Created page with \"== Lemma for Primitive of Reciprocal of $x \\left({x^2 + a^2}\\right)^2$ ==  :$\\dfrac 1 {x \\left({x^2 + a^2}...\" wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x \\left({x^2 + a^2}\\right)^2$ == :$\\dfrac 1 {x \\left({x^2 + a^2}\\right)^2} \\equiv \\dfrac 1 {a^4 x} - \\dfrac x {a^4 \\left({x^2 + a^2}\\right)} - \\dfrac x {a^2 \\left({x^2 + a^2}\\right)^2}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x \\left({x^2 + a^2}\\right)^2}       | o = \\equiv       | r = \\dfrac A x + \\dfrac {B x + C} {x^2 + a^2} + \\dfrac {D x + E} {\\left({x^2 + a^2}\\right)^2}       | c =  }} {{eqn | ll= \\implies       | l = 1       | o = \\equiv       | r = A \\left({x^2 + a^2}\\right)^2 + \\left({B x + C}\\right) x \\left({x^2 + a^2}\\right) + \\left({D x + E}\\right) x       | c = multiplying through by $x \\left({x^2 + a^2}\\right)^2$ }} {{eqn | n = 1       | o = \\equiv       | r = A x^4 + 2 A a^2 x^2 + A a^4 + B x^4 + B x^2 a^2 + C x^3 + C x a^2 + D x^2 + E x       | c = multiplying everything out }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = A a^4       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = A       | r = \\frac 1 {a^4}       | c =  }} {{end-eqn}} Equating coefficients of $x^4$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A + B       | c =  }} {{eqn | ll= \\implies       | l = B       | r = -\\frac 1 {a^4}       | c =  }} {{end-eqn}} Equating coefficients of $x^3$ in $(1)$: {{begin-eqn}} {{eqn | l = C       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = C + E       | c =  }} {{eqn | ll= \\implies       | l = E       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = 2 A a^2 + B a^2 + D       | c =  }} {{eqn | ll= \\implies       | l = D       | r = -\\frac 1 {a^2}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac 1 {a^4} }} {{eqn | l = B       | r = -\\frac 1 {a^4} }} {{eqn | l = C       | r = 0 }} {{eqn | l = D       | r = -\\frac 1 {a^2} }} {{eqn | l = E       | r = 0 }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving x squared plus a squared/Lemmata f4x9gicqomtsttw7s3vq6c7g1e6yhgq"}
+{"_id": "32491", "title": "Primitive of Reciprocal of x squared by x squared plus a squared squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x squared by x squared plus a squared squared/Partial Fraction Expansion 0 33962 437864 188358 2019-12-05T17:50:19Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x^2 \\paren {x^2 + a^2}^2$ == :$\\dfrac 1 {x^2 \\paren {x^2 + a^2}^2} \\equiv -\\dfrac 1 {a^4 x^2} - \\dfrac 1 {a^4 \\paren {x^2 + a^2} } - \\dfrac 1 {a^2 \\paren {x^2 + a^2}^2}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x^2 \\paren {x^2 + a^2}^2}       | o = \\equiv       | r = \\dfrac A x + \\dfrac B {x^2} + \\dfrac {C x + D} {x^2 + a^2} + \\dfrac {E x + F} {\\paren {x^2 + a^2}^2}       | c =  }} {{eqn | ll= \\leadsto       | l = 1       | o = \\equiv       | r = A x \\paren {x^2 + a^2}^2 + B \\paren {x^2 + a^2}^2       | c = multiplying through by $x^2 \\paren {x^2 + a^2}^2$ }} {{eqn | o =       | ro= +       | r = \\paren {C x + D} x^2 \\paren {x^2 + a^2} + \\paren {E x + F} x^2       | c =  }} {{eqn | n = 1       | o = \\equiv       | r = A x^5 + 2 A a^2 x^3 + A a^4 x + B x^4 + 2 B a^2 x^2 + B a^4       | c = multiplying everything out }} {{eqn | o =       | ro= +       | r = C x^5 + C x^3 a^2 + D x^4 + D x^2 a^2 + E x^3 + F x^2       | c =  }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = B a^4       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = B       | r = \\frac 1 {a^4}       | c =  }} {{end-eqn}} Equating coefficients of $x^4$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = B + D       | c =  }} {{eqn | ll= \\leadsto       | l = D       | r = -\\frac 1 {a^4}       | c =  }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = A       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^5$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A + C       | c =  }} {{eqn | ll= \\leadsto       | l = C       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^3$ in $(1)$: {{begin-eqn}} {{eqn | l = 2 A a^2 + C a^2 + E       | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = E       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = 2 B a^2 + D a^2 + F       | c =  }} {{eqn | ll= \\leadsto       | l = F       | r = -\\frac 1 {a^2}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = 0 }} {{eqn | l = B       | r = \\frac 1 {a^4} }} {{eqn | l = C       | r = 0 }} {{eqn | l = D       | r = -\\frac 1 {a^4} }} {{eqn | l = E       | r = 0 }} {{eqn | l = F       | r = -\\frac 1 {a^2} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving x squared plus a squared/Lemmata i0rxwsr3cu7dj83xtmmjzhct4fu84vg"}
+{"_id": "32492", "title": "Primitive of Reciprocal of x cubed by x squared plus a squared squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x cubed by x squared plus a squared squared/Partial Fraction Expansion 0 33964 188381 188367 2014-05-31T19:18:11Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x^3 \\left({x^2 + a^2}\\right)^2$ == :$\\dfrac 1 {x^3 \\left({x^2 + a^2}\\right)^2} \\equiv -\\dfrac 2 {a^6 x} + \\dfrac 1 {a^4 x^3} + \\dfrac {2 x} {a^6 \\left({x^2 + a^2}\\right)} + \\dfrac x {a^4 \\left({x^2 + a^2}\\right)^2}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x^3 \\left({x^2 + a^2}\\right)^2}       | o = \\equiv       | r = \\dfrac A x + \\dfrac B {x^2} + \\dfrac C {x^3} + \\dfrac {D x + E} {x^2 + a^2} + \\dfrac {F x + G} {\\left({x^2 + a^2}\\right)^2}       | c =  }} {{eqn | ll= \\implies       | l = 1       | o = \\equiv       | r = A x^2 \\left({x^2 + a^2}\\right)^2 + B x \\left({x^2 + a^2}\\right)^2 + C \\left({x^2 + a^2}\\right)^2       | c = multiplying through by $x^2 \\left({x^2 + a^2}\\right)^2$ }} {{eqn | o =       | ro= +       | r = \\left({D x + E}\\right) x^3 \\left({x^2 + a^2}\\right) + \\left({F x + G}\\right) x^3       | c =  }} {{eqn | n = 1       | o = \\equiv       | r = A x^6 + 2 A a^2 x^4 + A a^4 x^2 + B x^5 + 2 B a^2 x^3 + B a^4 x       | c = multiplying everything out }} {{eqn | o =       | ro= +       | r = C x^4 + 2 C a^2 x^2 + C a^4       | c =  }} {{eqn | o =       | ro= +       | r = D x^6 + D x^4 a^2 + E x^5 + E x^3 a^2 + F x^4 + G x^3       | c =  }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = C a^4       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = C       | r = \\frac 1 {a^4}       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A a^4 + 2 C a^2       | c =  }} {{eqn | ll= \\implies       | l = A       | r = -\\frac 2 {a^6}       | c =  }} {{end-eqn}} Equating coefficients of $x^6$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A + D       | c =  }} {{eqn | ll= \\implies       | l = D       | r = \\frac 2 {a^6}       | c =  }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = B       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^5$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = B + E       | c =  }} {{eqn | ll= \\implies       | l = E       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^3$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = 2 B a^2 + E a^2 + G       | c =  }} {{eqn | ll= \\implies       | l = G       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^4$ in $(1)$: {{begin-eqn}} {{eqn | l = 2 A a^2 + C + D a^2 + F       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = F       | r = \\frac 1 {a^4}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = -\\frac 2 {a^6} }} {{eqn | l = B       | r = 0 }} {{eqn | l = C       | r = \\frac 1 {a^4} }} {{eqn | l = D       | r = \\frac 2 {a^6} }} {{eqn | l = E       | r = 0 }} {{eqn | l = F       | r = \\frac 1 {a^4} }} {{eqn | l = G       | r = 0 }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving x squared plus a squared/Lemmata 6v2lbonc5pgtp69cwguk6924m9en808"}
+{"_id": "32493", "title": "Primitive of Reciprocal of x squared minus a squared/Logarithm Form/Proof 2/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x squared minus a squared/Logarithm Form/Proof 2/Partial Fraction Expansion 0 33971 188509 188406 2014-06-03T21:43:18Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $\\left({x^2 - a^2}\\right)$ == :$\\dfrac 1 {x^2 - a^2} \\equiv \\dfrac 1 {2 a \\left({x - a}\\right)} - \\dfrac 1 {2 a \\left({x + a}\\right)}$ == Proof == {{begin-eqn}} {{eqn | l = \\frac 1 {\\left({x^2 - a^2}\\right)}       | o = \\equiv       | r = \\frac A {x - a} + \\frac B {x + a}       | c = Difference of Two Squares }} {{eqn | n = 1       | ll= \\implies       | l = 1       | o = \\equiv       | r = A \\left({x + a}\\right) + B \\left({x - a}\\right)       | c = multiplying through by $\\left({x^2 - a^2}\\right)$ }} {{end-eqn}} Setting $x = a$ in $(1)$: {{begin-eqn}} {{eqn | l = A \\cdot 2 a + B \\cdot 0       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = A       | r = \\frac 1 {2 a}       | c =  }} {{end-eqn}} Setting $x = -a$ in $(1)$: {{begin-eqn}} {{eqn | l = A \\cdot 0 + B \\cdot \\left({-2 a}\\right)       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = B       | r = \\frac {-1} {2 a}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac 1 {2 a} }} {{eqn | l = B       | r = \\frac {-1} {2 a} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving x squared minus a squared/Lemmata na99tpvc4gybnvq4kqp4y4zcir52ex4"}
+{"_id": "32494", "title": "Primitive of Reciprocal of x by x squared minus a squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x by x squared minus a squared/Partial Fraction Expansion 0 33977 188429 188428 2014-06-01T20:23:54Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x \\left({x^2 - a^2}\\right)$ == :$\\dfrac 1 {x \\left({x^2 - a^2}\\right)} \\equiv \\dfrac {-1} {a^2 x} + \\dfrac x {a^2 \\left({x^2 - a^2}\\right)}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x \\left({x^2 - a^2}\\right)}       | o = \\equiv       | r = \\dfrac A x + \\dfrac {B x + C} {x^2 - a^2}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = 1       | o = \\equiv       | r = A \\left({x^2 - a^2}\\right) + B x^2 + C x       | c = multiplying through by $x \\left({x^2 - a^2}\\right)$ }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = -A a^2       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = A       | r = \\frac {-1} {a^2}       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A + B       | c =  }} {{eqn | ll= \\implies       | l = B       | r = \\frac 1 {a^2}       | c =  }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = C       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac {-1} {a^2} }} {{eqn | l = B       | r = \\frac 1 {a^2} }} {{eqn | l = C       | r = 0 }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving x squared minus a squared/Lemmata ns5f3xz1rqc7sjq55xthrwl7hjjtweb"}
+{"_id": "32495", "title": "Primitive of Reciprocal of x squared by x squared minus a squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x squared by x squared minus a squared/Partial Fraction Expansion 0 33979 188698 188431 2014-06-06T19:41:21Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x^2 \\left({x^2 - a^2}\\right)$ == :$\\dfrac 1 {x^2 \\left({x^2 - a^2}\\right)} \\equiv \\dfrac 1 {a^2 \\left({x^2 - a^2}\\right)} - \\dfrac 1 {a^2 x^2}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x^2 \\left({x^2 - a^2}\\right)}       | o = \\equiv       | r = \\dfrac A x + \\dfrac B {x^2} + \\dfrac {C x + D} {x^2 - a^2}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = 1       | o = \\equiv       | r = A x \\left({x^2 - a^2}\\right) + B \\left({x^2 - a^2}\\right) + C x^3 + D x^2       | c = multiplying through by $x^2 \\left({x^2 - a^2}\\right)$ }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = -B a^2       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = B       | r = -\\frac 1 {a^2}       | c =  }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = -A a^2       | c =  }} {{eqn | ll= \\implies       | l = A       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^3$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A + C       | c =  }} {{eqn | ll= \\implies       | l = C       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = B + D       | c =  }} {{eqn | ll= \\implies       | l = D       | r = \\frac 1 {a^2}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = 0 }} {{eqn | l = B       | r = \\frac {-1} {a^2} }} {{eqn | l = C       | r = 0 }} {{eqn | l = D       | r = \\frac 1 {a^2} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving x squared minus a squared/Lemmata 2ee6aeecv76b3a7by34d04guom9jqy6"}
+{"_id": "32496", "title": "Primitive of Reciprocal of x cubed by x squared minus a squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x cubed by x squared minus a squared/Partial Fraction Expansion 0 33981 188702 188434 2014-06-06T21:12:33Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x^3 \\left({x^2 - a^2}\\right)$ == :$\\dfrac 1 {x^3 \\left({x^2 - a^2}\\right)} \\equiv \\dfrac {-1} {a^2 x^3} - \\dfrac 1 {a^4 x} + \\dfrac x {a^4 \\left({x^2 - a^2}\\right)}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x^3 \\left({x^2 - a^2}\\right)}       | o = \\equiv       | r = \\dfrac A x + \\dfrac B {x^2} + \\dfrac C {x^3} + \\dfrac {D x + E} {x^2 - a^2}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = 1       | o = \\equiv       | r = A x^2 \\left({x^2 - a^2}\\right) + B x \\left({x^2 - a^2}\\right) + C \\left({x^2 - a^2}\\right) + D x^4 + E x^3       | c = multiplying through by $x^3 \\left({x^2 - a^2}\\right)$ }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = -C a^2       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = C       | r = \\frac {-1} {a^2}       | c =  }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = -B a^2       | c =  }} {{eqn | ll= \\implies       | l = B       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = -A a^2 + C       | c =  }} {{eqn | ll= \\implies       | l = A       | r = \\frac {-1} {a^4}       | c =  }} {{end-eqn}} Equating coefficients of $x^3$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = B + E       | c =  }} {{eqn | ll= \\implies       | l = E       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^4$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A + D       | c =  }} {{eqn | ll= \\implies       | l = D       | r = \\frac 1 {a^4}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac {-1} {a^4} }} {{eqn | l = B       | r = 0 }} {{eqn | l = C       | r = \\frac {-1} {a^2} }} {{eqn | l = D       | r = \\frac 1 {a^4} }} {{eqn | l = E       | r = 0 }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving x squared minus a squared/Lemmata c1gjx0v8dmhhjpcpuago13il6qci1py"}
+{"_id": "32497", "title": "Primitive of Reciprocal of x squared minus a squared squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x squared minus a squared squared/Partial Fraction Expansion 0 33983 188711 188708 2014-06-06T22:01:13Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $\\left({x^2 - a^2}\\right)^2$ == :$\\dfrac 1 {\\left({x^2 - a^2}\\right)^2} \\equiv \\dfrac 1 {4 a^3 \\left({x + a}\\right)} - \\dfrac 1 {4 a^3 \\left({x - a}\\right)} + \\dfrac 1 {4 a^2 \\left({x + a}\\right)^2} + \\dfrac 1 {4 a^2 \\left({x - a}\\right)^2}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {\\left({x^2 - a^2}\\right)^2}       | r = \\dfrac 1 {\\left({x + a}\\right)^2 \\left({x - a}\\right)^2}       | c = Difference of Two Squares }} {{eqn | o = \\equiv       | r = \\dfrac A {x + a} + \\dfrac B {\\left({x + a}\\right)^2} + \\dfrac C {x - a} + \\dfrac D {\\left({x - a}\\right)^2}       | c =  }} {{eqn | ll= \\implies       | l = 1       | o = \\equiv       | r = A \\left({x^2 - a^2}\\right) \\left({x - a}\\right) + B \\left({x - a}\\right)^2 + C \\left({x^2 - a^2}\\right) \\left({x + a}\\right) + D \\left({x + a}\\right)^2       | c = multiplying through by $\\left({x^2 - a^2}\\right)^2$ }} {{eqn | n = 1       | o = \\equiv       | r = A x^3 - A a x^2 - A a^2 x + A a^3 + B x^2 - 2 B a x + B a^2       | c = multiplying out }} {{eqn | o =        | ro= +       | r = C x^3 + C a x^2 - C a^2 x - C a^3 + D x^2 + 2 D a x + D a^2       | c =  }} {{end-eqn}} Setting $x = a$ in $(1)$: {{begin-eqn}} {{eqn | l = D \\left({2 a}\\right)^2       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = D       | r = \\frac 1 {4 a^2}       | c =  }} {{end-eqn}} Setting $x = -a$ in $(1)$: {{begin-eqn}} {{eqn | l = B \\left({-2 a}\\right)^2       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = B       | r = \\frac 1 {4 a^2}       | c =  }} {{end-eqn}} Equating coefficients of $x^3$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A + C       | c =  }} {{eqn | ll= \\implies       | l = A       | r = -C       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = - A a + C a + B + D       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = - A a + C a + \\frac 1 {4 a^2} + \\frac 1 {4 a^2}       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = C a + C a + \\frac 1 {2 a^2}       | r = 0       | c = as $A = - C$ }} {{eqn | ll= \\implies       | l = 2 C        | r = \\frac {-1} {2 a^3}       | c =  }} {{eqn | ll= \\implies       | l = C        | r = \\frac {-1} {4 a^3}       | c =  }} {{eqn | ll= \\implies       | l = A        | r = \\frac 1 {4 a^3}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac 1 {4 a^3} }} {{eqn | l = B       | r = \\frac 1 {4 a^2} }} {{eqn | l = C       | r = \\frac {-1} {4 a^3} }} {{eqn | l = D       | r = \\frac 1 {4 a^2} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving x squared minus a squared/Lemmata nr708mewttsrky3erlm3hpj1w4hjf3i"}
+{"_id": "32498", "title": "Primitive of Reciprocal of x by x squared minus a squared squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x by x squared minus a squared squared/Partial Fraction Expansion 0 34003 429199 188727 2019-10-02T12:57:03Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x \\paren {x^2 - a^2}^2$ == :$\\dfrac 1 {x \\paren {x^2 - a^2}^2} \\equiv \\dfrac 1 {a^4 x} + \\dfrac {-x} {a^4 \\paren {x^2 - a^2} } + \\dfrac x {a^2 \\paren {x^2 - a^2}^2}$ == Proof == {{begin-eqn}} {{eqn | l = \\frac 1 {x \\paren {x^2 - a^2}^2}       | r = \\frac 1 {x \\paren {x + a}^2 \\paren {x - a}^2}       | c = Difference of Two Squares }} {{eqn | o = \\equiv       | r = \\frac A {x + a} + \\frac B {\\paren {x + a}^2} + \\frac C {x - a} + \\frac D {\\paren {x - a}^2} + \\frac E x       | c =  }} {{eqn | ll= \\leadsto       | l = 1       | o = \\equiv       | r = A x \\paren {x^2 - a^2} \\paren {x - a} + B x \\paren {x - a}^2       | c = multiplying through by $x \\paren {x^2 - a^2}^2$ }} {{eqn | o =        | ro= +       | r = C x \\paren {x^2 - a^2} \\paren {x + a} + D x \\paren {x + a}^2 + E \\paren {x^2 - a^2}^2       | c =  }} {{eqn | n = 1       | o = \\equiv       | r = A x^4 - A a x^3 - A a^2 x^2 + A a^3 x + B x^3 - 2 B a x^2 + B a^2 x       | c = multiplying out }} {{eqn | o =        | ro= +       | r = C x^4 + C a x^3 - C a^2 x^2 - C a^3 x + D x^3 + 2 D a x^2 + D a^2 x       | c =  }} {{eqn | o =        | ro= +       | r = E x^4 - 2 E a^2 x^2 + E a^4       | c =  }} {{end-eqn}} Setting $x = a$ in $(1)$: {{begin-eqn}} {{eqn | l = D a \\paren {2 a}^2       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = D       | r = \\frac 1 {4 a^3}       | c =  }} {{end-eqn}} Setting $x = -a$ in $(1)$: {{begin-eqn}} {{eqn | l = B \\paren {-a} \\paren {-2 a}^2       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = B       | r = \\frac {-1} {4 a^3}       | c =  }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = E a^4       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = E       | r = \\frac 1 {a^4}       | c =  }} {{end-eqn}} Equating coefficients of $x^4$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A + C + E       | c =  }} {{eqn | ll= \\leadsto       | r = C       | l = -A - \\frac 1 {a^4}       | c =  }} {{end-eqn}} Equating coefficients of $x^3$ in $(1)$: {{begin-eqn}} {{eqn | l = - A a + C a + B + D       | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = - A a + C a + \\frac 1 {4 a^3} + \\frac {-1} {4 a^3}       | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = -A a + \\paren {-A - \\frac 1 {a^4} } a       | r = 0       | c = as $C = -A  -\\dfrac 1 {a^4}$ }} {{eqn | ll= \\leadsto       | l = -2 A - \\frac 1 {a^4}       | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = A        | r = \\frac {-1} {2 a^4}       | c =  }} {{eqn | ll= \\leadsto       | l = C        | r = \\frac {-1} {2 a^4}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac {-1} {2 a^4} }} {{eqn | l = B       | r = \\frac {-1} {4 a^3} }} {{eqn | l = C       | r = \\frac {-1} {2 a^4} }} {{eqn | l = D       | r = \\frac 1 {4 a^3} }} {{eqn | l = E       | r = \\frac 1 {a^4} }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | l = \\frac 1 {x \\paren {x^2 - a^2}^2}       | o = \\equiv       | r = \\frac {-1} {2 a^4 \\paren {x + a} } + \\frac {-1} {4 a^3 \\paren {x + a}^2} + \\frac {-1} {2 a^4 \\paren {x - a} } + \\frac 1 {4 a^3 \\paren {x - a}^2} + \\frac 1 {a^4 x}       | c =  }} {{eqn | o = \\equiv       | r = - \\frac {\\paren {x - a} + \\paren {x + a} } {2 a^4 \\paren {x + a} \\paren {x - a} } + \\frac {\\paren {x + a}^2 - \\paren {x - a}^2} {4 a^3 \\paren {x + a}^2 \\paren {x - a}^2} + \\frac 1 {a^4 x}       | c = common denominators }} {{eqn | o = \\equiv       | r = \\frac {- 2 x} {2 a^4 \\paren {x^2 - a^2} } + \\frac {\\paren {x^2 + 2 a x + a^2} - \\paren {x^2 - 2 a x + a^2} } {4 a^3 \\paren {x^2 - a^2}^2} + \\frac 1 {a^4 x}       | c = simplifying }} {{eqn | o = \\equiv       | r = \\frac {-x} {a^4 \\paren {x^2 - a^2} } + \\frac {4 a x} {4 a^3 \\paren {x^2 - a^2}^2} + \\frac 1 {a^4 x}       | c = simplifying }} {{eqn | o = \\equiv       | r = \\dfrac 1 {a^4 x} + \\dfrac {-x} {a^4 \\paren {x^2 - a^2} } + \\dfrac x {a^2 \\paren {x^2 - a^2}^2}       | c = simplifying and rearranging }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving x squared minus a squared/Lemmata aeeicmwtpjlf71202sglewj9w43ev11"}
+{"_id": "32499", "title": "Primitive of Reciprocal of x squared by x squared minus a squared squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x squared by x squared minus a squared squared/Partial Fraction Expansion 0 34005 188492 2014-06-03T15:13:21Z Prime.mover 59 Created page with \"== Lemma for Primitive of Reciprocal of $x^2 \\left({x^2 - a^2}\\right)^2$ ==  :$\\dfrac 1 {x^2 \\lef...\" wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x^2 \\left({x^2 - a^2}\\right)^2$ == :$\\dfrac 1 {x^2 \\left({x^2 - a^2}\\right)^2} \\equiv \\dfrac 1 {a^4 x^2} + \\dfrac 3 {4 a^5 \\left({x + a}\\right)} - \\dfrac 3 {4 a^5 \\left({x - a}\\right)} + \\dfrac 1 {4 a^4 \\left({x + a}\\right)^2} + \\dfrac 1 {4 a^4 \\left({x - a}\\right)^2}$ == Proof == {{begin-eqn}} {{eqn | l = \\frac 1 {x^2 \\left({x^2 - a^2}\\right)^2}       | r = \\frac 1 {x^2 \\left({x + a}\\right)^2 \\left({x - a}\\right)^2}       | c = Difference of Two Squares }} {{eqn | o = \\equiv       | r = \\frac A {x + a} + \\frac B {\\left({x + a}\\right)^2} + \\frac C {x - a} + \\frac D {\\left({x - a}\\right)^2} + \\frac E x + \\frac F {x^2}       | c =  }} {{eqn | ll= \\implies       | l = 1       | o = \\equiv       | r = A x^2 \\left({x^2 - a^2}\\right) \\left({x - a}\\right) + B x^2 \\left({x - a}\\right)^2       | c = multiplying through by $x^2 \\left({x^2 - a^2}\\right)^2$ }} {{eqn | o =        | ro= +       | r = C x^2 \\left({x^2 - a^2}\\right) \\left({x + a}\\right) + D x^2 \\left({x + a}\\right)^2       | c =  }} {{eqn | o =        | ro= +       | r = E x \\left({x^2 - a^2}\\right)^2 + F \\left({x^2 - a^2}\\right)^2       | c =  }} {{eqn | n = 1       | o = \\equiv       | r = A x^5 - A a x^4 - A a^2 x^3 + A a^3 x^2 + B x^4 - 2 B a x^3 + B a^2 x^2       | c = multiplying out }} {{eqn | o =        | ro= +       | r = C x^5 + C a x^4 - C a^2 x^3 - C a^3 x^2 + D x^4 + 2 D a x^3 + D a^2 x^2       | c =  }} {{eqn | o =        | ro= +       | r = E x^5 - 2 E a^2 x^3 + E a^4 x + F x^4 - 2 F a^2 x^2 + F a^4       | c =  }} {{end-eqn}} Setting $x = a$ in $(1)$: {{begin-eqn}} {{eqn | l = D a^2 \\left({2 a}\\right)^2       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = D       | r = \\frac 1 {4 a^4}       | c =  }} {{end-eqn}} Setting $x = -a$ in $(1)$: {{begin-eqn}} {{eqn | l = B \\left({-a}\\right)^2 \\left({-2 a}\\right)^2       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = B       | r = \\frac 1 {4 a^4}       | c =  }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = F a^4       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = F       | r = \\frac 1 {a^4}       | c =  }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = E       | c =  }} {{end-eqn}} Equating coefficients of $x^5$ in $(1)$: {{begin-eqn}} {{eqn | n = 2       | l = 0       | r = A + C + E       | c =  }} {{eqn | ll= \\implies       | l = 0       | r = A + C       | c = as $E = 0$ }} {{eqn | ll= \\implies       | l = A       | r = - C       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = A a^3 + B a^2 - C a^3 + D a^2 - 2 F a^2       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = A a^3 + \\frac 1 {4 a^4} a^2 - C a^3 + \\frac 1 {4 a^4} a^2 - 2 \\frac 1 {a^4} a^2       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = A + \\frac 1 {4 a^5} - C + \\frac 1 {4 a^5} - \\frac 8 {4 a^5}       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = A - C       | r = \\frac {8 - 1 - 1} {4 a^5}       | c =  }} {{eqn | r = \\frac 3 {2 a^5}       | c =  }} {{eqn | ll= \\implies       | l = A       | r = \\frac 3 {4 a^5}       | c =  }} {{eqn | ll= \\implies       | l = C       | r = \\frac {-3} {4 a^5}       | c = as $A = - C$ }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac 3 {4 a^5} }} {{eqn | l = B       | r = \\frac 1 {4 a^4} }} {{eqn | l = C       | r = \\frac {-3} {4 a^5} }} {{eqn | l = D       | r = \\frac 1 {4 a^4} }} {{eqn | l = E       | r = 0 }} {{eqn | l = F       | r = \\frac 1 {a^4} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving x squared minus a squared/Lemmata 6kkxsyez5vos4oeben020jniion3cbp"}
+{"_id": "32500", "title": "Primitive of Reciprocal of x cubed by x squared minus a squared squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x cubed by x squared minus a squared squared/Partial Fraction Expansion 0 34006 442084 188493 2020-01-02T18:07:27Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x^3 \\paren {x^2 - a^2}^2$ == :$\\dfrac 1 {x^3 \\paren {x^2 - a^2}^2} \\equiv \\dfrac 1 {a^4 x^3} + \\dfrac 2 {a^6 x} - \\dfrac {2 x} {a^6 \\paren {x^2 - a^2} } + \\dfrac x {a^4 \\paren {x^2 - a^2}^2}$ == Proof == {{begin-eqn}} {{eqn | l = \\frac 1 {x^3 \\paren {x^2 - a^2}^2}       | r = \\frac 1 {x^3 \\paren {x + a}^2 \\paren {x - a}^2}       | c = Difference of Two Squares }} {{eqn | o = \\equiv       | r = \\frac A {x + a} + \\frac B {\\paren {x + a}^2} + \\frac C {x - a} + \\frac D {\\paren {x - a}^2} + \\frac E x + \\frac F {x^2} + \\frac G {x^3}       | c =  }} {{eqn | ll= \\leadsto       | l = 1       | o = \\equiv       | r = A x^3 \\paren {x^2 - a^2} \\paren {x - a} + B x^3 \\paren {x - a}^2       | c = multiplying through by $x^3 \\paren {x^2 - a^2}^2$ }} {{eqn | o =        | ro= +       | r = C x^3 \\paren {x^2 - a^2} \\paren {x + a} + D x^3 \\paren {x + a}^2       | c =  }} {{eqn | o =        | ro= +       | r = E x^2 \\paren {x^2 - a^2}^2 + F x \\paren {x^2 - a^2}^2 + G \\paren {x^2 - a^2}^2       | c =  }} {{eqn | n = 1       | o = \\equiv       | r = A x^6 - A a x^5 - A a^2 x^4 + A a^3 x^3 + B x^5 - 2 B a x^4 + B a^2 x^3       | c = multiplying out }} {{eqn | o =        | ro= +       | r = C x^6 + C a x^5 - C a^2 x^4 - C a^3 x^3 + D x^5 + 2 D a x^4 + D a^2 x^3       | c =  }} {{eqn | o =        | ro= +       | r = E x^6 - 2 E a^2 x^4 + E a^4 x^2 + F x^5 - 2 F a^2 x^3 + F x a^4       | c =  }} {{eqn | o =        | ro= +       | r = G x^4 - 2 G a^2 x^2 + G a^4       | c =  }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = G a^4       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = G       | r = \\frac 1 {a^4}       | c =  }} {{end-eqn}} Setting $x = a$ in $(1)$: {{begin-eqn}} {{eqn | l = D a^3 \\paren {2 a}^2       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = D       | r = \\frac 1 {4 a^5}       | c =  }} {{end-eqn}} Setting $x = -a$ in $(1)$: {{begin-eqn}} {{eqn | l = B \\paren {-a}^3 \\paren {-2 a}^2       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = B       | r = \\frac {-1} {4 a^5}       | c =  }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = F       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = E a^4 - 2 G a^2        | c =  }} {{eqn | ll= \\leadsto       | l = E a^4       | r = 2 \\frac 1 {a^4} a^2        | c =  }} {{eqn | ll= \\leadsto       | l = E       | r = \\frac 2 {a^6}       | c =  }} {{end-eqn}} Equating coefficients of $x^6$ in $(1)$: {{begin-eqn}} {{eqn | n = 2       | l = 0       | r = A + C + E       | c =  }} {{eqn | ll= \\leadsto       | l = 0       | r = A + C + \\frac 2 {a^6}       | c =  }} {{eqn | ll= \\leadsto       | l = A + C       | r = -\\frac 2 {a^6}       | c =  }} {{end-eqn}} Equating coefficients of $x^3$ in $(1)$: {{begin-eqn}} {{eqn | l = A a^3 + B a^2 - C a^3 + D a^2 - 2 F a^2       | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = A a^3 + \\frac {-1} {4 a^5} a^2 - C a^3 + \\frac 1 {4 a^5} a^2       | r = 0       | c = substituting for $B$, $D$ and $F$ }} {{eqn | ll= \\leadsto       | l = A - C       | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = A       | r = C       | c =  }} {{eqn | r = \\frac {-1} {a^6}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac {-1} {a^6} }} {{eqn | l = B       | r = \\frac {-1} {4 a^5} }} {{eqn | l = C       | r = \\frac {-1} {a^6} }} {{eqn | l = D       | r = \\frac 1 {4 a^5} }} {{eqn | l = E       | r = \\frac 2 {a^6} }} {{eqn | l = F       | r = 0 }} {{eqn | l = G       | r = \\frac 1 {a^4} }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | l = \\frac 1 {x^3 \\paren {x^2 - a^2}^2}       | o = \\equiv       | r = \\dfrac 1 {a^4 x^3} + \\dfrac 2 {a^6 x} - \\dfrac 1 {a^6 \\paren {x + a} } - \\dfrac 1 {a^6 \\paren {x - a} } - \\dfrac 1 {4 a^5 \\paren {x + a}^2} + \\dfrac 1 {4 a^5 \\paren {x - a}^2}       | c =  }} {{eqn | o = \\equiv       | r = -\\frac {\\paren {x - a} + \\paren {x + a} } {a^6 \\paren {x + a} \\paren {x - a} } + \\frac {\\paren {x + a}^2 - \\paren {x - a}^2} {4 a^5 \\paren {x + a}^2 \\paren {x - a}^2} + \\dfrac 1 {a^4 x^3} + \\dfrac 2 {a^6 x}       | c = common denominators }} {{eqn | o = \\equiv       | r = \\frac {- 2 x} {a^6 \\paren {x^2 - a^2} } + \\frac {\\paren {x^2 + 2 a x + a^2} - \\paren {x^2 - 2 a x + a^2} } {4 a^5 \\paren {x^2 - a^2}^2} + \\dfrac 1 {a^4 x^3} + \\dfrac 2 {a^6 x}       | c = simplifying }} {{eqn | o = \\equiv       | r = \\frac {-2 x} {a^6 \\paren {x^2 - a^2} } + \\frac {4 a x} {4 a^5 \\paren {x^2 - a^2}^2} + \\frac 1 {a^4 x^3} + \\frac 2 {a^6 x}       | c = simplifying }} {{eqn | o = \\equiv       | r = \\frac 1 {a^4 x^3} + \\frac 2 {a^6 x} - \\frac {2 x} {a^6 \\paren {x^2 - a^2} } + \\frac x {a^4 \\paren {x^2 - a^2}^2}       | c = simplifying and rearranging }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving x squared minus a squared/Lemmata qozn8wt7hcbtz4e5sddnxwxqvrri9r2"}
+{"_id": "32501", "title": "Primitive of Reciprocal of a squared minus x squared/Logarithm Form/Partial Fraction Expansion", "text": "Primitive of Reciprocal of a squared minus x squared/Logarithm Form/Partial Fraction Expansion 0 34015 491278 491233 2020-09-27T20:40:40Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of $\\dfrac 1 {a^2 - x^2}$ == :$\\dfrac 1 {a^2 - x^2} \\equiv \\dfrac 1 {2 a \\paren {a + x} } + \\dfrac 1 {2 a \\paren {a - x} }$ == Proof == {{begin-eqn}} {{eqn | l = \\frac 1 {\\paren {a^2 - x^2} }       | o = \\equiv       | r = \\frac A {a - x} + \\frac B {a + x}       | c = Difference of Two Squares }} {{eqn | n = 1       | ll= \\leadsto       | l = 1       | o = \\equiv       | r = A \\paren {a + x} + B \\paren {a - x}       | c = multiplying through by $\\paren {a^2 - x^2}$ }} {{end-eqn}} Setting $x = a$ in $(1)$: {{begin-eqn}} {{eqn | l = A \\cdot 2 a + B \\cdot 0       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = A       | r = \\frac 1 {2 a}       | c =  }} {{end-eqn}} Setting $x = -a$ in $(1)$: {{begin-eqn}} {{eqn | l = A \\cdot 0 + B \\cdot \\paren {2 a}       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = B       | r = \\frac 1 {2 a}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac 1 {2 a} }} {{eqn | l = B       | r = \\frac 1 {2 a} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a squared minus x squared/Lemmata d534fuov29yxzvvm6akh93eluv35zge"}
+{"_id": "32502", "title": "Primitive of Reciprocal/Corollary", "text": "Primitive of Reciprocal/Corollary 0 34019 407574 220559 2019-06-12T19:52:11Z Prime.mover 59 wikitext text/x-wiki == Corollary to Primitive of Reciprocal == <onlyinclude> :$\\displaystyle \\int \\frac {\\d x} x = \\ln x + C$ for $x > 0$. </onlyinclude> == Proof == From Primitive of Reciprocal: :$\\displaystyle \\int \\frac {\\d x} x = \\ln \\size x + C$ for $x \\ne 0$. By definition of absolute value: :$\\forall x \\in \\R_{>0}: \\size x = x$ Hence the result. {{qed}} Category:Primitive of Reciprocal 2jrck878i0wl0as8k9diho1rzp3s0ai"}
+{"_id": "32503", "title": "Primitive of Reciprocal of x by a squared minus x squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x by a squared minus x squared/Partial Fraction Expansion 0 34052 188687 2014-06-06T18:57:30Z Prime.mover 59 Created page with \"== Lemma for Primitive of Reciprocal of $x \\left({a^2 - x^2}\\right)$ ==  :$\\dfrac 1 {x \\left({a^2 - x^2}\\right)} ...\" wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x \\left({a^2 - x^2}\\right)$ == :$\\dfrac 1 {x \\left({a^2 - x^2}\\right)} \\equiv \\dfrac 1 {a^2 x} + \\dfrac x {a^2 \\left({a^2 - x^2}\\right)}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x \\left({a^2 - x^2}\\right)}       | o = \\equiv       | r = \\dfrac A x + \\dfrac {B x + C} {a^2 - x^2}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = 1       | o = \\equiv       | r = A \\left({a^2 - x^2}\\right) + B x^2 + C x       | c = multiplying through by $x \\left({a^2 - x^2}\\right)$ }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = A a^2       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = A       | r = \\frac 1 {a^2}       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = -A + B       | c =  }} {{eqn | ll= \\implies       | l = B       | r = \\frac 1 {a^2}       | c =  }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = C       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac 1 {a^2} }} {{eqn | l = B       | r = \\frac 1 {a^2} }} {{eqn | l = C       | r = 0 }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a squared minus x squared/Lemmata svznimzdo5r7p2mkavnrwderha3qx8k"}
+{"_id": "32504", "title": "Primitive of Reciprocal of x by a squared minus x squared", "text": "Primitive of Reciprocal of x by a squared minus x squared 0 34053 188690 2014-06-06T19:03:04Z Prime.mover 59 Created page with \"<onlyinclude> :$\\displaystyle \\int \\frac {\\mathrm d x} {x \\left({a^2 - x^2}\\right)} = \\frac 1 {2 a^2} \\ln \\left({\\frac {x^2} {a^2 - x^2} }\\right) + C$ for $x^2 < a^2$. </onlyi...\" wikitext text/x-wiki <onlyinclude> :$\\displaystyle \\int \\frac {\\mathrm d x} {x \\left({a^2 - x^2}\\right)} = \\frac 1 {2 a^2} \\ln \\left({\\frac {x^2} {a^2 - x^2} }\\right) + C$ for $x^2 < a^2$. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\int \\frac {\\mathrm d x} {x \\left({a^2 - x^2}\\right)}       | r = \\int \\left({\\frac 1 {a^2 x} + \\frac x {a^2 \\left({a^2 - x^2}\\right)} }\\right) \\ \\mathrm d x       | c = Partial Fraction Expansion }} {{eqn | r = \\frac 1 {a^2} \\int \\frac {\\mathrm d x} x + \\frac 1 {a^2} \\int \\frac {x \\ \\mathrm d x} {a^2 - x^2}       | c = Linear Combination of Integrals }} {{eqn | r = \\frac 1 {a^2} \\ln \\left\\vert{x}\\right\\vert + \\frac 1 {a^2} \\int \\frac {x \\ \\mathrm d x} {a^2 - x^2} + C       | c = Primitive of Reciprocal }} {{eqn | r = \\frac 1 {a^2} \\ln \\left\\vert{x}\\right\\vert + \\frac 1 {a^2} \\left({-\\frac 1 2 \\ln \\left ({a^2 - x^2}\\right) }\\right) + C       | c = Primitive of $\\dfrac x {a^2 - x^2}$ }} {{eqn | r = \\frac 1 {2 a^2} \\ln \\left\\vert{x^2}\\right\\vert - \\frac 1 {2 a^2} \\ln \\left ({a^2 - x^2}\\right) + C       | c = Logarithm of Power }} {{eqn | r = \\frac 1 {2 a^2} \\ln \\left({x^2}\\right) - \\frac 1 {2 a^2} \\ln \\left ({a^2 - x^2}\\right) + C       | c = as $x^2 > 0$ }} {{eqn | r = \\frac 1 {2 a^2} \\ln \\left({\\frac {x^2} {x^2 - a^2} }\\right) + C       | c = Difference of Logarithms }} {{end-eqn}} {{qed}} == Also see == * Primitive of $\\dfrac 1 {x \\left({x^2 - a^2}\\right)}$ == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev=Primitive of x cubed over a squared minus x squared|next=Primitive of Reciprocal of x squared by a squared minus x squared}}: $\\S 14$: Integrals involving $a^2 - x^2$, $x^2 < a^2$: $14.167$ Category:Primitives involving a squared minus x squared oknobe5myf7a8jvth8l65ngmd1nmvvg"}
+{"_id": "32505", "title": "Primitive of Reciprocal of x squared by a squared minus x squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x squared by a squared minus x squared/Partial Fraction Expansion 0 34054 188699 2014-06-06T19:42:33Z Prime.mover 59 Created page with \"== Lemma for Primitive of Reciprocal of $x^2 \\left({a^2 - x^2}\\right)$ ==  :$\\dfrac 1 {x^2 \\left({a^2 - x...\" wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x^2 \\left({a^2 - x^2}\\right)$ == :$\\dfrac 1 {x^2 \\left({a^2 - x^2}\\right)} \\equiv \\dfrac 1 {a^2 \\left({a^2 - x^2}\\right)} + \\dfrac 1 {a^2 x^2}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {x^2 \\left({a^2 - x^2}\\right)}       | o = \\equiv       | r = \\dfrac A x + \\dfrac B {x^2} + \\dfrac {C x + D} {a^2 - x^2}       | c =  }} {{eqn | n = 1       | ll= \\implies       | l = 1       | o = \\equiv       | r = A x \\left({a^2 - x^2}\\right) + B \\left({a^2 - x^2}\\right) + C x^3 + D x^2       | c = multiplying through by $x^2 \\left({a^2 - x^2}\\right)$ }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = B a^2       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = B       | r = \\frac 1 {a^2}       | c =  }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A a^2       | c =  }} {{eqn | ll= \\implies       | l = A       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^3$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = -A + C       | c =  }} {{eqn | ll= \\implies       | l = C       | r = 0       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = -B + D       | c =  }} {{eqn | ll= \\implies       | l = D       | r = \\frac 1 {a^2}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = 0 }} {{eqn | l = B       | r = \\frac 1 {a^2} }} {{eqn | l = C       | r = 0 }} {{eqn | l = D       | r = \\frac 1 {a^2} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a squared minus x squared/Lemmata 8i2q2a06gzxgi1c2b8y5yr13jl8leog"}
+{"_id": "32507", "title": "Primitive of Reciprocal of a squared minus x squared squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of a squared minus x squared squared/Partial Fraction Expansion 0 34059 188714 188712 2014-06-06T22:11:34Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $\\left({a^2 - x^2}\\right)^2$ == :$\\dfrac 1 {\\left({a^2 - x^2}\\right)^2} \\equiv \\dfrac 1 {4 a^3 \\left({a + x}\\right)} + \\dfrac 1 {4 a^3 \\left({a - x}\\right)} + \\dfrac 1 {4 a^2 \\left({a + x}\\right)^2} + \\dfrac 1 {4 a^2 \\left({a - x}\\right)^2}$ == Proof == {{begin-eqn}} {{eqn | l = \\dfrac 1 {\\left({a^2 - x^2}\\right)^2}       | r = \\dfrac 1 {\\left({a + x}\\right)^2 \\left({a - x}\\right)^2}       | c = Difference of Two Squares }} {{eqn | o = \\equiv       | r = \\dfrac A {a + x} + \\dfrac B {\\left({a + x}\\right)^2} + \\dfrac C {a - x} + \\dfrac D {\\left({a - x}\\right)^2}       | c =  }} {{eqn | ll= \\implies       | l = 1       | o = \\equiv       | r = A \\left({a^2 - x^2}\\right) \\left({a - x}\\right) + B \\left({a - x}\\right)^2 + C \\left({a^2 - x^2}\\right) \\left({a + x}\\right) + D \\left({a + x}\\right)^2       | c = multiplying through by $\\left({a^2 - x^2}\\right)^2$ }} {{eqn | n = 1       | o = \\equiv       | r = A a^3 - A a^2 x - A a x^2 + A x^3 + B a^2 - 2 B a x + B x^2       | c = multiplying out }} {{eqn | o =        | ro= +       | r = C a^3 + C a^2 x - C a x^2 - C x^3 + D a^2 + 2 D a x + D x^2       | c =  }} {{end-eqn}} Setting $x = a$ in $(1)$: {{begin-eqn}} {{eqn | l = D \\left({2 a}\\right)^2       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = D       | r = \\frac 1 {4 a^2}       | c =  }} {{end-eqn}} Setting $x = -a$ in $(1)$: {{begin-eqn}} {{eqn | l = B \\left({2 a}\\right)^2       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = B       | r = \\frac 1 {4 a^2}       | c =  }} {{end-eqn}} Equating coefficients of $x^3$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A - C       | c =  }} {{eqn | ll= \\implies       | l = A       | r = C       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = - A a - C a + B + D       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = - A a - C a + \\frac 1 {4 a^2} + \\frac 1 {4 a^2}       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = -C a - C a + \\frac 1 {2 a^2}       | r = 0       | c = as $A = C$ }} {{eqn | ll= \\implies       | l = 2 C        | r = \\frac 1 {2 a^3}       | c =  }} {{eqn | ll= \\implies       | l = C        | r = \\frac 1 {4 a^3}       | c =  }} {{eqn | ll= \\implies       | l = A        | r = \\frac 1 {4 a^3}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac 1 {4 a^3} }} {{eqn | l = B       | r = \\frac 1 {4 a^2} }} {{eqn | l = C       | r = \\frac 1 {4 a^3} }} {{eqn | l = D       | r = \\frac 1 {4 a^2} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a squared minus x squared/Lemmata trze76y5ilafa2hh7z3bpu3so6s4wb9"}
+{"_id": "32508", "title": "Primitive of Reciprocal of a squared minus x squared squared", "text": "Primitive of Reciprocal of a squared minus x squared squared 0 34060 188713 2014-06-06T22:10:40Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> :$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({a^2 - x^2}\\right)^2} = \\frac x {2 a^2 \\left({a^2 - x^2}\\right)} + \\frac 1 {4 a^3} \\ln \\left({\\frac...\" wikitext text/x-wiki == Theorem == <onlyinclude> :$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({a^2 - x^2}\\right)^2} = \\frac x {2 a^2 \\left({a^2 - x^2}\\right)} + \\frac 1 {4 a^3} \\ln \\left({\\frac {a + x} {a - x} }\\right) + C$ for $x^2 < a^2$. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\int \\frac {\\mathrm d x} {\\left({a^2 - x^2}\\right)^2}       | r = \\int \\left({\\frac 1 {4 a^3 \\left({a + x}\\right)} + \\frac 1 {4 a^3 \\left({a - x}\\right)} + \\frac 1 {4 a^2 \\left({a + x}\\right)^2} + \\frac 1 {4 a^2 \\left({a - x}\\right)^2} }\\right) \\ \\mathrm d x       | c = Partial Fraction Expansion }} {{eqn | r = \\frac 1 {4 a^3} \\int \\frac {\\mathrm d x} {a + x} + \\frac 1 {4 a^3} \\int \\frac {\\mathrm d x} {a - x} + \\frac 1 {4 a^2} \\int \\frac {\\mathrm d x} {\\left({a + x}\\right)^2} + \\frac 1 {4 a^2} \\int \\frac {\\mathrm d x} {\\left({a - x}\\right)^2}       | c = Linear Combination of Integrals }} {{eqn | r = \\frac 1 {4 a^3} \\ln \\left\\vert{a + x}\\right\\vert - \\frac 1 {4 a^3} \\ln \\left\\vert{a - x}\\right\\vert + \\frac 1 {4 a^2} \\int \\frac {\\mathrm d x} {\\left({a + x}\\right)^2} + \\frac 1 {4 a^2} \\int \\frac {\\mathrm d x} {\\left({a - x}\\right)^2} + C       | c = Primitive of Function of $a x + b$ and Primitive of Reciprocal }} {{eqn | r = \\frac 1 {4 a^3} \\ln \\left\\vert{\\frac {a + x} {a - x} }\\right\\vert + \\frac 1 {4 a^2} \\int \\frac {\\mathrm d x} {\\left({a + x}\\right)^2} + \\frac 1 {4 a^2} \\int \\frac {\\mathrm d x} {\\left({a - x}\\right)^2} + C       | c = Difference of Logarithms }} {{eqn | r = \\frac 1 {4 a^3} \\ln \\left({\\frac {a + x} {a - x} }\\right) + \\frac 1 {4 a^2} \\int \\frac {\\mathrm d x} {\\left({a + x}\\right)^2} + \\frac 1 {4 a^2} \\int \\frac {\\mathrm d x} {\\left({a - x}\\right)^2} + C       | c = Sign of Quotient of Factors of Difference of Squares }} {{eqn | r = \\frac 1 {4 a^3} \\ln \\left({\\frac {a + x} {a - x} }\\right) + \\frac 1 {4 a^2} \\frac {-1} {\\left({a + x}\\right)} + \\frac 1 {4 a^2} \\frac 1 {\\left({a - x}\\right)} + C       | c = Primitive of Function of $a x + b$ and Primitive of Power }} {{eqn | r = \\frac 1 {4 a^3} \\ln \\left({\\frac {a + x} {a - x} }\\right) + \\left({\\frac 1 {4 a^2 \\left({a - x}\\right)} - \\frac 1  {4 a^2 \\left({a + x}\\right)} }\\right) + C       | c = simplifying }} {{eqn | r = \\frac 1 {4 a^3} \\ln \\left({\\frac {a + x} {a - x} }\\right) + \\left({\\frac {\\left({a + x}\\right) - \\left({a - x}\\right)} {4 a^2 \\left({a - x}\\right) \\left({a + x}\\right)} }\\right) + C       | c = simplifying }} {{eqn | r = \\frac x {2 a^2 \\left({a^2 - x^2}\\right)} + \\frac 1 {4 a^3} \\ln \\left({\\frac {a + x} {a - x} }\\right) + C       | c = Difference of Two Squares }} {{end-eqn}} {{qed}} == Also see == * Primitive of $\\dfrac 1 {\\left({x^2 - a^2}\\right)^2}$ == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev=Primitive of Reciprocal of x cubed by a squared minus x squared|next=Primitive of x over a squared minus x squared squared}}: $\\S 14$: Integrals involving $a^2 - x^2$, $x^2 < a^2$: $14.170$ Category:Primitives involving a squared minus x squared a35z20n0qx2m1syyf00x7pdmwa8p5pm"}
+{"_id": "32509", "title": "Primitive of Reciprocal of x by a squared minus x squared squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x by a squared minus x squared squared/Partial Fraction Expansion 0 34064 188730 188728 2014-06-07T12:50:03Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x \\left({a^2 - x^2}\\right)^2$ == :$\\dfrac 1 {x \\left({a^2 - x^2}\\right)^2} \\equiv \\dfrac 1 {a^4 x} + \\dfrac x {a^4 \\left({a^2 - x^2}\\right)} + \\dfrac x {a^2 \\left({a^2 - x^2}\\right)^2}$ == Proof == {{begin-eqn}} {{eqn | l = \\frac 1 {x \\left({a^2 - x^2}\\right)^2}       | r = \\frac 1 {x \\left({a + x}\\right)^2 \\left({a - x}\\right)^2}       | c = Difference of Two Squares }} {{eqn | o = \\equiv       | r = \\frac A {a + x} + \\frac B {\\left({a + x}\\right)^2} + \\frac C {a - x} + \\frac D {\\left({a - x}\\right)^2} + \\frac E x       | c =  }} {{eqn | ll= \\implies       | l = 1       | o = \\equiv       | r = A x \\left({a^2 - x^2}\\right) \\left({a - x}\\right) + B x \\left({a - x}\\right)^2       | c = multiplying through by $x \\left({a^2 - x^2}\\right)^2$ }} {{eqn | o =        | ro= +       | r = C x \\left({a^2 - x^2}\\right) \\left({a + x}\\right) + D x \\left({a + x}\\right)^2 + E \\left({a^2 - x^2}\\right)^2       | c =  }} {{eqn | n = 1       | o = \\equiv       | r = A x^4 - A a x^3 - A a^2 x^2 + A a^3 x + B x^3 - 2 B a x^2 + B a^2 x       | c = multiplying out }} {{eqn | o =        | ro= -       | r = C x^4 - C a x^3 + C a^2 x^2 + C a^3 x + D x^3 + 2 D a x^2 + D a^2 x       | c =  }} {{eqn | o =        | ro= +       | r = E x^4 - 2 E a^2 x^2 + E a^4       | c =  }} {{end-eqn}} Setting $x = a$ in $(1)$: {{begin-eqn}} {{eqn | l = D a \\left({2 a}\\right)^2       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = D       | r = \\frac 1 {4 a^3}       | c =  }} {{end-eqn}} Setting $x = -a$ in $(1)$: {{begin-eqn}} {{eqn | l = B \\left({-a}\\right) \\left({2 a}\\right)^2       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = B       | r = \\frac {-1} {4 a^3}       | c =  }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = E a^4       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = E       | r = \\frac 1 {a^4}       | c =  }} {{end-eqn}} Equating coefficients of $x^4$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = A - C + E       | c =  }} {{eqn | ll= \\implies       | r = C       | l = A + \\frac 1 {a^4}       | c =  }} {{end-eqn}} Equating coefficients of $x^3$ in $(1)$: {{begin-eqn}} {{eqn | l = - A a - C a + B + D       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = - A a - C a + \\frac 1 {4 a^3} + \\frac {-1} {4 a^3}       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = -A a - \\left({A + \\frac 1 {a^4} }\\right) a       | r = 0       | c = as $C = A + \\dfrac 1 {a^4}$ }} {{eqn | ll= \\implies       | l = -2 A - \\frac 1 {a^4}       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = A        | r = \\frac {-1} {2 a^4}       | c =  }} {{eqn | ll= \\implies       | l = C        | r = \\frac 1 {2 a^4}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac {-1} {2 a^4} }} {{eqn | l = B       | r = \\frac {-1} {4 a^3} }} {{eqn | l = C       | r = \\frac 1 {2 a^4} }} {{eqn | l = D       | r = \\frac 1 {4 a^3} }} {{eqn | l = E       | r = \\frac 1 {a^4} }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | l = \\frac 1 {x \\left({a^2 - x^2}\\right)^2}       | o = \\equiv       | r = \\frac {-1} {2 a^4 \\left({a + x}\\right)} + \\frac {-1} {4 a^3 \\left({a + x}\\right)^2} + \\frac 1 {2 a^4 \\left({a - x}\\right)} + \\frac 1 {4 a^3 \\left({a - x}\\right)^2} + \\frac 1 {a^4 x}       | c =  }} {{eqn | o = \\equiv       | r = \\frac {\\left({a + x}\\right) - \\left({a - x}\\right)} {2 a^4 \\left({a - x}\\right) \\left({a + x}\\right)} + \\frac {\\left({a + x}\\right)^2 - \\left({a - x}\\right)^2} {4 a^3 \\left({a + x}\\right)^2 \\left({a - x}\\right)^2} + \\frac 1 {a^4 x}       | c = common denominators }} {{eqn | o = \\equiv       | r = \\frac {2 x} {2 a^4 \\left({a^2 - x^2}\\right)} + \\frac {\\left({a^2 + 2 a x + x^2}\\right) - \\left({a^2 - 2 a x + x^2}\\right)} {4 a^3 \\left({a^2 - x^2}\\right)^2} + \\frac 1 {a^4 x}       | c = simplifying }} {{eqn | o = \\equiv       | r = \\frac x {a^4 \\left({a^2 - x^2}\\right)} + \\frac {4 a x} {4 a^3 \\left({a^2 - x^2}\\right)^2} + \\frac 1 {a^4 x}       | c = simplifying }} {{eqn | o = \\equiv       | r = \\dfrac 1 {a^4 x} + \\dfrac x {a^4 \\left({a^2 - x^2}\\right)} + \\dfrac x {a^2 \\left({a^2 - x^2}\\right)^2}       | c = simplifying and rearranging }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a squared minus x squared/Lemmata eyarnq5typ38bc2c0fs9p2k1b5ri6b9"}
+{"_id": "32510", "title": "Primitive of Reciprocal of x squared by a squared minus x squared squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x squared by a squared minus x squared squared/Partial Fraction Expansion 0 34066 476805 188735 2020-07-03T16:25:35Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x^2 \\paren {a^2 - x^2}^2$ == :$\\dfrac 1 {x^2 \\paren {a^2 - x^2}^2} \\equiv \\dfrac 1 {a^4 x^2} + \\dfrac 3 {4 a^5 \\paren {a + x} } - \\dfrac 3 {4 a^5 \\paren {a - x} } + \\dfrac 1 {4 a^4 \\paren {a + x}^2} + \\dfrac 1 {4 a^4 \\paren {a - x}^2}$ == Proof == {{begin-eqn}} {{eqn | l = \\frac 1 {x^2 \\paren {a^2 - x^2}^2}       | r = \\frac 1 {x^2 \\paren {a + x}^2 \\paren {a - x}^2}       | c = Difference of Two Squares }} {{eqn | o = \\equiv       | r = \\frac A {a + x} + \\frac B {\\paren {a + x}^2} + \\frac C {a - x} + \\frac D {\\paren {a - x}^2} + \\frac E x + \\frac F {x^2}       | c =  }} {{eqn | ll= \\leadsto       | l = 1       | o = \\equiv       | r = A x^2 \\paren {a^2 - x^2} \\paren {a - x} + B x^2 \\paren {a - x}^2       | c = multiplying through by $x^2 \\paren {a^2 - x^2}^2$ }} {{eqn | o =        | ro= +       | r = C x^2 \\paren {a^2 - x^2} \\paren {a + x} + D x^2 \\paren {a + x}^2       | c =  }} {{eqn | o =        | ro= +       | r = E x \\paren {a^2 - x^2}^2 + F \\paren {a^2 - x^2}^2       | c =  }} {{eqn | n = 1       | o = \\equiv       | r = A x^5 - A a x^4 - A a^2 x^3 + A a^3 x^2 + B x^4 - 2 B a x^3 + B a^2 x^2       | c = multiplying out }} {{eqn | o =        | ro= -       | r = C x^5 - C a x^4 + C a^2 x^3 + C a^3 x^2 + D x^4 + 2 D a x^3 + D a^2 x^2       | c =  }} {{eqn | o =        | ro= +       | r = E x^5 - 2 E a^2 x^3 + E a^4 x + F x^4 - 2 F a^2 x^2 + F a^4       | c =  }} {{end-eqn}} Setting $x = a$ in $(1)$: {{begin-eqn}} {{eqn | l = D a^2 \\paren {2 a}^2       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = D       | r = \\frac 1 {4 a^4}       | c =  }} {{end-eqn}} Setting $x = -a$ in $(1)$: {{begin-eqn}} {{eqn | l = B \\paren {-a}^2 \\paren {2 a}^2       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = B       | r = \\frac 1 {4 a^4}       | c =  }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = F a^4       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = F       | r = \\frac 1 {a^4}       | c =  }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = E       | c =  }} {{end-eqn}} Equating coefficients of $x^5$ in $(1)$: {{begin-eqn}} {{eqn | n = 2       | l = 0       | r = A - C + E       | c =  }} {{eqn | ll= \\leadsto       | l = 0       | r = A - C       | c = as $E = 0$ }} {{eqn | ll= \\leadsto       | l = A       | r = C       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = A a^3 + B a^2 + C a^3 + D a^2 - 2 F a^2       | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = A a^3 + \\frac 1 {4 a^4} a^2 + C a^3 + \\frac 1 {4 a^4} a^2 - 2 \\frac 1 {a^4} a^2       | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = A + \\frac 1 {4 a^5} + C + \\frac 1 {4 a^5} - \\frac 8 {4 a^5}       | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = A + C       | r = \\frac {8 - 1 - 1} {4 a^5}       | c =  }} {{eqn | r = \\frac 3 {2 a^5}       | c =  }} {{eqn | ll= \\leadsto       | l = A       | r = \\frac 3 {4 a^5}       | c =  }} {{eqn | ll= \\leadsto       | l = C       | r = \\frac 3 {4 a^5}       | c = as $A = C$ }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac 3 {4 a^5} }} {{eqn | l = B       | r = \\frac 1 {4 a^4} }} {{eqn | l = C       | r = \\frac 3 {4 a^5} }} {{eqn | l = D       | r = \\frac 1 {4 a^4} }} {{eqn | l = E       | r = 0 }} {{eqn | l = F       | r = \\frac 1 {a^4} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a squared minus x squared/Lemmata om4bfn722d26r1bdgs42ux9vdreolbk"}
+{"_id": "32511", "title": "Primitive of Reciprocal of x cubed by a squared minus x squared squared/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x cubed by a squared minus x squared squared/Partial Fraction Expansion 0 34068 394763 188738 2019-03-05T17:00:36Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x^3 \\paren {a^2 - x^2}^2$ == :$\\dfrac 1 {x^3 \\paren {a^2 - x^2}^2} \\equiv \\dfrac 1 {a^4 x^3} + \\dfrac 2 {a^6 x} + \\dfrac {2 x} {a^6 \\paren {a^2 - x^2} } + \\dfrac x {a^4 \\paren {a^2 - x^2}^2}$ == Proof == {{begin-eqn}} {{eqn | l = \\frac 1 {x^3 \\paren {a^2 - x^2}^2}       | r = \\frac 1 {x^3 \\paren {a + x}^2 \\paren {a - x}^2}       | c = Difference of Two Squares }} {{eqn | o = \\equiv       | r = \\frac A {a + x} + \\frac B {\\paren {a + x}^2} + \\frac C {a - x} + \\frac D {\\paren {a - x}^2} + \\frac E x + \\frac F {x^2} + \\frac G {x^3}       | c =  }} {{eqn | ll= \\leadsto       | l = 1       | o = \\equiv       | r = A x^3 \\paren {a^2 - x^2} \\paren {a - x} + B x^3 \\paren {a - x}^2       | c = multiplying through by $x^3 \\paren {a^2 - x^2}^2$ }} {{eqn | o =        | ro= +       | r = C x^3 \\paren {a^2 - x^2} \\paren {a + x} + D x^3 \\paren {a + x}^2       | c =  }} {{eqn | o =        | ro= +       | r = E x^2 \\paren {a^2 - x^2}^2 + F x \\paren {a^2 - x^2}^2 + G \\paren {a^2 - x^2}^2       | c =  }} {{eqn | n = 1       | o = \\equiv       | r = A x^6 - A a x^5 - A a^2 x^4 + A a^3 x^3 + B x^5 - 2 B a x^4 + B a^2 x^3       | c = multiplying out }} {{eqn | o =        | ro= -       | r = C x^6 - C a x^5 + C a^2 x^4 + C a^3 x^3 + D x^5 + 2 D a x^4 + D a^2 x^3       | c =  }} {{eqn | o =        | ro= +       | r = E x^6 - 2 E a^2 x^4 + E a^4 x^2 + F x^5 - 2 F a^2 x^3 + F x a^4       | c =  }} {{eqn | o =        | ro= +       | r = G x^4 - 2 G a^2 x^2 + G a^4       | c =  }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = G a^4       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = G       | r = \\frac 1 {a^4}       | c =  }} {{end-eqn}} Setting $x = a$ in $(1)$: {{begin-eqn}} {{eqn | l = D a^3 \\paren {2 a}^2       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = D       | r = \\frac 1 {4 a^5}       | c =  }} {{end-eqn}} Setting $x = -a$ in $(1)$: {{begin-eqn}} {{eqn | l = B \\paren {-a}^3 \\paren {2 a}^2       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = B       | r = \\frac {-1} {4 a^5}       | c =  }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = F       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = 0       | r = E a^4 - 2 G a^2        | c =  }} {{eqn | ll= \\leadsto       | l = E a^4       | r = 2 \\frac 1 {a^4} a^2        | c =  }} {{eqn | ll= \\leadsto       | l = E       | r = \\frac 2 {a^6}       | c =  }} {{end-eqn}} Equating coefficients of $x^6$ in $(1)$: {{begin-eqn}} {{eqn | n = 2       | l = 0       | r = A - C + E       | c =  }} {{eqn | ll= \\leadsto       | l = 0       | r = A - C + \\frac 2 {a^6}       | c =  }} {{eqn | ll= \\leadsto       | l = A       | r = C - \\frac 2 {a^6}       | c =  }} {{end-eqn}} Equating coefficients of $x^3$ in $(1)$: {{begin-eqn}} {{eqn | l = A a^3 + B a^2 + C a^3 + D a^2 - 2 F a^2       | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = A a^3 + \\frac {-1} {4 a^5} a^2 + C a^3 + \\frac 1 {4 a^5} a^2       | r = 0       | c = substituting for $B$, $D$ and $F$ }} {{eqn | ll= \\leadsto       | l = A + C       | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = A       | r = -C       | c =  }} {{eqn | r = \\frac {-1} {a^6}       | c =  }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac {-1} {a^6} }} {{eqn | l = B       | r = \\frac {-1} {4 a^5} }} {{eqn | l = C       | r = \\frac 1 {a^6} }} {{eqn | l = D       | r = \\frac 1 {4 a^5} }} {{eqn | l = E       | r = \\frac 2 {a^6} }} {{eqn | l = F       | r = 0 }} {{eqn | l = G       | r = \\frac 1 {a^4} }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | l = \\frac 1 {x^3 \\paren {a^2 - x^2}^2}       | o = \\equiv       | r = \\dfrac 1 {a^4 x^3} + \\dfrac 2 {a^6 x} - \\dfrac 1 {a^6 \\paren {a + x} } + \\dfrac 1 {a^6 \\paren {a - x} } - \\dfrac 1 {4 a^5 \\paren {a + x}^2} + \\dfrac 1 {4 a^5 \\paren {a - x}^2}       | c =  }} {{eqn | o = \\equiv       | r = \\frac {\\paren {a + x} - \\paren {a - x} } {a^6 \\paren {a + x} \\paren {a - x} } + \\frac {\\paren {a + x}^2 - \\paren {a - x}^2} {4 a^5 \\paren {a + x}^2 \\paren {a - x}^2} + \\dfrac 1 {a^4 x^3} + \\dfrac 2 {a^6 x}       | c = common denominators }} {{eqn | o = \\equiv       | r = \\frac {2 x} {a^6 \\paren {a^2 - x^2} } + \\frac {\\paren {a^2 + 2 a x + x^2} - \\paren {a^2 - 2 a x + x^2} } {4 a^5 \\paren {a^2 - x^2}^2} + \\dfrac 1 {a^4 x^3} + \\dfrac 2 {a^6 x}       | c = simplifying }} {{eqn | o = \\equiv       | r = \\frac {2 x} {a^6 \\paren {a^2 - x^2} } + \\frac {4 a x} {4 a^5 \\paren {a^2 - x^2}^2} + \\frac 1 {a^4 x^3} + \\frac 2 {a^6 x}       | c = simplifying }} {{eqn | o = \\equiv       | r = \\frac 1 {a^4 x^3} + \\frac 2 {a^6 x} + \\frac {2 x} {a^6 \\paren {a^2 - x^2} } + \\frac x {a^4 \\paren {a^2 - x^2}^2}       | c = simplifying and rearranging }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving a squared minus x squared/Lemmata mg0zd2ych3vpf10mlszcsga1v0g3v4h"}
+{"_id": "32512", "title": "Primitive of x squared over Root of a squared minus x squared", "text": "Primitive of x squared over Root of a squared minus x squared 0 34139 188942 2014-06-12T20:57:02Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> :$\\displaystyle \\int \\frac {x^2 \\ \\mathrm d x} {\\sqrt {a^2 - x^2} } = \\frac {-x \\sqrt {a^2 - x^2} } 2 + \\frac {a^2} 2 \\arcsin \\frac x a + C$ </only...\" wikitext text/x-wiki == Theorem == <onlyinclude> :$\\displaystyle \\int \\frac {x^2 \\ \\mathrm d x} {\\sqrt {a^2 - x^2} } = \\frac {-x \\sqrt {a^2 - x^2} } 2 + \\frac {a^2} 2 \\arcsin \\frac x a + C$ </onlyinclude> == Proof == With a view to expressing the problem in the form: :$\\displaystyle \\int u \\frac {\\mathrm d v}{\\mathrm d x} \\mathrm d x = u v - \\int v \\frac {\\mathrm d u}{\\mathrm d x} \\mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = x       | c =  }} {{eqn | ll= \\implies       | l = \\frac {\\mathrm d u}{\\mathrm d x}       | r = 1       | c = Power Rule for Derivatives }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \\frac {\\mathrm d v}{\\mathrm d x}       | r = \\frac x {\\sqrt {a^2 - x^2} }       | c =  }} {{eqn | ll= \\implies       | l = v       | r = -\\sqrt {a^2 - x^2}       | c = Primitive of $\\dfrac x {\\sqrt {a^2 - x^2} }$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \\int \\frac {x^2 \\ \\mathrm d x} {\\sqrt {a^2 - x^2} }       | r = \\int x \\frac {x \\ \\mathrm d x} {\\sqrt {a^2 - x^2} }       | c =  }} {{eqn | r = -x \\sqrt {a^2 - x^2} - \\int \\left({-\\sqrt {a^2 - x^2} }\\right) \\ \\mathrm d x       | c = Integration by Parts }} {{eqn | r = -x \\sqrt {a^2 - x^2} + \\int \\left({\\sqrt {a^2 - x^2} }\\right) \\ \\mathrm d x       | c = simplifying }} {{eqn | r = -x \\sqrt {a^2 - x^2} + \\left({\\frac {x \\sqrt {a^2 - x^2} } 2 + \\frac {a^2} 2 \\arcsin \\frac x a}\\right) + C       | c = Primitive of $\\sqrt {a^2 - x^2}$ }} {{eqn | r = \\frac {-x \\sqrt {a^2 - x^2} } 2 + \\frac {a^2} 2 \\arcsin \\frac x a + C       | c =  }} {{end-eqn}} {{qed}} == Also see == * Primitive of $\\dfrac {x^2} {\\sqrt {x^2 + a^2} }$ * Primitive of $\\dfrac {x^2} {\\sqrt {x^2 - a^2} }$ == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev=Primitive of x over Root of a squared minus x squared|next=Primitive of x cubed over Root of a squared minus x squared}}: $\\S 14$: Integrals involving $\\sqrt {a^2 - x^2}$: $14.239$ Category:Primitives involving Root of a squared minus x squared jqy62n9bfz203t5e1r181cpnio7poo3"}
+{"_id": "32515", "title": "Sum of Infinite Geometric Sequence/Corollary 2", "text": "Sum of Infinite Geometric Sequence/Corollary 2 0 34268 456673 456640 2020-03-20T13:09:22Z Prime.mover 59 wikitext text/x-wiki == Corollary to Sum of Infinite Geometric Sequence == Let $S$ be a standard number field, that is $\\Q$, $\\R$ or $\\C$. Let $z \\in S$. Let $\\size z < 1$, where $\\size z$ denotes: :the absolute value of $z$, for real and rational $z$ :the complex modulus of $z$ for complex $z$. Then: <onlyinclude> :$\\displaystyle \\sum_{n \\mathop = 0}^\\infty a z^n = \\frac a {1 - z}$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\sum_{n \\mathop = 0}^\\infty a z^n       | r = a \\sum_{n \\mathop = 0}^\\infty z^n       | c = }} {{eqn | r = a \\frac 1 {1 - z}       | c = Sum of Infinite Geometric Sequence }} {{eqn | r = \\frac a {1 - z}       | c = }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Handbook of Mathematical Functions|1964|Milton Abramowitz|author2 = Irene A. Stegun|prev = Sum of Geometric Sequence/Corollary 1|next = Definition:Arithmetic Mean}}: $3.1.10$: Sum of Geometric Progression to $n$ Terms * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Sum of Geometric Sequence/Corollary 1|next = Definition:Arithmetic-Geometric Series}}: $\\S 19$: Geometric Series: $19.5$ * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Sum of Geometric Sequence/Corollary 1|next = Sum of Infinite Geometric Sequence/Corollary 2/Mistake|entry = geometric series}} * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Geometric Series|next = Definition:Geometry|entry = geometric series}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Geometric Series|next = Definition:Geometry|entry = geometric series}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Sum of Geometric Sequence/Corollary 1|next = Definition:Geometry|entry = geometric series}} Category:Sum of Infinite Geometric Sequence ptukb8u6dds04fc9iy9h57wenboim4t"}
+{"_id": "32516", "title": "Perpendicular in Right-Angled Triangle makes two Similar Triangles/Porism", "text": "Perpendicular in Right-Angled Triangle makes two Similar Triangles/Porism 0 34269 286669 196141 2017-03-02T06:58:12Z Prime.mover 59 wikitext text/x-wiki == Porism to Perpendicular in Right-Angled Triangle makes two Similar Triangles == <onlyinclude>{{:Euclid:Proposition/VI/8/Porism}}</onlyinclude> == Proof == Follows directly from Perpendicular in Right-Angled Triangle makes two Similar Triangles. {{qed}} {{Euclid Note|8|VI}} == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 2|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Perpendicular in Right-Angled Triangle makes two Similar Triangles|next = Construction of Part of Line}}: Book $\\text{VI}$. Propositions Category:Right Triangles r54mxsgaiufhrpyzuf5f7djwbbpd39l"}
+{"_id": "32517", "title": "Primitive of x over Root of a x squared plus b x plus c", "text": "Primitive of x over Root of a x squared plus b x plus c 0 34340 189802 2014-07-06T05:51:32Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> Let $a \\in \\R_{\\ne 0}$.  Then: :$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\sqrt {a x^2 + b x + c} } = \\frac {\\sqrt {a x^2 + b x + c} } a - \\frac...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\sqrt {a x^2 + b x + c} } = \\frac {\\sqrt {a x^2 + b x + c} } a - \\frac b {2 a} \\int \\frac {\\mathrm d x} {\\sqrt {a x^2 + b x + c} }$ </onlyinclude> == Proof == First: {{begin-eqn}} {{eqn | l = z       | r = a x^2 + b x + c       | c =  }} {{eqn | ll= \\implies       | l = \\frac {\\mathrm d z} {\\mathrm d x}       | r = 2 a x + b       | c = Derivative of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \\int \\frac {x \\ \\mathrm d x} {\\sqrt {a x^2 + b x + c} }       | r = \\frac 1 {2 a} \\int \\frac {2 a x \\ \\mathrm d x} {\\sqrt {a x^2 + b x + c} }       | c =  }} {{eqn | r = \\frac 1 {2 a} \\int \\frac {\\left({2 a x + b - b}\\right) \\ \\mathrm d x} {\\sqrt {a x^2 + b x + c} }       | c =  }} {{eqn | r = \\frac 1 {2 a} \\int \\frac {\\left({2 a x + b}\\right) \\ \\mathrm d x} {\\sqrt {a x^2 + b x + c} } - \\frac b {2 a} \\int \\frac {\\mathrm d x} {\\sqrt {a x^2 + b x + c} }        | c = Linear Combination of Integrals }} {{eqn | r = \\frac 1 {2 a} \\int \\frac {\\mathrm d z} {\\sqrt z} - \\frac b {2 a} \\int \\frac {\\mathrm d x} {\\sqrt {a x^2 + b x + c} }        | c = Integration by Substitution }} {{eqn | r = \\frac 1 {2 a} 2 \\sqrt z - \\frac b {2 a} \\int \\frac {\\mathrm d x} {\\sqrt {a x^2 + b x + c} }       | c = Primitive of Power }} {{eqn | r = \\frac {\\sqrt {a x^2 + b x + c} } a - \\frac b {2 a} \\int \\frac {\\mathrm d x} {\\sqrt {a x^2 + b x + c} }       | c = substituting for $z$ }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev=Primitive of Reciprocal of Root of a x squared plus b x plus c|next=Primitive of x squared over Root of a x squared plus b x plus c}}: $\\S 14$: Integrals involving $\\sqrt {a x^2 + bx + c}$: $14.281$ Category:Primitives involving Root of a x squared plus b x plus c tld44ihdhusqw66zxp0tyyqvk373rwz"}
+{"_id": "32518", "title": "Primitive of Sine Function/Corollary", "text": "Primitive of Sine Function/Corollary 0 34448 485084 484694 2020-09-03T20:32:38Z Prime.mover 59 wikitext text/x-wiki == Corollary to Primitive of Sine Function == <onlyinclude> :$\\displaystyle \\int \\sin a x \\rd x = - \\frac {\\cos a x} a + C$ </onlyinclude> where $a$ is a non-zero constant. == Proof == {{begin-eqn}} {{eqn | l = \\int \\sin x \\rd x       | r = -\\cos x + C       | c = Primitive of $\\sin x$ }} {{eqn | ll= \\leadsto       | l = \\int \\sin a x \\rd x       | r = \\frac 1 a \\paren {-\\cos a x} + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = -\\frac {\\cos a x} a + C       | c = simplifying }} {{end-eqn}} {{qed}} == Also see == * Primitive of $\\cos a x$ * Primitive of $\\tan a x$ * Primitive of $\\cot a x$ * Primitive of $\\sec a x$ * Primitive of $\\csc a x$ == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Primitive of Power of x over Odd Power of x minus Odd Power of a|next = Primitive of x by Sine of a x}}: $\\S 14$: Integrals involving $\\sin a x$: $14.339$ * {{BookReference|Calculus|1983|K.G. Binmore|prev = Primitive of Cosine of a x|next = Primitive of Reciprocal of x squared plus a squared/Arctangent Form}}: $9$ Sums and Integrals: $9.8$ Standard Integrals Category:Primitives involving Sine Function t41fjeldr3kefxenh7brlgw05q2l91m"}
+{"_id": "32519", "title": "Primitive of Cosine Function/Corollary", "text": "Primitive of Cosine Function/Corollary 0 34467 484693 484690 2020-09-02T06:31:44Z Prime.mover 59 wikitext text/x-wiki == Corollary to Primitive of Cosine Function == <onlyinclude> :$\\displaystyle \\int \\cos a x \\rd x = \\frac {\\sin a x} a + C$ </onlyinclude> where $a$ is a non-zero constant. == Proof == {{begin-eqn}} {{eqn | l = \\int \\cos x \\rd x       | r = \\sin x + C       | c = Primitive of $\\cos x$ }} {{eqn | ll= \\leadsto       | l = \\int \\cos a x \\rd x       | r = \\frac 1 a \\paren {\\sin a x} + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = \\frac {\\sin a x} a + C       | c = simplifying }} {{end-eqn}} {{qed}} == Also see == * Primitive of $\\sin a x$ * Primitive of $\\tan a x$ * Primitive of $\\cot a x$ * Primitive of $\\sec a x$ * Primitive of $\\csc a x$ == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Primitive of x over Power of Sine of a x|next = Primitive of x by Cosine of a x}}: $\\S 14$: Integrals involving $\\cos a x$: $14.369$ * {{BookReference|Calculus|1983|K.G. Binmore|prev = Primitive of Exponential of a x|next = Primitive of Sine of a x}}: $9$ Sums and Integrals: $9.8$ Standard Integrals Category:Primitives involving Cosine Function huni9yvmkvxlhknwmr1qvrlgdn37axi"}
+{"_id": "32520", "title": "Weierstrass Substitution", "text": "Weierstrass Substitution 0 34494 484485 419749 2020-09-01T06:33:47Z Prime.mover 59 wikitext text/x-wiki {{POTW Candidate}} == Proof Technique == The '''Weierstrass Substitution''' is an application of Integration by Substitution. The substitution is: :$u \\leftrightarrow \\tan \\dfrac \\theta 2$ for $-\\pi < \\theta < \\pi$, $u \\in \\R$. It yields: {{begin-eqn}} {{eqn | l = \\sin \\theta       | r = \\frac {2 u} {1 + u^2} }} {{eqn | l = \\cos \\theta       | r = \\frac {1 - u^2} {1 + u^2} }} {{eqn | l = \\frac {\\d \\theta} {\\d u}       | r = \\frac 2 {1 + u^2} }} {{end-eqn}} This can be stated: :$\\displaystyle \\int \\map F {\\sin \\theta, \\cos \\theta} \\rd \\theta = 2 \\int \\map F {\\frac {2 u} {1 + u^2}, \\frac {1 - u^2} {1 + u^2} } \\frac {d u} {1 + u^2}$ where $u = \\tan \\dfrac \\theta 2$. == Proof == Let $u = \\tan \\dfrac \\theta 2$ for $-\\pi < \\theta < \\pi$. From Shape of Tangent Function, this substitution is valid for all real $u$. Then: {{begin-eqn}} {{eqn | l = u       | r = \\tan \\dfrac \\theta 2       | c =  }} {{eqn | ll= \\leadsto       | l = \\theta       | r = 2 \\tan^{-1} u       | c = Definition of Inverse Tangent }} {{eqn | ll= \\leadsto       | l = \\dfrac {\\d \\theta} {\\d u}       | r = \\dfrac 2 {1 + u^2}       | c = Derivative of Arctangent Function and Derivative of Constant Multiple }} {{eqn | l = \\sin \\theta       | r = \\dfrac {2 u} {1 + u^2}       | c = Tangent Half-Angle Substitution for Sine }} {{eqn | l = \\cos \\theta       | r = \\dfrac {1 - u^2} {1 + u^2}       | c = Tangent Half-Angle Substitution for Cosine }} {{end-eqn}} The result follows from Integration by Substitution. {{qed}} == Also known as == This technique is also known as '''tangent half-angle subsitution'''. == Also see == * Hyperbolic Tangent Half-Angle Substitution {{Namedfor|Karl Theodor Wilhelm Weierstrass|cat = Weierstrass}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Primitive of Function of Arcsine|next = Primitive of Reciprocal of a x + b}}: $\\S 14$: Important Transformations: $14.58$ * {{Planetmath|url = weierstrasssubstitutionformulas|title = Weierstrass substitution formulas}} * {{MathWorld|Weierstrass Substitution|WeierstrassSubstitution}} Category:Proof Techniques Category:Integral Substitutions Category:Primitives involving Sine Function Category:Primitives involving Cosine Function h31fkbztzhk6dycqjlbbqutnziypfx6"}
+{"_id": "32521", "title": "Primitive of x by Root of a x squared plus b x plus c/Lemma", "text": "Primitive of x by Root of a x squared plus b x plus c/Lemma 0 34503 190359 2014-07-19T10:53:22Z Prime.mover 59 Created page with \"== Lemma for Primitive of $x \\sqrt {a x^2 + b x + c}$ == <onlyinclude> Let $a \\in \\R_{\\ne 0}$.  Then: :$\\displaystyle...\" wikitext text/x-wiki == Lemma for Primitive of $x \\sqrt {a x^2 + b x + c}$ == <onlyinclude> Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int x \\sqrt {a x^2 + b x + c} \\ \\mathrm d x = \\frac {\\left({\\sqrt {a x^2 + b x + c} }\\right)^3} {3 a} - \\frac b {2 a} \\int \\sqrt {a x^2 + b x + c} \\ \\mathrm d x$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | o =        | r = \\int x \\sqrt {a x^2 + b x + c} \\ \\mathrm d x }} {{eqn | r = \\int \\frac {2 a x \\sqrt {a x^2 + b x + c} \\ \\mathrm d x} {2 a}       | c = multiplying top and bottom by $2 a$ }} {{eqn | r = \\int \\frac {\\left({2 a x + b - b}\\right) \\sqrt {a x^2 + b x + c} \\ \\mathrm d x} {2 a} }} {{eqn | n = 1       | r = \\frac 1 {2 a} \\int \\left({2 a x + b}\\right) \\sqrt {a x^2 + b x + c} \\ \\mathrm d x - \\frac b {2 a} \\int \\sqrt {a x^2 + b x + c} \\ \\mathrm d x       | c = Linear Combination of Integrals }} {{end-eqn}} Let: {{begin-eqn}} {{eqn | l = z       | r = a x^2 + b x + c }} {{eqn | ll= \\implies       | l = \\frac {\\mathrm d z} {\\mathrm d x}       | r = 2 a x + b       | c = Derivative of Power }} {{eqn | ll= \\implies       | l = \\int \\left({2 a x + b}\\right) \\sqrt {a x^2 + b x + c} \\ \\mathrm d x       | r = \\int \\sqrt z \\ \\mathrm d z       | c = Integration by Substitution }} {{eqn | r = \\frac {2 \\left({\\sqrt z}\\right)^3} 3       | c = Primitive of Power }} {{eqn | n = 2       | r = \\frac {2 \\left({\\sqrt {a x^2 + b x + c} }\\right)^3} 3       | c = substituting for $z$ }} {{end-eqn}} So: {{begin-eqn}} {{eqn | o =        | r = \\int x \\sqrt {a x^2 + b x + c} \\ \\mathrm d x }} {{eqn | r = \\frac 1 {2 a} \\int \\left({2 a x + b}\\right) \\sqrt {a x^2 + b x + c} \\ \\mathrm d x - \\frac b {2 a} \\int \\sqrt {a x^2 + b x + c} \\ \\mathrm d x       | c = from $(1)$ }} {{eqn | r = \\frac 1 {2 a} \\left({\\frac {2 \\left({\\sqrt {a x^2 + b x + c} }\\right)^3} 3}\\right) - \\frac b {2 a} \\int \\sqrt {a x^2 + b x + c} \\ \\mathrm d x       | c = from $(2)$ }} {{eqn | r = \\frac {\\left({\\sqrt {a x^2 + b x + c} }\\right)^3} {3 a} - \\frac b {2 a} \\int \\sqrt {a x^2 + b x + c} \\ \\mathrm d x       | c = simplifying }} {{end-eqn}} {{qed}} Category:Primitives involving Root of a x squared plus b x plus c/Lemmata 4u1cbx5qhf0q9wmlflbfd0jq9ms9cch"}
+{"_id": "32522", "title": "Primitive of Reciprocal of x cubed plus a cubed/Lemma", "text": "Primitive of Reciprocal of x cubed plus a cubed/Lemma 0 34512 481683 295964 2020-08-14T22:14:42Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x^3 + a^3$ == <onlyinclude> :$\\displaystyle \\int \\frac {\\d x} {x^2 - a x + a^2} = \\frac 2 {a \\sqrt 3} \\map \\arctan {\\frac {2 x - a} {a \\sqrt 3} }$ </onlyinclude> == Proof == The discriminant of $x^2 - a x + a^2$ is: {{begin-eqn}} {{eqn | l = \\map {\\mathrm {Disc} } {x^2 - a x + a^2}       | r = \\paren {-a}^2 - 4 \\times 1 \\times a^2       | c =  }} {{eqn | r = a^2 - 4 a^2       | c =  }} {{eqn | r = - 3 a^2       | c =  }} {{eqn | r = < 0       | c =  }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | l = \\int \\frac {\\d x} {x^2 - a x + a^2}       | r = \\frac 2 {\\sqrt {4 a^2 - \\paren {-a}^2} } \\map \\arctan {\\frac {2 x - a} {\\sqrt {4 a^2 - \\paren {-a}^2} } }       | c = Primitive of $\\dfrac 1 {a x^2 + b x + c}$ }} {{eqn | r = \\frac 2 {a \\sqrt 3} \\map \\arctan {\\frac {2 x - a} {a \\sqrt 3} }       | c = simplifying }} {{end-eqn}} {{qed}} Category:Primitives involving x cubed plus a cubed/Lemmata 8jvvghwfpa4szi3v1kui5n4qqhao8o7"}
+{"_id": "32523", "title": "Primitive of Reciprocal of x cubed plus a cubed/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x cubed plus a cubed/Partial Fraction Expansion 0 34514 481681 190410 2020-08-14T22:05:46Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x^3 + a^3$ == :$\\dfrac 1 {x^3 + a^3} = \\dfrac 1 {3 a^2 \\paren {x + a} } - \\dfrac {x - 2 a} {3 a^2 \\paren {x^2 - a x + a^2} }$ == Proof == {{begin-eqn}} {{eqn | l = \\frac 1 {x^3 + a^3}       | o = \\equiv       | r = \\frac 1 {\\paren {x + a} \\paren {x^2 - a x + a^2} }       | c = Sum of Two Cubes }} {{eqn | o = \\equiv       | r = \\frac A {x + a} + \\frac {B x + C} {x^2 - a x + a^2}       | c =  }} {{eqn | ll= \\leadsto       | l = 1       | o = \\equiv       | r = A \\paren {x^2 - a x + a^2} + \\paren {B x + C} \\paren {x + a}       | c = multiplying through by $x^3 + a^3$ }} {{eqn | n = 1       | o = \\equiv       | r = A x^2 - A a x + A a^2 + B x^2 + B a x + C x + C a       | c = multiplying out }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = A + B       | r = 0       | c =  }} {{eqn | n = 2       | ll= \\leadsto       | l = -A       | r = B       | c =  }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = -A a + B a + C       | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = -A a + -A a + C       | r = 0       | c = from $(2)$ }} {{eqn | n = 3       | ll= \\leadsto       | l = 2 A a       | r = C }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = A a^2 + C a       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = A a^2 + \\paren {2 A a} a       | r = 1       | c = from $(3)$ }} {{eqn | ll= \\leadsto       | l = A a^2 + \\paren {2 A a} a       | r = \\frac 1 {3 a^2}       | c =  }} {{eqn | ll= \\leadsto       | l = C       | r = 1 - \\frac a {3 a^2}       | c = from $(3)$ }} {{eqn | r = \\frac {2 a} {3 a^2}       | c =  }} {{eqn | ll= \\leadsto       | l = B       | r = -\\frac {-1} {3 a^2}       | c = from $(2)$ }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac 1 {3 a^2} }} {{eqn | l = B       | r = -\\frac 1 {3 a^2} }} {{eqn | l = C       | r = \\frac {2 a} {3 a^2} }} {{end-eqn}} Hence the result. {{qed}} Category:Primitives involving x cubed plus a cubed/Lemmata 9lbkep0ytd8y4bd8lal1r9n6lm2o0ie"}
+{"_id": "32524", "title": "Primitive of Reciprocal of x fourth plus a fourth/Partial Fraction Expansion", "text": "Primitive of Reciprocal of x fourth plus a fourth/Partial Fraction Expansion 0 34521 407615 220600 2019-06-12T20:44:32Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x^4 + a^4$ == :$\\dfrac 1 {x^4 + a^4} = \\dfrac {x + a \\sqrt 2} {2 a^3 \\sqrt 2 \\left({x^2 + a x \\sqrt 2 + a^2}\\right)} - \\dfrac {x - a \\sqrt 2} {2 a^3 \\sqrt 2 \\left({x^2 - a x \\sqrt 2 + a^2}\\right)}$ == Proof == {{begin-eqn}} {{eqn | l = \\frac 1 {x^4 + a^4}       | o = \\equiv       | r = \\frac 1 {\\left({x^2 + a x \\sqrt 2 + a^2}\\right) \\left({x^2 - a x \\sqrt 2 + a^2}\\right)}       | c = Sum of Two Fourth Powers }} {{eqn | o = \\equiv       | r = \\frac {A x + B} {x^2 + a x \\sqrt 2 + a^2} + \\frac {C x + D} {x^2 - a x \\sqrt 2 + a^2}       | c =  }} {{eqn | ll= \\implies       | l = 1       | o = \\equiv       | r = \\left({A x + B}\\right) \\left({x^2 - a x \\sqrt 2 + a^2}\\right) + \\left({C x + D}\\right) \\left({x^2 + a x \\sqrt 2 + a^2}\\right)       | c =  }} {{eqn | n = 1       | o = \\equiv       | r = A x^3 + B x^2 - A a x^2 \\sqrt 2 - B a x \\sqrt 2 + A x a^2 + B a^2       | c =  }} {{eqn | o =        | ro= +       | r = C x^3 + D x^2 + C a x^2 \\sqrt 2 + D a x \\sqrt 2 + C x a^2 + D a^2       | c =  }} {{end-eqn}} Equating coefficients of $x^3$ in $(1)$: {{begin-eqn}} {{eqn | l = A + C       | r = 0       | c =  }} {{eqn | n = 2       | ll= \\implies       | l = -A       | r = C       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = -A a \\sqrt 2 + B + C a \\sqrt 2 + D       | r = 0       | c =  }} {{eqn | n = 3       | ll= \\implies       | l = 2 C a \\sqrt 2 + B + D       | r = 0       | c = substituting for $A$ from $(2)$ }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = -B a \\sqrt 2 + A a^2 + D a \\sqrt 2 + C a^2       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = -B a \\sqrt 2 - C a^2 + D a \\sqrt 2 + C a^2       | r = 0       | c = substituting for $A$ from $(2)$ }} {{eqn | ll= \\implies       | l = -B + D       | r = 0 }} {{eqn | n = 4       | ll= \\implies       | l = B       | r = D }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = B a^2 + D a^2       | r = 1       | c =  }} {{eqn | ll= \\implies       | l = 2 D a^2       | r = 1       | c = substituting for $B$ from $(4)$ }} {{eqn | ll= \\implies       | l = D       | r = \\frac 1 {2 a^2}       | c =  }} {{eqn | ll= \\implies       | l = B       | r = \\frac 1 {2 a^2}       | c = from $(4)$ }} {{eqn | ll= \\implies       | l = 2 C a \\sqrt 2 + \\frac 1 {a^2}       | r = 0       | c = substituting for $B$ and $D$ in $(3)$ }} {{eqn | ll= \\implies       | l = C       | r = \\frac {-1} {2 a^3 \\sqrt 2}       | c = substituting for $B$ and $D$ in $(3)$ }} {{eqn | ll= \\implies       | l = A       | r = \\frac 1 {2 a^3 \\sqrt 2}       | c = from $(2)$ }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac 1 {2 a^3 \\sqrt 2} }} {{eqn | l = B       | r = \\frac 1 {2 a^2} }} {{eqn | l = C       | r = \\frac {-1} {2 a^3 \\sqrt 2} }} {{eqn | l = D       | r = \\frac 1 {2 a^2} }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | l = \\frac 1 {x^4 + a^4}       | r = \\frac {\\frac 1 {2 a^3 \\sqrt 2} x + \\frac 1 {2 a^2} } {x^2 + a x \\sqrt 2 + a^2} + \\frac {\\frac {-1} {2 a^3 \\sqrt 2} x + \\frac 1 {2 a^2} } {x^2 - a x \\sqrt 2 + a^2} }} {{eqn | r = \\frac {x + a \\sqrt 2} {2 a^3 \\sqrt 2 \\left({x^2 + a x \\sqrt 2 + a^2}\\right)} - \\frac {x - a \\sqrt 2} {2 a^3 \\sqrt 2 \\left({x^2 - a x \\sqrt 2 + a^2}\\right)} }} {{end-eqn}} {{qed}} Category:Primitives involving x to the fourth plus or minus a to the fourth/Lemmata j8cyzt8lqhoaai6e99d6zbcf1e2lln2"}
+{"_id": "32525", "title": "Primitive of Reciprocal of x fourth plus a fourth/Lemma 1", "text": "Primitive of Reciprocal of x fourth plus a fourth/Lemma 1 0 34524 436976 436975 2019-11-27T16:02:34Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x^4 + a^4$ == :$\\displaystyle \\int \\frac {\\d x} {x^2 + a x \\sqrt 2 + a^2} = \\frac {\\sqrt 2} a \\, \\map \\arctan {1 + \\frac {x \\sqrt 2} a}$ == Proof == The discriminant of $x^2 + a x \\sqrt 2 + a^2$ is: {{begin-eqn}} {{eqn | l = \\map {\\operatorname {Disc} } {x^2 + a x \\sqrt 2 + a^2}       | r = \\paren {a \\sqrt 2}^2 - 4 \\times 1 \\times a^2       | c =  }} {{eqn | r = 2 a^2 - 4 a^2       | c =  }} {{eqn | r = - 2 a^2       | c =  }} {{eqn | r = < 0       | c =  }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | l = \\int \\frac {\\d x} {x^2 + a x \\sqrt 2 + a^2}       | r = \\frac 2 {\\sqrt {4 a^2 - \\paren {a \\sqrt 2}^2} } \\map \\arctan {\\frac {2 x + a \\sqrt 2} {\\sqrt {4 a^2 - \\paren {a \\sqrt 2}^2} } }       | c = Primitive of $\\dfrac 1 {a x^2 + b x + c}$ }} {{eqn | r = \\frac 2 {a \\sqrt 2} \\, \\map \\arctan {\\frac {2 x + a \\sqrt 2} {a \\sqrt 2} }       | c = simplifying }} {{eqn | r = \\frac {\\sqrt 2} a \\, \\map \\arctan {1 + \\frac {x \\sqrt 2} a}       | c = simplifying }} {{end-eqn}} {{qed}} Category:Primitives involving x to the fourth plus or minus a to the fourth/Lemmata me6as1uajdq23e9etlwa77ztuuqvnce"}
+{"_id": "32526", "title": "Primitive of Reciprocal of x fourth plus a fourth/Lemma 2", "text": "Primitive of Reciprocal of x fourth plus a fourth/Lemma 2 0 34525 295966 190446 2017-05-02T21:35:43Z Barto 3079 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $x^4 + a^4$ == :$\\displaystyle \\int \\frac {\\mathrm d x} {x^2 - a x \\sqrt 2 + a^2} = \\frac {-\\sqrt 2} a \\arctan \\left({1 - \\frac {x \\sqrt 2} a}\\right)$ == Proof == The discriminant of $x^2 - a x \\sqrt 2 + a^2$ is: {{begin-eqn}} {{eqn | l = \\operatorname{Disc} \\left({x^2 - a x \\sqrt 2 + a^2}\\right)       | r = \\left({-a \\sqrt 2}\\right)^2 - 4 \\times 1 \\times a^2       | c =  }} {{eqn | r = 2 a^2 - 4 a^2       | c =  }} {{eqn | r = - 2 a^2       | c =  }} {{eqn | r = < 0       | c =  }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | l = \\int \\frac {\\mathrm d x} {x^2 - a x \\sqrt 2 + a^2}       | r = \\frac 2 {\\sqrt {4 a^2 - \\left({-a \\sqrt 2}\\right)^2} } \\arctan \\left({\\frac {2 x - a \\sqrt 2} {\\sqrt {4 a^2 - \\left({-a \\sqrt 2}\\right)^2} } }\\right)       | c = Primitive of $\\dfrac 1 {a x^2 + b x + c}$ }} {{eqn | r = \\frac 2 {a \\sqrt 2} \\arctan \\left({\\frac {2 x - a \\sqrt 2} {a \\sqrt 2} }\\right)       | c = simplifying }} {{eqn | r = \\frac {\\sqrt 2} a \\arctan \\left({\\frac {x \\sqrt 2} a - 1}\\right)       | c = simplifying }} {{eqn | r = \\frac {-\\sqrt 2} a \\arctan \\left({1 - \\frac {x \\sqrt 2} a}\\right)       | c = Arctangent is Odd Function }} {{end-eqn}} {{qed}} Category:Primitives involving x to the fourth plus or minus a to the fourth/Lemmata 81h5r7qmelpci09qmsm9pr12wm50mp2"}
+{"_id": "32527", "title": "Primitive of x squared over x fourth plus a fourth/Partial Fraction Expansion", "text": "Primitive of x squared over x fourth plus a fourth/Partial Fraction Expansion 0 34526 190454 2014-07-22T20:43:09Z Prime.mover 59 Created page with \"== Lemma for Primitive of $\\dfrac {x^2} {x^4 + a^4}$ ==  :$\\dfrac {x^2} {x^4 + a^4} = \\dfrac x {2 a \\sqrt 2 \\left({x^2 -...\" wikitext text/x-wiki == Lemma for Primitive of $\\dfrac {x^2} {x^4 + a^4}$ == :$\\dfrac {x^2} {x^4 + a^4} = \\dfrac x {2 a \\sqrt 2 \\left({x^2 - a x \\sqrt 2 + a^2}\\right)} - \\dfrac x {2 a \\sqrt 2 \\left({x^2 + a x \\sqrt 2 + a^2}\\right)}$ == Proof == {{begin-eqn}} {{eqn | l = \\frac {x^2} {x^4 + a^4}       | o = \\equiv       | r = \\frac {x^2} {\\left({x^2 + a x \\sqrt 2 + a^2}\\right) \\left({x^2 - a x \\sqrt 2 + a^2}\\right)}       | c = Sum of Two Fourth Powers }} {{eqn | o = \\equiv       | r = \\frac {A x + B} {x^2 + a x \\sqrt 2 + a^2} + \\frac {C x + D} {x^2 - a x \\sqrt 2 + a^2}       | c =  }} {{eqn | ll= \\implies       | l = x^2       | o = \\equiv       | r = \\left({A x + B}\\right) \\left({x^2 - a x \\sqrt 2 + a^2}\\right) + \\left({C x + D}\\right) \\left({x^2 + a x \\sqrt 2 + a^2}\\right)       | c =  }} {{eqn | n = 1       | o = \\equiv       | r = A x^3 + B x^2 - A a x^2 \\sqrt 2 - B a x \\sqrt 2 + A x a^2 + B a^2       | c =  }} {{eqn | o =        | ro= +       | r = C x^3 + D x^2 + C a x^2 \\sqrt 2 + D a x \\sqrt 2 + C x a^2 + D a^2       | c =  }} {{end-eqn}} Equating coefficients of $x^3$ in $(1)$: {{begin-eqn}} {{eqn | l = A + C       | r = 0       | c =  }} {{eqn | n = 2       | ll= \\implies       | l = -A       | r = C       | c =  }} {{end-eqn}} Equating coefficients of $x^2$ in $(1)$: {{begin-eqn}} {{eqn | l = -A a \\sqrt 2 + B + C a \\sqrt 2 + D       | r = 1       | c =  }} {{eqn | n = 3       | ll= \\implies       | l = 2 C a \\sqrt 2 + B + D       | r = 1       | c = substituting for $A$ from $(2)$ }} {{end-eqn}} Equating coefficients of $x$ in $(1)$: {{begin-eqn}} {{eqn | l = -B a \\sqrt 2 + A a^2 + D a \\sqrt 2 + C a^2       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = -B a \\sqrt 2 - C a^2 + D a \\sqrt 2 + C a^2       | r = 0       | c = substituting for $A$ from $(2)$ }} {{eqn | ll= \\implies       | l = -B + D       | r = 0 }} {{eqn | n = 4       | ll= \\implies       | l = B       | r = D }} {{end-eqn}} Setting $x = 0$ in $(1)$: {{begin-eqn}} {{eqn | l = B a^2 + D a^2       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = 2 D a^2       | r = 0       | c = substituting for $B$ from $(4)$ }} {{eqn | ll= \\implies       | l = D       | r = 0       | c =  }} {{eqn | ll= \\implies       | l = B       | r = 0       | c = from $(4)$ }} {{eqn | ll= \\implies       | l = 2 C a \\sqrt 2       | r = 1       | c = substituting for $B$ and $D$ in $(3)$ }} {{eqn | ll= \\implies       | l = C       | r = \\frac 1 {2 a \\sqrt 2}       | c =  }} {{eqn | ll= \\implies       | l = A       | r = \\frac {-1} {2 a \\sqrt 2}       | c = from $(2)$ }} {{end-eqn}} Summarising: {{begin-eqn}} {{eqn | l = A       | r = \\frac {-1} {2 a \\sqrt 2} }} {{eqn | l = B       | r = 0 }} {{eqn | l = C       | r = \\frac 1 {2 a \\sqrt 2} }} {{eqn | l = D       | r = 0 }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | l = \\frac 1 {x^4 + a^4}       | r = \\frac {\\frac {-1} {2 a \\sqrt 2} x} {x^2 + a x \\sqrt 2 + a^2} + \\frac {\\frac 1 {2 a \\sqrt 2} x} {x^2 - a x \\sqrt 2 + a^2} }} {{eqn | r = \\frac x {2 a \\sqrt 2 \\left({x^2 - a x \\sqrt 2 + a^2}\\right)} - \\frac x {2 a \\sqrt 2 \\left({x^2 + a x \\sqrt 2 + a^2}\\right)} }} {{end-eqn}} {{qed}} Category:Primitives involving x to the fourth plus or minus a to the fourth/Lemmata r1y6egprou63bvbk2vptndw85mpst2f"}
+{"_id": "32528", "title": "Primitive of Reciprocal of p plus q by Sine of a x/Weierstrass Substitution", "text": "Primitive of Reciprocal of p plus q by Sine of a x/Weierstrass Substitution 0 34543 191926 191924 2014-08-22T21:43:43Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $p + q \\sin a x$ == The Weierstrass Substitution of $\\displaystyle \\int \\frac {\\mathrm d x} {p + q \\sin a x}$ is: :$\\displaystyle \\frac 2 a \\int \\frac {\\mathrm d u} {p u^2 + 2 q u + p}$ where $u = \\tan \\dfrac {a x} 2$. == Proof == {{begin-eqn}} {{eqn | l = \\int \\frac {\\mathrm d x} {p + q \\sin a x}       | r = \\frac 1 a \\int \\frac {\\mathrm d z} {p + q \\sin z}       | c = Primitive of Function of Constant Multiple: $z = a x$ }} {{eqn | r = \\frac 1 a \\int \\frac 1 {p + q \\frac {2 u} {u^2 + 1} } \\frac {2 \\ \\mathrm d u} {u^2 + 1}       | c = Weierstrass Substitution: $u = \\tan \\dfrac z 2 = \\tan \\dfrac {a x} 2$ }} {{eqn | r = \\frac 1 a \\int \\frac {2 \\ \\mathrm d u} {\\left({u^2 + 1}\\right) \\frac {p \\left({u^2 + 1}\\right) + 2 q u} {u^2 + 1} }       | c = common denominator }} {{eqn | r = \\frac 2 a \\int \\frac {\\mathrm d u} {p u^2 + 2 q u + p}       | c = simplifying }} {{end-eqn}} {{qed}} Category:Primitives involving Sine Function Category:Weierstrass Substitutions 217lo91fbifomc7bs74shv7co8s8a71"}
+{"_id": "32529", "title": "Primitive of Reciprocal of square of p plus q by Sine of a x/Weierstrass Substitution", "text": "Primitive of Reciprocal of square of p plus q by Sine of a x/Weierstrass Substitution 0 34545 454008 192065 2020-03-12T08:10:10Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $\\paren {p + q \\sin a x}^2$ == The Weierstrass Substitution of $\\displaystyle \\int \\frac {\\d x} {\\paren {p + q \\sin a x}^2}$ is: :$\\displaystyle \\frac 2 a \\int \\frac {\\paren {u^2 + 1} \\rd u} {\\paren {p u^2 + 2 q u + p}^2}$ where $u = \\tan \\dfrac {a x} 2$. == Proof == {{begin-eqn}} {{eqn | l = \\int \\frac {\\d x} {\\paren {p + q \\sin a x}^2}       | r = \\frac 1 a \\int \\frac {\\d z} {\\paren {p + q \\sin z}^2}       | c = Primitive of Function of Constant Multiple: $z = a x$ }} {{eqn | r = \\frac 1 a \\int \\frac 1 {\\paren {p + q \\frac {2 u} {u^2 + 1} }^2} \\frac {2 \\rd u} {u^2 + 1}       | c = Weierstrass Substitution: $u = \\tan \\dfrac z 2 = \\tan \\dfrac {a x} 2$ }} {{eqn | r = \\frac 1 a \\int \\frac {2 \\rd u} {\\paren {u^2 + 1} \\paren {\\frac {p \\paren {u^2 + 1} + 2 q u} {u^2 + 1} }^2}       | c = common denominator }} {{eqn | r = \\frac 2 a \\int \\frac {\\paren {u^2 + 1} \\rd u} {\\paren {p u^2 + 2 q u + p}^2}       | c = simplifying }} {{end-eqn}} {{qed}} Category:Primitives involving Sine Function Category:Weierstrass Substitutions 25g9i93frf69kzc9w7ufgd6qeccuw85"}
+{"_id": "32530", "title": "Primitive of Exponential of a x by Sine of b x/Lemma", "text": "Primitive of Exponential of a x by Sine of b x/Lemma 0 34772 405304 191960 2019-05-19T08:26:14Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of $e^{a x} \\sin b x$ == <onlyinclude> :$\\displaystyle \\int e^{a x} \\sin b x \\rd x = \\frac {e^{a x} \\sin b x} a - \\frac b a \\int e^{a x} \\cos b x \\rd x$ </onlyinclude> == Proof == With a view to expressing the primitive in the form: :$\\displaystyle \\int u \\frac {\\d v} {\\d x} \\rd x = u v - \\int v \\frac {\\d u} {\\d x} \\rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \\sin b x       | c =  }} {{eqn | ll= \\leadsto       | l = \\frac {\\d u} {\\d x}       | r = b \\cos b x       | c = Derivative of $\\sin a x$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \\frac {\\d v} {\\ d x}       | r = e^{a x}       | c =  }} {{eqn | ll= \\leadsto       | l = v       | r = \\frac {e^{a x} } a       | c = Primitive of $e^{a x}$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \\int e^{a x} \\sin b x \\rd x       | r = \\sin b x \\paren {\\frac {e^{a x} } a} - \\int \\paren {\\frac {e^{a x} } a} \\paren {b \\cos b x} \\rd x + C       | c = Integration by Parts }} {{eqn | r = \\frac {e^{a x} \\sin b x} a - \\frac b a \\int e^{a x} \\cos b x \\rd x       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} {{qed}} Category:Primitive of Exponential of a x by Sine of b x 6osx4jjp5kxtrrj01u72mbjop8nfqsi"}
+{"_id": "32531", "title": "Primitive of Exponential of a x by Cosine of b x/Lemma", "text": "Primitive of Exponential of a x by Cosine of b x/Lemma 0 34773 405305 191965 2019-05-19T08:27:37Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of $e^{a x} \\cos b x$ == <onlyinclude> :$\\displaystyle \\int e^{a x} \\cos b x \\rd x = \\frac {e^{a x} \\cos b x} a + \\frac b a \\int e^{a x} \\sin b x \\rd x$ </onlyinclude> == Proof == With a view to expressing the primitive in the form: :$\\displaystyle \\int u \\frac {\\d v}{\\d x} \\rd x = u v - \\int v \\frac {\\d u} {\\d x} \\rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \\cos b x       | c =  }} {{eqn | ll= \\leadsto       | l = \\frac {\\d u} {\\d x}       | r = -b \\sin b x       | c = Derivative of $\\cos a x$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \\frac {\\d v} {\\d x}       | r = e^{a x}       | c =  }} {{eqn | ll= \\leadsto       | l = v       | r = \\frac {e^{a x} } a       | c = Primitive of $e^{a x}$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \\int e^{a x} \\cos b x \\rd x       | r = \\cos b x \\paren {\\frac {e^{a x} } a} - \\int \\paren {\\frac {e^{a x} } a} \\paren {-b \\sin b x} \\rd x + C       | c = Integration by Parts }} {{eqn | r = \\frac {e^{a x} \\cos b x} a + \\frac b a \\int e^{a x} \\sin b x \\rd x       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} {{qed}} Category:Primitive of Exponential of a x by Cosine of b x j0ddqk8949wn098gpmtnh5cdyjfilus"}
+{"_id": "32532", "title": "Primitive of Exponential of a x by Power of Sine of b x/Lemma 1", "text": "Primitive of Exponential of a x by Power of Sine of b x/Lemma 1 0 34784 447531 289822 2020-02-08T19:50:21Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of $e^{a x} \\sin^n b x \\cos b x$ == :$\\displaystyle \\int e^{a x} \\sin^{n - 1} b x \\cos b x \\rd x = \\frac {e^{a x} \\sin^{n - 1} b x \\paren {a \\cos b x + b \\sin b x} } {a^2 + n b^2} + \\frac {\\paren {n - 1} a b} {a^2 + n b^2} \\paren {\\int e^{a x} \\sin^n b x \\rd x - \\int e^{a x} \\sin^{n - 2} b x \\rd x} + C$ == Proof == With a view to expressing the primitive in the form: :$\\displaystyle \\int u \\frac {\\d v} {\\d x} \\rd x = u v - \\int v \\frac {\\d u} {\\d x} \\rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \\sin^{n - 1} b x       | c =  }} {{eqn | ll= \\leadsto       | l = \\frac {\\d u} {\\d x}       | r = \\paren {n - 1} b \\sin^{n - 2} b x \\cos b x       | c = Derivative of $\\sin a x$, Derivative of Power, Chain Rule for Derivatives }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \\frac {\\d v} {\\d x}       | r = e^{a x} \\cos b x       | c =  }} {{eqn | ll= \\leadsto       | l = v       | r = \\frac {e^{a x} \\paren {a \\cos b x + b \\sin b x} } {a^2 + b^2}       | c = Primitive of $e^{a x} \\cos b x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | r = \\int e^{a x} \\sin^{n - 1} b x \\cos b x \\rd x       | o =        | c =  }} {{eqn | r = \\int \\paren {\\sin^{n - 1} b x} \\paren {e^{a x} \\cos b x} \\rd x       | c =  }} {{eqn | r = \\paren {\\sin^{n - 1} b x} \\paren {\\frac {e^{a x} \\paren {a \\cos b x + b \\sin b x} } {a^2 + b^2} }       | c = Integration by Parts }} {{eqn | o =        | ro= -       | r = \\int \\paren {\\frac {e^{a x} \\paren {a \\cos b x + b \\sin b x} } {a^2 + b^2} } \\paren {\\paren {n - 1} b \\sin^{n - 2} b x \\cos b x} \\rd x + C       | c =  }} {{eqn | r = \\frac {e^{a x} \\sin^{n - 1} b x \\paren {a \\cos b x + b \\sin b x} } {a^2 + b^2}       | c =  }} {{eqn | o =        | ro= -       | r = \\frac {\\paren {n - 1} a b} {a^2 + b^2} \\int e^{a x} \\sin^{n - 2} b x \\cos^2 b x \\rd x + C       | c = Linear Combination of Integrals }} {{eqn | o =        | ro= -       | r = \\frac {\\paren {n - 1} b^2} {a^2 + b^2} \\int e^{a x} \\sin^{n - 1} b x \\cos b x \\rd x + C       | c =  }} {{eqn | ll= \\leadsto       | r = \\paren {1 + \\frac {\\paren {n - 1} b^2} {a^2 + b^2} } \\int e^{a x} \\sin^{n - 1} b x \\cos b x \\rd x       | o =        | c = gathering terms }} {{eqn | r = \\frac {e^{a x} \\sin^{n - 1} b x \\paren {a \\cos b x + b \\sin b x} } {a^2 + b^2}       | c =  }} {{eqn | o =        | ro= -       | r = \\frac {\\paren {n - 1} a b} {a^2 + b^2} \\int e^{a x} \\sin^{n - 2} b x \\cos^2 b x \\rd x + C       | c =  }} {{eqn | ll= \\leadsto       | r = \\paren {a^2 + n b^2} \\int e^{a x} \\sin^{n - 1} b x \\cos b x \\rd x       | o =        | c =  }} {{eqn | r = e^{a x} \\sin^{n - 1} b x \\paren {a \\cos b x + b \\sin b x}       | c = simplifying }} {{eqn | o =        | ro= -       | r = \\paren {n - 1} a b \\int e^{a x} \\sin^{n - 2} b x \\cos^2 b x \\rd x + C       | c =  }} {{eqn | r = e^{a x} \\sin^{n - 1} b x \\paren {a \\cos b x + b \\sin b x}       | c =  }} {{eqn | o =        | ro= -       | r = \\paren {n - 1} a b \\int e^{a x} \\sin^{n - 2} b x \\paren {1 - \\sin^2 b x} \\rd x + C       | c = Sum of Squares of Sine and Cosine }} {{eqn | r = e^{a x} \\sin^{n - 1} b x \\paren {a \\cos b x + b \\sin b x}       | c = simplifying }} {{eqn | o =        | ro= -       | r = \\paren {n - 1} a b \\int e^{a x} \\sin^{n - 2} b x \\rd x       | c =  }} {{eqn | o =        | ro= +       | r = \\paren {n - 1} a b \\int e^{a x} \\sin^n b x \\rd x + C       | c =  }} {{eqn | ll= \\leadsto       | r = \\int e^{a x} \\sin^{n - 1} b x \\cos b x \\rd x       | o =        | c =  }} {{eqn | r = \\frac {e^{a x} \\sin^{n - 1} b x \\paren {a \\cos b x + b \\sin b x} } {a^2 + n b^2}       | c = simplifying }} {{eqn | o =        | ro= -       | r = \\frac {\\paren {n - 1} a b} {a^2 + n b^2} \\int e^{a x} \\sin^{n - 2} b x \\rd x       | c =  }} {{eqn | o =        | ro= +       | r = \\frac {\\paren {n - 1} a b} {a^2 + n b^2} \\int e^{a x} \\sin^n b x \\rd x + C       | c =  }} {{end-eqn}} and so rearranging: :$\\displaystyle \\int e^{a x} \\sin^{n - 1} b x \\cos b x \\rd x = \\frac {e^{a x} \\sin^{n - 1} b x \\paren {a \\cos b x + b \\sin b x} } {a^2 + n b^2} + \\frac {\\paren {n - 1} a b} {a^2 + n b^2} \\paren {\\int e^{a x} \\sin^n b x \\rd x - \\int e^{a x} \\sin^{n - 2} b x \\rd x} + C$ {{qed}} Category:Primitive of Exponential of a x by Power of Sine of b x t1hfy192myllxo19rvpgybts5swykcq"}
+{"_id": "32533", "title": "Primitive of Exponential of a x by Power of Sine of b x/Lemma 2", "text": "Primitive of Exponential of a x by Power of Sine of b x/Lemma 2 0 34788 191953 191154 2014-08-23T12:28:51Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of $e^{a x} \\sin^n b x$ == :$\\dfrac {a^2 + n b^2} a e^{a x} \\sin^n b x - \\dfrac {n b} a e^{a x} \\sin^{n - 1} b x \\left({a \\cos b x + b \\sin b x}\\right) = e^{a x} \\sin^{n - 1} b x \\left({a \\sin b x - n b \\cos b x}\\right)$ == Proof == {{begin-eqn}} {{eqn | r = \\frac {a^2 + n b^2} a e^{a x} \\sin^n b x - \\frac {n b} a e^{a x} \\sin^{n - 1} b x \\left({a \\cos b x + b \\sin b x}\\right)       | c =  }} {{eqn | r = \\frac {a^2 + n b^2} a e^{a x} \\sin^{n - 1} b x \\sin b x - \\frac {n b} a e^{a x} \\sin^{n - 1} b x a \\cos b x - \\frac {n b} a e^{a x} \\sin^{n - 1} b x b \\sin b x       | c =  }} {{eqn | r = e^{a x} \\sin^{n - 1} b x \\left({\\frac {a^2 + n b^2} a \\sin b x - \\frac {n b} a a \\cos b x - \\frac {n b} a b \\sin b x}\\right)       | c =  }} {{eqn | r = e^{a x} \\sin^{n - 1} b x \\left({a \\sin b x + \\frac {n b^2} a \\sin b x - n b \\cos b x - \\frac {n b^2} a \\sin b x}\\right)       | c =  }} {{eqn | r = e^{a x} \\sin^{n - 1} b x \\left({a \\sin b x - n b \\cos b x}\\right)       | c =  }} {{end-eqn}} {{qed}} Category:Primitive of Exponential of a x by Power of Sine of b x j3w4z35t9a8n9n75vht7p4s0dj5af6j"}
+{"_id": "32534", "title": "Primitive of Exponential of a x by Power of Cosine of b x/Lemma 1", "text": "Primitive of Exponential of a x by Power of Cosine of b x/Lemma 1 0 34789 447530 289821 2020-02-08T19:43:36Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of $e^{a x} \\cos b x$ == :$\\displaystyle \\int e^{a x} \\cos^{n - 1} b x \\sin b x \\rd x = \\frac {e^{a x} \\cos^{n - 1} b x \\paren {a \\sin b x - b \\cos b x} } {a^2 + n b^2} - \\frac {\\paren {n - 1} a b} {a^2 + n b^2} \\paren {\\int e^{a x} \\cos^n b x \\rd x - \\int e^{a x} \\cos^{n - 2} b x \\rd x} + C$ == Proof == With a view to expressing the primitive in the form: :$\\displaystyle \\int u \\frac {\\d v} {\\d x} \\rd x = u v - \\int v \\frac {\\d u} {\\d x} \\rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \\cos^{n - 1} b x       | c =  }} {{eqn | ll= \\leadsto       | l = \\frac {\\d u} {\\d x}       | r = -\\paren {n - 1} b \\cos^{n - 2} b x \\sin b x       | c = Derivative of $\\cos a x$, Derivative of Power, Chain Rule for Derivatives }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \\frac {\\d v} {\\d x}       | r = e^{a x} \\sin b x       | c =  }} {{eqn | ll= \\leadsto       | l = v       | r = \\frac {e^{a x} \\paren {a \\sin b x - b \\cos b x} } {a^2 + b^2}       | c = Primitive of $e^{a x} \\sin b x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | r = \\int e^{a x} \\cos^{n - 1} b x \\sin b x \\rd x       | o =        | c =  }} {{eqn | r = \\int \\paren {\\cos^{n - 1} b x} \\paren {e^{a x} \\sin b x} \\rd x       | c =  }} {{eqn | r = \\paren {\\cos^{n - 1} b x} \\paren {\\frac {e^{a x} \\paren {a \\sin b x - b \\cos b x} } {a^2 + b^2} }       | c = Integration by Parts }} {{eqn | o =        | ro= -       | r = \\int \\paren {\\frac {e^{a x} \\paren {a \\sin b x - b \\cos b x} } {a^2 + b^2} } \\paren {-\\paren {n - 1} b \\cos^{n - 2} b x \\sin b x} \\rd x + C       | c =  }} {{eqn | r = \\frac {e^{a x} \\cos^{n - 1} b x \\paren {a \\sin b x - b \\cos b x} } {a^2 + b^2}       | c =  }} {{eqn | o =        | ro= +       | r = \\frac {\\paren {n - 1} a b} {a^2 + b^2} \\int e^{a x} \\cos^{n - 2} b x \\sin^2 b x \\rd x + C       | c = Linear Combination of Integrals }} {{eqn | o =        | ro= -       | r = \\frac {\\paren {n - 1} b^2} {a^2 + b^2} \\int e^{a x} \\cos^{n - 1} b x \\sin b x \\rd x + C       | c =  }} {{eqn | ll= \\leadsto       | r = \\paren {1 + \\frac {\\paren {n - 1} b^2} {a^2 + b^2} } \\int e^{a x} \\cos^{n - 1} b x \\sin b x \\rd x       | o =        | c = gathering terms }} {{eqn | r = \\frac {e^{a x} \\cos^{n - 1} b x \\paren {a \\sin b x - b \\cos b x} } {a^2 + b^2}       | c =  }} {{eqn | o =        | ro= +       | r = \\frac {\\paren {n - 1} a b} {a^2 + b^2} \\int e^{a x} \\cos^{n - 2} b x \\sin^2 b x \\rd x + C       | c =  }} {{eqn | ll= \\leadsto       | r = \\paren {a^2 + n b^2} \\int e^{a x} \\cos^{n - 1} b x \\sin b x \\rd x       | o =        | c =  }} {{eqn | r = e^{a x} \\cos^{n - 1} b x \\paren {a \\sin b x - b \\cos b x}       | c = simplifying }} {{eqn | o =        | ro= +       | r = \\paren {n - 1} a b \\int e^{a x} \\cos^{n - 2} b x \\sin^2 b x \\rd x + C       | c =  }} {{eqn | r = e^{a x} \\cos^{n - 1} b x \\paren {a \\sin b x - b \\cos b x}       | c =  }} {{eqn | o =        | ro= +       | r = \\paren {n - 1} a b \\int e^{a x} \\cos^{n - 2} b x \\paren {1 - \\cos^2 b x} \\rd x + C       | c = Sum of Squares of Sine and Cosine }} {{eqn | r = e^{a x} \\cos^{n - 1} b x \\paren {a \\sin b x - b \\cos b x}       | c = simplifying }} {{eqn | o =        | ro= +       | r = \\paren {n - 1} a b \\int e^{a x} \\cos^{n - 2} b x \\rd x       | c =  }} {{eqn | o =        | ro= -       | r = \\paren {n - 1} a b \\int e^{a x} \\cos^n b x \\rd x + C       | c =  }} {{eqn | ll= \\leadsto       | r = \\int e^{a x} \\cos^{n - 1} b x \\sin b x \\rd x       | o =        | c =  }} {{eqn | r = \\frac {e^{a x} \\cos^{n - 1} b x \\paren {a \\sin b x - b \\cos b x} } {a^2 + n b^2}       | c = simplifying }} {{eqn | o =        | ro= +       | r = \\frac {\\paren {n - 1} a b} {a^2 + n b^2} \\int e^{a x} \\cos^{n - 2} b x \\rd x       | c =  }} {{eqn | o =        | ro= -       | r = \\frac {\\paren {n - 1} a b} {a^2 + n b^2} \\int e^{a x} \\cos^n b x \\rd x + C       | c =  }} {{end-eqn}} and so rearranging: :$\\displaystyle \\int e^{a x} \\cos^{n - 1} b x \\sin b x \\rd x = \\frac {e^{a x} \\cos^{n - 1} b x \\paren {a \\sin b x - b \\cos b x} } {a^2 + n b^2} - \\frac {\\paren {n - 1} a b} {a^2 + n b^2} \\paren {\\int e^{a x} \\cos^n b x \\rd x - \\int e^{a x} \\cos^{n - 2} b x \\rd x} + C$ {{qed}} Category:Primitive of Exponential of a x by Power of Cosine of b x a68b4cw9jmb8vmt4983mb5qbxkyqqnh"}
+{"_id": "32535", "title": "Primitive of Exponential of a x by Power of Cosine of b x/Lemma 2", "text": "Primitive of Exponential of a x by Power of Cosine of b x/Lemma 2 0 34790 438886 191959 2019-12-10T17:12:58Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of $e^{a x} \\cos^n b x$ == :$\\dfrac {a^2 + n b^2} a e^{a x} \\cos^n b x + \\dfrac {n b} a e^{a x} \\cos^{n - 1} b x \\paren {a \\sin b x - b \\cos b x} = e^{a x} \\cos^{n - 1} b x \\paren {a \\cos b x + n b \\sin b x}$ == Proof == {{begin-eqn}} {{eqn | r = \\dfrac {a^2 + n b^2} a e^{a x} \\cos^n b x + \\dfrac {n b} a e^{a x} \\cos^{n - 1} b x \\paren {a \\sin b x - b \\cos b x}       | c =  }} {{eqn | r = \\frac {a^2 + n b^2} a e^{a x} \\cos^{n - 1} b x \\cos b x + \\frac {n b} a e^{a x} \\cos^{n - 1} b x a \\sin b x - \\frac {n b} a e^{a x} \\cos^{n - 1} b x b \\cos b x       | c =  }} {{eqn | r = e^{a x} \\cos^{n - 1} b x \\paren {\\frac {a^2 + n b^2} a \\cos b x + \\frac {n b} a a \\sin b x - \\frac {n b} a b \\cos b x}       | c =  }} {{eqn | r = e^{a x} \\cos^{n - 1} b x \\paren {a \\cos b x + \\frac {n b^2} a \\cos b x + n b \\sin b x - \\frac {n b^2} a \\cos b x}       | c =  }} {{eqn | r = e^{a x} \\cos^{n - 1} b x \\paren {a \\cos b x + n b \\sin b x}       | c =  }} {{end-eqn}} {{qed}} Category:Primitive of Exponential of a x by Power of Cosine of b x dn1an85z8kzr7d5l6xopz30ave4yqhd"}
+{"_id": "32536", "title": "Primitive of Power of Hyperbolic Tangent of a x", "text": "Primitive of Power of Hyperbolic Tangent of a x 0 34887 191348 2014-08-12T07:05:34Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> :$\\displaystyle \\int \\tanh^n a x \\ \\mathrm d x = \\frac {-\\tanh^{n - 1} a x} {a \\left({n - 1}\\right)} + \\int \\tanh^{n - 2} a x \\ \\mathrm d x + C$ </...\" wikitext text/x-wiki == Theorem == <onlyinclude> :$\\displaystyle \\int \\tanh^n a x \\ \\mathrm d x = \\frac {-\\tanh^{n - 1} a x} {a \\left({n - 1}\\right)} + \\int \\tanh^{n - 2} a x \\ \\mathrm d x + C$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\int \\tanh^n a x \\ \\mathrm d x       | r = \\int \\tanh^{n - 2} a x \\tanh^2 a x \\ \\mathrm d x       | c =  }} {{eqn | r = \\int \\tanh^{n - 2} a x \\left({1 - \\operatorname{sech}^2 a x}\\right) \\ \\mathrm d x       | c = Sum of Squares of Hyperbolic Secant and Tangent }} {{eqn | r = -\\int \\tanh^{n - 2} a x \\operatorname{sech}^2 a x \\ \\mathrm d x + \\int \\tanh^{n - 2} \\ \\mathrm d x       | c = Linear Combination of Integrals }} {{eqn | r = \\frac {-\\tanh^{n - 1} a x} {a \\left({n - 1}\\right)} + \\int \\tanh^{n - 2} a x \\ \\mathrm d x + C       | c = Primitive of $\\tanh^n a x \\operatorname{sech}^2 a x$ }} {{end-eqn}} {{qed}} == Also see == * Primitive of $\\sinh^n a x$ * Primitive of $\\cosh^n a x$ * Primitive of $\\coth^n a x$ * Primitive of $\\operatorname{sech}^n a x$ * Primitive of $\\operatorname{csch}^n a x$ == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev=Primitive of Reciprocal of p plus q by Hyperbolic Tangent of a x|next=Primitive of Hyperbolic Cotangent of a x}}: $\\S 14$: Integrals involving $\\tanh a x$: $14.614$ Category:Primitives involving Hyperbolic Tangent Function cv5oudgmta7w63zco5lo5gycx68zy61"}
+{"_id": "32537", "title": "Derivative of Tangent Function/Corollary", "text": "Derivative of Tangent Function/Corollary 0 34889 492829 485862 2020-10-06T06:37:48Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Tangent Function == <onlyinclude> :$\\map {\\dfrac \\d {\\d x} } {\\tan a x} = a \\sec^2 a x$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\map {\\dfrac \\d {\\d x} } {\\tan x}       | r = \\sec^2 x       | c = Derivative of $\\tan x$ }} {{eqn | ll= \\leadsto       | l = \\map {\\dfrac \\d {\\d x} } {\\tan a x}       | r = a \\sec^2 a x       | c = Derivative of Function of Constant Multiple }} {{end-eqn}} {{qed}} == Also see == * Derivative of $\\sin a x$ * Derivative of $\\cos a x$ * Derivative of $\\cot a x$ * Derivative of $\\sec a x$ * Derivative of $\\csc a x$ Category:Derivatives of Trigonometric Functions Category:Tangent Function qknl3w7pk5kotj74ifd5vefncx0gpv4"}
+{"_id": "32538", "title": "Derivative of Cotangent Function/Corollary", "text": "Derivative of Cotangent Function/Corollary 0 34891 485863 417400 2020-09-06T19:35:35Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Cotangent Function == <onlyinclude> :$\\map {\\dfrac \\d {\\d x} } {\\cot a x} = -a \\csc^2 a x$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\map {\\dfrac \\d {\\d x} } {\\cot x}       | r = -\\csc^2 x       | c = Derivative of $\\cot x$ }} {{eqn | ll= \\leadsto       | l = \\map {\\dfrac \\d {\\d x} } {\\cot a x}       | r = -a \\csc^2 a x       | c = Derivative of Function of Constant Multiple }} {{end-eqn}} {{qed}} == Also see == * Derivative of $\\sin a x$ * Derivative of $\\cos a x$ * Derivative of $\\tan a x$ * Derivative of $\\sec a x$ * Derivative of $\\csc a x$ Category:Derivatives of Trigonometric Functions Category:Cotangent Function j58vp1ogd2cewdu3mxdgtpjdh36ayc7"}
+{"_id": "32539", "title": "Hyperbolic Tangent Half-Angle Substitution for Cosine", "text": "Hyperbolic Tangent Half-Angle Substitution for Cosine 0 35045 494645 494644 2020-10-14T06:21:02Z Prime.mover 59 wikitext text/x-wiki == Corollary to Double Angle Formula for Hyperbolic Cosine == <onlyinclude> :$\\cosh 2 x = \\dfrac {1 + \\tanh^2 x}{1 - \\tanh^2 x}$ </onlyinclude> where $\\cosh$ and $\\tanh$ denote hyperbolic cosine and hyperbolic tangent respectively. == Proof == {{begin-eqn}} {{eqn | l = \\cosh 2 x       | r = \\cosh^2 x + \\sinh^2 x       | c = Double Angle Formula for Hyperbolic Cosine }} {{eqn | r = \\paren {\\cosh^2 x + \\sinh^2 x} \\frac {\\cosh^2 x} {\\cosh^2 x}       | c =  }} {{eqn | r = \\paren {1 + \\tanh^2 x} \\cosh^2 x       | c = {{Defof|Hyperbolic Tangent|index = 2}} }} {{eqn | r = \\frac {1 + \\tanh^2 x} {\\sech^2 x}       | c = {{Defof|Hyperbolic Secant|index = 2}} }} {{eqn | r = \\frac {1 + \\tanh^2 x} {1 - \\tanh^2 x}       | c = Sum of Squares of Hyperbolic Secant and Tangent }} {{end-eqn}} {{qed}} Category:Hyperbolic Cosine Function Category:Hyperbolic Tangent Half-Angle Substitutions cypg11rly023i8qefiqt1phsmly7j2a"}
+{"_id": "32540", "title": "Hyperbolic Tangent Half-Angle Substitution for Sine", "text": "Hyperbolic Tangent Half-Angle Substitution for Sine 0 35046 494642 494641 2020-10-14T06:18:15Z Prime.mover 59 wikitext text/x-wiki == Corollary to Double Angle Formula for Hyperbolic Sine == <onlyinclude> :$\\sinh 2 x = \\dfrac {2 \\tanh x} {1 - \\tanh^2 x}$ </onlyinclude> where $\\sin$ and $\\tan$ denote hyperbolic sine and hyperbolic tangent respectively. == Proof == {{begin-eqn}} {{eqn | l = \\sinh 2 x       | r = 2 \\sinh x \\cosh x       | c = Double Angle Formula for Hyperbolic Sine }} {{eqn | r = 2 \\sinh x \\cosh x \\frac {\\cosh x} {\\cosh x}       | c =  }} {{eqn | r = 2 \\tanh x \\cosh^2 x       | c = {{Defof|Hyperbolic Tangent|index = 2}} }} {{eqn | r = \\frac {2 \\tanh x} {\\operatorname{sech}^2 x}       | c = {{Defof|Hyperbolic Secant|index = 2}} }} {{eqn | r = \\frac {2 \\tanh x} {1 - \\tanh^2 x}       | c = Sum of Squares of Hyperbolic Secant and Tangent }} {{end-eqn}} {{qed}} Category:Hyperbolic Sine Function Category:Hyperbolic Tangent Half-Angle Substitutions o62ilw2z556idlwxcm8zfgjl0zsc3qb"}
+{"_id": "32541", "title": "Square Modulo 3/Corollary 1", "text": "Square Modulo 3/Corollary 1 0 35054 439743 367192 2019-12-16T16:15:20Z Prime.mover 59 wikitext text/x-wiki == Corollary to Square Modulo 3 == <onlyinclude> Let $x, y \\in \\Z$ be integers. Then: :$3 \\divides \\paren {x^2 + y^2} \\iff 3 \\divides x \\land 3 \\divides y$ where $3 \\divides x$ denotes that $3$ divides $x$. </onlyinclude> == Proof == === Sufficient Condition === Let $3 \\divides x \\land 3 \\divides y$. Then by definition of divisibility: :$x \\equiv 0 \\pmod 3$ and :$y \\equiv 0 \\pmod 3$ From Square Modulo 3: :$x^2 \\equiv 0 \\pmod 3$ and :$y^2 \\equiv 0 \\pmod 3$ Then from Modulo Addition is Well-Defined: :$\\paren {x^2 + y^2} \\equiv 0 \\pmod 3$ Thus: :$3 \\divides \\paren {x^2 + y^2}$ {{qed|lemma}} === Necessary Condition === Now suppose $3 \\divides \\paren {x^2 + y^2}$. Then by definition of divisibility: :$\\paren {x^2 + y^2} \\pmod 3$ From Square Modulo 3: : $x^2 \\equiv 0 \\pmod 3$ or $x^2 \\equiv 1 \\pmod 3$ and : $y^2 \\equiv 0 \\pmod 3$ or $y^2 \\equiv 1 \\pmod 3$ Thus the only way $\\paren {x^2 + y^2} \\equiv 0 \\pmod 3$ is for: : $x^2 \\equiv 0 \\pmod 3$ and: : $y^2 \\equiv 0 \\pmod 3$ {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Square Modulo 3|next = Square Modulo 3/Corollary 2}}: Chapter $2$: Some Properties of $\\Z$: Exercise $2.9$ Category:Square Modulo 3 1ypv1hvnzyj2k4taolsech0kh83dxlj"}
+{"_id": "32542", "title": "Primitive of Reciprocal of p plus q by Cosine of a x/Weierstrass Substitution", "text": "Primitive of Reciprocal of p plus q by Cosine of a x/Weierstrass Substitution 0 35065 435007 222169 2019-11-12T16:54:27Z Prime.mover 59 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $p + q \\cos a x$ == The Weierstrass Substitution of $\\displaystyle \\int \\frac {\\d x} {p + q \\cos a x}$ is: :$\\displaystyle \\frac 2 {a \\paren {p - q} } \\int \\frac {\\d u} {u^2 + \\dfrac {p + q} {p - q} }$ where $u = \\tan \\dfrac {a x} 2$. == Proof == {{begin-eqn}} {{eqn | l = \\int \\frac {\\d x} {p + q \\cos a x}       | r = \\frac 1 a \\int \\frac {\\d z} {p + q \\cos z}       | c = Primitive of Function of Constant Multiple: $z = a x$ }} {{eqn | r = \\frac 1 a \\int \\frac 1 {p + q \\paren {\\frac {1 - u^2} {1 + u^2} } } \\frac {2 \\rd u} {1 + u^2}       | c = Weierstrass Substitution: $u = \\tan \\dfrac z 2 = \\tan \\dfrac {a x} 2$ }} {{eqn | r = \\frac 1 a \\int \\frac {2 \\rd u} {\\paren {1 + u^2} \\paren {\\frac {p \\paren {1 + u^2} + q \\paren {1 - u^2} } {1 + u^2} } }       | c = common denominator }} {{eqn | r = \\frac 2 a \\int \\frac {\\d u} {\\paren {p - q} u^2 + \\paren {p + q} }       | c = simplifying }} {{eqn | r = \\frac 2 {a \\paren {p - q} } \\int \\frac {\\d u} {u^2 + \\dfrac {p + q} {p - q} }       | c = simplifying further }} {{end-eqn}} {{qed}} Category:Primitives involving Cosine Function Category:Weierstrass Substitutions a8btjt59amzygnx0tcakkfjvju57f9u"}
+{"_id": "32543", "title": "Primitive of Reciprocal of p plus q by Hyperbolic Sine of a x/Hyperbolic Tangent Half-Angle Substitution", "text": "Primitive of Reciprocal of p plus q by Hyperbolic Sine of a x/Hyperbolic Tangent Half-Angle Substitution 0 35075 191968 2014-08-23T15:33:05Z Prime.mover 59 Created page with \"== Lemma for Primitive of Reciprocal of $p + q \\sinh a x$ ==  The Hyperbolic Tangent Half-Angle Substitutio...\" wikitext text/x-wiki == Lemma for Primitive of Reciprocal of $p + q \\sinh a x$ == The Hyperbolic Tangent Half-Angle Substitution of $\\displaystyle \\int \\frac {\\mathrm d x} {p + q \\sinh a x}$ is: :$\\displaystyle \\frac 2 a \\int \\frac {\\mathrm d u} {p u^2 + 2 q u - p}$ where $u = \\tanh \\dfrac {a x} 2$. == Proof == {{begin-eqn}} {{eqn | l = \\int \\frac {\\mathrm d x} {p + q \\sinh a x}       | r = \\frac 1 a \\int \\frac {\\mathrm d z} {p + q \\sinh z}       | c = Primitive of Function of Constant Multiple: $z = a x$ }} {{eqn | r = \\frac 1 a \\int \\frac {\\dfrac {2 \\ \\mathrm d u} {u^2 - 1} } {p + q \\dfrac {2 u} {u^2 - 1} }       | c = Hyperbolic Tangent Half-Angle Substitution: $u = \\tanh \\dfrac z 2 = \\tanh \\dfrac {a x} 2$ }} {{eqn | r = \\frac 1 a \\int \\frac {2 \\ \\mathrm d u} {p \\left({u^2 - 1}\\right) + 2 q u}       | c = multiplying top and bottom by $u^2 - 1$ }} {{eqn | r = \\frac 2 a \\int \\frac {\\mathrm d u} {p u^2 + 2 q u - p}       | c = simplifying }} {{end-eqn}} {{qed}} Category:Primitives involving Hyperbolic Sine Function Category:Hyperbolic Tangent Half-Angle Substitutions 2jhgnn99npdwlb6twjpyt584qgwpeso"}
+{"_id": "32544", "title": "Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x/Lemma", "text": "Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x/Lemma 0 35117 257910 192217 2016-06-05T13:48:49Z Z423x5c6 2799 wikitext text/x-wiki == Lemma for Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x == :$\\displaystyle \\frac 1 2 \\left({\\arctan \\dfrac {-p} q}\\right) + \\frac \\pi 4 = \\frac {\\arctan \\dfrac q p} 2$ == Proof == {{begin-eqn}} {{eqn | l = y       | r = \\frac 1 2 \\left({\\arctan \\dfrac {-p} q}\\right) + \\frac \\pi 4       | c = making a definition }} {{eqn | ll= \\implies       | l = 2 y - \\frac \\pi 2       | r = \\arctan \\dfrac {-p} q       | c = rearranging }} {{eqn | ll= \\implies       | l = \\tan \\left({2 y - \\frac \\pi 2}\\right)       | r = \\dfrac {-p} q       | c = Definition of Arctangent }} {{eqn | ll= \\implies       | l = \\tan \\left({2 y + \\frac {3 \\pi} 2}\\right)       | r = \\dfrac {-p} q       | c = Tangent Function is Periodic on Reals }} {{eqn | ll= \\implies       | l = -\\cot \\left({2 y}\\right)       | r = \\dfrac {-p} q       | c = Tangent of Angle plus Three Right Angles }} {{eqn | ll= \\implies       | l = \\tan \\left({2 y}\\right)       | r = \\dfrac q p       | c = Cotangent is Reciprocal of Tangent }} {{eqn | ll= \\implies       | l = y       | r = \\frac {\\arctan \\dfrac q p} 2       | c = Definition of Arctangent }} {{end-eqn}} {{qed}} Category:Primitives involving Sine Function and Cosine Function iax7us9hl47oqc56p5un76hncihps5n"}
+{"_id": "32545", "title": "Cartesian Product of Countable Sets is Countable/Corollary/Proof 2", "text": "Cartesian Product of Countable Sets is Countable/Corollary/Proof 2 0 35214 340928 331093 2018-01-31T22:28:40Z Prime.mover 59 wikitext text/x-wiki == Corollary to Cartesian Product of Countable Sets is Countable == {{:Cartesian Product of Countable Sets is Countable/Corollary}} == Proof == <onlyinclude> Proof by induction: === Basis for the Induction === When $k = 2$, the case is the same as Cartesian Product of Countable Sets is Countable. So shown for basis for the induction. === Induction Hypothesis === This is our induction hypothesis: :$\\exists f_k: S_1 \\times S_2 \\times \\cdots \\times S_k \\to \\N$ where $f_k$ is an injection. Now we need to show that for $n = k + 1$: :$\\exists f_{k+1}: S_1 \\times S_2 \\times \\cdots \\times S_k \\times S_{k+1} \\to \\N$ where $f_{k+1}$ is an injection. === Induction Step === This is our induction step: By the induction hypothesis: :$\\exists f_k: S_1 \\times S_2 \\times \\cdots \\times S_k \\to \\N$ where $f_k$ is an injection. Thus by definition, $S_1 \\times S_2 \\times \\cdots \\times S_k$ is countable. By hypothesis $S_{k + 1}$ is countable. So by the basis for the induction: :$\\exists g: \\left({S_1 \\times S_2 \\times \\cdots \\times S_k}\\right) \\times S_{k+1} \\to \\N \\times \\N$ where $g$ is an injection. By Cartesian Product of Countable Sets is Countable, :$\\exists r: \\N \\times \\N \\to \\N$ where $r$ is an injection. Therefore, by Composite of Injections is Injection: :$f_{k+1} = r \\circ g: S_1 \\times S_2 \\times \\cdots \\times S_k \\times S_{k+1} \\to \\N$ is an injection. The result follows by induction. {{qed}} </onlyinclude> == Sources == * {{BookReference|Topology|2000|James R. Munkres|ed = 2nd|edpage = Second Edition|prev = Cartesian Product of Countable Sets is Countable|next = }}: $1$: Set Theory and Logic: $\\S 7$: Countable and Uncountable Sets: Theorem $7.6$ Category:Cartesian Product of Countable Sets is Countable fcwzd4m1hzib8tl1g1is5ax6aiuy7cb"}
+{"_id": "32546", "title": "Finding Center of Circle/Porism", "text": "Finding Center of Circle/Porism 0 35372 376817 196554 2018-11-15T20:19:14Z Caliburn 3218 wikitext text/x-wiki == Porism to Finding Center of Circle == <onlyinclude> {{:Euclid:Proposition/III/1/Porism}} </onlyinclude> == Proof == See Perpendicular Bisector of Chord Passes Through Center, where the proof has been extracted from the proof given by Euclid. == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 2|1926|ed=2nd|edpage=Second Edition|Sir Thomas L. Heath|prev=Finding Center of Circle/Proof 1|next=Perpendicular Bisector of Chord Passes Through Center}}: Book $\\text{III}$. Propositions Category:Finding Center of Circle 1j7b6jitih9ygbfsdl0qz4cynob7yev"}
+{"_id": "32547", "title": "Ratio of Areas of Similar Triangles/Porism", "text": "Ratio of Areas of Similar Triangles/Porism 0 35434 475770 197161 2020-06-24T06:09:11Z Prime.mover 59 wikitext text/x-wiki == Porism to Ratio of Areas of Similar Triangles == <onlyinclude>{{:Euclid:Proposition/VI/19/Porism}}</onlyinclude> == Proof == Follows immediately from Ratio of Areas of Similar Triangles. {{qed}} {{Euclid Note|19|VI}} == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 2|1926|ed=2nd|edpage=Second Edition|Sir Thomas L. Heath|prev=Ratio of Areas of Similar Triangles|next=Similar Polygons are composed of Similar Triangles}}: Book $\\text{VI}$. Propositions Category:Ratio of Areas of Similar Triangles f018i7q1se5o6mrq7n38ui9pdmd5epx"}
+{"_id": "32548", "title": "Euclidean Algorithm/Euclid's Proof/Porism", "text": "Euclidean Algorithm/Euclid's Proof/Porism 0 35502 267938 197317 2016-08-28T09:04:24Z Prime.mover 59 wikitext text/x-wiki == Porism to Euclidean Algorithm == <onlyinclude>{{:Euclid:Proposition/VII/2/Porism}}</onlyinclude> == Proof == An algebraic proof of this is given in Common Divisor Divides GCD. {{qed}} {{Euclid Note|2|VII}} == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 2|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Euclidean Algorithm/Euclid's Proof|next = Greatest Common Divisor of Three Numbers}}: Book $\\text{VII}$. Propositions Category:Euclidean Algorithm pvrckxq7tdbvicpi0q1bef5u23jmjho"}
+{"_id": "32549", "title": "Axiom:Euclid's Common Notion 1", "text": "Axiom:Euclid's Common Notion 1 100 35775 430024 227425 2019-10-08T12:22:09Z Prime.mover 59 wikitext text/x-wiki == Common Notion == <onlyinclude> {{EuclidCommonNotionStatement|1|Things which are equal to the same thing are also equal to each other.}} </onlyinclude> == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 1|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Axiom:Euclid's Fifth Postulate|next = Axiom:Euclid's Common Notion 2}}: Book $\\text{I}$. Common Notions * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Axiom:Euclid's Postulates|next = Axiom:Euclid's Fourth Postulate}}: Chapter $2$: The Logic of Shape: Euclid Category:Euclid's Common Notions h0nhjl1y9uz9fo7c996us571zkruoef"}
+{"_id": "32550", "title": "Axiom:Euclid's Common Notion 4", "text": "Axiom:Euclid's Common Notion 4 100 35778 196479 196470 2014-10-12T11:02:04Z Prime.mover 59 wikitext text/x-wiki == Common Notion == <onlyinclude> {{EuclidCommonNotionStatement|4|Things which coincide with one another are equal to one another.}} </onlyinclude> == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 1|1926|ed=2nd|edpage=Second Edition|Sir Thomas L. Heath|prev=Axiom:Euclid's Common Notion 3|next=Axiom:Euclid's Common Notion 5}}: Book $\\text{I}$. Common Notions Category:Euclid's Common Notions 66tjpc0xsmt6pe35ctcrjtz4brpd1y6"}
+{"_id": "32551", "title": "Similar Polygons are composed of Similar Triangles/Porism", "text": "Similar Polygons are composed of Similar Triangles/Porism 0 35954 197167 2014-10-17T22:56:20Z Prime.mover 59 Created page with \"== Porism to Similar Polygons are composed of Similar Triangles ==  <onlyinclude>{{:Euclid:Proposition/VI/20/Porism}}</onlyinclude>  == Proof ==  Follows immediately from...\" wikitext text/x-wiki == Porism to Similar Polygons are composed of Similar Triangles == <onlyinclude>{{:Euclid:Proposition/VI/20/Porism}}</onlyinclude> == Proof == Follows immediately from Similar Polygons are composed of Similar Triangles. {{qed}} {{Euclid Note|20|VI}} == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 2|1926|ed=2nd|edpage=Second Edition|Sir Thomas L. Heath|prev=Similar Polygons are composed of Similar Triangles|next=Similarity of Polygons is Equivalence Relation}}: Book $\\text{VI}$. Propositions Category:Triangles jnr9ony766y8a4etcrnn63xcl4dp3q5"}
+{"_id": "32552", "title": "Commensurability of Squares on Proportional Straight Lines/Lemma", "text": "Commensurability of Squares on Proportional Straight Lines/Lemma 0 36119 207209 207023 2015-02-01T13:01:40Z Prime.mover 59 wikitext text/x-wiki == Lemma to Commensurability of Squares on Proportional Straight Lines == <onlyinclude>{{:Euclid:Proposition/X/14/Lemma}}</onlyinclude> == Proof == :400px Let $AB$ and $C$ be two unequal straight lines. Let $AB > C$. Let the semicircle $ADB$ be described with $AB$ as the diameter. Using Fitting Chord Into Circle, let $AD$ be fitted into $ADB$ equal to $C$. Let $DB$ be joined. From Relative Sizes of Angles in Segments, $\\angle ADB$ is a right angle. From Pythagoras's Theorem: :$AB^2 = AD^2 + DB^2$ and so $AB^2$ is greater than $AD^2$, that is, $C^2$, by $DB^2$. Conversely, given two straight lines $A$ and $B$ the same technique can be used to find the straight line the square of whose length equals the sum of the squares on $A$ and $B$. {{qed}} {{Euclid Note|14|X}} == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 3|1926|ed=2nd|edpage=Second Edition|Sir Thomas L. Heath|prev=Commensurable Magnitudes are Incommensurable with Same Magnitude|next=Commensurability of Squares on Proportional Straight Lines}}: Book $\\text{X}$. Propositions Category:Euclidean Number Theory fqppdzd3vhirugs7oq6dpubculyyg5o"}
+{"_id": "32553", "title": "Straight Line Commensurable with Medial Straight Line is Medial/Porism", "text": "Straight Line Commensurable with Medial Straight Line is Medial/Porism 0 36145 479396 207087 2020-07-22T15:23:21Z Prime.mover 59 wikitext text/x-wiki == Porism to Straight Line Commensurable with Medial Straight Line is Medial == <onlyinclude>{{:Euclid:Proposition/X/23/Porism}}</onlyinclude> == Proof == {{EuclidSaid}} :''And in the same way as was explained in the case of the rationals it follows, as regards medials, that a straight line commensurable in length with a medial straight line is called '''''medial''''' and commensurable with it not only in length but in square also, since, in general, straight lines commensurable in length are always commensurable in square also.'' :''But, if any straight line be commensurable in square with a medial straight line , then, if it is also commensurable in length with it, the straight lines are called, in this case too, medial and commensurable in length and in square, but, if in square only, they are called medial straight lines commensurable in square only.'' {{Euclid Note|23|X}} == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 3|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Straight Line Commensurable with Medial Straight Line is Medial|next = Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial}}: Book $\\text{X}$. Propositions Category:Euclidean Number Theory tghou6sxgavec5f2u5avpf2nrx2iu9q"}
+{"_id": "32554", "title": "Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Greater/Lemma 1", "text": "Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Greater/Lemma 1 0 36151 207098 207097 2015-01-28T07:00:36Z Prime.mover 59 wikitext text/x-wiki == Lemma to Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Greater == <onlyinclude>{{:Euclid:Proposition/X/29/Lemma 1}}</onlyinclude> == Proof == Let $a$ and $b$ be two natural numbers such that $a > b$. Let $a$ and $b$ be either both even or both odd. From: :Even Number minus Even Number is Even and: :Odd Number minus Odd Number is Even their difference $c = a - b$ is even. Let $d = \\dfrac c 2$. Let $a$ and $b$ be similar plane numbers. By definition of similar plane numbers it is noted that $a$ and $b$ may both be square. Then by Square of Sum less Square: :$a b + d^2 = \\left({a - d}\\right)^2$ From Product of Similar Plane Numbers is Square, $a b$ is a square number. So two square numbers $a b$ and $d^2$ have been found whose sum is square. Similarly: :$a b = \\left({a - d}\\right)^2 - d^2$ from which it is noted that two square numbers $\\left({a - d}\\right)^2$ and $d^2$ have been found whose difference is square. {{qed}} {{Euclid Note|29|X}} == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 3|1926|ed=2nd|edpage=Second Edition|Sir Thomas L. Heath|prev=Construction of Components of Second Bimedial|next=Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Greater/Lemma 2}}: Book $\\text{X}$. Propositions Category:Euclidean Number Theory 3mec0xytbdrz345iqxzu3w1ctfr23pm"}
+{"_id": "32555", "title": "Axiom:Peano's Axioms/Formulation 1", "text": "Axiom:Peano's Axioms/Formulation 1 100 36585 488563 475174 2020-09-17T07:35:37Z Prime.mover 59 wikitext text/x-wiki == Axioms == '''Peano's Axioms''' are intended to reflect the intuition behind $\\N$, the mapping $s: \\N \\to \\N: \\map s n = n + 1$ and $0$ as an element of $\\N$. Let there be given a set $P$, a mapping $s: P \\to P$, and a distinguished element $0$. Historically, the existence of $s$ and the existence of $0$ were considered the first two of '''Peano's Axioms''': {{begin-axiom}} {{axiom | n = \\text P 1         | q =          | m = 0 \\in P         | t = $0$ is an element of $P$ }} {{axiom | n = \\text P 2         | q = \\forall n \\in P         | m = \\map s n \\in P         | t = For all $n \\in P$, its successor $\\map s n$ is also in $P$ }} {{end-axiom}} The other three are as follows: <onlyinclude> {{begin-axiom}} {{axiom | n = \\text P 3         | q = \\forall m, n \\in P         | m = \\map s m = \\map s n \\implies m = n         | t = $s$ is injective }} {{axiom | n = \\text P 4         | q = \\forall n \\in P         | m = \\map s n \\ne 0         | t = $0$ is not in the image of $s$ }} {{axiom | n = \\text P 5         | q = \\forall A \\subseteq P         | m = \\paren {0 \\in A \\land \\paren {\\forall z \\in A: \\map s z \\in A} } \\implies A = P         | t = Principle of Mathematical Induction: }} {{axiom | t = Any subset $A$ of $P$, containing $0$ and }} {{axiom | t = closed under $s$, is equal to $P$ }} {{end-axiom}} </onlyinclude> == Also defined as == {{:Axiom:Peano's Axioms/Also defined as}} == Also known as == {{:Axiom:Peano's Axioms/Also known as}} == Also see == * Equivalence of Formulations of Peano's Axioms {{LinkToCategory|Peano's Axioms|Peano's axioms}} {{NamedforAxiom|Giuseppe Peano}} == Historical Note == {{:Axiom:Peano's Axioms/Historical Note}} == Sources == * {{BookReference|Naive Set Theory|1960|Paul R. Halmos|prev = Definition:Transitive Set|next = Axiom:Peano's Axioms/Historical Note}}: $\\S 12$: The Peano Axioms * {{BookReference|Number Theory|1964|J. Hunter|prev = Definition:Successor Mapping|next = Axiom:Peano's Axioms/Also defined as}}: Chapter $\\text {I}$: Number Systems and Algebraic Structures: $2$. The positive integers * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Mathematician:Giuseppe Peano|next = Definition:Successor Mapping}}: $\\S 16$ * {{BookReference|A Handbook of Terms used in Algebra and Analysis|1972|A.G. Howson|prev = Definition:Natural Numbers|next = Principle of Finite Induction}}: $\\S 4$: Number systems $\\text{I}$: Peano's Axioms * {{BookReference|Introduction to Topology|1975|Bert Mendelson|ed = 3rd|edpage = Third Edition|prev = Definition:Natural Numbers|next = Principle of Mathematical Induction for Peano Structure}}: Chapter $1$: Theory of Sets: $\\S 1$: Introduction * {{BookReference|Numbers, Sets and Axioms|1982|Alan G. Hamilton|prev = Natural Number Multiplication is Cancellable|next = Axiom:Peano's Axioms/Historical Note}}: $\\S 1$: Numbers: $1.1$ Natural Numbers and Integers * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Peano Curve|next = Mathematician:Karl Pearson|entry = Peano's postulates}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Peano Curve|next = Mathematician:Karl Pearson|entry = Peano's postulates}} * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Basic Universe|next = Axiom:Peano's Axioms/Historical Note}}: Chapter $3$: The Natural Numbers: $\\S 1$ Preliminaries * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Axiom:Peano's Axioms/Historical Note|next = Definition:Peano Curve|entry = Peano's axioms}} Category:Axioms/Peano's Axioms 3vdnds24cohvzd8e49ex12rsgr7lw5e"}
+{"_id": "32556", "title": "Axiom:Peano's Axioms/Formulation 2", "text": "Axiom:Peano's Axioms/Formulation 2 100 36586 493102 475173 2020-10-07T21:53:59Z Prime.mover 59 wikitext text/x-wiki == Axioms == '''Peano's Axioms''' are intended to reflect the intuition behind $\\N$, the mapping $s: \\N \\to \\N: \\map s n = n + 1$ and $0$ as an element of $\\N$. Let there be given a set $P$, a mapping $s: P \\to P$, and a distinguished element $0$. Historically, the existence of $s$ and the existence of $0$ were considered the first two of '''Peano's Axioms''': {{begin-axiom}} {{axiom | n = \\text P 1         | q =          | m = 0 \\in P         | t = $0$ is an element of $P$ }} {{axiom | n = \\text P 2         | q = \\forall n \\in P         | m = \\map s n \\in P         | t = For all $n \\in P$, its successor $\\map s n$ is also in $P$ }} {{end-axiom}} The other three are as follows: <onlyinclude> {{begin-axiom}} {{axiom | n = \\text P 3         | q = \\forall m, n \\in P         | m = \\map s m = \\map s n \\implies m = n         | t = $s$ is injective }} {{axiom | n = \\text P 4         | m = \\Img s \\ne P         | t = $s$ is not surjective }} {{axiom | n = \\text P 5         | q = \\forall A \\subseteq P         | m = \\paren {\\paren {\\exists x \\in A: \\neg \\exists y \\in P: x = \\map s y} \\land \\paren {\\forall z \\in A: \\map s z \\in A} }         | t = Principle of Mathematical Induction: }} {{axiom | m = \\implies A = P         | t = Any subset $A$ of $P$, containing an element not }} {{axiom | t = in the image of $s$ and closed under $s$, }} {{axiom | t = is equal to $P$ }} {{end-axiom}} </onlyinclude> == Also defined as == {{:Axiom:Peano's Axioms/Also defined as}} == Also see == * Equivalence of Formulations of Peano's Axioms {{LinkToCategory|Peano's Axioms|Peano's axioms}} {{NamedforAxiom|Giuseppe Peano|name2 = Richard Dedekind}} == Historical Note == {{:Axiom:Peano's Axioms/Historical Note}} == Sources == * {{BookReference|Lectures in Abstract Algebra|1951|Nathan Jacobson|volume = I|subtitle = Basic Concepts|prev = Definition:Natural Numbers|next = Axiom:Peano's Axioms/Also defined as}}: Introduction $\\S 4$: The natural numbers Category:Axioms/Peano's Axioms adsnjp6vlat6hxl3jpkxiaeu2moj83e"}
+{"_id": "32557", "title": "Greatest Power of Two not Divisor", "text": "Greatest Power of Two not Divisor 0 37121 401274 255679 2019-04-18T22:22:41Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $S = \\set {1, 2, 3, 4, \\ldots, n}$ be a set of integers from $1$ through $n$. Let $2^t$ be the greatest power of $2$ in $S$. Then $2^t$ does not divide any other integer in $S$. That is, no other member of $S$ is a multiple of $2^t$. == Proof == Let $k$ be a multiple of $2^t$ in $S$. Then $k = 2^t \\times \\ell$ for some $\\ell \\in \\Z, \\ell \\ge 1$. If $\\ell = 2$, then $k = 2 \\times 2^t = 2^{t + 1}$, which would contradict $2^t$ being the highest power of $2$ in $S$. Otherwise, if $\\ell > 2$, then we would have $k = \\ell \\times 2^t > 2\\times 2^t = 2^{t+1}$. As $S$ contains all integers up to $k$, we would have $2^{t+1} \\in S$, contradicting $2^t$ being the highest power of $2$ in $S$. So $\\ell = 1$, that is, $2^t$ is the only multiple of $2^t$ in $S$. {{qed}}  == Also see == * Harmonic Number is not Integer Category:Number Theory 56o43b5ryyhluqucd74ea6gduw1e9ii"}
+{"_id": "32558", "title": "Bertrand-Chebyshev Theorem/Lemma 1", "text": "Bertrand-Chebyshev Theorem/Lemma 1 0 37634 310909 204675 2017-08-15T20:51:08Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> For all $n \\in \\N$: :$\\dbinom {2 n} n \\ge \\dfrac {2^{2 n}} {2 n + 1}$ where $\\dbinom {2 n} n$ denotes a binomial coefficient. </onlyinclude> == Proof == From Sequence of Binomial Coefficients is Strictly Increasing to Half Upper Index, $\\dbinom n k$ increases for $k < \\dfrac n 2$. From Sequence of Binomial Coefficients is Strictly Decreasing from Half Upper Index, $\\dbinom n k$ decreases for $k > \\dfrac n 2$. Therefore, $\\dbinom {2 n} n$ is the largest term in the sequence $\\dbinom {2 n} 0, \\dbinom {2 n} 1, \\ldots, \\dbinom {2 n} {2 n}$. Finally, observe that the mean of the terms in the sequence is $\\dfrac {2^{2 n}} {2 n + 1}$, by Sum of Binomial Coefficients over Lower Index. {{qed|lemma}} Category:Bertrand-Chebyshev Theorem cg5y0b16819x7n1i2i5r70t3ldykyww"}
+{"_id": "32559", "title": "Bertrand-Chebyshev Theorem/Lemma 2", "text": "Bertrand-Chebyshev Theorem/Lemma 2 0 37635 310912 204653 2017-08-15T20:52:31Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> For all $m \\in \\N$: :$\\displaystyle \\prod_{p \\mathop \\le m} p \\le 2^{2 m}$ where the product is taken over all prime numbers $p \\le m$. </onlyinclude> == Proof == It is plainly true for $m \\le 2$; we will proceed by strong induction. If $m > 2$ is even then: {{begin-eqn}} {{eqn | l = \\prod_{p \\mathop \\le m} p       | r = \\prod_{p \\mathop \\le m - 1} p       | c =  }} {{eqn | r = 2^{2 \\left({m - 1}\\right)}       | o = \\le        | c = by the induction hypothesis }} {{eqn | r = 2^{2 m}       | o = <       | c =  }} {{end-eqn}} By Sum of Binomial Coefficients over Lower Index: :$\\displaystyle \\sum_{r \\mathop = 0}^{2 k + 1} \\binom {2 k + 1} r = 2^{2 k + 1}$ Therefore: {{begin-eqn}} {{eqn | l = \\binom {2 k + 1} k + \\binom {2 k + 1} {k + 1}        | r = 2^{2 k + 1}       | o = \\le       | c =  }} {{eqn | ll = \\implies       | n = 1       | l = \\binom {2 k + 1} k        | r = 2^{2 k}       | o = \\le       | c = since $\\displaystyle \\binom {2 k + 1} k = \\binom {2 k + 1} {k + 1}$ }} {{end-eqn}} If $m = 2 k + 1$ is odd, then all the prime numbers $k + 2 \\le p \\le 2 k + 1$ divide: :$\\dbinom {2 k + 1} k = \\dfrac {\\left({2 k + 1}\\right)!} {k! \\left({k + 1}\\right)!}$ Thus: {{begin-eqn}} {{eqn | l = \\prod_{p \\mathop \\le m} p       | r = \\prod_{p \\mathop \\le k + 1} p \\prod_{k + 2 \\mathop \\le p \\mathop \\le 2 k + 1} p       | c =  }} {{eqn | r = \\binom {2 k + 1} k \\prod_{p \\mathop \\le k + 1} p        | o = \\le       | c =  }} {{eqn | r = 2^{2 k} \\prod_{p \\mathop \\le k + 1} p       | o = \\le       | c = by $(1)$ }} {{eqn | r = 2^{2 k} 2^{2 \\left({k + 1}\\right)}       | o = \\le       | c = by the induction hypothesis }} {{eqn | r = 2^{2 m}       | c =  }} {{end-eqn}} {{qed|lemma}} Category:Bertrand-Chebyshev Theorem gloahkyqwuyc7t0w9lrh07b5nugudqi"}
+{"_id": "32560", "title": "Bertrand-Chebyshev Theorem/Lemma 3", "text": "Bertrand-Chebyshev Theorem/Lemma 3 0 37636 310913 298898 2017-08-15T20:52:55Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> If $p$ is a prime number and $p^k \\mathrel \\backslash \\dbinom {2 n} n$, then $p^k \\le 2 n$. </onlyinclude> == Proof  == Let $l$ be the largest power of $p$ with $p^l \\le 2 n$. By De Polignac's Formula, the largest power of $p$ dividing $n!$ is $\\displaystyle \\sum_{i \\mathop \\ge 1} \\left \\lfloor{\\frac n {p^i}}\\right \\rfloor$. So: {{begin-eqn}} {{eqn | l = k       | r = \\sum_{i \\mathop \\ge 1} \\left \\lfloor{\\frac {2 n} {p^i} }\\right \\rfloor - 2 \\sum_{i \\mathop \\ge 1} \\left \\lfloor{\\frac n {p^i} }\\right \\rfloor       | o = \\le       | c = Largest power of $p$ dividing $\\dbinom {2 n} n$ }} {{eqn | r = \\sum_{i \\mathop = 1}^l \\left({\\left \\lfloor{\\frac {2 n} {p^i} }\\right \\rfloor - 2 \\left \\lfloor{\\frac n {p^i} }\\right \\rfloor}\\right)       | c = as terms with $i > l$ are zero }} {{eqn | r = \\sum_{i \\mathop = 1}^l 1       | o = \\le       | c = since $\\left \\lfloor{2 x}\\right \\rfloor - 2 \\left \\lfloor{x}\\right \\rfloor \\le 1$ }} {{eqn | r = l       | c =  }} {{end-eqn}} {{qed|lemma}} Category:Bertrand-Chebyshev Theorem fyirzym68wzdvkpndgukykiboi5d1w7"}
+{"_id": "32561", "title": "Sequence of Binomial Coefficients is Strictly Decreasing from Half Upper Index", "text": "Sequence of Binomial Coefficients is Strictly Decreasing from Half Upper Index 0 37642 204674 2014-12-20T15:04:02Z Prime.mover 59 Created page with \"== Corollary to Sequence of Binomial Coefficients is Strictly Increasing to Half Upper Index == <onlyinclude> Let $n \\in \\Z_{>0}$ be a Definition:Strictly Positive Integ...\" wikitext text/x-wiki == Corollary to Sequence of Binomial Coefficients is Strictly Increasing to Half Upper Index == <onlyinclude> Let $n \\in \\Z_{>0}$ be a strictly positive integer. Let $\\dbinom n k$ be the binomial coefficient of $n$ over $k$ for a positive integer $k \\in \\Z_{\\ge 0}$. Let $S_n = \\left\\langle{x_k}\\right\\rangle$ be the sequence defined as: :$x_k = \\dbinom n k$ Then $S_n$ is strictly decreasing exactly where $\\dfrac n 2 < k \\le n$. </onlyinclude> == Proof == If $k > \\dfrac n 2$ then it follows that $n - k < \\dfrac n 2$. Thus: {{begin-eqn}} {{eqn | l = \\binom n {k + 1}       | r = \\binom n {n - \\left({k + 1}\\right)}       | c = Symmetry Rule for Binomial Coefficients }} {{eqn | r = \\binom n {n - k - 1}       | c =  }} {{eqn | o = <       | r = \\binom n {n - k}       | c = Sequence of Binomial Coefficients is Strictly Increasing to Half Upper Index }} {{eqn | r = \\binom n k       | c = Symmetry Rule for Binomial Coefficients }} {{end-eqn}} Hence the result. {{Qed}} Category:Binomial Coefficients oiqxsersvz5a2zobgc675r70q44aeop"}
+{"_id": "32562", "title": "Isometry between Metric Spaces is Continuous/Corollary", "text": "Isometry between Metric Spaces is Continuous/Corollary 0 37658 440595 204984 2019-12-23T11:19:29Z Prime.mover 59 wikitext text/x-wiki == Corollary to Isometry between Metric Spaces is Continuous == <onlyinclude> Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces. Let $\\phi: M_1 \\to M_2$ be an isometry. Then its inverse $\\phi^{-1}: M_2 \\to M_1$ is a continuous mapping. </onlyinclude> == Proof == From Inverse of Isometry of Metric Spaces is Isometry, $\\phi^{-1}$ is an isometry. The result follows from Isometry between Metric Spaces is Continuous. {{qed}} == Sources == * {{BookReference|Introduction to Topology|1962|Bert Mendelson|prev = Isometry between Metric Spaces is Continuous|next = Definition:Homeomorphism/Metric Spaces/Definition 1}}: $\\S 2.7$: Subspaces and Equivalence of Metric Spaces: Lemma $7.5$ Category:Isometries fdtfx2mvzpjttuy7bpupoee5lcwjuj6"}
+{"_id": "32563", "title": "Axiom:Neighborhood Space Axioms", "text": "Axiom:Neighborhood Space Axioms 100 38011 206189 206186 2015-01-08T06:15:51Z Prime.mover 59 wikitext text/x-wiki == Axioms == A neighborhood space is a set $S$ such that, for each $x \\in S$, there exists a set of subsets $\\mathcal N_x$ of $S$ satisfying the following conditions: <onlyinclude> {{begin-axiom}} {{axiom|n = N1        |lc= There exists at least one element in $\\mathcal N_x$        |q = \\forall x \\in S        |m = \\mathcal N_x \\ne \\varnothing }} {{axiom|n = N2        |lc= Each element $x$ is in its own $\\mathcal N_x$        |q = \\forall x \\in S        |m = \\forall N \\in \\mathcal N_x: x \\in N }} {{axiom|n = N3        |lc= Each superset of $N \\in \\mathcal N_x$ is also in $\\mathcal N_x$        |q = \\forall x \\in S: \\forall N \\in \\mathcal N_x        |m = N' \\supseteq N \\implies N' \\in \\mathcal N_x }} {{axiom|n = N4        |lc= Intersection of $2$ elements of $\\mathcal N_x$ is also in $\\mathcal N_x$        |q = \\forall x \\in S: \\forall M, N \\in \\mathcal N_x        |m = M \\cap N \\in N_x }} {{axiom|n = N5        |lc= Exists $N' \\subseteq N \\in \\mathcal N_x$ which is $\\mathcal N_y$ of each $y \\in N'$        |q = \\forall x \\in S: \\forall N \\in \\mathcal N_x        |m = \\exists N' \\in \\mathcal N_x, N' \\subseteq N: \\forall y \\in N': N' \\in \\mathcal N_y }} {{end-axiom}} </onlyinclude> These stipulations are called the '''neighborhood space Axioms'''. Each element of $\\mathcal N_x$ is called a '''neighborhood''' of $x$. == Also see == * Basic Properties of Neighborhood in Topological Space == Sources == * {{BookReference|Introduction to Topology|1962|Bert Mendelson|prev=Definition:Neighborhood Space|next=Definition:Neighborhood (Neighborhood Space)}}: $\\S 3.3$: Neighborhoods and Neighborhood Spaces: Definition $3.4$ Category:Axioms hfetij0zdmhwht9wcsvkgiyegltpx9t"}
+{"_id": "32564", "title": "Finite Intersection of Open Sets of Neighborhood Space is Open", "text": "Finite Intersection of Open Sets of Neighborhood Space is Open 0 38022 206235 206234 2015-01-09T18:40:52Z Prime.mover 59 wikitext text/x-wiki == Corollary to Intersection of two Open Sets of Neighborhood Space is Open == <onlyinclude> Let $\\left({S, \\mathcal N}\\right)$ be a neighborhood space. Let $n \\in \\N_{>0}$ be a natural number. Let $\\displaystyle \\bigcap_{i \\mathop = 1}^n U_i$ be a finite intersection of open sets of $\\left({S, \\mathcal N}\\right)$. Then $\\displaystyle \\bigcap_{i \\mathop = 1}^n U_i$ is an open set of $\\left({S, \\mathcal N}\\right)$. </onlyinclude> == Proof == Proof by induction: Let $U_1, U_2, \\ldots$ be open sets of $\\left({S, \\mathcal N}\\right)$. For all $n \\in \\N_{> 0}$, let $P \\left({n}\\right)$ be the proposition: :$\\displaystyle \\bigcap_{i \\mathop = 1}^n U_i$ is an open set of $\\left({S, \\mathcal N}\\right)$. $P \\left({1}\\right)$ is true, as this just says: :$U_1$ is an open set of $\\left({S, \\mathcal N}\\right)$. === Basis for the Induction === $P \\left({2}\\right)$ is the case: :$U_1 \\cap U_2$ is an open set of $\\left({S, \\mathcal N}\\right)$ which has been proved above. This is our basis for the induction. === Induction Hypothesis === Now we need to show that, if $P \\left({k}\\right)$ is true, where $k \\ge 2$, then it logically follows that $P \\left({k+1}\\right)$ is true. So this is our induction hypothesis: :$\\displaystyle \\bigcap_{i \\mathop = 1}^k U_i$ is an open set of $\\left({S, \\mathcal N}\\right)$. Then we need to show: :$\\displaystyle \\bigcap_{i \\mathop = 1}^{k+1} U_i$ is an open set of $\\left({S, \\mathcal N}\\right)$. === Induction Step === This is our induction step: We have that: :$\\displaystyle \\bigcap_{i \\mathop = 1}^{k+1} U_i = \\bigcap_{i \\mathop = 1}^k U_i \\cap U_{k + 1}$ From the induction hypothesis: :$\\displaystyle \\bigcap_{i \\mathop = 1}^k U_i$ is an open set of $\\left({S, \\mathcal N}\\right)$. From the basis for the induction: :$\\displaystyle \\bigcap_{i \\mathop = 1}^k U_i \\cap U_{k + 1}$ is an open set of $\\left({S, \\mathcal N}\\right)$. So $P \\left({k}\\right) \\implies P \\left({k+1}\\right)$ and the result follows by the Principle of Mathematical Induction. Therefore: :$ \\displaystyle\\forall n \\in \\N_{>0}: \\bigcap_{i \\mathop = 1}^n U_i$ is an open set of $\\left({S, \\mathcal N}\\right)$. {{qed}} == Sources == * {{BookReference|Introduction to Topology|1962|Bert Mendelson|prev=Intersection of two Open Sets of Neighborhood Space is Open|next=Union of Open Sets of Neighborhood Space is Open}}: $\\S 3.3$: Neighborhoods and Neighborhood Spaces: Lemma $3.6$ Category:Neighborhood Spaces f9v794ni1r6prl1imt2ae75s9q9wv77"}
+{"_id": "32565", "title": "Form of Geometric Sequence of Integers/Corollary", "text": "Form of Geometric Sequence of Integers/Corollary 0 38081 456119 456010 2020-03-19T08:16:00Z Prime.mover 59 wikitext text/x-wiki == Corollary to Form of Geometric Sequence of Integers == <onlyinclude> Let $p$ and $q$ be integers. Then the finite sequence $P = \\sequence {a_j}_{0 \\mathop \\le j \\mathop \\le n}$ defined as: :$a_j = p^j q^{n - j}$ is a geometric sequence whose common ratio is $\\dfrac p q$. </onlyinclude> == Proof == Let the greatest common divisor of $p$ and $q$ be $d$. Then by Integers Divided by GCD are Coprime: :$p = d r$ :$q = d s$ where $r$ and $s$ are coprime integers. Thus: :$a_j = p^j q^{n - j}$ {{begin-eqn}} {{eqn | l = a_j       | r = p^j q^{n - j}       | c =  }} {{eqn | r = \\paren {d r}^j \\paren {d s}^{n - j}       | c =  }} {{eqn | r = d^n r^j s^{n - j}       | c =  }} {{end-eqn}} and so by Form of Geometric Sequence of Integers it follows that $P$ is a geometric sequence whose common ratio is $\\dfrac r s$. Then: {{begin-eqn}} {{eqn | l = \\dfrac r s       | r = \\paren {\\dfrac p d} / \\paren {\\dfrac q d}       | c =  }} {{eqn | r = \\dfrac p d \\dfrac d q       | c =  }} {{eqn | r = \\dfrac p q       | c =  }} {{end-eqn}} Hence the result. {{qed}} Category:Geometric Sequences of Integers 71grpt00w5ulx7b185qefx22vp26lp5"}
+{"_id": "32566", "title": "Greatest Common Measure of Commensurable Magnitudes/Porism", "text": "Greatest Common Measure of Commensurable Magnitudes/Porism 0 38120 206969 206915 2015-01-22T22:57:40Z Prime.mover 59 wikitext text/x-wiki == Porism to Greatest Common Measure of Commensurable Magnitudes == <onlyinclude>{{:Euclid:Proposition/X/3/Porism}}</onlyinclude> == Proof == Apparent from the construction. {{qed}} {{Euclid Note|3|X}} == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 3|1926|ed=2nd|edpage=Second Edition|Sir Thomas L. Heath|prev=Greatest Common Measure of Commensurable Magnitudes|next=Greatest Common Measure of Three Commensurable Magnitudes}}: Book $\\text{X}$. Propositions Category:Euclidean Number Theory kexhfffj0zodfj5xyoh3174xcbplrsm"}
+{"_id": "32567", "title": "Magnitudes with Rational Ratio are Commensurable/Porism", "text": "Magnitudes with Rational Ratio are Commensurable/Porism 0 38127 367972 206965 2018-09-29T08:04:39Z Prime.mover 59 wikitext text/x-wiki == Porism to Magnitudes with Rational Ratio are Commensurable == <onlyinclude>{{:Euclid:Proposition/X/6/Porism}}</onlyinclude> == Proof == Apparent from the construction. {{qed}} {{Euclid Note|6|X}} == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 3|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Magnitudes with Rational Ratio are Commensurable|next = Incommensurable Magnitudes have Irrational Ratio}}: Book $\\text{X}$. Propositions Category:Euclidean Number Theory ee42pvp66zptdd39xh88xd5afjam95d"}
+{"_id": "32568", "title": "Ratios of Equal Magnitudes/Porism", "text": "Ratios of Equal Magnitudes/Porism 0 38129 206964 206949 2015-01-22T22:53:35Z Prime.mover 59 wikitext text/x-wiki == Porism to Ratios of Equal Magnitudes == <onlyinclude>{{:Euclid:Proposition/V/7/Porism}}</onlyinclude> == Proof == Apparent from the construction. {{qed}} {{Euclid Note|7|V}} == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 2|1926|ed=2nd|edpage=Second Edition|Sir Thomas L. Heath|prev=Ratios of Equal Magnitudes|next=Relative Sizes of Ratios on Unequal Magnitudes}}: Book $\\text{V}$. Propositions Category:Ratios 2q9s5dp5m68bw4wissdq5bt1y0kl4bl"}
+{"_id": "32569", "title": "Construction of Incommensurable Lines/Lemma", "text": "Construction of Incommensurable Lines/Lemma 0 38133 242124 206984 2015-12-30T12:36:11Z Prime.mover 59 wikitext text/x-wiki == Lemma to Construction of Incommensurable Lines == <onlyinclude>{{:Euclid:Proposition/X/10/Lemma}}</onlyinclude> == Proof == The propositions referred to are: :{{EuclidPropLink|book = VIII|prop = 26|title = Similar Plane Numbers have Same Ratio as between Two Squares}} and its converse, which {{AuthorRef|Euclid}} does not explicitly prove. {{EuclidSaid}} :''For, if they have, they will be similar plane numbers: which is contrary to the hypothesis.'' :''Therefore numbers which are not similar plane numbers have not to one another the ratio which a square number has to a square number.'' {{qed}} {{Euclid Note|10|X|It was suggested by {{AuthorRef|Johan Ludvig Heiberg|Heiberg}} that this lemma is a later interpolation, and not part of the original work by {{AuthorRef|Euclid}}. The theorem Construction of Incommensurable Lines that it supports is itself also questionable.<br/><br/>Moreover, there is no reason why numbers which are not similar plane numbers should be introduced here.}} == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 3|1926|ed = 2nd|edpage =Second Edition|Sir Thomas L. Heath|prev = Commensurability of Squares/Porism|next = Construction of Incommensurable Lines}}: Book $\\text{X}$. Propositions Category:Euclidean Number Theory k1cfmilxg0xpqwkrpdlgs4cttxdw5gs"}
+{"_id": "32570", "title": "Sum of Möbius Function over Divisors/Lemma", "text": "Sum of Möbius Function over Divisors/Lemma 0 38589 457776 209811 2020-03-27T07:31:28Z Prime.mover 59 wikitext text/x-wiki == Lemma to Sum of Möbius Function over Divisors == Let $n \\in \\Z_{>0}$, i.e. let $n$ be a strictly positive integer. Let $\\displaystyle \\sum_{d \\mathop \\divides n}$ denote the sum over all of the divisors of $n$. Let $\\map \\mu d$ be the Möbius function. Then: <onlyinclude> :$\\displaystyle \\sum_{d \\mathop \\divides n} \\map \\mu d = \\floor {\\frac 1 n}$ That is: :$\\mu * u = \\iota$ where $u$ and $\\iota$ are the unit arithmetic function and identity arithmetic function respectively. </onlyinclude> == Proof == The lemma is clearly true if $n=1$. Assume, then, that $n > 1$ and write, by the Fundamental Theorem of Arithmetic: :$n = p_1^{a_1} p_2^{a_2} \\dots p_k^{a_k}$ In the sum $\\displaystyle \\sum_{d \\mathop \\divides n} \\map \\mu d$ the only non-zero terms come from $d = 1$ and the divisors of $n$ which are products of distinct primes. Thus: {{begin-eqn}} {{eqn | l = \\sum_{d \\mathop \\divides n} \\map \\mu d       | r = \\map \\mu 1 + \\map \\mu {p_1} + \\dotsb + \\map \\mu {p_k} + \\map \\mu {p_1 p_2} + \\dotsb + \\map \\mu {p_{k - 1} p_k} + \\dotsb + \\map \\mu {p_1 p_2 \\dotsm p_k}       | c =  }} {{eqn | r = \\dbinom k 0 + \\dbinom k 1 \\paren {-1} + \\dbinom k 2 \\paren {-1}^2 + \\cdots + \\dbinom k k \\paren {-1}^k       | c =  }} {{eqn | r = 0       | c = Alternating Sum and Difference of Binomial Coefficients for Given n }} {{end-eqn}} {{Handwaving|The jump from the expression in $\\mu$ to the one in binomial coefficients is far too big. Intermediate steps need to be included.}} Hence, the sum is $1$ for $n = 1$, and $0$ for $n > 1$, which are precisely the values of $\\floor {\\dfrac 1 n}$. {{qed}} Category:Euler Phi Function Category:Möbius Function cx9ok1t5lpk4hncdanse65yjsjy18ma"}
+{"_id": "32571", "title": "Symmetry Group of Square/Cayley Table", "text": "Symmetry Group of Square/Cayley Table 0 38643 385558 372598 2018-12-31T12:06:43Z Prime.mover 59 wikitext text/x-wiki == Cayley Table of Symmetry Group of Square == <onlyinclude> The Cayley table of the symmetry group of the square can be written: :$\\begin{array}{c|cccccc}        & e & r & r^2 & r^3 & t_x & t_y & t_{AC} & t_{BD} \\\\ \\hline e      & e & r & r^2 & r^3 & t_x & t_y & t_{AC} & t_{BD} \\\\ r      & r & r^2 & r^3 & e & t_{AC} & t_{BD} & t_y & t_x \\\\ r^2    & r^2 & r^3 & e & r & t_y & t_x & t_{BD} & t_{AC} \\\\ r^3    & r^3 & e & r & r^2 & t_{BD} & t_{AC} & t_x & t_y \\\\ t_x    & t_x & t_{BD} & t_y & t_{AC} & e & r^2 & r^3 & r \\\\ t_y    & t_y & t_{AC} & t_x & t_{BD} & r^2 & e & r & r^3 \\\\ t_{AC} & t_{AC} & t_x & t_{BD} & t_y & r & r^3 & e & r^2 \\\\ t_{BD} & t_{BD} & t_y & t_{AC} & t_x & r^3 & r & r^2 & e\\\\ \\end{array}$ </onlyinclude> where the various symmetry mappings of the square $\\mathcal S = ABCD$ are: :the identity mapping $e$ :the rotations $r, r^2, r^3$ of $90^\\circ, 180^\\circ, 270^\\circ$ counterclockwise respectively :the reflections $t_x$ and $t_y$ are reflections about the $x$ and $y$ axis respectively :the reflection $t_{AC}$ is a reflection about the diagonal through vertices $A$ and $C$ :the reflection $t_{BD}$ is a reflection about the diagonal through vertices $B$ and $D$. == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Composition of Symmetries is Associative|next = Count of Binary Operations on Set}}: $\\S 2$: Example $2.5$ * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Right Operation is Associative|next = Symmetry Group of Equilateral Triangle/Cayley Table}}: Exercise $2.10$ * {{BookReference|Elements of Abstract Algebra|1966|Richard A. Dean|prev = Symmetry Group of Equilateral Triangle/Cayley Table|next = All Elements Self-Inverse then Abelian}}: $\\S 1.4$: Exercise $2.1$ * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Dihedral Group D4/Matrix Representation/Formulation 2/Cayley Table|next = Definition:Symmetry Group of Rectangle}}: Chapter $1$: Definitions and Examples: Exercise $3$ Category:Examples of Cayley Tables qcis72b8caopz6ltd4xlp826o8ww80r"}
+{"_id": "32572", "title": "Symmetry Group of Equilateral Triangle/Cayley Table", "text": "Symmetry Group of Equilateral Triangle/Cayley Table 0 38649 387966 387965 2019-01-16T07:15:29Z Prime.mover 59 wikitext text/x-wiki == Cayley Table of Symmetry Group of Equilateral Triangle == The Cayley table of the symmetry group of the equilateral triangle can be written: <onlyinclude> :$\\begin{array}{c|ccc|ccc} \\circ & e & p & q & r & s & t \\\\ \\hline e & e & p & q & r & s & t \\\\ p & p & q & e & s & t & r \\\\ q & q & e & p & t & r & s \\\\ \\hline r & r & t & s & e & q & p \\\\ s & s & r & t & p & e & q \\\\ t & t & s & r & q & p & e \\\\ \\end{array}$ </onlyinclude> where: {{begin-eqn}} {{eqn | l = e       | o = :       | r = (A) (B) (C)       | c = Identity mapping }} {{eqn | l = p       | o = :       | r = (ABC)       | c = Rotation of $120 \\degrees$ anticlockwise about center }} {{eqn | l = q       | o = :       | r = (ACB)       | c = Rotation of $120 \\degrees$ clockwise about center }} {{eqn | l = r       | o = :       | r = (BC)       | c = Reflection in line $r$ }} {{eqn | l = s       | o = :       | r = (AC)       | c = Reflection in line $s$ }} {{eqn | l = t       | o = :       | r = (AB)       | c = Reflection in line $t$ }} {{end-eqn}} :400px == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Symmetry Group of Square/Cayley Table|next = Symmetry Group of Rectangle/Cayley Table}}: Exercise $2.11$ * {{BookReference|Elements of Abstract Algebra|1966|Richard A. Dean|prev = Cayley Table for Commutative Operation is Symmetrical about Main Diagonal|next = Symmetry Group of Square/Cayley Table}}: $\\S 1.4$: Exercise $2.1$ * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Definition:Symmetry Group of Equilateral Triangle|next = Definition:Finite Group}}: $\\S 34$. Examples of groups: $(5)$ * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Definition:Symmetry Group of Equilateral Triangle|next = Symmetry Group of Equilateral Triangle is Group}}: Chapter $1$: Definitions and Examples: Example $1.9$ Category:Examples of Cayley Tables Category:Symmetry Group of Equilateral Triangle f61utmd07tf309erhwam2gxgb1ihniy"}
+{"_id": "32573", "title": "Symmetry Group of Rectangle/Cayley Table", "text": "Symmetry Group of Rectangle/Cayley Table 0 38688 388187 372564 2019-01-17T16:32:55Z Ascii 3554 wikitext text/x-wiki == Cayley Table of Symmetry Group of Rectangle == === Definition === {{:Definition:Symmetry Group of Rectangle}} === Cayley Table === <onlyinclude> The Cayley table of the symmetry group of the (non-square) rectangle can be written: :$\\begin{array}{c|cccc}   & e & r & h & v \\\\ \\hline e & e & r & h & v \\\\ r & r & e & v & h \\\\ h & h & v & e & r \\\\ v & v & h & r & e \\\\ \\end{array}$ </onlyinclude> == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Symmetry Group of Equilateral Triangle/Cayley Table|next = Max is Associative}}: Exercise $2.11$ Category:Examples of Cayley Tables rr9cqxbt1gykhasan8wkbrzy27b2c0s"}
+{"_id": "32574", "title": "Parity Ring/Cayley Tables", "text": "Parity Ring/Cayley Tables 0 38730 371035 371033 2018-10-14T07:02:20Z Prime.mover 59 wikitext text/x-wiki == Cayley Tables for Parity Ring == The parity ring can be described completely by showing its Cayley tables: <onlyinclude> :$\\begin{array}{r|rr} + & \\text{even} &  \\text{odd} \\\\ \\hline \\text{even} & \\text{even} & \\text{odd} \\\\ \\text{odd} & \\text{odd} & \\text{even} \\\\ \\end{array} \\qquad \\begin{array}{r|rr} \\times & \\text{even} &  \\text{odd} \\\\ \\hline \\text{even} & \\text{even} & \\text{even} \\\\ \\text{odd} & \\text{even} & \\text{odd} \\\\ \\end{array}$ </onlyinclude> == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Definition:Parity Ring|next = Parity Addition is Associative}}: $\\S 2$: Example $2.2$ Category:Examples of Cayley Tables Category:Parity Ring 5sx34n05ujc4mptjwuhrcpy1r9p0e4e"}
+{"_id": "32575", "title": "Ring of Integers Modulo 2/Cayley Tables", "text": "Ring of Integers Modulo 2/Cayley Tables 0 38733 371029 371027 2018-10-14T06:56:57Z Prime.mover 59 wikitext text/x-wiki {{refactor|Transclude the separate tables for Addition and Multiplication}} == Cayley Tables for Ring of Integers Modulo $2$ == The Ring of Integers Modulo $2$: :$\\struct {\\Z_2, +_2, \\times_2}$ can be described completely by showing its Cayley tables: <onlyinclude> :$\\begin{array} {r|rr} \\struct {\\Z_2, +_2} & \\eqclass 0 2 & \\eqclass 1 2 \\\\ \\hline \\eqclass 0 2 & \\eqclass 0 2 & \\eqclass 1 2 \\\\ \\eqclass 1 2 & \\eqclass 1 2 & \\eqclass 0 2 \\\\ \\end{array} \\qquad \\begin{array}{r|rr} \\struct {\\Z_2, \\times_2} & \\eqclass 0 2 & \\eqclass 1 2 \\\\ \\hline \\eqclass 0 2 & \\eqclass 0 2 & \\eqclass 0 2 \\\\ \\eqclass 1 2 & \\eqclass 0 2 & \\eqclass 1 2 \\\\ \\end{array}$ They can be presented more simply as: :$\\begin{array}{r|rr} + & 0 & 1 \\\\ \\hline 0 & 0 & 1 \\\\ 1 & 1 & 0 \\\\ \\end{array} \\qquad \\begin{array}{r|rr} \\times & 0 & 1 \\\\ \\hline 0 & 0 & 0 \\\\ 1 & 0 & 1 \\\\ \\end{array}$ </onlyinclude> == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Isomorphism between Ring of Integers Modulo 2 and Parity Ring|next = Isomorphism between Roots of Unity under Multiplication and Integers under Modulo Addition}}: $\\S 6$: Example $6.1$ Category:Examples of Cayley Tables b2p640i0ujnz408d3pofryxivukop17"}
+{"_id": "32576", "title": "Group of Gaussian Integer Units/Cayley Table", "text": "Group of Gaussian Integer Units/Cayley Table 0 38755 379140 370979 2018-11-30T08:00:20Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Group of Gaussian Integer Units == The group of Gaussian integer units: :$\\struct {U_\\C, \\times}$ can be described completely by showing its Cayley table: <onlyinclude> :$\\begin{array}{r|rrrr} \\times & 1 & i & -1 & -i \\\\ \\hline 1 & 1 & i & -1 & -i \\\\ i & i & -1 & -i & 1 \\\\ -1 & -1 & -i & 1 & i \\\\ -i & -i & 1 & i & -1 \\\\ \\end{array}$ </onlyinclude> == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Modulo Addition/Cayley Table/Modulo 4|next = Set of Integers under Addition is Isomorphic to Set of Even Integers under Addition}}: $\\S 6$: Example $6.2$ * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Units of Gaussian Integers form Group|next = Quaternion Group/Complex Matrices}}: $\\S 34$. Examples of groups: $(6) \\ \\text{(i)}$ Category:Examples of Cayley Tables qkddxegk81xc7tizvsj0uc76fk5m7su"}
+{"_id": "32577", "title": "Modulo Addition/Cayley Table/Modulo 4", "text": "Modulo Addition/Cayley Table/Modulo 4 0 38770 399538 385902 2019-04-09T06:08:50Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Addition Modulo $4$ == The additive group of integers modulo $4$ can be described by showing its Cayley table: <onlyinclude> :$\\begin{array}{r|rrrr} \\struct {\\Z_4, +_4} & \\eqclass 0 4 & \\eqclass 1 4 & \\eqclass 2 4 & \\eqclass 3 4 \\\\ \\hline \\eqclass 0 4 & \\eqclass 0 4 & \\eqclass 1 4 & \\eqclass 2 4 & \\eqclass 3 4 \\\\ \\eqclass 1 4 & \\eqclass 1 4 & \\eqclass 2 4 & \\eqclass 3 4 & \\eqclass 0 4  \\\\ \\eqclass 2 4 & \\eqclass 2 4 & \\eqclass 3 4 & \\eqclass 0 4 & \\eqclass 1 4 \\\\ \\eqclass 3 4 & \\eqclass 3 4 & \\eqclass 0 4 & \\eqclass 1 4 & \\eqclass 2 4 \\\\ \\end{array}$ </onlyinclude> It can also be presented: :$\\begin{array}{r|rrrr} +_4 & 0 & 1 & 2 & 3 \\\\ \\hline 0 & 0 & 1 & 2 & 3  \\\\ 1 & 1 & 2 & 3 & 0 \\\\ 2 & 2 & 3 & 0 & 1 \\\\ 3 & 3 & 0 & 1 & 2 \\\\ \\end{array}$ == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Isomorphism between Gaussian Integer Units and Integers Modulo 4 under Addition/Proof 2|next = Group of Gaussian Integer Units/Cayley Table}}: $\\S 6$: Example $6.2$ * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Modulo Multiplication has Identity|next = Modulo Multiplication/Cayley Table/Modulo 4}}: Chapter $1$: Integral Domains: $\\S 6$. The Residue Classes * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Equivalence Relation/Examples/Non-Equivalence/Sum of Integers is Divisible by 3|next = Multiplicative Group of Reduced Residues Modulo 5/Cayley Table}}: Chapter $2$: Maps and relations on sets: Exercise $5$ Category:Examples of Cayley Tables Category:Cyclic Group of Order 4 Category:Examples of Additive Groups of Integers Modulo m lm0id19hwsdk2j663vq0mtie6ol9llg"}
+{"_id": "32578", "title": "Symmetric Group on 3 Letters/Cayley Table", "text": "Symmetric Group on 3 Letters/Cayley Table 0 38831 464009 385973 2020-04-24T13:57:30Z Prime.mover 59 wikitext text/x-wiki == Cayley Table of Symmetric Group on $3$ Letters == The Cayley table of the symmetric group on $3$ letters can be presented in cycle notation as: <onlyinclude> :$\\begin{array}{c|cccccc} \\circ & e & (123) & (132) & (23) & (13) & (12) \\\\ \\hline e & e & (123) & (132) & (23) & (13) & (12) \\\\ (123) & (123) & (132) & e & (13) & (12) & (23) \\\\ (132) & (132) & e & (123) & (12) & (23) & (13) \\\\ (23) & (23) & (12) & (13) & e & (132) & (123) \\\\ (13) & (13) & (23) & (12) & (123) & e & (132) \\\\ (12) & (12) & (13) & (23) & (132) & (123) & e \\\\ \\end{array}$ </onlyinclude> It can also often be seen presented as: :$\\begin{array}{c|cccccc}   & e & p & q & r & s & t \\\\ \\hline e & e & p & q & r & s & t \\\\ p & p & q & e & s & t & r \\\\ q & q & e & p & t & r & s \\\\ r & r & t & s & e & q & p \\\\ s & s & r & t & p & e & q \\\\ t & t & s & r & q & p & e \\\\ \\end{array}$ == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Center of Symmetric Group is Trivial|next = Modulo Addition/Cayley Table/Modulo 6}}: Tables: $1$. Symmetric Group $\\map S 3$ * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Left and Right Operation Closed for All Subsets|next = Centralizer of Group Element is Subgroup}}: Exercise $8.6$ * {{BookReference|Topology: An Introduction with Application to Topological Groups|1967|George McCarty|prev = Definition:Cayley Table|next = Cayley Table for Commutative Operation is Symmetrical about Main Diagonal}}: Chapter $\\text{II}$: Groups: Exercise $\\text{A}$ * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Order of Symmetric Group/Examples/Degree 3|next = Symmetry Group of Equilateral Triangle is Symmetric Group}}: Chapter $2$: Maps and relations on sets: Example $2.19$ Category:Examples of Cayley Tables Category:Symmetric Group on 3 Letters or9ffoyl7er55nqxdc1xxcfeakzw7k8"}
+{"_id": "32579", "title": "Lower Bound is Upper Bound for Inverse Ordering", "text": "Lower Bound is Upper Bound for Inverse Ordering 0 38951 488978 212640 2020-09-18T23:05:43Z Prime.mover 59 wikitext text/x-wiki == Definition == Let $\\struct {S, \\preceq}$ be an ordered set. Let $T \\subseteq S$. Let $m$ be a lower bound for $\\struct {T, \\preceq}$. Let $\\succeq$ be the dual ordering of $\\preceq$. Then $m$ is an upper bound for $\\struct {T, \\succeq}$. == Proof == Let $m$ be a lower bound for $\\struct {T, \\preceq}$. That is: :$\\forall a \\in T: m \\preceq a$ By definition of dual ordering, it follows that: :$\\forall a \\in T: a \\succeq m$ That is, $M$ is an upper bound for $\\struct {T, \\succeq}$. {{qed}} == Also see == * Upper Bound is Lower Bound for Inverse Ordering Category:Order Theory 3mg7i2sfm0uwu3tj54x0onxipf4xey5"}
+{"_id": "32580", "title": "Real Addition Identity is Zero/Corollary", "text": "Real Addition Identity is Zero/Corollary 0 39204 484166 473670 2020-08-30T13:20:59Z Prime.mover 59 wikitext text/x-wiki == Corollary to Real Addition Identity is Zero == <onlyinclude> :$\\forall x, y \\in \\R: x + y = x \\implies y = 0$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = x + y       | r = x       | c =  }} {{eqn | ll= \\leadsto       | l = \\paren {-x} + \\paren {x + y}       | r = \\paren {-x} + x       | c =  }} {{eqn | ll= \\leadsto       | l = \\paren {\\paren {-x} + x} + y       | r = \\paren {-x} + x       | c = Real Number Axioms: $\\R \\text A 1$ }} {{eqn | ll= \\leadsto       | l = 0 + y       | r = 0       | c = Real Number Axioms: $\\R \\text A 4$ }} {{eqn | ll= \\leadsto       | l = y       | r = 0       | c = Real Addition Identity is Zero }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Calculus|1967|Michael Spivak|prev = Definition:Additive Inverse of Number|next = Definition:Subtraction}}: Part $\\text I$: Prologue: Chapter $1$: Basic Properties of Numbers * {{BookReference|Topology|2000|James R. Munkres|ed = 2nd|edpage = Second Edition|prev = Archimedean Principle|next = Real Zero is Zero Element}}: $1$: Set Theory and Logic: $\\S 4$: The Integers and the Real Numbers: Exercise $1 \\ \\text{(a)}$ Category:Real Addition 0adpu22uwwmjuqy2w48prxp6xac52q9"}
+{"_id": "32581", "title": "Multiplication by Negative Real Number/Corollary", "text": "Multiplication by Negative Real Number/Corollary 0 39213 473617 406778 2020-06-12T08:28:10Z Prime.mover 59 wikitext text/x-wiki == Corollary to Multiplication by Negative Real Number == <onlyinclude> : $\\forall x \\in \\R: \\paren {-1} \\times x = -x$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\paren {-1} \\times x       | r = -\\paren {1 \\times x}       | c = Multiplication by Negative Real Number }} {{eqn | r = -x       | c = Real Number Axioms: $\\R M 3$: Identity for Multiplication }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Topology|2000|James R. Munkres|ed = 2nd|edpage = Second Edition|prev = Multiplication by Negative Real Number|next = Multiplication of Real Numbers is Left Distributive over Subtraction/Proof 2}}: $1$: Set Theory and Logic: $\\S 4$: The Integers and the Real Numbers: Exercise $1 \\ \\text{(f)}$ Category:Real Multiplication rr96djmwtyqep33grsxamtcvprif4hj"}
+{"_id": "32582", "title": "Cartesian Product of Bijections is Bijection/General Result", "text": "Cartesian Product of Bijections is Bijection/General Result 0 39315 215582 2015-04-30T08:06:02Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> Let $I$ be an indexing set.  Let $\\left\\langle{S_i}\\right\\rangle_{i \\mathop \\in I}$ and $\\left\\langle{T_i}\\right\\rangle...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $I$ be an indexing set. Let $\\left\\langle{S_i}\\right\\rangle_{i \\mathop \\in I}$ and $\\left\\langle{T_i}\\right\\rangle_{i \\mathop \\in I}$ be families of sets indexed by $I$. Let $\\displaystyle \\mathcal S := \\prod_{i \\mathop \\in I} S_i$ and $\\displaystyle \\mathcal T := \\prod_{i \\mathop \\in I} T_i$ be the Cartesian products of $\\left\\langle{S_i}\\right\\rangle_{i \\mathop \\in I}$ and $\\left\\langle{T_i}\\right\\rangle_{i \\mathop \\in I}$ respectively. For all $i \\in I$, let $f_i: S_i \\to T_i$ be a bijection. Let $\\displaystyle \\mathcal F: \\mathcal S \\to \\mathcal T := \\prod_{i \\mathop \\in I} \\left({f_i: S_i \\to T_i}\\right)$ be the Cartesian product of $\\left\\langle{f_i}\\right\\rangle_{i \\mathop \\in I}$ defined as: :$\\displaystyle \\forall \\mathbf s \\in \\mathcal S: \\mathcal F \\left({\\mathbf s}\\right) := \\prod_{i \\mathop \\in I} \\left({f_i \\left({s_i}\\right)}\\right)$ where $\\mathbf s := \\left\\langle{s_i}\\right\\rangle_{i \\mathop \\in I}$ is an arbitrary element of $\\left\\langle{S_i}\\right\\rangle_{i \\mathop \\in I}$. Then $\\mathcal F$ is a bijection. </onlyinclude> == Proof == Because $f_i$ are bijections, it follows by definition that they are surjections. Let $\\mathbf t := \\left\\langle{t_i}\\right\\rangle_{i \\mathop \\in I} \\in \\mathcal T$. Then as $f_i$ is a surjection: :$\\forall i \\in I: \\exists s_i \\in S_i: f_i \\left({s_i}\\right) = t_i$ Thus: :$\\exists \\mathbf s \\in \\mathcal S: \\mathcal F \\left({\\mathbf s}\\right) = \\mathbf t$ So $\\mathcal F$ is a surjection. Because $f_i$ are bijections, it follows by definition that they are injections. Let: :$\\mathbf t_1 := \\left\\langle{t_{i 1} }\\right\\rangle_{i \\mathop \\in I} \\in \\mathcal T$ :$\\mathbf t_2 := \\left\\langle{t_{i 2} }\\right\\rangle_{i \\mathop \\in I} \\in \\mathcal T$ Let: : $\\mathcal F \\left({\\mathbf s_1}\\right) = \\left({\\mathbf t_1}\\right), \\mathcal F \\left({\\mathbf s_2}\\right) = \\left({\\mathbf t_2}\\right)$ for some $\\mathbf s_1, \\mathbf s_2 \\in \\mathcal S$ where: :$\\mathbf s_1 := \\left\\langle{s_{i 1} }\\right\\rangle_{i \\mathop \\in I}$ :$\\mathbf s_2 := \\left\\langle{s_{i 2} }\\right\\rangle_{i \\mathop \\in I}$ Suppose $\\mathbf t_1 = \\mathbf t_2$. Then: : $\\forall i \\in I: t_{i 1} = t_{i 2}$ By definition of $\\mathcal F$: :$\\forall i \\in I: f_i \\left({s_{i 1} }\\right) = t_{i 1}$ :$\\forall i \\in I: f_i \\left({s_{i 2} }\\right) = t_{i 2}$ As $f_i$ is an injection for all $i \\in I$: :$\\forall i \\in I: t_{i 1} = t_{i 2 }\\implies s_{i 1} = s_{i 2}$ Thus it follows that: :$\\mathbf t_1 = \\mathbf t_2 \\implies \\mathbf s_1 = \\mathbf s_2$ and so $\\mathcal F$ is an injection. So $\\mathcal F$ is a surjection and also an injection. Hence by definition, $\\mathcal F$ is a bijection. {{qed}} Category:Bijections Category:Cartesian Product Category:Indexed Families 33dptureruoso9fnoj579ovc3zn9efq"}
+{"_id": "32583", "title": "Invertible Element of Monoid is Cancellable", "text": "Invertible Element of Monoid is Cancellable 0 39369 330008 215967 2017-11-25T22:26:53Z Prime.mover 59 wikitext text/x-wiki == Corollary to Invertible Element of Associative Structure is Cancellable == <onlyinclude> Let $\\left({S, \\circ}\\right)$ be a monoid whose identity is $e_S$. An element of $\\left({S, \\circ}\\right)$ which is invertible is also cancellable. </onlyinclude> == Proof == By definition of monoid, $\\left({S, \\circ}\\right)$ is an algebraic structure in which $\\circ$ is associative. The result follows from Invertible Element of Associative Structure is Cancellable. {{Qed}} Category:Monoids Category:Inverse Elements Category:Cancellability 8yx9w4i5jqf0d66ape2klha7p8t23ge"}
+{"_id": "32584", "title": "Group/Examples/inv x = 1 - x/Lemma 1", "text": "Group/Examples/inv x = 1 - x/Lemma 1 0 40103 374165 374161 2018-10-31T07:46:24Z Prime.mover 59 wikitext text/x-wiki == Lemma for Group Example: $x^{-1} = 1 - x$ == Define $f: \\openint 0 1 \\to \\R$ by: :$\\map f x := \\map \\ln {\\dfrac {1 - x} x}$ and $g: \\R \\to \\openint 0 1$: :$\\map g z := \\dfrac 1 {1 + \\exp z}$ Then: <onlyinclude> :$\\map {f \\circ g} x = x$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\map {f \\circ g} x       | r = \\map f {\\map g x}       | c = }} {{eqn | r = \\map \\ln {\\frac {1 - \\map g x} {\\map g x} }       | c =  }} {{eqn | r = \\map \\ln {\\frac {1 - \\frac 1 {1 + \\exp x} } {\\frac 1 {1 + \\exp x} } }       | c =  }} {{eqn | r = \\map \\ln {\\frac {1 + \\exp x - 1} 1}       | c = multiplying both numerator and denominator by $1 + \\exp x$ }} {{eqn | r = \\map \\ln {\\exp x}       | c = Simplifying }} {{eqn | r = x       | c = Exponential of Natural Logarithm }} {{end-eqn}} {{qed|lemma}} Category:Group/Examples/inv x = 1 - x j1s1au4kihahej6uaqr5f92l1g7qboy"}
+{"_id": "32585", "title": "Group/Examples/inv x = 1 - x/Lemma 2", "text": "Group/Examples/inv x = 1 - x/Lemma 2 0 40104 436941 374163 2019-11-27T13:33:25Z Prime.mover 59 wikitext text/x-wiki == Lemma for Group Example: $x^{-1} = 1 - x$ == Define $f: \\openint 0 1 \\to \\R$ by: :$\\map f x := \\map \\ln {\\dfrac {1 - x} x}$ and $g: \\R \\to \\openint 0 1$: :$\\map g z := \\dfrac 1 {1 + \\exp z}$ Then: <onlyinclude> :$\\map {g \\circ f} x = x$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\map {g \\circ f} x       | r = \\map g {\\map f x}       | c = }} {{eqn | r = \\frac 1 {1 + \\exp \\map f x}       | c = }} {{eqn | r = \\frac 1 {1 + \\map \\exp {\\ln \\dfrac {1 - x} x} }       | c = }} {{eqn | r = \\frac 1 {1 + \\dfrac {1 - x} x}       | c = Exponential of Natural Logarithm }} {{eqn | r = \\frac x {x + 1 - x}       | c = multiplying both numerator and denominator by $x$ }} {{eqn | r = x       | c = simplifying }} {{end-eqn}} {{qed|lemma}} Category:Group/Examples/inv x = 1 - x iqjd5dbz1lzzzm513bbk5nvn3zhg34c"}
+{"_id": "32586", "title": "Non-Abelian Order 8 Group with One Order 2 Element is Quaternion Group/Lemma 2", "text": "Non-Abelian Order 8 Group with One Order 2 Element is Quaternion Group/Lemma 2 0 40263 370920 248745 2018-10-13T20:39:30Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> :$\\paren {\\pm a}^2 = \\paren {\\pm b}^2 = \\paren {\\pm c}^2 = -1$ </onlyinclude> == Proof == {{WLOG}}, $a$ is checked. The proofs for other $5$ elements are similar. {{begin-eqn}} {{eqn | l = \\paren {a^2}^2       | r = a^4       | c = Powers of Group Elements: Product of Indices }} {{eqn | r = 1       | c = Definition of Order of Group Element }} {{end-eqn}} So $a^2 = 1$ or $a^2 = -1$. As the order of $a = 4$: :$a^2 \\ne 1$ Hence: :$a^2 = -1$ as required. {{qed}} Category:Non-Abelian Order 8 Group with One Order 2 Element is Quaternion Group cf4jnmridoq9xljcikvp5r2l4rtp8qr"}
+{"_id": "32587", "title": "Euler Triangle Formula/Lemma", "text": "Euler Triangle Formula/Lemma 0 43708 359230 238850 2018-06-18T20:55:31Z Prime.mover 59 wikitext text/x-wiki == Lemma to Euler Triangle Formula == <onlyinclude> :400px Let the bisector of angle $C$ of triangle $\\triangle ABC$ be produced to the circumcircle at $P$. Let $I$ be the incenter of $\\triangle ABC$. Then: :$AP = BP = IP$ </onlyinclude> == Proof == {{WLOG}}, it will be demonstrated that $BP = IP$. Let $CP$ be the bisector of $\\angle ACB$. We have therefore that: :$\\angle ACP = \\angle ICB$ From Angles on Equal Arcs are Equal: :$\\angle ACP = \\angle ABP$ and so: :$\\angle ABP = \\angle ICB$ By the construction of the incircle, $IB$ is the bisector of $B$. Then: :$\\angle IBA = \\angle IBC$ and so: {{begin-eqn}} {{eqn | l = \\angle IBP       | r = \\angle IBA + \\angle ABP       | c =  }} {{eqn | r = \\angle IBC + \\angle ICB       | c =  }} {{end-eqn}} By Sum of Angles of Triangle equals Two Right Angles: :$\\angle CIB + \\angle IBC + \\angle ICB = \\angle CIB + \\angle PIB$ as $\\angle CIB$ and $\\angle PIB$ are supplementary. Thus: {{begin-eqn}} {{eqn | l = \\angle PIB       | r = \\angle IBC + \\angle ICB       | c =  }} {{eqn | r = \\angle IBP       | c =  }} {{end-eqn}} and from Triangle with Two Equal Angles is Isosceles: :$IP = BP$ The proof that $IP = AP$ follows the same lines. {{qed}} Category:Triangles t682guz08q3t6tq3k9v2gbiq9n6yll4"}
+{"_id": "32588", "title": "Cassini's Identity/Lemma", "text": "Cassini's Identity/Lemma 0 43712 359832 359831 2018-06-30T09:25:43Z Prime.mover 59 wikitext text/x-wiki == Lemma for Cassini's Identity == <onlyinclude> :$\\forall n \\in \\Z_{>1}: \\begin{bmatrix} F_{n + 1} & F_n \\\\ F_n       & F_{n - 1} \\end{bmatrix} = \\begin{bmatrix} 1 & 1 \\\\ 1 & 0 \\end{bmatrix}^n$ </onlyinclude> == Proof == === Basis for the Induction === :$\\begin{bmatrix} F_2 & F_1     \\\\ F_1 & F_0 \\end{bmatrix} = \\begin{bmatrix} 1 & 1 \\\\ 1 & 0 \\end{bmatrix} = \\begin{bmatrix} 1 & 1 \\\\ 1 & 0 \\end{bmatrix}^1$ === Induction Hypothesis === For $k \\in \\Z_{>1}$, it is assumed that: :$\\begin{bmatrix} F_{k + 1} & F_k \\\\ F_k       & F_{k - 1} \\end{bmatrix} = \\begin{bmatrix} 1 & 1 \\\\ 1 & 0 \\end{bmatrix}^k$ It remains to be shown that: :$\\begin{bmatrix} F_{k + 2} & F_{k + 1}  \\\\ F_{k + 1} & F_k \\end{bmatrix} = \\begin{bmatrix} 1 & 1 \\\\ 1 & 0 \\end{bmatrix}^{k + 1}$ === Induction Step === The induction step follows from conventional matrix multiplication: {{begin-eqn}} {{eqn | l = \\begin{bmatrix} 1 & 1 \\\\ 1 & 0 \\end{bmatrix}^{k+1}       | r = \\begin{bmatrix} F_{k + 1} & F_k \\\\ F_k       & F_{k - 1} \\end{bmatrix} \\begin{bmatrix} 1 & 1 \\\\ 1 & 0 \\end{bmatrix}       | c = by the induction hypothesis }} {{eqn | r = \\begin{bmatrix} F_{k + 1} + F_k & F_{k + 1} \\\\ F_k + F_{k - 1} & F_k \\end{bmatrix}       | c = Definition of Matrix Product }} {{eqn | r = \\begin{bmatrix} F_{k + 2} & F_{k + 1} \\\\ F_{k + 1} & F_k \\end{bmatrix} }} {{end-eqn}} So by induction: :$\\begin{bmatrix} F_{n + 1} & F_n \\\\ F_n       & F_{n - 1} \\end{bmatrix} = \\begin{bmatrix} 1 & 1 \\\\ 1 & 0 \\end{bmatrix}^n$ {{qed}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Fibonacci Numbers which equal the Square of their Index|next = Prime Fibonacci Number has Prime Index except for 3}}: $\\S 1.2.8$: Fibonacci Numbers: Exercise $6$ Category:Cassini's Identity 6ycdp5tauruav9p4rowavwxc8r7nfmy"}
+{"_id": "32589", "title": "Real Numbers are Uncountable/Proof 2 using Ternary Notation/Lemma", "text": "Real Numbers are Uncountable/Proof 2 using Ternary Notation/Lemma 0 43940 301553 301548 2017-06-17T07:00:16Z Prime.mover 59 wikitext text/x-wiki == Lemma to Real Numbers are Uncountable: Proof 2 using Ternary Notation == <onlyinclude> Let $\\left\\langle{d_n}\\right\\rangle$ and $\\left\\langle{e_n}\\right\\rangle$ be infinite sequences in $\\left\\{{0, 1}\\right\\}$ such that: :$\\exists m \\in \\N: d_m \\ne e_m$ That is, the sequences $\\left\\langle{d_n}\\right\\rangle$ and $\\left\\langle{e_n}\\right\\rangle$ are different in at least one term. Then the ternary representations $D = 0.d_1 d_2 \\ldots$ and $E = 0.e_1 e_2 \\dots$ represent distinct real numbers. </onlyinclude> == Proof == Let $\\left\\langle{d_n}\\right\\rangle \\ne \\left\\langle{e_n}\\right\\rangle$. By the Well-Ordering Principle, there is a smallest $n \\in \\N_{>0}$ such that $d_n \\ne e_n$. {{WLOG}}, suppose that $d_n = 0$ and $e_n = 1$. Let $K = 0.d_1 d_2 \\ldots d_{n-1} = \\sum_{i \\mathop = 1}^{n-1} d_i 3^{-i}$. Let: : $\\displaystyle D := K + \\sum_{i \\mathop = n+1}^\\infty d_i 3^{-i}$ : $\\displaystyle E := K + 3^{-n} + \\sum_{i \\mathop = n+1}^\\infty e_i 3^{-i} \\ge K + 3^{-n}$ But then: {{begin-eqn}} {{eqn | l = D \\le K + \\sum_{i \\mathop = n + 1}^\\infty 3^{-i}       | r = K + 3^{-n-1} \\sum_{i \\mathop = 0}^\\infty 3^{-i}       | c =  }} {{eqn | r = K + 3^{-n - 1} \\dfrac 3 2       | c =  }} {{eqn | r = K + \\frac {3^{-n} } 2       | c =  }} {{end-eqn}} Thus $D < E$, so $D \\ne E$. {{qed}} Category:Real Numbers are Uncountable 6hhvrfh9ocg3rm55ze949678cd1ywcl"}
+{"_id": "32590", "title": "Complex Sequence is Cauchy iff Convergent/Lemma 1", "text": "Complex Sequence is Cauchy iff Convergent/Lemma 1 0 43973 364168 364167 2018-09-03T06:22:47Z Prime.mover 59 wikitext text/x-wiki {{refactor|Extract this into a second definition of Definition:Complex Cauchy Sequence.}} == Lemma for Complex Sequence is Cauchy iff Convergent == <onlyinclude> Let $\\sequence {z_n}$ be a complex sequence. Let $\\mathcal N$ be the domain of $\\sequence {z_n}$. Let $x_n = \\Re \\paren {z_n}$ for every $n \\in \\mathcal N$. Let $y_n = \\Im \\paren {z_n}$ for every $n \\in \\mathcal N$. Then $\\sequence {z_n}$ is a (complex) Cauchy sequence {{iff}} $\\sequence {x_n}$ and $\\sequence {y_n}$ are (real) Cauchy sequences. </onlyinclude> == Proof == === Necessary Condition === Let $\\sequence {z_n}$ be a Cauchy sequence. This means that, for a given $\\epsilon > 0$: :$\\exists N: \\forall m, n \\in \\mathcal N: m, n \\ge N: \\cmod {z_n - z_m} < \\epsilon$ We have, for every $m, n \\ge N$: {{begin-eqn}} {{eqn | l = \\cmod {x_n − x_m}       | r = \\cmod {\\Re \\paren {z_n − z_m} }       | c =  }} {{eqn | o = \\le       | r = \\cmod {z_n − z_m}       | c =  }} {{eqn | o = <       | r = \\epsilon       | c =  }} {{end-eqn}} Thus $\\sequence {x_n}$ is a Cauchy sequence by definition. A similar argument shows that $\\sequence {y_n}$ is a Cauchy sequence. {{qed|lemma}} === Sufficient Condition === Let $\\sequence {x_n}$ and $\\sequence {y_n}$ be Cauchy sequences. This means for $\\sequence {x_n}$ that, for a given $\\epsilon > 0$: : $\\exists N_1: \\forall m, n \\in \\mathcal N: m, n \\ge N_1: \\cmod {x_n - x_m} < \\dfrac \\epsilon 2$ Also, for $\\sequence {y_n}$: : $\\exists N_2: \\forall m, n \\in \\mathcal N: m, n \\ge N_2: \\cmod {y_n - y_m} < \\dfrac \\epsilon 2$ Let $N = \\max \\paren {N_1, N_2}$. Let $i = \\sqrt {-1}$ denote the imaginary unit. We have, for every $m, n \\ge N$: {{begin-eqn}} {{eqn | l = \\cmod {z_n − z_m}       | r = \\cmod {x_n + i y_n − \\paren {x_m + i y_m} }       | c = as $z_n = \\Re \\paren {z_n} + i \\Im \\paren {z_n}$ and $z_m = \\Re \\paren {z_m} + i \\Im \\paren {z_m}$ }} {{eqn | r = \\cmod {x_n − x_m + i \\paren {y_n − y_m} }       | c =  }} {{eqn | o = \\le       | r = \\cmod {x_n − x_m} + \\cmod {i\\paren {y_n − y_m} }       | c = Triangle Inequality for Complex Numbers }} {{eqn | r = \\cmod {x_n − x_m} + \\cmod {y_n − y_m}       | c = {{Defof|Complex Modulus}} }} {{eqn | o = <       | r = \\frac \\epsilon 2 + \\cmod {y_n − y_m}       | c = since $\\cmod {x_n - x_m} < \\dfrac \\epsilon 2$ }} {{eqn | o = <       | r = \\frac \\epsilon 2 + \\frac \\epsilon 2       | c = since $\\cmod {y_n - y_m} < \\dfrac \\epsilon 2$ }} {{eqn | r = \\epsilon       | c =  }} {{end-eqn}} Thus $\\sequence {z_n}$ is a Cauchy sequence by definition. {{qed}} Category:Complex Sequence is Cauchy iff Convergent d0t0livlp8jmcvldtfykw3pbyrvay3l"}
+{"_id": "32591", "title": "Field Adjoined Set/Corollary", "text": "Field Adjoined Set/Corollary 0 43999 352118 330667 2018-05-01T06:07:37Z Prime.mover 59 wikitext text/x-wiki == Corollary to Field Adjoined Set == Let $F$ be a field. Let $S \\subseteq F$ be a subset of $F$. Ket $K \\le F$ be a subfield of $F$. <onlyinclude> Let $A = K \\left[{X_1, \\ldots, X_n}\\right]$ be the ring of polynomial functions in $n$ indeterminates over $K$.  Let $B = K \\left({X_1, \\ldots, X_n}\\right)$ be the field of rational functions in $n$ indeterminates over $K$.  Let $\\alpha_1, \\ldots, \\alpha_n \\in F$. Then: :$(1): \\quad x \\in K \\left[{\\alpha_1, \\ldots, \\alpha_n}\\right] \\iff x = f \\left({\\alpha_1, \\ldots, \\alpha_n}\\right)$ for some $f \\in A$ :$(2): \\quad x \\in K \\left({\\alpha_1, \\ldots, \\alpha_n}\\right) \\iff x = f \\left({\\alpha_1, \\ldots, \\alpha_n}\\right)$ for some $f \\in B$ :$(3): \\quad x \\in K \\left[{S}\\right] \\iff x \\in K \\left[{\\alpha_1, \\ldots, \\alpha_n}\\right]$ for some $\\alpha_1, \\ldots, \\alpha_n \\in S$ :$(4): \\quad x \\in K \\left({S}\\right) \\iff x \\in K \\left({\\alpha_1, \\ldots, \\alpha_n}\\right)$ for some $\\alpha_1, \\ldots, \\alpha_n \\in S$ </onlyinclude> == Proof == {{proof wanted}} Category:Field Extensions 9btb2at5tqm0twod4kmx52gpfrxehxx"}
+{"_id": "32592", "title": "Rule of Conjunction/Proof Rule", "text": "Rule of Conjunction/Proof Rule 0 44547 493660 413771 2020-10-10T12:45:45Z Prime.mover 59 wikitext text/x-wiki == Proof Rule == The Rule of Conjunction is a valid deduction sequent in propositional logic. As a proof rule it is expressed in the form: <onlyinclude> :If we can conclude both $\\phi$ and $\\psi$, we may infer the compound statement $\\phi \\land \\psi$. </onlyinclude> It can be written: :$\\ds {\\phi \\qquad \\psi \\over \\phi \\land \\psi} \\land_i$ === Tableau Form === {{:Rule of Conjunction/Proof Rule/Tableau Form}} == Explanation == {{:Rule of Conjunction/Explanation}} == Also known as == {{:Rule of Conjunction/Also known as}} == Also see == * This is a rule of inference of the following proof system: ** Definition:Hilbert Proof System/Instance 2 * Rule of Simplification == Technical Note == {{:Rule of Conjunction/Proof Rule/Technical Note}} == Sources == * {{BookReference|Introduction to Symbolic Logic|1959|A.H. Basson|author2 = D.J. O'Connor|ed = 3rd|edpage = Third Edition|prev = Modus Ponendo Ponens/Proof Rule|next = Definition:Axiom (Formal Systems)}}: $\\S 4.2$: The Construction of an Axiom System: $RST \\, 4$ * {{BookReference|Logic: Techniques of Formal Reasoning|1964|Donald Kalish|author2 = Richard Montague|prev = Rule of Simplification/Proof Rule|next = Rule of Addition/Proof Rule}}: $\\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\\S 3$ * {{BookReference|Beginning Logic|1965|E.J. Lemmon|prev = Definition:Disjunct|next = Rule of Conjunction/Sequent Form/Formulation 1/Proof 1}}: $\\S 1.3$: Conjunction and Disjunction * {{BookReference|Symbolic Logic|1973|Irving M. Copi|ed = 4th|edpage = Fourth Edition|prev = Rule of Simplification|next = Rule of Addition/Proof Rule}}: $3.1$: Formal Proof of Validity * {{BookReference|Elementary Logic|1980|D.J. O'Connor|author2 = Betty Powell|prev = Rule of Addition/Proof Rule|next = Definition:Logical Equivalence}}: $\\S \\text{II}$: The Logic of Statements $(2): \\ 1$: Decision procedures and proofs: $9$ * {{BookReference|Logic in Computer Science: Modelling and reasoning about systems|2000|Michael R.A. Huth|author2 = Mark D. Ryan|prev = Definition:Valid Argument|next = Rule of Simplification/Proof Rule}}: $\\S 1.2.1$: Rules for natural deduction Category:Proof Rules Category:Rule of Conjunction qu0o2hcxjq6fbts7tehnh9grskpfnbl"}
+{"_id": "32593", "title": "Rule of Simplification/Proof Rule", "text": "Rule of Simplification/Proof Rule 0 44549 493659 244443 2020-10-10T12:45:26Z Prime.mover 59 wikitext text/x-wiki == Proof Rule == The Rule of Simplification is a valid deduction sequent in propositional logic. As a proof rule it is expressed in either of the two forms: <onlyinclude> :$(1): \\quad$ If we can conclude $\\phi \\land \\psi$, then we may infer $\\phi$. :$(2): \\quad$ If we can conclude $\\phi \\land \\psi$, then we may infer $\\psi$. </onlyinclude> It can be written: :$\\ds {\\phi \\land \\psi \\over \\phi} \\land_{e_1} \\qquad \\qquad {\\phi \\land \\psi \\over \\psi} \\land_{e_2}$ === Tableau Form === {{:Rule of Simplification/Proof Rule/Tableau Form}} == Explanation == {{:Rule of Simplification/Explanation}} == Also known as == {{:Rule of Simplification/Also known as}} == Also see == * Rule of Conjunction == Technical Note == {{:Rule of Simplification/Proof Rule/Technical Note}} == Sources == * {{BookReference|Introduction to Logic and to the Methodology of Deductive Sciences|1946|Alfred Tarski|ed = 2nd|edpage = Second Edition|prev = Definition:Statement Variable|next = Law of Identity}}: $\\S \\text{II}.12$: Laws of sentential calculus * {{BookReference|Logic: Techniques of Formal Reasoning|1964|Donald Kalish|author2 = Richard Montague|prev = Definition:Language of Propositional Logic|next = Rule of Conjunction/Proof Rule}}: $\\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\\S 3$ * {{BookReference|Beginning Logic|1965|E.J. Lemmon|prev = Rule of Exportation/Forward Implication/Formulation 1/Proof|next = Rule of Simplification/Sequent Form/Formulation 1/Form 1/Proof 1}}: $\\S 1.3$: Conjunction and Disjunction * {{BookReference|Symbolic Logic|1973|Irving M. Copi|ed = 4th|edpage = Fourth Edition|prev = Destructive Dilemma/Formulation 2|next = Rule of Conjunction/Proof Rule}}: $3.1$: Formal Proof of Validity * {{BookReference|Elementary Logic|1980|D.J. O'Connor|author2 = Betty Powell|prev = Hypothetical Syllogism/Formulation 1|next = Rule of Addition}}: $\\S \\text{II}$: The Logic of Statements $(2): \\ 1$: Decision procedures and proofs: $7$ * {{BookReference|Logic in Computer Science: Modelling and reasoning about systems|2000|Michael R.A. Huth|author2 = Mark D. Ryan|prev = Rule of Conjunction/Proof Rule|next = Definition:Tableau Proof (Formal Systems)}}: $\\S 1.2.1$: Rules for natural deduction Category:Proof Rules Category:Rule of Simplification 62l6x8nvxht42wkim5k8bl4c5aeviun"}
+{"_id": "32594", "title": "Quadratic Residue/Examples/11", "text": "Quadratic Residue/Examples/11 0 44728 433459 431422 2019-11-01T13:30:08Z Prime.mover 59 wikitext text/x-wiki == Example of Quadratic Residues == <onlyinclude> The set of quadratic residues modulo $11$ is: :$\\set {1, 3, 4, 5, 9}$ </onlyinclude> {{OEIS|A010375}} == Proof == To list the quadratic residues of $11$ it is enough to work out the squares $1^2, 2^2, \\ldots, 10^2$ modulo $11$. {{begin-eqn}} {{eqn | l = 1^2       | o = \\equiv       | r = 1       | rr= \\pmod {11} }} {{eqn | l = 2^2       | o = \\equiv       | r = 4       | rr= \\pmod {11} }} {{eqn | l = 3^2       | o = \\equiv       | r = 9       | rr= \\pmod {11} }} {{eqn | l = 4^2       | o = \\equiv       | r = 5       | rr= \\pmod {11} }} {{eqn | l = 5^2       | o = \\equiv       | r = 3       | rr= \\pmod {11} }} {{eqn | l = 6^2       | o = \\equiv       | r = 3       | rr= \\pmod {11} }} {{eqn | l = 7^2       | o = \\equiv       | r = 5       | rr= \\pmod {11} }} {{eqn | l = 8^2       | o = \\equiv       | r = 9       | rr= \\pmod {11} }} {{eqn | l = 9^2       | o = \\equiv       | r = 4       | rr= \\pmod {11} }} {{eqn | l = 10^2       | o = \\equiv       | r = 1       | rr= \\pmod {11} }} {{end-eqn}} So the set of quadratic residues modulo $11$ is: :$\\set {1, 3, 4, 5, 9}$ The set of quadratic non-residues of $11$ therefore consists of all the other non-zero least positive residues: :$\\set {2, 6, 7, 8, 10}$ {{qed}} == Sources == * {{BookReference|Number Theory|1971|George E. Andrews|prev = Definition:Quadratic Residue|next = Quadratic Residue/Examples/3}}: $\\text {3-5}$ The Use of Computers in Number Theory: Exercise $6$ * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Law of Quadratic Reciprocity|next = Definition:Quadratic Residue}}: Chapter $7$: Patterns in Numbers: Gauss Category:Examples of Quadratic Residues Category:11 td5wn2d2avl2viu77ndm01a48raj3jj"}
+{"_id": "32595", "title": "Kepler's Laws of Planetary Motion/First Law", "text": "Kepler's Laws of Planetary Motion/First Law 0 44852 447517 446166 2020-02-08T19:19:48Z Prime.mover 59 wikitext text/x-wiki == Physical Law == '''Kepler's first law of planetary motion''' is one of the three physical laws of celestial mechanics deduced by {{AuthorRef|Johannes Kepler}}: <onlyinclude> : Planets move around the sun in elliptical orbits, with the sun at one focus. </onlyinclude> == Proof == Consider a planet $p$ of mass $m$ moving around the sun in the plane under the influence of the force $\\mathbf F$ imparted by the gravitational field which the two bodies give rise to. Let the position of $p$ at time $t$ be given in polar coordinates as $\\polar {r, \\theta}$. Let $\\mathbf F$ be expressed as: :$\\mathbf F = F_r \\mathbf u_r + F_\\theta \\mathbf u_\\theta$ where: :$\\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$ :$\\mathbf u_\\theta$ is the unit vector in the direction of the angular coordinate of $p$ :$F_r$ and $F_\\theta$ are the magnitudes of the components of $\\mathbf F$ in the directions of $\\mathbf u_r$ and $\\mathbf u_\\theta$ respectively. Let $\\mathbf r$ be the radius vector from the origin to $p$. From Motion of Particle in Polar Coordinates, the second order ordinary differential equations governing the motion of $m$ under the force $\\mathbf F$ are: {{begin-eqn}} {{eqn | n = 1       | l = F_\\theta       | r = m \\paren {r \\dfrac {\\d^2 \\theta} {\\d t^2} + 2 \\dfrac {\\d r} {\\d t} \\dfrac {\\d \\theta} {\\d t} }       | c =  }} {{eqn | n = 2       | l = F_r       | r = m \\paren {\\dfrac {\\d^2 r} {\\d t^2} - r \\paren {\\dfrac {\\d \\theta} {\\d t} }^2}       | c =  }} {{end-eqn}} By Newton's Law of Universal Gravitation, $\\mathbf F$ is a central force of value: :$\\mathbf F = -G \\dfrac {M m} {r^3} \\mathbf r$ where: :$G$ is the gravitational constant :$M$ and $m$ are the masses of the bodies :the minus sign indicates that the central force is in the opposite direction to that of the radius vector $\\mathbf r$. :600px Hence we have: :$F_\\theta = 0$ :$F_r = -G \\dfrac {M m} {r^2} \\dfrac {\\size {\\mathbf r} } r = -G \\dfrac {M m} {r^2}$ As $G$ and $M$ are both constants, so we can express this as: :$F_r = -\\dfrac {k m} {r^2}$ So from $(2)$: {{begin-eqn}} {{eqn | l = -\\dfrac {k m} {r^2}       | r = m \\paren {\\dfrac {\\d^2 r} {\\d t^2} - r \\paren {\\dfrac {\\d \\theta} {\\d t} }^2}       | c =  }} {{eqn | n = 3       | ll= \\leadsto       | l = \\dfrac {\\d^2 r} {\\d t^2} - r \\paren {\\dfrac {\\d \\theta} {\\d t} }^2       | r = -\\dfrac k {r^2}       | c =  }} {{end-eqn}} Let $z = \\dfrac 1 r$. Then: {{begin-eqn}} {{eqn | l = \\dfrac {\\d r} {\\d t}       | r = \\dfrac \\d {\\d t} \\dfrac 1 z       | c =  }} {{eqn | r = -\\dfrac 1 {z^2} \\dfrac {\\d z} {\\d t}       | c = Power Rule for Derivatives }} {{eqn | r = -\\dfrac 1 {z^2} \\dfrac {\\d z} {\\d \\theta} \\dfrac {\\d \\theta} {\\d t}       | c = Chain Rule for Derivatives }} {{eqn | r = -\\dfrac 1 {z^2} \\dfrac {\\d z} {\\d \\theta} \\dfrac h {r^2}       | c = Derivative of Angular Component under Central Force }} {{eqn | r = -h \\dfrac {\\d z} {\\d \\theta}       | c = as $r = \\dfrac 1 z$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \\dfrac {\\d^2 r} {\\d t^2}       | r = -h \\map {\\dfrac \\d {\\d t} } {\\dfrac {\\d z} {\\d \\theta} }       | c =  }} {{eqn | r = -h \\map {\\dfrac \\d {\\d \\theta} } {\\dfrac {\\d z} {\\d \\theta} } \\dfrac {\\d \\theta} {\\d t}       | c = Chain Rule for Derivatives }} {{eqn | r = -h \\dfrac {\\d^2 z} {\\d \\theta^2} \\dfrac {\\d \\theta} {\\d t}       | c = Chain Rule for Derivatives }} {{eqn | r = -h \\dfrac {\\d^2 z} {\\d \\theta^2} \\dfrac h {r^2}       | c = Derivative of Angular Component under Central Force }} {{eqn | n = 4       | r = -h^2 z^2 \\dfrac {\\d^2 z} {\\d \\theta^2}       | c = as $r = \\dfrac 1 z$ }} {{end-eqn}} Substituting from $(4)$ into $(3)$: {{begin-eqn}} {{eqn | l = -h^2 z^2 \\dfrac {\\d^2 z} {\\d \\theta^2} - r \\paren {\\dfrac {\\d \\theta} {\\d t} }^2       | r = -\\dfrac k {r^2}       | c =  }} {{eqn | ll= \\leadsto       | l = -h^2 z^2 \\dfrac {\\d^2 z} {\\d \\theta^2} - r \\paren {\\dfrac h {r^2} }^2       | r = -\\dfrac k {r^2}       | c = Derivative of Angular Component under Central Force }} {{eqn | ll= \\leadsto       | l = -h^2 z^2 \\dfrac {\\d^2 z} {\\d \\theta^2} - \\frac 1 z \\paren {\\dfrac h {z^2} }^2       | r = -k z^2       | c = substituting $r = \\dfrac 1 z$ }} {{eqn | n = 5       | ll= \\leadsto       | l = \\dfrac {\\d^2 z} {\\d \\theta^2} + z       | r = \\dfrac k {h^2}       | c = simplifying }} {{end-eqn}} From Linear Second Order ODE: $y'' + y = K$, $(5)$ has the general solution: :$(6): \\quad z = A \\sin \\theta + B \\cos \\theta + \\dfrac k {h^2}$ {{WLOG}}, we shift the polar axis so as to make $r$ a minimum when $\\theta = 0$. That is, when $\\theta = 0$, $p$ is at its closest point to the origin. This means that $z = \\dfrac 1 z$ is a  maximum at this point. Thus at $\\theta = 0$: From Derivative at Maximum or Minimum: :$\\dfrac {\\d z} {\\d \\theta} = 0$ From Second Derivative of Strictly Concave Real Function is Strictly Negative: :$\\dfrac {\\d^2 z} {\\d \\theta^2} < 0$ Thus: :$A = 0$ :$B > 0$ Substituting these and $z = \\dfrac 1 r$ back into $(6)$: :$r = \\dfrac 1 {B \\cos \\theta + \\dfrac k {h^2} }$ {{begin-eqn}} {{eqn | l = r       | r = \\dfrac 1 {B \\cos \\theta + k / h^2}       | c =  }} {{eqn | r = \\dfrac {h^2 / k} {1 + \\paren {B h^2 / k} \\cos \\theta}       | c =  }} {{end-eqn}} Setting $B h^2 / k = e$: :$r = \\dfrac {h^2 / k} {1 + e \\cos \\theta}$ where $e \\in \\R_{>0}$ and is constant. This can be expressed as: :$(7): \\quad r = \\dfrac {p e} {1 + e \\cos \\theta}$ by setting $p = \\dfrac 1 B$. From Equation of Conic Section in Polar Form, $(7)$ is the equation of a conic section with one focus at the origin. As the planets remain in the solar system it follows that their orbits are stable and elliptical. {{qed}} == Also see == * Kepler's Second Law of Planetary Motion * Kepler's Third Law of Planetary Motion {{Namedfor|Johannes Kepler|cat = Kepler}} == Historical Note == {{:Kepler's Laws of Planetary Motion/Historical Note}} == Sources == * {{BookReference|Men of Mathematics|1937|Eric Temple Bell|prev = Kepler's Laws of Planetary Motion|next = Definition:Equidistance Property of Ellipse}}: Chapter $\\text{VI}$: On the Seashore * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Definition:Gravitational Constant|next = Definition:Eccentricity of Conic Section}}: $\\S 3.21$: Newton's Law of Gravitation * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Definition:Focus/Linguistic Note|next = Kepler's Laws of Planetary Motion/Historical Note}}: Chapter $\\text {A}.10$: Kepler ({{DateRange|1571|1630}}) * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Definition:Gravitational Constant|next = Kepler's Third Law of Planetary Motion}}: Chapter $\\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Kepler's Laws of Planetary Motion/Historical Note|next = Kepler's Second Law of Planetary Motion|entry = Kepler's laws}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Kepler's Laws of Planetary Motion/Historical Note|next = Kepler's Second Law of Planetary Motion|entry = Kepler's laws}} * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Kepler's Laws of Planetary Motion|next = Kepler's Second Law of Planetary Motion}}: Chapter $8$: The System of the World: Kepler * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Kepler's Laws of Planetary Motion|next = Kepler's Second Law of Planetary Motion|entry = Kepler's laws of planetary motion|subentry = i}} Category:Kepler's Laws of Planetary Motion geui0bvlqyjp7nwlo7jqqf5vx3pd8cm"}
+{"_id": "32596", "title": "Kepler's Laws of Planetary Motion/Second Law", "text": "Kepler's Laws of Planetary Motion/Second Law 0 44856 446167 431497 2020-02-04T09:51:30Z Prime.mover 59 wikitext text/x-wiki == Physical Law == '''Kepler's second law of planetary motion''' is one of the three physical laws of celestial mechanics deduced by {{AuthorRef|Johannes Kepler}}: <onlyinclude> : The line joining the sun and a planet sweeps out equal areas in equal times. :500px </onlyinclude> == Proof == Consider a planet $p$ of mass $m$ moving around the sun in the plane under the influence of the force $\\mathbf F$ imparted by the gravitational field which the two bodies give rise to. Let the position of $p$ at time $t$ be given in polar coordinates as $\\polar {r, \\theta}$. By definition of the gravitational field, $\\mathbf F$ is a central force. From Derivative of Angular Component under Central Force: :$r^2 \\dfrac {\\d \\theta} {\\d t} = h$ for some constant $h$. {{WLOG}}, assume that $h > 0$, which means that $p$ is travelling in the direction of positive $\\theta$. Let $\\map A t$ be the area swept out by $\\mathbf r$ in time $t$ relative to some fixed point of reference. For a small angle $\\delta \\theta$, the area $\\delta A$ can be approximated to the area of a sector of a circle. Thus: :$\\delta A = \\dfrac {r^2 \\delta \\theta} 2$ and so in the limit: {{begin-eqn}} {{eqn | l = \\dfrac {\\d A} {\\d \\theta}       | r = \\dfrac {r^2} 2       | c =  }} {{eqn | ll= \\leadsto       | l = \\int \\rd A       | r = \\dfrac 1 2 \\int r^2 \\rd \\theta       | c =  }} {{eqn | r = \\dfrac 1 2 \\int r^2 \\dfrac {\\d \\theta} {\\d t} \\rd t       | c =  }} {{eqn | r = \\dfrac 1 2 \\int h \\rd t       | c =  }} {{eqn | ll= \\leadsto       | l = \\int_{t_1}^{t_2} \\rd A       | r = \\dfrac 1 2 \\int_{t_1}^{t_2} h \\rd t       | c = integrating between times $t_1$ and $t_2$ }} {{eqn | ll= \\leadsto       | l = \\map A {t_2} - \\map A {t_1}       | r = \\dfrac h 2 \\paren {t_2 - t_1}       | c =  }} {{end-eqn}} That is, given a time interval $t_2 - t_1$, the area $\\map A {t_2} - \\map A {t_1}$ is the same, whatever the physical position of $p$. {{qed}} == Also see == * Kepler's First Law of Planetary Motion * Kepler's Third Law of Planetary Motion {{Namedfor|Johannes Kepler|cat = Kepler}} == Historical Note == {{:Kepler's Laws of Planetary Motion/Historical Note}} == Sources == * {{BookReference|Men of Mathematics|1937|Eric Temple Bell|prev = Definition:Equidistance Property of Ellipse|next = Kepler's Third Law of Planetary Motion}}: Chapter $\\text{VI}$: On the Seashore * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Derivative of Angular Component under Central Force|next = Kepler's Laws of Planetary Motion/Historical Note}}: $\\S 3.21$: Newton's Law of Gravitation: $(10)$ * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Book:Johannes Kepler/Astronomia Nova|next = Book:Johannes Kepler/Epitome Astronomiae Copernicanae}}: Chapter $\\text {A}.10$: Kepler ({{DateRange|1571|1630}}) * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Derivative of Angular Component under Central Force|next = Definition:Gravitational Constant}}: Chapter $\\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Kepler's First Law of Planetary Motion|next = Kepler's Third Law of Planetary Motion|entry = Kepler's laws}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Kepler's First Law of Planetary Motion|next = Kepler's Third Law of Planetary Motion|entry = Kepler's laws}} * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Kepler's First Law of Planetary Motion|next = Kepler's Third Law of Planetary Motion}}: Chapter $8$: The System of the World: Kepler * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Kepler's First Law of Planetary Motion|next = Kepler's Third Law of Planetary Motion|entry = Kepler's laws of planetary motion|subentry = ii}} Category:Kepler's Laws of Planetary Motion e5e7gaohjfdqoc3w21h4niy5irqebyk"}
+{"_id": "32597", "title": "Kepler's Laws of Planetary Motion/Third Law", "text": "Kepler's Laws of Planetary Motion/Third Law 0 44857 446168 431498 2020-02-04T09:52:56Z Prime.mover 59 wikitext text/x-wiki == Physical Law == '''Kepler's third law of planetary motion''' is one of the three physical laws of celestial mechanics deduced by {{AuthorRef|Johannes Kepler}}: <onlyinclude> :The square of the period of the orbit of a planet around the sun is proportional to the cube of its average distance from the sun. </onlyinclude> == Proof == Consider a planet $p$ of mass $m$ orbiting a star $S$ of mass $M$ under the influence of the gravitational field which the two bodies give rise to. From Kepler's First Law of Planetary Motion, $p$ travels in an elliptical orbit around $S$: :$(1): \\quad r = \\dfrac {h^2 / k} {1 + e \\cos \\theta}$ where $k = G M$. From Equation of Ellipse in Reduced Form: Cartesian Frame, the equation of the orbit can also be given as: :$(2): \\quad \\dfrac {x^2} {a^2} + \\dfrac {y^2} {b^2} = 1$ where the foci are placed at $\\tuple {\\pm c, 0}$. From Focus of Ellipse from Major and Minor Axis: :$a^2 - b^2 = c^2$ and also: :$e = \\dfrac c a$ {{explain|The above needs to be established -- I'm not sure if this has been done}} Thus: {{begin-eqn}} {{eqn | l = e^2       | r = \\dfrac {a^2 - b^2} {a^2}       | c =  }} {{eqn | n = 3       | ll= \\leadsto       | l = b^2       | r = a^2 \\paren {1 - e^2}       | c =  }} {{end-eqn}} :400px Then mean distance $a$ of $p$ from the focus $F$ is half the sum of the least and greatest values of $r$. So $(1)$ and $(3)$ give: {{begin-eqn}} {{eqn | l = a       | r = \\dfrac 1 2 \\paren {\\dfrac {h^2 / k} {1 + e} + \\dfrac {h^2 / k} {1 - e} }       | c =  }} {{eqn | r = \\dfrac {h^2} {k \\paren {1 - e^2} }       | c =  }} {{eqn | r = \\dfrac {h^2 a^2} {k b^2}       | c =  }} {{eqn | n = 4       | ll= \\leadsto       | l = b^2       | r = \\dfrac {h^2 a} k       | c =  }} {{end-eqn}} Let $T$ be the orbital period of $p$. From Area of Ellipse, the area $\\mathcal A$ of the orbit is given by: :$\\mathcal A = \\pi a b$ From Kepler's Second Law of Planetary Motion it follows that: :$\\dfrac {h T} 2 = \\pi a b$ and so from $(4)$: {{begin-eqn}} {{eqn | l = T^2       | r = \\dfrac {4 \\pi^2 a^2 b^2} {h^2}       | c =  }} {{eqn | r = \\paren {\\dfrac {4 \\pi^2} k} a^3       | c =  }} {{end-eqn}} We have that: :$k = G M$ where $G$ is the gravitational constant and $M$ is the mass of $S$. Hence the result. {{qed}} == Astronomical Units == {{:Kepler's Laws of Planetary Motion/Third Law/Astronomical Units}} == Examples == {{:Kepler's Laws of Planetary Motion/Third Law/Examples}} == Also see == * Kepler's First Law of Planetary Motion * Kepler's Second Law of Planetary Motion {{Namedfor|Johannes Kepler|cat = Kepler}} == Historical Note == {{:Kepler's Laws of Planetary Motion/Historical Note}} == Sources == * {{BookReference|Men of Mathematics|1937|Eric Temple Bell|prev = Kepler's Second Law of Planetary Motion|next = Newton's Law of Universal Gravitation}}: Chapter $\\text{VI}$: On the Seashore * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Eccentricity of Orbit indicates its Total Energy|next = Mathematician:Leonhard Paul Euler}}: $\\S 3.21$: Newton's Law of Gravitation: $(22)$ * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Book:Johannes Kepler/Epitome Astronomiae Copernicanae|next = Book:Johannes Kepler/Harmonices Mundi}}: Chapter $\\text {A}.10$: Kepler ({{DateRange|1571|1630}}) * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Kepler's First Law of Planetary Motion|next = Definition:Astronomical Unit}}: Chapter $\\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Kepler's Second Law of Planetary Motion|next = Definition:Kernel of Group Homomorphism|entry = Kepler's laws}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Kepler's Second Law of Planetary Motion|next = Definition:Kernel of Group Homomorphism|entry = Kepler's laws}} * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Kepler's Second Law of Planetary Motion|next = Book:Johannes Kepler/Mysterium Cosmographicum}}: Chapter $8$: The System of the World: Kepler * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Kepler's Second Law of Planetary Motion|next = Symbols:K/kg|entry = Kepler's laws of planetary motion|subentry = iii}} Category:Kepler's Laws of Planetary Motion jnhutz4crv3ed9uuctowyv7hxt26i2t"}
+{"_id": "32598", "title": "Equation of Catenary/Formulation 1", "text": "Equation of Catenary/Formulation 1 0 44943 494367 248173 2020-10-13T06:10:39Z Prime.mover 59 wikitext text/x-wiki == Curve == Consider a '''catenary'''. Let a cartesian plane be arranged so that the y-axis passes through the lowest point of the catenary. <onlyinclude> The '''catenary''' is described by the equation: :$y = \\dfrac {e^{ax} + e^{-ax}} {2 a} = \\dfrac {\\cosh a x} a$ where $a$ is a constant. The lowest point of the catenary is at $\\left({0, \\dfrac 1 a}\\right)$. </onlyinclude> == Proof == {{:Equation of Catenary/Formulation 1/Proof}} == Historical Note == {{:Definition:Catenary/Historical Note}} == Lingustic Note == {{:Definition:Catenary/Linguistic Note}} Category:Catenary f4loomt5a2ca6k6pdwyboub26r0czn9"}
+{"_id": "32599", "title": "Principle of Conservation of Energy", "text": "Principle of Conservation of Energy 0 45101 486181 254735 2020-09-07T21:34:55Z Prime.mover 59 wikitext text/x-wiki == Physical Law == Let $B$ be a body in a force field. Let $B$ be subject only to forces within that force field. Let $P$ be the potential energy of $B$. Let $K$ be the kinetic energy of $B$. Then: :$P + K = \\text{constant}$ == Historical Note == {{:Principle of Conservation of Energy/Historical Note}} == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Speed of Body under Free Fall from Height/Proof 2|next = Kinetic Energy of Motion}}: $1$: The Nature of Differential Equations: $\\S 5$: Falling Bodies and Other Rate Problems Category:Physics Category:Mechanics cw4ntztqn5pw7dyz6wefsk4lseopbpj"}
+{"_id": "32600", "title": "Kinetic Energy of Motion", "text": "Kinetic Energy of Motion 0 45105 486182 330581 2020-09-07T21:35:10Z Prime.mover 59 wikitext text/x-wiki == Physical Law == Let $B$ be a body of mass $m$ moving with speed $v$ such that $v$ is very much less than the speed of light. Then its kinetic energy $K$ is given by: :$K = \\dfrac {m v^2} 2$ == Proof == {{ProofWanted}} == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Principle of Conservation of Energy|next = Potential Energy of Position}}: $1$: The Nature of Differential Equations: $\\S 5$: Falling Bodies and Other Rate Problems Category:Physics Category:Mechanics 8yqgf2igf92s7zdgpimbft443q0b75q"}
+{"_id": "32601", "title": "Potential Energy of Position", "text": "Potential Energy of Position 0 45106 486183 329808 2020-09-07T21:35:22Z Prime.mover 59 wikitext text/x-wiki == Physical Law == Let $B$ be a body of mass $m$ in a gravitational field. Let the Acceleration Due to Gravity be of constant magnitude $g$. Let $B$ be at a height $h$ above a reference level $h_0$. Then the potential energy $P$ of $B$ due to the position of $B$ relative to $h_0$ is given by: :$P = m g h$ == Proof == {{proof wanted}} == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Kinetic Energy of Motion|next = Speed of Body under Free Fall from Height/Proof 3}}: $1$: The Nature of Differential Equations: $\\S 5$: Falling Bodies and Other Rate Problems Category:Physics Category:Mechanics r2pi2yt5umi2omx3f08map4eewwzjb1"}
+{"_id": "32602", "title": "Equation of Confocal Ellipses/Formulation 2", "text": "Equation of Confocal Ellipses/Formulation 2 0 45235 486234 486230 2020-09-07T22:14:04Z Prime.mover 59 wikitext text/x-wiki == Definition == <onlyinclude> The equation: :$(1): \\quad \\dfrac {x^2} {a^2} + \\dfrac {y^2} {a^2 - c^2} = 1$ where: :$\\tuple {x, y}$ denotes an arbitrary point in the cartesian plane :$c$ is a (strictly) positive constant :$a$ is a (strictly) positive parameter such that $a > c$ defines the set of all confocal ellipses whose foci are at $\\tuple {\\pm c, 0}$. </onlyinclude> == Proof == Let $a$ and $c$ be arbitrary (strictly) positive real numbers fulfilling the constraints as defined. Let $E$ be the locus of the equation: :$(1): \\quad \\dfrac {x^2} {a^2} + \\dfrac {y^2} {a^2 - c^2} = 1$ As $a > c$ it follows that: :$a^2 > c^2$ and so: :$a^2 - c^2 > 0$ Thus $(1)$ is in the form: :$\\dfrac {x^2} {a^2} + \\dfrac {y^2} {b^2} = 1$ From Equation of Ellipse in Reduced Form, this is the equation of an ellipse in reduced form. Thus: :$\\tuple {\\pm a, 0}$ are the positions of the vertices of $E$ :$\\tuple {0, \\pm b}$ are the positions of the covertices of $E$ From Focus of Ellipse from Major and Minor Axis: :$\\tuple {\\pm c, 0}$ are the positions of the foci of $E$. Hence the result. {{qed}} == Also see == * Equation of Confocal Conics * Equation of Confocal Hyperbolas == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Equation of Confocal Conics/Formulation 2|next = Equation of Confocal Hyperbolas/Formulation 2}}: $1$: The Nature of Differential Equations: Miscellaneous Problems for Chapter $1$: $6$ Category:Ellipses Category:Confocal Conics jkeyk4cycjfs0pi68zeyfamhyjypwpb"}
+{"_id": "32603", "title": "Equation of Confocal Hyperbolas/Formulation 2", "text": "Equation of Confocal Hyperbolas/Formulation 2 0 45236 486231 436926 2020-09-07T22:13:17Z Prime.mover 59 wikitext text/x-wiki == Definition == <onlyinclude> The equation: :$\\dfrac {x^2} {a^2} + \\dfrac {y^2} {a^2 - c^2} = 1$ where: :$\\tuple {x, y}$ denotes an arbitrary point in the cartesian plane :$c$ is a (strictly) positive constant :$a$ is a (strictly) positive parameter such that $a < c$ defines the set of all confocal hyperbolas whose foci are at $\\tuple {\\pm c, 0}$. </onlyinclude> == Proof == Let $a$ and $c$ be arbitrary (strictly) positive real numbers fulfilling the constraints as defined. Let $H$ be the locus of the equation: :$(1): \\quad \\dfrac {x^2} {a^2} + \\dfrac {y^2} {a^2 - c^2} = 1$ As $a < c$ it follows that: :$a^2 < c^2$ and so: :$a^2 - c^2 < 0$ Thus $(1)$ is in the form: :$\\dfrac {x^2} {a^2} - \\dfrac {y^2} {b^2} = 1$ Thus from Equation of Hyperbola in Reduced Form, $H$ defines an hyperbola where: :$\\tuple {\\pm a, 0}$ are the positions of the vertices of $H$ :the transverse axis of $H$ has length $2 a$ :the conjugate axis of $H$ has length $2 b$ From Focus of Hyperbola from Transverse and Conjugate Axis :$\\tuple {\\pm c, 0}$ are the positions of the foci of $H$. Hence the result. {{qed}} == Also see == * Equation of Confocal Conics * Equation of Confocal Ellipses == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Equation of Confocal Ellipses/Formulation 2|next = Confocal Conics are Self-Orthogonal}}: $1$: The Nature of Differential Equations: : Miscellaneous Problems for Chapter $1$: $6$ Category:Hyperbolas Category:Confocal Conics dx6hs47nfc8tr32sy0wo19x7sehutn3"}
+{"_id": "32604", "title": "Solution to Linear First Order Ordinary Differential Equation/Solution by Integrating Factor", "text": "Solution to Linear First Order Ordinary Differential Equation/Solution by Integrating Factor 0 45466 437267 437259 2019-11-30T20:50:23Z Prime.mover 59 wikitext text/x-wiki == Proof Technique == <onlyinclude> The technique to solve a linear first order ordinary differential equation in the form: :$\\dfrac {\\d y} {\\d x} + \\map P x y = \\map Q x$ It immediately follows from Integrating Factor for First Order ODE that: :$e^{\\int \\map P x \\rd x}$ is an integrating factor for $(1)$. Multiplying it by: :$e^{\\int \\map P x \\rd x}$ to reduce it to a form: :$\\dfrac {\\d y} {\\d x} e^{\\int \\map P x \\rd x} y = e^{\\int \\map P x \\rd x} \\map Q x$ is known as '''Solution by Integrating Factor'''. It is remembered by the procedure: :''Multiply by $e^{\\int \\map P x \\rd x}$ and integrate.'' </onlyinclude> == Examples == {{:Solution by Integrating Factor/Examples}} == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Linear First Order ODE/y' + (y over x) = 3 x|next = Solution to Linear First Order Ordinary Differential Equation/Proof 1}}: $\\S 2.10$ Category:Linear First Order ODEs Category:Proof Techniques 3y48xcqvjezllsl3jz1rl5t4z8aeo2a"}
+{"_id": "32605", "title": "Ohm's Law", "text": "Ohm's Law 0 45555 462359 462358 2020-04-16T13:42:51Z Prime.mover 59 wikitext text/x-wiki == Physical Law == Let a resistor $r$ oppose a current $I$ with resistance $R$. Then the drop in electromotive force produced by $r$ is given by: :$E_R = R I$ {{Namedfor|Georg Simon Ohm|cat = Ohm, Georg}} == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Pursuit Curve of Boat in River|next = Mathematician:Georg Simon Ohm}}: $\\S 2.13$: Simple Electric Circuits Category:Electromagnetism Category:Resistance 3wcowasbdslr0qjjpyb19ewzpck91me"}
+{"_id": "32606", "title": "Drop in EMF caused by Inductance is proportional to Rate of Change of Current", "text": "Drop in EMF caused by Inductance is proportional to Rate of Change of Current 0 45562 462380 462360 2020-04-16T16:19:13Z Prime.mover 59 wikitext text/x-wiki == Physical Law == The drop in EMF $E_L$ caused by an inductor is proportional to the rate of change of current $I$ through the inductor: :$E_L = L \\dfrac {\\d I} {\\d t}$ The constant of proportion $L$ is known as the inductance of the inductor. == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Mathematician:Georg Simon Ohm|next = Drop in EMF caused by Capacitance is proportional to Accumulated Charge}}: $\\S 2.13$: Simple Electric Circuits Category:Inductance 7wueewrc5dmtep0aoebrrfbcrres0s5"}
+{"_id": "32607", "title": "Drop in EMF caused by Capacitance is proportional to Accumulated Charge", "text": "Drop in EMF caused by Capacitance is proportional to Accumulated Charge 0 45564 462382 462381 2020-04-16T16:19:51Z Prime.mover 59 wikitext text/x-wiki == Physical Law == The drop in EMF $E_C$ caused by a capacitor is proportional to the electric charge $Q$ that has accumulated on the capacitor: :$E_C = \\dfrac 1 C Q$ The reciprocal of the constant of proportion $\\dfrac 1 C$ is known as the capacitance of the capacitor. == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Drop in EMF caused by Inductance is proportional to Rate of Change of Current|next = Definition:Electric Current}}: $\\S 2.13$: Simple Electric Circuits Category:Capacitance qufcm0jhh59pteff0or2dg18loelq44"}
+{"_id": "32608", "title": "Kirchhoff's Voltage Law", "text": "Kirchhoff's Voltage Law 0 45566 495622 495621 2020-10-20T22:07:45Z Prime.mover 59 wikitext text/x-wiki == Physical Law == The algebraic sum of the electromotive forces around a closed electrical circuit is equal to zero. {{Namedfor|Gustav Robert Kirchhoff|cat = Kirchhoff}} == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Definition:Electric Current|next = Mathematician:Gustav Robert Kirchhoff}}: $\\S 2.13$: Simple Electric Circuits Category:Electromagnetism q40s0ki9ihl3qyiorjjrbr749hfnyu6"}
+{"_id": "32609", "title": "Non-Abelian Order 8 Group with One Order 2 Element is Quaternion Group/Lemma 1", "text": "Non-Abelian Order 8 Group with One Order 2 Element is Quaternion Group/Lemma 1 0 45715 370919 321896 2018-10-13T20:38:35Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> :$\\set {\\pm 1}$ is a normal subgroup of $G$. </onlyinclude> == Proof == {{begin-eqn}} {{eqn| ll= \\forall g \\in G:      | l = g \\circ 1 \\circ g^{-1}      | r = g \\circ g^{-1}      | c = {{Defof|Identity Element}} }} {{eqn| r = 1      | c = {{Defof|Inverse Element}} }} {{end-eqn}} {{begin-eqn}} {{eqn| ll= \\forall g \\in G:      | l = \\paren {g \\circ \\paren {-1} \\circ g^{-1} }^2      | r = g \\circ \\paren {-1} \\circ g^{-1} \\circ g \\circ \\paren {-1} \\circ g^{-1}      | c =  }} {{eqn| r = g \\circ \\paren {-1} \\circ \\paren {-1} \\circ g^{-1}      | c = {{Defof|Inverse Element}} }} {{eqn| r = g \\circ g^{-1}      | c = Order of $-1$ is $2$ }} {{eqn| r = 1      | c = {{Defof|Inverse Element}} }} {{end-eqn}} So: : $g \\circ \\paren {-1} \\circ g^{-1} = 1$ or: : $g \\circ \\paren {-1} \\circ g^{-1} = -1$ Suppose $g \\circ \\paren {-1} \\circ g^{-1} = 1$. Then: {{begin-eqn}} {{eqn| l = g \\circ \\paren {-1} \\circ g^{-1}      | r = 1      | c =  }} {{eqn| l = g^{-1} \\circ g \\circ \\paren {-1} \\circ g^{-1} \\circ g      | r = g^{-1} \\circ 1 \\circ g      | c = applying same action on both sides }} {{eqn| l = -1      | r = 1      | c = {{Defof|Inverse Element}}, {{Defof|Identity Element}} }} {{end-eqn}} which is a contradiction. So it is the other case: : $g \\circ \\paren {-1} \\circ g^{-1} = -1$ Therefore: : $\\forall g \\in G: g \\circ \\set {\\pm 1} \\circ g^{-1} = \\set {\\pm 1}$ By definition, $\\set {\\pm 1}$ is a normal subgroup. {{qed}} Category:Non-Abelian Order 8 Group with One Order 2 Element is Quaternion Group qmac4lua36vm0o22dwomik7p1fcutrh"}
+{"_id": "32610", "title": "Method of Undetermined Coefficients/Exponential", "text": "Method of Undetermined Coefficients/Exponential 0 45825 439206 437485 2019-12-10T23:01:16Z Prime.mover 59 wikitext text/x-wiki == Proof Technique == Consider the nonhomogeneous linear second order ODE with constant coefficients: :$(1): \\quad y'' + p y' + q y = \\map R x$ Let $\\map R x$ be of the form of an exponential function: :$\\map R x = K e^{a x}$ The '''Method of Undetermined Coefficients''' can be used to solve $(1)$ in the following manner. == Method and Proof == Let $\\map {y_g} x$ be the general solution to: :$y'' + p y' + q y = 0$ From Solution of Constant Coefficient Homogeneous LSOODE, $\\map {y_g} x$ can be found systematically. Let $\\map {y_p} x$ be a particular solution to $(1)$. Then from General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution: :$\\map {y_g} x + \\map {y_p} x$ is the general solution to $(1)$. It remains to find $\\map {y_p} x$. <onlyinclude> Let $\\map R x = K e^{a x}$. Consider the auxiliary equation to $(1)$: :$(2): \\quad m^2 + p m + q = 0$ There are three cases to consider. ;$a$ is not a root of $(2)$ Assume that there is a particular solution to $(1)$ of the form: :$y_p = A e^{a x}$ We have: {{begin-eqn}} {{eqn | l = \\frac {\\d} {\\d x} y_p       | r = a A e^{a x}       | c = Derivative of Exponential Function }} {{eqn | l = \\frac {\\d^2} {\\d x^2} y_p       | r = a^2 A e^{a x}       | c = Derivative of Exponential Function }} {{end-eqn}} Inserting into $(1)$: {{begin-eqn}} {{eqn | l = a^2 A e^{a x} + a p A e^{a x} + q A e^{a x}       | r = K e^{a x}       | c =  }} {{eqn | ll= \\leadsto       | l = A \\left({a^2 + p a + q}\\right) e^{a x}       | r = K e^{a x}       | c =  }} {{eqn | ll= \\leadsto       | l = A       | r = \\frac K {a^2 + p a + q}       | c =  }} {{end-eqn}} Hence: :$y_p = \\dfrac {K e^{a x} } {a^2 + p a + q}$ From Solution of Constant Coefficient Homogeneous LSOODE, $y_g$ depends on whether $(2)$ has equal or unequal roots. Let $m_1$ and $m_2$ be the roots of $(2)$. Then: :$y = \\begin{cases} C_1 e^{m_1 x} + C_2 e^{m_2 x} + \\dfrac {K e^{a x} } {a^2 + p a + q} & : m_1 \\ne m_2: m_1, m_2 \\in \\R \\\\ C_1 e^{m_1 x} + C_2 x e^{m_1 x} + \\dfrac {K e^{a x} } {a^2 + p a + q} & : m_1 = m_2 \\\\ e^{r x} \\paren {C_1 \\cos s x + C_2 \\sin s x} + \\dfrac {K e^{a x} } {a^2 + p a + q} & : m_1 = r + i s, m_2 = r - i s \\end{cases}$ is the general solution to $(1)$. {{qed|lemma}} ;$a$ is a root of $(2)$ If $a$ is a root of $(2)$, then $a^2 + p a + q = 0$ and so $\\dfrac {K e^{a x} } {a^2 + p a + q}$ is undefined. Let the auxiliary equation to $(2)$ have two unequal real roots $a$ and $b$. Assume that there is a particular solution to $(1)$ of the form: :$y_p = A x e^{a x}$ We have: {{begin-eqn}} {{eqn | l = \\frac {\\d} {\\d x} y_p       | r = a A x e^{a x} + A e^{a x}       | c = Product Rule and Derivative of Exponential Function }} {{eqn | l = \\frac {\\d^2} {\\d x^2} y_p       | r = a^2 A x e^{a x} + a A e^{a x} + a A e^{a x}       | c = Product Rule and Derivative of Exponential Function }} {{eqn | r = a^2 A x e^{a x} + 2 a A e^{a x}       | c =  }} {{end-eqn}} Inserting into $(1)$: {{begin-eqn}} {{eqn | l = a^2 A x e^{a x} + 2 a A e^{a x} + p \\paren {a A x e^{a x} + A e^{a x} } + q A x e^{a x}       | r = K e^{a x}       | c =  }} {{eqn | ll= \\leadsto       | l = A \\paren {a^2 + p a + q} x e^{a x} + A \\paren {2 a + p} e^{a x}       | r = K e^{a x}       | c =  }} {{eqn | ll= \\leadsto       | l = A \\paren {2 a + p} e^{a x}       | r = K e^{a x}       | c = as $a^2 + p a + q = 0$ }} {{eqn | ll= \\leadsto       | l = A       | r = \\frac K {2 a + p}       | c =  }} {{end-eqn}} Hence: :$y_p = \\dfrac {K x e^{a x} } {2 a + p}$ and so from Solution of Constant Coefficient Homogeneous LSOODE: :$y = C_1 e^{a x} + C_2 e^{b x} + \\dfrac {K x e^{a x} } {2 a + p}$ is the general solution to $(1)$. {{qed|lemma}} ;$a$ is a repeated root of $(2)$ If the auxiliary equation to $(2)$ has two equal real roots $a$, then: :$a = - \\dfrac p 2$ and so not only: :$a^2 + p a + q = 0$ but also: :$2 a + p = 0$ and so neither of the above expressions involving $e^{a x}$ and $x e^{a x}$ will work as particular solution to $(1)$. So, assume that there is a particular solution to $(1)$ of the form: :$y_p = A x^2 e^{a x}$ We have: {{begin-eqn}} {{eqn | l = \\frac {\\d} {\\d x} y_p       | r = a A x^2 e^{a x} + 2 A x e^{a x}       | c = Product Rule and Derivative of Exponential Function }} {{eqn | l = \\frac {\\d^2} {\\d x^2} y_p       | r = a^2 A x^2 e^{a x} + 2 a x A e^{a x} + 2 a x A e^{a x} + 2 A e^{a x}       | c = Product Rule and Derivative of Exponential Function }} {{eqn | r = a^2 A x e^{a x} + 4 a x A e^{a x} + 2 A e^{a x}       | c = Derivative of Exponential Function }} {{end-eqn}} Inserting into $(1)$: {{begin-eqn}} {{eqn | l = a^2 A x e^{a x} + 4 a x A e^{a x} + 2 A e^{a x} + p \\paren {a A x^2 e^{a x} + 2 A x e^{a x} } + q A x^2 e^{a x}       | r = K e^{a x}       | c =  }} {{eqn | ll= \\leadsto       | l = A \\paren {a^2 + p a + q} x e^{a x} + 2 A \\paren {2 a + p} x e^{a x} + 2 A e^{a x}       | r = K e^{a x}       | c =  }} {{eqn | ll= \\leadsto       | l = 2 A e^{a x}       | r = K e^{a x}       | c = as $a^2 + p a + q = 0$ and $2 a + p = 0$ }} {{eqn | ll= \\leadsto       | l = A       | r = \\frac K 2       | c =  }} {{end-eqn}} Hence: :$y_p = \\dfrac {K x^2 e^{a x} } 2$ and so: :$y = y_g + \\dfrac {K x^2 e^{a x} } 2$ and so from Solution of Constant Coefficient Homogeneous LSOODE: :$y = C_1 e^{a x} + C_2 x e^{a x} + \\dfrac {K x^2 e^{a x} } 2$ is the general solution to $(1)$. </onlyinclude>{{qed}} == Sources == * {{BookReference|Elementary Differential Equations & Operators|1958|G.E.H. Reuter|prev = Method of Undetermined Coefficients/Polynomial|next = Linear Second Order ODE/y'' - 3 y' + 2 y = 5 exp 3 x}}: Chapter $1$: Linear Differential Equations with Constant Coefficients: $\\S 2$. The second order equation: $\\S 2.4$ Particular solution: exponential $\\map f x$ * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Method of Undetermined Coefficients|next = Method of Undetermined Coefficients/Sine and Cosine}}: $\\S 3.18$: The Method of Undetermined Coefficients Category:Method of Undetermined Coefficients tp4r9q6kqfv0pe8z3y74t7jauq0mdu1"}
+{"_id": "32611", "title": "Method of Undetermined Coefficients/Sine and Cosine", "text": "Method of Undetermined Coefficients/Sine and Cosine 0 45827 439029 437648 2019-12-10T20:58:06Z Prime.mover 59 wikitext text/x-wiki == Proof Technique == Consider the nonhomogeneous linear second order ODE with constant coefficients: :$(1): \\quad y'' + p y' + q y = \\map R x$ Let $\\map R x$ be a linear combination of sine and cosine: :$\\map R x = \\alpha \\sin b x + \\beta \\cos b x$ The '''Method of Undetermined Coefficients''' can be used to solve $(1)$ in the following manner. == Method and Proof == Let $\\map {y_g} x$ be the general solution to: :$y'' + p y' + q y = 0$ From Solution of Constant Coefficient Homogeneous LSOODE, $\\map {y_g} x$ can be found systematically. Let $\\map {y_p} x$ be a particular solution to $(1)$. Then from General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution: :$\\map {y_g} x + \\map {y_p} x$ is the general solution to $(1)$. It remains to find $\\map {y_p} x$. <onlyinclude> Let $\\map R x = \\alpha \\sin b x + \\beta \\cos b x$. Consider the auxiliary equation to $(1)$: :$(2): \\quad m^2 + p m + q = 0$ There are two cases which may apply. === $i b$ is not Root of Auxiliary Equation === First we investigate the case where $i b$ is not a root of the auxiliary equation to $(1)$. {{:Method of Undetermined Coefficients/Sine and Cosine/Particular Solution/i b is not Root of Auxiliary Equation}} === $i b$ is Root of Auxiliary Equation === Now suppose that $(1)$ is of the form $y'' + b^2 y = A \\sin b x + B \\cos b x$. Thus one of the $i b$ is one of the roots of the auxiliary equation to $(1)$. From Linear Second Order ODE: $y'' + k^2 y = 0$ the general solution to $(2)$ is: :$y = C_1 \\sin b x + C_2 \\cos b x$ and it can be seen that an expression of the form $A \\sin b x + B \\cos b x$ is already a particular solution of $(2)$. Thus we have: :$\\paren {q - b^2}^2 + b^2 p^2 = 0$. But using the Method of Undetermined Coefficients in the above manner, this would result in an attempt to calculate: :$\\dfrac {\\alpha \\paren {q - b^2} + \\beta b p} {\\paren {q - b^2}^2 + b^2 p^2}$ and: :$\\dfrac {\\beta \\paren {q - b^2} - \\alpha b p} {\\paren {q - b^2}^2 + b^2 p^2}$ both of which are are undefined. {{:Method of Undetermined Coefficients/Sine and Cosine/Particular Solution/i b is Root of Auxiliary Equation}}</onlyinclude>{{qed}} == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Method of Undetermined Coefficients/Exponential|next = Method of Undetermined Coefficients/Sine and Cosine/Particular Solution/i b is not Root of Auxiliary Equation/Trigonometric Form}}: $\\S 3.18$: The Method of Undetermined Coefficients Category:Method of Undetermined Coefficients g6izvwvjf8ft04tg89p1m64x74am18j"}
+{"_id": "32612", "title": "Method of Undetermined Coefficients/Polynomial", "text": "Method of Undetermined Coefficients/Polynomial 0 45828 439212 438979 2019-12-10T23:02:46Z Prime.mover 59 wikitext text/x-wiki == Proof Technique == Consider the nonhomogeneous linear second order ODE with constant coefficients: :$(1): \\quad y'' + p y' + q y = \\map R x$ Let $\\map R x$ be a polynomial in $x$: :$\\displaystyle \\map R x = \\sum_{j \\mathop = 0}^n a_j x^j$ The '''Method of Undetermined Coefficients''' can be used to solve $(1)$ in the following manner. == Method and Proof == Let $\\map {y_g} x$ be the general solution to: :$y'' + p y' + q y = 0$ From Solution of Constant Coefficient Homogeneous LSOODE, $\\map {y_g} x$ can be found systematically. Let $\\map {y_p} x$ be a particular solution to $(1)$. Then from General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution: :$\\map {y_g} x + \\map {y_p} x$ is the general solution to $(1)$. It remains to find $\\map {y_p} x$. <onlyinclude> Let $\\displaystyle \\map R x = \\sum_{j \\mathop = 0}^n a_j x^j$. Assume that there is a particular solution to $(1)$ of the form: :$\\displaystyle y_p = \\sum_{j \\mathop = 0}^n A_j x^j$ We have: {{begin-eqn}} {{eqn | l = \\frac {\\d} {\\d x} y_p       | r = \\sum_{j \\mathop = 1}^n j A_j x^{j - 1}       | c = Power Rule for Derivatives }} {{eqn | l = \\frac {\\d^2} {\\d x^2} y_p       | r = \\sum_{j \\mathop = 2}^n j \\paren {j - 1} A_j x^{j - 2}       | c = Power Rule for Derivatives }} {{end-eqn}} Inserting into $(1)$: {{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 2}^n j \\paren {j - 1} A_j x^{j - 2} + p \\sum_{j \\mathop = 1}^n j A_j x^{j - 1} + q \\sum_{j \\mathop = 0}^n A_j x^j       | r = \\sum_{j \\mathop = 0}^n a_j x^j       | c =  }} {{end-eqn}} The coefficients $A_0$ to $A_n$ can hence be solved in terms of $a_0$ to $a_n$ using the techniques of simultaneous equations. The general form is tedious and unenlightening to evaluate. </onlyinclude>{{qed}} == Sources == * {{BookReference|Elementary Differential Equations & Operators|1958|G.E.H. Reuter|prev = Linear Second Order ODE/y'' - 7 y' - 5 y = x^3 - 1|next = Method of Undetermined Coefficients/Exponential}}: Chapter $1$: Linear Differential Equations with Constant Coefficients: $\\S 2$. The second order equation: $\\S 2.3$ Particular solution: polynomial $\\map f x$ * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Method of Undetermined Coefficients/Sine and Cosine/Particular Solution/i b is Root of Auxiliary Equation/Trigonometric Form|next = Linear Second Order ODE/y'' + 3 y' - 10 y = 6 exp 4 x}}: $\\S 3.18$: The Method of Undetermined Coefficients Category:Method of Undetermined Coefficients bshbbr9yjc5lemdexfw8a5git12pmve"}
+{"_id": "32613", "title": "Method of Variation of Parameters", "text": "Method of Variation of Parameters 0 45895 439155 438423 2019-12-10T21:43:59Z Prime.mover 59 wikitext text/x-wiki == Proof Technique == The '''method of variation of parameters''' is a technique for finding a particular solution to a nonhomogeneous linear second order ODE: :$(1): \\quad y'' + \\map P x y' + \\map Q x y = \\map R x$ provided that the general solution of the corresponding homogeneous linear second order ODE: :$(2): \\quad y'' + \\map P x y' + \\map Q x y = 0$ is already known. == Method == Let the general solution of $(2)$ be: :$y = C_1 \\, \\map {y_1} x + C_2 \\, \\map {y_2} x$ Then a particular solution of $(1)$ is: :$\\displaystyle y = y_1 \\int -\\frac {\\map {y_2} x \\, \\map R x} {\\map W {y_1, y_2} } \\rd x + y_2 \\int \\frac {\\map {y_1} x \\, \\map R x} {\\map W {y_1, y_2} } \\rd x$ where $\\map W {y_1, y_2}$ denotes the Wronskian of $\\map {y_1} x$ and $\\map {y_2} x$. == Proof == Let the general solution of $(2)$ be: :$(3): \\quad y = C_1 \\, \\map {y_1} x + C_2 \\, \\map {y_2} x$ Let the arbitrary constants $C_1$ and $C_2$ be replaced by functions $\\map {v_1} x$ and $\\map {v_2} x$. It is required that $v_1$ and $v_2$ be determined so as to make: :$(4): \\quad y = \\map {v_1} x \\, \\map {y_1} x + \\map {v_2} x \\, \\map {y_2} x$ a particular solution of $(1)$. Then: {{begin-eqn}} {{eqn | l = y'       | r = \\paren {v_1 {y_1}' + {v_1}' y_1} + \\paren {v_2 {y_2}' + {v_2}' y_2}       | c = Product Rule for Derivatives }} {{eqn | n = 5       | r = \\paren {v_1 {y_1}' + v_2 {y_2}'} + \\paren { {v_1}' y_1 + {v_2}' y_2}       | c =  }} {{end-eqn}} Suppose ${v_1}' y_1 + {v_2}' y_2$ were made to vanish: :$(6): \\quad {v_1}' y_1 + {v_2}' y_2 = 0$ Then: {{begin-eqn}} {{eqn | n = 7       | l = y'       | r = v_1 {y_1}' + v_2 {y_2}'       | c =  }} {{eqn | n = 8       | ll= \\leadsto       | l = y''       | r = \\paren {v_1 {y_1}'' + {v_1}' {y_1}'} + \\paren {v_2 {y_2}'' + {v_2}' {y_2}'}       | c = Product Rule for Derivatives }} {{end-eqn}} Hence: {{begin-eqn}} {{eqn | l = y'' + \\map P x y' + \\map Q x y       | r = \\map R x       | c = $(1):$ given }} {{eqn | ll= \\leadsto       | l = \\paren {v_1 {y_1}'' + {v_1}' {y_1}'} + \\paren {v_2 {y_2}'' + {v_2}' {y_2}'} + \\map P x y' + \\map Q x y       | r = \\map R x       | c = substituting from $(8)$ }} {{eqn | ll= \\leadsto       | l = \\paren {v_1 {y_1}'' + {v_1}' {y_1}'} + \\paren {v_2 {y_2}'' + {v_2}' {y_2}'} + \\map P x \\paren {v_1 {y_1}' + v_2 {y_2}'} + \\map Q x y       | r = \\map R x       | c = substituting from $(7)$ }} {{eqn | ll= \\leadsto       | l = \\paren {v_1 {y_1}'' + {v_1}' {y_1}'} + \\paren {v_2 {y_2}'' + {v_2}' {y_2}'} + \\map P x \\paren {v_1 {y_1}' + v_2 {y_2}'} + \\map Q x \\paren {v_1 y_1 + v_2 y_2}       | r = \\map R x       | c = substituting from $(4)$ }} {{eqn | n = 9       | ll= \\leadsto       | l = v_1 \\paren { {y_1}'' + \\map P x {y_1}' + \\map Q x y_1} + v_2 \\paren { {y_2}'' + \\map P x {y_2}' + \\map Q x y_2} + {v_1}' {y_1}' + {v_2}' {y_2}'       | r = \\map R x       | c = rearranging }} {{end-eqn}} Because $y_1$ and $y_2$ are both particular solutions of $(2)$: :${y_1}'' + \\map P x {y_1}' + \\map Q x y_1 = {y_2}'' + \\map P x {y_2}' + \\map Q x y_2 = 0$ and so from $(9)$: :$(10): \\quad {v_1}' {y_1}' + {v_2}' {y_2}' = \\map R x$ In summary: {{begin-eqn}} {{eqn | n = 6       | l = {v_1}' y_1 + {v_2}' y_2       | r = 0 }} {{eqn | n = 10       | l = {v_1}' {y_1}' + {v_2}' {y_2}'       | r = \\map R x }} {{eqn | ll= \\leadsto       | l = {v_1}'       | r = \\frac {y_2 \\map R x} {y_2 {y_1}' - y_1 {y_2}'}       | c =  }} {{eqn | l = {v_2}'       | r = \\frac {y_1 \\map R x} {y_1 {y_2}' - y_2 {y_1}'}       | c =  }} {{eqn | n = 11       | ll= \\leadsto       | l = {v_1}'       | r = -\\frac {y_2 \\map R x} {\\map W {y_1, y_2} }       | c =  }} {{eqn | l = {v_2}'       | r = \\frac {y_1 \\map R x} {\\map W {y_1, y_2} }       | c =  }} {{end-eqn}} We started with the assumption that: :$(3): \\quad y = C_1 \\, \\map {y_1} x + C_2 \\, \\map {y_2} x$ and so $y_1$ and $y_2$ are linearly independent. Thus by Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent: :$\\map W {y_1, y_2} \\ne 0$ and so $(11)$ is defined. Thus: {{begin-eqn}} {{eqn | l = v_1       | r = \\int -\\frac {y_2 \\map R x} {\\map W {y_1, y_2} } \\rd x       | c =  }} {{eqn | l = v_2       | r = \\int \\frac {y_1 \\map R x} {\\map W {y_1, y_2} } \\rd x       | c =  }} {{end-eqn}} and so as required: :$\\displaystyle y = y_1 \\int -\\frac {\\map {y_2} x \\map R x} {\\map W {y_1, y_2} } \\rd x + y_2 \\int \\frac {\\map {y_1} x \\map R x} {\\map W {y_1, y_2} } \\rd x$ {{qed}} == Source of Name == The name '''method of variation of parameters''' derives from the method of operation: the parameters $C_1$ and $C_2$ are made to '''vary''' by replacing them with the functions $\\map {v_1} x$ and $\\map {v_2} x$. == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Linear Second Order ODE/y'' + 4 y = 4 cosine 2 x + 6 cosine x + 8 x^2 - 4 x|next = Linear Second Order ODE/y'' + y = cosecant x}}: $\\S 3.19$: The Method of Variation of Parameters Category:Linear Second Order ODEs Category:Proof Techniques 049iojp3clzezm3eijf0cdbu0mri6xq"}
+{"_id": "32614", "title": "6", "text": "6 0 46741 481579 481569 2020-08-13T21:08:55Z Prime.mover 59 wikitext text/x-wiki {{NumberPageLink|prev = 5|next = 7}} == Number == $6$ ('''six''') is: :$2 \\times 3$ :The only triangular number with less than $660$ digits whose square is also triangular: ::$6^2 = 36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = \\dfrac {8 \\paren {8 + 1} } 2$ :The only positive integer which is the sum of exactly $3$ of its distinct coprime divisors === $1$st Term === :The smallest positive integer which can be expressed as the sum of $2$ odd primes in $1$ way: ::$6 = 3 + 3$ :The $1$st triangular number which can be expressed as the product of $3$ consecutive integers: ::$6 = T_3 = 1 \\times 2 \\times 3$ :The $1$st power of $6$ after the zeroth $1$: ::$6 = 6^1$ :The $1$st: :: perfect number :: semiperfect number :: primitive semiperfect number :and the only number which is the sum and product of the same $3$ distinct positive integers: ::$6 = 1 + 2 + 3 = 1 \\times 2 \\times 3$ :The $1$st unitary perfect number: ::$6 = 1 + 2 + 3$ === $2$nd Term === :The $2$nd Ore number after $1$: ::$\\dfrac {6 \\times \\map \\tau 6} {\\map \\sigma 6} = 2$ :and the $2$nd after $1$ whose divisors also have an arithmetic mean which is an integer: ::$\\dfrac {\\map \\sigma 6} {\\map \\tau 6} = 3$ :The $2$nd semiprime after $4$: ::$6 = 2 \\times 3$ :The $2$nd primorial after $1$, $2$ (counting $1$ as the zeroth): ::$6 = p_2 \\# = 3 \\# = 2 \\times 3$ :The $2$nd hexagonal number after $1$: ::$6 = 1 + 5 = 2 \\paren {2 \\times 2 - 1}$ :The $2$nd pentagonal pyramidal number after $1$: ::$6 = 1 + 5 = \\dfrac {2^2 \\paren {2 + 1} } 2$ :The $2$nd composite number, and the first with distinct prime factors: ::$6 = 2 \\times 3$ :Hence the $1$st positive integer after $1$ which is not the power of a prime number. {{WIP|Set up the above as a sequence in its own right}} :The $2$nd primorial which can be expressed as the product of consecutive integers: ::$3 \\# = 6 = 2 \\times 3$ :The $2$nd central binomial coefficient after $2$: ::$6 = \\dbinom {2 \\times 2} 2 := \\dfrac {4!} {\\paren {2!}^2}$ :The $2$nd of the $3$rd pair of consecutive integers whose product is a primorial: ::$5 \\times 6 = 30 = 5 \\#$ === $3$rd Term === :The $3$rd special highly composite number after $1$, $2$ :The $3$rd factorial after $1$, $2$: ::$6 = 3! = 3 \\times 2 \\times 1$ :The $3$rd triangular number after $1$, $3$: ::$6 = 1 + 2 + 3 = \\dfrac {3 \\paren {3 + 1} } 2$ :The $3$rd automorphic number after $1$, $5$: ::$6^2 = 3 \\mathbf 6$ :The $3$rd even number after $2$, $4$ which cannot be expressed as the sum of $2$ composite odd numbers :The index (after $2, 3$) of the $3$rd Woodall prime: ::$6 \\times 2^6 - 1 = 383$ :The $3$rd positive integer which is not the sum of $1$ or more distinct squares: ::$2$, $3$, $6$, $\\ldots$ :The $3$rd integer $m$ after $3$, $4$ such that $m! - 1$ (its factorial minus $1$) is prime: ::$6! - 1 = 720 - 1 = 719$ === $4$th Term === :The $4$th highly composite number after $1$, $2$, $4$: ::$\\map \\tau 6 = 4$ :The $4$th superabundant number after $1$, $2$, $4$: ::$\\dfrac {\\map \\sigma 6} 6 = \\dfrac {12} 6 = 2$ :The $4$th trimorphic number after $1$, $4$, $5$: ::$6^3 = 21 \\mathbf 6$ :The $4$th palindromic triangular number after $0$, $1$, $3$ :The $4$th palindromic triangular number after $0$, $1$, $3$ whose index is itself palindromic: ::$6 = T_3$ :The $4$th integer $n$ after $3$, $4$, $5$ such that $m = \\displaystyle \\sum_{k \\mathop = 0}^{n - 1} \\paren {-1}^k \\paren {n - k}! = n! - \\paren {n - 1}! + \\paren {n - 2}! - \\paren {n - 3}! + \\cdots \\pm 1$ is prime: ::$6! - 5! + 4! - 3! + 2! - 1! = 619$ === $5$th Term === :The $5$th highly abundant number after $1$, $2$, $3$, $4$: ::$\\map \\sigma 6 = 12$ :The $5$th Ulam number after $1$, $2$, $3$, $4$: ::$6 = 2 + 4$ :The $5$th (strictly) positive integer after $1$, $2$, $3$, $4$ which cannot be expressed as the sum of exactly $5$ non-zero squares. :The $5$th positive integer after $1$, $2$, $3$, $4$ such that all smaller positive integers coprime to it are prime :The $5$th after $1$, $2$, $4$, $5$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes === $6$th Term === :The $6$th of the (trivial $1$-digit) pluperfect digital invariants after $1$, $2$, $3$, $4$, $5$: ::$6^1 = 6$ :The $6$th of the (trivial $1$-digit) Zuckerman numbers after $1$, $2$, $3$, $4$, $5$: ::$6 = 1 \\times 6$ :The $6$th of the (trivial $1$-digit) harshad numbers after $1$, $2$, $3$, $4$, $5$: ::$6 = 1 \\times 6$ === $7$th Term === :The $7$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$ such that $2^n$ contains no zero in its decimal representation: ::$2^6 = 64$ :The $7$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$ such that $5^n$ contains no zero in its decimal representation: ::$5^6 = 15 \\, 625$ :The $7$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$ such that both $2^n$ and $5^n$ have no zeroes: ::$2^6 = 64$, $5^6 = 15 \\, 625$ :The $7$th integer after $0$, $1$, $2$, $3$, $4$, $5$ which is (trivially) the sum of the increasing powers of its digits taken in order: ::$6^1 = 6$ === Miscellaneous === :$6 = \\sqrt {1^3 + 2^3 + 3^3}$ :The number of faces of a cube :The number of vertices of its dual, the regular octahedron :The number of edges of a tetrahedron :The area and semiperimeter of the $3-4-5$ triangle: ::$6 = \\dfrac {3 \\times 4} 2 = \\dfrac {3 + 4 + 5} 2$ {{ArithmeticFunctionTable|n = 6|sigma = 12|tau = 4|phi = 2}} == Also see == * Perfect Number is Sum of Successive Odd Cubes except 6 * Prime equals Plus or Minus One modulo 6 * Only Number which is Sum of 3 Factors is 6 * Triangular Number whose Square is Triangular === Previous in Sequence: $1$ === * {{NumberPageLink|prev = 1|next = 15|type = Hexagonal Number|cat = Hexagonal Numbers}} * {{NumberPageLink|prev = 1|next = 18|type = Pentagonal Pyramidal Number|cat = Pyramidal Numbers}} * {{NumberPageLink|prev = 1|next = 28|type = Ore Number|cat = Ore Numbers}} * {{NumberPageLink|prev = 1|next = 36|result = Sequence of Powers of 6|cat = Powers of 6}} * {{NumberPageLink|prev = 1|next = 140|result = Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors}} === Previous in Sequence: $2$ === * {{NumberPageLink|prev = 2|next = 12|type = Special Highly Composite Number|cat = Special Highly Composite Numbers}} * {{NumberPageLink|prev = 2|next = 20|type = Central Binomial Coefficient|cat = Central Binomial Coefficients}} * {{NumberPageLink|prev = 2|next = 24|type = Factorial|cat = Factorials}} * {{NumberPageLink|prev = 2|next = 30|type = Primorial|cat = Primorials}} * {{NumberPageLink|prev = 2|next = 30|result = Primorials which are Product of Consecutive Integers}} === Previous in Sequence: $3$ === * {{NumberPageLink|prev = 3|next = 7|result = Numbers not Sum of Distinct Squares}} * {{NumberPageLink|prev = 3|next = 10|type = Triangular Number|cat = Triangular Numbers}} * {{NumberPageLink|prev = 3|next = 30|type = Woodall Prime|cat = Woodall Primes}} * {{NumberPageLink|prev = 3|next = 55|result = Palindromic Triangular Numbers}} * {{NumberPageLink|prev = 3|next = 66|result = Palindromic Triangular Numbers with Palindromic Index}} === Previous in Sequence: $4$ === * {{NumberPageLink|prev = 4|next = 7|result = Sequence of Integers whose Factorial minus 1 is Prime}} * {{NumberPageLink|prev = 4|next = 7|result = Integer not Expressible as Sum of 5 Non-Zero Squares}} * {{NumberPageLink|prev = 4|next = 8|type = Ulam Number|cat = Ulam Numbers}} * {{NumberPageLink|prev = 4|next = 8|type = Highly Abundant Number|cat = Highly Abundant Numbers}} * {{NumberPageLink|prev = 4|next = 8|result = Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers}} * {{NumberPageLink|prev = 4|next = 8|result = Integers such that all Coprime and Less are Prime}} * {{NumberPageLink|prev = 4|next = 9|type = Semiprime Number|cat = Semiprimes}} * {{NumberPageLink|prev = 4|next = 12|type = Highly Composite Number|cat = Highly Composite Numbers}} * {{NumberPageLink|prev = 4|next = 12|type = Superabundant Number|cat = Superabundant Numbers}} === Previous in Sequence: $5$ === * {{NumberPageLink|prev = 5|next = 7|type = Pluperfect Digital Invariant|cat = Pluperfect Digital Invariants}} * {{NumberPageLink|prev = 5|next = 7|type = Zuckerman Number|cat = Zuckerman Numbers}} * {{NumberPageLink|prev = 5|next = 7|type = Harshad Number|cat = Harshad Numbers}} * {{NumberPageLink|prev = 5|next = 7|result = Powers of 2 with no Zero in Decimal Representation}} * {{NumberPageLink|prev = 5|next = 7|result = Powers of 5 with no Zero in Decimal Representation}} * {{NumberPageLink|prev = 5|next = 7|result = Powers of 2 and 5 without Zeroes}} * {{NumberPageLink|prev = 5|next = 7|result = Numbers which are Sum of Increasing Powers of Digits}} * {{NumberPageLink|prev = 5|next = 7|result = Sum of Sequence of Alternating Positive and Negative Factorials being Prime}} * {{NumberPageLink|prev = 5|next = 8|result = Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes}} * {{NumberPageLink|prev = 5|next = 9|type = Trimorphic Number|cat = Trimorphic Numbers}} * {{NumberPageLink|prev = 5|next = 14|result = Consecutive Integers whose Product is Primorial}} * {{NumberPageLink|prev = 5|next = 25|type = Automorphic Number|cat = Automorphic Numbers}} === Next in Sequence: $10$ and above === * {{NumberPageLink|next = 10|result = Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways}} * {{NumberPageLink|next = 12|type = Semiperfect Number|cat = Primitive Semiperfect Numbers}} * {{NumberPageLink|next = 20|type = Primitive Semiperfect Number|cat = Semiperfect Numbers}} * {{NumberPageLink|next = 28|type = Perfect Number|cat = Perfect Numbers}} * {{NumberPageLink|next = 60|type = Unitary Perfect Number|cat = Unitary Perfect Numbers}} * {{NumberPageLink|next = 120|result = Triangular Numbers which are Product of 3 Consecutive Integers}} * {{NumberPageLink|next = 140|result = Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors}} == Historical Note == {{:6/Historical Note}} == Linguistic Note == {{:6/Linguistic Note}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Definition:Automorphic Number/Sequence|next = Prime equals Plus or Minus One modulo 6}}: $5$ * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Volume of Unit Hypersphere/Sequence|next = 6/Historical Note}}: $6$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Definition:Automorphic Number/Sequence|next = Prime equals Plus or Minus One modulo 6}}: $5$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Volume of Unit Hypersphere/Sequence|next = 6/Historical Note}}: $6$ Category:Specific Numbers Category:6 ejp9al0qz8hkby09qu128kg0lw3tj1v"}
+{"_id": "32615", "title": "Up-Complete Product/Lemma 1", "text": "Up-Complete Product/Lemma 1 0 47356 256329 2016-05-19T16:27:54Z GrzegorzBancerek 2860 Created page with \"== Theorem == {{:Up-Complete Product}} <onlyinclude> Let $X$ be a directed subset of $S$.  Let $Y$ be a Definition:Directed Subset|directed su...\" wikitext text/x-wiki == Theorem == {{:Up-Complete Product}} <onlyinclude> Let $X$ be a directed subset of $S$. Let $Y$ be a directed subset of $T$. Then $X \\times Y$ is also a directed subset of $S \\times T$. </onlyinclude> == Proof == Let $\\left({s_1, t_1}\\right)$, $\\left({s_2, t_2}\\right) \\in X \\times Y$. By definition of Cartesian product: :$s_1, s_2 \\in X$ and $t_1, t_2 \\in Y$ By definition of directed subset: :$\\exists h_1 \\in X: s_1 \\preceq_1 h_1 \\land s_2 \\preceq_1 h_1$ and :$\\exists h_2 \\in X: t_1 \\preceq_2 h_2 \\land t_2 \\preceq_2 h_2$ By definition of Cartesian product of ordered sets: :$\\exists \\left({h_1, h_2}\\right) \\in X \\times Y: \\left({s_1, t_1}\\right) \\preceq \\left({h_1, h_2}\\right) \\land \\left({s_2, t_2}\\right) \\preceq \\left({h_1, h_2}\\right)$ Thus by definition: :$X \\times Y$ is a directed subset of $S \\times T$. {{qed}} == Sources == * {{BookReference|A Compendium of Continuous Lattices|1980|G. Gierz|author2 = K.H. Hofmann|author3 = K. Keimel|author4 = J.D. Lawson|author5 = M.W. Mislove|author6 = D.S. Scott}} * {{Mizar|link = yellow_3|sublink = FC9|display = YELLOW_3:funcreg 9}} Category:Order Theory jul4rr7gh926ejar25yrngrnpac5iyr"}
+{"_id": "32616", "title": "Dirichlet's Theorem on Arithmetic Sequences/Lemma 1", "text": "Dirichlet's Theorem on Arithmetic Sequences/Lemma 1 0 47367 456516 455650 2020-03-19T22:26:35Z Prime.mover 59 wikitext text/x-wiki == Lemma for Dirichlet's Theorem on Arithmetic Sequences == Let $a, q$ be coprime integers. Let $\\PP_{a, q}$ be the set of primes $p$ such that $p \\equiv a \\pmod q$. {{MissingLinks|trivial character}} <onlyinclude> Let $\\chi$ be a Dirichlet character modulo $q$. Let: :$\\displaystyle \\map f s = \\sum_p \\map \\chi p p^{-s}$ If $\\chi$ is non-trivial then $\\map f s$ is bounded as $s \\to 1$. If $\\chi$ is the trivial character then: :$\\map f s \\sim \\map \\ln {\\dfrac 1 {s - 1} }$ as $s \\to 1$. </onlyinclude> == Proof == By Logarithm of Dirichlet L-Functions: :$(1):\\quad \\displaystyle \\sum_p \\map \\chi p p^{-s} = \\ln \\map L {s, \\chi} - \\sum_p \\sum_{n \\mathop \\ge 2} \\frac {\\map \\chi p^n} {n p^{n s} }$ If $\\chi$ is non-trivial, then by L-Function does not Vanish at One, $\\ln \\map L {s, \\chi}$ is bounded as $s \\to 1$. If $\\chi$ is trivial, then by Analytic Continuation of Dirichlet L-Function, $\\map L {s, \\chi}$ has a simple pole at $s = 1$. Therefore, in this case: :$\\map L {s, \\chi} \\sim \\dfrac \\lambda {s - 1}$ where $\\lambda$ is the residue of $\\map L {s, \\chi}$ at $1$, and: :$\\ln \\map L {s, \\chi} \\sim \\map \\ln {\\dfrac \\lambda {s - 1} } \\sim \\map \\ln {\\dfrac 1 {s - 1} }$ Thus if we can show that the second term of $(1)$ is bounded, the result holds. On $\\map \\Re s > 1$: {{begin-eqn}} {{eqn | l = \\size {\\sum_p \\sum_{n \\mathop \\ge 2} \\frac {\\map \\chi p^n} {n p^{n s} } }       | o = \\le       | r = \\sum_p \\sum_{n \\mathop \\ge 2} \\frac 1 {\\size {p^s}^n} }} {{eqn | r = \\sum_p \\frac 1 {\\size {p^{2 s} } \\, \\size {p^s - 1}^n}       | c = Sum of Infinite Geometric Sequence }} {{eqn | o = \\le       | r = \\sum_p \\frac 1 {p^2}       | c = because $\\map \\Re s > 1$ }} {{eqn | o = \\le       | r = \\sum_n \\frac 1 {n^2} }} {{end-eqn}} This last is $\\map \\zeta 2$ where $\\zeta$ is the Riemann zeta function, so is finite. {{qed}} Category:Dirichlet's Theorem on Arithmetic Sequences ax5uhccn3jldi6qc30a9840yrqwyec9"}
+{"_id": "32617", "title": "Dirichlet's Theorem on Arithmetic Sequences/Lemma 2", "text": "Dirichlet's Theorem on Arithmetic Sequences/Lemma 2 0 47370 455651 455645 2020-03-18T17:55:54Z Prime.mover 59 wikitext text/x-wiki == Lemma for Dirichlet's Theorem on Arithmetic Sequences == Let $a, q$ be coprime integers. Let $\\PP_{a, q}$ be the set of primes $p$ such that $p \\equiv a \\pmod q$. Define: :$\\eta_{a, q} : n \\mapsto \\begin{cases} 1 & : n \\equiv a \\pmod q\\\\ 0 & : \\text{otherwise} \\end{cases}$ <onlyinclude> Let $G = \\paren {\\Z / q \\Z}^\\times$. Let $G^*$ be the dual group of characters on $G$. {{explain|Explain the notation $\\paren {\\Z / q \\Z}^\\times$}} Then for all $n \\in \\N$: :$\\displaystyle \\map {\\eta_{a, q} } n = \\sum_{\\chi \\mathop \\in G^*} \\frac {\\map {\\overline \\chi} a} {\\map \\phi q} \\, \\map \\chi n$ </onlyinclude> == Proof == There is only one $x \\in G$ such that $\\map \\eta x \\ne 0$, and this equals $\\map \\eta a = 1$. {{explain|Back up the above statement by a link or an explanation. Are we sure that $\\map \\eta x$ means $\\map {\\eta_{a, q} } x$ here?}} So: :$\\displaystyle \\sum_{x \\mathop \\in G} \\map {\\eta_{a, q} } x \\, \\map {\\overline \\chi} x = \\map {\\overline \\chi} a$ Therefore, by Discrete Fourier Transform on Abelian Group we have for all $x \\in G$: :$\\displaystyle \\map \\eta x = \\frac 1 {\\map \\phi q} \\sum_{\\chi \\mathop \\in G^*} \\map {\\overline \\chi} a \\, \\map \\chi x$ as required. {{qed}} Category:Dirichlet's Theorem on Arithmetic Sequences djw33ie4ddkrld71wuqv6baweck1a68"}
+{"_id": "32618", "title": "Squaring the Circle", "text": "Squaring the Circle 0 47413 452698 275657 2020-03-05T16:15:35Z Prime.mover 59 wikitext text/x-wiki == Classic Problem == '''Squaring the Circle''' is creating a square with the same area as that of a given circle. == Solution == The problem of '''squaring the circle''' reduces to: :starting with a line segment of given length ($1$, say) :constructing a line segment of length $\\sqrt \\pi$ of the first. == Fallacious Proofs == {{:Squaring the Circle/Fallacious Proofs}} == Also see == * Squaring the Circle by Archimedean Spiral * Squaring the Circle by Compass and Straightedge Construction is Impossible * Trisecting the Angle * Doubling the Cube == Historical Note == {{:Squaring the Circle/Historical Note}} == Sources == * {{BookReference|Men of Mathematics|1937|Eric Temple Bell|prev = Doubling the Cube|next = Trisecting the Angle/Historical Note}}: Chapter $\\text{II}$: Modern Minds in Ancient Bodies * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Definition:Compass and Straightedge Construction|next = Squaring the Circle by Compass and Straightedge Construction is Impossible/Historical Note}}: Chapter $\\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Positive Square Root|next = Squaring the Circle/Historical Note|entry = squaring the circle}} Category:Classic Problems Category:Euclidean Geometry Category:Circles Category:Squaring the Circle eflxogcb0bd0glkrrarijc17w1kk9xd"}
+{"_id": "32619", "title": "Liouville's Constant is Transcendental/Corollary", "text": "Liouville's Constant is Transcendental/Corollary 0 47418 468551 441910 2020-05-17T13:07:44Z RandomUndergrad 3904 wikitext text/x-wiki == Corollary to Liouville's Constant is Transcendental == <onlyinclude> All real numbers of the form: {{begin-eqn}} {{eqn | l = \\sum_{n \\mathop \\ge 1} \\frac {a_n} {10^{n!} }       | r = \\frac {a_1} {10^1} + \\frac {a_2} {10^2} + \\frac {a_3} {10^6} + \\frac {a_4} {10^{24} } + \\cdots       | c =  }} {{end-eqn}} where :$a_1, a_2, a_3, \\ldots \\in \\set {1, 2, \\ldots, 9}$ are transcendental. </onlyinclude> == Proof == Let $n \\in \\N$. For $n = 1$, let $p = a_1$ and $q = 10$. Then: :$\\size {L - \\dfrac p q} = \\displaystyle \\sum_{k \\mathop \\ge 2} \\dfrac {a_k} {10^{k!} } < \\dfrac 1 {10} = \\dfrac 1 q$ For $n > 1$, let $q = 10^{n!}$ and write: :$\\displaystyle L = \\frac p q + \\sum_{k \\mathop = n + 1}^\\infty \\frac {a_k} {10^{k!} }$ for some suitable $p \\in \\Z$. Then: {{begin-eqn}} {{eqn | l = \\size {L - \\frac p q}       | r = \\sum_{k \\mathop = n + 1}^\\infty \\frac {a_k} {10^{k!} }       | c =   }} {{eqn | r = \\sum_{k \\mathop = n + 1}^\\infty \\frac 9 {10^{k!} }       | o = \\le       | c =  }} {{eqn | r = \\frac {18} {10^{\\paren {n + 1}!} }       | o = \\le       | c =  }} {{eqn | r = \\frac {18} {q^{n + 1} }       | c =  }} {{eqn | r = \\frac 1 {q^n}       | o = <       | c = as $q \\ge 100$ for all $n > 1$ }} {{end-eqn}} Thus, by definition, $L$ is a Liouville number. Therefore, by Liouville's Theorem, $L$ is transcendental. {{qed}} == Sources == * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Liouville's Constant is Transcendental|next = Euler's Number is Transcendental/Historical Note}}: Chapter $\\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental Category:Transcendental Number Theory Category:Transcendental Numbers 0alyacskwwju8zw0k0p1qhj80ivn46c"}
+{"_id": "32620", "title": "Product of Sums/Corollary", "text": "Product of Sums/Corollary 0 47758 353398 329927 2018-05-03T08:01:18Z Prime.mover 59 wikitext text/x-wiki == Corollary to Product of Sums == <onlyinclude> Let $\\displaystyle \\sum_{i \\mathop \\in X} a_{ij}$ be absolutely convergent sequences for all $j \\in Y$. Then: :$\\displaystyle \\prod_{j \\mathop \\in Y} \\left({\\sum_{i \\mathop \\in X} a_{ij}}\\right) = \\sum_{f: Y \\mathop \\to X} \\left( \\prod_{j \\mathop \\in Y} a_{f \\left({j}\\right) j} \\right)$ where $f$ runs over all mappings from $Y$ to $X$. </onlyinclude> == Proof == We will prove the case $X = Y = \\N$ to avoid the notational inconvenience of enumerating the elements of $Y$ as $j_1, j_2, j_3 \\dots$. The general case where $X, Y$ are arbitrary sets has the same proof, but with more indices and notational distractions. Consider that by the main theorem: :$\\displaystyle \\prod_{j \\mathop = 1, 2} \\left({\\sum_{i \\mathop \\in \\N} a_{ij} }\\right) = \\sum_{x, y \\mathop \\in \\N} a_{x_1}a_{y_2}$ and continuing in this vein: :$\\displaystyle \\prod_{j \\mathop = 1, 2, 3} \\left({\\sum_{i \\mathop \\in \\N} a_{ij} }\\right) = \\left({\\sum_{x, y \\mathop \\in \\N} a_{x_1} a_{y_2} }\\right) \\left({\\sum_{z \\mathop \\in \\N} a_{z_3} }\\right) = \\sum_{x, y, z \\mathop \\in \\N} a_{x_1} a_{y_2} a_{z_3}$ For an inductive proof of this concept for finite $n$, we assume that for some $n \\in \\N$: :$\\displaystyle \\prod_{j \\mathop = 1}^n \\left({ \\sum_{i \\mathop \\in \\N} a_{ij} }\\right)  = \\sum_{u, v, \\dots, x, y \\mathop \\in \\N} a_{u_1} a_{v_2}\\dots a_{x_{(n-1)}} a_{y_n}$ Then: :$\\displaystyle \\prod_{j \\mathop = 1}^{n+1} \\left({ \\sum_{i \\mathop \\in \\N} a_{ij} }\\right) = \\left({ \\sum_{u, v, \\dots, x, y  \\mathop \\in \\N} a_{u_1} a_{v_2}\\dots a_{x_{(n-1)}} a_{y_n} }\\right) \\left({\\sum_{z \\mathop \\in \\N} a_{z_n} }\\right)$ which by Product of Sums is simply: :$\\displaystyle \\sum_{u, v, \\ldots, x, y, z \\mathop \\in \\N} a_{u_1} a_{v_2} \\ldots a_{x_{(n-1}} a_{y_n} a_{z_{(n+1)}}$ completing the induction for finite $n$. {{finish}} Category:Number Theory Category:Analysis 3j7c7lbps4uu24cf3jzt9ytaxdr66d3"}
+{"_id": "32621", "title": "Real Function/Examples/Square Root", "text": "Real Function/Examples/Square Root 0 48059 465506 465501 2020-05-02T15:00:47Z Prime.mover 59 wikitext text/x-wiki == Example of Real Function == <onlyinclude> The '''(real) square root function''' is the real function $f: \\R \\to \\R$ defined on the positive real numbers as: :$\\forall x \\in \\R_{\\ge 0}: \\map f x = \\sqrt x$ </onlyinclude> :400px == Also see == * Domain of Real Square Root Function * Image of Real Square Root Function == Sources == * {{BookReference|Limits and Continuity|1964|William K. Smith|prev = Image of Real Square Function|next = Domain of Real Square Root Function}}: $\\S 2.2$: Functions Category:Examples of Real Functions j51cw9n2k2c634qth6wol2nx83s8aqd"}
+{"_id": "32622", "title": "Real Number to Negative Power/Positive Integer", "text": "Real Number to Negative Power/Positive Integer 0 48238 260871 2016-06-29T17:14:12Z Prime.mover 59 Created page with \"== Theorem ==  Let $r \\in \\R_{> 0}$ be a positive real number. <onlyinclude> Let $n \\in \\Z_{\\ge 0}$ be a Definition:Positive Integer|posi...\" wikitext text/x-wiki == Theorem == Let $r \\in \\R_{> 0}$ be a positive real number. <onlyinclude> Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Let $r^n$ be defined as $r$ to the power of $n$. Then: :$r^{-n} = \\dfrac 1 {r^n}$ </onlyinclude> == Proof == Proof by induction on $m$: For all $n \\in \\Z_{\\ge 0}$, let $P \\left({n}\\right)$ be the proposition: :$r^{-n} = \\dfrac 1 {r^n}$ $P \\left({0}\\right)$ is the case: {{begin-eqn}} {{eqn | l = r^{-0}       | r = r^0       | c =  }} {{eqn | r = 1       | c = Definition of Integer Power }} {{eqn | r = \\dfrac 1 1       | c =  }} {{eqn | r = \\dfrac 1 {r^0}       | c = Definition of Integer Power }} {{end-eqn}} === Basis for the Induction === $P \\left({1}\\right)$ is the case: {{begin-eqn}} {{eqn | l = r^{-1}       | r = \\dfrac {r^{-1 + 1} } r       | c = Definition of Integer Power }} {{eqn | r = \\dfrac {r^0} r       | c =  }} {{eqn | r = \\dfrac 1 r       | c = Definition of Integer Power }} {{eqn | r = \\dfrac 1 {r^1}       | c = Definition of Integer Power }} {{end-eqn}} This is our basis for the induction. === Induction Hypothesis === Now we need to show that, if $P \\left({k}\\right)$ is true, where $k \\ge 1$, then it logically follows that $P \\left({k+1}\\right)$ is true. So this is our induction hypothesis: :$r^{- k} = \\dfrac 1 {r^k}$ Then we need to show: :$r^{- \\left({k + 1}\\right)} = \\dfrac 1 {r^{k + 1} }$ === Induction Step === This is our induction step: {{begin-eqn}} {{eqn | l = r^{- \\left({k + 1}\\right)}       | r = \\dfrac {r^{-\\left({k + 1}\\right) + 1} } r       | c = Definition of Integer Power }} {{eqn | r = \\dfrac {r^{-k} } r       | c = simplification }} {{eqn | r = \\dfrac 1 {r^k \\times r}       | c = Induction Hypothesis }} {{eqn | r = \\dfrac 1 {r^{k + 1} }       | c = Definition of Integer Power }} {{end-eqn}} So $P \\left({k}\\right) \\implies P \\left({k+1}\\right)$ and the result follows by the Principle of Mathematical Induction. Therefore: :$\\forall n \\in \\Z_{\\ge 0}: r^{-n} = \\dfrac 1 {r^n}$ {{qed}} Category:Powers hwjw4kv1xbs5pk3wtufm0089r6bd0yu"}
+{"_id": "32623", "title": "Sum of Summations over Overlapping Domains/Example", "text": "Sum of Summations over Overlapping Domains/Example 0 48365 361114 361002 2018-07-28T11:26:35Z Prime.mover 59 wikitext text/x-wiki == Example of Sum of Summations over Overlapping Domains == <onlyinclude> :$\\displaystyle \\sum_{1 \\mathop \\le j \\mathop \\le m} a_j + \\sum_{m \\mathop \\le j \\mathop \\le n} a_j = \\left({\\sum_{1 \\mathop \\le j \\mathop \\le n} a_j}\\right) + a_m$ </onlyinclude> == Proof == Let $R \\left({j}\\right)$ be the propositional function $1 \\mathop \\le j \\mathop \\le m$. Let $S \\left({j}\\right)$ be the propositional function $m \\mathop \\le j \\mathop \\le n$. Then we have: {{begin-eqn}} {{eqn | l = R \\left({j}\\right) \\lor S \\left({j}\\right)       | r = \\left({1 \\mathop \\le j \\mathop \\le m}\\right) \\lor \\left({m \\mathop \\le j \\mathop \\le n}\\right)       | c =  }} {{eqn | r = \\left({1 \\mathop \\le j \\mathop \\le n}\\right)       | c =  }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = R \\left({j}\\right) \\land S \\left({j}\\right)       | r = \\left({1 \\mathop \\le j \\mathop \\le m}\\right) \\land \\left({m \\mathop \\le j \\mathop \\le n}\\right)       | c =  }} {{eqn | r = \\left({j = m}\\right)       | c =  }} {{end-eqn}} The result follows from Sum of Summations over Overlapping Domains. {{qed}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Sum of Summations over Overlapping Domains|next = Summation of Products of n Numbers taken m at a time with Repetitions/Examples/Order 2/Proof 1}}: $\\S 1.2.3$: Sums and Products: $(12)$ Category:Sum of Summations over Overlapping Domains 90i5vzypn2dvf587wlc21bcxt9gl4th"}
+{"_id": "32624", "title": "Factorial/Examples", "text": "Factorial/Examples 0 48835 433303 385347 2019-11-01T09:02:03Z Prime.mover 59 wikitext text/x-wiki == Examples of Factorials == <onlyinclude> The factorials of the first few positive integers are as follows: $\\begin{array}{r|r} n & n! \\\\ \\hline 0 & 1 \\\\ 1 & 1 \\\\ 2 & 2 \\\\ 3 & 6 \\\\ 4 & 24 \\\\ 5 & 120 \\\\ 6 & 720 \\\\ 7 & 5 \\, 040 \\\\ 8 & 40 \\, 320 \\\\ 9 & 362 \\, 880 \\\\ 10 & 3 \\, 628 \\, 800 \\\\ \\end{array}$ </onlyinclude> {{OEIS|A000142}} === Factorial of $0$ === {{:Factorial/Examples/0}} === Factorial of $1$ === {{:Factorial/Examples/1}} === Factorial of $10$ === {{:Factorial/Examples/10}} === Factorial of $11$ === {{:Factorial/Examples/11}} === Factorial of $12$ === {{:Factorial/Examples/12}} === Factorial of $13$ === {{:Factorial/Examples/13}} === Factorial of $14$ === {{:Factorial/Examples/14}} === Factorial of $15$ === {{:Factorial/Examples/15}} === Factorial of $16$ === {{:Factorial/Examples/16}} === Factorial of $17$ === {{:Factorial/Examples/17}} === Factorial of $18$ === {{:Factorial/Examples/18}} === Factorial of $19$ === {{:Factorial/Examples/19}} === Factorial of $20$ === {{:Factorial/Examples/20}} === Factorial of $21$ === {{:Factorial/Examples/21}} === Factorial of $22$ === {{:Factorial/Examples/22}} === Factorial of $23$ === {{:Factorial/Examples/23}} === Factorial of $24$ === {{:Factorial/Examples/24}} === Factorial of $25$ === {{:Factorial/Examples/25}} === Prime Factors of $39!$ === {{:Prime Factors of 39 Factorial}} === Factorial of $52$ === {{:Factorial/Examples/52}} === Factorial of $450$ === {{:Factorial/Examples/450}} === Factorial of $1\\,000$ === {{:Factorial/Examples/1000}} === Factorial of $1\\,000\\,000$ === {{:Factorial/Examples/1,000,000}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Definition:Prime Number/Sequence|next = Reciprocals of Prime Numbers}}: Tables: $5$ The Factorials of the Numbers $1$ to $20$ * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Definition:Factorial/Definition 1|next = Factorial/Examples/1000}}: $\\S 1.2.5$: Permutations and Factorials * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Definition:Prime Number/Sequence|next = Reciprocals of Prime Numbers}}: Tables: $5$ The Factorials of the Numbers $1$ to $20$ Category:Factorials/Examples d7h41nrft4lnbuea0crnzgroxevvwqd"}
+{"_id": "32625", "title": "Factorial/Examples/0", "text": "Factorial/Examples/0 0 48836 492902 394672 2020-10-06T09:58:31Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> The factorial of $0$ is $1$: :$0! = 1$ </onlyinclude> == Proof == From the definition of factorial: :$n! = \\displaystyle \\prod_{k \\mathop = 1}^n k$ where $\\prod$ denotes product notation. When $n = 0$ we have: :$0! = \\displaystyle \\prod_{k \\mathop = 1}^0 k$ Hence the result, by definition of vacuous product. {{qed}} == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Cardinality of Power Set of Finite Set/Proof 1|next = Definition:Factorial/Definition 2}}: $\\S 19$ * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Definition:Factorial/Definition 2|next = Binomial Theorem/Integral Index}}: $\\S 3$: The Binomial Formula and Binomial Coefficients: $3.2$ * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Definition:Factorial/Definition 2|next = Quintuple Angle Formulas/Cosine/Proof 2}}: $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $21$ * {{BookReference|Special Functions of Mathematics for Engineers|1992|Larry C. Andrews|ed = 2nd|edpage = Second Edition|prev = Definition:Factorial/Definition 1|next = Binomial Theorem/Integral Index}}: $\\S 1.2.4$: Factorials and binomial coefficients * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Definition:Factorial/Definition 2|next = Definition:Factorial/Definition 1}}: $\\S 1.2.5$: Permutations and Factorials: $(5)$ Category:Factorials/Examples lngw6wmaj8gpip8lsyzya4wm9s5cf22"}
+{"_id": "32626", "title": "Common Logarithm/Examples/e", "text": "Common Logarithm/Examples/e 0 48843 433234 433024 2019-11-01T08:09:29Z Prime.mover 59 wikitext text/x-wiki == Example of Common Logarithm == <onlyinclude> The common logarithm of Euler's number $e$ is: :$\\log_{10} e = 0 \\cdotp 43429 \\, 44819 \\, 03251 \\, 82765 \\, 11289 \\, 18916 \\, 60508 \\, 22943 \\, 97005 \\, 803 \\ldots$ </onlyinclude> {{OEIS|A002285}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Common Logarithm of 3|next = Common Logarithm of Pi}}: $\\S 1$: Special Constants: $1.15$ * {{BookReference|Les Nombres Remarquables|1983|François Le Lionnais|author2 = Jean Brette|prev = Definition:Bloch's Constant/Historical Note|next = Change of Base of Logarithm/Base e to Base 10/Form 1}}: $0,43429 44819 \\ldots$ * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Envelope Problem|next = One Half as Pandigital Fraction}}: $0 \\cdotp 434 \\, 294 \\, 481 \\, 903 \\, 251 \\, 827 \\, 651 \\, 128 \\, 918 \\, 916 \\, 605 \\, 082 \\, 294 \\, 397 \\, 005 \\, 803 \\ldots$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Envelope Problem|next = One Half as Pandigital Fraction}}: $0 \\cdotp 43429 \\, 44819 \\, 03251 \\, 82765 \\, 11289 \\, 18916 \\, 60508 \\, 22943 \\, 97005 \\, 803 \\ldots$ Category:Examples of Common Logarithms Category:Euler's Number 5o0xza90mr8tpyg38zeer7glp0ow4ps"}
+{"_id": "32627", "title": "Common Logarithm/Examples/pi", "text": "Common Logarithm/Examples/pi 0 48845 433237 433235 2019-11-01T08:10:12Z Prime.mover 59 wikitext text/x-wiki == Example of Common Logarithm == <onlyinclude> The common logarithm of $\\pi$ is: :$\\log_{10} \\pi = 0.49714 \\, 98726 \\, 94133 \\, 85435 \\, 12683 \\ldots$ </onlyinclude> {{OEIS|A053511}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Common Logarithm of e|next = Natural Logarithm of 10}}: $\\S 1$: Special Constants: $1.16$ Category:Examples of Common Logarithms f2x26ka1sftbcfjmv4v8j8a2qps5198"}
+{"_id": "32628", "title": "Natural Logarithm/Examples/10", "text": "Natural Logarithm/Examples/10 0 48849 433418 432995 2019-11-01T12:59:21Z Prime.mover 59 wikitext text/x-wiki == Example of Natural Logarithm == <onlyinclude> The natural logarithm of $10$ is approximately: :$\\ln 10 \\approx 2 \\cdotp 30258 \\, 50929 \\, 94045 \\, 68401 \\, 7991 \\ldots$ </onlyinclude> {{OEIS|A002392}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Common Logarithm of Pi|next = Natural Logarithm of 2}}: $\\S 1$: Special Constants: $1.17$ * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Square Root of 5|next = Definition:Stirling's Constant}}: $2 \\cdotp 302 \\, 585 \\, 092 \\, 994 \\, 045 \\, 684 \\, 017 \\, 991 \\, 454 \\, 684 \\, 364 \\, 207 \\, 601 \\ldots$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Definition:Wallis's Number/Historical Note|next = Definition:Stirling's Constant}}: $2 \\cdotp 30258 \\, 50929 \\, 94045 \\, 68401 \\, 79914 \\, 54684 \\, 36420 \\, 7601 \\ldots$ Category:Examples of Natural Logarithms sqzz4hegej02mjzduxzvjdoysvr5ugq"}
+{"_id": "32629", "title": "Common Logarithm/Examples/2", "text": "Common Logarithm/Examples/2 0 48852 433232 432975 2019-11-01T08:08:56Z Prime.mover 59 wikitext text/x-wiki == Example of Common Logarithm == <onlyinclude> The common logarithm of $2$ is: :$\\log_{10} 2 \\approx 0.30102 \\, 99956 \\, 63981 \\, 19521 \\, 37389 \\ldots$ </onlyinclude> {{OEIS|A007524}} == Also see == * Logarithm Base 10 of 2 is Irrational == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Euler's Number to Power of Itself|next = Common Logarithm of 3}}: $\\S 1$: Special Constants: $1.13$ * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Anomalous Cancellation/Variants/3 + 25 + 38 over 7 + 20 + 39|next = Number of Digits in Power of 2}}: $0 \\cdotp 301 \\, 029 \\, 995 \\, 663 \\, 981 \\ldots$ * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Definition:Base of Logarithm|next = Sum of Logarithms/General Logarithm}}: $\\S 1.2.2$: Numbers, Powers, and Logarithms: $(10)$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Anomalous Cancellation/Variants/3 + 25 + 38 over 7 + 20 + 39|next = Number of Digits in Power of 2}}: $0 \\cdotp 30102 \\, 99956 \\, 63981 \\ldots$ Category:Examples of Common Logarithms rg51xodj34e98e8q0kyr4c0nhhsomwt"}
+{"_id": "32630", "title": "De Polignac's Formula/Examples/3 in 1000", "text": "De Polignac's Formula/Examples/3 in 1000 0 48858 366427 298881 2018-09-19T12:42:27Z Prime.mover 59 wikitext text/x-wiki == Example of Use of De Polignac's Formula == <onlyinclude> The prime factor $3$ appears in $1000!$ to the power of $498$. That is: :$3^{498} \\divides 1000!$ but: :$3^{499} \\nmid 1000!$ </onlyinclude> == Proof == Let $\\mu$ denote the power of $3$ which divides $1000!$ {{begin-eqn}} {{eqn | l = \\mu       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {1000} {3^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {1000} 3} + \\floor {\\frac {1000} 9} + \\floor {\\frac {1000} {27} } + \\floor {\\frac {1000} {81} } + \\floor {\\frac {1000} {243} } + \\floor {\\frac {1000} {729} }       | c =  }} {{eqn | r = 333 + 111 + 37 + 12 + 4 + 1       | c =  }} {{eqn | r = 498       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = De Polignac's Formula|next = De Polignac's Formula/Technique}}: $\\S 1.2.5$: Permutations and Factorials Category:De Polignac's Formula cxmcv5bldadrqcq4nkiaz91z076af8t"}
+{"_id": "32631", "title": "Negated Upper Index of Binomial Coefficient/Corollary 2", "text": "Negated Upper Index of Binomial Coefficient/Corollary 2 0 48908 417835 391833 2019-08-11T14:05:13Z Danhook 3709 Remove incorrectly duplicated comment wikitext text/x-wiki == Corollary to Negated Upper Index of Binomial Coefficient == Let $n, m \\in \\Z$. Then: <onlyinclude> :$\\dbinom n m = \\paren {-1}^{n - m} \\dbinom {-\\paren {m + 1} } {n - m}$ </onlyinclude> where $\\dbinom n m$ is a binomial coefficient. == Proof == {{begin-eqn}} {{eqn | l = \\dbinom r k       | r = \\paren {-1}^k \\dbinom {k - r - 1} k       | c = Negated Upper Index of Binomial Coefficient }} {{eqn | ll= \\leadsto       | l = \\dbinom n {n - m}       | r = \\paren {-1}^{n - m} \\dbinom {\\paren {n - m} - n - 1} {n - m}       | c = setting $r = n$ and $k = n - m$ }} {{eqn | ll= \\leadsto       | l = \\dbinom n m       | r = \\paren {-1}^{n - m} \\dbinom {-m - 1} {n - m}       | c = Symmetry Rule for Binomial Coefficients }} {{eqn | r = \\paren {-1}^{n - m} \\dbinom {- \\paren {m + 1} } {n - m} }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Alternating Sum and Difference of r Choose k up to n/Proof 1|next = Product of r Choose m with m Choose k/Proof 1}}: $\\S 1.2.6$: Binomial Coefficients: $(19)$ Category:Binomial Coefficients Category:Negated Upper Index of Binomial Coefficient 1rdl6gyubi8v4fhw6qqxn3qmnv6bi4f"}
+{"_id": "32632", "title": "Sum over k of r Choose k by s+k Choose n by -1^r-k/Corollary", "text": "Sum over k of r Choose k by s+k Choose n by -1^r-k/Corollary 0 48983 496845 357669 2020-10-28T22:07:25Z Prime.mover 59 wikitext text/x-wiki == Corollary to Sum over $k$ of $\\dbinom r k \\dbinom {s + k} n \\paren {-1}^{r - k}$ == Let $r \\in \\Z_{\\ge 0}, n \\in \\Z$. Then: <onlyinclude> :$\\ds \\sum_k \\binom r k \\binom k n \\paren {-1}^{r - k} = \\delta_{n r}$ where $\\delta_{n r}$ is the Kronecker delta. </onlyinclude> == Proof == From Sum over $k$ of $\\dbinom r k \\dbinom {s + k} n \\paren {-1}^{r - k}$: :$\\ds \\sum_k \\binom r k \\binom {s + k} n \\paren {-1}^{r - k} = \\binom s {n - r}$ which holds for $s \\in \\R, r \\in \\Z_{\\ge 0}, n \\in \\Z$. Setting $s = 0$: :$\\ds \\sum_k \\binom r k \\binom k n \\paren {-1}^{r - k} = \\binom 0 {n - r}$ We have by definition of binomial coefficient that: :$\\dbinom 0 0 = 1$ and: :$\\forall n \\in \\Z_{\\ne 0}: \\dbinom 0 n = 0$ So, using Iverson's convention: :$\\dbinom 0 {n - r} = \\sqrbk {n = r}$ The result follows by definition of the Kronecker delta. {{qed}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Factorial as Sum of Series of Subfactorial by Falling Factorial over Factorial/Proof|next = Sum over k of r Choose k by -1^r-k by Polynomial/Proof 1}}: $\\S 1.2.6$: Binomial Coefficients: $(33)$ Category:Sum over k of r Choose k by s+k Choose n by -1^r-k endpvg3aubb87ul64mh39oqtkmwx386"}
+{"_id": "32633", "title": "Condition for Infimum of Subset to equal Infimum of Set", "text": "Condition for Infimum of Subset to equal Infimum of Set 0 49120 265908 2016-08-02T09:05:03Z Ivar Sand 2302 Created page with \"== Lemma ==  Let $S$ be a real set.  Let $T$ be a subset of $S$.  Let $S$ and $T$ admit Definition:Infimum of Subset of Real...\" wikitext text/x-wiki == Lemma == Let $S$ be a real set. Let $T$ be a subset of $S$. Let $S$ and $T$ admit infima. Then: : $\\inf S = \\inf T \\iff \\forall \\epsilon \\in \\R_{>0}: \\forall s \\in S: \\exists t \\in T: s + \\epsilon > t$ == Proof == === Necessary Condition === Let $\\inf S = \\inf T$. The aim is to establish that $\\forall \\epsilon \\in \\R_{>0}: \\forall s \\in S: \\exists t \\in T: s + \\epsilon > t$. We have: {{begin-eqn}} {{eqn | l = \\inf S       | r = \\inf T }} {{eqn | ll = \\implies       | l = \\inf S       | o = \\ge       | r = \\inf T }} {{eqn | ll = \\iff       | l = \\forall \\epsilon \\in \\R_{>0}: \\forall s \\in S: \\exists t \\in T: s + \\epsilon       | o = >       | r = t       | c = Infima of two Real Sets }} {{end-eqn}} === Sufficient Condition === Let $\\forall \\epsilon \\in \\R_{>0}: \\forall s \\in S: \\exists t \\in T: s + \\epsilon > t$. The aim is to establish that $\\inf S = \\inf T$. We have: {{begin-eqn}} {{eqn | l = \\forall \\epsilon \\in \\R_{>0}: \\forall s \\in S: \\exists t \\in T: s + \\epsilon       | o = >       | r = t }} {{eqn | ll = \\iff       | l = \\inf S       | o = \\ge       | r = \\inf T       | c = Infima of two Real Sets }} {{eqn | ll = \\iff       | l = \\inf S       | o = \\ge       | r = \\inf T \\ge \\inf S       | c = as $\\inf T \\ge \\inf S$ is true by Infimum of Subset }} {{eqn | ll = \\iff       | l = \\inf S       | r = \\inf T       | c =  }} {{end-eqn}} {{qed}} Category:Real Analysis dv10hz54fb12g0za3wp8mwk2s3tdzx3"}
+{"_id": "32634", "title": "Limit of Monotone Real Function/Increasing/Corollary", "text": "Limit of Monotone Real Function/Increasing/Corollary 0 49157 454676 354386 2020-03-16T06:18:14Z Prime.mover 59 wikitext text/x-wiki == Corollary to Limit of Increasing Function == <onlyinclude> Let $f$ be a real function which is increasing on the open interval $\\openint a b$. If $\\xi \\in \\openint a b$, then: :$\\map f {\\xi^-}$ and $\\map f {\\xi^+}$ both exist and: :$\\map f x \\le \\map f {\\xi^-} \\le \\map f \\xi \\le \\map f {\\xi^+} \\le \\map f y$ provided that $a < x < \\xi < y < b$. </onlyinclude> == Proof == $f$ is bounded above on $\\openint a b$ by $\\map f \\xi$. By Limit of Increasing Function, the supremum is $\\map f {\\xi^-}$. So it follows that: :$\\forall x \\in \\openint a \\xi: \\map f x \\le \\map f {\\xi^-} \\le \\map f \\xi$ A similar argument for $\\openint \\xi b$ holds for the other inequalities. {{qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Limit of Decreasing Function|next = Derivative of Monotone Function}}: $\\S 12.5$ Category:Limits of Functions hhyl57m26twizx6dylbouij3p8zx4l7"}
+{"_id": "32635", "title": "Limit of Monotone Real Function/Decreasing/Corollary", "text": "Limit of Monotone Real Function/Decreasing/Corollary 0 49159 447658 354387 2020-02-09T08:47:12Z Prime.mover 59 wikitext text/x-wiki == Corollary to Limit of Decreasing Function == <onlyinclude> Let $f$ be a real function which is decreasing on the open interval $\\openint a b$. If $\\xi \\in \\openint a b$, then: :$\\map f {\\xi^-}$ and $\\map f {\\xi^+}$ both exist and: :$\\map f x \\ge \\map f {\\xi^-} \\ge \\map f \\xi \\ge \\map f {\\xi^+} \\ge \\map f y$ provided that $a < x < \\xi < y < b$. </onlyinclude> == Proof == $f$ is bounded below on $\\openint a \\xi$ by $\\map f \\xi$. By Limit of Decreasing Function, the infimum is $\\map f {\\xi^-}$. So it follows that: :$\\forall x \\in \\openint a \\xi: \\map f x \\ge \\map f {\\xi^-} \\ge \\map f \\xi$ A similar argument for $\\openint \\xi b$ holds for the other inequalities. {{qed}} Category:Limits of Functions h4jsquhqhqxxu35tt2yubcvdpl9pfi2"}
+{"_id": "32636", "title": "Prime Factors of 52 Factorial", "text": "Prime Factors of 52 Factorial 0 49189 475913 447243 2020-06-25T07:01:05Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> The prime decomposition of $52!$ is given as: :$52! = 2^{49} \\times 3^{23} \\times 5^{12} \\times 7^8 \\times 11^4 \\times 13^4 \\times 17^3 \\times 19^2 \\times 23^2 \\times 29 \\times 31 \\times 37 \\times 41 \\times 43 \\times 47$ </onlyinclude> == Proof == For each prime factor $p$ of $52!$, let $a_p$ be the integer such that: :$p^{a_p} \\divides 52!$ :$p^{a_p + 1} \\nmid 52!$ Taking the prime factors in turn: {{begin-eqn}} {{eqn | l = a_2       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {52} {2^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {52} 2} + \\floor {\\frac {52} 4} + \\floor {\\frac {52} 8 } + \\floor {\\frac {52} {16} } + \\floor {\\frac {52} {32} }       | c =  }} {{eqn | r = 26 + 13 + 6 + 3 + 1       | c =  }} {{eqn | r = 49       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_3       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {52} {3^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {52} 3} + \\floor {\\frac {52} 9} + \\floor {\\frac {52} {27} }       | c =  }} {{eqn | r = 17 + 5 + 1       | c =  }} {{eqn | r = 23       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_5       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {52} {5^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {52} 5} + \\floor {\\frac {52} {25} }       | c =  }} {{eqn | r = 10 + 2       | c =  }} {{eqn | r = 12       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_7       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {52} {7^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {52} 7} + \\floor {\\frac {52} {49} }       | c =  }} {{eqn | r = 7 + 1       | c =  }} {{eqn | r = 8       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{11}       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {52} {11^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {52} {11} }       | c =  }} {{eqn | r = 4       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{13}       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {52} {13^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {52} {13} }       | c =  }} {{eqn | r = 4       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{17}       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {52} {17^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {52} {17} }       | c =  }} {{eqn | r = 3       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{19}       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {52} {19^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {52} {19} }       | c =  }} {{eqn | r = 2       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{23}       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {52} {23^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {52} {23} }       | c =  }} {{eqn | r = 2       | c =  }} {{end-eqn}} Similarly: :$a_{29} = 1$ :$a_{31} = 1$ :$a_{37} = 1$ :$a_{41} = 1$ :$a_{43} = 1$ :$a_{47} = 1$ Hence the result. {{qed}} Category:Factorials/Examples Category:52 dydr7sxaqe3lc78afr39ts81wcply67"}
+{"_id": "32637", "title": "Prime Factors of 13 Factorial", "text": "Prime Factors of 13 Factorial 0 49190 475909 307335 2020-06-25T06:59:05Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> The prime decomposition of $13!$ is given as: :$13! = 2^{10} \\times 3^5 \\times 5^2 \\times 7 \\times 11 \\times 13$ </onlyinclude> == Proof == For each prime factor $p$ of $13!$, let $a_p$ be the integer such that: :$p^{a_p} \\divides 13!$ :$p^{a_p + 1} \\nmid 13!$ Taking the prime factors in turn: {{begin-eqn}} {{eqn | l = a_2       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {13} {2^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {13} 2} + \\floor {\\frac {13} 4} + \\floor {\\frac {13} 8}       | c =  }} {{eqn | r = 6 + 3 + 1       | c =  }} {{eqn | r = 10       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_3       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {13} {3^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {13} 3} + \\floor {\\frac {13} 9}       | c =  }} {{eqn | r = 4 + 1       | c =  }} {{eqn | r = 5       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_5       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {13} {5^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {13} 5}       | c =  }} {{eqn | r = 2       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_7       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {13} {7^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {13} 7}       | c =  }} {{eqn | r = 1       | c =  }} {{end-eqn}} Similarly: :$a_{11} = 1$ :$a_{13} = 1$ Hence the result. {{qed}} Category:Factorials/Examples Category:13 nawfhubalbhphu2ruckw39pdtg157lr"}
+{"_id": "32638", "title": "Prime Factors of 39 Factorial", "text": "Prime Factors of 39 Factorial 0 49191 475912 453485 2020-06-25T07:00:40Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> The prime decomposition of $39!$ is given as: :$39! = 2^{35} \\times 3^{18} \\times 5^8 \\times 7^5 \\times 11^3 \\times 13^3 \\times 17^2 \\times 19^2 \\times 23 \\times 29 \\times 31 \\times 37$ </onlyinclude> == Proof == For each prime factor $p$ of $39!$, let $a_p$ be the integer such that: :$p^{a_p} \\divides 39!$ :$p^{a_p + 1} \\nmid 39!$ Taking the prime factors in turn: {{begin-eqn}} {{eqn | l = a_2       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {39} {2^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {39} 2} + \\floor {\\frac {39} 4} + \\floor {\\frac {39} 8 } + \\floor {\\frac {39} {16} } + \\floor {\\frac {39} {32} }       | c =  }} {{eqn | r = 19 + 9 + 4 + 2 + 1       | c =  }} {{eqn | r = 35       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_3       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {39} {3^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {39} 3} + \\floor {\\frac {39} 9} + \\floor {\\frac {39} {27} }       | c =  }} {{eqn | r = 13 + 4 + 1       | c =  }} {{eqn | r = 18       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_5       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {39} {5^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {39} 5} + \\floor {\\frac {39} {25} }       | c =  }} {{eqn | r = 7 + 1       | c =  }} {{eqn | r = 8       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_7       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {39} {7^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {39} 7}       | c =  }} {{eqn | r = 5       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{11}       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {39} {11^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {39} {11} }       | c =  }} {{eqn | r = 3       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{13}       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {39} {13^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {39} {13} }       | c =  }} {{eqn | r = 3       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{17}       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {39} {17^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {39} {17} }       | c =  }} {{eqn | r = 2       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{19}       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {39} {19^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {39} {19} }       | c =  }} {{eqn | r = 2       | c =  }} {{end-eqn}} Similarly: :$a_{23} = 1$ :$a_{29} = 1$ :$a_{31} = 1$ :$a_{37} = 1$ Hence the result. {{qed}} Category:Factorials/Examples Category:39 kj1lusrp5gnw9m7w8puaq3gvq5xprdp"}
+{"_id": "32639", "title": "Lucas' Theorem/Corollary", "text": "Lucas' Theorem/Corollary 0 49210 484319 465328 2020-08-31T10:46:07Z Prime.mover 59 wikitext text/x-wiki == Corollary to Lucas' Theorem == Let $p$ be a prime number. Let $n, k \\in \\Z$. <onlyinclude> Let the representations of $n$ and $k$ to the base $p$ be given by: :$n = a_r p^r + \\cdots + a_1 p + a_0$ :$k = b_r p^r + \\cdots + b_1 p + b_0$ Then: :$\\displaystyle \\dbinom n k \\equiv \\prod_{j \\mathop = 0}^r \\dbinom {a_j} {b_j} \\pmod p$ </onlyinclude> where $\\dbinom n k$ denotes a binomial coefficient. == Proof == Consider the representations of $n$ and $k$ to the base $p$: :$n = a_r p^r + \\cdots + a_1 p + a_0$ :$k = b_r p^r + \\cdots + b_1 p + b_0$ Let: :$n_1 = \\floor {n / p}$ :$k_1 = \\floor {k / p}$ We have that: :$n \\bmod p = a_0$ :$k \\bmod p = b_0$ :$n_1 = a_r p^{r - 1} + a_{r - 1} p^{r - 2} \\cdots + a_1$ :$k_1 = b_r p^{r - 1} + b_{r - 1} p^{r - 2} \\cdots + b_1$ It follows from Lucas' Theorem that: :$\\dbinom n k \\equiv \\dbinom {n_1} {k_1} \\dbinom {a_0} {b_0} \\pmod p$ Now we do the same again to the representation to the base $p$ of $n_1$ and $n_2$. Thus: :$\\dbinom n k \\equiv \\dbinom {\\floor {n_1 / p}} {\\floor {k_1 / p} } \\dbinom {a_1} {b_1} \\dbinom {a_0} {b_0} \\pmod p$ and so on until: :$\\floor {n_r / p}$ and: :$\\floor {k_r / p}$ Hence the result. {{qed}} {{Namedfor|François Édouard Anatole Lucas|cat = Lucas}} == Sources == * {{citation|date = 1947|title = Binomial Coefficients Modulo a Prime|journal = American Mathematical Monthly|abbr = Amer. Math. Monthly|volume = 54|startpage = 589|endpage = 592|author = N.J. Fine|jstor = 2304500}} * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Lucas' Theorem|next = Kummer's Theorem}}: $\\S 1.2.6$: Binomial Coefficients: Exercise $10 \\ \\text{(f)}$ Category:Binomial Coefficients Category:Number Theory Category:Prime Numbers a9gkjsyp1fk0vetvl25z9ir6ks8mecg"}
+{"_id": "32640", "title": "De Polignac's Formula/Examples/2 in 1000", "text": "De Polignac's Formula/Examples/2 in 1000 0 49319 447241 298883 2020-02-08T00:17:04Z Prime.mover 59 wikitext text/x-wiki == Example of Use of De Polignac's Formula == <onlyinclude> The prime factor $2$ appears in $1000!$ to the power of $994$. That is: :$2^{994} \\divides 1000!$ but: :$2^{995} \\nmid 1000!$ </onlyinclude> == Proof == Let $\\mu$ denote the power of $2$ which divides $1000!$ {{begin-eqn}} {{eqn | l = \\mu       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {1000} {2^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {1000} 2} + \\floor {\\frac {1000} 4} + \\floor {\\frac {1000} 8} + \\floor {\\frac {1000} {16} } + \\floor {\\frac {1000} {32} }       | c =  }} {{eqn | o =        | ro= +       | r = \\floor {\\frac {1000} {64} } + \\floor {\\frac {1000} {128} } + \\floor {\\frac {1000} {256} } + \\floor {\\frac {1000} {512} }       | c =  }} {{eqn | r = 500 + 250 + 125 + 62 + 31 + 15 + 7 + 3 + 1       | c =  }} {{eqn | r = 994       | c =  }} {{end-eqn}} {{qed}} Category:De Polignac's Formula hyh7qxmaoutkcvc1jokk2e0hz3mw8uf"}
+{"_id": "32641", "title": "De Polignac's Formula/Examples/5 in 1000", "text": "De Polignac's Formula/Examples/5 in 1000 0 49320 371895 298893 2018-10-19T08:48:45Z Prime.mover 59 wikitext text/x-wiki == Example of Use of De Polignac's Formula == <onlyinclude> The prime factor $5$ appears in $1000!$ to the power of $249$. That is: :$5^{249} \\divides 1000!$ but: :$5^{250} \\nmid 1000!$ </onlyinclude> == Proof == Let $\\mu$ denote the power of $5$ which divides $1000!$ {{begin-eqn}} {{eqn | l = \\mu       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {1000} {5^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {1000} 5} + \\floor {\\frac {1000} {25} } + \\floor {\\frac {1000} {125} } + \\floor {\\frac {1000} {625} }       | c =  }} {{eqn | r = 200 + 40 + 8 + 1       | c =  }} {{eqn | r = 249       | c =  }} {{end-eqn}} {{qed}} Category:De Polignac's Formula 3qhncqeqbifm0j1axs8y8kc7x5lpvoi"}
+{"_id": "32642", "title": "De Polignac's Formula/Examples/7 in 1000", "text": "De Polignac's Formula/Examples/7 in 1000 0 49321 457791 298892 2020-03-27T07:59:04Z Prime.mover 59 wikitext text/x-wiki == Example of Use of De Polignac's Formula == <onlyinclude> The prime factor $7$ appears in $1000!$ to the power of $164$. That is: :$7^{164} \\divides 1000!$ but: :$7^{165} \\nmid 1000!$ </onlyinclude> == Proof == Let $\\mu$ denote the power of $7$ which divides $1000!$ {{begin-eqn}} {{eqn | l = \\mu       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {1000} {7^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {1000} 7} + \\floor {\\frac {1000} {49} } + \\floor {\\frac {1000} {343} }       | c =  }} {{eqn | r = 142 + 20 + 2       | c =  }} {{eqn | r = 164       | c =  }} {{end-eqn}} {{qed}} Category:De Polignac's Formula 3uy57p7sp7vx5rkrfwudk3v311e6odb"}
+{"_id": "32643", "title": "De Polignac's Formula/Examples/11 in 1000", "text": "De Polignac's Formula/Examples/11 in 1000 0 49322 376170 298887 2018-11-12T14:41:11Z Prime.mover 59 wikitext text/x-wiki == Example of Use of De Polignac's Formula == <onlyinclude> The prime factor $11$ appears in $1000!$ to the power of $98$. That is: :$11^{98} \\divides 1000!$ but: :$11^{99} \\nmid 1000!$ </onlyinclude> == Proof == Let $\\mu$ denote the power of $11$ which divides $1000!$ {{begin-eqn}} {{eqn | l = \\mu       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {1000} {11^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {1000} {11} } + \\floor {\\frac {1000} {121} }       | c =  }} {{eqn | r = 90 + 8       | c =  }} {{eqn | r = 98       | c =  }} {{end-eqn}} {{qed}} Category:De Polignac's Formula pejdiyzux29elx765xubzzatd3st0ru"}
+{"_id": "32644", "title": "De Polignac's Formula/Examples/13 in 1000", "text": "De Polignac's Formula/Examples/13 in 1000 0 49323 457790 298886 2020-03-27T07:58:15Z Prime.mover 59 wikitext text/x-wiki == Example of Use of De Polignac's Formula == <onlyinclude> The prime factor $13$ appears in $1000!$ to the power of $81$. That is: :$13^{81} \\divides 1000!$ but: :$13^{82} \\nmid 1000!$ </onlyinclude> == Proof == Let $\\mu$ denote the power of $13$ which divides $1000!$ {{begin-eqn}} {{eqn | l = \\mu       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {1000} {13^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {1000} {13} } + \\floor {\\frac {1000} {169} }       | c =  }} {{eqn | r = 76 + 5       | c =  }} {{eqn | r = 81       | c =  }} {{end-eqn}} {{qed}} Category:De Polignac's Formula asv117cxyp8yjv4onegahw8c5c8g1fu"}
+{"_id": "32645", "title": "Prime Factors of 1000 Factorial", "text": "Prime Factors of 1000 Factorial 0 49324 475910 437932 2020-06-25T06:59:38Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> The prime decomposition of $1000!$ is given as: {{begin-eqn}} {{eqn | l = 1000!       | r = 2^{994} \\times 3^{498} \\times 5^{249} \\times 7^{164} \\times 11^{98} \\times 13^{81} \\times 17^{61} \\times 19^{54} \\times 23^{44} \\times 29^{35} \\times 31^{33} \\times 37^{27} }} {{eqn | o =        | ro= \\times       | r = 41^{24} \\times 43^{23} \\times 47^{21} \\times 53^{18} \\times 59^{16} \\times 61^{16} \\times 67^{14} \\times 71^{14} \\times 73^{13} \\times 79^{13} \\times 83^{12} \\times 89^{11}       | c =  }} {{eqn | o =        | ro= \\times       | r = 97^{10} \\times 101^9 \\times 103^9 \\times 107^9 \\times 109^9 \\times 113^8 \\times 127^7 \\times 131^7 \\times 137^7 \\times 139^7 \\times 149^6 \\times 151^6 }} {{eqn | o =        | ro= \\times       | r = 157^6 \\times 163^6 \\times 167^5 \\times 173^5 \\times 179^5 \\times 181^5 \\times 191^5 \\times 193^5 \\times 197^5 \\times 199^5 \\times 211^4 \\times 223^4 }} {{eqn | o =        | ro= \\times       | r = 227^4 \\times 229^4 \\times 233^4 \\times 239^4 \\times 241^4 \\times 251^3 \\times 257^3 \\times 263^3 \\times 269^3 \\times 271^3 \\times 277^3 \\times 281^3 }} {{eqn | o =        | ro= \\times       | r = 283^3 \\times 293^3 \\times 307^3 \\times 311^3 \\times 313^3 \\times 317^3 \\times 331^3 \\times 337^2 \\times 347^2 \\times 349^2 \\times 353^2 \\times 359^2 }} {{eqn | o =        | ro= \\times       | r = 367^2 \\times 373^2 \\times 379^2 \\times 383^2 \\times 389^2 \\times 397^2 \\times 401^2 \\times 409^2 \\times 419^2 \\times 421^2 \\times 431^2 \\times 433^2 }} {{eqn | o =        | ro= \\times       | r = 439^2 \\times 443^2 \\times 449^2 \\times 457^2 \\times 461^2 \\times 463^2 \\times 467^2 \\times 479^2 \\times 487^2 \\times 491^2 \\times 499^2 \\times 503 }} {{eqn | o =        | ro= \\times       | r = 509 \\times 521 \\times 523 \\times 541 \\times 547 \\times 557 \\times 563 \\times 569 \\times 571 \\times 577 \\times 587 \\times 593 }} {{eqn | o =        | ro= \\times       | r = 599 \\times 601 \\times 607 \\times 613 \\times 617 \\times 619 \\times 631 \\times 641 \\times 643 \\times 647 \\times 653 \\times 659 }} {{eqn | o =        | ro= \\times       | r = 661 \\times 673 \\times 677 \\times 683 \\times 691 \\times 701 \\times 709 \\times 719 \\times 727 \\times 733 \\times 739 \\times 743 }} {{eqn | o =        | ro= \\times       | r = 751 \\times 757 \\times 761 \\times 769 \\times 773 \\times 787 \\times 797 \\times 809 \\times 811 \\times 821 \\times 823 \\times 827 }} {{eqn | o =        | ro= \\times       | r = 829 \\times 839 \\times 853 \\times 857 \\times 859 \\times 863 \\times 877 \\times 881 \\times 883 \\times 887 \\times 907 \\times 911 }} {{eqn | o =        | ro= \\times       | r = 919 \\times 929 \\times 937 \\times 941 \\times 947 \\times 953 \\times 967 \\times 971 \\times 977 \\times 983 \\times 991 \\times 997       | c =  }} {{end-eqn}} </onlyinclude> == Proof == For each prime factor $p$ of $1000!$, let $a_p$ be the integer such that: :$p^{a_p} \\divides 1000!$ :$p^{a_p + 1} \\nmid 1000!$ Taking the prime factors in turn: === Multiplicity of $2$ === {{:De Polignac's Formula/Examples/2 in 1000}} === Multiplicity of $3$ === {{:De Polignac's Formula/Examples/3 in 1000}} === Multiplicity of $5$ === {{:De Polignac's Formula/Examples/5 in 1000}} === Multiplicity of $7$ === {{:De Polignac's Formula/Examples/7 in 1000}} === Multiplicity of $11$ === {{:De Polignac's Formula/Examples/11 in 1000}} === Multiplicity of $13$ === {{:De Polignac's Formula/Examples/13 in 1000}} And so on: for the remaining factors: {{begin-eqn}} {{eqn | l = a_{17}       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {1000} {17^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {1000} {17} } + \\floor {\\frac {1000} {289} }       | c =  }} {{eqn | r = 58 + 3       | c =  }} {{eqn | r = 61       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{19}       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {1000} {19^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {1000} {19} } + \\floor {\\frac {1000} {361} }       | c =  }} {{eqn | r = 52 + 2       | c =  }} {{eqn | r = 54       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{23}       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {1000} {23^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {1000} {23} } + \\floor {\\frac {1000} {529} }       | c =  }} {{eqn | r = 43 + 1       | c =  }} {{eqn | r = 44       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{29}       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {1000} {29^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {1000} {29} } + \\floor {\\frac {1000} {841} }       | c =  }} {{eqn | r = 34 + 1       | c =  }} {{eqn | r = 35       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{31}       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {1000} {31^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {1000} {31} } + \\floor {\\frac {1000} {961} }       | c =  }} {{eqn | r = 32 + 1       | c =  }} {{eqn | r = 33       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{37}       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {1000} {37^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {1000} {37} }       | c =  }} {{eqn | r = 27       | c =  }} {{end-eqn}} The pattern is clear: the remaining indices are calculated in the same way. {{qed}} Category:Factorials/Examples Category:1000 stulnrpncxqchhg3iv0vt628gszvv2g"}
+{"_id": "32646", "title": "Integral Form of Gamma Function equivalent to Euler Form/Lemma", "text": "Integral Form of Gamma Function equivalent to Euler Form/Lemma 0 49351 300866 266937 2017-06-12T11:29:10Z Prime.mover 59 wikitext text/x-wiki == Lemma for Integral Form of Gamma Function equivalent to Euler Form == <onlyinclude> Let $0 \\le t \\le m$. Then: :$0 \\le e^{-t} - \\left({1 - \\dfrac t m}\\right)^m \\le t^2 \\dfrac {e^{-t} } m$ </onlyinclude> == Proof == From Exponential of x not less than 1+x: :$1 + x \\le e^x$ Let $x = \\pm \\dfrac t m$. Then: {{begin-eqn}} {{eqn | l = 1 - \\dfrac t m       | o = \\le       | r = e^{-t/m} }} {{eqn | ll= \\leadsto       | l = \\left({1 - \\dfrac t m}\\right)^m       | o = \\le       | r = e^{-t}       | c =  }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = 1 + \\dfrac t m       | o = \\le       | r = e^{t/m} }} {{eqn | ll= \\leadsto       | l = \\left({1 + \\dfrac t m}\\right)^m       | o = \\le       | r = e^t       | c =  }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = e^{-t}       | o = \\ge       | r = \\left({1 - \\dfrac t m}\\right)^m }} {{eqn | r = e^t \\left({1 - \\dfrac t m}\\right)^m e^{-t}       | c =  }} {{eqn | o = \\ge       | r = e^{-t} \\left({1 - \\dfrac t m}\\right)^m \\left({1 + \\dfrac t m}\\right)^m       | c =  }} {{eqn | r = e^{-t} \\left({1 - \\dfrac {t^2} {m^2} }\\right)^m       | c = Difference of Two Squares }} {{eqn | o = \\ge       | r = e^{-t} \\left({1 - \\dfrac {t^2} {m^2} }\\right)       | c = Corollary to Bernoulli's Inequality }} {{end-eqn}} Hence: {{begin-eqn}} {{eqn | l = 0       | o = \\le       | r = e^{-t} - \\left({1 - \\dfrac t m}\\right)^m }} {{eqn | o = \\le       | r = t^2 \\dfrac {e^{-t} } m       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Partial Gamma Function expressed as Integral/Lemma|next = Integral Form of Gamma Function equivalent to Euler Form/Proof 2}}: $\\S 1.2.5$: Permutations and Factorials: Exercise $20$ Category:Integral Form of Gamma Function equivalent to Euler Form b37ssvfo7qtkuriuvsk52ms7298rtzq"}
+{"_id": "32647", "title": "Faà di Bruno's Formula/Example/0", "text": "Faà di Bruno's Formula/Example/0 0 49370 267024 267023 2016-08-17T19:14:08Z Prime.mover 59 wikitext text/x-wiki == Example of use of Faà di Bruno's Formula: $n = 0$ == Consider Faà di Bruno's Formula: {{:Faà di Bruno's Formula}} When $n = 0$ we have: <onlyinclude> :$D_x^0 w = w$ </onlyinclude> == Proof == {{:Faà di Bruno's Formula/Example/0/Proof}} Category:Faà di Bruno's Formula rrbp367adksn7fc0otxjxc9ykz4b9yq"}
+{"_id": "32648", "title": "Faà di Bruno's Formula/Example/1", "text": "Faà di Bruno's Formula/Example/1 0 49372 267044 267035 2016-08-17T21:00:53Z Prime.mover 59 wikitext text/x-wiki == Example of use of Faà di Bruno's Formula: $n = 1$ == Consider Faà di Bruno's Formula: {{:Faà di Bruno's Formula}} When $n = 1$ we have: <onlyinclude> :$D_x^1 w = D_u^1 w D_x^1 u$ </onlyinclude> == Proof == {{:Faà di Bruno's Formula/Example/1/Proof}} Category:Faà di Bruno's Formula orpdb3msjh87gcr1juse7hv8af8k8q7"}
+{"_id": "32649", "title": "Faà di Bruno's Formula/Example/2", "text": "Faà di Bruno's Formula/Example/2 0 49374 267042 2016-08-17T20:45:57Z Prime.mover 59 Created page with \"== Example of use of Faà di Bruno's Formula: $n = 2$ ==  Consider Faà di Bruno's Formula: {{:Faà di Bruno's Formula}}  When $n = 2$ we have: <onlyinclude> :$D_x^2 w...\" wikitext text/x-wiki == Example of use of Faà di Bruno's Formula: $n = 2$ == Consider Faà di Bruno's Formula: {{:Faà di Bruno's Formula}} When $n = 2$ we have: <onlyinclude> :$D_x^2 w = D_u^2 w \\left({D_x^1 u}\\right)^2 + D_u^1 w D_x^2 u$ </onlyinclude> == Proof == {{:Faà di Bruno's Formula/Example/2/Proof}} Category:Faà di Bruno's Formula frd5i1ungjfiggnq778rl14v58spnjp"}
+{"_id": "32650", "title": "Ceiling of x+m over n/Corollary", "text": "Ceiling of x+m over n/Corollary 0 49513 316257 267646 2017-09-08T16:29:41Z Caliburn 3218 cat wikitext text/x-wiki == Corollary to Ceiling of $\\frac {x + m} n$ == <onlyinclude> Let $n \\in \\Z$ such that $n > 0$. Let $x \\in \\R$. Then: :$\\left \\lceil{\\dfrac x n}\\right \\rceil = \\left \\lceil{\\dfrac {\\left \\lceil{x}\\right \\rceil} n}\\right \\rceil$ where $\\left\\lceil{x}\\right\\rceil$ denotes the ceiling of $x$. </onlyinclude> == Proof == This is a special case of Ceiling of $\\dfrac {x + m} n$: :$\\left \\lceil{\\dfrac {x + m} n}\\right \\rceil = \\left \\lceil{\\dfrac {\\left \\lceil{x}\\right \\rceil + m} n}\\right \\rceil$ where $m = 0$. == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Ceiling of x+m over n|next = Ceiling of x+m over n/Proof 1}}: $\\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $35$ Category:Ceiling Function Category:Ceiling of x+m over n el4z1w1znfboid67kkdqimm9yqbclef"}
+{"_id": "32651", "title": "Floor of x+m over n/Corollary", "text": "Floor of x+m over n/Corollary 0 49514 496877 346261 2020-10-29T22:07:19Z Prime.mover 59 wikitext text/x-wiki == Corollary to Floor of $\\frac {x + m} n$ == <onlyinclude> Let $n \\in \\Z$ such that $n > 0$. Let $x \\in \\R$. Then: :$\\floor {\\dfrac x n} = \\floor {\\dfrac {\\floor x} n}$ where $\\floor x$ denotes the floor of $x$. </onlyinclude> == Proof == This is a special case of Floor of $\\dfrac {x + m} n$: :$\\floor {\\dfrac {x + m} n} = \\floor {\\dfrac {\\floor x + m} n}$ where $m = 0$. == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Floor of x+m over n|next = Floor of x+m over n/Proof 1}}: $\\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $35$ Category:Floor Function Category:Floor of x+m over n if4kgg6dh24exg2tdlh26agh3y7y4ib"}
+{"_id": "32652", "title": "Euclidean Algorithm/Demonstration", "text": "Euclidean Algorithm/Demonstration 0 49575 493848 390997 2020-10-11T06:58:28Z Prime.mover 59 wikitext text/x-wiki == Example of use of Euclidean Algorithm == <onlyinclude> Using the '''Euclidean Algorithm''', we can investigate in detail what happens when we apply the Division Theorem repeatedly to $a$ and $b$. {{begin-eqn}} {{eqn | l = a       | r = q_1 b + r_1       | c =  }} {{eqn | l = b       | r = q_2 r_1 + r_2       | c =  }} {{eqn | l = r_1       | r = q_3 r_2 + r_3       | c =  }} {{eqn | l = \\cdots       | o =        | c =  }} {{eqn | l = r_{n - 2}       | r = q_n r_{n - 1} + r_n       | c =  }} {{eqn | l = r_{n - 1}       | r = q_{n + 1} r_n + 0       | c =  }} {{end-eqn}} From the Division Theorem, we know that the remainder is always strictly less than the divisor. That is, in $a = q b + r$: :$0 \\le r < \\size b$ So we know that: :$b > r_1 > r_2 > \\ldots > r_{n - 2} > r_{n - 1} > r_n > 0$ So the algorithm ''has'' to terminate. {{qed}} </onlyinclude> == Sources == * {{BookReference|Number Theory|1971|George E. Andrews|prev = Euclidean Algorithm|next = Existence of Greatest Common Divisor/Proof 4}}: $\\text {2-2}$ Divisibility: Theorem $\\text {2-2}$ * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Euclidean Algorithm|next = Definition:Integer Combination}}: $\\S 12$: Highest common factors and Euclid's algorithm * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Euclidean Algorithm|next = Definition:Linearly Independent Set}}: $\\text{A}.1$: Number theory Category:Euclidean Algorithm gqtjjtsmaz86hb0cj0udaq6807ouum7"}
+{"_id": "32653", "title": "Right Regular Representation wrt Right Cancellable Element on Finite Semigroup is Bijection", "text": "Right Regular Representation wrt Right Cancellable Element on Finite Semigroup is Bijection 0 49666 268458 2016-08-30T05:32:36Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> Let $\\left({S, \\circ}\\right)$ be a finite semigroup.  Let $a \\in S$ be Definition:Right Cancellabl...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $\\left({S, \\circ}\\right)$ be a finite semigroup. Let $a \\in S$ be right cancellable. Then the right regular representation $\\rho_a$ of $\\left({S, \\circ}\\right)$ with respect to $a$ is a bijection. </onlyinclude> == Proof == By Right Cancellable iff Right Regular Representation Injective, $\\rho_a$ is an injection. By hypothesis, $S$ is finite. From Injection from Finite Set to Itself is Surjection, $\\rho_a$ is a surjection. Thus $\\rho_a$ is injective and surjective, and therefore a bijection. {{qed}} Category:Semigroups Category:Regular Representations Category:Cancellability 3dwl17i5fe551prpra9brqpnd0whzgb"}
+{"_id": "32654", "title": "Euler Phi Function/Examples/9", "text": "Euler Phi Function/Examples/9 0 49689 487588 357187 2020-09-13T21:27:28Z Prime.mover 59 wikitext text/x-wiki == Example of Use of Euler $\\phi$ Function == <onlyinclude> :$\\map \\phi 9 = 6$ </onlyinclude> where $\\phi$ denotes the Euler $\\phi$ function. == Proof == From Euler Phi Function of Prime Power: :$\\map \\phi {3^k} = 2 \\times 3^{k - 1}$ Thus: :$\\map \\phi 9 = \\map \\phi {3^2} = 2 \\times 3 = 6$ They can be enumerated as: :$1, 2, 4, 5, 7, 8$ {{qed}} == Sources == * {{BookReference|Introduction to the Theory of Finite Groups|1964|Walter Ledermann|ed = 5th|edpage = Fifth Edition|prev = Definition:Euler Phi Function|next = Euler Phi Function of Prime}}: Chapter $\\text {I}$: The Group Concept: $\\S 6$: Examples of Finite Groups: $\\text{(iii)}$ Category:Examples of Euler Phi Function Category:9 p5i3t2xd8nx03vkk15bcgmdgnhfbckz"}
+{"_id": "32655", "title": "Group of Rotation Matrices Order 4/Cayley Table", "text": "Group of Rotation Matrices Order 4/Cayley Table 0 49776 370971 370969 2018-10-14T05:06:03Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Group of Rotation Matrices Order $4$ == Consider the group of rotation matrices order $4$ :$R_4 = \\set {\\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}, \\begin{bmatrix} 0 & 1 \\\\ -1 & 0 \\end{bmatrix}, \\begin{bmatrix} -1 & 0 \\\\ 0 & -1 \\end{bmatrix}, \\begin{bmatrix} 0 & -1 \\\\ 1 & 0 \\end{bmatrix} }$ $R_4$ can be described completely by showing its Cayley table. Let: :$r_0 = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}$ :$r_1 = \\begin{bmatrix} 0 & 1 \\\\ -1 & 0 \\end{bmatrix}$ :$r_2 = \\begin{bmatrix} -1 & 0 \\\\ 0 & -1 \\end{bmatrix}$ :$r_3 = \\begin{bmatrix} 0 & -1 \\\\ 1 & 0 \\end{bmatrix}$ Then we have: <onlyinclude> :$\\begin{array}{r|rrrr} \\times & r_0 & r_1 & r_2 & r_3 \\\\ \\hline r_0 & r_0 & r_1 & r_2 & r_3 \\\\ r_1 & r_1 & r_2 & r_3 & r_0 \\\\ r_2 & r_2 & r_3 & r_0 & r_1 \\\\ r_3 & r_3 & r_0 & r_1 & r_2 \\\\ \\end{array}$ </onlyinclude> Category:Examples of Cayley Tables h310fikahfzikm7xwk68bhgocwhub0j"}
+{"_id": "32656", "title": "Multiplicative Group of Reduced Residues Modulo 5/Cayley Table", "text": "Multiplicative Group of Reduced Residues Modulo 5/Cayley Table 0 49801 385903 371167 2019-01-02T22:00:46Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Multiplicative Group of Reduced Residues Modulo 5 == The multiplicative group of reduced residues modulo $5$: :$\\Z'_5 = \\set {\\eqclass 1 5, \\eqclass 2 5, \\eqclass 3 5, \\eqclass 4 5}$ can be described completely by showing its Cayley table: <onlyinclude> :$\\begin{array}{r|rrrr} \\times_5 & \\eqclass 1 5 & \\eqclass 2 5 & \\eqclass 3 5 & \\eqclass 4 5 \\\\ \\hline \\eqclass 1 5 & \\eqclass 1 5 & \\eqclass 2 5 & \\eqclass 3 5 & \\eqclass 4 5 \\\\ \\eqclass 2 5 & \\eqclass 2 5 & \\eqclass 4 5 & \\eqclass 1 5 & \\eqclass 3 5 \\\\ \\eqclass 3 5 & \\eqclass 3 5 & \\eqclass 1 5 & \\eqclass 4 5 & \\eqclass 2 5 \\\\ \\eqclass 4 5 & \\eqclass 4 5 & \\eqclass 3 5 & \\eqclass 2 5 & \\eqclass 1 5 \\\\ \\end{array}$ By arranging the rows and columns into a different order, its cyclic nature becomes clear: :$\\begin{array}{r|rrrr} \\times_5 & \\eqclass 1 5 & \\eqclass 2 5 & \\eqclass 4 5 & \\eqclass 3 5 \\\\ \\hline \\eqclass 1 5 & \\eqclass 1 5 & \\eqclass 2 5 & \\eqclass 4 5 & \\eqclass 3 5 \\\\ \\eqclass 2 5 & \\eqclass 2 5 & \\eqclass 4 5 & \\eqclass 3 5 & \\eqclass 1 5 \\\\ \\eqclass 4 5 & \\eqclass 4 5 & \\eqclass 3 5 & \\eqclass 1 5 & \\eqclass 2 5 \\\\ \\eqclass 3 5 & \\eqclass 3 5 & \\eqclass 1 5 & \\eqclass 2 5 & \\eqclass 4 5 \\\\ \\end{array}$ </onlyinclude> == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Modulo Addition/Cayley Table/Modulo 4|next = Modulo Multiplication is Well-Defined}}: Chapter $2$: Maps and relations on sets: Exercise $5$ Category:Examples of Cayley Tables Category:Multiplicative Group of Reduced Residues Modulo 5 9uajnfql30wlvvgzmphmjtvembcikit"}
+{"_id": "32657", "title": "Group Generated by Reciprocal of z and 1 minus z/Cayley Table", "text": "Group Generated by Reciprocal of z and 1 minus z/Cayley Table 0 49814 487572 371069 2020-09-13T21:16:39Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Group Generated by Reciprocal of $1 / z$ and $1 - z$ == We have: {{begin-eqn}} {{eqn | l = \\map {f_1} z       | r = z }} {{eqn | l = \\map {f_2} z       | r = \\dfrac 1 {1 - z} }} {{eqn | l = \\map {f_3} z       | r = \\dfrac {z - 1} z }} {{eqn | l = \\map {f_4} z       | r = \\dfrac 1 z }} {{eqn | l = \\map {f_5} z       | r = 1 - z }} {{eqn | l = \\map {f_6} z       | r = \\dfrac z {z - 1} }} {{end-eqn}} Hence from Group Generated by Reciprocal of z and 1 minus z: <onlyinclude> :$\\begin{array}{r|rrrrrr} \\circ & f_1 & f_2 & f_3 & f_4 & f_5 & f_6 \\\\ \\hline f_1 & f_1 & f_2 & f_3 & f_4 & f_5 & f_6 \\\\ f_2 & f_2 & f_3 & f_1 & f_6 & f_4 & f_5 \\\\ f_3 & f_3 & f_1 & f_2 & f_5 & f_6 & f_4 \\\\ f_4 & f_4 & f_5 & f_6 & f_1 & f_2 & f_3 \\\\ f_5 & f_5 & f_6 & f_4 & f_3 & f_1 & f_2 \\\\ f_6 & f_6 & f_4 & f_5 & f_2 & f_3 & f_1 \\\\ \\end{array}$ Expressing the elements in full: :$\\begin{array}{c|cccccc} \\circ & z & \\dfrac 1 {1 - z} & \\dfrac {z - 1} z & \\dfrac 1 z & 1 - z & \\dfrac z {z - 1} \\\\ \\hline z & z & \\dfrac 1 {1 - z} & \\dfrac {z - 1} z & \\dfrac 1 z & 1 - z & \\dfrac z {z - 1} \\\\ \\dfrac 1 {1 - z} & \\dfrac 1 {1 - z} & \\dfrac {z - 1} z & z & \\dfrac z {z - 1} & \\dfrac 1 z & 1 - z \\\\ \\dfrac {z - 1} z & \\dfrac {z - 1} z & z & \\dfrac 1 {1 - z} & 1 - z & \\dfrac z {z - 1} & \\dfrac 1 z \\\\ \\dfrac 1 z & \\dfrac 1 z & 1 - z & \\dfrac z {z - 1} & z & \\dfrac 1 {1 - z} & \\dfrac {z - 1} z \\\\ 1 - z & 1 - z & \\dfrac z {z - 1} & \\dfrac 1 z & \\dfrac {z - 1} z & z & \\dfrac 1 {1 - z} \\\\ \\dfrac z {z - 1} & \\dfrac z {z - 1} & \\dfrac 1 z & 1 - z & \\dfrac 1 {1 - z} & \\dfrac {z - 1} z & z \\\\ \\end{array}$ </onlyinclude> == Sources == * {{BookReference|Introduction to the Theory of Finite Groups|1964|Walter Ledermann|ed = 5th|edpage = Fifth Edition|prev = Group Generated by Reciprocal of z and 1 minus z|next = Definition:Congruence (Number Theory)/Integers/Integer Multiple}}: Chapter $\\text {I}$: The Group Concept: $\\S 6$: Examples of Finite Groups: $\\text{(ii)}$ * {{BookReference|Elements of Abstract Algebra|1971|Allan Clark|prev = Group Generated by Reciprocal of z and 1 minus z|next = Isomorphism between Group Generated by Reciprocal of z and 1 minus z and Symmetric Group on 3 Letters}}: Introduction Category:Examples of Cayley Tables t51dc10j2jycz7bfmz2foh7jztds4aw"}
+{"_id": "32658", "title": "Group Generated by Reciprocal of z and Minus z/Cayley Table", "text": "Group Generated by Reciprocal of z and Minus z/Cayley Table 0 49823 372457 371010 2018-10-22T06:21:26Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Group Generated by Reciprocal of $1 / z$ and $-z$ == We have: :$\\map {f_1} z = z$ :$\\map {f_2} z = -z$ :$\\map {f_3} z = \\dfrac 1 z$ :$\\map {f_4} z = -\\dfrac 1 z$ Hence from Group Generated by Reciprocal of z and Minus z: <onlyinclude> :$\\begin{array}{r|rrrr} \\circ & f_1 & f_2 & f_3 & f_4 \\\\ \\hline f_1 & f_1 & f_2 & f_3 & f_4 \\\\ f_2 & f_2 & f_1 & f_4 & f_3 \\\\ f_3 & f_3 & f_4 & f_1 & f_2 \\\\ f_4 & f_4 & f_3 & f_2 & f_1 \\\\ \\end{array}$ Expressing the elements in full: :$\\begin{array}{c|cccc} \\circ & z & -z & \\dfrac 1 z & -\\dfrac 1 z \\\\ \\hline z & z & -z & \\dfrac 1 z & -\\dfrac 1 z \\\\ -z & -z & z & -\\dfrac 1 z & \\dfrac 1 z \\\\ \\dfrac 1 z & \\dfrac 1 z & -\\dfrac 1 z & z & -z \\\\ -\\dfrac 1 z & -\\dfrac 1 z & \\dfrac 1 z & -z & z \\\\ \\end{array}$ </onlyinclude> Category:Examples of Cayley Tables Category:Group Generated by Reciprocal of z and Minus z to60cmgij1pjl5e8utjqi95ilhndakp"}
+{"_id": "32659", "title": "Group of Reflection Matrices Order 4/Cayley Table", "text": "Group of Reflection Matrices Order 4/Cayley Table 0 49826 372448 370987 2018-10-22T06:17:56Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Group of Reflection Matrices Order $4$ == Consider the group of reflection matrices order $4$ :$R_4 = \\set {\\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}, \\begin{bmatrix} 1 & 0 \\\\ 0 & -1 \\end{bmatrix}, \\begin{bmatrix} -1 & 0 \\\\ 0 & 1 \\end{bmatrix}, \\begin{bmatrix} -1 & 0 \\\\ 0 & -1 \\end{bmatrix} }$ $R_4$ can be described completely by showing its Cayley table. Let: :$r_0 = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}$ :$r_1 = \\begin{bmatrix} 1 & 0 \\\\ 0 & -1 \\end{bmatrix}$ :$r_2 = \\begin{bmatrix} -1 & 0 \\\\ 0 & 1 \\end{bmatrix}$ :$r_3 = \\begin{bmatrix} -1 & 0 \\\\ 0 & -1 \\end{bmatrix}$ Then we have: <onlyinclude> :$\\begin{array}{r|rrrr} \\times & r_0 & r_1 & r_2 & r_3 \\\\ \\hline r_0 & r_0 & r_1 & r_2 & r_3 \\\\ r_1 & r_1 & r_0 & r_3 & r_2 \\\\ r_2 & r_2 & r_3 & r_0 & r_1 \\\\ r_3 & r_3 & r_2 & r_1 & r_0 \\\\ \\end{array}$ </onlyinclude> Category:Examples of Cayley Tables dgdam5yfreyiuqlpsfcllilq96gisd4"}
+{"_id": "32660", "title": "Multiplicative Group of Reduced Residues Modulo 8/Cayley Table", "text": "Multiplicative Group of Reduced Residues Modulo 8/Cayley Table 0 49828 371171 371165 2018-10-14T09:38:19Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Multiplicative Group of Reduced Residues Modulo 8 == The multiplicative group of reduced residues modulo $8$: :$\\Z'_8 = \\set {\\eqclass 1 8, \\eqclass 3 8, \\eqclass 5 8, \\eqclass 7 8}$ can be described completely by showing its Cayley table: <onlyinclude> :$\\begin{array}{r|rrrr} \\times_8 & \\eqclass 1 8 & \\eqclass 3 8 & \\eqclass 5 8 & \\eqclass 7 8 \\\\ \\hline \\eqclass 1 8 & \\eqclass 1 8 & \\eqclass 3 8 & \\eqclass 5 8 & \\eqclass 7 8 \\\\ \\eqclass 3 8 & \\eqclass 3 8 & \\eqclass 1 8 & \\eqclass 7 8 & \\eqclass 5 8 \\\\ \\eqclass 5 8 & \\eqclass 5 8 & \\eqclass 7 8 & \\eqclass 1 8 & \\eqclass 3 8 \\\\ \\eqclass 7 8 & \\eqclass 7 8 & \\eqclass 5 8 & \\eqclass 3 8 & \\eqclass 1 8 \\\\ \\end{array}$ </onlyinclude> Category:Examples of Cayley Tables Category:Multiplicative Group of Reduced Residues Modulo 8 tkrj64cqoh4o7k1sjya20y02wwf7qde"}
+{"_id": "32661", "title": "Klein Four-Group/Cayley Table", "text": "Klein Four-Group/Cayley Table 0 49829 379885 374806 2018-12-03T05:45:56Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Klein $4$-Group == The Klein $4$-group can be described completely by showing its Cayley table: <onlyinclude> :$\\begin{array}{c|cccc}   & e & a & b & c \\\\ \\hline e & e & a & b & c \\\\ a & a & e & c & b \\\\ b & b & c & e & a \\\\ c & c & b & a & e \\\\ \\end{array}$ </onlyinclude> {{NamedforDef|Felix Christian Klein}} == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Definition:Klein Four-Group|next = Symmetric Group on 3 Letters/Group Presentation}}: $\\S 25$ * {{BookReference|Elements of Abstract Algebra|1966|Richard A. Dean|prev = Exponential on Real Numbers is Group Isomorphism/Proof 1|next = Definition:Klein Four-Group}}: $\\S 1.5$: Example $15$ * {{BookReference|Elements of Abstract Algebra|1971|Allan Clark|prev = Definition:Klein Four-Group|next = Klein Four-Group is Group}}: Chapter $2$: The Definition of Group Structure: $\\S 26 \\iota$ * {{BookReference|Algebra|1974|Thomas W. Hungerford|prev = Definition:Klein Four-Group|next = Reduced Residue System under Multiplication forms Abelian Group/Corollary}}: $\\text{I}$: Groups: $\\S 1$: Semigroups, Monoids and Groups: Exercise $6$ * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Definition:Klein Four-Group|next = Definition:Symmetry Group of Rectangle}}: $\\S 44.5$ Some consequences of Lagrange's Theorem Category:Klein Four-Group Category:Examples of Cayley Tables rmuh1ij005q9d890mjwk5plu3oio1hv"}
+{"_id": "32662", "title": "Complex Roots of Unity/Examples/Cube Roots", "text": "Complex Roots of Unity/Examples/Cube Roots 0 49894 451602 451599 2020-03-01T14:12:50Z Prime.mover 59 wikitext text/x-wiki == Example of Complex Roots of Unity == <onlyinclude> The '''complex cube roots of unity''' are the elements of the set: :$U_3 = \\set {z \\in \\C: z^3 = 1}$ They are: {{begin-eqn}} {{eqn | m = e^{0 i \\pi / 3}       | o =       | r = 1       | mo= =       | c =  }} {{eqn | l = \\omega        | mo= =       | m = e^{2 i \\pi / 3}       | r = -\\frac 1 2 + \\frac {i \\sqrt 3} 2 }} {{eqn | l = \\omega^2        | mo= =       | m = e^{4 i \\pi / 3}       | r = -\\frac 1 2 - \\frac {i \\sqrt 3} 2 }} {{end-eqn}} The notation $\\omega$ for, specifically, the complex '''cube''' roots of unity is conventional. </onlyinclude> === Conjugate Form === {{:Complex Roots of Unity/Examples/Cube Roots/Conjugate Form}} == Proof == {{:Complex Roots of Unity/Examples/Cube Roots/Proof}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Sum of Powers of Primitive Complex Roots of Unity|next = Difference of Two Cubes/Corollary}}: $\\S 3$. Roots of Unity * {{BookReference|Elements of Abstract Algebra|1971|Allan Clark|prev = Roots of Complex Number/Examples/Cube Roots|next = Roots of Complex Number/Corollary/Examples/Cube Roots}}: Introduction * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Cube Root|next = Definition:Cubic (Geometry)|entry = cube root of unity}} Category:Examples of Complex Roots of Unity Category:Cube Roots of Unity plam55cl80oep7dnkq3wkwwvzv93wau"}
+{"_id": "32663", "title": "Primitive Complex Roots of Unity/Examples/Cube Roots", "text": "Primitive Complex Roots of Unity/Examples/Cube Roots 0 49902 431571 363692 2019-10-16T01:33:03Z Caliburn 3218 wikitext text/x-wiki == Examples of Primitive Complex Roots of Unity == <onlyinclude> The '''primitive complex cube roots of unity''' are: {{begin-eqn}} {{eqn | l = \\omega        | mo= =       | m = e^{2 \\pi i / 3}       | r = -\\frac 1 2 + \\frac {i \\sqrt 3} 2 }} {{eqn | l = \\omega^2        | mo= =       | m = e^{4 \\pi i / 3}       | r = -\\frac 1 2 - \\frac {i \\sqrt 3} 2 }} {{end-eqn}} </onlyinclude> == Proof == There are $3$ (complex) cube roots of unity: :$1, \\omega, \\omega^2$ We have: {{begin-eqn}} {{eqn | l = \\omega        | r = e^{2 \\pi i / 3} }} {{eqn | l = \\omega^2        | r = e^{4 \\pi i / 3} }} {{eqn | l = \\omega^3        | r = 1       | c = Cube Roots of Unity }} {{end-eqn}} Also: {{begin-eqn}} {{eqn | l = \\omega^2       | r = e^{4 \\pi i / 3} }} {{eqn | l = \\paren {\\omega^2}^2       | r = \\omega^3 \\times \\omega }} {{eqn | r = 1 \\times \\omega       | c = Cube Roots of Unity }} {{eqn | r = \\omega       | c =  }} {{eqn | l = \\paren {\\omega^2}^3        | r = \\paren {\\omega^3}^2       | c = Cube Roots of Unity }} {{eqn | r = 1 \\times 1       | c = Cube Roots of Unity }} {{eqn | r = 1       | c =  }} {{end-eqn}} Trivially, $1$ is not a primitive complex cube root of unity because you cannot make either $\\omega$ or $\\omega^2$ by multiplying $1$ by itself as many times as you like. Hence the result. {{qed}} Category:Cube Roots of Unity thejark721e79ld0xl97nug8xzhwxdl"}
+{"_id": "32664", "title": "Symmetric Group on 3 Letters", "text": "Symmetric Group on 3 Letters 0 49917 388523 388518 2019-01-19T12:30:30Z Prime.mover 59 wikitext text/x-wiki == Group Example == <onlyinclude> Let $S_3$ denote the set of permutations on $3$ letters. The '''symmetric group on $3$ letters''' is the algebraic structure: :$\\struct {S_3, \\circ}$ where $\\circ$ denotes composition of mappings. </onlyinclude> It is usually denoted, when the context is clear, without the operator: $S_3$. === Cycle Notation === It can be expressed in the form of permutations given in cycle notation as follows: {{:Symmetric Group on 3 Letters/Cycle Notation}} === Cayley Table === {{:Symmetric Group on 3 Letters/Cayley Table}} === Group Presentation === Its group presentation is: {{:Symmetric Group on 3 Letters/Group Presentation}} == Order of Elements == {{:Symmetric Group on 3 Letters/Order of Elements}} == Subgroups == {{:Symmetric Group on 3 Letters/Subgroups}} == Normal Subgroups == {{:Symmetric Group on 3 Letters/Normal Subgroups}} == Generators == {{:Symmetric Group on 3 Letters/Generators}} == Centralizers == {{:Symmetric Group on 3 Letters/Centralizers}} == Normalizers of Subgroups == {{:Symmetric Group on 3 Letters/Normalizers}} == Center == {{:Symmetric Group on 3 Letters/Center}} == Conjugacy Classes == {{:Symmetric Group on 3 Letters/Conjugacy Classes}} == Also see == * Symmetric Group is Group, which demonstrates that this is a (finite) group. Category:Groups of Order 6 Category:Symmetric Group on 3 Letters Category:Symmetric Groups h2sn5oa3n3faf7jeos3g7n2x850c8ve"}
+{"_id": "32665", "title": "Solution to Card Game with Bluffing", "text": "Solution to Card Game with Bluffing 0 50132 270689 270687 2016-09-13T06:35:01Z Prime.mover 59 wikitext text/x-wiki == Solution to Card Game with Bluffing == {{:Definition:Card Game with Bluffing}} == Proof == <onlyinclude> From the payoff table: {{:Definition:Card Game with Bluffing/Payoff Table}} The solution is: : $A$ takes strategy $A_1$ for $2/3$ of the time, and $A_2$ for $1/3$ of the time. : $B$ takes strategy $B_1$ for $2/3$ of the time, and $B_2$ for $1/3$ of the time. {{explain|It is not made clear in the source work why this is. You just get: \"This is indeed a solution, because no player can do better if the other sticks to his strategy, but could gain or lose more than $1/3$ (which is what $A$ gains now) if he departed from the optimal solution and his opponent took advantage of it.\" This is handwaving. A source work with a higher level of precision is needed here.}} </onlyinclude> == Sources == * {{BookReference|The Theory of Games and Linear Programming|1956|Steven Vajda|prev = Definition:Card Game with Bluffing/Payoff Table|next = Two-Person Zero-Sum Game with Multiple Solutions}}: Chapter $\\text{I}$: An Outline of the Theory of Games: $3$ Category:Examples of Two-Person Zero-Sum Games 4ksn9gq6ze2pq47senvjeub2xvbd4sp"}
+{"_id": "32666", "title": "Induction of Finite Set", "text": "Induction of Finite Set 0 50174 414435 270828 2019-07-24T17:03:06Z Prime.mover 59 wikitext text/x-wiki == Theorem Scheme == Let $A$ be finite set. Let $\\map P -$ be a predicate. Let $\\map P \\O$. Let :$\\forall B \\subseteq A, x \\in A: \\paren {\\map P B \\implies \\map P {B \\cup \\set x} }$ Then: :$\\map P A$ == Proof == We will prove the result by induction on cardinality of argument. === Base Case === :$\\forall X \\subseteq A: \\paren {\\size X = 0 \\implies \\map P X}$ Let $X \\subseteq A$ such that: :$\\size X = 0$ By Cardinality of Empty Set: :$X = \\O$ Thus by assumption: :$\\map P X$ === Induction Hypothesis === :$\\forall X \\subseteq A: \\paren {\\size X = n \\implies \\map P X}$ === Induction Step === :$\\forall X \\subseteq A: \\paren {\\size X = n + 1 \\implies \\map P X}$ Let $X \\subseteq A$ such that: :$\\size X = n + 1$ By definition of cardinality: :$X = \\set {x_1, \\dots, x_n, x_{n + 1} }$ By Union of Unordered Tuples: :$X = \\set {x_1, \\dots, x_n} \\cup \\set {x_{n + 1} }$ By definition of cardinality: :$\\size {\\set {x_1, \\dots, x_n} } = n$ By Set is Subset of Union: :$\\set {x_1, \\dots, x_n} \\subseteq X \\subseteq A$ Then by Induction Hypothesis: :$\\map P {\\set {x_1, \\dots, x_n} }$ By definition of subset: :$x_{n + 1} \\in A$ Thus by assumption: :$\\map P X$ {{qed|lemma}} By the Principle of Mathematical Induction: :$\\forall X \\subseteq A: \\paren {\\size X = \\size A \\implies \\map P X}$ Hence: :$\\map P A$ {{qed}} == Sources == * {{Mizar|link = finset_1|sublink = S2|display = FINSET_1:sch 2}} Category:Set Theory kvjbq876k6dqr3q8pdqldwexmetyrrf"}
+{"_id": "32667", "title": "Infimum of Real Subset", "text": "Infimum of Real Subset 0 50378 407208 271659 2019-06-10T07:08:22Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $S$ be a set of extended real numbers. Let $S$ be bounded below (in $\\R$). Let $T = S \\cap \\R$. Then: :$S$ admits an infimum (in $\\R$) if and only if $T$ admits an infimum (in $\\R$) and, if $\\inf S$ and $\\inf T$ exist as real numbers: :$\\inf S = \\inf T$ == Proof == We observe that $T$ constitutes the real numbers of $S$. Since there is a real number that is a lower bound for $S$, $-\\infty$ is not an element of $S$. Accordingly, $\\infty$ is the only possible element of $S \\setminus T$. Therefore: :$S$ is a subset of $T \\cup \\set \\infty$ First, we show that $S$ and $T$ have the same set of lower bounds. Let $b$ be a lower bound (in $\\R$) for $S$. Then $b$ is a lower bound for $T$ as $T$ is a subset of $S$. Therefore: :the set of lower bounds for $S$ is a subset of the set of lower bounds for $T$ Assume that $c$ is a lower bound (in $\\R$) for $T$. Then $c$ is a lower bound for $T \\cup \\left\\{{\\infty}\\right\\}$ as well since $c < \\infty$. Accordingly, $c$ is a lower bound for $S$ since $S$ is a subset of $T \\cup \\set \\infty$. Therefore: :the set of lower bounds for $T$ is a subset of the set of lower bounds for $S$ We have: :the set of lower bounds for $T$ is a subset of the set of lower bounds for $S$ :the set of lower bounds for $S$ is a subset of the set of lower bounds for $T$ Therefore: :the set of lower bounds for $T$ equals the set of lower bounds for $S$ by definition Next, we show that $S$ and $T$ have the same infima. We have that $S$ and $T$ have the same set of lower bounds. Therefore, $S$ and $T$ have the same greatest lower bound in $\\overline \\R$. Accordingly, as a corollary, if one of the sets $S$ and $T$ admits an infimum (in $\\R$), so does the other. Furthermore, these infima are equal. {{qed}} Category:Extended Real Numbers kx4t3fbsz4b7rckt7fjm04bbiubtkgz"}
+{"_id": "32668", "title": "Infimum of Set of Oscillations on Set", "text": "Infimum of Set of Oscillations on Set 0 50466 407204 272152 2019-06-10T06:32:13Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $f: D \\to \\R$ be a real function where $D \\subseteq \\R$. Let $x$ be a point in $D$. Let $S_x$ be a set of real sets that contain (as an element) $x$. Let: :$\\map {\\omega_f} x = \\inf \\set {\\map {\\omega_f} I: I \\in S_x}$ where $\\map {\\omega_f} I$ denotes the oscillation of $f$ on the set $I$: :$\\map {\\omega_f} I = \\sup \\set {\\size {\\map f y - \\map f z}: y, z \\in I \\cap D}$ Then: :$\\map {\\omega_f} x \\in \\R$ {{iff}} $\\set {\\map {\\omega_f} I: I \\in S_x}$ contains a real number. == Proof == Let: :$S = \\set {\\map {\\omega_f} I: I \\in S_x}$ We observe that: :$\\inf S = \\map {\\omega_f} x$ === Necessary Condition === Let $\\inf S \\in \\R$. We need to prove that $S$ contains a real number. Note that $S$ is non-empty as the empty set does not admit an infimum (in $\\R$). Therefore, $S$ has at least one element. Accordingly, there is an $I \\in S_x$ such that $\\map {\\omega_f} I \\in S$. Let $I \\in S_x$. Therefore, $x \\in I$. From this follows by Oscillation on Set is an Extended Real Number that $\\map {\\omega_f} I$ is an extended real number. Therefore $S$ is a set of extended real numbers as $S = \\set {\\map {\\omega_f} I: I \\in S_x}$. Accordingly, $S$ contains a real number by Infimum of Subset of Extended Real Numbers is Arbitrarily Close as $\\inf S \\in \\R$. {{qed|lemma}} === Sufficient Condition === Let $S$ contain a real number. We need to prove that $\\inf S \\in \\R$. We have: :$S \\cap \\R$ is non-empty as $S$ contains a real number. Let $I \\in S_x$. Therefore, $x \\in I$. From this follows by Oscillation on Set is an Extended Real Number that $\\map {\\omega_f} I \\in \\overline \\R_{\\ge 0}$. Therefore: :$S$ is a subset of $\\overline \\R_{\\ge 0}$ as $S = \\set {\\map {\\omega_f} I: I \\in S_x}$ Accordingly: :$S$ is bounded below. From this follows that: :$S \\cap \\R$ is bounded below as $SR$ is a subset of $S$ We have: :$S \\cap \\R$ is bounded below :$S \\cap \\R$ is not empty Therefore: :$\\inf S \\cap \\R \\in \\R$ Continuum Property We have: :$S$ is a set of extended real numbers as $S$ is a subset of $\\overline \\R_{\\ge 0}$ :$S$ is bounded below Therefore: :$\\inf S \\in \\R$ by Infimum of Real Subset as $\\inf S \\cap \\R \\in \\R$ {{qed}} Category:Real Analysis Category:Oscillation 6csqfmfq0gbqeu6e9pqlkytunxctepo"}
+{"_id": "32669", "title": "Infimum of Set of Oscillations on Set is Arbitrarily Close", "text": "Infimum of Set of Oscillations on Set is Arbitrarily Close 0 50507 272883 272273 2016-10-10T07:37:43Z Ivar Sand 2302 Generalized wikitext text/x-wiki == Lemma == Let $f: D \\to \\R$ be a real function where $D \\subseteq \\R$. Let $x$ be a point in $D$. Let $S_x$ be a set of real sets that contain (as an element) $x$. Let: :$\\omega_f \\left({x}\\right) = \\displaystyle \\inf \\left\\{{\\omega_f \\left({I}\\right): I \\in S_x}\\right\\}$ where $\\omega_f \\left({I}\\right)$ is the oscillation of $f$ on a real set $I$: :$\\omega_f \\left({I}\\right) = \\displaystyle \\sup \\left\\{{\\left\\vert{f \\left({y}\\right) - f \\left({z}\\right)}\\right\\vert: y, z \\in I \\cap D}\\right\\}$ Let $\\epsilon \\in \\R_{>0}$. Let $\\omega_f \\left({x}\\right) \\in \\R$. Then an $I \\in S_x$ exists such that: :$\\omega_f \\left({I}\\right) - \\omega_f \\left({x}\\right) < \\epsilon$ == Proof == Let $\\epsilon \\in \\R_{>0}$. Let $\\omega_f \\left({x}\\right) \\in \\R$. We need to prove that an $I \\in S_x$ exists such that: :$\\omega_f \\left({I}\\right) - \\omega_f \\left({x}\\right) < \\epsilon$ We have that $\\omega_f \\left({I}\\right) \\in \\overline{\\R}_{\\ge 0}$ for every $I \\in S_x$ by Oscillation on Set is an Extended Real Number. Therefore: :$\\left\\{{\\omega_f \\left({I}\\right): I \\in S_x}\\right\\}$ is a subset of $\\overline{\\R}$ We have also: :$\\displaystyle \\inf \\left\\{{\\omega_f \\left({I}\\right): I \\in S_x}\\right\\} \\in \\R$ as $\\displaystyle \\inf \\left\\{{\\omega_f \\left({I}\\right): I \\in S_x}\\right\\} = \\omega_f \\left({x}\\right)$ Therefore, an $I \\in S_x$ exists such that: {{begin-eqn}} {{eqn | l = \\omega_f \\left({I}\\right) - \\displaystyle \\inf \\left\\{ {\\omega_f \\left({I'}\\right): I' \\in S_x}\\right\\}       | o = <        | r = \\epsilon       | c = by Infimum of Subset of Extended Real Numbers is Arbitrarily Close }} {{eqn | ll = \\iff       | l = \\omega_f \\left({I}\\right) - \\omega_f \\left({x}\\right)       | o = <        | r = \\epsilon       | c = as $\\omega_f \\left({x}\\right) = \\displaystyle \\inf \\left\\{ {\\omega_f \\left({I'}\\right): I' \\in S_x}\\right\\}$ }} {{end-eqn}} {{qed|lemma}} Category:Real Analysis ez6vb772zpbteefj77vvv2y8o5rk4rh"}
+{"_id": "32670", "title": "Reciprocal/Examples/Euler's Number", "text": "Reciprocal/Examples/Euler's Number 0 50525 433471 379617 2019-11-01T13:37:32Z Prime.mover 59 wikitext text/x-wiki == Example of Reciprocal == The reciprocal of Euler's Number $e$ is approximately: <onlyinclude> :$\\dfrac 1 e \\approx 0 \\cdotp 36787 \\, 94411 \\, 71442 \\, 32159 \\, 55237 \\, 70161 \\, 46086 \\, 74458 \\, 11131 \\, 031 \\ldots$ </onlyinclude> {{OEIS|A068985}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Reciprocal/Examples/Pi|next = Envelope Problem}}: $0 \\cdotp 367 \\, 879 \\, 441 \\, 171 \\, 442 \\, 321 \\, 595 \\, 523 \\, 770 \\, 161 \\, 460 \\, 867 \\, 445 \\, 811 \\, 131 \\, 031$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = One Third as Quotient of Sequences of Odd Numbers|next = Euler's Number as Sum of Egyptian Fractions}}: $0 \\cdotp 36787 \\, 94411 \\, 71442 \\, 32159 \\, 55237 \\, 70161 \\, 46086 \\, 74458 \\, 11131 \\, 031$ Category:Examples of Reciprocals Category:Euler's Number fkt7k64j5ggfv11l4uh6q12mhgl4a9m"}
+{"_id": "32671", "title": "Continued Fraction Expansion of Irrational Square Root/Examples/2", "text": "Continued Fraction Expansion of Irrational Square Root/Examples/2 0 50722 470631 433258 2020-05-26T06:33:17Z Prime.mover 59 Prime.mover moved page Continued Fraction Expansion of Irrational Square Root/Example/2 to Continued Fraction Expansion of Irrational Square Root/Examples/2 wikitext text/x-wiki == Examples of Continued Fraction Expansion of Irrational Square Root == <onlyinclude> The continued fraction expansion of the square root of $2$ is given by: :$\\sqrt 2 = \\sqbrk {1, \\sequence 2}$ </onlyinclude> {{OEIS|A040000}} == Proof == {{begin-eqn}} {{eqn | l = \\sqrt 2       | r = 1 + \\paren {\\sqrt 2 − 1}       | c =  }} {{eqn | r = 1 + \\frac {\\paren {\\sqrt 2 − 1} \\paren {\\sqrt 2 + 1} } {\\sqrt 2 + 1}       | c = multiplying top and bottom by $\\sqrt 2 + 1$ }} {{eqn | r = 1 + \\frac {\\paren {\\sqrt 2}^2 − 1^2} {\\sqrt 2 + 1}       | c = Difference of Two Squares }} {{eqn | r = 1 + \\frac 1 {1 + \\sqrt 2}       | c = as $\\paren {\\sqrt 2}^2 − 1^2 = 2 - 1 = 1$ }} {{end-eqn}} Thus it is possible to replace $\\sqrt 2$ recursively: {{begin-eqn}} {{eqn | l = \\sqrt 2       | r = 1 + \\frac 1 {1 + \\sqrt 2}       | c =  }} {{eqn | r = 1 + \\frac 1 {1 + \\paren {1 + \\cfrac 1 {1 + \\sqrt 2} } }       | c =  }} {{eqn | r = 1 + \\frac 1 {2 + \\cfrac 1 {1 + \\sqrt 2} }       | c =  }} {{eqn | r = 1 + \\frac 1 {2 + \\cfrac 1 {1 + \\paren {1 + \\cfrac 1 {1 + \\sqrt 2} } } }       | c =  }} {{eqn | r = 1 + \\frac 1 {2 + \\cfrac 1 {2 + \\cfrac 1 {1 + \\sqrt 2} } }       | c =  }} {{end-eqn}} The pattern repeats indefinitely, producing the continued fraction expansion: :$\\sqrt 2 = \\sqbrk {1, 2, 2, 2, \\ldots} = \\sqbrk {1, \\sequence 2}$ {{handwaving}} {{qed}} Category:Continued Fractions 4kyf0xltted86u8fkvn1ttmfsyps42z"}
+{"_id": "32672", "title": "Riemann Zeta Function at Even Integers/Examples/2", "text": "Riemann Zeta Function at Even Integers/Examples/2 0 50830 433362 393171 2019-11-01T09:37:43Z Prime.mover 59 wikitext text/x-wiki == Example of Riemann Zeta Function at Even Integers == <onlyinclude> The Riemann zeta function of $2$ is given by: {{begin-eqn}} {{eqn | l = \\map \\zeta 2       | r = \\dfrac 1 {1^2} + \\dfrac 1 {2^2} + \\dfrac 1 {3^2} + \\dfrac 1 {4^2} + \\cdots       | c =  }} {{eqn | r = \\dfrac {\\pi^2} 6       | c =  }} {{eqn | o = \\approx       | r = 1 \\cdotp 64493 \\, 4066 \\ldots       | c =  }} {{end-eqn}} </onlyinclude> {{OEIS|A013661}} == Proof == {{:Basel Problem/Proof 6}} The decimal expansion can be found by an application of arithmetic. == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Definition:Wythoff's Game|next = Square Root/Examples/3}}: $1 \\cdotp 644 \\, 934 \\, 066 \\ldots$ * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Riemann Zeta Function at Even Integers|next = Riemann Zeta Function of 4}}: $\\S 1.2.7$: Harmonic Numbers: $(7)$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Definition:Wythoff's Game|next = Zeta of 2 as Product of Fractions with Prime Numerators}}: $1 \\cdotp 64493 \\, 4066 \\ldots$ Category:Riemann Zeta Function at Even Integers 4ddaa3tqvt1zokkd9f2ykl46o2pmv2k"}
+{"_id": "32673", "title": "Axiom:Real Number as Complex Number", "text": "Axiom:Real Number as Complex Number 100 51043 274515 274511 2016-10-30T14:33:07Z Prime.mover 59 wikitext text/x-wiki {{refactor|obviously needs refactoring}} {{MissingLinks|Needs a category}} == Axiom == Let $x \\in \\R$ be a real number. Then: : $x = \\left({x, 0}\\right)$ where $\\left({a, b}\\right)$ where $a, b \\in R$ denotes a complex number. Also: : $x = x + 0 i$ where $a + b i$ where $a, b \\in R$ denotes a complex number. 0ceu0zey8cofgzlvd94lyogii14qmr8"}
+{"_id": "32674", "title": "Magic Square/Examples/Order 3", "text": "Magic Square/Examples/Order 3 0 51180 446278 424228 2020-02-04T15:47:05Z Prime.mover 59 wikitext text/x-wiki == Example of Order $3$ Magic Square == Order $3$ magic square: <onlyinclude> :$\\begin{array}{|c|c|c|} \\hline 2 & 7 & 6 \\\\ \\hline 9 & 5 & 1 \\\\ \\hline 4 & 3 & 8 \\\\ \\hline \\end{array}$ </onlyinclude> == Also known as == {{:Magic Square/Examples/Order 3/Also known as}} == Also see == * Smallest Magic Square is of Order 3 * Magic Constant of Order 3 Magic Square == Historical Note == {{:Magic Square/Examples/Order 3/Historical Note}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Hilbert-Waring Theorem/Particular Cases/4/Historical Note|next = Magic Constant of Order 3 Magic Square}}: $9$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Hilbert-Waring Theorem/Particular Cases/4/Historical Note|next = Magic Constant of Order 3 Magic Square}}: $9$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Magic Square|next = Magic Square/Examples/Order 3/Also known as|entry = magic square}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Magic Square|next = Magic Square/Examples/Order 3/Also known as|entry = magic square}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Magic Square|next = Magic Square/Examples/Order 4/Dürer|entry = magic square}} Category:Magic Squares izm9s929wxgdxosf4i8k13lsxtsnq01"}
+{"_id": "32675", "title": "Sum of Integrals on Adjacent Intervals for Integrable Functions/Corollary", "text": "Sum of Integrals on Adjacent Intervals for Integrable Functions/Corollary 0 51241 468131 459088 2020-05-16T09:47:50Z Prime.mover 59 wikitext text/x-wiki == Corollary to Sum of Integrals on Adjacent Intervals for Integrable Functions == Let $f$ be a real function which is Darboux integrable on any closed interval $\\mathbb I$. <onlyinclude> Let $a_0, a_1, \\ldots, a_n$ be real numbers, where $n \\in \\N$ and $n \\ge 2$. Then: :$\\displaystyle \\int_{a_0}^{a_n} \\map f t \\rd t = \\sum_{i \\mathop = 0}^{n - 1} \\int_{a_i}^{a_{i + 1} } \\map f t \\rd t$ </onlyinclude> == Proof == Proof by induction: === Basis for the Induction === According to Sum of Integrals on Adjacent Intervals for Integrable Functions, $n = 2$ holds. This is the basis for the induction. === Induction Hypothesis === This is our induction hypothesis: :$\\displaystyle \\int_{a_0}^{a_k} \\map f t \\rd x = \\sum_{i \\mathop = 0}^{k - 1} \\int_{a_i}^{a_{i + 1} } \\map f t \\rd t$ Now we need to show true for $n = k + 1$: :$\\displaystyle \\int_{a_0}^{a_{k + 1} } \\map f t \\rd t = \\sum_{i \\mathop = 0}^k \\int_{a_i}^{a_{i + 1} } \\map f t \\rd t$ === Induction Step === This is our induction step: {{begin-eqn}} {{eqn | l = \\int_{a_0}^{a_{k + 1} } \\map f t \\rd t       | r = \\int_{a_0}^{a_k} \\map f t \\rd t + \\int_{a_k}^{a_{k + 1} } \\map f t \\rd t       | c = Sum of Integrals on Adjacent Intervals for Integrable Functions }} {{eqn | r = \\sum_{i \\mathop = 0}^{k - 1} \\int_{a_i}^{a_{i + 1} } \\map f t \\rd t + \\int_{a_k}^{a_{k + 1} } \\map f t \\rd t       | c = Induction Hypothesis }} {{eqn | r = \\sum_{i \\mathop = 0}^k \\int_{a_i}^{a_{i + 1} } \\map f t \\rd t       | c =  }} {{end-eqn}} The result follows by induction. {{qed}} == Sources == * {{BookReference|Fourier Series|1961|I.N. Sneddon|prev = Piecewise Continuous Function with One-Sided Limits is Darboux Integrable|next = Definition:Real Right-Hand Derivative}}: Chapter Two: $\\S 1$. Piecewise-Continuous Functions: $(5)$ * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Definition:Improper Integral on Open Below Interval|next = Definite Integral of Constant Multiple of Real Function}}: $\\S 15$: General Formulas involving Definite Integrals: $15.7$ Category:Proofs by Induction Category:Definite Integrals k1fa60kmb57gjadcvaqfg3nbpzzr9dn"}
+{"_id": "32676", "title": "Integration by Substitution/Corollary", "text": "Integration by Substitution/Corollary 0 51242 483482 468279 2020-08-28T12:48:52Z Prime.mover 59 wikitext text/x-wiki == Corollary to Integration by Substitution == <onlyinclude> Let $f : \\R \\to \\R$ be a real function. Let $f$ be integrable. Let $a$, $b$, and $c$ be real numbers. Then: :$\\displaystyle \\int_{a - c}^{b - c} \\map f t \\rd t = \\int_a^b \\map f {t - c} \\rd t$ </onlyinclude> == Proof == Let $\\map \\phi u = u - c$. By Sum Rule for Derivatives, Derivative of Identity Function, and Derivative of Constant, we have: :$\\map {\\phi'} u = 1$ By Integration by Substitution: {{begin-eqn}} {{eqn | l = \\int_{\\map \\phi a}^{\\map \\phi b} \\map f t \\rd t       | r = \\int_a^b \\map f {\\map \\phi u} \\map {\\phi'} u \\rd u       | c =  }} {{eqn | ll = \\leadsto       | l = \\int_{a - c}^{b - c} \\map f t \\rd t       | r = \\int_a^b \\map f {u - c} \\paren 1 \\rd u       | c =  }} {{eqn | r = \\int_a^b \\map f {u - c} \\rd u       | c =  }} {{eqn | r = \\int_a^b \\map f {t - c} \\rd t       | c =  }} {{end-eqn}} {{qed}} Category:Integration by Substitution 1ipmzu7snocnfv8io33p1jmkhas10ci"}
+{"_id": "32677", "title": "Finite Ordinal Times Ordinal/Lemma", "text": "Finite Ordinal Times Ordinal/Lemma 0 51259 275440 2016-11-06T09:54:20Z Kc kennylau 2331 Created page with \"== Lemma ==  Let $m$ be a finite ordinal.  Let $m \\ne 0$, where $0$ is the zero ordinal.   Then:  : $m \\times \\omeg...\" wikitext text/x-wiki == Lemma == Let $m$ be a finite ordinal. Let $m \\ne 0$, where $0$ is the zero ordinal. Then: : $m \\times \\omega = \\omega$ where $\\omega$ denotes the minimal infinite successor set. == Proof == {{begin-eqn}} {{eqn | ll = \\forall n \\in \\omega       | l = m \\times n       | o = \\in       | r = \\omega       | c = Natural Number Multiplication is Closed }} {{eqn | l = \\bigcup_{n \\mathop \\in \\omega} \\left({ m \\times n }\\right)       | o = \\le       | r = \\omega       | c = Supremum Inequality for Ordinals }} {{eqn | ll = \\implies       | l = m \\times \\omega       | o = \\le       | r = \\omega       | c = Definition of Ordinal Multiplication }} {{end-eqn}} Also, $\\omega \\le \\left({ m \\times \\omega }\\right)$ by Subset is Right Compatible with Ordinal Multiplication. The lemma follows from the definition of equality. {{qed|lemma}} Category:Ordinal Arithmetic Category:Transfinite Arithmetic Category:Finite Ordinals Category:Minimal Infinite Successor Set 18ma4g26xnn9ic9e0uzk74435l9nfut"}
+{"_id": "32678", "title": "Binary Logarithm/Examples/10", "text": "Binary Logarithm/Examples/10 0 51349 433259 433010 2019-11-01T08:32:16Z Prime.mover 59 wikitext text/x-wiki == Example of Binary Logarithm == <onlyinclude> The binary logarithm of $10$ is: :$\\log_2 10 \\approx 3 \\cdotp 32192 \\, 80948 \\, 87362 \\, 34787 \\, 0319 \\ldots$ </onlyinclude> {{OEIS|A020862}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Square Root/Examples/10|next = Number of Binary Digits in Power of 10}}: $3 \\cdotp 321 \\, 928 \\ldots$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Square Root/Examples/10|next = Number of Binary Digits in Power of 10}}: $3 \\cdotp 32192 \\, 8 \\ldots$ Category:Examples of Binary Logarithms 9g9r9j5lxwxupvxo71z87icnax3idki"}
+{"_id": "32679", "title": "Bound for Analytic Function and Derivatives", "text": "Bound for Analytic Function and Derivatives 0 51357 435939 435768 2019-11-21T10:51:19Z Ivar Sand 2302 Refactor template removed wikitext text/x-wiki == Lemma == Let $f$ be a complex function. Let $z_0$ be a point in $\\C$. Let $r$ be a real number in $\\R_{>0}$. Let $\\Gamma$ be a circle in $\\C$ with center at $z_0$ and radius $r$. Let $f$ be analytic on $\\Gamma$ and its interior. Let $t \\in \\C$ be such that $\\cmod {t - z_0} < r$. Then a real number $M$ exists such that, for every $n \\in \\N$: :$\\displaystyle \\cmod {\\map {f^{\\paren n} } t} \\le \\frac {M r \\, n!} {\\paren {r - \\cmod {t - z_0} }^\\paren {n + 1} }$ == Proof == === Lemma (Analytic Function Bounded on Circle) === {{:Bound for Analytic Function and Derivatives/Analytic Function Bounded on Circle}}{{qed|lemma}} We have: :$f$ is analytic on $\\Gamma$ and its interior :$t$ is in the interior of $\\Gamma$ Therefore: :$\\displaystyle \\map {f^{\\paren n} } t = \\frac {n!} {2 \\pi i} \\int_\\Gamma \\frac {\\map f z} {\\paren {z - t}^{\\paren {n + 1} } } \\rd z$ by Cauchy's Integral Formula for Derivatives where $\\Gamma$ is traversed counterclockwise. We have that $f$ is bounded on $\\Gamma$ by Lemma (Analytic Function Bounded on Circle). Therefore, there is a positive real number $M$ that satisfies: :$M \\ge \\cmod {\\map f z}$ for every $z$ on $\\Gamma$ We have $\\cmod {t - z_0} < r$. Therefore: :$0 < r - \\cmod {t - z_0}$ We observe that $r - \\cmod {t - z_0}$ is the minimum distance between $t$ and $\\Gamma$. Therefore: :$\\paren {r - \\cmod {t - z_0} } \\le \\cmod {z - t}$ for every $z$ on $\\Gamma$ We get: {{begin-eqn}} {{eqn | l = \\cmod {\\map {f^{\\paren n} } t}       | r = \\cmod {\\frac {n!} {2 \\pi i} \\int_\\Gamma \\frac {\\map f z} {\\paren {z - t}^{\\paren {n + 1} } } \\rd z} }} {{eqn | o = \\le       | r = \\frac {n!} {2 \\pi} \\int_\\Gamma \\frac {\\cmod {\\map f z} } {\\cmod {z - t}^{\\paren {n + 1} } } \\cmod {\\d z} }} {{eqn | o = \\le       | r = \\frac {n!} {2 \\pi} \\int_\\Gamma \\frac M {\\cmod {z - t}^{\\paren {n + 1} } } \\cmod {\\d z}       | c = as $M \\ge \\cmod {\\map f z}$ for every $z$ on $\\Gamma$ }} {{eqn | o = \\le       | r = \\frac {n!} {2 \\pi} \\int_\\Gamma \\frac M {\\paren {r - \\cmod {t - z_0} }^{\\paren {n + 1} } } \\cmod {\\d z}       | c = as $0 < \\paren {r - \\cmod {t - z_0} } \\le \\cmod {z - t}$ for every $z$ on $\\Gamma$ }} {{eqn | r = \\frac {n!} {2 \\pi} \\frac M {\\paren {r - \\cmod {t - z_0} }^{\\paren {n + 1} } } \\int_\\Gamma \\cmod {\\d z} }} {{eqn | r = \\frac {n!} {2 \\pi} \\frac M {\\paren {r - \\cmod {t - z_0} }^{\\paren {n + 1} } } 2 \\pi r }} {{eqn | r = \\frac {M r \\, n!} {\\paren {r - \\cmod {t - z_0} }^{\\paren {n + 1} } } }} {{end-eqn}} {{qed}} {{explain|The notation $\\cmod {\\d z}$ has not been raised before on {{ProofWiki}} -- we need to do something about that}} Category:Complex Analysis kfj9fmwm0855gzst2c2sez0pfyzc49o"}
+{"_id": "32680", "title": "Pythagorean Triangle/Examples/3-4-5", "text": "Pythagorean Triangle/Examples/3-4-5 0 51433 478808 478785 2020-07-18T13:12:02Z Prime.mover 59 wikitext text/x-wiki == Example of Primitive Pythagorean Triangle == <onlyinclude> The triangle whose sides are of length $3$, $4$ and $5$ is a primitive Pythagorean triangle. </onlyinclude> :300px == Proof == {{begin-eqn}} {{eqn | l = 3^2 + 4^2       | r = 9 + 16       | c =  }} {{eqn | r = 25       | c =  }} {{eqn | r = 5^2       | c =  }} {{end-eqn}} It follows by Pythagoras's Theorem that $3$, $4$ and $5$ form a Pythagorean triple. Note that $3$ and $4$ are coprime. Hence, by definition, $3$, $4$ and $5$ form a primitive Pythagorean triple. The result follows by definition of a primitive Pythagorean triangle. {{qed}} == Also see == * Smallest Pythagorean Triangle is 3-4-5 * Pythagorean Triangle whose Area is Half Perimeter: its area and semiperimeter are both $6$ * Pythagorean Triangle with Sides in Arithmetic Sequence == Historical Note == {{:Pythagorean Triangle/Examples/3-4-5/Historical Note}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Definition:Pythagorean Triangle|next = Pythagoras's Theorem}}: $5$ * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = 6/Historical Note|next = Definition:Perfect Number}}: $6$ * {{BookReference|Curious and Interesting Puzzles|1992|David Wells|prev = Definition:Pythagorean Triple|next = Babylonian Mathematics/Examples/Sliding Ladder}}: Pythagorean Triples * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Definition:Pythagorean Triangle|next = Pythagoras's Theorem}}: $5$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = 6/Historical Note|next = Definition:Perfect Number}}: $6$ Category:Examples of Pythagorean Triangles lwzjl2zodgtbfkortdr19jog0vbvdmh"}
+{"_id": "32681", "title": "Volume of Unit Hypersphere/Sequence", "text": "Volume of Unit Hypersphere/Sequence 0 51628 381091 276914 2018-12-10T23:29:46Z Prime.mover 59 wikitext text/x-wiki == Sequence of Volumes of Unit Hyperspheres == <onlyinclude> The sequence of volumes of the unit sphere in $n$-dimensional space begins as follows: {{begin-eqn}} {{eqn | l = n = 1       | o = :       | r = \\map V 1 = 2 }} {{eqn | l = n = 2       | o = :       | r = \\map V 2 = 3.1 }} {{eqn | l = n = 3       | o = :       | r = \\map V 3 = 4.2 }} {{eqn | l = n = 4       | o = :       | r = \\map V 4 = 4.9 }} {{eqn | l = n = 5       | o = :       | r = \\map V 5 = 5.264 }} {{eqn | l = n = 6       | o = :       | r = \\map V 6 = 5.2 }} {{eqn | l = n = 7       | o = :       | r = \\map V 7 = 4.7 }} {{end-eqn}} </onlyinclude> == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Maximum Volume of Unit Radius Sphere in Fractional Dimensions|next = 6}}: $5 \\cdotp 256 \\, 946 \\, 404 \\, 860 \\ldots$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Maximum Volume of Unit Radius Sphere in Fractional Dimensions|next = 6}}: $5 \\cdotp 25694 \\, 64048 \\, 60 \\ldots$ Category:Spheres 2132urzjbd5jd3ru1q589926yp8zeyw"}
+{"_id": "32682", "title": "Complex Power by Complex Exponential is Analytic", "text": "Complex Power by Complex Exponential is Analytic 0 51901 405960 353765 2019-05-30T07:10:27Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $\\psi, \\eta \\in \\C$ be constant. Let $\\map f z = z^\\psi \\, \\map \\exp {-\\eta z}$, where: :$z^\\psi$ denotes $z$ to the power of $\\psi$, defined on its principal branch :$\\map \\exp {-\\eta z}$ denotes the complex exponential function. Then $f$ is analytic on any simply connected domain that does not contain the origin nor any points on the negative real axis. == Proof == Let $z$ be written in exponential form: :$z = r \\map \\exp {i \\theta}$  where: :$r > 0$ :$\\theta \\in \\hointl {-\\pi} \\pi$ Let $\\psi = a + i b, \\eta = c + i d$. By the definition of $f$: {{begin-eqn}} {{eqn | l = \\map f {r \\, \\map \\exp {i \\theta} }       | r = z^\\psi \\, \\map \\exp {-\\eta z} }} {{eqn | r = \\paren {r e^{i \\theta} }^\\psi \\, \\map \\exp {-\\eta r \\, \\map \\exp {i \\theta} } }} {{eqn | r = \\map \\exp {\\psi \\, \\map \\Log {r \\, \\map \\exp {i \\theta} } } \\, \\map \\exp {-\\eta r \\paren {\\cos \\theta + i \\sin \\theta} } }} {{eqn | r = \\map \\exp {\\psi \\ln r + i \\psi \\theta - \\eta r \\cos \\theta - i \\eta r \\sin \\theta} }} {{eqn | r = \\map \\exp {\\paren {a + i b} \\ln r + i \\paren {a + i b} \\theta - \\paren {c + i d} r \\cos \\theta - i \\paren {c + i d} r \\sin \\theta} }} {{eqn | r = \\map \\exp {a \\ln r + i b \\ln r + i a \\theta - b \\theta - c r \\cos \\theta - i d r \\cos \\theta - i c r \\sin \\theta + d r \\sin \\theta} }} {{eqn | r = \\map \\exp {a \\ln r - b \\theta - c r \\cos \\theta + d r \\sin \\theta} \\, \\map \\exp {i \\paren {b \\ln r + a \\theta - d r \\cos \\theta - c r \\sin \\theta} } }} {{end-eqn}} Define: {{begin-eqn}} {{eqn | l = \\map g {r, \\theta}       | r = a \\ln r - b \\theta - c r \\cos \\theta + d r \\sin \\theta }} {{eqn | l = \\map h {r, \\theta}       | r = b \\ln r + a \\theta - d r \\cos \\theta - c r \\sin \\theta }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \\map f {r \\, \\map \\exp {i \\theta} }       | r = \\map \\exp g \\, \\map \\exp {i h} }} {{eqn | r = \\map \\exp g \\, \\map \\cos h + i \\map \\exp g \\, \\map \\sin h }} {{end-eqn}} Define: {{begin-eqn}} {{eqn | l = \\map u {r, \\theta}       | r = \\map \\exp g \\, \\map \\cos h }} {{eqn | l = \\map v {r, \\theta}       | r = \\map \\exp g \\, \\map \\sin h }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \\map f {r \\, \\map \\exp {i \\theta } }        | r = u + iv }} {{end-eqn}} We check the Polar Form of Cauchy-Riemann Equations. As a preliminary:  {{begin-eqn}} {{eqn | l = \\frac {\\partial g} {\\partial r}       | r = \\frac a r - c \\cos \\theta + d \\sin \\theta }} {{eqn | l = \\frac {\\partial h} {\\partial r}       | r = \\frac b r - d \\cos \\theta - \\sin \\theta }} {{eqn | l = \\frac {\\partial g} {\\partial \\theta}       | r = -b + c r \\sin \\theta + d r \\cos \\theta }} {{eqn | l = \\frac {\\partial h} {\\partial \\theta}       | r = a + d r \\sin \\theta - c r \\cos \\theta }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \\frac {\\partial u} {\\partial r}       | r = \\map \\exp g \\frac {\\partial g} {\\partial r} \\map \\cos h - \\map \\exp g \\, \\map \\sin f \\frac {\\partial h} {\\partial r} }} {{eqn | r = \\frac 1 r \\map \\exp g \\paren {\\paren {a - r c \\cos \\theta + r d \\sin \\theta} \\, \\map \\cos h + \\map \\sin h \\paren {- b + r d \\cos \\theta + r c \\sin \\theta} } }} {{eqn | r = \\frac 1 r \\map \\exp g \\paren {\\frac {\\partial h} {\\partial \\theta} \\map \\cos h + \\, \\map \\sin h \\frac {\\partial g} {\\partial \\theta} } }} {{eqn | r = \\frac 1 r \\frac {\\partial v} {\\partial \\theta} }} {{eqn | l = \\frac {\\partial u} {\\partial \\theta}       | r = \\map \\exp g \\frac {\\partial g} {\\partial \\theta} \\, \\map \\cos h - \\map \\exp g \\, \\map \\sin h \\frac {\\partial h} {\\partial \\theta} }} {{eqn | r = -r \\, \\map \\exp g \\paren {\\paren {\\frac b r - c \\sin \\theta - d \\cos \\theta} \\, \\map \\cos h + \\map \\sin h \\paren {\\frac a r + d \\sin \\theta - c  \\cos \\theta} }       | c = Sine Function is Odd }} {{eqn | r = -r \\, \\map \\exp g \\paren {\\frac {\\partial h} {\\partial r} \\, \\map \\cos h + \\map \\sin h \\frac {\\partial g} {\\partial r} } }} {{eqn | r = -r \\frac {\\partial v} {\\partial r} }} {{end-eqn}} {{qed}} Category:Complex Analysis oaqckgvsft6v52oxrbqf9p5bjzs3sz8"}
+{"_id": "32683", "title": "X to the x is not of Exponential Order/Lemma", "text": "X to the x is not of Exponential Order/Lemma 0 52223 357277 357276 2018-05-27T05:21:45Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $f: \\R \\to \\R$ be defined on $\\left [{0 \\,.\\,.\\, \\to} \\right)$ with $f \\left({x}\\right) = x^x$. Suppose there exist strictly positive real constants $M, K, a \\in \\R_{> 0}$ such that: :$\\forall t \\ge M: \\left\\vert {f \\left({t}\\right)} \\right \\vert < K e^{a t}$ Then there exists a constant $C$ such that: :$\\forall t > C: \\left\\vert {f \\left({t}\\right)} \\right \\vert > K e^{a t}$ </onlyinclude> == Proof == By the definition of power: :$f \\left({t}\\right) = \\exp \\left({t \\ln t}\\right)$  By Exponential of Real Number is Strictly Positive, we can reduce the lemma into the existence of $C$ such that: :$\\forall t > C: f \\left({t}\\right) > K e^{a t}$ We will divide into two cases. === Case 1: $K > 1$ === Assume that $t > K e^a$. {{begin-eqn}} {{eqn | l = a       | o = >       | r = 0       | c =  }} {{eqn | l = e^a       | o = >       | r = e^0       | c = Exponential is Strictly Increasing }} {{eqn | l = e^a       | o = >       | r = 1       | c = Exponential of Zero }} {{eqn | l = K e^a       | o = >       | r = 1       | c = As $K > 1$ }} {{eqn | l = t       | o = >       | r = 1       | c = As $t > K e^a$ }} {{eqn | n = 1       | l = t \\ln K       | o = >       | r = \\ln K       | c =  }} {{eqn | l = t       | o = >       | r = K e^a       | c = Assumption }} {{eqn | l = \\ln t       | o = >       | r = \\ln \\left({K e^a}\\right)       | c = Logarithm is Strictly Increasing }} {{eqn | l = \\ln t       | o = >       | r = a + \\ln K       | c =  }} {{eqn | l = t \\ln t       | o = >       | r = a t + t \\ln K       | c =  }} {{eqn | l = t \\ln t       | o = >       | r = a t + \\ln K       | c = from $(1)$ }} {{eqn | l = \\exp \\left({t \\ln t}\\right)       | o = >       | r = \\exp \\left({a t + \\ln K}\\right)       | c = Exponential is Strictly Increasing }} {{eqn | l = f \\left({t}\\right)       | o = >       | r = K e^{a t}       | c = Exponential is Strictly Increasing }} {{end-eqn}} Here, $C = K e^a$. === Case 2: $K \\le 1$ === Assume that $t > e^a$. {{begin-eqn}} {{eqn | l = \\ln K       | o = \\le       | r = \\ln 1       | c = Logarithm is Strictly Increasing }} {{eqn | n = 1       | l = \\ln K       | o = \\le       | r = 0       | c = Logarithm of 1 is 0 }} {{eqn | l = t       | o = >       | r = e^a       | c = by assumption }} {{eqn | l = \\ln t       | o = >       | r = a       | c = Logarithm is Strictly Increasing }} {{eqn | l = t \\ln t       | o = >       | r = a t       | c =  }} {{eqn | l = t \\ln t       | o = >       | r = a t + \\ln k       | c = from $(1)$ }} {{eqn | l = \\exp \\left({t \\ln t}\\right)       | o = >       | r = \\exp \\left({a t + \\ln K}\\right)       | c = Exponential is Strictly Increasing }} {{eqn | l = f \\left({t}\\right)       | o = >       | r = K e^{a t}       | c = Exponential is Strictly Increasing }} {{end-eqn}} Here, $C = e^a$. {{qed}} Category:Exponential Order tnwa4ejjlezlzh9pfl6um4dtvf4c64u"}
+{"_id": "32684", "title": "Schanuel's Conjecture", "text": "Schanuel's Conjecture 0 52233 440312 353764 2019-12-20T13:58:41Z Prime.mover 59 wikitext text/x-wiki == Conjecture == Let $z_1, \\cdots, z_n$ be complex numbers that are linearly independent over the rational numbers $\\Q$. Then: :the extension field $\\map \\Q {z_1, \\cdots, z_n, e^{z_1}, \\cdots, e^{z_n} }$ has transcendence degree at least $n$ over $\\Q$ where $e^z$ is the complex exponential of $z$. {{Namedfor|Stephen Hoel Schanuel|cat = Schanuel}} Category:Transcendental Number Theory Category:Unproven Hypotheses Category:Schanuel's Conjecture aw8yqn1akkivluwqxl046e7ci2s6qy4"}
+{"_id": "32685", "title": "Characterization of Cosine Integral Function", "text": "Characterization of Cosine Integral Function 0 52237 417225 405915 2019-08-08T14:18:35Z Caliburn 3218 wikitext text/x-wiki == Definition == Let $\\Ci: \\R_{>0}: \\R$ denote the cosine integral function: :$\\map \\Ci x = \\displaystyle \\int_{t \\mathop = x}^{t \\mathop \\to +\\infty} \\frac {\\cos t} t \\rd t$ Then: :$\\map \\Ci x = -\\gamma - \\ln x + \\displaystyle \\int_{t \\mathop \\to 0}^{t \\mathop = x} \\frac{1 - \\cos t} t \\rd t$ where $\\gamma$ is the Euler-Macheroni constant. == Proof == {{proof wanted|Laplace transform to the one, then invert to the other?}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Definition:Cosine Integral Function|next = Power Series Expansion for Cosine Integral Function plus Logarithm}}: $\\S 35$: Miscellaneous Special Functions: Cosine Integral: $35.14$ * {{MathWorld|Cosine Integral|CosineIntegral}} Category:Cosine Integral Function py644aus0b491ucur1l08l2rlwh2zqg"}
+{"_id": "32686", "title": "Sum of Terms of Magic Square/Sequence", "text": "Sum of Terms of Magic Square/Sequence 0 52261 433514 382814 2019-11-01T14:01:42Z Prime.mover 59 wikitext text/x-wiki == Sequence of Sums of Terms of Magic Squares == <onlyinclude> The sequence of the sum totals of all the entries in magic squares of order $n$ begins: :$1, \\paren {10,} \\, 45, 136, 325, 666, 1225, 2080, 3321, 5050, 7381, 10 \\, 440, 14 \\, 365, 19 \\, 306, 25 \\, 425, 32 \\, 896, \\ldots$ However, note that while $10 = \\dfrac {2^2 \\paren {2^2 + 1} } 2$, a magic square of order $2$ does not actually exist. </onlyinclude> {{OEIS|A037270}} Category:Magic Squares oqru1w4drzmzya0javj3gt9qjay04bc"}
+{"_id": "32687", "title": "Magic Constant of Magic Square/Sequence", "text": "Magic Constant of Magic Square/Sequence 0 52274 433416 382848 2019-11-01T12:58:36Z Prime.mover 59 wikitext text/x-wiki == Sequence of Magic Constants of Magic Squares == <onlyinclude> The sequence of the magic constants of magic squares of order $n$ begins: :$1, (5,) \\, 15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, 2056, 2465, 2925, 3439, \\ldots$ However, note that while $5 = \\dfrac {2 \\paren {2^2 + 1} } 2$, a magic square of order $2$ does not actually exist. </onlyinclude> {{OEIS|A006003}} Category:Magic Squares b0xwscvu10axivs1fnjg6ulrk1283ei"}
+{"_id": "32688", "title": "Schanuel's Conjecture Implies Transcendence of Pi to the power of Euler's Number/Lemma", "text": "Schanuel's Conjecture Implies Transcendence of Pi to the power of Euler's Number/Lemma 0 52293 442788 363086 2020-01-07T13:01:25Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let Schanuel's Conjecture be true. Let $z_1 = \\ln \\ln \\pi$, $z_2 = 1 + \\ln \\ln \\pi$, $z_3 = \\ln \\pi$, $z_4 = e \\ln \\pi$, and $z_5 = i \\pi$. Then, $z_1$, $z_2$, $z_3$, $z_4$, and $z_5$ are linearly independent over the rational numbers $\\Q$. == Proof == Assume the truth of Schanuel's Conjecture. Now, we will prove that $z_1$ and $z_3$ are linearly independent over the rational numbers $\\Q$. Equivalently, they are linearly independent over the integers $\\Z$. Let $a, b \\in \\Z$ such that: :$a z_1 + b z_3 = 0$ Substituting: :$a \\ln \\ln \\pi + b \\ln \\pi = 0$ Applying the exponential function to both sides: :$\\paren {\\ln \\pi}^a \\pi^b = 1$ By Schanuel's Conjecture Implies Algebraic Independence of Pi and Log of Pi over the Rationals, the above equation only has solution when $a = b = 0$. Thus, $z_1$ and $z_3$ are linearly independent over the rational numbers $\\Q$. Since $z_5$ is wholly imaginary, $z_1$, $z_3$, and $z_5$ are linearly independent over the rational numbers $\\Q$. By Schanuel's Conjecture, the extension field $\\map \\Q {z_1, z_3, z_5, e^{z_1}, e^{z_3}, e^{z_5} }$ has transcendence degree at least $3$ over the rational numbers $\\Q$. That is, the extension field $\\map \\Q {\\ln \\ln \\pi, \\ln \\pi, i \\pi, \\ln \\pi, \\pi, e^{i \\pi} }$ has transcendence degree at least $3$ over $\\Q$. However, by Euler's Identity, $e^{i \\pi} = -1$ is algebraic. Also, $\\pi$ and $i \\pi$ are not algebraically independent over $\\Q$. Therefore, $\\ln \\ln \\pi$, $\\ln \\pi$, and $i \\pi$ must be algebraically independent over $\\Q$. That is, $z_1$, $z_3$, and $z_5$ must be algebraically independent over rational numbers $\\Q$. It follows that $z_1$, $z_3$, $z_5$, and $1$ are linearly independent over the rational numbers $\\Q$. By Schanuel's Conjecture, the extension field $\\map \\Q {z_1, z_3, z_5, 1, e^{z_1}, e^{z_3}, e^{z_5}, e}$ has transcendence degree at least $4$ over the rational numbers $\\Q$. That is, the extension field $\\map \\Q {\\ln \\ln \\pi, \\ln \\pi, i \\pi, 1, \\ln \\pi, \\pi, e^{i \\pi}, e}$ has transcendence degree at least $4$ over $\\Q$. However, $1$ is algebraic. Moreover, by Euler's Identity, $e^{i \\pi} = -1$ is algebraic. Also, $\\pi$ and $i \\pi$ are not algebraically independent over $\\Q$. Therefore, $\\ln \\ln \\pi$, $\\ln \\pi$, $i \\pi$, and $e$ must be algebraically independent over $\\Q$. That is, $z_1$, $z_3$, $z_5$, and $e$ must be algebraically independent over rational numbers $\\Q$. It follows that $z_1$, $z_3$, $e z_3$, and $z_5$ must be algebraically independent over rational numbers $\\Q$. That is, $z_1$, $z_3$, $z_4$, and $z_5$ must be algebraically independent over $\\Q$. Therefore, $z_1$, $1$, $z_3$, $z_4$, and $z_5$ must be linearly independent over $\\Q$. Hence, $z_1$, $1 + z_1$, $z_3$, $z_4$, and $z_5$ must be linearly independent over $\\Q$. That is, if Schanuel's Conjecture holds, then $z_1$, $z_2$, $z_3$, $z_4$, and $z_5$ are linearly independent over the rational numbers $\\Q$. {{qed|lemma}} Category:Transcendental Numbers Category:Logarithms Category:Pi Category:Euler's Number Category:Schanuel's Conjecture bx4wk4y7r4gw7tfasu9nwmzru2ii363"}
+{"_id": "32689", "title": "Schanuel's Conjecture Implies Transcendence of 2 to the power of Euler's Number/Lemma", "text": "Schanuel's Conjecture Implies Transcendence of 2 to the power of Euler's Number/Lemma 0 52294 353794 280546 2018-05-04T18:15:23Z Prime.mover 59 wikitext text/x-wiki {{MissingLinks|or rather, correct the existing links}} == Lemma == Let: :$z_1 = \\ln \\ln 2$ :$z_2 = 1 + \\ln \\ln 2$ :$z_3 = \\ln 2$ :$z_4 = e \\ln 2$ Let Schanuel's Conjecture be true. Then $z_1$, $z_2$, $z_3$ and $z_4$ are linearly independent over the rational numbers $\\Q$. == Proof == Assume the truth of Schanuel's Conjecture. $2$ is algebraic. Hence, by the Corollary of the weaker Hermite-Lindemann-Weierstrass theorem, $\\ln 2$ is transcendental. Now, we will prove that $z_1$ and $z_3$ are linearly independent over the rational numbers $\\Q$. Equivalently, by Linearly Independent over the Rational Numbers iff Linearly Independent over the Integers, they are linearly independent over the integers $\\Z$. Let $a, b \\in \\Z$ such that: :$a z_1 + b z_3 = 0$ Substituting: :$a \\ln \\ln 2 + b \\ln 2 = 0$ Applying the exponential function to both sides: :$\\left({\\ln 2}\\right)^a 2^b = 1$ Since $\\ln 2$ is transcendental, the above equation only has solution when $a = b = 0$. Thus, $z_1$ and $z_3$ are linearly independent over the rational numbers $\\Q$. By Schanuel's Conjecture, the extension field $\\Q \\left({z_1, z_3, e^{z_1}, e^{z_3}}\\right)$ has transcendence degree at least $2$ over the rational numbers $\\Q$. That is, the extension field $\\Q \\left({\\ln \\ln 2, \\ln 2, \\ln 2, 2}\\right)$ has transcendence degree at least $2$ over $\\Q$. However, $2$ is algebraic. Therefore, $\\ln \\ln 2$ and $\\ln 2$ must be algebraically independent over $\\Q$. That is, $z_1$, $z_3$ must be algebraically independent over rational numbers $\\Q$. It follows that $z_1$, $z_3$, and $1$ are linearly independent over the rational numbers $\\Q$. By Schanuel's Conjecture, the extension field $\\Q \\left({z_1, z_3, 1, e^{z_1}, e^{z_3}, e}\\right)$ has transcendence degree at least $3$ over the rational numbers $\\Q$. That is, the extension field $\\Q \\left({\\ln \\ln 2, \\ln 2, 1, \\ln 2, 2, e}\\right)$ has transcendence degree at least $3$ over $\\Q$. However, $1$ and $2$ are algebraic. Therefore, $\\ln \\ln 2$, $\\ln 2$, $e$ must be algebraically independent over $\\Q$. That is, $z_1$, $z_3$, and $e$ must be algebraically independent over rational numbers $\\Q$. It follows that $z_1$, $z_3$, $e z_3$ must be algebraically independent over rational numbers $\\Q$. That is, $z_1$, $z_3$, $z_4$ must be algebraically independent over $\\Q$. Therefore, $z_1$, $1$, $z_3$, $z_4$ must be linearly independent over $\\Q$. Hence, $z_1$, $1 + z_1$, $z_3$, $z_4$ must be linearly independent over $\\Q$. That is, if Schanuel's Conjecture holds, then $z_1$, $z_2$, $z_3$, $z_4$ are linearly independent over the rational numbers $\\Q$. {{qed|lemma}} Category:Transcendental Numbers Category:2 Category:Euler's Number Category:Schanuel's Conjecture bafqcvc586o7i7lkv4u9rm9xwz2e8sp"}
+{"_id": "32690", "title": "Divisor Counting Function/Examples/12", "text": "Divisor Counting Function/Examples/12 0 52622 451779 292229 2020-03-02T09:43:28Z Prime.mover 59 Prime.mover moved page Tau Function/Examples/12 to Divisor Counting Function/Examples/12 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({12}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$12 = 2^2 \\times 3$ Thus: :$\\tau \\left({12}\\right) = \\tau \\left({2^2 \\times 3^1}\\right) = \\left({2 + 1}\\right) \\left({1 + 1}\\right) = 6$ The divisors of $12$ can be enumerated as: :$1, 2, 3, 4, 6, 12$ {{qed}} Category:Tau Function Category:12 hhyg9izsflqs51fkdhzqv7zjj2p2zhl"}
+{"_id": "32691", "title": "Differential of Differentiable Functional is Unique/Lemma", "text": "Differential of Differentiable Functional is Unique/Lemma 0 52631 282479 282477 2017-01-17T23:18:53Z Prime.mover 59 wikitext text/x-wiki {{MissingLinks}} == Lemma == <onlyinclude> Let $\\phi \\left[{y; h}\\right]$ be a linear functional w.r.t. $h$. Let: :$\\displaystyle \\lim_{\\left \\vert{h}\\right\\vert \\to 0} \\frac {\\phi \\left[{y; h}\\right]} {\\left\\vert{h}\\right\\vert} = 0$ Then : :$\\phi \\left[{y; h}\\right] = 0$ </onlyinclude> == Proof == This will be a proof by contradiction. {{AimForCont}} there exists a linear functional satisfying $\\phi \\left[{y; h_0}\\right] \\ne 0$ for some $h_0 \\ne 0$. Also suppose: :$\\displaystyle \\lim_{\\left \\vert{h_0}\\right\\vert \\to 0} \\frac{\\phi \\left[{y; h_0}\\right]} {\\left\\vert{h_0}\\right\\vert} = 0$ Now, define: :$h_n = \\dfrac {h_0} n$ and: :$m = \\dfrac{\\phi \\left[{y; h_0}\\right]} {\\left\\vert{h_0}\\right\\vert}$ Notice that $\\left\\vert{h_n}\\right\\vert \\to 0$ as $n \\to \\infty$.  Hence, from the assumption of the limit it should hold that: :$\\displaystyle \\lim_{n \\to \\infty} \\frac{\\phi \\left[{y; h_n}\\right]}{\\left\\vert{h_n}\\right\\vert} = \\lim_{\\left\\vert{h_n}\\right\\vert \\to 0} \\frac{ \\left[{y; h_n}\\right]} {\\left\\vert{h_n}\\right\\vert} = 0$ However, using the linearity of $\\phi \\left[{y; h_0}\\right]$ w.r.t. $h_0$: {{begin-eqn}} {{eqn | l = \\lim_{n \\to \\infty} \\frac {\\phi \\left[{y; h_n}\\right]} {\\left\\vert{h_n}\\right\\vert}       | r = \\lim_{n \\to \\infty} \\frac {\\phi \\left[{y; \\frac {h_0} n}\\right]} { \\left\\vert{\\frac {h_0} n}\\right\\vert}       | c = Definition of $h_n$ }} {{eqn | r = \\lim_{n \\to \\infty} \\frac {n \\, \\phi \\left[{y; h_0}\\right]} {n \\, \\left\\vert{h_0}\\right\\vert}       | c = extract of $n$ through linearity }} {{eqn | r = \\lim_{n \\to \\infty} \\frac {\\phi \\left[{y; h_0}\\right]} {\\left\\vert{h_0}\\right\\vert}       | c = cancel $n$ }} {{eqn | r = \\frac {\\phi \\left[{y; h_0}\\right]} {\\left\\vert{h_0}\\right\\vert}       | c = the limit does not depend on $n$ }} {{eqn | r = m       | c = definition of $m$ }} {{end-eqn}} However, by hypothesis: : $m \\ne 0$ Hence, the contradiction is achieved and the initial statement of the lemma holds. {{qed}} == Sources == * {{BookReference|Calculus of Variations|1963|I.M. Gelfand|author2 = S.V. Fomin}}: $\\S 1.3$: The Variation of a Functional. A Necessary Condition for an Extremum Category:Calculus of Variations 818us82q2tkfdakeemo573ft46fcnqt"}
+{"_id": "32692", "title": "Continued Fraction Expansion of Irrational Square Root/Examples/13", "text": "Continued Fraction Expansion of Irrational Square Root/Examples/13 0 52897 470666 470627 2020-05-26T07:57:26Z Prime.mover 59 wikitext text/x-wiki == Example of Continued Fraction Expansion of Irrational Square Root == <onlyinclude> The continued fraction expansion of the square root of $13$ is given by: :$\\sqrt {13} = \\sqbrk {3, \\sequence {1, 1, 1, 1, 6} }$ </onlyinclude> {{OEIS|A010122}} === Convergents === {{:Continued Fraction Expansion of Irrational Square Root/Examples/13/Convergents}} == Proof == Let $\\sqrt {13} = \\sqbrk {a_0, a_1, a_2, a_3, \\ldots}$ From Partial Quotients of Continued Fraction Expansion of Irrational Square Root, the partial quotients of this continued fraction expansion can be calculated as: :$a_r = \\floor {\\dfrac {\\floor {\\sqrt {13} } + P_r} {Q_r} }$ where: :$P_r = \\begin {cases} 0 & : r = 0 \\\\ a_{r - 1} Q_{r - 1} - P_{r - 1} & : r > 0 \\\\ \\end {cases}$ :$Q_r = \\begin {cases} 1 & : r = 0 \\\\ \\dfrac {n - {P_r}^2} {Q_{r - 1} } & : r > 0 \\\\ \\end {cases}$ {{PartialQuotientCalculator-Start | n = 13}} {{PartialQuotientCalculator | n = 13 | r = 1 | ar-1 = 3 | Qr-1 = 1 | Pr-1 = 0}} {{PartialQuotientCalculator | n = 13 | r = 2 | ar-1 = 1 | Qr-1 = 4 | Pr-1 = 3}} {{PartialQuotientCalculator | n = 13 | r = 3 | ar-1 = 1 | Qr-1 = 3 | Pr-1 = 1}} {{PartialQuotientCalculator | n = 13 | r = 4 | ar-1 = 1 | Qr-1 = 3 | Pr-1 = 2}} {{PartialQuotientCalculator | n = 13 | r = 5 | ar-1 = 1 | Qr-1 = 4 | Pr-1 = 1}} |} and the cycle is complete: :$\\sequence {1, 1, 1, 1, 6}$ {{qed}} Category:Examples of Continued Fractions Category:13 7armlnz95rqtjn6vf9wz1v5rpx5spzv"}
+{"_id": "32693", "title": "Continued Fraction Expansion of Irrational Square Root/Examples/29", "text": "Continued Fraction Expansion of Irrational Square Root/Examples/29 0 52898 470654 470633 2020-05-26T06:45:22Z Prime.mover 59 wikitext text/x-wiki == Examples of Continued Fraction Expansion of Irrational Square Root == <onlyinclude> The continued fraction expansion of the square root of $29$ is given by: :$\\sqrt {29} = \\sqbrk {5, \\sequence {2, 1, 1, 2, 10} }$ </onlyinclude> {{OEIS|A010128}} === Convergents === {{:Continued Fraction Expansion of Irrational Square Root/Examples/29/Convergents}} == Proof == Let $\\sqrt {29} = \\sqbrk {a_0, a_1, a_2, a_3, \\ldots}$ From Partial Quotients of Continued Fraction Expansion of Irrational Square Root, the partial quotients of this continued fraction expansion can be calculated as: :$a_r = \\floor {\\dfrac {\\floor {\\sqrt {29} } + P_r} {Q_r} }$ where: :$P_r = \\begin {cases} 0 & : r = 0 \\\\ a_{r - 1} Q_{r - 1} - P_{r - 1} & : r > 0 \\\\ \\end {cases}$ :$Q_r = \\begin {cases} 1 & : r = 0 \\\\ \\dfrac {n - {P_r}^2} {Q_{r - 1} } & : r > 0 \\\\ \\end {cases}$ {{PartialQuotientCalculator-Start | n = 29}} {{PartialQuotientCalculator | n = 29 | r = 1 | ar-1 = 5 | Qr-1 = 1 | Pr-1 = 0}} {{PartialQuotientCalculator | n = 29 | r = 2 | ar-1 = 2 | Qr-1 = 4 | Pr-1 = 5}} {{PartialQuotientCalculator | n = 29 | r = 3 | ar-1 = 1 | Qr-1 = 5 | Pr-1 = 3}} {{PartialQuotientCalculator | n = 29 | r = 4 | ar-1 = 1 | Qr-1 = 5 | Pr-1 = 2}} {{PartialQuotientCalculator | n = 29 | r = 5 | ar-1 = 2 | Qr-1 = 4 | Pr-1 = 3}} |} and the cycle is complete: :$\\sequence {2, 1, 1, 2, 10}$ {{qed}} Category:Examples of Continued Fractions Category:29 876zglxnx6xicqdyytjb2cwqplr12yr"}
+{"_id": "32694", "title": "Continued Fraction Expansion of Irrational Square Root/Examples/13/Convergents", "text": "Continued Fraction Expansion of Irrational Square Root/Examples/13/Convergents 0 52899 470657 470629 2020-05-26T06:50:08Z Prime.mover 59 wikitext text/x-wiki == Convergents to Continued Fraction Expansion of $\\sqrt {13}$ == <onlyinclude> The sequence of convergents to the continued fraction expansion of the square root of $13$ begins: :$\\dfrac 3 1, \\dfrac 4 1, \\dfrac 7 2, \\dfrac {11} 3, \\dfrac {18} 5, \\dfrac {119} {33}, \\dfrac {137} {38}, \\dfrac {256} {71}, \\dfrac {393} {109}, \\dfrac {649} {180}, \\ldots$ </onlyinclude> {{OEIS-Numerators|A041018}} {{OEIS-Denominators|A041019}} == Proof == Let $\\sqbrk {a_0, a_1, a_2, \\ldots}$ be its continued fraction expansion. Let $\\sequence {p_n}_{n \\mathop \\ge 0}$ and $\\sequence {q_n}_{n \\mathop \\ge 0}$ be its numerators and denominators. Then the $n$th convergent is $\\dfrac {p_n} {q_n}$. By definition: :$p_k = \\begin {cases} a_0 & : k = 0 \\\\ a_0 a_1 + 1 & : k = 1 \\\\ a_k p_{k - 1} + p_{k - 2} & : k > 1 \\end {cases}$ :$q_k = \\begin {cases} 1 & : k = 0 \\\\ a_1 & : k = 1 \\\\ a_k q_{k - 1} + q_{k - 2} & : k > 1 \\end {cases}$ From Continued Fraction Expansion of $\\sqrt {13}$: :$\\sqrt {13} = \\sqbrk {3, \\sequence {1, 1, 1, 1, 6} }$ Thus the convergents are assembled: {{ConvergentCalculator-Start | a0 = 3 | a1 = 1}} {{ConvergentCalculator | k = 2 | ak = 1 | pk-1 = 4 | pk-2 = 3 | qk-1 = 1 | qk-2 = 1}} {{ConvergentCalculator | k = 3 | ak = 1 | pk-1 = 7 | pk-2 = 4 | qk-1 = 2 | qk-2 = 1}} {{ConvergentCalculator | k = 4 | ak = 1 | pk-1 = 11 | pk-2 = 7 | qk-1 = 3 | qk-2 = 2 }} {{ConvergentCalculator | k = 5 | ak = 6 | pk-1 = 18 | pk-2 = 11 | qk-1 = 5 | qk-2 = 3}} {{ConvergentCalculator | k = 6 | ak = 1 | pk-1 = 119 | pk-2 = 18 | qk-1 = 33 | qk-2 = 5}} {{ConvergentCalculator | k = 7 | ak = 1 | pk-1 = 137 | pk-2 = 119 | qk-1 = 38 | qk-2 = 33}} {{ConvergentCalculator | k = 8 | ak = 1 | pk-1 = 256 | pk-2 = 137 | qk-1 = 71 | qk-2 = 38}} {{ConvergentCalculator | k = 9 | ak = 1 | pk-1 = 393 | pk-2 = 256 | qk-1 = 109 | qk-2 = 71}} |} {{qed}} Category:Continued Fractions Category:13 6k36tbcv7u9lpzztikxt314un6ggl5x"}
+{"_id": "32695", "title": "Continued Fraction Expansion of Irrational Square Root/Examples/29/Convergents", "text": "Continued Fraction Expansion of Irrational Square Root/Examples/29/Convergents 0 52904 470653 470635 2020-05-26T06:44:59Z Prime.mover 59 wikitext text/x-wiki == Convergents to Continued Fraction Expansion of $\\sqrt {29}$ == <onlyinclude> The sequence of convergents to the continued fraction expansion of the square root of $29$ begins: :$\\dfrac 5 1, \\dfrac {11} 2, \\dfrac {16} 3, \\dfrac {27} 5, \\dfrac {70} {13}, \\dfrac {727} {135}, \\dfrac {1524} {283}, \\dfrac {2251} {418}, \\dfrac {3775} {701}, \\dfrac {9801} {1820}, \\ldots$ </onlyinclude> {{OEIS-Numerators|A041046}} {{OEIS-Denominators|A041047}} == Proof == Let $\\sqbrk {a_0, a_1, a_2, \\ldots}$ be its continued fraction expansion. Let $\\sequence {p_n}_{n \\mathop \\ge 0}$ and $\\sequence {q_n}_{n \\mathop \\ge 0}$ be its numerators and denominators. Then the $n$th convergent is $p_n / q_n$. By definition: :$p_k = \\begin {cases} a_0 & : k = 0 \\\\ a_0 a_1 + 1 & : k = 1 \\\\ a_k p_{k - 1} + p_{k - 2} & : k > 1\\end {cases}$ :$q_k = \\begin {cases} 1 & : k = 0 \\\\ a_1 & : k = 1 \\\\ a_k q_{k - 1} + q_{k - 2} & : k > 1\\end {cases}$ From Continued Fraction Expansion of $\\sqrt {29}$: :$\\sqrt {29} = \\sqbrk {5, \\sequence {2, 1, 1, 2, 10} }$ Thus the convergents are assembled: {{ConvergentCalculator-Start | a0 = 5 | a1 = 2}} {{ConvergentCalculator | k = 2 | ak =  1 | pk-1 =   11 | pk-2 =    5 | qk-1 =   2 | qk-2 =   1}} {{ConvergentCalculator | k = 3 | ak =  1 | pk-1 =   16 | pk-2 =   11 | qk-1 =   3 | qk-2 =   2}} {{ConvergentCalculator | k = 4 | ak =  2 | pk-1 =   27 | pk-2 =   16 | qk-1 =   5 | qk-2 =   3}} {{ConvergentCalculator | k = 5 | ak = 10 | pk-1 =   70 | pk-2 =   27 | qk-1 =  13 | qk-2 =   5}} {{ConvergentCalculator | k = 6 | ak =  2 | pk-1 =  727 | pk-2 =   70 | qk-1 = 135 | qk-2 =  13}} {{ConvergentCalculator | k = 7 | ak =  1 | pk-1 = 1524 | pk-2 =  727 | qk-1 = 283 | qk-2 = 135}} {{ConvergentCalculator | k = 8 | ak =  1 | pk-1 = 2251 | pk-2 = 1524 | qk-1 = 418 | qk-2 = 283}} {{ConvergentCalculator | k = 9 | ak =  2 | pk-1 = 3775 | pk-2 = 2251 | qk-1 = 701 | qk-2 = 418}} |} {{qed}} Category:Continued Fractions Category:29 3yesbydecvf35st0npptdo09kpr9pum"}
+{"_id": "32696", "title": "Lifting The Exponent Lemma/Lemma", "text": "Lifting The Exponent Lemma/Lemma 0 52990 457792 312888 2020-03-27T08:04:40Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $x, y \\in \\Z$ be distinct integers. Let $p$ be an odd prime. Let: :$p \\divides x - y$ and: :$p \\nmid x y$. Then  :$\\map {\\nu_p} {x^p - y^p} = \\map {\\nu_p} {x - y} + 1$ </onlyinclude> where $\\nu_p$ denotes $p$-adic valuation. == Proof == Let $\\map {\\nu_p} {x - y} = k$. Then $x=p^k m + y$ where $p \\nmid m$. We have: {{begin-eqn}} {{eqn | l = x^p - y^p       | r = (p^k m + y)^p - y^p       | c =  }} {{eqn | r = \\sum_{i \\mathop = 0}^p \\paren {\\binom p i \\paren {p^k m}^{p - i} y^i} - y^p       | c = Binomial Theorem }} {{eqn | r = \\sum_{i \\mathop = 0}^{p - 2} \\paren {\\binom p i \\paren {p^k m}^{p - i} y^i} + \\binom p {p - 1} \\paren {p^k m} y^{p - 1}       | c = picking out the last two terms from the summation }} {{eqn | r = \\sum_{i \\mathop = 0}^{p - 2} \\paren {\\binom p i \\paren {p^k m}^{p - i} y^i} + p^{k + 1} m y^{p - 1}       | c =  }} {{end-eqn}} Note that all terms in the above expression have a factor of $p$ to the order at least $k+1$. So, $p^{k + 1} \\mid x^p - y^p$. Also note that all terms in the summation have a factor of $p$ to the order at least $k + 2$. But in the term $p^{k + 1} m y^{p - 1}$, since $p \\nmid m$ and $p \\nmid y$, we have: :$p^{k + 2} \\nmid p^{k + 1} m y^{p - 1}$ So: :$p^{k + 2} \\nmid x^p - y^p$ So by definition of $p$-adic valuation: :$\\map {\\nu_p} {x^p - y^p} = k + 1$ {{qed}} == Sources == * {{citation|date = July 1904|title = On the Integral Divisors of $a^n - b^n$|journal = Annals of Mathematics|abbr = Ann. Math.|volume = 5|startpage = 173|endpage = 180|jstor = 2007263|author = George David Birkhoff|author2 = Harry Schultz Vandiver|jstorcat = yes}}: Theorem $\\text{III}$. Category:P-adic Valuations d4e0h0xni7916lg5nyycr7wdwh3khto"}
+{"_id": "32697", "title": "Solutions of Pythagorean Equation/Sequence", "text": "Solutions of Pythagorean Equation/Sequence 0 53030 283072 283069 2017-01-24T23:19:35Z Prime.mover 59 wikitext text/x-wiki == Sequence == <onlyinclude> The sequence of solutions of the Pythagorean equation can be tabulated as follows: :$\\begin{array} {r r | r r | r r r | c} m & n & m^2 & n^2 & 2 m n & m^2 - n^2 & m^2 + n^2 \\\\ \\hline 2 & 1 & 4 & 1 & 4 & 3 & 5 & \\text{Primitive} \\\\ \\hline 3 & 1 & 9 & 1 & 6 & 8 & 10 \\\\ 3 & 2 & 9 & 4 & 12 & 5 & 13 & \\text{Primitive} \\\\ \\hline 4 & 1 & 16 & 1 & 8 & 15 & 17 & \\text{Primitive} \\\\ 4 & 2 & 16 & 4 & 16 & 12 & 20 \\\\ 4 & 3 & 16 & 9 & 24 & 7 & 25 & \\text{Primitive} \\\\ \\hline 5 & 1 & 25 & 1 & 10 & 24 & 26 \\\\ 5 & 2 & 25 & 4 & 20 & 21 & 29 & \\text{Primitive} \\\\ 5 & 3 & 25 & 9 & 30 & 16 & 34 \\\\ 5 & 4 & 25 & 16 & 40 & 9 & 41 & \\text{Primitive} \\\\ \\hline 6 & 1 & 36 & 1 & 12 & 35 & 37 & \\text{Primitive} \\\\ 6 & 2 & 36 & 4 & 24 & 32 & 40 \\\\ 6 & 3 & 36 & 9 & 36 & 27 & 45 \\\\ 6 & 4 & 36 & 16 & 48 & 20 & 52 \\\\ 6 & 5 & 36 & 25 & 60 & 11 & 61 & \\text{Primitive} \\\\ \\hline 7 & 1 & 49 & 1 & 14 & 48 & 50 \\\\ 7 & 2 & 49 & 4 & 28 & 45 & 53 & \\text{Primitive} \\\\ 7 & 3 & 49 & 9 & 42 & 40 & 58 \\\\ 7 & 4 & 49 & 16 & 56 & 33 & 65 & \\text{Primitive} \\\\ 7 & 5 & 49 & 25 & 70 & 24 & 74 \\\\ 7 & 6 & 49 & 36 & 84 & 13 & 85 & \\text{Primitive} \\\\ \\hline \\end{array}$ </onlyinclude> Category:Pythagorean Triples Category:Solutions of Pythagorean Equation dqkjdu0c3nvufai4wklffejw24x8wb2"}
+{"_id": "32698", "title": "Pythagorean Triangle/Examples/5-12-13", "text": "Pythagorean Triangle/Examples/5-12-13 0 53032 478775 478750 2020-07-18T12:13:55Z Prime.mover 59 wikitext text/x-wiki == Example of Primitive Pythagorean Triangle == <onlyinclude> The triangle whose sides are of length $5$, $12$ and $13$ is a primitive Pythagorean triangle. </onlyinclude> :400px == Proof == {{begin-eqn}} {{eqn | l = 5^2 + 12^2       | r = 25 + 144       | c =  }} {{eqn | r = 169       | c =  }} {{eqn | r = 13^2       | c =  }} {{end-eqn}} It follows by Pythagoras's Theorem that $5$, $12$ and $13$ form a Pythagorean triple. Note that $5$ and $12$ are coprime. Hence, by definition, $5$, $12$ and $13$ form a primitive Pythagorean triple. The result follows by definition of a primitive Pythagorean triangle. {{qed}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Smallest Pythagorean Triangle is 3-4-5|next = Pythagorean Triangle/Examples/6-8-10}}: $13$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Smallest Pythagorean Triangle is 3-4-5|next = Pythagorean Triangle/Examples/6-8-10}}: $13$ Category:Examples of Pythagorean Triangles 3tbh5ki2lweolv39z8bwoob7o159qdm"}
+{"_id": "32699", "title": "Dirichlet Series Convergence Lemma/Lemma", "text": "Dirichlet Series Convergence Lemma/Lemma 0 53046 357559 350951 2018-05-28T17:18:39Z AliceInNumberland 3357 Fixed the proof, it follows the broad strokes of the cited source, with a few modifications to make it more elementary, as that is the objective of proving it in the ordinary sense wikitext text/x-wiki == Lemma to Dirichlet Series Convergence Lemma == <onlyinclude> Let $\\displaystyle f \\left({s}\\right) = \\sum_{n \\mathop = 1}^\\infty \\frac{a_n}{n^s}$ be a Dirichlet series. Suppose that for some $s_0 = \\sigma_0 + i t_0 \\in \\C$, $f \\left({s_0}\\right)$ has bounded partial sums: :$(1): \\quad \\displaystyle \\left\\vert{\\sum_{n \\mathop = 1}^N a_n n^{-s_0} }\\right\\vert \\le M$ for some $M \\in \\R$ and all $N \\ge 1$. Then for every $s = \\sigma + i t \\in \\C$ with $\\sigma > \\sigma_0$: :$\\displaystyle \\left\\vert{\\sum_{n \\mathop = m}^N a_n n^{-s} }\\right\\vert \\le 2 M m^{\\sigma_0 - \\sigma} \\left( 1+ \\frac {\\left\\vert s - s_0\\right\\vert} {\\sigma-\\sigma_0} \\right)$ </onlyinclude> == Proof == We have the Summation by Parts formula: :$\\displaystyle \\sum_{n \\mathop = m}^N f_n g_n = f_N G_N - f_m G_{m-1} - \\sum_{n \\mathop = m}^{N - 1} G_n \\left({f_{n+1} - f_n}\\right)$ We let $g_n = a_n n^{-s_0}$ and $f_n = n^{s_0 - s}$.  For $N \\ge 1$, the quantities $G_N$ are the partial sums $(1)$ Thus $G_N \\le M$ for all $N \\ge 1$. We have: {{begin-eqn}} {{eqn | l = \\left\\vert{\\sum_{n \\mathop = m}^N \\frac {a_n} {n^s} }\\right\\vert       | r = \\left\\vert{\\sum_{n \\mathop = m}^N f_n g_n}\\right\\vert }} {{eqn | r = \\left\\vert{f_N G_N}\\right\\vert + \\left\\vert{f_m G_{m-1} }\\right\\vert + \\sum_{n \\mathop = m}^{N - 1} \\left\\vert{G_n \\left({f_{n+1} - f_n}\\right)}\\right\\vert       | o = \\le       | c = using partial summation and the Triangle Inequality }} {{eqn | r = M \\left\\vert{N^{s_0 - s} }\\right\\vert + M \\left\\vert{m^{s_0 - s} }\\right\\vert + M \\sum_{n \\mathop = m}^{N - 1} \\left\\vert{\\left({\\left({n + 1}\\right)^{s_0 - s} - n^{s_0 - s} }\\right) }\\right\\vert       | o = \\le       | c = using the given bound on the partial sums }} {{eqn |r = M N^{\\sigma_0 - \\sigma}  + M m^{\\sigma_0 - \\sigma} + M \\sum_{n \\mathop = m}^{N - 1} \\left\\vert{ \\left(s - s_0 \\right) \\int_n^{n+1} t^{s_0 - s -1} }\\right\\vert }} {{eqn |o= \\le       |r = M N^{\\sigma_0 - \\sigma}  + M m^{\\sigma_0 - \\sigma} + M \\sum_{n \\mathop = m}^{N - 1} \\left\\vert{ s - s_0 }\\right\\vert \\int_n^{n+1} \\left\\vert{ t^{s_0 - s -1} }\\right\\vert       |c = Modulus of Complex Integral }} {{eqn |r =  M N^{\\sigma_0 - \\sigma}  + M m^{\\sigma_0 - \\sigma} + M \\sum_{n \\mathop = m}^{N - 1} \\left\\vert{s - s_0 }\\right\\vert \\int_n^{n+1}  t^{\\sigma_0 - \\sigma -1} }} {{eqn |r =  M N^{\\sigma_0 - \\sigma}  + M m^{\\sigma_0 - \\sigma} + M \\left\\vert{ s - s_0 }\\right\\vert \\int_m^{N}  t^{\\sigma_0 - \\sigma -1} }} {{eqn |r =  M N^{\\sigma_0 - \\sigma}  + M m^{\\sigma_0 - \\sigma} + M \\frac{\\left\\vert{ s - s_0 }\\right\\vert}{\\sigma - \\sigma_0} \\left( m^{\\sigma_0 - \\sigma}- N^{\\sigma_0 - \\sigma} \\right) }} {{eqn |o= \\le       |r =  M N^{\\sigma_0 - \\sigma}  + M m^{\\sigma_0 - \\sigma} + M \\frac{\\left\\vert{ s - s_0 }\\right\\vert}{\\sigma - \\sigma_0} \\left( m^{\\sigma_0 - \\sigma} + N^{\\sigma_0 - \\sigma} \\right) }} {{end-eqn}} Finally, because $N \\ge m$ and $\\sigma_0 - \\sigma < 0$, we have: :$ N^{\\sigma_0 - \\sigma}  +  m^{\\sigma_0 - \\sigma} \\le 2 m^{\\sigma_0 - \\sigma}$ Therefore: {{begin-eqn}} {{eqn |l= \\left\\vert{\\sum_{n \\mathop = m}^N a_n n^{-s} }\\right\\vert       |o= \\le       |r= M N^{\\sigma_0 - \\sigma}  + M m^{\\sigma_0 - \\sigma} + M \\frac{\\left\\vert{ s - s_0 }\\right\\vert}{\\sigma - \\sigma_0} \\left( m^{\\sigma_0 - \\sigma} + N^{\\sigma_0 - \\sigma} \\right) }} {{eqn |o=\\leq       |r=   2 M m^{\\sigma_0 - \\sigma} + 2 M \\frac{\\left\\vert{ s - s_0 }\\right\\vert}{\\sigma - \\sigma_0} m^{\\sigma_0 - \\sigma} }} {{eqn |r = 2 M m^{\\sigma_0 - \\sigma} \\left(1 + \\frac{\\left\\vert{ s - s_0 }\\right\\vert}{\\sigma - \\sigma_0} \\right) }} {{end-eqn}} Hence the result. {{qed|lemma}} == Sources == {{SourceReview|wrt Dirichlet Series Convergence Lemma}} * {{BookReference|Introduction to Analytic Number Theory|1976|Tom M. Apostol}}: $\\S 11.6$: Lemma $2$, Theorem $11.8$ Category:Dirichlet Series jqn0qz0s10dk2k2et5rlpomi63xtihe"}
+{"_id": "32700", "title": "Pythagorean Triangle/Examples/6-8-10", "text": "Pythagorean Triangle/Examples/6-8-10 0 53055 478773 478752 2020-07-18T12:12:27Z Prime.mover 59 wikitext text/x-wiki == Example of Pythagorean Triangle == <onlyinclude> The triangle whose sides are of length $6$, $8$ and $10$ is a Pythagorean triangle. This is not a primitive Pythagorean triangle. </onlyinclude> :300px == Proof == {{begin-eqn}} {{eqn | l = 6^2 + 8^2       | r = 2^2 \\times 3^2 + 2^2 \\times 4^2       | c =  }} {{eqn | r = 4 \\times \\paren {9 + 16}       | c =  }} {{eqn | r = 4 \\times 25       | c =  }} {{eqn | r = 2^2 \\times 5^2       | c =  }} {{eqn | r = 10^2       | c =  }} {{end-eqn}} It follows by Pythagoras's Theorem that $6$, $8$ and $10$ form a Pythagorean triple. Note that $6$ and $8$ are not coprime as $\\gcd \\set {6, 8} = 2$. Hence, by definition, $6$, $8$ and $10$ do not form a primitive Pythagorean triple. The result follows by definition of a primitive Pythagorean triangle. {{qed}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Pythagorean Triangle/Examples/5-12-13|next = Pythagorean Triangle/Examples/7-24-25}}: $13$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Pythagorean Triangle/Examples/5-12-13|next = Pythagorean Triangle/Examples/7-24-25}}: $13$ Category:Examples of Pythagorean Triangles fxd28zo105jua0jquhuvakzl9v7vg0d"}
+{"_id": "32701", "title": "Convergence of Complex Sequence in Polar Form/Corollary", "text": "Convergence of Complex Sequence in Polar Form/Corollary 0 53060 407029 362409 2019-06-08T13:31:11Z Prime.mover 59 wikitext text/x-wiki == Corollary to Convergence of Complex Sequence in Polar Form == <onlyinclude> Let $\\left\\langle{z_n}\\right\\rangle$ be a sequence of nonzero complex numbers. Let $z \\ne 0$ be a complex number with modulus $r$ and argument $\\theta$. Let $I$ be a real interval of length at most $2 \\pi$ that contains $\\theta$. Suppose $\\theta$ is not an endpoint of $I$. Suppose each $z_n$ admits an argument $\\theta_n \\in I$. Let $r_n$ be the modulus of $z_n$. Then $z_n$ converges to $z$ {{iff}} $r_n$ converges to $r$ and $\\theta_n$ converges to $\\theta$. </onlyinclude> == Proof == Suppose $r_n \\to r$ and $\\theta_n \\to \\theta$. Then by Convergence of Complex Sequence in Polar Form, $z_n \\to z$. Conversely, suppose $z_n \\to z$. By Convergence of Complex Sequence in Polar Form, we have: :$(1): \\quad r_n \\to r$ :$(2): \\quad$ There exists a sequence $\\left\\langle{k_n}\\right\\rangle$ of integers such that $\\theta_n + 2 k_n \\pi$ converges to $\\theta$. It remains to be proved that $\\theta_n \\to \\theta$. Let $N \\in \\N$ such that $\\left\\vert{\\theta_N + 2 k_N \\pi - \\theta}\\right\\vert \\le \\pi / 2$ for all $n \\ge N$. By the Triangle Inequality for Real Numbers: :$\\left\\vert{2 k_n \\pi - 2 k_N \\pi}\\right\\vert \\le \\left\\vert{\\theta_n + 2 k_n \\pi - \\theta}\\right\\vert + \\left\\vert{\\theta_N + 2 k_N \\pi - \\theta}\\right\\vert \\le \\pi$ for all $n \\ge N$. Thus $\\left\\vert{k_n - k_N}\\right\\vert \\le 1 / 2$, so $k_n = k_N$ for all $n \\ge N$. By the Triangle Inequality for Real Numbers: :$\\left\\vert{2 \\pi k_N}\\right\\vert \\le \\left\\vert{\\theta_n - \\theta}\\right\\vert + \\left\\vert{\\theta_n + 2 \\pi k_N - \\theta}\\right\\vert$ for all $n \\in \\N$. Because $\\theta_n \\in I$ and $\\theta$ is not an endpoint of $I$: :$\\left\\vert{\\theta_n - \\theta}\\right\\vert < 2 \\pi$ for all $n \\in \\N$. Because $\\theta_n + 2 \\pi k_N - \\theta \\to 0$: :$\\left\\vert{2 \\pi k_N}\\right\\vert < 2 \\pi$ Thus $k_n = 0$ for all $n \\ge N$. Thus $\\theta_n \\to \\theta$. {{qed}} Category:Complex Analysis i7cch7fbmgc5k27c7k67jp1ygqqypw0"}
+{"_id": "32702", "title": "Pythagorean Triangle/Examples/693-1924-2045", "text": "Pythagorean Triangle/Examples/693-1924-2045 0 53074 478760 478754 2020-07-18T12:06:32Z Prime.mover 59 wikitext text/x-wiki == Example of Primitive Pythagorean Triangle == <onlyinclude> The triangle whose sides are of length $693$, $1924$ and $2045$ is a primitive Pythagorean triangle. </onlyinclude> :700px == Proof == {{begin-eqn}} {{eqn | l = 693^2 + 1924^2       | r = 480 \\, 249 + 3 \\, 701 \\, 776       | c =  }} {{eqn | r = 4 \\, 182 \\, 025       | c =  }} {{eqn | r = 2045^2       | c =  }} {{end-eqn}} It follows by Pythagoras's Theorem that $693$, $1924$ and $2045$  form a Pythagorean triple. Note that $693$ and $1924$ are coprime. Hence, by definition, $693$, $1924$ and $2045$ form a primitive Pythagorean triple. The result follows by definition of a primitive Pythagorean triangle. {{qed}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Pythagorean Triangle whose Area is Half Perimeter|next = Pythagorean Triangles whose Areas are Repdigit Numbers}}: $13$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Pythagorean Triangle whose Area is Half Perimeter|next = Pythagorean Triangles whose Areas are Repdigit Numbers}}: $13$ Category:Examples of Pythagorean Triangles h1hz8c88k0y8dp6iz2daoaftzcivnpz"}
+{"_id": "32703", "title": "693", "text": "693 0 53097 478771 306362 2020-07-18T12:10:45Z Prime.mover 59 wikitext text/x-wiki {{NumberPageLink|prev = 692|next = 694}} == Number == $693$ ('''six hundred and ninety three''') is: :$3^2 \\times 7 \\times 11$ :One of the legs of the Pythagorean triangle $693-1924-2045$, whose area is $666 \\, 666$. == Also see == * Pythagorean Triangles whose Areas are Repdigit Numbers Category:Specific Numbers Category:693 lrnrx7zujqpwoskv5ajj4nun7p1y6me"}
+{"_id": "32704", "title": "1924", "text": "1924 0 53099 478770 303430 2020-07-18T12:10:34Z Prime.mover 59 wikitext text/x-wiki {{NumberPageLink|prev = 1923|next = 1925}} == Number == $1924$ ('''one thousand nine hundred and twenty four''') is: :$2^2 \\times 13 \\times 37$ :One of the legs of the Pythagorean triangle $693-1924-2045$, whose area is $666 \\, 666$. == Also see == * Pythagorean Triangles whose Areas are Repdigit Numbers Category:Specific Numbers Category:1924 ny180aji87241vumpmtwksdgvejdo3i"}
+{"_id": "32705", "title": "111", "text": "111 0 53106 478342 461296 2020-07-16T05:33:30Z Prime.mover 59 wikitext text/x-wiki {{NumberPageLink|prev = 110|next = 112}} == Number == $111$ ('''one hundred and eleven''') is: :$3 \\times 37$ :The magic constant of the smallest prime magic square. :The $3$rd repuint after $1$, $11$ :The $4$th of the $17$ positive integers for which the value of the Euler $\\phi$ function is $72$: ::$73$, $91$, $95$, $111$, $117$, $135$, $146$, $148$, $152$, $182$, $190$, $216$, $222$, $228$, $234$, $252$, $270$ :The $6$th positive integer after $1$, $2$, $7$, $11$, $101$ whose cube is palindromic: ::$111^3 = 1 \\, 367 \\, 631$ :The magic constant of a magic square of order $6$, after $1$, $(5)$, $15$, $34$, $65$: ::$111 = \\displaystyle \\dfrac 1 6 \\sum_{k \\mathop = 1}^{6^2} k = \\dfrac {6 \\paren {6^2 + 1} } 2$ :The $7$th palindromic lucky number: ::$1$, $3$, $7$, $9$, $33$, $99$, $111$, $\\ldots$ :The $8$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $22$, $101$ whose square is also palindromic integer ::$111^2 = 12 \\, 321$ :The $15$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$, $36$: ::$111 = 111 \\times 1 = 111 \\times \\paren {1 \\times 1 \\times 1}$ :The $24$th lucky number: ::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $51$, $63$, $67$, $73$, $75$, $79$, $87$, $93$, $99$, $105$, $111$, $\\ldots$ :The $35$th semiprime: ::$111 = 3 \\times 37$ == Also see == * Magic Constant of Smallest Prime Magic Square * {{NumberPageLink|prev = 11|next = 1111|type = Repunit|cat = Repunits}} * {{NumberPageLink|prev = 36|next = 112|type = Zuckerman Number|cat = Zuckerman Numbers}} * {{NumberPageLink|prev = 65|next = 175|result = Magic Constant of Magic Square}} * {{NumberPageLink|prev = 95|next = 117|result = Numbers with Euler Phi Value of 72}} * {{NumberPageLink|prev = 99|next = 141|result = Sequence of Palindromic Lucky Numbers}} * {{NumberPageLink|prev = 101|next = 121|result = Square of Small-Digit Palindromic Number is Palindromic}} * {{NumberPageLink|prev = 101|next = 1001|result = Sequence of Integers whose Cube is Palindromic}} * {{NumberPageLink|prev = 105|next = 115|type = Lucky Number|cat = Lucky Numbers}} * {{NumberPageLink|prev = 106|next = 115|type = Semiprime Number|cat = Semiprimes}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Magic Constant of Smallest Prime Magic Square|next = 112/Historical Note}}: $111$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Magic Constant of Smallest Prime Magic Square|next = Sequence of Palindromic Lucky Numbers}}: $111$ Category:Specific Numbers Category:111 nj1uqqd71djtmooa94n27t3kops38hp"}
+{"_id": "32706", "title": "1001", "text": "1001 0 53108 478349 461059 2020-07-16T05:49:34Z Prime.mover 59 wikitext text/x-wiki {{NumberPageLink|prev = 1000|next = 1002}} == Number == $1001$ ('''one thousand and one''') is: :$7 \\times 11 \\times 13$ :The $4$th pentagonal number after $1$, $5$, $22$ which is also palindromic: ::$1001 = \\displaystyle \\sum_{k \\mathop = 1}^{26} \\paren {3 k - 2} = \\dfrac {26 \\paren {3 \\times 26 - 1} } 2$ :The $7$th positive integer after $1$, $2$, $7$, $11$, $101$, $111$ whose cube is palindromic: ::$1001^3 = 1 \\, 003 \\, 003 \\, 001$ :The $11$th pentatope number after $1$, $5$, $15$, $35$, $70$, $126$, $210$, $330$, $495$, $715$: ::$1001 = \\displaystyle \\sum_{k \\mathop = 1}^{11} \\dfrac {k \\paren {k + 1} \\paren {k + 2} } 6 = \\dfrac {11 \\paren {11 + 1} \\paren {11 + 2} \\paren {11 + 3} } {24}$ :The $12$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $22$, $101$, $111$, $121$, $202$, $212$ whose square is also palindromic integer ::$1001^2 = 1 \\, 002 \\, 001$ :The $26$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $\\ldots$, $477$, $532$, $590$, $651$, $715$, $782$, $852$, $925$: ::$1001 = \\displaystyle \\sum_{k \\mathop = 1}^{26} \\paren {3 k - 2} = \\dfrac {26 \\paren {3 \\times 26 - 1} } 2$ :The $51$st generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\\ldots$, $610$, $651$, $672$, $715$, $737$, $782$, $805$, $852$, $876$, $925$, $950$: ::$1001 = \\displaystyle \\sum_{k \\mathop = 1}^{26} \\paren {3 k - 2} = \\dfrac {26 \\paren {3 \\times 26 - 1} } 2$ == Also see == * {{NumberPageLink|prev = 22|next = 2882|result = Sequence of Palindromic Pentagonal Numbers}} * {{NumberPageLink|prev = 101|next = 10,001|result = Prime Factors of One More than Power of 10}} * {{NumberPageLink|prev = 111|next = 2201|result = Sequence of Integers whose Cube is Palindromic}} * {{NumberPageLink|prev = 212|next = 1111|result = Square of Small-Digit Palindromic Number is Palindromic}} * {{NumberPageLink|prev = 715|next = 1365|type = Pentatope Number|cat = Pentatope Numbers}} * {{NumberPageLink|prev = 925|next = 1080|type = Pentagonal Number|cat = Pentagonal Numbers}} * {{NumberPageLink|prev = 950|next = 1027|type = Generalized Pentagonal Number|cat = Generalized Pentagonal Numbers}} == Historical Note == {{:1001/Historical Note}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = 1000|next = Divisibility Test for 7, 11 and 13}}: $1001$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = 1000|next = Divisibility Test for 7, 11 and 13}}: $1001$ Category:Specific Numbers Category:1001 25posi7kny0a9yy14labtbcg1ve4mdt"}
+{"_id": "32707", "title": "Brahmagupta-Fibonacci Identity/General", "text": "Brahmagupta-Fibonacci Identity/General 0 53203 283820 283813 2017-01-29T09:29:18Z Prime.mover 59 wikitext text/x-wiki == General version of Brahmagupta-Fibonacci Identity == <onlyinclude> Let $a, b, c, d, n$ be numbers. :$\\left({a^2 + n b^2}\\right) \\left({c^2 + n d^2}\\right) = \\left({a c + n b d}\\right)^2 + n \\left({a d - b c}\\right)^2$ </onlyinclude> === Corollary === {{:Brahmagupta-Fibonacci Identity/General/Corollary}} === Extension === {{:Brahmagupta-Fibonacci Identity/General/Extension}} == Proof == {{begin-eqn}} {{eqn | o =        | r = \\left({a c + n b d}\\right)^2 + n \\left({a d - b c}\\right)^2       | c =  }} {{eqn | r = \\left({\\left({a c}\\right)^2 + 2 \\left({a c}\\right) \\left({n b d}\\right) + \\left({n b d}\\right)^2}\\right) + n \\left({\\left({a d}\\right)^2 - 2 \\left({a b}\\right) \\left({c d}\\right) + \\left({b c}\\right)^2}\\right)       | c = Square of Sum, Square of Difference }} {{eqn | r = a^2 c^2 + 2 n a b c d + n^2 b^2 d^2 + n a^2 d^2 - 2 n a b c d + n b^2 c^2       | c = multiplying out }} {{eqn | r = a^2 c^2 + n a^2 d^2 + n b^2 c^2 + n^2 b^2 d^2       | c = simplifying }} {{eqn | r = \\left({a^2 + n b^2}\\right) \\left({c^2 + n d^2}\\right)       | c =  }} {{end-eqn}} {{qed}} {{Namedfor|Brahmagupta|name2 = Leonardo Fibonacci}} Category:Brahmagupta-Fibonacci Identity 1usmy94pvju94p1q87n512l2liguijm"}
+{"_id": "32708", "title": "Brahmagupta-Fibonacci Identity/Extension/General", "text": "Brahmagupta-Fibonacci Identity/Extension/General 0 53211 437759 283952 2019-12-04T16:28:06Z Prime.mover 59 wikitext text/x-wiki == Extension to Brahmagupta-Fibonacci Identity == <onlyinclude> Let $a_1, a_2, \\ldots, a_n, b_1, b_2, \\ldots, b_n, m$ be integers. Then: :$\\displaystyle \\prod_{j \\mathop = 1}^n \\paren { {a_j}^2 + m {b_j}^2} = c^2 + m d^2$ for some $c, d \\in \\Z$. That is: the set of all integers of the form $a^2 + m b^2$ is closed under multiplication. </onlyinclude> == Proof == The proof proceeds by induction. For all $n \\in \\Z_{> 0}$, let $\\map P n$ be the proposition: :$\\displaystyle \\prod_{j \\mathop = 1}^n \\paren { {a_j}^2 + m {b_j}^2} = c^2 + m d^2$ for some $c, d \\in \\Z$. $\\map P 1$ is the trivial case: {{begin-eqn}} {{eqn | l = \\prod_{j \\mathop = 1}^1 \\paren { {a_j}^2 + m {b_j}^2}       | r = {a_1}^2 + m {b_1}^2       | c =  }} {{eqn | r = c^2 + m d^2       | c = setting $c = a_1$ and $d = b_1$ }} {{end-eqn}} Thus $\\map P 1$ is seen to hold. === Basis for the Induction === $\\map P 2$ is the case: {{begin-eqn}} {{eqn | l = \\prod_{j \\mathop = 1}^2 \\paren { {a_j}^2 + m {b_j}^2}       | r = \\paren { {a_1}^2 + m {b_1}^2} \\paren { {a_2}^2 + m {b_2}^2}       | c =  }} {{eqn | r = \\paren {a_1 a_2 + m b_1 b_2}^2 + m \\paren {a_1 b_2 - b_1 a_2}^2       | c = General Brahmagupta-Fibonacci Identity }} {{eqn | r = c^2 + m d^2       | c = setting $c = a_1 a_2 + m b_1 b_2$ and $d = a_1 b_2 - b_1 a_2$ }} {{end-eqn}} Thus $\\map P 2$ is seen to hold. This is the basis for the induction. === Induction Hypothesis === Now it needs to be shown that, if $\\map P k$ is true, where $k \\ge 2$, then it logically follows that $\\map P {k + 1}$ is true. So this is the induction hypothesis: :$\\displaystyle \\prod_{j \\mathop = 1}^k \\paren { {a_j}^2 + m {b_j}^2} = c^2 + m d^2$ for some $c, d \\in \\Z$. from which it is to be shown that: :$\\displaystyle \\prod_{j \\mathop = 1}^{k + 1} \\paren { {a_j}^2 + m {b_j}^2} = c^2 + m d^2$ for some $c, d \\in \\Z$. === Induction Step === This is the induction step: {{begin-eqn}} {{eqn | l = \\prod_{j \\mathop = 1}^{k + 1} \\paren { {a_j}^2 + m {b_j}^2}       | r = \\prod_{j \\mathop = 1}^k \\paren { {a_j}^2 + m {b_j}^2} \\paren { {a_{k + 1} }^2 + m {b_{k + 1} }^2}       | c =  }} {{eqn | r = \\paren { {c'}^2 + m {d'}^2} \\paren { {a_{k + 1} }^2 + m {b_{k + 1} }^2}       | c = Induction Hypothesis: for some $c', d' \\in \\Z$ }} {{eqn | r = c^2 + m d^2       | c = Basis for the Induction: for some $c, d \\in \\Z$ }} {{end-eqn}} So $\\map P k \\implies \\map P {k + 1}$ and the result follows by the Principle of Mathematical Induction. Therefore, for all $n \\in \\Z_{> 0}$: :$\\displaystyle \\prod_{j \\mathop = 1}^n \\paren { {a_j}^2 + m {b_j}^2} = c^2 + m d^2$ for some $c, d \\in \\Z$. {{qed}} Category:Brahmagupta-Fibonacci Identity 6fkhzc2kgy79r4dx5r34g6nzf2kxisq"}
+{"_id": "32709", "title": "Brahmagupta-Fibonacci Identity/Extension/Proof 2", "text": "Brahmagupta-Fibonacci Identity/Extension/Proof 2 0 53226 283951 283948 2017-01-29T15:26:25Z Prime.mover 59 wikitext text/x-wiki == Extension to Brahmagupta-Fibonacci Identity == {{:Brahmagupta-Fibonacci Identity/Extension}} == Proof == <onlyinclude> The proof proceeds by induction. For all $n \\in \\Z_{> 0}$, let $P \\left({n}\\right)$ be the proposition: :$\\displaystyle \\prod_{j \\mathop = 1}^n \\left({ {a_j}^2 + {b_j}^2}\\right) = c^2 + d^2$ for some $c, d \\in \\Z$. $P \\left({1}\\right)$ is the trivial case: {{begin-eqn}} {{eqn | l = \\prod_{j \\mathop = 1}^1 \\left({ {a_j}^2 + {b_j}^2}\\right)       | r = {a_1}^2 + {b_1}^2       | c =  }} {{eqn | r = c^2 + d^2       | c = setting $c = a_1$ and $d = b_1$ }} {{end-eqn}} Thus $P \\left({1}\\right)$ is seen to hold. === Basis for the Induction === $P \\left({2}\\right)$ is the case: {{begin-eqn}} {{eqn | l = \\prod_{j \\mathop = 1}^2 \\left({ {a_j}^2 + {b_j}^2}\\right)       | r = \\left({ {a_1}^2 + {b_1}^2}\\right) \\left({ {a_2}^2 + {b_2}^2}\\right)       | c =  }} {{eqn | r = \\left({a_1 a_2 + b_1 b_2}\\right)^2 + \\left({a_1 b_2 - b_1 a_2}\\right)^2       | c = Brahmagupta-Fibonacci Identity }} {{eqn | r = c^2 + d^2       | c = setting $c = a_1 a_2 + b_1 b_2$ and $d = a_1 b_2 - b_1 a_2$ }} {{end-eqn}} Thus $P \\left({2}\\right)$ is seen to hold. This is the basis for the induction. === Induction Hypothesis === Now it needs to be shown that, if $P \\left({k}\\right)$ is true, where $k \\ge 2$, then it logically follows that $P \\left({k + 1}\\right)$ is true. So this is the induction hypothesis: :$\\displaystyle \\prod_{j \\mathop = 1}^k \\left({ {a_j}^2 + {b_j}^2}\\right) = c^2 + d^2$ for some $c, d \\in \\Z$. from which it is to be shown that: :$\\displaystyle \\prod_{j \\mathop = 1}^{k + 1} \\left({ {a_j}^2 + {b_j}^2}\\right) = c^2 + d^2$ for some $c, d \\in \\Z$. === Induction Step === This is the induction step: {{begin-eqn}} {{eqn | l = \\prod_{j \\mathop = 1}^{k + 1} \\left({ {a_j}^2 + {b_j}^2}\\right)       | r = \\prod_{j \\mathop = 1}^k \\left({ {a_j}^2 + {b_j}^2}\\right) \\left({ {a_{k + 1} }^2 + {b_{k + 1} }^2}\\right)       | c =  }} {{eqn | r = \\left({ {c'}^2 + {d'}^2}\\right) \\left({ {a_{k + 1} }^2 + {b_{k + 1} }^2}\\right)       | c = Induction Hypothesis: for some $c', d' \\in \\Z$ }} {{eqn | r = c^2 + m d^2       | c = Basis for the Induction: for some $c, d \\in \\Z$ }} {{end-eqn}} So $P \\left({k}\\right) \\implies P \\left({k + 1}\\right)$ and the result follows by the Principle of Mathematical Induction. Therefore, for all $n \\in \\Z_{> 0}$: :$\\displaystyle \\prod_{j \\mathop = 1}^n \\left({ {a_j}^2 + {b_j}^2}\\right) = c^2 + d^2$ for some $c, d \\in \\Z$. {{qed}} </onlyinclude> Category:Brahmagupta-Fibonacci Identity ezs3g9lp2x0ml1h4z22zb4yg5j92d8a"}
+{"_id": "32710", "title": "Magic Square/Examples/Order 4", "text": "Magic Square/Examples/Order 4 0 53567 310268 300669 2017-08-10T05:40:04Z Prime.mover 59 wikitext text/x-wiki == Examples of Order $4$ Magic Squares == <onlyinclude> There are many order $4$ magic squares. ==== Dürer's Order $4$ Magic Square ==== This is one of the more famous ones, due to {{AuthorRef|Albrecht Dürer}}: {{:Magic Square/Examples/Order 4/Dürer}} ==== Moessner's Order $4$ Magic Square ==== This one, created by {{AuthorRef|Alfred Moessner}}, has extra interesting properties: {{:Magic Square/Examples/Order 4/Alfred Moessner}}</onlyinclude> == Also see == * Magic Constant of Order 4 Magic Square * Number of Magic Squares of Order 4 Category:Magic Squares q21ghuew2eagljs5i7h9xx2q5uuvfhz"}
+{"_id": "32711", "title": "Magic Square/Examples/Order 4/Alfred Moessner", "text": "Magic Square/Examples/Order 4/Alfred Moessner 0 53573 381665 381661 2018-12-12T23:57:37Z Prime.mover 59 wikitext text/x-wiki == Example of Order $4$ Magic Square == This example of an order $4$ magic square is due to {{AuthorRef|Alfred Moessner}}: <onlyinclude> :$\\begin{array}{|c|c|c|c|} \\hline 12 & 13 & 1 & 8 \\\\ \\hline 6 & 3 & 15 & 10 \\\\ \\hline 7 & 2 & 14 & 11 \\\\ \\hline 9 & 16 & 4 & 5 \\\\ \\hline \\end{array}$ </onlyinclude> == Also see == * Properties of Moessner's Order $4$ Magic Square == Historical Note == {{:Magic Square/Examples/Order 4/Alfred Moessner/Historical Note}} == Sources == * {{citation|author = Alfred Moessner|title = A Curious Magic Square|journal = Scripta Mathematica|volume = 13|date = 1947?|startpage = ???|endpage = ???}} * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Magic Square/Examples/Order 4/Dürer/Historical Note|next = Magic Square/Examples/Order 4/Alfred Moessner/Historical Note}}: $16$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Magic Square/Examples/Order 4/Dürer/Historical Note|next = Magic Square/Examples/Order 4/Alfred Moessner/Historical Note}}: $16$ Category:Magic Squares 30gjmsv91qxdg3xjo5bcb8iiqyk24oo"}
+{"_id": "32712", "title": "Euler Lucky Number/Examples/41", "text": "Euler Lucky Number/Examples/41 0 53646 382719 291883 2018-12-17T06:43:57Z Prime.mover 59 wikitext text/x-wiki == Example of Euler Lucky Number == <onlyinclude> The expression: :$n^2 + n + 41$ yields primes for $n = 0$ to $n = 39$. It also generates the same set of primes for $n = -1 \\to n = -40$. These are not the only primes generated by this formula. No other quadratic function of the form $x^2 + a x + b$, where $a, b \\in \\Z_{>0}$ and $a, b < 10000$ generates a longer sequence of primes. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = 0^2 + 0 + 41       | r = 0 + 0 + 41       | rr= = 41       | c = which is prime }} {{eqn | l = 1^2 + 1 + 41       | r = 1 + 1 + 41       | rr= = 43       | c = which is prime }} {{eqn | l = 2^2 + 2 + 41       | r = 4 + 2 + 41       | rr= = 47       | c = which is prime }} {{eqn | l = 3^2 + 3 + 41       | r = 9 + 3 + 41       | rr= = 53       | c = which is prime }} {{eqn | l = 4^2 + 4 + 41       | r = 16 + 4 + 41       | rr= = 61       | c = which is prime }} {{eqn | l = 5^2 + 5 + 41       | r = 25 + 5 + 41       | rr= = 71       | c = which is prime }} {{eqn | l = 6^2 + 6 + 41       | r = 36 + 6 + 41       | rr= = 83       | c = which is prime }} {{eqn | l = 7^2 + 7 + 41       | r = 49 + 7 + 41       | rr= = 97       | c = which is prime }} {{eqn | l = 8^2 + 8 + 41       | r = 64 + 8 + 41       | rr= = 113       | c = which is prime }} {{eqn | l = 9^2 + 9 + 41       | r = 81 + 9 + 41       | rr= = 131       | c = which is prime }} {{eqn | l = 10^2 + 10 + 41       | r = 100 + 10 + 41       | rr= = 151       | c = which is prime }} {{eqn | l = 11^2 + 11 + 17       | r = 121 + 11 + 17       | rr= = 173       | c = which is prime }} {{eqn | l = 12^2 + 12 + 41       | r = 144 + 12 + 41       | rr= = 197       | c = which is prime }} {{eqn | l = 13^2 + 13 + 17       | r = 169 + 13 + 17       | rr= = 223       | c = which is prime }} {{eqn | l = 14^2 + 14 + 41       | r = 196 + 14 + 41       | rr= = 251       | c = which is prime }} {{eqn | l = 15^2 + 15 + 41       | r = 225 + 15 + 41       | rr= = 281       | c = which is prime }} {{eqn | l = 16^2 + 16 + 41       | r = 256 + 16 + 41       | rr= = 313       | c = which is prime }} {{eqn | l = 17^2 + 17 + 41       | r = 289 + 17 + 41       | rr= = 347       | c = which is prime }} {{eqn | l = 18^2 + 18 + 41       | r = 324 + 18 + 41       | rr= = 383       | c = which is prime }} {{eqn | l = 19^2 + 19 + 41       | r = 361 + 19 + 41       | rr= = 421       | c = which is prime }} {{eqn | l = 20^2 + 20 + 41       | r = 400 + 20 + 41       | rr= = 461       | c = which is prime }} {{eqn | l = 21^2 + 21 + 41       | r = 441 + 21 + 41       | rr= = 503       | c = which is prime }} {{eqn | l = 22^2 + 22 + 41       | r = 484 + 22 + 41       | rr= = 547       | c = which is prime }} {{eqn | l = 23^2 + 23 + 41       | r = 529 + 23 + 41       | rr= = 593       | c = which is prime }} {{eqn | l = 24^2 + 24 + 41       | r = 576 + 24 + 41       | rr= = 641       | c = which is prime }} {{eqn | l = 25^2 + 25 + 41       | r = 625 + 25 + 41       | rr= = 691       | c = which is prime }} {{eqn | l = 26^2 + 26 + 41       | r = 676 + 26 + 41       | rr= = 743       | c = which is prime }} {{eqn | l = 27^2 + 27 + 41       | r = 729 + 27 + 41       | rr= = 797       | c = which is prime }} {{eqn | l = 28^2 + 28 + 41       | r = 784 + 28 + 41       | rr= = 853       | c = which is prime }} {{eqn | l = 29^2 + 29 + 41       | r = 841 + 29 + 41       | rr= = 911       | c = which is prime }} {{eqn | l = 30^2 + 30 + 41       | r = 900 + 30 + 41       | rr= = 971       | c = which is prime }} {{eqn | l = 31^2 + 31 + 41       | r = 961 + 31 + 41       | rr= = 1033       | c = which is prime }} {{eqn | l = 32^2 + 32 + 41       | r = 1024 + 32 + 41       | rr= = 1097       | c = which is prime }} {{eqn | l = 33^2 + 33 + 41       | r = 1089 + 33 + 41       | rr= = 1163       | c = which is prime }} {{eqn | l = 34^2 + 34 + 41       | r = 1156 + 34 + 41       | rr= = 1231       | c = which is prime }} {{eqn | l = 35^2 + 35 + 41       | r = 1225 + 35 + 41       | rr= = 1301       | c = which is prime }} {{eqn | l = 36^2 + 36 + 41       | r = 1296 + 36 + 41       | rr= = 1373       | c = which is prime }} {{eqn | l = 37^2 + 37 + 41       | r = 1369 + 37 + 41       | rr= = 1447       | c = which is prime }} {{eqn | l = 38^2 + 38 + 41       | r = 1444 + 38 + 41       | rr= = 1523       | c = which is prime }} {{eqn | l = 39^2 + 39 + 41       | r = 1521 + 39 + 41       | rr= = 1601       | c = which is prime }} {{eqn | l = 40^2 + 40 + 41       | r = 1600 + 40+ 41       | rr= = 1681       | c = which is not prime: $1681 = 41^2$ }} {{end-eqn}} {{OEIS|A005846}} Then we have: {{begin-eqn}} {{eqn | l = \\paren {-\\paren  {n + 1} }^2 + \\paren {-\\paren {n + 1} }       | r = n^2 + 2 n + 1 - \\paren {n + 1}       | c =  }} {{eqn | r = n^2 + n       | c =  }} {{end-eqn}} and so replacing $0$ to $39$ with $-1$ to $-40$ yields exactly the same sequence of primes. We note in addition the example: :$581^2 + 581 + 41 = 338 \\, 183$ which is prime. {{qed}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = 41|next = Prime-Generating Quadratic of form x squared - 79 x + 1601}}: $41$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = 41|next = Euler Lucky Number/Examples/41/Mistake}}: $41$ Category:Euler Lucky Numbers Category:41 lk4bfvl2uaiw5v6frkad6sln0swla02"}
+{"_id": "32713", "title": "Feit-Thompson Conjecture", "text": "Feit-Thompson Conjecture 0 53680 346749 285703 2018-03-11T10:32:43Z Prime.mover 59 wikitext text/x-wiki == Conjecture == There exist no distinct prime numbers $p$ and $q$ such that: :$\\dfrac {p^q - 1} {p - 1}$ divides $\\dfrac {q^p - 1} {q - 1}$ === Stronger Feit-Thompson Conjecture === {{:Feit-Thompson Conjecture/Stronger}} {{Namedfor|Walter Feit|name2 = John Griggs Thompson|cat = Feit|cat2 = Thompson}} Category:Prime Numbers Category:Divisors Category:Unproven Hypotheses Category:Feit-Thompson Conjecture mdebsx8acjyb9iigtnnvjvbmi14qoh6"}
+{"_id": "32714", "title": "Factorial/Examples/11", "text": "Factorial/Examples/11 0 53752 307317 285906 2017-07-30T05:19:59Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> :$11! = 39 \\, 916 \\, 800$ </onlyinclude> == Proof == From Examples of Factorials: :$10! = 3 \\, 628 \\, 800$ Then: <pre>  3 628 800 x       11 ----------  3 628 800 36 288 000 ---------- 39 916 800 </pre>{{qed}} Category:Factorials/Examples Category:11 6ra9v68ayfjn5rouvqw5o43wmkd2r37"}
+{"_id": "32715", "title": "Factorial/Examples/12", "text": "Factorial/Examples/12 0 53753 307318 285905 2017-07-30T05:20:19Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> :$12! = 479 \\, 001 \\, 600$ </onlyinclude> == Proof == From $11$ factorial: :$11! = 39 \\, 916 \\, 800$ Then: <pre>  39 916 800 x        12 -----------  79 833 600 399 168 000 ----------- 479 001 600 </pre>{{qed}} Category:Factorials/Examples Category:12 c3mh9jjeduydt0onxjery7csw6wgrbe"}
+{"_id": "32716", "title": "Factorial/Examples/13", "text": "Factorial/Examples/13 0 53754 307319 285911 2017-07-30T05:20:39Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> :$13! = 6 \\, 227 \\, 020 \\, 800$ </onlyinclude> === Prime Factors of $13!$ === {{:Prime Factors of 13 Factorial}} == Proof == From $12$ factorial: :$12! = 479 \\, 001 \\, 600$ Then: <pre>   479 001 600 x          13 ------------- 1 437 004 800 4 790 016 000 ------------- 6 227 020 800 </pre>{{qed}} Category:Factorials/Examples Category:13 7d7m48618hj5aze0mz9oc74dnppukox"}
+{"_id": "32717", "title": "Factorial/Examples/14", "text": "Factorial/Examples/14 0 53755 307320 285903 2017-07-30T05:20:57Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> :$14! = 87 \\, 178 \\, 291 \\, 200$ </onlyinclude> == Proof == From $13$ factorial: :$13! = 6 \\, 227 \\, 020 \\, 800$ Then: <pre>  6 227 020 800 x           14 --------------- 24 908 083 200 62 270 208 000 -------------- 87 178 291 200 </pre>{{qed}} Category:Factorials/Examples Category:14 0n3ojsabmqnri974kjbmklppvl0jc9q"}
+{"_id": "32718", "title": "Factorial/Examples/15", "text": "Factorial/Examples/15 0 53757 307321 285907 2017-07-30T05:21:10Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> :$15! = 1 \\, 307 \\, 674 \\, 368 \\, 000$ </onlyinclude> == Proof == From $14$ Factorial: :$14! = 87 \\, 178 \\, 291 \\, 200$ Then: <pre>    87 178 291 200 x              15 ------------------   435 891 456 000   871 782 912 000 ----------------- 1 307 674 368 000 </pre>{{qed}} Category:Factorials/Examples Category:15 lqtdq70q03acamfxx2281pfzhlhabvk"}
+{"_id": "32719", "title": "Factorial/Examples/16", "text": "Factorial/Examples/16 0 53758 307322 285908 2017-07-30T05:21:22Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> :$16! = 20 \\, 922 \\, 789 \\, 888 \\, 000$ </onlyinclude> == Proof == From $15$ Factorial: :$15! = 1 \\, 307 \\, 674 \\, 368 \\, 000$ Then: <pre>  1 307 674 368 000 x               16 ------------------  7 846 046 208 000 13 076 743 680 000 ------------------ 20 922 789 888 000 </pre>{{qed}} Category:Factorials/Examples Category:16 d1yze2qeyc9jpylsn7didmwujg4epuj"}
+{"_id": "32720", "title": "Factorial/Examples/17", "text": "Factorial/Examples/17 0 53759 307323 285909 2017-07-30T05:21:32Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> :$17! = 355 \\, 687 \\, 428 \\, 096 \\, 000$ </onlyinclude> == Proof == From $16$ Factorial: :$16! = 20 \\, 922 \\, 789 \\, 888 \\, 000$ Then: <pre>  20 922 789 888 000 x                17 ------------------- 146 459 529 216 000 209 227 898 880 000 ------------------- 355 687 428 096 000 </pre>{{qed}} Category:Factorials/Examples Category:17 aa7clqqtqsra93zliurbbr6ut34uy58"}
+{"_id": "32721", "title": "Factorial/Examples/18", "text": "Factorial/Examples/18 0 53760 307324 285910 2017-07-30T05:21:42Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> :$18! = 6 \\, 402 \\, 373 \\, 705 \\, 728 \\, 000$ </onlyinclude> == Proof == From $17$ Factorial: :$17! = 355 \\, 687 \\, 428 \\, 096 \\, 000$ Then: <pre>   355 687 428 096 000 x                  18 --------------------- 2 845 499 424 768 000 3 556 874 280 960 000 --------------------- 6 402 373 705 728 000 </pre>{{qed}} Category:Factorials/Examples Category:18 iy07acl27x5ararkfo5bnxgb1fezbbe"}
+{"_id": "32722", "title": "Factorial/Examples/19", "text": "Factorial/Examples/19 0 53761 307325 285915 2017-07-30T05:21:53Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> :$19! = 121 \\, 645 \\, 100 \\, 408 \\, 832 \\, 000$ </onlyinclude> == Proof == From $18$ Factorial: :$18! = 6 \\, 402 \\, 373 \\, 705 \\, 728 \\, 000$ Then: <pre>   6 402 373 705 728 000 x                    19 -----------------------  57 621 363 351 552 000  64 023 737 057 280 000 ----------------------- 121 645 100 408 832 000 </pre>{{qed}} Category:Factorials/Examples Category:19 8bw062ee8izvjerlllqkr0j857z6fw4"}
+{"_id": "32723", "title": "Factorial/Examples/20", "text": "Factorial/Examples/20 0 53762 382273 307326 2018-12-14T22:42:48Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> :$20! = 2 \\, 432 \\, 902 \\, 008 \\, 176 \\, 640 \\, 000$ </onlyinclude> === Prime Factors of $20!$ === {{:Prime Factors of 20 Factorial}} == Proof == From $19$ Factorial: :$19! = 121 \\, 645 \\, 100 \\, 408 \\, 832 \\, 000$ Then: <pre>   121 645 100 408 832 000 x                      20 ------------------------ 2 432 902 008 176 640 000 </pre>{{qed}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Cross-Sections of Leech Lattice|next = Factorial/Examples/450/Historical Note}}: $24$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Definition:Factorial|next = Factorial/Examples/1,000,000}}: $24$ Category:Factorials/Examples Category:20 sflj3v0q5ogye6so4mmykwgq7oynkkw"}
+{"_id": "32724", "title": "Factorial/Examples/21", "text": "Factorial/Examples/21 0 53836 307327 286137 2017-07-30T05:22:17Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> :$21! = 51 \\, 090 \\, 942 \\, 171 \\, 709 \\, 440 \\, 000$ </onlyinclude> == Proof == From $20$ Factorial: :$20! = 2 \\, 432 \\, 902 \\, 008 \\, 176 \\, 640 \\, 000$ Then: <pre>   2 432 902 008 176 640 000 x                        21 ---------------------------   2 432 902 008 176 640 000  48 658 040 163 532 800 000 ---------------------------  51 090 942 171 709 440 000    </pre>{{qed}} Category:Factorials/Examples Category:21 0a57vhpp7r4z699dh61a3eb8dtpaaw8"}
+{"_id": "32725", "title": "Factorial/Examples/22", "text": "Factorial/Examples/22 0 53837 307328 286139 2017-07-30T05:22:27Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> :$22! = 1 \\, 124 \\, 000 \\, 727 \\, 777 \\, 607 \\, 680 \\, 000$ </onlyinclude> == Proof == From $21$ Factorial: :$21! = 51 \\, 090 \\, 942 \\, 171 \\, 709 \\, 440 \\, 000$ Then: <pre>    51 090 942 171 709 440 000 x                          22 -----------------------------   102 181 884 343 418 880 000 1 021 818 843 434 188 800 000 ----------------------------- 1 124 000 727 777 607 680 000    </pre>{{qed}} Category:Factorials/Examples Category:22 s6zkqhpng4z50u0nvcp4kzeyag2zrkm"}
+{"_id": "32726", "title": "Factorial/Examples/23", "text": "Factorial/Examples/23 0 53838 307329 286140 2017-07-30T05:22:38Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> :$23! = 25 \\, 852 \\, 016 \\, 738 \\, 884 \\, 976 \\, 640 \\, 000$ </onlyinclude> == Proof == From $22$ Factorial: :$22! = 1 \\, 124 \\, 000 \\, 727 \\, 777 \\, 607 \\, 680 \\, 000$ Then: <pre>  1 124 000 727 777 607 680 000 x                           23 ------------------------------  3 372 002 183 332 823 040 000 22 480 014 555 552 153 600 000 ------------------------------ 25 852 016 738 884 976 640 000    </pre>{{qed}} Category:Factorials/Examples Category:23 fpy1sfyi1q72rxgk957fgb4wwlyhqyc"}
+{"_id": "32727", "title": "Factorial/Examples/24", "text": "Factorial/Examples/24 0 53839 307330 286141 2017-07-30T05:22:50Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> :$24! = 620 \\, 448 \\, 401 \\, 733 \\, 239 \\, 439 \\, 360 \\, 000$ </onlyinclude> == Proof == From $23$ Factorial: :$23! = 25 \\, 852 \\, 016 \\, 738 \\, 884 \\, 976 \\, 640 \\, 000$ Then: <pre>  25 852 016 738 884 976 640 000 x                            24 ------------------------------- 103 408 066 955 539 906 560 000 517 040 334 777 699 532 800 000 ------------------------------- 620 448 401 733 239 439 360 000    </pre>{{qed}} Category:Factorials/Examples Category:24 4tg05bm160modl1t8ynopwd8so39p4d"}
+{"_id": "32728", "title": "Factorial/Examples/25", "text": "Factorial/Examples/25 0 53840 307331 286143 2017-07-30T05:23:00Z Prime.mover 59 wikitext text/x-wiki == Example of Factorial == <onlyinclude> :$25! = 15 \\, 511 \\, 210 \\, 043 \\, 330 \\, 985 \\, 984 \\, 000 \\, 000$ </onlyinclude> == Proof == From $24$ Factorial: :$24! = 620 \\, 448 \\, 401 \\, 733 \\, 239 \\, 439 \\, 360 \\, 000$ Then: <pre>    620 448 401 733 239 439 360 000 x                               25 ----------------------------------  3 102 242 008 666 197 196 800 000 12 408 968 034 664 788 787 200 000 ---------------------------------- 15 511 210 043 330 985 984 000 000    </pre>{{qed}} Category:Factorials/Examples Category:25 3rsp94dts04c0kf6b2axggtlhyfstfn"}
+{"_id": "32729", "title": "Square of Small-Digit Palindromic Number is Palindromic/Examples/11", "text": "Square of Small-Digit Palindromic Number is Palindromic/Examples/11 0 53853 383304 382082 2018-12-20T06:39:14Z Prime.mover 59 wikitext text/x-wiki == Example of Square of Small-Digit Palindromic Number is Palindromic == <onlyinclude> ::$11^2 = 121$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = 11^2       | r = \\left({10 + 1}\\right)^2       | c =  }} {{eqn | r = 10^2 + 2 \\times 10 \\times 1 + 1^2       | c =  }} {{eqn | r = 100 + 20 + 1       | c =  }} {{eqn | r = 121       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Square of Small-Digit Palindromic Number is Palindromic/Examples/22|next = Square of Small-Digit Palindromic Number is Palindromic/Examples/111}}: $22$ * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = 121|next = 121 is Square Number in All Bases greater than 2}}: $121$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Square of Small-Digit Palindromic Number is Palindromic/Examples/22|next = Square of Small-Digit Palindromic Number is Palindromic/Examples/111}}: $22$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = 121|next = 121 is Square Number in All Bases greater than 2}}: $121$ Category:Square of Small-Digit Palindromic Number is Palindromic Category:11 Category:121 k0ar8p9ci8onod77ll6uy8e8c8s2snc"}
+{"_id": "32730", "title": "Square of Small-Digit Palindromic Number is Palindromic/Examples/121", "text": "Square of Small-Digit Palindromic Number is Palindromic/Examples/121 0 53859 382085 298747 2018-12-14T06:17:48Z Prime.mover 59 wikitext text/x-wiki == Example of Square of Small-Digit Palindromic Number is Palindromic == <onlyinclude> ::$121^2 = 14641$ </onlyinclude> == Proof == <pre>   121 x 121 -----   121  2420 12100 ------- 14641 </pre>{{qed}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Square of Small-Digit Palindromic Number is Palindromic/Examples/1111|next = Square of Small-Digit Palindromic Number is Palindromic/Examples/212}}: $22$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Square of Small-Digit Palindromic Number is Palindromic/Examples/1111|next = Square of Small-Digit Palindromic Number is Palindromic/Examples/212}}: $22$ Category:Square of Small-Digit Palindromic Number is Palindromic Category:121 tdhaur6apnd1st1t64fjvusyxar35xc"}
+{"_id": "32731", "title": "Sigma Function of Square-Free Integer/Examples/70", "text": "Sigma Function of Square-Free Integer/Examples/70 0 53870 492965 317776 2020-10-06T18:06:59Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Square-Free Integer == <onlyinclude> :$\\map \\sigma {70} = 144$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof 1 == {{:Sigma Function of Square-Free Integer/Examples/70/Proof 1}} == Proof 2 == {{:Sigma Function of Square-Free Integer/Examples/70/Proof 2}} Category:Sigma Function of Square-Free Integer Category:70 Category:Sigma Function of Square-Free Integer/Examples/70 ld8kh8u994m7leko1i8hf74cwmhm918"}
+{"_id": "32732", "title": "Birthday Paradox", "text": "Birthday Paradox 0 53960 478611 478607 2020-07-17T10:49:09Z Prime.mover 59 wikitext text/x-wiki == Paradox == Let there be $23$ or more people in a room. The probability that at least $2$ of them have the same birthday is greater than $50 \\%$. == Proof == Let there be $n$ people in the room. Let $\\map p n$ be the probability that no two people in the room have the same birthday. For simplicity, let us ignore leap years and assume there are $365$ days in the year. Let the birthday of person $1$ be established. The probability that person $2$ shares person $1$'s birthday is $\\dfrac 1 {365}$. Thus, the probability that person $2$ does not share person $1$'s birthday is $\\dfrac {364} {365}$. Similarly, the probability that person $3$ does not share the birthday of either person $1$ or person $2$ is $\\dfrac {363} {365}$. And further, the probability that person $n$ does not share the birthday of any of the people indexed $1$ to $n - 1$ is $\\dfrac {365 - \\paren {n - 1} } {365}$. Hence the total probability that none of the $n$ people share a birthday is given by: :$\\map p n = \\dfrac {364} {365} \\dfrac {363} {365} \\dfrac {362} {365} \\cdots \\dfrac {365 - n + 1} {365}$ {{begin-eqn}} {{eqn | l = \\map p n       | r = \\dfrac {364} {365} \\dfrac {363} {365} \\dfrac {362} {365} \\cdots \\dfrac {365 - n + 1} {365}       | c =  }} {{eqn | r = \\dfrac {365!} {365^n} \\binom {365} n       | c =  }} {{end-eqn}} Setting $n = 23$ and evaluating the above gives: :$\\map p {23} \\approx 0.493$ Hence the probability that at least $2$ people share a birthday is $1 = 0.492 = 0.507 = 50.7 \\%$ {{qed}} == Conclusion == This is a veridical paradox. Counter-intuitively, the probability of a shared birthday amongst such a small group of people is surprisingly high. == General Birthday Paradox == {{:Birthday Paradox/General}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = 23 is Largest Integer not Sum of Distinct Perfect Powers/Mistake|next = Harmonic Series is Divergent}}: $23$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = 23 is Largest Integer not Sum of Distinct Perfect Powers/Mistake|next = Birthday Paradox/General/3}}: $23$ Category:Probability Theory Category:Combinatorics Category:Veridical Paradoxes Category:Birthday Paradox n82iwsg3p01ongoa3zlquh7ap1exnev"}
+{"_id": "32733", "title": "Legendre's Condition/Lemma 1", "text": "Legendre's Condition/Lemma 1 0 54016 496498 457855 2020-10-25T13:41:04Z Prime.mover 59 wikitext text/x-wiki {{MissingLinks}} == Lemma == Let $y = \\map y x$ be a real function, such that: :$\\map y a = A$ :$\\map y b = B$ Let $J \\sqbrk y$ be a functional, such that: :$\\ds J \\sqbrk y = \\int_a^b \\map F {x, y, y'} \\rd x$ where: :$F \\in C^2 \\closedint a b$  {{WRT}} all its variables. Then: :$\\ds \\delta^2 J \\sqbrk {y; h} = \\int_a^b \\paren {\\map P {x, \\map y x} h'^2 + \\map Q {x, \\map y x} h^2} \\rd x$ where: {{begin-eqn}} {{eqn | l = \\map P {x, \\map y x}       | r = \\frac 1 2 F_{y'y'} }} {{eqn | l = \\map Q {x, \\map y x}       | r = \\frac 1 2 \\paren {F_{yy} - \\frac \\d {\\d x} F_{yy} } }} {{end-eqn}} == Proof == The minimising function $y$ has fixed end-points. Therefore, consider an increment of a functional with $h$ such that: :$h \\in C^1 \\closedint a b: \\paren {\\map h a = 0} \\land \\paren {\\map h b = 0}$ Then: {{begin-eqn}} {{eqn | l = \\Delta J \\sqbrk {y; h}       | r = J \\sqbrk {y + h} - J \\sqbrk y       | c = {{Defof|Increment of Functional}} }} {{eqn | r = \\int_a^b \\paren {\\map F {x, y + h, y' + h'} - \\map F {x, y, y'} } \\rd x       | c = form of $J$ }} {{eqn | r = \\int_a^b \\paren {F + \\paren {F_y h + F_{y'} h'} + \\frac 1 2 \\paren {\\overline F_{yy} h^2 + \\overline F_{yy'} h h' + \\overline F_{y'y'} h'^2} - F} \\rd x       | c = Taylor's Theorem }} {{eqn | r = \\int_a^b \\paren {F_y h + F_{y'} h'} \\rd x + \\frac 1 2 \\int_a^b \\paren {\\overline F_{yy} h^2 + \\overline F_{yy'} h h' + \\overline F_{y'y'} h'^2} \\rd x       | c = cancel $F$ }} {{end-eqn}} where omitted variables are $\\paren {x, y, y'}$, and the overbar indicates derivatives being taken along some intermediate curves: {{begin-eqn}} {{eqn | l = \\overline {\\map {F_{yy} } {x,y,y'} }       | r = \\map {F_{yy} } {x, y + \\theta h, y' + \\theta h'} }} {{eqn | l = \\overline {\\map {F_{yy'} } {x,y,y'} }       | r = \\map {F_{yy'} } {x, y + \\theta h, y' + \\theta h'} }} {{eqn | l = \\overline {\\map {F_{y'y'} } {x,y,y'} }       | r = \\map {F_{y'y'} } {x, y + \\theta h, y' + \\theta h'} }} {{end-eqn}} with $0 < \\theta < 1$. If $\\overline F_{yy}$, $\\overline F_{yy'}$, $\\overline F_{y'y'} $ are to be replaced by $F_{yy}$, $F_{yy}$, $F_{y'y'}$ evaluated at the point $\\tuple {x, \\map y x, \\map {y'} x}$, then: :$\\ds \\Delta J \\sqbrk {y; h} = \\int_a^b \\paren {\\map {F_y} {x, y, y'} h + \\map {F_{y'} } {x, y, y'} h'} \\rd x + \\frac 1 2 \\int_a^b \\paren {\\map {F_{yy} } {x, y, y'} h^2 + 2 \\map {F_{yy'} } {x, y, y'} h h'+ \\map {F_{y'y'} } {x, y, y'} h'^2} \\rd x + \\epsilon$ where: :$\\ds \\epsilon = \\int_a^b \\paren {\\epsilon_1 h^2 + \\epsilon_2 h h' + \\epsilon_3 h'^2}$ By continuity of $F_{yy}$, $F_{yy}$, $F_{y'y'}$: :$\\size h_1 \\to 0 \\implies \\epsilon_1, \\epsilon_2, \\epsilon_3 \\to 0$ {{explain|What does $\\size h_1$ mean?}} Thus, $\\epsilon$ is an infinitesimal of the order higher than 2 {{WRT}} $\\size h$. {{Stub|Expand on steps including $ \\epsilon $}} The first and second term on the {{RHS}} of $\\Delta J \\sqbrk {y; h}$ are $\\delta J \\sqbrk {y; h}$ and $\\delta^2 J \\sqbrk {y; h}$ respectively. Integrate the second term of $\\delta^2 J \\sqbrk {y; h}$ by parts: {{begin-eqn}} {{eqn | l = \\int_a^b 2 F_{yy'} h h' \\rd x       | r = \\int_a^b 2 F_{yy'} h \\rd h }} {{eqn | r = \\int_a^b F_{yy'} \\rd h^2 }} {{eqn | r = \\bigintlimits {F_{yy'} h^2} {x \\mathop = a} {x \\mathop = b} - \\int_a^b \\map {\\frac \\d {\\d x} } {F_{yy'} } h^2 \\rd x }} {{eqn | r = -\\int_a^b \\map {\\frac \\d {\\d x} } {F_{yy'} } h^2 \\rd x }} {{end-eqn}} {{explain|Why does $\\bigintlimits {F_{yy'} h^2} {x \\mathop {{=}} a} {x \\mathop {{=}} b}$ vanish?}} Therefore: :$\\ds \\delta^2 J \\sqbrk {y; h} = \\int_a^b \\paren {\\frac 1 2 F_{y'y'} h'^2 + \\frac 1 2 \\paren {F_{yy} - \\frac \\d {\\d x} F_{yy'} } h^2} \\rd x$ {{qed|lemma}} {{explain|Review use of square brackets. If they are being used purely for parenthesis, better to replace with round ones, as square ones have conventional meanings, so use of them for parenthesis may confuse.}} == Mistake == {{BookReference|Calculus of Variations|1963|I.M. Gelfand|author2 = S.V. Fomin}}: $\\S 5.25$: The Formula for the Second Variation. Legendre's Condition p. 102 states that :$P = \\dfrac 1 2 F_{y'y'} \\quad Q = \\dfrac 1 2 \\paren {F_{yy'} - \\dfrac \\d {\\d x} F_{yy'} }$ This is a mistake, since the second variation should contain both pure and mixed partial derivatives of the order 2. However, $F_{yy} $ is missing and could not have been lost during derivation of the proof. It should be: :$Q = \\dfrac 1 2 \\paren {F_{yy} - \\dfrac \\d {\\d x} F_{yy'} }$ == Sources == {{BookReference|Calculus of Variations|1963|I.M. Gelfand|author2 = S.V. Fomin|prev = Legendre's Condition|next = Legendre's Condition/Lemma 2}}: $\\S 5.25$: The Formula for the Second Variation. Legendre's Condition  Category:Calculus of Variations 4f4ujy87m15vkm3s39frexhfs87749v"}
+{"_id": "32734", "title": "Legendre's Condition/Lemma 2", "text": "Legendre's Condition/Lemma 2 0 54021 458074 458070 2020-03-28T22:13:00Z Julius 3095 wikitext text/x-wiki {{MissingLinks}} == Lemma == Let $ h$ be a real function such that: :$h \\in C^1 \\openint a b$ :$\\map h a = 0$ :$\\map h b = 0$ Let: :$\\displaystyle \\delta^2 J \\sqbrk {y; h} = \\int_a^b \\paren {\\map P {x, \\map y x} h'^2 + \\map Q {x, \\map y x} h^2} \\rd x$ where $P \\in C^0 \\closedint a b$.  Then a necessary condition for: :$\\delta^2 J \\sqbrk {y; h} \\ge 0$  is: :$\\forall x\\in\\closedint a b: \\map P {x, \\map y x} \\ge 0$ == Proof == Assume that above is not true. Then: :$\\paren {\\exists x_0 \\in \\closedint a b} \\land \\paren {\\exists \\beta \\in \\R_{<0} }: \\map P {x_0} = -2 \\beta$ $P$ is continuous. Thus: :$\\exists \\alpha \\in \\R_{>0}: \\paren {a \\ge x_0 - \\alpha} \\land \\paren {x_0 + \\alpha \\ge b}$ and: :$\\forall x \\in \\openint {x_0 - \\alpha} {x_0 + \\alpha}: \\map P x < -\\beta$ In other words: $\\map P x \\begin {cases} = 0 & : x \\in \\closedint a {x_0 - \\alpha} \\lor \\closedint {x_0 + \\alpha} b \\\\ < 0 & : x \\in \\closedint {x_0 - \\alpha} {x_0 + \\alpha} \\end {cases}$ Let :$h = \\begin {cases} \\sin^2 \\paren {\\dfrac {\\map \\pi {x - x_0} } \\alpha} & : x_0 - \\alpha \\ge x \\ge x_0 + \\alpha \\\\ 0 & : \\text {otherwise} \\end{cases}$ It belongs to $C^1 \\openint a b$ because: {{explain|The derivation and significance of all the below: what they mean, what depends on what, etc.}} {{begin-eqn}} {{eqn | l = \\lim_{x \\to x_0 - \\alpha + 0^+} h       | r = \\lim_{x \\to x_0 - \\alpha + 0^+} \\map {\\sin^2} {\\frac {\\map \\pi {x - x_0} } \\alpha} }} {{eqn | r = \\map {\\sin^2} {\\map \\pi {\\frac {0^+} \\alpha - 1} } }} {{eqn | r = 0 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \\lim_{x \\to x_0 - \\alpha + 0^+}h'       | r = \\lim_{x \\to x_0 - \\alpha + 0^+} \\map \\sin {\\frac {2 \\map \\pi {x - x_0} } \\alpha} \\frac \\pi \\alpha }} {{eqn | r = \\map \\sin {2 \\map \\pi {\\frac {0^+} \\alpha - 1} } \\frac \\pi \\alpha }} {{eqn | r = 0 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \\lim_{x \\to x_0 - \\alpha + 0^+}h''       | r = \\lim_{x \\to x_0 - \\alpha + 0^+} \\map \\cos {\\frac {2 \\map \\pi {x - x_0} } \\alpha} \\frac {2 \\pi^2} {\\alpha^2} }} {{eqn | r = \\map \\cos {2 \\map \\pi {\\frac {0^+} \\alpha - 1} } \\frac {2 \\pi^2} {\\alpha^2} }} {{eqn | r = \\frac {2 \\pi^2} {\\alpha^2} }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \\lim_{x \\to x_0 + \\alpha + 0^-} h       | r = \\lim_{x \\to x_0 + \\alpha + 0^-} \\map {\\sin^2} {\\frac {\\map \\pi {x - x_0} } \\alpha} }} {{eqn | r = \\map {\\sin^2} {\\map \\pi {\\frac {0^-} \\alpha + 1} } }} {{eqn | r = 0 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \\lim_{x \\to x_0 + \\alpha + 0^-}h'       | r = \\lim_{x \\to x_0 - \\alpha + 0^-} \\map \\sin {\\frac {2 \\map \\pi {x - x_0} } \\alpha} \\frac \\pi \\alpha }} {{eqn | r = \\map \\sin {2 \\map \\pi {\\frac {0^-} \\alpha + 1} } \\frac \\pi \\alpha }} {{eqn | r = 0 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \\lim_{x \\to x_0 + \\alpha + 0^-}h''       | r = \\lim_{x \\to x_0 - \\alpha + 0^-} \\map \\cos {\\frac {2 \\map \\pi {x - x_0} } \\alpha} \\frac {2 \\pi^2} {\\alpha^2} }} {{eqn | r = \\map \\cos {2 \\map \\pi {\\frac {0^-} \\alpha + 1} } \\frac {2 \\pi^2} {\\alpha^2} }} {{eqn | r = \\frac {2\\pi^2} {\\alpha^2} }} {{end-eqn}} In other words, only $h$ and $h'$ are continuous in $\\closedint a b$ Then: {{begin-eqn}} {{eqn | l = \\int_a^b \\paren {P h'^2 + Q h^2} \\rd x       | r = \\int_{x_0 - \\alpha}^{x_0 + \\alpha} P \\frac {\\pi^2} {\\alpha^2} \\map {\\sin^2} {\\frac {2 \\map \\pi {x - x_0} } \\alpha} \\rd x + \\int_{x_0 - \\alpha}^{x_0 + \\alpha} Q \\map {\\sin^4} {\\frac {\\map \\pi {x - x_0} } \\alpha} \\rd x }} {{eqn | o = <       | r = -\\beta \\frac {\\pi^2} {\\alpha^2} \\int_{x_0 - \\alpha}^{x_0 + \\alpha} \\map {\\sin^2} {\\frac {2 \\map \\pi {x - x_0} } \\alpha} \\rd x + \\max_{a \\le x \\le b} \\size {\\map Q x} \\int_{x_0 - \\alpha}^{x_0 + \\alpha} \\map {\\sin^4} {\\frac {\\map \\pi {x - x_0} } \\alpha}\\rd x       | c = as $\\displaystyle \\map P x < -\\beta, \\quad \\map Q x < \\max_{a \\mathop \\le x \\mathop \\le b} \\size {\\map Q x}$ }} {{eqn | r = -\\beta \\frac {\\pi^2} \\alpha + \\max_{a \\mathop \\le x \\mathop \\le b} \\size {\\map Q x} \\int_{x_0 - \\alpha}^{x_0 + \\alpha} \\map {\\sin^4} {\\frac {\\map \\pi {x - x_0} } \\alpha} \\rd x }} {{eqn | r = -\\beta \\frac {\\pi^2} \\alpha + \\frac 3 4 \\alpha M  }} {{end-eqn}} where: :$\\displaystyle M = \\max_{a \\mathop \\le x \\mathop \\le b} \\size {\\map Q x}$ For sufficiently small $\\alpha$ the {{RHS}} is negative. Hence, $\\delta^2 J$ is negative for the corresponding $h$. To conclude, it has been shown that :$P \\ge 0 \\quad \\neg \\forall x \\in \\closedint a b \\implies \\delta^2 J<0$ Then, by contrapositive statement this is equivalent to: :$\\forall x \\in \\closedint a b: \\delta^2 J \\ge 0 \\implies P \\ge 0$ {{qed}} == Sources == {{BookReference|Calculus of Variations|1963|I.M. Gelfand|author2 = S.V. Fomin|prev = Legendre's Condition/Lemma 1|next = Definition:Conjugate Point}}: $\\S 5.25$: The Formula for the Second Variation. Legendre's Condition  Category:Calculus of Variations r3a5e2emmox1zng2irquahrotbcr76s"}
+{"_id": "32735", "title": "Sigma Function of Integer/Examples/20", "text": "Sigma Function of Integer/Examples/20 0 54025 317610 317522 2017-09-16T11:58:00Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\sigma \\left({20}\\right) = 42$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == From Sigma Function of Integer :$\\displaystyle \\sigma \\left({n}\\right) = \\prod_{1 \\mathop \\le i \\mathop \\le r} \\frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \\displaystyle \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$20 = 2^2 \\times 5$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({20}\\right)       | r = \\frac {2^3 - 1} {2 - 1} \\times \\frac {5^2 - 1} {5 - 1}       | c =  }} {{eqn | r = \\frac 7 1 \\times \\frac {6 \\times 4} 4       | c =  }} {{eqn | r = 7 \\times 6       | c =  }} {{eqn | r = 42       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:20 svrer95xyqtzsimn8y4q3fhlh5itsor"}
+{"_id": "32736", "title": "Sigma Function of Non-Square Semiprime/Examples/35", "text": "Sigma Function of Non-Square Semiprime/Examples/35 0 54077 484228 352264 2020-08-30T21:18:49Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Non-Square Semiprime == <onlyinclude> :$\\map \\sigma {35} = 48$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$35 = 5 \\times 7$ and so by definition is a semiprime whose prime factors are distinct. Hence: {{begin-eqn}} {{eqn | l = \\map \\sigma {35}       | r = \\paren {5 + 1} \\paren {7 + 1}       | c = Sigma Function of Non-Square Semiprime }} {{eqn | r = 6 \\times 8       | c =  }} {{eqn | r = 48       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Non-Square Semiprime Category:35 f563vp1dfsoqnxbfeuiwl1ldgq3e6hn"}
+{"_id": "32737", "title": "Binomial Theorem/Examples/Cube of Sum", "text": "Binomial Theorem/Examples/Cube of Sum 0 54109 412151 393428 2019-07-08T08:23:28Z Prime.mover 59 wikitext text/x-wiki == Example of Use of Binomial Theorem == <onlyinclude> :$\\paren {x + y}^3 = x^3 + 3 x^2 y + 3 x y^2 + y^3$ </onlyinclude> === Corollary === {{:Cube of Sum/Corollary}} == Proof == Follows directly from the Binomial Theorem: :$\\displaystyle \\forall n \\in \\Z_{\\ge 0}: \\paren {x + y}^n = \\sum_{k \\mathop = 0}^n \\binom n k x^{n - k} y^k$ putting $n = 3$. {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Square of Difference|next = Cube of Difference}}: $\\S 2$: Special Products and Factors: $2.3$ * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Square of Sum/Algebraic Proof 2|next = Fourth Power of Sum}}: $\\S 20$: Binomial Series: $20.6$ * {{BookReference|Elementary Number Theory|1980|David M. Burton|ed = revised|edpage = Revised Printing|prev = Square of Sum|next = Binomial Theorem/Examples/4th Power of Sum}}: Chapter $1$: Some Preliminary Considerations: $1.2$ The Binomial Theorem Category:Examples of Use of Binomial Theorem Category:Third Powers jux8rm3q0livgh17bgq7gn9mq7opwy1"}
+{"_id": "32738", "title": "Taylor Expansion for Polynomials/Order 1", "text": "Taylor Expansion for Polynomials/Order 1 0 54559 288840 2017-03-15T22:09:30Z Barto 3079 Created page with \"== Theorem == <onlyinclude> Let $R$ be a commutative ring with unity.  Let $f(X)\\in R[X]$ be a Definition:Polynomial (Abstract Al...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $R$ be a commutative ring with unity. Let $f(X)\\in R[X]$ be a polynomial. Let $a\\in R$. Then there exists a polynomial $g(X)\\in R[X]$ such that: :$f(X+a) = f(X) + a f'(X) +a^2 g(X)$ where $f'$ denotes the formal derivative of $f$. </onlyinclude> == Proof == By linearity, it suffices to prove this for $f(X)=X^n$. This is now a direct consequence of the Binomial Theorem. {{qed}} Category:Polynomial Theory 2y2stiza6omv9tu8mo9wsygmj8frhvf"}
+{"_id": "32739", "title": "Smallest Set of Weights for One-Pan Balance", "text": "Smallest Set of Weights for One-Pan Balance 0 54967 481405 390665 2020-08-11T21:23:51Z Prime.mover 59 wikitext text/x-wiki == Classic Problem == Consider a set of balance scales for determining the weight of a physical object. Let this set of scales be such that weights may be placed in one of the pans. What is the smallest set of weights needed to weigh any given integer weight up to a given amount? == Solution == A set of $m$ weights in the sequence $\\sequence {2^n}$: :$1, 2, 4, 8, 16, \\ldots$ allows one to weigh any given integer weight up to $2^m - 1$. == Proof == This is equivalent to the statement that every positive integer can be expressed uniquely in binary notation. This in turn is an application of the Basis Representation Theorem. {{qed}} == Examples == {{:Smallest Set of Weights for One-Pan Balance/Examples}} == Also see == * Smallest Set of Weights for Two-Pan Balance == Historical Note == {{:Smallest Set of Weights for One-Pan Balance/Historical Note}} == Sources == * {{BookReference|Mathematical Recreations and Essays|1974|W.W. Rouse Ball|author2 = H.S.M. Coxeter|ed = 12th|edpage = Twelfth Edition}} * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = 18,446,744,073,709,551,615|next = Smallest Set of Weights for One-Pan Balance/Historical Note}}: $31$ * {{BookReference|Curious and Interesting Puzzles|1992|David Wells|prev = Smallest Set of Weights for One-Pan Balance/Historical Note|next = Smallest Set of Weights for Two-Pan Balance}}: Bachet: $108$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = 18,446,744,073,709,551,615|next = Smallest Set of Weights for One-Pan Balance/Historical Note}}: $31$ Category:Classic Problems Category:Binary Notation Category:Smallest Set of Weights for One-Pan Balance pep184wcvd7st290r60wp69n9qxgoza"}
+{"_id": "32740", "title": "Smallest Set of Weights for Two-Pan Balance", "text": "Smallest Set of Weights for Two-Pan Balance 0 54969 481412 390754 2020-08-11T22:01:32Z Prime.mover 59 wikitext text/x-wiki == Classic Problem == Consider a set of balance scales for determining the weight of a physical object. Let this set of scales be such that weights may be placed in either of the two pans. What is the smallest set of weights needed to weigh any given integer weight up to a given amount? == Solution == A set of weights up to $3^m$ in the sequence $\\sequence {3^n}$: :$1, 3, 9, 27, \\ldots$ allows one to weigh any given integer weight up to $\\dfrac {3^{m + 1} - 1} 2$. == Proof == Place the item to be weighed in the left hand pan of the balance. Let it weigh $n$. Let $n$ be expressed in balanced ternary representation. From Representation of Integers in Balanced Ternary, $n$ can be uniquely so represented. With $m$ digits, we can count up to $\\dfrac {3^{m + 1} - 1} 2$. We use the balanced ternary representation to model how to place the weights. Let the digits of $n$ in such a representation be numbered $0, 1, \\ldots, m$ from the least significant digit to the most significant digit. The $k$th digit represents the weight which weighs $3^k$. When the $k$th digit is $1$, place weight $3^k$ in the right hand pan. When the $k$th digit is $\\underline 1$, place weight $3^k$ in the left hand pan. When the $k$th digit is $$, place weight $3^k$ in neither pan. This will make the scales balance. {{qed}} == Examples == {{:Smallest Set of Weights for Two-Pan Balance/Examples}} == Also see == * Smallest Set of Weights for One-Pan Balance == Historical Note == {{:Smallest Set of Weights for Two-Pan Balance/Historical Note}} == Sources == * {{BookReference|Problèmes Plaisans et Delectables qui se font par les Nombres|1612|Claude-Gaspar Bachet}} * {{BookReference|Mathematical Recreations and Essays|1974|W.W. Rouse Ball|author2 = H.S.M. Coxeter|ed = 12th|edpage = Twelfth Edition}} * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Smallest Set of Weights for One-Pan Balance/Historical Note|next = Smallest Set of Weights for Two-Pan Balance/Historical Note}}: $31$ * {{BookReference|Curious and Interesting Puzzles|1992|David Wells|prev = Smallest Set of Weights for One-Pan Balance|next = Book:Claudio Gaspare Bacheto/Diophanti Alexandrini Arithmeticorum}}: Bachet: $108$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Smallest Set of Weights for One-Pan Balance/Historical Note|next = Smallest Set of Weights for Two-Pan Balance/Historical Note}}: $31$ Category:Classic Problems Category:Ternary Notation Category:Smallest Set of Weights for Two-Pan Balance ck5ksbidrvet8wqobildwk75wly3x9a"}
+{"_id": "32741", "title": "Sigma Function of 217", "text": "Sigma Function of 217 0 55223 352260 317842 2018-05-01T14:11:33Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Non-Square Semiprime == <onlyinclude> :$\\sigma \\left({217}\\right) = 256$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$217 = 7 \\times 31$ and so by definition is a semiprime whose prime factors are distinct. Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({217}\\right)       | r = \\left({7 + 1}\\right) \\left({31 + 1}\\right)       | c = Sigma Function of Non-Square Semiprime }} {{eqn | r = 8 \\times 32       | c =  }} {{eqn | r = 2^3 \\times 2^5       | c =  }} {{eqn | r = 2^8       | c =  }} {{eqn | r = \\left({2^4}\\right)^2       | c =  }} {{eqn | r = 16^2       | c =  }} {{eqn | r = 256       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Non-Square Semiprime Category:217 t1dzyhch4vk8p05ofatyl44vay5sw3p"}
+{"_id": "32742", "title": "Sigma Function of 214", "text": "Sigma Function of 214 0 55226 392679 352261 2019-02-18T14:54:06Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Non-Square Semiprime == <onlyinclude> :$\\map \\sigma {214} = 324$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$214 = 2 \\times 107$ and so by definition is a semiprime whose prime factors are distinct. Hence: {{begin-eqn}} {{eqn | l = \\map \\sigma {214}       | r = \\paren {2 + 1} \\paren {107 + 1}       | c = Sigma Function of Non-Square Semiprime }} {{eqn | r = 3 \\times 108       | c =  }} {{eqn | r = 3 \\times \\paren {2^2 \\times 3^3}       | c =  }} {{eqn | r = 2^3 \\times 3^4       | c =  }} {{eqn | r = \\paren {2 \\times 3^2}^2       | c =  }} {{eqn | r = 18^2       | c =  }} {{eqn | r = 324       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Non-Square Semiprime Category:214 llhx6cmvjx6219ej3u92ldhuco7nalq"}
+{"_id": "32743", "title": "Sigma Function of 210", "text": "Sigma Function of 210 0 55249 444501 444500 2020-01-22T15:27:26Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Square-Free Integer == <onlyinclude> :$\\map \\sigma {210} = 576$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$210 = 2 \\times 3 \\times 5 \\times 7$ Hence: {{begin-eqn}} {{eqn | l = \\map \\sigma {210}       | r = \\paren {2 + 1} \\paren {3 + 1} \\paren {5 + 1} \\paren {7 + 1}       | c = Sigma Function of Square-Free Integer }} {{eqn | r = 3 \\times 4 \\times 6 \\times 8       | c =  }} {{eqn | r = 3 \\times 2^2 \\times \\paren {2 \\times 3} \\times 2^3       | c =  }} {{eqn | r = 2^6 \\times 3^2       | c =  }} {{eqn | r = \\paren {2^3 \\times 3}^2       | c =  }} {{eqn | r = 24^2       | c =  }} {{eqn | r = 576       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Square-Free Integer Category:210 dcu9qi73ill3ucy6qxc63z0usb33us4"}
+{"_id": "32744", "title": "Sigma Function of 265", "text": "Sigma Function of 265 0 55252 317910 317908 2017-09-16T13:26:59Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Non-Square Semiprime == <onlyinclude> :$\\sigma \\left({265}\\right) = 324$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$265 = 5 \\times 53$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({265}\\right)       | r = \\left({5 + 1}\\right) \\left({53 + 1}\\right)       | c = Sigma Function of Non-Square Semiprime }} {{eqn | r = 6 \\times 54       | c =  }} {{eqn | r = \\left({2 \\times 3}\\right) \\times \\left({2 \\times 3^3}\\right)       | c =  }} {{eqn | r = 2^2 \\times 3^4       | c =  }} {{eqn | r = \\left({2 \\times 3^2}\\right)^2       | c =  }} {{eqn | r = 18^2       | c =  }} {{eqn | r = 324       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Non-Square Semiprime Category:265 27lfvdolzh8xkyhjnkvfr2k2qer0quv"}
+{"_id": "32745", "title": "Sigma Function of 282", "text": "Sigma Function of 282 0 55254 439920 317859 2019-12-17T13:28:01Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Square-Free Integer == <onlyinclude> :$\\map \\sigma {282} = 576$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$282 = 2 \\times 3 \\times 47$ Hence: {{begin-eqn}} {{eqn | l = \\map \\sigma {282}       | r = \\paren {2 + 1} \\paren {3 + 1} \\paren {47 + 1}       | c = Sigma Function of Square-Free Integer }} {{eqn | r = 3 \\times 4 \\times 48       | c =  }} {{eqn | r = 3 \\times 2^2 \\times \\paren {2^4 \\times 3}       | c =  }} {{eqn | r = 2^6 \\times 3^2       | c =  }} {{eqn | r = \\paren {2^3 \\times 3}^2       | c =  }} {{eqn | r = 24^2       | c =  }} {{eqn | r = 576       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Square-Free Integer Category:282 mxl5x6554hbloriiuetd6t7yb5ft1f2"}
+{"_id": "32746", "title": "Sigma Function of 310", "text": "Sigma Function of 310 0 55257 317871 317869 2017-09-16T13:20:45Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Square-Free Integer == <onlyinclude> :$\\sigma \\left({310}\\right) = 576$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$310 = 2 \\times 5 \\times 31$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({310}\\right)       | r = \\left({2 + 1}\\right) \\left({5 + 1}\\right) \\left({31 + 1}\\right)       | c = Sigma Function of Square-Free Integer }} {{eqn | r = 3 \\times 6 \\times 32       | c =  }} {{eqn | r = 3 \\times \\left({2 \\times 3}\\right) \\times 2^5       | c =  }} {{eqn | r = 2^6 \\times 3^2       | c =  }} {{eqn | r = \\left({2^3 \\times 3}\\right)^2       | c =  }} {{eqn | r = 24^2       | c =  }} {{eqn | r = 576       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Square-Free Integer Category:310 fgx8h5izqq9d9m76ftnwn651hr8qtts"}
+{"_id": "32747", "title": "Sigma Function of 322", "text": "Sigma Function of 322 0 55260 317874 317872 2017-09-16T13:21:13Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Square-Free Integer == <onlyinclude> :$\\sigma \\left({322}\\right) = 576$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$322 = 2 \\times 7 \\times 23$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({322}\\right)       | r = \\left({2 + 1}\\right) \\left({7 + 1}\\right) \\left({23 + 1}\\right)       | c = Sigma Function of Square-Free Integer }} {{eqn | r = 3 \\times 8 \\times 24       | c =  }} {{eqn | r = 3 \\times 2^3 \\times \\left({2^3 \\times 3}\\right)       | c =  }} {{eqn | r = 2^6 \\times 3^2       | c =  }} {{eqn | r = \\left({2^3 \\times 3}\\right)^2       | c =  }} {{eqn | r = 24^2       | c =  }} {{eqn | r = 576       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Square-Free Integer Category:322 7tw0yluzqzp8h0teneku39m8ua1bta0"}
+{"_id": "32748", "title": "Sigma Function of 343", "text": "Sigma Function of 343 0 55263 383904 383903 2018-12-22T16:29:10Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Power of Prime == <onlyinclude> :$\\map \\sigma {343} = 400$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == From Sigma Function of Power of Prime: :$\\map \\sigma {p^k} = \\dfrac {p^{k + 1} - 1} {p_i - 1}$ We have that: :$343 = 7^3$ Hence: {{begin-eqn}} {{eqn | l = \\map \\sigma {343}       | r = \\frac {7^4 - 1} {7 - 1}       | c =  }} {{eqn | r = \\frac {2400} 6       | c =  }} {{eqn | r = 400       | c =  }} {{eqn | r = 20^2       | c =  }} {{end-eqn}} Thus we have that: :$7^0 + 7^2 + 7^2 + 7^3 = 20^2$ {{qed}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = 400|next = Numbers whose Sigma is Square/Examples/400}}: $400$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = 400|next = Numbers whose Sigma is Square/Examples/400}}: $400$ Category:Sigma Function of Power of Prime Category:343 Category:400 ipq88t9fu4afs6dd70zbupj0gzvi375"}
+{"_id": "32749", "title": "Sigma Function of 345", "text": "Sigma Function of 345 0 55267 317877 317875 2017-09-16T13:21:35Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Square-Free Integer == <onlyinclude> :$\\sigma \\left({345}\\right) = 576$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$345 = 3 \\times 5 \\times 23$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({345}\\right)       | r = \\left({3 + 1}\\right) \\left({5 + 1}\\right) \\left({23 + 1}\\right)       | c = Sigma Function of Square-Free Integer }} {{eqn | r = 4 \\times 6 \\times 24       | c =  }} {{eqn | r = 2^2 \\times \\left({2 \\times 3}\\right) \\times \\left({2^3 \\times 3}\\right)       | c =  }} {{eqn | r = 2^6 \\times 3^2       | c =  }} {{eqn | r = \\left({2^3 \\times 3}\\right)^2       | c =  }} {{eqn | r = 24^2       | c =  }} {{eqn | r = 576       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Square-Free Integer Category:345 k9hdfy09zns4sbi7wj6r2bxagpmucpf"}
+{"_id": "32750", "title": "Sigma Function of 382", "text": "Sigma Function of 382 0 55282 352262 318092 2018-05-01T14:12:21Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Non-Square Semiprime == <onlyinclude> :$\\sigma \\left({382}\\right) = 576$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$382 = 2 \\times 191$ and so by definition is a semiprime whose prime factors are distinct. Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({382}\\right)       | r = \\left({2 + 1}\\right) \\left({191 + 1}\\right)       | c = Sigma Function of Non-Square Semiprime }} {{eqn | r = 3 \\times 192       | c =  }} {{eqn | r = 3 \\times \\left({2^6 \\times 3}\\right)       | c =  }} {{eqn | r = 2^6 \\times 3^2       | c =  }} {{eqn | r = \\left({2^3 \\times 3}\\right)^2       | c =  }} {{eqn | r = 24^2       | c =  }} {{eqn | r = 576       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Non-Square Semiprime Category:382 8acbm1vdulf64bmj1lrz6oshkujbmsa"}
+{"_id": "32751", "title": "Legendre's Condition/Lemma 1/Dependent on N Functions", "text": "Legendre's Condition/Lemma 1/Dependent on N Functions 0 55435 496503 470503 2020-10-25T13:44:39Z Prime.mover 59 wikitext text/x-wiki {{MissingLinks}} == Lemma == Let $\\mathbf y = \\paren {\\sequence {\\map {y_i} x}_{1 \\mathop \\le i \\mathop \\le N} }$ be a vector real function, such that: :$\\map {\\mathbf y} a = A$ :$\\map {\\mathbf y} b = B$ Let $J \\sqbrk {\\mathbf y}$ be a functional, such that: :$\\ds J \\sqbrk {\\mathbf y} = \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ where: :$F \\in C^2 \\closedint a b$  {{WRT}} all its variables. Then: :$\\ds \\delta^2 J \\sqbrk {\\mathbf y; \\mathbf h} = \\int_a^b \\paren {\\mathbf h' \\mathbf P \\mathbf h' + \\mathbf h \\mathbf Q \\mathbf h} \\rd x$ where: :$\\mathbf P = \\dfrac 1 2 F_{y_i'y_j'}$ :$\\mathbf Q = \\dfrac 1 2 \\paren {F_{ y_i y_j} - \\dfrac \\d {\\d x} F_{y_i y_j'} }$ == Proof == {{ProofWanted}} == Sources == {{BookReference|Calculus of Variations|1963|I.M. Gelfand|author2 = S.V. Fomin|prev = Definition:Twice Differentiable/Functional/Dependent on N functions|next = Legendre's Condition/Lemma 2/Dependent on N Functions}}: $\\S 5.29$: Generalization to n Unknown Functions Category:Calculus of Variations nom8ugjj6nbhrzoyzl65vluwk9memcs"}
+{"_id": "32752", "title": "Divisor Counting Function/Examples/1", "text": "Divisor Counting Function/Examples/1 0 55440 451771 292209 2020-03-02T09:43:28Z Prime.mover 59 Prime.mover moved page Tau Function/Examples/1 to Divisor Counting Function/Examples/1 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> The value of the $\\tau$ function for the integer $1$ is $1$. </onlyinclude> == Proof == By definition, the $\\tau$ function of an integer $n$ is the number of positive integer divisors of $n$. There is only one positive integer which is a divisor of $1$, and that is $1$ itself. Hence the result. {{qed}} Category:Tau Function p6llhtqkzlgph5ch80n8f8yooghmbhd"}
+{"_id": "32753", "title": "Sigma Function of 1", "text": "Sigma Function of 1 0 55443 379705 317546 2018-12-02T09:42:58Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\map \\sigma 1 = 1$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == By definition, the $\\sigma$ function of an integer $n$ is the sum of the positive integer divisors of $n$. There is only one positive integer which is a divisor of $1$, and that is $1$ itself. Hence the result. {{qed}} Category:Sigma Function of Integer lqa21r4lnidrvam1iqgugfvslb3sys4"}
+{"_id": "32754", "title": "Parenthesization/Examples/4", "text": "Parenthesization/Examples/4 0 55471 392225 382884 2019-02-12T21:45:51Z Prime.mover 59 wikitext text/x-wiki == Example of Parenthesization == <onlyinclude> A word of $4$ elements can be parenthesized in $5$ distinct ways: :$\\quad a_1 \\paren {a_2 \\paren {a_3 a_4} }$ :$\\quad a_1 \\paren {\\paren {a_2 a_3} a_4}$ :$\\quad \\paren {a_1 a_2} \\paren {a_3 a_4}$ :$\\quad \\paren {a_1 \\paren {a_2 a_3} } a_4$ :$\\quad \\paren {\\paren {a_1 a_2} a_3} a_4$ </onlyinclude> == Proof == From Number of Distinct Parenthesizations on Word, the number of distinct parenthesizations of a word $w$ of $n$ elements is the Catalan number $C_{n - 1}$: :$C_{n - 1} = \\dfrac 1 n \\dbinom {2 \\paren {n - 1} } {n - 1}$ For $n = 4$ we have: {{begin-eqn}} {{eqn | l = C_4       | r = \\dfrac 1 4 \\dbinom {2 \\times 3} 3       | c =  }} {{eqn | r = \\dfrac 1 4 \\times \\dfrac {6!} {3! \\times 3!}       | c = {{Defof|Binomial Coefficient}} }} {{eqn | r = \\dfrac 1 4 \\times \\dfrac {6 \\times 5 \\times 4} {3 \\times 2 \\times 1}       | c = {{Defof|Factorial}} }} {{eqn | r = 5       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Number Theory|1971|George E. Andrews|prev = Power Series Expansion for Exponential Function|next = Number of Distinct Parenthesizations on Word}}: $\\text {3-4}$ Generating Functions * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Parenthesization/Examples/3|next = Number of Paths on Graph along X-axis using Diagonal Steps}}: $42$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Parenthesization/Examples/3|next = Number of Paths on Graph along X-axis using Diagonal Steps}}: $42$ Category:Parenthesization Category:Catalan Numbers Category:5 0zj4kbos3u8p9pg4ng0n991004dq51p"}
+{"_id": "32755", "title": "Complex Exponential is Uniformly Continuous on Half-Planes/Corollary", "text": "Complex Exponential is Uniformly Continuous on Half-Planes/Corollary 0 55581 454378 437840 2020-03-14T09:10:09Z Prime.mover 59 wikitext text/x-wiki == Corollary to Complex Exponential is Uniformly Continuous on Half-Planes == <onlyinclude> Let $X$ be a set. Let $\\family {g_n}$ be a family of mappings $g_n : X \\to \\C$. Let $g_n$ converge uniformly to $g: X \\to \\C$. Let there be a constant $a \\in \\R$ such that $\\map \\Re {\\map g x} \\le a$ for all $x \\in X$. Then $\\exp g_n$ converges uniformly to $\\exp g$. </onlyinclude> == Proof == By uniform convergence, there exists $N > 0$ such that $\\cmod {\\map {g_n} x - \\map g x} \\le 1$ for all $n > N$. Then $\\map \\Re {\\map {g_n} x} \\le a + 1$. The result now follows from: :Complex Exponential is Uniformly Continuous on Half-Planes, applied to the half-plane $\\set {z \\in \\C : \\map \\Re z \\le a + 1}$ :Uniformly Continuous Function Preserves Uniform Convergence {{qed}} == Sources == * {{BookReference|Functions of One Complex Variable|1973|John B. Conway|next = Equivalence of Definitions of Uniform Absolute Convergence of Product of Complex Functions}} $\\text {VII}$: Compact and Convergence in the Space of Analytic Functions: $\\S 5$: Weierstrass Factorization Theorem: Lemma $5.7$ Category:Exponential Function Category:Uniform Continuity 3aina9ra6ungqocjtrdi605lkz8q7ml"}
+{"_id": "32756", "title": "Quasiamicable Numbers/Examples/48,75", "text": "Quasiamicable Numbers/Examples/48,75 0 55614 461023 318187 2020-04-10T11:22:48Z Prime.mover 59 wikitext text/x-wiki == Examples of Quasiamicable Pair == <onlyinclude> $48$ and $75$ form a quasiamicable pair. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\map \\sigma {48}       | r = 124       | c = {{SigmaLink|48}} }} {{eqn | l = \\map \\sigma {75}       | r = 124       | c = {{SigmaLink|75}} }} {{eqn | l = 48 + 75 + 1       | r = 124       | c =  }} {{end-eqn}} Hence the result by definition of quasiamicable pair. {{qed}} Category:Quasiamicable Numbers Category:48 Category:75 m2x4gsxvwcsq82aaucf03d9ukeig1tr"}
+{"_id": "32757", "title": "Sigma Function of 140", "text": "Sigma Function of 140 0 55646 415146 317552 2019-07-29T12:31:00Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\map \\sigma {140} = 336$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == From Sigma Function of Integer :$\\displaystyle \\map \\sigma n = \\prod_{1 \\mathop \\le i \\mathop \\le r} \\frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \\displaystyle \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$140 = 2^2 \\times 5 \\times 7$ Hence: {{begin-eqn}} {{eqn | l = \\map \\sigma {140}       | r = \\frac {2^3 - 1} {2 - 1} \\times \\frac {5^2 - 1} {5 - 1} \\times \\frac {7^2 - 1} {7 - 1}       | c =  }} {{eqn | r = \\frac 7 1 \\times \\frac {6 \\times 4} 4 \\times \\frac {8 \\times 6} 6       | c =  }} {{eqn | r = 7 \\times 6 \\times 8       | c =  }} {{eqn | r = 7 \\times \\paren {2 \\times 3} \\times 2^3       | c =  }} {{eqn | r = 2^4 \\times 3 \\times 7       | c =  }} {{eqn | r = 336       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:140 hi2p6sdll9aixcvg1wj6hxm44t6lo32"}
+{"_id": "32758", "title": "Partial Products of Uniformly Convergent Product Converge Uniformly", "text": "Partial Products of Uniformly Convergent Product Converge Uniformly 0 55686 390030 375099 2019-01-26T23:41:40Z Leigh.Samphier 3031 wikitext text/x-wiki == Definition == Let $X$ be a set. Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,}}$ be a valued field. Let $\\left\\langle{f_n}\\right\\rangle$ be a sequence of bounded mappings $f_n:X\\to \\mathbb K$. Let the product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ converge uniformly. Then the sequence of partial products converges uniformly. == Proof == Let $n_0\\in\\N$ be such that the sequence of partial products of $\\displaystyle \\prod_{n \\mathop = n_0}^\\infty f_n$ converges uniformly. By Product of Bounded Functions is Bounded, $\\displaystyle \\prod_{n \\mathop = 1}^{n_0-1}f_n$ is bounded. By Uniformly Convergent Sequence Multiplied with Function, the sequence of partial products converges uniformly. {{qed}} == Also see == * Uniform Product of Continuous Functions is Continuous * Infinite Product of Analytic Functions is Analytic Category:Uniform Convergence Category:Infinite Products hb2okugmrd5y4few6v4pdi5lygaqlxu"}
+{"_id": "32759", "title": "Logarithm of Infinite Product of Complex Functions/Corollary", "text": "Logarithm of Infinite Product of Complex Functions/Corollary 0 55695 458325 340546 2020-03-29T14:19:39Z Prime.mover 59 wikitext text/x-wiki {{MissingLinks}} == Corollary to Logarithm of Infinite Product of Complex Functions == <onlyinclude> Let $X$ be a locally compact and locally connected metric space. Let $\\sequence {f_n}$ be a sequence of continuous mappings $f_n: X \\to \\C$. Let the product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty f_n$ converge locally uniformly to $f$. Let $x_0 \\in X$. Then there exist $n_0 \\in \\N$, $k \\in \\Z$ and a neighborhood $U$ of $x_0$ such that: :$(1): \\quad \\map {f_n} x \\ne 0$ for $n \\ge n_0$ and $x \\in U$ :$(2): \\quad$ The series $\\displaystyle \\sum_{n \\mathop = n_0}^\\infty \\ln f_n$ converges uniformly on $U$ to $\\ln g + 2 k \\pi i$, where $g = \\displaystyle \\prod_{n \\mathop = n_0}^\\infty f_n$. </onlyinclude> == Outline of Proof == We construct a neighborhood of $x_0$ on which $\\displaystyle \\sum_{n \\mathop = n_0}^\\infty \\ln f_n$ and $\\ln g$ are continuous, so that $k$ is continuous and thus constant. == Proof == Let $K$ be a compact neighborhood of $x_0$. By Tail of Uniformly Convergent Product Converges Uniformly to One, there exists $N \\in \\N$ such that $\\displaystyle \\prod_{n \\mathop = N}^\\infty \\map {f_n} x \\notin \\R^-$ for all $x \\in K$. By Factors in Uniformly Convergent Product Converge Uniformly to One, there exists $M \\in \\N$ such that $\\size {\\map {f_n} x - 1} \\le \\dfrac 1 2$ for $n \\ge M$ and $x \\in K$. Let $n_0 = \\map \\max {N, M}$. Let $g = \\displaystyle \\prod_{n \\mathop = n_0}^\\infty f_n$. By Logarithm of Infinite Product of Complex Functions, there exists $k: K \\to \\Z$ such that $\\displaystyle \\sum_{n \\mathop = n_0}^\\infty \\ln f_n = \\ln g + 2 k \\pi i$ uniformly on $K$. We show that $k$ is constant on some neighborhood $U \\subset K$. By Uniform Product of Continuous Functions is Continuous and Complex Logarithm is Continuous Outside Branch, $\\ln g$ is continuous on $K$. By the Heine-Cantor Theorem, $\\ln$ is uniformly continuous on $\\map {\\overline B} {1, \\dfrac 1 2}$. By Uniformly Continuous Function Preserves Uniform Convergence, $\\displaystyle \\sum_{n \\mathop = n_0}^\\infty \\ln f_n$ converges uniformly on $K$. {{questionable|we're using here that partial products belong to $\\map {\\overline B} {1, \\dfrac 1 2}$. Need to use the Cauchy criterion for uniform products here}} By the Uniform Limit Theorem, $\\displaystyle \\sum_{n \\mathop = n_0}^\\infty \\ln f_n$ is continuous. Because $\\ln g$ and $\\displaystyle \\sum_{n \\mathop = n_0}^\\infty \\ln f_n$ are continuous, so is $k$. Let $U\\subset K$ be a connected neighborhood of $x_0$. By Continuous Mapping from Connected to Discrete Space is Constant $k$ is constant on $U$. {{qed}} Category:Infinite Products aua5s84b252zwf90pyleti5pwtb2l3v"}
+{"_id": "32760", "title": "Divisor Counting Function/Examples/60", "text": "Divisor Counting Function/Examples/60 0 56068 451785 319151 2020-03-02T09:43:29Z Prime.mover 59 Prime.mover moved page Tau Function/Examples/60 to Divisor Counting Function/Examples/60 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({60}\\right) = 12$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$60 = 2^2 \\times 3 \\times 5$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({60}\\right)       | r = \\tau \\left({2^2 \\times 3^1 \\times 5^1}\\right)       | c =  }} {{eqn | r = \\left({2 + 1}\\right) \\left({1 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 12       | c =  }} {{end-eqn}} The divisors of $60$ can be enumerated as: :$1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60$ {{OEIS|A018266}}{{qed}} Category:Tau Function Category:60 sb03tbvnrfiqeamd4dl20bo0izexchv"}
+{"_id": "32761", "title": "Sigma Function of 104", "text": "Sigma Function of 104 0 56216 479684 317755 2020-07-24T19:45:17Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\map \\sigma {104} = 210$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$104 = 2^3 \\times 13$ Hence: {{begin-eqn}} {{eqn | l = \\map \\sigma {104}       | r = \\frac {2^4 - 1} {2 - 1} \\times \\frac {13^2 - 1} {13 - 1}       | c = Sigma Function of Integer }} {{eqn | r = \\frac {16 - 1} 1 \\times \\frac {169 - 1} {12}       | c =  }} {{eqn | r = 15 \\times 14       | c =  }} {{eqn | r = 210       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:104 6cyvd2sqdk5b0s78odnr4n8pvjo8lkb"}
+{"_id": "32762", "title": "Sigma Function of 105", "text": "Sigma Function of 105 0 56217 442172 317715 2020-01-03T23:38:18Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Square-Free Integer == <onlyinclude> :$\\map \\sigma {105} = 192$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$105 = 3 \\times 5 \\times 7$ Hence: {{begin-eqn}} {{eqn | l = \\map \\sigma {105}       | r = \\paren {3 + 1} \\paren {5 + 1} \\paren {7 + 1}       | c = Sigma Function of Square-Free Integer }} {{eqn | r = 4 \\times 6 \\times 8       | c =  }} {{eqn | r = 2^2 \\times \\paren {2 \\times 3} \\times 2^3       | c =  }} {{eqn | r = 2^6 \\times 3       | c =  }} {{eqn | r = 192       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Square-Free Integer Category:105 bfc5pmjgfhijkmj2camptgunxtax9kh"}
+{"_id": "32763", "title": "Continued Fraction Expansion of Irrational Square Root/Examples/61", "text": "Continued Fraction Expansion of Irrational Square Root/Examples/61 0 56247 470651 470637 2020-05-26T06:42:40Z Prime.mover 59 wikitext text/x-wiki == Examples of Continued Fraction Expansion of Irrational Square Root == <onlyinclude> The continued fraction expansion of the square root of $61$ is given by: :$\\sqrt {61} = \\sqbrk {7, \\sequence {1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14} }$ </onlyinclude> {{OEIS|A010145}} === Convergents === {{:Continued Fraction Expansion of Irrational Square Root/Examples/61/Convergents}} == Proof == Let $\\sqrt {61} = \\sqbrk {a_0, a_1, a_2, a_3, \\ldots}$ From Partial Quotients of Continued Fraction Expansion of Irrational Square Root, the partial quotients of this continued fraction can be calculated as: :$a_r = \\floor {\\dfrac {\\floor {\\sqrt {61} } + P_r} {Q_r} }$ where: :$P_r = \\begin {cases} 0 & : r = 0 \\\\ a_{r - 1} Q_{r - 1} - P_{r - 1} & : r > 0 \\\\ \\end {cases}$ :$Q_r = \\begin {cases} 1 & : r = 0 \\\\ \\dfrac {n - {P_r}^2} {Q_{r - 1} } & : r > 0 \\\\ \\end {cases}$ {{PartialQuotientCalculator-Start | n = 61}} {{PartialQuotientCalculator | n = 61 | r =  1 | ar-1 =  7 | Qr-1 =  1 | Pr-1 = 0}} {{PartialQuotientCalculator | n = 61 | r =  2 | ar-1 =  1 | Qr-1 = 12 | Pr-1 = 7}} {{PartialQuotientCalculator | n = 61 | r =  3 | ar-1 =  4 | Qr-1 =  3 | Pr-1 = 5}} {{PartialQuotientCalculator | n = 61 | r =  4 | ar-1 =  3 | Qr-1 =  4 | Pr-1 = 7}} {{PartialQuotientCalculator | n = 61 | r =  5 | ar-1 =  1 | Qr-1 =  9 | Pr-1 = 5}} {{PartialQuotientCalculator | n = 61 | r =  6 | ar-1 =  2 | Qr-1 =  5 | Pr-1 = 4}} {{PartialQuotientCalculator | n = 61 | r =  7 | ar-1 =  2 | Qr-1 =  5 | Pr-1 = 6}} {{PartialQuotientCalculator | n = 61 | r =  8 | ar-1 =  1 | Qr-1 =  9 | Pr-1 = 4}} {{PartialQuotientCalculator | n = 61 | r =  9 | ar-1 =  3 | Qr-1 =  4 | Pr-1 = 5}} {{PartialQuotientCalculator | n = 61 | r = 10 | ar-1 =  4 | Qr-1 =  3 | Pr-1 = 7}} {{PartialQuotientCalculator | n = 61 | r = 11 | ar-1 =  1 | Qr-1 = 12 | Pr-1 = 5}} |} and the cycle is complete: :$\\sequence {1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14}$ {{qed}} Category:Continued Fractions Category:61 7tgz1nya9cmwb8mbta2ho1dtiwyxxbm"}
+{"_id": "32764", "title": "Continued Fraction Expansion of Irrational Square Root/Examples/61/Convergents", "text": "Continued Fraction Expansion of Irrational Square Root/Examples/61/Convergents 0 56249 470652 470639 2020-05-26T06:43:41Z Prime.mover 59 wikitext text/x-wiki == Convergents to Continued Fraction Expansion of $\\sqrt {61}$ == <onlyinclude> The sequence of convergents to the continued fraction expansion of the square root of $61$ begins: :$\\dfrac 7 1, \\dfrac {8} 1, \\dfrac {39} 5, \\dfrac {125} {16}, \\dfrac {164} {21}, \\dfrac {453} {58}, \\dfrac {1070} {137}, \\dfrac {1523} {195}, \\dfrac {5639} {722}, \\dfrac {24079} {3083}, \\ldots$ </onlyinclude> {{OEIS-Numerators|A041106}} {{OEIS-Denominators|A041107}} == Proof == Let $\\sqbrk {a_0, a_1, a_2, \\ldots}$ be its continued fraction expansion. Let $\\sequence {p_n}_{n \\ge \\mathop 0}$ and $\\sequence {q_n}_{n \\ge \\mathop 0}$ be its numerators and denominators. Then the $n$th convergent is $p_n / q_n$. By definition: :$p_k = \\begin {cases} a_0 & : k = 0 \\\\ a_0 a_1 + 1 & : k = 1 \\\\ a_k p_{k - 1} + p_{k - 2} & : k > 1 \\end {cases}$ :$q_k = \\begin {cases} 1 & : k = 0 \\\\ a_1 & : k = 1 \\\\ a_k q_{k - 1} + q_{k - 2} & : k > 1 \\end {cases}$ From Continued Fraction Expansion of $\\sqrt {61}$: :$\\sqrt {61} = \\sqbrk {7, \\sequence {1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14} }$ Thus the convergents are assembled: {{ConvergentCalculator-Start | a0 = 7 | a1 = 1}} {{ConvergentCalculator | k =  2 | ak =  4 | pk-1 =          8 | pk-2 =          7 | qk-1 =         1 | qk-2 =         1}} {{ConvergentCalculator | k =  3 | ak =  3 | pk-1 =         39 | pk-2 =          8 | qk-1 =         5 | qk-2 =         1}} {{ConvergentCalculator | k =  4 | ak =  1 | pk-1 =        125 | pk-2 =         39 | qk-1 =        16 | qk-2 =         5}} {{ConvergentCalculator | k =  5 | ak =  2 | pk-1 =        164 | pk-2 =        125 | qk-1 =        21 | qk-2 =        16}} {{ConvergentCalculator | k =  6 | ak =  2 | pk-1 =        453 | pk-2 =        164 | qk-1 =        58 | qk-2 =        21}} {{ConvergentCalculator | k =  7 | ak =  1 | pk-1 =       1070 | pk-2 =        453 | qk-1 =       137 | qk-2 =        58}} {{ConvergentCalculator | k =  8 | ak =  3 | pk-1 =       1523 | pk-2 =       1070 | qk-1 =       195 | qk-2 =       137}} {{ConvergentCalculator | k =  9 | ak =  4 | pk-1 =       5639 | pk-2 =       1523 | qk-1 =       722 | qk-2 =       195}} {{ConvergentCalculator | k = 10 | ak =  1 | pk-1 =      24079 | pk-2 =       5639 | qk-1 =      3083 | qk-2 =       722}} {{ConvergentCalculator | k = 11 | ak = 14 | pk-1 =      29718 | pk-2 =      24079 | qk-1 =      3805 | qk-2 =      3083}} {{ConvergentCalculator | k = 12 | ak =  1 | pk-1 =     440131 | pk-2 =      29718 | qk-1 =     56353 | qk-2 =      3805}} {{ConvergentCalculator | k = 13 | ak =  4 | pk-1 =     469849 | pk-2 =     440131 | qk-1 =     60158 | qk-2 =     56353}} {{ConvergentCalculator | k = 14 | ak =  3 | pk-1 =    2319527 | pk-2 =     469849 | qk-1 =    296985 | qk-2 =     60158}} {{ConvergentCalculator | k = 15 | ak =  1 | pk-1 =    7428430 | pk-2 =    2319527 | qk-1 =    951113 | qk-2 =    296985}} {{ConvergentCalculator | k = 16 | ak =  2 | pk-1 =    9747957 | pk-2 =    7428430 | qk-1 =   1248098 | qk-2 =    951113}} {{ConvergentCalculator | k = 17 | ak =  2 | pk-1 =   26924344 | pk-2 =    9747957 | qk-1 =   3447309 | qk-2 =   1248098}} {{ConvergentCalculator | k = 18 | ak =  1 | pk-1 =   63596645 | pk-2 =   26924344 | qk-1 =   8142716 | qk-2 =   3447309}} {{ConvergentCalculator | k = 19 | ak =  3 | pk-1 =   90520989 | pk-2 =   63596645 | qk-1 =  11590025 | qk-2 =   8142716}} {{ConvergentCalculator | k = 20 | ak =  4 | pk-1 =  335159612 | pk-2 =   90520989 | qk-1 =  42912791 | qk-2 =  11590025}} {{ConvergentCalculator | k = 21 | ak =  1 | pk-1 = 1431159437 | pk-2 =  335159612 | qk-1 = 183241189 | qk-2 =  42912791}} |} {{qed}} Category:Continued Fractions Category:61 5f0pmtfjlgu37o58x5s1l6eti82bo01"}
+{"_id": "32765", "title": "Magic Square/Examples/Order 5", "text": "Magic Square/Examples/Order 5 0 56292 300654 300567 2017-06-11T11:02:42Z Prime.mover 59 Prime.mover moved page Magic Square/Example/Order 5 to Magic Square/Examples/Order 5 wikitext text/x-wiki == Examples of Order $5$ Magic Squares == <onlyinclude> :$\\begin{array}{|c|c|c|c|c|} \\hline 23 &  6 & 19 &  2 & 15 \\\\ \\hline 10 & 18 &  1 & 14 & 22 \\\\ \\hline 17 &  5 & 13 & 21 &  9 \\\\ \\hline  4 & 12 & 25 &  8 & 16 \\\\ \\hline 11 & 23 &  7 & 20 &  3 \\\\ \\hline \\end{array}$ </onlyinclude> == Also see == * Magic Constant of Order 5 Magic Square Category:Magic Squares ahx3qr6080nia7iuamy6fgpvaedp5ti"}
+{"_id": "32766", "title": "Numbers with Euler Phi Value of 72", "text": "Numbers with Euler Phi Value of 72 0 56365 475849 455827 2020-06-24T16:21:09Z Prime.mover 59 wikitext text/x-wiki == Example of Use of Euler $\\phi$ Function == <onlyinclude> There are $17$  positive integers for which the value of the Euler $\\phi$ function is $72$: :$73, 91, 95, 111, 117, 135, 146, 148, 152, 182, 190, 216, 222, 228, 234, 252, 270$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = 72       | r = \\map \\phi {73}       | c = Euler Phi Function of Prime }} {{eqn | r = \\map \\phi {91}       | c = {{EulerPhiLink|91}} }} {{eqn | r = \\map \\phi {95}       | c = {{EulerPhiLink|95}} }} {{eqn | r = \\map \\phi {111}       | c = {{EulerPhiLink|111}} }} {{eqn | r = \\map \\phi {117}       | c = {{EulerPhiLink|117}} }} {{eqn | r = \\map \\phi {135}       | c = {{EulerPhiLink|135}} }} {{eqn | r = \\map \\phi {146}       | c = {{EulerPhiLink|146}} }} {{eqn | r = \\map \\phi {148}       | c = {{EulerPhiLink|148}} }} {{eqn | r = \\map \\phi {152}       | c = {{EulerPhiLink|152}} }} {{eqn | r = \\map \\phi {182}       | c = {{EulerPhiLink|182}} }} {{eqn | r = \\map \\phi {190}       | c = {{EulerPhiLink|190}} }} {{eqn | r = \\map \\phi {216}       | c = {{EulerPhiLink|216}} }} {{eqn | r = \\map \\phi {222}       | c = {{EulerPhiLink|222}} }} {{eqn | r = \\map \\phi {228}       | c = {{EulerPhiLink|228}} }} {{eqn | r = \\map \\phi {234}       | c = {{EulerPhiLink|234}} }} {{eqn | r = \\map \\phi {252}       | c = {{EulerPhiLink|252}} }} {{eqn | r = \\map \\phi {270}       | c = {{EulerPhiLink|270}} }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = 4 Positive Integers in Arithmetic Sequence which have Same Euler Phi Value/Mistake|next = Positive Integers which are Euler Phi Value for 17 Integers}}: $72$ Category:Euler Phi Function Category:72 36cpgnjrma7zyvwd49i6mkrq7mpnlfy"}
+{"_id": "32767", "title": "Square of 1 Less than Number Base/Examples/5", "text": "Square of 1 Less than Number Base/Examples/5 0 56424 296925 296921 2017-05-10T13:33:26Z Prime.mover 59 wikitext text/x-wiki == Example of Square of 1 Less than Number Base == <onlyinclude> The square of $5$ is expressed in base $6$ as: :$5^2 = \\left[{41}\\right]_6$ </onlyinclude> == Proof == In base $10$: {{begin-eqn}} {{eqn | l = 5^2       | r = \\left({6 - 1}\\right)^2       | c =  }} {{eqn | r = 6^2 - 2 \\times 6 + 1       | c =  }} {{eqn | r = 4 \\times 6 + 1       | c =  }} {{end-eqn}} Hence the result. {{qed}} Category:Square of 1 Less than Number Base etjz2uat44omvcyi8tz7rc2lkk253mj"}
+{"_id": "32768", "title": "Sigma Function of 836", "text": "Sigma Function of 836 0 56474 488970 318343 2020-09-18T22:38:27Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\map \\sigma {836} = 1680$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$836 = 2^2 \\times 11 \\times 19$ Hence: {{begin-eqn}} {{eqn | l = \\map \\sigma {836}       | r = \\frac {2^3 - 1} {2 - 1} \\times \\paren {11 + 1} \\times \\paren {19 + 1}       | c = Sigma Function of Integer }} {{eqn | r = \\frac 7 1 \\times 12 \\times 20       | c =  }} {{eqn | r = 7 \\times \\paren {2^2 \\times 3} \\times \\paren {2^2 \\times 5}       | c =  }} {{eqn | r = 2^4 \\times 3 \\times 5 \\times 7       | c =  }} {{eqn | r = 1680       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:836 r1t5ilql58pwkirxakqji1rijx8wigf"}
+{"_id": "32769", "title": "Euler Phi Function of 104", "text": "Euler Phi Function of 104 0 56574 415511 297622 2019-07-31T15:42:32Z Prime.mover 59 wikitext text/x-wiki == Example of Use of Euler $\\phi$ Function == <onlyinclude> :$\\map \\phi {104} = 48$ </onlyinclude> where $\\phi$ denotes the Euler $\\phi$ Function. == Proof == From Euler Phi Function of Integer: :$\\displaystyle \\map \\phi n = n \\prod_{p \\mathop \\divides n} \\paren {1 - \\frac 1 p}$ where $p \\divides n$ denotes the primes which divide $n$. We have that: :$104 = 2^3 \\times 13$ Thus: {{begin-eqn}} {{eqn | l = \\map \\phi {104}       | r = 104 \\paren {1 - \\dfrac 1 2} \\paren {1 - \\dfrac 1 {13} }       | c =  }} {{eqn | r = 104 \\times \\frac 1 2 \\times \\frac {12} {13}       | c =  }} {{eqn | r = 4 \\times 1 \\times 12       | c =  }} {{eqn | r = 48       | c =  }} {{end-eqn}} {{qed}} Category:Euler Phi Function Category:104 c5bntb3ll1mwk80jojr9g504forcbok"}
+{"_id": "32770", "title": "Euler Phi Function of 105", "text": "Euler Phi Function of 105 0 56575 394760 297623 2019-03-05T16:45:05Z Prime.mover 59 wikitext text/x-wiki == Example of Euler $\\phi$ Function of Square-Free Integer == <onlyinclude> :$\\map \\phi {105} = 48$ </onlyinclude> where $\\phi$ denotes the Euler $\\phi$ Function. == Proof == From Euler Phi Function of Square-Free Integer: :$\\displaystyle \\map \\phi n = \\prod_{\\substack {p \\mathop \\divides n \\\\ p \\mathop > 2} } \\paren {p - 1}$ where $p \\divides n$ denotes the primes which divide $n$. We have that: :$105 = 3 \\times 5 \\times 7$ and so is square-free. Thus: {{begin-eqn}} {{eqn | l = \\map \\phi {105}       | r = \\paren {3 - 1} \\paren {5 - 1} \\paren {7 - 1}       | c =  }} {{eqn | r = 2 \\times 4 \\times 6       | c =  }} {{eqn | r = 48       | c =  }} {{end-eqn}} {{qed}} Category:Euler Phi Function of Square-Free Integer Category:105 p9zwsy94v795fymhtmghu8rg3t14cai"}
+{"_id": "32771", "title": "Euler Phi Function of 15", "text": "Euler Phi Function of 15 0 56579 297640 2017-05-16T20:46:52Z Prime.mover 59 Created page with \"== Example of Euler $\\phi$ Function of Non-Square Semiprime == <onlyinclude> :$\\phi \\left({15}\\right) = 8$ </onlyinclude> where...\" wikitext text/x-wiki == Example of Euler $\\phi$ Function of Non-Square Semiprime == <onlyinclude> :$\\phi \\left({15}\\right) = 8$ </onlyinclude> where $\\phi$ denotes the Euler $\\phi$ Function. == Proof == We have that: :$15 = 3 \\times 5$ Thus: {{begin-eqn}} {{eqn | l = \\phi \\left({15}\\right)       | r = \\left({3 - 1}\\right) \\left({5 - 1}\\right)       | c = Euler $\\phi$ Function of Non-Square Semiprime }} {{eqn | r = 2 \\times 4       | c =  }} {{eqn | r = 8       | c =  }} {{end-eqn}} {{qed}} Category:Euler Phi Function of Non-Square Semiprime Category:15 38qrkw3pc8084fyd3flsrr96rpcrmzl"}
+{"_id": "32772", "title": "Divisor Counting Function/Examples/105", "text": "Divisor Counting Function/Examples/105 0 56595 451773 438890 2020-03-02T09:43:28Z Prime.mover 59 Prime.mover moved page Tau Function/Examples/105 to Divisor Counting Function/Examples/105 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {105} = 8$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$105 = 3 \\times 5 \\times 7$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {105}       | r = \\map \\tau {3^1 \\times 5^1 \\times 7^1}       | c =  }} {{eqn | r = \\paren {1 + 1} \\paren {1 + 1} \\paren {1 + 1}       | c =  }} {{eqn | r = 8       | c =  }} {{end-eqn}} The divisors of $105$ can be enumerated as: :$1, 3, 5, 7, 15, 21, 35, 105$ {{OEIS|A018286}}{{qed}} Category:Tau Function Category:105 2htkwufti6wb7pp0xrnn69im8wdkrev"}
+{"_id": "32773", "title": "Euler Phi Function of 165", "text": "Euler Phi Function of 165 0 56604 297726 2017-05-17T20:19:14Z Prime.mover 59 Created page with \"== Example of Euler $\\phi$ Function of Square-Free Integer == <onlyinclude> :$\\phi \\left({165}\\right) = 80$ </onlyinclude> where...\" wikitext text/x-wiki == Example of Euler $\\phi$ Function of Square-Free Integer == <onlyinclude> :$\\phi \\left({165}\\right) = 80$ </onlyinclude> where $\\phi$ denotes the Euler $\\phi$ Function. == Proof == From Euler Phi Function of Square-Free Integer: :$\\displaystyle \\phi \\left({n}\\right) = \\prod_{\\substack {p \\mathop \\backslash n \\\\ p \\mathop > 2} } \\left({p - 1}\\right)$ where $p \\mathop \\backslash n$ denotes the primes which divide $n$. We have that: :$165 = 3 \\times 5 \\times 11$ and so is square-free. Thus: {{begin-eqn}} {{eqn | l = \\phi \\left({165}\\right)       | r = \\left({3 - 1}\\right) \\left({5 - 1}\\right) \\left({11 - 1}\\right)       | c =  }} {{eqn | r = 2 \\times 4 \\times 10       | c =  }} {{eqn | r = 80       | c =  }} {{end-eqn}} {{qed}} Category:Euler Phi Function of Square-Free Integer Category:165 jiw6cuy5f1dkk50qucprga9b4gcmgmf"}
+{"_id": "32774", "title": "Euler Phi Function of 35", "text": "Euler Phi Function of 35 0 56612 440076 297741 2019-12-18T14:17:40Z Prime.mover 59 wikitext text/x-wiki == Example of Euler $\\phi$ Function of Non-Square Semiprime == <onlyinclude> :$\\map \\phi {35} = 24$ </onlyinclude> where $\\phi$ denotes the Euler $\\phi$ Function. == Proof == We have that: :$35 = 5 \\times 7$ Thus: {{begin-eqn}} {{eqn | l = \\map \\phi {35}       | r = \\paren {5 - 1} \\paren {7 - 1}       | c = Euler $\\phi$ Function of Non-Square Semiprime }} {{eqn | r = 4 \\times 6       | c =  }} {{eqn | r = 24       | c =  }} {{end-eqn}} {{qed}} Category:Euler Phi Function of Non-Square Semiprime Category:35 mjjv5cjj5ypy7kqkvzap4kg15ce1q85"}
+{"_id": "32775", "title": "Tau Function of 35", "text": "Tau Function of 35 0 56613 297742 2017-05-17T20:54:37Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({35}\\right) = 4$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Functi...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({35}\\right) = 4$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$35 = 5 \\times 7$ Thus: :$\\tau \\left({35}\\right) = \\tau \\left({5^1 \\times 7^1}\\right) = \\left({1 + 1}\\right) \\left({1 + 1}\\right) = 4$ The divisors of $35$ can be enumerated as: :$1, 5, 7, 35$ {{qed}} Category:Tau Function Category:35 148wl4amvfuqejw7csdysz46kyd7184"}
+{"_id": "32776", "title": "Prime Magic Square/Examples/Order 3/Smallest", "text": "Prime Magic Square/Examples/Order 3/Smallest 0 56666 481551 299484 2020-08-13T10:19:40Z RandomUndergrad 3904 wikitext text/x-wiki == Example of Order $3$ Prime Magic Square == This order $3$ prime magic square has the smallest elements: <onlyinclude> :$\\begin{array}{|c|c|c|} \\hline 67 & 1 & 43 \\\\ \\hline 13 & 37 & 61 \\\\ \\hline 31 & 73 & 7 \\\\ \\hline \\end{array}$ </onlyinclude> == Proof == For the purpose of this magic square only, we consider $1$ as a prime. A simple parity argument can show that $2$ cannot be included in a prime magic square: If it is, the row containing $2$ sum to an even number, while a row not containing $2$ will sum to an odd number. {{Improve|I'm drawing a blank on how to present the following result clearly}} We aim to show that all elements of an order $3$ prime magic square has the same remainder when divided by $3$. There are two parts to this: === $3$ cannot be used === For simplicity, we denote the numbers in the cells by their remainders when divided by $3$. Note that $3$ is the only prime divisible by $3$. We define the off-diagonals as: :$\\begin{array}{|c|c|c|} \\hline * &  &  \\\\ \\hline  & * &  \\\\ \\hline  &  & * \\\\ \\hline \\end{array} \\begin{array}{|c|c|c|} \\hline  & * &  \\\\ \\hline  &  & * \\\\ \\hline * &  &  \\\\ \\hline \\end{array} \\begin{array}{|c|c|c|} \\hline  &  & * \\\\ \\hline * &  &  \\\\ \\hline  & * &  \\\\ \\hline \\end{array} \\begin{array}{|c|c|c|} \\hline  &  & * \\\\ \\hline  & * &  \\\\ \\hline * &  &  \\\\ \\hline \\end{array} \\begin{array}{|c|c|c|} \\hline  & * &  \\\\ \\hline * &  &  \\\\ \\hline  &  & * \\\\ \\hline \\end{array} \\begin{array}{|c|c|c|} \\hline * &  &  \\\\ \\hline  &  & * \\\\ \\hline  & * &  \\\\ \\hline \\end{array}$ We also observe that, by switching rows and columns, the numbers in each row and column remains unchanged, while the two diagonals become two off-diagonals sharing one cell. Therefore the position of the numbers do not matter in the most part. Suppose $3$ is used in the square. {{WLOG}} there are only two cases: ==== Case $1$: The row containing $3$ has numbers with all remainders ==== We have: :$\\begin{array}{|c|c|c|} \\hline 0 & 1 & 2 \\\\ \\hline  &  &  \\\\ \\hline  &  &  \\\\ \\hline \\end{array}$ Hence the row sum is divisible by $3$. Since: :$1 + 1 \\equiv 2 \\pmod 3$ :$2 + 2 \\equiv 1 \\pmod 3$ :$1 + 2 \\equiv 0 \\pmod 3$ there is a unique way to fill in the columns: :$\\begin{array}{|c|c|c|} \\hline 0 & 1 & 2 \\\\ \\hline 1 & 1 & 2 \\\\ \\hline 2 & 1 & 2 \\\\ \\hline \\end{array}$ Note that the order of $1$ and $2$ in the leftmost column do not matter due to symmetry. The sums of rows $2$ and $3$ are not divisible by $3$. Hence this case cannot occur. {{qed|lemma}} ==== Case $2$: The row containing $3$ leave out numbers with some remainder ==== {{WLOG}} suppose $2$ is not used. Then: :$\\begin{array}{|c|c|c|} \\hline 0 & 1 & 1 \\\\ \\hline  &  &  \\\\ \\hline  &  &  \\\\ \\hline \\end{array}$ Filling in the columns: :$\\begin{array}{|c|c|c|} \\hline 0 & 1 & 1 \\\\ \\hline 1 & 2 & 2 \\\\ \\hline 1 & 2 & 2 \\\\ \\hline \\end{array}$ All off-diagonals sum to $1$, which is not $1 + 1 = 2$. Hence this case cannot occur. {{qed|lemma}} === Primes of remainder $1, 2$ cannot be mixed === {{WLOG}} suppose there are $2$ $1$'s and $1$ $2$. Then the row sum is not divisible by $3$. We have: :$\\begin{array}{|c|c|c|} \\hline 1 & 1 & 2 \\\\ \\hline  &  &  \\\\ \\hline  &  &  \\\\ \\hline \\end{array}$ Filling in the first and third columns: :$\\begin{array}{|c|c|c|} \\hline 1 & 1 & 2 \\\\ \\hline 1 &  & 1 \\\\ \\hline 2 &  & 1 \\\\ \\hline \\end{array}$ Finally, filling up the rows: :$\\begin{array}{|c|c|c|} \\hline 1 & 1 & 2 \\\\ \\hline 1 & 2 & 1 \\\\ \\hline 2 & 1 & 1 \\\\ \\hline \\end{array}$ There must be an off-diagonal with sum divisible by $3$. Hence this case cannot occur. {{qed|lemma}} Using this result, we divide the primes $\\le 73$ into two groups: :Remainder of $1$: $\\set {1, 7, 13, 19, 31, 37, 43, 61, 67, 73}$ :Remainder of $2$: $\\set {5, 11, 17, 23, 29, 41, 47, 53, 59, 71}$ We only need to show these primes cannot form a smaller magic square. Consider: :$\\begin{array}{|c|c|c|} \\hline a & b & c \\\\ \\hline d & e & f \\\\ \\hline g & h & i \\\\ \\hline \\end{array}$ Let $C$ be the magic constant. Then: {{begin-eqn}} {{eqn | l = 4C       | r = \\paren {a + e + i} + \\paren {b + e + h} + \\paren {c + e + g} + \\paren {d + e + f}       | c = These are all lines passing through the center }} {{eqn | r = \\paren {a + b + c + d + e + f + g + h + i} + 3 e       | c = Center counted $4$ times }} {{eqn | r = 3 C + 3 e }} {{end-eqn}} Hence $e = \\dfrac C 3$, which is $\\dfrac 1 9$ of the sum of all numbers in the square. We have: :$1 + 7 + 13 + 19 + 31 + 37 + 43 + 61 + 67 + 73 = 352$ :$5 + 11 + 17 + 23 + 29 + 41 + 47 + 53 + 59 + 71 = 356$ $352$ and $356$ have remainders $1$ and $5$ when divided by $9$. In the lists: :$1, 19, 37, 73$ have a remainder of $1$ when divided by $9$. :$5, 23, 41, 59$ have a remainder of $5$ when divided by $9$. Omitting each number gives the corresponding center square values: :$39, 37, 35, 31$ for the first list :$39, 37, 35, 33$ for the second list Only $31, 37$ of the first list are possible candidates. However: :$73 + 31 > 93$ Hence $31$ fail to produce a magic square. This leaves $37$, which possibility is demonstrated above. {{qed}} == Also see == * Magic Constant of Smallest Prime Magic Square == Sources == * {{MathWorld|Prime Magic Square|PrimeMagicSquare}} Category:Prime Magic Squares rm8quqhjni9pd5p3v8lxkgilju1wocj"}
+{"_id": "32777", "title": "Tau Function of 120", "text": "Tau Function of 120 0 56751 311826 298498 2017-08-19T11:36:27Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({120}\\right) = 16$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$120 = 2^3 \\times 3 \\times 5$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({120}\\right)       | r = \\tau \\left({2^3 \\times 3^1 \\times 5^1}\\right)       | c =  }} {{eqn | r = \\left({3 + 1}\\right) \\left({1 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 16       | c =  }} {{end-eqn}} The divisors of $120$ can be enumerated as: :$1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120$ {{OEIS|A018293}}{{qed}} Category:Tau Function Category:120 b9jhzy1sjyd0lx0pivvzu4lw82nskgs"}
+{"_id": "32778", "title": "Tau Function of 6", "text": "Tau Function of 6 0 56966 304284 299213 2017-07-10T11:16:00Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({6}\\right) = 4$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$6 = 2 \\times 3$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({6}\\right)       | r = \\tau \\left({2^1 \\times 3^1}\\right)       | c =  }} {{eqn | r = \\left({1 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 4       | c =  }} {{end-eqn}} The divisors of $6$ can be enumerated as: :$1, 2, 3, 6$ {{qed}} Category:Tau Function Category:6 m4moxy6pt99zs65yhbvjapb73oboovj"}
+{"_id": "32779", "title": "Tau Function of 28", "text": "Tau Function of 28 0 56967 304291 299215 2017-07-10T13:15:30Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({28}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$28 = 2^2 \\times 7$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({28}\\right)       | r = \\tau \\left({2^2 \\times 7^1}\\right)       | c =  }} {{eqn | r = \\left({2 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $28$ can be enumerated as: :$1, 2, 4, 7, 14, 28$ {{OEIS|A018254}}{{qed}} Category:Tau Function Category:28 kozulcl7ntshl4lk0ccw3yp6kdnjcfa"}
+{"_id": "32780", "title": "Tau Function of 140", "text": "Tau Function of 140 0 56970 300366 299221 2017-06-07T20:22:04Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({140}\\right) = 12$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$140 = 2^2 \\times 5 \\times 7$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({140}\\right)       | r = \\tau \\left({2^2 \\times 5^1 \\times 7^1}\\right)       | c =  }} {{eqn | r = \\left({2 + 1}\\right) \\left({1 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 12       | c =  }} {{end-eqn}} The divisors of $140$ can be enumerated as: :$1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140$ {{OEIS|A018301}}{{qed}} Category:Tau Function Category:140 03lobprrp437yfmyxrzusvctn5mqow1"}
+{"_id": "32781", "title": "Tau Function of 270", "text": "Tau Function of 270 0 56972 299223 2017-05-26T21:24:58Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({270}\\right) = 16$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Func...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({270}\\right) = 16$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$270 = 2 \\times 3^3 \\times 5$ Thus: :$\\tau \\left({270}\\right) = \\tau \\left({2^1 \\times 3^3 \\times 5^1}\\right) = \\left({1 + 1}\\right) \\left({3 + 1}\\right) \\left({1 + 1}\\right) = 16$ The divisors of $270$ can be enumerated as: :$1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135, 270$ {{qed}} Category:Tau Function Category:270 53q4gcguc1el4g5mz3s3spxvtkbucc3"}
+{"_id": "32782", "title": "Sigma Function of 270", "text": "Sigma Function of 270 0 56973 318217 318215 2017-09-16T18:25:41Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\sigma \\left({270}\\right) = 720$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == From Sigma Function of Integer :$\\displaystyle \\sigma \\left({n}\\right) = \\prod_{1 \\mathop \\le i \\mathop \\le r} \\frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \\displaystyle \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$270 = 2 \\times 3^3 \\times 5$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({270}\\right)       | r = \\left({2 + 1}\\right) \\times \\frac {3^4 - 1} {3 - 1} \\times \\left({5 + 1}\\right)       | c =  }} {{eqn | r = 3 \\times \\frac {81 - 1} 2 \\times 6       | c =  }} {{eqn | r = 3 \\times 40 \\times 6       | c =  }} {{eqn | r = 3 \\times \\left({2^3 \\times 5}\\right) \\times \\left({2 \\times 3}\\right)       | c =  }} {{eqn | r = 2^4 \\times 3^2 \\times 5       | c =  }} {{eqn | r = 720       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:270 8f9ogza4vhxcmt0zwjuesjjyatmwm20"}
+{"_id": "32783", "title": "Tau Function of 496", "text": "Tau Function of 496 0 56975 304296 301321 2017-07-10T21:04:44Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({496}\\right) = 10$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$496 = 2^4 \\times 31$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({496}\\right)       | r = \\tau \\left({2^4 \\times 31^1}\\right)       | c =  }} {{eqn | r = \\left({4 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 10       | c =  }} {{end-eqn}} The divisors of $496$ can be enumerated as: :$1, 2, 4, 8, 16, 31, 62, 124, 248, 496$ {{OEIS|A018487}}{{qed}} Category:Tau Function Category:496 21pb8k2a3ndqsh3wik0lkwnekrrraa4"}
+{"_id": "32784", "title": "Sigma Function of 496", "text": "Sigma Function of 496 0 56976 317812 317810 2017-09-16T13:05:07Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\sigma \\left({496}\\right) = 992$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == From Sigma Function of Integer :$\\displaystyle \\sigma \\left({n}\\right) = \\prod_{1 \\mathop \\le i \\mathop \\le r} \\frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \\displaystyle \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$496 = 2^4 \\times 31$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({496}\\right)       | r = \\left({2^5 - 1}\\right) \\times \\left({31 + 1}\\right)       | c =  }} {{eqn | r = 31 \\times 32       | c =  }} {{eqn | r = 31 \\times 2^5       | c =  }} {{eqn | r = 992       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:496 nchj6b8510ltl0p3vdpo9zxxjul9xx3"}
+{"_id": "32785", "title": "Prime Magic Square/Examples/Order 12/Smallest with Consecutive Primes from 3", "text": "Prime Magic Square/Examples/Order 12/Smallest with Consecutive Primes from 3 0 57043 462309 462295 2020-04-16T08:09:19Z RandomUndergrad 3904 wikitext text/x-wiki == Example of Order $12$ Prime Magic Square == This order $12$ prime magic square is the smallest whose elements are consecutive odd primes starting from $3$ (including $1$). The primes themselves are the $143$ consecutive odd primes from $3$ up to $827$. This magic square has magic constant $4514$. <onlyinclude> :$\\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \\hline   1 & 823 & 821 & 809 & 811 & 797 &  19 &  29 & 313 &  31 &  23 &  37 \\\\ \\hline  89 &  83 & 211 &  79 & 641 & 631 & 619 & 709 & 617 &  53 &  43 & 739 \\\\ \\hline  97 & 227 & 103 & 107 & 193 & 557 & 719 & 727 & 607 & 139 & 757 & 281 \\\\ \\hline 223 & 653 & 499 & 197 & 109 & 113 & 563 & 479 & 173 & 761 & 587 & 157 \\\\ \\hline 367 & 379 & 521 & 383 & 241 & 467 & 257 & 263 & 269 & 167 & 601 & 599 \\\\ \\hline 349 & 359 & 353 & 647 & 389 & 331 & 317 & 311 & 409 & 307 & 293 & 449 \\\\ \\hline 503 & 523 & 233 & 337 & 547 & 397 & 421 &  17 & 401 & 271 & 431 & 433 \\\\ \\hline 229 & 491 & 373 & 487 & 461 & 251 & 443 & 463 & 137 & 439 & 457 & 283 \\\\ \\hline 509 & 199 &  73 & 541 & 347 & 191 & 181 & 569 & 577 & 571 & 163 & 593 \\\\ \\hline 661 & 101 & 643 & 239 & 691 & 701 & 127 & 131 & 179 & 613 & 277 & 151 \\\\ \\hline 659 & 673 & 677 & 683 &  71 &  67 &  61 &  47 &  59 & 743 & 733 &  41 \\\\ \\hline 827 &   3 &   7 &   5 &  13 &  11 & 787 & 769 & 773 & 419 & 149 & 751 \\\\ \\hline \\end{array}$ </onlyinclude> == Proof == It is sufficient to show that for $n \\le 11$, there is no order $n$ prime magic square. We will show this fact regardless of whether $1$ is included in the magic square. === Order $2$ === First the order $2$ magic square is eliminated. Consider: :$\\begin{array}{|c|c|} \\hline  a & b \\\\ \\hline  c & d \\\\ \\hline \\end{array}$ Then we must have $a + b = a + c$. So $b = c$, so they are not distinct, so this array cannot be a magic square. {{qed|lemma}} Next, by definition of magic square, each row adds up to the magic constant. Hence the sum of all entries of the magic square of order $n$ must be divisible by $n$. Here is a list of: :$1 + $ the sums of the first $n^2 - 1$ odd primes :sums of the first $n^2$ odd primes :their divisibility by $n$: $\\begin{array}{|c|c|c|} \\hline    & \\text{Including } 1 & \\text{Divisible by } n? & \\text{Not including } 1 & \\text{Divisible by } n? \\\\ \\hline  3 &    99 & \\text{Yes} &   127 & \\text{No} \\\\ \\hline  4 &   380 & \\text{Yes} &   438 & \\text{No} \\\\ \\hline  5 &  1059 & \\text{No}  &  1159 & \\text{No} \\\\ \\hline  6 &  2426 & \\text{No}  &  2582 & \\text{No} \\\\ \\hline  7 &  4887 & \\text{No}  &  5115 & \\text{No} \\\\ \\hline  8 &  8892 & \\text{No}  &  9204 & \\text{No} \\\\ \\hline  9 & 15115 & \\text{No}  & 15535 & \\text{No} \\\\ \\hline 10 & 24132 & \\text{No}  & 24678 & \\text{No} \\\\ \\hline 11 & 36887 & \\text{No}  & 37559 & \\text{No} \\\\ \\hline \\end{array}$ So the only potential magic squares are of order $3$ or $4$. These magic squares, if they exist, must have magic constants $33$ and $95$. === Order $3$ === The first $8$ primes are $3, 5, 7, 11, 13, 17, 19, 23$. Because every prime and $1$ appears exactly once in a magic square, each number contributes to at least $2$ sums: the row and column sums. However, there is only one way to express $32$ as a sum of $2$ primes less than $23$: :$32 = 19 + 13$ and so $33$ cannot be made from a sum that includes $1$ in $2$ distinct ways. Thus an order $3$ prime magic square cannot be made. === Order $4$ === Every row of an order $4$ magic square contains $4$ odd numbers. These sum to an even number. But the magic constant of an order $4$ prime magic square, as shown above, is $95$. Hence it is not possible to create an order $4$ prime magic square. {{qed|lemma}} Hence there can be no prime magic square whose order is less than $12$. Thus the order $12$ prime magic square is the smallest whose elements are consecutive odd primes starting from $3$ (including or not including $1$). {{qed}} == Also see == * Magic Constant of Smallest Prime Magic Square with Consecutive Primes from 3 == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Square of Reversal of Small-Digit Number/Examples/12|next = Prime Magic Square/Examples/Order 12/Smallest with Consecutive Primes/Mistake}}: $144$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Square of Reversal of Small-Digit Number/Examples/12|next = Prime Magic Square/Examples/Order 12/Smallest with Consecutive Primes/Mistake}}: $144$ * {{MathWorld|Prime Magic Square|PrimeMagicSquare}} Category:Prime Magic Squares aritkhflesvmf45cqcy250otnkv5tim"}
+{"_id": "32786", "title": "Euler Phi Function of 164", "text": "Euler Phi Function of 164 0 57145 437843 299795 2019-12-05T15:22:47Z Prime.mover 59 wikitext text/x-wiki == Example of Use of Euler $\\phi$ Function == <onlyinclude> :$\\map \\phi {164} = 80$ </onlyinclude> where $\\phi$ denotes the Euler $\\phi$ Function. == Proof == From Euler Phi Function of Integer: :$\\displaystyle \\map \\phi n = n \\prod_{p \\mathop \\divides n} \\paren {1 - \\frac 1 p}$ where $p \\divides n$ denotes the primes which divide $n$. We have that: :$164 = 2^2 \\times 41$ Thus: {{begin-eqn}} {{eqn | l = \\map \\phi {164}       | r = 164 \\paren {1 - \\dfrac 1 2} \\paren {1 - \\dfrac 1 {41} }       | c =  }} {{eqn | r = 164 \\times \\frac 1 2 \\times \\frac {40} {41}       | c =  }} {{eqn | r = 2 \\times 1 \\times 40       | c =  }} {{eqn | r = 80       | c =  }} {{end-eqn}} {{qed}} Category:Euler Phi Function Category:164 p4uxlt8z13je88kx51h1ub82f2kob5w"}
+{"_id": "32787", "title": "Tau Function of 18", "text": "Tau Function of 18 0 57163 437769 299847 2019-12-04T16:52:45Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {18} = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$18 = 2 \\times 3^2$ Thus: :$\\map \\tau {18} = \\map \\tau {2^1 \\times 3^2} = \\paren {1 + 1} \\paren {2 + 1} = 6$ The divisors of $18$ can be enumerated as: :$1, 2, 3, 6, 9, 18$ {{OEIS|A018251}}{{qed}} Category:Tau Function Category:18 6mn7ak5k9wqy7vn876hfo5jym5mrp6f"}
+{"_id": "32788", "title": "Tau Function of 27", "text": "Tau Function of 27 0 57164 299848 2017-06-01T17:38:30Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({27}\\right) = 4$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Functi...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({27}\\right) = 4$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == {{begin-eqn}} {{eqn | l = \\tau \\left({27}\\right)       | r = \\tau \\left({3^3}\\right)       | c =  }} {{eqn | r = 3 + 1       | c = Tau of Power of Prime }} {{eqn | r = 4       | c =  }} {{end-eqn}} The divisors of $27$ can be enumerated as: :$1, 3, 9, 27$ {{qed}} Category:Tau Function Category:27 2wzd93gzsz3x4pam1hs2hjn0merpq37"}
+{"_id": "32789", "title": "Tau Function of 24", "text": "Tau Function of 24 0 57165 439339 299851 2019-12-12T16:51:03Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {24} = 8$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$24 = 2^3 \\times 3$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {24}       | r = \\map \\tau {2^3 \\times 3^1}       | c =  }} {{eqn | r = \\paren {3 + 1} \\paren {1 + 1}       | c =  }} {{eqn | r = 8       | c =  }} {{end-eqn}} The divisors of $24$ can be enumerated as: :$1, 2, 3, 4, 6, 8, 12, 24$ {{OEIS|A018253}}{{qed}} Category:Tau Function Category:24 12jn4nnaall53zx6orjfk1t5ittt0va"}
+{"_id": "32790", "title": "Tau Function of 32", "text": "Tau Function of 32 0 57166 299850 2017-06-01T17:45:57Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({32}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Functi...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({32}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == {{begin-eqn}} {{eqn | l = \\tau \\left({32}\\right)       | r = \\tau \\left({2^5}\\right)       | c =  }} {{eqn | r = 5 + 1       | c = Tau of Power of Prime }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $32$ can be enumerated as: :$1, 2, 4, 8, 16, 32$ {{qed}} Category:Tau Function Category:32 4pgdl9biib1b9f07pkgmtu3idyvrlv2"}
+{"_id": "32791", "title": "Tau Function of 56", "text": "Tau Function of 56 0 57167 478128 300358 2020-07-13T20:19:32Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {56} = 8$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$56 = 2^3 \\times 7$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {56}       | r = \\map \\tau {2^3 \\times 7^1}       | c =  }} {{eqn | r = \\paren {3 + 1} \\paren {1 + 1}       | c =  }} {{eqn | r = 8       | c =  }} {{end-eqn}} The divisors of $56$ can be enumerated as: :$1, 2, 4, 7, 8, 14, 28, 56$ {{OEIS|A018265}}{{qed}} Category:Tau Function Category:56 1ohrcsenhcfs5cro7d76r5byksnjw0r"}
+{"_id": "32792", "title": "Tau Function of 64", "text": "Tau Function of 64 0 57168 299853 2017-06-01T17:51:50Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({64}\\right) = 7$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Functi...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({64}\\right) = 7$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == {{begin-eqn}} {{eqn | l = \\tau \\left({64}\\right)       | r = \\tau \\left({2^6}\\right)       | c =  }} {{eqn | r = 6 + 1       | c = Tau of Power of Prime }} {{eqn | r = 7       | c =  }} {{end-eqn}} The divisors of $64$ can be enumerated as: :$1, 2, 4, 8, 16, 32, 64$ {{qed}} Category:Tau Function Category:64 929fehhxkun7m814g7j21flfet4nklw"}
+{"_id": "32793", "title": "Euler Phi Function of 194", "text": "Euler Phi Function of 194 0 57256 300154 300152 2017-06-05T05:53:24Z Prime.mover 59 wikitext text/x-wiki == Example of Use of Euler $\\phi$ Function == <onlyinclude> :$\\phi \\left({194}\\right) = 96$ </onlyinclude> where $\\phi$ denotes the Euler $\\phi$ Function. == Proof == We have that: :$194 = 2 \\times 97$ Thus: {{begin-eqn}} {{eqn | l = \\phi \\left({194}\\right)       | r = 97 - 1       | c = Euler Phi Function of 2 times Odd Prime }} {{eqn | r = 96       | c =  }} {{end-eqn}} {{qed}} Category:Euler Phi Function of 2 times Odd Prime Category:194 p556xb2odwsuha7fesumz8jcf821x1g"}
+{"_id": "32794", "title": "Euler Phi Function of 195", "text": "Euler Phi Function of 195 0 57257 442119 300153 2020-01-03T17:19:26Z Prime.mover 59 wikitext text/x-wiki == Example of Euler $\\phi$ Function of Square-Free Integer == <onlyinclude> :$\\map \\phi {195} = 96$ </onlyinclude> where $\\phi$ denotes the Euler $\\phi$ Function. == Proof == From Euler Phi Function of Square-Free Integer: :$\\displaystyle \\map \\phi n = \\prod_{\\substack {p \\mathop \\divides n \\\\ p \\mathop > 2} } \\paren {p - 1}$ where $p \\divides n$ denotes the primes which divide $n$. We have that: :$195 = 3 \\times 5 \\times 13$ and so is square-free. Thus: {{begin-eqn}} {{eqn | l = \\map \\phi {195}       | r = \\paren {3 - 1} \\paren {5 - 1} \\paren {13 - 1}       | c =  }} {{eqn | r = 2 \\times 4 \\times 12       | c =  }} {{eqn | r = 96       | c =  }} {{end-eqn}} {{qed}} Category:Euler Phi Function of Square-Free Integer Category:195 2rfjcc03p4hfc5wxk9g35vnqcr4w4qe"}
+{"_id": "32795", "title": "Tau Function of 70", "text": "Tau Function of 70 0 57331 419360 300361 2019-08-20T15:16:23Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {70} = 8$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$70 = 2 \\times 5 \\times 7$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {70}       | r = \\map \\tau {2^1 \\times 5^1 \\times 7^1}       | c =  }} {{eqn | r = \\paren {1 + 1} \\paren {1 + 1} \\paren {1 + 1}       | c =  }} {{eqn | r = 8       | c =  }} {{end-eqn}} The divisors of $70$ can be enumerated as: :$1, 2, 5, 7, 10, 14, 35, 70$ {{OEIS|A018270}}{{qed}} Category:Tau Function Category:70 srddfybjijycf4kl1td3oi05t3fmzmt"}
+{"_id": "32796", "title": "Amicable Pair/Examples/220-284", "text": "Amicable Pair/Examples/220-284 0 57457 495109 445192 2020-10-16T22:16:34Z Prime.mover 59 wikitext text/x-wiki == Example of Amicable Pair == <onlyinclude> $220$ and $284$ are the smallest amicable pair: :$\\map \\sigma {220} = \\map \\sigma {284} = 504 = 220 + 284$ </onlyinclude> == Proof == Let $\\map s n$ denote the aliquot sum of $n$. By definition: :$\\map s n = \\map \\sigma n - n$ where $\\map \\sigma n$ denotes the $\\sigma$ function. Thus: {{begin-eqn}} {{eqn | l = \\map s {220}       | r = \\map \\sigma {220} - 220       | c =  }} {{eqn | r = 504 - 220       | c = $\\sigma$ of $220$ }} {{eqn | r = 284       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \\map s {284}       | r = \\map \\sigma {284} - 284       | c =  }} {{eqn | r = 504 - 284       | c = $\\sigma$ of $284$ }} {{eqn | r = 220       | c =  }} {{end-eqn}} It can be determined by inspection of the aliquot sums of all smaller integers that there is no smaller amicable pair. {{qed}} == Historical Note == {{:Amicable Pair/Examples/220-284/Historical Note}} == Sources == * {{BookReference|History of the Theory of Numbers|1919|Leonard Eugene Dickson|volume = I|prev = Definition:Aliquot Part|next = Amicable Pair/Examples/220-284/Historical Note}}: Preface * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = 230|next = 220}}: $220$ * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Definition:Amicable Pair/Definition 1|next = Symbols:Abbreviations/A/amp|entry = amicable numbers}} * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = 230|next = 220}}: $220$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Amicable Pair/Historical Note|next = Mathematician:André-Marie Ampère|entry = amicable numbers}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Amicable Pair/Historical Note|next = Mathematician:André-Marie Ampère|entry = amicable numbers}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Amicable Pair/Definition 1|next = Amicable Pair/Examples/220-284/Historical Note|entry = amicable numbers}} * {{MathWorld|Amicable Pair|AmicablePair}} Category:Amicable Pairs 156m1wuwhpnymqw0qxn9gftolka1wxn"}
+{"_id": "32797", "title": "Sigma Function of 220", "text": "Sigma Function of 220 0 57461 454512 318160 2020-03-15T06:22:18Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\map \\sigma {220} = 504$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == From Sigma Function of Integer :$\\displaystyle \\map \\sigma n = \\prod_{1 \\mathop \\le i \\mathop \\le r} \\frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \\displaystyle \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$220 = 2^2 \\times 5 \\times 11$ Hence: {{begin-eqn}} {{eqn | l = \\map \\sigma {220}       | r = \\frac {2^3 - 1} {2 - 1} \\times \\frac {5^2 - 1} {5 - 1} \\times \\frac {11^2 - 1} {11 - 1}       | c =  }} {{eqn | r = \\frac 7 1 \\times \\frac {6 \\times 4} 4 \\times \\frac {12 \\times 10} {10}       | c =  }} {{eqn | r = 7 \\times 6 \\times 12       | c =  }} {{eqn | r = 7 \\times \\paren {2 \\times 3} \\times \\paren {2^2 \\times 3}       | c =  }} {{eqn | r = 2^3 \\times 3^2 \\times 7       | c =  }} {{eqn | r = 504       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:220 1vcippmcl9c5totyrn7swo390so3i5n"}
+{"_id": "32798", "title": "Sigma Function of 284", "text": "Sigma Function of 284 0 57462 428849 318226 2019-09-30T15:44:51Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\map \\sigma {284} = 504$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == From Sigma Function of Integer :$\\displaystyle \\map \\sigma n = \\prod_{1 \\mathop \\le i \\mathop \\le r} \\frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \\displaystyle \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$284 = 2^2 \\times 71$ Hence: {{begin-eqn}} {{eqn | l = \\map \\sigma {284}       | r = \\frac {2^3 - 1} {2 - 1} \\times \\paren {71 + 1}       | c =  }} {{eqn | r = 7 \\times 72       | c =  }} {{eqn | r = 7 \\times \\paren {2^3 \\times 3^2}       | c =  }} {{eqn | r = 2^3 \\times 3^2 \\times 7       | c =  }} {{eqn | r = 504       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:284 6rdlg5a1yonk3gmbtsgxoqqmutizo8w"}
+{"_id": "32799", "title": "Tau Function of 242", "text": "Tau Function of 242 0 57624 301348 301347 2017-06-16T06:38:54Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({242}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$242 = 2 \\times 11^2$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({242}\\right)       | r = \\tau \\left({2^1 \\times 11^2}\\right)       | c =  }} {{eqn | r = \\left({1 + 1}\\right) \\left({2 + 1}\\right)       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $242$ can be enumerated as: :$1, 2, 11, 22, 121, 242$ {{OEIS|A018351}}{{qed}} Category:Tau Function Category:242 cw3x17ukjsvscw56toml4uol58puatc"}
+{"_id": "32800", "title": "Tau Function of 243", "text": "Tau Function of 243 0 57625 301350 301349 2017-06-16T06:40:14Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({243}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$243 = 3^5$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({243}\\right)       | r = \\tau \\left({3^5}\\right)       | c =  }} {{eqn | r = 5 + 1       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $243$ can be enumerated as: :$1, 3, 9, 27, 81, 243$ {{qed}} Category:Tau Function Category:243 gbe4y7w8ust9kft8qcmx5c47zsdlomu"}
+{"_id": "32801", "title": "Tau Function of 244", "text": "Tau Function of 244 0 57627 442166 301352 2020-01-03T23:24:31Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {244} = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$244 = 2^2 \\times 61$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {244}       | r = \\map \\tau {2^2 \\times 61^1}       | c =  }} {{eqn | r = \\paren {2 + 1} \\paren {1 + 1}       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $244$ can be enumerated as: :$1, 2, 4, 61, 122, 244$ {{OEIS|A018352}}{{qed}} Category:Tau Function Category:244 tt5qjt0kh52cpbdoy10801body7eao2"}
+{"_id": "32802", "title": "Tau Function of 245", "text": "Tau Function of 245 0 57628 437662 301353 2019-12-03T22:27:03Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {245} = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$245 = 5 \\times 7^2$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {245}       | r = \\map \\tau {5^1 \\times 7^2}       | c =  }} {{eqn | r = \\paren {1 + 1} \\paren {2 + 1}       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $245$ can be enumerated as: :$1, 5, 7, 35, 49, 245$ {{OEIS|A018353}}{{qed}} Category:Tau Function Category:245 ax6xkkjmionh27wol5xl7cb1o3aikwv"}
+{"_id": "32803", "title": "Tau Function of 44", "text": "Tau Function of 44 0 57631 392274 301359 2019-02-13T13:43:41Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {44} = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$44 = 2^2 \\times 11$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {44}       | r = \\map \\tau {2^2 \\times 11^1}       | c =  }} {{eqn | r = \\paren {2 + 1} \\paren {1 + 1}       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $44$ can be enumerated as: :$1, 2, 4, 11, 22, 44$ {{OEIS|A018259}}{{qed}} Category:Tau Function Category:44 obchaa6rttld5l92qyw5jn6low7xme2"}
+{"_id": "32804", "title": "Tau Function of 45", "text": "Tau Function of 45 0 57632 498640 301360 2020-11-11T15:10:51Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {45} = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\ds \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$45 = 3^2 \\times 5$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {45}       | r = \\map \\tau {3^2 \\times 5^1}       | c =  }} {{eqn | r = \\paren {2 + 1} \\paren {1 + 1}       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $45$ can be enumerated as: :$1, 3, 5, 9, 15, 45$ {{OEIS|A018260}}{{qed}} Category:Tau Function Category:45 pi8h7w7wkxq59n6ftpw5q77k2hovj9c"}
+{"_id": "32805", "title": "Tau Function of 75", "text": "Tau Function of 75 0 57633 462846 301361 2020-04-18T19:11:38Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {75} = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$75 = 3 \\times 5^2$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {75}       | r = \\map \\tau {3^1 \\times 5^2}       | c =  }} {{eqn | r = \\paren {1 + 1} \\paren {2 + 1}       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $75$ can be enumerated as: :$1, 3, 5, 15, 25, 75$ {{OEIS|A018272}}{{qed}} Category:Tau Function Category:75 4n8i6x28vhggiyuf2krmi2wt5vf3pbe"}
+{"_id": "32806", "title": "Tau Function of 76", "text": "Tau Function of 76 0 57634 301362 2017-06-16T07:04:24Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({76}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Functi...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({76}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$76 = 2^2 \\times 19$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({76}\\right)       | r = \\tau \\left({2^2 \\times 19^1}\\right)       | c =  }} {{eqn | r = \\left({2 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $76$ can be enumerated as: :$1, 2, 4, 19, 38, 76$ {{OEIS|A018273}}{{qed}} Category:Tau Function Category:76 0v0qr9pw59m0qr4ynosy2tmi4b9xbfs"}
+{"_id": "32807", "title": "Tau Function of 98", "text": "Tau Function of 98 0 57635 493092 301363 2020-10-07T21:00:26Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {98} = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\ds \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$98 = 2 \\times 7^2$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {98}       | r = \\map \\tau {2^1 \\times 7^2}       | c =  }} {{eqn | r = \\paren {1 + 1} \\paren {2 + 1}       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $98$ can be enumerated as: :$1, 2, 7, 14, 49, 98$ {{OEIS|A018281}}{{qed}} Category:Tau Function Category:98 emweidxaurus9g2kcqsmh9li2ba1aip"}
+{"_id": "32808", "title": "Tau Function of 99", "text": "Tau Function of 99 0 57636 440180 301364 2019-12-19T14:26:33Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {99} = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$99 = 3^2 \\times 11$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {99}       | r = \\map \\tau {3^2 \\times 11^1}       | c =  }} {{eqn | r = \\paren {2 + 1} \\paren {1 + 1}       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $99$ can be enumerated as: :$1, 3, 9, 11, 33, 99$ {{OEIS|A018282}}{{qed}} Category:Tau Function Category:99 jmexhprzj9jku6bnx4x88o7tbhrbct4"}
+{"_id": "32809", "title": "Tau Function of 116", "text": "Tau Function of 116 0 57637 301365 2017-06-16T07:12:51Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({116}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Funct...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({116}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$116 = 2^2 \\times 29$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({116}\\right)       | r = \\tau \\left({2^2 \\times 29^1}\\right)       | c =  }} {{eqn | r = \\left({2 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $116$ can be enumerated as: :$1, 2, 4, 29, 58, 116$ {{OEIS|A018291}}{{qed}} Category:Tau Function Category:116 lb1er71ruxb8hxim1qr1w8j9alfsg7c"}
+{"_id": "32810", "title": "Tau Function of 117", "text": "Tau Function of 117 0 57638 301366 2017-06-16T07:14:55Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({117}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Funct...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({117}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$117 = 3^2 \\times 13$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({117}\\right)       | r = \\tau \\left({3^2 \\times 13^1}\\right)       | c =  }} {{eqn | r = \\left({2 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $117$ can be enumerated as: :$1, 3, 9, 13, 39, 117$ {{OEIS|A018292}}{{qed}} Category:Tau Function Category:117 sa61trstzw6njn8hwvtrkn1aovqug3j"}
+{"_id": "32811", "title": "Tau Function of 147", "text": "Tau Function of 147 0 57639 301367 2017-06-16T07:17:02Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({147}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Funct...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({147}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$147 = 3 \\times 7^2$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({147}\\right)       | r = \\tau \\left({3^1 \\times 7^2}\\right)       | c =  }} {{eqn | r = \\left({1 + 1}\\right) \\left({2 + 1}\\right)       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $147$ can be enumerated as: :$1, 3, 7, 21, 49, 147$ {{OEIS|A018303}}{{qed}} Category:Tau Function Category:147 ltd5c69ddqp6j2mj4cioynjxd2z1e5o"}
+{"_id": "32812", "title": "Tau Function of 148", "text": "Tau Function of 148 0 57640 301368 2017-06-16T07:18:53Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({148}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Funct...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({148}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$148 = 2^2 \\times 37$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({148}\\right)       | r = \\tau \\left({2^2 \\times 37^1}\\right)       | c =  }} {{eqn | r = \\left({2 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $148$ can be enumerated as: :$1, 2, 4, 37, 74, 148$ {{OEIS|A018304}}{{qed}} Category:Tau Function Category:148 bd1lmvoloq9hgkal0gdmfacusjavqru"}
+{"_id": "32813", "title": "Tau Function of 171", "text": "Tau Function of 171 0 57641 301369 2017-06-16T07:20:53Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({171}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Funct...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({171}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$171 = 3^2 \\times 19$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({171}\\right)       | r = \\tau \\left({3^2 \\times 19^1}\\right)       | c =  }} {{eqn | r = \\left({2 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $171$ can be enumerated as: :$1, 3, 9, 19, 57, 171$ {{OEIS|A018316}}{{qed}} Category:Tau Function Category:171 mxk5jkfzab71vxjdtrxj3uanp3verfz"}
+{"_id": "32814", "title": "Tau Function of 172", "text": "Tau Function of 172 0 57642 301370 2017-06-16T07:22:22Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({172}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Funct...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({172}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$172 = 2^2 \\times 43$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({172}\\right)       | r = \\tau \\left({2^2 \\times 43^1}\\right)       | c =  }} {{eqn | r = \\left({2 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $172$ can be enumerated as: :$1, 2, 4, 43, 86, 172$ {{OEIS|A018317}}{{qed}} Category:Tau Function Category:172 f4fyyhnchmz2lbtz7jghqb1nermgh2d"}
+{"_id": "32815", "title": "Tau Function of 332", "text": "Tau Function of 332 0 57643 453229 301371 2020-03-07T22:54:29Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {332} = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$332 = 2^2 \\times 83$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {332}       | r = \\map \\tau {2^2 \\times 83^1}       | c =  }} {{eqn | r = \\paren {2 + 1} \\paren {1 + 1}       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $332$ can be enumerated as: :$1, 2, 4, 83, 166, 332$ {{OEIS|A018397}}{{qed}} Category:Tau Function Category:332 t66rqz4j7izzms654yr849sg0ltcmsr"}
+{"_id": "32816", "title": "Tau Function of 333", "text": "Tau Function of 333 0 57645 479809 301373 2020-07-26T19:08:37Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {333} = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$333 = 3^2 \\times 37$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {333}       | r = \\map \\tau {3^2 \\times 37^1}       | c =  }} {{eqn | r = \\paren {2 + 1} \\paren {1 + 1}       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $333$ can be enumerated as: :$1, 3, 9, 37, 111, 333$ {{OEIS|A018398}}{{qed}} Category:Tau Function Category:333 1024mwyu3a23yig2nixfbw8izpf3zfr"}
+{"_id": "32817", "title": "Euler Phi Function of 255", "text": "Euler Phi Function of 255 0 57677 301456 2017-06-17T05:05:03Z Prime.mover 59 Created page with \"== Example of Euler $\\phi$ Function of Square-Free Integer == <onlyinclude> :$\\phi \\left({255}\\right) = 128$ </onlyinclude> where...\" wikitext text/x-wiki == Example of Euler $\\phi$ Function of Square-Free Integer == <onlyinclude> :$\\phi \\left({255}\\right) = 128$ </onlyinclude> where $\\phi$ denotes the Euler $\\phi$ Function. == Proof == From Euler Phi Function of Square-Free Integer: :$\\displaystyle \\phi \\left({n}\\right) = \\prod_{\\substack {p \\mathop \\backslash n \\\\ p \\mathop > 2} } \\left({p - 1}\\right)$ where $p \\mathop \\backslash n$ denotes the primes which divide $n$. We have that: :$255 = 3 \\times 5 \\times 17$ and so is square-free. Thus: {{begin-eqn}} {{eqn | l = \\phi \\left({255}\\right)       | r = \\left({3 - 1}\\right) \\left({5 - 1}\\right) \\left({17 - 1}\\right)       | c =  }} {{eqn | r = 2 \\times 4 \\times 16       | c =  }} {{eqn | r = 2 \\times 2^2 \\times 2^4       | c =  }} {{eqn | r = 2^7       | c =  }} {{eqn | r = 128       | c =  }} {{end-eqn}} {{qed}} Category:Euler Phi Function of Square-Free Integer Category:255 15jcgmcyt9gmekph9k99y8vohk7gi3y"}
+{"_id": "32818", "title": "Euler Phi Function of 256", "text": "Euler Phi Function of 256 0 57678 301457 2017-06-17T05:08:31Z Prime.mover 59 Created page with \"== Example of Use of Euler $\\phi$ Function == <onlyinclude> The value of the Euler $\\phi$ function for the...\" wikitext text/x-wiki == Example of Use of Euler $\\phi$ Function == <onlyinclude> The value of the Euler $\\phi$ function for the integer $256$ is $128$. </onlyinclude> == Proof == From the corollary to Euler Phi Function of Prime Power: :$\\phi \\left({2^k}\\right) = 2^{k-1}$ Thus: {{begin-eqn}} {{eqn | l = \\phi \\left({256}\\right)       | r = \\phi \\left({2^8}\\right)       | c =  }} {{eqn | r = 2^7       | c =  }} {{eqn | r = 128       | c =  }} {{end-eqn}} {{qed}} Category:Euler Phi Function Category:256 9v9f6py2viisafgp53w6dr79zy9yg8k"}
+{"_id": "32819", "title": "Tau Function of 20", "text": "Tau Function of 20 0 58327 304238 303115 2017-07-10T09:17:26Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({20}\\right) = 6$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$20 = 2^2 \\times 5$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({20}\\right)       | r = \\tau \\left({2^2 \\times 5^1}\\right)       | c =  }} {{eqn | r = \\left({2 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $20$ can be enumerated as: :$1, 2, 4, 5, 10, 20$ {{OEIS|A005018}}{{qed}} Category:Tau Function Category:20 aiul2j4zaqmzkql5jarvm0pqntnpuo0"}
+{"_id": "32820", "title": "Non-Square Positive Integers not Sum of Square and Prime", "text": "Non-Square Positive Integers not Sum of Square and Prime 0 58473 326104 303396 2017-11-11T09:49:55Z Barto 3079 \"work to do\" template wikitext text/x-wiki == Conjecture == The sequence of (strictly) positive integers which are not square and not the sum of a square and a prime is believed to be complete: :$10, 34, 58, 85, 91, 130, 214, 226, 370, 526, 706, 730, 771, 1255, 1351, 1414, 1906, 2986, 3676, 9634, 21679$ {{OEIS|A020495}} == Progress == From Square of n such that 2n-1 is Composite is not Sum of Square and Prime, $n^2$ is the sum of a square and a prime {{iff}} $2 n - 1$ composite. Hence the question is specifically about non-squares. No prime number is in this sequence, as trivially: :$p = p + 0^2$ and so is the sum of a prime (itself), and $0^2$, which is square. Each non-square composite $n \\in \\Z$ can be tested by subtracting successive squares less than $n$ and investigating whether a prime can result. In the following, the smallest $m$ such that $n - m^2 = p$ is shown where such a $p$ exists. Otherwise the nonexistence of such a $p$ is demonstrated explicitly. As follows: {{begin-eqn}} {{eqn | l = 4 - 1^2       | r = 3       | c = which is prime }} {{eqn | l = 6 - 1^2       | r = 5       | c = which is prime }} {{eqn | l = 8 - 1^2       | r = 7       | c = which is prime }} {{end-eqn}} $10$ cannot be expressed as $10 = m^2 + p$. Thus $10$ is seen to be in this sequence. {{begin-eqn}} {{eqn | l = 12 - 1^2       | r = 11       | c = which is prime }} {{eqn | l = 14 - 1^2       | r = 13       | c = which is prime }} {{eqn | l = 15 - 2^2       | r = 11       | c = which is prime }} {{eqn | l = 16 - 3^2       | r = 7       | c = which is prime }} {{eqn | l = 18 - 1^2       | r = 17       | c = which is prime }} {{eqn | l = 20 - 1^2       | r = 19       | c = which is prime }} {{eqn | l = 21 - 2^2       | r = 17       | c = which is prime }} {{eqn | l = 22 - 3^2       | r = 13       | c = which is prime }} {{eqn | l = 24 - 1^2       | r = 23       | c = which is prime }} {{eqn | l = 26 - 3^2       | r = 17       | c = which is prime }} {{eqn | l = 27 - 2^2       | r = 23       | c = which is prime }} {{eqn | l = 28 - 3^2       | r = 19       | c = which is prime }} {{eqn | l = 30 - 1^2       | r = 29       | c = which is prime }} {{eqn | l = 32 - 1^2       | r = 31       | c = which is prime }} {{eqn | l = 33 - 2^2       | r = 29       | c = which is prime }} {{end-eqn}} $34$ cannot be expressed as $34 = m^2 + p$. Thus $34$ is seen to be in this sequence. Similarly: $58$ cannot be expressed as $58 = m^2 + p$. Thus $58$ is seen to be in this sequence. This establishes the pattern. The algorithm for determining whether a particular $n$ belongs to this sequence can be defined in pseudocode as follows: <pre> For n := 1, loop indefinitely, incrementing by 1:   Is n prime? If so, continue to the next n   Is n square? If so, continue to the next n   For m := 1, incrementing by 1 until m^2 > n:     Is n - n^2 prime? If so, continue to the next n   Next m   Add n to the sequence Next n </pre> {{wtd|Design (or adoption) of a rigorous pseudocode needs to be done. Alternatively we may seek to stick with our existing technique for implement algorithms, and/or use a flow chart.}} {{finish|Tidy up the algorithm above.  Do we demonstrate an instance of an computer program here?}} == Examples == {{:Non-Square Positive Integers not Sum of Square and Prime/Examples}} == Historical Note == {{:Non-Square Positive Integers not Sum of Square and Prime/Historical Note}} Category:Numbers not Sum of Square and Prime Category:Unproven Hypotheses 200j0yl2xxwh429avnf174txlp4nyt8"}
+{"_id": "32821", "title": "Tau Function of 836", "text": "Tau Function of 836 0 58989 488971 304141 2020-09-18T22:40:48Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {836} = 12$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$836 = 2^2 \\times 11 \\times 19$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {836}       | r = \\map \\tau {2^2 \\times 11^1 \\times 19^1}       | c =  }} {{eqn | r = \\paren {2 + 1} \\paren {1 + 1} \\paren {1 + 1}       | c =  }} {{eqn | r = 12       | c =  }} {{end-eqn}} The divisors of $836$ can be enumerated as: :$1, 2, 4, 11, 19, 22, 38, 44, 76, 209, 418, 836$ {{OEIS|A018674}}{{qed}} Category:Tau Function Category:836 3p7ep19hp9lks5cd2e5mlmuyd81v6bc"}
+{"_id": "32822", "title": "Sigma Function of 4030", "text": "Sigma Function of 4030 0 58996 393901 317889 2019-02-25T15:01:57Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Square-Free Integer == <onlyinclude> :$\\map \\sigma {4030} = 8064$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == <onlyinclude> We have that: :$4030 = 2 \\times 5 \\times 13 \\times 31$ Hence: {{begin-eqn}} {{eqn | l = \\map \\sigma {4030}       | r = \\paren {2 + 1} \\paren {5 + 1} \\paren {13 + 1} \\paren {31 + 1}       | c = Sigma Function of Square-Free Integer }} {{eqn | r = 3 \\times 6 \\times 14 \\times 32       | c =  }} {{eqn | r = 3 \\times \\paren {2 \\times 3} \\times \\paren {2 \\times 7} \\times 2^5       | c =  }} {{eqn | r = 2^7 \\times 3^2 \\times 7       | c =  }} {{eqn | r = 8064       | c =  }} {{end-eqn}} {{qed}} </onlyinclude> Category:Sigma Function of Square-Free Integer Category:4030 94fs2bv2gjbu850omhvswl2iscnyp4p"}
+{"_id": "32823", "title": "Tau Function of 4030", "text": "Tau Function of 4030 0 58997 304156 2017-07-09T19:08:53Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({4030}\\right) = 16$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Fun...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({4030}\\right) = 16$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$4030 = 2 \\times 5 \\times 13 \\times 31$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({4030}\\right)       | r = \\tau \\left({2^1 \\times 5^1 \\times 13^1 \\times 31^1}\\right)       | c =  }} {{eqn | r = \\left({1 + 1}\\right) \\left({1 + 1}\\right) \\left({1 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 16       | c =  }} {{end-eqn}} The divisors of $4030$ can be enumerated as: :$1, 2, 5, 10, 13, 26, 31, 62, 65, 130, 155, 310, 403, 806, 2015, 4030$ {{qed}} Category:Tau Function Category:4030 h53efoeraud09v5hpgi203y0gm5a3ve"}
+{"_id": "32824", "title": "Tau Function of 490", "text": "Tau Function of 490 0 59007 304197 2017-07-09T22:41:58Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({490}\\right) = 12$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Func...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({490}\\right) = 12$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$490 = 2 \\times 5 \\times 7^2$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({490}\\right)       | r = \\tau \\left({2^1 \\times 5^1 \\times 7^2}\\right)       | c =  }} {{eqn | r = \\left({1 + 1}\\right) \\left({1 + 1}\\right) \\left({2 + 1}\\right)       | c =  }} {{eqn | r = 12       | c =  }} {{end-eqn}} The divisors of $490$ can be enumerated as: :$1, 2, 5, 7, 10, 14, 35, 49, 70, 98, 245, 490$ {{OEIS|A018483}}{{qed}} Category:Tau Function Category:490 jchifntti6yp12ggx2eiwo8aqnefwvb"}
+{"_id": "32825", "title": "Tau Function of 550", "text": "Tau Function of 550 0 59008 304202 2017-07-09T22:56:05Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({550}\\right) = 12$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Func...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({550}\\right) = 12$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$550 = 2 \\times 5^2 \\times 11$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({550}\\right)       | r = \\tau \\left({2^1 \\times 5^2 \\times 11^1}\\right)       | c =  }} {{eqn | r = \\left({1 + 1}\\right) \\left({2 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 12       | c =  }} {{end-eqn}} The divisors of $550$ can be enumerated as: :$1, 2, 5, 10, 11, 22, 25, 50, 55, 110, 275, 550$ {{OEIS|A018514}}{{qed}} Category:Tau Function Category:550 npbvgbi3gczo3qlsits8hp9d7wm6zvo"}
+{"_id": "32826", "title": "Tau Function of 572", "text": "Tau Function of 572 0 59010 304205 2017-07-09T23:06:44Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({572}\\right) = 12$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Func...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({572}\\right) = 12$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$572 = 2^2 \\times 11 \\times 13$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({572}\\right)       | r = \\tau \\left({2^2 \\times 11^1 \\times 13^1}\\right)       | c =  }} {{eqn | r = \\left({2 + 1}\\right) \\left({1 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 12       | c =  }} {{end-eqn}} The divisors of $572$ can be enumerated as: :$1, 2, 4, 11, 13, 22, 26, 44, 52, 143, 286, 572$ {{OEIS|A018525}}{{qed}} Category:Tau Function Category:572 rhv4hqbtrzva88kvoefc2n6ngmd4zed"}
+{"_id": "32827", "title": "Tau Function of 650", "text": "Tau Function of 650 0 59015 304223 2017-07-09T23:30:22Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({650}\\right) = 12$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Func...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({650}\\right) = 12$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$650 = 2 \\times 5^2 \\times 13$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({650}\\right)       | r = \\tau \\left({2^1 \\times 5^2 \\times 13^1}\\right)       | c =  }} {{eqn | r = \\left({1 + 1}\\right) \\left({2 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 12       | c =  }} {{end-eqn}} The divisors of $650$ can be enumerated as: :$1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650$ {{OEIS|A018571}}{{qed}} Category:Tau Function Category:650 5gefu5m1ce8n0dmit0o9j7j003lfdhc"}
+{"_id": "32828", "title": "Tau Function of 88", "text": "Tau Function of 88 0 59022 304235 2017-07-10T09:09:04Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({88}\\right) = 8$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Functi...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({88}\\right) = 8$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$88 = 2^3 \\times 11$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({88}\\right)       | r = \\tau \\left({2^3 \\times 11^1}\\right)       | c =  }} {{eqn | r = \\left({3 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 8       | c =  }} {{end-eqn}} The divisors of $88$ can be enumerated as: :$1, 2, 4, 8, 11, 22, 44, 88$ {{OEIS|A018277}}{{qed}} Category:Tau Function Category:88 qds6jo0jge6b258ki5stwt2s5aodcq0"}
+{"_id": "32829", "title": "Tau Function of 104", "text": "Tau Function of 104 0 59024 334107 304239 2018-01-02T16:23:08Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({104}\\right) = 8$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$104 = 2^3 \\times 13$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({104}\\right)       | r = \\tau \\left({2^3 \\times 13^1}\\right)       | c =  }} {{eqn | r = \\left({3 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 8       | c =  }} {{end-eqn}} The divisors of $104$ can be enumerated as: :$1, 2, 4, 8, 13, 26, 52, 104$ {{OEIS|A018285}}{{qed}} Category:Tau Function Category:104 euuy536l81o3uo5z62ubdnadh6md7m3"}
+{"_id": "32830", "title": "Sigma Function of 272", "text": "Sigma Function of 272 0 59026 318220 318218 2017-09-16T18:26:03Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\sigma \\left({272}\\right) = 558$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$272 = 2^4 \\times 17$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({272}\\right)       | r = \\frac {2^5 - 1} {2 - 1} \\times \\frac {17^2 - 1} {17 - 1}       | c = Sigma Function of Integer }} {{eqn | r = \\frac {31 - 1} 1 \\times \\frac {289 - 1} {16}       | c =  }} {{eqn | r = 31 \\times 18       | c =  }} {{eqn | r = \\left({2 \\times 3^2}\\right) \\times 31       | c =  }} {{eqn | r = 558       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:272 rx3scmsglraze6gjw7jksv67w3pcnw9"}
+{"_id": "32831", "title": "Tau Function of 272", "text": "Tau Function of 272 0 59027 304242 2017-07-10T09:26:58Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({272}\\right) = 10$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Func...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({272}\\right) = 10$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$272 = 2^4 \\times 17$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({272}\\right)       | r = \\tau \\left({2^4 \\times 17^1}\\right)       | c =  }} {{eqn | r = \\left({4 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 10       | c =  }} {{end-eqn}} The divisors of $272$ can be enumerated as: :$1, 2, 4, 8, 16, 17, 34, 68, 136, 272$ {{OEIS|A018366}}{{qed}} Category:Tau Function Category:272 s7nn4xxhrkckargd8ql0zio5nglgyc7"}
+{"_id": "32832", "title": "Tau Function of 304", "text": "Tau Function of 304 0 59029 437827 304244 2019-12-05T13:17:44Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {304} = 10$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$304 = 2^4 \\times 19$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {304}       | r = \\map \\tau {2^4 \\times 19^1}       | c =  }} {{eqn | r = \\paren {4 + 1} \\paren {1 + 1}       | c =  }} {{eqn | r = 10       | c =  }} {{end-eqn}} The divisors of $304$ can be enumerated as: :$1, 2, 4, 8, 16, 19, 38, 76, 152, 304$ {{OEIS|A018383}}{{qed}} Category:Tau Function Category:304 4sp5aqq0s67es5to2ipj3cfkzxgnnaa"}
+{"_id": "32833", "title": "Sigma Function of 304", "text": "Sigma Function of 304 0 59031 437054 318249 2019-11-28T11:59:16Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\map \\sigma {304} = 620$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$304 = 2^4 \\times 19$ Hence: {{begin-eqn}} {{eqn | l = \\map \\sigma {304}       | r = \\frac {2^5 - 1} {2 - 1} \\times \\frac {19^2 - 1} {19 - 1}       | c = Sigma Function of Integer }} {{eqn | r = \\frac {31 - 1} 1 \\times \\frac {361 - 1} {18}       | c =  }} {{eqn | r = 31 \\times 20       | c =  }} {{eqn | r = \\paren {2^2 \\times 5} \\times 31       | c =  }} {{eqn | r = 620       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:304 scg937vsfb5vp31ewr73chnst9ufmdk"}
+{"_id": "32834", "title": "Sigma Function of 748", "text": "Sigma Function of 748 0 59032 318334 318332 2017-09-16T18:43:23Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\sigma \\left({748}\\right) = 1512$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$748 = 2^2 \\times 11 \\times 17$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({748}\\right)       | r = \\frac {2^3 - 1} {2 - 1} \\times \\left({11 + 1}\\right) \\times \\left({17 + 1}\\right)       | c = Sigma Function of Integer }} {{eqn | r = 7 \\times 12 \\times 18       | c =  }} {{eqn | r = 7 \\times \\left({2^2 \\times 3}\\right) \\times \\left({2 \\times 3^2}\\right)       | c =  }} {{eqn | r = 2^3 \\times 3^3 \\times 7       | c =  }} {{eqn | r = 1512       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:748 bak7x1fyh5pcawhxicsevj5k53a0qix"}
+{"_id": "32835", "title": "Tau Function of 748", "text": "Tau Function of 748 0 59034 304251 304250 2017-07-10T09:47:44Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({748}\\right) = 12$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$748 = 2^2 \\times 11 \\times 17$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({748}\\right)       | r = \\tau \\left({2^2 \\times 11^1 \\times 17^1}\\right)       | c =  }} {{eqn | r = \\left({2 + 1}\\right) \\left({1 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 12       | c =  }} {{end-eqn}} The divisors of $748$ can be enumerated as: :$1, 2, 4, 11, 17, 22, 34, 44, 68, 187, 374, 748$ {{OEIS|A018625}}{{qed}} Category:Tau Function Category:748 gy36kbpwjvlwbg7l8qemzf3uswc8ul5"}
+{"_id": "32836", "title": "Perfect Number/Examples/6", "text": "Perfect Number/Examples/6 0 59047 304290 304286 2017-07-10T13:10:44Z Prime.mover 59 wikitext text/x-wiki == Example of Perfect Number == <onlyinclude> $6$ is a perfect number: :$1 + 2 + 3 = 6$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = 6       | r = 2 \\times 3       | c =  }} {{eqn | r = 2^{2 - 1} \\left({2^2 - 1}\\right)       | c =  }} {{end-eqn}} Thus $6$ is in the form $2^{n - 1} \\left({2^n - 1}\\right)$. $\\left({2^2 - 1}\\right) = 3$ is prime. So $6$ is perfect by the Theorem of Even Perfect Numbers. The aliquot parts of $6$ are enumerated at $\\tau$ of $6$: :$1, 2, 3$ {{qed}} Category:Perfect Numbers/Examples Category:6 l5ifi04j4tscc66zegdw575qgp0xjl4"}
+{"_id": "32837", "title": "Perfect Number/Examples/28", "text": "Perfect Number/Examples/28 0 59049 304301 304289 2017-07-10T21:11:53Z Prime.mover 59 wikitext text/x-wiki == Example of Perfect Number == <onlyinclude> $28$ is a perfect number: :$1 + 2 + 4 + 7 + 14 = 28$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = 28       | r = 4 \\times 7       | c =  }} {{eqn | r = 2^{3 - 1} \\left({2^3 - 1}\\right)       | c =  }} {{end-eqn}} Thus $28$ is in the form $2^{p - 1} \\left({2^p - 1}\\right)$. $\\left({2^3 - 1}\\right) = 7$ is prime. So $28$ is perfect by the Theorem of Even Perfect Numbers. The aliquot parts of $28$ are enumerated at $\\tau$ of $28$: :$1, 2, 4, 7, 14$ {{qed}} Category:Perfect Numbers/Examples Category:28 ko1oaen6qi02suw50mk42r15d4yc14d"}
+{"_id": "32838", "title": "Perfect Number/Examples/496", "text": "Perfect Number/Examples/496 0 59052 304297 2017-07-10T21:05:03Z Prime.mover 59 Created page with \"== Example of Perfect Number == <onlyinclude> $496$ is a perfect number: :$1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248...\" wikitext text/x-wiki == Example of Perfect Number == <onlyinclude> $496$ is a perfect number: :$1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = 496       | r = 16 \\times 31       | c =  }} {{eqn | r = 2^{5 - 1} \\left({2^5 - 1}\\right)       | c =  }} {{end-eqn}} Thus $496$ is in the form $2^{p - 1} \\left({2^p - 1}\\right)$. $\\left({2^5 - 1}\\right) = 31$ is prime. So $496$ is perfect by the Theorem of Even Perfect Numbers. The aliquot parts of $496$ are enumerated at $\\tau$ of $496$: :$1, 2, 4, 8, 16, 31, 62, 124, 248$ {{qed}} Category:Perfect Numbers/Examples Category:496 5caq0phszs33pcx9ubnbtaw1dnhv8f4"}
+{"_id": "32839", "title": "Tau Function of 8128", "text": "Tau Function of 8128 0 59053 304298 2017-07-10T21:08:43Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({8128}\\right) = 10$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Fun...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({8128}\\right) = 10$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$8128 = 2^6 \\times 127$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({8128}\\right)       | r = \\tau \\left({2^6 \\times 127^1}\\right)       | c =  }} {{eqn | r = \\left({6 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 14       | c =  }} {{end-eqn}} The divisors of $496$ can be enumerated as: :$1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, 8128$ {{OEIS|A133024}}{{qed}} Category:Tau Function Category:8128 hm2zpslcr20akwp3rts5nourkv1yngr"}
+{"_id": "32840", "title": "Tau Function of 350", "text": "Tau Function of 350 0 59059 304308 2017-07-10T22:22:33Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({350}\\right) = 12$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Func...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({350}\\right) = 12$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$350 = 2 \\times 5^2 \\times 7$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({350}\\right)       | r = \\tau \\left({2^1 \\times 5^2 \\times 7^1}\\right)       | c =  }} {{eqn | r = \\left({1 + 1}\\right) \\left({2 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 12       | c =  }} {{end-eqn}} The divisors of $350$ can be enumerated as: :$1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175, 350$ {{OEIS|A018406}}{{qed}} Category:Tau Function Category:350 pwa2hv67xd0ko87ic5yzleh47df80ss"}
+{"_id": "32841", "title": "Tau Function of 368", "text": "Tau Function of 368 0 59061 304310 2017-07-10T22:28:37Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({368}\\right) = 10$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Func...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({368}\\right) = 10$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$368 = 2^4 \\times 23$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({368}\\right)       | r = \\tau \\left({2^4 \\times 23^1}\\right)       | c =  }} {{eqn | r = \\left({4 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 10       | c =  }} {{end-eqn}} The divisors of $368$ can be enumerated as: :$1, 2, 4, 8, 16, 23, 46, 92, 184, 368$ {{OEIS|A018416}}{{qed}} Category:Tau Function Category:368 6k9v95xl1u1mmviudnaqyvxwui8t5yl"}
+{"_id": "32842", "title": "Tau Function of 464", "text": "Tau Function of 464 0 59063 473808 304313 2020-06-12T15:22:36Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {464} = 10$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$464 = 2^4 \\times 29$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {464}       | r = \\map \\tau {2^4 \\times 29^1}       | c =  }} {{eqn | r = \\paren {4 + 1} \\paren {1 + 1}       | c =  }} {{eqn | r = 10       | c =  }} {{end-eqn}} The divisors of $464$ can be enumerated as: :$1, 2, 4, 8, 16, 29, 58, 116, 232, 464$ {{OEIS|A018469}}{{qed}} Category:Tau Function Category:464 cm87fogt027tziyuwpzb01bo5l7fiqw"}
+{"_id": "32843", "title": "Sigma Function of 464", "text": "Sigma Function of 464 0 59074 318279 318277 2017-09-16T18:35:27Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\sigma \\left({464}\\right) = 930$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$464 = 2^4 \\times 29$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({464}\\right)       | r = \\frac {2^5 - 1} {2 - 1} \\times \\left({29 + 1}\\right)       | c = Sigma Function of Integer }} {{eqn | r = \\frac {31 - 1} 1 \\times 30       | c =  }} {{eqn | r = 31 \\times 30       | c =  }} {{eqn | r = \\left({2 \\times 3 \\times 5}\\right) \\times 31       | c =  }} {{eqn | r = 930       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:464 pwwssq4d21pngtykzhxl1bdd7lri6rr"}
+{"_id": "32844", "title": "Sigma Function of 550", "text": "Sigma Function of 550 0 59076 318303 318301 2017-09-16T18:38:48Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\sigma \\left({550}\\right) = 1116$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$550 = 2 \\times 5^2 \\times 11$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({550}\\right)       | r = \\left({2 + 1}\\right) \\times \\frac {5^3 - 1} {5 - 1} \\times \\left({11 + 1}\\right)       | c = Sigma Function of Integer }} {{eqn | r = 3 \\times \\frac {125 - 1} 4 \\times 12       | c =  }} {{eqn | r = 3 \\times 31 \\times \\left({2^2 \\times 3}\\right)       | c =  }} {{eqn | r = 2^2 \\times 3^2 \\times 31       | c =  }} {{eqn | r = 1116       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:550 czyhqlmea1f0bdvkvlv8w7iocze8mwu"}
+{"_id": "32845", "title": "Sigma Function of 572", "text": "Sigma Function of 572 0 59078 318309 318307 2017-09-16T18:39:36Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\sigma \\left({572}\\right) = 1176$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$572 = 2^2 \\times 11 \\times 13$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({572}\\right)       | r = \\frac {2^3 - 1} {2 - 1} \\times \\left({11 + 1}\\right) \\times \\left({13 + 1}\\right)       | c = Sigma Function of Integer }} {{eqn | r = \\frac {7 - 1} 1 \\times 12 \\times 14       | c =  }} {{eqn | r = 7 \\times \\left({2^2 \\times 3}\\right) \\times \\left({2 \\times 7}\\right)       | c =  }} {{eqn | r = 2^3 \\times 3 \\times 7^2       | c =  }} {{eqn | r = 1176       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:572 f9klaav8ehkgcafcylzae71k86l4ud0"}
+{"_id": "32846", "title": "Sigma Function of 650", "text": "Sigma Function of 650 0 59080 435882 318324 2019-11-20T19:00:14Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\map \\sigma {650} = 1302$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$650 = 2 \\times 5^2 \\times 13$ Hence: {{begin-eqn}} {{eqn | l = \\map \\sigma {650}       | r = \\paren {2 + 1} \\times \\frac {5^3 - 1} {5 - 1} \\times \\paren {13 + 1}       | c = Sigma Function of Integer }} {{eqn | r = 3 \\times \\frac {125 - 1} 4 \\times 14       | c =  }} {{eqn | r = 3 \\times 31 \\times \\paren {2 \\times 7}       | c =  }} {{eqn | r = 2 \\times 3 \\times 7 \\times 31       | c =  }} {{eqn | r = 1302       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:650 bm6g40cfm16s71b0igpb06njswgr2vd"}
+{"_id": "32847", "title": "Tau Function of 770", "text": "Tau Function of 770 0 59567 305915 305909 2017-07-22T08:34:49Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({770}\\right) = 16$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$770 = 2 \\times 5 \\times 7 \\times 11$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({770}\\right)       | r = \\tau \\left({2^1 \\times 5^1 \\times 7^1 \\times 11^1}\\right)       | c =  }} {{eqn | r = \\left({1 + 1}\\right) \\left({1 + 1}\\right) \\left({1 + 1}\\right) \\left({1 + 1}\\right)       | c =  }} {{eqn | r = 16       | c =  }} {{end-eqn}} The divisors of $770$ can be enumerated as: :$1, 2, 5, 7, 10, 11, 14, 22, 35, 55, 70, 77, 110, 154, 385, 770$ {{OEIS|A018636}}{{qed}} Category:Tau Function Category:770 6jgxiy0fxo1q0xqw1xnjov11x3mhfz9"}
+{"_id": "32848", "title": "Multiplicative Persistence/Examples/25", "text": "Multiplicative Persistence/Examples/25 0 60112 308235 308234 2017-08-02T06:02:20Z Prime.mover 59 wikitext text/x-wiki == Examples of Multiplicative Persistence == <onlyinclude> $25$ is the smallest positive integer which has a multiplicative persistence of $2$. </onlyinclude> == Proof == Trivially: {{begin-eqn}} {{eqn | n = 1       | l = 2 \\times 5       | r = 10 }} {{eqn | n = 2       | l = 1 \\times 0       | r = 0 }} {{end-eqn}} All positive integers between $10$ and $19$ are seen to have a multiplicative persistence of $1$: :$1 \\times n = n$ where $n$ is a single digit. Then for 2-digit positive integers starting with $2$: :$2 \\times n > 9 \\implies n > 4$ by inspection. Hence the result. {{qed}} Category:Multiplicative Persistence Category:25 q4xeibo3s0mj1uxi8vb11276c4hz6vy"}
+{"_id": "32849", "title": "De Polignac's Formula/Examples/2 in 720 Factorial", "text": "De Polignac's Formula/Examples/2 in 720 Factorial 0 60227 391883 308633 2019-02-09T23:45:11Z Prime.mover 59 wikitext text/x-wiki == Example of Use of De Polignac's Formula == <onlyinclude> The prime factor $2$ appears in $720!$ to the power of $716$. That is: :$2^{716} \\divides 720!$ but: :$2^{717} \\nmid 720!$ </onlyinclude> == Proof == Let $\\mu$ denote the power of $2$ which divides $720!$ {{begin-eqn}} {{eqn | l = \\mu       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {720} {2^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {720} 2} + \\floor {\\frac {720} 4} + \\floor {\\frac {720} 8} + \\floor {\\frac {720} {16} } + \\floor {\\frac {720} {32} }       | c =  }} {{eqn | o =        | r = + \\floor {\\frac {720} {64} } + \\floor {\\frac {720} {128} } + \\floor {\\frac {720} {256} } + \\floor {\\frac {720} {512} }       | c =  }} {{eqn | r = 360 + 180 + 90 + 45 + 22 + 11 + 5 + 2 + 1       | c =  }} {{eqn | r = 716       | c =  }} {{end-eqn}} {{qed}} Category:De Polignac's Formula maxgk7rg0b1l1jnx4zh0pbqnazmfcbw"}
+{"_id": "32850", "title": "De Polignac's Formula/Examples/3 in 720 Factorial", "text": "De Polignac's Formula/Examples/3 in 720 Factorial 0 60228 391885 334099 2019-02-09T23:46:07Z Prime.mover 59 wikitext text/x-wiki == Example of Use of De Polignac's Formula == <onlyinclude> The prime factor $3$ appears in $720!$ to the power of $356$. That is: :$3^{356} \\divides 720!$ but: :$3^{357} \\nmid 720!$ </onlyinclude> == Proof == Let $\\mu$ denote the power of $3$ which divides $720!$ {{begin-eqn}} {{eqn | l = \\mu       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {720} {3^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {720} 3} + \\floor {\\frac {720} 9} + \\floor {\\frac {720} {27} } + \\floor {\\frac {720} {81} } + \\floor {\\frac {720} {243} }       | c =  }} {{eqn | r = 240 + 80 + 26 + 8 + 2       | c =  }} {{eqn | r = 356       | c =  }} {{end-eqn}} {{qed}} Category:De Polignac's Formula aku8yebg8tzo3oxc1cpymz97023rxyq"}
+{"_id": "32851", "title": "De Polignac's Formula/Examples/5 in 720 Factorial", "text": "De Polignac's Formula/Examples/5 in 720 Factorial 0 60229 391886 358683 2019-02-09T23:47:00Z Prime.mover 59 wikitext text/x-wiki == Example of Use of De Polignac's Formula == <onlyinclude> The prime factor $5$ appears in $720!$ to the power of $178$. That is: :$5^{178} \\divides 720!$ but: :$5^{179} \\nmid 720!$ </onlyinclude> == Proof == Let $\\mu$ denote the power of $5$ which divides $720!$ {{begin-eqn}} {{eqn | l = \\mu       | r = \\sum_{k \\mathop > 0} \\floor {\\frac {720} {5^k} }       | c = De Polignac's Formula }} {{eqn | r = \\floor {\\frac {720} 5} + \\floor {\\frac {720} {25} } + \\floor {\\frac {720} {125} } + \\floor {\\frac {720} {625} }       | c =  }} {{eqn | r = 144 + 28 + 5 + 1       | c =  }} {{eqn | r = 178       | c =  }} {{end-eqn}} {{qed}} Category:De Polignac's Formula 45qodtu6gqlwu69yvt7btcb8fpudk4c"}
+{"_id": "32852", "title": "Sigma Function of 1184", "text": "Sigma Function of 1184 0 60464 454635 317765 2020-03-15T15:40:51Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\map \\sigma {1184} = 2394$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == From Sigma Function of Integer :$\\displaystyle \\map \\sigma n = \\prod_{1 \\mathop \\le i \\mathop \\le r} \\frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \\displaystyle \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$1184 = 2^5 \\times 37$ Hence: {{begin-eqn}} {{eqn | l = \\map \\sigma {1184}       | r = \\frac {2^6 - 1} {2 - 1} \\times \\frac {37^2 - 1} {37 - 1}       | c =  }} {{eqn | r = \\frac {63} 1 \\times \\frac {38 \\times 36} {36}       | c =  }} {{eqn | r = 63 \\times 38       | c =  }} {{eqn | r = \\paren {3^2 \\times 7} \\times \\paren {2 \\times 19}       | c =  }} {{eqn | r = 2 \\times 3^2 \\times 7 \\times 19       | c =  }} {{eqn | r = 2394       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:1184 ganlt3wy11purs9zxg4h9t2r84dq0mg"}
+{"_id": "32853", "title": "Sigma Function of 1210", "text": "Sigma Function of 1210 0 60465 317791 317789 2017-09-16T13:01:46Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\sigma \\left({1210}\\right) = 2394$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == From Sigma Function of Integer :$\\displaystyle \\sigma \\left({n}\\right) = \\prod_{1 \\mathop \\le i \\mathop \\le r} \\frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \\displaystyle \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$1210 = 2 \\times 5 \\times 11^2$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({1210}\\right)       | r = \\left({2 + 1}\\right) \\times \\left({5 + 1}\\right) \\times \\frac {11^3 - 1} {11 - 1}       | c =  }} {{eqn | r = 3 \\times 6 \\times \\frac {1330} {10}       | c =  }} {{eqn | r = 3 \\times 6 \\times 133       | c =  }} {{eqn | r = 3 \\times \\left({2 \\times 3}\\right) \\times \\left({7 \\times 19}\\right)       | c =  }} {{eqn | r = 2 \\times 3^2 \\times 7 \\times 19       | c =  }} {{eqn | r = 2394       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:1210 6eiwqik7ft4fpi17l6dtmvwllrjpuuv"}
+{"_id": "32854", "title": "Sigma Function of 1638", "text": "Sigma Function of 1638 0 60492 318129 318127 2017-09-16T14:56:56Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\sigma \\left({1638}\\right) = 4368$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == We have that: :$1638 = 2 \\times 3^2 \\times 7 \\times 13$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({1638}\\right)       | r = \\left({2 + 1}\\right) \\frac {3^3 - 1} {3 - 1} \\times \\left({7 + 1}\\right) \\times \\left({13 + 1}\\right)       | c = Sigma Function of Integer }} {{eqn | r = 3 \\times \\frac {26} 2 \\times 8 \\times 14       | c =  }} {{eqn | r = 3 \\times 13 \\times 2^3 \\times \\left({2 \\times 7}\\right)       | c =  }} {{eqn | r = 2^4 \\times 3 \\times 7 \\times 13       | c =  }} {{eqn | r = 4368       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:1638 piry1g4q0g9krzjvna8gqdbqybd7dko"}
+{"_id": "32855", "title": "Tau Function of 1638", "text": "Tau Function of 1638 0 60493 445384 309876 2020-01-31T11:05:00Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {1638} = 24$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$1638 = 2 \\times 3^2 \\times 7 \\times 13$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {1638}       | r = \\map \\tau {2^1 \\times 3^2 \\times 7^1 \\times 13^1}       | c =  }} {{eqn | r = \\paren {1 + 1} \\paren {2 + 1} \\paren {1 + 1} \\paren {1 + 1}       | c =  }} {{eqn | r = 24       | c =  }} {{end-eqn}} The divisors of $1638$ can be enumerated as: :$1, 2, 3, 6, 7, 9, 13, 14, 18, 21, 26, 39, 42, 63, 78, 91, 117, 126, 182, 234, 273, 546, 819, 1638$ {{qed}} Category:Tau Function Category:1638 s9kup1bhls2umalmu3n3pig5qpa3a4y"}
+{"_id": "32856", "title": "Tau Function of 945", "text": "Tau Function of 945 0 60606 425091 310276 2019-09-13T12:10:32Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {945} = 16$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$945 = 3^3 \\times 5 \\times 7$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {945}       | r = \\map \\tau {3^3 \\times 5^1 \\times 7^1}       | c =  }} {{eqn | r = \\paren {3 + 1} \\paren {1 + 1} \\paren {1 + 1}       | c =  }} {{eqn | r = 16       | c =  }} {{end-eqn}} The divisors of $945$ can be enumerated as: :$1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, 315, 945$ {{OEIS|A018736}}{{qed}} Category:Tau Function Category:945 87e3zg4t9qjr3unoyfn33x5h0mnybkn"}
+{"_id": "32857", "title": "Tau Function of 910", "text": "Tau Function of 910 0 60608 439494 310279 2019-12-14T10:31:38Z Prime.mover 59 wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\map \\tau {910} = 16$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$910 = 2 \\times 5 \\times 7 \\times 13$ Thus: {{begin-eqn}} {{eqn | l = \\map \\tau {910}       | r = \\map \\tau {2^1 \\times 5^1 \\times 7^1 \\times 13^1}       | c =  }} {{eqn | r = \\paren {1 + 1} \\paren {1 + 1} \\paren {1 + 1} \\paren {1 + 1}       | c =  }} {{eqn | r = 16       | c =  }} {{end-eqn}} The divisors of $910$ can be enumerated as: :$1, 2, 5, 7, 10, 13, 14, 26, 35, 65, 70, 91, 130, 182, 455, 910$ {{OEIS|A018717}}{{qed}} Category:Tau Function Category:910 ccz2ot8ebhj08bja1wyq1810eollxc4"}
+{"_id": "32858", "title": "Tau Function of 1184", "text": "Tau Function of 1184 0 60611 310282 2017-08-10T06:33:21Z Prime.mover 59 Created page with \"== Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({1184}\\right) = 12$ </onlyinclude> where $\\tau$ denotes the Definition:Tau Fun...\" wikitext text/x-wiki == Example of Use of $\\tau$ Function == <onlyinclude> :$\\tau \\left({1184}\\right) = 12$ </onlyinclude> where $\\tau$ denotes the $\\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\\displaystyle \\tau \\left({n}\\right) = \\prod_{j \\mathop = 1}^r \\left({k_j + 1}\\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$1184 = 2^5 \\times 37$ Thus: {{begin-eqn}} {{eqn | l = \\tau \\left({1184}\\right)       | r = \\tau \\left({2^5 \\times 37^1}\\right)       | c =  }} {{eqn | r = \\left({5 + 1}\\right) \\left({2 + 1}\\right)       | c =  }} {{eqn | r = 12       | c =  }} {{end-eqn}} The divisors of $1184$ can be enumerated as: :$1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592, 1184$ {{qed}} Category:Tau Function Category:1184 hp5q548fdculppmqc91vdbj1uf8rot5"}
+{"_id": "32859", "title": "Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999/Examples", "text": "Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999/Examples 0 60630 478390 478377 2020-07-16T06:53:03Z Prime.mover 59 wikitext text/x-wiki == Examples of Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999 == <onlyinclude> {{begin-eqn}} {{eqn | l = 4 \\times 999       | r = 3996       | c =  }} {{eqn | ll= \\leadsto       | l = 3 + 996       | r = 999       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 15 \\times 999       | r = 14 \\, 985       | c =  }} {{eqn | ll= \\leadsto       | l = 14 + 985       | r = 999       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 47 \\times 999       | r = 46 \\, 953       | c =  }} {{eqn | ll= \\leadsto       | l = 46 + 953       | r = 999       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 57 \\times 999       | r = 56 \\, 943       | c =  }} {{eqn | ll= \\leadsto       | l = 56 + 943       | r = 999       | c =  }} {{end-eqn}} </onlyinclude> Category:Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999 k75tr675sa2cztel9d4thg7069dew4m"}
+{"_id": "32860", "title": "Carmichael Number/Examples/1729", "text": "Carmichael Number/Examples/1729 0 60981 323299 312511 2017-10-26T20:16:49Z Prime.mover 59 wikitext text/x-wiki == Example of Carmichael Number== $1729$ is a Carmichael number: <onlyinclude> :$\\forall a \\in \\Z: a \\perp 1729: a^{1729} \\equiv a \\pmod {1729}$ while $1729$ is composite. </onlyinclude> == Proof == We have that: :$1729 = 7 \\times 13 \\times 19$ and so: {{begin-eqn}} {{eqn | l = 7^2       | o = \\nmid       | r = 1729       | c =  }} {{eqn | l = 13^2       | o = \\nmid       | r = 1729       | c =  }} {{eqn | l = 19^2       | o = \\nmid       | r = 1729       | c =  }} {{end-eqn}} We also have that: {{begin-eqn}} {{eqn | l = 1728       | r = 288 \\times 6       | c =  }} {{eqn | r = 144 \\times 12       | c =  }} {{eqn | r = 96 \\times 18       | c =  }} {{end-eqn}} The result follows by Korselt's Theorem. {{qed}} == Sources == * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = 509,033,161|next = Carmichael Number/Examples/294,409}}: $509,033,161$ Category:Carmichael Numbers Category:1729 qgour7zwj5qk61w91iqa7mzf20wlqhx"}
+{"_id": "32861", "title": "Highly Composite Number/Examples/1", "text": "Highly Composite Number/Examples/1 0 61035 492964 312820 2020-10-06T18:06:40Z Prime.mover 59 wikitext text/x-wiki == Example of Highly Composite Number == <onlyinclude> $1$ is a highly composite number, being the smallest positive integer with $1$ divisor or more. </onlyinclude> == Proof == From Tau Function of 1: :$\\map \\tau 1 = 1$ The positive integer $1$ has $1$ divisor, that is, $1$ itself. Vacuously, no smaller positive integer has a greater number of divisors. Thus, despite not actually being composite, $1$ is a highly composite number. {{qed}} Category:Highly Composite Numbers 0p8650qep457aueoerbrrp7pp93pyns"}
+{"_id": "32862", "title": "Special Highly Composite Number/Examples/1", "text": "Special Highly Composite Number/Examples/1 0 61038 312830 2017-08-23T21:58:22Z Prime.mover 59 Created page with \"== Example of Special Highly Composite Number == <onlyinclude> $1$ is a Definition:Special Highly Composite Number|special hig...\" wikitext text/x-wiki == Example of Special Highly Composite Number == <onlyinclude> $1$ is a special highly composite number, being a highly composite number which is a divisor of all larger highly composite numbers. </onlyinclude> == Proof == We have that $1$ is highly composite. From One Divides all Integers, it follows trivially that $1$ is a divisor of all larger highly composite numbers. Thus, by definition, $1$ is a special highly composite number. {{qed}} == Sources == * {{Citation|title = An Interesting Subset of the Highly Composite Numbers|author = Steven Ratering|journal = Mathematics Magazine|abbr = Math. Mag.|volume = 64|issue = 5|date = Dec. 1991|startpage = 343|endpage = 346|jstor = 2690653}} Category:Special Highly Composite Numbers i3t0aihfktpa57wuu91qmdax7ttfb3j"}
+{"_id": "32863", "title": "Highly Composite Number/Examples/2", "text": "Highly Composite Number/Examples/2 0 61039 312834 312833 2017-08-23T22:04:28Z Prime.mover 59 wikitext text/x-wiki == Example of Highly Composite Number == <onlyinclude> $2$ is a highly composite number, being the smallest positive integer with $2$ divisors or more. </onlyinclude> == Proof == From Tau of Prime Number: :$\\tau \\left({2}\\right) = 2$ From Tau Function of 1: :$\\tau \\left({1}\\right) = 1$ That is, the only positive integer smaller than $2$ has a smaller number of divisors. Thus, despite not actually being composite, $2$ is a highly composite number. {{qed}} Category:Highly Composite Numbers opsdh13u7m4uhchr7v0zb1rvxf5e8ar"}
+{"_id": "32864", "title": "Special Highly Composite Number/Examples/2", "text": "Special Highly Composite Number/Examples/2 0 61065 412551 313064 2019-07-12T16:23:03Z Prime.mover 59 wikitext text/x-wiki == Example of Special Highly Composite Number == <onlyinclude> $2$ is a special highly composite number, being a highly composite number which is a divisor of all larger highly composite numbers. </onlyinclude> == Proof == We have that $2$ is highly composite. Let $n > 2$ be a highly composite number. From Prime Decomposition of Highly Composite Number, the multiplicity of $2$ in $n$ is at least as high as the multiplicity of any other prime $p$ in $n$. Thus if $p \\divides n$ it follows that $2 \\divides n$. Thus, by definition, $2$ is a special highly composite number. {{qed}} == Sources == * {{Citation|title = An Interesting Subset of the Highly Composite Numbers|author = Steven Ratering|journal = Mathematics Magazine|abbr = Math. Mag.|volume = 64|issue = 5|date = Dec. 1991|startpage = 343|endpage = 346|jstor = 2690653}} Category:Special Highly Composite Numbers Category:2 3m6blgzzkw9e3zqsh0mzkup9sr9v5cc"}
+{"_id": "32865", "title": "Special Highly Composite Number/Examples/6", "text": "Special Highly Composite Number/Examples/6 0 61067 313065 312947 2017-08-26T06:11:36Z Prime.mover 59 wikitext text/x-wiki == Example of Special Highly Composite Number == <onlyinclude> $6$ is a special highly composite number, being a highly composite number which is a divisor of all larger highly composite numbers. </onlyinclude> == Proof == By inspection of the sequence of highly composite numbers, $6$ is highly composite. {{AimForCont}} $n > 6$ is a highly composite number which is not divisible by $6$. We have that $2$ is a special highly composite number. Therefore $2$ is a divisor of $n$. As $6$ is not a divisor of $n$, it follows that $3$ is also not a divisor of $n$. By Prime Decomposition of Highly Composite Number, that means $n = 2^k$ for some $k \\ge 3$. Then: {{begin-eqn}} {{eqn | l = 2^{k - 2} \\times 3       | o = <       | r = 2^k       | c = because $3 < 2^2 = 4$ }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^{k - 2} \\times 3}\\right)       | o = <       | r = \\tau \\left({2^k}\\right)       | c = as $2^k$ is highly composite }} {{eqn | ll= \\leadsto       | l = \\left({k - 1}\\right) \\left({1 + 1}\\right)       | o = <       | r = k + 1       | c = {{Defof|Tau Function}} }} {{eqn | ll= \\leadsto       | l = k       | o = <       | r = 3       | c = after algebra }} {{end-eqn}} But this contradicts our deduction that $n = 2^k$ where $k \\ge 3$. The result follows by Proof by Contradiction. {{qed}} == Sources == * {{Citation|title = An Interesting Subset of the Highly Composite Numbers|author = Steven Ratering|journal = Mathematics Magazine|abbr = Math. Mag.|volume = 64|issue = 5|date = Dec. 1991|startpage = 343|endpage = 346|jstor = 2690653}} Category:Special Highly Composite Numbers Category:6 85yg2crs8eimf36fbfbd8u16gudlxwr"}
+{"_id": "32866", "title": "Special Highly Composite Number/Examples/12", "text": "Special Highly Composite Number/Examples/12 0 61069 313066 312950 2017-08-26T06:13:05Z Prime.mover 59 wikitext text/x-wiki == Example of Special Highly Composite Number == <onlyinclude> $12$ is a special highly composite number, being a highly composite number which is a divisor of all larger highly composite numbers. </onlyinclude> == Proof == By inspection of the sequence of highly composite numbers, $12$ is highly composite. {{AimForCont}} $n > 12$ is a highly composite number which is not divisible by $12$. We have that $6$ is a special highly composite number. Therefore $6$ is a divisor of $n$. As $12$ is not a divisor of $n$, it follows that the multiplicity of $2$ in $n$ is $1$. From Prime Decomposition of Highly Composite Number, that means: : $n = 2 \\times 3 \\times 5 \\times r$ where $r$ is a possibly vacuous square-free product of prime numbers strictly greater than $5$. Then: {{begin-eqn}} {{eqn | l = 2^3 \\times 3 \\times r       | o = <       | r = 2 \\times 3 \\times 5 \\times r       | c = because $4 = 2^2 < 5$ }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^3 \\times 3 \\times r}\\right)       | o = <       | r = \\tau \\left({2 \\times 3 \\times 5 \\times r}\\right)       | c = as $2 \\times 3 \\times 5 \\times r$ is highly composite }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^3 \\times 3}\\right) \\times \\tau \\left({r}\\right)       | o = <       | r = \\tau \\left({2 \\times 3 \\times 5}\\right) \\times \\tau \\left({r}\\right)       | c = Tau Function is Multiplicative }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^3 \\times 3}\\right)       | o = <       | r = \\tau \\left({2 \\times 3 \\times 5}\\right)       | c = simplifying }} {{eqn | ll= \\leadsto       | l = \\left({3 + 1}\\right) \\left({1 + 1}\\right)       | o = <       | r = \\left({1 + 1}\\right) \\left({1 + 1}\\right) \\left({1 + 1}\\right)       | c = {{Defof|Tau Function}} }} {{eqn | ll= \\leadsto       | l = 8       | o = <       | r = 8       | c = which is a falsehood }} {{end-eqn}} The result follows by Proof by Contradiction. {{qed}} == Sources == * {{Citation|title = An Interesting Subset of the Highly Composite Numbers|author = Steven Ratering|journal = Mathematics Magazine|abbr = Math. Mag.|volume = 64|issue = 5|date = Dec. 1991|startpage = 343|endpage = 346|jstor = 2690653}} Category:Special Highly Composite Numbers Category:12 ne9hzwbf2tx192vwctgj5lqk14clq3z"}
+{"_id": "32867", "title": "Special Highly Composite Number/Examples/60", "text": "Special Highly Composite Number/Examples/60 0 61070 313067 312953 2017-08-26T06:17:38Z Prime.mover 59 wikitext text/x-wiki == Example of Special Highly Composite Number == <onlyinclude> $60$ is a special highly composite number, being a highly composite number which is a divisor of all larger highly composite numbers. </onlyinclude> == Proof == By inspection of the sequence of highly composite numbers, $60$ is highly composite. {{AimForCont}} $n > 60$ is a highly composite number which is not divisible by $60$. We have that $12$ is a special highly composite number. Therefore $12$ is a divisor of $n$. As $60$ is not a divisor of $n$, it follows that while $3$ is a divisor of $n$, $5$ is not. From Prime Decomposition of Highly Composite Number, no prime number greater than $5$ is a divisor of $n$. Thus: :$n = 2^a \\times 3^b$ where $a \\ge b \\ge 1$. This will be investigated on a case-by-case basis. :$(1): \\quad b = 1$ That is, $n = 2^a \\times 3$. We have that $n > 60$. Therefore: :$(1 \\text a): \\quad a \\ge 5$ as $2^4 \\times 3^1 = 48$. Then: {{begin-eqn}} {{eqn | l = 2^{a - 3} \\times 3 \\times 5       | o = <       | r = 2^a \\times 3       | c = because $5 < 2^3 = 8$ }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^{a - 3} \\times 3 \\times 5}\\right)       | o = <       | r = \\tau \\left({2^a \\times 3}\\right)       | c = as $2^a \\times 3$ is highly composite }} {{eqn | ll= \\leadsto       | l = \\left({\\left({a - 3}\\right) + 1}\\right) \\left({1 + 1}\\right) \\left({1 + 1}\\right)       | o = <       | r = \\left({a + 1}\\right) \\left({1 + 1}\\right)       | c = {{Defof|Tau Function}} }} {{eqn | ll= \\leadsto       | l = 4 \\left({a - 2}\\right)       | o = <       | r = 2 \\left({a + 1}\\right)       | c = simplifying }} {{eqn | ll= \\leadsto       | l = a       | o = <       | r = 5       | c = which is a contradiction of $(1 \\text a)$ }} {{end-eqn}} It follows by Proof by Contradiction that $b \\ne 1$. {{qed|lemma}} :$(2): \\quad b = 2$ That is, $n = 2^a \\times 3^2$. We have that $n > 60$. Therefore: :$(2 \\text a): \\quad a \\ge 3$ as $2^2 \\times 3^2 = 36$. Then: {{begin-eqn}} {{eqn | l = 2^{a - 1} \\times 3 \\times 5       | o = <       | r = 2^a \\times 3^2       | c = because $5 < 2 \\times 3 = 6$ }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^{a - 1} \\times 3 \\times 5}\\right)       | o = <       | r = \\tau \\left({2^a \\times 3^2}\\right)       | c = as $2^a \\times 3^2$ is highly composite }} {{eqn | ll= \\leadsto       | l = \\left({\\left({a - 1}\\right) + 1}\\right) \\left({1 + 1}\\right) \\left({1 + 1}\\right)       | o = <       | r = \\left({a + 1}\\right) \\left({2 + 1}\\right)       | c = {{Defof|Tau Function}} }} {{eqn | ll= \\leadsto       | l = 4 a       | o = <       | r = 3 \\left({a + 1}\\right)       | c = simplifying }} {{eqn | ll= \\leadsto       | l = a       | o = <       | r = 3       | c = which is a contradiction of $(2 \\text a)$ }} {{end-eqn}} It follows by Proof by Contradiction that $b \\ne 2$. {{qed|lemma}} :$(3): \\quad b \\ge 3$ By Prime Decomposition of Highly Composite Number we have that $a \\ge 3$. Then: {{begin-eqn}} {{eqn | l = 2^{a - 1} \\times 3^{b - 1} \\times 5       | o = <       | r = 2^a \\times 3^b       | c = because $5 < 2 \\times 3 = 6$ }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^{a - 1} \\times 3^{b - 1} \\times 5}\\right)       | o = <       | r = \\tau \\left({2^a \\times 3^b}\\right)       | c = as $2^a \\times 3^b$ is highly composite }} {{eqn | ll= \\leadsto       | l = \\left({\\left({a - 1}\\right) + 1}\\right) \\left({\\left({b - 1}\\right) + 1}\\right) \\left({1 + 1}\\right)       | o = <       | r = \\left({a + 1}\\right) \\left({b + 1}\\right)       | c = {{Defof|Tau Function}} }} {{eqn | ll= \\leadsto       | l = 2 a b       | o = <       | r = \\left({a + 1}\\right) \\left({b + 1}\\right)       | c =  }} {{eqn | ll= \\leadsto       | l = a b       | o = <       | r = a + b + 1       | c =  }} {{eqn | ll= \\leadsto       | l = 3 a       | o = <       | r = a + b + 1       | c = as $b \\ge 3$ }} {{eqn | ll= \\leadsto       | l = 2 a       | o = <       | r = b + 1       | c =  }} {{eqn | ll= \\leadsto       | l = 2 b       | o = <       | r = b + 1       | c = as $a \\ge b$ }} {{eqn | ll= \\leadsto       | l = b       | o = <       | r = 1       | c = which is a contradiction of $(3)$ }} {{end-eqn}} {{qed|lemma}} By Proof by Cases it is seen that the existence of a highly composite $n$ not divisible by $60$ leads to a contradiction. The result then follows by Proof by Contradiction. {{qed}} == Sources == * {{Citation|title = An Interesting Subset of the Highly Composite Numbers|author = Steven Ratering|journal = Mathematics Magazine|abbr = Math. Mag.|volume = 64|issue = 5|date = Dec. 1991|startpage = 343|endpage = 346|jstor = 2690653}} Category:Special Highly Composite Numbers Category:60 2scpj1d25yf7x7ls4qlytolch75s6en"}
+{"_id": "32868", "title": "Special Highly Composite Number/Examples/2520", "text": "Special Highly Composite Number/Examples/2520 0 61071 322270 321092 2017-10-22T12:17:49Z Prime.mover 59 wikitext text/x-wiki == Example of Special Highly Composite Number == <onlyinclude> $2520$ is a special highly composite number, being a highly composite number which is a divisor of all larger highly composite numbers. </onlyinclude> == Proof == By inspection of the sequence of highly composite numbers, $2520$ is highly composite. For reference, the prime decomposition of $2520$ is: :$2520 = 2^3 \\times 3^2 \\times 5 \\times 7$ {{AimForCont}} $n > 2520$ is a highly composite number which is not divisible by $2520$. We have that $60$ is a special highly composite number. Therefore $60$ is a divisor of $n$. It follows that $3$, $4$ and $5$ are all divisors of $n$. But as $2520$ is not a divisor of $n$, it follows that at least one of $7$, $8$ and $9$ is not a divisor of $n$. These will be investigated on a case-by-case basis. :$(1): \\quad 7$  is not a divisor of $n$. By Prime Decomposition of Highly Composite Number we have that: :$n = 2^a \\times 3^b \\times 5^c$ where $a \\ge b \\ge c \\ge 1$. Suppose that $a < 3$. Then: :$n \\le 2^2 \\times 3^2 \\times 5^2 = 900$ which is too small. So we have that $a \\ge 3$. Then: {{begin-eqn}} {{eqn | l = 2^{a - 3} \\times 3^b \\times 5^c \\times 7       | o = <       | r = 2^a \\times 3^b \\times 5^c       | c = because $7 < 2^3 = 8$ }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^{a - 3} \\times 3^b \\times 5^c \\times 7}\\right)       | o = <       | r = \\tau \\left({2^a \\times 3^b \\times 5^c}\\right)       | c = as $2^a \\times 3^b \\times 5^c$ is highly composite }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^{a - 3} \\times 7}\\right) \\times \\tau \\left({3^b \\times 5^c}\\right)       | o = <       | r = \\tau \\left({2^a}\\right) \\times \\tau \\left({3^b \\times 5^c}\\right)       | c = Tau Function is Multiplicative }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^{a - 3} \\times 7}\\right)       | o = <       | r = \\tau \\left({2^a}\\right)       | c = simplifying }} {{eqn | ll= \\leadsto       | l = \\left({\\left({a - 3}\\right) + 1}\\right) \\left({1 + 1}\\right)       | o = <       | r = \\left({a + 1}\\right)       | c = {{Defof|Tau Function}} }} {{eqn | ll= \\leadsto       | l = 2 \\left({a - 2}\\right)       | o = <       | r = a + 1       | c =  }} {{eqn | ll= \\leadsto       | l = a       | o = <       | r = 5       | c =  }} {{end-eqn}} Now suppose that $b < 2$. Then: :$n \\le 2^5 \\times 3 \\times 5 = 480$ which is too small. So we have that $b \\ge 2$. Then: {{begin-eqn}} {{eqn | l = 2^a \\times 3^{b - 2} \\times 5^c \\times 7       | o = <       | r = 2^a \\times 3^b \\times 5^c       | c = because $7 < 3^2 = 9$ }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^a \\times 3^{b - 2} \\times 5^c \\times 7}\\right)       | o = <       | r = \\tau \\left({2^a \\times 3^b \\times 5^c}\\right)       | c = as $2^a \\times 3^b \\times 5^c$ is highly composite }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^a \\times 5^c}\\right) \\times \\tau \\left({3^{b - 2} \\times 7}\\right)       | o = <       | r = \\tau \\left({2^a \\times 5^c}\\right) \\times \\tau \\left({3^b}\\right)       | c = Tau Function is Multiplicative }} {{eqn | ll= \\leadsto       | l = \\tau \\left({3^{b - 2} \\times 7}\\right)       | o = <       | r = \\tau \\left({3^b}\\right)       | c = simplifying }} {{eqn | ll= \\leadsto       | l = \\left({\\left({b - 2}\\right) + 1}\\right) \\left({1 + 1}\\right)       | o = <       | r = \\left({b + 1}\\right)       | c = {{Defof|Tau Function}} }} {{eqn | ll= \\leadsto       | l = 2 \\left({b - 1}\\right)       | o = <       | r = b + 1       | c =  }} {{eqn | ll= \\leadsto       | l = b       | o = <       | r = 3       | c =  }} {{end-eqn}} But $c \\le b$ and so $c < 3$ as well. Thus we have upper bounds on $a$, $b$ and $c$. Since $2^a \\times 3^b \\times 5^c > 2520$, it must be the case that: :$n = 2^4 \\times 3^2 \\times 5^2$ which gives that $n = 3600$. But: : from {{TauLink|3600}} we have that $\\tau \\left({3600}\\right) = 45$ : from {{TauLink|2520}} we have that $\\tau \\left({2520}\\right) = 48$ This contradicts our hypothesis that $3600$ is highly composite. By Proof by Contradiction it follows that $7$ must be a divisor of $n$. {{qed|lemma}} :$(2): \\quad 9$ is not a divisor of $n$, but $7$ is. By Prime Decomposition of Highly Composite Number we have that: :$n = 2^a \\times 3^1 \\times 5^1 \\times 7^1 \\times 11^e \\times r$ where: :$e$ is either $0$ or $1$ :$r$ is a possibly vacuous square-free product of prime numbers strictly greater than $11$. Suppose $e = 1$. Then: {{begin-eqn}} {{eqn | l = 2^a \\times 3^3 \\times 5 \\times 7 \\times r       | o = <       | r = 2^a \\times 3 \\times 5 \\times 7 \\times 11 \\times r       | c = because $9 = 3^2 < 11$ }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^a \\times 3^3 \\times 5 \\times 7 \\times r}\\right)       | o = <       | r = \\tau \\left({2^a \\times 3 \\times 5 \\times 7 \\times 11 \\times r}\\right)       | c = as $2^a \\times 3 \\times 5 \\times 7 \\times 11 \\times r$ is highly composite }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^a \\times 5 \\times 7 \\times r}\\right) \\times \\tau \\left({3^3}\\right)       | o = <       | r = \\tau \\left({2^a \\times 5 \\times 7 \\times r}\\right) \\times \\tau \\left({3 \\times 11}\\right)       | c = Tau Function is Multiplicative }} {{eqn | ll= \\leadsto       | l = \\tau \\left({3^3}\\right)       | o = <       | r = \\tau \\left({3 \\times 11}\\right)       | c = simplifying }} {{eqn | ll= \\leadsto       | l = 3 + 1       | o = <       | r = \\left({1 + 1}\\right) \\left({1 + 1}\\right)       | c = {{Defof|Tau Function}} }} {{eqn | ll= \\leadsto       | l = 4       | o = <       | r = 4       | c = which is an absurdity }} {{end-eqn}} So $e = 0$ and so by Prime Decomposition of Highly Composite Number $r = 1$. Thus: :$n = 2^a \\times 3 \\times 5 \\times 7$ We have that $n > 2520$, so: :$(2 \\text a): a \\ge 5$ Then: {{begin-eqn}} {{eqn | l = 2^{a - 2} \\times 3^2 \\times 5 \\times 7       | o = <       | r = 2^a \\times 3 \\times 5 \\times 7       | c = because $3 < 2^2 = 4$ }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^{a - 2} \\times 3^2 \\times 5 \\times 7}\\right)       | o = <       | r = \\tau \\left({2^a \\times 3 \\times 5 \\times 7}\\right)       | c = as $2^a \\times 3 \\times 5 \\times 7$ is highly composite }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^{a - 2} \\times 3^2}\\right) \\times \\tau \\left({5 \\times 7}\\right)       | o = <       | r = \\tau \\left({2^a \\times 3}\\right) \\times \\tau \\left({5 \\times 7}\\right)       | c = Tau Function is Multiplicative }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^{a - 2} \\times 3^2}\\right)       | o = <       | r = \\tau \\left({2^a \\times 3}\\right)       | c = simplifying }} {{eqn | ll= \\leadsto       | l = \\left({\\left({a - 2}\\right) + 1}\\right) \\left({2 + 1}\\right)       | o = <       | r = \\left({a + 1}\\right) \\left({1 + 1}\\right)       | c = {{Defof|Tau Function}} }} {{eqn | ll= \\leadsto       | l = 3 \\left({a - 1}\\right)       | o = <       | r = 2 \\left({a + 1}\\right)       | c = simplifying }} {{eqn | ll= \\leadsto       | l = a       | o = <       | r = 5       | c = which is a contradiction of $(2 \\text a)$ }} {{end-eqn}} It follows by Proof by Contradiction that $9$ is a divisor of $n$. {{qed|lemma}} :$(3): \\quad 8$ is not a divisor of $n$, but $7$ and $9$ both are. By Prime Decomposition of Highly Composite Number we have that: :$n = 2^2 \\times 3^2 \\times 5^c \\times 7^d \\times 11^e \\times r$ where $r$ is a possibly vacuous product of prime numbers strictly greater than $11$. Suppose: :$(3 \\text a): \\quad e > 0$ Then: {{begin-eqn}} {{eqn | l = 2^5 \\times 3^2 \\times 5^c \\times 7^d \\times 11^{e - 1} \\times r       | o = <       | r = 2^2 \\times 3^2 \\times 5^c \\times 7^d \\times 11^e \\times r       | c = because $8 = 2^3 < 11$ }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^5 \\times 3^2 \\times 5^c \\times 7^d \\times 11^{e - 1} \\times r}\\right)       | o = <       | r = \\tau \\left({2^2 \\times 3^2 \\times 5^c \\times 7^d \\times 11^e \\times r}\\right)       | c = as $2^2 \\times 3^2 \\times 5^c \\times 7^d \\times 11^e \\times r$ is highly composite }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^5 \\times 11^{e - 1} }\\right) \\times \\tau \\left({3^2 \\times 5^c \\times 7^d \\times r}\\right)       | o = <       | r = \\tau \\left({2^2 \\times 11^e}\\right) \\times \\tau \\left({3^2 \\times 5^c \\times 7^d \\times r}\\right)       | c = Tau Function is Multiplicative }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^5 \\times 11^{e - 1} }\\right)       | o = <       | r = \\tau \\left({2^2 \\times 11^e}\\right)       | c = simplifying }} {{eqn | ll= \\leadsto       | l = \\left({5 + 1}\\right) \\left({\\left({e - 1}\\right) + 1}\\right)       | o = <       | r = \\left({2 + 1}\\right) \\left({e + 1}\\right)       | c = {{Defof|Tau Function}} }} {{eqn | ll= \\leadsto       | l = 6 e       | o = <       | r = 3 \\left({e + 1}\\right)       | c = simplifying }} {{eqn | ll= \\leadsto       | l = e       | o = <       | r = 1       | c = which is a contradiction of $(3 \\text a)$ }} {{end-eqn}} So $e = 0$ and so Prime Decomposition of Highly Composite Number $r = 1$. Thus: :$n = 2^2 \\times 3^2 \\times 5^c \\times 7^d$ Suppose $c = 2$. Then: {{begin-eqn}} {{eqn | l = 2^4 \\times 3^2 \\times 5^1 \\times 7^d       | o = <       | r = 2^2 \\times 3^2 \\times 5^2 \\times 7^d       | c = because $4 = 2^2 < 5$ }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^4 \\times 3^2 \\times 5^1 \\times 7^d}\\right)       | o = <       | r = \\tau \\left({2^2 \\times 3^2 \\times 5^2 \\times 7^d}\\right)       | c = as $2^2 \\times 3^2 \\times 5^2 \\times 7^d$ is highly composite }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^4 \\times 5}\\right) \\times \\tau \\left({3^2 \\times 7^d}\\right)       | o = <       | r = \\tau \\left({2^2 \\times 5^2}\\right) \\times \\tau \\left({3^2 \\times 7^d}\\right)       | c = Tau Function is Multiplicative }} {{eqn | ll= \\leadsto       | l = \\tau \\left({2^4 \\times 5}\\right)       | o = <       | r = \\tau \\left({2^2 \\times 5^2}\\right)       | c = simplifying }} {{eqn | ll= \\leadsto       | l = \\left({4 + 1}\\right) \\left({1 + 1}\\right)       | o = <       | r = \\left({2 + 1}\\right) \\left({2 + 1}\\right)       | c = {{Defof|Tau Function}} }} {{eqn | ll= \\leadsto       | l = 10       | o = <       | r = 9       | c = which is an absurdity }} {{end-eqn}} The remaining possibility is that $c = 1$ and $d = 1$: Thus: :$n = 2^2 \\times 3^2 \\times 5 \\times 7 = 1260$ But this is a contradiction of our supposition that $n > 2520$. It follows by Proof by Contradiction that $8$ is a divisor of $n$. {{qed|lemma}} By Proof by Cases it is seen that the existence of a highly composite $n$ not divisible by $2520$ leads to a contradiction. The result then follows by Proof by Contradiction. {{qed}} == Sources == * {{Citation|title = An Interesting Subset of the Highly Composite Numbers|author = Steven Ratering|journal = Mathematics Magazine|abbr = Math. Mag.|volume = 64|issue = 5|date = Dec. 1991|startpage = 343|endpage = 346|jstor = 2690653}} Category:Special Highly Composite Numbers Category:2520 h5re8uupp5lyxmk06qhw4oe3ena9n0u"}
+{"_id": "32869", "title": "Sigma Function of 2620", "text": "Sigma Function of 2620 0 61119 318211 318209 2017-09-16T18:24:46Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\sigma \\left({2620}\\right) = 5544$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == From Sigma Function of Integer :$\\displaystyle \\sigma \\left({n}\\right) = \\prod_{1 \\mathop \\le i \\mathop \\le r} \\frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \\displaystyle \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$2620 = 2^2 \\times 5 \\times 131$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({2620}\\right)       | r = \\frac {2^3 - 1} {2 - 1} \\times \\left({5 + 1}\\right) \\times \\left({131 + 1}\\right)       | c =  }} {{eqn | r = \\frac 7 1 \\times 6 \\times 132       | c =  }} {{eqn | r = 7 \\times \\left({2 \\times 3}\\right) \\times \\left({2^2 \\times 3 \\times 11}\\right)       | c =  }} {{eqn | r = 2^3 \\times 3^2 \\times 7 \\times 11       | c =  }} {{eqn | r = 5544       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:2620 oo4wii402j4f183zvi1wjki1geo26ro"}
+{"_id": "32870", "title": "Sigma Function of 2924", "text": "Sigma Function of 2924 0 61123 318232 318230 2017-09-16T18:27:48Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\sigma \\left({2924}\\right) = 5544$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == From Sigma Function of Integer :$\\displaystyle \\sigma \\left({n}\\right) = \\prod_{1 \\mathop \\le i \\mathop \\le r} \\frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \\displaystyle \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$2924 = 2^2 \\times 17 \\times 43$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({2924}\\right)       | r = \\frac {2^3 - 1} {2 - 1} \\times \\left({17 + 1}\\right) \\times \\left({43 + 1}\\right)       | c =  }} {{eqn | r = \\frac 7 1 \\times 18 \\times 44       | c =  }} {{eqn | r = 7 \\times \\left({2 \\times 3^2}\\right) \\times \\left({2^2 \\times 11}\\right)       | c =  }} {{eqn | r = 2^3 \\times 3^2 \\times 7 \\times 11       | c =  }} {{eqn | r = 5544       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:2924 lpi8bmft2jg3kv0jl0f7apaxhn8xjbk"}
+{"_id": "32871", "title": "Sigma Function of 5020", "text": "Sigma Function of 5020 0 61419 318288 318286 2017-09-16T18:36:38Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\sigma \\left({5020}\\right) = 10 \\, 584$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == From Sigma Function of Integer :$\\displaystyle \\sigma \\left({n}\\right) = \\prod_{1 \\mathop \\le i \\mathop \\le r} \\frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \\displaystyle \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$5020 = 2^2 \\times 5 \\times 251$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({5020}\\right)       | r = \\frac {2^3 - 1} {2 - 1} \\times \\left({5 + 1}\\right) \\times \\left({251 + 1}\\right)       | c =  }} {{eqn | r = \\frac 7 1 \\times 6 \\times 252       | c =  }} {{eqn | r = 7 \\times \\left({2 \\times 3}\\right) \\times \\left({2^2 \\times 3^2 \\times 7}\\right)       | c =  }} {{eqn | r = 2^3 \\times 3^3 \\times 7^2       | c =  }} {{eqn | r = 10 \\, 584       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:5020 3fvyay6mnnqrsutdmzfx7aen7y3uq12"}
+{"_id": "32872", "title": "Sigma Function of 5564", "text": "Sigma Function of 5564 0 61422 449562 318246 2020-02-17T10:32:33Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\map \\sigma {5564} = 10 \\, 584$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == From Sigma Function of Integer :$\\map \\sigma n = \\displaystyle \\prod_{1 \\mathop \\le i \\mathop \\le r} \\frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \\displaystyle \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$5564 = 2^2 \\times 13 \\times 107$ Hence: {{begin-eqn}} {{eqn | l = \\map \\sigma {5020}       | r = \\frac {2^3 - 1} {2 - 1} \\times \\paren {13 + 1} \\times \\paren {107 + 1}       | c =  }} {{eqn | r = \\frac 7 1 \\times 14 \\times 108       | c =  }} {{eqn | r = 7 \\times \\paren {2 \\times 7} \\times \\paren {2^2 \\times 3^3}       | c =  }} {{eqn | r = 2^3 \\times 3^3 \\times 7^2       | c =  }} {{eqn | r = 10 \\, 584       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:5564 dkj32zjugpiuvxaye2xd0q0esa16uhu"}
+{"_id": "32873", "title": "Euler Phi Function of 5186", "text": "Euler Phi Function of 5186 0 61432 313964 2017-08-29T07:04:25Z Prime.mover 59 Created page with \"== Example of Euler Phi Function of 2 times Odd Prime == <onlyinclude> :$\\phi \\left({5186}\\right) = 2592$ </onlyinclude> where $\\phi$ denotes the Definition:Euler Phi Fu...\" wikitext text/x-wiki == Example of Euler Phi Function of 2 times Odd Prime == <onlyinclude> :$\\phi \\left({5186}\\right) = 2592$ </onlyinclude> where $\\phi$ denotes the Euler $\\phi$ Function. == Proof == We have that: :$5186 = 2 \\times 2593$ Thus: {{begin-eqn}} {{eqn | l = \\phi \\left({5186}\\right)       | r = 2593 - 1       | c = Euler Phi Function of 2 times Odd Prime }} {{eqn | r = 2592       | c =  }} {{end-eqn}} {{qed}} Category:Euler Phi Function of 2 times Odd Prime Category:5186 6u42hodcge8arxocxxfdwq9byioakzg"}
+{"_id": "32874", "title": "Euler Phi Function of 5187", "text": "Euler Phi Function of 5187 0 61433 313968 2017-08-29T07:09:59Z Prime.mover 59 Created page with \"== Example of Euler $\\phi$ Function of Square-Free Integer == <onlyinclude> :$\\phi \\left({5187}\\right) = 2592$ </onlyinclude> whe...\" wikitext text/x-wiki == Example of Euler $\\phi$ Function of Square-Free Integer == <onlyinclude> :$\\phi \\left({5187}\\right) = 2592$ </onlyinclude> where $\\phi$ denotes the Euler $\\phi$ Function. == Proof == From Euler Phi Function of Square-Free Integer: :$\\displaystyle \\phi \\left({n}\\right) = \\prod_{\\substack {p \\mathop \\backslash n \\\\ p \\mathop > 2} } \\left({p - 1}\\right)$ where $p \\mathop \\backslash n$ denotes the primes which divide $n$. We have that: :$5187 = 3 \\times 7 \\times 13 \\times 19$ and so is square-free. Thus: {{begin-eqn}} {{eqn | l = \\phi \\left({5187}\\right)       | r = \\left({2 - 1}\\right) \\left({7 - 1}\\right) \\left({13 - 1}\\right) \\left({19 - 1}\\right)       | c =  }} {{eqn | r = 2 \\times 6 \\times 12 \\times 18       | c =  }} {{eqn | r = 2 \\times \\left({2 \\times 3}\\right) \\times \\left({2^2 \\times 3}\\right) \\times \\left({2 \\times 3^2}\\right)       | c =  }} {{eqn | r = 2^5 \\times 3^4       | c =  }} {{eqn | r = 2592       | c =  }} {{end-eqn}} {{qed}} Category:Euler Phi Function of Square-Free Integer Category:5187 pfzyjagdkn9eh3xfjg84vlul4jf99nt"}
+{"_id": "32875", "title": "Euler Phi Function of 5188", "text": "Euler Phi Function of 5188 0 61434 313969 2017-08-29T07:12:17Z Prime.mover 59 Created page with \"== Example of Use of Euler $\\phi$ Function == <onlyinclude> :$\\phi \\left({5188}\\right) = 2592$ </onlyinclude> where $\\phi$ denotes the Defi...\" wikitext text/x-wiki == Example of Use of Euler $\\phi$ Function == <onlyinclude> :$\\phi \\left({5188}\\right) = 2592$ </onlyinclude> where $\\phi$ denotes the Euler $\\phi$ Function. == Proof == From Euler Phi Function of Integer: :$\\displaystyle \\phi \\left({n}\\right) = n \\prod_{p \\mathop \\backslash n} \\left({1 - \\frac 1 p}\\right)$ where $p \\mathop \\backslash n$ denotes the primes which divide $n$. We have that: :$5188 = 2^2 \\times 1297$ Thus: {{begin-eqn}} {{eqn | l = \\phi \\left({5188}\\right)       | r = 5188 \\left({1 - \\dfrac 1 2}\\right) \\left({1 - \\dfrac 1 {1297} }\\right)       | c =  }} {{eqn | r = 5188 \\times \\frac 1 2 \\times \\frac {1296} {1297}       | c =  }} {{eqn | r = 2 \\times 1 \\times 1296       | c =  }} {{eqn | r = 2592       | c =  }} {{end-eqn}} {{qed}} Category:Euler Phi Function Category:5188 ksc0vo75vdfeh9ceq0pw48wdpg33m9c"}
+{"_id": "32876", "title": "Hurwitz's Theorem (Number Theory)/Lemma 1", "text": "Hurwitz's Theorem (Number Theory)/Lemma 1 0 61447 408768 314012 2019-06-17T20:24:04Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $\\xi$ be an irrational number. Let $A \\in \\R$ be a real number strictly greater than $\\sqrt 5$. Then there may exist at most a finite number of relatively prime integers $p, q \\in \\Z$ such that: :$\\size {\\xi - \\dfrac p q} < \\dfrac 1 {A \\, q^2}$ </onlyinclude> == Proof == We will take as our example of such an irrational number: :$\\xi = \\dfrac {\\sqrt 5 - 1} 2$ This is equal to $1 - \\phi$, where $\\phi$ is the Golden mean. {{AimForCont}} that there exist an infinite number of $p, q$ with $p \\perp q$ such that: :$\\size {\\xi - \\dfrac p q} < \\dfrac 1 {A \\, q^2}$ Then there exist an infinite number of $p, q$ with $p \\perp q$ such that: :$\\xi = \\dfrac p q + \\dfrac \\delta {q^2}$ where: :$\\size \\delta < \\dfrac 1 A < \\dfrac 1 {\\sqrt 5}$ Hence: {{begin-eqn}} {{eqn | l = \\dfrac \\delta q       | r = q \\xi - p       | c =  }} {{eqn | ll= \\leadsto       | l = \\dfrac \\delta q - \\dfrac {q \\sqrt 5} 2       | r = -\\dfrac q 2 - p       | c =  }} {{eqn | n = 1       | ll= \\leadsto       | l = \\dfrac {\\delta^2} {q^2} - \\delta \\sqrt 5       | r = \\paren {\\dfrac q 2 - p}^2 - \\dfrac {5 q^2} 4       | c =  }} {{eqn | r = p^2 + p q - q^2       | c =  }} {{end-eqn}} When $q$ is large, the {{LHS}} of $(1)$ becomes less than $1$. At the same time, the {{RHS}} is always an integer. Thus: :$p^2  + p q - q^2 = 0$ or: :$\\paren {2 p + q}^2 = 5 q^2$ which would lead to: :$p = 2 q$ which contradicts the stipulation that $p$ and $q$ are coprime. Hence by Proof by Contradiction there cannot be an infinite number of such $p, q$. Hence the result. {{qed}} {{Namedfor|Adolf Hurwitz}} == Sources == * {{BookReference|An Introduction to the Theory of Numbers|1979|G.H. Hardy|author2 = E.M. Wright|ed = 5th|edpage = Fifth Edition}}: $11.8$: The measure of the closest approximation to an arbitrary irrational: Theorem $194$ Category:Hurwitz's Theorem (Number Theory) 3jryiijqyoi0w0w4np98gsgoqlqeckx"}
+{"_id": "32877", "title": "Hurwitz's Theorem (Number Theory)/Lemma 2", "text": "Hurwitz's Theorem (Number Theory)/Lemma 2 0 61450 314015 2017-08-29T21:47:57Z Prime.mover 59 Created page with \"{{MissingLinks|Much of the notation used below needs to be explained in its context.}} == Lemma == <onlyinclude> Let $\\xi$ be an Definition:Irrational Number|irrational numb...\" wikitext text/x-wiki {{MissingLinks|Much of the notation used below needs to be explained in its context.}} == Lemma == <onlyinclude> Let $\\xi$ be an irrational number. Let there be $3$ consecutive convergents of the continued fraction to $\\xi$. Then at least one of them, $\\dfrac p q$, say, satisfies: :$\\left|{\\xi - \\dfrac p q}\\right| < \\dfrac 1 {A \\, q^2}$ </onlyinclude> == Proof == Let $\\dfrac {p_k} {q_k}$ be an arbitrary convergent to $\\xi$. Let: :$\\dfrac {q_{n - 1} } {q_n} = b_{n + 1}$ Then: {{begin-eqn}} {{eqn | l = \\left\\lvert{\\dfrac {p_n} {q_n} - \\xi}\\right\\rvert       | r = \\dfrac 1 {q_n q'_{n + 1} }       | c =  }} {{eqn | r = \\dfrac 1 { {q_n}^2} \\dfrac 1 {a'_{n + 1} + b_{n + 1} }       | c =  }} {{end-eqn}} It is sufficient to prove that: :$(1): \\quad a'_i + b_i \\le 5$ cannot be true for all of $n - 1$, $n$ and $n + 1$ of $i$. Suppose $(1)$ is true for $i = n - 1$ and $i = n$. Then: :$a'_{n - 1} = a_{n - 1} + \\dfrac 1 {a'_n}$ and: :$\\dfrac 1 {b_n} = \\dfrac {q_{n - 1} } {q_{n - 2} } = a_{n - 1} + b_{n - 1}$ Hence: :$\\dfrac 1 {a'_n} + \\dfrac 1 {b_n} = a'_{n - 1} + b_{n - 1} \\le \\sqrt 5$ and: :$1 = a'_n \\dfrac 1 {a'_n} \\le \\left({\\sqrt 5 - b_n}\\right) \\left({\\sqrt 5 - \\dfrac 1 {b_n} }\\right)$ or: :$b_n + \\dfrac 1 {b_n} \\le \\sqrt 5$ As $b_n$ is rational, the equality cannot happen. We also have that $b_n < 1$. Thus: :${b_n}^2 -b_n \\sqrt 5 + 1 < 0$ :$\\left({\\dfrac {\\sqrt 5} 2 - b_n}\\right)^2 < \\dfrac 1 4$ and so: :$b_n > \\dfrac {\\sqrt 5 - 1} 2$ If $(1)$ were also true for $i = n + 1$, it could be proved similarly that: :$b_{n + 1} > \\dfrac {\\sqrt 5 - 1} 2$ and we would then be able to substitute $n + 1$ for $n$ in the above equations, to get: :$a_n = \\dfrac 1 {b_{n + 1} } - b_n < \\dfrac {\\sqrt 5 + 1} 2 - \\dfrac {\\sqrt 5 - 1} 2 = 1$ from which a contradiction is apparent. {{qed}} == Sources == * {{BookReference|An Introduction to the Theory of Numbers|1979|G.H. Hardy|author2 = E.M. Wright|ed = 5th|edpage = Fifth Edition}}: $11.8$: The measure of the closest approximation to an arbitrary irrational: Theorem $195$ Category:Hurwitz's Theorem (Number Theory) mtf3n0u2276ibpmkdowb12f8np3vhc4"}
+{"_id": "32878", "title": "Sigma Function of 6232", "text": "Sigma Function of 6232 0 62161 318239 318237 2017-09-16T18:29:08Z Prime.mover 59 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\sigma \\left({6232}\\right) = 12 \\, 600$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == From Sigma Function of Integer :$\\displaystyle \\sigma \\left({n}\\right) = \\prod_{1 \\mathop \\le i \\mathop \\le r} \\frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \\displaystyle \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$6232 = 2^3 \\times 19 \\times 41$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({6232}\\right)       | r = \\frac {2^4 - 1} {2 - 1} \\times \\left({19 + 1}\\right) \\times \\left({41 + 1}\\right)       | c =  }} {{eqn | r = \\frac {15} 1 \\times 20 \\times 42       | c =  }} {{eqn | r = \\left({3 \\times 5}\\right) \\times \\left({2^2 \\times 5}\\right) \\times \\left({2 \\times 3 \\times 7}\\right)       | c =  }} {{eqn | r = 2^3 \\times 3^2 \\times 5^2 \\times 7       | c =  }} {{eqn | r = 12 \\, 600       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:6232 3wiisvb0sskiv79patt7x501veejja0"}
+{"_id": "32879", "title": "Sigma Function of 6368", "text": "Sigma Function of 6368 0 62162 318240 318236 2017-09-16T18:29:20Z Prime.mover 59 Prime.mover moved page Sigma of 6368 to Sigma Function of 6368 wikitext text/x-wiki == Example of Sigma Function of Integer == <onlyinclude> :$\\sigma \\left({6368}\\right) = 12 \\, 600$ </onlyinclude> where $\\sigma$ denotes the $\\sigma$ function. == Proof == From Sigma Function of Integer :$\\displaystyle \\sigma \\left({n}\\right) = \\prod_{1 \\mathop \\le i \\mathop \\le r} \\frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \\displaystyle \\prod_{1 \\mathop \\le i \\mathop \\le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$6368 = 2^5 \\times 199$ Hence: {{begin-eqn}} {{eqn | l = \\sigma \\left({6368}\\right)       | r = \\frac {2^6 - 1} {2 - 1} \\times \\left({199 + 1}\\right)       | c =  }} {{eqn | r = \\frac {63} 1 \\times 200       | c =  }} {{eqn | r = \\left({3^2 \\times 7}\\right) \\times \\left({2^3 \\times 5^2}\\right)       | c =  }} {{eqn | r = 2^3 \\times 3^2 \\times 5^2 \\times 7       | c =  }} {{eqn | r = 12 \\, 600       | c =  }} {{end-eqn}} {{qed}} Category:Sigma Function of Integer Category:6368 iidrztpqoybupgrji3pcirgmyhsiy7p"}
+{"_id": "32880", "title": "Completion Theorem (Measure Spaces)/Lemma", "text": "Completion Theorem (Measure Spaces)/Lemma 0 62230 317250 317249 2017-09-13T21:08:39Z Lord Farin 560 wikitext text/x-wiki == Lemma == Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $\\mathcal N$ and $\\Sigma^*$ be defined as: :$\\mathcal N := \\left\\{{N \\subseteq X: \\exists M \\in \\Sigma: \\mu \\left({M}\\right) = 0, N \\subseteq M}\\right\\}$ :$\\Sigma^* := \\left\\{{E \\cup N: E \\in \\Sigma, N \\in \\mathcal N}\\right\\}$ Next, define $\\bar \\mu: \\Sigma^* \\to \\overline{\\R}_{\\ge 0}$ by: :$\\bar \\mu \\left({E \\cup N}\\right) := \\mu \\left({E}\\right)$ <onlyinclude> The mapping $\\bar \\mu$ is well-defined, i.e.: :$\\forall E, F \\in \\Sigma: \\forall N, M \\in \\mathcal N: E \\cup N = F \\cup M \\implies \\mu \\left({E}\\right) = \\mu \\left({F}\\right)$ </onlyinclude> == Proof == Let $N_0, M_0 \\in \\Sigma$ be null sets such that $N \\subseteq N_0, M \\subseteq M_0$. Then: :$E \\subseteq E \\cup N = F \\cup M \\subseteq F \\cup M_0$ so that: :$\\mu \\left({E}\\right) \\le \\mu \\left({F \\cup M_0}\\right) \\le \\mu \\left({F}\\right) + \\mu \\left({M_0}\\right) = \\mu \\left({F}\\right) + 0$ Analogously: :$F \\subseteq F \\cup M = E \\cup N \\subseteq E \\cup N_0$ so that: :$\\mu \\left({F}\\right) \\le \\mu \\left({E \\cup N_0}\\right) \\le \\mu \\left({E}\\right) + \\mu \\left({N_0}\\right) = \\mu \\left({E}\\right) + 0$ In total: :$\\mu \\left({E}\\right) = \\mu \\left({F}\\right)$ {{qed}} Category:Measure Theory q3ci6501bmbncb84lqewf3oofmb6nkt"}
+{"_id": "32881", "title": "Language of Propositional Logic has Unique Parsability/Lemma", "text": "Language of Propositional Logic has Unique Parsability/Lemma 0 62976 352583 319218 2018-05-01T21:04:07Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $\\mathcal L_0$ be the language of propositional logic. <onlyinclude> Let $\\mathbf A$ be a WFF. Suppose that $\\mathbf A = \\left({B \\circ C}\\right) = \\left({D * E}\\right)$. Then $\\mathbf B = \\mathbf D$, ${\\circ} = {*}$, and $\\mathbf C = \\mathbf E$. </onlyinclude> == Proof == The WFFs $\\mathbf B$ and $\\mathbf D$ are strings which both start in the same place, right after the first left bracket in $\\mathbf A$. By Initial Part of WFF of PropLog is not WFF, neither $\\mathbf B$ nor $\\mathbf D$ can be an initial part of the other. Therefore $\\mathbf B = \\mathbf D$. It follows that $* = \\circ$ and $\\mathbf C = \\mathbf E$. Hence the result. {{qed}} == Sources == * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin|prev = Definition:Main Connective/Propositional Logic/Definition 2|next = Definition:Abbreviation of WFFs of Propositional Logic}}: $\\S 1.4$: Main Connective: Theorem $1.4.2$ Category:Language of Propositional Logic j1eji49p7l5w74u7w2cl9j60e14lacs"}
+{"_id": "32882", "title": "Finished Branch Lemma/Corollary", "text": "Finished Branch Lemma/Corollary 0 63153 319762 2017-10-04T20:59:11Z Lord Farin 560 Created page with \"== Corollary to Finished Branch Lemma == <onlyinclude> Let $\\Gamma$ be a finished branch of a Definition:Proposit...\" wikitext text/x-wiki == Corollary to Finished Branch Lemma == <onlyinclude> Let $\\Gamma$ be a finished branch of a propositional tableau $\\left({T, \\mathbf H, \\Phi}\\right)$. Then $\\Phi \\left[{\\Gamma}\\right]$, the image of $\\Gamma$ under $\\Phi$, is satisfiable for boolean interpretations. </onlyinclude> == Proof == {{Finish|Branch not contradictory. See Finished Set Lemma}} Category:Propositional Tableaus 74txt8zkki3759oge83rh2k2wi40qnn"}
+{"_id": "32883", "title": "Pythagorean Triangle/Examples/4485-5852-7373", "text": "Pythagorean Triangle/Examples/4485-5852-7373 0 63453 478778 478748 2020-07-18T12:16:53Z Prime.mover 59 wikitext text/x-wiki == Example of Primitive Pythagorean Triangle == <onlyinclude> The triangle whose sides are of length $4485$, $5852$ and $7373$ is a primitive Pythagorean triangle. </onlyinclude> :700px It has generator $\\tuple {77, 38}$. == Proof == We have: {{begin-eqn}} {{eqn | l = 77^2 - 38^2       | r = 5929 - 1444       | c =  }} {{eqn | r = 4485       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 2 \\times 77 \\times 38       | r = 5852       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 77^2 + 38^2       | r = 5929 + 1444       | c =  }} {{eqn | r = 7373       | c =  }} {{end-eqn}} Hence: {{begin-eqn}} {{eqn | l = 4485^2 + 5852^2       | r = 20 \\, 115 \\, 225 + 34 \\, 245 \\, 904       | c =  }} {{eqn | r = 54 \\, 361 \\, 129       | c =  }} {{eqn | r = 7373^2       | c =  }} {{end-eqn}} It follows by Pythagoras's Theorem that $4485$, $5852$ and $7373$  form a Pythagorean triple. We have that: {{begin-eqn}} {{eqn | l = 4485       | r = 3 \\times 5 \\times 13 \\times 23       | c =  }} {{eqn | l = 5852       | r = 2^2 \\times 7 \\times 11 \\times 19       | c =  }} {{end-eqn}} It is seen that $4485$ and $5852$ share no prime factors. That is, $4485$ and $5852$ are coprime. Hence, by definition, $693$, $1924$ and $2045$ form a primitive Pythagorean triple. The result follows by definition of a primitive Pythagorean triangle. {{qed}} Category:Examples of Pythagorean Triangles rsjb3kgh1ab6emmlccixjpdz13jyq52"}
+{"_id": "32884", "title": "Pythagorean Triangle/Examples/3059-8580-9109", "text": "Pythagorean Triangle/Examples/3059-8580-9109 0 63454 478782 478746 2020-07-18T12:20:06Z Prime.mover 59 wikitext text/x-wiki == Example of Primitive Pythagorean Triangle == <onlyinclude> The triangle whose sides are of length $3059$, $8580$ and $9109$ is a primitive Pythagorean triangle. </onlyinclude> :700px It has generator $\\tuple {78, 55}$. == Proof == We have: {{begin-eqn}} {{eqn | l = 78^2 - 55^2       | r = 6084 - 3025       | c =  }} {{eqn | r = 3059       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 2 \\times 78 \\times 55       | r = 8580       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 78^2 + 55^2       | r = 6084 + 3025       | c =  }} {{eqn | r = 9109       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 3059^2 + 8580^2       | r = 9 \\, 357 \\, 481 + 73 \\, 616 \\, 400       | c =  }} {{eqn | r = 82 \\, 973 \\, 881       | c =  }} {{eqn | r = 9109^2       | c =  }} {{end-eqn}} It follows by Pythagoras's Theorem that $3059$, $8580$ and $9109$  form a Pythagorean triple. We have that: {{begin-eqn}} {{eqn | l = 3059       | r = 7 \\times 19 \\times 23       | c =  }} {{eqn | l = 8580       | r = 2^2 \\times 3 \\times 5 \\times 11 \\times 13       | c =  }} {{end-eqn}} It is seen that $3059$ and $8580$ share no prime factors. That is, $3059$ and $8580$ are coprime. Hence, by definition, $3059$, $8580$ and $9109$ form a primitive Pythagorean triple. The result follows by definition of a primitive Pythagorean triangle. {{qed}} Category:Examples of Pythagorean Triangles s2evzbxc685l7qigf75rmq6fh6k46es"}
+{"_id": "32885", "title": "Pythagorean Triangle/Examples/1380-19,019-19,069", "text": "Pythagorean Triangle/Examples/1380-19,019-19,069 0 63455 478798 478740 2020-07-18T12:30:57Z Prime.mover 59 wikitext text/x-wiki == Example of Primitive Pythagorean Triangle == <onlyinclude> The triangle whose sides are of length $1380$, $19 \\, 019$ and $19 \\, 069$ is a primitive Pythagorean triangle. </onlyinclude> :700px It has generator $\\left({138, 5}\\right)$. == Proof == We have: {{begin-eqn}} {{eqn | l = 138^2 - 5^2       | r = 19 \\, 044 - 25       | c =  }} {{eqn | r = 19 \\, 019       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 2 \\times 138 \\times 5       | r = 1380       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 138^2 + 5^2       | r = 19 \\, 044 + 25       | c =  }} {{eqn | r = 19 \\, 069       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 1380^2 + 19 \\, 019^2       | r = 1 \\, 904 \\, 400 + 361 \\, 722 \\, 361       | c =  }} {{eqn | r = 363 \\, 626 \\, 761       | c =  }} {{eqn | r = 19 \\, 069^2       | c =  }} {{end-eqn}} It follows by Pythagoras's Theorem that $1380$, $19 \\, 019$ and $19 \\, 069$  form a Pythagorean triple. We have that: {{begin-eqn}} {{eqn | l = 1380       | r = 2^2 \\times 3 \\times 5 \\times 23       | c =  }} {{eqn | l = 19 \\, 019       | r = 7 \\times 11 \\times 13 \\times 19       | c =  }} {{end-eqn}} It is seen that $1380$ and $19 \\, 019$ share no prime factors. That is, $1380$ and $19 \\, 019$ are coprime. Hence, by definition, $1380$, $19 \\, 019$ and $19 \\, 069$ form a primitive Pythagorean triple. The result follows by definition of a primitive Pythagorean triangle. {{qed}} Category:Examples of Pythagorean Triangles k7a305zpkonaow2uzak0kn3stqp6v1m"}
+{"_id": "32886", "title": "Carmichael Number/Examples/294,409", "text": "Carmichael Number/Examples/294,409 0 64010 392221 333769 2019-02-12T16:07:42Z Prime.mover 59 wikitext text/x-wiki == Example of Carmichael Number == $294 \\, 409$ is a Carmichael number: <onlyinclude> :$\\forall a \\in \\Z: a \\perp 294 \\, 409: a^{294 \\, 409} \\equiv a \\pmod {294 \\, 409}$ while $294 \\, 409$ is composite. </onlyinclude> == Proof == We have that: :$294 \\, 409 = 37 \\times 73 \\times 109$ First note that $294 \\, 409$ is square-free. Hence the square of none of its prime factors is a divisor of $294 \\, 409$: :$\\forall p \\divides 294 \\, 409: p^2 \\nmid 294 \\, 409$ We also see that: {{begin-eqn}} {{eqn | l = 294 \\, 408       | r = 2^3 \\times 3^3 \\times 29 \\times 47       | c =  }} {{eqn | r = 8178 \\times 36       | c =  }} {{eqn | r = 4089 \\times 72       | c =  }} {{eqn | r = 2726 \\times 108       | c =  }} {{end-eqn}} Thus $294 \\, 409$ is a Carmichael number by Korselt's Theorem. {{qed}} == Sources == * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Carmichael Number/Examples/1729|next = Smallest Integer which is Sum of 2 Fourth Powers in 2 Ways}}: $509,033,161$ Category:Carmichael Numbers Category:294,409 78fchhna1ilh2bjp5ltl9itndk8c6wg"}
+{"_id": "32887", "title": "Euler Phi Function of 6", "text": "Euler Phi Function of 6 0 64278 391878 324134 2019-02-09T23:41:24Z Prime.mover 59 wikitext text/x-wiki == Example of Euler $\\phi$ Function of Square-Free Integer == <onlyinclude> :$\\map \\phi 6 = 2$ </onlyinclude> where $\\phi$ denotes the Euler $\\phi$ Function. == Proof == From Euler Phi Function of Square-Free Integer: :$\\displaystyle \\map \\phi n = \\prod_{\\substack {p \\mathop \\divides n \\\\ p \\mathop > 2} } \\paren {p - 1}$ where $p \\divides n$ denotes the primes which divide $n$. We have that: :$6 = 2 \\times 3$ and so is square-free. Thus: {{begin-eqn}} {{eqn | l = \\map \\phi 6       | r = \\paren {3 - 1}       | c =  }} {{eqn | r = 2       | c =  }} {{end-eqn}} {{qed}} Category:Euler Phi Function of Square-Free Integer Category:6 gi0ykexw3gggcps7d1ep2kk89o3valv"}
+{"_id": "32888", "title": "Odd Amicable Pair/Examples/29,912,035,725-34,883,817,075", "text": "Odd Amicable Pair/Examples/29,912,035,725-34,883,817,075 0 64360 467815 324454 2020-05-14T06:00:22Z Prime.mover 59 wikitext text/x-wiki == Example of Odd Amicable Pair == <onlyinclude> $29 \\, 912 \\, 035 \\, 725$ and $34 \\, 883 \\, 817 \\, 075$ are an odd amicable pair: :$\\map \\sigma {29 \\, 912 \\, 035 \\, 725} = \\map \\sigma {34 \\, 883 \\, 817 \\, 075} = 64 \\, 795 \\, 852 \\, 800 = 29 \\, 912 \\, 035 \\, 725 + 34 \\, 883 \\, 817 \\, 075$ </onlyinclude> == Proof == By definition, $m$ and $n$ form an amicable pair {{iff}}: :$\\map \\sigma m = \\map \\sigma n = m + n$ where $\\map \\sigma n$ denotes the $\\sigma$ function of $n$. Thus: {{begin-eqn}} {{eqn | l = \\map \\sigma {29 \\, 912 \\, 035 \\, 725}       | r = 64 \\, 795 \\, 852 \\, 800       | c = {{SigmaLink|29,912,035,725|29 \\, 912 \\, 035 \\, 725}} }} {{eqn | r = 29 \\, 912 \\, 035 \\, 725 + 34 \\, 883 \\, 817 \\, 075       | c =  }} {{eqn | r = \\map \\sigma {34 \\, 883 \\, 817 \\, 075}       | c = {{SigmaLink|34,883,817,075|34 \\, 883 \\, 817 \\, 075}} }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = 64,795,852,800|next = Odd Amicable Pair/Examples/31,695,652,275-33,100,200,525}}: $64,795,852,800$ Category:Amicable Pairs sl6nmndx3i6chz3kyt71ni5m1qdfbg6"}
+{"_id": "32889", "title": "Odd Amicable Pair/Examples/31,695,652,275-33,100,200,525", "text": "Odd Amicable Pair/Examples/31,695,652,275-33,100,200,525 0 64361 324455 2017-11-03T07:11:08Z Prime.mover 59 Created page with \"== Example of Odd Amicable Pair == <onlyinclude> $31 \\, 695 \\, 652 \\, 275$ and $33 \\, 100 \\, 200 \\, 525$ are an Defin...\" wikitext text/x-wiki == Example of Odd Amicable Pair == <onlyinclude> $31 \\, 695 \\, 652 \\, 275$ and $33 \\, 100 \\, 200 \\, 525$ are an odd amicable pair: :$\\sigma \\left({31 \\, 695 \\, 652 \\, 275}\\right) = \\sigma \\left({33 \\, 100 \\, 200 \\, 525}\\right) = 64 \\, 795 \\, 852 \\, 800 = 31 \\, 695 \\, 652 \\, 275 + 33 \\, 100 \\, 200 \\, 525$ </onlyinclude> == Proof == By definition, $m$ and $n$ form an amicable pair {{iff}}: :$\\sigma \\left({m}\\right) = \\sigma \\left({n}\\right) = m + n$ where $\\sigma \\left({n}\\right)$ denotes the $\\sigma$ function. Thus: {{begin-eqn}} {{eqn | l = \\sigma \\left({31 \\, 695 \\, 652 \\, 275}\\right)       | r = 64 \\, 795 \\, 852 \\, 800       | c = {{SigmaLink|31,695,652,275|31 \\, 695 \\, 652 \\, 275}} }} {{eqn | r = 31 \\, 695 \\, 652 \\, 275 + 33 \\, 100 \\, 200 \\, 525       | c =  }} {{eqn | r = \\sigma \\left({33 \\, 100 \\, 200 \\, 525}\\right)       | c = {{SigmaLink|33,100,200,525|33 \\, 100 \\, 200 \\, 525}} }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Odd Amicable Pair/Examples/29,912,035,725-34,883,817,075|next = Odd Amicable Pair/Examples/32,129,958,525-32,665,894,275}}: $64,795,852,800$ Category:Amicable Pairs l41u780ot9j3ot1lx6zoqja2377ajfo"}
+{"_id": "32890", "title": "Odd Amicable Pair/Examples/32,129,958,525-32,665,894,275", "text": "Odd Amicable Pair/Examples/32,129,958,525-32,665,894,275 0 64362 324456 2017-11-03T07:15:34Z Prime.mover 59 Created page with \"== Example of Odd Amicable Pair == <onlyinclude> $32 \\, 129 \\, 958 \\, 525$ and $32 \\, 665 \\, 894 \\, 275$ are an Defin...\" wikitext text/x-wiki == Example of Odd Amicable Pair == <onlyinclude> $32 \\, 129 \\, 958 \\, 525$ and $32 \\, 665 \\, 894 \\, 275$ are an odd amicable pair: :$\\sigma \\left({32 \\, 129 \\, 958 \\, 525}\\right) = \\sigma \\left({32 \\, 665 \\, 894 \\, 275}\\right) = 64 \\, 795 \\, 852 \\, 800 = 32 \\, 129 \\, 958 \\, 525 + 32 \\, 665 \\, 894 \\, 275$ </onlyinclude> == Proof == By definition, $m$ and $n$ form an amicable pair {{iff}}: :$\\sigma \\left({m}\\right) = \\sigma \\left({n}\\right) = m + n$ where $\\sigma \\left({n}\\right)$ denotes the $\\sigma$ function. Thus: {{begin-eqn}} {{eqn | l = \\sigma \\left({32 \\, 129 \\, 958 \\, 525}\\right)       | r = 64 \\, 795 \\, 852 \\, 800       | c = {{SigmaLink|32,129,958,525|32 \\, 129 \\, 958 \\, 525}} }} {{eqn | r = 32 \\, 129 \\, 958 \\, 525 + 32 \\, 665 \\, 894 \\, 275       | c =  }} {{eqn | r = \\sigma \\left({32 \\, 665 \\, 894 \\, 275}\\right)       | c = {{SigmaLink|32,665,894,275|32 \\, 665 \\, 894 \\, 275}} }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Odd Amicable Pair/Examples/31,695,652,275-33,100,200,525|next = Multiply Perfect Number of Order 6}}: $64,795,852,800$ Category:Amicable Pairs h8hkza7rcn54glac3ff8jchw53909h2"}
+{"_id": "32891", "title": "Aurifeuillian Factorization/Examples/2^4n+2 + 1", "text": "Aurifeuillian Factorization/Examples/2^4n+2 + 1 0 65177 385072 385071 2018-12-28T15:20:25Z Prime.mover 59 wikitext text/x-wiki == Example of Aurifeuillian Factorizations == <onlyinclude> :$2^{4 n + 2} + 1 = \\paren {2^{2 n + 1} - 2^{n + 1} + 1} \\paren {2^{2 n + 1} + 2^{n + 1} + 1}$ </onlyinclude> == Proof == From Sum of Squares as Product of Factors with Square Roots: :$x^2 + y^2 = \\paren {x + \\sqrt {2 x y} + y} \\paren {x - \\sqrt {2 x y} + y}$ Let $x = 2^{2 n + 1}$ and $y = 1$. Then: {{begin-eqn}} {{eqn | l = 2^{4 n + 2} + 1       | r = \\paren {2^{2 n + 1} }^2 + 1^2       | c =  }} {{eqn | r = \\paren {2^{2 n + 1} + \\sqrt {2 \\times \\paren {2^{2 n + 1} } \\times 1} + 1} \\paren {2^{2 n + 1} - \\sqrt {2 \\times \\paren {2^{2 n + 1} } \\times 1} + 1}       | c =  }} {{eqn | r = \\paren {2^{2 n + 1} + \\sqrt {2^{2 n + 2} } + 1} \\paren {2^{2 n + 1} - \\sqrt {2^{2 n + 2} } + 1}       | c =  }} {{eqn | r = \\paren {2^{2 n + 1} - 2^{n + 1} + 1} \\paren {2^{2 n + 1} + 2^{n + 1} + 1}       | c =  }} {{end-eqn}} {{qed}} == Historical Note == {{:Aurifeuillian Factorization/Examples/2^4n+2 + 1/Historical Note}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Prime Decomposition of 2^58+1/Historical Note|next = Aurifeuillian Factorization/Examples/2^4n+2 + 1/Historical Note}}: $2^{58} + 1$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Prime Decomposition of 2^58+1/Historical Note|next = Aurifeuillian Factorization/Examples/2^4n+2 + 1/Historical Note}}: $2^{58} + 1$ Category:Aurifeuillian Factorizations 0c3oxexmsom4rnbzn48k0x3fupkqo47"}
+{"_id": "32892", "title": "Tutte's Wheel Theorem/Lemma", "text": "Tutte's Wheel Theorem/Lemma 0 66822 340403 340082 2018-01-28T07:02:36Z Stixme 3311 wikitext text/x-wiki {{MissingLinks}} {{tidy}} == Lemma == If a graph $G$ is 3-connected with $|V\\left({G}\\right)| > 4$ then $\\exists e, \\: e \\in E\\left({G}\\right)$ such that $G \\thinspace / \\thinspace e$ is also 3-connected. == Proof == Suppose that no such edge $e$ exists.  Then $\\forall e, \\: e = xy \\in E\\left({G}\\right)$, $\\: G \\thinspace / \\thinspace e$ contains a vertex cut $S, \\: |S| ≤ 2$. Since $\\kappa\\left({G}\\right) \\geq 3$, the contracted vertex $v_{x,y}$ of $G \\thinspace / \\thinspace e$ lies in $S$ (i.e. $\\exists z, \\: z \\in G, \\: z \\notin \\{x,y\\}$.)   Let $S = \\{v_{x,y}, z\\}$. Then $T = \\{x, y, z\\}$ is a vertex cut of $G$.  Thus, every vertex in $T$ has an edge to every component of $G'=G-T$.  Let $C$ be the smallest component of $G'$.  Let $u \\in N\\left({z}\\right) \\cap C$ where $N\\left({z}\\right)$ is the set of all neighbours of the vertex $z$. By assumption, $G \\thinspace / \\thinspace zu$ is again not 3-connected, so again $\\exists w$ such that $\\{w, z, u\\}$ is a vertex cut of $G$.  It also follows that every vertex in $\\{w, z, u\\}$ has an edge to every component of $G'' = G - \\{w, z, u\\}$.  Since $x,y$ are connected, $\\exists D$, $D$ is a component of $G''$ and $D \\cap \\{x,y\\} = \\emptyset$.  Then, $D \\subseteq N\\left({z}\\right) \\cap V\\left({C}\\right) = \\emptyset$.  Hence $D \\varsubsetneqq C$ by the choice of $D$, which contradicts the assumption that $C$ was the smallest component.  {{qed|lemma}} Category:Graph Theory b4if14xx8b5hx14s014gee025jvwj56"}
+{"_id": "32893", "title": "Completion Theorem (Metric Space)/Lemma 1", "text": "Completion Theorem (Metric Space)/Lemma 1 0 66833 453813 340173 2020-03-10T15:50:34Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $M = \\struct {A, d}$ be a metric space. Let $\\CC \\sqbrk A$ denote the set of all Cauchy sequences in $A$. Define the equivalence relation $\\sim$ on $\\CC \\sqbrk A$ by: :$\\displaystyle \\sequence {x_n} \\sim \\sequence {y_n} \\iff \\lim_{n \\mathop \\to \\infty} \\map d {x_n, y_n} = 0$ Denote the equivalence class of $\\sequence {x_n} \\in \\CC \\sqbrk A$ by $\\sqbrk {x_n}$. Denote the set of equivalence classes under $\\sim$ by $\\tilde A$. Define $\\tilde d: \\tilde A \\to \\R_{\\ge 0}$ by: :$\\displaystyle \\map {\\tilde d} {\\sqbrk {x_n}, \\sqbrk {y_n} } = \\lim_{n \\mathop \\to \\infty} \\map d {x_n, y_n}$ Then: <onlyinclude> :$\\tilde d$ is well-defined on $\\tilde A$. </onlyinclude> == Proof == Let $\\sequence {x_n}$, $\\sequence {\\hat x_n}$, $\\sequence {y_n}$, $\\sequence {\\hat y_n} \\in \\CC \\sqbrk A$ be such that: :$\\sequence {x_n} \\sim \\sequence {\\hat x_n}$ :$\\sequence {y_n} \\sim \\sequence {\\hat y_n}$ We have: {{begin-eqn}} {{eqn | l = \\map d {x_n, y_n} - \\map d {\\hat x_n, \\hat y_n}       | o = \\le       | r = \\map d {x_n, \\hat x_n} + \\map d {\\hat x_n, y_n} - \\map d {\\hat x_n, \\hat y_n}       | c = Triangle Inequality }} {{eqn | o = \\le       | r = \\map d {x_n, \\hat x_n} + \\map d {\\hat x_n, \\hat y_n} + \\map d {\\hat y_n, y_n} - \\map d {\\hat x_n, \\hat y_n}       | c = Triangle Inequality }} {{eqn | r = \\map d {x_n, \\hat x_n} + \\map d {\\hat y_n, y_n}       | c =  }} {{end-eqn}} By an identical argument, we can also show that: :$\\map d {\\hat x_n, \\hat y_n} - \\map d {x_n, y_n} \\le \\map d {x_n, \\hat x_n} + \\map d {\\hat y_n, y_n}$  and therefore: :$\\displaystyle 0 \\le \\size {\\map d {x_n, y_n} - \\map d {\\hat x_n, \\hat y_n} } \\le \\map d {x_n, \\hat x_n} + \\map d {\\hat y_n, y_n}$ Passing to the limit $n \\to \\infty$ and using the Combination Theorem for Sequences we have shown that: $\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map d {x_n, y_n} = \\lim_{n \\mathop \\to \\infty} \\map d {\\hat x_n, \\hat y_n}$ But this precisely means that: :$\\map {\\tilde d} {\\sqbrk {x_n}, \\sqbrk {y_n} } = \\map {\\tilde d} {\\sqbrk {\\hat x_n}, \\sqbrk {\\hat y_n} }$ {{qed}} Category:Completion Theorem 20d0c7shgcq6vlnumi2nzcsg5cx4w0g"}
+{"_id": "32894", "title": "Completion Theorem (Metric Space)/Lemma 2", "text": "Completion Theorem (Metric Space)/Lemma 2 0 66834 364362 340176 2018-09-04T21:11:34Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $M = \\left({A, d}\\right)$ be a metric space. Let $\\mathcal C \\left[{A}\\right]$ denote the set of all Cauchy sequences in $A$. Define the equivalence relation $\\sim$ on $\\mathcal C \\left[{A}\\right]$ by: :$\\displaystyle \\left\\langle{x_n}\\right\\rangle \\sim \\left\\langle{y_n}\\right\\rangle \\iff \\lim_{n \\mathop \\to \\infty} d \\left({x_n, y_n}\\right) = 0$ Denote the equivalence class of $\\left\\langle{x_n}\\right\\rangle \\in \\mathcal C \\left[{A}\\right]$ by $\\left[{x_n}\\right]$. Denote the set of equivalence classes under $\\sim$ by $\\tilde A$. Define $\\tilde d: \\tilde A \\to \\R_{\\ge 0}$ by: :$\\displaystyle \\tilde d \\left({\\left[{x_n}\\right], \\left[{y_n}\\right]}\\right) = \\lim_{n \\mathop \\to \\infty} d \\left({x_n, y_n}\\right)$ Then: <onlyinclude> :$\\tilde d$ is a metric on $\\tilde A$. </onlyinclude> == Proof == To prove $\\tilde d$ is a metric, we verify that it satisfies the axioms $M1$, $M2$, $M3$ and $M4$. === Proof of $M4$ === Let $\\tilde d \\left({\\left[{x_n}\\right], \\left[{y_n}\\right]}\\right) = \\infty$. Then $\\left\\langle{x_n}\\right\\rangle$ and $\\left\\langle y_n \\right\\rangle$ cannot both be Cauchy. So $\\tilde d \\left({\\left[{x_n}\\right], \\left[{y_n}\\right]}\\right) < \\infty$ for $\\left[{x_n}\\right], \\left[{y_n}\\right] \\in \\tilde A$. By the definition of $\\tilde d$, for any $\\left[{x_n}\\right], \\left[{y_n}\\right] \\in \\tilde A$, $\\tilde d \\left({\\left[{x_n}\\right], \\left[{y_n}\\right]}\\right)$ must be a limit point of $R_{\\ge 0}$. The closure of $\\R_{\\ge 0}$ is $\\R_{\\ge 0}$, so $\\tilde d: \\tilde A \\times \\tilde A \\to \\R_{\\ge 0}$. So axiom $M4$ holds for $\\tilde d$. === Proof of $M1$ === Let $\\tilde d \\left({\\left[{x_n}\\right], \\left[{y_n}\\right]}\\right) = 0$, which means that: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} d\\left({x_n, y_n}\\right) = 0$ So by definition: :$\\left\\langle{x_n}\\right\\rangle \\sim \\left\\langle{y_n}\\right\\rangle$ and: :$\\left[{x_n}\\right] = \\left[{y_n}\\right]$ As $d$ is a metric, we also find immediately: :$\\tilde d \\left({\\left[{x_n}\\right], \\left[{x_n}\\right]}\\right) = 0$ So axiom $M1$ holds for $\\tilde d$. {{qed|lemma}} === Proof of $M3$ === We have that: {{begin-eqn}} {{eqn | l = \\tilde d \\left({\\left[{x_n}\\right], \\left[{y_n}\\right]}\\right)       | r = \\lim_{n \\mathop \\to \\infty} d \\left({x_n, y_n}\\right)       | c = }} {{eqn | r = \\lim_{n \\mathop \\to \\infty} d \\left({y_n, x_n}\\right)       | c = $d$ is a Metric }} {{eqn | r = \\tilde d \\left({\\left[{y_n}\\right], \\left[{x_n}\\right]}\\right)       | c = }} {{end-eqn}} So axiom $M3$ holds for $\\tilde d$. {{qed|lemma}} === Proof of $M2$ === We have that: {{begin-eqn}} {{eqn | l = \\tilde d \\left({\\left[{x_n}\\right], \\left[{z_n}\\right]}\\right)       | r = \\lim_{n \\mathop \\to \\infty} d \\left({x_n, z_n}\\right)       | c = }} {{eqn | o = \\le       | r = \\lim_{n \\mathop \\to \\infty} \\left\\{ {d \\left({x_n, y_n}\\right) + d \\left({y_n, z_n}\\right)}\\right\\}       | c = $d$ is a metric, and using elementary properties of limits (Reference?) }} {{eqn | r = \\lim_{n \\mathop \\to \\infty} d\\left({x_n, y_n}\\right) +  \\lim_{n \\to \\infty} d \\left({y_n, z_n}\\right)       | c = Sum Rule for Real Sequences }} {{eqn | r = \\tilde d \\left({\\left[{x_n}\\right], \\left[{y_n}\\right]}\\right) + \\tilde d \\left({\\left[{y_n}\\right], \\left[{z_n}\\right]}\\right) }} {{end-eqn}} So axiom $M2$ holds for $\\tilde d$. {{qed|lemma}} Thus $\\tilde d$ satisfies all the metric space axioms and so is a metric. {{qed}} Category:Completion Theorem irpehes3ivyz25mgvi3adr78b3pr3ht"}
+{"_id": "32895", "title": "Completion Theorem (Metric Space)/Lemma 3", "text": "Completion Theorem (Metric Space)/Lemma 3 0 66835 340181 340180 2018-01-27T11:17:19Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $M = \\left({A, d}\\right)$ be a metric space. Let $\\mathcal C \\left[{A}\\right]$ denote the set of all Cauchy sequences in $A$. Define the equivalence relation $\\sim$ on $\\mathcal C \\left[{A}\\right]$ by: :$\\displaystyle \\left\\langle{x_n}\\right\\rangle \\sim \\left\\langle{y_n}\\right\\rangle \\iff \\lim_{n \\mathop \\to \\infty} d \\left({x_n, y_n}\\right) = 0$ Denote the equivalence class of $\\left\\langle{x_n}\\right\\rangle \\in \\mathcal C \\left[{A}\\right]$ by $\\left[{x_n}\\right]$. Denote the set of equivalence classes under $\\sim$ by $\\tilde A$. Define $\\tilde d: \\tilde A \\to \\R_{\\ge 0}$ by: :$\\displaystyle \\tilde d \\left({\\left[{x_n}\\right], \\left[{y_n}\\right]}\\right) = \\lim_{n \\mathop \\to \\infty} d \\left({x_n, y_n}\\right)$ Then: <onlyinclude> :$\\tilde M = \\left({\\tilde A, \\tilde d}\\right)$ is a completion of $M$. </onlyinclude> == Proof == We are to show that: : $(1): \\quad \\tilde M$ is a complete metric space : $(2): \\quad A \\subseteq \\tilde A$ : $(3): \\quad A$ is dense in $\\tilde M$ : $(4): \\quad \\forall x, y \\in A : \\tilde d \\left({x, y}\\right) = d \\left({x, y}\\right)$ For $x \\in A$, let $\\hat x = \\left({x, x, x, \\ldots}\\right)$ be the constant sequence with value $x$. Let $\\phi: A \\to \\tilde A: x = \\left[{\\hat x}\\right]$. We first demonstrate that $(2)$ holds, by showing that $A \\subseteq \\tilde A$. {{begin-eqn}} {{eqn | l = \\phi \\left({x}\\right)       | r = \\phi \\left({y}\\right) }} {{eqn | ll= \\implies       | l = \\left[{\\hat x}\\right]       | r = \\left[{\\hat y}\\right] }} {{eqn | ll= \\implies       | l = \\lim_{n \\mathop \\to \\infty} d \\left({x, y}\\right)       | r = 0 }} {{eqn | ll= \\implies       | l = d \\left({x, y}\\right)       | r = 0 }} {{eqn | ll= \\implies       | l = x       | r = y }} {{end-eqn}} Thus: :$A \\subseteq \\tilde A$ {{qed|lemma}} Henceforth we identify $A$ with its isomorphic copy in $\\tilde A$ when it is convenient. Now we demonstrate that $(4)$ holds, by showing that $\\phi$ is an injection from $A$ into $\\tilde A$. For any $x, y \\in A$: {{begin-eqn}} {{eqn | l = \\tilde d \\left({\\left[{\\hat x}\\right], \\left[{\\hat y}\\right]}\\right)       | r = \\lim_{n \\mathop \\to \\infty} d \\left({x, y}\\right) }} {{eqn | r = d \\left({x, y}\\right) }} {{end-eqn}} That is: :$\\forall x, y \\in A : \\tilde d \\left({x, y}\\right) = d \\left({x, y}\\right)$ {{qed|lemma}} Now we demonstrate that $(3)$ holds, by showing that $A$ is dense in $\\tilde A$. Recall that the closure of $A$ is the union of $A$ and the limit points of $A$. Let $\\left[{x_n}\\right] \\in \\tilde A$ and $\\epsilon > 0$ be arbitrary. If we can find $x \\in A$ such that $\\tilde d \\left({\\left[{\\hat x}\\right], \\left[{x_n}\\right]}\\right) < \\epsilon$ then we have shown that $A$ is dense in $\\tilde A$. Since $\\left\\langle{x_n}\\right\\rangle$ is Cauchy, there exists $N \\in \\N$ such that: :$\\forall m, n \\ge N: d \\left({x_m, x_n}\\right) < \\epsilon$ Then we have: {{begin-eqn}} {{eqn | l = \\tilde d \\left({\\left[{\\hat x_N}\\right], \\left[{x_n}\\right]}\\right)       | r = \\lim_{n \\mathop \\to \\infty} d \\left({x_N, x_n}\\right) }} {{eqn | o = <       | r = \\epsilon }} {{end-eqn}} and therefore $A$ is dense in $\\tilde A$. {{qed|lemma}} Finally we demonstrate that $(1)$ holds, by showing that $\\left({\\tilde A, \\tilde d}\\right)$ is complete. By the completeness criterion it is sufficient to show that every Cauchy sequence in $\\phi \\left({A}\\right)$ converges in $\\tilde A$. Let $\\left\\langle{\\hat w_n}\\right\\rangle$ be a Cauchy sequence in $\\phi \\left({A}\\right)$, so each $\\hat w_n$ has the form $\\left\\langle{w_n, w_n, w_n, \\ldots}\\right\\rangle$. Since $\\phi$ is an isometry: :$\\forall m, n \\in \\N: \\tilde d \\left({\\hat w_n, \\hat w_m}\\right) = d \\left({w_n, w_m}\\right)$ Therefore, $\\left\\langle{w_1, w_2, w_3,\\ldots}\\right\\rangle$ is Cauchy in $A$. Let $W = \\left[{\\left\\langle{w_1, w_2, w_3, \\ldots}\\right\\rangle}\\right] \\in \\tilde A$. We claim that $\\left\\langle{\\hat w_n}\\right\\rangle$ converges to $W$ in $\\tilde A$. Let $\\epsilon > 0$ be arbitrary. Since $\\left\\langle{w_1, w_2, w_3, \\ldots}\\right\\rangle$ is Cauchy in $A$, there exists $N \\in \\N$ such that for all $m, n \\ge N$, we have $d \\left({w_n, w_m}\\right) < \\epsilon$. Thus for all $n > N$: :$\\displaystyle \\tilde d \\left({w_n, W}\\right) = \\lim_{n \\mathop \\to \\infty} d \\left({w_n, W}\\right) < \\epsilon$  Therefore, $\\left\\langle{\\hat w_n}\\right\\rangle \\to W$ as $N \\to \\infty$, and $\\tilde A$ is complete. {{qed}} {{ACC|Completeness Criterion (Metric Spaces)}} Category:Completion Theorem iunu09pstzvf3vvstpmluzt6nz0bbt7"}
+{"_id": "32896", "title": "Heine-Borel Theorem/Euclidean Space/Necessary Condition", "text": "Heine-Borel Theorem/Euclidean Space/Necessary Condition 0 67083 342472 2018-02-12T08:09:32Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> For any natural number $n \\ge 1$, a closed and Definition:Bounded Metric Spac...\" wikitext text/x-wiki == Theorem == <onlyinclude> For any natural number $n \\ge 1$, a closed and bounded subspace of the Euclidean space $\\R^n$ is compact. </onlyinclude> == Proof 1 == {{:Heine-Borel Theorem/Euclidean Space/Necessary Condition/Proof 1}} == Proof 2 == {{:Heine-Borel Theorem/Euclidean Space/Necessary Condition/Proof 2}} {{Namedfor|Heinrich Eduard Heine|name2 = Émile Borel}} Category:Heine-Borel Theorem d74r4b7lk9ga1i4go85xka0z7cy390g"}
+{"_id": "32897", "title": "Separated Subsets of Linearly Ordered Space under Order Topology/Lemma", "text": "Separated Subsets of Linearly Ordered Space under Order Topology/Lemma 0 67102 471130 342563 2020-05-28T07:05:25Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $T = \\struct {S, \\preceq, \\tau}$ be a linearly ordered space. Let $A$ and $B$ be separated sets of $T$. Let $A^*$ and $B^*$ be defined as: :$A^* := \\displaystyle \\bigcup \\set {\\closedint a b: a, b \\in A, \\closedint a b \\cap B^- = \\O}$ :$B^* := \\displaystyle \\bigcup \\set {\\closedint a b: a, b \\in B, \\closedint a b \\cap A^- = \\O}$ where $A^-$ and $B^-$ denote the closure of $A$ and $B$ in $T$. Then: :$(1): \\quad A \\subseteq A^*$ :$(2): \\quad B \\subseteq B^*$ :$(3): \\quad A^* \\cap B^* = \\O$ == Proof == Let $a \\in A$. Then: {{begin-eqn}} {{eqn | l = \\closedint a a       | r = \\set a       | c =  }} {{eqn | ll= \\leadsto       | l = \\closedint a a \\cap B^-       | r = \\O       | c = {{Defof|Separated Sets}} }} {{eqn | ll= \\leadsto       | l = \\closedint a a       | o = \\subseteq       | r = A^*       | c = Definition of $A^*$ }} {{eqn | ll= \\leadsto       | l = a       | o = \\in       | r = A^*       | c = Definition of $\\closedint a a$ }} {{eqn | ll= \\leadsto       | l = A       | o = \\subseteq       | r = A^*       | c = {{Defof|Subset}} }} {{end-eqn}} Similarly, $B \\subseteq B^-$. {{qed|lemma}} {{AimForCont}} $A^* \\cap B^* \\ne \\O$. Then: :$\\exists p: p \\in A^* \\cap B^*$ Hence: :$\\exists a, b \\in A, c, d \\in B: p \\in \\closedint a b \\cap \\closedint c d$ But because $A$ and $B$ are separated sets: :$c, d \\notin A$ and: :$a, b \\notin B$ and so: :$\\closedint a b \\cap \\closedint c d = \\O$ Thus $p \\notin \\closedint a b \\cap \\closedint c d$ It follows by Proof by Contradiction that $A^* \\cap B^* = \\O$. {{qed}} == Sources == * {{BookReference|Counterexamples in Topology|1978|Lynn Arthur Steen|author2 = J. Arthur Seebach, Jr.|ed = 2nd|edpage = Second Edition|prev = Definition:Convex Component|next = Separated Subsets of Linearly Ordered Space under Order Topology}}: Part $\\text {II}$: Counterexamples: $39$. Order Topology: $3$ Category:Linearly Ordered Spaces i2v65eccpzydkuytk9n5k95db573kxl"}
+{"_id": "32898", "title": "Power Series Expansion for Logarithm of 1 + x/Corollary", "text": "Power Series Expansion for Logarithm of 1 + x/Corollary 0 67448 363669 349899 2018-08-26T11:27:37Z Prime.mover 59 wikitext text/x-wiki == Corollary to Power Series Expansion for $\\ln \\paren {1 + x}$ == <onlyinclude> {{begin-eqn}} {{eqn | l = \\ln \\paren {1 - x}       | r = -\\sum_{n \\mathop = 1}^\\infty \\frac {x^n} n       | c =  }} {{eqn | r = -x - \\frac {x^2} 2 - \\frac {x^3} 3 - \\frac {x^4} 4 - \\cdots       | c =  }} {{end-eqn}} valid for $-1 < x < 1$. </onlyinclude> == Proof == By Power Series Expansion for $\\ln \\paren {1 + x}$: :$\\displaystyle \\ln \\paren {1 + x} = \\sum_{n \\mathop = 1}^\\infty \\paren {-1}^{n - 1} \\frac {x^n} n$ Then: {{begin-eqn}} {{eqn | l = \\ln \\paren {1 - x}       | r = \\sum_{n \\mathop = 1}^\\infty \\paren {-1}^{n - 1} \\frac {\\paren {-x}^n} n       | c = substituting $x \\to -x$ }} {{eqn | r = -\\sum_{n \\mathop = 1}^\\infty \\paren {-1}^{2 n} \\frac {x^n} n  }} {{eqn | r = -\\sum_{n \\mathop = 1}^\\infty \\frac {x^n} n  }} {{end-eqn}} {{qed}} Category:Examples of Power Series Category:Logarithms jjoh7jh41havb8b3zshple7iibi2yvy"}
+{"_id": "32899", "title": "Fourier's Theorem/Lemma 1", "text": "Fourier's Theorem/Lemma 1 0 67846 459093 459092 2020-04-02T21:20:26Z Prime.mover 59 wikitext text/x-wiki == Lemma for Fourier's Theorem == <onlyinclude> Let $\\psi$ be a real function defined on a closed interval $\\closedint a b$. Let $\\psi$ be piecewise continuous with one-sided limits on $\\closedint a b$. Then: :$\\displaystyle \\lim_{N \\mathop \\to \\infty} \\int_a^b \\map \\psi u \\sin N u \\rd u = 0$ </onlyinclude> == Proof == We are given that $\\psi$ is piecewise continuous with one-sided limits on $\\closedint a b$. Therefore, there exists a finite subdivision $\\set {x_0, x_1, \\ldots, x_m}$ of $\\closedint a b$, where $x_0 = a$ and $x_m = b$, such that for all $i \\in \\set {1, 2, \\ldots, m}$: :$\\psi$ is continuous on $\\openint {x_{i - 1} } {x_i}$  :$\\displaystyle \\lim_{x \\mathop \\to {x_{i - 1} }^+} \\map \\psi x$ and $\\displaystyle \\lim_{x \\mathop \\to {x_i}^-} \\map \\psi x$ exist. From the corollary to Sum of Integrals on Adjacent Intervals for Integrable Functions: :$\\displaystyle \\int_a^b \\map \\psi u \\sin N u \\rd u = \\sum_{r \\mathop = 0}^{m - 1} \\int_{x_r}^{x_{r + 1} } \\map \\psi u \\sin N u \\rd u$ Then: {{begin-eqn}} {{eqn | n = 1       | l = \\int_{x_r}^{x_{r + 1} } \\map \\psi u \\sin N u \\rd u       | r = \\intlimits {-\\map \\psi u \\frac {\\cos N u} N} {x_r} {x_{r + 1} }       | c = Integration by Parts }} {{eqn | o =        | ro= +       | r = \\frac 1 N \\int_{x_r}^{x_{r + 1} } \\map {\\psi'} u \\cos N u \\rd u       | c =  }} {{end-eqn}} The last integral is bounded. Thus $(1)$ is less than $\\dfrac {M_r} N$ for $M_r \\in \\R$. Let $M = \\max \\set {\\size {M_0}, \\size {M_1}, \\dotsc, \\size {M_{m - 1} } }$. Then: :$\\displaystyle \\size {\\int_a^b \\map \\psi u \\sin N u \\rd u} < \\dfrac {M m} N$ As $M$ and $m$ are finite: :$\\displaystyle \\lim_{N \\mathop \\to \\infty} \\dfrac {M m} N = 0$ Hence the result. {{qed}} == Sources == * {{BookReference|Fourier Series|1961|I.N. Sneddon|prev = Definition:Real Left-Hand Derivative|next = Fourier's Theorem/Lemma 1/Mistake 1}}: Chapter Two: $\\S 2$. Some Important Limits: Lemma $(1)$ Category:Fourier's Theorem l5xmrwqfroh93mu5ef8de049918xstw"}
+{"_id": "32900", "title": "Fourier's Theorem/Lemma 2", "text": "Fourier's Theorem/Lemma 2 0 67850 459098 348113 2020-04-02T22:11:31Z Prime.mover 59 wikitext text/x-wiki == Lemma for Fourier's Theorem == <onlyinclude> Let $\\psi$ be a real function defined on a half-open interval $\\hointl 0 a$. Let $\\psi$ and its derivative $\\psi'$ be piecewise continuous with one-sided limits on $\\hointl 0 a$. Let $\\map \\psi u$ have a right-hand derivative at $u = 0$. Then: :$\\displaystyle \\lim_{N \\mathop \\to \\infty} \\int_0^a \\map \\psi u \\frac {\\sin N u} u \\rd u = \\frac \\pi 2 \\map \\psi {0^+}$ where $\\map \\psi {0^+}$ denotes the limit of $\\psi$ at $0$ from the right. </onlyinclude> == Proof == We have: :$\\map \\psi u = \\map \\psi {0^+} + \\paren {\\map \\psi u - \\map \\psi {0^+} }$ from which: :$(1): \\quad \\displaystyle \\int_0^a \\map \\psi u \\frac {\\sin N u} u \\rd u = \\map \\psi {0^+} \\int_0^a \\frac {\\sin N u} u \\rd u + \\int_0^a \\map \\phi u \\sin N u \\rd u$ where: :$\\map \\phi u = \\dfrac {\\map \\psi u - \\map \\psi {0^+} } u$ Let $\\xi = N u$. Then: {{begin-eqn}} {{eqn | l = \\int_0^a \\frac {\\sin N u} u \\rd u       | r = \\int_0^{N a} \\frac {\\sin \\xi} \\xi \\rd \\xi       | c =  }} {{eqn | o = \\to       | r = \\int_0^\\infty \\frac {\\sin \\xi} \\xi \\rd \\xi       | c =  }} {{eqn | r = \\frac \\pi 2       | c = Integral to Infinity of $\\dfrac {\\sin \\xi} \\xi$ }} {{end-eqn}} We have that $\\map \\psi u$ is piecewise continuous with one-sided limits on $\\hointl 0 a$. Hence it follows that $\\map \\phi u = \\dfrac {\\map \\psi u - \\map \\psi {0^+} } u$ is also piecewise continuous with one-sided limits on $\\hointl 0 a$. We also have that $\\map \\psi u$ has a right-hand derivative at $u = 0$. It follows that $\\map \\phi u$ is piecewise continuous with one-sided limits on $\\hointl 0 a$. Thus from Lemma 1 for Fourier's Theorem: :$\\displaystyle \\lim_{N \\mathop \\to \\infty} \\int_0^a \\map \\phi u \\sin N u \\rd u = 0$ and letting $N \\to \\infty$ in $(1)$ above: :$\\displaystyle \\lim_{N \\mathop \\to \\infty} \\int_0^a \\map \\psi u \\frac {\\sin N u} u \\rd u = \\frac \\pi 2 \\map \\psi {0^+}$ {{qed}} {{Proofread}} == Sources == * {{BookReference|Fourier Series|1961|I.N. Sneddon|prev = Fourier's Theorem/Lemma 1/Mistake 2|next = Fourier's Theorem/Lemma 2/Mistake}}: Chapter Two: $\\S 2$. Some Important Limits: Lemma $(2)$ Category:Fourier's Theorem 5mirn4oim09fohb29j85c0esawwisnn"}
+{"_id": "32901", "title": "Fourier's Theorem/Lemma 3", "text": "Fourier's Theorem/Lemma 3 0 67853 459101 348117 2020-04-02T22:20:37Z Prime.mover 59 wikitext text/x-wiki == Lemma for Fourier's Theorem == <onlyinclude> Let $\\psi$ be a real function defined on an open interval $\\openint a b$. Let $\\psi$ and its derivative $\\psi'$ be piecewise continuous with one-sided limits on $\\openint a b$. Let $\\map \\psi u$ have both right-hand derivative and left-hand derivative at a point $u = x$ where $x \\in \\openint a b$. Then: :$\\displaystyle \\lim_{N \\mathop \\to \\infty} \\int_a^b \\map \\psi u \\frac {\\sin N \\paren {u - x} } {u - x} \\rd u = \\frac \\pi 2 \\paren {\\map \\psi {x^+} + \\map \\psi {x^-} }$ where: :$\\map \\psi {x^+}$ denotes the limit of $\\psi$ at $x$ from the right :$\\map \\psi {x^-}$ denotes the limit of $\\psi$ at $x$ from the left. </onlyinclude> == Proof == From Sum of Integrals on Adjacent Intervals for Integrable Functions, we have: :$\\displaystyle \\int_a^b \\map \\psi u \\frac {\\sin N \\paren {u - x} } {u - x} \\rd u = \\int_a^x \\map \\psi u \\frac {\\sin N \\paren {u - x} } {u - x} \\rd u + \\int_x^b \\map \\psi u \\frac {\\sin N \\paren {u - x} } {u - x} \\rd u$ Let $u = x - \\xi$. Then by Integration by Substitution: :$\\displaystyle \\int_a^x \\map \\psi u \\frac {\\sin N \\paren {u - x} } {u - x} \\rd u = \\int_0^{x - a} \\map \\phi \\xi \\frac {\\sin N \\xi} \\xi \\rd \\xi$ where: :$\\map \\phi \\xi = \\map \\psi {u - \\xi}$ {{explain|It needs to be established how the limits change from $a \\to x$ to $0 \\to x - a$ as this is not obvious.}} By Fourier's Theorem: Lemma 2: {{begin-eqn}} {{eqn | l = \\lim_{N \\mathop \\to \\infty} \\int_0^{x - a} \\map \\phi \\xi \\frac {\\sin N \\xi} \\xi \\rd \\xi       | r = \\frac \\pi 2 \\map \\phi {0^+}       | c =  }} {{eqn | r = \\frac \\pi 2 \\map \\psi {x^-}       | c =  }} {{end-eqn}} Similarly, substituting $u = x + \\eta$: :$\\displaystyle \\int_x^b \\map \\psi u \\frac {\\sin N \\paren {u - x} } {u - x} \\rd u = \\int_0^{b - x} \\map \\chi \\eta \\frac {\\sin N \\eta} \\eta \\rd \\eta$ where: :$\\map \\chi \\xi = \\map \\psi {x + \\eta}$ By Fourier's Theorem: Lemma 2: {{begin-eqn}} {{eqn | l = \\lim_{N \\mathop \\to \\infty} \\int_0^{b - x} \\map \\chi \\eta \\frac {\\sin N \\eta} \\eta \\rd \\eta       | r = \\frac \\pi 2 \\map \\phi {0^+}       | c =  }} {{eqn | r = \\frac \\pi 2 \\map \\psi {x^+}       | c =  }} {{end-eqn}} The result follows by adding the two limits. {{qed}} {{Proofread}} == Sources == * {{BookReference|Fourier Series|1961|I.N. Sneddon|prev = Fourier's Theorem/Lemma 2/Mistake|next = Fourier's Theorem}}: Chapter Two: $\\S 2$. Some Important Limits: Lemma $(3)$ Category:Fourier's Theorem rneh9nk5wbf5h4nyhwnhdivs7nsu33f"}
+{"_id": "32902", "title": "Sum of Sequence of Products of Consecutive Odd and Consecutive Even Reciprocals/Corollary", "text": "Sum of Sequence of Products of Consecutive Odd and Consecutive Even Reciprocals/Corollary 0 68118 349357 349294 2018-03-31T20:15:11Z Prime.mover 59 wikitext text/x-wiki == Corollary to Sum of Sequence of Products of Consecutive Odd and Consecutive Even Reciprocals == <onlyinclude> {{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 1}^\\infty \\frac 1 {j \\left({j + 2}\\right)}       | r = \\frac 1 {1 \\times 3} + \\frac 1 {2 \\times 4} + \\frac 1 {3 \\times 5} + \\frac 1 {4 \\times 6} + \\cdots       | c =  }} {{eqn | r = \\frac 3 4       | c =  }} {{end-eqn}} </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 1}^\\infty \\frac 1 {j \\left({j + 2}\\right)}       | r = \\lim_{n \\mathop \\to \\infty} \\sum_{j \\mathop = 1}^n \\frac 1 {j \\left({j + 2}\\right)}       | c =  }} {{eqn | r = \\lim_{n \\mathop \\to \\infty} \\left({\\frac 3 4 - \\frac {2 n + 3} {2 \\left({n + 1}\\right) \\left({n + 2}\\right)} }\\right)       | c = Sum of Sequence of Products of Consecutive Odd Reciprocals }} {{eqn | r = \\frac 3 4 - \\lim_{n \\mathop \\to \\infty} \\frac {\\frac 2 n + \\frac 3 {n^2} } {2 \\left({1 + \\frac 1 n}\\right) \\left({1 + \\frac 2 n}\\right)}       | c = dividing top and bottom by $n^2$ }} {{eqn | r = \\frac 3 4       | c = Basic Null Sequences }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Sum of Sequence of Products of Consecutive Odd Reciprocals/Corollary|next = Sum of Sequence of Products of Squares of Consecutive Odd Reciprocals}}: $\\S 19$: Series involving Reciprocals of Powers of Positive Integers: $19.31$ Category:Sums of Sequences Category:Reciprocals o91i9uxaoxvvhgqx6lu7gh97c6hthz3"}
+{"_id": "32903", "title": "Binomial Theorem/Examples/4th Power of Sum", "text": "Binomial Theorem/Examples/4th Power of Sum 0 68172 417824 412152 2019-08-11T12:46:23Z Caliburn 3218 wikitext text/x-wiki == Example of Use of Binomial Theorem == <onlyinclude> :$\\paren {x + y}^4 = x^4 + 4 x^3 y + 6 x^2 y^2 + 4 x y^3 + y^4$ </onlyinclude> == Proof == Follows directly from the Binomial Theorem: :$\\displaystyle \\forall n \\in \\Z_{\\ge 0}: \\paren {x + y}^n = \\sum_{k \\mathop = 0}^n \\binom n k x^{n - k} y^k$ putting $n = 4$. {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Cube of Difference|next = Fourth Power of Difference}}: $\\S 2$: Special Products and Factors: $2.5$ * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Cube of Sum|next = Power Series Expansion of Reciprocal of 1 + x/Proof 2}}: $\\S 20$: Binomial Series: $20.7$ * {{BookReference|Elementary Number Theory|1980|David M. Burton|ed = revised|edpage = Revised Printing|prev = Binomial Theorem/Examples/Cube of Sum|next = Definition:Pascal's Triangle/Historical Note}}: Chapter $1$: Some Preliminary Considerations: $1.2$ The Binomial Theorem Category:Examples of Use of Binomial Theorem Category:Fourth Powers iohl4jm3n8273acnkq9fyb0me4cvzvp"}
+{"_id": "32904", "title": "Power Series Expansion of Reciprocal of Square of 1 + x", "text": "Power Series Expansion of Reciprocal of Square of 1 + x 0 68177 349714 2018-04-03T15:07:34Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> Let $x \\in \\R$ such that $-1 < x < 1$.  Then:  {{begin-eqn}} {{eqn | l = \\dfrac 1 {\\left({1 + x}\\right)^2}       | r = \\sum_{k \\mathop = 0}^\\infty...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $x \\in \\R$ such that $-1 < x < 1$. Then: {{begin-eqn}} {{eqn | l = \\dfrac 1 {\\left({1 + x}\\right)^2}       | r = \\sum_{k \\mathop = 0}^\\infty \\left({-1}\\right)^k \\left({k + 1}\\right) x^k       | c =  }} {{eqn | r = 1 - 2 x + 3 x^2 - 4 x^3 + 5 x^4 - \\cdots       | c =  }} {{end-eqn}} </onlyinclude> == Proof 1 == {{:Power Series Expansion of Reciprocal of Square of 1 + x/Proof 1}} == Proof 2 == {{:Power Series Expansion of Reciprocal of Square of 1 + x/Proof 2}} Category:Examples of Power Series Category:Power Series Expansion of Reciprocal of Square of 1 + x 7robieaxmfj2d4h8hycx1hom9mur1me"}
+{"_id": "32905", "title": "Harmonic Numbers/Examples/H1", "text": "Harmonic Numbers/Examples/H1 0 69012 355776 355774 2018-05-16T20:19:47Z Prime.mover 59 wikitext text/x-wiki == Example of Harmonic Number == <onlyinclude> :$H_1 = 1$ </onlyinclude> where $H_1$ denotes the first harmonic number. == Proof == {{begin-eqn}} {{eqn | l = H_1       | r = \\sum_{k \\mathop = 1}^1 \\frac 1 k       | c = {{Defof|Harmonic Number}} }} {{eqn | r = \\frac 1 1       | c =  }} {{eqn | r = 1       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Harmonic Numbers/Examples/H0|next = Harmonic Numbers/Examples/H2}}: $\\S 1.2.7$: Harmonic Numbers: Exercise $1$ Category:Examples of Harmonic Numbers atu4v1wo1h1vtcjcwzb01gicuko4wv2"}
+{"_id": "32906", "title": "Harmonic Numbers/Examples/H2", "text": "Harmonic Numbers/Examples/H2 0 69015 359009 355779 2018-06-15T06:22:29Z Prime.mover 59 wikitext text/x-wiki == Example of Harmonic Number == <onlyinclude> :$H_2 = \\dfrac 3 2$ </onlyinclude> where $H_2$ denotes the second harmonic number. == Proof == {{begin-eqn}} {{eqn | l = H_2       | r = \\sum_{k \\mathop = 1}^2 \\frac 1 k       | c = {{Defof|Harmonic Number}} }} {{eqn | r = \\frac 1 1 + \\frac 1 2       | c =  }} {{eqn | r = \\dfrac 3 2       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Harmonic Number H1|next = Upper Bound for Harmonic Number}}: $\\S 1.2.7$: Harmonic Numbers: Exercise $1$ Category:Examples of Harmonic Numbers p9ega59o43niojrowf6ddsxvx97aerz"}
+{"_id": "32907", "title": "Sum of Arithmetic Sequence/Examples/Sum of j from m to n", "text": "Sum of Arithmetic Sequence/Examples/Sum of j from m to n 0 69146 456256 455622 2020-03-19T10:13:27Z Prime.mover 59 wikitext text/x-wiki == Example of Sum of Arithmetic Sequence == <onlyinclude> {{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = m}^n j       | r = m \\paren {n - m + 1} + \\frac 1 2 \\paren {n - m} \\paren {n - m + 1}       | c =  }} {{eqn | r = \\frac {n \\paren {n + 1} } 2 - \\frac {\\paren {m - 1} m} 2       | c =  }} {{end-eqn}} </onlyinclude> == Proof 1 == {{:Sum of Arithmetic Sequence/Examples/Sum of j from m to n/Proof 1}} == Proof 2 == {{:Sum of Arithmetic Sequence/Examples/Sum of j from m to n/Proof 2}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Sum of Geometric Sequence/Examples/One Seventh from 1 to n|next = General Distributivity Theorem/Examples/Sum of j from m to n by Sum of k from r to s}}: $\\S 1.2.3$: Sums and Products: Exercise $13$ Category:Examples of Sum of Arithmetic Sequence mn2s6399j6zegqolarfea9gebtt16cc"}
+{"_id": "32908", "title": "Factorial/Examples/1", "text": "Factorial/Examples/1 0 69333 357503 2018-05-28T07:04:05Z Prime.mover 59 Created page with \"== Example of Factorial == <onlyinclude> The factorial of $1$ is $1$: :$1! = 1$ </onlyinclude>  == Proof ==  From the definit...\" wikitext text/x-wiki == Example of Factorial == <onlyinclude> The factorial of $1$ is $1$: :$1! = 1$ </onlyinclude> == Proof == From the definition of factorial: :$n! = \\displaystyle \\prod_{k \\mathop = 1}^n k$ where $\\prod$ denotes product notation. When $n = 1$ we have: :$1! = \\displaystyle \\prod_{k \\mathop = 1}^1 k$ {{begin-eqn}} {{eqn | l = 1!       | r = \\prod_{k \\mathop = 1}^1 k       | c =  }} {{eqn | r = 1       | c =  }} {{end-eqn}} {{qed}} Category:Factorials/Examples ozkkkc9bs06qczcuw9c9pcuox40ln6s"}
+{"_id": "32909", "title": "Factors of Binomial Coefficient/Corollary 1", "text": "Factors of Binomial Coefficient/Corollary 1 0 69484 358308 2018-06-04T21:51:29Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> For all $r \\in \\R, k \\in \\Z$: :$\\left ({r - k}\\right) \\dbinom r k = r \\dbinom {r - 1} k$ from which: :$\\dbinom r k = \\dfrac r {r - k} \\dbinom {r -...\" wikitext text/x-wiki == Theorem == <onlyinclude> For all $r \\in \\R, k \\in \\Z$: :$\\left ({r - k}\\right) \\dbinom r k = r \\dbinom {r - 1} k$ from which: :$\\dbinom r k = \\dfrac r {r - k} \\dbinom {r - 1} k$ (if $r \\ne k$) </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = r \\binom {r - 1} k       | r = r \\frac {\\left({r - 1}\\right) \\left({\\left({r - 1}\\right) - 1}\\right) \\cdots \\left({\\left({r - 1}\\right) - k + 2}\\right) \\left({\\left({r - 1}\\right) - k + 1}\\right)} {k \\left({k - 1}\\right) \\left({k - 2}\\right) \\cdots 1}       | c =  }} {{eqn | r = \\frac {r \\left({r - 1}\\right) \\left({r - 2}\\right) \\cdots \\left({r - k + 1}\\right) \\left({r - k}\\right)} {k \\left({k - 1}\\right) \\left({k - 2}\\right) \\cdots 1}       | c =  }} {{eqn | r = \\left({r - k}\\right) \\frac {r \\left({r - 1}\\right) \\left({r - 2}\\right) \\cdots \\left({r - k + 1}\\right)} {k \\left({k - 1}\\right) \\left({k - 2}\\right) \\cdots 1}       | c =  }} {{eqn | r = \\left({r - k}\\right) \\binom r k       | c =  }} {{end-eqn}} {{qed|lemma}} Then: :$\\dbinom r k = \\dfrac r {r - k} \\dbinom {r - 1} k$ follows from the  :$\\left ({r - k}\\right) \\dbinom r k = r \\dbinom {r - 1} k$ by dividing both sides by $r - k$. {{qed}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Factors of Binomial Coefficient|next = Pascal's Rule/Real Numbers}}: $\\S 1.2.6$: Binomial Coefficients: $\\text{C}$ Category:Binomial Coefficients pckf7nqveg56hvodmowqbga3in6ilal"}
+{"_id": "32910", "title": "Numbers Expressed as Sums of Binomial Coefficients", "text": "Numbers Expressed as Sums of Binomial Coefficients 0 69501 465323 358420 2020-05-02T06:29:16Z RandomUndergrad 3904 wikitext text/x-wiki == Definition == Let $n \\in \\Z_{> 0}$ be a (strictly) positive integer. Then for all $k \\in \\Z_{> 0}$, it is possible to express $k$ uniquely in the form: {{begin-eqn}} {{eqn | l = k       | r = \\sum_{j \\mathop = 1}^n \\dbinom {k_j} j       | c =  }} {{eqn | r = \\dbinom {k_1} 1 + \\dbinom {k_2} 2 + \\cdots + \\dbinom {k_n} n       | c =  }} {{end-eqn}} such that $0 \\le k_1 < k_2 < \\cdots < k_n$. == Proof == === Existence of Representation === Proof by induction: For all $k \\in \\Z_{\\ge 0}$, let $\\map P k$ be the proposition that it is possible to express $k$ in the form: :$\\displaystyle k = \\sum_{j \\mathop = 1}^n \\dbinom {k_j} j$ such that $0 \\le k_1 < k_2 < \\cdots < k_n$. ==== Basis for the Induction ==== $\\map P 0$ is true, as $\\displaystyle 0 = \\sum_{j \\mathop = 1}^n \\dbinom {j - 1} j$. This is our basis for the induction. ==== Induction Hypothesis ==== Now we need to show that, if $\\map P m$ is true, where $r \\ge 2$, then it logically follows that $\\map P {m + 1}$ is true. So this is our induction hypothesis: $m$ can be expressed in the form: :$\\displaystyle m = \\sum_{j \\mathop = 1}^n \\dbinom {m_j} j$ where $0 \\le m_1 < m_2 < \\cdots < m_n$. Then we need to show: $m + 1$ can be expressed in the form: :$\\displaystyle m = \\sum_{j \\mathop = 1}^n \\dbinom {k_j} j$ where $0 \\le k_1 < k_2 < \\cdots < k_n$. ==== Induction Step ==== This is our induction step: Suppose the first $c$ $m_j$ are consecutive, that is: :$\\forall \\, j \\in \\N: 1 \\le j < c: m_{j + 1} - m_j = 1$ and $m_{c + 1} - m_c > 1$ Then: {{begin-eqn}} {{eqn | l = m + 1       | r = \\sum_{j \\mathop = 1}^n \\dbinom {m_j} j + 1       | c = from the induction hypothesis }} {{eqn | r = \\sum_{j \\mathop = c + 1}^n \\dbinom {m_j} j + \\sum_{j \\mathop = 1}^c \\dbinom {m_j} j + 1       | c =  }} {{eqn | r = \\sum_{j \\mathop = c + 1}^n \\dbinom {m_j} j + \\sum_{j \\mathop = 1}^c \\dbinom {m_1 + j - 1} j + \\dbinom {m_1 + 0 - 1} 0       | c = Binomial Coefficient with Zero }} {{eqn | r = \\sum_{j \\mathop = c + 1}^n \\dbinom {m_j} j + \\dbinom {m_1 + c} c       | c = Sum of r+k Choose k up to n }} {{eqn | r = \\sum_{j \\mathop = c + 1}^n \\dbinom {m_j} j + \\dbinom {m_c - 1} c       | c =  }} {{eqn | r = \\sum_{j \\mathop = c + 1}^n \\dbinom {m_j} j + \\dbinom {m_c - 1} c + \\sum_{j = 1}^{c - 1} \\dbinom {j - 1} j       | c =  }} {{end-eqn}} Since $0 \\le m_1 < m_2 < \\cdots < m_n$, we must have $m_j \\ge j - 1$ for each $j \\le n$. Hence $0 \\le 0 < 1 < \\cdots < c - 2 < m_c - 1 < m_{c + 1} < \\cdots < m_n$. Thus the expression above satisfy the conditions. So $\\map P m \\implies \\map P {m + 1}$ and the result follows by the Principle of Mathematical Induction. Therefore: :$\\displaystyle \\forall n \\in \\Z_{\\ge 0}: \\sum_{k \\mathop = 0}^n \\binom {r + k} k = \\binom {r + n + 1} n$ {{qed|lemma}} === Uniqueness of Representation === Suppose $k$ can be expressed in the form: :$\\displaystyle k = \\sum_{j \\mathop = 1}^n \\dbinom {k_j} j = \\sum_{j \\mathop = 1}^n \\dbinom {m_j} j$ where $0 \\le k_1 < k_2 < \\cdots < k_n$ and $0 \\le m_1 < m_2 < \\cdots < m_n$. {{AimForCont}} $k_i$ and $m_i$ are not all equal. Let $c$ be the largest integer such that $k_c \\ne m_c$. {{WLOG}} assume $k_c < m_c$. Then: {{begin-eqn}} {{eqn | l = k       | r = \\sum_{j \\mathop = 1}^n \\dbinom {k_j} j }} {{eqn | r = \\sum_{j \\mathop = c + 1}^n \\dbinom {m_j} j + \\sum_{j \\mathop = 1}^c \\dbinom {k_j} j }} {{eqn | o = \\le       | r = \\sum_{j \\mathop = c + 1}^n \\dbinom {m_j} j + \\sum_{j \\mathop = 1}^c \\dbinom {k_c - c + j} j       | c = $0 \\le k_1 < k_2 < \\cdots < k_n$ }} {{eqn | r = \\sum_{j \\mathop = c + 1}^n \\dbinom {m_j} j + \\dbinom {k_c - c + c + 1} c - \\dbinom {k_c - c} 0       | c = Sum of r+k Choose k up to n }} {{eqn | o = \\le       | r = \\sum_{j \\mathop = c + 1}^n \\dbinom {m_j} j + \\dbinom {m_c} c - 1       | c = $k_c + 1 \\le m_c$ }} {{eqn | o = <       | r = \\sum_{j \\mathop = c + 1}^n \\dbinom {m_j} j + \\dbinom {m_c} c       | c =  }} {{eqn | o = \\le       | r = \\sum_{j \\mathop = 1}^n \\dbinom {m_j} j       | c = $\\dbinom {m_j} j \\ge 0$ }} {{eqn | r = k       | c =  }} {{end-eqn}} which is a contradiction. Thus the representation is unique. {{qed}} == Examples == {{:Numbers Expressed as Sums of Binomial Coefficients/Examples}} == Also see == * Definition:Combinatorial Number System == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Inverse of Stirling's Triangle expressed as Matrix|next = Numbers Expressed as Sums of Binomial Coefficients/Examples/n = 3}}: $\\S 1.2.6$: Binomial Coefficients: Exercise $56$ Category:Numbers Expressed as Sums of Binomial Coefficients Category:Binomial Coefficients jgdt82aonbbr8gwtxj44wswha7lvase"}
+{"_id": "32911", "title": "Abel's Lemma/Formulation 1", "text": "Abel's Lemma/Formulation 1 0 69643 454400 442776 2020-03-14T13:19:32Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $\\sequence a$ and $\\sequence b$ be sequences in an arbitrary ring $R$. Then: <onlyinclude> :$\\displaystyle \\sum_{k \\mathop = m}^n a_k \\paren {b_{k + 1} - b_k} = a_{n + 1} b_{n + 1} - a_m b_m - \\sum_{k \\mathop = m}^n \\paren {a_{k + 1} - a_k} b_{k + 1}$ </onlyinclude> Note that although proved for the general ring, this result is usually applied to one of the conventional number fields $\\Z, \\Q, \\R$ and $\\C$. === Corollary === {{:Abel's Lemma/Formulation 1/Corollary}} == Proof == {{begin-eqn}} {{eqn | l = \\sum_{k \\mathop = m}^n a_k \\paren {b_{k + 1} - b_k}        | r = \\sum_{k \\mathop = m}^n a_k b_{k + 1} - \\sum_{k \\mathop = m}^n a_k b_k        | c = }} {{eqn | r = \\sum_{k \\mathop = m}^n a_k b_{k + 1} - \\paren {a_m b_m + \\sum_{k \\mathop = m}^n a_{k + 1} b_{k + 1} - a_{n + 1} b_{n + 1} }        | c = }} {{eqn | r = a_{n + 1} b_{n + 1} - a_m b_m + \\sum_{k \\mathop = m}^n a_k b_{k + 1} - \\sum_{k \\mathop = m}^n a_{k + 1} b_{k + 1}        | c = }} {{eqn | r = a_{n + 1} b_{n + 1} - a_m b_m - \\sum_{k \\mathop = m}^n \\paren {a_{k + 1} - a_k} b_{k + 1}        | c = }} {{end-eqn}} {{qed}} == Also reported as == Some sources give this as: :$\\displaystyle \\sum_{k \\mathop = m}^n \\paren {a_{k + 1} - a_k} b_k = a_{n + 1} b_{n + 1} - a_m b_m - \\sum_{k \\mathop = m}^n a_{k + 1} \\paren {b_{k + 1} - b_k}$ which is obtained from the main result by interchanging $a$ and $b$. Others take the upper index to $n - 1$: :$\\displaystyle \\sum_{k \\mathop = m}^{n - 1} \\paren {a_{k + 1} - a_k} b_k = a_n b_n - a_m b_m - \\sum_{k \\mathop = m}^{n - 1} a_{k + 1} \\paren {b_{k + 1} - b_k}$ == Also known as == {{:Abel's Lemma/Also known as}} {{Namedfor|Niels Henrik Abel}} Category:Abel's Lemma 9vb9u0cd2cxfgnvmw9v6ewlc4gbrmt3"}
+{"_id": "32912", "title": "Abel's Lemma/Formulation 2", "text": "Abel's Lemma/Formulation 2 0 69646 454403 413263 2020-03-14T13:21:00Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $\\sequence a$ and $\\sequence b$ be sequences in an arbitrary ring $R$. <onlyinclude> Let $\\displaystyle A_n = \\sum_{i \\mathop = m}^n {a_i}$ be the partial sum of $\\sequence a$ from $m$ to $n$. Then: :$\\displaystyle \\sum_{k \\mathop = m}^n a_k b_k = \\sum_{k \\mathop = m}^{n - 1} A_k \\paren {b_k - b_{k + 1} } + A_n b_n$ </onlyinclude> Note that although proved for the general ring, this result is usually applied to one of the conventional number fields $\\Z, \\Q, \\R$ and $\\C$. === Corollary === {{:Abel's Lemma/Formulation 2/Corollary}} == Proof 1 == {{:Abel's Lemma/Formulation 2/Proof 1}} == Proof 2 == {{:Abel's Lemma/Formulation 2/Proof 2}} == Also known as == {{:Abel's Lemma/Also known as}} {{Namedfor|Niels Henrik Abel}} == Sources == * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Symmetric Group is not Abelian|next = Definition:Abel Summation Method|entry = Abel's partial summation formula}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Abel's Limit Theorem|next = Definition:Abel Summation Method|entry = Abel's partial summation formula}} Category:Abel's Lemma rx01s1cgpt2yy19j756dqwulgh7zwhd"}
+{"_id": "32913", "title": "Sum of Angles between Straight Lines at Point form Four Right Angles", "text": "Sum of Angles between Straight Lines at Point form Four Right Angles 0 69677 359191 359186 2018-06-18T15:58:12Z Prime.mover 59 wikitext text/x-wiki == Corollary to Two Angles on Straight Line make Two Right Angles == <onlyinclude> If any number of straight lines are drawn from a given point, the sum of the consecutive angles so formed is $4$ right angles. </onlyinclude> == Proof == Let $OA_1, OA_2, \\ldots, OA_n$ be straight lines drawn from a point $O$ to points $A_1, A_2, \\ldots, A_n$. Let $OA_1$ be produced past $O$ to $B$. Then $OB$ either coincides with $OA_j$ for some $j$ between $1$ and $n$, or $OB$ divides angle $A_j O A_k$ for some $j, k$ between $1$ and $n$. First suppose $OB$ coincides with $OA_j$. {{finish|etc.}} == Sources == * {{BookReference|Problems & Solutions in Euclidean Geometry|1968|M.N. Aref|author2 = William Wernick|prev = Two Angles on Straight Line make Two Right Angles|next = Bisectors of Adjacent Angles between Straight Lines Meeting at Point are Perpendicular}}: Chapter $1$: Triangles and Polygons: Theorems and Corollaries $1.1$: Corollary $1$ Category:Angles ekhdayhfzl4qf346826w1frlt9g85hq"}
+{"_id": "32914", "title": "Characterization of Strictly Increasing Mapping on Woset", "text": "Characterization of Strictly Increasing Mapping on Woset 0 69689 360226 360224 2018-07-09T20:07:20Z GFauxPas 522 wikitext text/x-wiki {{tidy}} == Lemma == Let $J$ and $E$ be well-ordered sets. Let $h: J \\to E$ be a mapping. Let $S_\\alpha$ denote an initial segment determined by $\\alpha$. {{TFAE}} :$(1):\\quad$ $h$ is strictly increasing and its image is either all of $E$ or an initial segment of $E$ :$(2):\\quad$ $\\forall \\alpha \\in J: h\\left({\\alpha}\\right) = \\min \\left({E\\setminus h\\left[{S_\\alpha}\\right]}\\right)$, and $h[S_\\alpha] = S_{h(\\alpha)}$ where: :$h\\left[{S_\\alpha}\\right]$ denotes the image of $S_\\alpha$ under $h$ :$\\min$ denotes the smallest element of the set. == Proof == === $(1)$ implies $(2)$ === Suppose $h$ satisfies: :$h$ is strictly increasing and its image is either all of $E$ or an initial segment of $E$ Then for any $x,y \\in J$: {{begin-eqn}} {{eqn | l = x       | o = \\prec       | r = y }} {{eqn | ll = \\implies       | l = h(x)       | o = \\prec       | r = h(y)       | c = {{Defof|Strictly Increasing Mapping|strictly increasing}} }} {{eqn | l = h[S_y]       | r = \\left\\{ { h(x) \\in E: \\exists x \\in J: h(x) \\prec h(y) } \\right\\} }} {{eqn | r = S_{h(y)} }} {{eqn | l = \\min\\left({E \\setminus h\\left[{S_y}\\right] }\\right)       | r = \\min\\left({E \\setminus S_{h(y)} }\\right) }} {{eqn | r = h(y)       | c = {{Defof|Smallest Element|smallest}} and of initial segment }} {{end-eqn}} {{qed|lemma}} === $(2)$ implies $(1)$ === Suppose $h$ satisfies: :$h(\\alpha)  = \\min\\left({E \\setminus h\\left[{S_\\alpha}\\right] }\\right)$ By the Principle of Recursive Definition for Well-Ordered Sets, $h$ is thus uniquely determined. Then: {{begin-eqn}} {{eqn | l = h(y)       | r = \\min\\left({E \\setminus h\\left[{S_y}\\right] }\\right) }} {{eqn | l = h[S_y]       | r = \\left\\{ { h(x) \\in E: \\exists x \\in J: h(x) =              \\min\\left({E \\setminus h\\left[{S_y }\\right] }\\right)} \\right\\} }} {{eqn | r =  \\left\\{ { h(x) \\in E: \\exists x \\in J: h(x) \\prec h(y)} \\right\\} }} {{eqn | r = S_{h(y)}       | c = {{Defof|Initial Segment|initial segment}} }} {{end-eqn}} Thus for every $x \\in S_y$, we have that $h(x) \\in S_{h(y)}$. Therefore $h$ is an strictly increasing mapping. Furthermore, the image set of $h$ is the union of initial segments in $E$. By Union of Initial Segments is Initial Segment or All of Woset, $h[J]$ is an initial segment of $E$ or all of $E$. {{qed}} {{proofread}} == Sources == * {{BookReference|Topology|2000|James R. Munkres|ed = 2nd|edpage = Second Edition}}: Supplementary Exercise $1.2$ Category:Well-Orderings 4nlrguvf1tumug3w9gmy3u63fs0vcml"}
+{"_id": "32915", "title": "Strictly Increasing Mapping Between Wosets Implies Order Isomorphism", "text": "Strictly Increasing Mapping Between Wosets Implies Order Isomorphism 0 69763 359719 359526 2018-06-28T07:19:25Z Prime.mover 59 wikitext text/x-wiki {{tidy}} == Lemma == Let $J$ and $E$ be well-ordered sets. Let there exist a mapping $k: J \\to E$ which is strictly increasing. Then $J$ is order isomorphic to $E$ or an initial segment of $E$. == Proof == If the sets considered are empty or singletons, the theorem holds vacuously or trivially. Suppose $J,E$ both have at least two elements. Let $e_0 = \\min E$, the smallest element of $E$. Define the mapping: :$h: J \\to E$: :$h(\\alpha) =  \\begin{cases} \\min \\left({E \\setminus h\\left[ {S_\\alpha} \\right]}\\right) & \\text{ if } h\\left[ {S_\\alpha} \\right] \\ne E \\\\ e_0 & \\text{ if } h\\left[ {S_\\alpha} \\right] = E \\end{cases}$ where $S_\\alpha$ is the initial segment determined by $\\alpha$ and $h\\left[ {S_\\alpha} \\right]$ is the image of $S_\\alpha$ under $h$. By the Principle of Recursive Definition for Well-Ordered Sets, this construction is well-defined and uniquely determined. Observe that: {{begin-eqn}} {{eqn | l = h[S_\\alpha]       | r = \\left\\{ { h(x) \\in E: \\exists x \\in J: h(x) =              \\min\\left({E \\setminus h\\left[{S_\\alpha}\\right] }\\right)} \\right\\}       | c = {{Defof|Image of Subset under Mapping|image of a subset}} }} {{eqn | r =  \\left\\{ { h(x) \\in E: \\exists x \\in J: h(x) \\prec h(\\alpha)} \\right\\} }} {{eqn | r = S_{h(\\alpha)}       | c = {{Defof|Initial Segment|initial segment}} }} {{end-eqn}} This equality also holds if $h(\\alpha) = e_0$, by Initial Segment Determined by Smallest Element is Empty. We claim that $h(\\alpha) \\preceq k(\\alpha)$ for all $\\alpha \\in J$. {{AimForCont}} there is some $a \\in J$ such that $h(a) \\not \\preceq k(a)$. Then $k(a) \\prec h(a)$ by the trichotomy law. Because $h(a)$ has an element preceding it, $h(a) \\ne e_0$. Thus $k(a) \\prec \\min \\left({E \\setminus h\\left[ {S_a} \\right]}\\right)$ by the construction of $k$. Then $k(a) \\in h\\left[ {S_a} \\right]$, because it precedes the smallest element that isn't in $h\\left[ {S_a} \\right]$. Recall that $h[S_a] = S_{h(a)}$. Then $k(a) \\in S_{h(a)}$. This implies that $h(a) \\prec k(a)$, contradicting the assumption that $h(a) \\not \\preceq k(a)$ From this contradiction we can conclude: :$h(\\alpha) \\preceq k(\\alpha)$ for all $\\alpha \\in J$. {{AimForCont}} there is some $\\alpha \\in J$ such that $h[S_\\alpha] = E$. Recall that $h[S_\\alpha] = S_{h(\\alpha)}$. Thus, were such an $\\alpha$ to exist, then it would succeed all elements in $E$. It particular, it would also succeed $k(\\alpha)$. But we showed above that $h(\\alpha) \\preceq k(\\alpha)$. From this contradiction we see that there cannot be any $\\alpha \\in J$ with $h[S_\\alpha] = E$. Thus the definition of $h$ can be simplified: :$h: J \\to E$: :$h(\\alpha) = \\min \\left({E \\setminus h\\left[ {S_\\alpha} \\right]}\\right)$ Then the hypotheses of Characterization of Strictly Increasing Mapping on Woset are satisfied. Thus $h$ is a strictly increasing mapping and its image is $E$ or an initial segment of $E$. From Strictly Monotone Mapping with Totally Ordered Domain is Injective, $h$ is also injective. From Injection to Image is Bijection, $h$ is also bijective to its image. We conclude that there is an order isomorphism from $J$ to $E$, or from $J$ to an initial segment of $E$. {{qed}}{{proofread}} == Sources == * {{BookReference|Topology|2000|James R. Munkres|ed = 2nd|edpage = Second Edition}}: Supplementary Exercises $1.3$ Category:Well-Orderings 26s8gwwcf2jck79dirajwntol68aa7w"}
+{"_id": "32916", "title": "Equality to Initial Segment Imposes Well-Ordering", "text": "Equality to Initial Segment Imposes Well-Ordering 0 69778 483120 359717 2020-08-27T07:42:20Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $X$ be a set. Let $\\AA$ be the set of all ordered pairs $\\struct {A, <}$ such that $A$ is a subset of $X$ and $<$ is a strict well-ordering of $A$. Define $\\prec$ as: :$\\struct {A, <} \\prec \\struct {A', <'}$ {{iff}}  :$\\struct {A, <}$ equals an initial segment of $\\struct {A', <'}$. Let $\\BB$ be a set of ordered pairs in $\\AA$ such that $\\BB$ is ordered by $\\prec$. Let $B'$ be the union of the sets $B$ for all $\\struct {B, <} \\in \\BB$. Let $<'$ be the union of the relations $<$ for all $\\struct {B, <}$. Then $\\struct {B', <'}$ is strictly well-ordered set. == Proof == If the set $X$ considered is empty or a singleton, the lemma holds vacuously or trivially. Thus assume $X$ contains at least two elements. We first prove that $\\prec$ is a strict partial ordering on $\\AA$. From the definition of initial segment, no $\\struct {A, <}$ can equal an initial segment of itself. Thus $\\prec$ is antireflexive. Suppose $\\struct {A, <_A}$ equals an initial segment of $\\struct {B, <_B}$ and $\\struct {B, <_B}$ equals an initial segment of $\\struct {C, <_C}$. Then $\\struct {A, <_A}$ equals an initial segment of $\\struct {C, <'}$ from Equality is Transitive. Thus $\\prec$ is a strict partial ordering on $\\AA$. We then prove that any $\\struct {B', <'}$ is a strictly well-ordered set. Let $x_1,x_2 \\in B'$. That is, let $x_i \\in \\struct {A_i, <_i}$ for $i = 1, 2$. Suppose $x_2 \\prec x_1$. That is, $\\struct {A_2, <_2}$ equals an initial segment in $\\struct {A_1, <_1}$ By the definition of initial segment, both $x_1$ and $x_2$ are in $\\struct {A_1, <_1}$. Thus $<'$ is connected, as all $<_i$ are strict well-orderings by hypothesis. For any $x_i \\in \\struct {A_i, <_i}$, $x_i \\nprec x_i$ as all $<_i$ are strict well-orderings by hypothesis. Thus $<'$ is antireflexive. To show that $<'$ is transitive, consider $x_i \\in \\struct {B', <'}$ for $i = 1, 2, 3$. Suppose $x_1 <' x_2 <' x_3$. Then $x_1 <_1 x_2$ and $x_2 <_2 x_3$, from the definition of $<'$ as a union of relations $<_i$. Then $\\struct {A_j, <_j}$ is an initial segment of $\\struct {A_i, <_i}$ for $j  = 1, 2; j < i$ Thus $x_1 <_i x_2 <_i x_3$. Then $x_1 <_i x_3$, as all $<_i$ as all $<_i$ are strict well-orderings by hypothesis. Conclude that $<'$ is itself a strict ordering. It remains to be shown that $<'$ is a well-ordering. Let $A$ be an arbitrary non-empty subset of $B'$. Let $x \\in A$ and $x \\in \\struct {B, <}$ for $\\struct {B, <} \\in \\BB$. Then for all $y \\in A$, $y <' x$ {{iff}} $y < x$ and $y \\in B$. As $<$ is a well-ordering, $\\struct {B \\cap A, <}$ has a smallest element $b$. This $b$ is then a smallest element of $<'$ in $A$. Conclude that $<'$ is a strict well-ordering on $B'$. {{qed}} == Sources == * {{BookReference|Topology|2000|James R. Munkres|ed = 2nd|edpage = Second Edition}}: Supplementary Exercises $1.5$ Category:Well-Orderings Category:Set Equality rezg3r5cjiygyv3pkro2xkyvwjxxk0r"}
+{"_id": "32917", "title": "Euler-Binet Formula/Corollary 1", "text": "Euler-Binet Formula/Corollary 1 0 69795 380982 380978 2018-12-10T07:48:51Z Prime.mover 59 wikitext text/x-wiki == Corollary to Euler-Binet Formula == <onlyinclude> :$F_n = \\dfrac {\\phi^n} {\\sqrt 5}$ rounded to the nearest integer </onlyinclude> where: :$F_n$ denotes the $n$th Fibonacci number :$\\phi$ denotes the golden mean. == Proof == By definition of $n$th Fibonacci number, $F_n$ is an integer. From Euler-Binet Formula: :$F_n = \\dfrac {\\phi^n - \\hat \\phi^n} {\\sqrt 5} = \\dfrac {\\phi^n } {\\sqrt 5} - \\dfrac {\\hat \\phi^n} {\\sqrt 5}$ But $\\size {\\dfrac {\\hat \\phi^n} {\\sqrt 5} } < \\dfrac 1 2$ for all $n \\ge 0$. Thus $\\dfrac {\\phi^n } {\\sqrt 5}$ differs from $F_n$ by a number less than $\\dfrac 1 2$. Thus the nearest integer to $\\dfrac {\\phi^n } {\\sqrt 5}$ is $F_n$. {{qed}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Euler-Binet Formula/Historical Note|next = Euler-Binet Formula/Corollary 1/Mistake}}: $5$ * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Definition:Golden Mean/One Minus Golden Mean/Decimal Expansion|next = Summation over k to n of Product of kth with n-kth Fibonacci Numbers}}: $\\S 1.2.8$: Fibonacci Numbers: $(15)$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Euler-Binet Formula/Historical Note|next = Cassini's Identity}}: $5$ Category:Euler-Binet Formula qs30a3prs4nh9ygu017vhmfe9ne526h"}
+{"_id": "32918", "title": "Fibonacci Number plus Arbitrary Function in terms of Fibonacci Numbers/Lemma", "text": "Fibonacci Number plus Arbitrary Function in terms of Fibonacci Numbers/Lemma 0 69841 359979 359974 2018-07-03T07:15:12Z Prime.mover 59 wikitext text/x-wiki == Lemma for Fibonacci Number plus Arbitrary Function in terms of Fibonacci Numbers == <onlyinclude> Let $f \\left({n}\\right)$ be an arbitrary arithmetic function. Let $\\left\\langle{a_n}\\right\\rangle$ be the sequence defined as: :$a_n = \\begin{cases} 0 & : n = 0 \\\\ 1 & : n = 1 \\\\ a_{n - 1} + a_{n - 2} + f \\left({n - 2}\\right) & : n > 1 \\end{cases}$ Then: :$a_n = F_n + \\displaystyle \\sum_{k \\mathop = 0}^{n - 2} F_{n - k - 1} f \\left({k}\\right)$ </onlyinclude> == Proof == Trying out a few values: {{begin-eqn}} {{eqn | l = a_0       | r = 0       | c =  }} {{eqn | r = F_0       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_1       | r = 1       | c =  }} {{eqn | r = F_1       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_2       | r = a_1 + a_0 + f \\left({0}\\right)       | c =  }} {{eqn | r = F_1 + F_0 + f \\left({0}\\right)       | c =  }} {{eqn | r = F_2 + F_1 f \\left({0}\\right)       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_3       | r = a_2 + a_1 + f \\left({1}\\right)       | c =  }} {{eqn | r = F_2 + F_1 f \\left({0}\\right) + F_1 + f \\left({1}\\right)       | c =  }} {{eqn | r = F_3 + F_2 f \\left({0}\\right) + F_1 f \\left({1}\\right)       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_4       | r = a_3 + a_2 + f \\left({2}\\right)       | c =  }} {{eqn | r = F_3 + F_2 f \\left({0}\\right) + F_1 f \\left({1}\\right) + F_2 + F_1 f \\left({0}\\right) + f \\left({2}\\right)       | c =  }} {{eqn | r = F_4 + F_3 f \\left({0}\\right) + F_2 f \\left({1}\\right) + F_1 f \\left({2}\\right)       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_5       | r = a_4 + a_3 + f \\left({3}\\right)       | c =  }} {{eqn | r = F_4 + F_3 f \\left({0}\\right) + F_2 f \\left({1}\\right) + F_1 f \\left({2}\\right) + F_3 + F_2 f \\left({0}\\right) + F_1 f \\left({1}\\right) + f \\left({3}\\right)       | c =  }} {{eqn | r = F_5 + F_4 f \\left({0}\\right) + F_3 f \\left({1}\\right) + F_2 f \\left({2}\\right) + F_1 f \\left({3}\\right)       | c =  }} {{end-eqn}} The proof proceeds by induction. For all $n \\in \\Z_{\\ge 0}$, let $P \\left({n}\\right)$ be the proposition: :$a_n = F_n + \\displaystyle \\sum_{k \\mathop = 1}^{n - 1} F_{n - k - 1} f \\left({k}\\right)$ $P \\left({0}\\right)$ is the case: {{begin-eqn}} {{eqn | l = F_0 + \\sum_{k \\mathop = 0}^{0 - 2} F_{0 - k - 1} f \\left({k}\\right)       | r = F_0       | c = as the summation is vacuous }} {{eqn | r = 0       | c = {{Defof|Fibonacci Number}}: $F_0 = 0$ }} {{eqn | r = a_0       | c = by definition }} {{end-eqn}} Thus $P \\left({0}\\right)$ is seen to hold. === Basis for the Induction === $P \\left({1}\\right)$ is the case: {{begin-eqn}} {{eqn | l = F_1 + \\sum_{k \\mathop = 0}^{1 - 2} F_{1 - k - 1} f \\left({k}\\right)       | r = F_1       | c = as the summation is vacuous }} {{eqn | r = 1       | c = {{Defof|Fibonacci Number}}: $F_1 = 1$ }} {{eqn | r = a_1       | c = by definition }} {{end-eqn}} Thus $P \\left({1}\\right)$ is seen to hold. $P \\left({2}\\right)$ is the case: {{begin-eqn}} {{eqn | l = F_2 + \\sum_{k \\mathop = 0}^{2 - 2} F_{2 - k - 1} f \\left({k}\\right)       | r = F_2 + F_1 f \\left({0}\\right)       | c =  }} {{eqn | r = F_2 + f \\left({0}\\right)       | c = {{Defof|Fibonacci Number}}: $F_1 = 1$ }} {{eqn | r = F_1 + F_0 + f \\left({0}\\right)       | c = {{Defof|Fibonacci Number}} }} {{eqn | r = 1 + 0 + f \\left({0}\\right)       | c = {{Defof|Fibonacci Number}}: $F_1 = 1, F_0 = 0$ }} {{eqn | r = a_1 + a_0 + f \\left({0}\\right)       | c = by definition }} {{end-eqn}} Thus $P \\left({2}\\right)$ is seen to hold. This is the basis for the induction. === Induction Hypothesis === Now it needs to be shown that, if $P \\left({r}\\right)$ is true, where $r \\ge 1$, then it logically follows that $P \\left({r + 1}\\right)$ is true. So this is the induction hypothesis: :$a_r = F_r + \\displaystyle \\sum_{k \\mathop = 0}^{r - 2} F_{r - k - 1} f \\left({k}\\right)$ and: :$a_{r - 1} = F_{r - 1} + \\displaystyle \\sum_{k \\mathop = 0}^{r - 3} F_{r - k - 2} f \\left({k}\\right)$ from which it is to be shown that: :$a_{r + 1} = F_{r + 1} + \\displaystyle \\sum_{k \\mathop = 0}^{r - 1} F_{r - k} f \\left({k}\\right)$ === Induction Step === This is the induction step: {{begin-eqn}} {{eqn | l = a_{r + 1}       | r = a_{r - 1} + a_r + f \\left({r - 1}\\right)       | c =  }} {{eqn | r = \\left({F_{r - 1} + \\sum_{k \\mathop = 0}^{r - 3} F_{r - k - 2} f \\left({k}\\right)}\\right) + \\left({F_r + \\sum_{k \\mathop = 0}^{r - 2} F_{r - k - 1} f \\left({k}\\right)}\\right) + f \\left({r - 1}\\right)       | c = Induction Hypothesis }} {{eqn | r = F_{r + 1} + \\sum_{k \\mathop = 0}^{r - 3} F_{r - k - 2} f \\left({k}\\right) + \\sum_{k \\mathop = 0}^{r - 2} F_{r - k - 1} f \\left({k}\\right) + F_1 f \\left({r - 1}\\right)       | c = {{Defof|Fibonacci Number}} }} {{eqn | r = F_{r + 1} + \\sum_{k \\mathop = 0}^{r - 3} \\left({F_{r - k - 2} + F_{r - k - 1} }\\right) f \\left({k}\\right) + F_{r - \\left({r - 2}\\right) - 1} f \\left({r - 2}\\right) + F_1 f \\left({r - 1}\\right)       | c = factoring out the summation }} {{eqn | r = F_{r + 1} + \\sum_{k \\mathop = 0}^{r - 3} F_{r - k} f \\left({k}\\right) + F_1 f \\left({r - 2}\\right) + F_1 f \\left({r - 1}\\right)       | c = {{Defof|Fibonacci Number}} and simplifying }} {{eqn | r = F_{r + 1} + \\sum_{k \\mathop = 0}^{r - 3} F_{r - k} f \\left({k}\\right) + F_2 f \\left({r - 2}\\right) + F_1 f \\left({r - 1}\\right)       | c = {{Defof|Fibonacci Number}}: $F_1 = F_2 = 1$ }} {{eqn | r = F_{r + 1} + \\sum_{k \\mathop = 0}^{r - 1} F_{r - k} f \\left({k}\\right)       | c =  }} {{end-eqn}} So $P \\left({k}\\right) \\implies P \\left({k + 1}\\right)$ and the result follows by the Principle of Mathematical Induction. Therefore: :$a_n = F_n + \\displaystyle \\sum_{k \\mathop = 0}^{n - 2} F_{n - k - 1} f \\left({k}\\right)$ Category:Fibonacci Number plus Arbitrary Function in terms of Fibonacci Numbers q6p3nz26e1plob0y403qqx7k2xtpvru"}
+{"_id": "32919", "title": "Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers/Lemma 1", "text": "Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers/Lemma 1 0 69897 360163 2018-07-08T11:52:00Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> :$\\displaystyle \\sum_{k \\mathop \\in \\Z} \\dbinom m k_\\mathcal F \\left({-1}\\right)^{\\left\\lceil{\\left({m - k}\\right) / 2}\\right\\rceil} {F_{n + k} }^{...\" wikitext text/x-wiki == Theorem == <onlyinclude> :$\\displaystyle \\sum_{k \\mathop \\in \\Z} \\dbinom m k_\\mathcal F \\left({-1}\\right)^{\\left\\lceil{\\left({m - k}\\right) / 2}\\right\\rceil} {F_{n + k} }^{m - 2} F_k = F_m \\sum_{k \\mathop \\in \\Z} \\dbinom {m - 1} {k - 1}_\\mathcal F \\left({-1}\\right)^{\\left\\lceil{\\left({m - k}\\right) / 2}\\right\\rceil} {F_{n + k} }^{m - 2} = 0$ </onlyinclude> where: :$\\dbinom m k_\\mathcal F$ denotes a Fibonomial coefficient :$F_{n + k}$ denotes the $n + k$th Fibonacci number :$\\left\\lceil{\\, \\cdot \\,}\\right\\rceil$ denotes the ceiling function == Proof == {{ProofWanted}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers|next = Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers/Lemma 2}}: $\\S 1.2.8$: Fibonacci Numbers: Exercise $30$: Solution: $\\text{(a)}$ Category:Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers dexfwbxjyaq6c8wlvcct67e1mfwm1ik"}
+{"_id": "32920", "title": "Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers/Lemma 2", "text": "Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers/Lemma 2 0 69904 360188 2018-07-08T13:24:12Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> :$\\displaystyle \\sum_{k \\mathop \\in \\Z} \\dbinom m k_\\mathcal F \\left({-1}\\right)^{\\left\\lceil{\\left({m - k}\\right) / 2}\\right\\rceil} {F_{n + k} }^{...\" wikitext text/x-wiki == Theorem == <onlyinclude> :$\\displaystyle \\sum_{k \\mathop \\in \\Z} \\dbinom m k_\\mathcal F \\left({-1}\\right)^{\\left\\lceil{\\left({m - k}\\right) / 2}\\right\\rceil} {F_{n + k} }^{m - 2} \\left({-1}\\right)^k F_{m - k} = \\left({-1}\\right)^m F_m \\sum_{k \\mathop \\in \\Z} \\dbinom {m - 1} k_\\mathcal F \\left({-1}\\right)^{\\left\\lceil{\\left({m - 1 - k}\\right) / 2}\\right\\rceil} {F_{n + k} }^{m - 2} = 0$ </onlyinclude> where: :$\\dbinom m k_\\mathcal F$ denotes a Fibonomial coefficient :$F_{n + k}$ denotes the $n + k$th Fibonacci number :$\\left\\lceil{\\, \\cdot \\,}\\right\\rceil$ denotes the ceiling function == Proof == {{ProofWanted}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers/Lemma 1|next = Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers/Examples/m = 3}}: $\\S 1.2.8$: Fibonacci Numbers: Exercise $30$: Solution: $\\text{(b)}$ Category:Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers gaub0lzv715zubeg5xxu0wdrs9ldap5"}
+{"_id": "32921", "title": "Addition of 1 in Golden Mean Number System", "text": "Addition of 1 in Golden Mean Number System 0 69969 360433 360432 2018-07-13T19:59:10Z Prime.mover 59 wikitext text/x-wiki == Algorithm == Let $x \\in \\R$ be a real number. The following algorithm performs the operation of addition of $1$ to $x$ in the golden mean number system. Let $S$ be the representation of $x$ in the golden mean number system in its simplest form. :'''Step $1$''': Is the digit immediately to the left of the radix point a zero? :: If '''Yes''', replace that $0$ with $1$. Go to '''Step $4$'''. :: If '''No''', set $m = 2$ and go to '''Step $2$'''. :'''Step $2$''': Does the $m$th place after the radix point contain $0$? :: If '''Yes''', expand the $100$ in the $3$ places ending in the $m$th place with $011$. Subtract $2$ from $m$. Go to '''Step $3$'''. :: If '''No''', add $2$ to $m$. Repeat '''Step $2$'''. :'''Step $3$''': Is $m = 0$? :: If '''Yes''', set the digit immediately to the left of the radix point from $0$ to $1$. Go to '''Step $4$'''. :: If '''No''', go to '''Step $2$'''. :'''Step $4$''': Convert $S$ to its simplest form. '''Stop'''. == Proof == The above constitutes an algorithm, for the following reasons: === Finiteness === The only case in which it is possible for the process not to terminate is if the $m$th place never contains $0$. This can only happen if $S$ ends in an infinite string $01010101 \\ldots$ But if this is the case, $S$ is not in its simplest form. === Definiteness === Each step can be seen to be precisely defined. === Inputs === The only input to the algorithm is the representation $S$ of $x$. === Outputs === The only output from the algorithm is the representation $S$ of $x + 1$. All operations that change $S$ are of the following nature: :$(1): \\quad$ Simplification of $S$, which does not change $x$, which happens if at all in '''Step $4$. :$(2): \\quad$ Expansion of $S$, which does not change $x$, which happens if at all in ''Step $2$. :$(3): \\quad$ Setting the digit corresponding to $\\phi^0$ to $1$ from $0$, which happens either in '''Step $1$''' or in '''Step $3$'''. :::In either step, it happens only once, after which the algorithm terminates. === Effective === Each step is basic enough to be done exactly and predictably. {{qed}} == Sources == * {{citation|author = George Bergman|title = Number System with an Irrational Base|journal = Mathematics Magazine|abbr = Math. Mag.|volume = 31|issue = 2|date = 1957|startpage = 98|endpage = 110|jstor = 3029218}} Category:Golden Mean Number System 5gqhubgrxtldabakec720486ayfx61p"}
+{"_id": "32922", "title": "Fibonacci String/Examples/S3", "text": "Fibonacci String/Examples/S3 0 69980 360465 2018-07-14T08:42:36Z Prime.mover 59 Created page with \"== Example of Fibonacci Strings == <onlyinclude> The Fibonacci string $S_3$ is $\\text{ba}$. </onlyinclude>  ==...\" wikitext text/x-wiki == Example of Fibonacci Strings == <onlyinclude> The Fibonacci string $S_3$ is $\\text{ba}$. </onlyinclude> == Proof == By definition of Fibonacci string: {{begin-eqn}} {{eqn | l = S_1       | r = \\text{a} }} {{eqn | l = S_2       | r = \\text{b} }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = S_3       | r = S_2 S_1       | c =  }} {{eqn | r = \\text{ba}       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Definition:Fibonacci String|next = Fibonacci String/Examples/S4}}: $\\S 1.2.8$: Fibonacci Numbers: Exercise $36$ Category:Fibonacci Strings gj4xfcodh6c7x5k8syqbowp5jacm53y"}
+{"_id": "32923", "title": "Fibonacci String/Examples/S4", "text": "Fibonacci String/Examples/S4 0 69981 360466 2018-07-14T08:45:06Z Prime.mover 59 Created page with \"== Example of Fibonacci Strings == <onlyinclude> The Fibonacci string $S_4$ is $\\text{bab}$. </onlyinclude>  ==...\" wikitext text/x-wiki == Example of Fibonacci Strings == <onlyinclude> The Fibonacci string $S_4$ is $\\text{bab}$. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = S_4       | r = S_3 S_2       | c = {{Defof|Fibonacci String}} }} {{eqn | r = \\text{ba} \\ \\text{b}       | c = Definition of Fibonacci String $S_3$ }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Fibonacci String/Examples/S3|next = Fibonacci String/Examples/S5}}: $\\S 1.2.8$: Fibonacci Numbers: Exercise $36$ Category:Fibonacci Strings q4xbjhm8j9hrnwcdze0jpptvzajqfnc"}
+{"_id": "32924", "title": "Complex Modulus/Examples/i", "text": "Complex Modulus/Examples/i 0 70270 362517 361893 2018-08-16T07:11:17Z Prime.mover 59 wikitext text/x-wiki == Example of Complex Modulus == <onlyinclude> :$\\cmod i = \\cmod {-i} = 1$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\cmod i       | r = \\cmod {0 + 1 i}       | c =  }} {{eqn | r = \\sqrt {0^2 + 1^2}       | c = {{Defof|Complex Modulus}} }} {{eqn | r = \\sqrt 1       | c =  }} {{eqn | r = 1       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \\cmod {-i}       | r = \\cmod {0 + \\left({-1}\\right) i}       | c =  }} {{eqn | r = \\sqrt {0^2 + \\left({-1}\\right)^2}       | c = {{Defof|Complex Modulus}} }} {{eqn | r = \\sqrt 1       | c =  }} {{eqn | r = 1       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Power of Complex Modulus equals Complex Modulus of Power|next = Complex Modulus/Examples/-5}}: $\\S 1.2$. The Algebraic Theory: Examples Category:Examples of Complex Modulus 60nu6xawnfq6iw0ersbqw2abszqxyfw"}
+{"_id": "32925", "title": "Complex Modulus/Examples/1+i", "text": "Complex Modulus/Examples/1+i 0 70274 496584 361891 2020-10-25T23:13:47Z Prime.mover 59 wikitext text/x-wiki == Example of Complex Modulus == <onlyinclude> :$\\cmod {1 + i} = \\sqrt 2$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\cmod {1 + i}       | r = \\cmod {1 + 1 i}       | c =  }} {{eqn | r = \\sqrt {1^2 + 1^2}       | c = {{Defof|Complex Modulus}} }} {{eqn | r = \\sqrt 2       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Complex Modulus/Examples/-5|next = Power of Complex Modulus equals Complex Modulus of Power/Examples/(1+i)^4}}: $\\S 1.2$. The Algebraic Theory: Examples Category:Examples of Complex Modulus 2e8ljo6803ducgre7b0p9mds7jhi34i"}
+{"_id": "32926", "title": "Polar Form of Complex Number/Examples/i", "text": "Polar Form of Complex Number/Examples/i 0 70398 395502 394578 2019-03-13T08:13:05Z Prime.mover 59 wikitext text/x-wiki == Example of Polar Form of Complex Number == <onlyinclude> The imaginary unit $i$ can be expressed in polar form as $\\polar {1, \\dfrac \\pi 2}$. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\cmod i       | r = \\sqrt {0^2 + 1^2}       | c = {{Defof|Complex Modulus}} }} {{eqn | r = 1       | c =  }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \\map \\cos {\\map \\arg i}       | r = \\dfrac 0 1       | c = {{Defof|Argument of Complex Number}} }} {{eqn | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = \\map \\arg i       | r = \\pm \\dfrac \\pi 2       | c = Cosine of Half-Integer Multiple of Pi }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \\map \\sin {\\map \\arg i}       | r = \\dfrac 1 1       | c = {{Defof|Argument of Complex Number}} }} {{eqn | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = \\map \\arg i       | r = \\dfrac \\pi 2       | c = Sine of Half-Integer Multiple of Pi }} {{end-eqn}} Hence: :$\\map \\arg i = \\dfrac \\pi 2$ and hence the result. {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Argument of Quotient equals Difference of Arguments|next = Multiplication by Imaginary Unit is Equivalent to Rotation through Right Angle}}: $\\S 2$. Geometrical Representations Category:Examples of Polar Form of Complex Number l0ukan4ajbvpnrlz8ckq40uydle6ia3"}
+{"_id": "32927", "title": "Euler's Formula/Examples/e^i pi by 2", "text": "Euler's Formula/Examples/e^i pi by 2 0 70472 363046 363041 2018-08-20T06:08:50Z Prime.mover 59 wikitext text/x-wiki == Example of Use of Euler's Formula == <onlyinclude> :$e^{i \\pi / 2} = i$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = e^{i \\pi / 2}       | r = \\cos \\frac \\pi 2 + i \\sin \\frac \\pi 2       | c = Euler's Formula }} {{eqn | r = 0 + i \\times 1       | c = Cosine of $\\dfrac \\pi 2$, Sine of $\\dfrac \\pi 2$ }} {{eqn | r = i       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Euler's Formula/Examples/e^i pi by 4|next = Euler's Formula/Examples/e^-i pi by 2}}: $\\S 2$. Geometrical Representations: $(2.19)$ Category:Examples of Euler's Formula b95q3mt20dmvzmum5n3wqjfcme18b7i"}
+{"_id": "32928", "title": "Euler's Formula/Examples/e^-i pi by 2", "text": "Euler's Formula/Examples/e^-i pi by 2 0 70473 363045 363043 2018-08-20T06:08:37Z Prime.mover 59 wikitext text/x-wiki == Example of Use of Euler's Formula == <onlyinclude> :$e^{-i \\pi / 2} = -i$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = e^{-i \\pi / 2}       | r = \\cos \\frac {-\\pi} 2 + i \\sin \\frac {-\\pi} 2       | c = Euler's Formula }} {{eqn | r = \\cos \\frac {3 \\pi} 2 + i \\sin \\frac {3 \\pi} 2       | c = Cosine of Angle plus Full Angle, Sine of Angle plus Full Angle }} {{eqn | r = 0 + i \\times \\paren {-1}       | c = Cosine of $\\dfrac {3 \\pi} 2$, Sine of $\\dfrac {3 \\pi} 2$ }} {{eqn | r = -i       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Euler's Formula/Examples/e^i pi by 2|next = Euler's Formula/Examples/e^i pi}}: $\\S 2$. Geometrical Representations: $(2.19)$ Category:Examples of Euler's Formula lha2db5g6v7msclv5qua2zyz4xsk0kh"}
+{"_id": "32929", "title": "Euler's Formula/Examples/e^2 i pi", "text": "Euler's Formula/Examples/e^2 i pi 0 70477 365064 363049 2018-09-09T00:08:20Z Prime.mover 59 wikitext text/x-wiki == Example of Use of Euler's Formula == <onlyinclude> :$e^{2 i \\pi} = 1$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = e^{2 i \\pi}       | r = \\cos 2 \\pi + i \\sin 2 \\pi       | c = Euler's Formula }} {{eqn | r = 1 + i \\times 0       | c = Cosine of $2 \\pi$, Sine of $2 \\pi$ }} {{eqn | r = 1       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Euler's Formula/Examples/e^i pi|next = Exponential of Complex Number plus 2 pi i}}: $\\S 2$. Geometrical Representations: $(2.19)$ * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Euler's Formula/Proof|next = Period of Complex Exponential Function}}: $\\S 4.5$. The Functions $e^z$, $\\cos z$, $\\sin z$: $\\text{(ii)}$ Category:Examples of Euler's Formula sy6uwbgoocnlk19yrkru0px3asxjluz"}
+{"_id": "32930", "title": "Euler's Formula/Examples/e^2 k i pi", "text": "Euler's Formula/Examples/e^2 k i pi 0 70480 363057 2018-08-20T07:07:01Z Prime.mover 59 Created page with \"== Example of Use of Euler's Formula == <onlyinclude> :$\\forall k \\in \\Z: e^{2 k i \\pi} = 1$ </onlyinclude>  == Proof ==  {{begin-eqn}} {{eqn | l = e^{2 k i \\pi}       | r...\" wikitext text/x-wiki == Example of Use of Euler's Formula == <onlyinclude> :$\\forall k \\in \\Z: e^{2 k i \\pi} = 1$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = e^{2 k i \\pi}       | r = \\cos 2 k \\pi + i \\sin 2 k \\pi       | c = Euler's Formula }} {{eqn | r = 1 + i \\times 0       | c = Cosine of Multiple of $\\pi$, Sine of Multiple of $\\pi$ }} {{eqn | r = 1       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Exponential of Complex Number plus 2 pi i|next = Period of Complex Exponential Function}}: $\\S 2$. Geometrical Representations Category:Examples of Euler's Formula mbx3yw6qpt5at8mltyx1f02mmfwtoca"}
+{"_id": "32931", "title": "Euler's Theorem/Corollary 1", "text": "Euler's Theorem/Corollary 1 0 70492 374816 363115 2018-11-03T21:59:13Z Prime.mover 59 wikitext text/x-wiki == Corollary to Euler's Theorem == <onlyinclude> Let $p^n$ be a prime power for some prime number $p > 1$. Let $a$ be an integer not divisible by $p: p \\nmid a$. Then: : $a^{\\paren {p - 1} p^{n - 1} } \\equiv 1 \\pmod {p^n}$ </onlyinclude> == Proof == We have that Divisor Relation is Transitive. Since $p \\divides p^n$, it follows that $p^n \\nmid a$. From Euler's Theorem: : $a^{\\map \\phi {p^n} } \\equiv 1 \\pmod {p^n}$ From Euler Phi Function of Prime Power: :$\\map \\phi {p^n} = \\paren {p - 1} p^{n - 1}$ Then: : $a^{\\paren {p - 1} p^{n - 1} } \\equiv 1 \\pmod {p^n}$ {{qed}} Category:Prime Numbers Category:Number Theory 0uc9jenglnm484q1xhjndaxfnpeos3l"}
+{"_id": "32932", "title": "Argument of Negative Real Number is Pi", "text": "Argument of Negative Real Number is Pi 0 70516 363220 2018-08-22T21:43:48Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> Let $x \\in \\R_{>0}$ be a positive real number.  Then: :$\\arg \\paren {-x} = \\pi$  where $\\arg$ denotes the Def...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $x \\in \\R_{>0}$ be a positive real number. Then: :$\\arg \\paren {-x} = \\pi$ where $\\arg$ denotes the argument of a complex number. </onlyinclude> == Proof == We have that: :$-x = -x + 0 i$ and so: {{begin-eqn}} {{eqn | l = \\cmod {-x}       | r = \\sqrt {\\paren {-x}^2 + 0^2}       | c = {{Defof|Complex Modulus}} }} {{eqn | r = x       | c =  }} {{end-eqn}} Hence: {{begin-eqn}} {{eqn | l = \\cos \\paren {\\arg \\paren {-x} }       | r = \\dfrac {-x} x       | c = {{Defof|Argument of Complex Number}} }} {{eqn | r = -1       | c =  }} {{eqn | ll= \\leadsto       | l = \\arg \\paren {-x}       | r = \\pi       | c = Cosine of Multiple of Pi }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \\sin \\paren {\\arg \\paren {-x} }       | r = \\dfrac 0 x       | c = {{Defof|Argument of Complex Number}} }} {{eqn | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = \\arg \\paren {-x}       | r = 0 \\text { or } \\pi       | c = Sine of Multiple of Pi }} {{end-eqn}} Hence: : $\\arg \\paren {-x} = \\pi$ {{qed}} Category:Examples of Arguments of Complex Numbers 10dadpk3wkk9w8kfmpi2lwet2shamug"}
+{"_id": "32933", "title": "Equation of Unit Circle in Complex Plane/Corollary 1", "text": "Equation of Unit Circle in Complex Plane/Corollary 1 0 70543 363328 2018-08-24T22:57:32Z Prime.mover 59 Created page with \"== Corollary to Equation of Unit Circle in Complex Plane ==  Consider the unit circle $C$ whose center is at $\\l...\" wikitext text/x-wiki == Corollary to Equation of Unit Circle in Complex Plane == Consider the unit circle $C$ whose center is at $\\left({0, 0}\\right)$ on the complex plane. <onlyinclude> The equation of $C$ can be given by: :$z \\overline z = 1$ where $\\overline z$ denotes the complex conjugate of $z$. </onlyinclude> == Proof == From Equation of Unit Circle in Complex Plane, the equation of $C$ can also be given by: :$\\cmod z = 1$ where $\\cmod z$ denotes the complex modulus of $z$. Thus: {{begin-eqn}} {{eqn | l = \\cmod z       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = \\cmod z^2       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = z \\overline z       | r = 1       | c = Modulus in Terms of Conjugate }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Equation of Unit Circle|next = Equation of Unit Circle in Complex Plane/Corollary 2}}: $\\S 3$. Roots of Unity Category:Equation of Unit Circle in Complex Plane 0khohopk8187y9269w6uf1hjfjdwohv"}
+{"_id": "32934", "title": "Complex Roots of Unity/Examples/4th Roots", "text": "Complex Roots of Unity/Examples/4th Roots 0 70575 445882 395931 2020-02-03T09:26:52Z Prime.mover 59 wikitext text/x-wiki == Example of Complex Roots of Unity == <onlyinclude> The '''complex $4$th roots of unity''' are the elements of the set: :$U_n = \\set {z \\in \\C: z^4 = 1}$ They are: {{begin-eqn}} {{eqn | l = e^{0 i \\pi / 4}       | r = 1 }} {{eqn | l = e^{i \\pi / 2}       | r = i }} {{eqn | l = e^{i \\pi}       | r = -1 }} {{eqn | l = e^{3 i \\pi / 2}       | r = -i }} {{end-eqn}} </onlyinclude> == Proof == By definition, the first complex $4$th root of unity $\\alpha$ is given by: {{begin-eqn}} {{eqn | l = \\alpha       | r = e^{2 i \\pi / 4}       | c =  }} {{eqn | r = e^{i \\pi / 2}       | c =  }} {{eqn | r = \\cos \\frac \\pi 2 + i \\sin \\frac \\pi 2       | c =  }} {{eqn | r = 0 + i \\times 1       | c = Cosine of $\\dfrac \\pi 2$, Sine of $\\dfrac \\pi 2$ }} {{eqn | r = i       | c =  }} {{end-eqn}} We have that: :$e^{0 i \\pi / 4} = e^0 = 1$ which gives us, as always, the zeroth complex $n$th root of unity for all $n$. The remaining complex $4$th roots of unity can be expressed as $e^{4 i \\pi / 4} = e^{i \\pi}$ and $e^{6 i \\pi / 4} = e^{3 i \\pi / 2}$, but it is simpler to calculate them as follows: {{begin-eqn}} {{eqn | l = \\alpha^2       | r = i^2       | c =  }} {{eqn | r = -1       | c = {{Defof|Imaginary Unit}} }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \\alpha^3       | r = \\alpha^2 \\times \\alpha       | c =  }} {{eqn | r = \\paren {-1} \\times i       | c =  }} {{eqn | r = -i       | c =  }} {{end-eqn}} :500px {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Complex Roots of Unity are Vertices of Regular Polygon Inscribed in Circle|next = Complex 5th Roots of Unity}}: $\\S 3$. Roots of Unity: Example $1$. * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Viète's Formulas/Examples/Sum 4, Product 8|next = Complex Roots of Unity/Examples/7th Roots}}: $1$: Complex Numbers: Supplementary Problems: The $n$th Roots of Unity: $105 \\ \\text {(a)}$ * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Four Squares Theorem|next = Definition:Fractal|entry = fourth root of unity}} Category:Examples of Complex Roots of Unity 626ort6gwxv170mtalcoktj0m3j1ma8"}
+{"_id": "32935", "title": "Complex Roots of Unity/Examples/5th Roots", "text": "Complex Roots of Unity/Examples/5th Roots 0 70576 437891 395936 2019-12-06T06:38:43Z Prime.mover 59 wikitext text/x-wiki == Example of Complex Roots of Unity == <onlyinclude> The '''complex $5$th roots of unity''' are the elements of the set: :$U_n = \\set {z \\in \\C: z^5 = 1}$ They are: {{begin-eqn}} {{eqn | l = e^{0 \\pi / 5}       | r = 1 }} {{eqn | l = e^{2 \\pi / 5}       | r = \\dfrac {\\sqrt 5 - 1} 4 + i \\dfrac {\\sqrt {10 + 2 \\sqrt 5} } 4 }} {{eqn | l = e^{4 \\pi / 5}       | r = -\\dfrac {1 + \\sqrt 5} 4 + i \\sqrt {\\dfrac 5 8 - \\dfrac {\\sqrt 5} 8} }} {{eqn | l = e^{6 \\pi / 5}       | r = -\\dfrac {1 + \\sqrt 5} 4 - i \\sqrt {\\dfrac 5 8 - \\dfrac {\\sqrt 5} 8} }} {{eqn | l = e^{8 \\pi / 5}       | r = \\dfrac {\\sqrt 5 - 1} 4 - i \\dfrac {\\sqrt {10 + 2 \\sqrt 5} } 4 }} {{end-eqn}} </onlyinclude> == Proof == By definition, the first complex $5$th root of unity $\\alpha$ is given by: {{begin-eqn}} {{eqn | l = \\alpha       | r = e^{2 i \\pi / 5}       | c =  }} {{eqn | r = \\cos \\frac {2 \\pi} 5 + i \\sin \\frac {2 \\pi} 5       | c =  }} {{eqn | r = \\dfrac {\\sqrt 5 - 1} 4 + i \\dfrac {\\sqrt {10 + 2 \\sqrt 5} } 4       | c = Cosine of $\\dfrac {2 \\pi} 5$, Sine of $\\dfrac {2 \\pi} 5$ }} {{end-eqn}} We have that: :$e^{0 i \\pi / 5} = e^0 = 1$ which gives us, as always, the zeroth complex $n$th root of unity for all $n$. The remaining complex $5$th roots of unity follow: {{begin-eqn}} {{eqn | l = \\alpha^2       | r = e^{4 i \\pi / 5}       | c =  }} {{eqn | r = -\\dfrac {1 + \\sqrt 5} 4 + i \\sqrt {\\dfrac 5 8 - \\dfrac {\\sqrt 5} 8}       | c =  }} {{eqn | r = \\dfrac {\\sqrt 5 - 1} 4 + i \\dfrac {\\sqrt {10 + 2 \\sqrt 5} } 4       | c = Cosine of $\\dfrac {4 \\pi} 5$, Sine of $\\dfrac {4 \\pi} 5$ }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \\alpha^3       | r = \\overline {\\alpha^{5 - 3} }       | c = Complex Roots of Unity occur in Conjugate Pairs }} {{eqn | r = \\overline {\\alpha^2}       | c =  }} {{eqn | r = \\overline {-\\dfrac {1 + \\sqrt 5} 4 + i \\sqrt {\\dfrac 5 8 - \\dfrac {\\sqrt 5} 8} }       | c =  }} {{eqn | r = -\\dfrac {1 + \\sqrt 5} 4 - i \\sqrt {\\dfrac 5 8 - \\dfrac {\\sqrt 5} 8}       | c = {{Defof|Complex Conjugate}} }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \\alpha^4       | r = \\overline {\\alpha^{4 - 1} }       | c = Complex Roots of Unity occur in Conjugate Pairs }} {{eqn | r = \\overline \\alpha       | c =  }} {{eqn | r = \\overline {\\dfrac {\\sqrt 5 - 1} 4 + i \\dfrac {\\sqrt {10 + 2 \\sqrt 5} } 4}       | c =  }} {{eqn | r = \\dfrac {\\sqrt 5 - 1} 4 - i \\dfrac {\\sqrt {10 + 2 \\sqrt 5} } 4       | c = {{Defof|Complex Conjugate}} }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Complex 4th Roots of Unity|next = Complex Roots of Unity/Examples/6th Roots/Illustration}}: $\\S 3$. Roots of Unity: Example $2$. * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs/Proof 2|next = Sum of Cosines of Fractions of Pi}}: $1$: Complex Numbers: Solved Problems: The $n$th Roots of Unity: $37$ Category:Examples of Complex Roots of Unity Category:Complex 5th Roots of Unity s7scye8xf3d942lgzdqkq63ufkrm3ja"}
+{"_id": "32936", "title": "Polynomial Factor Theorem/Corollary/Complex Numbers", "text": "Polynomial Factor Theorem/Corollary/Complex Numbers 0 70604 399146 363764 2019-04-05T22:07:27Z Prime.mover 59 wikitext text/x-wiki == Corollary to Polynomial Factor Theorem == <onlyinclude> Let $\\map P z$ be a polynomial in $z$ over the complex numbers $\\C$ of degree $n$. Suppose there exists $\\zeta \\in \\C: \\map P \\xi = 0$. Then: :$\\map P z = \\paren {x - \\zeta} \\map Q z$ where $\\map Q z$ is a polynomial of degree $n - 1$. Hence, if $\\zeta_1, \\zeta_2, \\ldots, \\zeta_n \\in \\C$ such that all are different, and $\\map P {\\zeta_1} = \\map P {\\zeta_2} = \\dotsb = \\map P {\\zeta_n} = 0$, then: :$\\displaystyle \\map P z = k \\prod_{j \\mathop = 1}^n \\paren {z - \\zeta_j}$ where $k \\in \\C$. </onlyinclude> == Proof == Recall that Complex Numbers form Field. The result then follows from the Polynomial Factor Theorem. {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Roots of Complex Number/Corollary/Examples/Cube Roots of 8i|next = Power of Complex Number minus 1}}: $\\S 3$. Roots of Unity Category:Polynomial Theory 5a1t2sdcgh059t3b0jzevzvi5rwhxlk"}
+{"_id": "32937", "title": "Complex Algebra/Examples/z^6 + z^3 + 1", "text": "Complex Algebra/Examples/z^6 + z^3 + 1 0 70620 456565 363910 2020-03-19T23:14:08Z Prime.mover 59 wikitext text/x-wiki == Example of Complex Algebra == <onlyinclude> :$z^6 + z^3 + 1 = \\paren {z^2 - 2 z \\cos \\dfrac {2 \\pi} 9 + 1} \\paren {z^2 - 2 z \\cos \\dfrac {4 \\pi} 9 + 1} \\paren {z^2 - 2 z \\cos \\dfrac {8 \\pi} 9 + 1}$ </onlyinclude> == Proof == From Sum of Geometric Sequence or Difference of Two Cubes: :$z^6 + z^3 + 1 = \\dfrac {z^9 - 1} {z^3 - 1}$ Then from Factorisation of $x^{2 n + 1} - 1$ in Real Domain: :$z^9 - 1 = \\paren {z - 1} \\displaystyle \\prod_{k \\mathop = 1}^4 \\paren {z^2 - 2 \\cos \\dfrac {2 \\pi k} 9 + 1}$ and: {{begin-eqn}} {{eqn | l = z^3 - 1       | r = \\paren {z - 1} \\prod_{k \\mathop = 1}^1 \\paren {z^2 - 2 \\cos \\dfrac {2 \\pi k} 3 + 1}       | c =  }} {{eqn | r = \\paren {z - 1} \\paren {z^2 - 2 \\cos \\dfrac {2 \\pi} 3 + 1}       | c =  }} {{eqn | r = \\paren {z - 1} \\paren {z^2 - 2 \\cos \\dfrac {6 \\pi} 9 + 1}       | c =  }} {{end-eqn}} Thus: :$\\paren {z^3 - 1} \\paren {z^6 + z^3 + 1} = \\paren {\\paren {z - 1} \\paren {z^2 - 2 \\cos \\dfrac {6 \\pi} 9 + 1} } \\paren {\\paren {z^2 - 2 \\cos \\dfrac {2 \\pi} 9 + 1} \\paren {z^2 - 2 \\cos \\dfrac {4 \\pi} 9 + 1} \\paren {z^2 - 2 \\cos \\dfrac {8 \\pi} 9 + 1} }$ from which: :$\\paren {z^6 + z^3 + 1} = \\paren {z^2 - 2 \\cos \\dfrac {2 \\pi} 9 + 1} \\paren {z^2 - 2 \\cos \\dfrac {4 \\pi} 9 + 1} \\paren {z^2 - 2 \\cos \\dfrac {8 \\pi} 9 + 1}$ {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Factorisation of z^n-a|next = Triple Angle Formulas/Cosine/2 cos 3 theta + 1}}: $\\S 3$. Roots of Unity: Example $6$: $(3.14)$ Category:Complex Roots Category:Examples of Complex Algebra 7rexsnnxj4bf04rohyd7kfxfpkltee3"}
+{"_id": "32938", "title": "Roots of Complex Number/Examples/z^8 + 1 = 0/Illustration", "text": "Roots of Complex Number/Examples/z^8 + 1 = 0/Illustration 0 70649 364035 2018-09-02T06:13:38Z Prime.mover 59 Created page with \"== Illustration of Roots of $z^8 + 1 = 0$ == <onlyinclude> The roots of the Definition:Pol...\" wikitext text/x-wiki == Illustration of Roots of $z^8 + 1 = 0$ == <onlyinclude> The roots of the polynomial: :$z^8 + 1 = 0$ are illustrated below: :620px </onlyinclude> == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Roots of Complex Number/Examples/z^8 + 1 = 0|next = Complex Algebra/Examples/z^8 + 1}}: $\\S 3$. Roots of Unity: Exercise $8$ Category:Examples of Complex Roots 576jivb1jsfmev9aqjwjg3o0oeu5huo"}
+{"_id": "32939", "title": "Complex Algebra/Examples/z^8 + 1", "text": "Complex Algebra/Examples/z^8 + 1 0 70650 364074 364042 2018-09-02T10:06:57Z Prime.mover 59 wikitext text/x-wiki == Example of Complex Algebra == <onlyinclude> :$z^8 + 1 = \\paren {z^2 - 2 z \\cos \\dfrac \\pi 8 + 1} \\paren {z^2 - 2 z \\cos \\dfrac {3 \\pi} 8 + 1} \\paren {z^2 - 2 z \\cos \\dfrac {5 \\pi} 8 + 1} \\paren {z^2 - 2 z \\cos \\dfrac {7 \\pi} 8 + 1}$ </onlyinclude> == Proof == From Roots of $z^8 + 1 = 0$ and the corollary to the Polynomial Factor Theorem: :$z^8 + 1 = \\displaystyle \\prod_{k \\mathop = 0}^7 \\paren {z - \\paren {\\cos \\dfrac {\\paren {2 k + 1} \\pi} 8 + i \\sin \\dfrac {\\paren {2 k + 1} \\pi} 8} }$ Hence: {{begin-eqn}} {{eqn | l = z^8 + 1       | r = \\prod_{k \\mathop = 0}^7 \\paren {z - \\exp \\dfrac {\\paren {2 k + 1} i \\pi} 8}       | c = {{Defof|Exponential Form of Complex Number}} }} {{eqn | r = \\prod_{k \\mathop = 0}^3 \\paren {z - \\exp \\dfrac {\\paren {2 k + 1} i \\pi} 8} \\paren {z - \\exp \\dfrac {-\\paren {2 k + 1} i \\pi} 8}       | c = Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs }} {{eqn | r = \\prod_{k \\mathop = 0}^3 \\paren {z^2 - z \\paren {\\exp \\dfrac {\\paren {2 k + 1} i \\pi} 8 + \\exp \\dfrac {-\\paren {2 k + 1} i \\pi} 8} + \\exp \\dfrac {\\paren {2 k + 1} i \\pi} 8 \\exp \\dfrac {-\\paren {2 k + 1} i \\pi} 8}       | c = multiplying out }} {{eqn | r = \\prod_{k \\mathop = 0}^3 \\paren {z^2 - z \\paren {\\exp \\dfrac {\\paren {2 k + 1} i \\pi} 8 + \\exp \\dfrac {-\\paren {2 k + 1} i \\pi} 8} + 1}       | c = simplifying }} {{eqn | r = \\prod_{k \\mathop = 0}^3 \\paren {z^2 - z \\paren {\\cos \\dfrac {\\paren {2 k + 1} \\pi} 8 + i \\sin \\dfrac {\\paren {2 k + 1} \\pi} 8 + \\cos \\dfrac {\\paren {2 k + 1} \\pi} 8 - i \\sin \\dfrac {\\paren {2 k + 1} \\pi} 8} + 1}       | c = {{Defof|Exponential Form of Complex Number}} }} {{eqn | r = \\prod_{k \\mathop = 0}^3 \\paren {z^2 - 2 z \\cos \\dfrac {\\paren {2 k + 1} \\pi} 8 + 1}       | c = simplifying }} {{end-eqn}} Hence the result. {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Roots of Complex Number/Examples/z^8 + 1 = 0/Illustration|next = Quadruple Angle Formulas/Cosine/Factor Form}}: $\\S 3$. Roots of Unity: Exercise $8$ Category:Complex Roots Category:Examples of Complex Algebra tcb2reog2caqly9vaw987knentssqgp"}
+{"_id": "32940", "title": "Euler's Formula/Corollary", "text": "Euler's Formula/Corollary 0 70856 365723 365141 2018-09-14T22:06:16Z Prime.mover 59 wikitext text/x-wiki == Corollary to Euler's Formula == Let $z \\in \\C$ be a complex number. Then: <onlyinclude> :$e^{-i z} = \\cos z - i \\sin z$ </onlyinclude> where: : $e^{-i z}$ denotes the complex exponential function : $\\cos z$ denotes the complex cosine function : $\\sin z$ denotes complex sine function : $i$ denotes the imaginary unit. === Corollary === This result is often presented and proved separately for arguments in the real domain: {{:Euler's Formula/Real Domain/Corollary}} == Proof == {{begin-eqn}} {{eqn | l = e^{-i z}       | r = \\cos \\paren {-z} + i \\sin \\paren {-z}       | c = Euler's Formula }} {{eqn | r = \\cos z + i \\sin \\paren {-z}       | c = Cosine Function is Even }} {{eqn | r = \\cos z - i \\sin z       | c = Sine Function is Odd }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Sine Function is Odd|next = Cosine Exponential Formulation/Proof 3}}: $\\S 4.5$. The Functions $e^z$, $\\cos z$, $\\sin z$: $\\text{(ii)}$: $(4.16)$ Category:Euler's Formula od8drfqw7i7eigin9lgee53gjmm4c2s"}
+{"_id": "32941", "title": "Complex Natural Logarithm/Examples/i", "text": "Complex Natural Logarithm/Examples/i 0 70907 365496 365494 2018-09-11T22:18:14Z Prime.mover 59 wikitext text/x-wiki == Example of Complex Natural Logarithm == <onlyinclude> :$\\ln \\paren i = \\paren {4 k + 1} \\dfrac {\\pi i} 2$ for all $k \\in \\Z$. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = i       | r = \\exp \\paren {\\dfrac {i \\pi} 2}       | c = Polar Form of  $i$ }} {{eqn | ll= \\leadsto       | l = \\ln \\paren i       | r = \\ln \\paren {\\exp \\paren {\\dfrac {i \\pi} 2 + 2 k \\pi i} }       | c =  }} {{eqn | r = \\dfrac {i \\pi + 4 k \\pi i} 2       | c = {{Defof|Complex Natural Logarithm}} }} {{eqn | r = \\paren {4 k + 1} \\dfrac {\\pi i} 2       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Complex Natural Logarithm/Examples/-2|next = Complex Cosine Function/Examples/4 cos z = 3+i}}: $\\S 4.6$. The Logarithm: Examples: $\\text {(ii)}$ Category:Examples of Complex Natural Logarithms 4vjtql85ggcsbb4s8ujl8vs7hcm55md"}
+{"_id": "32942", "title": "Square Root of Complex Number in Cartesian Form/Examples/-8+6i", "text": "Square Root of Complex Number in Cartesian Form/Examples/-8+6i 0 70916 365526 365505 2018-09-12T06:24:03Z Prime.mover 59 wikitext text/x-wiki == Example of Square Root of Complex Number in Cartesian Form == <onlyinclude> :$\\sqrt {-8 + 6 i} = \\pm \\paren {1 + 3 i}$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\paren {x + i y}^2       | r = -8 + 6 i       | c =  }} {{eqn | ll= \\leadsto       | l = x^2       | r = \\dfrac {-8 + \\sqrt {\\paren {-8}^2 + 6^2} } 2       | c = Square Root of Complex Number in Cartesian Form }} {{eqn | r = \\dfrac {-8 + \\sqrt {100} } 2       | c =  }} {{eqn | r = \\dfrac {-8 + 10} 2       | c =  }} {{eqn | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = x       | r = \\pm 1       | c =  }} {{eqn | ll= \\leadsto       | l = y       | r = \\pm \\dfrac 6 {2 \\times 1}       | c =  }} {{eqn | r = \\pm 3       | c =  }} {{end-eqn}} As $2 x y = 6$ it follows that the two solutions are: :$1 + 3 i$ :$-1 - 3 i$ {{Qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Complex Cosine Function/Examples/4 cos z = 3+i|next = Inverse Tangent of i}}: $\\S 4.6$. The Logarithm: Examples: $\\text {(iii)}$ Category:Examples of Square Roots lgeddlb38hbut62yl9p26gre49nmkar"}
+{"_id": "32943", "title": "Proof by Contraposition", "text": "Proof by Contraposition 0 70942 370643 370486 2018-10-12T07:57:47Z KarlFrei 3474 fixed this as well wikitext text/x-wiki == Proof Technique == '''Proof by contraposition''' is a rule of inference used in proofs.   This rule infers a conditional statement from its contrapositive. It is based on the Rule of Transposition, which says that a conditional statement and its contrapositive have the same truth value: :$p \\implies q \\dashv \\vdash \\neg q \\implies \\neg p$ In other words, the conclusion \"if A, then B\" is drawn from the single premise \"if not B, then not A.\"  == Explanation == '''Proof by Contraposition''' can be expressed in natural language as follows: If we know that by making an assumption  :$\\neg q$  we can deduce  :$\\neg p$ then it must be the case that  :$p \\implies q$. Thus it provides a means of proving a logical implication. This proof is often confused with Reductio ad Absurdum, which also starts with an assumption $\\neg q$. Reductio ad Absurdum itself is often confused with Proof by Contradiction. Unlike Reductio ad Absurdum however, Proof by Contraposition ''can'' be a valid proof in intuitionistic logic, just like Proof by Contradiction. Specifically, suppose :$p \\implies q$ is true.  Suppose furthermore that we have a proof of :$\\neg q$. Then if we had a proof of $p$, it could be turned into a proof of $q$. This would imply :$q\\land \\neg q$ which is impossible. Therefore :$\\neg p$. However, now suppose :$\\neg q \\implies \\neg p$ is true. Suppose furthermore that we have a proof of :$p$. Then if we had a proof of $\\neg q$, it could be turned into a proof of $\\neg p$. This would imply :$p\\land\\neg p$ which is impossible. Thus it is not possible to prove $\\neg q$. That means in this case we only know :$\\neg \\neg q$. Category:Proof Techniques dfilcz1dhbzddxx88jkwh9kofb4jzyy"}
+{"_id": "32944", "title": "Hyperbolic Sine of Complex Number", "text": "Hyperbolic Sine of Complex Number 0 70971 365755 2018-09-15T09:06:14Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> Let $a$ and $b$ be real numbers.  Let $i$ be the imaginary unit.  Then: :$\\sinh \\paren {a...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\\sinh \\paren {a + b i} = \\sinh a \\cos b + i \\cosh a \\sin b$ where: :$\\sin$ denotes the real sine function :$\\cos$ denotes the real cosine function :$\\sinh$ denotes the hyperbolic sine function :$\\cosh$ denotes the hyperbolic cosine function. </onlyinclude> == Proof 1 == {{:Hyperbolic Sine of Complex Number/Proof 1}} == Proof 2 == {{:Hyperbolic Sine of Complex Number/Proof 2}} == Also see == * Hyperbolic Cosine of Complex Number * Hyperbolic Tangent of Complex Number * Hyperbolic Cosecant of Complex Number * Hyperbolic Secant of Complex Number * Hyperbolic Cotangent of Complex Number Category:Hyperbolic Sine Function Category:Complex Numbers Category:Hyperbolic Sine of Complex Number a6nti2dzyb53lz3n90ykjnnj4vsqqjc"}
+{"_id": "32945", "title": "Hyperbolic Cotangent of Complex Number/Formulation 1", "text": "Hyperbolic Cotangent of Complex Number/Formulation 1 0 70986 365788 2018-09-15T10:49:12Z Prime.mover 59 Created page with \"== Theorem ==  Let $a$ and $b$ be real numbers.  Let $i$ be the imaginary unit.   Then: <onlyinclude> :$\\coth \\paren {...\" wikitext text/x-wiki == Theorem == Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: <onlyinclude> :$\\coth \\paren {a + b i} = \\dfrac {\\cosh a \\cos b + i \\sinh a \\sin b} {\\sinh a \\cos b + i \\cosh a \\sin b}$ </onlyinclude> where: :$\\coth$ denotes the hyperbolic cotangent function :$\\sin$ denotes the real sine function :$\\cos$ denotes the real cosine function :$\\sinh$ denotes the hyperbolic sine function :$\\cosh$ denotes the hyperbolic cosine function. == Proof == {{begin-eqn}} {{eqn | l = \\coth \\paren {a + b i}       | r = \\frac {\\cosh \\paren {a + b i} } {\\sinh \\paren {a + b i} }       | c = {{Defof|Hyperbolic Cotangent}} }} {{eqn | r = \\dfrac {\\cosh a \\cos b + i \\sinh a \\sin b} {\\sinh a \\cos b + i \\cosh a \\sin b}       | c = Hyperbolic Sine of Complex Number and Hyperbolic Cosine of Complex Number }} {{end-eqn}} {{qed}} == Also see == * Hyperbolic Sine of Complex Number * Hyperbolic Cosine of Complex Number * Hyperbolic Tangent of Complex Number * Hyperbolic Cosecant of Complex Number * Hyperbolic Secant of Complex Number Category:Hyperbolic Cotangent of Complex Number dpmhmf06af3ewir8mn3uwbn7sinxmtn"}
+{"_id": "32946", "title": "Cardinality/Examples/0 less than x less than 6", "text": "Cardinality/Examples/0 less than x less than 6 0 71072 366321 366318 2018-09-18T07:13:08Z Prime.mover 59 wikitext text/x-wiki == Example of Cardinality == Let $S_2 = \\set {x \\in \\Z: 0 < x < 6}$. <onlyinclude> The cardinality of $S_2$ is given by: :$\\card {S_2} = 5$ </onlyinclude> == Proof == The elements of $S_2$, by definition, are those integers greater than $0$ and less than $6$. That is: :$S_2 = \\set {1, 2, 3, 4, 5}$ Thus $S_2$ has $5$ elements: $1, 2, 3, 4, 5$. Hence the result by definition of cardinality. {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Cardinality/Examples/-1,0,1|next = Cardinality/Examples/x^2-x}}: Chapter $1$: Sets and Logic: Exercise $4$ Category:Examples of Cardinality 2pb843cw778el57i367ci63qz0be32w"}
+{"_id": "32947", "title": "Fermat's Little Theorem/Corollary 3", "text": "Fermat's Little Theorem/Corollary 3 0 71091 392010 366591 2019-02-10T22:10:48Z Prime.mover 59 wikitext text/x-wiki {{Proofread}} == Corollary to Fermat's Little Theorem == <onlyinclude> Let $p^k$ be a prime power for some prime number $p$ and $k \\in \\Z_{\\gt 0}$. Then: :$\\forall n \\in \\Z_{\\gt 0}: n^{p^k} \\equiv n \\pmod p$ </onlyinclude> == Proof == The proof proceeds by induction. For all $k \\in \\Z_{\\ge 1}$, let $P \\paren {k}$ be the proposition: :$\\forall n \\in \\Z_{\\gt 0}: n^{p^k} \\equiv n \\pmod p$ === Basis for the Induction === $P \\paren {1}$ is the case: :$\\forall n \\in \\Z_{\\gt 0}: n^p \\equiv n \\pmod p$ which follows from the corollary 1 to Fermat's Little Theorem. This is the basis for the induction. === Induction Hypothesis === Now it needs to be shown that, if $P \\paren{k-1}$ is true, where $k \\ge 2$, then it logically follows that $P \\paren {k}$ is true. So this is the induction hypothesis: :$\\forall n \\in \\Z_{\\gt 0}: n^{p^{k - 1} } \\equiv n \\pmod p$ from which it is to be shown that: :$\\forall n \\in \\Z_{\\gt 0}: n^{p^k} \\equiv n \\pmod p$ === Induction Step === This is the induction step: For any $n \\in \\Z_{\\gt 0}$ then: {{begin-eqn}} {{eqn | l = n^{p^k}       | r = \\paren {n^{p^{k - 1} } }^p }} {{eqn | r = n^{p^{k - 1} } \\pmod p       | o = \\equiv       | c = Corollary 1 to Fermat's Little Theorem }} {{eqn | r = n \\pmod p       | o = \\equiv       | c = Induction Hypothesis }} {{end-eqn}} {{qed}} == Sources == Category:Fermat's Little Theorem 84bvsuw33h9sxbgpby1ep4ig4816wh1"}
+{"_id": "32948", "title": "Fermat's Little Theorem/Corollary 4", "text": "Fermat's Little Theorem/Corollary 4 0 71092 392011 366414 2019-02-10T22:11:15Z Prime.mover 59 wikitext text/x-wiki {{Proofread}} == Corollary to Fermat's Little Theorem == <onlyinclude> Let $p^k$ be a prime power for some prime number $p$ and $k \\in \\Z_{\\gt 0}$. Let $n \\in \\Z_{\\gt 0}$ with $p \\nmid n$. Then: :$n^{p^k - 1} \\equiv 1 \\pmod p$ </onlyinclude> == Proof == By corollary 3 of Fermat's Little Theorem: :$n^{p^k} \\equiv n \\pmod p$ That is: :$p \\divides \\paren {n^{p^k} - n} = n \\paren {n^{p^k - 1} - 1}$ Since $p \\nmid n$, by Corollary to Divisors of Product of Coprime Integers: :$p \\divides \\paren {n^{p^k - 1} - 1}$ That is: :$n^{p^k-1} \\equiv 1 \\pmod p$ {{qed}} == Sources == Category:Fermat's Little Theorem q0leb0mj7psa4cwwhoy40z4g2khedcq"}
+{"_id": "32949", "title": "Total Number of Set Partitions/Examples", "text": "Total Number of Set Partitions/Examples 0 71112 366790 366778 2018-09-22T08:10:30Z Prime.mover 59 wikitext text/x-wiki == Examples of Total Number of Set Partitions == <onlyinclude> === Example: $\\card S = 2$ === {{:Total Number of Set Partitions/Examples/2}} === Example: $\\card S = 3$ === {{:Total Number of Set Partitions/Examples/3}} === Example: $\\card S = 4$ === {{:Total Number of Set Partitions/Examples/4}}</onlyinclude> Category:Examples of Set Partitions to3rp6r8nn76o1kly0i4xiycq66lq29"}
+{"_id": "32950", "title": "1", "text": "1 0 71115 478436 478434 2020-07-16T10:43:54Z Prime.mover 59 wikitext text/x-wiki {{NumberPageLink|prev = 0|next = 2}} == Number == $1$ ('''one''') is: :The immediate successor element of zero in the set of natural numbers $\\N$ :The only (strictly) positive integer which is neither prime nor composite :The only (strictly) positive integer which is a divisor of every integer === $0$th Term === :The $0$th (zeroth) power of every non-non-zero number: ::$\\forall n: n \\ne 0 \\implies n^1 = 1$ :The $0$th term of Göbel's sequence, by definition :The $0$th term of the $3$-Göbel sequence, by definition :The $0$th and $1$st Catalan numbers: ::$1 = \\dfrac 1 {0 + 1} \\dbinom {2 \\times 0} 1 = \\dfrac 1 1 \\times 1$ ::$1 = \\dfrac 1 {1 + 1} \\dbinom {2 \\times 1} 1 = \\dfrac 1 2 \\times 2$ :The $0$th and $1$st Bell numbers === $1$st Term === :The $1$st (strictly) positive integer :The $1$st (strictly) positive integer :The $1$st (positive) odd number ::$1 = 0 \\times 2 + 1$ :The $1$st number to be both square and triangular: ::$1 = 1^2 = \\dfrac {1 \\times \\paren {1 + 1}} 2$ :The $1$st square number to be the $\\sigma$ (sigma) value of some (strictly) positive integer: ::$1 = \\map \\sigma 1$ :The $1$st generalized pentagonal number: ::$1 = \\dfrac {1 \\paren {3 \\times 1 - 1} } 2$ :The $1$st highly composite number: ::$\\map \\tau 1 = 1$ :The $1$st special highly composite number :The $1$st highly abundant number: ::$\\map \\sigma 1 = 1$ :The $1$st superabundant number: ::$\\dfrac {\\map \\sigma 1} 1 = \\dfrac 1 1 = 1$ :The $1$st almost perfect number: ::$\\map \\sigma 1 = 1 = 2 - 1$ :The $1$st factorial: ::$1 = 1!$ :The $1$st superfactorial: ::$1 = 1\\$ = 1!$ :The $1$st Lucas number after the zeroth $(2)$ :The $1$st Ulam number :The $1$st (strictly) positive integer which cannot be expressed as the sum of exactly $5$ non-zero squares :The $1$st of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes :The $1$st of the $5$ known powers of $2$ whose digits are also all powers of $2$ :The $1$st factorion base $10$: ::$1 = 1!$ :The $1$st of the trivial $1$-digit pluperfect digital invariants: ::$1^1 = 1$ :The $1$st of the $1$st pair of consecutive integers whose product is a primorial: ::$1 \\times 2 = 2 = 2 \\#$ :The $1$st of the (trivial $1$-digit) Zuckerman numbers: ::$1 = 1 \\times 1$ :The $1$st of the (trivial $1$-digit) harshad numbers: ::$1 = 1 \\times 1$ :The $1$st positive integer whose cube is palindromic (in this case trivially): ::$1^3 = 1$ :The $1$st lucky number :The $1$st palindromic lucky number :The $1$st Stern number :The $1$st Cullen number: ::$1 = 0 \\times 2^0 + 1$ :The $1$st number whose $\\sigma$ value is square: ::$\\map \\sigma 1 = 1 = 1^2$ :The $1$st positive integer after $1$ of which the product of its Euler $\\phi$ function and its $\\tau$ function equals its $\\sigma$ function: ::$\\map \\phi 1 \\map \\tau 1 = 1 \\times 1 = 1 = \\map \\sigma 1$ :The $1$st positive integer solution to $\\map \\phi n = \\map \\phi {n + 1}$: ::$\\map \\phi 1 = 1 = \\map \\phi 2$ :The $1$st element of the Fermat set :The $1$st integer $n$ with the property that $\\map \\tau n \\divides \\map \\phi n \\divides \\map \\sigma n$: ::$\\map \\tau 1 = 1$, $\\map \\phi 1 = 1$, $\\map \\sigma 1 = 1$ :The $1$st Lucas number which is also triangular :The $1$st tetrahedral number: ::$1 = \\dfrac {1 \\left({1 + 1}\\right) \\left({1 + 2}\\right)} 6$ :The $1$st of the $3$ tetrahedral numbers which are also square :The $1$st trimorphic number: ::$1^3 = \\mathbf 1$ :The $1$st powerful number (vacuously) :The $1$st integer which equals the number of digits in its factorial: ::$1! = 1$ :which has $1$ digit :The $1$st power of $2$ which is the sum of distinct powers of $3$: ::$1 = 2^0 = 3^0$ :The $1$st square which has no more than $2$ distinct digits :The $1$st pentagonal number: ::$1 = \\dfrac {1 \\left({1 \\times 3 - 1}\\right)} 2$ :The $1$st pentagonal number which is also palindromic: ::$1 = \\dfrac {1 \\left({1 \\times 3 - 1}\\right)} 2$ :The $1$st square pyramidal number: ::$1 = \\dfrac {1 \\paren {1 + 1} \\paren {2 \\times 1 + 1} } 6$ :The $1$st pentatope number: ::$1 = \\dfrac {1 \\paren {1 + 1} \\paren {1 + 2} \\paren {1 + 3} } {24}$ :The $1$st automorphic number: ::$1^2 = \\mathbf 1$ :The $1$st number such that $2 n^2 - 1$ is square: ::$2 \\times 1^2 - 1 = 2 \\times 1 - 1 = 1 = 1^2$ {{WIP|Find out what sequence this belongs to, because it has not been set up on the $5$ page}} :The $1$st Ore number: ::$\\dfrac {1 \\times \\map \\tau 1} {\\map \\sigma 1} = 1$ :and the $1$st whose divisors also have an arithmetic mean which is an integer: ::$\\dfrac {\\map \\sigma 1} {\\map \\tau 1} = 1$ :The $1$st hexagonal number: ::$1 = 1 \\paren {2 \\times 1 - 1}$ :The $1$st pentagonal pyramidal number: ::$1 = \\dfrac {1^2 \\paren {1 + 1} } 2$ :The $1$st heptagonal number: ::$1 = \\dfrac {1 \\paren {5 \\times 1 - 3} } 2$ :The $1$st centered hexagonal number: ::$1 = 1^3 - 0^3$ :The $1$st hexagonal pyramidal number: :The $1$st Woodall number: ::$1 = 1 \\times 2^1 - 1$ :The $1$st happy number: ::$1 \\to 1^2 = 1$ :The $1$st positive integer the sum of whose divisors is a cube: ::$\\map \\sigma 1 = 1 = 1^3$ :The $1$st cube number: ::$1 = 1^3$ :The $1$st of the only two cubic Fibonacci numbers :The $1$st octagonal number: ::$1 = 1 \\paren {3 \\times 1 - 2}$ :The $1$st heptagonal pyramidal number: ::$1 = \\dfrac {1 \\paren {1 + 1} \\paren {5 \\times 1 - 2} } 6$ :The $1$st Kaprekar triple: ::$1^3 = 1 \\to 0 + 0 + 1 = 1$ :The $1$st palindromic cube: ::$1 = 1^3$ :The $1$st Kaprekar number: ::$1^2 = 01 \\to 0 + 1 = 1$ :The $1$st number whose square has a $\\sigma$ value which is itself square: ::$\\map \\sigma 1 = 1 = 1^2$ :The $1$st of the $5$ tetrahedral numbers which are also triangular :The $1$st positive integer which cannot be expressed as the sum of a square and a prime :The $1$st positive integer such that all smaller positive integers coprime to it are prime :The (trivial) $1$st repunit :The $1$st fourth power: ::$1 = 1 \\times 1 \\times 1 \\times 1$ :The $1$st integer $m$ whose cube can be expressed (trivially) as the sum of $m$ consecutive squares: ::$1^3 = \\displaystyle \\sum_{k \\mathop = 1}^1 \\left({0 + k}\\right)^2$ :The $1$st and $2$nd Fibonacci numbers after the zeroth ($0$): ::$1 = 0 + 1$ :The $1$st positive integer whose $\\sigma$ value of its Euler $\\phi$ value equals its $\\sigma$ value: ::$\\map \\sigma {\\map \\phi 1} = \\map \\sigma 1 = 1 = \\map \\sigma 1$ :The $1$st square pyramorphic number: ::$1 = \\displaystyle \\sum_{k \\mathop = 1}^1 k^2 = \\dfrac {1 \\paren {1 + 1} \\paren {2 \\times 1 + 1} } 6$ :The $1$st of the $4$ square pyramidal numbers which are also triangular :The $1$st Wonderful Demlo number :The $1$st obstinate number :The index of the $1$st Cullen prime: ::$1 \\times 2^1 + 1 = 3$ :The index of the $1$st Mersenne number which {{AuthorRef|Marin Mersenne}} asserted to be prime ::($1$ itself was classified as a prime number in those days) :The number of different representations of $1$ as the sum of $1$ unit fractions (degenerate case) :The $1$st centered hexagonal number which is also square :The $1$st pentagonal number which is also triangular: ::$1 = \\dfrac {1 \\paren {3 \\times 1 - 1} } 2 = \\dfrac {1 \\times \\paren {1 + 1} } 2$ :The $1$st odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime :The $1$st odd number which cannot be expressed as the sum of an integer power and a prime number :The number of distinct free monominoes === $2$nd Term === :The $2$nd after $0$ of the $5$ Fibonacci numbers which are also triangular :The $2$nd palindromic triangular number after $0$ :The $2$nd integer $n$ after $0$ such that $2^n$ contains no zero in its decimal representation: ::$2^1 = 2$ :The $2$nd integer $n$ after $0$ such that $5^n$ contains no zero in its decimal representation: ::$5^1 = 5$ :The $2$nd integer $n$ after $0$ such that both $2^n$ and $5^n$ have no zeroes: ::$2^1 = 2, 5^1 = 5$ :The $2$nd palindromic integer which is the index of a palindromic triangular number after $0$: ::$T_1 = 1$ :The $1$st palindromic integer after $0$ whose square is also palindromic integer ::$1^2 = 1$ :The $2$nd Dudeney number after $0$: ::$1^3 = 1$ :The $2$nd number after $0$ which is (trivially) the sum of the increasing powers of its digits taken in order: ::$1^1 = 1$ :The $2$nd non-negative integer $n$ after $0$ such that the Fibonacci number $F_n$ ends in $n$ :The $2$nd after $0$ of the $3$ Fibonacci numbers which equals its index :The $2$nd subfactorial after $0$: ::$1 = 2! \\paren {1 - \\dfrac 1 {1!} + \\dfrac 1 {2!} }$ :The $2$nd integer $m$ after $0$ such that $m^2 = \\dbinom n 0 + \\dbinom n 1 + \\dbinom n 2 + \\dbinom n 3$ for integer $n$: ::$1^2 = \\dbinom 0 0 + \\dbinom 0 1 + \\dbinom 0 2 + \\dbinom 0 3$ :The $2$nd integer $m$ after $0$ such that $m! + 1$ (its factorial plus $1$) is prime: ::$1! + 1 = 1 + 1 = 2$ :The $2$nd integer after $0$ such that its  double factorial plus $1$ is prime: ::$1!! + 1 = 2$ :The $2$nd integer after $0$ which is palindromic in both decimal and binary: ::$1_{10} = 1_2$ :The $2$nd integer after $0$ which is palindromic in both decimal and ternary: ::$1_{10} = 1_3$ :The $2$nd Ramanujan-Nagell number after $0$: ::$1 = 2^1 - 1 = \\dfrac {1 \\left({1 + 1}\\right)} 2$ :The number of different representations of $1$ as the sum of $2$ unit fractions: ::$1 = \\dfrac 1 2 + \\dfrac 1 2$ :The number of distinct free dominoes === Miscellaneous === :The total of all the entries in the trivial magic square of order $1$: ::$1 = \\displaystyle \\sum_{k \\mathop = 1}^{1^2} k = \\dfrac {1^2 \\paren {1^2 + 1} } 2$ :The total of all the entries in the trivial magic cube of order $1$: ::$1 = \\displaystyle \\sum_{k \\mathop = 1}^{1^3} k = \\dfrac {1^3 \\paren {1^3 + 1} } 2$ :The magic constant of the trivial magic square of order $1$: ::$1 = \\displaystyle \\dfrac 1 1 \\sum_{k \\mathop = 1}^{1^2} k = \\dfrac {1 \\paren {1^2 + 1} } 2$ :The magic constant of the trivial magic cube of order $1$: ::$1 = \\displaystyle \\dfrac 1 {1^2} \\sum_{k \\mathop = 1}^{1^3} k = \\dfrac {1 \\paren {1^3 + 1} } 2$ {{ArithmeticFunctionTable|n = 1|tau = 1|phi = 1|sigma = 1}} == Also see == * Definition:Unity * One is not Prime * Divisors of One * One Divides all Integers === Previous in sequence: $0$ === ==== Next in sequence: $2$ ==== * {{NumberPageLink|prev = 0|next = 2|type = Subfactorial|cat = Subfactorials}} * {{NumberPageLink|prev = 0|next = 2|result = Sequence of Integers whose Factorial plus 1 is Prime}} * {{NumberPageLink|prev = 0|next = 2|result = Prime Values of Double Factorial plus 1}} * {{NumberPageLink|prev = 0|next = 2|type = Fibonacci Number|cat = Fibonacci Numbers}} * {{NumberPageLink|prev = 0|next = 2|result = Powers of 2 with no Zero in Decimal Representation}} * {{NumberPageLink|prev = 0|next = 2|result = Powers of 5 with no Zero in Decimal Representation}} * {{NumberPageLink|prev = 0|next = 2|result = Powers of 2 and 5 without Zeroes}} * {{NumberPageLink|prev = 0|next = 2|result = Palindromes in Base 10 and Base 3}} * {{NumberPageLink|prev = 0|next = 2|result = Numbers which are Sum of Increasing Powers of Digits}} * {{NumberPageLink|prev = 0|next = 2|result = Square Formed from Sum of 4 Consecutive Binomial Coefficients}} * {{NumberPageLink|prev = 0|next = 2|result = Palindromic Indices of Palindromic Triangular Numbers}} * {{NumberPageLink|prev = 0|next = 2|result = Square of Small-Digit Palindromic Number is Palindromic}} ==== Next in sequence: $3$ ==== * {{NumberPageLink|prev = 0|next = 3|type = Triangular Number|cat = Triangular Numbers}} * {{NumberPageLink|prev = 0|next = 3|result = Palindromes in Base 10 and Base 2}} * {{NumberPageLink|prev = 0|next = 3|result = Triangular Fibonacci Numbers}} * {{NumberPageLink|prev = 0|next = 3|result = Palindromic Triangular Numbers}} * {{NumberPageLink|prev = 0|next = 3|type = Ramanujan-Nagell Number|cat = Ramanujan-Nagell Numbers}} ==== Next in sequence: $5$ ==== * {{NumberPageLink|prev = 0|next = 5|result = Fibonacci Numbers which equal their Index}} * {{NumberPageLink|prev = 0|next = 5|result = Sequence of Fibonacci Numbers ending in Index}} ==== Next in sequence: $8$ ==== * {{NumberPageLink|prev = 0|next = 8|type = Dudeney Number|cat = Dudeney Numbers}} === Previous in sequence: $2$ === * {{NumberPageLink|prev = 2|next = 3|type = Lucas Number|cat = Lucas Numbers}} === Next in sequence: $2$ === * {{NumberPageLink|next = 2|type = Generalized Pentagonal Number|cat = Generalized Pentagonal Numbers}} * {{NumberPageLink|next = 2|type = Highly Composite Number|cat = Highly Composite Numbers}} * {{NumberPageLink|next = 2|type = Special Highly Composite Number|cat = Special Highly Composite Numbers}} * {{NumberPageLink|next = 2|type = Highly Abundant Number|cat = Highly Abundant Numbers}} * {{NumberPageLink|next = 2|type = Superabundant Number|cat = Superabundant Numbers}} * {{NumberPageLink|next = 2|type = Almost Perfect Number|cat = Almost Perfect Numbers}} * {{NumberPageLink|next = 2|type = Factorial|cat = Factorials}} * {{NumberPageLink|next = 2|type = Superfactorial|cat = Superfactorials}} * {{NumberPageLink|next = 2|type = Fibonacci Number|cat = Fibonacci Numbers}} * {{NumberPageLink|next = 2|type = Catalan Number|cat = Catalan Numbers}} * {{NumberPageLink|next = 2|type = Ulam Number|cat = Ulam Numbers}} * {{NumberPageLink|next = 2|result = Sequence of Powers of 2|cat = Powers of 2}} * {{NumberPageLink|next = 2|result = Powers of 2 with no Zero in Decimal Representation}} * {{NumberPageLink|next = 2|result = Powers of 2 and 5 without Zeroes}} * {{NumberPageLink|next = 2|result = Integer not Expressible as Sum of 5 Non-Zero Squares}} * {{NumberPageLink|next = 2|result = Integers such that all Coprime and Less are Prime}} * {{NumberPageLink|next = 2|type = Göbel's Sequence|cat = Göbel's Sequence}} * {{NumberPageLink|next = 2|type = 3-Göbel Sequence|cat = Göbel's Sequence}} * {{NumberPageLink|next = 2|result = Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes}} * {{NumberPageLink|next = 2|result = Powers of 2 whose Digits are Powers of 2}} * {{NumberPageLink|next = 2|type = Pluperfect Digital Invariant|cat = Pluperfect Digital Invariants}} * {{NumberPageLink|next = 2|result = Factorions Base 10|cat = Factorions}} * {{NumberPageLink|next = 2|result = Consecutive Integers whose Product is Primorial}} * {{NumberPageLink|next = 2|type = Zuckerman Number|cat = Zuckerman Numbers}} * {{NumberPageLink|next = 2|type = Harshad Number|cat = Harshad Numbers}} * {{NumberPageLink|next = 2|result = Square Formed from Sum of 4 Consecutive Binomial Coefficients}} * {{NumberPageLink|next = 2|type = Bell Number|cat = Bell Numbers}} * {{NumberPageLink|next = 2|result = Sequence of Integers whose Cube is Palindromic}} * {{NumberPageLink|next = 2|result = Number of Free Polyominoes}} * {{NumberPageLink|next = 2|type = Mersenne Prime/Historical Note|cat = Mersenne's Assertion}} === Next in sequence: $3$ === * {{NumberPageLink|next = 3|result = Sequence of Powers of 3|cat = Powers of 3}} * {{NumberPageLink|next = 3|type = Lucky Number|cat = Lucky Numbers}} * {{NumberPageLink|next = 3|result = Sequence of Palindromic Lucky Numbers}} * {{NumberPageLink|next = 3|type = Stern Number|cat = Stern Numbers}} * {{NumberPageLink|next = 3|type = Cullen Number|cat = Cullen Numbers}} * {{NumberPageLink|next = 3|result = Numbers whose Sigma is Square|cat = Numbers whose Sigma is Square}} * {{NumberPageLink|next = 3||result = Integers whose Phi times Tau equal Sigma}} * {{NumberPageLink|next = 3|result = Consecutive Integers with Same Euler Phi Value}} * {{NumberPageLink|next = 3|type = Fermat Set}} * {{NumberPageLink|next = 3|result = Numbers such that Tau divides Phi divides Sigma}} * {{NumberPageLink|next = 3|result = Triangular Lucas Numbers}} * {{NumberPageLink|next = 3|result = Representation of 1 as Sum of n Unit Fractions}} * {{NumberPageLink|next = 3|result = Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares}} === Next in sequence: $4$ === * {{NumberPageLink|next = 4|result = Sequence of Powers of 4|cat = Powers of 4}} * {{NumberPageLink|next = 4|type = Tetrahedral Number|cat = Tetrahedral Numbers}} * {{NumberPageLink|next = 4|result = Square and Tetrahedral Numbers}} * {{NumberPageLink|next = 4|type = Trimorphic Number|cat = Trimorphic Numbers}} * {{NumberPageLink|next = 4|type = Powerful Number|cat = Powerful Numbers}} * {{NumberPageLink|next = 4|result = Powers of 2 which are Sum of Distinct Powers of 3}} * {{NumberPageLink|next = 4|result = Squares with No More than 2 Distinct Digits}} === Next in sequence: $5$ === * {{NumberPageLink|next = 5|result = Sequence of Powers of 5|cat = Powers of 5}} * {{NumberPageLink|next = 5|type = Pentagonal Number|cat = Pentagonal Numbers}} * {{NumberPageLink|next = 5|result = Sequence of Palindromic Pentagonal Numbers}} * {{NumberPageLink|next = 5|type = Square Pyramidal Number|cat = Pyramidal Numbers}} * {{NumberPageLink|next = 5|type = Pentatope Number|cat = Pentatope Numbers}} * {{NumberPageLink|next = 5|type = Automorphic Number|cat = Automorphic Numbers}} * {{NumberPageLink|next = 5|result = Magic Constant of Magic Square}} * {{NumberPageLink|next = 5|result = Odd Numbers not Sum of Prime and Power}} === Next in sequence: $6$ === * {{NumberPageLink|next = 6|result = Sequence of Powers of 6|cat = Powers of 6}} * {{NumberPageLink|next = 6|type = Ore Number|cat = Ore Numbers}} * {{NumberPageLink|next = 6|result = Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors}} * {{NumberPageLink|next = 6|type = Hexagonal Number|cat = Hexagonal Numbers}} * {{NumberPageLink|next = 6|type = Pentagonal Pyramidal Number|cat = Pyramidal Numbers}} === Next in sequence: $7$ === * {{NumberPageLink|next = 7|cat = Powers of 7|result = Sequence of Powers of 7}} * {{NumberPageLink|next = 7|type = Centered Hexagonal Number|cat = Centered Hexagonal Numbers}} * {{NumberPageLink|next = 7|type = Hexagonal Pyramidal Number|cat = Pyramidal Numbers}} * {{NumberPageLink|next = 7|type = Heptagonal Number|cat = Heptagonal Numbers}} * {{NumberPageLink|next = 7|type = Woodall Number|cat = Woodall Numbers}} * {{NumberPageLink|next = 7|type = Happy Number|cat = Happy Numbers}} * {{NumberPageLink|next = 7|result = Integers whose Sigma Value is Cube}} === Next in sequence: $8$ === * {{NumberPageLink|next = 8|result = Sequence of Powers of 8|cat = Powers of 8}} * {{NumberPageLink|next = 8|type = Cube Number|cat = Cube Numbers}} * {{NumberPageLink|next = 8|result = Cubic Fibonacci Numbers}} * {{NumberPageLink|next = 8|type = Octagonal Number|cat = Octagonal Numbers}} * {{NumberPageLink|next = 8|type = Heptagonal Pyramidal Number|cat = Pyramidal Numbers}} * {{NumberPageLink|next = 8|type = Kaprekar Triple|cat = Kaprekar Numbers}} * {{NumberPageLink|next = 8|result = Sequence of Palindromic Cubes}} === Next in sequence: $9$ === * {{NumberPageLink|next = 9|result = Sequence of Powers of 9|cat = Powers of 9}} * {{NumberPageLink|next = 9|type = Kaprekar Number|cat = Kaprekar Numbers}} * {{NumberPageLink|next = 9|result = Square Numbers whose Sigma is Square|cat = Square Numbers whose Sigma is Square}} * {{NumberPageLink|next = 9|result = Magic Constant of Magic Cube}} === Next in sequence: $10$ === * {{NumberPageLink|next = 10|result = Sum of Terms of Magic Square}} * {{NumberPageLink|next = 10|cat = Powers of 10|result = Sequence of Powers of 10}} * {{NumberPageLink|next = 10|result = Tetrahedral and Triangular Numbers}} * {{NumberPageLink|next = 10|result = Numbers not Sum of Square and Prime}} === Next in sequence: $11$ === * {{NumberPageLink|next = 11|result = Sequence of Powers of 11|cat = Powers of 11}} * {{NumberPageLink|next = 11|type = Repunit|cat = Repunits}} === Next in sequence: $12$ and above === * {{NumberPageLink|next = 12|result = Sequence of Powers of 12|cat = Powers of 12}} * {{NumberPageLink|next = 13|result = Sequence of Powers of 13|cat = Powers of 13}} * {{NumberPageLink|next = 14|result = Sequence of Powers of 14|cat = Powers of 14}} * {{NumberPageLink|next = 15|result = Sequence of Powers of 15|cat = Powers of 15}} * {{NumberPageLink|next = 16|type = Fourth Power|cat = Fourth Powers}} * {{NumberPageLink|next = 16|result = Sequence of Powers of 16|cat = Powers of 16}} * {{NumberPageLink|next = 22|result = Numbers Equal to Number of Digits in Factorial}} * {{NumberPageLink|next = 36|result = Integer both Square and Triangular}} * {{NumberPageLink|next = 36|result = Sum of Terms of Magic Cube}} * {{NumberPageLink|next = 47|result = Numbers whose Cube equals Sum of Sequence of that many Squares|cat = Numbers whose Cube equals Sum of Sequence of that many Squares}} * {{NumberPageLink|next = 55|result = Square Pyramidal and Triangular Numbers}} * {{NumberPageLink|next = 55|type = Square Pyramorphic Number|cat = Square Pyramorphic Numbers}} * {{NumberPageLink|next = 87|result = Integers for which Sigma of Phi equals Sigma}} * {{NumberPageLink|next = 121|type = Wonderful Demlo Number|cat = Wonderful Demlo Numbers}} * {{NumberPageLink|next = 127|type = Obstinate Number|cat = Obstinate Numbers}} * {{NumberPageLink|next = 141|type = Cullen Prime|cat = Cullen Primes}} * {{NumberPageLink|next = 169|result = Sequence of Square Centered Hexagonal Numbers}} * {{NumberPageLink|next = 210|result = Triangular Numbers which are also Pentagonal}} == Historical Note == {{:1/Historical Note}} == Linguistic Note == {{:1/Linguistic Note}} == Sources == * {{BookReference|Les Nombres Remarquables|1983|François Le Lionnais|author2 = Jean Brette|prev = Integral from 0 to 1 of Complete Elliptic Integral of First Kind|next = Trivial Group is Smallest Group}}: $1$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Unit Vector|next = Definition:Universal Quantifier|entry = unity}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Unit Vector|next = Universal Instantiation|entry = unity}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Unit Vector|next = Definition:Universal Gravitational Constant|entry = unity}} Category:Specific Numbers Category:1 f3jrybgjko4vrtj7j704f8zzrwv4hbg"}
+{"_id": "32951", "title": "Total Number of Set Partitions/Examples/2/Illustration", "text": "Total Number of Set Partitions/Examples/2/Illustration 0 71136 366804 2018-09-22T08:51:38Z Prime.mover 59 Created page with \"== Example of Total Number of Set Partitions == <onlyinclude> Let a set $S$ of cardinality $2$ be exemplified by $S = \\set {a...\" wikitext text/x-wiki == Example of Total Number of Set Partitions == <onlyinclude> Let a set $S$ of cardinality $2$ be exemplified by $S = \\set {a, b}$. Then the partitions of $S$ are: :$\\set {a, b}$ :$\\set {a \\mid b}$ </onlyinclude> Category:Examples of Set Partitions m79iolwjmaoknsukrz8pneqzcwq0imf"}
+{"_id": "32952", "title": "Total Number of Set Partitions/Examples/3/Illustration", "text": "Total Number of Set Partitions/Examples/3/Illustration 0 71137 366808 2018-09-22T08:54:10Z Prime.mover 59 Created page with \"== Example of Total Number of Set Partitions == <onlyinclude> Let a set $S$ of cardinality $3$ be exemplified by $S = \\set {a...\" wikitext text/x-wiki == Example of Total Number of Set Partitions == <onlyinclude> Let a set $S$ of cardinality $3$ be exemplified by $S = \\set {a, b, c}$. Then the partitions of $S$ are: :$\\set {a, b, c}$ :$\\set {a, b \\mid c}$ :$\\set {a, c \\mid b}$ :$\\set {b, c \\mid a}$ :$\\set {a \\mid b \\mid c}$ </onlyinclude> Category:Examples of Set Partitions awrrsx8y7vinkak3nia95du5zdfolgp"}
+{"_id": "32953", "title": "Total Number of Set Partitions/Examples/4/Illustration", "text": "Total Number of Set Partitions/Examples/4/Illustration 0 71138 418216 418215 2019-08-15T06:28:33Z Prime.mover 59 wikitext text/x-wiki == Example of Total Number of Set Partitions == <onlyinclude> Let a set $S$ of cardinality $4$ be exemplified by $S = \\set {a, b, c, d}$. Then the partitions of $S$ are: :$\\set {a, b, c, d}$ :$\\set {\\set a, \\set {b, c, d} }$ :$\\set {\\set b, \\set {a, c, d} }$ :$\\set {\\set c, \\set {a, b, d} }$ :$\\set {\\set d, \\set {a, b, c} }$ :$\\set {\\set {a, b}, \\set {c, d} }$ :$\\set {\\set {a, c}, \\set {b, d} }$ :$\\set {\\set {a, d}, \\set {b, c} }$ :$\\set {\\set a, \\set b, \\set {c, d} }$ :$\\set {\\set a, \\set c, \\set {b, d} }$ :$\\set {\\set a, \\set d, \\set {b, c} }$ :$\\set {\\set b, \\set c, \\set {a, d} }$ :$\\set {\\set b, \\set d, \\set {a, c} }$ :$\\set {\\set c, \\set d, \\set {a, b} }$ :$\\set {\\set a, \\set b, \\set c, \\set d}$ </onlyinclude> == Sources == * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth|prev = Definition:Set Partition/Definition 1|next = Set Partition/Examples/Partition into Singletons}}: $\\S 6$. Indexed families; partitions; equivalence relations: Example $6.2$ Category:Examples of Set Partitions g7vdhma5nj19r1y3o9j4rnx5w4mq25j"}
+{"_id": "32954", "title": "Modulo Addition/Cayley Table/Modulo 6", "text": "Modulo Addition/Cayley Table/Modulo 6 0 71211 379077 378169 2018-11-29T13:28:19Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Modulo Addition == The additive group of integers modulo $m$ can be described by showing its Cayley table. This one is for modulo $6$: <onlyinclude> :$\\begin{array}{r|rrrrrr} \\struct {\\Z_6, +_6} & \\eqclass 0 6 & \\eqclass 1 6 & \\eqclass 2 6 & \\eqclass 3 6 & \\eqclass 4 6 & \\eqclass 5 6 \\\\ \\hline \\eqclass 0 6 & \\eqclass 0 6 & \\eqclass 1 6 & \\eqclass 2 6 & \\eqclass 3 6 & \\eqclass 4 6 & \\eqclass 5 6 \\\\ \\eqclass 1 6 & \\eqclass 1 6 & \\eqclass 2 6 & \\eqclass 3 6 & \\eqclass 4 6 & \\eqclass 5 6 & \\eqclass 0 6 \\\\ \\eqclass 2 6 & \\eqclass 2 6 & \\eqclass 3 6 & \\eqclass 4 6 & \\eqclass 5 6 & \\eqclass 0 6 & \\eqclass 1 6 \\\\ \\eqclass 3 6 & \\eqclass 3 6 & \\eqclass 4 6 & \\eqclass 5 6 & \\eqclass 0 6 & \\eqclass 1 6 & \\eqclass 2 6 \\\\ \\eqclass 4 6 & \\eqclass 4 6 & \\eqclass 5 6 & \\eqclass 0 6 & \\eqclass 1 6 & \\eqclass 2 6 & \\eqclass 3 6 \\\\ \\eqclass 5 6 & \\eqclass 5 6 & \\eqclass 0 6 & \\eqclass 1 6 & \\eqclass 2 6 & \\eqclass 3 6 & \\eqclass 4 6 \\\\ \\end{array}$ </onlyinclude> which can also be presented: :$\\begin{array}{r|rrrrrr} +_6 & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline 0 & 0 & 1 & 2 & 3 & 4 & 5  \\\\ 1 & 1 & 2 & 3 & 4 & 5 & 0 \\\\ 2 & 2 & 3 & 4 & 5 & 0 & 1 \\\\ 3 & 3 & 4 & 5 & 0 & 1 & 2 \\\\ 4 & 4 & 5 & 0 & 1 & 2 & 3 \\\\ 5 & 5 & 0 & 1 & 2 & 3 & 4 \\\\ \\end{array}$ == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Symmetric Group on 3 Letters/Cayley Table|next = Alternating Group on 4 Letters/Cayley Table}}: Tables: $2$. Cyclic group of order $6$ * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Definition:Multiplication/Modulo Multiplication/Definition 3|next = Modulo Multiplication/Cayley Table/Modulo 6}}: $\\S 2$: Example $2.3$ Category:Examples of Cayley Tables Category:Groups of Order 6 Category:Examples of Additive Groups of Integers Modulo m hjmtao7z2dr3n1jhme9h0i1x6pjfg32"}
+{"_id": "32955", "title": "Modulo Multiplication/Cayley Table/Modulo 6", "text": "Modulo Multiplication/Cayley Table/Modulo 6 0 71213 397332 370857 2019-03-25T21:58:59Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Modulo Multiplication == The multiplicative monoid of integers modulo $m$ can be described by showing its Cayley table. This one is for modulo $6$: <onlyinclude> :$\\begin{array} {r|rrrrrr} \\struct {\\Z_6, \\times_6} & \\eqclass 0 6 & \\eqclass 1 6 & \\eqclass 2 6 & \\eqclass 3 6 & \\eqclass 4 6 & \\eqclass 5 6 \\\\ \\hline \\eqclass 0 6 & \\eqclass 0 6 & \\eqclass 0 6 & \\eqclass 0 6 & \\eqclass 0 6 & \\eqclass 0 6 & \\eqclass 0 6  \\\\ \\eqclass 1 6 & \\eqclass 0 6 & \\eqclass 1 6 & \\eqclass 2 6 & \\eqclass 3 6 & \\eqclass 4 6 & \\eqclass 5 6 \\\\ \\eqclass 2 6 & \\eqclass 0 6 & \\eqclass 2 6 & \\eqclass 4 6 & \\eqclass 0 6 & \\eqclass 2 6 & \\eqclass 4 6 \\\\ \\eqclass 3 6 & \\eqclass 0 6 & \\eqclass 3 6 & \\eqclass 0 6 & \\eqclass 3 6 & \\eqclass 0 6 & \\eqclass 3 6 \\\\ \\eqclass 4 6 & \\eqclass 0 6 & \\eqclass 4 6 & \\eqclass 2 6 & \\eqclass 0 6 & \\eqclass 4 6 & \\eqclass 2 6 \\\\ \\eqclass 5 6 & \\eqclass 0 6 & \\eqclass 5 6 & \\eqclass 4 6 & \\eqclass 3 6 & \\eqclass 2 6 & \\eqclass 1 6 \\\\ \\end{array}$ which can also be presented: :$\\begin{array} {r|rrrrrr} \\times_6 & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline 0 & 0 & 0 & 0 & 0 & 0 & 0  \\\\ 1 & 0 & 1 & 2 & 3 & 4 & 5 \\\\ 2 & 0 & 2 & 4 & 0 & 2 & 4 \\\\ 3 & 0 & 3 & 0 & 3 & 0 & 3 \\\\ 4 & 0 & 4 & 2 & 0 & 4 & 2 \\\\ 5 & 0 & 5 & 4 & 3 & 2 & 1 \\\\ \\end{array}$ </onlyinclude> == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Modulo Addition/Cayley Table/Modulo 6|next = Modulo Multiplication is Associative/Proof 2}}: $\\S 2$: Example $2.3$ * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Equivalence Relation on Square Matrices induced by Positive Integer Powers|next = Modulo Arithmetic/Examples/Solutions to x^2 = x Modulo 6}}: Chapter $3$: Equivalence Relations and Equivalence Classes: Exercise $7$ Category:Examples of Cayley Tables Category:Modulo Multiplication qdh7ih2623tp3ft6811sln6l30hf97q"}
+{"_id": "32956", "title": "Cauchy Sequences form Ring with Unity/Corollary", "text": "Cauchy Sequences form Ring with Unity/Corollary 0 71225 472816 472815 2020-06-09T06:45:06Z Prime.mover 59 wikitext text/x-wiki == Corollary to Cauchy Sequences form Ring with Unity == <onlyinclude> Let $\\struct {F, +, \\circ, \\norm {\\, \\cdot \\,} }$ be a valued field. Let $\\struct {F^\\N, +, \\circ}$ be the commutative ring of sequences over $F$ with unity $\\tuple {1, 1, 1, \\dotsc}$. {{explain|Has it been proved that $\\struct {F^\\N, +, \\circ}$ is actually a commutative ring?}} Let $\\CC \\subset F^\\N$ be the set of Cauchy sequences on $F$. Then: :$\\struct {\\CC, +, \\circ}$ is a commutative subring of $F^\\N$ with unity $\\tuple {1, 1, 1, \\dotsc}$. </onlyinclude> == Proof == By Cauchy Sequences form Ring with Unity, $\\struct {\\CC, +, \\circ}$ is a subring of $F^\\N$. We have that $\\circ$ is commutative on $F^\\N$. Hence by Restriction of Commutative Operation is Commutative the restriction of $\\circ$ to $\\CC$ is commutative. == Sources == * {{BookReference|p-adic Numbers: An Introduction|1997|Fernando Q. Gouvea}}: $\\S 3.2$: Completions Category:Cauchy Sequences in Normed Division Rings shjh29azsntz6pkub11ghpuvur8rmlw"}
+{"_id": "32957", "title": "Modulo Arithmetic/Examples/Multiplicative Inverse of 41 Modulo 97", "text": "Modulo Arithmetic/Examples/Multiplicative Inverse of 41 Modulo 97 0 71245 367522 367490 2018-09-27T15:38:12Z Prime.mover 59 wikitext text/x-wiki == Example of Modulo Arithmetic == <onlyinclude> The inverse of $41$ under multiplication modulo $97$ is given by: :${\\eqclass {41} {97} }^{-1} = 71$ </onlyinclude> === Solution to $41 x \\equiv 2 \\pmod {97}$ === {{:Modulo Arithmetic/Examples/Multiplicative Inverse of 41 Modulo 97/41x = 2 Modulo 97}} == Proof == From Ring of Integers Modulo Prime is Field, multiplication modulo $97$ has an inverse for all $x \\in \\Z_{97}$ where $x \\ne 0$. Using Euclid's Algorithm: {{begin-eqn}} {{eqn | n = 1       | l = 97       | r = 2 \\times 41 + 15 }} {{eqn | n = 2       | l = 41       | r = 2 \\times 15 + 11 }} {{eqn | n = 3       | l = 15       | r = 11 + 4 }} {{eqn | n = 4       | l = 11       | r = 2 \\times 4 + 3 }} {{eqn | n = 5       | l = 4       | r = 3 + 1 }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = 1       | r = 4 - 3       | c = from $(5)$ }} {{eqn | r = 4 - \\paren {11 - 2 \\times 4}       | c = from $(4)$ }} {{eqn | r = 3 \\times 4 - 11       | c =  }} {{eqn | r = 3 \\times \\paren {15 - 11} - 11       | c = from $(3)$ }} {{eqn | r = 3 \\times 15 - 4 \\times 11       | c =  }} {{eqn | r = 3 \\times 15 - 4 \\times \\paren {41 - 2 \\times 15}       | c = from $(2)$ }} {{eqn | r = 11 \\times 15 - 4 \\times 41       | c =  }} {{eqn | r = 11 \\times \\paren {97 - 2 \\times 14} - 4 \\times 41       | c = from $(1)$ }} {{eqn | r = 11 \\times 97 - 26 \\times 41       | c =  }} {{end-eqn}} So: : $\\paren {-26} \\times 41 \\equiv 1 \\pmod {97}$ {{begin-eqn}} {{eqn | l = \\paren {-26} \\times 41       | o = \\equiv       | r = 1       | rr= \\pmod {97}       | c =  }} {{eqn | l = 71 \\times 41       | o = \\equiv       | r = 1       | rr= \\pmod {97}       | c = as $-26 \\equiv 71 \\pmod {97}$ because $26 + 71 = 97$ }} {{eqn | ll= \\leadsto       | l = {\\eqclass {41} {97} }^{-1}       | r = \\eqclass {71} {97}       | c =  }} {{end-eqn}} Hence the result. {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Modulo Arithmetic/Examples/Solutions to x^2 = x Modulo 6|next = Modulo Arithmetic/Examples/Multiplicative Inverse of 41 Modulo 97/41x = 2 Modulo 97}}: Chapter $3$: Equivalence Relations and Equivalence Classes: Exercise $8$ Category:Examples of Modulo Arithmetic Category:Multiplicative Inverse of 41 Modulo 97 aj7quf3yc5k62f9zgwb5kwrh5za2ik7"}
+{"_id": "32958", "title": "Real Square Function is not Injective", "text": "Real Square Function is not Injective 0 71342 376794 376744 2018-11-15T16:13:56Z Prime.mover 59 wikitext text/x-wiki == Example of Mapping which is Not an Injection == <onlyinclude> Let $f: \\R \\to \\R$ be the real square function: :$\\forall x \\in \\R: \\map f x = x^2$ Then $f$ is not an injection. </onlyinclude> == Proof == For $f$ to be an injection, it would be necessary that: :$\\forall x_1, x_2 \\in \\R: \\map f {x_1} = \\map f {x_2} \\implies x_1 = x_2$ By definition of the squaring operation, we have: :$\\map f x = \\map f {-x}$ But unless $x = 0$ it is not the case that $x = -x$. Hence $f$ is not an injection. {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Definition:Injection/Definition 1 a|next = Injection/Examples/Cube Function}}: $\\S 22$: Injections; bijections; inverse of a bijection Category:Examples of Injections Category:Square Function rnjglt6paz96657wm169cjbl4y054mu"}
+{"_id": "32959", "title": "Image of Mapping/Examples/Image of x^2-4x+5", "text": "Image of Mapping/Examples/Image of x^2-4x+5 0 71455 427644 368749 2019-09-25T15:48:30Z Prime.mover 59 wikitext text/x-wiki == Example of Image of Element under Mapping == <onlyinclude> Let $f: \\R \\to \\R$ be the mapping defined as: :$\\forall x \\in \\R: \\map f x = x^2 - 4 x + 5$ The image of $f$ is the unbounded closed interval: :$\\Img f = \\hointr 1 \\to$ and so $f$ is not a surjection. </onlyinclude> === Graphical Representation of $\\map f x = x^2 - 4 x + 5$ === {{:Image of Mapping/Examples/Image of x^2-4x+5/Graph}} == Proof == By differentiating $x^2 - 4 x + 5$ twice {{WRT|Differentiation}} $x$: :$f' = 2 x - 4$ :$f' = 2 x - 4$ {{begin-eqn}} {{eqn | l = f'       | r = 2 x - 4 }} {{eqn | l = f''       | r = 2 }} {{end-eqn}} Equating $f'$ to $0$, a stationary point is found at $x = 2$. Inspecting the sign of $f''$, it is noted that $f'$ is increasing everywhere. Hence the stationary point at $x = 2$ is a minimum of $\\Img f$. This is the only stationary point, so it can be stated that '''the''' minimum of $f$ occurs at $x = 2$. We have that: :$f \\paren 2 = 2^2 - 4 \\times 2 + 5 = 4 - 8 + 5 = 1$ As $f$ is strictly increasing on $x > 2$ and strictly decreasing on $x < 2$, it is seen that $f$ is unbounded above. Thus: :$\\Img f = \\hointr 1 \\to$ {{qed}} == Also see == * Bijective Restrictions of $\\map f x = x^2 - 4 x + 5$ == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Cantor's Theorem|next = Bijective Restriction/Examples/x^2-4x+5}}: Chapter $4$: Mappings: Exercise $12 \\ \\text{(ii)}$ Category:Examples of Images under Mappings 4c0225x2ed0rbwxsccxhz3tccv0cdge"}
+{"_id": "32960", "title": "Subsemigroup/Examples/2x2 Matrices with One Non-Zero Entry", "text": "Subsemigroup/Examples/2x2 Matrices with One Non-Zero Entry 0 71509 369174 369160 2018-10-05T22:25:59Z Prime.mover 59 wikitext text/x-wiki == Example of Subsemigroup == <onlyinclude> Let $\\struct {S, \\times}$ be the semigroup formed by the set of order $2$ square matrices over the real numbers $R$ under (conventional) matrix multiplication. Let $T$ be the subset of $S$ consisting of the matrices of the form $\\begin{bmatrix} x & 0 \\\\ 0 & 0 \\end{bmatrix}$ for $x \\in \\R$. Then $\\struct {T, \\times}$ is a subsemigroup of $\\struct {S, \\times}$. </onlyinclude> == Proof == From the Subsemigroup Closure Test it is sufficient to demonstrate that $\\struct {T, \\times}$ is closed. Let $A = \\begin{bmatrix} x & 0 \\\\ 0 & 0 \\end{bmatrix}$ and $B = \\begin{bmatrix} y & 0 \\\\ 0 & 0 \\end{bmatrix}$. Then: {{begin-eqn}} {{eqn | l = A B       | r = \\begin{bmatrix} x & 0 \\\\ 0 & 0 \\end{bmatrix} \\begin{bmatrix} y & 0 \\\\ 0 & 0 \\end{bmatrix}       | c =  }} {{eqn | r = \\begin{bmatrix} x y + 0 \\times 0 & x \\times 0 + 0 \\times 0 \\\\ 0 \\times y + 0 \\times 0 & 0 \\times 0 + 0 \\times 0 \\end{bmatrix}       | c = {{Defof|Matrix Product (Conventional)}} }} {{eqn | r = \\begin{bmatrix} x y & 0 \\\\ 0 & 0 \\end{bmatrix}       | c =  }} {{eqn | 0 = \\in       | r = T       | c =  }} {{end-eqn}} Hence the result. {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Subsemigroup Closure Test|next = Identity of Submonoid is not necessarily Identity of Monoid}}: $\\S 32$ Identity element and inverses Category:Examples of Subsemigroups 18jb5pq679bn9stqcgsimhlezkawejn"}
+{"_id": "32961", "title": "Semigroup/Examples/x+y-xy on Integers", "text": "Semigroup/Examples/x+y-xy on Integers 0 71513 369176 369175 2018-10-05T22:31:28Z Prime.mover 59 wikitext text/x-wiki == Example of Semigroup == <onlyinclude> Let $\\circ: \\Z \\times \\Z$ be the operation defined on the integers $\\Z$ as: :$\\forall x, y \\in \\Z: x \\circ y := x + y - x y$ Then $\\struct {\\Z, \\circ}$ is a semigroup. </onlyinclude> == Proof == We have that: :$\\forall x, y \\in \\Z: x \\circ y \\in \\Z$ and so $\\struct {\\Z, \\circ}$ is closed. Now let $x, y, z \\in \\Z$. We have: {{begin-eqn}} {{eqn | l = x \\circ \\paren {y \\circ z}       | r = x + \\paren {y \\circ z} - x \\paren {y \\circ z}       | c = Definition of $\\circ$ }} {{eqn | r = x + \\paren {y + z - y z} - x \\paren {y + z - y z}       | c = Definition of $\\circ$ }} {{eqn | r = x + y + z - y z - x y - x z + x y z       | c =  }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = \\paren {x \\circ y} \\circ z       | r = \\paren {x \\circ y} + z - \\paren {x \\circ y} z       | c = Definition of $\\circ$ }} {{eqn | r = \\paren {x + y - x y} + z - \\paren {x + y - x y} z       | c = Definition of $\\circ$ }} {{eqn | r = x + y - x y + z - x z - y z + x y z       | c =  }} {{end-eqn}} As can be seen by inspection: :$x \\circ \\paren {y \\circ z} = \\paren {x \\circ y} \\circ z$ and so $\\circ$ is associative. The result follows by definition of semigroup. {{qed}} == Also see == * Inclusion-Exclusion Principle (think about why) == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Identity of Submonoid is not necessarily Identity of Monoid|next = Subsemigroup/Examples/x+y-xy on Integers}}: Chapter $5$: Semigroups: Exercise $1$ Category:Examples of Semigroups 29jgsmiq5wfzvoahsghgaqhysfrstze"}
+{"_id": "32962", "title": "Order of Group Element/Examples/Element of Multiplicative Group of Real Numbers", "text": "Order of Group Element/Examples/Element of Multiplicative Group of Real Numbers 0 71674 370495 370432 2018-10-11T19:59:37Z Prime.mover 59 wikitext text/x-wiki == Examples of Order of Group Element == <onlyinclude> Consider the multiplicative group of real numbers $\\struct {\\R_{\\ne 0}, \\times}$. The order of $2$ in $\\struct {\\R_{\\ne 0}, \\times}$ is infinite. </onlyinclude> == Proof == From Real Multiplication Identity is One, the identity of $\\struct {\\R_{\\ne 0}, \\times}$ is $1$. There exists no $n \\in \\Z_{\\ge 0}$ such that $2^n = 1$. Hence the result by definition of infinite order element. {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Definition:Order of Group Element/Also known as|next = Order of Group Element/Examples/Imaginary Unit in Multiplicative Group of Complex Numbers}}: $\\S 38$. Period of an element: Illustrations: $\\text{(i)}$ Category:Examples of Order of Group Elements Category:Multiplicative Group of Real Numbers r8303q13pbsr1eujhsy2zscbynoqbsf"}
+{"_id": "32963", "title": "Order of Group Element/Examples/Imaginary Unit in Multiplicative Group of Complex Numbers", "text": "Order of Group Element/Examples/Imaginary Unit in Multiplicative Group of Complex Numbers 0 71675 370340 370337 2018-10-11T06:38:37Z Prime.mover 59 Prime.mover moved page Order of Group Element/Examples/Gaussian Integer in Multiplicative Group of Complex Numbers to Order of Group Element/Examples/Imaginary Unit in Multiplicative Group of Complex Numbers wikitext text/x-wiki == Examples of Order of Group Element == <onlyinclude> Consider the multiplicative group of complex numbers $\\struct {\\C_{\\ne 0}, \\times}$. The order of $i$ in $\\struct {\\C_{\\ne 0}, \\times}$ is $4$. </onlyinclude> == Proof == We have: {{begin-eqn}} {{eqn | l = i^1       | r = 1       | c =  }} {{eqn | l = i^2       | r = -1       | c =  }} {{eqn | l = i^3       | r = -i       | c =  }} {{eqn | l = i^4       | r = 1       | c =  }} {{end-eqn}} Hence the result by definition of order of group element. {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Order of Group Element/Examples/Element of Multiplicative Group of Real Numbers|next = Identity is Only Group Element of Order 1}}: $\\S 38$. Period of an element: Illustrations: $\\text{(ii)}$ Category:Examples of Order of Group Elements oh68f40teq2kjhn3mrnyhe3jlurecwi"}
+{"_id": "32964", "title": "Quaternion Group/Cayley Table", "text": "Quaternion Group/Cayley Table 0 71742 387867 374905 2019-01-15T13:02:12Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Quaternion Group== The Cayley table for the quaternion group given with the group presentation: :$Q = \\Dic 2 = \\gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ can be presented as: <onlyinclude> :$\\begin{array}{r|rrrrrrrr}       & e     & a     & a^2   & a^3   & b     & a b   & a^2 b & a^3 b \\\\ \\hline e     & e     & a     & a^2   & a^3   & b     & a b   & a^2 b & a^3 b \\\\ a     & a     & a^2   & a^3   & e     & a b   & a^2 b & a^3 b & b     \\\\ a^2   & a^2   & a^3   & e     & a     & a^2 b & a^3 b & b     & a b   \\\\ a^3   & a^3   & e     & a     & a^2   & a^3 b & b     & a b   & a^2 b \\\\ b     & b     & a^3 b & a^2 b & a b   & a^2   & a     & e     & a^3   \\\\ a b   & a b   & b     & a^3 b & a^2 b & a^3   & a^2   & a     & e     \\\\ a^2 b & a^2 b & a b   & b     & a^3 b & e     & a^3   & a^2   & a     \\\\ a^3 b & a^3 b & a^2 b & a b   & b     & a     & e     & a^3   & a^2  \\end{array}$ </onlyinclude> === Coset Decomposition of $\\set {e, a^2}$ === Presenting the above Cayley table with respect to the coset decomposition of the normal subgroup $\\gen a^2$ gives: {{:Quaternion Group/Cayley Table/Coset Decomposition of (e, a^2)}} == Sources == * {{BookReference|Elements of Abstract Algebra|1971|Allan Clark|prev = Definition:Quaternion Group|next = Quaternion Group is Hamiltonian}}: Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\\S 46 \\iota$ Category:Quaternion Group Category:Examples of Cayley Tables lnm51jeg7pcggtfeaiqu9z7kxcezvi6"}
+{"_id": "32965", "title": "Quaternion Group/Complex Matrices/Cayley Table", "text": "Quaternion Group/Complex Matrices/Cayley Table 0 71744 370945 370909 2018-10-13T22:01:19Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Quaternion Group== The Cayley table for the quaternion group: :$Q = \\Dic 2 = \\set {\\mathbf 1, -\\mathbf 1, \\mathbf i, -\\mathbf i, \\mathbf j, -\\mathbf j, \\mathbf k, -\\mathbf k}$ under the operation of conventional matrix multiplication, where: :$\\mathbf 1 = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} \\qquad \\mathbf i = \\begin{bmatrix} i & 0 \\\\ 0 & -i \\end{bmatrix} \\qquad \\mathbf j = \\begin{bmatrix} 0 & 1 \\\\ -1 & 0 \\end{bmatrix} \\qquad \\mathbf k = \\begin{bmatrix} 0 & i \\\\ i & 0 \\end{bmatrix}$ can be presented as: <onlyinclude> :$\\begin{array}{r|rrrrrrrr}       &  \\mathbf 1 &  \\mathbf i & -\\mathbf 1 & -\\mathbf i &  \\mathbf j &  \\mathbf k & -\\mathbf j & -\\mathbf k \\\\ \\hline  \\mathbf 1 &  \\mathbf 1 &  \\mathbf i & -\\mathbf 1 & -\\mathbf i &  \\mathbf j &  \\mathbf k & -\\mathbf j & -\\mathbf k \\\\  \\mathbf i &  \\mathbf i & -\\mathbf 1 & -\\mathbf i &  \\mathbf 1 &  \\mathbf k & -\\mathbf j & -\\mathbf k &  \\mathbf j \\\\ -\\mathbf 1 & -\\mathbf 1 & -\\mathbf i &  \\mathbf 1 &  \\mathbf i & -\\mathbf j & -\\mathbf k &  \\mathbf j &  \\mathbf k \\\\ -\\mathbf i & -\\mathbf i &  \\mathbf 1 &  \\mathbf i & -\\mathbf 1 & -\\mathbf k &  \\mathbf j &  \\mathbf k & -\\mathbf j \\\\  \\mathbf j &  \\mathbf j & -\\mathbf k & -\\mathbf j &  \\mathbf k & -\\mathbf 1 &  \\mathbf i &  \\mathbf 1 & -\\mathbf i \\\\  \\mathbf k &  \\mathbf k &  \\mathbf j & -\\mathbf k & -\\mathbf j & -\\mathbf i & -\\mathbf 1 &  \\mathbf i &  \\mathbf 1 \\\\ -\\mathbf j & -\\mathbf j &  \\mathbf k &  \\mathbf j & -\\mathbf k &  \\mathbf 1 & -\\mathbf i & -\\mathbf 1 &  \\mathbf i \\\\ -\\mathbf k & -\\mathbf k & -\\mathbf j &  \\mathbf k &  \\mathbf j &  \\mathbf i &  \\mathbf 1 & -\\mathbf i & -\\mathbf 1 \\end{array}$ </onlyinclude> == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Order of Power of Group Element/Examples/Powers of Element of Order 20|next = Order of Elements in Quaternion Group}}: Chapter $6$: An Introduction to Groups: Exercise $14$ Category:Quaternion Group Category:Examples of Cayley Tables a8334fcqkltaejdxgcf2c1z0aauwyi2"}
+{"_id": "32966", "title": "Order of Subgroup Product/Lemma", "text": "Order of Subgroup Product/Lemma 0 71989 372771 2018-10-23T14:20:28Z Prime.mover 59 Created page with \"== Lemma for Order of Subgroup Product == <onlyinclude> Let $h_1, h_2 \\in H$.  Then: :$h_1 K = h_2 K$ {{iff}}: :$h_1$ and $h_2$ are in the same Definition:Left Coset|lef...\" wikitext text/x-wiki == Lemma for Order of Subgroup Product == <onlyinclude> Let $h_1, h_2 \\in H$. Then: :$h_1 K = h_2 K$ {{iff}}: :$h_1$ and $h_2$ are in the same left coset of $H \\cap K$. </onlyinclude> == Proof == Let $h_1, h_2 \\in H$. Then: {{begin-eqn}} {{eqn | l = h_1 K       | r = h_2 K       | c =  }} {{eqn | ll= \\iff       | l = h_1^{-1} h_2       | o = \\in       | r = K       | c = Left Cosets are Equal iff Product with Inverse in Subgroup }} {{eqn | ll= \\iff       | l = h_1^{-1} h_2       | o = \\in       | r = H \\cap K       | c = {{Defof|Set Intersection}} }} {{eqn | ll= \\iff       | l = h_1 \\paren {H \\cap K}       | r = h_2 \\paren {H \\cap K}       | c = Left Cosets are Equal iff Product with Inverse in Subgroup }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Intersection of Left Cosets of Subgroups is Left Coset of Intersection|next = Order of Subgroup Product/Proof 2}}: Chapter $7$: Cosets and Lagrange's Theorem: Exercise $10$ Category:Order of Subgroup Product 8ppal07pfp8shxynzdiwus7k1ph9xcu"}
+{"_id": "32967", "title": "Symmetry Group of Regular Hexagon/Group Action on Vertices", "text": "Symmetry Group of Regular Hexagon/Group Action on Vertices 0 72174 444487 374648 2020-01-22T13:40:32Z Prime.mover 59 wikitext text/x-wiki == Group Action of Symmetry Group of Regular Hexagon == Let $\\HH = ABCDEF$ be a regular hexagon. Let $D_6$ denote the symmetry group of $\\HH$. :520px Let $e$ denote the identity mapping Let $\\alpha$ denote rotation of $\\HH$ anticlockwise through $\\dfrac \\pi 3$ radians ($60 \\degrees$) Let $\\beta$ denote reflection of $\\HH$ in the $AD$ axis. <onlyinclude> $D_6$ acts on the vertices of $\\HH$ according to this table: :$\\begin{array}{cccccccccccc}  e & \\alpha & \\alpha^2 & \\alpha^3 & \\alpha^4 & \\alpha^5 & \\beta & \\alpha \\beta & \\alpha^2 \\beta & \\alpha^3 \\beta & \\alpha^4 \\beta & \\alpha^5 \\beta \\\\ \\hline A & B & C & D & E & F & A & B & C & D & E & F \\\\ B & C & D & E & F & A & F & A & B & C & D & E \\\\ C & D & E & F & A & B & E & F & A & B & C & D \\\\ D & E & F & A & B & C & D & E & F & A & B & C \\\\ E & F & A & B & C & D & C & D & E & F & A & B \\\\ F & A & B & C & D & E & B & C & D & E & F & A \\\\ \\end{array}$ </onlyinclude> == Sources == * {{BookReference|Elements of Abstract Algebra|1971|Allan Clark|prev = Definition:Symmetry Group of Regular Hexagon|next = Symmetry Group of Regular Hexagon/Examples/Subgroup that Fixes C}}: Chapter $2$: Subgroups and Cosets: $\\S 35 \\zeta$ Category:Symmetry Group of Regular Hexagon m0iwr36mc45sx35m6unnqrg85efbqc2"}
+{"_id": "32968", "title": "Composition of Distance-Preserving Mappings is Distance-Preserving", "text": "Composition of Distance-Preserving Mappings is Distance-Preserving 0 72279 375364 2018-11-08T10:30:08Z Leigh.Samphier 3031 Created page with \"{{Proofread}} == Theorem == <onlyinclude> Let: : $\\struct{X_1, d_1}$ : $\\struct{X_2, d_2}$ : $\\struct{X_3, d_3}$ be metric spaces.  Let: : $\\phi: \\...\" wikitext text/x-wiki {{Proofread}} == Theorem == <onlyinclude> Let: : $\\struct{X_1, d_1}$ : $\\struct{X_2, d_2}$ : $\\struct{X_3, d_3}$ be metric spaces. Let: : $\\phi: \\struct{X_1, d_1} \\to \\struct{X_2, d_2}$ : $\\psi: \\struct{X_2, d_2} \\to \\struct{X_3, d_3}$ be distance-preserving mappings. Then the composite of $\\phi$ and $\\psi$ is also a distance-preserving mapping. </onlyinclude> == Proof == Let $x,y \\in X_1$ then: {{begin-eqn}} {{eqn|l= d_1 \\paren {x,y}      |r= d_2 \\paren {\\map \\phi x, \\map \\phi y }      |c= $\\phi$ is a distance-preserving mapping }} {{eqn|r= d_3 \\paren {\\map \\psi {\\map \\phi x}, \\map \\psi {\\map \\phi y} }      |c= $\\psi$ is a distance-preserving mapping }} {{eqn|r= d_3 \\paren {\\map {\\psi \\circ \\phi} x, \\map {\\psi \\circ \\phi} y }      |c= Definition of composite mappings }} {{end-eqn}} By the definition of a distance-preserving mapping then $\\psi \\circ \\phi$ is distance-preserving. {{qed}} Category:Metric Spaces hfxs3l40pav13ef7tjwiauyc64ky2dk"}
+{"_id": "32969", "title": "First Sylow Theorem/Corollary", "text": "First Sylow Theorem/Corollary 0 72298 375516 375507 2018-11-09T08:07:40Z Prime.mover 59 wikitext text/x-wiki == Corollary to First Sylow Theorem == <onlyinclude> Let $p$ be a prime number. Let $G$ be a group. Let: :$p^n \\divides \\order G$ where: :$\\order G$ denotes the order of $G$ :$n$ is a positive integer. Then $G$ has at least one subgroup of order $p$. </onlyinclude> == Proof 1 == {{:First Sylow Theorem/Corollary/Proof 1}} == Proof 2 == This result can also be proved directly: {{:First Sylow Theorem/Corollary/Proof 2}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Group does not Necessarily have Subgroup of Order of Divisor of its Order/Proof 1|next = Sylow Theorems/Historical Note}}: $\\S 44$. Some consequences of Lagrange's Theorem Category:Sylow Theorems Category:First Sylow Theorem ney8asccuu12b8eb2wiwnl2i3np323q"}
+{"_id": "32970", "title": "Real Square Function is not Surjective", "text": "Real Square Function is not Surjective 0 72448 376766 376755 2018-11-15T12:35:50Z Prime.mover 59 wikitext text/x-wiki == Example of Mapping which is Not a Surjection == <onlyinclude> Let $f: \\R \\to \\R$ be the real square function: :$\\forall x \\in \\R: \\map f x = x^2$ Then $f$ is not a surjection. </onlyinclude> == Proof == For $f$ to be a surjection, it would be necessary that: :$\\forall y \\in \\R: \\exists x \\in \\R: \\map f x = y$ However from Square of Real Number is Non-Negative: :$\\forall y \\in \\R_{< 0}: \\nexists x \\in \\R: \\map f x = y$ Hence $f$ is not a surjection. {{qed}} Category:Examples of Surjections Category:Square Function jirhg29kr6a3z2p55pv7z9u7rl9en6t"}
+{"_id": "32971", "title": "Monoid/Examples/x+y+xy on Reals", "text": "Monoid/Examples/x+y+xy on Reals 0 72644 377613 377611 2018-11-20T22:53:40Z Prime.mover 59 wikitext text/x-wiki == Example of Monoid == <onlyinclude> Let $\\circ: \\R \\times \\R$ be the operation defined on the real numbers $\\R$ as: :$\\forall x, y \\in \\R: x \\circ y := x + y + x y$ Then $\\struct {\\R, \\circ}$ is a monoid whose identity is $0$. </onlyinclude> == Proof == We have that: :$\\forall x, y \\in \\R: x \\circ y \\in \\R$ and so $\\struct {\\R, \\circ}$ is closed. Now let $x, y, z \\in \\R$. We have: {{begin-eqn}} {{eqn | l = x \\circ \\paren {y \\circ z}       | r = x + \\paren {y \\circ z} + x \\paren {y \\circ z}       | c = Definition of $\\circ$ }} {{eqn | r = x + \\paren {y + z + y z} + x \\paren {y + z + y z}       | c = Definition of $\\circ$ }} {{eqn | r = x + y + z + y z + x y + x z + x y z       | c =  }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = \\paren {x \\circ y} \\circ z       | r = \\paren {x \\circ y} + z + \\paren {x \\circ y} z       | c = Definition of $\\circ$ }} {{eqn | r = \\paren {x + y + x y} + z + \\paren {x + y + x y} z       | c = Definition of $\\circ$ }} {{eqn | r = x + y + x y + z + x z + y z + x y z       | c =  }} {{end-eqn}} As can be seen by inspection: :$x \\circ \\paren {y \\circ z} = \\paren {x \\circ y} \\circ z$ and so $\\circ$ is associative. Then we have: {{begin-eqn}} {{eqn | l = x \\circ 0       | r = x + 0 + x \\times 0       | c = Definition of $\\circ$ }} {{eqn | r = x       | c =  }} {{eqn | r = 0 + x + 0 \\times x       | c =  }} {{eqn | r = 0 \\circ x       | c = Definition of $\\circ$ }} {{end-eqn}} The result follows by definition of monoid. {{qed}} == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Product of Semigroup Element with Right Inverse is Idempotent|next = Group/Examples/x+y+xy over Reals less -1}}: Chapter $4$. Groups: Exercise $11$ Category:Examples of Monoids 8k8qism2j4hs6tzrbri531k0xz0ye4e"}
+{"_id": "32972", "title": "Symmetric Group on 3 Letters/Subgroups", "text": "Symmetric Group on 3 Letters/Subgroups 0 72657 387850 386199 2019-01-15T10:15:05Z Prime.mover 59 wikitext text/x-wiki == Subgroups of the Symmetric Group on $3$ Letters == Let $S_3$ denote the Symmetric Group on $3$ Letters, whose Cayley table is given as: {{:Symmetric Group on 3 Letters/Cayley Table}} <onlyinclude> The subsets of $S_3$ which form subgroups of $S_3$ are: {{begin-eqn}} {{eqn | o =        | r = S_3 }} {{eqn | o =        | r = \\set e }} {{eqn | o =        | r = \\set {e, \\tuple {123}, \\tuple {132} } }} {{eqn | o =        | r = \\set {e, \\tuple {12} } }} {{eqn | o =        | r = \\set {e, \\tuple {13} } }} {{eqn | o =        | r = \\set {e, \\tuple {23} } }} {{end-eqn}} </onlyinclude> == Examples == {{:Symmetric Group on 3 Letters/Subgroups/Examples}} == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Integer Multiples under Addition form Subgroup of Integers|next = Intersection of Subgroups/General Result}}: $\\S 5.2$. Subgroups: Example $93$ * {{BookReference|Problems in Group Theory|1967|John D. Dixon|prev = Symmetric Group on 3 Letters/Group Presentation|next = Quaternion Group/Complex Matrices}}: $1$: Subgroups: Problem $1.1$ * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Conjugate of Subgroup is Subgroup|next = Symmetric Group on 3 Letters/Normal Subgroups}}: Chapter $7$: Normal subgroups and quotient groups: Exercise $2$ Category:Symmetric Group on 3 Letters ox31arglp9woaqb3klh0cjskly338vd"}
+{"_id": "32973", "title": "Alternating Group on 4 Letters", "text": "Alternating Group on 4 Letters 0 72718 378592 378497 2018-11-26T21:28:05Z Prime.mover 59 wikitext text/x-wiki == Group Example == Let $S_4$ denote the symmetric group on $4$ letters. <onlyinclude> The '''alternating group on $4$ letters''' $A_4$ is the kernel of the mapping $\\sgn: S_4 \\to C_2$. </onlyinclude> === Cycle Notation === It can be expressed in the form of permutations given in cycle notation as follows: {{:Alternating Group on 4 Letters/Cycle Notation}} === Cayley Table === The Cayley table of $A_4$ can be written: {{:Alternating Group on 4 Letters/Cayley Table}} == Order of Elements == {{:Alternating Group on 4 Letters/Order of Elements}} == Subgroups == {{:Alternating Group on 4 Letters/Subgroups}} == Normality of Subgroups == {{:Alternating Group on 4 Letters/Normality of Subgroups}} == Conjugacy Classes == {{:Alternating Group on 4 Letters/Conjugacy Classes}} Category:Groups of Order 12 Category:Alternating Group on 4 Letters Category:Examples of Alternating Groups b8c53avi3mg9mezem94330pcr7iayd8"}
+{"_id": "32974", "title": "Alternating Group on 4 Letters/Cayley Table", "text": "Alternating Group on 4 Letters/Cayley Table 0 72721 379079 378599 2018-11-29T13:33:18Z Prime.mover 59 wikitext text/x-wiki == Cayley Table of Alternating Group on $4$ Letters == The Cayley table of the alternating group on $4$ letters can be written: <onlyinclude> :$\\begin{array}{c|cccc|cccc|cccc} \\circ & e & t & u & v & a & b & c & d & p & q & r & s \\\\ \\hline e & e & t & u & v & a & b & c & d & p & q & r & s \\\\ t & t & e & v & u & b & a & d & c & q & p & s & r \\\\ u & u & v & e & t & c & d & a & b & r & s & p & q \\\\ v & v & u & t & e & d & c & b & a & s & r & q & p \\\\ \\hline a & a & c & d & b & p & r & s & q & e & u & v & t \\\\ b & b & d & c & a & q & s & r & p & t & v & u & e \\\\ c & c & a & b & d & r & p & q & s & u & e & t & v \\\\ d & d & b & a & c & s & q & p & r & v & t & e & u \\\\ \\hline p & p & s & q & r & e & v & t & u & a & d & b & c \\\\ q & q & r & p & s & t & u & e & v & b & c & a & d \\\\ r & r & q & s & p & u & t & v & e & c & b & d & a \\\\ s & s & p & r & q & v & e & u & t & d & a & c & b \\\\ \\end{array}$ </onlyinclude> where the expression for $A_4$ in cycle notation is given as: {{:Alternating Group on 4 Letters/Cycle Notation}} == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Modulo Addition/Cayley Table/Modulo 6|next = Definition:Klein Four-Group}}: Tables: $3$. Alternating group $\\map A 4$ Category:Examples of Cayley Tables Category:Alternating Group on 4 Letters nwy2h2z2dzkaiehq36hkylmxreidu1z"}
+{"_id": "32975", "title": "Alternating Group on 4 Letters/Cycle Notation", "text": "Alternating Group on 4 Letters/Cycle Notation 0 72722 388451 388450 2019-01-19T06:15:16Z Prime.mover 59 wikitext text/x-wiki == Cycle Notation for Alternating Group on $4$ Letters == The alternating group on $4$ letters can be given in cycle notation as follows: <onlyinclude> {{begin-eqn}} {{eqn | l = e       | o = :=       | r = \\text { the identity mapping} }} {{eqn | l = t       | o = :=       | r = \\tuple {1 2} \\tuple {3 4} }} {{eqn | l = u       | o = :=       | r = \\tuple {1 3} \\tuple {2 4} }} {{eqn | l = v       | o = :=       | r = \\tuple {1 4} \\tuple {2 3} }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a       | o = :=       | r = \\tuple {1 2 3} }} {{eqn | l = b       | o = :=       | r = \\tuple {1 3 4} }} {{eqn | l = c       | o = :=       | r = \\tuple {2 4 3} }} {{eqn | l = d       | o = :=       | r = \\tuple {1 4 2} }} {{end-eqn}} {{begin-eqn}} {{eqn | l = p       | o = :=       | r = \\tuple {1 3 2} }} {{eqn | l = q       | o = :=       | r = \\tuple {2 3 4} }} {{eqn | l = r       | o = :=       | r = \\tuple {1 2 4} }} {{eqn | l = s       | o = :=       | r = \\tuple {1 4 3} }} {{end-eqn}} </onlyinclude> == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Symmetric Group on 4 Letters/Subgroups/Examples/Disjoint Transpositions|next = Alternating Group on 4 Letters/Order of Elements}}: Chapter $9$: Permutations: Exercise $3$ Category:Alternating Group on 4 Letters 00uol1mkvs05aj9c2bxpvntjgd8kzuy"}
+{"_id": "32976", "title": "Symmetric Group on 3 Letters/Normal Subgroups", "text": "Symmetric Group on 3 Letters/Normal Subgroups 0 72767 387861 387857 2019-01-15T11:18:01Z Prime.mover 59 wikitext text/x-wiki == Normal Subgroups of the Symmetric Group on 3 Letters == Let $S_3$ denote the Symmetric Group on 3 Letters, whose Cayley table is given as: {{:Symmetric Group on 3 Letters/Cayley Table}} <onlyinclude> Consider the subgroups of $S_3$: {{:Symmetric Group on 3 Letters/Subgroups}} Of those, the normal subgroups in $S_3$ are: :$S_3, \\set e, \\set {e, \\tuple {123}, \\tuple {132} }$ </onlyinclude> == Proof == $S_3$ itself is normal in $S_3$ by Group is Normal in Itself. $\\set e$ is normal in $S_3$ by Trivial Subgroup is Normal. $\\set {e, \\tuple {12} }$: {{begin-eqn}} {{eqn | l = \\tuple {123} \\tuple {13} \\tuple {123}^{-1}       | r = \\tuple {123} \\tuple {13} \\tuple {132}       | c =  }} {{eqn | r = \\tuple {123} \\tuple {12}       | c =  }} {{eqn | r = \\tuple {23}       | c =  }} {{eqn | o = \\notin       | r = \\set {e, \\tuple {12} }       | c =  }} {{end-eqn}} Hence $\\set {e, \\tuple {12} }$ is not normal in $S_3$. $\\set {e, \\tuple {23} }$: {{begin-eqn}} {{eqn | l = \\tuple {123} \\tuple {23} \\tuple {123}^{-1}       | r = \\tuple {123} \\tuple {23} \\tuple {132}       | c =  }} {{eqn | r = \\tuple {123} \\tuple {13}       | c =  }} {{eqn | r = \\tuple {12}       | c =  }} {{eqn | o = \\notin       | r = \\set {e, \\tuple {23} }       | c =  }} {{end-eqn}} Hence $\\set {e, \\tuple {23} }$ is not normal in $S_3$. $\\set {e, \\tuple {13} }$: {{begin-eqn}} {{eqn | l = \\tuple {123} \\tuple {13} \\tuple {123}^{-1}       | r = \\tuple {123} \\tuple {13} \\tuple {132}       | c =  }} {{eqn | r = \\tuple {123} \\tuple {12}       | c =  }} {{eqn | r = \\tuple {23}       | c =  }} {{eqn | o = \\notin       | r = \\set {e, \\tuple {13} }       | c =  }} {{end-eqn}} Hence $\\set {e, \\tuple {13} }$ is not normal in $S_3$. $\\set {e, \\tuple {123}, \\tuple {132} }$: We have that $\\set {e, \\tuple {123}, \\tuple {132} }$ is the set of even permutations of $S_3$. Any permutation of the form $\\alpha \\pi \\alpha^{-1}$, for $\\pi$ even, is also even. Thus: :$\\forall \\alpha \\in S_3: \\alpha \\pi \\alpha^{-1} \\in \\set {e, \\tuple {123}, \\tuple {132} }$ Hence $\\set {e, \\tuple {123}, \\tuple {132} }$ is normal in $S_3$. {{qed}} == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Subgroup is Superset of Conjugate iff Normal|next = Group is Normal in Itself}}: $\\S 6.6$. Normal subgroups: Example $122$ * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Symmetric Group on 3 Letters/Subgroups|next = Subgroups of Quaternion Group}}: Chapter $7$: Normal subgroups and quotient groups: Exercise $2$ Category:Symmetric Group on 3 Letters Category:Examples of Normal Subgroups gr8pg2feru0djipvuiyeu3gr5xtagfi"}
+{"_id": "32977", "title": "Modulo Addition/Cayley Table/Modulo 3", "text": "Modulo Addition/Cayley Table/Modulo 3 0 72769 463911 463905 2020-04-24T06:18:00Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Addition Modulo $3$ == The additive group of integers modulo $3$ can be described by showing its Cayley table: <onlyinclude> :$\\begin{array}{r|rrr} \\struct {\\Z_3, +_3} & \\eqclass 0 3 & \\eqclass 1 3 & \\eqclass 2 3 \\\\ \\hline \\eqclass 0 3 & \\eqclass 0 3 & \\eqclass 1 3 & \\eqclass 2 3 \\\\ \\eqclass 1 3 & \\eqclass 1 3 & \\eqclass 2 3 & \\eqclass 0 0 \\\\ \\eqclass 2 3 & \\eqclass 2 3 & \\eqclass 0 3 & \\eqclass 1 3 \\\\ \\end{array}$ </onlyinclude> It can also be presented: :$\\begin{array}{r|rrr} +_3 & 0 & 1 & 2 \\\\ \\hline 0 & 0 & 1 & 2 \\\\ 1 & 1 & 2 & 0 \\\\ 2 & 2 & 0 & 1 \\\\ \\end{array}$ == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Quotient Group of Integers by Multiples|next = Klein Four-Group is Normal in A4}}: $\\S 6.7$. Quotient groups: Example $126$ * {{BookReference|Topology: An Introduction with Application to Topological Groups|1967|George McCarty|prev = Definition:Parity of Integer|next = Definition:Quotient Group/Motivation}}: Chapter $\\text{II}$: Groups: A Little Number Theory * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Integers Modulo m under Addition form Abelian Group|next = Mathematician:Arthur Cayley}}: Chapter $2$: Maps and relations on sets: Example $2.33$ Category:Examples of Cayley Tables Category:Examples of Additive Groups of Integers Modulo m Category:Cyclic Group of Order 3 b59mrv9uwib7stlw48ftagmhpq895h6"}
+{"_id": "32978", "title": "Alternating Group on 4 Letters/Subgroups", "text": "Alternating Group on 4 Letters/Subgroups 0 72772 388453 378472 2019-01-19T06:19:04Z Prime.mover 59 wikitext text/x-wiki == Subgroups of the Alternating Group on $4$ Letters == Let $A_4$ denote the alternating group on $4$ letters, whose Cayley table is given as: {{:Alternating Group on 4 Letters/Cayley Table}} <onlyinclude> The subsets of $A_4$ which form subgroups of $A_4$ are as follows: Trivial: {{begin-eqn}} {{eqn | o =        | r = \\set e       | c = Trivial Subgroup is Subgroup }} {{eqn | o =        | r = A_4       | c = Group is Subgroup of Itself }} {{end-eqn}} Order $2$ subgroups: {{begin-eqn}} {{eqn | o =        | r = \\set {e, t}       | c = as $t^2 = e$ }} {{eqn | o =        | r = \\set {e, u}       | c = as $u^2 = e$ }} {{eqn | o =        | r = \\set {e, v}       | c = as $v^2 = e$ }} {{end-eqn}} Order $3$ subgroups: {{begin-eqn}} {{eqn | o =        | r = \\set {e, a, p}       | c = as $a^2 = p$, $a^3 = a p = e$ }} {{eqn | o =        | r = \\set {e, b, s}       | c = as $b^2 = s$, $b^3 = b s = e$ }} {{eqn | o =        | r = \\set {e, c, q}       | c = as $c^2 = q$, $c^3 = c q = e$ }} {{eqn | o =        | r = \\set {e, d, r}       | c = as $d^2 = r$, $d^3 = d r = e$ }} {{end-eqn}} Order $4$ subgroup: {{begin-eqn}} {{eqn | o =        | r = \\set {e, t, u, v}       | c = Klein $4$-Group }} {{end-eqn}} </onlyinclude> == Examples of Subgroups == {{:Alternating Group on 4 Letters/Subgroups/Examples}} == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Alternating Group on 4 Letters/Order of Elements|next = Group does not Necessarily have Subgroup of Order of Divisor of its Order/Proof 2}}: Chapter $9$: Permutations: Exercise $3$ Category:Alternating Group on 4 Letters fcb3cssioaoq68bpr5xonor2rjqx8ny"}
+{"_id": "32979", "title": "Alternating Group on 4 Letters/Normality of Subgroups", "text": "Alternating Group on 4 Letters/Normality of Subgroups 0 72794 387204 385380 2019-01-11T14:09:46Z Prime.mover 59 wikitext text/x-wiki == Normality of Subgroups of the Alternating Group on $4$ Letters == Let $A_4$ denote the alternating group on $4$ letters, whose Cayley table is given as: {{:Alternating Group on 4 Letters/Cayley Table}} <onlyinclude> The normality status of the non-trivial proper subgroups of $A_4$ is as follows: Order $2$ subgroups: {{begin-eqn}} {{eqn | l = T       | o = :=       | r = \\set {e, t}       | c = Not normal }} {{eqn | l = U       | o = :=       | r = \\set {e, u}       | c = Not normal }} {{eqn | l = V       | o = :=       | r = \\set {e, v}       | c = Not normal }} {{end-eqn}} Order $3$ subgroups: {{begin-eqn}} {{eqn | l = P       | o = :=       | r = \\set {e, a, p}       | c = Not normal }} {{eqn | l = Q       | o = :=       | r = \\set {e, c, q}       | c = Not normal }} {{eqn | l = R       | o = :=       | r = \\set {e, d, r}       | c = Not normal }} {{eqn | l = S       | o = :=       | r = \\set {e, b, s}       | c = Not normal }} {{end-eqn}} Order $4$ subgroup: {{begin-eqn}} {{eqn | l = K       | o = :=       | r = \\set {e, t, u, v}       | c = Normal }} {{end-eqn}} </onlyinclude> == Proof == Testing one of the left cosets of $T = \\set {e, t}$ against its corresponding right coset: {{begin-eqn}} {{eqn | l = a T       | r = \\set {a \\circ e, a \\circ t}       | c =  }} {{eqn | r = \\set {a, c}       | c =  }} {{eqn | l = T a       | r = \\set {e \\circ a, t \\circ a}       | c =  }} {{eqn | r = \\set {a, b}       | c =  }} {{eqn | o = \\ne       | r = a T       | c =  }} {{end-eqn}} The left coset does not equal the right coset and so $T$ is not normal in $A_4$. {{qed|lemma}} Testing one of the left cosets of $U = \\set {e, u}$ against its corresponding right coset: {{begin-eqn}} {{eqn | l = b U       | r = \\set {b \\circ e, b \\circ u}       | c =  }} {{eqn | r = \\set {b, c}       | c =  }} {{eqn | l = U b       | r = \\set {e \\circ b, u \\circ b}       | c =  }} {{eqn | r = \\set {b, d}       | c =  }} {{eqn | o = \\ne       | r = b U       | c =  }} {{end-eqn}} The left coset does not equal the right coset and so $U$ is not normal in $A_4$. {{qed|lemma}} Testing one of the left cosets of $V = \\set {e, v}$ against its corresponding right coset: {{begin-eqn}} {{eqn | l = c V       | r = \\set {c \\circ e, c \\circ v}       | c =  }} {{eqn | r = \\set {c, d}       | c =  }} {{eqn | l = V c       | r = \\set {e \\circ c, v \\circ c}       | c =  }} {{eqn | r = \\set {c, b}       | c =  }} {{eqn | o = \\ne       | r = c V       | c =  }} {{end-eqn}} The left coset does not equal the right coset and so $V$ is not normal in $A_4$. {{qed|lemma}} Testing one of the left cosets of $P = \\set {e, a, p}$ against its corresponding right coset: {{begin-eqn}} {{eqn | l = t P       | r = \\set {t \\circ e, t \\circ a, t \\circ p}       | c =  }} {{eqn | r = \\set {t, b, q}       | c =  }} {{eqn | l = P t       | r = \\set {e \\circ t, a \\circ t, p \\circ t}       | c =  }} {{eqn | r = \\set {t, c, s}       | c =  }} {{eqn | o = \\ne       | r = t P       | c =  }} {{end-eqn}} The left coset does not equal the right coset and so $P$ is not normal in $A_4$. {{qed|lemma}} Testing one of the left cosets of $Q = \\set {e, c, q}$ against its corresponding right coset: {{begin-eqn}} {{eqn | l = t Q       | r = \\set {t \\circ e, t \\circ c, t \\circ q}       | c =  }} {{eqn | r = \\set {t, d, p}       | c =  }} {{eqn | l = Q t       | r = \\set {e \\circ t, c \\circ t, q \\circ t}       | c =  }} {{eqn | r = \\set {t, a, r}       | c =  }} {{eqn | o = \\ne       | r = t Q       | c =  }} {{end-eqn}} The left coset does not equal the right coset and so $Q$ is not normal in $A_4$. {{qed|lemma}} Testing one of the left cosets of $R = \\set {e, d, r}$ against its corresponding right coset: {{begin-eqn}} {{eqn | l = t R       | r = \\set {t \\circ e, t \\circ d, t \\circ r}       | c =  }} {{eqn | r = \\set {t, c, s}       | c =  }} {{eqn | l = R t       | r = \\set {e \\circ t, d \\circ t, r \\circ t}       | c =  }} {{eqn | r = \\set {t, b, q}       | c =  }} {{eqn | o = \\ne       | r = t R       | c =  }} {{end-eqn}} The left coset does not equal the right coset and so $R$ is not normal in $A_4$. {{qed|lemma}} Testing one of the left cosets of $S = \\set {e, b, s}$ against its corresponding right coset: {{begin-eqn}} {{eqn | l = t S       | r = \\set {t \\circ e, t \\circ b, t \\circ s}       | c =  }} {{eqn | r = \\set {t, a, r}       | c =  }} {{eqn | l = S t       | r = \\set {e \\circ t, b \\circ t, s \\circ t}       | c =  }} {{eqn | r = \\set {t, d, p}       | c =  }} {{eqn | o = \\ne       | r = t S       | c =  }} {{end-eqn}} The left coset does not equal the right coset and so $S$ is not normal in $A_4$. {{qed|lemma}} The cosets of $K = \\set {e, t, u, v}$ are as follows: {{begin-eqn}} {{eqn | l = a K       | r = \\set {a \\circ e, a \\circ t, a \\circ u, a \\circ v}       | c =  }} {{eqn | r = \\set {a, c, d, b}       | c =  }} {{eqn | l = K a       | r = \\set {e \\circ a, t \\circ a, u \\circ a, v \\circ a}       | c =  }} {{eqn | r = \\set {a, b, c, d}       | c =  }} {{eqn | r = a K       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = p K       | r = \\set {p \\circ e, p \\circ t, p \\circ u, p \\circ v}       | c =  }} {{eqn | r = \\set {p, s, q, r}       | c =  }} {{eqn | l = K p       | r = \\set {e \\circ p, t \\circ p, u \\circ p, v \\circ p}       | c =  }} {{eqn | r = \\set {p, q, r, s}       | c =  }} {{eqn | r = p K       | c =  }} {{end-eqn}} The left cosets equal the right cosets and so $K$ is normal in $A_4$. {{qed|lemma}} == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Stabilizer of Polynomial|next = Normality Relation is not Transitive/Proof 1}}: Chapter $6$: Cosets: Exercise $11$ Category:Alternating Group on 4 Letters bichruyi89pkryb6htqxbl6dk2nznh0"}
+{"_id": "32980", "title": "Dihedral Group D4/Matrix Representation/Formulation 1", "text": "Dihedral Group D4/Matrix Representation/Formulation 1 0 72869 389014 385553 2019-01-22T07:52:21Z Prime.mover 59 wikitext text/x-wiki == Matrix Representation of Dihedral Group $D_4$ == <onlyinclude> Let $\\mathbf I, \\mathbf A, \\mathbf B, \\mathbf C$ denote the following four elements of the matrix space $\\map {\\mathcal M_\\Z} 2$: :$\\mathbf I = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} \\qquad \\mathbf A = \\begin{bmatrix} 1 & 0 \\\\ 0 & -1 \\end{bmatrix} \\qquad \\mathbf B = \\begin{bmatrix} 0 & -1 \\\\ 1 & 0 \\end{bmatrix} \\qquad \\mathbf C = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix}$ The set: :$D_4 = \\set {\\mathbf I, -\\mathbf I, \\mathbf A, -\\mathbf A, \\mathbf B, -\\mathbf B, \\mathbf C, -\\mathbf C}$ under the operation of conventional matrix multiplication, forms the '''dihedral group $D_4$'''. </onlyinclude> === Cayley Table === Its Cayley table is given by: {{:Dihedral Group D4/Matrix Representation/Formulation 1/Cayley Table}} == Also see == * Dihedral Group D4 Defined by Matrices where it is shown that these have the appropriate properties. == Sources == {{SourceReview}} Category:Dihedral Group D4 fotfa45owyoai8vixxeyk8z782wx8gs"}
+{"_id": "32981", "title": "Dihedral Group D4/Matrix Representation/Formulation 1/Cayley Table", "text": "Dihedral Group D4/Matrix Representation/Formulation 1/Cayley Table 0 72870 385551 385533 2018-12-31T09:57:02Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Dihedral Group $D_4$ == The Cayley table for the dihedral group $D_4$: :$D_4 = \\set {\\mathbf I, -\\mathbf I, \\mathbf A, -\\mathbf A, \\mathbf B, -\\mathbf B, \\mathbf C, -\\mathbf C}$ under the operation of conventional matrix multiplication, where: :$\\mathbf I = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} \\qquad \\mathbf A = \\begin{bmatrix} 1 & 0 \\\\ 0 & -1 \\end{bmatrix} \\qquad \\mathbf B = \\begin{bmatrix} 0 & -1 \\\\ 1 & 0 \\end{bmatrix} \\qquad \\mathbf C = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix}$ can be presented as: <onlyinclude> :$\\begin{array}{r|rrrrrrrr}            &  \\mathbf I &  \\mathbf A &  \\mathbf B &  \\mathbf C & -\\mathbf I & -\\mathbf A & -\\mathbf B & -\\mathbf C \\\\ \\hline  \\mathbf I &  \\mathbf I &  \\mathbf A &  \\mathbf B &  \\mathbf C & -\\mathbf I & -\\mathbf A & -\\mathbf B & -\\mathbf C \\\\  \\mathbf A &  \\mathbf A &  \\mathbf I & -\\mathbf C & -\\mathbf B & -\\mathbf A & -\\mathbf I &  \\mathbf C &  \\mathbf B \\\\  \\mathbf B &  \\mathbf B &  \\mathbf C & -\\mathbf I & -\\mathbf A & -\\mathbf B & -\\mathbf C &  \\mathbf I &  \\mathbf A \\\\  \\mathbf C &  \\mathbf C &  \\mathbf B &  \\mathbf A &  \\mathbf I & -\\mathbf C & -\\mathbf B & -\\mathbf A & -\\mathbf I \\\\ -\\mathbf I & -\\mathbf I & -\\mathbf A & -\\mathbf B & -\\mathbf C &  \\mathbf I &  \\mathbf A &  \\mathbf B &  \\mathbf C \\\\ -\\mathbf A & -\\mathbf A & -\\mathbf I &  \\mathbf C &  \\mathbf B &  \\mathbf A &  \\mathbf I & -\\mathbf C & -\\mathbf B \\\\ -\\mathbf B & -\\mathbf B & -\\mathbf C &  \\mathbf I &  \\mathbf A &  \\mathbf B & -\\mathbf C & -\\mathbf I & -\\mathbf A \\\\ -\\mathbf C & -\\mathbf C & -\\mathbf B & -\\mathbf A & -\\mathbf I &  \\mathbf C &  \\mathbf B &  \\mathbf A &  \\mathbf I \\end{array}$ </onlyinclude> == Sources == {{SourceReview}} Category:Dihedral Group D4 Category:Examples of Cayley Tables f7uc1dmd4ox2xz595bc5jvw9rsw7tsk"}
+{"_id": "32982", "title": "Quaternion Group/Group Presentation", "text": "Quaternion Group/Group Presentation 0 72932 445965 387854 2020-02-03T12:02:51Z Prime.mover 59 wikitext text/x-wiki == Group Presentation of Quaternion Group == The group presentation of the quaternion group is given by: <onlyinclude> :$\\Dic 2 = \\gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ </onlyinclude> == Proof == Let $G = \\gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$. It is to be demonstrated that $\\Dic 2$ is isomorphic to $G$. Consider the Cayley table for $\\Dic 2$: {{:Quaternion Group/Cayley Table}} We have that: :$a^4 = e$ :$b^2 = a^2$ :$\\paren {a b} a = b$ demonstrating that $\\Dic 2$ has the same group presentation as $G$. Hence the result. {{qed}} == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Generated Subgroups/B, F|next = Definition:Left Coset}}: Chapter $4$: Subgroups: Exercise $4$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Generator of Group|next = Definition:Cyclic Group/Definition 2|entry = generator|index = 2}} Category:Quaternion Group Category:Group Presentations gyoyxuok1r5eambe6mx37emrp4yjam8"}
+{"_id": "32983", "title": "Bound for Analytic Function and Derivatives/Analytic Function Bounded on Circle", "text": "Bound for Analytic Function and Derivatives/Analytic Function Bounded on Circle 0 72944 413234 379993 2019-07-17T18:36:53Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $f$ be a complex function. Let $z_0$ be a point in $\\C$. Let $\\Gamma$ be a circle in $\\C$ with center at $z_0$ and radius in $\\R_{>0}$. Let $f$ be analytic on $\\Gamma$. Then $f$ is bounded on $\\Gamma$. </onlyinclude> == Proof == Let: :$\\map {f_{\\Re} } z = \\map \\Re {\\map f z}$ :$\\map {f_{\\Im} } z = \\map \\Im {\\map f z}$ Let $\\closedint a b$, $a < b$, be a closed real interval. Let $p$ be a continuous complex-valued function defined such that: :$\\Gamma = \\set {\\map p u: u \\in \\closedint a b}$ $f$ is continuous on $\\Gamma$ as $f$ is analytic on $\\Gamma$ by the definition of analytic. Also, Real and Imaginary Part Projections are Continuous. Therefore, $f_{\\Re}$ and $f_{\\Im}$ are continuous by the corollary to Composite of Continuous Mappings is Continuous. Observe that $f_{\\Re}$ and $f_{\\Im}$ are real-valued functions that are continuous. Also, $p$ is a continuous function defined on a set of real numbers. Therefore $\\map {f_{\\Re} } {\\map p u}$ and $\\map {f_{\\Im} } {\\map p u}$ are continuous real functions by the corollary to Composite of Continuous Mappings is Continuous. $\\map {f_{\\Re} } {\\map p u}$ and $\\map {f_{\\Im} } {\\map p u}$ are bounded on $\\closedint a b$ by Continuous Real Function is Bounded. Therefore $\\map f {\\map p u}$ is bounded on $\\closedint a b$ as $\\map f {\\map p u} = \\map {f_{\\Re} } {\\map p u} + i \\map {f_{\\Im} } {\\map p u}$ where $i = \\sqrt {-1}$. Accordingly, $f$ is bounded on $\\Gamma$ as $\\Gamma = \\set {\\map p u: u \\in \\closedint a b}$. {{qed}} Category:Complex Analysis gokqfygmi1t231kgrg5l4ovtp6573nv"}
+{"_id": "32984", "title": "Convergence of Taylor Series of Function Analytic on Disk/Lemma", "text": "Convergence of Taylor Series of Function Analytic on Disk/Lemma 0 72959 435251 380103 2019-11-14T15:05:50Z Prime.mover 59 wikitext text/x-wiki {{refactor|This (or something very similar) already exists somewhere as a basic null sequence}} == Lemma == <onlyinclude> Let $y > 1$. Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\frac n {y^n} = 0$ </onlyinclude> == Proof == Note that $\\ln y > 0$ as $y > 1$. {{begin-eqn}} {{eqn | l = \\lim_{n \\mathop \\to \\infty} \\frac n {y^n}       | r = \\lim_{n \\mathop \\to \\infty} \\frac n {\\left({e^{\\ln y} }\\right)^n} }} {{eqn | r = \\lim_{n \\mathop \\to \\infty} \\frac n {e^{\\left({\\ln y}\\right) n} } }} {{eqn | r = 0       | c = as $\\displaystyle \\lim_{x \\mathop \\to \\infty} \\frac x {e^{\\left({\\ln y}\\right) x} } = 0$ by Limit at Infinity of Polynomial over Complex Exponential as $\\ln y > 0$ }} {{end-eqn}} {{qed}} Category:Real Analysis 00nqt72unqz24foiob6y1l26q9rmv46"}
+{"_id": "32985", "title": "Oscillation at Point (Infimum) equals Oscillation at Point (Limit)/Lemma", "text": "Oscillation at Point (Infimum) equals Oscillation at Point (Limit)/Lemma 0 72976 437673 435381 2019-12-03T22:41:36Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $f: D \\to \\R$ be a real function where $D \\subseteq \\R$. Let $x$ be a point in $D$. Let $N_x$ be the set of open subset neighborhoods of $x$. Let $\\map {\\omega_f} x$ be the oscillation of $f$ at $x$ as defined by: :$\\map {\\omega_f} x = \\displaystyle \\inf \\set {\\map {\\omega_f} I: I \\in N_x}$ where $\\map {\\omega_f} I$ is the oscillation of $f$ on a real set $I$: :$\\map {\\omega_f} I = \\displaystyle \\sup \\set {\\size {\\map f y - \\map f z}: y, z \\in I \\cap D}$ Let $\\map {\\omega^L_f} x$ be the oscillation of $f$ at $x$ as defined by: :$\\map {\\omega^L_f} x = \\displaystyle \\lim_{h \\mathop \\to 0^+} \\map {\\omega_f} {\\openint {x - h} {x + h} }$ Let $\\map {\\omega^L_f} x \\in \\R$. Let $\\map {\\omega_f} x \\in \\R$. Then $\\map {\\omega^L_f} x = \\map {\\omega_f} x$. </onlyinclude> == Proof == We know that $\\map {\\omega^L_f} x$ and $\\map {\\omega_f} x$ are real numbers. We need to prove that $\\map {\\omega^L_f} x = \\map {\\omega_f} x$. Let $\\epsilon \\in \\R_{>0}$. First, we aim to prove that $\\size {\\map {\\omega_f} {\\openint {x - h} {x + h} } - \\map {\\omega_f} x} < \\epsilon$ for a small enough $h \\in R_{>0}$. $\\map {\\omega^L_f} x = \\displaystyle \\lim_{h \\mathop \\to 0^+} \\map {\\omega_f} {\\openint {x - h} {x + h} }$ means by the definition of limit from the right that a strictly positive real number $h_1$ exists such that: :$\\size {\\map {\\omega_f} {\\openint {x - h} {x + h} } - \\map {\\omega^L_f} x)} < \\epsilon$ for every $h$ that satisfies: $0 < h < h_1$. This means in particular that $\\map {\\omega_f} {\\openint {x - h} {x + h} } \\in \\R$ for every $h$ that satisfies: $0 < h < h_1$. Let $h'$ be a real number that satisfies: $0 < h' < h_1$. We observe that $\\openint {x - h'} {x + h'} \\in N_x$. Therefore, $\\map {\\omega_f} {\\openint {x - h'} {x + h'} } \\in \\set {\\map {\\omega_f} I: I \\in N_x}$. By definition, $\\map {\\omega_f} x$ is a lower bound for $\\set {\\map {\\omega_f} I: I \\in N_x}$. Accordingly: :$\\map {\\omega_f} {\\openint {x - h'} {x + h'} } \\ge \\map {\\omega_f} x$ The fact that $\\map {\\omega_f} x \\in \\R$ implies that: :$\\map {\\omega_f} I - \\map {\\omega_f} x < \\epsilon$ by Infimum of Set of Oscillations on Set is Arbitrarily Close for an $I \\in N_x$. Let $I$ be such an element of $N_x$. We observe in particular that $\\map {\\omega_f} I \\in \\R$. A neighborhood in $N_x$ contains an open subset that contains the point $x$. So, $I$ contains such an open subset as $I \\in N_x$. Therefore, a number $h_2 \\in \\R_{>0}$ exists such that $\\openint {x - h_2} {x + h_2}$ is a subset of $I$. Let $h''$ be a real number that satisfies: $0 < h'' < h_2$. We observe that $\\openint {x - h''} {x + h''}$ is a subset of $I$. We have: :$\\map {\\omega_f} I \\in \\R$ :$\\openint {x - h''} {x + h''}$ is a subset of $I$ Therefore: :$\\map {\\omega_f} {\\openint {x - h''} {x + h''} } \\le \\map {\\omega_f} I$ by Oscillation on Subset Putting all this together, we get for every $h$ that satisfies: $0 < h < \\min \\set {h_1, h_2}$: {{begin-eqn}} {{eqn | l = \\map {\\omega_f} {\\openint {x - h} {x + h} }       | o = \\le       | r = \\map {\\omega_f} I }} {{eqn | ll= \\leadsto       | l = \\map {\\omega_f} x \\le \\map {\\omega_f} {\\openint {x - h} {x + h} }       | o = \\le       | r = \\map {\\omega_f} I       | c = as $\\map {\\omega_f} x \\le \\map {\\omega_f} {\\openint {x - h} {x + h} }$ is true }} {{eqn | ll= \\leadsto       | l = \\map {\\omega_f} x \\le \\map {\\omega_f} {\\openint {x - h} {x + h} }       | o = \\le       | r = \\map {\\omega_f} I < \\map {\\omega_f} x + \\epsilon       | c = as $\\map {\\omega_f} I < \\map {\\omega_f} x + \\epsilon$ is true }} {{eqn | ll= \\leadsto       | l = \\map {\\omega_f} x \\le \\map {\\omega_f} {\\openint {x - h} {x + h} }       | o = <       | r = \\map {\\omega_f} x + \\epsilon       | c =  }} {{eqn | ll= \\leadsto       | l = 0 \\le \\map {\\omega_f} {\\openint {x - h} {x + h} } - \\map {\\omega_f} x       | o = <       | r = \\epsilon }} {{eqn | ll= \\leadsto       | l = \\size {\\map {\\omega_f} {\\openint {x - h} {x + h} } - \\map {\\omega_f} x}       | o = <       | r = \\epsilon }} {{end-eqn}} Thus, we achieved our first aim. Next, we get for every $h$ that satisfies: $0 < h < \\min \\set {h_1, h_2}$: {{begin-eqn}} {{eqn | l = \\size {\\map {\\omega^L_f} x - \\map {\\omega_f} x}       | r = \\size {\\map {\\omega^L_f} x - \\map {\\omega_f} {\\openint {x - h} {x + h} } + \\map {\\omega_f} {\\openint {x - h} {x + h} } - \\map {\\omega_f} x} }} {{eqn | o = \\le       | r = \\size {\\map {\\omega^L_f} x - \\map {\\omega_f} {\\openint {x - h} {x + h} } } + \\size {\\map {\\omega_f} {\\openint {x - h} {x + h} } - \\map {\\omega_f} x}       | c = Triangle Inequality for Real Numbers }} {{eqn | o = <       | r = \\epsilon + \\epsilon }} {{eqn | r = 2 \\epsilon }} {{end-eqn}} This holds for every $\\epsilon \\in \\R_{>0}$. Therefore, $\\size {\\map {\\omega^L_f} x - \\map {\\omega_f} x} = 0$ as $\\size {\\map {\\omega^L_f} x - \\map {\\omega_f} x}$ is independent of $\\epsilon$. Accordingly: :$\\map {\\omega^L_f} x = \\map {\\omega_f} x$ {{qed}} {{Improve|Implement a better category than this}} Category:Real Analysis 3pmygvx7f48jhe9eidgjxzeyt2q2wiy"}
+{"_id": "32986", "title": "Sum of Terms of Magic Cube/Sequence", "text": "Sum of Terms of Magic Cube/Sequence 0 73169 433513 382796 2019-11-01T14:01:22Z Prime.mover 59 wikitext text/x-wiki == Sequence of Sums of Terms of Magic Cubes == <onlyinclude> The sequence of the sum totals of all the entries in magic cubes of order $n$ begins: :$1, \\paren {36,} \\, 378, 2080, 7875, 23 \\, 436, 58 \\, 996, 131 \\, 328, \\ldots$ However, note that while $36 = \\dfrac {2^3 \\paren {2^3 + 1} } 2$, a magic cube of order $2$ does not actually exist. </onlyinclude> {{OEIS|A037270}} Category:Magic Cubes 2y8fzpedt6pmmf8b0svb3wy0m1xikwm"}
+{"_id": "32987", "title": "Magic Constant of Magic Cube/Sequence", "text": "Magic Constant of Magic Cube/Sequence 0 73171 433415 382868 2019-11-01T12:58:11Z Prime.mover 59 wikitext text/x-wiki == Sequence of Magic Constants of Magic Cubes == <onlyinclude> The sequence of the magic constants of magic cubes of order $n$ begins: :$1, (9,) \\, 42, 130, 315, 651, 1204, 2052, 3285, 5005, 7326, 10 \\, 374, 14 \\, 287, 19 \\, 215, 25 \\, 320, 32 \\, 776, \\ldots$ However, note that while $9 = \\dfrac {2 \\paren {2^3 + 1} } 2$, a magic cube of order $2$ does not actually exist. </onlyinclude> {{OEIS|A027441}} Category:Magic Cubes 9gyu6xzac1eu8s55wz15feg80loz1xg"}
+{"_id": "32988", "title": "Sequence Lemma", "text": "Sequence Lemma 0 73179 490698 490697 2020-09-25T16:19:10Z Prime.mover 59 wikitext text/x-wiki {{tidy}} {{MissingLinks}} == Lemma == Let $A$ be a subset of a topological space $X$. If there is a sequence of points of $A$ converging to $x$, then $x \\in \\bar A$. The converse holds if $X$ is first-countable. == Proof == Assume the sequence of points of $A$ that converges to $x$ is $\\sequence {x_i}$. Then for any open set $U$ of $x$, there exists a positive natural number $N$ such that when $i > N$, $x_i \\in U$. Thus $U \\cap A$ is nonempty, $x \\in \\bar A$. Let the topological space $X$ be first-countable. Then there is a countable collection of open neighbourhood $\\family {U_i}_{i \\mathop \\in \\Bbb Z_+}$ of $x$ such that any open neighbourhood $U$ of $x$ contains at least one of the sets $U_i$. Because $x \\in \\bar A$, $U_1 \\cap A$ is nonempty, we can select a point $x_1$ in it. In a similar manner, $U_1 \\cap U_2 \\cap A$ is nonempty. Hence we can select a point $x_2$ in it. The point $x_i$ is selected from: :$U_1 \\cap U_2 \\cap \\cdots \\cap U_i \\cap A$ We then obtain a sequence $\\sequence {x_i}$. For any open neighbourhood $U$ of $x$, it contains at least one of the set $U_N$, $N \\in \\Bbb Z_+$ of $\\family {U_i}_{i \\mathop \\in \\Bbb Z_+}$. Thus it contains the set: :$U_1 \\cap U_2 \\cap \\cdots \\cap U_N \\cap A$ :$U_1 \\cap U_2 \\cap \\cdots \\cap U_N \\cap U_{N + 1}\\cap A$ :$U_1 \\cap U_2 \\cap \\cdots \\cap U_N \\cap U_{N + 1} \\cap U_{N + 2} \\cap A$ :$\\ldots$ or the set $U_1 \\cap U_2 \\cap \\cdots \\cap U_i \\cap A$ with $i > N$, hence the points $x_i$ with $i > N$. The sequence $\\sequence {x_i}$ converges to $x$. {{qed}} == Sources == * {{BookReference|Topology|2004|James R. Munkres}} P130, Lemma 21.2 Category:Topology Category:Countability Axioms Category:First-Countable Spaces ta4s3r4pyrg9tkz7apjkpwtszep4f0q"}
+{"_id": "32989", "title": "Group Direct Product/Examples/C2 x C2", "text": "Group Direct Product/Examples/C2 x C2 0 73396 389230 386523 2019-01-23T15:31:00Z Prime.mover 59 wikitext text/x-wiki == Example of Group Direct Product == <onlyinclude> The direct product of $C_2$, the cyclic group of order $2$, with itself is as follows. Let us represent $C_2$ as the group $\\struct {\\set {1, -1}, \\times}$: :$\\begin {array} {r|rr} \\struct {\\set {1, -1} , \\times} & 1 & -1 \\\\ \\hline 1 & 1 & -1 \\\\ -1 & -1 & 1 \\\\ \\end{array}$ Then the Cayley table for $C_2 \\times C_2$ can be portrayed as: :$\\begin {array} {c|cccc} C_2 \\times C_2  & \\tuple { 1,  1} & \\tuple { 1, -1} & \\tuple {-1,  1} & \\tuple {-1, -1} \\\\ \\hline \\tuple { 1,  1} & \\tuple { 1,  1} & \\tuple { 1, -1} & \\tuple {-1,  1} & \\tuple {-1, -1} \\\\ \\tuple { 1, -1} & \\tuple { 1, -1} & \\tuple { 1,  1} & \\tuple {-1, -1} & \\tuple {-1,  1} \\\\ \\tuple {-1,  1} & \\tuple {-1,  1} & \\tuple {-1, -1} & \\tuple { 1,  1} & \\tuple { 1, -1} \\\\ \\tuple {-1, -1} & \\tuple {-1, -1} & \\tuple {-1,  1} & \\tuple { 1, -1} & \\tuple { 1,  1} \\\\ \\end{array}$ </onlyinclude> This is seen by inspection to be an instance of the Klein $4$-group. == Subgroups == {{:Group Direct Product/Examples/C2 x C2/Subgroups}} == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = External Direct Product of Groups is Group|next = Definition:Multiplicative Notation}}: Chapter $1$: Definitions and Examples: Example $1.10$ Category:Examples of Group Direct Products Category:Klein Four-Group prtc0kzrbj0gjde092r3sgogxcwrelf"}
+{"_id": "32990", "title": "Dihedral Group D4/Matrix Representation/Formulation 2/Cayley Table", "text": "Dihedral Group D4/Matrix Representation/Formulation 2/Cayley Table 0 73423 385557 385555 2018-12-31T12:05:18Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Dihedral Group $D_4$ == The Cayley table for the dihedral group $D_4$: :$D_4 = \\set {\\mathbf I, \\mathbf A, \\mathbf B, \\mathbf C, \\mathbf D, \\mathbf E, \\mathbf F, \\mathbf G}$ under the operation of conventional matrix multiplication, where: :$\\mathbf I = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} \\qquad \\mathbf A = \\begin{bmatrix} i & 0 \\\\ 0 & -i \\end{bmatrix} \\qquad \\mathbf B = \\begin{bmatrix} -1 & 0 \\\\ 0 & -1 \\end{bmatrix} \\qquad \\mathbf C = \\begin{bmatrix} -i & 0 \\\\ 0 & i \\end{bmatrix}$ :$\\mathbf D = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix} \\qquad \\mathbf E = \\begin{bmatrix} 0 & i \\\\ -i & 0 \\end{bmatrix} \\qquad \\mathbf F = \\begin{bmatrix} 0 & -1 \\\\ -1 & 0 \\end{bmatrix} \\qquad \\mathbf G = \\begin{bmatrix} 0 & -i \\\\ i & 0 \\end{bmatrix}$ can be presented as: <onlyinclude> :$\\begin{array}{r|rrrrrrrr}           & \\mathbf I & \\mathbf A & \\mathbf B & \\mathbf C & \\mathbf D & \\mathbf E & \\mathbf F & \\mathbf G \\\\ \\hline \\mathbf I & \\mathbf I & \\mathbf A & \\mathbf B & \\mathbf C & \\mathbf D & \\mathbf E & \\mathbf F & \\mathbf G \\\\ \\mathbf A & \\mathbf A & \\mathbf B & \\mathbf C & \\mathbf I & \\mathbf E & \\mathbf F & \\mathbf G & \\mathbf D \\\\ \\mathbf B & \\mathbf B & \\mathbf C & \\mathbf I & \\mathbf A & \\mathbf F & \\mathbf G & \\mathbf D & \\mathbf E \\\\ \\mathbf C & \\mathbf C & \\mathbf I & \\mathbf A & \\mathbf B & \\mathbf G & \\mathbf D & \\mathbf E & \\mathbf F \\\\ \\mathbf D & \\mathbf D & \\mathbf G & \\mathbf F & \\mathbf E & \\mathbf I & \\mathbf C & \\mathbf B & \\mathbf A \\\\ \\mathbf E & \\mathbf E & \\mathbf D & \\mathbf G & \\mathbf F & \\mathbf A & \\mathbf I & \\mathbf C & \\mathbf B \\\\ \\mathbf F & \\mathbf F & \\mathbf E & \\mathbf D & \\mathbf G & \\mathbf B & \\mathbf A & \\mathbf I & \\mathbf C \\\\ \\mathbf G & \\mathbf G & \\mathbf F & \\mathbf E & \\mathbf D & \\mathbf C & \\mathbf B & \\mathbf A & \\mathbf I \\end{array}$ </onlyinclude> == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Square Matrices with +1 or -1 Determinant under Multiplication forms Group|next = Symmetry Group of Square/Cayley Table}}: Chapter $1$: Definitions and Examples: Exercise $2$ Category:Dihedral Group D4 Category:Examples of Cayley Tables guk2cvj7qlekm5o2z8a17vgul9mj7g8"}
+{"_id": "32991", "title": "Dihedral Group D4/Group Presentation", "text": "Dihedral Group D4/Group Presentation 0 73538 386381 386233 2019-01-05T08:46:01Z Prime.mover 59 wikitext text/x-wiki == Group Presentation of Dihedral Group $D_4$ == The group presentation of the dihedral group $D_4$ is given by: <onlyinclude> :$D_4 = \\gen {a, b: a^4 = b^2 = e, a b = b a^{-1} }$ </onlyinclude> == Proof == We have that the group presentation of the dihedral group $D_n$ is: :$D_n = \\gen {\\alpha, \\beta: \\alpha^n = \\beta^2 = e, \\beta \\alpha \\beta = \\alpha^{−1} }$ Setting $n = 4, \\alpha = a, \\beta = b$, we get: :$D_4 = \\gen {a, b: a^4 = b^2 = e, b a b = a^{−1} }$ from which the result follows. {{qed}} Category:Dihedral Group D4 oh3atztie8viahgrxt735dwnqryv7xg"}
+{"_id": "32992", "title": "Dihedral Group D3/Group Presentation", "text": "Dihedral Group D3/Group Presentation 0 73560 386380 386377 2019-01-05T08:45:49Z Prime.mover 59 wikitext text/x-wiki == Group Presentation of Dihedral Group $D_3$ == The group presentation of the dihedral group $D_3$ is given by: <onlyinclude> :$D_3 = \\gen {a, b: a^3 = b^2 = e, a b = b a^{-1} }$ </onlyinclude> == Proof == We have that the group presentation of the dihedral group $D_n$ is: :$D_n = \\gen {\\alpha, \\beta: \\alpha^n = \\beta^2 = e, \\beta \\alpha \\beta = \\alpha^{−1} }$ Setting $n = 3, \\alpha = a, \\beta = b$, we get: :$D_3 = \\gen {a, b: a^3 = b^2 = e, b a b = a^{−1} }$ from which the result follows. {{qed}} Category:Dihedral Group D3 675nb0ghqc462n8hcjps9rwo03tf0nt"}
+{"_id": "32993", "title": "Dihedral Group D3/Cayley Table", "text": "Dihedral Group D3/Cayley Table 0 73563 386386 2019-01-05T09:08:29Z Prime.mover 59 Created page with \"== Cayley Table of Dihedral Group $D_3$ ==  The Cayley table of the Definition:Dihed...\" wikitext text/x-wiki == Cayley Table of Dihedral Group $D_3$ == The Cayley table of the dihedral group $D_3$ can be written: <onlyinclude> :$\\begin{array}{c|cccccc}       & e     & a     & a^2   & b     & a b   & a^2 b \\\\ \\hline e     & e     & a     & a^2   & b     & a b   & a^2 b \\\\ a     & a     & a^2   & e     & a b   & a^2 b & b     \\\\ a^2   & a^2   & e     & a     & a^2 b & b     & a b   \\\\ b     & b     & a^2 b & a b   & e     & a^2   & a     \\\\ a b   & a b   & b     & a^2 b & a     & e     & a^2   \\\\ a^2 b & a^2 b & a b   & b     & a^2   & a     & e     \\\\ \\end{array}$ </onlyinclude> where the group presentation of $D_3$ is given as: {{:Group Presentation of Dihedral Group D3}} Category:Examples of Cayley Tables Category:Dihedral Group D3 89aknx2cuypw4ge7xzprwq9gksdrk06"}
+{"_id": "32994", "title": "Standard Parity Check Matrix/Examples/(6, 3) code in Z2", "text": "Standard Parity Check Matrix/Examples/(6, 3) code in Z2 0 73781 387552 387549 2019-01-13T10:35:07Z Prime.mover 59 wikitext text/x-wiki == Example of Standard Parity Check Matrix == <onlyinclude> Let $C$ be the linear $\\tuple {6, 3}$-code in $\\Z_2$ whose standard generator matrix $G$ is given by: :$G := \\begin{pmatrix} 1 & 0 & 0 & 1 & 1 & 0 \\\\  0 & 1 & 0 & 1 & 0 & 1 \\\\ 0 & 0 & 1 & 0 & 1 & 1 \\end{pmatrix}$ Its standard parity check matrix $P$ is given by: :$P := \\begin{pmatrix} 1 & 1 & 0 & 1 & 0 & 0 \\\\  1 & 0 & 1 & 0 & 1 & 0 \\\\ 0 & 1 & 1 & 0 & 0 & 1 \\end{pmatrix}$ </onlyinclude> == Proof == Expressing $G$ in the form: :$G = \\paren {\\begin{array} {c|c} \\mathbf I_k & \\mathbf A \\end{array} }$ it is seen that: :$\\mathbf A = \\begin{pmatrix} 1 & 1 & 0 \\\\  1 & 0 & 1 \\\\ 0 & 1 & 1 \\end{pmatrix}$ It is noted that $\\mathbf A^\\intercal$ is: :$\\mathbf A^\\intercal =  \\begin{pmatrix} 1 & 1 & 0 \\\\  1 & 0 & 1 \\\\ 0 & 1 & 1 \\end{pmatrix}$ as $\\mathbf A$ is symmetrical about the main diagonal. Then each of the elements of $\\Z_2$ is self-inverse, so: :$-\\mathbf A^\\intercal = \\mathbf A^\\intercal$ {{Qed}} == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Definition:Standard Parity Check Matrix|next = Definition:Syndrome}}: Chapter $6$: Error-correcting codes: Example $6.18$ Category:Examples of Linear Codes qr1jfa03pvbul95tdgxdv9khqnvkcg1"}
+{"_id": "32995", "title": "Syndrome Decoding/Examples/(6, 3) code in Z2", "text": "Syndrome Decoding/Examples/(6, 3) code in Z2 0 73802 387561 2019-01-13T13:16:59Z Prime.mover 59 Created page with \"== Example of Syndrome Decoding == <onlyinclude> Let $C$ be the linear code:  :$C = \\set {000000, 100110, 010101, 110011, 001011, 101101, 011110...\" wikitext text/x-wiki == Example of Syndrome Decoding == <onlyinclude> Let $C$ be the linear code: :$C = \\set {000000, 100110, 010101, 110011, 001011, 101101, 011110, 111000}$ Then the Syndrome Decoding table $T$ for $C$ is: :$\\begin{array} {cc} 000000 & 000 \\\\ 100000 & 110 \\\\ 010000 & 101 \\\\ 001000 & 011 \\\\ 000100 & 100 \\\\ 000010 & 010 \\\\ 000001 & 001 \\\\ 100001 & 111 \\\\ \\end{array}$ </onlyinclude> To find the $8$th row, it is not necessary to hunt directly an element $v$ of $\\map V {6, 2}$ of weight $2$ which does not exist anywhere in the other $7$ rows. What you do is identify the remaining syndrome of $3$ digits that has not been used yet (that is: $111$). Then you work out what combinations of coset leaders and their own syndromes which when added together make that last syndrome. The fact that in this case that combination is not unique (here we get $100001$, $010010$, $001100$ and $000111$) means that an element of $\\map V {6, 2}$ which is more than $2$ distant from a codeword cannot be uniquely decoded. == Example == {{:Syndrome Decoding/Examples/(6, 3) code in Z2/Example}} == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Syndrome Decoding|next = Syndrome Decoding/Examples/(6, 3) code in Z2/Example}}: Chapter $6$: Error-correcting codes: Example $6.22$ Category:Examples of Linear Codes qdysn310ovs76bzkwix1spvm2ut3zse"}
+{"_id": "32996", "title": "Dihedral Group D4/Cayley Table", "text": "Dihedral Group D4/Cayley Table 0 73840 387808 387779 2019-01-15T07:52:51Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Dihedral Group $D_4$ == The Cayley table for the dihedral group $D_4$, whose group presentation is: {{:Group Presentation of Dihedral Group D4}} can be presented as: <onlyinclude> :$\\begin{array}{l|cccccccc}        &     e &     a &   a^2 &   a^3 &     b &   b a & b a^2 & b a^3 \\\\ \\hline  e     &     e &     a &   a^2 &   a^3 &     b &   b a & b a^2 & b a^3 \\\\  a     &     a &   a^2 &   a^3 &     e & b a^3 &     b &   b a & b a^2 \\\\  a^2   &   a^2 &   a^3 &     e &     a & b a^2 & b a^3 &     b &   b a \\\\  a^3   &   a^3 &     e &     a &   a^2 &   b a & b a^2 & b a^3 &     b \\\\  b     &     b &   b a & b a^2 & b a^3 &     e &     a &   a^2 &   a^3 \\\\  b a   &   b a & b a^2 & b a^3 &     b &   a^3 &     e &     a &   a^2 \\\\  b a^2 & b a^2 & b a^3 &     b &   b a &   a^2 &   a^3 &     e &     a \\\\  b a^3 & b a^3 &     b &   b a & b a^2 &     a &   a^2 &   a^3 &     e \\end{array}$ </onlyinclude> === Coset Decomposition of $\\set {e, a^2}$ === Presenting the above Cayley table with respect to the coset decomposition of the normal subgroup $\\gen a^2$ gives: {{:Dihedral Group D4/Cayley Table/Coset Decomposition of (e, a^2)}} === Coset Decomposition of $\\set {e, a, a^2, a^3}$ === Presenting the above Cayley table with respect to the coset decomposition of the normal subgroup $\\gen a$ gives: {{:Dihedral Group D4/Cayley Table/Coset Decomposition of (e, a, a^2, a^3)}} == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Dihedral Group D4/Normal Subgroups/Subgroup Generated by a^2/Quotient Group|next = Dihedral Group D4/Cayley Table/Coset Decomposition of (e, a^2)}}: Chapter $7$: Normal subgroups and quotient groups: Example $7.13$ Category:Dihedral Group D4 Category:Examples of Cayley Tables lk9n3g9c3rc1tk0fyai15cfnc7c0q8i"}
+{"_id": "32997", "title": "Quaternion Group/Subgroups", "text": "Quaternion Group/Subgroups 0 73851 387863 387860 2019-01-15T11:20:53Z Prime.mover 59 wikitext text/x-wiki == Subgroups of the Quaternion Group == Let $Q$ denote the quaternion group, whose group presentation is given as: {{:Group Presentation of Quaternion Group}} <onlyinclude> The subsets of $Q$ which form subgroups of $Q$ are: {{begin-eqn}} {{eqn | o =        | r = Q }} {{eqn | o =        | r = \\set e }} {{eqn | o =        | r = \\set {e, a^2} }} {{eqn | o =        | r = \\set {e, a, a^2, a^3} }} {{eqn | o =        | r = \\set {e, b, a^2, a^2 b} }} {{eqn | o =        | r = \\set {e, a b, a^2, a^3 b} }} {{end-eqn}} From Quaternion Group is Hamiltonian we have that all of these subgroups of $Q$ are normal. </onlyinclude> == Proof == Consider the Cayley table for $Q$: {{:Quaternion Group/Cayley Table}} We have that: :$a^4 = e$ and so $\\gen a = \\set {e, a, a^2, a^3}$ forms a subgroup of $Q$ which is cyclic. We have that: :$b^2 = a^2$ and so $\\gen b = \\set {e, b, a^2, a^2 b}$ forms a subgroup of $Q$ which is cyclic. We have that: :$\\paren {a b}^2 = a^2$ and so $\\gen {a b} = \\set {e, a b, a^2, a^3 b}$ forms a subgroup of $Q$ which is cyclic. We have that: :$\\paren {a^2}^2 = e$ and so $\\gen {a^2} = \\set {e, a^2}$ forms a subgroup of $Q$ which is also a subgroup of $\\gen a$, $\\gen b$ and $\\gen {a b}$. That exhausts all elements of $Q$. Any subgroup generated by any $2$ elements of $Q$ which are not both in the same subgroup as described above will generate the whole of $Q$. {{qed}} == Also see == * Quaternion Group is Hamiltonian == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Symmetric Group on 3 Letters/Normal Subgroups|next = Quaternion Group is Hamiltonian}}: Chapter $7$: Normal subgroups and quotient groups: Exercise $3$ Category:Quaternion Group 7liu2qz525m458f5b51dpwqwtyj0y46"}
+{"_id": "32998", "title": "Dihedral Group D4/Subgroups", "text": "Dihedral Group D4/Subgroups 0 73855 497569 497568 2020-11-02T16:20:32Z Prime.mover 59 wikitext text/x-wiki == Subgroups of the Dihedral Group $D_4$ == Let the dihedral group $D_4$ be represented by its group presentation: {{:Group Presentation of Dihedral Group D4}} <onlyinclude> The subsets of $D_4$ which form subgroups of $D_4$ are: {{begin-eqn}} {{eqn | o =        | r = D_4 }} {{eqn | o =        | r = \\set e }} {{eqn | o =        | r = \\set {e, a, a^2, a^3} }} {{eqn | o =        | r = \\set {e, a^2} }} {{eqn | o =        | r = \\set {e, b} }} {{eqn | o =        | r = \\set {e, b a} }} {{eqn | o =        | r = \\set {e, b a^2} }} {{eqn | o =        | r = \\set {e, b a^3} }} {{eqn | o =        | r = \\set {e, a^2, b, b a^2} }} {{eqn | o =        | r = \\set {e, a^2, b a, b a^3} }} {{end-eqn}} </onlyinclude> == Proof == Consider the Cayley table for $D_4$: {{:Dihedral Group D4/Cayley Table}} We have that: :$a^4 = e$ and so $\\gen a = \\set {e, a, a^2, a^3}$ forms a subgroup of $D_4$ which is cyclic. We have that: :$\\paren {a^2}^2 = e$ and so $\\gen {a^2} = \\set {e, a^2}$ forms a subgroup of $D_4$ which is cyclic, and also a subgroup of $\\gen a$. We have that: :$b^2 = e$ and so $\\gen b = \\set {e, b}$ forms a subgroup of $D_4$ which is cyclic. We have that: :$\\paren {b a}^2 = e$ and so $\\gen {b a} = \\set {e, b a}$ forms a subgroup of $D_4$ which is cyclic. We have that: :$\\paren {b a^2}^2 = e$ and so $\\gen {b a^2} = \\set {e, b a^2}$ forms a subgroup of $D_4$ which is cyclic. We have that: :$\\paren {b a^3}^2 = e$ and so $\\gen {b a^3} = \\set {e, b a^3}$ forms a subgroup of $D_4$ which is cyclic. Then we have that: :$b a^2 = a^2 b$ and so $\\gen {a^2, b} = \\set {e, a^2, b, b a^2}$ forms a subgroup of $D_4$ which is not cyclic, but which has subgroups $\\set {e, a^2}$, $\\set {e, b}$, $\\set {e, b a^2}$. Then we have that: :$b a^3 = a^2 b a$ and so $\\gen {a^2, b a} = \\set {e, a^2, b a, b a^3}$ forms a subgroup of $D_4$ which is not cyclic, but which has subgroups $\\set {e, a^2}$, $\\set {e, b}$, $\\set {e, b a^2}$. That exhausts all elements of $D_4$. Any subgroup generated by any $2$ elements of $Q$ which are not both in the same subgroup as described above generate the whole of $D^4$. {{qed}} Category:Dihedral Group D4 h8f00ilfqs8wjrsvc502y7dpox94vsu"}
+{"_id": "32999", "title": "Dihedral Group D4/Center", "text": "Dihedral Group D4/Center 0 73863 387887 387886 2019-01-15T21:08:09Z Prime.mover 59 wikitext text/x-wiki == Center of the Dihedral Group $D_4$ == Let $D_4$ denote the dihedral group $D_4$, whose group presentation is given as: {{:Group Presentation of Dihedral Group D4}} <onlyinclude> The center of $D_4$ is given by: :$\\map Z {D_4} = \\set {e, a^2}$ </onlyinclude> == Proof == From Center of Dihedral Group: :$\\map Z {D_n} = \\begin{cases} e & : n \\text { odd} \\\\ \\set {e, \\alpha^{n / 2} } & : n \\text { even} \\end{cases}$ Hence the result. {{qed}} == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Symmetric Group on 3 Letters/Center|next = Definition:Group Homomorphism}}: Chapter $7$: Normal subgroups and quotient groups: Exercise $5$ Category:Centers of Groups Category:Dihedral Group D4 cswd14eda2ndj39sr0wh4es0d3hluuv"}
+{"_id": "33000", "title": "Group Action of Symmetric Group on Complex Vector Space/Orbit/Examples/Example 1", "text": "Group Action of Symmetric Group on Complex Vector Space/Orbit/Examples/Example 1 0 73994 417080 388751 2019-08-07T15:42:10Z Prime.mover 59 wikitext text/x-wiki == Example of Orbit of Group Action of Symmetric Group on Complex Vector Space == Let $S_n$ denote the symmetric group on $n$ letters. Let $V$ denote a vector space over the complex numbers $\\C$. Let $V$ have a basis: :$\\mathcal B := \\set {v_1, v_2, \\ldots, v_n}$ Let $*: S_n \\times V \\to V$ be a group action of $S_n$ on $V$ defined as: :$\\forall \\tuple {\\rho, v} \\in S_n \\times V: \\rho * v := \\lambda_1 v_{\\map \\rho 1} + \\lambda_2 v_{\\map \\rho 2} + \\dotsb + \\lambda_n v_{\\map \\rho n}$ where: :$v = \\lambda_1 v_1 + \\lambda_2 v_2 + \\dotsb + \\lambda_n v_n$ <onlyinclude> Let $n = 4$. Let $v = v_1 + v_2 + v_3 + v_4$. The orbit of $v$ is: :$\\Orb v = \\set v$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\Orb v       | r = \\set {w \\in V: \\exists \\rho \\in S_4: w = \\rho * v}       | c = {{Defof|Orbit (Group Theory)|Orbit}} }} {{eqn | r = \\set {w \\in V: \\exists \\rho \\in S_4: w = \\rho * \\sum_{k \\mathop = 1}^4 v_k}       | c = Definition of $v$ }} {{eqn | r = \\set {w \\in V: \\exists \\rho \\in S_4: w = \\sum_{k \\mathop = 1}^4 v_\\map \\rho k}       | c = Definition of $*$ }} {{eqn | r = \\set {w \\in V: \\exists \\rho \\in S_4: w = \\sum_{k \\mathop = 1}^4 v_k}       | c = Permutation of Indices of Summation }} {{eqn | r = \\set {w \\in V: \\exists \\rho \\in S_4: w = v}       | c = Definition of $v$ }} {{eqn | r = \\set v       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Group Action of Symmetric Group on Complex Vector Space/Stabilizer|next = Group Action of Symmetric Group on Complex Vector Space/Stabilizer/Examples/Example 1}}: Chapter $10$: The Orbit-Stabiliser Theorem: Exercise $1 \\ \\text {(a)}$ Category:Group Action of Symmetric Group on Complex Vector Space bvxiiee6p1aslbmoipu19e3qdvvc2ia"}
+{"_id": "33001", "title": "Sylow Theorems/Examples/Sylow 3-Subgroups in Group of Order 12", "text": "Sylow Theorems/Examples/Sylow 3-Subgroups in Group of Order 12 0 74043 388972 2019-01-21T21:07:43Z Prime.mover 59 Created page with \"== Example of Use of Sylow Theorems == <onlyinclude> In a group of order $12$, there are either $1$ or $4$ Definition:...\" wikitext text/x-wiki == Example of Use of Sylow Theorems == <onlyinclude> In a group of order $12$, there are either $1$ or $4$ Sylow $3$-subgroups. </onlyinclude> == Proof == Let $G$ be a group of order $12$. Let $n_3$ be the number of Sylow $3$-subgroups in $G$. From the Fourth Sylow Theorem, $n_3$ is congruent to $1$ modulo $3$, that is, in $\\set {1, 4, 7, \\ldots}$ Let $H$ be a Sylow $3$-subgroup of $G$. We have that: :$12 = 4 \\times 3$ and so the order of $H$ is $3$. Thus: {{begin-eqn}} {{eqn | l = \\index G H       | r = \\dfrac {12} {3}       | c =  }} {{eqn | r = 4       | c =  }} {{end-eqn}} From the Fifth Sylow Theorem: :$n_3 \\divides 4$ where $\\divides$ denotes divisibility. Thus there may be $1$ or $4$ Sylow $3$-subgroups of $G$. {{qed}} == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Sylow Theorems/Examples/Sylow 2-Subgroups in Group of Order 12|next = Direct Product of Normal Subgroups is Normal}}: Chapter $11$: The Sylow Theorems: Exercise $1 \\ \\text{(e)}$ Category:Sylow Theorems Category:Groups of Order 12 i082pikg0cnwnvas2h0g72fnb5xdgqw"}
+{"_id": "33002", "title": "Pullback is Subgroup", "text": "Pullback is Subgroup 0 74142 389466 389454 2019-01-24T16:18:20Z Prime.mover 59 wikitext text/x-wiki {{disambig}} * Pullback of Quotient Group Isomorphism is Subgroup Category:Group Theory 1bgtgm1hui7kmw21di56qb30x84d37i"}
+{"_id": "33003", "title": "Pullback of Quotient Group Isomorphism/Examples/Subgroups of Index 2", "text": "Pullback of Quotient Group Isomorphism/Examples/Subgroups of Index 2 0 74155 389510 389500 2019-01-24T21:43:39Z Prime.mover 59 wikitext text/x-wiki == Example of Pullback of Quotient Group Isomorphism == <onlyinclude> Let $G$ and $H$ be groups. Let $N$ and $K$ be normal subgroups of $G$ and $H$ respectively such that: :their quotient groups $G / N$ and $H / K$ are isomorphic :their indices are $2$: ::$\\index G N = \\index H K = 2$ Let $\\theta: G / N \\to H / K$ be an isomorphism. The pullback of $G$ and $H$ by $\\theta$ is a subset of $G \\times H$ of the form: :$G \\times^\\theta H = \\set {\\tuple {g, h}: \\paren {g \\in N, h \\in K} \\text { or } \\paren {g \\notin N, h \\notin K} }$ </onlyinclude> == Proof == As $\\index G N = \\index H K = 2$, it follows that: :$\\order {G / N} = \\order {H / K} = 2$ and they are the cyclic group of order $2$. Let: : $x \\in G: x \\notin N$ : $y \\in H: y \\notin K$ Then: :$G / N = \\gen {x N}$ :$H / K = \\gen {y K}$ and we have: {{begin-eqn}} {{eqn | l = \\map \\theta N       | r = K }} {{eqn | l = \\map \\theta {x N}       | r = y K }} {{end-eqn}} Let $\\tuple {g, h} \\in G \\times^\\theta H$. By definition: :$G \\times^\\theta H = \\set {\\tuple {g, h}: \\map \\theta {g N} = h K}$ Let $g \\in N$. Then $g N = N$ and so: : $\\map \\theta {g N} = h K = K$ So $g \\in N \\implies h \\in K$. Let $g \\notin N$. Then $g N = x N$ and so: : $\\map \\theta {g N} = h K = x K$ So $g \\notin N \\implies h \\notin K$. Hence the result. {{qed}} == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Pullback of Quotient Group Isomorphism is Subgroup|next = Pullback of Quotient Group Isomorphism/Examples/Alternating Subgroups of Symmetric Groups}}: Chapter $13$: Direct products: Example $13.12$ Category:Examples of Pullbacks of Quotient Group Isomorphisms tgclnb62ygmyfrq9ftv21h2wws2sqvg"}
+{"_id": "33004", "title": "Arcsin as an Integral/Lemma 1", "text": "Arcsin as an Integral/Lemma 1 0 74302 390719 390718 2019-01-30T14:50:47Z Pelliott 3564 sin_A inverse arcsin_a wikitext text/x-wiki == Lemma == <onlyinclude> Let $sin_A$ be the analytic sine function for real numbers, the one defined by Definition:Sine/Real Numbers. $\\arcsin_A$ is the inverse of this function. :$\\displaystyle \\map {\\arcsin_A} x = \\int_0^x \\frac {\\d x} {\\sqrt {1 - x^2} }$ </onlyinclude> == Proof == For this proof only, let $\\sin_A$ be the analytic sine function from Definition:Sine/Real Numbers. Consider: :$\\displaystyle \\int_0^x \\frac {\\d x} {\\sqrt {1 - x^2} }$ Let: :$x = \\sin_A \\theta \\iff x = \\map {\\arcsin_A} \\theta$ Then: {{begin-eqn}} {{eqn | l = \\d x       | r = \\cos_A \\theta \\rd \\theta       | c = Derivative of Sine Function }} {{eqn | l = \\int \\frac {\\d x} {\\sqrt {1 - x^2} }       | r = \\int \\frac {\\d x} {\\cos_A \\theta} \\cos_A \\theta \\rd \\theta       | c = Integration by Substitution }} {{eqn | r = \\int 1 \\rd \\theta }} {{eqn | r = \\theta + C }} {{eqn | r = \\map {\\arcsin_A} x + C }} {{eqn | ll= \\leadsto       | l = \\int_0^x \\frac {\\d x} {\\sqrt {1 - x^2} }       | r = \\map {\\arcsin_A} x       | c = Fundamental Theorem of Calculus: Second Part }} {{end-eqn}} {{qed|lemma}} chldcq0gp73np0qqonhd2zb3nt5sh13"}
+{"_id": "33005", "title": "Arcsin as an Integral/Lemma 2", "text": "Arcsin as an Integral/Lemma 2 0 74304 390774 390773 2019-01-31T06:27:22Z Prime.mover 59 Reverted edits by Prime.mover (talk) to last revision by Pelliott wikitext text/x-wiki == Lemma == <onlyinclude> Let, $\\sin_G$ be the Geometric Sine from Definition:Sine/Definition from Circle. $\\arcsin_G$ is the inverse of this function. :$\\displaystyle \\map {\\arcsin_G} x = \\int_0^x \\frac {\\d x} {\\sqrt {1 - x^2} }$ </onlyinclude> == Proof == This result will be used in proving Derivative of Sine Function in the geometric case. So we can not use the same reasoning as Arcsin as an Integral/Lemma 1 because our logic would be circular. :640px {{improve|Might be worth revisiting the diagram to make all the text the same size}} Let $\\theta$ be the length of the arc associated with the angle on the circle of radius $1$. By definition of arcsine: :$y = \\sin \\theta \\iff \\theta = \\arcsin y$ We have that arc length is always positive.  For negative $y$, the $\\arcsin$ function is defined as being the negative of the arc length. This makes the $\\arcsin$ function and the $\\sin$ function odd, and puts us in line with mathematical convention: :Inverse Sine is Odd Function. :Sine Function is Odd Without this convention, the derivative of the $\\sin$ function would not be continuous.  Now: {{begin-eqn}} {{eqn | l = x^2 + y^2       | r = 1       | c = Equation of Circle       | n = 1 }} {{eqn | l = \\dfrac {\\d x} {\\d y}       | r = -\\dfrac y x       | c = Implicit Differentiation }} {{eqn | r = -\\dfrac y {\\sqrt {1 - y^2} }       | c = substituting for $x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \\arcsin_G y       | r = \\int_0^y \\sqrt {1 + \\paren {\\dfrac {\\d x} {\\d y} }^2} \\rd y       | c = {{Defof|Arc Length}} }} {{eqn | r = \\int_0^y \\sqrt {1 + \\paren {-\\dfrac y x}^2}       | c = substituting for $\\dfrac {\\d x} {\\d y}$ }} {{eqn | r = \\int_0^y \\sqrt {1 + \\dfrac {y^2} {x^2} } \\rd y }} {{eqn | r = \\int_0^y \\sqrt {\\dfrac {x^2} {x^2} + \\dfrac {y^2} {x^2} } \\rd y       | c = rewriting $1$ to create common denominator }} {{eqn | r = \\int_0^y \\sqrt {\\dfrac {x^2 + y^2} {x^2} } \\rd y       | c = combining terms with common denominator }} {{eqn | r = \\int_0^y \\sqrt {\\dfrac 1 {x^2} } \\rd y       | c = Equation of Circle $(1)$ }} {{eqn | r = \\int_0^y \\dfrac 1 x \\rd y       | c = in Quadrant $\\text I$ and Quadrant $\\text {IV}$ }} {{eqn | r = \\int_0^y \\dfrac 1 {\\sqrt {1 - y^2} } \\rd y       | c = substituting for $x$ in Quadrant $\\text I$ and Quadrant $\\text {IV}$ }} {{end-eqn}} {{qed|lemma}} l2gxb81370tgmmd6hllwvmdl8u05l8q"}
+{"_id": "33006", "title": "Modulo Arithmetic/Examples/n(n^2-1)(3n-2) Modulo 24", "text": "Modulo Arithmetic/Examples/n(n^2-1)(3n-2) Modulo 24 0 74319 390579 2019-01-29T21:13:22Z Prime.mover 59 Created page with \"== Example of Modulo Arithmetic == <onlyinclude> :$n \\paren {n^2 - 1} \\paren {3 n + 2} \\equiv 0 \\pmod {24}$ </onlyinclude>  == Proof ==  The p...\" wikitext text/x-wiki == Example of Modulo Arithmetic == <onlyinclude> :$n \\paren {n^2 - 1} \\paren {3 n + 2} \\equiv 0 \\pmod {24}$ </onlyinclude> == Proof == The proof proceeds by induction. For all $n \\in \\Z_{> 0}$, let $\\map P n$ be the proposition: :$n \\paren {n^2 - 1} \\paren {3 n + 2} \\equiv 0 \\pmod {24}$ === Basis for the Induction === $\\map P 1$ is the case: {{begin-eqn}} {{eqn | l = 1 \\paren {1^2 - 1} \\paren {3 \\times 1 + 2}       | r = 0       | c =  }} {{eqn | o = \\equiv       | r = 0       | rr= \\pmod {24}       | c =  }} {{end-eqn}} Thus $\\map P 1$ is seen to hold. This is the basis for the induction. === Induction Hypothesis === Now it needs to be shown that, if $\\map P k$ is true, where $k \\ge 1$, then it logically follows that $\\map P {k + 1}$ is true. So this is the induction hypothesis: :$k \\paren {k^2 - 1} \\paren {3 k + 2} \\equiv 0 \\pmod {24}$ from which it is to be shown that: :$\\paren {k + 1} \\paren {\\paren {k + 1}^2 - 1} \\paren {3 \\paren {k + 1} + 2} \\equiv 0 \\pmod {24}$ === Induction Step === This is the induction step: {{begin-eqn}} {{eqn | l = \\paren {k + 1} \\paren {\\paren {k + 1}^2 - 1} \\paren {3 \\paren {k + 1} - 2}       | r = k \\paren {k + 1} \\paren {k + 2} \\paren {3 k + 5}       | c =  }} {{eqn | r = k \\paren {k + 1} \\paren {k - 1 + 3} \\paren {3 k + 2 + 3}       | c =  }} {{eqn | r = k \\paren {k + 1} \\paren {k - 1} \\paren {3 k + 2 + 3} + 3 k \\paren {k + 1} \\paren {3 k + 2 + 3}       | c =  }} {{eqn | r = k \\paren {k + 1} \\paren {k - 1} \\paren {3 k + 2} + 3 k \\paren {k + 1} \\paren {k - 1} + 3 k \\paren {k + 1} \\paren {3 k + 5}       | c =  }} {{eqn | r = k \\paren {k^2 - 1} \\paren {3 k + 2} + 3 k \\paren {k + 1} \\paren {k - 1 + 3 k + 5}       | c =  }} {{eqn | r = k \\paren {k^2 - 1} \\paren {3 k + 2} + 3 k \\paren {k + 1} \\paren {4 k + 4}       | c =  }} {{eqn | r = k \\paren {k^2 - 1} \\paren {3 k + 2} + 12 k \\paren {k + 1}^2       | c =  }} {{end-eqn}} By the induction hypothesis: :$k \\paren {k^2 - 1} \\paren {3 k + 2} = 24 r$ for some $r \\in \\Z$. Take $12 k \\paren {k + 1}^2$. If $k$ is even, then $12 k$ and so $12 k \\paren {k + 1}^2$ is divisible by $24$. If $k$ is odd, then $k + 1$ is even and so $12 k \\paren {k + 1}^2$ is again divisible by $24$. Thus: :$12 k \\paren {k + 1}^2 = 24 s$ for some $s \\in \\Z$. Thus: :$k \\paren {k^2 - 1} \\paren {3 k + 2} + 12 k \\paren {k + 1}^2 = 24 \\paren {r + s}$ So $\\map P k \\implies \\map P {k + 1}$ and the result follows by the Principle of Mathematical Induction. Therefore: :$\\forall n \\in \\Z_{>0}: n \\paren {n^2 - 1} \\paren {3 n + 2} \\equiv 0 \\pmod {24}$ {{qed}} == Sources == * {{BookReference|Number Theory|1971|George E. Andrews|prev = Sum of Sequence of Product of Lucas Numbers with Powers of 2|next = Sum of Odd Positive Powers}}: $\\text {1-1}$ Principle of Mathematical Induction: Exercise $18$ Category:Examples of Modulo Arithmetic 040thy9kquhouj6rzuegnhscy352s0e"}
+{"_id": "33007", "title": "Euclidean Algorithm/Examples/341 and 527", "text": "Euclidean Algorithm/Examples/341 and 527 0 74394 493846 391022 2020-10-11T06:57:10Z Prime.mover 59 wikitext text/x-wiki == Examples of Use of Euclidean Algorithm == <onlyinclude> The GCD of $341$ and $527$ is found to be: :$\\gcd \\set {341, 527} = 31$ ==== Integer Combination ==== {{:Euclidean Algorithm/Examples/341 and 527/Integer Combination}}</onlyinclude> == Proof == {{begin-eqn}} {{eqn | n = 1       | l = 527       | r = 1 \\times 341 + 186 }} {{eqn | n = 2       | l = 341       | r = 1 \\times 186 + 155 }} {{eqn | n = 3       | l = 186       | r = 1 \\times 155 + 31 }} {{eqn | n =        | l = 155       | r = 5 \\times 31 }} {{end-eqn}} Thus: :$\\gcd \\set {341, 527} = 31$ {{qed}} == Sources == * {{BookReference|Number Theory|1971|George E. Andrews|prev = GCD of Integer and Divisor|next = Euclidean Algorithm}}: $\\text {2-2}$ Divisibility: Example $\\text {2-7}$ Category:Examples of Euclidean Algorithm 1tp7629hnw1wa04yq9vndlb9a8x1e6g"}
+{"_id": "33008", "title": "Lowest Common Multiple/Examples/n and n+1", "text": "Lowest Common Multiple/Examples/n and n+1 0 74418 391109 391100 2019-02-02T13:17:31Z Prime.mover 59 wikitext text/x-wiki == Example of Lowest Common Multiple of Integers == Let $n \\in \\Z_{>0}$ be a strictly positive integer. <onlyinclude> The lowest common multiple of $n$ and $n + 1$ is: :$\\lcm \\set {n, n + 1} = n \\paren {n + 1}$ </onlyinclude> == Proof == We find the greatest common divisor of $n$ and $n + 1$ using the Euclidean Algorithm: {{begin-eqn}} {{eqn | n = 1       | l = n + 1       | r = 1 \\times n + 1 }} {{eqn | n = 2       | l = n       | r = n \\times 1 }} {{end-eqn}} Thus $\\gcd \\set {n, n + 1} = 1$. Hence by definition $n$ and $n + 1$ are coprime. The result follows from LCM of Coprime Integers. {{qed}} == Sources == * {{BookReference|Number Theory|1971|George E. Andrews|prev = Lowest Common Multiple/Examples/28 and 29|next = Lowest Common Multiple of Consecutive Odd Integers}}: $\\text {2-2}$ Divisibility: Exercise $5 \\ \\text {(e)}$ Category:Examples of Lowest Common Multiples tenqrtcuqhx6ex6zgf2ko269dorzlak"}
+{"_id": "33009", "title": "Area of Triangle in Determinant Form with Vertex at Origin", "text": "Area of Triangle in Determinant Form with Vertex at Origin 0 74481 391400 391394 2019-02-04T07:47:41Z Prime.mover 59 wikitext text/x-wiki == Example of Area of Triangle in Determinant Form == <onlyinclude> Let $A = \\tuple {0, 0}, B = \\tuple {b, a}, C = \\tuple {x, y}$ be points in the Cartesian plane. Let $T$ the triangle whose vertices are at $A$, $B$ and $C$. Then the area $\\mathcal A$ of $T$ is: :$\\map \\Area T = \\dfrac {\\size {b y - a x} } 2$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\map \\Area T       | r = \\dfrac 1 2 \\size {\\paren {\\begin{vmatrix} 0 & 0 & 1 \\\\ b & a & 1 \\\\ x & y & 1 \\end{vmatrix} } }       | c = Area of Triangle in Determinant Form }} {{eqn | r = \\dfrac 1 2 \\size {b y - a x}       | c = {{Defof|Determinant of Order 3}} }} {{end-eqn}} {{qed}} == Example == {{:Area of Triangle in Determinant Form with Vertex at Origin/Example}} == Sources == * {{BookReference|Number Theory|1971|George E. Andrews|prev = Linear Diophantine Equation/Examples/15x + 51y = 41/Graph|next = Area of Triangle in Determinant Form with Vertex at Origin/Example}}: $\\text {2-3}$ The Linear Diophantine Equation: Exercise $4$ Category:Area of Triangle in Determinant Form e4vesh61jg84w2k8j6f36tt46abd0bs"}
+{"_id": "33010", "title": "Square Modulo 3/Corollary 3", "text": "Square Modulo 3/Corollary 3 0 74617 476278 462461 2020-06-28T15:53:22Z RandomUndergrad 3904 wikitext text/x-wiki == Corollary to Square Modulo 3 == <onlyinclude> Let $n \\in \\Z$ be an integer such that: :$3 \\nmid n$ where $\\nmid$ denotes non-divisibility. Then: :$3 \\divides n^2 - 1$ where $\\divides$ denotes divisibility. </onlyinclude> == Proof == From Square Modulo 3: :$n \\equiv 0 \\pmod 3 \\iff n^2 \\equiv 0 \\pmod 3$ Hence also from Square Modulo 3: :$n \\not \\equiv 0 \\pmod 3 \\iff n^2 \\equiv 1 \\pmod 3$ That is: $3 \\nmid n \\iff 3 \\divides n^2 - 1$ {{qed}} == Sources == * {{BookReference|Number Theory|1971|George E. Andrews|prev = Integer and Fifth Power have same Last Digit|next = Sufficient Condition for 5 to divide n^2+1}}: $\\text {3-2}$ Fermat's Little Theorem: Exercise $5$ Category:Square Modulo 3 n39xgma4sjtzzqmqv79rj9wyejujslo"}
+{"_id": "33011", "title": "Quadratic Residue/Examples/3", "text": "Quadratic Residue/Examples/3 0 74687 392366 392356 2019-02-14T07:45:47Z Prime.mover 59 wikitext text/x-wiki == Example of Quadratic Residues == <onlyinclude> There exists exactly $1$ quadratic residue modulo $3$, and that is $1$. </onlyinclude> == Proof == To list the quadratic residues of $3$ it is enough to work out the squares $1^2$ and $2^2$ modulo $3$. {{begin-eqn}} {{eqn | l = 1^2       | o = \\equiv       | r = 1       | rr= \\pmod 3 }} {{eqn | l = 2^2       | o = \\equiv       | r = 1       | rr= \\pmod 3 }} {{end-eqn}} So the set of quadratic residues modulo $3$ is: :$\\set 1$ The set of quadratic non-residues of $3$ therefore consists of all the other non-zero least positive residues: :$\\set 2$ {{qed}} == Sources == * {{BookReference|Number Theory|1971|George E. Andrews|prev = Quadratic Residue/Examples/11|next = Quadratic Residue/Examples/5}}: $\\text {3-5}$ The Use of Computers in Number Theory: Exercise $6$ Category:Examples of Quadratic Residues Category:3 hychsymwxsahklszp3zsz49zhllthfr"}
+{"_id": "33012", "title": "Quadratic Residue/Examples/5", "text": "Quadratic Residue/Examples/5 0 74688 392365 392355 2019-02-14T07:45:36Z Prime.mover 59 wikitext text/x-wiki == Example of Quadratic Residues == <onlyinclude> The set of quadratic residues modulo $5$ is: :$\\set {1, 4}$ </onlyinclude> == Proof == To list the quadratic residues of $5$ it is enough to work out the squares $1^2, 2^2, 3^2, 4^2$ modulo $5$. {{begin-eqn}} {{eqn | l = 1^2       | o = \\equiv       | r = 1       | rr= \\pmod 5 }} {{eqn | l = 2^2       | o = \\equiv       | r = 4       | rr= \\pmod 5 }} {{eqn | l = 3^2       | o = \\equiv       | r = 4       | rr= \\pmod 5 }} {{eqn | l = 4^2       | o = \\equiv       | r = 1       | rr= \\pmod 5 }} {{end-eqn}} So the set of quadratic residues modulo $5$ is: :$\\set {1, 4}$ The set of quadratic non-residues of $5$ therefore consists of all the other non-zero least positive residues: :$\\set {2, 3}$ {{qed}} == Sources == * {{BookReference|Number Theory|1971|George E. Andrews|prev = Quadratic Residue/Examples/3|next = Quadratic Residue/Examples/7}}: $\\text {3-5}$ The Use of Computers in Number Theory: Exercise $6$ Category:Examples of Quadratic Residues Category:5 siq4yv3zvjp16uejhkb1053p13l75s9"}
+{"_id": "33013", "title": "Quadratic Residue/Examples/7", "text": "Quadratic Residue/Examples/7 0 74689 392364 392354 2019-02-14T07:45:22Z Prime.mover 59 wikitext text/x-wiki == Example of Quadratic Residues == <onlyinclude> The set of quadratic residues modulo $7$ is: :$\\set {1, 2, 4}$ </onlyinclude> == Proof == To list the quadratic residues of $7$ it is enough to work out the squares $1^2, 2^2, \\dotsc, 6^2$ modulo $7$. {{begin-eqn}} {{eqn | l = 1^2       | o = \\equiv       | r = 1       | rr= \\pmod 7 }} {{eqn | l = 2^2       | o = \\equiv       | r = 4       | rr= \\pmod 7 }} {{eqn | l = 3^2       | o = \\equiv       | r = 2       | rr= \\pmod 7 }} {{eqn | l = 4^2       | o = \\equiv       | r = 2       | rr= \\pmod 7 }} {{eqn | l = 5^2       | o = \\equiv       | r = 4       | rr= \\pmod 7 }} {{eqn | l = 6^2       | o = \\equiv       | r = 1       | rr= \\pmod 7 }} {{end-eqn}} So the set of quadratic residues modulo $7$ is: :$\\set {1, 2, 4}$ The set of quadratic non-residues of $7$ therefore consists of all the other non-zero least positive residues: :$\\set {3, 5, 6}$ {{qed}} == Sources == * {{BookReference|Number Theory|1971|George E. Andrews|prev = Quadratic Residue/Examples/5|next = Quadratic Residue/Examples/17}}: $\\text {3-5}$ The Use of Computers in Number Theory: Exercise $6$ Category:Examples of Quadratic Residues Category:7 3igdqrcv9cp30opd1bnv3momeyleese"}
+{"_id": "33014", "title": "Quadratic Residue/Examples/17", "text": "Quadratic Residue/Examples/17 0 74690 433460 392363 2019-11-01T13:30:35Z Prime.mover 59 wikitext text/x-wiki == Example of Quadratic Residues == <onlyinclude> The set of quadratic residues modulo $17$ is: :$\\set {1, 2, 4, 8, 9, 13, 15, 16}$ </onlyinclude> {{OEIS|A010379}} == Proof == To list the quadratic residues of $17$ it is enough to work out the squares $1^2, 2^2, \\dotsc, 16^2$ modulo $17$. {{begin-eqn}} {{eqn | l = 1^2       | o = \\equiv       | r = 1       | rr= \\pmod {17} }} {{eqn | l = 2^2       | o = \\equiv       | r = 4       | rr= \\pmod {17} }} {{eqn | l = 3^2       | o = \\equiv       | r = 9       | rr= \\pmod {17} }} {{eqn | l = 4^2       | o = \\equiv       | r = 16       | rr= \\pmod {17} }} {{eqn | l = 5^2       | o = \\equiv       | r = 8       | rr= \\pmod {17} }} {{eqn | l = 6^2       | o = \\equiv       | r = 2       | rr= \\pmod {17} }} {{eqn | l = 7^2       | o = \\equiv       | r = 15       | rr= \\pmod {17} }} {{eqn | l = 8^2       | o = \\equiv       | r = 13       | rr= \\pmod {17} }} {{eqn | l = 9^2       | o = \\equiv       | r = 13       | rr= \\pmod {17} }} {{eqn | l = 10^2       | o = \\equiv       | r = 15       | rr= \\pmod {17} }} {{eqn | l = 11^2       | o = \\equiv       | r = 2       | rr= \\pmod {17} }} {{eqn | l = 12^2       | o = \\equiv       | r = 8       | rr= \\pmod {17} }} {{eqn | l = 13^2       | o = \\equiv       | r = 16       | rr= \\pmod {17} }} {{eqn | l = 14^2       | o = \\equiv       | r = 9       | rr= \\pmod {17} }} {{eqn | l = 15^2       | o = \\equiv       | r = 4       | rr= \\pmod {17} }} {{eqn | l = 16^2       | o = \\equiv       | r = 1       | rr= \\pmod {17} }} {{end-eqn}} So the set of quadratic residues modulo $17$ is: :$\\set {1, 2, 4, 8, 9, 13, 15, 16}$ The set of quadratic non-residues of $17$ therefore consists of all the other non-zero least positive residues: :$\\set {3, 5, 6, 7, 10, 11, 12, 14}$ {{OEIS|A028730}}{{qed}} == Sources == * {{BookReference|Number Theory|1971|George E. Andrews|prev = Quadratic Residue/Examples/7|next = Quadratic Residue/Examples/29}}: $\\text {3-5}$ The Use of Computers in Number Theory: Exercise $6$ Category:Examples of Quadratic Residues Category:17 5bdaz6nnpoy6ye4vre6u1ec5wxq45oc"}
+{"_id": "33015", "title": "Quadratic Residue/Examples/29", "text": "Quadratic Residue/Examples/29 0 74691 433461 392370 2019-11-01T13:31:01Z Prime.mover 59 wikitext text/x-wiki == Example of Quadratic Residues == <onlyinclude> The set of quadratic residues modulo $29$ is: :$\\set {1, 4, 5, 6, 7, 9, 13, 16, 20, 22, 23, 24, 25, 28}$ </onlyinclude> {{OEIS|A010391}} == Proof == From Square Modulo n Congruent to Square of Inverse Modulo n, to list the quadratic residues of $29$ it is sufficient to work out the squares $1^2, 2^2, \\dotsc, \\paren {\\dfrac {28} 2}^2$ modulo $29$. So: {{begin-eqn}} {{eqn | l = 1^2       | o = \\equiv       | r = 1       | rr= \\pmod {29} }} {{eqn | l = 2^2       | o = \\equiv       | r = 4       | rr= \\pmod {29} }} {{eqn | l = 3^2       | o = \\equiv       | r = 9       | rr= \\pmod {29} }} {{eqn | l = 4^2       | o = \\equiv       | r = 16       | rr= \\pmod {29} }} {{eqn | l = 5^2       | o = \\equiv       | r = 25       | rr= \\pmod {29} }} {{eqn | l = 6^2       | o = \\equiv       | r = 7       | rr= \\pmod {29} }} {{eqn | l = 7^2       | o = \\equiv       | r = 20       | rr= \\pmod {29} }} {{eqn | l = 8^2       | o = \\equiv       | r = 6       | rr= \\pmod {29} }} {{eqn | l = 9^2       | o = \\equiv       | r = 23       | rr= \\pmod {29} }} {{eqn | l = 10^2       | o = \\equiv       | r = 13       | rr= \\pmod {29} }} {{eqn | l = 11^2       | o = \\equiv       | r = 5       | rr= \\pmod {29} }} {{eqn | l = 12^2       | o = \\equiv       | r = 28       | rr= \\pmod {29} }} {{eqn | l = 13^2       | o = \\equiv       | r = 24       | rr= \\pmod {29} }} {{eqn | l = 14^2       | o = \\equiv       | r = 22       | rr= \\pmod {29} }} {{end-eqn}} So the set of quadratic residues modulo $29$ is: :$\\set {1, 4, 5, 6, 7, 9, 13, 16, 20, 22, 23, 24, 25, 28}$ The set of quadratic non-residues of $29$ therefore consists of all the other non-zero least positive residues: :$\\set {2, 3, 8, 10, 11, 12, 14, 15, 17, 18, 19, 21, 26, 27}$ {{OEIS|A028742}}{{qed}} == Sources == * {{BookReference|Number Theory|1971|George E. Andrews|prev = Quadratic Residue/Examples/17|next = Quadratic Residue/Examples/61}}: $\\text {3-5}$ The Use of Computers in Number Theory: Exercise $6$ Category:Examples of Quadratic Residues Category:29 fwkl4anq98hwx4k3p3b5ma1cxx5z5p1"}
+{"_id": "33016", "title": "Reduced Residue System/Examples/Modulo 18/Least Positive Residues", "text": "Reduced Residue System/Examples/Modulo 18/Least Positive Residues 0 74833 392896 392895 2019-02-20T07:49:41Z Prime.mover 59 wikitext text/x-wiki == Examples of Reduced Residue Systems == <onlyinclude> The least positive reduced residue system of $18$ is the set of positive integers: :$\\set {1, 5, 7, 11, 13, 17}$ </onlyinclude> == Proof == The least positive residues of $18$ are: :$S := \\set {0, 1, 2, \\dotsc, 17}$ We have that: :$18 = 2 \\times 3^2$ so the least positive reduced residue system of $18$ is the set of elements of $S$ which have neither $2$ or $3$ as a prime factor. That is: :$S \\setminus \\set {0, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16}$ Hence the result. {{Qed}} == Sources == * {{BookReference|Number Theory|1971|George E. Andrews|prev = Reduced Residue System/Examples/Modulo 18/Square Numbers|next = Congruence Modulo Power of p as Linear Combination of Congruences Modulo p}}: $\\text {4-2}$ Residue Systems: Exercise $2 \\ \\text {(d)}$ Category:Examples of Residue Systems 48gahudxbhrdrfntwdnfmgl02jw3rqk"}
+{"_id": "33017", "title": "Binomial Theorem/Examples/Cube of Difference", "text": "Binomial Theorem/Examples/Cube of Difference 0 74934 393424 393370 2019-02-23T11:47:30Z Prime.mover 59 wikitext text/x-wiki == Example of Use of Binomial Theorem == <onlyinclude> :$\\paren {x - y}^3 = x^3 - 3 x^2 y + 3 x y^2 - y^3$ </onlyinclude> == Proof == Follows directly from the Binomial Theorem: :$\\displaystyle \\forall n \\in \\Z_{\\ge 0}: \\paren {x + \\paren {-y} }^n = \\sum_{k \\mathop = 0}^n \\binom n k x^{n - k} \\paren {-y}^k$ putting $n = 3$. {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Cube of Sum|next = Fourth Power of Sum}}: $\\S 2$: Special Products and Factors: $2.4$ * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Square of Sum/Algebraic Proof 2|next = Fourth Power of Sum}}: $\\S 20$: Binomial Series: $20.6$ Category:Examples of Use of Binomial Theorem Category:Third Powers slkcwun4hpi3tv9f8xfnyqmchcwkcuv"}
+{"_id": "33018", "title": "Binomial Theorem/Examples/4th Power of Difference", "text": "Binomial Theorem/Examples/4th Power of Difference 0 74937 393380 393379 2019-02-23T11:05:01Z Prime.mover 59 Prime.mover moved page Fourth Power of Difference to Binomial Theorem/Examples/4th Power of Difference wikitext text/x-wiki == Example of Use of Binomial Theorem == <onlyinclude> :$\\paren {x - y}^4 = x^4 - 4 x^3 y + 6 x^2 y^2 - 4 x y^3 + y^4$ </onlyinclude> == Proof == Follows directly from the Binomial Theorem: :$\\displaystyle \\forall n \\in \\Z_{\\ge 0}: \\paren {x + \\paren {-y} }^n = \\sum_{k \\mathop = 0}^n \\binom n k x^{n - k} \\paren {-y}^k$ putting $n = 4$. {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Fourth Power of Sum|next = Fifth Power of Sum}}: $\\S 2$: Special Products and Factors: $2.6$ Category:Examples of Use of Binomial Theorem Category:Fourth Powers mfi4lv1t7avb4bdvyt7g8fh69w8a8ku"}
+{"_id": "33019", "title": "Square Root of Complex Number in Cartesian Form/Examples/-15-8i", "text": "Square Root of Complex Number in Cartesian Form/Examples/-15-8i 0 75200 394768 2019-03-05T20:37:56Z Prime.mover 59 Created page with \"== Example of Square Root of Complex Number in Cartesian Form == <onlyinclude> :$\\sqrt {-15 - 8 i} = \\pm \\paren {1 - 4 i}$ </onlyinclude>  == Square Root of Complex Numb...\" wikitext text/x-wiki == Example of Square Root of Complex Number in Cartesian Form == <onlyinclude> :$\\sqrt {-15 - 8 i} = \\pm \\paren {1 - 4 i}$ </onlyinclude> == Proof 1 == {{:Square Root of Complex Number in Cartesian Form/Examples/-15-8i/Proof 1}} == Proof 2 == {{:Square Root of Complex Number in Cartesian Form/Examples/-15-8i/Proof 2}} Category:Examples of Square Roots 98oypb55b6zaacw4n4kdz9vczrzctm0"}
+{"_id": "33020", "title": "Complex Dot Product/Examples/3-4i dot -4+3i", "text": "Complex Dot Product/Examples/3-4i dot -4+3i 0 75209 394832 394815 2019-03-05T23:07:31Z Prime.mover 59 wikitext text/x-wiki == Examples of Complex Dot Product == <onlyinclude> Let: :$z_1 = 3 - 4 i$ :$z_2 = -4 + 3 i$ Then: :$z_1 \\circ z_2 = -24$ where $\\circ$ denotes (complex) dot product. </onlyinclude> == Proof 1 == {{:Complex Dot Product/Examples/3-4i dot -4+3i/Proof 1}} == Proof 2 == {{:Complex Dot Product/Examples/3-4i dot -4+3i/Proof 2}} Category:Examples of Complex Dot Product eg5l3le7w8vqt9naxty6h6qiylu358n"}
+{"_id": "33021", "title": "Condition for Points in Complex Plane to form Isosceles Triangle", "text": "Condition for Points in Complex Plane to form Isosceles Triangle 0 75372 395335 2019-03-11T07:44:51Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> Let $A = z_1 = x_1 + i y_1$, $B = z_2 = x_2 + i y_2$ and $C = z_3 = x_3 + i y_3$ represent on the complex plane the ...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $A = z_1 = x_1 + i y_1$, $B = z_2 = x_2 + i y_2$ and $C = z_3 = x_3 + i y_3$ represent on the complex plane the vertices of a triangle. Then $\\triangle ABC$ is isosceles, where $A$ is the apex, {{iff}}: :${x_2}^2 + {y_2}^2 - 2 \\paren {x_1 x_2 + y_1 y_2} = {x_3}^2 + {y_3}^2 - 2 \\paren {x_1 x_3 + y_1 y_3}$ </onlyinclude> == Proof == By definition of isosceles triangle: :$\\triangle ABC$ is isosceles, where $A$ is the apex, {{iff}} $AB = AC$. Hence: {{begin-eqn}} {{eqn | l = \\cmod {z_1 - z_2}       | r = \\cmod {z_1 - z_3}       | c =  }} {{eqn | ll= \\leadstoandfrom       | l = \\cmod {x_1 + i y_1 - x_2 + i y_2}^2       | r = \\cmod {x_1 + i y_1 - x_3 + i y_3}^2       | c =  }} {{eqn | ll= \\leadstoandfrom       | l = \\paren {x_1 - x_2}^2 + \\paren {y_1 - y_2}^2       | r = \\paren {x_1 - x_3}^2 + \\paren {y_1 - y_3}^2       | c = {{Defof|Complex Modulus}} }} {{eqn | ll= \\leadstoandfrom       | l = {x_1}^2 - 2 x_1 x_2 + {x_2}^2 + {y_1}^2 - 2 y_1 y_2 + {y_2}^2       | r = {x_1}^2 - 2 x_1 x_3 + {x_3}^2 + {y_1}^2 - 2 y_1 y_3 + {y_3}^2       | c =  }} {{eqn | ll= \\leadstoandfrom       | l = {x_2}^2 + {y_2}^2 - 2 \\paren {x_1 x_2 + y_1 y_2}       | r = {x_3}^2 + {y_3}^2 - 2 \\paren {x_1 x_3 + y_1 y_3}       | c =  }} {{end-eqn}} {{qed}} == Examples == {{:Condition for Points in Complex Plane to form Isosceles Triangle/Examples}} Category:Isosceles Triangles Category:Geometry of Complex Plane 0lnphteume90914gxuq2b9xzokin3qx"}
+{"_id": "33022", "title": "Equation of Hyperbola in Complex Plane", "text": "Equation of Hyperbola in Complex Plane 0 75404 395426 2019-03-12T06:57:38Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> Let $\\C$ be the complex plane.  Let $H$ be a hyperbola in $\\C$ whose Definition:Major Axis...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $\\C$ be the complex plane. Let $H$ be a hyperbola in $\\C$ whose major axis is $d \\in \\R_{>0}$ and whose foci are at $\\alpha, \\beta \\in \\C$. Then $C$ may be written as: :$\\cmod {z - \\alpha} - \\cmod {z - \\beta} = d$ where $\\cmod {\\, \\cdot \\,}$ denotes complex modulus. </onlyinclude> == Proof == By definition of complex modulus: :$\\cmod {z - \\alpha}$ is the distance from $z$ to $\\alpha$ :$\\cmod {z - \\beta}$ is the distance from $z$ to $\\beta$. Thus $\\cmod {z - \\alpha} - \\cmod {z - \\beta}$ is the difference of the distance from $z$ to $\\alpha$ and from $z$ to $\\beta$. This is precisely the equidistance property of the hyperbola. From Equidistance of Hyperbola equals Transverse Axis, the constant distance $d$ is equal to the transverse axis of $H$. {{qed}} == Examples == {{:Equation of Hyperbola in Complex Plane/Examples}} Category:Hyperbolas Category:Geometry of Complex Plane Category:Equation of Hyperbola in Complex Plane d6w7jvqz2etr8cp7z27p3nl5n2t38kf"}
+{"_id": "33023", "title": "Complex Algebra/Examples/z^4 - 3z^2 + 1 = 0", "text": "Complex Algebra/Examples/z^4 - 3z^2 + 1 = 0 0 75476 441412 441410 2019-12-29T23:47:35Z Prime.mover 59 wikitext text/x-wiki == Example of Complex Algebra == <onlyinclude> The roots of the equation: :$z^4 - 3z^2 + 1 = 0$ are: :$2 \\cos 36 \\degrees, 2 \\cos 72 \\degrees, 2 \\cos 216 \\degrees, 2 \\cos 252 \\degrees$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = z^4 - 3z^2 + 1       | r = z^4 - 2z^2 + 1 - z^2       | c = Separating the $z^2$ term }} {{eqn | r = (z^2 - 1)^2 - z^2       | c = Completing the Square }} {{eqn | r = (z^2 - 1 - z)(z^2 - 1 + z)       | c = Difference of Two Squares }} {{end-eqn}} From the Quadratic Formula applied to each of the above quadratic factors we can easily see that the four roots are: :$\\dfrac {\\pm 1 \\pm \\sqrt 5} 2$ $360 \\degrees = 2 \\pi \\radians$, so $72 \\degrees = 2 \\pi / 5 \\radians$ From De Moivre's Formula, the roots of $x^5 - 1 = 0$ are: :$(\\cos(2 n \\pi) + i \\sin(2 n \\pi))^{1/5}= \\cos \\dfrac {2 n \\pi} 5 + i \\sin \\dfrac {2 n \\pi} 5$ However, the coefficient of $x^4$ is $0$ and therefore, by Viète's Formulas, the sum of the roots of $x^5 - 1 = 0$ are also $0$ which means that the sum of the real parts of the roots are also $0$: {{begin-eqn}} {{eqn | l = 0       | r = \\cos 0 + \\cos \\dfrac {2 \\pi} 5 + \\cos \\dfrac {4 \\pi} 5 + \\cos \\dfrac {6 \\pi} 5 + \\cos \\dfrac {8 \\pi} 5       | c }} {{eqn | r = 1 + \\cos \\dfrac {2 \\pi} 5 + \\cos \\dfrac {4 \\pi} 5 + \\cos \\dfrac {6 \\pi} 5 + \\cos \\dfrac {8 \\pi} 5       | c }} {{end-eqn}} Now: {{begin-eqn}} {{eqn | l = \\cos(\\pi + x)       | r = -\\cos(x)       | c = Cosine of Angle plus Straight Angle }} {{eqn | r = \\cos(-x)       | c = Cosine Function is Even }} {{eqn | r = -\\cos(\\pi - x)       | c = Cosine of Angle plus Straight Angle }} {{eqn | r = \\cos(\\pi - x)       | c = Cosine Function is Even }} {{end-eqn}} Hence: {{begin-eqn}} {{eqn | l = \\cos \\dfrac {6 \\pi} 5       | r = \\map {\\cos} {\\pi + \\dfrac \\pi 5}       | c = }} {{eqn | r = \\map {\\cos} {\\pi - \\dfrac \\pi 5}       | c = }} {{eqn | r = \\cos \\dfrac {4 \\pi} 5       | c = }} {{end-eqn}} and {{begin-eqn}} {{eqn | l = \\cos \\dfrac {8 \\pi} 5       | r = \\map {\\cos} {\\pi + \\dfrac {3 \\pi} 5}       | c = }} {{eqn | r = \\map {\\cos} {\\pi - \\dfrac {3 \\pi} 5}       | c = }} {{eqn | r = \\cos \\dfrac {2 \\pi} 5       | c = }} {{end-eqn}} We can now simplify the sum of the real parts of the roots of $x^5 - 1 = 0$: {{begin-eqn}} {{eqn | l = 0       | r = 1 + \\cos \\dfrac {2 \\pi} 5 + \\cos \\dfrac {4 \\pi} 5 + \\cos \\dfrac {6 \\pi} 5 + \\cos \\dfrac {8 \\pi} 5       | c = }} {{eqn | r = 1 + 2 \\cos \\dfrac {2 \\pi} 5 + 2 \\cos \\dfrac {4 \\pi} 5       | c = $\\cos(\\pi + x) = \\cos(\\pi - x)$ }} {{eqn | r = 1 + 2 \\cos \\dfrac {2 \\pi} 5 + 4 \\cos^2 \\dfrac {2 \\pi} 5 - 2       | c = Double Angle Formula for Cosine, $\\cos 2 \\theta = 2 \\cos^2 \\theta - 1$ }} {{eqn | r = 4 \\cos^2 \\dfrac {2 \\pi} 5 + 2 \\cos \\dfrac {2 \\pi} 5 - 1       | c = simplifying }} {{end-eqn}} From the Quadratic Formula we then have two '''potential''' values: :$\\cos \\dfrac {2 \\pi} 5 = \\dfrac {-1 \\pm \\sqrt 5} 4$ $0 < 2 \\pi / 5 < \\pi / 2$, so we know that $\\cos \\dfrac {2 \\pi} 5 > 0$, hence: :$2 \\cos 72 \\degrees = 2 \\cos \\dfrac {2 \\pi} 5 = \\dfrac {-1 + \\sqrt 5} 2$ From Cosine of Angle plus Straight Angle, $\\map \\cos {x + 180 \\degrees} = -\\cos x$, hence: :$2 \\cos 252 \\degrees = \\dfrac {1 - \\sqrt 5} 2$ Now: {{begin-eqn}} {{eqn | l = 2 \\cos 36 \\degrees       | r = 2 \\cos \\dfrac \\pi 5       | c = }} {{eqn | r = 2 \\map {\\cos} {\\dfrac {- \\pi} 5}       | c = Cosine Function is Even }} {{eqn | r = - 2 \\map {\\cos} {\\pi - \\dfrac \\pi 5}       | c = Cosine of Angle plus Straight Angle }} {{eqn | r = - 2 \\map {\\cos} {\\dfrac {4 \\pi} 5}       | c = simplifying }} {{eqn | r = 1 + 2 \\cos \\dfrac {2 \\pi} 5       | c = Sum of the real parts, $1 + 2 \\cos \\dfrac {2 \\pi} 5 + 2 \\cos \\dfrac {4 \\pi} 5 = 0$ }} {{eqn | r = 1 + \\dfrac {-1 + \\sqrt 5} 2       | c =  }} {{eqn | r = \\dfrac {1 + \\sqrt 5} 2       | c =  }} {{end-eqn}} From Cosine of Angle plus Straight Angle, $\\map \\cos {x + 180 \\degrees} = -\\cos x$, hence: :$2 \\cos 216 \\degrees = \\dfrac {- 1 - \\sqrt 5} 2$ We have therefore shown that the four roots of $z^4 - 3z^2 + 1 = 0$ are $\\dfrac {\\pm 1 \\pm \\sqrt 5} 2$ and that these four values are also equal to $2 \\cos 36 \\degrees, 2 \\cos 72 \\degrees, 2 \\cos 216 \\degrees, 2 \\cos 252 \\degrees$ {{qed}} == Sources == * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Triple Angle Formula for Cosine|next = Cosine of 36 Degrees/Proof 2}}: $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $91$ Category:Examples of Complex Algebra khrbdqo2y3dxrz51j3mg5370ewlxrmp"}
+{"_id": "33024", "title": "Complex Dot Product/Examples/2+5i dot 3-i", "text": "Complex Dot Product/Examples/2+5i dot 3-i 0 75561 395995 395983 2019-03-17T12:19:53Z Prime.mover 59 wikitext text/x-wiki == Example of Complex Dot Product == <onlyinclude> Let: :$z_1 = 2 + 5 i$ :$z_2 = 3 - i$ Then: :$z_1 \\circ z_2 = 1$ where $\\circ$ denotes (complex) dot product. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = z_1 \\circ z_2       | r = \\paren {2 + 5 i} \\circ \\paren {3 - i}       | c =  }} {{eqn | r = 2 \\times 3 + 5 \\times \\paren {-1}       | c = {{Defof|Dot Product|subdef = Complex|index = 1}} }} {{eqn | r = 6 - 5       | c =  }} {{eqn | r = 1       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Complex Algebra/Examples/(1+z)^5 = (1-z)^5|next = Complex Cross Product/Examples/2+5i cross 3-i}}: $1$: Complex Numbers: Supplementary Problems: The Dot and Cross Product: $110 \\ \\text {(a)}$ Category:Examples of Complex Dot Product 0r746nc4d54hjyo2e5id9idadr1g0o9"}
+{"_id": "33025", "title": "Complex Cross Product/Examples/2+5i cross 3-i", "text": "Complex Cross Product/Examples/2+5i cross 3-i 0 75562 395984 2019-03-17T11:32:37Z Prime.mover 59 Created page with \"== Example of Complex Cross Product == <onlyinclude> Let: :$z_1 = 2 + 5 i$ :$z_2 = 3 - i$ Then: :$z_1 \\times z_2 = -17$ where $\\times$ den...\" wikitext text/x-wiki == Example of Complex Cross Product == <onlyinclude> Let: :$z_1 = 2 + 5 i$ :$z_2 = 3 - i$ Then: :$z_1 \\times z_2 = -17$ where $\\times$ denotes (complex) cross product. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = z_1 \\circ z_2       | r = \\paren {2 + 5 i} \\times \\paren {3 - i}       | c =  }} {{eqn | r = 2 \\times \\paren {-1} - 3 \\times 5       | c = {{Defof|Vector Cross Product|subdef = Complex|index = 1|Complex Cross Product}} }} {{eqn | r = -2 - 15       | c =  }} {{eqn | r = -17       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Complex Dot Product/Examples/2+5i dot 3-i|next = Complex Dot Product/Examples/3-i dot 2+5i}}: $1$: Complex Numbers: Supplementary Problems: The Dot and Cross Product: $110 \\ \\text {(b)}$ Category:Examples of Complex Cross Product 1so6w2l42oop38p1lok9kwclfkrmzy5"}
+{"_id": "33026", "title": "Complex Dot Product/Examples/3-i dot 2+5i", "text": "Complex Dot Product/Examples/3-i dot 2+5i 0 75563 395985 2019-03-17T11:33:50Z Prime.mover 59 Created page with \"== Examples of Complex Dot Product == <onlyinclude> Let: :$z_1 = 3 - i$ :$z_1 = 2 + 5 i$ Then: :$z_1 \\circ z_2 = 1$ where $\\circ$ denotes ...\" wikitext text/x-wiki == Examples of Complex Dot Product == <onlyinclude> Let: :$z_1 = 3 - i$ :$z_1 = 2 + 5 i$ Then: :$z_1 \\circ z_2 = 1$ where $\\circ$ denotes (complex) dot product. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = z_1 \\circ z_2       | r = \\paren {3 - i} \\circ \\paren {2 + 5 i}       | c =  }} {{eqn | r = 3 \\times 2 + \\paren {-1} \\times 5       | c = {{Defof|Dot Product|subdef = Complex|index = 1}} }} {{eqn | r = 6 - 5       | c =  }} {{eqn | r = 1       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Complex Cross Product/Examples/2+5i cross 3-i|next = Complex Cross Product/Examples/3-i cross 2+5i}}: $1$: Complex Numbers: Supplementary Problems: The Dot and Cross Product: $110 \\ \\text {(c)}$ Category:Examples of Complex Dot Product jnyt7mr50lf1umwf9f3uzoywtl1p3jo"}
+{"_id": "33027", "title": "Complex Cross Product/Examples/3-i cross 2+5i", "text": "Complex Cross Product/Examples/3-i cross 2+5i 0 75568 395997 395996 2019-03-17T12:24:37Z Prime.mover 59 wikitext text/x-wiki == Example of Complex Cross Product == <onlyinclude> Let: :$z_1 = 3 - i$ :$z_2 = 2 + 5 i$ Then: :$z_1 \\times z_2 = 17$ where $\\times$ denotes (complex) cross product. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = z_1 \\circ z_2       | r = \\paren {3 - i} \\times \\paren {2 + 5 i}       | c =  }} {{eqn | r = 3 \\times \\paren 5 - \\paren {-1} \\times 2       | c = {{Defof|Vector Cross Product|subdef = Complex|index = 1|Complex Cross Product}} }} {{eqn | r = 15 - \\paren {-2}       | c =  }} {{eqn | r = 17       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Complex Dot Product/Examples/3-i dot 2+5i|next = Complex Dot Product/Examples/Size of 2+5i dot 3-i}}: $1$: Complex Numbers: Supplementary Problems: The Dot and Cross Product: $110 \\ \\text {(d)}$ Category:Examples of Complex Cross Product 3diaq2k5gsoeu2jimbxz9j8ddhrl80x"}
+{"_id": "33028", "title": "Property of Group Automorphism which Fixes Identity Only/Corollary 2", "text": "Property of Group Automorphism which Fixes Identity Only/Corollary 2 0 75766 397082 397081 2019-03-24T13:41:56Z Prime.mover 59 wikitext text/x-wiki == Corollary to Property of Group Automorphism which Fixes Identity Only == Let $G$ be a finite group whose identity is $e$. Let $\\phi: G \\to G$ be a group automorphism. Let $\\phi$ have the property that: :$\\forall g \\in G \\setminus \\set e: \\map \\phi t \\ne t$ That is, the only fixed element of $\\phi$ is $e$. <onlyinclude> Let: :$\\phi^2 = I_G$ where $I_G$ denotes the identity mapping on $G$. Then: :$\\forall g \\in G: \\map \\phi g = g^{-1}$ </onlyinclude> == Proof == Let $g \\in G$. Then: {{begin-eqn}} {{eqn | lo= \\exists x \\in G:       | l = \\map \\phi g       | r = \\map \\phi {x^{-1} \\, \\map \\phi x}       | c = Corollary 1 }} {{eqn | r = \\paren {\\map \\phi x}^{-1} x       | c =  }} {{eqn | r = g^{-1}       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Property of Group Automorphism which Fixes Identity Only/Corollary 1|next = Property of Group Automorphism which Fixes Identity Only/Corollary 3}}: Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $26$ Category:Property of Group Automorphism which Fixes Identity Only 5gql12bgeelztra6b58ljh7glyqgfui"}
+{"_id": "33029", "title": "Ideal of Ring/Examples/Order 2 Matrices with some Zero Entries", "text": "Ideal of Ring/Examples/Order 2 Matrices with some Zero Entries 0 75836 398716 398710 2019-04-01T06:57:59Z Prime.mover 59 wikitext text/x-wiki == Example of Ideal of Ring == <onlyinclude> Let $R$ be the set of all order $2$ square matrices of the form $\\begin{pmatrix} x & y \\\\ 0 & z \\end{pmatrix}$ with $x, y, z \\in \\R$. Let $S$ be the set of all order $2$ square matrices of the form $\\begin{pmatrix} x & y \\\\ 0 & 0 \\end{pmatrix}$ with $x, y \\in \\R$. Then $R$ is a ring and $S$ is an ideal of $R$. </onlyinclude> === Corollary === {{:Ideal of Ring/Examples/Order 2 Matrices with some Zero Entries/Corollary}} == Proof 1 == {{:Ideal of Ring/Examples/Order 2 Matrices with some Zero Entries/Proof 1}} == Proof 2 == {{:Ideal of Ring/Examples/Order 2 Matrices with some Zero Entries/Proof 2}} Category:Examples of Ideals of Rings 2k524j7dyy588n7efhx04zowmioa2vr"}
+{"_id": "33030", "title": "Cyclotomic Ring/Examples/5th", "text": "Cyclotomic Ring/Examples/5th 0 76116 399083 399078 2019-04-05T05:01:24Z Prime.mover 59 wikitext text/x-wiki == Examples of Cyclotomic Rings == <onlyinclude> The '''$5$th cyclotomic ring''' is the algebraic structure: :$\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ where $\\Z \\sqbrk {i \\sqrt 5}$ is the set $\\set {a + i b \\sqrt 5: a, b \\in \\Z}$. $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ is a ring. </onlyinclude> == Proof == We have that $\\Z \\sqbrk {i \\sqrt 5}$ is a subset of the Field of Complex Numbers $\\struct {\\C, +, \\times}$. So to prove that $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ is a ring it is sufficient to demonstrate that $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ fulfils the conditions of the Subring Test. First we note that setting $a = 1, b = 0$ we have that $1 + 0 i \\in \\Z \\sqbrk {i \\sqrt 5}$ and so $\\Z \\sqbrk {i \\sqrt 5} \\ne \\O$. Let $z_1 = a_1 + i b_1 \\sqrt 5$ and $z_2 = a_2 + i b_2 \\sqrt 5$ be arbitrary elements of $\\Z \\sqbrk {i \\sqrt 5}$ Then: {{begin-eqn}} {{eqn | l = z_1 - z_2       | r = \\paren {a_1 + i b_1 \\sqrt 5} - \\paren {a_2 + i b_2 \\sqrt 5}       | c =  }} {{eqn | r = \\paren {a_1 - a_2} + i \\paren {b_1 - b_2} \\sqrt 5       | c = {{Defof|Complex Addition}} }} {{eqn | o = \\in       | r = \\Z \\sqbrk {i \\sqrt 5}       | c = as $a_1 - a_2$ and $b_1 - b_2$ are both integers }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = z_1 z_2       | r = \\paren {a_1 + i b_1 \\sqrt 5} \\paren {a_2 + i b_2 \\sqrt 5}       | c =  }} {{eqn | r = \\paren {a_1 a_2 - 5 b_1 b_2} + i \\sqrt 5 \\paren {a_1 b_2 + a_2 b_1}       | c = {{Defof|Complex Multiplication}} }} {{eqn | o = \\in       | r = \\Z \\sqbrk {i \\sqrt 5}       | c = as $a_1 a_2 - 5 b_1 b_2$ and $a_1 b_2 + a_2 b_1$ are both integers }} {{end-eqn}} The Subring Test is satisfied, and so $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ is a ring. {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Field Norm of Complex Number is Multiplicative Function|next = Field Norm on 5th Cyclotomic Ring}}: Chapter $9$: Rings: Exercise $19$ Category:Cyclotomic Rings png51kxp7hzzwt12dzlxh9d5ya3oj6z"}
+{"_id": "33031", "title": "Matrix Entrywise Addition/Examples/Real 2 x 2", "text": "Matrix Entrywise Addition/Examples/Real 2 x 2 0 76177 399417 2019-04-07T13:43:52Z Prime.mover 59 Created page with \"== Example of Matrix Entrywise Addition == <onlyinclude> Let $\\mathbf A = \\begin {pmatrix} p & q \\\\ r & s \\end {pmatrix}$ and $\\mathbf...\" wikitext text/x-wiki == Example of Matrix Entrywise Addition == <onlyinclude> Let $\\mathbf A = \\begin {pmatrix} p & q \\\\ r & s \\end {pmatrix}$ and $\\mathbf B = \\begin {pmatrix} w & x \\\\ y & z \\end {pmatrix}$ be order $2$ square matrices over the real numbers. Then the matrix sum of $\\mathbf A$ and $\\mathbf B$ is given by: :$\\mathbf A + \\mathbf B = \\begin {pmatrix} p + w & q + x \\\\ r + y & s + z \\end {pmatrix}$ </onlyinclude> == Sources == * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Square Matrix/Examples/Real 2 x 2|next = Matrix Product (Conventional)/Examples/2 x 2 Real Matrices}}: Chapter $1$: Integral Domains: $\\S 3$. Definition of an Integral Domain: Example $3$ Category:Examples of Matrix Entrywise Addition 4chl61eqf35mue4u9kw7nimhso3gh3z"}
+{"_id": "33032", "title": "Matrix Product (Conventional)/Examples/2 x 2 Real Matrices", "text": "Matrix Product (Conventional)/Examples/2 x 2 Real Matrices 0 76178 399456 399418 2019-04-07T22:37:11Z Prime.mover 59 wikitext text/x-wiki == Example of (Conventional) Matrix Product == <onlyinclude> Let $\\mathbf A = \\begin {pmatrix} p & q \\\\ r & s \\end {pmatrix}$ and $\\mathbf B = \\begin {pmatrix} w & x \\\\ y & z \\end {pmatrix}$ be order $2$ square matrices over the real numbers. Then the matrix product of $\\mathbf A$ with $\\mathbf B$ is given by: :$\\mathbf A \\mathbf B = \\begin {pmatrix} p w + q y & p x + q z \\\\ r w + s y & r x + s z \\end {pmatrix}$ </onlyinclude> == Sources == * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Matrix Entrywise Addition/Examples/Real 2 x 2|next = Ring of Square Matrices over Real Numbers/Examples/2 x 2}}: Chapter $1$: Integral Domains: $\\S 3$. Definition of an Integral Domain: Example $3$ Category:Examples of Matrix Product 8z73wwbi4jtocrp4s7psqg6r3kraskx"}
+{"_id": "33033", "title": "Irreducible Polynomial/Examples/8 x^3 - 6 x - 1 in Rationals", "text": "Irreducible Polynomial/Examples/8 x^3 - 6 x - 1 in Rationals 0 76645 402320 402308 2019-04-23T15:08:49Z Prime.mover 59 wikitext text/x-wiki == Examples of Irreducible Polynomials == <onlyinclude> Consider the polynomial: :$\\map P x = 8 x^3 - 6 x - 1$ over the ring of polynomials $\\Q \\sqbrk X$ over the rational numbers. Then $\\map P x$ is irreducible. </onlyinclude> == Proof == {{AimForCont}} $\\map P x$ has proper factors. Then one of these has to be of degree $1$. Thus from Factors of Polynomial with Integer Coefficients have Integer Coefficients we have: :$8 x^3 - 6 x - 1 = \\paren {a x + b} \\paren {c^2 + d x + e}$ for some $a, b, c, d, e \\in \\Z$. Hence: {{begin-eqn}} {{eqn | l = a c       | r = 8       | c =  }} {{eqn | ll= \\leadsto       | l = a       | o = \\divides       | r = 8       | c =  }} {{eqn | l = b e       | r = -1       | c =  }} {{eqn | ll= \\leadsto       | l = b       | o = \\divides       | r = 1       | c =  }} {{end-eqn}} The only possible degree $1$ factors with integer coefficients are: :$x \\pm 1, 2 x \\pm 1, 4 x \\pm 1, 8 x \\pm 1$ By trying each of these possibilities, it is determined that no integer value of $d$ gives the correct values. Hence the result. {{qed}} == Sources == * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Polynomial which is Irreducible over Integers is Irreducible over Rationals|next = Square Root of Prime is Irrational/Proof 2}}: Chapter $6$: Polynomials and Euclidean Rings: $\\S 31$. Polynomials with Integer Coefficients: Example $59$ Category:Examples of Irreducible Polynomials iid2qj8cu0183a2m7m4cpq3hogfbkko"}
+{"_id": "33034", "title": "Vector Space over Division Subring/Examples/Real Numbers in Complex Numbers", "text": "Vector Space over Division Subring/Examples/Real Numbers in Complex Numbers 0 76773 403683 403680 2019-05-01T11:44:59Z Prime.mover 59 wikitext text/x-wiki == Example of Vector Space over Division Subring == <onlyinclude> Consider the field of complex numbers $\\struct {\\C, +, \\times}$, which is a ring with unity whose unity is $1$. Consider the field of real numbers $\\struct {\\R, +, \\times}$, which is a division subring of $\\struct {\\C, +, \\circ}$ such that $1 \\in \\R$. Then $\\struct {\\C, +, \\times_\\R}_\\R$ is an $\\R$-vector space, where $\\times_\\R$ is the restriction of $\\times$ to $\\R \\times \\C$. $\\struct {\\C, +, \\times_\\R}_\\R$ is of dimension $2$. The set $\\paren {1 + 0 i, 0 + i}$ forms a basis of $\\struct {\\C, +, \\times_\\R}_\\R$, as do any two complex numbers which are not real multiples of each other. </onlyinclude> == Sources == * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Vector Space over Division Subring is Vector Space|next = Definition:Generator of Vector Space}}: Chapter $7$: Vector Spaces: $\\S 32$. Definition of a Vector Space: Example $64$ * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Dimension of Vector Space of Polynomial Functions|next = Linearly Independent Set is Contained in some Basis/Proof 2}}: Chapter $7$: Vector Spaces: $\\S 34$. Dimension: Example $68$ Category:Examples of Vector Spaces hhi02na2gim992gybj174abzqorm2nq"}
+{"_id": "33035", "title": "Smallest Field containing Subfield and Complex Number/Examples/Numbers of Type Rational a plus b root 2", "text": "Smallest Field containing Subfield and Complex Number/Examples/Numbers of Type Rational a plus b root 2 0 76845 458060 458058 2020-03-28T20:26:37Z Prime.mover 59 wikitext text/x-wiki == Example of Smallest Field containing Subfield and Complex Number == <onlyinclude> Let $\\Q \\sqbrk {\\sqrt 2}$ denote the set: :$\\Q \\sqbrk {\\sqrt 2} := \\set {a + b \\sqrt 2: a, b \\in \\Q}$ that is, all numbers of the form $a + b \\sqrt 2$ where $a$ and $b$ are rational numbers. Then $\\Q \\sqbrk {\\sqrt 2}$ is the smallest field containing $\\Q$ and $\\sqrt 2$. </onlyinclude> {{refactor|The following material needs to go on a page explaining splitting fields.}} Formally, $\\Q \\sqbrk {\\sqrt 2}$ is the field extension of $\\Q$ for the minimal polynomial of $\\sqrt 2$, the second-degree polynomial $x^2 - 2$. Therefore, $\\Q \\sqbrk {\\sqrt 2}$ is the vector space of dimension $2$ isomorphic to $\\Q \\sqbrk x / \\left\\langle x^2 - 2 \\right\\rangle$. == Sources == * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Definition:Smallest Field containing Subfield and Complex Number/General Definition|next = Smallest Field containing Subfield and Complex Number/Examples/Complex Numbers}}: Chapter $8$: Field Extensions: $\\S 36$. The Degree of a Field Extension: Example $72$ * {{BookReference|Contemporary Abstract Algebra|2017|Joseph A. Gallian|ed = 9th|edpage = Ninth Edition|prev = Definition:Vector Space|next = Definition:Algebraic Field Extension}}: Chapter $20$: Extension Fields: $\\S 1$. Splitting Fields Category:Examples of Field Extensions ftu2qn8d0qoqq44yx3cl24kh3zvomzr"}
+{"_id": "33036", "title": "Algebraic Number/Examples/Root of (2 plus Root 3)", "text": "Algebraic Number/Examples/Root of (2 plus Root 3) 0 76862 403850 2019-05-02T17:55:58Z Prime.mover 59 Created page with \"== Example of Algebraic Number == <onlyinclude> :$\\sqrt {2 + \\sqrt 3}$ is an algebraic number. </onlyinclude>...\" wikitext text/x-wiki == Example of Algebraic Number == <onlyinclude> :$\\sqrt {2 + \\sqrt 3}$ is an algebraic number. </onlyinclude> == Proof == Let $x = \\sqrt {2 + \\sqrt 3}$. We have that: {{begin-eqn}} {{eqn | l = x^2       | r = 2 + \\sqrt 3       | c =  }} {{eqn | ll= \\leadsto       | l = \\paren {x^2 - 2}^2       | r = 3       | c =  }} {{eqn | ll= \\leadsto       | l = x^4 - 4 x^2 + 4       | r = 3       | c =  }} {{eqn | ll= \\leadsto       | l = x^4 - 4 x^2 + 1       | r = 0       | c =  }} {{end-eqn}} Thus $\\sqrt {2 + \\sqrt 3}$ is a root of $x^4 - 4 x^2 + 1 = 0$. Hence the result by definition of algebraic number. {{qed}} </onlyinclude> == Sources == * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Algebraic Number/Examples/Imaginary Unit|next = Pi is Transcendental}}: Chapter $8$: Field Extensions: $\\S 38$. Simple Algebraic Extensions: Example $76$ Category:Examples of Algebraic Numbers qzt6n7kxbx8nk1c3ui3cbzc2ln82ngy"}
+{"_id": "33037", "title": "Group of Order 30 is not Simple", "text": "Group of Order 30 is not Simple 0 76938 404179 2019-05-04T14:22:49Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> Let $G$ be a group of order $30$.  Then $G$ is not simple. </onlyinc...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $G$ be a group of order $30$. Then $G$ is not simple. </onlyinclude> == Proof == From Group of Order 30 has Normal Cyclic Subgroup of Order 15, $G$ has a normal subgroup of order $15$. Hence the result, by definition of simple group. {{qed}} == Sources == * {{BookReference|Elements of Abstract Algebra|1971|Allan Clark|prev = Prime Group is Simple|next = Group of Order p^2 q is not Simple}}: Chapter $2$: The Sylow Theorems: $\\S 59 \\epsilon$ Category:Groups of Order 30 Category:Simple Groups qegxmxy6exuousudjz43olsur3diedn"}
+{"_id": "33038", "title": "Bessel Function of the First Kind/Instances/Order 0", "text": "Bessel Function of the First Kind/Instances/Order 0 0 77298 405970 405969 2019-05-30T08:42:23Z Prime.mover 59 wikitext text/x-wiki == Specific Instance of Bessel Functions of the First Kind == Let $\\map {J_n} x$ denote the Bessel function of the first kind of order $n$. <onlyinclude> {{begin-eqn}} {{eqn | l = \\map {J_0} x       | r = \\sum_{k \\mathop = 0}^\\infty \\dfrac {\\paren {-1}^k} {\\paren {k!}^2} \\paren {\\dfrac x 2}^{2 k}       | c =  }} {{eqn | r = 1 - \\dfrac {x^2} {2^2} + \\dfrac {x^4} {2^2 \\times 4^2} - \\dfrac {x^6} {2^2 \\times 4^2 \\times 6^2} + \\dotsb       | c =  }} {{end-eqn}} </onlyinclude> == Proof == From Series Expansion of Bessel Function of the First Kind: {{begin-eqn}} {{eqn | l = \\map {J_n} x       | r = \\dfrac {x^n} {2^n \\, \\map \\Gamma {n + 1} } \\paren {1 - \\dfrac {x^2} {2 \\paren {2 n + 2} } + \\dfrac {x^4} {2 \\times 4 \\paren {2 n + 2} \\paren {2 n + 4} } - \\cdots}       | c =  }} {{eqn | r = \\sum_{k \\mathop = 0}^\\infty \\dfrac {\\paren {-1}^k} {k! \\, \\map \\Gamma {n + k + 1} } \\paren {\\dfrac x 2}^{n + 2 k}       | c =  }} {{end-eqn}} where $n$ is not a (strictly) negative integer. $0$ fits that category, and so: {{begin-eqn}} {{eqn | l = \\map {J_0} x       | r = \\sum_{k \\mathop = 0}^\\infty \\dfrac {\\paren {-1}^k} {k! \\, \\map \\Gamma {0 + k + 1} } \\paren {\\dfrac x 2}^{0 + 2 k}       | c =  }} {{eqn | r = \\sum_{k \\mathop = 0}^\\infty \\dfrac {\\paren {-1}^k} {k! \\, \\map \\Gamma {k + 1} } \\paren {\\dfrac x 2}^{2 k}       | c =  }} {{eqn | r = \\sum_{k \\mathop = 0}^\\infty \\dfrac {\\paren {-1}^k} {\\paren {k!}^2} \\paren {\\dfrac x 2}^{2 k}       | c = Gamma Function Extends Factorial }} {{end-eqn}} Or working directly upon the terms themselves: {{begin-eqn}} {{eqn | l = \\map {J_0} x       | r = \\dfrac {x^0} {2^0 \\, \\map \\Gamma {0 + 1} } \\paren {1 - \\dfrac {x^2} {2 \\paren {2 \\times 0 + 2} } + \\dfrac {x^4} {2 \\times 4 \\paren {2 \\times 0 + 2} \\paren {2 \\times 0 + 4} } - \\cdots}       | c =  }} {{eqn | r = \\dfrac 1 {1 \\times \\map \\Gamma 1} \\paren {1 - \\dfrac {x^2} {2 \\times 2} + \\dfrac {x^4} {2 \\times 4 \\times 2 \\times 4} - \\cdots}       | c =  }} {{eqn | r = 1 - \\dfrac {x^2} {2^2} + \\dfrac {x^4} {2^2 \\times 4^2} - \\cdots       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Bessel Function of the First Kind for Negative Non-Integer Order is Unbounded|next = Bessel Function of the First Kind/Instances/Order 1}}: $\\S 24$: Bessel Functions: Bessel Function of the First Kind of Order $n$: $24.5$ Category:Examples of Bessel Functions 5wp5vajmwmg214g25lzd0sog675f43v"}
+{"_id": "33039", "title": "Laplace Transform of Dirac Delta Function/Lemma", "text": "Laplace Transform of Dirac Delta Function/Lemma 0 77407 406282 2019-06-03T06:43:29Z Prime.mover 59 Created page with \"== Lemma for Laplace Transform of Dirac Delta Function == <onlyinclude> Let $F_\\epsilon: \\R \\to \\R$ be the real function defined as:  :$\\map {...\" wikitext text/x-wiki == Lemma for Laplace Transform of Dirac Delta Function == <onlyinclude> Let $F_\\epsilon: \\R \\to \\R$ be the real function defined as: :$\\map {F_\\epsilon} t = \\begin{cases} 0 & : x < 0 \\\\ \\dfrac 1 \\epsilon & : 0 \\le t \\le \\epsilon \\\\ 0 & : t > \\epsilon \\end{cases}$ Then: :$\\laptrans {\\map {F_\\epsilon} t} = \\dfrac {1 - e^{-s \\epsilon} } {\\epsilon s}$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\laptrans {\\map {F_\\epsilon} t}       | r = \\int_0^\\infty e^{-s t} \\map {F_\\epsilon} t \\rd t       | c =  }} {{eqn | r = \\int_0^\\epsilon e^{-s t} \\map {F_\\epsilon} t \\rd t + \\int_\\epsilon^\\infty e^{-s t} \\map {F_\\epsilon} t \\rd t       | c = Sum of Integrals on Adjacent Intervals for Integrable Functions }} {{eqn | r = \\int_0^\\epsilon e^{-s t} \\dfrac 1 \\epsilon \\rd t + \\int_\\epsilon^\\infty  e^{-s t} \\times 0 \\rd t       | c = Definition of $F_\\epsilon$ }} {{eqn | r = \\dfrac 1 \\epsilon \\int_0^\\epsilon e^{-s t} \\rd t       | c = simplification }} {{eqn | r = \\dfrac 1 \\epsilon \\intlimits {\\dfrac {e^{-s t} } {-s} } 0 \\epsilon       | c = Primitive of $e^{a x}$ }} {{eqn | r = \\dfrac 1 \\epsilon \\paren {\\dfrac {e^{-s \\epsilon} - e^{-s \\times 0} } {-s} }       | c =  }} {{eqn | r = \\dfrac {1 - e^{-s \\epsilon} } {\\epsilon s}       | c = simplification }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Theory and Problems of Laplace Transforms|1965|Murray R. Spiegel|prev = Laplace Transform of Heaviside Step Function/Proof 2|next = Laplace Transform of Dirac Delta Function/Proof 2}}: Chapter $1$: The Laplace Transform: Solved Problems: Impulse Functions. The Dirac Delta Function: $41$ Category:Laplace Transform of Dirac Delta Function dec9evt703r3ni8ww9fpol3yp13c8wl"}
+{"_id": "33040", "title": "Infimum of Subset of Real Numbers/Examples/Example 3", "text": "Infimum of Subset of Real Numbers/Examples/Example 3 0 77566 407912 407276 2019-06-14T14:53:18Z Prime.mover 59 wikitext text/x-wiki == Example of Infimum of Subset of Real Numbers == <onlyinclude> The subset $V$ of the real numbers $\\R$ defined as: :$V := \\set {x \\in \\R: x > 0}$ admits an infimum: :$\\inf V = 0$ </onlyinclude> == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Supremum of Subset of Real Numbers/Examples/Example 3|next = Definition:Supremum of Subset of Real Numbers}}: $\\S 2$: Continuum Property: $\\S 2.5$: Examples: $\\text{(iii)}$ Category:Examples of Infima erold9knxsoe1w6d6sh0ju16cv8vx7x"}
+{"_id": "33041", "title": "Real Null Sequence/Examples/n^alpha x^n/Lemma", "text": "Real Null Sequence/Examples/n^alpha x^n/Lemma 0 77733 408456 408444 2019-06-16T09:09:59Z Prime.mover 59 wikitext text/x-wiki == Lemma for Real Null Sequence: $n^\\alpha x^n$ == Let $\\alpha \\in \\Q$ be a (strictly) positive rational number. Let $x \\in \\R$ be a real number such that $\\size x < 1$. <onlyinclude> There exists $N \\in \\N$ such that: :$\\paren {1 + \\dfrac 1 N}^{\\alpha + 1} \\, \\size x \\le 1$ </onlyinclude> == Proof == {{AimForCont}}: :$\\forall n \\in \\N: \\paren {1 + \\dfrac 1 n}^{\\alpha + 1} \\, \\size x > 1$ Then: {{begin-eqn}} {{eqn | lo= \\forall n \\in \\N:       | l = \\dfrac 1 n       | o = >       | r = \\paren {\\dfrac 1 {\\size x} }^{1 / \\paren {\\alpha + 1} } - 1       | c =  }} {{eqn | o = >       | r = 0       | c =  }} {{eqn | ll= \\leadsto       | l = \\paren {\\dfrac 1 {\\size x} }^{1 / \\paren {\\alpha + 1} }       | o = >       | r = 1       | c = as $\\dfrac 1 n > 0$ for all $n \\in \\N$ }} {{end-eqn}} But this contradicts Sequence of Powers of Reciprocals is Null Sequence. Hence by Proof by Contradiction: :$\\exists N \\in \\N: \\paren {1 + \\dfrac 1 N}^{\\alpha + 1} \\, \\size x \\le 1$ {{qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Power over Factorial|next = Real Null Sequence/Examples/n^alpha x^n}}: $\\S 4$: Convergent Sequences: Exercise $\\S 4.20 \\ (5)$ Category:Real Null Sequence/Examples/n^alpha x^n 9krbowsncd6pnacbratgp3py4jh6cx1"}
+{"_id": "33042", "title": "Hero's Method/Lemma 1", "text": "Hero's Method/Lemma 1 0 77808 419809 419795 2019-08-23T07:17:28Z Prime.mover 59 wikitext text/x-wiki == Lemma for Hero's Method == Let $a \\in \\R$ be a real number such that $a > 0$. Let $x_1 \\in \\R$ be a real number such that $x_1 > 0$. Let $\\sequence {x_n}$ be the sequence in $\\R$ defined recursively by: :$\\forall n \\in \\N_{>0}: x_{n + 1} = \\dfrac {x_n + \\dfrac a {x_n} } 2$ Then: <onlyinclude> :$\\forall n \\in \\N_{>0}: x_n > 0$ </onlyinclude> == Proof == The proof proceeds by induction. For all $n \\in \\Z_{>0}$, let $\\map P n$ be the proposition: :$x_n > 0$ === Basis for the Induction === $\\map P 1$ is the case: :$x_1 > 0$ which is assumed. Thus $\\map P 1$ is seen to hold. This is the basis for the induction. === Induction Hypothesis === Now it needs to be shown that if $\\map P k$ is true, where $k \\ge 1$, then it logically follows that $\\map P {k + 1}$ is true. So this is the induction hypothesis: :$x_k > 0$ from which it is to be shown that: :$x_{k + 1} > 0$ === Induction Step === This is the induction step: We have that: :$x_{k + 1} = \\dfrac {x_k + \\dfrac a {x_k} } 2$ But as $x_k > 0$ and $a > 0$, it follows that: :$\\dfrac a {x_k} > 0$ Then as $x_k > 0$ and $\\dfrac a {x_k} > 0$, it follows that: :$\\dfrac 1 2 \\paren {x_k + \\dfrac a {x_k} } > 0$ So $\\map P k \\implies \\map P {k + 1}$ and the result follows by the Principle of Mathematical Induction. Therefore: :$\\forall n \\in \\N_{>0}: x_n > 0$ {{qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Hero's Method|next = Hero's Method/Lemma 2}}: $\\S 5$: Subsequences: $\\S 5.5$: Example Category:Hero's Method 6oibp2a6jgh5au897um27rh5vca257g"}
+{"_id": "33043", "title": "Hero's Method/Lemma 2", "text": "Hero's Method/Lemma 2 0 77814 419810 419797 2019-08-23T07:18:13Z Prime.mover 59 wikitext text/x-wiki == Lemma for Hero's Method == Let $a \\in \\R$ be a real number such that $a > 0$. Let $x_1 \\in \\R$ be a real number such that $x_1 > 0$. Let $\\sequence {x_n}$ be the sequence in $\\R$ defined recursively by: :$\\forall n \\in \\N_{>0}: x_{n + 1} = \\dfrac {x_n + \\dfrac a {x_n} } 2$ Then: <onlyinclude> :$\\forall n \\ge 2: x_n \\ge \\sqrt a$ </onlyinclude> == Proof == === Lemma 1 === {{:Hero's Method/Lemma 1}}{{qed|lemma}} We have: {{begin-eqn}} {{eqn | l = x_{n + 1}       | r = \\frac {x_n + \\dfrac a {x_n} } 2       | c =  }} {{eqn | ll= \\leadstoandfrom       | l = 2 x_n x_{n + 1}       | r = x_n^2 + a       | c =  }} {{eqn | ll= \\leadstoandfrom       | l = x_n^2 - 2 x_n x_{n + 1} + a       | r = 0       | c =  }} {{end-eqn}} This is a quadratic equation in $x_n$. We know that this equation must have a real solution with respect to $x_n$, because $x_n$ has been explicitly constructed by the iterative process. Thus its discriminant is $b^2 - 4 a c \\ge 0$, where: :$a = 1$ :$b = -2 x_{n + 1}$ :$c = a$ Thus $x_{n + 1}^2 \\ge a$. From Lemma 1: :$x_{n + 1} > 0$  It follows that: :$\\forall n \\ge 1: x_{n + 1} \\ge \\sqrt a$ for $n \\ge 1$ Thus: :$\\forall n \\ge 2: x_n \\ge \\sqrt a$ for $n \\ge 2$ {{qed}} == Sources == {{SourceReview|username = Prime.mover}} * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Hero's Method/Lemma 1|next = Hero's Method/Proof 1}}: $\\S 5$: Subsequences: $\\S 5.5$: Example Category:Hero's Method fnm4qakm9gr0gsrk7pwre9y0zlkta6c"}
+{"_id": "33044", "title": "Convergent Real Sequence/Examples/x (n+1) = x n^2 + k/Lemma 1", "text": "Convergent Real Sequence/Examples/x (n+1) = x n^2 + k/Lemma 1 0 77817 408882 408871 2019-06-18T14:21:24Z Prime.mover 59 wikitext text/x-wiki == Example of Convergent Real Sequence == Let $\\sequence {x_n}$ be the real sequence defined as: :$x_n = \\begin {cases} h & : n = 1 \\\\ {x_{n - 1} }^2 + k & : n > 1 \\end {cases}$ where: :$0 < k < \\dfrac 1 4$ :$a < h < b$, where $a$ and $b$ are the roots of the quadratic equation $x^2 - x + k = 0$. Then: <onlyinclude> :$\\forall n \\in \\N_{>0}: a < x_n < b$ </onlyinclude> == Proof == The proof proceeds by induction. For all $n \\in \\Z_{\\ge 0}$, let $\\map P n$ be the proposition: :$a < x_n < b$ === Basis for the Induction === $\\map P 1$ is the case: :$a < x_1 < b$ By assertion: :$a < h < b$ and: :$x_1 = h$ Thus $\\map P 1$ is seen to hold. This is the basis for the induction. === Induction Hypothesis === Now it needs to be shown that if $\\map P k$ is true, where $k \\ge 1$, then it logically follows that $\\map P {k + 1}$ is true. So this is the induction hypothesis: :$a < x_k < b$ from which it is to be shown that: :$a < x_{k + 1} < b$ === Induction Step === This is the induction step: {{begin-eqn}} {{eqn | l = x_{k + 1} - a       | r = {x_k}^2 + k - a       | c = Definition of $x_{k + 1}$ }} {{eqn | o = >       | r = a^2 - a + k       | c = as $x_k > a$ }} {{eqn | r = 0       | c = Definition of $a$ }} {{eqn | ll= \\leadsto       | l = x_{k + 1}       | o = >       | r = a       | c =  }} {{end-eqn}} Similarly: {{begin-eqn}} {{eqn | l = x_{k + 1} - b       | r = {x_k}^2 + k - b       | c = Definition of $x_{k + 1}$ }} {{eqn | o = <       | r = b^2 - b + k       | c = as $x_k < b$ }} {{eqn | r = 0       | c = Definition of $b$ }} {{eqn | ll= \\leadsto       | l = x_{k + 1}       | o = <       | r = b       | c =  }} {{end-eqn}} So $\\map P k \\implies \\map P {k + 1}$ and the result follows by the Principle of Mathematical Induction. Therefore: :$\\forall n \\in \\N_{>0}: a < x_n < b$ {{qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Convergent Real Sequence/Examples/x (n+1) = x n^2 + k|next = Convergent Real Sequence/Examples/x (n+1) = x n^2 + k/Lemma 2}}: $\\S 5$: Subsequences: Exercise $\\S 5.7 \\ (2)$ Category:Convergent Real Sequence/Examples/x (n+1) = x n^2 + k a8stp465ywvsogvr8u3ug937ibusiqx"}
+{"_id": "33045", "title": "Convergent Real Sequence/Examples/x (n+1) = x n^2 + k/Lemma 2", "text": "Convergent Real Sequence/Examples/x (n+1) = x n^2 + k/Lemma 2 0 77819 408885 408864 2019-06-18T14:29:03Z Prime.mover 59 wikitext text/x-wiki == Example of Convergent Real Sequence == <onlyinclude> Let $a$ and $b$ be the roots of the quadratic equation: :$(1): \\quad x^2 - x + k = 0$ Let: :$0 < k < \\dfrac 1 4$ Then $a$ and $b$ are both strictly positive real numbers. </onlyinclude> == Proof == First we investigate the consequences of the condition $k < \\dfrac 1 4$. By Solution to Quadratic Equation with Real Coefficients: :In order for the quadratic equation $a x^2 + b x + c$ to have real roots, its discriminant $b^2 - 4 a c$ needs to be strictly positive. The discriminant $D$ of $(1)$ is: {{begin-eqn}} {{eqn | l = D       | r = \\paren {-1}^2 - 4 \\times 1 \\times k       | c =  }} {{eqn | r = 1 - 4 k       | c =  }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | l = k       | o = <       | r = \\dfrac 1 4       | c =  }} {{eqn | ll= \\leadsto       | l = 1 - 4 k       | o = >       | r = 0       | c =  }} {{end-eqn}} and so when $k < \\dfrac 1 4$, $(1)$ has real roots. {{qed|lemma}} Next we investigate the consequences of the condition $0 < k$. By Solution to Quadratic Equation: {{begin-eqn}} {{eqn | l = x       | r = \\dfrac {-b \\pm \\sqrt {b^2 - 4 a c} } {2 a}       | c = where $b = -1$, $a = 1$, $c = k$ in $(1)$ }} {{eqn | r = \\dfrac {-\\paren {-1} \\pm \\sqrt {\\paren {-1}^2 - 4 \\times 1 \\times k} } {2 \\times 1}       | c =  }} {{eqn | r = \\dfrac {1 \\pm \\sqrt {1 - 4 k} } 2       | c =  }} {{end-eqn}} We have that: {{begin-eqn}} {{eqn | l = 0       | o = <       | r = k       | c =  }} {{eqn | ll= \\leadsto       | l = 1       | o = >       | r = 1 - 4 k       | c =  }} {{eqn | ll= \\leadsto       | l = 1       | o = >       | r = +\\sqrt {1 - 4 k}       | c =  }} {{eqn | l = -1       | o = <       | r = -\\sqrt {1 - 4 k}       | c =  }} {{eqn | ll= \\leadsto       | l = \\dfrac {1 \\pm \\sqrt {1 - 4 k} } 2       | o = >       | r = 0       | c =  }} {{end-eqn}} That is, when $0 < k$ both roots of $(1)$ are strictly positive. {{qed|lemma}} Hence when $0 < k < \\dfrac 1 4$, both roots of $(1)$ are strictly positive real numbers. {{qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Convergent Real Sequence/Examples/x (n+1) = x n^2 + k/Lemma 1|next = Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n}}: $\\S 5$: Subsequences: Exercise $\\S 5.7 \\ (2)$ Category:Convergent Real Sequence/Examples/x (n+1) = x n^2 + k 47tbne0b2az5ehu854zf2el1ldm973l"}
+{"_id": "33046", "title": "Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n/Lemma 1", "text": "Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n/Lemma 1 0 77823 408895 2019-06-18T15:50:09Z Prime.mover 59 Created page with \"== Example of Convergent Real Sequence ==  Let $h, k \\in \\R_{>0}$.  Let $\\sequence {x_n}$ be the Definition:Real Sequence|real sequen...\" wikitext text/x-wiki == Example of Convergent Real Sequence == Let $h, k \\in \\R_{>0}$. Let $\\sequence {x_n}$ be the real sequence defined as: :$x_n = \\begin {cases} h & : n = 1 \\\\ \\dfrac k {1 + x_{n - 1} } & : n > 1 \\end {cases}$ Then: <onlyinclude> :$\\forall n \\in \\N_{>1}: k > x_n > 0$ </onlyinclude> == Proof == The proof proceeds by induction. For all $n \\in \\Z_{>1}$, let $\\map P n$ be the proposition: :$k > x_n > 0$ === Basis for the Induction === $\\map P 2$ is the case: :$k > x_2 > 0$ We have: {{begin-eqn}} {{eqn | l = x_2       | r = \\dfrac k {1 + x_1}       | c =  }} {{eqn | o = <       | r = \\dfrac k 1       | c = as $x_1 > 0$ }} {{eqn | r = k       | c =  }} {{end-eqn}} Also, as $k > 0$ and $x_1 > 0$ we have that: :$\\dfrac k {1 + x_1} > 0$ Thus $\\map P 2$ is seen to hold. This is the basis for the induction. === Induction Hypothesis === Now it needs to be shown that if $\\map P r$ is true, where $r \\ge 2$, then it logically follows that $\\map P {r + 1}$ is true. So this is the induction hypothesis: :$k > x_r > 0$ from which it is to be shown that: :$k > x_{r + 1} > 0$ === Induction Step === This is the induction step: {{begin-eqn}} {{eqn | l = x_{r + 1}       | r = \\dfrac k {1 + x_r}       | c =  }} {{eqn | o = <       | r = \\dfrac k 1       | c = Induction Hypothesis: $x_r > 0$ }} {{eqn | r = k       | c =  }} {{end-eqn}} Also, as $k > 0$ and $x_r > 0$ we have that: :$\\dfrac k {1 + x_r} > 0$ So $\\map P r \\implies \\map P {r + 1}$ and the result follows by the Principle of Mathematical Induction. Therefore: :$\\forall n \\in \\N_{>1}: k > x_n > 0$ {{qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n|next = Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n/Mistake}}: $\\S 5$: Subsequences: Exercise $\\S 5.7 \\ (3)$ Category:Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n bb2tocxnxw2uu7d3q7cfbgoncgs8myz"}
+{"_id": "33047", "title": "Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n/Lemma 2", "text": "Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n/Lemma 2 0 77831 408946 408930 2019-06-18T22:26:35Z Prime.mover 59 wikitext text/x-wiki == Example of Convergent Real Sequence == Let $h, k \\in \\R_{>0}$. Let $\\sequence {x_n}$ be the real sequence defined as: :$x_n = \\begin {cases} h & : n = 1 \\\\ \\dfrac k {1 + x_{n - 1} } & : n > 1 \\end {cases}$ <onlyinclude> Consider the subsequences $\\sequence {x_{2 n} }$ and $\\sequence {x_{2 n - 1} }$. One of them is strictly increasing and the other is strictly decreasing. </onlyinclude> == Proof == We have that: {{begin-eqn}} {{eqn | l = x_{n + 1} - x_{n - 1}       | r = \\dfrac k {1 + x_n} - \\dfrac k {1 + x_{n - 2} }       | c =  }} {{eqn | r = \\dfrac {k \\paren {x_{n - 2} - x_n} } {\\paren {1 + x_n} \\paren {1 + x_{n - 2} } }       | c =  }} {{end-eqn}} and so $x_{n + 1} - x_{n - 1}$ has the opposite sign to $x_{n - 2} - x_n$. It can be proved by induction that one of the sequences $\\sequence {x_{2 n} }$ and $\\sequence {x_{2 n - 1} }$ increases and one decreases. {{finish|Provide the workings for the above}} In fact: :$\\sequence {x_{2 n - 1} }$ is strictly increasing {{iff}} $x_3 > x_1$ and is strictly decreasing {{iff}} $x_3 < x_1$. {{finish|Prove the above as well.}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n/Mistake|next = Convergent Real Sequence/Examples/x n = root x n-1 y n-1, 1 over y n = half (1 over x n + 1 over y n-1)}}: $\\S 5$: Subsequences: Exercise $\\S 5.7 \\ (3)$ Category:Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n 50e8ta82z9l1obkx7f6th0hiv29yzvx"}
+{"_id": "33048", "title": "Square of Chi Random Variable has Chi-Squared Distribution", "text": "Square of Chi Random Variable has Chi-Squared Distribution 0 77952 409556 2019-06-22T10:16:19Z Caliburn 3218 Created page with \"== Theorem ==  <onlyinclude> Let $n$ be a strictly positive integer.  Let $X \\sim \\chi_n$ where $\\chi_n$ is the Definition:Chi Distr...\" wikitext text/x-wiki == Theorem ==  <onlyinclude> Let $n$ be a strictly positive integer. Let $X \\sim \\chi_n$ where $\\chi_n$ is the chi distribution with $n$ degrees of freedom.  Then $X^2 \\sim \\chi^2_n$ where $\\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom. </onlyinclude> == Proof ==  Let $Y \\sim \\chi^2_n$.  We aim to show that:  :$\\map \\Pr {Y < x^2} = \\map \\Pr {X < x}$ for all $x \\in \\hointr 0 \\infty$. We have: {{begin-eqn}} {{eqn\t| l = \\map \\Pr {Y < x^2} \t| r = \\int_0^{x^2} \\frac 1 {2^{n / 2} \\map \\Gamma {n / 2} } t^{\\paren {n / 2} - 1} e^{- t / 2} \\rd t \t| c = {{Defof|Chi-Squared Distribution}} }} {{eqn\t| r = \\frac 2 {2^{n / 2} \\map \\Gamma {n / 2} } \\int_0^x u \\paren {u^2}^{\\paren {n / 2} - 1} e^{- u^2 / 2} \\rd u \t| c = substituting $t = u^2$ }} {{eqn\t| r = \\frac 1 {2^{\\paren {n / 2} - 1} \\map \\Gamma {n / 2} } \\int_0^x u u^{2 \\paren {\\paren {n / 2} - 1} } e^{- u^2 / 2} \\rd u }} {{eqn\t| r = \\frac 1 {2^{\\paren {n / 2} - 1} \\map \\Gamma {n / 2} } \\int_0^x u^{n - 1} e^{- u^2 / 2} \\rd u }} {{eqn\t| r = \\map \\Pr {X < x} \t| c = {{Defof|Chi Distribution}} }} {{end-eqn}} {{qed}} Category:Chi Distribution Category:Chi-Squared Distribution 4plq49xte1ivxn4xpdvm9dx8g2ibaq9"}
+{"_id": "33049", "title": "Multiple of Chi-Squared Random Variable has Gamma Distribution", "text": "Multiple of Chi-Squared Random Variable has Gamma Distribution 0 77989 409843 2019-06-23T13:22:53Z Caliburn 3218 Created page with \"== Theorem ==  <onlyinclude> Let $n$ be a strictly positive integer.   Let $k > 0$ be a real number.   Let...\" wikitext text/x-wiki == Theorem ==  <onlyinclude> Let $n$ be a strictly positive integer.  Let $k > 0$ be a real number.  Let $X \\sim \\chi^2_n$ where $\\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then:  :$k X \\sim \\map \\Gamma {\\dfrac n 2, \\dfrac 1 {2 k}}$ where $\\map \\Gamma {\\dfrac n 2, \\dfrac 1 {2 k}}$ is the gamma distribution with parameters $\\dfrac n 2$ and $\\dfrac 1 {2 k}$. </onlyinclude> == Proof == Let: :$Y \\sim \\map \\Gamma {\\dfrac n 2, \\dfrac 1 {2 k}}$ We aim to show that:  :$\\map \\Pr {Y < k x} = \\map \\Pr {X < x}$ for all real $x \\ge 0$. We have:  {{begin-eqn}} {{eqn\t| l = \\map \\Pr {Y < k x} \t| r = \\frac 1 {\\map \\Gamma {n / 2} } \\paren {\\frac 1 {2 k} }^{n / 2} \\int_0^{k x} t^{\\paren {n / 2} - 1} e^{-\\paren {1 / k} t / 2} \\rd t \t| c = {{Defof|Gamma Distribution}} }} {{eqn\t| r = \\frac 1 {\\map \\Gamma {n / 2} } \\paren {\\frac 1 {2 k} }^{n / 2} \\int_0^x k \\paren {k u}^{\\paren {n / 2} - 1} e^{-u / 2} \\rd u \t| c = substituting $t = k u$ }} {{eqn\t| r = \\frac {k^{n / 2} } {\\map \\Gamma {n / 2} \\paren {2 k}^{n / 2} } \\int_0^x u^{\\paren {n / 2} - 1} e^{-u / 2} \\rd u }} {{eqn\t| r = \\frac 1 {2^{n / 2} \\map \\Gamma {n / 2} } \\int_0^x u^{\\paren {n / 2} - 1} e^{-u / 2} \\rd u  }} {{eqn\t| r = \\map \\Pr {X < x} \t| c = {{Defof|Chi-Squared Distribution}} }} {{end-eqn}} {{qed}} Category:Chi-Squared Distribution Category:Gamma Distribution iemi4sla1j0pqk0tb8sxnp2gkq4hd72"}
+{"_id": "33050", "title": "Expectation of Student's t-Distribution", "text": "Expectation of Student's t-Distribution 0 78121 410154 2019-06-25T13:45:04Z Caliburn 3218 Created page with \"== Theorem == <onlyinclude> Let $k$ be a strictly positive integer.   Let $X \\sim t_k$ where $t_k$ is the Definition:Student's t-Dis...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $k$ be a strictly positive integer.  Let $X \\sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom. Then the expectation of $X$ is equal to $0$ for $k > 1$, and does not exist otherwise. </onlyinclude> == Proof == {{ProofWanted}} Category:Expectation Category:Student's t-Distribution n1kupkw20s6aloggikkrhn7j4h9r356"}
+{"_id": "33051", "title": "Raw Moment of Erlang Distribution", "text": "Raw Moment of Erlang Distribution 0 78142 410309 2019-06-26T11:05:45Z Caliburn 3218 Created page with \"== Theorem == <onlyinclude> Let $n, k$ be strictly positive integers.   Let $\\lambda$ be a Definition:Strictly Positive Real Number|...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $n, k$ be strictly positive integers.  Let $\\lambda$ be a strictly positive real number.  Let $X$ have a continuous random variable with an Erlang distribution with parameters $k$ and $\\lambda$. Then the $n$th raw moment of $X$ is given by: :$\\displaystyle \\expect {X^n} = \\frac 1 {\\lambda^n} \\prod_{m \\mathop = 0}^{n - 1} \\paren {k + m}$ </onlyinclude>  == Proof == From the definition of the Erlang distribution, $X$ has probability density function: :$\\map {f_X} x = \\dfrac {\\lambda^k x^{k - 1} e^{- \\lambda x} } {\\map \\Gamma k}$ From the definition of the expected value of a continuous random variable:  :$\\displaystyle \\expect {X^n} = \\int_0^\\infty x^n \\map {f_X} x \\rd x$ So: {{begin-eqn}} {{eqn\t| l = \\expect {X^n} \t| r = \\frac {\\lambda^k} {\\map \\Gamma k} \\int_0^\\infty x^{n + k - 1} e^{- \\lambda x} \\rd x }} {{eqn\t| r = \\frac {\\lambda^k} {\\lambda \\map \\Gamma k} \\int_0^\\infty \\paren {\\frac u \\lambda}^{n + k - 1} e^{- u} \\rd u \t| c = substituting $u = \\lambda x$ }} {{eqn\t| r = \\frac {\\lambda^k} {\\lambda^{n + k} \\map \\Gamma k} \\int_0^\\infty u^{n + k - 1} e^{- u} \\rd u }} {{eqn\t| r = \\frac 1 {\\lambda^n \\map \\Gamma k} \\map \\Gamma {n + k} \t| c = {{Defof|Gamma Function}} }} {{eqn\t| r = \\frac 1 {\\lambda^n} \\frac {\\map \\Gamma k} {\\map \\Gamma k} \\prod_{m \\mathop = 0}^{n - 1} \\paren {k + m} \t| c = Gamma Difference Equation }} {{eqn\t| r = \\frac 1 {\\lambda^n} \\prod_{m \\mathop = 0}^{n - 1} \\paren {k + m} }} {{end-eqn}} {{qed}} Category:Raw Moments Category:Erlang Distribution jou9tst4u65j3aafpjsdh7jycni2zu7"}
+{"_id": "33052", "title": "Linear Transformation of Gaussian Random Variable", "text": "Linear Transformation of Gaussian Random Variable 0 78298 411601 2019-07-04T13:50:30Z Caliburn 3218 Created page with \"== Theorem ==  <onlyinclude> Let $\\mu$, $\\alpha$ and $\\beta$ be real numbers.   Let $\\sigma$ be a Definition:Positive Real Number|positive real nu...\" wikitext text/x-wiki == Theorem ==  <onlyinclude> Let $\\mu$, $\\alpha$ and $\\beta$ be real numbers.  Let $\\sigma$ be a positive real number.  Let $X \\sim \\Gaussian \\mu {\\sigma^2}$ where $\\Gaussian \\mu {\\sigma^2}$ is the Gaussian distribution with parameters $\\mu$ and $\\sigma^2$.  Then:  :$\\alpha X + \\beta \\sim \\Gaussian {\\alpha \\mu + \\beta} {\\alpha^2 \\sigma^2}$ </onlyinclude>  == Proof == Let $Z = \\alpha X + \\beta$.  Let $M_Z$ be the moment generating function of $Z$.  We aim to show that:  :$Z \\sim \\Gaussian {\\alpha \\mu + \\beta} {\\alpha^2 \\sigma^2}$ By Moment Generating Function of Gaussian Distribution and Moment Generating Function is Unique, it is sufficient to show that: :$\\displaystyle \\map {M_Z} t = \\map \\exp {\\paren {\\alpha \\mu + \\beta} t + \\frac 1 2 \\alpha^2 \\sigma^2 t^2}$ We also have, by Moment Generating Function of Gaussian Distribution, that the moment generating function of $X$, $M_X$, is given by:  :$\\displaystyle \\map {M_X} t = \\map \\exp {\\mu t + \\frac 1 2 \\sigma^2 t^2}$ We have: {{begin-eqn}} {{eqn\t| l = \\map {M_Z} t \t| r = e^{\\beta t} \\map {M_X} {\\alpha t} \t| c = Moment Generating Function of Linear Combination of Independent Random Variables }} {{eqn\t| r = e^{\\beta t} \\map \\exp {\\alpha \\mu t + \\frac 1 2 \\sigma^2 \\paren {\\alpha t}^2} }} {{eqn\t| r = \\map \\exp {\\paren {\\alpha \\mu + \\beta} t + \\frac 1 2 \\sigma^2 \\alpha^2 t^2} \t| c = Exponential of Sum }} {{end-eqn}} {{qed}} Category:Gaussian Distribution nc5ethpa6uic5hj0lh5och2t610pcii"}
+{"_id": "33053", "title": "Quotient of Independent Random Variables with Chi-Squared Distribution Divided by Degrees of Freedom has F-Distribution", "text": "Quotient of Independent Random Variables with Chi-Squared Distribution Divided by Degrees of Freedom has F-Distribution 0 78327 411751 2019-07-05T11:40:51Z Caliburn 3218 Created page with \"== Theorem == <onlyinclude> Let $n$ and $m$ be strictly positive integers.  Let $X$ and $Y$ be Definition:Independent Random Variabl...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $n$ and $m$ be strictly positive integers. Let $X$ and $Y$ be independent random variables.  Let $X \\sim \\chi^2_n$ where $\\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Let $Y \\sim \\chi^2_m$ where $\\chi^2_m$ is the chi-squared distribution with $m$ degrees of freedom. Then:  :$\\dfrac {X / n} {Y / m} \\sim F_{n, m}$ where $F_{n, m}$ is the F-distribution with $\\tuple {n, m}$ degrees of freedom. </onlyinclude>  == Proof == {{ProofWanted}} Category:Chi-Squared Distribution Category:F-Distribution cta2r07liq22w6ls6deby0820pfi5tc"}
+{"_id": "33054", "title": "Derivative of Function plus Constant", "text": "Derivative of Function plus Constant 0 78816 414752 2019-07-27T07:52:21Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> Let $f$ be a real function which is differentiable on $\\R$.  Let $c \\in \\R$...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $f$ be a real function which is differentiable on $\\R$. Let $c \\in \\R$ be a constant. Then: :$\\map {D_x} {\\map f x + c} = \\map {D_x} {\\map f x}$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\map {D_x} {\\map f x + c}       | r = \\map {D_x} {\\map f x} + \\map f x \\, c       | c = Sum Rule for Derivatives }} {{eqn | r = \\map {D_x} {\\map f x} + 0       | c = Derivative of Constant }} {{eqn | r = \\map {D_x} {\\map f x}       | c = }} {{end-eqn}} {{qed}} Category:Differential Calculus stxghwu9hz138be4dsaxr1oyx8br1bi"}
+{"_id": "33055", "title": "Definite Integral from 0 to Half Pi of Logarithm of Sine x/Lemma", "text": "Definite Integral from 0 to Half Pi of Logarithm of Sine x/Lemma 0 78911 415395 2019-07-30T15:31:34Z Caliburn 3218 need this for another page or two wikitext text/x-wiki == Lemma for Definite Integral from 0 to Half Pi of Logarithm of Sine x == <onlyinclude> :$\\displaystyle \\int_0^\\pi \\map \\ln {\\sin x} \\rd x = 2 \\int_0^{\\pi/2} \\map \\ln {\\sin x} \\rd x$ </onlyinclude> == Proof == We have:  {{begin-eqn}} {{eqn\t| l = \\int_{\\pi/2}^\\pi \\map \\ln {\\sin x} \\rd x \t| r = -\\int_{\\pi/2}^0 \\map \\ln {\\map \\sin {\\pi - x} } \\rd x \t| c = substituting $x \\mapsto \\pi - x$ }} {{eqn\t| r = \\int_0^{\\pi/2} \\map \\ln {\\sin x} \\rd x \t| c = Reversal of Limits of Definite Integral, Sine of Supplementary Angle }} {{end-eqn}} We can therefore write:  {{begin-eqn}} {{eqn\t| l = \\int_0^\\pi \\map \\ln {\\sin x} \\rd x \t| r = \\int_0^{\\pi/2} \\map \\ln {\\sin x} \\rd x + \\int_{\\pi/2}^\\pi \\map \\ln {\\sin x} \\rd x \t| c = Sum of Integrals on Adjacent Intervals for Integrable Functions }} {{eqn\t| r = 2 \\int_0^{\\pi/2} \\map \\ln {\\sin x} \\rd x }} {{end-eqn}} {{qed|lemma}} Category:Definite Integral from 0 to Half Pi of Logarithm of Sine x 5q2g8m27o01zps1mk2jo1crqxcaa6a5"}
+{"_id": "33056", "title": "Lowest Common Multiple/Examples/6 and 15", "text": "Lowest Common Multiple/Examples/6 and 15 0 79059 416693 2019-08-05T07:01:34Z Prime.mover 59 Created page with \"== Example of Lowest Common Multiple of Integers == <onlyinclude> The Definition:Lowest Common Multiple|lowest common multi...\" wikitext text/x-wiki == Example of Lowest Common Multiple of Integers == <onlyinclude> The lowest common multiple of $6$ and $15$ is: :$\\lcm \\set {6, 15} = 30$ </onlyinclude> == Proof == We find the greatest common divisor of $6$ and $15$ using the Euclidean Algorithm: {{begin-eqn}} {{eqn | n = 1       | l = 15       | r = 2 \\times 6 + 3 }} {{eqn | n = 2       | l = 6       | r = 2 \\times 3 }} {{end-eqn}} Thus $\\gcd \\set {6, 15} = 3$. Then: {{begin-eqn}} {{eqn | l = \\lcm \\set {6, 15}       | r = \\dfrac {6 \\times 15} {\\gcd \\set {6, 15} }       | c = Product of GCD and LCM }} {{eqn | r = \\dfrac {\\paren {2 \\times 3} \\times \\paren {3 \\times 5} } 3       | c =  }} {{eqn | r = 2 \\times 3 \\times 5       | c =  }} {{eqn | r = 30       | c =  }} {{end-eqn}} {{qed}} Category:Examples of Lowest Common Multiples Category:30 k9gdjvcsr46fx5lmvholwp81g8j1236"}
+{"_id": "33057", "title": "Definite Integral from 0 to Half Pi of Logarithm of Sine x/Proof 1", "text": "Definite Integral from 0 to Half Pi of Logarithm of Sine x/Proof 1 0 79137 417073 2019-08-07T15:17:48Z Caliburn 3218 Created page with \"== Theorem == {{:Definite Integral from 0 to Half Pi of Logarithm of Sine x}}  == Proof == <onlyinclude> By Definite Integral from 0 to Half Pi of Logarithm of Sine x/Lemma|...\" wikitext text/x-wiki == Theorem == {{:Definite Integral from 0 to Half Pi of Logarithm of Sine x}} == Proof == <onlyinclude> By Definite Integral from $0$ to $\\dfrac \\pi 2$ of $\\map \\ln {\\sin x}$: Lemma, we have:  :$\\displaystyle \\int_0^\\pi \\map \\ln {\\sin x} \\rd x = 2 \\int_0^{\\pi/2} \\map \\ln {\\sin x} \\rd x$ We also have:  {{begin-eqn}} {{eqn\t| l = \\int_0^{\\pi/2} \\map \\ln {\\sin x} \\rd x \t| r = \\int_0^{\\pi/2} \\map \\ln {\\map \\sin {\\frac \\pi 2 - x} } \\rd x \t| c = Integral between Limits is Independent of Direction }} {{eqn\t| r = \\int_0^{\\pi/2} \\map \\ln {\\cos x} \\rd x \t| c = Sine of Complement equals Cosine }} {{end-eqn}} giving:  {{begin-eqn}} {{eqn\t| l = 2 \\int_0^{\\pi/2} \\map \\ln {\\sin x} \\rd x \t| r = \\int_0^{\\pi/2} \\map \\ln {\\sin x \\cos x} \\rd x \t| c = Sum of Logarithms }} {{eqn\t| r = \\int_0^{\\pi/2} \\map \\ln {\\frac 1 2 \\sin 2 x} \\rd x \t| c = Double Angle Formula for Sine }} {{eqn\t| r = \\frac \\pi 2 \\map \\ln {\\frac 1 2} + \\int_0^{\\pi/2} \\map \\ln {\\sin 2 x} \\rd x \t| c = Primitive of Constant, Sum of Logarithms }} {{eqn\t| r = \\frac \\pi 2 \\map \\ln {\\frac 1 2} + \\frac 1 2 \\int_0^\\pi \\map \\ln {\\sin u} \\rd u \t| c = substituting $u = 2 x$ }} {{eqn\t| r = -\\frac \\pi 2 \\ln 2 + \\int_0^{\\pi/2} \\map \\ln {\\sin u} \\rd u \t| c = Logarithm of Reciprocal }} {{end-eqn}} Therefore:  :$\\displaystyle \\int_0^{\\pi/2} \\map \\ln {\\sin x} \\rd x = -\\frac \\pi 2 \\ln 2$ {{qed}} </onlyinclude> Category:Definite Integral from 0 to Half Pi of Logarithm of Sine x mybzaitrvxdq39bjnc0ju2ey2qdui8b"}
+{"_id": "33058", "title": "Equivalence Class/Examples/Months that Start on the Same Day of the Week", "text": "Equivalence Class/Examples/Months that Start on the Same Day of the Week 0 79374 418399 418395 2019-08-16T05:31:13Z Prime.mover 59 wikitext text/x-wiki == Examples of Equivalence Class == <onlyinclude> Let $M$ be the set of months of the (calendar) year according to the (usual) Gregorian calendar. Let $\\sim$ be the relation on $M$ defined as: :$\\forall x, y \\in M: x \\sim y \\iff \\text {$x$ and $y$ both start on the same day of the week}$ The set of equivalence classes under $\\sim$ depends on whether the year is a leap year. For a non-leap year, the set of equivalence classes is: :$\\set {\\set {\\text {January}, \\text {October} }, \\set {\\text {February}, \\text {March}, \\text {November} }, \\set {\\text {April}, \\text {July} }, \\set {\\text {May} }, \\set {\\text {June} }, \\set {\\text {August} }, \\set {\\text {September}, \\text {December} } }$ For a leap year, the set of equivalence classes is: :$\\set {\\set {\\text {January}, \\text {April}, \\text {July} }, \\set {\\text {February}, \\text {August} }, \\set {\\text {March}, \\text {November} }, \\set {\\text {May} }, \\set {\\text {June} }, \\set {\\text {September}, \\text {December} }, \\set {\\text {October} } }$ </onlyinclude> == Proof == We have that: :The months with $30$ days are: ::$\\text {April}, \\text {June}, \\text {September}, \\text {November}$ :The months with $31$ days are: ::$\\text {January}, \\text {March}, \\text {May}, \\text {July}, \\text {August}, \\text {October}, \\text {December}$ :In a non-leap year, $\\text {February}$ has $28$ days :In a leap year, $\\text {February}$ has $29$ days. Let month $m$ have $m_d$ days in it. Let month $m$ start on day $d$, where $d$ is in the range $0$ to $6$ (which day of the week corresponds to which number is irrelevant at this stage). Then month $m + 1$ starts on day $\\paren {d + m_d} \\pmod 7$. For reference: {{begin-eqn}} {{eqn | l = 28       | o = \\equiv       | r = 0 \\pmod 7       | c =  }} {{eqn | l = 29       | o = \\equiv       | r = 0 \\pmod 7       | c =  }} {{eqn | l = 30       | o = \\equiv       | r = 0 \\pmod 7       | c =  }} {{eqn | l = 31       | o = \\equiv       | r = 0 \\pmod 7       | c =  }} {{end-eqn}} {{WLOG}}, let $\\text {January}$ start on day $0$. Then the sequence of the days which are the $1$st of the month are as follows: For a non-leap year: :$\\tuple {0, 3, 3, 6, 1, 4, 6, 2, 5, 0, 3, 5}$ {{OEIS|A189915}} For a leap year: :$\\tuple {0, 3, 4, 0, 2, 5, 0, 3, 6, 1, 4, 6}$ {{OEIS|A189916}} The result follows. {{qed}} == Sources == * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth|prev = Equivalence Relation/Examples/Months that Start on the Same Day of the Week|next = Number of Friday 13ths in a Year}}: $\\S 6$. Indexed families; partitions; equivalence relations: Exercise $7$ Category:Examples of Equivalence Classes Category:Calendars i37jokwxlf2u7mc5q9xhufclfbsaq5v"}
+{"_id": "33059", "title": "Symbols:Abbreviations/W/WFF", "text": "Symbols:Abbreviations/W/WFF 104 79554 446989 424930 2020-02-06T23:42:49Z Prime.mover 59 wikitext text/x-wiki == Abbreviation: WFF == <onlyinclude> :Well-formed formula. </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Mathematician:Hermann Klaus Hugo Weyl|next = Mathematician:Alfred North Whitehead|entry = wff}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Mathematician:Hermann Klaus Hugo Weyl|next = Mathematician:Alfred North Whitehead|entry = wff}} Category:Symbols/Abbreviations/W oip6gzhhqge2br7jpl3lq08vts6f7ml"}
+{"_id": "33060", "title": "Symbols:Number Theory/Does Not Divide", "text": "Symbols:Number Theory/Does Not Divide 104 80006 421036 421033 2019-08-26T16:02:36Z Prime.mover 59 wikitext text/x-wiki == Is Not a Divisor == <onlyinclude> :$x \\nmid y$ This means '''$x$ is not a divisor of $y$'''. {{LatexFor|for = x \\nmid y}} </onlyinclude> == Also denoted as == This symbol is preferable to $x \\mathrel {\\not \\backslash} y$ due to the somewhat confusing appearance of this symbol: {{LatexFor|for = x \\mathrel {\\not \\backslash} y}} Category:Symbols/Number Theory f7u01z8p54adxwuyz5rv8ksyyd8bar0"}
+{"_id": "33061", "title": "Principle of Stationary Action", "text": "Principle of Stationary Action 0 80401 479412 479410 2020-07-22T16:15:04Z Prime.mover 59 wikitext text/x-wiki == Physical Law == The '''principle of stationary actions''' states that the equations of motion of a physical system can be acquired by finding a stationary point of the action. In other words, the first variation of the action has to vanish. == Also known as == The '''principle of stationary action''' is also known as the '''principle of least action'''. == Notes == {{refactor|Extract the appropriate bit of this into a historical note, and the other bits into separate pages of their own as statement and proof. Hence lose the \"notes\" section.}} The '''principle of least action''' attained its name due to classical problems of minimization. However, if broken trajectories are allowed, the action can sometimes acquire lower values than for any allowed smooth trajectory. Since smooth trajectories are more realistic, leastness has been weakened to stationarity. Category:Mechanics Category:Physics Category:Applied Mathematics Category:Lagrangian Mechanics 0d1283hbybpfcwj6pcfhzchm2qcxr43"}
+{"_id": "33062", "title": "Powers of 3 Modulo 8", "text": "Powers of 3 Modulo 8 0 80449 424107 2019-09-08T12:06:35Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> Let $n \\in \\Z_{\\ge 0}$ be a strictly positive integer.  Then: :$3^n \\equiv \\begin {cases} 1 \\pmod 8 & : \\t...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $n \\in \\Z_{\\ge 0}$ be a strictly positive integer. Then: :$3^n \\equiv \\begin {cases} 1 \\pmod 8 & : \\text {$n$ even} \\\\ 3 \\pmod 8 & : \\text {$n$ odd} \\end {cases}$ </onlyinclude> == Proof 1 == {{:Powers of 3 Modulo 8/Proof 1}} == Proof 2 == {{:Powers of 3 Modulo 8/Proof 2}} Category:Powers of 3 Category:Modulo Arithmetic Category:Powers of 3 Modulo 8 lxmoxenu4w6om1ahif62euwc57p46is"}
+{"_id": "33063", "title": "Common Logarithm/Examples/2.36", "text": "Common Logarithm/Examples/2.36 0 80916 432976 426975 2019-10-30T18:28:15Z Prime.mover 59 wikitext text/x-wiki == Example of Common Logarithm == <onlyinclude> The common logarithm of $2 \\cdotp 36$ is: :$\\log_{10} 2 \\cdotp 36 = 0 \\cdotp 3729$ </onlyinclude> == Sources == * {{BookReference|Theory and Problems of Statistics|1972|Murray R. Spiegel|author2 = R.W. Boxer|ed = SI|edpage = SI Edition|prev = Range of Common Logarithm of Number between 1 and 10|next = Common Logarithm/Examples/23.6}}: Chapter $1$: Logarithms: '''Example 1.''' Category:Examples of Common Logarithms k8050bu2cizg984eixbrl73qdt96g8s"}
+{"_id": "33064", "title": "Jensen's Inequality (Real Analysis)/Corollary", "text": "Jensen's Inequality (Real Analysis)/Corollary 0 81312 429062 429061 2019-10-01T19:51:22Z Caliburn 3218 wikitext text/x-wiki == Corollary to Jensen's Inequality: Real Analysis == <onlyinclude> Let $I$ be a real interval. Let $\\phi: I \\to \\R$ be a concave function. Let $x_1, x_2, \\ldots, x_n \\in I$. Let $\\lambda_1, \\lambda_2, \\ldots, \\lambda_n \\ge 0$ be real numbers, at least one of which is non-zero. Then: :$\\displaystyle \\map \\phi {\\frac {\\sum_{k \\mathop = 1}^n \\lambda_k x_k} {\\sum_{k \\mathop = 1}^n \\lambda_k} } \\ge \\frac {\\sum_{k \\mathop = 1}^n \\lambda_k \\map \\phi {x_k} } {\\sum_{k \\mathop = 1}^n \\lambda_k}$ For $\\phi$ strictly concave, equality holds {{iff}} $x_1 = x_2 = \\cdots = x_n$. </onlyinclude> == Proof == By Real Function is Concave iff its Negative is Convex, $-\\phi: I \\to \\R$ is a convex function. Therefore, we can apply Jensen's Inequality: Real Analysis with $-\\phi$ to obtain: :$\\displaystyle -\\map \\phi {\\frac {\\sum_{k \\mathop = 1}^n \\lambda_k x_k} {\\sum_{k \\mathop = 1}^n \\lambda_k} } \\le -\\frac {\\sum_{k \\mathop = 1}^n \\lambda_k \\map \\phi {x_k} } {\\sum_{k \\mathop = 1}^n \\lambda_k}$ with equality for $-\\phi$ strictly convex {{iff}} $x_1 = x_2 = \\cdots = x_n$. That is, for $\\phi$ strictly concave, equality holds {{iff}} $x_1 = x_2 = \\cdots = x_n$. With that, we have established the equality case. Multiplying through $-1$ in our inequality gives:  :$\\displaystyle \\map \\phi {\\frac {\\sum_{k \\mathop = 1}^n \\lambda_k x_k} {\\sum_{k \\mathop = 1}^n \\lambda_k} } \\ge \\frac {\\sum_{k \\mathop = 1}^n \\lambda_k \\map \\phi {x_k} } {\\sum_{k \\mathop = 1}^n \\lambda_k}$ as required. {{qed}} Category:Jensen's Inequality (Real Analysis) 8y2kpnwksu76razdzqnd66qcoxwmfd2"}
+{"_id": "33065", "title": "Axiom:Axioms of Uncertainty/Axiom 5", "text": "Axiom:Axioms of Uncertainty/Axiom 5 100 81786 431708 2019-10-18T00:07:10Z Prime.mover 59 Created page with \"== Axiom == {{:Axiom:Axioms of Uncertainty}}  ${H_n}$ fulfils the following axiom: <onlyinclude> :$\\map {H_n} {\\dfrac 1 n, \\dfrac 1 n, \\dotsc, \\dfrac 1 n}...\" wikitext text/x-wiki == Axiom == {{:Axiom:Axioms of Uncertainty}} ${H_n}$ fulfils the following axiom: <onlyinclude> :$\\map {H_n} {\\dfrac 1 n, \\dfrac 1 n, \\dotsc, \\dfrac 1 n} \\le \\map {H_{n + 1} } {\\dfrac 1 {n + 1}, \\dfrac 1 {n + 1}, \\dotsc, \\dfrac 1 {n + 1} }$ </onlyinclude> Thus, for example, a $2$-horse race is less uncertain than a $3$-horse race. == Sources == * {{BookReference|Codes and Cryptography|1988|Dominic Welsh|prev = Axiom:Axioms of Uncertainty/Axiom 4|next = Axiom:Axioms of Uncertainty/Axiom 6}}: $\\S 1$: Entropy = uncertainty = information: $1.1$ Uncertainty Category:Axioms/Axioms of Uncertainty d9zz9602wmis3uhox6g9pqlboesvvc1"}
+{"_id": "33066", "title": "Axiom:Axioms of Uncertainty/Axiom 6", "text": "Axiom:Axioms of Uncertainty/Axiom 6 100 81787 431709 2019-10-18T00:08:51Z Prime.mover 59 Created page with \"== Axiom == {{:Axiom:Axioms of Uncertainty}}  ${H_n}$ fulfils the following axiom: <onlyinclude> :$H_n$ is a Definition:Continuous Real-Valued Function|...\" wikitext text/x-wiki == Axiom == {{:Axiom:Axioms of Uncertainty}} ${H_n}$ fulfils the following axiom: <onlyinclude> :$H_n$ is a continuous function of its arguments. </onlyinclude> == Sources == * {{BookReference|Codes and Cryptography|1988|Dominic Welsh|prev = Axiom:Axioms of Uncertainty/Axiom 5|next = Axiom:Axioms of Uncertainty/Axiom 7}}: $\\S 1$: Entropy = uncertainty = information: $1.1$ Uncertainty Category:Axioms/Axioms of Uncertainty 93a1kzzyzfb8d1h9vfh8lseoqttcern"}
+{"_id": "33067", "title": "Axiom:Axioms of Uncertainty/Axiom 7", "text": "Axiom:Axioms of Uncertainty/Axiom 7 100 81788 431710 2019-10-18T00:15:50Z Prime.mover 59 Created page with \"== Axiom == {{:Axiom:Axioms of Uncertainty}}  ${H_n}$ fulfils the following axiom: <onlyinclude> :$\\map {H_{m n} } {\\dfrac 1 {m n}, \\dfrac 1 {m n}, \\dotsc...\" wikitext text/x-wiki == Axiom == {{:Axiom:Axioms of Uncertainty}} ${H_n}$ fulfils the following axiom: <onlyinclude> :$\\map {H_{m n} } {\\dfrac 1 {m n}, \\dfrac 1 {m n}, \\dotsc, \\dfrac 1 {m n} } = \\map {H_m} {\\dfrac 1 m, \\dfrac 1 m, \\dotsc, \\dfrac 1 m} + \\map {H_n} {\\dfrac 1 n, \\dfrac 1 n, \\dotsc, \\dfrac 1 n}$ </onlyinclude> == Sources == * {{BookReference|Codes and Cryptography|1988|Dominic Welsh|prev = Axiom:Axioms of Uncertainty/Axiom 6|next = Axiom:Axioms of Uncertainty/Axiom 8}}: $\\S 1$: Entropy = uncertainty = information: $1.1$ Uncertainty Category:Axioms/Axioms of Uncertainty ehf4tp5ajzwggteo2hcaxjxu9x0wfrm"}
+{"_id": "33068", "title": "Axiom:Axioms of Uncertainty/Axiom 8", "text": "Axiom:Axioms of Uncertainty/Axiom 8 100 81789 431757 431711 2019-10-18T23:15:59Z Prime.mover 59 wikitext text/x-wiki == Axiom == {{:Axiom:Axioms of Uncertainty}} ${H_n}$ fulfils the following axiom: <onlyinclude> Let: :$p = p_1 + p_2 + \\dotsb + p_m$ :$q = q_1 + q_2 + \\dotsb + q_n$ such that: :each of $p_i$ and $q_j$ are non-negative :$p + q = 1$ Then: :$\\map {H_{m + n} } {p_1, p_2, \\dotsc, p_m, q_1, q_2, \\dotsc q_n} = \\map {H_2} {p, q} + p \\map {H_m} {\\dfrac {p_1} p, \\dfrac {p_2} p, \\dotsc, \\dfrac {p_m} p} + q \\map {H_n} {\\dfrac {q_1} q, \\dfrac {q_2} q, \\dotsc, \\dfrac {q_n} q}$ </onlyinclude> == Sources == * {{BookReference|Codes and Cryptography|1988|Dominic Welsh|prev = Axiom:Axioms of Uncertainty/Axiom 7|next = Function that Satisfies Axioms of Uncertainty}}: $\\S 1$: Entropy = uncertainty = information: $1.1$ Uncertainty Category:Axioms/Axioms of Uncertainty drq6oflopludhg4ygoda2ost1fg9qs1"}
+{"_id": "33069", "title": "Complex Exponential Function is Entire", "text": "Complex Exponential Function is Entire 0 81953 432464 2019-10-27T15:09:41Z Caliburn 3218 Created page with \"== Theorem == <onlyinclude> Let $\\exp: \\C \\to \\C$ be the complex exponential function.   Then $\\exp$ is Definition:Entire Functio...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $\\exp: \\C \\to \\C$ be the complex exponential function.  Then $\\exp$ is entire.  </onlyinclude> == Proof == By the definition of the complex exponential function, $\\exp$ admits a power series expansion about $0$:  :$\\displaystyle \\exp z = \\sum_{n \\mathop = 0}^\\infty \\frac {z^n} {n!}$ By Complex Function is Entire iff it has Everywhere Convergent Power Series, to show that $\\exp$ is entire it suffices to show that this series is everywhere convergent. Note that this power series is of the form:  :$\\displaystyle \\sum_{n \\mathop = 0}^\\infty \\frac {\\paren {z - \\xi}^n} {n!}$ with $\\xi = 0$.  Therefore, by Radius of Convergence of Power Series over Factorial: Complex Case, we have that the former power series is everywhere convergent.  Hence the result. {{qed}} Category:Exponential Function bs3ajz8lpqth8r3s5pmgrrjvyc98dq8"}
+{"_id": "33070", "title": "Elementary Symmetric Function/Examples/Recursion", "text": "Elementary Symmetric Function/Examples/Recursion 0 82170 493064 493063 2020-10-07T14:06:14Z Prime.mover 59 wikitext text/x-wiki == Example of Elementary Symmetric Function: Recursion == <onlyinclude> Let $\\set {z_1, z_2, \\ldots, z_{n + 1} }$ be a set of $n + 1$ values, duplicate values permitted. Then for $1 \\le m \\le n$: :$\\map {e_m} {\\set {z_1, \\ldots, z_n, z_{n + 1} } } = z_{n + 1} \\map {e_{m - 1} } {\\set {z_1, \\ldots, z_n} } + \\map {e_m} {\\set {z_1, \\ldots, z_n} }$ </onlyinclude> == Proof == Case $m = 1$ holds because $e_0$ is $1$ and $e_1$ is the sum of the elements. Assume $2 \\le m \\le n$. Define four sets: :$A = \\set {\\set {p_1, \\ldots, p_m} : 1 \\le p_1 < \\cdots < p_m \\le n + 1}$ :$B = \\set {\\set {p_1, \\ldots, p_m} : 1 \\le p_1 < \\cdots < p_{m - 1} \\le n, p_m = n + 1}$ :$C = \\set {\\set {p_1, \\ldots, p_m} : 1 \\le p_1 < \\cdots < p_m \\le n}$ :$D = \\set {\\set {p_1, \\ldots, p_{m - 1} } : 1 \\le p_1 < \\cdots < p_{m - 1} \\le n}$ Then $A = B \\cup C$ and $B \\cap C = \\O$ implies: :$\\ds \\sum_A z_{p_1} \\cdots z_{p_m} = \\sum_B z_{p_1} \\cdots z_{p_m} + \\sum_C z_{p_1} \\cdots z_{p_m}$ Simplify: :$\\ds \\sum_B z_{p_1} \\cdots z_{p_m} = z_{n + 1} \\sum_D z_{p_1} \\cdots z_{p_{m - 1} }$ Notation for Elementary Symmetric Functions: :$\\ds \\map {e_m} {\\set {z_1, \\ldots, z_n, z_{n + 1} } } = \\sum_A z_{p_1} \\cdots z_{p_m}$ :$\\ds \\sum_D z_{p_1} \\cdots z_{p_{m - 1} } = \\map {e_{m - 1} } {\\set {z_1, \\ldots, z_n} }$ :$\\ds \\sum_C z_{p_1} \\cdots z_{p_m} = \\map {e_m} {\\set {z_1, \\ldots, z_n} }$ Assemble the preceding equations: {{begin-eqn}} {{eqn | l = \\map {e_m} {\\set {z_1, \\ldots, z_n, z_{n + 1} } }        | r = \\sum_A z_{p_1} \\cdots z_{p_m} }} {{eqn | r = \\sum_B z_{p_1} \\cdots z_{p_m} + \\sum_C z_{p_1} \\cdots z_{p_m} }} {{eqn | r = z_{n + 1} \\sum_D z_{p_1} \\cdots z_{p_m} + \\sum_C z_{p_1} \\cdots z_{p_m} }} {{eqn | r = z_{n + 1} \\map {e_{m - 1} } {\\set {z_1, \\ldots, z_n} } + \\map {e_m} {\\set {z_1, \\ldots, z_n} } }} {{end-eqn}} {{qed}} Category:Elementary Symmetric Functions mlp1kflfs6ugu7nw9q8rvovs64gt3mb"}
+{"_id": "33071", "title": "Inverse of Vandermonde Matrix/Corollary", "text": "Inverse of Vandermonde Matrix/Corollary 0 82308 443085 436852 2020-01-09T20:59:23Z Prime.mover 59 wikitext text/x-wiki {{refactor}} == Corollary to Inverse of Vandermonde Matrix == <onlyinclude> Define for variables $\\set {y_1,\\ldots, y_k}$ elementary symmetric functions: {{begin-eqn}} {{eqn | l = \\map {e_m} {\\set {y_1, \\ldots, y_k} }       | r = \\sum_{1 \\mathop \\le j_1 \\mathop < j_2 \\mathop < \\mathop \\cdots \\mathop < j_m \\mathop \\le k } y_{j_1} y_{j_2} \\cdots y_{j_m}       | c =  for $m = 0, 1, \\ldots, k$ }} {{end-eqn}} Let $\\set {x_1, \\ldots, x_n}$ be a set of distinct values. Let $W_n$ and $V_n$ be Vandermonde matrices of order $n$: :$W_n = \\begin{bmatrix}   1         &  x_1      & \\cdots & x_1^{n-1} \\\\   1         & x_2       & \\cdots & x_2^{n-1} \\\\ \\vdots      & \\vdots    & \\ddots & \\vdots    \\\\   1         & x_1^{n-1} & \\cdots & x_n^{n-1} \\\\ \\end{bmatrix}, \\quad V_n = \\begin{bmatrix}   x_1   & x_2    & \\cdots & x_n    \\\\   x_1^2 & x_2^2  & \\cdots & x_n^2  \\\\ \\vdots  & \\vdots & \\ddots & \\vdots \\\\   x_1^n & x_2^n  & \\cdots & x_n^n  \\\\ \\end{bmatrix}$ Let their matrix inverses be written as $W_n^{-1} = \\begin{bmatrix} b_{ij} \\end{bmatrix}$ $V_n^{-1} = \\begin{bmatrix} c_{ij} \\end{bmatrix}$. Then: {{begin-eqn}} {{eqn   | l = b_{ij}   | r = \\dfrac {\\paren {-1}^{n - i} \\map {e_{n - i} } {\\set {x_1, \\ldots, x_n} \\setminus \\set {x_j} } } {\\prod_{m \\mathop = 1, m \\mathop \\ne j }^n \\paren {x_j - x_m} }   | c = for $i, j = 1, \\ldots, n$ }} {{eqn   | l = c_{ij}   | r = \\dfrac 1 {x_i} \\, b_{j i}   | c = for $i, j = 1, \\ldots, n$ }} {{end-eqn}} </onlyinclude> == Proof == The details appear in Inverse of Vandermonde Matrix/Proof 1, same notation.{{qed}} Category:Inverse of Vandermonde Matrix 2g40bipql82wprdkeuo5qloqoohp17t"}
+{"_id": "33072", "title": "Equation of Cardioid/Parametric", "text": "Equation of Cardioid/Parametric 0 82574 435490 2019-11-17T00:32:52Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> Let $C$ be a cardioid embedded in a Cartesian coordinate plane such that: :its Definition:...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $C$ be a cardioid embedded in a Cartesian coordinate plane such that: :its stator of radius $a$ is positioned with its center at $\\tuple {a, 0}$ :there is a cusp at the origin. Then $C$ can be expressed by the parametric equation: :$\\begin {cases} x = 2 a \\cos t \\paren {1 + \\cos t} \\\\ y = 2 a \\sin t \\paren {1 + \\cos t} \\end {cases}$ </onlyinclude> == Proof == :525px Let $P = \\polar {x, y}$ be an arbitrary point on $C$. From Polar Form of Equation of Cardioid, $C$ is expressed as a polar equation as: :$r = 2 a \\paren {1 + \\cos \\theta}$ We have that: :$x = r \\cos \\theta$ :$y = r \\sin \\theta$ Replace $\\theta$ with $t$ and the required parametric equation is the result. {{qed}} == Sources == * {{MathWorld|Cardioid|Cardioid}} Category:Cardioids 6d8qpudg46rkavewi2qwje22robgjgh"}
+{"_id": "33073", "title": "Irrational Number divided by Rational Number is Irrational", "text": "Irrational Number divided by Rational Number is Irrational 0 82682 435876 2019-11-20T16:53:26Z Caliburn 3218 Created page with \"== Theorem == <onlyinclude> Let $x$ be a irrational number.   Let $y$ be a non-zero Definition:Rational Number|rational...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $x$ be a irrational number.  Let $y$ be a non-zero rational number. Then:  :$\\dfrac x y$ is irrational. </onlyinclude> == Proof == {{AimForCont}} $\\dfrac x y$ is rational number. Then there exists an integer $p_1$ and a natural number $q_1$ such that:  :$\\dfrac x y = \\dfrac {p_1} {q_1}$ That is:  :$x = \\dfrac {p_1} {q_1} y$ From the fact that $y$ is rational, we similarly have that there exists an integer $p_2$ and a natural number $q_2$ such that: :$y = \\dfrac {p_2} {q_2}$ Then:  :$x = \\dfrac {p_1 p_2} {q_1 q_2}$ From Integer Multiplication is Closed, we have that $p_1 p_2$ is an integer.  From Natural Number Multiplication is Closed, we have that $q_1 q_2$ is a natural number. Let $p_3 = p_1 p_2$ and $q_3 = q_1 q_2$. Then $x$ is expressible in the form:  :$x = \\dfrac {p_3} {q_3}$ where $p_3$ is an integer and $q_3$ is a natural number. This, however, implies that $x$ is rational, which is a contradiction. By Proof by Contradiction, we conclude that $\\dfrac x y$ is irrational. {{qed}} Category:Rational Numbers Category:Irrational Numbers 8lpmk0l0uozjamjcnib2a455dg7oqoi"}
+{"_id": "33074", "title": "Equation of Confocal Ellipses/Formulation 1", "text": "Equation of Confocal Ellipses/Formulation 1 0 82816 436936 436930 2019-11-27T12:50:20Z Prime.mover 59 wikitext text/x-wiki == Definition == <onlyinclude> The equation: :$(1): \\quad \\dfrac {x^2} {a^2 + \\lambda} + \\dfrac {y^2} {b^2 + \\lambda} = 1$ where: :$\\tuple {x, y}$ denotes an arbitrary point in the cartesian plane :$a$ and $b$ are (strictly) positive constants such that $a^2 > b^2$ :$\\lambda$ is a (strictly) positive parameter such that $b^2 > -\\lambda$ defines the set of all confocal ellipses whose foci are at $\\tuple {\\pm \\sqrt {a^2 - b^2}, 0}$. </onlyinclude> == Proof == Let $a$ and $b$ be arbitrary (strictly) positive real numbers fulfilling the constraints as defined. Let $E$ be the locus of the equation: :$(1): \\quad \\dfrac {x^2} {a^2 + \\lambda} + \\dfrac {y^2} {b^2  + \\lambda} = 1$ As $b^2 > -\\lambda$ it follows that: :$b^2 + \\lambda > 0$ and as $a^2 > b^2$: :$a^2 + \\lambda > 0$ Thus $(1)$ is in the form: :$\\dfrac {x^2} {r^2} + \\dfrac {y^2} {s^2} = 1$ where: :$r^2 = a^2 + \\lambda$ :$s^2 = b^2 + \\lambda$ From Equation of Ellipse in Reduced Form, this is the equation of an ellipse in reduced form. It follows that: :$\\tuple {\\pm \\sqrt {a^2 + \\lambda}, 0}$ are the positions of the vertices of $E$ :$\\tuple {0, \\pm \\sqrt {b^2 + \\lambda} }$ are the positions of the covertices of $E$ From Focus of Ellipse from Major and Minor Axis, the positions of the foci of $E$ are given by: :$\\paren {a^2 + \\lambda} - \\paren {b^2 + \\lambda} = c^2$ where $\\tuple {\\pm c, 0}$ are the positions of the foci of $E$. Thus we have: {{begin-eqn}} {{eqn | l = c^2       | r = \\paren {a^2 + \\lambda} - \\paren {b^2 + \\lambda}       | c = Focus of Ellipse from Major and Minor Axis }} {{eqn | r = a^2 - b^2       | c =  }} {{end-eqn}} Hence the result. {{qed}} == Also see == * Equation of Confocal Conics * Equation of Confocal Hyperbolas Category:Ellipses Category:Confocal Conics 1sp1wxwjfbglclrywxkh8n4hhl7u2v3"}
+{"_id": "33075", "title": "Equation of Confocal Hyperbolas/Formulation 1", "text": "Equation of Confocal Hyperbolas/Formulation 1 0 82817 436938 436937 2019-11-27T13:01:49Z Prime.mover 59 wikitext text/x-wiki == Definition == <onlyinclude> The equation: :$(1): \\quad \\dfrac {x^2} {a^2 + \\lambda} + \\dfrac {y^2} {b^2 + \\lambda} = 1$ where: :$\\tuple {x, y}$ denotes an arbitrary point in the cartesian plane :$a$ and $b$ are (strictly) positive constants such that $a^2 > b^2$ :$\\lambda$ is a (strictly) positive parameter such that $b^2 < -\\lambda < a^2$ defines the set of all confocal hyperbolas whose foci are at $\\tuple {\\pm \\sqrt {a^2 + b^2}, 0}$. </onlyinclude> == Proof == Let $a$ and $b$ be arbitrary (strictly) positive real numbers fulfilling the constraints as defined. Let $E$ be the locus of the equation: :$(1): \\quad \\dfrac {x^2} {a^2 + \\lambda} + \\dfrac {y^2} {b^2 + \\lambda} = 1$ As $b^2 < -\\lambda$ it follows that: :$b^2 + \\lambda < 0$ and as $-\\lambda < a^2$: :$a^2 + \\lambda > 0$ Thus $(1)$ is in the form: :$\\dfrac {x^2} {r^2} - \\dfrac {y^2} {s^2} = 1$ where: :$r^2 = a^2 + \\lambda$ :$s^2 = -\\lambda + b^2$ From Equation of Hyperbola in Reduced Form, this is the equation of an hyperbola in reduced form. It follows that: :$\\tuple {\\pm \\sqrt {a^2 + \\lambda}, 0}$ are the positions of the vertices of $E$ :$\\tuple {0, \\pm \\sqrt {b^2 - \\lambda} }$ are the positions of the covertices of $E$ From Focus of Hyperbola from Transverse and Conjugate Axis, the positions of the foci of $E$ are given by: :$\\paren {a^2 + \\lambda} + \\paren {b^2 - \\lambda} = c^2$ where $\\tuple {\\pm c, 0}$ are the positions of the foci of $E$. Thus we have: {{begin-eqn}} {{eqn | l = c^2       | r = \\paren {a^2 + \\lambda} + \\paren {b^2 - \\lambda}       | c = Focus of Hyperbola from Transverse and Conjugate Axis }} {{eqn | r = a^2 + b^2       | c =  }} {{end-eqn}} Hence the result. {{qed}} == Also see == * Equation of Confocal Conics * Equation of Confocal Ellipses Category:Hyperbolas Category:Confocal Conics 87l6gatjd3w8a9fg9zqn0l1bxm9d1ra"}
+{"_id": "33076", "title": "Method of Undetermined Coefficients/Exponential of Sine and Cosine", "text": "Method of Undetermined Coefficients/Exponential of Sine and Cosine 0 82927 439147 437659 2019-12-10T21:32:12Z Prime.mover 59 wikitext text/x-wiki == Proof Technique == Consider the nonhomogeneous linear second order ODE with constant coefficients: :$(1): \\quad y'' + p y' + q y = \\map R x$ Let $\\map R x$ be of the form: :$\\map R x = e^{a x} \\paren {\\alpha \\sin b x + \\beta \\cos b x}$ The '''Method of Undetermined Coefficients''' can be used to solve $(1)$ in the following manner. == Method and Proof == Let $\\map {y_g} x$ be the general solution to: :$(2): \\quad y'' + p y' + q y = 0$ From Solution of Constant Coefficient Homogeneous LSOODE, $\\map {y_g} x$ can be found systematically. Let $\\map {y_p} x$ be a particular solution to $(1)$. Then from General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution: :$\\map {y_g} x + \\map {y_p} x$ is the general solution to $(1)$. It remains to find $\\map {y_p} x$. <onlyinclude> Substitute a trial solution of similar form, either: :$e^{a x} \\paren {A \\sin b x + B \\cos b x}$ or replace the {{RHS}} of $(1)$ by: :$\\paren {\\alpha - i \\beta} e^{i \\paren {a + i b} x}$ find a solution, and take the real part. If $e^{a x} \\sin b x$ and $e^{a x} \\cos b x$ appear in the general solution to $(2)$, then insert a factor of $x$: :$x e^{a x} \\paren {A \\sin b x + B \\cos b x}$ or: :$x \\paren {\\alpha - i \\beta} e^{i \\paren {a + i b} x}$ </onlyinclude>{{qed}} == Sources == * {{BookReference|Elementary Differential Equations & Operators|1958|G.E.H. Reuter|prev = Linear Second Order ODE/y'' + 4 y = 3 sin 2 x|next = Method of Undetermined Coefficients/Product of Polynomial and Exponential}}: Chapter $1$: Linear Differential Equations with Constant Coefficients: $\\S 2$. The second order equation: $\\S 2.6$ Particular solution: some further cases $\\text{(i)}$ Category:Method of Undetermined Coefficients jdbghwqd5u95hfnwjk7vhou2369z49v"}
+{"_id": "33077", "title": "Method of Undetermined Coefficients/Product of Polynomial and Exponential", "text": "Method of Undetermined Coefficients/Product of Polynomial and Exponential 0 82928 437658 437656 2019-12-03T22:24:07Z Prime.mover 59 wikitext text/x-wiki == Proof Technique == Consider the nonhomogeneous linear second order ODE with constant coefficients: :$(1): \\quad y'' + p y' + q y = \\map R x$ Let $\\map R x$ be of the form: :$\\map R x = e^{a x} \\paren {\\map f x}$ where $\\map f x$ is a real polynomial function. The '''Method of Undetermined Coefficients''' can be used to solve $(1)$ in the following manner. == Method and Proof == Let $\\map {y_g} x$ be the general solution to: :$(2): \\quad y'' + p y' + q y = 0$ From Solution of Constant Coefficient Homogeneous LSOODE, $\\map {y_g} x$ can be found systematically. Let $\\map {y_p} x$ be a particular solution to $(1)$. Then from General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution: :$\\map {y_g} x + \\map {y_p} x$ is the general solution to $(1)$. It remains to find $\\map {y_p} x$. <onlyinclude> Substitute a trial solution of similar form: :$e^{a x} \\paren {\\map g x}$ where $\\map g x$ is a real polynomial function with undetermined coefficients of as high a degree as $f$. Then: :differentiate twice {{WRT|Differentiation}} $x$ :establish a set of simultaneous equations by equating powers :solve these equations for the coefficients. If $e^{a x} \\paren {\\map g x}$ appears in the general solution to $(2)$, then add a further degree to $g$. The last step may need to be repeated if that last polynomial also appears as a general solution to $(2)$. </onlyinclude>{{qed}} == Sources == * {{BookReference|Elementary Differential Equations & Operators|1958|G.E.H. Reuter|prev = Method of Undetermined Coefficients/Exponential of Sine and Cosine|next = Method of Undetermined Coefficients/Product of Polynomial and Function of Sine and Cosine}}: Chapter $1$: Linear Differential Equations with Constant Coefficients: $\\S 2$. The second order equation: $\\S 2.6$ Particular solution: some further cases $\\text{(ii)}$ Category:Method of Undetermined Coefficients mctg71208riib0jzabmvpl29c71c0n5"}
+{"_id": "33078", "title": "Method of Undetermined Coefficients/Product of Polynomial and Function of Sine and Cosine", "text": "Method of Undetermined Coefficients/Product of Polynomial and Function of Sine and Cosine 0 82929 437677 437657 2019-12-03T22:44:13Z Prime.mover 59 wikitext text/x-wiki == Proof Technique == Consider the nonhomogeneous linear second order ODE with constant coefficients: :$(1): \\quad y'' + p y' + q y = \\map R x$ Let $\\map R x$ be of the form: :$\\map R x = \\paren {\\alpha \\cos b x + \\beta \\sin b x} \\paren {\\map f x}$ where $\\map f x$ is a real polynomial function. The '''Method of Undetermined Coefficients''' can be used to solve $(1)$ in the following manner. == Method and Proof == Let $\\map {y_g} x$ be the general solution to: :$(2): \\quad y'' + p y' + q y = 0$ From Solution of Constant Coefficient Homogeneous LSOODE, $\\map {y_g} x$ can be found systematically. Let $\\map {y_p} x$ be a particular solution to $(1)$. Then from General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution: :$\\map {y_g} x + \\map {y_p} x$ is the general solution to $(1)$. It remains to find $\\map {y_p} x$. <onlyinclude> Substitute a trial solution of similar form, either: :$\\paren {\\alpha \\cos b x + \\beta \\sin b x} \\paren {\\map g x}$ or replace the {{RHS}} of $(1)$ by: :$\\paren {\\alpha - i \\beta} e^{i \\paren {a + i b} x} \\paren {\\map g x}$ find a solution, and take the real part. In the above, $\\map g x$ is a real polynomial function with undetermined coefficients of as high a degree as $f$. Then: :differentiate twice {{WRT|Differentiation}} $x$ :establish a set of simultaneous equations by equating powers :solve these equations for the coefficients. If $\\paren {\\alpha \\cos b x + \\beta \\sin b x} \\paren {\\map g x}$ appears in the general solution to $(2)$, then add a further degree to $g$. The last step may need to be repeated if that last polynomial also appears as a general solution to $(2)$. </onlyinclude>{{qed}} == Sources == * {{BookReference|Elementary Differential Equations & Operators|1958|G.E.H. Reuter|prev = Method of Undetermined Coefficients/Product of Polynomial and Exponential|next = Method of Undetermined Coefficients/Sum of Several Terms}}: Chapter $1$: Linear Differential Equations with Constant Coefficients: $\\S 2$. The second order equation: $\\S 2.6$ Particular solution: some further cases $\\text{(iii)}$ Category:Method of Undetermined Coefficients b3b0v1ep33jfqbzqz6q6wud2gav5a6p"}
+{"_id": "33079", "title": "Solution of Constant Coefficient Linear nth Order ODE", "text": "Solution of Constant Coefficient Linear nth Order ODE 0 82960 437885 437884 2019-12-06T06:33:33Z Prime.mover 59 wikitext text/x-wiki {{refactor|Extract the homogeneous case into its own page and improve its rigour and its structure. Then address the particular case by means of Method of Undetermined Coefficients as per usual.}} == Proof Technique == Consider the linear second order ODE with constant coefficients: :$(1): \\quad \\displaystyle \\sum_{k \\mathop = 0}^n a_k \\dfrac {\\d^k y} {d x^k} = \\map R x$ where $a_k$ is a constant for $0 \\le k \\le n$ and $\\map R x$ is a function of $x$. The general solution to $(1)$ can be found as follows. ;Find the roots $m_1, m_2, \\ldots, m_n$ of the auxiliary equation: :$(2): \\quad \\displaystyle \\sum_{k \\mathop = 0}^n a_k m^k = 0$. If $(2)$ has distinct roots, then the general solution $\\map {y_g} x$ is of the form: :$y_g = \\sum_{k \\mathop = 0}^n A_k e^{m_k x}$ If there are repeated roots of $(2)$, further needs to be done. Let $m_j$ be a repeated root of $(2)$ with multiplicity $r$. Then the {{LHS}} of $(2)$ can be written: :$\\map P m \\paren {m - m_j}^r$ where $\\map P m$ is a polynomial which does not contain the factor $m - m_j$. Then the $r$ instances of $m_j$ give rise to the solutions: :$(4): \\quad y = A_{j_0} e^{m_j x} + A_{j_1} x e^{m_j x} + \\dotsb + A_{j_{r - 1} } x^{r - 1} e^{m_j x}$ == Proof == Let the reduced equation of $(1)$ be written: :$(3): \\quad \\paren {a_n D^n + a_{n - 1} D^{n - 1} + \\dotsb + a_1 D + a_0} y = 0$ By factoring the {{LHS}} we get: :$\\map P D \\paren {D - m_j}^r y = 0$ It remains to be shown that $(4)$ is a solution to $(3)$. Each of the terms is of the form $p e^{m_j x} u$, where $u$ is a power of $x$ and $p$ is a constant. So: {{begin-eqn}} {{eqn | l = \\map {\\paren {D - m_j} } {p e^{m_j x} u}       | r = \\map D {p e^{m_j x} u} - m_j p e^{m_j x} u       | c =  }} {{eqn | r = p e^{m_j x} D u       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \\map {\\paren {D - m_j}^2} {p e^{m_j x} u}       | r = \\map {\\paren {D - m_j} } {p e^{m_j x} D u}       | c =  }} {{eqn | r = p e^{m_j x} \\map D {D u}       | c =  }} {{eqn | r = p e^{m_j x} \\map {D^2} u       | c =  }} {{end-eqn}} and so on until: {{begin-eqn}} {{eqn | l = \\map {\\paren {D - m_j}^r} {p e^{m_j x} u}       | r = p e^{m_j x} D^r u       | c =  }} {{eqn | r = 0       | c =  }} {{end-eqn}} because $D^r u = 0$ when $u = 1, x, x^2, \\ldots, x^{r - 1}$. {{finish}} == Sources == * {{BookReference|Elementary Differential Equations & Operators|1958|G.E.H. Reuter|prev = Definition:Reduced Equation of Linear ODE with Constant Coefficients|next = ODE/(D^4 - 1) y = sin x}}: Chapter $1$: Linear Differential Equations with Constant Coefficients: $\\S 3$. Equations of higher order and systems of first order equations: $\\S 3.1$ The $n$th order equation Category:Linear ODEs f3keoplpt0ksewiw6w9xeg7ybcn2q48"}
+{"_id": "33080", "title": "Lindelöf's Lemma/Lemma/Lemma/Lemma", "text": "Lindelöf's Lemma/Lemma/Lemma/Lemma 0 83033 443077 438449 2020-01-09T12:14:37Z Prime.mover 59 wikitext text/x-wiki {{rename|This is such a basic result it really should not be hidden as obscurely as a lemma of a lemma of a lemma of a lemma.}} == Lemma == <onlyinclude> Let $S$ be countable set. Let $T$ be a set. Let $T$ be in one-to-one correspondence with $S$. Then $T$ is countable. </onlyinclude> == Proof == $S$ is countable. Therefore, $S$ is in one-to-one correspondence with a subset of the natural numbers by a definition of countable set. $T$ is in one-to-one correspondence with $S$. Therefore, $T$ is in one-to-one correspondence with a subset of the natural numbers by Composite of Bijections is Bijection. Accordingly, $T$ is countable by a definition of countable. {{qed}} Category:Real Analysis sj6s0ge2aqhc7uapyw70f858jj3xbgs"}
+{"_id": "33081", "title": "Lindelöf's Lemma/Lemma/Lemma", "text": "Lindelöf's Lemma/Lemma/Lemma 0 83034 443076 439243 2020-01-09T12:12:24Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $R$ be a set of real intervals with rational numbers as endpoints. Let every interval in $R$ be of the same type of which there are four: $\\openint \\ldots \\ldots$, $\\closedint \\ldots  \\ldots$, $\\hointr \\ldots \\ldots$, and $\\hointl \\ldots \\ldots$. Then $R$ is countable. </onlyinclude> == Proof == === Lemma 2 === {{:Lindelöf's Lemma/Lemma/Lemma/Lemma|Lemma 2}} {{qed|lemma}} By Rational Numbers are Countably Infinite, the rationals are countable. By Subset of Countably Infinite Set is Countable, a subset of the rationals is countable. The endpoint of an interval in $R$ is characterized by a rational number as every interval in $R$ is of the same type. Therefore, the set consisting of the left hand endpoints of every interval in $R$ is countable. Also, the set consisting of the right hand endpoints of every interval in $R$ is countable. The cartesian product of countable sets is countable. Therefore, the cartesian product of the sets consisting of the respectively left hand and  right hand endpoints of every interval in $R$ is countable. A subset of this cartesian product is in one-to-one correspondence with $R$. This subset is countable by Subset of Countably Infinite Set is Countable. $R$ is countable by Lemma 2 as $R$ is in one-to-one correspondence with a countable set. {{qed}} Category:Real Analysis t5y1vlhdt2cj7xhliqjwgn4fq4etsog"}
+{"_id": "33082", "title": "Absolute Value of Cut is Greater Than or Equal To Zero Cut", "text": "Absolute Value of Cut is Greater Than or Equal To Zero Cut 0 83472 441244 441243 2019-12-28T20:44:19Z Prime.mover 59 wikitext text/x-wiki == Definition == Let $\\alpha$ be a cut. Let $\\size \\alpha$ denote the '''absolute value of $\\alpha$'''. Then: :$\\size \\alpha \\ge 0^*$ where: :$0^*$ denotes the rational cut associated with the (rational) number $0$ :$\\ge$ denotes the ordering on cuts. == Proof == Let $\\alpha \\ge 0^*$. Then by definition $\\size \\alpha = \\alpha \\ge 0^*$. Let $\\alpha < 0^*$. Then: :$\\exists \\beta: \\beta + \\alpha = 0^*$ Thus: :$\\alpha = -\\beta$ and it follows that $\\beta > 0^*$. The result follows. {{qed}} == Sources == * {{BookReference|Principles of Mathematical Analysis|1964|ed = 2nd|edpage = Second Edition|Walter Rudin|prev = Definition:Absolute Value of Cut|next = Absolute Value of Cut is Zero iff Cut is Zero}}: Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.24$. Definition Category:Cuts Category:Absolute Value Function 5zo6m47unf76etmjv8jv6zu8jj659cg"}
+{"_id": "33083", "title": "Absolute Value of Cut is Zero iff Cut is Zero", "text": "Absolute Value of Cut is Zero iff Cut is Zero 0 83473 441246 441245 2019-12-28T20:47:06Z Prime.mover 59 wikitext text/x-wiki == Definition == Let $\\alpha$ be a cut. Let $\\size \\alpha$ denote the '''absolute value of $\\alpha$'''. Then: :$\\size \\alpha = 0^*$ {{iff}} $\\alpha = 0^*$ where $0^*$ denotes the rational cut associated with the (rational) number $0$. == Proof == Let $\\alpha 0^*$. Then by definition $\\size \\alpha = \\alpha = 0^*$. Let $\\alpha \\ne 0^*$. Then either: :$\\alpha > 0^*$ in which case $\\size \\alpha = \\alpha > 0^*$ or: :$\\alpha < 0^*$ in which case $\\size \\alpha = -\\alpha > 0^*$ In either case $\\size \\alpha \\ne 0^*$. The result follows. {{qed}} == Sources == * {{BookReference|Principles of Mathematical Analysis|1964|ed = 2nd|edpage = Second Edition|Walter Rudin|prev = Absolute Value of Cut is Greater Than or Equal To Zero Cut|next = Definition:Multiplication of Cuts}}: Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.24$. Definition Category:Cuts Category:Absolute Value Function nlxiqqazy04xx2adz93ayrrs0e4pfko"}
+{"_id": "33084", "title": "Dedekind's Theorem/Corollary", "text": "Dedekind's Theorem/Corollary 0 83566 441930 2020-01-01T22:30:08Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> Let $\\tuple {L, R}$ be a Dedekind cut of the set of real numbers $\\R$.   Then either $L$ con...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $\\tuple {L, R}$ be a Dedekind cut of the set of real numbers $\\R$. Then either $L$ contains a largest number or $R$ contains a smallest number. </onlyinclude> == Proof == From Dedekind's Theorem, there exists a unique real number such that: :$l \\le \\gamma$ for all $l \\in L$ :$\\gamma \\le r$ for all $r \\in R$. Let $\\gamma \\in L$. Then by definition $\\gamma$ is the largest number in $L$. Let $\\gamma \\in R$. Then by definition $\\gamma$ is the smallest number in $R$. By the definition of Dedekind cut, $\\tuple {L, R}$ is a partition of $\\R$. Hence $\\gamma$ is either in $L$ or $R$, but not both. That is, $\\gamma$ is either: :the largest number in $L$ or: :the smallest number in $R$. Hence the result. {{Qed}} == Sources == * {{BookReference|Principles of Mathematical Analysis|1964|ed = 2nd|edpage = Second Edition|Walter Rudin|prev = Dedekind's Theorem|next = Dedekind's Theorem/Proof 3}}: Chapter $1$: The Real and Complex Number Systems: Real Numbers: $1.32$. Corollary Category:Real Analysis Category:Dedekind's Theorem i5bjas4p73f66davhsz1m87fy9g50kb"}
+{"_id": "33085", "title": "Uniqueness of Positive Root of Positive Real Number/Negative Exponent", "text": "Uniqueness of Positive Root of Positive Real Number/Negative Exponent 0 83583 442142 2020-01-03T21:18:38Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> Let $x \\in \\R$ be a real number such that $x > 0$.  Let $n \\in \\Z$ be an integer such that $n < 0...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $x \\in \\R$ be a real number such that $x > 0$. Let $n \\in \\Z$ be an integer such that $n < 0$. Then there is at most one $y \\in \\R: y \\ge 0$ such that $y^n = x$. </onlyinclude> == Proof == Let $m = -n$. Let $g$ be the real function defined on $\\hointr 0 \\to$ defined by: :$\\map g y = y^m$ From the definition of power: :$\\map g y = \\dfrac 1 {\\map f y}$ Hence $\\map g y$ is strictly decreasing. {{explain|Needs to invoke a result about reciprocals inverting the sense.}} From Strictly Monotone Mapping with Totally Ordered Domain is Injective: :there is at most one $y \\in \\R: y \\ge 0$ such that $y^n = x$. {{qed}} Category:Uniqueness of Positive Root of Positive Real Number fgjw4321lcbb33b8bbthpra853zansn"}
+{"_id": "33086", "title": "Russell's Paradox/Corollary", "text": "Russell's Paradox/Corollary 0 83748 444615 444300 2020-01-23T20:40:50Z Prime.mover 59 wikitext text/x-wiki == Corollary to Russell's Paradox == <onlyinclude> :$\\not \\exists x: \\forall y: \\paren {\\map \\RR {x, y} \\iff \\neg \\map \\RR {y, y} }$ </onlyinclude> Given a relation $\\RR$, there cannot exist an element $x$ that bears $\\RR$ to all $y$ that do not bear $\\RR$ to $y$. == Proof == {{AimForCont}} there does exist such an $x$. Let $\\RR$ be such that $\\map \\RR {x, x}$. Then $\\neg \\map \\RR {x, x}$. Hence it cannot be the case that $\\map \\RR {x, x}$. Now suppose that $\\neg \\map \\RR {x, x}$. Then by definition of $x$ it follows that $\\map \\RR {x, x}$. In both cases a contradiction results. Hence there can be no such $x$. {{qed}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Barber Paradox|next = Axiom:Axiom of Specification/Set Theory}}: Chapter $1$: General Background: $\\S 8$ Russell's paradox Category:Russell's Paradox 4m9sguygtvjcki18ilwru953leixhdu"}
+{"_id": "33087", "title": "Axiom:Axiom of Extension/Class Theory", "text": "Axiom:Axiom of Extension/Class Theory 100 83823 444630 444266 2020-01-23T20:48:15Z Prime.mover 59 Prime.mover moved page Axiom:Axiom of Extension/Classes to Axiom:Axiom of Extension/Class Theory wikitext text/x-wiki == Axiom == <onlyinclude> Let $A$ and $B$ be classes. Then: :$\\forall x: \\paren {x \\in A \\iff x \\in B} \\iff A = B$ </onlyinclude> Hence the order in which the elements are listed in the sets is immaterial. == Also known as == {{:Axiom:Axiom of Extension/Also known as}} == Also see == * Definition:Class Equality * Definition:Equals == Linguistic Note == {{:Axiom:Axiom of Extension/Linguistic Note}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Subclass|next = Definition:Class Equality/Definition 2}}: Chapter $2$: Some Basics of Class-Set Theory: $\\S 1$ Extensionality and separation Extension Extension Extension d49c7lcjvs32dhy04hu4mqht7odovxl"}
+{"_id": "33088", "title": "Axiom:Axiom of Extension/Set Theory", "text": "Axiom:Axiom of Extension/Set Theory 100 83836 497668 491976 2020-11-03T06:13:01Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> Let $A$ and $B$ be sets. The '''axiom of extension''' states that $A$ and $B$ are equal {{iff}} they contain the same elements. That is, {{iff}}: :every element of $A$ is also an element of $B$ and: :every element of $B$ is also an element of $A$. This can be formulated as follows: </onlyinclude> === Formulation 1 === <onlyinclude>{{:Axiom:Axiom of Extension/Set Theory/Formulation 1}}</onlyinclude> === Formulation 2 === In set theories that define $=$ instead of admitting it as a primitive, the '''axiom of extension''' can be formulated as: {{:Axiom:Axiom of Extension/Set Theory/Formulation 2}} The order of the elements in the sets is immaterial. Hence a set is completely and uniquely determined by its elements. == Also known as == {{:Axiom:Axiom of Extension/Also known as}} == Also see == * Definition:Set Equality * Definition:Equals * Axiom:Axiom of Extension (Classes) == Linguistic Note == {{:Axiom:Axiom of Extension/Linguistic Note}} == Sources == * {{BookReference|Naive Set Theory|1960|Paul R. Halmos|prev = Definition:Set Equality/Definition 1|next = Definition:Subset}}: $\\S 1$: The Axiom of Extension * {{BookReference|Abstract Algebra|1964|W.E. Deskins|prev = Definition:Uniqueness of Set Elements|next = Axiom:Axiom of Extension/Also known as}}: Chapter $1$: A Common Language: $\\S 1.1$ Sets * {{BookReference|Elements of Abstract Algebra|1966|Richard A. Dean|prev = Definition:Set Equality/Definition 1|next = Definition:Explicit Set Definition}}: $\\S 0.2$. Sets * {{BookReference|The Joy of Sets: Fundamentals of Contemporary Set Theory|1993|Keith Devlin|ed = 2nd|edpage = Second Edition|prev = Definition:Set Equality/Definition 1|next = Axiom:Axiom of Extension/Set Theory/Formulation 1}}: $\\S 1$: Naive Set Theory: $\\S 1.1$: What is a Set? Extension Extension Extension Extension ezzwxmd0yb8s2uo5azlvo4angx795ly"}
+{"_id": "33089", "title": "Axiom:Axiom of Specification/Class Theory", "text": "Axiom:Axiom of Specification/Class Theory 100 83846 444613 444609 2020-01-23T20:39:57Z Prime.mover 59 wikitext text/x-wiki == Axiom == The '''axiom of specification''' is an axiom schema which can be formally stated as follows: <onlyinclude> Let $\\map \\phi {A_1, A_2, \\ldots, A_n, x}$ be a function of propositional logic such that: :$A_1, A_2, \\ldots, A_n$ are a finite number of free variables whose domain ranges over all classes :$x$ is a free variable whose domain ranges over all sets. Then the '''axiom of specification''' gives that: :$\\forall A_1, A_2, \\ldots, A_n: \\exists B: \\forall x: \\paren {x \\in B \\iff \\paren {x \\in B \\land \\phi {A_1, A_2, \\ldots, A_n, x} } }$ where each of $B$ ranges over arbitrary classes. </onlyinclude> This means that for any finite number $A_1, A_2, \\ldots, A_n$ of subclasses of the universal class $V$, the class  $B$ exists (or can be formed) of all sets $x \\in V$ that satisfy the function $\\map \\phi {A_1, A_2, \\ldots, A_n, x}$. == Also known as == {{:Axiom:Axiom of Specification/Also known as}} == Also see == * Axiom:Axiom of Specification/Set Theory == Historical Note == {{:Axiom:Axiom of Specification/Historical Note}} {{Languages|Axiom of specification}} {{language|German|Aussonderungsaxiom|lit = axiom of segregation}} {{end-languages}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Axiom:Axiom of Specification/Also known as|next = Not Every Class is a Set/Proof 1}}: Chapter $2$: Some Basics of Class-Set Theory: $\\S 1$ Extensionality and separation Specification Specification Specification 4ucgl8faz73dvwupsgvw7quikszyn4c"}
+{"_id": "33090", "title": "Axiom:Axiom of Transitivity", "text": "Axiom:Axiom of Transitivity 100 83863 449994 449988 2020-02-19T10:27:16Z Prime.mover 59 wikitext text/x-wiki == Axiom == Let $V$ be a basic universe. <onlyinclude> :$V$ is a transitive class. </onlyinclude> That is, every set $S$ which is an element of $V$ is a subclass of $V$. Briefly: :Every set is a class. == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Basic Universe Axioms|next = Axiom:Axiom of Swelledness}}: Chapter $2$: Some Basics of Class-Set Theory: $\\S 2$ Transitivity and supercompleteness Transitivity scdhzlpwz76qf6x9m0n8b032aanphlz"}
+{"_id": "33091", "title": "Axiom:Axiom of Swelledness", "text": "Axiom:Axiom of Swelledness 100 83864 449996 449987 2020-02-19T10:27:52Z Prime.mover 59 wikitext text/x-wiki == Axiom == Let $V$ be a basic universe. <onlyinclude> :$V$ is a swelled class. </onlyinclude> That is, every subclass of a set which is an element of $V$ is a set in $V$. Briefly: :Every subclass of a set is a set. == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Axiom:Axiom of Transitivity|next = Basic Universe is Supercomplete}}: Chapter $2$: Some Basics of Class-Set Theory: $\\S 2$ Transitivity and supercompleteness Swelledness qts7d0jpsbn35fi3ipb1mrhwouvkz41"}
+{"_id": "33092", "title": "Axiom:Axiom of Empty Set/Class Theory", "text": "Axiom:Axiom of Empty Set/Class Theory 100 83881 449995 449989 2020-02-19T10:27:32Z Prime.mover 59 wikitext text/x-wiki == Axiom == Let $V$ be a basic universe. <onlyinclude> The empty class $\\O$ is a set, that is: :$\\O \\in V$ </onlyinclude> == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Empty Class is Supercomplete|next = Basic Universe is not Empty}}: Chapter $2$: Some Basics of Class-Set Theory: $\\S 3$ Axiom of the empty set Empty Set Empty Set Empty Set 5q3ryuct4vv49oes7o7j8g6p3i4z23x"}
+{"_id": "33093", "title": "Axiom:Axiom of Pairing/Set Theory/Formulation 1", "text": "Axiom:Axiom of Pairing/Set Theory/Formulation 1 100 83902 491372 444586 2020-09-28T07:53:10Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> For any two sets, there exists a set to which only those two sets are elements: :$\\forall A: \\forall B: \\exists x: \\forall y: \\paren {y \\in x \\iff y = A \\lor y = B}$ Thus it is possible to create a set that contains as elements any two sets that have already been created. </onlyinclude> == Also known as == {{:Axiom:Axiom of Pairing/Also known as}} == Also see == * Equivalence of Definitions of Axiom of Pairing == Sources == * {{BookReference|Point Set Topology|1964|Steven A. Gaal|prev = Set Definition by Predicate/Examples/Sums of Two Squares|next = Definition:Doubleton}}: Introduction to Set Theory: $1$. Elementary Operations on Sets * {{MathWorld|Zermelo-Fraenkel Axioms|Zermelo-FraenkelAxioms}} * {{MathWorld|Axiom of the Unordered Pair|AxiomoftheUnorderedPair}} Pairing 0vps8rk1gp0i13mvkkqa7sam5szgj7u"}
+{"_id": "33094", "title": "Axiom:Axiom of Pairing/Set Theory/Formulation 2", "text": "Axiom:Axiom of Pairing/Set Theory/Formulation 2 100 83903 444588 444570 2020-01-23T20:23:15Z Prime.mover 59 Prime.mover moved page Axiom:Axiom of Pairing/Sets/Formulation 2 to Axiom:Axiom of Pairing/Set Theory/Formulation 2 wikitext text/x-wiki == Axiom == <onlyinclude> For any two sets, there exists a set containing those two sets as elements: :$\\forall A: \\forall B: \\exists x: \\forall y: \\paren {y \\in x \\implies y = A \\lor y = B}$ Thus it is possible to create a set that contains as elements '''at least''' two sets that have already been created. </onlyinclude> == Also known as == {{:Axiom:Axiom of Pairing/Also known as}} == Also see == * Equivalence of Definitions of Axiom of Pairing == Sources == * {{BookReference|Naive Set Theory|1960|Paul R. Halmos|prev = Empty Set is Subset of All Sets/Proof 2|next = Definition:Doubleton}}: $\\S 3$: Unordered Pairs Pairing 4kdr8tfqdpzc7yqjhyixitdy2i6nkbp"}
+{"_id": "33095", "title": "Axiom:Axiom of Pairing/Class Theory", "text": "Axiom:Axiom of Pairing/Class Theory 100 83907 449997 449990 2020-02-19T10:28:05Z Prime.mover 59 wikitext text/x-wiki == Axiom == === Formulation 1 === <onlyinclude>{{:Axiom:Axiom of Pairing/Class Theory/Formulation 1}}</onlyinclude> === Formulation 2 === {{:Axiom:Axiom of Pairing/Class Theory/Formulation 2}} == Also known as == {{:Axiom:Axiom of Pairing/Also known as}} == Also see == * Equivalence of Definitions of Axiom of Pairing for Classes * Definition:Doubleton Class * Doubleton Class can be Formed from Two Sets, which demonstrates the fact that the class $\\set {a, b}$ can be created in the first place Pairing Pairing Pairing Pairing 0dvx4rpabm6uigw4fmmjdmfferroicn"}
+{"_id": "33096", "title": "Axiom:Axiom of Pairing/Class Theory/Formulation 1", "text": "Axiom:Axiom of Pairing/Class Theory/Formulation 1 100 83923 444674 444667 2020-01-23T23:19:24Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> Let $a$ and $b$ be sets. Then the class $\\set {a, b}$ is likewise a set. </onlyinclude> == Also known as == {{:Axiom:Axiom of Pairing/Also known as}} == Also see == * Equivalence of Definitions of Axiom of Pairing for Classes == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Doubleton Class of Equal Sets is Singleton Class|next = Axiom:Axiom of Pairing/Class Theory/Formulation 2}}: Chapter $2$: Some Basics of Class-Set Theory: $\\S 4$ The pairing axiom Pairing b0jm1h7muszglmtaayg9uzvuujle2at"}
+{"_id": "33097", "title": "Axiom:Axiom of Pairing/Class Theory/Formulation 2", "text": "Axiom:Axiom of Pairing/Class Theory/Formulation 2 100 83924 444676 444675 2020-01-23T23:20:12Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> Let $a$ and $b$ be sets. Then there exists a set $c$ such that $a \\in c$ and $b \\in c$. </onlyinclude> == Also known as == {{:Axiom:Axiom of Pairing/Also known as}} == Also see == * Equivalence of Definitions of Axiom of Pairing for Classes == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Axiom:Axiom of Pairing/Class Theory/Formulation 1|next = Equivalence of Definitions of Axiom of Pairing for Classes}}: Chapter $2$: Some Basics of Class-Set Theory: $\\S 4$ The pairing axiom Pairing b0z2difchl5tiyxr25667hu45jlh9bi"}
+{"_id": "33098", "title": "Axiom:Axiom of Unions/Class Theory", "text": "Axiom:Axiom of Unions/Class Theory 100 83953 449993 449991 2020-02-19T10:26:56Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> Let $x$ be a set (of sets). Then its union $\\displaystyle \\bigcup x$ is also a set. </onlyinclude> == Also known as == {{:Axiom:Axiom of Unions/Also known as}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Set Union/General Definition|next = Definition:Intersection of Class}}: Chapter $2$: Some Basics of Class-Set Theory: $\\S 5$ The union axiom Unions Unions Unions Unions 1r5hxyqtplv3nii7ppvijkwqih0qqga"}
+{"_id": "33099", "title": "Axiom:Axiom of Powers/Class Theory", "text": "Axiom:Axiom of Powers/Class Theory 100 83988 473014 449992 2020-06-10T02:05:34Z RandomUndergrad 3904 wikitext text/x-wiki == Axiom == <onlyinclude> Let $x$ be a set. Then its power set $\\powerset x$ is also a set. </onlyinclude> == Also known as == {{:Axiom:Axiom of Powers/Also known as}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Power Set Exists and is Unique|next = Axiom:Axiom of Powers/Set Theory}}: Chapter $2$: Some Basics of Class-Set Theory: $\\S 6$ The power axiom :::''although there exists a misprint: it is referred to as the '''Power aet axiom'''.'' Powers Powers Powers Powers lbb7ba64npqzd7u1kflb2nwjvfmvn82"}
+{"_id": "33100", "title": "Chain Rule", "text": "Chain Rule 0 84211 447653 447652 2020-02-09T01:09:31Z Prime.mover 59 wikitext text/x-wiki {{Disambiguation}} * Chain Rule for Derivatives * Chain Rule for Probability ropx1clmxaww5s5fmzg03to31b4a8kc"}
+{"_id": "33101", "title": "Projection from Product Topology is Continuous/General Result/Proof", "text": "Projection from Product Topology is Continuous/General Result/Proof 0 84475 449480 2020-02-16T13:02:03Z Prime.mover 59 Created page with \"== Theorem == {{:Projection from Product Topology is Continuous/General Result}}  == Proof == <onlyinclude> By definition of the Definition:Tychonoff Topology|Tychonoff topo...\" wikitext text/x-wiki == Theorem == {{:Projection from Product Topology is Continuous/General Result}} == Proof == <onlyinclude> By definition of the Tychonoff topology on $S$: :$\\tau$ is the initial topology on $S$ with respect to $\\family {\\pr_i}_{i \\mathop \\in I}$ By definition of the Initial Topoplogy:Definition 2: :$\\tau$ is the coarsest topology on $S$ such that each $\\pr_i: S \\to S_i$ is a $\\struct{\\tau, \\tau_i}$-continuous. </onlyinclude>{{qed}} Category:Projection from Product Topology is Open and Continuous t3mxbrerdivq2amckxjvljr3exiz6zm"}
+{"_id": "33102", "title": "Projection from Product Topology is Open/General Result/Proof", "text": "Projection from Product Topology is Open/General Result/Proof 0 84476 449481 2020-02-16T13:08:46Z Prime.mover 59 Created page with \"== Theorem == {{:Projection from Product Topology is Open/General Result}}  == Proof == <onlyinclude> Let $U \\in \\tau$.  It follows from the definition of Definition:Tychono...\" wikitext text/x-wiki == Theorem == {{:Projection from Product Topology is Open/General Result}} == Proof == <onlyinclude> Let $U \\in \\tau$. It follows from the definition of Tychonoff topology that $U$ can be expressed as: :$\\displaystyle U = \\bigcup_{j \\mathop \\in J} \\bigcap_{k \\mathop = 1}^{n_j} \\map {\\pr_{i_{k, j} }^{-1} } {U_{k, j} }$ where: :$J$ is an arbitrary index set :$n_j \\in \\N$ :$i_{k, j} \\in I$ :$U_{k, j} \\in \\tau_{i_{k, j} }$. For all $i' \\in I$, define $V_{i', k, j} \\in \\tau_{i'}$ by: :$V_{i', k, j} = \\begin {cases} U_{k, j} & : i' = i_{k, j} \\\\ S_{i'} & : i' \\ne i_{k, j} \\end {cases}$ For all $i \\in I$, we have: {{begin-eqn}} {{eqn | l = \\map {\\pr_i} U       | r = \\bigcup_{j \\mathop \\in J} \\map {\\pr_i} {\\bigcap_{k \\mathop = 1}^{n_j} \\map {\\pr_{i_{k,j} }^{-1} } { U_{k,j} } }       | c = Image of Union under Relation: Family of Sets }} {{eqn | r = \\bigcup_{j \\mathop \\in J} \\map {\\pr_i} {\\bigcap_{k \\mathop = 1}^{n_j} \\prod_{i' \\mathop \\in I} V_{i', k, j} }       | c = {{Defof|Projection (Mapping Theory)|Projection}} }} {{eqn | r = \\bigcup_{j \\mathop \\in J} \\map {\\pr_i} {\\prod_{i' \\mathop \\in I} \\bigcap_{k \\mathop = 1}^{n_j} V_{i', k, j} }       | c = Cartesian Product of Intersections: General Case }} {{eqn | r = \\bigcup_{j \\mathop \\in J} \\bigcap_{k \\mathop = 1}^{n_j} V_{i,k,j}       | c = {{Defof|Projection (Mapping Theory)|Projection}} }} {{end-eqn}} As: :$\\displaystyle \\bigcup_{j \\mathop \\in J} \\bigcap_{k \\mathop = 1}^{n_j} V_{i, k, j} \\in \\tau_i$ it follows that $\\pr_i$ is open. </onlyinclude>{{qed}} Category:Projection from Product Topology is Open and Continuous id4pkftu0oqbob7rnlrhlpcasyrdq3a"}
+{"_id": "33103", "title": "Axiom:Axiom of Infinity/Class Theory", "text": "Axiom:Axiom of Infinity/Class Theory 100 84547 449998 449984 2020-02-19T10:28:41Z Prime.mover 59 wikitext text/x-wiki == Axiom == === Formulation 1 === <onlyinclude>{{:Axiom:Axiom of Infinity/Class Theory/Formulation 1}}</onlyinclude> === Formulation 2 === {{:Axiom:Axiom of Infinity/Class Theory/Formulation 2}} === Formulation 3 === {{:Axiom:Axiom of Infinity/Class Theory/Formulation 3}} == Also see == * Equivalence of Formulations of Axiom of Infinity for Zermelo Universe Infinity Infinity Infinity klx7xc3gld1khpe2k745q44yanxdsls"}
+{"_id": "33104", "title": "Axiom:Axiom of Infinity/Class Theory/Formulation 1", "text": "Axiom:Axiom of Infinity/Class Theory/Formulation 1 100 84548 450002 449985 2020-02-19T10:36:44Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> Let $\\omega$ be the class of natural numbers as constructed by the Von Neumann construction: {{begin-eqn}} {{eqn | l = 0       | o = :=       | r = \\O       | c =  }} {{eqn | l = 1       | o = :=       | r = 0 \\cup \\set 0       | c =  }} {{eqn | l = 2       | o = :=       | r = 1 \\cup \\set 1       | c =  }} {{eqn | l = 3       | o = :=       | r = 2 \\cup \\set 2       | c =  }} {{eqn | o = \\vdots       | c =  }} {{eqn | l = n + 1       | o = :=       | r = n \\cup \\set n       | c =  }} {{eqn | o = \\vdots       | c =  }} {{end-eqn}} Then $\\omega$ is a set. </onlyinclude> == Also see == * Equivalence of Formulations of Axiom of Infinity for Zermelo Universe == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Inductive Set Definition for Natural Numbers|next = Definition:Zermelo Universe}}: Chapter $3$: The Natural Numbers: $\\S 2$ Definition of the Natural Numbers: Axiom $A_7$ Infinity Infinity mhsvoa9flwdf1bidqd73qx0ix0q0dsl"}
+{"_id": "33105", "title": "Axiom:Axiom of Infinity/Class Theory/Formulation 2", "text": "Axiom:Axiom of Infinity/Class Theory/Formulation 2 100 84551 450010 450004 2020-02-19T11:04:10Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> There exists an inductive set. </onlyinclude> == Also see == * Equivalence of Formulations of Axiom of Infinity for Zermelo Universe == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Zermelo Universe|next = Axiom:Axiom of Infinity/Class Theory/Formulation 3/Historical Note}}: Chapter $3$: The Natural Numbers: $\\S 2$ Definition of the Natural Numbers Infinity Infinity lu67jeskhv591puclw4kt2ka86u3h0s"}
+{"_id": "33106", "title": "Axiom:Axiom of Infinity/Class Theory/Formulation 3", "text": "Axiom:Axiom of Infinity/Class Theory/Formulation 3 100 84554 450009 450007 2020-02-19T11:03:55Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> Not every set is a natural number. </onlyinclude> == Also see == * Equivalence of Formulations of Axiom of Infinity for Zermelo Universe == Historical Note == {{:Axiom:Axiom of Infinity/Class Theory/Formulation 3/Historical Note}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Axiom:Axiom of Infinity/Class Theory/Formulation 3/Historical Note|next = Equivalence of Formulations of Axiom of Infinity for Zermelo Universe}}: Chapter $3$: The Natural Numbers: $\\S 2$ Definition of the Natural Numbers Infinity Infinity 5rohtd602vj2snp31dz1wzeorlmvcax"}
+{"_id": "33107", "title": "Condition for Expectation of Non-Negative Random Variable to be Zero", "text": "Condition for Expectation of Non-Negative Random Variable to be Zero 0 85335 455455 2020-03-18T11:29:40Z Caliburn 3218 Created page with \"== Theorem == <onlyinclude> Let $X$ be a random variable.   Let:   :$\\map \\Pr {X \\ge 0} = 1$   Then $\\expect X = 0$ {{iff}} $\\map \\Pr {X = 0} =...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $X$ be a random variable.  Let:  :$\\map \\Pr {X \\ge 0} = 1$ Then $\\expect X = 0$ {{iff}} $\\map \\Pr {X = 0} = 1$. </onlyinclude> == Proof == {{ProofWanted}} Category:Expectation fnf3epu60to075znwj0nr2hoeytp6ot"}
+{"_id": "33108", "title": "Covariance of Linear Combination of Random Variables with Another", "text": "Covariance of Linear Combination of Random Variables with Another 0 85343 455483 2020-03-18T12:17:31Z Caliburn 3218 Created page with \"== Theorem == <onlyinclude> Let $X, Y, Z$ be random variables.   Let $a, b$ be real numbers.   Then:   :$\\cov {a X +...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $X, Y, Z$ be random variables.  Let $a, b$ be real numbers. Then:  :$\\cov {a X + b Y, Z} = a \\cov {X, Z} + b \\cov {Y, Z}$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn\t| l = \\cov {a X + b Y, Z} \t| r = \\expect {\\paren {a X + b Y} Z} - \\expect {a X + b Y} \\expect Z \t| c = Covariance as Expectation of Product minus Product of Expectations }} {{eqn\t| r = a \\expect {X Z} + b \\expect {Y Z} - \\paren {a \\expect X + b \\expect Y} \\expect Z \t| c = Linearity of Expectation Function }} {{eqn\t| r = a \\paren {\\expect {X Z} - \\expect X \\expect Z} + b \\paren {\\expect {Y Z} - \\expect Y \\expect Z} }} {{eqn\t| r = a \\cov {X, Z} + b \\cov {Y, Z} \t| c = Covariance as Expectation of Product minus Product of Expectations }} {{end-eqn}} {{qed}} Category:Covariance clomckc88pxcsw54tysvlx7zl4j66xw"}
+{"_id": "33109", "title": "Covariance of Sums of Random Variables/Lemma", "text": "Covariance of Sums of Random Variables/Lemma 0 85345 455505 2020-03-18T14:14:08Z Caliburn 3218 Created page with \"== Theorem == <onlyinclude> Let $n$ be a strictly positive integer.  Let $\\sequence {X_i}_{1 \\le i \\le n}$ be a Definition:Sequence|...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $n$ be a strictly positive integer. Let $\\sequence {X_i}_{1 \\le i \\le n}$ be a sequence of random variables. Let $Y$ be a random variable. Then:  :$\\displaystyle \\cov {\\sum_{i \\mathop = 1}^n X_i, Y} = \\sum_{i \\mathop = 1}^n \\cov {X_i, Y}$ </onlyinclude>  == Proof == Proof by induction: For all $n \\in \\N$, let $\\map P n$ be the proposition:  :$\\displaystyle \\cov {\\sum_{i \\mathop = 1}^n X_i, Y} = \\sum_{i \\mathop = 1}^n \\cov {X_i, Y}$ === Basis for the Induction === We have that:  :$\\displaystyle \\cov {\\sum_{i \\mathop = 1}^n X_i, Y} = \\cov {X_1, Y} = \\sum_{i \\mathop = 1}^1 \\cov {X_i, Y}$ We therefore have that $\\map P 1$ is true.  This is our base case. === Induction Hypothesis === Suppose that $\\map P n$ is true for some fixed $n \\in \\N$.  That is:  :$\\displaystyle \\cov {\\sum_{i \\mathop = 1}^n X_i, Y} = \\sum_{i \\mathop = 1}^n \\cov {X_i, Y}$ We aim to show that it logically follows that $\\map P {n + 1}$ is true.  That is:  :$\\displaystyle \\cov {\\sum_{i \\mathop = 1}^{n + 1} X_i, Y} = \\sum_{i \\mathop = 1}^{n + 1} \\cov {X_i, Y}$ === Induction Step === This is our induction step: We have:  {{begin-eqn}} {{eqn\t| l = \\cov {\\sum_{i \\mathop = 1}^{n + 1} X_i, Y} \t| r = \\cov {\\sum_{i \\mathop = 1}^n X_i + X_{n + 1}, Y} \t| c = splitting up the sum }} {{eqn\t| r = \\cov {\\sum_{i \\mathop = 1}^n X_i, Y} + \\cov {X_{n + 1}, Y} \t| c = Covariance of Linear Combination of Random Variables with Another }} {{eqn\t| r = \\sum_{i \\mathop = 1}^n \\cov {X_i, Y} + \\cov {X_{n + 1}, Y} \t| c = induction hypothesis }} {{eqn\t| r = \\sum_{i \\mathop = 1}^{n + 1} \\cov {X_i, Y} }} {{end-eqn}} Hence the result by induction. {{qed}} Category:Covariance Category:Covariance of Sums of Random Variables di2bhly3lry6nhiskt1rq3obzfzf9i3"}
+{"_id": "33110", "title": "Expectation of Linear Transformation of Random Variable", "text": "Expectation of Linear Transformation of Random Variable 0 85348 455529 2020-03-18T16:10:35Z Caliburn 3218 Created page with \"== Theorem == <onlyinclude> Let $X$ be a random variable.   Let $a, b$ be real numbers.    Then we have:   :$\\expect...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $X$ be a random variable.  Let $a, b$ be real numbers.  Then we have:  :$\\expect {a X + b} = a \\expect X + b$ where $\\expect X$ denotes the expectation of $X$.  </onlyinclude> == Proof == === Discrete Random Variable === {{:Expectation of Linear Transformation of Random Variable/Discrete}} === Continuous Random Variable === {{:Expectation of Linear Transformation of Random Variable/Continuous}} Category:Expectation Category:Expectation of Linear Transformation of Random Variable s4z0ah7c7vdt0d5628c128pjokk1yvv"}
+{"_id": "33111", "title": "Definite Integral of Odd Function/Corollary", "text": "Definite Integral of Odd Function/Corollary 0 85650 457201 2020-03-23T11:39:22Z Prime.mover 59 Created page with \"== Corollary to Definite Integral of Odd Function == <onlyinclude> Let $f$ be an odd function with a Definition:Primitive (Calculus)|primitiv...\" wikitext text/x-wiki == Corollary to Definite Integral of Odd Function == <onlyinclude> Let $f$ be an odd function with a primitive on the open interval $\\openint {-a} a$, where $a > 0$. Then the improper integral of $f$ on $\\openint {-a} a$ is: :$\\displaystyle \\int_{\\mathop \\to -a}^{\\mathop \\to a} \\map f x \\rd x = 0$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \\int_{\\mathop \\to -a}^{\\mathop \\to a} \\map f x \\rd x       | r = \\lim_{y \\mathop \\to a} \\int_{-y}^y \\map f x \\rd x       | c = {{Defof|Improper Integral over Open Interval}} }} {{eqn | r = \\lim_{y \\mathop \\to a} 0       | c = Definite Integral of Odd Function }} {{eqn | r = 0       | c =  }} {{end-eqn}} {{qed}} Category:Integral Calculus Category:Odd Functions Category:Improper Integrals ds0u8xk48wh9kabypnu744covo100ww"}
+{"_id": "33112", "title": "Coulomb's Law of Electrostatics", "text": "Coulomb's Law of Electrostatics 0 86443 462481 462451 2020-04-17T07:38:30Z Prime.mover 59 wikitext text/x-wiki == Physical Law == :400pxrightthumbForce Between two Like Charges Let $a$ and $b$ be stationary particles in a vacuum, each carrying an electric charge of $q_a$ and $q_b$ respectively. Then $a$ and $b$ exert a force upon each other whose magnitude and direction are given by '''Coulomb's law (of electrostatics)''': :$\\mathbf F_{a b} \\propto \\dfrac {q_a q_b {\\mathbf r_{a b} } } {r^3}$ where: :$\\mathbf F_{a b}$ is the force exerted on $b$ by the electric charge on $a$ :$\\mathbf r_{a b}$ is the displacement vector from $a$ to $b$ :$r$ is the distance between $a$ and $b$. :the constant of proportion is defined as being positive. By exchanging $a$ and $b$ in the above, it is seen that $b$ exerts the same force on $a$ as $a$ does on $b$, but in the opposite direction. == SI Units == {{:Coulomb's Law of Electrostatics/SI Units}} Thus the equation becomes: :$\\mathbf F_{a b} = \\dfrac 1 {4 \\pi \\varepsilon_0} \\dfrac {q_a q_b {\\mathbf r_{a b} } } {r^3}$ == Also presented as == :$\\mathbf F_{a b} \\propto \\dfrac {q_a q_b \\hat {\\mathbf r}_{a b} } {r^2}$ where $\\hat {\\mathbf r}_{a b}$ is the unit vector in the direction from $a$ to $b$. == Also known as == '''Coulomb's Law of Electrostatics''' is also known as just '''Coulomb's Law'''. {{Namedfor|Charles-Augustin de Coulomb|cat = Coulomb}} == Historical Note == {{:Coulomb's Law of Electrostatics/Historical Note}} == Sources == * {{BookReference|Electromagnetism|1990|I.S. Grant|author2 = W.R. Phillips|ed = 2nd|edpage = Second Edition|prev = Newton's Law of Universal Gravitation|next = Newton's Law of Universal Gravitation/Historical Note}}: Chapter $1$: Force and energy in electrostatics * {{BookReference|Electromagnetism|1990|I.S. Grant|author2 = W.R. Phillips|ed = 2nd|edpage = Second Edition|prev = Definition:Negative Electric Charge|next = Definition:Unit Vector}}: Chapter $1$: Force and energy in electrostatics: $1.1$ Electric Charge Category:Electrostatics Category:Coulomb's Law of Electrostatics 4l3qloy2d1vr2or0tyutli4spkychjd"}
+{"_id": "33113", "title": "Bijection/Examples/Real Cube Function", "text": "Bijection/Examples/Real Cube Function 0 86552 463120 2020-04-20T09:42:42Z Prime.mover 59 Created page with \"== Example of Bijection == <onlyinclude> Let $f: \\R \\to \\R$ be the mapping defined on the Definition:Real Numbers|set of real...\" wikitext text/x-wiki == Example of Bijection == <onlyinclude> Let $f: \\R \\to \\R$ be the mapping defined on the set of real numbers as: :$\\forall x \\in \\R: \\map f x = x^3$ Then $f$ is a bijection. </onlyinclude> == Proof == A direct application of Integer Power Function is Bijective iff Index is Odd. {{qed}} == Sources == * {{BookReference|Topology: An Introduction with Application to Topological Groups|1967|George McCarty|prev = Inverse of Identity Mapping|next = Inverse Mapping/Examples/Real Cube Function}}: Chapter $\\text{I}$: Sets and Functions: Composition of Functions Category:Examples of Bijections r50xzx9y89yeysojmrvmxud5nr5raov"}
+{"_id": "33114", "title": "Limit of Subsequence equals Limit of Sequence/Normed Vector Space", "text": "Limit of Subsequence equals Limit of Sequence/Normed Vector Space 0 86622 463604 2020-04-22T17:22:12Z Julius 3095 Created page with \"== Theorem == <onlyinclude> Let $\\struct {X, \\norm {\\, \\cdot \\,} }$ be a normed vector space.  Let $\\sequence {x_n}$ be a Definition:Seque...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $\\struct {X, \\norm {\\, \\cdot \\,} }$ be a normed vector space. Let $\\sequence {x_n}$ be a sequence in $X$. Let $\\sequence {x_n}$ be convergent in the norm $\\norm {\\, \\cdot \\,}$ to the following limit: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ Let $\\sequence {x_{n_r} }$ be a subsequence of $\\sequence {x_n}$. Then: :$\\sequence {x_{n_r} }$ is convergent and $\\displaystyle \\lim_{r \\mathop \\to \\infty} x_{n_r} = l$ </onlyinclude> That is, the limit of a convergent sequence equals the limit of a subsequence of it. == Proof == Let $\\epsilon > 0$. Since $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$, it follows from the definition of limit that: :$\\exists N \\in \\N : \\forall n \\in \\N : n > N \\implies \\norm{x_n - l} < \\epsilon$ Now let $R = N$. Then from Strictly Increasing Sequence of Natural Numbers: : $\\forall r > R: n_r \\ge r$ Thus $n_r > N$ and so: :$\\norm {x_n - l} < \\epsilon$ The result follows. {{qed}} Category:Normed Vector Spaces Category:Convergence Category:Limits of Sequences 8oblsbs01cotfach8fs1lr7skim9xsd"}
+{"_id": "33115", "title": "Natural Numbers under Addition do not form Group/Corollary", "text": "Natural Numbers under Addition do not form Group/Corollary 0 86692 492381 464126 2020-10-03T15:43:14Z Prime.mover 59 wikitext text/x-wiki == Corollary to Natural Numbers under Addition do not form Group == <onlyinclude> The algebraic structure $\\struct {\\Z_{\\ge 0}, +}$ consisting of the set of non-negative integers $\\Z_{\\ge 0}$ under addition $+$ does not form a subgroup of the additive group of integers. </onlyinclude> == Proof == By Natural Numbers are Non-Negative Integers, $\\struct {\\Z_{\\ge 0}, +}$ and $\\struct {\\N, +}$ are the same (or if not exactly the same, at least isomorphic). The result follows from Natural Numbers under Addition do not form Group. {{qed}} == Sources == * {{BookReference|Topology: An Introduction with Application to Topological Groups|1967|George McCarty|prev = Group Product Identity therefore Inverses/Part 2/Proof 2|next = Subgroups of Additive Group of Integers/Examples/Even Integers}}: Chapter $\\text{II}$: Groups: Exercise $\\text{E i}$ Category:Natural Numbers 8drkvaruqw9i95ilcv9v2r7eibuglwc"}
+{"_id": "33116", "title": "Existence and Uniqueness of Direct Limit of Sequence of Groups/Lemma 1", "text": "Existence and Uniqueness of Direct Limit of Sequence of Groups/Lemma 1 0 87099 466013 465985 2020-05-05T09:53:58Z Prime.mover 59 wikitext text/x-wiki {{MissingLinks}} == Lemma == On $\\widehat G_\\infty := \\displaystyle \\coprod_{n \\mathop \\in \\N} G_n$ the relation: :$\\tuple {x_n, n} \\sim \\tuple {y_m, m} \\iff \\exists k \\ge n, m: \\map {g_{n, k} } {x_n} = \\map {g_{m, k} } {y_m}$ is an equivalence relation. == Proof == === Reflexivity === Since $g_{n,n} = \\mathop {Id}_{G_n}$ we have: :$\\forall \\tuple {x_n, n} \\in \\widehat G_\\infty: \\map {g_{n, n} } {x_n} = \\map {g_{n, n} } {x_n}$ Hence: :$\\tuple {x_n, n} \\sim \\tuple {x_n, n}$ {{qed|lemma}} === Symmetry === Let $\\tuple {x_n, n} \\sim \\tuple {y_m, m}$. Then there exists a $k \\ge n, m$ such that: :$\\map {g_{n, k} } {x_n} = \\map {g_{m, k} } {x_m}$ Hence also: :$\\map {g_{m, k} } {x_m} = \\map {g_{n, k} } {x_n}$ That is: :$\\tuple {y_m, m} \\sim \\tuple {x_n, n}$ {{qed|lemma}} === Transitivity === Let $\\tuple {x_n, n} \\sim \\tuple {y_m, m}$ and  $\\tuple {y_m, m} \\sim \\tuple {z_r, r}$. Then there exist $k \\ge m, n$ and $l \\ge n, r$ such that: :$\\map {g_{n, k} } {x_n} = \\map {g_{m, k} } {y_m}$ :$\\map {g_{m, l} } {y_m} = \\map {g_{r, l} } {z_r}$ Let $q:= \\max \\set {k, l}$. Then we have: {{begin-eqn}} {{eqn | l = \\map {g_{n, q} } {x_n}       | r = \\map {g_{k, q} } {\\map {g_{m, k} } {y_m} }       | c =  }} {{eqn | r = \\map {g_{m, q} } {y_m}       | c =  }} {{eqn | r = \\map {g_{l,q} } {\\map {g_{m, l} } {y_m} }       | c =  }} {{eqn | r = \\map {g_{l, q} } {\\map {g_{r, l} } {z_r} }       | c =  }} {{eqn | r = \\map {g_{r,q} } {z_r}       | c =  }} {{end-eqn}} that is: :$\\tuple {x_n, n} \\sim \\tuple {z_r, r}$ {{qed}} Category:Existence and Uniqueness of Direct Limit of Sequence of Groups fir5rrdz7eqpwgerchypsxfad5j9us8"}
+{"_id": "33117", "title": "Existence and Uniqueness of Direct Limit of Sequence of Groups/Lemma 2", "text": "Existence and Uniqueness of Direct Limit of Sequence of Groups/Lemma 2 0 87100 466052 466043 2020-05-05T12:42:39Z Prime.mover 59 wikitext text/x-wiki {{tidy}} {{MissingLinks}} == Lemma == The following defines a group structure on $G_\\infty$:  Let $\\struct {G_\\infty, \\cdot}$ be the algebraic structure defined as follows. Let $\\eqclass {\\tuple {x_n, n} } {}, \\eqclass {\\tuple {y_m, m} } {} \\in G_\\infty$ be arbitrary elements of $G_\\infty$. Let $l := \\max \\set {m, n}$. Let the operation $\\cdot$ on $G_\\infty$ be defined as: :$\\tuple {\\eqclass {\\tuple {x_n, n} } {} \\cdot \\eqclass {\\tuple {y_m, m} } {} } := \\eqclass {\\tuple {\\map {g_{n l} } {x_n} \\map {g_{m l} } {y_m}, l} } {}$ Then $\\struct {G_\\infty, \\cdot}$ is a group. == Proof == === Well-Definedness === The definition depends on the choice $\\tuple {x_n, n}$ and $\\tuple {y_m, m}$ of representatives of $\\eqclass {\\tuple {x_n, n} } {}$ and $\\eqclass {\\tuple {y_m, m} } {}$.  We have to show that the product element is independent of this choice. Let $\\tuple {x_{n'}, n'}$ and $\\tuple {y_{m'}, m'}$ be different representatives of the chosen equivalence classes.  Let $l' := \\max \\set {n', m'}$. {{WLOG}}, suppose that $l' \\ge l$. We have that: :$\\tuple {x_n, n} \\sim \\tuple {x_{n'}, n'}$ and: :$\\tuple {y_m, m} \\sim \\tuple {y_{m'}, m'}$ \t\t and so: :$\\map {g_{n, l'} } {x_n} = \\map {g_{n', l'} } {x_{n'} }$ and: :$\\map {g_{m, l'} } {y_m} = \\map {g_{m', l'} } {y_{m'} }$ \t\t Then we have, since all our maps are group homomorphisms: {{begin-eqn}} {{eqn | l = \\map {g_{ l, l'} } {\\map {g_{n, l} } {x_n} \\map {g_{m, l} } {y_m} }       | r = \\map {g_{ l, l'} } {\\map {g_{n, l} } {x_n} } \\map{ g_{ l,l' } } { \\map{ g_{ m,l } } {y_m} }       | c = }} {{eqn | r = \\map {g_{n, l'} } {x_n} \\map {g_{m, l'} } {y_m}        | c =  }} {{eqn | r = \\map{ g_{ n',l' } } { x_{ n' } } \\map{ g_{m',l'} } { y_{ m' } }       | c = }} \t\t {{end-eqn}}\t that is: :$\\map {g_{n, l} } {x_n} \\map {g_{m,l} } {y_m} \\sim \\map {g_{ n', l'} } {x_{n'} } \\map {g_{ m',l' } } {y_{m'} }$ This proves that our definition is independent of the choice of representative. {{qed|lemma}} === Group Axioms === By the definition of the group operation, we may assume, without loss of generality, that the representatives are always in the same group $G_l \\in \\sequence {G_n}_{n \\mathop \\in \\N}$. To see this we note that we always consider a finite collection of group elements  :$\\{ \\eqclass{ \\tuple{ x_{n_1}, {n_1} } }{}, \\dots, \\eqclass{ \\tuple{ x_{n_k}, {n_k} } }{} \\} \\subset G_\\infty$. Define $l:= \\max\\{n_1,\\dots, n_k\\}$. Then  :$\\forall i \\in \\{1,\\dots, k\\} :  \\map{ g_{n_i,l} } { x_{n_i} } \\in G_n \\land \\tuple{ x_{n_1}, {n_1} } \\sim \\tuple{ \\map{ g_{n_i,l}  } { x_{n_i} }, l}$ ==== $\\text G 1$: Associativity ==== Let $\\eqclass{\\tuple{x_n, n}}{},\\eqclass{\\tuple{y_m, m}}{},\\eqclass{\\tuple{y_n, n}}{},\\eqclass{\\tuple{z_n, n}}{} \\in G_\\infty$.  Then:  {{begin-eqn}} {{eqn | l = \\paren{\\eqclass{\\tuple{x_n, n} }{} \\cdot \\eqclass{ \\tuple{y_n, n} }{} } \\cdot \\eqclass{ \\tuple{z_n, n} }{}       | r = \\eqclass{ \\tuple{ x_n y_n, n} }{} \\cdot \\eqclass{ \\tuple{z_n, n} }{}          | c =  }} {{eqn | r = \\eqclass{ \\tuple{ \\paren{x_n y_n}z_n, n} }{}       | c =  }} {{eqn | r = \\eqclass{ \\tuple{ x_n \\paren{y_n z_n}, n} }{}       | c = $G_n$ is a group }} {{eqn | r =\\eqclass{ \\tuple{ x_n, n} }{} \\cdot \\eqclass{ \\tuple{ y_n z_n, n} }{}        | c =  }} {{eqn | r = \\eqclass{ \\tuple{ x_n, n} }{} \\cdot \\paren{\\eqclass{ \\tuple{ y_n, n} }{} \\cdot \\eqclass{ \\tuple{ z_n, n} }{} }       | c =  }} {{end-eqn}} ==== $\\text G 2$: Identity ==== Let $\\eqclass{\\tuple{x_n, n}}{} \\in G_\\infty$ and let $1_n$ be the identity of $G_n$.  Note that :$\\forall k, n \\in \\N : \\paren{ k \\ge n \\implies  \\map{ g_{nk} }{1_n} = 1_k}$ because the maps $g_{nk}$ are group homomorphisms. Then:  {{begin-eqn}} {{eqn | l = \\eqclass{\\tuple{x_n, n} }{} \\cdot \\eqclass{ \\tuple{1_n, n} }{}        | r = \\eqclass{\\tuple{x_n 1_n, n} }{}        | c =  }} {{eqn | r = \\eqclass{\\tuple{x_n, n} }{}       | c =  }} {{end-eqn}} Similarly we also find that $\\eqclass{\\tuple{1_n, n} }{} \\cdot \\eqclass{\\tuple{x_n, n} }{} = \\eqclass{\\tuple{x_n, n} }{}$. Thus $\\eqclass{\\tuple{1_n, n} }{}$ is the identity of $G_\\infty$. ==== $\\text G 3$: Inverses ==== Let $\\eqclass{\\tuple{x_n, n}}{} \\in G_\\infty$. Then : {{begin-eqn}} {{eqn | l = \\eqclass{\\tuple{x_n, n} }{} \\cdot \\eqclass{ \\tuple{x_n^{-1}, n} }{}        | r = \\eqclass{\\tuple{x_n x^{-1}_n, n} }{}        | c =  }} {{eqn | r = \\eqclass{\\tuple{1_n, n} }{}       | c =  }} {{end-eqn}} Similarly we also find that $\\eqclass{\\tuple{x^{-1}_n, n} }{} \\cdot \\eqclass{\\tuple{x_n, n} }{} = \\eqclass{\\tuple{1_n, n} }{}$. Thus $\\eqclass{\\tuple{x_n, n} }{}$ has an inverse, that is: :$\\eqclass{\\tuple{x_n^{-1}, n} }{}$ in $G_\\infty$. {{qed}} Category:Existence and Uniqueness of Direct Limit of Sequence of Groups 656h564i43fjh6iurxncjg9e44p4vb4"}
+{"_id": "33118", "title": "Existence and Uniqueness of Direct Limit of Sequence of Groups/Lemma 3", "text": "Existence and Uniqueness of Direct Limit of Sequence of Groups/Lemma 3 0 87101 466053 466031 2020-05-05T12:44:48Z Prime.mover 59 wikitext text/x-wiki {{MissingLinks|It is important for all pages to be self-contained, so we need to include on this page the required definitions from its parent page.}} == Lemma == Let $h_\\infty: G_\\infty \\to H$ be the mapping defined as: :$\\eqclass {\\tuple {x_n, n} } {} \\mapsto \\map {h_n} {x_n}$ Then $h_\\infty$ is a well-defined group homomorphism. == Proof == === Well-Definedness of $h_\\infty$ === Let $\\tuple {x_n, n}, \\tuple{x_{n'}, n'} \\in \\eqclass {\\tuple {x_n, n} } {}$. {{WLOG}}, let $n' \\ge n$.  Then we have: :$\\map {g_{n, n'} } {x_n} = x_{n'}$ and: {{begin-eqn}} {{eqn | l = \\map {h_{n'} } {x_{n'} }        | r = \\map {h_{n'} } {\\map {g_{n,n'} } {x_n} } }} {{eqn | r = \\map {\\paren {h_{n'} \\circ g_{n, n'} } } {x_n}        | c = because $h_{n'} \\circ g_{n, n'} = h_n$ }} {{eqn | r = \\map {h_n} {x_n} }} {{end-eqn}} This proves that $h_\\infty$ is independent of the representative chosen. That is, $h_\\infty$ is well-defined. {{qed|lemma}} === Homomorphism Property === Let $\\eqclass {\\tuple{x_n, n} } {}, \\eqclass {\\tuple {y_m, m} } {} \\in G_\\infty$. By the definition of the group operation, we may assume, without loss of generality, that $n = m$. See Lemma 2 for details. {{handwaving|Can that be demonstrated?}} It follows that: {{begin-eqn}} {{eqn | l = \\map {h_{\\infty} } {\\eqclass {\\tuple {x_n, n} } {} \\circ \\eqclass {\\tuple {y_n, n} } {} }       | r = \\map {h_n} {x_n y_n}       | c = Definition of $h_\\infty$ }} {{eqn | r = \\map {h_n} {x_n} \\map {h_n} {y_n}       | c = $h_n$ is a homomorphism }} {{eqn | r = \\map {h_\\infty} {\\eqclass {\\tuple {x_n, n} } {} } \\circ \\map {h_\\infty} {\\eqclass {\\tuple {y_n, n} } {} }       | c = Definition of $h_\\infty$ }} {{end-eqn}} Thus $h_\\infty$ is a homomorphism. {{qed}} Category:Existence and Uniqueness of Direct Limit of Sequence of Groups nswa1slfide05n79yjxj9m0v5r6vdcj"}
+{"_id": "33119", "title": "Schwarz's Lemma/Lemma", "text": "Schwarz's Lemma/Lemma 0 87118 466065 2020-05-05T14:18:32Z Caliburn 3218 Created page with \"== Theorem == <onlyinclude> Let $D$ be the unit disk centred at $0$.  Let $g : D \\to \\C$ be a complex function with:...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $D$ be the unit disk centred at $0$. Let $g : D \\to \\C$ be a complex function with:  :$\\map g z = \\begin{cases}\\frac {\\map f z} z & z \\ne 0 \\\\ \\map {f'} 0 & z = 0\\end{cases}$ Then $g$ is holomorphic on $D$. </onlyinclude> == Proof == By Differentiable Function is Continuous, $f$ is continuous, so by Combination Theorem for Continuous Functions: Quotient Rule: :$g$ is continuous on $D \\setminus \\set 0$. We aim to show that $f$ is continuous on $D$.  Note that since $f$ is holomorphic on $D$ and $0 \\in D$ we have, by the definition of the complex derivative:  :$\\displaystyle \\lim_{z \\mathop \\to 0} \\frac {\\map f z - \\map f 0} z = \\map {f'} 0 \\in \\C$ Since $\\map f 0 = 0$, we furthermore have:  :$\\displaystyle \\map {f'} 0 = \\lim_{z \\mathop \\to 0} \\frac {\\map f z} z$ That is:  :$\\displaystyle \\map g 0 = \\lim_{z \\mathop \\to 0} \\map g z$ so $g$ is continuous at $0$. Since $f$ is holomorphic on $D$, by the Combination Theorem for Complex Derivatives: Quotient Rule: :$g$ is differentiable on $D \\setminus \\set 0$. It remains to show that $g$ is differentiable at $0$.  Take $z \\ne 0$ and consider:  :$\\dfrac {\\map g z - \\map g 0} z$ We have:  {{begin-eqn}} {{eqn\t| l = \\frac {\\map g z - \\map g 0} z \t| r = \\frac {\\frac {\\map f z} z - \\map {f'} 0} z \t| c = as $\\map g z = \\dfrac {\\map f z} z$ for $z \\ne 0$ and $\\map g 0 = \\map {f'} 0$ }} {{eqn\t| r = \\frac {\\map f z - z \\map {f'} 0} {z^2} }} {{end-eqn}} Since $f$ is holomorphic on $D$, by Holomorphic Function is Analytic, there exists a positive real number $R$ such that the series:  :$\\displaystyle \\sum_{n \\mathop = 0}^\\infty \\frac {\\map {f^{\\paren n} } 0} {n!} z^n$ converges to $\\map f z$ on $\\cmod z < R$. Note that since $\\map f 0 = 0$, the first term of this series is zero. With that, we have:  {{begin-eqn}} {{eqn\t| l = \\frac {\\map f z - z \\map {f'} 0} {z^2} \t| r = \\frac {\\sum_{n \\mathop = 1}^\\infty \\frac {\\map {f^{\\paren n} } 0} {n!} z^n - z \\map {f'} 0} {z^2} }} {{eqn\t| r = \\frac {z \\map {f'} 0 + \\frac {z^2} 2 \\map {f''} 0 + \\sum_{n \\mathop = 3}^\\infty \\frac {\\map {f^{\\paren n} } 0} {n!} z^n - z \\map {f'} 0} {z^2} }} {{eqn\t| r = \\frac 1 2 \\map {f''} 0 + \\sum_{n \\mathop = 3}^\\infty \\frac {\\map {f^{\\paren n} } 0} {n!} z^{n - 2} }} {{end-eqn}} Taking $z \\to 0$ we have:  :$\\displaystyle \\lim_{z \\mathop \\to 0} \\frac {\\map g z - \\map g 0} z = \\frac 1 2 \\map {f''} 0$ so $g$ is indeed differentiable at $0$ and hence holomorphic on $D$. {{qed}} Category:Complex Analysis Category:Schwarz's Lemma 1o8sfwvysfvchvd5xuw4lup45d07t6p"}
+{"_id": "33120", "title": "Partial Derivative/Examples/x^(x y)/wrt x", "text": "Partial Derivative/Examples/x^(x y)/wrt x 0 87256 466578 2020-05-08T07:34:37Z Prime.mover 59 Created page with \"== Example of Partial Derivative ==  Let $\\map f {x, y} = x^{x y}$ be a Definition:Real Function of Two Variables|real function of $2$ vari...\" wikitext text/x-wiki == Example of Partial Derivative == Let $\\map f {x, y} = x^{x y}$ be a real function of $2$ variables such that $x, y \\in \\R_{>0}$. Then: :$\\dfrac {\\partial f} {\\partial x} = x^{x y} \\paren {y \\ln x + y}$ == Proof == <onlyinclude> By definition, the partial derivative {{WRT|Differentiation}} $x$ is obtained by holding $y$ constant. Hence Derivative of $x^{a x}$ can be directly used: :$\\dfrac \\d {\\d x} x^{y x} = y x^{y x} \\paren {\\ln x + 1}$ The result can then be rearranged to match the form given. {{qed}} </onlyinclude>  == Sources == * {{BookReference|Advanced Calculus|1961|David V. Widder|ed = 2nd|edpage = Second Edition|prev = Partial Derivative/Examples/x^(x y)|next = Partial Derivative/Examples/x^(x y)/wrt y}}: $1$ Partial Differentiation: $\\S 1$. Introduction: $1.1$ Partial Derivatives: Example $\\text A$ Category:Examples of Partial Derivatives okref889tzskah8eegtcifw5thw7tbe"}
+{"_id": "33121", "title": "Partial Derivative/Examples/x^(x y)/wrt y", "text": "Partial Derivative/Examples/x^(x y)/wrt y 0 87257 466680 466583 2020-05-08T14:23:04Z Prime.mover 59 wikitext text/x-wiki == Example of Partial Derivative == Let $\\map f {x, y} = x^{x y}$ be a real function of $2$ variables such that $x, y \\in \\R_{>0}$. Then: :$\\dfrac {\\partial f} {\\partial y} = x^{x y + 1} \\ln x$ == Proof == <onlyinclude> By definition, the partial derivative {{WRT|Differentiation}} $y$ is obtained by holding $x$ constant. From Derivative of Power of Constant: :$\\map {D_y} {x^y} = x^y \\ln x$ for constant $a$. Then: {{begin-eqn}} {{eqn | l = \\map {D_y} {x^{x y} }       | r = x \\map {D_{x y} } {x^{x y} }       | c = Derivative of Function of Constant Multiple }} {{eqn | r = x \\paren {x^{x y} } \\ln x       | c =  }} {{eqn | r = x^{x y + 1} \\ln x       | c =  }} {{end-eqn}} {{qed}} </onlyinclude>  == Sources == * {{BookReference|Advanced Calculus|1961|David V. Widder|ed = 2nd|edpage = Second Edition|prev = Partial Derivative/Examples/x^(x y)/wrt x|next = Partial Derivative/Examples/x sine y z}}: $1$ Partial Differentiation: $\\S 1$. Introduction: $1.1$ Partial Derivatives: Example $\\text A$ Category:Examples of Partial Derivatives lleozosukca2zilh8np4wdog51r2uuu"}
+{"_id": "33122", "title": "Partial Derivative/Examples/u + ln u = x y", "text": "Partial Derivative/Examples/u + ln u = x y 0 87272 466715 2020-05-08T22:40:08Z Prime.mover 59 Created page with \"== Example of Partial Derivative == <onlyinclude> Let $u + \\ln u = x y$ be an implicit function.  Th...\" wikitext text/x-wiki == Example of Partial Derivative == <onlyinclude> Let $u + \\ln u = x y$ be an implicit function. Then: {{begin-eqn}} {{eqn | l = \\dfrac {\\partial u} {\\partial x}       | r = \\dfrac {u y} {u + 1}       | c =  }} {{eqn | l = \\dfrac {\\partial u} {\\partial y}       | r = \\dfrac {u x} {u + 1}       | c =  }} {{end-eqn}} </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = u + \\ln u       | r = x y       | c =  }} {{eqn | ll= \\leadsto       | l = \\dfrac \\partial {\\partial x} u + \\dfrac \\partial {\\partial x} \\ln u       | r = \\dfrac \\partial {\\partial x} x y       | c =  }} {{eqn | ll= \\leadsto       | l = \\dfrac {\\partial u} {\\partial x} + \\dfrac 1 u \\dfrac {\\partial u} {\\partial x}       | r = y       | c =  }} {{eqn | ll= \\leadsto       | l = \\dfrac {\\partial u} {\\partial x} \\paren {1 + \\dfrac 1 u}       | r = y       | c =  }} {{eqn | ll= \\leadsto       | l = \\dfrac {\\partial u} {\\partial x}       | r = \\dfrac {u y} {u + 1}       | c =  }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = u + \\ln u       | r = x y       | c =  }} {{eqn | ll= \\leadsto       | l = \\dfrac \\partial {\\partial y} u + \\dfrac \\partial {\\partial y} \\ln u       | r = \\dfrac \\partial {\\partial y} x y       | c =  }} {{eqn | ll= \\leadsto       | l = \\dfrac {\\partial u} {\\partial y} + \\dfrac 1 u \\dfrac {\\partial u} {\\partial y}       | r = x       | c =  }} {{eqn | ll= \\leadsto       | l = \\dfrac {\\partial u} {\\partial y} \\paren {1 + \\dfrac 1 u}       | r = x       | c =  }} {{eqn | ll= \\leadsto       | l = \\dfrac {\\partial u} {\\partial y}       | r = \\dfrac {u x} {u + 1}       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Advanced Calculus|1961|David V. Widder|ed = 2nd|edpage = Second Edition|prev = Partial Derivative/Examples/u^2 + x^2 + y^2 = a^2|next = Partial Derivative/Examples/v + ln u = x y, u + ln v = x - y}}: $1$ Partial Differentiation: $\\S 1$. Introduction: $1.2$ Implicit Functions Category:Examples of Partial Derivatives fvlsrhhtykoswl8uglatw8x5ai8pi4y"}
+{"_id": "33123", "title": "Function of Bounded Variation is Bounded", "text": "Function of Bounded Variation is Bounded 0 87489 467886 2020-05-14T12:35:35Z Caliburn 3218 Created page with \"== Theorem == <onlyinclude> Let $a, b$ be real numbers with $a < b$.  Let $f : \\closedint a b \\to \\R$ be a function of...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $a, b$ be real numbers with $a < b$. Let $f : \\closedint a b \\to \\R$ be a function of bounded variation. Then $f$ is bounded. </onlyinclude> == Proof == We use the notation from the definition of bounded variation. Since $f$ is of bounded variation, there exists $M \\in \\R$ such that: :$\\map {V_f} P \\le M$ for all finite subdivisions $P$ of $\\closedint a b$. Let $x$ be a real number with: :$a < x < b$ Then $\\set {a, x, b}$ is a finite subdivision of $\\closedint a b$. We have:  :$\\map {V_f} {\\set {a, x, b} } = \\size {\\map f x - \\map f a} + \\size {\\map f b - \\map f x}$ Since $x \\in \\openint a b$ was arbitrary, we therefore have:  :$\\size {\\map f x - \\map f a} + \\size {\\map f b - \\map f x} \\le M$ for all $x \\in \\openint a b$. We have: {{begin-eqn}} {{eqn\t| l = \\size {\\map f x - \\map f a} + \\size {\\map f b - \\map f x} \t| o = \\ge \t| r = \\size {\\map f x - \\map f a} }} {{eqn\t| o = \\ge \t| r = \\size {\\size {\\map f x} - \\size {\\map f a} } \t| c = Reverse Triangle Inequality: Real and Complex Fields }} {{eqn\t| o = \\ge \t| r = \\size {\\map f x} - \\size {\\map f a} }} {{end-eqn}} So for all $x \\in \\openint a b$, we have:  :$\\size {\\map f x} \\le \\size {\\map f a} + M$ Since $M \\ge 0$, this inequality is also satisfied for $x = a$.  We therefore have:  :$\\size {\\map f x} \\le \\map \\max {\\size {\\map f a} + M, \\size {\\map f b} }$ for all $x \\in \\closedint a b$. So $f$ is is bounded. {{qed}} == Sources == * {{BookReference|Mathematical Analysis|1973|Tom M. Apostol|prev = Differentiable Function with Bounded Derivative is of Bounded Variation|next = Definition:Total Variation|ed = 2nd|edpage = Second Edition}}: $\\S 6.3$: Functions of Bounded Variation: Theorem $6.7$ Category:Bounded Variation atydoyyyc1561qo943yi4zocd8iv4p0"}
+{"_id": "33124", "title": "Multiple of Function of Bounded Variation is of Bounded Variation", "text": "Multiple of Function of Bounded Variation is of Bounded Variation 0 87633 469076 2020-05-19T14:59:04Z Caliburn 3218 Created page with \"== Theorem == <onlyinclude> Let $a, b, k$ be real numbers with $a < b$.  Let $f : \\closedint a b \\to \\R$ be a functions...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $a, b, k$ be real numbers with $a < b$. Let $f : \\closedint a b \\to \\R$ be a functions of bounded variation. Let the total variations of $f$ be $V_f$. Then $k f$ is of bounded variation with:  :$V_{k f} = \\size k V_f$ where $V_{k f}$ is the total variation of $k f$. </onlyinclude> == Proof == For each finite subdivision $P$ of $\\closedint a b$, write:  :$P = \\set {x_0, x_1, \\ldots, x_n }$ with: :$a = x_0 < x_1 < x_2 < \\cdots < x_{n - 1} < x_n = b$ Then: {{begin-eqn}} {{eqn\t| l = \\map {V_{k f} } P \t| r = \\sum_{i \\mathop = 1}^n \\size {k \\map f {x_i} - k \\map f {x_{i - 1} } } \t| c = using the notation from the definition of bounded variation }} {{eqn\t| r = \\size k \\sum_{i \\mathop = 1}^n \\size {\\map f {x_i} - \\map f {x_{i - 1} } } }} {{eqn\t| r = \\size k \\map {V_f} P }} {{end-eqn}} Since $f$ is of bounded variation, there exists $M \\in \\R$ such that:  :$\\map {V_f} P \\le M$ for all finite subdivisions $P$.  So: :$\\map {V_{k f} } P \\le \\size k M$ So $k f$ is of bounded variation. We then have:  {{begin-eqn}} {{eqn\t| l = V_{k f} \t| r = \\sup_P \\paren {\\map {V_{k f} } P} \t| c = {{Defof|Total Variation}} }} {{eqn\t| r = \\sup_P \\paren {\\size k \\map {V_f} P} }} {{eqn\t| r = \\size k \\sup_P \\paren {\\map {V_f} P} \t| c = Multiple of Supremum }} {{eqn\t| r = \\size k V_f }} {{end-eqn}} {{qed}} Category:Bounded Variation Category:Total Variation tm2b1jjxz079gs06iegfka06ihtvebc"}
+{"_id": "33125", "title": "Constant Real Function is Absolutely Continuous", "text": "Constant Real Function is Absolutely Continuous 0 87716 470257 2020-05-24T14:39:47Z Caliburn 3218 Created page with \"== Theorem == <onlyinclude> Let $I \\subseteq \\R$ be a real interval.  Let $f : I \\to \\R$ be an Definition:Constant Mapping|constant real functio...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $I \\subseteq \\R$ be a real interval. Let $f : I \\to \\R$ be an constant real function. Then $f$ is absolutely continuous. </onlyinclude> == Proof == Let $\\delta, \\varepsilon$ be positive real numbers. Let $\\closedint {a_1} {b_1}, \\dotsc, \\closedint {a_n} {b_n} \\subseteq I$ be a collection of disjoint closed real intervals with:  :$\\displaystyle \\sum_{i \\mathop = 1}^n \\paren {b_i - a_i} < \\delta$ Since $f$ is constant, for all $i \\in \\set {1, 2, \\ldots, n}$ we have: :$\\size {\\map f {b_i} - \\map f {a_i} } = 0$ so: :$\\displaystyle \\sum_{i = 1}^n \\size {\\map f {b_i} - \\map f {a_i} } = 0 < \\varepsilon$ Since $\\varepsilon$ was arbitrary:  :$f$ is absolutely continuous. {{qed}} Category:Absolutely Continuous Functions gxiz5ue31jl2hsjw5qjzh2h1iwsifgs"}
+{"_id": "33126", "title": "Multiple of Absolutely Continuous Function is Absolutely Continuous", "text": "Multiple of Absolutely Continuous Function is Absolutely Continuous 0 87718 470270 2020-05-24T15:56:02Z Caliburn 3218 Created page with \"== Theorem == <onlyinclude> Let $k$ be a real number.  Let $I \\subseteq \\R$ be a real interval.  Let $f : I \\to \\R$ be...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $k$ be a real number. Let $I \\subseteq \\R$ be a real interval. Let $f : I \\to \\R$ be absolutely continuous function. Then $k f$ is absolutely continuous. </onlyinclude> == Proof == Note that if $k = 0$, then $k f$ is constant.  Hence, by Constant Real Function is Absolutely Continuous:  :$k f$ is absolutely continuous if $k = 0$. Take now $k \\ne 0$. Let $\\varepsilon$ be a positive real number. Since $f$ is absolutely continuous, there exists real $\\delta > 0$ such that for all collections of disjoint closed real intervals $\\closedint {a_1} {b_1}, \\dotsc, \\closedint {a_n} {b_n} \\subseteq I$ with:  :$\\displaystyle \\sum_{i \\mathop = 1}^n \\paren {b_i - a_i} < \\delta$ we have:  :$\\displaystyle \\sum_{i \\mathop = 1}^n \\size {\\map f {b_i} - \\map f {a_i} } < \\frac {\\varepsilon} {\\size k}$ Then: {{begin-eqn}} {{eqn\t| l = \\sum_{i \\mathop = 1}^n \\size {\\map {\\paren {k f} } {b_i} - \\map {\\paren {k f} } {a_i} } \t| r = \\size k \\sum_{i \\mathop = 1}^n \\size {\\map f {b_i} - \\map f {a_i} } }} {{eqn\t| o = < \t| r = \\size k \\times \\frac {\\varepsilon} {\\size k} }} {{eqn\t| r = \\varepsilon }} {{end-eqn}} whenever: :$\\displaystyle \\sum_{i \\mathop = 1}^n \\paren {b_i - a_i} < \\delta$ Since $\\varepsilon$ was arbitrary:  :$k f$ is absolutely continuous if $k \\ne 0$. Therefore:  :$k f$ is absolutely continuous for all $k \\in \\R$. {{qed}} Category:Absolutely Continuous Functions 0fnxe36o4cbzvbtdwfv9uhnt7ari2tz"}
+{"_id": "33127", "title": "Elementary Row Operation/Examples/r3 + 2r2", "text": "Elementary Row Operation/Examples/r3 + 2r2 0 88375 476229 476182 2020-06-26T21:46:13Z Prime.mover 59 wikitext text/x-wiki == Examples of Elementary Row Operations == <onlyinclude> Consider the elementary row operation $e$ defined as: :$e := r_3 \\to r_3 + 2 r_2$ acting on a matrix space $\\map \\MM {3, n}$ for some $n \\in \\Z_{>0}$. The elementary row matrix corresponding to $e$ is: :$\\begin {pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 2 & 1 \\end {pmatrix}$ </onlyinclude> == Proof == Let $\\mathbf E$ denote the elementary row matrix corresponding to $e$. From Elementary Matrix corresponding to Elementary Row Operation: Scale Row and Add: :$E_{a b} = \\delta_{a b} + 2 \\delta_{a 3} \\cdot \\delta_{2 b}$ where: :$E_{a b}$ denotes the element of $\\mathbf E$ whose indices are $\\tuple {a, b}$ :$\\delta_{a b}$ is the Kronecker delta: ::$\\delta_{a b} = \\begin {cases} 1 & : \\text {if $a = b$} \\\\ 0 & : \\text {if $a \\ne b$} \\end {cases}$ That is: :When $a \\ne 3$ and $b \\ne 2$, elements of $\\mathbf E$ are $0$ except for those on the main diagonal :When $a = 3$ and $b = 2$, $E_{a b}$ equals $2 + \\delta_{a b}$. But as $a \\ne b$ it follows that $\\delta_{a b} = 0$. Hence $\\mathbf E$ can be constructed as described. {{qed}} == Sources == * {{BookReference|Linear Algebra|1998|Richard Kaye|author2 = Robert Wilson|prev = Elementary Row Operations as Matrix Multiplications|next = Elementary Row Operation/Examples/lambda r2}}: Part $\\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: $1$ Category:Examples of Elementary Row Operations pze5znfymx9cgsoxx9krw861l622ojf"}
+{"_id": "33128", "title": "Elementary Row Operation/Examples/lambda r2", "text": "Elementary Row Operation/Examples/lambda r2 0 88376 476177 474167 2020-06-26T19:39:05Z Prime.mover 59 wikitext text/x-wiki == Examples of Elementary Row Operations == <onlyinclude> Consider the elementary row operation $e$ defined as: :$e := r_2 \\to \\lambda r_2$ acting on a matrix space $\\map \\MM {3, n}$ for some $n \\in \\Z_{>0}$. The elementary row matrix corresponding to $e$ is: :$\\begin {pmatrix} 1 & 0 & 0 \\\\ 0 & \\lambda & 0 \\\\ 0 & 0 & 1 \\end {pmatrix}$ </onlyinclude> == Proof == Let $\\mathbf E$ denote the elementary row matrix corresponding to $e$. From Elementary Matrix corresponding to Elementary Row Operation: Scale Row and Add: :$E_{a b} = \\begin {cases} \\delta_{a b} & : a \\ne 2 \\\\ \\lambda \\cdot \\delta_{a b} & : a = 2 \\end{cases}$ where: :$E_{a b}$ denotes the element of $\\mathbf E$ whose indices are $\\tuple {a, b}$ :$\\delta_{a b}$ is the Kronecker delta: ::$\\delta_{a b} = \\begin {cases} 1 & : \\text {if $a = b$} \\\\ 0 & : \\text {if $a \\ne b$} \\end {cases}$ That is: :When $a \\ne 2$, the elements of $\\mathbf E$ are $0$ except for those on the main diagonal, when they are $1$ :When $a = 2$, the elements of $\\mathbf E$ are $0$ except for those on the main diagonal, when they are $\\lambda$. Hence $\\mathbf E$ can be constructed as described. {{qed}} == Sources == * {{BookReference|Linear Algebra|1998|Richard Kaye|author2 = Robert Wilson|prev = Elementary Row Operation/Examples/r3 + 2r2|next = Elementary Row Operation/Examples/Swap r1 and r2}}: Part $\\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: $2$ Category:Examples of Elementary Row Operations aqxgu67irgyx1epnqekbhtdfw41zxwb"}
+{"_id": "33129", "title": "Elementary Row Operation/Examples/Swap r1 and r2", "text": "Elementary Row Operation/Examples/Swap r1 and r2 0 88377 476183 474168 2020-06-26T19:42:13Z Prime.mover 59 wikitext text/x-wiki == Examples of Elementary Row Operations == <onlyinclude> Consider the elementary row operation $e$ defined as: :$e := r_1 \\leftrightarrow r_2$ acting on a matrix space $\\map \\MM {3, n}$ for some $n \\in \\Z_{>0}$. The elementary row matrix corresponding to $e$ is: :$\\begin {pmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 1 \\end {pmatrix}$ </onlyinclude> == Proof == By definition, the elementary row matrix corresponding to $e$ is found by applying $e$ to the unit matrix. By definition of unit matrix: :$\\mathbf I = \\begin {pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end {pmatrix}$ Let $\\mathbf E$ denote the elementary row matrix corresponding to $e$. $\\mathbf E$ is constructed by exchanging row $1$ with row $2$. {{qed}} == Sources == * {{BookReference|Linear Algebra|1998|Richard Kaye|author2 = Robert Wilson|prev = Elementary Row Operation/Examples/lambda r2|next = Definition:Row Operation}}: Part $\\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: $3$ Category:Examples of Elementary Row Operations ge9rfuj2jkwfytemgp4pnri8bevnl66"}
+{"_id": "33130", "title": "Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 1", "text": "Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 1 0 88443 474449 2020-06-17T06:35:30Z Prime.mover 59 Created page with \"== Examples of Use of Matrix is Row Equivalent to Echelon Matrix == <onlyinclude> Let $\\mathbf A = \\begin {bmatrix} 0 & 1 & 1 \\\\ 0 & 1 & 0 \\\\ 1 & 1 & 0 \\\\ \\end {bmatrix}$...\" wikitext text/x-wiki == Examples of Use of Matrix is Row Equivalent to Echelon Matrix == <onlyinclude> Let $\\mathbf A = \\begin {bmatrix} 0 & 1 & 1 \\\\ 0 & 1 & 0 \\\\ 1 & 1 & 0 \\\\ \\end {bmatrix}$ This can be converted into the echelon form: :$\\mathbf E = \\begin {bmatrix} 1 & 1 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ \\end {bmatrix}$ </onlyinclude> == Proof == Using Row Operation to Clear First Column of Matrix we obtain: :$\\mathbf B = \\begin {bmatrix} 1 & 1 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 1 & 1 \\\\ \\end {bmatrix}$ which happens on the first step, exchanging row $1$ with row $3$. Then we investigate the submatrix: :$\\mathbf B' = \\begin {bmatrix} 1 & 0 \\\\ 1 & 1 \\\\ \\end {bmatrix}$ Using Row Operation to Clear First Column of Matrix we obtain: :$\\mathbf C' = \\begin {bmatrix} 1 & 0 \\\\ 0 & 1 \\\\ \\end {bmatrix}$ which is obtained by adding $-1$ of row $1$ of $\\mathbf B'$ to row $2$ of $\\mathbf B'$. This is the same as adding $-1$ of row $2$ of $\\mathbf B$ to row $3$ of $\\mathbf B$. Thus we are left with: :$\\mathbf E = \\begin {bmatrix} 1 & 1 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ \\end {bmatrix}$ {{qed}} == Sources == * {{BookReference|Linear Algebra|1998|Richard Kaye|author2 = Robert Wilson|prev = Matrix is Row Equivalent to Echelon Matrix/Examples|next = Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 2}}: Part $\\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: Exercise $1.4 \\ \\text {(a)}$ Category:Matrix is Row Equivalent to Echelon Matrix q1s1xe6u108ddrmiocao6mc40fo9ieu"}
+{"_id": "33131", "title": "Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 2", "text": "Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 2 0 88444 474450 2020-06-17T06:39:44Z Prime.mover 59 Created page with \"== Examples of Use of Matrix is Row Equivalent to Echelon Matrix == <onlyinclude> Let $\\mathbf A = \\begin {bmatrix} 1 & 1 & 1 & 1 \\\\ 2 & 3 & 4 & 5 \\\\ 3 & 4 & 5 & 6 \\\\ \\end...\" wikitext text/x-wiki == Examples of Use of Matrix is Row Equivalent to Echelon Matrix == <onlyinclude> Let $\\mathbf A = \\begin {bmatrix} 1 & 1 & 1 & 1 \\\\ 2 & 3 & 4 & 5 \\\\ 3 & 4 & 5 & 6 \\\\ \\end {bmatrix}$ This can be converted into the echelon form: :$\\mathbf E = \\begin {bmatrix} 1 & 1 & 1 & 1 \\\\ 0 & 1 & 2 & 3 \\\\ 0 & 0 & 0 & 0 \\\\ \\end {bmatrix}$ </onlyinclude> == Proof == Using Row Operation to Clear First Column of Matrix we obtain: :$\\mathbf B = \\begin {bmatrix} 1 & 1 & 1 & 1 \\\\ 0 & 1 & 2 & 3 \\\\ 0 & 1 & 2 & 3 \\\\ \\end {bmatrix}$ which is obtained by: :adding $-2$ of row $1$ to row $2$ :adding $-3$ of row $1$ to row $3$. Then we investigate the submatrix: :$\\mathbf B' = \\begin {bmatrix} 1 & 2 & 3 \\\\ 1 & 2 & 3 \\\\ \\end {bmatrix}$ Using Row Operation to Clear First Column of Matrix we obtain: :$\\mathbf C' = \\begin {bmatrix} 1 & 2 & 3 \\\\ 0 & 0 & 0 \\\\ \\end {bmatrix}$ which is obtained by adding $-1$ of row $1$ of $\\mathbf B'$ to row $2$ of $\\mathbf B'$. Thus we are left with: :$\\mathbf E = \\begin {bmatrix} 1 & 1 & 1 & 1 \\\\ 0 & 1 & 2 & 3 \\\\ 0 & 0 & 0 & 0 \\\\ \\end {bmatrix}$ {{qed}} == Sources == * {{BookReference|Linear Algebra|1998|Richard Kaye|author2 = Robert Wilson|prev = Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 1|next = Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 3}}: Part $\\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: Exercise $1.4 \\ \\text {(b)}$ Category:Matrix is Row Equivalent to Echelon Matrix 4umk6y8jvs2i4g6ijtc9oo6yuahan7o"}
+{"_id": "33132", "title": "Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 3", "text": "Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 3 0 88445 474451 2020-06-17T06:52:35Z Prime.mover 59 Created page with \"== Examples of Use of Matrix is Row Equivalent to Echelon Matrix == <onlyinclude> Let $\\mathbf A = \\begin {bmatrix} 1 & 2 & 3 & 5 \\\\ 1 & 2 & 3 & 4 \\\\ 0 & 0 & 1 & 1 \\\\ \\end...\" wikitext text/x-wiki == Examples of Use of Matrix is Row Equivalent to Echelon Matrix == <onlyinclude> Let $\\mathbf A = \\begin {bmatrix} 1 & 2 & 3 & 5 \\\\ 1 & 2 & 3 & 4 \\\\ 0 & 0 & 1 & 1 \\\\ \\end {bmatrix}$ This can be converted into the echelon form: :$\\mathbf E = \\begin {bmatrix} 1 & 2 & 3 & 5 \\\\ 0 & 0 & 1 & 1 \\\\ 0 & 0 & 0 & 1 \\\\ \\end {bmatrix}$ </onlyinclude> == Proof == Using Row Operation to Clear First Column of Matrix we obtain: :$\\mathbf B = \\begin {bmatrix} 1 & 2 & 3 & 5 \\\\ 0 & 0 & 0 & -1 \\\\ 0 & 0 & 1 & 1 \\\\ \\end {bmatrix}$ which is obtained by adding $-2$ of row $1$ to row $2$. Then we investigate the submatrix: :$\\mathbf B' = \\begin {bmatrix} 0 & 0 & -1 \\\\ 0 & 1 & 1 \\\\ \\end {bmatrix}$ Using Row Operation to Clear First Column of Matrix we obtain: :$\\mathbf C' = \\begin {bmatrix} 0 & 1 & 1 \\\\ 0 & 0 & 1 \\\\ \\end {bmatrix}$ which is obtained by: :$(1): \\quad$ exchanging row $1$ of $\\mathbf B'$ with row $2$ of $\\mathbf B'$. :$(2): \\quad$ multiplying row $2$ of $\\mathbf B'$ by $-1$. Thus we are left with: :$\\mathbf E = \\begin {bmatrix} 1 & 2 & 3 & 5 \\\\ 0 & 0 & 1 & 1 \\\\ 0 & 0 & 0 & 1 \\\\ \\end {bmatrix}$ {{qed}} == Also presented as == Some sources use the non-unity variant of the echelon matrix. Such sources do not require that the leading coefficients necessarily have to equal to $1$. Hence they consider the final step to convert row $3$ of $\\mathbf E$ from $\\begin {bmatrix} 0 & 0 & 0 & -1 \\end {bmatrix}$ to $\\begin {bmatrix} 0 & 0 & 0 & 1 \\end {bmatrix}$ as optional. == Sources == * {{BookReference|Linear Algebra|1998|Richard Kaye|author2 = Robert Wilson|prev = Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 2|next = Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 4}}: Part $\\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: Exercise $1.4 \\ \\text {(c)}$ Category:Matrix is Row Equivalent to Echelon Matrix j5ktt0est6bisyso96lj4ls8hqanuxe"}
+{"_id": "33133", "title": "Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 4", "text": "Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 4 0 88446 474452 2020-06-17T06:57:05Z Prime.mover 59 Created page with \"== Examples of Use of Matrix is Row Equivalent to Echelon Matrix == <onlyinclude> Let $\\mathbf A = \\begin {bmatrix} 1 & 1 & 2 & 3 \\\\ 0 & 0 & 1 & 1 \\\\ 0 & 0 & 3 & 3 \\\\ \\end...\" wikitext text/x-wiki == Examples of Use of Matrix is Row Equivalent to Echelon Matrix == <onlyinclude> Let $\\mathbf A = \\begin {bmatrix} 1 & 1 & 2 & 3 \\\\ 0 & 0 & 1 & 1 \\\\ 0 & 0 & 3 & 3 \\\\ \\end {bmatrix}$ This can be converted into the echelon form: :$\\mathbf E = \\begin {bmatrix} 1 & 1 & 2 & 3 \\\\ 0 & 0 & 1 & 1 \\\\ 0 & 0 & 0 & 0 \\\\ \\end {bmatrix}$ </onlyinclude> == Proof == It is noted that $\\mathbf A$ is already most of the way there. It remains to use the elementary row operation: :$e := r_3 \\to r_3 - 3 r_2$ to convert $\\mathbf A$ to the form: :$\\mathbf E = \\begin {bmatrix} 1 & 1 & 2 & 3 \\\\ 0 & 0 & 1 & 1 \\\\ 0 & 0 & 0 & 0 \\\\ \\end {bmatrix}$ {{qed}} == Sources == * {{BookReference|Linear Algebra|1998|Richard Kaye|author2 = Robert Wilson|prev = Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 3|next = Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 5}}: Part $\\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: Exercise $1.4 \\ \\text {(d)}$ Category:Matrix is Row Equivalent to Echelon Matrix gjjde47xvgm9q7oghx3f8qxbkms6pr4"}
+{"_id": "33134", "title": "Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 5", "text": "Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 5 0 88447 475261 474454 2020-06-21T21:00:54Z Prime.mover 59 wikitext text/x-wiki == Examples of Use of Matrix is Row Equivalent to Echelon Matrix == <onlyinclude> Let $\\mathbf A = \\begin {bmatrix} -1 & 0 & 1 & 2 & 3 \\\\ 0 & 1 & 2 & 3 & 4 \\\\ -1 & -2 & -3 & -4 & -5 \\\\ \\end {bmatrix}$ This can be converted into the echelon form: :$\\mathbf E = \\begin {bmatrix} 1 & 0 & -1 & -2 & -3 \\\\ 0 & 1 & 2 & 3 & 4 \\\\ 0 & 0 & 0 & 0 & 0 \\\\ \\end {bmatrix}$ </onlyinclude> == Proof == This matrix can more easily be handled by direct application of elementary row operations, as follows. Let $e_1$ be the elementary row operation: :$e_1 := r_3 \\to r_3 - r_1$ which leaves: $\\mathbf A_1 = \\begin {bmatrix} -1 & 0 & 1 & 2 & 3 \\\\ 0 & 1 & 2 & 3 & 4 \\\\ 0 & -2 & -4 & -6 & -8 \\\\ \\end {bmatrix}$ Let $e_2$ be the elementary row operation: :$e_2 := r_3 \\to r_3 + 2 r_2$ which leaves: :$\\mathbf A_2 = \\begin {bmatrix} -1 & 0 & 1 & 2 & 3 \\\\ 0 & 1 & 2 & 3 & 4 \\\\ 0 & 0 & 0 & 0 & 0 \\\\ \\end {bmatrix}$ $\\mathbf A_2$ is in non-unity echelon form. It remains to perform the elementary row operation $e_3$ to convert it into echelon form: :$e_3 := r_1 \\to -r_1$ which leaves: :$\\mathbf E = \\begin {bmatrix} 1 & 0 & -1 & -2 & -3 \\\\ 0 & 1 & 2 & 3 & 4 \\\\ 0 & 0 & 0 & 0 & 0 \\\\ \\end {bmatrix}$ {{qed}} == Also presented as == Some sources use the non-unity variant of the echelon matrix. Such sources do not require that the leading coefficients necessarily have to equal to $1$. Hence they consider the final step to convert row $1$ of $\\mathbf E$ from $\\begin {bmatrix} -1 & 0 & 1 & 2 & 3 \\end {bmatrix}$ to $\\begin {bmatrix} 1 & 0 & -1 & -2 & -3 \\end {bmatrix}$ as optional. == Sources == * {{BookReference|Linear Algebra|1998|Richard Kaye|author2 = Robert Wilson|prev = Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 4|next = Definition:Rank/Matrix/Definition 2}}: Part $\\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: Exercise $1.4 \\ \\text {(e)}$ Category:Matrix is Row Equivalent to Echelon Matrix 5x7co12j9kvmvmkvqmyiia2t825x5kb"}
+{"_id": "33135", "title": "Simultaneous Linear Equations/Examples/Arbitrary System 1", "text": "Simultaneous Linear Equations/Examples/Arbitrary System 1 0 88460 475448 474529 2020-06-23T06:30:57Z Prime.mover 59 wikitext text/x-wiki == Example of Simultaneous Linear Equations == <onlyinclude> The system of simultaneous linear equations: {{begin-eqn}} {{eqn | n = 1       | l = x_1 - 2 x_2 + x_3       | r = 1 }} {{eqn | n = 2       | l = 2 x_1 - x_2 + x_3       | r = 2 }} {{eqn | n = 3       | l = 4 x_1 + x_2 - x_3       | r = 1 }} {{end-eqn}} has as its solution set: {{begin-eqn}} {{eqn | l = x_1       | r = -\\dfrac 1 2 }} {{eqn | l = x_2       | r = \\dfrac 1 2 }} {{eqn | l = x_3       | r = \\dfrac 3 2 }} {{end-eqn}} </onlyinclude> == Proof == Subtract $2 \\times$ equation $(1)$ from equation $(2)$. Subtract $4 \\times$ equation $(1)$ from equation $(3)$. This gives us: {{begin-eqn}} {{eqn | n = 1       | l = x_1 - 2 x_2 + x_3       | r = 1 }} {{eqn | n = 2'       | l = 3 x_2 - x_3       | r = 0 }} {{eqn | n = 3'       | l = 9 x_2 - 5 x_3       | r = -3 }} {{end-eqn}} Divide equation $(2')$ by $3$ to get $(2'')$. Add $2 \\times$ equation $(2'')$ to equation $(1)$. Subtract $9 \\times$  equation $(2'')$ from equation $(3')$. This gives us: {{begin-eqn}} {{eqn | n = 1'       | l = x_1 + \\dfrac {x_3} 3       | r = 1 }} {{eqn | n = 2''       | l = x_2 - \\dfrac {x_3} 3       | r = 0 }} {{eqn | n = 3''       | l = -2 x_3       | r = -3 }} {{end-eqn}} From equation $(3'')$ we have directly that $x_3 = \\dfrac 3 2$. Substituting for $x_3$ in equation $(1')$ and equation $(2'')$ gives the single solution: {{begin-eqn}} {{eqn | l = x_1       | r = -\\dfrac 1 2 }} {{eqn | l = x_2       | r = \\dfrac 1 2 }} {{eqn | l = x_3       | r = \\dfrac 3 2 }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Linear Algebra: An Introduction|1982|A.O. Morris|ed = 2nd|edpage = Second Edition|next = Simultaneous Linear Equations/Examples/Arbitrary System 2}}: Chapter $1$: Linear Equations and Matrices: $1.1$ Introduction: Example $\\text {(i)}$ Category:Examples of Simultaneous Linear Equations 13dg15ivlcdbesrvf4ppbezrsjuf304"}
+{"_id": "33136", "title": "Simultaneous Linear Equations/Examples/Arbitrary System 2", "text": "Simultaneous Linear Equations/Examples/Arbitrary System 2 0 88461 475449 474528 2020-06-23T06:31:10Z Prime.mover 59 wikitext text/x-wiki == Example of Simultaneous Linear Equations == <onlyinclude> The system of simultaneous linear equations: {{begin-eqn}} {{eqn | n = 1       | l = x_1 + x_2       | r = 2 }} {{eqn | n = 2       | l = 2 x_1 + 2 x_2       | r = 3 }} {{end-eqn}} has no solutions. </onlyinclude> == Proof == {{AimForCont}} $(1)$ and $(2)$ together have a solution. Subtract $2 \\times$ equation $(1)$ from equation $(2)$. {{begin-eqn}} {{eqn | n = 1       | l = x_1 - 2 x_2 + x_3       | r = 1 }} {{eqn | n = 2'       | l = 0       | r = -1 }} {{end-eqn}} which is an inconsistency. Hence there is no such solution. {{qed}} == Sources == * {{BookReference|Linear Algebra: An Introduction|1982|A.O. Morris|ed = 2nd|edpage = Second Edition|prev = Simultaneous Linear Equations/Examples/Arbitrary System 1|next = Simultaneous Linear Equations/Examples/Arbitrary System 3}}: Chapter $1$: Linear Equations and Matrices: $1.1$ Introduction: Example $\\text {(ii)}$ Category:Examples of Simultaneous Linear Equations 93lvo8ybsxkb39742mc7kad62bzvkfx"}
+{"_id": "33137", "title": "Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 1", "text": "Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 1 0 88536 476404 476375 2020-06-30T06:03:14Z Prime.mover 59 wikitext text/x-wiki == Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations == <onlyinclude> Let $S$ be a system of simultaneous linear equations: :$\\displaystyle \\forall i \\in \\set {1, 2, \\ldots, m}: \\sum_{j \\mathop = 1}^n \\alpha_{i j} x_j = \\beta_i$ Let $\\begin {pmatrix} \\mathbf A & \\mathbf b \\end {pmatrix}$ denote the augmented matrix of $S$. Let $\\begin {pmatrix} \\mathbf R & \\mathbf s \\end {pmatrix}$ be a reduced echelon matrix derived from $\\begin {pmatrix} \\mathbf A & \\mathbf b \\end {pmatrix}$. Let $S'$ be the system of simultaneous linear equations: :$\\displaystyle \\forall i \\in \\set {1, 2, \\ldots, m}: \\sum_{j \\mathop = 1}^n \\rho_{i j} x_j = \\sigma_i$ whose augmented matrix is $\\begin {pmatrix} \\mathbf R & \\mathbf s \\end {pmatrix}$. Then $S$ and $S'$ are equivalent. </onlyinclude> == Proof == By Matrix is Row Equivalent to Reduced Echelon Matrix, $\\begin {pmatrix} \\mathbf R & \\mathbf s \\end {pmatrix}$ can be obtained from $\\begin {pmatrix} \\mathbf A & \\mathbf b \\end {pmatrix}$ by means of a finite sequence of elementary row operations. By Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations, the system of simultaneous linear equations whose augmented matrix is obtained from $\\begin {pmatrix} \\mathbf A & \\mathbf b \\end {pmatrix}$ in this way is equivalent to $S$ after each elementary row operation. Hence the entire sequence of such systems of simultaneous linear equations are equivalent to $S$. In particular, this applies to $S'$. {{qed}} == Sources == * {{BookReference|Linear Algebra: An Introduction|1982|A.O. Morris|ed = 2nd|edpage = Second Edition|prev = Definition:Inverse of Elementary Row Operation|next = Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 2}}: Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations: Corollary $1$ Category:Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations oam0mx3c7tnxpnhhn57ngl7tev13wwu"}
+{"_id": "33138", "title": "Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 3", "text": "Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 3 0 88538 475394 474883 2020-06-22T16:20:15Z Prime.mover 59 wikitext text/x-wiki == Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations == <onlyinclude> Let $S$ be a system of homogeneous simultaneous linear equations: :$\\displaystyle \\forall i \\in \\set {1, 2, \\ldots, m}: \\sum_{j \\mathop = 1}^n \\alpha_{i j} x_j = 0$ If $m < n$, then $S$ has at least one non-trivial solution. </onlyinclude> == Proof == Let $\\begin {pmatrix} \\mathbf A & \\mathbf b \\end {pmatrix}$ denote the augmented matrix of $S$. Because $S$ is homogeneous, we have that $\\mathbf b = \\mathbf 0$, and so its augmented matrix is $\\begin {pmatrix} \\mathbf A & \\mathbf 0 \\end {pmatrix}$. Let $\\begin {pmatrix} \\mathbf R & \\mathbf 0 \\end {pmatrix}$ be a reduced echelon matrix derived from $\\begin {pmatrix} \\mathbf A & \\mathbf 0 \\end {pmatrix}$. Let $S'$ be the system of simultaneous linear equations: :$\\displaystyle \\forall i \\in \\set {1, 2, \\ldots, m}: \\sum_{j \\mathop = 1}^n \\rho_{i j} x_j = 0$ whose augmented matrix is $\\begin {pmatrix} \\mathbf R & \\mathbf 0 \\end {pmatrix}$. By Corollary 1, $S$ and $S'$ are equivalent. Hence any every solution to $S'$ is also a solution to $S$. Consider the structure of $\\begin {pmatrix} \\mathbf R & \\mathbf 0 \\end {pmatrix}$. Suppose the leading coefficients appear in columns which we name $j_1, j_2, \\ldots, j_l$. Let the remaining columns be named $j_{l + 1}, j_{l + 2}, \\ldots, j_n$. Then we have that $S'$ can be expressed as: {{begin-eqn}} {{eqn | l = x_{j_1} + \\sum_{k \\mathop = l + 1}^n \\rho_{i j_k} x_{j_k}       | r = 0 }} {{eqn | l = x_{j_2} + \\sum_{k \\mathop = l + 1}^n \\rho_{2 j_k} x_{j_k}       | r = 0 }} {{eqn | o = \\cdots }} {{eqn | l = x_{j_l} + \\sum_{k \\mathop = l + 1}^n \\rho_{2 j_k} x_{j_k}       | r = 0 }} {{end-eqn}} where $l + 1 \\le m$. Setting arbitrary values to $x_{j_k}$ for $l < k \\le n$ gives us a non-trivial solution for $S$. {{qed}} == Sources == * {{BookReference|Linear Algebra: An Introduction|1982|A.O. Morris|ed = 2nd|edpage = Second Edition|prev = Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 2|next = Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 4}}: Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations: Corollary $3$ Category:Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations 8ifa6ehw6jyhti1mfzf2hng29hhikxb"}
+{"_id": "33139", "title": "Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 4", "text": "Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 4 0 88539 474887 2020-06-20T12:54:48Z Prime.mover 59 Created page with \"== Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations == <onlyinclude> Let $S$ be a system of Definiti...\" wikitext text/x-wiki == Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations == <onlyinclude> Let $S$ be a system of homogeneous simultaneous linear equations: :$\\displaystyle \\forall i \\in \\set {1, 2, \\ldots, m}: \\sum_{j \\mathop = 1}^n \\alpha_{i j} x_j = 0$ Let $\\begin {pmatrix} \\mathbf R & \\mathbf 0 \\end {pmatrix}$ be a reduced echelon matrix derived from $\\begin {pmatrix} \\mathbf A & \\mathbf 0 \\end {pmatrix}$. Let the number of non-zero rows of $\\begin {pmatrix} \\mathbf R & \\mathbf 0 \\end {pmatrix}$ be $l$. If $l = n$, then the only solution to $S$ is the trivial solution. </onlyinclude> == Proof == Consider the structure of $\\begin {pmatrix} \\mathbf R & \\mathbf 0 \\end {pmatrix}$. Suppose the leading coefficients appear in columns which we name $j_1, j_2, \\ldots, j_n$. As there are $n$ columns as well as $n$ non-zero rows: :each row has exactly one $1$ in it :each column has exactly one $1$ in it. Thus $S'$ can be expressed as: {{begin-eqn}} {{eqn | l = x_{j_1}       | r = 0 }} {{eqn | l = x_{j_2}       | r = 0 }} {{eqn | o = \\cdots }} {{eqn | l = x_{j_n}       | r = 0 }} {{end-eqn}} and the result follows. {{qed}} == Sources == * {{BookReference|Linear Algebra: An Introduction|1982|A.O. Morris|ed = 2nd|edpage = Second Edition|prev = Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 3|next = Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 5}}: Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations: Corollary $3$ Category:Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations 6gal5yyvtfrmls1qx2ti5kldjih0to5"}
+{"_id": "33140", "title": "Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse", "text": "Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse 0 88767 476442 476440 2020-06-30T09:23:35Z Prime.mover 59 Replaced content with \"== Theorem == <onlyinclude> Let $e$ be an elementary row operation.  Let $\\mathbf E$ be the Definition:Elementary Row Matrix|elem...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $e$ be an elementary row operation. Let $\\mathbf E$ be the elementary row matrix corresponding to $e$. Let $e'$ be the inverse of $e$. Then the elementary row matrix corresponding to $e'$ is the inverse of $\\mathbf E$. </onlyinclude> == Proof 1 == {{:Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse/Proof 1}} == Proof 2 == {{:Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse/Proof 2}} == Also see == * Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse Category:Elementary Row Operations Category:Elementary Matrices Category:Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse b7onpxovwpfws6crk4vyoqqino8jaqy"}
+{"_id": "33141", "title": "Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse", "text": "Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse 0 88788 476460 2020-06-30T10:39:05Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> Let $e$ be an elementary column operation.  Let $\\mathbf E$ be the Definition:Elementary Column Matrix...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $e$ be an elementary column operation. Let $\\mathbf E$ be the elementary column matrix corresponding to $e$. Let $e'$ be the inverse of $e$. Then the elementary column matrix corresponding to $e'$ is the inverse of $\\mathbf E$. </onlyinclude> == Proof 1 == {{:Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse/Proof 1}} == Proof 2 == {{:Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse/Proof 2}} == Also see == * Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse Category:Elementary Column Operations Category:Elementary Matrices Category:Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse hcex4o5gt7xef84ftxuu3z9n252qmd2"}
+{"_id": "33142", "title": "Elementary Column Operations as Matrix Multiplications/Corollary", "text": "Elementary Column Operations as Matrix Multiplications/Corollary 0 88796 476469 2020-06-30T13:33:38Z Prime.mover 59 Created page with \"== Theorem == <onlyinclude> Let $\\mathbf X$ and $\\mathbf Y$ be two $m \\times n$ matrices that differ by exactly one Definition:Elementary Column Operat...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $\\mathbf X$ and $\\mathbf Y$ be two $m \\times n$ matrices that differ by exactly one elementary column operation. Then there exists an elementary column matrix $\\mathbf E$ of order $n$ such that: :$\\mathbf X \\mathbf E = \\mathbf Y$ </onlyinclude> == Proof == Let $e$ be the elementary column operation such that $e \\paren {\\mathbf X} = \\mathbf Y$. Then this result follows immediately from Elementary Column Operations as Matrix Multiplications: :$e \\paren {\\mathbf X} = \\mathbf X \\mathbf E = \\mathbf Y$ where $\\mathbf E = e \\paren {\\mathbf I}$. {{qed}} Category:Conventional Matrix Multiplication Category:Elementary Column Operations ntseqkowo0j4dpokncrjvwc10rc0lfv"}
+{"_id": "33143", "title": "Fibonacci's Greedy Algorithm", "text": "Fibonacci's Greedy Algorithm 0 89051 478027 477999 2020-07-12T15:37:32Z Prime.mover 59 wikitext text/x-wiki == Algorithm == Let $\\dfrac p q$ denote a proper fraction expressed in canonical form. '''Fibonacci's greedy algorithm''' is a greedy algorithm which calculates a sequence of distinct unit fractions which together sum to $\\dfrac p q$: {{begin-eqn}} {{eqn | l = \\dfrac p q       | r = \\sum_{\\substack {1 \\mathop \\le k \\mathop \\le m \\\\ n_j \\mathop \\le n_{j + 1} } } \\dfrac 1 {n_k}       | c =  }} {{eqn | r = \\dfrac 1 {n_1} + \\dfrac 1 {n_2} + \\dotsb + \\dfrac 1 {n_m}       | c =  }} {{end-eqn}} <onlyinclude> '''Fibonacci's Greedy Algorithm''' is as follows: :$(1) \\quad$ Let $p = x_0$ and $q = y_0$ and set $k = 0$. :$(2) \\quad$ Is $x_k = 1$? If so, the algorithm has finished. :$(3) \\quad$ Find the largest unit fraction $\\dfrac 1 {m_k}$ less than $\\dfrac {x_k} {y_k}$.  :$(4) \\quad$ Calculate $\\dfrac {x_{k + 1} } {y_{k + 1} } = \\dfrac {x_k} {y_k} - \\dfrac 1 {m_k}$ expressed in canonical form. :$(5) \\quad$ Go to step $(2)$. </onlyinclude> == Also see == * Proper Fraction can be Expressed as Finite Sum of Unit Fractions/Fibonacci's Greedy Algorithm, which proves that Fibonacci's Greedy Algorithm works as expected {{Namedfor|Leonardo Fibonacci|cat = Fibonacci}} == Sources == * {{BookReference|Liber Abaci|1202|Leonardo Fibonacci}} * {{BookReference|Curious and Interesting Puzzles|1992|David Wells|prev = Proper Fraction can be Expressed as Finite Sum of Unit Fractions|next = Proper Fraction can be Expressed as Finite Sum of Unit Fractions/Fibonacci's Greedy Algorithm}}: Egyptian Fractions Category:Greedy Algorithms Category:Fibonacci's Greedy Algorithm o09ig10g3tj3495ctsekudc30kafuya"}
+{"_id": "33144", "title": "Largest Number not Expressible as Sum of Multiples of Coprime Integers/Generalization", "text": "Largest Number not Expressible as Sum of Multiples of Coprime Integers/Generalization 0 89295 479146 2020-07-22T04:02:15Z RandomUndergrad 3904 Created page with \"== Theorem ==  <onlyinclude> Let $a, b$ be integers greater than $1$.  Let $d = \\gcd \\set {a, b}$.  Then the largest multiple of $d$ not expressible as...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $a, b$ be integers greater than $1$. Let $d = \\gcd \\set {a, b}$. Then the largest multiple of $d$ not expressible as a sum of multiples of $a$ and $b$ is the number: :$\\dfrac {a b} d - a - b$ </onlyinclude> == Proof == By Integers Divided by GCD are Coprime: :$\\dfrac a d \\perp \\dfrac b d$ By Largest Number not Expressible as Sum of Multiples of Coprime Integers, the largest number not expressible as a sum of multiples of $\\dfrac a d$ and $\\dfrac b d$ is the number: :$\\dfrac {a b} {d^2} - \\dfrac a d - \\dfrac b d$ Let $k d$ be a multiple of $d$ expressible as a sum of multiples of $a$ and $b$: :$\\exists s, t \\in \\N: s a + t b = k d$ Then: :$s \\dfrac a d + t \\dfrac b d = k$ showing that $k$ is a sum of multiples of $\\dfrac a d$ and $\\dfrac b d$. This argument reverses. Hence the largest multiple of $d$ not expressible as a sum of multiples of $a$ and $b$ is the number: :$d \\paren {\\dfrac {a b} {d^2} - \\dfrac a d - \\dfrac b d} = \\dfrac {a b} d - a - b$ {{qed}} Category:Largest Number not Expressible as Sum of Multiples of Coprime Integers Category:Integer Combinations 9j77t0y7ednibo94kkth5ti58vhi8i0"}
+{"_id": "33145", "title": "Inscribing Equilateral Triangle inside Square with a Coincident Vertex/Construction 4", "text": "Inscribing Equilateral Triangle inside Square with a Coincident Vertex/Construction 4 0 89408 479724 479717 2020-07-25T12:59:29Z Prime.mover 59 wikitext text/x-wiki == Construction for Inscribing Equilateral Triangle inside Square with a Coincident Vertex == {{:Inscribing Equilateral Triangle inside Square with a Coincident Vertex}} == Construction == <onlyinclude> :300px By Construction of Equilateral Triangle, let an equilateral triangle $\\triangle ABN$ be constructed on $AB$ such that $N$ is inside $\\Box ABCD$. Let $DN$ be produced to cut $BC$ at $H$. Construct $H$ on $BC$ such that $DH = DG$. Then $DGH$ is the required equilateral triangle. </onlyinclude> == Proof == First a lemma: === Lemma === {{:Inscribing Equilateral Triangle inside Square with a Coincident Vertex/Lemma}}{{qed|lemma}} Because $\\triangle ABN$ is equilateral: :$AB = AN$ :$\\angle BAN = 60 \\degrees$ Thus $\\triangle ADN$ is isosceles with apex at $A$. Then $\\angle DAN = 90 \\degrees - 60 \\degrees = 30 \\degrees$. From Sum of Angles of Triangle equals Two Right Angles: :$\\angle ADN + \\angle AND = 180 \\degrees - 30 \\degrees = 150 \\degrees$ From Isosceles Triangle has Two Equal Angles: :$\\angle ADN = \\angle AND = \\dfrac {150 \\degrees} 2 = 75 \\degrees$ Thus: :$\\angle CDH = 15 \\degrees$ The result follows from the lemma. {{qed}} Category:Inscribing Equilateral Triangle inside Square with a Coincident Vertex qskjduxorus4x54p5ch2u4abeqe9wnc"}
+{"_id": "33146", "title": "Inscribing Equilateral Triangle inside Square with a Coincident Vertex/Lemma", "text": "Inscribing Equilateral Triangle inside Square with a Coincident Vertex/Lemma 0 89410 479728 479723 2020-07-25T13:15:46Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $\\Box ABCD$ be a square. Let $\\triangle DGH$ be an isosceles triangle inscribed within $\\Box ABCD$ such that the apex $D$ of $\\triangle DGH$ coincides with vertex $D$ of $\\Box ABCD$. :300px Then: :$\\triangle DGH$ is equilateral triangle {{iff}}: :$\\angle ADG = 15 \\degrees \\text { or } \\angle CDH = 15 \\degrees$ (and in fact both are the case). </onlyinclude> == Proof == First note that $\\triangle DGH$ is isosceles. First we note that: {{begin-eqn}} {{eqn | l = CD       | r = AD       | c = as they are the sides of a square }} {{eqn | l = DH       | r = DG       | c = as $\\triangle DGH$ is isosceles }} {{eqn | l = \\angle DCH       | r = \\angle DAG = 90 \\degrees       | c = as $\\Box ABCD$ is a square }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = CH^2       | r = DH^2 - CD^2       | c = Pythagoras's Theorem }} {{eqn | r = DG^2 - AD^2       | c =  }} {{eqn | r = AG^2       | c =  }} {{eqn | ll= \\leadsto       | l = CH       | r = AG       | c =  }} {{end-eqn}} So by Triangle Side-Side-Side Equality: :$\\triangle GAD = \\triangle CDH$ and in particular: :$\\angle GAD = \\angle CDH$ === Necessary Condition === Let $\\angle GAD = \\angle CDH = 15 \\degrees$. Then: :$\\angle GDH = 90 \\degrees - 2 \\times 15 \\degrees = 60 \\degrees$ We have that $\\triangle DGH$ is isosceles. Hence from Isosceles Triangle has Two Equal Angles: :$\\angle DGH = \\angle DHG$ From Sum of Angles of Triangle equals Two Right Angles it follows that: :$\\angle DGH + \\angle DHG = 180 \\degrees - 60 \\degrees = 120 \\degrees$ from which: :$\\angle DGH = \\angle DHG = 60 \\degrees$ Hence all the vertices of $\\triangle DGH$ equal $60 \\degrees$. It follows from Equiangular Triangle is Equilateral that $\\triangle DGH$ is equilateral. {{qed|lemma}} === Sufficient Condition === Let $\\triangle DGH$ be equilateral. From Internal Angle of Equilateral Triangle: :$\\angle GDH = 60 \\degrees$ Because $\\Box ABCD$ is a square: :$\\angle ADC = 90 \\degrees$ Thus: {{begin-eqn}} {{eqn | l = \\angle GAD + \\angle GDH + \\angle CDH       | r = 90 \\degrees       | c =  }} {{eqn | ll= \\leadsto       | l = \\angle GAD + \\angle CDH       | r = 90 \\degrees - \\angle GDH       | c =  }} {{eqn | r = 90 \\degrees - 60 \\degrees       | c =  }} {{eqn | r = 30 \\degrees       | c =  }} {{end-eqn}} But we have that: :$\\angle GAD = \\angle CDH$ and so: :$\\angle GAD = \\angle CDH = 15 \\degrees$ {{qed}} Category:Inscribing Equilateral Triangle inside Square with a Coincident Vertex lnchnr9pz5v2faxis0mmygg664pqlz9"}
+{"_id": "33147", "title": "Returning Explorer Puzzle", "text": "Returning Explorer Puzzle 0 90240 482712 482685 2020-08-23T09:36:27Z Prime.mover 59 wikitext text/x-wiki == Puzzle == An explorer walks: :one mile due south,  :one mile due east, :and one mile due north. He finds himself back where he started. He shoots a {{WP|Bear|bear}}. What colour is that {{WP|Bear|bear}}? === Variant === {{:Returning Explorer Puzzle/Variant}} == Solution == White. == Proof == The explorer starts from the North Pole. Every direction from the North Pole is due south. Walking due east keeps him the same distance (one mile) from the North Pole. Walking due north takes him back to the North Pole again. The bear, as a consequence, must be a {{WP|Polar_bear|polar bear}}. {{qed}} == Sources == * {{BookReference|Mathematical Puzzles and Diversions|1965|Martin Gardner|prev = |next = Returning Explorer Puzzle/Variant}}: Nine Problems: $1$ * {{BookReference|Hexaflexagons and Other Mathematical Diversions|1988|Martin Gardner|prev = |next = Returning Explorer Puzzle/Variant}}: Nine Problems: $1$ Category:Geography Puzzles 9udj5kjwnpqk3j4x5bwj5pxk5aku2a4"}
+{"_id": "33148", "title": "Rationals are Everywhere Dense in Reals/Topology", "text": "Rationals are Everywhere Dense in Reals/Topology 0 90798 486361 2020-09-08T09:33:47Z Julius 3095 Created page with \"== Theorem == <onlyinclude> Let $\\struct {\\R, \\tau_d}$ denote the real number line with the usual (Euclidean) topology....\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $\\struct {\\R, \\tau_d}$ denote the real number line with the usual (Euclidean) topology. Let $\\Q$ be the set of rational numbers. Then $\\Q$ is everywhere dense in $\\struct {\\R, \\tau_d}$. </onlyinclude> == Proof == Let $x \\in \\R$. Let $U \\subseteq \\R$ be an open set of $\\struct {\\R, \\tau_d}$ such that $x \\in U$. From Basis for Euclidean Topology on Real Number Line, there exists an open interval $V = \\openint {x - \\epsilon} {x + \\epsilon} \\subseteq U$ for some $\\epsilon > 0$ such that $x \\in V$. Now consider the open interval $\\openint x {x + \\epsilon} \\subseteq V$. By Subset Relation is Transitive it follows that $\\openint x {x + \\epsilon} \\subseteq U$. Note that $x \\notin \\openint x {x + \\epsilon}$. From Between two Real Numbers exists Rational Number, there exists $y \\in \\Q: y \\in \\openint x {x + \\epsilon}$. As $x \\notin \\openint x {x + \\epsilon}$, it must be the case that $x \\ne y$. That is, $V$ is an open set of $\\struct {\\R, \\tau_d}$ containing $x$ which also contains an element of $\\Q$ other than $x$. As $V$ is arbitrary, it follows that every open set of $\\struct {\\R, \\tau_d}$ containing $x$ also contains an element of $\\Q$ other than $x$. That is, $x$ is by definition a limit point of $\\Q$. As $x$ is arbitrary, it follows that all elements of $\\R$ are limit points of $\\Q$. The result follows from the definition of everywhere dense. {{qed}} Category:Real Analysis Category:Rational Number Space Category:Real Number Line with Euclidean Topology Category:Denseness gqg5ugj8zepiyrt2c4szbkgnh6767q7"}
+{"_id": "33149", "title": "Convergent Sequence is Cauchy Sequence/Metric Space", "text": "Convergent Sequence is Cauchy Sequence/Metric Space 0 91017 487795 2020-09-14T11:31:55Z Julius 3095 Created page with \"== Theorem == <onlyinclude> Let $M = \\struct {A, d}$ be a metric space.   Every Definition:Convergent Sequence in Metric Space|convergent sequenc...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $M = \\struct {A, d}$ be a metric space. Every convergent sequence in $M$ is a Cauchy sequence. </onlyinclude> == Proof == Let $\\sequence {x_n}$ be a sequence in $A$ that converges to the limit $l \\in A$. Let $\\epsilon > 0$.  Then also $\\dfrac \\epsilon 2 > 0$. Because $\\sequence {x_n}$ converges to $l$, we have: :$\\exists N: \\forall n > N: \\map d {x_n, l} < \\dfrac \\epsilon 2$ So if $m > N$ and $n > N$, then: {{begin-eqn}} {{eqn | l = \\map d {x_n, x_m}       | o = \\le       | r = \\map d {x_n, l} + \\map d {l, x_m}       | c = Triangle Inequality }} {{eqn | o = <       | r = \\frac \\epsilon 2 + \\frac \\epsilon 2       | c = (by choice of $N$) }} {{eqn | r = \\epsilon       | c =  }} {{end-eqn}} Thus $\\sequence {x_n}$ is a Cauchy sequence. {{qed}} == Also see == * Definition:Complete Metric Space, where the converse is true. * Real Convergent Sequence is Cauchy Sequence * Complex Sequence is Cauchy iff Convergent == Sources == * {{BookReference|Principles of Mathematical Analysis|1953|Walter Rudin|next = Compact Metric Space is Complete}}: $3.11a$ * {{BookReference|Introduction to Metric and Topological Spaces|1975|W.A. Sutherland|prev = Definition:Cauchy Sequence (Metric Space)|next = Subsequence of Sequence in Metric Space with Limit}}: $7.2$: Sequential compactness: Proposition $7.2.4$ * {{BookReference|Counterexamples in Topology|1978|Lynn Arthur Steen|author2 = J. Arthur Seebach, Jr.|ed = 2nd|edpage = Second Edition|prev = Definition:Cauchy Sequence (Metric Space)|next = Cauchy Sequence in Metric Space is not necessarily Convergent}}: Part $\\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces Category:Cauchy Sequences 3fp1bcpai3vu42ov3pmvs4jpzk155qx"}
+{"_id": "33150", "title": "Cauchy Sequence is Bounded/Metric Space", "text": "Cauchy Sequence is Bounded/Metric Space 0 91058 488083 2020-09-15T11:26:19Z Julius 3095 Created page with \"== Theorem == <onlyinclude> Let $M = \\struct {A, d}$ be a metric space.   Then every Cauchy sequence...\" wikitext text/x-wiki == Theorem == <onlyinclude> Let $M = \\struct {A, d}$ be a metric space. Then every Cauchy sequence in $M$ is bounded. </onlyinclude> == Proof == Let $\\sequence {x_n}$ be a Cauchy sequence in $M$. By definition: :$\\forall \\epsilon > 0: \\exists N \\in \\N: \\forall m, n > N: \\map d {x_n, x_m} < \\epsilon$ Particularly, setting $\\epsilon = 1$: :$\\exists N_1: \\forall m, n > N_1: \\map d {x_n, x_m} < 1$ Note that since $N_1 \\ge N_1$, this means that: :$\\forall n \\ge N_1: \\map d {x_n, x_{N_1} } < 1$ To show $\\sequence {x_n}$ is bounded, we need to show that there exists $a \\in A$ and $K \\in \\R$ such that $\\map d {x_n, a} \\le K$ for all $x_n \\in \\sequence {x_n}$. Let $K'$ be the maximum distance from $x_{N_1}$ to any of the earlier terms in the sequence. That is, $K' = \\max \\set {\\map d {x_{N_1}, x_1}, \\map d {x_{N_1},  x_2}, \\ldots, \\map d {x_{N_1}, x_{N_1 - 1} } }$ Then: :Each $x_n$ for $n \\ge N_1$ satisfies $\\map d {x_{N_1}, x_n} \\le 1$ by choice of $N_1$ as mentioned above :Each $x_n$ for $n < N_1$ satisfies $\\map d {x_{N_1}, x_n} \\le K'$ by choice of $K'$. Thus, taking $a = x_{N_1}$ and $K = \\max \\set {K', 1}$, we have shown that each $x_n$ satisfies $\\map d {a, x_n} \\le K$. So, $\\sequence {x_n}$ is bounded. {{qed}} == Sources == * {{BookReference|Mathematical Analysis|1957|Tom M. Apostol|prev = Real Convergent Sequence is Cauchy Sequence|next = Unbounded Sequence is Divergent}}: $\\S 12$-$2$: Convergent and divergent sequences Category:Cauchy Sequences s8kcwka7sx73viuxqvaryz87ba99gi3"}
+{"_id": "33151", "title": "Autocorrelation of Strictly Stationary Stochastic Process", "text": "Autocorrelation of Strictly Stationary Stochastic Process 0 91323 493070 489826 2020-10-07T15:20:48Z Prime.mover 59 wikitext text/x-wiki == Example of Strictly Stationary Stochastic Process == Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. <onlyinclude> It is necessary that: :The autocorrelation between every two observations $z_t, z_{t + k}$ separated by a given lag $k$  is the same as: :the autocorrelation between every other two observations $z_{t + m}, z_{t + m + k}$ separated by a given lag $k$ </onlyinclude> For such a strictly stationary stochastic process: :$\\rho_k = \\dfrac {\\gamma_k} {\\gamma_0}$ where $\\gamma_k$ denotes the autocovariance of $S$. == Proof == The autocorrelation is defined as: :$\\rho_k := \\dfrac {\\expect {\\paren {z_t - \\mu} \\paren {z_{t + k} - \\mu} } } {\\sqrt {\\expect {\\paren {z_t - \\mu}^2} \\expect {\\paren {z_{t + k} - \\mu}^2} } }$ The autocovariance is defined as: :$\\gamma_k := \\expect {\\paren {z_t - \\mu} \\paren {z_{t - k} - \\mu} }$ Hence: :$\\rho_k := \\dfrac {\\gamma_k} {\\sqrt {\\expect {\\paren {z_t - \\mu}^2} \\expect {\\paren {z_{t + k} - \\mu}^2} } }$ Then we have that for a strictly stationary stochastic process: :$\\expect {\\paren {z_t - \\mu}^2} = \\sigma_z^2$ where $\\sigma_z^2$ is the variance of $S$ and, for a strictly stationary stochastic process, is constant. Thus: :$\\rho_k := \\dfrac {\\gamma_k} {\\sigma_t^2}$ Then from Autocovariance at Zero Lag for Strictly Stationary Stochastic Process is Variance: :$\\sigma_z^2 = \\gamma_0$ Hence: :$\\rho_k = \\dfrac {\\gamma_k} {\\gamma_0}$ {{qed}} == Sources == * {{BookReference|Time Series Analysis: Forecasting and Control|1994|George E.P. Box|author2 = Gwilym M. Jenkins|author3 = Gregory C. Reinsel|ed = 3rd|edpage = Third Edition|prev = Definition:Autocorrelation|next = Autocovariance at Zero Lag for Strictly Stationary Stochastic Process is Variance}}: ::Part $\\text {I}$: Stochastic Models and their Forecasting: :::$2$: Autocorrelation Function and Spectrum of Stationary Processes: ::::$2.1$ Autocorrelation Properties of Stationary Models: :::::$2.1.2$ Stationary Stochastic Processes: Autocovariance and autocorrelation coefficients: $(2.1.6)$ Category:Autocorrelation Category:Stationary Stochastic Processes oxk3s6u49i9odf3w3jttrcnsq6frhb7"}
+{"_id": "33152", "title": "Autocovariance at Zero Lag for Strictly Stationary Stochastic Process is Variance", "text": "Autocovariance at Zero Lag for Strictly Stationary Stochastic Process is Variance 0 91332 489817 489808 2020-09-21T15:36:53Z Prime.mover 59 wikitext text/x-wiki == Example of Strictly Stationary Stochastic Process == Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Then the autocovariance at zero lag is given by: :$\\gamma_0 = \\sigma_z^2$ where $\\sigma_z^2$ is the variance of $S$. == Proof == By definition, the '''autocovariance''' of $S$ at lag $k$ is defined as: :$\\gamma_k := \\cov {z_t, z_{t + k} } = \\expect {\\paren {z_t - \\mu} \\paren {z_{t - k} - \\mu} }$ where: :$z_t$ is the observation at time $t$ :$\\mu$ is the mean of $S$ :$\\expect \\cdot$ is the expectation. For a strictly stationary stochastic process: :$\\expect {\\paren {z_t - \\mu}^2} = \\sigma_z^2$ where: :$\\mu$ is the constant mean level of $S$ :$\\expect {\\paren {z_t - \\mu}^2}$ is the expectation of $\\paren {z_t - \\mu}^2$ :$\\sigma_z^2$ is the variance of $S$ and, for a strictly stationary stochastic process, is constant. Hence we have that: {{begin-eqn}} {{eqn | l = \\gamma_0       | r = \\expect {\\paren {z_t - \\mu} \\paren {z_{t + 0} - \\mu} }       | c =  }} {{eqn | r = \\expect {\\paren {z_t - \\mu}^2}       | c =  }} {{eqn | r = \\sigma_z^2       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Time Series Analysis: Forecasting and Control|1994|George E.P. Box|author2 = Gwilym M. Jenkins|author3 = Gregory C. Reinsel|ed = 3rd|edpage = Third Edition|prev = Autocorrelation of Strictly Stationary Stochastic Process|next = Autocorrelation at Zero Lag for Strictly Stationary Stochastic Process is 1}}: ::Part $\\text {I}$: Stochastic Models and their Forecasting: :::$2$: Autocorrelation Function and Spectrum of Stationary Processes: ::::$2.1$ Autocorrelation Properties of Stationary Models: :::::$2.1.2$ Stationary Stochastic Processes: Autocovariance and autocorrelation coefficients Category:Stationary Stochastic Processes Category:Autocovariance csnx3gtbincn8tjz9921ydy7z84u086"}
+{"_id": "33153", "title": "Determinant of Autocorrelation Matrix is Strictly Positive/Examples/Order 2", "text": "Determinant of Autocorrelation Matrix is Strictly Positive/Examples/Order 2 0 91368 490102 2020-09-22T10:34:28Z Prime.mover 59 Created page with \"== Example of Use of Determinant of Autocorrelation Matrix is Strictly Positive == <onlyinclude> Let $\\rho_1$ be the autocorrelation of a ...\" wikitext text/x-wiki == Example of Use of Determinant of Autocorrelation Matrix is Strictly Positive == <onlyinclude> Let $\\rho_1$ be the autocorrelation of a strictly stationary stochastic process $S$ at lag $1$. Then: :$-1 < \\rho_1 < 1$ </onlyinclude> == Proof == Consider the autocorrelation matrix of order $2$: {{begin-eqn}} {{eqn | l = \\map \\det {\\mathbf P_2}       | o = >       | r = 0       | c = Determinant of Autocorrelation Matrix is Strictly Positive }} {{eqn | l = \\begin {vmatrix} 1 & \\rho_1 \\\\ \\rho_1 & 1 \\end {vmatrix}       | o = >       | r = 0       | c = {{Defof|Autocorrelation Matrix}} }} {{eqn | l = 1 - \\rho_1^2       | o = >       | r = 0       | c = {{Defof|Determinant}} }} {{eqn | ll= \\leadsto       | l = \\rho_1^2       | o = <       | r = 1       | c =  }} {{eqn | ll= \\leadsto       | l = \\size {\\rho_1}       | o = <       | r = 1       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Time Series Analysis: Forecasting and Control|1994|George E.P. Box|author2 = Gwilym M. Jenkins|author3 = Gregory C. Reinsel|ed = 3rd|edpage = Third Edition|prev = Determinant of Autocorrelation Matrix is Strictly Positive|next = Determinant of Autocorrelation Matrix is Strictly Positive/Examples/Order 3}}: ::Part $\\text {I}$: Stochastic Models and their Forecasting: :::$2$: Autocorrelation Function and Spectrum of Stationary Processes: ::::$2.1$ Autocorrelation Properties of Stationary Models: :::::$2.1.3$ Positive Definiteness and the Autocovariance Matrix: Conditions satisfied by the autocorrelations of a stationary process Category:Determinant of Autocorrelation Matrix is Strictly Positive nsyymouh42ctehv27y9wj35pcqp9uyw"}
+{"_id": "33154", "title": "Negative of Logarithm of x plus Root x squared plus a squared/Corollary", "text": "Negative of Logarithm of x plus Root x squared plus a squared/Corollary 0 91523 490958 2020-09-26T13:22:45Z Prime.mover 59 Created page with \"== Theorem ==  Let $x \\in \\R: \\size x > 1$.  Let $x > 1$.  Then: <onlyinclude> :$-\\map \\ln {x + \\sqrt {x^2 + a^2} } = \\ln \\size {x - \\sqrt {x^2 + a^2} } - \\map \\ln {a^2}$ </on...\" wikitext text/x-wiki == Theorem == Let $x \\in \\R: \\size x > 1$. Let $x > 1$. Then: <onlyinclude> :$-\\map \\ln {x + \\sqrt {x^2 + a^2} } = \\ln \\size {x - \\sqrt {x^2 + a^2} } - \\map \\ln {a^2}$ </onlyinclude> == Proof == We have that $\\sqrt {x^2 + a^2} > x$ for all $x$. Hence for all $x$: :$-x + \\sqrt {x^2 + a^2} > 0$ and so: :$x - \\sqrt {x^2 + a^2} < 0$ Hence: :$-x + \\sqrt {x^2 + a^2} = \\size {x - \\sqrt {x^2 + a^2} }$ Then we have: {{begin-eqn}} {{eqn | l = -\\map \\ln {x + \\sqrt {x^2 - a^2} }       | r = \\map \\ln {-x + \\sqrt {x^2 + a^2} } - \\map \\ln {a^2}       | c = Negative of Logarithm of x plus Root x squared plus a squared }} {{eqn | ll= \\leadsto       | l = -\\map \\ln {x + \\sqrt {x^2 - a^2} }       | r = \\ln \\size {x - \\sqrt {x^2 + a^2} } - \\map \\ln {a^2}       | c =  }} {{end-eqn}} {{qed}} == Also see == * Negative of Logarithm of x plus Root x squared plus a squared Category:Logarithms bq1gb4ofmsp608jt6gjqt15zeqxcjlc"}
+{"_id": "33155", "title": "Primitive of Reciprocal of a squared minus x squared/Logarithm Form/Lemma", "text": "Primitive of Reciprocal of a squared minus x squared/Logarithm Form/Lemma 0 91571 491217 2020-09-27T16:21:54Z Prime.mover 59 Created page with \"== Lemma == <onlyinclude> Let $a \\in \\R_{>0}$ be a strictly positive real constant.  Then: :$\\map \\ln {\\df...\" wikitext text/x-wiki == Lemma == <onlyinclude> Let $a \\in \\R_{>0}$ be a strictly positive real constant. Then: :$\\map \\ln {\\dfrac {a + x} {a - x} }$ is defined {{iff}} $\\size x < a$ :$\\map \\ln {\\dfrac {x + a} {x - a} }$ is defined {{iff}} $\\size x > a$ </onlyinclude> == Proof == We have that the real natural logarithm is defined only on the strictly positive real numbers. Hence: :$\\map \\ln {\\dfrac {a + x} {a - x} }$ is defined {{iff}} $\\dfrac {a + x} {a - x} > 0$ :$\\map \\ln {\\dfrac {x + a} {x - a} }$ is defined {{iff}} $\\dfrac {x + a} {x - a} > 0$ First we note that if $\\size x = a$, then either the numerator or denominator of the arguments of the logarithm functions in question are either $0$ or undefined. Hence the expressions have no meaning unless $\\size x \\ne a$. The following table indicates whether each of $x + a$, $a - x$ and $x - a$ are positive $(+)$ or negative $(-)$ on the domains in question. $\\begin {array} {c|ccc|cc} & x + a & a - x & x - a & \\dfrac {a + x} {a - x} & \\dfrac {x + a} {x - a} \\\\ \\hline  a < x      & + & - & + & + & - \\\\ 0 < x < a  & + & + & - & - & + \\\\ -a < x < 0 & + & + & - & - & + \\\\ x < -a     & - & + & - & + & - \\\\ \\end {array}$ Hence: :$\\map \\ln {\\dfrac {a + x} {a - x} }$ is defined {{iff}} $-a < x < a$ :$\\map \\ln {\\dfrac {x + a} {x - a} }$ is defined {{iff}} $x > a$ or $x < -a$ as we were required to show. {{qed}} Category:Logarithms Category:Primitive of Reciprocal of a squared minus x squared czlzx7khcp4z5tjwrrh1qrxjlihs7d6"}
+{"_id": "33156", "title": "Primitive of Reciprocal of x squared minus a squared/Logarithm Form/Lemma", "text": "Primitive of Reciprocal of x squared minus a squared/Logarithm Form/Lemma 0 91592 491313 2020-09-27T22:16:21Z Prime.mover 59 Created page with \"== Lemma == <onlyinclude> Let $a \\in \\R_{>0}$ be a strictly positive real constant.  Then: :$\\map \\ln {\\df...\" wikitext text/x-wiki == Lemma == <onlyinclude> Let $a \\in \\R_{>0}$ be a strictly positive real constant. Then: :$\\map \\ln {\\dfrac {x - a} {x + a} }$ is defined {{iff}} $\\size x > a$ :$\\map \\ln {\\dfrac {a - x} {a + x} }$ is defined {{iff}} $\\size x < a$ </onlyinclude> == Proof == We have that the real natural logarithm is defined only on the strictly positive real numbers. Hence: :$\\map \\ln {\\dfrac {x - a} {x + a} }$ is defined {{iff}} $\\dfrac {x - a} {x + a} > 0$ :$\\map \\ln {\\dfrac {a - x} {a + x} }$ is defined {{iff}} $\\dfrac {a - x} {a + x} > 0$ First we note that if $\\size x = a$, then either the numerator or denominator of the arguments of the logarithm functions in question are either $0$ or undefined. Hence the expressions have no meaning unless $\\size x \\ne a$. The following table indicates whether each of $x + a$, $a - x$ and $x - a$ are positive $(+)$ or negative $(-)$ on the domains in question. $\\begin {array} {c|ccc|cc} & x + a & a - x & x - a & \\dfrac {x - a} {x + a} & \\dfrac {a - x} {a + x} \\\\ \\hline  a < x      & + & - & + & + & - \\\\ 0 < x < a  & + & + & - & - & + \\\\ -a < x < 0 & + & + & - & - & + \\\\ x < -a     & - & + & - & + & - \\\\ \\end {array}$ Hence: :$\\map \\ln {\\dfrac {x - a} {x + a} }$ is defined {{iff}} $x > a$ or $x < -a$ :$\\map \\ln {\\dfrac {a - x} {a + x} }$ is defined {{iff}} $-a < x < a$ as we were required to show. {{qed}} Category:Logarithms Category:Primitive of Reciprocal of x squared minus a squared 2du9lctlzc521fqaiwvjzyok1zbdl2o"}
+{"_id": "33157", "title": "Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Tangent Form/Proof", "text": "Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Tangent Form/Proof 0 91596 491322 2020-09-27T22:30:09Z Prime.mover 59 Created page with \"== Theorem == {{:Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Tangent Form}}  == Proof == <onlyinclude> Let $\\size x < a$.  Let:  {{begin-eqn}} {{eq...\" wikitext text/x-wiki == Theorem == {{:Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Tangent Form}} == Proof == <onlyinclude> Let $\\size x < a$. Let: {{begin-eqn}} {{eqn | l = u       | r = \\tanh^{-1} {\\frac x a}       | c = {{Defof|Real Inverse Hyperbolic Tangent}}, which is defined where $\\size {\\dfrac x a} < 1$ }} {{eqn | ll= \\leadsto       | l = x       | r = a \\tanh u       | c =  }} {{eqn | ll= \\leadsto       | l = \\frac {\\d x} {\\d u}       | r = a \\sech^2 u       | c = Derivative of Hyperbolic Cotangent }} {{eqn | ll= \\leadsto       | l = \\int \\frac 1 {x^2 - a^2} \\rd x       | r = \\int \\frac {a \\sech^2 u} {a^2 \\tanh^2 u - a^2} \\rd u       | c = Integration by Substitution }} {{eqn | r = \\frac a {a^2} \\int \\frac {\\sech^2 u} {-\\paren {1 - \\tanh^2 u} } \\rd u       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \\frac 1 a \\int \\frac {\\sech^2 u} {-\\sech^2 u} \\rd u       | c = Sum of Squares of Hyperbolic Secant and Tangent }} {{eqn | r = -\\frac 1 a \\int \\rd u }} {{eqn | r = -\\frac 1 a u + C       | c = Integral of Constant }} {{eqn | r = -\\frac 1 a \\tanh^{-1} {\\frac x a} + C       | c = Definition of $u$ }} {{end-eqn}} </onlyinclude>{{qed}} Category:Primitive of Reciprocal of x squared minus a squared cum8etbjs33ubz9sorsuh7ts10svvrq"}
+{"_id": "33158", "title": "Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 1", "text": "Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 1 0 91645 491748 491696 2020-09-30T06:34:44Z Leigh.Samphier 3031 wikitext text/x-wiki {{Proofread}} == Theoren == Let $U, V$ be finite sets. Let $\\card V < \\card U$. Then: <onlyinclude> :$\\card{V \\setminus U} < \\card{U \\setminus V}$ </onlyinclude> == Proof == We have: {{begin-eqn}} {{eqn | l = \\card{U \\setminus V}       | r = \\card U - \\card{U \\cap V}       | c = Cardinality of Set Difference }} {{eqn | o = >        | r = \\card V - \\card{U \\cap V}       | c = As $\\card V < \\card U$ }} {{eqn | r = \\card{V \\setminus U}       | c = Cardinality of Set Difference }} {{end-eqn}} {{qed}} Category:Matroid Satisfies Base Axiom ncp1vmjeal1kfh62yi04bort1blni90"}
+{"_id": "33159", "title": "Transitivity of Integrality/Lemma", "text": "Transitivity of Integrality/Lemma 0 92667 497829 2020-11-05T14:11:07Z Prime.mover 59 Created page with \"== Lemma for Transitivity of Integrality == <onlyinclude> Let $A \\subseteq B$ be a ring extension.  Let $x_1, \\dotsc, x_n \\in B$ be Definit...\" wikitext text/x-wiki == Lemma for Transitivity of Integrality == <onlyinclude> Let $A \\subseteq B$ be a ring extension. Let $x_1, \\dotsc, x_n \\in B$ be integral over $A$. Let $A \\sqbrk {x_1, \\dotsc, x_n}$ be the subring of $B$ generated by $A \\cup \\set {x_1, \\dotsc, x_n}$ over $A$. Then $A \\sqbrk {x_1, \\dotsc, x_n}$ is integral over $A$. </onlyinclude> == Proof == Let $C$ be the integral closure of $A$ in $B$. Since the $x_i$ are integral over $A$, they lie in $C$. So by Integral Closure is Subring, all sums of the form: :$\\ds \\sum_{\\text {finite} } r x_1^{\\alpha_1} \\dotsm x_n^{\\alpha_n}, \\quad r \\in A,\\ \\alpha_j \\in \\N \\cup \\set 0$ lie in $C$. That is, they are integral over $A$.  But the set of such sums is precisely $A \\sqbrk {x_1, \\dotsc, x_n}$. {{qed}} == Linguistic Note == {{:Transitivity of Integrality/Linguistic Note}} Category:Transitivity of Integrality qnem75iriv50mo5gee7s0cgqervax5e"}
+{"_id": "11", "title": "Schur-Zassenhaus Theorem", "text": "Let $G$ be a finite group and $N$ be a normal subgroup in $G$. Let $N$ be a Hall subgroup of $G$. Then there exists $H$, a complement of $N$, such that $G$ is the semidirect product of $N$ and $H$."}
+{"_id": "8228", "title": "Non-Zero Integer has Unique Positive Integer Associate", "text": "Let $a \\in \\Z$ be an integer such that $a \\ne 0$. Then $a$ has a unique associate $b \\in \\Z_{>0}$."}
+{"_id": "16436", "title": "Element of Cyclic Group is not necessarily Generator", "text": "Let $\\gen g = G$ be a cyclic group. Let $a \\in G$ Then it is not necessarily the case that $a$ is also a generator of $G$."}
+{"_id": "16444", "title": "Equivalent Statements for Congruence Modulo Subgroup/Left", "text": "Let $x \\equiv^l y \\pmod H$ denote that $x$ is left congruent modulo $H$ to $y$. Then the following statements are equivalent: {{begin-eqn}} {{eqn | n = 1       | l = x       | o = \\equiv^l       | r = y \\pmod H }} {{eqn | n = 2       | l = x^{-1} y       | o = \\in       | r = H }} {{eqn | n = 3       | l = \\exists h \\in H: x^{-1} y       | r = h }} {{eqn | n = 4       | l = \\exists h \\in H: y       | r = x h }} {{end-eqn}}"}
+{"_id": "16448", "title": "Element of Group is in Unique Coset of Subgroup/Left", "text": "There exists a exactly one left coset of $H$ containing $x$, that is: $x H$"}
+{"_id": "16453", "title": "Order of Product of Commuting Group Elements of Coprime Order is Product of Orders", "text": "Let $G$ be a group. Let $g_1, g_2 \\in G$ be commuting elements such that: {{begin-eqn}} {{eqn | l = \\order {g_1}       | r = n_1 }} {{eqn | l = \\order {g_1}       | r = n_2 }} {{end-eqn}} where $\\order {g_1}$ denotes the order of $g_1$ in $G$. Let $n_1$ and $n_2$ be coprime. Then: :$\\order {g_1 g_2} = n_1 n_2$"}
+{"_id": "8268", "title": "Trivial Norm on Division Ring is Norm", "text": "Let $\\struct {R, +, \\circ}$ be a division ring, and denote its ring zero by $0_R$. Then the trivial norm $\\norm {\\, \\cdot \\,}: R \\to \\R_{\\ge 0}$, which is given by: :$\\norm x = \\begin{cases}   0 & \\text { if } x = 0_R\\\\   1 & \\text { otherwise} \\end{cases}$ defines a norm on $R$."}
+{"_id": "16469", "title": "Coset of Subgroup of Subgroup", "text": "Let $G$ be a group. Let $H, K \\le G$ be subgroups of $G$. Let $K \\subseteq H$. Let $x \\in G$. Then either: :$x K \\subseteq H$ or: :$x K \\cap H = \\O$ where $x K$ denotes the left coset of $K$ by $x$."}
+{"_id": "8278", "title": "Equivalence of Definitions of Characteristic of Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. {{TFAE|def = Characteristic of Ring}}"}
+{"_id": "16485", "title": "Non-Cyclic Group of Order p^2 has p+3 Subgroups", "text": "Let $p$ be a prime number. Let $G$ be a non-cyclic group whose order is $p^2$. Then $G$ has exactly $p + 3$ subgroups."}
+{"_id": "16488", "title": "General Morphism Property for Groups", "text": "Let $\\struct {G, \\circ}$ and $\\struct {H, *}$ be groups. Let $\\phi: G \\to H$ be a homomorphism. Then: :$\\forall g_k \\in H: \\map \\phi {g_1 \\circ g_2 \\circ \\cdots \\circ g_n} = \\map \\phi {g_1} * \\map \\phi {g_2} * \\cdots * \\map \\phi {g_n}$"}
+{"_id": "111", "title": "Set Difference is Anticommutative", "text": "Set difference is an anticommutative operation: :$S = T \\iff S \\setminus T = T \\setminus S = \\varnothing$"}
+{"_id": "8313", "title": "Integers under Addition form Monoid", "text": "The set of integers under addition $\\struct {\\Z, +}$ forms a monoid."}
+{"_id": "16512", "title": "Klein Four-Group is Group", "text": "The Klein $4$-group $K_4$ is a group."}
+{"_id": "16515", "title": "Dihedral Group is Non-Abelian", "text": "Let $n \\in \\N$ be a natural number such that $n > 2$. Let $D_n$ denote the dihedral group of order $2 n$. Then $D_n$ is not abelian."}
+{"_id": "16518", "title": "Summation Formula (Complex Analysis)", "text": ":$\\displaystyle \\sum_{n \\in \\Z \\setminus X} \\map f n = - \\sum_{z_0 \\mathop \\in X} \\Res {\\pi \\cot \\paren {\\pi z} \\map f z} {z_0}$"}
+{"_id": "136", "title": "Set with Complement forms Partition", "text": "Let $\\varnothing \\subset S \\subset \\mathbb U$. Then $S$ and its complement $\\complement \\left({S}\\right)$ form a partition of the universal set $\\mathbb U$."}
+{"_id": "16523", "title": "Subgroups of Additive Group of Integers Modulo m", "text": "Let $n \\in \\Z_{> 0}$ be a (strictly) positive integer. Let $\\struct {\\Z_m, +_m}$ denote the additive group of integers modulo $m$. The subgroups of $\\struct {\\Z_m, +_m}$ are the additive groups of integers modulo $k$ where: :$k \\divides m$"}
+{"_id": "8336", "title": "Finite Monoid with Left Cancellable Operation is Group", "text": "Let $\\left({S, \\circ}\\right)$ be a finite monoid. Let $\\circ$ be a left cancellable operation. Then $\\left({S, \\circ}\\right)$ is a group."}
+{"_id": "145", "title": "Cartesian Product is Anticommutative", "text": "Let $S, T \\ne \\O$. Then: :$S \\times T = T \\times S \\implies S = T$"}
+{"_id": "16535", "title": "Groups of Order 6", "text": "There exist exactly $2$ groups of order $6$, up to isomorphism: :$C_6$, the cyclic group of order $6$ :$S_3$, the symmetric group on $3$ letters."}
+{"_id": "16541", "title": "Normalizer of Rotation in Dihedral Group", "text": "Let $n \\in \\N$ be a natural number such that $n \\ge 3$. Let $D_n$ be the dihedral group of order $2 n$, given by: :$D_n = \\gen {\\alpha, \\beta: \\alpha^n = \\beta^2 = e, \\beta \\alpha \\beta = \\alpha^{−1} }$ Let $\\map {N_{D_n} } {\\set \\alpha}$ denote the normalizer of the singleton containing the rotation element $\\alpha$. Then: :$\\map {N_{D_n} } {\\set \\alpha} = \\gen \\alpha$ where $\\gen \\alpha$ is the subgroup generated by $\\alpha$."}
+{"_id": "8355", "title": "Möbius Strip has Euler Characteristic Zero", "text": "Let $M$ be a Möbius Strip. Then: :$\\map \\chi M = 0$ where $\\map \\chi M$ denotes the Euler characteristic of the graph $M$."}
+{"_id": "8359", "title": "Group Action of Symmetric Group Acts Transitively", "text": "Let $S$ be a set. Let $\\struct {\\map \\Gamma S, \\circ}$ be the symmetric group on $S$. Let $*: \\map \\Gamma S \\times S \\to S$ be the group action defined as: :$\\forall \\pi \\in \\map \\Gamma S, \\forall s \\in S: \\pi * s = \\map \\pi s$ Then $*$ is a transitive group action. In other words, $\\struct {\\map \\Gamma S, \\circ}$ acts transitively on $S$ by $*$."}
+{"_id": "8361", "title": "Orbit of Trivial Group Action is Singleton", "text": "Let $\\left({G, \\circ}\\right)$ be a group whose identity is $e$. Let $S$ be a set. Let $*: G \\times S \\to S$ be the trivial group action: :$\\forall \\left({g, s}\\right) \\in G \\times S: g * s = s$ Let $s \\in S$. Then the orbit of $s$ under $*$ is $\\left\\{{s}\\right\\}$."}
+{"_id": "16559", "title": "Length of Orbit of Subgroup Action on Left Coset Space", "text": "Let $G$ be a group. Let $H$ and $K$ be subgroups of $G$. Let $K$ act on the left coset space $G / H^l$ by: :$\\forall \\tuple {k, g H} \\in K \\times G / H^l: k * g H := \\paren {k g} H$ The length of the orbit of $g H$ is $\\index K {K \\cap H^g}$."}
+{"_id": "8367", "title": "Conjugacy Action on Abelian Group is Trivial", "text": "Let $\\struct {G, \\circ}$ be an abelian group whose identity is $e$. Let $*: G \\times G \\to G$ be the conjugacy group action: : $\\forall g, h \\in G: g * h = g \\circ h \\circ g^{-1}$ Then $*$ is a trivial group action."}
+{"_id": "179", "title": "Relation Symmetry", "text": "Every non-null relation has exactly one of these properties: it is either: :symmetric, :asymmetric or :non-symmetric."}
+{"_id": "8379", "title": "Stabilizer of Cartesian Product of Group Actions", "text": "Let $\\struct {G, \\circ}$ be a group. Let $S$ and $T$ be sets. Let $*_S: G \\times S \\to S$ and $*_T: G \\times T \\to T$ be group actions. Let the group action $*: G \\times \\paren {S \\times T} \\to S \\times T$ be defined as: :$\\forall \\tuple {g, \\tuple {s, t} } \\in G \\times \\paren {S \\times T}: g * \\tuple {s, t} = \\tuple {g *_S s, g *_T t}$ Then the stabilizer of $\\tuple {s, t} \\in S \\times T$ is given by: :$\\Stab {s, t} = \\Stab s \\cap \\Stab t$ where $\\Stab s$ and $\\Stab t$ are the stabilizers of $s$ and $t$ under $*_S$ and $*_T$ respectively."}
+{"_id": "16572", "title": "Divisibility by 12", "text": "Let $N \\in \\N$ be expressed as: :$N = a_0 + a_1 10 + a_2 10^2 + \\cdots + a_n 10^n$ Then $N$ is divisible by $12$ {{iff}} $a_0 - 2 a_1 + 4 \\paren {\\displaystyle \\sum_{r \\mathop = 2}^n a_r}$ is  divisible by $12$."}
+{"_id": "192", "title": "Composite Relation with Inverse is Symmetric", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation. Then the composition of $\\mathcal R$ with its inverse $\\mathcal R^{-1}$ is symmetric: * $\\mathcal R^{-1} \\circ \\mathcal R$ is a symmetric relation on $S$ * $\\mathcal R \\circ \\mathcal R^{-1}$ is a symmetric relation on $T$."}
+{"_id": "16576", "title": "Minimum Rule for Real Sequences", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\min \\set {x_n, y_n} = \\min \\set {l, m}$"}
+{"_id": "209", "title": "Union of Equivalences", "text": "The union of two equivalence relations is '''not''' necessarily an equivalence relation itself."}
+{"_id": "16604", "title": "Topological Properties of Non-Archimedean Division Rings/Intersection of Open Balls", "text": ":$\\map {B_r} x \\cap \\map {B_s} y \\ne \\O \\iff \\map {B_r} x \\subseteq \\map {B_s} y$ or $\\map {B_s} y \\subseteq \\map {B_r} x$"}
+{"_id": "8416", "title": "Universal Negative implies Particular Negative iff First Predicate is not Vacuous", "text": "Consider the categorical statements: :$\\map {\\mathbf E} {S, P}: \\quad$ The universal negative: $\\forall x: \\map S x \\implies \\neg \\map P x$ :$\\map {\\mathbf O} {S, P}: \\quad$ The particular negative: $\\exists x: \\map S x \\land \\neg \\map P x$ Then: :$\\map {\\mathbf E} {S, P} \\implies \\map {\\mathbf O} {S, P}$ {{iff}}: :$\\exists x: \\map S x$ Using the symbology of predicate logic: :$\\exists x: \\map S x \\iff \\paren {\\paren {\\forall x: \\map S x \\implies \\neg \\map P x} \\implies \\paren {\\exists x: \\map S x \\land \\neg \\map P x} }$"}
+{"_id": "16615", "title": "Addition on Numbers has no Zero Element", "text": "On all the number systems: * natural numbers $\\N$ * integers $\\Z$ * rational numbers $\\Q$ * real numbers $\\R$ * complex numbers $\\C$ there exists no zero element for addition."}
+{"_id": "16618", "title": "Group has Latin Square Property/Additive Notation", "text": "Let $\\struct {G, +}$ be a group. Then $G$ satisfies the Latin square property. That is, for all $a, b \\in G$, there exists a unique $g \\in G$ such that $a + g = b$. Similarly, there exists a unique $h \\in G$ such that $h + a = b$."}
+{"_id": "16626", "title": "Normed Division Ring Operations are Continuous/Inversion", "text": ":$\\iota : \\struct {R^* ,d^*} \\to \\struct {R, d} : \\map \\iota x = x^{-1}$  is continuous."}
+{"_id": "16641", "title": "Intersection Operation on Supersets of Subset is Closed", "text": "Let $S$ be a set. Let $T \\subseteq S$ be a given subset of $S$. Let $\\powerset S$ denote the power set of $S$ Let $\\mathscr S$ be the subset of $\\powerset S$ defined as: :$\\mathscr S = \\set {Y \\in \\powerset S: T \\subseteq Y}$ Then the algebraic structure $\\struct {\\mathscr S, \\cap}$ is closed."}
+{"_id": "259", "title": "Composite of Quotient Mappings", "text": "Let $S$ be a set. Let $\\mathcal R_1$ be an equivalence on $S$, and $\\mathcal R_2$ be an equivalence on the quotient set $S / \\mathcal R_1$. We can find an equivalence $\\mathcal R_3$ on $S$ such that $\\paren {S / \\mathcal R_1} / \\mathcal R_2$ is in one-to-one correspondence with $S / \\mathcal R_3$ under the mapping: :$\\phi: \\paren {S / \\mathcal R_1} / \\mathcal R_2 \\to S / \\mathcal R_3: \\eqclass {\\eqclass x {\\mathcal R_1} } {\\mathcal R_2} \\mapsto \\eqclass x {\\mathcal R_3}$."}
+{"_id": "16648", "title": "Subgroup Generated by Commuting Elements is Abelian", "text": "Let $\\struct {G, \\circ}$ be a group. Let $S \\subseteq G$ such that: :$\\forall x, y \\in S: x \\circ y = y \\circ x$ Then the subgroup generated by $S$ is abelian."}
+{"_id": "16665", "title": "Stabilizer is Normal iff Stabilizer of Each Element of Orbit", "text": "Let $\\struct {G, \\circ}$ be a group. Let $S$ be a set. Let $*: G \\times S \\to S$ be a group action. Let $x \\in S$. Let $\\Stab x$ denote the stabilizer of $x$ under $*$. Let $\\Orb x$ denote the orbit of $x$ under $*$. Then $\\Stab x$ is normal in $G$ {{iff}} $\\Stab x$ is also the stabilizer of every element in $\\Orb x$."}
+{"_id": "16670", "title": "Additive Group of Integers is Normal Subgroup of Rationals", "text": "Let $\\struct {\\Z, +}$ be the additive group of integers. Let $\\struct {\\Q, +}$ be the additive group of rational numbers. Then $\\struct {\\Z, +}$ is a normal subgroup of $\\struct {\\Q, +}$."}
+{"_id": "8480", "title": "Empty Set and Set form Algebra of Sets", "text": "Let $S$ be any non-empty set. Then $\\left\\{{S, \\varnothing}\\right\\}$ is (trivially) an algebra of sets, where $S$ is the unit."}
+{"_id": "8481", "title": "Closure of Union and Complement imply Closure of Set Difference", "text": "Let $\\RR$ be a system of sets on a universe $\\mathbb U$ such that for all $A, B \\in \\RR$: :$(1): \\quad A \\cup B \\in \\RR$ :$(2): \\quad \\map \\complement A \\in \\RR$ where $\\cup$ denotes set union and $\\complement$ denotes complement (relative to $\\mathbb U$). Then: :$\\forall A, B \\in \\RR: A \\setminus B \\in \\RR$ where $\\setminus$ denotes set difference."}
+{"_id": "16676", "title": "Order of Alternating Group", "text": "Let $n \\in \\Z$ be an integer such that $n > 1$. Let $A_n$ be the alternating group on $n$ letters. Then: :$\\order {A_n} = \\dfrac {n!} 2$ where $\\order {A_n}$ denotes the order of $A_n$."}
+{"_id": "8488", "title": "Plane contains Infinite Number of Lines", "text": "A plane contains an infinite number of distinct lines."}
+{"_id": "8490", "title": "Three Non-Collinear Planes have One Point in Common", "text": "Three planes which are not collinear have exactly one point in all three planes."}
+{"_id": "8494", "title": "Union of Mappings which Agree is Mapping", "text": "Let $A, B, Y$ be sets. Let $f: A \\to Y$ and $g: B \\to Y$ be mappings. Let $X = A \\cup B$. Let $f$ and $g$ agree on $A \\cap B$. Then $f \\cup g: X \\to Y$ is a mapping."}
+{"_id": "307", "title": "Right Operation is Idempotent", "text": "The right operation is idempotent: :$\\forall x: x \\rightarrow x = x$"}
+{"_id": "16693", "title": "Index of Intersection of Subgroups/Corollary", "text": "Let $H$ be a subgroup of $G$. Let $K$ be a subgroup of finite index of $G$. Then: :$\\index H {H \\cap K} \\le \\index G K$"}
+{"_id": "16705", "title": "Centralizer of Self-Inverse Element of Non-Abelian Finite Simple Group is not That Group", "text": "Let $G$ be a non-abelian finite simple group. Let $t \\in G$ be a self-inverse element of $G$. Then: :$\\map {C_G} t \\ne G$ where $\\map {C_G} t$ denotes the centralizer of $t$ in $G$."}
+{"_id": "16715", "title": "Sum of Sequence of Squares of Fibonacci Numbers", "text": ":$\\forall n \\ge 1: \\displaystyle \\sum_{j \\mathop = 1}^n {F_j}^2 = F_n F_{n + 1}$ That is: :${F_1}^2 + {F_2}^2 + {F_3}^2 + \\cdots + {F_n}^2 = F_n F_{n + 1}$"}
+{"_id": "16725", "title": "Properties of Norm on Division Ring/Norm of Integer", "text": "For all $n \\in \\N_{\\gt 0}$, let $n \\cdot 1_R$ denote the sum of $1_R$ with itself $n$-times. That is: :$n \\cdot 1_R = \\underbrace {1_R + 1_R + \\dots + 1_R}_{n \\, times}$ Then: :$\\norm {n \\cdot 1_R} \\le n$."}
+{"_id": "359", "title": "Equivalence Relation is Congruence for Constant Operation", "text": "Every equivalence relation is a congruence relation for the constant operation."}
+{"_id": "361", "title": "Quotient Structure on Subset Product", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure. Let $\\mathcal R$ be a congruence for $\\circ$ on $S$. Then: :$\\forall X, Y \\in S / \\mathcal R: X \\circ_\\mathcal P Y \\subseteq X \\circ_\\mathcal R Y$ where: : $S / \\mathcal R$ is the quotient of $S$ by $\\mathcal R$ : $\\circ_\\mathcal P$ is the operation induced on $\\mathcal P \\left({S}\\right)$ by $\\circ$ : $\\circ_\\mathcal R$ is the operation induced on $S / \\mathcal R$ by $\\circ$"}
+{"_id": "16749", "title": "Even Power of 3 as Sum of Consecutive Positive Integers", "text": "Take the positive integers and group them in sets such that the $n$th set contains the next $3^n$ positive integers: :$\\set 1, \\set {2, 3, 4}, \\set {5, 6, \\ldots, 13}, \\set {14, 15, \\cdots, 40}, \\ldots$ Let the $n$th such set be denoted $S_{n - 1}$, that is, letting $S_0 := \\set 1$ be considered as the zeroth. Then the sum of all the elements of $S_n$ is $3^{2 n}$."}
+{"_id": "8564", "title": "Exterior of Union of Singleton Rationals is Empty", "text": "Let $B_\\alpha$ be the singleton containing the rational number $\\alpha$. Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology $\\tau_d$. Then the exterior in $\\struct {\\R, \\tau_d}$ of the union of all $B_\\alpha$ is the empty set: :$\\displaystyle \\paren {\\bigcup_{\\alpha \\mathop \\in \\Q} B_\\alpha}^e = \\O$"}
+{"_id": "16773", "title": "Smallest 18 Primes in Arithmetic Sequence", "text": "The smallest $18$ primes in arithmetic sequence are: :$107\\,928\\,278\\,317 + 9\\,922\\,782\\,870 n$ for $n = 0, 1, \\ldots, 16$."}
+{"_id": "8582", "title": "Irrational Number Space is Non-Meager", "text": "Let $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\R \\setminus \\Q, \\tau_d}$ is non-meager."}
+{"_id": "394", "title": "Quotient Structure of Inverse Completion", "text": "Let $\\left({T, \\circ'}\\right)$ be an inverse completion of a commutative semigroup $\\left({S, \\circ}\\right)$, where $C$ is the set of cancellable elements of $S$. Let $f: S \\times C: T$ be the mapping defined as: :$\\forall x \\in S, y \\in C: f \\left({x, y}\\right) = x \\circ' y^{-1}$ Then the mapping $g: \\left({S \\times C}\\right) / \\mathcal R_f \\to T$ defined by $g \\left({\\left[\\!\\left[{x, y}\\right]\\!\\right]_{\\mathcal R_f}}\\right) = x \\circ' y^{-1}$, where $\\left({S \\times C}\\right) / \\mathcal R_f$ is a quotient structure, is an isomorphism."}
+{"_id": "16785", "title": "Square Matrices with +1 or -1 Determinant under Multiplication forms Group", "text": "Let $n \\in \\Z_{> 0}$ be a strictly positive integer. Let $S$ be the set of square matrices of order $n$ of real numbers whose determinant is either $1$ or $-1$. Let $\\struct {S, \\times}$ denote the algebraic structure formed by $S$ whose operation is (conventional) matrix multiplication. Then $\\struct {S, \\times}$ is a group."}
+{"_id": "8593", "title": "Rational Number Space is not Weakly Sigma-Locally Compact", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\Q, \\tau_d}$ is not weakly $\\sigma$-locally compact."}
+{"_id": "16792", "title": "Group of Order 3 is Unique", "text": "There exists exactly $1$ group of order $3$, up to isomorphism: :$C_3$, the cyclic group of order $3$."}
+{"_id": "8606", "title": "Integer Reciprocal Space is Topological Space", "text": "Let $\\struct {\\R, \\tau_d}$ be the real number line $\\R$ under the usual (Euclidean) topology $\\tau_d$. Let $A \\subseteq \\R$ be the set of all points on $\\R$ defined as: :$A := \\set {\\dfrac 1 n: n \\in \\Z_{>0} }$ Then the integer reciprocal space $\\struct {A, \\tau_d}$ is a topological space."}
+{"_id": "8619", "title": "Integer Reciprocal Space with Zero is not Extremally Disconnected", "text": "Let $A \\subseteq \\R$ be the set of all points on $\\R$ defined as: :$A := \\set 0 \\cup \\set {\\dfrac 1 n : n \\in \\Z_{>0} }$ Let $\\struct {A, \\tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology. Then $A$ is not extremally disconnected."}
+{"_id": "8617", "title": "Components of Integer Reciprocal Space with Zero are Single Points", "text": "Let $A \\subseteq \\R$ be the set of all points on $\\R$ defined as: :$A := \\set 0 \\cup \\set {\\dfrac 1 n : n \\in \\Z_{>0} }$ Let $\\struct {A, \\tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology. Then the components of $A$ are singletons."}
+{"_id": "8637", "title": "Non-Homeomorphic Sets may be Homeomorphic to Subsets of Each Other", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ and $T_2 = \\struct {S_2, \\tau_2}$ be topological spaces. Let $H_1 \\subseteq S_1$ and $H_2 \\subseteq S_2$. Then it is possible for: :$(1): \\quad T_1$ to be homeomorphic to $H_2$ :$(2): \\quad T_2$ to be homeomorphic to $H_1$ but: :$(3): \\quad T_1$ and $T_2$ to not be homeomorphic."}
+{"_id": "8638", "title": "Superspace of Homeomorphic Subspaces may not have Homeomorphism to Itself containing Subspace Homeomorphism", "text": "Let $T_1 = \\struct {S_1, \\tau_1}$ and $T_2 = \\struct {S_2, \\tau_2}$ be topological spaces. Let $H_1 \\subseteq S_1$ and $H_2 \\subseteq S_2$. Let $H_1$ and $H_2$ be a homeomorphic. Then it may be the case that there does not exist a homeomorphism $g: T_1 \\to T_2$ such that: :$g \\restriction_{H_1} = f$ where: :$g \\restriction_{H_1}$ is the restriction of $g$ to $H_1$ :$f: H_1 \\to H_2$ is a homeomorphism."}
+{"_id": "8640", "title": "Finite Group is p-Group iff Order is Power of p", "text": "Let $p$ be a prime number. Let $G$ be a finite group. Then $G$ is a $p$-group {{iff}} the order of $G$ is a power of $p$."}
+{"_id": "448", "title": "Trivial Ring from Abelian Group", "text": "Any abelian group $\\struct {G, +}$ may be turned into a trivial ring by defining the ring product to be: :$\\forall x, y \\in G: x \\circ y = e_G$"}
+{"_id": "16837", "title": "Product with Inverse on Homomorphic Image is Group Homomorphism", "text": "Let $G$ be a group. Let $H$ be an abelian group. Let $\\theta: G \\to H$ be a (group) homomorphism. Let $\\phi: G \\times G \\to H$ be the mapping defined as: :$\\forall \\tuple {g_1, g_2} \\in G \\times G: \\map \\phi {g_1, g_2} = \\map \\theta {g_1} \\map \\theta {g_2}^{-1}$ Then $\\phi$ is a homomorphism."}
+{"_id": "8645", "title": "Divisor of Product may not be Divisor of Factors", "text": "Let $a, b, c \\in \\Z_{>0}$ be (strictly) positive integers. Let: :$c \\divides a b$ where $\\divides$ expresses the relation of divisibility. Then it is not necessarily the case that either $c \\divides a$ or $c \\divides b$."}
+{"_id": "8671", "title": "Equality of Complex Numbers", "text": "Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers. Then $z_1 = z_2$ {{iff}} $a_1 = a_2$ and $b_1 = b_2$."}
+{"_id": "482", "title": "Divisor of Unit is Unit", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose unity is $1_D$. Let $\\struct {U_D, \\circ}$ be the group of units of $\\struct {D, +, \\circ}$. Then: :$x \\in D, u \\in U_D: x \\divides u \\implies x \\in U_D$ That is, if $x$ is a divisor of a unit, $x$ must itself be a unit."}
+{"_id": "483", "title": "Associatehood is Equivalence Relation", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $\\sim$ be the relation defined on $D$ as: $\\forall x, y \\in D: x \\sim y$ {{iff}} $x$ is an associate of $y$ Then $\\sim$ is an equivalence relation."}
+{"_id": "490", "title": "Field is Subfield of Itself", "text": "Let $\\struct {F, +, \\circ}$ be a field. Then $\\struct {F, +, \\circ}$ is a subfield of $\\struct {F, +, \\circ}$."}
+{"_id": "8693", "title": "Tangent Exponential Formulation/Formulation 1", "text": ":$\\tan z = i \\dfrac {1 - e^{2 i z} } {1 + e^{2 i z} }$"}
+{"_id": "8695", "title": "Tangent Exponential Formulation/Formulation 3", "text": ":$\\tan z = -i \\paren {\\dfrac {e^{i z} - e^{-i z} } {e^{i z} + e^{-i z} } }$"}
+{"_id": "8705", "title": "Roots of Complex Number/Exponential Form", "text": ":$z^{1 / n} = \\set {r^{1 / n} e^{i \\paren {\\theta + 2 \\pi k} / n}: k \\in \\set {0, 1, 2, \\ldots, n - 1} }$"}
+{"_id": "523", "title": "Normalizer of Center is Group", "text": "Let $G$ be a group. Let $\\map Z G$ be the center of $G$. Let $x \\in G$. Let $\\map {N_G} x$ be the normalizer of $x$ in $G$. Then: :$\\map Z G = \\set {x \\in G: \\map {N_G} x = G}$ That is, the center of a group $G$ is the set of elements $x$ of $G$ such that the normalizer of $x$ is the whole of $G$. Thus: :$x \\in \\map Z G \\iff \\map {N_G} x = G$ and so: :$\\index G {\\map {N_G} x} = 1$ where $\\index G {\\map {N_G} x}$ is the index of $\\map {N_G} x$ in $G$."}
+{"_id": "16908", "title": "Characterisation of Non-Archimedean Division Ring Norms/Corollary 4", "text": "Let $R$ have characteristic $p > 0$. Then $\\norm {\\,\\cdot\\,}$ is a non_Archimedean norm on $R$."}
+{"_id": "16912", "title": "Relations with Combinations of Reflexivity, Symmetry and Transitivity Properties", "text": "Let $S$ be a set which has at least $3$ elements. Then it is possible to set up a relation $\\circledcirc$ on $S$ which has any combination of the $3$ properties: :Reflexivity :Symmetry :Transitivity but this is not possible for a set which has fewer than $3$ elements."}
+{"_id": "8725", "title": "Half Angle Formulas/Hyperbolic Tangent/Corollary 1", "text": ":$\\tanh \\dfrac x 2 = \\dfrac {\\sinh x} {\\cosh x + 1}$"}
+{"_id": "16918", "title": "Even Integer Plus 5 is Odd", "text": "Let $x \\in \\Z$ be an even integer. Then $x + 5$ is odd."}
+{"_id": "16938", "title": "Minimal Smooth Surface Spanned by Contour", "text": "Let $\\map z {x, y}: \\R^2 \\to \\R$ be a real-valued function. Let $\\Gamma$ be a closed contour in $3$-dimensional Euclidean space. Suppose this surface is smooth for every $x$ and $y$. Then it has to satisfy the following Euler's equation: :$r \\paren {1 + q^2} - 2 s p q + t \\paren {1 + p^2} = 0$ where: {{begin-eqn}} {{eqn | l = p       | r = z_x }} {{eqn | l = q       | r = z_y }} {{eqn | l = r       | r = z_{xx} }} {{eqn | l = s       | r = z_{xy} }} {{eqn | l = t       | r = z_{yy} }} {{end-eqn}} with subscript denoting respective partial derivatives. In other words, its mean curvature has to vanish."}
+{"_id": "560", "title": "Principle of Induction applied to Interval of Naturally Ordered Semigroup", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a naturally ordered semigroup. Let $\\left[{p \\,.\\,.\\, q}\\right]$ be a closed interval of $\\left({S, \\circ, \\preceq}\\right)$. Let $T \\subseteq \\left[{p \\,.\\,.\\, q}\\right]$ such that the minimal element of $\\left[{p \\,.\\,.\\, q}\\right]$ is in $T$. Let: : $x \\in T: x \\prec q \\implies x \\circ 1 \\in T$ Then: : $T = \\left[{p \\,.\\,.\\, q}\\right]$"}
+{"_id": "16944", "title": "Lowest Common Multiple of Integers with Common Divisor", "text": "Let $b, d \\in \\Z_{>0}$ be (strictly) positive integers Then: :$\\lcm \\set {a b, a d} = a \\lcm \\set {b, d}$ where: :$a \\in \\Z_{>0}$ :$\\lcm \\set {b, d}$ denotes the lowest common multiple of $m$ and $n$."}
+{"_id": "16946", "title": "GCD of Sum and Difference of Integers", "text": ":$\\gcd \\set {a + b, a - b} \\ge \\gcd \\set {a, b}$"}
+{"_id": "8757", "title": "Inverse Hyperbolic Cotangent is Odd Function", "text": ":$\\map {\\coth^{-1} } {-x} = -\\coth^{-1} x$"}
+{"_id": "8783", "title": "Inverse Cosine of Imaginary Number", "text": ":$\\cos^{-1} x = \\pm \\, i \\cosh^{-1} x$"}
+{"_id": "8785", "title": "Inverse Tangent of Imaginary Number", "text": ":$\\tan^{-1} \\left({i x}\\right) = i \\tanh^{-1} x$"}
+{"_id": "16982", "title": "LCM of 3 Integers in terms of GCDs of Pairs of those Integers", "text": "Let $a, b, c \\in \\Z_{>0}$ be strictly positive integers. Then: :$\\lcm \\set {a, b, c} = \\dfrac {a b c \\gcd \\set {a, b, c} } {d_1 d_2 d_3}$ where: :$\\gcd$ denotes greatest common divisor :$\\lcm$ denotes lowest common multiple :$d_1 = \\gcd \\set {a, b}$ :$d_2 = \\gcd \\set {b, c}$ :$d_3 = \\gcd \\set {a, c}$"}
+{"_id": "599", "title": "Integer Multiplication is Well-Defined", "text": "Integer multiplication is well-defined."}
+{"_id": "8790", "title": "Inverse Hyperbolic Secant of Imaginary Number", "text": ":$\\sech^{-1} x = \\pm \\, i \\sec^{-1} x$"}
+{"_id": "16987", "title": "Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient", "text": "Let $F_n$ denote the $n$th Fibonacci number. Then: {{begin-eqn}} {{eqn | lo= \\forall n \\in \\Z_{>0}:       | l = F_{2 n}       | r = \\sum_{k \\mathop = 1}^n \\dbinom n k F_k       | c =  }} {{eqn | r = \\dbinom n 1 F_1 + \\dbinom n 2 F_2 + \\dbinom n 3 F_3 + \\dotsb + \\dbinom n {n - 1} F_{n - 1} + \\dbinom n n F_n       | c =  }} {{end-eqn}} where $\\dbinom n k$ denotes a binomial coefficient."}
+{"_id": "8800", "title": "Cardano's Formula/Real Coefficients", "text": ":$(1): \\quad$ If $D > 0$, then one root is real and two are complex conjugates. :$(2): \\quad$ If $D = 0$, then all roots are real, and at least two are equal. :$(3): \\quad$ If $D < 0$, then all roots are real and unequal."}
+{"_id": "8801", "title": "Cardano's Formula/Trigonometric Form", "text": "Let $a, b, c, d \\in \\R$. Let the discriminant $D < 0$, where $D := Q^3 + R^2$. Then the solutions of $P$ can be expressed as: :$x_1 = 2 \\sqrt {-Q} \\map \\cos {\\dfrac \\theta 3} - \\dfrac b {3 a}$ :$x_2 = 2 \\sqrt {-Q} \\map \\cos {\\dfrac \\theta 3 + \\dfrac {2 \\pi} 3} - \\dfrac b {3 a}$ :$x_3 = 2 \\sqrt {-Q} \\map \\cos {\\dfrac \\theta 3 + \\dfrac {4 \\pi} 3} - \\dfrac b {3 a}$ where: : $\\cos \\theta = \\dfrac R {\\sqrt{-Q^3} }$"}
+{"_id": "8809", "title": "Triangle Inequality/Complex Numbers/General Result", "text": "Let $z_1, z_2, \\dotsc, z_n \\in \\C$ be complex numbers. Let $\\cmod z$ be the modulus of $z$. Then: :$\\cmod {z_1 + z_2 + \\dotsb + z_n} \\le \\cmod {z_1} + \\cmod {z_2} + \\dotsb + \\cmod {z_n}$"}
+{"_id": "8821", "title": "Substitution for Equivalent Subformula is Equivalent", "text": "Let $\\mathbf B$ a WFF of propositional logic. Let $\\mathbf A, \\mathbf A'$ be equivalent WFFs. Let $\\mathbf A$ be a subformula of $\\mathbf B$. Let $\\mathbf B' = \\mathbf B \\left({\\mathbf A \\,//\\, \\mathbf A'}\\right)$ be the substitution of $\\mathbf A'$ for $\\mathbf A$ in $\\mathbf B$. Then $\\mathbf B$ and $\\mathbf B'$ are equivalent."}
+{"_id": "17017", "title": "Congruence Modulo Power of p as Linear Combination of Congruences Modulo p", "text": "Let $p$ be a prime number. Let $S = \\set {a_1, a_2, \\ldots, a_p}$ be a complete residue system modulo $p$. Then for all integers $n \\in \\Z$ and non-negative integer $s \\in \\Z_{\\ge 0}$, there exists a congruence of the form: :$n \\equiv \\displaystyle \\sum_{j \\mathop = 0}^s b_j p^j \\pmod {p^{s + 1} }$ where $b_j \\in S$."}
+{"_id": "17023", "title": "Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 62 Cards", "text": "Let $D$ be a deck of $62$ cards. Let $D$ be given a sequence of modified perfect faro shuffles. Then after $6$ such shuffles, the cards of $D$ will be in the same order they started in."}
+{"_id": "17025", "title": "Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 12 Cards", "text": "Let $D$ be a deck of $12$ cards. Let $D$ be given a sequence of modified perfect faro shuffles. Then after $6$ such shuffles, the cards of $D$ will be in the same order they started in."}
+{"_id": "8842", "title": "Exclusive Or with Tautology", "text": ":$p \\oplus \\top \\dashv \\vdash \\neg p$"}
+{"_id": "17038", "title": "Difference of Two Even-Times Odd Powers", "text": "Let $\\F$ be one of the standard number systems, that is $\\Z, \\Q, \\R$ and so on. Let $n \\in \\Z_{> 0}$ be a (strictly) positive odd integer. Then: {{begin-eqn}} {{eqn | l = a^{2 n} - b^{2 n}       | r = \\paren {a - b} \\paren {a + b} \\paren {\\sum_{j \\mathop = 0}^{n - 1} a^{n - j - 1} b^j} \\paren {\\sum_{j \\mathop = 0}^{n - 1} \\paren {-1}^j a^{n - j - 1} b^j}       | c =  }} {{eqn | r = \\paren {a - b} \\paren {a + b} \\paren {a^{n - 1} + a^{n - 2} b + a^{n - 3} b^2 + \\dotsb + a b^{n - 2} + b^{n - 1} } \\paren {a^{n - 1} - a^{n - 2} b + a^{n - 3} b^2 - \\dotsb - a b^{n - 2} + b^{n - 1} }       | c =  }} {{end-eqn}}"}
+{"_id": "17046", "title": "Existence of Real Polynomial with no Real Root", "text": "There exist polynomials in real numbers $\\R$ which have no roots in $\\R$."}
+{"_id": "673", "title": "GCD from Congruence Modulo m", "text": "Let $a, b \\in \\Z, m \\in \\N$.   Let $a$ be congruent to $b$ modulo $m$. Then the GCD of $a$ and $m$ is equal to the GCD of $b$ and $m$. That is: :$a \\equiv b \\pmod m \\implies \\gcd \\set {a, m} = \\gcd \\set {b, m}$"}
+{"_id": "8877", "title": "Equivalence of Definitions of Closed Set in Metric Space", "text": "{{TFAE|def = Closed Set (Metric Space)|view = Closed Set|context = Metric Space|contextview = Metric Spaces}} Let $M = \\left({A, d}\\right)$ be a metric space. Let $H \\subseteq A$."}
+{"_id": "8881", "title": "Bolzano-Weierstrass Theorem/General Form", "text": "Every infinite bounded space in a real Euclidean space has at least one limit point."}
+{"_id": "693", "title": "Sum of Euler Phi Function over Divisors", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive integer. Then $\\displaystyle \\sum_{d \\mathop \\divides n} \\map \\phi d = n$ where: :$\\displaystyle \\sum_{d \\mathop \\divides n}$ denotes the sum over all of the divisors of $n$ :$\\map \\phi d$ is the Euler $\\phi$ function, the number of integers less than $d$ that are prime to $d$. That is, the total of all the totients of all divisors of a number equals that number."}
+{"_id": "17078", "title": "Valuation Ideal is Maximal Ideal of Induced Valuation Ring/Corollary 1", "text": ":$\\OO$ is a local ring."}
+{"_id": "17089", "title": "Bessel's Correction", "text": "Let $X_1, X_2, \\ldots, X_n$ form a random sample from a population with mean $\\mu$ and variance $\\sigma^2$. Let:  :$\\displaystyle \\bar X = \\frac 1 n \\sum_{i \\mathop = 1}^n X_i$ Then: :$\\displaystyle \\hat {\\sigma^2} = \\frac 1 {n - 1} \\sum_{i \\mathop = 1}^n \\paren {X_i - \\bar X}^2$ is an unbiased estimator of $\\sigma^2$."}
+{"_id": "8898", "title": "Sum of Cosines of Fractions of Pi", "text": "Let $n \\in \\Z$ such that $n > 1$. Then: :$\\displaystyle \\sum_{k \\mathop = 1}^{n - 1} \\cos \\frac {2 k \\pi} n = -1$"}
+{"_id": "17105", "title": "Area of Quadrilateral in Determinant Form", "text": "Let $A = \\tuple {x_1, y_1}$, $B = \\tuple {x_2, y_2}$, $C = \\tuple {x_3, y_3}$ and $D = \\tuple {x_4, y_4}$ be points in the Cartesian plane. Let $A$, $B$, $C$ and $D$ form the vertices of a quadrilateral. The area $\\mathcal A$ of $\\Box ABCD$ is given by: :$\\mathcal A = \\dfrac 1 2 \\paren {\\size {\\paren {\\begin{vmatrix} x_1 & y_1 & 1 \\\\ x_2 & y_2 & 1 \\\\ x_3 & y_3 & 1 \\\\ \\end{vmatrix} } } + \\size {\\paren {\\begin{vmatrix} x_1 & y_1 & 1 \\\\ x_4 & y_4 & 1 \\\\ x_3 & y_3 & 1 \\\\ \\end{vmatrix} } } }$"}
+{"_id": "17117", "title": "Equation for Perpendicular Bisector of Two Points", "text": "Let $\\tuple {x_1, y_1}$ and $\\tuple {y_1, y_2}$ be two points in the cartesian plane. Let $L$ be the perpendicular bisector of the straight line through $z_1$ and $z_2$ in the complex plane. $L$ can be expressed by the equation: :$y - \\dfrac {y_1 + y_2} 2 = \\dfrac {x_1 - x_2} {y_2 - y_1} \\paren {x - \\dfrac {x_1 + x_2} 2}$"}
+{"_id": "17122", "title": "Finite Order Elements of Infinite Abelian Group form Normal Subgroup", "text": "Let $G$ be an infinite  abelian group. Let $H \\subseteq G$ be the subset of $G$ defined as: :$H := \\set {x \\in G: x \\text { is of finite order in } G}$ Then $H$ forms a normal subgroup of $G$."}
+{"_id": "17131", "title": "Subgroup of Index 3 does not necessarily contain all Cubes of Group Elements", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$ whose index is $3$. Then it is not necessarily the case that: :$\\forall x \\in G: x^3 \\in H$"}
+{"_id": "8951", "title": "Complex Cross Product Distributes over Addition", "text": "Let $z_1, z_2, z_3 \\in \\C$ be complex numbers. Then: :$z_1 \\times \\paren {z_2 + z_3} = z_1 \\times z_2 + z_1 \\times z_3$ where $\\times$ denotes cross product."}
+{"_id": "8954", "title": "Finite Union of Open Sets in Complex Plane is Open", "text": "Let $S_1, S_2, \\ldots, S_n$ be open sets of $\\C$. Then $\\displaystyle \\bigcup_{k \\mathop = 1}^n S_k$ is an open set of $\\C$."}
+{"_id": "8955", "title": "Limit Point of Set in Complex Plane not Element is Boundary Point", "text": "Let $S \\subseteq \\C$ be a subset of the complex plane. Let $z \\in \\C$ be a limit point of $S$ such that $z \\notin S$. Then $z$ is a boundary point of $S$."}
+{"_id": "768", "title": "Order of Isomorphic Image of Group Element", "text": "Let $G$ and $H$ be groups whose identities are $e_G$ and $e_H$. Let $\\phi: G \\to H$ be a group isomorphism. Then: :$a \\in G \\implies \\order {\\map \\phi a} = \\order a$"}
+{"_id": "17151", "title": "Ring is Subring of Itself", "text": "Let $R$ be a ring. Then $R$ is a subring of itself."}
+{"_id": "776", "title": "Cyclic Group Elements whose Powers equal Identity", "text": "Let $G$ be a cyclic group whose identity is $e$ and whose order is $n$. Let $d \\divides n$. Then there exist exactly $d$ elements $x \\in G$ satisfying the equation $x^d = e$. These are the elements of the group $G_d$ generated by $g^{n / d}$:  :$G_d = \\gen {g^{n / d} }$"}
+{"_id": "8971", "title": "Conditional iff Biconditional of Antecedent with Conjunction", "text": ":$p \\implies q \\dashv \\vdash p \\iff \\left({p \\land q}\\right)$"}
+{"_id": "17164", "title": "Element in Integral Domain is Unit iff Principal Ideal is Whole Domain", "text": ":$x \\in U_D \\iff \\ideal x = D$"}
+{"_id": "8973", "title": "Conjunction iff Biconditional of Biconditional with Disjunction", "text": ":$p \\land q \\dashv \\vdash \\left({p \\iff q}\\right) \\iff \\left({p \\lor q}\\right)$"}
+{"_id": "785", "title": "Group Generated by Normal Intersection is Normal", "text": "Let $I$ be an indexing set, and $\\left\\{{N_i: i \\in I}\\right\\}$ be a set of normal subgroups of the group $G$. Then $\\left \\langle {N_i: i \\in I} \\right \\rangle$ is a normal subgroup of $G$."}
+{"_id": "796", "title": "Abelian Quotient Group", "text": "Let $G$ be a group. Let $H$ be a normal subgroup of $G$. Let $G / H$ denote the quotient group of $G$ by $H$. Then $G / H$ is abelian {{iff}} $H$ contains every element of $G$ of the form $a b a^{-1} b^{-1}$ where $a, b \\in G$."}
+{"_id": "798", "title": "Generator of Quotient Groups", "text": "Let $N \\lhd G$ be a normal subgroup of $G$. Let: :$N \\le A \\le G$ :$N \\le B \\le G$ For a subgroup $H$ of $G$, let $\\alpha$ be the bijection defined as: :$\\map \\alpha H = \\set {h N: h \\in H}$ Then: :$\\map \\alpha {\\gen {A, B} } = \\gen {\\map \\alpha A, \\map \\alpha B}$ where $\\gen {A, B}$ denotes the subgroup of $G$ generated by $\\set {A, B}$."}
+{"_id": "808", "title": "Inner Automorphism Group is Isomorphic to Quotient Group with Center", "text": "Let $G$ be a group. Let $\\Inn G$ be the inner automorphism group of $G$. Let $\\map Z G$ be the center of $G$. Let $G / \\map Z G$ be the quotient group of $G$ by $\\map Z G$. Then $G / \\map Z G \\cong \\Inn G$."}
+{"_id": "9001", "title": "Real Natural Logarithm is Restriction of Complex Natural Logarithm", "text": "Let $\\ln: \\C_{\\ne 0} \\to \\C$ be the complex natural logarithm. Let $\\ln': \\R_{>0} \\to \\R$ be the real natural logarithm. Then: :$\\ln' = \\ln \\restriction_{\\R_{>0} \\times \\R}$ That is, the real natural logarithm is the restriction of the complex natural logarithm."}
+{"_id": "17202", "title": "Inverse of Central Unit of Ring is in Center", "text": "Let $R$ be a ring. Let $\\map Z R$ denote the center of $R$. Let $u \\in R$ be a unit of $R$. Then: :$u \\in \\map Z R \\implies u^{-1} \\in \\map Z R$"}
+{"_id": "821", "title": "Commutativity of Group Direct Product", "text": "Let $\\struct {G, \\circ_g}$ and $\\struct {H, \\circ_h}$ be groups. Let $\\struct {G \\times H, \\circ}$ be the group direct product of $\\struct {G, \\circ_g}$ and $\\struct {H, \\circ_h}$, where the operation $\\circ$ is defined as: :$\\tuple {g_1, h_1} \\circ \\tuple {g_2, h_2} = \\tuple {g_1 \\circ_g g_2, h_1 \\circ_h h_2}$ Let $\\struct {H \\times G, \\star}$ be the group direct product of $\\struct {H, \\circ_h}$ and $\\struct {G, \\circ_g}$, where the operation $\\star$ is defined as: :$\\tuple {h_1, g_1} \\star \\tuple {h_2, g_2} = \\tuple {h_1 \\circ_h h_2, g_1 \\circ_g g_2}$ The group direct product $\\struct {G \\times H, \\circ}$ is isomorphic to the $\\struct {H \\times G, \\star}$."}
+{"_id": "823", "title": "Associativity of Group Direct Product", "text": "The group direct product $G \\times \\paren {H \\times K}$ is (group) isomorphic to $\\paren {G \\times H} \\times K$."}
+{"_id": "833", "title": "Cyclic Group of Order 6", "text": "Let $C_n$ be the cyclic group of order $n$. Then: : $C_2 \\times C_3 \\cong C_6$ : $C_6$ is the internal group direct product of $C_2$ and $C_3$."}
+{"_id": "837", "title": "Direct Product of Central Subgroup with Inverse Isomorphism is Central Subgroup", "text": "Let $G$ and $H$ be groups. Let $\\map Z G$ denote the center of $G$. Let $Z$ and $W$ be central subgroups of $G$ and $H$ respectively. Let: :$Z \\cong W$ where $\\cong$ denotes isomorphism. Let such a group isomorphism be $\\theta: Z \\to W$. Let $X$ be the set defined as: :$X = \\set {\\tuple {x, \\map \\theta x^{-1} }: x \\in Z}$ Then $X$ is a central subgroup of $G \\times H$."}
+{"_id": "17224", "title": "Non-Commutative Ring with Unity and 2 Ideals not necessarily Division Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose zero is $0_F$ and whose unity is $1_F$. Let $\\struct {R, +, \\circ}$ specifically not be commutative. Let $\\struct {R, +, \\circ}$ be such that the only ideals of $\\struct {R, +, \\circ}$ are $\\set {0_R}$ and $R$ itself. Then it is not necessarily the case that $\\struct {R, +, \\circ}$ is a division ring."}
+{"_id": "9035", "title": "Equivalence of Definitions of Complex Inverse Hyperbolic Tangent", "text": "{{TFAE|def = Complex Inverse Hyperbolic Tangent}} Let $S$ be the subset of the complex plane: :$S = \\C \\setminus \\left\\{{-1 + 0 i, 1 + 0 i}\\right\\}$"}
+{"_id": "9037", "title": "Natural Numbers form Inductive Set", "text": "Let $\\N$ denote the natural numbers as subset of the real numbers $\\R$. Then $\\N$ is an inductive set."}
+{"_id": "851", "title": "Additive Group of Reals is Normal Subgroup of Complex", "text": "Let $\\struct {\\R, +}$ be the additive group of real numbers. Let $\\struct {\\C, +}$ be the additive group of complex numbers. Then $\\struct {\\R, +}$ is a normal subgroup of $\\struct {\\C, +}$."}
+{"_id": "857", "title": "Generators of Additive Group of Integers", "text": "The only generators of the additive group of integers $\\struct {\\Z, +}$ are $1$ and $-1$."}
+{"_id": "17244", "title": "Prime Power Mapping on Galois Field is Automorphism", "text": "Let $\\GF$ be a Galois field whose zero is $0_\\GF$ and whose characteristic is $p$. Let $\\sigma: \\GF \\to \\GF$ be defined as: :$\\forall x \\in \\GF: \\map \\sigma x = x^p$ Then $\\sigma$ is an automorphism of $\\GF$."}
+{"_id": "17246", "title": "Principal Ideal Domain cannot have Infinite Strictly Increasing Sequence of Ideals", "text": "Let $\\struct {D, +, \\circ}$ be a principal ideal domain. Then $D$ cannot have an infinite sequence of ideals $\\sequence {j_n}_{n \\mathop \\in \\N}$ such that: :$\\forall n \\in \\N: J_n \\subsetneq j_{n + 1}$"}
+{"_id": "17249", "title": "Field Norm of Complex Number is not Norm", "text": "Let $\\C$ denote the set of complex numbers. Let $N: \\C \\to \\R_{\\ge 0}$ denote the field norm on complex numbers: :$\\forall z \\in \\C: \\map N z = \\cmod z^2$ where $\\cmod z$ denotes the complex modulus of $z$. Then $N$ is not a norm on $\\C$."}
+{"_id": "874", "title": "Internal Angles of Square", "text": "The internal angles of a square are right angles."}
+{"_id": "17261", "title": "Ideal of Ring of Polynomials over Field has Unique Monic Polynomial forming Principal Ideal", "text": "Let $F$ be a field. Let $F \\sqbrk X$ be the ring of polynomials in $X$ over $F$. Let $J$ be a non-null ideal of $F \\sqbrk X$. Then there exists exactly one monic polynomial $f \\in F \\sqbrk X$ such that: :$J = \\ideal f$ where $\\ideal f$ is the principal ideal generated by $f$ in $F \\sqbrk X$."}
+{"_id": "878", "title": "Fixed Elements form 1-Cycles", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\pi \\in S_n$. Let $\\Fix \\pi$ be the set of elements fixed by $\\pi$. For any $\\pi \\in S_n$, all the elements of $\\Fix \\pi$ form $1$-cycles."}
+{"_id": "17263", "title": "Polynomial X^2 + 1 is Irreducible in Ring of Real Polynomials", "text": "Let $\\R \\sqbrk X$ be the ring of polynomials in $X$ over the real numbers $\\R$. Then the polynomial $X^2 + 1$ is an irreducible element of $\\R \\sqbrk X$."}
+{"_id": "9073", "title": "Equivalence of Definitions of Concave Real Function", "text": "Let $f$ be a real function which is defined on a real interval $I$. {{TFAE|def = Concave Real Function}}"}
+{"_id": "17267", "title": "Ring Subtraction equals Zero iff Elements are Equal", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$ Then: :$\\forall a, b \\in R: a - b = 0_R \\iff a = b$ where $a - b$ denotes ring subtraction."}
+{"_id": "884", "title": "Powers of Disjoint Permutations", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\rho, \\sigma$ be disjoint permutations. Then: : $\\forall k \\in \\Z: \\paren {\\sigma \\rho}^k = \\sigma^k \\rho^k$"}
+{"_id": "9078", "title": "Equivalence of Definitions of Strictly Concave Real Function", "text": "Let $f$ be a real function which is defined on a real interval $I$. {{TFAE|def = Strictly Concave Real Function}}"}
+{"_id": "9079", "title": "Equivalence of Definitions of Strictly Convex Real Function", "text": "Let $f$ be a real function which is defined on a real interval $I$. {{TFAE|def = Strictly Convex Real Function}}"}
+{"_id": "896", "title": "Group Action on Subgroup of Symmetric Group", "text": "Let $S_n$ be the symmetric group of $n$ elements. Let $H$ be a subgroup of $S_n$. Let $X$ be any set with $n$ elements. Then $H$ acts on $X$ as a group of transformations on $X$."}
+{"_id": "897", "title": "Kernel of Group Action is Normal Subgroup", "text": "Let $G$ be a group whose identity is $e$. Let $X$ be a set. Let $\\phi: G \\times X \\to X$ be a group action. Let $G_0$ denote the kernel of $\\phi$. Then $G_0$ is a normal subgroup of $G$."}
+{"_id": "17289", "title": "Equivalence of Definitions of Well-Ordered Integral Domain", "text": "{{TFAE|def = Well-Ordered Integral Domain}} Let $\\struct {D, +, \\times \\le}$ be an ordered integral domain whose zero is $0_D$."}
+{"_id": "17294", "title": "Second Principle of Finite Induction/Zero-Based", "text": "Let $S \\subseteq \\N$ be a subset of the natural numbers. Suppose that: :$(1): \\quad 0 \\in S$ :$(2): \\quad \\forall n \\in \\N: \\paren {\\forall k: 0 \\le k \\le n \\implies k \\in S} \\implies n + 1 \\in S$ Then: :$S = \\N$"}
+{"_id": "9104", "title": "Integral of Exponent of Half Square over Reals", "text": ":$\\displaystyle \\int_{\\mathop \\to -\\infty}^{\\mathop \\to +\\infty} e^{- x^2 / 2} \\rd x = \\sqrt {2 \\pi}$"}
+{"_id": "920", "title": "Cayley's Representation Theorem", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Every finite group is isomorphic to a subgroup of $S_n$ for some $n \\in \\Z$."}
+{"_id": "17310", "title": "Coprimality Relation is not Antisymmetric", "text": ":$\\perp$ is not antisymmetric."}
+{"_id": "17311", "title": "Coprimality Relation is Non-Transitive", "text": ":$\\perp$ is non-transitive."}
+{"_id": "932", "title": "Canonical Injection into Cartesian Product of Modules", "text": "Let $G$ be the cartesian product of a sequence $\\sequence {G_n}$ of $R$-modules. Then for each $j \\in \\closedint 1 n$, the canonical injection $\\inj_j$ from $G_j$ into $G$ is a monomorphism."}
+{"_id": "17318", "title": "Gaussian Integers does not form Subfield of Complex Numbers", "text": "The ring of Gaussian integers: :$\\struct {\\Z \\sqbrk i, +, \\times}$ is not a subfield of $\\C$."}
+{"_id": "17316", "title": "Even Integers form Commutative Ring", "text": "Let $2 \\Z$ be the set of even integers. Then $\\struct {2 \\Z, +, \\times}$ is a commutative ring. However, $\\struct {2 \\Z, +, \\times}$ is not an integral domain."}
+{"_id": "9142", "title": "Leibniz's Rule/One Variable/Second Derivative", "text": "Let $f$ and $g$ be real functions defined on the open interval $I$. Let $x \\in I$ be a point in $I$ at which both $f$ and $g$ are twice differentiable. Then: :$\\paren {\\map f x \\, \\map g x}'' = \\map f x \\, \\map {g''} x + 2 \\map {f'} x \\, \\map {g'} x + \\map {f''} x \\, \\map g x$"}
+{"_id": "17357", "title": "Vector Space on Cartesian Product is Vector Space", "text": "Let $\\struct {K, +, \\circ}$ be a division ring. Let $n \\in \\N_{>0}$. Let $\\struct {K^n, +, \\times}_K$ be the '''$K$-vector space $K^n$'''. Then $\\struct {K^n, +, \\times}_K$ is a $K$-vector space."}
+{"_id": "977", "title": "Ring of Linear Operators", "text": "Let $\\map {\\LL_R} G$ be the set of all linear operators on $G$. {{explain|the precise nature of $G$}} Let $\\phi \\circ \\psi$ denote the composition of the two linear operators $\\phi$ and $\\psi$. Then $\\struct {\\map {\\LL_R} G, +, \\circ}$ is a ring."}
+{"_id": "17362", "title": "No Non-Trivial Norm on Rational Numbers is Complete", "text": "No non-trivial norm on the set of the rational numbers is complete."}
+{"_id": "980", "title": "Inverse Evaluation Isomorphism of Annihilator", "text": "Let $R$ be a commutative ring.  Let $G$ be a module over $R$ whose dimension is finite. Let $G^*$ be the algebraic dual of $G$. Let $G^{**}$ be the algebraic dual of $G^*$. Let $N$ be a submodule of $G^*$. Let $J$ be the evaluation isomorphism from $G$ onto $G^{**}$. Let $N^\\circ$ be the annihilator of $N$. Then: :$J^{-1} \\left({N^\\circ}\\right) = \\left\\{{x \\in G: \\forall t' \\in N: t' \\left({x}\\right) = 0}\\right\\}$"}
+{"_id": "987", "title": "Complex Numbers form Vector Space over Reals", "text": "Let $\\R$ be the set of real numbers. Let $\\C$ be the set of complex numbers. Then the $\\R$-module $\\C$ is a vector space."}
+{"_id": "17377", "title": "Dimension of Vector Space on Cartesian Product", "text": "Let $\\struct {K, +, \\circ}$ be a division ring. Let $n \\in \\N_{>0}$. Let $\\mathbf V := \\struct {K^n, +, \\times}_K$ be the '''$K$-vector space $K^n$'''. Then the dimension of $\\mathbf V$ is $n$."}
+{"_id": "995", "title": "Finitely Generated Vector Space has Basis", "text": "Let $K$ be a division ring. Let $V$ be a finitely generated vector space over $K$. Then $V$ has a finite basis."}
+{"_id": "9197", "title": "Primitive of Hyperbolic Cosecant Function/Logarithm Form", "text": ":$\\displaystyle \\int \\csch x \\rd x = -\\ln \\size {\\csch x + \\coth x} + C$ where $\\csch x + \\coth x \\ne 0$."}
+{"_id": "9199", "title": "Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent Form", "text": ":$\\displaystyle \\int \\csch x \\rd x = -2 \\map {\\coth^{-1} } {e^x} + C$"}
+{"_id": "9210", "title": "Primitive of Product of Hyperbolic Secant and Tangent", "text": ":$\\ds \\int \\sech x \\tanh x \\rd x = -\\sech x + C$ where $C$ is an arbitrary constant."}
+{"_id": "17404", "title": "Existence of Subgroup whose Index is Prime Power", "text": "Let $G$ be a finite group. Let $H$ be a normal subgroup of $G$ which has a finite index in $G$. Let: :$p^k \\divides \\index G H$ where: :$p$ is a prime number :$k \\in \\Z_{>0}$ is a (strictly) positive integer :$\\divides$ denotes divisibility. Then $G$ contains a subgroup $K$ such that: :$\\index K H = p^k$"}
+{"_id": "1021", "title": "Condition for Planes to be Parallel", "text": "Let $P: \\alpha_1 x_1 + \\alpha_2 x_2 + \\alpha_3 x_3 = \\gamma$ be a plane in $\\R^3$. Then the plane $P'$ is parallel to $P$ iff there is a $\\gamma' \\in \\R$ such that: :$P' = \\left\\{{ \\left({x_1, x_2, x_3}\\right) \\in \\R^3 : \\alpha_1 x_1 + \\alpha_2 x_2 + \\alpha_3 x_3 = \\gamma' }\\right\\}$"}
+{"_id": "9214", "title": "Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2", "text": ":$\\ds \\int \\frac {\\d x} {x^2 - a^2} = \\frac 1 {2 a} \\ln \\size {\\frac {x - a} {x + a} } + C$"}
+{"_id": "1036", "title": "Invertible Matrix Corresponds with Change of Basis", "text": "Let $R$ be a commutative ring with unity. Let $G$ be an $n$-dimensional unitary $R$-module. Let $\\left \\langle {a_n} \\right \\rangle$ be an ordered basis of $G$. Let $\\mathbf P = \\left[{\\alpha}\\right]_{n}$ be a square matrix of order $n$ over $R$. Let $\\displaystyle \\forall j \\in \\left[{1 \\,.\\,.\\, n}\\right]: b_j = \\sum_{i \\mathop = 1}^n \\alpha_{i j} a_i$. Then $\\left \\langle {b_n} \\right \\rangle$ is an ordered basis of $G$ iff $\\mathbf P$ is invertible."}
+{"_id": "1039", "title": "Matrix Similarity is Equivalence Relation", "text": "Matrix similarity is an equivalence relation."}
+{"_id": "1040", "title": "Similar Matrices are Equivalent", "text": "If two square matrices over a ring with unity $R$ are similar, then they are equivalent. It follows directly that every equivalence class for the relation of similarity on $\\mathcal M_R \\left({n}\\right)$ is contained in an equivalence class for the relation of matrix equivalence. Here, $\\mathcal M_R \\left({n}\\right)$ denotes the $n \\times n$ matrix space over $R$."}
+{"_id": "1044", "title": "Rank is Dimension of Subspace", "text": "Let $K$ be a field. Let $\\mathbf A$ be an $m \\times n$ matrix over $K$. Then the rank of $\\mathbf A$ is the dimension of the subspace of $K^n$ generated by the rows of $\\mathbf A$."}
+{"_id": "9238", "title": "Primitive of Function of Nth Root of a x + b", "text": ":$\\displaystyle \\int F \\left({\\sqrt [n] {a x + b}}\\right) \\ \\mathrm d x = \\frac n a \\int u^{n-1} F \\left({u}\\right) \\ \\mathrm d u$ where $u = \\sqrt [n] {a x + b}$."}
+{"_id": "9240", "title": "Primitive of Function of Root of a squared plus x squared", "text": ":$\\displaystyle \\int F \\left({\\sqrt {a^2 + x^2}}\\right) \\ \\mathrm d x = a \\int \\sec^2 u \\ F \\left({a \\sec u}\\right) \\ \\mathrm d u$ where $x = a \\tan u$."}
+{"_id": "9243", "title": "Primitive of Function of Natural Logarithm", "text": ":$\\displaystyle \\int F \\left({\\ln x}\\right) \\rd x = \\int F \\left({u}\\right) e^u \\rd u$ where $u = \\ln x$."}
+{"_id": "9250", "title": "Primitive of Function of Arccosecant", "text": ":$\\displaystyle \\int F \\left({\\operatorname{arccsc} \\frac x a}\\right) \\ \\mathrm d x = -a \\int F \\left({u}\\right) \\left\\vert{\\csc u}\\right\\vert \\cot u \\ \\mathrm d u$ where $u = \\operatorname{arccsc} \\dfrac x a$."}
+{"_id": "17456", "title": "Subtraction of Subring is Subtraction of Ring", "text": "Let $\\struct {R, +, \\circ}$ be an ring. For each $x, y \\in R$ let $x - y$ denote the subtraction of $x$ and $y$ in $R$. Let $\\struct {S, + {\\restriction_S}, \\circ {\\restriction_S}}$ be a subring of $R$. For each $x, y \\in S$ let $x \\sim y$ denote the subtraction of $x$ and $y$ in $S$. Then: :$\\forall x, y \\in S: x \\sim y = x - y$"}
+{"_id": "17463", "title": "Area between Smooth Curve and Line is Maximized by Semicircle", "text": "Let $y$ be a smooth curve, embedded in $2$-dimensional Euclidean space. Let $y$ have a total length of $l$. Let it be contained in the upper half-plane with an exception of endpoints, which are on the $x$-axis. Suppose, $y$, together with a line segment connecting $y$'s endpoints, maximizes the enclosed area. Then $y$ is a semicircle."}
+{"_id": "1082", "title": "Unique Representation in Polynomial Forms", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\\struct {D, +, \\circ}$ be an integral subdomain of $R$. Let $X \\in R$ be transcendental over $D$. Let $D \\sqbrk X$ be the ring of polynomials in $X$ over $D$. Then each non-zero member of $D \\left[{X}\\right]$ can be expressed in just one way in the form: :$\\ds f \\in D \\sqbrk X: f = \\sum_{k \\mathop = 0}^n {a_k \\circ X^k}$"}
+{"_id": "1084", "title": "Rings of Polynomials in Ring Elements are Isomorphic", "text": "Let $R_1, R_2$ be commutative rings with unity. Let $D$ be an integral subdomain of both $R_1$ and $R_2$. Let $X_1, X_2 \\in R$ be transcendental over $D$. Let $D \\sqbrk {X_1}, D \\sqbrk {X_2}$ be the rings of polynomials in $X_1$ and $X_2$ over $D$. Then $D \\sqbrk {X_1}$ is isomorphic to $D \\sqbrk {X_2}$."}
+{"_id": "9278", "title": "Laplace Transform of Higher Order Derivatives", "text": "{{begin-eqn}} {{eqn | l = \\laptrans {\\map {f^{\\paren n} } t}       | r = s^n \\laptrans {\\map f t} - \\sum_{j \\mathop = 1}^n s^{j - 1} \\map {f^{\\paren {n - j} } } 0 }} {{eqn | r = s^n \\map F s - s^{n - 1} \\, \\map f 0 - s^{n - 2} \\, \\map {f'} 0 - s^{n - 3} \\, \\map {f''} 0 - \\ldots - s \\, \\map {f^{\\paren {n - 2} } } 0 - \\map {f^{\\paren {n - 1} } } 0 }} {{end-eqn}}"}
+{"_id": "17481", "title": "Laplace Transform of Error Function", "text": ":$\\laptrans {\\map \\erf t} = \\dfrac 1 s \\, \\map \\exp {\\dfrac {s^2} 4} \\, \\map \\erfc {\\dfrac s 2}$ where: :$\\laptrans f$ denotes the Laplace transform of the function $f$ :$\\erf$ denotes the error function :$\\erfc$ denotes the complementary error function :$\\exp$ denotes the exponential function."}
+{"_id": "17485", "title": "Laplace Transform of Exponential Integral Function", "text": ":$\\laptrans {\\map \\Ei t} = \\dfrac {\\map \\ln {s + 1} } s$ where: :$\\laptrans f$ denotes the Laplace transform of the function $f$ :$\\Ei$ denotes the exponential integral function."}
+{"_id": "17488", "title": "Laplace Transform of Shifted Dirac Delta Function", "text": "Let $\\map \\delta t$ denote the Dirac delta function. The Laplace transform of $\\map \\delta {t - a}$ is given by: :$\\laptrans {\\map \\delta {t - a} } = e^{-a s}$"}
+{"_id": "9301", "title": "Primitive of x over Root of a x + b", "text": ":$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\sqrt{a x + b} } = \\frac {2 \\left({a x - 2 b}\\right) \\sqrt{a x + b} } {3 a^2}$"}
+{"_id": "17495", "title": "Laplace Transform of t by Sine a t", "text": "Let $\\sin$ denote the real sine function. Let $\\laptrans f$ denote the Laplace transform of a real function $f$. Then: :$\\laptrans {t \\sin a t} = \\dfrac {2 a s} {\\paren {s^2 + a^2}^2}$"}
+{"_id": "17494", "title": "Laplace Transform of Sine of t over t/Corollary", "text": ":$\\laptrans {\\dfrac {\\sin a t} t} = \\arctan \\dfrac a s$"}
+{"_id": "17529", "title": "Laplace Transform of Natural Logarithm", "text": ":$\\laptrans {\\ln t} = \\dfrac {\\map {\\Gamma'} 1 - \\ln s} s = -\\dfrac {\\gamma + \\ln s} s$ where: :$\\laptrans f$ denotes the Laplace transform of the function $f$ :$\\Gamma$ denotes the Gamma function :$\\gamma$ denotes the Euler-Mascheroni constant."}
+{"_id": "17533", "title": "Convolution Theorem", "text": "Let $\\mathbb F \\in \\set {\\R, \\C}$.  Let $f: \\R \\to \\F$ and $g: \\R \\to \\F$ be functions. Let their Laplace transforms $\\laptrans {\\map f t} = \\map F s$ and $\\laptrans {\\map g t} = \\map G s$ exist. Then: :$\\map F s \\map G s = \\displaystyle \\laptrans {\\int_0^t \\map f u \\map g {t - u} \\rd u}$"}
+{"_id": "9352", "title": "Primitive of p x + q over Root of a x + b", "text": ":$\\displaystyle \\int \\frac {p x + q} {\\sqrt {a x + b} } \\rd x = \\frac {2 \\paren {a p x + 3 a q - 2 b p} } {3 a^2} \\sqrt{a x + b}$"}
+{"_id": "1206", "title": "Negative of Supremum is Infimum of Negatives", "text": "Let $S$ be a subset of the real numbers $\\R$. Let $S$ be bounded above. Then: :$(1): \\quad \\set {x \\in \\R: -x \\in S}$ is bounded below :$(2): \\quad \\displaystyle -\\sup_{x \\mathop \\in S} x = \\map {\\inf_{x \\mathop \\in S} } {-x}$ where $\\sup$ and $\\inf$ denote the supremum and infimum respectively."}
+{"_id": "17590", "title": "Propagation of Light in Inhomogeneous Medium", "text": "Let $v: \\R^3 \\to \\R$ be a real function. Let $M$ be a 3-dimensional Euclidean space. Let $\\gamma:t \\in \\R \\to M$ be a smooth curve embedded in $M$, where $t$ is time. Denote its derivative {{WRT}} time by $v$. Suppose $M$ is filled with an optically inhomogeneous medium such that at each point speed of light is $v = \\map v {x, y, z}$ Suppose $\\map y x$ and $\\map z x$ are real functions. Let the light move according to Fermat's principle. Then equations of motion have the following form: :$\\dfrac {\\partial v} {\\partial y} \\dfrac {\\sqrt {1 + y'^2 + z'^2} } {v^2} + \\dfrac \\d {\\d x} \\dfrac {y'} {v \\sqrt {1 + y'^2 + z'^2} } = 0$ :$\\dfrac {\\partial v} {\\partial z} \\dfrac {\\sqrt {1 + y'^2 + z'^2} } {v^2} + \\dfrac \\d {\\d x} \\dfrac {z'} {v \\sqrt {1 + y'^2 + z'^2} } = 0$"}
+{"_id": "9403", "title": "Primitive of Reciprocal of x by Power of x squared minus a squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x \\left({x^2 - a^2}\\right)^n} = \\frac {-1} {2 \\left({n - 1}\\right) a^2 \\left({x^2 - a^2}\\right)^{n - 1} } - \\frac 1 {a^2} \\int \\frac {\\mathrm d x} {x \\left({x^2 - a^2}\\right)^{n - 1} }$ for $x^2 > a^2$."}
+{"_id": "9405", "title": "Primitive of Reciprocal of Power of x by Power of x squared minus a squared", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^m \\paren {x^2 - a^2}^n} = \\frac 1 {a^2} \\int \\frac {\\d x} {x^{m - 2} \\paren {x^2 - a^2}^n} - \\frac 1 {a^2} \\int \\frac {\\d x} {x^m \\paren {x^2 - a^2}^{n - 1} }$ for $x^2 > a^2$."}
+{"_id": "9408", "title": "Signum Function is Quotient of Number with Absolute Value", "text": "Let $x \\in \\R_{\\ne 0}$ be a non-zero real number. Then: :$\\operatorname{sgn} \\left({x}\\right) = \\dfrac x {\\left\\vert{x}\\right\\vert} = \\dfrac {\\left\\vert{x}\\right\\vert} x$ where: :$\\operatorname{sgn} \\left({x}\\right)$ denotes the signum function of $x$ :$\\left\\vert{x}\\right\\vert$ denotes the absolute value of $x$."}
+{"_id": "17602", "title": "Local Basis Generated from Neighborhood Basis", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $x$ be an element of $S$. Let $\\BB$ be a neighborhood basis of $x$. For any subset $A \\subseteq S$, let $A^\\circ$ denote the interior of $A$. Then the set: :$\\BB' = \\set {H^\\circ: H \\in B}$ is a local basis of $x$."}
+{"_id": "17609", "title": "Excess Kurtosis of Bernoulli Distribution", "text": "Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$. Then the excess kurtosis $\\gamma_2$ of $X$ is given by:  :$\\gamma_2 = \\dfrac {1 - 6 p q} {p q}$ where $q = 1 - p$."}
+{"_id": "17612", "title": "Skewness of Beta Distribution", "text": "Let $X \\sim \\BetaDist \\alpha \\beta$ for some $\\alpha, \\beta > 0$, where $\\operatorname {Beta}$ denotes the Beta distribution.  Then the skewness $\\gamma_1$ of $X$ is given by:  :$\\gamma_1 = \\dfrac {2 \\paren {\\beta - \\alpha} \\sqrt {\\alpha + \\beta + 1} } {\\paren {\\alpha + \\beta + 2} \\sqrt {\\alpha \\beta} }$"}
+{"_id": "9423", "title": "Primitive of x over Power of a squared minus x squared", "text": ":$\\displaystyle \\int \\frac {x \\rd x} {\\paren {a^2 - x^2}^n} = \\frac 1 {2 \\paren {n - 1} \\paren {a^2 - x^2}^{n - 1} }$ for $x^2 < a^2$."}
+{"_id": "9424", "title": "Primitive of Reciprocal of x by Power of a squared minus x squared", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {x \\left({a^2 - x^2}\\right)^n} = \\frac 1 {2 \\left({n - 1}\\right) a^2 \\left({a^2 - x^2}\\right)^{n - 1} } + \\frac 1 {a^2} \\int \\frac {\\mathrm d x} {x \\left({a^2 - x^2}\\right)^{n - 1} }$ for $x^2 < a^2$."}
+{"_id": "9434", "title": "Primitive of x squared by Root of x squared plus a squared", "text": ":$\\displaystyle \\int x^2 \\sqrt {x^2 + a^2} \\rd x = \\frac {x \\paren {\\sqrt {x^2 + a^2} }^3} 4 - \\frac {a^2 x \\sqrt {x^2 + a^2} } 8 - \\frac {a^4} 8 \\map \\ln {x + \\sqrt {x^2 + a^2} } + C$"}
+{"_id": "17628", "title": "Sum of Squares of Standard Gaussian Random Variables has Chi-Squared Distribution", "text": "Let $X_1, X_2, \\ldots, X_n$ be independent random variables. Let $X_i \\sim \\Gaussian 0 1$ for $1 \\le i \\le n$ where $\\Gaussian 0 1$ is the standard Gaussian Distribution. Then:  :$\\displaystyle \\sum_{i \\mathop = 1}^n X^2_i \\sim \\chi^2_n$ where $\\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom."}
+{"_id": "9441", "title": "Primitive of x squared over Root of x squared plus a squared cubed", "text": ":$\\displaystyle \\int \\frac {x^2 \\ \\mathrm d x} {\\left({\\sqrt {x^2 + a^2} }\\right)^3} = \\frac {-x} {\\sqrt {x^2 + a^2} } + \\ln \\left({x + \\sqrt {x^2 + a^2} }\\right) + C$"}
+{"_id": "9444", "title": "Primitive of Reciprocal of x squared by Root of x squared plus a squared cubed", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^2 \\paren {\\sqrt {x^2 + a^2} }^3} = \\frac {-\\sqrt {x^2 + a^2} } {a^4 x} - \\frac x {a^4 \\sqrt {x^2 + a^2} } + C$"}
+{"_id": "17641", "title": "Variance of Student's t-Distribution", "text": "Let $k$ be a strictly positive integer.  Let $X \\sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom. Then the variance of $X$ is given by:  :$\\var X = \\dfrac k {k - 2}$ for $k > 2$, and does not exist otherwise."}
+{"_id": "9450", "title": "Primitive of Root of x squared plus a squared cubed over x", "text": ":$\\ds \\int \\frac {\\paren {\\sqrt {x^2 + a^2} }^3} x \\rd x = \\frac {\\paren {\\sqrt {x^2 + a^2} }^3} 3 + a^2 \\sqrt {x^2 + a^2} - a^3 \\map \\ln {\\frac {a + \\sqrt {x^2 + a^2} } x} + C$"}
+{"_id": "9451", "title": "Primitive of Root of x squared plus a squared cubed over x squared", "text": ":$\\displaystyle \\int \\frac{\\left({\\sqrt {x^2 + a^2} }\\right)^3} {x^2} \\ \\mathrm d x = \\frac {-\\left({\\sqrt {x^2 + a^2} }\\right)^3} x + \\frac{3 x \\sqrt {x^2 + a^2} } 2 + \\frac {3 a^2} 2 \\ln \\left({x + \\sqrt {x^2 + a^2} }\\right) + C$"}
+{"_id": "1258", "title": "Weak Whitney Immersion Theorem", "text": "Every $k$-dimensional manifold $X$ admits a one-to-one immersion in $\\R^{2 k + 1}$. {{MissingLinks|$k$-dimensional}}"}
+{"_id": "9452", "title": "Primitive of Root of x squared plus a squared cubed over x cubed", "text": ":$\\displaystyle \\int \\frac {\\paren {\\sqrt {x^2 + a^2} }^3} {x^3} \\rd x = \\frac {-\\paren {\\sqrt {x^2 + a^2} }^3} {2 x^2} + \\frac {3 \\sqrt {x^2 + a^2} } 2 - \\frac {3 a} 2 \\map \\ln {\\frac {a + \\sqrt {x^2 + a^2} } x} + C$"}
+{"_id": "1263", "title": "Homotopy Group is Homeomorphism Invariant", "text": "Let $X$ and $Y$ be two topological spaces. Let $\\phi: X \\to Y$ be a homeomorphism. Let $x_0 \\in X$, $y_0 \\in Y$. Then for all $n \\in \\N$ the induced mapping: :$\\phi_* : \\pi_n \\left({X, x_0}\\right) \\to \\pi_n \\left({Y, y_0}\\right):$ ::$\\left[{\\!\\left[{\\, c \\,}\\right]\\!}\\right] \\mapsto \\left[{\\!\\left[{\\, \\phi \\circ c \\,}\\right]\\!}\\right]$ is an isomorphism, where $\\pi_n$ denotes the $n$th homotopy group."}
+{"_id": "9461", "title": "Primitive of x cubed by Root of x squared minus a squared", "text": ":$\\displaystyle \\int x^3 \\sqrt {x^2 - a^2} \\rd x = \\frac {\\paren {\\sqrt {x^2 - a^2} }^5} 5 + \\frac {a^2 \\paren {\\sqrt {x^2 - a^2} }^3} 3 + C$"}
+{"_id": "9470", "title": "Primitive of Reciprocal of x squared by Root of x squared minus a squared cubed", "text": ":$\\displaystyle \\int \\frac {\\d x} {x^2 \\paren {\\sqrt {x^2 - a^2} }^3} = \\frac {-\\sqrt {x^2 - a^2} } {a^4 x} - \\frac x {a^4 \\sqrt {x^2 - a^2} } + C$"}
+{"_id": "9474", "title": "Primitive of x squared by Root of x squared minus a squared cubed", "text": ":$\\displaystyle \\int x^2 \\paren {\\sqrt {x^2 - a^2} }^3 \\rd x = \\frac {x \\paren {\\sqrt {x^2 - a^2} }^5} 6 + \\frac {a^2 x \\paren {\\sqrt {x^2 - a^2} }^3} {24} - \\frac {a^4 x \\sqrt {x^2 - a^2} } {16} + \\frac {a^6} {16} \\ln \\size {x + \\sqrt {x^2 - a^2} } + C$"}
+{"_id": "17668", "title": "Binomial Coefficient 2 n Choose n is Divisible by All Primes between n and 2 n", "text": "Let $\\dbinom {2 n} n$ denote a binomial coefficient. Then for all prime numbers $p$ such that $n < p < 2 n$: :$p \\divides \\dbinom {2 n} n$ where $\\divides$ denotes divisibility."}
+{"_id": "17674", "title": "Linear Combination of Gaussian Random Variables", "text": "Let $X_1, X_2, X_3, \\ldots, X_n$ be independent random variables.  Let $\\sequence {\\alpha_i}_{1 \\le i \\le n}$ and $\\sequence {\\mu_i}_{1 \\le i \\le n}$ be sequences of real numbers. Let $\\sequence {\\sigma_i}_{1 \\le i \\le n}$ be a sequence of positive real numbers. Let $X_i \\sim \\Gaussian {\\mu_i} {\\sigma^2_i}$ for $1 \\le i \\le n$, where $\\Gaussian {\\mu_i} {\\sigma^2_i}$ is the Gaussian distribution with parameters $\\mu_i$ and $\\sigma^2_i$. Then:  :$\\displaystyle \\sum_{i \\mathop = 1}^n \\alpha_i X_i \\sim \\Gaussian {\\sum_{i \\mathop = 1}^n \\alpha_i \\mu_i} {\\sum_{i \\mathop = 1}^n \\alpha^2_i \\sigma^2_i}$"}
+{"_id": "9485", "title": "Primitive of x squared by Root of a squared minus x squared", "text": ":$\\displaystyle \\int x^2 \\sqrt {a^2 - x^2} \\rd x = \\frac {-x \\paren {\\sqrt {a^2 - x^2} }^3} 4 + \\frac {a^2 x \\sqrt {a^2 - x^2} } 8 + \\frac {a^4} 8 \\sinh^{-1} \\frac x a + C$"}
+{"_id": "1303", "title": "Existence of Non-Measurable Subset of Real Numbers", "text": "There exists a subset of the real numbers which is not measurable."}
+{"_id": "17696", "title": "Cube as Difference between Two Squares", "text": "A cube number can be expressed as the difference between two squares."}
+{"_id": "17698", "title": "Factorial Greater than Cube for n Greater than 5", "text": "Let $n \\in \\Z$ be an integer such that $n > 5$. Then $n! > n^3$."}
+{"_id": "1328", "title": "Existence of Euler-Mascheroni Constant", "text": "The real sequence: :$\\displaystyle \\sequence {\\sum_{k \\mathop = 1}^n \\frac 1 k - \\ln n}$ converges to a limit. This limit is known as the Euler-Mascheroni constant."}
+{"_id": "17721", "title": "Square Product of Three Consecutive Triangular Numbers", "text": "Let $T_n$ denote the $n$th triangular number for $n \\in \\Z_{>0}$ a (strictly) positive integer. Let $T_n \\times T_{n + 1} \\times T_{n + 2}$ be a square number. Then at least one value of $n$ fulfils this condition: :$n = 3$"}
+{"_id": "17727", "title": "Cube Modulo 9", "text": "Let $x \\in \\Z$ be an integer. Then one of the following holds: {{begin-eqn}} {{eqn | l = x^3       | o = \\equiv       | r = 0 \\pmod 9       | c =  }} {{eqn | l = x^3       | o = \\equiv       | r = 1 \\pmod 9       | c =  }} {{eqn | l = x^3       | o = \\equiv       | r = 8 \\pmod 9       | c =  }} {{end-eqn}}"}
+{"_id": "17748", "title": "Acceleration of Particle moving in Circle", "text": "Let $P$ be a particle moving in a circular path $C$. Then the acceleration of $P$ is given as: :$\\mathbf a = -\\dfrac {\\size {\\mathbf v}^2 \\mathbf r} {\\size {\\mathbf r}^2}$ where: :$\\mathbf v$ is the instantaneous velocity of $P$ :$\\mathbf r$ is the vector whose magnitude equals the length of the radius of $C$ and whose direction is from the center of $C$ to $P$ :$\\size {\\, \\cdot \\,}$ denotes the magnitude of a vector."}
+{"_id": "9568", "title": "Primitive of Reciprocal of Root of a x squared plus b x plus c/a equal to 0", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\sqrt {a x^2 + b x + c} } = \\frac {2 \\sqrt {b x + c} } b + C$ when $a = 0$."}
+{"_id": "9592", "title": "Primitive of x squared over Cube of Root of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {x^2 \\ \\mathrm d x} {\\left({\\sqrt {a x^2 + b x + c} }\\right)^3} = \\frac {\\left({2 b^2 - 4 a c}\\right) x + 2 b c} {a \\left({4 a c - b^2}\\right) \\sqrt {a x^2 + b x + c} } + \\frac 1 a \\int \\frac {\\mathrm d x} {\\sqrt {a x^2 + b x + c} }$"}
+{"_id": "9594", "title": "Primitive of Reciprocal of x squared by Cube of Root of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {\\d x} {x^2 \\paren {\\sqrt {a x^2 + b x + c} }^3} = -\\frac {a x^2 + 2 b x + c} {c^2 x \\sqrt {a x^2 + b x + c} } + \\frac {b^2 - 2 a c} {2 c^2} \\int \\frac {\\d x} {\\paren {\\sqrt {a x^2 + b x + c} }^3} - \\frac {3 b} {2 c^2} \\int \\frac {\\d x} {x \\sqrt {a x^2 + b x + c} }$"}
+{"_id": "17787", "title": "Dirichlet Beta Function at Odd Positive Integers", "text": ":$\\map \\beta {2 n + 1} = \\paren {-1}^n \\dfrac {E_{2 n} \\pi^{2 n + 1} } {4^{n + 1} \\paren {2 n}!}$"}
+{"_id": "9597", "title": "Primitive of Reciprocal of Half Integer Power of a x squared plus b x plus c", "text": "Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({a x^2 + b x + c}\\right)^{n + \\frac 1 2} } = \\frac {2 \\left({2 a x + b}\\right)} {\\left({2 n - 1}\\right) \\left({4 a c - b^2}\\right) \\left({a x^2 + b x + c}\\right)^{n - \\frac 1 2} } + \\frac {8 a \\left({n - 1}\\right)} {\\left({2 n - 1}\\right) \\left({4 a c - b^2}\\right)} \\int \\frac {\\mathrm d x} {\\left({a x^2 + b x + c}\\right)^{n - \\frac 1 2} }$"}
+{"_id": "9601", "title": "Primitive of x squared over x cubed plus a cubed", "text": ":$\\displaystyle \\int \\frac {x^2 \\rd x} {x^3 + a^3} = \\frac 1 3 \\ln \\size {x^3 + a^3} + C$"}
+{"_id": "9605", "title": "Primitive of x over x cubed plus a cubed squared", "text": ":$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\left({x^3 + a^3}\\right)^2} = \\frac {x^2} {3 a^3 \\left({x^3 + a^3}\\right)} + \\frac 1 {18 a^4} \\ln \\left({\\frac {x^2 - a x + a^2} {\\left({x + a}\\right)^2} }\\right) + \\frac 1 {3 a^4 \\sqrt 3} \\arctan \\frac {2 x - a} {a \\sqrt 3}$"}
+{"_id": "1437", "title": "Compact First-Countable Space is Sequentially Compact", "text": "Let $T = \\struct {S, \\tau}$ be a compact first-countable topological space. Then every infinite sequence in $S$ has a convergent subsequence; that is, $T$ is sequentially compact."}
+{"_id": "1441", "title": "Convergence in Indiscrete Space", "text": "Let $\\left({S, \\left\\{{S, \\varnothing}\\right\\}}\\right)$ be an indiscrete space. Let $\\left \\langle {x_n} \\right \\rangle$ be any sequence in $S$. Then $\\left \\langle {x_n} \\right \\rangle$ converges to any point $x$ of $S$."}
+{"_id": "17828", "title": "Möbius Transformation is Bijection/Restriction to Reals", "text": "Let $a, b, c, d \\in \\R$ be real numbers. Let $f: \\R^* \\to \\R^*$ be the Möbius transformation restricted to the real numbers: :$\\map f x = \\begin {cases} \\dfrac {a x + b} {c x + d} & : x \\ne -\\dfrac d c \\\\ \\infty & : x = -\\dfrac d c \\\\ \\dfrac a c & : x = \\infty \\\\ \\infty & : x = \\infty \\text { and } c = 0 \\end {cases}$ Then: :$f: \\R^* \\to \\R^*$ is a bijection {{iff}}: :$a c - b d \\ne 0$"}
+{"_id": "17831", "title": "Definite Integral from 0 to 1 of Logarithm of x over One plus x", "text": ":$\\displaystyle \\int_0^1 \\frac {\\ln x} {1 + x} \\rd x = -\\frac {\\pi^2} {12}$"}
+{"_id": "9640", "title": "Primitive of Reciprocal of x by Root of Power of x minus Power of a", "text": ":$\\displaystyle \\int \\frac {\\d x} {x \\sqrt {x^n - a^n} } = \\frac 2 {n \\sqrt {a^n} } \\arccos \\sqrt {\\frac {a^n} {x^n} }$"}
+{"_id": "17838", "title": "Set of Sets can be Defined as Family", "text": "Let $\\Bbb S$ be a set of sets. Then $\\Bbb S$ can be defined as an indexed family of sets."}
+{"_id": "9648", "title": "Primitive of Sine of a x over x", "text": "{{begin-eqn}} {{eqn | l = \\int \\frac {\\sin a x \\d x} x       | r = \\sum_{k \\mathop \\ge 0} \\frac {\\paren {-1}^k \\paren {a x}^{2 k + 1} } {\\paren {2 k + 1} \\paren {2 k + 1}!} + C }} {{eqn | r = a x - \\frac {\\paren {a x}^3} {3 \\times 3!} + \\frac {\\paren {a x}^5} {5 \\times 5!} - \\cdots }} {{end-eqn}}"}
+{"_id": "9650", "title": "Primitive of Reciprocal of Sine of a x/Logarithm of Cosecant minus Cotangent Form", "text": ":$\\ds \\int \\frac {\\d x} {\\sin a x} = \\frac 1 a \\ln \\size {\\csc a x - \\cot a x} + C$"}
+{"_id": "9666", "title": "Primitive of Reciprocal of Square of 1 minus Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({1 - \\sin a x}\\right)^2} = \\frac 1 {2 a} \\tan \\left({\\frac \\pi 4 + \\frac {a x} 2}\\right) + \\frac 1 {6 a} \\tan^3 \\left({\\frac \\pi 4 + \\frac {a x} 2}\\right) + C$"}
+{"_id": "1482", "title": "Converse Hinge Theorem", "text": "If two triangles have two pairs of sides which are the same length, the triangle in which the third side is longer also has the larger angle contained by the first two sides. {{:Euclid:Proposition/I/25}}"}
+{"_id": "17866", "title": "Definite Integral over Reals of Exponential of -(a x^2 plus b x plus c)", "text": ":$\\displaystyle \\int_{-\\infty}^\\infty \\map \\exp {-\\paren {a x^2 + b x + c} } \\rd x = \\sqrt {\\frac \\pi a} \\map \\exp {\\frac {b^2 - 4 a c} {4 a} }$"}
+{"_id": "17868", "title": "Definite Integral to Infinity of Exponential of -(a x^2 plus b over x^2)", "text": ":$\\displaystyle \\int_0^\\infty \\map \\exp {-\\paren {a x^2 + \\frac b {x^2} } } \\rd x = \\frac 1 2 \\sqrt {\\frac \\pi a} \\map \\exp {-2 \\sqrt {a b} }$"}
+{"_id": "1484", "title": "Invariance of Extremal Length under Conformal Mappings", "text": "Let $X, Y$ be Riemann surfaces (usually, subsets of the complex plane). Let $\\phi: X \\to Y$ be a conformal isomorphism between $X$ and $Y$. Let $\\Gamma$ be a family of rectifiable curves (or, more generally, of unions of rectifiable curves) in $X$. Let $\\Gamma'$ be the family of their images under $\\phi$. Then $\\Gamma$ and $\\Gamma'$ have the same extremal length: :$\\map \\lambda \\Gamma = \\map \\lambda {\\Gamma'}$"}
+{"_id": "17874", "title": "Set is Subset of Intersection of Supersets/Set of Sets", "text": "Let $T$ be a set. Let $\\mathbb S$ be a set of sets. Suppose that for each $S \\in \\mathbb S$, $T \\subseteq S$. Then: :$T \\subseteq \\displaystyle \\bigcap \\mathbb S$"}
+{"_id": "17887", "title": "Membership Relation is Not Symmetric", "text": "Let $\\Bbb S$ be a set of sets in the context of pure set theory Let $\\RR$ denote the membership relation on $\\Bbb S$: :$\\forall \\tuple {a, b} \\in \\Bbb S \\times \\Bbb S: \\tuple {a, b} \\in \\RR \\iff a \\in b$ $\\RR$ is not in general a symmetric relation."}
+{"_id": "17884", "title": "Complementary Error Function of Zero", "text": ":$\\displaystyle \\map \\erfc 0 = 1$"}
+{"_id": "9699", "title": "Primitive of Cosine of a x over x", "text": "{{begin-eqn}} {{eqn | l = \\int \\frac {\\cos a x \\rd x} x       | r = \\ln \\size x + \\sum_{k \\mathop \\ge 1} \\frac {\\paren {-1}^k \\paren {a x}^{2 k} } {\\paren {2 k} \\paren {2 k}!} + C }} {{eqn | r = \\ln \\size x - \\frac {\\paren {a x}^2} {2 \\times 2!} + \\frac {\\paren {a x}^4} {4 \\times 4!} - \\frac {\\paren {a x}^6} {6 \\times 6!} - \\cdots + C }} {{end-eqn}}"}
+{"_id": "9700", "title": "Primitive of Cosine of a x over x squared", "text": ":$\\displaystyle \\int \\frac {\\cos a x \\ \\mathrm d x} {x^2} = \\frac {-\\cos a x} x - a \\int \\frac {\\sin a x \\ \\mathrm d x} x$"}
+{"_id": "17900", "title": "Definite Integral to Infinity of x over Hyperbolic Sine of a x", "text": ":$\\displaystyle \\int_0^\\infty \\frac x {\\sinh a x} \\rd x = \\frac {\\pi^2} {4 a^2}$"}
+{"_id": "17904", "title": "Empty Set from Principle of Non-Contradiction", "text": "The empty set can be characterised as: :$\\O := \\set {x: x \\in E \\text { and } x \\notin E}$ where $E$ is an arbitrary set."}
+{"_id": "9721", "title": "Primitive of Reciprocal of Square of 1 plus Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({1 + \\cos a x}\\right)^2} = \\frac 1 {2a} \\tan \\frac {a x} 2 + \\frac 1 {6 a} \\tan^3 \\frac {a x} 2 + C$"}
+{"_id": "17914", "title": "Definite Integral from 0 to 2 Pi of Reciprocal of One minus 2 a Cosine x plus a Squared", "text": ":$\\displaystyle \\int_0^{2 \\pi} \\frac {\\d x} {1 - 2 a \\cos x + a^2} = \\frac {2 \\pi} {1 - a^2}$"}
+{"_id": "1530", "title": "Divisibility by 7", "text": "An integer $X$ with $n$ digits ($X_0$ in the ones place, $X_1$ in the tens place, and so on) is divisible by $7$ {{iff}}: :$\\displaystyle \\sum_{i \\mathop = 0}^{n - 1} \\paren {3^i X_i}$ is divisible by $7$."}
+{"_id": "17916", "title": "Intersection is Empty and Union is Universe if Sets are Complementary", "text": "Let $A$ and $B$ be subsets of a universe $\\Bbb U$. Then: :$A \\cap B = \\O$ and $A \\cup B = \\Bbb U$ {{iff}}: :$B = \\relcomp {\\Bbb U} A$ where $\\relcomp {\\Bbb U} A$ denotes the complement of $A$ with respect to $\\Bbb U$."}
+{"_id": "17925", "title": "Definite Integral to Infinity of Reciprocal of Exponential of x minus One minus Exponential of -x over x", "text": ":$\\displaystyle \\int_0^\\infty \\paren {\\frac 1 {e^x - 1} - \\frac {e^{-x} } x} \\rd x = \\gamma$"}
+{"_id": "17927", "title": "Set Intersection expressed as Intersection Complement", "text": "Let $A$ and $B$ be subsets of a universal set $\\Bbb U$. Let $\\uparrow$ denote the operation on $A$ and $B$ defined as: :$\\paren {A \\uparrow B} \\iff \\paren {\\relcomp {\\Bbb U} {A \\cap B} }$ where $\\relcomp {\\Bbb U} A$ denotes the complement of $A$ in $\\Bbb U$. Then: :$A \\cap B = \\paren {A \\uparrow B} \\uparrow \\paren {A \\uparrow B}$"}
+{"_id": "9738", "title": "Primitive of Power of Secant of a x by Tangent of a x", "text": ":$\\displaystyle \\int \\sec^n a x \\tan a x \\ \\mathrm d x = \\frac {\\sec^n a x} {n a} + C$"}
+{"_id": "9739", "title": "Primitive of Reciprocal of Square of Sine of a x by Square of Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\sin^2 a x \\cos^2 a x} = \\frac {-2 \\cot 2 a x} a + C$"}
+{"_id": "17930", "title": "Singleton of Subset is Element of Powerset of Powerset", "text": "Let $S \\subseteq T$ where $S$ and $T$ are both sets. Then: :$\\set S \\in \\powerset {\\powerset T}$ where $\\powerset T$ denotes the power set of $T$."}
+{"_id": "1558", "title": "Power Function is Completely Multiplicative", "text": "Let $K$ be a field. Let $z \\in K$. Let $f_z: K \\to K$ be the mapping defined as: :$\\forall x \\in K: f_z \\left({x}\\right) = x^z$ Then $f_z$ is completely multiplicative."}
+{"_id": "9753", "title": "Primitive of Cosine of a x over p plus q of Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\cos a x \\rd x} {p + q \\sin a x} = \\frac 1 {a q} \\ln \\left\\vert{p + q \\sin a x}\\right\\vert + C$"}
+{"_id": "9755", "title": "Primitive of Cosine of a x over Power of p plus q of Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\cos a x \\ \\mathrm d x} {\\left({p + q \\sin a x}\\right)^n} = \\frac {-1} {a q \\left({n - 1}\\right) \\left({p + q \\sin a x}\\right)^{n - 1} } + C$"}
+{"_id": "1570", "title": "Number of Quadratic Residues of Prime", "text": "Let $p$ be an odd prime. Then $p$ has $\\dfrac {p-1} 2$ quadratic residues and $\\dfrac {p-1} 2$ quadratic non-residues. The quadratic residues are congruent modulo $p$ to the integers $1^2, 2^2, \\ldots, \\left({\\dfrac {p-1} 2}\\right)$."}
+{"_id": "17967", "title": "Limit to Infinity of Fresnel Sine Integral Function", "text": ":$\\displaystyle \\lim_{x \\mathop \\to \\infty} \\map {\\operatorname S} x = \\frac 1 2$"}
+{"_id": "1584", "title": "Solution to Simultaneous Linear Congruences", "text": "Let: {{begin-eqn}} {{eqn | l = a_1 x       | o = \\equiv       | r = b_1       | rr= \\pmod {n_1}       | c =  }} {{eqn | l = a_2 x       | o = \\equiv       | r = b_2       | rr= \\pmod {n_2}       | c =  }} {{eqn | o = \\ldots       | c =  }} {{eqn | l = a_r x       | o = \\equiv       | r = b_r       | rr= \\pmod {n_r}       | c =  }} {{end-eqn}} be a system of simultaneous linear congruences. This system has a simultaneous solution {{iff}}: :$\\forall i, j: 1 \\le i, j \\le r: \\gcd \\set {n_i, n_j}$ divides $b_j - b_i$. If a solution exists then it is unique modulo $\\lcm \\set {n_1, n_2, \\ldots, n_r}$."}
+{"_id": "9784", "title": "Primitive of Reciprocal of Tangent of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\tan a x} = \\frac 1 a \\ln \\size {\\sin a x} + C$"}
+{"_id": "9789", "title": "Primitive of Power of Tangent of a x", "text": ":$\\displaystyle \\int \\tan^n a x \\ \\mathrm d x = \\frac {\\tan^{n - 1} a x} {\\left({n - 1}\\right) a} - \\int \\tan^{n - 2} a x \\ \\mathrm d x + C$"}
+{"_id": "17982", "title": "Logarithm of One plus x in terms of Gaussian Hypergeometric Function", "text": ":$\\displaystyle \\map \\ln {1 + x} = x \\, {}_2 \\map {F_1} {1, 1; 2; -x}$"}
+{"_id": "17984", "title": "Laplace Transform of Exponential times Cosine", "text": ":$\\map {\\laptrans {e^{b t} \\cos a t} } s = \\dfrac {s - b} {\\paren {s - b}^2 + a^2}$"}
+{"_id": "17985", "title": "Complement of Direct Image Mapping of Injection equals Direct Image of Complement", "text": "Let $f: S \\to T$ be an injection. Let $f^\\to: \\powerset S \\to \\powerset T$ denote the direct image mapping of $f$. Then: :$\\forall A \\in \\powerset S: \\map {\\paren {\\complement_{\\Img f} \\circ f^\\to} } A = \\map {\\paren {f^\\to \\circ \\complement_S} } A$ where $\\circ$ denotes composition of mappings."}
+{"_id": "17986", "title": "Direct Image of Inverse Image of Direct Image equals Direct Image Mapping", "text": "Let $f: S \\to T$ be a mapping. Let: :$f^\\to: \\powerset S \\to \\powerset T$ denote the direct image mapping of $f$ :$f^\\gets: \\powerset T \\to \\powerset S$ denote the inverse image mapping of $f$ where $\\powerset S$ denotes the power set of $S$. Then: :$f^\\to \\circ f^\\gets \\circ f^\\to = f^\\to$ where $\\circ$ denotes composition of mappings."}
+{"_id": "9795", "title": "Primitive of x by Cotangent of a x", "text": ":$\\displaystyle \\int x \\cot a x \\rd x = \\frac 1 {a ^ 2} \\paren {a x - \\frac {\\paren {a x}^3} 9 - \\frac {\\paren {a x}^5} {225} - \\cdots + \\frac {\\paren {-1}^n 2^{2 n} B_{2 n} \\paren {a x}^{2 n + 1} } {\\paren {2 n + 1} !} + \\cdots} + C$ where $B_{2 n}$ denotes the $2 n$th Bernoulli number."}
+{"_id": "9798", "title": "Primitive of Reciprocal of p plus q by Cotangent of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {p + q \\cot a x} = \\frac {p x} {p^2 + q^2} - \\frac q {a \\paren {p^2 + q^2} } \\ln \\size {p \\sin a x + q \\cos a x} + C$"}
+{"_id": "9801", "title": "Primitive of Reciprocal of Secant of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\sec a x} = \\frac {\\sin a x} a + C$"}
+{"_id": "9807", "title": "Primitive of Reciprocal of Cosecant of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\csc a x} = \\frac {-\\cos a x} a + C$"}
+{"_id": "9813", "title": "Primitive of x by Arcsine of x over a", "text": ":$\\displaystyle \\int x \\arcsin \\frac x a \\ \\mathrm d x = \\left({\\frac {x^2} 2 - \\frac {a^2} 4}\\right) \\arcsin \\frac x a + \\frac {x \\sqrt {a^2 - x^2} } 4 + C$"}
+{"_id": "1627", "title": "Elements of Primitive Pythagorean Triples Modulo 4", "text": "Let $x \\in \\Z: x > 2$. Then $x$ is an element of some primitive Pythagorean triple {{iff}} $x \\not \\equiv 2 \\pmod 4$."}
+{"_id": "18016", "title": "Ceva's Theorem", "text": "Let $\\triangle ABC$ be a triangle. Let $L$, $M$ and $N$ be points on the sides $BC$, $AC$ and $AB$ respectively. Then the lines $AL$, $BM$ and $CN$ are concurrent {{iff}}: :$\\dfrac {BL} {LC} \\times \\dfrac {CM} {MA} \\times \\dfrac {AN} {NB} = 1$"}
+{"_id": "9828", "title": "Primitive of Arccosine of x over a over x", "text": ":$\\displaystyle \\int \\frac {\\arccos \\frac x a \\rd x} x = \\frac \\pi 2 \\ln \\size x - \\int \\frac {\\arcsin \\frac x a \\rd x} x + C$"}
+{"_id": "18025", "title": "Left Ideal is Left Module over Ring/Ring is Left Module over Ring", "text": "Let $\\struct {R, +, \\times}$ be a ring. Then $\\struct {R, +, \\times}$ is a left module over $\\struct {R, +, \\times}$."}
+{"_id": "9837", "title": "Primitive of x squared by Arccotangent of x over a", "text": ":$\\displaystyle \\int x^2 \\operatorname{arccot} \\frac x a \\ \\mathrm d x = \\frac {x^3} 3 \\operatorname{arccot} \\frac x a + \\frac {a x^2} 6 - \\frac {a^3} 6 \\ln \\left({x^2 + a^2}\\right) + C$"}
+{"_id": "9839", "title": "Primitive of Arccotangent of x over a over x", "text": ":$\\displaystyle \\int \\frac {\\operatorname{arccot} \\frac x a \\ \\mathrm d x} x = \\frac \\pi 2 \\ln \\left\\vert{x}\\right\\vert - \\int \\frac {\\arctan \\frac x a \\ \\mathrm d x} x$"}
+{"_id": "9852", "title": "Primitive of Arcsecant of x over a over x squared", "text": ":$\\displaystyle \\int \\frac {\\operatorname{arcsec} \\frac x a} {x^2} \\ \\mathrm d x = \\begin{cases} \\displaystyle \\frac {-\\operatorname{arcsec} \\frac x a} x + \\frac {\\sqrt{x^2 - a^2} } {a x} + C & : 0 < \\operatorname{arcsec} \\dfrac x a < \\dfrac \\pi 2 \\\\ \\displaystyle \\frac {-\\operatorname{arcsec} \\frac x a} x - \\frac {\\sqrt{x^2 - a^2} } {a x} + C & : \\dfrac \\pi 2 < \\operatorname{arcsec} \\dfrac x a < \\pi \\\\ \\end{cases}$"}
+{"_id": "9855", "title": "Primitive of Power of x by Arcsecant of x over a", "text": ":$\\displaystyle \\int x^m \\operatorname{arcsec} \\frac x a \\ \\mathrm d x = \\begin{cases} \\displaystyle \\frac {x^{m + 1} } {m + 1} \\operatorname{arcsec} \\frac x a - \\frac a {m + 1} \\int \\frac {x^m \\ \\mathrm d x} {\\sqrt {x^2 - a^2} } + C & : 0 < \\operatorname{arcsec} \\dfrac x a < \\dfrac \\pi 2 \\\\ \\displaystyle \\frac {x^{m + 1} } {m + 1} \\operatorname{arcsec} \\frac x a + \\frac a {m + 1} \\int \\frac {x^m \\ \\mathrm d x} {\\sqrt {x^2 - a^2} } + C & : \\dfrac \\pi 2 < \\operatorname{arcsec} \\dfrac x a < \\pi \\\\ \\end{cases}$"}
+{"_id": "1670", "title": "Linear Function is Primitive Recursive", "text": "The function $f: \\N \\to \\N$, defined as: :$\\map f n = a n + b$ where $a$ and $b$ are constants, is primitive recursive."}
+{"_id": "1671", "title": "Substitution of Constant yields Primitive Recursive Function", "text": "Let $f: \\N^{k+1} \\to \\N$ be a primitive recursive function. Then $g: \\N^k \\to \\N$ given by: :$g \\left({n_1, n_2, \\ldots, n_k}\\right) = f \\left({n_1, n_2, \\ldots, n_{i-1}, a, n_i \\ldots, n_k}\\right)$ is primitive recursive."}
+{"_id": "9865", "title": "Primitive of Reciprocal of Square of p plus q by Exponential of a x", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\paren {p + q e^{a x} }^2} = \\frac x {p^2} + \\frac 1 {a p \\paren {p + q e^{a x} } } - \\frac 1 {a p^2} \\ln \\size {p + q e^{a x} } + C$"}
+{"_id": "1678", "title": "Minimum Function is Primitive Recursive", "text": "The minimum function $\\min: \\N^2 \\to \\N$, defined as: :$\\min \\left({n, m}\\right) = \\begin{cases} n: & n \\le m \\\\ m: & m \\le n \\end{cases}$ is primitive recursive."}
+{"_id": "18070", "title": "Additive Regular Representations of Topological Ring are Homeomorphisms", "text": "Let $\\struct {R, + , \\circ, \\tau}$ be a topological ring. Let $x \\in R$. Let $\\lambda_x$ and $\\rho_x$ be the left and right regular representations of $\\struct {R, +}$ with respect to $x$. Then $\\lambda_x, \\,\\rho_x: \\struct {R, \\tau} \\to \\struct {R, \\tau}$ are homeomorphisms with inverses $\\lambda_{-x}, \\,\\rho_{-x}: \\struct {R, \\tau} \\to \\struct {R, \\tau}$ respectively."}
+{"_id": "18071", "title": "Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms", "text": "Let $\\struct{R, + , \\circ, \\tau}$ be a topological ring with unity $1_R$. For all $y \\in R$, let $\\lambda_y$ and $\\rho_y$ denote the left and right regular representations of $\\struct{R, \\circ}$ with respect to $y$. Let $x \\in R$ be a unit of $R$ with product inverse $x^{-1}$. Then $\\lambda_x, \\, \\rho_x: \\struct{R, \\tau} \\to \\struct{R, \\tau}$ are homeomorphisms with inverse mappings $\\lambda_{x^{-1} }, \\, \\rho_{x^{-1} }: \\struct{R, \\tau} \\to \\struct{R, \\tau}$ respectively."}
+{"_id": "9882", "title": "Primitive of Logarithm of x over x squared", "text": ":$\\displaystyle \\int \\frac {\\ln x} {x^2} \\ \\mathrm d x = \\frac {-\\ln x} x - \\frac 1 x + C$"}
+{"_id": "1691", "title": "Sylow Subgroup is Hall Subgroup", "text": "Let $G$ be a group. Let $H$ be a Sylow $p$-subgroup of $G$. Then $H$ is a Hall subgroup of $G$."}
+{"_id": "9886", "title": "Primitive of Reciprocal of Logarithm of x", "text": ":$\\displaystyle \\int \\frac {\\d x} {\\ln x} = \\map \\ln {\\ln x} + \\ln x + \\sum_{k \\mathop \\ge 2}^n \\frac {\\paren {\\ln x}^k} {k \\times k!} + C$"}
+{"_id": "9888", "title": "Primitive of Power of Logarithm of x", "text": ":$\\displaystyle \\int \\ln^n x \\rd x = x \\ln^n x - n \\int \\ln^{n - 1} x \\rd x + C$"}
+{"_id": "9891", "title": "Primitive of Logarithm of x squared minus a squared", "text": ":$\\displaystyle \\int \\ln \\left({x^2 - a^2}\\right) \\rd x = x \\ln \\left({x^2 - a^2}\\right) - 2 x + a \\ln \\left({\\frac {x + a} {x - a} }\\right) + C$ for $x^2 > a^2$."}
+{"_id": "18091", "title": "Conservation of Energy", "text": "Let $P$ be a physical system. Let it have the action $S$: :$\\displaystyle S = \\int_{t_0}^{t_1} L \\rd t$ where $L$ is the standard Lagrangian, and $t$ is time. Suppose $L$ does not depend on time explicitly: :$\\dfrac {\\partial L} {\\partial t} = 0$ Then the total energy of $P$ is conserved."}
+{"_id": "9902", "title": "Primitive of x squared by Hyperbolic Sine of a x", "text": ":$\\displaystyle \\int x^2 \\sinh a x \\ \\mathrm d x = \\left({\\frac {x^2} a + \\frac 2 {a^3} }\\right) \\cosh a x - \\frac {2 x \\sinh a x} {a^2} + C$"}
+{"_id": "18101", "title": "Poisson Brackets of Harmonic Oscillator", "text": "Let $P$ be a classical harmonic oscillator. Let the real-valued function $\\map x t$ be the position of $P$, where $t$ is time. Then $P$ has the following Poisson brackets: :$\\sqbrk {x, p} = 1$ :$\\sqbrk {x, H} = \\dfrac p m$ :$\\sqbrk {p, H} = - k x$"}
+{"_id": "1719", "title": "Primitive Recursive Function is Total Recursive Function", "text": "Every primitive recursive function is a total recursive function."}
+{"_id": "9918", "title": "Primitive of Reciprocal of Hyperbolic Secant of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\operatorname{sech} a x} = \\frac {\\sinh a x} a + C$"}
+{"_id": "1738", "title": "Triangle Angle-Side-Angle and Side-Angle-Angle Equality", "text": "=== Triangle Angle-Side-Angle Equality === {{:Triangle Angle-Side-Angle Equality}} === Triangle Side-Angle-Angle Equality === {{:Triangle Side-Angle-Angle Equality}}"}
+{"_id": "9935", "title": "Primitive of x by Square of Hyperbolic Cosecant of a x", "text": ":$\\displaystyle \\int x \\csch^2 a x \\rd x = \\frac {-x \\coth a x} a + \\frac 1 {a^2} \\ln \\size {\\sinh a x} + C$"}
+{"_id": "9936", "title": "Primitive of Reciprocal of Square of Hyperbolic Sine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\sinh^2 a x} = - \\frac {\\coth a x} a + C$"}
+{"_id": "9937", "title": "Primitive of Reciprocal of Square of Hyperbolic Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\cosh^2 a x} = \\frac {\\tanh a x} a + C$"}
+{"_id": "1748", "title": "Sum of Sequence of Odd Index Fibonacci Numbers", "text": "{{begin-eqn}} {{eqn | lo= \\forall n \\ge 1:       | l = \\sum_{j \\mathop = 1}^n F_{2 j - 1}       | r = F_1 + F_3 + F_5 + \\cdots + F_{2 n - 1}       | c =  }} {{eqn | r = F_{2 n}       | c =  }} {{end-eqn}}"}
+{"_id": "9941", "title": "Primitive of Hyperbolic Sine of a x by Cosine of p x", "text": ":$\\displaystyle \\int \\sinh a x \\cos p x \\ \\mathrm d x = \\frac {a \\cosh a x \\cos p x + p \\sinh a x \\sin p x} {a^2 + p^2} + C$"}
+{"_id": "9956", "title": "Primitive of Power of x by Hyperbolic Sine of a x", "text": ":$\\displaystyle \\int x^m \\sinh a x \\ \\mathrm d x = \\frac {x^m \\cosh a x} a - \\frac m a \\int x^{m - 1} \\cosh a x \\ \\mathrm d x + C$"}
+{"_id": "18153", "title": "Summary of Topology on P-adic Numbers", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\tau_p$ be the topology induced by the non-Archimedean norm $\\norm {\\,\\cdot\\,}_p$. Then $\\struct{\\Q_p, \\tau_p}$ is: :$(1): \\quad$ Hausdorff :$(2): \\quad$ second-countable :$(3): \\quad$ totally disconnected :$(4): \\quad$ locally compact"}
+{"_id": "18154", "title": "Napier's Cosine Rule for Right Spherical Triangles", "text": "Let $\\triangle ABC$ be a right spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let the angle $\\sphericalangle C$ be a right angle. Let the remaining parts of $\\triangle ABC$ be arranged according to the '''interior''' of this circle, where the symbol $\\Box$ denotes a right angle. :410px Let one of the parts of this circle be called a '''middle part'''. Let the two parts which do not neighbor the '''middle part''' be called '''opposite parts'''. Then the sine of the '''middle part''' equals the product of the cosine of the '''opposite parts'''."}
+{"_id": "18157", "title": "Napier's Tangent Rule for Quadrantal Triangles", "text": "Let $\\triangle ABC$ be a quadrantal triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let the side $c$ be a right angle. Let the remaining parts of $\\triangle ABC$ be arranged according to the '''exterior''' of this circle, where the symbol $\\Box$ denotes a right angle. :410px Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Then the sine of the '''middle part''' equals the product of the tangents of the '''adjacent parts'''."}
+{"_id": "18158", "title": "Reciprocal of 7", "text": "The decimal expansion of the reciprocal of $7$ has the maximum period, that is: $6$: :$\\dfrac 1 {7} = 0 \\cdotp \\dot 14285 \\, \\dot 7$ {{OEIS|A020806}}"}
+{"_id": "9974", "title": "Primitive of Reciprocal of Hyperbolic Cosine of a x minus 1", "text": ":$\\ds \\int \\frac {\\d x} {\\cosh a x - 1} = \\frac {-1} a \\coth \\frac {a x} 2 + C$"}
+{"_id": "9975", "title": "Primitive of Reciprocal of Square of Hyperbolic Cosine of a x plus 1", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({\\cosh a x + 1}\\right)^2} = \\frac 1 {2 a} \\tanh \\frac {a x} 2 - \\frac 1 {6 a} \\tanh^3 \\frac {a x} 2 + C$"}
+{"_id": "18168", "title": "Equation of Witch of Agnesi/Cartesian", "text": "The equation of the Witch of Agnesi is given in cartesian coordinates as: :$y = \\dfrac {8 a^3} {x^2 + 4 a^2}$"}
+{"_id": "9978", "title": "Primitive of Hyperbolic Sine of p x by Hyperbolic Cosine of q x", "text": ":$\\displaystyle \\int \\sinh p x \\cosh q x \\ \\mathrm d x = \\frac {\\cosh \\left({p + q}\\right) x} {2 \\left({p + q}\\right)} + \\frac {\\cosh \\left({p - q}\\right) x} {2 \\left({p - q}\\right)} + C$"}
+{"_id": "1789", "title": "Proof of Theorem by Truth Table", "text": "Let $\\phi$ be a propositional formula whose atoms are $p_1, p_2, \\ldots, p_n$. Let $l$ be the line number of any row in the truth table of $\\phi$. For all $i: 1 \\le i \\ne n$, let $\\hat {p_i}$ be defined as: : $\\hat {p_i} = \\begin{cases} p_i & : \\text {the entry in line } l \\text { of } p_i \\text { is } T \\\\ \\neg p_i & : \\text {the entry in line } l \\text { of } p_i \\text { is } F \\end{cases}$ Then: * $\\hat {p_1}, \\hat {p_2}, \\ldots, \\hat {p_n} \\vdash \\phi$ is provable if the entry for $\\phi$ in line $l$ is $T$ * $\\hat {p_1}, \\hat {p_2}, \\ldots, \\hat {p_n} \\vdash \\neg \\phi$ is provable if the entry for $\\phi$ in line $l$ is $F$"}
+{"_id": "1796", "title": "Soundness Theorem for Propositional Tableaus and Boolean Interpretations", "text": "Tableau proofs (in terms of propositional tableaus) are a sound proof system for boolean interpretations. That is, for every WFF $\\mathbf A$: :$\\vdash_{\\mathrm{PT}} \\mathbf A$ implies $\\models_{\\mathrm{BI}} \\mathbf A$"}
+{"_id": "9990", "title": "Primitive of Reciprocal of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x", "text": ":$\\displaystyle \\int \\frac {\\mathrm d x} {\\sinh a x \\cosh^2 a x} = \\frac 1 a \\ln \\left\\vert{\\tanh \\frac {a x} 2}\\right\\vert + \\frac {\\operatorname{sech} a x} a + C$"}
+{"_id": "18186", "title": "Celestial Equator is Parallel to Geographical Equator", "text": "Consider the celestial sphere with observer $O$. The plane of the celestial equator is parallel to the plane of the geographical equator."}
+{"_id": "1809", "title": "Disjunction of Conjunctions", "text": ": $\\left({p \\land q}\\right) \\lor \\left({r \\land s}\\right) \\vdash p \\lor r$"}
+{"_id": "1830", "title": "Borel-Cantelli Lemma", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $E_n \\subseteq \\Sigma$ be a countable collection of measurable sets. If: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\map \\mu {E_n} < \\infty$ then: :$\\displaystyle \\map \\mu {\\limsup_{n \\mathop \\to \\infty} } {E_n} = 0$ where $\\limsup$ denotes limit superior of sets."}
+{"_id": "10022", "title": "Primitive of Inverse Hyperbolic Cosine of x over a", "text": ":$\\displaystyle \\int \\cosh^{-1} \\frac x a \\ \\mathrm d x = \\begin{cases} x \\cosh^{-1} \\dfrac x a - \\sqrt {x^2 - a^2} + C & : \\cosh^{-1} \\dfrac x a > 0 \\\\ x \\cosh^{-1} \\dfrac x a + \\sqrt {x^2 - a^2} + C & : \\cosh^{-1} \\dfrac x a < 0 \\end{cases}$"}
+{"_id": "18221", "title": "Vandermonde Matrix Identity for Hilbert Matrix", "text": "Define polynomial root sets $\\set {1,2,\\ldots, n}$ and $\\set { 0,-1,\\ldots,-n+1}$ for Definition:Cauchy Matrix. Let: {{begin-eqn}} {{eqn   | l = H   | r = \\paren {\\begin{smallmatrix}\\displaystyle         1 & \\dfrac {1} {2} & \\cdots & \\dfrac {1} {n} \\\\         \\dfrac {1} {2} & \\dfrac 1 {3}   & \\cdots & \\dfrac {1} {n+1} \\\\         \\vdots                 & \\vdots                 & \\cdots & \\vdots \\\\         \\dfrac {1} {n} & \\dfrac {1} {n+1} & \\cdots & \\dfrac {1} {2n-1} \\\\ \\end{smallmatrix} }   | c = Hilbert matrix of order $n$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn   | l = H   | r = -P V_x^{-1} V_y Q^{-1}   | c = Vandermonde Matrix Identity for Cauchy Matrix and Hilbert Matrix is Cauchy Matrix }} {{end-eqn}} Definitions of Vandermonde matrices $V_x$, $V_y$ and diagonal matrices $P$, $Q$: :$\\displaystyle V_x=\\paren {\\begin{smallmatrix} 1         & 1         & \\cdots & 1 \\\\ 1       & 2       & \\cdots & n \\\\ \\vdots    & \\vdots    & \\ddots & \\vdots \\\\ 1 & 2^{n-1} & \\cdots & n^{n-1} \\\\ \\end{smallmatrix} },\\quad  V_y=\\paren {\\begin{smallmatrix} \\displaystyle 1         & 1         & \\cdots & 1 \\\\ 0       & -1       & \\cdots & -n+1 \\\\ \\vdots    & \\vdots    & \\ddots & \\vdots \\\\ 0 & \\paren {-1}^{n-1} & \\cdots & \\paren {-n+1}^{n-1} \\\\ \\end{smallmatrix} }$ Vandermonde matrices :$\\displaystyle P= \\paren {\\begin{smallmatrix} p_1(1) &  \\cdots & 0 \\\\ \\vdots   & \\ddots  & \\vdots \\\\ 0        & \\cdots  & p_n(n) \\\\ \\end{smallmatrix} }, \\quad Q= \\paren {\\begin{smallmatrix} p(0)  & \\cdots  & 0 \\\\ \\vdots  & \\ddots  & \\vdots \\\\ 0       & \\cdots  & p(-n+1) \\\\ \\end{smallmatrix} }$  Diagonal matrices Definitions of polynomials $p$, $p_1$, $\\ldots$, $p_n$: :$\\displaystyle p(x) = \\prod_{i \\mathop = 1}^n \\paren {x - i}$ :$\\displaystyle p_k(x) = \\dfrac{ \\map p x}{x-k} = \\prod_{i \\mathop = 1,i \\mathop \\ne k}^n \\, \\paren {x - i}$, $1 \\mathop \\le k \\mathop \\le n$"}
+{"_id": "1840", "title": "Egorov's Theorem", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $D \\in \\Sigma$ be such that $\\map \\mu D < +\\infty$. Let $\\sequence {f_n}_{n \\mathop \\in \\N}, f_n: D \\to \\R$ be a sequence of $\\Sigma$-measurable functions. Suppose that $f_n$ converges a.e. to $f$, for some $\\Sigma$-measurable function $f: D \\to \\R$. Then $f_n$ converges a.u. to $f$."}
+{"_id": "10034", "title": "Primitive of Root of x squared plus a squared/Logarithm Form", "text": ":$\\displaystyle \\int \\sqrt {x^2 + a^2} \\rd x = \\frac {x \\sqrt {x^2 + a^2} } 2 + \\frac {a^2} 2 \\map \\ln {x + \\sqrt {x^2 + a^2} } + C$"}
+{"_id": "1843", "title": "Existence of Solution of 2nd Order Linear ODE", "text": "Let $P \\left({x}\\right)$, $Q \\left({x}\\right)$ and $R \\left({x}\\right)$ be continuous real functions on a closed interval $I = \\left[{a . . b}\\right]$. Let $x_0 \\in I$, and let $y_0 \\in \\R$ and $y_0' \\in \\R$ be arbitrary. Then the initial value problem: :$\\displaystyle \\frac {d^2y}{dx^2} + P \\left({x}\\right) \\frac {dy}{dx} + Q \\left({x}\\right) y = R \\left({x}\\right), y \\left({x_0}\\right) = y_0, y' \\left({x_0}\\right) = y_0'$ has one and only one solution $y = y \\left({x}\\right)$ on the interval $a \\le x \\le b$."}
+{"_id": "1845", "title": "Cartesian Metric is Rotation Invariant", "text": "The cartesian metric does not change under rotation."}
+{"_id": "10048", "title": "Primitive of x squared by Inverse Hyperbolic Cosine of x over a", "text": ":$\\displaystyle \\int x^2 \\cosh^{-1} \\frac x a \\ \\mathrm d x = \\begin{cases} \\displaystyle \\frac {x^3} 3 \\cosh^{-1} \\frac x a - \\frac {\\left({x^2 + 2 a^2}\\right) \\sqrt {x^2 - a^2} } 9 + C & : \\cosh^{-1} \\frac x a > 0 \\\\ \\displaystyle \\frac {x^3} 3 \\cosh^{-1} \\frac x a - \\frac {\\left({x^2 + 2 a^2}\\right) \\sqrt {x^2 - a^2} } 9 + C & : \\cosh^{-1} \\frac x a < 0 \\end{cases}$"}
+{"_id": "1869", "title": "Odd Vertices Determines Edge-Disjoint Trails", "text": "Let $G$ be a loop-multigraph with $2 n$ odd vertices, $n > 0$. Then $G$ has $n$ edge-disjoint trails such that every edge of $G$ is contained in one of these trails. Each of these trails starts and ends on an odd vertex."}
+{"_id": "18256", "title": "Yff's Conjecture", "text": "Let $\\triangle ABC$ be a triangle. Let $\\omega$ be the Brocard angle of $\\triangle ABC$. Then: :$8 \\omega^3 < ABC$ where $A, B, C$ are measured in radians."}
+{"_id": "18261", "title": "Reciprocal of One minus x in terms of Gaussian Hypergeometric Function", "text": ":$\\dfrac 1 {1 - x} = {}_2 \\map {F_1} {1, p; p; x}$"}
+{"_id": "18268", "title": "Rational Number Expressible as Sum of Reciprocals of Distinct Squares", "text": "Let $x$ be a rational number such that $0 < x < \\dfrac {\\pi^2} 6 - 1$. Then $x$ can be expressed as the sum of a finite number of reciprocals of distinct squares."}
+{"_id": "1886", "title": "Adding Edge to Tree Creates One Cycle", "text": "Adding a new edge to a tree can create no more than one cycle."}
+{"_id": "1891", "title": "Sophie Germain's Identity", "text": ":$x^4 + 4 y^4 = \\paren {x^2 + 2 y^2 + 2 x y} \\paren {x^2 + 2 y^2 - 2 x y}$"}
+{"_id": "1894", "title": "Coreflexive Relation Subset of Diagonal Relation", "text": "A coreflexive relation is a subset of the diagonal relation."}
+{"_id": "1899", "title": "Strict Weak Ordering Induces Partition", "text": "Let $\\struct {S, \\prec}$ be a relational structure such that $\\prec$ is a strict weak ordering on $S$. Then $S$ can be partitioned into equivalence classes whose equivalence relation is \"is non-comparable\". That is, each of the partitions $A$ of $S$ is a relational structure $\\struct {\\mathbb S, <}$ such that: :$\\mathbb S$ is the set of these partitions of $S$; :$<$ is the strict total ordering on $\\mathbb S$ '''induced by''' $\\prec$."}
+{"_id": "18286", "title": "Coherent Sequence Converges to P-adic Integer", "text": "Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\sequence {\\alpha_n}$ be a coherent sequence.  Let $\\Z_p$ be the $p$-adic integers. Then the sequence $\\sequence {\\alpha_n}$ converges to some $x \\in \\Z_p$. That is, there exists $x \\in \\Z_p$ such that: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\alpha_n = x$"}
+{"_id": "10101", "title": "Space in which All Convergent Sequences have Unique Limit not necessarily Hausdorff", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $T$ be such that all convergent sequences have a unique limit point. Then it is not necessarily the case that $T$ is a Hausdorff space."}
+{"_id": "1929", "title": "Equal Sized Triangles on Equal Base have Same Height", "text": "Triangles of equal area which are on equal bases, and on the same side of it, are also in the same parallels. {{:Euclid:Proposition/I/40}}"}
+{"_id": "10142", "title": "Characteristic Function of Union/Variant 3", "text": ":$\\chi_{A \\mathop \\cup B} = \\max \\set {\\chi_A, \\chi_B}$"}
+{"_id": "10144", "title": "Characteristic Function of Intersection/Variant 2", "text": ":$\\chi_{A \\cap B} = \\min \\left\\{{\\chi_A, \\chi_B}\\right\\}$"}
+{"_id": "10145", "title": "Tangent Space is Vector Space", "text": "Let $M$ be a smooth manifold of dimension $n \\in \\N$.  Let $m \\in M$ be a point.  Let $\\left({U, \\kappa}\\right)$ be a chart with $m \\in U$.  Let $T_m M$ be the tangent space at $m$.  Then $T_m M$ is a real vector space of dimension $n$, spanned by the basis:  :$\\left\\{ {\\left.{\\dfrac \\partial {\\partial \\kappa^i} }\\right\\vert_m : i \\in \\left\\{{1, \\dotsc, n}\\right\\} }\\right\\}$ that is, the set of partial derivatives with respect to the $i$th coordinate function $\\kappa^i$ evaluated at $m$."}
+{"_id": "18337", "title": "Parametric Equation of Involute of Circle", "text": "Let $C$ be a circle of radius $a$ whose center is at the origin of a cartesian plane. The involute $V$ of $C$ can be described by the parametric equation: :$\\begin {cases} x = a \\paren {\\cos \\theta + \\theta \\sin \\theta} \\\\ y = a \\paren {\\sin \\theta - \\theta \\cos \\theta} \\end {cases}$"}
+{"_id": "1974", "title": "Basis Expansion of Irrational Number", "text": "A basis expansion of an irrational number never terminates and does not recur."}
+{"_id": "10171", "title": "Composite Number has Prime Factor", "text": "Let $a$ be a composite number. Then there exists a prime number $p$ such that: :$p \\divides a$ where $\\divides$ means '''is a divisor of'''. {{:Euclid:Proposition/VII/31}}"}
+{"_id": "10184", "title": "General Associativity Theorem/Formulation 3", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Let $a_i$ denote elements of $S$. Let $\\circ$ be associative. Let $n \\in \\Z$ be a positive integer such that $n \\ge 3$. Then all possible parenthesizations of the expression: :$a_1 \\circ a_2 \\circ \\cdots \\circ a_n$ are equivalent."}
+{"_id": "18384", "title": "Partial Differential Equation of Planes in 3-Space", "text": "The set of planes in real Cartesian $3$-dimensional space can be described by the system of partial differential equations: {{begin-eqn}} {{eqn | l = \\dfrac {\\partial^2 z} {\\partial x^2}       | r = 0 }} {{eqn | l = \\dfrac {\\partial^2 z} {\\partial x \\partial y}       | r = 0 }} {{eqn | l = \\dfrac {\\partial^2 z} {\\partial y^2}       | r = 0 }} {{end-eqn}}"}
+{"_id": "2002", "title": "Equivalence of Definitions of Congruence", "text": "{{TFAE|def = Congruence (Number Theory)|view = Congruence|context = Number Theory}} Let $z \\in \\R$."}
+{"_id": "2004", "title": "Congruence Modulo Zero is Diagonal Relation", "text": "Congruence modulo zero is the diagonal relation. That is: :$x \\equiv y \\pmod 0 \\iff x = y$"}
+{"_id": "10202", "title": "Numbers between which exist two Mean Proportionals are Similar Solid", "text": "Let $a, b \\in \\Z$ be the extremes of a geometric sequence of integers whose length is $4$: :$\\tuple {a, m_1, m_2, b}$ That is, such that $a$ and $b$ have $2$ mean proportionals. Then $a$ and $b$ are similar solid numbers. {{:Euclid:Proposition/VIII/21}}"}
+{"_id": "10203", "title": "If First of Three Numbers in Geometric Sequence is Square then Third is Square", "text": "Let $P = \\tuple {a, b, c}$ be a geometric sequence of integers. Let $a$ be a square number. Then $c$ is also a square number. {{:Euclid:Proposition/VIII/22}}"}
+{"_id": "2012", "title": "Finite Fourier Series", "text": "Let $\\map a n$ be any finite periodic function on $\\Z$ with period $b$. Let $\\xi = e^{2 \\pi i/ b}$ be the first $b$th root of unity. Then: :$\\displaystyle \\map a n = \\sum_{k \\mathop = 0}^{b - 1} \\map {a_*} k \\xi^{n k}$ where: :$\\displaystyle \\map {a_*} n = \\frac 1 b \\sum_{k \\mathop = 0}^{b - 1} \\map a k \\xi^{-n k}$"}
+{"_id": "18408", "title": "Linear Second Order ODE/y'' - y = 3 exp -x", "text": "The second order ODE: :$(1): \\quad y'' - y = 3 e^{-x}$ has the general solution: :$y = C_1 e^x + C_2 e^{-x} - \\dfrac {3 x e^{-x} } 2$"}
+{"_id": "10220", "title": "Elements of Geometric Sequence from One which Divide Later Elements", "text": "Let $G_n = \\sequence {a_n}_{0 \\mathop \\le i \\mathop \\le n}$ be a geometric Sequence of integers. Let $a_0 = 1$. Let $m \\in \\Z_{> 0}$. Then: :$\\forall r \\in \\set {0, 1, \\ldots, m}: a_k \\divides a_m$ where $\\divides$ denotes divisibility. {{:Euclid:Proposition/IX/11}}"}
+{"_id": "10221", "title": "Elements of Geometric Sequence from One Divisible by Prime", "text": "Let $G_n = \\sequence {a_n}_{0 \\mathop \\le i \\mathop \\le n}$ be a geometric sequence of integers. Let $a_0 = 1$. Let $p$ be a prime number such that: :$p \\divides a_n$ where $\\divides$ denotes divisibility. Then $p \\divides a_1$. {{:Euclid:Proposition/IX/12}}"}
+{"_id": "10222", "title": "Divisibility of Elements of Geometric Sequence from One where First Element is Prime", "text": "Let $Q_n = \\sequence {a_j}_{0 \\mathop \\le j \\mathop \\le n}$ be a geometric sequence of length $n$ consisting of integers only. Let $a_0 = 1$. Let $a_1$ be a prime number. Then the only divisors of $a_n$ are $a_j$ for $j \\in \\set {1, 2, \\ldots, n}$. {{:Euclid:Proposition/IX/13}}"}
+{"_id": "18409", "title": "Linear Second Order ODE/y'' - 2 y' + y = exp x", "text": "The second order ODE: :$(1): \\quad y'' - 2 y' + y = e^x$ has the general solution: :$y = C_1 e^x + C_2 x e^x + \\dfrac {x^2 e^x} 2$"}
+{"_id": "18428", "title": "Linear First Order ODE/y' - y = x^2", "text": "The linear first order ODE: :$(1): \\quad \\dfrac {\\d y} {\\d x} - y = x^2$ has the general solution: :$y = C e^x - \\paren {x^2 + 2 x + 2}$"}
+{"_id": "18432", "title": "Linear Second Order ODE/y'' + y = exp -x cos x", "text": "The second order ODE: :$(1): \\quad y'' + y = e^{-x} \\cos x$ has the general solution: :$y = \\dfrac {e^{-x} } 5 \\paren {\\cos x - 2 \\sin x} + C_1 \\sin x + C_2 \\cos x$"}
+{"_id": "10252", "title": "Restriction of Homeomorphism is Homeomorphism", "text": "Let $T_1 = \\left({S_1, \\tau_1}\\right)$, $T_2 = \\left({S_2, \\tau_2}\\right)$ be topological spaces. Let $f: S_1 \\to S_2$ be a homeomorphism between $T_1$ and $T_2$. Let $S$ be a subset of $S_1$. Let $f {\\restriction_{S \\times f\\left[{S}\\right]}} : S \\to f \\left[{S}\\right]$ be the restriction of $f$ to $S \\times f \\left[{S}\\right]$. Let $S$ and $f \\left[{S}\\right]$ bear their respective subspace topologies. Then $f {\\restriction_{S \\times f \\left[{S}\\right]}}$ is a homeomorphism."}
+{"_id": "2073", "title": "Disjoint Independent Events means One is Void", "text": "Let $A$ and $B$ be events in a probability space. Suppose $A$ and $B$ are: :disjoint :independent. Then either $\\map \\Pr A = 0$ or $\\map \\Pr B = 0$. That is, if two events are disjoint and independent, at least one of them can't happen."}
+{"_id": "18502", "title": "Product of Rational Cuts is Rational Cut", "text": "Let $p \\in\\ Q$ and $q \\in \\Q$ be rational numbers. Let $p^*$ and $q^*$ denote the rational cuts associated with $p$ and $q$. Then: :$p^* q^* = \\paren {p q}^*$ Thus the operation of multiplication on the set of rational cuts is closed."}
+{"_id": "18508", "title": "Set of Rational Cuts forms Ordered Field", "text": "Let $\\RR$ denote the set of rational cuts. Let $\\struct {\\RR, +, \\times, \\le}$ denote the ordered structure formed from $\\RR$ and: :the operation $+$ of addition of cuts :the operation $\\times$ of multiplication of cuts :the ordering $\\le$ of cuts. Then $\\struct {\\RR, + \\times, \\le}$ is an ordered field."}
+{"_id": "18513", "title": "Infimum of Subset of Real Numbers May or May Not be in Subset", "text": "Let $S \\subset \\R$ be a proper subset of the set $\\R$ of real numbers. Let $S$ admit an infimum $m$. Then $m$ may or may not be an element of $S$."}
+{"_id": "10327", "title": "Apotome is Irrational", "text": "Every apotome is irrational, i.e.: : $\\displaystyle \\forall a, b \\in \\set {x \\in \\R_{>0} : x^2 \\in \\Q}: \\paren {\\frac a b \\notin \\Q \\land \\paren {\\frac a b}^2 \\in \\Q} \\implies \\paren {\\paren {a - b} \\notin \\Q \\land \\paren {a - b}^2 \\notin \\Q}$ {{:Euclid:Proposition/X/73}}"}
+{"_id": "18519", "title": "Basis Condition for Coarser Topology/Corollary 1", "text": "If $\\BB_1$ and $\\BB_2$ satisfy: :$\\forall U \\in \\BB_1: \\forall x \\in U: \\exists V \\in \\BB_2: x \\in V \\subseteq U$ then $\\tau_1$ is coarser than $\\tau_2$."}
+{"_id": "2138", "title": "Derivatives of PGF of Discrete Uniform Distribution", "text": "Let $X$ be a discrete random variable with the discrete uniform distribution with parameter $n$. Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are: :$\\dfrac {\\d^m} {\\d s^m} \\map {\\Pi_X} s = \\begin{cases} \\displaystyle \\dfrac 1 n \\sum_{k \\mathop = m}^n k^{\\underline m} s^{k - m} & : m \\le n \\\\ 0 & : k > n \\end{cases}$ where $k^{\\underline m}$ is the falling factorial."}
+{"_id": "18532", "title": "Set of Doubletons of Natural Numbers is Countable", "text": "Let $S$ be the set defined as: :$S = \\set {\\set {n_1, n_2}: n_1, n_2 \\in \\N, n_1 \\ne n_2}$ where $\\N$ denotes the set of natural numbers. Then $S$ is countably infinite."}
+{"_id": "18545", "title": "Frege Set Theory is Logically Inconsistent", "text": "The system of axiomatic set theory that is Frege set theory is inconsistent."}
+{"_id": "18557", "title": "Empty Class is Unique", "text": "There is exactly one empty class."}
+{"_id": "18561", "title": "Existence of Set is Equivalent to Existence of Empty Set", "text": "Let $V$ be a basic universe. Let $P$ be the axiom: :$V$ has at least one element. Then $P$ is equivalent to the axiom of the empty set: :The empty class $\\O$ is a set."}
+{"_id": "2178", "title": "Associativity on Four Elements", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Let $a, b, c, d \\in S$. Then: :$a \\circ b \\circ c \\circ d$ gives a unique answer no matter how the elements are associated."}
+{"_id": "18581", "title": "Intersection of Class is Subset of Intersection of Subclass", "text": "Let $V$ be a basic universe. Let $A$ and $B$ be classes of $V$: :$A \\subseteq V, B \\subseteq V$ such that it is not the case that $A = B = \\O$. Let $\\displaystyle \\bigcap A$ and $\\displaystyle \\bigcap B$ denote the intersection of $A$ and intersection of $B$ respectively. Let $A$ be a subclass of $B$: :$A \\subseteq B$ Then $\\displaystyle \\bigcap B$ is a subset of $\\displaystyle \\bigcap A$: :$\\displaystyle \\bigcap B \\subseteq \\displaystyle \\bigcap A$"}
+{"_id": "2212", "title": "Frobenius's Theorem", "text": "An algebraic associative real division algebra $A$ is isomorphic to $\\R, \\C$ or $\\Bbb H$."}
+{"_id": "10417", "title": "Areas of Circles are as Squares on Diameters/Lemma", "text": "{{:Euclid:Proposition/XII/2/Lemma}}"}
+{"_id": "2239", "title": "Dominance Relation is Ordering", "text": "Let $S$ and $T$ be cardinals. Let $S \\preccurlyeq T$ denote that $S$ is dominated by $T$. Let $\\mathbb S$ be any set of cardinals. Then the relational structure $\\struct {\\mathbb S, \\preccurlyeq}$ is an ordered set. That is, $\\preccurlyeq$ is an ordering (at least partial) on $\\mathbb S$."}
+{"_id": "18628", "title": "Basic Universe is Inductive", "text": "Let $V$ be a basic universe. Then $V$ is an inductive class."}
+{"_id": "2274", "title": "Image of Subset is Image of Restriction", "text": "Let $f: S \\to T$ be a mapping. Let $X \\subseteq S$. Let $f {\\restriction_X}$ be the restriction of $f$ to $X$. Then: :$f \\sqbrk X = \\Img {f {\\restriction_X} }$ where $\\Img f$ denotes the image of $f$, defined as: :$\\Img f = \\set {t \\in T: \\exists s \\in S: t = \\map f s}$"}
+{"_id": "2275", "title": "Binomial Distribution PMF", "text": "The probability mass function (pmf) of a binomially distributed random variable $X$ is equal to: :$\\displaystyle \\Pr \\left({X = x}\\right) = \\binom n x p^x(1-p)^{n-x}$ where $n$ is the number of trials and $p$ is the probability of success."}
+{"_id": "10478", "title": "Successor Mapping of Peano Structure has no Fixed Point", "text": "Let $\\PP = \\struct {P, s, 0}$ be a Peano structure. Then: :$\\forall n \\in P: \\map s n \\ne n$ That is, the successor mapping has no fixed points."}
+{"_id": "2296", "title": "Closed Set is F-Sigma Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $V$ be a closed set of $T$. Then $V$ is an $F_\\sigma$ set of $T$."}
+{"_id": "10488", "title": "Union of Open Intervals of Positive Reals is Set of Strictly Positive Reals", "text": "Let $\\R_{> 0}$ be the set of strictly positive real numbers. For all $x \\in \\R_{> 0}$, let $A_x$ be the open real interval $\\openint 0 x$. Then: :$\\displaystyle \\bigcup_{x \\mathop \\in \\R_{> 0} } A_x = \\R_{> 0}$"}
+{"_id": "10489", "title": "Intersection of Closed Intervals of Positive Reals is Zero", "text": "Let $\\R_{> 0}$ be the set of strictly positive real numbers. For all $x \\in \\R_{> 0}$, let $B_x$ be the closed real interval $\\closedint 0 x$. Then: :$\\displaystyle \\bigcap_{x \\mathop \\in \\R_{> 0} } B_x = \\set 0$"}
+{"_id": "18684", "title": "Minimally Closed Class under Progressing Mapping is Well-Ordered", "text": "$N$ is well-ordered under the inclusion relation."}
+{"_id": "2301", "title": "Sum of Independent Poisson Random Variables is Poisson", "text": "Let $X$ and $Y$ be independent discrete random variables with: :$X \\sim \\Poisson {\\lambda_1}$ and :$Y \\sim \\Poisson {\\lambda_2}$ for some $\\lambda_1, \\lambda_2 \\in \\R_{> 0}$. Then their sum $X + Y$ is distributed: :$X + Y \\sim \\Poisson {\\lambda_1 + \\lambda_2}$"}
+{"_id": "2312", "title": "Order of General Linear Group over Galois Field", "text": "Let $\\GF$ be a Galois field with $p$ elements. Then the order of the general linear group $\\GL {n, \\GF}$ is: :$\\displaystyle \\prod_{j \\mathop = 1}^n \\paren {p^n - p^{j - 1} }$"}
+{"_id": "2313", "title": "Existence of Latin Squares", "text": "For each $n \\in \\Z_{>0}$ there exists at least one Latin square of order $n$."}
+{"_id": "2315", "title": "Lipschitz Condition implies Uniform Continuity", "text": "Let $\\left({M_1, d_1}\\right)$ and $\\left({M_2, d_2}\\right)$ be metric spaces. Let $g: M_1 \\to M_2$ satisfy the Lipschitz condition. Then $g$ is uniformly continuous on $M_1$."}
+{"_id": "10511", "title": "Repeated Composition of Injection is Injection", "text": "Let $S$ be a set. Let $f: S \\to S$ be an injection. Let the sequence of mappings: :$f^0, f^1, f^2, \\ldots, f^n, \\ldots$ be defined as: :$\\forall n \\in \\N: \\map {f^n} x = \\begin {cases} x & : n = 0 \\\\ \\map f x & : n = 1 \\\\ \\map f {\\map {f^{n - 1} } x} & : n > 1 \\end{cases}$ Then for all $n \\in \\N$, $f^n$ is an injection."}
+{"_id": "10519", "title": "Renaming Mapping from Set of Mappings on Single Element", "text": "Let $X$ and $Y$ be sets. Let $E$ be the set of all mappings from $X$ to $Y$. Let $b \\in X$. Let $\\mathcal R \\subseteq E \\times E$ be the relation on $E$ defined as: :$\\mathcal R := \\set {\\tuple {f, g} \\in \\mathcal R: \\map f b = \\map g b}$ Let $e_b: E / \\mathcal R \\to Y$ be the renaming mapping induced by $\\mathcal R$. Then $e_b$ is a bijection."}
+{"_id": "18722", "title": "Power of Random Variable with Continuous Uniform Distribution has Beta Distribution", "text": "Let $X \\sim \\ContinuousUniform 0 1$ where $\\ContinuousUniform 0 1$ is the continuous uniform distribution on $\\closedint 0 1$. Let $n$ be a positive real number.  Then: :$X^n \\sim \\BetaDist {\\dfrac 1 n} 1$ where $\\operatorname {Beta}$ is the beta distribution."}
+{"_id": "18735", "title": "Variance of Linear Transformation of Random Variable", "text": "Let $X$ be a random variable.  Let $a, b$ be real numbers.  Then we have:  :$\\var {a X + b} = a^2 \\var X$ where $\\var X$ denotes the variance of $X$."}
+{"_id": "18754", "title": "Versed Sine Function is Even", "text": "The versed sine is an even function: :$\\forall \\theta \\in \\R: \\map \\vers {-\\theta} = \\vers \\theta$"}
+{"_id": "10563", "title": "Supremum Metric on Continuous Real Functions is Metric", "text": "Let $\\left[{a \\,.\\,.\\, b}\\right] \\subseteq \\R$ be a closed real interval. Let $\\mathscr C \\left[{a \\,.\\,.\\, b}\\right]$ be the set of all continuous functions $f: \\left[{a \\,.\\,.\\, b}\\right] \\to \\R$. Let $d$ be the supremum metric on  $\\mathscr C \\left[{a \\,.\\,.\\, b}\\right]$. Then $d$ is a metric."}
+{"_id": "18758", "title": "P-Norm of Real Sequence is Strictly Decreasing Function of P", "text": "Let $p \\ge 1$ be a real number. Let $\\ell^p$ denote the $p$-sequence space. Let $\\mathbf x = \\sequence {x_n} \\in \\ell^p$. Suppose $\\mathbf x$ is not a sequence of zero elements. Let $\\norm {\\mathbf x}_p$ denote the $p$-norm. Then the mapping $p \\to \\norm {\\mathbf x}_p$ is strictly decreasing {{WRT}} $p$."}
+{"_id": "18760", "title": "Extension of Half-Range Fourier Cosine Function to Symmetric Range", "text": "Let $\\map f x$ be a real function defined on the interval $\\openint 0 \\lambda$. Let $\\map f x$ be represented by the half-range Fourier cosine series $\\map S x$: :$\\map f x \\sim \\map S x = \\dfrac {a_0} 2 + \\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n \\cos \\frac {n \\pi x} \\lambda$ where for all $n \\in \\Z_{> 0}$: :$a_n = \\displaystyle \\frac 2 \\lambda \\int_0^\\lambda \\map f x \\cos \\frac {n \\pi x} \\lambda \\rd x$ Then $\\map S x$ also represents the extension to the even function $g: \\openint {-\\lambda} \\lambda \\to \\R$ of $f$, defined as: :$\\forall x \\in \\openint {-\\lambda} \\lambda: \\map g x = \\begin {cases} \\map f x & : x > 0 \\\\ \\map f {-x} & : x < 0 \\\\ \\displaystyle \\lim_{x \\mathop \\to 0} \\map g x & : x = 0 \\end {cases}$"}
+{"_id": "10572", "title": "P-adic Metric is Metric", "text": "Let $p \\in \\N$ be a prime. Let $\\norm{\\,\\cdot\\,}_p: \\Q \\to \\R_{\\ge 0}$ be the $p$-adic norm on $\\Q$. Let $d_p$ be the $p$-adic metric on $\\Q$: :$\\forall x, y \\in \\Q: \\map {d_p} {x, y} = \\norm{x - y}_p$ Then $d_p$ is a metric."}
+{"_id": "18769", "title": "Sum over Integers of Sine of n + alpha of theta over n + alpha", "text": "For $0 < \\theta < 2 \\pi$: :$\\displaystyle \\sum_{n \\mathop \\in \\Z} \\dfrac {\\sin \\paren {n + \\alpha} \\theta} {n + \\alpha} = \\pi$"}
+{"_id": "18773", "title": "Mittag-Leffler Expansion for Cotangent Function/Real Domain", "text": ":$\\displaystyle \\dfrac 1 \\alpha + \\sum_{n \\mathop \\ge 1} \\dfrac {2 \\alpha} {\\alpha^2 - n^2} = \\pi \\cot \\pi \\alpha$"}
+{"_id": "18779", "title": "Sum of Cosines of Arithmetic Sequence of Angles/Formulation 1", "text": "{{begin-eqn}} {{eqn | l = \\sum_{k \\mathop = 0}^n \\map \\cos {\\theta + k \\alpha}       | r = \\cos \\theta + \\map \\cos {\\theta + \\alpha} + \\map \\cos {\\theta + 2 \\alpha} + \\map \\cos {\\theta + 3 \\alpha} + \\dotsb }} {{eqn | r = \\frac {\\map \\sin {\\alpha \\paren {n + 1} / 2} } {\\map \\sin {\\alpha / 2} } \\map \\cos {\\theta + \\frac {n \\alpha} 2} }} {{end-eqn}}"}
+{"_id": "2394", "title": "Count of Binary Operations Without Identity", "text": "Let $S$ be a set whose cardinality is $n$. The number $N$ of possible different binary operations which do not have an identity element that can be applied to $S$ is given by: :$N = n^{\\left({\\left({n-1}\\right)^2 + 1}\\right)} \\left({n^{2 \\left({n-1}\\right)} - 1}\\right)$"}
+{"_id": "10589", "title": "Addition of Coordinates on Cartesian Plane under Chebyshev Distance is Continuous Function", "text": "Let $\\R^2$ be the real number plane. Let $d_\\infty$ be the Chebyshev distance on $\\R^2$. Let $f: \\R^2 \\to \\R$ be the real-valued function defined as: :$\\forall \\left({x_1, x_2}\\right) \\in \\R^2: f \\left({x_1, x_2}\\right) = x_1 + x_2$ Then $f$ is continuous."}
+{"_id": "2399", "title": "Equivalence of Definitions of Order of Group Element", "text": "{{TFAE|def = Order of Group Element}} Let $G$ be a group whose identity is $e$. Let $x \\in G$."}
+{"_id": "18802", "title": "Maximum Rule for Continuous Functions", "text": "Let $\\struct {S, \\tau}$ be a topological space. Let $f, g: S \\to \\R$ be continuous real-valued functions. Let $\\max \\set {f, g}: S \\to \\R$ denote the pointwise maximum of $f$ and $g$. Then: :$\\max \\set {f, g}$ is continuous."}
+{"_id": "10621", "title": "Continuity of Mapping to Cartesian Product under Chebyshev Distance", "text": "Let $M_1 = \\left({A_1, d_1}\\right), M_2 = \\left({A_2, d_2}\\right), \\ldots, M_n = \\left({A_n, d_n}\\right)$ be metric spaces. Let $\\displaystyle \\mathcal A = \\prod_{i \\mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \\ldots, A_n$. Let $d_\\infty: \\mathcal A \\times \\mathcal A \\to \\R$ be the Chebyshev distance on $\\mathcal A$: : $\\displaystyle d_\\infty \\left({x, y}\\right) = \\max_{i \\mathop = 1}^n \\left\\{ {d_i \\left({x_i, y_i}\\right)}\\right\\}$ where $x = \\left({x_1, x_2, \\ldots, x_n}\\right), y = \\left({y_1, y_2, \\ldots, y_n}\\right) \\in \\mathcal A$. For all $i \\in \\left\\{ {1, 2, \\ldots, n}\\right\\}$, let $\\operatorname{pr}_i: \\mathcal A \\to A_i$ be the $i$th projection on $\\mathcal A$: :$\\forall a \\in \\mathcal A: \\operatorname{pr}_i \\left({a}\\right) = a_i$ where $a = \\left({a_1, a_2, \\ldots, a_n}\\right) \\in \\mathcal A$. Let $M' = \\left({X, d'}\\right)$ be a metric space. Let $f: X \\to \\mathcal A$ be a mapping. Then $f$ is continuous on $X$ {{iff}} each of $\\operatorname{pr}_i \\circ f: X \\to A_i$ is continuous on $X$."}
+{"_id": "2437", "title": "Union of Mappings with Disjoint Domains is Mapping", "text": "Let $S_1, S_2, T_1, T_2$ be sets. Let $f: S_1 \\to T_1$ and $g: S_2 \\to T_2$ be mappings. Let $h = f \\cup g$ be their union. If $S_1 \\cap S_2 = \\O$, then $h: S_1 \\cup S_2 \\to T_1 \\cup T_2$ is a mapping whose domain is $S_1 \\cup S_2$."}
+{"_id": "18825", "title": "Natural Number m is Less than n iff m is an Element of n", "text": "Let $\\omega$ be the set of natural numbers defined as the von Neumann construction. Let $m, n \\in \\omega$. Then: :$m < n \\iff m \\in n$ That is, every natural number is the set of all smaller natural numbers."}
+{"_id": "10638", "title": "Limit of Image of Sequence/Real Number Line", "text": "Let $f$ be a real function which is continuous on the interval $\\Bbb I$. Let $\\left \\langle {x_n} \\right \\rangle$ be a sequence of points in $\\Bbb I$ such that: : $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = \\xi$ where: : $(1): \\quad \\xi \\in \\Bbb I$ : $(2): \\quad \\displaystyle \\lim_{n \\mathop \\to \\infty} x_n$ denotes the limit of $x_n$. Then: : $\\displaystyle \\lim_{n \\mathop \\to \\infty} f \\left({x_n}\\right) = f \\left({\\xi}\\right)$ That is: : $\\displaystyle \\lim_{n \\mathop \\to \\infty} f \\left({x_n}\\right) = f \\left({\\lim_{n \\mathop \\to \\infty} x_n}\\right)$"}
+{"_id": "18836", "title": "Like Electric Charges Repel", "text": "Let $a$ and $b$ be stationary particles, each carrying an electric charge of $q_a$ and $q_b$ respectively. Let $q_a$ and $q_b$ be of the same sign. That is, let $q_a$ and $q_b$ be like charges. Then the forces exerted by $a$ on $b$, and by $b$ on $a$, are such as to cause $a$ and $b$ to repel each other."}
+{"_id": "18847", "title": "P-Sequence Space with P-Norm forms Banach Space", "text": "Let $\\ell^p$ be a p-sequence space. Let $\\norm {\\, \\cdot \\,}_p$ be a p-norm. Then $\\struct {\\ell^p, \\norm {\\, \\cdot \\,}_p}$ is a Banach space."}
+{"_id": "10658", "title": "Neighbourhood of Point Contains Point of Subset iff Distance is Zero", "text": "Let $M = \\struct {X, d}$ be a metric space. Let $A \\subseteq X$ be a non-empty subset of $X$. Let $x \\in X$. Then every neighborhood of $x$ contains a point of $A$ {{iff}}: :$\\map d {x, A} = 0$ where $\\map d {x, A}$ denotes the distance from $x$ to $A$."}
+{"_id": "10660", "title": "Set is Open iff Union of Open Balls", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $U \\subseteq A$. Then $U$ is open in $M$ {{iff}} it is a union of open balls."}
+{"_id": "10661", "title": "Empty Set is Open and Closed in Metric Space", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Then the empty set $\\varnothing$ is both open and closed in $M$."}
+{"_id": "2479", "title": "Equivalence of Definitions of Independent Subgroups", "text": "{{TFAE|def = Independent Subgroups}} Let $G$ be a group whose identity is $e$. Let $\\left \\langle {H_n} \\right \\rangle$ be a sequence of independent subgroups of $G$."}
+{"_id": "18866", "title": "Condition for Factoring of Quotient Mapping between Modulo Addition Groups", "text": "Let $m, n \\in \\Z_{>0}$ be strictly positive integers. Let $\\struct {\\Z, +}$ denote the additive group of integers. Let $\\struct {\\Z_m, +_m}$ and $\\struct {\\Z_n, +_n}$ denote the additive groups of integers modulo $m$ and $n$ respectively. Let $f: \\Z \\to \\Z_n$ be the quotient epimorphism from $\\struct {\\Z, +}$ to $\\struct {\\Z_n, +_n}$. Let $q: \\Z \\to \\Z_m$ be the quotient epimorphism from $\\struct {\\Z, +}$ to $\\struct {\\Z_m, +_m}$. Then Then $N \\subseteq K$ Then: :there exists a group homomorphism $\\psi: \\struct {\\Z_m, +_m} \\to \\struct {\\Z_n, +_n}$ {{iff}}  :$m \\divides n$ where $\\divides$ denotes divisibility."}
+{"_id": "2494", "title": "Construction of Inverse Completion/Quotient Structure", "text": "Let the quotient structure defined by $\\boxtimes$ be: : $\\displaystyle \\left({T', \\oplus'}\\right) := \\left({\\frac {S \\times C} \\boxtimes, \\oplus_\\boxtimes}\\right)$ where $\\oplus_\\boxtimes$ is the operation induced on $\\displaystyle \\frac {S \\times C} \\boxtimes$ by $\\oplus$. === Quotient Structure is Commutative Semigroup === {{:Construction of Inverse Completion/Quotient Structure is Commutative Semigroup}} === Quotient Mapping is Injective === {{:Construction of Inverse Completion/Quotient Mapping is Injective}} === Quotient Mapping is Monomorphism === {{:Construction of Inverse Completion/Quotient Mapping is Monomorphism}} === Image of Quotient Mapping is Subsemigroup === {{:Construction of Inverse Completion/Image of Quotient Mapping is Subsemigroup}} === Quotient Mapping to Image is Isomorphism === {{:Construction of Inverse Completion/Quotient Mapping to Image is Isomorphism}} === Image of Cancellable Elements in Quotient Mapping === {{:Construction of Inverse Completion/Quotient Mapping/Image of Cancellable Elements}}"}
+{"_id": "10691", "title": "Inclusion Mapping on Metric Space is Continuous", "text": "Let $M = \\left({A, d}\\right)$ be a metric space. Let $\\left({H, d_H}\\right)$ be a metric subspace of $M$. Then the inclusion mapping $i_H: H \\to A$ is continuous."}
+{"_id": "18884", "title": "Set of Transpositions is not Subgroup of Symmetric Group", "text": "Let $S$ be a finite set with $n$ elements such that $n > 2$. Let $G = \\struct {\\map \\Gamma S, \\circ}$ denote the symmetric group on $S$. Let $H \\subseteq G$ denote the set of all transpositions of $S$ along with the identity mapping which moves no elements of $S$. Then $H$ does not form a subgroup of $G$."}
+{"_id": "18886", "title": "Homomorphism from Reals to Circle Group/Corollary", "text": "Let $\\struct {\\R, +}$ be the additive group of real numbers. Let $\\struct {C_{\\ne 0}, \\times}$ be the multiplicative group of complex numbers. Let $\\phi: \\struct {\\R, +} \\to \\struct {C_{\\ne 0}, \\times}$ be the mapping defined as: :$\\forall x \\in \\R: \\map \\phi x = \\cos x + i \\sin x$ Then $\\phi$ is a (group) homomorphism."}
+{"_id": "2508", "title": "Projection onto Ideal of External Direct Sum of Rings", "text": "Let $\\left({R_1, +_1, \\circ_1}\\right), \\left({R_2, +_2, \\circ_2}\\right), \\ldots, \\left({R_n, +_n, \\circ_n}\\right)$ be rings. Let $\\displaystyle \\left({R, +, \\circ}\\right) = \\prod_{k \\mathop = 1}^n \\left({R_k, +_k, \\circ_k}\\right)$ be their direct product. For each $k \\in \\left[{1 \\,.\\,.\\, n}\\right]$, let: :$R'_k = \\left\\{{\\left({x_1, \\ldots, x_n}\\right) \\in R: \\forall j \\ne k: x_j = 0}\\right\\}$ Let $\\operatorname{pr}_k: R \\to R'_k$ be the projection on the $k$th coordinate of $\\left({R, +, \\circ}\\right)$ onto $R'_k$. Then $\\operatorname{pr}_k$ is an epimorphism."}
+{"_id": "18893", "title": "Edgeless Graph is Bipartite", "text": "Let $N_n$ denote the edgeless graph with $n$ vertices. Then $N_n$ is a bipartite graph."}
+{"_id": "2513", "title": "Set of Subfields forms Complete Lattice", "text": "Let $\\struct {F, +, \\circ}$ be a field. Let $\\mathbb F$ be the set of all subfields of $F$. Then $\\struct {\\mathbb F, \\subseteq}$ is a complete lattice."}
+{"_id": "18905", "title": "Simple Graph of Maximum Size is Complete Graph", "text": "Let $G$ be a simple graph of order $n$ such that $n \\ge 1$. Let $G$ have the largest size of all simple graphs of order $n$. Then: :$G$ is the complete graph $K_n$ :its size is $\\dfrac {n \\paren {n - 1} } 2$."}
+{"_id": "10717", "title": "Sum of Integrals on Complementary Sets", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $A, E \\in \\Sigma$ with $A \\subseteq E$. Let $f$ be a $\\mu$-integrable function on $X$. Then  :$\\displaystyle \\int_E f \\rd \\mu = \\int_A f \\rd \\mu + \\int_{E \\mathop \\setminus A} f \\rd \\mu$"}
+{"_id": "18915", "title": "Characteristics of Cycle Graph", "text": "Let $G = \\struct {V, E}$ be an (undirected) graph whose order is greater than $2$. Then $G$ is a cycle graph {{iff}}: :$G$ is connected :every vertex of $G$ is adjacent to $2$ other vertices :every edge of $G$ is adjacent to $2$ other edges."}
+{"_id": "10726", "title": "Neighborhood of Point in Metrizable Space contains Closed Neighborhood", "text": "Let $T = \\struct {S, \\tau}$ be a metrizable topological space. Let $x \\in S$ be an arbitrary point of $T$. Let $N$ be a neighborhood of $x$. Then $N$ has as a subset a neighborhood $V$ of $x$ such that $V$ is closed."}
+{"_id": "10741", "title": "Induced Neighborhood Space is Neighborhood Space", "text": "Let $S$ be a set. Let $\\tau$ be a topology on $S$, thus forming the topological space $\\left({S, \\tau}\\right)$. Let $\\left({S, \\mathcal N}\\right)$ be the neighborhood space induced by $\\left({S, \\tau}\\right)$. Then $\\left({S, \\mathcal N}\\right)$ is a neighborhood space."}
+{"_id": "18933", "title": "Order 1 Simple Graph is Unique up to Isomorphism", "text": "Let $G_1 = \\struct {\\map V {G_1}, \\map E {G_1} }$ and $G_2 = \\struct {\\map V {G_2}, \\map E {G_2} }$ be simple graphs of order $1$. Then $G_1$ and $G_2$ are isomorphic."}
+{"_id": "2564", "title": "Definition of Polynomial from Polynomial Ring over Sequence", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity. Let $\\struct {P \\sqbrk R, \\oplus, \\odot}$ be the polynomial ring over the set of all sequences in $R$: :$P \\sqbrk R = \\set {\\sequence {r_0, r_1, r_2, \\ldots} }$ where the operations $\\oplus$ and $\\odot$ on $P \\sqbrk R$ be defined as: {{:Definition:Operations on Polynomial Ring of Sequences}} Let $\\struct {R \\sqbrk X, +, \\circ}$ be the ring of polynomials over $R$ in $X$. {{explain|Strictly speaking the definition in terms of Definition:Polynomial Form is needed here, with $X$ specifically being an Definition:Indeterminate (Polynomial Theory) or Definition:Transcendental over Integral Domain.}} Then $\\struct {R \\sqbrk X, +, \\circ}$ and $\\struct {P \\sqbrk R, \\oplus, \\odot}$ are isomorphic."}
+{"_id": "10757", "title": "Bertrand's Theorem", "text": "Let $U: \\R_{>0} \\to \\R$ be analytic for $r > 0$. Let $M > 0$ be a nonvanishing angular momentum such that a stable circular orbit exists. Suppose that every orbit sufficiently close to the circular orbit is closed. Then $U$ is either $k r^2$ or $-\\dfrac k r$ (for $k > 0$) up to an additive constant."}
+{"_id": "10758", "title": "Cube Root of 2 is Irrational", "text": ":$\\sqrt [3] 2$ is irrational."}
+{"_id": "2565", "title": "Integers under Multiplication form Countably Infinite Semigroup", "text": "The set of integers under multiplication $\\struct {\\Z, \\times}$ is a countably infinite semigroup."}
+{"_id": "18955", "title": "Third Derivative of Natural Logarithm Function", "text": "Let $\\ln x$ be the natural logarithm function. Then: :$\\map {D^3_x} {\\ln x} = \\dfrac 2 {x^3}$"}
+{"_id": "10766", "title": "Bounded Piecewise Continuous Function may not have One-Sided Limits", "text": "Let $f$ be a real function defined on a closed interval $\\left[{a \\,.\\,.\\, b}\\right]$, $a < b$.  Let $f$ be a bounded piecewise continuous function. {{:Definition:Bounded Piecewise Continuous Function}} Then it is not necessarily the case that $f$ is a piecewise continuous function with one-sided limits: {{:Definition:Piecewise Continuous Function with One-Sided Limits}}"}
+{"_id": "18959", "title": "Linear Second Order ODE/y'' = 1 over 1 - x^2", "text": "The second order ODE: :$(1): \\quad y'' = \\dfrac 1 {1 - x^2}$ has the general solution: :$y = x \\tanh^{-1} x + \\map \\ln {1 - x^2} + C x + D$"}
+{"_id": "2598", "title": "Universal Property of Polynomial Ring/Free Monoid on Set", "text": "Let $R, S$ be commutative and unitary rings. Let $\\left\\langle{s_j}\\right\\rangle_{j \\mathop \\in J}$ be an indexed family of elements of $S$. Let $\\psi: R \\to S$ be a ring homomorphism. Let $R \\left[{\\left\\{{X_j: j \\in J}\\right\\}}\\right]$ be a polynomial ring. Then there exists a unique evaluation homomorphism $\\phi: R \\left[{\\left\\{{X_j: j \\in J}\\right\\}}\\right] \\to S$ at $\\langle s_j\\rangle_{j \\in J}$ extending $\\psi$."}
+{"_id": "2601", "title": "Polynomial Addition is Associative", "text": "Addition of polynomials is an associative operation."}
+{"_id": "2602", "title": "Null Polynomial is Additive Identity", "text": "The set of polynomial forms has an additive identity. {{explain|Context}}"}
+{"_id": "10800", "title": "De Morgan's Laws (Set Theory)/Proof by Induction/Difference with Union/Proof", "text": "Let $\\mathbb T = \\set {T_i: i \\mathop \\in I}$, where each $T_i$ is a set and $I$ is some finite indexing set. Then: : $\\displaystyle S \\setminus \\bigcup_{i \\mathop \\in I} T_i = \\bigcap_{i \\mathop \\in I} \\paren {S \\setminus T_i}$"}
+{"_id": "10808", "title": "Parity Addition is Commutative", "text": "Let $R := \\struct {\\set {\\text{even}, \\text{odd} }, +, \\times}$ be the parity ring. The operation $+$ is commutative: :$\\forall a, b \\in R: a + b = b + a$"}
+{"_id": "10809", "title": "Parity Multiplication is Associative", "text": "Let $R := \\struct {\\set {\\text{even}, \\text{odd} }, +, \\times}$ be the parity ring. The operation $\\times$ is associative: :$\\forall a, b, c \\in R: \\paren {a \\times b} \\times c = a \\times \\paren {b \\times c}$"}
+{"_id": "19007", "title": "Events One of Which equals Union", "text": "Let the probability space of an experiment $\\EE$ be $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $A, B \\in \\Sigma$ be events of $\\EE$, so that $A \\subseteq \\Omega$ and $B \\subseteq \\Omega$. Let $A$ and $B$ be such that: :$A \\cup B = A$ Then whenever $B$ occurs, it is always the case that $A$ occurs as well."}
+{"_id": "2641", "title": "Direct Product iff Nontrivial Idempotent", "text": "Let $R$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Then $R$ is the direct product of two non-trivial rings {{iff}} $R$ contains an idempotent element not equal to $0_R$ or $1_R$."}
+{"_id": "10833", "title": "Real Numbers under Multiplication do not form Group", "text": "The algebraic structure $\\struct {\\R, \\times}$ consisting of the set of real numbers $\\R$ under multiplication $\\times$ is not a group."}
+{"_id": "19027", "title": "Total Variation is Non-Negative", "text": "Let $a, b$ be real numbers with $a < b$. Let $f : \\closedint a b \\to \\R$ be a function of bounded variation. Let $V_f$ be the total variation of $f$ on $\\closedint a b$. Then:  :$V_f \\ge 0$ with equality {{iff}} $f$ is constant."}
+{"_id": "2650", "title": "Numbers of Type Integer a plus b root 2 Form Ordered Integral Domain", "text": "Let $\\Z \\sqbrk {\\sqrt 2}$ denote the set: :$\\Z \\sqbrk {\\sqrt 2} := \\set {a + b \\sqrt 2: a, b \\in \\Z}$ that is, all numbers of the form $a + b \\sqrt 2$ where $a$ and $b$ are integers. Then the algebraic structure: :$\\struct {\\Z \\sqbrk {\\sqrt 2}, +, \\times}$ where $+$ and $\\times$ are conventional addition and multiplication on real numbers, is an ordered integral domain."}
+{"_id": "19044", "title": "Basis Test for Isolated Point", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\BB$ be a synthetic basis of $T$. Let $H \\subseteq S$. Then $x \\in H$ is an isolated point of $H$ {{iff}}: :$\\exists U \\in \\BB : U \\cap H = \\set x$"}
+{"_id": "19050", "title": "Boundary of Boundary is not necessarily Equal to Boundary", "text": "Let $T$ be a topological space. Let $H \\subseteq T$. Let $\\partial H$ denote the boundary of $H$. While it is true that: :$\\map \\partial {\\partial H} \\subseteq \\partial H$ it is not necessarily the case that: :$\\map \\partial {\\partial H} = \\partial H$"}
+{"_id": "10861", "title": "Right Congruence Class Modulo Subgroup is Right Coset", "text": "Let $\\mathcal R^r_H$ be the equivalence defined as right congruence modulo $H$. The equivalence class $\\eqclass g {\\mathcal R^r_H}$ of an element $g \\in G$ is the right coset $H g$. This is known as the '''right congruence class of $g \\bmod H$'''."}
+{"_id": "19067", "title": "Equivalence of Definitions of Filter Basis", "text": "{{TFAE|def = Filter Basis}} Let $S$ be a set. Let $\\FF$ be a filter on $S$."}
+{"_id": "10877", "title": "Lexicographic Order on Pair of Well-Ordered Sets is Well-Ordering", "text": "Let $\\struct {S_1, \\preceq_1}$ and $\\struct {S_2, \\preceq_2}$ be ordered sets. Let $\\preccurlyeq$ be the lexicographic order on $S_1 \\times S_2$''': :$\\tuple {x_1, x_2} \\preccurlyeq \\tuple {y_1, y_2} \\iff \\tuple {x_1 \\prec_1 y_1} \\lor \\tuple {x_1 = y_1 \\land x_2 \\preceq_2 y_2}$ Then: :$\\preccurlyeq$ is a well-ordering on $S_1 \\times S_2$ {{iff}} :both $\\preceq_1$ and $\\preceq_2$ are well-orderings."}
+{"_id": "19070", "title": "Product of Functions of Bounded Variation is of Bounded Variation", "text": "Let $a, b$ be real numbers with $a < b$. Let $f, g : \\closedint a b \\to \\R$ be functions of bounded variation. Let $V_f$ and $V_g$ be the total variations of $f$ and $g$ respectively.  Then $f \\times g$ is of bounded variation with: :$V_{f \\times g} \\le A V_f + B V_g$ where: :$V_{f \\times g}$ denotes the total variation of $f \\times g$ :$A, B$ are non-negative real numbers."}
+{"_id": "19069", "title": "Difference of Functions of Bounded Variation is of Bounded Variation", "text": "Let $a, b$ be real numbers with $a < b$. Let $f, g : \\closedint a b \\to \\R$ be functions of bounded variation. Let $V_f$ and $V_g$ be the total variations of $f$ and $g$ respectively.  Then $f - g$ is of bounded variation with: :$V_{f - g} \\le V_f + V_g$ where $V_{f - g}$ denotes the total variation of $f - g$."}
+{"_id": "2692", "title": "Approximation to Stirling's Formula for Gamma Function", "text": "Let: :$D_\\epsilon = \\left\\{{z \\in \\C : \\left\\vert{\\operatorname{Arg} \\left({z}\\right)}\\right\\vert < \\pi - \\epsilon,\\ \\left\\vert{z}\\right\\vert > 1}\\right\\}$ where: :$\\left\\vert{\\operatorname{Arg} \\left({z}\\right)}\\right\\vert$ denotes the absolute value of the principal argument of $z$ :$\\left\\vert{z}\\right\\vert$ denotes the modulus of $z$ :$\\epsilon \\in \\R_{>0}$. Then for all $z \\in D_\\epsilon$, the gamma function of $z$ satisfies: :$\\Gamma \\left({z}\\right) = \\sqrt{\\dfrac {2 \\pi} z} \\left({\\dfrac z e}\\right)^z\\left({1 + \\mathcal O \\left({z^{-1} }\\right)}\\right)$ where $\\mathcal O \\left({z^{-1} }\\right)$ denotes big-O of $z^{-1}$."}
+{"_id": "19082", "title": "Cauchy Sequence in Metric Space is not necessarily Convergent", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $\\sequence {x_n}$ be a Cauchy sequence in $M$. Then it is not necessarily the case that $M$ is a convergent sequence in $M$."}
+{"_id": "10892", "title": "Integers are not Close Packed", "text": "The integers $\\Z$ are not close packed. That is: :$\\forall n \\in \\Z: \\not \\exists m \\in \\Z: n < m < n + 1$"}
+{"_id": "2700", "title": "Linear Combination of Integers is Ideal", "text": "Let $a, b$ be any integers. Let $\\Bbb S = \\set {a x + b y: x, y \\in \\Z}$. Then the algebraic structure: :$\\struct {\\Bbb S, +, \\times}$ is an ideal of $\\Z$."}
+{"_id": "10900", "title": "Negative of Sum of Real Numbers/Corollary", "text": ":$\\forall x, y \\in \\R: -\\paren {x - y} = -x + y$"}
+{"_id": "10905", "title": "Real Division by One", "text": ":$\\forall x, y \\in \\R: \\dfrac x 1 = x$"}
+{"_id": "10909", "title": "Reciprocal of Real Number is Non-Zero", "text": ":$\\forall x \\in \\R: x \\ne 0 \\implies \\dfrac 1 x \\ne 0$"}
+{"_id": "19102", "title": "Differentiable Function with Bounded Derivative is Absolutely Continuous", "text": "Let $a, b$ be real numbers with $a < b$. Let $f: \\closedint a b \\to \\R$ be a continuous function. Let $f$ be differentiable on $\\openint a b$, with bounded derivative. Then $f$ is absolutely continuous."}
+{"_id": "19101", "title": "Limit Points in Open Extension Space/Subset", "text": "Let $U \\subseteq S^*_p$. Then $p$ is a limit point of $U$."}
+{"_id": "2723", "title": "Polynomial Forms is PID Implies Coefficient Ring is Field", "text": "Let $D$ be an integral domain. Let $D \\sqbrk X$ be the ring of polynomial forms in $X$ over $D$. Let $D sqbrk X$ be a principal ideal domain; Then $D$ is a field."}
+{"_id": "10916", "title": "Sum of Strictly Positive Real Numbers is Strictly Positive", "text": ":$x, y \\in \\R_{>0} \\implies x + y \\in \\R_{>0}$"}
+{"_id": "19116", "title": "Lagrange's Theorem (Number Theory)", "text": "Let $f$ be a polynomial in one variable of degree $n$ over $\\Z_p$ for some prime $p$. Then $f$ has at most $n$ roots in $\\Z_p$."}
+{"_id": "2735", "title": "Finite Intersection of Regular Open Sets is Regular Open", "text": "Let $T$ be a topological space. Let $n \\in \\N$. Suppose that: :$\\forall i \\in \\set {1, 2, \\dotsc, n}: H_i \\subseteq T$ where all the $H_i$ are regular open in $T$. That is: :$\\forall i \\in \\set {1, 2, \\dotsc, n}: H_i = H_i^{- \\circ}$ where $H_i^{- \\circ}$ denotes the interior of the closure of $H_i$. Then $\\displaystyle \\bigcap_{i \\mathop = 1}^n H_i$ is regular open in $T$. That is: :$\\displaystyle \\bigcap_{i \\mathop = 1}^n H_i = \\paren {\\bigcap_{i \\mathop = 1}^n H_i}^{- \\circ}$"}
+{"_id": "19153", "title": "Closure of Subspace of Normed Vector Space is Subspace", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,}}$ be a normed vector space. Let $Y \\subseteq X$ be a subspace of $X$. Let $Y^-$ be the closure of $Y$. Then $Y^- \\subseteq X$ is also a subspace of $X$."}
+{"_id": "10963", "title": "Not every Open Set is F-Sigma Set", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $V$ be an open set of $T$. Then it is not necessarily the case that $V$ is an $F_\\sigma$ set of $T$."}
+{"_id": "19160", "title": "Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,}}$ is a normed vector space. Let $D \\subseteq X$ be a subset of $X$. Let $D^-$ be the closure of $D$. Then $D$ is dense iff $D^- = X$."}
+{"_id": "2787", "title": "Tychonoff Space is Preserved under Homeomorphism", "text": "If $T_A$ is a Tychonoff (completely regular) space, then so is $T_B$."}
+{"_id": "2788", "title": "Normal Space is Preserved under Homeomorphism", "text": "If $T_A$ is a normal space, then so is $T_B$."}
+{"_id": "19192", "title": "Left and Right Inverses of Square Matrix over Field are Equal", "text": "Let $\\Bbb F$ be a field, usually one of the standard number fields $\\Q$, $\\R$ or $\\C$. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\map \\MM n$ denote the matrix space of order $n$ square matrices over $\\Bbb F$. Let $\\mathbf B$ be a left inverse matrix of $\\mathbf A$. Then $\\mathbf B$ is also a right inverse matrix of $\\mathbf A$. Similarly, let $\\mathbf B$ be a right inverse matrix of $\\mathbf A$. Then $\\mathbf B$ is also a right inverse matrix of $\\mathbf A$."}
+{"_id": "11018", "title": "Principle of Finite Induction/Peano Structure", "text": "Let $\\struct {P, s, 0}$ be a Peano structure. Let $S \\subseteq P$. Suppose that: :$(1): \\quad 0 \\in S$ :$(2): \\quad \\forall n: n \\in S \\implies \\map s n \\in S$ Then: :$S = P$"}
+{"_id": "19214", "title": "System of Simultaneous Equations may have Multiple Solutions", "text": "Let $S$ be a system of simultaneous equations. Then it is possible that $S$ may have a solution set which is a singleton."}
+{"_id": "11027", "title": "Fibonacci Number as Sum of Binomial Coefficients", "text": "{{begin-eqn}} {{eqn | lo= \\forall n \\in \\Z_{>0}:       | l = F_n       | r = \\sum_{k \\mathop = 0}^{\\floor {\\frac {n - 1} 2} } \\dbinom {n - k - 1} k       | c =  }} {{eqn | r = \\binom {n - 1} 0 + \\binom {n - 2} 1 + \\binom {n - 3} 2 + \\dotsb + \\binom {n - j} {j - 1} + \\binom {n - j - 1} j       | c = where $j = \\floor {\\frac {n - 1} 2}$ }} {{end-eqn}}"}
+{"_id": "2852", "title": "Countably Metacompact Lindelöf Space is Metacompact", "text": "Let $T = \\struct {S, \\tau}$ be a Lindelöf space which is also countably metacompact. Then $T$ is metacompact."}
+{"_id": "2854", "title": "Dirichlet's Approximation Theorem", "text": "Let $\\alpha, x \\in \\R$. Then there exist integers $a, q$ such that: : $\\gcd \\left\\{{a, q}\\right\\} = 1$, $1 \\le q \\le x$ and: :$\\left|{\\alpha - \\dfrac a q}\\right| \\le \\dfrac 1 {q x}$"}
+{"_id": "11059", "title": "Secant in terms of Tangent", "text": "{{begin-eqn}} {{eqn | l = \\sec x       | r = +\\sqrt {\\tan ^2 x + 1}       | c = if there exists an integer $n$ such that $\\paren {2 n - \\dfrac 1 2} \\pi < x < \\paren {2 n + \\dfrac 1 2} \\pi$ }} {{eqn | l = \\sec x       | r = -\\sqrt {\\tan ^2 x + 1}       | c = if there exists an integer $n$ such that $\\paren {2 n + \\dfrac 1 2} \\pi < x < \\paren {2 n + \\dfrac 3 2} \\pi$ }} {{end-eqn}}"}
+{"_id": "2869", "title": "Compact Hausdorff Topology is Minimal Hausdorff", "text": "Let $T = \\struct {S, \\tau}$ be a Hausdorff space which is compact. Then $\\tau$ is the minimal subset of the power set $\\powerset S$ such that $T$ is a Hausdorff space."}
+{"_id": "2870", "title": "Compact Hausdorff Topology is Maximally Compact", "text": "Let $T = \\struct {S, \\tau}$ be a Hausdorff space which is compact. Then $\\tau$ is maximally compact."}
+{"_id": "11063", "title": "Cosine is Reciprocal of Secant", "text": ":$\\cos \\theta = \\dfrac 1 {\\sec \\theta}$"}
+{"_id": "19259", "title": "Column Equivalence is Equivalence Relation", "text": "Column equivalence is an equivalence relation."}
+{"_id": "11067", "title": "Cosine in terms of Sine", "text": "{{begin-eqn}} {{eqn | l = \\cos x       | r = +\\sqrt {1 - \\sin^2 x}       | c = if there exists an integer $n$ such that $\\paren {2 n - \\dfrac 1 2} \\pi < x < \\paren {2 n + \\dfrac 1 2} \\pi$ }} {{eqn | l = \\cos x       | r = -\\sqrt {1 - \\sin^2 x}       | c = if there exists an integer $n$ such that $\\paren {2 n + \\dfrac 1 2} \\pi < x < \\paren {2 n + \\dfrac 3 2} \\pi$ }} {{end-eqn}}"}
+{"_id": "11077", "title": "Sign of Cosecant", "text": "Let $x$ be a real number. Then: {{begin-eqn}} {{eqn | l = \\csc x       | o = >       | r = 0       | c = if there exists an integer $n$ such that $2 n \\pi < x < \\paren {2 n + 1} \\pi$ }} {{eqn | l = \\csc x       | o = <       | r = 0       | c = if there exists an integer $n$ such that $\\paren {2 n + 1} \\pi < x < \\paren {2 n + 2} \\pi$ }} {{end-eqn}} where $\\csc$ is the real cosecant function."}
+{"_id": "19282", "title": "Square Root of Number Minus Square Root/Proof 2", "text": "Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$. Then: {{:Square Root of Number Minus Square Root}}"}
+{"_id": "2900", "title": "Atlas Belongs to Unique Differentiable Structure", "text": "Let $M$ be a locally Euclidean space of dimension $d$. Let $\\mathcal A$ be an atlas on $M$. Then there exists a unique differentiable structure $\\mathcal F$ on $M$ with $\\mathcal A \\in \\mathcal F$."}
+{"_id": "2905", "title": "Topological Space with One Quasicomponent is Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which has one quasicomponent. Then $T$ is connected."}
+{"_id": "19289", "title": "Multiple of Column Added to Column of Determinant", "text": "Let $\\mathbf A = \\begin {bmatrix} a_{1 1} & \\cdots & a_{1 r} & \\cdots & a_{1 s} & \\cdots & a_{1 n} \\\\ a_{2 1} & \\cdots & a_{2 r} & \\cdots & a_{2 s} & \\cdots & a_{2 n} \\\\  \\vdots & \\ddots &  \\vdots & \\ddots &  \\vdots & \\ddots &  \\vdots \\\\ a_{n 1} & \\cdots & a_{n r} & \\cdots & a_{n s} & \\cdots & a_{n n} \\\\ \\end {bmatrix}$ be a square matrix of order $n$. Let $\\map \\det {\\mathbf A}$ denote the determinant of $\\mathbf A$. Let $\\mathbf B = \\begin{bmatrix} a_{1 1} & \\cdots & a_{1 r} + \\lambda a_{1 s} & \\cdots & a_{1 s} & \\cdots & a_{1 n} \\\\ a_{2 1} & \\cdots & a_{2 r} + \\lambda a_{2 s} & \\cdots & a_{2 s} & \\cdots & a_{2 n} \\\\  \\vdots & \\ddots &                    \\vdots & \\ddots &  \\vdots & \\ddots &  \\vdots \\\\ a_{n 1} & \\cdots & a_{n r} + \\lambda a_{n s} & \\cdots & a_{n s} & \\cdots & a_{n n} \\\\ \\end{bmatrix}$. Then $\\map \\det {\\mathbf B} = \\map \\det {\\mathbf A}$. That is, the value of a determinant remains unchanged if a constant multiple of any column is added to any other column."}
+{"_id": "2908", "title": "Integral Closure is Integrally Closed", "text": "Let $A \\subseteq B$ be an extension of commutative rings with unity. Let $C$ be the integral closure of $A$ in $B$. Then $C$ is integrally closed."}
+{"_id": "19297", "title": "Determinant of Lower Triangular Matrix", "text": "Let $\\mathbf T_n$ be a lower triangular matrix of order $n$. Let $\\map \\det {\\mathbf T_n}$ be the determinant of $\\mathbf T_n$. Then $\\map \\det {\\mathbf T_n}$ is equal to the product of all the diagonal elements of $\\mathbf T_n$. That is: :$\\displaystyle \\map \\det {\\mathbf T_n} = \\prod_{k \\mathop = 1}^n a_{k k}$"}
+{"_id": "2916", "title": "Relationship between Component Types", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $p \\in S$. Let: : $A$ be the arc component of $p$ : $P$ be the path component of $p$ : $C$ be the component of $p$ : $Q$ be the quasicomponent of $p$. Then: :$A \\subseteq P \\subseteq C \\subseteq Q$ In general, the inclusions do not hold in the other direction."}
+{"_id": "2928", "title": "Ultraconnected Space is T4", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is ultraconnected. Then $T$ is a $T_4$ space."}
+{"_id": "19317", "title": "Squares Ending in n Occurrences of m-Digit Pattern", "text": "Suppose there exists some integer $x$ such that $x^2$ ends in some $m$-digit pattern ending in an odd number not equal to $5$ and is preceded by another odd number, i.e.: :$\\exists x \\in \\Z: x^2 \\equiv \\sqbrk {1 a_1 a_2 \\cdots a_m} \\pmod {2 \\times 10^m}$ where $a_m$ is odd, $a_m \\ne 5$ and $m \\ge 1$. Then for any $n \\ge 1$, there exists some integer with not more than $m n$-digits such that its square ends in $n$ occurrences of the $m$-digit pattern."}
+{"_id": "2935", "title": "Locally Arc-Connected Space is Locally Path-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is locally arc-connected. Then $T$ is also locally path-connected."}
+{"_id": "2939", "title": "Unique Factorization Domain is Integrally Closed", "text": "Let $A$ be a unique factorization domain (UFD). Then $A$ is integrally closed."}
+{"_id": "11136", "title": "Lindelöf's Lemma", "text": "Let $C$ be a set of open real sets. Let $S$ be a real set that is covered by $C$. Then there exists a countable subset of $C$ that covers $S$."}
+{"_id": "11146", "title": "Mellin Transform of Heaviside Step Function", "text": "Let $c$ be a constant real number. Let $u_c \\left({t}\\right)$ be the Heaviside step function. Let $\\mathcal M$ be the Mellin transform. Then: :$\\mathcal M \\left\\{ {u_c \\left({t}\\right)}\\right\\} \\left({s}\\right) = - \\dfrac {c^s} s$ for $c > 0, \\Re \\left({s}\\right) < 0$."}
+{"_id": "11162", "title": "Complex Numbers as External Direct Product", "text": "Let $\\struct {\\C_{\\ne 0}, \\times}$ be the group of non-zero complex numbers under multiplication. Let $\\struct {\\R_{> 0}, \\times}$ be the group of positive real numbers under multiplication. Let $\\struct {K, \\times}$ be the circle group. Then: :$\\struct {\\C_{\\ne 0}, \\times} \\cong \\struct {\\R_{> 0}, \\times} \\times \\struct {K, \\times}$ {{explain|It is apparent that the second $\\times$ is Cartesian product, but this is not obvious.}}"}
+{"_id": "19359", "title": "Definite Integral from 0 to Half Pi of Square of Logarithm of Cosine x", "text": ":$\\displaystyle \\int_0^{\\pi/2} \\paren {\\map \\ln {\\cos x} }^2 \\rd x = \\frac \\pi 2 \\paren {\\ln 2}^2 + \\frac {\\pi^3} {24}$"}
+{"_id": "2994", "title": "Metric Space Completeness is Preserved by Isometry", "text": "Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces. Let $\\phi: M_1 \\to M_2$ be an isometry. If $M_1$ is complete then so is $M_2$."}
+{"_id": "19378", "title": "3 Proper Integer Heronian Triangles whose Area and Perimeter are Equal", "text": "There are exactly $3$ proper integer Heronian triangles whose area and perimeter are equal. These are the triangles whose sides are: :$\\tuple {6, 25, 29}$ :$\\tuple {7, 15, 20}$ :$\\tuple {9, 10, 17}$"}
+{"_id": "3004", "title": "Multiplication is Superfunction", "text": "The function $f: \\C \\to \\C$, defined as: :$\\map f z = z \\times c$ is a superfunction for any complex number $c$."}
+{"_id": "19417", "title": "Integration by Substitution/Definite Integral", "text": "The definite integral of $f$ from $a$ to $b$ can be evaluated by: :$\\displaystyle \\int_{\\map \\phi a}^{\\map \\phi b} \\map f t \\rd t = \\int_a^b \\map f {\\map \\phi u} \\dfrac \\d {\\d u} \\map \\phi u \\rd u$ where $x = \\map \\phi u$."}
+{"_id": "3039", "title": "Discrete Space is Non-Meager", "text": "Let $T = \\left({S, \\tau}\\right)$ be a discrete topological space. Then $T$ is non-meager."}
+{"_id": "3053", "title": "Subset of Indiscrete Space is Dense-in-itself", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Let $H \\subseteq S$ be a subset of $S$ containing more than one point. Then $H$ is dense-in-itself."}
+{"_id": "3054", "title": "Indiscrete Space is Non-Meager", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Then $T$ is non-meager."}
+{"_id": "11245", "title": "Equivalence of Definitions of Minimal Element", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $T \\subseteq S$ be a subset of $S$. {{TFAE|def = Minimal Element}}"}
+{"_id": "11249", "title": "Non-Zero Natural Numbers under Addition form Semigroup", "text": "Let $\\N_{>0}$ be the set of natural numbers without zero, that is: :$\\N_{>0} = \\N \\setminus \\set 0$ Let $+$ denote natural number addition. The structure $\\struct {\\N_{>0}, +}$ forms a semigroup."}
+{"_id": "3063", "title": "Indiscrete Space is Second-Countable", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Then  $T$ is a second-countable space."}
+{"_id": "11255", "title": "Doubling the Cube by Compass and Straightedge Construction is Impossible", "text": "There is no compass and straightedge construction to allow a cube to be constructed whose volume is double that of a given cube."}
+{"_id": "19445", "title": "Derivative of Inverse Hyperbolic Cotangent Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\coth^{-1} u} = \\dfrac {-1} {u^2 - 1} \\dfrac {\\d u} {\\d x}$  where $\\size u > 1$"}
+{"_id": "3072", "title": "Indiscrete Space is T4", "text": "Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Then $T$ is a $T_4$ space."}
+{"_id": "19456", "title": "Derivative of Cosine of Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\cos u} = -\\sin u \\dfrac {\\d u} {\\d x}$"}
+{"_id": "11272", "title": "Equation of Circle/Polar", "text": ": $r^2 - 2 r r_0 \\map \\cos {\\theta - \\varphi} + \\paren {r_0}^2 = R^2$"}
+{"_id": "3080", "title": "Singleton Partition yields Indiscrete Topology", "text": "Let $S$ be a set which is not empty. Let $\\PP$ be the (trivial) singleton partition $\\set S$ on $S$. Then the partition topology on $\\PP$ is the indiscrete topology."}
+{"_id": "19471", "title": "Derivative of Even Function is Odd", "text": "Let $f$ be a differentiable real function such that $f$ is even. Then its derivative $f'$ is an odd function."}
+{"_id": "11281", "title": "Graph of Quadratic describes Parabola/Corollary 2", "text": "The locus of the equation of the square root function on the non-negative reals: :$\\forall x \\in \\R_{\\ge 0}: \\map f x = \\sqrt x$ describes half of a parabola."}
+{"_id": "11286", "title": "Westwood's Puzzle", "text": ":500px Take any rectangle $ABCD$ and draw the diagonal $AC$. Inscribe a circle $GFJ$ in one of the resulting triangles $\\triangle ABC$. Drop perpendiculars $IEF$ and $HEJ$ from the center of this incircle $E$ to the sides of the rectangle. Then the area of the rectangle $DHEI$ equals half the area of the rectangle $ABCD$."}
+{"_id": "19486", "title": "Number of Parameters of ARMA Model", "text": "Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \\dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \\dotsc$ Let $\\tilde z_t, \\tilde z_{t - 1}, \\tilde z_{t - 2}, \\dotsc$ be deviations from a constant mean level $\\mu$: :$\\tilde z_t = z_t - \\mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \\dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \\dotsc$ Let $M$ be an '''ARMA model''' on $S$ of order $p$: :$\\tilde z_t = \\phi_1 \\tilde z_{t - 1} + \\phi_2 \\tilde z_{t - 2} + \\dotsb + \\phi_p \\tilde z_{t - p} + a_t - \\theta_1 a_{t - 1} - \\theta_2 a_{t - 2} - \\dotsb - \\theta_q a_{t - q}$ Then $M$ has $p + q + 2$ parameters. </onlyinclude>"}
+{"_id": "11295", "title": "Summation Formula for Polygonal Numbers", "text": "Let $P \\left({k, n}\\right)$ be the $n$th $k$-gonal number. Then: : $\\displaystyle P \\left({k, n}\\right) = \\sum_{j \\mathop = 1}^n \\left({\\left({k - 2}\\right) \\left({j - 1}\\right) + 1}\\right)$"}
+{"_id": "19493", "title": "ARIMA Model subsumes Moving Average Model", "text": "Let $S$ be a stochastic process based on an equispaced time series. Let $M$ be a moving average model for $S$. Then $M$ is also an implementation of an ARIMA model."}
+{"_id": "3115", "title": "Infinite Particular Point Space is not Strongly Locally Compact", "text": "Let $T = \\struct {S, \\tau_p}$ be an infinite particular point space. Then $T$ is not strongly locally compact."}
+{"_id": "3124", "title": "Kronecker’s Theorem", "text": "Let $K$ be a field. Let $f$ be a polynomial over $K$ of degree $n \\ge 1$. Then there exists a finite extension $L / K$ of $K$ such that $f$ has at least one root in $L$. Moreover, we can choose $L$ such that the degree $\\index L K$ of $L / K$ satisfies $\\index L K \\le n$. {{explain|Work out exactly which definitions of Polynomial and Degree are appropriate here.}}"}
+{"_id": "19511", "title": "Cancellation Law for Field Product", "text": "Let $\\struct {F, +, \\times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $a, b, c \\in F$. Then: :$a \\times b = a \\times c \\implies a = 0_F \\text { or } b = c$"}
+{"_id": "11323", "title": "Edge is Bridge iff in All Spanning Trees", "text": "Let $G$ be a simple graph. Let $e$ be an edge of $G$. Then $e$ is a bridge in $G$ {{iff}} $e$ belongs to every spanning tree for $G$."}
+{"_id": "19520", "title": "Leigh.Samphier/Sandbox/Set Difference of Distinct Equal Cardinality Sets is Not Empty", "text": "Let $S$ and $T$ be distinct finite sets. Let $\\card S = \\card T$. Then: :$S \\setminus T \\ne \\O$"}
+{"_id": "3144", "title": "Particular Point Space is Locally Path-Connected", "text": "Let $T = \\struct {S, \\tau_p}$ be a particular point space. Then $T$ is locally path-connected."}
+{"_id": "19537", "title": "Derivative of Inverse Hyperbolic Sine of x over a/Corollary 2", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\ln \\size {x - \\sqrt {x^2 + a^2} } } = -\\dfrac 1 {\\sqrt {x^2 + a^2} }$"}
+{"_id": "3159", "title": "Tarski-Vaught Test", "text": "Let $\\mathcal M, \\mathcal N$ be $\\mathcal L$-structures such that $\\mathcal M$ is a substructure of $\\mathcal N$. {{wtd|The page Definition:Structure is a disambiguation page, in which the form in which it is used here may not be included. The level of clarity in this page generally needs improving. Hence the invocation of the Disambiguate template.}} {{Disambiguate|Definition:Structure}} Then $\\mathcal M$ is an elementary substructure of $\\mathcal N$ {{iff}}: :for every $\\mathcal L$-formula $\\phi \\left({x, \\bar v}\\right)$ and for every $\\bar a$ in $\\mathcal M$: ::if there exists an $n$ in $\\mathcal N$ such that $\\mathcal N \\models \\phi \\left({n, \\bar a}\\right)$ ::then there exists an $m$ in $\\mathcal M$ such that $\\mathcal N \\models \\phi \\left({m, \\bar a}\\right)$. {{wtd|Before sense can be made of this page, Definition:Substructure and  Definition:Elementary Substructure need to be written.}} {{Disambiguate|Definition:Logical Formula}} The condition on the right side of the {{iff}} statement above can be rephrased as: :Every existential statement with parameters from $\\mathcal M$ which is satisfied in $\\mathcal N$ can be witnessed by an element from the substructure $\\mathcal M$."}
+{"_id": "11360", "title": "Conditional and Inverse are not Equivalent", "text": "A conditional statement: :$p \\implies q$ is not logically equivalent to its inverse: :$\\lnot p \\implies \\lnot q$"}
+{"_id": "11361", "title": "Weight of Discrete Topology equals Cardinality of Space", "text": "Let $T = \\left({S, \\tau}\\right)$ be a discrete topological space. Then: :$w \\left({T}\\right) = \\left\\vert{S}\\right\\vert$ where  :$w \\left({T}\\right)$ denotes the weight of $T$ :$\\left\\vert{S}\\right\\vert$ denotes the cardinality of $S$."}
+{"_id": "19559", "title": "Primitive of Reciprocal of a squared minus x squared/Logarithm Form 2", "text": ":$\\displaystyle \\int \\frac {\\d x} {a^2 - x^2} = \\dfrac 1 {2 a} \\ln \\size {\\dfrac {a + x} {a - x} } + C$"}
+{"_id": "3176", "title": "Open Extension Topology is Topology", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\tau^*_{\\bar p}$ be the open extension topology of $\\tau$. Then $\\tau^*_{\\bar p}$ is a topology on $S^*_p = S \\cup \\set p$."}
+{"_id": "19567", "title": "Symmetric Difference with Intersection forms Boolean Ring", "text": "Let $S$ be a set. Then $\\struct {\\powerset S, *, \\cap}$ is a Boolean ring."}
+{"_id": "19571", "title": "1-Seminorm on Continuous on Closed Interval Real-Valued Functions is Norm", "text": "Let $\\CC \\closedint a b$ be the space of continuous on closed interval real-valued functions. Let $x \\in \\CC \\closedint a b$ be a continuous real valued function. Let $\\displaystyle \\norm x_1 := \\int_a^b \\size {\\map x t} \\rd t$ be the 1-seminorm. Then $\\norm {\\, \\cdot \\,}_1$ is a norm on $\\CC \\closedint a b$."}
+{"_id": "19573", "title": "Perpendicularity is Symmetric Relation", "text": "Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \\in S$, let $l_1 \\perp l_2$ denote that $l_1$ is perpendicular to $l_2$. Then $\\perp$ is a symmetric relation on $S$."}
+{"_id": "19576", "title": "Perpendicularity is Antitransitive Relation", "text": "Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \\in S$, let $l_1 \\perp l_2$ denote that $l_1$ is perpendicular to $l_2$. Then $\\perp$ is an antitransitive relation on $S$."}
+{"_id": "11393", "title": "Multiplication using Parabola", "text": ":500pxrightthumb Let the parabola $P$ defined as $y = x^2$ be plotted on the Cartesian plane. Let $A = \\tuple {x_a, y_a}$ and $B = \\tuple {x_b, y_b}$ be points on the curve $\\map f x$ so that $x_a < x_b$. Then the line segment joining $A B$ will cross the $y$-axis at $-x_a x_b$. Thus $P$ can be used as a nomogram to calculate the product of two numbers $x_a$ and $x^b$, as follows: :$(1) \\quad$ Find the points $-x_a$ and $x_b$ on the $x$-axis. :$(2) \\quad$ Find the points $A$ and $B$ where the lines $x = -x_a$ and $x = x_b$ cut $P$. :$(3) \\quad$ Lay a straightedge on the straight line joining $A$ and $B$ and locate its $y$-intercept $c$. Then $x_a x_b$ can be read off from the $y$-axis as the position of $c$."}
+{"_id": "3212", "title": "Excluded Point Space is Locally Path-Connected", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be an excluded point space. Then $T$ is locally path-connected."}
+{"_id": "11405", "title": "Occurrence in Polish Notation has Unique Scope", "text": "Let $\\mathcal F$ be a formal language in Polish notation. Let $\\mathbf A$ be a well-formed formula of $\\mathcal F$. Let $a$ be an occurrence in $\\mathbf A$. Then $a$ has a unique scope."}
+{"_id": "11407", "title": "Exclusive Or as Conjunction of Disjunctions", "text": ": $p \\oplus q \\dashv \\vdash \\left({p \\lor q}\\right) \\land \\left({\\neg p \\lor \\neg q}\\right)$"}
+{"_id": "19602", "title": "Space of Almost-Zero Sequences is Everywhere Dense in 2-Sequence Space", "text": "Let $\\struct {\\ell^2, \\norm {\\, \\cdot \\,}_2}$ be the 2-sequence space equipped with Euclidean norm. Let $c_{00}$ be the space of almost-zero sequences. Then $c_{00}$ is everywhere dense in $\\struct {\\ell^2, \\norm {\\, \\cdot \\,}_2}$"}
+{"_id": "19605", "title": "Ordering of Integers is Reversed by Negation", "text": "Let $x, y \\in \\Z$ such that $x > y$. Then: :$-x < -y$"}
+{"_id": "3226", "title": "Excluded Point Space is Sequentially Compact", "text": "Let $T = \\struct {S, \\tau_{\\bar p} }$ be an excluded point space. Then $T$ is a sequentially compact space."}
+{"_id": "19611", "title": "Sufficient Conditions for Weak Stationarity of Order 2", "text": "Let $S$ be a stochastic process giving rise to a time series $T$. Let the mean of $S$ be fixed. Let the autocovariance matrix of $S$ be of the form: :$\\boldsymbol \\Gamma_n = \\begin {pmatrix} \\gamma_0 & \\gamma_1 & \\gamma_2 & \\cdots & \\gamma_{n - 1} \\\\ \\gamma_1 & \\gamma_0 & \\gamma_1 & \\cdots & \\gamma_{n - 2} \\\\ \\gamma_2 & \\gamma_1 & \\gamma_0 & \\cdots & \\gamma_{n - 3} \\\\ \\vdots   & \\vdots   & \\vdots   & \\ddots & \\vdots \\\\ \\gamma_{n - 1} & \\gamma_{n - 2} & \\gamma_{n - 3} & \\cdots & \\gamma_0 \\end {pmatrix} = \\sigma_z^2 \\mathbf P_n = \\begin {pmatrix} 1 & \\rho_1 & \\rho_2 & \\cdots & \\rho_{n - 1} \\\\ \\rho_1 & 1 & \\rho_1 & \\cdots & \\rho_{n - 2} \\\\ \\rho_2 & \\rho_1 & 1 & \\cdots & \\rho_{n - 3} \\\\ \\vdots   & \\vdots   & \\vdots   & \\ddots & \\vdots \\\\ \\rho_{n - 1} & \\rho_{n - 2} & \\rho_{n - 3} & \\cdots & 1 \\end {pmatrix}$ Then $S$ is weakly stationary of order $2$."}
+{"_id": "19616", "title": "Strict Ordering on Integers is Trichotomy", "text": "Let $\\eqclass {a, b} {}$ and $\\eqclass {c, d} {}$ be integers, as defined by the formal definition of integers. Then exactly one of the following holds: {{begin-eqn}} {{eqn | l = \\eqclass {a, b} {}       | o = <       | r = \\eqclass {c, d} {}       | c =  }} {{eqn | l = \\eqclass {a, b} {}       | o = =       | r = \\eqclass {c, d} {}       | c =  }} {{eqn | l = \\eqclass {a, b} {}       | o = >       | r = \\eqclass {c, d} {}       | c =  }} {{end-eqn}} That is, strict ordering is a trichotomy."}
+{"_id": "3233", "title": "Excluded Set Topology is not T0", "text": "Let $T = \\struct {S, \\tau_{\\bar H} }$ be an excluded set space where $H$ has at least two distinct points. Then $T$ is not a $T_0$ (Kolmogorov) space."}
+{"_id": "3236", "title": "Boubaker's Theorem", "text": "Let $\\left({R, +, \\circ}\\right)$ be a commutative ring. Let $\\left({D, +, \\circ}\\right)$ be an integral subdomain of $R$ whose zero is $0_D$ and whose unity is $1_D$. Let $X \\in R$ be transcendental over $D$. Let $D \\left[{X}\\right]$ be the ring of polynomial forms in $X$ over $D$. Finally, consider the following properties: {{begin-eqn}} {{eqn | n = 1       | l = \\sum_{k \\mathop = 1}^N {p_n \\left({0}\\right)}       | r = -2N }} {{eqn | n = 2       | l = \\sum_{k \\mathop = 1}^N {p_n \\left({\\alpha_k}\\right)}       | r = 0 }} {{eqn | n = 3       | l = \\left.{\\sum_{k \\mathop = 1}^N \\frac {\\mathrm d p_x \\left({x}\\right)} {\\mathrm d x} }\\right\\vert_{x \\mathop = 0}       | r = 0 }} {{eqn | n = 4       | l = \\left.{\\sum_{k \\mathop = 1}^N \\frac {\\mathrm d {p_n}^2 \\left({x}\\right)} {\\mathrm d x^2} }\\right\\vert_{x \\mathop = 0}       | r = \\frac 8 3 N \\left({N^2 - 1}\\right) }} {{end-eqn}} where, for a given positive integer $n$, $p_n \\in D \\left[{X}\\right]$ is a non-null polynomial such that $p_n$ has $N$ roots $\\alpha_k$ in $F$. Then the subsequence $\\left \\langle {B_{4 n} \\left({x}\\right)}\\right \\rangle$ of the Boubaker polynomials is the unique polynomial sequence of $D \\left[{X}\\right]$ which verifies simultaneously the four properties $(1) - (4)$."}
+{"_id": "3238", "title": "Either-Or Topology is T0", "text": "Let $T = \\struct {S, \\tau}$ be the either-or space. Then $T$ is a $T_0$ (Kolmogorov) space."}
+{"_id": "3239", "title": "Either-Or Topology is not T1", "text": "Let $T = \\struct {S, \\tau}$ be the either-or space. Then $T$ is not a $T_1$ (Fréchet) space."}
+{"_id": "19627", "title": "Cardinality of Set of Self-Mappings on Finite Set", "text": "Let $S$ be a finite set. Let the cardinality of $S$ be $n$. The cardinality of the set of all mappings from $S$ to itself (that is, the total number of self-maps on $S$) is: :$\\card {S^S} = n^n$"}
+{"_id": "3246", "title": "Cover of Interval By Closed Intervals is not Pairwise Disjoint", "text": "Let $\\closedint a b$ be a closed interval in $\\R$. {{explain|Title mentions only \"interval\"; this does not affect truth of statement so may \"closed\" above line be removed as superfluous?}} Let $\\JJ$ be a set of two or more closed intervals contained in $\\closedint a b$ such that $\\displaystyle \\bigcup \\JJ = \\closedint a b$. Then the intervals in $\\JJ$ are not pairwise disjoint."}
+{"_id": "11447", "title": "Factorization of Natural Numbers within 4 n + 1 not Unique", "text": "Let: :$S = \\set {4 n + 1: n \\in \\N} = \\set {1, 5, 9, 13, 17, \\ldots}$ be the set of natural numbers of the form $4 n + 1$. Then not all elements of $S$ have a complete factorization by other elements of $S$ which is unique."}
+{"_id": "3256", "title": "Either-Or Topology is not T3", "text": "Let $T = \\struct {S, \\tau}$ be the either-or space. Then $T$ is not a $T_3$ space."}
+{"_id": "19643", "title": "Definite Integral to Infinity of Reciprocal of x Squared plus a Squared/Corollary", "text": ":$\\ds \\int_0^\\infty \\dfrac {\\d x} {1 + x^2} = \\frac \\pi 2$ for $a \\ne 0$."}
+{"_id": "3262", "title": "Either-Or Topology is First-Countable", "text": "Let $T = \\struct {S, \\tau}$ be the either-or space. Then $T$ is a first-countable space."}
+{"_id": "11457", "title": "Construction of Regular Heptadecagon", "text": "It is possible to construct a regular hepadecagon (that is, a regular polygon with $17$ sides) using a compass and straightedge construction."}
+{"_id": "3268", "title": "Countable Stability implies Stability for All Infinite Cardinalities", "text": "Let $T$ be a complete $\\mathcal L$-theory whose language $\\mathcal L$ is countable. If $T$ is $\\omega$-stable, then $T$ is $\\kappa$-stable for all infinite $\\kappa$."}
+{"_id": "3269", "title": "Finite Complement Topology is Topology", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement space. Then $\\tau$ is a topology on $T$."}
+{"_id": "19654", "title": "Event Space of Experiment with Final Sample Space has Even Cardinality", "text": "Let $\\EE$ be an experiment with a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $\\Omega$ be a finite set. Then the event space $\\Sigma$ consists of an even number of subsets of $\\Omega$."}
+{"_id": "19653", "title": "Set of Elementary Events belonging to k Events is Event", "text": "Let $\\EE$ be an experiment with a probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $A_1, A_2, \\ldots, A_m$ be events in the event space $\\Sigma$ of $\\EE$. Let $S$ denote the set of all elementary events of $\\EE$ which are elements of exactly $k$ of the events $A_1, A_2, \\ldots, A_m$. Then $S$ is an event of $\\Sigma$."}
+{"_id": "19659", "title": "Probability of Set Difference of Events", "text": "Let $\\EE$ be an experiment with probability space $\\struct {\\Omega, \\Sigma, \\Pr}$. Let $A, B \\in \\Sigma$ be events of $\\EE$. Let $\\map \\Pr A$ denote the probability of event $A$ occurring. Then: :$\\map \\pr {A \\setminus B} = \\map \\Pr A - \\map \\Pr {A \\cap B}$"}
+{"_id": "11468", "title": "Distance Moved by Body from Rest under Constant Acceleration", "text": "Let a body $B$ be stationary. Let $B$ be subject to a constant acceleration. Then the distance travelled by $B$ is proportional to the square of the length of time $B$ is under the acceleration."}
+{"_id": "19685", "title": "Set of Endomorphisms of Non-Abelian Group is not Ring", "text": "Let $\\struct {G, \\oplus}$ be a group which is non-abelian. Let $\\mathbb G$ be the set of all group endomorphisms of $\\struct {G, \\oplus}$. Let $*: \\mathbb G \\times \\mathbb G \\to \\mathbb G$ be the operation defined as: :$\\forall u, v \\in \\mathbb G: u * v = u \\circ v$ where $u \\circ v$ is defined as composition of mappings. Then the algebraic structure $\\struct {\\mathbb G, \\oplus, *}$ is not a ring."}
+{"_id": "11498", "title": "Density not greater than Weight", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Then :$d \\left({T}\\right) \\leq w \\left({T}\\right)$ where :$d \\left({T}\\right)$ denotes the density of $T$, :$w \\left({T}\\right)$ denotes the weight of $T$."}
+{"_id": "3316", "title": "Countable Complement Space is not Countably Compact", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not a countably compact space."}
+{"_id": "3319", "title": "Closed Form for Number of Derangements on Finite Set", "text": "The number of derangements $D_n$ on a finite set $S$ of cardinality $n$ is: :$D_n = n! \\paren {1 - \\dfrac 1 {1!} + \\dfrac 1 {2!} - \\dfrac 1 {3!} + \\cdots + \\dfrac {\\paren {-1}^n} {n!} }$"}
+{"_id": "19704", "title": "Characteristics of Birkhoff-James Orthogonality", "text": "Let $\\struct {V, \\norm {\\,\\cdot\\,} }$ be a normed linear space. Let $x, y \\in V$. Then $x$ and $y$ are '''Birkhoff-James orthogonal''' {{iff}} either: :$(1): \\quad x = 0$ or: :$(2): \\quad$ there exists a continuous functional $ f$ on $\\struct {V, \\norm {\\,\\cdot\\,} }$ such that: ::::$\\norm f = 1$ ::::$\\map f x = \\norm x$ ::::$\\map f y = 0$"}
+{"_id": "3325", "title": "Countable Complement Space is Pseudocompact", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is pseudocompact."}
+{"_id": "3326", "title": "Countable Complement Space is not Countably Metacompact", "text": "Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not countably metacompact."}
+{"_id": "19711", "title": "Direct Product of Banach Spaces is Banach Space", "text": "Let $\\struct {X, \\norm {\\, \\cdot \\,}}$ and $\\struct {Y, \\norm {\\, \\cdot \\,}}$ be normed vector spaces. Let $V = X \\times Y$ be a direct product of vector spaces $X$ and $Y$ together with induced component-wise operations. Let $\\norm {\\tuple {x, y} }$ be the direct product norm. Suppose $X$ and $Y$ are Banach spaces. Then $V$ is a Banach space."}
+{"_id": "11527", "title": "Rational Numbers are F-Sigma Set in Real Line", "text": "Let $\\left({\\R, \\tau}\\right)$ be the real number line considered asa  topological space with the usual (Euclidean) topology. Then: :$\\Q$ is an $F_\\sigma$ set in $\\left({\\R, \\tau}\\right)$."}
+{"_id": "11529", "title": "Confocal Conics are Self-Orthogonal", "text": "The confocal conics defined by: :$\\quad \\dfrac {x^2} {a^2} + \\dfrac {y^2} {a^2 - c^2} = 1$ forms a family of orthogonal trajectories which is self-orthogonal. :500px"}
+{"_id": "3339", "title": "Compact Complement Topology is Connected", "text": "Let $T = \\struct {\\R, \\tau}$ be the compact complement topology. Then $T$ is a connected space."}
+{"_id": "3341", "title": "Compact Complement Topology is not Ultraconnected", "text": "Let $T = \\struct {\\R, \\tau}$ be the compact complement topology on $\\R$. Then $T$ is not an ultraconnected space."}
+{"_id": "19726", "title": "Partial Derivatives of x tan^-1 (x^2 + y)", "text": "Let: :$\\map f {x, y} = x \\map \\arctan {x^2 + y}$ Then: {{begin-eqn}} {{eqn | l = \\map {f_1} {1, 0}       | r = \\dfrac \\pi 4 + 1 }} {{eqn | l = \\map {f_2} {x, y}       | r = \\dfrac x {1 + \\paren {x^2 + y}^2} }} {{end-eqn}}"}
+{"_id": "19727", "title": "Partial Derivatives of x ln y^2 + y e^z", "text": "Let: :$\\map f {x, y, z} = x \\ln y^2 + y e^z$ Then: {{begin-eqn}} {{eqn | l = \\map {f_1} {1, -1, 0}       | r = 0 }} {{eqn | l = \\map {f_2} {x, x y, y + z}       | r = \\dfrac 2 y + e^{y + z} }} {{end-eqn}}"}
+{"_id": "19725", "title": "Partial Derivative/Examples/u - v + 2 w, 2 u + v + 2 w, u - v + w", "text": "Let: {{begin-eqn}} {{eqn | l = u - v + 2 w       | r = x + 2 z }} {{eqn | l = 2 u + v - 2 w       | r = 2 x - 2 z }} {{eqn | l = u - v + w       | r = z - y }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \\dfrac {\\partial u} {\\partial y}       | r = 0 }} {{eqn | l = \\dfrac {\\partial v} {\\partial y}       | r = 2 }} {{eqn | l = \\dfrac {\\partial w} {\\partial y}       | r = 1 }} {{end-eqn}}"}
+{"_id": "3346", "title": "Compact Complement Topology is Sequentially Compact", "text": "Let $T = \\struct {\\R, \\tau}$ be the compact complement topology on $\\R$. Then $T$ is a sequentially compact space."}
+{"_id": "3348", "title": "Theories with Infinite Models have Models with Order Indiscernibles", "text": "Let $T$ be an $\\LL$-theory with infinite models. Let $\\struct {I, <$ be an infinite strict linearly ordered set. There is a model $\\MM \\models T$ containing an order indiscernible set $\\set {x_i : i \\in I}$."}
+{"_id": "3358", "title": "Fort Space is Sequentially Compact", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a sequentially compact space."}
+{"_id": "11553", "title": "First Order ODE/x y' = y + 2 x exp (- y over x)", "text": "is a homogeneous differential equation with solution: :$e^{y / x} = \\ln x^2 + C$"}
+{"_id": "3365", "title": "Fort Space is not Extremally Disconnected", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is not an extremally disconnected space."}
+{"_id": "3368", "title": "Non-Forking Types have Non-Forking Completions", "text": "Let $T$ be a complete $\\mathcal L$-theory. Let $\\mathfrak C$ be a monster model for $T$. Let $A\\subseteq B$ be subsets of the universe of $\\mathfrak C$. Let $\\pi(\\bar x)$ be an $n$-type over $B$. If $\\pi$ does not fork over $A$, then there is a complete $n$-type $p$ over $B$ such that $\\pi \\subseteq p$ and $p$ does not fork over $A$."}
+{"_id": "3373", "title": "Fortissimo Space is Completely Normal", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fortissimo space. Then $T$ is a completely normal space. Consequently, $T$ satisfies all weaker separation axioms."}
+{"_id": "11568", "title": "First Order ODE/-1 over y sine x over y dx + x over y^2 sine x over y dy", "text": "is an exact differential equation with solution: :$\\dfrac x y = C$"}
+{"_id": "11570", "title": "First Order ODE/dx = (y over (1 - x^2 y^2)) dx + (x over (1 - x^2 y^2)) dy", "text": "is an exact differential equation with solution: :$\\map \\ln {\\dfrac {1 + x y} {1 - x y} } - 2 x = C$"}
+{"_id": "3399", "title": "First-Countability is not Continuous Invariant", "text": "Let $T_A = \\struct {A, \\tau_A}$ and $T_B = \\struct {B, \\tau_B}$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a continuous mapping. If $T_A$ is a first-countable space, then it does not necessarily follow that $T_B$ is also first-countable."}
+{"_id": "11591", "title": "First Order ODE/(x y - 1) dx + (x^2 - x y) dy = 0", "text": "The first order ODE: :$(1): \\quad \\paren {x y - 1} \\rd x + \\paren {x^2 - x y} \\rd y = 0$ has the general solution: :$x y - \\ln x - \\dfrac {y^2} 2 + C$"}
+{"_id": "11605", "title": "First Order ODE/x dy = (x^5 + x^3 y^2 + y) dx", "text": "The first order ODE: :$(1): \\quad x \\rd y = \\paren {x^5 + x^3 y^2 + y} \\rd x$ has the general solution: :$\\arctan \\dfrac x y = -\\dfrac {x^4} 4 + C$"}
+{"_id": "11617", "title": "Linear First Order ODE/y' + y = 2 x exp -x + x^2", "text": "The linear first order ODE: :$(1): \\quad y' + y = 2 x e^{-x} + x^2$ has the general solution: :$y = x^2 e^{-x} + x^2 - 2 x + 2 + C e^{-x}$"}
+{"_id": "11623", "title": "First Order ODE/(exp y - 2 x y) y' = y^2", "text": "The first order ODE: :$(1): \\quad \\paren {e^y - 2 x y} y' = y^2$ has the general solution: :$x y^2 = e^y + C$"}
+{"_id": "11627", "title": "Differential Equation of Family of Linear Combination of Functions is Linear", "text": "Consider the one-parameter family of curves: :$(1): \\quad y = C \\map f x + \\map g x$ The differential equation that describes $(1)$ is linear and of first order."}
+{"_id": "3436", "title": "Trivial Topological Space is Non-Meager", "text": "Let $T = \\struct {S, \\tau}$ be a trivial topological space. Then $T$ is non-meager."}
+{"_id": "3452", "title": "Hermitian Operators have Orthogonal Eigenvectors", "text": "The eigenvectors of a Hermitian operation are orthogonal."}
+{"_id": "11668", "title": "First Order ODE/(exp x - 3 x^2 y^2) y' + y exp x = 2 x y^3", "text": "The first order ordinary differential equation: :$(1): \\quad \\paren {e^x - 3 x^2 y^2} y' + y e^x = 2 x y^3$ is an exact differential equation with solution: :$y e^x - x^2 y^3 = C$"}
+{"_id": "11704", "title": "First Order ODE/(y^2 exp x y + cosine x) dx + (exp x y + x y exp x y) dy = 0", "text": "The first order ordinary differential equation: :$(1): \\quad \\paren {y^2 e^{x y} + \\cos x} \\rd x + \\paren {e^{x y} + x y e^{x y} } \\rd y = 0$ is an exact differential equation with solution: :$y e^{x y} + \\sin x = C$"}
+{"_id": "11707", "title": "First Order ODE/(y^2 - 3 x y - 2 x^2) dx = (x^2 - x y) dy", "text": "The first order ordinary differential equation: :$(1): \\quad \\paren {y^2 - 3 x y - 2 x^2} \\rd x = \\paren {x^2 - x y} \\rd y$ is a homogeneous differential equation with solution: :$y^2 x^2 - 2 y x^3 + x^4 = C$"}
+{"_id": "11709", "title": "First Order ODE/exp x sine y dx + exp x cos y dy = y sine x y dx + x sine x y dy", "text": "The first order ordinary differential equation: :$(1): \\quad e^x \\sin y \\rd x + e^x \\cos y \\rd y = y \\sin x y \\rd x + x \\sin x y \\rd y$ is an exact differential equation with solution: :$e^x \\sin y + \\cos x y = C$"}
+{"_id": "11711", "title": "Linear Second Order ODE/(1 + x^2) y'' + x y' = 0", "text": "The second order ODE: :$\\paren {1 + x^2} y'' + x y' = 0$ has the general solution: :$y = C_1 \\map \\ln {x + \\sqrt {x^2 + 1} } + C_2$"}
+{"_id": "3520", "title": "Circumscribing Circle about Regular Pentagon", "text": "About any given regular pentagon it is possible to circumscribe a circle. {{:Euclid:Proposition/IV/14}}"}
+{"_id": "3527", "title": "Pólya-Vinogradov Inequality", "text": "Let $p$ be a positive odd prime. Then: :$\\forall m, n \\in \\N: \\displaystyle \\size {\\sum_{k \\mathop = m}^{m + n} \\paren {\\frac k p} } < \\sqrt p \\, \\ln p$ where $\\paren {\\dfrac k p}$ is the Legendre symbol."}
+{"_id": "3528", "title": "Multiplication of Real Numbers is Left Distributive over Subtraction", "text": "{{:Euclid:Proposition/V/5}} That is, for any numbers $a, b$ and for any integer $m$: : $m a - m b = m \\paren {a - b}$"}
+{"_id": "11739", "title": "Linear Second Order ODE/y'' + y = 0/y(0) = 2, y'(0) = 3", "text": "The second order ODE: :$(1): \\quad y'' + y = 0$ with initial conditions: :$\\map y 0 = 2$ :$\\map {y'} 0 = 3$ has the particular solution: :$y = 3 \\sin x + 2 \\cos x$"}
+{"_id": "11754", "title": "Legendre's Differential Equation/(1 - x^2) y'' - 2 x y' + 2 y = 0", "text": "The special case of Legendre's differential equation: :$(1): \\quad \\paren {1 - x^2} y'' - 2 x y' + 2 y = 0$ has the general solution: :$y = C_1 x + C_2 \\paren {\\dfrac x 2 \\, \\map \\ln {\\dfrac {1 + x} {1 - x} } - 1}$"}
+{"_id": "11762", "title": "Second Order ODE/y'' - f(x) y' + (f(x) - 1) y = 0", "text": "The second order ODE: :$(1): \\quad y'' - \\map f x y' + \\paren {\\map f x - 1} y = 0$ has the general solution: :$\\displaystyle y = C_1 e^x + C_2 e^x \\int e^{-2 x + \\int \\map f x \\rd x} \\rd x$"}
+{"_id": "11781", "title": "Linear Second Order ODE/2 x^2 y'' + 10 x y' + 8 y = 0", "text": "The second order ODE: :$(1): \\quad 2 x^2 y'' + 10 x y' + 8 y = 0$ has the general solution: :$y = C_1 x^{-2} + C_2 x^{-2} \\ln x$"}
+{"_id": "3594", "title": "Divisors obey Distributive Law", "text": "{{:Euclid:Proposition/VII/5}} In modern algebraic language: :$\\displaystyle a = \\frac 1 n b, c = \\frac 1 n d \\implies a + c = \\frac 1 n \\left({b + d}\\right)$"}
+{"_id": "3597", "title": "Real Numbers under Addition Modulo 1 form Group", "text": "Let $S = \\left\\{{x \\in \\R: 0 \\le x < 1}\\right\\}$. Let $\\circ: S \\times S \\to S$ be the operation defined as: : $x \\circ y = x + y - \\left \\lfloor {x + y} \\right \\rfloor$ That is, $\\circ$ is defined as addition modulo $1$. Then $\\left({S, \\circ}\\right)$ is a group."}
+{"_id": "11821", "title": "Linear Second Order ODE/y'' + y = 2 cosine x", "text": "The second order ODE: :$(1): \\quad y'' + y = 2 \\cos x$ has the general solution: :$y = C_1 \\sin x + C_2 \\cos x + x \\cos x$"}
+{"_id": "11831", "title": "Linear Second Order ODE/y'' + y = cosecant x", "text": "The second order ODE: :$(1): \\quad y'' + y = \\csc x$ has the general solution: :$y = C_1 \\sin x + C_2 \\cos x - x \\cos x + \\sin x \\map \\ln {\\sin x}$"}
+{"_id": "3645", "title": "Quaternions Defined by Ordered Pairs", "text": "Consider the quaternions $\\Bbb H$ as numbers in the form: : $a \\mathbf 1 + b \\mathbf i + c \\mathbf j + d \\mathbf k$ where: : $a, b, c, d$ are real numbers; : $\\mathbf 1, \\mathbf i, \\mathbf j, \\mathbf k$ are entities related to each other in the following way: {{begin-eqn}}      {{eqn | l = \\mathbf i \\mathbf j = - \\mathbf j \\mathbf i       | r = \\mathbf k       | c =  }} {{eqn | l = \\mathbf j \\mathbf k = - \\mathbf k \\mathbf j       | r = \\mathbf i       | c =  }} {{eqn | l = \\mathbf k \\mathbf i = - \\mathbf i \\mathbf k       | r = \\mathbf j       | c =  }} {{eqn | l = \\mathbf i^2 = \\mathbf j^2 = \\mathbf k^2 = \\mathbf i \\mathbf j \\mathbf k       | r = - \\mathbf 1       | c =  }} {{end-eqn}} Now consider the quaternions $\\Bbb H$ defined as ordered pairs $\\left({x, y}\\right)$ where $x, y \\in \\C$ are complex numbers, on which the operation of multiplication is defined as follows: Let $w = a_1 + b_1 i, x = c_1 + d_1 i, y = a_2 + b_2 i, z = c_2 + d_2 i$ be complex numbers. Then $\\left({w, x}\\right) \\left({y, z}\\right)$ is defined as: :$\\left({w, x}\\right) \\left({y, z}\\right) := \\left({w y - z \\overline x, \\overline w z + x y}\\right)$ where $\\overline w$ and $\\overline x$ are the complex conjugates of $w$ and $x$ respectively. These two definitions are equivalent."}
+{"_id": "11845", "title": "Linear Second Order ODE/y'' - 2 y' - 3 y = 64 x exp -x", "text": "The second order ODE: :$(1): \\quad y'' - 2 y' - 3 y = 64 x e^{-x}$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-x} - e^{-x} \\paren {8 x^2 + 4 x + 1}$"}
+{"_id": "11854", "title": "Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = x (x + 1)^2", "text": "The second order ODE: :$(1): \\quad \\paren {x^2 + x} y'' + \\paren {2 - x^2} y' - \\paren {2 + x} y = x \\paren {x + 1}^2$ has the general solution: :$y = C_1 e^x + \\dfrac {C_2} x - x - 1 - \\dfrac {x^2} 3$"}
+{"_id": "11870", "title": "Position of Cart attached to Wall by Spring under Damping/Critically Damped/x = x0 at t = 0", "text": "Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the horizontal position of $C$ at time $t$ can be expressed as: :$x = x_0 e^{-a t} \\left({1 + a t}\\right)$"}
+{"_id": "3685", "title": "Ring of Square Matrices over Real Numbers", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\struct {\\map {\\mathcal M_\\R} n, +, \\times}$ denote the ring of square matrices of order $n$ over $\\R$. Then $\\struct {\\map {\\mathcal M_\\R} n, +, \\times}$ is a ring with unity, but is not a commutative ring."}
+{"_id": "11886", "title": "Resonance Frequency is less than Natural Frequency", "text": "Consider a physical system $S$ whose behaviour is defined by the second order ODE: :$(1): \\quad \\dfrac {\\d^2 y} {\\d x^2} + 2 b \\dfrac {\\d y} {\\d x} + a^2 x = K \\cos \\omega x$ where: :$K \\in \\R: k > 0$ :$a, b \\in \\R_{>0}: b < a$ Then the resonance frequency of $S$ is smaller than the natural frequency of the associated second order ODE: :$(2): \\quad \\dfrac {\\d^2 y} {\\d x^2} + 2 b \\dfrac {\\d y} {\\d x} + a^2 x = 0$"}
+{"_id": "11903", "title": "Circle is Bisected by Diameter", "text": "A circle is bisected by a diameter."}
+{"_id": "3718", "title": "Power to Characteristic Power of Field is Monomorphism", "text": "Let $F$ be a field whose characteristic is $p$ where $p \\ne 0$. Let $n \\in \\Z_{\\ge 0}$ be any positive integer. Let $\\phi_n: F \\to F$ be the mapping on $F$ defined as: :$\\forall x \\in F: \\map {\\phi_n} x = x^{p^n}$ Then $\\phi_n$ is a (field) monomorphism."}
+{"_id": "11949", "title": "Logarithmic Spiral is Equiangular", "text": "The logarithmic spiral is '''equiangular''', in the following sense: Let $P = \\left\\langle{r, \\theta}\\right\\rangle$ be a point on a logarithmic spiral $S$ expressed in polar coordinates as: :$r = a e^{b \\theta}$ Then the angle $\\psi$ that the tangent makes to the radius vector of $S$ is constant."}
+{"_id": "11997", "title": "Area Enclosed by First Turn of Archimedean Spiral", "text": "Let $S$ be the Archimedean spiral defined by the equation: :$r = a \\theta$ The area $\\mathcal A$ enclosed by the first turn of $S$ and the polar axis is given by: :$\\mathcal A = \\dfrac {4 \\pi^3 a^2} 3$ :500px"}
+{"_id": "11998", "title": "Trisecting the Angle/Archimedean Spiral", "text": "Let $\\alpha$ be an angle which is to be trisected. This can be achieved by means of an Archimedean spiral."}
+{"_id": "12003", "title": "Upper Adjoint Preserves All Infima", "text": "Let $\\left({S, \\preceq}\\right)$, $\\left({T, \\precsim}\\right)$ be ordered sets. Let $g: S \\to T$ be an upper adjoint of Galois connection. Then $g$ preserves all infima."}
+{"_id": "12006", "title": "Trisecting the Angle/Parabola", "text": "Let $\\alpha$ be an angle which is to be trisected. This can be achieved by means of a parabola."}
+{"_id": "3818", "title": "Cardinality of Set of Induced Equivalence Classes of Injection", "text": "Let $f: S \\to T$ be a mapping. Let $\\mathcal R_f \\subseteq S \\times S$ be the relation induced by $f$: :$\\tuple {s_1, s_2} \\in \\mathcal R_f \\iff \\map f {s_1} = \\map f {s_2}$ Let $f$ be an injection. Then there are $\\card S$ different $\\mathcal R_f$-classes."}
+{"_id": "3836", "title": "Multiplicative Group of Positive Rationals is Non-Cyclic", "text": "Let $\\struct {\\Q_{>0}, \\times}$ be the multiplicative group of positive rational numbers. Then $\\struct {\\Q_{>0}, \\times}$ is not a cyclic group."}
+{"_id": "12052", "title": "Top equals to Relative Pseudocomplement in Brouwerian Lattice", "text": "Let $\\struct {S, \\vee, \\wedge, \\preceq}$ be a Brouwerian lattice with greatest element $\\top$. Let $a, b \\in S$. Then :$\\top = a \\to b$ {{iff}} $a \\preceq b$"}
+{"_id": "12071", "title": "Approximate Value of Nth Prime Number", "text": "The $n$th prime number is approximately $n \\ln n$."}
+{"_id": "3885", "title": "Fortissimo Space is not Compact", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fortissimo space. Then $T$ is not a compact space."}
+{"_id": "3886", "title": "Fortissimo Space is not Sequentially Compact", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fortissimo space. Then $T$ is not sequentially compact."}
+{"_id": "12081", "title": "Rational Number is Algebraic of Degree 1", "text": "Let $r \\in \\Q$ be a rational number. Then $r$ is an algebraic number of degree $1$."}
+{"_id": "3890", "title": "Double Pointed Fortissimo Space is Lindelöf", "text": "Let $T = \\struct {S, \\tau}$ be a Fortissimo space. Let $T \\times D$ be the double pointed topology on $T$. Then $T \\times D$ is a Lindelöf space."}
+{"_id": "3889", "title": "Double Pointed Fortissimo Space is Weakly Countably Compact", "text": "Let $T = \\struct {S, \\tau_p}$ be a Fortissimo space. Let $T \\times D$ be the double pointed topology on $T$. Then $T \\times D$ is weakly countably compact."}
+{"_id": "3892", "title": "Double Pointed Fortissimo Space is not Pseudocompact", "text": "Let $T = \\struct {S, \\tau}$ be a Fortissimo space. Let $T \\times D$ be the double pointed topology on $T$. Then $T \\times D$ is not pseudocompact."}
+{"_id": "3897", "title": "Modified Fort Space is not Locally Connected", "text": "Let $T = \\struct {S, \\tau_{a, b} }$ be a modified Fort space. Then $T$ is not locally connected."}
+{"_id": "3900", "title": "Clopen Sets in Modified Fort Space", "text": "Let $T = \\struct {S, \\tau_{a, b} }$ be a modified Fort space. Let $A \\subseteq S$ be both closed and open in $T$. If $a \\in A$, then $b \\in A$ as well. That is, any clopen set of $T$ must contain '''both''' or '''neither''' of $a$ and $b$."}
+{"_id": "12120", "title": "Maximum Rate of Change of Y Coordinate of Cycloid", "text": "Let a circle $C$ of radius $a$ roll without slipping along the x-axis of a cartesian plane at a constant speed such that the center moves with a velocity $\\mathbf v_0$ in the direction of increasing $x$. Consider a point $P$ on the circumference of this circle. Let $\\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The maximum rate of change of $y$ is $\\mathbf v_0$, which happens when $\\theta = \\dfrac \\pi 2 + 2 n \\pi$ where $n \\in \\Z$."}
+{"_id": "3931", "title": "Vertices in Locally Finite Graph", "text": "Let $G$ be a locally finite graph. Then if $G$ is infinite, it contains an infinite number of vertices."}
+{"_id": "12135", "title": "Relation between Equations for Hypocycloid and Epicycloid", "text": "Consider the hypocycloid defined by the equations: :$x = \\paren {a - b} \\cos \\theta + b \\map \\cos {\\paren {\\dfrac {a - b} b} \\theta}$ :$y = \\paren {a - b} \\sin \\theta - b \\map \\sin {\\paren {\\dfrac {a - b} b} \\theta}$ By replacing $b$ with $-b$, this converts to the equations which define an epicycloid: :$x = \\paren {a + b} \\cos \\theta - b \\map \\cos {\\paren {\\dfrac {a + b} b} \\theta}$ :$y = \\paren {a + b} \\sin \\theta - b \\map \\sin {\\paren {\\dfrac {a + b} b} \\theta}$"}
+{"_id": "3946", "title": "Derivative of Hyperbolic Sine Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\sinh u} = \\cosh u \\dfrac {\\d u} {\\d x}$"}
+{"_id": "12144", "title": "Radius of Curvature in Cartesian Form", "text": "Let $C$ be a curve defined by a real function which is twice differentiable. Let $C$ be embedded in a cartesian plane. The '''radius of curvature''' $\\rho$ of $C$ at a point $P = \\tuple {x, y}$ is given by: :$\\rho = \\dfrac {\\paren {1 + y'^2}^{3/2} } {\\size {y''} }$ where: :$y' = \\dfrac {\\d y} {\\d x}$ is the derivative of $y$ {{WRT|Differentiation}} $x$ at $P$ :$y'' = \\dfrac {\\d^2 y} {\\d x^2}$ is the second derivative of $y$ {{WRT|Differentiation}} $x$ at $P$."}
+{"_id": "3954", "title": "Derivative of Hyperbolic Tangent Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\tanh u} = \\sech^2 u \\dfrac {\\d u} {\\d x}$"}
+{"_id": "12159", "title": "Quaternion Multplication is not Commutative", "text": "The operation of multplication on the quaternions $H$ is not commutative."}
+{"_id": "12166", "title": "Vectors in Three Dimensional Space with Cross Product forms Lie Algebra", "text": "Let $S$ be the set of vectors in $3$ dimensional Euclidean space. Let $\\times$ denote the vector cross product on $S$. Then $\\struct {S, \\times}$ is a Lie algebra."}
+{"_id": "4012", "title": "Tangent Line to Convex Graph", "text": "Let $f$ be a real function that is continuous on some closed interval $\\left[{a \\,.\\,.\\, b}\\right]$ and differentiable and convex on the open interval $\\left({a \\,.\\,.\\, b}\\right)$.  Then all the tangent lines to $f$ are below the graph of $f$. {{explain|\"below\"}}"}
+{"_id": "4013", "title": "Characterization of Lower Semicontinuity", "text": "Let $f: S \\to \\overline \\R$ be an extended real valued function. Let $S$ be endowed with a topology $\\tau$. The following are equivalent: :$(1): \\quad$ $f$ is lower semicontinuous (LSC) on $S$. :$(2): \\quad$ The epigraph $\\map {\\operatorname{epi}} f$ of $f$ is a closed set in $S \\times \\R$ with the product topology. :$(3): \\quad$ All lower level sets of $f$ are closed in $S$."}
+{"_id": "12204", "title": "Power is Well-Defined/Rational", "text": "Let $x \\in \\R_{> 0}$ be a (strictly) positive real number. Let $q$ be a rational number. Then $x^q$ is  well-defined."}
+{"_id": "4019", "title": "Integral of Arcsine Function", "text": ":$\\displaystyle \\int \\arcsin x \\rd x = x \\arcsin x + \\sqrt {1 - x^2} + C$ for $x \\in \\closedint {-1} 1$."}
+{"_id": "4024", "title": "Ordinal is Subset of Ordinal Class", "text": "Suppose $A$ is an ordinal. Then: :$A \\subseteq \\On$ where $\\On$ represents the class of all ordinals."}
+{"_id": "12223", "title": "Power Function on Base Greater than One is Strictly Increasing/Integer", "text": "Let $a \\in \\R$ be a real number such that $a > 1$. Let $f: \\Z \\to \\R$ be the real-valued function defined as: :$\\map f k = a^k$ where $a^k$ denotes $a$ to the power of $k$. Then $f$ is strictly decreasing."}
+{"_id": "4074", "title": "Subset Products of Normal Subgroup with Normal Subgroup of Subgroup", "text": "Let $G$ be a group. Let: :$(1): \\quad H$ be a subgroup of $G$ :$(2): \\quad K$ be a normal subgroup of $H$ :$(3): \\quad N$ be a normal subgroup of $G$ Then: :$N K \\lhd N H$ where: : $N K$ and $N H$ denote subset product : $\\lhd$ denotes the relation of being a normal subgroup."}
+{"_id": "4078", "title": "Intersection with Normal Subgroup is Normal/Examples/Subset Product of Normal Subgroup with Intersection", "text": "Let $\\struct G$ be a group whose identity is $e$. Let $H_1, H_2$ be subgroups of $G$. Let: : $N_1 \\lhd H_1$ : $N_2 \\lhd H_2$ where $\\lhd$ denotes the relation of being a normal subgroup. Then: :$N_1 \\paren {H_1 \\cap N_2} \\lhd N_1 \\paren {H_1 \\cap H_2}$"}
+{"_id": "4079", "title": "Quotient Group of Direct Products", "text": "Let $G$ and $G'$ be groups. Let: :$H \\lhd G$ :$H' \\lhd G'$ where $\\lhd$ denotes the relation of being a normal subgroup. Then: :$\\paren {G \\times G'} / \\paren {H \\times H'}$ is isomorphic to $\\paren {G / H} \\times \\paren {G' / H'}$ where: :$G \\times G'$ denotes the group direct product of $H$ and $H'$ :$G / H$ denotes the quotient group of $G$ by $H$."}
+{"_id": "12275", "title": "Set of Non-Negative Real Numbers is not Well-Ordered by Usual Ordering", "text": "The set of non-negative real numbers $\\R_{\\ge 0}$ is not well-ordered under the usual ordering $\\le$."}
+{"_id": "4091", "title": "Characteristic Subgroup is Transitive", "text": "Let $G$ be a group. Let $H$ be a characteristic subgroup of $G$. Let $K$ be a characteristic subgroup of $H$. Then $K$ is a characteristic subgroup of $G$."}
+{"_id": "4096", "title": "Integral Resulting in Arcsecant", "text": ":$\\displaystyle \\int \\frac 1 {x \\sqrt{x^2 - a^2} }\\ \\mathrm dx = \\begin{cases} \\dfrac 1 {\\left\\vert{a}\\right\\vert} \\operatorname {arcsec} \\dfrac x {\\left\\vert{a}\\right\\vert} + C & : x > \\left\\vert{a}\\right\\vert \\\\ -\\dfrac 1 {\\left\\vert{a}\\right\\vert} \\operatorname {arcsec} \\dfrac x {\\left\\vert{a}\\right\\vert} + C & : x < -\\left\\vert{a}\\right\\vert \\end{cases}$ where $a$ is a constant."}
+{"_id": "12316", "title": "Upper Bound of Natural Logarithm/Corollary", "text": ":$\\forall s \\in \\R_{>0}: \\ln x \\le \\dfrac {x^s} s$"}
+{"_id": "4125", "title": "Finite Abelian Group is Solvable", "text": "Let $G$ be a finite abelian group. Then $G$ is solvable."}
+{"_id": "12317", "title": "Powers Drown Logarithms/Corollary", "text": ":$\\displaystyle \\lim_{y \\mathop \\to 0_+} y^r \\ln y = 0$"}
+{"_id": "12319", "title": "Logarithm of Logarithm in terms of Natural Logarithms", "text": "Let $b, x \\in \\R_{>0}$ be (strictly) positive real numbers. Then: :$\\map {\\log_b} {\\log_b x} = \\dfrac {\\map \\ln {\\ln x} - \\map \\ln {\\ln b} } {\\ln b}$ where $\\ln x$ denotes the natural logarithm of $x$."}
+{"_id": "4146", "title": "Intersection is Subset of Union of Intersections with Complements", "text": "Let $R, S, T$ be sets. Then: :$S \\cap T \\subseteq \\paren {R \\cap S} \\cup \\paren {\\overline R \\cap T}$ where $\\overline R$ denotes the complement of $R$."}
+{"_id": "4241", "title": "Adjoining Commutes with Inverting", "text": "Let $H$ be a Hilbert space. Let $A \\in B \\left({H, K}\\right)$ be a bounded linear operator. Let $A^{-1} \\in B \\left({K, H}\\right)$ be an inverse for $A$. Then the adjoint of $A$, $A^*$, is invertible. Furthermore, $\\left({A^*}\\right)^{-1} = \\left({A^{-1}}\\right)^*$."}
+{"_id": "12444", "title": "Sum of Elements in Inverse of Combinatorial Matrix", "text": "Let $C_n$ be the combinatorial matrix of order $n$ given by: :$C_n = \\begin{bmatrix} x + y & y & \\cdots & y \\\\ y & x + y & \\cdots & y \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ y & y & \\cdots & x + y \\end{bmatrix}$ Let $C_n^{-1}$ be its inverse, from Inverse of Combinatorial Matrix: :$b_{i j} = \\dfrac {-y + \\delta_{i j} \\left({x + n y}\\right)} {x \\left({x + n y}\\right)}$ where $\\delta_{i j}$ is the Kronecker delta. The sum of all the elements of $C_n^{-1}$ is: :$\\displaystyle \\sum_{1 \\mathop \\le i, \\ j \\mathop \\le n} b_{i j} = \\dfrac n {x + n y}$"}
+{"_id": "12449", "title": "Elements of Inverse of Hilbert Matrix are Integers", "text": "Let $H_n$ be the Hilbert matrix of order $n$: :$\\begin{bmatrix} a_{i j} \\end{bmatrix} = \\begin{bmatrix} \\dfrac 1 {i + j - 1} \\end{bmatrix}$ Consider its inverse $H_n^{-1}$. All the elements of $H_n^{-1}$ are integers."}
+{"_id": "12461", "title": "Auxiliary Relation is Transitive", "text": "Let $\\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let $\\mathcal R$ be relation on $S$ satisfying conditions $(i)$ and $(ii)$ of auxiliary relation. Then :$\\mathcal R$ is a transitive relation."}
+{"_id": "12473", "title": "Ordered Set of Auxiliary Relations is Complete Lattice", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let $\\operatorname {Aux} \\left({L}\\right)$ be the set of all auxiliary relations on $S$. Let $P = \\left({\\operatorname {Aux} \\left({L}\\right), \\precsim}\\right)$ be an ordered set where $\\precsim \\mathop = \\subseteq \\restriction_{\\operatorname {Aux} \\left({L}\\right) \\times \\operatorname {Aux} \\left({L}\\right)}$ Then :$P$ is a complete lattice."}
+{"_id": "4283", "title": "Zorn's Lemma Implies Axiom of Choice", "text": "If Zorn's Lemma is true, then so must the Axiom of Choice be."}
+{"_id": "12503", "title": "Number minus Modulo is Integer Multiple", "text": "Let $x, y \\in \\R$ be real numbers. Let $x \\bmod y$ denote the modulo operation: :$x \\bmod y := \\begin{cases} x - y \\left \\lfloor {\\dfrac x y}\\right \\rfloor & : y \\ne 0 \\\\ x & : y = 0 \\end{cases}$ where $\\left \\lfloor {\\dfrac x y}\\right \\rfloor$ denotes the floor of $\\dfrac x y$. Let $y < 0$. Then: :$x - \\left({x \\bmod y}\\right)$ is an integer multiple of $y$."}
+{"_id": "12504", "title": "Modulo Operation/Examples/5 mod 3", "text": ":$5 \\bmod 3 = 2$"}
+{"_id": "12521", "title": "Modulo Multiplication is Well-Defined/Warning", "text": "Let $z \\in \\R$ be a real number. Let: :$a \\equiv b \\pmod z$ and: :$x \\equiv y \\pmod z$ where $a, b, x, y \\in \\R$. Then it does '''not''' necessarily hold that: : $a x \\equiv b y \\pmod z$"}
+{"_id": "12528", "title": "Constant to Power of Number of Distinct Prime Divisors is Multiplicative Function", "text": "Let $c \\in \\R$ be a constant. Let $f: \\N \\to \\R$ denotes the mapping defined as: :$\\forall n \\in \\N: f \\left({n}\\right) = c^k$ where $k$ is number of distinct primes that divide $n$. Then $f$ is multiplicative."}
+{"_id": "4338", "title": "Clopen Sets in Finite Complement Topology", "text": "Let $T = \\struct {S, \\tau}$ be a finite complement topology on an infinite set $S$. Then the only clopen sets of $T$ are $S$ and $\\O$."}
+{"_id": "12534", "title": "Number of Digits in Factorial", "text": "Let $n!$ denote the factorial of $n$. The number of digits in $n!$ is approximately: :$1 + \\left\\lfloor{\\dfrac 1 2 \\left({\\log_{10} 2 + \\log_{10} \\pi}\\right) + \\dfrac 1 2 \\log_{10} n + n \\left({\\log_{10} n - \\log_{10} e}\\right)}\\right\\rfloor$ when $n!$ is shown in decimal notation. This evaluates to: :$1 + \\left\\lfloor{\\left({n + \\dfrac 1 2}\\right) \\log_{10} n - 0.43429 \\ 4481 \\, n + 0.39908 \\ 9934}\\right\\rfloor$"}
+{"_id": "4354", "title": "Transformation of Unit Matrix into Inverse", "text": "Let $\\mathbf A$ be a square matrix of order $n$ of the matrix space $\\map {\\MM_\\R} n$. Let $\\mathbf I$ be the unit matrix of order $n$. Suppose there exists a sequence of elementary row operations that reduces $\\mathbf A$ to $\\mathbf I$. Then $\\mathbf A$ is invertible. Futhermore, the same sequence, when performed on $\\mathbf I$, results in the inverse of $\\mathbf A$."}
+{"_id": "12566", "title": "Sum over k of r-kt Choose k by r over r-kt by s-(n-k)t Choose n-k by s over s-(n-k)t", "text": "For $n \\in \\Z_{\\ge 0}$: :$\\displaystyle \\sum_k A_k \\left({r, t}\\right) A_{n - k} \\left({s, t}\\right) = A_n \\left({r + s, t}\\right)$ where $A_n \\left({x, t}\\right)$ is the polynomial of degree $n$ defined as: :$A_n \\left({x, t}\\right) = \\dbinom {x - n t} n \\dfrac x {x - n t}$ where $x \\ne n t$."}
+{"_id": "4414", "title": "Integer Multiples Greater than Positive Integer Closed under Addition", "text": "Let $n \\Z$ be the set of integer multiples of $n$. Let $p \\in \\Z: p \\ge 0$ be a positive integer. Let $S \\subseteq n \\Z$  be defined as: :$S := \\set {x \\in n \\Z: x > p}$ that is, the set of integer multiples of $n$ greater than $p$. Then the algebraic structure $\\struct {S, +}$ is closed under addition."}
+{"_id": "12617", "title": "Signed Stirling Number of the First Kind of n+1 with 0", "text": ":$\\map s {n + 1, 0} = 0$"}
+{"_id": "4429", "title": "Subset of Natural Numbers under Max Operation is Monoid", "text": "Let $S \\subseteq \\N$ be a subset of the natural numbers $\\N$. Let $\\left({S, \\max}\\right)$ denote the algebraic structure formed from $S$ and the max operation. Then $\\left({S, \\max}\\right)$ is a monoid. Its identity element is the smallest element of $S$."}
+{"_id": "12621", "title": "Stirling Number of the Second Kind of n+1 with 2", "text": ":$\\displaystyle {n + 1 \\brace 2} = 2^n - 1$"}
+{"_id": "12625", "title": "Preceding is Approximating Relation", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Then $\\preceq$ is an approximating relation on $S$."}
+{"_id": "12628", "title": "Substitution Instance of WFF is WFF", "text": "Let $\\mathbf A$ be a WFF of predicate logic. Let $\\tau$ be a term of predicate logic. Let $x \\in \\mathrm{VAR}$ be a variable. Let $\\mathbf A \\left({x \\gets \\tau}\\right)$ be the substitution instance of $\\mathbf A$ substituting $\\tau$ for $x$. Then $\\mathbf A \\left({x \\gets \\tau}\\right)$ is a WFF."}
+{"_id": "4446", "title": "Sequence Converges to Within Half Limit/Real Numbers", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let $\\sequence {x_n}$ be convergent to the limit $l$. That is, let $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$. Suppose $l > 0$. Then: : $\\exists N: \\forall n > N: x_n > \\dfrac l 2$ Similarly, suppose $l < 0$. Then: : $\\exists N: \\forall n > N: x_n < \\dfrac l 2$"}
+{"_id": "12652", "title": "Binomial Coefficient/Examples/Number of Bridge Hands", "text": "The total number $N$ of possible different hands for a game of [https://en.wikipedia.org/wiki/Contract_bridge bridge] is: :$N = \\dfrac {52!} {13! \\, 39!} = 635 \\ 013 \\ 559 \\ 600$"}
+{"_id": "4474", "title": "Symmetric Difference is Subset of Union of Symmetric Differences", "text": "Let $R, S, T$ be sets. Then: :$R * S \\subseteq \\left({R * T}\\right) \\cup \\left({S * T}\\right)$ where $R * S$ denotes the symmetric difference between $R$ and $S$."}
+{"_id": "12680", "title": "Continuous iff Meet-Continuous and There Exists Smallest Auxiliary Approximating Relation", "text": "Let $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice. Then: :$L$ is continuous {{iff}} :$L$ is meet-continuous and there exists the smallest auxiliary approximating relation on $S$ That is: :$L$ is continuous {{iff}} :$L$ is meet-continuous and there exists an auxiliary approximating relation $\\mathcal R$ on $S$ ::for every auxiliary approximating relation $\\mathcal Q$ on $S$: $\\mathcal R \\subseteq \\mathcal Q$"}
+{"_id": "12696", "title": "Sum over k of r+tk choose k by s-tk choose n-k", "text": "Let $n \\in \\Z_{\\ge 0}$ be a non-negative integer. Then: :$\\displaystyle \\sum_k \\dbinom {r + t k} k \\dbinom {s - t k} {n - k} = \\sum_{k \\mathop \\ge 0} \\dbinom {r + s - k} {n - k} t^k$ where $\\dbinom {r + t k} k$ etc. denotes a binomial coefficient."}
+{"_id": "4523", "title": "Homogeneous System has Zero Vector as Solution", "text": "Every homogeneous system of linear equations has the zero vector as a solution."}
+{"_id": "12720", "title": "Summation over Lower Index of Unsigned Stirling Numbers of the First Kind", "text": "Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Then: :$\\displaystyle \\sum_k \\left[{n \\atop k}\\right] = n!$ where: :$\\displaystyle \\left[{n \\atop k}\\right]$ denotes an unsigned Stirling number of the first kind :$n!$ denotes the factorial of $n$."}
+{"_id": "4531", "title": "Co-Countable Measure is Probability Measure", "text": "Let $X$ be an uncountable set. Let $\\mathcal A$ be the $\\sigma$-algebra of countable sets on $X$. Then the co-countable measure $\\mu$ on $X$ is a probability measure."}
+{"_id": "12744", "title": "Supremum of Ideals is Upper Adjoint", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below continuous join semilattice. Let $\\mathit{Ids}\\left({L}\\right)$ be the set of all ideals in $L$. Let $P = \\left({\\mathit{Ids}\\left({L}\\right), \\precsim}\\right)$ be an ordered set where $\\mathord \\precsim = \\subseteq\\restriction_{\\mathit{Ids}\\left({L}\\right)\\times \\mathit{Ids}\\left({L}\\right)}$ Let $f: \\mathit{Ids}\\left({L}\\right) \\to S$ be a mapping such that :$\\forall I \\in \\mathit{Ids}\\left({L}\\right): f\\left({I}\\right) = \\sup I$ Then $f$ is an upper adjoint of Galois connection."}
+{"_id": "12749", "title": "Summation of Summation over Divisors of Function of Two Variables", "text": "Let $c, d, n \\in \\Z$. Then: :$\\displaystyle \\sum_{d \\mathop \\divides n} \\sum_{c \\mathop \\divides d} \\map f {c, d} = \\sum_{c \\mathop \\divides n} \\sum_{d \\mathop \\divides \\paren {n / c} } \\map f {c, c d}$ where $c \\divides d$ denotes that $c$ is a divisor of $d$."}
+{"_id": "12766", "title": "Summation over k of Ceiling of k over 2", "text": ":$\\displaystyle \\sum_{k \\mathop = 1}^n \\left \\lceil{\\dfrac k 2}\\right \\rceil = \\left \\lceil{\\dfrac {n \\left({n + 2}\\right)} 4}\\right \\rceil$"}
+{"_id": "4580", "title": "Open Rectangles Closed under Intersection", "text": "Let $\\left(({\\mathbf a \\,.\\,.\\, \\mathbf b}\\right))$ and $\\left(({\\mathbf c \\,.\\,.\\, \\mathbf d}\\right))$ be open $n$-rectangles. Then $\\left(({\\mathbf a \\,.\\,.\\, \\mathbf b}\\right)) \\cap \\left(({\\mathbf c \\,.\\,.\\, \\mathbf d}\\right))$ is also an open $n$-rectangle."}
+{"_id": "4581", "title": "Euclidean Borel Sigma-Algebra Closed under Scalar Multiplication", "text": "Let $\\mathcal B \\left({\\R^n}\\right)$ be the Borel $\\sigma$-algebra on $\\R^n$. Let $B \\in \\mathcal B$, and let $t \\in \\R_{>0}$. Then also $t \\cdot B := \\left\\{{t \\mathbf b: \\mathbf b \\in B}\\right\\} \\in \\mathcal B$."}
+{"_id": "4582", "title": "Lebesgue Measure of Scalar Multiple", "text": "Let $\\lambda^n$ be the $n$-dimensional Lebesgue measure on $\\R^n$ equipped with the Borel $\\sigma$-algebra $\\mathcal B \\left({\\R^n}\\right)$. Let $B \\in \\mathcal B$, and let $t \\in \\R_{>0}$. Then $\\lambda^n \\left({t \\cdot B}\\right) = t^n \\lambda^n \\left({B}\\right)$, where $t \\cdot B$ is the set $\\left\\{{t \\mathbf b: \\mathbf b \\in B}\\right\\}$."}
+{"_id": "12779", "title": "Membership of Set of Strictly Positive Integers is Replicative Function", "text": "Let $f: \\R \\to \\R$ be the real function defined as: :$\\forall x \\in \\R: \\map f x = \\sqbrk {x \\in \\Z_{> 0} }$ where $\\sqbrk \\cdots$ is Iverson's convention. Then $f$ is a replicative function."}
+{"_id": "12780", "title": "Membership of Equivalence Class of m mod pi is Replicative Function", "text": "Let $f: \\R \\to \\R$ be the real function defined as: :$\\forall x \\in \\R: \\map f x = \\sqbrk {\\exists r \\in \\Q, \\exists m \\in \\Z: x = r \\pi + m}$ where $\\sqbrk {\\cdots}$ is Iverson's convention. Then $f$ is a replicative function."}
+{"_id": "12783", "title": "Sum of Replicative Functions is Replicative", "text": "Let $f: \\R \\to \\R$ and $g: \\R \\to \\R$ be real functions. Let $f$ and $g$ both be replicative functions. Then the pointwise sum of $f$ and $g$ is also a replicative function."}
+{"_id": "12794", "title": "Meet Irreducible iff Finite Infimum equals Element", "text": "Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Let $x \\in S$. Then :$x$ is meet irreducible {{iff}} :for every non-empty finite subset $A$ of $S$: $x = \\inf A \\implies x \\in A$"}
+{"_id": "12796", "title": "Sum over j of Function of Floor of mj over n", "text": "Let $f$ be a real function. Then: :$\\displaystyle \\sum_{0 \\mathop \\le j \\mathop < n} \\map f {\\floor {\\dfrac {m j} n} } = \\sum_{0 \\mathop \\le r \\mathop < m} \\ceiling {\\dfrac {r n} m} \\paren {\\map f {r - 1} - \\map f r} + n \\map f {m - 1}$"}
+{"_id": "4632", "title": "Isomorphism of External Direct Products/General Result", "text": "Let: : $(1): \\quad \\displaystyle \\left({S, \\circ}\\right) = \\prod_{k \\mathop = 1}^n S_k = \\left({S_1, \\circ_1}\\right) \\times \\left({S_2, \\circ_2}\\right) \\times \\cdots \\times \\left({S_n, \\circ_n}\\right)$ : $(2): \\quad \\displaystyle \\left({T, \\ast}\\right) = \\prod_{k \\mathop = 1}^n T_k = \\left({T_1, \\ast_1}\\right) \\times \\left({T_2, \\ast_2}\\right) \\times \\cdots \\times \\left({T_n, \\ast_n}\\right)$ be external direct products of algebraic structures. Let $\\phi_k: \\left({S_k, \\circ_k}\\right) \\to \\left({T_k, \\ast_k}\\right)$ be an isomorphism for each $k \\in \\left[{1 \\,.\\,.\\, n}\\right]$. Then: :$\\phi: \\left({s_1, \\ldots, s_n}\\right) \\to \\left({\\phi_1 \\left({s_1}\\right), \\ldots, \\phi_n \\left({s_n}\\right)}\\right)$ is an isomorphism from $\\left({S, \\circ}\\right)$ to $\\left({T, \\ast}\\right)$."}
+{"_id": "12836", "title": "Cancellable Infinite Semigroup is not necessarily Group", "text": "Let $\\struct {S, \\circ}$ be a semigroup whose underlying set is infinite. Let $\\struct {S, \\circ}$ be such that all elements of $S$ are cancellable. Then it is not necessarily the case that $\\struct {S, \\circ}$ is a group."}
+{"_id": "12840", "title": "Latin Square is not necessarily Cayley Table of Group", "text": "While it is true that the Cayley table of a (finite) group is in the form of a Latin square it is not necessarily the case that a Latin square is the Cayley table of a group."}
+{"_id": "4652", "title": "Projection is Epimorphism/General Result", "text": "Let $\\left({S, \\circ}\\right)$ be the external direct product of the algebraic structures $\\left({S_1, \\circ_1}\\right), \\left({S_2, \\circ_2}\\right), \\ldots, \\left({S_k, \\circ_k}\\right), \\ldots, \\left({S_n, \\circ_n}\\right)$. Then: :for each $j \\in \\left[{1 \\,.\\,.\\, n}\\right]$, $\\operatorname{pr}_j$ is an epimorphism from $\\left({S, \\circ}\\right)$ to $\\left({S_j, \\circ_j}\\right)$ where $\\operatorname{pr}_j: \\left({S, \\circ}\\right) \\to \\left({S_j, \\circ_j}\\right)$ is the $j$th projection from $\\left({S, \\circ}\\right)$ to $\\left({S_j, \\circ_j}\\right)$."}
+{"_id": "4653", "title": "External Direct Product of Projection with Canonical Injection/General Result", "text": "Let $\\struct {S_1, \\circ_1}, \\struct {S_2, \\circ_2}, \\dotsc, \\struct {S_j, \\circ_j}, \\dotsc, \\struct {S_n, \\circ_n}$ be algebraic structures with identities $e_1, e_2, \\dotsc, e_j, \\dotsc, e_n$ respectively. Let $\\displaystyle \\struct {S, \\circ} = \\prod_{i \\mathop = 1}^n \\struct {S_i, \\circ_i}$ be the external direct product of $\\struct {S_1, \\circ_1}, \\struct {S_2, \\circ_2}, \\dotsc, \\struct {S_j, \\circ_j}, \\dotsc, \\struct {S_n, \\circ_n}$. Let $\\pr_j: \\struct {S, \\circ} \\to \\struct {S_j, \\circ_j}$ be the $j$th projection from $\\struct {S, \\circ}$ to $\\struct {S_j, \\circ_j}$. Let $\\inj_j: \\struct {S_j, \\circ_j} \\to \\struct {S, \\circ}$ be the canonical injection from $\\struct {S_j, \\circ_j}$ to $\\struct {S, \\circ}$. Then: :$\\pr_j \\circ \\inj_j = I_{S_j}$ where $I_{S_j}$ is the identity mapping from $S_j$ to $S_j$."}
+{"_id": "4668", "title": "Pre-Image Sigma-Algebra on Codomain is Sigma-Algebra", "text": "Let $X, X'$ be sets, and let $f: X \\to X'$ be a mapping. Let $\\Sigma$ be a $\\sigma$-algebra on $X$. Denote with $\\Sigma'$ the pre-image $\\sigma$-algebra on the domain of $f$. Then $\\Sigma'$ is a $\\sigma$-algebra on $X'$."}
+{"_id": "12873", "title": "Multiplicative Group of Reduced Residues Modulo 8 is Klein Four-Group", "text": "Let $K_4$ denote the Klein $4$-group. Let $R_4$ be the multiplicative group of reduced residues Modulo $8$. Then $K_4$ and $R_4$ are isomorphic algebraic structures."}
+{"_id": "4685", "title": "Properties of Relation Not Preserved by Restriction", "text": "If a relation is: * serial, * non-reflexive, * non-symmetric, * non-transitive or * non-connected it is impossible to state without further information whether or not any restriction of that relation has the same properties."}
+{"_id": "4703", "title": "Totally Ordered Set is Well-Ordered iff Subsets Contain Infima", "text": "Let $\\left({S, \\preccurlyeq}\\right)$ be a totally ordered set. Then $\\left({S, \\preccurlyeq}\\right)$ is a well-ordered set iff every non-empty subset of $T \\subseteq S$ has an infimum such that $\\inf \\left({T}\\right) \\in T$."}
+{"_id": "4708", "title": "Cantor Set has Zero Lebesgue Measure", "text": "Let $\\mathcal C$ be the Cantor set. Let $\\lambda$ be the Lebesgue measure on the Borel $\\sigma$-algebra $\\mathcal B \\left({\\R}\\right)$ on $\\R$. Then $\\mathcal C$ is $\\mathcal B \\left({\\R}\\right)$-measurable, and $\\lambda \\left({\\mathcal C}\\right) = 0$. That is, $\\mathcal C$ is a $\\lambda$-null set."}
+{"_id": "4709", "title": "Factorization Lemma/Real-Valued Function", "text": "Then a mapping $g: X \\to \\R$ is $\\map \\sigma f \\, / \\, \\map {\\mathcal B} \\R$-measurable {{iff}}: :There exists a $\\Sigma \\, / \\, \\map {\\mathcal B} \\R$-measurable mapping $\\tilde g: Y \\to \\R$ such that $g = \\tilde g \\circ f$ where: :$\\map \\sigma f$ denotes the $\\sigma$-algebra generated by $f$ :$\\map {\\mathcal B} \\R$ denotes the Borel $\\sigma$-algebra on $\\R$"}
+{"_id": "12904", "title": "Non-Zero-Sum Game as Zero-Sum Game", "text": "Let $G$ be a non-zero-sum game for $n$ players. Then $G$ can be modelled as a zero-sum game for $n + 1$ players."}
+{"_id": "12905", "title": "Two-Person Zero-Sum Game is Non-Cooperative", "text": "A two-person zero-sum game necessarily has to be non-cooperative."}
+{"_id": "12906", "title": "Simple Graph with Finite Vertex Set is Finite", "text": "Let $G$ be a simple graph. Suppose that the vertex set of $G$ is finite. Then $G$ is a finite graph. That is to say, its edge set is also finite."}
+{"_id": "12916", "title": "Tarski's Geometry is Complete/Corollary", "text": "Tarski's geometry does not contain minimal arithmetic."}
+{"_id": "12917", "title": "Symmetric Closure of Symmetric Relation", "text": "Let $\\mathcal R$ be a relation on a set $S$. Let $\\mathcal R^\\leftrightarrow$ be the symmetric closure of $\\mathcal R$. Then $\\mathcal R = \\mathcal R^\\leftrightarrow$."}
+{"_id": "12919", "title": "Eluding Game has no Saddle Point", "text": "The eluding game has no saddle point."}
+{"_id": "12928", "title": "Prime Element iff Element Greater is Top", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a Boolean lattice. Let $p \\in S$ such that :$p \\ne \\top$ Then :$p$ is prime element {{iff}} :$\\forall x \\in S: \\left({ p \\prec x \\implies x = \\top }\\right)$"}
+{"_id": "4738", "title": "Relation Isomorphism Preserves Antisymmetry", "text": "Let $\\left({S, \\mathcal R_1}\\right)$ and $\\left({T, \\mathcal R_2}\\right)$ be relational structures. Let $\\left({S, \\mathcal R_1}\\right)$ and $\\left({T, \\mathcal R_2}\\right)$ be (relationally) isomorphic. Then $\\mathcal R_1$ is an antisymmetric relation {{iff}} $\\mathcal R_2$ is also an antisymmetric relation."}
+{"_id": "12947", "title": "Anomalous Cancellation on 2-Digit Numbers", "text": "There are exactly four anomalously cancelling vulgar fractions having two-digit numerator and denominator when expressed in base $10$ notation: {{:Anomalous Cancellation on 2-Digit Numbers/Examples}}"}
+{"_id": "4774", "title": "Intersection of Strict Lower Closures in Toset", "text": "Let $\\left({S, \\preceq}\\right)$ be a totally ordered set. Let $a,b \\in S$. Then: :$a^\\prec \\cap b^\\prec = \\left({\\min \\left({a, b}\\right)}\\right)^\\prec$ where: : $a^\\prec$ denotes strict lower closure of $a$ : $\\min$ denotes the min operation."}
+{"_id": "4818", "title": "Intermediate Value Theorem (Topology)", "text": "Let $X$ be a connected topological space. Let $\\struct {Y, \\preceq, \\tau}$ be a totally ordered set equipped with the order topology. Let $f: X \\to Y$ be a continuous mapping. Let $a$ and $b$ are two points of $a, b \\in X$ such that: :$\\map f a \\prec \\map f b$ Let: :$r \\in Y: \\map f a \\prec r \\prec \\map f b$ Then there exists a point $c$ of $X$ such that: :$\\map f c = r$"}
+{"_id": "13016", "title": "Equivalence of Definitions of Upper Wythoff Sequence", "text": "The following definitions of the upper Wythoff sequence are equivalent:"}
+{"_id": "13034", "title": "Positive Integer is Sum of Consecutive Positive Integers iff not Power of 2", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then $n$ can be expressed as the sum of $2$ or more consecutive (strictly) positive integers {{iff}} $n$ is not a power of $2$."}
+{"_id": "4885", "title": "Baire-Osgood Theorem", "text": "Let $X$ be a Baire space. Let $Y$ be a metrizable topological space Let $f: X \\to Y$ be a mapping which is the pointwise limit of a sequence $\\left \\langle{f_n}\\right\\rangle$ in $C \\left({X, Y}\\right)$. {{explain|$C \\left({X, Y}\\right)$}} Let $D \\left({f}\\right)$ be the set of points where $f$ is discontinuous. Then $D \\left({f}\\right)$ is a meager subset of $X$."}
+{"_id": "13089", "title": "Ramanujan's Infinite Nested Roots", "text": ":$3 = \\sqrt {1 + 2 \\sqrt {1 + 3 \\sqrt { 1 + \\cdots} } }$"}
+{"_id": "13090", "title": "Product of Three Consecutive Integers is never Perfect Power", "text": "Let $n \\in \\Z_{> 1}$ be a (strictly) positive integer. Then: :$\\paren {n - 1} n \\paren {n + 1}$ cannot be expressed in the form $a^k$ for $a, k \\in \\Z$ where $k \\ge 2$. That is, the product of $3$ consecutive (strictly) positive integers can never be a perfect power."}
+{"_id": "4899", "title": "Maximal Element need not be Greatest Element", "text": "Let $\\struct {S, \\preccurlyeq}$ be an ordered set. Let $M \\in $ be a maximal element of $S$. Then $M$ is not necessarily the greatest element of $S$."}
+{"_id": "4900", "title": "Maximal Ideal of Division Ring", "text": "Let $\\left({D, +, \\circ}\\right)$ be a Division Ring whose zero is $0$. :Let $\\left({J, +, \\circ}\\right)$ be a maximal ideal of $D$. Then $J = \\left\\{{0}\\right\\}$."}
+{"_id": "4908", "title": "Identity Mapping is Ordered Ring Automorphism", "text": "Let $\\struct {S, +, \\circ, \\preceq}$ be an ordered ring. Then the identity mapping $I_S: S \\to S$ is an ordered ring automorphism."}
+{"_id": "13117", "title": "Number of Binary Digits in Power of 10/Example/1000", "text": "When expressed in binary notation, the number of digits in $1000$ is $10$."}
+{"_id": "4934", "title": "Characteristic of Integral Domain is Zero or Prime", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain. Let $\\operatorname{Char} \\left({D}\\right)$ be the characteristic of $D$. Then $\\operatorname{Char} \\left({D}\\right)$ is either $0$ or a prime number."}
+{"_id": "13135", "title": "Number is Sum of Five Cubes", "text": "Let $n \\in \\Z$ be an integer. Then $n$ can be expressed as the sum of $5$ cubes (either positive or negative) in an infinite number of ways."}
+{"_id": "13146", "title": "Exponential is of Exponential Order Real Part of Index", "text": "Let $\\map f t = e^{\\psi t}$ be the complex exponential function, where $t \\in \\R, \\psi \\in \\C$. Let $a = \\map \\Re \\psi$. Then $e^{\\psi t}$ is of exponential order $a$."}
+{"_id": "4963", "title": "Scalar Product with Multiple of Unity", "text": ":$\\paren {n \\cdot 1_R} \\circ x = n \\cdot x$ that is: :$\\paren {\\map {\\paren {+_R}^n} {1_R} } \\circ x = \\map {\\paren {+_G}^n} x$"}
+{"_id": "13162", "title": "Pythagorean Triangle from Fibonacci Numbers", "text": "Take $4$ consecutive Fibonacci numbers: :$F_n, F_{n + 1}, F_{n + 2}, F_{n + 3}$ Let: :$a := F_n F_{n + 3}$ :$b := 2 F_{n + 1} F_{n + 2}$ :$c := F_{2 n + 3}$ Then: :$a^2 + b^2 = c^2$ and: :$\\dfrac {a b} 2 = F_n \\times F_{n + 1} \\times F_{n + 2} \\times F_{n + 3}$ That is, if the legs of a right triangle are the product of the outer terms and twice the inner terms, then: :the hypotenuse is the Fibonacci number whose index is half the sum of the indices of the four given Fibonacci numbers. :the area is the product of the four given Fibonacci numbers."}
+{"_id": "4989", "title": "Conditions for Homogeneity/Plane", "text": "The plane $P = \\alpha_1 x_1 + \\alpha_2 x_2 + \\alpha_3 x_3 = \\gamma$ is homogeneous iff $\\gamma = 0$."}
+{"_id": "5012", "title": "Field Homomorphism Preserves Subfields", "text": "Let $\\struct {F_1, +_1, \\circ_1}$ and $\\struct {F_2, +_2, \\circ_2}$ be fields. Let $\\phi: F_1 \\to F_2$ be a field homomorphism such that $\\phi$ is not the trivial homomorphism. If $K$ is a subfield of $F_1$, then $\\phi \\sqbrk K$ is a subfield of $F_2$."}
+{"_id": "13210", "title": "Fermat Quotient of 2 wrt p is Square iff p is 3 or 7", "text": "Let $p$ be a prime number. The Fermat quotient of $2$ with respect to $p$: :$\\map {q_p} 2 = \\dfrac {2^{p - 1} - 1} p$ is a square {{iff}} $p = 3$ or $p = 7$."}
+{"_id": "5049", "title": "Integer Addition is Well-Defined", "text": "Let $\\struct {\\N, +}$ be the semigroup of natural numbers under addition. Let $\\struct {\\N \\times \\N, \\oplus}$ be the (external) direct product of $\\struct {\\N, +}$ with itself, where $\\oplus$ is the operation on $\\N \\times \\N$ induced by $+$ on $\\N$. Let $\\boxtimes$ be the cross-relation defined on $\\N \\times \\N$ by: :$\\tuple {x_1, y_1} \\boxtimes \\tuple {x_2, y_2} \\iff x_1 + y_2 = x_2 + y_1$ Let $\\eqclass {x, y} {}$ denote the equivalence class of $\\tuple {x, y}$ under $\\boxtimes$. The operation $\\oplus$ on these equivalence classes is well-defined, in the sense that: {{begin-eqn}} {{eqn | l = \\eqclass {a_1, b_1} {}       | r = \\eqclass {a_2, b_2} {}       | c =  }} {{eqn | l = \\eqclass {c_1, d_1} {}       | r = \\eqclass {c_2, d_2} {}       | c =  }} {{eqn | ll= \\leadsto       | l = \\eqclass {a_1, b_1} {} \\oplus \\eqclass {c_1, d_1} {}       | r = \\eqclass {a_2, b_2} {} \\oplus \\eqclass {c_2, d_2} {}       | c =  }} {{end-eqn}}"}
+{"_id": "5060", "title": "Preimage of Serial Relation is Domain", "text": "Let $\\mathcal R$ be a serial relation on $S$. Then the preimage of $\\mathcal R$ is $S$ (the domain of $\\mathcal R$)."}
+{"_id": "13273", "title": "Closed Form for Octagonal Numbers", "text": "The closed-form expression for the $n$th octagonal number is: :$O_n = n \\left({3 n - 2}\\right)$"}
+{"_id": "13277", "title": "Number of Distinct Deltahedra is Unlimited", "text": "There are an unlimited number of distinct deltahedra."}
+{"_id": "5092", "title": "Set Intersection Preserves Subsets/Corollary/Proof 1", "text": "Let $A, B, S$ be sets. Then: :$A \\subseteq B \\implies A \\cap S \\subseteq B \\cap S$"}
+{"_id": "13323", "title": "Characteristic of Increasing Mapping from Toset to Order Complete Toset", "text": "Let $\\struct {S, \\preceq}$ and $\\struct {T, \\preccurlyeq}$ be tosets. Let $T$ be order complete. Let $H \\subseteq S$ be a subset of $S$. Let $f: H \\to T$ be an increasing mapping from $H$ to $T$. Then: :$f$ has an extension to $S$ which is increasing {{iff}}: :for all $A \\subseteq H$: if $A$ is bounded in $S$, then $f \\sqbrk A$ is bounded in $T$ where $f \\sqbrk A$ denotes the image set of $A$ under $f$."}
+{"_id": "5145", "title": "A.E. Equal Positive Measurable Functions have Equal Integrals", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f, g: X \\to \\overline \\R_{\\ge 0}$ be positive $\\mu$-measurable functions. Suppose that $f = g$ almost everywhere. Then: :$\\displaystyle \\int f \\rd \\mu = \\int g \\rd \\mu$"}
+{"_id": "13337", "title": "Characterization of Euler's Number by Inequality", "text": "Let $a$ be a (strictly) positive real number. Then: :$a = e \\iff \\forall x \\in \\R: a^x \\ge x + 1$ where $e$ denotes Euler's number."}
+{"_id": "13353", "title": "Limit of x to the x", "text": "Let $f: \\R \\to \\R$ be defined on $\\left [{0 \\,.\\,.\\, \\to} \\right)$ with $f \\left({x}\\right) = x^x$. Then: :$\\displaystyle \\lim_{x \\to 0^+} x^x = 1$ Equivalently, from the definition of power: :$\\displaystyle \\lim_{x \\to 0^+} \\exp \\left({x \\ln x}\\right) = 1$"}
+{"_id": "13356", "title": "Schanuel's Conjecture Implies Transcendence of Log Pi", "text": "Let Schanuel's Conjecture be true. Then the logarithm of $\\pi$ (pi): :$\\ln \\pi$ is transcendental."}
+{"_id": "5175", "title": "Cauchy-Bunyakovsky-Schwarz Inequality/Lebesgue 2-Space", "text": "Let $\\struct {X, \\Sigma, \\mu}$ be a measure space. Let $f, g: X \\to \\R$ be $\\mu$-square integrable functions, that is $f, g \\in \\map {\\LL^2} \\mu$, Lebesgue $2$-space. Then: :$\\displaystyle \\int \\size {f g} \\rd \\mu \\le \\norm f_2^2 \\cdot \\norm g_2^2$ where $\\norm {\\, \\cdot \\,}_2$ is the $2$-norm."}
+{"_id": "5180", "title": "Preimage of Intersection under Relation/Family of Sets", "text": "Let $S$ and $T$ be sets. Let $\\family {T_i}_{i \\mathop \\in I}$ be a family of subsets of $T$. Let $\\RR \\subseteq S \\times T$ be a relation. Then: :$\\ds \\RR^{-1} \\sqbrk {\\bigcap_{i \\mathop \\in I} T_i} \\subseteq \\bigcap_{i \\mathop \\in I} \\RR^{-1} \\sqbrk {T_i}$ where $\\ds \\bigcap_{i \\mathop \\in I} T_i$ denotes the intersection of $\\family {T_i}_{i \\mathop \\in I}$."}
+{"_id": "5181", "title": "Image of Intersection under Injection/Family of Sets", "text": "Let $S$ and $T$ be sets. Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of subsets of $S$. Let $f: S \\to T$ be a mapping. Then: :$\\displaystyle f \\sqbrk {\\bigcap_{i \\mathop \\in I} S_i} = \\bigcap_{i \\mathop \\in I} f \\sqbrk {S_i}$ {{iff}} $f$ is an injection."}
+{"_id": "13375", "title": "Lines through Center Square of Order 3 Magic Square are in Arithmetic Sequence", "text": "Consider the order 3 magic square: {{:Magic Square/Examples/Order 3}} Each of the lines through the center cell contain $3$ integers in arithmetic sequence."}
+{"_id": "13377", "title": "Omega Constant is Transcendental", "text": "The omega constant is transcendental."}
+{"_id": "13389", "title": "Divisibility by Power of 10", "text": "Let $r \\in \\Z_{\\ge 1}$ be a strictly positive integer. An integer $N$ expressed in decimal notation is divisible by $10^r$ {{iff}} the last $r$ digits of $N$ are all $0$. That is: :$N = \\sqbrk {a_n \\ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \\cdots + a_n 10^n$ is divisible by $10^r$  {{iff}}: :$a_0 + a_1 10 + a_2 10^2 + \\cdots + a_r 10^r = 0$"}
+{"_id": "5201", "title": "Correspondence Theorem (Set Theory)", "text": "Let $S$ be a set. Let $\\RR \\subseteq S \\times S$ be an equivalence relation on $S$. Let $\\mathscr A$ be the set of partitions of $S$ associated with equivalence relations $\\RR'$ on $S$ such that: :$\\tuple {x, y} \\in \\RR \\iff \\tuple {x, y} \\in \\RR'$ Then there exists a bijection $\\phi$ from $\\mathscr A$ onto the set of partitions of the quotient set $S / \\RR$."}
+{"_id": "13399", "title": "Factorial as Product of Three Factorials", "text": "This general pattern can be used to find a factorial which is the product of three factorials: :$\\left({\\left({n!}\\right)!}\\right)! = n! \\left({n! - 1}\\right)! \\left({\\left({n!}\\right)! - 1}\\right)!$ while there are instances of factorials which do not fit that pattern."}
+{"_id": "5215", "title": "Right Quasigroup if (1-3) Parastrophe of Magma is Magma", "text": "Let $\\struct {S, \\circ}$ be a magma. Let the $(1-3)$ parastrophe of $\\struct {S, \\circ}$ be a magma. Then $\\struct {S, \\circ}$ is a right quasigroup."}
+{"_id": "5216", "title": "Left Quasigroup if (2-3) Parastrophe of Magma is Magma", "text": "Let $\\struct {S, \\circ}$ be a magma. Let the $\\paren {2 - 3}$ parastrophe of $\\struct {S, \\circ}$ be a magma. Then $\\struct {S, \\circ}$ is a left quasigroup."}
+{"_id": "5221", "title": "Product of Cardinals is Associative", "text": "Let $\\mathbf a$, $\\mathbf b$ and $\\mathbf c$ be cardinals. Then: : $\\mathbf a \\left({\\mathbf b \\mathbf c}\\right) = \\left({\\mathbf a \\mathbf b}\\right) \\mathbf c$ where $\\mathbf a \\mathbf b$ denotes the product of $\\mathbf a$ and $\\mathbf b$."}
+{"_id": "13425", "title": "Fundamental Theorem of Line Integrals", "text": "Let $\\mathcal C$ be a smooth curve given by the vector function $\\mathbf r \\left({t}\\right)$ for $a \\le t \\le b$. Let $f$ be a differentiable function of two or three variables whose gradient vector $\\nabla f$ is continuous on $\\mathcal C$. Then: :$\\displaystyle \\int_\\mathcal C \\nabla f \\cdot d \\mathbf r = f \\left({\\mathbf r \\left({b}\\right)}\\right) - f \\left({\\mathbf r \\left({a}\\right)}\\right)$"}
+{"_id": "5235", "title": "Abelian Group Induces Commutative B-Algebra", "text": "Let $\\left({G, \\circ}\\right)$ be an abelian group whose identity element is $e$. Let $*$ be the binary operation on $G$ defined as: :$\\forall a, b \\in G: a * b = a \\circ b^{-1}$ where $b^{-1}$ is the inverse element of $b$ under the operation $\\circ$. Then the algebraic structure $\\left({G, *}\\right)$ is a commutative $B$-algebra. That is: :$\\forall a, b \\in G: a * \\left({0 * b}\\right) = b * \\left({0 * a}\\right)$"}
+{"_id": "5239", "title": "Sum of Cardinals is Associative", "text": "Let $\\mathbf a$, $\\mathbf b$ and $\\mathbf c$ be cardinals. Then: : $\\mathbf a + \\left({\\mathbf b + \\mathbf c}\\right) = \\left({\\mathbf a + \\mathbf b}\\right) + \\mathbf c$ where $\\mathbf a + \\mathbf b$ denotes the sum of $\\mathbf a$ and $\\mathbf b$."}
+{"_id": "13435", "title": "Lucas Number 2n in terms of Square of Lucas Number n", "text": "Let $L_n$ denote the $n$th Lucas number. Then: :$L_{2 n} = {L_n}^2 + 2 \\left({-1}\\right)^n$"}
+{"_id": "5252", "title": "Natural Numbers as Cardinals", "text": "The natural numbers $\\N = \\set {0, 1, 2, 3, \\ldots}$ can be defined as the set of cardinals."}
+{"_id": "5273", "title": "Young's Inequality for Convolutions", "text": "Let $p, q, r \\in \\R_{\\ge 1}$ satisfy: :$1 + \\dfrac 1 r = \\dfrac 1 p + \\dfrac 1 q$ Let $L^p \\left({\\R^n}\\right)$, $L^q \\left({\\R^n}\\right)$, and $L^r \\left({\\R^n}\\right)$ be Lebesgue spaces with seminorms $\\left\\Vert{\\cdot}\\right\\Vert_p$, $\\left\\Vert{\\cdot}\\right\\Vert_q$, and $\\left\\Vert{\\cdot}\\right\\Vert_r$ respectively. Let $f \\in L^p \\left({\\R^n}\\right)$ and $g \\in L^q \\left({\\R^n}\\right)$. Then the convolution $f * g$ is in $L^r \\left({\\R^n}\\right)$ and the following inequality is satisfied: :$\\left\\Vert{f * g}\\right\\Vert_r \\le \\left\\Vert{f}\\right\\Vert_p \\cdot \\left\\Vert{g}\\right\\Vert_q$"}
+{"_id": "13471", "title": "Conditions for C^1 Smooth Solution of Euler's Equation to have Second Derivative", "text": "Let $\\map y x:\\R \\to \\R$ be a real function. Let $\\map F {x, y, y'}:\\R^3 \\to \\R$ be a real function. Suppose $\\map F {x, y, y'}$ has continuous first and second derivatives {{WRT|Differentiation}} all its arguments. Suppose $y$ has a continuous first derivative and satisfies Euler's equation:  :$F_y - \\dfrac \\d {\\d x} F_{y'} = 0$ Suppose: :$\\map {F_{y' y'} } {x, \\map y x, \\map y x'} \\ne 0$ Then $\\map y x$ has continuous second derivatives."}
+{"_id": "13476", "title": "If Double Integral of a(x, y)h(x, y) vanishes for any C^2 h(x, y) then C^0 a(x, y) vanishes", "text": "Let $\\alpha \\left({x, y}\\right)$, $h \\left({x, y}\\right)$ be functions in $\\R$. Let $\\alpha \\in C^0$ in a closed region $R$ whose boundary is $\\Gamma$. Let $h \\in C^2$ in $R$ and $h = 0$ on $\\Gamma$. Let: : $\\displaystyle \\int \\int_R \\alpha \\left({x, y}\\right) h \\left({x, y}\\right) \\rd x \\rd y = 0$ Then $\\alpha \\left({x, y}\\right)$ vanishes everywhere in $R$."}
+{"_id": "13479", "title": "Simple Variable End Point Problem", "text": "Let $y$ and $F$ be mappings. {{explain|Define their domain and codomain}} Suppose the endpoints of $y$ lie on two given vertical lines $x = a$ and $x = b$. Suppose $J$ is a functional of the form :$(1): \\quad J \\sqbrk y = \\displaystyle \\int_a^b \\map F {x, y, y'} \\rd x$ and has an extremum for a certain function $\\hat y$. Then $y$ satisfies the system of equations :$\\begin {cases} F_y - \\dfrac \\d {\\d x} F_{y'} = 0 \\\\ \\bigvalueat {F_{y'} } {x \\mathop = a} = 0 \\\\ \\bigvalueat {F_{y'} } {x \\mathop = b} = 0 \\end {cases}$"}
+{"_id": "13480", "title": "Ordered Set of Closure Operators and Dual Ordered Set of Closure Systems are Isomorphic", "text": "Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Then $\\operatorname{Closure}\\left({L}\\right)$ and $\\operatorname{ClSystems}\\left({L}\\right)^{-1}$ are order isomorphic where :$\\operatorname{Closure}\\left({L}\\right)$ denotes the ordered set of closure operators of $L$, :$\\operatorname{ClSystems}\\left({L}\\right)$ denotes the ordered set of closure systems oj $L$, :$\\operatorname{ClSystems}\\left({L}\\right)^{-1}$ denotes the dual to $\\operatorname{ClSystems}\\left({L}\\right)$."}
+{"_id": "13486", "title": "Euler's Equation for Vanishing Variation is Invariant under Coordinate Transformations", "text": "Euler's Equation for Vanishing Variation is invariant under coordinate transformations."}
+{"_id": "5302", "title": "Minimal Infinite Successor Set is Limit Ordinal", "text": "Let $\\omega$ denote the minimal infinite successor set. Then $\\omega$ is a limit ordinal."}
+{"_id": "5305", "title": "Ordinals Isomorphic to the Same Well-Ordered Set", "text": "Let $A$ and $B$ be ordinals. Let $\\left({\\prec, S}\\right)$ be a strict well-ordering. Let $\\left({\\in, A}\\right)$ and $\\left({\\prec, S}\\right)$ be order isomorphic. Let $\\left({\\in, B}\\right)$ and $\\left({\\prec, S}\\right)$ be order isomorphic. Then: : $A = B$"}
+{"_id": "13500", "title": "Topological Group is T1 iff T2", "text": "Let $G$ be a topological group. Then $G$ is a $T_1$ space {{iff}} $G$ is Hausdorff."}
+{"_id": "13509", "title": "Logarithm of Infinite Product of Complex Numbers", "text": "Let $\\sequence {z_n}$ be a sequence of nonzero complex numbers. {{TFAE}} :$(1): \\quad$ The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty z_n$ converges to $z \\in \\C_{\\ne 0}$. :$(2): \\quad$ The series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\log z_n$ converges to $\\log z + 2 k \\pi i$ for some integer $k \\in \\Z$."}
+{"_id": "5344", "title": "Infinite Ordinal can be expressed Uniquely as Sum of Limit Ordinal plus Finite Ordinal", "text": "Let $x$ be an ordinal. Suppose $x$ satisfies $\\omega \\subseteq x$. Then $x$ has a unique representation as $\\paren {y + z}$ where $y$ is a limit ordinal and $z$ is a finite ordinal."}
+{"_id": "13552", "title": "Legs of Pythagorean Triangle used as Generator for another Pythagorean Triangle", "text": "Let $a$ and $b$ be the legs of a Pythagorean triangle $P_1$. Let $\\tuple {a, b}$ be used as the generator for a new Pythagorean triangle $P_2$. Then the hypotenuse of $P_2$ is the square of the hypotenuse of $P_1$."}
+{"_id": "13564", "title": "Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation", "text": "Let $\\Phi = \\map {\\Phi} {x, \\family {y_i}_{1 \\mathop \\le i \\mathop \\le n}, \\family {p_i}_{1 \\mathop \\le i \\mathop \\le n} }$ be a real function. Let $H$ be Hamiltonian. Then a necessary and sufficient condition for $\\Phi$ to be the first integral of Euler's Equations is :$\\dfrac {\\partial \\Phi} {\\partial x} + \\sqbrk{\\Phi, H} = 0$"}
+{"_id": "13568", "title": "Homotopic Paths Implies Homotopic Composition", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $f_1, f_2, g_1, g_2: \\left[{0 \\,.\\,.\\, 1}\\right] \\to S$ be paths in $T$. Let $f_1$ be homotopic to $f_2$ and $g_1$ be homotopic to $g_2$. Then the concatenated paths $f_1 * g_1$ and $f_2 * g_2$ are homotopic."}
+{"_id": "13589", "title": "Product of Two Triangular Numbers to make Square", "text": "Let $T_n$ be a triangular number. Then there is an infinite number of $m \\in \\Z_{>0}$ such that $T_n \\times T_m$ is a square number."}
+{"_id": "13616", "title": "Numbers of form 31 x 16^n are sum of 16 Powers of 4", "text": "Let $m \\in \\Z$ be an integer of the form $31 \\times 16^n$ for $n \\in \\Z_{\\ge 0}$. Then in order express $m$ as the sum of powers of $4$, you need $16$ of them."}
+{"_id": "13619", "title": "Element of Ordered Set of Topology is Dense iff is Everywhere Dense", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $P = \\struct {\\tau, \\preceq}$ be an ordered set where $\\mathord\\preceq = \\mathord\\subseteq \\cap \\paren {\\tau \\times \\tau}$ Let $A \\in \\tau$. Then $A$ is a dense element in $P$ {{iff}} $A$ is everywhere dense."}
+{"_id": "5431", "title": "Element Commutes with Product of Commuting Elements/General Theorem", "text": "Let $(S,\\circ)$ be a semigroup. Let $\\left \\langle {a_k} \\right \\rangle_{1 \\mathop \\le k \\mathop \\le n}$ be a sequence of terms of $S$. Let $b \\in S$. If $b$ commutes with $a_k$ for each $k \\in \\left[{1 \\,.\\,.\\, n}\\right]$, then $b$ commutes with $a_1 \\circ \\cdots \\circ a_n$."}
+{"_id": "5446", "title": "Set is Small Class", "text": "Let $x$ be a set. Then $x$ is a small class."}
+{"_id": "5450", "title": "Group Induced by B-Algebra Induced by Group", "text": "Let $\\left({S, \\circ}\\right)$ be a group. Let $\\left({S, *}\\right)$ be the $B$-algebra described on Group Induces $B$-Algebra. Let $\\left({S, \\circ'}\\right)$ be the group described on $B$-Algebra Induces Group. Then $\\left({S, \\circ'}\\right) = \\left({S, \\circ}\\right)$."}
+{"_id": "5458", "title": "Category of Ordered Sets is Category", "text": "Let $\\mathbf{OrdSet}$ be the category of ordered sets. Then $\\mathbf{OrdSet}$ is a metacategory."}
+{"_id": "13651", "title": "Magic Hexagon of Order 3 is Unique", "text": "Apart from the trivial order $1$ magic hexagon, there exists only one magic hexagon: the order $3$ magic hexagon: {{:Definition:Order 3 Magic Hexagon}}"}
+{"_id": "5463", "title": "Universal Class is Proper", "text": "Let $V$ denote the universal class. Then $V$ is a proper class."}
+{"_id": "13655", "title": "Squares Ending in 5 Occurrences of 2-Digit Pattern", "text": "Let $n$ be a square number whose decimal representation ends in the pattern $\\mathtt {xyxyxyxyxy}$. Then $\\mathtt {xy}$ is one of: :$21, 29, 61, 69, 84$ The smallest examples of such numbers are: {{begin-eqn}} {{eqn | l = 508 \\, 853 \\, 989^2       | r = \\phantom {0 \\,} 258 \\, 932 \\, 38 \\mathbf {2 \\, 121 \\, 212 \\, 121} }} {{eqn | l = 162 \\, 459 \\, 327^2       | r = \\phantom {0 \\, 0} 26 \\, 393 \\, 03 \\mathbf {2 \\, 929 \\, 292 \\, 929} }} {{eqn | l = 1 \\, 318 \\, 820 \\, 881^2       | r = 1 \\, 739 \\, 288 \\, 51 \\mathbf {6 \\, 161 \\, 616 \\, 161} }} {{eqn | l = 541 \\, 713 \\, 187^2       | r = \\phantom {0 \\,} 293 \\, 453 \\, 17 \\mathbf {6 \\, 969 \\, 696 \\, 969} }} {{eqn | l = 509 \\, 895 \\, 478^2       | r = \\phantom {0 \\,} 259 \\, 993 \\, 39 \\mathbf {8 \\, 484 \\, 848 \\, 484} }} {{end-eqn}}"}
+{"_id": "13663", "title": "Sufficient Condition for Twice Differentiable Functional to have Minimum", "text": "Let $J$ be a twice differentiable functional. Let $J$ have an extremum for $y=\\hat y$. Let the second variation $\\delta^2 J \\sqbrk {\\hat y; h}$ be strongly positive {{WRT}} $h$. Then $J$ acquires the minimum for $y = \\hat y$ ."}
+{"_id": "5476", "title": "Discrete Category is Order Category", "text": "Let $\\mathbf{Dis} \\left({S}\\right)$ be a discrete category. Then $\\mathbf{Dis} \\left({S}\\right)$ is also an order category."}
+{"_id": "5480", "title": "Category of Monoids is Category", "text": "Let $\\mathbf{Mon}$ be the category of monoids. Then $\\mathbf{Mon}$ is a metacategory."}
+{"_id": "5481", "title": "Cayley's Representation Theorem/General Case", "text": "Let $\\struct {G, \\cdot}$ be a group. Then there exists a permutation group $P$ on some set $S$ such that: :$G \\cong P$ That is, such that $G$ is isomorphic to $P$."}
+{"_id": "5482", "title": "Permutation of Cosets/Corollary 1", "text": "Let $G$ be a group. Let $H \\le G$ such that $\\index G H = n$ where $n \\in \\Z$. Then: : $\\exists N \\lhd G: N \\lhd H: n \\divides \\index G H \\divides n!$"}
+{"_id": "5493", "title": "Cartesian Product is Small iff Inverse is Small", "text": "Let $A$ and $B$ be classes. Then the Cartesian product $A \\times B$ is a small class {{iff}} $B \\times A$ is small."}
+{"_id": "5502", "title": "Order Isomorphism on Foundational Relation preserves Foundational Structure", "text": "Let $A_1$ and $A_2$ be classes. Let $\\prec_1$ and $\\prec_2$ be relations. Let $\\phi: \\left({A_1, \\prec_1}\\right) \\to \\left({A_2, \\prec_2}\\right)$ be an order isomorphism. Then $\\left({A_1, \\prec_1}\\right)$ is a foundational structure iff $\\left({A_2, \\prec_2}\\right)$ is also a foundational structure."}
+{"_id": "5527", "title": "Kleene Closure is Free Monoid", "text": "Let $S$ be a set. Let $S^*$ be its Kleene closure, and let $i: S \\to S^*$ be the insertion of generators. Then $\\left({S^*, i}\\right)$ is a free monoid over $S$."}
+{"_id": "5547", "title": "Transitive Closure Always Exists (Set Theory)", "text": "Let $S$ be a set. Let $G$ be a mapping such that $\\map G x = x \\cup \\bigcup x$. {{explain|Domain and range of $G$ needed}} Let $F$ be defined using the Principle of Recursive Definition: :$\\map F 0 = S$ :$\\map F {n^+} = \\map G {\\map F n}$ Let $\\displaystyle T = \\bigcup_{n \\mathop \\in \\omega} \\map F n$. Then: :$T$ is a set and is transitive :$S \\subseteq T$ :If $R$ is transitive and $S \\subseteq R$, then $T \\subseteq R$. That is, given any set $S$, there is an explicit construction for its transitive closure."}
+{"_id": "5553", "title": "Stabilizer of Polynomial", "text": "Let $n \\in \\Z: n > 0$. Let $\\map f {x_1, x_2, \\ldots, x_n}$ be a polynomial in $n$ variables $x_1, x_2, \\ldots, x_n$. Let $S_n$ denote the symmetric group on $n$ letters. Let $\\pi, \\rho \\in S_n$. Let the group action $\\pi * f$ be defined as the permutation on the polynomial $f$ by $\\pi$. Then the stabilizer of $f$ is the set of permutations on $n$ letters which fix $f$."}
+{"_id": "5554", "title": "Stabilizer of Element of Group Acting on Itself is Trivial", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $*$ be the group action of $\\struct {G, \\circ}$ on itself by the rule: :$\\forall g, h \\in G: g * h = g \\circ h$ Then the stabilizer of an element $x \\in G$ is given by: :$\\Stab x = \\set e$"}
+{"_id": "13753", "title": "Even Perfect Number except 6 is Congruent to 1 Modulo 9", "text": "Let $n$ be an even perfect number, but not $6$. Then: :$n \\equiv 1 \\pmod 9$"}
+{"_id": "13765", "title": "Divisibility of Sum of 3 Fourth Powers", "text": "Let $n \\in \\Z_{\\ge 0}$ be the sum of three $4$th powers. Then: :$n$ is divisible by $5$ {{iff}} all three addends are also divisible by $5$ :$n$ is divisible by $29$ {{iff}} all three addends are also divisible by $29$."}
+{"_id": "13767", "title": "Prime-Generating Quadratics of form 2 a squared plus p", "text": "The quadratic form: :$2 a^2 + p$ yields prime numbers for $a = 0, 1, \\ldots, p - 1$ for values of $p$: :$3, 5, 11, 29$"}
+{"_id": "5577", "title": "Trivial Group is Initial Object", "text": "Let $\\mathbf{Grp}$ be the category of groups. Let $1 = \\left\\{{e}\\right\\}$ be the trivial group. Then $1$ is an initial object of $\\mathbf{Grp}$."}
+{"_id": "5587", "title": "Rank is Ordinal", "text": "Let $S$ be a small class The rank of $S$ is an ordinal."}
+{"_id": "5589", "title": "Group Direct Product of Cyclic Groups/Corollary", "text": "Let $n_1, n_2, \\ldots, n_s$ be a finite sequence of integers, all greater than $1$, such that for any pair of them $n_i$ and $n_j$, $n_1 \\perp n_j$. Let $G_i$ be a cyclic group of order $n_i$ for each $i: 1 \\le i \\le s$. Then $G_1 \\times G_2 \\times \\cdots \\times G_s$ is cyclic of order $n_1 n_2 \\ldots n_s$."}
+{"_id": "13792", "title": "Compact Closure of Element is Principal Ideal on Compact Subset iff Element is Compact", "text": "Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below algebraic join semilattice. Let $P = \\left({K \\left({L} \\right), \\precsim}\\right)$ be an ordered subset of $L$ where $K \\left({L} \\right)$ denotes the compact subset of $L$. Let $x \\in S$. Then $x^{\\mathrm{compact} }$ is principal ideal in $P$ {{iff}} $x$ is a compact element."}
+{"_id": "13814", "title": "Arithmetic Sequence of 4 Terms with 3 Distinct Prime Factors", "text": "The arithmetic sequence: :$30, 66, 102, 138$ is the smallest of $4$ terms which consists entirely of positive integers each with $3$ distinct prime factors."}
+{"_id": "5623", "title": "Identity Morphism is Terminal Object in Slice Category", "text": "Let $\\mathbf C$ be a metacategory, and let $C \\in \\mathbf C_0$ be an object of $\\mathbf C$. Let $\\operatorname{id}_C: C \\to C$ be the identity morphism for $C$. Then $\\operatorname{id}_C$ is a terminal object in the slice category $\\mathbf C \\mathop / C$."}
+{"_id": "5624", "title": "Identity Morphism is Initial Object in Coslice Category", "text": "Let $\\mathbf C$ be a metacategory, and let $C \\in \\mathbf C_0$ be an object of $\\mathbf C$. Let $\\operatorname{id}_C: C \\to C$ be the identity morphism for $C$. Then $\\operatorname{id}_C$ is an initial object in the coslice category $C \\mathop / \\mathbf C$."}
+{"_id": "5652", "title": "Ordinal Subset is Well-Ordered", "text": "Let $S$ be a class. Let every element of $S$ be an ordinal. Then $\\struct {S, \\in}$ is a strict well-ordering."}
+{"_id": "13851", "title": "Real Symmetric Matrix is Hermitian", "text": "Every real symmetric matrix is Hermitian."}
+{"_id": "13863", "title": "Sufficient Conditions for Weak Extremum", "text": "Let $J$ be a functional such that: :$\\ds J \\sqbrk y = \\int_a^b \\map F {x, y, y'} \\rd x$ :$\\map y a = A$ :$\\map y b = B$ Let $y = \\map y x$ be an extremum. Let the strengthened Legendre's Condition hold. Let the strengthened Jacobi's Necessary Condition hold. {{explain|specific links to those strengthened versions}} Then the functional $J$ has a weak minimum for $y = \\map y x$."}
+{"_id": "13868", "title": "Equivalence Class of Fixed Element/Corollary", "text": ":$i \\notin \\Fix \\sigma$ {{iff}} $\\eqclass i {\\RR_\\sigma}$ contains more than one element"}
+{"_id": "13872", "title": "Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence", "text": "There are an infinite number of rows of Pascal's triangle which contain $3$ integers in arithmetic sequence."}
+{"_id": "13873", "title": "Rows in Pascal's Triangle containing Numbers in Geometric Sequence", "text": "There exist no rows of Pascal's triangle which contain $3$ integers in geometric sequence."}
+{"_id": "13874", "title": "Rows in Pascal's Triangle containing Numbers in Harmonic Sequence", "text": "There exist no rows of Pascal's triangle which contain $3$ integers in harmonic sequence."}
+{"_id": "13876", "title": "Element of Leibniz Harmonic Triangle is Sum of Numbers Below", "text": "The elements in the Leibniz harmonic triangle are the sum of the elements immediately below them. {{refactor|Rework this as another definition of LHT, establishing that column and diagonal $0$ are defined as the reciprocals.}}"}
+{"_id": "13888", "title": "Characterization of Prime Element in Inclusion Ordered Set of Topology", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $L = \\struct {\\tau, \\preceq}$  be an inclusion ordered set of $\\tau$. Let $Z \\in \\tau$. Then $Z$ is prime element in $L$ {{iff}}: :$\\forall X, Y \\in \\tau: X \\cap Y \\subseteq Z \\implies X \\subseteq Z \\lor Y \\subseteq Z$"}
+{"_id": "5698", "title": "Permutation is Cyclic iff At Most One Non-Trivial Orbit", "text": "Let $S$ be a set. Let $\\rho: S \\to S$ be a permutation on $S$. Then: :$\\rho$ is a cyclic permutation {{iff}}: :$S$ has no more than one orbit under $\\rho$ with more than one element."}
+{"_id": "5707", "title": "Class of Cardinals Contains Minimal Infinite Successor Set", "text": "Let $\\mathcal N$ denote the class of all cardinal numbers. Then: :$\\omega \\subseteq \\mathcal N$ Where $\\omega$ denotes the minimal infinite successor set."}
+{"_id": "5727", "title": "Nonlimit Ordinal Cofinal to One", "text": "Let $x$ be a nonlimit non-empty ordinal. Let $\\operatorname{cof}$ denote the cofinal relation.  Let $1$ denote the ordinal one. Then: :$\\operatorname{cof} \\left({x, 1}\\right)$"}
+{"_id": "13919", "title": "GCD of Polynomials does not depend on Base Field", "text": "Let $E / F$ be a field extension. Let $P, Q \\in F \\sqbrk X$ be polynomials. Let: :$\\gcd \\set {P, Q} = R$ in $F \\sqbrk X$ :$\\gcd \\set {P, Q} = S$ in $E \\sqbrk X$. Then $R = S$. In particular, $S \\in F \\sqbrk X$."}
+{"_id": "13935", "title": "Sequences of 4 Consecutive Integers with Falling Sigma", "text": "The following ordered quadruple of consecutive integers have sigma values which are strictly decreasing: :$44, 45, 46, 47$ :$104, 105, 106, 107$"}
+{"_id": "13936", "title": "Pairs of Consecutive Integers with 6 Divisors", "text": "The following sequence of integers are those $n$  which fulfil the equation: :$\\tau \\left({n}\\right) = \\tau \\left({n + 1}\\right) = 6$ where $\\tau \\left({n}\\right)$ denotes the $\\tau$ function. That is, they are the first of pairs of consecutive integers which each have $6$ divisors: :$44, 75, 98, 116, 147, 171, 242, 243, 244, 332, \\ldots$ {{OEIS|A049103}}"}
+{"_id": "5743", "title": "Intersection of Subsemigroups/General Result", "text": "Let $\\mathbb S$ be a set of subsemigroups of $\\left({S, \\circ}\\right)$, where $\\mathbb S \\ne \\varnothing$. Then the intersection $\\bigcap \\mathbb S$ of the members of $\\mathbb S$ is itself a subsemigroup of $\\left({S, \\circ}\\right)$. Also, $\\bigcap \\mathbb S$ is the largest subsemigroup of $\\left({S, \\circ}\\right)$ contained in each member of $\\mathbb S$."}
+{"_id": "5746", "title": "Abelian Group of Order Twice Odd has Exactly One Order 2 Element", "text": "Let $G$ be an abelian group whose identity element is $e$. Let the order of $G$ be $2 n$ such that $n$ is odd. Then there exists exactly one $g \\in G$ with $g \\ne e$ such that $g = g^{-1}$."}
+{"_id": "13946", "title": "Numbers which Multiplied by 2 are the Reverse of when Added to 2", "text": "{{begin-eqn}} {{eqn | l = 47 + 2       | r = 49 }} {{eqn | l = 47 \\times 2       | r = 94 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 497 + 2       | r = 499 }} {{eqn | l = 497 \\times 2       | r = 994 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 4997 + 2       | r = 4999 }} {{eqn | l = 4997 \\times 2       | r = 9994 }} {{end-eqn}} ... and so on:"}
+{"_id": "5757", "title": "Third Isomorphism Theorem/Groups/Corollary 1", "text": "Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $q: G \\to \\dfrac G N$ be the quotient epimorphism from $G$ to the quotient group $\\dfrac G N$. Let $K$ be the kernel of $q$. Then: :$\\dfrac G N \\cong \\dfrac {G / K} {N / K}$"}
+{"_id": "5761", "title": "Morphisms-Only Metacategory Induces Metacategory", "text": "Let $\\mathbf C$ be a morphisms-only metacategory. Then $\\mathbf C$ induces a metacategory $\\mathbf C'$, as follows (phrased to fit with Characterization of Metacategory via Equations): :Define $\\mathbf C'_1$ to be the collection $\\mathbf C_1$ of morphisms of $\\mathbf C$. :Define $\\mathbf C'_0$ to be the image of the operation $\\operatorname{dom}$ on $\\mathbf C_1$. :Define $\\operatorname{id}$ to be the identity on $\\mathbf C'_0$, and take $\\operatorname{dom}$ and $\\operatorname{cod}$ as in $\\mathbf C$. :Define $\\circ$ to be as in $\\mathbf C$; i.e., $g \\circ f$ is the unique element of $\\mathbf C_1$ with $R_\\circ \\left({g, f, g \\circ f}\\right)$."}
+{"_id": "13953", "title": "Smallest Pair of Quasiamicable Numbers", "text": "The smallest pair of quasiamicable numbers is $48$ and $75$."}
+{"_id": "13956", "title": "Equivalence of Definitions of Quasiamicable Numbers", "text": "Let $m \\in \\Z_{>0}$ and $n \\in \\Z_{>0}$ be (strictly) positive integers. {{TFAE|def = Quasiamicable Numbers}}"}
+{"_id": "5769", "title": "Duality Principle (Category Theory)/Formal Duality", "text": "=== Morphisms-Only Category Theory === Let $\\Sigma$ be a statement in the language of category theory. Suppose $\\Sigma$ is provable from the axioms for morphisms-only category theory $\\mathrm{MOCT}$: :$\\mathrm{MOCT} \\vdash \\Sigma$ Then the dual statement $\\Sigma^*$ is also provable from these axioms, i.e.: :$\\mathrm{MOCT} \\vdash \\Sigma^*$ === Object Category Theory === Let $\\mathrm{CT}$ be the collection of seven axioms on Characterization of Metacategory via Equations. Suppose a statement $\\Sigma$ about metacategories follows from the axioms $\\mathrm{CT}$. Then so does its dual statement $\\Sigma^*$."}
+{"_id": "13972", "title": "Logarithm of Divergent Product of Real Numbers/Zero", "text": "The following are equivalent: * The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ diverges to $0$. * The series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\log a_n$ diverges to $-\\infty$."}
+{"_id": "5796", "title": "Disjoint Union is Coproduct in Category of Sets", "text": "Let $\\mathbf{Set}$ be the category of sets. Let $S$ and $T$ be sets. Then their disjoint union $S \\sqcup T$ is a coproduct in $\\mathbf{Set}$."}
+{"_id": "14006", "title": "Continuous Implies Locally Bounded", "text": "Let $X$ be a topological space. Let $M$ be a metric space. Let $f: X \\to M$ be continuous. Then $f$ is locally bounded."}
+{"_id": "5828", "title": "Image of Canonical Injection is Normal Subgroup", "text": "Let $\\struct {G_1, \\circ_1}$ and $\\struct {G_2, \\circ_2}$ be groups with identity elements $e_1$ and $e_2$ respectively. Let $\\struct {G_1 \\times G_2, \\circ}$ be the group direct product of $\\struct {G_1, \\circ_1}$ and $\\struct {G_2, \\circ_2}$. Let: :$\\inj_1: \\struct {G_1, \\circ_1} \\to \\struct {G_1 \\times G_2, \\circ}$ be the canonical injection from $\\struct {G_1, \\circ_1}$ to $\\struct {G_1 \\times G_2, \\circ}$ :$\\inj_2: \\struct {G_2, \\circ_2} \\to \\struct {G_1 \\times G_2, \\circ}$ be the canonical injection from $\\struct {G_2, \\circ_2}$ to $\\struct {G_1 \\times G_2, \\circ}$. Then: :$(1): \\quad \\Img {\\inj_1} \\lhd \\struct {G_1 \\times G_2, \\circ}$ :$(2): \\quad \\Img {\\inj_2} \\lhd \\struct {G_1 \\times G_2, \\circ}$ That is, the images of the canonical injections are normal subgroups of the group direct product of $\\struct {G_1, \\circ_1}$ and $\\struct {G_2, \\circ_2}$."}
+{"_id": "14021", "title": "Period of Reciprocal of 53 is of Quarter Maximal Length", "text": "The decimal expansion of the reciprocal of $53$ has $\\dfrac 1 4$ the maximum period, that is: $13$: :$\\dfrac 1 {53} = 0 \\cdotp \\dot 01886 \\, 79245 \\, 28 \\dot 3$ {{OEIS|A007450}}"}
+{"_id": "5842", "title": "Pointwise Addition on Real-Valued Functions is Associative", "text": "Let $f, g, h: S \\to \\R$ be real-valued functions. Let $f + g: S \\to \\R$ denote the pointwise sum of $f$ and $g$. Then: :$\\paren {f + g} + h = f + \\paren {g + h}$"}
+{"_id": "5849", "title": "Pointwise Addition on Integer-Valued Functions is Associative", "text": "Let $f, g, h: S \\to \\Z$ be integer-valued functions. Let $f + g: S \\to \\Z$ denote the pointwise sum of $f$ and $g$. Then: :$\\paren {f + g} + h = f + \\paren {g + h}$"}
+{"_id": "5850", "title": "Pointwise Addition on Rational-Valued Functions is Associative", "text": "Let $f, g, h: S \\to \\Q$ be rational-valued functions. Let $f + g: S \\to \\Q$ denote the pointwise sum of $f$ and $g$. Then: :$\\paren {f + g} + h = f + \\paren {g + h}$"}
+{"_id": "14045", "title": "Dirichlet Convolution Preserves Multiplicativity", "text": "Let $f, g: \\N \\to \\C$ be multiplicative arithmetic functions. Then their Dirichlet convolution $f * g$ is again multiplicative."}
+{"_id": "14048", "title": "Closed Form for Pentatope Numbers", "text": "The closed-form expression for the $n$th pentatope number is: :$P_n = \\dfrac {n \\paren {n + 1} \\paren {n + 2} \\paren {n + 3} } {24}$"}
+{"_id": "14050", "title": "Necessary and Sufficient Condition for First Order System to be Field for Second Order System", "text": "Let $\\mathbf y$, $\\mathbf f$, $\\boldsymbol \\psi$ be N-dimensional vectors. Let $\\boldsymbol\\psi$ be continuously differentiable. Then $\\forall x \\in \\closedint a b$ the first-order system of differential equations: :$\\mathbf y' = \\map {\\boldsymbol \\psi} {x, \\mathbf y}$ is a field for the second-order system :$\\mathbf y'' = \\map {\\mathbf f} {x, \\mathbf y, \\mathbf y'}$ {{iff}} $\\boldsymbol \\psi$ satisfies : :$\\displaystyle \\frac {\\partial \\boldsymbol \\psi} {\\partial x} + \\sum_{i \\mathop = 1}^N \\frac {\\partial \\boldsymbol \\psi} {\\partial y_i} \\psi_i = \\map {\\mathbf f} {x, \\mathbf y, \\boldsymbol \\psi}$ That is, every solution to Hamilton-Jacobi system is a field for the original system."}
+{"_id": "14051", "title": "Tetrahedral Number as Sum of Squares", "text": ":$H_n = \\displaystyle \\sum_{k \\mathop = 0}^{n / 2} \\paren {n - 2 k}^2$ where $H_n$ denotes the $n$th tetrahedral number."}
+{"_id": "5864", "title": "Smooth Homotopy is an Equivalence Relation", "text": "Let $X$ and $Y$ be smooth manifolds. Let $K \\subseteq X$ be a (possibly empty) subset of $X$. Let $\\mathcal C^\\infty \\left({X, Y}\\right)$ be the set of all smooth mappings from $X$ to $Y$. Define a relation $\\sim$ on $\\mathcal C \\left({X, Y}\\right)$ by $f \\sim g$ if $f$ and $g$ are smoothly homotopic relative to $K$. Then $\\sim$ is an equivalence relation."}
+{"_id": "14105", "title": "Positive Even Integers as Sum of 2 Composite Odd Integers in 2 Ways", "text": "Let $n \\in \\Z_{>0}$ be a positive even integer. Let $n$ be such that it cannot be expressed as the sum of $2$ odd positive composite integers in at least $2$ different ways. Then $n$ belongs to the set: :$\\left\\{ {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 38, 40, 44, 46, 52, 56, 62, 68}\\right\\}$ {{OEIS|A284788}}"}
+{"_id": "5923", "title": "Continuous Image of Path-Connected Set is Path-Connected", "text": "Let $M_1, M_2$ be metric spaces whose metrics are $d_1, d_2$ respectively. Let $f: M_1 \\to M_2$ be a continuous mapping. Let $S \\subseteq M_1$ be a path-connected subspace of $M_1$. Then $f \\sqbrk S$ is a a path-connected subspace of $M_2$."}
+{"_id": "5937", "title": "Quotient Mapping is Coequalizer", "text": "Let $\\mathbf{Set}$ be the category of sets. Let $S$ be a Set, and let $\\mathcal R \\subseteq S \\times S$ be an equivalence relation on $S$. Let $r_1, r_2: \\mathcal R \\to S$ be the projections corresponding to the inclusion mapping $\\mathcal R \\hookrightarrow S \\times S$. Let $q: S \\to S / \\mathcal R$ be the quotient mapping induced by $\\mathcal R$. Then $q$ is a coequalizer of $r_1$ and $r_2$ in $\\mathbf{Set}$."}
+{"_id": "14132", "title": "Reciprocal of 81", "text": "The decimal expansion of the reciprocal of $81$ has a particularly interesting pattern: :$\\dfrac 1 {81} = 0 \\cdotp \\dot 01234 \\, 567 \\dot 9$"}
+{"_id": "5951", "title": "Open Real Interval is not Closed Set/Corollary", "text": "Let: :$I_a = \\openint \\gets a$ :$I_b = \\openint b \\to$ be unbounded open real intervals. Then neither $I_a$ nor $I_b$ are closed sets of $\\R$."}
+{"_id": "14144", "title": "Necessary and Sufficient Condition for First Order System to be Mutually Consistent", "text": "Let $\\mathbf y$, $\\boldsymbol \\psi$ be N-dimensional vectors. Let $g$ be a twice differentiable mapping. Let :$(1): \\quad \\map {\\boldsymbol \\psi} {x, \\mathbf y} = \\map {\\mathbf y'} {x, \\mathbf y}$ :$(2): \\quad \\mathbf p \\sqbrk {x, \\mathbf y, \\map {\\boldsymbol \\psi} {x, \\mathbf y} } = \\map {g_{\\mathbf y} } {x, \\mathbf y}$ where $\\mathbf p$ is a momentum. Then the boundary conditions defined by $(1)$ are mutually consistent {{iff}} the mapping $\\map g {x, \\mathbf y}$ satisfies the Hamilton-Jacobi equation: :$(3): \\quad \\dfrac {\\partial g} {\\partial x} + \\map H {x, \\mathbf y, \\dfrac {\\partial g} {\\partial \\mathbf y} } = 0$"}
+{"_id": "14146", "title": "Reciprocal of 89", "text": "The decimal expansion of the reciprocal of $89$ contains the Fibonacci sequence: :$\\dfrac 1 {89} = 0 \\cdotp \\dot 01123 \\, 59550 \\, 56179 \\, 77528 \\, 08988 \\, 76404 \\, 49438 \\, 20224 \\, 719 \\dot 1$"}
+{"_id": "14148", "title": "Reciprocal as Sum of Fibonacci Numbers by Negative Powers of 10", "text": ":$\\displaystyle \\sum_{k \\mathop \\ge 0} \\dfrac {F_k} {10^{k + 1} } = \\dfrac 1 {89}$ where $F_k$ is the $k$th Fibonacci number: :$F_0 = 0, F_1 = 1, F_k = F_{k - 1} + F_{k - 2}$ That is: <pre> 1 / 89 = 0.0        + 0.01        + 0.001        + 0.0002        + 0.00003        + 0.000005        + 0.0000008        + 0.00000013        + 0.000000021        + 0.0000000034        + 0.00000000055        + .............. </pre>"}
+{"_id": "5957", "title": "Continuous Function on Compact Space is Uniformly Continuous", "text": "Let $\\R^n$ be the $n$-dimensional Euclidean space. Let $S \\subseteq \\R^n$ be a compact subspace of $\\R^n$. Let $f: S \\to \\R$ be a continuous function. Then $f$ is uniformly continuous in $\\R$."}
+{"_id": "14150", "title": "91 is Pseudoprime to 35 Bases less than 91", "text": "$91$ is a Fermat pseudoprime in $35$ bases less than itself: :$3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 27, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 64, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88, 90$"}
+{"_id": "14152", "title": "Reciprocal of 97", "text": "The decimal expansion of the reciprocal of $97$ has the maximum period, that is: $96$: :$\\dfrac 1 {97} = 0 \\cdotp \\dot 01030 \\, 92783 \\, 50515 \\, 46391 \\, 75257 \\, 73195 \\, 87628 \\, 86597 \\, 93814 \\, 43298 \\, 96907 \\, 21649 \\, 48453 \\, 60824 \\, 74226 \\, 80412 \\, 37113 \\, 40206 \\, 18556 \\, \\dot 7$ {{OEIS|A007450}}"}
+{"_id": "14156", "title": "Reciprocal of 98", "text": "The decimal expansion of the reciprocal of $98$ starts with the powers of $2$: :$\\dfrac 1 {98} = 0 \\cdotp 0 \\dot 1020 \\, 40816 \\, 32653 \\, 06122 \\, 44897 \\, 95918 \\, 36734 \\, 69387 \\, 75 \\dot 5$ {{OEIS|A021102}}"}
+{"_id": "14166", "title": "Reciprocal of 103", "text": ":$\\dfrac 1 {103} = 0 \\cdotp \\dot 00970 \\, 87378 \\, 64077 \\, 66990 \\, 29126 \\, 21359 \\, 223 \\dot 3$"}
+{"_id": "6002", "title": "Infinite Set in Compact Space has Omega-Accumulation Point", "text": "Let $\\struct {X, \\tau}$ be a compact topological space. Let $A \\subseteq X$ be infinite. Then $A$ has an $\\omega$-accumulation point in $X$."}
+{"_id": "14209", "title": "Cube of 11 is Palindromic", "text": "::$11^3 = 1331$"}
+{"_id": "14210", "title": "Fourth Power of 11 is Palindromic", "text": "::$11^4 = 14641$"}
+{"_id": "6023", "title": "Local Membership of Equalizer", "text": "Let $\\mathbf C$ be a metacategory. Let $e: E \\to C$ be the equalizer of $f,g : C \\to D$. Then a variable element $z: Z \\to C$ is a local member of $e$ iff $f \\circ z = g \\circ z$: :$z \\in_C e \\iff f \\circ z = g \\circ z$"}
+{"_id": "6026", "title": "Topological Equivalence is Equivalence Relation", "text": "Let $A$ be a set. Let $\\mathcal D$ be the set of all metrics on $A$. Let $\\sim$ be the relation on $\\mathcal D$ defined as: :$\\forall d_1, d_2 \\in \\mathcal D: d_1 \\sim d_2 \\iff d_1$ is topologically equivalent to $d_2$ Then $\\sim$ is an equivalence relation."}
+{"_id": "14227", "title": "Prime Gaps of 10", "text": "The following pairs of consecutive prime numbers are those whose difference is $10$: :$\\tuple {139, 149}, \\tuple {181, 191}, \\tuple {241, 251}, \\tuple {283, 293}, \\ldots$ {{OEIS|A031928|order = lower}}"}
+{"_id": "6043", "title": "Euclidean Plane is Abstract Geometry", "text": "The Euclidean plane $\\left({\\R^2, L_E}\\right)$ is an abstract geometry."}
+{"_id": "14237", "title": "Factorions Base 10", "text": "The following positive integers are the only factorions base $10$: :$1, 2, 145, 40 \\, 585$"}
+{"_id": "6056", "title": "Strong Separation Theorem", "text": "Let $C \\subset \\R^\\ell$ be closed and convex. Let $D = \\set {\\mathbf v} \\subset C^c$. Then $C$ and $D$ can be strongly separated."}
+{"_id": "14252", "title": "Pairs of Integers whose Product with Tau Value are Equal", "text": "Let $\\tau \\left({n}\\right)$ denote the $\\tau$ function: the number of divisors of $n$. The following pairs of integers $T$ have the property that $m \\tau \\left({m}\\right)$ is equal for each $m \\in T$: :$\\left\\{ {18, 27}\\right\\}$ :$\\left\\{ {24, 32}\\right\\}$ :$\\left\\{ {56, 64}\\right\\}$"}
+{"_id": "14270", "title": "Numbers that cannot be made Prime by changing 1 Digit", "text": "The following positive integers cannot be made into prime numbers by changing just one digit: :$200, 202, 204, 205, 206, 208, \\ldots$ {{OEIS|A192545}}"}
+{"_id": "14273", "title": "Triangular Numbers which are also Pentagonal", "text": "The sequence of triangular numbers which are also pentagonal begins: :$1, 210, 40 \\, 755, 7 \\, 906 \\, 276, 1 \\, 533 \\, 776 \\, 805, 297 \\, 544 \\, 793 \\, 910, \\ldots$ {{OEIS|A014979}}"}
+{"_id": "14283", "title": "For Complete Ritz Sequence Continuous Functional approaches its Minimal Value", "text": "Let $J$ be a continuous functional. Let $\\sequence {\\phi_n}$ be a complete Ritz sequence. {{explain|The concept of $\\sequence {\\phi_n}$ does not appear to be related in any way to the statement of the theorem.}} Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\mu_n = \\mu$ where $\\displaystyle \\mu = \\inf_y J \\sqbrk y$."}
+{"_id": "14285", "title": "17 Wallpaper Groups", "text": "There are $17$ wallpaper groups."}
+{"_id": "6096", "title": "Closure of Infinite Union may not equal Union of Closures", "text": "Let $T$ be a topological space. Let $I$ be an infinite indexing set. Let $\\family {H_i}_{i \\mathop \\in I}$ be an indexed family of subsets of a set $S$. Let $\\displaystyle H = \\bigcup_{i \\mathop \\in I} H_i$ be the union of $\\family {H_i}_{i \\mathop \\in I}$. Then it is not always the case that: :$\\displaystyle \\bigcup_{i \\mathop \\in I} \\map \\cl {H_i} = \\map \\cl {\\bigcup_{i \\mathop \\in I} H_i}$ where $\\map \\cl {H_i}$ denotes the closure of $H_i$."}
+{"_id": "6097", "title": "Set of Reciprocals of Positive Integers is Nowhere Dense in Reals", "text": "Let $N$ be the set defined as: :$N := \\set {\\dfrac 1 n: n \\in \\Z_{>0} }$ where $\\Z_{>0}$ is the set of (strictly) positive integers. Let $\\R$ denote the real number line with the usual (Euclidean) metric. Then $N$ is nowhere dense in $\\R$."}
+{"_id": "14291", "title": "Numbers in Even-Even Amicable Pair are not Divisible by 3", "text": "Let $\\tuple {m_1, m_2}$ be an amicable pair such that both $m_1$ and $m_2$ are even. Then neither $m_1$ nor $m_2$ is divisible by $3$."}
+{"_id": "14320", "title": "Construction of Regular 257-Gon", "text": "It is possible to construct a regular polygon with $257$ sides) using a compass and straightedge construction."}
+{"_id": "6138", "title": "Young's Inequality for Increasing Functions/Equality", "text": "Let $a_0$ and $b_0$ be strictly positive real numbers. Let $f: \\closedint 0 {a_0} \\to \\closedint 0 {b_0}$ be a strictly increasing bijection. Let $a$ and $b$ be real numbers such that $0 \\le a \\le a_0$ and $0 \\le b \\le b_0$. Then $b = \\map f a$ {{iff}}: :$\\displaystyle a b = \\int_0^a \\map f u \\rd u + \\int_0^b \\map {f^{-1} } v \\rd v$ where $\\displaystyle \\int$ denotes the Darboux integral."}
+{"_id": "14336", "title": "Filters of Lattice of Power Set form Bounded Below Ordered Set", "text": "Let $X$ be a set. Let $L = \\left({\\mathcal P\\left({X}\\right), \\cup, \\cap, \\subseteq}\\right)$ be an inclusion lattice of power set of $X$. Let $F = \\left({\\mathit{Filt}\\left({L}\\right), \\subseteq}\\right)$ be an inclusion ordered set, where $\\mathit{Filt}\\left({L}\\right)$ denotes the set of all filters on $L$. Then $F$ is bounded below and $\\bot_F = \\left\\{{X}\\right\\}$ where $\\bot_F$ denotes the smallest element of $F$."}
+{"_id": "6151", "title": "Deterministic Time Hierarchy Theorem", "text": "Let $\\map f n$ be a time-constructible function. Then there exists a decision problem which: :can be solved in worst-case deterministic time $\\map f {2 n + 1}^3$ but: :cannot be solved in worst-case deterministic time $\\map f n$. In other words, the complexity class $\\map {\\mathsf {DTIME} } {\\map f n} \\subsetneq \\map {\\mathsf {DTIME} } {\\map f {2 n + 1}^3}$."}
+{"_id": "6184", "title": "Points in Product Spaces are Near Open Sets", "text": "Let $\\family {X_i}_{i \\mathop \\in I}$ be an indexed family of topological spaces, where $I$ is an arbitrary index set. Let $X = \\displaystyle \\prod_{i \\mathop \\in I} X_i$ be the product space of $\\family {X_i}_{i \\mathop \\in I}$. Let $U$ be nonempty open subset of $X$. Let $x$ be a point in $X$. For each point $y$ in $X$, let $\\map K y = \\set {i \\in I : y_i \\ne x_i}$. Then there exists a point $u$ in $U$ such that $\\map K u$ is finite."}
+{"_id": "6193", "title": "Equivalence of Definitions of Limit of Function in Metric Space", "text": "{{TFAE|def = Limit of Function (Metric Space)|view = Limit of Function|context = Metric Space|contextview = Metric Spaces}} Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $c$ be a limit point of $M_1$. Let $f: A_1 \\to A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ ''except possibly'' at $c$. Let $L \\in M_2$."}
+{"_id": "14401", "title": "Equivalence of Definitions of Field of Quotients", "text": "Let $D$ be an integral domain. Let $F$ be a field. {{TFAE|def = Field of Quotients}}"}
+{"_id": "14406", "title": "Products of 2-Digit Pairs which Reversed reveal Same Product", "text": "The following positive integers can be expressed as the product of $2$ two-digit numbers in $2$ ways such that the factors in one of those pairs is the reversal of each of the factors in the other: :$504, 756, 806, 1008, 1148, 1209, 1472, 1512, 2016, 2208, 2418, 2924, 3024, 4416$ <!-- fascists won't include the damn thing {{OEIS|A289978}}  SIEG HEIL! SIEG HEIL!-->"}
+{"_id": "14414", "title": "Poulet Numbers which are also Magic Constant for Magic Square", "text": "The sequence of Poulet numbers which are also the magic constant of a magic square begins: :$1105, 2465, \\ldots$"}
+{"_id": "6238", "title": "Length of Contour is Well-Defined", "text": "Let $C_1, \\ldots, C_n$ be directed smooth curves. Let $C_i$ be parameterized by the smooth path $\\gamma_i: \\closedint {a_i} {b_i} \\to \\C$ for all $i \\in \\set {1, \\ldots, n}$. Let $C$ be the contour defined by the finite sequence $C_1, \\ldots, C_n$. Suppose that $\\sigma_i: \\closedint {c_i} {d_i} \\to \\C$ is a reparameterization of $C_i$ for all $i \\in \\set {1, \\ldots, n}$ Then: :$\\displaystyle \\sum_{i \\mathop = 1}^n \\int_{a_i }^{b_i} \\size {\\map {\\gamma_i'} t} \\rd t = \\sum_{i \\mathop = 1}^n \\int_{c_i}^{d_i} \\size {\\map {\\sigma_i'} t} \\rd t$ and all real integrals in the equation are defined."}
+{"_id": "14448", "title": "Equivalence of Definitions of Change of Basis Matrix", "text": "Let $R$ be a ring with unity. Let $G$ be a finite-dimensional unitary $R$-module. Let $A = \\sequence {a_n}$ and $B = \\sequence {b_n}$ be ordered bases of $G$. {{TFAE|def = Change of Basis Matrix}}"}
+{"_id": "14460", "title": "Numbers not Expressible as Sum of Fewer than 19 Fourth Powers", "text": "The following positive integer are the only ones which cannot be expressed as the sum of fewer than $19$ fourth powers: :$79, 159, 239, 319, 399, 479, 559$ {{OEIS|A046050}}"}
+{"_id": "14464", "title": "Free Module on Set is Free", "text": "Let $R$ be a ring with unity. Let $I$ be a set. Let $R^{\\paren I}$ be the free $R$-module on $I$. Then $R^{\\paren I}$ is a free $R$-module."}
+{"_id": "14475", "title": "Pandigital Square Equation", "text": "The following equations, which include each digit from $1$ to $9$ inclusive, are the only ones of their kind: {{begin-eqn}} {{eqn | l = 567^2       | r = 321 \\, 489 }} {{eqn | l = 854^2       | r = 729 \\, 316 }} {{end-eqn}}"}
+{"_id": "14477", "title": "Sequence of 11 Primes by Trebling and Adding 16", "text": "The process of multiplication by $3$ and then adding $16$ produces a sequence of $11$ primes when starting from $587$: :$587, 1777, 5347, 16 \\, 057, 48 \\, 187, 144 \\, 577, 433 \\, 747, 1 \\, 301 \\, 257, 3 \\, 903 \\, 787, 11 \\, 711 \\, 377, 35 \\, 134 \\, 147$"}
+{"_id": "14487", "title": "Consecutive Sophie Germain Primes cannot be Pair of Twin Primes", "text": "Let $p$ and $p + 2$ be twin primes. Then unless $p = 3$ it is not possible for both $p$ and $p + 2$ to be Sophie Germain primes."}
+{"_id": "6310", "title": "Ring Without Unity may have Quotient Ring with Unity", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $I$ be an ideal of $R$. Let $\\struct {R / I, +, \\circ}$ be the associated quotient ring. Then $\\struct {R / I, +, \\circ}$ may have a unity even if $\\struct {R, +, \\circ}$ has not."}
+{"_id": "6309", "title": "Stone Space is Topological Space", "text": "Let $\\struct {B, \\preceq, \\wedge, \\vee}$ be a non-empty Boolean algebra. Let $\\struct {U, \\tau}$ be the Stone space of $B$. Then $\\struct {U, \\tau}$ is a topological space."}
+{"_id": "14514", "title": "Sum of 714 and 715", "text": "The sum of $714$ and $715$ is a $4$-digit integer which has $6$ anagrams which are prime."}
+{"_id": "6327", "title": "Limits of Real and Imaginary Parts", "text": "Let $f: D \\to \\C$ be a complex function, where $D \\subseteq \\C$. Let $z_o \\in D$ be a complex number. Suppose $f$ is continuous at $z_0$. Then: :$(1): \\quad \\displaystyle \\lim_{z \\to z_o} \\operatorname{Re} \\left({f \\left({z}\\right) }\\right) = \\operatorname{Re} \\left({ \\lim_{z \\to z_o} f \\left({z}\\right) }\\right)$ :$(2): \\quad \\displaystyle \\lim_{z \\to z_o} \\operatorname{Im} \\left({f \\left({z}\\right) }\\right) = \\operatorname{Im} \\left({ \\lim_{z \\to z_o} f \\left({z}\\right) }\\right)$ Here, $\\operatorname{Re} \\left({f \\left({z}\\right) }\\right) $ denotes the real part of $f \\left({z}\\right)$, and $\\operatorname{Im} \\left({f \\left({z}\\right) }\\right) $ denotes the imaginary part of $f \\left({z}\\right)$."}
+{"_id": "14519", "title": "Equivalence of Definitions of Synthetic Basis", "text": "Let $S$ be a set. {{TFAE|def = Synthetic Basis}}"}
+{"_id": "14523", "title": "Continuous iff Mapping at Element is Supremum of Compact Elements", "text": "Let $L = \\left({S, \\preceq_1, \\tau_1}\\right)$ and $R = \\left({T, \\preceq_2, \\tau_2}\\right)$ be complete algebraic topological lattices with Scott topologies. Let $f: S \\to T$ be a mapping. Then $f$ is continuous {{iff}} :$\\forall x \\in S: f \\left({x}\\right) = \\sup \\left\\{ {f \\left({w}\\right): w \\in S \\land w \\preceq_1 x \\land w}\\right.$ is compact$\\left.{}\\right\\}$"}
+{"_id": "14542", "title": "Sum of Pandigital Triplet of 3-Digit Primes", "text": "The smallest integer which is the sum of a set of $3$ three-digit primes using all $9$ digits from $1$ to $9$ once each is $999$: :$149 + 263 + 587 = 999$"}
+{"_id": "14546", "title": "Reciprocal of 1089", "text": ":$\\dfrac 1 {1089} = 0 \\cdotp \\dot 00091 \\, 82736 \\, 45546 \\, 37281 \\, 9 \\dot 1$"}
+{"_id": "14557", "title": "Square Numbers which are Sum of Sequence of Odd Cubes", "text": "The sequence of square numbers which can be expressed as the sum of a sequence of odd cubes from $1$ begins: :$1, 1225, 1 \\, 413 \\, 721, 1 \\, 631 \\, 432 \\, 881, \\dotsc$ {{OEIS|A046177}} The sequence of square roots of this sequence is: :$1, 35, 1189, 40 \\, 391, \\dotsc$ {{OEIS|A046176}}"}
+{"_id": "14563", "title": "Smallest Triplet of Consecutive Integers Divisible by Cube", "text": "The smallest sequence of triplets of consecutive integers each of which is divisible by a cube greater than $1$ is: :$\\tuple {1375, 1376, 1377}$"}
+{"_id": "14582", "title": "Sequence of Composite Mersenne Numbers", "text": "The sequence of Mersenne numbers which are composite begins: :$2047, 8 \\, 388 \\, 607, 536 \\, 870 \\, 911, 137 \\, 438 \\, 953 \\, 471, 2 \\, 199 \\, 023 \\, 255 \\, 551,\\ldots$ {{OEIS|A065341}} The sequence of corresponding indices $p$ such that $2^p - 1$ is composite begins: :$11, 23, 29, 37, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, \\ldots$ {{OEIS|A054723}} The sequence of corresponding integers $n$ such that the $n$th prime number $p \\left({n}\\right)$ is such that $2^{p \\left({n}\\right)} - 1$ is composite begins: :$5, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, \\ldots$ {{OEIS|A135980}}"}
+{"_id": "14584", "title": "Numbers Reversed when Multiplying by 4", "text": "Numbers of the form $\\sqbrk {21 \\paren 9 78}_{10}$ are reversed when they are multiplied by $4$: {{begin-eqn}} {{eqn | l = 2178 \\times 4       | r = 8712 }} {{eqn | l = 21 \\, 978 \\times 4       | r = 87 \\, 912 }} {{eqn | l = 219 \\, 978 \\times 4       | r = 879 \\, 912 }} {{end-eqn}} and so on."}
+{"_id": "14586", "title": "Relational Structure admits Lower Topology", "text": "Let $R = \\left({S, \\preceq}\\right)$ be a relational structure. Then there exists a relational structure with lower topology $T = \\left({S, \\preceq, \\tau}\\right)$ such that $T$ is a topological space."}
+{"_id": "14599", "title": "2520 equals Sum of 4 Divisors in 6 Ways", "text": "The number $2520$ can be expressed as the sum of $4$ of its divisors in $6$ different ways: {{begin-eqn}} {{eqn | l = 2520       | r = 1260 + 630 + 504 + 126 }} {{eqn | r = 1260 + 630 + 421 + 210 }} {{eqn | r = 1260 + 840 + 360 + 60 }} {{eqn | r = 1260 + 840 + 315 + 105 }} {{eqn | r = 1260 + 840 + 280 + 140 }} {{eqn | r = 1260 + 840 + 252 + 168 }} {{end-eqn}} This is the maximum possible number of ways it is possible to express an integer as the sum of $4$ of its divisors."}
+{"_id": "6416", "title": "Operating on Ordered Group Inequalities", "text": "If $x \\prec y$ and $z \\prec w$, then $x \\circ z \\prec y \\circ w$. If $x \\prec y$ and $z \\preceq w$, then $x \\circ z \\prec y \\circ w$. If $x \\preceq y$ and $z \\prec w$, then $x \\circ z \\prec y \\circ w$. If $x \\preceq y$ and $z \\preceq w$, then $x \\circ z \\preceq y \\circ w$."}
+{"_id": "6428", "title": "Reflexive Closure of Antisymmetric Relation is Antisymmetric", "text": "Let $S$ be a set. Let $\\mathcal R$ be an antisymmetric relation on $S$. Let $\\mathcal R^=$ be the reflexive closure of $\\mathcal R$. Then $\\mathcal R^=$ is also antisymmetric."}
+{"_id": "6436", "title": "Conditional is not Left Self-Distributive/Formulation 2", "text": "While this holds: :$\\vdash \\paren {\\paren {p \\implies q} \\implies r} \\implies \\paren {\\paren {p \\implies r} \\implies \\paren {q \\implies r} }$ its converse does not: :$\\not \\vdash \\paren {\\paren {p \\implies r} \\implies \\paren {q \\implies r} } \\implies \\paren {\\paren {p \\implies q} \\implies r}$"}
+{"_id": "6439", "title": "Self-Distributive Law for Conditional/Formulation 2", "text": ":$\\vdash \\paren {p \\implies \\paren {q \\implies r} } \\iff \\paren {\\paren {p \\implies q} \\implies \\paren {p \\implies r} }$"}
+{"_id": "6477", "title": "Rule of Exportation/Formulation 2", "text": ":$\\vdash \\paren {\\paren {p \\land q} \\implies r} \\iff \\paren {p \\implies \\paren {q \\implies r} }$"}
+{"_id": "14691", "title": "Cube of 20 is Sum of Sequence of 4 Consecutive Cubes", "text": ":$20^3 = \\displaystyle \\sum_{k \\mathop = 11}^{14} k^3$ That is: :$20^3 = 11^3 + 12^3 + 13^3 + 14^3$"}
+{"_id": "14719", "title": "Largest Integer not Sum of Two Abundant Numbers", "text": "The largest integer which is not the sum of $2$ abundant numbers is $20 \\, 161$."}
+{"_id": "14720", "title": "Smallest Integer using Three Words in English Description", "text": "The smallest integer which uses exactly $3$ words in its standard (British) English description is: :$21 \\, 000$: '''twenty-one thousand''' counting hyphenations as separate words."}
+{"_id": "14736", "title": "Carmichael Number with 4 Prime Factors", "text": "$41 \\, 041$ is the smallest Carmichael number with $4$ prime factors: :$41 \\, 041 = 7 \\times 11 \\times 13 \\times 41$"}
+{"_id": "14752", "title": "Tableau Extension Lemma/General Statement/Proof 1", "text": "Let $T$ be a finite propositional tableau. Let its hypothesis set $\\mathbf H$ be finite. {{:Tableau Extension Lemma/General Statement}}"}
+{"_id": "14761", "title": "Reciprocal of 142,857", "text": ":$\\dfrac 1 {142 \\, 857} = 0 \\cdotp \\dot 00000 \\, \\dot 7$"}
+{"_id": "14784", "title": "Numbers whose Fourth Root equals Number of Divisors", "text": "There are $4$ positive integers whose $4$th root equals the number of its divisors: {{begin-eqn}} {{eqn | l = 1       | r = 1^4       | c =  }} {{eqn | l = 625       | r = 5^4       | c =  }} {{eqn | l = 6561       | r = 9^4       | c =  }} {{eqn | l = 4 \\, 100 \\, 625       | r = 45^4       | c =  }} {{end-eqn}} {{OEIS|A143026}}"}
+{"_id": "14790", "title": "Smallest 5 Consecutive Primes in Arithmetic Sequence", "text": "The smallest $5$ consecutive primes in arithmetic sequence are: :$9 \\, 843 \\, 019 + 30 n$ for $n = 0, 1, 2, 3, 4$. Note that while there are many longer arithmetic sequences of far smaller primes, those primes are not consecutive."}
+{"_id": "14793", "title": "Smallest Triplet of Primitive Pythagorean Triangles with Same Area", "text": "The smallest set of $3$ primitive Pythagorean triangles which all have the same area are: :the $4485-5852-7373$ triangle :the $3059-8580-9109$ triangle :the $1380-19 \\, 019-19 \\, 069$ triangle. That area is $13 \\, 123 \\, 110$."}
+{"_id": "6610", "title": "Biconditional is Transitive/Formulation 1/Proof 2", "text": ":$p \\iff q, q \\iff r \\vdash p \\iff r$"}
+{"_id": "6618", "title": "Law of Identity/Formulation 2/Proof 2", "text": ": $\\vdash p \\implies p$"}
+{"_id": "14827", "title": "Exchange of Order of Indexed Summations/Rectangular Domain", "text": "Let $D = \\closedint a b \\times \\closedint c d$ be the cartesian product. Let $f: D \\to \\mathbb A$ be a mapping Then we have an equality of indexed summations: :$\\displaystyle \\sum_{i \\mathop = a}^b \\sum_{j \\mathop = c}^d \\map f {i, j} = \\sum_{j \\mathop = c}^d \\sum_{i \\mathop = a}^b \\map f {i, j}$"}
+{"_id": "14832", "title": "Sum over Union of Finite Sets", "text": "Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $S$ and $T$ be finite sets. Let $f: S \\cup T \\to \\mathbb A$ be a mapping. Then we have the equality of summations over finite sets: :$\\displaystyle \\sum_{u \\mathop \\in S \\mathop \\cup T} \\map f u = \\sum_{s \\mathop \\in S} \\map f s + \\sum_{t \\mathop \\in T} \\map f t - \\sum_{v \\mathop \\in S \\mathop \\cap T} \\map f v$"}
+{"_id": "6644", "title": "Rule of Idempotence/Conjunction/Formulation 1/Proof", "text": ": $p \\dashv \\vdash p \\land p$"}
+{"_id": "14838", "title": "Exchange of Order of Summations over Finite Sets/Subset of Cartesian Product", "text": "Let $D\\subset S \\times T$ be a subset. Let $\\pi_1 : D \\to S$ and $\\pi_2 : D \\to T$ be the restrictions of the projections of $S\\times T$. Then we have an equality of summations over finite sets: :$\\displaystyle \\sum_{s \\mathop \\in S} \\sum_{t \\mathop \\in \\pi_2 \\left({\\pi_1^{-1} \\left({s}\\right)}\\right)} f \\left({s, t}\\right) = \\sum_{t \\mathop \\in T} \\sum_{s \\mathop \\in \\pi_1 \\left({\\pi_2^{-1} \\left({t}\\right)}\\right)} f \\left({s, t}\\right)$"}
+{"_id": "14856", "title": "Canonical Homomorphism to Polynomial Ring is Ring Monomorphism", "text": "Let $R$ be a commutative ring with unity. Let $(R[X], \\iota, X)$ be a polynomial ring over $R$ in one indeterminate $X$. Then the canonical homomorphism $\\iota : R \\to R[X]$ is a ring monomorphism."}
+{"_id": "14868", "title": "Equivalence of Definitions of Consistent Proof System", "text": "{{TFAE|def = Consistent (Logic)/Proof System/Propositional Logic|view = Consistent Proof System for Propositional Logic}} Let $\\LL_0$ be the language of propositional logic. Let $\\mathscr P$ be a proof system for $\\LL_0$."}
+{"_id": "14884", "title": "Irreducible Component is Closed", "text": "Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $Y$ be an irreducible component of $T$. Then $Y$ is closed in $T$."}
+{"_id": "6697", "title": "Homotopy Characterisation of Simply Connected Sets", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $X$ be a subset of $S$. Then $X$ is simply connected {{iff}} the following conditions hold: :$(1): \\quad $ $X$ is path-connected. :$(2): \\quad $ All paths in $X$ with the same initial points and final points are freely homotopic."}
+{"_id": "14890", "title": "Largest Pandigital Square including Zero", "text": "The largest pandigital square (in the sense where pandigital includes the zero) is $9 \\, 814 \\, 072 \\, 356$: :$9 \\, 814 \\, 072 \\, 356 = 99 \\, 066^2$"}
+{"_id": "6714", "title": "Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 1/Proof", "text": ":$p \\lor \\paren {q \\land r} \\dashv \\vdash \\paren {p \\lor q} \\land \\paren {p \\lor r}$"}
+{"_id": "14906", "title": "Multiply Perfect Number of Order 6", "text": "The number defined as: :$n = 2^{36} \\times 3^8 \\times 5^5 \\times 7^7 \\times 11 \\times 13^2 \\times 19 \\times 31^2$ ::$\\times \\ 43 \\times 61 \\times 83 \\times 223 \\times 331 \\times 379 \\times 601 \\times 757 \\times 1201$ ::$\\times \\ 7019 \\times 112 \\, 303 \\times 898 \\, 423 \\times 616 \\, 318 \\, 177$ is multiply perfect of order $6$."}
+{"_id": "6718", "title": "Rule of Distribution/Disjunction Distributes over Conjunction/Right Distributive/Formulation 1/Proof", "text": ":$\\left({q \\land r}\\right) \\lor p \\dashv \\vdash \\left({q \\lor p}\\right) \\land \\left({r \\lor p}\\right)$"}
+{"_id": "14948", "title": "Functional Equation for Completed Riemann Zeta Function", "text": "Let $\\xi : \\C \\to \\C$ be the completed Riemann zeta function. Let $s\\in \\C$ be a complex number. Then: :$\\map \\xi s = \\map \\xi {1 - s}$"}
+{"_id": "14984", "title": "Largest Integer whose Digits taken in Pairs all form Distinct Primes", "text": "The largest integer which has the property that every pair of its digits taken together is a distinct prime number is $619 \\, 737 \\, 131 \\, 179$."}
+{"_id": "14990", "title": "Smallest 22 Primes in Arithmetic Sequence", "text": "The smallest $22$ primes in arithmetic sequence are: :$11 \\, 410 \\, 337 \\, 850 \\, 553 + 4 \\, 609 \\, 098 \\, 694 \\, 200 n$ for $n = 0, 1, \\ldots, 21$."}
+{"_id": "14995", "title": "Smallest Cunningham Chain of the Second Kind of Length 13", "text": "The smallest Cunningham chain of the second kind of length $13$ is: :$758 \\, 083 \\, 947 \\, 856 \\, 951$, $1 \\, 516 \\, 167 \\, 895 \\, 713 \\, 901$, $3 \\, 032 \\, 335 \\, 791 \\, 427 \\, 801$, $6 \\, 064 \\, 671 \\, 582 \\, 855 \\, 601$, $12 \\, 129 \\, 343 \\, 165 \\, 711 \\, 201$, $24 \\, 258 \\, 686 \\, 331 \\, 422 \\, 401$, $48 \\, 517 \\, 372 \\, 662 \\, 844 \\, 801$, $97 \\, 034 \\, 745 \\, 325 \\, 689 \\, 601$, $194 \\, 069 \\, 490 \\, 651 \\, 379 \\, 201$, $388 \\, 138 \\, 981 \\, 302 \\, 758 \\, 401$, $776 \\, 277 \\, 962 \\, 605 \\, 516 \\, 801$, $1 \\, 552 \\, 555 \\, 925 \\, 211 \\, 033 \\, 601$, $3 \\, 105 \\, 111 \\, 850 \\, 422 \\, 067 \\, 201$"}
+{"_id": "15005", "title": "Repunit Expressed using Power of 10", "text": "The repunit number $R_n$ can be expressed as: :$R_n = \\dfrac {10^n - 1} 9$"}
+{"_id": "15012", "title": "Repunit in Base 9 is Triangular", "text": "Let $m$ be a repunit base $9$. Then $m$ is a triangular number."}
+{"_id": "15017", "title": "Largest nth Power which has n Digits", "text": "The largest $n$th power which has $n$ digits is $9^{21}$: :$9^{21} = 109 \\, 418 \\, 989 \\, 131 \\, 512 \\, 359 \\, 209$"}
+{"_id": "15021", "title": "Left-Truncatable Prime/Examples/357,686,312,646,216,567,629,137", "text": "The largest left-truncatable prime is $357 \\, 686 \\, 312 \\, 646 \\, 216 \\, 567 \\, 629 \\, 137$."}
+{"_id": "15027", "title": "Smallest Integer Divisible by All Numbers from 1 to 100", "text": "The smallest positive integer which is divisible by each of the integers from $1$ to $100$ is: :$69 \\, 720 \\, 375 \\, 229 \\, 712 \\, 477 \\, 164 \\, 533 \\, 808 \\, 935 \\, 312 \\, 303 \\, 556 \\, 800$"}
+{"_id": "15028", "title": "Integer which is Multiplied by 9 when moving Last Digit to First", "text": "Let $N$ be the positive integer: :$N = 10 \\, 112 \\, 359 \\, 550 \\, 561 \\, 797 \\, 752 \\, 808 \\, 988 \\, 764 \\, 044 \\, 943 \\, 820 \\, 224 \\, 719$ $N$ is the smallest positive integer $N$ such that if you move the last digit to the front, the result is the positive integer $9 N$."}
+{"_id": "15049", "title": "Yoneda Embedding Theorem", "text": "Let $C$ be a locally small category. Let $\\mathbf{Set}$ be the category of sets. Let $[C^{\\operatorname{op}}, \\mathbf{Set}]$ be the contravariant functor category. Then the Yoneda embedding $h_- : C \\to [C^{\\operatorname{op}}, \\mathbf{Set}]$ is a fully faithful embedding."}
+{"_id": "15059", "title": "Titanic Prime whose Digits are all Odd", "text": "The integer defined as: :$1358 \\times 10^{3821} - 1$ is a titanic prime all of whose digits are odd. That is: :$1357 \\paren 9_{3821}$ where $\\paren a_b$ means $b$ instances of $a$ in a string."}
+{"_id": "15061", "title": "Gigantic Palindromic Prime", "text": "The integer defined as: :$10^{11 \\, 810} + 1 \\, 465 \\, 641 \\times 10^{5902} + 1$ is a gigantic prime which is also palindromic. That is: :$1(0)_{5901}1465641(0)_{5901}1$ where $\\left({a}\\right)_b$ means $b$ instances of $a$ in a string."}
+{"_id": "15084", "title": "Existence of Topological Space which satisfies no Separation Axioms", "text": "There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied."}
+{"_id": "6893", "title": "Relation Compatible with Group Operation is Reflexive or Antireflexive", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\RR$ be a relation on $G$ that is compatible with $\\circ$. Then $\\RR$ is reflexive or antireflexive."}
+{"_id": "15091", "title": "Existence of Compact Hausdorff Space which is not T5", "text": "There exists at least one example of a compact $T_2$ (Hausdorff) space which is not a $T_5$ space."}
+{"_id": "15097", "title": "Existence of Hausdorff Space which is not Completely Hausdorff", "text": "There exists at least one example of a topological space which is a $T_2$ (Hausdorff) space, but is not also a completely Hausdorff space."}
+{"_id": "15104", "title": "Existence of Semiregular Topological Space which is not Urysohn Space", "text": "There exists at least one example of a semiregular topological space which is not a Urysohn space."}
+{"_id": "15107", "title": "Existence of Compact Space which is not Sequentially Compact", "text": "There exists at least one example of a compact topological space which is not also a sequentially compact space."}
+{"_id": "6918", "title": "Norm of Vector Cross Product", "text": "Let $\\mathbf a$ and $\\mathbf b$ be vectors in the Euclidean space $\\R^3$. Let $\\times$ denote the vector cross product. Then: :$(1): \\quad$ $\\left\\Vert{ \\mathbf a \\times \\mathbf b }\\right\\Vert^2 = \\left\\Vert{\\mathbf a}\\right\\Vert^2 \\left\\Vert{\\mathbf b}\\right\\Vert^2 - \\left({\\mathbf a \\cdot \\mathbf b}\\right)^2$ :$(2): \\quad$ $\\left\\Vert{ \\mathbf a \\times \\mathbf b }\\right\\Vert = \\left\\Vert{\\mathbf a}\\right\\Vert \\left\\Vert{\\mathbf b}\\right\\Vert \\left\\vert{\\sin \\theta}\\right\\vert$ where $\\theta$ is the angle between $\\mathbf a$ and $\\mathbf b$, or an arbitrary number if $\\mathbf a$ or $\\mathbf b$ is the zero vector."}
+{"_id": "15119", "title": "Existence of Separable Space which is not Second-Countable", "text": "There exists at least one example of a separable topological space which is not also a second-countable space."}
+{"_id": "15125", "title": "Existence of Paracompact Space which is not Compact", "text": "There exists at least one example of a paracompact topological space which is not also a compact topological space."}
+{"_id": "15137", "title": "Uncountable Product of First-Countable Spaces is not always First-Countable", "text": "Let $I$ be an indexing set with uncountable cardinality. Let $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be a family of topological spaces indexed by $I$. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let each of $\\struct {S_\\alpha, \\tau_\\alpha}$ be a first-countable space. Then it is not necessarily the case that $\\struct {S, \\tau}$ is also a first-countable space."}
+{"_id": "15139", "title": "Uncountable Product of Separable Spaces is not always Separable", "text": "Let $I$ be an indexing set with uncountable cardinality. Let $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be a family of topological spaces indexed by $I$. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let each of $\\struct {S_\\alpha, \\tau_\\alpha}$ be a separable space. Then it is not necessarily the case that $\\struct {S, \\tau}$ is also a separable space."}
+{"_id": "15141", "title": "Product of Metacompact Spaces is not always Metacompact", "text": "Let $I$ be an indexing set. Let $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be a family of topological spaces indexed by $I$. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let each of $\\struct {S_\\alpha, \\tau_\\alpha}$ be a metacompact space. Then it is not necessarily the case that $\\struct {S, \\tau}$ is also metacompact space."}
+{"_id": "15162", "title": "Finite Irreducible Space is Path-Connected", "text": "Let $T = \\left({S, \\tau}\\right)$ be a finite irreducible topological space. Then $T$ is path-connected."}
+{"_id": "15173", "title": "Equivalence of Definitions of Convergent of Continued Fraction", "text": "Let $F$ be a field, such as the field of real numbers. Let $n \\in \\N \\cup \\set \\infty$ be an extended natural number. Let $C = \\sqbrk {a_0, a_1, a_2, \\dotsc}$ be a continued fraction in $F$ of length $n$. Let $k \\le n$ be a natural number. {{TFAE|def = Convergent of Continued Fraction}}"}
+{"_id": "15175", "title": "Locally Connected Space is not necessarily Locally Path-Connected", "text": "Let $T = \\struct {S, \\tau}$ be a topological space which is locally connected. Then it is not necessarily the case that $T$ is also an locally patj-connected space."}
+{"_id": "15190", "title": "Binary Product in Preadditive Category is Biproduct", "text": "Let $A$ be a preadditive category. Let $a_1, a_2$ be objects of $A$. Let $(a_1 \\times a_2, p_1, p_2)$ be their binary product, assuming it exists. Let $i_1 : a_1 \\to a_1 \\times a_2$ be the unique morphism with: :$p_1 \\circ i_1 = 1 : a_1 \\to a_1$ :$p_2 \\circ i_1 = 0 : a_1 \\to a_2$ Let $i_2 : a_1 \\to a_1 \\times a_2$ be the unique morphism with: :$p_1 \\circ i_2 = 0 : a_2 \\to a_1$ :$p_2 \\circ i_2 = 1 : a_2 \\to a_2$ where $1$ is the identity morphism and $0$ is the zero morphism. Then $(a_1 \\times a_2, i_1, i_2, p_1, p_2)$ is the binary biproduct of $a_1$ and $a_2$."}
+{"_id": "7007", "title": "Matrix Multiplication Interpretation of Relation Composition", "text": "Let $A$, $B$ and $C$ be finite non-empty sets that are initial segments of $\\N_{\\ne 0}$. Let $\\mathcal R \\subseteq B \\times A$ and $\\mathcal S \\subseteq C \\times B$ be relations. Let $\\mathbf R$ and $\\mathbf S$ be matrices which we define as follows: :$\\left[{r}\\right]_{i j} = \\begin{cases} T & : (i, j) \\in \\mathcal R \\\\ F & : (i, j) \\notin \\mathcal R\\\\ \\end{cases}$ :$\\left[{s}\\right]_{i j} = \\begin{cases} T & : (i, j) \\in \\mathcal S \\\\ F & : (i, j) \\notin \\mathcal S\\\\ \\end{cases}$ Then we can interpret the matrix product $\\mathbf R \\mathbf S$ as the composition $\\mathcal S \\circ \\mathcal R$. To do so we temporarily consider $\\left({\\left\\{ {T, F}\\right\\}, \\land, \\lor}\\right)$ to be our \"ring\" on which we are basing matrix multiplication. Then: :$\\left[{r s}\\right]_{i j} = T \\iff (i, j) \\in \\mathcal S \\circ \\mathcal R$"}
+{"_id": "7013", "title": "Set of Subsets is Cover iff Set of Complements is Free", "text": "Let $S$ be a set. Let $\\mathcal C$ be a set of sets. Then $\\mathcal C$ is a cover for $S$ {{iff}} $\\set {\\relcomp S X: X \\in \\mathcal C}$ is free."}
+{"_id": "15208", "title": "Metric Space is Perfectly Normal", "text": "Let $M = \\struct {A, d}$ be a metric space. Then $M$ is a perfectly normal space."}
+{"_id": "15211", "title": "Tutte's Wheel Theorem", "text": "Every 3-connected graph can be obtained by the following procedure: * Start with $G_0 := K_4$ * Given $G_i$ pick a vertex $v$ * Split into $v'$ and $v''$ and add edge $\\{v',v''\\}$ This procedure directly follows from the theorem:  :A graph $G$ is 3-connected ('''A''') iff there exists a sequence $G_0, G_1, . . . , G_n$ of graphs with the following properties ('''B'''):  :* $G_0 = K_4$ and $G_n = G$; :* $G_{i+1}$ has an edge $e = xy$ with $deg \\left({x}\\right), deg \\left({y}\\right) \\geq 3$ and $G_i = G_{i+1} \\thinspace / \\thinspace e$ for every $i < n$."}
+{"_id": "15223", "title": "G-Delta Sets in Indiscrete Topology", "text": "$H$ is a $G_\\delta$ ($G$-delta) set of $T$ {{iff}} either $H = S$ or $H = \\O$."}
+{"_id": "15225", "title": "Subset of Indiscrete Space is Sequentially Compact", "text": "$H$ is sequentially compact in $T$."}
+{"_id": "15238", "title": "Infinite Particular Point Space is not Metacompact", "text": "Let $T = \\left({S, \\tau_p}\\right)$ be an infinite particular point space. Then $T$ is not metacompact."}
+{"_id": "15244", "title": "Condition for Closed Extension Space to be T4 Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_p = \\struct {S^*_p, \\tau^*_p}$ be the closed extension space of $T$. Then: :$T^*_p$ is a $T_4$ space {{iff}} $T$ is a $T_4$ space vacuously and $T^*_p$ in this case is also a $T_4$ space vacuously."}
+{"_id": "15245", "title": "Condition for Closed Extension Space to be T5 Space", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_p = \\struct {S^*_p, \\tau^*_p}$ be the closed extension space of $T$. Then: :$T^*_p$ is a $T_5$ space {{iff}} $T$ is a $T_5$ space vacuously and $T^*_p$ in this case is also a $T_5$ space vacuously."}
+{"_id": "15260", "title": "Separation Properties of Alexandroff Extension of Rational Number Space", "text": "Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Let $p$ be a new element not in $\\Q$. Let $\\Q^* := \\Q \\cup \\set p$. Let $T^* = \\struct {\\Q^*, \\tau^*}$ be the Alexandroff extension on $\\struct {\\Q, \\tau_d}$. Then $T^*$ satisfies no Tychonoff separation axioms higher than a $T_1$ (Fréchet) space."}
+{"_id": "15269", "title": "Hilbert Sequence Space is not Locally Compact Hausdorff Space", "text": "Let $A$ be the set of all real sequences $\\sequence {x_i}$ such that the series $\\ds \\sum_{i \\mathop \\ge 0} x_i^2$ is convergent. Let $\\ell^2 = \\struct {A, d_2}$ be the Hilbert sequence space on $\\R$. Then $\\ell^2$ is not a locally compact Hausdorff space."}
+{"_id": "15302", "title": "Every Point except Endpoint in Connected Linearly Ordered Space is Cut Point", "text": "Let $T = \\struct {S, \\preceq, \\tau}$ be a linearly ordered space. Let $A \\subseteq S$ be a connected space. Let $p \\in A$ be a point of $A$ which is not an endpoint of $A$. Then $p$ is a cut point of $A$."}
+{"_id": "15336", "title": "Sum of Infinite Series of Product of Power and Sine", "text": "Let $r \\in \\R$ such that $\\size r < 1$. Let $x \\in \\R$ such that $x \\ne 2 m \\pi$ for any $m \\in \\Z$. Then: {{begin-eqn}} {{eqn | l = \\sum_{k \\mathop = 1}^\\infty r^k \\sin k x       | r = r \\sin x + r^2 \\sin 2 x + r^3 \\sin 3 x + \\cdots       | c =  }} {{eqn | r = \\dfrac {r \\sin x} {1 - 2 r \\cos x + r^2}       | c =  }} {{end-eqn}}"}
+{"_id": "15342", "title": "Coefficients of Cosine Terms in Convergent Trigonometric Series", "text": "Let $\\map S x$ be a trigonometric series which converges to $\\map f x$ on the interval $\\openint \\alpha {\\alpha + 2 \\pi}$: :$\\map f x = \\dfrac {a_0} 2 + \\displaystyle \\sum_{m \\mathop = 1}^\\infty \\paren {a_m \\cos m x + b_m \\sin m x}$ Then: :$\\forall n \\in \\Z_{\\ge 0}: a_n = \\dfrac 1 \\pi \\displaystyle \\int_\\alpha^{\\alpha + 2 \\pi} \\map f x \\cos n x \\rd x$"}
+{"_id": "7157", "title": "De Morgan's Laws (Logic)/Conjunction/Formulation 2/Proof 2", "text": ": $\\vdash \\left({p \\land q}\\right) \\iff \\left({\\neg \\left({\\neg p \\lor \\neg q}\\right)}\\right)$"}
+{"_id": "15357", "title": "Derivation of Fourier Series over General Range", "text": "Let $\\alpha \\in \\R$ be a real number. Let $\\lambda \\in \\R_{>0}$ be a strictly positive real number. Let $f: \\R \\to \\R$ be a function such that $\\displaystyle \\int_{\\mathop \\to \\alpha}^{\\mathop \\to \\alpha + 2 \\lambda} \\map f x \\rd x$ converges absolutely. Let: :$\\displaystyle f \\sim \\frac {a_0} 2 + \\sum_{n \\mathop = 1}^\\infty \\paren {a_n \\cos \\frac {n \\pi x} \\lambda + b_n \\sin \\frac {n \\pi x} \\lambda}$ The '''Fourier coefficients''' for $f$ are calculated by: {{begin-eqn}} {{eqn | l = a_n       | r = \\dfrac 1 \\lambda \\int_{\\mathop \\to \\alpha}^{\\mathop \\to \\alpha + 2 \\lambda} \\map f x \\cos \\frac {n \\pi x} \\lambda \\rd x }} {{eqn | l = b_n       | r = \\dfrac 1 \\lambda \\int_{\\mathop \\to \\alpha}^{\\mathop \\to \\alpha + 2 \\lambda} \\map f x \\sin \\frac {n \\pi x} \\lambda \\rd x }} {{end-eqn}}"}
+{"_id": "15370", "title": "Half-Range Fourier Sine Series/Cosine over 0 to Pi", "text": ":500pxrightthumb$\\map f x$ and its $7$th approximation On the interval $\\openint 0 \\pi$: {{begin-eqn}} {{eqn | l = \\cos x       | r = \\frac 8 \\pi \\sum_{m \\mathop = 1}^\\infty \\frac {m \\sin 2 m x} {4 m^2 - 1}       | c =  }} {{eqn | r = \\frac 8 \\pi \\paren {\\frac {\\sin 2 x} {1 \\times 3} + \\frac {2 \\sin 4 x} {3 \\times 5} + \\frac {3 \\sin 6 x} {5 \\times 7} + \\cdots}       | c =  }} {{end-eqn}}"}
+{"_id": "7194", "title": "Properties of Ordered Group/OG2/Proof 1", "text": "Let $\\left({G, \\circ, \\preceq}\\right)$ be an ordered group with identity $e$. Let $\\prec$ be the reflexive reduction of $\\preceq$. Let $x, y \\in G$. Then the following equivalences hold: :$(\\operatorname{OG}2.1):\\quad x \\preceq y \\iff e \\preceq y \\circ x^{-1}$ :$(\\operatorname{OG}2.2):\\quad x \\preceq y \\iff e \\preceq x^{-1} \\circ y$ :$(\\operatorname{OG}2.3):\\quad x \\preceq y \\iff x \\circ y^{-1} \\preceq e$ :$(\\operatorname{OG}2.4):\\quad x \\preceq y \\iff y^{-1} \\circ x \\preceq e$ :$(\\operatorname{OG}2.1'):\\quad x \\prec y \\iff e \\prec y \\circ x^{-1}$ :$(\\operatorname{OG}2.2'):\\quad x \\prec y \\iff e \\prec x^{-1} \\circ y$ :$(\\operatorname{OG}2.3'):\\quad x \\prec y \\iff x \\circ y^{-1} \\prec e$ :$(\\operatorname{OG}2.4'):\\quad x \\prec y \\iff y^{-1} \\circ x \\prec e$"}
+{"_id": "15400", "title": "Series Expansion for Pi over Root 2", "text": ":$\\displaystyle \\frac \\pi {\\sqrt 2} = \\sum_{r \\mathop = 1}^\\infty \\paren {-1}^{r - 1} \\frac {r - \\frac 1 2} {r^2 - r + \\frac 3 {16} }$"}
+{"_id": "15412", "title": "Leading Coefficient of Product of Polynomials over Integral Domain", "text": "Let $R$ be an integral domain. Let $f, g \\in R \\sqbrk x$ be polynomials. Let $c$ and $d$ be their leading coefficients. Then $f g$ has leading coefficient $c d$."}
+{"_id": "7220", "title": "Ordered Set is Upper Set in Itself", "text": "Let $(S, \\preceq)$ be an ordered set. Then $S$ is an upper set in $S$."}
+{"_id": "15419", "title": "Ideal Contained in Finite Union of Prime Ideals", "text": "Let $A$ be a commutative ring with unity. Let $\\mathfrak p_1, \\ldots, \\mathfrak p_n$ be prime ideals. Let $\\mathfrak a \\subseteq \\ds \\bigcup_{i \\mathop = 1}^n \\mathfrak p_i$ be an ideal contained in their union. Then $\\mathfrak a \\subseteq \\mathfrak p_i$ for some $i \\in \\{1, \\ldots, n\\}$."}
+{"_id": "15421", "title": "Equivalence of Definitions of Unital Associative Commutative Algebra/Correspondence", "text": "Let $B$ be a algebra over $A$ that is unital, associative and commutative. Let $(C, f)$ be a ring under $A$. {{TFAE}} #$C$ is the underlying ring of $B$ and $f : A \\to C$ is the canonical mapping to the unital algebra $B$. #$B$ is the algebra defined by $f$."}
+{"_id": "15430", "title": "Cayley-Hamilton Theorem/Matrices", "text": "Let $A$ be a commutative ring with unity. Let $\\mathbf N = \\left({a_{ij} }\\right)$ be an $n \\times n$ matrix with entries in $A$. Let $\\mathbf I_n$ denote the $n \\times n$ unit matrix. Let $p_N \\left({x}\\right)$ be the determinant $\\det \\left({x \\cdot \\mathbf I_n - \\mathbf N}\\right)$. Then: : $p_N \\left({N}\\right) = \\mathbf 0$ as an $n \\times n$ zero matrix. That is: :$ N^n + b_{n-1} N^{n-1} + \\cdots + b_1 N + b_0 = \\mathbf 0$ where the $b_i$ are the coefficients of $p_N \\left({x}\\right)$."}
+{"_id": "15435", "title": "Subalgebra of Algebraic Field Extension is Field", "text": "Let $E / F$ be an algebraic field extension. Let $A \\subseteq E$ be a unital subalgebra over $F$. Then $A$ is a field."}
+{"_id": "15443", "title": "Modulus of Gamma Function of One Half plus Imaginary Number", "text": "Let $t \\in \\R$ be a real number. Then:  :$\\cmod {\\map \\Gamma {\\dfrac 1 2 + i t} } = \\sqrt {\\pi \\map \\sech {\\pi t} }$ where: :$\\Gamma$ is the Gamma function :$\\sech$ is the hyperbolic secant function."}
+{"_id": "7254", "title": "Factor Principles/Conjunction on Right/Formulation 1/Proof 1", "text": ": $p \\implies q \\vdash \\left({p \\land r}\\right) \\implies \\left ({q \\land r}\\right)$"}
+{"_id": "7268", "title": "Equivalence of Definitions of Reflexive Closure", "text": "{{TFAE|def = Reflexive Closure}} Let $\\RR$ be a relation on a set $S$."}
+{"_id": "15463", "title": "Definite Integral of Constant Multiple of Real Function", "text": "Let $f$ be a real function which is integrable on the closed interval $\\closedint a b$. {{mistake|$f$ is only integrable here, but theorems used requiring that $f$ is continuous are used in both proofs}} Let $c \\in \\R$ be a real number. Then: :$\\displaystyle \\int_a^b c \\map f x \\rd x = c \\int_a^b \\map f x \\rd x$"}
+{"_id": "15467", "title": "Mean Value Theorem for Integrals/Generalization", "text": "Let $f$ and $g$ be continuous real functions on the closed interval $\\closedint a b$ such that: :$\\forall x \\in \\closedint a b: \\map g x \\ge 0$ Then there exists a real number $k \\in \\closedint a b$ such that: :$\\displaystyle \\int_a^b \\map f x \\map g x \\rd x = \\map f k \\int_a^b \\map g x \\rd x$"}
+{"_id": "7282", "title": "Barycenter Exists and is Well Defined", "text": "Let $\\mathcal E$ be an affine space over a field $k$. Let $p_1, \\ldots, p_n \\in \\mathcal E$ be points. Let $\\lambda_1, \\ldots, \\lambda_n \\in k$ such that $\\displaystyle \\sum_{i \\mathop = 1}^n \\lambda_i = 1$. Then the barycentre of $p_1, \\ldots, p_n$ with weights $\\lambda_1, \\ldots, \\lambda_n$ exists and is unique."}
+{"_id": "15478", "title": "Definite Integral to Infinity of Power of x over Power of x plus Power of a", "text": ":$\\displaystyle \\int_0^\\infty \\dfrac {x^m \\rd x} {x^n + a^n} = \\frac {\\pi a^{m + 1 - n} } {n \\map \\sin {\\paren {m + 1} \\frac \\pi n} }$ for $0 < m + 1 < n$."}
+{"_id": "15481", "title": "Definite Integral from 0 to a of Reciprocal of Root of a Squared minus x Squared", "text": ":$\\displaystyle \\int_0^a \\dfrac {\\d x} {\\sqrt {a^2 - x^2} } = \\frac \\pi 2$ for $a > 0$."}
+{"_id": "15491", "title": "Definite Integral from 0 to Pi of Sine of m x by Sine of n x", "text": "Let $m, n \\in \\Z$ be integers. Then: :$\\displaystyle \\int_0^\\pi \\sin m x \\sin n x \\rd x = \\begin{cases} 0 & : m \\ne n \\\\ \\dfrac \\pi 2 & : m = n \\end{cases}$ That is: :$\\displaystyle \\int_0^\\pi \\sin m x \\sin n x \\rd x = \\dfrac \\pi 2 \\delta_{m n}$ where $\\delta_{m n}$ is the Kronecker delta."}
+{"_id": "7306", "title": "Equivalence of Definitions of Strict Ordering", "text": "Let $S$ be a set. Let $\\RR$ be a relation on $S$. {{TFAE|def = Strict Ordering}}"}
+{"_id": "7309", "title": "Subband of Induced Operation is Set of Subbands", "text": "Let $\\left({S, \\circ}\\right)$ be a band. Let $\\left({\\mathcal P \\left({S}\\right), \\circ_\\mathcal P}\\right)$ be the algebraic structure consisting of: : the power set $\\mathcal P \\left({S}\\right)$ of $S$ and : the operation $\\circ_\\mathcal P$ induced on $\\mathcal P \\left({S}\\right)$ by $\\circ$. Let $T \\subseteq \\mathcal P \\left({S}\\right)$. Let $\\left({T, \\circ_\\mathcal P}\\right)$ be a subband of $\\left({\\mathcal P \\left({S}\\right), \\circ_\\mathcal P}\\right)$. Then every element of $T$ is a subband of $\\left({S, \\circ}\\right)$."}
+{"_id": "7321", "title": "Equivalence of Definitions of Closed Element", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Let $\\cl$ be a closure operator on $S$. Let $x \\in S$. {{TFAE|def = Closed Element}}"}
+{"_id": "15520", "title": "Sum of Reciprocals of Cubes of Odd Integers Alternating in Sign", "text": "{{begin-eqn}} {{eqn | l = \\sum_{n \\mathop = 0}^\\infty \\dfrac {\\paren {-1}^n} {\\paren {2 n + 1}^3}       | r = \\frac 1 {1^3} - \\frac 1 {3^3} + \\frac 1 {5^3} - \\frac 1 {7^3} + \\cdots       | c =  }} {{eqn | r = \\frac {\\pi^3} {32}       | c =  }} {{end-eqn}}"}
+{"_id": "7343", "title": "Closed Element of Composite Closure Operator", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $f, g: S \\to S$ be closure operators. Let $h = f \\circ g$, where $\\circ$ represents composition. Suppose that $h$ is also a closure operator. Then an element $x \\in S$ is closed with respect to $h$ iff it is closed with respect to $f$ and with respect to $g$."}
+{"_id": "15553", "title": "Power Series Expansion of Reciprocal of Square Root of 1 + x", "text": "Let $x \\in \\R$ such that $-1 < x \\le 1$. Then: {{begin-eqn}} {{eqn | l = \\dfrac 1 {\\sqrt {1 + x} }       | r = \\sum_{k \\mathop = 0}^\\infty \\left({-1}\\right)^k \\frac {\\left({2 k}\\right)!} {\\left({2^k k!}\\right)^2} x^k       | c =  }} {{eqn | r = 1 - \\frac 1 2 x + \\frac {1 \\times 3} {2 \\times 4} x^2 - \\frac {1 \\times 3 \\times 5} {2 \\times 4 \\times 6} x^3 + \\cdots       | c =  }} {{end-eqn}}"}
+{"_id": "7362", "title": "Inverse Image under Embedding of Image under Relation of Image of Point", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R_S$ and $\\mathcal R_t$ be relations on $S$ and $T$, respectively. Let $\\phi: S \\to T$ be a mapping with the property that: : $\\forall p, q \\in S: \\left({ p \\mathrel{\\mathcal R_S} q \\iff \\phi(p) \\mathrel{\\mathcal R_T} \\phi(q) }\\right)$ Then for each $p \\in S$: : $\\mathcal R_S (p) = \\phi^{-1}\\left({\\mathcal R_T \\left({ \\phi(p) }\\right) }\\right)$"}
+{"_id": "15556", "title": "Power Series Expansion of Cube Root of 1 + x", "text": "Let $x \\in \\R$ such that $-1 < x \\le 1$. Then: :$\\dfrac 1 {\\sqrt [3] {1 + x} } = 1 + \\dfrac 1 3 x - \\dfrac 2 {3 \\times 6} x^2 + \\dfrac {2 \\times 5} {3 \\times 6 \\times 9} x^3 - \\cdots$"}
+{"_id": "15565", "title": "Power Series Expansion for Real Arcsecant Function", "text": "The arcsecant function has a Taylor Series expansion: {{begin-eqn}} {{eqn | l = \\operatorname {arcsec} x       | r = \\frac \\pi 2 - \\sum_{n \\mathop = 0}^\\infty \\frac {\\left({2 n}\\right)!} {2^{2 n} \\left({n!}\\right)^2 \\left({2 n + 1}\\right) x^{2 n + 1} }       | c =  }} {{eqn | r = \\frac \\pi 2 - \\left({\\frac 1 x + \\frac 1 2 \\frac 1 {3 x^3} + \\frac {1 \\times 3} {2 \\times 4} \\frac 1 {5 x^5} + \\frac {1 \\times 3 \\times 5} {2 \\times 4 \\times 6} \\frac 1 {7 x^7} + \\cdots}\\right)       | c =  }} {{end-eqn}} which converges for $\\left\\lvert{x}\\right\\rvert \\ge 1$."}
+{"_id": "7377", "title": "Fuzzy Intersection is Commutative", "text": "Fuzzy intersection is commutative."}
+{"_id": "7382", "title": "Integral Domain is Reduced Ring", "text": "Let $\\left({D, +, \\circ}\\right)$ be an integral domain. Then $D$ is reduced."}
+{"_id": "7392", "title": "Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 2/Proof 2", "text": ":$\\vdash \\left({p \\land \\left({q \\lor r}\\right)}\\right) \\iff \\left({\\left({p \\land q}\\right) \\lor \\left({p \\land r}\\right)}\\right)$"}
+{"_id": "7403", "title": "Ring of Polynomial Functions is Commutative Ring with Unity", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $R \\sqbrk {\\set {X_j: j \\in J} }$ be the ring of polynomial forms over $R$ in the indeterminates $\\set {X_j: j \\in J}$. Let $R^J$ be the free module on $J$. Let $A$ be the set of all polynomial functions $R^J \\to R$. Let $\\struct {A, +, \\circ}$ be the ring of polynomial functions on $R$. Then $\\struct {A, +, \\circ}$ is a commutative ring with unity."}
+{"_id": "15608", "title": "Moment Generating Function of Continuous Uniform Distribution", "text": "Let $X \\sim \\operatorname U \\left[{a \\,.\\,.\\, b}\\right]$ for some $a, b \\in \\R$, $a \\ne b$, where $\\operatorname U$ is the continuous uniform distribution.  Then the moment generating function of $X$, $M_X$ is given by:  :$\\displaystyle M_X \\left({t}\\right) = \\begin{cases} \\dfrac{ e^{t b} - e^{t a} } {t \\left({b - a}\\right)} & t \\ne 0 \\\\ 1 & t = 0 \\end{cases}$"}
+{"_id": "15615", "title": "Sum of Reciprocals of Odd Powers of Odd Integers Alternating in Sign/Corollary", "text": "Let $n \\in \\Z_{> 0}$ be a (strictly) positive integer. {{begin-eqn}} {{eqn | l = E_{2 n}       | r = \\paren {-1}^{n + 1} \\dfrac {2^{2 n + 2} \\paren {2 n}!} {\\pi^{2 n + 1} } \\sum_{j \\mathop = 0}^\\infty \\frac {\\paren {-1}^j} {\\paren {2 j + 1}^{2 n + 1} }       | c =  }} {{eqn | r = \\paren {-1}^{n + 1} \\dfrac {2^{2 n + 2} \\paren {2 n}!} {\\pi^{2 n + 1} } \\paren {\\frac 1 {1^{2 n + 1} } - \\frac 1 {3^{2 n + 1} } + \\frac 1 {5^{2 n + 1} } - \\frac 1 {7^{2 n + 1} } + \\cdots}       | c =  }} {{end-eqn}}"}
+{"_id": "7425", "title": "Natural Number has Same Prime Factors as Integer Power", "text": "Let $x$ be a natural number such that $x > 1$. Let $n \\ge 1$ be a (strictly) positive integer. The $n$th power of $x$ has the same prime factors as $x$."}
+{"_id": "7460", "title": "Equivalence of Definitions of Well-Ordering", "text": "{{TFAE|def = Well-Ordering}} Let $\\left({S, \\preceq}\\right)$ be a ordered set."}
+{"_id": "15656", "title": "Contour Integral of Gamma Function", "text": "Let $\\Gamma$ denote the gamma function. Let $y$ be a positive number. Then for any positive number $c$: :$\\displaystyle \\frac 1 {2 \\pi i} \\int_{c - i \\infty}^{c + i \\infty} \\Gamma \\left({t}\\right) y^{-t} \\rd t = e^{-y}$"}
+{"_id": "15660", "title": "Sign of Composition of Permutations", "text": "Let $n \\in \\N$ be a natural number. Let $N_n$ denote the set of natural numbers $\\set {1, 2, \\ldots, n}$. Let $S_n$ denote the set of permutations on $N_n$. Let $\\map \\sgn \\pi$ denote the sign of $\\pi$ of a permutation $\\pi$ of $N_n$. Let $\\pi_1, \\pi_2 \\in S_n$. Then: :$\\map \\sgn {\\pi_1} \\, \\map \\sgn {\\pi_2} = \\map \\sgn {\\pi_1 \\circ \\pi_2}$ where $\\pi_1 \\circ \\pi_2$ denotes the composite of $\\pi_1$ and $\\pi_2$."}
+{"_id": "7472", "title": "Relational Closure from Transitive Closure", "text": "Let $A$ be a set or class. Let $\\RR$ be a relation on $A$. Let $\\RR^+$ be the transitive closure of $\\RR$. Let $B \\subseteq A$. Let $B' = B \\cup \\map {\\paren {\\RR^+}^{-1} } B$. Let $C$ be an $\\RR$-transitive subset or subclass of $A$ such that $B \\subseteq C$. Then: :$B'$ is $\\RR$-transitive :$B' \\subseteq C$ :If $B$ is a set and $\\RR$ is set-like then $B'$ is a set. That is, $B'$ is the relational closure of $B$ under $\\RR$."}
+{"_id": "15680", "title": "Product of r Choose m with m Choose k/Complex Numbers", "text": "For all $z, w \\in \\C$ such that it is not the case that $z$ is a negative integer and $t, w$ integers: :$\\dbinom z t \\dbinom t w = \\dbinom z w \\dbinom {z - w} {t - w}$ where $\\dbinom z w$ is a binomial coefficient."}
+{"_id": "15691", "title": "Approximation to 2n Choose n", "text": ":$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\dbinom {2 n} n = \\dfrac {4^n} {\\sqrt {n \\pi} }$"}
+{"_id": "15704", "title": "Upper Bound for Harmonic Number", "text": ":$H_{2^m} \\le 1 + m$ where $H_{2^m}$ denotes the $2^m$th harmonic number."}
+{"_id": "15708", "title": "Smallest Strictly Positive Rational Number does not Exist", "text": "There exists no smallest element of the set of strictly positive rational numbers."}
+{"_id": "7541", "title": "Generating Function for Powers of Two", "text": "Let $\\sequence {a_n}$ be the sequence defined as: :$\\forall n \\in \\N: a_n = 2^n$ That is: :$\\sequence {a_n} = 1, 2, 4, 8, \\ldots$ Then the generating function for $\\sequence {a_n}$ is given as: :$\\displaystyle \\map G z = \\frac 1 {1 - 2 z}$ for $\\size z < \\dfrac 1 2$"}
+{"_id": "15738", "title": "Exchange of Order of Summation with Dependency on Both Indices/Example", "text": "Let $n \\in \\Z$ be an integer. Let $R: \\Z \\to \\set {\\mathrm T, \\mathrm F}$ be the propositional function on the set of integers defining: :$\\forall i \\in \\Z: \\map R 1 := \\paren {n = k i \\text { for some } k \\in \\Z}$ Let $S: \\Z \\times \\Z \\to \\set {\\mathrm T, \\mathrm F}$ be a propositional function on the Cartesian product of the set of integers with itself defining: :$\\forall i, j \\in \\Z: \\map S {i, j} := \\paren {1 \\le j < i}$ Consider the summation: :$\\displaystyle \\sum_{\\map R i} \\sum_{\\map S {i, j} } a_{i j}$ Then: :$\\displaystyle \\sum_{\\map R i} \\sum_{\\map S {i, j} } a_{i j} = \\sum_{\\map {S'} j} \\sum_{\\map {R'} {i, j} } a_{i j}$ where: :$\\map {S'} j$ denotes the propositional function: ::$\\forall j \\in \\Z: \\map {S'} j := \\paren {1 < j \\le n}$ :$\\map {R'} {i, j}$ denotes the propositional function: ::$\\forall i, j \\in \\Z: \\map {R'} {i, j} := \\paren {n = k i \\text { for some } k \\in \\Z \\text { and } i > j}$"}
+{"_id": "7552", "title": "Probability Generating Function defines Probability Distribution", "text": "Let $X$ and $Y$ be discrete random variables  whose codomain, $\\Omega_X$, is a subset of the natural numbers $\\N$. Let the probability generating functions of $X$ and $Y$ be $\\map {\\Pi_X} s$ and $\\map {\\Pi_Y} s$ respectively. Then: :$\\forall s \\in \\closedint {-1} 1: \\map {\\Pi_X} s = \\map {\\Pi_Y} s$ {{iff}}: :$\\forall k \\in \\N: \\Pr \\left({X = k}\\right) = \\map \\Pr {Y = k}$ That is, discrete random variables which are integer-valued have the same PGFs {{iff}} they have the same PMF."}
+{"_id": "7560", "title": "Expectation of Square of Discrete Random Variable", "text": "Let $X$ be a discrete random variable whose probability generating function is $\\Pi_X \\paren s$. Then the square of the expectation of $X$ is given by the expression: :$\\expect {X^2} = \\Pi''_X \\paren 1 + \\Pi'_X \\paren 1$ where $\\Pi''_X \\paren 1$ and $\\Pi'_X \\paren 1$ denote the second and first derivative respectively of the PGF $\\Pi_X \\paren s$ evaluated at $1$."}
+{"_id": "15763", "title": "Modulo Operation/Examples/100 mod 7", "text": ":$100 \\bmod 7 = 2$"}
+{"_id": "7572", "title": "Expectation and Variance of Poisson Distribution equal its Parameter", "text": "Let $X$ be a discrete random variable with the Poisson distribution with parameter $\\lambda$. Then the expectation of $X$ equals the variance of $X$, that is, $\\lambda$ itself."}
+{"_id": "15773", "title": "Modulo Operation as Integer Difference by Quotient", "text": "Let $x, y, z \\in \\R$ be real numbers. Let $y > 0$. Let $0 \\le z < y$. Let: :$\\dfrac {x - z} y = k$ for some integer $k$. Then: :$z = x \\bmod y$ where $\\bmod$ denotes the modulo operation."}
+{"_id": "15777", "title": "Dirichlet Series Convergence Lemma/General", "text": "Let $\\displaystyle f \\left({s}\\right) = \\sum_{n \\mathop = 1}^\\infty a_n e^{-\\lambda_n \\left({s}\\right)}$ be a general Dirichlet series. Let $f \\left({s}\\right)$ converge at $s_0 = \\sigma_0 + i t_0$. Then $f \\left({s}\\right)$ converge for all $s = \\sigma + i t$ where $\\sigma > \\sigma_0$."}
+{"_id": "7589", "title": "Second Derivative of PGF of Negative Binomial Distribution/First Form", "text": "Let $X$ be a discrete random variable with the negative binomial distribution (first form) with parameters $n$ and $p$. Then the second derivative of the PGF of $X$ {{WRT|Differentiation}} $s$ is: :$\\dfrac {\\mathrm d^2} {\\mathrm d s^2} \\Pi_X \\left({s}\\right) = \\dfrac {n \\left({n + 1}\\right) p^2} {q^2} \\left({\\dfrac q {1 - p s} }\\right)^{n + 2}$ where $q = 1 - p$."}
+{"_id": "15790", "title": "Stirling's Formula/Refinement", "text": "A refinement of Stirling's Formula is: :$n! \\sim \\sqrt {2 \\pi n} \\paren {\\dfrac n e}^n \\paren {1 + \\dfrac 1 {12 n} }$"}
+{"_id": "15794", "title": "Legendre's Theorem/Corollary", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $B$ be the binary representation of $n$. Let $r$ be the number of unit digits in $B$. Let $n!$ denote the factorial of $n$. Then $2^{n - r}$ is a divisor of $n!$, but $2^{n - r + 1}$ is not."}
+{"_id": "15821", "title": "Falling Factorial of Complex Number as Summation of Unsigned Stirling Numbers of First Kind", "text": "Let $z \\in \\C$ be a complex number whose real part is positive. Then: :$z^{\\underline r} = \\displaystyle \\sum_{k \\mathop = 0}^m \\left[{r \\atop r - k}\\right] \\left({-1}\\right)^k z^{r - k} + \\mathcal O \\left({z^{r - m - 1} }\\right)$ where: :$\\displaystyle \\left[{r \\atop r - k}\\right]$ denotes the extension of the unsigned Stirling numbers of the first kind to the complex plane :$z^{\\underline r}$ denotes $z$ to the $r$ falling :$\\mathcal O \\left({z^{r - m - 1} }\\right)$ denotes big-$\\mathcal O$ notation."}
+{"_id": "7631", "title": "Galois Field of Order q Exists iff q is Prime Power", "text": "Let $q \\ge 0$ be a positive integer. Then there exists a Galois field of order $q$ {{iff}} $q$ is a prime power."}
+{"_id": "15825", "title": "Product of r Choose k with r Minus Half Choose k/Formulation 2", "text": "Let $k \\in \\Z$, $r \\in \\R$. :$\\dbinom r k \\dbinom {r - \\frac 1 2} k = \\dfrac {\\dbinom {2 r} {2 k} \\dbinom {2 k} k} {4^k}$ where $\\dbinom r k$ denotes a binomial coefficient."}
+{"_id": "15835", "title": "Existence of Minimal Uncountable Well-Ordered Set", "text": "There exists a minimal uncountable well-ordered set. That is, there exists an uncountable well-ordered set $\\Omega$ with the property that every initial segment in $\\Omega$ is countable."}
+{"_id": "15848", "title": "Inverse of Stirling's Triangle expressed as Matrix", "text": "Consider Stirling's triangle of the first kind (signed) expressed as a (square) matrix $\\mathbf A$, with the top left element holding $s \\left({0, 0}\\right)$. :$\\begin{pmatrix} 1 &     0 &       0 &      0 &      0 &     0 & \\cdots \\\\ 0 &     1 &       0 &      0 &      0 &     0 & \\cdots \\\\ 0 &    -1 &       1 &      0 &      0 &     0 & \\cdots \\\\ 0 &     2 &      -3 &      1 &      0 &     0 & \\cdots \\\\ 0 &    -6 &      11 &     -6 &      1 &     0 & \\cdots \\\\ 0 &    24 &     -50 &     35 &    -10 &     1 & \\cdots \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots &  \\ddots \\\\ \\end{pmatrix}$ Thenconsider Stirling's triangle of the second kind expressed as a (square) matrix $\\mathbf B$, with the top left element holding $\\displaystyle \\left \\{ {0 \\atop 0}\\right\\}$. :$\\begin{pmatrix} 1 & 0 &   0 &    0 &    0 &    0 & \\cdots \\\\ 0 & 1 &   0 &    0 &    0 &    0 & \\cdots \\\\ 0 & 1 &   1 &    0 &    0 &    0 & \\cdots \\\\ 0 & 1 &   3 &    1 &    0 &    0 & \\cdots \\\\ 0 & 1 &   7 &    6 &    1 &    0 & \\cdots \\\\ 0 & 1 &  15 &   25 &   10 &    1 & \\cdots \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\\\ \\end{pmatrix}$ Then: : $\\mathbf A = \\mathbf B^{-1}$ that is: : $\\mathbf B = \\mathbf A^{-1}$"}
+{"_id": "15861", "title": "Reflection Rule for Gaussian Binomial Coefficients", "text": "Let $q \\in \\R_{\\ne 1}, n \\in \\Z_{>0}, k \\in \\Z$. Then: :$\\dbinom n k_q = q^{k \\paren {n - k} } \\dbinom n k_{q^{-1} }$ where $\\dbinom n k_q$ is a Gaussian binomial coefficient."}
+{"_id": "7670", "title": "Rooted Tree Corresponds to Arborescence", "text": "Let $T = \\struct {V, E}$ be a rooted tree with root $r$. Then there is a unique orientation of $T$ which is an $r$-arborescence."}
+{"_id": "7673", "title": "Equivalence of Definitions of Arborescence", "text": "Let $G = \\struct {V, A}$ be a directed graph. Let $r \\in V$. {{TFAE|def = Arborescence}}"}
+{"_id": "15883", "title": "Inductive Construction of Sigma-Algebra Generated by Collection of Subsets", "text": "Let $\\EE$ be a set of sets which are subsets of some set $X$. Let $\\map \\sigma \\EE$ be the $\\sigma$-algebra generated by $\\EE$. Then $\\map \\sigma \\EE$  can be constructed inductively. The construction is as follows: Let $\\Omega$ denote the minimal uncountable well-ordered set. Let $\\alpha$ be an arbitrary initial segment in $\\Omega$. Considering separately the cases whether or not $\\alpha$ has an immediate predecessor $\\beta$, we define: :$\\EE_1 = \\EE$ :$\\EE_\\alpha = \\begin{cases} \\set {\\SS \\in \\powerset {\\EE_\\beta}: \\SS \\text { is countable or } \\SS^\\complement \\text{ is countable} } & \\alpha \\text{ has an immediate predecessor } \\beta \\\\ \\ds \\bigcup_{\\beta \\mathop \\prec \\alpha} \\EE_\\beta & \\text { otherwise} \\end{cases}$ :$\\EE_\\Omega = \\ds \\bigcup_{\\alpha \\mathop \\in \\Omega} \\EE_\\alpha$ Then $\\map \\sigma \\EE = \\EE_\\Omega$."}
+{"_id": "7694", "title": "Equivalence of Definitions of Set Partition", "text": "Let $S$ be a set {{TFAE|def = Set Partition}}"}
+{"_id": "15899", "title": "Summation to n of Reciprocal of k by k-1 of Harmonic Number", "text": ":$\\displaystyle \\sum_{1 \\mathop < k \\mathop \\le n} \\dfrac 1 {k \\paren {k - 1} } H_k = 2 - \\dfrac {H_n} n - \\dfrac 1 n$ where $H_n$ denotes the $n$th harmonic number."}
+{"_id": "15900", "title": "Riemann Zeta Function of 1000", "text": "To at least $100$ decimal places: :$\\zeta \\left({1000}\\right) \\approx 1$ where $\\zeta$ denotes the Riemann zeta function."}
+{"_id": "7715", "title": "Logarithmic Integral as Non-Convergent Series", "text": "The logarithmic integral can be defined in terms of a non-convergent series. That is: :$\\displaystyle \\operatorname {li} \\left({z}\\right) = \\sum_{i \\mathop = 0}^{+\\infty} \\frac {i! \\, z} {\\ln^{i + 1} z}  = \\frac z {\\ln z} \\left({\\sum_{i \\mathop = 0}^{+\\infty} \\frac {i!} {\\ln^i z} }\\right)$"}
+{"_id": "7726", "title": "Bridge divides Graph into Two Components", "text": "Let $G$ be a connected graph. Let $e$ be a bridge of $G$. Then the edge deletion $G - e$ contains exactly $2$ components."}
+{"_id": "7727", "title": "Connected Graph with only Even Vertices has no Bridge", "text": "Let $G$ be a connected graph whose vertices are all even. Then no edge of $G$ is a bridge."}
+{"_id": "7725", "title": "Cut-Vertex divides Graph into Two or More Components", "text": "Let $G$ be a graph. Let $v$ be a cut-vertex of $G$. Then the vertex deletion $G - v$ contains $2$ or more components."}
+{"_id": "15930", "title": "Median to Hypotenuse of Right Triangle equals Half Hypotenuse", "text": "Let $\\triangle ABC$ be a right triangle such that $BC$ is the hypotenuse. Let $AD$ be the median to $BC$. Then $AD$ is half of $BC$."}
+{"_id": "15932", "title": "Summation to n of Square of kth Harmonic Number", "text": ":$\\displaystyle \\sum_{k \\mathop = 1}^n {H_k}^2 = \\paren {n + 1} {H_n}^2 - \\paren {2 n + 1} H_n + 2 n$ where $H_k$ denotes the $k$th harmonic number."}
+{"_id": "7741", "title": "1+1 = 2", "text": "Define $0$ as the only element in the set $P \\setminus s \\left({P}\\right)$, where: :$P$ is the Peano Structure :$s \\left({P}\\right)$ is the image of the mapping $s$ defined in Peano structure : $\\setminus$ denotes the set difference. The theorem to be proven is: :$1 + 1 = 2$ where: :$1 := s \\left({0}\\right)$ :$2 := s \\left({1}\\right) = s \\left({s \\left({0}\\right)}\\right)$ :$+$ denotes addition :$=$ denotes equality :$s \\left({n}\\right)$ denotes the successor function as defined by Peano"}
+{"_id": "15938", "title": "Highest Power of 2 Dividing Numerator of Sum of Odd Reciprocals", "text": "Let: : $S = \\dfrac p q = \\displaystyle \\sum_{k \\mathop = 1}^n \\dfrac 1 {2 k - 1}$ where $\\dfrac p q$ is the canonical form of $S$. Let $n = 2^k m$ where $m$ is odd. Then the largest power of $2$ that divides $p$ is $2^{2 k}$."}
+{"_id": "7749", "title": "Two Paths between Vertices in Cycle Graph", "text": "Let $G$ be a simple graph. Let $u, v$ be vertices in $G$ such that $u \\ne v$. Then: :for any two vertices $u, v$ in $G$ such that $u \\ne v$ there exists exactly two paths between $u$ and $v$ {{iff}}: :$G$ is a cycle graph."}
+{"_id": "7754", "title": "Value of Radian in Degrees", "text": "The value of a radian in degrees is given by: :$1 \\radians = \\dfrac {180 \\degrees} {\\pi} \\approx 57.29577 \\ 95130 \\ 8232 \\ldots \\degrees$  {{OEIS|A072097}}"}
+{"_id": "15954", "title": "Cassini's Identity/Negative Indices", "text": "Let $n \\in \\Z_{<0}$ be a negative integer. Let $F_n$ be the $n$th Fibonacci number (as extended to negative integers). Then Cassini's Identity: :$F_{n + 1} F_{n - 1} - F_n^2 = \\left({-1}\\right)^n$ continues to hold."}
+{"_id": "15977", "title": "Sum over k of n Choose k by Fibonacci Number with index m+k", "text": ":$\\displaystyle \\sum_{k \\mathop \\ge 0} \\binom n k F_{m + k} = F_{m + 2 n}$ where: : $\\dbinom n k$ denotes a binomial coefficient : $F_n$ denotes the $n$th Fibonacci number."}
+{"_id": "7786", "title": "Area of Regular Polygon", "text": "Let $P$ be a regular $n$-sided polygon whose side length is $b$. Then the area of $P$ is given by: :$\\Box P = \\dfrac 1 4 n b^2 \\cot \\dfrac \\pi n$ where $\\cot$ denotes cotangent."}
+{"_id": "15984", "title": "Fibonacci Number whose Index is Plus or Minus Prime p is Multiple of p", "text": "Let $p$ be a prime number distinct from $5$. Let $F_n$ denote the $n$th Fibonacci number. Then either $F_{p - 1}$ or $F_{p + 1}$ (but not both) is a multiple of $p$."}
+{"_id": "7796", "title": "Perimeter of Regular Polygon by Circumradius", "text": "Let $P$ be a regular $n$-gon. Let $C$ be a circumcircle of $P$. Let the radius of $C$ be $r$. Then the perimeter $\\mathcal P$ of $P$ is given by: :$\\mathcal P = 2 n r \\sin \\dfrac \\pi n$"}
+{"_id": "7797", "title": "Area of Regular Polygon by Inradius", "text": "Let $P$ be a regular $n$-gon. Let $C$ be an incircle of $P$. Let the radius of $C$ be $r$. Then the area $\\AA$ of $P$ is given by: :$\\AA = n r^2 \\tan \\dfrac \\pi n$"}
+{"_id": "16018", "title": "Zeckendorf Representation of Integer shifted Right", "text": "Let $f \\left({x}\\right)$ be the real function defined as: :$\\forall x \\in \\R: f \\left({x}\\right) = \\left\\lfloor{x + \\phi^{-1} }\\right\\rfloor$ where: :$\\left\\lfloor{\\, \\cdot \\,}\\right\\rfloor$ denotes the floor function :$\\phi$ denotes the golden mean. Let $n \\in \\Z_{\\ge 0}$ be a positive integer. Let $n$ be expressed in Zeckendorf representation: :$n = F_{k_1} + F_{k_2} + \\cdots + F_{k_r}$ with the appropriate restrictions on $k_1, k_2, \\ldots, k_r$. Then: :$F_{k_1 - 1} + F_{k_2 - 1} + \\cdots + F_{k_r - 1} = f \\left({\\phi^{-1} n}\\right)$"}
+{"_id": "16021", "title": "Linear Combination of Generating Functions", "text": "Let $G \\left({z}\\right)$ be the generating function for the sequence $\\left\\langle{a_n}\\right\\rangle$. and $H \\left({z}\\right)$ be the generating function for the sequence $\\left\\langle{b_n}\\right\\rangle$. Then $\\alpha G \\left({z}\\right) + \\beta H \\left({z}\\right)$ is the generating function for the sequence $\\left\\langle{\\alpha a_n + \\beta b_n}\\right\\rangle$."}
+{"_id": "7832", "title": "Semigroup is Group Iff Latin Square Property Holds", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroup. Then $\\left({S, \\circ}\\right)$ is a group {{iff}} for all $a, b \\in S$ the Latin square property holds in $S$: :$a \\circ x = b$ :$y \\circ a = b$ for $x$ and $y$ each unique in $S$."}
+{"_id": "16032", "title": "Generating Function for Even Terms of Sequence", "text": "Let $G \\left({z}\\right)$ be the generating function for the sequence $\\left\\langle{a_n}\\right\\rangle$. Consider the subsequence $\\left\\langle{b_n}\\right\\rangle := \\left({a_0, a_2, a_4, \\ldots}\\right)$ Then the generating function for $\\left\\langle{b_n}\\right\\rangle$ is: :$\\dfrac 1 2 \\left({G \\left({z}\\right) + G \\left({-z}\\right)}\\right)$"}
+{"_id": "16047", "title": "Ultraproduct is Well-Defined", "text": "'''Definition:Ultraproduct is well-defined.''' More specificly, following the definitions on Definition:Ultraproduct, we are going to prove that: :(1) $f^\\mathcal M$ is well-defined :(2) $R^\\mathcal M$ is well-defined"}
+{"_id": "16115", "title": "Equation of Circular Arc in Complex Plane", "text": "Let $a, b \\in \\C$ be complex constants representing the points $A$ and $B$ respectively in the complex plane. Let $z \\in \\C$ be a complex variable representing the point $Z$ in the complex plane. Let $\\lambda \\in \\R$ be a real constant such that $-\\pi < \\lambda < \\pi$. Then the equation: :$\\arg \\dfrac {z - b} {z - a} = \\lambda$ represents the arc of a circle with $AB$ as a chord subtending an angle $\\lambda$ at $Z$ on the circumference."}
+{"_id": "16128", "title": "Exponential of Complex Number plus 2 pi i", "text": ":$\\map \\exp {z + 2 \\pi i} = \\map \\exp z$"}
+{"_id": "16143", "title": "Sextuple Angle Formula for Sine", "text": ":$\\dfrac {\\map \\sin {6 \\theta} } {\\sin \\theta} = 32 \\cos^5 \\theta - 32 \\cos^3 \\theta + 6 \\cos \\theta$"}
+{"_id": "16151", "title": "Real Complex Roots of Unity for Odd Index", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer such that $n$ is odd. Let $U_n = \\set {z \\in \\C: z^n = 1}$ be the set of complex $n$th roots of unity. The only $x \\in U_n$ such that $x \\in \\R$ is: :$x = 1$ That is, $1$ is the only complex $n$th root of unity which is a real number."}
+{"_id": "16163", "title": "Unit Vectors in Complex Plane which are Vertices of Equilateral Triangle", "text": "Let $\\epsilon_1, \\epsilon_2, \\epsilon_3$ be complex numbers embedded in the complex plane such that: :$\\epsilon_1, \\epsilon_2, \\epsilon_3$ all have modulus $1$ :$\\epsilon_1 + \\epsilon_2 + \\epsilon_3 = 0$ Then: :$\\paren {\\dfrac {\\epsilon_2} {\\epsilon_1} }^3 = \\paren {\\dfrac {\\epsilon_3} {\\epsilon_2} }^2 = \\paren {\\dfrac {\\epsilon_1} {\\epsilon_3} }^2 = 1$"}
+{"_id": "16171", "title": "Factorisation of x^(2n)-1 in Real Domain", "text": "Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: :$z^{2 n} - 1 = \\paren {z - 1} \\paren {z + 1} \\displaystyle \\prod_{k \\mathop = 1}^n \\paren {z^2 - 2 \\cos \\dfrac {k \\pi} n + 1}$"}
+{"_id": "16187", "title": "4 Sine Pi over 10 by Cosine Pi over 5", "text": ":$4 \\sin \\dfrac \\pi {10} \\cos \\dfrac \\pi 5 = 1$"}
+{"_id": "8007", "title": "Secant of Complement equals Cosecant", "text": ":$\\map \\sec {\\dfrac \\pi 2 - \\theta} = \\csc \\theta$ for $\\theta \\ne n \\pi$ where $\\sec$ and $\\csc$ are secant and cosecant respectively. That is, the cosecant of an angle is the secant of its complement. This relation is defined wherever $\\sin \\theta \\ne 0$."}
+{"_id": "8008", "title": "Cosecant of Complement equals Secant", "text": ":$\\map \\csc {\\dfrac \\pi 2 - \\theta} = \\sec \\theta$ for $\\theta \\ne \\paren {2 n + 1} \\dfrac \\pi 2$ where $\\csc$ and $\\sec$ are cosecant and secant respectively. That is, the secant of an angle is the cosecant of its complement. This relation is defined wherever $\\cos \\theta \\ne 0$."}
+{"_id": "8009", "title": "Secant of Supplementary Angle", "text": ":$\\map \\sec {\\pi - \\theta} = -\\sec \\theta$ where $\\sec$ denotes secant. That is, the secant of an angle is the negative of its supplement."}
+{"_id": "16217", "title": "Absolutely Convergent Series is Convergent/Complex Numbers", "text": "Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty z_n$ be an absolutely convergent series in $\\C$. Then $\\displaystyle \\sum_{n \\mathop = 1}^\\infty z_n$ is convergent."}
+{"_id": "16225", "title": "Existence of Radius of Convergence of Complex Power Series/Absolute Convergence", "text": "Let $B_R \\paren \\xi$ denote the open $R$-ball of $\\xi$. Let $z \\in B_R \\paren \\xi$. Then $S \\paren z$ converges absolutely. If $R = +\\infty$, we define $B_R \\paren \\xi = \\C$."}
+{"_id": "16255", "title": "Imaginary Part of Complex Exponential Function", "text": "Let $z = x + i y \\in \\C$ be a complex number, where $x, y \\in \\R$. Let $\\exp z$ denote the complex exponential function. Then: :$\\Im \\paren {\\exp z} = e^x \\sin y$ where: :$\\Im z$ denotes the imaginary part of a complex number $z$ :$e^x$ denotes the real exponential function of $x$ :$\\sin y$ denotes the real sine function of $y$."}
+{"_id": "16259", "title": "Modulus of Sine of Complex Number", "text": "Let $z = x + i y \\in \\C$ be a complex number, where $x, y \\in \\R$. Let $\\sin z$ denote the complex sine function. Then: :$\\cmod {\\sin z} = \\sqrt {\\sin^2 x + \\sinh^2 y}$ where: :$\\cmod z$ denotes the modulus of a complex number $z$ :$\\sin x$ denotes the real sine function :$\\sinh$ denotes the hyperbolic sine function."}
+{"_id": "16267", "title": "Bounds for Modulus of e^z on Circle x^2 + y^2 - 2x - 2y - 2 = 0", "text": "Consider the circle $C$ embedded in the complex plane defined by the equation: :$x^2 + y^2 - 2 x - 2 y - 2 = 0$ Let $z = x + i y \\in \\C$ be a point lying on $C$. Then: :$e^{-1} \\le \\cmod {e^z} \\le e^3$"}
+{"_id": "8091", "title": "Secant of Angle plus Three Right Angles", "text": ":$\\map \\sec {x + \\dfrac {3 \\pi} 2} = \\csc x$"}
+{"_id": "8092", "title": "Cosecant of Angle plus Three Right Angles", "text": ":$\\map \\csc {x + \\dfrac {3 \\pi} 2} = -\\sec x$"}
+{"_id": "8095", "title": "Secant of Three Right Angles less Angle", "text": ":$\\map \\sec {\\dfrac {3 \\pi} 2 - \\theta} = -\\csc \\theta$ where $\\sec$ and $\\csc$ are secant and cosecant respectively."}
+{"_id": "16300", "title": "Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind/Corollary", "text": ":$B_n = \\displaystyle \\sum_{k \\mathop = 1}^n {n \\brace k}$"}
+{"_id": "16324", "title": "Structure Induced by Commutative Ring Operations is Commutative Ring", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring. Let $S$ be a set. Let $\\struct {R^S, +', \\circ'}$ be the structure on $R^S$ induced by $+'$ and $\\circ'$. Then $\\struct {R^S, +', \\circ'}$ is a commutative ring."}
+{"_id": "16326", "title": "Equivalence Relation on Power Set induced by Intersection with Subset/Cardinality of Set of Equivalence Classes", "text": "Let $A$ be finite with $\\card A = n$, where $\\card {\\, \\cdot \\,}$ denotes cardinality. The cardinality of the set of $\\alpha$-equivalence classes is given by: :$\\card {\\set {\\eqclass X \\alpha: X \\in S} } = 2^n$"}
+{"_id": "8141", "title": "Cosine to Power of Even Integer", "text": "{{begin-eqn}} {{eqn | l = \\cos^{2 n} \\theta       | r = \\frac 1 {2^{2 n} } \\binom {2 n} n + \\frac 1 {2^{2 n - 1} } \\paren {\\cos 2 n \\theta + \\binom {2 n} 1 \\map \\cos {2 n - 2} \\theta + \\cdots + \\binom {2 n} {n - 1} \\cos 2 \\theta}       | c =  }} {{eqn | r = \\frac 1 {2^{2 n} } \\binom {2 n} n + \\frac 1 {2^{2 n - 1} } \\sum_{k \\mathop = 0}^{n - 1} \\binom {2 n} k \\map \\cos {2 n - 2 k} \\theta       | c =  }} {{end-eqn}}"}
+{"_id": "8155", "title": "Law of Tangents", "text": "Let $\\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$. Then: :$\\dfrac {a + b} {a - b} = \\dfrac {\\tan \\frac 1 2 \\paren {A + B} } {\\tan \\frac 1 2 \\paren {A - B} }$"}
+{"_id": "8164", "title": "Functionally Complete Singleton Sets", "text": "The only binary logical connectives that form singleton sets which are functionally complete are NAND and NOR."}